# Properties

 Label 960.4.a.j.1.1 Level $960$ Weight $4$ Character 960.1 Self dual yes Analytic conductor $56.642$ 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] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) 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)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -5.00000 q^{5} +32.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -5.00000 q^{5} +32.0000 q^{7} +9.00000 q^{9} +60.0000 q^{11} +34.0000 q^{13} +15.0000 q^{15} +42.0000 q^{17} +76.0000 q^{19} -96.0000 q^{21} +25.0000 q^{25} -27.0000 q^{27} -6.00000 q^{29} -232.000 q^{31} -180.000 q^{33} -160.000 q^{35} -134.000 q^{37} -102.000 q^{39} +234.000 q^{41} +412.000 q^{43} -45.0000 q^{45} -360.000 q^{47} +681.000 q^{49} -126.000 q^{51} -222.000 q^{53} -300.000 q^{55} -228.000 q^{57} -660.000 q^{59} +490.000 q^{61} +288.000 q^{63} -170.000 q^{65} -812.000 q^{67} +120.000 q^{71} +746.000 q^{73} -75.0000 q^{75} +1920.00 q^{77} +152.000 q^{79} +81.0000 q^{81} +804.000 q^{83} -210.000 q^{85} +18.0000 q^{87} -678.000 q^{89} +1088.00 q^{91} +696.000 q^{93} -380.000 q^{95} +194.000 q^{97} +540.000 q^{99} +O(q^{100})$$

## 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
$$7$$ 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
$$11$$ 60.0000 1.64461 0.822304 0.569049i $$-0.192689\pi$$
0.822304 + 0.569049i $$0.192689\pi$$
$$12$$ 0 0
$$13$$ 34.0000 0.725377 0.362689 0.931910i $$-0.381859\pi$$
0.362689 + 0.931910i $$0.381859\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ 0 0
$$17$$ 42.0000 0.599206 0.299603 0.954064i $$-0.403146\pi$$
0.299603 + 0.954064i $$0.403146\pi$$
$$18$$ 0 0
$$19$$ 76.0000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ −96.0000 −0.997567
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ −6.00000 −0.0384197 −0.0192099 0.999815i $$-0.506115\pi$$
−0.0192099 + 0.999815i $$0.506115\pi$$
$$30$$ 0 0
$$31$$ −232.000 −1.34414 −0.672071 0.740486i $$-0.734595\pi$$
−0.672071 + 0.740486i $$0.734595\pi$$
$$32$$ 0 0
$$33$$ −180.000 −0.949514
$$34$$ 0 0
$$35$$ −160.000 −0.772712
$$36$$ 0 0
$$37$$ −134.000 −0.595391 −0.297695 0.954661i $$-0.596218\pi$$
−0.297695 + 0.954661i $$0.596218\pi$$
$$38$$ 0 0
$$39$$ −102.000 −0.418797
$$40$$ 0 0
$$41$$ 234.000 0.891333 0.445667 0.895199i $$-0.352967\pi$$
0.445667 + 0.895199i $$0.352967\pi$$
$$42$$ 0 0
$$43$$ 412.000 1.46115 0.730575 0.682833i $$-0.239252\pi$$
0.730575 + 0.682833i $$0.239252\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 0 0
$$47$$ −360.000 −1.11726 −0.558632 0.829416i $$-0.688674\pi$$
−0.558632 + 0.829416i $$0.688674\pi$$
$$48$$ 0 0
$$49$$ 681.000 1.98542
$$50$$ 0 0
$$51$$ −126.000 −0.345952
$$52$$ 0 0
$$53$$ −222.000 −0.575359 −0.287680 0.957727i $$-0.592884\pi$$
−0.287680 + 0.957727i $$0.592884\pi$$
$$54$$ 0 0
$$55$$ −300.000 −0.735491
$$56$$ 0 0
$$57$$ −228.000 −0.529813
$$58$$ 0 0
$$59$$ −660.000 −1.45635 −0.728175 0.685391i $$-0.759631\pi$$
−0.728175 + 0.685391i $$0.759631\pi$$
$$60$$ 0 0
$$61$$ 490.000 1.02849 0.514246 0.857642i $$-0.328072\pi$$
0.514246 + 0.857642i $$0.328072\pi$$
$$62$$ 0 0
$$63$$ 288.000 0.575946
$$64$$ 0 0
$$65$$ −170.000 −0.324399
$$66$$ 0 0
$$67$$ −812.000 −1.48062 −0.740310 0.672265i $$-0.765321\pi$$
−0.740310 + 0.672265i $$0.765321\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 120.000 0.200583 0.100291 0.994958i $$-0.468022\pi$$
0.100291 + 0.994958i $$0.468022\pi$$
$$72$$ 0 0
$$73$$ 746.000 1.19606 0.598032 0.801472i $$-0.295949\pi$$
0.598032 + 0.801472i $$0.295949\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ 1920.00 2.84161
$$78$$ 0 0
$$79$$ 152.000 0.216473 0.108236 0.994125i $$-0.465480\pi$$
0.108236 + 0.994125i $$0.465480\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 804.000 1.06326 0.531629 0.846977i $$-0.321580\pi$$
0.531629 + 0.846977i $$0.321580\pi$$
$$84$$ 0 0
$$85$$ −210.000 −0.267973
$$86$$ 0 0
$$87$$ 18.0000 0.0221816
$$88$$ 0 0
$$89$$ −678.000 −0.807504 −0.403752 0.914868i $$-0.632294\pi$$
−0.403752 + 0.914868i $$0.632294\pi$$
$$90$$ 0 0
$$91$$ 1088.00 1.25333
$$92$$ 0 0
$$93$$ 696.000 0.776041
$$94$$ 0 0
$$95$$ −380.000 −0.410391
$$96$$ 0 0
$$97$$ 194.000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 540.000 0.548202
$$100$$ 0 0
$$101$$ −798.000 −0.786178 −0.393089 0.919500i $$-0.628594\pi$$
−0.393089 + 0.919500i $$0.628594\pi$$
$$102$$ 0 0
$$103$$ 1088.00 1.04081 0.520407 0.853918i $$-0.325780\pi$$
0.520407 + 0.853918i $$0.325780\pi$$
$$104$$ 0 0
$$105$$ 480.000 0.446126
$$106$$ 0 0
$$107$$ −1716.00 −1.55039 −0.775196 0.631721i $$-0.782349\pi$$
−0.775196 + 0.631721i $$0.782349\pi$$
$$108$$ 0 0
$$109$$ 970.000 0.852378 0.426189 0.904634i $$-0.359856\pi$$
0.426189 + 0.904634i $$0.359856\pi$$
$$110$$ 0 0
$$111$$ 402.000 0.343749
$$112$$ 0 0
$$113$$ 426.000 0.354643 0.177322 0.984153i $$-0.443257\pi$$
0.177322 + 0.984153i $$0.443257\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 306.000 0.241792
$$118$$ 0 0
$$119$$ 1344.00 1.03533
$$120$$ 0 0
$$121$$ 2269.00 1.70473
$$122$$ 0 0
$$123$$ −702.000 −0.514611
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 200.000 0.139741 0.0698706 0.997556i $$-0.477741\pi$$
0.0698706 + 0.997556i $$0.477741\pi$$
$$128$$ 0 0
$$129$$ −1236.00 −0.843595
$$130$$ 0 0
$$131$$ −60.0000 −0.0400170 −0.0200085 0.999800i $$-0.506369\pi$$
−0.0200085 + 0.999800i $$0.506369\pi$$
$$132$$ 0 0
$$133$$ 2432.00 1.58557
$$134$$ 0 0
$$135$$ 135.000 0.0860663
$$136$$ 0 0
$$137$$ 642.000 0.400363 0.200182 0.979759i $$-0.435847\pi$$
0.200182 + 0.979759i $$0.435847\pi$$
$$138$$ 0 0
$$139$$ 2836.00 1.73055 0.865275 0.501298i $$-0.167144\pi$$
0.865275 + 0.501298i $$0.167144\pi$$
$$140$$ 0 0
$$141$$ 1080.00 0.645053
$$142$$ 0 0
$$143$$ 2040.00 1.19296
$$144$$ 0 0
$$145$$ 30.0000 0.0171818
$$146$$ 0 0
$$147$$ −2043.00 −1.14628
$$148$$ 0 0
$$149$$ 1554.00 0.854420 0.427210 0.904152i $$-0.359496\pi$$
0.427210 + 0.904152i $$0.359496\pi$$
$$150$$ 0 0
$$151$$ −2272.00 −1.22446 −0.612228 0.790682i $$-0.709726\pi$$
−0.612228 + 0.790682i $$0.709726\pi$$
$$152$$ 0 0
$$153$$ 378.000 0.199735
$$154$$ 0 0
$$155$$ 1160.00 0.601119
$$156$$ 0 0
$$157$$ −1694.00 −0.861120 −0.430560 0.902562i $$-0.641684\pi$$
−0.430560 + 0.902562i $$0.641684\pi$$
$$158$$ 0 0
$$159$$ 666.000 0.332184
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 52.0000 0.0249874 0.0124937 0.999922i $$-0.496023\pi$$
0.0124937 + 0.999922i $$0.496023\pi$$
$$164$$ 0 0
$$165$$ 900.000 0.424636
$$166$$ 0 0
$$167$$ −1200.00 −0.556041 −0.278020 0.960575i $$-0.589678\pi$$
−0.278020 + 0.960575i $$0.589678\pi$$
$$168$$ 0 0
$$169$$ −1041.00 −0.473828
$$170$$ 0 0
$$171$$ 684.000 0.305888
$$172$$ 0 0
$$173$$ −54.0000 −0.0237315 −0.0118657 0.999930i $$-0.503777\pi$$
−0.0118657 + 0.999930i $$0.503777\pi$$
$$174$$ 0 0
$$175$$ 800.000 0.345568
$$176$$ 0 0
$$177$$ 1980.00 0.840824
$$178$$ 0 0
$$179$$ −876.000 −0.365784 −0.182892 0.983133i $$-0.558546\pi$$
−0.182892 + 0.983133i $$0.558546\pi$$
$$180$$ 0 0
$$181$$ −3854.00 −1.58268 −0.791341 0.611375i $$-0.790617\pi$$
−0.791341 + 0.611375i $$0.790617\pi$$
$$182$$ 0 0
$$183$$ −1470.00 −0.593801
$$184$$ 0 0
$$185$$ 670.000 0.266267
$$186$$ 0 0
$$187$$ 2520.00 0.985458
$$188$$ 0 0
$$189$$ −864.000 −0.332522
$$190$$ 0 0
$$191$$ −2784.00 −1.05468 −0.527338 0.849656i $$-0.676810\pi$$
−0.527338 + 0.849656i $$0.676810\pi$$
$$192$$ 0 0
$$193$$ 914.000 0.340887 0.170443 0.985367i $$-0.445480\pi$$
0.170443 + 0.985367i $$0.445480\pi$$
$$194$$ 0 0
$$195$$ 510.000 0.187292
$$196$$ 0 0
$$197$$ 5202.00 1.88136 0.940678 0.339300i $$-0.110190\pi$$
0.940678 + 0.339300i $$0.110190\pi$$
$$198$$ 0 0
$$199$$ 3152.00 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$200$$ 0 0
$$201$$ 2436.00 0.854837
$$202$$ 0 0
$$203$$ −192.000 −0.0663830
$$204$$ 0 0
$$205$$ −1170.00 −0.398616
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 4560.00 1.50920
$$210$$ 0 0
$$211$$ −740.000 −0.241439 −0.120720 0.992687i $$-0.538520\pi$$
−0.120720 + 0.992687i $$0.538520\pi$$
$$212$$ 0 0
$$213$$ −360.000 −0.115807
$$214$$ 0 0
$$215$$ −2060.00 −0.653446
$$216$$ 0 0
$$217$$ −7424.00 −2.32246
$$218$$ 0 0
$$219$$ −2238.00 −0.690548
$$220$$ 0 0
$$221$$ 1428.00 0.434650
$$222$$ 0 0
$$223$$ −520.000 −0.156151 −0.0780757 0.996947i $$-0.524878\pi$$
−0.0780757 + 0.996947i $$0.524878\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −396.000 −0.115786 −0.0578930 0.998323i $$-0.518438\pi$$
−0.0578930 + 0.998323i $$0.518438\pi$$
$$228$$ 0 0
$$229$$ 1330.00 0.383794 0.191897 0.981415i $$-0.438536\pi$$
0.191897 + 0.981415i $$0.438536\pi$$
$$230$$ 0 0
$$231$$ −5760.00 −1.64061
$$232$$ 0 0
$$233$$ 4866.00 1.36816 0.684082 0.729405i $$-0.260203\pi$$
0.684082 + 0.729405i $$0.260203\pi$$
$$234$$ 0 0
$$235$$ 1800.00 0.499656
$$236$$ 0 0
$$237$$ −456.000 −0.124981
$$238$$ 0 0
$$239$$ −1824.00 −0.493660 −0.246830 0.969059i $$-0.579389\pi$$
−0.246830 + 0.969059i $$0.579389\pi$$
$$240$$ 0 0
$$241$$ 6482.00 1.73254 0.866270 0.499575i $$-0.166511\pi$$
0.866270 + 0.499575i $$0.166511\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ −3405.00 −0.887908
$$246$$ 0 0
$$247$$ 2584.00 0.665652
$$248$$ 0 0
$$249$$ −2412.00 −0.613873
$$250$$ 0 0
$$251$$ −1476.00 −0.371172 −0.185586 0.982628i $$-0.559418\pi$$
−0.185586 + 0.982628i $$0.559418\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 630.000 0.154714
$$256$$ 0 0
$$257$$ 4314.00 1.04708 0.523541 0.852001i $$-0.324611\pi$$
0.523541 + 0.852001i $$0.324611\pi$$
$$258$$ 0 0
$$259$$ −4288.00 −1.02874
$$260$$ 0 0
$$261$$ −54.0000 −0.0128066
$$262$$ 0 0
$$263$$ −5280.00 −1.23794 −0.618971 0.785414i $$-0.712450\pi$$
−0.618971 + 0.785414i $$0.712450\pi$$
$$264$$ 0 0
$$265$$ 1110.00 0.257309
$$266$$ 0 0
$$267$$ 2034.00 0.466213
$$268$$ 0 0
$$269$$ −5526.00 −1.25251 −0.626257 0.779617i $$-0.715414\pi$$
−0.626257 + 0.779617i $$0.715414\pi$$
$$270$$ 0 0
$$271$$ 2024.00 0.453687 0.226844 0.973931i $$-0.427159\pi$$
0.226844 + 0.973931i $$0.427159\pi$$
$$272$$ 0 0
$$273$$ −3264.00 −0.723613
$$274$$ 0 0
$$275$$ 1500.00 0.328921
$$276$$ 0 0
$$277$$ −2054.00 −0.445534 −0.222767 0.974872i $$-0.571509\pi$$
−0.222767 + 0.974872i $$0.571509\pi$$
$$278$$ 0 0
$$279$$ −2088.00 −0.448048
$$280$$ 0 0
$$281$$ −7302.00 −1.55018 −0.775090 0.631850i $$-0.782296\pi$$
−0.775090 + 0.631850i $$0.782296\pi$$
$$282$$ 0 0
$$283$$ 3724.00 0.782222 0.391111 0.920344i $$-0.372091\pi$$
0.391111 + 0.920344i $$0.372091\pi$$
$$284$$ 0 0
$$285$$ 1140.00 0.236940
$$286$$ 0 0
$$287$$ 7488.00 1.54008
$$288$$ 0 0
$$289$$ −3149.00 −0.640953
$$290$$ 0 0
$$291$$ −582.000 −0.117242
$$292$$ 0 0
$$293$$ 7218.00 1.43918 0.719591 0.694399i $$-0.244330\pi$$
0.719591 + 0.694399i $$0.244330\pi$$
$$294$$ 0 0
$$295$$ 3300.00 0.651300
$$296$$ 0 0
$$297$$ −1620.00 −0.316505
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 13184.0 2.52463
$$302$$ 0 0
$$303$$ 2394.00 0.453900
$$304$$ 0 0
$$305$$ −2450.00 −0.459956
$$306$$ 0 0
$$307$$ −2540.00 −0.472200 −0.236100 0.971729i $$-0.575869\pi$$
−0.236100 + 0.971729i $$0.575869\pi$$
$$308$$ 0 0
$$309$$ −3264.00 −0.600914
$$310$$ 0 0
$$311$$ 1560.00 0.284436 0.142218 0.989835i $$-0.454577\pi$$
0.142218 + 0.989835i $$0.454577\pi$$
$$312$$ 0 0
$$313$$ −934.000 −0.168667 −0.0843335 0.996438i $$-0.526876\pi$$
−0.0843335 + 0.996438i $$0.526876\pi$$
$$314$$ 0 0
$$315$$ −1440.00 −0.257571
$$316$$ 0 0
$$317$$ 1674.00 0.296597 0.148298 0.988943i $$-0.452620\pi$$
0.148298 + 0.988943i $$0.452620\pi$$
$$318$$ 0 0
$$319$$ −360.000 −0.0631854
$$320$$ 0 0
$$321$$ 5148.00 0.895119
$$322$$ 0 0
$$323$$ 3192.00 0.549869
$$324$$ 0 0
$$325$$ 850.000 0.145075
$$326$$ 0 0
$$327$$ −2910.00 −0.492120
$$328$$ 0 0
$$329$$ −11520.0 −1.93045
$$330$$ 0 0
$$331$$ 3988.00 0.662237 0.331118 0.943589i $$-0.392574\pi$$
0.331118 + 0.943589i $$0.392574\pi$$
$$332$$ 0 0
$$333$$ −1206.00 −0.198464
$$334$$ 0 0
$$335$$ 4060.00 0.662154
$$336$$ 0 0
$$337$$ 2.00000 0.000323285 0 0.000161642 1.00000i $$-0.499949\pi$$
0.000161642 1.00000i $$0.499949\pi$$
$$338$$ 0 0
$$339$$ −1278.00 −0.204753
$$340$$ 0 0
$$341$$ −13920.0 −2.21059
$$342$$ 0 0
$$343$$ 10816.0 1.70265
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −1764.00 −0.272901 −0.136450 0.990647i $$-0.543569\pi$$
−0.136450 + 0.990647i $$0.543569\pi$$
$$348$$ 0 0
$$349$$ −4310.00 −0.661057 −0.330529 0.943796i $$-0.607227\pi$$
−0.330529 + 0.943796i $$0.607227\pi$$
$$350$$ 0 0
$$351$$ −918.000 −0.139599
$$352$$ 0 0
$$353$$ 138.000 0.0208074 0.0104037 0.999946i $$-0.496688\pi$$
0.0104037 + 0.999946i $$0.496688\pi$$
$$354$$ 0 0
$$355$$ −600.000 −0.0897034
$$356$$ 0 0
$$357$$ −4032.00 −0.597748
$$358$$ 0 0
$$359$$ −11976.0 −1.76064 −0.880319 0.474382i $$-0.842672\pi$$
−0.880319 + 0.474382i $$0.842672\pi$$
$$360$$ 0 0
$$361$$ −1083.00 −0.157895
$$362$$ 0 0
$$363$$ −6807.00 −0.984228
$$364$$ 0 0
$$365$$ −3730.00 −0.534896
$$366$$ 0 0
$$367$$ 9704.00 1.38023 0.690115 0.723699i $$-0.257560\pi$$
0.690115 + 0.723699i $$0.257560\pi$$
$$368$$ 0 0
$$369$$ 2106.00 0.297111
$$370$$ 0 0
$$371$$ −7104.00 −0.994128
$$372$$ 0 0
$$373$$ 8122.00 1.12746 0.563728 0.825960i $$-0.309367\pi$$
0.563728 + 0.825960i $$0.309367\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 0 0
$$377$$ −204.000 −0.0278688
$$378$$ 0 0
$$379$$ −3404.00 −0.461350 −0.230675 0.973031i $$-0.574093\pi$$
−0.230675 + 0.973031i $$0.574093\pi$$
$$380$$ 0 0
$$381$$ −600.000 −0.0806796
$$382$$ 0 0
$$383$$ −2520.00 −0.336204 −0.168102 0.985770i $$-0.553764\pi$$
−0.168102 + 0.985770i $$0.553764\pi$$
$$384$$ 0 0
$$385$$ −9600.00 −1.27081
$$386$$ 0 0
$$387$$ 3708.00 0.487050
$$388$$ 0 0
$$389$$ −1566.00 −0.204111 −0.102056 0.994779i $$-0.532542\pi$$
−0.102056 + 0.994779i $$0.532542\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 180.000 0.0231038
$$394$$ 0 0
$$395$$ −760.000 −0.0968095
$$396$$ 0 0
$$397$$ 4354.00 0.550431 0.275215 0.961383i $$-0.411251\pi$$
0.275215 + 0.961383i $$0.411251\pi$$
$$398$$ 0 0
$$399$$ −7296.00 −0.915431
$$400$$ 0 0
$$401$$ −8046.00 −1.00199 −0.500995 0.865450i $$-0.667033\pi$$
−0.500995 + 0.865450i $$0.667033\pi$$
$$402$$ 0 0
$$403$$ −7888.00 −0.975011
$$404$$ 0 0
$$405$$ −405.000 −0.0496904
$$406$$ 0 0
$$407$$ −8040.00 −0.979184
$$408$$ 0 0
$$409$$ −2806.00 −0.339237 −0.169618 0.985510i $$-0.554253\pi$$
−0.169618 + 0.985510i $$0.554253\pi$$
$$410$$ 0 0
$$411$$ −1926.00 −0.231150
$$412$$ 0 0
$$413$$ −21120.0 −2.51634
$$414$$ 0 0
$$415$$ −4020.00 −0.475504
$$416$$ 0 0
$$417$$ −8508.00 −0.999133
$$418$$ 0 0
$$419$$ −11580.0 −1.35017 −0.675084 0.737741i $$-0.735892\pi$$
−0.675084 + 0.737741i $$0.735892\pi$$
$$420$$ 0 0
$$421$$ 370.000 0.0428330 0.0214165 0.999771i $$-0.493182\pi$$
0.0214165 + 0.999771i $$0.493182\pi$$
$$422$$ 0 0
$$423$$ −3240.00 −0.372421
$$424$$ 0 0
$$425$$ 1050.00 0.119841
$$426$$ 0 0
$$427$$ 15680.0 1.77707
$$428$$ 0 0
$$429$$ −6120.00 −0.688756
$$430$$ 0 0
$$431$$ 5040.00 0.563267 0.281634 0.959522i $$-0.409124\pi$$
0.281634 + 0.959522i $$0.409124\pi$$
$$432$$ 0 0
$$433$$ −3742.00 −0.415310 −0.207655 0.978202i $$-0.566583\pi$$
−0.207655 + 0.978202i $$0.566583\pi$$
$$434$$ 0 0
$$435$$ −90.0000 −0.00991993
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −6208.00 −0.674924 −0.337462 0.941339i $$-0.609568\pi$$
−0.337462 + 0.941339i $$0.609568\pi$$
$$440$$ 0 0
$$441$$ 6129.00 0.661808
$$442$$ 0 0
$$443$$ 15564.0 1.66923 0.834614 0.550835i $$-0.185691\pi$$
0.834614 + 0.550835i $$0.185691\pi$$
$$444$$ 0 0
$$445$$ 3390.00 0.361127
$$446$$ 0 0
$$447$$ −4662.00 −0.493300
$$448$$ 0 0
$$449$$ −15774.0 −1.65795 −0.828977 0.559283i $$-0.811076\pi$$
−0.828977 + 0.559283i $$0.811076\pi$$
$$450$$ 0 0
$$451$$ 14040.0 1.46589
$$452$$ 0 0
$$453$$ 6816.00 0.706940
$$454$$ 0 0
$$455$$ −5440.00 −0.560508
$$456$$ 0 0
$$457$$ 9722.00 0.995133 0.497567 0.867426i $$-0.334227\pi$$
0.497567 + 0.867426i $$0.334227\pi$$
$$458$$ 0 0
$$459$$ −1134.00 −0.115317
$$460$$ 0 0
$$461$$ 10890.0 1.10021 0.550106 0.835095i $$-0.314587\pi$$
0.550106 + 0.835095i $$0.314587\pi$$
$$462$$ 0 0
$$463$$ 15128.0 1.51848 0.759242 0.650809i $$-0.225570\pi$$
0.759242 + 0.650809i $$0.225570\pi$$
$$464$$ 0 0
$$465$$ −3480.00 −0.347056
$$466$$ 0 0
$$467$$ −10668.0 −1.05708 −0.528540 0.848909i $$-0.677260\pi$$
−0.528540 + 0.848909i $$0.677260\pi$$
$$468$$ 0 0
$$469$$ −25984.0 −2.55827
$$470$$ 0 0
$$471$$ 5082.00 0.497168
$$472$$ 0 0
$$473$$ 24720.0 2.40302
$$474$$ 0 0
$$475$$ 1900.00 0.183533
$$476$$ 0 0
$$477$$ −1998.00 −0.191786
$$478$$ 0 0
$$479$$ 15264.0 1.45601 0.728006 0.685571i $$-0.240447\pi$$
0.728006 + 0.685571i $$0.240447\pi$$
$$480$$ 0 0
$$481$$ −4556.00 −0.431883
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −970.000 −0.0908153
$$486$$ 0 0
$$487$$ −5776.00 −0.537445 −0.268722 0.963218i $$-0.586601\pi$$
−0.268722 + 0.963218i $$0.586601\pi$$
$$488$$ 0 0
$$489$$ −156.000 −0.0144265
$$490$$ 0 0
$$491$$ −14244.0 −1.30921 −0.654606 0.755971i $$-0.727165\pi$$
−0.654606 + 0.755971i $$0.727165\pi$$
$$492$$ 0 0
$$493$$ −252.000 −0.0230213
$$494$$ 0 0
$$495$$ −2700.00 −0.245164
$$496$$ 0 0
$$497$$ 3840.00 0.346575
$$498$$ 0 0
$$499$$ 17116.0 1.53551 0.767753 0.640746i $$-0.221375\pi$$
0.767753 + 0.640746i $$0.221375\pi$$
$$500$$ 0 0
$$501$$ 3600.00 0.321030
$$502$$ 0 0
$$503$$ −16848.0 −1.49347 −0.746735 0.665122i $$-0.768380\pi$$
−0.746735 + 0.665122i $$0.768380\pi$$
$$504$$ 0 0
$$505$$ 3990.00 0.351589
$$506$$ 0 0
$$507$$ 3123.00 0.273565
$$508$$ 0 0
$$509$$ 3834.00 0.333868 0.166934 0.985968i $$-0.446613\pi$$
0.166934 + 0.985968i $$0.446613\pi$$
$$510$$ 0 0
$$511$$ 23872.0 2.06660
$$512$$ 0 0
$$513$$ −2052.00 −0.176604
$$514$$ 0 0
$$515$$ −5440.00 −0.465466
$$516$$ 0 0
$$517$$ −21600.0 −1.83746
$$518$$ 0 0
$$519$$ 162.000 0.0137014
$$520$$ 0 0
$$521$$ −18822.0 −1.58274 −0.791369 0.611338i $$-0.790631\pi$$
−0.791369 + 0.611338i $$0.790631\pi$$
$$522$$ 0 0
$$523$$ 15340.0 1.28255 0.641273 0.767313i $$-0.278407\pi$$
0.641273 + 0.767313i $$0.278407\pi$$
$$524$$ 0 0
$$525$$ −2400.00 −0.199513
$$526$$ 0 0
$$527$$ −9744.00 −0.805418
$$528$$ 0 0
$$529$$ −12167.0 −1.00000
$$530$$ 0 0
$$531$$ −5940.00 −0.485450
$$532$$ 0 0
$$533$$ 7956.00 0.646553
$$534$$ 0 0
$$535$$ 8580.00 0.693357
$$536$$ 0 0
$$537$$ 2628.00 0.211185
$$538$$ 0 0
$$539$$ 40860.0 3.26524
$$540$$ 0 0
$$541$$ −18950.0 −1.50596 −0.752980 0.658044i $$-0.771384\pi$$
−0.752980 + 0.658044i $$0.771384\pi$$
$$542$$ 0 0
$$543$$ 11562.0 0.913762
$$544$$ 0 0
$$545$$ −4850.00 −0.381195
$$546$$ 0 0
$$547$$ 10036.0 0.784476 0.392238 0.919864i $$-0.371701\pi$$
0.392238 + 0.919864i $$0.371701\pi$$
$$548$$ 0 0
$$549$$ 4410.00 0.342831
$$550$$ 0 0
$$551$$ −456.000 −0.0352564
$$552$$ 0 0
$$553$$ 4864.00 0.374030
$$554$$ 0 0
$$555$$ −2010.00 −0.153729
$$556$$ 0 0
$$557$$ −10326.0 −0.785506 −0.392753 0.919644i $$-0.628477\pi$$
−0.392753 + 0.919644i $$0.628477\pi$$
$$558$$ 0 0
$$559$$ 14008.0 1.05988
$$560$$ 0 0
$$561$$ −7560.00 −0.568954
$$562$$ 0 0
$$563$$ −4524.00 −0.338657 −0.169328 0.985560i $$-0.554160\pi$$
−0.169328 + 0.985560i $$0.554160\pi$$
$$564$$ 0 0
$$565$$ −2130.00 −0.158601
$$566$$ 0 0
$$567$$ 2592.00 0.191982
$$568$$ 0 0
$$569$$ 16362.0 1.20550 0.602751 0.797929i $$-0.294071\pi$$
0.602751 + 0.797929i $$0.294071\pi$$
$$570$$ 0 0
$$571$$ −6620.00 −0.485181 −0.242591 0.970129i $$-0.577997\pi$$
−0.242591 + 0.970129i $$0.577997\pi$$
$$572$$ 0 0
$$573$$ 8352.00 0.608918
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 8834.00 0.637373 0.318687 0.947860i $$-0.396758\pi$$
0.318687 + 0.947860i $$0.396758\pi$$
$$578$$ 0 0
$$579$$ −2742.00 −0.196811
$$580$$ 0 0
$$581$$ 25728.0 1.83714
$$582$$ 0 0
$$583$$ −13320.0 −0.946240
$$584$$ 0 0
$$585$$ −1530.00 −0.108133
$$586$$ 0 0
$$587$$ −3636.00 −0.255662 −0.127831 0.991796i $$-0.540802\pi$$
−0.127831 + 0.991796i $$0.540802\pi$$
$$588$$ 0 0
$$589$$ −17632.0 −1.23347
$$590$$ 0 0
$$591$$ −15606.0 −1.08620
$$592$$ 0 0
$$593$$ 6570.00 0.454971 0.227485 0.973782i $$-0.426950\pi$$
0.227485 + 0.973782i $$0.426950\pi$$
$$594$$ 0 0
$$595$$ −6720.00 −0.463014
$$596$$ 0 0
$$597$$ −9456.00 −0.648255
$$598$$ 0 0
$$599$$ 16584.0 1.13123 0.565613 0.824671i $$-0.308640\pi$$
0.565613 + 0.824671i $$0.308640\pi$$
$$600$$ 0 0
$$601$$ −502.000 −0.0340716 −0.0170358 0.999855i $$-0.505423\pi$$
−0.0170358 + 0.999855i $$0.505423\pi$$
$$602$$ 0 0
$$603$$ −7308.00 −0.493540
$$604$$ 0 0
$$605$$ −11345.0 −0.762380
$$606$$ 0 0
$$607$$ −18568.0 −1.24160 −0.620801 0.783969i $$-0.713192\pi$$
−0.620801 + 0.783969i $$0.713192\pi$$
$$608$$ 0 0
$$609$$ 576.000 0.0383263
$$610$$ 0 0
$$611$$ −12240.0 −0.810438
$$612$$ 0 0
$$613$$ 13114.0 0.864061 0.432031 0.901859i $$-0.357797\pi$$
0.432031 + 0.901859i $$0.357797\pi$$
$$614$$ 0 0
$$615$$ 3510.00 0.230141
$$616$$ 0 0
$$617$$ 5250.00 0.342556 0.171278 0.985223i $$-0.445210\pi$$
0.171278 + 0.985223i $$0.445210\pi$$
$$618$$ 0 0
$$619$$ 10804.0 0.701534 0.350767 0.936463i $$-0.385921\pi$$
0.350767 + 0.936463i $$0.385921\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −21696.0 −1.39524
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ −13680.0 −0.871334
$$628$$ 0 0
$$629$$ −5628.00 −0.356762
$$630$$ 0 0
$$631$$ −27088.0 −1.70896 −0.854482 0.519481i $$-0.826125\pi$$
−0.854482 + 0.519481i $$0.826125\pi$$
$$632$$ 0 0
$$633$$ 2220.00 0.139395
$$634$$ 0 0
$$635$$ −1000.00 −0.0624942
$$636$$ 0 0
$$637$$ 23154.0 1.44018
$$638$$ 0 0
$$639$$ 1080.00 0.0668609
$$640$$ 0 0
$$641$$ 18930.0 1.16644 0.583222 0.812313i $$-0.301792\pi$$
0.583222 + 0.812313i $$0.301792\pi$$
$$642$$ 0 0
$$643$$ −20108.0 −1.23325 −0.616627 0.787256i $$-0.711501\pi$$
−0.616627 + 0.787256i $$0.711501\pi$$
$$644$$ 0 0
$$645$$ 6180.00 0.377267
$$646$$ 0 0
$$647$$ −7152.00 −0.434581 −0.217291 0.976107i $$-0.569722\pi$$
−0.217291 + 0.976107i $$0.569722\pi$$
$$648$$ 0 0
$$649$$ −39600.0 −2.39512
$$650$$ 0 0
$$651$$ 22272.0 1.34087
$$652$$ 0 0
$$653$$ 31626.0 1.89528 0.947642 0.319333i $$-0.103459\pi$$
0.947642 + 0.319333i $$0.103459\pi$$
$$654$$ 0 0
$$655$$ 300.000 0.0178961
$$656$$ 0 0
$$657$$ 6714.00 0.398688
$$658$$ 0 0
$$659$$ −28092.0 −1.66056 −0.830280 0.557347i $$-0.811819\pi$$
−0.830280 + 0.557347i $$0.811819\pi$$
$$660$$ 0 0
$$661$$ 13186.0 0.775909 0.387955 0.921678i $$-0.373182\pi$$
0.387955 + 0.921678i $$0.373182\pi$$
$$662$$ 0 0
$$663$$ −4284.00 −0.250945
$$664$$ 0 0
$$665$$ −12160.0 −0.709090
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 1560.00 0.0901541
$$670$$ 0 0
$$671$$ 29400.0 1.69147
$$672$$ 0 0
$$673$$ 5138.00 0.294287 0.147144 0.989115i $$-0.452992\pi$$
0.147144 + 0.989115i $$0.452992\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ −6078.00 −0.345047 −0.172523 0.985005i $$-0.555192\pi$$
−0.172523 + 0.985005i $$0.555192\pi$$
$$678$$ 0 0
$$679$$ 6208.00 0.350871
$$680$$ 0 0
$$681$$ 1188.00 0.0668491
$$682$$ 0 0
$$683$$ −32244.0 −1.80642 −0.903208 0.429203i $$-0.858795\pi$$
−0.903208 + 0.429203i $$0.858795\pi$$
$$684$$ 0 0
$$685$$ −3210.00 −0.179048
$$686$$ 0 0
$$687$$ −3990.00 −0.221584
$$688$$ 0 0
$$689$$ −7548.00 −0.417353
$$690$$ 0 0
$$691$$ −4484.00 −0.246859 −0.123429 0.992353i $$-0.539389\pi$$
−0.123429 + 0.992353i $$0.539389\pi$$
$$692$$ 0 0
$$693$$ 17280.0 0.947205
$$694$$ 0 0
$$695$$ −14180.0 −0.773925
$$696$$ 0 0
$$697$$ 9828.00 0.534092
$$698$$ 0 0
$$699$$ −14598.0 −0.789910
$$700$$ 0 0
$$701$$ 30426.0 1.63934 0.819668 0.572839i $$-0.194158\pi$$
0.819668 + 0.572839i $$0.194158\pi$$
$$702$$ 0 0
$$703$$ −10184.0 −0.546368
$$704$$ 0 0
$$705$$ −5400.00 −0.288476
$$706$$ 0 0
$$707$$ −25536.0 −1.35839
$$708$$ 0 0
$$709$$ −13262.0 −0.702489 −0.351245 0.936284i $$-0.614241\pi$$
−0.351245 + 0.936284i $$0.614241\pi$$
$$710$$ 0 0
$$711$$ 1368.00 0.0721575
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −10200.0 −0.533508
$$716$$ 0 0
$$717$$ 5472.00 0.285015
$$718$$ 0 0
$$719$$ 13920.0 0.722014 0.361007 0.932563i $$-0.382433\pi$$
0.361007 + 0.932563i $$0.382433\pi$$
$$720$$ 0 0
$$721$$ 34816.0 1.79836
$$722$$ 0 0
$$723$$ −19446.0 −1.00028
$$724$$ 0 0
$$725$$ −150.000 −0.00768395
$$726$$ 0 0
$$727$$ −9376.00 −0.478317 −0.239159 0.970981i $$-0.576872\pi$$
−0.239159 + 0.970981i $$0.576872\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 17304.0 0.875529
$$732$$ 0 0
$$733$$ −6014.00 −0.303045 −0.151523 0.988454i $$-0.548418\pi$$
−0.151523 + 0.988454i $$0.548418\pi$$
$$734$$ 0 0
$$735$$ 10215.0 0.512634
$$736$$ 0 0
$$737$$ −48720.0 −2.43504
$$738$$ 0 0
$$739$$ 7468.00 0.371739 0.185869 0.982574i $$-0.440490\pi$$
0.185869 + 0.982574i $$0.440490\pi$$
$$740$$ 0 0
$$741$$ −7752.00 −0.384314
$$742$$ 0 0
$$743$$ 31248.0 1.54290 0.771452 0.636287i $$-0.219531\pi$$
0.771452 + 0.636287i $$0.219531\pi$$
$$744$$ 0 0
$$745$$ −7770.00 −0.382108
$$746$$ 0 0
$$747$$ 7236.00 0.354420
$$748$$ 0 0
$$749$$ −54912.0 −2.67883
$$750$$ 0 0
$$751$$ 32840.0 1.59567 0.797835 0.602875i $$-0.205978\pi$$
0.797835 + 0.602875i $$0.205978\pi$$
$$752$$ 0 0
$$753$$ 4428.00 0.214297
$$754$$ 0 0
$$755$$ 11360.0 0.547593
$$756$$ 0 0
$$757$$ 19066.0 0.915410 0.457705 0.889104i $$-0.348672\pi$$
0.457705 + 0.889104i $$0.348672\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6858.00 0.326678 0.163339 0.986570i $$-0.447773\pi$$
0.163339 + 0.986570i $$0.447773\pi$$
$$762$$ 0 0
$$763$$ 31040.0 1.47277
$$764$$ 0 0
$$765$$ −1890.00 −0.0893243
$$766$$ 0 0
$$767$$ −22440.0 −1.05640
$$768$$ 0 0
$$769$$ 22178.0 1.04000 0.519999 0.854167i $$-0.325932\pi$$
0.519999 + 0.854167i $$0.325932\pi$$
$$770$$ 0 0
$$771$$ −12942.0 −0.604533
$$772$$ 0 0
$$773$$ −14286.0 −0.664724 −0.332362 0.943152i $$-0.607846\pi$$
−0.332362 + 0.943152i $$0.607846\pi$$
$$774$$ 0 0
$$775$$ −5800.00 −0.268829
$$776$$ 0 0
$$777$$ 12864.0 0.593943
$$778$$ 0 0
$$779$$ 17784.0 0.817943
$$780$$ 0 0
$$781$$ 7200.00 0.329880
$$782$$ 0 0
$$783$$ 162.000 0.00739388
$$784$$ 0 0
$$785$$ 8470.00 0.385105
$$786$$ 0 0
$$787$$ 18868.0 0.854602 0.427301 0.904109i $$-0.359465\pi$$
0.427301 + 0.904109i $$0.359465\pi$$
$$788$$ 0 0
$$789$$ 15840.0 0.714726
$$790$$ 0 0
$$791$$ 13632.0 0.612766
$$792$$ 0 0
$$793$$ 16660.0 0.746045
$$794$$ 0 0
$$795$$ −3330.00 −0.148557
$$796$$ 0 0
$$797$$ 21690.0 0.963989 0.481994 0.876174i $$-0.339913\pi$$
0.481994 + 0.876174i $$0.339913\pi$$
$$798$$ 0 0
$$799$$ −15120.0 −0.669471
$$800$$ 0 0
$$801$$ −6102.00 −0.269168
$$802$$ 0 0
$$803$$ 44760.0 1.96706
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 16578.0 0.723139
$$808$$ 0 0
$$809$$ −24726.0 −1.07456 −0.537281 0.843404i $$-0.680548\pi$$
−0.537281 + 0.843404i $$0.680548\pi$$
$$810$$ 0 0
$$811$$ 2644.00 0.114480 0.0572401 0.998360i $$-0.481770\pi$$
0.0572401 + 0.998360i $$0.481770\pi$$
$$812$$ 0 0
$$813$$ −6072.00 −0.261936
$$814$$ 0 0
$$815$$ −260.000 −0.0111747
$$816$$ 0 0
$$817$$ 31312.0 1.34084
$$818$$ 0 0
$$819$$ 9792.00 0.417778
$$820$$ 0 0
$$821$$ 37842.0 1.60864 0.804321 0.594195i $$-0.202529\pi$$
0.804321 + 0.594195i $$0.202529\pi$$
$$822$$ 0 0
$$823$$ −880.000 −0.0372720 −0.0186360 0.999826i $$-0.505932\pi$$
−0.0186360 + 0.999826i $$0.505932\pi$$
$$824$$ 0 0
$$825$$ −4500.00 −0.189903
$$826$$ 0 0
$$827$$ 12876.0 0.541406 0.270703 0.962663i $$-0.412744\pi$$
0.270703 + 0.962663i $$0.412744\pi$$
$$828$$ 0 0
$$829$$ 25498.0 1.06825 0.534127 0.845404i $$-0.320641\pi$$
0.534127 + 0.845404i $$0.320641\pi$$
$$830$$ 0 0
$$831$$ 6162.00 0.257229
$$832$$ 0 0
$$833$$ 28602.0 1.18968
$$834$$ 0 0
$$835$$ 6000.00 0.248669
$$836$$ 0 0
$$837$$ 6264.00 0.258680
$$838$$ 0 0
$$839$$ −40584.0 −1.66998 −0.834991 0.550263i $$-0.814527\pi$$
−0.834991 + 0.550263i $$0.814527\pi$$
$$840$$ 0 0
$$841$$ −24353.0 −0.998524
$$842$$ 0 0
$$843$$ 21906.0 0.894997
$$844$$ 0 0
$$845$$ 5205.00 0.211902
$$846$$ 0 0
$$847$$ 72608.0 2.94550
$$848$$ 0 0
$$849$$ −11172.0 −0.451616
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 25738.0 1.03312 0.516561 0.856251i $$-0.327212\pi$$
0.516561 + 0.856251i $$0.327212\pi$$
$$854$$ 0 0
$$855$$ −3420.00 −0.136797
$$856$$ 0 0
$$857$$ 13314.0 0.530686 0.265343 0.964154i $$-0.414515\pi$$
0.265343 + 0.964154i $$0.414515\pi$$
$$858$$ 0 0
$$859$$ −24524.0 −0.974096 −0.487048 0.873375i $$-0.661926\pi$$
−0.487048 + 0.873375i $$0.661926\pi$$
$$860$$ 0 0
$$861$$ −22464.0 −0.889165
$$862$$ 0 0
$$863$$ 5592.00 0.220572 0.110286 0.993900i $$-0.464823\pi$$
0.110286 + 0.993900i $$0.464823\pi$$
$$864$$ 0 0
$$865$$ 270.000 0.0106130
$$866$$ 0 0
$$867$$ 9447.00 0.370054
$$868$$ 0 0
$$869$$ 9120.00 0.356012
$$870$$ 0 0
$$871$$ −27608.0 −1.07401
$$872$$ 0 0
$$873$$ 1746.00 0.0676897
$$874$$ 0 0
$$875$$ −4000.00 −0.154542
$$876$$ 0 0
$$877$$ 14386.0 0.553912 0.276956 0.960883i $$-0.410674\pi$$
0.276956 + 0.960883i $$0.410674\pi$$
$$878$$ 0 0
$$879$$ −21654.0 −0.830912
$$880$$ 0 0
$$881$$ 47106.0 1.80141 0.900705 0.434432i $$-0.143051\pi$$
0.900705 + 0.434432i $$0.143051\pi$$
$$882$$ 0 0
$$883$$ −51548.0 −1.96458 −0.982292 0.187354i $$-0.940009\pi$$
−0.982292 + 0.187354i $$0.940009\pi$$
$$884$$ 0 0
$$885$$ −9900.00 −0.376028
$$886$$ 0 0
$$887$$ 34080.0 1.29007 0.645036 0.764152i $$-0.276842\pi$$
0.645036 + 0.764152i $$0.276842\pi$$
$$888$$ 0 0
$$889$$ 6400.00 0.241450
$$890$$ 0 0
$$891$$ 4860.00 0.182734
$$892$$ 0 0
$$893$$ −27360.0 −1.02527
$$894$$ 0 0
$$895$$ 4380.00 0.163584
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1392.00 0.0516416
$$900$$ 0 0
$$901$$ −9324.00 −0.344759
$$902$$ 0 0
$$903$$ −39552.0 −1.45759
$$904$$ 0 0
$$905$$ 19270.0 0.707797
$$906$$ 0 0
$$907$$ −25748.0 −0.942611 −0.471306 0.881970i $$-0.656217\pi$$
−0.471306 + 0.881970i $$0.656217\pi$$
$$908$$ 0 0
$$909$$ −7182.00 −0.262059
$$910$$ 0 0
$$911$$ −24768.0 −0.900769 −0.450384 0.892835i $$-0.648713\pi$$
−0.450384 + 0.892835i $$0.648713\pi$$
$$912$$ 0 0
$$913$$ 48240.0 1.74864
$$914$$ 0 0
$$915$$ 7350.00 0.265556
$$916$$ 0 0
$$917$$ −1920.00 −0.0691428
$$918$$ 0 0
$$919$$ −31264.0 −1.12220 −0.561101 0.827747i $$-0.689622\pi$$
−0.561101 + 0.827747i $$0.689622\pi$$
$$920$$ 0 0
$$921$$ 7620.00 0.272625
$$922$$ 0 0
$$923$$ 4080.00 0.145498
$$924$$ 0 0
$$925$$ −3350.00 −0.119078
$$926$$ 0 0
$$927$$ 9792.00 0.346938
$$928$$ 0 0
$$929$$ −6174.00 −0.218043 −0.109022 0.994039i $$-0.534772\pi$$
−0.109022 + 0.994039i $$0.534772\pi$$
$$930$$ 0 0
$$931$$ 51756.0 1.82195
$$932$$ 0 0
$$933$$ −4680.00 −0.164219
$$934$$ 0 0
$$935$$ −12600.0 −0.440710
$$936$$ 0 0
$$937$$ 28922.0 1.00837 0.504184 0.863596i $$-0.331793\pi$$
0.504184 + 0.863596i $$0.331793\pi$$
$$938$$ 0 0
$$939$$ 2802.00 0.0973800
$$940$$ 0 0
$$941$$ −29238.0 −1.01289 −0.506446 0.862272i $$-0.669041\pi$$
−0.506446 + 0.862272i $$0.669041\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 4320.00 0.148709
$$946$$ 0 0
$$947$$ 2868.00 0.0984134 0.0492067 0.998789i $$-0.484331\pi$$
0.0492067 + 0.998789i $$0.484331\pi$$
$$948$$ 0 0
$$949$$ 25364.0 0.867598
$$950$$ 0 0
$$951$$ −5022.00 −0.171240
$$952$$ 0 0
$$953$$ 24018.0 0.816390 0.408195 0.912895i $$-0.366158\pi$$
0.408195 + 0.912895i $$0.366158\pi$$
$$954$$ 0 0
$$955$$ 13920.0 0.471666
$$956$$ 0 0
$$957$$ 1080.00 0.0364801
$$958$$ 0 0
$$959$$ 20544.0 0.691763
$$960$$ 0 0
$$961$$ 24033.0 0.806720
$$962$$ 0 0
$$963$$ −15444.0 −0.516797
$$964$$ 0 0
$$965$$ −4570.00 −0.152449
$$966$$ 0 0
$$967$$ 25712.0 0.855059 0.427530 0.904001i $$-0.359384\pi$$
0.427530 + 0.904001i $$0.359384\pi$$
$$968$$ 0 0
$$969$$ −9576.00 −0.317467
$$970$$ 0 0
$$971$$ 12396.0 0.409688 0.204844 0.978795i $$-0.434331\pi$$
0.204844 + 0.978795i $$0.434331\pi$$
$$972$$ 0 0
$$973$$ 90752.0 2.99011
$$974$$ 0 0
$$975$$ −2550.00 −0.0837593
$$976$$ 0 0
$$977$$ −46614.0 −1.52642 −0.763211 0.646150i $$-0.776378\pi$$
−0.763211 + 0.646150i $$0.776378\pi$$
$$978$$ 0 0
$$979$$ −40680.0 −1.32803
$$980$$ 0 0
$$981$$ 8730.00 0.284126
$$982$$ 0 0
$$983$$ −672.000 −0.0218041 −0.0109021 0.999941i $$-0.503470\pi$$
−0.0109021 + 0.999941i $$0.503470\pi$$
$$984$$ 0 0
$$985$$ −26010.0 −0.841368
$$986$$ 0 0
$$987$$ 34560.0 1.11455
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −38776.0 −1.24295 −0.621473 0.783435i $$-0.713466\pi$$
−0.621473 + 0.783435i $$0.713466\pi$$
$$992$$ 0 0
$$993$$ −11964.0 −0.382342
$$994$$ 0 0
$$995$$ −15760.0 −0.502136
$$996$$ 0 0
$$997$$ −30422.0 −0.966374 −0.483187 0.875517i $$-0.660521\pi$$
−0.483187 + 0.875517i $$0.660521\pi$$
$$998$$ 0 0
$$999$$ 3618.00 0.114583
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.j.1.1 1
4.3 odd 2 960.4.a.s.1.1 1
8.3 odd 2 240.4.a.c.1.1 1
8.5 even 2 30.4.a.a.1.1 1
24.5 odd 2 90.4.a.d.1.1 1
24.11 even 2 720.4.a.b.1.1 1
40.3 even 4 1200.4.f.u.49.1 2
40.13 odd 4 150.4.c.a.49.2 2
40.19 odd 2 1200.4.a.bk.1.1 1
40.27 even 4 1200.4.f.u.49.2 2
40.29 even 2 150.4.a.e.1.1 1
40.37 odd 4 150.4.c.a.49.1 2
56.13 odd 2 1470.4.a.a.1.1 1
72.5 odd 6 810.4.e.e.541.1 2
72.13 even 6 810.4.e.m.541.1 2
72.29 odd 6 810.4.e.e.271.1 2
72.61 even 6 810.4.e.m.271.1 2
120.29 odd 2 450.4.a.b.1.1 1
120.53 even 4 450.4.c.k.199.1 2
120.77 even 4 450.4.c.k.199.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
30.4.a.a.1.1 1 8.5 even 2
90.4.a.d.1.1 1 24.5 odd 2
150.4.a.e.1.1 1 40.29 even 2
150.4.c.a.49.1 2 40.37 odd 4
150.4.c.a.49.2 2 40.13 odd 4
240.4.a.c.1.1 1 8.3 odd 2
450.4.a.b.1.1 1 120.29 odd 2
450.4.c.k.199.1 2 120.53 even 4
450.4.c.k.199.2 2 120.77 even 4
720.4.a.b.1.1 1 24.11 even 2
810.4.e.e.271.1 2 72.29 odd 6
810.4.e.e.541.1 2 72.5 odd 6
810.4.e.m.271.1 2 72.61 even 6
810.4.e.m.541.1 2 72.13 even 6
960.4.a.j.1.1 1 1.1 even 1 trivial
960.4.a.s.1.1 1 4.3 odd 2
1200.4.a.bk.1.1 1 40.19 odd 2
1200.4.f.u.49.1 2 40.3 even 4
1200.4.f.u.49.2 2 40.27 even 4
1470.4.a.a.1.1 1 56.13 odd 2