# Properties

 Label 2496.4.a.m.1.1 Level $2496$ Weight $4$ Character 2496.1 Self dual yes Analytic conductor $147.269$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2496.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: $$147.268767374$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2496.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +2.00000 q^{5} +32.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +2.00000 q^{5} +32.0000 q^{7} +9.00000 q^{9} -68.0000 q^{11} -13.0000 q^{13} +6.00000 q^{15} -14.0000 q^{17} +4.00000 q^{19} +96.0000 q^{21} -72.0000 q^{23} -121.000 q^{25} +27.0000 q^{27} -102.000 q^{29} +136.000 q^{31} -204.000 q^{33} +64.0000 q^{35} +386.000 q^{37} -39.0000 q^{39} +250.000 q^{41} -140.000 q^{43} +18.0000 q^{45} +296.000 q^{47} +681.000 q^{49} -42.0000 q^{51} -526.000 q^{53} -136.000 q^{55} +12.0000 q^{57} +332.000 q^{59} +410.000 q^{61} +288.000 q^{63} -26.0000 q^{65} +596.000 q^{67} -216.000 q^{69} +880.000 q^{71} +506.000 q^{73} -363.000 q^{75} -2176.00 q^{77} +640.000 q^{79} +81.0000 q^{81} +1380.00 q^{83} -28.0000 q^{85} -306.000 q^{87} +1450.00 q^{89} -416.000 q^{91} +408.000 q^{93} +8.00000 q^{95} -446.000 q^{97} -612.000 q^{99} +O(q^{100})$$

## 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
$$5$$ 2.00000 0.178885 0.0894427 0.995992i $$-0.471491\pi$$
0.0894427 + 0.995992i $$0.471491\pi$$
$$6$$ 0 0
$$7$$ 32.0000 1.72784 0.863919 0.503631i $$-0.168003\pi$$
0.863919 + 0.503631i $$0.168003\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −68.0000 −1.86389 −0.931944 0.362602i $$-0.881889\pi$$
−0.931944 + 0.362602i $$0.881889\pi$$
$$12$$ 0 0
$$13$$ −13.0000 −0.277350
$$14$$ 0 0
$$15$$ 6.00000 0.103280
$$16$$ 0 0
$$17$$ −14.0000 −0.199735 −0.0998676 0.995001i $$-0.531842\pi$$
−0.0998676 + 0.995001i $$0.531842\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.0482980 0.0241490 0.999708i $$-0.492312\pi$$
0.0241490 + 0.999708i $$0.492312\pi$$
$$20$$ 0 0
$$21$$ 96.0000 0.997567
$$22$$ 0 0
$$23$$ −72.0000 −0.652741 −0.326370 0.945242i $$-0.605826\pi$$
−0.326370 + 0.945242i $$0.605826\pi$$
$$24$$ 0 0
$$25$$ −121.000 −0.968000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −102.000 −0.653135 −0.326568 0.945174i $$-0.605892\pi$$
−0.326568 + 0.945174i $$0.605892\pi$$
$$30$$ 0 0
$$31$$ 136.000 0.787946 0.393973 0.919122i $$-0.371100\pi$$
0.393973 + 0.919122i $$0.371100\pi$$
$$32$$ 0 0
$$33$$ −204.000 −1.07612
$$34$$ 0 0
$$35$$ 64.0000 0.309085
$$36$$ 0 0
$$37$$ 386.000 1.71508 0.857541 0.514416i $$-0.171991\pi$$
0.857541 + 0.514416i $$0.171991\pi$$
$$38$$ 0 0
$$39$$ −39.0000 −0.160128
$$40$$ 0 0
$$41$$ 250.000 0.952279 0.476140 0.879370i $$-0.342036\pi$$
0.476140 + 0.879370i $$0.342036\pi$$
$$42$$ 0 0
$$43$$ −140.000 −0.496507 −0.248253 0.968695i $$-0.579857\pi$$
−0.248253 + 0.968695i $$0.579857\pi$$
$$44$$ 0 0
$$45$$ 18.0000 0.0596285
$$46$$ 0 0
$$47$$ 296.000 0.918639 0.459320 0.888271i $$-0.348093\pi$$
0.459320 + 0.888271i $$0.348093\pi$$
$$48$$ 0 0
$$49$$ 681.000 1.98542
$$50$$ 0 0
$$51$$ −42.0000 −0.115317
$$52$$ 0 0
$$53$$ −526.000 −1.36324 −0.681619 0.731707i $$-0.738724\pi$$
−0.681619 + 0.731707i $$0.738724\pi$$
$$54$$ 0 0
$$55$$ −136.000 −0.333422
$$56$$ 0 0
$$57$$ 12.0000 0.0278849
$$58$$ 0 0
$$59$$ 332.000 0.732588 0.366294 0.930499i $$-0.380626\pi$$
0.366294 + 0.930499i $$0.380626\pi$$
$$60$$ 0 0
$$61$$ 410.000 0.860576 0.430288 0.902692i $$-0.358412\pi$$
0.430288 + 0.902692i $$0.358412\pi$$
$$62$$ 0 0
$$63$$ 288.000 0.575946
$$64$$ 0 0
$$65$$ −26.0000 −0.0496139
$$66$$ 0 0
$$67$$ 596.000 1.08676 0.543381 0.839487i $$-0.317144\pi$$
0.543381 + 0.839487i $$0.317144\pi$$
$$68$$ 0 0
$$69$$ −216.000 −0.376860
$$70$$ 0 0
$$71$$ 880.000 1.47094 0.735470 0.677557i $$-0.236961\pi$$
0.735470 + 0.677557i $$0.236961\pi$$
$$72$$ 0 0
$$73$$ 506.000 0.811272 0.405636 0.914035i $$-0.367050\pi$$
0.405636 + 0.914035i $$0.367050\pi$$
$$74$$ 0 0
$$75$$ −363.000 −0.558875
$$76$$ 0 0
$$77$$ −2176.00 −3.22050
$$78$$ 0 0
$$79$$ 640.000 0.911464 0.455732 0.890117i $$-0.349378\pi$$
0.455732 + 0.890117i $$0.349378\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 1380.00 1.82500 0.912498 0.409081i $$-0.134151\pi$$
0.912498 + 0.409081i $$0.134151\pi$$
$$84$$ 0 0
$$85$$ −28.0000 −0.0357297
$$86$$ 0 0
$$87$$ −306.000 −0.377088
$$88$$ 0 0
$$89$$ 1450.00 1.72696 0.863481 0.504381i $$-0.168279\pi$$
0.863481 + 0.504381i $$0.168279\pi$$
$$90$$ 0 0
$$91$$ −416.000 −0.479216
$$92$$ 0 0
$$93$$ 408.000 0.454921
$$94$$ 0 0
$$95$$ 8.00000 0.00863982
$$96$$ 0 0
$$97$$ −446.000 −0.466850 −0.233425 0.972375i $$-0.574993\pi$$
−0.233425 + 0.972375i $$0.574993\pi$$
$$98$$ 0 0
$$99$$ −612.000 −0.621296
$$100$$ 0 0
$$101$$ 610.000 0.600963 0.300482 0.953788i $$-0.402853\pi$$
0.300482 + 0.953788i $$0.402853\pi$$
$$102$$ 0 0
$$103$$ 1352.00 1.29336 0.646682 0.762760i $$-0.276156\pi$$
0.646682 + 0.762760i $$0.276156\pi$$
$$104$$ 0 0
$$105$$ 192.000 0.178450
$$106$$ 0 0
$$107$$ −732.000 −0.661356 −0.330678 0.943744i $$-0.607277\pi$$
−0.330678 + 0.943744i $$0.607277\pi$$
$$108$$ 0 0
$$109$$ 1514.00 1.33041 0.665206 0.746660i $$-0.268344\pi$$
0.665206 + 0.746660i $$0.268344\pi$$
$$110$$ 0 0
$$111$$ 1158.00 0.990203
$$112$$ 0 0
$$113$$ −1518.00 −1.26373 −0.631865 0.775079i $$-0.717710\pi$$
−0.631865 + 0.775079i $$0.717710\pi$$
$$114$$ 0 0
$$115$$ −144.000 −0.116766
$$116$$ 0 0
$$117$$ −117.000 −0.0924500
$$118$$ 0 0
$$119$$ −448.000 −0.345110
$$120$$ 0 0
$$121$$ 3293.00 2.47408
$$122$$ 0 0
$$123$$ 750.000 0.549799
$$124$$ 0 0
$$125$$ −492.000 −0.352047
$$126$$ 0 0
$$127$$ 96.0000 0.0670758 0.0335379 0.999437i $$-0.489323\pi$$
0.0335379 + 0.999437i $$0.489323\pi$$
$$128$$ 0 0
$$129$$ −420.000 −0.286658
$$130$$ 0 0
$$131$$ −2548.00 −1.69939 −0.849694 0.527276i $$-0.823213\pi$$
−0.849694 + 0.527276i $$0.823213\pi$$
$$132$$ 0 0
$$133$$ 128.000 0.0834512
$$134$$ 0 0
$$135$$ 54.0000 0.0344265
$$136$$ 0 0
$$137$$ −230.000 −0.143432 −0.0717162 0.997425i $$-0.522848\pi$$
−0.0717162 + 0.997425i $$0.522848\pi$$
$$138$$ 0 0
$$139$$ 516.000 0.314867 0.157434 0.987530i $$-0.449678\pi$$
0.157434 + 0.987530i $$0.449678\pi$$
$$140$$ 0 0
$$141$$ 888.000 0.530377
$$142$$ 0 0
$$143$$ 884.000 0.516950
$$144$$ 0 0
$$145$$ −204.000 −0.116836
$$146$$ 0 0
$$147$$ 2043.00 1.14628
$$148$$ 0 0
$$149$$ 1842.00 1.01277 0.506384 0.862308i $$-0.330982\pi$$
0.506384 + 0.862308i $$0.330982\pi$$
$$150$$ 0 0
$$151$$ 528.000 0.284556 0.142278 0.989827i $$-0.454557\pi$$
0.142278 + 0.989827i $$0.454557\pi$$
$$152$$ 0 0
$$153$$ −126.000 −0.0665784
$$154$$ 0 0
$$155$$ 272.000 0.140952
$$156$$ 0 0
$$157$$ 1306.00 0.663886 0.331943 0.943299i $$-0.392296\pi$$
0.331943 + 0.943299i $$0.392296\pi$$
$$158$$ 0 0
$$159$$ −1578.00 −0.787066
$$160$$ 0 0
$$161$$ −2304.00 −1.12783
$$162$$ 0 0
$$163$$ −3772.00 −1.81255 −0.906276 0.422687i $$-0.861087\pi$$
−0.906276 + 0.422687i $$0.861087\pi$$
$$164$$ 0 0
$$165$$ −408.000 −0.192502
$$166$$ 0 0
$$167$$ −384.000 −0.177933 −0.0889665 0.996035i $$-0.528356\pi$$
−0.0889665 + 0.996035i $$0.528356\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 36.0000 0.0160993
$$172$$ 0 0
$$173$$ −1462.00 −0.642508 −0.321254 0.946993i $$-0.604104\pi$$
−0.321254 + 0.946993i $$0.604104\pi$$
$$174$$ 0 0
$$175$$ −3872.00 −1.67255
$$176$$ 0 0
$$177$$ 996.000 0.422960
$$178$$ 0 0
$$179$$ −1332.00 −0.556192 −0.278096 0.960553i $$-0.589703\pi$$
−0.278096 + 0.960553i $$0.589703\pi$$
$$180$$ 0 0
$$181$$ −2030.00 −0.833639 −0.416820 0.908989i $$-0.636855\pi$$
−0.416820 + 0.908989i $$0.636855\pi$$
$$182$$ 0 0
$$183$$ 1230.00 0.496854
$$184$$ 0 0
$$185$$ 772.000 0.306803
$$186$$ 0 0
$$187$$ 952.000 0.372284
$$188$$ 0 0
$$189$$ 864.000 0.332522
$$190$$ 0 0
$$191$$ 16.0000 0.00606136 0.00303068 0.999995i $$-0.499035\pi$$
0.00303068 + 0.999995i $$0.499035\pi$$
$$192$$ 0 0
$$193$$ −2078.00 −0.775014 −0.387507 0.921867i $$-0.626664\pi$$
−0.387507 + 0.921867i $$0.626664\pi$$
$$194$$ 0 0
$$195$$ −78.0000 −0.0286446
$$196$$ 0 0
$$197$$ −3486.00 −1.26075 −0.630374 0.776292i $$-0.717098\pi$$
−0.630374 + 0.776292i $$0.717098\pi$$
$$198$$ 0 0
$$199$$ −568.000 −0.202334 −0.101167 0.994869i $$-0.532258\pi$$
−0.101167 + 0.994869i $$0.532258\pi$$
$$200$$ 0 0
$$201$$ 1788.00 0.627442
$$202$$ 0 0
$$203$$ −3264.00 −1.12851
$$204$$ 0 0
$$205$$ 500.000 0.170349
$$206$$ 0 0
$$207$$ −648.000 −0.217580
$$208$$ 0 0
$$209$$ −272.000 −0.0900222
$$210$$ 0 0
$$211$$ 3804.00 1.24113 0.620564 0.784156i $$-0.286904\pi$$
0.620564 + 0.784156i $$0.286904\pi$$
$$212$$ 0 0
$$213$$ 2640.00 0.849248
$$214$$ 0 0
$$215$$ −280.000 −0.0888179
$$216$$ 0 0
$$217$$ 4352.00 1.36144
$$218$$ 0 0
$$219$$ 1518.00 0.468388
$$220$$ 0 0
$$221$$ 182.000 0.0553966
$$222$$ 0 0
$$223$$ −5912.00 −1.77532 −0.887661 0.460498i $$-0.847671\pi$$
−0.887661 + 0.460498i $$0.847671\pi$$
$$224$$ 0 0
$$225$$ −1089.00 −0.322667
$$226$$ 0 0
$$227$$ −3308.00 −0.967223 −0.483612 0.875283i $$-0.660675\pi$$
−0.483612 + 0.875283i $$0.660675\pi$$
$$228$$ 0 0
$$229$$ 6050.00 1.74583 0.872915 0.487872i $$-0.162227\pi$$
0.872915 + 0.487872i $$0.162227\pi$$
$$230$$ 0 0
$$231$$ −6528.00 −1.85935
$$232$$ 0 0
$$233$$ 4794.00 1.34792 0.673960 0.738768i $$-0.264592\pi$$
0.673960 + 0.738768i $$0.264592\pi$$
$$234$$ 0 0
$$235$$ 592.000 0.164331
$$236$$ 0 0
$$237$$ 1920.00 0.526234
$$238$$ 0 0
$$239$$ 4440.00 1.20167 0.600836 0.799372i $$-0.294834\pi$$
0.600836 + 0.799372i $$0.294834\pi$$
$$240$$ 0 0
$$241$$ 1330.00 0.355489 0.177744 0.984077i $$-0.443120\pi$$
0.177744 + 0.984077i $$0.443120\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 1362.00 0.355163
$$246$$ 0 0
$$247$$ −52.0000 −0.0133955
$$248$$ 0 0
$$249$$ 4140.00 1.05366
$$250$$ 0 0
$$251$$ −5116.00 −1.28653 −0.643265 0.765644i $$-0.722421\pi$$
−0.643265 + 0.765644i $$0.722421\pi$$
$$252$$ 0 0
$$253$$ 4896.00 1.21664
$$254$$ 0 0
$$255$$ −84.0000 −0.0206286
$$256$$ 0 0
$$257$$ 642.000 0.155824 0.0779122 0.996960i $$-0.475175\pi$$
0.0779122 + 0.996960i $$0.475175\pi$$
$$258$$ 0 0
$$259$$ 12352.0 2.96338
$$260$$ 0 0
$$261$$ −918.000 −0.217712
$$262$$ 0 0
$$263$$ −4264.00 −0.999732 −0.499866 0.866103i $$-0.666617\pi$$
−0.499866 + 0.866103i $$0.666617\pi$$
$$264$$ 0 0
$$265$$ −1052.00 −0.243864
$$266$$ 0 0
$$267$$ 4350.00 0.997062
$$268$$ 0 0
$$269$$ 3850.00 0.872634 0.436317 0.899793i $$-0.356283\pi$$
0.436317 + 0.899793i $$0.356283\pi$$
$$270$$ 0 0
$$271$$ −2936.00 −0.658115 −0.329058 0.944310i $$-0.606731\pi$$
−0.329058 + 0.944310i $$0.606731\pi$$
$$272$$ 0 0
$$273$$ −1248.00 −0.276675
$$274$$ 0 0
$$275$$ 8228.00 1.80424
$$276$$ 0 0
$$277$$ 2066.00 0.448137 0.224068 0.974573i $$-0.428066\pi$$
0.224068 + 0.974573i $$0.428066\pi$$
$$278$$ 0 0
$$279$$ 1224.00 0.262649
$$280$$ 0 0
$$281$$ −214.000 −0.0454312 −0.0227156 0.999742i $$-0.507231\pi$$
−0.0227156 + 0.999742i $$0.507231\pi$$
$$282$$ 0 0
$$283$$ −2620.00 −0.550328 −0.275164 0.961397i $$-0.588732\pi$$
−0.275164 + 0.961397i $$0.588732\pi$$
$$284$$ 0 0
$$285$$ 24.0000 0.00498820
$$286$$ 0 0
$$287$$ 8000.00 1.64538
$$288$$ 0 0
$$289$$ −4717.00 −0.960106
$$290$$ 0 0
$$291$$ −1338.00 −0.269536
$$292$$ 0 0
$$293$$ 1154.00 0.230094 0.115047 0.993360i $$-0.463298\pi$$
0.115047 + 0.993360i $$0.463298\pi$$
$$294$$ 0 0
$$295$$ 664.000 0.131049
$$296$$ 0 0
$$297$$ −1836.00 −0.358705
$$298$$ 0 0
$$299$$ 936.000 0.181038
$$300$$ 0 0
$$301$$ −4480.00 −0.857883
$$302$$ 0 0
$$303$$ 1830.00 0.346966
$$304$$ 0 0
$$305$$ 820.000 0.153944
$$306$$ 0 0
$$307$$ −4076.00 −0.757751 −0.378876 0.925448i $$-0.623689\pi$$
−0.378876 + 0.925448i $$0.623689\pi$$
$$308$$ 0 0
$$309$$ 4056.00 0.746724
$$310$$ 0 0
$$311$$ 6456.00 1.17713 0.588563 0.808451i $$-0.299694\pi$$
0.588563 + 0.808451i $$0.299694\pi$$
$$312$$ 0 0
$$313$$ −5526.00 −0.997917 −0.498958 0.866626i $$-0.666284\pi$$
−0.498958 + 0.866626i $$0.666284\pi$$
$$314$$ 0 0
$$315$$ 576.000 0.103028
$$316$$ 0 0
$$317$$ 10458.0 1.85293 0.926467 0.376377i $$-0.122830\pi$$
0.926467 + 0.376377i $$0.122830\pi$$
$$318$$ 0 0
$$319$$ 6936.00 1.21737
$$320$$ 0 0
$$321$$ −2196.00 −0.381834
$$322$$ 0 0
$$323$$ −56.0000 −0.00964682
$$324$$ 0 0
$$325$$ 1573.00 0.268475
$$326$$ 0 0
$$327$$ 4542.00 0.768114
$$328$$ 0 0
$$329$$ 9472.00 1.58726
$$330$$ 0 0
$$331$$ 2348.00 0.389903 0.194951 0.980813i $$-0.437545\pi$$
0.194951 + 0.980813i $$0.437545\pi$$
$$332$$ 0 0
$$333$$ 3474.00 0.571694
$$334$$ 0 0
$$335$$ 1192.00 0.194406
$$336$$ 0 0
$$337$$ 5298.00 0.856381 0.428191 0.903688i $$-0.359151\pi$$
0.428191 + 0.903688i $$0.359151\pi$$
$$338$$ 0 0
$$339$$ −4554.00 −0.729615
$$340$$ 0 0
$$341$$ −9248.00 −1.46864
$$342$$ 0 0
$$343$$ 10816.0 1.70265
$$344$$ 0 0
$$345$$ −432.000 −0.0674148
$$346$$ 0 0
$$347$$ 9876.00 1.52787 0.763936 0.645292i $$-0.223264\pi$$
0.763936 + 0.645292i $$0.223264\pi$$
$$348$$ 0 0
$$349$$ 5370.00 0.823638 0.411819 0.911266i $$-0.364894\pi$$
0.411819 + 0.911266i $$0.364894\pi$$
$$350$$ 0 0
$$351$$ −351.000 −0.0533761
$$352$$ 0 0
$$353$$ 7330.00 1.10520 0.552601 0.833446i $$-0.313635\pi$$
0.552601 + 0.833446i $$0.313635\pi$$
$$354$$ 0 0
$$355$$ 1760.00 0.263130
$$356$$ 0 0
$$357$$ −1344.00 −0.199249
$$358$$ 0 0
$$359$$ −7488.00 −1.10084 −0.550420 0.834888i $$-0.685532\pi$$
−0.550420 + 0.834888i $$0.685532\pi$$
$$360$$ 0 0
$$361$$ −6843.00 −0.997667
$$362$$ 0 0
$$363$$ 9879.00 1.42841
$$364$$ 0 0
$$365$$ 1012.00 0.145125
$$366$$ 0 0
$$367$$ −1504.00 −0.213919 −0.106959 0.994263i $$-0.534111\pi$$
−0.106959 + 0.994263i $$0.534111\pi$$
$$368$$ 0 0
$$369$$ 2250.00 0.317426
$$370$$ 0 0
$$371$$ −16832.0 −2.35546
$$372$$ 0 0
$$373$$ −6702.00 −0.930339 −0.465169 0.885222i $$-0.654007\pi$$
−0.465169 + 0.885222i $$0.654007\pi$$
$$374$$ 0 0
$$375$$ −1476.00 −0.203254
$$376$$ 0 0
$$377$$ 1326.00 0.181147
$$378$$ 0 0
$$379$$ −5700.00 −0.772531 −0.386266 0.922388i $$-0.626235\pi$$
−0.386266 + 0.922388i $$0.626235\pi$$
$$380$$ 0 0
$$381$$ 288.000 0.0387262
$$382$$ 0 0
$$383$$ −2328.00 −0.310588 −0.155294 0.987868i $$-0.549633\pi$$
−0.155294 + 0.987868i $$0.549633\pi$$
$$384$$ 0 0
$$385$$ −4352.00 −0.576100
$$386$$ 0 0
$$387$$ −1260.00 −0.165502
$$388$$ 0 0
$$389$$ 11554.0 1.50594 0.752971 0.658054i $$-0.228620\pi$$
0.752971 + 0.658054i $$0.228620\pi$$
$$390$$ 0 0
$$391$$ 1008.00 0.130375
$$392$$ 0 0
$$393$$ −7644.00 −0.981142
$$394$$ 0 0
$$395$$ 1280.00 0.163048
$$396$$ 0 0
$$397$$ −6486.00 −0.819957 −0.409979 0.912095i $$-0.634464\pi$$
−0.409979 + 0.912095i $$0.634464\pi$$
$$398$$ 0 0
$$399$$ 384.000 0.0481806
$$400$$ 0 0
$$401$$ 7698.00 0.958653 0.479326 0.877637i $$-0.340881\pi$$
0.479326 + 0.877637i $$0.340881\pi$$
$$402$$ 0 0
$$403$$ −1768.00 −0.218537
$$404$$ 0 0
$$405$$ 162.000 0.0198762
$$406$$ 0 0
$$407$$ −26248.0 −3.19672
$$408$$ 0 0
$$409$$ 3338.00 0.403554 0.201777 0.979432i $$-0.435328\pi$$
0.201777 + 0.979432i $$0.435328\pi$$
$$410$$ 0 0
$$411$$ −690.000 −0.0828107
$$412$$ 0 0
$$413$$ 10624.0 1.26579
$$414$$ 0 0
$$415$$ 2760.00 0.326465
$$416$$ 0 0
$$417$$ 1548.00 0.181789
$$418$$ 0 0
$$419$$ −52.0000 −0.00606293 −0.00303146 0.999995i $$-0.500965\pi$$
−0.00303146 + 0.999995i $$0.500965\pi$$
$$420$$ 0 0
$$421$$ 5858.00 0.678151 0.339075 0.940759i $$-0.389886\pi$$
0.339075 + 0.940759i $$0.389886\pi$$
$$422$$ 0 0
$$423$$ 2664.00 0.306213
$$424$$ 0 0
$$425$$ 1694.00 0.193344
$$426$$ 0 0
$$427$$ 13120.0 1.48694
$$428$$ 0 0
$$429$$ 2652.00 0.298461
$$430$$ 0 0
$$431$$ −8840.00 −0.987953 −0.493977 0.869475i $$-0.664457\pi$$
−0.493977 + 0.869475i $$0.664457\pi$$
$$432$$ 0 0
$$433$$ 11346.0 1.25925 0.629624 0.776900i $$-0.283209\pi$$
0.629624 + 0.776900i $$0.283209\pi$$
$$434$$ 0 0
$$435$$ −612.000 −0.0674555
$$436$$ 0 0
$$437$$ −288.000 −0.0315261
$$438$$ 0 0
$$439$$ −16456.0 −1.78907 −0.894535 0.446997i $$-0.852493\pi$$
−0.894535 + 0.446997i $$0.852493\pi$$
$$440$$ 0 0
$$441$$ 6129.00 0.661808
$$442$$ 0 0
$$443$$ −3788.00 −0.406260 −0.203130 0.979152i $$-0.565111\pi$$
−0.203130 + 0.979152i $$0.565111\pi$$
$$444$$ 0 0
$$445$$ 2900.00 0.308929
$$446$$ 0 0
$$447$$ 5526.00 0.584722
$$448$$ 0 0
$$449$$ 546.000 0.0573883 0.0286941 0.999588i $$-0.490865\pi$$
0.0286941 + 0.999588i $$0.490865\pi$$
$$450$$ 0 0
$$451$$ −17000.0 −1.77494
$$452$$ 0 0
$$453$$ 1584.00 0.164289
$$454$$ 0 0
$$455$$ −832.000 −0.0857248
$$456$$ 0 0
$$457$$ 3546.00 0.362965 0.181482 0.983394i $$-0.441910\pi$$
0.181482 + 0.983394i $$0.441910\pi$$
$$458$$ 0 0
$$459$$ −378.000 −0.0384391
$$460$$ 0 0
$$461$$ −12918.0 −1.30510 −0.652550 0.757746i $$-0.726301\pi$$
−0.652550 + 0.757746i $$0.726301\pi$$
$$462$$ 0 0
$$463$$ −18328.0 −1.83969 −0.919843 0.392287i $$-0.871684\pi$$
−0.919843 + 0.392287i $$0.871684\pi$$
$$464$$ 0 0
$$465$$ 816.000 0.0813787
$$466$$ 0 0
$$467$$ 11980.0 1.18708 0.593542 0.804803i $$-0.297729\pi$$
0.593542 + 0.804803i $$0.297729\pi$$
$$468$$ 0 0
$$469$$ 19072.0 1.87775
$$470$$ 0 0
$$471$$ 3918.00 0.383295
$$472$$ 0 0
$$473$$ 9520.00 0.925434
$$474$$ 0 0
$$475$$ −484.000 −0.0467525
$$476$$ 0 0
$$477$$ −4734.00 −0.454413
$$478$$ 0 0
$$479$$ 12344.0 1.17748 0.588739 0.808323i $$-0.299625\pi$$
0.588739 + 0.808323i $$0.299625\pi$$
$$480$$ 0 0
$$481$$ −5018.00 −0.475678
$$482$$ 0 0
$$483$$ −6912.00 −0.651153
$$484$$ 0 0
$$485$$ −892.000 −0.0835126
$$486$$ 0 0
$$487$$ 80.0000 0.00744383 0.00372192 0.999993i $$-0.498815\pi$$
0.00372192 + 0.999993i $$0.498815\pi$$
$$488$$ 0 0
$$489$$ −11316.0 −1.04648
$$490$$ 0 0
$$491$$ −15660.0 −1.43936 −0.719680 0.694306i $$-0.755712\pi$$
−0.719680 + 0.694306i $$0.755712\pi$$
$$492$$ 0 0
$$493$$ 1428.00 0.130454
$$494$$ 0 0
$$495$$ −1224.00 −0.111141
$$496$$ 0 0
$$497$$ 28160.0 2.54155
$$498$$ 0 0
$$499$$ −60.0000 −0.00538270 −0.00269135 0.999996i $$-0.500857\pi$$
−0.00269135 + 0.999996i $$0.500857\pi$$
$$500$$ 0 0
$$501$$ −1152.00 −0.102730
$$502$$ 0 0
$$503$$ −12248.0 −1.08571 −0.542854 0.839827i $$-0.682656\pi$$
−0.542854 + 0.839827i $$0.682656\pi$$
$$504$$ 0 0
$$505$$ 1220.00 0.107504
$$506$$ 0 0
$$507$$ 507.000 0.0444116
$$508$$ 0 0
$$509$$ 90.0000 0.00783729 0.00391864 0.999992i $$-0.498753\pi$$
0.00391864 + 0.999992i $$0.498753\pi$$
$$510$$ 0 0
$$511$$ 16192.0 1.40175
$$512$$ 0 0
$$513$$ 108.000 0.00929496
$$514$$ 0 0
$$515$$ 2704.00 0.231364
$$516$$ 0 0
$$517$$ −20128.0 −1.71224
$$518$$ 0 0
$$519$$ −4386.00 −0.370952
$$520$$ 0 0
$$521$$ 9818.00 0.825594 0.412797 0.910823i $$-0.364552\pi$$
0.412797 + 0.910823i $$0.364552\pi$$
$$522$$ 0 0
$$523$$ −20252.0 −1.69323 −0.846614 0.532208i $$-0.821363\pi$$
−0.846614 + 0.532208i $$0.821363\pi$$
$$524$$ 0 0
$$525$$ −11616.0 −0.965645
$$526$$ 0 0
$$527$$ −1904.00 −0.157381
$$528$$ 0 0
$$529$$ −6983.00 −0.573929
$$530$$ 0 0
$$531$$ 2988.00 0.244196
$$532$$ 0 0
$$533$$ −3250.00 −0.264115
$$534$$ 0 0
$$535$$ −1464.00 −0.118307
$$536$$ 0 0
$$537$$ −3996.00 −0.321118
$$538$$ 0 0
$$539$$ −46308.0 −3.70061
$$540$$ 0 0
$$541$$ 12634.0 1.00403 0.502013 0.864860i $$-0.332593\pi$$
0.502013 + 0.864860i $$0.332593\pi$$
$$542$$ 0 0
$$543$$ −6090.00 −0.481302
$$544$$ 0 0
$$545$$ 3028.00 0.237991
$$546$$ 0 0
$$547$$ 11756.0 0.918922 0.459461 0.888198i $$-0.348043\pi$$
0.459461 + 0.888198i $$0.348043\pi$$
$$548$$ 0 0
$$549$$ 3690.00 0.286859
$$550$$ 0 0
$$551$$ −408.000 −0.0315452
$$552$$ 0 0
$$553$$ 20480.0 1.57486
$$554$$ 0 0
$$555$$ 2316.00 0.177133
$$556$$ 0 0
$$557$$ −17622.0 −1.34052 −0.670259 0.742128i $$-0.733817\pi$$
−0.670259 + 0.742128i $$0.733817\pi$$
$$558$$ 0 0
$$559$$ 1820.00 0.137706
$$560$$ 0 0
$$561$$ 2856.00 0.214938
$$562$$ 0 0
$$563$$ −23092.0 −1.72862 −0.864309 0.502961i $$-0.832244\pi$$
−0.864309 + 0.502961i $$0.832244\pi$$
$$564$$ 0 0
$$565$$ −3036.00 −0.226063
$$566$$ 0 0
$$567$$ 2592.00 0.191982
$$568$$ 0 0
$$569$$ −1302.00 −0.0959274 −0.0479637 0.998849i $$-0.515273\pi$$
−0.0479637 + 0.998849i $$0.515273\pi$$
$$570$$ 0 0
$$571$$ 24868.0 1.82258 0.911290 0.411765i $$-0.135087\pi$$
0.911290 + 0.411765i $$0.135087\pi$$
$$572$$ 0 0
$$573$$ 48.0000 0.00349953
$$574$$ 0 0
$$575$$ 8712.00 0.631853
$$576$$ 0 0
$$577$$ 2562.00 0.184848 0.0924241 0.995720i $$-0.470538\pi$$
0.0924241 + 0.995720i $$0.470538\pi$$
$$578$$ 0 0
$$579$$ −6234.00 −0.447455
$$580$$ 0 0
$$581$$ 44160.0 3.15330
$$582$$ 0 0
$$583$$ 35768.0 2.54092
$$584$$ 0 0
$$585$$ −234.000 −0.0165380
$$586$$ 0 0
$$587$$ 13484.0 0.948116 0.474058 0.880494i $$-0.342789\pi$$
0.474058 + 0.880494i $$0.342789\pi$$
$$588$$ 0 0
$$589$$ 544.000 0.0380562
$$590$$ 0 0
$$591$$ −10458.0 −0.727893
$$592$$ 0 0
$$593$$ −16974.0 −1.17544 −0.587722 0.809063i $$-0.699975\pi$$
−0.587722 + 0.809063i $$0.699975\pi$$
$$594$$ 0 0
$$595$$ −896.000 −0.0617352
$$596$$ 0 0
$$597$$ −1704.00 −0.116818
$$598$$ 0 0
$$599$$ −3864.00 −0.263571 −0.131785 0.991278i $$-0.542071\pi$$
−0.131785 + 0.991278i $$0.542071\pi$$
$$600$$ 0 0
$$601$$ 17546.0 1.19088 0.595438 0.803401i $$-0.296979\pi$$
0.595438 + 0.803401i $$0.296979\pi$$
$$602$$ 0 0
$$603$$ 5364.00 0.362254
$$604$$ 0 0
$$605$$ 6586.00 0.442577
$$606$$ 0 0
$$607$$ −9296.00 −0.621603 −0.310801 0.950475i $$-0.600597\pi$$
−0.310801 + 0.950475i $$0.600597\pi$$
$$608$$ 0 0
$$609$$ −9792.00 −0.651547
$$610$$ 0 0
$$611$$ −3848.00 −0.254785
$$612$$ 0 0
$$613$$ 6914.00 0.455553 0.227776 0.973713i $$-0.426854\pi$$
0.227776 + 0.973713i $$0.426854\pi$$
$$614$$ 0 0
$$615$$ 1500.00 0.0983510
$$616$$ 0 0
$$617$$ −25446.0 −1.66032 −0.830160 0.557525i $$-0.811751\pi$$
−0.830160 + 0.557525i $$0.811751\pi$$
$$618$$ 0 0
$$619$$ −11236.0 −0.729585 −0.364792 0.931089i $$-0.618860\pi$$
−0.364792 + 0.931089i $$0.618860\pi$$
$$620$$ 0 0
$$621$$ −1944.00 −0.125620
$$622$$ 0 0
$$623$$ 46400.0 2.98391
$$624$$ 0 0
$$625$$ 14141.0 0.905024
$$626$$ 0 0
$$627$$ −816.000 −0.0519743
$$628$$ 0 0
$$629$$ −5404.00 −0.342562
$$630$$ 0 0
$$631$$ 29424.0 1.85634 0.928170 0.372156i $$-0.121381\pi$$
0.928170 + 0.372156i $$0.121381\pi$$
$$632$$ 0 0
$$633$$ 11412.0 0.716566
$$634$$ 0 0
$$635$$ 192.000 0.0119989
$$636$$ 0 0
$$637$$ −8853.00 −0.550657
$$638$$ 0 0
$$639$$ 7920.00 0.490314
$$640$$ 0 0
$$641$$ −5054.00 −0.311421 −0.155711 0.987803i $$-0.549767\pi$$
−0.155711 + 0.987803i $$0.549767\pi$$
$$642$$ 0 0
$$643$$ −1132.00 −0.0694273 −0.0347136 0.999397i $$-0.511052\pi$$
−0.0347136 + 0.999397i $$0.511052\pi$$
$$644$$ 0 0
$$645$$ −840.000 −0.0512790
$$646$$ 0 0
$$647$$ −1464.00 −0.0889579 −0.0444790 0.999010i $$-0.514163\pi$$
−0.0444790 + 0.999010i $$0.514163\pi$$
$$648$$ 0 0
$$649$$ −22576.0 −1.36546
$$650$$ 0 0
$$651$$ 13056.0 0.786029
$$652$$ 0 0
$$653$$ −5494.00 −0.329245 −0.164622 0.986357i $$-0.552641\pi$$
−0.164622 + 0.986357i $$0.552641\pi$$
$$654$$ 0 0
$$655$$ −5096.00 −0.303996
$$656$$ 0 0
$$657$$ 4554.00 0.270424
$$658$$ 0 0
$$659$$ 11580.0 0.684511 0.342256 0.939607i $$-0.388809\pi$$
0.342256 + 0.939607i $$0.388809\pi$$
$$660$$ 0 0
$$661$$ 1298.00 0.0763787 0.0381894 0.999271i $$-0.487841\pi$$
0.0381894 + 0.999271i $$0.487841\pi$$
$$662$$ 0 0
$$663$$ 546.000 0.0319832
$$664$$ 0 0
$$665$$ 256.000 0.0149282
$$666$$ 0 0
$$667$$ 7344.00 0.426328
$$668$$ 0 0
$$669$$ −17736.0 −1.02498
$$670$$ 0 0
$$671$$ −27880.0 −1.60402
$$672$$ 0 0
$$673$$ 16162.0 0.925705 0.462852 0.886435i $$-0.346826\pi$$
0.462852 + 0.886435i $$0.346826\pi$$
$$674$$ 0 0
$$675$$ −3267.00 −0.186292
$$676$$ 0 0
$$677$$ 9890.00 0.561453 0.280726 0.959788i $$-0.409425\pi$$
0.280726 + 0.959788i $$0.409425\pi$$
$$678$$ 0 0
$$679$$ −14272.0 −0.806641
$$680$$ 0 0
$$681$$ −9924.00 −0.558427
$$682$$ 0 0
$$683$$ 27100.0 1.51823 0.759116 0.650955i $$-0.225631\pi$$
0.759116 + 0.650955i $$0.225631\pi$$
$$684$$ 0 0
$$685$$ −460.000 −0.0256580
$$686$$ 0 0
$$687$$ 18150.0 1.00796
$$688$$ 0 0
$$689$$ 6838.00 0.378094
$$690$$ 0 0
$$691$$ −11132.0 −0.612853 −0.306426 0.951894i $$-0.599133\pi$$
−0.306426 + 0.951894i $$0.599133\pi$$
$$692$$ 0 0
$$693$$ −19584.0 −1.07350
$$694$$ 0 0
$$695$$ 1032.00 0.0563252
$$696$$ 0 0
$$697$$ −3500.00 −0.190204
$$698$$ 0 0
$$699$$ 14382.0 0.778222
$$700$$ 0 0
$$701$$ −1862.00 −0.100323 −0.0501617 0.998741i $$-0.515974\pi$$
−0.0501617 + 0.998741i $$0.515974\pi$$
$$702$$ 0 0
$$703$$ 1544.00 0.0828351
$$704$$ 0 0
$$705$$ 1776.00 0.0948766
$$706$$ 0 0
$$707$$ 19520.0 1.03837
$$708$$ 0 0
$$709$$ 7938.00 0.420477 0.210238 0.977650i $$-0.432576\pi$$
0.210238 + 0.977650i $$0.432576\pi$$
$$710$$ 0 0
$$711$$ 5760.00 0.303821
$$712$$ 0 0
$$713$$ −9792.00 −0.514324
$$714$$ 0 0
$$715$$ 1768.00 0.0924748
$$716$$ 0 0
$$717$$ 13320.0 0.693786
$$718$$ 0 0
$$719$$ 24240.0 1.25730 0.628651 0.777688i $$-0.283608\pi$$
0.628651 + 0.777688i $$0.283608\pi$$
$$720$$ 0 0
$$721$$ 43264.0 2.23472
$$722$$ 0 0
$$723$$ 3990.00 0.205242
$$724$$ 0 0
$$725$$ 12342.0 0.632235
$$726$$ 0 0
$$727$$ 13720.0 0.699927 0.349963 0.936763i $$-0.386194\pi$$
0.349963 + 0.936763i $$0.386194\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 1960.00 0.0991699
$$732$$ 0 0
$$733$$ −21958.0 −1.10646 −0.553231 0.833028i $$-0.686605\pi$$
−0.553231 + 0.833028i $$0.686605\pi$$
$$734$$ 0 0
$$735$$ 4086.00 0.205054
$$736$$ 0 0
$$737$$ −40528.0 −2.02560
$$738$$ 0 0
$$739$$ 13348.0 0.664430 0.332215 0.943204i $$-0.392204\pi$$
0.332215 + 0.943204i $$0.392204\pi$$
$$740$$ 0 0
$$741$$ −156.000 −0.00773388
$$742$$ 0 0
$$743$$ −20304.0 −1.00253 −0.501266 0.865293i $$-0.667132\pi$$
−0.501266 + 0.865293i $$0.667132\pi$$
$$744$$ 0 0
$$745$$ 3684.00 0.181170
$$746$$ 0 0
$$747$$ 12420.0 0.608332
$$748$$ 0 0
$$749$$ −23424.0 −1.14272
$$750$$ 0 0
$$751$$ 3952.00 0.192025 0.0960123 0.995380i $$-0.469391\pi$$
0.0960123 + 0.995380i $$0.469391\pi$$
$$752$$ 0 0
$$753$$ −15348.0 −0.742779
$$754$$ 0 0
$$755$$ 1056.00 0.0509030
$$756$$ 0 0
$$757$$ 22386.0 1.07481 0.537406 0.843324i $$-0.319404\pi$$
0.537406 + 0.843324i $$0.319404\pi$$
$$758$$ 0 0
$$759$$ 14688.0 0.702425
$$760$$ 0 0
$$761$$ 12458.0 0.593433 0.296716 0.954966i $$-0.404108\pi$$
0.296716 + 0.954966i $$0.404108\pi$$
$$762$$ 0 0
$$763$$ 48448.0 2.29874
$$764$$ 0 0
$$765$$ −252.000 −0.0119099
$$766$$ 0 0
$$767$$ −4316.00 −0.203183
$$768$$ 0 0
$$769$$ −28126.0 −1.31892 −0.659460 0.751740i $$-0.729215\pi$$
−0.659460 + 0.751740i $$0.729215\pi$$
$$770$$ 0 0
$$771$$ 1926.00 0.0899652
$$772$$ 0 0
$$773$$ 35778.0 1.66474 0.832371 0.554219i $$-0.186983\pi$$
0.832371 + 0.554219i $$0.186983\pi$$
$$774$$ 0 0
$$775$$ −16456.0 −0.762732
$$776$$ 0 0
$$777$$ 37056.0 1.71091
$$778$$ 0 0
$$779$$ 1000.00 0.0459932
$$780$$ 0 0
$$781$$ −59840.0 −2.74167
$$782$$ 0 0
$$783$$ −2754.00 −0.125696
$$784$$ 0 0
$$785$$ 2612.00 0.118760
$$786$$ 0 0
$$787$$ 7252.00 0.328470 0.164235 0.986421i $$-0.447484\pi$$
0.164235 + 0.986421i $$0.447484\pi$$
$$788$$ 0 0
$$789$$ −12792.0 −0.577196
$$790$$ 0 0
$$791$$ −48576.0 −2.18352
$$792$$ 0 0
$$793$$ −5330.00 −0.238681
$$794$$ 0 0
$$795$$ −3156.00 −0.140795
$$796$$ 0 0
$$797$$ −4454.00 −0.197953 −0.0989766 0.995090i $$-0.531557\pi$$
−0.0989766 + 0.995090i $$0.531557\pi$$
$$798$$ 0 0
$$799$$ −4144.00 −0.183485
$$800$$ 0 0
$$801$$ 13050.0 0.575654
$$802$$ 0 0
$$803$$ −34408.0 −1.51212
$$804$$ 0 0
$$805$$ −4608.00 −0.201752
$$806$$ 0 0
$$807$$ 11550.0 0.503816
$$808$$ 0 0
$$809$$ −34118.0 −1.48273 −0.741363 0.671104i $$-0.765820\pi$$
−0.741363 + 0.671104i $$0.765820\pi$$
$$810$$ 0 0
$$811$$ 16428.0 0.711301 0.355650 0.934619i $$-0.384259\pi$$
0.355650 + 0.934619i $$0.384259\pi$$
$$812$$ 0 0
$$813$$ −8808.00 −0.379963
$$814$$ 0 0
$$815$$ −7544.00 −0.324239
$$816$$ 0 0
$$817$$ −560.000 −0.0239803
$$818$$ 0 0
$$819$$ −3744.00 −0.159739
$$820$$ 0 0
$$821$$ 18738.0 0.796542 0.398271 0.917268i $$-0.369610\pi$$
0.398271 + 0.917268i $$0.369610\pi$$
$$822$$ 0 0
$$823$$ −13928.0 −0.589914 −0.294957 0.955510i $$-0.595305\pi$$
−0.294957 + 0.955510i $$0.595305\pi$$
$$824$$ 0 0
$$825$$ 24684.0 1.04168
$$826$$ 0 0
$$827$$ 41804.0 1.75776 0.878880 0.477043i $$-0.158291\pi$$
0.878880 + 0.477043i $$0.158291\pi$$
$$828$$ 0 0
$$829$$ 43226.0 1.81098 0.905489 0.424369i $$-0.139504\pi$$
0.905489 + 0.424369i $$0.139504\pi$$
$$830$$ 0 0
$$831$$ 6198.00 0.258732
$$832$$ 0 0
$$833$$ −9534.00 −0.396559
$$834$$ 0 0
$$835$$ −768.000 −0.0318296
$$836$$ 0 0
$$837$$ 3672.00 0.151640
$$838$$ 0 0
$$839$$ −10240.0 −0.421364 −0.210682 0.977555i $$-0.567568\pi$$
−0.210682 + 0.977555i $$0.567568\pi$$
$$840$$ 0 0
$$841$$ −13985.0 −0.573414
$$842$$ 0 0
$$843$$ −642.000 −0.0262297
$$844$$ 0 0
$$845$$ 338.000 0.0137604
$$846$$ 0 0
$$847$$ 105376. 4.27481
$$848$$ 0 0
$$849$$ −7860.00 −0.317732
$$850$$ 0 0
$$851$$ −27792.0 −1.11950
$$852$$ 0 0
$$853$$ 25682.0 1.03087 0.515437 0.856928i $$-0.327630\pi$$
0.515437 + 0.856928i $$0.327630\pi$$
$$854$$ 0 0
$$855$$ 72.0000 0.00287994
$$856$$ 0 0
$$857$$ −21558.0 −0.859285 −0.429643 0.902999i $$-0.641360\pi$$
−0.429643 + 0.902999i $$0.641360\pi$$
$$858$$ 0 0
$$859$$ −14060.0 −0.558465 −0.279232 0.960224i $$-0.590080\pi$$
−0.279232 + 0.960224i $$0.590080\pi$$
$$860$$ 0 0
$$861$$ 24000.0 0.949963
$$862$$ 0 0
$$863$$ −42008.0 −1.65697 −0.828487 0.560008i $$-0.810798\pi$$
−0.828487 + 0.560008i $$0.810798\pi$$
$$864$$ 0 0
$$865$$ −2924.00 −0.114935
$$866$$ 0 0
$$867$$ −14151.0 −0.554317
$$868$$ 0 0
$$869$$ −43520.0 −1.69887
$$870$$ 0 0
$$871$$ −7748.00 −0.301413
$$872$$ 0 0
$$873$$ −4014.00 −0.155617
$$874$$ 0 0
$$875$$ −15744.0 −0.608279
$$876$$ 0 0
$$877$$ −23734.0 −0.913843 −0.456921 0.889507i $$-0.651048\pi$$
−0.456921 + 0.889507i $$0.651048\pi$$
$$878$$ 0 0
$$879$$ 3462.00 0.132845
$$880$$ 0 0
$$881$$ −37550.0 −1.43597 −0.717986 0.696057i $$-0.754936\pi$$
−0.717986 + 0.696057i $$0.754936\pi$$
$$882$$ 0 0
$$883$$ 12556.0 0.478531 0.239266 0.970954i $$-0.423093\pi$$
0.239266 + 0.970954i $$0.423093\pi$$
$$884$$ 0 0
$$885$$ 1992.00 0.0756614
$$886$$ 0 0
$$887$$ −37368.0 −1.41454 −0.707269 0.706945i $$-0.750073\pi$$
−0.707269 + 0.706945i $$0.750073\pi$$
$$888$$ 0 0
$$889$$ 3072.00 0.115896
$$890$$ 0 0
$$891$$ −5508.00 −0.207099
$$892$$ 0 0
$$893$$ 1184.00 0.0443685
$$894$$ 0 0
$$895$$ −2664.00 −0.0994946
$$896$$ 0 0
$$897$$ 2808.00 0.104522
$$898$$ 0 0
$$899$$ −13872.0 −0.514635
$$900$$ 0 0
$$901$$ 7364.00 0.272287
$$902$$ 0 0
$$903$$ −13440.0 −0.495299
$$904$$ 0 0
$$905$$ −4060.00 −0.149126
$$906$$ 0 0
$$907$$ 33364.0 1.22143 0.610713 0.791852i $$-0.290883\pi$$
0.610713 + 0.791852i $$0.290883\pi$$
$$908$$ 0 0
$$909$$ 5490.00 0.200321
$$910$$ 0 0
$$911$$ −50432.0 −1.83412 −0.917062 0.398745i $$-0.869446\pi$$
−0.917062 + 0.398745i $$0.869446\pi$$
$$912$$ 0 0
$$913$$ −93840.0 −3.40159
$$914$$ 0 0
$$915$$ 2460.00 0.0888799
$$916$$ 0 0
$$917$$ −81536.0 −2.93627
$$918$$ 0 0
$$919$$ −11864.0 −0.425851 −0.212926 0.977068i $$-0.568299\pi$$
−0.212926 + 0.977068i $$0.568299\pi$$
$$920$$ 0 0
$$921$$ −12228.0 −0.437488
$$922$$ 0 0
$$923$$ −11440.0 −0.407966
$$924$$ 0 0
$$925$$ −46706.0 −1.66020
$$926$$ 0 0
$$927$$ 12168.0 0.431121
$$928$$ 0 0
$$929$$ −5950.00 −0.210133 −0.105066 0.994465i $$-0.533505\pi$$
−0.105066 + 0.994465i $$0.533505\pi$$
$$930$$ 0 0
$$931$$ 2724.00 0.0958920
$$932$$ 0 0
$$933$$ 19368.0 0.679614
$$934$$ 0 0
$$935$$ 1904.00 0.0665962
$$936$$ 0 0
$$937$$ −20806.0 −0.725403 −0.362701 0.931905i $$-0.618146\pi$$
−0.362701 + 0.931905i $$0.618146\pi$$
$$938$$ 0 0
$$939$$ −16578.0 −0.576148
$$940$$ 0 0
$$941$$ 22346.0 0.774133 0.387066 0.922052i $$-0.373488\pi$$
0.387066 + 0.922052i $$0.373488\pi$$
$$942$$ 0 0
$$943$$ −18000.0 −0.621591
$$944$$ 0 0
$$945$$ 1728.00 0.0594834
$$946$$ 0 0
$$947$$ 31636.0 1.08557 0.542783 0.839873i $$-0.317370\pi$$
0.542783 + 0.839873i $$0.317370\pi$$
$$948$$ 0 0
$$949$$ −6578.00 −0.225006
$$950$$ 0 0
$$951$$ 31374.0 1.06979
$$952$$ 0 0
$$953$$ −23190.0 −0.788245 −0.394123 0.919058i $$-0.628951\pi$$
−0.394123 + 0.919058i $$0.628951\pi$$
$$954$$ 0 0
$$955$$ 32.0000 0.00108429
$$956$$ 0 0
$$957$$ 20808.0 0.702850
$$958$$ 0 0
$$959$$ −7360.00 −0.247828
$$960$$ 0 0
$$961$$ −11295.0 −0.379141
$$962$$ 0 0
$$963$$ −6588.00 −0.220452
$$964$$ 0 0
$$965$$ −4156.00 −0.138639
$$966$$ 0 0
$$967$$ −304.000 −0.0101096 −0.00505480 0.999987i $$-0.501609\pi$$
−0.00505480 + 0.999987i $$0.501609\pi$$
$$968$$ 0 0
$$969$$ −168.000 −0.00556960
$$970$$ 0 0
$$971$$ −53372.0 −1.76394 −0.881972 0.471302i $$-0.843784\pi$$
−0.881972 + 0.471302i $$0.843784\pi$$
$$972$$ 0 0
$$973$$ 16512.0 0.544039
$$974$$ 0 0
$$975$$ 4719.00 0.155004
$$976$$ 0 0
$$977$$ 13650.0 0.446983 0.223491 0.974706i $$-0.428255\pi$$
0.223491 + 0.974706i $$0.428255\pi$$
$$978$$ 0 0
$$979$$ −98600.0 −3.21887
$$980$$ 0 0
$$981$$ 13626.0 0.443471
$$982$$ 0 0
$$983$$ −34992.0 −1.13537 −0.567686 0.823245i $$-0.692161\pi$$
−0.567686 + 0.823245i $$0.692161\pi$$
$$984$$ 0 0
$$985$$ −6972.00 −0.225529
$$986$$ 0 0
$$987$$ 28416.0 0.916405
$$988$$ 0 0
$$989$$ 10080.0 0.324090
$$990$$ 0 0
$$991$$ −18096.0 −0.580059 −0.290029 0.957018i $$-0.593665\pi$$
−0.290029 + 0.957018i $$0.593665\pi$$
$$992$$ 0 0
$$993$$ 7044.00 0.225110
$$994$$ 0 0
$$995$$ −1136.00 −0.0361946
$$996$$ 0 0
$$997$$ 18914.0 0.600815 0.300407 0.953811i $$-0.402877\pi$$
0.300407 + 0.953811i $$0.402877\pi$$
$$998$$ 0 0
$$999$$ 10422.0 0.330068
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 2496.4.a.m.1.1 1
4.3 odd 2 2496.4.a.d.1.1 1
8.3 odd 2 156.4.a.b.1.1 1
8.5 even 2 624.4.a.b.1.1 1
24.5 odd 2 1872.4.a.i.1.1 1
24.11 even 2 468.4.a.a.1.1 1
104.51 odd 2 2028.4.a.b.1.1 1
104.83 even 4 2028.4.b.d.337.2 2
104.99 even 4 2028.4.b.d.337.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
156.4.a.b.1.1 1 8.3 odd 2
468.4.a.a.1.1 1 24.11 even 2
624.4.a.b.1.1 1 8.5 even 2
1872.4.a.i.1.1 1 24.5 odd 2
2028.4.a.b.1.1 1 104.51 odd 2
2028.4.b.d.337.1 2 104.99 even 4
2028.4.b.d.337.2 2 104.83 even 4
2496.4.a.d.1.1 1 4.3 odd 2
2496.4.a.m.1.1 1 1.1 even 1 trivial