# Properties

 Label 825.4.c.s Level $825$ Weight $4$ Character orbit 825.c Analytic conductor $48.677$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 43 x^{8} + 631 x^{6} + 3625 x^{4} + 7104 x^{2} + 900$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + 3 \beta_{2} q^{3} + ( -1 + \beta_{3} ) q^{4} -3 \beta_{4} q^{6} + ( 4 \beta_{2} - \beta_{7} - \beta_{9} ) q^{7} + ( 2 \beta_{1} + \beta_{5} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + 3 \beta_{2} q^{3} + ( -1 + \beta_{3} ) q^{4} -3 \beta_{4} q^{6} + ( 4 \beta_{2} - \beta_{7} - \beta_{9} ) q^{7} + ( 2 \beta_{1} + \beta_{5} ) q^{8} -9 q^{9} -11 q^{11} + ( -3 \beta_{2} + 3 \beta_{7} ) q^{12} + ( 4 \beta_{1} - 6 \beta_{2} - 3 \beta_{5} - \beta_{7} - \beta_{9} ) q^{13} + ( 1 - \beta_{3} - 11 \beta_{4} + \beta_{6} ) q^{14} + ( -27 + 5 \beta_{3} + 2 \beta_{6} ) q^{16} + ( 10 \beta_{2} - 6 \beta_{7} + 7 \beta_{9} ) q^{17} -9 \beta_{1} q^{18} + ( 18 - 9 \beta_{3} - 16 \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{19} + ( -12 + 3 \beta_{3} + 3 \beta_{6} ) q^{21} -11 \beta_{1} q^{22} + ( 12 \beta_{1} - 29 \beta_{2} + 2 \beta_{5} - 15 \beta_{7} + 4 \beta_{9} ) q^{23} + ( -6 \beta_{4} + 3 \beta_{8} ) q^{24} + ( -32 + 18 \beta_{3} - \beta_{4} - 5 \beta_{6} ) q^{26} -27 \beta_{2} q^{27} + ( 4 \beta_{1} - 66 \beta_{2} - 2 \beta_{5} + 2 \beta_{7} - 7 \beta_{9} ) q^{28} + ( 22 + 6 \beta_{3} - 4 \beta_{4} + \beta_{6} + 11 \beta_{8} ) q^{29} + ( -58 + 3 \beta_{3} + 3 \beta_{6} + 11 \beta_{8} ) q^{31} + ( -40 \beta_{1} + 2 \beta_{2} + 11 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} ) q^{32} -33 \beta_{2} q^{33} + ( -7 + 7 \beta_{3} - 26 \beta_{4} - 7 \beta_{6} - 13 \beta_{8} ) q^{34} + ( 9 - 9 \beta_{3} ) q^{36} + ( -24 \beta_{1} - 39 \beta_{2} - 11 \beta_{5} - 15 \beta_{7} - 5 \beta_{9} ) q^{37} + ( 59 \beta_{1} - 141 \beta_{2} - 11 \beta_{5} + 19 \beta_{7} ) q^{38} + ( 18 + 3 \beta_{3} - 12 \beta_{4} + 3 \beta_{6} - 9 \beta_{8} ) q^{39} + ( -16 + 4 \beta_{3} - 68 \beta_{4} - 7 \beta_{6} - 12 \beta_{8} ) q^{41} + ( -33 \beta_{1} + 3 \beta_{2} - 3 \beta_{7} + 3 \beta_{9} ) q^{42} + ( 12 \beta_{1} - 62 \beta_{2} + 11 \beta_{5} - 45 \beta_{7} + 3 \beta_{9} ) q^{43} + ( 11 - 11 \beta_{3} ) q^{44} + ( -114 + 6 \beta_{3} - 38 \beta_{4} - 19 \beta_{8} ) q^{46} + ( -4 \beta_{1} - 33 \beta_{2} - 11 \beta_{5} + 33 \beta_{7} - 33 \beta_{9} ) q^{47} + ( -81 \beta_{2} + 15 \beta_{7} + 6 \beta_{9} ) q^{48} + ( 94 - 18 \beta_{3} - 8 \beta_{4} - 11 \beta_{6} + 11 \beta_{8} ) q^{49} + ( -30 + 18 \beta_{3} - 21 \beta_{6} ) q^{51} + ( -80 \beta_{1} - 62 \beta_{2} - \beta_{5} - 2 \beta_{7} - 13 \beta_{9} ) q^{52} + ( 24 \beta_{1} - 182 \beta_{2} + 26 \beta_{5} + 24 \beta_{7} - 14 \beta_{9} ) q^{53} + 27 \beta_{4} q^{54} + ( -19 - \beta_{3} - 26 \beta_{4} + 11 \beta_{6} + 9 \beta_{8} ) q^{56} + ( -48 \beta_{1} + 54 \beta_{2} + 3 \beta_{5} - 27 \beta_{7} + 6 \beta_{9} ) q^{57} + ( -10 \beta_{1} - 46 \beta_{2} + 5 \beta_{5} - 52 \beta_{7} + 23 \beta_{9} ) q^{58} + ( 235 + 5 \beta_{3} - 44 \beta_{4} - 5 \beta_{6} - 11 \beta_{8} ) q^{59} + ( -24 - 53 \beta_{3} - 116 \beta_{4} + 22 \beta_{6} + 2 \beta_{8} ) q^{61} + ( -79 \beta_{1} - 8 \beta_{2} - 58 \beta_{7} + 25 \beta_{9} ) q^{62} + ( -36 \beta_{2} + 9 \beta_{7} + 9 \beta_{9} ) q^{63} + ( 131 - 53 \beta_{3} - 8 \beta_{4} + 36 \beta_{6} - 4 \beta_{8} ) q^{64} + 33 \beta_{4} q^{66} + ( -160 \beta_{1} - 150 \beta_{2} + 20 \beta_{5} - 19 \beta_{7} + 22 \beta_{9} ) q^{67} + ( -28 \beta_{1} - 148 \beta_{2} + 14 \beta_{5} + 50 \beta_{7} + 23 \beta_{9} ) q^{68} + ( 87 + 45 \beta_{3} - 36 \beta_{4} - 12 \beta_{6} + 6 \beta_{8} ) q^{69} + ( -95 - 39 \beta_{3} - 132 \beta_{4} - 2 \beta_{6} + 22 \beta_{8} ) q^{71} + ( -18 \beta_{1} - 9 \beta_{5} ) q^{72} + ( -8 \beta_{1} - 64 \beta_{2} - 38 \beta_{5} + 22 \beta_{7} - 44 \beta_{9} ) q^{73} + ( 232 + 26 \beta_{3} - 46 \beta_{4} - 17 \beta_{6} - 10 \beta_{8} ) q^{74} + ( -376 + 42 \beta_{3} + 108 \beta_{4} - 6 \beta_{6} + 11 \beta_{8} ) q^{76} + ( -44 \beta_{2} + 11 \beta_{7} + 11 \beta_{9} ) q^{77} + ( -3 \beta_{1} - 96 \beta_{2} + 54 \beta_{7} - 15 \beta_{9} ) q^{78} + ( 520 + 2 \beta_{3} - 116 \beta_{4} - 39 \beta_{6} - 6 \beta_{8} ) q^{79} + 81 q^{81} + ( -22 \beta_{1} - 607 \beta_{2} + 11 \beta_{5} + 135 \beta_{7} - 31 \beta_{9} ) q^{82} + ( 32 \beta_{1} - 330 \beta_{2} - 35 \beta_{5} - 2 \beta_{7} - 15 \beta_{9} ) q^{83} + ( 198 - 6 \beta_{3} - 12 \beta_{4} + 21 \beta_{6} - 6 \beta_{8} ) q^{84} + ( -122 - 40 \beta_{3} - 157 \beta_{4} + 19 \beta_{6} - 48 \beta_{8} ) q^{86} + ( -12 \beta_{1} + 66 \beta_{2} - 33 \beta_{5} + 18 \beta_{7} + 3 \beta_{9} ) q^{87} + ( -22 \beta_{1} - 11 \beta_{5} ) q^{88} + ( 494 + 14 \beta_{3} - 28 \beta_{4} + 47 \beta_{6} + 35 \beta_{8} ) q^{89} + ( -142 - 35 \beta_{3} - 40 \beta_{4} - 44 \beta_{6} - 16 \beta_{8} ) q^{91} + ( -48 \beta_{1} - 555 \beta_{2} + 22 \beta_{5} + 13 \beta_{7} - 6 \beta_{9} ) q^{92} + ( -174 \beta_{2} - 33 \beta_{5} + 9 \beta_{7} + 9 \beta_{9} ) q^{93} + ( 80 + 18 \beta_{3} + 132 \beta_{4} + 11 \beta_{6} + 66 \beta_{8} ) q^{94} + ( -6 + 6 \beta_{3} + 120 \beta_{4} - 6 \beta_{6} + 33 \beta_{8} ) q^{96} + ( -112 \beta_{1} - 393 \beta_{2} - 4 \beta_{5} + 90 \beta_{7} + 58 \beta_{9} ) q^{97} + ( 206 \beta_{1} - 94 \beta_{2} - 7 \beta_{5} - 36 \beta_{7} + 11 \beta_{9} ) q^{98} + 99 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 6q^{4} - 6q^{6} - 90q^{9} + O(q^{10})$$ $$10q - 6q^{4} - 6q^{6} - 90q^{9} - 110q^{11} - 16q^{14} - 250q^{16} + 114q^{19} - 108q^{21} - 18q^{24} - 250q^{26} + 214q^{29} - 590q^{31} - 68q^{34} + 54q^{36} + 186q^{39} - 256q^{41} + 66q^{44} - 1154q^{46} + 830q^{49} - 228q^{51} + 54q^{54} - 264q^{56} + 2304q^{59} - 688q^{61} + 1090q^{64} + 66q^{66} + 966q^{69} - 1414q^{71} + 2352q^{74} - 3398q^{76} + 4988q^{79} + 810q^{81} + 1944q^{84} - 1598q^{86} + 4870q^{89} - 1608q^{91} + 1004q^{94} + 138q^{96} + 990q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 43 x^{8} + 631 x^{6} + 3625 x^{4} + 7104 x^{2} + 900$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{9} + 37 \nu^{7} + 445 \nu^{5} + 1891 \nu^{3} + 2202 \nu$$$$)/720$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 9$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{8} + 31 \nu^{6} + 289 \nu^{4} + 817 \nu^{2} + 150$$$$)/120$$ $$\beta_{5}$$ $$=$$ $$\nu^{3} + 14 \nu$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{4} + 19 \nu^{2} + 46$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{9} + 49 \nu^{7} + 757 \nu^{5} + 4039 \nu^{3} + 6306 \nu$$$$)/240$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{8} - 46 \nu^{6} - 619 \nu^{4} - 2242 \nu^{2} - 600$$$$)/60$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{9} + 55 \nu^{7} + 1093 \nu^{5} + 8893 \nu^{3} + 22398 \nu$$$$)/360$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 9$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - 14 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{6} - 19 \beta_{3} + 125$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{9} - 2 \beta_{7} - 21 \beta_{5} + 2 \beta_{2} + 216 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-4 \beta_{8} - 44 \beta_{6} - 8 \beta_{4} + 323 \beta_{3} - 1925$$ $$\nu^{7}$$ $$=$$ $$-52 \beta_{9} + 72 \beta_{7} + 367 \beta_{5} - 112 \beta_{2} - 3452 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$124 \beta_{8} + 786 \beta_{6} + 368 \beta_{4} - 5339 \beta_{3} + 30753$$ $$\nu^{9}$$ $$=$$ $$1034 \beta_{9} - 1774 \beta_{7} - 6125 \beta_{5} + 3974 \beta_{2} + 55876 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 4.08549i − 3.98707i − 2.51748i − 1.98454i − 0.368634i 0.368634i 1.98454i 2.51748i 3.98707i 4.08549i
4.08549i 3.00000i −8.69126 0 −12.2565 7.95915i 2.82414i −9.00000 0
199.2 3.98707i 3.00000i −7.89672 0 11.9612 12.5627i 0.411800i −9.00000 0
199.3 2.51748i 3.00000i 1.66228 0 −7.55245 18.4627i 24.3246i −9.00000 0
199.4 1.98454i 3.00000i 4.06159 0 5.95363 5.59777i 23.9367i −9.00000 0
199.5 0.368634i 3.00000i 7.86411 0 −1.10590 26.5824i 5.84806i −9.00000 0
199.6 0.368634i 3.00000i 7.86411 0 −1.10590 26.5824i 5.84806i −9.00000 0
199.7 1.98454i 3.00000i 4.06159 0 5.95363 5.59777i 23.9367i −9.00000 0
199.8 2.51748i 3.00000i 1.66228 0 −7.55245 18.4627i 24.3246i −9.00000 0
199.9 3.98707i 3.00000i −7.89672 0 11.9612 12.5627i 0.411800i −9.00000 0
199.10 4.08549i 3.00000i −8.69126 0 −12.2565 7.95915i 2.82414i −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.10 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 825.4.c.s 10
5.b even 2 1 inner 825.4.c.s 10
5.c odd 4 1 825.4.a.x 5
5.c odd 4 1 825.4.a.y yes 5
15.e even 4 1 2475.4.a.bi 5
15.e even 4 1 2475.4.a.bj 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.x 5 5.c odd 4 1
825.4.a.y yes 5 5.c odd 4 1
825.4.c.s 10 1.a even 1 1 trivial
825.4.c.s 10 5.b even 2 1 inner
2475.4.a.bi 5 15.e even 4 1
2475.4.a.bj 5 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{10} + 43 T_{2}^{8} + 631 T_{2}^{6} + 3625 T_{2}^{4} + 7104 T_{2}^{2} + 900$$ $$T_{7}^{10} + 1300 T_{7}^{8} + 522294 T_{7}^{6} + 78865816 T_{7}^{4} + 4405598425 T_{7}^{2} + 75458991204$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$900 + 7104 T^{2} + 3625 T^{4} + 631 T^{6} + 43 T^{8} + T^{10}$$
$3$ $$( 9 + T^{2} )^{5}$$
$5$ $$T^{10}$$
$7$ $$75458991204 + 4405598425 T^{2} + 78865816 T^{4} + 522294 T^{6} + 1300 T^{8} + T^{10}$$
$11$ $$( 11 + T )^{10}$$
$13$ $$25911930697436224 + 69886893907728 T^{2} + 74058213313 T^{4} + 38577043 T^{6} + 9891 T^{8} + T^{10}$$
$17$ $$8989649377695360000 + 10356565119872400 T^{2} + 3788205649704 T^{4} + 591042217 T^{6} + 40518 T^{8} + T^{10}$$
$19$ $$( 92396817 + 3849987 T - 252712 T^{2} - 11988 T^{3} - 57 T^{4} + T^{5} )^{2}$$
$23$ $$13299414298805129616 + 18434318630344248 T^{2} + 7551384503569 T^{4} + 1179799355 T^{6} + 64211 T^{8} + T^{10}$$
$29$ $$( -16041054816 + 396997440 T + 2770092 T^{2} - 60112 T^{3} - 107 T^{4} + T^{5} )^{2}$$
$31$ $$( 9419016168 + 261146472 T - 7920319 T^{2} - 33433 T^{3} + 295 T^{4} + T^{5} )^{2}$$
$37$ $$13\!\cdots\!16$$$$+ 4717474279099681968 T^{2} + 508282630702540 T^{4} + 18140901625 T^{6} + 238398 T^{8} + T^{10}$$
$41$ $$( 144243119304 + 3215304180 T - 188058 T^{2} - 184077 T^{3} + 128 T^{4} + T^{5} )^{2}$$
$43$ $$11\!\cdots\!04$$$$+$$$$13\!\cdots\!76$$$$T^{2} + 6012855295602169 T^{4} + 91651241931 T^{6} + 545979 T^{8} + T^{10}$$
$47$ $$98\!\cdots\!00$$$$+$$$$36\!\cdots\!44$$$$T^{2} + 50483210112941520 T^{4} + 319610068801 T^{6} + 928978 T^{8} + T^{10}$$
$53$ $$42\!\cdots\!24$$$$+$$$$36\!\cdots\!36$$$$T^{2} + 77578583189829376 T^{4} + 486382853984 T^{6} + 1187744 T^{8} + T^{10}$$
$59$ $$( 849531990720 - 6150914688 T - 35969354 T^{2} + 416773 T^{3} - 1152 T^{4} + T^{5} )^{2}$$
$61$ $$( 105472451720 - 5498810188 T - 146032310 T^{2} - 621055 T^{3} + 344 T^{4} + T^{5} )^{2}$$
$67$ $$11\!\cdots\!00$$$$+$$$$38\!\cdots\!56$$$$T^{2} + 335025092514202156 T^{4} + 1168843160425 T^{6} + 1790126 T^{8} + T^{10}$$
$71$ $$( 13467667404912 + 113335765416 T - 283106455 T^{2} - 621005 T^{3} + 707 T^{4} + T^{5} )^{2}$$
$73$ $$79\!\cdots\!56$$$$+ 36338954132823652608 T^{2} + 205783209482291584 T^{4} + 1660481463904 T^{6} + 2445732 T^{8} + T^{10}$$
$79$ $$( -57405986771584 - 322674693392 T + 336709760 T^{2} + 1537579 T^{3} - 2494 T^{4} + T^{5} )^{2}$$
$83$ $$49\!\cdots\!56$$$$+$$$$29\!\cdots\!40$$$$T^{2} + 307503627025814352 T^{4} + 1153179964200 T^{6} + 1803841 T^{8} + T^{10}$$
$89$ $$( 205516733649744 - 1628635153200 T + 2330847928 T^{2} + 528072 T^{3} - 2435 T^{4} + T^{5} )^{2}$$
$97$ $$21\!\cdots\!25$$$$+$$$$46\!\cdots\!25$$$$T^{2} + 16220519649170023306 T^{4} + 16900938077482 T^{6} + 6962517 T^{8} + T^{10}$$