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Properties

Label 213.3.p.a
Level $213$
Weight $3$
Character orbit 213.p
Analytic conductor $5.804$
Analytic rank $0$
Dimension $576$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,3,Mod(7,213)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(213, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 213.p (of order \(70\), degree \(24\), 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: \(5.80382963087\)
Analytic rank: \(0\)
Dimension: \(576\)
Relative dimension: \(24\) over \(\Q(\zeta_{70})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{70}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 576 q - 6 q^{2} + 66 q^{4} + 28 q^{8} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 576 q - 6 q^{2} + 66 q^{4} + 28 q^{8} + 72 q^{9} + 98 q^{10} - 24 q^{12} + 80 q^{14} - 60 q^{15} + 382 q^{16} + 70 q^{17} - 30 q^{18} - 8 q^{19} - 256 q^{20} - 118 q^{22} - 780 q^{25} + 112 q^{26} + 70 q^{28} + 196 q^{29} + 192 q^{30} + 98 q^{31} - 556 q^{32} - 36 q^{33} - 128 q^{35} + 108 q^{36} + 436 q^{37} - 626 q^{38} + 128 q^{40} + 14 q^{41} - 30 q^{42} + 22 q^{43} + 792 q^{44} - 290 q^{46} - 966 q^{47} + 336 q^{48} - 420 q^{49} + 118 q^{50} + 224 q^{52} - 150 q^{53} - 230 q^{55} - 576 q^{56} + 576 q^{57} - 1162 q^{58} - 1032 q^{59} - 324 q^{60} - 1054 q^{61} - 642 q^{62} - 2640 q^{64} - 670 q^{65} - 180 q^{66} + 128 q^{67} - 968 q^{68} + 246 q^{69} + 458 q^{71} + 24 q^{72} + 1000 q^{73} + 1212 q^{74} - 924 q^{75} + 336 q^{76} + 1138 q^{77} + 1264 q^{79} + 3416 q^{80} + 216 q^{81} + 2462 q^{82} + 578 q^{83} - 540 q^{84} + 1950 q^{85} + 2736 q^{86} + 306 q^{87} + 2170 q^{88} - 414 q^{89} - 84 q^{90} - 532 q^{91} - 174 q^{92} - 504 q^{93} + 1484 q^{94} + 752 q^{95} + 300 q^{96} + 238 q^{97} - 314 q^{98} - 240 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −3.72675 − 1.02852i 0.680739 + 1.59267i 9.39705 + 5.61448i −2.76955 − 8.52380i −0.898858 − 6.63564i 1.90127 + 0.0853860i −18.5591 − 19.4113i −2.07319 + 2.16838i 1.55455 + 34.6146i
7.2 −3.47138 − 0.958041i −0.680739 − 1.59267i 7.69887 + 4.59986i −0.545020 − 1.67740i 0.837265 + 6.18094i 5.18205 + 0.232726i −12.3644 − 12.9321i −2.07319 + 2.16838i 0.284956 + 6.34505i
7.3 −3.26395 − 0.900793i −0.680739 − 1.59267i 6.40815 + 3.82869i 2.36861 + 7.28982i 0.787234 + 5.81160i −9.46539 − 0.425091i −8.10735 − 8.47962i −2.07319 + 2.16838i −1.16439 − 25.9272i
7.4 −2.98971 − 0.825108i 0.680739 + 1.59267i 4.82379 + 2.88208i 1.89643 + 5.83662i −0.721091 − 5.32331i 7.95980 + 0.357475i −3.47043 − 3.62979i −2.07319 + 2.16838i −0.853945 − 19.0146i
7.5 −2.78464 − 0.768512i 0.680739 + 1.59267i 3.72982 + 2.22847i −0.640326 − 1.97072i −0.671630 − 4.95817i −2.46896 − 0.110881i −0.688405 − 0.720015i −2.07319 + 2.16838i 0.268555 + 5.97985i
7.6 −2.70943 − 0.747756i −0.680739 − 1.59267i 3.34809 + 2.00039i −2.91829 − 8.98158i 0.653490 + 4.82426i −11.3565 − 0.510021i 0.193917 + 0.202822i −2.07319 + 2.16838i 1.19089 + 26.5172i
7.7 −2.19990 − 0.607133i −0.680739 − 1.59267i 1.03714 + 0.619660i −0.810967 − 2.49590i 0.530594 + 3.91700i 11.3156 + 0.508185i 4.40302 + 4.60520i −2.07319 + 2.16838i 0.268701 + 5.98308i
7.8 −1.46618 − 0.404639i 0.680739 + 1.59267i −1.44785 − 0.865052i −0.424213 − 1.30559i −0.353628 − 2.61059i 0.834337 + 0.0374701i 5.97717 + 6.25163i −2.07319 + 2.16838i 0.0936773 + 2.08589i
7.9 −1.27286 − 0.351287i −0.680739 − 1.59267i −1.93703 − 1.15732i 0.967530 + 2.97775i 0.307002 + 2.26638i −3.91248 − 0.175710i 5.70905 + 5.97120i −2.07319 + 2.16838i −0.185485 − 4.13014i
7.10 −0.949639 − 0.262084i 0.680739 + 1.59267i −2.60067 − 1.55383i −1.90796 − 5.87211i −0.229044 − 1.69087i −2.60420 − 0.116955i 4.78564 + 5.00539i −2.07319 + 2.16838i 0.272893 + 6.07643i
7.11 −0.924489 − 0.255143i 0.680739 + 1.59267i −2.64421 − 1.57984i 2.72598 + 8.38970i −0.222978 − 1.64609i −11.5907 − 0.520541i 4.69252 + 4.90799i −2.07319 + 2.16838i −0.379566 − 8.45170i
7.12 −0.0913600 − 0.0252138i −0.680739 − 1.59267i −3.42608 − 2.04699i −2.06890 − 6.36742i 0.0220352 + 0.162670i −0.542163 − 0.0243486i 0.523378 + 0.547411i −2.07319 + 2.16838i 0.0284681 + 0.633892i
7.13 0.0855200 + 0.0236020i 0.680739 + 1.59267i −3.42704 − 2.04756i 1.11154 + 3.42096i 0.0206266 + 0.152272i 9.79031 + 0.439683i −0.489990 − 0.512490i −2.07319 + 2.16838i 0.0143171 + 0.318795i
7.14 0.326703 + 0.0901642i −0.680739 − 1.59267i −3.33519 − 1.99268i 1.29645 + 3.99005i −0.0787976 − 0.581707i 10.0213 + 0.450056i −1.84680 − 1.93160i −2.07319 + 2.16838i 0.0637927 + 1.42045i
7.15 0.535617 + 0.147821i −0.680739 − 1.59267i −3.16876 − 1.89325i 2.30380 + 7.09037i −0.129186 − 0.953688i −1.03898 − 0.0466607i −2.95331 − 3.08892i −2.07319 + 2.16838i 0.185850 + 4.13827i
7.16 1.07920 + 0.297841i 0.680739 + 1.59267i −2.35783 − 1.40874i 1.19032 + 3.66343i 0.260293 + 1.92156i −10.0557 − 0.451600i −5.21970 − 5.45938i −2.07319 + 2.16838i 0.193477 + 4.30811i
7.17 1.20823 + 0.333449i 0.680739 + 1.59267i −2.08517 − 1.24583i −1.83589 − 5.65028i 0.291413 + 2.15130i −1.49765 − 0.0672595i −5.56863 − 5.82434i −2.07319 + 2.16838i −0.334085 − 7.43898i
7.18 1.72584 + 0.476302i −0.680739 − 1.59267i −0.682128 − 0.407553i 0.300950 + 0.926229i −0.416257 − 3.07293i −12.0694 − 0.542038i −5.93214 − 6.20453i −2.07319 + 2.16838i 0.0782274 + 1.74187i
7.19 2.30859 + 0.637131i −0.680739 − 1.59267i 1.48987 + 0.890156i −1.78832 − 5.50389i −0.556811 − 4.11054i 0.962728 + 0.0432362i −3.74774 − 3.91983i −2.07319 + 2.16838i −0.621808 − 13.8456i
7.20 2.57490 + 0.710627i 0.680739 + 1.59267i 2.69132 + 1.60799i 2.13902 + 6.58322i 0.621042 + 4.58471i 0.615905 + 0.0276603i −1.59656 − 1.66987i −2.07319 + 2.16838i 0.829541 + 18.4712i
See next 80 embeddings (of 576 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.h odd 70 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.3.p.a ✓ 576
71.h odd 70 1 inner 213.3.p.a ✓ 576
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.3.p.a ✓ 576 1.a even 1 1 trivial
213.3.p.a ✓ 576 71.h odd 70 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(213, [\chi])\).