# Properties

 Label 315.2.bv.a Level 315 Weight 2 Character orbit 315.bv Analytic conductor 2.515 Analytic rank 0 Dimension 176 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 315.bv (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.51528766367$$ Analytic rank: $$0$$ Dimension: $$176$$ Relative dimension: $$44$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$176q - 2q^{3} - 6q^{5} - 24q^{6} - 2q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$176q - 2q^{3} - 6q^{5} - 24q^{6} - 2q^{7} - 4q^{10} - 24q^{11} + 26q^{12} - 4q^{13} - 14q^{15} - 136q^{16} + 18q^{17} - 10q^{18} - 12q^{20} - 16q^{21} + 4q^{22} - 30q^{23} + 2q^{25} - 32q^{27} - 4q^{28} + 10q^{30} - 8q^{31} - 34q^{33} + 8q^{36} - 4q^{37} - 30q^{38} + 18q^{40} - 36q^{41} + 8q^{42} - 4q^{43} + 22q^{45} + 4q^{46} + 38q^{48} + 36q^{50} - 40q^{51} + 26q^{52} + 4q^{55} + 24q^{56} + 32q^{57} + 6q^{58} + 22q^{60} + 16q^{61} + 14q^{63} + 4q^{66} - 4q^{67} + 114q^{68} + 18q^{70} - 46q^{72} - 4q^{73} + 6q^{75} - 24q^{76} - 54q^{77} + 54q^{78} - 36q^{80} - 64q^{81} - 8q^{82} - 12q^{83} - 4q^{85} - 120q^{86} - 28q^{87} - 6q^{88} - 24q^{90} - 16q^{91} + 72q^{92} - 38q^{93} + 192q^{96} - 4q^{97} - 12q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1 −1.86946 1.86946i −0.784062 1.54442i 4.98975i −0.167803 2.22976i −1.42147 + 4.35301i −2.39984 1.11389i 5.58921 5.58921i −1.77049 + 2.42185i −3.85475 + 4.48215i
23.2 −1.77797 1.77797i 0.265860 1.71153i 4.32238i −0.447954 + 2.19074i −3.51574 + 2.57035i 1.37680 + 2.25930i 4.12912 4.12912i −2.85864 0.910053i 4.69153 3.09862i
23.3 −1.76616 1.76616i −1.54103 + 0.790710i 4.23867i 1.75658 + 1.38362i 4.11824 + 1.32519i −2.15527 + 1.53454i 3.95386 3.95386i 1.74955 2.43702i −0.658708 5.54612i
23.4 −1.76155 1.76155i 1.72928 + 0.0979388i 4.20612i −2.22663 + 0.205270i −2.87369 3.21874i 0.0218295 2.64566i 3.88619 3.88619i 2.98082 + 0.338727i 4.28391 + 3.56072i
23.5 −1.69531 1.69531i 1.72524 0.153414i 3.74812i 1.48276 1.67374i −3.18490 2.66473i 1.46770 + 2.20133i 2.96360 2.96360i 2.95293 0.529352i −5.35124 + 0.323772i
23.6 −1.57216 1.57216i 0.440043 + 1.67522i 2.94339i 0.402568 + 2.19953i 1.94190 3.32554i 2.54364 0.727930i 1.48316 1.48316i −2.61272 + 1.47434i 2.82512 4.09092i
23.7 −1.41417 1.41417i −1.46976 0.916412i 1.99973i 2.20169 + 0.390582i 0.782522 + 3.37444i 1.74936 1.98488i −0.000387571 0 0.000387571i 1.32038 + 2.69381i −2.56121 3.66590i
23.8 −1.33173 1.33173i −1.22217 + 1.22732i 1.54698i −2.16580 + 0.556151i 3.26204 0.00685671i −1.84223 1.89899i −0.603293 + 0.603293i −0.0126118 2.99997i 3.62489 + 2.14361i
23.9 −1.29695 1.29695i −1.63045 0.584483i 1.36418i −2.10879 0.743641i 1.35657 + 2.87267i 1.22224 + 2.34651i −0.824634 + 0.824634i 2.31676 + 1.90595i 1.77054 + 3.69947i
23.10 −1.14944 1.14944i 0.768701 + 1.55213i 0.642437i −0.945461 2.02635i 0.900504 2.66766i −1.70140 + 2.02614i −1.56044 + 1.56044i −1.81820 + 2.38624i −1.24242 + 3.41593i
23.11 −1.14308 1.14308i 1.67242 + 0.450568i 0.613246i 0.197501 + 2.22733i −1.39667 2.42674i −2.27793 + 1.34575i −1.58517 + 1.58517i 2.59398 + 1.50708i 2.32025 2.77176i
23.12 −1.08313 1.08313i 1.10330 1.33519i 0.346328i 2.23510 0.0656858i −2.64119 + 0.251168i −2.63546 0.233113i −1.79114 + 1.79114i −0.565465 2.94623i −2.49205 2.34975i
23.13 −1.02458 1.02458i 0.918977 1.46816i 0.0995230i −0.891651 2.05060i −2.44581 + 0.562677i 2.22810 1.42674i −1.94719 + 1.94719i −1.31096 2.69840i −1.18743 + 3.01457i
23.14 −0.902350 0.902350i −0.320197 1.70220i 0.371529i −0.953884 + 2.02240i −1.24705 + 1.82491i −0.594335 2.57813i −2.13995 + 2.13995i −2.79495 + 1.09008i 2.68565 0.964176i
23.15 −0.710963 0.710963i −0.818385 + 1.52651i 0.989064i 2.18368 0.481173i 1.66714 0.503454i 1.29429 + 2.30756i −2.12511 + 2.12511i −1.66049 2.49855i −1.89461 1.21042i
23.16 −0.481194 0.481194i 1.08354 + 1.35127i 1.53691i −2.19786 + 0.411594i 0.128828 1.17162i 2.63650 + 0.221076i −1.70194 + 1.70194i −0.651863 + 2.92832i 1.25565 + 0.859540i
23.17 −0.477311 0.477311i −1.72825 + 0.114750i 1.54435i 0.814847 2.08231i 0.879681 + 0.770138i −2.51050 0.835101i −1.69176 + 1.69176i 2.97366 0.396634i −1.38284 + 0.604974i
23.18 −0.421918 0.421918i 1.58110 + 0.707197i 1.64397i 2.16393 0.563383i −0.368715 0.965474i 1.26919 2.32145i −1.53746 + 1.53746i 1.99974 + 2.23630i −1.15070 0.675301i
23.19 −0.391477 0.391477i −1.67401 + 0.444638i 1.69349i −0.676575 + 2.13125i 0.829400 + 0.481269i 2.43542 1.03378i −1.44592 + 1.44592i 2.60459 1.48865i 1.09920 0.569474i
23.20 −0.0820754 0.0820754i −0.222382 1.71772i 1.98653i −2.22963 0.169605i −0.122730 + 0.159234i −1.67271 + 2.04989i −0.327196 + 0.327196i −2.90109 + 0.763977i 0.169077 + 0.196918i
See next 80 embeddings (of 176 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 263.44 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.c odd 4 1 inner
63.j odd 6 1 inner
315.bv even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bv.a 176
3.b odd 2 1 945.2.by.a 176
5.c odd 4 1 inner 315.2.bv.a 176
7.c even 3 1 315.2.bx.a yes 176
9.c even 3 1 945.2.ca.a 176
9.d odd 6 1 315.2.bx.a yes 176
15.e even 4 1 945.2.by.a 176
21.h odd 6 1 945.2.ca.a 176
35.l odd 12 1 315.2.bx.a yes 176
45.k odd 12 1 945.2.ca.a 176
45.l even 12 1 315.2.bx.a yes 176
63.h even 3 1 945.2.by.a 176
63.j odd 6 1 inner 315.2.bv.a 176
105.x even 12 1 945.2.ca.a 176
315.bt odd 12 1 945.2.by.a 176
315.bv even 12 1 inner 315.2.bv.a 176

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bv.a 176 1.a even 1 1 trivial
315.2.bv.a 176 5.c odd 4 1 inner
315.2.bv.a 176 63.j odd 6 1 inner
315.2.bv.a 176 315.bv even 12 1 inner
315.2.bx.a yes 176 7.c even 3 1
315.2.bx.a yes 176 9.d odd 6 1
315.2.bx.a yes 176 35.l odd 12 1
315.2.bx.a yes 176 45.l even 12 1
945.2.by.a 176 3.b odd 2 1
945.2.by.a 176 15.e even 4 1
945.2.by.a 176 63.h even 3 1
945.2.by.a 176 315.bt odd 12 1
945.2.ca.a 176 9.c even 3 1
945.2.ca.a 176 21.h odd 6 1
945.2.ca.a 176 45.k odd 12 1
945.2.ca.a 176 105.x even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(315, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database