# Properties

 Label 425.4.b.f Level $425$ Weight $4$ Character orbit 425.b Analytic conductor $25.076$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$425 = 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 425.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.0758117524$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.27793984.1 Defining polynomial: $$x^{6} - 2x^{3} + 49x^{2} - 14x + 2$$ x^6 - 2*x^3 + 49*x^2 - 14*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 17) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} - \beta_{3}) q^{2} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{3} + ( - 3 \beta_{2} - \beta_1 - 8) q^{4} + ( - 6 \beta_{2} - 2 \beta_1 - 24) q^{6} + ( - 4 \beta_{5} - 6 \beta_{4} + \beta_{3}) q^{7} + (9 \beta_{5} - 16 \beta_{4} + 5 \beta_{3}) q^{8} + ( - 8 \beta_{2} - 2 \beta_1 - 19) q^{9}+O(q^{10})$$ q + (-b5 - b3) * q^2 + (-2*b5 + 2*b4 - b3) * q^3 + (-3*b2 - b1 - 8) * q^4 + (-6*b2 - 2*b1 - 24) * q^6 + (-4*b5 - 6*b4 + b3) * q^7 + (9*b5 - 16*b4 + 5*b3) * q^8 + (-8*b2 - 2*b1 - 19) * q^9 $$q + ( - \beta_{5} - \beta_{3}) q^{2} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{3} + ( - 3 \beta_{2} - \beta_1 - 8) q^{4} + ( - 6 \beta_{2} - 2 \beta_1 - 24) q^{6} + ( - 4 \beta_{5} - 6 \beta_{4} + \beta_{3}) q^{7} + (9 \beta_{5} - 16 \beta_{4} + 5 \beta_{3}) q^{8} + ( - 8 \beta_{2} - 2 \beta_1 - 19) q^{9} + ( - 11 \beta_{2} + 2 \beta_1 - 10) q^{11} + (26 \beta_{5} - 16 \beta_{4} + 26 \beta_{3}) q^{12} + ( - 6 \beta_{5} + 12 \beta_{4} + 8 \beta_{3}) q^{13} + (4 \beta_{2} - 20 \beta_1 - 24) q^{14} + (11 \beta_{2} - 7 \beta_1 + 48) q^{16} + 17 \beta_{4} q^{17} + (41 \beta_{5} - 48 \beta_{4} + 33 \beta_{3}) q^{18} + (22 \beta_{2} - 8 \beta_1 - 24) q^{19} + (12 \beta_{2} - 30 \beta_1 - 54) q^{21} + (26 \beta_{5} - 104 \beta_{4} + 34 \beta_{3}) q^{22} + (4 \beta_{5} + 46 \beta_{4} - 39 \beta_{3}) q^{23} + (46 \beta_{2} - 6 \beta_1 + 224) q^{24} + ( - 2 \beta_{2} - 22 \beta_1 + 16) q^{26} + (8 \beta_{5} + 4 \beta_{4} + 40 \beta_{3}) q^{27} + (44 \beta_{5} + 144 \beta_{4} + 4 \beta_{3}) q^{28} + ( - 16 \beta_{2} + 30 \beta_1 + 142) q^{29} + ( - 39 \beta_{2} - 16 \beta_1 + 82) q^{31} + (23 \beta_{5} + 16 \beta_{4} - 37 \beta_{3}) q^{32} + (34 \beta_{5} - 122 \beta_{4} + 76 \beta_{3}) q^{33} + ( - 17 \beta_{2} + 17 \beta_1) q^{34} + (91 \beta_{2} - 7 \beta_1 + 440) q^{36} + ( - 50 \beta_{5} - 102 \beta_{4} + 28 \beta_{3}) q^{37} + (4 \beta_{5} + 240 \beta_{4} - 28 \beta_{3}) q^{38} + (36 \beta_{2} - 16 \beta_1 - 84) q^{39} + (52 \beta_{2} + 60 \beta_1 - 118) q^{41} + (120 \beta_{5} + 336 \beta_{4}) q^{42} + ( - 56 \beta_{5} + 204 \beta_{4} + 2 \beta_{3}) q^{43} + (110 \beta_{2} - 78 \beta_1 + 400) q^{44} + ( - 120 \beta_{2} + 136 \beta_1 - 280) q^{46} + (44 \beta_{5} - 228 \beta_{4} + 48 \beta_{3}) q^{47} + ( - 90 \beta_{5} + 288 \beta_{4} - 114 \beta_{3}) q^{48} + (20 \beta_{2} - 94 \beta_1 + 121) q^{49} + ( - 17 \beta_{2} + 34 \beta_1 - 34) q^{51} + (6 \beta_{5} + 256 \beta_{4} + 30 \beta_{3}) q^{52} + (8 \beta_{5} + 98 \beta_{4} + 116 \beta_{3}) q^{53} + (84 \beta_{2} - 52 \beta_1 + 384) q^{54} + ( - 60 \beta_{2} + 108 \beta_1 + 192) q^{56} + (36 \beta_{5} + 228 \beta_{4} - 108 \beta_{3}) q^{57} + ( - 200 \beta_{5} - 368 \beta_{4} - 80 \beta_{3}) q^{58} + ( - 130 \beta_{2} - 212) q^{59} + (64 \beta_{2} - 78 \beta_1 - 2) q^{61} + (44 \beta_{5} - 184 \beta_{4} - 20 \beta_{3}) q^{62} + (96 \beta_{5} + 342 \beta_{4} + 9 \beta_{3}) q^{63} + (21 \beta_{2} + 103 \beta_1 + 272) q^{64} + (308 \beta_{2} - 172 \beta_1 + 880) q^{66} + ( - 132 \beta_{5} - 292 \beta_{4} - 24 \beta_{3}) q^{67} + ( - 17 \beta_{5} - 136 \beta_{4} + 51 \beta_{3}) q^{68} + ( - 280 \beta_{2} + 186 \beta_1 - 254) q^{69} + ( - 185 \beta_{2} - 72 \beta_1 - 110) q^{71} + ( - 273 \beta_{5} + 400 \beta_{4} - 365 \beta_{3}) q^{72} + (16 \beta_{5} + 274 \beta_{4} + 16 \beta_{3}) q^{73} + (108 \beta_{2} - 308 \beta_1 - 176) q^{74} + ( - 116 \beta_{2} + 244 \beta_1 - 384) q^{76} + ( - 18 \beta_{5} + 174 \beta_{4}) q^{77} + (60 \beta_{5} + 416 \beta_{4} - 4 \beta_{3}) q^{78} + ( - 267 \beta_{2} + 180 \beta_1 + 138) q^{79} + (20 \beta_{2} - 94 \beta_1 - 137) q^{81} + ( - 166 \beta_{5} - 64 \beta_{4} + 74 \beta_{3}) q^{82} + ( - 128 \beta_{5} - 756 \beta_{4} + 82 \beta_{3}) q^{83} + ( - 120 \beta_{2} + 456 \beta_1 + 528) q^{84} + ( - 256 \beta_{2} + 32 \beta_1 - 432) q^{86} + ( - 372 \beta_{5} - 352 \beta_{4} - 46 \beta_{3}) q^{87} + ( - 178 \beta_{5} + 672 \beta_{4} - 426 \beta_{3}) q^{88} + ( - 276 \beta_{2} + 110 \beta_1 + 20) q^{89} + (64 \beta_{2} - 44 \beta_1 - 324) q^{91} + (144 \beta_{5} - 1680 \beta_{4} + 344 \beta_{3}) q^{92} + ( - 22 \beta_{5} + 218 \beta_{4} + 152 \beta_{3}) q^{93} + (368 \beta_{2} - 192 \beta_1 + 736) q^{94} + ( - 238 \beta_{2} + 198 \beta_1 + 160) q^{96} + (120 \beta_{5} + 50 \beta_{4} - 140 \beta_{3}) q^{97} + (121 \beta_{5} + 912 \beta_{4} - 255 \beta_{3}) q^{98} + (281 \beta_{2} - 206 \beta_1 + 1042) q^{99}+O(q^{100})$$ q + (-b5 - b3) * q^2 + (-2*b5 + 2*b4 - b3) * q^3 + (-3*b2 - b1 - 8) * q^4 + (-6*b2 - 2*b1 - 24) * q^6 + (-4*b5 - 6*b4 + b3) * q^7 + (9*b5 - 16*b4 + 5*b3) * q^8 + (-8*b2 - 2*b1 - 19) * q^9 + (-11*b2 + 2*b1 - 10) * q^11 + (26*b5 - 16*b4 + 26*b3) * q^12 + (-6*b5 + 12*b4 + 8*b3) * q^13 + (4*b2 - 20*b1 - 24) * q^14 + (11*b2 - 7*b1 + 48) * q^16 + 17*b4 * q^17 + (41*b5 - 48*b4 + 33*b3) * q^18 + (22*b2 - 8*b1 - 24) * q^19 + (12*b2 - 30*b1 - 54) * q^21 + (26*b5 - 104*b4 + 34*b3) * q^22 + (4*b5 + 46*b4 - 39*b3) * q^23 + (46*b2 - 6*b1 + 224) * q^24 + (-2*b2 - 22*b1 + 16) * q^26 + (8*b5 + 4*b4 + 40*b3) * q^27 + (44*b5 + 144*b4 + 4*b3) * q^28 + (-16*b2 + 30*b1 + 142) * q^29 + (-39*b2 - 16*b1 + 82) * q^31 + (23*b5 + 16*b4 - 37*b3) * q^32 + (34*b5 - 122*b4 + 76*b3) * q^33 + (-17*b2 + 17*b1) * q^34 + (91*b2 - 7*b1 + 440) * q^36 + (-50*b5 - 102*b4 + 28*b3) * q^37 + (4*b5 + 240*b4 - 28*b3) * q^38 + (36*b2 - 16*b1 - 84) * q^39 + (52*b2 + 60*b1 - 118) * q^41 + (120*b5 + 336*b4) * q^42 + (-56*b5 + 204*b4 + 2*b3) * q^43 + (110*b2 - 78*b1 + 400) * q^44 + (-120*b2 + 136*b1 - 280) * q^46 + (44*b5 - 228*b4 + 48*b3) * q^47 + (-90*b5 + 288*b4 - 114*b3) * q^48 + (20*b2 - 94*b1 + 121) * q^49 + (-17*b2 + 34*b1 - 34) * q^51 + (6*b5 + 256*b4 + 30*b3) * q^52 + (8*b5 + 98*b4 + 116*b3) * q^53 + (84*b2 - 52*b1 + 384) * q^54 + (-60*b2 + 108*b1 + 192) * q^56 + (36*b5 + 228*b4 - 108*b3) * q^57 + (-200*b5 - 368*b4 - 80*b3) * q^58 + (-130*b2 - 212) * q^59 + (64*b2 - 78*b1 - 2) * q^61 + (44*b5 - 184*b4 - 20*b3) * q^62 + (96*b5 + 342*b4 + 9*b3) * q^63 + (21*b2 + 103*b1 + 272) * q^64 + (308*b2 - 172*b1 + 880) * q^66 + (-132*b5 - 292*b4 - 24*b3) * q^67 + (-17*b5 - 136*b4 + 51*b3) * q^68 + (-280*b2 + 186*b1 - 254) * q^69 + (-185*b2 - 72*b1 - 110) * q^71 + (-273*b5 + 400*b4 - 365*b3) * q^72 + (16*b5 + 274*b4 + 16*b3) * q^73 + (108*b2 - 308*b1 - 176) * q^74 + (-116*b2 + 244*b1 - 384) * q^76 + (-18*b5 + 174*b4) * q^77 + (60*b5 + 416*b4 - 4*b3) * q^78 + (-267*b2 + 180*b1 + 138) * q^79 + (20*b2 - 94*b1 - 137) * q^81 + (-166*b5 - 64*b4 + 74*b3) * q^82 + (-128*b5 - 756*b4 + 82*b3) * q^83 + (-120*b2 + 456*b1 + 528) * q^84 + (-256*b2 + 32*b1 - 432) * q^86 + (-372*b5 - 352*b4 - 46*b3) * q^87 + (-178*b5 + 672*b4 - 426*b3) * q^88 + (-276*b2 + 110*b1 + 20) * q^89 + (64*b2 - 44*b1 - 324) * q^91 + (144*b5 - 1680*b4 + 344*b3) * q^92 + (-22*b5 + 218*b4 + 152*b3) * q^93 + (368*b2 - 192*b1 + 736) * q^94 + (-238*b2 + 198*b1 + 160) * q^96 + (120*b5 + 50*b4 - 140*b3) * q^97 + (121*b5 + 912*b4 - 255*b3) * q^98 + (281*b2 - 206*b1 + 1042) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 50 q^{4} - 148 q^{6} - 118 q^{9}+O(q^{10})$$ 6 * q - 50 * q^4 - 148 * q^6 - 118 * q^9 $$6 q - 50 q^{4} - 148 q^{6} - 118 q^{9} - 56 q^{11} - 184 q^{14} + 274 q^{16} - 160 q^{19} - 384 q^{21} + 1332 q^{24} + 52 q^{26} + 912 q^{29} + 460 q^{31} + 34 q^{34} + 2626 q^{36} - 536 q^{39} - 588 q^{41} + 2244 q^{44} - 1408 q^{46} + 538 q^{49} - 136 q^{51} + 2200 q^{54} + 1368 q^{56} - 1272 q^{59} - 168 q^{61} + 1838 q^{64} + 4936 q^{66} - 1152 q^{69} - 804 q^{71} - 1672 q^{74} - 1816 q^{76} + 1188 q^{79} - 1010 q^{81} + 4080 q^{84} - 2528 q^{86} + 340 q^{89} - 2032 q^{91} + 4032 q^{94} + 1356 q^{96} + 5840 q^{99}+O(q^{100})$$ 6 * q - 50 * q^4 - 148 * q^6 - 118 * q^9 - 56 * q^11 - 184 * q^14 + 274 * q^16 - 160 * q^19 - 384 * q^21 + 1332 * q^24 + 52 * q^26 + 912 * q^29 + 460 * q^31 + 34 * q^34 + 2626 * q^36 - 536 * q^39 - 588 * q^41 + 2244 * q^44 - 1408 * q^46 + 538 * q^49 - 136 * q^51 + 2200 * q^54 + 1368 * q^56 - 1272 * q^59 - 168 * q^61 + 1838 * q^64 + 4936 * q^66 - 1152 * q^69 - 804 * q^71 - 1672 * q^74 - 1816 * q^76 + 1188 * q^79 - 1010 * q^81 + 4080 * q^84 - 2528 * q^86 + 340 * q^89 - 2032 * q^91 + 4032 * q^94 + 1356 * q^96 + 5840 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{3} + 49x^{2} - 14x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 49\nu^{4} + 7\nu^{3} - \nu^{2} + 1696 ) / 342$$ (v^5 + 49*v^4 + 7*v^3 - v^2 + 1696) / 342 $$\beta_{2}$$ $$=$$ $$( -7\nu^{5} - \nu^{4} - 49\nu^{3} + 7\nu^{2} + 98 ) / 342$$ (-7*v^5 - v^4 - 49*v^3 + 7*v^2 + 98) / 342 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} + \nu^{4} + 49\nu^{3} - 7\nu^{2} + 684\nu - 98 ) / 342$$ (7*v^5 + v^4 + 49*v^3 - 7*v^2 + 684*v - 98) / 342 $$\beta_{4}$$ $$=$$ $$( -49\nu^{5} - 7\nu^{4} - \nu^{3} + 49\nu^{2} - 2394\nu + 344 ) / 342$$ (-49*v^5 - 7*v^4 - v^3 + 49*v^2 - 2394*v + 344) / 342 $$\beta_{5}$$ $$=$$ $$( -245\nu^{5} - 35\nu^{4} - 5\nu^{3} + 587\nu^{2} - 11970\nu + 1720 ) / 342$$ (-245*v^5 - 35*v^4 - 5*v^3 + 587*v^2 - 11970*v + 1720) / 342
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 5\beta_{4}$$ b5 - 5*b4 $$\nu^{3}$$ $$=$$ $$( 2\beta_{4} + 7\beta_{3} - 7\beta_{2} + 2 ) / 2$$ (2*b4 + 7*b3 - 7*b2 + 2) / 2 $$\nu^{4}$$ $$=$$ $$\beta_{2} + 7\beta _1 - 35$$ b2 + 7*b1 - 35 $$\nu^{5}$$ $$=$$ $$( 2\beta_{5} - 24\beta_{4} - 49\beta_{3} - 49\beta_{2} - 2\beta _1 + 24 ) / 2$$ (2*b5 - 24*b4 - 49*b3 - 49*b2 - 2*b1 + 24) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$$.

 $$n$$ $$52$$ $$326$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 1.79483 + 1.79483i 0.143705 − 0.143705i −1.93854 + 1.93854i −1.93854 − 1.93854i 0.143705 + 0.143705i 1.79483 − 1.79483i
5.03251i 8.47535i −17.3261 0 −42.6523 3.81828i 46.9339i −44.8316 0
324.2 4.67129i 7.62999i −13.8209 0 −35.6419 26.1222i 27.1912i −31.2167 0
324.3 1.36122i 3.15463i 6.14708 0 4.29415 7.94049i 19.2573i 17.0483 0
324.4 1.36122i 3.15463i 6.14708 0 4.29415 7.94049i 19.2573i 17.0483 0
324.5 4.67129i 7.62999i −13.8209 0 −35.6419 26.1222i 27.1912i −31.2167 0
324.6 5.03251i 8.47535i −17.3261 0 −42.6523 3.81828i 46.9339i −44.8316 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 324.6 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 425.4.b.f 6
5.b even 2 1 inner 425.4.b.f 6
5.c odd 4 1 17.4.a.b 3
5.c odd 4 1 425.4.a.g 3
15.e even 4 1 153.4.a.g 3
20.e even 4 1 272.4.a.h 3
35.f even 4 1 833.4.a.d 3
40.i odd 4 1 1088.4.a.v 3
40.k even 4 1 1088.4.a.x 3
55.e even 4 1 2057.4.a.e 3
60.l odd 4 1 2448.4.a.bi 3
85.f odd 4 1 289.4.b.b 6
85.g odd 4 1 289.4.a.b 3
85.i odd 4 1 289.4.b.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.b 3 5.c odd 4 1
153.4.a.g 3 15.e even 4 1
272.4.a.h 3 20.e even 4 1
289.4.a.b 3 85.g odd 4 1
289.4.b.b 6 85.f odd 4 1
289.4.b.b 6 85.i odd 4 1
425.4.a.g 3 5.c odd 4 1
425.4.b.f 6 1.a even 1 1 trivial
425.4.b.f 6 5.b even 2 1 inner
833.4.a.d 3 35.f even 4 1
1088.4.a.v 3 40.i odd 4 1
1088.4.a.x 3 40.k even 4 1
2057.4.a.e 3 55.e even 4 1
2448.4.a.bi 3 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 $$S_{4}^{\mathrm{new}}(425, [\chi])$$:

 $$T_{2}^{6} + 49T_{2}^{4} + 640T_{2}^{2} + 1024$$ T2^6 + 49*T2^4 + 640*T2^2 + 1024 $$T_{3}^{6} + 140T_{3}^{4} + 5476T_{3}^{2} + 41616$$ T3^6 + 140*T3^4 + 5476*T3^2 + 41616

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 49 T^{4} + 640 T^{2} + \cdots + 1024$$
$3$ $$T^{6} + 140 T^{4} + 5476 T^{2} + \cdots + 41616$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 760 T^{4} + 53892 T^{2} + \cdots + 627264$$
$11$ $$(T^{3} + 28 T^{2} - 1366 T - 4692)^{2}$$
$13$ $$T^{6} + 3844 T^{4} + \cdots + 88209664$$
$17$ $$(T^{2} + 289)^{3}$$
$19$ $$(T^{3} + 80 T^{2} - 4632 T - 340128)^{2}$$
$23$ $$T^{6} + 51704 T^{4} + \cdots + 2561741095936$$
$29$ $$(T^{3} - 456 T^{2} + 53908 T - 1518624)^{2}$$
$31$ $$(T^{3} - 230 T^{2} - 11586 T - 81608)^{2}$$
$37$ $$T^{6} + 162664 T^{4} + \cdots + 38152265269504$$
$41$ $$(T^{3} + 294 T^{2} - 86564 T - 1638744)^{2}$$
$43$ $$T^{6} + 206944 T^{4} + \cdots + 52856854953984$$
$47$ $$T^{6} + 238944 T^{4} + \cdots + 2792802484224$$
$53$ $$T^{6} + \cdots + 329860859333184$$
$59$ $$(T^{3} + 636 T^{2} - 101768 T - 49419072)^{2}$$
$61$ $$(T^{3} + 84 T^{2} - 124412 T - 6792784)^{2}$$
$67$ $$T^{6} + 885984 T^{4} + \cdots + 586682466304$$
$71$ $$(T^{3} + 402 T^{2} - 589874 T - 274866016)^{2}$$
$73$ $$T^{6} + \cdots + 398302285230144$$
$79$ $$(T^{3} - 594 T^{2} - 1121274 T + 742135824)^{2}$$
$83$ $$T^{6} + 2763040 T^{4} + \cdots + 20\!\cdots\!16$$
$89$ $$(T^{3} - 170 T^{2} - 1072304 T + 446571376)^{2}$$
$97$ $$T^{6} + 1245100 T^{4} + \cdots + 42\!\cdots\!00$$