# Properties

 Label 4600.2.e.w 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: $$x^{10} + 18 x^{8} + 117 x^{6} + 333 x^{4} + 396 x^{2} + 144$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -\beta_{4} + \beta_{6} ) q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -\beta_{4} + \beta_{6} ) q^{7} + ( -1 + \beta_{2} ) q^{9} -\beta_{5} q^{11} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{13} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{9} ) q^{17} + \beta_{5} q^{19} + ( 1 - 2 \beta_{2} + \beta_{5} + \beta_{8} ) q^{21} -\beta_{4} q^{23} + ( \beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{27} + ( -2 - \beta_{2} - \beta_{5} - 3 \beta_{7} ) q^{29} + ( -3 - \beta_{2} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{31} + ( \beta_{4} + \beta_{6} + \beta_{9} ) q^{33} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} ) q^{37} + ( -1 - 2 \beta_{2} + \beta_{5} - 3 \beta_{7} ) q^{39} + ( -1 + 2 \beta_{5} + \beta_{7} + \beta_{8} ) q^{41} + ( 2 \beta_{1} + 2 \beta_{4} - \beta_{9} ) q^{43} + ( \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - \beta_{6} - \beta_{9} ) q^{47} + ( -3 + \beta_{5} - 2 \beta_{7} ) q^{49} + ( -2 + 2 \beta_{2} + 2 \beta_{5} + 4 \beta_{7} ) q^{51} + ( 2 \beta_{3} + 3 \beta_{4} - \beta_{6} - 3 \beta_{9} ) q^{53} + ( -\beta_{4} - \beta_{6} - \beta_{9} ) q^{57} + ( -2 \beta_{5} - \beta_{8} ) q^{59} + ( 1 + 2 \beta_{2} - 3 \beta_{5} - \beta_{8} ) q^{61} + ( 4 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{63} + ( 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{9} ) q^{67} + \beta_{7} q^{69} + ( 3 \beta_{2} + 2 \beta_{5} + 4 \beta_{7} - 2 \beta_{8} ) q^{71} + ( -2 \beta_{1} - 5 \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{9} ) q^{73} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{9} ) q^{77} + ( -1 + 2 \beta_{2} - 2 \beta_{5} - \beta_{8} ) q^{79} + ( -6 + 2 \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} ) q^{81} + ( -4 \beta_{3} - 2 \beta_{4} + 3 \beta_{9} ) q^{83} + ( -4 \beta_{1} - 3 \beta_{3} - 9 \beta_{4} + \beta_{9} ) q^{87} + ( -2 - 2 \beta_{2} - 4 \beta_{7} ) q^{89} + ( -8 - 2 \beta_{2} + \beta_{5} + 2 \beta_{8} ) q^{91} + ( -2 \beta_{1} + \beta_{3} - \beta_{6} + 2 \beta_{9} ) q^{93} + ( -\beta_{4} - \beta_{6} - 3 \beta_{9} ) q^{97} + ( 2 - 2 \beta_{2} - \beta_{5} - 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 6 q^{9} + O(q^{10})$$ $$10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 18 x^{8} + 117 x^{6} + 333 x^{4} + 396 x^{2} + 144$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 9 \nu^{3} + 15 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{9} + 14 \nu^{7} + 61 \nu^{5} + 81 \nu^{3}$$$$)/24$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} + 12 \nu^{4} + 39 \nu^{2} + 27$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{9} + 14 \nu^{7} + 61 \nu^{5} + 93 \nu^{3} + 60 \nu$$$$)/12$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{8} + 14 \nu^{6} + 63 \nu^{4} + 99 \nu^{2} + 36$$$$)/6$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{8} - 16 \nu^{6} - 81 \nu^{4} - 135 \nu^{2} - 48$$$$)/6$$ $$\beta_{9}$$ $$=$$ $$($$$$-3 \nu^{9} - 50 \nu^{7} - 279 \nu^{5} - 579 \nu^{3} - 336 \nu$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} - 2 \beta_{4} - 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{8} + \beta_{7} + \beta_{5} - 7 \beta_{2} + 21$$ $$\nu^{5}$$ $$=$$ $$-9 \beta_{6} + 18 \beta_{4} + 3 \beta_{3} + 30 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-12 \beta_{8} - 12 \beta_{7} - 9 \beta_{5} + 45 \beta_{2} - 123$$ $$\nu^{7}$$ $$=$$ $$-3 \beta_{9} + 66 \beta_{6} - 141 \beta_{4} - 36 \beta_{3} - 192 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$105 \beta_{8} + 111 \beta_{7} + 63 \beta_{5} - 288 \beta_{2} + 759$$ $$\nu^{9}$$ $$=$$ $$42 \beta_{9} - 456 \beta_{6} + 1062 \beta_{4} + 321 \beta_{3} + 1263 \beta_{1}$$

## 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
 − 2.61696i − 2.29544i − 1.83957i − 1.36629i − 0.794805i 0.794805i 1.36629i 1.83957i 2.29544i 2.61696i
0 2.61696i 0 0 0 3.83744i 0 −3.84849 0
4049.2 0 2.29544i 0 0 0 1.61758i 0 −2.26905 0
4049.3 0 1.83957i 0 0 0 3.97272i 0 −0.384010 0
4049.4 0 1.36629i 0 0 0 3.28093i 0 1.13327 0
4049.5 0 0.794805i 0 0 0 2.47193i 0 2.36829 0
4049.6 0 0.794805i 0 0 0 2.47193i 0 2.36829 0
4049.7 0 1.36629i 0 0 0 3.28093i 0 1.13327 0
4049.8 0 1.83957i 0 0 0 3.97272i 0 −0.384010 0
4049.9 0 2.29544i 0 0 0 1.61758i 0 −2.26905 0
4049.10 0 2.61696i 0 0 0 3.83744i 0 −3.84849 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.w 10
5.b even 2 1 inner 4600.2.e.w 10
5.c odd 4 1 4600.2.a.bd 5
5.c odd 4 1 4600.2.a.bf yes 5
20.e even 4 1 9200.2.a.ct 5
20.e even 4 1 9200.2.a.cv 5

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

 $$T_{3}^{10} + 18 T_{3}^{8} + 117 T_{3}^{6} + 333 T_{3}^{4} + 396 T_{3}^{2} + 144$$ $$T_{7}^{10} + 50 T_{7}^{8} + 937 T_{7}^{6} + 8056 T_{7}^{4} + 30800 T_{7}^{2} + 40000$$ $$T_{11}^{5} - 15 T_{11}^{3} - 6 T_{11}^{2} + 48 T_{11} + 24$$ $$T_{13}^{10} + 80 T_{13}^{8} + 2326 T_{13}^{6} + 29425 T_{13}^{4} + 145523 T_{13}^{2} + 120409$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$144 + 396 T^{2} + 333 T^{4} + 117 T^{6} + 18 T^{8} + T^{10}$$
$5$ $$T^{10}$$
$7$ $$40000 + 30800 T^{2} + 8056 T^{4} + 937 T^{6} + 50 T^{8} + T^{10}$$
$11$ $$( 24 + 48 T - 6 T^{2} - 15 T^{3} + T^{5} )^{2}$$
$13$ $$120409 + 145523 T^{2} + 29425 T^{4} + 2326 T^{6} + 80 T^{8} + T^{10}$$
$17$ $$147456 + 175104 T^{2} + 60480 T^{4} + 5280 T^{6} + 132 T^{8} + T^{10}$$
$19$ $$( -24 + 48 T + 6 T^{2} - 15 T^{3} + T^{5} )^{2}$$
$23$ $$( 1 + T^{2} )^{5}$$
$29$ $$( 2787 - 273 T - 339 T^{2} + 12 T^{4} + T^{5} )^{2}$$
$31$ $$( -228 - 516 T + 45 T^{2} + 93 T^{3} + 18 T^{4} + T^{5} )^{2}$$
$37$ $$369664 + 1464320 T^{2} + 198784 T^{4} + 9088 T^{6} + 164 T^{8} + T^{10}$$
$41$ $$( 453 + 1311 T - 237 T^{2} - 66 T^{3} + 6 T^{4} + T^{5} )^{2}$$
$43$ $$2166784 + 1048832 T^{2} + 148192 T^{4} + 7345 T^{6} + 146 T^{8} + T^{10}$$
$47$ $$16 + 109148 T^{2} + 155221 T^{4} + 21985 T^{6} + 290 T^{8} + T^{10}$$
$53$ $$65536 + 290816 T^{2} + 284416 T^{4} + 17872 T^{6} + 284 T^{8} + T^{10}$$
$59$ $$( 3004 + 1322 T - 107 T^{2} - 83 T^{3} - T^{4} + T^{5} )^{2}$$
$61$ $$( 7648 + 12800 T + 1072 T^{2} - 200 T^{3} - 10 T^{4} + T^{5} )^{2}$$
$67$ $$94633984 + 28909568 T^{2} + 2370112 T^{4} + 56464 T^{6} + 440 T^{8} + T^{10}$$
$71$ $$( 8084 + 5960 T + 689 T^{2} - 179 T^{3} - 8 T^{4} + T^{5} )^{2}$$
$73$ $$816187761 + 106077267 T^{2} + 4279401 T^{4} + 70566 T^{6} + 480 T^{8} + T^{10}$$
$79$ $$( 1728 + 5184 T + 36 T^{2} - 153 T^{3} + T^{5} )^{2}$$
$83$ $$5721664 + 5594384 T^{2} + 1446280 T^{4} + 41977 T^{6} + 386 T^{8} + T^{10}$$
$89$ $$( -3104 - 4864 T - 1280 T^{2} - 44 T^{3} + 14 T^{4} + T^{5} )^{2}$$
$97$ $$9216 + 175104 T^{2} + 830592 T^{4} + 31680 T^{6} + 324 T^{8} + T^{10}$$