# Properties

 Label 124.4.d.c Level $124$ Weight $4$ Character orbit 124.d Analytic conductor $7.316$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 124.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.31623684071$$ Analytic rank: $$0$$ Dimension: $$40$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 94 q^{8} + 536 q^{9}+O(q^{10})$$ 40 * q - 2 * q^2 + 10 * q^4 - 4 * q^5 + 94 * q^8 + 536 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 94 q^{8} + 536 q^{9} + 228 q^{10} - 104 q^{14} - 78 q^{16} + 114 q^{18} - 44 q^{20} + 28 q^{25} + 48 q^{28} - 602 q^{32} - 136 q^{33} - 482 q^{36} + 420 q^{38} - 516 q^{40} - 4 q^{41} - 1596 q^{45} + 1876 q^{49} - 662 q^{50} + 1576 q^{56} - 838 q^{62} - 302 q^{64} - 3900 q^{66} - 872 q^{69} - 912 q^{70} - 2166 q^{72} + 3220 q^{76} - 476 q^{78} + 572 q^{80} - 2056 q^{81} + 3096 q^{82} - 6220 q^{90} - 2904 q^{93} + 6408 q^{94} - 1836 q^{97} - 1358 q^{98}+O(q^{100})$$ 40 * q - 2 * q^2 + 10 * q^4 - 4 * q^5 + 94 * q^8 + 536 * q^9 + 228 * q^10 - 104 * q^14 - 78 * q^16 + 114 * q^18 - 44 * q^20 + 28 * q^25 + 48 * q^28 - 602 * q^32 - 136 * q^33 - 482 * q^36 + 420 * q^38 - 516 * q^40 - 4 * q^41 - 1596 * q^45 + 1876 * q^49 - 662 * q^50 + 1576 * q^56 - 838 * q^62 - 302 * q^64 - 3900 * q^66 - 872 * q^69 - 912 * q^70 - 2166 * q^72 + 3220 * q^76 - 476 * q^78 + 572 * q^80 - 2056 * q^81 + 3096 * q^82 - 6220 * q^90 - 2904 * q^93 + 6408 * q^94 - 1836 * q^97 - 1358 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
123.1 −2.73824 0.708563i −3.81956 6.99588 + 3.88043i −9.46257 10.4589 + 2.70640i 9.26693i −16.4068 15.5825i −12.4110 25.9108 + 6.70483i
123.2 −2.73824 0.708563i 3.81956 6.99588 + 3.88043i −9.46257 −10.4589 2.70640i 9.26693i −16.4068 15.5825i −12.4110 25.9108 + 6.70483i
123.3 −2.73824 + 0.708563i −3.81956 6.99588 3.88043i −9.46257 10.4589 2.70640i 9.26693i −16.4068 + 15.5825i −12.4110 25.9108 6.70483i
123.4 −2.73824 + 0.708563i 3.81956 6.99588 3.88043i −9.46257 −10.4589 + 2.70640i 9.26693i −16.4068 + 15.5825i −12.4110 25.9108 6.70483i
123.5 −2.59794 1.11835i −8.21784 5.49858 + 5.81081i 6.37189 21.3495 + 9.19042i 24.3788i −7.78647 21.2455i 40.5329 −16.5538 7.12601i
123.6 −2.59794 1.11835i 8.21784 5.49858 + 5.81081i 6.37189 −21.3495 9.19042i 24.3788i −7.78647 21.2455i 40.5329 −16.5538 7.12601i
123.7 −2.59794 + 1.11835i −8.21784 5.49858 5.81081i 6.37189 21.3495 9.19042i 24.3788i −7.78647 + 21.2455i 40.5329 −16.5538 + 7.12601i
123.8 −2.59794 + 1.11835i 8.21784 5.49858 5.81081i 6.37189 −21.3495 + 9.19042i 24.3788i −7.78647 + 21.2455i 40.5329 −16.5538 + 7.12601i
123.9 −1.77211 2.20446i −7.49955 −1.71925 + 7.81308i −18.0979 13.2900 + 16.5324i 21.7591i 20.2703 10.0556i 29.2432 32.0715 + 39.8960i
123.10 −1.77211 2.20446i 7.49955 −1.71925 + 7.81308i −18.0979 −13.2900 16.5324i 21.7591i 20.2703 10.0556i 29.2432 32.0715 + 39.8960i
123.11 −1.77211 + 2.20446i −7.49955 −1.71925 7.81308i −18.0979 13.2900 16.5324i 21.7591i 20.2703 + 10.0556i 29.2432 32.0715 39.8960i
123.12 −1.77211 + 2.20446i 7.49955 −1.71925 7.81308i −18.0979 −13.2900 + 16.5324i 21.7591i 20.2703 + 10.0556i 29.2432 32.0715 39.8960i
123.13 −1.63773 2.30605i −0.254294 −2.63571 + 7.55335i 3.31209 0.416464 + 0.586414i 7.54711i 21.7349 6.29225i −26.9353 −5.42430 7.63784i
123.14 −1.63773 2.30605i 0.254294 −2.63571 + 7.55335i 3.31209 −0.416464 0.586414i 7.54711i 21.7349 6.29225i −26.9353 −5.42430 7.63784i
123.15 −1.63773 + 2.30605i −0.254294 −2.63571 7.55335i 3.31209 0.416464 0.586414i 7.54711i 21.7349 + 6.29225i −26.9353 −5.42430 + 7.63784i
123.16 −1.63773 + 2.30605i 0.254294 −2.63571 7.55335i 3.31209 −0.416464 + 0.586414i 7.54711i 21.7349 + 6.29225i −26.9353 −5.42430 + 7.63784i
123.17 −0.546214 2.77518i −7.92016 −7.40330 + 3.03169i 15.6847 4.32610 + 21.9799i 9.09573i 12.4573 + 18.8896i 35.7289 −8.56720 43.5279i
123.18 −0.546214 2.77518i 7.92016 −7.40330 + 3.03169i 15.6847 −4.32610 21.9799i 9.09573i 12.4573 + 18.8896i 35.7289 −8.56720 43.5279i
123.19 −0.546214 + 2.77518i −7.92016 −7.40330 3.03169i 15.6847 4.32610 21.9799i 9.09573i 12.4573 18.8896i 35.7289 −8.56720 + 43.5279i
123.20 −0.546214 + 2.77518i 7.92016 −7.40330 3.03169i 15.6847 −4.32610 + 21.9799i 9.09573i 12.4573 18.8896i 35.7289 −8.56720 + 43.5279i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 123.40 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
31.b odd 2 1 inner
124.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.4.d.c 40
4.b odd 2 1 inner 124.4.d.c 40
31.b odd 2 1 inner 124.4.d.c 40
124.d even 2 1 inner 124.4.d.c 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.d.c 40 1.a even 1 1 trivial
124.4.d.c 40 4.b odd 2 1 inner
124.4.d.c 40 31.b odd 2 1 inner
124.4.d.c 40 124.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(124, [\chi])$$:

 $$T_{3}^{20} - 404 T_{3}^{18} + 69148 T_{3}^{16} - 6576608 T_{3}^{14} + 382699024 T_{3}^{12} - 14122671680 T_{3}^{10} + 330615016768 T_{3}^{8} - 4740799173632 T_{3}^{6} + \cdots + 8218725875712$$ T3^20 - 404*T3^18 + 69148*T3^16 - 6576608*T3^14 + 382699024*T3^12 - 14122671680*T3^10 + 330615016768*T3^8 - 4740799173632*T3^6 + 37885977842176*T3^4 - 129526401245184*T3^2 + 8218725875712 $$T_{5}^{10} + T_{5}^{9} - 628 T_{5}^{8} - 466 T_{5}^{7} + 128573 T_{5}^{6} + 104761 T_{5}^{5} - 10043754 T_{5}^{4} - 7181688 T_{5}^{3} + 293435776 T_{5}^{2} + 170815696 T_{5} - 2515695776$$ T5^10 + T5^9 - 628*T5^8 - 466*T5^7 + 128573*T5^6 + 104761*T5^5 - 10043754*T5^4 - 7181688*T5^3 + 293435776*T5^2 + 170815696*T5 - 2515695776