Properties

Label 300.2.r.a
Level $300$
Weight $2$
Character orbit 300.r
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.r (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.

\(\operatorname{Tr}(f)(q) = \) \( 224q - 6q^{4} - 7q^{6} - 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 224q - 6q^{4} - 7q^{6} - 6q^{9} - 5q^{12} - 20q^{13} + 6q^{16} - 24q^{21} - 10q^{22} - 28q^{24} - 20q^{25} - 50q^{28} - 25q^{30} - 10q^{33} - 14q^{34} - 25q^{36} - 60q^{37} - 20q^{40} - 85q^{42} - 30q^{45} + 6q^{46} - 70q^{48} + 88q^{49} + 40q^{52} - 46q^{54} + 50q^{58} + 10q^{60} - 28q^{61} + 18q^{64} + 64q^{66} - 42q^{69} - 40q^{70} - 5q^{72} - 20q^{73} - 80q^{76} - 85q^{78} + 18q^{81} - 48q^{84} - 20q^{85} + 50q^{88} - 60q^{90} + 60q^{94} + 44q^{96} - 120q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −1.41059 0.101112i −1.18137 + 1.26664i 1.97955 + 0.285257i −0.243022 + 2.22282i 1.79450 1.66726i −4.47645 −2.76350 0.602540i −0.208752 2.99273i 0.567561 3.11093i
59.2 −1.41029 0.105278i 1.45041 0.946744i 1.97783 + 0.296946i −2.23180 0.138020i −2.14516 + 1.18249i −1.71470 −2.75806 0.627002i 1.20735 2.74632i 3.13296 + 0.429608i
59.3 −1.40665 + 0.146030i 1.00033 + 1.41398i 1.95735 0.410828i −1.66487 1.49272i −1.61361 1.84290i −0.571505 −2.69332 + 0.863724i −0.998661 + 2.82890i 2.55988 + 1.85662i
59.4 −1.37885 0.314289i 1.03587 1.38816i 1.80244 + 0.866714i 1.94623 1.10099i −1.86458 + 1.58850i 2.63015 −2.21290 1.76156i −0.853964 2.87589i −3.02959 + 0.906413i
59.5 −1.37668 + 0.323633i −1.44316 0.957755i 1.79052 0.891082i 2.16605 + 0.555182i 2.29674 + 0.851472i 0.146941 −2.17660 + 1.80621i 1.16541 + 2.76438i −3.16164 0.0633046i
59.6 −1.32465 + 0.495274i 1.60045 + 0.662243i 1.50941 1.31213i 0.462356 + 2.18774i −2.44803 0.0845826i 2.28606 −1.34958 + 2.48569i 2.12287 + 2.11977i −1.69599 2.66901i
59.7 −1.30486 0.545281i 0.0653839 + 1.73082i 1.40534 + 1.42303i 2.14307 + 0.638169i 0.858463 2.29413i 4.06778 −1.05782 2.62317i −2.99145 + 0.226335i −2.44843 2.00130i
59.8 −1.29016 0.579209i −1.71582 0.236546i 1.32903 + 1.49455i −0.101169 2.23378i 2.07668 + 1.29900i −0.903444 −0.849014 2.69799i 2.88809 + 0.811741i −1.16330 + 2.94053i
59.9 −1.28571 0.589029i −0.760326 1.55625i 1.30609 + 1.51464i −1.70605 + 1.44547i 0.0608822 + 2.44873i 1.06519 −0.787085 2.71671i −1.84381 + 2.36651i 3.04491 0.853547i
59.10 −1.26137 + 0.639492i −0.689827 + 1.58875i 1.18210 1.61327i 1.00094 1.99953i −0.145871 2.44514i −0.923130 −0.459388 + 2.79087i −2.04828 2.19193i 0.0161381 + 3.16224i
59.11 −1.19645 + 0.753994i −0.269355 1.71098i 0.862987 1.80423i −1.27460 1.83722i 1.61234 + 1.84401i 4.55385 0.327858 + 2.80936i −2.85490 + 0.921722i 2.91025 + 1.23711i
59.12 −1.12183 + 0.861108i 1.64941 0.528616i 0.516985 1.93203i 1.99646 1.00704i −1.39516 + 2.01334i −3.00727 1.08372 + 2.61258i 2.44113 1.74381i −1.37251 + 2.84890i
59.13 −1.06241 + 0.933425i 0.154412 1.72515i 0.257435 1.98336i −0.778418 + 2.09620i 1.44625 + 1.97695i −3.14870 1.57782 + 2.34744i −2.95231 0.532768i −1.12965 2.95362i
59.14 −0.957505 1.04076i 0.760326 + 1.55625i −0.166367 + 1.99307i −1.70605 + 1.44547i 0.891666 2.28143i −1.06519 2.23361 1.73523i −1.84381 + 2.36651i 3.13794 + 0.391539i
59.15 −0.949542 1.04803i 1.71582 + 0.236546i −0.196739 + 1.99030i −0.101169 2.23378i −1.38134 2.02285i 0.903444 2.27271 1.68369i 2.88809 + 0.811741i −2.24500 + 2.22710i
59.16 −0.932419 + 1.06329i −1.73059 + 0.0711045i −0.261191 1.98287i −1.84864 1.25798i 1.53803 1.90643i −2.80461 2.35192 + 1.57114i 2.98989 0.246105i 3.06132 0.792685i
59.17 −0.921818 1.07250i −0.0653839 1.73082i −0.300505 + 1.97730i 2.14307 + 0.638169i −1.79603 + 1.66562i −4.06778 2.39766 1.50042i −2.99145 + 0.226335i −1.29108 2.88671i
59.18 −0.724994 1.21424i −1.03587 + 1.38816i −0.948767 + 1.76064i 1.94623 1.10099i 2.43656 + 0.251385i −2.63015 2.82569 0.124420i −0.853964 2.87589i −2.74787 1.56499i
59.19 −0.723120 + 1.21536i −1.44187 + 0.959691i −0.954195 1.75770i 1.84864 + 1.25798i −0.123723 2.44636i 2.80461 2.82623 + 0.111339i 1.15799 2.76750i −2.86569 + 1.33709i
59.20 −0.559437 + 1.29886i 1.13894 + 1.30492i −1.37406 1.45326i 0.778418 2.09620i −2.33207 + 0.749304i 3.14870 2.65627 0.971702i −0.405623 + 2.97245i 2.28719 + 2.18375i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.56
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.e even 10 1 inner
75.h odd 10 1 inner
100.h odd 10 1 inner
300.r 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.r.a 224
3.b odd 2 1 inner 300.2.r.a 224
4.b odd 2 1 inner 300.2.r.a 224
12.b even 2 1 inner 300.2.r.a 224
25.e even 10 1 inner 300.2.r.a 224
75.h odd 10 1 inner 300.2.r.a 224
100.h odd 10 1 inner 300.2.r.a 224
300.r even 10 1 inner 300.2.r.a 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.r.a 224 1.a even 1 1 trivial
300.2.r.a 224 3.b odd 2 1 inner
300.2.r.a 224 4.b odd 2 1 inner
300.2.r.a 224 12.b even 2 1 inner
300.2.r.a 224 25.e even 10 1 inner
300.2.r.a 224 75.h odd 10 1 inner
300.2.r.a 224 100.h odd 10 1 inner
300.2.r.a 224 300.r even 10 1 inner

Hecke kernels

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