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Properties

Label 46.4.c.b.9.1
Level $46$
Weight $4$
Character 46.9
Analytic conductor $2.714$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [46,4,Mod(3,46)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("46.3"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(46, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 46 = 2 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 46.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.71408786026\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 9.1
Character \(\chi\) \(=\) 46.9
Dual form 46.4.c.b.41.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.91899 - 0.563465i) q^{2} +(-3.58705 - 4.13968i) q^{3} +(3.36501 - 2.16256i) q^{4} +(-1.27675 - 8.88002i) q^{5} +(-9.21607 - 5.92281i) q^{6} +(1.52711 - 3.34391i) q^{7} +(5.23889 - 6.04600i) q^{8} +(-0.427496 + 2.97330i) q^{9} +(-7.45366 - 16.3212i) q^{10} +(15.1410 + 4.44581i) q^{11} +(-21.0228 - 6.17285i) q^{12} +(22.1543 + 48.5111i) q^{13} +(1.04633 - 7.27738i) q^{14} +(-32.1806 + 37.1384i) q^{15} +(6.64664 - 14.5541i) q^{16} +(75.1229 + 48.2785i) q^{17} +(0.854991 + 5.94659i) q^{18} +(-31.8031 + 20.4386i) q^{19} +(-23.4999 - 27.1203i) q^{20} +(-19.3205 + 5.67302i) q^{21} +31.5605 q^{22} +(-47.5380 - 99.5346i) q^{23} -43.8206 q^{24} +(42.7119 - 12.5414i) q^{25} +(69.8481 + 80.6090i) q^{26} +(-110.575 + 71.0622i) q^{27} +(-2.09266 - 14.5548i) q^{28} +(123.602 + 79.4344i) q^{29} +(-40.8280 + 89.4008i) q^{30} +(-29.4250 + 33.9582i) q^{31} +(4.55407 - 31.6743i) q^{32} +(-35.9074 - 78.6263i) q^{33} +(171.363 + 50.3167i) q^{34} +(-31.6437 - 9.29143i) q^{35} +(4.99141 + 10.9297i) q^{36} +(7.66648 - 53.3215i) q^{37} +(-49.5132 + 57.1413i) q^{38} +(121.352 - 265.723i) q^{39} +(-60.3774 - 38.8022i) q^{40} +(-35.1771 - 244.662i) q^{41} +(-33.8793 + 21.7729i) q^{42} +(-270.256 - 311.893i) q^{43} +(60.5641 - 17.7832i) q^{44} +26.9487 q^{45} +(-147.309 - 164.220i) q^{46} +64.2627 q^{47} +(-84.0912 + 24.6914i) q^{48} +(215.768 + 249.009i) q^{49} +(74.8970 - 48.1334i) q^{50} +(-69.6120 - 484.162i) q^{51} +(179.458 + 115.331i) q^{52} +(-34.9398 + 76.5074i) q^{53} +(-172.151 + 198.672i) q^{54} +(20.1475 - 140.129i) q^{55} +(-12.2169 - 26.7513i) q^{56} +(198.689 + 58.3402i) q^{57} +(281.950 + 82.7879i) q^{58} +(352.579 + 772.040i) q^{59} +(-27.9741 + 194.564i) q^{60} +(-578.678 + 667.830i) q^{61} +(-37.3318 + 81.7453i) q^{62} +(9.28959 + 5.97006i) q^{63} +(-9.10815 - 63.3486i) q^{64} +(402.494 - 258.667i) q^{65} +(-113.209 - 130.650i) q^{66} +(-729.127 + 214.091i) q^{67} +357.195 q^{68} +(-241.520 + 553.828i) q^{69} -65.9592 q^{70} +(613.322 - 180.087i) q^{71} +(15.7369 + 18.1614i) q^{72} +(-610.359 + 392.254i) q^{73} +(-15.3330 - 106.643i) q^{74} +(-205.127 - 131.827i) q^{75} +(-62.8181 + 137.552i) q^{76} +(37.9884 - 43.8409i) q^{77} +(83.1465 - 578.297i) q^{78} +(-65.1569 - 142.674i) q^{79} +(-137.727 - 40.4403i) q^{80} +(768.632 + 225.691i) q^{81} +(-205.363 - 449.681i) q^{82} +(-57.2679 + 398.307i) q^{83} +(-52.7456 + 60.8716i) q^{84} +(332.801 - 728.732i) q^{85} +(-694.359 - 446.237i) q^{86} +(-114.535 - 796.609i) q^{87} +(106.201 - 68.2515i) q^{88} +(892.963 + 1030.53i) q^{89} +(51.7143 - 15.1847i) q^{90} +196.049 q^{91} +(-375.216 - 232.132i) q^{92} +246.125 q^{93} +(123.319 - 36.2098i) q^{94} +(222.100 + 256.317i) q^{95} +(-147.457 + 94.7649i) q^{96} +(-153.264 - 1065.97i) q^{97} +(554.363 + 356.267i) q^{98} +(-19.6914 + 43.1182i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q + 6 q^{2} + 2 q^{3} - 12 q^{4} - 4 q^{6} - 115 q^{7} + 24 q^{8} - 83 q^{9} - 30 q^{11} + 52 q^{12} + 104 q^{13} - 56 q^{14} + 492 q^{15} - 48 q^{16} + 274 q^{17} + 166 q^{18} - 381 q^{19} - 176 q^{20}+ \cdots + 7587 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/46\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(e\left(\frac{5}{11}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.91899 − 0.563465i 0.678464 − 0.199215i
\(3\) −3.58705 − 4.13968i −0.690328 − 0.796681i 0.297084 − 0.954851i \(-0.403986\pi\)
−0.987412 + 0.158170i \(0.949441\pi\)
\(4\) 3.36501 − 2.16256i 0.420627 − 0.270320i
\(5\) −1.27675 − 8.88002i −0.114196 − 0.794253i −0.963761 − 0.266768i \(-0.914044\pi\)
0.849564 − 0.527485i \(-0.176865\pi\)
\(6\) −9.21607 − 5.92281i −0.627074 − 0.402996i
\(7\) 1.52711 − 3.34391i 0.0824562 − 0.180554i −0.863913 − 0.503641i \(-0.831993\pi\)
0.946369 + 0.323087i \(0.104721\pi\)
\(8\) 5.23889 − 6.04600i 0.231528 − 0.267198i
\(9\) −0.427496 + 2.97330i −0.0158332 + 0.110122i
\(10\) −7.45366 − 16.3212i −0.235705 − 0.516123i
\(11\) 15.1410 + 4.44581i 0.415017 + 0.121860i 0.482574 − 0.875855i \(-0.339702\pi\)
−0.0675564 + 0.997715i \(0.521520\pi\)
\(12\) −21.0228 − 6.17285i −0.505730 − 0.148496i
\(13\) 22.1543 + 48.5111i 0.472653 + 1.03497i 0.984419 + 0.175841i \(0.0562644\pi\)
−0.511765 + 0.859125i \(0.671008\pi\)
\(14\) 1.04633 − 7.27738i 0.0199745 − 0.138926i
\(15\) −32.1806 + 37.1384i −0.553934 + 0.639274i
\(16\) 6.64664 − 14.5541i 0.103854 − 0.227408i
\(17\) 75.1229 + 48.2785i 1.07176 + 0.688780i 0.952640 − 0.304099i \(-0.0983555\pi\)
0.119123 + 0.992880i \(0.461992\pi\)
\(18\) 0.854991 + 5.94659i 0.0111957 + 0.0778681i
\(19\) −31.8031 + 20.4386i −0.384007 + 0.246786i −0.718375 − 0.695657i \(-0.755114\pi\)
0.334368 + 0.942443i \(0.391477\pi\)
\(20\) −23.4999 − 27.1203i −0.262737 − 0.303215i
\(21\) −19.3205 + 5.67302i −0.200766 + 0.0589502i
\(22\) 31.5605 0.305851
\(23\) −47.5380 − 99.5346i −0.430972 − 0.902365i
\(24\) −43.8206 −0.372702
\(25\) 42.7119 − 12.5414i 0.341696 − 0.100331i
\(26\) 69.8481 + 80.6090i 0.526859 + 0.608028i
\(27\) −110.575 + 71.0622i −0.788154 + 0.506516i
\(28\) −2.09266 − 14.5548i −0.0141241 − 0.0982354i
\(29\) 123.602 + 79.4344i 0.791461 + 0.508641i 0.872819 − 0.488044i \(-0.162289\pi\)
−0.0813578 + 0.996685i \(0.525926\pi\)
\(30\) −40.8280 + 89.4008i −0.248471 + 0.544076i
\(31\) −29.4250 + 33.9582i −0.170480 + 0.196744i −0.834560 − 0.550917i \(-0.814278\pi\)
0.664080 + 0.747662i \(0.268823\pi\)
\(32\) 4.55407 − 31.6743i 0.0251579 − 0.174977i
\(33\) −35.9074 − 78.6263i −0.189415 − 0.414760i
\(34\) 171.363 + 50.3167i 0.864368 + 0.253801i
\(35\) −31.6437 − 9.29143i −0.152822 − 0.0448725i
\(36\) 4.99141 + 10.9297i 0.0231084 + 0.0506003i
\(37\) 7.66648 − 53.3215i 0.0340638 − 0.236919i −0.965675 − 0.259752i \(-0.916359\pi\)
0.999739 + 0.0228325i \(0.00726844\pi\)
\(38\) −49.5132 + 57.1413i −0.211371 + 0.243935i
\(39\) 121.352 − 265.723i 0.498252 − 1.09102i
\(40\) −60.3774 − 38.8022i −0.238662 − 0.153379i
\(41\) −35.1771 − 244.662i −0.133993 − 0.931945i −0.940276 − 0.340414i \(-0.889433\pi\)
0.806282 − 0.591531i \(-0.201476\pi\)
\(42\) −33.8793 + 21.7729i −0.124469 + 0.0799912i
\(43\) −270.256 − 311.893i −0.958459 − 1.10612i −0.994285 − 0.106762i \(-0.965952\pi\)
0.0358259 − 0.999358i \(-0.488594\pi\)
\(44\) 60.5641 − 17.7832i 0.207509 − 0.0609300i
\(45\) 26.9487 0.0892729
\(46\) −147.309 − 164.220i −0.472164 − 0.526366i
\(47\) 64.2627 0.199440 0.0997200 − 0.995016i \(-0.468205\pi\)
0.0997200 + 0.995016i \(0.468205\pi\)
\(48\) −84.0912 + 24.6914i −0.252865 + 0.0742478i
\(49\) 215.768 + 249.009i 0.629060 + 0.725974i
\(50\) 74.8970 − 48.1334i 0.211841 − 0.136142i
\(51\) −69.6120 − 484.162i −0.191130 − 1.32934i
\(52\) 179.458 + 115.331i 0.478583 + 0.307567i
\(53\) −34.9398 + 76.5074i −0.0905537 + 0.198285i −0.949490 − 0.313798i \(-0.898399\pi\)
0.858936 + 0.512083i \(0.171126\pi\)
\(54\) −172.151 + 198.672i −0.433829 + 0.500665i
\(55\) 20.1475 − 140.129i 0.0493943 − 0.343545i
\(56\) −12.2169 − 26.7513i −0.0291527 − 0.0638355i
\(57\) 198.689 + 58.3402i 0.461701 + 0.135568i
\(58\) 281.950 + 82.7879i 0.638307 + 0.187424i
\(59\) 352.579 + 772.040i 0.777997 + 1.70358i 0.708196 + 0.706016i \(0.249509\pi\)
0.0698010 + 0.997561i \(0.477764\pi\)
\(60\) −27.9741 + 194.564i −0.0601907 + 0.418635i
\(61\) −578.678 + 667.830i −1.21462 + 1.40175i −0.324592 + 0.945854i \(0.605227\pi\)
−0.890033 + 0.455897i \(0.849319\pi\)
\(62\) −37.3318 + 81.7453i −0.0764701 + 0.167446i
\(63\) 9.28959 + 5.97006i 0.0185774 + 0.0119390i
\(64\) −9.10815 − 63.3486i −0.0177894 − 0.123728i
\(65\) 402.494 − 258.667i 0.768050 − 0.493596i
\(66\) −113.209 − 130.650i −0.211137 − 0.243665i
\(67\) −729.127 + 214.091i −1.32951 + 0.390379i −0.867915 − 0.496713i \(-0.834540\pi\)
−0.461593 + 0.887092i \(0.652722\pi\)
\(68\) 357.195 0.637004
\(69\) −241.520 + 553.828i −0.421386 + 0.966276i
\(70\) −65.9592 −0.112623
\(71\) 613.322 − 180.087i 1.02518 − 0.301020i 0.274431 − 0.961607i \(-0.411510\pi\)
0.750750 + 0.660586i \(0.229692\pi\)
\(72\) 15.7369 + 18.1614i 0.0257586 + 0.0297270i
\(73\) −610.359 + 392.254i −0.978591 + 0.628902i −0.929083 − 0.369872i \(-0.879402\pi\)
−0.0495079 + 0.998774i \(0.515765\pi\)
\(74\) −15.3330 − 106.643i −0.0240868 − 0.167527i
\(75\) −205.127 − 131.827i −0.315814 − 0.202961i
\(76\) −62.8181 + 137.552i −0.0948122 + 0.207610i
\(77\) 37.9884 − 43.8409i 0.0562231 − 0.0648849i
\(78\) 83.1465 − 578.297i 0.120699 − 0.839477i
\(79\) −65.1569 − 142.674i −0.0927940 − 0.203190i 0.857544 − 0.514411i \(-0.171989\pi\)
−0.950338 + 0.311221i \(0.899262\pi\)
\(80\) −137.727 − 40.4403i −0.192479 − 0.0565170i
\(81\) 768.632 + 225.691i 1.05436 + 0.309589i
\(82\) −205.363 − 449.681i −0.276567 − 0.605598i
\(83\) −57.2679 + 398.307i −0.0757345 + 0.526745i 0.916272 + 0.400557i \(0.131183\pi\)
−0.992006 + 0.126188i \(0.959726\pi\)
\(84\) −52.7456 + 60.8716i −0.0685121 + 0.0790671i
\(85\) 332.801 − 728.732i 0.424674 − 0.929907i
\(86\) −694.359 − 446.237i −0.870635 − 0.559523i
\(87\) −114.535 − 796.609i −0.141143 − 0.981672i
\(88\) 106.201 − 68.2515i 0.128649 − 0.0826777i
\(89\) 892.963 + 1030.53i 1.06353 + 1.22737i 0.972836 + 0.231496i \(0.0743620\pi\)
0.0906906 + 0.995879i \(0.471093\pi\)
\(90\) 51.7143 − 15.1847i 0.0605685 − 0.0177845i
\(91\) 196.049 0.225840
\(92\) −375.216 − 232.132i −0.425206 − 0.263059i
\(93\) 246.125 0.274430
\(94\) 123.319 − 36.2098i 0.135313 − 0.0397315i
\(95\) 222.100 + 256.317i 0.239863 + 0.276817i
\(96\) −147.457 + 94.7649i −0.156768 + 0.100749i
\(97\) −153.264 − 1065.97i −0.160428 − 1.11580i −0.897828 − 0.440347i \(-0.854855\pi\)
0.737399 − 0.675457i \(-0.236054\pi\)
\(98\) 554.363 + 356.267i 0.571419 + 0.367229i
\(99\) −19.6914 + 43.1182i −0.0199905 + 0.0437731i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 46.4.c.b.9.1 ✓ 30
23.8 even 11 1058.4.a.u.1.12 15
23.15 odd 22 1058.4.a.t.1.12 15
23.18 even 11 inner 46.4.c.b.41.1 yes 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
46.4.c.b.9.1 ✓ 30 1.1 even 1 trivial
46.4.c.b.41.1 yes 30 23.18 even 11 inner
1058.4.a.t.1.12 15 23.15 odd 22
1058.4.a.u.1.12 15 23.8 even 11