# Properties

 Label 225.10.b.m Level $225$ Weight $10$ Character orbit 225.b Analytic conductor $115.883$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$115.883063137$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Defining polynomial: $$x^{6} + 1305 x^{4} + 433104 x^{2} + 16000000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}\cdot 5^{8}$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -114 + \beta_{1} ) q^{4} + ( -140 \beta_{2} - 4 \beta_{3} + 26 \beta_{4} ) q^{7} + ( 279 \beta_{2} - 27 \beta_{3} - 33 \beta_{4} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -114 + \beta_{1} ) q^{4} + ( -140 \beta_{2} - 4 \beta_{3} + 26 \beta_{4} ) q^{7} + ( 279 \beta_{2} - 27 \beta_{3} - 33 \beta_{4} ) q^{8} + ( 18264 - 76 \beta_{1} - 17 \beta_{5} ) q^{11} + ( -928 \beta_{2} + 832 \beta_{3} - 460 \beta_{4} ) q^{13} + ( 90870 + 282 \beta_{1} + 30 \beta_{5} ) q^{14} + ( -233112 + 95 \beta_{1} - 6 \beta_{5} ) q^{16} + ( -3974 \beta_{2} + 1521 \beta_{3} - 168 \beta_{4} ) q^{17} + ( -273096 + 92 \beta_{1} + 351 \beta_{5} ) q^{19} + ( 28107 \beta_{2} - 3558 \beta_{3} + 4310 \beta_{4} ) q^{22} + ( -4764 \beta_{2} - 11160 \beta_{3} + 3426 \beta_{4} ) q^{23} + ( 428628 - 4588 \beta_{1} - 1292 \beta_{5} ) q^{26} + ( -15778 \beta_{2} + 238 \beta_{3} + 826 \beta_{4} ) q^{28} + ( 728268 - 9832 \beta_{1} + 508 \beta_{5} ) q^{29} + ( 1423984 + 2648 \beta_{1} - 534 \beta_{5} ) q^{31} + ( -101287 \beta_{2} - 18369 \beta_{3} - 19395 \beta_{4} ) q^{32} + ( 2279959 + 775 \beta_{1} - 1689 \beta_{5} ) q^{34} + ( 137524 \beta_{2} + 15866 \beta_{3} + 37572 \beta_{4} ) q^{37} + ( -300541 \beta_{2} + 113346 \beta_{3} - 40242 \beta_{4} ) q^{38} + ( -1952709 - 73088 \beta_{1} + 4904 \beta_{5} ) q^{41} + ( 34976 \beta_{2} - 83000 \beta_{3} + 2732 \beta_{4} ) q^{43} + ( -7346514 + 44675 \beta_{1} - 836 \beta_{5} ) q^{44} + ( 4736994 - 2322 \beta_{1} + 14586 \beta_{5} ) q^{46} + ( 894880 \beta_{2} + 121500 \beta_{3} - 9384 \beta_{4} ) q^{47} + ( 2188595 - 93952 \beta_{1} - 17904 \beta_{5} ) q^{49} + ( 560188 \beta_{2} + 123500 \beta_{3} + 52836 \beta_{4} ) q^{52} + ( 208484 \beta_{2} - 428022 \beta_{3} - 314856 \beta_{4} ) q^{53} + ( 56459448 + 143838 \beta_{1} + 15948 \beta_{5} ) q^{56} + ( 1874400 \beta_{2} + 433104 \beta_{3} + 270608 \beta_{4} ) q^{58} + ( 1818504 + 193768 \beta_{1} + 21680 \beta_{5} ) q^{59} + ( 41881302 - 359960 \beta_{1} + 22980 \beta_{5} ) q^{61} + ( 1133970 \beta_{2} - 247716 \beta_{3} - 30780 \beta_{4} ) q^{62} + ( -55688032 - 474207 \beta_{1} - 4098 \beta_{5} ) q^{64} + ( 6018994 \beta_{2} - 906256 \beta_{3} - 322747 \beta_{4} ) q^{67} + ( 232429 \beta_{2} + 200457 \beta_{3} + 67443 \beta_{4} ) q^{68} + ( -98905212 - 786920 \beta_{1} - 48040 \beta_{5} ) q^{71} + ( 10112822 \beta_{2} - 497441 \beta_{3} + 56864 \beta_{4} ) q^{73} + ( -84128214 + 855578 \beta_{1} + 21706 \beta_{5} ) q^{74} + ( 29919854 - 370821 \beta_{1} + 26124 \beta_{5} ) q^{76} + ( 2388876 \beta_{2} + 1613286 \beta_{3} - 701052 \beta_{4} ) q^{77} + ( 102948380 + 1255264 \beta_{1} - 75234 \beta_{5} ) q^{79} + ( 6514275 \beta_{2} + 3591696 \beta_{3} + 1892080 \beta_{4} ) q^{82} + ( 10762890 \beta_{2} + 1204200 \beta_{3} + 165897 \beta_{4} ) q^{83} + ( -11233116 - 333580 \beta_{1} + 85732 \beta_{5} ) q^{86} + ( 1767237 \beta_{2} - 3303801 \beta_{3} + 821061 \beta_{4} ) q^{88} + ( -367582761 - 1107576 \beta_{1} - 4176 \beta_{5} ) q^{89} + ( 393981224 + 796432 \beta_{1} + 447552 \beta_{5} ) q^{91} + ( 1888602 \beta_{2} - 837846 \beta_{3} + 284622 \beta_{4} ) q^{92} + ( -576367700 + 1342852 \beta_{1} - 130884 \beta_{5} ) q^{94} + ( 29036476 \beta_{2} - 2021122 \beta_{3} - 62616 \beta_{4} ) q^{97} + ( 14210371 \beta_{2} - 3371616 \beta_{3} + 4998240 \beta_{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 682q^{4} + O(q^{10})$$ $$6q - 682q^{4} + 109398q^{11} + 545844q^{14} - 1398494q^{16} - 1637690q^{19} + 2560008q^{26} + 4350960q^{29} + 8548132q^{31} + 13677926q^{34} - 11852622q^{41} - 43991406q^{44} + 28446492q^{46} + 12907858q^{49} + 339076260q^{56} + 11341920q^{59} + 250613852q^{61} - 335084802q^{64} - 595101192q^{71} - 503014716q^{74} + 178829730q^{76} + 620050340q^{79} - 67894392q^{86} - 2207720070q^{89} + 2366375312q^{91} - 3455782264q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 1305 x^{4} + 433104 x^{2} + 16000000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-35 \nu^{4} - 17275 \nu^{2} + 2252704$$$$)/13392$$ $$\beta_{2}$$ $$=$$ $$($$$$77 \nu^{5} + 71485 \nu^{3} + 13296008 \nu$$$$)/3348000$$ $$\beta_{3}$$ $$=$$ $$($$$$-491 \nu^{5} - 908755 \nu^{3} - 378730064 \nu$$$$)/13392000$$ $$\beta_{4}$$ $$=$$ $$($$$$-1177 \nu^{5} - 1291985 \nu^{3} - 198655408 \nu$$$$)/13392000$$ $$\beta_{5}$$ $$=$$ $$($$$$-185 \nu^{4} - 171025 \nu^{2} - 22789928$$$$)/6696$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$25 \beta_{4} - 11 \beta_{3} + 78 \beta_{2}$$$$)/250$$ $$\nu^{2}$$ $$=$$ $$($$$$-21 \beta_{5} + 222 \beta_{1} - 108817$$$$)/250$$ $$\nu^{3}$$ $$=$$ $$($$$$-16225 \beta_{4} - 253 \beta_{3} - 62406 \beta_{2}$$$$)/250$$ $$\nu^{4}$$ $$=$$ $$($$$$2073 \beta_{5} - 41046 \beta_{1} + 13959941$$$$)/50$$ $$\nu^{5}$$ $$=$$ $$($$$$10746025 \beta_{4} + 2134309 \beta_{3} + 55337718 \beta_{2}$$$$)/250$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 27.7229i 22.2334i − 6.48955i 6.48955i − 22.2334i 27.7229i
31.7828i 0 −498.143 0 0 637.237i 440.406i 0 0
199.2 21.4187i 0 53.2406 0 0 9905.49i 12106.7i 0 0
199.3 20.2014i 0 103.903 0 0 4010.25i 12442.1i 0 0
199.4 20.2014i 0 103.903 0 0 4010.25i 12442.1i 0 0
199.5 21.4187i 0 53.2406 0 0 9905.49i 12106.7i 0 0
199.6 31.7828i 0 −498.143 0 0 637.237i 440.406i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.6 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 225.10.b.m 6
3.b odd 2 1 25.10.b.c 6
5.b even 2 1 inner 225.10.b.m 6
5.c odd 4 1 225.10.a.m 3
5.c odd 4 1 225.10.a.p 3
12.b even 2 1 400.10.c.q 6
15.d odd 2 1 25.10.b.c 6
15.e even 4 1 25.10.a.c 3
15.e even 4 1 25.10.a.d yes 3
60.h even 2 1 400.10.c.q 6
60.l odd 4 1 400.10.a.u 3
60.l odd 4 1 400.10.a.y 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.10.a.c 3 15.e even 4 1
25.10.a.d yes 3 15.e even 4 1
25.10.b.c 6 3.b odd 2 1
25.10.b.c 6 15.d odd 2 1
225.10.a.m 3 5.c odd 4 1
225.10.a.p 3 5.c odd 4 1
225.10.b.m 6 1.a even 1 1 trivial
225.10.b.m 6 5.b even 2 1 inner
400.10.a.u 3 60.l odd 4 1
400.10.a.y 3 60.l odd 4 1
400.10.c.q 6 12.b even 2 1
400.10.c.q 6 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{10}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{6} + 1877 T_{2}^{4} + 1062868 T_{2}^{2} + 189117504$$ $$T_{11}^{3} - 54699 T_{11}^{2} - 1257655633 T_{11} +$$$$75\!\cdots\!63$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$189117504 + 1062868 T^{2} + 1877 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$64\!\cdots\!44$$$$+ 1624330022979888 T^{2} + 114606892 T^{4} + T^{6}$$
$11$ $$( 75283351667163 - 1257655633 T - 54699 T^{2} + T^{3} )^{2}$$
$13$ $$22\!\cdots\!56$$$$+$$$$64\!\cdots\!88$$$$T^{2} + 47782585008 T^{4} + T^{6}$$
$17$ $$31\!\cdots\!49$$$$+$$$$11\!\cdots\!03$$$$T^{2} + 113679545147 T^{4} + T^{6}$$
$19$ $$( -226123024842854125 - 499889436625 T + 818845 T^{2} + T^{3} )^{2}$$
$23$ $$14\!\cdots\!56$$$$+$$$$81\!\cdots\!88$$$$T^{2} + 5844421656108 T^{4} + T^{6}$$
$29$ $$( 3964526545895424000 - 11063731072000 T - 2175480 T^{2} + T^{3} )^{2}$$
$31$ $$( -817098195664566648 + 3532627327452 T - 4274066 T^{2} + T^{3} )^{2}$$
$37$ $$29\!\cdots\!84$$$$+$$$$59\!\cdots\!08$$$$T^{2} + 204279936810732 T^{4} + T^{6}$$
$41$ $$( -$$$$15\!\cdots\!47$$$$- 750102076057093 T + 5926311 T^{2} + T^{3} )^{2}$$
$43$ $$10\!\cdots\!56$$$$+$$$$12\!\cdots\!88$$$$T^{2} + 260277885634608 T^{4} + T^{6}$$
$47$ $$21\!\cdots\!64$$$$+$$$$11\!\cdots\!48$$$$T^{2} + 2150337413915312 T^{4} + T^{6}$$
$53$ $$16\!\cdots\!56$$$$+$$$$10\!\cdots\!88$$$$T^{2} + 18968285309086508 T^{4} + T^{6}$$
$59$ $$( -$$$$49\!\cdots\!00$$$$- 6651292273432000 T - 5670960 T^{2} + T^{3} )^{2}$$
$61$ $$($$$$62\!\cdots\!12$$$$- 12890308075143508 T - 125306926 T^{2} + T^{3} )^{2}$$
$67$ $$77\!\cdots\!49$$$$+$$$$21\!\cdots\!03$$$$T^{2} + 110005113588027547 T^{4} + T^{6}$$
$71$ $$( -$$$$11\!\cdots\!32$$$$- 50509406137014928 T + 297550596 T^{2} + T^{3} )^{2}$$
$73$ $$19\!\cdots\!81$$$$+$$$$11\!\cdots\!63$$$$T^{2} + 197376584603920683 T^{4} + T^{6}$$
$79$ $$($$$$26\!\cdots\!00$$$$- 183462234827962500 T - 310025170 T^{2} + T^{3} )^{2}$$
$83$ $$70\!\cdots\!81$$$$+$$$$16\!\cdots\!63$$$$T^{2} + 285265670458184883 T^{4} + T^{6}$$
$89$ $$( -$$$$19\!\cdots\!75$$$$+ 269595002285863875 T + 1103860035 T^{2} + T^{3} )^{2}$$
$97$ $$11\!\cdots\!64$$$$+$$$$85\!\cdots\!48$$$$T^{2} + 1696277618616206412 T^{4} + T^{6}$$