# Properties

 Label 3626.2.a.bh Level $3626$ Weight $2$ Character orbit 3626.a Self dual yes Analytic conductor $28.954$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3626,2,Mod(1,3626)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3626, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3626.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3626 = 2 \cdot 7^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3626.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: $$28.9537557729$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 32x^{8} + 323x^{6} - 1412x^{4} + 2721x^{2} - 1800$$ x^10 - 32*x^8 + 323*x^6 - 1412*x^4 + 2721*x^2 - 1800 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta_{8} q^{3} + q^{4} - \beta_1 q^{5} + \beta_{8} q^{6} + q^{8} + ( - \beta_{3} - \beta_{2} + 4) q^{9}+O(q^{10})$$ q + q^2 + b8 * q^3 + q^4 - b1 * q^5 + b8 * q^6 + q^8 + (-b3 - b2 + 4) * q^9 $$q + q^{2} + \beta_{8} q^{3} + q^{4} - \beta_1 q^{5} + \beta_{8} q^{6} + q^{8} + ( - \beta_{3} - \beta_{2} + 4) q^{9} - \beta_1 q^{10} + ( - \beta_{5} + 1) q^{11} + \beta_{8} q^{12} + ( - \beta_{9} - \beta_1) q^{13} + (\beta_{6} + \beta_{5} - \beta_{3} + 1) q^{15} + q^{16} + ( - \beta_{7} + 2 \beta_{4}) q^{17} + ( - \beta_{3} - \beta_{2} + 4) q^{18} + (\beta_{7} - 2 \beta_{4}) q^{19} - \beta_1 q^{20} + ( - \beta_{5} + 1) q^{22} + ( - \beta_{6} - \beta_{5} - \beta_{2}) q^{23} + \beta_{8} q^{24} + (\beta_{5} + 2 \beta_{3} + \beta_{2} + 1) q^{25} + ( - \beta_{9} - \beta_1) q^{26} + (3 \beta_{8} + \beta_{7} + \cdots - \beta_1) q^{27}+ \cdots + ( - 6 \beta_{5} - 3 \beta_{2} - 3) q^{99}+O(q^{100})$$ q + q^2 + b8 * q^3 + q^4 - b1 * q^5 + b8 * q^6 + q^8 + (-b3 - b2 + 4) * q^9 - b1 * q^10 + (-b5 + 1) * q^11 + b8 * q^12 + (-b9 - b1) * q^13 + (b6 + b5 - b3 + 1) * q^15 + q^16 + (-b7 + 2*b4) * q^17 + (-b3 - b2 + 4) * q^18 + (b7 - 2*b4) * q^19 - b1 * q^20 + (-b5 + 1) * q^22 + (-b6 - b5 - b2) * q^23 + b8 * q^24 + (b5 + 2*b3 + b2 + 1) * q^25 + (-b9 - b1) * q^26 + (3*b8 + b7 + 2*b4 - b1) * q^27 + (-b5 - b2 + 4) * q^29 + (b6 + b5 - b3 + 1) * q^30 + (b9 + b8 - b4 + 2*b1) * q^31 + q^32 + (2*b9 - b8 - 2*b7 - 3*b4 + b1) * q^33 + (-b7 + 2*b4) * q^34 + (-b3 - b2 + 4) * q^36 - q^37 + (b7 - 2*b4) * q^38 + (3*b5 + 4*b2 - 1) * q^39 - b1 * q^40 + (-b8 + 2*b4) * q^41 + (2*b3 + 2*b2) * q^43 + (-b5 + 1) * q^44 + (-2*b9 + 3*b8 + 2*b7 - 3*b4 - b1) * q^45 + (-b6 - b5 - b2) * q^46 + (2*b8 + 4*b4) * q^47 + b8 * q^48 + (b5 + 2*b3 + b2 + 1) * q^50 + (-4*b6 - 2*b5 + 2*b2 + 2) * q^51 + (-b9 - b1) * q^52 + (-2*b6 - 2*b5 + 2*b3) * q^53 + (3*b8 + b7 + 2*b4 - b1) * q^54 + (b9 - 2*b8 - b7 - 3*b4 - b1) * q^55 + (4*b6 + 2*b5 - 2*b2 - 2) * q^57 + (-b5 - b2 + 4) * q^58 + (2*b8 + b7) * q^59 + (b6 + b5 - b3 + 1) * q^60 + (b9 - b8 - b4 + 2*b1) * q^61 + (b9 + b8 - b4 + 2*b1) * q^62 + q^64 + (-2*b6 + b2 + 5) * q^65 + (2*b9 - b8 - 2*b7 - 3*b4 + b1) * q^66 + (2*b6 + b5 - 2*b2 - 1) * q^67 + (-b7 + 2*b4) * q^68 + (2*b9 - b7 + 5*b4 + 3*b1) * q^69 + (2*b6 + 4) * q^71 + (-b3 - b2 + 4) * q^72 + (-b9 - b8 - b7 - 3*b4) * q^73 - q^74 + (-b9 + b7 + 2*b1) * q^75 + (b7 - 2*b4) * q^76 + (3*b5 + 4*b2 - 1) * q^78 + (b5 - b2) * q^79 - b1 * q^80 + (b6 + 3*b5 - b3 - 2*b2 + 8) * q^81 + (-b8 + 2*b4) * q^82 + (-2*b8 + b4 + 2*b1) * q^83 + (2*b5 + 2*b3 + 4*b2 - 2) * q^85 + (2*b3 + 2*b2) * q^86 + (3*b9 + 3*b8 - b7 - 2*b4 + 2*b1) * q^87 + (-b5 + 1) * q^88 + (-2*b8 - b7 - 2*b1) * q^89 + (-2*b9 + 3*b8 + 2*b7 - 3*b4 - b1) * q^90 + (-b6 - b5 - b2) * q^92 + (-4*b5 - 5*b2 + 7) * q^93 + (2*b8 + 4*b4) * q^94 + (-2*b5 - 2*b3 - 4*b2 + 2) * q^95 + b8 * q^96 + (b7 + 4*b4 - 2*b1) * q^97 + (-6*b5 - 3*b2 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 10 q^{2} + 10 q^{4} + 10 q^{8} + 38 q^{9}+O(q^{10})$$ 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^8 + 38 * q^9 $$10 q + 10 q^{2} + 10 q^{4} + 10 q^{8} + 38 q^{9} + 8 q^{11} + 12 q^{15} + 10 q^{16} + 38 q^{18} + 8 q^{22} - 4 q^{23} + 14 q^{25} + 36 q^{29} + 12 q^{30} + 10 q^{32} + 38 q^{36} - 10 q^{37} + 4 q^{39} + 4 q^{43} + 8 q^{44} - 4 q^{46} + 14 q^{50} + 20 q^{51} - 4 q^{53} - 20 q^{57} + 36 q^{58} + 12 q^{60} + 10 q^{64} + 52 q^{65} - 12 q^{67} + 40 q^{71} + 38 q^{72} - 10 q^{74} + 4 q^{78} + 82 q^{81} - 8 q^{85} + 4 q^{86} + 8 q^{88} - 4 q^{92} + 52 q^{93} + 8 q^{95} - 48 q^{99}+O(q^{100})$$ 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^8 + 38 * q^9 + 8 * q^11 + 12 * q^15 + 10 * q^16 + 38 * q^18 + 8 * q^22 - 4 * q^23 + 14 * q^25 + 36 * q^29 + 12 * q^30 + 10 * q^32 + 38 * q^36 - 10 * q^37 + 4 * q^39 + 4 * q^43 + 8 * q^44 - 4 * q^46 + 14 * q^50 + 20 * q^51 - 4 * q^53 - 20 * q^57 + 36 * q^58 + 12 * q^60 + 10 * q^64 + 52 * q^65 - 12 * q^67 + 40 * q^71 + 38 * q^72 - 10 * q^74 + 4 * q^78 + 82 * q^81 - 8 * q^85 + 4 * q^86 + 8 * q^88 - 4 * q^92 + 52 * q^93 + 8 * q^95 - 48 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 32x^{8} + 323x^{6} - 1412x^{4} + 2721x^{2} - 1800$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 2\nu^{8} - 76\nu^{6} + 925\nu^{4} - 3949\nu^{2} + 4887 ) / 177$$ (2*v^8 - 76*v^6 + 925*v^4 - 3949*v^2 + 4887) / 177 $$\beta_{3}$$ $$=$$ $$( 16\nu^{8} - 431\nu^{6} + 2975\nu^{4} - 7166\nu^{2} + 4758 ) / 177$$ (16*v^8 - 431*v^6 + 2975*v^4 - 7166*v^2 + 4758) / 177 $$\beta_{4}$$ $$=$$ $$( 97\nu^{9} - 2624\nu^{7} + 18401\nu^{5} - 47714\nu^{3} + 43647\nu ) / 5310$$ (97*v^9 - 2624*v^7 + 18401*v^5 - 47714*v^3 + 43647*v) / 5310 $$\beta_{5}$$ $$=$$ $$( -34\nu^{8} + 938\nu^{6} - 6875\nu^{4} + 18458\nu^{2} - 15465 ) / 177$$ (-34*v^8 + 938*v^6 - 6875*v^4 + 18458*v^2 - 15465) / 177 $$\beta_{6}$$ $$=$$ $$( -37\nu^{8} + 1052\nu^{6} - 8174\nu^{4} + 22523\nu^{2} - 17574 ) / 177$$ (-37*v^8 + 1052*v^6 - 8174*v^4 + 22523*v^2 - 17574) / 177 $$\beta_{7}$$ $$=$$ $$( 157\nu^{9} - 4904\nu^{7} + 46151\nu^{5} - 166184\nu^{3} + 190257\nu ) / 5310$$ (157*v^9 - 4904*v^7 + 46151*v^5 - 166184*v^3 + 190257*v) / 5310 $$\beta_{8}$$ $$=$$ $$( 209\nu^{9} - 5818\nu^{7} + 43297\nu^{5} - 114868\nu^{3} + 87219\nu ) / 1770$$ (209*v^9 - 5818*v^7 + 43297*v^5 - 114868*v^3 + 87219*v) / 1770 $$\beta_{9}$$ $$=$$ $$( 341\nu^{9} - 9772\nu^{7} + 77443\nu^{5} - 222397\nu^{3} + 190281\nu ) / 2655$$ (341*v^9 - 9772*v^7 + 77443*v^5 - 222397*v^3 + 190281*v) / 2655
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{3} + \beta_{2} + 6$$ b5 + 2*b3 + b2 + 6 $$\nu^{3}$$ $$=$$ $$-3\beta_{9} + 4\beta_{8} + 2\beta_{7} - 8\beta_{4} + 12\beta_1$$ -3*b9 + 4*b8 + 2*b7 - 8*b4 + 12*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{6} + 19\beta_{5} + 42\beta_{3} + 24\beta_{2} + 67$$ 2*b6 + 19*b5 + 42*b3 + 24*b2 + 67 $$\nu^{5}$$ $$=$$ $$-69\beta_{9} + 88\beta_{8} + 49\beta_{7} - 163\beta_{4} + 193\beta_1$$ -69*b9 + 88*b8 + 49*b7 - 163*b4 + 193*b1 $$\nu^{6}$$ $$=$$ $$50\beta_{6} + 337\beta_{5} + 775\beta_{3} + 454\beta_{2} + 1041$$ 50*b6 + 337*b5 + 775*b3 + 454*b2 + 1041 $$\nu^{7}$$ $$=$$ $$-1313\beta_{9} + 1651\beta_{8} + 942\beta_{7} - 2965\beta_{4} + 3366\beta_1$$ -1313*b9 + 1651*b8 + 942*b7 - 2965*b4 + 3366*b1 $$\nu^{8}$$ $$=$$ $$975\beta_{6} + 5993\beta_{5} + 13974\beta_{3} + 8215\beta_{2} + 17974$$ 975*b6 + 5993*b5 + 13974*b3 + 8215*b2 + 17974 $$\nu^{9}$$ $$=$$ $$-23905\beta_{9} + 29936\beta_{8} + 17171\beta_{7} - 53167\beta_{4} + 59896\beta_1$$ -23905*b9 + 29936*b8 + 17171*b7 - 53167*b4 + 59896*b1

## 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
 2.25483 1.85747 −2.04532 1.17019 −4.23236 4.23236 −1.17019 2.04532 −1.85747 −2.25483
1.00000 −3.27400 1.00000 −2.25483 −3.27400 0 1.00000 7.71911 −2.25483
1.2 1.00000 −3.18017 1.00000 −1.85747 −3.18017 0 1.00000 7.11351 −1.85747
1.3 1.00000 −2.85745 1.00000 2.04532 −2.85745 0 1.00000 5.16501 2.04532
1.4 1.00000 −2.04457 1.00000 −1.17019 −2.04457 0 1.00000 1.18025 −1.17019
1.5 1.00000 −0.906713 1.00000 4.23236 −0.906713 0 1.00000 −2.17787 4.23236
1.6 1.00000 0.906713 1.00000 −4.23236 0.906713 0 1.00000 −2.17787 −4.23236
1.7 1.00000 2.04457 1.00000 1.17019 2.04457 0 1.00000 1.18025 1.17019
1.8 1.00000 2.85745 1.00000 −2.04532 2.85745 0 1.00000 5.16501 −2.04532
1.9 1.00000 3.18017 1.00000 1.85747 3.18017 0 1.00000 7.11351 1.85747
1.10 1.00000 3.27400 1.00000 2.25483 3.27400 0 1.00000 7.71911 2.25483
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$
$$37$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3626.2.a.bh 10
7.b odd 2 1 inner 3626.2.a.bh 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3626.2.a.bh 10 1.a even 1 1 trivial
3626.2.a.bh 10 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3626))$$:

 $$T_{3}^{10} - 34T_{3}^{8} + 427T_{3}^{6} - 2378T_{3}^{4} + 5385T_{3}^{2} - 3042$$ T3^10 - 34*T3^8 + 427*T3^6 - 2378*T3^4 + 5385*T3^2 - 3042 $$T_{5}^{10} - 32T_{5}^{8} + 323T_{5}^{6} - 1412T_{5}^{4} + 2721T_{5}^{2} - 1800$$ T5^10 - 32*T5^8 + 323*T5^6 - 1412*T5^4 + 2721*T5^2 - 1800 $$T_{11}^{5} - 4T_{11}^{4} - 29T_{11}^{3} + 144T_{11}^{2} - 171T_{11} + 54$$ T11^5 - 4*T11^4 - 29*T11^3 + 144*T11^2 - 171*T11 + 54

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{10}$$
$3$ $$T^{10} - 34 T^{8} + \cdots - 3042$$
$5$ $$T^{10} - 32 T^{8} + \cdots - 1800$$
$7$ $$T^{10}$$
$11$ $$(T^{5} - 4 T^{4} - 29 T^{3} + \cdots + 54)^{2}$$
$13$ $$T^{10} - 104 T^{8} + \cdots - 131072$$
$17$ $$T^{10} - 130 T^{8} + \cdots - 1782272$$
$19$ $$T^{10} - 130 T^{8} + \cdots - 1782272$$
$23$ $$(T^{5} + 2 T^{4} + \cdots - 216)^{2}$$
$29$ $$(T^{5} - 18 T^{4} + \cdots + 1858)^{2}$$
$31$ $$T^{10} - 224 T^{8} + \cdots - 32902272$$
$37$ $$(T + 1)^{10}$$
$41$ $$T^{10} - 74 T^{8} + \cdots - 2$$
$43$ $$(T^{5} - 2 T^{4} + \cdots + 7520)^{2}$$
$47$ $$T^{10} - 296 T^{8} + \cdots - 346112$$
$53$ $$(T^{5} + 2 T^{4} + \cdots + 46208)^{2}$$
$59$ $$T^{10} - 194 T^{8} + \cdots - 1384448$$
$61$ $$T^{10} - 256 T^{8} + \cdots - 109512$$
$67$ $$(T^{5} + 6 T^{4} + \cdots + 3812)^{2}$$
$71$ $$(T^{5} - 20 T^{4} + \cdots - 17920)^{2}$$
$73$ $$T^{10} + \cdots - 615233042$$
$79$ $$(T^{5} - 81 T^{3} + \cdots - 56)^{2}$$
$83$ $$T^{10} - 314 T^{8} + \cdots - 913952$$
$89$ $$T^{10} - 242 T^{8} + \cdots - 61649408$$
$97$ $$T^{10} - 466 T^{8} + \cdots - 54080000$$