# Properties

 Label 4600.2.a.bi Level $4600$ Weight $2$ Character orbit 4600.a Self dual yes Analytic conductor $36.731$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.7311849298$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 7 x^{5} + 24 x^{4} + x^{3} - 35 x^{2} + 17 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 1 + \beta_{3} ) q^{7} + ( \beta_{2} + \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 1 + \beta_{3} ) q^{7} + ( \beta_{2} + \beta_{5} + \beta_{6} ) q^{9} + ( -1 - \beta_{5} - \beta_{6} ) q^{11} + ( -\beta_{1} - \beta_{2} - \beta_{5} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{17} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{19} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{21} - q^{23} + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{27} + ( -1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{29} + ( -\beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{31} + ( -1 - 4 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{33} + ( -3 - \beta_{2} - 2 \beta_{6} ) q^{37} + ( -1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{39} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{6} ) q^{41} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{43} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{47} + ( -2 - \beta_{1} + \beta_{4} ) q^{49} + ( -3 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{51} + ( -2 + \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{53} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{57} + ( -3 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{59} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{61} + ( 3 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{63} + ( 1 - 4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{67} -\beta_{1} q^{69} + ( -4 + \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{71} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{73} + ( -3 + 3 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{6} ) q^{77} + ( -5 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{79} + ( -3 + \beta_{2} - 4 \beta_{3} + \beta_{5} ) q^{81} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{83} + ( 4 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{87} + ( -3 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{89} + ( -1 - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{91} + ( -\beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} ) q^{93} + ( 4 - \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{97} + ( -6 - 5 \beta_{2} + 3 \beta_{3} - 4 \beta_{5} - 4 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 3q^{3} + 4q^{7} + 2q^{9} + O(q^{10})$$ $$7q + 3q^{3} + 4q^{7} + 2q^{9} - 7q^{11} - 7q^{13} - 7q^{19} - 6q^{21} - 7q^{23} - 11q^{29} - 10q^{31} - 19q^{33} - 19q^{37} - 24q^{39} - 16q^{41} + 6q^{43} + 6q^{47} - 17q^{49} - 7q^{51} - 15q^{53} - 8q^{57} - 11q^{59} + 5q^{61} + 13q^{63} + 9q^{67} - 3q^{69} - 14q^{71} - 10q^{73} - 6q^{77} - 32q^{79} - 5q^{81} + q^{83} + 10q^{87} - 24q^{89} - 7q^{91} - 26q^{93} + 7q^{97} - 61q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3 x^{6} - 7 x^{5} + 24 x^{4} + x^{3} - 35 x^{2} + 17 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 11 \nu^{4} - 3 \nu^{3} + 26 \nu^{2} + 7 \nu - 6$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{5} - 9 \nu^{4} + 15 \nu^{3} + 16 \nu^{2} - 21 \nu - 2$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - \nu^{4} - 10 \nu^{3} + 6 \nu^{2} + 20 \nu - 6$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{6} - 8 \nu^{5} - 23 \nu^{4} + 63 \nu^{3} + 20 \nu^{2} - 91 \nu + 22$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$-2 \nu^{6} + 4 \nu^{5} + 17 \nu^{4} - 30 \nu^{3} - 22 \nu^{2} + 42 \nu - 11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 6 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{6} + 10 \beta_{5} - 4 \beta_{3} + 10 \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$13 \beta_{6} + 14 \beta_{5} - 9 \beta_{4} - 14 \beta_{3} + 24 \beta_{2} + 40 \beta_{1} - 7$$ $$\nu^{6}$$ $$=$$ $$76 \beta_{6} + 87 \beta_{5} - 3 \beta_{4} - 47 \beta_{3} + 92 \beta_{2} + 11 \beta_{1} + 90$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.49931 −1.40334 0.189375 0.356372 1.58319 1.78665 2.98707
0 −2.49931 0 0 0 2.92777 0 3.24657 0
1.2 0 −1.40334 0 0 0 1.57117 0 −1.03063 0
1.3 0 0.189375 0 0 0 −1.65661 0 −2.96414 0
1.4 0 0.356372 0 0 0 −2.46376 0 −2.87300 0
1.5 0 1.58319 0 0 0 2.84620 0 −0.493499 0
1.6 0 1.78665 0 0 0 1.75578 0 0.192116 0
1.7 0 2.98707 0 0 0 −0.980560 0 5.92257 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$1$$

## 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 4600.2.a.bi 7
4.b odd 2 1 9200.2.a.cz 7
5.b even 2 1 4600.2.a.bh 7
5.c odd 4 2 920.2.e.b 14
20.d odd 2 1 9200.2.a.dc 7
20.e even 4 2 1840.2.e.g 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.b 14 5.c odd 4 2
1840.2.e.g 14 20.e even 4 2
4600.2.a.bh 7 5.b even 2 1
4600.2.a.bi 7 1.a even 1 1 trivial
9200.2.a.cz 7 4.b odd 2 1
9200.2.a.dc 7 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4600))$$:

 $$T_{3}^{7} - 3 T_{3}^{6} - 7 T_{3}^{5} + 24 T_{3}^{4} + T_{3}^{3} - 35 T_{3}^{2} + 17 T_{3} - 2$$ $$T_{7}^{7} - 4 T_{7}^{6} - 8 T_{7}^{5} + 41 T_{7}^{4} + 10 T_{7}^{3} - 116 T_{7}^{2} + 12 T_{7} + 92$$ $$T_{11}^{7} + 7 T_{11}^{6} - 27 T_{11}^{5} - 198 T_{11}^{4} + 248 T_{11}^{3} + 1224 T_{11}^{2} - 1780 T_{11} + 128$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7}$$
$3$ $$-2 + 17 T - 35 T^{2} + T^{3} + 24 T^{4} - 7 T^{5} - 3 T^{6} + T^{7}$$
$5$ $$T^{7}$$
$7$ $$92 + 12 T - 116 T^{2} + 10 T^{3} + 41 T^{4} - 8 T^{5} - 4 T^{6} + T^{7}$$
$11$ $$128 - 1780 T + 1224 T^{2} + 248 T^{3} - 198 T^{4} - 27 T^{5} + 7 T^{6} + T^{7}$$
$13$ $$-2 - 161 T + 135 T^{2} + 129 T^{3} - 92 T^{4} - 13 T^{5} + 7 T^{6} + T^{7}$$
$17$ $$52 - 184 T - 1050 T^{2} + 502 T^{3} + 105 T^{4} - 58 T^{5} + T^{7}$$
$19$ $$1936 - 3828 T + 1104 T^{2} + 1030 T^{3} - 238 T^{4} - 53 T^{5} + 7 T^{6} + T^{7}$$
$23$ $$( 1 + T )^{7}$$
$29$ $$9244 + 38290 T + 15243 T^{2} - 971 T^{3} - 930 T^{4} - 56 T^{5} + 11 T^{6} + T^{7}$$
$31$ $$-225251 + 21800 T + 38532 T^{2} + 2165 T^{3} - 1243 T^{4} - 110 T^{5} + 10 T^{6} + T^{7}$$
$37$ $$42832 + 13432 T - 10948 T^{2} - 5832 T^{3} - 700 T^{4} + 68 T^{5} + 19 T^{6} + T^{7}$$
$41$ $$110153 + 76850 T + 8034 T^{2} - 4287 T^{3} - 955 T^{4} + 12 T^{5} + 16 T^{6} + T^{7}$$
$43$ $$64 - 336 T - 168 T^{2} + 188 T^{3} + 64 T^{4} - 30 T^{5} - 6 T^{6} + T^{7}$$
$47$ $$5752 - 21784 T - 39856 T^{2} + 6697 T^{3} + 1130 T^{4} - 178 T^{5} - 6 T^{6} + T^{7}$$
$53$ $$156352 + 154304 T + 38096 T^{2} - 2704 T^{3} - 1596 T^{4} - 60 T^{5} + 15 T^{6} + T^{7}$$
$59$ $$486592 - 430528 T + 70592 T^{2} + 16480 T^{3} - 2288 T^{4} - 252 T^{5} + 11 T^{6} + T^{7}$$
$61$ $$2669336 - 157108 T - 209468 T^{2} + 25754 T^{3} + 2048 T^{4} - 315 T^{5} - 5 T^{6} + T^{7}$$
$67$ $$462832 - 114808 T - 72148 T^{2} + 11872 T^{3} + 1732 T^{4} - 244 T^{5} - 9 T^{6} + T^{7}$$
$71$ $$632317 + 554654 T + 127100 T^{2} - 211 T^{3} - 2711 T^{4} - 156 T^{5} + 14 T^{6} + T^{7}$$
$73$ $$16 - 1460 T + 2516 T^{2} - 391 T^{3} - 466 T^{4} - 30 T^{5} + 10 T^{6} + T^{7}$$
$79$ $$473312 + 422784 T + 29312 T^{2} - 22724 T^{3} - 2876 T^{4} + 166 T^{5} + 32 T^{6} + T^{7}$$
$83$ $$-11984 - 68664 T - 20548 T^{2} + 5720 T^{3} + 1070 T^{4} - 222 T^{5} - T^{6} + T^{7}$$
$89$ $$311456 + 288384 T + 48656 T^{2} - 10836 T^{3} - 2260 T^{4} + 46 T^{5} + 24 T^{6} + T^{7}$$
$97$ $$-4038856 - 1487436 T + 53776 T^{2} + 45720 T^{3} + 964 T^{4} - 369 T^{5} - 7 T^{6} + T^{7}$$
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