# Properties

 Label 575.6.a.b Level $575$ Weight $6$ Character orbit 575.a Self dual yes Analytic conductor $92.221$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$92.2206963925$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.7925.1 Defining polynomial: $$x^{3} - x^{2} - 13x + 12$$ x^3 - x^2 - 13*x + 12 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 23) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} + 2 \beta_1 + 7) q^{3} + (4 \beta_{2} - 6 \beta_1 + 2) q^{4} + ( - 10 \beta_{2} + \beta_1 - 73) q^{6} + (4 \beta_{2} - 17 \beta_1 + 87) q^{7} + (16 \beta_{2} - 8 \beta_1 + 112) q^{8} + (41 \beta_{2} + 25 \beta_1 + 35) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b2 + 2*b1 + 7) * q^3 + (4*b2 - 6*b1 + 2) * q^4 + (-10*b2 + b1 - 73) * q^6 + (4*b2 - 17*b1 + 87) * q^7 + (16*b2 - 8*b1 + 112) * q^8 + (41*b2 + 25*b1 + 35) * q^9 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} + 2 \beta_1 + 7) q^{3} + (4 \beta_{2} - 6 \beta_1 + 2) q^{4} + ( - 10 \beta_{2} + \beta_1 - 73) q^{6} + (4 \beta_{2} - 17 \beta_1 + 87) q^{7} + (16 \beta_{2} - 8 \beta_1 + 112) q^{8} + (41 \beta_{2} + 25 \beta_1 + 35) q^{9} + (28 \beta_{2} + 3 \beta_1 + 37) q^{11} + ( - 16 \beta_{2} + 34 \beta_1 - 190) q^{12} + (99 \beta_{2} - 45 \beta_1 + 324) q^{13} + (60 \beta_{2} - 180 \beta_1 + 592) q^{14} + ( - 128 \beta_{2} + 8 \beta_1 + 88) q^{16} + (146 \beta_{2} + 57 \beta_1 + 269) q^{17} + ( - 182 \beta_{2} + 8 \beta_1 - 1364) q^{18} + (200 \beta_{2} + 40 \beta_1 + 498) q^{19} + ( - 52 \beta_{2} + 193 \beta_1 - 475) q^{21} + ( - 68 \beta_{2} - 78 \beta_1 - 454) q^{22} + 529 q^{23} + (216 \beta_{2} + 360 \beta_1 + 1248) q^{24} + ( - 18 \beta_{2} - 747 \beta_1 + 423) q^{26} + (559 \beta_{2} - 22 \beta_1 + 3373) q^{27} + (472 \beta_{2} - 1068 \beta_1 + 2908) q^{28} + (523 \beta_{2} - 745 \beta_1 - 704) q^{29} + (479 \beta_{2} - 914 \beta_1 - 1471) q^{31} + ( - 288 \beta_{2} + 464 \beta_1 - 1968) q^{32} + (406 \beta_{2} + 329 \beta_1 + 2431) q^{33} + ( - 520 \beta_{2} - 276 \beta_1 - 3656) q^{34} + ( - 980 \beta_{2} + 968 \beta_1 - 200) q^{36} + ( - 1662 \beta_{2} + 1117 \beta_1 - 2053) q^{37} + ( - 560 \beta_{2} - 698 \beta_1 - 3622) q^{38} + (1017 \beta_{2} + 1494 \beta_1 + 5499) q^{39} + ( - 1847 \beta_{2} + 1171 \beta_1 - 3258) q^{41} + ( - 668 \beta_{2} + 1544 \beta_1 - 6116) q^{42} + (1150 \beta_{2} - 2050 \beta_1 + 4510) q^{43} + ( - 448 \beta_{2} + 104 \beta_1 + 1888) q^{44} + ( - 529 \beta_1 + 529) q^{46} + (1021 \beta_{2} + 682 \beta_1 - 7613) q^{47} + ( - 1360 \beta_{2} - 968 \beta_1 - 7576) q^{48} + (1428 \beta_{2} - 4306 \beta_1 - 949) q^{49} + (2648 \beta_{2} + 1909 \beta_1 + 16517) q^{51} + ( - 144 \beta_{2} - 2682 \beta_1 + 14958) q^{52} + ( - 534 \beta_{2} - 4444 \beta_1 - 7006) q^{53} + ( - 1030 \beta_{2} - 4601 \beta_1 - 3727) q^{54} + (1408 \beta_{2} - 3432 \beta_1 + 12600) q^{56} + (3338 \beta_{2} + 2836 \beta_1 + 20486) q^{57} + (1934 \beta_{2} - 4067 \beta_1 + 16559) q^{58} + (1980 \beta_{2} + 934 \beta_1 + 17710) q^{59} + ( - 42 \beta_{2} + 1322 \beta_1 - 23320) q^{61} + (2698 \beta_{2} - 4057 \beta_1 + 21985) q^{62} + (52 \beta_{2} + 2906 \beta_1 - 12614) q^{63} + (2816 \beta_{2} + 4608 \beta_1 - 16064) q^{64} + ( - 2128 \beta_{2} - 1598 \beta_1 - 14110) q^{66} + (1436 \beta_{2} + 4323 \beta_1 + 21949) q^{67} + ( - 2528 \beta_{2} + 1492 \beta_1 + 4124) q^{68} + (529 \beta_{2} + 1058 \beta_1 + 3703) q^{69} + (2227 \beta_{2} - 1628 \beta_1 + 31515) q^{71} + (3912 \beta_{2} + 6744 \beta_1 + 25224) q^{72} + (21 \beta_{2} - 3241 \beta_1 + 26170) q^{73} + ( - 1144 \beta_{2} + 10962 \beta_1 - 15646) q^{74} + ( - 2488 \beta_{2} - 28 \beta_1 + 11316) q^{76} + (876 \beta_{2} - 532 \beta_1 - 368) q^{77} + ( - 8010 \beta_{2} - 63 \beta_1 - 58041) q^{78} + ( - 6662 \beta_{2} - 4778 \beta_1 + 20036) q^{79} + ( - 124 \beta_{2} + 5680 \beta_1 + 51917) q^{81} + ( - 990 \beta_{2} + 12807 \beta_1 - 16043) q^{82} + (13758 \beta_{2} - 5867 \beta_1 - 9839) q^{83} + ( - 3176 \beta_{2} + 8996 \beta_1 - 32516) q^{84} + (5900 \beta_{2} - 17060 \beta_1 + 56060) q^{86} + ( - 2623 \beta_{2} + 2554 \beta_1 - 28441) q^{87} + (2656 \beta_{2} + 2024 \beta_1 + 19256) q^{88} + (8666 \beta_{2} - 5964 \beta_1 + 4304) q^{89} + (6984 \beta_{2} - 14229 \beta_1 + 43587) q^{91} + (2116 \beta_{2} - 3174 \beta_1 + 1058) q^{92} + ( - 5777 \beta_{2} + 455 \beta_1 - 50366) q^{93} + ( - 4770 \beta_{2} + 8981 \beta_1 - 44413) q^{94} + ( - 320 \beta_{2} - 6064 \beta_1 + 3472) q^{96} + ( - 618 \beta_{2} - 3351 \beta_1 + 90313) q^{97} + (14368 \beta_{2} - 23437 \beta_1 + 121157) q^{98} + (4118 \beta_{2} + 8116 \beta_1 + 62360) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b2 + 2*b1 + 7) * q^3 + (4*b2 - 6*b1 + 2) * q^4 + (-10*b2 + b1 - 73) * q^6 + (4*b2 - 17*b1 + 87) * q^7 + (16*b2 - 8*b1 + 112) * q^8 + (41*b2 + 25*b1 + 35) * q^9 + (28*b2 + 3*b1 + 37) * q^11 + (-16*b2 + 34*b1 - 190) * q^12 + (99*b2 - 45*b1 + 324) * q^13 + (60*b2 - 180*b1 + 592) * q^14 + (-128*b2 + 8*b1 + 88) * q^16 + (146*b2 + 57*b1 + 269) * q^17 + (-182*b2 + 8*b1 - 1364) * q^18 + (200*b2 + 40*b1 + 498) * q^19 + (-52*b2 + 193*b1 - 475) * q^21 + (-68*b2 - 78*b1 - 454) * q^22 + 529 * q^23 + (216*b2 + 360*b1 + 1248) * q^24 + (-18*b2 - 747*b1 + 423) * q^26 + (559*b2 - 22*b1 + 3373) * q^27 + (472*b2 - 1068*b1 + 2908) * q^28 + (523*b2 - 745*b1 - 704) * q^29 + (479*b2 - 914*b1 - 1471) * q^31 + (-288*b2 + 464*b1 - 1968) * q^32 + (406*b2 + 329*b1 + 2431) * q^33 + (-520*b2 - 276*b1 - 3656) * q^34 + (-980*b2 + 968*b1 - 200) * q^36 + (-1662*b2 + 1117*b1 - 2053) * q^37 + (-560*b2 - 698*b1 - 3622) * q^38 + (1017*b2 + 1494*b1 + 5499) * q^39 + (-1847*b2 + 1171*b1 - 3258) * q^41 + (-668*b2 + 1544*b1 - 6116) * q^42 + (1150*b2 - 2050*b1 + 4510) * q^43 + (-448*b2 + 104*b1 + 1888) * q^44 + (-529*b1 + 529) * q^46 + (1021*b2 + 682*b1 - 7613) * q^47 + (-1360*b2 - 968*b1 - 7576) * q^48 + (1428*b2 - 4306*b1 - 949) * q^49 + (2648*b2 + 1909*b1 + 16517) * q^51 + (-144*b2 - 2682*b1 + 14958) * q^52 + (-534*b2 - 4444*b1 - 7006) * q^53 + (-1030*b2 - 4601*b1 - 3727) * q^54 + (1408*b2 - 3432*b1 + 12600) * q^56 + (3338*b2 + 2836*b1 + 20486) * q^57 + (1934*b2 - 4067*b1 + 16559) * q^58 + (1980*b2 + 934*b1 + 17710) * q^59 + (-42*b2 + 1322*b1 - 23320) * q^61 + (2698*b2 - 4057*b1 + 21985) * q^62 + (52*b2 + 2906*b1 - 12614) * q^63 + (2816*b2 + 4608*b1 - 16064) * q^64 + (-2128*b2 - 1598*b1 - 14110) * q^66 + (1436*b2 + 4323*b1 + 21949) * q^67 + (-2528*b2 + 1492*b1 + 4124) * q^68 + (529*b2 + 1058*b1 + 3703) * q^69 + (2227*b2 - 1628*b1 + 31515) * q^71 + (3912*b2 + 6744*b1 + 25224) * q^72 + (21*b2 - 3241*b1 + 26170) * q^73 + (-1144*b2 + 10962*b1 - 15646) * q^74 + (-2488*b2 - 28*b1 + 11316) * q^76 + (876*b2 - 532*b1 - 368) * q^77 + (-8010*b2 - 63*b1 - 58041) * q^78 + (-6662*b2 - 4778*b1 + 20036) * q^79 + (-124*b2 + 5680*b1 + 51917) * q^81 + (-990*b2 + 12807*b1 - 16043) * q^82 + (13758*b2 - 5867*b1 - 9839) * q^83 + (-3176*b2 + 8996*b1 - 32516) * q^84 + (5900*b2 - 17060*b1 + 56060) * q^86 + (-2623*b2 + 2554*b1 - 28441) * q^87 + (2656*b2 + 2024*b1 + 19256) * q^88 + (8666*b2 - 5964*b1 + 4304) * q^89 + (6984*b2 - 14229*b1 + 43587) * q^91 + (2116*b2 - 3174*b1 + 1058) * q^92 + (-5777*b2 + 455*b1 - 50366) * q^93 + (-4770*b2 + 8981*b1 - 44413) * q^94 + (-320*b2 - 6064*b1 + 3472) * q^96 + (-618*b2 - 3351*b1 + 90313) * q^97 + (14368*b2 - 23437*b1 + 121157) * q^98 + (4118*b2 + 8116*b1 + 62360) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 4 q^{2} + 20 q^{3} + 16 q^{4} - 230 q^{6} + 282 q^{7} + 360 q^{8} + 121 q^{9}+O(q^{10})$$ 3 * q + 4 * q^2 + 20 * q^3 + 16 * q^4 - 230 * q^6 + 282 * q^7 + 360 * q^8 + 121 * q^9 $$3 q + 4 q^{2} + 20 q^{3} + 16 q^{4} - 230 q^{6} + 282 q^{7} + 360 q^{8} + 121 q^{9} + 136 q^{11} - 620 q^{12} + 1116 q^{13} + 2016 q^{14} + 128 q^{16} + 896 q^{17} - 4282 q^{18} + 1654 q^{19} - 1670 q^{21} - 1352 q^{22} + 1587 q^{23} + 3600 q^{24} + 1998 q^{26} + 10700 q^{27} + 10264 q^{28} - 844 q^{29} - 3020 q^{31} - 6656 q^{32} + 7370 q^{33} - 11212 q^{34} - 2548 q^{36} - 8938 q^{37} - 10728 q^{38} + 16020 q^{39} - 12792 q^{41} - 20560 q^{42} + 16730 q^{43} + 5112 q^{44} + 2116 q^{46} - 22500 q^{47} - 23120 q^{48} + 2887 q^{49} + 50290 q^{51} + 47412 q^{52} - 17108 q^{53} - 7610 q^{54} + 42640 q^{56} + 61960 q^{57} + 55678 q^{58} + 54176 q^{59} - 71324 q^{61} + 72710 q^{62} - 40696 q^{63} - 49984 q^{64} - 42860 q^{66} + 62960 q^{67} + 8352 q^{68} + 10580 q^{69} + 98400 q^{71} + 72840 q^{72} + 81772 q^{73} - 59044 q^{74} + 31488 q^{76} + 304 q^{77} - 182070 q^{78} + 58224 q^{79} + 149947 q^{81} - 61926 q^{82} - 9892 q^{83} - 109720 q^{84} + 191140 q^{86} - 90500 q^{87} + 58400 q^{88} + 27542 q^{89} + 151974 q^{91} + 8464 q^{92} - 157330 q^{93} - 146990 q^{94} + 16160 q^{96} + 273672 q^{97} + 401276 q^{98} + 183082 q^{99}+O(q^{100})$$ 3 * q + 4 * q^2 + 20 * q^3 + 16 * q^4 - 230 * q^6 + 282 * q^7 + 360 * q^8 + 121 * q^9 + 136 * q^11 - 620 * q^12 + 1116 * q^13 + 2016 * q^14 + 128 * q^16 + 896 * q^17 - 4282 * q^18 + 1654 * q^19 - 1670 * q^21 - 1352 * q^22 + 1587 * q^23 + 3600 * q^24 + 1998 * q^26 + 10700 * q^27 + 10264 * q^28 - 844 * q^29 - 3020 * q^31 - 6656 * q^32 + 7370 * q^33 - 11212 * q^34 - 2548 * q^36 - 8938 * q^37 - 10728 * q^38 + 16020 * q^39 - 12792 * q^41 - 20560 * q^42 + 16730 * q^43 + 5112 * q^44 + 2116 * q^46 - 22500 * q^47 - 23120 * q^48 + 2887 * q^49 + 50290 * q^51 + 47412 * q^52 - 17108 * q^53 - 7610 * q^54 + 42640 * q^56 + 61960 * q^57 + 55678 * q^58 + 54176 * q^59 - 71324 * q^61 + 72710 * q^62 - 40696 * q^63 - 49984 * q^64 - 42860 * q^66 + 62960 * q^67 + 8352 * q^68 + 10580 * q^69 + 98400 * q^71 + 72840 * q^72 + 81772 * q^73 - 59044 * q^74 + 31488 * q^76 + 304 * q^77 - 182070 * q^78 + 58224 * q^79 + 149947 * q^81 - 61926 * q^82 - 9892 * q^83 - 109720 * q^84 + 191140 * q^86 - 90500 * q^87 + 58400 * q^88 + 27542 * q^89 + 151974 * q^91 + 8464 * q^92 - 157330 * q^93 - 146990 * q^94 + 16160 * q^96 + 273672 * q^97 + 401276 * q^98 + 183082 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 13x + 12$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 9$$ v^2 + v - 9
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} - \beta _1 + 17 ) / 2$$ (2*b2 - b1 + 17) / 2

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.65736 0.917748 −3.57511
−5.31473 27.6631 −3.75366 0 −147.022 11.7843 190.021 522.249 0
1.2 0.164504 1.43100 −31.9729 0 0.235406 43.8366 −10.5238 −240.952 0
1.3 9.15022 −9.09413 51.7266 0 −83.2134 226.379 180.503 −160.297 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$23$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.6.a.b 3
5.b even 2 1 23.6.a.a 3
15.d odd 2 1 207.6.a.b 3
20.d odd 2 1 368.6.a.e 3
115.c odd 2 1 529.6.a.a 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.6.a.a 3 5.b even 2 1
207.6.a.b 3 15.d odd 2 1
368.6.a.e 3 20.d odd 2 1
529.6.a.a 3 115.c odd 2 1
575.6.a.b 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 4T_{2}^{2} - 48T_{2} + 8$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(575))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 4 T^{2} - 48 T + 8$$
$3$ $$T^{3} - 20 T^{2} - 225 T + 360$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 282 T^{2} + 13108 T - 116944$$
$11$ $$T^{3} - 136 T^{2} - 43688 T + 840152$$
$13$ $$T^{3} - 1116 T^{2} + \cdots + 255615102$$
$17$ $$T^{3} - 896 T^{2} + \cdots - 220718408$$
$19$ $$T^{3} - 1654 T^{2} + \cdots + 460771768$$
$23$ $$(T - 529)^{3}$$
$29$ $$T^{3} + 844 T^{2} + \cdots - 33789223458$$
$31$ $$T^{3} + 3020 T^{2} + \cdots - 117638912880$$
$37$ $$T^{3} + 8938 T^{2} + \cdots - 1048082031344$$
$41$ $$T^{3} + 12792 T^{2} + \cdots - 1564944049486$$
$43$ $$T^{3} - 16730 T^{2} + \cdots + 95315904000$$
$47$ $$T^{3} + 22500 T^{2} + \cdots - 916008439440$$
$53$ $$T^{3} + 17108 T^{2} + \cdots - 7849670295504$$
$59$ $$T^{3} - 54176 T^{2} + \cdots - 1725012447168$$
$61$ $$T^{3} + 71324 T^{2} + \cdots + 11439907465152$$
$67$ $$T^{3} - 62960 T^{2} + \cdots + 11971711378840$$
$71$ $$T^{3} - 98400 T^{2} + \cdots - 24837760695040$$
$73$ $$T^{3} - 81772 T^{2} + \cdots - 7199078503954$$
$79$ $$T^{3} + \cdots + 235690469012368$$
$83$ $$T^{3} + \cdots + 297029282761704$$
$89$ $$T^{3} + \cdots + 125799322340896$$
$97$ $$T^{3} + \cdots - 693755159518744$$