LMFDB - Newform orbit 7400.2.a.u

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7400,2,Mod(1,7400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7400 = 2^{3} \cdot 5^{2} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7400.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:	$59.0892974957$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{7} - 12x^{6} + 8x^{5} + 39x^{4} - 14x^{3} - 36x^{2} + 11x + 6$ x^8 - x^7 - 12x^6 + 8x^5 + 39x^4 - 14x^3 - 36x^2 + 11x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{3} - \beta_{7} q^{7} + (\beta_{7} - \beta_{4} + \beta_{3} - 1) q^{9} + (\beta_{5} + \beta_{4} + \beta_1) q^{11} + ( - \beta_{6} + \beta_{5} - \beta_{3} - 1) q^{13} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{17}+ \cdots + (\beta_{5} - \beta_{4} - 2 \beta_{3} + \cdots - 2) q^{99}+O(q^{100})$ q - b1 * q^3 - b7 * q^7 + (b7 - b4 + b3 - 1) * q^9 + (b5 + b4 + b1) * q^11 + (-b6 + b5 - b3 - 1) * q^13 + (-b7 + b6 - b5 - b3 - b1 + 1) * q^17 + (2b7 - b4 + b3 - b1 - 1) q^19 + (b6 + b1 + 1) * q^21 + (-b7 + b6 - 2b5 - b4 - b2) q^23 + (b7 - 2b6 + b5 + b4 - b3 - b2 - 2) q^27 + (b7 - b6 + b5 + b4 + b2 + b1 - 1) * q^29 + (-b6 - b5 + b4 + b1 + 1) * q^31 + (-2b7 + b6 - 2b5 + b4 + b3 + b1 - 1) * q^33 + q^37 + (b6 - 2b5 - b4 + b3 + b2 + 2b1 + 1) * q^39 + (-b7 - b5 + 2b4 + b2 + 2b1 + 1) * q^41 + (-b7 + b6 - b3 + 3b2 + 3b1 + 2) * q^43 + (b7 + b5 + 2b4 - 2b3 + 2b2) q^47 + (-b6 - b5 + b4 - b3 - 2b2 - 2) q^49 + (-b7 + 2b6 - 2b4 + b2 + 3b1 + 4) q^51 + (-b5 - b4 + 3b3 - 3) q^53 + (2b7 - 3b6 + b5 - b2 - 4b1 - 1) q^57 + (b7 + b6 + b5 - b4 - b3 - b2 - b1 - 2) * q^59 + (2b7 + b6 + b5 + b4 - 2b2 - b1 - 3) * q^61 + (b6 - b5 + b4 - b3 - 1) * q^63 + (2b6 - b4 + b3 + b2 + 2b1 - 2) * q^67 + (-b7 + b6 + 2b5 - 2b3 + b1 - 1) * q^69 + (b7 - b6 + b5 - 4b4 - b2 + b1 + 3) q^71 + (-b7 - b6 + b5 - b3 - b2 + 2b1 - 4) q^73 + (-b7 + b6 - 2b5 + b3 - 2b1 + 1) * q^77 + (-4b5 + b4 + 2b3 - b2 - b1 - 1) * q^79 + (-2b7 - 2b5 + 2b4 + b2 + 2b1 + 1) * q^81 + (2b7 - 4b6 + b5 - b3 - 3b2 - 4) q^83 + (b7 - 2b6 + b4 - 2) q^87 + (-b7 - 2b6 - 2b5 + b4 + 3b1 - 2) q^89 + (-2b6 - 2b5 + b4 - b3 - b2 + b1 + 2) * q^91 + (2b7 - 2b6 + 3b5 + b4 - b3 - b1 - 4) q^93 + (-b7 + 4b6 - 2b5 + 3b3 + b2 - 3b1) * q^97 + (b5 - b4 - 2b3 - b2 - 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q - q^{3} - 4 q^{7} + q^{9} - 9 q^{13} - 4 q^{17} + 4 q^{19} + 6 q^{21} - 2 q^{23} - 7 q^{27} - 5 q^{29} + 11 q^{31} - 15 q^{33} + 8 q^{37} + 8 q^{39} - 4 q^{43} - 14 q^{47} - 10 q^{49} + 23 q^{51}+ \cdots - 19 q^{99}+O(q^{100})$ 8 * q - q^3 - 4 * q^7 + q^9 - 9 * q^13 - 4 * q^17 + 4 * q^19 + 6 * q^21 - 2 * q^23 - 7 * q^27 - 5 * q^29 + 11 * q^31 - 15 * q^33 + 8 * q^37 + 8 * q^39 - 4 * q^43 - 14 * q^47 - 10 * q^49 + 23 * q^51 - 11 * q^53 + 9 * q^57 - 15 * q^59 - 13 * q^61 - 16 * q^63 - 19 * q^67 - 22 * q^69 + 40 * q^71 - 31 * q^73 + 3 * q^77 + 2 * q^79 - 4 * q^81 - 4 * q^83 - 7 * q^87 - 12 * q^89 + 22 * q^91 - 24 * q^93 - 11 * q^97 - 19 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - x^{7} - 12x^{6} + 8x^{5} + 39x^{4} - 14x^{3} - 36x^{2} + 11x + 6$ x^8 - x^7 - 12*x^6 + 8*x^5 + 39*x^4 - 14*x^3 - 36*x^2 + 11*x + 6 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$2\nu^{7} - 2\nu^{6} - 22\nu^{5} + 13\nu^{4} + 58\nu^{3} - 5\nu^{2} - 30\nu - 4$ 2v^7 - 2v^6 - 22v^5 + 13v^4 + 58v^3 - 5v^2 - 30*v - 4
$\beta_{3}$	$=$	$3\nu^{7} + \nu^{6} - 39\nu^{5} - 21\nu^{4} + 132\nu^{3} + 81\nu^{2} - 90\nu - 30$ 3v^7 + v^6 - 39v^5 - 21v^4 + 132v^3 + 81v^2 - 90v - 30
$\beta_{4}$	$=$	$-5\nu^{7} + 2\nu^{6} + 59\nu^{5} - \nu^{4} - 174\nu^{3} - 62\nu^{2} + 99\nu + 34$ -5v^7 + 2v^6 + 59v^5 - v^4 - 174v^3 - 62v^2 + 99v + 34
$\beta_{5}$	$=$	$4\nu^{7} - 50\nu^{5} - 15\nu^{4} + 161\nu^{3} + 82\nu^{2} - 104\nu - 34$ 4v^7 - 50v^5 - 15v^4 + 161v^3 + 82v^2 - 104v - 34
$\beta_{6}$	$=$	$-7\nu^{7} + 2\nu^{6} + 84\nu^{5} + 6\nu^{4} - 254\nu^{3} - 99\nu^{2} + 149\nu + 47$ -7v^7 + 2v^6 + 84v^5 + 6v^4 - 254v^3 - 99v^2 + 149*v + 47
$\beta_{7}$	$=$	$-8\nu^{7} + \nu^{6} + 98\nu^{5} + 20\nu^{4} - 306\nu^{3} - 142\nu^{2} + 189\nu + 62$ -8v^7 + v^6 + 98v^5 + 20v^4 - 306v^3 - 142v^2 + 189v + 62

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{7} - \beta_{4} + \beta_{3} + 2$ b7 - b4 + b3 + 2
$\nu^{3}$	$=$	$-\beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 2$ -b7 + 2b6 - b5 - b4 + b3 + b2 + 6b1 + 2
$\nu^{4}$	$=$	$7\beta_{7} - 2\beta_{5} - 7\beta_{4} + 9\beta_{3} + \beta_{2} + 2\beta _1 + 10$ 7b7 - 2b5 - 7b4 + 9b3 + b2 + 2*b1 + 10
$\nu^{5}$	$=$	$-10\beta_{7} + 19\beta_{6} - 14\beta_{5} - 11\beta_{4} + 12\beta_{3} + 9\beta_{2} + 42\beta _1 + 21$ -10b7 + 19b6 - 14b5 - 11b4 + 12b3 + 9b2 + 42*b1 + 21
$\nu^{6}$	$=$	$45\beta_{7} + 6\beta_{6} - 30\beta_{5} - 54\beta_{4} + 76\beta_{3} + 12\beta_{2} + 27\beta _1 + 72$ 45b7 + 6b6 - 30b5 - 54b4 + 76b3 + 12b2 + 27*b1 + 72
$\nu^{7}$	$=$	$-79\beta_{7} + 157\beta_{6} - 142\beta_{5} - 103\beta_{4} + 123\beta_{3} + 76\beta_{2} + 317\beta _1 + 187$ -79b7 + 157b6 - 142b5 - 103b4 + 123b3 + 76b2 + 317*b1 + 187

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

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

2.95803
2.07336
0.819833
0.800175
−0.305014
−1.31809
−1.42472
−2.60358

−2.95803

−0.939702

5.74994

1.2

−2.07336

−2.31363

1.29882

1.3

−0.819833

3.46626

−2.32787

1.4

−0.800175

−4.46764

−2.35972

1.5

0.305014

0.258050

−2.90697

1.6

1.31809

2.07329

−1.26265

1.7

1.42472

−0.242089

−0.970174

1.8

2.60358

−1.83454

3.77862

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$-1$
$37$	$-1$

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	7400.2.a.u	✓	8
5.b	even	2	1		7400.2.a.v	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7400.2.a.u	✓	8	1.a	even	1	1	trivial
7400.2.a.v	yes	8	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7400))$ :

T_{3}^{8} + T_{3}^{7} - 12T_{3}^{6} - 8T_{3}^{5} + 39T_{3}^{4} + 14T_{3}^{3} - 36T_{3}^{2} - 11T_{3} + 6

T_{7}^{8} + 4T_{7}^{7} - 15T_{7}^{6} - 62T_{7}^{5} + 18T_{7}^{4} + 195T_{7}^{3} + 124T_{7}^{2} - 14T_{7} - 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$T^{8} + T^{7} - 12 T^{6} + \cdots + 6$ T^8 + T^7 - 12T^6 - 8T^5 + 39T^4 + 14T^3 - 36T^2 - 11T + 6
$5$	$T^{8}$ T^8
$7$	$T^{8} + 4 T^{7} + \cdots - 8$ T^8 + 4T^7 - 15T^6 - 62T^5 + 18T^4 + 195T^3 + 124T^2 - 14*T - 8
$11$	$T^{8} - 48 T^{6} + \cdots - 288$ T^8 - 48T^6 + 21T^5 + 536T^4 + 28T^3 - 1821T^2 - 1443T - 288
$13$	$T^{8} + 9 T^{7} + \cdots - 1068$ T^8 + 9T^7 - 5T^6 - 185T^5 - 126T^4 + 1077T^3 + 614T^2 - 1554*T - 1068
$17$	$T^{8} + 4 T^{7} + \cdots - 7528$ T^8 + 4T^7 - 65T^6 - 331T^5 + 706T^4 + 5416T^3 + 4318T^2 - 7663*T - 7528
$19$	$T^{8} - 4 T^{7} + \cdots - 1641$ T^8 - 4T^7 - 63T^6 + 176T^5 + 663T^4 - 1505T^3 - 1220T^2 + 3681*T - 1641
$23$	$T^{8} + 2 T^{7} + \cdots - 41788$ T^8 + 2T^7 - 90T^6 - 21T^5 + 2579T^4 - 3412T^3 - 21715T^2 + 60516*T - 41788
$29$	$T^{8} + 5 T^{7} + \cdots + 1264$ T^8 + 5T^7 - 45T^6 - 215T^5 + 368T^4 + 1775T^3 - 1034T^2 - 3928*T + 1264
$31$	$T^{8} - 11 T^{7} + \cdots - 54888$ T^8 - 11T^7 - 17T^6 + 482T^5 - 513T^4 - 6589T^3 + 11402T^2 + 28686*T - 54888
$37$	$(T - 1)^{8}$ (T - 1)^8
$41$	$T^{8} - 104 T^{6} + \cdots + 14499$ T^8 - 104T^6 - 141T^5 + 3145T^4 + 6193T^3 - 29413T^2 - 64836T + 14499
$43$	$T^{8} + 4 T^{7} + \cdots - 931052$ T^8 + 4T^7 - 268T^6 - 1011T^5 + 22861T^4 + 86244T^3 - 596065T^2 - 2615436*T - 931052
$47$	$T^{8} + 14 T^{7} + \cdots + 255168$ T^8 + 14T^7 - 173T^6 - 2404T^5 + 9724T^4 + 106017T^3 - 271126T^2 - 838560*T + 255168
$53$	$T^{8} + 11 T^{7} + \cdots - 65412$ T^8 + 11T^7 - 195T^6 - 2346T^5 + 6102T^4 + 96124T^3 - 4457T^2 - 624652*T - 65412
$59$	$T^{8} + 15 T^{7} + \cdots + 585268$ T^8 + 15T^7 - 65T^6 - 1531T^5 + 160T^4 + 43471T^3 - 12319T^2 - 431496*T + 585268
$61$	$T^{8} + 13 T^{7} + \cdots - 24587796$ T^8 + 13T^7 - 325T^6 - 4386T^5 + 31249T^4 + 464689T^3 - 631964T^2 - 15457794*T - 24587796
$67$	$T^{8} + 19 T^{7} + \cdots + 13278$ T^8 + 19T^7 + 23T^6 - 1350T^5 - 7275T^4 - 618T^3 + 48175T^2 + 68185*T + 13278
$71$	$T^{8} - 40 T^{7} + \cdots - 22260148$ T^8 - 40T^7 + 331T^6 + 4912T^5 - 72858T^4 - 42309T^3 + 2787714T^2 - 173714*T - 22260148
$73$	$T^{8} + 31 T^{7} + \cdots + 2429664$ T^8 + 31T^7 + 275T^6 - 575T^5 - 21972T^4 - 105384T^3 + 16552T^2 + 1269312*T + 2429664
$79$	$T^{8} - 2 T^{7} + \cdots + 298156$ T^8 - 2T^7 - 391T^6 + 1423T^5 + 36104T^4 - 139918T^3 - 500037T^2 - 69148*T + 298156
$83$	$T^{8} + 4 T^{7} + \cdots + 35726136$ T^8 + 4T^7 - 363T^6 - 1467T^5 + 42016T^4 + 125978T^3 - 2041698T^2 - 3062845*T + 35726136
$89$	$T^{8} + 12 T^{7} + \cdots - 380742$ T^8 + 12T^7 - 253T^6 - 3051T^5 + 15458T^4 + 212906T^3 + 47828T^2 - 2130741*T - 380742
$97$	$T^{8} + 11 T^{7} + \cdots + 2651024$ T^8 + 11T^7 - 435T^6 - 4330T^5 + 27319T^4 + 117667T^3 - 517564T^2 - 727700*T + 2651024
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