# Properties

 Label 361.3.f.d Level $361$ Weight $3$ Character orbit 361.f Analytic conductor $9.837$ Analytic rank $0$ Dimension $12$ CM no Inner twists $12$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 361.f (of order $$18$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.83653754341$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{18})$$ Coefficient field: 12.0.7659539263855005696.1 Defining polynomial: $$x^{12} - 2197 x^{6} + 4826809$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{7} ) q^{3} + 9 \beta_{2} q^{4} + ( -4 \beta_{4} + 4 \beta_{10} ) q^{5} + ( 13 \beta_{2} - 13 \beta_{8} ) q^{6} + 5 \beta_{6} q^{7} + 5 \beta_{3} q^{8} -4 \beta_{8} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{7} ) q^{3} + 9 \beta_{2} q^{4} + ( -4 \beta_{4} + 4 \beta_{10} ) q^{5} + ( 13 \beta_{2} - 13 \beta_{8} ) q^{6} + 5 \beta_{6} q^{7} + 5 \beta_{3} q^{8} -4 \beta_{8} q^{9} + ( -4 \beta_{5} + 4 \beta_{11} ) q^{10} + ( 10 - 10 \beta_{6} ) q^{11} + ( 9 \beta_{3} - 9 \beta_{9} ) q^{12} + \beta_{5} q^{13} + 5 \beta_{7} q^{14} + 4 \beta_{11} q^{15} + 29 \beta_{4} q^{16} -15 \beta_{10} q^{17} -4 \beta_{9} q^{18} -36 q^{20} + 5 \beta_{1} q^{21} + ( 10 \beta_{1} - 10 \beta_{7} ) q^{22} -35 \beta_{2} q^{23} + ( 65 \beta_{4} - 65 \beta_{10} ) q^{24} + ( -9 \beta_{2} + 9 \beta_{8} ) q^{25} + 13 \beta_{6} q^{26} + 5 \beta_{3} q^{27} + 45 \beta_{8} q^{28} + ( -5 \beta_{5} + 5 \beta_{11} ) q^{29} + ( -52 + 52 \beta_{6} ) q^{30} + ( -10 \beta_{3} + 10 \beta_{9} ) q^{31} + 9 \beta_{5} q^{32} -10 \beta_{7} q^{33} -15 \beta_{11} q^{34} -20 \beta_{4} q^{35} -36 \beta_{10} q^{36} -6 \beta_{9} q^{37} + 13 q^{39} -20 \beta_{1} q^{40} + ( -10 \beta_{1} + 10 \beta_{7} ) q^{41} + 65 \beta_{2} q^{42} + ( 20 \beta_{4} - 20 \beta_{10} ) q^{43} + ( 90 \beta_{2} - 90 \beta_{8} ) q^{44} + 16 \beta_{6} q^{45} -35 \beta_{3} q^{46} + 10 \beta_{8} q^{47} + ( 29 \beta_{5} - 29 \beta_{11} ) q^{48} + ( 24 - 24 \beta_{6} ) q^{49} + ( -9 \beta_{3} + 9 \beta_{9} ) q^{50} -15 \beta_{5} q^{51} + 9 \beta_{7} q^{52} + 21 \beta_{11} q^{53} + 65 \beta_{4} q^{54} + 40 \beta_{10} q^{55} + 25 \beta_{9} q^{56} -65 q^{58} + 5 \beta_{1} q^{59} + ( -36 \beta_{1} + 36 \beta_{7} ) q^{60} + 40 \beta_{2} q^{61} + ( -130 \beta_{4} + 130 \beta_{10} ) q^{62} + ( 20 \beta_{2} - 20 \beta_{8} ) q^{63} + \beta_{6} q^{64} -4 \beta_{3} q^{65} -130 \beta_{8} q^{66} + ( -11 \beta_{5} + 11 \beta_{11} ) q^{67} + ( 135 - 135 \beta_{6} ) q^{68} + ( -35 \beta_{3} + 35 \beta_{9} ) q^{69} -20 \beta_{5} q^{70} -30 \beta_{7} q^{71} -20 \beta_{11} q^{72} + 105 \beta_{4} q^{73} -78 \beta_{10} q^{74} + 9 \beta_{9} q^{75} + 50 q^{77} + 13 \beta_{1} q^{78} + ( 10 \beta_{1} - 10 \beta_{7} ) q^{79} -116 \beta_{2} q^{80} + ( 101 \beta_{4} - 101 \beta_{10} ) q^{81} + ( -130 \beta_{2} + 130 \beta_{8} ) q^{82} + 40 \beta_{6} q^{83} + 45 \beta_{3} q^{84} + 60 \beta_{8} q^{85} + ( 20 \beta_{5} - 20 \beta_{11} ) q^{86} + ( -65 + 65 \beta_{6} ) q^{87} + ( 50 \beta_{3} - 50 \beta_{9} ) q^{88} + 16 \beta_{7} q^{90} + 5 \beta_{11} q^{91} -315 \beta_{4} q^{92} + 130 \beta_{10} q^{93} + 10 \beta_{9} q^{94} + 117 q^{96} + 34 \beta_{1} q^{97} + ( 24 \beta_{1} - 24 \beta_{7} ) q^{98} -40 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 30q^{7} + O(q^{10})$$ $$12q + 30q^{7} + 60q^{11} - 432q^{20} + 78q^{26} - 312q^{30} + 156q^{39} + 96q^{45} + 144q^{49} - 780q^{58} + 6q^{64} + 810q^{68} + 600q^{77} + 240q^{83} - 390q^{87} + 1404q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2197 x^{6} + 4826809$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/13$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/13$$ $$\beta_{4}$$ $$=$$ $$\nu^{4}$$$$/169$$ $$\beta_{5}$$ $$=$$ $$\nu^{5}$$$$/169$$ $$\beta_{6}$$ $$=$$ $$\nu^{6}$$$$/2197$$ $$\beta_{7}$$ $$=$$ $$\nu^{7}$$$$/2197$$ $$\beta_{8}$$ $$=$$ $$\nu^{8}$$$$/28561$$ $$\beta_{9}$$ $$=$$ $$\nu^{9}$$$$/28561$$ $$\beta_{10}$$ $$=$$ $$\nu^{10}$$$$/371293$$ $$\beta_{11}$$ $$=$$ $$\nu^{11}$$$$/371293$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$13 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$13 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$169 \beta_{4}$$ $$\nu^{5}$$ $$=$$ $$169 \beta_{5}$$ $$\nu^{6}$$ $$=$$ $$2197 \beta_{6}$$ $$\nu^{7}$$ $$=$$ $$2197 \beta_{7}$$ $$\nu^{8}$$ $$=$$ $$28561 \beta_{8}$$ $$\nu^{9}$$ $$=$$ $$28561 \beta_{9}$$ $$\nu^{10}$$ $$=$$ $$371293 \beta_{10}$$ $$\nu^{11}$$ $$=$$ $$371293 \beta_{11}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/361\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 −3.55077 − 0.626097i 3.55077 + 0.626097i −2.31760 − 2.76201i 2.31760 + 2.76201i −1.23317 + 3.38811i 1.23317 − 3.38811i −1.23317 − 3.38811i 1.23317 + 3.38811i −2.31760 + 2.76201i 2.31760 − 2.76201i −3.55077 + 0.626097i 3.55077 − 0.626097i
−3.55077 0.626097i −2.31760 + 2.76201i 8.45723 + 3.07818i −3.75877 + 1.36808i 9.95858 8.35624i 2.50000 + 4.33013i −15.6125 9.01388i −0.694593 3.93923i 14.2031 2.50439i
116.2 3.55077 + 0.626097i 2.31760 2.76201i 8.45723 + 3.07818i −3.75877 + 1.36808i 9.95858 8.35624i 2.50000 + 4.33013i 15.6125 + 9.01388i −0.694593 3.93923i −14.2031 + 2.50439i
127.1 −2.31760 2.76201i 1.23317 3.38811i −1.56283 + 8.86327i 0.694593 + 3.93923i −12.2160 + 4.44626i 2.50000 4.33013i 15.6125 9.01388i −3.06418 2.57115i 9.27041 11.0481i
127.2 2.31760 + 2.76201i −1.23317 + 3.38811i −1.56283 + 8.86327i 0.694593 + 3.93923i −12.2160 + 4.44626i 2.50000 4.33013i −15.6125 + 9.01388i −3.06418 2.57115i −9.27041 + 11.0481i
262.1 −1.23317 + 3.38811i −3.55077 + 0.626097i −6.89440 5.78509i 3.06418 2.57115i 2.25743 12.8025i 2.50000 4.33013i 15.6125 9.01388i 3.75877 1.36808i 4.93268 + 13.5524i
262.2 1.23317 3.38811i 3.55077 0.626097i −6.89440 5.78509i 3.06418 2.57115i 2.25743 12.8025i 2.50000 4.33013i −15.6125 + 9.01388i 3.75877 1.36808i −4.93268 13.5524i
299.1 −1.23317 3.38811i −3.55077 0.626097i −6.89440 + 5.78509i 3.06418 + 2.57115i 2.25743 + 12.8025i 2.50000 + 4.33013i 15.6125 + 9.01388i 3.75877 + 1.36808i 4.93268 13.5524i
299.2 1.23317 + 3.38811i 3.55077 + 0.626097i −6.89440 + 5.78509i 3.06418 + 2.57115i 2.25743 + 12.8025i 2.50000 + 4.33013i −15.6125 9.01388i 3.75877 + 1.36808i −4.93268 + 13.5524i
307.1 −2.31760 + 2.76201i 1.23317 + 3.38811i −1.56283 8.86327i 0.694593 3.93923i −12.2160 4.44626i 2.50000 + 4.33013i 15.6125 + 9.01388i −3.06418 + 2.57115i 9.27041 + 11.0481i
307.2 2.31760 2.76201i −1.23317 3.38811i −1.56283 8.86327i 0.694593 3.93923i −12.2160 4.44626i 2.50000 + 4.33013i −15.6125 9.01388i −3.06418 + 2.57115i −9.27041 11.0481i
333.1 −3.55077 + 0.626097i −2.31760 2.76201i 8.45723 3.07818i −3.75877 1.36808i 9.95858 + 8.35624i 2.50000 4.33013i −15.6125 + 9.01388i −0.694593 + 3.93923i 14.2031 + 2.50439i
333.2 3.55077 0.626097i 2.31760 + 2.76201i 8.45723 3.07818i −3.75877 1.36808i 9.95858 + 8.35624i 2.50000 4.33013i 15.6125 9.01388i −0.694593 + 3.93923i −14.2031 2.50439i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 333.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner
19.c even 3 2 inner
19.d odd 6 2 inner
19.e even 9 3 inner
19.f odd 18 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.3.f.d 12
19.b odd 2 1 inner 361.3.f.d 12
19.c even 3 2 inner 361.3.f.d 12
19.d odd 6 2 inner 361.3.f.d 12
19.e even 9 1 19.3.b.b 2
19.e even 9 2 361.3.d.b 4
19.e even 9 3 inner 361.3.f.d 12
19.f odd 18 1 19.3.b.b 2
19.f odd 18 2 361.3.d.b 4
19.f odd 18 3 inner 361.3.f.d 12
57.j even 18 1 171.3.c.b 2
57.l odd 18 1 171.3.c.b 2
76.k even 18 1 304.3.e.d 2
76.l odd 18 1 304.3.e.d 2
95.o odd 18 1 475.3.c.b 2
95.p even 18 1 475.3.c.b 2
95.q odd 36 2 475.3.d.b 4
95.r even 36 2 475.3.d.b 4
152.s odd 18 1 1216.3.e.g 2
152.t even 18 1 1216.3.e.g 2
152.u odd 18 1 1216.3.e.h 2
152.v even 18 1 1216.3.e.h 2
228.u odd 18 1 2736.3.o.d 2
228.v even 18 1 2736.3.o.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 19.e even 9 1
19.3.b.b 2 19.f odd 18 1
171.3.c.b 2 57.j even 18 1
171.3.c.b 2 57.l odd 18 1
304.3.e.d 2 76.k even 18 1
304.3.e.d 2 76.l odd 18 1
361.3.d.b 4 19.e even 9 2
361.3.d.b 4 19.f odd 18 2
361.3.f.d 12 1.a even 1 1 trivial
361.3.f.d 12 19.b odd 2 1 inner
361.3.f.d 12 19.c even 3 2 inner
361.3.f.d 12 19.d odd 6 2 inner
361.3.f.d 12 19.e even 9 3 inner
361.3.f.d 12 19.f odd 18 3 inner
475.3.c.b 2 95.o odd 18 1
475.3.c.b 2 95.p even 18 1
475.3.d.b 4 95.q odd 36 2
475.3.d.b 4 95.r even 36 2
1216.3.e.g 2 152.s odd 18 1
1216.3.e.g 2 152.t even 18 1
1216.3.e.h 2 152.u odd 18 1
1216.3.e.h 2 152.v even 18 1
2736.3.o.d 2 228.u odd 18 1
2736.3.o.d 2 228.v even 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 2197 T_{2}^{6} + 4826809$$ acting on $$S_{3}^{\mathrm{new}}(361, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4826809 - 2197 T^{6} + T^{12}$$
$3$ $$4826809 - 2197 T^{6} + T^{12}$$
$5$ $$( 4096 + 64 T^{3} + T^{6} )^{2}$$
$7$ $$( 25 - 5 T + T^{2} )^{6}$$
$11$ $$( 100 - 10 T + T^{2} )^{6}$$
$13$ $$4826809 - 2197 T^{6} + T^{12}$$
$17$ $$( 11390625 + 3375 T^{3} + T^{6} )^{2}$$
$19$ $$T^{12}$$
$23$ $$( 1838265625 + 42875 T^{3} + T^{6} )^{2}$$
$29$ $$1178420166015625 - 34328125 T^{6} + T^{12}$$
$31$ $$( 1690000 - 1300 T^{2} + T^{4} )^{3}$$
$37$ $$( 468 + T^{2} )^{6}$$
$41$ $$4826809000000000000 - 2197000000 T^{6} + T^{12}$$
$43$ $$( 64000000 - 8000 T^{3} + T^{6} )^{2}$$
$47$ $$( 1000000 + 1000 T^{3} + T^{6} )^{2}$$
$53$ $$35\!\cdots\!69$$$$- 188428167837 T^{6} + T^{12}$$
$59$ $$1178420166015625 - 34328125 T^{6} + T^{12}$$
$61$ $$( 4096000000 - 64000 T^{3} + T^{6} )^{2}$$
$67$ $$15148594334612313289 - 3892119517 T^{6} + T^{12}$$
$71$ $$25\!\cdots\!00$$$$- 1601613000000 T^{6} + T^{12}$$
$73$ $$( 1340095640625 + 1157625 T^{3} + T^{6} )^{2}$$
$79$ $$4826809000000000000 - 2197000000 T^{6} + T^{12}$$
$83$ $$( 1600 - 40 T + T^{2} )^{6}$$
$89$ $$T^{12}$$
$97$ $$11\!\cdots\!04$$$$- 3393935301952 T^{6} + T^{12}$$
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