Properties

 Label 3750.2.c.b Level 3750 Weight 2 Character orbit 3750.c Analytic conductor 29.944 Analytic rank 0 Dimension 4 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$3750 = 2 \cdot 3 \cdot 5^{4}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3750.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$29.9439007580$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 150) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -\beta_{3} q^{3} - q^{4} - q^{6} + ( \beta_{1} - \beta_{3} ) q^{7} + \beta_{3} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} -\beta_{3} q^{3} - q^{4} - q^{6} + ( \beta_{1} - \beta_{3} ) q^{7} + \beta_{3} q^{8} - q^{9} + ( 2 - \beta_{2} ) q^{11} + \beta_{3} q^{12} + 4 \beta_{1} q^{13} + ( -1 + \beta_{2} ) q^{14} + q^{16} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{17} + \beta_{3} q^{18} + ( -4 - 6 \beta_{2} ) q^{19} + ( -1 + \beta_{2} ) q^{21} + ( \beta_{1} - 2 \beta_{3} ) q^{22} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{23} + q^{24} + 4 \beta_{2} q^{26} + \beta_{3} q^{27} + ( -\beta_{1} + \beta_{3} ) q^{28} + ( -2 + 4 \beta_{2} ) q^{29} + ( 5 - \beta_{2} ) q^{31} -\beta_{3} q^{32} + ( \beta_{1} - 2 \beta_{3} ) q^{33} + ( 2 + 2 \beta_{2} ) q^{34} + q^{36} -8 \beta_{3} q^{37} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{38} + 4 \beta_{2} q^{39} + ( -4 - 6 \beta_{2} ) q^{41} + ( -\beta_{1} + \beta_{3} ) q^{42} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -2 + \beta_{2} ) q^{44} + ( -2 - 4 \beta_{2} ) q^{46} + ( 6 \beta_{1} + 8 \beta_{3} ) q^{47} -\beta_{3} q^{48} + ( 5 + 3 \beta_{2} ) q^{49} + ( 2 + 2 \beta_{2} ) q^{51} -4 \beta_{1} q^{52} + ( 5 \beta_{1} + 6 \beta_{3} ) q^{53} + q^{54} + ( 1 - \beta_{2} ) q^{56} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{57} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{58} + ( 2 - \beta_{2} ) q^{59} + ( 6 + 2 \beta_{2} ) q^{61} + ( \beta_{1} - 5 \beta_{3} ) q^{62} + ( -\beta_{1} + \beta_{3} ) q^{63} - q^{64} + ( -2 + \beta_{2} ) q^{66} + ( 4 \beta_{1} + 8 \beta_{3} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{68} + ( -2 - 4 \beta_{2} ) q^{69} + ( 12 + 4 \beta_{2} ) q^{71} -\beta_{3} q^{72} + ( 4 \beta_{1} + 10 \beta_{3} ) q^{73} -8 q^{74} + ( 4 + 6 \beta_{2} ) q^{76} + ( 4 \beta_{1} - 3 \beta_{3} ) q^{77} -4 \beta_{1} q^{78} + ( 4 - \beta_{2} ) q^{79} + q^{81} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{82} + ( 3 \beta_{1} + 7 \beta_{3} ) q^{83} + ( 1 - \beta_{2} ) q^{84} + ( 2 + 6 \beta_{2} ) q^{86} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{87} + ( -\beta_{1} + 2 \beta_{3} ) q^{88} + ( -10 - 4 \beta_{2} ) q^{89} + ( -4 + 8 \beta_{2} ) q^{91} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{92} + ( \beta_{1} - 5 \beta_{3} ) q^{93} + ( 8 + 6 \beta_{2} ) q^{94} - q^{96} + ( \beta_{1} + 5 \beta_{3} ) q^{97} + ( -3 \beta_{1} - 5 \beta_{3} ) q^{98} + ( -2 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{6} - 4q^{9} + 10q^{11} - 6q^{14} + 4q^{16} - 4q^{19} - 6q^{21} + 4q^{24} - 8q^{26} - 16q^{29} + 22q^{31} + 4q^{34} + 4q^{36} - 8q^{39} - 4q^{41} - 10q^{44} + 14q^{49} + 4q^{51} + 4q^{54} + 6q^{56} + 10q^{59} + 20q^{61} - 4q^{64} - 10q^{66} + 40q^{71} - 32q^{74} + 4q^{76} + 18q^{79} + 4q^{81} + 6q^{84} - 4q^{86} - 32q^{89} - 32q^{91} + 20q^{94} - 4q^{96} - 10q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2 \beta_{1}$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times$$.

 $$n$$ $$2501$$ $$3127$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 − 1.61803i 0.618034i − 0.618034i 1.61803i
1.00000i 1.00000i −1.00000 0 −1.00000 2.61803i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 −1.00000 0.381966i 1.00000i −1.00000 0
1249.3 1.00000i 1.00000i −1.00000 0 −1.00000 0.381966i 1.00000i −1.00000 0
1249.4 1.00000i 1.00000i −1.00000 0 −1.00000 2.61803i 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
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3750.2.c.b 4
5.b even 2 1 inner 3750.2.c.b 4
5.c odd 4 1 3750.2.a.d 2
5.c odd 4 1 3750.2.a.f 2
25.d even 5 2 750.2.h.b 8
25.e even 10 2 750.2.h.b 8
25.f odd 20 2 150.2.g.a 4
25.f odd 20 2 750.2.g.b 4
75.l even 20 2 450.2.h.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.a 4 25.f odd 20 2
450.2.h.c 4 75.l even 20 2
750.2.g.b 4 25.f odd 20 2
750.2.h.b 8 25.d even 5 2
750.2.h.b 8 25.e even 10 2
3750.2.a.d 2 5.c odd 4 1
3750.2.a.f 2 5.c odd 4 1
3750.2.c.b 4 1.a even 1 1 trivial
3750.2.c.b 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 7 T_{7}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(3750, [\chi])$$.

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ 
$7$ $$1 - 21 T^{2} + 197 T^{4} - 1029 T^{6} + 2401 T^{8}$$
$11$ $$( 1 - 5 T + 27 T^{2} - 55 T^{3} + 121 T^{4} )^{2}$$
$13$ $$1 - 4 T^{2} + 22 T^{4} - 676 T^{6} + 28561 T^{8}$$
$17$ $$1 - 56 T^{2} + 1342 T^{4} - 16184 T^{6} + 83521 T^{8}$$
$19$ $$( 1 + 2 T - 6 T^{2} + 38 T^{3} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 26 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 8 T + 54 T^{2} + 232 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 11 T + 91 T^{2} - 341 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 2 T + 38 T^{2} + 82 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 80 T^{2} + 5118 T^{4} - 147920 T^{6} + 3418801 T^{8}$$
$47$ $$1 - 48 T^{2} + 494 T^{4} - 106032 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 125 T^{2} + 7993 T^{4} - 351125 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 - 5 T + 123 T^{2} - 295 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 10 T + 142 T^{2} - 610 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 - 156 T^{2} + 12182 T^{4} - 700284 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 - 124 T^{2} + 9382 T^{4} - 660796 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 9 T + 177 T^{2} - 711 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 - 249 T^{2} + 27917 T^{4} - 1715361 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 + 16 T + 222 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 - 345 T^{2} + 48473 T^{4} - 3246105 T^{6} + 88529281 T^{8}$$