Properties

Label 525.2.w.a
Level 525
Weight 2
Character orbit 525.w
Analytic conductor 4.192
Analytic rank 0
Dimension 304
CM no
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(304\)
Relative dimension: \(76\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 304q - 84q^{4} - 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 304q - 84q^{4} - 6q^{9} - 36q^{15} - 60q^{16} - 18q^{21} - 40q^{22} - 8q^{25} + 60q^{28} + 14q^{30} + 28q^{36} - 20q^{37} - 2q^{39} - 35q^{42} - 24q^{46} - 20q^{49} - 52q^{51} - 80q^{58} + 4q^{60} + 55q^{63} - 76q^{64} - 20q^{67} + 6q^{70} + 30q^{72} - 40q^{78} - 68q^{79} - 10q^{81} + 80q^{84} - 88q^{85} - 20q^{88} - 82q^{91} - 36q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
104.1 −0.840823 + 2.58779i −0.654037 + 1.60382i −4.37163 3.17618i 1.62287 + 1.53827i −3.60041 3.04104i −0.317962 + 2.62658i 7.49243 5.44357i −2.14447 2.09792i −5.34527 + 2.90624i
104.2 −0.840823 + 2.58779i 0.654037 1.60382i −4.37163 3.17618i −1.62287 1.53827i 3.60041 + 3.04104i 0.317962 + 2.62658i 7.49243 5.44357i −2.14447 2.09792i 5.34527 2.90624i
104.3 −0.825889 + 2.54182i −1.65709 0.504029i −4.16074 3.02296i −1.82317 + 1.29462i 2.64973 3.79577i 2.62130 0.358847i 6.79573 4.93738i 2.49191 + 1.67044i −1.78496 5.70340i
104.4 −0.825889 + 2.54182i 1.65709 + 0.504029i −4.16074 3.02296i 1.82317 1.29462i −2.64973 + 3.79577i −2.62130 0.358847i 6.79573 4.93738i 2.49191 + 1.67044i 1.78496 + 5.70340i
104.5 −0.780529 + 2.40222i −0.282161 + 1.70891i −3.54341 2.57444i −0.607661 2.15192i −3.88495 2.01167i 1.54353 2.14884i 4.86320 3.53332i −2.84077 0.964377i 5.64368 + 0.219898i
104.6 −0.780529 + 2.40222i 0.282161 1.70891i −3.54341 2.57444i 0.607661 + 2.15192i 3.88495 + 2.01167i −1.54353 2.14884i 4.86320 3.53332i −2.84077 0.964377i −5.64368 0.219898i
104.7 −0.742792 + 2.28608i −1.73121 0.0538885i −3.05638 2.22059i 2.20023 + 0.398729i 1.40912 3.91766i −0.0744001 2.64471i 3.45740 2.51195i 2.99419 + 0.186585i −2.54584 + 4.73373i
104.8 −0.742792 + 2.28608i 1.73121 + 0.0538885i −3.05638 2.22059i −2.20023 0.398729i −1.40912 + 3.91766i 0.0744001 2.64471i 3.45740 2.51195i 2.99419 + 0.186585i 2.54584 4.73373i
104.9 −0.715634 + 2.20250i −1.15693 1.28900i −2.72082 1.97679i 1.79617 1.33182i 3.66695 1.62568i −1.56332 + 2.13449i 2.55388 1.85551i −0.323024 + 2.98256i 1.64793 + 4.90917i
104.10 −0.715634 + 2.20250i 1.15693 + 1.28900i −2.72082 1.97679i −1.79617 + 1.33182i −3.66695 + 1.62568i 1.56332 + 2.13449i 2.55388 1.85551i −0.323024 + 2.98256i −1.64793 4.90917i
104.11 −0.677246 + 2.08435i −1.04597 + 1.38056i −2.26781 1.64766i −2.02988 + 0.937853i −2.16919 3.11514i −2.59371 0.522162i 1.42406 1.03464i −0.811893 2.88805i −0.580080 4.86614i
104.12 −0.677246 + 2.08435i 1.04597 1.38056i −2.26781 1.64766i 2.02988 0.937853i 2.16919 + 3.11514i 2.59371 0.522162i 1.42406 1.03464i −0.811893 2.88805i 0.580080 + 4.86614i
104.13 −0.605685 + 1.86411i −0.698093 1.58514i −1.49001 1.08256i −1.90667 1.16816i 3.37770 0.341225i −2.25871 1.37777i −0.250932 + 0.182313i −2.02533 + 2.21315i 3.33242 2.84671i
104.14 −0.605685 + 1.86411i 0.698093 + 1.58514i −1.49001 1.08256i 1.90667 + 1.16816i −3.37770 + 0.341225i 2.25871 1.37777i −0.250932 + 0.182313i −2.02533 + 2.21315i −3.33242 + 2.84671i
104.15 −0.588826 + 1.81222i −1.26929 1.17851i −1.31939 0.958593i −0.430450 2.19425i 2.88312 1.60630i 2.48611 0.905127i −0.569061 + 0.413447i 0.222220 + 2.99176i 4.22991 + 0.511957i
104.16 −0.588826 + 1.81222i 1.26929 + 1.17851i −1.31939 0.958593i 0.430450 + 2.19425i −2.88312 + 1.60630i −2.48611 0.905127i −0.569061 + 0.413447i 0.222220 + 2.99176i −4.22991 0.511957i
104.17 −0.534897 + 1.64624i −1.66454 + 0.478872i −0.805968 0.585570i −2.04228 0.910555i 0.102015 2.99638i 0.846893 + 2.50655i −1.40566 + 1.02127i 2.54136 1.59420i 2.59140 2.87503i
104.18 −0.534897 + 1.64624i 1.66454 0.478872i −0.805968 0.585570i 2.04228 + 0.910555i −0.102015 + 2.99638i −0.846893 + 2.50655i −1.40566 + 1.02127i 2.54136 1.59420i −2.59140 + 2.87503i
104.19 −0.507173 + 1.56092i −0.407162 1.68351i −0.561203 0.407738i 0.134898 + 2.23200i 2.83433 + 0.218287i 1.99984 + 1.73224i −1.73452 + 1.26020i −2.66844 + 1.37092i −3.55238 0.921442i
104.20 −0.507173 + 1.56092i 0.407162 + 1.68351i −0.561203 0.407738i −0.134898 2.23200i −2.83433 0.218287i −1.99984 + 1.73224i −1.73452 + 1.26020i −2.66844 + 1.37092i 3.55238 + 0.921442i
See next 80 embeddings (of 304 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.76
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner
175.m odd 10 1 inner
525.w 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.w.a 304
3.b odd 2 1 inner 525.2.w.a 304
7.b odd 2 1 inner 525.2.w.a 304
21.c even 2 1 inner 525.2.w.a 304
25.e even 10 1 inner 525.2.w.a 304
75.h odd 10 1 inner 525.2.w.a 304
175.m odd 10 1 inner 525.2.w.a 304
525.w even 10 1 inner 525.2.w.a 304
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.w.a 304 1.a even 1 1 trivial
525.2.w.a 304 3.b odd 2 1 inner
525.2.w.a 304 7.b odd 2 1 inner
525.2.w.a 304 21.c even 2 1 inner
525.2.w.a 304 25.e even 10 1 inner
525.2.w.a 304 75.h odd 10 1 inner
525.2.w.a 304 175.m odd 10 1 inner
525.2.w.a 304 525.w even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(525, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database