# Properties

 Label 25.4.a.a.1.1 Level $25$ Weight $4$ Character 25.1 Self dual yes Analytic conductor $1.475$ Analytic rank $1$ 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] = [25,4,Mod(1,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("25.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 25.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: $$1.47504775014$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 25.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -7.00000 q^{3} -7.00000 q^{4} +7.00000 q^{6} -6.00000 q^{7} +15.0000 q^{8} +22.0000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -7.00000 q^{3} -7.00000 q^{4} +7.00000 q^{6} -6.00000 q^{7} +15.0000 q^{8} +22.0000 q^{9} -43.0000 q^{11} +49.0000 q^{12} +28.0000 q^{13} +6.00000 q^{14} +41.0000 q^{16} -91.0000 q^{17} -22.0000 q^{18} -35.0000 q^{19} +42.0000 q^{21} +43.0000 q^{22} -162.000 q^{23} -105.000 q^{24} -28.0000 q^{26} +35.0000 q^{27} +42.0000 q^{28} +160.000 q^{29} +42.0000 q^{31} -161.000 q^{32} +301.000 q^{33} +91.0000 q^{34} -154.000 q^{36} +314.000 q^{37} +35.0000 q^{38} -196.000 q^{39} -203.000 q^{41} -42.0000 q^{42} -92.0000 q^{43} +301.000 q^{44} +162.000 q^{46} -196.000 q^{47} -287.000 q^{48} -307.000 q^{49} +637.000 q^{51} -196.000 q^{52} -82.0000 q^{53} -35.0000 q^{54} -90.0000 q^{56} +245.000 q^{57} -160.000 q^{58} -280.000 q^{59} -518.000 q^{61} -42.0000 q^{62} -132.000 q^{63} -167.000 q^{64} -301.000 q^{66} -141.000 q^{67} +637.000 q^{68} +1134.00 q^{69} +412.000 q^{71} +330.000 q^{72} +763.000 q^{73} -314.000 q^{74} +245.000 q^{76} +258.000 q^{77} +196.000 q^{78} +510.000 q^{79} -839.000 q^{81} +203.000 q^{82} -777.000 q^{83} -294.000 q^{84} +92.0000 q^{86} -1120.00 q^{87} -645.000 q^{88} -945.000 q^{89} -168.000 q^{91} +1134.00 q^{92} -294.000 q^{93} +196.000 q^{94} +1127.00 q^{96} -1246.00 q^{97} +307.000 q^{98} -946.000 q^{99} +O(q^{100})$$

## 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 $$-0.556567\pi$$
−0.176777 + 0.984251i $$0.556567\pi$$
$$3$$ −7.00000 −1.34715 −0.673575 0.739119i $$-0.735242\pi$$
−0.673575 + 0.739119i $$0.735242\pi$$
$$4$$ −7.00000 −0.875000
$$5$$ 0 0
$$6$$ 7.00000 0.476290
$$7$$ −6.00000 −0.323970 −0.161985 0.986793i $$-0.551790\pi$$
−0.161985 + 0.986793i $$0.551790\pi$$
$$8$$ 15.0000 0.662913
$$9$$ 22.0000 0.814815
$$10$$ 0 0
$$11$$ −43.0000 −1.17864 −0.589318 0.807901i $$-0.700603\pi$$
−0.589318 + 0.807901i $$0.700603\pi$$
$$12$$ 49.0000 1.17876
$$13$$ 28.0000 0.597369 0.298685 0.954352i $$-0.403452\pi$$
0.298685 + 0.954352i $$0.403452\pi$$
$$14$$ 6.00000 0.114541
$$15$$ 0 0
$$16$$ 41.0000 0.640625
$$17$$ −91.0000 −1.29828 −0.649139 0.760669i $$-0.724871\pi$$
−0.649139 + 0.760669i $$0.724871\pi$$
$$18$$ −22.0000 −0.288081
$$19$$ −35.0000 −0.422608 −0.211304 0.977420i $$-0.567771\pi$$
−0.211304 + 0.977420i $$0.567771\pi$$
$$20$$ 0 0
$$21$$ 42.0000 0.436436
$$22$$ 43.0000 0.416710
$$23$$ −162.000 −1.46867 −0.734333 0.678789i $$-0.762505\pi$$
−0.734333 + 0.678789i $$0.762505\pi$$
$$24$$ −105.000 −0.893043
$$25$$ 0 0
$$26$$ −28.0000 −0.211202
$$27$$ 35.0000 0.249472
$$28$$ 42.0000 0.283473
$$29$$ 160.000 1.02453 0.512263 0.858829i $$-0.328807\pi$$
0.512263 + 0.858829i $$0.328807\pi$$
$$30$$ 0 0
$$31$$ 42.0000 0.243336 0.121668 0.992571i $$-0.461176\pi$$
0.121668 + 0.992571i $$0.461176\pi$$
$$32$$ −161.000 −0.889408
$$33$$ 301.000 1.58780
$$34$$ 91.0000 0.459011
$$35$$ 0 0
$$36$$ −154.000 −0.712963
$$37$$ 314.000 1.39517 0.697585 0.716502i $$-0.254258\pi$$
0.697585 + 0.716502i $$0.254258\pi$$
$$38$$ 35.0000 0.149414
$$39$$ −196.000 −0.804747
$$40$$ 0 0
$$41$$ −203.000 −0.773251 −0.386625 0.922237i $$-0.626359\pi$$
−0.386625 + 0.922237i $$0.626359\pi$$
$$42$$ −42.0000 −0.154303
$$43$$ −92.0000 −0.326276 −0.163138 0.986603i $$-0.552162\pi$$
−0.163138 + 0.986603i $$0.552162\pi$$
$$44$$ 301.000 1.03131
$$45$$ 0 0
$$46$$ 162.000 0.519252
$$47$$ −196.000 −0.608288 −0.304144 0.952626i $$-0.598370\pi$$
−0.304144 + 0.952626i $$0.598370\pi$$
$$48$$ −287.000 −0.863018
$$49$$ −307.000 −0.895044
$$50$$ 0 0
$$51$$ 637.000 1.74898
$$52$$ −196.000 −0.522698
$$53$$ −82.0000 −0.212520 −0.106260 0.994338i $$-0.533888\pi$$
−0.106260 + 0.994338i $$0.533888\pi$$
$$54$$ −35.0000 −0.0882018
$$55$$ 0 0
$$56$$ −90.0000 −0.214763
$$57$$ 245.000 0.569317
$$58$$ −160.000 −0.362225
$$59$$ −280.000 −0.617846 −0.308923 0.951087i $$-0.599968\pi$$
−0.308923 + 0.951087i $$0.599968\pi$$
$$60$$ 0 0
$$61$$ −518.000 −1.08726 −0.543632 0.839324i $$-0.682951\pi$$
−0.543632 + 0.839324i $$0.682951\pi$$
$$62$$ −42.0000 −0.0860323
$$63$$ −132.000 −0.263975
$$64$$ −167.000 −0.326172
$$65$$ 0 0
$$66$$ −301.000 −0.561372
$$67$$ −141.000 −0.257103 −0.128551 0.991703i $$-0.541033\pi$$
−0.128551 + 0.991703i $$0.541033\pi$$
$$68$$ 637.000 1.13599
$$69$$ 1134.00 1.97852
$$70$$ 0 0
$$71$$ 412.000 0.688668 0.344334 0.938847i $$-0.388105\pi$$
0.344334 + 0.938847i $$0.388105\pi$$
$$72$$ 330.000 0.540151
$$73$$ 763.000 1.22332 0.611660 0.791121i $$-0.290502\pi$$
0.611660 + 0.791121i $$0.290502\pi$$
$$74$$ −314.000 −0.493267
$$75$$ 0 0
$$76$$ 245.000 0.369782
$$77$$ 258.000 0.381842
$$78$$ 196.000 0.284521
$$79$$ 510.000 0.726323 0.363161 0.931726i $$-0.381697\pi$$
0.363161 + 0.931726i $$0.381697\pi$$
$$80$$ 0 0
$$81$$ −839.000 −1.15089
$$82$$ 203.000 0.273385
$$83$$ −777.000 −1.02755 −0.513776 0.857924i $$-0.671754\pi$$
−0.513776 + 0.857924i $$0.671754\pi$$
$$84$$ −294.000 −0.381881
$$85$$ 0 0
$$86$$ 92.0000 0.115356
$$87$$ −1120.00 −1.38019
$$88$$ −645.000 −0.781332
$$89$$ −945.000 −1.12550 −0.562752 0.826626i $$-0.690257\pi$$
−0.562752 + 0.826626i $$0.690257\pi$$
$$90$$ 0 0
$$91$$ −168.000 −0.193530
$$92$$ 1134.00 1.28508
$$93$$ −294.000 −0.327811
$$94$$ 196.000 0.215062
$$95$$ 0 0
$$96$$ 1127.00 1.19817
$$97$$ −1246.00 −1.30425 −0.652124 0.758112i $$-0.726122\pi$$
−0.652124 + 0.758112i $$0.726122\pi$$
$$98$$ 307.000 0.316446
$$99$$ −946.000 −0.960369
$$100$$ 0 0
$$101$$ 1302.00 1.28271 0.641356 0.767244i $$-0.278372\pi$$
0.641356 + 0.767244i $$0.278372\pi$$
$$102$$ −637.000 −0.618357
$$103$$ −532.000 −0.508927 −0.254464 0.967082i $$-0.581899\pi$$
−0.254464 + 0.967082i $$0.581899\pi$$
$$104$$ 420.000 0.396004
$$105$$ 0 0
$$106$$ 82.0000 0.0751372
$$107$$ 1269.00 1.14653 0.573266 0.819370i $$-0.305676\pi$$
0.573266 + 0.819370i $$0.305676\pi$$
$$108$$ −245.000 −0.218288
$$109$$ 1070.00 0.940251 0.470126 0.882599i $$-0.344209\pi$$
0.470126 + 0.882599i $$0.344209\pi$$
$$110$$ 0 0
$$111$$ −2198.00 −1.87950
$$112$$ −246.000 −0.207543
$$113$$ 503.000 0.418746 0.209373 0.977836i $$-0.432858\pi$$
0.209373 + 0.977836i $$0.432858\pi$$
$$114$$ −245.000 −0.201284
$$115$$ 0 0
$$116$$ −1120.00 −0.896460
$$117$$ 616.000 0.486745
$$118$$ 280.000 0.218441
$$119$$ 546.000 0.420603
$$120$$ 0 0
$$121$$ 518.000 0.389181
$$122$$ 518.000 0.384406
$$123$$ 1421.00 1.04169
$$124$$ −294.000 −0.212919
$$125$$ 0 0
$$126$$ 132.000 0.0933293
$$127$$ 874.000 0.610669 0.305334 0.952245i $$-0.401232\pi$$
0.305334 + 0.952245i $$0.401232\pi$$
$$128$$ 1455.00 1.00473
$$129$$ 644.000 0.439543
$$130$$ 0 0
$$131$$ 1092.00 0.728309 0.364155 0.931339i $$-0.381358\pi$$
0.364155 + 0.931339i $$0.381358\pi$$
$$132$$ −2107.00 −1.38932
$$133$$ 210.000 0.136912
$$134$$ 141.000 0.0908996
$$135$$ 0 0
$$136$$ −1365.00 −0.860645
$$137$$ −411.000 −0.256307 −0.128154 0.991754i $$-0.540905\pi$$
−0.128154 + 0.991754i $$0.540905\pi$$
$$138$$ −1134.00 −0.699511
$$139$$ −595.000 −0.363074 −0.181537 0.983384i $$-0.558107\pi$$
−0.181537 + 0.983384i $$0.558107\pi$$
$$140$$ 0 0
$$141$$ 1372.00 0.819456
$$142$$ −412.000 −0.243481
$$143$$ −1204.00 −0.704081
$$144$$ 902.000 0.521991
$$145$$ 0 0
$$146$$ −763.000 −0.432509
$$147$$ 2149.00 1.20576
$$148$$ −2198.00 −1.22077
$$149$$ −3200.00 −1.75942 −0.879712 0.475507i $$-0.842265\pi$$
−0.879712 + 0.475507i $$0.842265\pi$$
$$150$$ 0 0
$$151$$ 202.000 0.108864 0.0544322 0.998517i $$-0.482665\pi$$
0.0544322 + 0.998517i $$0.482665\pi$$
$$152$$ −525.000 −0.280152
$$153$$ −2002.00 −1.05786
$$154$$ −258.000 −0.135002
$$155$$ 0 0
$$156$$ 1372.00 0.704153
$$157$$ −406.000 −0.206384 −0.103192 0.994661i $$-0.532906\pi$$
−0.103192 + 0.994661i $$0.532906\pi$$
$$158$$ −510.000 −0.256794
$$159$$ 574.000 0.286297
$$160$$ 0 0
$$161$$ 972.000 0.475803
$$162$$ 839.000 0.406902
$$163$$ 3803.00 1.82745 0.913724 0.406336i $$-0.133194\pi$$
0.913724 + 0.406336i $$0.133194\pi$$
$$164$$ 1421.00 0.676594
$$165$$ 0 0
$$166$$ 777.000 0.363295
$$167$$ −4116.00 −1.90722 −0.953610 0.301046i $$-0.902664\pi$$
−0.953610 + 0.301046i $$0.902664\pi$$
$$168$$ 630.000 0.289319
$$169$$ −1413.00 −0.643150
$$170$$ 0 0
$$171$$ −770.000 −0.344347
$$172$$ 644.000 0.285492
$$173$$ −1512.00 −0.664481 −0.332241 0.943195i $$-0.607805\pi$$
−0.332241 + 0.943195i $$0.607805\pi$$
$$174$$ 1120.00 0.487971
$$175$$ 0 0
$$176$$ −1763.00 −0.755063
$$177$$ 1960.00 0.832331
$$178$$ 945.000 0.397926
$$179$$ 2585.00 1.07940 0.539698 0.841859i $$-0.318538\pi$$
0.539698 + 0.841859i $$0.318538\pi$$
$$180$$ 0 0
$$181$$ −2758.00 −1.13260 −0.566300 0.824199i $$-0.691626\pi$$
−0.566300 + 0.824199i $$0.691626\pi$$
$$182$$ 168.000 0.0684230
$$183$$ 3626.00 1.46471
$$184$$ −2430.00 −0.973598
$$185$$ 0 0
$$186$$ 294.000 0.115899
$$187$$ 3913.00 1.53020
$$188$$ 1372.00 0.532252
$$189$$ −210.000 −0.0808214
$$190$$ 0 0
$$191$$ −2378.00 −0.900869 −0.450435 0.892809i $$-0.648731\pi$$
−0.450435 + 0.892809i $$0.648731\pi$$
$$192$$ 1169.00 0.439403
$$193$$ −3067.00 −1.14387 −0.571937 0.820298i $$-0.693808\pi$$
−0.571937 + 0.820298i $$0.693808\pi$$
$$194$$ 1246.00 0.461122
$$195$$ 0 0
$$196$$ 2149.00 0.783163
$$197$$ −2346.00 −0.848455 −0.424227 0.905556i $$-0.639454\pi$$
−0.424227 + 0.905556i $$0.639454\pi$$
$$198$$ 946.000 0.339542
$$199$$ 4900.00 1.74549 0.872743 0.488180i $$-0.162339\pi$$
0.872743 + 0.488180i $$0.162339\pi$$
$$200$$ 0 0
$$201$$ 987.000 0.346356
$$202$$ −1302.00 −0.453507
$$203$$ −960.000 −0.331915
$$204$$ −4459.00 −1.53036
$$205$$ 0 0
$$206$$ 532.000 0.179933
$$207$$ −3564.00 −1.19669
$$208$$ 1148.00 0.382690
$$209$$ 1505.00 0.498101
$$210$$ 0 0
$$211$$ 4307.00 1.40524 0.702621 0.711564i $$-0.252013\pi$$
0.702621 + 0.711564i $$0.252013\pi$$
$$212$$ 574.000 0.185955
$$213$$ −2884.00 −0.927739
$$214$$ −1269.00 −0.405360
$$215$$ 0 0
$$216$$ 525.000 0.165378
$$217$$ −252.000 −0.0788335
$$218$$ −1070.00 −0.332429
$$219$$ −5341.00 −1.64800
$$220$$ 0 0
$$221$$ −2548.00 −0.775552
$$222$$ 2198.00 0.664505
$$223$$ −2212.00 −0.664244 −0.332122 0.943236i $$-0.607765\pi$$
−0.332122 + 0.943236i $$0.607765\pi$$
$$224$$ 966.000 0.288141
$$225$$ 0 0
$$226$$ −503.000 −0.148049
$$227$$ −476.000 −0.139177 −0.0695886 0.997576i $$-0.522169\pi$$
−0.0695886 + 0.997576i $$0.522169\pi$$
$$228$$ −1715.00 −0.498152
$$229$$ −2940.00 −0.848387 −0.424194 0.905572i $$-0.639442\pi$$
−0.424194 + 0.905572i $$0.639442\pi$$
$$230$$ 0 0
$$231$$ −1806.00 −0.514399
$$232$$ 2400.00 0.679171
$$233$$ −1002.00 −0.281730 −0.140865 0.990029i $$-0.544988\pi$$
−0.140865 + 0.990029i $$0.544988\pi$$
$$234$$ −616.000 −0.172091
$$235$$ 0 0
$$236$$ 1960.00 0.540615
$$237$$ −3570.00 −0.978466
$$238$$ −546.000 −0.148706
$$239$$ 2480.00 0.671204 0.335602 0.942004i $$-0.391060\pi$$
0.335602 + 0.942004i $$0.391060\pi$$
$$240$$ 0 0
$$241$$ 1897.00 0.507039 0.253520 0.967330i $$-0.418412\pi$$
0.253520 + 0.967330i $$0.418412\pi$$
$$242$$ −518.000 −0.137596
$$243$$ 4928.00 1.30095
$$244$$ 3626.00 0.951356
$$245$$ 0 0
$$246$$ −1421.00 −0.368291
$$247$$ −980.000 −0.252453
$$248$$ 630.000 0.161311
$$249$$ 5439.00 1.38427
$$250$$ 0 0
$$251$$ −2373.00 −0.596743 −0.298371 0.954450i $$-0.596443\pi$$
−0.298371 + 0.954450i $$0.596443\pi$$
$$252$$ 924.000 0.230978
$$253$$ 6966.00 1.73102
$$254$$ −874.000 −0.215904
$$255$$ 0 0
$$256$$ −119.000 −0.0290527
$$257$$ 4494.00 1.09077 0.545385 0.838185i $$-0.316383\pi$$
0.545385 + 0.838185i $$0.316383\pi$$
$$258$$ −644.000 −0.155402
$$259$$ −1884.00 −0.451993
$$260$$ 0 0
$$261$$ 3520.00 0.834799
$$262$$ −1092.00 −0.257496
$$263$$ −722.000 −0.169279 −0.0846396 0.996412i $$-0.526974\pi$$
−0.0846396 + 0.996412i $$0.526974\pi$$
$$264$$ 4515.00 1.05257
$$265$$ 0 0
$$266$$ −210.000 −0.0484057
$$267$$ 6615.00 1.51622
$$268$$ 987.000 0.224965
$$269$$ −6160.00 −1.39621 −0.698107 0.715993i $$-0.745974\pi$$
−0.698107 + 0.715993i $$0.745974\pi$$
$$270$$ 0 0
$$271$$ −7238.00 −1.62243 −0.811213 0.584751i $$-0.801192\pi$$
−0.811213 + 0.584751i $$0.801192\pi$$
$$272$$ −3731.00 −0.831710
$$273$$ 1176.00 0.260713
$$274$$ 411.000 0.0906183
$$275$$ 0 0
$$276$$ −7938.00 −1.73120
$$277$$ −1776.00 −0.385233 −0.192616 0.981274i $$-0.561697\pi$$
−0.192616 + 0.981274i $$0.561697\pi$$
$$278$$ 595.000 0.128366
$$279$$ 924.000 0.198274
$$280$$ 0 0
$$281$$ 4542.00 0.964246 0.482123 0.876104i $$-0.339866\pi$$
0.482123 + 0.876104i $$0.339866\pi$$
$$282$$ −1372.00 −0.289721
$$283$$ −7077.00 −1.48652 −0.743258 0.669005i $$-0.766720\pi$$
−0.743258 + 0.669005i $$0.766720\pi$$
$$284$$ −2884.00 −0.602584
$$285$$ 0 0
$$286$$ 1204.00 0.248930
$$287$$ 1218.00 0.250510
$$288$$ −3542.00 −0.724703
$$289$$ 3368.00 0.685528
$$290$$ 0 0
$$291$$ 8722.00 1.75702
$$292$$ −5341.00 −1.07041
$$293$$ 4158.00 0.829054 0.414527 0.910037i $$-0.363947\pi$$
0.414527 + 0.910037i $$0.363947\pi$$
$$294$$ −2149.00 −0.426300
$$295$$ 0 0
$$296$$ 4710.00 0.924876
$$297$$ −1505.00 −0.294037
$$298$$ 3200.00 0.622050
$$299$$ −4536.00 −0.877337
$$300$$ 0 0
$$301$$ 552.000 0.105703
$$302$$ −202.000 −0.0384894
$$303$$ −9114.00 −1.72801
$$304$$ −1435.00 −0.270733
$$305$$ 0 0
$$306$$ 2002.00 0.374009
$$307$$ 2569.00 0.477591 0.238796 0.971070i $$-0.423247\pi$$
0.238796 + 0.971070i $$0.423247\pi$$
$$308$$ −1806.00 −0.334112
$$309$$ 3724.00 0.685602
$$310$$ 0 0
$$311$$ 2982.00 0.543710 0.271855 0.962338i $$-0.412363\pi$$
0.271855 + 0.962338i $$0.412363\pi$$
$$312$$ −2940.00 −0.533477
$$313$$ −2422.00 −0.437379 −0.218689 0.975795i $$-0.570178\pi$$
−0.218689 + 0.975795i $$0.570178\pi$$
$$314$$ 406.000 0.0729679
$$315$$ 0 0
$$316$$ −3570.00 −0.635532
$$317$$ 9484.00 1.68036 0.840181 0.542307i $$-0.182449\pi$$
0.840181 + 0.542307i $$0.182449\pi$$
$$318$$ −574.000 −0.101221
$$319$$ −6880.00 −1.20754
$$320$$ 0 0
$$321$$ −8883.00 −1.54455
$$322$$ −972.000 −0.168222
$$323$$ 3185.00 0.548663
$$324$$ 5873.00 1.00703
$$325$$ 0 0
$$326$$ −3803.00 −0.646100
$$327$$ −7490.00 −1.26666
$$328$$ −3045.00 −0.512598
$$329$$ 1176.00 0.197067
$$330$$ 0 0
$$331$$ −183.000 −0.0303885 −0.0151942 0.999885i $$-0.504837\pi$$
−0.0151942 + 0.999885i $$0.504837\pi$$
$$332$$ 5439.00 0.899108
$$333$$ 6908.00 1.13681
$$334$$ 4116.00 0.674304
$$335$$ 0 0
$$336$$ 1722.00 0.279592
$$337$$ −2861.00 −0.462459 −0.231229 0.972899i $$-0.574275\pi$$
−0.231229 + 0.972899i $$0.574275\pi$$
$$338$$ 1413.00 0.227388
$$339$$ −3521.00 −0.564113
$$340$$ 0 0
$$341$$ −1806.00 −0.286805
$$342$$ 770.000 0.121745
$$343$$ 3900.00 0.613936
$$344$$ −1380.00 −0.216292
$$345$$ 0 0
$$346$$ 1512.00 0.234930
$$347$$ 629.000 0.0973098 0.0486549 0.998816i $$-0.484507\pi$$
0.0486549 + 0.998816i $$0.484507\pi$$
$$348$$ 7840.00 1.20767
$$349$$ 5950.00 0.912597 0.456298 0.889827i $$-0.349175\pi$$
0.456298 + 0.889827i $$0.349175\pi$$
$$350$$ 0 0
$$351$$ 980.000 0.149027
$$352$$ 6923.00 1.04829
$$353$$ 11718.0 1.76682 0.883408 0.468604i $$-0.155243\pi$$
0.883408 + 0.468604i $$0.155243\pi$$
$$354$$ −1960.00 −0.294274
$$355$$ 0 0
$$356$$ 6615.00 0.984815
$$357$$ −3822.00 −0.566615
$$358$$ −2585.00 −0.381624
$$359$$ 8070.00 1.18640 0.593201 0.805054i $$-0.297864\pi$$
0.593201 + 0.805054i $$0.297864\pi$$
$$360$$ 0 0
$$361$$ −5634.00 −0.821403
$$362$$ 2758.00 0.400434
$$363$$ −3626.00 −0.524286
$$364$$ 1176.00 0.169338
$$365$$ 0 0
$$366$$ −3626.00 −0.517853
$$367$$ −8316.00 −1.18281 −0.591406 0.806374i $$-0.701427\pi$$
−0.591406 + 0.806374i $$0.701427\pi$$
$$368$$ −6642.00 −0.940865
$$369$$ −4466.00 −0.630056
$$370$$ 0 0
$$371$$ 492.000 0.0688500
$$372$$ 2058.00 0.286834
$$373$$ −12062.0 −1.67439 −0.837194 0.546906i $$-0.815805\pi$$
−0.837194 + 0.546906i $$0.815805\pi$$
$$374$$ −3913.00 −0.541006
$$375$$ 0 0
$$376$$ −2940.00 −0.403242
$$377$$ 4480.00 0.612021
$$378$$ 210.000 0.0285747
$$379$$ 1735.00 0.235148 0.117574 0.993064i $$-0.462488\pi$$
0.117574 + 0.993064i $$0.462488\pi$$
$$380$$ 0 0
$$381$$ −6118.00 −0.822663
$$382$$ 2378.00 0.318505
$$383$$ −7602.00 −1.01421 −0.507107 0.861883i $$-0.669285\pi$$
−0.507107 + 0.861883i $$0.669285\pi$$
$$384$$ −10185.0 −1.35352
$$385$$ 0 0
$$386$$ 3067.00 0.404420
$$387$$ −2024.00 −0.265855
$$388$$ 8722.00 1.14122
$$389$$ 3030.00 0.394928 0.197464 0.980310i $$-0.436729\pi$$
0.197464 + 0.980310i $$0.436729\pi$$
$$390$$ 0 0
$$391$$ 14742.0 1.90674
$$392$$ −4605.00 −0.593336
$$393$$ −7644.00 −0.981142
$$394$$ 2346.00 0.299974
$$395$$ 0 0
$$396$$ 6622.00 0.840323
$$397$$ 1204.00 0.152209 0.0761046 0.997100i $$-0.475752\pi$$
0.0761046 + 0.997100i $$0.475752\pi$$
$$398$$ −4900.00 −0.617123
$$399$$ −1470.00 −0.184441
$$400$$ 0 0
$$401$$ 1077.00 0.134122 0.0670609 0.997749i $$-0.478638\pi$$
0.0670609 + 0.997749i $$0.478638\pi$$
$$402$$ −987.000 −0.122455
$$403$$ 1176.00 0.145362
$$404$$ −9114.00 −1.12237
$$405$$ 0 0
$$406$$ 960.000 0.117350
$$407$$ −13502.0 −1.64440
$$408$$ 9555.00 1.15942
$$409$$ −3955.00 −0.478147 −0.239074 0.971001i $$-0.576844\pi$$
−0.239074 + 0.971001i $$0.576844\pi$$
$$410$$ 0 0
$$411$$ 2877.00 0.345285
$$412$$ 3724.00 0.445311
$$413$$ 1680.00 0.200163
$$414$$ 3564.00 0.423094
$$415$$ 0 0
$$416$$ −4508.00 −0.531305
$$417$$ 4165.00 0.489115
$$418$$ −1505.00 −0.176105
$$419$$ 6265.00 0.730466 0.365233 0.930916i $$-0.380989\pi$$
0.365233 + 0.930916i $$0.380989\pi$$
$$420$$ 0 0
$$421$$ −3788.00 −0.438517 −0.219259 0.975667i $$-0.570364\pi$$
−0.219259 + 0.975667i $$0.570364\pi$$
$$422$$ −4307.00 −0.496828
$$423$$ −4312.00 −0.495642
$$424$$ −1230.00 −0.140882
$$425$$ 0 0
$$426$$ 2884.00 0.328005
$$427$$ 3108.00 0.352240
$$428$$ −8883.00 −1.00321
$$429$$ 8428.00 0.948503
$$430$$ 0 0
$$431$$ −15258.0 −1.70523 −0.852613 0.522544i $$-0.824983\pi$$
−0.852613 + 0.522544i $$0.824983\pi$$
$$432$$ 1435.00 0.159818
$$433$$ 13573.0 1.50641 0.753206 0.657784i $$-0.228506\pi$$
0.753206 + 0.657784i $$0.228506\pi$$
$$434$$ 252.000 0.0278719
$$435$$ 0 0
$$436$$ −7490.00 −0.822720
$$437$$ 5670.00 0.620670
$$438$$ 5341.00 0.582655
$$439$$ −8120.00 −0.882794 −0.441397 0.897312i $$-0.645517\pi$$
−0.441397 + 0.897312i $$0.645517\pi$$
$$440$$ 0 0
$$441$$ −6754.00 −0.729295
$$442$$ 2548.00 0.274199
$$443$$ 6183.00 0.663122 0.331561 0.943434i $$-0.392425\pi$$
0.331561 + 0.943434i $$0.392425\pi$$
$$444$$ 15386.0 1.64457
$$445$$ 0 0
$$446$$ 2212.00 0.234846
$$447$$ 22400.0 2.37021
$$448$$ 1002.00 0.105670
$$449$$ −1975.00 −0.207586 −0.103793 0.994599i $$-0.533098\pi$$
−0.103793 + 0.994599i $$0.533098\pi$$
$$450$$ 0 0
$$451$$ 8729.00 0.911380
$$452$$ −3521.00 −0.366402
$$453$$ −1414.00 −0.146657
$$454$$ 476.000 0.0492066
$$455$$ 0 0
$$456$$ 3675.00 0.377407
$$457$$ −11831.0 −1.21101 −0.605504 0.795842i $$-0.707029\pi$$
−0.605504 + 0.795842i $$0.707029\pi$$
$$458$$ 2940.00 0.299950
$$459$$ −3185.00 −0.323885
$$460$$ 0 0
$$461$$ 1932.00 0.195189 0.0975946 0.995226i $$-0.468885\pi$$
0.0975946 + 0.995226i $$0.468885\pi$$
$$462$$ 1806.00 0.181867
$$463$$ 9228.00 0.926267 0.463133 0.886289i $$-0.346725\pi$$
0.463133 + 0.886289i $$0.346725\pi$$
$$464$$ 6560.00 0.656337
$$465$$ 0 0
$$466$$ 1002.00 0.0996068
$$467$$ −13916.0 −1.37892 −0.689460 0.724324i $$-0.742152\pi$$
−0.689460 + 0.724324i $$0.742152\pi$$
$$468$$ −4312.00 −0.425902
$$469$$ 846.000 0.0832935
$$470$$ 0 0
$$471$$ 2842.00 0.278031
$$472$$ −4200.00 −0.409578
$$473$$ 3956.00 0.384560
$$474$$ 3570.00 0.345940
$$475$$ 0 0
$$476$$ −3822.00 −0.368027
$$477$$ −1804.00 −0.173165
$$478$$ −2480.00 −0.237307
$$479$$ 2310.00 0.220348 0.110174 0.993912i $$-0.464859\pi$$
0.110174 + 0.993912i $$0.464859\pi$$
$$480$$ 0 0
$$481$$ 8792.00 0.833432
$$482$$ −1897.00 −0.179266
$$483$$ −6804.00 −0.640979
$$484$$ −3626.00 −0.340533
$$485$$ 0 0
$$486$$ −4928.00 −0.459956
$$487$$ 17114.0 1.59242 0.796211 0.605019i $$-0.206835\pi$$
0.796211 + 0.605019i $$0.206835\pi$$
$$488$$ −7770.00 −0.720761
$$489$$ −26621.0 −2.46185
$$490$$ 0 0
$$491$$ −17228.0 −1.58348 −0.791740 0.610858i $$-0.790825\pi$$
−0.791740 + 0.610858i $$0.790825\pi$$
$$492$$ −9947.00 −0.911474
$$493$$ −14560.0 −1.33012
$$494$$ 980.000 0.0892556
$$495$$ 0 0
$$496$$ 1722.00 0.155887
$$497$$ −2472.00 −0.223107
$$498$$ −5439.00 −0.489412
$$499$$ −12500.0 −1.12140 −0.560698 0.828020i $$-0.689467\pi$$
−0.560698 + 0.828020i $$0.689467\pi$$
$$500$$ 0 0
$$501$$ 28812.0 2.56931
$$502$$ 2373.00 0.210980
$$503$$ 868.000 0.0769428 0.0384714 0.999260i $$-0.487751\pi$$
0.0384714 + 0.999260i $$0.487751\pi$$
$$504$$ −1980.00 −0.174992
$$505$$ 0 0
$$506$$ −6966.00 −0.612009
$$507$$ 9891.00 0.866420
$$508$$ −6118.00 −0.534335
$$509$$ 13370.0 1.16427 0.582136 0.813091i $$-0.302217\pi$$
0.582136 + 0.813091i $$0.302217\pi$$
$$510$$ 0 0
$$511$$ −4578.00 −0.396319
$$512$$ −11521.0 −0.994455
$$513$$ −1225.00 −0.105429
$$514$$ −4494.00 −0.385646
$$515$$ 0 0
$$516$$ −4508.00 −0.384600
$$517$$ 8428.00 0.716950
$$518$$ 1884.00 0.159803
$$519$$ 10584.0 0.895156
$$520$$ 0 0
$$521$$ 21637.0 1.81945 0.909726 0.415210i $$-0.136292\pi$$
0.909726 + 0.415210i $$0.136292\pi$$
$$522$$ −3520.00 −0.295146
$$523$$ −287.000 −0.0239955 −0.0119977 0.999928i $$-0.503819\pi$$
−0.0119977 + 0.999928i $$0.503819\pi$$
$$524$$ −7644.00 −0.637270
$$525$$ 0 0
$$526$$ 722.000 0.0598492
$$527$$ −3822.00 −0.315918
$$528$$ 12341.0 1.01718
$$529$$ 14077.0 1.15698
$$530$$ 0 0
$$531$$ −6160.00 −0.503430
$$532$$ −1470.00 −0.119798
$$533$$ −5684.00 −0.461916
$$534$$ −6615.00 −0.536066
$$535$$ 0 0
$$536$$ −2115.00 −0.170437
$$537$$ −18095.0 −1.45411
$$538$$ 6160.00 0.493637
$$539$$ 13201.0 1.05493
$$540$$ 0 0
$$541$$ −5328.00 −0.423417 −0.211709 0.977333i $$-0.567903\pi$$
−0.211709 + 0.977333i $$0.567903\pi$$
$$542$$ 7238.00 0.573614
$$543$$ 19306.0 1.52578
$$544$$ 14651.0 1.15470
$$545$$ 0 0
$$546$$ −1176.00 −0.0921761
$$547$$ −71.0000 −0.00554980 −0.00277490 0.999996i $$-0.500883\pi$$
−0.00277490 + 0.999996i $$0.500883\pi$$
$$548$$ 2877.00 0.224269
$$549$$ −11396.0 −0.885919
$$550$$ 0 0
$$551$$ −5600.00 −0.432973
$$552$$ 17010.0 1.31158
$$553$$ −3060.00 −0.235306
$$554$$ 1776.00 0.136200
$$555$$ 0 0
$$556$$ 4165.00 0.317689
$$557$$ 18444.0 1.40305 0.701524 0.712646i $$-0.252503\pi$$
0.701524 + 0.712646i $$0.252503\pi$$
$$558$$ −924.000 −0.0701004
$$559$$ −2576.00 −0.194907
$$560$$ 0 0
$$561$$ −27391.0 −2.06141
$$562$$ −4542.00 −0.340912
$$563$$ −672.000 −0.0503045 −0.0251522 0.999684i $$-0.508007\pi$$
−0.0251522 + 0.999684i $$0.508007\pi$$
$$564$$ −9604.00 −0.717024
$$565$$ 0 0
$$566$$ 7077.00 0.525563
$$567$$ 5034.00 0.372854
$$568$$ 6180.00 0.456526
$$569$$ −10935.0 −0.805657 −0.402829 0.915275i $$-0.631973\pi$$
−0.402829 + 0.915275i $$0.631973\pi$$
$$570$$ 0 0
$$571$$ −13588.0 −0.995867 −0.497934 0.867215i $$-0.665908\pi$$
−0.497934 + 0.867215i $$0.665908\pi$$
$$572$$ 8428.00 0.616071
$$573$$ 16646.0 1.21361
$$574$$ −1218.00 −0.0885685
$$575$$ 0 0
$$576$$ −3674.00 −0.265770
$$577$$ −8701.00 −0.627777 −0.313889 0.949460i $$-0.601632\pi$$
−0.313889 + 0.949460i $$0.601632\pi$$
$$578$$ −3368.00 −0.242371
$$579$$ 21469.0 1.54097
$$580$$ 0 0
$$581$$ 4662.00 0.332896
$$582$$ −8722.00 −0.621200
$$583$$ 3526.00 0.250484
$$584$$ 11445.0 0.810955
$$585$$ 0 0
$$586$$ −4158.00 −0.293115
$$587$$ −11361.0 −0.798839 −0.399420 0.916768i $$-0.630788\pi$$
−0.399420 + 0.916768i $$0.630788\pi$$
$$588$$ −15043.0 −1.05504
$$589$$ −1470.00 −0.102836
$$590$$ 0 0
$$591$$ 16422.0 1.14300
$$592$$ 12874.0 0.893781
$$593$$ −11417.0 −0.790624 −0.395312 0.918547i $$-0.629364\pi$$
−0.395312 + 0.918547i $$0.629364\pi$$
$$594$$ 1505.00 0.103958
$$595$$ 0 0
$$596$$ 22400.0 1.53950
$$597$$ −34300.0 −2.35143
$$598$$ 4536.00 0.310185
$$599$$ −21050.0 −1.43586 −0.717930 0.696116i $$-0.754910\pi$$
−0.717930 + 0.696116i $$0.754910\pi$$
$$600$$ 0 0
$$601$$ 7427.00 0.504083 0.252041 0.967716i $$-0.418898\pi$$
0.252041 + 0.967716i $$0.418898\pi$$
$$602$$ −552.000 −0.0373718
$$603$$ −3102.00 −0.209491
$$604$$ −1414.00 −0.0952564
$$605$$ 0 0
$$606$$ 9114.00 0.610942
$$607$$ 4144.00 0.277100 0.138550 0.990355i $$-0.455756\pi$$
0.138550 + 0.990355i $$0.455756\pi$$
$$608$$ 5635.00 0.375871
$$609$$ 6720.00 0.447140
$$610$$ 0 0
$$611$$ −5488.00 −0.363373
$$612$$ 14014.0 0.925625
$$613$$ −30122.0 −1.98469 −0.992346 0.123489i $$-0.960592\pi$$
−0.992346 + 0.123489i $$0.960592\pi$$
$$614$$ −2569.00 −0.168854
$$615$$ 0 0
$$616$$ 3870.00 0.253128
$$617$$ 11934.0 0.778679 0.389339 0.921094i $$-0.372703\pi$$
0.389339 + 0.921094i $$0.372703\pi$$
$$618$$ −3724.00 −0.242397
$$619$$ 8540.00 0.554526 0.277263 0.960794i $$-0.410573\pi$$
0.277263 + 0.960794i $$0.410573\pi$$
$$620$$ 0 0
$$621$$ −5670.00 −0.366392
$$622$$ −2982.00 −0.192230
$$623$$ 5670.00 0.364629
$$624$$ −8036.00 −0.515541
$$625$$ 0 0
$$626$$ 2422.00 0.154637
$$627$$ −10535.0 −0.671017
$$628$$ 2842.00 0.180586
$$629$$ −28574.0 −1.81132
$$630$$ 0 0
$$631$$ −3158.00 −0.199236 −0.0996181 0.995026i $$-0.531762\pi$$
−0.0996181 + 0.995026i $$0.531762\pi$$
$$632$$ 7650.00 0.481488
$$633$$ −30149.0 −1.89307
$$634$$ −9484.00 −0.594097
$$635$$ 0 0
$$636$$ −4018.00 −0.250510
$$637$$ −8596.00 −0.534672
$$638$$ 6880.00 0.426931
$$639$$ 9064.00 0.561137
$$640$$ 0 0
$$641$$ −4278.00 −0.263605 −0.131803 0.991276i $$-0.542076\pi$$
−0.131803 + 0.991276i $$0.542076\pi$$
$$642$$ 8883.00 0.546081
$$643$$ 11508.0 0.705803 0.352901 0.935661i $$-0.385195\pi$$
0.352901 + 0.935661i $$0.385195\pi$$
$$644$$ −6804.00 −0.416328
$$645$$ 0 0
$$646$$ −3185.00 −0.193982
$$647$$ 8204.00 0.498505 0.249252 0.968439i $$-0.419815\pi$$
0.249252 + 0.968439i $$0.419815\pi$$
$$648$$ −12585.0 −0.762941
$$649$$ 12040.0 0.728215
$$650$$ 0 0
$$651$$ 1764.00 0.106201
$$652$$ −26621.0 −1.59902
$$653$$ 5518.00 0.330683 0.165342 0.986236i $$-0.447127\pi$$
0.165342 + 0.986236i $$0.447127\pi$$
$$654$$ 7490.00 0.447832
$$655$$ 0 0
$$656$$ −8323.00 −0.495364
$$657$$ 16786.0 0.996780
$$658$$ −1176.00 −0.0696736
$$659$$ 13295.0 0.785887 0.392944 0.919563i $$-0.371457\pi$$
0.392944 + 0.919563i $$0.371457\pi$$
$$660$$ 0 0
$$661$$ −9968.00 −0.586551 −0.293276 0.956028i $$-0.594745\pi$$
−0.293276 + 0.956028i $$0.594745\pi$$
$$662$$ 183.000 0.0107440
$$663$$ 17836.0 1.04479
$$664$$ −11655.0 −0.681177
$$665$$ 0 0
$$666$$ −6908.00 −0.401921
$$667$$ −25920.0 −1.50469
$$668$$ 28812.0 1.66882
$$669$$ 15484.0 0.894837
$$670$$ 0 0
$$671$$ 22274.0 1.28149
$$672$$ −6762.00 −0.388169
$$673$$ 15738.0 0.901419 0.450710 0.892671i $$-0.351171\pi$$
0.450710 + 0.892671i $$0.351171\pi$$
$$674$$ 2861.00 0.163504
$$675$$ 0 0
$$676$$ 9891.00 0.562756
$$677$$ 19824.0 1.12540 0.562702 0.826660i $$-0.309762\pi$$
0.562702 + 0.826660i $$0.309762\pi$$
$$678$$ 3521.00 0.199444
$$679$$ 7476.00 0.422537
$$680$$ 0 0
$$681$$ 3332.00 0.187493
$$682$$ 1806.00 0.101401
$$683$$ 11073.0 0.620346 0.310173 0.950680i $$-0.399613\pi$$
0.310173 + 0.950680i $$0.399613\pi$$
$$684$$ 5390.00 0.301304
$$685$$ 0 0
$$686$$ −3900.00 −0.217059
$$687$$ 20580.0 1.14291
$$688$$ −3772.00 −0.209021
$$689$$ −2296.00 −0.126953
$$690$$ 0 0
$$691$$ −6503.00 −0.358011 −0.179006 0.983848i $$-0.557288\pi$$
−0.179006 + 0.983848i $$0.557288\pi$$
$$692$$ 10584.0 0.581421
$$693$$ 5676.00 0.311130
$$694$$ −629.000 −0.0344042
$$695$$ 0 0
$$696$$ −16800.0 −0.914946
$$697$$ 18473.0 1.00389
$$698$$ −5950.00 −0.322652
$$699$$ 7014.00 0.379533
$$700$$ 0 0
$$701$$ −10148.0 −0.546768 −0.273384 0.961905i $$-0.588143\pi$$
−0.273384 + 0.961905i $$0.588143\pi$$
$$702$$ −980.000 −0.0526891
$$703$$ −10990.0 −0.589610
$$704$$ 7181.00 0.384438
$$705$$ 0 0
$$706$$ −11718.0 −0.624664
$$707$$ −7812.00 −0.415559
$$708$$ −13720.0 −0.728290
$$709$$ −9980.00 −0.528641 −0.264321 0.964435i $$-0.585148\pi$$
−0.264321 + 0.964435i $$0.585148\pi$$
$$710$$ 0 0
$$711$$ 11220.0 0.591818
$$712$$ −14175.0 −0.746110
$$713$$ −6804.00 −0.357380
$$714$$ 3822.00 0.200329
$$715$$ 0 0
$$716$$ −18095.0 −0.944472
$$717$$ −17360.0 −0.904214
$$718$$ −8070.00 −0.419456
$$719$$ −27510.0 −1.42691 −0.713456 0.700700i $$-0.752871\pi$$
−0.713456 + 0.700700i $$0.752871\pi$$
$$720$$ 0 0
$$721$$ 3192.00 0.164877
$$722$$ 5634.00 0.290410
$$723$$ −13279.0 −0.683059
$$724$$ 19306.0 0.991025
$$725$$ 0 0
$$726$$ 3626.00 0.185363
$$727$$ 17024.0 0.868480 0.434240 0.900797i $$-0.357017\pi$$
0.434240 + 0.900797i $$0.357017\pi$$
$$728$$ −2520.00 −0.128293
$$729$$ −11843.0 −0.601687
$$730$$ 0 0
$$731$$ 8372.00 0.423597
$$732$$ −25382.0 −1.28162
$$733$$ 34748.0 1.75095 0.875475 0.483263i $$-0.160549\pi$$
0.875475 + 0.483263i $$0.160549\pi$$
$$734$$ 8316.00 0.418187
$$735$$ 0 0
$$736$$ 26082.0 1.30624
$$737$$ 6063.00 0.303030
$$738$$ 4466.00 0.222758
$$739$$ −12020.0 −0.598326 −0.299163 0.954202i $$-0.596707\pi$$
−0.299163 + 0.954202i $$0.596707\pi$$
$$740$$ 0 0
$$741$$ 6860.00 0.340092
$$742$$ −492.000 −0.0243422
$$743$$ −28642.0 −1.41423 −0.707115 0.707098i $$-0.750004\pi$$
−0.707115 + 0.707098i $$0.750004\pi$$
$$744$$ −4410.00 −0.217310
$$745$$ 0 0
$$746$$ 12062.0 0.591986
$$747$$ −17094.0 −0.837265
$$748$$ −27391.0 −1.33892
$$749$$ −7614.00 −0.371441
$$750$$ 0 0
$$751$$ 8752.00 0.425253 0.212627 0.977134i $$-0.431798\pi$$
0.212627 + 0.977134i $$0.431798\pi$$
$$752$$ −8036.00 −0.389685
$$753$$ 16611.0 0.803902
$$754$$ −4480.00 −0.216382
$$755$$ 0 0
$$756$$ 1470.00 0.0707188
$$757$$ −10256.0 −0.492418 −0.246209 0.969217i $$-0.579185\pi$$
−0.246209 + 0.969217i $$0.579185\pi$$
$$758$$ −1735.00 −0.0831373
$$759$$ −48762.0 −2.33195
$$760$$ 0 0
$$761$$ 33957.0 1.61753 0.808765 0.588132i $$-0.200136\pi$$
0.808765 + 0.588132i $$0.200136\pi$$
$$762$$ 6118.00 0.290855
$$763$$ −6420.00 −0.304613
$$764$$ 16646.0 0.788261
$$765$$ 0 0
$$766$$ 7602.00 0.358579
$$767$$ −7840.00 −0.369082
$$768$$ 833.000 0.0391384
$$769$$ 27965.0 1.31137 0.655685 0.755034i $$-0.272380\pi$$
0.655685 + 0.755034i $$0.272380\pi$$
$$770$$ 0 0
$$771$$ −31458.0 −1.46943
$$772$$ 21469.0 1.00089
$$773$$ −9912.00 −0.461203 −0.230601 0.973048i $$-0.574069\pi$$
−0.230601 + 0.973048i $$0.574069\pi$$
$$774$$ 2024.00 0.0939938
$$775$$ 0 0
$$776$$ −18690.0 −0.864603
$$777$$ 13188.0 0.608902
$$778$$ −3030.00 −0.139628
$$779$$ 7105.00 0.326782
$$780$$ 0 0
$$781$$ −17716.0 −0.811688
$$782$$ −14742.0 −0.674134
$$783$$ 5600.00 0.255591
$$784$$ −12587.0 −0.573387
$$785$$ 0 0
$$786$$ 7644.00 0.346886
$$787$$ 25564.0 1.15789 0.578944 0.815367i $$-0.303465\pi$$
0.578944 + 0.815367i $$0.303465\pi$$
$$788$$ 16422.0 0.742398
$$789$$ 5054.00 0.228045
$$790$$ 0 0
$$791$$ −3018.00 −0.135661
$$792$$ −14190.0 −0.636641
$$793$$ −14504.0 −0.649498
$$794$$ −1204.00 −0.0538141
$$795$$ 0 0
$$796$$ −34300.0 −1.52730
$$797$$ −12446.0 −0.553149 −0.276575 0.960992i $$-0.589199\pi$$
−0.276575 + 0.960992i $$0.589199\pi$$
$$798$$ 1470.00 0.0652098
$$799$$ 17836.0 0.789728
$$800$$ 0 0
$$801$$ −20790.0 −0.917077
$$802$$ −1077.00 −0.0474192
$$803$$ −32809.0 −1.44185
$$804$$ −6909.00 −0.303062
$$805$$ 0 0
$$806$$ −1176.00 −0.0513931
$$807$$ 43120.0 1.88091
$$808$$ 19530.0 0.850325
$$809$$ 33970.0 1.47629 0.738147 0.674640i $$-0.235701\pi$$
0.738147 + 0.674640i $$0.235701\pi$$
$$810$$ 0 0
$$811$$ 18732.0 0.811060 0.405530 0.914082i $$-0.367087\pi$$
0.405530 + 0.914082i $$0.367087\pi$$
$$812$$ 6720.00 0.290426
$$813$$ 50666.0 2.18565
$$814$$ 13502.0 0.581382
$$815$$ 0 0
$$816$$ 26117.0 1.12044
$$817$$ 3220.00 0.137887
$$818$$ 3955.00 0.169051
$$819$$ −3696.00 −0.157691
$$820$$ 0 0
$$821$$ 6162.00 0.261943 0.130972 0.991386i $$-0.458190\pi$$
0.130972 + 0.991386i $$0.458190\pi$$
$$822$$ −2877.00 −0.122077
$$823$$ 25388.0 1.07530 0.537649 0.843169i $$-0.319313\pi$$
0.537649 + 0.843169i $$0.319313\pi$$
$$824$$ −7980.00 −0.337374
$$825$$ 0 0
$$826$$ −1680.00 −0.0707684
$$827$$ −25201.0 −1.05964 −0.529821 0.848109i $$-0.677741\pi$$
−0.529821 + 0.848109i $$0.677741\pi$$
$$828$$ 24948.0 1.04710
$$829$$ −19740.0 −0.827019 −0.413509 0.910500i $$-0.635697\pi$$
−0.413509 + 0.910500i $$0.635697\pi$$
$$830$$ 0 0
$$831$$ 12432.0 0.518967
$$832$$ −4676.00 −0.194845
$$833$$ 27937.0 1.16202
$$834$$ −4165.00 −0.172928
$$835$$ 0 0
$$836$$ −10535.0 −0.435838
$$837$$ 1470.00 0.0607057
$$838$$ −6265.00 −0.258259
$$839$$ 29680.0 1.22130 0.610648 0.791902i $$-0.290909\pi$$
0.610648 + 0.791902i $$0.290909\pi$$
$$840$$ 0 0
$$841$$ 1211.00 0.0496535
$$842$$ 3788.00 0.155039
$$843$$ −31794.0 −1.29898
$$844$$ −30149.0 −1.22959
$$845$$ 0 0
$$846$$ 4312.00 0.175236
$$847$$ −3108.00 −0.126083
$$848$$ −3362.00 −0.136146
$$849$$ 49539.0 2.00256
$$850$$ 0 0
$$851$$ −50868.0 −2.04904
$$852$$ 20188.0 0.811772
$$853$$ 1218.00 0.0488904 0.0244452 0.999701i $$-0.492218\pi$$
0.0244452 + 0.999701i $$0.492218\pi$$
$$854$$ −3108.00 −0.124536
$$855$$ 0 0
$$856$$ 19035.0 0.760050
$$857$$ −38731.0 −1.54379 −0.771894 0.635752i $$-0.780690\pi$$
−0.771894 + 0.635752i $$0.780690\pi$$
$$858$$ −8428.00 −0.335346
$$859$$ −23555.0 −0.935607 −0.467803 0.883833i $$-0.654954\pi$$
−0.467803 + 0.883833i $$0.654954\pi$$
$$860$$ 0 0
$$861$$ −8526.00 −0.337474
$$862$$ 15258.0 0.602888
$$863$$ −24872.0 −0.981058 −0.490529 0.871425i $$-0.663196\pi$$
−0.490529 + 0.871425i $$0.663196\pi$$
$$864$$ −5635.00 −0.221883
$$865$$ 0 0
$$866$$ −13573.0 −0.532597
$$867$$ −23576.0 −0.923510
$$868$$ 1764.00 0.0689793
$$869$$ −21930.0 −0.856069
$$870$$ 0 0
$$871$$ −3948.00 −0.153585
$$872$$ 16050.0 0.623305
$$873$$ −27412.0 −1.06272
$$874$$ −5670.00 −0.219440
$$875$$ 0 0
$$876$$ 37387.0 1.44200
$$877$$ 17124.0 0.659335 0.329667 0.944097i $$-0.393063\pi$$
0.329667 + 0.944097i $$0.393063\pi$$
$$878$$ 8120.00 0.312115
$$879$$ −29106.0 −1.11686
$$880$$ 0 0
$$881$$ −658.000 −0.0251630 −0.0125815 0.999921i $$-0.504005\pi$$
−0.0125815 + 0.999921i $$0.504005\pi$$
$$882$$ 6754.00 0.257845
$$883$$ −33727.0 −1.28540 −0.642698 0.766120i $$-0.722185\pi$$
−0.642698 + 0.766120i $$0.722185\pi$$
$$884$$ 17836.0 0.678608
$$885$$ 0 0
$$886$$ −6183.00 −0.234449
$$887$$ −36036.0 −1.36412 −0.682058 0.731298i $$-0.738915\pi$$
−0.682058 + 0.731298i $$0.738915\pi$$
$$888$$ −32970.0 −1.24595
$$889$$ −5244.00 −0.197838
$$890$$ 0 0
$$891$$ 36077.0 1.35648
$$892$$ 15484.0 0.581214
$$893$$ 6860.00 0.257067
$$894$$ −22400.0 −0.837996
$$895$$ 0 0
$$896$$ −8730.00 −0.325501
$$897$$ 31752.0 1.18190
$$898$$ 1975.00 0.0733927
$$899$$ 6720.00 0.249304
$$900$$ 0 0
$$901$$ 7462.00 0.275910
$$902$$ −8729.00 −0.322222
$$903$$ −3864.00 −0.142399
$$904$$ 7545.00 0.277592
$$905$$ 0 0
$$906$$ 1414.00 0.0518510
$$907$$ −39156.0 −1.43347 −0.716733 0.697348i $$-0.754363\pi$$
−0.716733 + 0.697348i $$0.754363\pi$$
$$908$$ 3332.00 0.121780
$$909$$ 28644.0 1.04517
$$910$$ 0 0
$$911$$ 43532.0 1.58318 0.791591 0.611051i $$-0.209253\pi$$
0.791591 + 0.611051i $$0.209253\pi$$
$$912$$ 10045.0 0.364718
$$913$$ 33411.0 1.21111
$$914$$ 11831.0 0.428156
$$915$$ 0 0
$$916$$ 20580.0 0.742339
$$917$$ −6552.00 −0.235950
$$918$$ 3185.00 0.114511
$$919$$ −28610.0 −1.02694 −0.513469 0.858108i $$-0.671640\pi$$
−0.513469 + 0.858108i $$0.671640\pi$$
$$920$$ 0 0
$$921$$ −17983.0 −0.643388
$$922$$ −1932.00 −0.0690098
$$923$$ 11536.0 0.411389
$$924$$ 12642.0 0.450099
$$925$$ 0 0
$$926$$ −9228.00 −0.327485
$$927$$ −11704.0 −0.414682
$$928$$ −25760.0 −0.911221
$$929$$ −24290.0 −0.857835 −0.428918 0.903344i $$-0.641105\pi$$
−0.428918 + 0.903344i $$0.641105\pi$$
$$930$$ 0 0
$$931$$ 10745.0 0.378253
$$932$$ 7014.00 0.246514
$$933$$ −20874.0 −0.732459
$$934$$ 13916.0 0.487522
$$935$$ 0 0
$$936$$ 9240.00 0.322670
$$937$$ −34461.0 −1.20149 −0.600743 0.799442i $$-0.705128\pi$$
−0.600743 + 0.799442i $$0.705128\pi$$
$$938$$ −846.000 −0.0294487
$$939$$ 16954.0 0.589215
$$940$$ 0 0
$$941$$ −40628.0 −1.40748 −0.703738 0.710460i $$-0.748487\pi$$
−0.703738 + 0.710460i $$0.748487\pi$$
$$942$$ −2842.00 −0.0982987
$$943$$ 32886.0 1.13565
$$944$$ −11480.0 −0.395807
$$945$$ 0 0
$$946$$ −3956.00 −0.135963
$$947$$ 20904.0 0.717306 0.358653 0.933471i $$-0.383236\pi$$
0.358653 + 0.933471i $$0.383236\pi$$
$$948$$ 24990.0 0.856158
$$949$$ 21364.0 0.730774
$$950$$ 0 0
$$951$$ −66388.0 −2.26370
$$952$$ 8190.00 0.278823
$$953$$ −1807.00 −0.0614213 −0.0307106 0.999528i $$-0.509777\pi$$
−0.0307106 + 0.999528i $$0.509777\pi$$
$$954$$ 1804.00 0.0612229
$$955$$ 0 0
$$956$$ −17360.0 −0.587304
$$957$$ 48160.0 1.62674
$$958$$ −2310.00 −0.0779047
$$959$$ 2466.00 0.0830358
$$960$$ 0 0
$$961$$ −28027.0 −0.940787
$$962$$ −8792.00 −0.294663
$$963$$ 27918.0 0.934211
$$964$$ −13279.0 −0.443660
$$965$$ 0 0
$$966$$ 6804.00 0.226620
$$967$$ 57584.0 1.91497 0.957485 0.288482i $$-0.0931505\pi$$
0.957485 + 0.288482i $$0.0931505\pi$$
$$968$$ 7770.00 0.257993
$$969$$ −22295.0 −0.739132
$$970$$ 0 0
$$971$$ 27237.0 0.900182 0.450091 0.892983i $$-0.351392\pi$$
0.450091 + 0.892983i $$0.351392\pi$$
$$972$$ −34496.0 −1.13833
$$973$$ 3570.00 0.117625
$$974$$ −17114.0 −0.563006
$$975$$ 0 0
$$976$$ −21238.0 −0.696528
$$977$$ 13649.0 0.446950 0.223475 0.974710i $$-0.428260\pi$$
0.223475 + 0.974710i $$0.428260\pi$$
$$978$$ 26621.0 0.870394
$$979$$ 40635.0 1.32656
$$980$$ 0 0
$$981$$ 23540.0 0.766131
$$982$$ 17228.0 0.559845
$$983$$ −16002.0 −0.519211 −0.259606 0.965715i $$-0.583593\pi$$
−0.259606 + 0.965715i $$0.583593\pi$$
$$984$$ 21315.0 0.690546
$$985$$ 0 0
$$986$$ 14560.0 0.470269
$$987$$ −8232.00 −0.265479
$$988$$ 6860.00 0.220896
$$989$$ 14904.0 0.479191
$$990$$ 0 0
$$991$$ 37022.0 1.18672 0.593362 0.804936i $$-0.297800\pi$$
0.593362 + 0.804936i $$0.297800\pi$$
$$992$$ −6762.00 −0.216425
$$993$$ 1281.00 0.0409379
$$994$$ 2472.00 0.0788804
$$995$$ 0 0
$$996$$ −38073.0 −1.21123
$$997$$ −18396.0 −0.584360 −0.292180 0.956363i $$-0.594381\pi$$
−0.292180 + 0.956363i $$0.594381\pi$$
$$998$$ 12500.0 0.396474
$$999$$ 10990.0 0.348056
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 25.4.a.a.1.1 1
3.2 odd 2 225.4.a.e.1.1 1
4.3 odd 2 400.4.a.s.1.1 1
5.2 odd 4 25.4.b.b.24.1 2
5.3 odd 4 25.4.b.b.24.2 2
5.4 even 2 25.4.a.b.1.1 yes 1
7.6 odd 2 1225.4.a.h.1.1 1
8.3 odd 2 1600.4.a.h.1.1 1
8.5 even 2 1600.4.a.bt.1.1 1
15.2 even 4 225.4.b.f.199.2 2
15.8 even 4 225.4.b.f.199.1 2
15.14 odd 2 225.4.a.c.1.1 1
20.3 even 4 400.4.c.e.49.2 2
20.7 even 4 400.4.c.e.49.1 2
20.19 odd 2 400.4.a.c.1.1 1
35.34 odd 2 1225.4.a.i.1.1 1
40.19 odd 2 1600.4.a.bs.1.1 1
40.29 even 2 1600.4.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
25.4.a.a.1.1 1 1.1 even 1 trivial
25.4.a.b.1.1 yes 1 5.4 even 2
25.4.b.b.24.1 2 5.2 odd 4
25.4.b.b.24.2 2 5.3 odd 4
225.4.a.c.1.1 1 15.14 odd 2
225.4.a.e.1.1 1 3.2 odd 2
225.4.b.f.199.1 2 15.8 even 4
225.4.b.f.199.2 2 15.2 even 4
400.4.a.c.1.1 1 20.19 odd 2
400.4.a.s.1.1 1 4.3 odd 2
400.4.c.e.49.1 2 20.7 even 4
400.4.c.e.49.2 2 20.3 even 4
1225.4.a.h.1.1 1 7.6 odd 2
1225.4.a.i.1.1 1 35.34 odd 2
1600.4.a.h.1.1 1 8.3 odd 2
1600.4.a.i.1.1 1 40.29 even 2
1600.4.a.bs.1.1 1 40.19 odd 2
1600.4.a.bt.1.1 1 8.5 even 2