# Properties

 Label 2366.2.d.b Level $2366$ Weight $2$ Character orbit 2366.d Analytic conductor $18.893$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2366 = 2 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2366.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.8926051182$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -i q^{2} -2 q^{3} - q^{4} + 2 i q^{6} -i q^{7} + i q^{8} + q^{9} +O(q^{10})$$ $$q -i q^{2} -2 q^{3} - q^{4} + 2 i q^{6} -i q^{7} + i q^{8} + q^{9} + 2 q^{12} - q^{14} + q^{16} -6 q^{17} -i q^{18} + 2 i q^{19} + 2 i q^{21} -2 i q^{24} + 5 q^{25} + 4 q^{27} + i q^{28} -6 q^{29} -4 i q^{31} -i q^{32} + 6 i q^{34} - q^{36} -2 i q^{37} + 2 q^{38} + 6 i q^{41} + 2 q^{42} -8 q^{43} + 12 i q^{47} -2 q^{48} - q^{49} -5 i q^{50} + 12 q^{51} + 6 q^{53} -4 i q^{54} + q^{56} -4 i q^{57} + 6 i q^{58} + 6 i q^{59} + 8 q^{61} -4 q^{62} -i q^{63} - q^{64} -4 i q^{67} + 6 q^{68} + i q^{72} -2 i q^{73} -2 q^{74} -10 q^{75} -2 i q^{76} + 8 q^{79} -11 q^{81} + 6 q^{82} -6 i q^{83} -2 i q^{84} + 8 i q^{86} + 12 q^{87} + 6 i q^{89} + 8 i q^{93} + 12 q^{94} + 2 i q^{96} -10 i q^{97} + i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 4q^{3} - 2q^{4} + 2q^{9} + 4q^{12} - 2q^{14} + 2q^{16} - 12q^{17} + 10q^{25} + 8q^{27} - 12q^{29} - 2q^{36} + 4q^{38} + 4q^{42} - 16q^{43} - 4q^{48} - 2q^{49} + 24q^{51} + 12q^{53} + 2q^{56} + 16q^{61} - 8q^{62} - 2q^{64} + 12q^{68} - 4q^{74} - 20q^{75} + 16q^{79} - 22q^{81} + 12q^{82} + 24q^{87} + 24q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$2199$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 1.00000i − 1.00000i
1.00000i −2.00000 −1.00000 0 2.00000i 1.00000i 1.00000i 1.00000 0
337.2 1.00000i −2.00000 −1.00000 0 2.00000i 1.00000i 1.00000i 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2366.2.d.b 2
13.b even 2 1 inner 2366.2.d.b 2
13.d odd 4 1 14.2.a.a 1
13.d odd 4 1 2366.2.a.j 1
39.f even 4 1 126.2.a.b 1
52.f even 4 1 112.2.a.c 1
65.f even 4 1 350.2.c.d 2
65.g odd 4 1 350.2.a.f 1
65.k even 4 1 350.2.c.d 2
91.i even 4 1 98.2.a.a 1
91.z odd 12 2 98.2.c.b 2
91.bb even 12 2 98.2.c.a 2
104.j odd 4 1 448.2.a.g 1
104.m even 4 1 448.2.a.a 1
117.y odd 12 2 1134.2.f.l 2
117.z even 12 2 1134.2.f.f 2
143.g even 4 1 1694.2.a.e 1
156.l odd 4 1 1008.2.a.h 1
195.j odd 4 1 3150.2.g.j 2
195.n even 4 1 3150.2.a.i 1
195.u odd 4 1 3150.2.g.j 2
208.l even 4 1 1792.2.b.g 2
208.m odd 4 1 1792.2.b.c 2
208.r odd 4 1 1792.2.b.c 2
208.s even 4 1 1792.2.b.g 2
221.g odd 4 1 4046.2.a.f 1
247.i even 4 1 5054.2.a.c 1
260.l odd 4 1 2800.2.g.h 2
260.s odd 4 1 2800.2.g.h 2
260.u even 4 1 2800.2.a.g 1
273.o odd 4 1 882.2.a.i 1
273.cb odd 12 2 882.2.g.d 2
273.cd even 12 2 882.2.g.c 2
299.g even 4 1 7406.2.a.a 1
312.w odd 4 1 4032.2.a.r 1
312.y even 4 1 4032.2.a.w 1
364.p odd 4 1 784.2.a.b 1
364.bw odd 12 2 784.2.i.i 2
364.ce even 12 2 784.2.i.c 2
455.n odd 4 1 2450.2.c.c 2
455.u even 4 1 2450.2.a.t 1
455.w odd 4 1 2450.2.c.c 2
728.x odd 4 1 3136.2.a.z 1
728.ba even 4 1 3136.2.a.e 1
1092.u even 4 1 7056.2.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 13.d odd 4 1
98.2.a.a 1 91.i even 4 1
98.2.c.a 2 91.bb even 12 2
98.2.c.b 2 91.z odd 12 2
112.2.a.c 1 52.f even 4 1
126.2.a.b 1 39.f even 4 1
350.2.a.f 1 65.g odd 4 1
350.2.c.d 2 65.f even 4 1
350.2.c.d 2 65.k even 4 1
448.2.a.a 1 104.m even 4 1
448.2.a.g 1 104.j odd 4 1
784.2.a.b 1 364.p odd 4 1
784.2.i.c 2 364.ce even 12 2
784.2.i.i 2 364.bw odd 12 2
882.2.a.i 1 273.o odd 4 1
882.2.g.c 2 273.cd even 12 2
882.2.g.d 2 273.cb odd 12 2
1008.2.a.h 1 156.l odd 4 1
1134.2.f.f 2 117.z even 12 2
1134.2.f.l 2 117.y odd 12 2
1694.2.a.e 1 143.g even 4 1
1792.2.b.c 2 208.m odd 4 1
1792.2.b.c 2 208.r odd 4 1
1792.2.b.g 2 208.l even 4 1
1792.2.b.g 2 208.s even 4 1
2366.2.a.j 1 13.d odd 4 1
2366.2.d.b 2 1.a even 1 1 trivial
2366.2.d.b 2 13.b even 2 1 inner
2450.2.a.t 1 455.u even 4 1
2450.2.c.c 2 455.n odd 4 1
2450.2.c.c 2 455.w odd 4 1
2800.2.a.g 1 260.u even 4 1
2800.2.g.h 2 260.l odd 4 1
2800.2.g.h 2 260.s odd 4 1
3136.2.a.e 1 728.ba even 4 1
3136.2.a.z 1 728.x odd 4 1
3150.2.a.i 1 195.n even 4 1
3150.2.g.j 2 195.j odd 4 1
3150.2.g.j 2 195.u odd 4 1
4032.2.a.r 1 312.w odd 4 1
4032.2.a.w 1 312.y even 4 1
4046.2.a.f 1 221.g odd 4 1
5054.2.a.c 1 247.i even 4 1
7056.2.a.bd 1 1092.u even 4 1
7406.2.a.a 1 299.g even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2366, [\chi])$$:

 $$T_{3} + 2$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( 2 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$4 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$36 + T^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$144 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$36 + T^{2}$$
$61$ $$( -8 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$36 + T^{2}$$
$97$ $$100 + T^{2}$$