# Properties

 Label 70.4.c.a Level $70$ Weight $4$ Character orbit 70.c Analytic conductor $4.130$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$70 = 2 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 70.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.13013370040$$ 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: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} + 7 i q^{3} - 4 q^{4} + (5 i + 10) q^{5} - 14 q^{6} + 7 i q^{7} - 8 i q^{8} - 22 q^{9} +O(q^{10})$$ q + 2*i * q^2 + 7*i * q^3 - 4 * q^4 + (5*i + 10) * q^5 - 14 * q^6 + 7*i * q^7 - 8*i * q^8 - 22 * q^9 $$q + 2 i q^{2} + 7 i q^{3} - 4 q^{4} + (5 i + 10) q^{5} - 14 q^{6} + 7 i q^{7} - 8 i q^{8} - 22 q^{9} + (20 i - 10) q^{10} - 37 q^{11} - 28 i q^{12} - 51 i q^{13} - 14 q^{14} + (70 i - 35) q^{15} + 16 q^{16} + 41 i q^{17} - 44 i q^{18} + 108 q^{19} + ( - 20 i - 40) q^{20} - 49 q^{21} - 74 i q^{22} + 70 i q^{23} + 56 q^{24} + (100 i + 75) q^{25} + 102 q^{26} + 35 i q^{27} - 28 i q^{28} + 249 q^{29} + ( - 70 i - 140) q^{30} - 134 q^{31} + 32 i q^{32} - 259 i q^{33} - 82 q^{34} + (70 i - 35) q^{35} + 88 q^{36} - 334 i q^{37} + 216 i q^{38} + 357 q^{39} + ( - 80 i + 40) q^{40} + 206 q^{41} - 98 i q^{42} + 376 i q^{43} + 148 q^{44} + ( - 110 i - 220) q^{45} - 140 q^{46} - 287 i q^{47} + 112 i q^{48} - 49 q^{49} + (150 i - 200) q^{50} - 287 q^{51} + 204 i q^{52} + 6 i q^{53} - 70 q^{54} + ( - 185 i - 370) q^{55} + 56 q^{56} + 756 i q^{57} + 498 i q^{58} + 2 q^{59} + ( - 280 i + 140) q^{60} - 940 q^{61} - 268 i q^{62} - 154 i q^{63} - 64 q^{64} + ( - 510 i + 255) q^{65} + 518 q^{66} + 106 i q^{67} - 164 i q^{68} - 490 q^{69} + ( - 70 i - 140) q^{70} + 456 q^{71} + 176 i q^{72} - 650 i q^{73} + 668 q^{74} + (525 i - 700) q^{75} - 432 q^{76} - 259 i q^{77} + 714 i q^{78} + 1239 q^{79} + (80 i + 160) q^{80} - 839 q^{81} + 412 i q^{82} - 428 i q^{83} + 196 q^{84} + (410 i - 205) q^{85} - 752 q^{86} + 1743 i q^{87} + 296 i q^{88} + 220 q^{89} + ( - 440 i + 220) q^{90} + 357 q^{91} - 280 i q^{92} - 938 i q^{93} + 574 q^{94} + (540 i + 1080) q^{95} - 224 q^{96} - 1055 i q^{97} - 98 i q^{98} + 814 q^{99} +O(q^{100})$$ q + 2*i * q^2 + 7*i * q^3 - 4 * q^4 + (5*i + 10) * q^5 - 14 * q^6 + 7*i * q^7 - 8*i * q^8 - 22 * q^9 + (20*i - 10) * q^10 - 37 * q^11 - 28*i * q^12 - 51*i * q^13 - 14 * q^14 + (70*i - 35) * q^15 + 16 * q^16 + 41*i * q^17 - 44*i * q^18 + 108 * q^19 + (-20*i - 40) * q^20 - 49 * q^21 - 74*i * q^22 + 70*i * q^23 + 56 * q^24 + (100*i + 75) * q^25 + 102 * q^26 + 35*i * q^27 - 28*i * q^28 + 249 * q^29 + (-70*i - 140) * q^30 - 134 * q^31 + 32*i * q^32 - 259*i * q^33 - 82 * q^34 + (70*i - 35) * q^35 + 88 * q^36 - 334*i * q^37 + 216*i * q^38 + 357 * q^39 + (-80*i + 40) * q^40 + 206 * q^41 - 98*i * q^42 + 376*i * q^43 + 148 * q^44 + (-110*i - 220) * q^45 - 140 * q^46 - 287*i * q^47 + 112*i * q^48 - 49 * q^49 + (150*i - 200) * q^50 - 287 * q^51 + 204*i * q^52 + 6*i * q^53 - 70 * q^54 + (-185*i - 370) * q^55 + 56 * q^56 + 756*i * q^57 + 498*i * q^58 + 2 * q^59 + (-280*i + 140) * q^60 - 940 * q^61 - 268*i * q^62 - 154*i * q^63 - 64 * q^64 + (-510*i + 255) * q^65 + 518 * q^66 + 106*i * q^67 - 164*i * q^68 - 490 * q^69 + (-70*i - 140) * q^70 + 456 * q^71 + 176*i * q^72 - 650*i * q^73 + 668 * q^74 + (525*i - 700) * q^75 - 432 * q^76 - 259*i * q^77 + 714*i * q^78 + 1239 * q^79 + (80*i + 160) * q^80 - 839 * q^81 + 412*i * q^82 - 428*i * q^83 + 196 * q^84 + (410*i - 205) * q^85 - 752 * q^86 + 1743*i * q^87 + 296*i * q^88 + 220 * q^89 + (-440*i + 220) * q^90 + 357 * q^91 - 280*i * q^92 - 938*i * q^93 + 574 * q^94 + (540*i + 1080) * q^95 - 224 * q^96 - 1055*i * q^97 - 98*i * q^98 + 814 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4} + 20 q^{5} - 28 q^{6} - 44 q^{9}+O(q^{10})$$ 2 * q - 8 * q^4 + 20 * q^5 - 28 * q^6 - 44 * q^9 $$2 q - 8 q^{4} + 20 q^{5} - 28 q^{6} - 44 q^{9} - 20 q^{10} - 74 q^{11} - 28 q^{14} - 70 q^{15} + 32 q^{16} + 216 q^{19} - 80 q^{20} - 98 q^{21} + 112 q^{24} + 150 q^{25} + 204 q^{26} + 498 q^{29} - 280 q^{30} - 268 q^{31} - 164 q^{34} - 70 q^{35} + 176 q^{36} + 714 q^{39} + 80 q^{40} + 412 q^{41} + 296 q^{44} - 440 q^{45} - 280 q^{46} - 98 q^{49} - 400 q^{50} - 574 q^{51} - 140 q^{54} - 740 q^{55} + 112 q^{56} + 4 q^{59} + 280 q^{60} - 1880 q^{61} - 128 q^{64} + 510 q^{65} + 1036 q^{66} - 980 q^{69} - 280 q^{70} + 912 q^{71} + 1336 q^{74} - 1400 q^{75} - 864 q^{76} + 2478 q^{79} + 320 q^{80} - 1678 q^{81} + 392 q^{84} - 410 q^{85} - 1504 q^{86} + 440 q^{89} + 440 q^{90} + 714 q^{91} + 1148 q^{94} + 2160 q^{95} - 448 q^{96} + 1628 q^{99}+O(q^{100})$$ 2 * q - 8 * q^4 + 20 * q^5 - 28 * q^6 - 44 * q^9 - 20 * q^10 - 74 * q^11 - 28 * q^14 - 70 * q^15 + 32 * q^16 + 216 * q^19 - 80 * q^20 - 98 * q^21 + 112 * q^24 + 150 * q^25 + 204 * q^26 + 498 * q^29 - 280 * q^30 - 268 * q^31 - 164 * q^34 - 70 * q^35 + 176 * q^36 + 714 * q^39 + 80 * q^40 + 412 * q^41 + 296 * q^44 - 440 * q^45 - 280 * q^46 - 98 * q^49 - 400 * q^50 - 574 * q^51 - 140 * q^54 - 740 * q^55 + 112 * q^56 + 4 * q^59 + 280 * q^60 - 1880 * q^61 - 128 * q^64 + 510 * q^65 + 1036 * q^66 - 980 * q^69 - 280 * q^70 + 912 * q^71 + 1336 * q^74 - 1400 * q^75 - 864 * q^76 + 2478 * q^79 + 320 * q^80 - 1678 * q^81 + 392 * q^84 - 410 * q^85 - 1504 * q^86 + 440 * q^89 + 440 * q^90 + 714 * q^91 + 1148 * q^94 + 2160 * q^95 - 448 * q^96 + 1628 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/70\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$57$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 − 1.00000i 1.00000i
2.00000i 7.00000i −4.00000 10.0000 5.00000i −14.0000 7.00000i 8.00000i −22.0000 −10.0000 20.0000i
29.2 2.00000i 7.00000i −4.00000 10.0000 + 5.00000i −14.0000 7.00000i 8.00000i −22.0000 −10.0000 + 20.0000i
 $$n$$: 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 70.4.c.a 2
3.b odd 2 1 630.4.g.a 2
4.b odd 2 1 560.4.g.c 2
5.b even 2 1 inner 70.4.c.a 2
5.c odd 4 1 350.4.a.i 1
5.c odd 4 1 350.4.a.m 1
7.b odd 2 1 490.4.c.a 2
15.d odd 2 1 630.4.g.a 2
20.d odd 2 1 560.4.g.c 2
35.c odd 2 1 490.4.c.a 2
35.f even 4 1 2450.4.a.c 1
35.f even 4 1 2450.4.a.bn 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.c.a 2 1.a even 1 1 trivial
70.4.c.a 2 5.b even 2 1 inner
350.4.a.i 1 5.c odd 4 1
350.4.a.m 1 5.c odd 4 1
490.4.c.a 2 7.b odd 2 1
490.4.c.a 2 35.c odd 2 1
560.4.g.c 2 4.b odd 2 1
560.4.g.c 2 20.d odd 2 1
630.4.g.a 2 3.b odd 2 1
630.4.g.a 2 15.d odd 2 1
2450.4.a.c 1 35.f even 4 1
2450.4.a.bn 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 49$$ acting on $$S_{4}^{\mathrm{new}}(70, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2} + 49$$
$5$ $$T^{2} - 20T + 125$$
$7$ $$T^{2} + 49$$
$11$ $$(T + 37)^{2}$$
$13$ $$T^{2} + 2601$$
$17$ $$T^{2} + 1681$$
$19$ $$(T - 108)^{2}$$
$23$ $$T^{2} + 4900$$
$29$ $$(T - 249)^{2}$$
$31$ $$(T + 134)^{2}$$
$37$ $$T^{2} + 111556$$
$41$ $$(T - 206)^{2}$$
$43$ $$T^{2} + 141376$$
$47$ $$T^{2} + 82369$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 2)^{2}$$
$61$ $$(T + 940)^{2}$$
$67$ $$T^{2} + 11236$$
$71$ $$(T - 456)^{2}$$
$73$ $$T^{2} + 422500$$
$79$ $$(T - 1239)^{2}$$
$83$ $$T^{2} + 183184$$
$89$ $$(T - 220)^{2}$$
$97$ $$T^{2} + 1113025$$