# Properties

 Label 475.2.p.e Level $475$ Weight $2$ Character orbit 475.p Analytic conductor $3.793$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.p (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: 16.0.14096583954457373039394816.1 Defining polynomial: $$x^{16} - 127 x^{12} + 13728 x^{8} - 304927 x^{4} + 5764801$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{14} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} -\beta_{5} q^{4} -\beta_{6} q^{6} + ( -\beta_{9} - 2 \beta_{14} ) q^{7} + ( -3 \beta_{3} + 3 \beta_{7} ) q^{8} + ( -4 \beta_{5} - \beta_{10} ) q^{9} +O(q^{10})$$ $$q + \beta_{14} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} -\beta_{5} q^{4} -\beta_{6} q^{6} + ( -\beta_{9} - 2 \beta_{14} ) q^{7} + ( -3 \beta_{3} + 3 \beta_{7} ) q^{8} + ( -4 \beta_{5} - \beta_{10} ) q^{9} + ( -1 + \beta_{4} ) q^{11} + ( \beta_{8} - \beta_{9} ) q^{12} + ( -\beta_{7} - \beta_{11} ) q^{13} + ( -\beta_{5} - \beta_{12} ) q^{14} -\beta_{2} q^{16} + ( -\beta_{8} + \beta_{9} - \beta_{15} ) q^{17} + ( -\beta_{1} - 5 \beta_{3} + 4 \beta_{7} + \beta_{11} ) q^{18} + ( -5 \beta_{5} + 2 \beta_{12} ) q^{19} + ( -\beta_{4} + \beta_{6} ) q^{21} + ( -\beta_{8} + \beta_{9} - \beta_{14} + \beta_{15} ) q^{22} + ( -2 \beta_{3} + \beta_{7} ) q^{23} + 3 \beta_{13} q^{24} + ( 1 - \beta_{4} ) q^{26} + ( 2 \beta_{8} - 9 \beta_{9} ) q^{27} + ( 2 \beta_{3} - \beta_{7} ) q^{28} + ( \beta_{10} - 2 \beta_{13} ) q^{29} + ( -2 + 4 \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{31} + ( 5 \beta_{9} + 5 \beta_{14} ) q^{32} + ( -2 \beta_{1} - 9 \beta_{3} ) q^{33} + ( \beta_{10} - 2 \beta_{13} ) q^{34} + ( 4 \beta_{2} - \beta_{4} - \beta_{6} ) q^{36} + ( -2 \beta_{8} - 2 \beta_{9} ) q^{37} + ( -3 \beta_{3} + 5 \beta_{7} ) q^{38} + ( 2 \beta_{10} + 7 \beta_{12} - 2 \beta_{13} ) q^{39} + ( -2 + \beta_{2} - \beta_{4} + \beta_{6} ) q^{41} + ( 2 \beta_{8} - 2 \beta_{9} - \beta_{15} ) q^{42} + ( -\beta_{1} + 2 \beta_{3} - 6 \beta_{7} + 2 \beta_{11} ) q^{43} + ( \beta_{5} + \beta_{10} ) q^{44} + ( 1 - 2 \beta_{2} ) q^{46} + ( -4 \beta_{8} + 3 \beta_{9} + \beta_{14} + 2 \beta_{15} ) q^{47} -\beta_{11} q^{48} -4 \beta_{12} q^{49} + ( -7 - 7 \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{51} + ( -\beta_{8} + \beta_{9} - \beta_{14} + \beta_{15} ) q^{52} + 7 \beta_{7} q^{53} + ( 7 \beta_{5} - 7 \beta_{12} + 2 \beta_{13} ) q^{54} + ( -3 + 6 \beta_{2} ) q^{56} + ( 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{15} ) q^{57} + ( \beta_{1} + \beta_{3} + \beta_{11} ) q^{58} + ( \beta_{5} - 4 \beta_{10} + \beta_{12} + 2 \beta_{13} ) q^{59} + ( -3 + 3 \beta_{2} - 3 \beta_{6} ) q^{61} + ( \beta_{8} + 3 \beta_{9} + 2 \beta_{14} + \beta_{15} ) q^{62} + ( 2 \beta_{1} + 10 \beta_{3} - 4 \beta_{7} - \beta_{11} ) q^{63} + 7 \beta_{12} q^{64} + ( 7 - 7 \beta_{2} + 2 \beta_{6} ) q^{66} + ( -2 \beta_{9} - 2 \beta_{14} - 4 \beta_{15} ) q^{67} + ( -\beta_{1} - \beta_{3} - \beta_{11} ) q^{68} + ( \beta_{10} + \beta_{13} ) q^{69} + ( 10 - 5 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{71} + ( 12 \beta_{9} + 12 \beta_{14} - 3 \beta_{15} ) q^{72} + ( \beta_{1} + 5 \beta_{3} - 8 \beta_{7} - 2 \beta_{11} ) q^{73} + ( 4 \beta_{5} - 4 \beta_{12} - 2 \beta_{13} ) q^{74} + ( 2 + 3 \beta_{2} ) q^{76} + ( \beta_{8} + 2 \beta_{14} - 2 \beta_{15} ) q^{77} + ( 2 \beta_{1} + 9 \beta_{3} ) q^{78} + ( 3 \beta_{5} - 6 \beta_{10} + 3 \beta_{12} + 3 \beta_{13} ) q^{79} + ( 2 \beta_{2} - 6 \beta_{4} - 6 \beta_{6} ) q^{81} + ( 2 \beta_{8} - \beta_{9} - \beta_{14} - \beta_{15} ) q^{82} + ( \beta_{1} + 2 \beta_{3} + \beta_{7} + \beta_{11} ) q^{83} + ( -\beta_{10} - \beta_{13} ) q^{84} + ( 3 + 3 \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{86} + ( -\beta_{8} - 6 \beta_{9} - 14 \beta_{14} + 2 \beta_{15} ) q^{87} + ( 3 \beta_{1} + 6 \beta_{3} - 3 \beta_{7} - 3 \beta_{11} ) q^{88} + ( 3 \beta_{5} - 6 \beta_{12} ) q^{89} + ( -1 - \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{91} + ( 2 \beta_{9} + \beta_{14} ) q^{92} + ( -\beta_{1} + 6 \beta_{3} - 14 \beta_{7} + 2 \beta_{11} ) q^{93} + ( 2 \beta_{5} - 2 \beta_{10} - \beta_{12} - 2 \beta_{13} ) q^{94} + 5 \beta_{4} q^{96} + ( \beta_{8} - \beta_{9} + 4 \beta_{14} - \beta_{15} ) q^{97} -4 \beta_{3} q^{98} + ( 11 \beta_{5} + 6 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{6} + O(q^{10})$$ $$16 q - 4 q^{6} - 24 q^{11} - 8 q^{16} + 12 q^{21} + 24 q^{26} + 36 q^{36} - 12 q^{41} - 180 q^{51} - 36 q^{61} + 64 q^{66} + 96 q^{71} + 56 q^{76} + 40 q^{81} + 60 q^{86} - 36 q^{91} - 40 q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 127 x^{12} + 13728 x^{8} - 304927 x^{4} + 5764801$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$127 \nu^{12} - 13728 \nu^{8} + 1743456 \nu^{4} - 5764801$$$$)/32960928$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{13} + 669761 \nu$$$$)/1441440$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{12} - 669761$$$$)/205920$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{14} - 2111201 \nu^{2}$$$$)/10090080$$ $$\beta_{6}$$ $$=$$ $$($$$$-827 \nu^{12} + 121836 \nu^{8} - 11353056 \nu^{4} + 252174629$$$$)/61801740$$ $$\beta_{7}$$ $$=$$ $$($$$$9017 \nu^{13} - 974688 \nu^{9} + 90824448 \nu^{5} - 409300871 \nu$$$$)/ 3460897440$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{15} + 2111201 \nu^{3}$$$$)/10090080$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{15} - 849941 \nu^{3}$$$$)/8828820$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{14} - 849941 \nu^{2}$$$$)/1261260$$ $$\beta_{11}$$ $$=$$ $$($$$$127 \nu^{13} - 13728 \nu^{9} + 1556178 \nu^{5} - 5764801 \nu$$$$)/19664190$$ $$\beta_{12}$$ $$=$$ $$($$$$-127 \nu^{14} + 13728 \nu^{10} - 1556178 \nu^{6} + 5764801 \nu^{2}$$$$)/ 137649330$$ $$\beta_{13}$$ $$=$$ $$($$$$66263 \nu^{14} - 9238944 \nu^{10} + 909658464 \nu^{6} - 20205377801 \nu^{2}$$$$)/ 24226282080$$ $$\beta_{14}$$ $$=$$ $$($$$$85471 \nu^{15} - 9238944 \nu^{11} + 909658464 \nu^{7} - 3879711073 \nu^{3}$$$$)/ 169583974560$$ $$\beta_{15}$$ $$=$$ $$($$$$-\nu^{15} + 127 \nu^{11} - 13728 \nu^{7} + 304927 \nu^{3}$$$$)/823543$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} - 8 \beta_{5}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{9} + 8 \beta_{8}$$ $$\nu^{4}$$ $$=$$ $$15 \beta_{6} + 15 \beta_{4} + 71 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$71 \beta_{11} - 176 \beta_{7}$$ $$\nu^{6}$$ $$=$$ $$-176 \beta_{13} - 673 \beta_{12} + 176 \beta_{10}$$ $$\nu^{7}$$ $$=$$ $$-673 \beta_{15} - 1905 \beta_{14} - 673 \beta_{9} + 673 \beta_{8}$$ $$\nu^{8}$$ $$=$$ $$1905 \beta_{6} + 6616 \beta_{2} - 6616$$ $$\nu^{9}$$ $$=$$ $$6616 \beta_{11} - 19951 \beta_{7} + 13335 \beta_{3} - 6616 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-19951 \beta_{13} - 66263 \beta_{12} + 66263 \beta_{5}$$ $$\nu^{11}$$ $$=$$ $$-66263 \beta_{15} - 205920 \beta_{14} - 205920 \beta_{9}$$ $$\nu^{12}$$ $$=$$ $$-205920 \beta_{4} - 669761$$ $$\nu^{13}$$ $$=$$ $$1441440 \beta_{3} - 669761 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-2111201 \beta_{10} + 6799528 \beta_{5}$$ $$\nu^{15}$$ $$=$$ $$-14778407 \beta_{9} - 6799528 \beta_{8}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$\beta_{12}$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −0.826301 + 3.08380i 0.567482 − 2.11787i −0.567482 + 2.11787i 0.826301 − 3.08380i −0.826301 − 3.08380i 0.567482 + 2.11787i −0.567482 − 2.11787i 0.826301 + 3.08380i −3.08380 + 0.826301i 2.11787 − 0.567482i −2.11787 + 0.567482i 3.08380 − 0.826301i −3.08380 − 0.826301i 2.11787 + 0.567482i −2.11787 − 0.567482i 3.08380 + 0.826301i
−0.965926 0.258819i −0.567482 + 2.11787i −0.866025 0.500000i 0 1.09629 1.89883i 1.22474 + 1.22474i 2.12132 + 2.12132i −1.56527 0.903709i 0
107.2 −0.965926 0.258819i 0.826301 3.08380i −0.866025 0.500000i 0 −1.59629 + 2.76486i 1.22474 + 1.22474i 2.12132 + 2.12132i −6.22896 3.59629i 0
107.3 0.965926 + 0.258819i −0.826301 + 3.08380i −0.866025 0.500000i 0 −1.59629 + 2.76486i −1.22474 1.22474i −2.12132 2.12132i −6.22896 3.59629i 0
107.4 0.965926 + 0.258819i 0.567482 2.11787i −0.866025 0.500000i 0 1.09629 1.89883i −1.22474 1.22474i −2.12132 2.12132i −1.56527 0.903709i 0
293.1 −0.965926 + 0.258819i −0.567482 2.11787i −0.866025 + 0.500000i 0 1.09629 + 1.89883i 1.22474 1.22474i 2.12132 2.12132i −1.56527 + 0.903709i 0
293.2 −0.965926 + 0.258819i 0.826301 + 3.08380i −0.866025 + 0.500000i 0 −1.59629 2.76486i 1.22474 1.22474i 2.12132 2.12132i −6.22896 + 3.59629i 0
293.3 0.965926 0.258819i −0.826301 3.08380i −0.866025 + 0.500000i 0 −1.59629 2.76486i −1.22474 + 1.22474i −2.12132 + 2.12132i −6.22896 + 3.59629i 0
293.4 0.965926 0.258819i 0.567482 + 2.11787i −0.866025 + 0.500000i 0 1.09629 + 1.89883i −1.22474 + 1.22474i −2.12132 + 2.12132i −1.56527 + 0.903709i 0
407.1 −0.258819 0.965926i −2.11787 + 0.567482i 0.866025 0.500000i 0 1.09629 + 1.89883i 1.22474 + 1.22474i −2.12132 2.12132i 1.56527 0.903709i 0
407.2 −0.258819 0.965926i 3.08380 0.826301i 0.866025 0.500000i 0 −1.59629 2.76486i 1.22474 + 1.22474i −2.12132 2.12132i 6.22896 3.59629i 0
407.3 0.258819 + 0.965926i −3.08380 + 0.826301i 0.866025 0.500000i 0 −1.59629 2.76486i −1.22474 1.22474i 2.12132 + 2.12132i 6.22896 3.59629i 0
407.4 0.258819 + 0.965926i 2.11787 0.567482i 0.866025 0.500000i 0 1.09629 + 1.89883i −1.22474 1.22474i 2.12132 + 2.12132i 1.56527 0.903709i 0
468.1 −0.258819 + 0.965926i −2.11787 0.567482i 0.866025 + 0.500000i 0 1.09629 1.89883i 1.22474 1.22474i −2.12132 + 2.12132i 1.56527 + 0.903709i 0
468.2 −0.258819 + 0.965926i 3.08380 + 0.826301i 0.866025 + 0.500000i 0 −1.59629 + 2.76486i 1.22474 1.22474i −2.12132 + 2.12132i 6.22896 + 3.59629i 0
468.3 0.258819 0.965926i −3.08380 0.826301i 0.866025 + 0.500000i 0 −1.59629 + 2.76486i −1.22474 + 1.22474i 2.12132 2.12132i 6.22896 + 3.59629i 0
468.4 0.258819 0.965926i 2.11787 + 0.567482i 0.866025 + 0.500000i 0 1.09629 1.89883i −1.22474 + 1.22474i 2.12132 2.12132i 1.56527 + 0.903709i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 468.4 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
5.c odd 4 2 inner
19.d odd 6 1 inner
95.h odd 6 1 inner
95.l even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.p.e 16
5.b even 2 1 inner 475.2.p.e 16
5.c odd 4 2 inner 475.2.p.e 16
19.d odd 6 1 inner 475.2.p.e 16
95.h odd 6 1 inner 475.2.p.e 16
95.l even 12 2 inner 475.2.p.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.p.e 16 1.a even 1 1 trivial
475.2.p.e 16 5.b even 2 1 inner
475.2.p.e 16 5.c odd 4 2 inner
475.2.p.e 16 19.d odd 6 1 inner
475.2.p.e 16 95.h odd 6 1 inner
475.2.p.e 16 95.l even 12 2 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$:

 $$T_{2}^{8} - T_{2}^{4} + 1$$ $$T_{3}^{16} - 127 T_{3}^{12} + 13728 T_{3}^{8} - 304927 T_{3}^{4} + 5764801$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{4} + T^{8} )^{2}$$
$3$ $$5764801 - 304927 T^{4} + 13728 T^{8} - 127 T^{12} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 9 + T^{4} )^{4}$$
$11$ $$( -5 + 3 T + T^{2} )^{8}$$
$13$ $$390625 - 194375 T^{4} + 96096 T^{8} - 311 T^{12} + T^{16}$$
$17$ $$37822859361 - 222291783 T^{4} + 1111968 T^{8} - 1143 T^{12} + T^{16}$$
$19$ $$( 361 - 37 T^{2} + T^{4} )^{4}$$
$23$ $$( 81 - 9 T^{4} + T^{8} )^{2}$$
$29$ $$( 194481 + 19845 T^{2} + 1584 T^{4} + 45 T^{6} + T^{8} )^{2}$$
$31$ $$( 225 + 57 T^{2} + T^{4} )^{4}$$
$37$ $$( 160000 + 4976 T^{4} + T^{8} )^{2}$$
$41$ $$( 441 - 63 T - 18 T^{2} + 3 T^{3} + T^{4} )^{4}$$
$43$ $$6561 - 529983 T^{4} + 42810768 T^{8} - 6543 T^{12} + T^{16}$$
$47$ $$( 57289761 - 7569 T^{4} + T^{8} )^{2}$$
$53$ $$( 5764801 - 2401 T^{4} + T^{8} )^{2}$$
$59$ $$( 7569 + 87 T^{2} + T^{4} )^{4}$$
$61$ $$( 2025 - 405 T + 126 T^{2} + 9 T^{3} + T^{4} )^{4}$$
$67$ $$( 181063936 - 13456 T^{4} + T^{8} )^{2}$$
$71$ $$( 1521 + 936 T + 153 T^{2} - 24 T^{3} + T^{4} )^{4}$$
$73$ $$5352009260481 - 55945943703 T^{4} + 582504048 T^{8} - 24183 T^{12} + T^{16}$$
$79$ $$( 1275989841 + 14467005 T^{2} + 128304 T^{4} + 405 T^{6} + T^{8} )^{2}$$
$83$ $$( 50625 + 2799 T^{4} + T^{8} )^{2}$$
$89$ $$( 729 + 27 T^{2} + T^{4} )^{4}$$
$97$ $$815730721 - 76743407 T^{4} + 7191408 T^{8} - 2687 T^{12} + T^{16}$$