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Properties

Label 35.5.l.a.2.6
Level $35$
Weight $5$
Character 35.2
Analytic conductor $3.618$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [35,5,Mod(2,35)] 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("35.2"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.l (of order \(12\), degree \(4\), 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: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 2.6
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.6

$q$-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+(-1.93554 + 0.518627i) q^{2} +(0.952096 + 0.255113i) q^{3} +(-10.3791 + 5.99235i) q^{4} +(3.12714 - 24.8036i) q^{5} -1.97513 q^{6} +(-47.7288 - 11.0890i) q^{7} +(39.6520 - 39.6520i) q^{8} +(-69.3067 - 40.0142i) q^{9} +(6.81113 + 49.6304i) q^{10} +(-46.1603 - 79.9519i) q^{11} +(-11.4106 + 3.05746i) q^{12} +(-64.8218 + 64.8218i) q^{13} +(98.1321 - 3.29027i) q^{14} +(9.30508 - 22.8177i) q^{15} +(39.6941 - 68.7522i) q^{16} +(-17.2363 + 64.3266i) q^{17} +(154.898 + 41.5049i) q^{18} +(270.008 + 155.889i) q^{19} +(116.175 + 276.177i) q^{20} +(-42.6134 - 22.7340i) q^{21} +(130.810 + 130.810i) q^{22} +(225.082 + 840.019i) q^{23} +(47.8682 - 27.6367i) q^{24} +(-605.442 - 155.129i) q^{25} +(91.8471 - 159.084i) q^{26} +(-112.234 - 112.234i) q^{27} +(561.828 - 170.915i) q^{28} -1133.18i q^{29} +(-6.17651 + 48.9905i) q^{30} +(-858.076 - 1486.23i) q^{31} +(-273.391 + 1020.31i) q^{32} +(-23.5522 - 87.8980i) q^{33} -133.446i q^{34} +(-424.301 + 1149.17i) q^{35} +959.117 q^{36} +(-370.889 + 99.3793i) q^{37} +(-603.460 - 161.697i) q^{38} +(-78.2535 + 45.1797i) q^{39} +(-859.516 - 1107.51i) q^{40} -1219.29 q^{41} +(94.2706 + 21.9022i) q^{42} +(1625.68 - 1625.68i) q^{43} +(958.200 + 553.217i) q^{44} +(-1209.23 + 1593.93i) q^{45} +(-871.313 - 1509.16i) q^{46} +(-272.196 + 72.9348i) q^{47} +(55.3322 - 55.3322i) q^{48} +(2155.07 + 1058.52i) q^{49} +(1252.31 - 13.7398i) q^{50} +(-32.8211 + 56.8479i) q^{51} +(284.354 - 1061.22i) q^{52} +(812.315 + 217.659i) q^{53} +(275.441 + 159.026i) q^{54} +(-2127.45 + 894.922i) q^{55} +(-2332.24 + 1452.84i) q^{56} +(217.304 + 217.304i) q^{57} +(587.697 + 2193.31i) q^{58} +(77.1325 - 44.5324i) q^{59} +(40.1536 + 292.585i) q^{60} +(1434.14 - 2484.01i) q^{61} +(2431.64 + 2431.64i) q^{62} +(2864.20 + 2678.37i) q^{63} -846.429i q^{64} +(1405.11 + 1810.52i) q^{65} +(91.1726 + 157.916i) q^{66} +(946.688 - 3533.09i) q^{67} +(-206.571 - 770.935i) q^{68} +857.200i q^{69} +(225.262 - 2444.32i) q^{70} +5389.96 q^{71} +(-4334.79 + 1161.50i) q^{72} +(-9116.45 - 2442.75i) q^{73} +(666.330 - 384.706i) q^{74} +(-536.863 - 302.154i) q^{75} -3736.57 q^{76} +(1316.59 + 4327.88i) q^{77} +(128.032 - 128.032i) q^{78} +(-2335.70 - 1348.52i) q^{79} +(-1581.18 - 1199.56i) q^{80} +(3162.93 + 5478.35i) q^{81} +(2359.99 - 632.357i) q^{82} +(4334.36 - 4334.36i) q^{83} +(578.517 - 19.3971i) q^{84} +(1541.63 + 628.680i) q^{85} +(-2303.45 + 3989.69i) q^{86} +(289.089 - 1078.89i) q^{87} +(-5000.60 - 1339.91i) q^{88} +(-12188.7 - 7037.13i) q^{89} +(1513.86 - 3712.26i) q^{90} +(3812.67 - 2375.06i) q^{91} +(-7369.83 - 7369.83i) q^{92} +(-437.813 - 1633.94i) q^{93} +(489.022 - 282.337i) q^{94} +(4710.97 - 6209.69i) q^{95} +(-520.589 + 901.686i) q^{96} +(-2885.72 - 2885.72i) q^{97} +(-4720.21 - 931.143i) q^{98} +7388.27i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8} - 66 q^{10} + 296 q^{11} - 358 q^{12} - 8 q^{13} - 68 q^{15} + 468 q^{16} + 28 q^{17} - 868 q^{18} - 1032 q^{20} + 1280 q^{21} + 56 q^{22}+ \cdots - 78606 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.93554 + 0.518627i −0.483886 + 0.129657i −0.492511 − 0.870306i \(-0.663921\pi\)
0.00862561 + 0.999963i \(0.497254\pi\)
\(3\) 0.952096 + 0.255113i 0.105788 + 0.0283459i 0.311325 − 0.950304i \(-0.399227\pi\)
−0.205536 + 0.978649i \(0.565894\pi\)
\(4\) −10.3791 + 5.99235i −0.648691 + 0.374522i
\(5\) 3.12714 − 24.8036i 0.125086 − 0.992146i
\(6\) −1.97513 −0.0548648
\(7\) −47.7288 − 11.0890i −0.974056 − 0.226305i
\(8\) 39.6520 − 39.6520i 0.619562 − 0.619562i
\(9\) −69.3067 − 40.0142i −0.855638 − 0.494003i
\(10\) 6.81113 + 49.6304i 0.0681113 + 0.496304i
\(11\) −46.1603 − 79.9519i −0.381490 − 0.660760i 0.609786 − 0.792566i \(-0.291256\pi\)
−0.991275 + 0.131807i \(0.957922\pi\)
\(12\) −11.4106 + 3.05746i −0.0792401 + 0.0212323i
\(13\) −64.8218 + 64.8218i −0.383561 + 0.383561i −0.872383 − 0.488822i \(-0.837427\pi\)
0.488822 + 0.872383i \(0.337427\pi\)
\(14\) 98.1321 − 3.29027i 0.500674 − 0.0167871i
\(15\) 9.30508 − 22.8177i 0.0413559 − 0.101412i
\(16\) 39.6941 − 68.7522i 0.155055 − 0.268563i
\(17\) −17.2363 + 64.3266i −0.0596410 + 0.222583i −0.989314 − 0.145804i \(-0.953423\pi\)
0.929673 + 0.368387i \(0.120090\pi\)
\(18\) 154.898 + 41.5049i 0.478082 + 0.128102i
\(19\) 270.008 + 155.889i 0.747944 + 0.431826i 0.824951 − 0.565205i \(-0.191203\pi\)
−0.0770064 + 0.997031i \(0.524536\pi\)
\(20\) 116.175 + 276.177i 0.290438 + 0.690443i
\(21\) −42.6134 − 22.7340i −0.0966291 − 0.0515510i
\(22\) 130.810 + 130.810i 0.270270 + 0.270270i
\(23\) 225.082 + 840.019i 0.425486 + 1.58794i 0.762859 + 0.646565i \(0.223795\pi\)
−0.337372 + 0.941371i \(0.609538\pi\)
\(24\) 47.8682 − 27.6367i 0.0831045 − 0.0479804i
\(25\) −605.442 − 155.129i −0.968707 − 0.248206i
\(26\) 91.8471 − 159.084i 0.135868 − 0.235331i
\(27\) −112.234 − 112.234i −0.153956 − 0.153956i
\(28\) 561.828 − 170.915i 0.716618 − 0.218003i
\(29\) − 1133.18i − 1.34742i −0.738997 − 0.673708i \(-0.764700\pi\)
0.738997 − 0.673708i \(-0.235300\pi\)
\(30\) −6.17651 + 48.9905i −0.00686279 + 0.0544338i
\(31\) −858.076 − 1486.23i −0.892899 − 1.54655i −0.836383 − 0.548145i \(-0.815334\pi\)
−0.0565160 − 0.998402i \(-0.517999\pi\)
\(32\) −273.391 + 1020.31i −0.266983 + 0.996395i
\(33\) −23.5522 − 87.8980i −0.0216274 − 0.0807144i
\(34\) − 133.446i − 0.115438i
\(35\) −424.301 + 1149.17i −0.346368 + 0.938099i
\(36\) 959.117 0.740059
\(37\) −370.889 + 99.3793i −0.270919 + 0.0725926i −0.391721 − 0.920084i \(-0.628120\pi\)
0.120802 + 0.992677i \(0.461453\pi\)
\(38\) −603.460 − 161.697i −0.417909 − 0.111978i
\(39\) −78.2535 + 45.1797i −0.0514487 + 0.0297039i
\(40\) −859.516 − 1107.51i −0.537198 − 0.692194i
\(41\) −1219.29 −0.725336 −0.362668 − 0.931918i \(-0.618134\pi\)
−0.362668 + 0.931918i \(0.618134\pi\)
\(42\) 94.2706 + 21.9022i 0.0534414 + 0.0124162i
\(43\) 1625.68 − 1625.68i 0.879221 − 0.879221i −0.114233 − 0.993454i \(-0.536441\pi\)
0.993454 + 0.114233i \(0.0364411\pi\)
\(44\) 958.200 + 553.217i 0.494938 + 0.285753i
\(45\) −1209.23 + 1593.93i −0.597151 + 0.787125i
\(46\) −871.313 − 1509.16i −0.411774 − 0.713213i
\(47\) −272.196 + 72.9348i −0.123222 + 0.0330171i −0.319903 − 0.947450i \(-0.603650\pi\)
0.196681 + 0.980467i \(0.436984\pi\)
\(48\) 55.3322 − 55.3322i 0.0240157 − 0.0240157i
\(49\) 2155.07 + 1058.52i 0.897572 + 0.440868i
\(50\) 1252.31 − 13.7398i 0.500925 − 0.00549590i
\(51\) −32.8211 + 56.8479i −0.0126187 + 0.0218562i
\(52\) 284.354 − 1061.22i 0.105161 − 0.392465i
\(53\) 812.315 + 217.659i 0.289183 + 0.0774863i 0.400494 − 0.916299i \(-0.368838\pi\)
−0.111311 + 0.993786i \(0.535505\pi\)
\(54\) 275.441 + 159.026i 0.0944586 + 0.0545357i
\(55\) −2127.45 + 894.922i −0.703289 + 0.295842i
\(56\) −2332.24 + 1452.84i −0.743698 + 0.463278i
\(57\) 217.304 + 217.304i 0.0668833 + 0.0668833i
\(58\) 587.697 + 2193.31i 0.174702 + 0.651996i
\(59\) 77.1325 − 44.5324i 0.0221581 − 0.0127930i −0.488880 − 0.872351i \(-0.662594\pi\)
0.511038 + 0.859558i \(0.329261\pi\)
\(60\) 40.1536 + 292.585i 0.0111538 + 0.0812736i
\(61\) 1434.14 − 2484.01i 0.385419 − 0.667565i −0.606408 − 0.795153i \(-0.707390\pi\)
0.991827 + 0.127588i \(0.0407236\pi\)
\(62\) 2431.64 + 2431.64i 0.632582 + 0.632582i
\(63\) 2864.20 + 2678.37i 0.721644 + 0.674822i
\(64\) − 846.429i − 0.206648i
\(65\) 1405.11 + 1810.52i 0.332571 + 0.428527i
\(66\) 91.1726 + 157.916i 0.0209303 + 0.0362524i
\(67\) 946.688 − 3533.09i 0.210891 − 0.787054i −0.776682 − 0.629892i \(-0.783099\pi\)
0.987573 − 0.157162i \(-0.0502344\pi\)
\(68\) −206.571 − 770.935i −0.0446737 − 0.166725i
\(69\) 857.200i 0.180046i
\(70\) 225.262 − 2444.32i 0.0459719 − 0.498842i
\(71\) 5389.96 1.06922 0.534612 − 0.845098i \(-0.320458\pi\)
0.534612 + 0.845098i \(0.320458\pi\)
\(72\) −4334.79 + 1161.50i −0.836186 + 0.224055i
\(73\) −9116.45 − 2442.75i −1.71072 − 0.458387i −0.735123 − 0.677934i \(-0.762875\pi\)
−0.975602 + 0.219547i \(0.929542\pi\)
\(74\) 666.330 − 384.706i 0.121682 − 0.0702531i
\(75\) −536.863 − 302.154i −0.0954424 − 0.0537163i
\(76\) −3736.57 −0.646913
\(77\) 1316.59 + 4327.88i 0.222059 + 0.729950i
\(78\) 128.032 − 128.032i 0.0210440 − 0.0210440i
\(79\) −2335.70 − 1348.52i −0.374251 − 0.216074i 0.301063 − 0.953604i \(-0.402659\pi\)
−0.675314 + 0.737530i \(0.735992\pi\)
\(80\) −1581.18 − 1199.56i −0.247059 − 0.187431i
\(81\) 3162.93 + 5478.35i 0.482080 + 0.834987i
\(82\) 2359.99 − 632.357i 0.350980 − 0.0940447i
\(83\) 4334.36 − 4334.36i 0.629171 − 0.629171i −0.318689 − 0.947859i \(-0.603242\pi\)
0.947859 + 0.318689i \(0.103242\pi\)
\(84\) 578.517 − 19.3971i 0.0819894 − 0.00274902i
\(85\) 1541.63 + 628.680i 0.213375 + 0.0870146i
\(86\) −2303.45 + 3989.69i −0.311445 + 0.539439i
\(87\) 289.089 − 1078.89i 0.0381938 − 0.142541i
\(88\) −5000.60 − 1339.91i −0.645738 − 0.173025i
\(89\) −12188.7 − 7037.13i −1.53878 − 0.888414i −0.998911 − 0.0466649i \(-0.985141\pi\)
−0.539868 − 0.841749i \(-0.681526\pi\)
\(90\) 1513.86 − 3712.26i 0.186897 − 0.458303i
\(91\) 3812.67 − 2375.06i 0.460412 − 0.286808i
\(92\) −7369.83 − 7369.83i −0.870726 − 0.870726i
\(93\) −437.813 − 1633.94i −0.0506201 − 0.188917i
\(94\) 489.022 − 282.337i 0.0553443 − 0.0319530i
\(95\) 4710.97 − 6209.69i 0.521991 − 0.688055i
\(96\) −520.589 + 901.686i −0.0564875 + 0.0978392i
\(97\) −2885.72 − 2885.72i −0.306698 − 0.306698i 0.536929 − 0.843627i \(-0.319584\pi\)
−0.843627 + 0.536929i \(0.819584\pi\)
\(98\) −4720.21 − 931.143i −0.491484 − 0.0969536i
\(99\) 7388.27i 0.753828i
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 35.5.l.a.2.6 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.9 yes 56
7.4 even 3 inner 35.5.l.a.32.9 yes 56
35.18 odd 12 inner 35.5.l.a.18.6 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.6 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.6 yes 56 35.18 odd 12 inner
35.5.l.a.23.9 yes 56 5.3 odd 4 inner
35.5.l.a.32.9 yes 56 7.4 even 3 inner