# Properties

 Label 100.4.e.b Level $100$ Weight $4$ Character orbit 100.e Analytic conductor $5.900$ Analytic rank $0$ Dimension $2$ CM discriminant -20 Inner twists $4$

# 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: $$2$$ 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: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (2 i - 2) q^{2} + (7 i + 7) q^{3} - 8 i q^{4} - 28 q^{6} + (9 i - 9) q^{7} + (16 i + 16) q^{8} + 71 i q^{9} +O(q^{10})$$ q + (2*i - 2) * q^2 + (7*i + 7) * q^3 - 8*i * q^4 - 28 * q^6 + (9*i - 9) * q^7 + (16*i + 16) * q^8 + 71*i * q^9 $$q + (2 i - 2) q^{2} + (7 i + 7) q^{3} - 8 i q^{4} - 28 q^{6} + (9 i - 9) q^{7} + (16 i + 16) q^{8} + 71 i q^{9} + ( - 56 i + 56) q^{12} - 36 i q^{14} - 64 q^{16} + ( - 142 i - 142) q^{18} - 126 q^{21} + (67 i + 67) q^{23} + 224 i q^{24} + (308 i - 308) q^{27} + (72 i + 72) q^{28} - 306 i q^{29} + ( - 128 i + 128) q^{32} + 568 q^{36} + 252 q^{41} + ( - 252 i + 252) q^{42} + (297 i + 297) q^{43} - 268 q^{46} + ( - 301 i + 301) q^{47} + ( - 448 i - 448) q^{48} + 181 i q^{49} - 1232 i q^{54} - 288 q^{56} + (612 i + 612) q^{58} + 952 q^{61} + ( - 639 i - 639) q^{63} + 512 i q^{64} + (549 i - 549) q^{67} + 938 i q^{69} + (1136 i - 1136) q^{72} - 2395 q^{81} + (504 i - 504) q^{82} + (77 i + 77) q^{83} + 1008 i q^{84} - 1188 q^{86} + ( - 2142 i + 2142) q^{87} - 1386 i q^{89} + ( - 536 i + 536) q^{92} + 1204 i q^{94} + 1792 q^{96} + ( - 362 i - 362) q^{98} +O(q^{100})$$ q + (2*i - 2) * q^2 + (7*i + 7) * q^3 - 8*i * q^4 - 28 * q^6 + (9*i - 9) * q^7 + (16*i + 16) * q^8 + 71*i * q^9 + (-56*i + 56) * q^12 - 36*i * q^14 - 64 * q^16 + (-142*i - 142) * q^18 - 126 * q^21 + (67*i + 67) * q^23 + 224*i * q^24 + (308*i - 308) * q^27 + (72*i + 72) * q^28 - 306*i * q^29 + (-128*i + 128) * q^32 + 568 * q^36 + 252 * q^41 + (-252*i + 252) * q^42 + (297*i + 297) * q^43 - 268 * q^46 + (-301*i + 301) * q^47 + (-448*i - 448) * q^48 + 181*i * q^49 - 1232*i * q^54 - 288 * q^56 + (612*i + 612) * q^58 + 952 * q^61 + (-639*i - 639) * q^63 + 512*i * q^64 + (549*i - 549) * q^67 + 938*i * q^69 + (1136*i - 1136) * q^72 - 2395 * q^81 + (504*i - 504) * q^82 + (77*i + 77) * q^83 + 1008*i * q^84 - 1188 * q^86 + (-2142*i + 2142) * q^87 - 1386*i * q^89 + (-536*i + 536) * q^92 + 1204*i * q^94 + 1792 * q^96 + (-362*i - 362) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 14 q^{3} - 56 q^{6} - 18 q^{7} + 32 q^{8}+O(q^{10})$$ 2 * q - 4 * q^2 + 14 * q^3 - 56 * q^6 - 18 * q^7 + 32 * q^8 $$2 q - 4 q^{2} + 14 q^{3} - 56 q^{6} - 18 q^{7} + 32 q^{8} + 112 q^{12} - 128 q^{16} - 284 q^{18} - 252 q^{21} + 134 q^{23} - 616 q^{27} + 144 q^{28} + 256 q^{32} + 1136 q^{36} + 504 q^{41} + 504 q^{42} + 594 q^{43} - 536 q^{46} + 602 q^{47} - 896 q^{48} - 576 q^{56} + 1224 q^{58} + 1904 q^{61} - 1278 q^{63} - 1098 q^{67} - 2272 q^{72} - 4790 q^{81} - 1008 q^{82} + 154 q^{83} - 2376 q^{86} + 4284 q^{87} + 1072 q^{92} + 3584 q^{96} - 724 q^{98}+O(q^{100})$$ 2 * q - 4 * q^2 + 14 * q^3 - 56 * q^6 - 18 * q^7 + 32 * q^8 + 112 * q^12 - 128 * q^16 - 284 * q^18 - 252 * q^21 + 134 * q^23 - 616 * q^27 + 144 * q^28 + 256 * q^32 + 1136 * q^36 + 504 * q^41 + 504 * q^42 + 594 * q^43 - 536 * q^46 + 602 * q^47 - 896 * q^48 - 576 * q^56 + 1224 * q^58 + 1904 * q^61 - 1278 * q^63 - 1098 * q^67 - 2272 * q^72 - 4790 * q^81 - 1008 * q^82 + 154 * q^83 - 2376 * q^86 + 4284 * q^87 + 1072 * q^92 + 3584 * q^96 - 724 * q^98

## Character values

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

 $$n$$ $$51$$ $$77$$ $$\chi(n)$$ $$-1$$ $$i$$

## 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
 1.00000i − 1.00000i
−2.00000 + 2.00000i 7.00000 + 7.00000i 8.00000i 0 −28.0000 −9.00000 + 9.00000i 16.0000 + 16.0000i 71.0000i 0
43.1 −2.00000 2.00000i 7.00000 7.00000i 8.00000i 0 −28.0000 −9.00000 9.00000i 16.0000 16.0000i 71.0000i 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 14T_{3} + 98$$ acting on $$S_{4}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4T + 8$$
$3$ $$T^{2} - 14T + 98$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 18T + 162$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 134T + 8978$$
$29$ $$T^{2} + 93636$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T - 252)^{2}$$
$43$ $$T^{2} - 594T + 176418$$
$47$ $$T^{2} - 602T + 181202$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 952)^{2}$$
$67$ $$T^{2} + 1098 T + 602802$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 154T + 11858$$
$89$ $$T^{2} + 1920996$$
$97$ $$T^{2}$$