# Properties

 Label 525.4.a.k.1.1 Level $525$ Weight $4$ Character 525.1 Self dual yes Analytic conductor $30.976$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(1,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, 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("525.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.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: $$30.9760027530$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$4.53113$$ of defining polynomial Character $$\chi$$ $$=$$ 525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.53113 q^{2} -3.00000 q^{3} +12.5311 q^{4} +13.5934 q^{6} +7.00000 q^{7} -20.5311 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.53113 q^{2} -3.00000 q^{3} +12.5311 q^{4} +13.5934 q^{6} +7.00000 q^{7} -20.5311 q^{8} +9.00000 q^{9} -19.0623 q^{11} -37.5934 q^{12} +2.93774 q^{13} -31.7179 q^{14} -7.21984 q^{16} +6.49806 q^{17} -40.7802 q^{18} -5.43580 q^{19} -21.0000 q^{21} +86.3735 q^{22} -49.3774 q^{23} +61.5934 q^{24} -13.3113 q^{26} -27.0000 q^{27} +87.7179 q^{28} -291.494 q^{29} +244.307 q^{31} +196.963 q^{32} +57.1868 q^{33} -29.4436 q^{34} +112.780 q^{36} +193.121 q^{37} +24.6303 q^{38} -8.81323 q^{39} +315.113 q^{41} +95.1537 q^{42} +300.996 q^{43} -238.872 q^{44} +223.735 q^{46} -86.5058 q^{47} +21.6595 q^{48} +49.0000 q^{49} -19.4942 q^{51} +36.8132 q^{52} -509.677 q^{53} +122.340 q^{54} -143.718 q^{56} +16.3074 q^{57} +1320.80 q^{58} -83.3852 q^{59} -5.25291 q^{61} -1106.99 q^{62} +63.0000 q^{63} -834.706 q^{64} -259.121 q^{66} -205.992 q^{67} +81.4281 q^{68} +148.132 q^{69} +1004.31 q^{71} -184.780 q^{72} +1007.29 q^{73} -875.055 q^{74} -68.1168 q^{76} -133.436 q^{77} +39.9339 q^{78} -863.237 q^{79} +81.0000 q^{81} -1427.82 q^{82} -1334.72 q^{83} -263.154 q^{84} -1363.85 q^{86} +874.483 q^{87} +391.370 q^{88} +326.249 q^{89} +20.5642 q^{91} -618.755 q^{92} -732.922 q^{93} +391.969 q^{94} -590.889 q^{96} -1526.77 q^{97} -222.025 q^{98} -171.560 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 6 q^{3} + 17 q^{4} + 3 q^{6} + 14 q^{7} - 33 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - q^2 - 6 * q^3 + 17 * q^4 + 3 * q^6 + 14 * q^7 - 33 * q^8 + 18 * q^9 $$2 q - q^{2} - 6 q^{3} + 17 q^{4} + 3 q^{6} + 14 q^{7} - 33 q^{8} + 18 q^{9} - 22 q^{11} - 51 q^{12} + 22 q^{13} - 7 q^{14} - 87 q^{16} - 116 q^{17} - 9 q^{18} + 102 q^{19} - 42 q^{21} + 76 q^{22} - 260 q^{23} + 99 q^{24} + 54 q^{26} - 54 q^{27} + 119 q^{28} - 196 q^{29} + 150 q^{31} + 15 q^{32} + 66 q^{33} - 462 q^{34} + 153 q^{36} + 96 q^{37} + 404 q^{38} - 66 q^{39} - 176 q^{41} + 21 q^{42} + 344 q^{43} - 252 q^{44} - 520 q^{46} - 560 q^{47} + 261 q^{48} + 98 q^{49} + 348 q^{51} + 122 q^{52} - 326 q^{53} + 27 q^{54} - 231 q^{56} - 306 q^{57} + 1658 q^{58} - 844 q^{59} - 204 q^{61} - 1440 q^{62} + 126 q^{63} - 839 q^{64} - 228 q^{66} + 104 q^{67} - 466 q^{68} + 780 q^{69} + 1670 q^{71} - 297 q^{72} + 386 q^{73} - 1218 q^{74} + 412 q^{76} - 154 q^{77} - 162 q^{78} - 888 q^{79} + 162 q^{81} - 3162 q^{82} - 928 q^{83} - 357 q^{84} - 1212 q^{86} + 588 q^{87} + 428 q^{88} + 588 q^{89} + 154 q^{91} - 1560 q^{92} - 450 q^{93} - 1280 q^{94} - 45 q^{96} - 522 q^{97} - 49 q^{98} - 198 q^{99}+O(q^{100})$$ 2 * q - q^2 - 6 * q^3 + 17 * q^4 + 3 * q^6 + 14 * q^7 - 33 * q^8 + 18 * q^9 - 22 * q^11 - 51 * q^12 + 22 * q^13 - 7 * q^14 - 87 * q^16 - 116 * q^17 - 9 * q^18 + 102 * q^19 - 42 * q^21 + 76 * q^22 - 260 * q^23 + 99 * q^24 + 54 * q^26 - 54 * q^27 + 119 * q^28 - 196 * q^29 + 150 * q^31 + 15 * q^32 + 66 * q^33 - 462 * q^34 + 153 * q^36 + 96 * q^37 + 404 * q^38 - 66 * q^39 - 176 * q^41 + 21 * q^42 + 344 * q^43 - 252 * q^44 - 520 * q^46 - 560 * q^47 + 261 * q^48 + 98 * q^49 + 348 * q^51 + 122 * q^52 - 326 * q^53 + 27 * q^54 - 231 * q^56 - 306 * q^57 + 1658 * q^58 - 844 * q^59 - 204 * q^61 - 1440 * q^62 + 126 * q^63 - 839 * q^64 - 228 * q^66 + 104 * q^67 - 466 * q^68 + 780 * q^69 + 1670 * q^71 - 297 * q^72 + 386 * q^73 - 1218 * q^74 + 412 * q^76 - 154 * q^77 - 162 * q^78 - 888 * q^79 + 162 * q^81 - 3162 * q^82 - 928 * q^83 - 357 * q^84 - 1212 * q^86 + 588 * q^87 + 428 * q^88 + 588 * q^89 + 154 * q^91 - 1560 * q^92 - 450 * q^93 - 1280 * q^94 - 45 * q^96 - 522 * q^97 - 49 * q^98 - 198 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.53113 −1.60200 −0.800998 0.598667i $$-0.795697\pi$$
−0.800998 + 0.598667i $$0.795697\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 12.5311 1.56639
$$5$$ 0 0
$$6$$ 13.5934 0.924913
$$7$$ 7.00000 0.377964
$$8$$ −20.5311 −0.907356
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −19.0623 −0.522499 −0.261249 0.965271i $$-0.584135\pi$$
−0.261249 + 0.965271i $$0.584135\pi$$
$$12$$ −37.5934 −0.904356
$$13$$ 2.93774 0.0626756 0.0313378 0.999509i $$-0.490023\pi$$
0.0313378 + 0.999509i $$0.490023\pi$$
$$14$$ −31.7179 −0.605498
$$15$$ 0 0
$$16$$ −7.21984 −0.112810
$$17$$ 6.49806 0.0927066 0.0463533 0.998925i $$-0.485240\pi$$
0.0463533 + 0.998925i $$0.485240\pi$$
$$18$$ −40.7802 −0.533999
$$19$$ −5.43580 −0.0656347 −0.0328173 0.999461i $$-0.510448\pi$$
−0.0328173 + 0.999461i $$0.510448\pi$$
$$20$$ 0 0
$$21$$ −21.0000 −0.218218
$$22$$ 86.3735 0.837041
$$23$$ −49.3774 −0.447648 −0.223824 0.974630i $$-0.571854\pi$$
−0.223824 + 0.974630i $$0.571854\pi$$
$$24$$ 61.5934 0.523862
$$25$$ 0 0
$$26$$ −13.3113 −0.100406
$$27$$ −27.0000 −0.192450
$$28$$ 87.7179 0.592040
$$29$$ −291.494 −1.86652 −0.933261 0.359200i $$-0.883050\pi$$
−0.933261 + 0.359200i $$0.883050\pi$$
$$30$$ 0 0
$$31$$ 244.307 1.41545 0.707724 0.706489i $$-0.249722\pi$$
0.707724 + 0.706489i $$0.249722\pi$$
$$32$$ 196.963 1.08808
$$33$$ 57.1868 0.301665
$$34$$ −29.4436 −0.148516
$$35$$ 0 0
$$36$$ 112.780 0.522130
$$37$$ 193.121 0.858077 0.429038 0.903286i $$-0.358853\pi$$
0.429038 + 0.903286i $$0.358853\pi$$
$$38$$ 24.6303 0.105147
$$39$$ −8.81323 −0.0361858
$$40$$ 0 0
$$41$$ 315.113 1.20030 0.600151 0.799887i $$-0.295107\pi$$
0.600151 + 0.799887i $$0.295107\pi$$
$$42$$ 95.1537 0.349584
$$43$$ 300.996 1.06748 0.533738 0.845650i $$-0.320787\pi$$
0.533738 + 0.845650i $$0.320787\pi$$
$$44$$ −238.872 −0.818437
$$45$$ 0 0
$$46$$ 223.735 0.717130
$$47$$ −86.5058 −0.268472 −0.134236 0.990949i $$-0.542858\pi$$
−0.134236 + 0.990949i $$0.542858\pi$$
$$48$$ 21.6595 0.0651309
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ −19.4942 −0.0535242
$$52$$ 36.8132 0.0981745
$$53$$ −509.677 −1.32093 −0.660467 0.750855i $$-0.729642\pi$$
−0.660467 + 0.750855i $$0.729642\pi$$
$$54$$ 122.340 0.308304
$$55$$ 0 0
$$56$$ −143.718 −0.342948
$$57$$ 16.3074 0.0378942
$$58$$ 1320.80 2.99016
$$59$$ −83.3852 −0.183997 −0.0919985 0.995759i $$-0.529326\pi$$
−0.0919985 + 0.995759i $$0.529326\pi$$
$$60$$ 0 0
$$61$$ −5.25291 −0.0110257 −0.00551283 0.999985i $$-0.501755\pi$$
−0.00551283 + 0.999985i $$0.501755\pi$$
$$62$$ −1106.99 −2.26754
$$63$$ 63.0000 0.125988
$$64$$ −834.706 −1.63029
$$65$$ 0 0
$$66$$ −259.121 −0.483266
$$67$$ −205.992 −0.375611 −0.187806 0.982206i $$-0.560138\pi$$
−0.187806 + 0.982206i $$0.560138\pi$$
$$68$$ 81.4281 0.145215
$$69$$ 148.132 0.258450
$$70$$ 0 0
$$71$$ 1004.31 1.67872 0.839362 0.543573i $$-0.182929\pi$$
0.839362 + 0.543573i $$0.182929\pi$$
$$72$$ −184.780 −0.302452
$$73$$ 1007.29 1.61499 0.807494 0.589876i $$-0.200823\pi$$
0.807494 + 0.589876i $$0.200823\pi$$
$$74$$ −875.055 −1.37464
$$75$$ 0 0
$$76$$ −68.1168 −0.102810
$$77$$ −133.436 −0.197486
$$78$$ 39.9339 0.0579695
$$79$$ −863.237 −1.22939 −0.614695 0.788765i $$-0.710721\pi$$
−0.614695 + 0.788765i $$0.710721\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ −1427.82 −1.92288
$$83$$ −1334.72 −1.76512 −0.882560 0.470200i $$-0.844182\pi$$
−0.882560 + 0.470200i $$0.844182\pi$$
$$84$$ −263.154 −0.341815
$$85$$ 0 0
$$86$$ −1363.85 −1.71009
$$87$$ 874.483 1.07764
$$88$$ 391.370 0.474093
$$89$$ 326.249 0.388565 0.194283 0.980946i $$-0.437762\pi$$
0.194283 + 0.980946i $$0.437762\pi$$
$$90$$ 0 0
$$91$$ 20.5642 0.0236892
$$92$$ −618.755 −0.701192
$$93$$ −732.922 −0.817210
$$94$$ 391.969 0.430091
$$95$$ 0 0
$$96$$ −590.889 −0.628202
$$97$$ −1526.77 −1.59815 −0.799075 0.601232i $$-0.794677\pi$$
−0.799075 + 0.601232i $$0.794677\pi$$
$$98$$ −222.025 −0.228857
$$99$$ −171.560 −0.174166
$$100$$ 0 0
$$101$$ 96.8716 0.0954365 0.0477182 0.998861i $$-0.484805\pi$$
0.0477182 + 0.998861i $$0.484805\pi$$
$$102$$ 88.3307 0.0857455
$$103$$ −1321.99 −1.26466 −0.632329 0.774700i $$-0.717901\pi$$
−0.632329 + 0.774700i $$0.717901\pi$$
$$104$$ −60.3152 −0.0568691
$$105$$ 0 0
$$106$$ 2309.41 2.11613
$$107$$ −1745.71 −1.57724 −0.788619 0.614883i $$-0.789203\pi$$
−0.788619 + 0.614883i $$0.789203\pi$$
$$108$$ −338.340 −0.301452
$$109$$ 476.856 0.419032 0.209516 0.977805i $$-0.432811\pi$$
0.209516 + 0.977805i $$0.432811\pi$$
$$110$$ 0 0
$$111$$ −579.362 −0.495411
$$112$$ −50.5389 −0.0426382
$$113$$ −1641.65 −1.36666 −0.683332 0.730108i $$-0.739470\pi$$
−0.683332 + 0.730108i $$0.739470\pi$$
$$114$$ −73.8910 −0.0607064
$$115$$ 0 0
$$116$$ −3652.75 −2.92370
$$117$$ 26.4397 0.0208919
$$118$$ 377.829 0.294763
$$119$$ 45.4864 0.0350398
$$120$$ 0 0
$$121$$ −967.630 −0.726995
$$122$$ 23.8016 0.0176631
$$123$$ −945.339 −0.692994
$$124$$ 3061.45 2.21715
$$125$$ 0 0
$$126$$ −285.461 −0.201833
$$127$$ 844.016 0.589719 0.294859 0.955541i $$-0.404727\pi$$
0.294859 + 0.955541i $$0.404727\pi$$
$$128$$ 2206.46 1.52363
$$129$$ −902.988 −0.616308
$$130$$ 0 0
$$131$$ −2796.20 −1.86493 −0.932463 0.361265i $$-0.882345\pi$$
−0.932463 + 0.361265i $$0.882345\pi$$
$$132$$ 716.615 0.472525
$$133$$ −38.0506 −0.0248076
$$134$$ 933.377 0.601728
$$135$$ 0 0
$$136$$ −133.413 −0.0841179
$$137$$ 2057.13 1.28287 0.641433 0.767179i $$-0.278340\pi$$
0.641433 + 0.767179i $$0.278340\pi$$
$$138$$ −671.206 −0.414035
$$139$$ −1745.12 −1.06489 −0.532444 0.846465i $$-0.678726\pi$$
−0.532444 + 0.846465i $$0.678726\pi$$
$$140$$ 0 0
$$141$$ 259.517 0.155002
$$142$$ −4550.65 −2.68931
$$143$$ −56.0000 −0.0327479
$$144$$ −64.9786 −0.0376033
$$145$$ 0 0
$$146$$ −4564.15 −2.58720
$$147$$ −147.000 −0.0824786
$$148$$ 2420.02 1.34408
$$149$$ −1173.57 −0.645254 −0.322627 0.946526i $$-0.604566\pi$$
−0.322627 + 0.946526i $$0.604566\pi$$
$$150$$ 0 0
$$151$$ 1540.07 0.829994 0.414997 0.909823i $$-0.363783\pi$$
0.414997 + 0.909823i $$0.363783\pi$$
$$152$$ 111.603 0.0595540
$$153$$ 58.4826 0.0309022
$$154$$ 604.615 0.316372
$$155$$ 0 0
$$156$$ −110.440 −0.0566811
$$157$$ 2544.53 1.29348 0.646738 0.762712i $$-0.276133\pi$$
0.646738 + 0.762712i $$0.276133\pi$$
$$158$$ 3911.44 1.96948
$$159$$ 1529.03 0.762642
$$160$$ 0 0
$$161$$ −345.642 −0.169195
$$162$$ −367.021 −0.178000
$$163$$ 594.708 0.285774 0.142887 0.989739i $$-0.454361\pi$$
0.142887 + 0.989739i $$0.454361\pi$$
$$164$$ 3948.72 1.88014
$$165$$ 0 0
$$166$$ 6047.81 2.82772
$$167$$ −928.498 −0.430236 −0.215118 0.976588i $$-0.569014\pi$$
−0.215118 + 0.976588i $$0.569014\pi$$
$$168$$ 431.154 0.198001
$$169$$ −2188.37 −0.996072
$$170$$ 0 0
$$171$$ −48.9222 −0.0218782
$$172$$ 3771.82 1.67209
$$173$$ 315.642 0.138716 0.0693578 0.997592i $$-0.477905\pi$$
0.0693578 + 0.997592i $$0.477905\pi$$
$$174$$ −3962.39 −1.72637
$$175$$ 0 0
$$176$$ 137.626 0.0589431
$$177$$ 250.156 0.106231
$$178$$ −1478.28 −0.622480
$$179$$ −1445.49 −0.603581 −0.301791 0.953374i $$-0.597584\pi$$
−0.301791 + 0.953374i $$0.597584\pi$$
$$180$$ 0 0
$$181$$ −1843.81 −0.757180 −0.378590 0.925564i $$-0.623591\pi$$
−0.378590 + 0.925564i $$0.623591\pi$$
$$182$$ −93.1790 −0.0379499
$$183$$ 15.7587 0.00636567
$$184$$ 1013.77 0.406176
$$185$$ 0 0
$$186$$ 3320.97 1.30917
$$187$$ −123.868 −0.0484391
$$188$$ −1084.02 −0.420532
$$189$$ −189.000 −0.0727393
$$190$$ 0 0
$$191$$ −244.074 −0.0924637 −0.0462318 0.998931i $$-0.514721\pi$$
−0.0462318 + 0.998931i $$0.514721\pi$$
$$192$$ 2504.12 0.941246
$$193$$ 1733.03 0.646355 0.323178 0.946338i $$-0.395249\pi$$
0.323178 + 0.946338i $$0.395249\pi$$
$$194$$ 6918.01 2.56023
$$195$$ 0 0
$$196$$ 614.025 0.223770
$$197$$ −358.230 −0.129557 −0.0647787 0.997900i $$-0.520634\pi$$
−0.0647787 + 0.997900i $$0.520634\pi$$
$$198$$ 777.362 0.279014
$$199$$ −3203.63 −1.14120 −0.570601 0.821227i $$-0.693290\pi$$
−0.570601 + 0.821227i $$0.693290\pi$$
$$200$$ 0 0
$$201$$ 617.977 0.216859
$$202$$ −438.938 −0.152889
$$203$$ −2040.46 −0.705479
$$204$$ −244.284 −0.0838398
$$205$$ 0 0
$$206$$ 5990.12 2.02598
$$207$$ −444.397 −0.149216
$$208$$ −21.2100 −0.00707044
$$209$$ 103.619 0.0342940
$$210$$ 0 0
$$211$$ 4943.16 1.61280 0.806401 0.591369i $$-0.201412\pi$$
0.806401 + 0.591369i $$0.201412\pi$$
$$212$$ −6386.83 −2.06910
$$213$$ −3012.92 −0.969211
$$214$$ 7910.05 2.52673
$$215$$ 0 0
$$216$$ 554.340 0.174621
$$217$$ 1710.15 0.534989
$$218$$ −2160.70 −0.671288
$$219$$ −3021.86 −0.932414
$$220$$ 0 0
$$221$$ 19.0896 0.00581044
$$222$$ 2625.16 0.793646
$$223$$ −3160.15 −0.948965 −0.474482 0.880265i $$-0.657365\pi$$
−0.474482 + 0.880265i $$0.657365\pi$$
$$224$$ 1378.74 0.411255
$$225$$ 0 0
$$226$$ 7438.51 2.18939
$$227$$ 3651.11 1.06755 0.533773 0.845628i $$-0.320774\pi$$
0.533773 + 0.845628i $$0.320774\pi$$
$$228$$ 204.350 0.0593571
$$229$$ −4083.70 −1.17842 −0.589210 0.807980i $$-0.700561\pi$$
−0.589210 + 0.807980i $$0.700561\pi$$
$$230$$ 0 0
$$231$$ 400.307 0.114019
$$232$$ 5984.70 1.69360
$$233$$ 3682.51 1.03540 0.517702 0.855561i $$-0.326788\pi$$
0.517702 + 0.855561i $$0.326788\pi$$
$$234$$ −119.802 −0.0334687
$$235$$ 0 0
$$236$$ −1044.91 −0.288211
$$237$$ 2589.71 0.709789
$$238$$ −206.105 −0.0561336
$$239$$ 2658.78 0.719591 0.359796 0.933031i $$-0.382846\pi$$
0.359796 + 0.933031i $$0.382846\pi$$
$$240$$ 0 0
$$241$$ −4820.39 −1.28842 −0.644209 0.764850i $$-0.722813\pi$$
−0.644209 + 0.764850i $$0.722813\pi$$
$$242$$ 4384.46 1.16464
$$243$$ −243.000 −0.0641500
$$244$$ −65.8249 −0.0172705
$$245$$ 0 0
$$246$$ 4283.45 1.11017
$$247$$ −15.9690 −0.00411369
$$248$$ −5015.91 −1.28432
$$249$$ 4004.17 1.01909
$$250$$ 0 0
$$251$$ 1672.27 0.420530 0.210265 0.977644i $$-0.432567\pi$$
0.210265 + 0.977644i $$0.432567\pi$$
$$252$$ 789.461 0.197347
$$253$$ 941.245 0.233896
$$254$$ −3824.34 −0.944727
$$255$$ 0 0
$$256$$ −3320.09 −0.810569
$$257$$ 3697.74 0.897506 0.448753 0.893656i $$-0.351868\pi$$
0.448753 + 0.893656i $$0.351868\pi$$
$$258$$ 4091.56 0.987322
$$259$$ 1351.84 0.324323
$$260$$ 0 0
$$261$$ −2623.45 −0.622174
$$262$$ 12670.0 2.98760
$$263$$ −7319.00 −1.71600 −0.858002 0.513646i $$-0.828294\pi$$
−0.858002 + 0.513646i $$0.828294\pi$$
$$264$$ −1174.11 −0.273717
$$265$$ 0 0
$$266$$ 172.412 0.0397416
$$267$$ −978.747 −0.224338
$$268$$ −2581.32 −0.588354
$$269$$ −815.097 −0.184749 −0.0923743 0.995724i $$-0.529446\pi$$
−0.0923743 + 0.995724i $$0.529446\pi$$
$$270$$ 0 0
$$271$$ −5106.02 −1.14453 −0.572267 0.820068i $$-0.693936\pi$$
−0.572267 + 0.820068i $$0.693936\pi$$
$$272$$ −46.9150 −0.0104582
$$273$$ −61.6926 −0.0136769
$$274$$ −9321.13 −2.05515
$$275$$ 0 0
$$276$$ 1856.26 0.404833
$$277$$ −1398.72 −0.303398 −0.151699 0.988427i $$-0.548474\pi$$
−0.151699 + 0.988427i $$0.548474\pi$$
$$278$$ 7907.39 1.70595
$$279$$ 2198.77 0.471816
$$280$$ 0 0
$$281$$ −7102.38 −1.50780 −0.753901 0.656988i $$-0.771830\pi$$
−0.753901 + 0.656988i $$0.771830\pi$$
$$282$$ −1175.91 −0.248313
$$283$$ −4465.18 −0.937907 −0.468953 0.883223i $$-0.655369\pi$$
−0.468953 + 0.883223i $$0.655369\pi$$
$$284$$ 12585.1 2.62954
$$285$$ 0 0
$$286$$ 253.743 0.0524621
$$287$$ 2205.79 0.453671
$$288$$ 1772.67 0.362692
$$289$$ −4870.78 −0.991405
$$290$$ 0 0
$$291$$ 4580.32 0.922692
$$292$$ 12622.5 2.52970
$$293$$ −7590.61 −1.51348 −0.756738 0.653718i $$-0.773208\pi$$
−0.756738 + 0.653718i $$0.773208\pi$$
$$294$$ 666.076 0.132130
$$295$$ 0 0
$$296$$ −3964.98 −0.778581
$$297$$ 514.681 0.100555
$$298$$ 5317.61 1.03369
$$299$$ −145.058 −0.0280566
$$300$$ 0 0
$$301$$ 2106.97 0.403468
$$302$$ −6978.26 −1.32965
$$303$$ −290.615 −0.0551003
$$304$$ 39.2456 0.00740425
$$305$$ 0 0
$$306$$ −264.992 −0.0495052
$$307$$ −9480.12 −1.76241 −0.881203 0.472737i $$-0.843266\pi$$
−0.881203 + 0.472737i $$0.843266\pi$$
$$308$$ −1672.10 −0.309340
$$309$$ 3965.98 0.730151
$$310$$ 0 0
$$311$$ 7078.01 1.29054 0.645268 0.763956i $$-0.276746\pi$$
0.645268 + 0.763956i $$0.276746\pi$$
$$312$$ 180.945 0.0328334
$$313$$ −5593.84 −1.01017 −0.505084 0.863070i $$-0.668539\pi$$
−0.505084 + 0.863070i $$0.668539\pi$$
$$314$$ −11529.6 −2.07214
$$315$$ 0 0
$$316$$ −10817.3 −1.92571
$$317$$ −3567.81 −0.632139 −0.316070 0.948736i $$-0.602363\pi$$
−0.316070 + 0.948736i $$0.602363\pi$$
$$318$$ −6928.24 −1.22175
$$319$$ 5556.54 0.975255
$$320$$ 0 0
$$321$$ 5237.14 0.910618
$$322$$ 1566.15 0.271050
$$323$$ −35.3222 −0.00608477
$$324$$ 1015.02 0.174043
$$325$$ 0 0
$$326$$ −2694.70 −0.457809
$$327$$ −1430.57 −0.241928
$$328$$ −6469.62 −1.08910
$$329$$ −605.541 −0.101473
$$330$$ 0 0
$$331$$ −4389.67 −0.728936 −0.364468 0.931216i $$-0.618749\pi$$
−0.364468 + 0.931216i $$0.618749\pi$$
$$332$$ −16725.6 −2.76487
$$333$$ 1738.09 0.286026
$$334$$ 4207.14 0.689236
$$335$$ 0 0
$$336$$ 151.617 0.0246172
$$337$$ −2348.83 −0.379671 −0.189835 0.981816i $$-0.560795\pi$$
−0.189835 + 0.981816i $$0.560795\pi$$
$$338$$ 9915.79 1.59570
$$339$$ 4924.94 0.789044
$$340$$ 0 0
$$341$$ −4657.05 −0.739570
$$342$$ 221.673 0.0350488
$$343$$ 343.000 0.0539949
$$344$$ −6179.79 −0.968581
$$345$$ 0 0
$$346$$ −1430.21 −0.222222
$$347$$ −558.436 −0.0863931 −0.0431965 0.999067i $$-0.513754\pi$$
−0.0431965 + 0.999067i $$0.513754\pi$$
$$348$$ 10958.3 1.68800
$$349$$ 3233.89 0.496006 0.248003 0.968759i $$-0.420226\pi$$
0.248003 + 0.968759i $$0.420226\pi$$
$$350$$ 0 0
$$351$$ −79.3190 −0.0120619
$$352$$ −3754.56 −0.568519
$$353$$ 7516.35 1.13330 0.566650 0.823959i $$-0.308239\pi$$
0.566650 + 0.823959i $$0.308239\pi$$
$$354$$ −1133.49 −0.170181
$$355$$ 0 0
$$356$$ 4088.27 0.608646
$$357$$ −136.459 −0.0202302
$$358$$ 6549.70 0.966935
$$359$$ −6577.76 −0.967021 −0.483511 0.875338i $$-0.660639\pi$$
−0.483511 + 0.875338i $$0.660639\pi$$
$$360$$ 0 0
$$361$$ −6829.45 −0.995692
$$362$$ 8354.56 1.21300
$$363$$ 2902.89 0.419731
$$364$$ 257.693 0.0371065
$$365$$ 0 0
$$366$$ −71.4048 −0.0101978
$$367$$ −8307.17 −1.18155 −0.590777 0.806835i $$-0.701179\pi$$
−0.590777 + 0.806835i $$0.701179\pi$$
$$368$$ 356.497 0.0504992
$$369$$ 2836.02 0.400101
$$370$$ 0 0
$$371$$ −3567.74 −0.499266
$$372$$ −9184.34 −1.28007
$$373$$ 4551.09 0.631760 0.315880 0.948799i $$-0.397700\pi$$
0.315880 + 0.948799i $$0.397700\pi$$
$$374$$ 561.261 0.0775992
$$375$$ 0 0
$$376$$ 1776.06 0.243599
$$377$$ −856.335 −0.116985
$$378$$ 856.383 0.116528
$$379$$ −1788.29 −0.242370 −0.121185 0.992630i $$-0.538669\pi$$
−0.121185 + 0.992630i $$0.538669\pi$$
$$380$$ 0 0
$$381$$ −2532.05 −0.340474
$$382$$ 1105.93 0.148126
$$383$$ 1358.47 0.181240 0.0906199 0.995886i $$-0.471115\pi$$
0.0906199 + 0.995886i $$0.471115\pi$$
$$384$$ −6619.37 −0.879670
$$385$$ 0 0
$$386$$ −7852.60 −1.03546
$$387$$ 2708.97 0.355825
$$388$$ −19132.2 −2.50333
$$389$$ 9722.54 1.26723 0.633615 0.773649i $$-0.281570\pi$$
0.633615 + 0.773649i $$0.281570\pi$$
$$390$$ 0 0
$$391$$ −320.858 −0.0414999
$$392$$ −1006.03 −0.129622
$$393$$ 8388.61 1.07672
$$394$$ 1623.19 0.207551
$$395$$ 0 0
$$396$$ −2149.84 −0.272812
$$397$$ 4788.04 0.605302 0.302651 0.953101i $$-0.402128\pi$$
0.302651 + 0.953101i $$0.402128\pi$$
$$398$$ 14516.1 1.82820
$$399$$ 114.152 0.0143227
$$400$$ 0 0
$$401$$ 9681.41 1.20565 0.602826 0.797873i $$-0.294041\pi$$
0.602826 + 0.797873i $$0.294041\pi$$
$$402$$ −2800.13 −0.347408
$$403$$ 717.712 0.0887141
$$404$$ 1213.91 0.149491
$$405$$ 0 0
$$406$$ 9245.58 1.13017
$$407$$ −3681.32 −0.448344
$$408$$ 400.238 0.0485655
$$409$$ 11113.1 1.34353 0.671767 0.740763i $$-0.265536\pi$$
0.671767 + 0.740763i $$0.265536\pi$$
$$410$$ 0 0
$$411$$ −6171.40 −0.740663
$$412$$ −16566.1 −1.98095
$$413$$ −583.696 −0.0695443
$$414$$ 2013.62 0.239043
$$415$$ 0 0
$$416$$ 578.627 0.0681959
$$417$$ 5235.37 0.614814
$$418$$ −469.510 −0.0549389
$$419$$ −1230.09 −0.143421 −0.0717107 0.997425i $$-0.522846\pi$$
−0.0717107 + 0.997425i $$0.522846\pi$$
$$420$$ 0 0
$$421$$ −12356.5 −1.43044 −0.715222 0.698897i $$-0.753674\pi$$
−0.715222 + 0.698897i $$0.753674\pi$$
$$422$$ −22398.1 −2.58370
$$423$$ −778.552 −0.0894906
$$424$$ 10464.2 1.19856
$$425$$ 0 0
$$426$$ 13651.9 1.55267
$$427$$ −36.7703 −0.00416731
$$428$$ −21875.7 −2.47057
$$429$$ 168.000 0.0189070
$$430$$ 0 0
$$431$$ −7375.27 −0.824256 −0.412128 0.911126i $$-0.635214\pi$$
−0.412128 + 0.911126i $$0.635214\pi$$
$$432$$ 194.936 0.0217103
$$433$$ 690.067 0.0765877 0.0382939 0.999267i $$-0.487808\pi$$
0.0382939 + 0.999267i $$0.487808\pi$$
$$434$$ −7748.92 −0.857051
$$435$$ 0 0
$$436$$ 5975.55 0.656369
$$437$$ 268.406 0.0293812
$$438$$ 13692.5 1.49372
$$439$$ 8408.79 0.914191 0.457095 0.889418i $$-0.348890\pi$$
0.457095 + 0.889418i $$0.348890\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ −86.4976 −0.00930830
$$443$$ −6568.55 −0.704473 −0.352236 0.935911i $$-0.614579\pi$$
−0.352236 + 0.935911i $$0.614579\pi$$
$$444$$ −7260.06 −0.776007
$$445$$ 0 0
$$446$$ 14319.0 1.52024
$$447$$ 3520.72 0.372537
$$448$$ −5842.94 −0.616190
$$449$$ 2954.55 0.310543 0.155271 0.987872i $$-0.450375\pi$$
0.155271 + 0.987872i $$0.450375\pi$$
$$450$$ 0 0
$$451$$ −6006.76 −0.627156
$$452$$ −20571.7 −2.14073
$$453$$ −4620.21 −0.479197
$$454$$ −16543.7 −1.71020
$$455$$ 0 0
$$456$$ −334.810 −0.0343835
$$457$$ 8144.84 0.833697 0.416849 0.908976i $$-0.363135\pi$$
0.416849 + 0.908976i $$0.363135\pi$$
$$458$$ 18503.8 1.88782
$$459$$ −175.448 −0.0178414
$$460$$ 0 0
$$461$$ 2495.26 0.252095 0.126048 0.992024i $$-0.459771\pi$$
0.126048 + 0.992024i $$0.459771\pi$$
$$462$$ −1813.84 −0.182657
$$463$$ 5755.66 0.577728 0.288864 0.957370i $$-0.406722\pi$$
0.288864 + 0.957370i $$0.406722\pi$$
$$464$$ 2104.54 0.210562
$$465$$ 0 0
$$466$$ −16685.9 −1.65871
$$467$$ 4143.73 0.410598 0.205299 0.978699i $$-0.434183\pi$$
0.205299 + 0.978699i $$0.434183\pi$$
$$468$$ 331.319 0.0327248
$$469$$ −1441.95 −0.141968
$$470$$ 0 0
$$471$$ −7633.60 −0.746789
$$472$$ 1711.99 0.166951
$$473$$ −5737.67 −0.557755
$$474$$ −11734.3 −1.13708
$$475$$ 0 0
$$476$$ 569.996 0.0548860
$$477$$ −4587.09 −0.440312
$$478$$ −12047.3 −1.15278
$$479$$ −6765.96 −0.645396 −0.322698 0.946502i $$-0.604590\pi$$
−0.322698 + 0.946502i $$0.604590\pi$$
$$480$$ 0 0
$$481$$ 567.339 0.0537805
$$482$$ 21841.8 2.06404
$$483$$ 1036.93 0.0976848
$$484$$ −12125.5 −1.13876
$$485$$ 0 0
$$486$$ 1101.06 0.102768
$$487$$ −6360.42 −0.591824 −0.295912 0.955215i $$-0.595623\pi$$
−0.295912 + 0.955215i $$0.595623\pi$$
$$488$$ 107.848 0.0100042
$$489$$ −1784.12 −0.164992
$$490$$ 0 0
$$491$$ −7072.54 −0.650060 −0.325030 0.945704i $$-0.605374\pi$$
−0.325030 + 0.945704i $$0.605374\pi$$
$$492$$ −11846.2 −1.08550
$$493$$ −1894.15 −0.173039
$$494$$ 72.3576 0.00659012
$$495$$ 0 0
$$496$$ −1763.86 −0.159677
$$497$$ 7030.15 0.634498
$$498$$ −18143.4 −1.63258
$$499$$ −18473.9 −1.65732 −0.828661 0.559751i $$-0.810897\pi$$
−0.828661 + 0.559751i $$0.810897\pi$$
$$500$$ 0 0
$$501$$ 2785.49 0.248397
$$502$$ −7577.28 −0.673687
$$503$$ −11379.2 −1.00869 −0.504347 0.863501i $$-0.668267\pi$$
−0.504347 + 0.863501i $$0.668267\pi$$
$$504$$ −1293.46 −0.114316
$$505$$ 0 0
$$506$$ −4264.90 −0.374700
$$507$$ 6565.11 0.575082
$$508$$ 10576.5 0.923730
$$509$$ 6064.48 0.528101 0.264051 0.964509i $$-0.414941\pi$$
0.264051 + 0.964509i $$0.414941\pi$$
$$510$$ 0 0
$$511$$ 7051.02 0.610408
$$512$$ −2607.89 −0.225105
$$513$$ 146.767 0.0126314
$$514$$ −16755.0 −1.43780
$$515$$ 0 0
$$516$$ −11315.5 −0.965379
$$517$$ 1649.00 0.140276
$$518$$ −6125.38 −0.519563
$$519$$ −946.926 −0.0800875
$$520$$ 0 0
$$521$$ 2682.88 0.225603 0.112801 0.993618i $$-0.464018\pi$$
0.112801 + 0.993618i $$0.464018\pi$$
$$522$$ 11887.2 0.996720
$$523$$ 4309.02 0.360268 0.180134 0.983642i $$-0.442347\pi$$
0.180134 + 0.983642i $$0.442347\pi$$
$$524$$ −35039.6 −2.92120
$$525$$ 0 0
$$526$$ 33163.3 2.74903
$$527$$ 1587.52 0.131221
$$528$$ −412.879 −0.0340308
$$529$$ −9728.87 −0.799611
$$530$$ 0 0
$$531$$ −750.467 −0.0613323
$$532$$ −476.817 −0.0388584
$$533$$ 925.720 0.0752296
$$534$$ 4434.83 0.359389
$$535$$ 0 0
$$536$$ 4229.25 0.340813
$$537$$ 4336.47 0.348478
$$538$$ 3693.31 0.295966
$$539$$ −934.051 −0.0746427
$$540$$ 0 0
$$541$$ 4081.47 0.324355 0.162178 0.986762i $$-0.448148\pi$$
0.162178 + 0.986762i $$0.448148\pi$$
$$542$$ 23136.0 1.83354
$$543$$ 5531.44 0.437158
$$544$$ 1279.88 0.100872
$$545$$ 0 0
$$546$$ 279.537 0.0219104
$$547$$ −8844.82 −0.691366 −0.345683 0.938351i $$-0.612353\pi$$
−0.345683 + 0.938351i $$0.612353\pi$$
$$548$$ 25778.2 2.00947
$$549$$ −47.2762 −0.00367522
$$550$$ 0 0
$$551$$ 1584.51 0.122509
$$552$$ −3041.32 −0.234506
$$553$$ −6042.66 −0.464666
$$554$$ 6337.80 0.486042
$$555$$ 0 0
$$556$$ −21868.4 −1.66803
$$557$$ 11144.7 0.847787 0.423894 0.905712i $$-0.360663\pi$$
0.423894 + 0.905712i $$0.360663\pi$$
$$558$$ −9962.90 −0.755848
$$559$$ 884.249 0.0669047
$$560$$ 0 0
$$561$$ 371.603 0.0279663
$$562$$ 32181.8 2.41549
$$563$$ 21857.5 1.63621 0.818104 0.575071i $$-0.195025\pi$$
0.818104 + 0.575071i $$0.195025\pi$$
$$564$$ 3252.05 0.242794
$$565$$ 0 0
$$566$$ 20232.3 1.50252
$$567$$ 567.000 0.0419961
$$568$$ −20619.6 −1.52320
$$569$$ 23496.4 1.73115 0.865573 0.500783i $$-0.166954\pi$$
0.865573 + 0.500783i $$0.166954\pi$$
$$570$$ 0 0
$$571$$ 11067.8 0.811164 0.405582 0.914059i $$-0.367069\pi$$
0.405582 + 0.914059i $$0.367069\pi$$
$$572$$ −701.743 −0.0512961
$$573$$ 732.222 0.0533839
$$574$$ −9994.72 −0.726780
$$575$$ 0 0
$$576$$ −7512.36 −0.543429
$$577$$ −20482.9 −1.47784 −0.738922 0.673791i $$-0.764665\pi$$
−0.738922 + 0.673791i $$0.764665\pi$$
$$578$$ 22070.1 1.58823
$$579$$ −5199.10 −0.373173
$$580$$ 0 0
$$581$$ −9343.07 −0.667153
$$582$$ −20754.0 −1.47815
$$583$$ 9715.60 0.690187
$$584$$ −20680.8 −1.46537
$$585$$ 0 0
$$586$$ 34394.1 2.42458
$$587$$ 23444.3 1.64847 0.824235 0.566248i $$-0.191606\pi$$
0.824235 + 0.566248i $$0.191606\pi$$
$$588$$ −1842.08 −0.129194
$$589$$ −1328.01 −0.0929025
$$590$$ 0 0
$$591$$ 1074.69 0.0748000
$$592$$ −1394.30 −0.0967996
$$593$$ −4404.69 −0.305024 −0.152512 0.988302i $$-0.548736\pi$$
−0.152512 + 0.988302i $$0.548736\pi$$
$$594$$ −2332.09 −0.161089
$$595$$ 0 0
$$596$$ −14706.2 −1.01072
$$597$$ 9610.89 0.658874
$$598$$ 657.277 0.0449466
$$599$$ 3327.05 0.226945 0.113472 0.993541i $$-0.463803\pi$$
0.113472 + 0.993541i $$0.463803\pi$$
$$600$$ 0 0
$$601$$ −14244.8 −0.966818 −0.483409 0.875395i $$-0.660602\pi$$
−0.483409 + 0.875395i $$0.660602\pi$$
$$602$$ −9546.97 −0.646354
$$603$$ −1853.93 −0.125204
$$604$$ 19298.8 1.30010
$$605$$ 0 0
$$606$$ 1316.81 0.0882704
$$607$$ −11446.5 −0.765402 −0.382701 0.923872i $$-0.625006\pi$$
−0.382701 + 0.923872i $$0.625006\pi$$
$$608$$ −1070.65 −0.0714156
$$609$$ 6121.38 0.407308
$$610$$ 0 0
$$611$$ −254.132 −0.0168266
$$612$$ 732.852 0.0484049
$$613$$ 19436.4 1.28063 0.640316 0.768111i $$-0.278803\pi$$
0.640316 + 0.768111i $$0.278803\pi$$
$$614$$ 42955.6 2.82337
$$615$$ 0 0
$$616$$ 2739.59 0.179190
$$617$$ 20530.1 1.33956 0.669781 0.742558i $$-0.266388\pi$$
0.669781 + 0.742558i $$0.266388\pi$$
$$618$$ −17970.4 −1.16970
$$619$$ 5833.35 0.378776 0.189388 0.981902i $$-0.439350\pi$$
0.189388 + 0.981902i $$0.439350\pi$$
$$620$$ 0 0
$$621$$ 1333.19 0.0861499
$$622$$ −32071.4 −2.06744
$$623$$ 2283.74 0.146864
$$624$$ 63.6301 0.00408212
$$625$$ 0 0
$$626$$ 25346.4 1.61829
$$627$$ −310.856 −0.0197997
$$628$$ 31885.9 2.02609
$$629$$ 1254.91 0.0795493
$$630$$ 0 0
$$631$$ 24776.6 1.56314 0.781568 0.623820i $$-0.214420\pi$$
0.781568 + 0.623820i $$0.214420\pi$$
$$632$$ 17723.2 1.11549
$$633$$ −14829.5 −0.931152
$$634$$ 16166.2 1.01268
$$635$$ 0 0
$$636$$ 19160.5 1.19460
$$637$$ 143.949 0.00895366
$$638$$ −25177.4 −1.56235
$$639$$ 9038.77 0.559574
$$640$$ 0 0
$$641$$ 27219.4 1.67723 0.838613 0.544728i $$-0.183367\pi$$
0.838613 + 0.544728i $$0.183367\pi$$
$$642$$ −23730.1 −1.45881
$$643$$ −7091.79 −0.434950 −0.217475 0.976066i $$-0.569782\pi$$
−0.217475 + 0.976066i $$0.569782\pi$$
$$644$$ −4331.28 −0.265026
$$645$$ 0 0
$$646$$ 160.049 0.00974777
$$647$$ −27773.0 −1.68758 −0.843792 0.536670i $$-0.819682\pi$$
−0.843792 + 0.536670i $$0.819682\pi$$
$$648$$ −1663.02 −0.100817
$$649$$ 1589.51 0.0961382
$$650$$ 0 0
$$651$$ −5130.46 −0.308876
$$652$$ 7452.37 0.447634
$$653$$ −21380.4 −1.28129 −0.640643 0.767839i $$-0.721332\pi$$
−0.640643 + 0.767839i $$0.721332\pi$$
$$654$$ 6482.09 0.387568
$$655$$ 0 0
$$656$$ −2275.06 −0.135406
$$657$$ 9065.59 0.538329
$$658$$ 2743.78 0.162559
$$659$$ −17232.3 −1.01863 −0.509315 0.860580i $$-0.670101\pi$$
−0.509315 + 0.860580i $$0.670101\pi$$
$$660$$ 0 0
$$661$$ 26577.7 1.56392 0.781962 0.623326i $$-0.214219\pi$$
0.781962 + 0.623326i $$0.214219\pi$$
$$662$$ 19890.1 1.16775
$$663$$ −57.2689 −0.00335466
$$664$$ 27403.4 1.60159
$$665$$ 0 0
$$666$$ −7875.49 −0.458212
$$667$$ 14393.2 0.835544
$$668$$ −11635.1 −0.673917
$$669$$ 9480.44 0.547885
$$670$$ 0 0
$$671$$ 100.132 0.00576090
$$672$$ −4136.22 −0.237438
$$673$$ 31695.2 1.81540 0.907698 0.419624i $$-0.137838\pi$$
0.907698 + 0.419624i $$0.137838\pi$$
$$674$$ 10642.9 0.608231
$$675$$ 0 0
$$676$$ −27422.7 −1.56024
$$677$$ −20440.3 −1.16039 −0.580195 0.814477i $$-0.697024\pi$$
−0.580195 + 0.814477i $$0.697024\pi$$
$$678$$ −22315.5 −1.26405
$$679$$ −10687.4 −0.604044
$$680$$ 0 0
$$681$$ −10953.3 −0.616348
$$682$$ 21101.7 1.18479
$$683$$ 22896.9 1.28276 0.641381 0.767223i $$-0.278362\pi$$
0.641381 + 0.767223i $$0.278362\pi$$
$$684$$ −613.051 −0.0342699
$$685$$ 0 0
$$686$$ −1554.18 −0.0864997
$$687$$ 12251.1 0.680361
$$688$$ −2173.14 −0.120422
$$689$$ −1497.30 −0.0827904
$$690$$ 0 0
$$691$$ −23764.0 −1.30829 −0.654143 0.756371i $$-0.726971\pi$$
−0.654143 + 0.756371i $$0.726971\pi$$
$$692$$ 3955.35 0.217283
$$693$$ −1200.92 −0.0658287
$$694$$ 2530.34 0.138401
$$695$$ 0 0
$$696$$ −17954.1 −0.977800
$$697$$ 2047.62 0.111276
$$698$$ −14653.2 −0.794600
$$699$$ −11047.5 −0.597791
$$700$$ 0 0
$$701$$ 26259.5 1.41485 0.707423 0.706791i $$-0.249858\pi$$
0.707423 + 0.706791i $$0.249858\pi$$
$$702$$ 359.405 0.0193232
$$703$$ −1049.77 −0.0563196
$$704$$ 15911.4 0.851822
$$705$$ 0 0
$$706$$ −34057.6 −1.81554
$$707$$ 678.101 0.0360716
$$708$$ 3134.73 0.166399
$$709$$ 12783.0 0.677116 0.338558 0.940945i $$-0.390061\pi$$
0.338558 + 0.940945i $$0.390061\pi$$
$$710$$ 0 0
$$711$$ −7769.14 −0.409797
$$712$$ −6698.26 −0.352567
$$713$$ −12063.3 −0.633623
$$714$$ 618.315 0.0324087
$$715$$ 0 0
$$716$$ −18113.6 −0.945444
$$717$$ −7976.35 −0.415456
$$718$$ 29804.7 1.54916
$$719$$ −27609.0 −1.43205 −0.716025 0.698075i $$-0.754040\pi$$
−0.716025 + 0.698075i $$0.754040\pi$$
$$720$$ 0 0
$$721$$ −9253.95 −0.477996
$$722$$ 30945.1 1.59509
$$723$$ 14461.2 0.743868
$$724$$ −23105.1 −1.18604
$$725$$ 0 0
$$726$$ −13153.4 −0.672407
$$727$$ −31306.2 −1.59709 −0.798544 0.601937i $$-0.794396\pi$$
−0.798544 + 0.601937i $$0.794396\pi$$
$$728$$ −422.206 −0.0214945
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 1955.89 0.0989621
$$732$$ 197.475 0.00997113
$$733$$ 15765.8 0.794441 0.397220 0.917723i $$-0.369975\pi$$
0.397220 + 0.917723i $$0.369975\pi$$
$$734$$ 37640.8 1.89285
$$735$$ 0 0
$$736$$ −9725.53 −0.487076
$$737$$ 3926.68 0.196256
$$738$$ −12850.4 −0.640959
$$739$$ 3966.51 0.197443 0.0987216 0.995115i $$-0.468525\pi$$
0.0987216 + 0.995115i $$0.468525\pi$$
$$740$$ 0 0
$$741$$ 47.9070 0.00237504
$$742$$ 16165.9 0.799823
$$743$$ −8224.50 −0.406094 −0.203047 0.979169i $$-0.565084\pi$$
−0.203047 + 0.979169i $$0.565084\pi$$
$$744$$ 15047.7 0.741500
$$745$$ 0 0
$$746$$ −20621.6 −1.01208
$$747$$ −12012.5 −0.588373
$$748$$ −1552.20 −0.0758745
$$749$$ −12220.0 −0.596140
$$750$$ 0 0
$$751$$ 18929.2 0.919754 0.459877 0.887983i $$-0.347894\pi$$
0.459877 + 0.887983i $$0.347894\pi$$
$$752$$ 624.558 0.0302863
$$753$$ −5016.82 −0.242793
$$754$$ 3880.16 0.187410
$$755$$ 0 0
$$756$$ −2368.38 −0.113938
$$757$$ 34906.8 1.67597 0.837984 0.545695i $$-0.183734\pi$$
0.837984 + 0.545695i $$0.183734\pi$$
$$758$$ 8102.96 0.388276
$$759$$ −2823.74 −0.135040
$$760$$ 0 0
$$761$$ −13683.4 −0.651803 −0.325902 0.945404i $$-0.605668\pi$$
−0.325902 + 0.945404i $$0.605668\pi$$
$$762$$ 11473.0 0.545438
$$763$$ 3337.99 0.158379
$$764$$ −3058.52 −0.144834
$$765$$ 0 0
$$766$$ −6155.42 −0.290345
$$767$$ −244.964 −0.0115321
$$768$$ 9960.28 0.467982
$$769$$ 41837.3 1.96189 0.980943 0.194294i $$-0.0622416\pi$$
0.980943 + 0.194294i $$0.0622416\pi$$
$$770$$ 0 0
$$771$$ −11093.2 −0.518175
$$772$$ 21716.9 1.01245
$$773$$ −19640.0 −0.913843 −0.456921 0.889507i $$-0.651048\pi$$
−0.456921 + 0.889507i $$0.651048\pi$$
$$774$$ −12274.7 −0.570031
$$775$$ 0 0
$$776$$ 31346.4 1.45009
$$777$$ −4055.53 −0.187248
$$778$$ −44054.1 −2.03010
$$779$$ −1712.89 −0.0787814
$$780$$ 0 0
$$781$$ −19144.4 −0.877131
$$782$$ 1453.85 0.0664827
$$783$$ 7870.34 0.359212
$$784$$ −353.772 −0.0161157
$$785$$ 0 0
$$786$$ −38009.9 −1.72489
$$787$$ −24935.3 −1.12941 −0.564705 0.825293i $$-0.691010\pi$$
−0.564705 + 0.825293i $$0.691010\pi$$
$$788$$ −4489.02 −0.202938
$$789$$ 21957.0 0.990736
$$790$$ 0 0
$$791$$ −11491.5 −0.516551
$$792$$ 3522.33 0.158031
$$793$$ −15.4317 −0.000691041 0
$$794$$ −21695.2 −0.969692
$$795$$ 0 0
$$796$$ −40145.1 −1.78757
$$797$$ 1168.33 0.0519251 0.0259625 0.999663i $$-0.491735\pi$$
0.0259625 + 0.999663i $$0.491735\pi$$
$$798$$ −517.237 −0.0229448
$$799$$ −562.120 −0.0248891
$$800$$ 0 0
$$801$$ 2936.24 0.129522
$$802$$ −43867.7 −1.93145
$$803$$ −19201.2 −0.843829
$$804$$ 7743.95 0.339686
$$805$$ 0 0
$$806$$ −3252.05 −0.142120
$$807$$ 2445.29 0.106665
$$808$$ −1988.88 −0.0865949
$$809$$ −35175.7 −1.52869 −0.764345 0.644807i $$-0.776938\pi$$
−0.764345 + 0.644807i $$0.776938\pi$$
$$810$$ 0 0
$$811$$ −15256.5 −0.660577 −0.330288 0.943880i $$-0.607146\pi$$
−0.330288 + 0.943880i $$0.607146\pi$$
$$812$$ −25569.3 −1.10506
$$813$$ 15318.1 0.660797
$$814$$ 16680.5 0.718245
$$815$$ 0 0
$$816$$ 140.745 0.00603806
$$817$$ −1636.16 −0.0700635
$$818$$ −50354.7 −2.15233
$$819$$ 185.078 0.00789639
$$820$$ 0 0
$$821$$ −15971.9 −0.678956 −0.339478 0.940614i $$-0.610250\pi$$
−0.339478 + 0.940614i $$0.610250\pi$$
$$822$$ 27963.4 1.18654
$$823$$ −2312.41 −0.0979409 −0.0489705 0.998800i $$-0.515594\pi$$
−0.0489705 + 0.998800i $$0.515594\pi$$
$$824$$ 27142.0 1.14750
$$825$$ 0 0
$$826$$ 2644.80 0.111410
$$827$$ −10422.4 −0.438238 −0.219119 0.975698i $$-0.570318\pi$$
−0.219119 + 0.975698i $$0.570318\pi$$
$$828$$ −5568.79 −0.233731
$$829$$ −13213.4 −0.553584 −0.276792 0.960930i $$-0.589271\pi$$
−0.276792 + 0.960930i $$0.589271\pi$$
$$830$$ 0 0
$$831$$ 4196.17 0.175167
$$832$$ −2452.15 −0.102179
$$833$$ 318.405 0.0132438
$$834$$ −23722.2 −0.984929
$$835$$ 0 0
$$836$$ 1298.46 0.0537179
$$837$$ −6596.30 −0.272403
$$838$$ 5573.67 0.229761
$$839$$ −10119.6 −0.416409 −0.208205 0.978085i $$-0.566762\pi$$
−0.208205 + 0.978085i $$0.566762\pi$$
$$840$$ 0 0
$$841$$ 60579.9 2.48390
$$842$$ 55988.7 2.29157
$$843$$ 21307.1 0.870530
$$844$$ 61943.4 2.52628
$$845$$ 0 0
$$846$$ 3527.72 0.143364
$$847$$ −6773.41 −0.274778
$$848$$ 3679.79 0.149015
$$849$$ 13395.5 0.541501
$$850$$ 0 0
$$851$$ −9535.80 −0.384116
$$852$$ −37755.3 −1.51816
$$853$$ −35378.1 −1.42007 −0.710037 0.704165i $$-0.751322\pi$$
−0.710037 + 0.704165i $$0.751322\pi$$
$$854$$ 166.611 0.00667602
$$855$$ 0 0
$$856$$ 35841.4 1.43112
$$857$$ −6697.57 −0.266960 −0.133480 0.991052i $$-0.542615\pi$$
−0.133480 + 0.991052i $$0.542615\pi$$
$$858$$ −761.230 −0.0302890
$$859$$ −24298.4 −0.965135 −0.482568 0.875859i $$-0.660296\pi$$
−0.482568 + 0.875859i $$0.660296\pi$$
$$860$$ 0 0
$$861$$ −6617.37 −0.261927
$$862$$ 33418.3 1.32045
$$863$$ −24942.9 −0.983853 −0.491926 0.870637i $$-0.663707\pi$$
−0.491926 + 0.870637i $$0.663707\pi$$
$$864$$ −5318.00 −0.209401
$$865$$ 0 0
$$866$$ −3126.78 −0.122693
$$867$$ 14612.3 0.572388
$$868$$ 21430.1 0.838002
$$869$$ 16455.3 0.642355
$$870$$ 0 0
$$871$$ −605.152 −0.0235417
$$872$$ −9790.39 −0.380212
$$873$$ −13741.0 −0.532716
$$874$$ −1216.18 −0.0470686
$$875$$ 0 0
$$876$$ −37867.4 −1.46052
$$877$$ 16276.6 0.626705 0.313353 0.949637i $$-0.398548\pi$$
0.313353 + 0.949637i $$0.398548\pi$$
$$878$$ −38101.3 −1.46453
$$879$$ 22771.8 0.873806
$$880$$ 0 0
$$881$$ 26636.5 1.01862 0.509311 0.860582i $$-0.329900\pi$$
0.509311 + 0.860582i $$0.329900\pi$$
$$882$$ −1998.23 −0.0762855
$$883$$ 21788.3 0.830392 0.415196 0.909732i $$-0.363713\pi$$
0.415196 + 0.909732i $$0.363713\pi$$
$$884$$ 239.215 0.00910142
$$885$$ 0 0
$$886$$ 29763.0 1.12856
$$887$$ 26813.2 1.01499 0.507496 0.861654i $$-0.330571\pi$$
0.507496 + 0.861654i $$0.330571\pi$$
$$888$$ 11895.0 0.449514
$$889$$ 5908.11 0.222893
$$890$$ 0 0
$$891$$ −1544.04 −0.0580554
$$892$$ −39600.2 −1.48645
$$893$$ 470.229 0.0176211
$$894$$ −15952.8 −0.596803
$$895$$ 0 0
$$896$$ 15445.2 0.575879
$$897$$ 435.174 0.0161985
$$898$$ −13387.4 −0.497488
$$899$$ −71214.2 −2.64196
$$900$$ 0 0
$$901$$ −3311.91 −0.122459
$$902$$ 27217.4 1.00470
$$903$$ −6320.92 −0.232942
$$904$$ 33704.8 1.24005
$$905$$ 0 0
$$906$$ 20934.8 0.767672
$$907$$ −15543.0 −0.569014 −0.284507 0.958674i $$-0.591830\pi$$
−0.284507 + 0.958674i $$0.591830\pi$$
$$908$$ 45752.6 1.67219
$$909$$ 871.844 0.0318122
$$910$$ 0 0
$$911$$ 48711.1 1.77154 0.885768 0.464128i $$-0.153632\pi$$
0.885768 + 0.464128i $$0.153632\pi$$
$$912$$ −117.737 −0.00427485
$$913$$ 25442.8 0.922273
$$914$$ −36905.3 −1.33558
$$915$$ 0 0
$$916$$ −51173.3 −1.84587
$$917$$ −19573.4 −0.704876
$$918$$ 794.976 0.0285818
$$919$$ 1030.47 0.0369883 0.0184941 0.999829i $$-0.494113\pi$$
0.0184941 + 0.999829i $$0.494113\pi$$
$$920$$ 0 0
$$921$$ 28440.4 1.01753
$$922$$ −11306.3 −0.403855
$$923$$ 2950.40 0.105215
$$924$$ 5016.30 0.178598
$$925$$ 0 0
$$926$$ −26079.6 −0.925518
$$927$$ −11897.9 −0.421553
$$928$$ −57413.6 −2.03092
$$929$$ 879.756 0.0310698 0.0155349 0.999879i $$-0.495055\pi$$
0.0155349 + 0.999879i $$0.495055\pi$$
$$930$$ 0 0
$$931$$ −266.354 −0.00937638
$$932$$ 46146.0 1.62185
$$933$$ −21234.0 −0.745092
$$934$$ −18775.8 −0.657776
$$935$$ 0 0
$$936$$ −542.836 −0.0189564
$$937$$ 18668.1 0.650864 0.325432 0.945565i $$-0.394490\pi$$
0.325432 + 0.945565i $$0.394490\pi$$
$$938$$ 6533.64 0.227432
$$939$$ 16781.5 0.583221
$$940$$ 0 0
$$941$$ 29613.4 1.02590 0.512948 0.858420i $$-0.328553\pi$$
0.512948 + 0.858420i $$0.328553\pi$$
$$942$$ 34588.8 1.19635
$$943$$ −15559.5 −0.537313
$$944$$ 602.028 0.0207567
$$945$$ 0 0
$$946$$ 25998.1 0.893521
$$947$$ −20738.9 −0.711640 −0.355820 0.934554i $$-0.615798\pi$$
−0.355820 + 0.934554i $$0.615798\pi$$
$$948$$ 32452.0 1.11181
$$949$$ 2959.15 0.101220
$$950$$ 0 0
$$951$$ 10703.4 0.364966
$$952$$ −933.888 −0.0317936
$$953$$ 45776.5 1.55598 0.777988 0.628279i $$-0.216240\pi$$
0.777988 + 0.628279i $$0.216240\pi$$
$$954$$ 20784.7 0.705377
$$955$$ 0 0
$$956$$ 33317.5 1.12716
$$957$$ −16669.6 −0.563064
$$958$$ 30657.4 1.03392
$$959$$ 14399.9 0.484878
$$960$$ 0 0
$$961$$ 29895.1 1.00349
$$962$$ −2570.68 −0.0861561
$$963$$ −15711.4 −0.525746
$$964$$ −60404.9 −2.01817
$$965$$ 0 0
$$966$$ −4698.44 −0.156491
$$967$$ −34461.0 −1.14601 −0.573005 0.819552i $$-0.694222\pi$$
−0.573005 + 0.819552i $$0.694222\pi$$
$$968$$ 19866.5 0.659643
$$969$$ 105.967 0.00351304
$$970$$ 0 0
$$971$$ −22762.8 −0.752309 −0.376154 0.926557i $$-0.622754\pi$$
−0.376154 + 0.926557i $$0.622754\pi$$
$$972$$ −3045.06 −0.100484
$$973$$ −12215.9 −0.402490
$$974$$ 28819.9 0.948099
$$975$$ 0 0
$$976$$ 37.9251 0.00124381
$$977$$ −4809.57 −0.157494 −0.0787470 0.996895i $$-0.525092\pi$$
−0.0787470 + 0.996895i $$0.525092\pi$$
$$978$$ 8084.10 0.264316
$$979$$ −6219.04 −0.203025
$$980$$ 0 0
$$981$$ 4291.70 0.139677
$$982$$ 32046.6 1.04139
$$983$$ 27591.6 0.895256 0.447628 0.894220i $$-0.352269\pi$$
0.447628 + 0.894220i $$0.352269\pi$$
$$984$$ 19408.9 0.628793
$$985$$ 0 0
$$986$$ 8582.63 0.277207
$$987$$ 1816.62 0.0585853
$$988$$ −200.109 −0.00644365
$$989$$ −14862.4 −0.477854
$$990$$ 0 0
$$991$$ −22263.4 −0.713643 −0.356822 0.934173i $$-0.616140\pi$$
−0.356822 + 0.934173i $$0.616140\pi$$
$$992$$ 48119.5 1.54012
$$993$$ 13169.0 0.420851
$$994$$ −31854.5 −1.01646
$$995$$ 0 0
$$996$$ 50176.8 1.59630
$$997$$ −30378.2 −0.964983 −0.482491 0.875901i $$-0.660268\pi$$
−0.482491 + 0.875901i $$0.660268\pi$$
$$998$$ 83707.4 2.65502
$$999$$ −5214.26 −0.165137
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 525.4.a.k.1.1 2
3.2 odd 2 1575.4.a.w.1.2 2
5.2 odd 4 525.4.d.h.274.1 4
5.3 odd 4 525.4.d.h.274.4 4
5.4 even 2 105.4.a.f.1.2 2
15.14 odd 2 315.4.a.i.1.1 2
20.19 odd 2 1680.4.a.bg.1.2 2
35.34 odd 2 735.4.a.p.1.2 2
105.104 even 2 2205.4.a.z.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.2 2 5.4 even 2
315.4.a.i.1.1 2 15.14 odd 2
525.4.a.k.1.1 2 1.1 even 1 trivial
525.4.d.h.274.1 4 5.2 odd 4
525.4.d.h.274.4 4 5.3 odd 4
735.4.a.p.1.2 2 35.34 odd 2
1575.4.a.w.1.2 2 3.2 odd 2
1680.4.a.bg.1.2 2 20.19 odd 2
2205.4.a.z.1.1 2 105.104 even 2