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Properties

Label 245.4.q.b
Level $245$
Weight $4$
Character orbit 245.q
Analytic conductor $14.455$
Analytic rank $0$
Dimension $348$
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] = [245,4,Mod(11,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 40]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.q (of order \(21\), degree \(12\), 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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(348\)
Relative dimension: \(29\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 348 q + 2 q^{2} - 6 q^{3} + 126 q^{4} - 145 q^{5} + 80 q^{6} + 62 q^{7} - 48 q^{8} + 487 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 348 q + 2 q^{2} - 6 q^{3} + 126 q^{4} - 145 q^{5} + 80 q^{6} + 62 q^{7} - 48 q^{8} + 487 q^{9} - 10 q^{10} + 39 q^{11} + 319 q^{12} - 66 q^{13} + 168 q^{14} - 60 q^{15} + 366 q^{16} + 158 q^{17} + 183 q^{18} - 19 q^{19} + 1260 q^{20} - 178 q^{21} - 898 q^{22} + 413 q^{23} + 1288 q^{24} + 725 q^{25} - 337 q^{26} + 252 q^{27} - 471 q^{28} + 672 q^{29} - 80 q^{30} - 1036 q^{31} - 256 q^{32} + 1660 q^{33} - 1322 q^{34} - 360 q^{35} - 2424 q^{36} - 687 q^{37} + 2043 q^{38} - 270 q^{39} - 120 q^{40} + 1522 q^{41} - 537 q^{42} - 974 q^{43} - 1293 q^{44} - 1525 q^{45} - 3521 q^{46} - 2349 q^{47} - 10092 q^{48} - 1474 q^{49} + 600 q^{50} - 354 q^{51} - 5605 q^{52} + 2222 q^{53} - 6112 q^{54} + 145 q^{55} + 2780 q^{56} - 2164 q^{57} + 2360 q^{58} + 643 q^{59} + 2045 q^{60} + 1000 q^{61} + 2984 q^{62} - 4657 q^{63} - 2564 q^{64} + 2915 q^{65} - 4235 q^{66} - 5644 q^{67} + 2930 q^{68} + 7256 q^{69} - 865 q^{70} - 5268 q^{71} - 1630 q^{72} + 626 q^{73} + 3387 q^{74} - 150 q^{75} - 3450 q^{76} + 305 q^{77} - 21498 q^{78} - 8160 q^{79} + 16650 q^{80} + 17103 q^{81} + 24069 q^{82} + 4446 q^{83} + 28160 q^{84} - 450 q^{85} - 7437 q^{86} + 2494 q^{87} - 10930 q^{88} + 6174 q^{89} + 1165 q^{90} + 3333 q^{91} - 6874 q^{92} + 2960 q^{93} - 7671 q^{94} - 1095 q^{95} - 26303 q^{96} - 564 q^{97} - 10307 q^{98} + 32670 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −4.04840 + 3.75637i 0.525570 − 0.0792170i 1.68142 − 22.4370i −1.82671 + 4.65437i −1.83015 + 2.29494i −12.0658 − 14.0505i 49.9278 + 62.6075i −25.5305 + 7.87512i −10.0883 − 25.7045i
11.2 −4.04212 + 3.75054i 9.56369 − 1.44149i 1.67435 − 22.3426i −1.82671 + 4.65437i −33.2512 + 41.6957i 11.3298 + 14.6504i 49.5250 + 62.1024i 63.5857 − 19.6136i −10.0726 − 25.6646i
11.3 −3.45204 + 3.20302i −7.01816 + 1.05782i 1.05937 − 14.1363i −1.82671 + 4.65437i 20.8387 − 26.1310i 16.8453 − 7.69659i 18.1331 + 22.7382i 22.3351 − 6.88947i −8.60220 − 21.9180i
11.4 −3.44695 + 3.19830i 0.844124 − 0.127231i 1.05449 − 14.0711i −1.82671 + 4.65437i −2.50273 + 3.13832i −13.5174 + 12.6601i 17.9149 + 22.4645i −25.1041 + 7.74359i −8.58952 − 21.8857i
11.5 −3.00515 + 2.78837i 4.90472 − 0.739268i 0.658066 − 8.78129i −1.82671 + 4.65437i −12.6781 + 15.8978i 14.7317 − 11.2239i 2.05990 + 2.58304i −2.29069 + 0.706586i −7.48858 − 19.0806i
11.6 −2.85640 + 2.65035i −4.30171 + 0.648379i 0.536812 − 7.16326i −1.82671 + 4.65437i 10.5690 − 13.2531i 13.6457 + 12.5218i −1.98404 − 2.48791i −7.71615 + 2.38012i −7.11791 − 18.1361i
11.7 −2.72164 + 2.52531i −3.68140 + 0.554881i 0.432278 − 5.76834i −1.82671 + 4.65437i 8.61818 − 10.8069i −9.84766 − 15.6851i −5.12853 − 6.43098i −12.5557 + 3.87291i −6.78210 − 17.2805i
11.8 −2.47374 + 2.29529i 8.91929 − 1.34437i 0.253168 − 3.37829i −1.82671 + 4.65437i −18.9783 + 23.7980i −9.92078 − 15.6390i −9.70421 − 12.1687i 51.9460 − 16.0232i −6.16436 − 15.7065i
11.9 −2.20101 + 2.04224i 5.87757 − 0.885901i 0.0758653 − 1.01235i −1.82671 + 4.65437i −11.1274 + 13.9533i −14.6756 + 11.2972i −13.0759 − 16.3967i 7.96050 − 2.45549i −5.48474 − 13.9749i
11.10 −1.98644 + 1.84314i −7.65696 + 1.15410i −0.0490894 + 0.655052i −1.82671 + 4.65437i 13.0829 − 16.4054i 0.268895 + 18.5183i −14.6262 − 18.3407i 31.4966 − 9.71540i −4.95004 − 12.6125i
11.11 −1.08337 + 1.00522i 4.20418 − 0.633679i −0.434618 + 5.79958i −1.82671 + 4.65437i −3.91770 + 4.91264i 16.0985 + 9.15636i −12.7306 − 15.9637i −8.52686 + 2.63019i −2.69967 − 6.87865i
11.12 −1.08151 + 1.00349i −9.88321 + 1.48965i −0.435179 + 5.80705i −1.82671 + 4.65437i 9.19392 − 11.5288i −11.7253 − 14.3359i −12.7156 − 15.9449i 69.6582 − 21.4867i −2.69503 − 6.86683i
11.13 −0.872013 + 0.809110i −3.69164 + 0.556425i −0.492093 + 6.56652i −1.82671 + 4.65437i 2.76895 − 3.47215i −13.6667 + 12.4989i −10.8174 − 13.5646i −12.4819 + 3.85015i −2.17299 − 5.53668i
11.14 −0.841581 + 0.780873i −1.88399 + 0.283965i −0.499345 + 6.66330i −1.82671 + 4.65437i 1.36379 − 1.71013i 10.4072 − 15.3196i −10.5093 − 13.1783i −22.3317 + 6.88842i −2.09715 − 5.34345i
11.15 0.286574 − 0.265901i 5.24489 − 0.790540i −0.586420 + 7.82523i −1.82671 + 4.65437i 1.29284 − 1.62117i −16.8067 − 7.78033i 3.86262 + 4.84358i 1.08344 − 0.334197i 0.714118 + 1.81954i
11.16 0.593269 − 0.550473i −7.73514 + 1.16588i −0.548893 + 7.32447i −1.82671 + 4.65437i −3.94723 + 4.94967i 17.9447 + 4.58133i 7.74307 + 9.70950i 32.6726 − 10.0782i 1.47838 + 3.76684i
11.17 0.742066 − 0.688537i 9.15994 − 1.38064i −0.521261 + 6.95575i −1.82671 + 4.65437i 5.84666 − 7.33148i 4.62452 + 17.9336i 9.45173 + 11.8521i 56.1978 − 17.3347i 1.84917 + 4.71160i
11.18 0.765929 − 0.710678i 2.05268 − 0.309392i −0.516257 + 6.88897i −1.82671 + 4.65437i 1.35233 − 1.69577i −18.2278 − 3.27812i 9.71205 + 12.1785i −21.6827 + 6.68823i 1.90863 + 4.86312i
11.19 1.26895 − 1.17741i −0.173241 + 0.0261119i −0.373909 + 4.98947i −1.82671 + 4.65437i −0.189090 + 0.237112i 16.4521 + 8.50465i 14.0345 + 17.5988i −25.7711 + 7.94934i 3.16212 + 8.05695i
11.20 1.42461 − 1.32184i −6.30691 + 0.950614i −0.315602 + 4.21142i −1.82671 + 4.65437i −7.72832 + 9.69100i −16.6385 + 8.13390i 14.8107 + 18.5720i 13.0730 − 4.03250i 3.55001 + 9.04527i
See next 80 embeddings (of 348 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.q.b ✓ 348
49.g even 21 1 inner 245.4.q.b ✓ 348
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.q.b ✓ 348 1.a even 1 1 trivial
245.4.q.b ✓ 348 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{348} - 2 T_{2}^{347} - 177 T_{2}^{346} + 394 T_{2}^{345} + 13981 T_{2}^{344} + \cdots + 40\!\cdots\!56 \) acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\). Copy content Toggle raw display