# Properties

 Label 100.4.e.d Level $100$ Weight $4$ Character orbit 100.e Analytic conductor $5.900$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 100.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.90019100057$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.2342560000.1 Defining polynomial: $$x^{8} - 4x^{7} + 24x^{6} - 58x^{5} + 141x^{4} - 190x^{3} + 186x^{2} - 100x + 20$$ x^8 - 4*x^7 + 24*x^6 - 58*x^5 + 141*x^4 - 190*x^3 + 186*x^2 - 100*x + 20 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + ( - \beta_{6} - \beta_{3}) q^{3} + ( - 3 \beta_{4} - \beta_1) q^{4} + ( - \beta_{7} + 5) q^{6} + (5 \beta_{5} - 5 \beta_{2}) q^{7} + ( - 8 \beta_{6} - 6 \beta_{3}) q^{8} + 17 \beta_{4} q^{9}+O(q^{10})$$ q + b5 * q^2 + (-b6 - b3) * q^3 + (-3*b4 - b1) * q^4 + (-b7 + 5) * q^6 + (5*b5 - 5*b2) * q^7 + (-8*b6 - 6*b3) * q^8 + 17*b4 * q^9 $$q + \beta_{5} q^{2} + ( - \beta_{6} - \beta_{3}) q^{3} + ( - 3 \beta_{4} - \beta_1) q^{4} + ( - \beta_{7} + 5) q^{6} + (5 \beta_{5} - 5 \beta_{2}) q^{7} + ( - 8 \beta_{6} - 6 \beta_{3}) q^{8} + 17 \beta_{4} q^{9} - 8 \beta_{7} q^{11} + (2 \beta_{5} + 8 \beta_{2}) q^{12} + (8 \beta_{6} - 8 \beta_{3}) q^{13} + (25 \beta_{4} - 5 \beta_1) q^{14} + ( - 6 \beta_{7} + 46) q^{16} + ( - 24 \beta_{5} - 24 \beta_{2}) q^{17} + 17 \beta_{3} q^{18} - 8 \beta_1 q^{19} + 50 q^{21} + ( - 24 \beta_{5} + 64 \beta_{2}) q^{22} + (47 \beta_{6} + 47 \beta_{3}) q^{23} + ( - 70 \beta_{4} - 2 \beta_1) q^{24} + ( - 8 \beta_{7} - 88) q^{26} + (44 \beta_{5} - 44 \beta_{2}) q^{27} + ( - 40 \beta_{6} + 10 \beta_{3}) q^{28} - 46 \beta_{4} q^{29} + 16 \beta_{7} q^{31} + (28 \beta_{5} + 48 \beta_{2}) q^{32} + (40 \beta_{6} - 40 \beta_{3}) q^{33} + (264 \beta_{4} + 24 \beta_1) q^{34} + (17 \beta_{7} + 51) q^{36} + ( - 64 \beta_{5} - 64 \beta_{2}) q^{37} + ( - 64 \beta_{6} - 24 \beta_{3}) q^{38} + 16 \beta_1 q^{39} - 188 q^{41} + 50 \beta_{5} q^{42} + ( - 55 \beta_{6} - 55 \beta_{3}) q^{43} + ( - 440 \beta_{4} + 24 \beta_1) q^{44} + (47 \beta_{7} - 235) q^{46} + (87 \beta_{5} - 87 \beta_{2}) q^{47} + ( - 16 \beta_{6} - 76 \beta_{3}) q^{48} - 93 \beta_{4} q^{49} + 48 \beta_{7} q^{51} + ( - 112 \beta_{5} + 64 \beta_{2}) q^{52} + (8 \beta_{6} - 8 \beta_{3}) q^{53} + (220 \beta_{4} - 44 \beta_1) q^{54} + (10 \beta_{7} + 350) q^{56} + (40 \beta_{5} + 40 \beta_{2}) q^{57} - 46 \beta_{3} q^{58} + 56 \beta_1 q^{59} + 72 q^{61} + (48 \beta_{5} - 128 \beta_{2}) q^{62} + (85 \beta_{6} + 85 \beta_{3}) q^{63} + ( - 468 \beta_{4} - 28 \beta_1) q^{64} + ( - 40 \beta_{7} - 440) q^{66} + ( - 267 \beta_{5} + 267 \beta_{2}) q^{67} + (192 \beta_{6} + 336 \beta_{3}) q^{68} + 470 \beta_{4} q^{69} - 96 \beta_{7} q^{71} + (102 \beta_{5} - 136 \beta_{2}) q^{72} + ( - 152 \beta_{6} + 152 \beta_{3}) q^{73} + (704 \beta_{4} + 64 \beta_1) q^{74} + ( - 24 \beta_{7} + 440) q^{76} + (200 \beta_{5} + 200 \beta_{2}) q^{77} + (128 \beta_{6} + 48 \beta_{3}) q^{78} - 112 \beta_1 q^{79} - 19 q^{81} - 188 \beta_{5} q^{82} + ( - 99 \beta_{6} - 99 \beta_{3}) q^{83} + ( - 150 \beta_{4} - 50 \beta_1) q^{84} + ( - 55 \beta_{7} + 275) q^{86} + ( - 46 \beta_{5} + 46 \beta_{2}) q^{87} + (192 \beta_{6} - 368 \beta_{3}) q^{88} - 726 \beta_{4} q^{89} - 80 \beta_{7} q^{91} + ( - 94 \beta_{5} - 376 \beta_{2}) q^{92} + ( - 80 \beta_{6} + 80 \beta_{3}) q^{93} + (435 \beta_{4} - 87 \beta_1) q^{94} + ( - 76 \beta_{7} - 100) q^{96} + (136 \beta_{5} + 136 \beta_{2}) q^{97} - 93 \beta_{3} q^{98} - 136 \beta_1 q^{99}+O(q^{100})$$ q + b5 * q^2 + (-b6 - b3) * q^3 + (-3*b4 - b1) * q^4 + (-b7 + 5) * q^6 + (5*b5 - 5*b2) * q^7 + (-8*b6 - 6*b3) * q^8 + 17*b4 * q^9 - 8*b7 * q^11 + (2*b5 + 8*b2) * q^12 + (8*b6 - 8*b3) * q^13 + (25*b4 - 5*b1) * q^14 + (-6*b7 + 46) * q^16 + (-24*b5 - 24*b2) * q^17 + 17*b3 * q^18 - 8*b1 * q^19 + 50 * q^21 + (-24*b5 + 64*b2) * q^22 + (47*b6 + 47*b3) * q^23 + (-70*b4 - 2*b1) * q^24 + (-8*b7 - 88) * q^26 + (44*b5 - 44*b2) * q^27 + (-40*b6 + 10*b3) * q^28 - 46*b4 * q^29 + 16*b7 * q^31 + (28*b5 + 48*b2) * q^32 + (40*b6 - 40*b3) * q^33 + (264*b4 + 24*b1) * q^34 + (17*b7 + 51) * q^36 + (-64*b5 - 64*b2) * q^37 + (-64*b6 - 24*b3) * q^38 + 16*b1 * q^39 - 188 * q^41 + 50*b5 * q^42 + (-55*b6 - 55*b3) * q^43 + (-440*b4 + 24*b1) * q^44 + (47*b7 - 235) * q^46 + (87*b5 - 87*b2) * q^47 + (-16*b6 - 76*b3) * q^48 - 93*b4 * q^49 + 48*b7 * q^51 + (-112*b5 + 64*b2) * q^52 + (8*b6 - 8*b3) * q^53 + (220*b4 - 44*b1) * q^54 + (10*b7 + 350) * q^56 + (40*b5 + 40*b2) * q^57 - 46*b3 * q^58 + 56*b1 * q^59 + 72 * q^61 + (48*b5 - 128*b2) * q^62 + (85*b6 + 85*b3) * q^63 + (-468*b4 - 28*b1) * q^64 + (-40*b7 - 440) * q^66 + (-267*b5 + 267*b2) * q^67 + (192*b6 + 336*b3) * q^68 + 470*b4 * q^69 - 96*b7 * q^71 + (102*b5 - 136*b2) * q^72 + (-152*b6 + 152*b3) * q^73 + (704*b4 + 64*b1) * q^74 + (-24*b7 + 440) * q^76 + (200*b5 + 200*b2) * q^77 + (128*b6 + 48*b3) * q^78 - 112*b1 * q^79 - 19 * q^81 - 188*b5 * q^82 + (-99*b6 - 99*b3) * q^83 + (-150*b4 - 50*b1) * q^84 + (-55*b7 + 275) * q^86 + (-46*b5 + 46*b2) * q^87 + (192*b6 - 368*b3) * q^88 - 726*b4 * q^89 - 80*b7 * q^91 + (-94*b5 - 376*b2) * q^92 + (-80*b6 + 80*b3) * q^93 + (435*b4 - 87*b1) * q^94 + (-76*b7 - 100) * q^96 + (136*b5 + 136*b2) * q^97 - 93*b3 * q^98 - 136*b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 40 q^{6}+O(q^{10})$$ 8 * q + 40 * q^6 $$8 q + 40 q^{6} + 368 q^{16} + 400 q^{21} - 704 q^{26} + 408 q^{36} - 1504 q^{41} - 1880 q^{46} + 2800 q^{56} + 576 q^{61} - 3520 q^{66} + 3520 q^{76} - 152 q^{81} + 2200 q^{86} - 800 q^{96}+O(q^{100})$$ 8 * q + 40 * q^6 + 368 * q^16 + 400 * q^21 - 704 * q^26 + 408 * q^36 - 1504 * q^41 - 1880 * q^46 + 2800 * q^56 + 576 * q^61 - 3520 * q^66 + 3520 * q^76 - 152 * q^81 + 2200 * q^86 - 800 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 24x^{6} - 58x^{5} + 141x^{4} - 190x^{3} + 186x^{2} - 100x + 20$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{6} + 3\nu^{5} - 17\nu^{4} + 29\nu^{3} - 34\nu^{2} + 20\nu + 70 ) / 10$$ (-v^6 + 3*v^5 - 17*v^4 + 29*v^3 - 34*v^2 + 20*v + 70) / 10 $$\beta_{2}$$ $$=$$ $$( 33\nu^{7} - 98\nu^{6} + 678\nu^{5} - 1190\nu^{4} + 3213\nu^{3} - 2756\nu^{2} + 2760\nu - 620 ) / 150$$ (33*v^7 - 98*v^6 + 678*v^5 - 1190*v^4 + 3213*v^3 - 2756*v^2 + 2760*v - 620) / 150 $$\beta_{3}$$ $$=$$ $$( -33\nu^{7} + 133\nu^{6} - 783\nu^{5} + 1885\nu^{4} - 4428\nu^{3} + 5746\nu^{2} - 5160\nu + 2020 ) / 150$$ (-33*v^7 + 133*v^6 - 783*v^5 + 1885*v^4 - 4428*v^3 + 5746*v^2 - 5160*v + 2020) / 150 $$\beta_{4}$$ $$=$$ $$( -22\nu^{7} + 77\nu^{6} - 487\nu^{5} + 1025\nu^{4} - 2547\nu^{3} + 2834\nu^{2} - 2540\nu + 830 ) / 50$$ (-22*v^7 + 77*v^6 - 487*v^5 + 1025*v^4 - 2547*v^3 + 2834*v^2 - 2540*v + 830) / 50 $$\beta_{5}$$ $$=$$ $$( 163\nu^{7} - 563\nu^{6} + 3613\nu^{5} - 7535\nu^{4} + 19108\nu^{3} - 21206\nu^{2} + 19960\nu - 6620 ) / 150$$ (163*v^7 - 563*v^6 + 3613*v^5 - 7535*v^4 + 19108*v^3 - 21206*v^2 + 19960*v - 6620) / 150 $$\beta_{6}$$ $$=$$ $$( 163\nu^{7} - 578\nu^{6} + 3658\nu^{5} - 7790\nu^{4} + 19543\nu^{3} - 22016\nu^{2} + 20560\nu - 6920 ) / 150$$ (163*v^7 - 578*v^6 + 3658*v^5 - 7790*v^4 + 19543*v^3 - 22016*v^2 + 20560*v - 6920) / 150 $$\beta_{7}$$ $$=$$ $$( 16\nu^{7} - 56\nu^{6} + 356\nu^{5} - 750\nu^{4} + 1876\nu^{3} - 2092\nu^{2} + 1880\nu - 615 ) / 5$$ (16*v^7 - 56*v^6 + 356*v^5 - 750*v^4 + 1876*v^3 - 2092*v^2 + 1880*v - 615) / 5
 $$\nu$$ $$=$$ $$( \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2$$ (b4 - b3 + b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 - 8 ) / 2$$ (-b6 + b5 + b4 - b3 + b2 + b1 - 8) / 2 $$\nu^{3}$$ $$=$$ $$( -3\beta_{7} + 6\beta_{5} - 23\beta_{4} + 16\beta_{3} - 16\beta_{2} + 3\beta _1 - 25 ) / 4$$ (-3*b7 + 6*b5 - 23*b4 + 16*b3 - 16*b2 + 3*b1 - 25) / 4 $$\nu^{4}$$ $$=$$ $$( -3\beta_{7} + 18\beta_{6} - 12\beta_{5} - 24\beta_{4} + 20\beta_{3} - 14\beta_{2} - 8\beta _1 + 59 ) / 2$$ (-3*b7 + 18*b6 - 12*b5 - 24*b4 + 20*b3 - 14*b2 - 8*b1 + 59) / 2 $$\nu^{5}$$ $$=$$ $$( 35\beta_{7} + 56\beta_{6} - 104\beta_{5} + 257\beta_{4} - 106\beta_{3} + 136\beta_{2} - 45\beta _1 + 337 ) / 4$$ (35*b7 + 56*b6 - 104*b5 + 257*b4 - 106*b3 + 136*b2 - 45*b1 + 337) / 4 $$\nu^{6}$$ $$=$$ $$( 60\beta_{7} - 188\beta_{6} + 101\beta_{5} + 446\beta_{4} - 253\beta_{3} + 196\beta_{2} + 58\beta _1 - 428 ) / 2$$ (60*b7 - 188*b6 + 101*b5 + 446*b4 - 253*b3 + 196*b2 + 58*b1 - 428) / 2 $$\nu^{7}$$ $$=$$ $$( - 287 \beta_{7} - 1136 \beta_{6} + 1454 \beta_{5} - 2123 \beta_{4} + 560 \beta_{3} - 1064 \beta_{2} + 567 \beta _1 - 4205 ) / 4$$ (-287*b7 - 1136*b6 + 1454*b5 - 2123*b4 + 560*b3 - 1064*b2 + 567*b1 - 4205) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$77$$ $$\chi(n)$$ $$-1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 − 3.27635i 0.5 − 1.04028i 0.5 + 0.0402784i 0.5 + 2.27635i 0.5 + 3.27635i 0.5 + 1.04028i 0.5 − 0.0402784i 0.5 − 2.27635i
−2.77635 0.540278i −2.23607 2.23607i 7.41620 + 3.00000i 0 5.00000 + 7.41620i −11.1803 + 11.1803i −18.9691 12.3359i 17.0000i 0
7.2 −0.540278 2.77635i 2.23607 + 2.23607i −7.41620 + 3.00000i 0 5.00000 7.41620i 11.1803 11.1803i 12.3359 + 18.9691i 17.0000i 0
7.3 0.540278 + 2.77635i −2.23607 2.23607i −7.41620 + 3.00000i 0 5.00000 7.41620i −11.1803 + 11.1803i −12.3359 18.9691i 17.0000i 0
7.4 2.77635 + 0.540278i 2.23607 + 2.23607i 7.41620 + 3.00000i 0 5.00000 + 7.41620i 11.1803 11.1803i 18.9691 + 12.3359i 17.0000i 0
43.1 −2.77635 + 0.540278i −2.23607 + 2.23607i 7.41620 3.00000i 0 5.00000 7.41620i −11.1803 11.1803i −18.9691 + 12.3359i 17.0000i 0
43.2 −0.540278 + 2.77635i 2.23607 2.23607i −7.41620 3.00000i 0 5.00000 + 7.41620i 11.1803 + 11.1803i 12.3359 18.9691i 17.0000i 0
43.3 0.540278 2.77635i −2.23607 + 2.23607i −7.41620 3.00000i 0 5.00000 + 7.41620i −11.1803 11.1803i −12.3359 + 18.9691i 17.0000i 0
43.4 2.77635 0.540278i 2.23607 2.23607i 7.41620 3.00000i 0 5.00000 7.41620i 11.1803 + 11.1803i 18.9691 12.3359i 17.0000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.4.e.d 8
4.b odd 2 1 inner 100.4.e.d 8
5.b even 2 1 inner 100.4.e.d 8
5.c odd 4 2 inner 100.4.e.d 8
20.d odd 2 1 inner 100.4.e.d 8
20.e even 4 2 inner 100.4.e.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.e.d 8 1.a even 1 1 trivial
100.4.e.d 8 4.b odd 2 1 inner
100.4.e.d 8 5.b even 2 1 inner
100.4.e.d 8 5.c odd 4 2 inner
100.4.e.d 8 20.d odd 2 1 inner
100.4.e.d 8 20.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 100$$ acting on $$S_{4}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 92T^{4} + 4096$$
$3$ $$(T^{4} + 100)^{2}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 62500)^{2}$$
$11$ $$(T^{2} + 3520)^{4}$$
$13$ $$(T^{4} + 1982464)^{2}$$
$17$ $$(T^{4} + 160579584)^{2}$$
$19$ $$(T^{2} - 3520)^{4}$$
$23$ $$(T^{4} + 487968100)^{2}$$
$29$ $$(T^{2} + 2116)^{4}$$
$31$ $$(T^{2} + 14080)^{4}$$
$37$ $$(T^{4} + 8120172544)^{2}$$
$41$ $$(T + 188)^{8}$$
$43$ $$(T^{4} + 915062500)^{2}$$
$47$ $$(T^{4} + 5728976100)^{2}$$
$53$ $$(T^{4} + 1982464)^{2}$$
$59$ $$(T^{2} - 172480)^{4}$$
$61$ $$(T - 72)^{8}$$
$67$ $$(T^{4} + 508212152100)^{2}$$
$71$ $$(T^{2} + 506880)^{4}$$
$73$ $$(T^{4} + 258356690944)^{2}$$
$79$ $$(T^{2} - 689920)^{4}$$
$83$ $$(T^{4} + 9605960100)^{2}$$
$89$ $$(T^{2} + 527076)^{4}$$
$97$ $$(T^{4} + 165577375744)^{2}$$