Show commands: Magma / Pari/GP / SageMath

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 41.7
Character \(\chi\) \(=\) 230.41
Dual form 230.4.g.d.101.7

$q$-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+(-1.91899 - 0.563465i) q^{2} +(4.61105 - 5.32144i) q^{3} +(3.36501 + 2.16256i) q^{4} +(-0.711574 + 4.94911i) q^{5} +(-11.8470 + 7.61360i) q^{6} +(4.61346 + 10.1021i) q^{7} +(-5.23889 - 6.04600i) q^{8} +(-3.21340 - 22.3497i) q^{9} +(4.15415 - 9.09632i) q^{10} +(42.6228 - 12.5152i) q^{11} +(27.0242 - 7.93502i) q^{12} +(-18.9954 + 41.5940i) q^{13} +(-3.16100 - 21.9853i) q^{14} +(23.0553 + 26.6072i) q^{15} +(6.64664 + 14.5541i) q^{16} +(63.5394 - 40.8343i) q^{17} +(-6.42680 + 44.6994i) q^{18} +(85.6830 + 55.0651i) q^{19} +(-13.0972 + 15.1150i) q^{20} +(75.0305 + 22.0309i) q^{21} -88.8444 q^{22} +(101.718 - 42.6667i) q^{23} -56.3302 q^{24} +(-23.9873 - 7.04331i) q^{25} +(59.8886 - 69.1151i) q^{26} +(26.1848 + 16.8279i) q^{27} +(-6.32201 + 43.9705i) q^{28} +(-213.599 + 137.272i) q^{29} +(-29.2505 - 64.0497i) q^{30} +(-120.562 - 139.136i) q^{31} +(-4.55407 - 31.6743i) q^{32} +(129.937 - 284.523i) q^{33} +(-144.940 + 42.5582i) q^{34} +(-53.2791 + 15.6441i) q^{35} +(37.5195 - 82.1562i) q^{36} +(-9.77094 - 67.9583i) q^{37} +(-133.397 - 153.949i) q^{38} +(133.751 + 292.875i) q^{39} +(33.6501 - 21.6256i) q^{40} +(31.2173 - 217.121i) q^{41} +(-131.569 - 84.5541i) q^{42} +(255.076 - 294.373i) q^{43} +(170.491 + 50.0607i) q^{44} +112.898 q^{45} +(-219.237 + 24.5622i) q^{46} +457.364 q^{47} +(108.097 + 31.7401i) q^{48} +(143.849 - 166.011i) q^{49} +(42.0627 + 27.0320i) q^{50} +(75.6863 - 526.410i) q^{51} +(-153.869 + 98.8858i) q^{52} +(202.863 + 444.207i) q^{53} +(-40.7663 - 47.0468i) q^{54} +(31.6097 + 219.850i) q^{55} +(36.9077 - 80.8166i) q^{56} +(688.115 - 202.049i) q^{57} +(487.242 - 143.067i) q^{58} +(-289.563 + 634.054i) q^{59} +(20.0415 + 139.392i) q^{60} +(-283.222 - 326.856i) q^{61} +(152.958 + 334.932i) q^{62} +(210.953 - 135.571i) q^{63} +(-9.10815 + 63.3486i) q^{64} +(-192.337 - 123.607i) q^{65} +(-409.666 + 472.780i) q^{66} +(-95.7923 - 28.1272i) q^{67} +302.118 q^{68} +(241.979 - 738.024i) q^{69} +111.057 q^{70} +(-204.193 - 59.9565i) q^{71} +(-118.291 + 136.516i) q^{72} +(-723.924 - 465.238i) q^{73} +(-19.5419 + 135.917i) q^{74} +(-148.087 + 95.1700i) q^{75} +(169.243 + 370.590i) q^{76} +(323.068 + 372.840i) q^{77} +(-91.6424 - 637.387i) q^{78} +(174.950 - 383.088i) q^{79} +(-76.7594 + 22.5386i) q^{80} +(795.240 - 233.503i) q^{81} +(-182.246 + 399.062i) q^{82} +(17.3511 + 120.680i) q^{83} +(204.835 + 236.393i) q^{84} +(156.880 + 343.520i) q^{85} +(-655.356 + 421.172i) q^{86} +(-254.434 + 1769.63i) q^{87} +(-298.963 - 192.132i) q^{88} +(-188.102 + 217.081i) q^{89} +(-216.649 - 63.6138i) q^{90} -507.820 q^{91} +(434.552 + 76.3976i) q^{92} -1296.32 q^{93} +(-877.676 - 257.709i) q^{94} +(-333.493 + 384.872i) q^{95} +(-189.552 - 121.818i) q^{96} +(-109.700 + 762.980i) q^{97} +(-369.586 + 237.519i) q^{98} +(-416.674 - 912.389i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{6}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.91899 − 0.563465i −0.678464 − 0.199215i
\(3\) 4.61105 − 5.32144i 0.887397 − 1.02411i −0.112140 − 0.993692i \(-0.535770\pi\)
0.999537 − 0.0304188i \(-0.00968409\pi\)
\(4\) 3.36501 + 2.16256i 0.420627 + 0.270320i
\(5\) −0.711574 + 4.94911i −0.0636451 + 0.442662i
\(6\) −11.8470 + 7.61360i −0.806085 + 0.518040i
\(7\) 4.61346 + 10.1021i 0.249104 + 0.545461i 0.992335 − 0.123573i \(-0.0394354\pi\)
−0.743232 + 0.669034i \(0.766708\pi\)
\(8\) −5.23889 − 6.04600i −0.231528 − 0.267198i
\(9\) −3.21340 − 22.3497i −0.119015 − 0.827766i
\(10\) 4.15415 − 9.09632i 0.131366 − 0.287651i
\(11\) 42.6228 − 12.5152i 1.16830 − 0.343043i 0.360645 − 0.932703i \(-0.382557\pi\)
0.807651 + 0.589661i \(0.200739\pi\)
\(12\) 27.0242 − 7.93502i 0.650101 − 0.190887i
\(13\) −18.9954 + 41.5940i −0.405259 + 0.887393i 0.591451 + 0.806341i \(0.298555\pi\)
−0.996710 + 0.0810520i \(0.974172\pi\)
\(14\) −3.16100 − 21.9853i −0.0603438 − 0.419701i
\(15\) 23.0553 + 26.6072i 0.396856 + 0.457996i
\(16\) 6.64664 + 14.5541i 0.103854 + 0.227408i
\(17\) 63.5394 − 40.8343i 0.906503 − 0.582574i −0.00220874 − 0.999998i \(-0.500703\pi\)
0.908712 + 0.417423i \(0.137067\pi\)
\(18\) −6.42680 + 44.6994i −0.0841561 + 0.585319i
\(19\) 85.6830 + 55.0651i 1.03458 + 0.664885i 0.943641 − 0.330972i \(-0.107376\pi\)
0.0909401 + 0.995856i \(0.471013\pi\)
\(20\) −13.0972 + 15.1150i −0.146431 + 0.168991i
\(21\) 75.0305 + 22.0309i 0.779666 + 0.228931i
\(22\) −88.8444 −0.860986
\(23\) 101.718 − 42.6667i 0.922160 − 0.386809i
\(24\) −56.3302 −0.479098
\(25\) −23.9873 − 7.04331i −0.191899 − 0.0563465i
\(26\) 59.8886 − 69.1151i 0.451735 − 0.521330i
\(27\) 26.1848 + 16.8279i 0.186639 + 0.119946i
\(28\) −6.32201 + 43.9705i −0.0426695 + 0.296773i
\(29\) −213.599 + 137.272i −1.36774 + 0.878992i −0.998727 − 0.0504324i \(-0.983940\pi\)
−0.369011 + 0.929425i \(0.620304\pi\)
\(30\) −29.2505 − 64.0497i −0.178013 − 0.389794i
\(31\) −120.562 − 139.136i −0.698501 − 0.806113i 0.290048 − 0.957012i \(-0.406329\pi\)
−0.988549 + 0.150899i \(0.951783\pi\)
\(32\) −4.55407 − 31.6743i −0.0251579 − 0.174977i
\(33\) 129.937 − 284.523i 0.685429 − 1.50088i
\(34\) −144.940 + 42.5582i −0.731088 + 0.214667i
\(35\) −53.2791 + 15.6441i −0.257309 + 0.0755527i
\(36\) 37.5195 − 82.1562i 0.173701 − 0.380353i
\(37\) −9.77094 − 67.9583i −0.0434144 − 0.301954i −0.999947 − 0.0103272i \(-0.996713\pi\)
0.956532 − 0.291626i \(-0.0941964\pi\)
\(38\) −133.397 − 153.949i −0.569471 − 0.657204i
\(39\) 133.751 + 292.875i 0.549163 + 1.20250i
\(40\) 33.6501 − 21.6256i 0.133014 − 0.0854828i
\(41\) 31.2173 − 217.121i 0.118910 − 0.827039i −0.839849 − 0.542820i \(-0.817357\pi\)
0.958759 − 0.284219i \(-0.0917343\pi\)
\(42\) −131.569 − 84.5541i −0.483369 − 0.310642i
\(43\) 255.076 − 294.373i 0.904621 − 1.04399i −0.0942049 − 0.995553i \(-0.530031\pi\)
0.998826 − 0.0484359i \(-0.0154237\pi\)
\(44\) 170.491 + 50.0607i 0.584148 + 0.171521i
\(45\) 112.898 0.373995
\(46\) −219.237 + 24.5622i −0.702710 + 0.0787283i
\(47\) 457.364 1.41944 0.709718 − 0.704486i \(-0.248823\pi\)
0.709718 + 0.704486i \(0.248823\pi\)
\(48\) 108.097 + 31.7401i 0.325051 + 0.0954435i
\(49\) 143.849 − 166.011i 0.419386 − 0.483997i
\(50\) 42.0627 + 27.0320i 0.118971 + 0.0764582i
\(51\) 75.6863 − 526.410i 0.207808 − 1.44534i
\(52\) −153.869 + 98.8858i −0.410343 + 0.263711i
\(53\) 202.863 + 444.207i 0.525761 + 1.15125i 0.967212 + 0.253970i \(0.0817364\pi\)
−0.441452 + 0.897285i \(0.645536\pi\)
\(54\) −40.7663 − 47.0468i −0.102733 − 0.118560i
\(55\) 31.6097 + 219.850i 0.0774954 + 0.538993i
\(56\) 36.9077 − 80.8166i 0.0880714 − 0.192849i
\(57\) 688.115 − 202.049i 1.59900 − 0.469509i
\(58\) 487.242 − 143.067i 1.10307 − 0.323891i
\(59\) −289.563 + 634.054i −0.638946 + 1.39910i 0.261958 + 0.965079i \(0.415632\pi\)
−0.900904 + 0.434018i \(0.857095\pi\)
\(60\) 20.0415 + 139.392i 0.0431225 + 0.299924i
\(61\) −283.222 − 326.856i −0.594474 − 0.686059i 0.376178 − 0.926547i \(-0.377238\pi\)
−0.970652 + 0.240488i \(0.922693\pi\)
\(62\) 152.958 + 334.932i 0.313318 + 0.686071i
\(63\) 210.953 − 135.571i 0.421867 − 0.271117i
\(64\) −9.10815 + 63.3486i −0.0177894 + 0.123728i
\(65\) −192.337 − 123.607i −0.367022 − 0.235871i
\(66\) −409.666 + 472.780i −0.764037 + 0.881745i
\(67\) −95.7923 − 28.1272i −0.174670 − 0.0512877i 0.193228 − 0.981154i \(-0.438104\pi\)
−0.367898 + 0.929866i \(0.619922\pi\)
\(68\) 302.118 0.538781
\(69\) 241.979 − 738.024i 0.422186 − 1.28765i
\(70\) 111.057 0.189626
\(71\) −204.193 − 59.9565i −0.341313 − 0.100219i 0.106584 − 0.994304i \(-0.466009\pi\)
−0.447897 + 0.894085i \(0.647827\pi\)
\(72\) −118.291 + 136.516i −0.193622 + 0.223452i
\(73\) −723.924 − 465.238i −1.16067 − 0.745917i −0.188934 − 0.981990i \(-0.560503\pi\)
−0.971735 + 0.236073i \(0.924140\pi\)
\(74\) −19.5419 + 135.917i −0.0306986 + 0.213513i
\(75\) −148.087 + 95.1700i −0.227995 + 0.146524i
\(76\) 169.243 + 370.590i 0.255440 + 0.559337i
\(77\) 323.068 + 372.840i 0.478143 + 0.551806i
\(78\) −91.6424 − 637.387i −0.133032 − 0.925255i
\(79\) 174.950 − 383.088i 0.249158 − 0.545579i −0.743186 − 0.669085i \(-0.766686\pi\)
0.992344 + 0.123505i \(0.0394136\pi\)
\(80\) −76.7594 + 22.5386i −0.107275 + 0.0314987i
\(81\) 795.240 − 233.503i 1.09086 − 0.320307i
\(82\) −182.246 + 399.062i −0.245435 + 0.537427i
\(83\) 17.3511 + 120.680i 0.0229462 + 0.159594i 0.998073 − 0.0620545i \(-0.0197652\pi\)
−0.975127 + 0.221649i \(0.928856\pi\)
\(84\) 204.835 + 236.393i 0.266064 + 0.307054i
\(85\) 156.880 + 343.520i 0.200189 + 0.438352i
\(86\) −655.356 + 421.172i −0.821731 + 0.528095i
\(87\) −254.434 + 1769.63i −0.313542 + 2.18073i
\(88\) −298.963 − 192.132i −0.362154 − 0.232742i
\(89\) −188.102 + 217.081i −0.224031 + 0.258546i −0.856627 − 0.515936i \(-0.827444\pi\)
0.632596 + 0.774482i \(0.281990\pi\)
\(90\) −216.649 − 63.6138i −0.253742 − 0.0745054i
\(91\) −507.820 −0.584989
\(92\) 434.552 + 76.3976i 0.492448 + 0.0865761i
\(93\) −1296.32 −1.44540
\(94\) −877.676 − 257.709i −0.963036 − 0.282773i
\(95\) −333.493 + 384.872i −0.360165 + 0.415653i
\(96\) −189.552 − 121.818i −0.201521 − 0.129510i
\(97\) −109.700 + 762.980i −0.114828 + 0.798649i 0.848283 + 0.529544i \(0.177637\pi\)
−0.963111 + 0.269105i \(0.913272\pi\)
\(98\) −369.586 + 237.519i −0.380958 + 0.244827i
\(99\) −416.674 − 912.389i −0.423003 − 0.926248i
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.41.7 ✓ 70
23.9 even 11 inner 230.4.g.d.101.7 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.41.7 ✓ 70 1.1 even 1 trivial
230.4.g.d.101.7 yes 70 23.9 even 11 inner