Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [312,4,Mod(181,312)] 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("312.181"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 312.m (of order \(2\), degree \(1\), 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: \(18.4085959218\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 181.74
Character \(\chi\) \(=\) 312.181
Dual form 312.4.m.a.181.73

$q$-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+(2.48687 + 1.34740i) q^{2} +3.00000i q^{3} +(4.36905 + 6.70160i) q^{4} -8.37088 q^{5} +(-4.04219 + 7.46061i) q^{6} +16.5971i q^{7} +(1.83557 + 22.5528i) q^{8} -9.00000 q^{9} +(-20.8173 - 11.2789i) q^{10} +23.2256 q^{11} +(-20.1048 + 13.1072i) q^{12} +(46.2169 + 7.80987i) q^{13} +(-22.3629 + 41.2748i) q^{14} -25.1126i q^{15} +(-25.8228 + 58.5592i) q^{16} -14.7565 q^{17} +(-22.3818 - 12.1266i) q^{18} -124.395 q^{19} +(-36.5728 - 56.0983i) q^{20} -49.7913 q^{21} +(57.7591 + 31.2941i) q^{22} -173.645 q^{23} +(-67.6585 + 5.50670i) q^{24} -54.9283 q^{25} +(104.413 + 81.6946i) q^{26} -27.0000i q^{27} +(-111.227 + 72.5136i) q^{28} -126.409i q^{29} +(33.8367 - 62.4519i) q^{30} -27.9669i q^{31} +(-143.120 + 110.836i) q^{32} +69.6768i q^{33} +(-36.6976 - 19.8829i) q^{34} -138.932i q^{35} +(-39.3215 - 60.3144i) q^{36} +362.661 q^{37} +(-309.355 - 167.610i) q^{38} +(-23.4296 + 138.651i) q^{39} +(-15.3653 - 188.787i) q^{40} +309.872i q^{41} +(-123.825 - 67.0886i) q^{42} +556.370i q^{43} +(101.474 + 155.649i) q^{44} +75.3379 q^{45} +(-431.834 - 233.969i) q^{46} -163.307i q^{47} +(-175.678 - 77.4683i) q^{48} +67.5362 q^{49} +(-136.600 - 74.0102i) q^{50} -44.2696i q^{51} +(149.586 + 343.849i) q^{52} -334.459i q^{53} +(36.3797 - 67.1455i) q^{54} -194.419 q^{55} +(-374.312 + 30.4651i) q^{56} -373.186i q^{57} +(170.323 - 314.363i) q^{58} +809.123 q^{59} +(168.295 - 109.718i) q^{60} -6.20734i q^{61} +(37.6825 - 69.5500i) q^{62} -149.374i q^{63} +(-505.261 + 82.7945i) q^{64} +(-386.877 - 65.3755i) q^{65} +(-93.8822 + 173.277i) q^{66} +252.494 q^{67} +(-64.4721 - 98.8923i) q^{68} -520.936i q^{69} +(187.197 - 345.507i) q^{70} +822.887i q^{71} +(-16.5201 - 202.976i) q^{72} +449.440i q^{73} +(901.892 + 488.648i) q^{74} -164.785i q^{75} +(-543.489 - 833.647i) q^{76} +385.478i q^{77} +(-245.084 + 313.238i) q^{78} -547.122 q^{79} +(216.159 - 490.192i) q^{80} +81.0000 q^{81} +(-417.521 + 770.613i) q^{82} +959.070 q^{83} +(-217.541 - 333.681i) q^{84} +123.525 q^{85} +(-749.651 + 1383.62i) q^{86} +379.227 q^{87} +(42.6322 + 523.803i) q^{88} +656.191i q^{89} +(187.356 + 101.510i) q^{90} +(-129.621 + 767.067i) q^{91} +(-758.666 - 1163.70i) q^{92} +83.9007 q^{93} +(220.039 - 406.123i) q^{94} +1041.30 q^{95} +(-332.507 - 429.361i) q^{96} +417.791i q^{97} +(167.954 + 90.9980i) q^{98} -209.030 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84 q - 756 q^{9} - 36 q^{10} + 12 q^{12} - 208 q^{14} - 148 q^{16} - 104 q^{17} + 620 q^{22} + 2188 q^{25} + 444 q^{26} - 204 q^{30} - 40 q^{38} - 1924 q^{40} + 192 q^{42} + 624 q^{48} - 3396 q^{49} - 1292 q^{52}+ \cdots + 2480 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(157\) \(209\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.48687 + 1.34740i 0.879242 + 0.476376i
\(3\) 3.00000i 0.577350i
\(4\) 4.36905 + 6.70160i 0.546131 + 0.837699i
\(5\) −8.37088 −0.748714 −0.374357 − 0.927285i \(-0.622137\pi\)
−0.374357 + 0.927285i \(0.622137\pi\)
\(6\) −4.04219 + 7.46061i −0.275036 + 0.507630i
\(7\) 16.5971i 0.896159i 0.893994 + 0.448080i \(0.147892\pi\)
−0.893994 + 0.448080i \(0.852108\pi\)
\(8\) 1.83557 + 22.5528i 0.0811214 + 0.996704i
\(9\) −9.00000 −0.333333
\(10\) −20.8173 − 11.2789i −0.658301 − 0.356670i
\(11\) 23.2256 0.636616 0.318308 − 0.947987i \(-0.396885\pi\)
0.318308 + 0.947987i \(0.396885\pi\)
\(12\) −20.1048 + 13.1072i −0.483646 + 0.315309i
\(13\) 46.2169 + 7.80987i 0.986021 + 0.166621i
\(14\) −22.3629 + 41.2748i −0.426909 + 0.787940i
\(15\) − 25.1126i − 0.432270i
\(16\) −25.8228 + 58.5592i −0.403481 + 0.914988i
\(17\) −14.7565 −0.210529 −0.105264 − 0.994444i \(-0.533569\pi\)
−0.105264 + 0.994444i \(0.533569\pi\)
\(18\) −22.3818 − 12.1266i −0.293081 − 0.158792i
\(19\) −124.395 −1.50201 −0.751006 − 0.660295i \(-0.770431\pi\)
−0.751006 + 0.660295i \(0.770431\pi\)
\(20\) −36.5728 − 56.0983i −0.408896 − 0.627198i
\(21\) −49.7913 −0.517398
\(22\) 57.7591 + 31.2941i 0.559740 + 0.303269i
\(23\) −173.645 −1.57424 −0.787121 − 0.616798i \(-0.788429\pi\)
−0.787121 + 0.616798i \(0.788429\pi\)
\(24\) −67.6585 + 5.50670i −0.575447 + 0.0468355i
\(25\) −54.9283 −0.439427
\(26\) 104.413 + 81.6946i 0.787577 + 0.616217i
\(27\) − 27.0000i − 0.192450i
\(28\) −111.227 + 72.5136i −0.750712 + 0.489421i
\(29\) − 126.409i − 0.809433i −0.914442 − 0.404716i \(-0.867370\pi\)
0.914442 − 0.404716i \(-0.132630\pi\)
\(30\) 33.8367 − 62.4519i 0.205923 − 0.380070i
\(31\) − 27.9669i − 0.162032i −0.996713 − 0.0810162i \(-0.974183\pi\)
0.996713 − 0.0810162i \(-0.0258165\pi\)
\(32\) −143.120 + 110.836i −0.790636 + 0.612287i
\(33\) 69.6768i 0.367551i
\(34\) −36.6976 − 19.8829i −0.185105 − 0.100291i
\(35\) − 138.932i − 0.670967i
\(36\) −39.3215 − 60.3144i −0.182044 − 0.279233i
\(37\) 362.661 1.61138 0.805692 − 0.592335i \(-0.201794\pi\)
0.805692 + 0.592335i \(0.201794\pi\)
\(38\) −309.355 − 167.610i −1.32063 − 0.715523i
\(39\) −23.4296 + 138.651i −0.0961985 + 0.569280i
\(40\) −15.3653 − 188.787i −0.0607368 − 0.746247i
\(41\) 309.872i 1.18034i 0.807279 + 0.590170i \(0.200939\pi\)
−0.807279 + 0.590170i \(0.799061\pi\)
\(42\) −123.825 − 67.0886i −0.454918 − 0.246476i
\(43\) 556.370i 1.97316i 0.163293 + 0.986578i \(0.447788\pi\)
−0.163293 + 0.986578i \(0.552212\pi\)
\(44\) 101.474 + 155.649i 0.347676 + 0.533293i
\(45\) 75.3379 0.249571
\(46\) −431.834 − 233.969i −1.38414 − 0.749932i
\(47\) − 163.307i − 0.506825i −0.967358 − 0.253412i \(-0.918447\pi\)
0.967358 − 0.253412i \(-0.0815530\pi\)
\(48\) −175.678 − 77.4683i −0.528269 − 0.232950i
\(49\) 67.5362 0.196899
\(50\) −136.600 − 74.0102i −0.386362 − 0.209332i
\(51\) − 44.2696i − 0.121549i
\(52\) 149.586 + 343.849i 0.398919 + 0.916986i
\(53\) − 334.459i − 0.866821i −0.901197 − 0.433411i \(-0.857310\pi\)
0.901197 − 0.433411i \(-0.142690\pi\)
\(54\) 36.3797 − 67.1455i 0.0916786 − 0.169210i
\(55\) −194.419 −0.476644
\(56\) −374.312 + 30.4651i −0.893206 + 0.0726977i
\(57\) − 373.186i − 0.867187i
\(58\) 170.323 − 314.363i 0.385595 − 0.711687i
\(59\) 809.123 1.78540 0.892702 − 0.450647i \(-0.148807\pi\)
0.892702 + 0.450647i \(0.148807\pi\)
\(60\) 168.295 − 109.718i 0.362113 − 0.236077i
\(61\) − 6.20734i − 0.0130290i −0.999979 − 0.00651450i \(-0.997926\pi\)
0.999979 − 0.00651450i \(-0.00207364\pi\)
\(62\) 37.6825 − 69.5500i 0.0771883 − 0.142466i
\(63\) − 149.374i − 0.298720i
\(64\) −505.261 + 82.7945i −0.986839 + 0.161708i
\(65\) −386.877 − 65.3755i −0.738248 − 0.124751i
\(66\) −93.8822 + 173.277i −0.175092 + 0.323166i
\(67\) 252.494 0.460403 0.230202 − 0.973143i \(-0.426061\pi\)
0.230202 + 0.973143i \(0.426061\pi\)
\(68\) −64.4721 − 98.8923i −0.114976 − 0.176360i
\(69\) − 520.936i − 0.908889i
\(70\) 187.197 − 345.507i 0.319633 − 0.589942i
\(71\) 822.887i 1.37548i 0.725959 + 0.687738i \(0.241396\pi\)
−0.725959 + 0.687738i \(0.758604\pi\)
\(72\) −16.5201 − 202.976i −0.0270405 − 0.332235i
\(73\) 449.440i 0.720589i 0.932839 + 0.360294i \(0.117324\pi\)
−0.932839 + 0.360294i \(0.882676\pi\)
\(74\) 901.892 + 488.648i 1.41679 + 0.767624i
\(75\) − 164.785i − 0.253703i
\(76\) −543.489 − 833.647i −0.820296 − 1.25823i
\(77\) 385.478i 0.570510i
\(78\) −245.084 + 313.238i −0.355773 + 0.454708i
\(79\) −547.122 −0.779191 −0.389595 − 0.920986i \(-0.627385\pi\)
−0.389595 + 0.920986i \(0.627385\pi\)
\(80\) 216.159 − 490.192i 0.302092 − 0.685065i
\(81\) 81.0000 0.111111
\(82\) −417.521 + 770.613i −0.562286 + 1.03780i
\(83\) 959.070 1.26833 0.634166 − 0.773197i \(-0.281344\pi\)
0.634166 + 0.773197i \(0.281344\pi\)
\(84\) −217.541 − 333.681i −0.282567 − 0.433424i
\(85\) 123.525 0.157626
\(86\) −749.651 + 1383.62i −0.939964 + 1.73488i
\(87\) 379.227 0.467326
\(88\) 42.6322 + 523.803i 0.0516432 + 0.634518i
\(89\) 656.191i 0.781529i 0.920491 + 0.390765i \(0.127789\pi\)
−0.920491 + 0.390765i \(0.872211\pi\)
\(90\) 187.356 + 101.510i 0.219434 + 0.118890i
\(91\) −129.621 + 767.067i −0.149319 + 0.883632i
\(92\) −758.666 − 1163.70i −0.859743 − 1.31874i
\(93\) 83.9007 0.0935494
\(94\) 220.039 − 406.123i 0.241439 − 0.445621i
\(95\) 1041.30 1.12458
\(96\) −332.507 − 429.361i −0.353504 − 0.456474i
\(97\) 417.791i 0.437322i 0.975801 + 0.218661i \(0.0701689\pi\)
−0.975801 + 0.218661i \(0.929831\pi\)
\(98\) 167.954 + 90.9980i 0.173121 + 0.0937978i
\(99\) −209.030 −0.212205
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 312.4.m.a.181.74 yes 84
4.3 odd 2 1248.4.m.a.337.57 84
8.3 odd 2 1248.4.m.a.337.60 84
8.5 even 2 inner 312.4.m.a.181.12 yes 84
13.12 even 2 inner 312.4.m.a.181.11 ✓ 84
52.51 odd 2 1248.4.m.a.337.58 84
104.51 odd 2 1248.4.m.a.337.59 84
104.77 even 2 inner 312.4.m.a.181.73 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.m.a.181.11 ✓ 84 13.12 even 2 inner
312.4.m.a.181.12 yes 84 8.5 even 2 inner
312.4.m.a.181.73 yes 84 104.77 even 2 inner
312.4.m.a.181.74 yes 84 1.1 even 1 trivial
1248.4.m.a.337.57 84 4.3 odd 2
1248.4.m.a.337.58 84 52.51 odd 2
1248.4.m.a.337.59 84 104.51 odd 2
1248.4.m.a.337.60 84 8.3 odd 2