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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(4,95)] 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("95.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(18)) 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.p (of order \(18\), 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: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(28\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 4.14
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.14

$q$-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+(-0.108808 + 0.298949i) q^{2} +(-4.85373 + 0.855843i) q^{3} +(6.05082 + 5.07724i) q^{4} +(4.62275 - 10.1799i) q^{5} +(0.272273 - 1.54414i) q^{6} +(5.90230 + 3.40770i) q^{7} +(-4.38032 + 2.52898i) q^{8} +(-2.54550 + 0.926487i) q^{9} +(2.54027 + 2.48962i) q^{10} +(18.3995 + 31.8689i) q^{11} +(-33.7144 - 19.4650i) q^{12} +(78.9228 + 13.9162i) q^{13} +(-1.66095 + 1.39370i) q^{14} +(-13.7252 + 53.3668i) q^{15} +(10.6935 + 60.6457i) q^{16} +(-39.1604 + 107.592i) q^{17} -0.861784i q^{18} +(78.5608 - 26.2146i) q^{19} +(79.6572 - 38.1259i) q^{20} +(-31.5646 - 11.4886i) q^{21} +(-11.5292 + 2.03291i) q^{22} +(26.8797 - 32.0340i) q^{23} +(19.0965 - 16.0238i) q^{24} +(-82.2604 - 94.1181i) q^{25} +(-12.7477 + 22.0797i) q^{26} +(126.806 - 73.2116i) q^{27} +(18.4121 + 50.5868i) q^{28} +(-232.931 + 84.7799i) q^{29} +(-14.4605 - 9.90987i) q^{30} +(63.1471 - 109.374i) q^{31} +(-59.1424 - 10.4284i) q^{32} +(-116.581 - 138.936i) q^{33} +(-27.9036 - 23.4139i) q^{34} +(61.9748 - 44.3319i) q^{35} +(-20.1064 - 7.31813i) q^{36} -178.985i q^{37} +(-0.711237 + 26.3380i) q^{38} -394.980 q^{39} +(5.49562 + 56.2820i) q^{40} +(-5.01270 - 28.4284i) q^{41} +(6.86900 - 8.18615i) q^{42} +(273.906 + 326.429i) q^{43} +(-50.4739 + 286.252i) q^{44} +(-2.33568 + 30.1958i) q^{45} +(6.65179 + 11.5212i) q^{46} +(-44.6081 - 122.560i) q^{47} +(-103.806 - 285.206i) q^{48} +(-148.275 - 256.820i) q^{49} +(37.0871 - 14.3508i) q^{50} +(97.9916 - 555.738i) q^{51} +(406.892 + 484.915i) q^{52} +(260.719 - 310.713i) q^{53} +(8.08893 + 45.8746i) q^{54} +(409.478 - 39.9832i) q^{55} -34.4720 q^{56} +(-358.877 + 194.474i) q^{57} -78.8591i q^{58} +(-584.526 - 212.750i) q^{59} +(-354.005 + 253.227i) q^{60} +(322.573 + 270.671i) q^{61} +(25.8263 + 30.7786i) q^{62} +(-18.1815 - 3.20589i) q^{63} +(-236.772 + 410.101i) q^{64} +(506.506 - 739.095i) q^{65} +(54.2197 - 19.7343i) q^{66} +(-188.908 - 519.021i) q^{67} +(-783.224 + 452.195i) q^{68} +(-103.051 + 178.489i) q^{69} +(6.50958 + 23.3510i) q^{70} +(476.061 - 399.463i) q^{71} +(8.80704 - 10.4958i) q^{72} +(-1119.26 + 197.355i) q^{73} +(53.5074 + 19.4751i) q^{74} +(479.820 + 386.422i) q^{75} +(608.456 + 240.252i) q^{76} +250.800i q^{77} +(42.9772 - 118.079i) q^{78} +(39.0355 + 221.381i) q^{79} +(666.799 + 171.491i) q^{80} +(-496.797 + 416.862i) q^{81} +(9.04407 + 1.59471i) q^{82} +(536.115 + 309.526i) q^{83} +(-132.662 - 229.777i) q^{84} +(914.248 + 896.020i) q^{85} +(-127.389 + 46.3657i) q^{86} +(1058.02 - 610.851i) q^{87} +(-161.191 - 93.0639i) q^{88} +(-75.5949 + 428.720i) q^{89} +(-8.77287 - 3.98381i) q^{90} +(418.404 + 351.083i) q^{91} +(325.289 - 57.3573i) q^{92} +(-212.892 + 584.915i) q^{93} +41.4928 q^{94} +(96.3042 - 920.924i) q^{95} +295.986 q^{96} +(-406.658 + 1117.28i) q^{97} +(92.9097 - 16.3825i) q^{98} +(-76.3621 - 64.0754i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 168 q + 6 q^{4} - 6 q^{5} + 6 q^{6} - 12 q^{9} + 3 q^{10} - 90 q^{11} + 162 q^{14} - 72 q^{15} + 426 q^{16} - 444 q^{19} - 966 q^{20} + 114 q^{21} + 108 q^{24} + 270 q^{25} - 642 q^{26} - 60 q^{29} + 396 q^{30}+ \cdots + 22692 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.108808 + 0.298949i −0.0384696 + 0.105694i −0.957440 − 0.288632i \(-0.906800\pi\)
0.918971 + 0.394326i \(0.129022\pi\)
\(3\) −4.85373 + 0.855843i −0.934100 + 0.164707i −0.619928 − 0.784659i \(-0.712838\pi\)
−0.314172 + 0.949366i \(0.601727\pi\)
\(4\) 6.05082 + 5.07724i 0.756353 + 0.634656i
\(5\) 4.62275 − 10.1799i 0.413471 − 0.910517i
\(6\) 0.272273 − 1.54414i 0.0185258 − 0.105065i
\(7\) 5.90230 + 3.40770i 0.318694 + 0.183998i 0.650811 − 0.759240i \(-0.274429\pi\)
−0.332116 + 0.943239i \(0.607763\pi\)
\(8\) −4.38032 + 2.52898i −0.193585 + 0.111766i
\(9\) −2.54550 + 0.926487i −0.0942778 + 0.0343143i
\(10\) 2.54027 + 2.48962i 0.0803304 + 0.0787288i
\(11\) 18.3995 + 31.8689i 0.504333 + 0.873530i 0.999987 + 0.00501032i \(0.00159484\pi\)
−0.495655 + 0.868520i \(0.665072\pi\)
\(12\) −33.7144 − 19.4650i −0.811042 − 0.468255i
\(13\) 78.9228 + 13.9162i 1.68379 + 0.296897i 0.931987 − 0.362492i \(-0.118074\pi\)
0.751802 + 0.659389i \(0.229185\pi\)
\(14\) −1.66095 + 1.39370i −0.0317076 + 0.0266059i
\(15\) −13.7252 + 53.3668i −0.236255 + 0.918616i
\(16\) 10.6935 + 60.6457i 0.167085 + 0.947588i
\(17\) −39.1604 + 107.592i −0.558693 + 1.53500i 0.262842 + 0.964839i \(0.415340\pi\)
−0.821535 + 0.570158i \(0.806882\pi\)
\(18\) − 0.861784i − 0.0112847i
\(19\) 78.5608 − 26.2146i 0.948583 − 0.316529i
\(20\) 79.6572 − 38.1259i 0.890595 − 0.426261i
\(21\) −31.5646 − 11.4886i −0.327998 − 0.119382i
\(22\) −11.5292 + 2.03291i −0.111729 + 0.0197008i
\(23\) 26.8797 − 32.0340i 0.243688 − 0.290416i −0.630312 − 0.776342i \(-0.717073\pi\)
0.874000 + 0.485926i \(0.161518\pi\)
\(24\) 19.0965 − 16.0238i 0.162419 − 0.136285i
\(25\) −82.2604 − 94.1181i −0.658083 − 0.752945i
\(26\) −12.7477 + 22.0797i −0.0961551 + 0.166545i
\(27\) 126.806 − 73.2116i 0.903847 − 0.521836i
\(28\) 18.4121 + 50.5868i 0.124270 + 0.341429i
\(29\) −232.931 + 84.7799i −1.49152 + 0.542870i −0.953850 − 0.300283i \(-0.902919\pi\)
−0.537673 + 0.843153i \(0.680697\pi\)
\(30\) −14.4605 − 9.90987i −0.0880039 − 0.0603096i
\(31\) 63.1471 − 109.374i 0.365857 − 0.633682i −0.623057 − 0.782177i \(-0.714109\pi\)
0.988913 + 0.148495i \(0.0474427\pi\)
\(32\) −59.1424 − 10.4284i −0.326719 − 0.0576093i
\(33\) −116.581 − 138.936i −0.614974 − 0.732897i
\(34\) −27.9036 − 23.4139i −0.140748 − 0.118101i
\(35\) 61.9748 − 44.3319i 0.299305 − 0.214099i
\(36\) −20.1064 − 7.31813i −0.0930851 − 0.0338802i
\(37\) − 178.985i − 0.795270i −0.917544 − 0.397635i \(-0.869831\pi\)
0.917544 − 0.397635i \(-0.130169\pi\)
\(38\) −0.711237 + 26.3380i −0.00303626 + 0.112437i
\(39\) −394.980 −1.62173
\(40\) 5.49562 + 56.2820i 0.0217233 + 0.222474i
\(41\) −5.01270 − 28.4284i −0.0190940 − 0.108287i 0.973771 − 0.227529i \(-0.0730647\pi\)
−0.992865 + 0.119242i \(0.961954\pi\)
\(42\) 6.86900 − 8.18615i 0.0252359 − 0.0300750i
\(43\) 273.906 + 326.429i 0.971403 + 1.15767i 0.987471 + 0.157802i \(0.0504407\pi\)
−0.0160681 + 0.999871i \(0.505115\pi\)
\(44\) −50.4739 + 286.252i −0.172937 + 0.980775i
\(45\) −2.33568 + 30.1958i −0.00773738 + 0.100030i
\(46\) 6.65179 + 11.5212i 0.0213207 + 0.0369286i
\(47\) −44.6081 − 122.560i −0.138442 − 0.380365i 0.851025 − 0.525125i \(-0.175981\pi\)
−0.989467 + 0.144760i \(0.953759\pi\)
\(48\) −103.806 − 285.206i −0.312149 − 0.857622i
\(49\) −148.275 − 256.820i −0.432289 − 0.748747i
\(50\) 37.0871 − 14.3508i 0.104898 − 0.0405902i
\(51\) 97.9916 − 555.738i 0.269050 − 1.52586i
\(52\) 406.892 + 484.915i 1.08511 + 1.29319i
\(53\) 260.719 − 310.713i 0.675708 − 0.805278i −0.313841 − 0.949476i \(-0.601616\pi\)
0.989549 + 0.144198i \(0.0460602\pi\)
\(54\) 8.08893 + 45.8746i 0.0203845 + 0.115606i
\(55\) 409.478 − 39.9832i 1.00389 − 0.0980243i
\(56\) −34.4720 −0.0822591
\(57\) −358.877 + 194.474i −0.833937 + 0.451908i
\(58\) − 78.8591i − 0.178530i
\(59\) −584.526 − 212.750i −1.28981 − 0.469452i −0.396145 − 0.918188i \(-0.629652\pi\)
−0.893665 + 0.448736i \(0.851874\pi\)
\(60\) −354.005 + 253.227i −0.761697 + 0.544858i
\(61\) 322.573 + 270.671i 0.677069 + 0.568129i 0.915148 − 0.403117i \(-0.132073\pi\)
−0.238079 + 0.971246i \(0.576518\pi\)
\(62\) 25.8263 + 30.7786i 0.0529023 + 0.0630465i
\(63\) −18.1815 − 3.20589i −0.0363596 − 0.00641118i
\(64\) −236.772 + 410.101i −0.462446 + 0.800979i
\(65\) 506.506 − 739.095i 0.966528 − 1.41036i
\(66\) 54.2197 − 19.7343i 0.101121 − 0.0368050i
\(67\) −188.908 − 519.021i −0.344460 − 0.946396i −0.984083 − 0.177708i \(-0.943132\pi\)
0.639623 − 0.768688i \(-0.279090\pi\)
\(68\) −783.224 + 452.195i −1.39676 + 0.806422i
\(69\) −103.051 + 178.489i −0.179795 + 0.311414i
\(70\) 6.50958 + 23.3510i 0.0111149 + 0.0398711i
\(71\) 476.061 − 399.463i 0.795748 − 0.667712i −0.151413 − 0.988471i \(-0.548382\pi\)
0.947161 + 0.320759i \(0.103938\pi\)
\(72\) 8.80704 − 10.4958i 0.0144156 − 0.0171798i
\(73\) −1119.26 + 197.355i −1.79451 + 0.316420i −0.968831 − 0.247722i \(-0.920318\pi\)
−0.825678 + 0.564142i \(0.809207\pi\)
\(74\) 53.5074 + 19.4751i 0.0840555 + 0.0305937i
\(75\) 479.820 + 386.422i 0.738731 + 0.594935i
\(76\) 608.456 + 240.252i 0.918350 + 0.362616i
\(77\) 250.800i 0.371186i
\(78\) 42.9772 − 118.079i 0.0623872 − 0.171408i
\(79\) 39.0355 + 221.381i 0.0555929 + 0.315283i 0.999905 − 0.0137675i \(-0.00438248\pi\)
−0.944312 + 0.329050i \(0.893271\pi\)
\(80\) 666.799 + 171.491i 0.931881 + 0.239666i
\(81\) −496.797 + 416.862i −0.681478 + 0.571828i
\(82\) 9.04407 + 1.59471i 0.0121799 + 0.00214764i
\(83\) 536.115 + 309.526i 0.708991 + 0.409336i 0.810687 − 0.585479i \(-0.199094\pi\)
−0.101696 + 0.994815i \(0.532427\pi\)
\(84\) −132.662 − 229.777i −0.172316 − 0.298461i
\(85\) 914.248 + 896.020i 1.16664 + 1.14338i
\(86\) −127.389 + 46.3657i −0.159729 + 0.0581366i
\(87\) 1058.02 − 610.851i 1.30382 − 0.752759i
\(88\) −161.191 − 93.0639i −0.195262 − 0.112735i
\(89\) −75.5949 + 428.720i −0.0900342 + 0.510609i 0.906122 + 0.423016i \(0.139029\pi\)
−0.996156 + 0.0875932i \(0.972082\pi\)
\(90\) −8.77287 − 3.98381i −0.0102749 − 0.00466589i
\(91\) 418.404 + 351.083i 0.481986 + 0.404434i
\(92\) 325.289 − 57.3573i 0.368628 − 0.0649990i
\(93\) −212.892 + 584.915i −0.237375 + 0.652182i
\(94\) 41.4928 0.0455282
\(95\) 96.3042 − 920.924i 0.104006 − 0.994577i
\(96\) 295.986 0.314677
\(97\) −406.658 + 1117.28i −0.425668 + 1.16951i 0.522748 + 0.852487i \(0.324907\pi\)
−0.948417 + 0.317027i \(0.897315\pi\)
\(98\) 92.9097 − 16.3825i 0.0957683 − 0.0168865i
\(99\) −76.3621 − 64.0754i −0.0775220 − 0.0650487i
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.p.a.4.14 ✓ 168
5.4 even 2 inner 95.4.p.a.4.15 yes 168
19.5 even 9 inner 95.4.p.a.24.15 yes 168
95.24 even 18 inner 95.4.p.a.24.14 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.14 ✓ 168 1.1 even 1 trivial
95.4.p.a.4.15 yes 168 5.4 even 2 inner
95.4.p.a.24.14 yes 168 95.24 even 18 inner
95.4.p.a.24.15 yes 168 19.5 even 9 inner