Properties

 Label 525.2.z.a Level 525 Weight 2 Character orbit 525.z Analytic conductor 4.192 Analytic rank 0 Dimension 56 CM no Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.z (of order $$10$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$14$$ over $$\Q(\zeta_{10})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$56q + 12q^{4} - 2q^{5} + 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q + 12q^{4} - 2q^{5} + 14q^{9} + 36q^{10} + 12q^{11} + 20q^{13} - 4q^{15} + 4q^{16} - 22q^{19} - 22q^{20} + 14q^{21} + 30q^{22} - 10q^{23} + 16q^{25} - 60q^{26} - 20q^{28} - 18q^{29} - 18q^{30} + 16q^{31} - 10q^{33} + 6q^{35} - 12q^{36} + 10q^{37} + 100q^{38} - 22q^{40} + 8q^{41} - 66q^{44} + 2q^{45} + 40q^{46} - 100q^{47} - 56q^{49} + 62q^{50} + 32q^{51} + 80q^{52} - 30q^{53} - 58q^{55} - 40q^{58} - 20q^{59} + 66q^{60} + 28q^{61} - 50q^{62} - 12q^{64} + 78q^{65} - 20q^{67} + 4q^{69} - 18q^{70} + 40q^{71} - 20q^{73} + 100q^{74} - 8q^{75} - 164q^{76} + 20q^{77} - 90q^{78} - 4q^{79} - 158q^{80} - 14q^{81} + 30q^{83} - 12q^{84} + 46q^{85} + 80q^{86} - 40q^{87} + 130q^{88} - 38q^{89} + 4q^{90} - 100q^{92} - 10q^{94} + 8q^{95} + 10q^{96} - 130q^{97} + 48q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1 −1.58398 2.18016i 0.951057 + 0.309017i −1.62607 + 5.00454i −0.548304 2.16780i −0.832747 2.56293i 1.00000i 8.36050 2.71649i 0.809017 + 0.587785i −3.85765 + 4.62914i
64.2 −1.23567 1.70075i −0.951057 0.309017i −0.747648 + 2.30102i 2.01518 + 0.969039i 0.649630 + 1.99936i 1.00000i 0.838609 0.272480i 0.809017 + 0.587785i −0.842003 4.62474i
64.3 −1.17586 1.61843i −0.951057 0.309017i −0.618636 + 1.90397i −0.258970 + 2.22102i 0.618185 + 1.90258i 1.00000i 0.00370838 0.00120493i 0.809017 + 0.587785i 3.89907 2.19248i
64.4 −1.00461 1.38273i 0.951057 + 0.309017i −0.284658 + 0.876087i −2.14464 + 0.632860i −0.528155 1.62549i 1.00000i −1.75363 + 0.569787i 0.809017 + 0.587785i 3.02960 + 2.32968i
64.5 −0.894373 1.23100i 0.951057 + 0.309017i −0.0974215 + 0.299833i 1.95748 + 1.08087i −0.470200 1.44713i 1.00000i −2.43803 + 0.792163i 0.809017 + 0.587785i −0.420165 3.37636i
64.6 −0.292589 0.402714i −0.951057 0.309017i 0.541463 1.66645i −1.58524 + 1.57703i 0.153823 + 0.473419i 1.00000i −1.77637 + 0.577177i 0.809017 + 0.587785i 1.09892 + 0.176977i
64.7 0.158974 + 0.218810i −0.951057 0.309017i 0.595429 1.83254i −0.756459 2.10423i −0.0835778 0.257226i 1.00000i 1.01009 0.328197i 0.809017 + 0.587785i 0.340167 0.500039i
64.8 0.341242 + 0.469679i 0.951057 + 0.309017i 0.513881 1.58156i 2.23604 + 0.0116437i 0.179402 + 0.552141i 1.00000i 2.02247 0.657140i 0.809017 + 0.587785i 0.757561 + 1.05419i
64.9 0.342442 + 0.471330i −0.951057 0.309017i 0.513148 1.57931i 2.09675 + 0.776951i −0.180032 0.554082i 1.00000i 2.02826 0.659022i 0.809017 + 0.587785i 0.351813 + 1.25432i
64.10 0.507633 + 0.698697i 0.951057 + 0.309017i 0.387548 1.19275i −2.23580 + 0.0345861i 0.266878 + 0.821367i 1.00000i 2.67284 0.868457i 0.809017 + 0.587785i −1.15913 1.54459i
64.11 0.975068 + 1.34207i −0.951057 0.309017i −0.232349 + 0.715098i −2.23159 0.141462i −0.512624 1.57769i 1.00000i 1.96912 0.639806i 0.809017 + 0.587785i −1.98610 3.13287i
64.12 1.14001 + 1.56909i 0.951057 + 0.309017i −0.544387 + 1.67545i −1.00833 + 1.99581i 0.599339 + 1.84458i 1.00000i 0.439612 0.142839i 0.809017 + 0.587785i −4.28112 + 0.693093i
64.13 1.22763 + 1.68969i −0.951057 0.309017i −0.729938 + 2.24652i 1.98040 1.03828i −0.645404 1.98635i 1.00000i −0.719317 + 0.233720i 0.809017 + 0.587785i 4.18557 + 2.07164i
64.14 1.49408 + 2.05642i 0.951057 + 0.309017i −1.37856 + 4.24278i 1.10152 1.94593i 0.785482 + 2.41746i 1.00000i −5.94967 + 1.93316i 0.809017 + 0.587785i 5.64741 0.642187i
169.1 −2.50474 + 0.813838i 0.587785 + 0.809017i 3.99333 2.90133i 0.517719 2.17531i −2.13065 1.54801i 1.00000i −4.54501 + 6.25567i −0.309017 + 0.951057i 0.473598 + 5.86991i
169.2 −2.47318 + 0.803584i −0.587785 0.809017i 3.85283 2.79924i −2.08479 0.808499i 2.10381 + 1.52851i 1.00000i −4.22228 + 5.81147i −0.309017 + 0.951057i 5.80574 + 0.324262i
169.3 −1.98360 + 0.644512i 0.587785 + 0.809017i 1.90126 1.38134i −1.68150 + 1.47397i −1.68735 1.22593i 1.00000i −0.429178 + 0.590713i −0.309017 + 0.951057i 2.38543 4.00751i
169.4 −1.32106 + 0.429238i −0.587785 0.809017i −0.0570843 + 0.0414741i 0.747101 2.10757i 1.12376 + 0.816459i 1.00000i 1.69053 2.32681i −0.309017 + 0.951057i −0.0823162 + 3.10490i
169.5 −0.752353 + 0.244454i −0.587785 0.809017i −1.11176 + 0.807738i 2.10831 + 0.745003i 0.639990 + 0.464980i 1.00000i 1.56894 2.15946i −0.309017 + 0.951057i −1.76831 0.0451201i
169.6 −0.748945 + 0.243347i 0.587785 + 0.809017i −1.11633 + 0.811064i −1.45716 1.69608i −0.637090 0.462873i 1.00000i 1.56445 2.15328i −0.309017 + 0.951057i 1.50407 + 0.915677i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 484.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.z.a 56
25.e even 10 1 inner 525.2.z.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.z.a 56 1.a even 1 1 trivial
525.2.z.a 56 25.e even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{56} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database