# Properties

 Label 325.4.b.b Level $325$ Weight $4$ Character orbit 325.b Analytic conductor $19.176$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 325.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1756207519$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) 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.

 $$f(q)$$ $$=$$ $$q + 5 i q^{2} - 7 i q^{3} - 17 q^{4} + 35 q^{6} + 13 i q^{7} - 45 i q^{8} - 22 q^{9} +O(q^{10})$$ q + 5*i * q^2 - 7*i * q^3 - 17 * q^4 + 35 * q^6 + 13*i * q^7 - 45*i * q^8 - 22 * q^9 $$q + 5 i q^{2} - 7 i q^{3} - 17 q^{4} + 35 q^{6} + 13 i q^{7} - 45 i q^{8} - 22 q^{9} - 26 q^{11} + 119 i q^{12} + 13 i q^{13} - 65 q^{14} + 89 q^{16} - 77 i q^{17} - 110 i q^{18} + 126 q^{19} + 91 q^{21} - 130 i q^{22} - 96 i q^{23} - 315 q^{24} - 65 q^{26} - 35 i q^{27} - 221 i q^{28} + 82 q^{29} + 196 q^{31} + 85 i q^{32} + 182 i q^{33} + 385 q^{34} + 374 q^{36} + 131 i q^{37} + 630 i q^{38} + 91 q^{39} + 336 q^{41} + 455 i q^{42} - 201 i q^{43} + 442 q^{44} + 480 q^{46} + 105 i q^{47} - 623 i q^{48} + 174 q^{49} - 539 q^{51} - 221 i q^{52} - 432 i q^{53} + 175 q^{54} + 585 q^{56} - 882 i q^{57} + 410 i q^{58} + 294 q^{59} - 56 q^{61} + 980 i q^{62} - 286 i q^{63} + 287 q^{64} - 910 q^{66} - 478 i q^{67} + 1309 i q^{68} - 672 q^{69} + 9 q^{71} + 990 i q^{72} + 98 i q^{73} - 655 q^{74} - 2142 q^{76} - 338 i q^{77} + 455 i q^{78} - 1304 q^{79} - 839 q^{81} + 1680 i q^{82} - 308 i q^{83} - 1547 q^{84} + 1005 q^{86} - 574 i q^{87} + 1170 i q^{88} + 1190 q^{89} - 169 q^{91} + 1632 i q^{92} - 1372 i q^{93} - 525 q^{94} + 595 q^{96} - 70 i q^{97} + 870 i q^{98} + 572 q^{99} +O(q^{100})$$ q + 5*i * q^2 - 7*i * q^3 - 17 * q^4 + 35 * q^6 + 13*i * q^7 - 45*i * q^8 - 22 * q^9 - 26 * q^11 + 119*i * q^12 + 13*i * q^13 - 65 * q^14 + 89 * q^16 - 77*i * q^17 - 110*i * q^18 + 126 * q^19 + 91 * q^21 - 130*i * q^22 - 96*i * q^23 - 315 * q^24 - 65 * q^26 - 35*i * q^27 - 221*i * q^28 + 82 * q^29 + 196 * q^31 + 85*i * q^32 + 182*i * q^33 + 385 * q^34 + 374 * q^36 + 131*i * q^37 + 630*i * q^38 + 91 * q^39 + 336 * q^41 + 455*i * q^42 - 201*i * q^43 + 442 * q^44 + 480 * q^46 + 105*i * q^47 - 623*i * q^48 + 174 * q^49 - 539 * q^51 - 221*i * q^52 - 432*i * q^53 + 175 * q^54 + 585 * q^56 - 882*i * q^57 + 410*i * q^58 + 294 * q^59 - 56 * q^61 + 980*i * q^62 - 286*i * q^63 + 287 * q^64 - 910 * q^66 - 478*i * q^67 + 1309*i * q^68 - 672 * q^69 + 9 * q^71 + 990*i * q^72 + 98*i * q^73 - 655 * q^74 - 2142 * q^76 - 338*i * q^77 + 455*i * q^78 - 1304 * q^79 - 839 * q^81 + 1680*i * q^82 - 308*i * q^83 - 1547 * q^84 + 1005 * q^86 - 574*i * q^87 + 1170*i * q^88 + 1190 * q^89 - 169 * q^91 + 1632*i * q^92 - 1372*i * q^93 - 525 * q^94 + 595 * q^96 - 70*i * q^97 + 870*i * q^98 + 572 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 34 q^{4} + 70 q^{6} - 44 q^{9}+O(q^{10})$$ 2 * q - 34 * q^4 + 70 * q^6 - 44 * q^9 $$2 q - 34 q^{4} + 70 q^{6} - 44 q^{9} - 52 q^{11} - 130 q^{14} + 178 q^{16} + 252 q^{19} + 182 q^{21} - 630 q^{24} - 130 q^{26} + 164 q^{29} + 392 q^{31} + 770 q^{34} + 748 q^{36} + 182 q^{39} + 672 q^{41} + 884 q^{44} + 960 q^{46} + 348 q^{49} - 1078 q^{51} + 350 q^{54} + 1170 q^{56} + 588 q^{59} - 112 q^{61} + 574 q^{64} - 1820 q^{66} - 1344 q^{69} + 18 q^{71} - 1310 q^{74} - 4284 q^{76} - 2608 q^{79} - 1678 q^{81} - 3094 q^{84} + 2010 q^{86} + 2380 q^{89} - 338 q^{91} - 1050 q^{94} + 1190 q^{96} + 1144 q^{99}+O(q^{100})$$ 2 * q - 34 * q^4 + 70 * q^6 - 44 * q^9 - 52 * q^11 - 130 * q^14 + 178 * q^16 + 252 * q^19 + 182 * q^21 - 630 * q^24 - 130 * q^26 + 164 * q^29 + 392 * q^31 + 770 * q^34 + 748 * q^36 + 182 * q^39 + 672 * q^41 + 884 * q^44 + 960 * q^46 + 348 * q^49 - 1078 * q^51 + 350 * q^54 + 1170 * q^56 + 588 * q^59 - 112 * q^61 + 574 * q^64 - 1820 * q^66 - 1344 * q^69 + 18 * q^71 - 1310 * q^74 - 4284 * q^76 - 2608 * q^79 - 1678 * q^81 - 3094 * q^84 + 2010 * q^86 + 2380 * q^89 - 338 * q^91 - 1050 * q^94 + 1190 * q^96 + 1144 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 − 1.00000i 1.00000i
5.00000i 7.00000i −17.0000 0 35.0000 13.0000i 45.0000i −22.0000 0
274.2 5.00000i 7.00000i −17.0000 0 35.0000 13.0000i 45.0000i −22.0000 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 325.4.b.b 2
5.b even 2 1 inner 325.4.b.b 2
5.c odd 4 1 13.4.a.a 1
5.c odd 4 1 325.4.a.d 1
15.e even 4 1 117.4.a.b 1
20.e even 4 1 208.4.a.g 1
35.f even 4 1 637.4.a.a 1
40.i odd 4 1 832.4.a.r 1
40.k even 4 1 832.4.a.a 1
55.e even 4 1 1573.4.a.a 1
60.l odd 4 1 1872.4.a.k 1
65.f even 4 1 169.4.b.a 2
65.h odd 4 1 169.4.a.e 1
65.k even 4 1 169.4.b.a 2
65.o even 12 2 169.4.e.e 4
65.q odd 12 2 169.4.c.e 2
65.r odd 12 2 169.4.c.a 2
65.t even 12 2 169.4.e.e 4
195.s even 4 1 1521.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 5.c odd 4 1
117.4.a.b 1 15.e even 4 1
169.4.a.e 1 65.h odd 4 1
169.4.b.a 2 65.f even 4 1
169.4.b.a 2 65.k even 4 1
169.4.c.a 2 65.r odd 12 2
169.4.c.e 2 65.q odd 12 2
169.4.e.e 4 65.o even 12 2
169.4.e.e 4 65.t even 12 2
208.4.a.g 1 20.e even 4 1
325.4.a.d 1 5.c odd 4 1
325.4.b.b 2 1.a even 1 1 trivial
325.4.b.b 2 5.b even 2 1 inner
637.4.a.a 1 35.f even 4 1
832.4.a.a 1 40.k even 4 1
832.4.a.r 1 40.i odd 4 1
1521.4.a.a 1 195.s even 4 1
1573.4.a.a 1 55.e even 4 1
1872.4.a.k 1 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 $$S_{4}^{\mathrm{new}}(325, [\chi])$$:

 $$T_{2}^{2} + 25$$ T2^2 + 25 $$T_{3}^{2} + 49$$ T3^2 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 25$$
$3$ $$T^{2} + 49$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 169$$
$11$ $$(T + 26)^{2}$$
$13$ $$T^{2} + 169$$
$17$ $$T^{2} + 5929$$
$19$ $$(T - 126)^{2}$$
$23$ $$T^{2} + 9216$$
$29$ $$(T - 82)^{2}$$
$31$ $$(T - 196)^{2}$$
$37$ $$T^{2} + 17161$$
$41$ $$(T - 336)^{2}$$
$43$ $$T^{2} + 40401$$
$47$ $$T^{2} + 11025$$
$53$ $$T^{2} + 186624$$
$59$ $$(T - 294)^{2}$$
$61$ $$(T + 56)^{2}$$
$67$ $$T^{2} + 228484$$
$71$ $$(T - 9)^{2}$$
$73$ $$T^{2} + 9604$$
$79$ $$(T + 1304)^{2}$$
$83$ $$T^{2} + 94864$$
$89$ $$(T - 1190)^{2}$$
$97$ $$T^{2} + 4900$$