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(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 4.6
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.6

$q$-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+(-1.17190 + 3.21976i) q^{2} +(-1.04177 + 0.183692i) q^{3} +(-2.86517 - 2.40416i) q^{4} +(8.45885 - 7.31080i) q^{5} +(0.629403 - 3.56952i) q^{6} +(-30.3869 - 17.5439i) q^{7} +(-12.6403 + 7.29787i) q^{8} +(-24.3202 + 8.85181i) q^{9} +(13.6261 + 35.8030i) q^{10} +(-22.7759 - 39.4489i) q^{11} +(3.42647 + 1.97827i) q^{12} +(32.5306 + 5.73602i) q^{13} +(92.0975 - 77.2790i) q^{14} +(-7.46924 + 9.17000i) q^{15} +(-13.8801 - 78.7182i) q^{16} +(20.5569 - 56.4797i) q^{17} -88.6785i q^{18} +(-6.63144 + 82.5532i) q^{19} +(-41.8123 + 0.610227i) q^{20} +(34.8789 + 12.6949i) q^{21} +(153.707 - 27.1027i) q^{22} +(-79.8017 + 95.1039i) q^{23} +(11.8277 - 9.92463i) q^{24} +(18.1044 - 123.682i) q^{25} +(-56.5911 + 98.0187i) q^{26} +(48.4452 - 27.9698i) q^{27} +(44.8852 + 123.321i) q^{28} +(-67.2479 + 24.4762i) q^{29} +(-20.7720 - 34.7955i) q^{30} +(-63.1459 + 109.372i) q^{31} +(154.728 + 27.2827i) q^{32} +(30.9737 + 36.9130i) q^{33} +(157.761 + 132.377i) q^{34} +(-385.299 + 73.7515i) q^{35} +(90.9625 + 33.1076i) q^{36} -344.348i q^{37} +(-258.030 - 118.095i) q^{38} -34.9430 q^{39} +(-53.5690 + 154.142i) q^{40} +(-0.865889 - 4.91070i) q^{41} +(-81.7489 + 97.4246i) q^{42} +(-58.5206 - 69.7422i) q^{43} +(-29.5849 + 167.785i) q^{44} +(-141.007 + 252.676i) q^{45} +(-212.693 - 368.394i) q^{46} +(32.6977 + 89.8363i) q^{47} +(28.9198 + 79.4566i) q^{48} +(444.077 + 769.164i) q^{49} +(377.010 + 203.234i) q^{50} +(-11.0407 + 62.6151i) q^{51} +(-79.4152 - 94.6433i) q^{52} +(-200.720 + 239.209i) q^{53} +(33.2834 + 188.760i) q^{54} +(-481.061 - 167.183i) q^{55} +512.133 q^{56} +(-8.25593 - 87.2196i) q^{57} -245.206i q^{58} +(-194.872 - 70.9278i) q^{59} +(43.4468 - 8.31632i) q^{60} +(-209.212 - 175.550i) q^{61} +(-278.151 - 331.488i) q^{62} +(894.310 + 157.691i) q^{63} +(50.5613 - 87.5747i) q^{64} +(317.106 - 189.304i) q^{65} +(-155.149 + 56.4696i) q^{66} +(-150.415 - 413.261i) q^{67} +(-194.685 + 112.402i) q^{68} +(65.6652 - 113.735i) q^{69} +(214.068 - 1327.00i) q^{70} +(-604.959 + 507.621i) q^{71} +(242.814 - 289.375i) q^{72} +(425.522 - 75.0311i) q^{73} +(1108.72 + 403.541i) q^{74} +(3.85882 + 132.174i) q^{75} +(217.471 - 220.585i) q^{76} +1598.31i q^{77} +(40.9497 - 112.508i) q^{78} +(-186.620 - 1058.37i) q^{79} +(-692.903 - 564.390i) q^{80} +(489.970 - 411.134i) q^{81} +(16.8260 + 2.96688i) q^{82} +(-312.556 - 180.455i) q^{83} +(-69.4133 - 120.227i) q^{84} +(-239.024 - 628.041i) q^{85} +(293.133 - 106.692i) q^{86} +(65.5608 - 37.8515i) q^{87} +(575.787 + 332.431i) q^{88} +(-242.212 + 1373.65i) q^{89} +(-648.311 - 750.119i) q^{90} +(-887.872 - 745.013i) q^{91} +(457.290 - 80.6326i) q^{92} +(45.6928 - 125.540i) q^{93} -327.570 q^{94} +(547.435 + 746.786i) q^{95} -166.202 q^{96} +(-340.020 + 934.196i) q^{97} +(-2996.94 + 528.441i) q^{98} +(903.107 + 757.797i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.17190 + 3.21976i −0.414328 + 1.13836i 0.540537 + 0.841320i \(0.318221\pi\)
−0.954866 + 0.297038i \(0.904001\pi\)
\(3\) −1.04177 + 0.183692i −0.200489 + 0.0353516i −0.272991 − 0.962017i \(-0.588013\pi\)
0.0725018 + 0.997368i \(0.476902\pi\)
\(4\) −2.86517 − 2.40416i −0.358146 − 0.300520i
\(5\) 8.45885 − 7.31080i 0.756583 − 0.653898i
\(6\) 0.629403 − 3.56952i 0.0428254 − 0.242875i
\(7\) −30.3869 − 17.5439i −1.64074 − 0.947282i −0.980571 − 0.196164i \(-0.937151\pi\)
−0.660169 − 0.751117i \(-0.729515\pi\)
\(8\) −12.6403 + 7.29787i −0.558627 + 0.322523i
\(9\) −24.3202 + 8.85181i −0.900747 + 0.327845i
\(10\) 13.6261 + 35.8030i 0.430896 + 1.13219i
\(11\) −22.7759 − 39.4489i −0.624289 − 1.08130i −0.988678 − 0.150053i \(-0.952055\pi\)
0.364389 − 0.931247i \(-0.381278\pi\)
\(12\) 3.42647 + 1.97827i 0.0824280 + 0.0475899i
\(13\) 32.5306 + 5.73602i 0.694028 + 0.122376i 0.509525 − 0.860456i \(-0.329821\pi\)
0.184503 + 0.982832i \(0.440932\pi\)
\(14\) 92.0975 − 77.2790i 1.75815 − 1.47526i
\(15\) −7.46924 + 9.17000i −0.128570 + 0.157846i
\(16\) −13.8801 − 78.7182i −0.216877 − 1.22997i
\(17\) 20.5569 − 56.4797i 0.293282 − 0.805785i −0.702299 − 0.711882i \(-0.747843\pi\)
0.995581 − 0.0939036i \(-0.0299345\pi\)
\(18\) − 88.6785i − 1.16121i
\(19\) −6.63144 + 82.5532i −0.0800714 + 0.996789i
\(20\) −41.8123 + 0.610227i −0.467476 + 0.00682254i
\(21\) 34.8789 + 12.6949i 0.362438 + 0.131917i
\(22\) 153.707 − 27.1027i 1.48957 − 0.262651i
\(23\) −79.8017 + 95.1039i −0.723470 + 0.862197i −0.994963 − 0.100243i \(-0.968038\pi\)
0.271493 + 0.962440i \(0.412482\pi\)
\(24\) 11.8277 − 9.92463i 0.100597 − 0.0844107i
\(25\) 18.1044 − 123.682i 0.144835 − 0.989456i
\(26\) −56.5911 + 98.0187i −0.426863 + 0.739348i
\(27\) 48.4452 − 27.9698i 0.345307 − 0.199363i
\(28\) 44.8852 + 123.321i 0.302947 + 0.832340i
\(29\) −67.2479 + 24.4762i −0.430608 + 0.156728i −0.548226 − 0.836331i \(-0.684696\pi\)
0.117618 + 0.993059i \(0.462474\pi\)
\(30\) −20.7720 − 34.7955i −0.126415 − 0.211759i
\(31\) −63.1459 + 109.372i −0.365850 + 0.633670i −0.988912 − 0.148502i \(-0.952555\pi\)
0.623062 + 0.782172i \(0.285888\pi\)
\(32\) 154.728 + 27.2827i 0.854758 + 0.150717i
\(33\) 30.9737 + 36.9130i 0.163389 + 0.194719i
\(34\) 157.761 + 132.377i 0.795757 + 0.667719i
\(35\) −385.299 + 73.7515i −1.86078 + 0.356179i
\(36\) 90.9625 + 33.1076i 0.421122 + 0.153276i
\(37\) − 344.348i − 1.53001i −0.644022 − 0.765007i \(-0.722735\pi\)
0.644022 − 0.765007i \(-0.277265\pi\)
\(38\) −258.030 − 118.095i −1.10153 − 0.504148i
\(39\) −34.9430 −0.143471
\(40\) −53.5690 + 154.142i −0.211750 + 0.609301i
\(41\) −0.865889 − 4.91070i −0.00329827 − 0.0187054i 0.983114 − 0.182993i \(-0.0585786\pi\)
−0.986412 + 0.164288i \(0.947467\pi\)
\(42\) −81.7489 + 97.4246i −0.300336 + 0.357927i
\(43\) −58.5206 − 69.7422i −0.207542 − 0.247339i 0.652225 − 0.758025i \(-0.273836\pi\)
−0.859767 + 0.510686i \(0.829391\pi\)
\(44\) −29.5849 + 167.785i −0.101366 + 0.574874i
\(45\) −141.007 + 252.676i −0.467112 + 0.837038i
\(46\) −212.693 − 368.394i −0.681735 − 1.18080i
\(47\) 32.6977 + 89.8363i 0.101478 + 0.278808i 0.980034 − 0.198832i \(-0.0637147\pi\)
−0.878556 + 0.477640i \(0.841493\pi\)
\(48\) 28.9198 + 79.4566i 0.0869628 + 0.238928i
\(49\) 444.077 + 769.164i 1.29468 + 2.24246i
\(50\) 377.010 + 203.234i 1.06635 + 0.574834i
\(51\) −11.0407 + 62.6151i −0.0303139 + 0.171919i
\(52\) −79.4152 − 94.6433i −0.211787 − 0.252397i
\(53\) −200.720 + 239.209i −0.520208 + 0.619959i −0.960630 − 0.277831i \(-0.910384\pi\)
0.440422 + 0.897791i \(0.354829\pi\)
\(54\) 33.2834 + 188.760i 0.0838759 + 0.475684i
\(55\) −481.061 − 167.183i −1.17939 − 0.409872i
\(56\) 512.133 1.22208
\(57\) −8.25593 − 87.2196i −0.0191846 − 0.202676i
\(58\) − 245.206i − 0.555123i
\(59\) −194.872 − 70.9278i −0.430004 − 0.156509i 0.117946 − 0.993020i \(-0.462369\pi\)
−0.547949 + 0.836511i \(0.684591\pi\)
\(60\) 43.4468 − 8.31632i 0.0934825 − 0.0178939i
\(61\) −209.212 − 175.550i −0.439129 − 0.368473i 0.396254 − 0.918141i \(-0.370310\pi\)
−0.835383 + 0.549668i \(0.814754\pi\)
\(62\) −278.151 − 331.488i −0.569762 − 0.679016i
\(63\) 894.310 + 157.691i 1.78845 + 0.315352i
\(64\) 50.5613 − 87.5747i 0.0987525 − 0.171044i
\(65\) 317.106 − 189.304i 0.605111 − 0.361236i
\(66\) −155.149 + 56.4696i −0.289356 + 0.105317i
\(67\) −150.415 − 413.261i −0.274270 − 0.753550i −0.997985 − 0.0634514i \(-0.979789\pi\)
0.723715 − 0.690099i \(-0.242433\pi\)
\(68\) −194.685 + 112.402i −0.347192 + 0.200451i
\(69\) 65.6652 − 113.735i 0.114567 − 0.198437i
\(70\) 214.068 − 1327.00i 0.365515 − 2.26581i
\(71\) −604.959 + 507.621i −1.01120 + 0.848500i −0.988497 − 0.151243i \(-0.951673\pi\)
−0.0227057 + 0.999742i \(0.507228\pi\)
\(72\) 242.814 − 289.375i 0.397444 − 0.473655i
\(73\) 425.522 − 75.0311i 0.682242 − 0.120298i 0.178223 − 0.983990i \(-0.442965\pi\)
0.504019 + 0.863693i \(0.331854\pi\)
\(74\) 1108.72 + 403.541i 1.74170 + 0.633928i
\(75\) 3.85882 + 132.174i 0.00594104 + 0.203495i
\(76\) 217.471 − 220.585i 0.328232 − 0.332933i
\(77\) 1598.31i 2.36551i
\(78\) 40.9497 − 112.508i 0.0594441 − 0.163321i
\(79\) −186.620 − 1058.37i −0.265777 − 1.50729i −0.766814 − 0.641869i \(-0.778159\pi\)
0.501038 − 0.865425i \(-0.332952\pi\)
\(80\) −692.903 − 564.390i −0.968361 − 0.788759i
\(81\) 489.970 − 411.134i 0.672113 − 0.563970i
\(82\) 16.8260 + 2.96688i 0.0226600 + 0.00399557i
\(83\) −312.556 − 180.455i −0.413344 − 0.238644i 0.278882 − 0.960325i \(-0.410036\pi\)
−0.692225 + 0.721681i \(0.743370\pi\)
\(84\) −69.4133 − 120.227i −0.0901620 − 0.156165i
\(85\) −239.024 − 628.041i −0.305009 − 0.801420i
\(86\) 293.133 − 106.692i 0.367551 − 0.133778i
\(87\) 65.5608 − 37.8515i 0.0807914 − 0.0466449i
\(88\) 575.787 + 332.431i 0.697489 + 0.402696i
\(89\) −242.212 + 1373.65i −0.288477 + 1.63603i 0.404117 + 0.914707i \(0.367579\pi\)
−0.692594 + 0.721328i \(0.743532\pi\)
\(90\) −648.311 − 750.119i −0.759311 − 0.878549i
\(91\) −887.872 − 745.013i −1.02279 − 0.858226i
\(92\) 457.290 − 80.6326i 0.518215 − 0.0913753i
\(93\) 45.6928 − 125.540i 0.0509475 − 0.139977i
\(94\) −327.570 −0.359428
\(95\) 547.435 + 746.786i 0.591218 + 0.806512i
\(96\) −166.202 −0.176697
\(97\) −340.020 + 934.196i −0.355915 + 0.977868i 0.624517 + 0.781011i \(0.285296\pi\)
−0.980432 + 0.196857i \(0.936926\pi\)
\(98\) −2996.94 + 528.441i −3.08915 + 0.544700i
\(99\) 903.107 + 757.797i 0.916825 + 0.769308i
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.4.6 ✓ 168
5.4 even 2 inner 95.4.p.a.4.23 yes 168
19.5 even 9 inner 95.4.p.a.24.23 yes 168
95.24 even 18 inner 95.4.p.a.24.6 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.6 ✓ 168 1.1 even 1 trivial
95.4.p.a.4.23 yes 168 5.4 even 2 inner
95.4.p.a.24.6 yes 168 95.24 even 18 inner
95.4.p.a.24.23 yes 168 19.5 even 9 inner