Properties

Label 211.4.f.a
Level $211$
Weight $4$
Character orbit 211.f
Analytic conductor $12.449$
Analytic rank $0$
Dimension $312$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [211,4,Mod(58,211)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(211, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([12]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("211.58");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 211.f (of order \(7\), degree \(6\), 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: \(12.4494030112\)
Analytic rank: \(0\)
Dimension: \(312\)
Relative dimension: \(52\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 2 q^{2} - 3 q^{3} - 246 q^{4} - 5 q^{5} - 29 q^{6} + q^{7} - 150 q^{8} - 517 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 312 q - 2 q^{2} - 3 q^{3} - 246 q^{4} - 5 q^{5} - 29 q^{6} + q^{7} - 150 q^{8} - 517 q^{9} + 106 q^{10} + 29 q^{11} + 83 q^{12} - 9 q^{13} + 148 q^{14} + 442 q^{15} - 126 q^{16} + 121 q^{17} + 87 q^{18} - 438 q^{19} + 257 q^{20} - 390 q^{21} - 461 q^{22} + 1352 q^{23} - 323 q^{24} - 1045 q^{25} + 377 q^{26} - 348 q^{27} + 1755 q^{28} + 105 q^{29} + 754 q^{30} - 697 q^{31} - 1152 q^{32} - 363 q^{33} - 1841 q^{34} + 675 q^{35} + 1541 q^{36} + 75 q^{37} + 387 q^{38} - 1385 q^{39} - 5123 q^{40} - 1183 q^{41} + 1439 q^{42} + 149 q^{43} + 259 q^{44} - 386 q^{45} + 3112 q^{46} - 499 q^{47} + 1767 q^{48} - 811 q^{49} - 3164 q^{50} - 102 q^{51} + 3968 q^{52} + 67 q^{53} - 3502 q^{54} + 538 q^{55} + 3388 q^{56} - 51 q^{57} - 19 q^{58} - 2695 q^{59} - 4141 q^{60} + 4100 q^{61} - 4982 q^{62} - 1130 q^{63} - 5120 q^{64} + 555 q^{65} - 565 q^{66} - 3768 q^{67} - 1527 q^{68} - 1732 q^{69} - 9089 q^{70} + 4640 q^{71} - 206 q^{72} + 1078 q^{73} + 6972 q^{74} + 350 q^{75} + 6704 q^{76} + 4898 q^{77} - 827 q^{78} - 1145 q^{79} + 3773 q^{80} + 5181 q^{81} - 2160 q^{82} - 2130 q^{83} + 6491 q^{84} + 2481 q^{85} - 2378 q^{86} - 3157 q^{87} - 4122 q^{88} + 2097 q^{89} - 5114 q^{90} + 5685 q^{91} - 2043 q^{92} + 4597 q^{93} + 1045 q^{94} - 318 q^{95} + 10909 q^{96} - 247 q^{97} - 1291 q^{98} + 9995 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1 −4.97605 + 2.39634i −1.94388 + 0.936125i 14.0307 17.5940i −2.40114 1.15633i 7.42959 9.31641i −15.4643 7.44721i −17.8246 + 78.0948i −13.9319 + 17.4700i 14.7192
58.2 −4.90121 + 2.36030i 4.10047 1.97468i 13.4629 16.8819i 14.3224 + 6.89730i −15.4364 + 19.3566i 24.7699 + 11.9285i −16.4541 + 72.0900i −3.91977 + 4.91524i −86.4767
58.3 −4.48223 + 2.15853i 7.21184 3.47304i 10.4432 13.0954i 1.78326 + 0.858773i −24.8285 + 31.1339i −22.0474 10.6174i −9.68599 + 42.4371i 23.1145 28.9846i −9.84667
58.4 −4.27434 + 2.05841i −7.67440 + 3.69580i 9.04499 11.3421i −12.8775 6.20149i 25.1955 31.5942i −1.72606 0.831228i −6.86931 + 30.0964i 28.4033 35.6166i 67.8082
58.5 −4.26590 + 2.05435i −6.92250 + 3.33370i 8.98962 11.2726i 12.9045 + 6.21448i 22.6821 28.4425i 20.2172 + 9.73607i −6.76219 + 29.6271i 19.9733 25.0457i −67.8160
58.6 −4.23580 + 2.03986i 0.891304 0.429229i 8.79311 11.0262i −14.0955 6.78802i −2.89982 + 3.63626i 16.2686 + 7.83456i −6.38475 + 27.9734i −16.2240 + 20.3443i 73.5522
58.7 −4.21502 + 2.02985i 6.96432 3.35384i 8.65822 10.8571i −14.7232 7.09032i −22.5470 + 28.2730i 6.24622 + 3.00802i −6.12821 + 26.8495i 20.4193 25.6050i 76.4509
58.8 −4.11840 + 1.98331i −6.02527 + 2.90162i 8.03973 10.0815i 13.2398 + 6.37594i 19.0596 23.9000i −24.3098 11.7070i −4.97871 + 21.8132i 11.0503 13.8566i −67.1721
58.9 −3.98998 + 1.92147i 0.271323 0.130662i 7.23994 9.07860i 3.61502 + 1.74090i −0.831507 + 1.04268i 4.01486 + 1.93345i −3.55938 + 15.5947i −16.7777 + 21.0385i −17.7689
58.10 −3.21096 + 1.54632i 3.63743 1.75169i 2.93123 3.67565i 17.1840 + 8.27538i −8.97095 + 11.2492i −17.4875 8.42155i 2.61598 11.4613i −6.67179 + 8.36616i −67.9734
58.11 −3.20031 + 1.54119i 0.0460954 0.0221984i 2.87883 3.60993i 7.24370 + 3.48838i −0.113308 + 0.142084i −7.06865 3.40408i 2.67375 11.7144i −16.8326 + 21.1074i −28.5584
58.12 −3.08366 + 1.48501i −4.08815 + 1.96875i 2.31577 2.90388i −0.273383 0.131654i 9.68284 12.1419i 17.9048 + 8.62251i 3.26406 14.3008i −3.99722 + 5.01235i 1.03853
58.13 −3.06711 + 1.47704i 7.77421 3.74386i 2.23761 2.80587i 4.61133 + 2.22070i −18.3145 + 22.9657i 22.9736 + 11.0635i 3.34152 14.6401i 29.5876 37.1016i −17.4235
58.14 −2.84004 + 1.36769i −2.86373 + 1.37910i 1.20732 1.51394i −12.4872 6.01350i 6.24692 7.83339i −27.9663 13.4678i 4.25320 18.6345i −10.5352 + 13.2107i 43.6886
58.15 −2.59971 + 1.25195i 4.51364 2.17366i 0.203164 0.254760i −11.4836 5.53021i −9.01283 + 11.3017i −23.1270 11.1374i 4.92737 21.5882i −1.18602 + 1.48723i 36.7775
58.16 −2.08199 + 1.00264i 5.67503 2.73295i −1.65849 + 2.07969i −2.24841 1.08278i −9.07523 + 11.3800i −0.706202 0.340089i 5.48149 24.0160i 7.90271 9.90969i 5.76681
58.17 −2.03194 + 0.978531i −7.76198 + 3.73797i −1.81666 + 2.27801i −5.73829 2.76341i 12.1142 15.1907i −2.97528 1.43282i 5.47701 23.9964i 29.4417 36.9187i 14.3640
58.18 −1.90830 + 0.918988i −5.61985 + 2.70637i −2.19085 + 2.74724i −8.82929 4.25196i 8.23722 10.3291i 26.0543 + 12.5471i 5.42661 23.7755i 7.42397 9.30937i 20.7564
58.19 −1.73917 + 0.837542i −0.342599 + 0.164987i −2.66467 + 3.34139i 16.3636 + 7.88027i 0.457655 0.573881i 19.5807 + 9.42957i 5.27209 23.0985i −16.7441 + 20.9964i −35.0591
58.20 −1.60260 + 0.771774i −7.73273 + 3.72389i −3.01521 + 3.78096i 10.4825 + 5.04811i 9.51851 11.9358i −10.0207 4.82572i 5.08063 22.2597i 29.0935 36.4822i −20.6953
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.f even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 211.4.f.a 312
211.f even 7 1 inner 211.4.f.a 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.4.f.a 312 1.a even 1 1 trivial
211.4.f.a 312 211.f even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(211, [\chi])\).