# Properties

 Label 471.2.b.a Level $471$ Weight $2$ Character orbit 471.b Analytic conductor $3.761$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$471 = 3 \cdot 157$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 471.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.76095393520$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 15 x^{10} + 77 x^{8} + 158 x^{6} + 111 x^{4} + 21 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} - q^{3} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{7} ) q^{5} -\beta_{1} q^{6} + \beta_{10} q^{7} + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} - q^{3} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{1} + \beta_{7} ) q^{5} -\beta_{1} q^{6} + \beta_{10} q^{7} + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{8} + q^{9} + ( 2 - \beta_{2} ) q^{10} + ( -\beta_{2} - \beta_{4} ) q^{11} + ( 1 - \beta_{2} ) q^{12} -\beta_{3} q^{13} -\beta_{3} q^{14} + ( \beta_{1} - \beta_{7} ) q^{15} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{16} + ( \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{17} + \beta_{1} q^{18} + ( -\beta_{3} - \beta_{8} ) q^{19} + ( 2 \beta_{1} - \beta_{6} + \beta_{7} ) q^{20} -\beta_{10} q^{21} + ( \beta_{1} - 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{22} + ( \beta_{9} + \beta_{11} ) q^{23} + ( \beta_{1} - \beta_{6} - \beta_{7} ) q^{24} + ( 1 - \beta_{5} ) q^{25} + ( -2 \beta_{10} - \beta_{11} ) q^{26} - q^{27} -\beta_{11} q^{28} + ( -\beta_{1} + 2 \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{29} + ( -2 + \beta_{2} ) q^{30} + ( -\beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{31} + ( -\beta_{1} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{32} + ( \beta_{2} + \beta_{4} ) q^{33} + ( -\beta_{1} + 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} ) q^{34} + ( \beta_{3} - \beta_{5} + \beta_{8} ) q^{35} + ( -1 + \beta_{2} ) q^{36} + ( -\beta_{2} - \beta_{4} + \beta_{8} ) q^{37} + ( \beta_{1} + \beta_{6} - \beta_{10} - \beta_{11} ) q^{38} + \beta_{3} q^{39} + ( -4 + 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{40} + ( \beta_{1} - 2 \beta_{6} + \beta_{7} ) q^{41} + \beta_{3} q^{42} + ( -\beta_{1} + 3 \beta_{7} - \beta_{9} + \beta_{11} ) q^{43} + ( -4 + 4 \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{8} ) q^{44} + ( -\beta_{1} + \beta_{7} ) q^{45} + ( -\beta_{3} + \beta_{4} ) q^{46} + ( 4 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{47} + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} ) q^{48} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} ) q^{49} + ( 2 \beta_{1} + \beta_{6} ) q^{50} + ( -\beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} ) q^{51} + ( \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{52} + ( 2 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{53} -\beta_{1} q^{54} + ( 2 \beta_{6} - 2 \beta_{7} - \beta_{10} + \beta_{11} ) q^{55} + ( \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{56} + ( \beta_{3} + \beta_{8} ) q^{57} + ( 4 - 4 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} ) q^{58} + ( \beta_{1} - 3 \beta_{7} + \beta_{10} ) q^{59} + ( -2 \beta_{1} + \beta_{6} - \beta_{7} ) q^{60} + ( \beta_{1} - 3 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{61} + ( -3 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{62} + \beta_{10} q^{63} + ( 3 + \beta_{3} + \beta_{4} + \beta_{8} ) q^{64} + ( \beta_{10} + \beta_{11} ) q^{65} + ( -\beta_{1} + 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{66} + ( -4 + \beta_{2} - \beta_{4} - 2 \beta_{8} ) q^{67} + ( 4 - 4 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{8} ) q^{68} + ( -\beta_{9} - \beta_{11} ) q^{69} + ( \beta_{10} + \beta_{11} ) q^{70} + ( -4 + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{8} ) q^{71} + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{72} + ( 3 \beta_{10} - \beta_{11} ) q^{73} + ( -3 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{74} + ( -1 + \beta_{5} ) q^{75} + ( -2 + \beta_{3} - \beta_{8} ) q^{76} + ( 2 \beta_{6} - 2 \beta_{7} - \beta_{10} - \beta_{11} ) q^{77} + ( 2 \beta_{10} + \beta_{11} ) q^{78} + ( -\beta_{1} - 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{79} + ( -2 \beta_{1} + 2 \beta_{6} + 4 \beta_{7} - \beta_{9} ) q^{80} + q^{81} + ( -6 + 5 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{82} + ( -2 \beta_{1} - \beta_{9} - 2 \beta_{10} ) q^{83} + \beta_{11} q^{84} + ( \beta_{1} - 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{85} + ( -3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} ) q^{86} + ( \beta_{1} - 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} ) q^{87} + ( -6 \beta_{1} + 4 \beta_{6} + 2 \beta_{7} + \beta_{10} + \beta_{11} ) q^{88} + ( \beta_{3} + 2 \beta_{5} - 2 \beta_{8} ) q^{89} + ( 2 - \beta_{2} ) q^{90} + ( -5 \beta_{1} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{91} + ( \beta_{1} + \beta_{6} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{92} + ( \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{8} ) q^{93} + ( 5 \beta_{1} - 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{94} + ( -2 \beta_{7} + 2 \beta_{10} ) q^{95} + ( \beta_{1} + \beta_{6} - \beta_{7} - \beta_{9} ) q^{96} + ( 5 \beta_{1} - 3 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{97} + ( 2 \beta_{1} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{98} + ( -\beta_{2} - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{3} - 6 q^{4} + 12 q^{9} + O(q^{10})$$ $$12 q - 12 q^{3} - 6 q^{4} + 12 q^{9} + 18 q^{10} - 2 q^{11} + 6 q^{12} + 4 q^{13} + 4 q^{14} + 10 q^{16} + 2 q^{17} + 4 q^{19} + 12 q^{25} - 12 q^{27} - 18 q^{30} + 2 q^{31} + 2 q^{33} - 4 q^{35} - 6 q^{36} - 2 q^{37} - 4 q^{39} - 40 q^{40} - 4 q^{42} - 36 q^{44} + 34 q^{47} - 10 q^{48} + 10 q^{49} - 2 q^{51} - 6 q^{52} + 2 q^{56} - 4 q^{57} + 24 q^{58} + 28 q^{64} - 38 q^{67} + 32 q^{68} - 26 q^{71} - 12 q^{75} - 28 q^{76} + 12 q^{81} - 50 q^{82} + 6 q^{86} - 4 q^{89} + 18 q^{90} - 2 q^{93} - 2 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 15 x^{10} + 77 x^{8} + 158 x^{6} + 111 x^{4} + 21 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{10} + 14 \nu^{8} + 65 \nu^{6} + 115 \nu^{4} + 62 \nu^{2} + 5$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$\nu^{10} + 15 \nu^{8} + 77 \nu^{6} + 157 \nu^{4} + 103 \nu^{2} + 9$$ $$\beta_{5}$$ $$=$$ $$\nu^{10} + 15 \nu^{8} + 77 \nu^{6} + 158 \nu^{4} + 110 \nu^{2} + 15$$ $$\beta_{6}$$ $$=$$ $$-\nu^{11} - 15 \nu^{9} - 77 \nu^{7} - 158 \nu^{5} - 110 \nu^{3} - 16 \nu$$ $$\beta_{7}$$ $$=$$ $$\nu^{11} + 15 \nu^{9} + 77 \nu^{7} + 158 \nu^{5} + 111 \nu^{3} + 21 \nu$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{10} - 44 \nu^{8} - 217 \nu^{6} - 409 \nu^{4} - 220 \nu^{2} - 13$$$$)/2$$ $$\beta_{9}$$ $$=$$ $$-2 \nu^{11} - 30 \nu^{9} - 154 \nu^{7} - 315 \nu^{5} - 213 \nu^{3} - 24 \nu$$ $$\beta_{10}$$ $$=$$ $$($$$$5 \nu^{11} + 74 \nu^{9} + 371 \nu^{7} + 725 \nu^{5} + 440 \nu^{3} + 43 \nu$$$$)/2$$ $$\beta_{11}$$ $$=$$ $$($$$$-9 \nu^{11} - 134 \nu^{9} - 677 \nu^{7} - 1335 \nu^{5} - 818 \nu^{3} - 81 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} - 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} - \beta_{4} - 7 \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$\beta_{9} - 7 \beta_{7} - 9 \beta_{6} + 27 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{8} - 10 \beta_{5} + 11 \beta_{4} + \beta_{3} + 46 \beta_{2} - 83$$ $$\nu^{7}$$ $$=$$ $$\beta_{11} + \beta_{10} - 11 \beta_{9} + 46 \beta_{7} + 66 \beta_{6} - 155 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-12 \beta_{8} + 78 \beta_{5} - 89 \beta_{4} - 14 \beta_{3} - 299 \beta_{2} + 485$$ $$\nu^{9}$$ $$=$$ $$-14 \beta_{11} - 16 \beta_{10} + 89 \beta_{9} - 299 \beta_{7} - 454 \beta_{6} + 928 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$103 \beta_{8} - 557 \beta_{5} + 646 \beta_{4} + 133 \beta_{3} + 1939 \beta_{2} - 2939$$ $$\nu^{11}$$ $$=$$ $$133 \beta_{11} + 163 \beta_{10} - 646 \beta_{9} + 1939 \beta_{7} + 3039 \beta_{6} - 5717 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/471\mathbb{Z}\right)^\times$$.

 $$n$$ $$158$$ $$319$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
313.1
 − 2.55080i − 2.06731i − 1.76113i − 0.927515i − 0.430002i − 0.269982i 0.269982i 0.430002i 0.927515i 1.76113i 2.06731i 2.55080i
2.55080i −1.00000 −4.50658 2.15877i 2.55080i 0.830530i 6.39378i 1.00000 5.50658
313.2 2.06731i −1.00000 −2.27376 1.58359i 2.06731i 2.74875i 0.565947i 1.00000 3.27376
313.3 1.76113i −1.00000 −1.10159 1.19332i 1.76113i 3.36519i 1.58223i 1.00000 2.10159
313.4 0.927515i −1.00000 1.13972 0.150634i 0.927515i 1.39352i 2.91213i 1.00000 −0.139715
313.5 0.430002i −1.00000 1.81510 1.89557i 0.430002i 3.40343i 1.64050i 1.00000 −0.815098
313.6 0.269982i −1.00000 1.92711 3.43396i 0.269982i 1.97607i 1.06025i 1.00000 −0.927110
313.7 0.269982i −1.00000 1.92711 3.43396i 0.269982i 1.97607i 1.06025i 1.00000 −0.927110
313.8 0.430002i −1.00000 1.81510 1.89557i 0.430002i 3.40343i 1.64050i 1.00000 −0.815098
313.9 0.927515i −1.00000 1.13972 0.150634i 0.927515i 1.39352i 2.91213i 1.00000 −0.139715
313.10 1.76113i −1.00000 −1.10159 1.19332i 1.76113i 3.36519i 1.58223i 1.00000 2.10159
313.11 2.06731i −1.00000 −2.27376 1.58359i 2.06731i 2.74875i 0.565947i 1.00000 3.27376
313.12 2.55080i −1.00000 −4.50658 2.15877i 2.55080i 0.830530i 6.39378i 1.00000 5.50658
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 313.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
157.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.b.a 12
3.b odd 2 1 1413.2.b.c 12
157.b even 2 1 inner 471.2.b.a 12
471.d odd 2 1 1413.2.b.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.b.a 12 1.a even 1 1 trivial
471.2.b.a 12 157.b even 2 1 inner
1413.2.b.c 12 3.b odd 2 1
1413.2.b.c 12 471.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 15 T_{2}^{10} + 77 T_{2}^{8} + 158 T_{2}^{6} + 111 T_{2}^{4} + 21 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(471, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 21 T^{2} + 111 T^{4} + 158 T^{6} + 77 T^{8} + 15 T^{10} + T^{12}$$
$3$ $$( 1 + T )^{12}$$
$5$ $$16 + 732 T^{2} + 1200 T^{4} + 722 T^{6} + 197 T^{8} + 24 T^{10} + T^{12}$$
$7$ $$5184 + 13104 T^{2} + 10172 T^{4} + 3339 T^{6} + 515 T^{8} + 37 T^{10} + T^{12}$$
$11$ $$( -1728 + 288 T + 504 T^{2} - 46 T^{3} - 45 T^{4} + T^{5} + T^{6} )^{2}$$
$13$ $$( -72 - 108 T + 94 T^{2} + 63 T^{3} - 36 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$17$ $$( -2088 + 204 T + 830 T^{2} + 21 T^{3} - 55 T^{4} - T^{5} + T^{6} )^{2}$$
$19$ $$( -544 - 128 T + 528 T^{2} + 68 T^{3} - 52 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$23$ $$36 + 2036660 T^{2} + 3945298 T^{4} + 366241 T^{6} + 12601 T^{8} + 187 T^{10} + T^{12}$$
$29$ $$1230045184 + 304888892 T^{2} + 27455284 T^{4} + 1164754 T^{6} + 24683 T^{8} + 253 T^{10} + T^{12}$$
$31$ $$( 7152 + 10276 T + 3712 T^{2} - 66 T^{3} - 127 T^{4} - T^{5} + T^{6} )^{2}$$
$37$ $$( 21072 + 15176 T + 2460 T^{2} - 396 T^{3} - 110 T^{4} + T^{5} + T^{6} )^{2}$$
$41$ $$3610000 + 14624700 T^{2} + 4068144 T^{4} + 413594 T^{6} + 17237 T^{8} + 240 T^{10} + T^{12}$$
$43$ $$23740646400 + 6736845312 T^{2} + 348441920 T^{4} + 7673200 T^{6} + 84688 T^{8} + 463 T^{10} + T^{12}$$
$47$ $$( 18432 - 4416 T - 7184 T^{2} + 2092 T^{3} - 63 T^{4} - 17 T^{5} + T^{6} )^{2}$$
$53$ $$1937664 + 8566400 T^{2} + 4558176 T^{4} + 655996 T^{6} + 24240 T^{8} + 282 T^{10} + T^{12}$$
$59$ $$238144 + 1137888 T^{2} + 1187556 T^{4} + 208700 T^{6} + 12086 T^{8} + 201 T^{10} + T^{12}$$
$61$ $$5308416 + 21003264 T^{2} + 11464064 T^{4} + 1082240 T^{6} + 33977 T^{8} + 362 T^{10} + T^{12}$$
$67$ $$( -115200 - 93168 T - 27272 T^{2} - 3192 T^{3} - 39 T^{4} + 19 T^{5} + T^{6} )^{2}$$
$71$ $$( -660960 + 124128 T + 19632 T^{2} - 2810 T^{3} - 253 T^{4} + 13 T^{5} + T^{6} )^{2}$$
$73$ $$215737344 + 206779392 T^{2} + 48122784 T^{4} + 2680804 T^{6} + 56173 T^{8} + 443 T^{10} + T^{12}$$
$79$ $$324000000 + 258496128 T^{2} + 65815424 T^{4} + 5566864 T^{6} + 93289 T^{8} + 538 T^{10} + T^{12}$$
$83$ $$132066064 + 69413708 T^{2} + 10700588 T^{4} + 706906 T^{6} + 21387 T^{8} + 269 T^{10} + T^{12}$$
$89$ $$( -253944 - 26148 T + 26218 T^{2} - 63 T^{3} - 358 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$97$ $$6737126400 + 5008822272 T^{2} + 479253024 T^{4} + 12314212 T^{6} + 131481 T^{8} + 614 T^{10} + T^{12}$$