↑

Properties

Label 751.2.d.c
Level $751$
Weight $2$
Character orbit 751.d
Analytic conductor $5.997$
Analytic rank $0$
Dimension $236$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [751,2,Mod(80,751)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(751, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("751.80");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 751.d (of order \(5\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.99676519180\)
Analytic rank: \(0\)
Dimension: \(236\)
Relative dimension: \(59\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 236 q + 2 q^{3} - 66 q^{4} - 4 q^{5} - 10 q^{6} + 9 q^{7} + 18 q^{8} - 65 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 236 q + 2 q^{3} - 66 q^{4} - 4 q^{5} - 10 q^{6} + 9 q^{7} + 18 q^{8} - 65 q^{9} - 11 q^{10} - 8 q^{11} + q^{12} - 20 q^{13} + 21 q^{14} + 9 q^{15} - 112 q^{16} - 7 q^{17} + 30 q^{18} + 13 q^{19} + 10 q^{20} + 14 q^{21} + 16 q^{22} + 36 q^{23} + 3 q^{24} - 57 q^{25} - 20 q^{26} + 2 q^{27} - 48 q^{28} + 48 q^{29} - 7 q^{30} + 4 q^{31} + 26 q^{32} - 32 q^{33} + 52 q^{34} + 49 q^{35} - 38 q^{36} - 38 q^{37} - 16 q^{38} - 9 q^{39} - 3 q^{40} - 90 q^{41} - 75 q^{42} + 48 q^{43} - 27 q^{44} - 9 q^{45} - 4 q^{46} - 3 q^{47} + 90 q^{48} + 12 q^{49} + 59 q^{50} - 76 q^{51} - 8 q^{52} - 38 q^{53} + 48 q^{54} + 51 q^{55} + 23 q^{56} - 46 q^{57} + 39 q^{58} + 16 q^{59} - 90 q^{60} + 68 q^{61} + 37 q^{62} + 5 q^{63} - 88 q^{64} - 54 q^{65} + 50 q^{66} - 13 q^{67} + 74 q^{68} + 71 q^{69} - 49 q^{70} - 41 q^{71} + 232 q^{72} - 130 q^{73} - 3 q^{74} + 46 q^{75} + 66 q^{76} - 55 q^{77} - 28 q^{78} + 56 q^{79} - 84 q^{80} - 82 q^{81} + 79 q^{82} - 132 q^{83} + 89 q^{84} - 48 q^{85} - 18 q^{86} + 41 q^{87} + 45 q^{88} - 32 q^{89} + 208 q^{90} + 26 q^{91} - 112 q^{92} - 115 q^{93} + 46 q^{94} + 10 q^{95} + 197 q^{96} + 55 q^{97} + 28 q^{98} - 57 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1 −2.27665 − 1.65408i 1.32225 + 0.960670i 1.82911 + 5.62944i −1.42261 − 1.03359i −1.42127 − 4.37422i 3.71046 + 2.69580i 3.40810 − 10.4890i −0.101596 − 0.312679i 1.52915 + 4.70623i
80.2 −2.24723 − 1.63271i 2.63504 + 1.91447i 1.76627 + 5.43601i 1.13617 + 0.825477i −2.79577 − 8.60449i −3.12410 − 2.26979i 3.18947 − 9.81618i 2.35120 + 7.23624i −1.20547 − 3.71007i
80.3 −2.16602 − 1.57371i −1.48005 − 1.07532i 1.59706 + 4.91524i 2.84073 + 2.06391i 1.51358 + 4.65832i 3.51197 + 2.55159i 2.62119 − 8.06718i 0.107184 + 0.329877i −2.90509 − 8.94093i
80.4 −2.08466 − 1.51459i −0.459110 − 0.333563i 1.43377 + 4.41269i −2.12337 − 1.54272i 0.451875 + 1.39073i 0.421548 + 0.306272i 2.10197 − 6.46920i −0.827533 − 2.54689i 2.08991 + 6.43209i
80.5 −2.03217 − 1.47646i −1.15815 − 0.841448i 1.33176 + 4.09874i −0.309303 − 0.224722i 1.11121 + 3.41994i −2.21190 − 1.60704i 1.79281 − 5.51771i −0.293765 − 0.904115i 0.296764 + 0.913346i
80.6 −1.93435 − 1.40539i 0.289760 + 0.210523i 1.14857 + 3.53493i 1.74071 + 1.26470i −0.264632 − 0.814452i −1.99270 − 1.44778i 1.26850 − 3.90405i −0.887410 − 2.73117i −1.58976 − 4.89276i
80.7 −1.93144 − 1.40328i −2.31494 − 1.68190i 1.14326 + 3.51859i −3.24593 − 2.35831i 2.11100 + 6.49699i 0.810088 + 0.588563i 1.25392 − 3.85916i 1.60309 + 4.93382i 2.95998 + 9.10989i
80.8 −1.88356 − 1.36849i 1.20426 + 0.874948i 1.05702 + 3.25317i 1.71286 + 1.24447i −1.07095 − 3.29604i 0.500165 + 0.363391i 1.02205 − 3.14554i −0.242336 − 0.745834i −1.52324 − 4.68807i
80.9 −1.87668 − 1.36349i −2.41211 − 1.75250i 1.04479 + 3.21554i 0.0616945 + 0.0448237i 2.13725 + 6.57777i 1.00149 + 0.727628i 0.989951 − 3.04675i 1.81997 + 5.60130i −0.0546642 − 0.168239i
80.10 −1.79052 − 1.30089i 0.520349 + 0.378056i 0.895615 + 2.75642i 0.500569 + 0.363685i −0.439887 − 1.35383i 1.67413 + 1.21633i 0.614343 − 1.89075i −0.799214 − 2.45973i −0.423166 − 1.30237i
80.11 −1.67633 − 1.21792i 2.12560 + 1.54434i 0.708701 + 2.18116i −2.73054 − 1.98385i −1.68231 − 5.17763i −0.305710 − 0.222111i 0.187868 − 0.578197i 1.20614 + 3.71212i 2.16110 + 6.65117i
80.12 −1.56828 − 1.13942i −1.82480 − 1.32579i 0.543179 + 1.67173i 3.52591 + 2.56173i 1.35115 + 4.15842i −2.50630 − 1.82094i −0.145106 + 0.446590i 0.645110 + 1.98544i −2.61072 − 8.03498i
80.13 −1.47288 − 1.07011i 2.45990 + 1.78722i 0.406210 + 1.25019i 2.77287 + 2.01461i −1.71062 − 5.26474i 1.72255 + 1.25150i −0.385642 + 1.18689i 1.92990 + 5.93962i −1.92826 − 5.93457i
80.14 −1.39830 − 1.01592i −0.378109 − 0.274712i 0.305105 + 0.939016i −1.94158 − 1.41064i 0.249622 + 0.768259i 3.15899 + 2.29514i −0.540864 + 1.66461i −0.859552 − 2.64543i 1.28181 + 3.94500i
80.15 −1.18390 − 0.860153i −0.246723 − 0.179255i 0.0437201 + 0.134557i −2.05782 − 1.49509i 0.137909 + 0.424440i −1.74835 − 1.27025i −0.840439 + 2.58661i −0.898311 − 2.76472i 1.15024 + 3.54008i
80.16 −1.17113 − 0.850878i 2.23971 + 1.62724i 0.0295256 + 0.0908705i −0.724964 − 0.526717i −1.23841 − 3.81144i −0.562486 − 0.408670i −0.851925 + 2.62196i 1.44132 + 4.43593i 0.400857 + 1.23371i
80.17 −1.16722 − 0.848032i 1.55558 + 1.13020i 0.0252000 + 0.0775576i 0.965576 + 0.701532i −0.857256 − 2.63836i −3.64351 − 2.64716i −0.855316 + 2.63239i 0.215439 + 0.663052i −0.532114 − 1.63768i
80.18 −1.15599 − 0.839877i 1.18089 + 0.857967i 0.0128895 + 0.0396697i −0.367796 − 0.267219i −0.644512 − 1.98361i 1.59495 + 1.15880i −0.864682 + 2.66122i −0.268657 − 0.826841i 0.200738 + 0.617807i
80.19 −1.06090 − 0.770789i 0.0460677 + 0.0334701i −0.0866413 − 0.266655i 2.65324 + 1.92769i −0.0230748 − 0.0710169i 3.20632 + 2.32953i −0.924072 + 2.84400i −0.926049 − 2.85009i −1.32898 − 4.09018i
80.20 −0.669284 − 0.486263i −2.01632 − 1.46494i −0.406545 − 1.25122i 1.12175 + 0.814997i 0.637143 + 1.96092i −0.146245 − 0.106253i −0.847614 + 2.60869i 0.992438 + 3.05441i −0.354464 − 1.09093i
See next 80 embeddings (of 236 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.59
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
751.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 751.2.d.c ✓ 236
751.d even 5 1 inner 751.2.d.c ✓ 236
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
751.2.d.c ✓ 236 1.a even 1 1 trivial
751.2.d.c ✓ 236 751.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(751, [\chi])\):

\( T_{2}^{236} + 92 T_{2}^{234} - 6 T_{2}^{233} + 4462 T_{2}^{232} - 586 T_{2}^{231} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
\( T_{3}^{236} - 2 T_{3}^{235} + 123 T_{3}^{234} - 258 T_{3}^{233} + 7991 T_{3}^{232} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display