# Properties

 Label 453.1.t.a Level $453$ Weight $1$ Character orbit 453.t Analytic conductor $0.226$ Analytic rank $0$ Dimension $20$ Projective image $D_{25}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$453 = 3 \cdot 151$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 453.t (of order $$50$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.226076450723$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\Q(\zeta_{50})$$ Defining polynomial: $$x^{20} - x^{15} + x^{10} - x^{5} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{25}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{25} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{50}^{11} q^{3} -\zeta_{50}^{15} q^{4} + ( \zeta_{50}^{6} - \zeta_{50}^{23} ) q^{7} + \zeta_{50}^{22} q^{9} +O(q^{10})$$ $$q -\zeta_{50}^{11} q^{3} -\zeta_{50}^{15} q^{4} + ( \zeta_{50}^{6} - \zeta_{50}^{23} ) q^{7} + \zeta_{50}^{22} q^{9} -\zeta_{50} q^{12} + ( \zeta_{50}^{18} + \zeta_{50}^{24} ) q^{13} -\zeta_{50}^{5} q^{16} + ( \zeta_{50}^{2} - \zeta_{50}^{3} ) q^{19} + ( -\zeta_{50}^{9} - \zeta_{50}^{17} ) q^{21} -\zeta_{50}^{19} q^{25} + \zeta_{50}^{8} q^{27} + ( -\zeta_{50}^{13} - \zeta_{50}^{21} ) q^{28} + ( \zeta_{50}^{16} + \zeta_{50}^{20} ) q^{31} + \zeta_{50}^{12} q^{36} + ( -\zeta_{50}^{7} + \zeta_{50}^{10} ) q^{37} + ( \zeta_{50}^{4} + \zeta_{50}^{10} ) q^{39} + ( -\zeta_{50}^{7} + \zeta_{50}^{14} ) q^{43} + \zeta_{50}^{16} q^{48} + ( \zeta_{50}^{4} + \zeta_{50}^{12} - \zeta_{50}^{21} ) q^{49} + ( \zeta_{50}^{8} + \zeta_{50}^{14} ) q^{52} + ( -\zeta_{50}^{13} + \zeta_{50}^{14} ) q^{57} + ( -\zeta_{50}^{9} - \zeta_{50}^{19} ) q^{61} + ( -\zeta_{50}^{3} + \zeta_{50}^{20} ) q^{63} + \zeta_{50}^{20} q^{64} + ( \zeta_{50}^{2} + \zeta_{50}^{4} ) q^{67} + ( \zeta_{50}^{12} - \zeta_{50}^{17} ) q^{73} -\zeta_{50}^{5} q^{75} + ( -\zeta_{50}^{17} + \zeta_{50}^{18} ) q^{76} + ( -\zeta_{50} - \zeta_{50}^{13} ) q^{79} -\zeta_{50}^{19} q^{81} + ( -\zeta_{50}^{7} + \zeta_{50}^{24} ) q^{84} + ( -\zeta_{50}^{5} + \zeta_{50}^{16} + \zeta_{50}^{22} + \zeta_{50}^{24} ) q^{91} + ( \zeta_{50}^{2} + \zeta_{50}^{6} ) q^{93} + ( \zeta_{50}^{8} - \zeta_{50}^{23} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 5q^{4} + O(q^{10})$$ $$20q - 5q^{4} - 5q^{16} - 5q^{31} - 5q^{37} - 5q^{39} - 5q^{63} - 5q^{64} - 5q^{75} - 5q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/453\mathbb{Z}\right)^\times$$.

 $$n$$ $$152$$ $$157$$ $$\chi(n)$$ $$-1$$ $$\zeta_{50}^{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
20.1
 0.637424 + 0.770513i 0.992115 − 0.125333i −0.728969 + 0.684547i −0.876307 − 0.481754i 0.637424 − 0.770513i −0.968583 + 0.248690i 0.187381 − 0.982287i 0.425779 − 0.904827i 0.992115 + 0.125333i 0.929776 + 0.368125i 0.187381 + 0.982287i −0.0627905 + 0.998027i −0.728969 − 0.684547i −0.876307 + 0.481754i −0.535827 + 0.844328i −0.968583 − 0.248690i 0.929776 − 0.368125i 0.425779 + 0.904827i −0.535827 − 0.844328i −0.0627905 − 0.998027i
0 0.968583 + 0.248690i −0.809017 0.587785i 0 0 0.348445 1.82662i 0 0.876307 + 0.481754i 0
29.1 0 −0.187381 + 0.982287i 0.309017 + 0.951057i 0 0 1.69755 0.435857i 0 −0.929776 0.368125i 0
44.1 0 −0.425779 0.904827i 0.309017 + 0.951057i 0 0 −0.124591 + 1.98031i 0 −0.637424 + 0.770513i 0
50.1 0 0.728969 0.684547i 0.309017 + 0.951057i 0 0 −0.456288 0.718995i 0 0.0627905 0.998027i 0
68.1 0 0.968583 0.248690i −0.809017 + 0.587785i 0 0 0.348445 + 1.82662i 0 0.876307 0.481754i 0
86.1 0 −0.929776 0.368125i −0.809017 + 0.587785i 0 0 0.939097 0.516273i 0 0.728969 + 0.684547i 0
98.1 0 0.876307 + 0.481754i 0.309017 + 0.951057i 0 0 −1.35556 0.536702i 0 0.535827 + 0.844328i 0
110.1 0 −0.992115 0.125333i 0.309017 0.951057i 0 0 0.238883 + 0.288760i 0 0.968583 + 0.248690i 0
125.1 0 −0.187381 0.982287i 0.309017 0.951057i 0 0 1.69755 + 0.435857i 0 −0.929776 + 0.368125i 0
242.1 0 0.535827 + 0.844328i −0.809017 + 0.587785i 0 0 0.0915446 + 0.0859661i 0 −0.425779 + 0.904827i 0
245.1 0 0.876307 0.481754i 0.309017 0.951057i 0 0 −1.35556 + 0.536702i 0 0.535827 0.844328i 0
275.1 0 −0.637424 + 0.770513i −0.809017 + 0.587785i 0 0 −1.92189 0.242791i 0 −0.187381 0.982287i 0
278.1 0 −0.425779 + 0.904827i 0.309017 0.951057i 0 0 −0.124591 1.98031i 0 −0.637424 0.770513i 0
299.1 0 0.728969 + 0.684547i 0.309017 0.951057i 0 0 −0.456288 + 0.718995i 0 0.0627905 + 0.998027i 0
311.1 0 0.0627905 + 0.998027i −0.809017 0.587785i 0 0 0.542804 + 1.15352i 0 −0.992115 + 0.125333i 0
374.1 0 −0.929776 + 0.368125i −0.809017 0.587785i 0 0 0.939097 + 0.516273i 0 0.728969 0.684547i 0
380.1 0 0.535827 0.844328i −0.809017 0.587785i 0 0 0.0915446 0.0859661i 0 −0.425779 0.904827i 0
383.1 0 −0.992115 + 0.125333i 0.309017 + 0.951057i 0 0 0.238883 0.288760i 0 0.968583 0.248690i 0
386.1 0 0.0627905 0.998027i −0.809017 + 0.587785i 0 0 0.542804 1.15352i 0 −0.992115 0.125333i 0
425.1 0 −0.637424 0.770513i −0.809017 0.587785i 0 0 −1.92189 + 0.242791i 0 −0.187381 + 0.982287i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 425.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
151.h even 25 1 inner
453.t odd 50 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 453.1.t.a 20
3.b odd 2 1 CM 453.1.t.a 20
151.h even 25 1 inner 453.1.t.a 20
453.t odd 50 1 inner 453.1.t.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
453.1.t.a 20 1.a even 1 1 trivial
453.1.t.a 20 3.b odd 2 1 CM
453.1.t.a 20 151.h even 25 1 inner
453.1.t.a 20 453.t odd 50 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(453, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$1 + T^{5} + T^{10} + T^{15} + T^{20}$$
$5$ $$T^{20}$$
$7$ $$1 - 15 T + 110 T^{2} - 300 T^{3} + 475 T^{4} - 122 T^{5} + 335 T^{6} - 465 T^{7} + 175 T^{8} + 275 T^{9} - 246 T^{10} + 95 T^{11} + 145 T^{12} - 75 T^{13} + 25 T^{14} + 2 T^{15} - 20 T^{16} + 5 T^{17} + T^{20}$$
$11$ $$T^{20}$$
$13$ $$1 + 10 T + 85 T^{2} + 200 T^{3} - 25 T^{4} - 122 T^{5} + 385 T^{6} - 615 T^{7} + 675 T^{8} - 225 T^{9} - 246 T^{10} + 145 T^{11} + 20 T^{12} - 75 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$
$17$ $$T^{20}$$
$19$ $$1 - 10 T + 100 T^{2} - 390 T^{3} + 800 T^{4} - 752 T^{5} + 740 T^{6} - 475 T^{7} + 510 T^{8} - 175 T^{9} + 374 T^{10} + 30 T^{11} + 275 T^{12} + 20 T^{13} + 75 T^{14} + 2 T^{15} + 20 T^{16} + 5 T^{18} + T^{20}$$
$23$ $$T^{20}$$
$29$ $$T^{20}$$
$31$ $$1 - 10 T + 70 T^{2} - 45 T^{3} - 340 T^{4} - 252 T^{5} + 665 T^{6} + 1520 T^{7} + 1655 T^{8} + 1235 T^{9} + 629 T^{10} + 205 T^{11} + 165 T^{12} + 185 T^{13} + 170 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20}$$
$37$ $$1 + 15 T + 120 T^{2} + 380 T^{3} + 435 T^{4} - 252 T^{5} - 735 T^{6} - 905 T^{7} - 645 T^{8} + 85 T^{9} + 629 T^{10} + 730 T^{11} + 540 T^{12} + 310 T^{13} + 195 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20}$$
$41$ $$T^{20}$$
$43$ $$1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$
$47$ $$T^{20}$$
$53$ $$T^{20}$$
$59$ $$T^{20}$$
$61$ $$1 - 7 T^{5} + 124 T^{10} - 18 T^{15} + T^{20}$$
$67$ $$1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$
$71$ $$T^{20}$$
$73$ $$1 - 7 T^{5} + 124 T^{10} - 18 T^{15} + T^{20}$$
$79$ $$1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$
$83$ $$T^{20}$$
$89$ $$T^{20}$$
$97$ $$1 + 18 T^{5} + 124 T^{10} + 7 T^{15} + T^{20}$$