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Properties

Label 213.4.m.a
Level $213$
Weight $4$
Character orbit 213.m
Analytic conductor $12.567$
Analytic rank $0$
Dimension $432$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,4,Mod(4,213)]
 
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, 12]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 213.m (of order \(35\), 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: \(12.5674068312\)
Analytic rank: \(0\)
Dimension: \(432\)
Relative dimension: \(18\) over \(\Q(\zeta_{35})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 432 q + 3 q^{2} - 54 q^{3} + 47 q^{4} + 4 q^{5} + 6 q^{6} - 4 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 432 q + 3 q^{2} - 54 q^{3} + 47 q^{4} + 4 q^{5} + 6 q^{6} - 4 q^{8} + 162 q^{9} - 77 q^{10} - 14 q^{11} - 603 q^{12} + 2 q^{13} + 91 q^{14} + 60 q^{15} - 361 q^{16} - 248 q^{17} + 45 q^{18} + 166 q^{19} - 886 q^{20} - 545 q^{22} - 166 q^{23} - 657 q^{24} - 2220 q^{25} - 820 q^{26} - 486 q^{27} + 323 q^{28} - 1398 q^{29} + 270 q^{30} + 470 q^{31} + 1485 q^{32} - 504 q^{33} + 614 q^{34} + 1054 q^{35} + 774 q^{36} + 62 q^{37} - 1571 q^{38} + 384 q^{39} - 4648 q^{40} + 144 q^{41} + 4074 q^{42} - 632 q^{43} - 12681 q^{44} + 36 q^{45} + 4737 q^{46} - 1386 q^{47} + 3075 q^{48} + 2562 q^{49} + 407 q^{50} + 420 q^{51} - 3818 q^{52} - 2374 q^{53} - 81 q^{54} + 5578 q^{55} + 12909 q^{56} - 1494 q^{57} - 1855 q^{58} - 3808 q^{59} + 2103 q^{60} + 3018 q^{61} + 933 q^{62} - 2194 q^{64} - 3572 q^{65} + 4449 q^{66} - 5130 q^{67} - 4742 q^{68} - 1812 q^{69} - 11398 q^{70} - 2098 q^{71} - 216 q^{72} - 3120 q^{73} - 921 q^{74} - 720 q^{75} + 1964 q^{76} - 7376 q^{77} - 960 q^{78} + 122 q^{79} + 3615 q^{80} + 1458 q^{81} + 9819 q^{82} + 10872 q^{83} + 1632 q^{84} + 10266 q^{85} + 9638 q^{86} - 1056 q^{87} + 10449 q^{88} - 460 q^{89} + 1260 q^{90} - 2974 q^{91} + 3431 q^{92} + 2118 q^{93} - 9910 q^{94} - 16016 q^{95} + 1302 q^{96} + 6692 q^{97} + 27423 q^{98} + 1512 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5.43388 − 0.489058i −0.402700 − 2.97285i 21.4165 + 3.88652i −9.05912 + 6.58184i 0.734327 + 16.3511i 0.490597 + 0.293118i −72.4001 − 19.9811i −8.67567 + 2.39433i 52.4451 − 31.3345i
4.2 −4.68743 − 0.421876i −0.402700 − 2.97285i 13.9226 + 2.52658i 16.4913 − 11.9816i 0.633452 + 14.1049i −21.1907 − 12.6608i −27.9011 − 7.70020i −8.67567 + 2.39433i −82.3564 + 49.2057i
4.3 −4.33056 − 0.389758i −0.402700 − 2.97285i 10.7304 + 1.94728i 3.33927 − 2.42612i 0.585226 + 13.0311i −9.01877 − 5.38846i −12.1788 − 3.36112i −8.67567 + 2.39433i −15.4065 + 9.20496i
4.4 −3.73368 − 0.336037i −0.402700 − 2.97285i 5.95599 + 1.08085i 6.04527 − 4.39214i 0.504564 + 11.2350i 29.7760 + 17.7903i 7.03484 + 1.94149i −8.67567 + 2.39433i −24.0470 + 14.3674i
4.5 −3.66855 − 0.330176i −0.402700 − 2.97285i 5.47781 + 0.994077i −9.21349 + 6.69399i 0.495763 + 11.0390i 3.68540 + 2.20192i 8.63770 + 2.38385i −8.67567 + 2.39433i 36.0104 − 21.5152i
4.6 −2.59228 − 0.233310i −0.402700 − 2.97285i −1.20593 − 0.218844i −10.1408 + 7.36773i 0.350317 + 7.80042i −26.7962 − 16.0100i 23.1468 + 6.38811i −8.67567 + 2.39433i 28.0068 − 16.7333i
4.7 −1.70935 − 0.153845i −0.402700 − 2.97285i −4.97322 − 0.902506i 11.3401 − 8.23906i 0.230999 + 5.14360i 8.57207 + 5.12157i 21.5974 + 5.96051i −8.67567 + 2.39433i −20.6518 + 12.3389i
4.8 −0.911003 − 0.0819917i −0.402700 − 2.97285i −7.04823 − 1.27907i 6.37931 − 4.63484i 0.123112 + 2.74129i 0.536951 + 0.320813i 13.3699 + 3.68985i −8.67567 + 2.39433i −6.19159 + 3.69930i
4.9 −0.550565 − 0.0495518i −0.402700 − 2.97285i −7.57077 − 1.37389i −13.7536 + 9.99259i 0.0744025 + 1.65670i 19.3325 + 11.5507i 8.36308 + 2.30806i −8.67567 + 2.39433i 8.06741 − 4.82006i
4.10 −0.115787 − 0.0104210i −0.402700 − 2.97285i −7.85814 − 1.42604i −5.28128 + 3.83708i 0.0156472 + 0.348412i −25.9966 − 15.5323i 1.79153 + 0.494430i −8.67567 + 2.39433i 0.651488 − 0.389246i
4.11 0.890832 + 0.0801763i −0.402700 − 2.97285i −7.08428 − 1.28561i 0.159358 − 0.115780i −0.120386 − 2.68060i 4.37686 + 2.61505i −13.1054 − 3.61687i −8.67567 + 2.39433i 0.151244 − 0.0903640i
4.12 2.22468 + 0.200225i −0.402700 − 2.97285i −2.96232 − 0.537582i 11.6358 − 8.45390i −0.300640 − 6.69427i −19.0425 − 11.3774i −23.7080 − 6.54299i −8.67567 + 2.39433i 27.5786 − 16.4775i
4.13 2.64882 + 0.238398i −0.402700 − 2.97285i −0.912008 − 0.165505i −7.90183 + 5.74101i −0.357958 − 7.97055i 14.4127 + 8.61118i −22.8858 − 6.31608i −8.67567 + 2.39433i −22.2992 + 13.3231i
4.14 2.71499 + 0.244354i −0.402700 − 2.97285i −0.559968 − 0.101619i 13.9720 − 10.1513i −0.366900 − 8.16966i 26.5880 + 15.8856i −22.5173 − 6.21438i −8.67567 + 2.39433i 40.4144 − 24.1465i
4.15 3.57339 + 0.321611i −0.402700 − 2.97285i 4.79424 + 0.870026i −2.26149 + 1.64307i −0.482902 − 10.7527i −16.7022 − 9.97911i −10.8164 − 2.98514i −8.67567 + 2.39433i −8.60961 + 5.14400i
4.16 4.09538 + 0.368591i −0.402700 − 2.97285i 8.76483 + 1.59058i −16.6924 + 12.1277i −0.553443 − 12.3234i −7.51255 − 4.48854i 3.59903 + 0.993269i −8.67567 + 2.39433i −72.8317 + 43.5149i
4.17 4.98782 + 0.448912i −0.402700 − 2.97285i 16.8054 + 3.04972i 11.4532 − 8.32121i −0.674046 − 15.0088i −3.80246 − 2.27186i 43.8330 + 12.0971i −8.67567 + 2.39433i 60.8618 − 36.3632i
4.18 5.22062 + 0.469864i −0.402700 − 2.97285i 19.1627 + 3.47751i −6.94593 + 5.04651i −0.705507 − 15.7093i 22.2909 + 13.3182i 57.9845 + 16.0027i −8.67567 + 2.39433i −38.6332 + 23.0823i
10.1 −5.32917 + 1.47076i 1.17908 − 2.75858i 19.3693 − 11.5726i −0.147237 + 0.453148i −2.22628 + 16.4351i −1.32230 − 29.4433i −55.6380 + 58.1928i −6.21956 − 6.50515i 0.118178 − 2.63145i
10.2 −4.95049 + 1.36625i 1.17908 − 2.75858i 15.7731 − 9.42398i −3.85365 + 11.8603i −2.06809 + 15.2672i 1.37440 + 30.6034i −36.8171 + 38.5076i −6.21956 − 6.50515i 2.87332 − 63.9794i
See next 80 embeddings (of 432 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.g even 35 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.4.m.a ✓ 432
71.g even 35 1 inner 213.4.m.a ✓ 432
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.4.m.a ✓ 432 1.a even 1 1 trivial
213.4.m.a ✓ 432 71.g even 35 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{432} - 3 T_{2}^{431} - 91 T_{2}^{430} + 294 T_{2}^{429} + 3306 T_{2}^{428} + \cdots + 34\!\cdots\!96 \) acting on \(S_{4}^{\mathrm{new}}(213, [\chi])\). Copy content Toggle raw display