# Properties

 Label 234.2.b.a Level $234$ Weight $2$ Character orbit 234.b Analytic conductor $1.868$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 234.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.86849940730$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + 2 i q^{5} + 2 i q^{7} - i q^{8} +O(q^{10})$$ q + i * q^2 - q^4 + 2*i * q^5 + 2*i * q^7 - i * q^8 $$q + i q^{2} - q^{4} + 2 i q^{5} + 2 i q^{7} - i q^{8} - 2 q^{10} + (2 i - 3) q^{13} - 2 q^{14} + q^{16} + 2 q^{17} + 6 i q^{19} - 2 i q^{20} - 4 q^{23} + q^{25} + ( - 3 i - 2) q^{26} - 2 i q^{28} + 10 q^{29} - 10 i q^{31} + i q^{32} + 2 i q^{34} - 4 q^{35} - 8 i q^{37} - 6 q^{38} + 2 q^{40} + 10 i q^{41} + 4 q^{43} - 4 i q^{46} - 12 i q^{47} + 3 q^{49} + i q^{50} + ( - 2 i + 3) q^{52} + 6 q^{53} + 2 q^{56} + 10 i q^{58} + 4 i q^{59} + 2 q^{61} + 10 q^{62} - q^{64} + ( - 6 i - 4) q^{65} + 2 i q^{67} - 2 q^{68} - 4 i q^{70} + 4 i q^{73} + 8 q^{74} - 6 i q^{76} + 2 i q^{80} - 10 q^{82} - 4 i q^{83} + 4 i q^{85} + 4 i q^{86} - 6 i q^{89} + ( - 6 i - 4) q^{91} + 4 q^{92} + 12 q^{94} - 12 q^{95} + 12 i q^{97} + 3 i q^{98} +O(q^{100})$$ q + i * q^2 - q^4 + 2*i * q^5 + 2*i * q^7 - i * q^8 - 2 * q^10 + (2*i - 3) * q^13 - 2 * q^14 + q^16 + 2 * q^17 + 6*i * q^19 - 2*i * q^20 - 4 * q^23 + q^25 + (-3*i - 2) * q^26 - 2*i * q^28 + 10 * q^29 - 10*i * q^31 + i * q^32 + 2*i * q^34 - 4 * q^35 - 8*i * q^37 - 6 * q^38 + 2 * q^40 + 10*i * q^41 + 4 * q^43 - 4*i * q^46 - 12*i * q^47 + 3 * q^49 + i * q^50 + (-2*i + 3) * q^52 + 6 * q^53 + 2 * q^56 + 10*i * q^58 + 4*i * q^59 + 2 * q^61 + 10 * q^62 - q^64 + (-6*i - 4) * q^65 + 2*i * q^67 - 2 * q^68 - 4*i * q^70 + 4*i * q^73 + 8 * q^74 - 6*i * q^76 + 2*i * q^80 - 10 * q^82 - 4*i * q^83 + 4*i * q^85 + 4*i * q^86 - 6*i * q^89 + (-6*i - 4) * q^91 + 4 * q^92 + 12 * q^94 - 12 * q^95 + 12*i * q^97 + 3*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} - 4 q^{10} - 6 q^{13} - 4 q^{14} + 2 q^{16} + 4 q^{17} - 8 q^{23} + 2 q^{25} - 4 q^{26} + 20 q^{29} - 8 q^{35} - 12 q^{38} + 4 q^{40} + 8 q^{43} + 6 q^{49} + 6 q^{52} + 12 q^{53} + 4 q^{56} + 4 q^{61} + 20 q^{62} - 2 q^{64} - 8 q^{65} - 4 q^{68} + 16 q^{74} - 20 q^{82} - 8 q^{91} + 8 q^{92} + 24 q^{94} - 24 q^{95}+O(q^{100})$$ 2 * q - 2 * q^4 - 4 * q^10 - 6 * q^13 - 4 * q^14 + 2 * q^16 + 4 * q^17 - 8 * q^23 + 2 * q^25 - 4 * q^26 + 20 * q^29 - 8 * q^35 - 12 * q^38 + 4 * q^40 + 8 * q^43 + 6 * q^49 + 6 * q^52 + 12 * q^53 + 4 * q^56 + 4 * q^61 + 20 * q^62 - 2 * q^64 - 8 * q^65 - 4 * q^68 + 16 * q^74 - 20 * q^82 - 8 * q^91 + 8 * q^92 + 24 * q^94 - 24 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/234\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$209$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 2.00000i 0 2.00000i 1.00000i 0 −2.00000
181.2 1.00000i 0 −1.00000 2.00000i 0 2.00000i 1.00000i 0 −2.00000
 $$n$$: 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 234.2.b.a 2
3.b odd 2 1 78.2.b.a 2
4.b odd 2 1 1872.2.c.b 2
12.b even 2 1 624.2.c.a 2
13.b even 2 1 inner 234.2.b.a 2
13.d odd 4 1 3042.2.a.c 1
13.d odd 4 1 3042.2.a.n 1
15.d odd 2 1 1950.2.b.c 2
15.e even 4 1 1950.2.f.d 2
15.e even 4 1 1950.2.f.g 2
21.c even 2 1 3822.2.c.d 2
24.f even 2 1 2496.2.c.m 2
24.h odd 2 1 2496.2.c.f 2
39.d odd 2 1 78.2.b.a 2
39.f even 4 1 1014.2.a.b 1
39.f even 4 1 1014.2.a.g 1
39.h odd 6 2 1014.2.i.c 4
39.i odd 6 2 1014.2.i.c 4
39.k even 12 2 1014.2.e.b 2
39.k even 12 2 1014.2.e.e 2
52.b odd 2 1 1872.2.c.b 2
156.h even 2 1 624.2.c.a 2
156.l odd 4 1 8112.2.a.g 1
156.l odd 4 1 8112.2.a.j 1
195.e odd 2 1 1950.2.b.c 2
195.s even 4 1 1950.2.f.d 2
195.s even 4 1 1950.2.f.g 2
273.g even 2 1 3822.2.c.d 2
312.b odd 2 1 2496.2.c.f 2
312.h even 2 1 2496.2.c.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 3.b odd 2 1
78.2.b.a 2 39.d odd 2 1
234.2.b.a 2 1.a even 1 1 trivial
234.2.b.a 2 13.b even 2 1 inner
624.2.c.a 2 12.b even 2 1
624.2.c.a 2 156.h even 2 1
1014.2.a.b 1 39.f even 4 1
1014.2.a.g 1 39.f even 4 1
1014.2.e.b 2 39.k even 12 2
1014.2.e.e 2 39.k even 12 2
1014.2.i.c 4 39.h odd 6 2
1014.2.i.c 4 39.i odd 6 2
1872.2.c.b 2 4.b odd 2 1
1872.2.c.b 2 52.b odd 2 1
1950.2.b.c 2 15.d odd 2 1
1950.2.b.c 2 195.e odd 2 1
1950.2.f.d 2 15.e even 4 1
1950.2.f.d 2 195.s even 4 1
1950.2.f.g 2 15.e even 4 1
1950.2.f.g 2 195.s even 4 1
2496.2.c.f 2 24.h odd 2 1
2496.2.c.f 2 312.b odd 2 1
2496.2.c.m 2 24.f even 2 1
2496.2.c.m 2 312.h even 2 1
3042.2.a.c 1 13.d odd 4 1
3042.2.a.n 1 13.d odd 4 1
3822.2.c.d 2 21.c even 2 1
3822.2.c.d 2 273.g even 2 1
8112.2.a.g 1 156.l odd 4 1
8112.2.a.j 1 156.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(234, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T - 10)^{2}$$
$31$ $$T^{2} + 100$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 100$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 144$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 144$$
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