# Properties

 Label 225.12.b.d Level $225$ Weight $12$ Character orbit 225.b Analytic conductor $172.877$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$172.877215626$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 12 \beta q^{2} + 1472 q^{4} - 8372 \beta q^{7} + 42240 \beta q^{8} +O(q^{10})$$ q + 12*b * q^2 + 1472 * q^4 - 8372*b * q^7 + 42240*b * q^8 $$q + 12 \beta q^{2} + 1472 q^{4} - 8372 \beta q^{7} + 42240 \beta q^{8} - 534612 q^{11} + 288869 \beta q^{13} + 401856 q^{14} + 987136 q^{16} + 3452967 \beta q^{17} - 10661420 q^{19} - 6415344 \beta q^{22} + 9321636 \beta q^{23} - 13865712 q^{26} - 12323584 \beta q^{28} + 128406630 q^{29} - 52843168 q^{31} + 98353152 \beta q^{32} - 165742416 q^{34} - 91106657 \beta q^{37} - 127937040 \beta q^{38} - 308120442 q^{41} + 8562854 \beta q^{43} - 786948864 q^{44} - 447438528 q^{46} - 1343674248 \beta q^{47} + 1696965207 q^{49} + 425215168 \beta q^{52} - 798027849 \beta q^{53} + 1414533120 q^{56} + 1540879560 \beta q^{58} - 5189203740 q^{59} + 6956478662 q^{61} - 634118016 \beta q^{62} - 2699296768 q^{64} - 7740913442 \beta q^{67} + 5082767424 \beta q^{68} - 9791485272 q^{71} - 731895661 \beta q^{73} + 4373119536 q^{74} - 15693610240 q^{76} + 4475771664 \beta q^{77} - 38116845680 q^{79} - 3697445304 \beta q^{82} - 14667549834 \beta q^{83} - 411016992 q^{86} - 22582010880 \beta q^{88} - 24992917110 q^{89} + 9673645072 q^{91} + 13721448192 \beta q^{92} + 64496363904 q^{94} + 37506784273 \beta q^{97} + 20363582484 \beta q^{98} +O(q^{100})$$ q + 12*b * q^2 + 1472 * q^4 - 8372*b * q^7 + 42240*b * q^8 - 534612 * q^11 + 288869*b * q^13 + 401856 * q^14 + 987136 * q^16 + 3452967*b * q^17 - 10661420 * q^19 - 6415344*b * q^22 + 9321636*b * q^23 - 13865712 * q^26 - 12323584*b * q^28 + 128406630 * q^29 - 52843168 * q^31 + 98353152*b * q^32 - 165742416 * q^34 - 91106657*b * q^37 - 127937040*b * q^38 - 308120442 * q^41 + 8562854*b * q^43 - 786948864 * q^44 - 447438528 * q^46 - 1343674248*b * q^47 + 1696965207 * q^49 + 425215168*b * q^52 - 798027849*b * q^53 + 1414533120 * q^56 + 1540879560*b * q^58 - 5189203740 * q^59 + 6956478662 * q^61 - 634118016*b * q^62 - 2699296768 * q^64 - 7740913442*b * q^67 + 5082767424*b * q^68 - 9791485272 * q^71 - 731895661*b * q^73 + 4373119536 * q^74 - 15693610240 * q^76 + 4475771664*b * q^77 - 38116845680 * q^79 - 3697445304*b * q^82 - 14667549834*b * q^83 - 411016992 * q^86 - 22582010880*b * q^88 - 24992917110 * q^89 + 9673645072 * q^91 + 13721448192*b * q^92 + 64496363904 * q^94 + 37506784273*b * q^97 + 20363582484*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2944 q^{4}+O(q^{10})$$ 2 * q + 2944 * q^4 $$2 q + 2944 q^{4} - 1069224 q^{11} + 803712 q^{14} + 1974272 q^{16} - 21322840 q^{19} - 27731424 q^{26} + 256813260 q^{29} - 105686336 q^{31} - 331484832 q^{34} - 616240884 q^{41} - 1573897728 q^{44} - 894877056 q^{46} + 3393930414 q^{49} + 2829066240 q^{56} - 10378407480 q^{59} + 13912957324 q^{61} - 5398593536 q^{64} - 19582970544 q^{71} + 8746239072 q^{74} - 31387220480 q^{76} - 76233691360 q^{79} - 822033984 q^{86} - 49985834220 q^{89} + 19347290144 q^{91} + 128992727808 q^{94}+O(q^{100})$$ 2 * q + 2944 * q^4 - 1069224 * q^11 + 803712 * q^14 + 1974272 * q^16 - 21322840 * q^19 - 27731424 * q^26 + 256813260 * q^29 - 105686336 * q^31 - 331484832 * q^34 - 616240884 * q^41 - 1573897728 * q^44 - 894877056 * q^46 + 3393930414 * q^49 + 2829066240 * q^56 - 10378407480 * q^59 + 13912957324 * q^61 - 5398593536 * q^64 - 19582970544 * q^71 + 8746239072 * q^74 - 31387220480 * q^76 - 76233691360 * q^79 - 822033984 * q^86 - 49985834220 * q^89 + 19347290144 * q^91 + 128992727808 * q^94

## 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
 − 1.00000i 1.00000i
24.0000i 0 1472.00 0 0 16744.0i 84480.0i 0 0
199.2 24.0000i 0 1472.00 0 0 16744.0i 84480.0i 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 225.12.b.d 2
3.b odd 2 1 25.12.b.b 2
5.b even 2 1 inner 225.12.b.d 2
5.c odd 4 1 9.12.a.b 1
5.c odd 4 1 225.12.a.b 1
15.d odd 2 1 25.12.b.b 2
15.e even 4 1 1.12.a.a 1
15.e even 4 1 25.12.a.b 1
20.e even 4 1 144.12.a.d 1
45.k odd 12 2 81.12.c.b 2
45.l even 12 2 81.12.c.d 2
60.l odd 4 1 16.12.a.a 1
105.k odd 4 1 49.12.a.a 1
105.w odd 12 2 49.12.c.c 2
105.x even 12 2 49.12.c.b 2
120.q odd 4 1 64.12.a.f 1
120.w even 4 1 64.12.a.b 1
165.l odd 4 1 121.12.a.b 1
195.s even 4 1 169.12.a.a 1
240.z odd 4 1 256.12.b.c 2
240.bb even 4 1 256.12.b.e 2
240.bd odd 4 1 256.12.b.c 2
240.bf even 4 1 256.12.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 15.e even 4 1
9.12.a.b 1 5.c odd 4 1
16.12.a.a 1 60.l odd 4 1
25.12.a.b 1 15.e even 4 1
25.12.b.b 2 3.b odd 2 1
25.12.b.b 2 15.d odd 2 1
49.12.a.a 1 105.k odd 4 1
49.12.c.b 2 105.x even 12 2
49.12.c.c 2 105.w odd 12 2
64.12.a.b 1 120.w even 4 1
64.12.a.f 1 120.q odd 4 1
81.12.c.b 2 45.k odd 12 2
81.12.c.d 2 45.l even 12 2
121.12.a.b 1 165.l odd 4 1
144.12.a.d 1 20.e even 4 1
169.12.a.a 1 195.s even 4 1
225.12.a.b 1 5.c odd 4 1
225.12.b.d 2 1.a even 1 1 trivial
225.12.b.d 2 5.b even 2 1 inner
256.12.b.c 2 240.z odd 4 1
256.12.b.c 2 240.bd odd 4 1
256.12.b.e 2 240.bb even 4 1
256.12.b.e 2 240.bf even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{12}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{2} + 576$$ T2^2 + 576 $$T_{11} + 534612$$ T11 + 534612

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 576$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 280361536$$
$11$ $$(T + 534612)^{2}$$
$13$ $$T^{2} + 333781196644$$
$17$ $$T^{2} + 47691924412356$$
$19$ $$(T + 10661420)^{2}$$
$23$ $$T^{2} + \cdots + 347571590865984$$
$29$ $$(T - 128406630)^{2}$$
$31$ $$(T + 52843168)^{2}$$
$37$ $$T^{2} + 33\!\cdots\!96$$
$41$ $$(T + 308120442)^{2}$$
$43$ $$T^{2} + \cdots + 293289874501264$$
$47$ $$T^{2} + 72\!\cdots\!16$$
$53$ $$T^{2} + 25\!\cdots\!04$$
$59$ $$(T + 5189203740)^{2}$$
$61$ $$(T - 6956478662)^{2}$$
$67$ $$T^{2} + 23\!\cdots\!56$$
$71$ $$(T + 9791485272)^{2}$$
$73$ $$T^{2} + 21\!\cdots\!84$$
$79$ $$(T + 38116845680)^{2}$$
$83$ $$T^{2} + 86\!\cdots\!24$$
$89$ $$(T + 24992917110)^{2}$$
$97$ $$T^{2} + 56\!\cdots\!16$$