Properties

 Label 325.4.b.e Level $325$ Weight $4$ Character orbit 325.b Analytic conductor $19.176$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (4 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} + 3) q^{4} + ( - \beta_{3} - 11) q^{6} + (10 \beta_{2} + 11 \beta_1) q^{7} + (4 \beta_{2} + 11 \beta_1) q^{8} + ( - 15 \beta_{3} - 10) q^{9}+O(q^{10})$$ q + b1 * q^2 + (4*b2 + 3*b1) * q^3 + (b3 + 3) * q^4 + (-b3 - 11) * q^6 + (10*b2 + 11*b1) * q^7 + (4*b2 + 11*b1) * q^8 + (-15*b3 - 10) * q^9 $$q + \beta_1 q^{2} + (4 \beta_{2} + 3 \beta_1) q^{3} + (\beta_{3} + 3) q^{4} + ( - \beta_{3} - 11) q^{6} + (10 \beta_{2} + 11 \beta_1) q^{7} + (4 \beta_{2} + 11 \beta_1) q^{8} + ( - 15 \beta_{3} - 10) q^{9} + ( - 12 \beta_{3} + 46) q^{11} + (28 \beta_{2} + 13 \beta_1) q^{12} - 13 \beta_{2} q^{13} + (\beta_{3} - 45) q^{14} + (15 \beta_{3} - 27) q^{16} + ( - 18 \beta_{2} - 17 \beta_1) q^{17} + ( - 60 \beta_{2} - 10 \beta_1) q^{18} + ( - 32 \beta_{3} + 58) q^{19} + ( - 41 \beta_{3} - 131) q^{21} + ( - 48 \beta_{2} + 46 \beta_1) q^{22} + (104 \beta_{2} + 12 \beta_1) q^{23} + ( - 23 \beta_{3} - 125) q^{24} + (13 \beta_{3} - 13) q^{26} + ( - 172 \beta_{2} - 9 \beta_1) q^{27} + (84 \beta_{2} + 43 \beta_1) q^{28} + (96 \beta_{3} - 26) q^{29} + (34 \beta_{3} - 60) q^{31} + (92 \beta_{2} + 61 \beta_1) q^{32} + ( - 8 \beta_{2} + 90 \beta_1) q^{33} + (\beta_{3} + 67) q^{34} + ( - 70 \beta_{3} - 90) q^{36} + ( - 102 \beta_{2} + 5 \beta_1) q^{37} + ( - 128 \beta_{2} + 58 \beta_1) q^{38} + (39 \beta_{3} + 13) q^{39} + ( - 22 \beta_{3} - 104) q^{41} + ( - 164 \beta_{2} - 131 \beta_1) q^{42} + (72 \beta_{2} - 143 \beta_1) q^{43} + ( - 2 \beta_{3} + 90) q^{44} + ( - 92 \beta_{3} + 44) q^{46} + ( - 278 \beta_{2} - 121 \beta_1) q^{47} + (132 \beta_{2} - 21 \beta_1) q^{48} + ( - 99 \beta_{3} - 142) q^{49} + (71 \beta_{3} + 205) q^{51} + ( - 52 \beta_{2} - 13 \beta_1) q^{52} + ( - 74 \beta_{2} - 30 \beta_1) q^{53} + (163 \beta_{3} - 127) q^{54} + ( - 33 \beta_{3} - 491) q^{56} + ( - 280 \beta_{2} + 46 \beta_1) q^{57} + (384 \beta_{2} - 26 \beta_1) q^{58} + (124 \beta_{3} + 122) q^{59} + (190 \beta_{3} - 624) q^{61} + (136 \beta_{2} - 60 \beta_1) q^{62} + ( - 910 \beta_{2} - 260 \beta_1) q^{63} + (89 \beta_{3} - 429) q^{64} + (98 \beta_{3} - 458) q^{66} + ( - 150 \beta_{2} - 232 \beta_1) q^{67} + ( - 140 \beta_{2} - 69 \beta_1) q^{68} + ( - 324 \beta_{3} - 236) q^{69} + (231 \beta_{3} - 181) q^{71} + ( - 760 \beta_{2} - 170 \beta_1) q^{72} + (98 \beta_{2} - 260 \beta_1) q^{73} + (107 \beta_{3} - 127) q^{74} + ( - 70 \beta_{3} + 46) q^{76} + ( - 188 \beta_{2} + 386 \beta_1) q^{77} + (156 \beta_{2} + 13 \beta_1) q^{78} + (40 \beta_{3} + 484) q^{79} + (120 \beta_{3} + 1) q^{81} + ( - 88 \beta_{2} - 104 \beta_1) q^{82} + (1070 \beta_{2} + 182 \beta_1) q^{83} + ( - 295 \beta_{3} - 557) q^{84} + ( - 215 \beta_{3} + 787) q^{86} + (1432 \beta_{2} + 306 \beta_1) q^{87} + ( - 392 \beta_{2} + 458 \beta_1) q^{88} + ( - 388 \beta_{3} + 554) q^{89} + (143 \beta_{3} - 13) q^{91} + (464 \beta_{2} + 140 \beta_1) q^{92} + (304 \beta_{2} - 44 \beta_1) q^{93} + (157 \beta_{3} + 327) q^{94} + ( - 337 \beta_{3} - 763) q^{96} + (718 \beta_{2} + 508 \beta_1) q^{97} + ( - 396 \beta_{2} - 142 \beta_1) q^{98} + ( - 390 \beta_{3} + 260) q^{99}+O(q^{100})$$ q + b1 * q^2 + (4*b2 + 3*b1) * q^3 + (b3 + 3) * q^4 + (-b3 - 11) * q^6 + (10*b2 + 11*b1) * q^7 + (4*b2 + 11*b1) * q^8 + (-15*b3 - 10) * q^9 + (-12*b3 + 46) * q^11 + (28*b2 + 13*b1) * q^12 - 13*b2 * q^13 + (b3 - 45) * q^14 + (15*b3 - 27) * q^16 + (-18*b2 - 17*b1) * q^17 + (-60*b2 - 10*b1) * q^18 + (-32*b3 + 58) * q^19 + (-41*b3 - 131) * q^21 + (-48*b2 + 46*b1) * q^22 + (104*b2 + 12*b1) * q^23 + (-23*b3 - 125) * q^24 + (13*b3 - 13) * q^26 + (-172*b2 - 9*b1) * q^27 + (84*b2 + 43*b1) * q^28 + (96*b3 - 26) * q^29 + (34*b3 - 60) * q^31 + (92*b2 + 61*b1) * q^32 + (-8*b2 + 90*b1) * q^33 + (b3 + 67) * q^34 + (-70*b3 - 90) * q^36 + (-102*b2 + 5*b1) * q^37 + (-128*b2 + 58*b1) * q^38 + (39*b3 + 13) * q^39 + (-22*b3 - 104) * q^41 + (-164*b2 - 131*b1) * q^42 + (72*b2 - 143*b1) * q^43 + (-2*b3 + 90) * q^44 + (-92*b3 + 44) * q^46 + (-278*b2 - 121*b1) * q^47 + (132*b2 - 21*b1) * q^48 + (-99*b3 - 142) * q^49 + (71*b3 + 205) * q^51 + (-52*b2 - 13*b1) * q^52 + (-74*b2 - 30*b1) * q^53 + (163*b3 - 127) * q^54 + (-33*b3 - 491) * q^56 + (-280*b2 + 46*b1) * q^57 + (384*b2 - 26*b1) * q^58 + (124*b3 + 122) * q^59 + (190*b3 - 624) * q^61 + (136*b2 - 60*b1) * q^62 + (-910*b2 - 260*b1) * q^63 + (89*b3 - 429) * q^64 + (98*b3 - 458) * q^66 + (-150*b2 - 232*b1) * q^67 + (-140*b2 - 69*b1) * q^68 + (-324*b3 - 236) * q^69 + (231*b3 - 181) * q^71 + (-760*b2 - 170*b1) * q^72 + (98*b2 - 260*b1) * q^73 + (107*b3 - 127) * q^74 + (-70*b3 + 46) * q^76 + (-188*b2 + 386*b1) * q^77 + (156*b2 + 13*b1) * q^78 + (40*b3 + 484) * q^79 + (120*b3 + 1) * q^81 + (-88*b2 - 104*b1) * q^82 + (1070*b2 + 182*b1) * q^83 + (-295*b3 - 557) * q^84 + (-215*b3 + 787) * q^86 + (1432*b2 + 306*b1) * q^87 + (-392*b2 + 458*b1) * q^88 + (-388*b3 + 554) * q^89 + (143*b3 - 13) * q^91 + (464*b2 + 140*b1) * q^92 + (304*b2 - 44*b1) * q^93 + (157*b3 + 327) * q^94 + (-337*b3 - 763) * q^96 + (718*b2 + 508*b1) * q^97 + (-396*b2 - 142*b1) * q^98 + (-390*b3 + 260) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 14 q^{4} - 46 q^{6} - 70 q^{9}+O(q^{10})$$ 4 * q + 14 * q^4 - 46 * q^6 - 70 * q^9 $$4 q + 14 q^{4} - 46 q^{6} - 70 q^{9} + 160 q^{11} - 178 q^{14} - 78 q^{16} + 168 q^{19} - 606 q^{21} - 546 q^{24} - 26 q^{26} + 88 q^{29} - 172 q^{31} + 270 q^{34} - 500 q^{36} + 130 q^{39} - 460 q^{41} + 356 q^{44} - 8 q^{46} - 766 q^{49} + 962 q^{51} - 182 q^{54} - 2030 q^{56} + 736 q^{59} - 2116 q^{61} - 1538 q^{64} - 1636 q^{66} - 1592 q^{69} - 262 q^{71} - 294 q^{74} + 44 q^{76} + 2016 q^{79} + 244 q^{81} - 2818 q^{84} + 2718 q^{86} + 1440 q^{89} + 234 q^{91} + 1622 q^{94} - 3726 q^{96} + 260 q^{99}+O(q^{100})$$ 4 * q + 14 * q^4 - 46 * q^6 - 70 * q^9 + 160 * q^11 - 178 * q^14 - 78 * q^16 + 168 * q^19 - 606 * q^21 - 546 * q^24 - 26 * q^26 + 88 * q^29 - 172 * q^31 + 270 * q^34 - 500 * q^36 + 130 * q^39 - 460 * q^41 + 356 * q^44 - 8 * q^46 - 766 * q^49 + 962 * q^51 - 182 * q^54 - 2030 * q^56 + 736 * q^59 - 2116 * q^61 - 1538 * q^64 - 1636 * q^66 - 1592 * q^69 - 262 * q^71 - 294 * q^74 + 44 * q^76 + 2016 * q^79 + 244 * q^81 - 2818 * q^84 + 2718 * q^86 + 1440 * q^89 + 234 * q^91 + 1622 * q^94 - 3726 * q^96 + 260 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 4$$ (v^3 + 5*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$4\beta_{2} - 5\beta_1$$ 4*b2 - 5*b1

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
 − 2.56155i − 1.56155i 1.56155i 2.56155i
2.56155i 3.68466i 1.43845 0 −9.43845 18.1771i 24.1771i 13.4233 0
274.2 1.56155i 8.68466i 5.56155 0 −13.5616 27.1771i 21.1771i −48.4233 0
274.3 1.56155i 8.68466i 5.56155 0 −13.5616 27.1771i 21.1771i −48.4233 0
274.4 2.56155i 3.68466i 1.43845 0 −9.43845 18.1771i 24.1771i 13.4233 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.e 4
5.b even 2 1 inner 325.4.b.e 4
5.c odd 4 1 13.4.a.b 2
5.c odd 4 1 325.4.a.f 2
15.e even 4 1 117.4.a.d 2
20.e even 4 1 208.4.a.h 2
35.f even 4 1 637.4.a.b 2
40.i odd 4 1 832.4.a.s 2
40.k even 4 1 832.4.a.z 2
55.e even 4 1 1573.4.a.b 2
60.l odd 4 1 1872.4.a.bb 2
65.f even 4 1 169.4.b.f 4
65.h odd 4 1 169.4.a.g 2
65.k even 4 1 169.4.b.f 4
65.o even 12 2 169.4.e.f 8
65.q odd 12 2 169.4.c.g 4
65.r odd 12 2 169.4.c.j 4
65.t even 12 2 169.4.e.f 8
195.s even 4 1 1521.4.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 5.c odd 4 1
117.4.a.d 2 15.e even 4 1
169.4.a.g 2 65.h odd 4 1
169.4.b.f 4 65.f even 4 1
169.4.b.f 4 65.k even 4 1
169.4.c.g 4 65.q odd 12 2
169.4.c.j 4 65.r odd 12 2
169.4.e.f 8 65.o even 12 2
169.4.e.f 8 65.t even 12 2
208.4.a.h 2 20.e even 4 1
325.4.a.f 2 5.c odd 4 1
325.4.b.e 4 1.a even 1 1 trivial
325.4.b.e 4 5.b even 2 1 inner
637.4.a.b 2 35.f even 4 1
832.4.a.s 2 40.i odd 4 1
832.4.a.z 2 40.k even 4 1
1521.4.a.r 2 195.s even 4 1
1573.4.a.b 2 55.e even 4 1
1872.4.a.bb 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 $$S_{4}^{\mathrm{new}}(325, [\chi])$$:

 $$T_{2}^{4} + 9T_{2}^{2} + 16$$ T2^4 + 9*T2^2 + 16 $$T_{3}^{4} + 89T_{3}^{2} + 1024$$ T3^4 + 89*T3^2 + 1024

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 9T^{2} + 16$$
$3$ $$T^{4} + 89T^{2} + 1024$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 1069 T^{2} + 244036$$
$11$ $$(T^{2} - 80 T + 988)^{2}$$
$13$ $$(T^{2} + 169)^{2}$$
$17$ $$T^{4} + 2637 T^{2} + \cdots + 1295044$$
$19$ $$(T^{2} - 84 T - 2588)^{2}$$
$23$ $$T^{4} + 20432 T^{2} + \cdots + 80856064$$
$29$ $$(T^{2} - 44 T - 38684)^{2}$$
$31$ $$(T^{2} + 86 T - 3064)^{2}$$
$37$ $$T^{4} + 22053 T^{2} + \cdots + 116942596$$
$41$ $$(T^{2} + 230 T + 11168)^{2}$$
$43$ $$T^{4} + 215001 T^{2} + \cdots + 4397811856$$
$47$ $$T^{4} + 219061 T^{2} + \cdots + 222546724$$
$53$ $$T^{4} + 14612 T^{2} + \cdots + 118336$$
$59$ $$(T^{2} - 368 T - 31492)^{2}$$
$61$ $$(T^{2} + 1058 T + 126416)^{2}$$
$67$ $$T^{4} + 459816 T^{2} + \cdots + 51799939216$$
$71$ $$(T^{2} + 131 T - 222494)^{2}$$
$73$ $$T^{4} + 678568 T^{2} + \cdots + 55373619856$$
$79$ $$(T^{2} - 1008 T + 247216)^{2}$$
$83$ $$T^{4} + 2198436 T^{2} + \cdots + 668574416896$$
$89$ $$(T^{2} - 720 T - 510212)^{2}$$
$97$ $$T^{4} + 2624136 T^{2} + \cdots + 776999938576$$