# Properties

 Label 76.7.j.a Level $76$ Weight $7$ Character orbit 76.j Analytic conductor $17.484$ Analytic rank $0$ Dimension $60$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 76.j (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.4841103551$$ Analytic rank: $$0$$ Dimension: $$60$$ Relative dimension: $$10$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$60q + 30q^{3} - 216q^{7} + 690q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$60q + 30q^{3} - 216q^{7} + 690q^{9} + 1680q^{11} - 2940q^{13} + 2496q^{15} - 5112q^{17} - 15792q^{19} - 32076q^{21} - 19272q^{23} + 58896q^{25} - 124830q^{27} + 126840q^{29} + 30780q^{31} - 274470q^{33} - 226284q^{35} + 178968q^{39} + 83394q^{41} + 418848q^{43} - 95472q^{45} - 498696q^{47} - 744330q^{49} + 1334538q^{51} + 458004q^{53} - 260136q^{55} - 981984q^{57} - 523362q^{59} - 644172q^{61} + 926832q^{63} + 1337220q^{65} + 1719114q^{67} + 1333800q^{69} - 1895220q^{71} - 1189704q^{73} + 1337256q^{77} + 147432q^{79} + 272130q^{81} + 442800q^{83} + 1479096q^{85} + 1300572q^{87} - 301596q^{89} + 661008q^{91} + 3576q^{93} - 5709984q^{95} + 1386630q^{97} + 3822798q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 0 −16.6973 + 45.8755i 0 9.11718 + 51.7061i 0 −320.368 + 554.894i 0 −1267.32 1063.40i 0
13.2 0 −13.8876 + 38.1558i 0 −5.33633 30.2638i 0 187.863 325.388i 0 −704.551 591.189i 0
13.3 0 −7.21536 + 19.8240i 0 −40.4206 229.237i 0 −48.6163 + 84.2059i 0 217.516 + 182.517i 0
13.4 0 −6.46108 + 17.7517i 0 14.8495 + 84.2156i 0 158.113 273.859i 0 285.070 + 239.202i 0
13.5 0 −3.50553 + 9.63135i 0 35.8010 + 203.037i 0 −107.477 + 186.155i 0 477.972 + 401.066i 0
13.6 0 −1.48422 + 4.07785i 0 −10.2967 58.3955i 0 −54.2532 + 93.9692i 0 544.020 + 456.487i 0
13.7 0 6.80225 18.6890i 0 −6.07240 34.4383i 0 −223.008 + 386.261i 0 255.437 + 214.337i 0
13.8 0 9.61067 26.4051i 0 26.4812 + 150.182i 0 197.423 341.946i 0 −46.4181 38.9494i 0
13.9 0 10.8061 29.6895i 0 −26.2824 149.055i 0 184.362 319.324i 0 −206.249 173.063i 0
13.10 0 17.6351 48.4521i 0 11.8840 + 67.3972i 0 −186.943 + 323.795i 0 −1478.16 1240.32i 0
21.1 0 −31.3285 + 37.3358i 0 37.0498 13.4850i 0 −148.463 257.146i 0 −285.900 1621.42i 0
21.2 0 −18.1708 + 21.6552i 0 40.6936 14.8113i 0 179.977 + 311.730i 0 −12.1770 69.0593i 0
21.3 0 −17.9376 + 21.3772i 0 −219.363 + 79.8415i 0 237.710 + 411.726i 0 −8.63778 48.9873i 0
21.4 0 −5.74881 + 6.85117i 0 55.2802 20.1204i 0 −190.315 329.635i 0 112.700 + 639.153i 0
21.5 0 −5.40108 + 6.43676i 0 −171.496 + 62.4193i 0 −231.676 401.275i 0 114.329 + 648.394i 0
21.6 0 1.99278 2.37490i 0 187.566 68.2686i 0 84.5123 + 146.380i 0 124.921 + 708.460i 0
21.7 0 15.2868 18.2181i 0 −90.7408 + 33.0269i 0 −57.6521 99.8564i 0 28.3767 + 160.932i 0
21.8 0 18.3534 21.8727i 0 −6.18195 + 2.25005i 0 237.753 + 411.801i 0 −14.9788 84.9490i 0
21.9 0 24.5774 29.2903i 0 −71.7580 + 26.1178i 0 38.3425 + 66.4112i 0 −127.279 721.834i 0
21.10 0 31.0369 36.9883i 0 186.326 67.8172i 0 −265.452 459.776i 0 −278.258 1578.08i 0
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.7.j.a 60
19.f odd 18 1 inner 76.7.j.a 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.j.a 60 1.a even 1 1 trivial
76.7.j.a 60 19.f odd 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(76, [\chi])$$.