# Properties

 Label 4650.2.d.bd Level $4650$ Weight $2$ Character orbit 4650.d Analytic conductor $37.130$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$37.1304369399$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 4$$ (v^3 + 5*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$4\beta_{2} - 5\beta_1$$ 4*b2 - 5*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$$.

 $$n$$ $$1801$$ $$2977$$ $$3101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3349.1
 − 2.56155i 1.56155i − 1.56155i 2.56155i
1.00000i 1.00000i −1.00000 0 −1.00000 2.56155i 1.00000i −1.00000 0
3349.2 1.00000i 1.00000i −1.00000 0 −1.00000 1.56155i 1.00000i −1.00000 0
3349.3 1.00000i 1.00000i −1.00000 0 −1.00000 1.56155i 1.00000i −1.00000 0
3349.4 1.00000i 1.00000i −1.00000 0 −1.00000 2.56155i 1.00000i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 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 4650.2.d.bd 4
5.b even 2 1 inner 4650.2.d.bd 4
5.c odd 4 1 930.2.a.p 2
5.c odd 4 1 4650.2.a.ce 2
15.e even 4 1 2790.2.a.be 2
20.e even 4 1 7440.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.p 2 5.c odd 4 1
2790.2.a.be 2 15.e even 4 1
4650.2.a.ce 2 5.c odd 4 1
4650.2.d.bd 4 1.a even 1 1 trivial
4650.2.d.bd 4 5.b even 2 1 inner
7440.2.a.bl 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4650, [\chi])$$:

 $$T_{7}^{4} + 9T_{7}^{2} + 16$$ T7^4 + 9*T7^2 + 16 $$T_{11}^{2} - T_{11} - 4$$ T11^2 - T11 - 4 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{17}^{4} + 36T_{17}^{2} + 256$$ T17^4 + 36*T17^2 + 256 $$T_{19}^{2} - 3T_{19} - 36$$ T19^2 - 3*T19 - 36 $$T_{29}^{2} + 6T_{29} - 8$$ T29^2 + 6*T29 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 9T^{2} + 16$$
$11$ $$(T^{2} - T - 4)^{2}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$T^{4} + 36T^{2} + 256$$
$19$ $$(T^{2} - 3 T - 36)^{2}$$
$23$ $$T^{4} + 33T^{2} + 64$$
$29$ $$(T^{2} + 6 T - 8)^{2}$$
$31$ $$(T - 1)^{4}$$
$37$ $$T^{4} + 36T^{2} + 256$$
$41$ $$(T^{2} - 6 T - 8)^{2}$$
$43$ $$T^{4} + 225 T^{2} + 10000$$
$47$ $$T^{4} + 36T^{2} + 256$$
$53$ $$T^{4} + 189T^{2} + 7396$$
$59$ $$(T^{2} - 18 T + 64)^{2}$$
$61$ $$(T - 6)^{4}$$
$67$ $$T^{4} + 324 T^{2} + 20736$$
$71$ $$(T^{2} + 3 T - 36)^{2}$$
$73$ $$T^{4} + 213 T^{2} + 11236$$
$79$ $$(T^{2} - 21 T + 72)^{2}$$
$83$ $$T^{4} + 208T^{2} + 1024$$
$89$ $$(T^{2} - 15 T + 18)^{2}$$
$97$ $$(T^{2} + 36)^{2}$$
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