# Properties

 Label 225.10.b.i Level $225$ Weight $10$ Character orbit 225.b Analytic conductor $115.883$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{241})$$ Defining polynomial: $$x^{4} + 121 x^{2} + 3600$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + 8 \beta_{2} ) q^{2} + ( -255 - 31 \beta_{3} ) q^{4} + ( 224 \beta_{1} - 3584 \beta_{2} ) q^{7} + ( 239 \beta_{1} - 6593 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} + 8 \beta_{2} ) q^{2} + ( -255 - 31 \beta_{3} ) q^{4} + ( 224 \beta_{1} - 3584 \beta_{2} ) q^{7} + ( 239 \beta_{1} - 6593 \beta_{2} ) q^{8} + ( 9572 + 2368 \beta_{3} ) q^{11} + ( 5344 \beta_{1} + 4735 \beta_{2} ) q^{13} + ( 225568 + 10528 \beta_{3} ) q^{14} + ( 193183 + 899 \beta_{3} ) q^{16} + ( -7520 \beta_{1} - 37359 \beta_{2} ) q^{17} + ( 45084 + 5728 \beta_{3} ) q^{19} + ( -47460 \beta_{1} + 737248 \beta_{2} ) q^{22} + ( 26272 \beta_{1} + 177248 \beta_{2} ) q^{23} + ( 2674238 + 70690 \beta_{3} ) q^{26} + ( -279328 \beta_{1} + 2906848 \beta_{2} ) q^{28} + ( -1254818 - 168576 \beta_{3} ) q^{29} + ( 5467584 - 152736 \beta_{3} ) q^{31} + ( -85199 \beta_{1} - 1579331 \beta_{2} ) q^{32} + ( -2842270 - 38082 \beta_{3} ) q^{34} + ( -198496 \beta_{1} - 5442459 \beta_{2} ) q^{37} + ( -136732 \beta_{1} + 1958784 \beta_{2} ) q^{38} + ( -13276234 + 492096 \beta_{3} ) q^{41} + ( 702336 \beta_{1} - 1973374 \beta_{2} ) q^{43} + ( -42227996 - 973980 \beta_{3} ) q^{44} + ( 8527904 + 39584 \beta_{3} ) q^{46} + ( -1970528 \beta_{1} - 7402428 \beta_{2} ) q^{47} + ( -35060921 - 3161088 \beta_{3} ) q^{49} + ( -1069150 \beta_{1} + 43540734 \beta_{2} ) q^{52} + ( -177728 \beta_{1} - 738347 \beta_{2} ) q^{53} + ( -118920480 - 4613280 \beta_{3} ) q^{56} + ( 3952034 \beta_{1} - 57071248 \beta_{2} ) q^{58} + ( -15573156 - 4348352 \beta_{3} ) q^{59} + ( 170273566 + 950208 \beta_{3} ) q^{61} + ( -3023808 \beta_{1} + 1127328 \beta_{2} ) q^{62} + ( 101389753 + 2340965 \beta_{3} ) q^{64} + ( 1026560 \beta_{1} + 71792314 \beta_{2} ) q^{67} + ( -398658 \beta_{1} - 52490846 \beta_{2} ) q^{68} + ( -107415672 + 4545280 \beta_{3} ) q^{71} + ( -1168192 \beta_{1} - 57873723 \beta_{2} ) q^{73} + ( 58666378 + 7907478 \beta_{3} ) q^{74} + ( -107738276 - 3035812 \beta_{3} ) q^{76} + ( 19117952 \beta_{1} - 186540032 \beta_{2} ) q^{77} + ( 21902080 - 19049120 \beta_{3} ) q^{79} + ( 5402698 \beta_{1} + 31084912 \beta_{2} ) q^{82} + ( 7260288 \beta_{1} + 91205466 \beta_{2} ) q^{83} + ( 429332292 + 14481788 \beta_{3} ) q^{86} + ( 33512156 \beta_{1} - 232093412 \beta_{2} ) q^{88} + ( -218344134 + 9049152 \beta_{3} ) q^{89} + ( -545935936 - 34987456 \beta_{3} ) q^{91} + ( 4290016 \beta_{1} + 170018144 \beta_{2} ) q^{92} + ( -816395416 - 14753064 \beta_{3} ) q^{94} + ( -13450880 \beta_{1} - 439552001 \beta_{2} ) q^{97} + ( 85638329 \beta_{1} - 1162430920 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 1082q^{4} + O(q^{10})$$ $$4q - 1082q^{4} + 43024q^{11} + 923328q^{14} + 774530q^{16} + 191792q^{19} + 10838332q^{26} - 5356424q^{29} + 21564864q^{31} - 11445244q^{34} - 52120744q^{41} - 170859944q^{44} + 34190784q^{46} - 146565860q^{49} - 484908480q^{56} - 70989328q^{59} + 682994680q^{61} + 410240942q^{64} - 420572128q^{71} + 250480468q^{74} - 437024728q^{76} + 49510080q^{79} + 1746292744q^{86} - 855278232q^{89} - 2253718656q^{91} - 3295087792q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 121 x^{2} + 3600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 241 \nu$$$$)/60$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 61 \nu$$$$)/30$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{2} + 182$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 182$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-241 \beta_{2} - 122 \beta_{1}$$$$)/6$$

## 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
 7.26209i 8.26209i − 8.26209i − 7.26209i
38.7863i 0 −992.374 0 0 12272.1i 18631.9i 0 0
199.2 7.78626i 0 451.374 0 0 1839.88i 7501.08i 0 0
199.3 7.78626i 0 451.374 0 0 1839.88i 7501.08i 0 0
199.4 38.7863i 0 −992.374 0 0 12272.1i 18631.9i 0 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 225.10.b.i 4
3.b odd 2 1 75.10.b.f 4
5.b even 2 1 inner 225.10.b.i 4
5.c odd 4 1 45.10.a.d 2
5.c odd 4 1 225.10.a.k 2
15.d odd 2 1 75.10.b.f 4
15.e even 4 1 15.10.a.d 2
15.e even 4 1 75.10.a.f 2
60.l odd 4 1 240.10.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.d 2 15.e even 4 1
45.10.a.d 2 5.c odd 4 1
75.10.a.f 2 15.e even 4 1
75.10.b.f 4 3.b odd 2 1
75.10.b.f 4 15.d odd 2 1
225.10.a.k 2 5.c odd 4 1
225.10.b.i 4 1.a even 1 1 trivial
225.10.b.i 4 5.b even 2 1 inner
240.10.a.r 2 60.l odd 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 1565 T_{2}^{2} + 91204$$ $$T_{11}^{2} - 21512 T_{11} - 2924934128$$