Properties

Label 225.4.q.a
Level $225$
Weight $4$
Character orbit 225.q
Analytic conductor $13.275$
Analytic rank $0$
Dimension $704$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,4,Mod(16,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 6]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.16");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.q (of order \(15\), degree \(8\), 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: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(704\)
Relative dimension: \(88\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 704 q - 3 q^{2} - 2 q^{3} + 341 q^{4} - 20 q^{5} - 22 q^{6} - 8 q^{7} - 44 q^{8} - 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 704 q - 3 q^{2} - 2 q^{3} + 341 q^{4} - 20 q^{5} - 22 q^{6} - 8 q^{7} - 44 q^{8} - 38 q^{9} - 32 q^{10} + 85 q^{11} - 172 q^{12} - 3 q^{13} + 157 q^{14} + 423 q^{15} + 1325 q^{16} + 84 q^{17} - 492 q^{18} - 12 q^{19} - 137 q^{20} + 123 q^{21} - 35 q^{22} - 131 q^{23} - 66 q^{24} + 284 q^{25} + 2432 q^{26} + 1060 q^{27} + 340 q^{28} + 345 q^{29} - 78 q^{30} - 39 q^{31} - 372 q^{32} + 16 q^{33} + 13 q^{34} + 1784 q^{35} + 584 q^{36} - 156 q^{37} + 1795 q^{38} + 392 q^{39} + 169 q^{40} + 653 q^{41} - 1328 q^{42} - 8 q^{43} - 2060 q^{44} - 2375 q^{45} - 44 q^{46} + 1343 q^{47} - 235 q^{48} - 14904 q^{49} - 909 q^{50} - 236 q^{51} - 283 q^{52} + 1788 q^{53} - 49 q^{54} - 1274 q^{55} + 72 q^{56} + 3236 q^{57} + 469 q^{58} + 915 q^{59} + 291 q^{60} - 3 q^{61} + 6088 q^{62} + 2881 q^{63} - 9692 q^{64} + 120 q^{65} + 2638 q^{66} - 921 q^{67} + 2484 q^{68} - 2624 q^{69} - 1762 q^{70} + 24 q^{71} - 2272 q^{72} - 12 q^{73} - 3798 q^{74} - 2717 q^{75} + 88 q^{76} + 4759 q^{77} + 2068 q^{78} - 255 q^{79} - 15302 q^{80} - 158 q^{81} - 3808 q^{82} - 1027 q^{83} - 3353 q^{84} + 949 q^{85} + 4545 q^{86} + 4366 q^{87} + 2869 q^{88} - 524 q^{89} - 139 q^{90} + 2046 q^{91} + 299 q^{92} + 1622 q^{93} - 2519 q^{94} - 2177 q^{95} + 4733 q^{96} + 1941 q^{97} + 2486 q^{98} - 470 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −4.99262 + 2.22286i −0.539608 5.16806i 14.6321 16.2506i −11.1000 + 1.33789i 14.1819 + 24.6027i 8.61806 14.9269i −23.4194 + 72.0775i −26.4176 + 5.57745i 52.4442 31.3533i
16.2 −4.95886 + 2.20783i 5.16428 + 0.574673i 14.3627 15.9514i 5.23221 9.88049i −26.8777 + 8.55210i 11.6110 20.1109i −22.5857 + 69.5116i 26.3395 + 5.93554i −4.13139 + 60.5477i
16.3 −4.88581 + 2.17530i 3.34361 + 3.97747i 13.7862 15.3111i −5.63199 + 9.65819i −24.9884 12.1598i −3.78510 + 6.55599i −20.8289 + 64.1048i −4.64059 + 26.5982i 6.50737 59.4394i
16.4 −4.86733 + 2.16708i −2.70456 + 4.43682i 13.6417 15.1506i 9.93766 5.12279i 3.54905 27.4565i −12.9967 + 22.5110i −20.3946 + 62.7682i −12.3708 23.9993i −37.2684 + 46.4700i
16.5 −4.73105 + 2.10640i −4.59183 2.43210i 12.5929 13.9858i 6.40846 + 9.16142i 26.8472 + 1.83415i −0.678500 + 1.17520i −17.3152 + 53.2908i 15.1698 + 22.3356i −49.6164 29.8444i
16.6 −4.55459 + 2.02783i −2.55636 + 4.52383i 11.2791 12.5267i 2.67963 + 10.8545i 2.46961 25.7880i 15.1279 26.2024i −13.6445 + 41.9935i −13.9300 23.1291i −34.2157 44.0038i
16.7 −4.46544 + 1.98814i −5.18899 + 0.272686i 10.6344 11.8107i −11.1764 + 0.297118i 22.6290 11.5341i −12.8663 + 22.2852i −11.9221 + 36.6924i 26.8513 2.82993i 49.3168 23.5470i
16.8 −4.41308 + 1.96483i 3.87578 3.46097i 10.2617 11.3968i −6.88435 8.80941i −10.3039 + 22.8888i −13.7602 + 23.8334i −10.9508 + 33.7031i 3.04337 26.8279i 47.6902 + 25.3501i
16.9 −4.40428 + 1.96091i −4.10868 + 3.18099i 10.1995 11.3277i −7.46697 8.32132i 11.8582 22.0668i 13.0363 22.5795i −10.7905 + 33.2096i 6.76255 26.1394i 49.2040 + 22.0074i
16.10 −4.39884 + 1.95849i 0.262179 5.18953i 10.1611 11.2850i 9.88111 5.23102i 9.01036 + 23.3414i −5.56077 + 9.63153i −10.6917 + 32.9056i −26.8625 2.72118i −33.2205 + 42.3625i
16.11 −4.21154 + 1.87510i 1.87188 + 4.84727i 8.86800 9.84891i −8.20714 7.59229i −16.9726 16.9045i −4.19737 + 7.27006i −7.48344 + 23.0317i −19.9921 + 18.1470i 48.8009 + 16.5860i
16.12 −4.11599 + 1.83256i 5.19523 + 0.0977072i 8.23008 9.14043i 9.68777 + 5.58096i −21.5626 + 9.11840i −10.5565 + 18.2844i −5.98634 + 18.4240i 26.9809 + 1.01522i −50.1022 5.21779i
16.13 −4.10662 + 1.82839i −3.62152 3.72620i 8.16829 9.07181i 2.22738 10.9562i 21.6852 + 8.68055i 5.37596 9.31143i −5.84444 + 17.9873i −0.769138 + 26.9890i 10.8852 + 49.0655i
16.14 −3.94276 + 1.75543i 4.68747 2.24224i 7.11079 7.89733i −7.42625 + 8.35768i −14.5455 + 17.0691i 3.16503 5.48200i −3.50346 + 10.7825i 16.9447 21.0208i 14.6086 45.9886i
16.15 −3.79552 + 1.68988i 2.86836 4.33273i 6.19727 6.88277i 9.82087 + 5.34326i −3.56515 + 21.2921i 17.2551 29.8867i −1.61984 + 4.98535i −10.5450 24.8556i −46.3048 3.68441i
16.16 −3.51417 + 1.56461i 3.51869 + 3.82345i 4.54836 5.05146i 9.92601 + 5.14531i −18.3475 7.93089i 3.48252 6.03190i 1.42952 4.39961i −2.23760 + 26.9071i −42.9321 2.55117i
16.17 −3.44830 + 1.53528i 0.918757 + 5.11428i 4.18063 4.64306i 6.75551 8.90859i −11.0200 16.2250i 5.43165 9.40790i 2.04374 6.28999i −25.3118 + 9.39757i −9.61783 + 41.0911i
16.18 −3.38912 + 1.50893i −0.512070 5.17086i 3.85619 4.28273i −0.951580 + 11.1398i 9.53794 + 16.7520i −8.94160 + 15.4873i 2.56453 7.89280i −26.4756 + 5.29569i −13.5841 39.1899i
16.19 −3.33000 + 1.48261i −5.18559 + 0.331159i 3.53770 3.92901i 10.7249 3.15860i 16.7770 8.79097i 5.87729 10.1798i 3.05594 9.40521i 26.7807 3.43451i −31.0309 + 26.4190i
16.20 −3.24605 + 1.44523i −4.88286 + 1.77700i 3.09509 3.43745i −4.43805 + 10.2618i 13.2818 12.8251i −4.89407 + 8.47677i 3.70520 11.4034i 20.6845 17.3537i −0.424499 39.7242i
See next 80 embeddings (of 704 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
25.d even 5 1 inner
225.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.q.a 704
9.c even 3 1 inner 225.4.q.a 704
25.d even 5 1 inner 225.4.q.a 704
225.q even 15 1 inner 225.4.q.a 704
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.q.a 704 1.a even 1 1 trivial
225.4.q.a 704 9.c even 3 1 inner
225.4.q.a 704 25.d even 5 1 inner
225.4.q.a 704 225.q even 15 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(225, [\chi])\).