# Properties

 Label 403.2.i.a Level 403 Weight 2 Character orbit 403.i Analytic conductor 3.218 Analytic rank 0 Dimension 68 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$68$$ Relative dimension: $$34$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$68q - 4q^{2} - 4q^{5} + 8q^{7} + 16q^{8} - 60q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$68q - 4q^{2} - 4q^{5} + 8q^{7} + 16q^{8} - 60q^{9} - 48q^{14} - 40q^{16} + 4q^{18} - 24q^{19} - 16q^{20} + 44q^{28} + 24q^{31} + 28q^{32} - 40q^{35} - 24q^{39} + 24q^{40} + 20q^{41} - 24q^{45} - 36q^{47} + 80q^{50} + 28q^{59} - 76q^{63} + 152q^{66} - 32q^{67} - 48q^{70} + 20q^{71} - 32q^{72} + 72q^{76} + 84q^{78} - 20q^{80} + 52q^{81} - 112q^{87} - 8q^{93} - 16q^{94} - 4q^{97} - 92q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
216.1 −1.86899 1.86899i 2.77664i 4.98628i 1.10552 + 1.10552i −5.18952 + 5.18952i 3.07074 3.07074i 5.58133 5.58133i −4.70973 4.13241i
216.2 −1.86899 1.86899i 2.77664i 4.98628i 1.10552 + 1.10552i 5.18952 5.18952i 3.07074 3.07074i 5.58133 5.58133i −4.70973 4.13241i
216.3 −1.78306 1.78306i 0.329930i 4.35859i −1.02687 1.02687i −0.588285 + 0.588285i −0.568114 + 0.568114i 4.20550 4.20550i 2.89115 3.66195i
216.4 −1.78306 1.78306i 0.329930i 4.35859i −1.02687 1.02687i 0.588285 0.588285i −0.568114 + 0.568114i 4.20550 4.20550i 2.89115 3.66195i
216.5 −1.50749 1.50749i 1.65703i 2.54505i 2.11743 + 2.11743i −2.49795 + 2.49795i −2.06359 + 2.06359i 0.821650 0.821650i 0.254264 6.38401i
216.6 −1.50749 1.50749i 1.65703i 2.54505i 2.11743 + 2.11743i 2.49795 2.49795i −2.06359 + 2.06359i 0.821650 0.821650i 0.254264 6.38401i
216.7 −1.30566 1.30566i 2.35582i 1.40948i −3.07515 3.07515i −3.07590 + 3.07590i 1.59752 1.59752i −0.771014 + 0.771014i −2.54991 8.03019i
216.8 −1.30566 1.30566i 2.35582i 1.40948i −3.07515 3.07515i 3.07590 3.07590i 1.59752 1.59752i −0.771014 + 0.771014i −2.54991 8.03019i
216.9 −1.12794 1.12794i 1.37229i 0.544506i 1.86354 + 1.86354i −1.54787 + 1.54787i 2.05385 2.05385i −1.64171 + 1.64171i 1.11681 4.20393i
216.10 −1.12794 1.12794i 1.37229i 0.544506i 1.86354 + 1.86354i 1.54787 1.54787i 2.05385 2.05385i −1.64171 + 1.64171i 1.11681 4.20393i
216.11 −0.840661 0.840661i 1.32208i 0.586577i −1.04050 1.04050i −1.11142 + 1.11142i −1.92533 + 1.92533i −2.17444 + 2.17444i 1.25210 1.74942i
216.12 −0.840661 0.840661i 1.32208i 0.586577i −1.04050 1.04050i 1.11142 1.11142i −1.92533 + 1.92533i −2.17444 + 2.17444i 1.25210 1.74942i
216.13 −0.471532 0.471532i 3.39270i 1.55532i 1.41859 + 1.41859i −1.59977 + 1.59977i −1.23146 + 1.23146i −1.67644 + 1.67644i −8.51044 1.33782i
216.14 −0.471532 0.471532i 3.39270i 1.55532i 1.41859 + 1.41859i 1.59977 1.59977i −1.23146 + 1.23146i −1.67644 + 1.67644i −8.51044 1.33782i
216.15 −0.459282 0.459282i 0.325710i 1.57812i −0.464951 0.464951i −0.149593 + 0.149593i 2.03738 2.03738i −1.64337 + 1.64337i 2.89391 0.427088i
216.16 −0.459282 0.459282i 0.325710i 1.57812i −0.464951 0.464951i 0.149593 0.149593i 2.03738 2.03738i −1.64337 + 1.64337i 2.89391 0.427088i
216.17 0.194750 + 0.194750i 2.63371i 1.92414i −1.36075 1.36075i 0.512915 0.512915i 2.99770 2.99770i 0.764228 0.764228i −3.93641 0.530014i
216.18 0.194750 + 0.194750i 2.63371i 1.92414i −1.36075 1.36075i −0.512915 + 0.512915i 2.99770 2.99770i 0.764228 0.764228i −3.93641 0.530014i
216.19 0.235416 + 0.235416i 2.22877i 1.88916i −2.22025 2.22025i 0.524689 0.524689i −2.84607 + 2.84607i 0.915572 0.915572i −1.96742 1.04537i
216.20 0.235416 + 0.235416i 2.22877i 1.88916i −2.22025 2.22025i −0.524689 + 0.524689i −2.84607 + 2.84607i 0.915572 0.915572i −1.96742 1.04537i
See all 68 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 278.34 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner
31.b odd 2 1 inner
403.i even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.i.a 68
13.d odd 4 1 inner 403.2.i.a 68
31.b odd 2 1 inner 403.2.i.a 68
403.i even 4 1 inner 403.2.i.a 68

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.i.a 68 1.a even 1 1 trivial
403.2.i.a 68 13.d odd 4 1 inner
403.2.i.a 68 31.b odd 2 1 inner
403.2.i.a 68 403.i even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database