Properties

Label 300.2.n.a
Level $300$
Weight $2$
Character orbit 300.n
Analytic conductor $2.396$
Analytic rank $0$
Dimension $224$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 300.n (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.39551206064\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(56\) over \(\Q(\zeta_{10})\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 224q - 6q^{4} + q^{6} - 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 224q - 6q^{4} + q^{6} - 6q^{9} - 8q^{10} - 9q^{12} - 12q^{13} - 18q^{16} - 26q^{18} + 12q^{21} - 6q^{22} - 16q^{24} - 12q^{25} + 2q^{28} - 13q^{30} + 6q^{33} - 30q^{34} + 35q^{36} + 12q^{37} - 24q^{40} - 13q^{42} - 6q^{45} - 18q^{46} - 34q^{48} - 168q^{49} - 28q^{52} - 38q^{54} - 44q^{57} - 34q^{58} - 76q^{60} + 4q^{61} + 18q^{64} - 46q^{66} - 18q^{69} + 72q^{70} - 29q^{72} - 20q^{73} + 16q^{76} + 5q^{78} - 30q^{81} - 20q^{82} - 18q^{84} - 76q^{85} + 6q^{88} + 2q^{90} - 52q^{93} + 96q^{94} - 50q^{96} - 72q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.40873 + 0.124406i 0.673811 1.59561i 1.96905 0.350510i −2.20342 0.380687i −0.750715 + 2.33161i 0.0822219i −2.73025 + 0.738736i −2.09196 2.15028i 3.15139 + 0.262166i
11.2 −1.40679 + 0.144680i −1.60959 + 0.639710i 1.95814 0.407070i 2.17904 + 0.501802i 2.17180 1.13282i 2.67645i −2.69580 + 0.855967i 2.18154 2.05934i −3.13805 0.390668i
11.3 −1.39445 + 0.235585i −1.04145 1.38397i 1.88900 0.657024i 1.89530 1.18653i 1.77830 + 1.68454i 4.72716i −2.47934 + 1.36121i −0.830762 + 2.88268i −2.36338 + 2.10106i
11.4 −1.39367 0.240196i −0.354401 + 1.69541i 1.88461 + 0.669506i −0.834949 + 2.07433i 0.901146 2.27770i 4.41884i −2.46571 1.38574i −2.74880 1.20171i 1.66189 2.69038i
11.5 −1.38528 0.284622i 1.64016 + 0.556662i 1.83798 + 0.788561i 1.45398 + 1.69881i −2.11364 1.23796i 2.78321i −2.32167 1.61550i 2.38025 + 1.82603i −1.53064 2.76716i
11.6 −1.31876 0.510760i −1.38129 1.04500i 1.47825 + 1.34714i −1.62369 + 1.53741i 1.28785 + 2.08361i 1.57078i −1.26139 2.53158i 0.815942 + 2.88691i 2.92650 1.19816i
11.7 −1.29913 0.558792i 1.06350 1.36711i 1.37550 + 1.45189i 1.51671 1.64305i −2.14555 + 1.18178i 1.88972i −0.975659 2.65482i −0.737954 2.90782i −2.88853 + 1.28702i
11.8 −1.25692 + 0.648193i 1.55000 0.772971i 1.15969 1.62945i 0.411823 + 2.19782i −1.44720 + 1.97626i 3.34784i −0.401439 + 2.79979i 1.80503 2.39622i −1.94224 2.49554i
11.9 −1.24841 + 0.664442i 1.66858 + 0.464581i 1.11703 1.65899i 0.345709 2.20918i −2.39175 + 0.528689i 0.309856i −0.292213 + 2.81329i 2.56833 + 1.55038i 1.03629 + 2.98766i
11.10 −1.24461 + 0.671529i −1.27306 + 1.17444i 1.09810 1.67158i −2.10329 0.759058i 0.795787 2.31662i 0.738888i −0.244189 + 2.81787i 0.241361 2.99028i 3.12750 0.467689i
11.11 −1.21555 0.722800i −1.65104 + 0.523516i 0.955121 + 1.75720i −0.573135 2.16137i 2.38532 + 0.557011i 1.39878i 0.109105 2.82632i 2.45186 1.72869i −0.865563 + 3.04151i
11.12 −1.03327 0.965583i 0.539210 + 1.64598i 0.135297 + 1.99542i −1.86451 1.23434i 1.03218 2.22140i 3.63805i 1.78694 2.19245i −2.41850 + 1.77506i 0.734688 + 3.07575i
11.13 −1.02327 + 0.976179i 0.339606 + 1.69843i 0.0941493 1.99778i 2.10329 + 0.759058i −2.00548 1.40643i 0.738888i 1.85385 + 2.13617i −2.76934 + 1.15359i −2.89320 + 1.27647i
11.14 −1.01770 + 0.981981i −1.62298 0.604914i 0.0714277 1.99872i −0.345709 + 2.20918i 2.24573 0.978119i 0.309856i 1.89002 + 2.10424i 2.26816 + 1.96353i −1.81755 2.58776i
11.15 −1.00488 + 0.995099i −0.799639 1.53642i 0.0195579 1.99990i −0.411823 2.19782i 2.33243 + 0.748191i 3.34784i 1.97045 + 2.02912i −1.72115 + 2.45716i 2.60088 + 1.79873i
11.16 −0.923787 1.07080i 1.20611 + 1.24310i −0.293236 + 1.97839i 2.16410 0.562738i 0.216925 2.43987i 3.72853i 2.38935 1.51361i −0.0905939 + 2.99863i −2.60175 1.79747i
11.17 −0.884246 1.10368i 0.846655 1.51102i −0.436219 + 1.95185i 0.448288 + 2.19067i −2.41633 + 0.401675i 2.41810i 2.53994 1.24447i −1.56635 2.55862i 2.02140 2.43186i
11.18 −0.776415 1.18202i −0.846655 + 1.51102i −0.794359 + 1.83548i 0.448288 + 2.19067i 2.44341 0.172411i 2.41810i 2.78634 0.486145i −1.56635 2.55862i 2.24137 2.23076i
11.19 −0.732928 1.20947i −1.20611 1.24310i −0.925633 + 1.77291i 2.16410 0.562738i −0.619499 + 2.36986i 3.72853i 2.82270 0.179888i −0.0905939 + 2.99863i −2.26674 2.20497i
11.20 −0.654964 + 1.25340i 1.65603 0.507508i −1.14204 1.64187i −1.89530 + 1.18653i −0.448527 + 2.40807i 4.72716i 2.80592 0.356077i 2.48487 1.68090i −0.245847 3.15271i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner
100.j odd 10 1 inner
300.n even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.n.a 224
3.b odd 2 1 inner 300.2.n.a 224
4.b odd 2 1 inner 300.2.n.a 224
12.b even 2 1 inner 300.2.n.a 224
25.d even 5 1 inner 300.2.n.a 224
75.j odd 10 1 inner 300.2.n.a 224
100.j odd 10 1 inner 300.2.n.a 224
300.n even 10 1 inner 300.2.n.a 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.n.a 224 1.a even 1 1 trivial
300.2.n.a 224 3.b odd 2 1 inner
300.2.n.a 224 4.b odd 2 1 inner
300.2.n.a 224 12.b even 2 1 inner
300.2.n.a 224 25.d even 5 1 inner
300.2.n.a 224 75.j odd 10 1 inner
300.2.n.a 224 100.j odd 10 1 inner
300.2.n.a 224 300.n even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(300, [\chi])\).