# Properties

 Label 1450.2.b.k Level $1450$ Weight $2$ Character orbit 1450.b Analytic conductor $11.578$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1450,2,Mod(349,1450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1450.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1450 = 2 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1450.b (of order $$2$$, degree $$1$$, 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: $$11.5783082931$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.399424.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$$ x^6 - 2*x^5 + 3*x^4 - 6*x^3 + 6*x^2 - 8*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} - \beta_{5} q^{3} - q^{4} + \beta_{3} q^{6} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{7} + \beta_{2} q^{8} - \beta_1 q^{9}+O(q^{10})$$ q - b2 * q^2 - b5 * q^3 - q^4 + b3 * q^6 + (b5 - b4 - b2) * q^7 + b2 * q^8 - b1 * q^9 $$q - \beta_{2} q^{2} - \beta_{5} q^{3} - q^{4} + \beta_{3} q^{6} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{7} + \beta_{2} q^{8} - \beta_1 q^{9} + ( - 2 \beta_{3} - \beta_1 + 1) q^{11} + \beta_{5} q^{12} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{13} + ( - \beta_{3} + \beta_1 - 1) q^{14} + q^{16} + (2 \beta_{5} + \beta_{4} - 3 \beta_{2}) q^{17} - \beta_{4} q^{18} - \beta_{3} q^{19} + (2 \beta_1 + 4) q^{21} + ( - 2 \beta_{5} - \beta_{4} - \beta_{2}) q^{22} + (\beta_{5} + 3 \beta_{4} - 3 \beta_{2}) q^{23} - \beta_{3} q^{24} + (\beta_{3} + \beta_1 + 1) q^{26} + ( - 2 \beta_{5} - \beta_{4} + \beta_{2}) q^{27} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{28} - q^{29} + ( - 4 \beta_{3} - 3 \beta_1) q^{31} - \beta_{2} q^{32} + (\beta_{4} - 5 \beta_{2}) q^{33} + ( - 2 \beta_{3} - \beta_1 - 3) q^{34} + \beta_1 q^{36} + (\beta_{5} + 2 \beta_{4} - 5 \beta_{2}) q^{37} - \beta_{5} q^{38} + ( - 2 \beta_{3} - 2) q^{39} + (\beta_{3} + 2 \beta_1 + 6) q^{41} + (2 \beta_{4} - 4 \beta_{2}) q^{42} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{43} + (2 \beta_{3} + \beta_1 - 1) q^{44} + ( - \beta_{3} - 3 \beta_1 - 3) q^{46} + (3 \beta_{4} - 4 \beta_{2}) q^{47} - \beta_{5} q^{48} + (2 \beta_{3} - 3) q^{49} + (4 \beta_{3} + \beta_1 + 5) q^{51} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{52} + ( - 3 \beta_{5} - 3 \beta_{4} - \beta_{2}) q^{53} + (2 \beta_{3} + \beta_1 + 1) q^{54} + (\beta_{3} - \beta_1 + 1) q^{56} + (\beta_{4} - 3 \beta_{2}) q^{57} + \beta_{2} q^{58} + (\beta_{3} + 2 \beta_1 - 5) q^{59} + ( - 5 \beta_{3} - 1) q^{61} + ( - 4 \beta_{5} - 3 \beta_{4}) q^{62} + ( - 3 \beta_{5} - \beta_{4} - 5 \beta_{2}) q^{63} - q^{64} + ( - \beta_1 - 5) q^{66} + ( - \beta_{5} - 3 \beta_{4}) q^{67} + ( - 2 \beta_{5} - \beta_{4} + 3 \beta_{2}) q^{68} + (6 \beta_{3} - 2 \beta_1) q^{69} + (\beta_{3} + 3 \beta_1 + 3) q^{71} + \beta_{4} q^{72} + (5 \beta_{5} - 6 \beta_{2}) q^{73} + ( - \beta_{3} - 2 \beta_1 - 5) q^{74} + \beta_{3} q^{76} + ( - 2 \beta_{5} - 6 \beta_{4} + 2 \beta_{2}) q^{77} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{78} + 6 \beta_1 q^{79} + ( - 2 \beta_{3} - 4 \beta_1 - 5) q^{81} + (\beta_{5} + 2 \beta_{4} - 6 \beta_{2}) q^{82} + (4 \beta_{5} + 3 \beta_{4} - 5 \beta_{2}) q^{83} + ( - 2 \beta_1 - 4) q^{84} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{86} + \beta_{5} q^{87} + (2 \beta_{5} + \beta_{4} + \beta_{2}) q^{88} + ( - \beta_{3} + 2 \beta_1 - 4) q^{89} + (4 \beta_{3} + 2 \beta_1) q^{91} + ( - \beta_{5} - 3 \beta_{4} + 3 \beta_{2}) q^{92} + (3 \beta_{5} + \beta_{4} - 9 \beta_{2}) q^{93} + ( - 3 \beta_1 - 4) q^{94} + \beta_{3} q^{96} + ( - 3 \beta_{5} - \beta_{4} - 7 \beta_{2}) q^{97} + (2 \beta_{5} + 3 \beta_{2}) q^{98} + ( - 4 \beta_1 + 2) q^{99}+O(q^{100})$$ q - b2 * q^2 - b5 * q^3 - q^4 + b3 * q^6 + (b5 - b4 - b2) * q^7 + b2 * q^8 - b1 * q^9 + (-2*b3 - b1 + 1) * q^11 + b5 * q^12 + (-b5 - b4 + b2) * q^13 + (-b3 + b1 - 1) * q^14 + q^16 + (2*b5 + b4 - 3*b2) * q^17 - b4 * q^18 - b3 * q^19 + (2*b1 + 4) * q^21 + (-2*b5 - b4 - b2) * q^22 + (b5 + 3*b4 - 3*b2) * q^23 - b3 * q^24 + (b3 + b1 + 1) * q^26 + (-2*b5 - b4 + b2) * q^27 + (-b5 + b4 + b2) * q^28 - q^29 + (-4*b3 - 3*b1) * q^31 - b2 * q^32 + (b4 - 5*b2) * q^33 + (-2*b3 - b1 - 3) * q^34 + b1 * q^36 + (b5 + 2*b4 - 5*b2) * q^37 - b5 * q^38 + (-2*b3 - 2) * q^39 + (b3 + 2*b1 + 6) * q^41 + (2*b4 - 4*b2) * q^42 + (2*b5 - 2*b4 - 2*b2) * q^43 + (2*b3 + b1 - 1) * q^44 + (-b3 - 3*b1 - 3) * q^46 + (3*b4 - 4*b2) * q^47 - b5 * q^48 + (2*b3 - 3) * q^49 + (4*b3 + b1 + 5) * q^51 + (b5 + b4 - b2) * q^52 + (-3*b5 - 3*b4 - b2) * q^53 + (2*b3 + b1 + 1) * q^54 + (b3 - b1 + 1) * q^56 + (b4 - 3*b2) * q^57 + b2 * q^58 + (b3 + 2*b1 - 5) * q^59 + (-5*b3 - 1) * q^61 + (-4*b5 - 3*b4) * q^62 + (-3*b5 - b4 - 5*b2) * q^63 - q^64 + (-b1 - 5) * q^66 + (-b5 - 3*b4) * q^67 + (-2*b5 - b4 + 3*b2) * q^68 + (6*b3 - 2*b1) * q^69 + (b3 + 3*b1 + 3) * q^71 + b4 * q^72 + (5*b5 - 6*b2) * q^73 + (-b3 - 2*b1 - 5) * q^74 + b3 * q^76 + (-2*b5 - 6*b4 + 2*b2) * q^77 + (-2*b5 + 2*b2) * q^78 + 6*b1 * q^79 + (-2*b3 - 4*b1 - 5) * q^81 + (b5 + 2*b4 - 6*b2) * q^82 + (4*b5 + 3*b4 - 5*b2) * q^83 + (-2*b1 - 4) * q^84 + (-2*b3 + 2*b1 - 2) * q^86 + b5 * q^87 + (2*b5 + b4 + b2) * q^88 + (-b3 + 2*b1 - 4) * q^89 + (4*b3 + 2*b1) * q^91 + (-b5 - 3*b4 + 3*b2) * q^92 + (3*b5 + b4 - 9*b2) * q^93 + (-3*b1 - 4) * q^94 + b3 * q^96 + (-3*b5 - b4 - 7*b2) * q^97 + (2*b5 + 3*b2) * q^98 + (-4*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 2 q^{6}+O(q^{10})$$ 6 * q - 6 * q^4 - 2 * q^6 $$6 q - 6 q^{4} - 2 q^{6} + 10 q^{11} - 4 q^{14} + 6 q^{16} + 2 q^{19} + 24 q^{21} + 2 q^{24} + 4 q^{26} - 6 q^{29} + 8 q^{31} - 14 q^{34} - 8 q^{39} + 34 q^{41} - 10 q^{44} - 16 q^{46} - 22 q^{49} + 22 q^{51} + 2 q^{54} + 4 q^{56} - 32 q^{59} + 4 q^{61} - 6 q^{64} - 30 q^{66} - 12 q^{69} + 16 q^{71} - 28 q^{74} - 2 q^{76} - 26 q^{81} - 24 q^{84} - 8 q^{86} - 22 q^{89} - 8 q^{91} - 24 q^{94} - 2 q^{96} + 12 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 - 2 * q^6 + 10 * q^11 - 4 * q^14 + 6 * q^16 + 2 * q^19 + 24 * q^21 + 2 * q^24 + 4 * q^26 - 6 * q^29 + 8 * q^31 - 14 * q^34 - 8 * q^39 + 34 * q^41 - 10 * q^44 - 16 * q^46 - 22 * q^49 + 22 * q^51 + 2 * q^54 + 4 * q^56 - 32 * q^59 + 4 * q^61 - 6 * q^64 - 30 * q^66 - 12 * q^69 + 16 * q^71 - 28 * q^74 - 2 * q^76 - 26 * q^81 - 24 * q^84 - 8 * q^86 - 22 * q^89 - 8 * q^91 - 24 * q^94 - 2 * q^96 + 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{4} + 2\nu^{3} - \nu^{2} + 2\nu - 2 ) / 2$$ (-v^4 + 2*v^3 - v^2 + 2*v - 2) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} - 3\nu^{3} + 4\nu^{2} - 2\nu + 8 ) / 4$$ (-v^5 - 3*v^3 + 4*v^2 - 2*v + 8) / 4 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 2\nu + 4 ) / 4$$ (-v^5 + 2*v^4 - 3*v^3 + 6*v^2 - 2*v + 4) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} + 3\nu^{3} - 6\nu^{2} + 10\nu - 8 ) / 4$$ (v^5 - 2*v^4 + 3*v^3 - 6*v^2 + 10*v - 8) / 4 $$\beta_{5}$$ $$=$$ $$( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 2\nu - 12 ) / 4$$ (3*v^5 - 2*v^4 + 5*v^3 - 6*v^2 + 2*v - 12) / 4
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} + 1 ) / 2$$ (b4 + b3 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta_1 ) / 2$$ (b5 + 2*b3 + b2 + b1) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 3 ) / 2$$ (-b4 + b3 - 2*b2 + 2*b1 + 3) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{5} + 2\beta_{3} - 5\beta_{2} - \beta _1 + 4 ) / 2$$ (-b5 + 2*b3 - 5*b2 - b1 + 4) / 2 $$\nu^{5}$$ $$=$$ $$( 4\beta_{5} + \beta_{4} + 3\beta_{3} + 2\beta_{2} - 2\beta _1 + 5 ) / 2$$ (4*b5 + b4 + 3*b3 + 2*b2 - 2*b1 + 5) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1277$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 −0.671462 − 1.24464i 0.264658 + 1.38923i 1.40680 − 0.144584i 1.40680 + 0.144584i 0.264658 − 1.38923i −0.671462 + 1.24464i
1.00000i 2.34292i −1.00000 0 −2.34292 3.83221i 1.00000i −2.48929 0
349.2 1.00000i 0.470683i −1.00000 0 −0.470683 3.30777i 1.00000i 2.77846 0
349.3 1.00000i 1.81361i −1.00000 0 1.81361 2.52444i 1.00000i −0.289169 0
349.4 1.00000i 1.81361i −1.00000 0 1.81361 2.52444i 1.00000i −0.289169 0
349.5 1.00000i 0.470683i −1.00000 0 −0.470683 3.30777i 1.00000i 2.77846 0
349.6 1.00000i 2.34292i −1.00000 0 −2.34292 3.83221i 1.00000i −2.48929 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1450.2.b.k 6
5.b even 2 1 inner 1450.2.b.k 6
5.c odd 4 1 1450.2.a.q 3
5.c odd 4 1 1450.2.a.s yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1450.2.a.q 3 5.c odd 4 1
1450.2.a.s yes 3 5.c odd 4 1
1450.2.b.k 6 1.a even 1 1 trivial
1450.2.b.k 6 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1450, [\chi])$$:

 $$T_{3}^{6} + 9T_{3}^{4} + 20T_{3}^{2} + 4$$ T3^6 + 9*T3^4 + 20*T3^2 + 4 $$T_{7}^{6} + 32T_{7}^{4} + 324T_{7}^{2} + 1024$$ T7^6 + 32*T7^4 + 324*T7^2 + 1024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$T^{6} + 9 T^{4} + \cdots + 4$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 32 T^{4} + \cdots + 1024$$
$11$ $$(T^{3} - 5 T^{2} - 8 T + 44)^{2}$$
$13$ $$T^{6} + 16 T^{4} + \cdots + 64$$
$17$ $$T^{6} + 49 T^{4} + \cdots + 16$$
$19$ $$(T^{3} - T^{2} - 4 T + 2)^{2}$$
$23$ $$T^{6} + 132 T^{4} + \cdots + 71824$$
$29$ $$(T + 1)^{6}$$
$31$ $$(T^{3} - 4 T^{2} + \cdots + 158)^{2}$$
$37$ $$T^{6} + 114 T^{4} + \cdots + 3364$$
$41$ $$(T^{3} - 17 T^{2} + 72 T + 2)^{2}$$
$43$ $$T^{6} + 128 T^{4} + \cdots + 65536$$
$47$ $$T^{6} + 174 T^{4} + \cdots + 58564$$
$53$ $$T^{6} + 144 T^{4} + \cdots + 1024$$
$59$ $$(T^{3} + 16 T^{2} + \cdots + 68)^{2}$$
$61$ $$(T^{3} - 2 T^{2} + \cdots + 146)^{2}$$
$67$ $$T^{6} + 111 T^{4} + \cdots + 14641$$
$71$ $$(T^{3} - 8 T^{2} + \cdots + 268)^{2}$$
$73$ $$T^{6} + 273 T^{4} + \cdots + 98596$$
$79$ $$(T^{3} - 252 T + 432)^{2}$$
$83$ $$T^{6} + 209 T^{4} + \cdots + 44944$$
$89$ $$(T^{3} + 11 T^{2} - 158)^{2}$$
$97$ $$T^{6} + 260 T^{4} + \cdots + 26896$$