# Properties

 Label 930.2.bf.a Level $930$ Weight $2$ Character orbit 930.bf Analytic conductor $7.426$ Analytic rank $0$ Dimension $256$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.bf (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$256$$ Relative dimension: $$64$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$256q + 8q^{3} - 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$256q + 8q^{3} - 4q^{7} - 8q^{10} - 8q^{12} + 8q^{15} - 256q^{16} - 4q^{22} - 16q^{25} - 40q^{27} - 4q^{28} - 48q^{30} + 32q^{31} - 8q^{33} + 8q^{37} - 12q^{42} - 40q^{45} + 16q^{46} - 8q^{48} - 16q^{51} + 20q^{55} + 40q^{57} + 56q^{58} - 8q^{60} - 32q^{61} + 56q^{63} + 32q^{66} - 24q^{67} + 24q^{70} + 8q^{73} - 4q^{75} + 16q^{76} - 88q^{78} - 48q^{81} + 64q^{85} - 20q^{87} - 4q^{88} - 8q^{90} - 68q^{93} + 152q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
377.1 −0.707107 + 0.707107i −1.72168 0.189281i 1.00000i 0.213393 + 2.22586i 1.35125 1.08357i 3.30534 0.885662i 0.707107 + 0.707107i 2.92835 + 0.651761i −1.72481 1.42303i
377.2 −0.707107 + 0.707107i −1.70669 + 0.295334i 1.00000i 2.22324 + 0.239148i 0.997976 1.41564i −3.38718 + 0.907593i 0.707107 + 0.707107i 2.82556 1.00809i −1.74117 + 1.40297i
377.3 −0.707107 + 0.707107i −1.65429 0.513158i 1.00000i 1.83992 1.27071i 1.53262 0.806900i 3.00096 0.804106i 0.707107 + 0.707107i 2.47334 + 1.69782i −0.402493 + 2.19955i
377.4 −0.707107 + 0.707107i −1.64704 0.535974i 1.00000i −0.601583 + 2.15362i 1.54362 0.785641i −4.77476 + 1.27939i 0.707107 + 0.707107i 2.42546 + 1.76554i −1.09746 1.94823i
377.5 −0.707107 + 0.707107i −1.64517 + 0.541693i 1.00000i −2.17658 0.512363i 0.780273 1.54634i 0.122813 0.0329078i 0.707107 + 0.707107i 2.41314 1.78235i 1.90137 1.17678i
377.6 −0.707107 + 0.707107i −1.45410 0.941054i 1.00000i −1.41418 1.73208i 1.69363 0.362781i −1.47385 + 0.394917i 0.707107 + 0.707107i 1.22883 + 2.73678i 2.22474 + 0.224784i
377.7 −0.707107 + 0.707107i −1.38753 + 1.03670i 1.00000i −0.924875 + 2.03583i 0.248075 1.71419i −0.101233 + 0.0271252i 0.707107 + 0.707107i 0.850495 2.87692i −0.785564 2.09354i
377.8 −0.707107 + 0.707107i −1.34147 + 1.09565i 1.00000i 2.02568 + 0.946905i 0.173818 1.72331i 2.29933 0.616104i 0.707107 + 0.707107i 0.599084 2.93957i −2.10193 + 0.762808i
377.9 −0.707107 + 0.707107i −1.27826 1.16878i 1.00000i −2.23600 0.0178176i 1.73032 0.0774131i 4.17555 1.11884i 0.707107 + 0.707107i 0.267899 + 2.98801i 1.59369 1.56849i
377.10 −0.707107 + 0.707107i −1.00727 + 1.40904i 1.00000i −0.926393 2.03514i −0.284092 1.70859i 3.45336 0.925324i 0.707107 + 0.707107i −0.970795 2.83858i 2.09412 + 0.784002i
377.11 −0.707107 + 0.707107i −1.00673 1.40943i 1.00000i 0.813777 2.08273i 1.70848 + 0.284755i −1.29509 + 0.347018i 0.707107 + 0.707107i −0.972997 + 2.83783i 0.897286 + 2.04814i
377.12 −0.707107 + 0.707107i −0.792487 1.54012i 1.00000i −0.947623 + 2.02534i 1.64940 + 0.528655i −1.77478 + 0.475552i 0.707107 + 0.707107i −1.74393 + 2.44105i −0.762062 2.10220i
377.13 −0.707107 + 0.707107i −0.537394 1.64657i 1.00000i 0.858744 + 2.06460i 1.54430 + 0.784309i 2.17271 0.582177i 0.707107 + 0.707107i −2.42242 + 1.76972i −2.06711 0.852667i
377.14 −0.707107 + 0.707107i −0.390383 + 1.68748i 1.00000i −1.57881 1.58347i −0.917189 1.46927i −4.83994 + 1.29686i 0.707107 + 0.707107i −2.69520 1.31753i 2.23607 + 0.00329236i
377.15 −0.707107 + 0.707107i −0.320052 + 1.70222i 1.00000i 1.97048 1.05698i −0.977343 1.42997i 1.65949 0.444658i 0.707107 + 0.707107i −2.79513 1.08960i −0.645943 + 2.14074i
377.16 −0.707107 + 0.707107i −0.136833 + 1.72664i 1.00000i −2.04548 + 0.903327i −1.12416 1.31767i −0.382918 + 0.102603i 0.707107 + 0.707107i −2.96255 0.472523i 0.807626 2.08512i
377.17 −0.707107 + 0.707107i −0.134174 1.72685i 1.00000i 2.18104 + 0.493020i 1.31594 + 1.12619i −3.79793 + 1.01765i 0.707107 + 0.707107i −2.96399 + 0.463396i −1.89085 + 1.19361i
377.18 −0.707107 + 0.707107i 0.321845 1.70189i 1.00000i −0.575509 2.16074i 0.975836 + 1.43099i 2.59908 0.696422i 0.707107 + 0.707107i −2.79283 1.09549i 1.93482 + 1.12093i
377.19 −0.707107 + 0.707107i 0.609255 + 1.62136i 1.00000i 1.73683 1.40835i −1.57728 0.715666i −3.28936 + 0.881380i 0.707107 + 0.707107i −2.25762 + 1.97564i −0.232271 + 2.22397i
377.20 −0.707107 + 0.707107i 0.661860 1.60061i 1.00000i −2.12466 + 0.697001i 0.663795 + 1.59981i 1.92566 0.515980i 0.707107 + 0.707107i −2.12388 2.11875i 1.00951 1.99522i
See next 80 embeddings (of 256 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 893.64 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
31.c even 3 1 inner
93.h odd 6 1 inner
155.o odd 12 1 inner
465.be even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bf.a 256
3.b odd 2 1 inner 930.2.bf.a 256
5.c odd 4 1 inner 930.2.bf.a 256
15.e even 4 1 inner 930.2.bf.a 256
31.c even 3 1 inner 930.2.bf.a 256
93.h odd 6 1 inner 930.2.bf.a 256
155.o odd 12 1 inner 930.2.bf.a 256
465.be even 12 1 inner 930.2.bf.a 256

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bf.a 256 1.a even 1 1 trivial
930.2.bf.a 256 3.b odd 2 1 inner
930.2.bf.a 256 5.c odd 4 1 inner
930.2.bf.a 256 15.e even 4 1 inner
930.2.bf.a 256 31.c even 3 1 inner
930.2.bf.a 256 93.h odd 6 1 inner
930.2.bf.a 256 155.o odd 12 1 inner
930.2.bf.a 256 465.be even 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(930, [\chi])$$.