# Properties

 Label 960.4.a.bd.1.1 Level $960$ Weight $4$ Character 960.1 Self dual yes Analytic conductor $56.642$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(1,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, 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("960.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 960.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: $$56.6418336055$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 960.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +5.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +5.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} +16.0000 q^{11} -58.0000 q^{13} +15.0000 q^{15} +38.0000 q^{17} +4.00000 q^{19} -60.0000 q^{21} +80.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -82.0000 q^{29} +8.00000 q^{31} +48.0000 q^{33} -100.000 q^{35} -426.000 q^{37} -174.000 q^{39} -246.000 q^{41} -524.000 q^{43} +45.0000 q^{45} +464.000 q^{47} +57.0000 q^{49} +114.000 q^{51} +702.000 q^{53} +80.0000 q^{55} +12.0000 q^{57} -592.000 q^{59} -574.000 q^{61} -180.000 q^{63} -290.000 q^{65} -172.000 q^{67} +240.000 q^{69} -768.000 q^{71} -558.000 q^{73} +75.0000 q^{75} -320.000 q^{77} -408.000 q^{79} +81.0000 q^{81} +164.000 q^{83} +190.000 q^{85} -246.000 q^{87} -510.000 q^{89} +1160.00 q^{91} +24.0000 q^{93} +20.0000 q^{95} +514.000 q^{97} +144.000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −20.0000 −1.07990 −0.539949 0.841698i $$-0.681557\pi$$
−0.539949 + 0.841698i $$0.681557\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 16.0000 0.438562 0.219281 0.975662i $$-0.429629\pi$$
0.219281 + 0.975662i $$0.429629\pi$$
$$12$$ 0 0
$$13$$ −58.0000 −1.23741 −0.618704 0.785624i $$-0.712342\pi$$
−0.618704 + 0.785624i $$0.712342\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ 0 0
$$17$$ 38.0000 0.542138 0.271069 0.962560i $$-0.412623\pi$$
0.271069 + 0.962560i $$0.412623\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.0482980 0.0241490 0.999708i $$-0.492312\pi$$
0.0241490 + 0.999708i $$0.492312\pi$$
$$20$$ 0 0
$$21$$ −60.0000 −0.623480
$$22$$ 0 0
$$23$$ 80.0000 0.725268 0.362634 0.931932i $$-0.381878\pi$$
0.362634 + 0.931932i $$0.381878\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −82.0000 −0.525070 −0.262535 0.964923i $$-0.584558\pi$$
−0.262535 + 0.964923i $$0.584558\pi$$
$$30$$ 0 0
$$31$$ 8.00000 0.0463498 0.0231749 0.999731i $$-0.492623\pi$$
0.0231749 + 0.999731i $$0.492623\pi$$
$$32$$ 0 0
$$33$$ 48.0000 0.253204
$$34$$ 0 0
$$35$$ −100.000 −0.482945
$$36$$ 0 0
$$37$$ −426.000 −1.89281 −0.946405 0.322982i $$-0.895315\pi$$
−0.946405 + 0.322982i $$0.895315\pi$$
$$38$$ 0 0
$$39$$ −174.000 −0.714418
$$40$$ 0 0
$$41$$ −246.000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −524.000 −1.85835 −0.929177 0.369634i $$-0.879483\pi$$
−0.929177 + 0.369634i $$0.879483\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ 0 0
$$47$$ 464.000 1.44003 0.720014 0.693959i $$-0.244135\pi$$
0.720014 + 0.693959i $$0.244135\pi$$
$$48$$ 0 0
$$49$$ 57.0000 0.166181
$$50$$ 0 0
$$51$$ 114.000 0.313004
$$52$$ 0 0
$$53$$ 702.000 1.81938 0.909690 0.415288i $$-0.136319\pi$$
0.909690 + 0.415288i $$0.136319\pi$$
$$54$$ 0 0
$$55$$ 80.0000 0.196131
$$56$$ 0 0
$$57$$ 12.0000 0.0278849
$$58$$ 0 0
$$59$$ −592.000 −1.30630 −0.653151 0.757228i $$-0.726553\pi$$
−0.653151 + 0.757228i $$0.726553\pi$$
$$60$$ 0 0
$$61$$ −574.000 −1.20481 −0.602403 0.798192i $$-0.705790\pi$$
−0.602403 + 0.798192i $$0.705790\pi$$
$$62$$ 0 0
$$63$$ −180.000 −0.359966
$$64$$ 0 0
$$65$$ −290.000 −0.553386
$$66$$ 0 0
$$67$$ −172.000 −0.313629 −0.156815 0.987628i $$-0.550122\pi$$
−0.156815 + 0.987628i $$0.550122\pi$$
$$68$$ 0 0
$$69$$ 240.000 0.418733
$$70$$ 0 0
$$71$$ −768.000 −1.28373 −0.641865 0.766818i $$-0.721839\pi$$
−0.641865 + 0.766818i $$0.721839\pi$$
$$72$$ 0 0
$$73$$ −558.000 −0.894643 −0.447322 0.894373i $$-0.647622\pi$$
−0.447322 + 0.894373i $$0.647622\pi$$
$$74$$ 0 0
$$75$$ 75.0000 0.115470
$$76$$ 0 0
$$77$$ −320.000 −0.473602
$$78$$ 0 0
$$79$$ −408.000 −0.581058 −0.290529 0.956866i $$-0.593831\pi$$
−0.290529 + 0.956866i $$0.593831\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 164.000 0.216884 0.108442 0.994103i $$-0.465414\pi$$
0.108442 + 0.994103i $$0.465414\pi$$
$$84$$ 0 0
$$85$$ 190.000 0.242452
$$86$$ 0 0
$$87$$ −246.000 −0.303149
$$88$$ 0 0
$$89$$ −510.000 −0.607415 −0.303707 0.952765i $$-0.598224\pi$$
−0.303707 + 0.952765i $$0.598224\pi$$
$$90$$ 0 0
$$91$$ 1160.00 1.33628
$$92$$ 0 0
$$93$$ 24.0000 0.0267600
$$94$$ 0 0
$$95$$ 20.0000 0.0215995
$$96$$ 0 0
$$97$$ 514.000 0.538029 0.269014 0.963136i $$-0.413302\pi$$
0.269014 + 0.963136i $$0.413302\pi$$
$$98$$ 0 0
$$99$$ 144.000 0.146187
$$100$$ 0 0
$$101$$ −666.000 −0.656133 −0.328067 0.944655i $$-0.606397\pi$$
−0.328067 + 0.944655i $$0.606397\pi$$
$$102$$ 0 0
$$103$$ 1100.00 1.05229 0.526147 0.850394i $$-0.323636\pi$$
0.526147 + 0.850394i $$0.323636\pi$$
$$104$$ 0 0
$$105$$ −300.000 −0.278829
$$106$$ 0 0
$$107$$ 1212.00 1.09503 0.547516 0.836795i $$-0.315573\pi$$
0.547516 + 0.836795i $$0.315573\pi$$
$$108$$ 0 0
$$109$$ −2078.00 −1.82602 −0.913011 0.407936i $$-0.866249\pi$$
−0.913011 + 0.407936i $$0.866249\pi$$
$$110$$ 0 0
$$111$$ −1278.00 −1.09281
$$112$$ 0 0
$$113$$ −1458.00 −1.21378 −0.606890 0.794786i $$-0.707583\pi$$
−0.606890 + 0.794786i $$0.707583\pi$$
$$114$$ 0 0
$$115$$ 400.000 0.324349
$$116$$ 0 0
$$117$$ −522.000 −0.412469
$$118$$ 0 0
$$119$$ −760.000 −0.585455
$$120$$ 0 0
$$121$$ −1075.00 −0.807663
$$122$$ 0 0
$$123$$ −738.000 −0.541002
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 2436.00 1.70205 0.851024 0.525127i $$-0.175982\pi$$
0.851024 + 0.525127i $$0.175982\pi$$
$$128$$ 0 0
$$129$$ −1572.00 −1.07292
$$130$$ 0 0
$$131$$ 2544.00 1.69672 0.848360 0.529420i $$-0.177590\pi$$
0.848360 + 0.529420i $$0.177590\pi$$
$$132$$ 0 0
$$133$$ −80.0000 −0.0521570
$$134$$ 0 0
$$135$$ 135.000 0.0860663
$$136$$ 0 0
$$137$$ 694.000 0.432791 0.216396 0.976306i $$-0.430570\pi$$
0.216396 + 0.976306i $$0.430570\pi$$
$$138$$ 0 0
$$139$$ 516.000 0.314867 0.157434 0.987530i $$-0.449678\pi$$
0.157434 + 0.987530i $$0.449678\pi$$
$$140$$ 0 0
$$141$$ 1392.00 0.831401
$$142$$ 0 0
$$143$$ −928.000 −0.542680
$$144$$ 0 0
$$145$$ −410.000 −0.234818
$$146$$ 0 0
$$147$$ 171.000 0.0959445
$$148$$ 0 0
$$149$$ −770.000 −0.423361 −0.211681 0.977339i $$-0.567894\pi$$
−0.211681 + 0.977339i $$0.567894\pi$$
$$150$$ 0 0
$$151$$ 424.000 0.228507 0.114254 0.993452i $$-0.463552\pi$$
0.114254 + 0.993452i $$0.463552\pi$$
$$152$$ 0 0
$$153$$ 342.000 0.180713
$$154$$ 0 0
$$155$$ 40.0000 0.0207282
$$156$$ 0 0
$$157$$ −922.000 −0.468685 −0.234343 0.972154i $$-0.575294\pi$$
−0.234343 + 0.972154i $$0.575294\pi$$
$$158$$ 0 0
$$159$$ 2106.00 1.05042
$$160$$ 0 0
$$161$$ −1600.00 −0.783215
$$162$$ 0 0
$$163$$ −3788.00 −1.82024 −0.910120 0.414345i $$-0.864011\pi$$
−0.910120 + 0.414345i $$0.864011\pi$$
$$164$$ 0 0
$$165$$ 240.000 0.113236
$$166$$ 0 0
$$167$$ 48.0000 0.0222416 0.0111208 0.999938i $$-0.496460\pi$$
0.0111208 + 0.999938i $$0.496460\pi$$
$$168$$ 0 0
$$169$$ 1167.00 0.531179
$$170$$ 0 0
$$171$$ 36.0000 0.0160993
$$172$$ 0 0
$$173$$ −3242.00 −1.42477 −0.712384 0.701790i $$-0.752384\pi$$
−0.712384 + 0.701790i $$0.752384\pi$$
$$174$$ 0 0
$$175$$ −500.000 −0.215980
$$176$$ 0 0
$$177$$ −1776.00 −0.754194
$$178$$ 0 0
$$179$$ −2728.00 −1.13911 −0.569554 0.821954i $$-0.692884\pi$$
−0.569554 + 0.821954i $$0.692884\pi$$
$$180$$ 0 0
$$181$$ 4090.00 1.67960 0.839799 0.542897i $$-0.182673\pi$$
0.839799 + 0.542897i $$0.182673\pi$$
$$182$$ 0 0
$$183$$ −1722.00 −0.695595
$$184$$ 0 0
$$185$$ −2130.00 −0.846490
$$186$$ 0 0
$$187$$ 608.000 0.237761
$$188$$ 0 0
$$189$$ −540.000 −0.207827
$$190$$ 0 0
$$191$$ 1480.00 0.560676 0.280338 0.959901i $$-0.409554\pi$$
0.280338 + 0.959901i $$0.409554\pi$$
$$192$$ 0 0
$$193$$ −1622.00 −0.604944 −0.302472 0.953158i $$-0.597812\pi$$
−0.302472 + 0.953158i $$0.597812\pi$$
$$194$$ 0 0
$$195$$ −870.000 −0.319497
$$196$$ 0 0
$$197$$ −2530.00 −0.915000 −0.457500 0.889210i $$-0.651255\pi$$
−0.457500 + 0.889210i $$0.651255\pi$$
$$198$$ 0 0
$$199$$ 2440.00 0.869181 0.434590 0.900628i $$-0.356893\pi$$
0.434590 + 0.900628i $$0.356893\pi$$
$$200$$ 0 0
$$201$$ −516.000 −0.181074
$$202$$ 0 0
$$203$$ 1640.00 0.567022
$$204$$ 0 0
$$205$$ −1230.00 −0.419058
$$206$$ 0 0
$$207$$ 720.000 0.241756
$$208$$ 0 0
$$209$$ 64.0000 0.0211817
$$210$$ 0 0
$$211$$ −148.000 −0.0482879 −0.0241439 0.999708i $$-0.507686\pi$$
−0.0241439 + 0.999708i $$0.507686\pi$$
$$212$$ 0 0
$$213$$ −2304.00 −0.741162
$$214$$ 0 0
$$215$$ −2620.00 −0.831081
$$216$$ 0 0
$$217$$ −160.000 −0.0500530
$$218$$ 0 0
$$219$$ −1674.00 −0.516523
$$220$$ 0 0
$$221$$ −2204.00 −0.670847
$$222$$ 0 0
$$223$$ 676.000 0.202997 0.101498 0.994836i $$-0.467636\pi$$
0.101498 + 0.994836i $$0.467636\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −6276.00 −1.83503 −0.917517 0.397696i $$-0.869810\pi$$
−0.917517 + 0.397696i $$0.869810\pi$$
$$228$$ 0 0
$$229$$ −6190.00 −1.78623 −0.893115 0.449828i $$-0.851485\pi$$
−0.893115 + 0.449828i $$0.851485\pi$$
$$230$$ 0 0
$$231$$ −960.000 −0.273434
$$232$$ 0 0
$$233$$ 5406.00 1.52000 0.759998 0.649926i $$-0.225200\pi$$
0.759998 + 0.649926i $$0.225200\pi$$
$$234$$ 0 0
$$235$$ 2320.00 0.644000
$$236$$ 0 0
$$237$$ −1224.00 −0.335474
$$238$$ 0 0
$$239$$ 600.000 0.162388 0.0811941 0.996698i $$-0.474127\pi$$
0.0811941 + 0.996698i $$0.474127\pi$$
$$240$$ 0 0
$$241$$ −1054.00 −0.281718 −0.140859 0.990030i $$-0.544986\pi$$
−0.140859 + 0.990030i $$0.544986\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 285.000 0.0743183
$$246$$ 0 0
$$247$$ −232.000 −0.0597644
$$248$$ 0 0
$$249$$ 492.000 0.125218
$$250$$ 0 0
$$251$$ −2232.00 −0.561285 −0.280643 0.959812i $$-0.590548\pi$$
−0.280643 + 0.959812i $$0.590548\pi$$
$$252$$ 0 0
$$253$$ 1280.00 0.318075
$$254$$ 0 0
$$255$$ 570.000 0.139980
$$256$$ 0 0
$$257$$ 3630.00 0.881063 0.440531 0.897737i $$-0.354790\pi$$
0.440531 + 0.897737i $$0.354790\pi$$
$$258$$ 0 0
$$259$$ 8520.00 2.04404
$$260$$ 0 0
$$261$$ −738.000 −0.175023
$$262$$ 0 0
$$263$$ −6960.00 −1.63183 −0.815916 0.578170i $$-0.803767\pi$$
−0.815916 + 0.578170i $$0.803767\pi$$
$$264$$ 0 0
$$265$$ 3510.00 0.813651
$$266$$ 0 0
$$267$$ −1530.00 −0.350691
$$268$$ 0 0
$$269$$ 2062.00 0.467369 0.233685 0.972312i $$-0.424922\pi$$
0.233685 + 0.972312i $$0.424922\pi$$
$$270$$ 0 0
$$271$$ 2544.00 0.570247 0.285124 0.958491i $$-0.407965\pi$$
0.285124 + 0.958491i $$0.407965\pi$$
$$272$$ 0 0
$$273$$ 3480.00 0.771499
$$274$$ 0 0
$$275$$ 400.000 0.0877124
$$276$$ 0 0
$$277$$ 694.000 0.150536 0.0752679 0.997163i $$-0.476019\pi$$
0.0752679 + 0.997163i $$0.476019\pi$$
$$278$$ 0 0
$$279$$ 72.0000 0.0154499
$$280$$ 0 0
$$281$$ −1982.00 −0.420769 −0.210385 0.977619i $$-0.567472\pi$$
−0.210385 + 0.977619i $$0.567472\pi$$
$$282$$ 0 0
$$283$$ 5228.00 1.09814 0.549068 0.835778i $$-0.314983\pi$$
0.549068 + 0.835778i $$0.314983\pi$$
$$284$$ 0 0
$$285$$ 60.0000 0.0124705
$$286$$ 0 0
$$287$$ 4920.00 1.01191
$$288$$ 0 0
$$289$$ −3469.00 −0.706086
$$290$$ 0 0
$$291$$ 1542.00 0.310631
$$292$$ 0 0
$$293$$ 7454.00 1.48624 0.743118 0.669160i $$-0.233346\pi$$
0.743118 + 0.669160i $$0.233346\pi$$
$$294$$ 0 0
$$295$$ −2960.00 −0.584196
$$296$$ 0 0
$$297$$ 432.000 0.0844013
$$298$$ 0 0
$$299$$ −4640.00 −0.897452
$$300$$ 0 0
$$301$$ 10480.0 2.00683
$$302$$ 0 0
$$303$$ −1998.00 −0.378819
$$304$$ 0 0
$$305$$ −2870.00 −0.538806
$$306$$ 0 0
$$307$$ −1316.00 −0.244652 −0.122326 0.992490i $$-0.539035\pi$$
−0.122326 + 0.992490i $$0.539035\pi$$
$$308$$ 0 0
$$309$$ 3300.00 0.607542
$$310$$ 0 0
$$311$$ 832.000 0.151699 0.0758495 0.997119i $$-0.475833\pi$$
0.0758495 + 0.997119i $$0.475833\pi$$
$$312$$ 0 0
$$313$$ 6770.00 1.22257 0.611283 0.791412i $$-0.290654\pi$$
0.611283 + 0.791412i $$0.290654\pi$$
$$314$$ 0 0
$$315$$ −900.000 −0.160982
$$316$$ 0 0
$$317$$ 6582.00 1.16619 0.583095 0.812404i $$-0.301842\pi$$
0.583095 + 0.812404i $$0.301842\pi$$
$$318$$ 0 0
$$319$$ −1312.00 −0.230276
$$320$$ 0 0
$$321$$ 3636.00 0.632217
$$322$$ 0 0
$$323$$ 152.000 0.0261842
$$324$$ 0 0
$$325$$ −1450.00 −0.247482
$$326$$ 0 0
$$327$$ −6234.00 −1.05425
$$328$$ 0 0
$$329$$ −9280.00 −1.55508
$$330$$ 0 0
$$331$$ 11292.0 1.87512 0.937560 0.347825i $$-0.113080\pi$$
0.937560 + 0.347825i $$0.113080\pi$$
$$332$$ 0 0
$$333$$ −3834.00 −0.630937
$$334$$ 0 0
$$335$$ −860.000 −0.140259
$$336$$ 0 0
$$337$$ −8006.00 −1.29411 −0.647054 0.762444i $$-0.723999\pi$$
−0.647054 + 0.762444i $$0.723999\pi$$
$$338$$ 0 0
$$339$$ −4374.00 −0.700776
$$340$$ 0 0
$$341$$ 128.000 0.0203272
$$342$$ 0 0
$$343$$ 5720.00 0.900440
$$344$$ 0 0
$$345$$ 1200.00 0.187263
$$346$$ 0 0
$$347$$ −316.000 −0.0488869 −0.0244435 0.999701i $$-0.507781\pi$$
−0.0244435 + 0.999701i $$0.507781\pi$$
$$348$$ 0 0
$$349$$ −4926.00 −0.755538 −0.377769 0.925900i $$-0.623309\pi$$
−0.377769 + 0.925900i $$0.623309\pi$$
$$350$$ 0 0
$$351$$ −1566.00 −0.238139
$$352$$ 0 0
$$353$$ 2438.00 0.367597 0.183798 0.982964i $$-0.441161\pi$$
0.183798 + 0.982964i $$0.441161\pi$$
$$354$$ 0 0
$$355$$ −3840.00 −0.574102
$$356$$ 0 0
$$357$$ −2280.00 −0.338012
$$358$$ 0 0
$$359$$ 3336.00 0.490438 0.245219 0.969468i $$-0.421140\pi$$
0.245219 + 0.969468i $$0.421140\pi$$
$$360$$ 0 0
$$361$$ −6843.00 −0.997667
$$362$$ 0 0
$$363$$ −3225.00 −0.466305
$$364$$ 0 0
$$365$$ −2790.00 −0.400097
$$366$$ 0 0
$$367$$ −44.0000 −0.00625826 −0.00312913 0.999995i $$-0.500996\pi$$
−0.00312913 + 0.999995i $$0.500996\pi$$
$$368$$ 0 0
$$369$$ −2214.00 −0.312348
$$370$$ 0 0
$$371$$ −14040.0 −1.96475
$$372$$ 0 0
$$373$$ 11966.0 1.66106 0.830531 0.556973i $$-0.188037\pi$$
0.830531 + 0.556973i $$0.188037\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 0 0
$$377$$ 4756.00 0.649725
$$378$$ 0 0
$$379$$ 12676.0 1.71800 0.859001 0.511975i $$-0.171086\pi$$
0.859001 + 0.511975i $$0.171086\pi$$
$$380$$ 0 0
$$381$$ 7308.00 0.982678
$$382$$ 0 0
$$383$$ −6672.00 −0.890139 −0.445070 0.895496i $$-0.646821\pi$$
−0.445070 + 0.895496i $$0.646821\pi$$
$$384$$ 0 0
$$385$$ −1600.00 −0.211801
$$386$$ 0 0
$$387$$ −4716.00 −0.619452
$$388$$ 0 0
$$389$$ −354.000 −0.0461401 −0.0230701 0.999734i $$-0.507344\pi$$
−0.0230701 + 0.999734i $$0.507344\pi$$
$$390$$ 0 0
$$391$$ 3040.00 0.393195
$$392$$ 0 0
$$393$$ 7632.00 0.979602
$$394$$ 0 0
$$395$$ −2040.00 −0.259857
$$396$$ 0 0
$$397$$ 5054.00 0.638924 0.319462 0.947599i $$-0.396498\pi$$
0.319462 + 0.947599i $$0.396498\pi$$
$$398$$ 0 0
$$399$$ −240.000 −0.0301129
$$400$$ 0 0
$$401$$ 10266.0 1.27845 0.639226 0.769019i $$-0.279255\pi$$
0.639226 + 0.769019i $$0.279255\pi$$
$$402$$ 0 0
$$403$$ −464.000 −0.0573536
$$404$$ 0 0
$$405$$ 405.000 0.0496904
$$406$$ 0 0
$$407$$ −6816.00 −0.830114
$$408$$ 0 0
$$409$$ −1526.00 −0.184489 −0.0922443 0.995736i $$-0.529404\pi$$
−0.0922443 + 0.995736i $$0.529404\pi$$
$$410$$ 0 0
$$411$$ 2082.00 0.249872
$$412$$ 0 0
$$413$$ 11840.0 1.41067
$$414$$ 0 0
$$415$$ 820.000 0.0969933
$$416$$ 0 0
$$417$$ 1548.00 0.181789
$$418$$ 0 0
$$419$$ 2064.00 0.240652 0.120326 0.992734i $$-0.461606\pi$$
0.120326 + 0.992734i $$0.461606\pi$$
$$420$$ 0 0
$$421$$ −4590.00 −0.531361 −0.265680 0.964061i $$-0.585597\pi$$
−0.265680 + 0.964061i $$0.585597\pi$$
$$422$$ 0 0
$$423$$ 4176.00 0.480010
$$424$$ 0 0
$$425$$ 950.000 0.108428
$$426$$ 0 0
$$427$$ 11480.0 1.30107
$$428$$ 0 0
$$429$$ −2784.00 −0.313317
$$430$$ 0 0
$$431$$ 5536.00 0.618700 0.309350 0.950948i $$-0.399889\pi$$
0.309350 + 0.950948i $$0.399889\pi$$
$$432$$ 0 0
$$433$$ 1850.00 0.205324 0.102662 0.994716i $$-0.467264\pi$$
0.102662 + 0.994716i $$0.467264\pi$$
$$434$$ 0 0
$$435$$ −1230.00 −0.135572
$$436$$ 0 0
$$437$$ 320.000 0.0350290
$$438$$ 0 0
$$439$$ −11704.0 −1.27244 −0.636220 0.771507i $$-0.719503\pi$$
−0.636220 + 0.771507i $$0.719503\pi$$
$$440$$ 0 0
$$441$$ 513.000 0.0553936
$$442$$ 0 0
$$443$$ 6948.00 0.745168 0.372584 0.927998i $$-0.378472\pi$$
0.372584 + 0.927998i $$0.378472\pi$$
$$444$$ 0 0
$$445$$ −2550.00 −0.271644
$$446$$ 0 0
$$447$$ −2310.00 −0.244428
$$448$$ 0 0
$$449$$ 12090.0 1.27074 0.635370 0.772208i $$-0.280848\pi$$
0.635370 + 0.772208i $$0.280848\pi$$
$$450$$ 0 0
$$451$$ −3936.00 −0.410951
$$452$$ 0 0
$$453$$ 1272.00 0.131929
$$454$$ 0 0
$$455$$ 5800.00 0.597600
$$456$$ 0 0
$$457$$ 11626.0 1.19002 0.595012 0.803717i $$-0.297147\pi$$
0.595012 + 0.803717i $$0.297147\pi$$
$$458$$ 0 0
$$459$$ 1026.00 0.104335
$$460$$ 0 0
$$461$$ −16314.0 −1.64820 −0.824098 0.566447i $$-0.808318\pi$$
−0.824098 + 0.566447i $$0.808318\pi$$
$$462$$ 0 0
$$463$$ 15756.0 1.58152 0.790760 0.612127i $$-0.209686\pi$$
0.790760 + 0.612127i $$0.209686\pi$$
$$464$$ 0 0
$$465$$ 120.000 0.0119675
$$466$$ 0 0
$$467$$ 5684.00 0.563221 0.281610 0.959529i $$-0.409131\pi$$
0.281610 + 0.959529i $$0.409131\pi$$
$$468$$ 0 0
$$469$$ 3440.00 0.338688
$$470$$ 0 0
$$471$$ −2766.00 −0.270596
$$472$$ 0 0
$$473$$ −8384.00 −0.815004
$$474$$ 0 0
$$475$$ 100.000 0.00965961
$$476$$ 0 0
$$477$$ 6318.00 0.606460
$$478$$ 0 0
$$479$$ 3368.00 0.321269 0.160634 0.987014i $$-0.448646\pi$$
0.160634 + 0.987014i $$0.448646\pi$$
$$480$$ 0 0
$$481$$ 24708.0 2.34218
$$482$$ 0 0
$$483$$ −4800.00 −0.452190
$$484$$ 0 0
$$485$$ 2570.00 0.240614
$$486$$ 0 0
$$487$$ 5588.00 0.519952 0.259976 0.965615i $$-0.416285\pi$$
0.259976 + 0.965615i $$0.416285\pi$$
$$488$$ 0 0
$$489$$ −11364.0 −1.05092
$$490$$ 0 0
$$491$$ 10584.0 0.972809 0.486405 0.873734i $$-0.338308\pi$$
0.486405 + 0.873734i $$0.338308\pi$$
$$492$$ 0 0
$$493$$ −3116.00 −0.284660
$$494$$ 0 0
$$495$$ 720.000 0.0653770
$$496$$ 0 0
$$497$$ 15360.0 1.38630
$$498$$ 0 0
$$499$$ −12220.0 −1.09628 −0.548139 0.836388i $$-0.684663\pi$$
−0.548139 + 0.836388i $$0.684663\pi$$
$$500$$ 0 0
$$501$$ 144.000 0.0128412
$$502$$ 0 0
$$503$$ −16152.0 −1.43177 −0.715887 0.698216i $$-0.753977\pi$$
−0.715887 + 0.698216i $$0.753977\pi$$
$$504$$ 0 0
$$505$$ −3330.00 −0.293432
$$506$$ 0 0
$$507$$ 3501.00 0.306676
$$508$$ 0 0
$$509$$ −10642.0 −0.926716 −0.463358 0.886171i $$-0.653356\pi$$
−0.463358 + 0.886171i $$0.653356\pi$$
$$510$$ 0 0
$$511$$ 11160.0 0.966124
$$512$$ 0 0
$$513$$ 108.000 0.00929496
$$514$$ 0 0
$$515$$ 5500.00 0.470600
$$516$$ 0 0
$$517$$ 7424.00 0.631542
$$518$$ 0 0
$$519$$ −9726.00 −0.822590
$$520$$ 0 0
$$521$$ 22882.0 1.92414 0.962072 0.272797i $$-0.0879487\pi$$
0.962072 + 0.272797i $$0.0879487\pi$$
$$522$$ 0 0
$$523$$ −10052.0 −0.840427 −0.420213 0.907425i $$-0.638045\pi$$
−0.420213 + 0.907425i $$0.638045\pi$$
$$524$$ 0 0
$$525$$ −1500.00 −0.124696
$$526$$ 0 0
$$527$$ 304.000 0.0251280
$$528$$ 0 0
$$529$$ −5767.00 −0.473987
$$530$$ 0 0
$$531$$ −5328.00 −0.435434
$$532$$ 0 0
$$533$$ 14268.0 1.15950
$$534$$ 0 0
$$535$$ 6060.00 0.489713
$$536$$ 0 0
$$537$$ −8184.00 −0.657664
$$538$$ 0 0
$$539$$ 912.000 0.0728806
$$540$$ 0 0
$$541$$ 6530.00 0.518940 0.259470 0.965751i $$-0.416452\pi$$
0.259470 + 0.965751i $$0.416452\pi$$
$$542$$ 0 0
$$543$$ 12270.0 0.969717
$$544$$ 0 0
$$545$$ −10390.0 −0.816621
$$546$$ 0 0
$$547$$ 16652.0 1.30162 0.650812 0.759239i $$-0.274429\pi$$
0.650812 + 0.759239i $$0.274429\pi$$
$$548$$ 0 0
$$549$$ −5166.00 −0.401602
$$550$$ 0 0
$$551$$ −328.000 −0.0253598
$$552$$ 0 0
$$553$$ 8160.00 0.627484
$$554$$ 0 0
$$555$$ −6390.00 −0.488721
$$556$$ 0 0
$$557$$ 12886.0 0.980247 0.490123 0.871653i $$-0.336952\pi$$
0.490123 + 0.871653i $$0.336952\pi$$
$$558$$ 0 0
$$559$$ 30392.0 2.29954
$$560$$ 0 0
$$561$$ 1824.00 0.137272
$$562$$ 0 0
$$563$$ −11108.0 −0.831521 −0.415761 0.909474i $$-0.636485\pi$$
−0.415761 + 0.909474i $$0.636485\pi$$
$$564$$ 0 0
$$565$$ −7290.00 −0.542819
$$566$$ 0 0
$$567$$ −1620.00 −0.119989
$$568$$ 0 0
$$569$$ −9214.00 −0.678859 −0.339430 0.940631i $$-0.610234\pi$$
−0.339430 + 0.940631i $$0.610234\pi$$
$$570$$ 0 0
$$571$$ −4052.00 −0.296972 −0.148486 0.988915i $$-0.547440\pi$$
−0.148486 + 0.988915i $$0.547440\pi$$
$$572$$ 0 0
$$573$$ 4440.00 0.323706
$$574$$ 0 0
$$575$$ 2000.00 0.145054
$$576$$ 0 0
$$577$$ −8446.00 −0.609379 −0.304689 0.952452i $$-0.598553\pi$$
−0.304689 + 0.952452i $$0.598553\pi$$
$$578$$ 0 0
$$579$$ −4866.00 −0.349264
$$580$$ 0 0
$$581$$ −3280.00 −0.234212
$$582$$ 0 0
$$583$$ 11232.0 0.797911
$$584$$ 0 0
$$585$$ −2610.00 −0.184462
$$586$$ 0 0
$$587$$ 2172.00 0.152722 0.0763612 0.997080i $$-0.475670\pi$$
0.0763612 + 0.997080i $$0.475670\pi$$
$$588$$ 0 0
$$589$$ 32.0000 0.00223860
$$590$$ 0 0
$$591$$ −7590.00 −0.528276
$$592$$ 0 0
$$593$$ −1218.00 −0.0843461 −0.0421731 0.999110i $$-0.513428\pi$$
−0.0421731 + 0.999110i $$0.513428\pi$$
$$594$$ 0 0
$$595$$ −3800.00 −0.261823
$$596$$ 0 0
$$597$$ 7320.00 0.501822
$$598$$ 0 0
$$599$$ −21240.0 −1.44882 −0.724410 0.689370i $$-0.757888\pi$$
−0.724410 + 0.689370i $$0.757888\pi$$
$$600$$ 0 0
$$601$$ 17626.0 1.19631 0.598153 0.801382i $$-0.295902\pi$$
0.598153 + 0.801382i $$0.295902\pi$$
$$602$$ 0 0
$$603$$ −1548.00 −0.104543
$$604$$ 0 0
$$605$$ −5375.00 −0.361198
$$606$$ 0 0
$$607$$ −2580.00 −0.172519 −0.0862594 0.996273i $$-0.527491\pi$$
−0.0862594 + 0.996273i $$0.527491\pi$$
$$608$$ 0 0
$$609$$ 4920.00 0.327370
$$610$$ 0 0
$$611$$ −26912.0 −1.78190
$$612$$ 0 0
$$613$$ 14166.0 0.933376 0.466688 0.884422i $$-0.345447\pi$$
0.466688 + 0.884422i $$0.345447\pi$$
$$614$$ 0 0
$$615$$ −3690.00 −0.241943
$$616$$ 0 0
$$617$$ −21426.0 −1.39802 −0.699010 0.715112i $$-0.746376\pi$$
−0.699010 + 0.715112i $$0.746376\pi$$
$$618$$ 0 0
$$619$$ 3668.00 0.238173 0.119087 0.992884i $$-0.462003\pi$$
0.119087 + 0.992884i $$0.462003\pi$$
$$620$$ 0 0
$$621$$ 2160.00 0.139578
$$622$$ 0 0
$$623$$ 10200.0 0.655946
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 192.000 0.0122293
$$628$$ 0 0
$$629$$ −16188.0 −1.02617
$$630$$ 0 0
$$631$$ −20032.0 −1.26381 −0.631903 0.775048i $$-0.717726\pi$$
−0.631903 + 0.775048i $$0.717726\pi$$
$$632$$ 0 0
$$633$$ −444.000 −0.0278790
$$634$$ 0 0
$$635$$ 12180.0 0.761179
$$636$$ 0 0
$$637$$ −3306.00 −0.205633
$$638$$ 0 0
$$639$$ −6912.00 −0.427910
$$640$$ 0 0
$$641$$ 7458.00 0.459553 0.229776 0.973243i $$-0.426201\pi$$
0.229776 + 0.973243i $$0.426201\pi$$
$$642$$ 0 0
$$643$$ 7092.00 0.434963 0.217481 0.976064i $$-0.430216\pi$$
0.217481 + 0.976064i $$0.430216\pi$$
$$644$$ 0 0
$$645$$ −7860.00 −0.479825
$$646$$ 0 0
$$647$$ −3384.00 −0.205624 −0.102812 0.994701i $$-0.532784\pi$$
−0.102812 + 0.994701i $$0.532784\pi$$
$$648$$ 0 0
$$649$$ −9472.00 −0.572894
$$650$$ 0 0
$$651$$ −480.000 −0.0288981
$$652$$ 0 0
$$653$$ 29398.0 1.76177 0.880883 0.473335i $$-0.156950\pi$$
0.880883 + 0.473335i $$0.156950\pi$$
$$654$$ 0 0
$$655$$ 12720.0 0.758796
$$656$$ 0 0
$$657$$ −5022.00 −0.298214
$$658$$ 0 0
$$659$$ −6624.00 −0.391554 −0.195777 0.980648i $$-0.562723\pi$$
−0.195777 + 0.980648i $$0.562723\pi$$
$$660$$ 0 0
$$661$$ −8646.00 −0.508760 −0.254380 0.967104i $$-0.581871\pi$$
−0.254380 + 0.967104i $$0.581871\pi$$
$$662$$ 0 0
$$663$$ −6612.00 −0.387313
$$664$$ 0 0
$$665$$ −400.000 −0.0233253
$$666$$ 0 0
$$667$$ −6560.00 −0.380816
$$668$$ 0 0
$$669$$ 2028.00 0.117200
$$670$$ 0 0
$$671$$ −9184.00 −0.528382
$$672$$ 0 0
$$673$$ 28698.0 1.64372 0.821862 0.569686i $$-0.192935\pi$$
0.821862 + 0.569686i $$0.192935\pi$$
$$674$$ 0 0
$$675$$ 675.000 0.0384900
$$676$$ 0 0
$$677$$ −19426.0 −1.10281 −0.551405 0.834238i $$-0.685908\pi$$
−0.551405 + 0.834238i $$0.685908\pi$$
$$678$$ 0 0
$$679$$ −10280.0 −0.581016
$$680$$ 0 0
$$681$$ −18828.0 −1.05946
$$682$$ 0 0
$$683$$ 8604.00 0.482025 0.241012 0.970522i $$-0.422521\pi$$
0.241012 + 0.970522i $$0.422521\pi$$
$$684$$ 0 0
$$685$$ 3470.00 0.193550
$$686$$ 0 0
$$687$$ −18570.0 −1.03128
$$688$$ 0 0
$$689$$ −40716.0 −2.25132
$$690$$ 0 0
$$691$$ 12980.0 0.714591 0.357296 0.933991i $$-0.383699\pi$$
0.357296 + 0.933991i $$0.383699\pi$$
$$692$$ 0 0
$$693$$ −2880.00 −0.157867
$$694$$ 0 0
$$695$$ 2580.00 0.140813
$$696$$ 0 0
$$697$$ −9348.00 −0.508007
$$698$$ 0 0
$$699$$ 16218.0 0.877570
$$700$$ 0 0
$$701$$ 19630.0 1.05765 0.528827 0.848730i $$-0.322632\pi$$
0.528827 + 0.848730i $$0.322632\pi$$
$$702$$ 0 0
$$703$$ −1704.00 −0.0914190
$$704$$ 0 0
$$705$$ 6960.00 0.371814
$$706$$ 0 0
$$707$$ 13320.0 0.708558
$$708$$ 0 0
$$709$$ −8030.00 −0.425350 −0.212675 0.977123i $$-0.568218\pi$$
−0.212675 + 0.977123i $$0.568218\pi$$
$$710$$ 0 0
$$711$$ −3672.00 −0.193686
$$712$$ 0 0
$$713$$ 640.000 0.0336160
$$714$$ 0 0
$$715$$ −4640.00 −0.242694
$$716$$ 0 0
$$717$$ 1800.00 0.0937549
$$718$$ 0 0
$$719$$ −22720.0 −1.17846 −0.589230 0.807965i $$-0.700569\pi$$
−0.589230 + 0.807965i $$0.700569\pi$$
$$720$$ 0 0
$$721$$ −22000.0 −1.13637
$$722$$ 0 0
$$723$$ −3162.00 −0.162650
$$724$$ 0 0
$$725$$ −2050.00 −0.105014
$$726$$ 0 0
$$727$$ −27116.0 −1.38332 −0.691662 0.722221i $$-0.743121\pi$$
−0.691662 + 0.722221i $$0.743121\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −19912.0 −1.00749
$$732$$ 0 0
$$733$$ −30882.0 −1.55614 −0.778071 0.628176i $$-0.783802\pi$$
−0.778071 + 0.628176i $$0.783802\pi$$
$$734$$ 0 0
$$735$$ 855.000 0.0429077
$$736$$ 0 0
$$737$$ −2752.00 −0.137546
$$738$$ 0 0
$$739$$ −13836.0 −0.688722 −0.344361 0.938837i $$-0.611904\pi$$
−0.344361 + 0.938837i $$0.611904\pi$$
$$740$$ 0 0
$$741$$ −696.000 −0.0345050
$$742$$ 0 0
$$743$$ −32712.0 −1.61519 −0.807595 0.589737i $$-0.799231\pi$$
−0.807595 + 0.589737i $$0.799231\pi$$
$$744$$ 0 0
$$745$$ −3850.00 −0.189333
$$746$$ 0 0
$$747$$ 1476.00 0.0722945
$$748$$ 0 0
$$749$$ −24240.0 −1.18252
$$750$$ 0 0
$$751$$ 8472.00 0.411648 0.205824 0.978589i $$-0.434013\pi$$
0.205824 + 0.978589i $$0.434013\pi$$
$$752$$ 0 0
$$753$$ −6696.00 −0.324058
$$754$$ 0 0
$$755$$ 2120.00 0.102192
$$756$$ 0 0
$$757$$ −9866.00 −0.473693 −0.236847 0.971547i $$-0.576114\pi$$
−0.236847 + 0.971547i $$0.576114\pi$$
$$758$$ 0 0
$$759$$ 3840.00 0.183641
$$760$$ 0 0
$$761$$ −3774.00 −0.179773 −0.0898866 0.995952i $$-0.528650\pi$$
−0.0898866 + 0.995952i $$0.528650\pi$$
$$762$$ 0 0
$$763$$ 41560.0 1.97192
$$764$$ 0 0
$$765$$ 1710.00 0.0808172
$$766$$ 0 0
$$767$$ 34336.0 1.61643
$$768$$ 0 0
$$769$$ −28670.0 −1.34443 −0.672215 0.740356i $$-0.734657\pi$$
−0.672215 + 0.740356i $$0.734657\pi$$
$$770$$ 0 0
$$771$$ 10890.0 0.508682
$$772$$ 0 0
$$773$$ 3246.00 0.151036 0.0755178 0.997144i $$-0.475939\pi$$
0.0755178 + 0.997144i $$0.475939\pi$$
$$774$$ 0 0
$$775$$ 200.000 0.00926995
$$776$$ 0 0
$$777$$ 25560.0 1.18013
$$778$$ 0 0
$$779$$ −984.000 −0.0452573
$$780$$ 0 0
$$781$$ −12288.0 −0.562995
$$782$$ 0 0
$$783$$ −2214.00 −0.101050
$$784$$ 0 0
$$785$$ −4610.00 −0.209602
$$786$$ 0 0
$$787$$ −19372.0 −0.877430 −0.438715 0.898626i $$-0.644566\pi$$
−0.438715 + 0.898626i $$0.644566\pi$$
$$788$$ 0 0
$$789$$ −20880.0 −0.942139
$$790$$ 0 0
$$791$$ 29160.0 1.31076
$$792$$ 0 0
$$793$$ 33292.0 1.49084
$$794$$ 0 0
$$795$$ 10530.0 0.469762
$$796$$ 0 0
$$797$$ 11814.0 0.525061 0.262530 0.964924i $$-0.415443\pi$$
0.262530 + 0.964924i $$0.415443\pi$$
$$798$$ 0 0
$$799$$ 17632.0 0.780695
$$800$$ 0 0
$$801$$ −4590.00 −0.202472
$$802$$ 0 0
$$803$$ −8928.00 −0.392357
$$804$$ 0 0
$$805$$ −8000.00 −0.350265
$$806$$ 0 0
$$807$$ 6186.00 0.269836
$$808$$ 0 0
$$809$$ −30054.0 −1.30611 −0.653055 0.757311i $$-0.726513\pi$$
−0.653055 + 0.757311i $$0.726513\pi$$
$$810$$ 0 0
$$811$$ 2852.00 0.123486 0.0617431 0.998092i $$-0.480334\pi$$
0.0617431 + 0.998092i $$0.480334\pi$$
$$812$$ 0 0
$$813$$ 7632.00 0.329232
$$814$$ 0 0
$$815$$ −18940.0 −0.814036
$$816$$ 0 0
$$817$$ −2096.00 −0.0897549
$$818$$ 0 0
$$819$$ 10440.0 0.445425
$$820$$ 0 0
$$821$$ −2170.00 −0.0922455 −0.0461227 0.998936i $$-0.514687\pi$$
−0.0461227 + 0.998936i $$0.514687\pi$$
$$822$$ 0 0
$$823$$ −19804.0 −0.838790 −0.419395 0.907804i $$-0.637758\pi$$
−0.419395 + 0.907804i $$0.637758\pi$$
$$824$$ 0 0
$$825$$ 1200.00 0.0506408
$$826$$ 0 0
$$827$$ 5508.00 0.231598 0.115799 0.993273i $$-0.463057\pi$$
0.115799 + 0.993273i $$0.463057\pi$$
$$828$$ 0 0
$$829$$ −33262.0 −1.39353 −0.696765 0.717299i $$-0.745378\pi$$
−0.696765 + 0.717299i $$0.745378\pi$$
$$830$$ 0 0
$$831$$ 2082.00 0.0869119
$$832$$ 0 0
$$833$$ 2166.00 0.0900930
$$834$$ 0 0
$$835$$ 240.000 0.00994676
$$836$$ 0 0
$$837$$ 216.000 0.00892001
$$838$$ 0 0
$$839$$ 4600.00 0.189284 0.0946422 0.995511i $$-0.469829\pi$$
0.0946422 + 0.995511i $$0.469829\pi$$
$$840$$ 0 0
$$841$$ −17665.0 −0.724302
$$842$$ 0 0
$$843$$ −5946.00 −0.242931
$$844$$ 0 0
$$845$$ 5835.00 0.237550
$$846$$ 0 0
$$847$$ 21500.0 0.872195
$$848$$ 0 0
$$849$$ 15684.0 0.634009
$$850$$ 0 0
$$851$$ −34080.0 −1.37279
$$852$$ 0 0
$$853$$ 4198.00 0.168507 0.0842537 0.996444i $$-0.473149\pi$$
0.0842537 + 0.996444i $$0.473149\pi$$
$$854$$ 0 0
$$855$$ 180.000 0.00719985
$$856$$ 0 0
$$857$$ −5826.00 −0.232220 −0.116110 0.993236i $$-0.537042\pi$$
−0.116110 + 0.993236i $$0.537042\pi$$
$$858$$ 0 0
$$859$$ −3004.00 −0.119319 −0.0596596 0.998219i $$-0.519002\pi$$
−0.0596596 + 0.998219i $$0.519002\pi$$
$$860$$ 0 0
$$861$$ 14760.0 0.584227
$$862$$ 0 0
$$863$$ 36936.0 1.45691 0.728457 0.685092i $$-0.240238\pi$$
0.728457 + 0.685092i $$0.240238\pi$$
$$864$$ 0 0
$$865$$ −16210.0 −0.637175
$$866$$ 0 0
$$867$$ −10407.0 −0.407659
$$868$$ 0 0
$$869$$ −6528.00 −0.254830
$$870$$ 0 0
$$871$$ 9976.00 0.388087
$$872$$ 0 0
$$873$$ 4626.00 0.179343
$$874$$ 0 0
$$875$$ −2500.00 −0.0965891
$$876$$ 0 0
$$877$$ −5434.00 −0.209228 −0.104614 0.994513i $$-0.533361\pi$$
−0.104614 + 0.994513i $$0.533361\pi$$
$$878$$ 0 0
$$879$$ 22362.0 0.858079
$$880$$ 0 0
$$881$$ −4758.00 −0.181954 −0.0909768 0.995853i $$-0.528999\pi$$
−0.0909768 + 0.995853i $$0.528999\pi$$
$$882$$ 0 0
$$883$$ 15476.0 0.589818 0.294909 0.955525i $$-0.404711\pi$$
0.294909 + 0.955525i $$0.404711\pi$$
$$884$$ 0 0
$$885$$ −8880.00 −0.337286
$$886$$ 0 0
$$887$$ 27440.0 1.03872 0.519360 0.854555i $$-0.326170\pi$$
0.519360 + 0.854555i $$0.326170\pi$$
$$888$$ 0 0
$$889$$ −48720.0 −1.83804
$$890$$ 0 0
$$891$$ 1296.00 0.0487291
$$892$$ 0 0
$$893$$ 1856.00 0.0695506
$$894$$ 0 0
$$895$$ −13640.0 −0.509424
$$896$$ 0 0
$$897$$ −13920.0 −0.518144
$$898$$ 0 0
$$899$$ −656.000 −0.0243368
$$900$$ 0 0
$$901$$ 26676.0 0.986356
$$902$$ 0 0
$$903$$ 31440.0 1.15865
$$904$$ 0 0
$$905$$ 20450.0 0.751139
$$906$$ 0 0
$$907$$ −48924.0 −1.79106 −0.895532 0.444997i $$-0.853205\pi$$
−0.895532 + 0.444997i $$0.853205\pi$$
$$908$$ 0 0
$$909$$ −5994.00 −0.218711
$$910$$ 0 0
$$911$$ 3440.00 0.125107 0.0625534 0.998042i $$-0.480076\pi$$
0.0625534 + 0.998042i $$0.480076\pi$$
$$912$$ 0 0
$$913$$ 2624.00 0.0951169
$$914$$ 0 0
$$915$$ −8610.00 −0.311080
$$916$$ 0 0
$$917$$ −50880.0 −1.83229
$$918$$ 0 0
$$919$$ 27184.0 0.975753 0.487877 0.872913i $$-0.337772\pi$$
0.487877 + 0.872913i $$0.337772\pi$$
$$920$$ 0 0
$$921$$ −3948.00 −0.141250
$$922$$ 0 0
$$923$$ 44544.0 1.58850
$$924$$ 0 0
$$925$$ −10650.0 −0.378562
$$926$$ 0 0
$$927$$ 9900.00 0.350764
$$928$$ 0 0
$$929$$ 42490.0 1.50059 0.750297 0.661101i $$-0.229911\pi$$
0.750297 + 0.661101i $$0.229911\pi$$
$$930$$ 0 0
$$931$$ 228.000 0.00802621
$$932$$ 0 0
$$933$$ 2496.00 0.0875835
$$934$$ 0 0
$$935$$ 3040.00 0.106330
$$936$$ 0 0
$$937$$ 37354.0 1.30235 0.651175 0.758928i $$-0.274276\pi$$
0.651175 + 0.758928i $$0.274276\pi$$
$$938$$ 0 0
$$939$$ 20310.0 0.705849
$$940$$ 0 0
$$941$$ 24470.0 0.847714 0.423857 0.905729i $$-0.360676\pi$$
0.423857 + 0.905729i $$0.360676\pi$$
$$942$$ 0 0
$$943$$ −19680.0 −0.679607
$$944$$ 0 0
$$945$$ −2700.00 −0.0929429
$$946$$ 0 0
$$947$$ −34100.0 −1.17012 −0.585059 0.810991i $$-0.698929\pi$$
−0.585059 + 0.810991i $$0.698929\pi$$
$$948$$ 0 0
$$949$$ 32364.0 1.10704
$$950$$ 0 0
$$951$$ 19746.0 0.673300
$$952$$ 0 0
$$953$$ 1878.00 0.0638346 0.0319173 0.999491i $$-0.489839\pi$$
0.0319173 + 0.999491i $$0.489839\pi$$
$$954$$ 0 0
$$955$$ 7400.00 0.250742
$$956$$ 0 0
$$957$$ −3936.00 −0.132950
$$958$$ 0 0
$$959$$ −13880.0 −0.467371
$$960$$ 0 0
$$961$$ −29727.0 −0.997852
$$962$$ 0 0
$$963$$ 10908.0 0.365011
$$964$$ 0 0
$$965$$ −8110.00 −0.270539
$$966$$ 0 0
$$967$$ −38484.0 −1.27980 −0.639898 0.768460i $$-0.721023\pi$$
−0.639898 + 0.768460i $$0.721023\pi$$
$$968$$ 0 0
$$969$$ 456.000 0.0151175
$$970$$ 0 0
$$971$$ 45272.0 1.49624 0.748119 0.663564i $$-0.230957\pi$$
0.748119 + 0.663564i $$0.230957\pi$$
$$972$$ 0 0
$$973$$ −10320.0 −0.340025
$$974$$ 0 0
$$975$$ −4350.00 −0.142884
$$976$$ 0 0
$$977$$ −25354.0 −0.830242 −0.415121 0.909766i $$-0.636261\pi$$
−0.415121 + 0.909766i $$0.636261\pi$$
$$978$$ 0 0
$$979$$ −8160.00 −0.266389
$$980$$ 0 0
$$981$$ −18702.0 −0.608674
$$982$$ 0 0
$$983$$ 18744.0 0.608180 0.304090 0.952643i $$-0.401648\pi$$
0.304090 + 0.952643i $$0.401648\pi$$
$$984$$ 0 0
$$985$$ −12650.0 −0.409201
$$986$$ 0 0
$$987$$ −27840.0 −0.897829
$$988$$ 0 0
$$989$$ −41920.0 −1.34780
$$990$$ 0 0
$$991$$ −59600.0 −1.91045 −0.955225 0.295880i $$-0.904387\pi$$
−0.955225 + 0.295880i $$0.904387\pi$$
$$992$$ 0 0
$$993$$ 33876.0 1.08260
$$994$$ 0 0
$$995$$ 12200.0 0.388710
$$996$$ 0 0
$$997$$ 17886.0 0.568160 0.284080 0.958801i $$-0.408312\pi$$
0.284080 + 0.958801i $$0.408312\pi$$
$$998$$ 0 0
$$999$$ −11502.0 −0.364271
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 960.4.a.bd.1.1 1
4.3 odd 2 960.4.a.q.1.1 1
8.3 odd 2 120.4.a.e.1.1 1
8.5 even 2 240.4.a.a.1.1 1
24.5 odd 2 720.4.a.s.1.1 1
24.11 even 2 360.4.a.m.1.1 1
40.3 even 4 600.4.f.f.49.2 2
40.13 odd 4 1200.4.f.h.49.1 2
40.19 odd 2 600.4.a.a.1.1 1
40.27 even 4 600.4.f.f.49.1 2
40.29 even 2 1200.4.a.bj.1.1 1
40.37 odd 4 1200.4.f.h.49.2 2
120.59 even 2 1800.4.a.e.1.1 1
120.83 odd 4 1800.4.f.k.649.1 2
120.107 odd 4 1800.4.f.k.649.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.e.1.1 1 8.3 odd 2
240.4.a.a.1.1 1 8.5 even 2
360.4.a.m.1.1 1 24.11 even 2
600.4.a.a.1.1 1 40.19 odd 2
600.4.f.f.49.1 2 40.27 even 4
600.4.f.f.49.2 2 40.3 even 4
720.4.a.s.1.1 1 24.5 odd 2
960.4.a.q.1.1 1 4.3 odd 2
960.4.a.bd.1.1 1 1.1 even 1 trivial
1200.4.a.bj.1.1 1 40.29 even 2
1200.4.f.h.49.1 2 40.13 odd 4
1200.4.f.h.49.2 2 40.37 odd 4
1800.4.a.e.1.1 1 120.59 even 2
1800.4.f.k.649.1 2 120.83 odd 4
1800.4.f.k.649.2 2 120.107 odd 4