# Properties

 Label 4600.2.e.v Level $4600$ Weight $2$ Character orbit 4600.e Analytic conductor $36.731$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 18 x^{8} + 103 x^{6} + 239 x^{4} + 197 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{5} ) q^{3} + \beta_{8} q^{7} + ( -2 + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{5} ) q^{3} + \beta_{8} q^{7} + ( -2 + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{9} + ( \beta_{4} + \beta_{7} + \beta_{9} ) q^{11} + ( \beta_{1} + \beta_{2} + 2 \beta_{5} ) q^{13} + ( 2 \beta_{1} + \beta_{5} + \beta_{6} ) q^{17} + ( -2 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{19} + ( -2 + 2 \beta_{3} ) q^{21} + \beta_{2} q^{23} + ( -2 \beta_{2} + 3 \beta_{6} + \beta_{8} ) q^{27} + ( 2 - \beta_{3} + \beta_{4} - 2 \beta_{7} ) q^{29} + ( 2 - 3 \beta_{3} - \beta_{4} + \beta_{9} ) q^{31} + ( 4 \beta_{2} + \beta_{5} - 3 \beta_{6} - 2 \beta_{8} ) q^{33} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{6} + 2 \beta_{8} ) q^{37} + ( -8 + 2 \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{9} ) q^{39} + ( -1 - 2 \beta_{3} - 2 \beta_{4} + \beta_{7} - \beta_{9} ) q^{41} + 3 \beta_{8} q^{43} + ( -\beta_{1} + \beta_{2} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} ) q^{47} + ( 1 + 2 \beta_{4} + \beta_{9} ) q^{49} + ( \beta_{4} + \beta_{7} + 2 \beta_{9} ) q^{51} + ( 4 \beta_{1} + 6 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} ) q^{53} + ( -2 \beta_{1} - 6 \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{57} + ( -5 + 3 \beta_{4} - 3 \beta_{7} + \beta_{9} ) q^{59} + ( -6 + 2 \beta_{3} ) q^{61} + ( -2 \beta_{5} + \beta_{8} ) q^{63} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{5} + \beta_{6} ) q^{67} + ( -1 - \beta_{7} ) q^{69} + ( 1 - 3 \beta_{3} + \beta_{9} ) q^{71} + ( -5 \beta_{1} + \beta_{5} - 4 \beta_{6} ) q^{73} + ( -2 \beta_{2} - 2 \beta_{6} - 3 \beta_{8} ) q^{77} + ( -8 - 2 \beta_{4} + 2 \beta_{7} + \beta_{9} ) q^{79} + ( 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{7} ) q^{81} + ( -2 \beta_{1} + 4 \beta_{2} - 3 \beta_{5} + \beta_{6} - \beta_{8} ) q^{83} + ( -\beta_{1} - 5 \beta_{2} + 3 \beta_{5} + 2 \beta_{8} ) q^{87} + ( -4 - 3 \beta_{4} + \beta_{7} - 4 \beta_{9} ) q^{89} + ( -6 + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} - \beta_{9} ) q^{91} + ( -3 \beta_{1} - 5 \beta_{2} + \beta_{5} + 2 \beta_{6} + 4 \beta_{8} ) q^{93} + ( 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{8} ) q^{97} + ( -10 - 4 \beta_{4} - 4 \beta_{7} - \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 8 q^{9} + O(q^{10})$$ $$10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 18 x^{8} + 103 x^{6} + 239 x^{4} + 197 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{9} + 4 \nu^{7} - 101 \nu^{5} - 493 \nu^{3} - 523 \nu$$$$)/74$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{8} - 102 \nu^{6} - 366 \nu^{4} - 360 \nu^{2} - 2$$$$)/37$$ $$\beta_{4}$$ $$=$$ $$($$$$-7 \nu^{8} - 102 \nu^{6} - 366 \nu^{4} - 323 \nu^{2} + 109$$$$)/37$$ $$\beta_{5}$$ $$=$$ $$($$$$7 \nu^{9} + 102 \nu^{7} + 329 \nu^{5} - 121 \nu^{3} - 1071 \nu$$$$)/74$$ $$\beta_{6}$$ $$=$$ $$($$$$11 \nu^{9} + 192 \nu^{7} + 1035 \nu^{5} + 2199 \nu^{3} + 1647 \nu$$$$)/74$$ $$\beta_{7}$$ $$=$$ $$($$$$16 \nu^{8} + 249 \nu^{6} + 1048 \nu^{4} + 1362 \nu^{2} + 105$$$$)/37$$ $$\beta_{8}$$ $$=$$ $$($$$$33 \nu^{9} + 502 \nu^{7} + 1995 \nu^{5} + 2231 \nu^{3} - 239 \nu$$$$)/74$$ $$\beta_{9}$$ $$=$$ $$($$$$-19 \nu^{8} - 298 \nu^{6} - 1263 \nu^{4} - 1585 \nu^{2} - 53$$$$)/37$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} - 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{8} + 2 \beta_{6} + \beta_{5} + 4 \beta_{2} - 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{9} + 4 \beta_{7} - 9 \beta_{4} + 10 \beta_{3} + 20$$ $$\nu^{5}$$ $$=$$ $$13 \beta_{8} - 25 \beta_{6} - 15 \beta_{5} - 49 \beta_{2} + 35 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-40 \beta_{9} - 51 \beta_{7} + 86 \beta_{4} - 94 \beta_{3} - 171$$ $$\nu^{7}$$ $$=$$ $$-137 \beta_{8} + 260 \beta_{6} + 166 \beta_{5} + 499 \beta_{2} - 300 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$426 \beta_{9} + 534 \beta_{7} - 834 \beta_{4} + 893 \beta_{3} + 1600$$ $$\nu^{9}$$ $$=$$ $$1368 \beta_{8} - 2579 \beta_{6} - 1686 \beta_{5} - 4899 \beta_{2} + 2793 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1151$$ $$1201$$ $$2301$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4049.1
 0.144312i 1.45894i − 1.64975i 3.11721i − 1.84717i 1.84717i − 3.11721i 1.64975i − 1.45894i − 0.144312i
0 3.08344i 0 0 0 0.555022i 0 −6.50759 0
4049.2 0 2.21042i 0 0 0 2.22487i 0 −1.88594 0
4049.3 0 1.51466i 0 0 0 3.49880i 0 0.705809 0
4049.4 0 1.33689i 0 0 0 3.16736i 0 1.21273 0
4049.5 0 0.724570i 0 0 0 2.33840i 0 2.47500 0
4049.6 0 0.724570i 0 0 0 2.33840i 0 2.47500 0
4049.7 0 1.33689i 0 0 0 3.16736i 0 1.21273 0
4049.8 0 1.51466i 0 0 0 3.49880i 0 0.705809 0
4049.9 0 2.21042i 0 0 0 2.22487i 0 −1.88594 0
4049.10 0 3.08344i 0 0 0 0.555022i 0 −6.50759 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4049.10 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 4600.2.e.v 10
5.b even 2 1 inner 4600.2.e.v 10
5.c odd 4 1 4600.2.a.bc 5
5.c odd 4 1 4600.2.a.bg yes 5
20.e even 4 1 9200.2.a.cs 5
20.e even 4 1 9200.2.a.cw 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.bc 5 5.c odd 4 1
4600.2.a.bg yes 5 5.c odd 4 1
4600.2.e.v 10 1.a even 1 1 trivial
4600.2.e.v 10 5.b even 2 1 inner
9200.2.a.cs 5 20.e even 4 1
9200.2.a.cw 5 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 $$S_{2}^{\mathrm{new}}(4600, [\chi])$$:

 $$T_{3}^{10} + 19 T_{3}^{8} + 119 T_{3}^{6} + 306 T_{3}^{4} + 321 T_{3}^{2} + 100$$ $$T_{7}^{10} + 33 T_{7}^{8} + 392 T_{7}^{6} + 2000 T_{7}^{4} + 3904 T_{7}^{2} + 1024$$ $$T_{11}^{5} + 4 T_{11}^{4} - 17 T_{11}^{3} - 88 T_{11}^{2} - 92 T_{11} - 8$$ $$T_{13}^{10} + 75 T_{13}^{8} + 2107 T_{13}^{6} + 26966 T_{13}^{4} + 146625 T_{13}^{2} + 206116$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$100 + 321 T^{2} + 306 T^{4} + 119 T^{6} + 19 T^{8} + T^{10}$$
$5$ $$T^{10}$$
$7$ $$1024 + 3904 T^{2} + 2000 T^{4} + 392 T^{6} + 33 T^{8} + T^{10}$$
$11$ $$( -8 - 92 T - 88 T^{2} - 17 T^{3} + 4 T^{4} + T^{5} )^{2}$$
$13$ $$206116 + 146625 T^{2} + 26966 T^{4} + 2107 T^{6} + 75 T^{8} + T^{10}$$
$17$ $$1024 + 12608 T^{2} + 9744 T^{4} + 1800 T^{6} + 81 T^{8} + T^{10}$$
$19$ $$( 64 - 272 T - 264 T^{2} - 53 T^{3} + 4 T^{4} + T^{5} )^{2}$$
$23$ $$( 1 + T^{2} )^{5}$$
$29$ $$( 256 - 551 T + 306 T^{2} - 11 T^{3} - 11 T^{4} + T^{5} )^{2}$$
$31$ $$( 400 + 802 T + 159 T^{2} - 75 T^{3} - 4 T^{4} + T^{5} )^{2}$$
$37$ $$28558336 + 27983872 T^{2} + 1704064 T^{4} + 35856 T^{6} + 316 T^{8} + T^{10}$$
$41$ $$( -2069 - 2097 T - 671 T^{2} - 52 T^{3} + 8 T^{4} + T^{5} )^{2}$$
$43$ $$60466176 + 25614144 T^{2} + 1458000 T^{4} + 31752 T^{6} + 297 T^{8} + T^{10}$$
$47$ $$34951744 + 21165556 T^{2} + 1547109 T^{4} + 37137 T^{6} + 338 T^{8} + T^{10}$$
$53$ $$490356736 + 119363584 T^{2} + 4532800 T^{4} + 66736 T^{6} + 428 T^{8} + T^{10}$$
$59$ $$( 5128 - 558 T - 1087 T^{2} + 53 T^{3} + 23 T^{4} + T^{5} )^{2}$$
$61$ $$( -256 + 880 T + 848 T^{2} + 236 T^{3} + 26 T^{4} + T^{5} )^{2}$$
$67$ $$262144 + 299008 T^{2} + 78336 T^{4} + 5392 T^{6} + 129 T^{8} + T^{10}$$
$71$ $$( -8 + 666 T - 133 T^{2} - 67 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$73$ $$2756355001 + 1246938683 T^{2} + 26292185 T^{4} + 211014 T^{6} + 752 T^{8} + T^{10}$$
$79$ $$( -45872 - 2080 T + 3292 T^{2} + 634 T^{3} + 43 T^{4} + T^{5} )^{2}$$
$83$ $$1086361600 + 180557056 T^{2} + 7391008 T^{4} + 104529 T^{6} + 562 T^{8} + T^{10}$$
$89$ $$( -7120 - 12608 T - 3188 T^{2} - 154 T^{3} + 15 T^{4} + T^{5} )^{2}$$
$97$ $$1048576 + 26599680 T^{2} + 4513216 T^{4} + 85872 T^{6} + 520 T^{8} + T^{10}$$