# Properties

 Label 74.3.d.b Level $74$ Weight $3$ Character orbit 74.d Analytic conductor $2.016$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 74.d (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01635395627$$ 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: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + i ) q^{2} + 4 i q^{3} + 2 i q^{4} + ( 3 - 3 i ) q^{5} + ( -4 + 4 i ) q^{6} -4 q^{7} + ( -2 + 2 i ) q^{8} -7 q^{9} +O(q^{10})$$ $$q + ( 1 + i ) q^{2} + 4 i q^{3} + 2 i q^{4} + ( 3 - 3 i ) q^{5} + ( -4 + 4 i ) q^{6} -4 q^{7} + ( -2 + 2 i ) q^{8} -7 q^{9} + 6 q^{10} + 4 i q^{11} -8 q^{12} + ( 3 - 3 i ) q^{13} + ( -4 - 4 i ) q^{14} + ( 12 + 12 i ) q^{15} -4 q^{16} + ( 23 - 23 i ) q^{17} + ( -7 - 7 i ) q^{18} + ( 10 - 10 i ) q^{19} + ( 6 + 6 i ) q^{20} -16 i q^{21} + ( -4 + 4 i ) q^{22} + ( -10 + 10 i ) q^{23} + ( -8 - 8 i ) q^{24} + 7 i q^{25} + 6 q^{26} + 8 i q^{27} -8 i q^{28} + ( -19 - 19 i ) q^{29} + 24 i q^{30} + ( -18 - 18 i ) q^{31} + ( -4 - 4 i ) q^{32} -16 q^{33} + 46 q^{34} + ( -12 + 12 i ) q^{35} -14 i q^{36} -37 i q^{37} + 20 q^{38} + ( 12 + 12 i ) q^{39} + 12 i q^{40} + 74 i q^{41} + ( 16 - 16 i ) q^{42} + ( 42 - 42 i ) q^{43} -8 q^{44} + ( -21 + 21 i ) q^{45} -20 q^{46} -44 q^{47} -16 i q^{48} -33 q^{49} + ( -7 + 7 i ) q^{50} + ( 92 + 92 i ) q^{51} + ( 6 + 6 i ) q^{52} -80 q^{53} + ( -8 + 8 i ) q^{54} + ( 12 + 12 i ) q^{55} + ( 8 - 8 i ) q^{56} + ( 40 + 40 i ) q^{57} -38 i q^{58} + ( -54 + 54 i ) q^{59} + ( -24 + 24 i ) q^{60} + ( -3 - 3 i ) q^{61} -36 i q^{62} + 28 q^{63} -8 i q^{64} -18 i q^{65} + ( -16 - 16 i ) q^{66} -12 i q^{67} + ( 46 + 46 i ) q^{68} + ( -40 - 40 i ) q^{69} -24 q^{70} + 124 q^{71} + ( 14 - 14 i ) q^{72} -10 i q^{73} + ( 37 - 37 i ) q^{74} -28 q^{75} + ( 20 + 20 i ) q^{76} -16 i q^{77} + 24 i q^{78} + ( 14 - 14 i ) q^{79} + ( -12 + 12 i ) q^{80} -95 q^{81} + ( -74 + 74 i ) q^{82} -64 q^{83} + 32 q^{84} -138 i q^{85} + 84 q^{86} + ( 76 - 76 i ) q^{87} + ( -8 - 8 i ) q^{88} + ( 17 + 17 i ) q^{89} -42 q^{90} + ( -12 + 12 i ) q^{91} + ( -20 - 20 i ) q^{92} + ( 72 - 72 i ) q^{93} + ( -44 - 44 i ) q^{94} -60 i q^{95} + ( 16 - 16 i ) q^{96} + ( 129 - 129 i ) q^{97} + ( -33 - 33 i ) q^{98} -28 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 6 q^{5} - 8 q^{6} - 8 q^{7} - 4 q^{8} - 14 q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} + 6 q^{5} - 8 q^{6} - 8 q^{7} - 4 q^{8} - 14 q^{9} + 12 q^{10} - 16 q^{12} + 6 q^{13} - 8 q^{14} + 24 q^{15} - 8 q^{16} + 46 q^{17} - 14 q^{18} + 20 q^{19} + 12 q^{20} - 8 q^{22} - 20 q^{23} - 16 q^{24} + 12 q^{26} - 38 q^{29} - 36 q^{31} - 8 q^{32} - 32 q^{33} + 92 q^{34} - 24 q^{35} + 40 q^{38} + 24 q^{39} + 32 q^{42} + 84 q^{43} - 16 q^{44} - 42 q^{45} - 40 q^{46} - 88 q^{47} - 66 q^{49} - 14 q^{50} + 184 q^{51} + 12 q^{52} - 160 q^{53} - 16 q^{54} + 24 q^{55} + 16 q^{56} + 80 q^{57} - 108 q^{59} - 48 q^{60} - 6 q^{61} + 56 q^{63} - 32 q^{66} + 92 q^{68} - 80 q^{69} - 48 q^{70} + 248 q^{71} + 28 q^{72} + 74 q^{74} - 56 q^{75} + 40 q^{76} + 28 q^{79} - 24 q^{80} - 190 q^{81} - 148 q^{82} - 128 q^{83} + 64 q^{84} + 168 q^{86} + 152 q^{87} - 16 q^{88} + 34 q^{89} - 84 q^{90} - 24 q^{91} - 40 q^{92} + 144 q^{93} - 88 q^{94} + 32 q^{96} + 258 q^{97} - 66 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$$.

 $$n$$ $$39$$ $$\chi(n)$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 1.00000i − 1.00000i
1.00000 + 1.00000i 4.00000i 2.00000i 3.00000 3.00000i −4.00000 + 4.00000i −4.00000 −2.00000 + 2.00000i −7.00000 6.00000
43.1 1.00000 1.00000i 4.00000i 2.00000i 3.00000 + 3.00000i −4.00000 4.00000i −4.00000 −2.00000 2.00000i −7.00000 6.00000
 $$n$$: 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
37.d odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.3.d.b 2
3.b odd 2 1 666.3.i.a 2
4.b odd 2 1 592.3.k.c 2
37.d odd 4 1 inner 74.3.d.b 2
111.g even 4 1 666.3.i.a 2
148.g even 4 1 592.3.k.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.3.d.b 2 1.a even 1 1 trivial
74.3.d.b 2 37.d odd 4 1 inner
592.3.k.c 2 4.b odd 2 1
592.3.k.c 2 148.g even 4 1
666.3.i.a 2 3.b odd 2 1
666.3.i.a 2 111.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 $$S_{3}^{\mathrm{new}}(74, [\chi])$$:

 $$T_{3}^{2} + 16$$ $$T_{5}^{2} - 6 T_{5} + 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 - 2 T + T^{2}$$
$3$ $$16 + T^{2}$$
$5$ $$18 - 6 T + T^{2}$$
$7$ $$( 4 + T )^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$18 - 6 T + T^{2}$$
$17$ $$1058 - 46 T + T^{2}$$
$19$ $$200 - 20 T + T^{2}$$
$23$ $$200 + 20 T + T^{2}$$
$29$ $$722 + 38 T + T^{2}$$
$31$ $$648 + 36 T + T^{2}$$
$37$ $$1369 + T^{2}$$
$41$ $$5476 + T^{2}$$
$43$ $$3528 - 84 T + T^{2}$$
$47$ $$( 44 + T )^{2}$$
$53$ $$( 80 + T )^{2}$$
$59$ $$5832 + 108 T + T^{2}$$
$61$ $$18 + 6 T + T^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$( -124 + T )^{2}$$
$73$ $$100 + T^{2}$$
$79$ $$392 - 28 T + T^{2}$$
$83$ $$( 64 + T )^{2}$$
$89$ $$578 - 34 T + T^{2}$$
$97$ $$33282 - 258 T + T^{2}$$