# Properties

 Label 117.4.a.c.1.2 Level $117$ Weight $4$ Character 117.1 Self dual yes Analytic conductor $6.903$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(1,117)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("117.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$6.90322347067$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 14$$ x^2 - 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.74166$$ of defining polynomial Character $$\chi$$ $$=$$ 117.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.74166 q^{2} -0.483315 q^{4} -19.4833 q^{5} +7.48331 q^{7} -23.2583 q^{8} +O(q^{10})$$ $$q+2.74166 q^{2} -0.483315 q^{4} -19.4833 q^{5} +7.48331 q^{7} -23.2583 q^{8} -53.4166 q^{10} -22.8999 q^{11} -13.0000 q^{13} +20.5167 q^{14} -59.8999 q^{16} -67.0334 q^{17} +16.5167 q^{19} +9.41657 q^{20} -62.7836 q^{22} +175.600 q^{23} +254.600 q^{25} -35.6415 q^{26} -3.61680 q^{28} -291.800 q^{29} +117.283 q^{31} +21.8418 q^{32} -183.783 q^{34} -145.800 q^{35} -154.766 q^{37} +45.2831 q^{38} +453.150 q^{40} +251.716 q^{41} -502.566 q^{43} +11.0679 q^{44} +481.434 q^{46} +281.733 q^{47} -287.000 q^{49} +698.025 q^{50} +6.28309 q^{52} -366.999 q^{53} +446.166 q^{55} -174.049 q^{56} -800.015 q^{58} +79.6663 q^{59} -194.865 q^{61} +321.550 q^{62} +539.082 q^{64} +253.283 q^{65} +400.082 q^{67} +32.3982 q^{68} -399.733 q^{70} -528.299 q^{71} -734.366 q^{73} -424.316 q^{74} -7.98276 q^{76} -171.367 q^{77} +113.266 q^{79} +1167.05 q^{80} +690.118 q^{82} +933.466 q^{83} +1306.03 q^{85} -1377.86 q^{86} +532.613 q^{88} -1190.91 q^{89} -97.2831 q^{91} -84.8699 q^{92} +772.415 q^{94} -321.800 q^{95} +557.165 q^{97} -786.856 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8 $$2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8} - 32 q^{10} + 44 q^{11} - 26 q^{13} + 56 q^{14} - 30 q^{16} - 164 q^{17} + 48 q^{19} - 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 26 q^{26} - 112 q^{28} - 404 q^{29} + 40 q^{31} + 126 q^{32} + 276 q^{34} - 112 q^{35} - 100 q^{37} - 104 q^{38} + 592 q^{40} - 200 q^{41} - 616 q^{43} + 980 q^{44} + 1352 q^{46} + 324 q^{47} - 574 q^{49} + 1194 q^{50} - 182 q^{52} + 164 q^{53} + 144 q^{55} + 56 q^{56} - 268 q^{58} - 140 q^{59} + 628 q^{61} + 688 q^{62} - 194 q^{64} + 312 q^{65} - 472 q^{67} - 1372 q^{68} - 560 q^{70} - 428 q^{71} - 900 q^{73} - 684 q^{74} + 448 q^{76} - 672 q^{77} - 432 q^{79} + 1032 q^{80} + 2832 q^{82} + 1388 q^{83} + 1744 q^{85} - 840 q^{86} - 1524 q^{88} - 960 q^{89} - 2744 q^{92} + 572 q^{94} - 464 q^{95} - 532 q^{97} + 574 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8 - 32 * q^10 + 44 * q^11 - 26 * q^13 + 56 * q^14 - 30 * q^16 - 164 * q^17 + 48 * q^19 - 56 * q^20 - 380 * q^22 - 8 * q^23 + 150 * q^25 + 26 * q^26 - 112 * q^28 - 404 * q^29 + 40 * q^31 + 126 * q^32 + 276 * q^34 - 112 * q^35 - 100 * q^37 - 104 * q^38 + 592 * q^40 - 200 * q^41 - 616 * q^43 + 980 * q^44 + 1352 * q^46 + 324 * q^47 - 574 * q^49 + 1194 * q^50 - 182 * q^52 + 164 * q^53 + 144 * q^55 + 56 * q^56 - 268 * q^58 - 140 * q^59 + 628 * q^61 + 688 * q^62 - 194 * q^64 + 312 * q^65 - 472 * q^67 - 1372 * q^68 - 560 * q^70 - 428 * q^71 - 900 * q^73 - 684 * q^74 + 448 * q^76 - 672 * q^77 - 432 * q^79 + 1032 * q^80 + 2832 * q^82 + 1388 * q^83 + 1744 * q^85 - 840 * q^86 - 1524 * q^88 - 960 * q^89 - 2744 * q^92 + 572 * q^94 - 464 * q^95 - 532 * q^97 + 574 * q^98

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.74166 0.969322 0.484661 0.874702i $$-0.338943\pi$$
0.484661 + 0.874702i $$0.338943\pi$$
$$3$$ 0 0
$$4$$ −0.483315 −0.0604143
$$5$$ −19.4833 −1.74264 −0.871320 0.490715i $$-0.836736\pi$$
−0.871320 + 0.490715i $$0.836736\pi$$
$$6$$ 0 0
$$7$$ 7.48331 0.404061 0.202031 0.979379i $$-0.435246\pi$$
0.202031 + 0.979379i $$0.435246\pi$$
$$8$$ −23.2583 −1.02788
$$9$$ 0 0
$$10$$ −53.4166 −1.68918
$$11$$ −22.8999 −0.627689 −0.313844 0.949474i $$-0.601617\pi$$
−0.313844 + 0.949474i $$0.601617\pi$$
$$12$$ 0 0
$$13$$ −13.0000 −0.277350
$$14$$ 20.5167 0.391665
$$15$$ 0 0
$$16$$ −59.8999 −0.935936
$$17$$ −67.0334 −0.956352 −0.478176 0.878264i $$-0.658702\pi$$
−0.478176 + 0.878264i $$0.658702\pi$$
$$18$$ 0 0
$$19$$ 16.5167 0.199431 0.0997155 0.995016i $$-0.468207\pi$$
0.0997155 + 0.995016i $$0.468207\pi$$
$$20$$ 9.41657 0.105280
$$21$$ 0 0
$$22$$ −62.7836 −0.608433
$$23$$ 175.600 1.59196 0.795979 0.605324i $$-0.206956\pi$$
0.795979 + 0.605324i $$0.206956\pi$$
$$24$$ 0 0
$$25$$ 254.600 2.03680
$$26$$ −35.6415 −0.268842
$$27$$ 0 0
$$28$$ −3.61680 −0.0244111
$$29$$ −291.800 −1.86848 −0.934239 0.356648i $$-0.883920\pi$$
−0.934239 + 0.356648i $$0.883920\pi$$
$$30$$ 0 0
$$31$$ 117.283 0.679505 0.339753 0.940515i $$-0.389657\pi$$
0.339753 + 0.940515i $$0.389657\pi$$
$$32$$ 21.8418 0.120660
$$33$$ 0 0
$$34$$ −183.783 −0.927013
$$35$$ −145.800 −0.704133
$$36$$ 0 0
$$37$$ −154.766 −0.687661 −0.343830 0.939032i $$-0.611724\pi$$
−0.343830 + 0.939032i $$0.611724\pi$$
$$38$$ 45.2831 0.193313
$$39$$ 0 0
$$40$$ 453.150 1.79123
$$41$$ 251.716 0.958815 0.479407 0.877592i $$-0.340852\pi$$
0.479407 + 0.877592i $$0.340852\pi$$
$$42$$ 0 0
$$43$$ −502.566 −1.78234 −0.891170 0.453669i $$-0.850115\pi$$
−0.891170 + 0.453669i $$0.850115\pi$$
$$44$$ 11.0679 0.0379214
$$45$$ 0 0
$$46$$ 481.434 1.54312
$$47$$ 281.733 0.874361 0.437181 0.899374i $$-0.355977\pi$$
0.437181 + 0.899374i $$0.355977\pi$$
$$48$$ 0 0
$$49$$ −287.000 −0.836735
$$50$$ 698.025 1.97431
$$51$$ 0 0
$$52$$ 6.28309 0.0167559
$$53$$ −366.999 −0.951154 −0.475577 0.879674i $$-0.657761\pi$$
−0.475577 + 0.879674i $$0.657761\pi$$
$$54$$ 0 0
$$55$$ 446.166 1.09384
$$56$$ −174.049 −0.415328
$$57$$ 0 0
$$58$$ −800.015 −1.81116
$$59$$ 79.6663 0.175791 0.0878955 0.996130i $$-0.471986\pi$$
0.0878955 + 0.996130i $$0.471986\pi$$
$$60$$ 0 0
$$61$$ −194.865 −0.409016 −0.204508 0.978865i $$-0.565559\pi$$
−0.204508 + 0.978865i $$0.565559\pi$$
$$62$$ 321.550 0.658660
$$63$$ 0 0
$$64$$ 539.082 1.05289
$$65$$ 253.283 0.483322
$$66$$ 0 0
$$67$$ 400.082 0.729519 0.364759 0.931102i $$-0.381151\pi$$
0.364759 + 0.931102i $$0.381151\pi$$
$$68$$ 32.3982 0.0577774
$$69$$ 0 0
$$70$$ −399.733 −0.682532
$$71$$ −528.299 −0.883065 −0.441532 0.897245i $$-0.645565\pi$$
−0.441532 + 0.897245i $$0.645565\pi$$
$$72$$ 0 0
$$73$$ −734.366 −1.17741 −0.588706 0.808347i $$-0.700362\pi$$
−0.588706 + 0.808347i $$0.700362\pi$$
$$74$$ −424.316 −0.666565
$$75$$ 0 0
$$76$$ −7.98276 −0.0120485
$$77$$ −171.367 −0.253625
$$78$$ 0 0
$$79$$ 113.266 0.161309 0.0806545 0.996742i $$-0.474299\pi$$
0.0806545 + 0.996742i $$0.474299\pi$$
$$80$$ 1167.05 1.63100
$$81$$ 0 0
$$82$$ 690.118 0.929400
$$83$$ 933.466 1.23447 0.617236 0.786778i $$-0.288252\pi$$
0.617236 + 0.786778i $$0.288252\pi$$
$$84$$ 0 0
$$85$$ 1306.03 1.66658
$$86$$ −1377.86 −1.72766
$$87$$ 0 0
$$88$$ 532.613 0.645191
$$89$$ −1190.91 −1.41839 −0.709195 0.705012i $$-0.750941\pi$$
−0.709195 + 0.705012i $$0.750941\pi$$
$$90$$ 0 0
$$91$$ −97.2831 −0.112066
$$92$$ −84.8699 −0.0961771
$$93$$ 0 0
$$94$$ 772.415 0.847538
$$95$$ −321.800 −0.347536
$$96$$ 0 0
$$97$$ 557.165 0.583211 0.291606 0.956539i $$-0.405811\pi$$
0.291606 + 0.956539i $$0.405811\pi$$
$$98$$ −786.856 −0.811066
$$99$$ 0 0
$$100$$ −123.052 −0.123052
$$101$$ 286.766 0.282518 0.141259 0.989973i $$-0.454885\pi$$
0.141259 + 0.989973i $$0.454885\pi$$
$$102$$ 0 0
$$103$$ −1911.36 −1.82847 −0.914234 0.405187i $$-0.867206\pi$$
−0.914234 + 0.405187i $$0.867206\pi$$
$$104$$ 302.358 0.285084
$$105$$ 0 0
$$106$$ −1006.19 −0.921975
$$107$$ −834.334 −0.753814 −0.376907 0.926251i $$-0.623012\pi$$
−0.376907 + 0.926251i $$0.623012\pi$$
$$108$$ 0 0
$$109$$ −1077.66 −0.946986 −0.473493 0.880798i $$-0.657007\pi$$
−0.473493 + 0.880798i $$0.657007\pi$$
$$110$$ 1223.23 1.06028
$$111$$ 0 0
$$112$$ −448.250 −0.378175
$$113$$ 166.065 0.138248 0.0691241 0.997608i $$-0.477980\pi$$
0.0691241 + 0.997608i $$0.477980\pi$$
$$114$$ 0 0
$$115$$ −3421.26 −2.77421
$$116$$ 141.031 0.112883
$$117$$ 0 0
$$118$$ 218.418 0.170398
$$119$$ −501.632 −0.386424
$$120$$ 0 0
$$121$$ −806.595 −0.606007
$$122$$ −534.254 −0.396468
$$123$$ 0 0
$$124$$ −56.6847 −0.0410519
$$125$$ −2525.03 −1.80676
$$126$$ 0 0
$$127$$ 1296.16 0.905637 0.452819 0.891603i $$-0.350419\pi$$
0.452819 + 0.891603i $$0.350419\pi$$
$$128$$ 1303.24 0.899934
$$129$$ 0 0
$$130$$ 694.415 0.468494
$$131$$ 197.201 0.131523 0.0657617 0.997835i $$-0.479052\pi$$
0.0657617 + 0.997835i $$0.479052\pi$$
$$132$$ 0 0
$$133$$ 123.600 0.0805823
$$134$$ 1096.89 0.707139
$$135$$ 0 0
$$136$$ 1559.09 0.983018
$$137$$ 546.915 0.341066 0.170533 0.985352i $$-0.445451\pi$$
0.170533 + 0.985352i $$0.445451\pi$$
$$138$$ 0 0
$$139$$ 609.666 0.372023 0.186012 0.982548i $$-0.440444\pi$$
0.186012 + 0.982548i $$0.440444\pi$$
$$140$$ 70.4672 0.0425397
$$141$$ 0 0
$$142$$ −1448.42 −0.855974
$$143$$ 297.699 0.174090
$$144$$ 0 0
$$145$$ 5685.23 3.25609
$$146$$ −2013.38 −1.14129
$$147$$ 0 0
$$148$$ 74.8009 0.0415446
$$149$$ 2165.08 1.19040 0.595202 0.803576i $$-0.297072\pi$$
0.595202 + 0.803576i $$0.297072\pi$$
$$150$$ 0 0
$$151$$ −846.549 −0.456233 −0.228116 0.973634i $$-0.573257\pi$$
−0.228116 + 0.973634i $$0.573257\pi$$
$$152$$ −384.151 −0.204992
$$153$$ 0 0
$$154$$ −469.830 −0.245844
$$155$$ −2285.06 −1.18413
$$156$$ 0 0
$$157$$ 1653.60 0.840581 0.420291 0.907390i $$-0.361928\pi$$
0.420291 + 0.907390i $$0.361928\pi$$
$$158$$ 310.536 0.156360
$$159$$ 0 0
$$160$$ −425.550 −0.210267
$$161$$ 1314.07 0.643248
$$162$$ 0 0
$$163$$ −2866.51 −1.37744 −0.688720 0.725027i $$-0.741827\pi$$
−0.688720 + 0.725027i $$0.741827\pi$$
$$164$$ −121.658 −0.0579262
$$165$$ 0 0
$$166$$ 2559.24 1.19660
$$167$$ −729.066 −0.337825 −0.168913 0.985631i $$-0.554026\pi$$
−0.168913 + 0.985631i $$0.554026\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 3580.69 1.61545
$$171$$ 0 0
$$172$$ 242.898 0.107679
$$173$$ 3834.83 1.68530 0.842650 0.538462i $$-0.180995\pi$$
0.842650 + 0.538462i $$0.180995\pi$$
$$174$$ 0 0
$$175$$ 1905.25 0.822990
$$176$$ 1371.70 0.587476
$$177$$ 0 0
$$178$$ −3265.08 −1.37488
$$179$$ 283.862 0.118530 0.0592649 0.998242i $$-0.481124\pi$$
0.0592649 + 0.998242i $$0.481124\pi$$
$$180$$ 0 0
$$181$$ 2363.60 0.970634 0.485317 0.874338i $$-0.338704\pi$$
0.485317 + 0.874338i $$0.338704\pi$$
$$182$$ −266.717 −0.108628
$$183$$ 0 0
$$184$$ −4084.15 −1.63635
$$185$$ 3015.36 1.19835
$$186$$ 0 0
$$187$$ 1535.06 0.600291
$$188$$ −136.166 −0.0528240
$$189$$ 0 0
$$190$$ −882.265 −0.336875
$$191$$ −2514.26 −0.952491 −0.476246 0.879312i $$-0.658003\pi$$
−0.476246 + 0.879312i $$0.658003\pi$$
$$192$$ 0 0
$$193$$ 2420.73 0.902839 0.451420 0.892312i $$-0.350918\pi$$
0.451420 + 0.892312i $$0.350918\pi$$
$$194$$ 1527.55 0.565320
$$195$$ 0 0
$$196$$ 138.711 0.0505508
$$197$$ 4633.65 1.67581 0.837903 0.545819i $$-0.183781\pi$$
0.837903 + 0.545819i $$0.183781\pi$$
$$198$$ 0 0
$$199$$ 3054.17 1.08796 0.543980 0.839098i $$-0.316917\pi$$
0.543980 + 0.839098i $$0.316917\pi$$
$$200$$ −5921.56 −2.09359
$$201$$ 0 0
$$202$$ 786.215 0.273851
$$203$$ −2183.63 −0.754979
$$204$$ 0 0
$$205$$ −4904.26 −1.67087
$$206$$ −5240.30 −1.77237
$$207$$ 0 0
$$208$$ 778.699 0.259582
$$209$$ −378.230 −0.125181
$$210$$ 0 0
$$211$$ −4031.60 −1.31539 −0.657694 0.753285i $$-0.728468\pi$$
−0.657694 + 0.753285i $$0.728468\pi$$
$$212$$ 177.376 0.0574634
$$213$$ 0 0
$$214$$ −2287.46 −0.730689
$$215$$ 9791.66 3.10598
$$216$$ 0 0
$$217$$ 877.666 0.274562
$$218$$ −2954.59 −0.917935
$$219$$ 0 0
$$220$$ −215.638 −0.0660834
$$221$$ 871.434 0.265244
$$222$$ 0 0
$$223$$ 3784.95 1.13659 0.568294 0.822826i $$-0.307604\pi$$
0.568294 + 0.822826i $$0.307604\pi$$
$$224$$ 163.449 0.0487539
$$225$$ 0 0
$$226$$ 455.292 0.134007
$$227$$ −2013.83 −0.588821 −0.294411 0.955679i $$-0.595123\pi$$
−0.294411 + 0.955679i $$0.595123\pi$$
$$228$$ 0 0
$$229$$ −3050.73 −0.880340 −0.440170 0.897915i $$-0.645082\pi$$
−0.440170 + 0.897915i $$0.645082\pi$$
$$230$$ −9379.93 −2.68910
$$231$$ 0 0
$$232$$ 6786.78 1.92058
$$233$$ −5587.49 −1.57103 −0.785513 0.618846i $$-0.787601\pi$$
−0.785513 + 0.618846i $$0.787601\pi$$
$$234$$ 0 0
$$235$$ −5489.09 −1.52370
$$236$$ −38.5039 −0.0106203
$$237$$ 0 0
$$238$$ −1375.30 −0.374570
$$239$$ 1335.69 0.361501 0.180750 0.983529i $$-0.442147\pi$$
0.180750 + 0.983529i $$0.442147\pi$$
$$240$$ 0 0
$$241$$ −571.558 −0.152769 −0.0763845 0.997078i $$-0.524338\pi$$
−0.0763845 + 0.997078i $$0.524338\pi$$
$$242$$ −2211.41 −0.587416
$$243$$ 0 0
$$244$$ 94.1813 0.0247104
$$245$$ 5591.71 1.45813
$$246$$ 0 0
$$247$$ −214.717 −0.0553122
$$248$$ −2727.81 −0.698452
$$249$$ 0 0
$$250$$ −6922.76 −1.75134
$$251$$ −4088.60 −1.02817 −0.514084 0.857740i $$-0.671868\pi$$
−0.514084 + 0.857740i $$0.671868\pi$$
$$252$$ 0 0
$$253$$ −4021.21 −0.999254
$$254$$ 3553.64 0.877854
$$255$$ 0 0
$$256$$ −739.607 −0.180568
$$257$$ −3050.23 −0.740342 −0.370171 0.928964i $$-0.620701\pi$$
−0.370171 + 0.928964i $$0.620701\pi$$
$$258$$ 0 0
$$259$$ −1158.17 −0.277857
$$260$$ −122.415 −0.0291996
$$261$$ 0 0
$$262$$ 540.659 0.127489
$$263$$ −5770.99 −1.35306 −0.676530 0.736415i $$-0.736517\pi$$
−0.676530 + 0.736415i $$0.736517\pi$$
$$264$$ 0 0
$$265$$ 7150.35 1.65752
$$266$$ 338.868 0.0781102
$$267$$ 0 0
$$268$$ −193.365 −0.0440734
$$269$$ 2079.40 0.471314 0.235657 0.971836i $$-0.424276\pi$$
0.235657 + 0.971836i $$0.424276\pi$$
$$270$$ 0 0
$$271$$ 6012.00 1.34761 0.673807 0.738908i $$-0.264658\pi$$
0.673807 + 0.738908i $$0.264658\pi$$
$$272$$ 4015.29 0.895084
$$273$$ 0 0
$$274$$ 1499.45 0.330603
$$275$$ −5830.30 −1.27847
$$276$$ 0 0
$$277$$ −735.201 −0.159473 −0.0797364 0.996816i $$-0.525408\pi$$
−0.0797364 + 0.996816i $$0.525408\pi$$
$$278$$ 1671.50 0.360610
$$279$$ 0 0
$$280$$ 3391.06 0.723767
$$281$$ 1902.92 0.403981 0.201990 0.979387i $$-0.435259\pi$$
0.201990 + 0.979387i $$0.435259\pi$$
$$282$$ 0 0
$$283$$ 2125.71 0.446502 0.223251 0.974761i $$-0.428333\pi$$
0.223251 + 0.974761i $$0.428333\pi$$
$$284$$ 255.335 0.0533498
$$285$$ 0 0
$$286$$ 816.187 0.168749
$$287$$ 1883.67 0.387420
$$288$$ 0 0
$$289$$ −419.527 −0.0853913
$$290$$ 15586.9 3.15620
$$291$$ 0 0
$$292$$ 354.930 0.0711325
$$293$$ 1641.03 0.327200 0.163600 0.986527i $$-0.447689\pi$$
0.163600 + 0.986527i $$0.447689\pi$$
$$294$$ 0 0
$$295$$ −1552.16 −0.306341
$$296$$ 3599.61 0.706835
$$297$$ 0 0
$$298$$ 5935.91 1.15389
$$299$$ −2282.79 −0.441530
$$300$$ 0 0
$$301$$ −3760.86 −0.720174
$$302$$ −2320.95 −0.442237
$$303$$ 0 0
$$304$$ −989.348 −0.186655
$$305$$ 3796.62 0.712767
$$306$$ 0 0
$$307$$ −3373.27 −0.627111 −0.313555 0.949570i $$-0.601520\pi$$
−0.313555 + 0.949570i $$0.601520\pi$$
$$308$$ 82.8242 0.0153226
$$309$$ 0 0
$$310$$ −6264.86 −1.14781
$$311$$ 868.525 0.158359 0.0791793 0.996860i $$-0.474770\pi$$
0.0791793 + 0.996860i $$0.474770\pi$$
$$312$$ 0 0
$$313$$ −4343.19 −0.784319 −0.392159 0.919897i $$-0.628272\pi$$
−0.392159 + 0.919897i $$0.628272\pi$$
$$314$$ 4533.59 0.814794
$$315$$ 0 0
$$316$$ −54.7431 −0.00974537
$$317$$ 3277.65 0.580730 0.290365 0.956916i $$-0.406223\pi$$
0.290365 + 0.956916i $$0.406223\pi$$
$$318$$ 0 0
$$319$$ 6682.18 1.17282
$$320$$ −10503.1 −1.83482
$$321$$ 0 0
$$322$$ 3602.72 0.623515
$$323$$ −1107.17 −0.190726
$$324$$ 0 0
$$325$$ −3309.79 −0.564906
$$326$$ −7859.00 −1.33518
$$327$$ 0 0
$$328$$ −5854.49 −0.985550
$$329$$ 2108.30 0.353295
$$330$$ 0 0
$$331$$ 5589.62 0.928197 0.464099 0.885784i $$-0.346378\pi$$
0.464099 + 0.885784i $$0.346378\pi$$
$$332$$ −451.158 −0.0745798
$$333$$ 0 0
$$334$$ −1998.85 −0.327461
$$335$$ −7794.92 −1.27129
$$336$$ 0 0
$$337$$ 901.544 0.145728 0.0728638 0.997342i $$-0.476786\pi$$
0.0728638 + 0.997342i $$0.476786\pi$$
$$338$$ 463.340 0.0745633
$$339$$ 0 0
$$340$$ −631.225 −0.100685
$$341$$ −2685.77 −0.426518
$$342$$ 0 0
$$343$$ −4714.49 −0.742153
$$344$$ 11688.9 1.83204
$$345$$ 0 0
$$346$$ 10513.8 1.63360
$$347$$ 812.318 0.125670 0.0628350 0.998024i $$-0.479986\pi$$
0.0628350 + 0.998024i $$0.479986\pi$$
$$348$$ 0 0
$$349$$ 4437.96 0.680683 0.340342 0.940302i $$-0.389457\pi$$
0.340342 + 0.940302i $$0.389457\pi$$
$$350$$ 5223.54 0.797743
$$351$$ 0 0
$$352$$ −500.174 −0.0757368
$$353$$ −7115.35 −1.07284 −0.536419 0.843952i $$-0.680223\pi$$
−0.536419 + 0.843952i $$0.680223\pi$$
$$354$$ 0 0
$$355$$ 10293.0 1.53886
$$356$$ 575.587 0.0856911
$$357$$ 0 0
$$358$$ 778.253 0.114894
$$359$$ −4693.98 −0.690081 −0.345040 0.938588i $$-0.612135\pi$$
−0.345040 + 0.938588i $$0.612135\pi$$
$$360$$ 0 0
$$361$$ −6586.20 −0.960227
$$362$$ 6480.17 0.940857
$$363$$ 0 0
$$364$$ 47.0184 0.00677042
$$365$$ 14307.9 2.05181
$$366$$ 0 0
$$367$$ 9243.98 1.31480 0.657400 0.753542i $$-0.271656\pi$$
0.657400 + 0.753542i $$0.271656\pi$$
$$368$$ −10518.4 −1.48997
$$369$$ 0 0
$$370$$ 8267.09 1.16158
$$371$$ −2746.37 −0.384324
$$372$$ 0 0
$$373$$ −4311.99 −0.598569 −0.299285 0.954164i $$-0.596748\pi$$
−0.299285 + 0.954164i $$0.596748\pi$$
$$374$$ 4208.60 0.581876
$$375$$ 0 0
$$376$$ −6552.64 −0.898741
$$377$$ 3793.40 0.518223
$$378$$ 0 0
$$379$$ −2382.73 −0.322936 −0.161468 0.986878i $$-0.551623\pi$$
−0.161468 + 0.986878i $$0.551623\pi$$
$$380$$ 155.531 0.0209962
$$381$$ 0 0
$$382$$ −6893.25 −0.923271
$$383$$ −4845.81 −0.646499 −0.323250 0.946314i $$-0.604775\pi$$
−0.323250 + 0.946314i $$0.604775\pi$$
$$384$$ 0 0
$$385$$ 3338.80 0.441976
$$386$$ 6636.81 0.875142
$$387$$ 0 0
$$388$$ −269.286 −0.0352343
$$389$$ −9561.50 −1.24624 −0.623120 0.782127i $$-0.714135\pi$$
−0.623120 + 0.782127i $$0.714135\pi$$
$$390$$ 0 0
$$391$$ −11771.0 −1.52247
$$392$$ 6675.14 0.860066
$$393$$ 0 0
$$394$$ 12703.9 1.62440
$$395$$ −2206.79 −0.281103
$$396$$ 0 0
$$397$$ −7440.11 −0.940575 −0.470287 0.882513i $$-0.655850\pi$$
−0.470287 + 0.882513i $$0.655850\pi$$
$$398$$ 8373.48 1.05458
$$399$$ 0 0
$$400$$ −15250.5 −1.90631
$$401$$ 8687.80 1.08192 0.540958 0.841050i $$-0.318062\pi$$
0.540958 + 0.841050i $$0.318062\pi$$
$$402$$ 0 0
$$403$$ −1524.68 −0.188461
$$404$$ −138.598 −0.0170681
$$405$$ 0 0
$$406$$ −5986.76 −0.731818
$$407$$ 3544.13 0.431637
$$408$$ 0 0
$$409$$ 2556.10 0.309024 0.154512 0.987991i $$-0.450619\pi$$
0.154512 + 0.987991i $$0.450619\pi$$
$$410$$ −13445.8 −1.61961
$$411$$ 0 0
$$412$$ 923.790 0.110466
$$413$$ 596.168 0.0710303
$$414$$ 0 0
$$415$$ −18187.0 −2.15124
$$416$$ −283.943 −0.0334650
$$417$$ 0 0
$$418$$ −1036.98 −0.121340
$$419$$ 3347.46 0.390296 0.195148 0.980774i $$-0.437481\pi$$
0.195148 + 0.980774i $$0.437481\pi$$
$$420$$ 0 0
$$421$$ −1854.48 −0.214684 −0.107342 0.994222i $$-0.534234\pi$$
−0.107342 + 0.994222i $$0.534234\pi$$
$$422$$ −11053.3 −1.27503
$$423$$ 0 0
$$424$$ 8535.79 0.977675
$$425$$ −17066.7 −1.94789
$$426$$ 0 0
$$427$$ −1458.24 −0.165267
$$428$$ 403.246 0.0455412
$$429$$ 0 0
$$430$$ 26845.4 3.01069
$$431$$ 14043.1 1.56945 0.784725 0.619844i $$-0.212804\pi$$
0.784725 + 0.619844i $$0.212804\pi$$
$$432$$ 0 0
$$433$$ 3086.47 0.342555 0.171278 0.985223i $$-0.445210\pi$$
0.171278 + 0.985223i $$0.445210\pi$$
$$434$$ 2406.26 0.266139
$$435$$ 0 0
$$436$$ 520.851 0.0572116
$$437$$ 2900.32 0.317486
$$438$$ 0 0
$$439$$ 2837.68 0.308508 0.154254 0.988031i $$-0.450703\pi$$
0.154254 + 0.988031i $$0.450703\pi$$
$$440$$ −10377.1 −1.12434
$$441$$ 0 0
$$442$$ 2389.17 0.257107
$$443$$ −18309.4 −1.96367 −0.981834 0.189744i $$-0.939234\pi$$
−0.981834 + 0.189744i $$0.939234\pi$$
$$444$$ 0 0
$$445$$ 23203.0 2.47174
$$446$$ 10377.0 1.10172
$$447$$ 0 0
$$448$$ 4034.12 0.425433
$$449$$ −13861.2 −1.45690 −0.728451 0.685098i $$-0.759759\pi$$
−0.728451 + 0.685098i $$0.759759\pi$$
$$450$$ 0 0
$$451$$ −5764.26 −0.601837
$$452$$ −80.2614 −0.00835217
$$453$$ 0 0
$$454$$ −5521.23 −0.570758
$$455$$ 1895.40 0.195291
$$456$$ 0 0
$$457$$ −8990.36 −0.920243 −0.460122 0.887856i $$-0.652194\pi$$
−0.460122 + 0.887856i $$0.652194\pi$$
$$458$$ −8364.05 −0.853333
$$459$$ 0 0
$$460$$ 1653.55 0.167602
$$461$$ 3406.90 0.344198 0.172099 0.985080i $$-0.444945\pi$$
0.172099 + 0.985080i $$0.444945\pi$$
$$462$$ 0 0
$$463$$ −7498.45 −0.752662 −0.376331 0.926485i $$-0.622814\pi$$
−0.376331 + 0.926485i $$0.622814\pi$$
$$464$$ 17478.8 1.74878
$$465$$ 0 0
$$466$$ −15319.0 −1.52283
$$467$$ −7711.38 −0.764112 −0.382056 0.924139i $$-0.624784\pi$$
−0.382056 + 0.924139i $$0.624784\pi$$
$$468$$ 0 0
$$469$$ 2993.94 0.294770
$$470$$ −15049.2 −1.47695
$$471$$ 0 0
$$472$$ −1852.91 −0.180693
$$473$$ 11508.7 1.11875
$$474$$ 0 0
$$475$$ 4205.14 0.406200
$$476$$ 242.446 0.0233456
$$477$$ 0 0
$$478$$ 3662.01 0.350411
$$479$$ 9439.82 0.900451 0.450226 0.892915i $$-0.351344\pi$$
0.450226 + 0.892915i $$0.351344\pi$$
$$480$$ 0 0
$$481$$ 2011.96 0.190723
$$482$$ −1567.02 −0.148082
$$483$$ 0 0
$$484$$ 389.839 0.0366115
$$485$$ −10855.4 −1.01633
$$486$$ 0 0
$$487$$ −6156.20 −0.572821 −0.286411 0.958107i $$-0.592462\pi$$
−0.286411 + 0.958107i $$0.592462\pi$$
$$488$$ 4532.25 0.420420
$$489$$ 0 0
$$490$$ 15330.6 1.41340
$$491$$ −3842.74 −0.353198 −0.176599 0.984283i $$-0.556510\pi$$
−0.176599 + 0.984283i $$0.556510\pi$$
$$492$$ 0 0
$$493$$ 19560.3 1.78692
$$494$$ −588.680 −0.0536153
$$495$$ 0 0
$$496$$ −7025.24 −0.635973
$$497$$ −3953.43 −0.356812
$$498$$ 0 0
$$499$$ −12842.4 −1.15211 −0.576056 0.817410i $$-0.695409\pi$$
−0.576056 + 0.817410i $$0.695409\pi$$
$$500$$ 1220.38 0.109154
$$501$$ 0 0
$$502$$ −11209.5 −0.996627
$$503$$ −8580.11 −0.760573 −0.380287 0.924869i $$-0.624175\pi$$
−0.380287 + 0.924869i $$0.624175\pi$$
$$504$$ 0 0
$$505$$ −5587.16 −0.492327
$$506$$ −11024.8 −0.968599
$$507$$ 0 0
$$508$$ −626.455 −0.0547135
$$509$$ 43.5957 0.00379635 0.00189818 0.999998i $$-0.499396\pi$$
0.00189818 + 0.999998i $$0.499396\pi$$
$$510$$ 0 0
$$511$$ −5495.49 −0.475746
$$512$$ −12453.7 −1.07496
$$513$$ 0 0
$$514$$ −8362.68 −0.717630
$$515$$ 37239.7 3.18636
$$516$$ 0 0
$$517$$ −6451.66 −0.548827
$$518$$ −3175.29 −0.269333
$$519$$ 0 0
$$520$$ −5890.94 −0.496798
$$521$$ −11368.1 −0.955939 −0.477969 0.878377i $$-0.658627\pi$$
−0.477969 + 0.878377i $$0.658627\pi$$
$$522$$ 0 0
$$523$$ −5229.53 −0.437230 −0.218615 0.975811i $$-0.570154\pi$$
−0.218615 + 0.975811i $$0.570154\pi$$
$$524$$ −95.3103 −0.00794590
$$525$$ 0 0
$$526$$ −15822.1 −1.31155
$$527$$ −7861.88 −0.649846
$$528$$ 0 0
$$529$$ 18668.2 1.53433
$$530$$ 19603.8 1.60667
$$531$$ 0 0
$$532$$ −59.7375 −0.00486832
$$533$$ −3272.31 −0.265927
$$534$$ 0 0
$$535$$ 16255.6 1.31363
$$536$$ −9305.24 −0.749860
$$537$$ 0 0
$$538$$ 5701.01 0.456855
$$539$$ 6572.27 0.525209
$$540$$ 0 0
$$541$$ −6567.99 −0.521959 −0.260980 0.965344i $$-0.584046\pi$$
−0.260980 + 0.965344i $$0.584046\pi$$
$$542$$ 16482.9 1.30627
$$543$$ 0 0
$$544$$ −1464.13 −0.115393
$$545$$ 20996.5 1.65026
$$546$$ 0 0
$$547$$ −13675.7 −1.06897 −0.534487 0.845177i $$-0.679495\pi$$
−0.534487 + 0.845177i $$0.679495\pi$$
$$548$$ −264.332 −0.0206053
$$549$$ 0 0
$$550$$ −15984.7 −1.23925
$$551$$ −4819.57 −0.372632
$$552$$ 0 0
$$553$$ 847.604 0.0651786
$$554$$ −2015.67 −0.154581
$$555$$ 0 0
$$556$$ −294.661 −0.0224755
$$557$$ −4527.96 −0.344445 −0.172222 0.985058i $$-0.555095\pi$$
−0.172222 + 0.985058i $$0.555095\pi$$
$$558$$ 0 0
$$559$$ 6533.36 0.494332
$$560$$ 8733.39 0.659023
$$561$$ 0 0
$$562$$ 5217.15 0.391588
$$563$$ −18441.8 −1.38051 −0.690256 0.723566i $$-0.742502\pi$$
−0.690256 + 0.723566i $$0.742502\pi$$
$$564$$ 0 0
$$565$$ −3235.49 −0.240917
$$566$$ 5827.96 0.432804
$$567$$ 0 0
$$568$$ 12287.4 0.907687
$$569$$ 13553.5 0.998578 0.499289 0.866436i $$-0.333595\pi$$
0.499289 + 0.866436i $$0.333595\pi$$
$$570$$ 0 0
$$571$$ 14815.5 1.08583 0.542915 0.839788i $$-0.317321\pi$$
0.542915 + 0.839788i $$0.317321\pi$$
$$572$$ −143.882 −0.0105175
$$573$$ 0 0
$$574$$ 5164.37 0.375534
$$575$$ 44707.6 3.24249
$$576$$ 0 0
$$577$$ 21596.2 1.55816 0.779081 0.626923i $$-0.215686\pi$$
0.779081 + 0.626923i $$0.215686\pi$$
$$578$$ −1150.20 −0.0827716
$$579$$ 0 0
$$580$$ −2747.75 −0.196714
$$581$$ 6985.42 0.498802
$$582$$ 0 0
$$583$$ 8404.23 0.597029
$$584$$ 17080.1 1.21024
$$585$$ 0 0
$$586$$ 4499.13 0.317163
$$587$$ 918.801 0.0646047 0.0323024 0.999478i $$-0.489716\pi$$
0.0323024 + 0.999478i $$0.489716\pi$$
$$588$$ 0 0
$$589$$ 1937.13 0.135514
$$590$$ −4255.50 −0.296943
$$591$$ 0 0
$$592$$ 9270.49 0.643606
$$593$$ −19816.0 −1.37226 −0.686128 0.727481i $$-0.740691\pi$$
−0.686128 + 0.727481i $$0.740691\pi$$
$$594$$ 0 0
$$595$$ 9773.45 0.673399
$$596$$ −1046.42 −0.0719175
$$597$$ 0 0
$$598$$ −6258.64 −0.427985
$$599$$ 5141.86 0.350736 0.175368 0.984503i $$-0.443889\pi$$
0.175368 + 0.984503i $$0.443889\pi$$
$$600$$ 0 0
$$601$$ 12380.9 0.840312 0.420156 0.907452i $$-0.361975\pi$$
0.420156 + 0.907452i $$0.361975\pi$$
$$602$$ −10311.0 −0.698081
$$603$$ 0 0
$$604$$ 409.150 0.0275630
$$605$$ 15715.1 1.05605
$$606$$ 0 0
$$607$$ −23717.0 −1.58590 −0.792951 0.609286i $$-0.791456\pi$$
−0.792951 + 0.609286i $$0.791456\pi$$
$$608$$ 360.754 0.0240633
$$609$$ 0 0
$$610$$ 10409.0 0.690901
$$611$$ −3662.53 −0.242504
$$612$$ 0 0
$$613$$ −26157.1 −1.72345 −0.861726 0.507373i $$-0.830617\pi$$
−0.861726 + 0.507373i $$0.830617\pi$$
$$614$$ −9248.36 −0.607872
$$615$$ 0 0
$$616$$ 3985.71 0.260696
$$617$$ −23613.9 −1.54077 −0.770387 0.637576i $$-0.779937\pi$$
−0.770387 + 0.637576i $$0.779937\pi$$
$$618$$ 0 0
$$619$$ 23345.4 1.51588 0.757940 0.652324i $$-0.226206\pi$$
0.757940 + 0.652324i $$0.226206\pi$$
$$620$$ 1104.40 0.0715387
$$621$$ 0 0
$$622$$ 2381.20 0.153501
$$623$$ −8911.99 −0.573116
$$624$$ 0 0
$$625$$ 17371.0 1.11174
$$626$$ −11907.5 −0.760258
$$627$$ 0 0
$$628$$ −799.207 −0.0507832
$$629$$ 10374.5 0.657645
$$630$$ 0 0
$$631$$ 15245.7 0.961841 0.480921 0.876764i $$-0.340302\pi$$
0.480921 + 0.876764i $$0.340302\pi$$
$$632$$ −2634.38 −0.165807
$$633$$ 0 0
$$634$$ 8986.20 0.562914
$$635$$ −25253.6 −1.57820
$$636$$ 0 0
$$637$$ 3731.00 0.232068
$$638$$ 18320.3 1.13684
$$639$$ 0 0
$$640$$ −25391.5 −1.56826
$$641$$ −10192.7 −0.628063 −0.314032 0.949413i $$-0.601680\pi$$
−0.314032 + 0.949413i $$0.601680\pi$$
$$642$$ 0 0
$$643$$ −5506.31 −0.337710 −0.168855 0.985641i $$-0.554007\pi$$
−0.168855 + 0.985641i $$0.554007\pi$$
$$644$$ −635.108 −0.0388614
$$645$$ 0 0
$$646$$ −3035.48 −0.184875
$$647$$ 13297.5 0.808005 0.404003 0.914758i $$-0.367619\pi$$
0.404003 + 0.914758i $$0.367619\pi$$
$$648$$ 0 0
$$649$$ −1824.35 −0.110342
$$650$$ −9074.32 −0.547576
$$651$$ 0 0
$$652$$ 1385.43 0.0832171
$$653$$ 12440.2 0.745519 0.372760 0.927928i $$-0.378412\pi$$
0.372760 + 0.927928i $$0.378412\pi$$
$$654$$ 0 0
$$655$$ −3842.14 −0.229198
$$656$$ −15077.7 −0.897389
$$657$$ 0 0
$$658$$ 5780.23 0.342457
$$659$$ 9562.87 0.565276 0.282638 0.959227i $$-0.408791\pi$$
0.282638 + 0.959227i $$0.408791\pi$$
$$660$$ 0 0
$$661$$ 2409.69 0.141795 0.0708973 0.997484i $$-0.477414\pi$$
0.0708973 + 0.997484i $$0.477414\pi$$
$$662$$ 15324.8 0.899722
$$663$$ 0 0
$$664$$ −21710.9 −1.26889
$$665$$ −2408.13 −0.140426
$$666$$ 0 0
$$667$$ −51239.9 −2.97454
$$668$$ 352.368 0.0204095
$$669$$ 0 0
$$670$$ −21371.0 −1.23229
$$671$$ 4462.40 0.256735
$$672$$ 0 0
$$673$$ 7929.02 0.454147 0.227074 0.973878i $$-0.427084\pi$$
0.227074 + 0.973878i $$0.427084\pi$$
$$674$$ 2471.72 0.141257
$$675$$ 0 0
$$676$$ −81.6802 −0.00464726
$$677$$ 2628.26 0.149206 0.0746030 0.997213i $$-0.476231\pi$$
0.0746030 + 0.997213i $$0.476231\pi$$
$$678$$ 0 0
$$679$$ 4169.44 0.235653
$$680$$ −30376.1 −1.71305
$$681$$ 0 0
$$682$$ −7363.46 −0.413433
$$683$$ −10021.5 −0.561437 −0.280719 0.959790i $$-0.590573\pi$$
−0.280719 + 0.959790i $$0.590573\pi$$
$$684$$ 0 0
$$685$$ −10655.7 −0.594356
$$686$$ −12925.5 −0.719385
$$687$$ 0 0
$$688$$ 30103.7 1.66816
$$689$$ 4770.99 0.263803
$$690$$ 0 0
$$691$$ 23987.2 1.32057 0.660286 0.751014i $$-0.270435\pi$$
0.660286 + 0.751014i $$0.270435\pi$$
$$692$$ −1853.43 −0.101816
$$693$$ 0 0
$$694$$ 2227.10 0.121815
$$695$$ −11878.3 −0.648303
$$696$$ 0 0
$$697$$ −16873.4 −0.916964
$$698$$ 12167.4 0.659801
$$699$$ 0 0
$$700$$ −920.835 −0.0497204
$$701$$ 3763.71 0.202787 0.101393 0.994846i $$-0.467670\pi$$
0.101393 + 0.994846i $$0.467670\pi$$
$$702$$ 0 0
$$703$$ −2556.23 −0.137141
$$704$$ −12344.9 −0.660890
$$705$$ 0 0
$$706$$ −19507.8 −1.03993
$$707$$ 2145.96 0.114155
$$708$$ 0 0
$$709$$ −36047.8 −1.90946 −0.954728 0.297479i $$-0.903854\pi$$
−0.954728 + 0.297479i $$0.903854\pi$$
$$710$$ 28219.9 1.49166
$$711$$ 0 0
$$712$$ 27698.7 1.45794
$$713$$ 20594.9 1.08174
$$714$$ 0 0
$$715$$ −5800.15 −0.303376
$$716$$ −137.195 −0.00716090
$$717$$ 0 0
$$718$$ −12869.3 −0.668910
$$719$$ 3944.18 0.204580 0.102290 0.994755i $$-0.467383\pi$$
0.102290 + 0.994755i $$0.467383\pi$$
$$720$$ 0 0
$$721$$ −14303.3 −0.738812
$$722$$ −18057.1 −0.930770
$$723$$ 0 0
$$724$$ −1142.36 −0.0586402
$$725$$ −74292.1 −3.80571
$$726$$ 0 0
$$727$$ −20447.8 −1.04315 −0.521573 0.853206i $$-0.674655\pi$$
−0.521573 + 0.853206i $$0.674655\pi$$
$$728$$ 2262.64 0.115191
$$729$$ 0 0
$$730$$ 39227.3 1.98886
$$731$$ 33688.7 1.70454
$$732$$ 0 0
$$733$$ −13536.2 −0.682089 −0.341045 0.940047i $$-0.610781\pi$$
−0.341045 + 0.940047i $$0.610781\pi$$
$$734$$ 25343.8 1.27447
$$735$$ 0 0
$$736$$ 3835.40 0.192085
$$737$$ −9161.83 −0.457911
$$738$$ 0 0
$$739$$ 15839.1 0.788433 0.394217 0.919018i $$-0.371016\pi$$
0.394217 + 0.919018i $$0.371016\pi$$
$$740$$ −1457.37 −0.0723972
$$741$$ 0 0
$$742$$ −7529.60 −0.372534
$$743$$ 1664.92 0.0822075 0.0411037 0.999155i $$-0.486913\pi$$
0.0411037 + 0.999155i $$0.486913\pi$$
$$744$$ 0 0
$$745$$ −42182.9 −2.07445
$$746$$ −11822.0 −0.580206
$$747$$ 0 0
$$748$$ −741.916 −0.0362662
$$749$$ −6243.58 −0.304587
$$750$$ 0 0
$$751$$ 22399.1 1.08835 0.544177 0.838970i $$-0.316842\pi$$
0.544177 + 0.838970i $$0.316842\pi$$
$$752$$ −16875.8 −0.818346
$$753$$ 0 0
$$754$$ 10400.2 0.502325
$$755$$ 16493.6 0.795050
$$756$$ 0 0
$$757$$ 23798.9 1.14265 0.571326 0.820723i $$-0.306429\pi$$
0.571326 + 0.820723i $$0.306429\pi$$
$$758$$ −6532.64 −0.313029
$$759$$ 0 0
$$760$$ 7484.53 0.357227
$$761$$ −13693.5 −0.652285 −0.326142 0.945321i $$-0.605749\pi$$
−0.326142 + 0.945321i $$0.605749\pi$$
$$762$$ 0 0
$$763$$ −8064.50 −0.382640
$$764$$ 1215.18 0.0575441
$$765$$ 0 0
$$766$$ −13285.5 −0.626666
$$767$$ −1035.66 −0.0487556
$$768$$ 0 0
$$769$$ 16299.9 0.764358 0.382179 0.924088i $$-0.375174\pi$$
0.382179 + 0.924088i $$0.375174\pi$$
$$770$$ 9153.84 0.428418
$$771$$ 0 0
$$772$$ −1169.97 −0.0545444
$$773$$ −33532.2 −1.56024 −0.780122 0.625628i $$-0.784843\pi$$
−0.780122 + 0.625628i $$0.784843\pi$$
$$774$$ 0 0
$$775$$ 29860.2 1.38401
$$776$$ −12958.7 −0.599473
$$777$$ 0 0
$$778$$ −26214.3 −1.20801
$$779$$ 4157.51 0.191217
$$780$$ 0 0
$$781$$ 12098.0 0.554290
$$782$$ −32272.1 −1.47577
$$783$$ 0 0
$$784$$ 17191.3 0.783130
$$785$$ −32217.5 −1.46483
$$786$$ 0 0
$$787$$ 16163.3 0.732097 0.366049 0.930596i $$-0.380710\pi$$
0.366049 + 0.930596i $$0.380710\pi$$
$$788$$ −2239.51 −0.101243
$$789$$ 0 0
$$790$$ −6050.27 −0.272480
$$791$$ 1242.71 0.0558607
$$792$$ 0 0
$$793$$ 2533.25 0.113441
$$794$$ −20398.2 −0.911720
$$795$$ 0 0
$$796$$ −1476.12 −0.0657284
$$797$$ 39636.4 1.76160 0.880798 0.473492i $$-0.157007\pi$$
0.880798 + 0.473492i $$0.157007\pi$$
$$798$$ 0 0
$$799$$ −18885.5 −0.836197
$$800$$ 5560.90 0.245760
$$801$$ 0 0
$$802$$ 23819.0 1.04872
$$803$$ 16816.9 0.739048
$$804$$ 0 0
$$805$$ −25602.4 −1.12095
$$806$$ −4180.15 −0.182679
$$807$$ 0 0
$$808$$ −6669.71 −0.290396
$$809$$ 23811.2 1.03481 0.517403 0.855742i $$-0.326899\pi$$
0.517403 + 0.855742i $$0.326899\pi$$
$$810$$ 0 0
$$811$$ 27218.6 1.17851 0.589256 0.807946i $$-0.299421\pi$$
0.589256 + 0.807946i $$0.299421\pi$$
$$812$$ 1055.38 0.0456116
$$813$$ 0 0
$$814$$ 9716.80 0.418395
$$815$$ 55849.2 2.40038
$$816$$ 0 0
$$817$$ −8300.73 −0.355454
$$818$$ 7007.94 0.299544
$$819$$ 0 0
$$820$$ 2370.30 0.100944
$$821$$ 43094.8 1.83193 0.915967 0.401253i $$-0.131425\pi$$
0.915967 + 0.401253i $$0.131425\pi$$
$$822$$ 0 0
$$823$$ 26541.1 1.12414 0.562068 0.827091i $$-0.310006\pi$$
0.562068 + 0.827091i $$0.310006\pi$$
$$824$$ 44455.1 1.87945
$$825$$ 0 0
$$826$$ 1634.49 0.0688512
$$827$$ 44898.7 1.88788 0.943942 0.330112i $$-0.107087\pi$$
0.943942 + 0.330112i $$0.107087\pi$$
$$828$$ 0 0
$$829$$ −7137.48 −0.299029 −0.149514 0.988760i $$-0.547771\pi$$
−0.149514 + 0.988760i $$0.547771\pi$$
$$830$$ −49862.6 −2.08525
$$831$$ 0 0
$$832$$ −7008.06 −0.292020
$$833$$ 19238.6 0.800213
$$834$$ 0 0
$$835$$ 14204.6 0.588708
$$836$$ 182.804 0.00756270
$$837$$ 0 0
$$838$$ 9177.59 0.378323
$$839$$ 4387.17 0.180527 0.0902634 0.995918i $$-0.471229\pi$$
0.0902634 + 0.995918i $$0.471229\pi$$
$$840$$ 0 0
$$841$$ 60758.1 2.49121
$$842$$ −5084.36 −0.208098
$$843$$ 0 0
$$844$$ 1948.53 0.0794683
$$845$$ −3292.68 −0.134049
$$846$$ 0 0
$$847$$ −6036.01 −0.244864
$$848$$ 21983.2 0.890219
$$849$$ 0 0
$$850$$ −46791.0 −1.88814
$$851$$ −27176.9 −1.09473
$$852$$ 0 0
$$853$$ −9328.85 −0.374459 −0.187230 0.982316i $$-0.559951\pi$$
−0.187230 + 0.982316i $$0.559951\pi$$
$$854$$ −3997.99 −0.160197
$$855$$ 0 0
$$856$$ 19405.2 0.774833
$$857$$ 5010.39 0.199710 0.0998552 0.995002i $$-0.468162\pi$$
0.0998552 + 0.995002i $$0.468162\pi$$
$$858$$ 0 0
$$859$$ 30233.4 1.20088 0.600438 0.799672i $$-0.294993\pi$$
0.600438 + 0.799672i $$0.294993\pi$$
$$860$$ −4732.45 −0.187646
$$861$$ 0 0
$$862$$ 38501.4 1.52130
$$863$$ −4334.93 −0.170988 −0.0854940 0.996339i $$-0.527247\pi$$
−0.0854940 + 0.996339i $$0.527247\pi$$
$$864$$ 0 0
$$865$$ −74715.2 −2.93687
$$866$$ 8462.05 0.332047
$$867$$ 0 0
$$868$$ −424.189 −0.0165875
$$869$$ −2593.78 −0.101252
$$870$$ 0 0
$$871$$ −5201.06 −0.202332
$$872$$ 25064.7 0.973391
$$873$$ 0 0
$$874$$ 7951.69 0.307746
$$875$$ −18895.6 −0.730043
$$876$$ 0 0
$$877$$ 34683.3 1.33543 0.667716 0.744416i $$-0.267272\pi$$
0.667716 + 0.744416i $$0.267272\pi$$
$$878$$ 7779.95 0.299044
$$879$$ 0 0
$$880$$ −26725.3 −1.02376
$$881$$ 18269.2 0.698642 0.349321 0.937003i $$-0.386412\pi$$
0.349321 + 0.937003i $$0.386412\pi$$
$$882$$ 0 0
$$883$$ −14592.0 −0.556128 −0.278064 0.960563i $$-0.589693\pi$$
−0.278064 + 0.960563i $$0.589693\pi$$
$$884$$ −421.177 −0.0160246
$$885$$ 0 0
$$886$$ −50198.0 −1.90343
$$887$$ −30459.3 −1.15301 −0.576507 0.817092i $$-0.695585\pi$$
−0.576507 + 0.817092i $$0.695585\pi$$
$$888$$ 0 0
$$889$$ 9699.60 0.365933
$$890$$ 63614.6 2.39592
$$891$$ 0 0
$$892$$ −1829.32 −0.0686662
$$893$$ 4653.30 0.174375
$$894$$ 0 0
$$895$$ −5530.57 −0.206555
$$896$$ 9752.58 0.363628
$$897$$ 0 0
$$898$$ −38002.6 −1.41221
$$899$$ −34223.2 −1.26964
$$900$$ 0 0
$$901$$ 24601.2 0.909638
$$902$$ −15803.6 −0.583374
$$903$$ 0 0
$$904$$ −3862.39 −0.142103
$$905$$ −46050.7 −1.69147
$$906$$ 0 0
$$907$$ −9364.89 −0.342840 −0.171420 0.985198i $$-0.554836\pi$$
−0.171420 + 0.985198i $$0.554836\pi$$
$$908$$ 973.313 0.0355733
$$909$$ 0 0
$$910$$ 5196.53 0.189300
$$911$$ −32479.8 −1.18123 −0.590616 0.806952i $$-0.701115\pi$$
−0.590616 + 0.806952i $$0.701115\pi$$
$$912$$ 0 0
$$913$$ −21376.3 −0.774864
$$914$$ −24648.5 −0.892012
$$915$$ 0 0
$$916$$ 1474.46 0.0531851
$$917$$ 1475.72 0.0531435
$$918$$ 0 0
$$919$$ 295.958 0.0106232 0.00531161 0.999986i $$-0.498309\pi$$
0.00531161 + 0.999986i $$0.498309\pi$$
$$920$$ 79572.9 2.85157
$$921$$ 0 0
$$922$$ 9340.56 0.333639
$$923$$ 6867.89 0.244918
$$924$$ 0 0
$$925$$ −39403.5 −1.40062
$$926$$ −20558.2 −0.729572
$$927$$ 0 0
$$928$$ −6373.42 −0.225450
$$929$$ 5620.38 0.198492 0.0992458 0.995063i $$-0.468357\pi$$
0.0992458 + 0.995063i $$0.468357\pi$$
$$930$$ 0 0
$$931$$ −4740.29 −0.166871
$$932$$ 2700.52 0.0949125
$$933$$ 0 0
$$934$$ −21142.0 −0.740670
$$935$$ −29908.0 −1.04609
$$936$$ 0 0
$$937$$ −32583.1 −1.13601 −0.568006 0.823024i $$-0.692285\pi$$
−0.568006 + 0.823024i $$0.692285\pi$$
$$938$$ 8208.35 0.285727
$$939$$ 0 0
$$940$$ 2652.96 0.0920532
$$941$$ −8812.99 −0.305308 −0.152654 0.988280i $$-0.548782\pi$$
−0.152654 + 0.988280i $$0.548782\pi$$
$$942$$ 0 0
$$943$$ 44201.2 1.52639
$$944$$ −4772.00 −0.164529
$$945$$ 0 0
$$946$$ 31552.9 1.08443
$$947$$ −13426.8 −0.460732 −0.230366 0.973104i $$-0.573992\pi$$
−0.230366 + 0.973104i $$0.573992\pi$$
$$948$$ 0 0
$$949$$ 9546.76 0.326555
$$950$$ 11529.1 0.393739
$$951$$ 0 0
$$952$$ 11667.1 0.397199
$$953$$ 13394.6 0.455293 0.227647 0.973744i $$-0.426897\pi$$
0.227647 + 0.973744i $$0.426897\pi$$
$$954$$ 0 0
$$955$$ 48986.2 1.65985
$$956$$ −645.560 −0.0218398
$$957$$ 0 0
$$958$$ 25880.7 0.872828
$$959$$ 4092.74 0.137812
$$960$$ 0 0
$$961$$ −16035.7 −0.538273
$$962$$ 5516.11 0.184872
$$963$$ 0 0
$$964$$ 276.243 0.00922944
$$965$$ −47163.8 −1.57332
$$966$$ 0 0
$$967$$ 45590.8 1.51613 0.758066 0.652178i $$-0.226144\pi$$
0.758066 + 0.652178i $$0.226144\pi$$
$$968$$ 18760.1 0.622904
$$969$$ 0 0
$$970$$ −29761.8 −0.985149
$$971$$ −264.763 −0.00875041 −0.00437521 0.999990i $$-0.501393\pi$$
−0.00437521 + 0.999990i $$0.501393\pi$$
$$972$$ 0 0
$$973$$ 4562.32 0.150320
$$974$$ −16878.2 −0.555248
$$975$$ 0 0
$$976$$ 11672.4 0.382812
$$977$$ −610.521 −0.0199921 −0.00999606 0.999950i $$-0.503182\pi$$
−0.00999606 + 0.999950i $$0.503182\pi$$
$$978$$ 0 0
$$979$$ 27271.8 0.890308
$$980$$ −2702.56 −0.0880918
$$981$$ 0 0
$$982$$ −10535.5 −0.342363
$$983$$ 57829.7 1.87638 0.938190 0.346121i $$-0.112501\pi$$
0.938190 + 0.346121i $$0.112501\pi$$
$$984$$ 0 0
$$985$$ −90278.8 −2.92033
$$986$$ 53627.7 1.73210
$$987$$ 0 0
$$988$$ 103.776 0.00334165
$$989$$ −88250.4 −2.83741
$$990$$ 0 0
$$991$$ −56780.7 −1.82008 −0.910039 0.414522i $$-0.863949\pi$$
−0.910039 + 0.414522i $$0.863949\pi$$
$$992$$ 2561.67 0.0819890
$$993$$ 0 0
$$994$$ −10838.9 −0.345866
$$995$$ −59505.3 −1.89592
$$996$$ 0 0
$$997$$ 18616.6 0.591369 0.295684 0.955286i $$-0.404452\pi$$
0.295684 + 0.955286i $$0.404452\pi$$
$$998$$ −35209.4 −1.11677
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 117.4.a.c.1.2 2
3.2 odd 2 39.4.a.b.1.1 2
4.3 odd 2 1872.4.a.t.1.1 2
12.11 even 2 624.4.a.r.1.2 2
13.12 even 2 1521.4.a.s.1.1 2
15.14 odd 2 975.4.a.j.1.2 2
21.20 even 2 1911.4.a.h.1.1 2
24.5 odd 2 2496.4.a.bc.1.1 2
24.11 even 2 2496.4.a.s.1.1 2
39.5 even 4 507.4.b.f.337.3 4
39.8 even 4 507.4.b.f.337.2 4
39.38 odd 2 507.4.a.f.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.1 2 3.2 odd 2
117.4.a.c.1.2 2 1.1 even 1 trivial
507.4.a.f.1.2 2 39.38 odd 2
507.4.b.f.337.2 4 39.8 even 4
507.4.b.f.337.3 4 39.5 even 4
624.4.a.r.1.2 2 12.11 even 2
975.4.a.j.1.2 2 15.14 odd 2
1521.4.a.s.1.1 2 13.12 even 2
1872.4.a.t.1.1 2 4.3 odd 2
1911.4.a.h.1.1 2 21.20 even 2
2496.4.a.s.1.1 2 24.11 even 2
2496.4.a.bc.1.1 2 24.5 odd 2