# Properties

 Label 975.4.a.j.1.2 Level $975$ Weight $4$ Character 975.1 Self dual yes Analytic conductor $57.527$ Analytic rank $0$ 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] = [975,4,Mod(1,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("975.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 975.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: $$57.5268622556$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 975.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.74166 q^{2} +3.00000 q^{3} -0.483315 q^{4} +8.22497 q^{6} -7.48331 q^{7} -23.2583 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+2.74166 q^{2} +3.00000 q^{3} -0.483315 q^{4} +8.22497 q^{6} -7.48331 q^{7} -23.2583 q^{8} +9.00000 q^{9} +22.8999 q^{11} -1.44994 q^{12} +13.0000 q^{13} -20.5167 q^{14} -59.8999 q^{16} -67.0334 q^{17} +24.6749 q^{18} +16.5167 q^{19} -22.4499 q^{21} +62.7836 q^{22} +175.600 q^{23} -69.7750 q^{24} +35.6415 q^{26} +27.0000 q^{27} +3.61680 q^{28} +291.800 q^{29} +117.283 q^{31} +21.8418 q^{32} +68.6997 q^{33} -183.783 q^{34} -4.34983 q^{36} +154.766 q^{37} +45.2831 q^{38} +39.0000 q^{39} -251.716 q^{41} -61.5501 q^{42} +502.566 q^{43} -11.0679 q^{44} +481.434 q^{46} +281.733 q^{47} -179.700 q^{48} -287.000 q^{49} -201.100 q^{51} -6.28309 q^{52} -366.999 q^{53} +74.0247 q^{54} +174.049 q^{56} +49.5501 q^{57} +800.015 q^{58} -79.6663 q^{59} -194.865 q^{61} +321.550 q^{62} -67.3498 q^{63} +539.082 q^{64} +188.351 q^{66} -400.082 q^{67} +32.3982 q^{68} +526.799 q^{69} +528.299 q^{71} -209.325 q^{72} +734.366 q^{73} +424.316 q^{74} -7.98276 q^{76} -171.367 q^{77} +106.925 q^{78} +113.266 q^{79} +81.0000 q^{81} -690.118 q^{82} +933.466 q^{83} +10.8504 q^{84} +1377.86 q^{86} +875.399 q^{87} -532.613 q^{88} +1190.91 q^{89} -97.2831 q^{91} -84.8699 q^{92} +351.849 q^{93} +772.415 q^{94} +65.5253 q^{96} -557.165 q^{97} -786.856 q^{98} +206.099 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 6 q^{3} + 14 q^{4} - 6 q^{6} - 54 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 6 * q^3 + 14 * q^4 - 6 * q^6 - 54 * q^8 + 18 * q^9 $$2 q - 2 q^{2} + 6 q^{3} + 14 q^{4} - 6 q^{6} - 54 q^{8} + 18 q^{9} - 44 q^{11} + 42 q^{12} + 26 q^{13} - 56 q^{14} - 30 q^{16} - 164 q^{17} - 18 q^{18} + 48 q^{19} + 380 q^{22} - 8 q^{23} - 162 q^{24} - 26 q^{26} + 54 q^{27} + 112 q^{28} + 404 q^{29} + 40 q^{31} + 126 q^{32} - 132 q^{33} + 276 q^{34} + 126 q^{36} + 100 q^{37} - 104 q^{38} + 78 q^{39} + 200 q^{41} - 168 q^{42} + 616 q^{43} - 980 q^{44} + 1352 q^{46} + 324 q^{47} - 90 q^{48} - 574 q^{49} - 492 q^{51} + 182 q^{52} + 164 q^{53} - 54 q^{54} - 56 q^{56} + 144 q^{57} + 268 q^{58} + 140 q^{59} + 628 q^{61} + 688 q^{62} - 194 q^{64} + 1140 q^{66} + 472 q^{67} - 1372 q^{68} - 24 q^{69} + 428 q^{71} - 486 q^{72} + 900 q^{73} + 684 q^{74} + 448 q^{76} - 672 q^{77} - 78 q^{78} - 432 q^{79} + 162 q^{81} - 2832 q^{82} + 1388 q^{83} + 336 q^{84} + 840 q^{86} + 1212 q^{87} + 1524 q^{88} + 960 q^{89} - 2744 q^{92} + 120 q^{93} + 572 q^{94} + 378 q^{96} + 532 q^{97} + 574 q^{98} - 396 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 6 * q^3 + 14 * q^4 - 6 * q^6 - 54 * q^8 + 18 * q^9 - 44 * q^11 + 42 * q^12 + 26 * q^13 - 56 * q^14 - 30 * q^16 - 164 * q^17 - 18 * q^18 + 48 * q^19 + 380 * q^22 - 8 * q^23 - 162 * q^24 - 26 * q^26 + 54 * q^27 + 112 * q^28 + 404 * q^29 + 40 * q^31 + 126 * q^32 - 132 * q^33 + 276 * q^34 + 126 * q^36 + 100 * q^37 - 104 * q^38 + 78 * q^39 + 200 * q^41 - 168 * q^42 + 616 * q^43 - 980 * q^44 + 1352 * q^46 + 324 * q^47 - 90 * q^48 - 574 * q^49 - 492 * q^51 + 182 * q^52 + 164 * q^53 - 54 * q^54 - 56 * q^56 + 144 * q^57 + 268 * q^58 + 140 * q^59 + 628 * q^61 + 688 * q^62 - 194 * q^64 + 1140 * q^66 + 472 * q^67 - 1372 * q^68 - 24 * q^69 + 428 * q^71 - 486 * q^72 + 900 * q^73 + 684 * q^74 + 448 * q^76 - 672 * q^77 - 78 * q^78 - 432 * q^79 + 162 * q^81 - 2832 * q^82 + 1388 * q^83 + 336 * q^84 + 840 * q^86 + 1212 * q^87 + 1524 * q^88 + 960 * q^89 - 2744 * q^92 + 120 * q^93 + 572 * q^94 + 378 * q^96 + 532 * q^97 + 574 * q^98 - 396 * q^99

## 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 $$0.338943\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −0.483315 −0.0604143
$$5$$ 0 0
$$6$$ 8.22497 0.559638
$$7$$ −7.48331 −0.404061 −0.202031 0.979379i $$-0.564754\pi$$
−0.202031 + 0.979379i $$0.564754\pi$$
$$8$$ −23.2583 −1.02788
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 22.8999 0.627689 0.313844 0.949474i $$-0.398383\pi$$
0.313844 + 0.949474i $$0.398383\pi$$
$$12$$ −1.44994 −0.0348802
$$13$$ 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 $$0.658702\pi$$
$$18$$ 24.6749 0.323107
$$19$$ 16.5167 0.199431 0.0997155 0.995016i $$-0.468207\pi$$
0.0997155 + 0.995016i $$0.468207\pi$$
$$20$$ 0 0
$$21$$ −22.4499 −0.233285
$$22$$ 62.7836 0.608433
$$23$$ 175.600 1.59196 0.795979 0.605324i $$-0.206956\pi$$
0.795979 + 0.605324i $$0.206956\pi$$
$$24$$ −69.7750 −0.593449
$$25$$ 0 0
$$26$$ 35.6415 0.268842
$$27$$ 27.0000 0.192450
$$28$$ 3.61680 0.0244111
$$29$$ 291.800 1.86848 0.934239 0.356648i $$-0.116080\pi$$
0.934239 + 0.356648i $$0.116080\pi$$
$$30$$ 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
$$33$$ 68.6997 0.362396
$$34$$ −183.783 −0.927013
$$35$$ 0 0
$$36$$ −4.34983 −0.0201381
$$37$$ 154.766 0.687661 0.343830 0.939032i $$-0.388276\pi$$
0.343830 + 0.939032i $$0.388276\pi$$
$$38$$ 45.2831 0.193313
$$39$$ 39.0000 0.160128
$$40$$ 0 0
$$41$$ −251.716 −0.958815 −0.479407 0.877592i $$-0.659148\pi$$
−0.479407 + 0.877592i $$0.659148\pi$$
$$42$$ −61.5501 −0.226128
$$43$$ 502.566 1.78234 0.891170 0.453669i $$-0.149885\pi$$
0.891170 + 0.453669i $$0.149885\pi$$
$$44$$ −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 $$0.355977\pi$$
$$48$$ −179.700 −0.540363
$$49$$ −287.000 −0.836735
$$50$$ 0 0
$$51$$ −201.100 −0.552150
$$52$$ −6.28309 −0.0167559
$$53$$ −366.999 −0.951154 −0.475577 0.879674i $$-0.657761\pi$$
−0.475577 + 0.879674i $$0.657761\pi$$
$$54$$ 74.0247 0.186546
$$55$$ 0 0
$$56$$ 174.049 0.415328
$$57$$ 49.5501 0.115141
$$58$$ 800.015 1.81116
$$59$$ −79.6663 −0.175791 −0.0878955 0.996130i $$-0.528014\pi$$
−0.0878955 + 0.996130i $$0.528014\pi$$
$$60$$ 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
$$63$$ −67.3498 −0.134687
$$64$$ 539.082 1.05289
$$65$$ 0 0
$$66$$ 188.351 0.351279
$$67$$ −400.082 −0.729519 −0.364759 0.931102i $$-0.618849\pi$$
−0.364759 + 0.931102i $$0.618849\pi$$
$$68$$ 32.3982 0.0577774
$$69$$ 526.799 0.919117
$$70$$ 0 0
$$71$$ 528.299 0.883065 0.441532 0.897245i $$-0.354435\pi$$
0.441532 + 0.897245i $$0.354435\pi$$
$$72$$ −209.325 −0.342628
$$73$$ 734.366 1.17741 0.588706 0.808347i $$-0.299638\pi$$
0.588706 + 0.808347i $$0.299638\pi$$
$$74$$ 424.316 0.666565
$$75$$ 0 0
$$76$$ −7.98276 −0.0120485
$$77$$ −171.367 −0.253625
$$78$$ 106.925 0.155216
$$79$$ 113.266 0.161309 0.0806545 0.996742i $$-0.474299\pi$$
0.0806545 + 0.996742i $$0.474299\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ −690.118 −0.929400
$$83$$ 933.466 1.23447 0.617236 0.786778i $$-0.288252\pi$$
0.617236 + 0.786778i $$0.288252\pi$$
$$84$$ 10.8504 0.0140937
$$85$$ 0 0
$$86$$ 1377.86 1.72766
$$87$$ 875.399 1.07877
$$88$$ −532.613 −0.645191
$$89$$ 1190.91 1.41839 0.709195 0.705012i $$-0.249059\pi$$
0.709195 + 0.705012i $$0.249059\pi$$
$$90$$ 0 0
$$91$$ −97.2831 −0.112066
$$92$$ −84.8699 −0.0961771
$$93$$ 351.849 0.392313
$$94$$ 772.415 0.847538
$$95$$ 0 0
$$96$$ 65.5253 0.0696630
$$97$$ −557.165 −0.583211 −0.291606 0.956539i $$-0.594189\pi$$
−0.291606 + 0.956539i $$0.594189\pi$$
$$98$$ −786.856 −0.811066
$$99$$ 206.099 0.209230
$$100$$ 0 0
$$101$$ −286.766 −0.282518 −0.141259 0.989973i $$-0.545115\pi$$
−0.141259 + 0.989973i $$0.545115\pi$$
$$102$$ −551.348 −0.535211
$$103$$ 1911.36 1.82847 0.914234 0.405187i $$-0.132794\pi$$
0.914234 + 0.405187i $$0.132794\pi$$
$$104$$ −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 $$0.623012\pi$$
$$108$$ −13.0495 −0.0116267
$$109$$ −1077.66 −0.946986 −0.473493 0.880798i $$-0.657007\pi$$
−0.473493 + 0.880798i $$0.657007\pi$$
$$110$$ 0 0
$$111$$ 464.299 0.397021
$$112$$ 448.250 0.378175
$$113$$ 166.065 0.138248 0.0691241 0.997608i $$-0.477980\pi$$
0.0691241 + 0.997608i $$0.477980\pi$$
$$114$$ 135.849 0.111609
$$115$$ 0 0
$$116$$ −141.031 −0.112883
$$117$$ 117.000 0.0924500
$$118$$ −218.418 −0.170398
$$119$$ 501.632 0.386424
$$120$$ 0 0
$$121$$ −806.595 −0.606007
$$122$$ −534.254 −0.396468
$$123$$ −755.147 −0.553572
$$124$$ −56.6847 −0.0410519
$$125$$ 0 0
$$126$$ −184.650 −0.130555
$$127$$ −1296.16 −0.905637 −0.452819 0.891603i $$-0.649581\pi$$
−0.452819 + 0.891603i $$0.649581\pi$$
$$128$$ 1303.24 0.899934
$$129$$ 1507.70 1.02903
$$130$$ 0 0
$$131$$ −197.201 −0.131523 −0.0657617 0.997835i $$-0.520948\pi$$
−0.0657617 + 0.997835i $$0.520948\pi$$
$$132$$ −33.2036 −0.0218939
$$133$$ −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 $$0.445451\pi$$
$$138$$ 1444.30 0.890921
$$139$$ 609.666 0.372023 0.186012 0.982548i $$-0.440444\pi$$
0.186012 + 0.982548i $$0.440444\pi$$
$$140$$ 0 0
$$141$$ 845.199 0.504813
$$142$$ 1448.42 0.855974
$$143$$ 297.699 0.174090
$$144$$ −539.099 −0.311979
$$145$$ 0 0
$$146$$ 2013.38 1.14129
$$147$$ −861.000 −0.483089
$$148$$ −74.8009 −0.0415446
$$149$$ −2165.08 −1.19040 −0.595202 0.803576i $$-0.702928\pi$$
−0.595202 + 0.803576i $$0.702928\pi$$
$$150$$ 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
$$153$$ −603.300 −0.318784
$$154$$ −469.830 −0.245844
$$155$$ 0 0
$$156$$ −18.8493 −0.00967404
$$157$$ −1653.60 −0.840581 −0.420291 0.907390i $$-0.638072\pi$$
−0.420291 + 0.907390i $$0.638072\pi$$
$$158$$ 310.536 0.156360
$$159$$ −1101.00 −0.549149
$$160$$ 0 0
$$161$$ −1314.07 −0.643248
$$162$$ 222.074 0.107702
$$163$$ 2866.51 1.37744 0.688720 0.725027i $$-0.258173\pi$$
0.688720 + 0.725027i $$0.258173\pi$$
$$164$$ 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 $$0.554026\pi$$
$$168$$ 522.148 0.239789
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 148.650 0.0664770
$$172$$ −242.898 −0.107679
$$173$$ 3834.83 1.68530 0.842650 0.538462i $$-0.180995\pi$$
0.842650 + 0.538462i $$0.180995\pi$$
$$174$$ 2400.05 1.04567
$$175$$ 0 0
$$176$$ −1371.70 −0.587476
$$177$$ −238.999 −0.101493
$$178$$ 3265.08 1.37488
$$179$$ −283.862 −0.118530 −0.0592649 0.998242i $$-0.518876\pi$$
−0.0592649 + 0.998242i $$0.518876\pi$$
$$180$$ 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
$$183$$ −584.596 −0.236145
$$184$$ −4084.15 −1.63635
$$185$$ 0 0
$$186$$ 964.650 0.380277
$$187$$ −1535.06 −0.600291
$$188$$ −136.166 −0.0528240
$$189$$ −202.049 −0.0777616
$$190$$ 0 0
$$191$$ 2514.26 0.952491 0.476246 0.879312i $$-0.341997\pi$$
0.476246 + 0.879312i $$0.341997\pi$$
$$192$$ 1617.25 0.607889
$$193$$ −2420.73 −0.902839 −0.451420 0.892312i $$-0.649082\pi$$
−0.451420 + 0.892312i $$0.649082\pi$$
$$194$$ −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 $$0.183781\pi$$
$$198$$ 565.053 0.202811
$$199$$ 3054.17 1.08796 0.543980 0.839098i $$-0.316917\pi$$
0.543980 + 0.839098i $$0.316917\pi$$
$$200$$ 0 0
$$201$$ −1200.25 −0.421188
$$202$$ −786.215 −0.273851
$$203$$ −2183.63 −0.754979
$$204$$ 97.1947 0.0333578
$$205$$ 0 0
$$206$$ 5240.30 1.77237
$$207$$ 1580.40 0.530653
$$208$$ −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
$$213$$ 1584.90 0.509838
$$214$$ −2287.46 −0.730689
$$215$$ 0 0
$$216$$ −627.975 −0.197816
$$217$$ −877.666 −0.274562
$$218$$ −2954.59 −0.917935
$$219$$ 2203.10 0.679779
$$220$$ 0 0
$$221$$ −871.434 −0.265244
$$222$$ 1272.95 0.384841
$$223$$ −3784.95 −1.13659 −0.568294 0.822826i $$-0.692396\pi$$
−0.568294 + 0.822826i $$0.692396\pi$$
$$224$$ −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 $$0.595123\pi$$
$$228$$ −23.9483 −0.00695620
$$229$$ −3050.73 −0.880340 −0.440170 0.897915i $$-0.645082\pi$$
−0.440170 + 0.897915i $$0.645082\pi$$
$$230$$ 0 0
$$231$$ −514.101 −0.146430
$$232$$ −6786.78 −1.92058
$$233$$ −5587.49 −1.57103 −0.785513 0.618846i $$-0.787601\pi$$
−0.785513 + 0.618846i $$0.787601\pi$$
$$234$$ 320.774 0.0896139
$$235$$ 0 0
$$236$$ 38.5039 0.0106203
$$237$$ 339.798 0.0931317
$$238$$ 1375.30 0.374570
$$239$$ −1335.69 −0.361501 −0.180750 0.983529i $$-0.557853\pi$$
−0.180750 + 0.983529i $$0.557853\pi$$
$$240$$ 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
$$243$$ 243.000 0.0641500
$$244$$ 94.1813 0.0247104
$$245$$ 0 0
$$246$$ −2070.36 −0.536590
$$247$$ 214.717 0.0553122
$$248$$ −2727.81 −0.698452
$$249$$ 2800.40 0.712723
$$250$$ 0 0
$$251$$ 4088.60 1.02817 0.514084 0.857740i $$-0.328132\pi$$
0.514084 + 0.857740i $$0.328132\pi$$
$$252$$ 32.5512 0.00813703
$$253$$ 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 $$0.620701\pi$$
$$258$$ 4133.59 0.997466
$$259$$ −1158.17 −0.277857
$$260$$ 0 0
$$261$$ 2626.20 0.622826
$$262$$ −540.659 −0.127489
$$263$$ −5770.99 −1.35306 −0.676530 0.736415i $$-0.736517\pi$$
−0.676530 + 0.736415i $$0.736517\pi$$
$$264$$ −1597.84 −0.372501
$$265$$ 0 0
$$266$$ −338.868 −0.0781102
$$267$$ 3572.74 0.818908
$$268$$ 193.365 0.0440734
$$269$$ −2079.40 −0.471314 −0.235657 0.971836i $$-0.575724\pi$$
−0.235657 + 0.971836i $$0.575724\pi$$
$$270$$ 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
$$273$$ −291.849 −0.0647015
$$274$$ 1499.45 0.330603
$$275$$ 0 0
$$276$$ −254.610 −0.0555279
$$277$$ 735.201 0.159473 0.0797364 0.996816i $$-0.474592\pi$$
0.0797364 + 0.996816i $$0.474592\pi$$
$$278$$ 1671.50 0.360610
$$279$$ 1055.55 0.226502
$$280$$ 0 0
$$281$$ −1902.92 −0.403981 −0.201990 0.979387i $$-0.564741\pi$$
−0.201990 + 0.979387i $$0.564741\pi$$
$$282$$ 2317.25 0.489326
$$283$$ −2125.71 −0.446502 −0.223251 0.974761i $$-0.571667\pi$$
−0.223251 + 0.974761i $$0.571667\pi$$
$$284$$ −255.335 −0.0533498
$$285$$ 0 0
$$286$$ 816.187 0.168749
$$287$$ 1883.67 0.387420
$$288$$ 196.576 0.0402200
$$289$$ −419.527 −0.0853913
$$290$$ 0 0
$$291$$ −1671.49 −0.336717
$$292$$ −354.930 −0.0711325
$$293$$ 1641.03 0.327200 0.163600 0.986527i $$-0.447689\pi$$
0.163600 + 0.986527i $$0.447689\pi$$
$$294$$ −2360.57 −0.468269
$$295$$ 0 0
$$296$$ −3599.61 −0.706835
$$297$$ 618.297 0.120799
$$298$$ −5935.91 −1.15389
$$299$$ 2282.79 0.441530
$$300$$ 0 0
$$301$$ −3760.86 −0.720174
$$302$$ −2320.95 −0.442237
$$303$$ −860.299 −0.163112
$$304$$ −989.348 −0.186655
$$305$$ 0 0
$$306$$ −1654.04 −0.309004
$$307$$ 3373.27 0.627111 0.313555 0.949570i $$-0.398480\pi$$
0.313555 + 0.949570i $$0.398480\pi$$
$$308$$ 82.8242 0.0153226
$$309$$ 5734.09 1.05567
$$310$$ 0 0
$$311$$ −868.525 −0.158359 −0.0791793 0.996860i $$-0.525230\pi$$
−0.0791793 + 0.996860i $$0.525230\pi$$
$$312$$ −907.075 −0.164593
$$313$$ 4343.19 0.784319 0.392159 0.919897i $$-0.371728\pi$$
0.392159 + 0.919897i $$0.371728\pi$$
$$314$$ −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 $$0.406223\pi$$
$$318$$ −3018.56 −0.532302
$$319$$ 6682.18 1.17282
$$320$$ 0 0
$$321$$ −2503.00 −0.435215
$$322$$ −3602.72 −0.623515
$$323$$ −1107.17 −0.190726
$$324$$ −39.1485 −0.00671271
$$325$$ 0 0
$$326$$ 7859.00 1.33518
$$327$$ −3232.99 −0.546743
$$328$$ 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
$$333$$ 1392.90 0.229220
$$334$$ −1998.85 −0.327461
$$335$$ 0 0
$$336$$ 1344.75 0.218340
$$337$$ −901.544 −0.145728 −0.0728638 0.997342i $$-0.523214\pi$$
−0.0728638 + 0.997342i $$0.523214\pi$$
$$338$$ 463.340 0.0745633
$$339$$ 498.194 0.0798176
$$340$$ 0 0
$$341$$ 2685.77 0.426518
$$342$$ 407.548 0.0644376
$$343$$ 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 $$0.479986\pi$$
$$348$$ −423.093 −0.0651730
$$349$$ 4437.96 0.680683 0.340342 0.940302i $$-0.389457\pi$$
0.340342 + 0.940302i $$0.389457\pi$$
$$350$$ 0 0
$$351$$ 351.000 0.0533761
$$352$$ 500.174 0.0757368
$$353$$ −7115.35 −1.07284 −0.536419 0.843952i $$-0.680223\pi$$
−0.536419 + 0.843952i $$0.680223\pi$$
$$354$$ −655.253 −0.0983794
$$355$$ 0 0
$$356$$ −575.587 −0.0856911
$$357$$ 1504.90 0.223102
$$358$$ −778.253 −0.114894
$$359$$ 4693.98 0.690081 0.345040 0.938588i $$-0.387865\pi$$
0.345040 + 0.938588i $$0.387865\pi$$
$$360$$ 0 0
$$361$$ −6586.20 −0.960227
$$362$$ 6480.17 0.940857
$$363$$ −2419.79 −0.349878
$$364$$ 47.0184 0.00677042
$$365$$ 0 0
$$366$$ −1602.76 −0.228901
$$367$$ −9243.98 −1.31480 −0.657400 0.753542i $$-0.728344\pi$$
−0.657400 + 0.753542i $$0.728344\pi$$
$$368$$ −10518.4 −1.48997
$$369$$ −2265.44 −0.319605
$$370$$ 0 0
$$371$$ 2746.37 0.384324
$$372$$ −170.054 −0.0237013
$$373$$ 4311.99 0.598569 0.299285 0.954164i $$-0.403252\pi$$
0.299285 + 0.954164i $$0.403252\pi$$
$$374$$ −4208.60 −0.581876
$$375$$ 0 0
$$376$$ −6552.64 −0.898741
$$377$$ 3793.40 0.518223
$$378$$ −553.951 −0.0753760
$$379$$ −2382.73 −0.322936 −0.161468 0.986878i $$-0.551623\pi$$
−0.161468 + 0.986878i $$0.551623\pi$$
$$380$$ 0 0
$$381$$ −3888.49 −0.522870
$$382$$ 6893.25 0.923271
$$383$$ −4845.81 −0.646499 −0.323250 0.946314i $$-0.604775\pi$$
−0.323250 + 0.946314i $$0.604775\pi$$
$$384$$ 3909.73 0.519577
$$385$$ 0 0
$$386$$ −6636.81 −0.875142
$$387$$ 4523.10 0.594113
$$388$$ 269.286 0.0352343
$$389$$ 9561.50 1.24624 0.623120 0.782127i $$-0.285865\pi$$
0.623120 + 0.782127i $$0.285865\pi$$
$$390$$ 0 0
$$391$$ −11771.0 −1.52247
$$392$$ 6675.14 0.860066
$$393$$ −591.604 −0.0759350
$$394$$ 12703.9 1.62440
$$395$$ 0 0
$$396$$ −99.6107 −0.0126405
$$397$$ 7440.11 0.940575 0.470287 0.882513i $$-0.344150\pi$$
0.470287 + 0.882513i $$0.344150\pi$$
$$398$$ 8373.48 1.05458
$$399$$ −370.799 −0.0465242
$$400$$ 0 0
$$401$$ −8687.80 −1.08192 −0.540958 0.841050i $$-0.681938\pi$$
−0.540958 + 0.841050i $$0.681938\pi$$
$$402$$ −3290.66 −0.408267
$$403$$ 1524.68 0.188461
$$404$$ 138.598 0.0170681
$$405$$ 0 0
$$406$$ −5986.76 −0.731818
$$407$$ 3544.13 0.431637
$$408$$ 4677.26 0.567546
$$409$$ 2556.10 0.309024 0.154512 0.987991i $$-0.450619\pi$$
0.154512 + 0.987991i $$0.450619\pi$$
$$410$$ 0 0
$$411$$ 1640.74 0.196915
$$412$$ −923.790 −0.110466
$$413$$ 596.168 0.0710303
$$414$$ 4332.90 0.514374
$$415$$ 0 0
$$416$$ 283.943 0.0334650
$$417$$ 1829.00 0.214788
$$418$$ 1036.98 0.121340
$$419$$ −3347.46 −0.390296 −0.195148 0.980774i $$-0.562519\pi$$
−0.195148 + 0.980774i $$0.562519\pi$$
$$420$$ 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
$$423$$ 2535.60 0.291454
$$424$$ 8535.79 0.977675
$$425$$ 0 0
$$426$$ 4345.25 0.494197
$$427$$ 1458.24 0.165267
$$428$$ 403.246 0.0455412
$$429$$ 893.096 0.100511
$$430$$ 0 0
$$431$$ −14043.1 −1.56945 −0.784725 0.619844i $$-0.787196\pi$$
−0.784725 + 0.619844i $$0.787196\pi$$
$$432$$ −1617.30 −0.180121
$$433$$ −3086.47 −0.342555 −0.171278 0.985223i $$-0.554790\pi$$
−0.171278 + 0.985223i $$0.554790\pi$$
$$434$$ −2406.26 −0.266139
$$435$$ 0 0
$$436$$ 520.851 0.0572116
$$437$$ 2900.32 0.317486
$$438$$ 6040.14 0.658925
$$439$$ 2837.68 0.308508 0.154254 0.988031i $$-0.450703\pi$$
0.154254 + 0.988031i $$0.450703\pi$$
$$440$$ 0 0
$$441$$ −2583.00 −0.278912
$$442$$ −2389.17 −0.257107
$$443$$ −18309.4 −1.96367 −0.981834 0.189744i $$-0.939234\pi$$
−0.981834 + 0.189744i $$0.939234\pi$$
$$444$$ −224.403 −0.0239858
$$445$$ 0 0
$$446$$ −10377.0 −1.10172
$$447$$ −6495.24 −0.687281
$$448$$ −4034.12 −0.425433
$$449$$ 13861.2 1.45690 0.728451 0.685098i $$-0.240241\pi$$
0.728451 + 0.685098i $$0.240241\pi$$
$$450$$ 0 0
$$451$$ −5764.26 −0.601837
$$452$$ −80.2614 −0.00835217
$$453$$ −2539.65 −0.263406
$$454$$ −5521.23 −0.570758
$$455$$ 0 0
$$456$$ −1152.45 −0.118352
$$457$$ 8990.36 0.920243 0.460122 0.887856i $$-0.347806\pi$$
0.460122 + 0.887856i $$0.347806\pi$$
$$458$$ −8364.05 −0.853333
$$459$$ −1809.90 −0.184050
$$460$$ 0 0
$$461$$ −3406.90 −0.344198 −0.172099 0.985080i $$-0.555055\pi$$
−0.172099 + 0.985080i $$0.555055\pi$$
$$462$$ −1409.49 −0.141938
$$463$$ 7498.45 0.752662 0.376331 0.926485i $$-0.377186\pi$$
0.376331 + 0.926485i $$0.377186\pi$$
$$464$$ −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 $$0.624784\pi$$
$$468$$ −56.5478 −0.00558531
$$469$$ 2993.94 0.294770
$$470$$ 0 0
$$471$$ −4960.79 −0.485310
$$472$$ 1852.91 0.180693
$$473$$ 11508.7 1.11875
$$474$$ 931.608 0.0902747
$$475$$ 0 0
$$476$$ −242.446 −0.0233456
$$477$$ −3302.99 −0.317051
$$478$$ −3662.01 −0.350411
$$479$$ −9439.82 −0.900451 −0.450226 0.892915i $$-0.648656\pi$$
−0.450226 + 0.892915i $$0.648656\pi$$
$$480$$ 0 0
$$481$$ 2011.96 0.190723
$$482$$ −1567.02 −0.148082
$$483$$ −3942.20 −0.371380
$$484$$ 389.839 0.0366115
$$485$$ 0 0
$$486$$ 666.223 0.0621821
$$487$$ 6156.20 0.572821 0.286411 0.958107i $$-0.407538\pi$$
0.286411 + 0.958107i $$0.407538\pi$$
$$488$$ 4532.25 0.420420
$$489$$ 8599.54 0.795265
$$490$$ 0 0
$$491$$ 3842.74 0.353198 0.176599 0.984283i $$-0.443490\pi$$
0.176599 + 0.984283i $$0.443490\pi$$
$$492$$ 364.974 0.0334437
$$493$$ −19560.3 −1.78692
$$494$$ 588.680 0.0536153
$$495$$ 0 0
$$496$$ −7025.24 −0.635973
$$497$$ −3953.43 −0.356812
$$498$$ 7677.73 0.690858
$$499$$ −12842.4 −1.15211 −0.576056 0.817410i $$-0.695409\pi$$
−0.576056 + 0.817410i $$0.695409\pi$$
$$500$$ 0 0
$$501$$ −2187.20 −0.195043
$$502$$ 11209.5 0.996627
$$503$$ −8580.11 −0.760573 −0.380287 0.924869i $$-0.624175\pi$$
−0.380287 + 0.924869i $$0.624175\pi$$
$$504$$ 1566.45 0.138443
$$505$$ 0 0
$$506$$ 11024.8 0.968599
$$507$$ 507.000 0.0444116
$$508$$ 626.455 0.0547135
$$509$$ −43.5957 −0.00379635 −0.00189818 0.999998i $$-0.500604\pi$$
−0.00189818 + 0.999998i $$0.500604\pi$$
$$510$$ 0 0
$$511$$ −5495.49 −0.475746
$$512$$ −12453.7 −1.07496
$$513$$ 445.951 0.0383805
$$514$$ −8362.68 −0.717630
$$515$$ 0 0
$$516$$ −728.693 −0.0621684
$$517$$ 6451.66 0.548827
$$518$$ −3175.29 −0.269333
$$519$$ 11504.5 0.973008
$$520$$ 0 0
$$521$$ 11368.1 0.955939 0.477969 0.878377i $$-0.341373\pi$$
0.477969 + 0.878377i $$0.341373\pi$$
$$522$$ 7200.14 0.603719
$$523$$ 5229.53 0.437230 0.218615 0.975811i $$-0.429846\pi$$
0.218615 + 0.975811i $$0.429846\pi$$
$$524$$ 95.3103 0.00794590
$$525$$ 0 0
$$526$$ −15822.1 −1.31155
$$527$$ −7861.88 −0.649846
$$528$$ −4115.10 −0.339180
$$529$$ 18668.2 1.53433
$$530$$ 0 0
$$531$$ −716.997 −0.0585970
$$532$$ 59.7375 0.00486832
$$533$$ −3272.31 −0.265927
$$534$$ 9795.24 0.793786
$$535$$ 0 0
$$536$$ 9305.24 0.749860
$$537$$ −851.586 −0.0684333
$$538$$ −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
$$543$$ 7090.79 0.560396
$$544$$ −1464.13 −0.115393
$$545$$ 0 0
$$546$$ −800.151 −0.0627166
$$547$$ 13675.7 1.06897 0.534487 0.845177i $$-0.320505\pi$$
0.534487 + 0.845177i $$0.320505\pi$$
$$548$$ −264.332 −0.0206053
$$549$$ −1753.79 −0.136339
$$550$$ 0 0
$$551$$ 4819.57 0.372632
$$552$$ −12252.5 −0.944745
$$553$$ −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 $$0.555095\pi$$
$$558$$ 2893.95 0.219553
$$559$$ 6533.36 0.494332
$$560$$ 0 0
$$561$$ −4605.17 −0.346578
$$562$$ −5217.15 −0.391588
$$563$$ −18441.8 −1.38051 −0.690256 0.723566i $$-0.742502\pi$$
−0.690256 + 0.723566i $$0.742502\pi$$
$$564$$ −408.497 −0.0304979
$$565$$ 0 0
$$566$$ −5827.96 −0.432804
$$567$$ −606.148 −0.0448957
$$568$$ −12287.4 −0.907687
$$569$$ −13553.5 −0.998578 −0.499289 0.866436i $$-0.666405\pi$$
−0.499289 + 0.866436i $$0.666405\pi$$
$$570$$ 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
$$573$$ 7542.79 0.549921
$$574$$ 5164.37 0.375534
$$575$$ 0 0
$$576$$ 4851.74 0.350965
$$577$$ −21596.2 −1.55816 −0.779081 0.626923i $$-0.784314\pi$$
−0.779081 + 0.626923i $$0.784314\pi$$
$$578$$ −1150.20 −0.0827716
$$579$$ −7262.19 −0.521254
$$580$$ 0 0
$$581$$ −6985.42 −0.498802
$$582$$ −4582.66 −0.326387
$$583$$ −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 $$0.489716\pi$$
$$588$$ 416.134 0.0291855
$$589$$ 1937.13 0.135514
$$590$$ 0 0
$$591$$ 13900.9 0.967527
$$592$$ −9270.49 −0.643606
$$593$$ −19816.0 −1.37226 −0.686128 0.727481i $$-0.740691\pi$$
−0.686128 + 0.727481i $$0.740691\pi$$
$$594$$ 1695.16 0.117093
$$595$$ 0 0
$$596$$ 1046.42 0.0719175
$$597$$ 9162.50 0.628134
$$598$$ 6258.64 0.427985
$$599$$ −5141.86 −0.350736 −0.175368 0.984503i $$-0.556111\pi$$
−0.175368 + 0.984503i $$0.556111\pi$$
$$600$$ 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
$$603$$ −3600.74 −0.243173
$$604$$ 409.150 0.0275630
$$605$$ 0 0
$$606$$ −2358.65 −0.158108
$$607$$ 23717.0 1.58590 0.792951 0.609286i $$-0.208544\pi$$
0.792951 + 0.609286i $$0.208544\pi$$
$$608$$ 360.754 0.0240633
$$609$$ −6550.89 −0.435887
$$610$$ 0 0
$$611$$ 3662.53 0.242504
$$612$$ 291.584 0.0192591
$$613$$ 26157.1 1.72345 0.861726 0.507373i $$-0.169383\pi$$
0.861726 + 0.507373i $$0.169383\pi$$
$$614$$ 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 $$0.779937\pi$$
$$618$$ 15720.9 1.02328
$$619$$ 23345.4 1.51588 0.757940 0.652324i $$-0.226206\pi$$
0.757940 + 0.652324i $$0.226206\pi$$
$$620$$ 0 0
$$621$$ 4741.19 0.306372
$$622$$ −2381.20 −0.153501
$$623$$ −8911.99 −0.573116
$$624$$ −2336.10 −0.149870
$$625$$ 0 0
$$626$$ 11907.5 0.760258
$$627$$ 1134.69 0.0722730
$$628$$ 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
$$633$$ −12094.8 −0.759439
$$634$$ 8986.20 0.562914
$$635$$ 0 0
$$636$$ 532.128 0.0331765
$$637$$ −3731.00 −0.232068
$$638$$ 18320.3 1.13684
$$639$$ 4754.69 0.294355
$$640$$ 0 0
$$641$$ 10192.7 0.628063 0.314032 0.949413i $$-0.398320\pi$$
0.314032 + 0.949413i $$0.398320\pi$$
$$642$$ −6862.37 −0.421863
$$643$$ 5506.31 0.337710 0.168855 0.985641i $$-0.445993\pi$$
0.168855 + 0.985641i $$0.445993\pi$$
$$644$$ 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 $$0.367619\pi$$
$$648$$ −1883.93 −0.114209
$$649$$ −1824.35 −0.110342
$$650$$ 0 0
$$651$$ −2633.00 −0.158518
$$652$$ −1385.43 −0.0832171
$$653$$ 12440.2 0.745519 0.372760 0.927928i $$-0.378412\pi$$
0.372760 + 0.927928i $$0.378412\pi$$
$$654$$ −8863.76 −0.529970
$$655$$ 0 0
$$656$$ 15077.7 0.897389
$$657$$ 6609.29 0.392470
$$658$$ −5780.23 −0.342457
$$659$$ −9562.87 −0.565276 −0.282638 0.959227i $$-0.591209\pi$$
−0.282638 + 0.959227i $$0.591209\pi$$
$$660$$ 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
$$663$$ −2614.30 −0.153139
$$664$$ −21710.9 −1.26889
$$665$$ 0 0
$$666$$ 3818.85 0.222188
$$667$$ 51239.9 2.97454
$$668$$ 352.368 0.0204095
$$669$$ −11354.9 −0.656209
$$670$$ 0 0
$$671$$ −4462.40 −0.256735
$$672$$ −490.346 −0.0281481
$$673$$ −7929.02 −0.454147 −0.227074 0.973878i $$-0.572916\pi$$
−0.227074 + 0.973878i $$0.572916\pi$$
$$674$$ −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 $$0.476231\pi$$
$$678$$ 1365.88 0.0773690
$$679$$ 4169.44 0.235653
$$680$$ 0 0
$$681$$ −6041.48 −0.339956
$$682$$ 7363.46 0.413433
$$683$$ −10021.5 −0.561437 −0.280719 0.959790i $$-0.590573\pi$$
−0.280719 + 0.959790i $$0.590573\pi$$
$$684$$ −71.8448 −0.00401616
$$685$$ 0 0
$$686$$ 12925.5 0.719385
$$687$$ −9152.18 −0.508264
$$688$$ −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
$$693$$ −1542.30 −0.0845415
$$694$$ 2227.10 0.121815
$$695$$ 0 0
$$696$$ −20360.3 −1.10885
$$697$$ 16873.4 0.916964
$$698$$ 12167.4 0.659801
$$699$$ −16762.5 −0.907032
$$700$$ 0 0
$$701$$ −3763.71 −0.202787 −0.101393 0.994846i $$-0.532330\pi$$
−0.101393 + 0.994846i $$0.532330\pi$$
$$702$$ 962.322 0.0517386
$$703$$ 2556.23 0.137141
$$704$$ 12344.9 0.660890
$$705$$ 0 0
$$706$$ −19507.8 −1.03993
$$707$$ 2145.96 0.114155
$$708$$ 115.512 0.00613163
$$709$$ −36047.8 −1.90946 −0.954728 0.297479i $$-0.903854\pi$$
−0.954728 + 0.297479i $$0.903854\pi$$
$$710$$ 0 0
$$711$$ 1019.39 0.0537696
$$712$$ −27698.7 −1.45794
$$713$$ 20594.9 1.08174
$$714$$ 4125.91 0.216258
$$715$$ 0 0
$$716$$ 137.195 0.00716090
$$717$$ −4007.08 −0.208713
$$718$$ 12869.3 0.668910
$$719$$ −3944.18 −0.204580 −0.102290 0.994755i $$-0.532617\pi$$
−0.102290 + 0.994755i $$0.532617\pi$$
$$720$$ 0 0
$$721$$ −14303.3 −0.738812
$$722$$ −18057.1 −0.930770
$$723$$ −1714.68 −0.0882012
$$724$$ −1142.36 −0.0586402
$$725$$ 0 0
$$726$$ −6634.22 −0.339145
$$727$$ 20447.8 1.04315 0.521573 0.853206i $$-0.325345\pi$$
0.521573 + 0.853206i $$0.325345\pi$$
$$728$$ 2262.64 0.115191
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −33688.7 −1.70454
$$732$$ 282.544 0.0142666
$$733$$ 13536.2 0.682089 0.341045 0.940047i $$-0.389219\pi$$
0.341045 + 0.940047i $$0.389219\pi$$
$$734$$ −25343.8 −1.27447
$$735$$ 0 0
$$736$$ 3835.40 0.192085
$$737$$ −9161.83 −0.457911
$$738$$ −6211.07 −0.309800
$$739$$ 15839.1 0.788433 0.394217 0.919018i $$-0.371016\pi$$
0.394217 + 0.919018i $$0.371016\pi$$
$$740$$ 0 0
$$741$$ 644.151 0.0319345
$$742$$ 7529.60 0.372534
$$743$$ 1664.92 0.0822075 0.0411037 0.999155i $$-0.486913\pi$$
0.0411037 + 0.999155i $$0.486913\pi$$
$$744$$ −8183.43 −0.403252
$$745$$ 0 0
$$746$$ 11822.0 0.580206
$$747$$ 8401.19 0.411491
$$748$$ 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
$$753$$ 12265.8 0.593613
$$754$$ 10400.2 0.502325
$$755$$ 0 0
$$756$$ 97.6535 0.00469792
$$757$$ −23798.9 −1.14265 −0.571326 0.820723i $$-0.693571\pi$$
−0.571326 + 0.820723i $$0.693571\pi$$
$$758$$ −6532.64 −0.313029
$$759$$ 12063.6 0.576920
$$760$$ 0 0
$$761$$ 13693.5 0.652285 0.326142 0.945321i $$-0.394251\pi$$
0.326142 + 0.945321i $$0.394251\pi$$
$$762$$ −10660.9 −0.506829
$$763$$ 8064.50 0.382640
$$764$$ −1215.18 −0.0575441
$$765$$ 0 0
$$766$$ −13285.5 −0.626666
$$767$$ −1035.66 −0.0487556
$$768$$ −2218.82 −0.104251
$$769$$ 16299.9 0.764358 0.382179 0.924088i $$-0.375174\pi$$
0.382179 + 0.924088i $$0.375174\pi$$
$$770$$ 0 0
$$771$$ −9150.68 −0.427437
$$772$$ 1169.97 0.0545444
$$773$$ −33532.2 −1.56024 −0.780122 0.625628i $$-0.784843\pi$$
−0.780122 + 0.625628i $$0.784843\pi$$
$$774$$ 12400.8 0.575887
$$775$$ 0 0
$$776$$ 12958.7 0.599473
$$777$$ −3474.50 −0.160421
$$778$$ 26214.3 1.20801
$$779$$ −4157.51 −0.191217
$$780$$ 0 0
$$781$$ 12098.0 0.554290
$$782$$ −32272.1 −1.47577
$$783$$ 7878.59 0.359589
$$784$$ 17191.3 0.783130
$$785$$ 0 0
$$786$$ −1621.98 −0.0736055
$$787$$ −16163.3 −0.732097 −0.366049 0.930596i $$-0.619290\pi$$
−0.366049 + 0.930596i $$0.619290\pi$$
$$788$$ −2239.51 −0.101243
$$789$$ −17313.0 −0.781189
$$790$$ 0 0
$$791$$ −1242.71 −0.0558607
$$792$$ −4793.52 −0.215064
$$793$$ −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 $$0.157007\pi$$
$$798$$ −1016.60 −0.0450969
$$799$$ −18885.5 −0.836197
$$800$$ 0 0
$$801$$ 10718.2 0.472797
$$802$$ −23819.0 −1.04872
$$803$$ 16816.9 0.739048
$$804$$ 580.096 0.0254458
$$805$$ 0 0
$$806$$ 4180.15 0.182679
$$807$$ −6238.21 −0.272113
$$808$$ 6669.71 0.290396
$$809$$ −23811.2 −1.03481 −0.517403 0.855742i $$-0.673101\pi$$
−0.517403 + 0.855742i $$0.673101\pi$$
$$810$$ 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
$$813$$ 18036.0 0.778045
$$814$$ 9716.80 0.418395
$$815$$ 0 0
$$816$$ 12045.9 0.516777
$$817$$ 8300.73 0.355454
$$818$$ 7007.94 0.299544
$$819$$ −875.548 −0.0373555
$$820$$ 0 0
$$821$$ −43094.8 −1.83193 −0.915967 0.401253i $$-0.868575\pi$$
−0.915967 + 0.401253i $$0.868575\pi$$
$$822$$ 4498.36 0.190874
$$823$$ −26541.1 −1.12414 −0.562068 0.827091i $$-0.689994\pi$$
−0.562068 + 0.827091i $$0.689994\pi$$
$$824$$ −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 $$0.107087\pi$$
$$828$$ −763.829 −0.0320590
$$829$$ −7137.48 −0.299029 −0.149514 0.988760i $$-0.547771\pi$$
−0.149514 + 0.988760i $$0.547771\pi$$
$$830$$ 0 0
$$831$$ 2205.60 0.0920717
$$832$$ 7008.06 0.292020
$$833$$ 19238.6 0.800213
$$834$$ 5014.49 0.208198
$$835$$ 0 0
$$836$$ −182.804 −0.00756270
$$837$$ 3166.64 0.130771
$$838$$ −9177.59 −0.378323
$$839$$ −4387.17 −0.180527 −0.0902634 0.995918i $$-0.528771\pi$$
−0.0902634 + 0.995918i $$0.528771\pi$$
$$840$$ 0 0
$$841$$ 60758.1 2.49121
$$842$$ −5084.36 −0.208098
$$843$$ −5708.76 −0.233239
$$844$$ 1948.53 0.0794683
$$845$$ 0 0
$$846$$ 6951.74 0.282513
$$847$$ 6036.01 0.244864
$$848$$ 21983.2 0.890219
$$849$$ −6377.12 −0.257788
$$850$$ 0 0
$$851$$ 27176.9 1.09473
$$852$$ −766.004 −0.0308015
$$853$$ 9328.85 0.374459 0.187230 0.982316i $$-0.440049\pi$$
0.187230 + 0.982316i $$0.440049\pi$$
$$854$$ 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 $$0.468162\pi$$
$$858$$ 2448.56 0.0974272
$$859$$ 30233.4 1.20088 0.600438 0.799672i $$-0.294993\pi$$
0.600438 + 0.799672i $$0.294993\pi$$
$$860$$ 0 0
$$861$$ 5651.01 0.223677
$$862$$ −38501.4 −1.52130
$$863$$ −4334.93 −0.170988 −0.0854940 0.996339i $$-0.527247\pi$$
−0.0854940 + 0.996339i $$0.527247\pi$$
$$864$$ 589.728 0.0232210
$$865$$ 0 0
$$866$$ −8462.05 −0.332047
$$867$$ −1258.58 −0.0493007
$$868$$ 424.189 0.0165875
$$869$$ 2593.78 0.101252
$$870$$ 0 0
$$871$$ −5201.06 −0.202332
$$872$$ 25064.7 0.973391
$$873$$ −5014.48 −0.194404
$$874$$ 7951.69 0.307746
$$875$$ 0 0
$$876$$ −1064.79 −0.0410684
$$877$$ −34683.3 −1.33543 −0.667716 0.744416i $$-0.732728\pi$$
−0.667716 + 0.744416i $$0.732728\pi$$
$$878$$ 7779.95 0.299044
$$879$$ 4923.08 0.188909
$$880$$ 0 0
$$881$$ −18269.2 −0.698642 −0.349321 0.937003i $$-0.613588\pi$$
−0.349321 + 0.937003i $$0.613588\pi$$
$$882$$ −7081.70 −0.270355
$$883$$ 14592.0 0.556128 0.278064 0.960563i $$-0.410307\pi$$
0.278064 + 0.960563i $$0.410307\pi$$
$$884$$ 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 $$0.695585\pi$$
$$888$$ −10798.8 −0.408091
$$889$$ 9699.60 0.365933
$$890$$ 0 0
$$891$$ 1854.89 0.0697432
$$892$$ 1829.32 0.0686662
$$893$$ 4653.30 0.174375
$$894$$ −17807.7 −0.666196
$$895$$ 0 0
$$896$$ −9752.58 −0.363628
$$897$$ 6848.38 0.254917
$$898$$ 38002.6 1.41221
$$899$$ 34223.2 1.26964
$$900$$ 0 0
$$901$$ 24601.2 0.909638
$$902$$ −15803.6 −0.583374
$$903$$ −11282.6 −0.415793
$$904$$ −3862.39 −0.142103
$$905$$ 0 0
$$906$$ −6962.84 −0.255326
$$907$$ 9364.89 0.342840 0.171420 0.985198i $$-0.445164\pi$$
0.171420 + 0.985198i $$0.445164\pi$$
$$908$$ 973.313 0.0355733
$$909$$ −2580.90 −0.0941727
$$910$$ 0 0
$$911$$ 32479.8 1.18123 0.590616 0.806952i $$-0.298885\pi$$
0.590616 + 0.806952i $$0.298885\pi$$
$$912$$ −2968.04 −0.107765
$$913$$ 21376.3 0.774864
$$914$$ 24648.5 0.892012
$$915$$ 0 0
$$916$$ 1474.46 0.0531851
$$917$$ 1475.72 0.0531435
$$918$$ −4962.13 −0.178404
$$919$$ 295.958 0.0106232 0.00531161 0.999986i $$-0.498309\pi$$
0.00531161 + 0.999986i $$0.498309\pi$$
$$920$$ 0 0
$$921$$ 10119.8 0.362062
$$922$$ −9340.56 −0.333639
$$923$$ 6867.89 0.244918
$$924$$ 248.473 0.00884649
$$925$$ 0 0
$$926$$ 20558.2 0.729572
$$927$$ 17202.3 0.609489
$$928$$ 6373.42 0.225450
$$929$$ −5620.38 −0.198492 −0.0992458 0.995063i $$-0.531643\pi$$
−0.0992458 + 0.995063i $$0.531643\pi$$
$$930$$ 0 0
$$931$$ −4740.29 −0.166871
$$932$$ 2700.52 0.0949125
$$933$$ −2605.58 −0.0914284
$$934$$ −21142.0 −0.740670
$$935$$ 0 0
$$936$$ −2721.23 −0.0950278
$$937$$ 32583.1 1.13601 0.568006 0.823024i $$-0.307715\pi$$
0.568006 + 0.823024i $$0.307715\pi$$
$$938$$ 8208.35 0.285727
$$939$$ 13029.6 0.452827
$$940$$ 0 0
$$941$$ 8812.99 0.305308 0.152654 0.988280i $$-0.451218\pi$$
0.152654 + 0.988280i $$0.451218\pi$$
$$942$$ −13600.8 −0.470422
$$943$$ −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 $$0.573992\pi$$
$$948$$ −164.229 −0.00562649
$$949$$ 9546.76 0.326555
$$950$$ 0 0
$$951$$ 9832.96 0.335285
$$952$$ −11667.1 −0.397199
$$953$$ 13394.6 0.455293 0.227647 0.973744i $$-0.426897\pi$$
0.227647 + 0.973744i $$0.426897\pi$$
$$954$$ −9055.67 −0.307325
$$955$$ 0 0
$$956$$ 645.560 0.0218398
$$957$$ 20046.5 0.677129
$$958$$ −25880.7 −0.872828
$$959$$ −4092.74 −0.137812
$$960$$ 0 0
$$961$$ −16035.7 −0.538273
$$962$$ 5516.11 0.184872
$$963$$ −7509.00 −0.251271
$$964$$ 276.243 0.00922944
$$965$$ 0 0
$$966$$ −10808.2 −0.359986
$$967$$ −45590.8 −1.51613 −0.758066 0.652178i $$-0.773856\pi$$
−0.758066 + 0.652178i $$0.773856\pi$$
$$968$$ 18760.1 0.622904
$$969$$ −3321.51 −0.110116
$$970$$ 0 0
$$971$$ 264.763 0.00875041 0.00437521 0.999990i $$-0.498607\pi$$
0.00437521 + 0.999990i $$0.498607\pi$$
$$972$$ −117.445 −0.00387558
$$973$$ −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 $$0.503182\pi$$
$$978$$ 23577.0 0.770868
$$979$$ 27271.8 0.890308
$$980$$ 0 0
$$981$$ −9698.98 −0.315662
$$982$$ 10535.5 0.342363
$$983$$ 57829.7 1.87638 0.938190 0.346121i $$-0.112501\pi$$
0.938190 + 0.346121i $$0.112501\pi$$
$$984$$ 17563.5 0.569007
$$985$$ 0 0
$$986$$ −53627.7 −1.73210
$$987$$ −6324.89 −0.203975
$$988$$ −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
$$993$$ 16768.9 0.535895
$$994$$ −10838.9 −0.345866
$$995$$ 0 0
$$996$$ −1353.47 −0.0430587
$$997$$ −18616.6 −0.591369 −0.295684 0.955286i $$-0.595548\pi$$
−0.295684 + 0.955286i $$0.595548\pi$$
$$998$$ −35209.4 −1.11677
$$999$$ 4178.69 0.132340
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 975.4.a.j.1.2 2
5.4 even 2 39.4.a.b.1.1 2
15.14 odd 2 117.4.a.c.1.2 2
20.19 odd 2 624.4.a.r.1.2 2
35.34 odd 2 1911.4.a.h.1.1 2
40.19 odd 2 2496.4.a.s.1.1 2
40.29 even 2 2496.4.a.bc.1.1 2
60.59 even 2 1872.4.a.t.1.1 2
65.34 odd 4 507.4.b.f.337.2 4
65.44 odd 4 507.4.b.f.337.3 4
65.64 even 2 507.4.a.f.1.2 2
195.194 odd 2 1521.4.a.s.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.1 2 5.4 even 2
117.4.a.c.1.2 2 15.14 odd 2
507.4.a.f.1.2 2 65.64 even 2
507.4.b.f.337.2 4 65.34 odd 4
507.4.b.f.337.3 4 65.44 odd 4
624.4.a.r.1.2 2 20.19 odd 2
975.4.a.j.1.2 2 1.1 even 1 trivial
1521.4.a.s.1.1 2 195.194 odd 2
1872.4.a.t.1.1 2 60.59 even 2
1911.4.a.h.1.1 2 35.34 odd 2
2496.4.a.s.1.1 2 40.19 odd 2
2496.4.a.bc.1.1 2 40.29 even 2