# Properties

 Label 825.4.a.f.1.1 Level $825$ Weight $4$ Character 825.1 Self dual yes Analytic conductor $48.677$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.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: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} +3.00000 q^{6} +26.0000 q^{7} -15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} +3.00000 q^{6} +26.0000 q^{7} -15.0000 q^{8} +9.00000 q^{9} +11.0000 q^{11} -21.0000 q^{12} +32.0000 q^{13} +26.0000 q^{14} +41.0000 q^{16} -74.0000 q^{17} +9.00000 q^{18} -60.0000 q^{19} +78.0000 q^{21} +11.0000 q^{22} +182.000 q^{23} -45.0000 q^{24} +32.0000 q^{26} +27.0000 q^{27} -182.000 q^{28} -90.0000 q^{29} -8.00000 q^{31} +161.000 q^{32} +33.0000 q^{33} -74.0000 q^{34} -63.0000 q^{36} +66.0000 q^{37} -60.0000 q^{38} +96.0000 q^{39} +422.000 q^{41} +78.0000 q^{42} -408.000 q^{43} -77.0000 q^{44} +182.000 q^{46} +506.000 q^{47} +123.000 q^{48} +333.000 q^{49} -222.000 q^{51} -224.000 q^{52} -348.000 q^{53} +27.0000 q^{54} -390.000 q^{56} -180.000 q^{57} -90.0000 q^{58} -200.000 q^{59} +132.000 q^{61} -8.00000 q^{62} +234.000 q^{63} -167.000 q^{64} +33.0000 q^{66} +1036.00 q^{67} +518.000 q^{68} +546.000 q^{69} +762.000 q^{71} -135.000 q^{72} +542.000 q^{73} +66.0000 q^{74} +420.000 q^{76} +286.000 q^{77} +96.0000 q^{78} -550.000 q^{79} +81.0000 q^{81} +422.000 q^{82} +132.000 q^{83} -546.000 q^{84} -408.000 q^{86} -270.000 q^{87} -165.000 q^{88} +570.000 q^{89} +832.000 q^{91} -1274.00 q^{92} -24.0000 q^{93} +506.000 q^{94} +483.000 q^{96} -14.0000 q^{97} +333.000 q^{98} +99.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.353553 0.176777 0.984251i $$-0.443433\pi$$
0.176777 + 0.984251i $$0.443433\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −7.00000 −0.875000
$$5$$ 0 0
$$6$$ 3.00000 0.204124
$$7$$ 26.0000 1.40387 0.701934 0.712242i $$-0.252320\pi$$
0.701934 + 0.712242i $$0.252320\pi$$
$$8$$ −15.0000 −0.662913
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ −21.0000 −0.505181
$$13$$ 32.0000 0.682708 0.341354 0.939935i $$-0.389115\pi$$
0.341354 + 0.939935i $$0.389115\pi$$
$$14$$ 26.0000 0.496342
$$15$$ 0 0
$$16$$ 41.0000 0.640625
$$17$$ −74.0000 −1.05574 −0.527872 0.849324i $$-0.677010\pi$$
−0.527872 + 0.849324i $$0.677010\pi$$
$$18$$ 9.00000 0.117851
$$19$$ −60.0000 −0.724471 −0.362235 0.932087i $$-0.617986\pi$$
−0.362235 + 0.932087i $$0.617986\pi$$
$$20$$ 0 0
$$21$$ 78.0000 0.810524
$$22$$ 11.0000 0.106600
$$23$$ 182.000 1.64998 0.824992 0.565145i $$-0.191180\pi$$
0.824992 + 0.565145i $$0.191180\pi$$
$$24$$ −45.0000 −0.382733
$$25$$ 0 0
$$26$$ 32.0000 0.241374
$$27$$ 27.0000 0.192450
$$28$$ −182.000 −1.22838
$$29$$ −90.0000 −0.576296 −0.288148 0.957586i $$-0.593039\pi$$
−0.288148 + 0.957586i $$0.593039\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −0.0463498 −0.0231749 0.999731i $$-0.507377\pi$$
−0.0231749 + 0.999731i $$0.507377\pi$$
$$32$$ 161.000 0.889408
$$33$$ 33.0000 0.174078
$$34$$ −74.0000 −0.373262
$$35$$ 0 0
$$36$$ −63.0000 −0.291667
$$37$$ 66.0000 0.293252 0.146626 0.989192i $$-0.453159\pi$$
0.146626 + 0.989192i $$0.453159\pi$$
$$38$$ −60.0000 −0.256139
$$39$$ 96.0000 0.394162
$$40$$ 0 0
$$41$$ 422.000 1.60745 0.803724 0.595003i $$-0.202849\pi$$
0.803724 + 0.595003i $$0.202849\pi$$
$$42$$ 78.0000 0.286563
$$43$$ −408.000 −1.44696 −0.723482 0.690344i $$-0.757459\pi$$
−0.723482 + 0.690344i $$0.757459\pi$$
$$44$$ −77.0000 −0.263822
$$45$$ 0 0
$$46$$ 182.000 0.583357
$$47$$ 506.000 1.57038 0.785188 0.619257i $$-0.212566\pi$$
0.785188 + 0.619257i $$0.212566\pi$$
$$48$$ 123.000 0.369865
$$49$$ 333.000 0.970845
$$50$$ 0 0
$$51$$ −222.000 −0.609534
$$52$$ −224.000 −0.597369
$$53$$ −348.000 −0.901915 −0.450957 0.892546i $$-0.648917\pi$$
−0.450957 + 0.892546i $$0.648917\pi$$
$$54$$ 27.0000 0.0680414
$$55$$ 0 0
$$56$$ −390.000 −0.930642
$$57$$ −180.000 −0.418273
$$58$$ −90.0000 −0.203751
$$59$$ −200.000 −0.441318 −0.220659 0.975351i $$-0.570821\pi$$
−0.220659 + 0.975351i $$0.570821\pi$$
$$60$$ 0 0
$$61$$ 132.000 0.277063 0.138532 0.990358i $$-0.455762\pi$$
0.138532 + 0.990358i $$0.455762\pi$$
$$62$$ −8.00000 −0.0163871
$$63$$ 234.000 0.467956
$$64$$ −167.000 −0.326172
$$65$$ 0 0
$$66$$ 33.0000 0.0615457
$$67$$ 1036.00 1.88907 0.944534 0.328414i $$-0.106514\pi$$
0.944534 + 0.328414i $$0.106514\pi$$
$$68$$ 518.000 0.923775
$$69$$ 546.000 0.952618
$$70$$ 0 0
$$71$$ 762.000 1.27370 0.636850 0.770987i $$-0.280237\pi$$
0.636850 + 0.770987i $$0.280237\pi$$
$$72$$ −135.000 −0.220971
$$73$$ 542.000 0.868990 0.434495 0.900674i $$-0.356927\pi$$
0.434495 + 0.900674i $$0.356927\pi$$
$$74$$ 66.0000 0.103680
$$75$$ 0 0
$$76$$ 420.000 0.633912
$$77$$ 286.000 0.423282
$$78$$ 96.0000 0.139357
$$79$$ −550.000 −0.783289 −0.391645 0.920117i $$-0.628094\pi$$
−0.391645 + 0.920117i $$0.628094\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 422.000 0.568318
$$83$$ 132.000 0.174565 0.0872824 0.996184i $$-0.472182\pi$$
0.0872824 + 0.996184i $$0.472182\pi$$
$$84$$ −546.000 −0.709208
$$85$$ 0 0
$$86$$ −408.000 −0.511579
$$87$$ −270.000 −0.332725
$$88$$ −165.000 −0.199876
$$89$$ 570.000 0.678875 0.339438 0.940629i $$-0.389763\pi$$
0.339438 + 0.940629i $$0.389763\pi$$
$$90$$ 0 0
$$91$$ 832.000 0.958432
$$92$$ −1274.00 −1.44374
$$93$$ −24.0000 −0.0267600
$$94$$ 506.000 0.555212
$$95$$ 0 0
$$96$$ 483.000 0.513500
$$97$$ −14.0000 −0.0146545 −0.00732724 0.999973i $$-0.502332\pi$$
−0.00732724 + 0.999973i $$0.502332\pi$$
$$98$$ 333.000 0.343246
$$99$$ 99.0000 0.100504
$$100$$ 0 0
$$101$$ 1702.00 1.67679 0.838393 0.545067i $$-0.183496\pi$$
0.838393 + 0.545067i $$0.183496\pi$$
$$102$$ −222.000 −0.215503
$$103$$ 1132.00 1.08291 0.541453 0.840731i $$-0.317874\pi$$
0.541453 + 0.840731i $$0.317874\pi$$
$$104$$ −480.000 −0.452576
$$105$$ 0 0
$$106$$ −348.000 −0.318875
$$107$$ −564.000 −0.509570 −0.254785 0.966998i $$-0.582005\pi$$
−0.254785 + 0.966998i $$0.582005\pi$$
$$108$$ −189.000 −0.168394
$$109$$ −320.000 −0.281197 −0.140598 0.990067i $$-0.544903\pi$$
−0.140598 + 0.990067i $$0.544903\pi$$
$$110$$ 0 0
$$111$$ 198.000 0.169309
$$112$$ 1066.00 0.899353
$$113$$ 2142.00 1.78321 0.891604 0.452817i $$-0.149581\pi$$
0.891604 + 0.452817i $$0.149581\pi$$
$$114$$ −180.000 −0.147882
$$115$$ 0 0
$$116$$ 630.000 0.504259
$$117$$ 288.000 0.227569
$$118$$ −200.000 −0.156030
$$119$$ −1924.00 −1.48212
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 132.000 0.0979567
$$123$$ 1266.00 0.928060
$$124$$ 56.0000 0.0405560
$$125$$ 0 0
$$126$$ 234.000 0.165447
$$127$$ 1606.00 1.12212 0.561061 0.827775i $$-0.310393\pi$$
0.561061 + 0.827775i $$0.310393\pi$$
$$128$$ −1455.00 −1.00473
$$129$$ −1224.00 −0.835405
$$130$$ 0 0
$$131$$ −1908.00 −1.27254 −0.636270 0.771466i $$-0.719524\pi$$
−0.636270 + 0.771466i $$0.719524\pi$$
$$132$$ −231.000 −0.152318
$$133$$ −1560.00 −1.01706
$$134$$ 1036.00 0.667886
$$135$$ 0 0
$$136$$ 1110.00 0.699866
$$137$$ 2186.00 1.36323 0.681615 0.731711i $$-0.261278\pi$$
0.681615 + 0.731711i $$0.261278\pi$$
$$138$$ 546.000 0.336801
$$139$$ 2740.00 1.67197 0.835985 0.548753i $$-0.184897\pi$$
0.835985 + 0.548753i $$0.184897\pi$$
$$140$$ 0 0
$$141$$ 1518.00 0.906657
$$142$$ 762.000 0.450321
$$143$$ 352.000 0.205844
$$144$$ 369.000 0.213542
$$145$$ 0 0
$$146$$ 542.000 0.307235
$$147$$ 999.000 0.560518
$$148$$ −462.000 −0.256596
$$149$$ −1310.00 −0.720264 −0.360132 0.932901i $$-0.617268\pi$$
−0.360132 + 0.932901i $$0.617268\pi$$
$$150$$ 0 0
$$151$$ −1198.00 −0.645641 −0.322821 0.946460i $$-0.604631\pi$$
−0.322821 + 0.946460i $$0.604631\pi$$
$$152$$ 900.000 0.480261
$$153$$ −666.000 −0.351914
$$154$$ 286.000 0.149653
$$155$$ 0 0
$$156$$ −672.000 −0.344891
$$157$$ −2114.00 −1.07462 −0.537311 0.843384i $$-0.680560\pi$$
−0.537311 + 0.843384i $$0.680560\pi$$
$$158$$ −550.000 −0.276934
$$159$$ −1044.00 −0.520721
$$160$$ 0 0
$$161$$ 4732.00 2.31636
$$162$$ 81.0000 0.0392837
$$163$$ −3868.00 −1.85868 −0.929341 0.369223i $$-0.879624\pi$$
−0.929341 + 0.369223i $$0.879624\pi$$
$$164$$ −2954.00 −1.40652
$$165$$ 0 0
$$166$$ 132.000 0.0617180
$$167$$ −2004.00 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ −1170.00 −0.537306
$$169$$ −1173.00 −0.533910
$$170$$ 0 0
$$171$$ −540.000 −0.241490
$$172$$ 2856.00 1.26609
$$173$$ −678.000 −0.297962 −0.148981 0.988840i $$-0.547599\pi$$
−0.148981 + 0.988840i $$0.547599\pi$$
$$174$$ −270.000 −0.117636
$$175$$ 0 0
$$176$$ 451.000 0.193156
$$177$$ −600.000 −0.254795
$$178$$ 570.000 0.240019
$$179$$ −1680.00 −0.701503 −0.350752 0.936469i $$-0.614074\pi$$
−0.350752 + 0.936469i $$0.614074\pi$$
$$180$$ 0 0
$$181$$ −4358.00 −1.78966 −0.894828 0.446412i $$-0.852702\pi$$
−0.894828 + 0.446412i $$0.852702\pi$$
$$182$$ 832.000 0.338857
$$183$$ 396.000 0.159963
$$184$$ −2730.00 −1.09379
$$185$$ 0 0
$$186$$ −24.0000 −0.00946110
$$187$$ −814.000 −0.318319
$$188$$ −3542.00 −1.37408
$$189$$ 702.000 0.270175
$$190$$ 0 0
$$191$$ −1778.00 −0.673568 −0.336784 0.941582i $$-0.609339\pi$$
−0.336784 + 0.941582i $$0.609339\pi$$
$$192$$ −501.000 −0.188315
$$193$$ 3962.00 1.47767 0.738837 0.673884i $$-0.235375\pi$$
0.738837 + 0.673884i $$0.235375\pi$$
$$194$$ −14.0000 −0.00518114
$$195$$ 0 0
$$196$$ −2331.00 −0.849490
$$197$$ −374.000 −0.135261 −0.0676304 0.997710i $$-0.521544\pi$$
−0.0676304 + 0.997710i $$0.521544\pi$$
$$198$$ 99.0000 0.0355335
$$199$$ 2100.00 0.748066 0.374033 0.927415i $$-0.377975\pi$$
0.374033 + 0.927415i $$0.377975\pi$$
$$200$$ 0 0
$$201$$ 3108.00 1.09065
$$202$$ 1702.00 0.592833
$$203$$ −2340.00 −0.809043
$$204$$ 1554.00 0.533342
$$205$$ 0 0
$$206$$ 1132.00 0.382865
$$207$$ 1638.00 0.549995
$$208$$ 1312.00 0.437360
$$209$$ −660.000 −0.218436
$$210$$ 0 0
$$211$$ 2232.00 0.728233 0.364117 0.931353i $$-0.381371\pi$$
0.364117 + 0.931353i $$0.381371\pi$$
$$212$$ 2436.00 0.789175
$$213$$ 2286.00 0.735372
$$214$$ −564.000 −0.180160
$$215$$ 0 0
$$216$$ −405.000 −0.127578
$$217$$ −208.000 −0.0650689
$$218$$ −320.000 −0.0994180
$$219$$ 1626.00 0.501712
$$220$$ 0 0
$$221$$ −2368.00 −0.720764
$$222$$ 198.000 0.0598599
$$223$$ −2128.00 −0.639020 −0.319510 0.947583i $$-0.603518\pi$$
−0.319510 + 0.947583i $$0.603518\pi$$
$$224$$ 4186.00 1.24861
$$225$$ 0 0
$$226$$ 2142.00 0.630459
$$227$$ −2964.00 −0.866641 −0.433321 0.901240i $$-0.642658\pi$$
−0.433321 + 0.901240i $$0.642658\pi$$
$$228$$ 1260.00 0.365989
$$229$$ −2550.00 −0.735846 −0.367923 0.929856i $$-0.619931\pi$$
−0.367923 + 0.929856i $$0.619931\pi$$
$$230$$ 0 0
$$231$$ 858.000 0.244382
$$232$$ 1350.00 0.382034
$$233$$ 3042.00 0.855314 0.427657 0.903941i $$-0.359339\pi$$
0.427657 + 0.903941i $$0.359339\pi$$
$$234$$ 288.000 0.0804579
$$235$$ 0 0
$$236$$ 1400.00 0.386154
$$237$$ −1650.00 −0.452232
$$238$$ −1924.00 −0.524010
$$239$$ 2700.00 0.730747 0.365373 0.930861i $$-0.380941\pi$$
0.365373 + 0.930861i $$0.380941\pi$$
$$240$$ 0 0
$$241$$ −578.000 −0.154491 −0.0772453 0.997012i $$-0.524612\pi$$
−0.0772453 + 0.997012i $$0.524612\pi$$
$$242$$ 121.000 0.0321412
$$243$$ 243.000 0.0641500
$$244$$ −924.000 −0.242430
$$245$$ 0 0
$$246$$ 1266.00 0.328119
$$247$$ −1920.00 −0.494602
$$248$$ 120.000 0.0307258
$$249$$ 396.000 0.100785
$$250$$ 0 0
$$251$$ 3752.00 0.943522 0.471761 0.881726i $$-0.343618\pi$$
0.471761 + 0.881726i $$0.343618\pi$$
$$252$$ −1638.00 −0.409462
$$253$$ 2002.00 0.497489
$$254$$ 1606.00 0.396730
$$255$$ 0 0
$$256$$ −119.000 −0.0290527
$$257$$ −674.000 −0.163591 −0.0817957 0.996649i $$-0.526065\pi$$
−0.0817957 + 0.996649i $$0.526065\pi$$
$$258$$ −1224.00 −0.295360
$$259$$ 1716.00 0.411687
$$260$$ 0 0
$$261$$ −810.000 −0.192099
$$262$$ −1908.00 −0.449911
$$263$$ 4352.00 1.02036 0.510182 0.860066i $$-0.329578\pi$$
0.510182 + 0.860066i $$0.329578\pi$$
$$264$$ −495.000 −0.115398
$$265$$ 0 0
$$266$$ −1560.00 −0.359585
$$267$$ 1710.00 0.391949
$$268$$ −7252.00 −1.65293
$$269$$ 500.000 0.113329 0.0566646 0.998393i $$-0.481953\pi$$
0.0566646 + 0.998393i $$0.481953\pi$$
$$270$$ 0 0
$$271$$ −6538.00 −1.46552 −0.732759 0.680489i $$-0.761768\pi$$
−0.732759 + 0.680489i $$0.761768\pi$$
$$272$$ −3034.00 −0.676336
$$273$$ 2496.00 0.553351
$$274$$ 2186.00 0.481975
$$275$$ 0 0
$$276$$ −3822.00 −0.833541
$$277$$ −124.000 −0.0268969 −0.0134484 0.999910i $$-0.504281\pi$$
−0.0134484 + 0.999910i $$0.504281\pi$$
$$278$$ 2740.00 0.591131
$$279$$ −72.0000 −0.0154499
$$280$$ 0 0
$$281$$ 3642.00 0.773180 0.386590 0.922252i $$-0.373653\pi$$
0.386590 + 0.922252i $$0.373653\pi$$
$$282$$ 1518.00 0.320552
$$283$$ −4648.00 −0.976307 −0.488154 0.872758i $$-0.662329\pi$$
−0.488154 + 0.872758i $$0.662329\pi$$
$$284$$ −5334.00 −1.11449
$$285$$ 0 0
$$286$$ 352.000 0.0727769
$$287$$ 10972.0 2.25664
$$288$$ 1449.00 0.296469
$$289$$ 563.000 0.114594
$$290$$ 0 0
$$291$$ −42.0000 −0.00846077
$$292$$ −3794.00 −0.760367
$$293$$ 3102.00 0.618501 0.309250 0.950981i $$-0.399922\pi$$
0.309250 + 0.950981i $$0.399922\pi$$
$$294$$ 999.000 0.198173
$$295$$ 0 0
$$296$$ −990.000 −0.194401
$$297$$ 297.000 0.0580259
$$298$$ −1310.00 −0.254652
$$299$$ 5824.00 1.12646
$$300$$ 0 0
$$301$$ −10608.0 −2.03135
$$302$$ −1198.00 −0.228269
$$303$$ 5106.00 0.968093
$$304$$ −2460.00 −0.464114
$$305$$ 0 0
$$306$$ −666.000 −0.124421
$$307$$ −1244.00 −0.231267 −0.115633 0.993292i $$-0.536890\pi$$
−0.115633 + 0.993292i $$0.536890\pi$$
$$308$$ −2002.00 −0.370372
$$309$$ 3396.00 0.625216
$$310$$ 0 0
$$311$$ 2082.00 0.379612 0.189806 0.981822i $$-0.439214\pi$$
0.189806 + 0.981822i $$0.439214\pi$$
$$312$$ −1440.00 −0.261295
$$313$$ −2378.00 −0.429433 −0.214716 0.976676i $$-0.568883\pi$$
−0.214716 + 0.976676i $$0.568883\pi$$
$$314$$ −2114.00 −0.379936
$$315$$ 0 0
$$316$$ 3850.00 0.685378
$$317$$ 496.000 0.0878806 0.0439403 0.999034i $$-0.486009\pi$$
0.0439403 + 0.999034i $$0.486009\pi$$
$$318$$ −1044.00 −0.184103
$$319$$ −990.000 −0.173760
$$320$$ 0 0
$$321$$ −1692.00 −0.294200
$$322$$ 4732.00 0.818957
$$323$$ 4440.00 0.764855
$$324$$ −567.000 −0.0972222
$$325$$ 0 0
$$326$$ −3868.00 −0.657143
$$327$$ −960.000 −0.162349
$$328$$ −6330.00 −1.06560
$$329$$ 13156.0 2.20460
$$330$$ 0 0
$$331$$ −2708.00 −0.449683 −0.224842 0.974395i $$-0.572186\pi$$
−0.224842 + 0.974395i $$0.572186\pi$$
$$332$$ −924.000 −0.152744
$$333$$ 594.000 0.0977507
$$334$$ −2004.00 −0.328305
$$335$$ 0 0
$$336$$ 3198.00 0.519242
$$337$$ −4034.00 −0.652065 −0.326033 0.945359i $$-0.605712\pi$$
−0.326033 + 0.945359i $$0.605712\pi$$
$$338$$ −1173.00 −0.188766
$$339$$ 6426.00 1.02954
$$340$$ 0 0
$$341$$ −88.0000 −0.0139750
$$342$$ −540.000 −0.0853797
$$343$$ −260.000 −0.0409291
$$344$$ 6120.00 0.959210
$$345$$ 0 0
$$346$$ −678.000 −0.105345
$$347$$ −11084.0 −1.71476 −0.857378 0.514687i $$-0.827908\pi$$
−0.857378 + 0.514687i $$0.827908\pi$$
$$348$$ 1890.00 0.291134
$$349$$ −3120.00 −0.478538 −0.239269 0.970953i $$-0.576908\pi$$
−0.239269 + 0.970953i $$0.576908\pi$$
$$350$$ 0 0
$$351$$ 864.000 0.131387
$$352$$ 1771.00 0.268167
$$353$$ 5622.00 0.847674 0.423837 0.905739i $$-0.360683\pi$$
0.423837 + 0.905739i $$0.360683\pi$$
$$354$$ −600.000 −0.0900837
$$355$$ 0 0
$$356$$ −3990.00 −0.594016
$$357$$ −5772.00 −0.855705
$$358$$ −1680.00 −0.248019
$$359$$ −8500.00 −1.24962 −0.624809 0.780778i $$-0.714823\pi$$
−0.624809 + 0.780778i $$0.714823\pi$$
$$360$$ 0 0
$$361$$ −3259.00 −0.475142
$$362$$ −4358.00 −0.632739
$$363$$ 363.000 0.0524864
$$364$$ −5824.00 −0.838628
$$365$$ 0 0
$$366$$ 396.000 0.0565553
$$367$$ −7144.00 −1.01611 −0.508057 0.861324i $$-0.669636\pi$$
−0.508057 + 0.861324i $$0.669636\pi$$
$$368$$ 7462.00 1.05702
$$369$$ 3798.00 0.535816
$$370$$ 0 0
$$371$$ −9048.00 −1.26617
$$372$$ 168.000 0.0234150
$$373$$ 632.000 0.0877312 0.0438656 0.999037i $$-0.486033\pi$$
0.0438656 + 0.999037i $$0.486033\pi$$
$$374$$ −814.000 −0.112543
$$375$$ 0 0
$$376$$ −7590.00 −1.04102
$$377$$ −2880.00 −0.393442
$$378$$ 702.000 0.0955211
$$379$$ −4220.00 −0.571944 −0.285972 0.958238i $$-0.592316\pi$$
−0.285972 + 0.958238i $$0.592316\pi$$
$$380$$ 0 0
$$381$$ 4818.00 0.647857
$$382$$ −1778.00 −0.238142
$$383$$ −8458.00 −1.12842 −0.564208 0.825632i $$-0.690819\pi$$
−0.564208 + 0.825632i $$0.690819\pi$$
$$384$$ −4365.00 −0.580079
$$385$$ 0 0
$$386$$ 3962.00 0.522437
$$387$$ −3672.00 −0.482321
$$388$$ 98.0000 0.0128227
$$389$$ 1740.00 0.226790 0.113395 0.993550i $$-0.463827\pi$$
0.113395 + 0.993550i $$0.463827\pi$$
$$390$$ 0 0
$$391$$ −13468.0 −1.74196
$$392$$ −4995.00 −0.643586
$$393$$ −5724.00 −0.734701
$$394$$ −374.000 −0.0478219
$$395$$ 0 0
$$396$$ −693.000 −0.0879408
$$397$$ 5126.00 0.648027 0.324013 0.946053i $$-0.394968\pi$$
0.324013 + 0.946053i $$0.394968\pi$$
$$398$$ 2100.00 0.264481
$$399$$ −4680.00 −0.587201
$$400$$ 0 0
$$401$$ −3098.00 −0.385802 −0.192901 0.981218i $$-0.561790\pi$$
−0.192901 + 0.981218i $$0.561790\pi$$
$$402$$ 3108.00 0.385604
$$403$$ −256.000 −0.0316433
$$404$$ −11914.0 −1.46719
$$405$$ 0 0
$$406$$ −2340.00 −0.286040
$$407$$ 726.000 0.0884189
$$408$$ 3330.00 0.404068
$$409$$ 6390.00 0.772531 0.386265 0.922388i $$-0.373765\pi$$
0.386265 + 0.922388i $$0.373765\pi$$
$$410$$ 0 0
$$411$$ 6558.00 0.787062
$$412$$ −7924.00 −0.947542
$$413$$ −5200.00 −0.619553
$$414$$ 1638.00 0.194452
$$415$$ 0 0
$$416$$ 5152.00 0.607206
$$417$$ 8220.00 0.965312
$$418$$ −660.000 −0.0772288
$$419$$ 9760.00 1.13796 0.568982 0.822350i $$-0.307337\pi$$
0.568982 + 0.822350i $$0.307337\pi$$
$$420$$ 0 0
$$421$$ −5138.00 −0.594800 −0.297400 0.954753i $$-0.596119\pi$$
−0.297400 + 0.954753i $$0.596119\pi$$
$$422$$ 2232.00 0.257469
$$423$$ 4554.00 0.523459
$$424$$ 5220.00 0.597891
$$425$$ 0 0
$$426$$ 2286.00 0.259993
$$427$$ 3432.00 0.388960
$$428$$ 3948.00 0.445873
$$429$$ 1056.00 0.118844
$$430$$ 0 0
$$431$$ −7008.00 −0.783210 −0.391605 0.920133i $$-0.628080\pi$$
−0.391605 + 0.920133i $$0.628080\pi$$
$$432$$ 1107.00 0.123288
$$433$$ −5578.00 −0.619080 −0.309540 0.950886i $$-0.600175\pi$$
−0.309540 + 0.950886i $$0.600175\pi$$
$$434$$ −208.000 −0.0230053
$$435$$ 0 0
$$436$$ 2240.00 0.246047
$$437$$ −10920.0 −1.19536
$$438$$ 1626.00 0.177382
$$439$$ −10430.0 −1.13393 −0.566967 0.823741i $$-0.691883\pi$$
−0.566967 + 0.823741i $$0.691883\pi$$
$$440$$ 0 0
$$441$$ 2997.00 0.323615
$$442$$ −2368.00 −0.254829
$$443$$ 4432.00 0.475329 0.237664 0.971347i $$-0.423618\pi$$
0.237664 + 0.971347i $$0.423618\pi$$
$$444$$ −1386.00 −0.148146
$$445$$ 0 0
$$446$$ −2128.00 −0.225928
$$447$$ −3930.00 −0.415845
$$448$$ −4342.00 −0.457902
$$449$$ −6290.00 −0.661121 −0.330561 0.943785i $$-0.607238\pi$$
−0.330561 + 0.943785i $$0.607238\pi$$
$$450$$ 0 0
$$451$$ 4642.00 0.484664
$$452$$ −14994.0 −1.56031
$$453$$ −3594.00 −0.372761
$$454$$ −2964.00 −0.306404
$$455$$ 0 0
$$456$$ 2700.00 0.277279
$$457$$ −3054.00 −0.312604 −0.156302 0.987709i $$-0.549957\pi$$
−0.156302 + 0.987709i $$0.549957\pi$$
$$458$$ −2550.00 −0.260161
$$459$$ −1998.00 −0.203178
$$460$$ 0 0
$$461$$ 12882.0 1.30146 0.650732 0.759308i $$-0.274462\pi$$
0.650732 + 0.759308i $$0.274462\pi$$
$$462$$ 858.000 0.0864021
$$463$$ −6148.00 −0.617110 −0.308555 0.951207i $$-0.599845\pi$$
−0.308555 + 0.951207i $$0.599845\pi$$
$$464$$ −3690.00 −0.369190
$$465$$ 0 0
$$466$$ 3042.00 0.302399
$$467$$ −5124.00 −0.507731 −0.253866 0.967240i $$-0.581702\pi$$
−0.253866 + 0.967240i $$0.581702\pi$$
$$468$$ −2016.00 −0.199123
$$469$$ 26936.0 2.65200
$$470$$ 0 0
$$471$$ −6342.00 −0.620433
$$472$$ 3000.00 0.292555
$$473$$ −4488.00 −0.436276
$$474$$ −1650.00 −0.159888
$$475$$ 0 0
$$476$$ 13468.0 1.29686
$$477$$ −3132.00 −0.300638
$$478$$ 2700.00 0.258358
$$479$$ −16520.0 −1.57582 −0.787910 0.615790i $$-0.788837\pi$$
−0.787910 + 0.615790i $$0.788837\pi$$
$$480$$ 0 0
$$481$$ 2112.00 0.200206
$$482$$ −578.000 −0.0546207
$$483$$ 14196.0 1.33735
$$484$$ −847.000 −0.0795455
$$485$$ 0 0
$$486$$ 243.000 0.0226805
$$487$$ −524.000 −0.0487571 −0.0243785 0.999703i $$-0.507761\pi$$
−0.0243785 + 0.999703i $$0.507761\pi$$
$$488$$ −1980.00 −0.183669
$$489$$ −11604.0 −1.07311
$$490$$ 0 0
$$491$$ −15028.0 −1.38127 −0.690636 0.723203i $$-0.742669\pi$$
−0.690636 + 0.723203i $$0.742669\pi$$
$$492$$ −8862.00 −0.812052
$$493$$ 6660.00 0.608421
$$494$$ −1920.00 −0.174868
$$495$$ 0 0
$$496$$ −328.000 −0.0296928
$$497$$ 19812.0 1.78811
$$498$$ 396.000 0.0356329
$$499$$ 9020.00 0.809200 0.404600 0.914494i $$-0.367411\pi$$
0.404600 + 0.914494i $$0.367411\pi$$
$$500$$ 0 0
$$501$$ −6012.00 −0.536120
$$502$$ 3752.00 0.333586
$$503$$ 14812.0 1.31299 0.656495 0.754330i $$-0.272038\pi$$
0.656495 + 0.754330i $$0.272038\pi$$
$$504$$ −3510.00 −0.310214
$$505$$ 0 0
$$506$$ 2002.00 0.175889
$$507$$ −3519.00 −0.308253
$$508$$ −11242.0 −0.981856
$$509$$ 12660.0 1.10245 0.551223 0.834358i $$-0.314161\pi$$
0.551223 + 0.834358i $$0.314161\pi$$
$$510$$ 0 0
$$511$$ 14092.0 1.21995
$$512$$ 11521.0 0.994455
$$513$$ −1620.00 −0.139424
$$514$$ −674.000 −0.0578383
$$515$$ 0 0
$$516$$ 8568.00 0.730979
$$517$$ 5566.00 0.473486
$$518$$ 1716.00 0.145553
$$519$$ −2034.00 −0.172028
$$520$$ 0 0
$$521$$ −3738.00 −0.314328 −0.157164 0.987573i $$-0.550235\pi$$
−0.157164 + 0.987573i $$0.550235\pi$$
$$522$$ −810.000 −0.0679171
$$523$$ 6352.00 0.531078 0.265539 0.964100i $$-0.414450\pi$$
0.265539 + 0.964100i $$0.414450\pi$$
$$524$$ 13356.0 1.11347
$$525$$ 0 0
$$526$$ 4352.00 0.360753
$$527$$ 592.000 0.0489334
$$528$$ 1353.00 0.111518
$$529$$ 20957.0 1.72245
$$530$$ 0 0
$$531$$ −1800.00 −0.147106
$$532$$ 10920.0 0.889929
$$533$$ 13504.0 1.09742
$$534$$ 1710.00 0.138575
$$535$$ 0 0
$$536$$ −15540.0 −1.25229
$$537$$ −5040.00 −0.405013
$$538$$ 500.000 0.0400679
$$539$$ 3663.00 0.292721
$$540$$ 0 0
$$541$$ −24728.0 −1.96514 −0.982569 0.185898i $$-0.940481\pi$$
−0.982569 + 0.185898i $$0.940481\pi$$
$$542$$ −6538.00 −0.518139
$$543$$ −13074.0 −1.03326
$$544$$ −11914.0 −0.938986
$$545$$ 0 0
$$546$$ 2496.00 0.195639
$$547$$ 22756.0 1.77875 0.889375 0.457178i $$-0.151140\pi$$
0.889375 + 0.457178i $$0.151140\pi$$
$$548$$ −15302.0 −1.19283
$$549$$ 1188.00 0.0923545
$$550$$ 0 0
$$551$$ 5400.00 0.417509
$$552$$ −8190.00 −0.631503
$$553$$ −14300.0 −1.09963
$$554$$ −124.000 −0.00950949
$$555$$ 0 0
$$556$$ −19180.0 −1.46297
$$557$$ 9526.00 0.724649 0.362325 0.932052i $$-0.381983\pi$$
0.362325 + 0.932052i $$0.381983\pi$$
$$558$$ −72.0000 −0.00546237
$$559$$ −13056.0 −0.987853
$$560$$ 0 0
$$561$$ −2442.00 −0.183781
$$562$$ 3642.00 0.273360
$$563$$ −12068.0 −0.903385 −0.451692 0.892174i $$-0.649180\pi$$
−0.451692 + 0.892174i $$0.649180\pi$$
$$564$$ −10626.0 −0.793325
$$565$$ 0 0
$$566$$ −4648.00 −0.345177
$$567$$ 2106.00 0.155985
$$568$$ −11430.0 −0.844352
$$569$$ 15090.0 1.11179 0.555893 0.831254i $$-0.312377\pi$$
0.555893 + 0.831254i $$0.312377\pi$$
$$570$$ 0 0
$$571$$ 4412.00 0.323356 0.161678 0.986844i $$-0.448309\pi$$
0.161678 + 0.986844i $$0.448309\pi$$
$$572$$ −2464.00 −0.180114
$$573$$ −5334.00 −0.388885
$$574$$ 10972.0 0.797844
$$575$$ 0 0
$$576$$ −1503.00 −0.108724
$$577$$ 3906.00 0.281818 0.140909 0.990023i $$-0.454998\pi$$
0.140909 + 0.990023i $$0.454998\pi$$
$$578$$ 563.000 0.0405151
$$579$$ 11886.0 0.853135
$$580$$ 0 0
$$581$$ 3432.00 0.245066
$$582$$ −42.0000 −0.00299133
$$583$$ −3828.00 −0.271937
$$584$$ −8130.00 −0.576065
$$585$$ 0 0
$$586$$ 3102.00 0.218673
$$587$$ 12016.0 0.844895 0.422448 0.906387i $$-0.361171\pi$$
0.422448 + 0.906387i $$0.361171\pi$$
$$588$$ −6993.00 −0.490453
$$589$$ 480.000 0.0335790
$$590$$ 0 0
$$591$$ −1122.00 −0.0780929
$$592$$ 2706.00 0.187865
$$593$$ 11342.0 0.785430 0.392715 0.919660i $$-0.371536\pi$$
0.392715 + 0.919660i $$0.371536\pi$$
$$594$$ 297.000 0.0205152
$$595$$ 0 0
$$596$$ 9170.00 0.630231
$$597$$ 6300.00 0.431896
$$598$$ 5824.00 0.398263
$$599$$ 20690.0 1.41130 0.705651 0.708559i $$-0.250654\pi$$
0.705651 + 0.708559i $$0.250654\pi$$
$$600$$ 0 0
$$601$$ −598.000 −0.0405872 −0.0202936 0.999794i $$-0.506460\pi$$
−0.0202936 + 0.999794i $$0.506460\pi$$
$$602$$ −10608.0 −0.718189
$$603$$ 9324.00 0.629689
$$604$$ 8386.00 0.564936
$$605$$ 0 0
$$606$$ 5106.00 0.342272
$$607$$ 166.000 0.0111001 0.00555003 0.999985i $$-0.498233\pi$$
0.00555003 + 0.999985i $$0.498233\pi$$
$$608$$ −9660.00 −0.644350
$$609$$ −7020.00 −0.467101
$$610$$ 0 0
$$611$$ 16192.0 1.07211
$$612$$ 4662.00 0.307925
$$613$$ −20108.0 −1.32488 −0.662442 0.749113i $$-0.730480\pi$$
−0.662442 + 0.749113i $$0.730480\pi$$
$$614$$ −1244.00 −0.0817651
$$615$$ 0 0
$$616$$ −4290.00 −0.280599
$$617$$ 2286.00 0.149159 0.0745793 0.997215i $$-0.476239\pi$$
0.0745793 + 0.997215i $$0.476239\pi$$
$$618$$ 3396.00 0.221047
$$619$$ −25660.0 −1.66618 −0.833088 0.553141i $$-0.813429\pi$$
−0.833088 + 0.553141i $$0.813429\pi$$
$$620$$ 0 0
$$621$$ 4914.00 0.317539
$$622$$ 2082.00 0.134213
$$623$$ 14820.0 0.953051
$$624$$ 3936.00 0.252510
$$625$$ 0 0
$$626$$ −2378.00 −0.151827
$$627$$ −1980.00 −0.126114
$$628$$ 14798.0 0.940294
$$629$$ −4884.00 −0.309599
$$630$$ 0 0
$$631$$ −11408.0 −0.719723 −0.359862 0.933006i $$-0.617176\pi$$
−0.359862 + 0.933006i $$0.617176\pi$$
$$632$$ 8250.00 0.519252
$$633$$ 6696.00 0.420446
$$634$$ 496.000 0.0310705
$$635$$ 0 0
$$636$$ 7308.00 0.455631
$$637$$ 10656.0 0.662804
$$638$$ −990.000 −0.0614333
$$639$$ 6858.00 0.424567
$$640$$ 0 0
$$641$$ −3378.00 −0.208148 −0.104074 0.994570i $$-0.533188\pi$$
−0.104074 + 0.994570i $$0.533188\pi$$
$$642$$ −1692.00 −0.104015
$$643$$ 11212.0 0.687649 0.343824 0.939034i $$-0.388278\pi$$
0.343824 + 0.939034i $$0.388278\pi$$
$$644$$ −33124.0 −2.02681
$$645$$ 0 0
$$646$$ 4440.00 0.270417
$$647$$ 86.0000 0.00522567 0.00261284 0.999997i $$-0.499168\pi$$
0.00261284 + 0.999997i $$0.499168\pi$$
$$648$$ −1215.00 −0.0736570
$$649$$ −2200.00 −0.133062
$$650$$ 0 0
$$651$$ −624.000 −0.0375676
$$652$$ 27076.0 1.62635
$$653$$ 4432.00 0.265601 0.132801 0.991143i $$-0.457603\pi$$
0.132801 + 0.991143i $$0.457603\pi$$
$$654$$ −960.000 −0.0573990
$$655$$ 0 0
$$656$$ 17302.0 1.02977
$$657$$ 4878.00 0.289663
$$658$$ 13156.0 0.779444
$$659$$ 4580.00 0.270731 0.135365 0.990796i $$-0.456779\pi$$
0.135365 + 0.990796i $$0.456779\pi$$
$$660$$ 0 0
$$661$$ 4282.00 0.251967 0.125984 0.992032i $$-0.459791\pi$$
0.125984 + 0.992032i $$0.459791\pi$$
$$662$$ −2708.00 −0.158987
$$663$$ −7104.00 −0.416133
$$664$$ −1980.00 −0.115721
$$665$$ 0 0
$$666$$ 594.000 0.0345601
$$667$$ −16380.0 −0.950879
$$668$$ 14028.0 0.812514
$$669$$ −6384.00 −0.368938
$$670$$ 0 0
$$671$$ 1452.00 0.0835378
$$672$$ 12558.0 0.720886
$$673$$ −8438.00 −0.483300 −0.241650 0.970363i $$-0.577689\pi$$
−0.241650 + 0.970363i $$0.577689\pi$$
$$674$$ −4034.00 −0.230540
$$675$$ 0 0
$$676$$ 8211.00 0.467171
$$677$$ −34494.0 −1.95822 −0.979108 0.203341i $$-0.934820\pi$$
−0.979108 + 0.203341i $$0.934820\pi$$
$$678$$ 6426.00 0.363996
$$679$$ −364.000 −0.0205730
$$680$$ 0 0
$$681$$ −8892.00 −0.500356
$$682$$ −88.0000 −0.00494090
$$683$$ 13712.0 0.768192 0.384096 0.923293i $$-0.374513\pi$$
0.384096 + 0.923293i $$0.374513\pi$$
$$684$$ 3780.00 0.211304
$$685$$ 0 0
$$686$$ −260.000 −0.0144706
$$687$$ −7650.00 −0.424841
$$688$$ −16728.0 −0.926961
$$689$$ −11136.0 −0.615744
$$690$$ 0 0
$$691$$ 11372.0 0.626066 0.313033 0.949742i $$-0.398655\pi$$
0.313033 + 0.949742i $$0.398655\pi$$
$$692$$ 4746.00 0.260717
$$693$$ 2574.00 0.141094
$$694$$ −11084.0 −0.606258
$$695$$ 0 0
$$696$$ 4050.00 0.220567
$$697$$ −31228.0 −1.69705
$$698$$ −3120.00 −0.169189
$$699$$ 9126.00 0.493815
$$700$$ 0 0
$$701$$ −6398.00 −0.344721 −0.172360 0.985034i $$-0.555139\pi$$
−0.172360 + 0.985034i $$0.555139\pi$$
$$702$$ 864.000 0.0464524
$$703$$ −3960.00 −0.212453
$$704$$ −1837.00 −0.0983445
$$705$$ 0 0
$$706$$ 5622.00 0.299698
$$707$$ 44252.0 2.35399
$$708$$ 4200.00 0.222946
$$709$$ −5830.00 −0.308816 −0.154408 0.988007i $$-0.549347\pi$$
−0.154408 + 0.988007i $$0.549347\pi$$
$$710$$ 0 0
$$711$$ −4950.00 −0.261096
$$712$$ −8550.00 −0.450035
$$713$$ −1456.00 −0.0764763
$$714$$ −5772.00 −0.302537
$$715$$ 0 0
$$716$$ 11760.0 0.613815
$$717$$ 8100.00 0.421897
$$718$$ −8500.00 −0.441807
$$719$$ 34530.0 1.79103 0.895516 0.445030i $$-0.146807\pi$$
0.895516 + 0.445030i $$0.146807\pi$$
$$720$$ 0 0
$$721$$ 29432.0 1.52026
$$722$$ −3259.00 −0.167988
$$723$$ −1734.00 −0.0891952
$$724$$ 30506.0 1.56595
$$725$$ 0 0
$$726$$ 363.000 0.0185567
$$727$$ 17316.0 0.883377 0.441688 0.897169i $$-0.354380\pi$$
0.441688 + 0.897169i $$0.354380\pi$$
$$728$$ −12480.0 −0.635357
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 30192.0 1.52762
$$732$$ −2772.00 −0.139967
$$733$$ 27072.0 1.36416 0.682079 0.731279i $$-0.261076\pi$$
0.682079 + 0.731279i $$0.261076\pi$$
$$734$$ −7144.00 −0.359250
$$735$$ 0 0
$$736$$ 29302.0 1.46751
$$737$$ 11396.0 0.569575
$$738$$ 3798.00 0.189439
$$739$$ −17320.0 −0.862147 −0.431073 0.902317i $$-0.641865\pi$$
−0.431073 + 0.902317i $$0.641865\pi$$
$$740$$ 0 0
$$741$$ −5760.00 −0.285559
$$742$$ −9048.00 −0.447658
$$743$$ −14588.0 −0.720299 −0.360149 0.932895i $$-0.617274\pi$$
−0.360149 + 0.932895i $$0.617274\pi$$
$$744$$ 360.000 0.0177396
$$745$$ 0 0
$$746$$ 632.000 0.0310176
$$747$$ 1188.00 0.0581883
$$748$$ 5698.00 0.278529
$$749$$ −14664.0 −0.715368
$$750$$ 0 0
$$751$$ 26152.0 1.27071 0.635353 0.772222i $$-0.280855\pi$$
0.635353 + 0.772222i $$0.280855\pi$$
$$752$$ 20746.0 1.00602
$$753$$ 11256.0 0.544743
$$754$$ −2880.00 −0.139103
$$755$$ 0 0
$$756$$ −4914.00 −0.236403
$$757$$ 1066.00 0.0511815 0.0255908 0.999673i $$-0.491853\pi$$
0.0255908 + 0.999673i $$0.491853\pi$$
$$758$$ −4220.00 −0.202213
$$759$$ 6006.00 0.287225
$$760$$ 0 0
$$761$$ −37518.0 −1.78716 −0.893578 0.448907i $$-0.851813\pi$$
−0.893578 + 0.448907i $$0.851813\pi$$
$$762$$ 4818.00 0.229052
$$763$$ −8320.00 −0.394763
$$764$$ 12446.0 0.589372
$$765$$ 0 0
$$766$$ −8458.00 −0.398956
$$767$$ −6400.00 −0.301292
$$768$$ −357.000 −0.0167736
$$769$$ −17290.0 −0.810785 −0.405392 0.914143i $$-0.632865\pi$$
−0.405392 + 0.914143i $$0.632865\pi$$
$$770$$ 0 0
$$771$$ −2022.00 −0.0944495
$$772$$ −27734.0 −1.29296
$$773$$ 17172.0 0.799009 0.399504 0.916731i $$-0.369182\pi$$
0.399504 + 0.916731i $$0.369182\pi$$
$$774$$ −3672.00 −0.170526
$$775$$ 0 0
$$776$$ 210.000 0.00971464
$$777$$ 5148.00 0.237688
$$778$$ 1740.00 0.0801825
$$779$$ −25320.0 −1.16455
$$780$$ 0 0
$$781$$ 8382.00 0.384035
$$782$$ −13468.0 −0.615876
$$783$$ −2430.00 −0.110908
$$784$$ 13653.0 0.621948
$$785$$ 0 0
$$786$$ −5724.00 −0.259756
$$787$$ 9536.00 0.431921 0.215960 0.976402i $$-0.430712\pi$$
0.215960 + 0.976402i $$0.430712\pi$$
$$788$$ 2618.00 0.118353
$$789$$ 13056.0 0.589108
$$790$$ 0 0
$$791$$ 55692.0 2.50339
$$792$$ −1485.00 −0.0666252
$$793$$ 4224.00 0.189153
$$794$$ 5126.00 0.229112
$$795$$ 0 0
$$796$$ −14700.0 −0.654557
$$797$$ 20516.0 0.911812 0.455906 0.890028i $$-0.349315\pi$$
0.455906 + 0.890028i $$0.349315\pi$$
$$798$$ −4680.00 −0.207607
$$799$$ −37444.0 −1.65791
$$800$$ 0 0
$$801$$ 5130.00 0.226292
$$802$$ −3098.00 −0.136402
$$803$$ 5962.00 0.262010
$$804$$ −21756.0 −0.954322
$$805$$ 0 0
$$806$$ −256.000 −0.0111876
$$807$$ 1500.00 0.0654306
$$808$$ −25530.0 −1.11156
$$809$$ 22470.0 0.976518 0.488259 0.872699i $$-0.337632\pi$$
0.488259 + 0.872699i $$0.337632\pi$$
$$810$$ 0 0
$$811$$ −3368.00 −0.145828 −0.0729140 0.997338i $$-0.523230\pi$$
−0.0729140 + 0.997338i $$0.523230\pi$$
$$812$$ 16380.0 0.707913
$$813$$ −19614.0 −0.846117
$$814$$ 726.000 0.0312608
$$815$$ 0 0
$$816$$ −9102.00 −0.390483
$$817$$ 24480.0 1.04828
$$818$$ 6390.00 0.273131
$$819$$ 7488.00 0.319477
$$820$$ 0 0
$$821$$ −10738.0 −0.456466 −0.228233 0.973607i $$-0.573295\pi$$
−0.228233 + 0.973607i $$0.573295\pi$$
$$822$$ 6558.00 0.278268
$$823$$ 15912.0 0.673946 0.336973 0.941514i $$-0.390597\pi$$
0.336973 + 0.941514i $$0.390597\pi$$
$$824$$ −16980.0 −0.717872
$$825$$ 0 0
$$826$$ −5200.00 −0.219045
$$827$$ −22924.0 −0.963900 −0.481950 0.876199i $$-0.660071\pi$$
−0.481950 + 0.876199i $$0.660071\pi$$
$$828$$ −11466.0 −0.481245
$$829$$ −41690.0 −1.74663 −0.873313 0.487159i $$-0.838033\pi$$
−0.873313 + 0.487159i $$0.838033\pi$$
$$830$$ 0 0
$$831$$ −372.000 −0.0155289
$$832$$ −5344.00 −0.222680
$$833$$ −24642.0 −1.02496
$$834$$ 8220.00 0.341289
$$835$$ 0 0
$$836$$ 4620.00 0.191132
$$837$$ −216.000 −0.00892001
$$838$$ 9760.00 0.402331
$$839$$ −16450.0 −0.676898 −0.338449 0.940985i $$-0.609902\pi$$
−0.338449 + 0.940985i $$0.609902\pi$$
$$840$$ 0 0
$$841$$ −16289.0 −0.667883
$$842$$ −5138.00 −0.210294
$$843$$ 10926.0 0.446396
$$844$$ −15624.0 −0.637204
$$845$$ 0 0
$$846$$ 4554.00 0.185071
$$847$$ 3146.00 0.127624
$$848$$ −14268.0 −0.577789
$$849$$ −13944.0 −0.563671
$$850$$ 0 0
$$851$$ 12012.0 0.483861
$$852$$ −16002.0 −0.643450
$$853$$ 30892.0 1.24000 0.620001 0.784601i $$-0.287132\pi$$
0.620001 + 0.784601i $$0.287132\pi$$
$$854$$ 3432.00 0.137518
$$855$$ 0 0
$$856$$ 8460.00 0.337800
$$857$$ 38906.0 1.55076 0.775381 0.631493i $$-0.217558\pi$$
0.775381 + 0.631493i $$0.217558\pi$$
$$858$$ 1056.00 0.0420178
$$859$$ −1020.00 −0.0405145 −0.0202572 0.999795i $$-0.506449\pi$$
−0.0202572 + 0.999795i $$0.506449\pi$$
$$860$$ 0 0
$$861$$ 32916.0 1.30287
$$862$$ −7008.00 −0.276907
$$863$$ −15078.0 −0.594741 −0.297370 0.954762i $$-0.596110\pi$$
−0.297370 + 0.954762i $$0.596110\pi$$
$$864$$ 4347.00 0.171167
$$865$$ 0 0
$$866$$ −5578.00 −0.218878
$$867$$ 1689.00 0.0661608
$$868$$ 1456.00 0.0569353
$$869$$ −6050.00 −0.236171
$$870$$ 0 0
$$871$$ 33152.0 1.28968
$$872$$ 4800.00 0.186409
$$873$$ −126.000 −0.00488483
$$874$$ −10920.0 −0.422625
$$875$$ 0 0
$$876$$ −11382.0 −0.438998
$$877$$ −22704.0 −0.874184 −0.437092 0.899417i $$-0.643992\pi$$
−0.437092 + 0.899417i $$0.643992\pi$$
$$878$$ −10430.0 −0.400906
$$879$$ 9306.00 0.357092
$$880$$ 0 0
$$881$$ −19358.0 −0.740281 −0.370141 0.928976i $$-0.620690\pi$$
−0.370141 + 0.928976i $$0.620690\pi$$
$$882$$ 2997.00 0.114415
$$883$$ 11252.0 0.428833 0.214417 0.976742i $$-0.431215\pi$$
0.214417 + 0.976742i $$0.431215\pi$$
$$884$$ 16576.0 0.630669
$$885$$ 0 0
$$886$$ 4432.00 0.168054
$$887$$ −43684.0 −1.65362 −0.826812 0.562478i $$-0.809848\pi$$
−0.826812 + 0.562478i $$0.809848\pi$$
$$888$$ −2970.00 −0.112237
$$889$$ 41756.0 1.57531
$$890$$ 0 0
$$891$$ 891.000 0.0335013
$$892$$ 14896.0 0.559142
$$893$$ −30360.0 −1.13769
$$894$$ −3930.00 −0.147023
$$895$$ 0 0
$$896$$ −37830.0 −1.41050
$$897$$ 17472.0 0.650360
$$898$$ −6290.00 −0.233742
$$899$$ 720.000 0.0267112
$$900$$ 0 0
$$901$$ 25752.0 0.952190
$$902$$ 4642.00 0.171354
$$903$$ −31824.0 −1.17280
$$904$$ −32130.0 −1.18211
$$905$$ 0 0
$$906$$ −3594.00 −0.131791
$$907$$ −45804.0 −1.67684 −0.838422 0.545022i $$-0.816521\pi$$
−0.838422 + 0.545022i $$0.816521\pi$$
$$908$$ 20748.0 0.758311
$$909$$ 15318.0 0.558928
$$910$$ 0 0
$$911$$ −15318.0 −0.557089 −0.278544 0.960423i $$-0.589852\pi$$
−0.278544 + 0.960423i $$0.589852\pi$$
$$912$$ −7380.00 −0.267956
$$913$$ 1452.00 0.0526333
$$914$$ −3054.00 −0.110522
$$915$$ 0 0
$$916$$ 17850.0 0.643865
$$917$$ −49608.0 −1.78648
$$918$$ −1998.00 −0.0718342
$$919$$ 11350.0 0.407401 0.203701 0.979033i $$-0.434703\pi$$
0.203701 + 0.979033i $$0.434703\pi$$
$$920$$ 0 0
$$921$$ −3732.00 −0.133522
$$922$$ 12882.0 0.460137
$$923$$ 24384.0 0.869566
$$924$$ −6006.00 −0.213834
$$925$$ 0 0
$$926$$ −6148.00 −0.218181
$$927$$ 10188.0 0.360969
$$928$$ −14490.0 −0.512562
$$929$$ 33030.0 1.16650 0.583250 0.812292i $$-0.301781\pi$$
0.583250 + 0.812292i $$0.301781\pi$$
$$930$$ 0 0
$$931$$ −19980.0 −0.703349
$$932$$ −21294.0 −0.748399
$$933$$ 6246.00 0.219169
$$934$$ −5124.00 −0.179510
$$935$$ 0 0
$$936$$ −4320.00 −0.150859
$$937$$ 10006.0 0.348860 0.174430 0.984670i $$-0.444192\pi$$
0.174430 + 0.984670i $$0.444192\pi$$
$$938$$ 26936.0 0.937624
$$939$$ −7134.00 −0.247933
$$940$$ 0 0
$$941$$ 2622.00 0.0908340 0.0454170 0.998968i $$-0.485538\pi$$
0.0454170 + 0.998968i $$0.485538\pi$$
$$942$$ −6342.00 −0.219356
$$943$$ 76804.0 2.65226
$$944$$ −8200.00 −0.282720
$$945$$ 0 0
$$946$$ −4488.00 −0.154247
$$947$$ 39876.0 1.36832 0.684158 0.729334i $$-0.260170\pi$$
0.684158 + 0.729334i $$0.260170\pi$$
$$948$$ 11550.0 0.395703
$$949$$ 17344.0 0.593267
$$950$$ 0 0
$$951$$ 1488.00 0.0507379
$$952$$ 28860.0 0.982519
$$953$$ −38918.0 −1.32285 −0.661426 0.750011i $$-0.730048\pi$$
−0.661426 + 0.750011i $$0.730048\pi$$
$$954$$ −3132.00 −0.106292
$$955$$ 0 0
$$956$$ −18900.0 −0.639403
$$957$$ −2970.00 −0.100320
$$958$$ −16520.0 −0.557137
$$959$$ 56836.0 1.91380
$$960$$ 0 0
$$961$$ −29727.0 −0.997852
$$962$$ 2112.00 0.0707834
$$963$$ −5076.00 −0.169857
$$964$$ 4046.00 0.135179
$$965$$ 0 0
$$966$$ 14196.0 0.472825
$$967$$ −1114.00 −0.0370464 −0.0185232 0.999828i $$-0.505896\pi$$
−0.0185232 + 0.999828i $$0.505896\pi$$
$$968$$ −1815.00 −0.0602648
$$969$$ 13320.0 0.441589
$$970$$ 0 0
$$971$$ −1688.00 −0.0557884 −0.0278942 0.999611i $$-0.508880\pi$$
−0.0278942 + 0.999611i $$0.508880\pi$$
$$972$$ −1701.00 −0.0561313
$$973$$ 71240.0 2.34722
$$974$$ −524.000 −0.0172382
$$975$$ 0 0
$$976$$ 5412.00 0.177494
$$977$$ 41826.0 1.36963 0.684817 0.728715i $$-0.259882\pi$$
0.684817 + 0.728715i $$0.259882\pi$$
$$978$$ −11604.0 −0.379402
$$979$$ 6270.00 0.204689
$$980$$ 0 0
$$981$$ −2880.00 −0.0937322
$$982$$ −15028.0 −0.488353
$$983$$ −978.000 −0.0317328 −0.0158664 0.999874i $$-0.505051\pi$$
−0.0158664 + 0.999874i $$0.505051\pi$$
$$984$$ −18990.0 −0.615223
$$985$$ 0 0
$$986$$ 6660.00 0.215109
$$987$$ 39468.0 1.27283
$$988$$ 13440.0 0.432777
$$989$$ −74256.0 −2.38747
$$990$$ 0 0
$$991$$ 47272.0 1.51528 0.757641 0.652671i $$-0.226352\pi$$
0.757641 + 0.652671i $$0.226352\pi$$
$$992$$ −1288.00 −0.0412238
$$993$$ −8124.00 −0.259625
$$994$$ 19812.0 0.632192
$$995$$ 0 0
$$996$$ −2772.00 −0.0881869
$$997$$ −51104.0 −1.62335 −0.811675 0.584109i $$-0.801444\pi$$
−0.811675 + 0.584109i $$0.801444\pi$$
$$998$$ 9020.00 0.286095
$$999$$ 1782.00 0.0564364
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 825.4.a.f.1.1 1
3.2 odd 2 2475.4.a.e.1.1 1
5.2 odd 4 825.4.c.f.199.2 2
5.3 odd 4 825.4.c.f.199.1 2
5.4 even 2 33.4.a.b.1.1 1
15.14 odd 2 99.4.a.a.1.1 1
20.19 odd 2 528.4.a.h.1.1 1
35.34 odd 2 1617.4.a.d.1.1 1
40.19 odd 2 2112.4.a.h.1.1 1
40.29 even 2 2112.4.a.u.1.1 1
55.54 odd 2 363.4.a.d.1.1 1
60.59 even 2 1584.4.a.l.1.1 1
165.164 even 2 1089.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.b.1.1 1 5.4 even 2
99.4.a.a.1.1 1 15.14 odd 2
363.4.a.d.1.1 1 55.54 odd 2
528.4.a.h.1.1 1 20.19 odd 2
825.4.a.f.1.1 1 1.1 even 1 trivial
825.4.c.f.199.1 2 5.3 odd 4
825.4.c.f.199.2 2 5.2 odd 4
1089.4.a.e.1.1 1 165.164 even 2
1584.4.a.l.1.1 1 60.59 even 2
1617.4.a.d.1.1 1 35.34 odd 2
2112.4.a.h.1.1 1 40.19 odd 2
2112.4.a.u.1.1 1 40.29 even 2
2475.4.a.e.1.1 1 3.2 odd 2