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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(31,230)] 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("230.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [70,-14,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(70\)
Relative dimension: \(7\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 81.4
Character \(\chi\) \(=\) 230.81
Dual form 230.4.g.d.71.4

$q$-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+(1.68251 - 1.08128i) q^{2} +(0.0867437 - 0.603316i) q^{3} +(1.66166 - 3.63853i) q^{4} +(-4.79746 + 1.40866i) q^{5} +(-0.506407 - 1.10888i) q^{6} +(-3.02881 - 3.49544i) q^{7} +(-1.13852 - 7.91857i) q^{8} +(25.5498 + 7.50211i) q^{9} +(-6.54861 + 7.55750i) q^{10} +(-47.2942 - 30.3941i) q^{11} +(-2.05104 - 1.31812i) q^{12} +(40.8418 - 47.1339i) q^{13} +(-8.87555 - 2.60610i) q^{14} +(0.433718 + 3.01658i) q^{15} +(-10.4778 - 12.0920i) q^{16} +(-45.2528 - 99.0897i) q^{17} +(51.0997 - 15.0042i) q^{18} +(-5.84951 + 12.8086i) q^{19} +(-2.84630 + 19.7964i) q^{20} +(-2.37158 + 1.52412i) q^{21} -112.437 q^{22} +(-109.995 + 8.25194i) q^{23} -4.87616 q^{24} +(21.0313 - 13.5160i) q^{25} +(17.7515 - 123.465i) q^{26} +(13.5789 - 29.7337i) q^{27} +(-17.7511 + 5.21220i) q^{28} +(-71.0633 - 155.607i) q^{29} +(3.99151 + 4.60644i) q^{30} +(10.9089 + 75.8732i) q^{31} +(-30.7038 - 9.01544i) q^{32} +(-22.4397 + 25.8968i) q^{33} +(-183.282 - 117.788i) q^{34} +(19.4545 + 12.5027i) q^{35} +(69.7518 - 80.4979i) q^{36} +(145.237 + 42.6454i) q^{37} +(4.00790 + 27.8756i) q^{38} +(-24.8939 - 28.7291i) q^{39} +(16.6166 + 36.3853i) q^{40} +(31.5360 - 9.25981i) q^{41} +(-2.34220 + 5.12870i) q^{42} +(-32.3482 + 224.986i) q^{43} +(-189.177 + 121.576i) q^{44} -133.142 q^{45} +(-176.145 + 132.820i) q^{46} +357.908 q^{47} +(-8.20417 + 5.27250i) q^{48} +(45.7696 - 318.335i) q^{49} +(20.7708 - 45.4816i) q^{50} +(-63.7078 + 18.7063i) q^{51} +(-103.633 - 226.925i) q^{52} +(-165.739 - 191.273i) q^{53} +(-9.30387 - 64.7098i) q^{54} +(269.707 + 79.1931i) q^{55} +(-24.2305 + 27.9635i) q^{56} +(7.22024 + 4.64017i) q^{57} +(-287.819 - 184.970i) q^{58} +(-60.7869 + 70.1518i) q^{59} +(11.6966 + 3.43443i) q^{60} +(61.9536 + 430.897i) q^{61} +(100.395 + 115.862i) q^{62} +(-51.1626 - 112.030i) q^{63} +(-61.4076 + 18.0309i) q^{64} +(-129.541 + 283.656i) q^{65} +(-9.75323 + 67.8352i) q^{66} +(255.732 - 164.349i) q^{67} -435.736 q^{68} +(-4.56285 + 67.0775i) q^{69} +46.2513 q^{70} +(511.993 - 329.038i) q^{71} +(30.3170 - 210.860i) q^{72} +(104.334 - 228.460i) q^{73} +(290.474 - 85.2909i) q^{74} +(-6.33009 - 13.8610i) q^{75} +(36.8847 + 42.5672i) q^{76} +(37.0045 + 257.372i) q^{77} +(-72.9483 - 21.4196i) q^{78} +(-224.412 + 258.985i) q^{79} +(67.3003 + 43.2513i) q^{80} +(588.074 + 377.932i) q^{81} +(43.0471 - 49.6790i) q^{82} +(655.197 + 192.383i) q^{83} +(1.60480 + 11.1616i) q^{84} +(356.683 + 411.634i) q^{85} +(188.848 + 413.519i) q^{86} +(-100.044 + 29.3757i) q^{87} +(-186.833 + 409.106i) q^{88} +(-99.1414 + 689.543i) q^{89} +(-224.013 + 143.964i) q^{90} -288.456 q^{91} +(-152.749 + 413.932i) q^{92} +46.7218 q^{93} +(602.183 - 386.999i) q^{94} +(10.0198 - 69.6890i) q^{95} +(-8.10252 + 17.7420i) q^{96} +(-694.554 + 203.939i) q^{97} +(-267.202 - 585.090i) q^{98} +(-980.338 - 1131.37i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 70 q - 14 q^{2} + 3 q^{3} - 28 q^{4} - 35 q^{5} + 6 q^{6} - 78 q^{7} - 56 q^{8} - 24 q^{9} - 70 q^{10} - 15 q^{11} - 120 q^{12} - 270 q^{13} + 64 q^{14} + 15 q^{15} - 112 q^{16} + 114 q^{17} - 48 q^{18}+ \cdots + 11285 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{10}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.68251 − 1.08128i 0.594856 − 0.382291i
\(3\) 0.0867437 − 0.603316i 0.0166938 − 0.116108i −0.979771 − 0.200124i \(-0.935866\pi\)
0.996464 + 0.0840158i \(0.0267746\pi\)
\(4\) 1.66166 − 3.63853i 0.207708 − 0.454816i
\(5\) −4.79746 + 1.40866i −0.429098 + 0.125995i
\(6\) −0.506407 − 1.10888i −0.0344567 − 0.0754495i
\(7\) −3.02881 − 3.49544i −0.163541 − 0.188736i 0.668064 − 0.744103i \(-0.267123\pi\)
−0.831605 + 0.555368i \(0.812578\pi\)
\(8\) −1.13852 − 7.91857i −0.0503159 − 0.349955i
\(9\) 25.5498 + 7.50211i 0.946291 + 0.277856i
\(10\) −6.54861 + 7.55750i −0.207085 + 0.238989i
\(11\) −47.2942 − 30.3941i −1.29634 − 0.833106i −0.303530 − 0.952822i \(-0.598165\pi\)
−0.992808 + 0.119716i \(0.961802\pi\)
\(12\) −2.05104 − 1.31812i −0.0493404 − 0.0317092i
\(13\) 40.8418 − 47.1339i 0.871344 − 1.00559i −0.128559 − 0.991702i \(-0.541035\pi\)
0.999904 − 0.0138832i \(-0.00441930\pi\)
\(14\) −8.87555 − 2.60610i −0.169435 − 0.0497506i
\(15\) 0.433718 + 3.01658i 0.00746571 + 0.0519251i
\(16\) −10.4778 − 12.0920i −0.163715 − 0.188937i
\(17\) −45.2528 − 99.0897i −0.645612 − 1.41369i −0.895342 − 0.445378i \(-0.853069\pi\)
0.249730 − 0.968315i \(-0.419658\pi\)
\(18\) 51.0997 − 15.0042i 0.669128 − 0.196474i
\(19\) −5.84951 + 12.8086i −0.0706299 + 0.154658i −0.941654 − 0.336582i \(-0.890729\pi\)
0.871024 + 0.491240i \(0.163456\pi\)
\(20\) −2.84630 + 19.7964i −0.0318226 + 0.221331i
\(21\) −2.37158 + 1.52412i −0.0246439 + 0.0158377i
\(22\) −112.437 −1.08962
\(23\) −109.995 + 8.25194i −0.997198 + 0.0748108i
\(24\) −4.87616 −0.0414726
\(25\) 21.0313 − 13.5160i 0.168251 − 0.108128i
\(26\) 17.7515 − 123.465i 0.133899 − 0.931285i
\(27\) 13.5789 − 29.7337i 0.0967876 − 0.211935i
\(28\) −17.7511 + 5.21220i −0.119809 + 0.0351790i
\(29\) −71.0633 − 155.607i −0.455038 − 0.996395i −0.988591 − 0.150628i \(-0.951870\pi\)
0.533552 − 0.845767i \(-0.320857\pi\)
\(30\) 3.99151 + 4.60644i 0.0242915 + 0.0280339i
\(31\) 10.9089 + 75.8732i 0.0632032 + 0.439588i 0.996711 + 0.0810322i \(0.0258217\pi\)
−0.933508 + 0.358556i \(0.883269\pi\)
\(32\) −30.7038 − 9.01544i −0.169616 − 0.0498038i
\(33\) −22.4397 + 25.8968i −0.118371 + 0.136608i
\(34\) −183.282 − 117.788i −0.924489 − 0.594133i
\(35\) 19.4545 + 12.5027i 0.0939547 + 0.0603810i
\(36\) 69.7518 − 80.4979i 0.322925 − 0.372675i
\(37\) 145.237 + 42.6454i 0.645320 + 0.189483i 0.587985 − 0.808872i \(-0.299922\pi\)
0.0573349 + 0.998355i \(0.481740\pi\)
\(38\) 4.00790 + 27.8756i 0.0171097 + 0.119000i
\(39\) −24.8939 − 28.7291i −0.102211 − 0.117957i
\(40\) 16.6166 + 36.3853i 0.0656829 + 0.143825i
\(41\) 31.5360 − 9.25981i 0.120124 − 0.0352717i −0.221118 − 0.975247i \(-0.570971\pi\)
0.341242 + 0.939975i \(0.389152\pi\)
\(42\) −2.34220 + 5.12870i −0.00860498 + 0.0188423i
\(43\) −32.3482 + 224.986i −0.114722 + 0.797909i 0.848499 + 0.529198i \(0.177507\pi\)
−0.963221 + 0.268712i \(0.913402\pi\)
\(44\) −189.177 + 121.576i −0.648169 + 0.416553i
\(45\) −133.142 −0.441060
\(46\) −176.145 + 132.820i −0.564590 + 0.425721i
\(47\) 357.908 1.11077 0.555386 − 0.831593i \(-0.312571\pi\)
0.555386 + 0.831593i \(0.312571\pi\)
\(48\) −8.20417 + 5.27250i −0.0246702 + 0.0158546i
\(49\) 45.7696 − 318.335i 0.133439 − 0.928089i
\(50\) 20.7708 − 45.4816i 0.0587486 − 0.128641i
\(51\) −63.7078 + 18.7063i −0.174919 + 0.0513609i
\(52\) −103.633 − 226.925i −0.276371 − 0.605169i
\(53\) −165.739 − 191.273i −0.429548 − 0.495725i 0.499174 − 0.866502i \(-0.333637\pi\)
−0.928722 + 0.370777i \(0.879091\pi\)
\(54\) −9.30387 − 64.7098i −0.0234462 − 0.163072i
\(55\) 269.707 + 79.1931i 0.661224 + 0.194153i
\(56\) −24.2305 + 27.9635i −0.0578203 + 0.0667282i
\(57\) 7.22024 + 4.64017i 0.0167780 + 0.0107825i
\(58\) −287.819 − 184.970i −0.651595 − 0.418755i
\(59\) −60.7869 + 70.1518i −0.134132 + 0.154796i −0.818842 − 0.574019i \(-0.805383\pi\)
0.684710 + 0.728816i \(0.259929\pi\)
\(60\) 11.6966 + 3.43443i 0.0251671 + 0.00738972i
\(61\) 61.9536 + 430.897i 0.130038 + 0.904437i 0.945499 + 0.325624i \(0.105575\pi\)
−0.815461 + 0.578812i \(0.803516\pi\)
\(62\) 100.395 + 115.862i 0.205647 + 0.237330i
\(63\) −51.1626 − 112.030i −0.102316 − 0.224040i
\(64\) −61.4076 + 18.0309i −0.119937 + 0.0352166i
\(65\) −129.541 + 283.656i −0.247194 + 0.541280i
\(66\) −9.75323 + 67.8352i −0.0181900 + 0.126514i
\(67\) 255.732 − 164.349i 0.466308 − 0.299678i −0.286308 − 0.958138i \(-0.592428\pi\)
0.752616 + 0.658460i \(0.228792\pi\)
\(68\) −435.736 −0.777069
\(69\) −4.56285 + 67.0775i −0.00796091 + 0.117032i
\(70\) 46.2513 0.0789726
\(71\) 511.993 − 329.038i 0.855807 − 0.549994i −0.0375736 − 0.999294i \(-0.511963\pi\)
0.893381 + 0.449300i \(0.148326\pi\)
\(72\) 30.3170 − 210.860i 0.0496236 − 0.345139i
\(73\) 104.334 − 228.460i 0.167280 − 0.366291i −0.807364 − 0.590053i \(-0.799107\pi\)
0.974644 + 0.223762i \(0.0718339\pi\)
\(74\) 290.474 − 85.2909i 0.456310 − 0.133985i
\(75\) −6.33009 − 13.8610i −0.00974581 − 0.0213404i
\(76\) 36.8847 + 42.5672i 0.0556706 + 0.0642473i
\(77\) 37.0045 + 257.372i 0.0547669 + 0.380912i
\(78\) −72.9483 − 21.4196i −0.105895 − 0.0310934i
\(79\) −224.412 + 258.985i −0.319599 + 0.368837i −0.892703 − 0.450646i \(-0.851194\pi\)
0.573104 + 0.819483i \(0.305739\pi\)
\(80\) 67.3003 + 43.2513i 0.0940550 + 0.0604455i
\(81\) 588.074 + 377.932i 0.806686 + 0.518426i
\(82\) 43.0471 − 49.6790i 0.0579727 − 0.0669040i
\(83\) 655.197 + 192.383i 0.866472 + 0.254419i 0.684614 − 0.728906i \(-0.259971\pi\)
0.181858 + 0.983325i \(0.441789\pi\)
\(84\) 1.60480 + 11.1616i 0.00208450 + 0.0144980i
\(85\) 356.683 + 411.634i 0.455149 + 0.525270i
\(86\) 188.848 + 413.519i 0.236790 + 0.518498i
\(87\) −100.044 + 29.3757i −0.123286 + 0.0362000i
\(88\) −186.833 + 409.106i −0.226323 + 0.495578i
\(89\) −99.1414 + 689.543i −0.118078 + 0.821252i 0.841590 + 0.540117i \(0.181620\pi\)
−0.959668 + 0.281135i \(0.909289\pi\)
\(90\) −224.013 + 143.964i −0.262367 + 0.168613i
\(91\) −288.456 −0.332290
\(92\) −152.749 + 413.932i −0.173100 + 0.469080i
\(93\) 46.7218 0.0520949
\(94\) 602.183 − 386.999i 0.660749 − 0.424638i
\(95\) 10.0198 − 69.6890i 0.0108211 − 0.0752625i
\(96\) −8.10252 + 17.7420i −0.00861416 + 0.0188624i
\(97\) −694.554 + 203.939i −0.727024 + 0.213473i −0.624242 − 0.781231i \(-0.714592\pi\)
−0.102781 + 0.994704i \(0.532774\pi\)
\(98\) −267.202 − 585.090i −0.275423 − 0.603092i
\(99\) −980.338 − 1131.37i −0.995229 − 1.14856i
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 230.4.g.d.81.4 yes 70
23.2 even 11 inner 230.4.g.d.71.4 ✓ 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.71.4 ✓ 70 23.2 even 11 inner
230.4.g.d.81.4 yes 70 1.1 even 1 trivial