# Properties

 Label 460.2.c.a Level $460$ Weight $2$ Character orbit 460.c Analytic conductor $3.673$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$460 = 2^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 460.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 24 x^{10} + 188 x^{8} + 530 x^{6} + 508 x^{4} + 80 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{5} + \beta_{6} ) q^{3} + \beta_{9} q^{5} + ( -\beta_{1} - \beta_{6} + \beta_{10} ) q^{7} + ( -2 + \beta_{2} + \beta_{7} + \beta_{10} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{5} + \beta_{6} ) q^{3} + \beta_{9} q^{5} + ( -\beta_{1} - \beta_{6} + \beta_{10} ) q^{7} + ( -2 + \beta_{2} + \beta_{7} + \beta_{10} ) q^{9} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} ) q^{11} + ( -\beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{13} + ( -\beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{15} + ( \beta_{5} - \beta_{11} ) q^{17} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{19} + ( 1 - 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{21} + \beta_{6} q^{23} + ( 1 + \beta_{1} - \beta_{2} + \beta_{6} + \beta_{8} - \beta_{11} ) q^{25} + ( 2 \beta_{1} - \beta_{5} - 5 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{27} + ( -2 + 2 \beta_{2} + \beta_{4} ) q^{29} + ( 2 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{31} + ( \beta_{1} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{33} + ( -\beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{11} ) q^{35} + ( \beta_{1} - 5 \beta_{6} - 2 \beta_{7} + \beta_{10} + 2 \beta_{11} ) q^{37} + ( 1 + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{8} - \beta_{9} ) q^{39} + ( -2 \beta_{2} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{41} + ( -3 \beta_{1} + \beta_{5} + 5 \beta_{6} + 2 \beta_{7} + \beta_{10} - \beta_{11} ) q^{43} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 5 \beta_{6} + \beta_{7} - 2 \beta_{9} + 2 \beta_{10} ) q^{45} + ( -\beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{47} + ( -4 + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{10} ) q^{49} + ( -2 - \beta_{8} + \beta_{9} ) q^{51} + ( 2 \beta_{1} - \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{53} + ( 3 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{9} + \beta_{11} ) q^{55} + ( -\beta_{1} - 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{57} + ( 3 + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} ) q^{59} + ( 1 + \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{8} + 2 \beta_{9} ) q^{61} + ( 2 \beta_{1} + \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - 4 \beta_{10} - 3 \beta_{11} ) q^{63} + ( 3 - \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{10} ) q^{65} + ( -\beta_{1} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{10} ) q^{67} + ( -1 + \beta_{2} ) q^{69} + ( -4 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{7} - 2 \beta_{10} ) q^{71} + ( \beta_{5} - \beta_{6} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{73} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 4 \beta_{10} - \beta_{11} ) q^{75} + ( -2 \beta_{1} + 4 \beta_{5} + 4 \beta_{7} - 2 \beta_{10} - 2 \beta_{11} ) q^{77} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{79} + ( 1 - 2 \beta_{2} + 4 \beta_{3} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{81} + ( -2 \beta_{1} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{83} + ( 1 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{85} + ( -\beta_{5} - 11 \beta_{6} + \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{87} + ( 5 - 3 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{89} + ( -1 + 7 \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} ) q^{91} + ( -4 \beta_{1} + 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{93} + ( 1 - 3 \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{95} + ( \beta_{1} - \beta_{5} - \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{97} + ( 4 - 4 \beta_{2} + 2 \beta_{8} - 2 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 20 q^{9} + O(q^{10})$$ $$12 q - 20 q^{9} + 4 q^{11} + 2 q^{15} - 8 q^{19} + 8 q^{25} - 10 q^{29} + 18 q^{31} - 10 q^{35} + 16 q^{39} - 2 q^{41} + 2 q^{45} - 38 q^{49} - 24 q^{51} + 16 q^{55} + 22 q^{59} - 8 q^{61} + 38 q^{65} - 8 q^{69} - 34 q^{71} + 16 q^{75} - 20 q^{79} + 28 q^{81} + 6 q^{85} + 48 q^{89} - 8 q^{91} + 12 q^{95} + 32 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 24 x^{10} + 188 x^{8} + 530 x^{6} + 508 x^{4} + 80 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{10} - 23 \nu^{8} - 8 \nu^{7} - 157 \nu^{6} - 144 \nu^{5} - 261 \nu^{4} - 752 \nu^{3} + 121 \nu^{2} - 1064 \nu + 159$$$$)/96$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{10} + 69 \nu^{8} + 503 \nu^{6} + 1215 \nu^{4} + 869 \nu^{2} + 83$$$$)/48$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{10} + 169 \nu^{8} + 1331 \nu^{6} + 3779 \nu^{4} + 3689 \nu^{2} + 415$$$$)/96$$ $$\beta_{4}$$ $$=$$ $$($$$$9 \nu^{10} + 215 \nu^{8} + 1661 \nu^{6} + 4493 \nu^{4} + 3895 \nu^{2} + 497$$$$)/96$$ $$\beta_{5}$$ $$=$$ $$($$$$-7 \nu^{11} - 169 \nu^{9} - 1339 \nu^{7} - 3875 \nu^{5} - 3913 \nu^{3} - 663 \nu$$$$)/48$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{11} + 24 \nu^{9} + 188 \nu^{7} + 529 \nu^{5} + 496 \nu^{3} + 58 \nu$$$$)/6$$ $$\beta_{7}$$ $$=$$ $$($$$$-27 \nu^{11} - \nu^{10} - 645 \nu^{9} - 23 \nu^{8} - 5007 \nu^{7} - 157 \nu^{6} - 13815 \nu^{5} - 261 \nu^{4} - 12597 \nu^{3} + 121 \nu^{2} - 1419 \nu + 159$$$$)/96$$ $$\beta_{8}$$ $$=$$ $$($$$$27 \nu^{11} + \nu^{10} + 645 \nu^{9} + 23 \nu^{8} + 5007 \nu^{7} + 173 \nu^{6} + 13815 \nu^{5} + 453 \nu^{4} + 12597 \nu^{3} + 231 \nu^{2} + 1515 \nu - 143$$$$)/96$$ $$\beta_{9}$$ $$=$$ $$($$$$27 \nu^{11} - \nu^{10} + 645 \nu^{9} - 23 \nu^{8} + 5007 \nu^{7} - 173 \nu^{6} + 13815 \nu^{5} - 453 \nu^{4} + 12597 \nu^{3} - 231 \nu^{2} + 1515 \nu + 143$$$$)/96$$ $$\beta_{10}$$ $$=$$ $$($$$$27 \nu^{11} - \nu^{10} + 645 \nu^{9} - 23 \nu^{8} + 5007 \nu^{7} - 157 \nu^{6} + 13815 \nu^{5} - 261 \nu^{4} + 12597 \nu^{3} + 121 \nu^{2} + 1419 \nu + 159$$$$)/96$$ $$\beta_{11}$$ $$=$$ $$($$$$-37 \nu^{11} - 891 \nu^{9} - 7017 \nu^{7} - 19993 \nu^{5} - 19547 \nu^{3} - 3245 \nu$$$$)/96$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{4} - 2 \beta_{3} - 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} + 4 \beta_{10} - 4 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{10} - 7 \beta_{9} + 7 \beta_{8} - 9 \beta_{7} - 22 \beta_{4} + 22 \beta_{3} + 2 \beta_{2} + 66$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$22 \beta_{11} - 73 \beta_{10} + 75 \beta_{9} + 75 \beta_{8} + 51 \beta_{7} - 56 \beta_{6} - 72 \beta_{5} + 22 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$46 \beta_{10} + 28 \beta_{9} - 28 \beta_{8} + 46 \beta_{7} + 110 \beta_{4} - 110 \beta_{3} - 12 \beta_{2} - 309$$ $$\nu^{7}$$ $$=$$ $$($$$$-208 \beta_{11} + 707 \beta_{10} - 731 \beta_{9} - 731 \beta_{8} - 475 \beta_{7} + 632 \beta_{6} + 732 \beta_{5} - 232 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-955 \beta_{10} - 483 \beta_{9} + 483 \beta_{8} - 955 \beta_{7} - 2146 \beta_{4} + 2170 \beta_{3} + 208 \beta_{2} + 5972$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$941 \beta_{11} - 3488 \beta_{10} + 3592 \beta_{9} + 3592 \beta_{8} + 2271 \beta_{7} - 3376 \beta_{6} - 3597 \beta_{5} + 1217 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$($$$$9895 \beta_{10} + 4265 \beta_{9} - 4265 \beta_{8} + 9895 \beta_{7} + 20802 \beta_{4} - 21354 \beta_{3} - 1538 \beta_{2} - 58206$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-16710 \beta_{11} + 69215 \beta_{10} - 70753 \beta_{9} - 70753 \beta_{8} - 43769 \beta_{7} + 70884 \beta_{6} + 70152 \beta_{5} - 25446 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/460\mathbb{Z}\right)^\times$$.

 $$n$$ $$231$$ $$277$$ $$281$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
369.1
 3.16223i − 1.65047i − 1.26443i 0.420790i 3.08006i 0.116918i − 0.116918i − 3.08006i − 0.420790i 1.26443i 1.65047i − 3.16223i
0 3.21923i 0 1.59013 + 1.57210i 0 2.43185i 0 −7.36343 0
369.2 0 2.80150i 0 −2.17393 + 0.523461i 0 4.50896i 0 −4.84843 0
369.3 0 2.40050i 0 −0.817027 2.08146i 0 4.41307i 0 −2.76241 0
369.4 0 1.73961i 0 1.77747 1.35668i 0 3.32224i 0 −0.0262434 0
369.5 0 0.873449i 0 −1.89824 + 1.18182i 0 0.992530i 0 2.23709 0
369.6 0 0.486391i 0 1.52160 + 1.63852i 0 1.80495i 0 2.76342 0
369.7 0 0.486391i 0 1.52160 1.63852i 0 1.80495i 0 2.76342 0
369.8 0 0.873449i 0 −1.89824 1.18182i 0 0.992530i 0 2.23709 0
369.9 0 1.73961i 0 1.77747 + 1.35668i 0 3.32224i 0 −0.0262434 0
369.10 0 2.40050i 0 −0.817027 + 2.08146i 0 4.41307i 0 −2.76241 0
369.11 0 2.80150i 0 −2.17393 0.523461i 0 4.50896i 0 −4.84843 0
369.12 0 3.21923i 0 1.59013 1.57210i 0 2.43185i 0 −7.36343 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 369.12 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 460.2.c.a 12
3.b odd 2 1 4140.2.f.b 12
4.b odd 2 1 1840.2.e.f 12
5.b even 2 1 inner 460.2.c.a 12
5.c odd 4 1 2300.2.a.n 6
5.c odd 4 1 2300.2.a.o 6
15.d odd 2 1 4140.2.f.b 12
20.d odd 2 1 1840.2.e.f 12
20.e even 4 1 9200.2.a.cx 6
20.e even 4 1 9200.2.a.cy 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.c.a 12 1.a even 1 1 trivial
460.2.c.a 12 5.b even 2 1 inner
1840.2.e.f 12 4.b odd 2 1
1840.2.e.f 12 20.d odd 2 1
2300.2.a.n 6 5.c odd 4 1
2300.2.a.o 6 5.c odd 4 1
4140.2.f.b 12 3.b odd 2 1
4140.2.f.b 12 15.d odd 2 1
9200.2.a.cx 6 20.e even 4 1
9200.2.a.cy 6 20.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(460, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$256 + 1604 T^{2} + 2497 T^{4} + 1296 T^{6} + 286 T^{8} + 28 T^{10} + T^{12}$$
$5$ $$15625 - 2500 T^{2} + 1500 T^{3} + 1075 T^{4} + 60 T^{5} - 48 T^{6} + 12 T^{7} + 43 T^{8} + 12 T^{9} - 4 T^{10} + T^{12}$$
$7$ $$82944 + 139536 T^{2} + 68992 T^{4} + 14312 T^{6} + 1380 T^{8} + 61 T^{10} + T^{12}$$
$11$ $$( -256 - 144 T + 212 T^{2} + 28 T^{3} - 32 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$13$ $$1401856 + 1726756 T^{2} + 551105 T^{4} + 70560 T^{6} + 4126 T^{8} + 108 T^{10} + T^{12}$$
$17$ $$256 + 2448 T^{2} + 4384 T^{4} + 2620 T^{6} + 628 T^{8} + 57 T^{10} + T^{12}$$
$19$ $$( 256 + 848 T + 236 T^{2} - 140 T^{3} - 42 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$23$ $$( 1 + T^{2} )^{6}$$
$29$ $$( 11862 + 9453 T + 1229 T^{2} - 450 T^{3} - 84 T^{4} + 5 T^{5} + T^{6} )^{2}$$
$31$ $$( 916 + 1987 T - 2651 T^{2} + 838 T^{3} - 58 T^{4} - 9 T^{5} + T^{6} )^{2}$$
$37$ $$3996262656 + 1041449616 T^{2} + 90354016 T^{4} + 3059272 T^{6} + 48220 T^{8} + 357 T^{10} + T^{12}$$
$41$ $$( -2 + 477 T + 257 T^{2} - 158 T^{3} - 64 T^{4} + T^{5} + T^{6} )^{2}$$
$43$ $$8399355904 + 1503406336 T^{2} + 96250128 T^{4} + 2919296 T^{6} + 44988 T^{8} + 340 T^{10} + T^{12}$$
$47$ $$215296 + 2925732 T^{2} + 6778057 T^{4} + 585160 T^{6} + 17974 T^{8} + 228 T^{10} + T^{12}$$
$53$ $$149426176 + 94470400 T^{2} + 17572096 T^{4} + 1019104 T^{6} + 25232 T^{8} + 273 T^{10} + T^{12}$$
$59$ $$( -360576 - 63024 T + 12832 T^{2} + 1692 T^{3} - 188 T^{4} - 11 T^{5} + T^{6} )^{2}$$
$61$ $$( -171088 + 10976 T + 10996 T^{2} - 532 T^{3} - 198 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$67$ $$55115776 + 103780240 T^{2} + 40158752 T^{4} + 2163464 T^{6} + 41820 T^{8} + 341 T^{10} + T^{12}$$
$71$ $$( -16108 + 20769 T + 569 T^{2} - 1570 T^{3} - 62 T^{4} + 17 T^{5} + T^{6} )^{2}$$
$73$ $$30294016 + 238969152 T^{2} + 74655361 T^{4} + 3161400 T^{6} + 51998 T^{8} + 376 T^{10} + T^{12}$$
$79$ $$( -128 + 584 T + 84 T^{2} - 208 T^{3} - 10 T^{4} + 10 T^{5} + T^{6} )^{2}$$
$83$ $$401124622336 + 40997671440 T^{2} + 1542179872 T^{4} + 26507740 T^{6} + 215860 T^{8} + 777 T^{10} + T^{12}$$
$89$ $$( 180432 - 167568 T - 3212 T^{2} + 5172 T^{3} - 142 T^{4} - 24 T^{5} + T^{6} )^{2}$$
$97$ $$2091049984 + 1007200576 T^{2} + 153522128 T^{4} + 8110928 T^{6} + 146088 T^{8} + 704 T^{10} + T^{12}$$