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Properties

Label 35.5.l.a.18.2
Level $35$
Weight $5$
Character 35.18
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 18.2
Character \(\chi\) \(=\) 35.18
Dual form 35.5.l.a.2.2

$q$-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+(-5.71329 - 1.53087i) q^{2} +(-15.4449 + 4.13845i) q^{3} +(16.4417 + 9.49261i) q^{4} +(-22.7561 + 10.3519i) q^{5} +94.5767 q^{6} +(-15.2970 - 46.5511i) q^{7} +(-12.4854 - 12.4854i) q^{8} +(151.270 - 87.3360i) q^{9} +(145.859 - 24.3067i) q^{10} +(-42.5401 + 73.6816i) q^{11} +(-293.225 - 78.5694i) q^{12} +(-10.7310 - 10.7310i) q^{13} +(16.1328 + 289.377i) q^{14} +(308.625 - 254.059i) q^{15} +(-99.6625 - 172.620i) q^{16} +(93.8444 + 350.232i) q^{17} +(-997.952 + 267.400i) q^{18} +(-81.5338 + 47.0736i) q^{19} +(-472.414 - 45.8122i) q^{20} +(428.911 + 655.671i) q^{21} +(355.841 - 355.841i) q^{22} +(143.194 - 534.406i) q^{23} +(244.507 + 141.166i) q^{24} +(410.677 - 471.136i) q^{25} +(44.8815 + 77.7370i) q^{26} +(-1059.10 + 1059.10i) q^{27} +(190.382 - 910.587i) q^{28} -228.863i q^{29} +(-2152.19 + 979.046i) q^{30} +(408.154 - 706.943i) q^{31} +(378.261 + 1411.69i) q^{32} +(352.100 - 1314.06i) q^{33} -2144.64i q^{34} +(829.991 + 900.966i) q^{35} +3316.19 q^{36} +(1937.56 + 519.167i) q^{37} +(537.890 - 144.127i) q^{38} +(210.149 + 121.330i) q^{39} +(413.367 + 154.872i) q^{40} -151.398 q^{41} +(-1446.74 - 4402.64i) q^{42} +(209.207 + 209.207i) q^{43} +(-1398.86 + 807.633i) q^{44} +(-2538.23 + 3553.36i) q^{45} +(-1636.21 + 2834.01i) q^{46} +(-1306.86 - 350.173i) q^{47} +(2253.66 + 2253.66i) q^{48} +(-1933.00 + 1424.19i) q^{49} +(-3067.57 + 2063.04i) q^{50} +(-2898.84 - 5020.93i) q^{51} +(-74.5705 - 278.301i) q^{52} +(1024.67 - 274.560i) q^{53} +(7672.29 - 4429.60i) q^{54} +(205.302 - 2117.07i) q^{55} +(-390.220 + 772.201i) q^{56} +(1064.47 - 1064.47i) q^{57} +(-350.359 + 1307.56i) q^{58} +(-1517.42 - 876.081i) q^{59} +(7485.99 - 1247.50i) q^{60} +(3536.71 + 6125.76i) q^{61} +(-3414.14 + 3414.14i) q^{62} +(-6379.58 - 5705.82i) q^{63} -5455.25i q^{64} +(355.281 + 133.109i) q^{65} +(-4023.30 + 6968.56i) q^{66} +(-786.257 - 2934.35i) q^{67} +(-1781.66 + 6649.24i) q^{68} +8846.46i q^{69} +(-3362.72 - 6418.09i) q^{70} +4893.58 q^{71} +(-2979.11 - 798.249i) q^{72} +(7641.14 - 2047.44i) q^{73} +(-10275.0 - 5932.30i) q^{74} +(-4393.10 + 8976.22i) q^{75} -1787.40 q^{76} +(4080.69 + 853.175i) q^{77} +(-1014.90 - 1014.90i) q^{78} +(4206.19 - 2428.45i) q^{79} +(4054.87 + 2896.47i) q^{80} +(4900.45 - 8487.83i) q^{81} +(864.978 + 231.770i) q^{82} +(-3185.61 - 3185.61i) q^{83} +(827.988 + 14851.8i) q^{84} +(-5761.09 - 6998.44i) q^{85} +(-874.990 - 1515.53i) q^{86} +(947.137 + 3534.76i) q^{87} +(1451.08 - 388.815i) q^{88} +(-1185.37 + 684.375i) q^{89} +(19941.4 - 16415.7i) q^{90} +(-335.387 + 663.692i) q^{91} +(7427.26 - 7427.26i) q^{92} +(-3378.25 + 12607.8i) q^{93} +(6930.42 + 4001.28i) q^{94} +(1368.09 - 1915.24i) q^{95} +(-11684.4 - 20238.0i) q^{96} +(3098.05 - 3098.05i) q^{97} +(13224.0 - 5177.62i) q^{98} +14861.1i 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 \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.71329 − 1.53087i −1.42832 − 0.382718i −0.539894 − 0.841733i \(-0.681536\pi\)
−0.888428 + 0.459016i \(0.848202\pi\)
\(3\) −15.4449 + 4.13845i −1.71610 + 0.459828i −0.976907 − 0.213664i \(-0.931460\pi\)
−0.739194 + 0.673492i \(0.764793\pi\)
\(4\) 16.4417 + 9.49261i 1.02761 + 0.593288i
\(5\) −22.7561 + 10.3519i −0.910243 + 0.414075i
\(6\) 94.5767 2.62713
\(7\) −15.2970 − 46.5511i −0.312185 − 0.950021i
\(8\) −12.4854 − 12.4854i −0.195085 − 0.195085i
\(9\) 151.270 − 87.3360i 1.86754 − 1.07822i
\(10\) 145.859 − 24.3067i 1.45859 − 0.243067i
\(11\) −42.5401 + 73.6816i −0.351571 + 0.608939i −0.986525 − 0.163611i \(-0.947686\pi\)
0.634954 + 0.772550i \(0.281019\pi\)
\(12\) −293.225 − 78.5694i −2.03629 − 0.545621i
\(13\) −10.7310 − 10.7310i −0.0634970 − 0.0634970i 0.674645 − 0.738142i \(-0.264297\pi\)
−0.738142 + 0.674645i \(0.764297\pi\)
\(14\) 16.1328 + 289.377i 0.0823101 + 1.47642i
\(15\) 308.625 − 254.059i 1.37167 − 1.12915i
\(16\) −99.6625 − 172.620i −0.389306 − 0.674299i
\(17\) 93.8444 + 350.232i 0.324721 + 1.21188i 0.914592 + 0.404378i \(0.132512\pi\)
−0.589871 + 0.807498i \(0.700821\pi\)
\(18\) −997.952 + 267.400i −3.08010 + 0.825310i
\(19\) −81.5338 + 47.0736i −0.225855 + 0.130398i −0.608659 − 0.793432i \(-0.708292\pi\)
0.382803 + 0.923830i \(0.374959\pi\)
\(20\) −472.414 − 45.8122i −1.18104 − 0.114530i
\(21\) 428.911 + 655.671i 0.972587 + 1.48678i
\(22\) 355.841 − 355.841i 0.735208 − 0.735208i
\(23\) 143.194 − 534.406i 0.270688 − 1.01022i −0.687989 − 0.725721i \(-0.741506\pi\)
0.958676 − 0.284499i \(-0.0918271\pi\)
\(24\) 244.507 + 141.166i 0.424491 + 0.245080i
\(25\) 410.677 − 471.136i 0.657084 − 0.753818i
\(26\) 44.8815 + 77.7370i 0.0663927 + 0.114996i
\(27\) −1059.10 + 1059.10i −1.45281 + 1.45281i
\(28\) 190.382 − 910.587i 0.242834 − 1.16146i
\(29\) − 228.863i − 0.272131i −0.990700 − 0.136066i \(-0.956554\pi\)
0.990700 − 0.136066i \(-0.0434458\pi\)
\(30\) −2152.19 + 979.046i −2.39133 + 1.08783i
\(31\) 408.154 − 706.943i 0.424718 − 0.735633i −0.571676 − 0.820479i \(-0.693707\pi\)
0.996394 + 0.0848463i \(0.0270399\pi\)
\(32\) 378.261 + 1411.69i 0.369395 + 1.37860i
\(33\) 352.100 − 1314.06i 0.323324 − 1.20666i
\(34\) − 2144.64i − 1.85523i
\(35\) 829.991 + 900.966i 0.677544 + 0.735482i
\(36\) 3316.19 2.55879
\(37\) 1937.56 + 519.167i 1.41531 + 0.379231i 0.883817 − 0.467833i \(-0.154965\pi\)
0.531492 + 0.847063i \(0.321632\pi\)
\(38\) 537.890 − 144.127i 0.372500 − 0.0998110i
\(39\) 210.149 + 121.330i 0.138165 + 0.0797696i
\(40\) 413.367 + 154.872i 0.258355 + 0.0967949i
\(41\) −151.398 −0.0900641 −0.0450320 − 0.998986i \(-0.514339\pi\)
−0.0450320 + 0.998986i \(0.514339\pi\)
\(42\) −1446.74 − 4402.64i −0.820149 − 2.49583i
\(43\) 209.207 + 209.207i 0.113146 + 0.113146i 0.761413 − 0.648267i \(-0.224506\pi\)
−0.648267 + 0.761413i \(0.724506\pi\)
\(44\) −1398.86 + 807.633i −0.722552 + 0.417166i
\(45\) −2538.23 + 3553.36i −1.25345 + 1.75474i
\(46\) −1636.21 + 2834.01i −0.773258 + 1.33932i
\(47\) −1306.86 − 350.173i −0.591609 − 0.158521i −0.0494204 − 0.998778i \(-0.515737\pi\)
−0.542188 + 0.840257i \(0.682404\pi\)
\(48\) 2253.66 + 2253.66i 0.978151 + 0.978151i
\(49\) −1933.00 + 1424.19i −0.805082 + 0.593164i
\(50\) −3067.57 + 2063.04i −1.22703 + 0.825217i
\(51\) −2898.84 − 5020.93i −1.11451 − 1.93039i
\(52\) −74.5705 − 278.301i −0.0275778 − 0.102922i
\(53\) 1024.67 − 274.560i 0.364781 − 0.0977428i −0.0717725 − 0.997421i \(-0.522866\pi\)
0.436554 + 0.899678i \(0.356199\pi\)
\(54\) 7672.29 − 4429.60i 2.63110 − 1.51907i
\(55\) 205.302 − 2117.07i 0.0678685 − 0.699859i
\(56\) −390.220 + 772.201i −0.124432 + 0.246237i
\(57\) 1064.47 − 1064.47i 0.327630 − 0.327630i
\(58\) −350.359 + 1307.56i −0.104150 + 0.388691i
\(59\) −1517.42 − 876.081i −0.435914 − 0.251675i 0.265949 − 0.963987i \(-0.414315\pi\)
−0.701863 + 0.712312i \(0.747648\pi\)
\(60\) 7485.99 − 1247.50i 2.07944 − 0.346527i
\(61\) 3536.71 + 6125.76i 0.950472 + 1.64627i 0.744405 + 0.667729i \(0.232733\pi\)
0.206068 + 0.978538i \(0.433933\pi\)
\(62\) −3414.14 + 3414.14i −0.888174 + 0.888174i
\(63\) −6379.58 − 5705.82i −1.60735 − 1.43760i
\(64\) − 5455.25i − 1.33185i
\(65\) 355.281 + 133.109i 0.0840902 + 0.0315052i
\(66\) −4023.30 + 6968.56i −0.923622 + 1.59976i
\(67\) −786.257 − 2934.35i −0.175152 − 0.653676i −0.996526 − 0.0832854i \(-0.973459\pi\)
0.821374 − 0.570390i \(-0.193208\pi\)
\(68\) −1781.66 + 6649.24i −0.385306 + 1.43798i
\(69\) 8846.46i 1.85811i
\(70\) −3362.72 − 6418.09i −0.686269 − 1.30981i
\(71\) 4893.58 0.970756 0.485378 − 0.874304i \(-0.338682\pi\)
0.485378 + 0.874304i \(0.338682\pi\)
\(72\) −2979.11 − 798.249i −0.574673 − 0.153983i
\(73\) 7641.14 − 2047.44i 1.43388 − 0.384207i 0.543494 − 0.839413i \(-0.317101\pi\)
0.890386 + 0.455206i \(0.150435\pi\)
\(74\) −10275.0 − 5932.30i −1.87638 − 1.08333i
\(75\) −4393.10 + 8976.22i −0.780996 + 1.59577i
\(76\) −1787.40 −0.309454
\(77\) 4080.69 + 853.175i 0.688260 + 0.143899i
\(78\) −1014.90 − 1014.90i −0.166815 − 0.166815i
\(79\) 4206.19 − 2428.45i 0.673961 − 0.389112i −0.123615 − 0.992330i \(-0.539449\pi\)
0.797576 + 0.603219i \(0.206115\pi\)
\(80\) 4054.87 + 2896.47i 0.633574 + 0.452573i
\(81\) 4900.45 − 8487.83i 0.746906 − 1.29368i
\(82\) 864.978 + 231.770i 0.128640 + 0.0344691i
\(83\) −3185.61 − 3185.61i −0.462420 − 0.462420i 0.437028 − 0.899448i \(-0.356031\pi\)
−0.899448 + 0.437028i \(0.856031\pi\)
\(84\) 827.988 + 14851.8i 0.117345 + 2.10485i
\(85\) −5761.09 − 6998.44i −0.797383 − 0.968642i
\(86\) −874.990 − 1515.53i −0.118306 − 0.204912i
\(87\) 947.137 + 3534.76i 0.125134 + 0.467005i
\(88\) 1451.08 − 388.815i 0.187381 − 0.0502086i
\(89\) −1185.37 + 684.375i −0.149649 + 0.0864001i −0.572955 − 0.819587i \(-0.694203\pi\)
0.423306 + 0.905987i \(0.360870\pi\)
\(90\) 19941.4 − 16415.7i 2.46190 − 2.02662i
\(91\) −335.387 + 663.692i −0.0405007 + 0.0801463i
\(92\) 7427.26 − 7427.26i 0.877511 − 0.877511i
\(93\) −3378.25 + 12607.8i −0.390594 + 1.45772i
\(94\) 6930.42 + 4001.28i 0.784339 + 0.452838i
\(95\) 1368.09 − 1915.24i 0.151589 − 0.212215i
\(96\) −11684.4 − 20238.0i −1.26784 − 2.19596i
\(97\) 3098.05 − 3098.05i 0.329264 − 0.329264i −0.523042 − 0.852307i \(-0.675203\pi\)
0.852307 + 0.523042i \(0.175203\pi\)
\(98\) 13224.0 − 5177.62i 1.37693 − 0.539110i
\(99\) 14861.1i 1.51629i
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.18.2 yes 56
5.2 odd 4 inner 35.5.l.a.32.13 yes 56
7.2 even 3 inner 35.5.l.a.23.13 yes 56
35.2 odd 12 inner 35.5.l.a.2.2 ✓ 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.2 ✓ 56 35.2 odd 12 inner
35.5.l.a.18.2 yes 56 1.1 even 1 trivial
35.5.l.a.23.13 yes 56 7.2 even 3 inner
35.5.l.a.32.13 yes 56 5.2 odd 4 inner