# Properties

 Label 605.2.m.a Level 605 Weight 2 Character orbit 605.m Analytic conductor 4.831 Analytic rank 0 Dimension 16 CM discriminant -11 Inner twists 16

# Related objects

## Newspace parameters

 Level: $$N$$ = $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 605.m (of order $$20$$, degree $$8$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{20})$$ Coefficient field: 16.0.3429742096000000000000.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{U}(1)[D_{20}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{3} + ( -2 \beta_{5} - 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{14} ) q^{4} + ( 2 \beta_{10} - \beta_{12} ) q^{5} + ( \beta_{3} + 2 \beta_{8} + 3 \beta_{14} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{3} + ( -2 \beta_{5} - 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{14} ) q^{4} + ( 2 \beta_{10} - \beta_{12} ) q^{5} + ( \beta_{3} + 2 \beta_{8} + 3 \beta_{14} ) q^{9} + ( 2 \beta_{6} + 2 \beta_{15} ) q^{12} + ( 2 \beta_{2} - 4 \beta_{5} - 4 \beta_{7} + 4 \beta_{10} + \beta_{11} - 3 \beta_{13} + 4 \beta_{14} ) q^{15} + 4 \beta_{9} q^{16} + ( 2 \beta_{3} - 2 \beta_{8} ) q^{20} + ( 4 - \beta_{6} - 4 \beta_{7} + \beta_{15} ) q^{23} + ( 1 + \beta_{3} - 3 \beta_{4} - \beta_{9} + \beta_{13} ) q^{25} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - 6 \beta_{9} - 5 \beta_{10} - \beta_{12} + \beta_{13} ) q^{27} + ( -6 \beta_{1} - 6 \beta_{5} - 6 \beta_{7} + 6 \beta_{10} + 6 \beta_{11} - 6 \beta_{12} + 3 \beta_{14} - 6 \beta_{15} ) q^{31} + ( -6 - 4 \beta_{1} - 6 \beta_{3} - 2 \beta_{5} + 6 \beta_{9} - 6 \beta_{13} ) q^{36} + ( 3 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{10} + 3 \beta_{11} - 5 \beta_{13} - 2 \beta_{14} ) q^{37} + ( -3 + 3 \beta_{6} - 4 \beta_{7} + 3 \beta_{15} ) q^{45} + ( 7 + 2 \beta_{1} + 7 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 7 \beta_{9} + 7 \beta_{13} ) q^{47} + ( -4 \beta_{2} + 4 \beta_{11} + 4 \beta_{13} ) q^{48} -7 \beta_{10} q^{49} + ( 4 \beta_{1} - \beta_{3} + 4 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} - 4 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} - 5 \beta_{14} + 4 \beta_{15} ) q^{53} + ( -2 \beta_{2} + \beta_{13} ) q^{59} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{8} + 6 \beta_{9} + 2 \beta_{10} + 4 \beta_{12} + 2 \beta_{13} ) q^{60} + 8 \beta_{14} q^{64} + ( 5 - 3 \beta_{6} + 5 \beta_{7} - 3 \beta_{15} ) q^{67} + ( -4 - 4 \beta_{3} + 8 \beta_{4} - \beta_{5} + 4 \beta_{9} - 4 \beta_{13} ) q^{69} + 3 \beta_{9} q^{71} + ( \beta_{1} + 7 \beta_{3} + \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - 9 \beta_{14} + \beta_{15} ) q^{75} + ( 4 \beta_{1} + 8 \beta_{5} ) q^{80} + ( 3 \beta_{5} + 3 \beta_{7} - 3 \beta_{10} - 6 \beta_{11} - 2 \beta_{13} - 3 \beta_{14} ) q^{81} + 9 \beta_{7} q^{89} + ( -2 \beta_{2} - 8 \beta_{5} - 8 \beta_{7} + 8 \beta_{10} - 2 \beta_{11} + 10 \beta_{13} + 8 \beta_{14} ) q^{92} + ( -3 + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} - 3 \beta_{8} - 15 \beta_{9} + 18 \beta_{10} - 3 \beta_{12} - 3 \beta_{13} ) q^{93} + ( 3 \beta_{1} - 10 \beta_{3} + 3 \beta_{5} + 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + 7 \beta_{14} + 3 \beta_{15} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 2q^{3} + O(q^{10})$$ $$16q + 2q^{3} - 16q^{12} + 8q^{15} + 16q^{16} - 12q^{20} + 72q^{23} - 2q^{25} - 22q^{27} - 24q^{36} + 14q^{37} - 72q^{45} + 24q^{47} - 8q^{48} + 12q^{53} + 28q^{60} + 104q^{67} + 12q^{71} - 32q^{75} + 8q^{81} - 36q^{92} - 66q^{93} + 34q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 5 x^{14} + 16 x^{12} + 35 x^{10} + 31 x^{8} + 315 x^{6} + 1296 x^{4} + 3645 x^{2} + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{14} + 718 \nu^{4}$$$$)/2511$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{12} - 160 \nu^{2}$$$$)/279$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{14} - 16 \nu^{12} - 35 \nu^{10} - 31 \nu^{8} + 403 \nu^{6} - 1296 \nu^{4} - 3645 \nu^{2} - 6561$$$$)/7533$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{11} - 253 \nu$$$$)/93$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{10} - 253$$$$)/31$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} - 718 \nu^{5}$$$$)/7533$$ $$\beta_{8}$$ $$=$$ $$($$$$-2 \nu^{12} - 599 \nu^{2}$$$$)/279$$ $$\beta_{9}$$ $$=$$ $$($$$$35 \nu^{14} + 175 \nu^{12} + 560 \nu^{10} + 496 \nu^{8} + 1085 \nu^{6} + 11025 \nu^{4} + 45360 \nu^{2} + 127575$$$$)/22599$$ $$\beta_{10}$$ $$=$$ $$($$$$2 \nu^{13} + 599 \nu^{3}$$$$)/837$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{15} + 5 \nu^{13} + 16 \nu^{11} + 35 \nu^{9} + 31 \nu^{7} + 315 \nu^{5} + 1296 \nu^{3} + 3645 \nu$$$$)/2187$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{13} - 160 \nu^{3}$$$$)/279$$ $$\beta_{13}$$ $$=$$ $$($$$$-5 \nu^{14} - 1079 \nu^{4}$$$$)/2511$$ $$\beta_{14}$$ $$=$$ $$($$$$65 \nu^{15} + 208 \nu^{13} + 455 \nu^{11} + 403 \nu^{9} + 2294 \nu^{7} + 16848 \nu^{5} + 47385 \nu^{3} + 85293 \nu$$$$)/67797$$ $$\beta_{15}$$ $$=$$ $$($$$$-5 \nu^{15} - 1079 \nu^{5}$$$$)/2511$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{8} + 2 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{12} + 3 \beta_{10}$$ $$\nu^{4}$$ $$=$$ $$\beta_{13} + 5 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$\beta_{15} - 15 \beta_{7}$$ $$\nu^{6}$$ $$=$$ $$-3 \beta_{13} + 3 \beta_{9} + 16 \beta_{4} - 3 \beta_{3} - 3$$ $$\nu^{7}$$ $$=$$ $$13 \beta_{15} + 35 \beta_{14} + 13 \beta_{12} - 13 \beta_{11} - 13 \beta_{10} + 13 \beta_{7} + 13 \beta_{5} + 13 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$35 \beta_{13} + 39 \beta_{9} + 35 \beta_{8} + 35 \beta_{6} - 35 \beta_{4} + 35 \beta_{3} - 35 \beta_{2} + 35$$ $$\nu^{9}$$ $$=$$ $$-31 \beta_{14} + 74 \beta_{11} - 31 \beta_{10} + 31 \beta_{7} + 31 \beta_{5}$$ $$\nu^{10}$$ $$=$$ $$-31 \beta_{6} - 253$$ $$\nu^{11}$$ $$=$$ $$-93 \beta_{5} - 253 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$160 \beta_{8} - 599 \beta_{3}$$ $$\nu^{13}$$ $$=$$ $$-599 \beta_{12} - 480 \beta_{10}$$ $$\nu^{14}$$ $$=$$ $$-718 \beta_{13} - 1079 \beta_{2}$$ $$\nu^{15}$$ $$=$$ $$-718 \beta_{15} + 3237 \beta_{7}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$\beta_{7}$$ $$-\beta_{13}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
112.1
 −1.04771 + 1.37924i 1.63550 − 0.570223i 0.0369185 + 1.73166i −0.987975 − 1.42264i −1.63550 + 0.570223i 1.04771 − 1.37924i 0.0369185 − 1.73166i −0.987975 + 1.42264i −0.0369185 + 1.73166i 0.987975 − 1.42264i −1.63550 − 0.570223i 1.04771 + 1.37924i −1.04771 − 1.37924i 1.63550 + 0.570223i −0.0369185 − 1.73166i 0.987975 + 1.42264i
0 −0.256255 + 1.61793i 1.17557 1.61803i 0.914138 2.04067i 0 0 0 0.301128 + 0.0978424i 0
112.2 0 0.477487 3.01474i 1.17557 1.61803i 1.93903 + 1.11362i 0 0 0 −6.00747 1.95194i 0
118.1 0 −0.743682 + 1.45956i −1.90211 0.618034i 2.22328 0.238794i 0 0 0 0.186107 + 0.256155i 0
118.2 0 1.38572 2.71963i −1.90211 0.618034i −0.459925 2.18826i 0 0 0 −3.71282 5.11026i 0
233.1 0 −1.61793 0.256255i −1.17557 + 1.61803i −1.93903 1.11362i 0 0 0 −0.301128 0.0978424i 0
233.2 0 3.01474 + 0.477487i −1.17557 + 1.61803i −0.914138 + 2.04067i 0 0 0 6.00747 + 1.95194i 0
282.1 0 −0.743682 1.45956i −1.90211 + 0.618034i 2.22328 + 0.238794i 0 0 0 0.186107 0.256155i 0
282.2 0 1.38572 + 2.71963i −1.90211 + 0.618034i −0.459925 + 2.18826i 0 0 0 −3.71282 + 5.11026i 0
403.1 0 −2.71963 + 1.38572i 1.90211 0.618034i −2.22328 0.238794i 0 0 0 3.71282 5.11026i 0
403.2 0 1.45956 0.743682i 1.90211 0.618034i 0.459925 2.18826i 0 0 0 −0.186107 + 0.256155i 0
457.1 0 −1.61793 + 0.256255i −1.17557 1.61803i −1.93903 + 1.11362i 0 0 0 −0.301128 + 0.0978424i 0
457.2 0 3.01474 0.477487i −1.17557 1.61803i −0.914138 2.04067i 0 0 0 6.00747 1.95194i 0
578.1 0 −0.256255 1.61793i 1.17557 + 1.61803i 0.914138 + 2.04067i 0 0 0 0.301128 0.0978424i 0
578.2 0 0.477487 + 3.01474i 1.17557 + 1.61803i 1.93903 1.11362i 0 0 0 −6.00747 + 1.95194i 0
602.1 0 −2.71963 1.38572i 1.90211 + 0.618034i −2.22328 + 0.238794i 0 0 0 3.71282 + 5.11026i 0
602.2 0 1.45956 + 0.743682i 1.90211 + 0.618034i 0.459925 + 2.18826i 0 0 0 −0.186107 0.256155i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 602.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
5.c odd 4 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.e even 4 1 inner
55.k odd 20 3 inner
55.l even 20 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.m.a 16
5.c odd 4 1 inner 605.2.m.a 16
11.b odd 2 1 CM 605.2.m.a 16
11.c even 5 1 55.2.e.b 4
11.c even 5 3 inner 605.2.m.a 16
11.d odd 10 1 55.2.e.b 4
11.d odd 10 3 inner 605.2.m.a 16
33.f even 10 1 495.2.k.a 4
33.h odd 10 1 495.2.k.a 4
44.g even 10 1 880.2.bd.d 4
44.h odd 10 1 880.2.bd.d 4
55.e even 4 1 inner 605.2.m.a 16
55.h odd 10 1 275.2.e.a 4
55.j even 10 1 275.2.e.a 4
55.k odd 20 1 55.2.e.b 4
55.k odd 20 1 275.2.e.a 4
55.k odd 20 3 inner 605.2.m.a 16
55.l even 20 1 55.2.e.b 4
55.l even 20 1 275.2.e.a 4
55.l even 20 3 inner 605.2.m.a 16
165.u odd 20 1 495.2.k.a 4
165.v even 20 1 495.2.k.a 4
220.v even 20 1 880.2.bd.d 4
220.w odd 20 1 880.2.bd.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.b 4 11.c even 5 1
55.2.e.b 4 11.d odd 10 1
55.2.e.b 4 55.k odd 20 1
55.2.e.b 4 55.l even 20 1
275.2.e.a 4 55.h odd 10 1
275.2.e.a 4 55.j even 10 1
275.2.e.a 4 55.k odd 20 1
275.2.e.a 4 55.l even 20 1
495.2.k.a 4 33.f even 10 1
495.2.k.a 4 33.h odd 10 1
495.2.k.a 4 165.u odd 20 1
495.2.k.a 4 165.v even 20 1
605.2.m.a 16 1.a even 1 1 trivial
605.2.m.a 16 5.c odd 4 1 inner
605.2.m.a 16 11.b odd 2 1 CM
605.2.m.a 16 11.c even 5 3 inner
605.2.m.a 16 11.d odd 10 3 inner
605.2.m.a 16 55.e even 4 1 inner
605.2.m.a 16 55.k odd 20 3 inner
605.2.m.a 16 55.l even 20 3 inner
880.2.bd.d 4 44.g even 10 1
880.2.bd.d 4 44.h odd 10 1
880.2.bd.d 4 220.v even 20 1
880.2.bd.d 4 220.w odd 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} - 4 T^{4} + 8 T^{6} - 16 T^{7} + 16 T^{8} )^{2}( 1 + 2 T + 2 T^{2} - 4 T^{4} + 8 T^{6} + 16 T^{7} + 16 T^{8} )^{2}$$
$3$ $$( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 15 T^{5} - 18 T^{6} - 27 T^{7} + 81 T^{8} )^{2}( 1 + 5 T^{2} + 16 T^{4} + 35 T^{6} + 31 T^{8} + 315 T^{10} + 1296 T^{12} + 3645 T^{14} + 6561 T^{16} )$$
$5$ $$1 + T^{2} - 24 T^{4} - 49 T^{6} + 551 T^{8} - 1225 T^{10} - 15000 T^{12} + 15625 T^{14} + 390625 T^{16}$$
$7$ $$( 1 - 49 T^{4} + 2401 T^{8} - 117649 T^{12} + 5764801 T^{16} )^{2}$$
$11$ 
$13$ $$( 1 - 169 T^{4} + 28561 T^{8} - 4826809 T^{12} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 - 289 T^{4} + 83521 T^{8} - 24137569 T^{12} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 - 9 T + 23 T^{2} )^{8}( 1 + 35 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 29 T^{2} + 841 T^{4} - 24389 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 + 37 T^{2} + 408 T^{4} - 20461 T^{6} - 1149145 T^{8} - 19663021 T^{10} + 376796568 T^{12} + 32837636197 T^{14} + 852891037441 T^{16} )^{2}$$
$37$ $$( 1 - 7 T + 12 T^{2} + 175 T^{3} - 1669 T^{4} + 6475 T^{5} + 16428 T^{6} - 354571 T^{7} + 1874161 T^{8} )^{2}( 1 + 25 T^{2} - 744 T^{4} - 52825 T^{6} - 302089 T^{8} - 72317425 T^{10} - 1394375784 T^{12} + 64143160225 T^{14} + 3512479453921 T^{16} )$$
$41$ $$( 1 + 41 T^{2} + 1681 T^{4} + 68921 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 + 1849 T^{4} )^{8}$$
$47$ $$( 1 - 12 T + 97 T^{2} - 600 T^{3} + 2641 T^{4} - 28200 T^{5} + 214273 T^{6} - 1245876 T^{7} + 4879681 T^{8} )^{2}( 1 - 50 T^{2} + 291 T^{4} + 95900 T^{6} - 5437819 T^{8} + 211843100 T^{10} + 1419987171 T^{12} - 538960766450 T^{14} + 23811286661761 T^{16} )$$
$53$ $$( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 22260 T^{5} - 47753 T^{6} - 893262 T^{7} + 7890481 T^{8} )^{2}( 1 + 70 T^{2} + 2091 T^{4} - 50260 T^{6} - 9391819 T^{8} - 141180340 T^{10} + 16498995771 T^{12} + 1551505279030 T^{14} + 62259690411361 T^{16} )$$
$59$ $$( 1 - 15 T + 166 T^{2} - 1605 T^{3} + 14281 T^{4} - 94695 T^{5} + 577846 T^{6} - 3080685 T^{7} + 12117361 T^{8} )^{2}( 1 + 15 T + 166 T^{2} + 1605 T^{3} + 14281 T^{4} + 94695 T^{5} + 577846 T^{6} + 3080685 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 61 T^{2} + 3721 T^{4} + 226981 T^{6} + 13845841 T^{8} )^{4}$$
$67$ $$( 1 - 13 T + 67 T^{2} )^{8}( 1 + 35 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 28329 T^{5} - 312542 T^{6} - 1073733 T^{7} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 5329 T^{4} + 28398241 T^{8} - 151334226289 T^{12} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 79 T^{2} + 6241 T^{4} - 493039 T^{6} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 - 6889 T^{4} + 47458321 T^{8} - 326940373369 T^{12} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 - 97 T^{2} + 7921 T^{4} )^{8}$$
$97$ $$( 1 - 17 T + 192 T^{2} - 1615 T^{3} + 8831 T^{4} - 156655 T^{5} + 1806528 T^{6} - 15515441 T^{7} + 88529281 T^{8} )^{2}( 1 - 95 T^{2} - 384 T^{4} + 930335 T^{6} - 84768769 T^{8} + 8753522015 T^{10} - 33995243904 T^{12} - 79132340468255 T^{14} + 7837433594376961 T^{16} )$$