# Properties

 Label 1925.2.b.h Level $1925$ Weight $2$ Character orbit 1925.b Analytic conductor $15.371$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1925 = 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1925.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.3712023891$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 77) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} - 3 q^{4} + ( - \beta_{3} - 5) q^{6} - \beta_1 q^{7} - \beta_{2} q^{8} + ( - 2 \beta_{3} - 3) q^{9}+O(q^{10})$$ q + b2 * q^2 + (b2 + b1) * q^3 - 3 * q^4 + (-b3 - 5) * q^6 - b1 * q^7 - b2 * q^8 + (-2*b3 - 3) * q^9 $$q + \beta_{2} q^{2} + (\beta_{2} + \beta_1) q^{3} - 3 q^{4} + ( - \beta_{3} - 5) q^{6} - \beta_1 q^{7} - \beta_{2} q^{8} + ( - 2 \beta_{3} - 3) q^{9} - q^{11} + ( - 3 \beta_{2} - 3 \beta_1) q^{12} + ( - \beta_{2} + \beta_1) q^{13} + \beta_{3} q^{14} - q^{16} + ( - \beta_{2} + \beta_1) q^{17} + ( - 3 \beta_{2} - 10 \beta_1) q^{18} + (2 \beta_{3} - 2) q^{19} + (\beta_{3} + 1) q^{21} - \beta_{2} q^{22} + ( - 2 \beta_{2} - 2 \beta_1) q^{23} + (\beta_{3} + 5) q^{24} + ( - \beta_{3} + 5) q^{26} + ( - 2 \beta_{2} - 10 \beta_1) q^{27} + 3 \beta_1 q^{28} + (2 \beta_{3} - 4) q^{29} + ( - \beta_{3} - 5) q^{31} - 3 \beta_{2} q^{32} + ( - \beta_{2} - \beta_1) q^{33} + ( - \beta_{3} + 5) q^{34} + (6 \beta_{3} + 9) q^{36} + ( - 2 \beta_{2} + 4 \beta_1) q^{37} + ( - 2 \beta_{2} + 10 \beta_1) q^{38} + 4 q^{39} + (\beta_{3} - 9) q^{41} + (\beta_{2} + 5 \beta_1) q^{42} + 8 \beta_1 q^{43} + 3 q^{44} + (2 \beta_{3} + 10) q^{46} + ( - \beta_{2} - 5 \beta_1) q^{47} + ( - \beta_{2} - \beta_1) q^{48} - q^{49} + 4 q^{51} + (3 \beta_{2} - 3 \beta_1) q^{52} + (2 \beta_{2} + 4 \beta_1) q^{53} + (10 \beta_{3} + 10) q^{54} - \beta_{3} q^{56} + 8 \beta_1 q^{57} + ( - 4 \beta_{2} + 10 \beta_1) q^{58} + ( - \beta_{3} - 1) q^{59} + (\beta_{3} - 5) q^{61} + ( - 5 \beta_{2} - 5 \beta_1) q^{62} + (2 \beta_{2} + 3 \beta_1) q^{63} + 13 q^{64} + (\beta_{3} + 5) q^{66} + (2 \beta_{2} - 10 \beta_1) q^{67} + (3 \beta_{2} - 3 \beta_1) q^{68} + (4 \beta_{3} + 12) q^{69} + (2 \beta_{3} - 6) q^{71} + (3 \beta_{2} + 10 \beta_1) q^{72} + ( - \beta_{2} - 3 \beta_1) q^{73} + ( - 4 \beta_{3} + 10) q^{74} + ( - 6 \beta_{3} + 6) q^{76} + \beta_1 q^{77} + 4 \beta_{2} q^{78} - 4 \beta_{3} q^{79} + (6 \beta_{3} + 11) q^{81} + ( - 9 \beta_{2} + 5 \beta_1) q^{82} + (6 \beta_{2} + 2 \beta_1) q^{83} + ( - 3 \beta_{3} - 3) q^{84} - 8 \beta_{3} q^{86} + ( - 2 \beta_{2} + 6 \beta_1) q^{87} + \beta_{2} q^{88} - 2 q^{89} + ( - \beta_{3} + 1) q^{91} + (6 \beta_{2} + 6 \beta_1) q^{92} + ( - 6 \beta_{2} - 10 \beta_1) q^{93} + (5 \beta_{3} + 5) q^{94} + (3 \beta_{3} + 15) q^{96} + (6 \beta_{2} - 4 \beta_1) q^{97} - \beta_{2} q^{98} + (2 \beta_{3} + 3) q^{99}+O(q^{100})$$ q + b2 * q^2 + (b2 + b1) * q^3 - 3 * q^4 + (-b3 - 5) * q^6 - b1 * q^7 - b2 * q^8 + (-2*b3 - 3) * q^9 - q^11 + (-3*b2 - 3*b1) * q^12 + (-b2 + b1) * q^13 + b3 * q^14 - q^16 + (-b2 + b1) * q^17 + (-3*b2 - 10*b1) * q^18 + (2*b3 - 2) * q^19 + (b3 + 1) * q^21 - b2 * q^22 + (-2*b2 - 2*b1) * q^23 + (b3 + 5) * q^24 + (-b3 + 5) * q^26 + (-2*b2 - 10*b1) * q^27 + 3*b1 * q^28 + (2*b3 - 4) * q^29 + (-b3 - 5) * q^31 - 3*b2 * q^32 + (-b2 - b1) * q^33 + (-b3 + 5) * q^34 + (6*b3 + 9) * q^36 + (-2*b2 + 4*b1) * q^37 + (-2*b2 + 10*b1) * q^38 + 4 * q^39 + (b3 - 9) * q^41 + (b2 + 5*b1) * q^42 + 8*b1 * q^43 + 3 * q^44 + (2*b3 + 10) * q^46 + (-b2 - 5*b1) * q^47 + (-b2 - b1) * q^48 - q^49 + 4 * q^51 + (3*b2 - 3*b1) * q^52 + (2*b2 + 4*b1) * q^53 + (10*b3 + 10) * q^54 - b3 * q^56 + 8*b1 * q^57 + (-4*b2 + 10*b1) * q^58 + (-b3 - 1) * q^59 + (b3 - 5) * q^61 + (-5*b2 - 5*b1) * q^62 + (2*b2 + 3*b1) * q^63 + 13 * q^64 + (b3 + 5) * q^66 + (2*b2 - 10*b1) * q^67 + (3*b2 - 3*b1) * q^68 + (4*b3 + 12) * q^69 + (2*b3 - 6) * q^71 + (3*b2 + 10*b1) * q^72 + (-b2 - 3*b1) * q^73 + (-4*b3 + 10) * q^74 + (-6*b3 + 6) * q^76 + b1 * q^77 + 4*b2 * q^78 - 4*b3 * q^79 + (6*b3 + 11) * q^81 + (-9*b2 + 5*b1) * q^82 + (6*b2 + 2*b1) * q^83 + (-3*b3 - 3) * q^84 - 8*b3 * q^86 + (-2*b2 + 6*b1) * q^87 + b2 * q^88 - 2 * q^89 + (-b3 + 1) * q^91 + (6*b2 + 6*b1) * q^92 + (-6*b2 - 10*b1) * q^93 + (5*b3 + 5) * q^94 + (3*b3 + 15) * q^96 + (6*b2 - 4*b1) * q^97 - b2 * q^98 + (2*b3 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{4} - 20 q^{6} - 12 q^{9}+O(q^{10})$$ 4 * q - 12 * q^4 - 20 * q^6 - 12 * q^9 $$4 q - 12 q^{4} - 20 q^{6} - 12 q^{9} - 4 q^{11} - 4 q^{16} - 8 q^{19} + 4 q^{21} + 20 q^{24} + 20 q^{26} - 16 q^{29} - 20 q^{31} + 20 q^{34} + 36 q^{36} + 16 q^{39} - 36 q^{41} + 12 q^{44} + 40 q^{46} - 4 q^{49} + 16 q^{51} + 40 q^{54} - 4 q^{59} - 20 q^{61} + 52 q^{64} + 20 q^{66} + 48 q^{69} - 24 q^{71} + 40 q^{74} + 24 q^{76} + 44 q^{81} - 12 q^{84} - 8 q^{89} + 4 q^{91} + 20 q^{94} + 60 q^{96} + 12 q^{99}+O(q^{100})$$ 4 * q - 12 * q^4 - 20 * q^6 - 12 * q^9 - 4 * q^11 - 4 * q^16 - 8 * q^19 + 4 * q^21 + 20 * q^24 + 20 * q^26 - 16 * q^29 - 20 * q^31 + 20 * q^34 + 36 * q^36 + 16 * q^39 - 36 * q^41 + 12 * q^44 + 40 * q^46 - 4 * q^49 + 16 * q^51 + 40 * q^54 - 4 * q^59 - 20 * q^61 + 52 * q^64 + 20 * q^66 + 48 * q^69 - 24 * q^71 + 40 * q^74 + 24 * q^76 + 44 * q^81 - 12 * q^84 - 8 * q^89 + 4 * q^91 + 20 * q^94 + 60 * q^96 + 12 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2\beta_1$$ -b2 + 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$$.

 $$n$$ $$276$$ $$1002$$ $$1751$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 − 0.618034i − 1.61803i 1.61803i 0.618034i
2.23607i 3.23607i −3.00000 0 −7.23607 1.00000i 2.23607i −7.47214 0
1849.2 2.23607i 1.23607i −3.00000 0 −2.76393 1.00000i 2.23607i 1.47214 0
1849.3 2.23607i 1.23607i −3.00000 0 −2.76393 1.00000i 2.23607i 1.47214 0
1849.4 2.23607i 3.23607i −3.00000 0 −7.23607 1.00000i 2.23607i −7.47214 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 1925.2.b.h 4
5.b even 2 1 inner 1925.2.b.h 4
5.c odd 4 1 77.2.a.d 2
5.c odd 4 1 1925.2.a.r 2
15.e even 4 1 693.2.a.h 2
20.e even 4 1 1232.2.a.m 2
35.f even 4 1 539.2.a.f 2
35.k even 12 2 539.2.e.j 4
35.l odd 12 2 539.2.e.i 4
40.i odd 4 1 4928.2.a.bm 2
40.k even 4 1 4928.2.a.bv 2
55.e even 4 1 847.2.a.f 2
55.k odd 20 2 847.2.f.a 4
55.k odd 20 2 847.2.f.n 4
55.l even 20 2 847.2.f.b 4
55.l even 20 2 847.2.f.m 4
105.k odd 4 1 4851.2.a.y 2
140.j odd 4 1 8624.2.a.ce 2
165.l odd 4 1 7623.2.a.bl 2
385.l odd 4 1 5929.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 5.c odd 4 1
539.2.a.f 2 35.f even 4 1
539.2.e.i 4 35.l odd 12 2
539.2.e.j 4 35.k even 12 2
693.2.a.h 2 15.e even 4 1
847.2.a.f 2 55.e even 4 1
847.2.f.a 4 55.k odd 20 2
847.2.f.b 4 55.l even 20 2
847.2.f.m 4 55.l even 20 2
847.2.f.n 4 55.k odd 20 2
1232.2.a.m 2 20.e even 4 1
1925.2.a.r 2 5.c odd 4 1
1925.2.b.h 4 1.a even 1 1 trivial
1925.2.b.h 4 5.b even 2 1 inner
4851.2.a.y 2 105.k odd 4 1
4928.2.a.bm 2 40.i odd 4 1
4928.2.a.bv 2 40.k even 4 1
5929.2.a.m 2 385.l odd 4 1
7623.2.a.bl 2 165.l odd 4 1
8624.2.a.ce 2 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1925, [\chi])$$:

 $$T_{2}^{2} + 5$$ T2^2 + 5 $$T_{3}^{4} + 12T_{3}^{2} + 16$$ T3^4 + 12*T3^2 + 16 $$T_{19}^{2} + 4T_{19} - 16$$ T19^2 + 4*T19 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 5)^{2}$$
$3$ $$T^{4} + 12T^{2} + 16$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T + 1)^{4}$$
$13$ $$T^{4} + 12T^{2} + 16$$
$17$ $$T^{4} + 12T^{2} + 16$$
$19$ $$(T^{2} + 4 T - 16)^{2}$$
$23$ $$T^{4} + 48T^{2} + 256$$
$29$ $$(T^{2} + 8 T - 4)^{2}$$
$31$ $$(T^{2} + 10 T + 20)^{2}$$
$37$ $$T^{4} + 72T^{2} + 16$$
$41$ $$(T^{2} + 18 T + 76)^{2}$$
$43$ $$(T^{2} + 64)^{2}$$
$47$ $$T^{4} + 60T^{2} + 400$$
$53$ $$T^{4} + 72T^{2} + 16$$
$59$ $$(T^{2} + 2 T - 4)^{2}$$
$61$ $$(T^{2} + 10 T + 20)^{2}$$
$67$ $$T^{4} + 240T^{2} + 6400$$
$71$ $$(T^{2} + 12 T + 16)^{2}$$
$73$ $$T^{4} + 28T^{2} + 16$$
$79$ $$(T^{2} - 80)^{2}$$
$83$ $$T^{4} + 368 T^{2} + 30976$$
$89$ $$(T + 2)^{4}$$
$97$ $$T^{4} + 392 T^{2} + 26896$$
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