# Properties

 Label 132.4.a.d.1.1 Level $132$ Weight $4$ Character 132.1 Self dual yes Analytic conductor $7.788$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [132,4,Mod(1,132)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(132, 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("132.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$132 = 2^{2} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 132.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: $$7.78825212076$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 132.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +10.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +10.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} -11.0000 q^{11} +18.0000 q^{13} +30.0000 q^{15} +46.0000 q^{17} +40.0000 q^{19} +24.0000 q^{21} +44.0000 q^{23} -25.0000 q^{25} +27.0000 q^{27} +186.000 q^{29} -72.0000 q^{31} -33.0000 q^{33} +80.0000 q^{35} -114.000 q^{37} +54.0000 q^{39} +174.000 q^{41} -416.000 q^{43} +90.0000 q^{45} -156.000 q^{47} -279.000 q^{49} +138.000 q^{51} -62.0000 q^{53} -110.000 q^{55} +120.000 q^{57} -348.000 q^{59} -446.000 q^{61} +72.0000 q^{63} +180.000 q^{65} -956.000 q^{67} +132.000 q^{69} -444.000 q^{71} +306.000 q^{73} -75.0000 q^{75} -88.0000 q^{77} -664.000 q^{79} +81.0000 q^{81} -124.000 q^{83} +460.000 q^{85} +558.000 q^{87} +602.000 q^{89} +144.000 q^{91} -216.000 q^{93} +400.000 q^{95} +1522.00 q^{97} -99.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 10.0000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ 8.00000 0.431959 0.215980 0.976398i $$-0.430705\pi$$
0.215980 + 0.976398i $$0.430705\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ 18.0000 0.384023 0.192012 0.981393i $$-0.438499\pi$$
0.192012 + 0.981393i $$0.438499\pi$$
$$14$$ 0 0
$$15$$ 30.0000 0.516398
$$16$$ 0 0
$$17$$ 46.0000 0.656273 0.328136 0.944630i $$-0.393579\pi$$
0.328136 + 0.944630i $$0.393579\pi$$
$$18$$ 0 0
$$19$$ 40.0000 0.482980 0.241490 0.970403i $$-0.422364\pi$$
0.241490 + 0.970403i $$0.422364\pi$$
$$20$$ 0 0
$$21$$ 24.0000 0.249392
$$22$$ 0 0
$$23$$ 44.0000 0.398897 0.199449 0.979908i $$-0.436085\pi$$
0.199449 + 0.979908i $$0.436085\pi$$
$$24$$ 0 0
$$25$$ −25.0000 −0.200000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 186.000 1.19101 0.595506 0.803351i $$-0.296952\pi$$
0.595506 + 0.803351i $$0.296952\pi$$
$$30$$ 0 0
$$31$$ −72.0000 −0.417148 −0.208574 0.978007i $$-0.566882\pi$$
−0.208574 + 0.978007i $$0.566882\pi$$
$$32$$ 0 0
$$33$$ −33.0000 −0.174078
$$34$$ 0 0
$$35$$ 80.0000 0.386356
$$36$$ 0 0
$$37$$ −114.000 −0.506527 −0.253263 0.967397i $$-0.581504\pi$$
−0.253263 + 0.967397i $$0.581504\pi$$
$$38$$ 0 0
$$39$$ 54.0000 0.221716
$$40$$ 0 0
$$41$$ 174.000 0.662786 0.331393 0.943493i $$-0.392481\pi$$
0.331393 + 0.943493i $$0.392481\pi$$
$$42$$ 0 0
$$43$$ −416.000 −1.47534 −0.737668 0.675164i $$-0.764073\pi$$
−0.737668 + 0.675164i $$0.764073\pi$$
$$44$$ 0 0
$$45$$ 90.0000 0.298142
$$46$$ 0 0
$$47$$ −156.000 −0.484148 −0.242074 0.970258i $$-0.577828\pi$$
−0.242074 + 0.970258i $$0.577828\pi$$
$$48$$ 0 0
$$49$$ −279.000 −0.813411
$$50$$ 0 0
$$51$$ 138.000 0.378899
$$52$$ 0 0
$$53$$ −62.0000 −0.160686 −0.0803430 0.996767i $$-0.525602\pi$$
−0.0803430 + 0.996767i $$0.525602\pi$$
$$54$$ 0 0
$$55$$ −110.000 −0.269680
$$56$$ 0 0
$$57$$ 120.000 0.278849
$$58$$ 0 0
$$59$$ −348.000 −0.767894 −0.383947 0.923355i $$-0.625435\pi$$
−0.383947 + 0.923355i $$0.625435\pi$$
$$60$$ 0 0
$$61$$ −446.000 −0.936138 −0.468069 0.883692i $$-0.655050\pi$$
−0.468069 + 0.883692i $$0.655050\pi$$
$$62$$ 0 0
$$63$$ 72.0000 0.143986
$$64$$ 0 0
$$65$$ 180.000 0.343481
$$66$$ 0 0
$$67$$ −956.000 −1.74319 −0.871597 0.490223i $$-0.836915\pi$$
−0.871597 + 0.490223i $$0.836915\pi$$
$$68$$ 0 0
$$69$$ 132.000 0.230303
$$70$$ 0 0
$$71$$ −444.000 −0.742156 −0.371078 0.928602i $$-0.621012\pi$$
−0.371078 + 0.928602i $$0.621012\pi$$
$$72$$ 0 0
$$73$$ 306.000 0.490611 0.245305 0.969446i $$-0.421112\pi$$
0.245305 + 0.969446i $$0.421112\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ −88.0000 −0.130241
$$78$$ 0 0
$$79$$ −664.000 −0.945644 −0.472822 0.881158i $$-0.656765\pi$$
−0.472822 + 0.881158i $$0.656765\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −124.000 −0.163985 −0.0819926 0.996633i $$-0.526128\pi$$
−0.0819926 + 0.996633i $$0.526128\pi$$
$$84$$ 0 0
$$85$$ 460.000 0.586988
$$86$$ 0 0
$$87$$ 558.000 0.687631
$$88$$ 0 0
$$89$$ 602.000 0.716987 0.358494 0.933532i $$-0.383290\pi$$
0.358494 + 0.933532i $$0.383290\pi$$
$$90$$ 0 0
$$91$$ 144.000 0.165882
$$92$$ 0 0
$$93$$ −216.000 −0.240840
$$94$$ 0 0
$$95$$ 400.000 0.431991
$$96$$ 0 0
$$97$$ 1522.00 1.59315 0.796576 0.604539i $$-0.206643\pi$$
0.796576 + 0.604539i $$0.206643\pi$$
$$98$$ 0 0
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ 1090.00 1.07385 0.536926 0.843629i $$-0.319585\pi$$
0.536926 + 0.843629i $$0.319585\pi$$
$$102$$ 0 0
$$103$$ −1392.00 −1.33163 −0.665815 0.746117i $$-0.731916\pi$$
−0.665815 + 0.746117i $$0.731916\pi$$
$$104$$ 0 0
$$105$$ 240.000 0.223063
$$106$$ 0 0
$$107$$ 308.000 0.278276 0.139138 0.990273i $$-0.455567\pi$$
0.139138 + 0.990273i $$0.455567\pi$$
$$108$$ 0 0
$$109$$ 978.000 0.859407 0.429704 0.902970i $$-0.358618\pi$$
0.429704 + 0.902970i $$0.358618\pi$$
$$110$$ 0 0
$$111$$ −342.000 −0.292443
$$112$$ 0 0
$$113$$ 1162.00 0.967361 0.483680 0.875245i $$-0.339300\pi$$
0.483680 + 0.875245i $$0.339300\pi$$
$$114$$ 0 0
$$115$$ 440.000 0.356784
$$116$$ 0 0
$$117$$ 162.000 0.128008
$$118$$ 0 0
$$119$$ 368.000 0.283483
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ 522.000 0.382660
$$124$$ 0 0
$$125$$ −1500.00 −1.07331
$$126$$ 0 0
$$127$$ −984.000 −0.687527 −0.343763 0.939056i $$-0.611702\pi$$
−0.343763 + 0.939056i $$0.611702\pi$$
$$128$$ 0 0
$$129$$ −1248.00 −0.851785
$$130$$ 0 0
$$131$$ 2012.00 1.34190 0.670951 0.741501i $$-0.265886\pi$$
0.670951 + 0.741501i $$0.265886\pi$$
$$132$$ 0 0
$$133$$ 320.000 0.208628
$$134$$ 0 0
$$135$$ 270.000 0.172133
$$136$$ 0 0
$$137$$ −1286.00 −0.801974 −0.400987 0.916084i $$-0.631333\pi$$
−0.400987 + 0.916084i $$0.631333\pi$$
$$138$$ 0 0
$$139$$ 464.000 0.283136 0.141568 0.989929i $$-0.454786\pi$$
0.141568 + 0.989929i $$0.454786\pi$$
$$140$$ 0 0
$$141$$ −468.000 −0.279523
$$142$$ 0 0
$$143$$ −198.000 −0.115787
$$144$$ 0 0
$$145$$ 1860.00 1.06527
$$146$$ 0 0
$$147$$ −837.000 −0.469623
$$148$$ 0 0
$$149$$ −2894.00 −1.59118 −0.795590 0.605836i $$-0.792839\pi$$
−0.795590 + 0.605836i $$0.792839\pi$$
$$150$$ 0 0
$$151$$ 976.000 0.525998 0.262999 0.964796i $$-0.415288\pi$$
0.262999 + 0.964796i $$0.415288\pi$$
$$152$$ 0 0
$$153$$ 414.000 0.218758
$$154$$ 0 0
$$155$$ −720.000 −0.373108
$$156$$ 0 0
$$157$$ 1646.00 0.836720 0.418360 0.908281i $$-0.362605\pi$$
0.418360 + 0.908281i $$0.362605\pi$$
$$158$$ 0 0
$$159$$ −186.000 −0.0927721
$$160$$ 0 0
$$161$$ 352.000 0.172307
$$162$$ 0 0
$$163$$ 3268.00 1.57037 0.785183 0.619264i $$-0.212569\pi$$
0.785183 + 0.619264i $$0.212569\pi$$
$$164$$ 0 0
$$165$$ −330.000 −0.155700
$$166$$ 0 0
$$167$$ 1608.00 0.745094 0.372547 0.928013i $$-0.378484\pi$$
0.372547 + 0.928013i $$0.378484\pi$$
$$168$$ 0 0
$$169$$ −1873.00 −0.852526
$$170$$ 0 0
$$171$$ 360.000 0.160993
$$172$$ 0 0
$$173$$ −4070.00 −1.78865 −0.894325 0.447418i $$-0.852343\pi$$
−0.894325 + 0.447418i $$0.852343\pi$$
$$174$$ 0 0
$$175$$ −200.000 −0.0863919
$$176$$ 0 0
$$177$$ −1044.00 −0.443344
$$178$$ 0 0
$$179$$ 52.0000 0.0217132 0.0108566 0.999941i $$-0.496544\pi$$
0.0108566 + 0.999941i $$0.496544\pi$$
$$180$$ 0 0
$$181$$ 1798.00 0.738366 0.369183 0.929357i $$-0.379638\pi$$
0.369183 + 0.929357i $$0.379638\pi$$
$$182$$ 0 0
$$183$$ −1338.00 −0.540480
$$184$$ 0 0
$$185$$ −1140.00 −0.453051
$$186$$ 0 0
$$187$$ −506.000 −0.197874
$$188$$ 0 0
$$189$$ 216.000 0.0831306
$$190$$ 0 0
$$191$$ −3852.00 −1.45927 −0.729636 0.683836i $$-0.760310\pi$$
−0.729636 + 0.683836i $$0.760310\pi$$
$$192$$ 0 0
$$193$$ −1958.00 −0.730259 −0.365129 0.930957i $$-0.618975\pi$$
−0.365129 + 0.930957i $$0.618975\pi$$
$$194$$ 0 0
$$195$$ 540.000 0.198309
$$196$$ 0 0
$$197$$ −1630.00 −0.589506 −0.294753 0.955573i $$-0.595237\pi$$
−0.294753 + 0.955573i $$0.595237\pi$$
$$198$$ 0 0
$$199$$ −4504.00 −1.60442 −0.802211 0.597040i $$-0.796343\pi$$
−0.802211 + 0.597040i $$0.796343\pi$$
$$200$$ 0 0
$$201$$ −2868.00 −1.00643
$$202$$ 0 0
$$203$$ 1488.00 0.514469
$$204$$ 0 0
$$205$$ 1740.00 0.592814
$$206$$ 0 0
$$207$$ 396.000 0.132966
$$208$$ 0 0
$$209$$ −440.000 −0.145624
$$210$$ 0 0
$$211$$ 3776.00 1.23199 0.615997 0.787749i $$-0.288754\pi$$
0.615997 + 0.787749i $$0.288754\pi$$
$$212$$ 0 0
$$213$$ −1332.00 −0.428484
$$214$$ 0 0
$$215$$ −4160.00 −1.31958
$$216$$ 0 0
$$217$$ −576.000 −0.180191
$$218$$ 0 0
$$219$$ 918.000 0.283254
$$220$$ 0 0
$$221$$ 828.000 0.252024
$$222$$ 0 0
$$223$$ −1728.00 −0.518903 −0.259452 0.965756i $$-0.583542\pi$$
−0.259452 + 0.965756i $$0.583542\pi$$
$$224$$ 0 0
$$225$$ −225.000 −0.0666667
$$226$$ 0 0
$$227$$ 4716.00 1.37891 0.689454 0.724330i $$-0.257851\pi$$
0.689454 + 0.724330i $$0.257851\pi$$
$$228$$ 0 0
$$229$$ 1766.00 0.509609 0.254805 0.966993i $$-0.417989\pi$$
0.254805 + 0.966993i $$0.417989\pi$$
$$230$$ 0 0
$$231$$ −264.000 −0.0751945
$$232$$ 0 0
$$233$$ 4382.00 1.23208 0.616039 0.787715i $$-0.288736\pi$$
0.616039 + 0.787715i $$0.288736\pi$$
$$234$$ 0 0
$$235$$ −1560.00 −0.433035
$$236$$ 0 0
$$237$$ −1992.00 −0.545968
$$238$$ 0 0
$$239$$ 1264.00 0.342098 0.171049 0.985263i $$-0.445284\pi$$
0.171049 + 0.985263i $$0.445284\pi$$
$$240$$ 0 0
$$241$$ 3010.00 0.804528 0.402264 0.915524i $$-0.368223\pi$$
0.402264 + 0.915524i $$0.368223\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −2790.00 −0.727537
$$246$$ 0 0
$$247$$ 720.000 0.185476
$$248$$ 0 0
$$249$$ −372.000 −0.0946769
$$250$$ 0 0
$$251$$ 5404.00 1.35895 0.679477 0.733697i $$-0.262207\pi$$
0.679477 + 0.733697i $$0.262207\pi$$
$$252$$ 0 0
$$253$$ −484.000 −0.120272
$$254$$ 0 0
$$255$$ 1380.00 0.338898
$$256$$ 0 0
$$257$$ 1618.00 0.392716 0.196358 0.980532i $$-0.437088\pi$$
0.196358 + 0.980532i $$0.437088\pi$$
$$258$$ 0 0
$$259$$ −912.000 −0.218799
$$260$$ 0 0
$$261$$ 1674.00 0.397004
$$262$$ 0 0
$$263$$ 48.0000 0.0112540 0.00562701 0.999984i $$-0.498209\pi$$
0.00562701 + 0.999984i $$0.498209\pi$$
$$264$$ 0 0
$$265$$ −620.000 −0.143722
$$266$$ 0 0
$$267$$ 1806.00 0.413953
$$268$$ 0 0
$$269$$ −1814.00 −0.411158 −0.205579 0.978641i $$-0.565908\pi$$
−0.205579 + 0.978641i $$0.565908\pi$$
$$270$$ 0 0
$$271$$ 1016.00 0.227740 0.113870 0.993496i $$-0.463675\pi$$
0.113870 + 0.993496i $$0.463675\pi$$
$$272$$ 0 0
$$273$$ 432.000 0.0957723
$$274$$ 0 0
$$275$$ 275.000 0.0603023
$$276$$ 0 0
$$277$$ 6914.00 1.49972 0.749859 0.661597i $$-0.230121\pi$$
0.749859 + 0.661597i $$0.230121\pi$$
$$278$$ 0 0
$$279$$ −648.000 −0.139049
$$280$$ 0 0
$$281$$ −6098.00 −1.29458 −0.647289 0.762245i $$-0.724097\pi$$
−0.647289 + 0.762245i $$0.724097\pi$$
$$282$$ 0 0
$$283$$ 4192.00 0.880525 0.440262 0.897869i $$-0.354885\pi$$
0.440262 + 0.897869i $$0.354885\pi$$
$$284$$ 0 0
$$285$$ 1200.00 0.249410
$$286$$ 0 0
$$287$$ 1392.00 0.286297
$$288$$ 0 0
$$289$$ −2797.00 −0.569306
$$290$$ 0 0
$$291$$ 4566.00 0.919806
$$292$$ 0 0
$$293$$ 9314.00 1.85710 0.928549 0.371210i $$-0.121057\pi$$
0.928549 + 0.371210i $$0.121057\pi$$
$$294$$ 0 0
$$295$$ −3480.00 −0.686825
$$296$$ 0 0
$$297$$ −297.000 −0.0580259
$$298$$ 0 0
$$299$$ 792.000 0.153186
$$300$$ 0 0
$$301$$ −3328.00 −0.637285
$$302$$ 0 0
$$303$$ 3270.00 0.619989
$$304$$ 0 0
$$305$$ −4460.00 −0.837308
$$306$$ 0 0
$$307$$ 1384.00 0.257293 0.128647 0.991690i $$-0.458937\pi$$
0.128647 + 0.991690i $$0.458937\pi$$
$$308$$ 0 0
$$309$$ −4176.00 −0.768817
$$310$$ 0 0
$$311$$ 7916.00 1.44333 0.721664 0.692243i $$-0.243377\pi$$
0.721664 + 0.692243i $$0.243377\pi$$
$$312$$ 0 0
$$313$$ 218.000 0.0393677 0.0196838 0.999806i $$-0.493734\pi$$
0.0196838 + 0.999806i $$0.493734\pi$$
$$314$$ 0 0
$$315$$ 720.000 0.128785
$$316$$ 0 0
$$317$$ 666.000 0.118001 0.0590005 0.998258i $$-0.481209\pi$$
0.0590005 + 0.998258i $$0.481209\pi$$
$$318$$ 0 0
$$319$$ −2046.00 −0.359103
$$320$$ 0 0
$$321$$ 924.000 0.160662
$$322$$ 0 0
$$323$$ 1840.00 0.316967
$$324$$ 0 0
$$325$$ −450.000 −0.0768046
$$326$$ 0 0
$$327$$ 2934.00 0.496179
$$328$$ 0 0
$$329$$ −1248.00 −0.209132
$$330$$ 0 0
$$331$$ −2500.00 −0.415143 −0.207572 0.978220i $$-0.566556\pi$$
−0.207572 + 0.978220i $$0.566556\pi$$
$$332$$ 0 0
$$333$$ −1026.00 −0.168842
$$334$$ 0 0
$$335$$ −9560.00 −1.55916
$$336$$ 0 0
$$337$$ −6678.00 −1.07945 −0.539724 0.841842i $$-0.681471\pi$$
−0.539724 + 0.841842i $$0.681471\pi$$
$$338$$ 0 0
$$339$$ 3486.00 0.558506
$$340$$ 0 0
$$341$$ 792.000 0.125775
$$342$$ 0 0
$$343$$ −4976.00 −0.783320
$$344$$ 0 0
$$345$$ 1320.00 0.205990
$$346$$ 0 0
$$347$$ −1892.00 −0.292703 −0.146351 0.989233i $$-0.546753\pi$$
−0.146351 + 0.989233i $$0.546753\pi$$
$$348$$ 0 0
$$349$$ −7350.00 −1.12733 −0.563663 0.826005i $$-0.690608\pi$$
−0.563663 + 0.826005i $$0.690608\pi$$
$$350$$ 0 0
$$351$$ 486.000 0.0739053
$$352$$ 0 0
$$353$$ 2370.00 0.357344 0.178672 0.983909i $$-0.442820\pi$$
0.178672 + 0.983909i $$0.442820\pi$$
$$354$$ 0 0
$$355$$ −4440.00 −0.663805
$$356$$ 0 0
$$357$$ 1104.00 0.163669
$$358$$ 0 0
$$359$$ 912.000 0.134077 0.0670383 0.997750i $$-0.478645\pi$$
0.0670383 + 0.997750i $$0.478645\pi$$
$$360$$ 0 0
$$361$$ −5259.00 −0.766730
$$362$$ 0 0
$$363$$ 363.000 0.0524864
$$364$$ 0 0
$$365$$ 3060.00 0.438816
$$366$$ 0 0
$$367$$ −8464.00 −1.20386 −0.601931 0.798548i $$-0.705602\pi$$
−0.601931 + 0.798548i $$0.705602\pi$$
$$368$$ 0 0
$$369$$ 1566.00 0.220929
$$370$$ 0 0
$$371$$ −496.000 −0.0694098
$$372$$ 0 0
$$373$$ 11890.0 1.65051 0.825256 0.564759i $$-0.191031\pi$$
0.825256 + 0.564759i $$0.191031\pi$$
$$374$$ 0 0
$$375$$ −4500.00 −0.619677
$$376$$ 0 0
$$377$$ 3348.00 0.457376
$$378$$ 0 0
$$379$$ −9556.00 −1.29514 −0.647571 0.762005i $$-0.724215\pi$$
−0.647571 + 0.762005i $$0.724215\pi$$
$$380$$ 0 0
$$381$$ −2952.00 −0.396944
$$382$$ 0 0
$$383$$ 5236.00 0.698556 0.349278 0.937019i $$-0.386427\pi$$
0.349278 + 0.937019i $$0.386427\pi$$
$$384$$ 0 0
$$385$$ −880.000 −0.116491
$$386$$ 0 0
$$387$$ −3744.00 −0.491778
$$388$$ 0 0
$$389$$ −8262.00 −1.07686 −0.538432 0.842669i $$-0.680983\pi$$
−0.538432 + 0.842669i $$0.680983\pi$$
$$390$$ 0 0
$$391$$ 2024.00 0.261785
$$392$$ 0 0
$$393$$ 6036.00 0.774748
$$394$$ 0 0
$$395$$ −6640.00 −0.845809
$$396$$ 0 0
$$397$$ −2402.00 −0.303660 −0.151830 0.988407i $$-0.548517\pi$$
−0.151830 + 0.988407i $$0.548517\pi$$
$$398$$ 0 0
$$399$$ 960.000 0.120451
$$400$$ 0 0
$$401$$ 6290.00 0.783311 0.391655 0.920112i $$-0.371903\pi$$
0.391655 + 0.920112i $$0.371903\pi$$
$$402$$ 0 0
$$403$$ −1296.00 −0.160194
$$404$$ 0 0
$$405$$ 810.000 0.0993808
$$406$$ 0 0
$$407$$ 1254.00 0.152724
$$408$$ 0 0
$$409$$ −6950.00 −0.840233 −0.420117 0.907470i $$-0.638011\pi$$
−0.420117 + 0.907470i $$0.638011\pi$$
$$410$$ 0 0
$$411$$ −3858.00 −0.463020
$$412$$ 0 0
$$413$$ −2784.00 −0.331699
$$414$$ 0 0
$$415$$ −1240.00 −0.146673
$$416$$ 0 0
$$417$$ 1392.00 0.163469
$$418$$ 0 0
$$419$$ 12660.0 1.47609 0.738045 0.674752i $$-0.235749\pi$$
0.738045 + 0.674752i $$0.235749\pi$$
$$420$$ 0 0
$$421$$ 5342.00 0.618416 0.309208 0.950994i $$-0.399936\pi$$
0.309208 + 0.950994i $$0.399936\pi$$
$$422$$ 0 0
$$423$$ −1404.00 −0.161383
$$424$$ 0 0
$$425$$ −1150.00 −0.131255
$$426$$ 0 0
$$427$$ −3568.00 −0.404374
$$428$$ 0 0
$$429$$ −594.000 −0.0668499
$$430$$ 0 0
$$431$$ −13560.0 −1.51546 −0.757729 0.652570i $$-0.773691\pi$$
−0.757729 + 0.652570i $$0.773691\pi$$
$$432$$ 0 0
$$433$$ 4658.00 0.516973 0.258486 0.966015i $$-0.416776\pi$$
0.258486 + 0.966015i $$0.416776\pi$$
$$434$$ 0 0
$$435$$ 5580.00 0.615036
$$436$$ 0 0
$$437$$ 1760.00 0.192660
$$438$$ 0 0
$$439$$ −6392.00 −0.694928 −0.347464 0.937693i $$-0.612957\pi$$
−0.347464 + 0.937693i $$0.612957\pi$$
$$440$$ 0 0
$$441$$ −2511.00 −0.271137
$$442$$ 0 0
$$443$$ 10772.0 1.15529 0.577645 0.816288i $$-0.303972\pi$$
0.577645 + 0.816288i $$0.303972\pi$$
$$444$$ 0 0
$$445$$ 6020.00 0.641293
$$446$$ 0 0
$$447$$ −8682.00 −0.918668
$$448$$ 0 0
$$449$$ −5606.00 −0.589228 −0.294614 0.955616i $$-0.595191\pi$$
−0.294614 + 0.955616i $$0.595191\pi$$
$$450$$ 0 0
$$451$$ −1914.00 −0.199838
$$452$$ 0 0
$$453$$ 2928.00 0.303685
$$454$$ 0 0
$$455$$ 1440.00 0.148370
$$456$$ 0 0
$$457$$ 17050.0 1.74522 0.872610 0.488418i $$-0.162426\pi$$
0.872610 + 0.488418i $$0.162426\pi$$
$$458$$ 0 0
$$459$$ 1242.00 0.126300
$$460$$ 0 0
$$461$$ −8438.00 −0.852488 −0.426244 0.904608i $$-0.640163\pi$$
−0.426244 + 0.904608i $$0.640163\pi$$
$$462$$ 0 0
$$463$$ 5064.00 0.508302 0.254151 0.967164i $$-0.418204\pi$$
0.254151 + 0.967164i $$0.418204\pi$$
$$464$$ 0 0
$$465$$ −2160.00 −0.215414
$$466$$ 0 0
$$467$$ −14268.0 −1.41380 −0.706900 0.707314i $$-0.749907\pi$$
−0.706900 + 0.707314i $$0.749907\pi$$
$$468$$ 0 0
$$469$$ −7648.00 −0.752989
$$470$$ 0 0
$$471$$ 4938.00 0.483081
$$472$$ 0 0
$$473$$ 4576.00 0.444830
$$474$$ 0 0
$$475$$ −1000.00 −0.0965961
$$476$$ 0 0
$$477$$ −558.000 −0.0535620
$$478$$ 0 0
$$479$$ −18312.0 −1.74676 −0.873379 0.487042i $$-0.838076\pi$$
−0.873379 + 0.487042i $$0.838076\pi$$
$$480$$ 0 0
$$481$$ −2052.00 −0.194518
$$482$$ 0 0
$$483$$ 1056.00 0.0994817
$$484$$ 0 0
$$485$$ 15220.0 1.42496
$$486$$ 0 0
$$487$$ 4376.00 0.407178 0.203589 0.979056i $$-0.434739\pi$$
0.203589 + 0.979056i $$0.434739\pi$$
$$488$$ 0 0
$$489$$ 9804.00 0.906651
$$490$$ 0 0
$$491$$ 17380.0 1.59745 0.798725 0.601696i $$-0.205508\pi$$
0.798725 + 0.601696i $$0.205508\pi$$
$$492$$ 0 0
$$493$$ 8556.00 0.781629
$$494$$ 0 0
$$495$$ −990.000 −0.0898933
$$496$$ 0 0
$$497$$ −3552.00 −0.320581
$$498$$ 0 0
$$499$$ 11324.0 1.01590 0.507948 0.861388i $$-0.330404\pi$$
0.507948 + 0.861388i $$0.330404\pi$$
$$500$$ 0 0
$$501$$ 4824.00 0.430180
$$502$$ 0 0
$$503$$ 2392.00 0.212036 0.106018 0.994364i $$-0.466190\pi$$
0.106018 + 0.994364i $$0.466190\pi$$
$$504$$ 0 0
$$505$$ 10900.0 0.960482
$$506$$ 0 0
$$507$$ −5619.00 −0.492206
$$508$$ 0 0
$$509$$ −1238.00 −0.107806 −0.0539031 0.998546i $$-0.517166\pi$$
−0.0539031 + 0.998546i $$0.517166\pi$$
$$510$$ 0 0
$$511$$ 2448.00 0.211924
$$512$$ 0 0
$$513$$ 1080.00 0.0929496
$$514$$ 0 0
$$515$$ −13920.0 −1.19105
$$516$$ 0 0
$$517$$ 1716.00 0.145976
$$518$$ 0 0
$$519$$ −12210.0 −1.03268
$$520$$ 0 0
$$521$$ 21738.0 1.82794 0.913972 0.405777i $$-0.132999\pi$$
0.913972 + 0.405777i $$0.132999\pi$$
$$522$$ 0 0
$$523$$ 22016.0 1.84071 0.920356 0.391081i $$-0.127899\pi$$
0.920356 + 0.391081i $$0.127899\pi$$
$$524$$ 0 0
$$525$$ −600.000 −0.0498784
$$526$$ 0 0
$$527$$ −3312.00 −0.273763
$$528$$ 0 0
$$529$$ −10231.0 −0.840881
$$530$$ 0 0
$$531$$ −3132.00 −0.255965
$$532$$ 0 0
$$533$$ 3132.00 0.254525
$$534$$ 0 0
$$535$$ 3080.00 0.248897
$$536$$ 0 0
$$537$$ 156.000 0.0125361
$$538$$ 0 0
$$539$$ 3069.00 0.245253
$$540$$ 0 0
$$541$$ 9490.00 0.754172 0.377086 0.926178i $$-0.376926\pi$$
0.377086 + 0.926178i $$0.376926\pi$$
$$542$$ 0 0
$$543$$ 5394.00 0.426296
$$544$$ 0 0
$$545$$ 9780.00 0.768677
$$546$$ 0 0
$$547$$ 21632.0 1.69089 0.845446 0.534061i $$-0.179335\pi$$
0.845446 + 0.534061i $$0.179335\pi$$
$$548$$ 0 0
$$549$$ −4014.00 −0.312046
$$550$$ 0 0
$$551$$ 7440.00 0.575235
$$552$$ 0 0
$$553$$ −5312.00 −0.408480
$$554$$ 0 0
$$555$$ −3420.00 −0.261569
$$556$$ 0 0
$$557$$ −4854.00 −0.369247 −0.184624 0.982809i $$-0.559107\pi$$
−0.184624 + 0.982809i $$0.559107\pi$$
$$558$$ 0 0
$$559$$ −7488.00 −0.566563
$$560$$ 0 0
$$561$$ −1518.00 −0.114242
$$562$$ 0 0
$$563$$ −5308.00 −0.397346 −0.198673 0.980066i $$-0.563663\pi$$
−0.198673 + 0.980066i $$0.563663\pi$$
$$564$$ 0 0
$$565$$ 11620.0 0.865234
$$566$$ 0 0
$$567$$ 648.000 0.0479955
$$568$$ 0 0
$$569$$ −12490.0 −0.920225 −0.460113 0.887861i $$-0.652191\pi$$
−0.460113 + 0.887861i $$0.652191\pi$$
$$570$$ 0 0
$$571$$ −7448.00 −0.545865 −0.272933 0.962033i $$-0.587994\pi$$
−0.272933 + 0.962033i $$0.587994\pi$$
$$572$$ 0 0
$$573$$ −11556.0 −0.842511
$$574$$ 0 0
$$575$$ −1100.00 −0.0797794
$$576$$ 0 0
$$577$$ 10994.0 0.793217 0.396608 0.917988i $$-0.370187\pi$$
0.396608 + 0.917988i $$0.370187\pi$$
$$578$$ 0 0
$$579$$ −5874.00 −0.421615
$$580$$ 0 0
$$581$$ −992.000 −0.0708349
$$582$$ 0 0
$$583$$ 682.000 0.0484486
$$584$$ 0 0
$$585$$ 1620.00 0.114494
$$586$$ 0 0
$$587$$ −5148.00 −0.361977 −0.180989 0.983485i $$-0.557930\pi$$
−0.180989 + 0.983485i $$0.557930\pi$$
$$588$$ 0 0
$$589$$ −2880.00 −0.201474
$$590$$ 0 0
$$591$$ −4890.00 −0.340351
$$592$$ 0 0
$$593$$ −22314.0 −1.54524 −0.772619 0.634870i $$-0.781054\pi$$
−0.772619 + 0.634870i $$0.781054\pi$$
$$594$$ 0 0
$$595$$ 3680.00 0.253555
$$596$$ 0 0
$$597$$ −13512.0 −0.926314
$$598$$ 0 0
$$599$$ −15588.0 −1.06329 −0.531643 0.846968i $$-0.678425\pi$$
−0.531643 + 0.846968i $$0.678425\pi$$
$$600$$ 0 0
$$601$$ −21638.0 −1.46861 −0.734303 0.678822i $$-0.762491\pi$$
−0.734303 + 0.678822i $$0.762491\pi$$
$$602$$ 0 0
$$603$$ −8604.00 −0.581065
$$604$$ 0 0
$$605$$ 1210.00 0.0813116
$$606$$ 0 0
$$607$$ −7496.00 −0.501241 −0.250620 0.968085i $$-0.580635\pi$$
−0.250620 + 0.968085i $$0.580635\pi$$
$$608$$ 0 0
$$609$$ 4464.00 0.297029
$$610$$ 0 0
$$611$$ −2808.00 −0.185924
$$612$$ 0 0
$$613$$ 2106.00 0.138761 0.0693805 0.997590i $$-0.477898\pi$$
0.0693805 + 0.997590i $$0.477898\pi$$
$$614$$ 0 0
$$615$$ 5220.00 0.342261
$$616$$ 0 0
$$617$$ 26962.0 1.75924 0.879619 0.475680i $$-0.157798\pi$$
0.879619 + 0.475680i $$0.157798\pi$$
$$618$$ 0 0
$$619$$ −17740.0 −1.15191 −0.575954 0.817482i $$-0.695369\pi$$
−0.575954 + 0.817482i $$0.695369\pi$$
$$620$$ 0 0
$$621$$ 1188.00 0.0767678
$$622$$ 0 0
$$623$$ 4816.00 0.309709
$$624$$ 0 0
$$625$$ −11875.0 −0.760000
$$626$$ 0 0
$$627$$ −1320.00 −0.0840761
$$628$$ 0 0
$$629$$ −5244.00 −0.332420
$$630$$ 0 0
$$631$$ 19360.0 1.22141 0.610705 0.791858i $$-0.290886\pi$$
0.610705 + 0.791858i $$0.290886\pi$$
$$632$$ 0 0
$$633$$ 11328.0 0.711292
$$634$$ 0 0
$$635$$ −9840.00 −0.614943
$$636$$ 0 0
$$637$$ −5022.00 −0.312369
$$638$$ 0 0
$$639$$ −3996.00 −0.247385
$$640$$ 0 0
$$641$$ −19158.0 −1.18049 −0.590246 0.807223i $$-0.700969\pi$$
−0.590246 + 0.807223i $$0.700969\pi$$
$$642$$ 0 0
$$643$$ 11228.0 0.688630 0.344315 0.938854i $$-0.388111\pi$$
0.344315 + 0.938854i $$0.388111\pi$$
$$644$$ 0 0
$$645$$ −12480.0 −0.761860
$$646$$ 0 0
$$647$$ −22628.0 −1.37496 −0.687480 0.726204i $$-0.741283\pi$$
−0.687480 + 0.726204i $$0.741283\pi$$
$$648$$ 0 0
$$649$$ 3828.00 0.231529
$$650$$ 0 0
$$651$$ −1728.00 −0.104033
$$652$$ 0 0
$$653$$ 28338.0 1.69824 0.849121 0.528199i $$-0.177132\pi$$
0.849121 + 0.528199i $$0.177132\pi$$
$$654$$ 0 0
$$655$$ 20120.0 1.20023
$$656$$ 0 0
$$657$$ 2754.00 0.163537
$$658$$ 0 0
$$659$$ −17052.0 −1.00797 −0.503985 0.863713i $$-0.668133\pi$$
−0.503985 + 0.863713i $$0.668133\pi$$
$$660$$ 0 0
$$661$$ −21354.0 −1.25654 −0.628271 0.777995i $$-0.716237\pi$$
−0.628271 + 0.777995i $$0.716237\pi$$
$$662$$ 0 0
$$663$$ 2484.00 0.145506
$$664$$ 0 0
$$665$$ 3200.00 0.186603
$$666$$ 0 0
$$667$$ 8184.00 0.475091
$$668$$ 0 0
$$669$$ −5184.00 −0.299589
$$670$$ 0 0
$$671$$ 4906.00 0.282256
$$672$$ 0 0
$$673$$ −22198.0 −1.27143 −0.635713 0.771925i $$-0.719294\pi$$
−0.635713 + 0.771925i $$0.719294\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ −32974.0 −1.87193 −0.935963 0.352099i $$-0.885468\pi$$
−0.935963 + 0.352099i $$0.885468\pi$$
$$678$$ 0 0
$$679$$ 12176.0 0.688177
$$680$$ 0 0
$$681$$ 14148.0 0.796112
$$682$$ 0 0
$$683$$ 22572.0 1.26456 0.632279 0.774740i $$-0.282120\pi$$
0.632279 + 0.774740i $$0.282120\pi$$
$$684$$ 0 0
$$685$$ −12860.0 −0.717307
$$686$$ 0 0
$$687$$ 5298.00 0.294223
$$688$$ 0 0
$$689$$ −1116.00 −0.0617071
$$690$$ 0 0
$$691$$ 2700.00 0.148644 0.0743219 0.997234i $$-0.476321\pi$$
0.0743219 + 0.997234i $$0.476321\pi$$
$$692$$ 0 0
$$693$$ −792.000 −0.0434136
$$694$$ 0 0
$$695$$ 4640.00 0.253245
$$696$$ 0 0
$$697$$ 8004.00 0.434969
$$698$$ 0 0
$$699$$ 13146.0 0.711341
$$700$$ 0 0
$$701$$ −17382.0 −0.936532 −0.468266 0.883587i $$-0.655121\pi$$
−0.468266 + 0.883587i $$0.655121\pi$$
$$702$$ 0 0
$$703$$ −4560.00 −0.244642
$$704$$ 0 0
$$705$$ −4680.00 −0.250013
$$706$$ 0 0
$$707$$ 8720.00 0.463860
$$708$$ 0 0
$$709$$ 20454.0 1.08345 0.541725 0.840556i $$-0.317771\pi$$
0.541725 + 0.840556i $$0.317771\pi$$
$$710$$ 0 0
$$711$$ −5976.00 −0.315215
$$712$$ 0 0
$$713$$ −3168.00 −0.166399
$$714$$ 0 0
$$715$$ −1980.00 −0.103563
$$716$$ 0 0
$$717$$ 3792.00 0.197510
$$718$$ 0 0
$$719$$ 10236.0 0.530930 0.265465 0.964121i $$-0.414475\pi$$
0.265465 + 0.964121i $$0.414475\pi$$
$$720$$ 0 0
$$721$$ −11136.0 −0.575210
$$722$$ 0 0
$$723$$ 9030.00 0.464494
$$724$$ 0 0
$$725$$ −4650.00 −0.238202
$$726$$ 0 0
$$727$$ 9672.00 0.493418 0.246709 0.969090i $$-0.420651\pi$$
0.246709 + 0.969090i $$0.420651\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −19136.0 −0.968222
$$732$$ 0 0
$$733$$ −28078.0 −1.41485 −0.707425 0.706789i $$-0.750143\pi$$
−0.707425 + 0.706789i $$0.750143\pi$$
$$734$$ 0 0
$$735$$ −8370.00 −0.420044
$$736$$ 0 0
$$737$$ 10516.0 0.525593
$$738$$ 0 0
$$739$$ −26776.0 −1.33284 −0.666422 0.745575i $$-0.732175\pi$$
−0.666422 + 0.745575i $$0.732175\pi$$
$$740$$ 0 0
$$741$$ 2160.00 0.107084
$$742$$ 0 0
$$743$$ 25280.0 1.24823 0.624114 0.781333i $$-0.285460\pi$$
0.624114 + 0.781333i $$0.285460\pi$$
$$744$$ 0 0
$$745$$ −28940.0 −1.42319
$$746$$ 0 0
$$747$$ −1116.00 −0.0546617
$$748$$ 0 0
$$749$$ 2464.00 0.120204
$$750$$ 0 0
$$751$$ 25160.0 1.22251 0.611253 0.791436i $$-0.290666\pi$$
0.611253 + 0.791436i $$0.290666\pi$$
$$752$$ 0 0
$$753$$ 16212.0 0.784592
$$754$$ 0 0
$$755$$ 9760.00 0.470467
$$756$$ 0 0
$$757$$ 28910.0 1.38805 0.694024 0.719952i $$-0.255836\pi$$
0.694024 + 0.719952i $$0.255836\pi$$
$$758$$ 0 0
$$759$$ −1452.00 −0.0694391
$$760$$ 0 0
$$761$$ 10278.0 0.489589 0.244794 0.969575i $$-0.421280\pi$$
0.244794 + 0.969575i $$0.421280\pi$$
$$762$$ 0 0
$$763$$ 7824.00 0.371229
$$764$$ 0 0
$$765$$ 4140.00 0.195663
$$766$$ 0 0
$$767$$ −6264.00 −0.294889
$$768$$ 0 0
$$769$$ −29558.0 −1.38607 −0.693036 0.720903i $$-0.743727\pi$$
−0.693036 + 0.720903i $$0.743727\pi$$
$$770$$ 0 0
$$771$$ 4854.00 0.226735
$$772$$ 0 0
$$773$$ 9762.00 0.454223 0.227112 0.973869i $$-0.427072\pi$$
0.227112 + 0.973869i $$0.427072\pi$$
$$774$$ 0 0
$$775$$ 1800.00 0.0834296
$$776$$ 0 0
$$777$$ −2736.00 −0.126324
$$778$$ 0 0
$$779$$ 6960.00 0.320113
$$780$$ 0 0
$$781$$ 4884.00 0.223769
$$782$$ 0 0
$$783$$ 5022.00 0.229210
$$784$$ 0 0
$$785$$ 16460.0 0.748385
$$786$$ 0 0
$$787$$ 14824.0 0.671434 0.335717 0.941963i $$-0.391021\pi$$
0.335717 + 0.941963i $$0.391021\pi$$
$$788$$ 0 0
$$789$$ 144.000 0.00649751
$$790$$ 0 0
$$791$$ 9296.00 0.417861
$$792$$ 0 0
$$793$$ −8028.00 −0.359499
$$794$$ 0 0
$$795$$ −1860.00 −0.0829779
$$796$$ 0 0
$$797$$ 14946.0 0.664259 0.332130 0.943234i $$-0.392233\pi$$
0.332130 + 0.943234i $$0.392233\pi$$
$$798$$ 0 0
$$799$$ −7176.00 −0.317733
$$800$$ 0 0
$$801$$ 5418.00 0.238996
$$802$$ 0 0
$$803$$ −3366.00 −0.147925
$$804$$ 0 0
$$805$$ 3520.00 0.154116
$$806$$ 0 0
$$807$$ −5442.00 −0.237382
$$808$$ 0 0
$$809$$ 14774.0 0.642060 0.321030 0.947069i $$-0.395971\pi$$
0.321030 + 0.947069i $$0.395971\pi$$
$$810$$ 0 0
$$811$$ 22760.0 0.985464 0.492732 0.870181i $$-0.335998\pi$$
0.492732 + 0.870181i $$0.335998\pi$$
$$812$$ 0 0
$$813$$ 3048.00 0.131486
$$814$$ 0 0
$$815$$ 32680.0 1.40458
$$816$$ 0 0
$$817$$ −16640.0 −0.712558
$$818$$ 0 0
$$819$$ 1296.00 0.0552941
$$820$$ 0 0
$$821$$ 16370.0 0.695879 0.347940 0.937517i $$-0.386881\pi$$
0.347940 + 0.937517i $$0.386881\pi$$
$$822$$ 0 0
$$823$$ 1784.00 0.0755605 0.0377803 0.999286i $$-0.487971\pi$$
0.0377803 + 0.999286i $$0.487971\pi$$
$$824$$ 0 0
$$825$$ 825.000 0.0348155
$$826$$ 0 0
$$827$$ −22052.0 −0.927235 −0.463617 0.886036i $$-0.653449\pi$$
−0.463617 + 0.886036i $$0.653449\pi$$
$$828$$ 0 0
$$829$$ −29738.0 −1.24589 −0.622945 0.782265i $$-0.714064\pi$$
−0.622945 + 0.782265i $$0.714064\pi$$
$$830$$ 0 0
$$831$$ 20742.0 0.865863
$$832$$ 0 0
$$833$$ −12834.0 −0.533820
$$834$$ 0 0
$$835$$ 16080.0 0.666433
$$836$$ 0 0
$$837$$ −1944.00 −0.0802801
$$838$$ 0 0
$$839$$ 43956.0 1.80874 0.904368 0.426753i $$-0.140343\pi$$
0.904368 + 0.426753i $$0.140343\pi$$
$$840$$ 0 0
$$841$$ 10207.0 0.418508
$$842$$ 0 0
$$843$$ −18294.0 −0.747424
$$844$$ 0 0
$$845$$ −18730.0 −0.762523
$$846$$ 0 0
$$847$$ 968.000 0.0392690
$$848$$ 0 0
$$849$$ 12576.0 0.508371
$$850$$ 0 0
$$851$$ −5016.00 −0.202052
$$852$$ 0 0
$$853$$ −6438.00 −0.258421 −0.129210 0.991617i $$-0.541244\pi$$
−0.129210 + 0.991617i $$0.541244\pi$$
$$854$$ 0 0
$$855$$ 3600.00 0.143997
$$856$$ 0 0
$$857$$ −2282.00 −0.0909587 −0.0454794 0.998965i $$-0.514482\pi$$
−0.0454794 + 0.998965i $$0.514482\pi$$
$$858$$ 0 0
$$859$$ −29972.0 −1.19049 −0.595245 0.803544i $$-0.702945\pi$$
−0.595245 + 0.803544i $$0.702945\pi$$
$$860$$ 0 0
$$861$$ 4176.00 0.165293
$$862$$ 0 0
$$863$$ −3716.00 −0.146575 −0.0732874 0.997311i $$-0.523349\pi$$
−0.0732874 + 0.997311i $$0.523349\pi$$
$$864$$ 0 0
$$865$$ −40700.0 −1.59982
$$866$$ 0 0
$$867$$ −8391.00 −0.328689
$$868$$ 0 0
$$869$$ 7304.00 0.285122
$$870$$ 0 0
$$871$$ −17208.0 −0.669427
$$872$$ 0 0
$$873$$ 13698.0 0.531050
$$874$$ 0 0
$$875$$ −12000.0 −0.463627
$$876$$ 0 0
$$877$$ 25194.0 0.970058 0.485029 0.874498i $$-0.338809\pi$$
0.485029 + 0.874498i $$0.338809\pi$$
$$878$$ 0 0
$$879$$ 27942.0 1.07220
$$880$$ 0 0
$$881$$ 27194.0 1.03994 0.519971 0.854184i $$-0.325943\pi$$
0.519971 + 0.854184i $$0.325943\pi$$
$$882$$ 0 0
$$883$$ −14300.0 −0.544998 −0.272499 0.962156i $$-0.587850\pi$$
−0.272499 + 0.962156i $$0.587850\pi$$
$$884$$ 0 0
$$885$$ −10440.0 −0.396539
$$886$$ 0 0
$$887$$ 4944.00 0.187151 0.0935757 0.995612i $$-0.470170\pi$$
0.0935757 + 0.995612i $$0.470170\pi$$
$$888$$ 0 0
$$889$$ −7872.00 −0.296984
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ 0 0
$$893$$ −6240.00 −0.233834
$$894$$ 0 0
$$895$$ 520.000 0.0194209
$$896$$ 0 0
$$897$$ 2376.00 0.0884418
$$898$$ 0 0
$$899$$ −13392.0 −0.496828
$$900$$ 0 0
$$901$$ −2852.00 −0.105454
$$902$$ 0 0
$$903$$ −9984.00 −0.367937
$$904$$ 0 0
$$905$$ 17980.0 0.660415
$$906$$ 0 0
$$907$$ −24332.0 −0.890773 −0.445386 0.895338i $$-0.646934\pi$$
−0.445386 + 0.895338i $$0.646934\pi$$
$$908$$ 0 0
$$909$$ 9810.00 0.357951
$$910$$ 0 0
$$911$$ 20796.0 0.756314 0.378157 0.925741i $$-0.376558\pi$$
0.378157 + 0.925741i $$0.376558\pi$$
$$912$$ 0 0
$$913$$ 1364.00 0.0494434
$$914$$ 0 0
$$915$$ −13380.0 −0.483420
$$916$$ 0 0
$$917$$ 16096.0 0.579647
$$918$$ 0 0
$$919$$ 15392.0 0.552487 0.276243 0.961088i $$-0.410910\pi$$
0.276243 + 0.961088i $$0.410910\pi$$
$$920$$ 0 0
$$921$$ 4152.00 0.148548
$$922$$ 0 0
$$923$$ −7992.00 −0.285005
$$924$$ 0 0
$$925$$ 2850.00 0.101305
$$926$$ 0 0
$$927$$ −12528.0 −0.443876
$$928$$ 0 0
$$929$$ −15150.0 −0.535043 −0.267522 0.963552i $$-0.586205\pi$$
−0.267522 + 0.963552i $$0.586205\pi$$
$$930$$ 0 0
$$931$$ −11160.0 −0.392862
$$932$$ 0 0
$$933$$ 23748.0 0.833306
$$934$$ 0 0
$$935$$ −5060.00 −0.176984
$$936$$ 0 0
$$937$$ 15610.0 0.544244 0.272122 0.962263i $$-0.412275\pi$$
0.272122 + 0.962263i $$0.412275\pi$$
$$938$$ 0 0
$$939$$ 654.000 0.0227289
$$940$$ 0 0
$$941$$ −45222.0 −1.56663 −0.783313 0.621627i $$-0.786472\pi$$
−0.783313 + 0.621627i $$0.786472\pi$$
$$942$$ 0 0
$$943$$ 7656.00 0.264384
$$944$$ 0 0
$$945$$ 2160.00 0.0743543
$$946$$ 0 0
$$947$$ −14820.0 −0.508538 −0.254269 0.967134i $$-0.581835\pi$$
−0.254269 + 0.967134i $$0.581835\pi$$
$$948$$ 0 0
$$949$$ 5508.00 0.188406
$$950$$ 0 0
$$951$$ 1998.00 0.0681279
$$952$$ 0 0
$$953$$ 5334.00 0.181307 0.0906533 0.995883i $$-0.471104\pi$$
0.0906533 + 0.995883i $$0.471104\pi$$
$$954$$ 0 0
$$955$$ −38520.0 −1.30521
$$956$$ 0 0
$$957$$ −6138.00 −0.207328
$$958$$ 0 0
$$959$$ −10288.0 −0.346420
$$960$$ 0 0
$$961$$ −24607.0 −0.825988
$$962$$ 0 0
$$963$$ 2772.00 0.0927585
$$964$$ 0 0
$$965$$ −19580.0 −0.653163
$$966$$ 0 0
$$967$$ 18400.0 0.611897 0.305948 0.952048i $$-0.401027\pi$$
0.305948 + 0.952048i $$0.401027\pi$$
$$968$$ 0 0
$$969$$ 5520.00 0.183001
$$970$$ 0 0
$$971$$ −14460.0 −0.477903 −0.238951 0.971032i $$-0.576804\pi$$
−0.238951 + 0.971032i $$0.576804\pi$$
$$972$$ 0 0
$$973$$ 3712.00 0.122303
$$974$$ 0 0
$$975$$ −1350.00 −0.0443432
$$976$$ 0 0
$$977$$ −9998.00 −0.327394 −0.163697 0.986511i $$-0.552342\pi$$
−0.163697 + 0.986511i $$0.552342\pi$$
$$978$$ 0 0
$$979$$ −6622.00 −0.216180
$$980$$ 0 0
$$981$$ 8802.00 0.286469
$$982$$ 0 0
$$983$$ −52548.0 −1.70501 −0.852503 0.522722i $$-0.824916\pi$$
−0.852503 + 0.522722i $$0.824916\pi$$
$$984$$ 0 0
$$985$$ −16300.0 −0.527270
$$986$$ 0 0
$$987$$ −3744.00 −0.120742
$$988$$ 0 0
$$989$$ −18304.0 −0.588507
$$990$$ 0 0
$$991$$ −7096.00 −0.227459 −0.113729 0.993512i $$-0.536280\pi$$
−0.113729 + 0.993512i $$0.536280\pi$$
$$992$$ 0 0
$$993$$ −7500.00 −0.239683
$$994$$ 0 0
$$995$$ −45040.0 −1.43504
$$996$$ 0 0
$$997$$ 45202.0 1.43587 0.717935 0.696111i $$-0.245088\pi$$
0.717935 + 0.696111i $$0.245088\pi$$
$$998$$ 0 0
$$999$$ −3078.00 −0.0974811
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 132.4.a.d.1.1 1
3.2 odd 2 396.4.a.c.1.1 1
4.3 odd 2 528.4.a.e.1.1 1
8.3 odd 2 2112.4.a.q.1.1 1
8.5 even 2 2112.4.a.e.1.1 1
11.10 odd 2 1452.4.a.i.1.1 1
12.11 even 2 1584.4.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
132.4.a.d.1.1 1 1.1 even 1 trivial
396.4.a.c.1.1 1 3.2 odd 2
528.4.a.e.1.1 1 4.3 odd 2
1452.4.a.i.1.1 1 11.10 odd 2
1584.4.a.f.1.1 1 12.11 even 2
2112.4.a.e.1.1 1 8.5 even 2
2112.4.a.q.1.1 1 8.3 odd 2