Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(11,95)] 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("95.11"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 11.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 95.11
Dual form 95.4.e.a.26.1

$q$-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+(0.500000 - 0.866025i) q^{2} +(2.50000 - 4.33013i) q^{3} +(3.50000 + 6.06218i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(-2.50000 - 4.33013i) q^{6} +22.0000 q^{7} +15.0000 q^{8} +(1.00000 + 1.73205i) q^{9} +(2.50000 + 4.33013i) q^{10} +9.00000 q^{11} +35.0000 q^{12} +(-27.0000 - 46.7654i) q^{13} +(11.0000 - 19.0526i) q^{14} +(12.5000 + 21.6506i) q^{15} +(-20.5000 + 35.5070i) q^{16} +(27.0000 - 46.7654i) q^{17} +2.00000 q^{18} +(-66.5000 - 49.3634i) q^{19} -35.0000 q^{20} +(55.0000 - 95.2628i) q^{21} +(4.50000 - 7.79423i) q^{22} +(46.0000 + 79.6743i) q^{23} +(37.5000 - 64.9519i) q^{24} +(-12.5000 - 21.6506i) q^{25} -54.0000 q^{26} +145.000 q^{27} +(77.0000 + 133.368i) q^{28} +(67.0000 + 116.047i) q^{29} +25.0000 q^{30} -252.000 q^{31} +(80.5000 + 139.430i) q^{32} +(22.5000 - 38.9711i) q^{33} +(-27.0000 - 46.7654i) q^{34} +(-55.0000 + 95.2628i) q^{35} +(-7.00000 + 12.1244i) q^{36} -236.000 q^{37} +(-76.0000 + 32.9090i) q^{38} -270.000 q^{39} +(-37.5000 + 64.9519i) q^{40} +(121.500 - 210.444i) q^{41} +(-55.0000 - 95.2628i) q^{42} +(-248.000 + 429.549i) q^{43} +(31.5000 + 54.5596i) q^{44} -10.0000 q^{45} +92.0000 q^{46} +(-251.000 - 434.745i) q^{47} +(102.500 + 177.535i) q^{48} +141.000 q^{49} -25.0000 q^{50} +(-135.000 - 233.827i) q^{51} +(189.000 - 327.358i) q^{52} +(-31.0000 - 53.6936i) q^{53} +(72.5000 - 125.574i) q^{54} +(-22.5000 + 38.9711i) q^{55} +330.000 q^{56} +(-380.000 + 164.545i) q^{57} +134.000 q^{58} +(-340.500 + 589.763i) q^{59} +(-87.5000 + 151.554i) q^{60} +(71.0000 + 122.976i) q^{61} +(-126.000 + 218.238i) q^{62} +(22.0000 + 38.1051i) q^{63} -167.000 q^{64} +270.000 q^{65} +(-22.5000 - 38.9711i) q^{66} +(-27.5000 - 47.6314i) q^{67} +378.000 q^{68} +460.000 q^{69} +(55.0000 + 95.2628i) q^{70} +(487.000 - 843.509i) q^{71} +(15.0000 + 25.9808i) q^{72} +(-347.500 + 601.888i) q^{73} +(-118.000 + 204.382i) q^{74} -125.000 q^{75} +(66.5000 - 575.907i) q^{76} +198.000 q^{77} +(-135.000 + 233.827i) q^{78} +(368.000 - 637.395i) q^{79} +(-102.500 - 177.535i) q^{80} +(335.500 - 581.103i) q^{81} +(-121.500 - 210.444i) q^{82} -63.0000 q^{83} +770.000 q^{84} +(135.000 + 233.827i) q^{85} +(248.000 + 429.549i) q^{86} +670.000 q^{87} +135.000 q^{88} +(-363.000 - 628.734i) q^{89} +(-5.00000 + 8.66025i) q^{90} +(-594.000 - 1028.84i) q^{91} +(-322.000 + 557.720i) q^{92} +(-630.000 + 1091.19i) q^{93} -502.000 q^{94} +(380.000 - 164.545i) q^{95} +805.000 q^{96} +(583.500 - 1010.65i) q^{97} +(70.5000 - 122.110i) q^{98} +(9.00000 + 15.5885i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{3} + 7 q^{4} - 5 q^{5} - 5 q^{6} + 44 q^{7} + 30 q^{8} + 2 q^{9} + 5 q^{10} + 18 q^{11} + 70 q^{12} - 54 q^{13} + 22 q^{14} + 25 q^{15} - 41 q^{16} + 54 q^{17} + 4 q^{18} - 133 q^{19}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 − 0.866025i 0.176777 − 0.306186i −0.763998 − 0.645219i \(-0.776766\pi\)
0.940775 + 0.339032i \(0.110100\pi\)
\(3\) 2.50000 − 4.33013i 0.481125 − 0.833333i −0.518640 − 0.854993i \(-0.673562\pi\)
0.999765 + 0.0216593i \(0.00689490\pi\)
\(4\) 3.50000 + 6.06218i 0.437500 + 0.757772i
\(5\) −2.50000 + 4.33013i −0.223607 + 0.387298i
\(6\) −2.50000 − 4.33013i −0.170103 − 0.294628i
\(7\) 22.0000 1.18789 0.593944 − 0.804506i \(-0.297570\pi\)
0.593944 + 0.804506i \(0.297570\pi\)
\(8\) 15.0000 0.662913
\(9\) 1.00000 + 1.73205i 0.0370370 + 0.0641500i
\(10\) 2.50000 + 4.33013i 0.0790569 + 0.136931i
\(11\) 9.00000 0.246691 0.123346 − 0.992364i \(-0.460638\pi\)
0.123346 + 0.992364i \(0.460638\pi\)
\(12\) 35.0000 0.841969
\(13\) −27.0000 − 46.7654i −0.576035 − 0.997722i −0.995928 − 0.0901482i \(-0.971266\pi\)
0.419894 − 0.907573i \(-0.362067\pi\)
\(14\) 11.0000 − 19.0526i 0.209991 − 0.363715i
\(15\) 12.5000 + 21.6506i 0.215166 + 0.372678i
\(16\) −20.5000 + 35.5070i −0.320312 + 0.554798i
\(17\) 27.0000 − 46.7654i 0.385204 − 0.667192i −0.606594 − 0.795012i \(-0.707465\pi\)
0.991797 + 0.127820i \(0.0407979\pi\)
\(18\) 2.00000 0.0261891
\(19\) −66.5000 − 49.3634i −0.802955 − 0.596040i
\(20\) −35.0000 −0.391312
\(21\) 55.0000 − 95.2628i 0.571523 − 0.989907i
\(22\) 4.50000 − 7.79423i 0.0436092 − 0.0755334i
\(23\) 46.0000 + 79.6743i 0.417029 + 0.722315i 0.995639 − 0.0932891i \(-0.0297381\pi\)
−0.578610 + 0.815604i \(0.696405\pi\)
\(24\) 37.5000 − 64.9519i 0.318944 − 0.552427i
\(25\) −12.5000 − 21.6506i −0.100000 − 0.173205i
\(26\) −54.0000 −0.407318
\(27\) 145.000 1.03353
\(28\) 77.0000 + 133.368i 0.519701 + 0.900149i
\(29\) 67.0000 + 116.047i 0.429020 + 0.743085i 0.996786 − 0.0801050i \(-0.0255256\pi\)
−0.567766 + 0.823190i \(0.692192\pi\)
\(30\) 25.0000 0.152145
\(31\) −252.000 −1.46002 −0.730009 − 0.683438i \(-0.760484\pi\)
−0.730009 + 0.683438i \(0.760484\pi\)
\(32\) 80.5000 + 139.430i 0.444704 + 0.770250i
\(33\) 22.5000 − 38.9711i 0.118689 − 0.205576i
\(34\) −27.0000 − 46.7654i −0.136190 − 0.235888i
\(35\) −55.0000 + 95.2628i −0.265620 + 0.460067i
\(36\) −7.00000 + 12.1244i −0.0324074 + 0.0561313i
\(37\) −236.000 −1.04860 −0.524299 − 0.851534i \(-0.675673\pi\)
−0.524299 + 0.851534i \(0.675673\pi\)
\(38\) −76.0000 + 32.9090i −0.324443 + 0.140488i
\(39\) −270.000 −1.10858
\(40\) −37.5000 + 64.9519i −0.148232 + 0.256745i
\(41\) 121.500 − 210.444i 0.462808 − 0.801606i −0.536292 − 0.844033i \(-0.680175\pi\)
0.999100 + 0.0424262i \(0.0135087\pi\)
\(42\) −55.0000 − 95.2628i −0.202064 − 0.349985i
\(43\) −248.000 + 429.549i −0.879527 + 1.52338i −0.0276654 + 0.999617i \(0.508807\pi\)
−0.851861 + 0.523768i \(0.824526\pi\)
\(44\) 31.5000 + 54.5596i 0.107927 + 0.186936i
\(45\) −10.0000 −0.0331269
\(46\) 92.0000 0.294884
\(47\) −251.000 − 434.745i −0.778981 − 1.34923i −0.932530 − 0.361094i \(-0.882403\pi\)
0.153548 − 0.988141i \(-0.450930\pi\)
\(48\) 102.500 + 177.535i 0.308221 + 0.533854i
\(49\) 141.000 0.411079
\(50\) −25.0000 −0.0707107
\(51\) −135.000 − 233.827i −0.370662 − 0.642006i
\(52\) 189.000 − 327.358i 0.504030 − 0.873006i
\(53\) −31.0000 − 53.6936i −0.0803430 − 0.139158i 0.823054 − 0.567963i \(-0.192268\pi\)
−0.903397 + 0.428805i \(0.858935\pi\)
\(54\) 72.5000 − 125.574i 0.182704 − 0.316452i
\(55\) −22.5000 + 38.9711i −0.0551618 + 0.0955431i
\(56\) 330.000 0.787466
\(57\) −380.000 + 164.545i −0.883022 + 0.382360i
\(58\) 134.000 0.303363
\(59\) −340.500 + 589.763i −0.751344 + 1.30137i 0.195827 + 0.980639i \(0.437261\pi\)
−0.947171 + 0.320728i \(0.896072\pi\)
\(60\) −87.5000 + 151.554i −0.188270 + 0.326093i
\(61\) 71.0000 + 122.976i 0.149027 + 0.258122i 0.930868 − 0.365356i \(-0.119053\pi\)
−0.781841 + 0.623477i \(0.785719\pi\)
\(62\) −126.000 + 218.238i −0.258097 + 0.447037i
\(63\) 22.0000 + 38.1051i 0.0439959 + 0.0762031i
\(64\) −167.000 −0.326172
\(65\) 270.000 0.515221
\(66\) −22.5000 − 38.9711i −0.0419630 − 0.0726821i
\(67\) −27.5000 − 47.6314i −0.0501442 − 0.0868523i 0.839864 − 0.542797i \(-0.182635\pi\)
−0.890008 + 0.455945i \(0.849301\pi\)
\(68\) 378.000 0.674106
\(69\) 460.000 0.802572
\(70\) 55.0000 + 95.2628i 0.0939108 + 0.162658i
\(71\) 487.000 − 843.509i 0.814032 − 1.40994i −0.0959890 − 0.995382i \(-0.530601\pi\)
0.910021 − 0.414562i \(-0.136065\pi\)
\(72\) 15.0000 + 25.9808i 0.0245523 + 0.0425259i
\(73\) −347.500 + 601.888i −0.557148 + 0.965009i 0.440585 + 0.897711i \(0.354771\pi\)
−0.997733 + 0.0672976i \(0.978562\pi\)
\(74\) −118.000 + 204.382i −0.185368 + 0.321067i
\(75\) −125.000 −0.192450
\(76\) 66.5000 − 575.907i 0.100369 − 0.869224i
\(77\) 198.000 0.293041
\(78\) −135.000 + 233.827i −0.195971 + 0.339432i
\(79\) 368.000 − 637.395i 0.524092 − 0.907753i −0.475515 − 0.879708i \(-0.657738\pi\)
0.999607 − 0.0280457i \(-0.00892838\pi\)
\(80\) −102.500 − 177.535i −0.143248 − 0.248113i
\(81\) 335.500 − 581.103i 0.460219 − 0.797124i
\(82\) −121.500 − 210.444i −0.163627 − 0.283411i
\(83\) −63.0000 −0.0833150 −0.0416575 − 0.999132i \(-0.513264\pi\)
−0.0416575 + 0.999132i \(0.513264\pi\)
\(84\) 770.000 1.00017
\(85\) 135.000 + 233.827i 0.172268 + 0.298377i
\(86\) 248.000 + 429.549i 0.310960 + 0.538598i
\(87\) 670.000 0.825650
\(88\) 135.000 0.163535
\(89\) −363.000 − 628.734i −0.432336 − 0.748828i 0.564738 − 0.825270i \(-0.308977\pi\)
−0.997074 + 0.0764421i \(0.975644\pi\)
\(90\) −5.00000 + 8.66025i −0.00585607 + 0.0101430i
\(91\) −594.000 − 1028.84i −0.684265 − 1.18518i
\(92\) −322.000 + 557.720i −0.364900 + 0.632026i
\(93\) −630.000 + 1091.19i −0.702451 + 1.21668i
\(94\) −502.000 −0.550823
\(95\) 380.000 − 164.545i 0.410391 − 0.177705i
\(96\) 805.000 0.855833
\(97\) 583.500 − 1010.65i 0.610778 − 1.05790i −0.380332 − 0.924850i \(-0.624190\pi\)
0.991110 − 0.133048i \(-0.0424765\pi\)
\(98\) 70.5000 − 122.110i 0.0726691 − 0.125867i
\(99\) 9.00000 + 15.5885i 0.00913671 + 0.0158252i
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 95.4.e.a.11.1 ✓ 2
19.7 even 3 inner 95.4.e.a.26.1 yes 2
19.8 odd 6 1805.4.a.g.1.1 1
19.11 even 3 1805.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.e.a.11.1 ✓ 2 1.1 even 1 trivial
95.4.e.a.26.1 yes 2 19.7 even 3 inner
1805.4.a.e.1.1 1 19.11 even 3
1805.4.a.g.1.1 1 19.8 odd 6