# Properties

 Label 1680.4.a.s.1.1 Level $1680$ Weight $4$ Character 1680.1 Self dual yes Analytic conductor $99.123$ 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] = [1680,4,Mod(1,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1680.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1680.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: $$99.1232088096$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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 Character $$\chi$$ $$=$$ 1680.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -42.0000 q^{11} +20.0000 q^{13} +15.0000 q^{15} +66.0000 q^{17} -38.0000 q^{19} -21.0000 q^{21} -12.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -258.000 q^{29} -146.000 q^{31} -126.000 q^{33} -35.0000 q^{35} +434.000 q^{37} +60.0000 q^{39} -282.000 q^{41} -20.0000 q^{43} +45.0000 q^{45} +72.0000 q^{47} +49.0000 q^{49} +198.000 q^{51} +336.000 q^{53} -210.000 q^{55} -114.000 q^{57} +360.000 q^{59} -682.000 q^{61} -63.0000 q^{63} +100.000 q^{65} -812.000 q^{67} -36.0000 q^{69} -810.000 q^{71} -124.000 q^{73} +75.0000 q^{75} +294.000 q^{77} -1136.00 q^{79} +81.0000 q^{81} -156.000 q^{83} +330.000 q^{85} -774.000 q^{87} -1038.00 q^{89} -140.000 q^{91} -438.000 q^{93} -190.000 q^{95} +1208.00 q^{97} -378.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −7.00000 −0.377964
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −42.0000 −1.15123 −0.575613 0.817723i $$-0.695236\pi$$
−0.575613 + 0.817723i $$0.695236\pi$$
$$12$$ 0 0
$$13$$ 20.0000 0.426692 0.213346 0.976977i $$-0.431564\pi$$
0.213346 + 0.976977i $$0.431564\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ 0 0
$$17$$ 66.0000 0.941609 0.470804 0.882238i $$-0.343964\pi$$
0.470804 + 0.882238i $$0.343964\pi$$
$$18$$ 0 0
$$19$$ −38.0000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −21.0000 −0.218218
$$22$$ 0 0
$$23$$ −12.0000 −0.108790 −0.0543951 0.998519i $$-0.517323\pi$$
−0.0543951 + 0.998519i $$0.517323\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −258.000 −1.65205 −0.826024 0.563635i $$-0.809403\pi$$
−0.826024 + 0.563635i $$0.809403\pi$$
$$30$$ 0 0
$$31$$ −146.000 −0.845883 −0.422942 0.906157i $$-0.639002\pi$$
−0.422942 + 0.906157i $$0.639002\pi$$
$$32$$ 0 0
$$33$$ −126.000 −0.664660
$$34$$ 0 0
$$35$$ −35.0000 −0.169031
$$36$$ 0 0
$$37$$ 434.000 1.92836 0.964178 0.265257i $$-0.0854567\pi$$
0.964178 + 0.265257i $$0.0854567\pi$$
$$38$$ 0 0
$$39$$ 60.0000 0.246351
$$40$$ 0 0
$$41$$ −282.000 −1.07417 −0.537085 0.843528i $$-0.680475\pi$$
−0.537085 + 0.843528i $$0.680475\pi$$
$$42$$ 0 0
$$43$$ −20.0000 −0.0709296 −0.0354648 0.999371i $$-0.511291\pi$$
−0.0354648 + 0.999371i $$0.511291\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ 0 0
$$47$$ 72.0000 0.223453 0.111726 0.993739i $$-0.464362\pi$$
0.111726 + 0.993739i $$0.464362\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 198.000 0.543638
$$52$$ 0 0
$$53$$ 336.000 0.870814 0.435407 0.900234i $$-0.356604\pi$$
0.435407 + 0.900234i $$0.356604\pi$$
$$54$$ 0 0
$$55$$ −210.000 −0.514844
$$56$$ 0 0
$$57$$ −114.000 −0.264906
$$58$$ 0 0
$$59$$ 360.000 0.794373 0.397187 0.917738i $$-0.369987\pi$$
0.397187 + 0.917738i $$0.369987\pi$$
$$60$$ 0 0
$$61$$ −682.000 −1.43149 −0.715747 0.698360i $$-0.753914\pi$$
−0.715747 + 0.698360i $$0.753914\pi$$
$$62$$ 0 0
$$63$$ −63.0000 −0.125988
$$64$$ 0 0
$$65$$ 100.000 0.190823
$$66$$ 0 0
$$67$$ −812.000 −1.48062 −0.740310 0.672265i $$-0.765321\pi$$
−0.740310 + 0.672265i $$0.765321\pi$$
$$68$$ 0 0
$$69$$ −36.0000 −0.0628100
$$70$$ 0 0
$$71$$ −810.000 −1.35393 −0.676967 0.736013i $$-0.736706\pi$$
−0.676967 + 0.736013i $$0.736706\pi$$
$$72$$ 0 0
$$73$$ −124.000 −0.198810 −0.0994048 0.995047i $$-0.531694\pi$$
−0.0994048 + 0.995047i $$0.531694\pi$$
$$74$$ 0 0
$$75$$ 75.0000 0.115470
$$76$$ 0 0
$$77$$ 294.000 0.435122
$$78$$ 0 0
$$79$$ −1136.00 −1.61785 −0.808924 0.587913i $$-0.799950\pi$$
−0.808924 + 0.587913i $$0.799950\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −156.000 −0.206304 −0.103152 0.994666i $$-0.532893\pi$$
−0.103152 + 0.994666i $$0.532893\pi$$
$$84$$ 0 0
$$85$$ 330.000 0.421100
$$86$$ 0 0
$$87$$ −774.000 −0.953810
$$88$$ 0 0
$$89$$ −1038.00 −1.23627 −0.618134 0.786073i $$-0.712111\pi$$
−0.618134 + 0.786073i $$0.712111\pi$$
$$90$$ 0 0
$$91$$ −140.000 −0.161275
$$92$$ 0 0
$$93$$ −438.000 −0.488371
$$94$$ 0 0
$$95$$ −190.000 −0.205196
$$96$$ 0 0
$$97$$ 1208.00 1.26447 0.632236 0.774776i $$-0.282137\pi$$
0.632236 + 0.774776i $$0.282137\pi$$
$$98$$ 0 0
$$99$$ −378.000 −0.383742
$$100$$ 0 0
$$101$$ 546.000 0.537911 0.268956 0.963153i $$-0.413322\pi$$
0.268956 + 0.963153i $$0.413322\pi$$
$$102$$ 0 0
$$103$$ 520.000 0.497448 0.248724 0.968574i $$-0.419989\pi$$
0.248724 + 0.968574i $$0.419989\pi$$
$$104$$ 0 0
$$105$$ −105.000 −0.0975900
$$106$$ 0 0
$$107$$ −1212.00 −1.09503 −0.547516 0.836795i $$-0.684427\pi$$
−0.547516 + 0.836795i $$0.684427\pi$$
$$108$$ 0 0
$$109$$ −1078.00 −0.947281 −0.473641 0.880718i $$-0.657060\pi$$
−0.473641 + 0.880718i $$0.657060\pi$$
$$110$$ 0 0
$$111$$ 1302.00 1.11334
$$112$$ 0 0
$$113$$ −1452.00 −1.20878 −0.604392 0.796687i $$-0.706584\pi$$
−0.604392 + 0.796687i $$0.706584\pi$$
$$114$$ 0 0
$$115$$ −60.0000 −0.0486524
$$116$$ 0 0
$$117$$ 180.000 0.142231
$$118$$ 0 0
$$119$$ −462.000 −0.355895
$$120$$ 0 0
$$121$$ 433.000 0.325319
$$122$$ 0 0
$$123$$ −846.000 −0.620173
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 1312.00 0.916702 0.458351 0.888771i $$-0.348440\pi$$
0.458351 + 0.888771i $$0.348440\pi$$
$$128$$ 0 0
$$129$$ −60.0000 −0.0409512
$$130$$ 0 0
$$131$$ 1356.00 0.904384 0.452192 0.891921i $$-0.350642\pi$$
0.452192 + 0.891921i $$0.350642\pi$$
$$132$$ 0 0
$$133$$ 266.000 0.173422
$$134$$ 0 0
$$135$$ 135.000 0.0860663
$$136$$ 0 0
$$137$$ −984.000 −0.613641 −0.306820 0.951767i $$-0.599265\pi$$
−0.306820 + 0.951767i $$0.599265\pi$$
$$138$$ 0 0
$$139$$ 394.000 0.240422 0.120211 0.992748i $$-0.461643\pi$$
0.120211 + 0.992748i $$0.461643\pi$$
$$140$$ 0 0
$$141$$ 216.000 0.129011
$$142$$ 0 0
$$143$$ −840.000 −0.491219
$$144$$ 0 0
$$145$$ −1290.00 −0.738818
$$146$$ 0 0
$$147$$ 147.000 0.0824786
$$148$$ 0 0
$$149$$ −1014.00 −0.557518 −0.278759 0.960361i $$-0.589923\pi$$
−0.278759 + 0.960361i $$0.589923\pi$$
$$150$$ 0 0
$$151$$ 1996.00 1.07571 0.537855 0.843037i $$-0.319235\pi$$
0.537855 + 0.843037i $$0.319235\pi$$
$$152$$ 0 0
$$153$$ 594.000 0.313870
$$154$$ 0 0
$$155$$ −730.000 −0.378290
$$156$$ 0 0
$$157$$ −2392.00 −1.21594 −0.607969 0.793960i $$-0.708016\pi$$
−0.607969 + 0.793960i $$0.708016\pi$$
$$158$$ 0 0
$$159$$ 1008.00 0.502765
$$160$$ 0 0
$$161$$ 84.0000 0.0411188
$$162$$ 0 0
$$163$$ −2036.00 −0.978355 −0.489177 0.872184i $$-0.662703\pi$$
−0.489177 + 0.872184i $$0.662703\pi$$
$$164$$ 0 0
$$165$$ −630.000 −0.297245
$$166$$ 0 0
$$167$$ 3936.00 1.82381 0.911907 0.410398i $$-0.134610\pi$$
0.911907 + 0.410398i $$0.134610\pi$$
$$168$$ 0 0
$$169$$ −1797.00 −0.817934
$$170$$ 0 0
$$171$$ −342.000 −0.152944
$$172$$ 0 0
$$173$$ 378.000 0.166120 0.0830601 0.996545i $$-0.473531\pi$$
0.0830601 + 0.996545i $$0.473531\pi$$
$$174$$ 0 0
$$175$$ −175.000 −0.0755929
$$176$$ 0 0
$$177$$ 1080.00 0.458631
$$178$$ 0 0
$$179$$ 222.000 0.0926987 0.0463493 0.998925i $$-0.485241\pi$$
0.0463493 + 0.998925i $$0.485241\pi$$
$$180$$ 0 0
$$181$$ −2590.00 −1.06361 −0.531804 0.846867i $$-0.678486\pi$$
−0.531804 + 0.846867i $$0.678486\pi$$
$$182$$ 0 0
$$183$$ −2046.00 −0.826474
$$184$$ 0 0
$$185$$ 2170.00 0.862387
$$186$$ 0 0
$$187$$ −2772.00 −1.08400
$$188$$ 0 0
$$189$$ −189.000 −0.0727393
$$190$$ 0 0
$$191$$ −2214.00 −0.838740 −0.419370 0.907815i $$-0.637749\pi$$
−0.419370 + 0.907815i $$0.637749\pi$$
$$192$$ 0 0
$$193$$ 4178.00 1.55823 0.779117 0.626879i $$-0.215668\pi$$
0.779117 + 0.626879i $$0.215668\pi$$
$$194$$ 0 0
$$195$$ 300.000 0.110172
$$196$$ 0 0
$$197$$ −3060.00 −1.10668 −0.553340 0.832955i $$-0.686647\pi$$
−0.553340 + 0.832955i $$0.686647\pi$$
$$198$$ 0 0
$$199$$ −2666.00 −0.949687 −0.474844 0.880070i $$-0.657495\pi$$
−0.474844 + 0.880070i $$0.657495\pi$$
$$200$$ 0 0
$$201$$ −2436.00 −0.854837
$$202$$ 0 0
$$203$$ 1806.00 0.624416
$$204$$ 0 0
$$205$$ −1410.00 −0.480384
$$206$$ 0 0
$$207$$ −108.000 −0.0362634
$$208$$ 0 0
$$209$$ 1596.00 0.528218
$$210$$ 0 0
$$211$$ 1348.00 0.439811 0.219906 0.975521i $$-0.429425\pi$$
0.219906 + 0.975521i $$0.429425\pi$$
$$212$$ 0 0
$$213$$ −2430.00 −0.781694
$$214$$ 0 0
$$215$$ −100.000 −0.0317207
$$216$$ 0 0
$$217$$ 1022.00 0.319714
$$218$$ 0 0
$$219$$ −372.000 −0.114783
$$220$$ 0 0
$$221$$ 1320.00 0.401777
$$222$$ 0 0
$$223$$ −3188.00 −0.957329 −0.478664 0.877998i $$-0.658879\pi$$
−0.478664 + 0.877998i $$0.658879\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ 3396.00 0.992953 0.496477 0.868050i $$-0.334627\pi$$
0.496477 + 0.868050i $$0.334627\pi$$
$$228$$ 0 0
$$229$$ 5294.00 1.52767 0.763837 0.645409i $$-0.223313\pi$$
0.763837 + 0.645409i $$0.223313\pi$$
$$230$$ 0 0
$$231$$ 882.000 0.251218
$$232$$ 0 0
$$233$$ 852.000 0.239555 0.119778 0.992801i $$-0.461782\pi$$
0.119778 + 0.992801i $$0.461782\pi$$
$$234$$ 0 0
$$235$$ 360.000 0.0999311
$$236$$ 0 0
$$237$$ −3408.00 −0.934065
$$238$$ 0 0
$$239$$ −4866.00 −1.31697 −0.658484 0.752595i $$-0.728802\pi$$
−0.658484 + 0.752595i $$0.728802\pi$$
$$240$$ 0 0
$$241$$ −2050.00 −0.547934 −0.273967 0.961739i $$-0.588336\pi$$
−0.273967 + 0.961739i $$0.588336\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 245.000 0.0638877
$$246$$ 0 0
$$247$$ −760.000 −0.195780
$$248$$ 0 0
$$249$$ −468.000 −0.119110
$$250$$ 0 0
$$251$$ 1152.00 0.289696 0.144848 0.989454i $$-0.453731\pi$$
0.144848 + 0.989454i $$0.453731\pi$$
$$252$$ 0 0
$$253$$ 504.000 0.125242
$$254$$ 0 0
$$255$$ 990.000 0.243122
$$256$$ 0 0
$$257$$ −6450.00 −1.56553 −0.782763 0.622321i $$-0.786190\pi$$
−0.782763 + 0.622321i $$0.786190\pi$$
$$258$$ 0 0
$$259$$ −3038.00 −0.728850
$$260$$ 0 0
$$261$$ −2322.00 −0.550683
$$262$$ 0 0
$$263$$ −1968.00 −0.461415 −0.230707 0.973023i $$-0.574104\pi$$
−0.230707 + 0.973023i $$0.574104\pi$$
$$264$$ 0 0
$$265$$ 1680.00 0.389440
$$266$$ 0 0
$$267$$ −3114.00 −0.713759
$$268$$ 0 0
$$269$$ −3894.00 −0.882607 −0.441304 0.897358i $$-0.645484\pi$$
−0.441304 + 0.897358i $$0.645484\pi$$
$$270$$ 0 0
$$271$$ −7094.00 −1.59015 −0.795073 0.606513i $$-0.792568\pi$$
−0.795073 + 0.606513i $$0.792568\pi$$
$$272$$ 0 0
$$273$$ −420.000 −0.0931119
$$274$$ 0 0
$$275$$ −1050.00 −0.230245
$$276$$ 0 0
$$277$$ −3310.00 −0.717973 −0.358987 0.933343i $$-0.616878\pi$$
−0.358987 + 0.933343i $$0.616878\pi$$
$$278$$ 0 0
$$279$$ −1314.00 −0.281961
$$280$$ 0 0
$$281$$ 7158.00 1.51961 0.759805 0.650151i $$-0.225294\pi$$
0.759805 + 0.650151i $$0.225294\pi$$
$$282$$ 0 0
$$283$$ 5164.00 1.08469 0.542346 0.840155i $$-0.317536\pi$$
0.542346 + 0.840155i $$0.317536\pi$$
$$284$$ 0 0
$$285$$ −570.000 −0.118470
$$286$$ 0 0
$$287$$ 1974.00 0.405998
$$288$$ 0 0
$$289$$ −557.000 −0.113373
$$290$$ 0 0
$$291$$ 3624.00 0.730043
$$292$$ 0 0
$$293$$ −8598.00 −1.71434 −0.857168 0.515037i $$-0.827778\pi$$
−0.857168 + 0.515037i $$0.827778\pi$$
$$294$$ 0 0
$$295$$ 1800.00 0.355254
$$296$$ 0 0
$$297$$ −1134.00 −0.221553
$$298$$ 0 0
$$299$$ −240.000 −0.0464199
$$300$$ 0 0
$$301$$ 140.000 0.0268089
$$302$$ 0 0
$$303$$ 1638.00 0.310563
$$304$$ 0 0
$$305$$ −3410.00 −0.640184
$$306$$ 0 0
$$307$$ 448.000 0.0832857 0.0416429 0.999133i $$-0.486741\pi$$
0.0416429 + 0.999133i $$0.486741\pi$$
$$308$$ 0 0
$$309$$ 1560.00 0.287202
$$310$$ 0 0
$$311$$ 5832.00 1.06335 0.531676 0.846948i $$-0.321562\pi$$
0.531676 + 0.846948i $$0.321562\pi$$
$$312$$ 0 0
$$313$$ 9848.00 1.77841 0.889204 0.457510i $$-0.151259\pi$$
0.889204 + 0.457510i $$0.151259\pi$$
$$314$$ 0 0
$$315$$ −315.000 −0.0563436
$$316$$ 0 0
$$317$$ −5616.00 −0.995035 −0.497517 0.867454i $$-0.665755\pi$$
−0.497517 + 0.867454i $$0.665755\pi$$
$$318$$ 0 0
$$319$$ 10836.0 1.90188
$$320$$ 0 0
$$321$$ −3636.00 −0.632217
$$322$$ 0 0
$$323$$ −2508.00 −0.432040
$$324$$ 0 0
$$325$$ 500.000 0.0853385
$$326$$ 0 0
$$327$$ −3234.00 −0.546913
$$328$$ 0 0
$$329$$ −504.000 −0.0844572
$$330$$ 0 0
$$331$$ −452.000 −0.0750579 −0.0375290 0.999296i $$-0.511949\pi$$
−0.0375290 + 0.999296i $$0.511949\pi$$
$$332$$ 0 0
$$333$$ 3906.00 0.642785
$$334$$ 0 0
$$335$$ −4060.00 −0.662154
$$336$$ 0 0
$$337$$ −2302.00 −0.372101 −0.186050 0.982540i $$-0.559569\pi$$
−0.186050 + 0.982540i $$0.559569\pi$$
$$338$$ 0 0
$$339$$ −4356.00 −0.697892
$$340$$ 0 0
$$341$$ 6132.00 0.973802
$$342$$ 0 0
$$343$$ −343.000 −0.0539949
$$344$$ 0 0
$$345$$ −180.000 −0.0280895
$$346$$ 0 0
$$347$$ −1584.00 −0.245054 −0.122527 0.992465i $$-0.539100\pi$$
−0.122527 + 0.992465i $$0.539100\pi$$
$$348$$ 0 0
$$349$$ 8174.00 1.25371 0.626854 0.779137i $$-0.284342\pi$$
0.626854 + 0.779137i $$0.284342\pi$$
$$350$$ 0 0
$$351$$ 540.000 0.0821170
$$352$$ 0 0
$$353$$ 8610.00 1.29820 0.649099 0.760704i $$-0.275146\pi$$
0.649099 + 0.760704i $$0.275146\pi$$
$$354$$ 0 0
$$355$$ −4050.00 −0.605498
$$356$$ 0 0
$$357$$ −1386.00 −0.205476
$$358$$ 0 0
$$359$$ 2154.00 0.316668 0.158334 0.987386i $$-0.449388\pi$$
0.158334 + 0.987386i $$0.449388\pi$$
$$360$$ 0 0
$$361$$ −5415.00 −0.789474
$$362$$ 0 0
$$363$$ 1299.00 0.187823
$$364$$ 0 0
$$365$$ −620.000 −0.0889104
$$366$$ 0 0
$$367$$ −6644.00 −0.944997 −0.472499 0.881331i $$-0.656648\pi$$
−0.472499 + 0.881331i $$0.656648\pi$$
$$368$$ 0 0
$$369$$ −2538.00 −0.358057
$$370$$ 0 0
$$371$$ −2352.00 −0.329137
$$372$$ 0 0
$$373$$ 7958.00 1.10469 0.552345 0.833615i $$-0.313733\pi$$
0.552345 + 0.833615i $$0.313733\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 0 0
$$377$$ −5160.00 −0.704917
$$378$$ 0 0
$$379$$ −3440.00 −0.466229 −0.233115 0.972449i $$-0.574892\pi$$
−0.233115 + 0.972449i $$0.574892\pi$$
$$380$$ 0 0
$$381$$ 3936.00 0.529258
$$382$$ 0 0
$$383$$ −12936.0 −1.72585 −0.862923 0.505336i $$-0.831369\pi$$
−0.862923 + 0.505336i $$0.831369\pi$$
$$384$$ 0 0
$$385$$ 1470.00 0.194593
$$386$$ 0 0
$$387$$ −180.000 −0.0236432
$$388$$ 0 0
$$389$$ −14862.0 −1.93710 −0.968552 0.248812i $$-0.919960\pi$$
−0.968552 + 0.248812i $$0.919960\pi$$
$$390$$ 0 0
$$391$$ −792.000 −0.102438
$$392$$ 0 0
$$393$$ 4068.00 0.522146
$$394$$ 0 0
$$395$$ −5680.00 −0.723524
$$396$$ 0 0
$$397$$ 10460.0 1.32235 0.661174 0.750232i $$-0.270058\pi$$
0.661174 + 0.750232i $$0.270058\pi$$
$$398$$ 0 0
$$399$$ 798.000 0.100125
$$400$$ 0 0
$$401$$ −9150.00 −1.13947 −0.569737 0.821827i $$-0.692955\pi$$
−0.569737 + 0.821827i $$0.692955\pi$$
$$402$$ 0 0
$$403$$ −2920.00 −0.360932
$$404$$ 0 0
$$405$$ 405.000 0.0496904
$$406$$ 0 0
$$407$$ −18228.0 −2.21997
$$408$$ 0 0
$$409$$ −4894.00 −0.591669 −0.295835 0.955239i $$-0.595598\pi$$
−0.295835 + 0.955239i $$0.595598\pi$$
$$410$$ 0 0
$$411$$ −2952.00 −0.354286
$$412$$ 0 0
$$413$$ −2520.00 −0.300245
$$414$$ 0 0
$$415$$ −780.000 −0.0922619
$$416$$ 0 0
$$417$$ 1182.00 0.138808
$$418$$ 0 0
$$419$$ 1668.00 0.194480 0.0972400 0.995261i $$-0.468999\pi$$
0.0972400 + 0.995261i $$0.468999\pi$$
$$420$$ 0 0
$$421$$ −12418.0 −1.43757 −0.718784 0.695233i $$-0.755301\pi$$
−0.718784 + 0.695233i $$0.755301\pi$$
$$422$$ 0 0
$$423$$ 648.000 0.0744843
$$424$$ 0 0
$$425$$ 1650.00 0.188322
$$426$$ 0 0
$$427$$ 4774.00 0.541054
$$428$$ 0 0
$$429$$ −2520.00 −0.283605
$$430$$ 0 0
$$431$$ −15186.0 −1.69718 −0.848589 0.529052i $$-0.822548\pi$$
−0.848589 + 0.529052i $$0.822548\pi$$
$$432$$ 0 0
$$433$$ −5704.00 −0.633064 −0.316532 0.948582i $$-0.602518\pi$$
−0.316532 + 0.948582i $$0.602518\pi$$
$$434$$ 0 0
$$435$$ −3870.00 −0.426557
$$436$$ 0 0
$$437$$ 456.000 0.0499163
$$438$$ 0 0
$$439$$ 17206.0 1.87061 0.935305 0.353843i $$-0.115125\pi$$
0.935305 + 0.353843i $$0.115125\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 0 0
$$443$$ −3456.00 −0.370654 −0.185327 0.982677i $$-0.559334\pi$$
−0.185327 + 0.982677i $$0.559334\pi$$
$$444$$ 0 0
$$445$$ −5190.00 −0.552875
$$446$$ 0 0
$$447$$ −3042.00 −0.321883
$$448$$ 0 0
$$449$$ 16074.0 1.68949 0.844743 0.535173i $$-0.179753\pi$$
0.844743 + 0.535173i $$0.179753\pi$$
$$450$$ 0 0
$$451$$ 11844.0 1.23661
$$452$$ 0 0
$$453$$ 5988.00 0.621061
$$454$$ 0 0
$$455$$ −700.000 −0.0721242
$$456$$ 0 0
$$457$$ 7526.00 0.770353 0.385177 0.922843i $$-0.374141\pi$$
0.385177 + 0.922843i $$0.374141\pi$$
$$458$$ 0 0
$$459$$ 1782.00 0.181213
$$460$$ 0 0
$$461$$ −2274.00 −0.229741 −0.114871 0.993380i $$-0.536645\pi$$
−0.114871 + 0.993380i $$0.536645\pi$$
$$462$$ 0 0
$$463$$ 10024.0 1.00617 0.503083 0.864238i $$-0.332199\pi$$
0.503083 + 0.864238i $$0.332199\pi$$
$$464$$ 0 0
$$465$$ −2190.00 −0.218406
$$466$$ 0 0
$$467$$ 2460.00 0.243759 0.121879 0.992545i $$-0.461108\pi$$
0.121879 + 0.992545i $$0.461108\pi$$
$$468$$ 0 0
$$469$$ 5684.00 0.559622
$$470$$ 0 0
$$471$$ −7176.00 −0.702023
$$472$$ 0 0
$$473$$ 840.000 0.0816559
$$474$$ 0 0
$$475$$ −950.000 −0.0917663
$$476$$ 0 0
$$477$$ 3024.00 0.290271
$$478$$ 0 0
$$479$$ −19320.0 −1.84291 −0.921454 0.388486i $$-0.872998\pi$$
−0.921454 + 0.388486i $$0.872998\pi$$
$$480$$ 0 0
$$481$$ 8680.00 0.822815
$$482$$ 0 0
$$483$$ 252.000 0.0237400
$$484$$ 0 0
$$485$$ 6040.00 0.565489
$$486$$ 0 0
$$487$$ 12544.0 1.16719 0.583596 0.812044i $$-0.301645\pi$$
0.583596 + 0.812044i $$0.301645\pi$$
$$488$$ 0 0
$$489$$ −6108.00 −0.564853
$$490$$ 0 0
$$491$$ 15510.0 1.42557 0.712787 0.701381i $$-0.247433\pi$$
0.712787 + 0.701381i $$0.247433\pi$$
$$492$$ 0 0
$$493$$ −17028.0 −1.55558
$$494$$ 0 0
$$495$$ −1890.00 −0.171615
$$496$$ 0 0
$$497$$ 5670.00 0.511739
$$498$$ 0 0
$$499$$ 14344.0 1.28682 0.643412 0.765520i $$-0.277518\pi$$
0.643412 + 0.765520i $$0.277518\pi$$
$$500$$ 0 0
$$501$$ 11808.0 1.05298
$$502$$ 0 0
$$503$$ 21384.0 1.89556 0.947779 0.318929i $$-0.103323\pi$$
0.947779 + 0.318929i $$0.103323\pi$$
$$504$$ 0 0
$$505$$ 2730.00 0.240561
$$506$$ 0 0
$$507$$ −5391.00 −0.472234
$$508$$ 0 0
$$509$$ −7134.00 −0.621236 −0.310618 0.950535i $$-0.600536\pi$$
−0.310618 + 0.950535i $$0.600536\pi$$
$$510$$ 0 0
$$511$$ 868.000 0.0751430
$$512$$ 0 0
$$513$$ −1026.00 −0.0883022
$$514$$ 0 0
$$515$$ 2600.00 0.222465
$$516$$ 0 0
$$517$$ −3024.00 −0.257244
$$518$$ 0 0
$$519$$ 1134.00 0.0959096
$$520$$ 0 0
$$521$$ −19122.0 −1.60797 −0.803983 0.594653i $$-0.797290\pi$$
−0.803983 + 0.594653i $$0.797290\pi$$
$$522$$ 0 0
$$523$$ 15640.0 1.30763 0.653814 0.756655i $$-0.273168\pi$$
0.653814 + 0.756655i $$0.273168\pi$$
$$524$$ 0 0
$$525$$ −525.000 −0.0436436
$$526$$ 0 0
$$527$$ −9636.00 −0.796491
$$528$$ 0 0
$$529$$ −12023.0 −0.988165
$$530$$ 0 0
$$531$$ 3240.00 0.264791
$$532$$ 0 0
$$533$$ −5640.00 −0.458341
$$534$$ 0 0
$$535$$ −6060.00 −0.489713
$$536$$ 0 0
$$537$$ 666.000 0.0535196
$$538$$ 0 0
$$539$$ −2058.00 −0.164461
$$540$$ 0 0
$$541$$ 2846.00 0.226172 0.113086 0.993585i $$-0.463926\pi$$
0.113086 + 0.993585i $$0.463926\pi$$
$$542$$ 0 0
$$543$$ −7770.00 −0.614075
$$544$$ 0 0
$$545$$ −5390.00 −0.423637
$$546$$ 0 0
$$547$$ 4444.00 0.347371 0.173685 0.984801i $$-0.444432\pi$$
0.173685 + 0.984801i $$0.444432\pi$$
$$548$$ 0 0
$$549$$ −6138.00 −0.477165
$$550$$ 0 0
$$551$$ 9804.00 0.758012
$$552$$ 0 0
$$553$$ 7952.00 0.611489
$$554$$ 0 0
$$555$$ 6510.00 0.497899
$$556$$ 0 0
$$557$$ 18552.0 1.41126 0.705631 0.708579i $$-0.250663\pi$$
0.705631 + 0.708579i $$0.250663\pi$$
$$558$$ 0 0
$$559$$ −400.000 −0.0302651
$$560$$ 0 0
$$561$$ −8316.00 −0.625850
$$562$$ 0 0
$$563$$ 16452.0 1.23156 0.615781 0.787918i $$-0.288841\pi$$
0.615781 + 0.787918i $$0.288841\pi$$
$$564$$ 0 0
$$565$$ −7260.00 −0.540585
$$566$$ 0 0
$$567$$ −567.000 −0.0419961
$$568$$ 0 0
$$569$$ 7722.00 0.568933 0.284467 0.958686i $$-0.408183\pi$$
0.284467 + 0.958686i $$0.408183\pi$$
$$570$$ 0 0
$$571$$ −2576.00 −0.188796 −0.0943978 0.995535i $$-0.530093\pi$$
−0.0943978 + 0.995535i $$0.530093\pi$$
$$572$$ 0 0
$$573$$ −6642.00 −0.484247
$$574$$ 0 0
$$575$$ −300.000 −0.0217580
$$576$$ 0 0
$$577$$ −2464.00 −0.177778 −0.0888888 0.996042i $$-0.528332\pi$$
−0.0888888 + 0.996042i $$0.528332\pi$$
$$578$$ 0 0
$$579$$ 12534.0 0.899646
$$580$$ 0 0
$$581$$ 1092.00 0.0779755
$$582$$ 0 0
$$583$$ −14112.0 −1.00250
$$584$$ 0 0
$$585$$ 900.000 0.0636076
$$586$$ 0 0
$$587$$ −1452.00 −0.102096 −0.0510481 0.998696i $$-0.516256\pi$$
−0.0510481 + 0.998696i $$0.516256\pi$$
$$588$$ 0 0
$$589$$ 5548.00 0.388118
$$590$$ 0 0
$$591$$ −9180.00 −0.638942
$$592$$ 0 0
$$593$$ 10698.0 0.740833 0.370417 0.928866i $$-0.379215\pi$$
0.370417 + 0.928866i $$0.379215\pi$$
$$594$$ 0 0
$$595$$ −2310.00 −0.159161
$$596$$ 0 0
$$597$$ −7998.00 −0.548302
$$598$$ 0 0
$$599$$ 8730.00 0.595489 0.297745 0.954646i $$-0.403766\pi$$
0.297745 + 0.954646i $$0.403766\pi$$
$$600$$ 0 0
$$601$$ 1910.00 0.129635 0.0648174 0.997897i $$-0.479354\pi$$
0.0648174 + 0.997897i $$0.479354\pi$$
$$602$$ 0 0
$$603$$ −7308.00 −0.493540
$$604$$ 0 0
$$605$$ 2165.00 0.145487
$$606$$ 0 0
$$607$$ 5596.00 0.374192 0.187096 0.982342i $$-0.440092\pi$$
0.187096 + 0.982342i $$0.440092\pi$$
$$608$$ 0 0
$$609$$ 5418.00 0.360506
$$610$$ 0 0
$$611$$ 1440.00 0.0953456
$$612$$ 0 0
$$613$$ 28586.0 1.88349 0.941744 0.336332i $$-0.109186\pi$$
0.941744 + 0.336332i $$0.109186\pi$$
$$614$$ 0 0
$$615$$ −4230.00 −0.277350
$$616$$ 0 0
$$617$$ −19236.0 −1.25513 −0.627563 0.778566i $$-0.715947\pi$$
−0.627563 + 0.778566i $$0.715947\pi$$
$$618$$ 0 0
$$619$$ −6734.00 −0.437257 −0.218629 0.975808i $$-0.570158\pi$$
−0.218629 + 0.975808i $$0.570158\pi$$
$$620$$ 0 0
$$621$$ −324.000 −0.0209367
$$622$$ 0 0
$$623$$ 7266.00 0.467265
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 4788.00 0.304967
$$628$$ 0 0
$$629$$ 28644.0 1.81576
$$630$$ 0 0
$$631$$ −7184.00 −0.453234 −0.226617 0.973984i $$-0.572767\pi$$
−0.226617 + 0.973984i $$0.572767\pi$$
$$632$$ 0 0
$$633$$ 4044.00 0.253925
$$634$$ 0 0
$$635$$ 6560.00 0.409962
$$636$$ 0 0
$$637$$ 980.000 0.0609561
$$638$$ 0 0
$$639$$ −7290.00 −0.451311
$$640$$ 0 0
$$641$$ 510.000 0.0314256 0.0157128 0.999877i $$-0.494998\pi$$
0.0157128 + 0.999877i $$0.494998\pi$$
$$642$$ 0 0
$$643$$ 20752.0 1.27275 0.636376 0.771379i $$-0.280433\pi$$
0.636376 + 0.771379i $$0.280433\pi$$
$$644$$ 0 0
$$645$$ −300.000 −0.0183139
$$646$$ 0 0
$$647$$ −21072.0 −1.28041 −0.640205 0.768204i $$-0.721151\pi$$
−0.640205 + 0.768204i $$0.721151\pi$$
$$648$$ 0 0
$$649$$ −15120.0 −0.914502
$$650$$ 0 0
$$651$$ 3066.00 0.184587
$$652$$ 0 0
$$653$$ 2892.00 0.173312 0.0866560 0.996238i $$-0.472382\pi$$
0.0866560 + 0.996238i $$0.472382\pi$$
$$654$$ 0 0
$$655$$ 6780.00 0.404453
$$656$$ 0 0
$$657$$ −1116.00 −0.0662699
$$658$$ 0 0
$$659$$ −750.000 −0.0443336 −0.0221668 0.999754i $$-0.507056\pi$$
−0.0221668 + 0.999754i $$0.507056\pi$$
$$660$$ 0 0
$$661$$ 30062.0 1.76895 0.884475 0.466587i $$-0.154517\pi$$
0.884475 + 0.466587i $$0.154517\pi$$
$$662$$ 0 0
$$663$$ 3960.00 0.231966
$$664$$ 0 0
$$665$$ 1330.00 0.0775567
$$666$$ 0 0
$$667$$ 3096.00 0.179727
$$668$$ 0 0
$$669$$ −9564.00 −0.552714
$$670$$ 0 0
$$671$$ 28644.0 1.64797
$$672$$ 0 0
$$673$$ 15446.0 0.884695 0.442347 0.896844i $$-0.354146\pi$$
0.442347 + 0.896844i $$0.354146\pi$$
$$674$$ 0 0
$$675$$ 675.000 0.0384900
$$676$$ 0 0
$$677$$ −25110.0 −1.42549 −0.712744 0.701424i $$-0.752548\pi$$
−0.712744 + 0.701424i $$0.752548\pi$$
$$678$$ 0 0
$$679$$ −8456.00 −0.477926
$$680$$ 0 0
$$681$$ 10188.0 0.573282
$$682$$ 0 0
$$683$$ −7968.00 −0.446394 −0.223197 0.974773i $$-0.571649\pi$$
−0.223197 + 0.974773i $$0.571649\pi$$
$$684$$ 0 0
$$685$$ −4920.00 −0.274429
$$686$$ 0 0
$$687$$ 15882.0 0.882003
$$688$$ 0 0
$$689$$ 6720.00 0.371570
$$690$$ 0 0
$$691$$ 14398.0 0.792657 0.396328 0.918109i $$-0.370284\pi$$
0.396328 + 0.918109i $$0.370284\pi$$
$$692$$ 0 0
$$693$$ 2646.00 0.145041
$$694$$ 0 0
$$695$$ 1970.00 0.107520
$$696$$ 0 0
$$697$$ −18612.0 −1.01145
$$698$$ 0 0
$$699$$ 2556.00 0.138307
$$700$$ 0 0
$$701$$ −9234.00 −0.497523 −0.248761 0.968565i $$-0.580023\pi$$
−0.248761 + 0.968565i $$0.580023\pi$$
$$702$$ 0 0
$$703$$ −16492.0 −0.884790
$$704$$ 0 0
$$705$$ 1080.00 0.0576953
$$706$$ 0 0
$$707$$ −3822.00 −0.203311
$$708$$ 0 0
$$709$$ 8030.00 0.425350 0.212675 0.977123i $$-0.431782\pi$$
0.212675 + 0.977123i $$0.431782\pi$$
$$710$$ 0 0
$$711$$ −10224.0 −0.539283
$$712$$ 0 0
$$713$$ 1752.00 0.0920237
$$714$$ 0 0
$$715$$ −4200.00 −0.219680
$$716$$ 0 0
$$717$$ −14598.0 −0.760352
$$718$$ 0 0
$$719$$ −27060.0 −1.40357 −0.701786 0.712388i $$-0.747614\pi$$
−0.701786 + 0.712388i $$0.747614\pi$$
$$720$$ 0 0
$$721$$ −3640.00 −0.188018
$$722$$ 0 0
$$723$$ −6150.00 −0.316350
$$724$$ 0 0
$$725$$ −6450.00 −0.330410
$$726$$ 0 0
$$727$$ 3724.00 0.189980 0.0949900 0.995478i $$-0.469718\pi$$
0.0949900 + 0.995478i $$0.469718\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −1320.00 −0.0667879
$$732$$ 0 0
$$733$$ −5668.00 −0.285610 −0.142805 0.989751i $$-0.545612\pi$$
−0.142805 + 0.989751i $$0.545612\pi$$
$$734$$ 0 0
$$735$$ 735.000 0.0368856
$$736$$ 0 0
$$737$$ 34104.0 1.70453
$$738$$ 0 0
$$739$$ 16072.0 0.800024 0.400012 0.916510i $$-0.369006\pi$$
0.400012 + 0.916510i $$0.369006\pi$$
$$740$$ 0 0
$$741$$ −2280.00 −0.113034
$$742$$ 0 0
$$743$$ 8256.00 0.407649 0.203825 0.979007i $$-0.434663\pi$$
0.203825 + 0.979007i $$0.434663\pi$$
$$744$$ 0 0
$$745$$ −5070.00 −0.249329
$$746$$ 0 0
$$747$$ −1404.00 −0.0687680
$$748$$ 0 0
$$749$$ 8484.00 0.413883
$$750$$ 0 0
$$751$$ 6352.00 0.308639 0.154319 0.988021i $$-0.450682\pi$$
0.154319 + 0.988021i $$0.450682\pi$$
$$752$$ 0 0
$$753$$ 3456.00 0.167256
$$754$$ 0 0
$$755$$ 9980.00 0.481072
$$756$$ 0 0
$$757$$ 11558.0 0.554931 0.277465 0.960736i $$-0.410506\pi$$
0.277465 + 0.960736i $$0.410506\pi$$
$$758$$ 0 0
$$759$$ 1512.00 0.0723085
$$760$$ 0 0
$$761$$ 7770.00 0.370121 0.185061 0.982727i $$-0.440752\pi$$
0.185061 + 0.982727i $$0.440752\pi$$
$$762$$ 0 0
$$763$$ 7546.00 0.358039
$$764$$ 0 0
$$765$$ 2970.00 0.140367
$$766$$ 0 0
$$767$$ 7200.00 0.338953
$$768$$ 0 0
$$769$$ 22646.0 1.06194 0.530972 0.847389i $$-0.321827\pi$$
0.530972 + 0.847389i $$0.321827\pi$$
$$770$$ 0 0
$$771$$ −19350.0 −0.903856
$$772$$ 0 0
$$773$$ −35502.0 −1.65190 −0.825950 0.563744i $$-0.809361\pi$$
−0.825950 + 0.563744i $$0.809361\pi$$
$$774$$ 0 0
$$775$$ −3650.00 −0.169177
$$776$$ 0 0
$$777$$ −9114.00 −0.420802
$$778$$ 0 0
$$779$$ 10716.0 0.492863
$$780$$ 0 0
$$781$$ 34020.0 1.55868
$$782$$ 0 0
$$783$$ −6966.00 −0.317937
$$784$$ 0 0
$$785$$ −11960.0 −0.543784
$$786$$ 0 0
$$787$$ 17080.0 0.773617 0.386808 0.922160i $$-0.373578\pi$$
0.386808 + 0.922160i $$0.373578\pi$$
$$788$$ 0 0
$$789$$ −5904.00 −0.266398
$$790$$ 0 0
$$791$$ 10164.0 0.456878
$$792$$ 0 0
$$793$$ −13640.0 −0.610808
$$794$$ 0 0
$$795$$ 5040.00 0.224843
$$796$$ 0 0
$$797$$ 5730.00 0.254664 0.127332 0.991860i $$-0.459359\pi$$
0.127332 + 0.991860i $$0.459359\pi$$
$$798$$ 0 0
$$799$$ 4752.00 0.210405
$$800$$ 0 0
$$801$$ −9342.00 −0.412089
$$802$$ 0 0
$$803$$ 5208.00 0.228875
$$804$$ 0 0
$$805$$ 420.000 0.0183889
$$806$$ 0 0
$$807$$ −11682.0 −0.509574
$$808$$ 0 0
$$809$$ 2550.00 0.110820 0.0554099 0.998464i $$-0.482353\pi$$
0.0554099 + 0.998464i $$0.482353\pi$$
$$810$$ 0 0
$$811$$ 27538.0 1.19234 0.596171 0.802857i $$-0.296688\pi$$
0.596171 + 0.802857i $$0.296688\pi$$
$$812$$ 0 0
$$813$$ −21282.0 −0.918072
$$814$$ 0 0
$$815$$ −10180.0 −0.437534
$$816$$ 0 0
$$817$$ 760.000 0.0325447
$$818$$ 0 0
$$819$$ −1260.00 −0.0537582
$$820$$ 0 0
$$821$$ −19242.0 −0.817966 −0.408983 0.912542i $$-0.634117\pi$$
−0.408983 + 0.912542i $$0.634117\pi$$
$$822$$ 0 0
$$823$$ 11752.0 0.497751 0.248875 0.968536i $$-0.419939\pi$$
0.248875 + 0.968536i $$0.419939\pi$$
$$824$$ 0 0
$$825$$ −3150.00 −0.132932
$$826$$ 0 0
$$827$$ 28692.0 1.20643 0.603216 0.797578i $$-0.293886\pi$$
0.603216 + 0.797578i $$0.293886\pi$$
$$828$$ 0 0
$$829$$ 28442.0 1.19159 0.595797 0.803135i $$-0.296836\pi$$
0.595797 + 0.803135i $$0.296836\pi$$
$$830$$ 0 0
$$831$$ −9930.00 −0.414522
$$832$$ 0 0
$$833$$ 3234.00 0.134516
$$834$$ 0 0
$$835$$ 19680.0 0.815634
$$836$$ 0 0
$$837$$ −3942.00 −0.162790
$$838$$ 0 0
$$839$$ −20172.0 −0.830053 −0.415027 0.909809i $$-0.636228\pi$$
−0.415027 + 0.909809i $$0.636228\pi$$
$$840$$ 0 0
$$841$$ 42175.0 1.72926
$$842$$ 0 0
$$843$$ 21474.0 0.877347
$$844$$ 0 0
$$845$$ −8985.00 −0.365791
$$846$$ 0 0
$$847$$ −3031.00 −0.122959
$$848$$ 0 0
$$849$$ 15492.0 0.626247
$$850$$ 0 0
$$851$$ −5208.00 −0.209786
$$852$$ 0 0
$$853$$ 19820.0 0.795573 0.397787 0.917478i $$-0.369778\pi$$
0.397787 + 0.917478i $$0.369778\pi$$
$$854$$ 0 0
$$855$$ −1710.00 −0.0683986
$$856$$ 0 0
$$857$$ −10290.0 −0.410151 −0.205076 0.978746i $$-0.565744\pi$$
−0.205076 + 0.978746i $$0.565744\pi$$
$$858$$ 0 0
$$859$$ 31606.0 1.25539 0.627697 0.778458i $$-0.283998\pi$$
0.627697 + 0.778458i $$0.283998\pi$$
$$860$$ 0 0
$$861$$ 5922.00 0.234403
$$862$$ 0 0
$$863$$ −23172.0 −0.914002 −0.457001 0.889466i $$-0.651076\pi$$
−0.457001 + 0.889466i $$0.651076\pi$$
$$864$$ 0 0
$$865$$ 1890.00 0.0742912
$$866$$ 0 0
$$867$$ −1671.00 −0.0654558
$$868$$ 0 0
$$869$$ 47712.0 1.86251
$$870$$ 0 0
$$871$$ −16240.0 −0.631770
$$872$$ 0 0
$$873$$ 10872.0 0.421491
$$874$$ 0 0
$$875$$ −875.000 −0.0338062
$$876$$ 0 0
$$877$$ −15550.0 −0.598730 −0.299365 0.954139i $$-0.596775\pi$$
−0.299365 + 0.954139i $$0.596775\pi$$
$$878$$ 0 0
$$879$$ −25794.0 −0.989772
$$880$$ 0 0
$$881$$ 28530.0 1.09103 0.545517 0.838100i $$-0.316334\pi$$
0.545517 + 0.838100i $$0.316334\pi$$
$$882$$ 0 0
$$883$$ 28780.0 1.09686 0.548428 0.836198i $$-0.315226\pi$$
0.548428 + 0.836198i $$0.315226\pi$$
$$884$$ 0 0
$$885$$ 5400.00 0.205106
$$886$$ 0 0
$$887$$ −22872.0 −0.865802 −0.432901 0.901441i $$-0.642510\pi$$
−0.432901 + 0.901441i $$0.642510\pi$$
$$888$$ 0 0
$$889$$ −9184.00 −0.346481
$$890$$ 0 0
$$891$$ −3402.00 −0.127914
$$892$$ 0 0
$$893$$ −2736.00 −0.102527
$$894$$ 0 0
$$895$$ 1110.00 0.0414561
$$896$$ 0 0
$$897$$ −720.000 −0.0268006
$$898$$ 0 0
$$899$$ 37668.0 1.39744
$$900$$ 0 0
$$901$$ 22176.0 0.819966
$$902$$ 0 0
$$903$$ 420.000 0.0154781
$$904$$ 0 0
$$905$$ −12950.0 −0.475660
$$906$$ 0 0
$$907$$ 10708.0 0.392010 0.196005 0.980603i $$-0.437203\pi$$
0.196005 + 0.980603i $$0.437203\pi$$
$$908$$ 0 0
$$909$$ 4914.00 0.179304
$$910$$ 0 0
$$911$$ −1326.00 −0.0482243 −0.0241122 0.999709i $$-0.507676\pi$$
−0.0241122 + 0.999709i $$0.507676\pi$$
$$912$$ 0 0
$$913$$ 6552.00 0.237502
$$914$$ 0 0
$$915$$ −10230.0 −0.369610
$$916$$ 0 0
$$917$$ −9492.00 −0.341825
$$918$$ 0 0
$$919$$ 13696.0 0.491610 0.245805 0.969319i $$-0.420948\pi$$
0.245805 + 0.969319i $$0.420948\pi$$
$$920$$ 0 0
$$921$$ 1344.00 0.0480850
$$922$$ 0 0
$$923$$ −16200.0 −0.577713
$$924$$ 0 0
$$925$$ 10850.0 0.385671
$$926$$ 0 0
$$927$$ 4680.00 0.165816
$$928$$ 0 0
$$929$$ 42354.0 1.49579 0.747895 0.663817i $$-0.231064\pi$$
0.747895 + 0.663817i $$0.231064\pi$$
$$930$$ 0 0
$$931$$ −1862.00 −0.0655474
$$932$$ 0 0
$$933$$ 17496.0 0.613926
$$934$$ 0 0
$$935$$ −13860.0 −0.484781
$$936$$ 0 0
$$937$$ 6644.00 0.231644 0.115822 0.993270i $$-0.463050\pi$$
0.115822 + 0.993270i $$0.463050\pi$$
$$938$$ 0 0
$$939$$ 29544.0 1.02676
$$940$$ 0 0
$$941$$ 1350.00 0.0467681 0.0233840 0.999727i $$-0.492556\pi$$
0.0233840 + 0.999727i $$0.492556\pi$$
$$942$$ 0 0
$$943$$ 3384.00 0.116859
$$944$$ 0 0
$$945$$ −945.000 −0.0325300
$$946$$ 0 0
$$947$$ −49320.0 −1.69238 −0.846190 0.532881i $$-0.821109\pi$$
−0.846190 + 0.532881i $$0.821109\pi$$
$$948$$ 0 0
$$949$$ −2480.00 −0.0848306
$$950$$ 0 0
$$951$$ −16848.0 −0.574484
$$952$$ 0 0
$$953$$ 5940.00 0.201905 0.100953 0.994891i $$-0.467811\pi$$
0.100953 + 0.994891i $$0.467811\pi$$
$$954$$ 0 0
$$955$$ −11070.0 −0.375096
$$956$$ 0 0
$$957$$ 32508.0 1.09805
$$958$$ 0 0
$$959$$ 6888.00 0.231934
$$960$$ 0 0
$$961$$ −8475.00 −0.284482
$$962$$ 0 0
$$963$$ −10908.0 −0.365011
$$964$$ 0 0
$$965$$ 20890.0 0.696863
$$966$$ 0 0
$$967$$ −47216.0 −1.57018 −0.785090 0.619382i $$-0.787383\pi$$
−0.785090 + 0.619382i $$0.787383\pi$$
$$968$$ 0 0
$$969$$ −7524.00 −0.249438
$$970$$ 0 0
$$971$$ −12552.0 −0.414843 −0.207422 0.978252i $$-0.566507\pi$$
−0.207422 + 0.978252i $$0.566507\pi$$
$$972$$ 0 0
$$973$$ −2758.00 −0.0908709
$$974$$ 0 0
$$975$$ 1500.00 0.0492702
$$976$$ 0 0
$$977$$ 46908.0 1.53605 0.768025 0.640420i $$-0.221240\pi$$
0.768025 + 0.640420i $$0.221240\pi$$
$$978$$ 0 0
$$979$$ 43596.0 1.42322
$$980$$ 0 0
$$981$$ −9702.00 −0.315760
$$982$$ 0 0
$$983$$ 46128.0 1.49670 0.748349 0.663305i $$-0.230847\pi$$
0.748349 + 0.663305i $$0.230847\pi$$
$$984$$ 0 0
$$985$$ −15300.0 −0.494922
$$986$$ 0 0
$$987$$ −1512.00 −0.0487614
$$988$$ 0 0
$$989$$ 240.000 0.00771644
$$990$$ 0 0
$$991$$ 12184.0 0.390552 0.195276 0.980748i $$-0.437440\pi$$
0.195276 + 0.980748i $$0.437440\pi$$
$$992$$ 0 0
$$993$$ −1356.00 −0.0433347
$$994$$ 0 0
$$995$$ −13330.0 −0.424713
$$996$$ 0 0
$$997$$ −5164.00 −0.164038 −0.0820188 0.996631i $$-0.526137\pi$$
−0.0820188 + 0.996631i $$0.526137\pi$$
$$998$$ 0 0
$$999$$ 11718.0 0.371112
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 1680.4.a.s.1.1 1
4.3 odd 2 105.4.a.a.1.1 1
12.11 even 2 315.4.a.d.1.1 1
20.3 even 4 525.4.d.f.274.1 2
20.7 even 4 525.4.d.f.274.2 2
20.19 odd 2 525.4.a.e.1.1 1
28.27 even 2 735.4.a.c.1.1 1
60.59 even 2 1575.4.a.f.1.1 1
84.83 odd 2 2205.4.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.a.1.1 1 4.3 odd 2
315.4.a.d.1.1 1 12.11 even 2
525.4.a.e.1.1 1 20.19 odd 2
525.4.d.f.274.1 2 20.3 even 4
525.4.d.f.274.2 2 20.7 even 4
735.4.a.c.1.1 1 28.27 even 2
1575.4.a.f.1.1 1 60.59 even 2
1680.4.a.s.1.1 1 1.1 even 1 trivial
2205.4.a.o.1.1 1 84.83 odd 2