Properties

Label 430.2.p.a
Level 430
Weight 2
Character orbit 430.p
Analytic conductor 3.434
Analytic rank 0
Dimension 132
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{14})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 132q + 22q^{4} - 4q^{5} - 4q^{6} + 12q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 132q + 22q^{4} - 4q^{5} - 4q^{6} + 12q^{9} + 8q^{11} - 10q^{14} - 20q^{15} - 22q^{16} - 4q^{19} + 4q^{20} - 24q^{21} + 4q^{24} - 16q^{25} + 12q^{26} + 40q^{29} + 24q^{31} - 4q^{35} + 128q^{36} - 56q^{39} - 28q^{41} - 8q^{44} - 80q^{45} - 12q^{46} - 136q^{49} - 56q^{50} + 172q^{51} + 16q^{54} - 4q^{56} + 16q^{59} - 8q^{60} - 44q^{61} + 22q^{64} + 42q^{65} + 8q^{66} - 62q^{69} + 12q^{70} - 8q^{71} - 4q^{74} - 122q^{75} - 52q^{76} - 64q^{79} - 4q^{80} - 56q^{81} + 24q^{84} - 72q^{85} - 10q^{86} + 16q^{89} + 2q^{90} - 24q^{91} - 20q^{94} + 106q^{95} - 4q^{96} - 104q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −0.974928 + 0.222521i −2.62517 0.599178i 0.900969 0.433884i −2.23603 + 0.0136065i 2.69268 3.67204i −0.781831 + 0.623490i 3.82959 + 1.84424i 2.17694 0.510828i
59.2 −0.974928 + 0.222521i −2.59733 0.592824i 0.900969 0.433884i −0.425312 2.19525i 2.66413 4.04951i −0.781831 + 0.623490i 3.69178 + 1.77787i 0.903137 + 2.04557i
59.3 −0.974928 + 0.222521i −2.18942 0.499720i 0.900969 0.433884i 2.05139 + 0.889832i 2.24572 1.14095i −0.781831 + 0.623490i 1.84092 + 0.886539i −2.19796 0.411045i
59.4 −0.974928 + 0.222521i −1.09644 0.250255i 0.900969 0.433884i 1.67031 1.48663i 1.12463 4.66029i −0.781831 + 0.623490i −1.56336 0.752874i −1.29763 + 1.82103i
59.5 −0.974928 + 0.222521i −0.468512 0.106935i 0.900969 0.433884i −2.22867 0.181698i 0.480560 0.219529i −0.781831 + 0.623490i −2.49484 1.20145i 2.21323 0.318784i
59.6 −0.974928 + 0.222521i 0.271317 + 0.0619263i 0.900969 0.433884i −0.882134 + 2.05471i −0.278294 4.13300i −0.781831 + 0.623490i −2.63313 1.26805i 0.402800 2.19949i
59.7 −0.974928 + 0.222521i 0.607117 + 0.138570i 0.900969 0.433884i −0.987643 2.00613i −0.622730 0.713157i −0.781831 + 0.623490i −2.35352 1.13339i 1.40929 + 1.73606i
59.8 −0.974928 + 0.222521i 1.43560 + 0.327666i 0.900969 0.433884i 0.699215 + 2.12393i −1.47252 4.43745i −0.781831 + 0.623490i −0.749332 0.360859i −1.15430 1.91509i
59.9 −0.974928 + 0.222521i 1.83835 + 0.419592i 0.900969 0.433884i 1.44701 1.70474i −1.88563 0.473994i −0.781831 + 0.623490i 0.500573 + 0.241063i −1.03139 + 1.98399i
59.10 −0.974928 + 0.222521i 2.10581 + 0.480636i 0.900969 0.433884i 2.17028 + 0.538429i −2.15996 1.96131i −0.781831 + 0.623490i 1.50050 + 0.722602i −2.23567 0.0419976i
59.11 −0.974928 + 0.222521i 3.15256 + 0.719552i 0.900969 0.433884i −1.55589 + 1.60599i −3.23364 0.962433i −0.781831 + 0.623490i 6.71798 + 3.23521i 1.15952 1.91194i
59.12 0.974928 0.222521i −3.15256 0.719552i 0.900969 0.433884i 1.91194 1.15952i −3.23364 0.962433i 0.781831 0.623490i 6.71798 + 3.23521i 1.60599 1.55589i
59.13 0.974928 0.222521i −2.10581 0.480636i 0.900969 0.433884i 0.0419976 + 2.23567i −2.15996 1.96131i 0.781831 0.623490i 1.50050 + 0.722602i 0.538429 + 2.17028i
59.14 0.974928 0.222521i −1.83835 0.419592i 0.900969 0.433884i −1.98399 + 1.03139i −1.88563 0.473994i 0.781831 0.623490i 0.500573 + 0.241063i −1.70474 + 1.44701i
59.15 0.974928 0.222521i −1.43560 0.327666i 0.900969 0.433884i 1.91509 + 1.15430i −1.47252 4.43745i 0.781831 0.623490i −0.749332 0.360859i 2.12393 + 0.699215i
59.16 0.974928 0.222521i −0.607117 0.138570i 0.900969 0.433884i −1.73606 1.40929i −0.622730 0.713157i 0.781831 0.623490i −2.35352 1.13339i −2.00613 0.987643i
59.17 0.974928 0.222521i −0.271317 0.0619263i 0.900969 0.433884i 2.19949 0.402800i −0.278294 4.13300i 0.781831 0.623490i −2.63313 1.26805i 2.05471 0.882134i
59.18 0.974928 0.222521i 0.468512 + 0.106935i 0.900969 0.433884i 0.318784 2.21323i 0.480560 0.219529i 0.781831 0.623490i −2.49484 1.20145i −0.181698 2.22867i
59.19 0.974928 0.222521i 1.09644 + 0.250255i 0.900969 0.433884i −1.82103 + 1.29763i 1.12463 4.66029i 0.781831 0.623490i −1.56336 0.752874i −1.48663 + 1.67031i
59.20 0.974928 0.222521i 2.18942 + 0.499720i 0.900969 0.433884i 0.411045 + 2.19796i 2.24572 1.14095i 0.781831 0.623490i 1.84092 + 0.886539i 0.889832 + 2.05139i
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
43.e even 7 1 inner
215.p even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.p.a 132
5.b even 2 1 inner 430.2.p.a 132
43.e even 7 1 inner 430.2.p.a 132
215.p even 14 1 inner 430.2.p.a 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.p.a 132 1.a even 1 1 trivial
430.2.p.a 132 5.b even 2 1 inner
430.2.p.a 132 43.e even 7 1 inner
430.2.p.a 132 215.p even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(430, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database