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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(2,95)] 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("95.2"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([9, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.r (of order \(36\), degree \(12\), 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: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 2.3
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.3

$q$-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+(-2.70630 - 3.86500i) q^{2} +(-0.855559 - 9.77909i) q^{3} +(-4.87798 + 13.4021i) q^{4} +(0.171615 - 11.1790i) q^{5} +(-35.4807 + 29.7719i) q^{6} +(0.905101 - 3.37788i) q^{7} +(28.5404 - 7.64737i) q^{8} +(-68.3088 + 12.0447i) q^{9} +(-43.6713 + 29.5905i) q^{10} +(27.2507 - 47.1997i) q^{11} +(135.234 + 36.2359i) q^{12} +(32.4287 + 2.83714i) q^{13} +(-15.5050 + 5.64335i) q^{14} +(-109.467 + 7.88608i) q^{15} +(-19.3916 - 16.2715i) q^{16} +(33.7234 - 23.6134i) q^{17} +(231.417 + 231.417i) q^{18} +(80.4728 - 19.5735i) q^{19} +(148.986 + 56.8311i) q^{20} +(-33.8070 - 5.96109i) q^{21} +(-256.175 + 22.4124i) q^{22} +(120.696 - 56.2816i) q^{23} +(-99.2022 - 272.556i) q^{24} +(-124.941 - 3.83698i) q^{25} +(-76.7961 - 133.015i) q^{26} +(107.630 + 401.680i) q^{27} +(40.8558 + 28.6075i) q^{28} +(17.2225 + 97.6735i) q^{29} +(326.731 + 401.749i) q^{30} +(-154.089 + 88.9631i) q^{31} +(10.1919 - 116.494i) q^{32} +(-484.884 - 226.105i) q^{33} +(-182.531 - 66.4359i) q^{34} +(-37.6061 - 10.6978i) q^{35} +(171.784 - 974.237i) q^{36} +(0.0229221 - 0.0229221i) q^{37} +(-293.435 - 258.056i) q^{38} -319.550i q^{39} +(-80.5921 - 320.366i) q^{40} +(-88.1048 + 104.999i) q^{41} +(68.4523 + 146.796i) q^{42} +(79.0426 - 169.507i) q^{43} +(499.648 + 595.457i) q^{44} +(122.925 + 765.692i) q^{45} +(-544.169 - 314.176i) q^{46} +(13.0538 - 18.6427i) q^{47} +(-142.530 + 203.553i) q^{48} +(286.456 + 165.385i) q^{49} +(323.298 + 493.281i) q^{50} +(-259.770 - 309.581i) q^{51} +(-196.210 + 420.774i) q^{52} +(40.4134 + 86.6669i) q^{53} +(1261.21 - 1503.05i) q^{54} +(-522.970 - 312.737i) q^{55} -103.328i q^{56} +(-260.260 - 770.205i) q^{57} +(330.899 - 330.899i) q^{58} +(-153.176 + 868.703i) q^{59} +(428.290 - 1505.57i) q^{60} +(-312.012 - 113.563i) q^{61} +(760.852 + 354.791i) q^{62} +(-21.1408 + 241.641i) q^{63} +(-653.210 + 377.131i) q^{64} +(37.2817 - 362.034i) q^{65} +(438.346 + 2485.99i) q^{66} +(157.673 + 110.404i) q^{67} +(151.968 + 567.151i) q^{68} +(-653.646 - 1132.15i) q^{69} +(60.4263 + 174.299i) q^{70} +(-10.6309 - 29.2083i) q^{71} +(-1857.45 + 866.142i) q^{72} +(-436.726 + 38.2085i) q^{73} +(-0.150628 - 0.0265598i) q^{74} +(69.3724 + 1225.09i) q^{75} +(-130.218 + 1173.99i) q^{76} +(-134.770 - 134.770i) q^{77} +(-1235.06 + 864.798i) q^{78} +(588.633 + 493.922i) q^{79} +(-185.227 + 213.987i) q^{80} +(2076.13 - 755.650i) q^{81} +(644.260 + 56.3654i) q^{82} +(-264.192 - 70.7902i) q^{83} +(244.801 - 424.008i) q^{84} +(-258.187 - 381.047i) q^{85} +(-869.058 + 153.238i) q^{86} +(940.423 - 251.986i) q^{87} +(416.793 - 1555.49i) q^{88} +(746.309 - 626.228i) q^{89} +(2626.73 - 2547.30i) q^{90} +(38.9348 - 106.972i) q^{91} +(165.540 + 1892.13i) q^{92} +(1001.81 + 1430.73i) q^{93} -107.381 q^{94} +(-205.002 - 902.967i) q^{95} -1147.92 q^{96} +(-854.622 - 1220.53i) q^{97} +(-136.022 - 1554.73i) q^{98} +(-1292.96 + 3552.38i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 336 q - 12 q^{2} - 12 q^{3} - 12 q^{5} + 12 q^{6} + 6 q^{7} - 18 q^{8} - 12 q^{10} - 12 q^{11} - 18 q^{12} - 12 q^{13} - 264 q^{15} - 996 q^{16} + 132 q^{17} + 1116 q^{20} - 600 q^{21} + 468 q^{22} + 564 q^{23}+ \cdots + 1290 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.70630 − 3.86500i −0.956821 − 1.36648i −0.930286 − 0.366835i \(-0.880441\pi\)
−0.0265354 − 0.999648i \(-0.508447\pi\)
\(3\) −0.855559 − 9.77909i −0.164652 − 1.88199i −0.409336 − 0.912383i \(-0.634240\pi\)
0.244684 − 0.969603i \(-0.421316\pi\)
\(4\) −4.87798 + 13.4021i −0.609747 + 1.67527i
\(5\) 0.171615 − 11.1790i 0.0153497 − 0.999882i
\(6\) −35.4807 + 29.7719i −2.41416 + 2.02572i
\(7\) 0.905101 − 3.37788i 0.0488709 − 0.182389i −0.937176 − 0.348857i \(-0.886570\pi\)
0.986047 + 0.166469i \(0.0532365\pi\)
\(8\) 28.5404 − 7.64737i 1.26132 − 0.337969i
\(9\) −68.3088 + 12.0447i −2.52995 + 0.446099i
\(10\) −43.6713 + 29.5905i −1.38101 + 0.935734i
\(11\) 27.2507 − 47.1997i 0.746946 − 1.29375i −0.202334 − 0.979317i \(-0.564853\pi\)
0.949280 − 0.314432i \(-0.101814\pi\)
\(12\) 135.234 + 36.2359i 3.25323 + 0.871699i
\(13\) 32.4287 + 2.83714i 0.691853 + 0.0605293i 0.427659 − 0.903940i \(-0.359338\pi\)
0.264194 + 0.964469i \(0.414894\pi\)
\(14\) −15.5050 + 5.64335i −0.295992 + 0.107732i
\(15\) −109.467 + 7.88608i −1.88429 + 0.135745i
\(16\) −19.3916 − 16.2715i −0.302994 − 0.254242i
\(17\) 33.7234 − 23.6134i 0.481125 − 0.336887i −0.307723 − 0.951476i \(-0.599567\pi\)
0.788848 + 0.614589i \(0.210678\pi\)
\(18\) 231.417 + 231.417i 3.03030 + 3.03030i
\(19\) 80.4728 − 19.5735i 0.971670 − 0.236340i
\(20\) 148.986 + 56.8311i 1.66571 + 0.635390i
\(21\) −33.8070 − 5.96109i −0.351300 − 0.0619436i
\(22\) −256.175 + 22.4124i −2.48258 + 0.217198i
\(23\) 120.696 − 56.2816i 1.09421 − 0.510240i 0.210180 − 0.977663i \(-0.432595\pi\)
0.884034 + 0.467422i \(0.154817\pi\)
\(24\) −99.2022 − 272.556i −0.843732 − 2.31813i
\(25\) −124.941 − 3.83698i −0.999529 − 0.0306958i
\(26\) −76.7961 − 133.015i −0.579268 − 1.00332i
\(27\) 107.630 + 401.680i 0.767161 + 2.86309i
\(28\) 40.8558 + 28.6075i 0.275751 + 0.193083i
\(29\) 17.2225 + 97.6735i 0.110280 + 0.625432i 0.988979 + 0.148055i \(0.0473012\pi\)
−0.878699 + 0.477377i \(0.841588\pi\)
\(30\) 326.731 + 401.749i 1.98842 + 2.44497i
\(31\) −154.089 + 88.9631i −0.892746 + 0.515427i −0.874840 − 0.484413i \(-0.839033\pi\)
−0.0179061 + 0.999840i \(0.505700\pi\)
\(32\) 10.1919 − 116.494i 0.0563027 − 0.643543i
\(33\) −484.884 − 226.105i −2.55780 − 1.19272i
\(34\) −182.531 − 66.4359i −0.920701 − 0.335108i
\(35\) −37.6061 − 10.6978i −0.181617 − 0.0516647i
\(36\) 171.784 − 974.237i 0.795298 − 4.51036i
\(37\) 0.0229221 − 0.0229221i 0.000101848 − 0.000101848i −0.707056 − 0.707158i \(-0.749977\pi\)
0.707158 + 0.707056i \(0.249977\pi\)
\(38\) −293.435 − 258.056i −1.25267 − 1.10164i
\(39\) − 319.550i − 1.31202i
\(40\) −80.5921 − 320.366i −0.318568 − 1.26636i
\(41\) −88.1048 + 104.999i −0.335602 + 0.399954i −0.907282 − 0.420522i \(-0.861847\pi\)
0.571681 + 0.820476i \(0.306292\pi\)
\(42\) 68.4523 + 146.796i 0.251486 + 0.539314i
\(43\) 79.0426 − 169.507i 0.280323 − 0.601154i −0.714852 − 0.699276i \(-0.753506\pi\)
0.995174 + 0.0981220i \(0.0312835\pi\)
\(44\) 499.648 + 595.457i 1.71193 + 2.04019i
\(45\) 122.925 + 765.692i 0.407213 + 2.53650i
\(46\) −544.169 − 314.176i −1.74420 − 1.00702i
\(47\) 13.0538 − 18.6427i 0.0405125 − 0.0578578i −0.798385 − 0.602148i \(-0.794312\pi\)
0.838897 + 0.544290i \(0.183201\pi\)
\(48\) −142.530 + 203.553i −0.428591 + 0.612092i
\(49\) 286.456 + 165.385i 0.835148 + 0.482173i
\(50\) 323.298 + 493.281i 0.914425 + 1.39521i
\(51\) −259.770 − 309.581i −0.713236 − 0.850001i
\(52\) −196.210 + 420.774i −0.523259 + 1.12213i
\(53\) 40.4134 + 86.6669i 0.104740 + 0.224615i 0.951692 − 0.307053i \(-0.0993430\pi\)
−0.846953 + 0.531669i \(0.821565\pi\)
\(54\) 1261.21 − 1503.05i 3.17832 − 3.78777i
\(55\) −522.970 − 312.737i −1.28213 − 0.766717i
\(56\) − 103.328i − 0.246567i
\(57\) −260.260 − 770.205i −0.604777 − 1.78976i
\(58\) 330.899 − 330.899i 0.749123 − 0.749123i
\(59\) −153.176 + 868.703i −0.337996 + 1.91687i 0.0573780 + 0.998353i \(0.481726\pi\)
−0.395374 + 0.918520i \(0.629385\pi\)
\(60\) 428.290 − 1505.57i 0.921533 − 3.23946i
\(61\) −312.012 − 113.563i −0.654902 − 0.238365i −0.00686799 − 0.999976i \(-0.502186\pi\)
−0.648034 + 0.761612i \(0.724408\pi\)
\(62\) 760.852 + 354.791i 1.55852 + 0.726750i
\(63\) −21.1408 + 241.641i −0.0422777 + 0.483236i
\(64\) −653.210 + 377.131i −1.27580 + 0.736584i
\(65\) 37.2817 − 362.034i 0.0711419 − 0.690843i
\(66\) 438.346 + 2485.99i 0.817526 + 4.63642i
\(67\) 157.673 + 110.404i 0.287504 + 0.201313i 0.708427 − 0.705784i \(-0.249405\pi\)
−0.420923 + 0.907096i \(0.638294\pi\)
\(68\) 151.968 + 567.151i 0.271012 + 1.01143i
\(69\) −653.646 − 1132.15i −1.14043 − 1.97528i
\(70\) 60.4263 + 174.299i 0.103176 + 0.297610i
\(71\) −10.6309 − 29.2083i −0.0177699 − 0.0488223i 0.930491 − 0.366316i \(-0.119381\pi\)
−0.948260 + 0.317494i \(0.897159\pi\)
\(72\) −1857.45 + 866.142i −3.04031 + 1.41772i
\(73\) −436.726 + 38.2085i −0.700204 + 0.0612599i −0.431698 − 0.902018i \(-0.642085\pi\)
−0.268506 + 0.963278i \(0.586530\pi\)
\(74\) −0.150628 − 0.0265598i −0.000236624 − 4.17232e-5i
\(75\) 69.3724 + 1225.09i 0.106806 + 1.88615i
\(76\) −130.218 + 1173.99i −0.196540 + 1.77192i
\(77\) −134.770 − 134.770i −0.199461 − 0.199461i
\(78\) −1235.06 + 864.798i −1.79286 + 1.25537i
\(79\) 588.633 + 493.922i 0.838308 + 0.703424i 0.957183 − 0.289485i \(-0.0934840\pi\)
−0.118874 + 0.992909i \(0.537928\pi\)
\(80\) −185.227 + 213.987i −0.258863 + 0.299056i
\(81\) 2076.13 − 755.650i 2.84792 − 1.03656i
\(82\) 644.260 + 56.3654i 0.867641 + 0.0759088i
\(83\) −264.192 − 70.7902i −0.349384 − 0.0936172i 0.0798590 − 0.996806i \(-0.474553\pi\)
−0.429243 + 0.903189i \(0.641220\pi\)
\(84\) 244.801 − 424.008i 0.317976 − 0.550751i
\(85\) −258.187 − 381.047i −0.329462 − 0.486239i
\(86\) −869.058 + 153.238i −1.08969 + 0.192141i
\(87\) 940.423 − 251.986i 1.15890 − 0.310525i
\(88\) 416.793 − 1555.49i 0.504889 − 1.88427i
\(89\) 746.309 − 626.228i 0.888861 − 0.745843i −0.0791202 − 0.996865i \(-0.525211\pi\)
0.967981 + 0.251022i \(0.0807666\pi\)
\(90\) 2626.73 − 2547.30i 3.07646 − 2.98343i
\(91\) 38.9348 − 106.972i 0.0448513 − 0.123228i
\(92\) 165.540 + 1892.13i 0.187595 + 2.14422i
\(93\) 1001.81 + 1430.73i 1.11702 + 1.59527i
\(94\) −107.381 −0.117825
\(95\) −205.002 − 902.967i −0.221398 − 0.975184i
\(96\) −1147.92 −1.22041
\(97\) −854.622 − 1220.53i −0.894574 − 1.27758i −0.960078 − 0.279732i \(-0.909754\pi\)
0.0655039 − 0.997852i \(-0.479135\pi\)
\(98\) −136.022 − 1554.73i −0.140207 − 1.60257i
\(99\) −1292.96 + 3552.38i −1.31260 + 3.60634i
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.r.a.2.3 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.3 yes 336
19.10 odd 18 inner 95.4.r.a.67.3 yes 336
95.48 even 36 inner 95.4.r.a.48.3 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.3 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.3 yes 336 95.48 even 36 inner
95.4.r.a.67.3 yes 336 19.10 odd 18 inner
95.4.r.a.78.3 yes 336 5.3 odd 4 inner