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Properties

Label 19.4.e.a.6.1
Level $19$
Weight $4$
Character 19.6
Analytic conductor $1.121$
Analytic rank $0$
Dimension $24$
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] = [19,4,Mod(4,19)] 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("19.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(19, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 19.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.12103629011\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 6.1
Character \(\chi\) \(=\) 19.6
Dual form 19.4.e.a.16.1

$q$-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+(-2.26193 + 1.89799i) q^{2} +(-8.82799 - 3.21312i) q^{3} +(0.124797 - 0.707761i) q^{4} +(-0.0925462 - 0.524856i) q^{5} +(26.0668 - 9.48752i) q^{6} +(-11.8949 + 20.6025i) q^{7} +(-10.7499 - 18.6194i) q^{8} +(46.9260 + 39.3756i) q^{9} +(1.20550 + 1.01154i) q^{10} +(-18.5849 - 32.1900i) q^{11} +(-3.37584 + 5.84712i) q^{12} +(-14.9925 + 5.45682i) q^{13} +(-12.1979 - 69.1778i) q^{14} +(-0.869430 + 4.93078i) q^{15} +(65.0577 + 23.6791i) q^{16} +(-47.7006 + 40.0256i) q^{17} -180.878 q^{18} +(-57.8848 + 59.2314i) q^{19} -0.383022 q^{20} +(171.206 - 143.659i) q^{21} +(103.134 + 37.5376i) q^{22} +(-3.18128 + 18.0420i) q^{23} +(35.0737 + 198.913i) q^{24} +(117.195 - 42.6554i) q^{25} +(23.5550 - 40.7984i) q^{26} +(-160.917 - 278.716i) q^{27} +(13.0972 + 10.9899i) q^{28} +(13.6165 + 11.4256i) q^{29} +(-7.39196 - 12.8033i) q^{30} +(-76.5514 + 132.591i) q^{31} +(-30.4725 + 11.0911i) q^{32} +(60.6368 + 343.888i) q^{33} +(31.9276 - 181.070i) q^{34} +(11.9142 + 4.33641i) q^{35} +(33.7248 - 28.2984i) q^{36} -41.5560 q^{37} +(18.5110 - 243.842i) q^{38} +149.887 q^{39} +(-8.77764 + 7.36532i) q^{40} +(-314.270 - 114.385i) q^{41} +(-114.594 + 649.894i) q^{42} +(-53.2608 - 302.057i) q^{43} +(-25.1022 + 9.13644i) q^{44} +(16.3237 - 28.2734i) q^{45} +(-27.0475 - 46.8477i) q^{46} +(37.2512 + 31.2575i) q^{47} +(-498.245 - 418.077i) q^{48} +(-111.476 - 193.082i) q^{49} +(-184.127 + 318.917i) q^{50} +(549.708 - 200.077i) q^{51} +(1.99110 + 11.2921i) q^{52} +(-84.6010 + 479.796i) q^{53} +(892.983 + 325.019i) q^{54} +(-15.1751 + 12.7334i) q^{55} +511.476 q^{56} +(701.324 - 336.903i) q^{57} -52.4854 q^{58} +(160.725 - 134.865i) q^{59} +(3.38131 + 1.23070i) q^{60} +(-36.3671 + 206.248i) q^{61} +(-78.5016 - 445.205i) q^{62} +(-1369.41 + 498.426i) q^{63} +(-229.056 + 396.736i) q^{64} +(4.25154 + 7.36388i) q^{65} +(-789.851 - 662.764i) q^{66} +(327.060 + 274.436i) q^{67} +(22.3756 + 38.7557i) q^{68} +(86.0554 - 149.052i) q^{69} +(-35.1795 + 12.8043i) q^{70} +(105.727 + 599.609i) q^{71} +(228.699 - 1297.02i) q^{72} +(-299.014 - 108.832i) q^{73} +(93.9969 - 78.8728i) q^{74} -1171.65 q^{75} +(34.6978 + 48.3605i) q^{76} +884.260 q^{77} +(-339.034 + 284.483i) q^{78} +(-245.166 - 89.2331i) q^{79} +(6.40725 - 36.3373i) q^{80} +(237.817 + 1348.72i) q^{81} +(927.958 - 337.749i) q^{82} +(328.838 - 569.565i) q^{83} +(-80.3103 - 139.101i) q^{84} +(25.4222 + 21.3317i) q^{85} +(693.772 + 582.144i) q^{86} +(-83.4947 - 144.617i) q^{87} +(-399.572 + 692.080i) q^{88} +(-1246.08 + 453.535i) q^{89} +(16.7395 + 94.9346i) q^{90} +(65.9094 - 373.791i) q^{91} +(12.3724 + 4.50318i) q^{92} +(1101.83 - 924.542i) q^{93} -143.586 q^{94} +(36.4449 + 24.8995i) q^{95} +304.647 q^{96} +(245.942 - 206.370i) q^{97} +(618.618 + 225.159i) q^{98} +(395.384 - 2242.34i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{2} - 3 q^{3} - 24 q^{4} - 6 q^{5} + 42 q^{6} + 3 q^{7} - 75 q^{8} - 51 q^{9} + 75 q^{10} + 39 q^{11} - 219 q^{12} - 156 q^{13} + 93 q^{14} - 192 q^{15} + 504 q^{16} + 12 q^{17} + 264 q^{18}+ \cdots + 492 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.26193 + 1.89799i −0.799713 + 0.671039i −0.948129 − 0.317886i \(-0.897027\pi\)
0.148416 + 0.988925i \(0.452583\pi\)
\(3\) −8.82799 − 3.21312i −1.69895 − 0.618366i −0.703242 − 0.710950i \(-0.748265\pi\)
−0.995705 + 0.0925844i \(0.970487\pi\)
\(4\) 0.124797 − 0.707761i 0.0155997 − 0.0884702i
\(5\) −0.0925462 − 0.524856i −0.00827759 − 0.0469445i 0.980389 − 0.197071i \(-0.0631430\pi\)
−0.988667 + 0.150127i \(0.952032\pi\)
\(6\) 26.0668 − 9.48752i 1.77362 − 0.645544i
\(7\) −11.8949 + 20.6025i −0.642263 + 1.11243i 0.342664 + 0.939458i \(0.388671\pi\)
−0.984926 + 0.172974i \(0.944662\pi\)
\(8\) −10.7499 − 18.6194i −0.475084 − 0.822870i
\(9\) 46.9260 + 39.3756i 1.73800 + 1.45835i
\(10\) 1.20550 + 1.01154i 0.0381213 + 0.0319876i
\(11\) −18.5849 − 32.1900i −0.509414 − 0.882331i −0.999941 − 0.0109047i \(-0.996529\pi\)
0.490527 − 0.871426i \(-0.336804\pi\)
\(12\) −3.37584 + 5.84712i −0.0812100 + 0.140660i
\(13\) −14.9925 + 5.45682i −0.319859 + 0.116419i −0.496960 − 0.867774i \(-0.665550\pi\)
0.177101 + 0.984193i \(0.443328\pi\)
\(14\) −12.1979 − 69.1778i −0.232859 − 1.32061i
\(15\) −0.869430 + 4.93078i −0.0149657 + 0.0848748i
\(16\) 65.0577 + 23.6791i 1.01653 + 0.369985i
\(17\) −47.7006 + 40.0256i −0.680535 + 0.571037i −0.916163 − 0.400807i \(-0.868730\pi\)
0.235627 + 0.971843i \(0.424285\pi\)
\(18\) −180.878 −2.36851
\(19\) −57.8848 + 59.2314i −0.698930 + 0.715190i
\(20\) −0.383022 −0.00428232
\(21\) 171.206 − 143.659i 1.77906 − 1.49281i
\(22\) 103.134 + 37.5376i 0.999464 + 0.363775i
\(23\) −3.18128 + 18.0420i −0.0288410 + 0.163566i −0.995827 − 0.0912656i \(-0.970909\pi\)
0.966986 + 0.254831i \(0.0820199\pi\)
\(24\) 35.0737 + 198.913i 0.298308 + 1.69179i
\(25\) 117.195 − 42.6554i 0.937557 − 0.341243i
\(26\) 23.5550 − 40.7984i 0.177674 − 0.307740i
\(27\) −160.917 − 278.716i −1.14698 − 1.98663i
\(28\) 13.0972 + 10.9899i 0.0883979 + 0.0741747i
\(29\) 13.6165 + 11.4256i 0.0871906 + 0.0731616i 0.685341 − 0.728222i \(-0.259653\pi\)
−0.598151 + 0.801384i \(0.704098\pi\)
\(30\) −7.39196 − 12.8033i −0.0449860 − 0.0779181i
\(31\) −76.5514 + 132.591i −0.443517 + 0.768195i −0.997948 − 0.0640356i \(-0.979603\pi\)
0.554430 + 0.832230i \(0.312936\pi\)
\(32\) −30.4725 + 11.0911i −0.168338 + 0.0612701i
\(33\) 60.6368 + 343.888i 0.319864 + 1.81404i
\(34\) 31.9276 − 181.070i 0.161045 − 0.913332i
\(35\) 11.9142 + 4.33641i 0.0575390 + 0.0209425i
\(36\) 33.7248 − 28.2984i 0.156133 − 0.131011i
\(37\) −41.5560 −0.184642 −0.0923212 − 0.995729i \(-0.529429\pi\)
−0.0923212 + 0.995729i \(0.529429\pi\)
\(38\) 18.5110 − 243.842i 0.0790234 − 1.04096i
\(39\) 149.887 0.615413
\(40\) −8.77764 + 7.36532i −0.0346967 + 0.0291140i
\(41\) −314.270 − 114.385i −1.19709 − 0.435706i −0.334883 − 0.942260i \(-0.608697\pi\)
−0.862208 + 0.506554i \(0.830919\pi\)
\(42\) −114.594 + 649.894i −0.421005 + 2.38764i
\(43\) −53.2608 − 302.057i −0.188888 − 1.07124i −0.920857 − 0.389901i \(-0.872509\pi\)
0.731969 − 0.681338i \(-0.238602\pi\)
\(44\) −25.1022 + 9.13644i −0.0860067 + 0.0313039i
\(45\) 16.3237 − 28.2734i 0.0540753 − 0.0936612i
\(46\) −27.0475 − 46.8477i −0.0866944 − 0.150159i
\(47\) 37.2512 + 31.2575i 0.115610 + 0.0970080i 0.698760 − 0.715356i \(-0.253736\pi\)
−0.583150 + 0.812364i \(0.698180\pi\)
\(48\) −498.245 − 418.077i −1.49824 − 1.25717i
\(49\) −111.476 − 193.082i −0.325003 − 0.562922i
\(50\) −184.127 + 318.917i −0.520790 + 0.902034i
\(51\) 549.708 − 200.077i 1.50930 − 0.549341i
\(52\) 1.99110 + 11.2921i 0.00530992 + 0.0301141i
\(53\) −84.6010 + 479.796i −0.219261 + 1.24349i 0.654096 + 0.756412i \(0.273049\pi\)
−0.873357 + 0.487080i \(0.838062\pi\)
\(54\) 892.983 + 325.019i 2.25036 + 0.819065i
\(55\) −15.1751 + 12.7334i −0.0372039 + 0.0312178i
\(56\) 511.476 1.22052
\(57\) 701.324 − 336.903i 1.62969 − 0.782875i
\(58\) −52.4854 −0.118822
\(59\) 160.725 − 134.865i 0.354655 − 0.297591i −0.448001 − 0.894033i \(-0.647864\pi\)
0.802656 + 0.596442i \(0.203419\pi\)
\(60\) 3.38131 + 1.23070i 0.00727543 + 0.00264804i
\(61\) −36.3671 + 206.248i −0.0763332 + 0.432907i 0.922559 + 0.385855i \(0.126094\pi\)
−0.998892 + 0.0470516i \(0.985017\pi\)
\(62\) −78.5016 − 445.205i −0.160802 − 0.911953i
\(63\) −1369.41 + 498.426i −2.73857 + 0.996759i
\(64\) −229.056 + 396.736i −0.447375 + 0.774876i
\(65\) 4.25154 + 7.36388i 0.00811290 + 0.0140520i
\(66\) −789.851 − 662.764i −1.47309 − 1.23607i
\(67\) 327.060 + 274.436i 0.596369 + 0.500413i 0.890276 − 0.455421i \(-0.150511\pi\)
−0.293907 + 0.955834i \(0.594956\pi\)
\(68\) 22.3756 + 38.7557i 0.0399036 + 0.0691151i
\(69\) 86.0554 − 149.052i 0.150143 − 0.260055i
\(70\) −35.1795 + 12.8043i −0.0600679 + 0.0218629i
\(71\) 105.727 + 599.609i 0.176726 + 1.00226i 0.936133 + 0.351646i \(0.114378\pi\)
−0.759407 + 0.650615i \(0.774511\pi\)
\(72\) 228.699 − 1297.02i 0.374340 − 2.12299i
\(73\) −299.014 − 108.832i −0.479411 − 0.174491i 0.0909999 − 0.995851i \(-0.470994\pi\)
−0.570411 + 0.821360i \(0.693216\pi\)
\(74\) 93.9969 − 78.8728i 0.147661 − 0.123902i
\(75\) −1171.65 −1.80387
\(76\) 34.6978 + 48.3605i 0.0523699 + 0.0729912i
\(77\) 884.260 1.30871
\(78\) −339.034 + 284.483i −0.492154 + 0.412966i
\(79\) −245.166 − 89.2331i −0.349156 − 0.127082i 0.161488 − 0.986875i \(-0.448371\pi\)
−0.510644 + 0.859792i \(0.670593\pi\)
\(80\) 6.40725 − 36.3373i 0.00895440 − 0.0507829i
\(81\) 237.817 + 1348.72i 0.326223 + 1.85010i
\(82\) 927.958 − 337.749i 1.24971 − 0.454856i
\(83\) 328.838 − 569.565i 0.434876 − 0.753227i −0.562410 − 0.826859i \(-0.690126\pi\)
0.997286 + 0.0736316i \(0.0234589\pi\)
\(84\) −80.3103 − 139.101i −0.104316 − 0.180681i
\(85\) 25.4222 + 21.3317i 0.0324402 + 0.0272206i
\(86\) 693.772 + 582.144i 0.869900 + 0.729933i
\(87\) −83.4947 − 144.617i −0.102892 − 0.178213i
\(88\) −399.572 + 692.080i −0.484029 + 0.838363i
\(89\) −1246.08 + 453.535i −1.48409 + 0.540165i −0.951886 − 0.306452i \(-0.900858\pi\)
−0.532204 + 0.846616i \(0.678636\pi\)
\(90\) 16.7395 + 94.9346i 0.0196056 + 0.111189i
\(91\) 65.9094 − 373.791i 0.0759251 − 0.430593i
\(92\) 12.3724 + 4.50318i 0.0140208 + 0.00510314i
\(93\) 1101.83 − 924.542i 1.22854 − 1.03087i
\(94\) −143.586 −0.157551
\(95\) 36.4449 + 24.8995i 0.0393597 + 0.0268909i
\(96\) 304.647 0.323885
\(97\) 245.942 − 206.370i 0.257440 − 0.216018i −0.504928 − 0.863161i \(-0.668481\pi\)
0.762368 + 0.647144i \(0.224037\pi\)
\(98\) 618.618 + 225.159i 0.637652 + 0.232086i
\(99\) 395.384 − 2242.34i 0.401390 − 2.27640i
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 19.4.e.a.6.1 ✓ 24
3.2 odd 2 171.4.u.b.82.4 24
19.4 even 9 361.4.a.n.1.3 12
19.15 odd 18 361.4.a.m.1.10 12
19.16 even 9 inner 19.4.e.a.16.1 yes 24
57.35 odd 18 171.4.u.b.73.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
19.4.e.a.6.1 ✓ 24 1.1 even 1 trivial
19.4.e.a.16.1 yes 24 19.16 even 9 inner
171.4.u.b.73.4 24 57.35 odd 18
171.4.u.b.82.4 24 3.2 odd 2
361.4.a.m.1.10 12 19.15 odd 18
361.4.a.n.1.3 12 19.4 even 9