# Properties

 Label 1216.4.a.a.1.1 Level $1216$ Weight $4$ Character 1216.1 Self dual yes Analytic conductor $71.746$ 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] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$71.7463225670$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1216.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.00000 q^{3} +12.0000 q^{5} -11.0000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q-5.00000 q^{3} +12.0000 q^{5} -11.0000 q^{7} -2.00000 q^{9} -54.0000 q^{11} -11.0000 q^{13} -60.0000 q^{15} -93.0000 q^{17} +19.0000 q^{19} +55.0000 q^{21} -183.000 q^{23} +19.0000 q^{25} +145.000 q^{27} +249.000 q^{29} -56.0000 q^{31} +270.000 q^{33} -132.000 q^{35} +250.000 q^{37} +55.0000 q^{39} +240.000 q^{41} -196.000 q^{43} -24.0000 q^{45} +168.000 q^{47} -222.000 q^{49} +465.000 q^{51} -435.000 q^{53} -648.000 q^{55} -95.0000 q^{57} +195.000 q^{59} +358.000 q^{61} +22.0000 q^{63} -132.000 q^{65} -961.000 q^{67} +915.000 q^{69} +246.000 q^{71} +353.000 q^{73} -95.0000 q^{75} +594.000 q^{77} +34.0000 q^{79} -671.000 q^{81} +234.000 q^{83} -1116.00 q^{85} -1245.00 q^{87} -168.000 q^{89} +121.000 q^{91} +280.000 q^{93} +228.000 q^{95} +758.000 q^{97} +108.000 q^{99} +O(q^{100})$$

## 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
$$3$$ −5.00000 −0.962250 −0.481125 0.876652i $$-0.659772\pi$$
−0.481125 + 0.876652i $$0.659772\pi$$
$$4$$ 0 0
$$5$$ 12.0000 1.07331 0.536656 0.843801i $$-0.319687\pi$$
0.536656 + 0.843801i $$0.319687\pi$$
$$6$$ 0 0
$$7$$ −11.0000 −0.593944 −0.296972 0.954886i $$-0.595977\pi$$
−0.296972 + 0.954886i $$0.595977\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.0740741
$$10$$ 0 0
$$11$$ −54.0000 −1.48015 −0.740073 0.672526i $$-0.765209\pi$$
−0.740073 + 0.672526i $$0.765209\pi$$
$$12$$ 0 0
$$13$$ −11.0000 −0.234681 −0.117340 0.993092i $$-0.537437\pi$$
−0.117340 + 0.993092i $$0.537437\pi$$
$$14$$ 0 0
$$15$$ −60.0000 −1.03280
$$16$$ 0 0
$$17$$ −93.0000 −1.32681 −0.663406 0.748259i $$-0.730890\pi$$
−0.663406 + 0.748259i $$0.730890\pi$$
$$18$$ 0 0
$$19$$ 19.0000 0.229416
$$20$$ 0 0
$$21$$ 55.0000 0.571523
$$22$$ 0 0
$$23$$ −183.000 −1.65905 −0.829525 0.558470i $$-0.811389\pi$$
−0.829525 + 0.558470i $$0.811389\pi$$
$$24$$ 0 0
$$25$$ 19.0000 0.152000
$$26$$ 0 0
$$27$$ 145.000 1.03353
$$28$$ 0 0
$$29$$ 249.000 1.59442 0.797209 0.603703i $$-0.206309\pi$$
0.797209 + 0.603703i $$0.206309\pi$$
$$30$$ 0 0
$$31$$ −56.0000 −0.324448 −0.162224 0.986754i $$-0.551867\pi$$
−0.162224 + 0.986754i $$0.551867\pi$$
$$32$$ 0 0
$$33$$ 270.000 1.42427
$$34$$ 0 0
$$35$$ −132.000 −0.637488
$$36$$ 0 0
$$37$$ 250.000 1.11080 0.555402 0.831582i $$-0.312564\pi$$
0.555402 + 0.831582i $$0.312564\pi$$
$$38$$ 0 0
$$39$$ 55.0000 0.225822
$$40$$ 0 0
$$41$$ 240.000 0.914188 0.457094 0.889418i $$-0.348890\pi$$
0.457094 + 0.889418i $$0.348890\pi$$
$$42$$ 0 0
$$43$$ −196.000 −0.695110 −0.347555 0.937660i $$-0.612988\pi$$
−0.347555 + 0.937660i $$0.612988\pi$$
$$44$$ 0 0
$$45$$ −24.0000 −0.0795046
$$46$$ 0 0
$$47$$ 168.000 0.521390 0.260695 0.965421i $$-0.416048\pi$$
0.260695 + 0.965421i $$0.416048\pi$$
$$48$$ 0 0
$$49$$ −222.000 −0.647230
$$50$$ 0 0
$$51$$ 465.000 1.27673
$$52$$ 0 0
$$53$$ −435.000 −1.12739 −0.563697 0.825982i $$-0.690621\pi$$
−0.563697 + 0.825982i $$0.690621\pi$$
$$54$$ 0 0
$$55$$ −648.000 −1.58866
$$56$$ 0 0
$$57$$ −95.0000 −0.220755
$$58$$ 0 0
$$59$$ 195.000 0.430285 0.215143 0.976583i $$-0.430978\pi$$
0.215143 + 0.976583i $$0.430978\pi$$
$$60$$ 0 0
$$61$$ 358.000 0.751430 0.375715 0.926735i $$-0.377397\pi$$
0.375715 + 0.926735i $$0.377397\pi$$
$$62$$ 0 0
$$63$$ 22.0000 0.0439959
$$64$$ 0 0
$$65$$ −132.000 −0.251886
$$66$$ 0 0
$$67$$ −961.000 −1.75231 −0.876155 0.482029i $$-0.839900\pi$$
−0.876155 + 0.482029i $$0.839900\pi$$
$$68$$ 0 0
$$69$$ 915.000 1.59642
$$70$$ 0 0
$$71$$ 246.000 0.411195 0.205597 0.978637i $$-0.434086\pi$$
0.205597 + 0.978637i $$0.434086\pi$$
$$72$$ 0 0
$$73$$ 353.000 0.565966 0.282983 0.959125i $$-0.408676\pi$$
0.282983 + 0.959125i $$0.408676\pi$$
$$74$$ 0 0
$$75$$ −95.0000 −0.146262
$$76$$ 0 0
$$77$$ 594.000 0.879124
$$78$$ 0 0
$$79$$ 34.0000 0.0484215 0.0242108 0.999707i $$-0.492293\pi$$
0.0242108 + 0.999707i $$0.492293\pi$$
$$80$$ 0 0
$$81$$ −671.000 −0.920439
$$82$$ 0 0
$$83$$ 234.000 0.309456 0.154728 0.987957i $$-0.450550\pi$$
0.154728 + 0.987957i $$0.450550\pi$$
$$84$$ 0 0
$$85$$ −1116.00 −1.42408
$$86$$ 0 0
$$87$$ −1245.00 −1.53423
$$88$$ 0 0
$$89$$ −168.000 −0.200089 −0.100045 0.994983i $$-0.531899\pi$$
−0.100045 + 0.994983i $$0.531899\pi$$
$$90$$ 0 0
$$91$$ 121.000 0.139387
$$92$$ 0 0
$$93$$ 280.000 0.312201
$$94$$ 0 0
$$95$$ 228.000 0.246235
$$96$$ 0 0
$$97$$ 758.000 0.793435 0.396718 0.917941i $$-0.370149\pi$$
0.396718 + 0.917941i $$0.370149\pi$$
$$98$$ 0 0
$$99$$ 108.000 0.109640
$$100$$ 0 0
$$101$$ 726.000 0.715245 0.357622 0.933866i $$-0.383588\pi$$
0.357622 + 0.933866i $$0.383588\pi$$
$$102$$ 0 0
$$103$$ −2.00000 −0.00191326 −0.000956630 1.00000i $$-0.500305\pi$$
−0.000956630 1.00000i $$0.500305\pi$$
$$104$$ 0 0
$$105$$ 660.000 0.613423
$$106$$ 0 0
$$107$$ 1413.00 1.27663 0.638317 0.769773i $$-0.279631\pi$$
0.638317 + 0.769773i $$0.279631\pi$$
$$108$$ 0 0
$$109$$ −389.000 −0.341830 −0.170915 0.985286i $$-0.554672\pi$$
−0.170915 + 0.985286i $$0.554672\pi$$
$$110$$ 0 0
$$111$$ −1250.00 −1.06887
$$112$$ 0 0
$$113$$ 342.000 0.284714 0.142357 0.989815i $$-0.454532\pi$$
0.142357 + 0.989815i $$0.454532\pi$$
$$114$$ 0 0
$$115$$ −2196.00 −1.78068
$$116$$ 0 0
$$117$$ 22.0000 0.0173838
$$118$$ 0 0
$$119$$ 1023.00 0.788053
$$120$$ 0 0
$$121$$ 1585.00 1.19083
$$122$$ 0 0
$$123$$ −1200.00 −0.879678
$$124$$ 0 0
$$125$$ −1272.00 −0.910169
$$126$$ 0 0
$$127$$ 1150.00 0.803512 0.401756 0.915747i $$-0.368400\pi$$
0.401756 + 0.915747i $$0.368400\pi$$
$$128$$ 0 0
$$129$$ 980.000 0.668870
$$130$$ 0 0
$$131$$ −1452.00 −0.968411 −0.484205 0.874954i $$-0.660891\pi$$
−0.484205 + 0.874954i $$0.660891\pi$$
$$132$$ 0 0
$$133$$ −209.000 −0.136260
$$134$$ 0 0
$$135$$ 1740.00 1.10930
$$136$$ 0 0
$$137$$ −1689.00 −1.05329 −0.526646 0.850085i $$-0.676551\pi$$
−0.526646 + 0.850085i $$0.676551\pi$$
$$138$$ 0 0
$$139$$ 2144.00 1.30829 0.654143 0.756371i $$-0.273030\pi$$
0.654143 + 0.756371i $$0.273030\pi$$
$$140$$ 0 0
$$141$$ −840.000 −0.501708
$$142$$ 0 0
$$143$$ 594.000 0.347362
$$144$$ 0 0
$$145$$ 2988.00 1.71131
$$146$$ 0 0
$$147$$ 1110.00 0.622798
$$148$$ 0 0
$$149$$ 3000.00 1.64946 0.824730 0.565527i $$-0.191327\pi$$
0.824730 + 0.565527i $$0.191327\pi$$
$$150$$ 0 0
$$151$$ 1006.00 0.542166 0.271083 0.962556i $$-0.412618\pi$$
0.271083 + 0.962556i $$0.412618\pi$$
$$152$$ 0 0
$$153$$ 186.000 0.0982824
$$154$$ 0 0
$$155$$ −672.000 −0.348234
$$156$$ 0 0
$$157$$ −2846.00 −1.44672 −0.723362 0.690469i $$-0.757404\pi$$
−0.723362 + 0.690469i $$0.757404\pi$$
$$158$$ 0 0
$$159$$ 2175.00 1.08483
$$160$$ 0 0
$$161$$ 2013.00 0.985383
$$162$$ 0 0
$$163$$ −1600.00 −0.768845 −0.384422 0.923157i $$-0.625599\pi$$
−0.384422 + 0.923157i $$0.625599\pi$$
$$164$$ 0 0
$$165$$ 3240.00 1.52869
$$166$$ 0 0
$$167$$ 2004.00 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −2076.00 −0.944925
$$170$$ 0 0
$$171$$ −38.0000 −0.0169938
$$172$$ 0 0
$$173$$ 462.000 0.203036 0.101518 0.994834i $$-0.467630\pi$$
0.101518 + 0.994834i $$0.467630\pi$$
$$174$$ 0 0
$$175$$ −209.000 −0.0902795
$$176$$ 0 0
$$177$$ −975.000 −0.414042
$$178$$ 0 0
$$179$$ 720.000 0.300644 0.150322 0.988637i $$-0.451969\pi$$
0.150322 + 0.988637i $$0.451969\pi$$
$$180$$ 0 0
$$181$$ 2338.00 0.960122 0.480061 0.877235i $$-0.340614\pi$$
0.480061 + 0.877235i $$0.340614\pi$$
$$182$$ 0 0
$$183$$ −1790.00 −0.723063
$$184$$ 0 0
$$185$$ 3000.00 1.19224
$$186$$ 0 0
$$187$$ 5022.00 1.96388
$$188$$ 0 0
$$189$$ −1595.00 −0.613858
$$190$$ 0 0
$$191$$ −2871.00 −1.08763 −0.543817 0.839204i $$-0.683022\pi$$
−0.543817 + 0.839204i $$0.683022\pi$$
$$192$$ 0 0
$$193$$ 1658.00 0.618370 0.309185 0.951002i $$-0.399944\pi$$
0.309185 + 0.951002i $$0.399944\pi$$
$$194$$ 0 0
$$195$$ 660.000 0.242377
$$196$$ 0 0
$$197$$ 4176.00 1.51029 0.755146 0.655556i $$-0.227566\pi$$
0.755146 + 0.655556i $$0.227566\pi$$
$$198$$ 0 0
$$199$$ 241.000 0.0858494 0.0429247 0.999078i $$-0.486332\pi$$
0.0429247 + 0.999078i $$0.486332\pi$$
$$200$$ 0 0
$$201$$ 4805.00 1.68616
$$202$$ 0 0
$$203$$ −2739.00 −0.946996
$$204$$ 0 0
$$205$$ 2880.00 0.981209
$$206$$ 0 0
$$207$$ 366.000 0.122893
$$208$$ 0 0
$$209$$ −1026.00 −0.339569
$$210$$ 0 0
$$211$$ −745.000 −0.243071 −0.121535 0.992587i $$-0.538782\pi$$
−0.121535 + 0.992587i $$0.538782\pi$$
$$212$$ 0 0
$$213$$ −1230.00 −0.395672
$$214$$ 0 0
$$215$$ −2352.00 −0.746070
$$216$$ 0 0
$$217$$ 616.000 0.192704
$$218$$ 0 0
$$219$$ −1765.00 −0.544601
$$220$$ 0 0
$$221$$ 1023.00 0.311377
$$222$$ 0 0
$$223$$ 1978.00 0.593976 0.296988 0.954881i $$-0.404018\pi$$
0.296988 + 0.954881i $$0.404018\pi$$
$$224$$ 0 0
$$225$$ −38.0000 −0.0112593
$$226$$ 0 0
$$227$$ 5355.00 1.56574 0.782872 0.622183i $$-0.213754\pi$$
0.782872 + 0.622183i $$0.213754\pi$$
$$228$$ 0 0
$$229$$ 6370.00 1.83817 0.919086 0.394057i $$-0.128929\pi$$
0.919086 + 0.394057i $$0.128929\pi$$
$$230$$ 0 0
$$231$$ −2970.00 −0.845938
$$232$$ 0 0
$$233$$ −2838.00 −0.797955 −0.398978 0.916961i $$-0.630635\pi$$
−0.398978 + 0.916961i $$0.630635\pi$$
$$234$$ 0 0
$$235$$ 2016.00 0.559614
$$236$$ 0 0
$$237$$ −170.000 −0.0465936
$$238$$ 0 0
$$239$$ 369.000 0.0998687 0.0499344 0.998753i $$-0.484099\pi$$
0.0499344 + 0.998753i $$0.484099\pi$$
$$240$$ 0 0
$$241$$ 6608.00 1.76622 0.883109 0.469167i $$-0.155446\pi$$
0.883109 + 0.469167i $$0.155446\pi$$
$$242$$ 0 0
$$243$$ −560.000 −0.147835
$$244$$ 0 0
$$245$$ −2664.00 −0.694680
$$246$$ 0 0
$$247$$ −209.000 −0.0538395
$$248$$ 0 0
$$249$$ −1170.00 −0.297774
$$250$$ 0 0
$$251$$ 4674.00 1.17538 0.587690 0.809086i $$-0.300038\pi$$
0.587690 + 0.809086i $$0.300038\pi$$
$$252$$ 0 0
$$253$$ 9882.00 2.45564
$$254$$ 0 0
$$255$$ 5580.00 1.37033
$$256$$ 0 0
$$257$$ 4512.00 1.09514 0.547570 0.836760i $$-0.315553\pi$$
0.547570 + 0.836760i $$0.315553\pi$$
$$258$$ 0 0
$$259$$ −2750.00 −0.659756
$$260$$ 0 0
$$261$$ −498.000 −0.118105
$$262$$ 0 0
$$263$$ −3768.00 −0.883440 −0.441720 0.897153i $$-0.645632\pi$$
−0.441720 + 0.897153i $$0.645632\pi$$
$$264$$ 0 0
$$265$$ −5220.00 −1.21005
$$266$$ 0 0
$$267$$ 840.000 0.192536
$$268$$ 0 0
$$269$$ −4758.00 −1.07844 −0.539220 0.842165i $$-0.681281\pi$$
−0.539220 + 0.842165i $$0.681281\pi$$
$$270$$ 0 0
$$271$$ 2041.00 0.457498 0.228749 0.973485i $$-0.426537\pi$$
0.228749 + 0.973485i $$0.426537\pi$$
$$272$$ 0 0
$$273$$ −605.000 −0.134126
$$274$$ 0 0
$$275$$ −1026.00 −0.224982
$$276$$ 0 0
$$277$$ −1964.00 −0.426012 −0.213006 0.977051i $$-0.568325\pi$$
−0.213006 + 0.977051i $$0.568325\pi$$
$$278$$ 0 0
$$279$$ 112.000 0.0240332
$$280$$ 0 0
$$281$$ −5496.00 −1.16678 −0.583388 0.812194i $$-0.698273\pi$$
−0.583388 + 0.812194i $$0.698273\pi$$
$$282$$ 0 0
$$283$$ 3098.00 0.650731 0.325366 0.945588i $$-0.394513\pi$$
0.325366 + 0.945588i $$0.394513\pi$$
$$284$$ 0 0
$$285$$ −1140.00 −0.236940
$$286$$ 0 0
$$287$$ −2640.00 −0.542977
$$288$$ 0 0
$$289$$ 3736.00 0.760432
$$290$$ 0 0
$$291$$ −3790.00 −0.763484
$$292$$ 0 0
$$293$$ −117.000 −0.0233284 −0.0116642 0.999932i $$-0.503713\pi$$
−0.0116642 + 0.999932i $$0.503713\pi$$
$$294$$ 0 0
$$295$$ 2340.00 0.461831
$$296$$ 0 0
$$297$$ −7830.00 −1.52977
$$298$$ 0 0
$$299$$ 2013.00 0.389347
$$300$$ 0 0
$$301$$ 2156.00 0.412856
$$302$$ 0 0
$$303$$ −3630.00 −0.688244
$$304$$ 0 0
$$305$$ 4296.00 0.806519
$$306$$ 0 0
$$307$$ −1420.00 −0.263986 −0.131993 0.991251i $$-0.542138\pi$$
−0.131993 + 0.991251i $$0.542138\pi$$
$$308$$ 0 0
$$309$$ 10.0000 0.00184104
$$310$$ 0 0
$$311$$ 6561.00 1.19627 0.598135 0.801395i $$-0.295909\pi$$
0.598135 + 0.801395i $$0.295909\pi$$
$$312$$ 0 0
$$313$$ −1483.00 −0.267809 −0.133904 0.990994i $$-0.542751\pi$$
−0.133904 + 0.990994i $$0.542751\pi$$
$$314$$ 0 0
$$315$$ 264.000 0.0472213
$$316$$ 0 0
$$317$$ 1239.00 0.219524 0.109762 0.993958i $$-0.464991\pi$$
0.109762 + 0.993958i $$0.464991\pi$$
$$318$$ 0 0
$$319$$ −13446.0 −2.35997
$$320$$ 0 0
$$321$$ −7065.00 −1.22844
$$322$$ 0 0
$$323$$ −1767.00 −0.304392
$$324$$ 0 0
$$325$$ −209.000 −0.0356715
$$326$$ 0 0
$$327$$ 1945.00 0.328926
$$328$$ 0 0
$$329$$ −1848.00 −0.309676
$$330$$ 0 0
$$331$$ −8899.00 −1.47774 −0.738872 0.673846i $$-0.764641\pi$$
−0.738872 + 0.673846i $$0.764641\pi$$
$$332$$ 0 0
$$333$$ −500.000 −0.0822818
$$334$$ 0 0
$$335$$ −11532.0 −1.88078
$$336$$ 0 0
$$337$$ 5816.00 0.940112 0.470056 0.882637i $$-0.344234\pi$$
0.470056 + 0.882637i $$0.344234\pi$$
$$338$$ 0 0
$$339$$ −1710.00 −0.273966
$$340$$ 0 0
$$341$$ 3024.00 0.480231
$$342$$ 0 0
$$343$$ 6215.00 0.978363
$$344$$ 0 0
$$345$$ 10980.0 1.71346
$$346$$ 0 0
$$347$$ −1578.00 −0.244125 −0.122063 0.992522i $$-0.538951\pi$$
−0.122063 + 0.992522i $$0.538951\pi$$
$$348$$ 0 0
$$349$$ −1658.00 −0.254300 −0.127150 0.991883i $$-0.540583\pi$$
−0.127150 + 0.991883i $$0.540583\pi$$
$$350$$ 0 0
$$351$$ −1595.00 −0.242549
$$352$$ 0 0
$$353$$ −11367.0 −1.71389 −0.856947 0.515405i $$-0.827641\pi$$
−0.856947 + 0.515405i $$0.827641\pi$$
$$354$$ 0 0
$$355$$ 2952.00 0.441341
$$356$$ 0 0
$$357$$ −5115.00 −0.758304
$$358$$ 0 0
$$359$$ −2553.00 −0.375326 −0.187663 0.982233i $$-0.560091\pi$$
−0.187663 + 0.982233i $$0.560091\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ 0 0
$$363$$ −7925.00 −1.14588
$$364$$ 0 0
$$365$$ 4236.00 0.607459
$$366$$ 0 0
$$367$$ 196.000 0.0278777 0.0139389 0.999903i $$-0.495563\pi$$
0.0139389 + 0.999903i $$0.495563\pi$$
$$368$$ 0 0
$$369$$ −480.000 −0.0677176
$$370$$ 0 0
$$371$$ 4785.00 0.669609
$$372$$ 0 0
$$373$$ −9353.00 −1.29834 −0.649169 0.760644i $$-0.724883\pi$$
−0.649169 + 0.760644i $$0.724883\pi$$
$$374$$ 0 0
$$375$$ 6360.00 0.875811
$$376$$ 0 0
$$377$$ −2739.00 −0.374180
$$378$$ 0 0
$$379$$ 3827.00 0.518680 0.259340 0.965786i $$-0.416495\pi$$
0.259340 + 0.965786i $$0.416495\pi$$
$$380$$ 0 0
$$381$$ −5750.00 −0.773180
$$382$$ 0 0
$$383$$ −5694.00 −0.759660 −0.379830 0.925056i $$-0.624018\pi$$
−0.379830 + 0.925056i $$0.624018\pi$$
$$384$$ 0 0
$$385$$ 7128.00 0.943575
$$386$$ 0 0
$$387$$ 392.000 0.0514896
$$388$$ 0 0
$$389$$ −1290.00 −0.168138 −0.0840689 0.996460i $$-0.526792\pi$$
−0.0840689 + 0.996460i $$0.526792\pi$$
$$390$$ 0 0
$$391$$ 17019.0 2.20125
$$392$$ 0 0
$$393$$ 7260.00 0.931854
$$394$$ 0 0
$$395$$ 408.000 0.0519714
$$396$$ 0 0
$$397$$ −6536.00 −0.826278 −0.413139 0.910668i $$-0.635568\pi$$
−0.413139 + 0.910668i $$0.635568\pi$$
$$398$$ 0 0
$$399$$ 1045.00 0.131116
$$400$$ 0 0
$$401$$ 2328.00 0.289912 0.144956 0.989438i $$-0.453696\pi$$
0.144956 + 0.989438i $$0.453696\pi$$
$$402$$ 0 0
$$403$$ 616.000 0.0761418
$$404$$ 0 0
$$405$$ −8052.00 −0.987919
$$406$$ 0 0
$$407$$ −13500.0 −1.64415
$$408$$ 0 0
$$409$$ −6676.00 −0.807107 −0.403554 0.914956i $$-0.632225\pi$$
−0.403554 + 0.914956i $$0.632225\pi$$
$$410$$ 0 0
$$411$$ 8445.00 1.01353
$$412$$ 0 0
$$413$$ −2145.00 −0.255565
$$414$$ 0 0
$$415$$ 2808.00 0.332143
$$416$$ 0 0
$$417$$ −10720.0 −1.25890
$$418$$ 0 0
$$419$$ −8136.00 −0.948615 −0.474307 0.880359i $$-0.657301\pi$$
−0.474307 + 0.880359i $$0.657301\pi$$
$$420$$ 0 0
$$421$$ 8665.00 1.00310 0.501551 0.865128i $$-0.332763\pi$$
0.501551 + 0.865128i $$0.332763\pi$$
$$422$$ 0 0
$$423$$ −336.000 −0.0386215
$$424$$ 0 0
$$425$$ −1767.00 −0.201676
$$426$$ 0 0
$$427$$ −3938.00 −0.446307
$$428$$ 0 0
$$429$$ −2970.00 −0.334249
$$430$$ 0 0
$$431$$ −750.000 −0.0838196 −0.0419098 0.999121i $$-0.513344\pi$$
−0.0419098 + 0.999121i $$0.513344\pi$$
$$432$$ 0 0
$$433$$ −4858.00 −0.539170 −0.269585 0.962977i $$-0.586887\pi$$
−0.269585 + 0.962977i $$0.586887\pi$$
$$434$$ 0 0
$$435$$ −14940.0 −1.64671
$$436$$ 0 0
$$437$$ −3477.00 −0.380612
$$438$$ 0 0
$$439$$ −6500.00 −0.706670 −0.353335 0.935497i $$-0.614952\pi$$
−0.353335 + 0.935497i $$0.614952\pi$$
$$440$$ 0 0
$$441$$ 444.000 0.0479430
$$442$$ 0 0
$$443$$ 3486.00 0.373871 0.186936 0.982372i $$-0.440144\pi$$
0.186936 + 0.982372i $$0.440144\pi$$
$$444$$ 0 0
$$445$$ −2016.00 −0.214759
$$446$$ 0 0
$$447$$ −15000.0 −1.58719
$$448$$ 0 0
$$449$$ −15030.0 −1.57975 −0.789877 0.613265i $$-0.789856\pi$$
−0.789877 + 0.613265i $$0.789856\pi$$
$$450$$ 0 0
$$451$$ −12960.0 −1.35313
$$452$$ 0 0
$$453$$ −5030.00 −0.521700
$$454$$ 0 0
$$455$$ 1452.00 0.149606
$$456$$ 0 0
$$457$$ −2959.00 −0.302880 −0.151440 0.988466i $$-0.548391\pi$$
−0.151440 + 0.988466i $$0.548391\pi$$
$$458$$ 0 0
$$459$$ −13485.0 −1.37130
$$460$$ 0 0
$$461$$ 156.000 0.0157606 0.00788031 0.999969i $$-0.497492\pi$$
0.00788031 + 0.999969i $$0.497492\pi$$
$$462$$ 0 0
$$463$$ −4484.00 −0.450085 −0.225042 0.974349i $$-0.572252\pi$$
−0.225042 + 0.974349i $$0.572252\pi$$
$$464$$ 0 0
$$465$$ 3360.00 0.335089
$$466$$ 0 0
$$467$$ 8766.00 0.868613 0.434306 0.900765i $$-0.356994\pi$$
0.434306 + 0.900765i $$0.356994\pi$$
$$468$$ 0 0
$$469$$ 10571.0 1.04077
$$470$$ 0 0
$$471$$ 14230.0 1.39211
$$472$$ 0 0
$$473$$ 10584.0 1.02886
$$474$$ 0 0
$$475$$ 361.000 0.0348712
$$476$$ 0 0
$$477$$ 870.000 0.0835106
$$478$$ 0 0
$$479$$ 18996.0 1.81200 0.906001 0.423275i $$-0.139119\pi$$
0.906001 + 0.423275i $$0.139119\pi$$
$$480$$ 0 0
$$481$$ −2750.00 −0.260684
$$482$$ 0 0
$$483$$ −10065.0 −0.948185
$$484$$ 0 0
$$485$$ 9096.00 0.851604
$$486$$ 0 0
$$487$$ 7450.00 0.693207 0.346603 0.938012i $$-0.387335\pi$$
0.346603 + 0.938012i $$0.387335\pi$$
$$488$$ 0 0
$$489$$ 8000.00 0.739821
$$490$$ 0 0
$$491$$ 6180.00 0.568023 0.284012 0.958821i $$-0.408335\pi$$
0.284012 + 0.958821i $$0.408335\pi$$
$$492$$ 0 0
$$493$$ −23157.0 −2.11549
$$494$$ 0 0
$$495$$ 1296.00 0.117679
$$496$$ 0 0
$$497$$ −2706.00 −0.244227
$$498$$ 0 0
$$499$$ 2576.00 0.231097 0.115549 0.993302i $$-0.463137\pi$$
0.115549 + 0.993302i $$0.463137\pi$$
$$500$$ 0 0
$$501$$ −10020.0 −0.893534
$$502$$ 0 0
$$503$$ 10545.0 0.934748 0.467374 0.884060i $$-0.345200\pi$$
0.467374 + 0.884060i $$0.345200\pi$$
$$504$$ 0 0
$$505$$ 8712.00 0.767681
$$506$$ 0 0
$$507$$ 10380.0 0.909254
$$508$$ 0 0
$$509$$ 14694.0 1.27957 0.639784 0.768555i $$-0.279024\pi$$
0.639784 + 0.768555i $$0.279024\pi$$
$$510$$ 0 0
$$511$$ −3883.00 −0.336152
$$512$$ 0 0
$$513$$ 2755.00 0.237108
$$514$$ 0 0
$$515$$ −24.0000 −0.00205353
$$516$$ 0 0
$$517$$ −9072.00 −0.771733
$$518$$ 0 0
$$519$$ −2310.00 −0.195371
$$520$$ 0 0
$$521$$ 10332.0 0.868816 0.434408 0.900716i $$-0.356958\pi$$
0.434408 + 0.900716i $$0.356958\pi$$
$$522$$ 0 0
$$523$$ 10937.0 0.914420 0.457210 0.889359i $$-0.348849\pi$$
0.457210 + 0.889359i $$0.348849\pi$$
$$524$$ 0 0
$$525$$ 1045.00 0.0868715
$$526$$ 0 0
$$527$$ 5208.00 0.430482
$$528$$ 0 0
$$529$$ 21322.0 1.75245
$$530$$ 0 0
$$531$$ −390.000 −0.0318730
$$532$$ 0 0
$$533$$ −2640.00 −0.214542
$$534$$ 0 0
$$535$$ 16956.0 1.37023
$$536$$ 0 0
$$537$$ −3600.00 −0.289295
$$538$$ 0 0
$$539$$ 11988.0 0.957996
$$540$$ 0 0
$$541$$ −18578.0 −1.47640 −0.738198 0.674584i $$-0.764323\pi$$
−0.738198 + 0.674584i $$0.764323\pi$$
$$542$$ 0 0
$$543$$ −11690.0 −0.923878
$$544$$ 0 0
$$545$$ −4668.00 −0.366890
$$546$$ 0 0
$$547$$ 21404.0 1.67307 0.836535 0.547914i $$-0.184578\pi$$
0.836535 + 0.547914i $$0.184578\pi$$
$$548$$ 0 0
$$549$$ −716.000 −0.0556614
$$550$$ 0 0
$$551$$ 4731.00 0.365785
$$552$$ 0 0
$$553$$ −374.000 −0.0287597
$$554$$ 0 0
$$555$$ −15000.0 −1.14723
$$556$$ 0 0
$$557$$ 3948.00 0.300327 0.150163 0.988661i $$-0.452020\pi$$
0.150163 + 0.988661i $$0.452020\pi$$
$$558$$ 0 0
$$559$$ 2156.00 0.163129
$$560$$ 0 0
$$561$$ −25110.0 −1.88974
$$562$$ 0 0
$$563$$ 5724.00 0.428486 0.214243 0.976780i $$-0.431271\pi$$
0.214243 + 0.976780i $$0.431271\pi$$
$$564$$ 0 0
$$565$$ 4104.00 0.305587
$$566$$ 0 0
$$567$$ 7381.00 0.546689
$$568$$ 0 0
$$569$$ −20592.0 −1.51716 −0.758578 0.651582i $$-0.774105\pi$$
−0.758578 + 0.651582i $$0.774105\pi$$
$$570$$ 0 0
$$571$$ 20684.0 1.51593 0.757967 0.652293i $$-0.226193\pi$$
0.757967 + 0.652293i $$0.226193\pi$$
$$572$$ 0 0
$$573$$ 14355.0 1.04658
$$574$$ 0 0
$$575$$ −3477.00 −0.252176
$$576$$ 0 0
$$577$$ −19573.0 −1.41219 −0.706096 0.708116i $$-0.749545\pi$$
−0.706096 + 0.708116i $$0.749545\pi$$
$$578$$ 0 0
$$579$$ −8290.00 −0.595027
$$580$$ 0 0
$$581$$ −2574.00 −0.183800
$$582$$ 0 0
$$583$$ 23490.0 1.66871
$$584$$ 0 0
$$585$$ 264.000 0.0186582
$$586$$ 0 0
$$587$$ 13524.0 0.950929 0.475464 0.879735i $$-0.342280\pi$$
0.475464 + 0.879735i $$0.342280\pi$$
$$588$$ 0 0
$$589$$ −1064.00 −0.0744335
$$590$$ 0 0
$$591$$ −20880.0 −1.45328
$$592$$ 0 0
$$593$$ 8994.00 0.622832 0.311416 0.950274i $$-0.399197\pi$$
0.311416 + 0.950274i $$0.399197\pi$$
$$594$$ 0 0
$$595$$ 12276.0 0.845827
$$596$$ 0 0
$$597$$ −1205.00 −0.0826087
$$598$$ 0 0
$$599$$ −10128.0 −0.690850 −0.345425 0.938446i $$-0.612265\pi$$
−0.345425 + 0.938446i $$0.612265\pi$$
$$600$$ 0 0
$$601$$ −22696.0 −1.54041 −0.770207 0.637794i $$-0.779847\pi$$
−0.770207 + 0.637794i $$0.779847\pi$$
$$602$$ 0 0
$$603$$ 1922.00 0.129801
$$604$$ 0 0
$$605$$ 19020.0 1.27814
$$606$$ 0 0
$$607$$ 5182.00 0.346509 0.173254 0.984877i $$-0.444572\pi$$
0.173254 + 0.984877i $$0.444572\pi$$
$$608$$ 0 0
$$609$$ 13695.0 0.911247
$$610$$ 0 0
$$611$$ −1848.00 −0.122360
$$612$$ 0 0
$$613$$ −10082.0 −0.664287 −0.332144 0.943229i $$-0.607772\pi$$
−0.332144 + 0.943229i $$0.607772\pi$$
$$614$$ 0 0
$$615$$ −14400.0 −0.944169
$$616$$ 0 0
$$617$$ −12174.0 −0.794338 −0.397169 0.917745i $$-0.630007\pi$$
−0.397169 + 0.917745i $$0.630007\pi$$
$$618$$ 0 0
$$619$$ 7490.00 0.486347 0.243173 0.969983i $$-0.421812\pi$$
0.243173 + 0.969983i $$0.421812\pi$$
$$620$$ 0 0
$$621$$ −26535.0 −1.71467
$$622$$ 0 0
$$623$$ 1848.00 0.118842
$$624$$ 0 0
$$625$$ −17639.0 −1.12890
$$626$$ 0 0
$$627$$ 5130.00 0.326750
$$628$$ 0 0
$$629$$ −23250.0 −1.47383
$$630$$ 0 0
$$631$$ −11072.0 −0.698525 −0.349263 0.937025i $$-0.613568\pi$$
−0.349263 + 0.937025i $$0.613568\pi$$
$$632$$ 0 0
$$633$$ 3725.00 0.233895
$$634$$ 0 0
$$635$$ 13800.0 0.862419
$$636$$ 0 0
$$637$$ 2442.00 0.151893
$$638$$ 0 0
$$639$$ −492.000 −0.0304589
$$640$$ 0 0
$$641$$ −18894.0 −1.16422 −0.582112 0.813108i $$-0.697774\pi$$
−0.582112 + 0.813108i $$0.697774\pi$$
$$642$$ 0 0
$$643$$ −19834.0 −1.21645 −0.608224 0.793765i $$-0.708118\pi$$
−0.608224 + 0.793765i $$0.708118\pi$$
$$644$$ 0 0
$$645$$ 11760.0 0.717906
$$646$$ 0 0
$$647$$ −3375.00 −0.205077 −0.102539 0.994729i $$-0.532697\pi$$
−0.102539 + 0.994729i $$0.532697\pi$$
$$648$$ 0 0
$$649$$ −10530.0 −0.636885
$$650$$ 0 0
$$651$$ −3080.00 −0.185430
$$652$$ 0 0
$$653$$ 24948.0 1.49509 0.747543 0.664214i $$-0.231234\pi$$
0.747543 + 0.664214i $$0.231234\pi$$
$$654$$ 0 0
$$655$$ −17424.0 −1.03941
$$656$$ 0 0
$$657$$ −706.000 −0.0419234
$$658$$ 0 0
$$659$$ −9879.00 −0.583962 −0.291981 0.956424i $$-0.594314\pi$$
−0.291981 + 0.956424i $$0.594314\pi$$
$$660$$ 0 0
$$661$$ 14155.0 0.832928 0.416464 0.909152i $$-0.363269\pi$$
0.416464 + 0.909152i $$0.363269\pi$$
$$662$$ 0 0
$$663$$ −5115.00 −0.299623
$$664$$ 0 0
$$665$$ −2508.00 −0.146250
$$666$$ 0 0
$$667$$ −45567.0 −2.64522
$$668$$ 0 0
$$669$$ −9890.00 −0.571554
$$670$$ 0 0
$$671$$ −19332.0 −1.11223
$$672$$ 0 0
$$673$$ 8948.00 0.512511 0.256256 0.966609i $$-0.417511\pi$$
0.256256 + 0.966609i $$0.417511\pi$$
$$674$$ 0 0
$$675$$ 2755.00 0.157096
$$676$$ 0 0
$$677$$ 11511.0 0.653477 0.326738 0.945115i $$-0.394050\pi$$
0.326738 + 0.945115i $$0.394050\pi$$
$$678$$ 0 0
$$679$$ −8338.00 −0.471256
$$680$$ 0 0
$$681$$ −26775.0 −1.50664
$$682$$ 0 0
$$683$$ −10476.0 −0.586900 −0.293450 0.955974i $$-0.594803\pi$$
−0.293450 + 0.955974i $$0.594803\pi$$
$$684$$ 0 0
$$685$$ −20268.0 −1.13051
$$686$$ 0 0
$$687$$ −31850.0 −1.76878
$$688$$ 0 0
$$689$$ 4785.00 0.264578
$$690$$ 0 0
$$691$$ 30098.0 1.65699 0.828496 0.559995i $$-0.189197\pi$$
0.828496 + 0.559995i $$0.189197\pi$$
$$692$$ 0 0
$$693$$ −1188.00 −0.0651203
$$694$$ 0 0
$$695$$ 25728.0 1.40420
$$696$$ 0 0
$$697$$ −22320.0 −1.21296
$$698$$ 0 0
$$699$$ 14190.0 0.767833
$$700$$ 0 0
$$701$$ 14700.0 0.792028 0.396014 0.918245i $$-0.370393\pi$$
0.396014 + 0.918245i $$0.370393\pi$$
$$702$$ 0 0
$$703$$ 4750.00 0.254836
$$704$$ 0 0
$$705$$ −10080.0 −0.538489
$$706$$ 0 0
$$707$$ −7986.00 −0.424815
$$708$$ 0 0
$$709$$ −31178.0 −1.65150 −0.825751 0.564035i $$-0.809248\pi$$
−0.825751 + 0.564035i $$0.809248\pi$$
$$710$$ 0 0
$$711$$ −68.0000 −0.00358678
$$712$$ 0 0
$$713$$ 10248.0 0.538276
$$714$$ 0 0
$$715$$ 7128.00 0.372828
$$716$$ 0 0
$$717$$ −1845.00 −0.0960987
$$718$$ 0 0
$$719$$ 33285.0 1.72645 0.863227 0.504815i $$-0.168439\pi$$
0.863227 + 0.504815i $$0.168439\pi$$
$$720$$ 0 0
$$721$$ 22.0000 0.00113637
$$722$$ 0 0
$$723$$ −33040.0 −1.69954
$$724$$ 0 0
$$725$$ 4731.00 0.242352
$$726$$ 0 0
$$727$$ 34729.0 1.77170 0.885851 0.463970i $$-0.153575\pi$$
0.885851 + 0.463970i $$0.153575\pi$$
$$728$$ 0 0
$$729$$ 20917.0 1.06269
$$730$$ 0 0
$$731$$ 18228.0 0.922280
$$732$$ 0 0
$$733$$ −4196.00 −0.211436 −0.105718 0.994396i $$-0.533714\pi$$
−0.105718 + 0.994396i $$0.533714\pi$$
$$734$$ 0 0
$$735$$ 13320.0 0.668457
$$736$$ 0 0
$$737$$ 51894.0 2.59368
$$738$$ 0 0
$$739$$ −10744.0 −0.534810 −0.267405 0.963584i $$-0.586166\pi$$
−0.267405 + 0.963584i $$0.586166\pi$$
$$740$$ 0 0
$$741$$ 1045.00 0.0518071
$$742$$ 0 0
$$743$$ 2208.00 0.109022 0.0545112 0.998513i $$-0.482640\pi$$
0.0545112 + 0.998513i $$0.482640\pi$$
$$744$$ 0 0
$$745$$ 36000.0 1.77039
$$746$$ 0 0
$$747$$ −468.000 −0.0229227
$$748$$ 0 0
$$749$$ −15543.0 −0.758249
$$750$$ 0 0
$$751$$ −13160.0 −0.639434 −0.319717 0.947513i $$-0.603588\pi$$
−0.319717 + 0.947513i $$0.603588\pi$$
$$752$$ 0 0
$$753$$ −23370.0 −1.13101
$$754$$ 0 0
$$755$$ 12072.0 0.581914
$$756$$ 0 0
$$757$$ −758.000 −0.0363936 −0.0181968 0.999834i $$-0.505793\pi$$
−0.0181968 + 0.999834i $$0.505793\pi$$
$$758$$ 0 0
$$759$$ −49410.0 −2.36294
$$760$$ 0 0
$$761$$ 4851.00 0.231076 0.115538 0.993303i $$-0.463141\pi$$
0.115538 + 0.993303i $$0.463141\pi$$
$$762$$ 0 0
$$763$$ 4279.00 0.203028
$$764$$ 0 0
$$765$$ 2232.00 0.105488
$$766$$ 0 0
$$767$$ −2145.00 −0.100980
$$768$$ 0 0
$$769$$ −33091.0 −1.55175 −0.775873 0.630890i $$-0.782690\pi$$
−0.775873 + 0.630890i $$0.782690\pi$$
$$770$$ 0 0
$$771$$ −22560.0 −1.05380
$$772$$ 0 0
$$773$$ −42357.0 −1.97086 −0.985430 0.170079i $$-0.945598\pi$$
−0.985430 + 0.170079i $$0.945598\pi$$
$$774$$ 0 0
$$775$$ −1064.00 −0.0493161
$$776$$ 0 0
$$777$$ 13750.0 0.634850
$$778$$ 0 0
$$779$$ 4560.00 0.209729
$$780$$ 0 0
$$781$$ −13284.0 −0.608629
$$782$$ 0 0
$$783$$ 36105.0 1.64788
$$784$$ 0 0
$$785$$ −34152.0 −1.55279
$$786$$ 0 0
$$787$$ −39877.0 −1.80618 −0.903089 0.429454i $$-0.858706\pi$$
−0.903089 + 0.429454i $$0.858706\pi$$
$$788$$ 0 0
$$789$$ 18840.0 0.850091
$$790$$ 0 0
$$791$$ −3762.00 −0.169104
$$792$$ 0 0
$$793$$ −3938.00 −0.176346
$$794$$ 0 0
$$795$$ 26100.0 1.16437
$$796$$ 0 0
$$797$$ 30033.0 1.33478 0.667392 0.744706i $$-0.267410\pi$$
0.667392 + 0.744706i $$0.267410\pi$$
$$798$$ 0 0
$$799$$ −15624.0 −0.691786
$$800$$ 0 0
$$801$$ 336.000 0.0148214
$$802$$ 0 0
$$803$$ −19062.0 −0.837713
$$804$$ 0 0
$$805$$ 24156.0 1.05762
$$806$$ 0 0
$$807$$ 23790.0 1.03773
$$808$$ 0 0
$$809$$ 585.000 0.0254234 0.0127117 0.999919i $$-0.495954\pi$$
0.0127117 + 0.999919i $$0.495954\pi$$
$$810$$ 0 0
$$811$$ 28361.0 1.22798 0.613989 0.789315i $$-0.289564\pi$$
0.613989 + 0.789315i $$0.289564\pi$$
$$812$$ 0 0
$$813$$ −10205.0 −0.440228
$$814$$ 0 0
$$815$$ −19200.0 −0.825211
$$816$$ 0 0
$$817$$ −3724.00 −0.159469
$$818$$ 0 0
$$819$$ −242.000 −0.0103250
$$820$$ 0 0
$$821$$ −25068.0 −1.06563 −0.532813 0.846233i $$-0.678865\pi$$
−0.532813 + 0.846233i $$0.678865\pi$$
$$822$$ 0 0
$$823$$ −10901.0 −0.461707 −0.230854 0.972989i $$-0.574152\pi$$
−0.230854 + 0.972989i $$0.574152\pi$$
$$824$$ 0 0
$$825$$ 5130.00 0.216489
$$826$$ 0 0
$$827$$ 12027.0 0.505707 0.252854 0.967505i $$-0.418631\pi$$
0.252854 + 0.967505i $$0.418631\pi$$
$$828$$ 0 0
$$829$$ 19339.0 0.810219 0.405109 0.914268i $$-0.367233\pi$$
0.405109 + 0.914268i $$0.367233\pi$$
$$830$$ 0 0
$$831$$ 9820.00 0.409930
$$832$$ 0 0
$$833$$ 20646.0 0.858753
$$834$$ 0 0
$$835$$ 24048.0 0.996665
$$836$$ 0 0
$$837$$ −8120.00 −0.335326
$$838$$ 0 0
$$839$$ 13188.0 0.542670 0.271335 0.962485i $$-0.412535\pi$$
0.271335 + 0.962485i $$0.412535\pi$$
$$840$$ 0 0
$$841$$ 37612.0 1.54217
$$842$$ 0 0
$$843$$ 27480.0 1.12273
$$844$$ 0 0
$$845$$ −24912.0 −1.01420
$$846$$ 0 0
$$847$$ −17435.0 −0.707289
$$848$$ 0 0
$$849$$ −15490.0 −0.626167
$$850$$ 0 0
$$851$$ −45750.0 −1.84288
$$852$$ 0 0
$$853$$ 4678.00 0.187775 0.0938873 0.995583i $$-0.470071\pi$$
0.0938873 + 0.995583i $$0.470071\pi$$
$$854$$ 0 0
$$855$$ −456.000 −0.0182396
$$856$$ 0 0
$$857$$ 15252.0 0.607933 0.303966 0.952683i $$-0.401689\pi$$
0.303966 + 0.952683i $$0.401689\pi$$
$$858$$ 0 0
$$859$$ −610.000 −0.0242293 −0.0121146 0.999927i $$-0.503856\pi$$
−0.0121146 + 0.999927i $$0.503856\pi$$
$$860$$ 0 0
$$861$$ 13200.0 0.522479
$$862$$ 0 0
$$863$$ −774.000 −0.0305299 −0.0152649 0.999883i $$-0.504859\pi$$
−0.0152649 + 0.999883i $$0.504859\pi$$
$$864$$ 0 0
$$865$$ 5544.00 0.217921
$$866$$ 0 0
$$867$$ −18680.0 −0.731726
$$868$$ 0 0
$$869$$ −1836.00 −0.0716709
$$870$$ 0 0
$$871$$ 10571.0 0.411234
$$872$$ 0 0
$$873$$ −1516.00 −0.0587730
$$874$$ 0 0
$$875$$ 13992.0 0.540590
$$876$$ 0 0
$$877$$ 31039.0 1.19511 0.597556 0.801827i $$-0.296139\pi$$
0.597556 + 0.801827i $$0.296139\pi$$
$$878$$ 0 0
$$879$$ 585.000 0.0224477
$$880$$ 0 0
$$881$$ 33678.0 1.28790 0.643950 0.765067i $$-0.277294\pi$$
0.643950 + 0.765067i $$0.277294\pi$$
$$882$$ 0 0
$$883$$ −42982.0 −1.63812 −0.819060 0.573708i $$-0.805504\pi$$
−0.819060 + 0.573708i $$0.805504\pi$$
$$884$$ 0 0
$$885$$ −11700.0 −0.444397
$$886$$ 0 0
$$887$$ −4494.00 −0.170117 −0.0850585 0.996376i $$-0.527108\pi$$
−0.0850585 + 0.996376i $$0.527108\pi$$
$$888$$ 0 0
$$889$$ −12650.0 −0.477241
$$890$$ 0 0
$$891$$ 36234.0 1.36238
$$892$$ 0 0
$$893$$ 3192.00 0.119615
$$894$$ 0 0
$$895$$ 8640.00 0.322685
$$896$$ 0 0
$$897$$ −10065.0 −0.374649
$$898$$ 0 0
$$899$$ −13944.0 −0.517306
$$900$$ 0 0
$$901$$ 40455.0 1.49584
$$902$$ 0 0
$$903$$ −10780.0 −0.397271
$$904$$ 0 0
$$905$$ 28056.0 1.03051
$$906$$ 0 0
$$907$$ −23839.0 −0.872724 −0.436362 0.899771i $$-0.643733\pi$$
−0.436362 + 0.899771i $$0.643733\pi$$
$$908$$ 0 0
$$909$$ −1452.00 −0.0529811
$$910$$ 0 0
$$911$$ 10332.0 0.375757 0.187878 0.982192i $$-0.439839\pi$$
0.187878 + 0.982192i $$0.439839\pi$$
$$912$$ 0 0
$$913$$ −12636.0 −0.458040
$$914$$ 0 0
$$915$$ −21480.0 −0.776073
$$916$$ 0 0
$$917$$ 15972.0 0.575182
$$918$$ 0 0
$$919$$ 14371.0 0.515838 0.257919 0.966166i $$-0.416963\pi$$
0.257919 + 0.966166i $$0.416963\pi$$
$$920$$ 0 0
$$921$$ 7100.00 0.254021
$$922$$ 0 0
$$923$$ −2706.00 −0.0964995
$$924$$ 0 0
$$925$$ 4750.00 0.168842
$$926$$ 0 0
$$927$$ 4.00000 0.000141723 0
$$928$$ 0 0
$$929$$ 26889.0 0.949623 0.474811 0.880088i $$-0.342516\pi$$
0.474811 + 0.880088i $$0.342516\pi$$
$$930$$ 0 0
$$931$$ −4218.00 −0.148485
$$932$$ 0 0
$$933$$ −32805.0 −1.15111
$$934$$ 0 0
$$935$$ 60264.0 2.10785
$$936$$ 0 0
$$937$$ 785.000 0.0273691 0.0136845 0.999906i $$-0.495644\pi$$
0.0136845 + 0.999906i $$0.495644\pi$$
$$938$$ 0 0
$$939$$ 7415.00 0.257699
$$940$$ 0 0
$$941$$ 18141.0 0.628459 0.314229 0.949347i $$-0.398254\pi$$
0.314229 + 0.949347i $$0.398254\pi$$
$$942$$ 0 0
$$943$$ −43920.0 −1.51668
$$944$$ 0 0
$$945$$ −19140.0 −0.658862
$$946$$ 0 0
$$947$$ 23100.0 0.792660 0.396330 0.918108i $$-0.370284\pi$$
0.396330 + 0.918108i $$0.370284\pi$$
$$948$$ 0 0
$$949$$ −3883.00 −0.132821
$$950$$ 0 0
$$951$$ −6195.00 −0.211237
$$952$$ 0 0
$$953$$ 45690.0 1.55304 0.776519 0.630094i $$-0.216984\pi$$
0.776519 + 0.630094i $$0.216984\pi$$
$$954$$ 0 0
$$955$$ −34452.0 −1.16737
$$956$$ 0 0
$$957$$ 67230.0 2.27089
$$958$$ 0 0
$$959$$ 18579.0 0.625597
$$960$$ 0 0
$$961$$ −26655.0 −0.894733
$$962$$ 0 0
$$963$$ −2826.00 −0.0945655
$$964$$ 0 0
$$965$$ 19896.0 0.663705
$$966$$ 0 0
$$967$$ −21584.0 −0.717781 −0.358891 0.933380i $$-0.616845\pi$$
−0.358891 + 0.933380i $$0.616845\pi$$
$$968$$ 0 0
$$969$$ 8835.00 0.292901
$$970$$ 0 0
$$971$$ −50556.0 −1.67087 −0.835437 0.549586i $$-0.814786\pi$$
−0.835437 + 0.549586i $$0.814786\pi$$
$$972$$ 0 0
$$973$$ −23584.0 −0.777049
$$974$$ 0 0
$$975$$ 1045.00 0.0343249
$$976$$ 0 0
$$977$$ 8568.00 0.280568 0.140284 0.990111i $$-0.455198\pi$$
0.140284 + 0.990111i $$0.455198\pi$$
$$978$$ 0 0
$$979$$ 9072.00 0.296162
$$980$$ 0 0
$$981$$ 778.000 0.0253207
$$982$$ 0 0
$$983$$ −29706.0 −0.963860 −0.481930 0.876210i $$-0.660064\pi$$
−0.481930 + 0.876210i $$0.660064\pi$$
$$984$$ 0 0
$$985$$ 50112.0 1.62102
$$986$$ 0 0
$$987$$ 9240.00 0.297986
$$988$$ 0 0
$$989$$ 35868.0 1.15322
$$990$$ 0 0
$$991$$ −30512.0 −0.978048 −0.489024 0.872270i $$-0.662647\pi$$
−0.489024 + 0.872270i $$0.662647\pi$$
$$992$$ 0 0
$$993$$ 44495.0 1.42196
$$994$$ 0 0
$$995$$ 2892.00 0.0921433
$$996$$ 0 0
$$997$$ −47756.0 −1.51700 −0.758499 0.651674i $$-0.774067\pi$$
−0.758499 + 0.651674i $$0.774067\pi$$
$$998$$ 0 0
$$999$$ 36250.0 1.14805
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 1216.4.a.a.1.1 1
4.3 odd 2 1216.4.a.f.1.1 1
8.3 odd 2 19.4.a.a.1.1 1
8.5 even 2 304.4.a.b.1.1 1
24.11 even 2 171.4.a.d.1.1 1
40.3 even 4 475.4.b.c.324.2 2
40.19 odd 2 475.4.a.e.1.1 1
40.27 even 4 475.4.b.c.324.1 2
56.27 even 2 931.4.a.a.1.1 1
88.43 even 2 2299.4.a.b.1.1 1
152.75 even 2 361.4.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
19.4.a.a.1.1 1 8.3 odd 2
171.4.a.d.1.1 1 24.11 even 2
304.4.a.b.1.1 1 8.5 even 2
361.4.a.b.1.1 1 152.75 even 2
475.4.a.e.1.1 1 40.19 odd 2
475.4.b.c.324.1 2 40.27 even 4
475.4.b.c.324.2 2 40.3 even 4
931.4.a.a.1.1 1 56.27 even 2
1216.4.a.a.1.1 1 1.1 even 1 trivial
1216.4.a.f.1.1 1 4.3 odd 2
2299.4.a.b.1.1 1 88.43 even 2