# Properties

 Label 240.4.a.e.1.1 Level $240$ Weight $4$ Character 240.1 Self dual yes Analytic conductor $14.160$ 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] = [240,4,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, 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("240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 240.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: $$14.1604584014$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 240.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +5.00000 q^{5} +24.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +5.00000 q^{5} +24.0000 q^{7} +9.00000 q^{9} -52.0000 q^{11} +22.0000 q^{13} -15.0000 q^{15} -14.0000 q^{17} +20.0000 q^{19} -72.0000 q^{21} +168.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +230.000 q^{29} +288.000 q^{31} +156.000 q^{33} +120.000 q^{35} -34.0000 q^{37} -66.0000 q^{39} +122.000 q^{41} +188.000 q^{43} +45.0000 q^{45} -256.000 q^{47} +233.000 q^{49} +42.0000 q^{51} -338.000 q^{53} -260.000 q^{55} -60.0000 q^{57} -100.000 q^{59} +742.000 q^{61} +216.000 q^{63} +110.000 q^{65} +84.0000 q^{67} -504.000 q^{69} +328.000 q^{71} -38.0000 q^{73} -75.0000 q^{75} -1248.00 q^{77} +240.000 q^{79} +81.0000 q^{81} -1212.00 q^{83} -70.0000 q^{85} -690.000 q^{87} +330.000 q^{89} +528.000 q^{91} -864.000 q^{93} +100.000 q^{95} +866.000 q^{97} -468.000 q^{99} +O(q^{100})$$

## 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
$$7$$ 24.0000 1.29588 0.647939 0.761692i $$-0.275631\pi$$
0.647939 + 0.761692i $$0.275631\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −52.0000 −1.42533 −0.712663 0.701506i $$-0.752511\pi$$
−0.712663 + 0.701506i $$0.752511\pi$$
$$12$$ 0 0
$$13$$ 22.0000 0.469362 0.234681 0.972072i $$-0.424595\pi$$
0.234681 + 0.972072i $$0.424595\pi$$
$$14$$ 0 0
$$15$$ −15.0000 −0.258199
$$16$$ 0 0
$$17$$ −14.0000 −0.199735 −0.0998676 0.995001i $$-0.531842\pi$$
−0.0998676 + 0.995001i $$0.531842\pi$$
$$18$$ 0 0
$$19$$ 20.0000 0.241490 0.120745 0.992684i $$-0.461472\pi$$
0.120745 + 0.992684i $$0.461472\pi$$
$$20$$ 0 0
$$21$$ −72.0000 −0.748176
$$22$$ 0 0
$$23$$ 168.000 1.52306 0.761531 0.648129i $$-0.224448\pi$$
0.761531 + 0.648129i $$0.224448\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 230.000 1.47276 0.736378 0.676570i $$-0.236535\pi$$
0.736378 + 0.676570i $$0.236535\pi$$
$$30$$ 0 0
$$31$$ 288.000 1.66859 0.834296 0.551317i $$-0.185875\pi$$
0.834296 + 0.551317i $$0.185875\pi$$
$$32$$ 0 0
$$33$$ 156.000 0.822913
$$34$$ 0 0
$$35$$ 120.000 0.579534
$$36$$ 0 0
$$37$$ −34.0000 −0.151069 −0.0755347 0.997143i $$-0.524066\pi$$
−0.0755347 + 0.997143i $$0.524066\pi$$
$$38$$ 0 0
$$39$$ −66.0000 −0.270986
$$40$$ 0 0
$$41$$ 122.000 0.464712 0.232356 0.972631i $$-0.425357\pi$$
0.232356 + 0.972631i $$0.425357\pi$$
$$42$$ 0 0
$$43$$ 188.000 0.666738 0.333369 0.942796i $$-0.391815\pi$$
0.333369 + 0.942796i $$0.391815\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ 0 0
$$47$$ −256.000 −0.794499 −0.397249 0.917711i $$-0.630035\pi$$
−0.397249 + 0.917711i $$0.630035\pi$$
$$48$$ 0 0
$$49$$ 233.000 0.679300
$$50$$ 0 0
$$51$$ 42.0000 0.115317
$$52$$ 0 0
$$53$$ −338.000 −0.875998 −0.437999 0.898976i $$-0.644313\pi$$
−0.437999 + 0.898976i $$0.644313\pi$$
$$54$$ 0 0
$$55$$ −260.000 −0.637425
$$56$$ 0 0
$$57$$ −60.0000 −0.139424
$$58$$ 0 0
$$59$$ −100.000 −0.220659 −0.110330 0.993895i $$-0.535191\pi$$
−0.110330 + 0.993895i $$0.535191\pi$$
$$60$$ 0 0
$$61$$ 742.000 1.55743 0.778716 0.627376i $$-0.215871\pi$$
0.778716 + 0.627376i $$0.215871\pi$$
$$62$$ 0 0
$$63$$ 216.000 0.431959
$$64$$ 0 0
$$65$$ 110.000 0.209905
$$66$$ 0 0
$$67$$ 84.0000 0.153168 0.0765838 0.997063i $$-0.475599\pi$$
0.0765838 + 0.997063i $$0.475599\pi$$
$$68$$ 0 0
$$69$$ −504.000 −0.879340
$$70$$ 0 0
$$71$$ 328.000 0.548260 0.274130 0.961693i $$-0.411610\pi$$
0.274130 + 0.961693i $$0.411610\pi$$
$$72$$ 0 0
$$73$$ −38.0000 −0.0609255 −0.0304628 0.999536i $$-0.509698\pi$$
−0.0304628 + 0.999536i $$0.509698\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ −1248.00 −1.84705
$$78$$ 0 0
$$79$$ 240.000 0.341799 0.170899 0.985288i $$-0.445333\pi$$
0.170899 + 0.985288i $$0.445333\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −1212.00 −1.60282 −0.801411 0.598114i $$-0.795917\pi$$
−0.801411 + 0.598114i $$0.795917\pi$$
$$84$$ 0 0
$$85$$ −70.0000 −0.0893243
$$86$$ 0 0
$$87$$ −690.000 −0.850296
$$88$$ 0 0
$$89$$ 330.000 0.393033 0.196516 0.980501i $$-0.437037\pi$$
0.196516 + 0.980501i $$0.437037\pi$$
$$90$$ 0 0
$$91$$ 528.000 0.608236
$$92$$ 0 0
$$93$$ −864.000 −0.963362
$$94$$ 0 0
$$95$$ 100.000 0.107998
$$96$$ 0 0
$$97$$ 866.000 0.906484 0.453242 0.891387i $$-0.350267\pi$$
0.453242 + 0.891387i $$0.350267\pi$$
$$98$$ 0 0
$$99$$ −468.000 −0.475109
$$100$$ 0 0
$$101$$ −1218.00 −1.19996 −0.599978 0.800017i $$-0.704824\pi$$
−0.599978 + 0.800017i $$0.704824\pi$$
$$102$$ 0 0
$$103$$ 88.0000 0.0841835 0.0420917 0.999114i $$-0.486598\pi$$
0.0420917 + 0.999114i $$0.486598\pi$$
$$104$$ 0 0
$$105$$ −360.000 −0.334594
$$106$$ 0 0
$$107$$ −36.0000 −0.0325257 −0.0162629 0.999868i $$-0.505177\pi$$
−0.0162629 + 0.999868i $$0.505177\pi$$
$$108$$ 0 0
$$109$$ −970.000 −0.852378 −0.426189 0.904634i $$-0.640144\pi$$
−0.426189 + 0.904634i $$0.640144\pi$$
$$110$$ 0 0
$$111$$ 102.000 0.0872199
$$112$$ 0 0
$$113$$ 1042.00 0.867461 0.433731 0.901043i $$-0.357197\pi$$
0.433731 + 0.901043i $$0.357197\pi$$
$$114$$ 0 0
$$115$$ 840.000 0.681134
$$116$$ 0 0
$$117$$ 198.000 0.156454
$$118$$ 0 0
$$119$$ −336.000 −0.258833
$$120$$ 0 0
$$121$$ 1373.00 1.03156
$$122$$ 0 0
$$123$$ −366.000 −0.268302
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −1936.00 −1.35269 −0.676347 0.736583i $$-0.736438\pi$$
−0.676347 + 0.736583i $$0.736438\pi$$
$$128$$ 0 0
$$129$$ −564.000 −0.384941
$$130$$ 0 0
$$131$$ −732.000 −0.488207 −0.244104 0.969749i $$-0.578494\pi$$
−0.244104 + 0.969749i $$0.578494\pi$$
$$132$$ 0 0
$$133$$ 480.000 0.312942
$$134$$ 0 0
$$135$$ −135.000 −0.0860663
$$136$$ 0 0
$$137$$ −2214.00 −1.38069 −0.690346 0.723479i $$-0.742542\pi$$
−0.690346 + 0.723479i $$0.742542\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −0.0122042 −0.00610208 0.999981i $$-0.501942\pi$$
−0.00610208 + 0.999981i $$0.501942\pi$$
$$140$$ 0 0
$$141$$ 768.000 0.458704
$$142$$ 0 0
$$143$$ −1144.00 −0.668994
$$144$$ 0 0
$$145$$ 1150.00 0.658637
$$146$$ 0 0
$$147$$ −699.000 −0.392194
$$148$$ 0 0
$$149$$ −1330.00 −0.731261 −0.365630 0.930760i $$-0.619147\pi$$
−0.365630 + 0.930760i $$0.619147\pi$$
$$150$$ 0 0
$$151$$ 1208.00 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ −126.000 −0.0665784
$$154$$ 0 0
$$155$$ 1440.00 0.746217
$$156$$ 0 0
$$157$$ −3514.00 −1.78629 −0.893146 0.449768i $$-0.851507\pi$$
−0.893146 + 0.449768i $$0.851507\pi$$
$$158$$ 0 0
$$159$$ 1014.00 0.505757
$$160$$ 0 0
$$161$$ 4032.00 1.97370
$$162$$ 0 0
$$163$$ 2068.00 0.993732 0.496866 0.867827i $$-0.334484\pi$$
0.496866 + 0.867827i $$0.334484\pi$$
$$164$$ 0 0
$$165$$ 780.000 0.368018
$$166$$ 0 0
$$167$$ 24.0000 0.0111208 0.00556041 0.999985i $$-0.498230\pi$$
0.00556041 + 0.999985i $$0.498230\pi$$
$$168$$ 0 0
$$169$$ −1713.00 −0.779700
$$170$$ 0 0
$$171$$ 180.000 0.0804967
$$172$$ 0 0
$$173$$ −618.000 −0.271593 −0.135797 0.990737i $$-0.543359\pi$$
−0.135797 + 0.990737i $$0.543359\pi$$
$$174$$ 0 0
$$175$$ 600.000 0.259176
$$176$$ 0 0
$$177$$ 300.000 0.127398
$$178$$ 0 0
$$179$$ −3340.00 −1.39466 −0.697328 0.716752i $$-0.745628\pi$$
−0.697328 + 0.716752i $$0.745628\pi$$
$$180$$ 0 0
$$181$$ −178.000 −0.0730974 −0.0365487 0.999332i $$-0.511636\pi$$
−0.0365487 + 0.999332i $$0.511636\pi$$
$$182$$ 0 0
$$183$$ −2226.00 −0.899184
$$184$$ 0 0
$$185$$ −170.000 −0.0675603
$$186$$ 0 0
$$187$$ 728.000 0.284688
$$188$$ 0 0
$$189$$ −648.000 −0.249392
$$190$$ 0 0
$$191$$ 1888.00 0.715240 0.357620 0.933867i $$-0.383588\pi$$
0.357620 + 0.933867i $$0.383588\pi$$
$$192$$ 0 0
$$193$$ 1922.00 0.716832 0.358416 0.933562i $$-0.383317\pi$$
0.358416 + 0.933562i $$0.383317\pi$$
$$194$$ 0 0
$$195$$ −330.000 −0.121189
$$196$$ 0 0
$$197$$ 2526.00 0.913554 0.456777 0.889581i $$-0.349004\pi$$
0.456777 + 0.889581i $$0.349004\pi$$
$$198$$ 0 0
$$199$$ 1160.00 0.413217 0.206609 0.978424i $$-0.433757\pi$$
0.206609 + 0.978424i $$0.433757\pi$$
$$200$$ 0 0
$$201$$ −252.000 −0.0884314
$$202$$ 0 0
$$203$$ 5520.00 1.90851
$$204$$ 0 0
$$205$$ 610.000 0.207826
$$206$$ 0 0
$$207$$ 1512.00 0.507687
$$208$$ 0 0
$$209$$ −1040.00 −0.344202
$$210$$ 0 0
$$211$$ 4468.00 1.45777 0.728886 0.684635i $$-0.240039\pi$$
0.728886 + 0.684635i $$0.240039\pi$$
$$212$$ 0 0
$$213$$ −984.000 −0.316538
$$214$$ 0 0
$$215$$ 940.000 0.298174
$$216$$ 0 0
$$217$$ 6912.00 2.16229
$$218$$ 0 0
$$219$$ 114.000 0.0351754
$$220$$ 0 0
$$221$$ −308.000 −0.0937481
$$222$$ 0 0
$$223$$ −6032.00 −1.81136 −0.905678 0.423965i $$-0.860638\pi$$
−0.905678 + 0.423965i $$0.860638\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −2636.00 −0.770738 −0.385369 0.922763i $$-0.625926\pi$$
−0.385369 + 0.922763i $$0.625926\pi$$
$$228$$ 0 0
$$229$$ 4830.00 1.39378 0.696889 0.717179i $$-0.254567\pi$$
0.696889 + 0.717179i $$0.254567\pi$$
$$230$$ 0 0
$$231$$ 3744.00 1.06639
$$232$$ 0 0
$$233$$ 2682.00 0.754093 0.377046 0.926194i $$-0.376940\pi$$
0.377046 + 0.926194i $$0.376940\pi$$
$$234$$ 0 0
$$235$$ −1280.00 −0.355311
$$236$$ 0 0
$$237$$ −720.000 −0.197338
$$238$$ 0 0
$$239$$ −2320.00 −0.627901 −0.313950 0.949439i $$-0.601653\pi$$
−0.313950 + 0.949439i $$0.601653\pi$$
$$240$$ 0 0
$$241$$ 2002.00 0.535104 0.267552 0.963543i $$-0.413785\pi$$
0.267552 + 0.963543i $$0.413785\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 1165.00 0.303792
$$246$$ 0 0
$$247$$ 440.000 0.113346
$$248$$ 0 0
$$249$$ 3636.00 0.925390
$$250$$ 0 0
$$251$$ −132.000 −0.0331943 −0.0165971 0.999862i $$-0.505283\pi$$
−0.0165971 + 0.999862i $$0.505283\pi$$
$$252$$ 0 0
$$253$$ −8736.00 −2.17086
$$254$$ 0 0
$$255$$ 210.000 0.0515714
$$256$$ 0 0
$$257$$ −7614.00 −1.84805 −0.924024 0.382335i $$-0.875120\pi$$
−0.924024 + 0.382335i $$0.875120\pi$$
$$258$$ 0 0
$$259$$ −816.000 −0.195767
$$260$$ 0 0
$$261$$ 2070.00 0.490919
$$262$$ 0 0
$$263$$ 4888.00 1.14603 0.573017 0.819543i $$-0.305773\pi$$
0.573017 + 0.819543i $$0.305773\pi$$
$$264$$ 0 0
$$265$$ −1690.00 −0.391758
$$266$$ 0 0
$$267$$ −990.000 −0.226918
$$268$$ 0 0
$$269$$ 1270.00 0.287856 0.143928 0.989588i $$-0.454027\pi$$
0.143928 + 0.989588i $$0.454027\pi$$
$$270$$ 0 0
$$271$$ −1072.00 −0.240293 −0.120146 0.992756i $$-0.538336\pi$$
−0.120146 + 0.992756i $$0.538336\pi$$
$$272$$ 0 0
$$273$$ −1584.00 −0.351165
$$274$$ 0 0
$$275$$ −1300.00 −0.285065
$$276$$ 0 0
$$277$$ −5394.00 −1.17001 −0.585007 0.811028i $$-0.698908\pi$$
−0.585007 + 0.811028i $$0.698908\pi$$
$$278$$ 0 0
$$279$$ 2592.00 0.556197
$$280$$ 0 0
$$281$$ 2442.00 0.518425 0.259213 0.965820i $$-0.416537\pi$$
0.259213 + 0.965820i $$0.416537\pi$$
$$282$$ 0 0
$$283$$ −2772.00 −0.582255 −0.291128 0.956684i $$-0.594030\pi$$
−0.291128 + 0.956684i $$0.594030\pi$$
$$284$$ 0 0
$$285$$ −300.000 −0.0623525
$$286$$ 0 0
$$287$$ 2928.00 0.602210
$$288$$ 0 0
$$289$$ −4717.00 −0.960106
$$290$$ 0 0
$$291$$ −2598.00 −0.523359
$$292$$ 0 0
$$293$$ 4542.00 0.905619 0.452810 0.891607i $$-0.350422\pi$$
0.452810 + 0.891607i $$0.350422\pi$$
$$294$$ 0 0
$$295$$ −500.000 −0.0986818
$$296$$ 0 0
$$297$$ 1404.00 0.274304
$$298$$ 0 0
$$299$$ 3696.00 0.714867
$$300$$ 0 0
$$301$$ 4512.00 0.864011
$$302$$ 0 0
$$303$$ 3654.00 0.692795
$$304$$ 0 0
$$305$$ 3710.00 0.696505
$$306$$ 0 0
$$307$$ −5116.00 −0.951093 −0.475546 0.879691i $$-0.657750\pi$$
−0.475546 + 0.879691i $$0.657750\pi$$
$$308$$ 0 0
$$309$$ −264.000 −0.0486034
$$310$$ 0 0
$$311$$ 2808.00 0.511984 0.255992 0.966679i $$-0.417598\pi$$
0.255992 + 0.966679i $$0.417598\pi$$
$$312$$ 0 0
$$313$$ −7318.00 −1.32153 −0.660763 0.750594i $$-0.729767\pi$$
−0.660763 + 0.750594i $$0.729767\pi$$
$$314$$ 0 0
$$315$$ 1080.00 0.193178
$$316$$ 0 0
$$317$$ 2246.00 0.397943 0.198971 0.980005i $$-0.436240\pi$$
0.198971 + 0.980005i $$0.436240\pi$$
$$318$$ 0 0
$$319$$ −11960.0 −2.09916
$$320$$ 0 0
$$321$$ 108.000 0.0187787
$$322$$ 0 0
$$323$$ −280.000 −0.0482341
$$324$$ 0 0
$$325$$ 550.000 0.0938723
$$326$$ 0 0
$$327$$ 2910.00 0.492120
$$328$$ 0 0
$$329$$ −6144.00 −1.02957
$$330$$ 0 0
$$331$$ −1332.00 −0.221188 −0.110594 0.993866i $$-0.535275\pi$$
−0.110594 + 0.993866i $$0.535275\pi$$
$$332$$ 0 0
$$333$$ −306.000 −0.0503564
$$334$$ 0 0
$$335$$ 420.000 0.0684987
$$336$$ 0 0
$$337$$ −11534.0 −1.86438 −0.932191 0.361966i $$-0.882106\pi$$
−0.932191 + 0.361966i $$0.882106\pi$$
$$338$$ 0 0
$$339$$ −3126.00 −0.500829
$$340$$ 0 0
$$341$$ −14976.0 −2.37829
$$342$$ 0 0
$$343$$ −2640.00 −0.415588
$$344$$ 0 0
$$345$$ −2520.00 −0.393253
$$346$$ 0 0
$$347$$ −11956.0 −1.84966 −0.924830 0.380382i $$-0.875793\pi$$
−0.924830 + 0.380382i $$0.875793\pi$$
$$348$$ 0 0
$$349$$ 4870.00 0.746949 0.373474 0.927640i $$-0.378166\pi$$
0.373474 + 0.927640i $$0.378166\pi$$
$$350$$ 0 0
$$351$$ −594.000 −0.0903287
$$352$$ 0 0
$$353$$ 10722.0 1.61664 0.808321 0.588742i $$-0.200377\pi$$
0.808321 + 0.588742i $$0.200377\pi$$
$$354$$ 0 0
$$355$$ 1640.00 0.245189
$$356$$ 0 0
$$357$$ 1008.00 0.149437
$$358$$ 0 0
$$359$$ −120.000 −0.0176417 −0.00882083 0.999961i $$-0.502808\pi$$
−0.00882083 + 0.999961i $$0.502808\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ 0 0
$$363$$ −4119.00 −0.595569
$$364$$ 0 0
$$365$$ −190.000 −0.0272467
$$366$$ 0 0
$$367$$ −3936.00 −0.559830 −0.279915 0.960025i $$-0.590306\pi$$
−0.279915 + 0.960025i $$0.590306\pi$$
$$368$$ 0 0
$$369$$ 1098.00 0.154904
$$370$$ 0 0
$$371$$ −8112.00 −1.13519
$$372$$ 0 0
$$373$$ 3022.00 0.419499 0.209750 0.977755i $$-0.432735\pi$$
0.209750 + 0.977755i $$0.432735\pi$$
$$374$$ 0 0
$$375$$ −375.000 −0.0516398
$$376$$ 0 0
$$377$$ 5060.00 0.691255
$$378$$ 0 0
$$379$$ 13340.0 1.80799 0.903997 0.427539i $$-0.140619\pi$$
0.903997 + 0.427539i $$0.140619\pi$$
$$380$$ 0 0
$$381$$ 5808.00 0.780979
$$382$$ 0 0
$$383$$ 1008.00 0.134481 0.0672407 0.997737i $$-0.478580\pi$$
0.0672407 + 0.997737i $$0.478580\pi$$
$$384$$ 0 0
$$385$$ −6240.00 −0.826026
$$386$$ 0 0
$$387$$ 1692.00 0.222246
$$388$$ 0 0
$$389$$ 9630.00 1.25517 0.627584 0.778549i $$-0.284044\pi$$
0.627584 + 0.778549i $$0.284044\pi$$
$$390$$ 0 0
$$391$$ −2352.00 −0.304209
$$392$$ 0 0
$$393$$ 2196.00 0.281867
$$394$$ 0 0
$$395$$ 1200.00 0.152857
$$396$$ 0 0
$$397$$ 7126.00 0.900866 0.450433 0.892810i $$-0.351270\pi$$
0.450433 + 0.892810i $$0.351270\pi$$
$$398$$ 0 0
$$399$$ −1440.00 −0.180677
$$400$$ 0 0
$$401$$ −8718.00 −1.08568 −0.542838 0.839837i $$-0.682650\pi$$
−0.542838 + 0.839837i $$0.682650\pi$$
$$402$$ 0 0
$$403$$ 6336.00 0.783173
$$404$$ 0 0
$$405$$ 405.000 0.0496904
$$406$$ 0 0
$$407$$ 1768.00 0.215323
$$408$$ 0 0
$$409$$ −10870.0 −1.31415 −0.657074 0.753826i $$-0.728206\pi$$
−0.657074 + 0.753826i $$0.728206\pi$$
$$410$$ 0 0
$$411$$ 6642.00 0.797143
$$412$$ 0 0
$$413$$ −2400.00 −0.285947
$$414$$ 0 0
$$415$$ −6060.00 −0.716804
$$416$$ 0 0
$$417$$ 60.0000 0.00704607
$$418$$ 0 0
$$419$$ 9700.00 1.13097 0.565484 0.824759i $$-0.308689\pi$$
0.565484 + 0.824759i $$0.308689\pi$$
$$420$$ 0 0
$$421$$ 862.000 0.0997893 0.0498947 0.998754i $$-0.484111\pi$$
0.0498947 + 0.998754i $$0.484111\pi$$
$$422$$ 0 0
$$423$$ −2304.00 −0.264833
$$424$$ 0 0
$$425$$ −350.000 −0.0399470
$$426$$ 0 0
$$427$$ 17808.0 2.01824
$$428$$ 0 0
$$429$$ 3432.00 0.386244
$$430$$ 0 0
$$431$$ −15792.0 −1.76490 −0.882452 0.470402i $$-0.844109\pi$$
−0.882452 + 0.470402i $$0.844109\pi$$
$$432$$ 0 0
$$433$$ 11602.0 1.28766 0.643830 0.765169i $$-0.277345\pi$$
0.643830 + 0.765169i $$0.277345\pi$$
$$434$$ 0 0
$$435$$ −3450.00 −0.380264
$$436$$ 0 0
$$437$$ 3360.00 0.367805
$$438$$ 0 0
$$439$$ 440.000 0.0478361 0.0239181 0.999714i $$-0.492386\pi$$
0.0239181 + 0.999714i $$0.492386\pi$$
$$440$$ 0 0
$$441$$ 2097.00 0.226433
$$442$$ 0 0
$$443$$ 10188.0 1.09266 0.546328 0.837571i $$-0.316025\pi$$
0.546328 + 0.837571i $$0.316025\pi$$
$$444$$ 0 0
$$445$$ 1650.00 0.175770
$$446$$ 0 0
$$447$$ 3990.00 0.422194
$$448$$ 0 0
$$449$$ −13310.0 −1.39897 −0.699485 0.714647i $$-0.746587\pi$$
−0.699485 + 0.714647i $$0.746587\pi$$
$$450$$ 0 0
$$451$$ −6344.00 −0.662367
$$452$$ 0 0
$$453$$ −3624.00 −0.375873
$$454$$ 0 0
$$455$$ 2640.00 0.272011
$$456$$ 0 0
$$457$$ 3226.00 0.330210 0.165105 0.986276i $$-0.447204\pi$$
0.165105 + 0.986276i $$0.447204\pi$$
$$458$$ 0 0
$$459$$ 378.000 0.0384391
$$460$$ 0 0
$$461$$ 6582.00 0.664977 0.332488 0.943107i $$-0.392112\pi$$
0.332488 + 0.943107i $$0.392112\pi$$
$$462$$ 0 0
$$463$$ −15072.0 −1.51286 −0.756431 0.654073i $$-0.773059\pi$$
−0.756431 + 0.654073i $$0.773059\pi$$
$$464$$ 0 0
$$465$$ −4320.00 −0.430828
$$466$$ 0 0
$$467$$ −476.000 −0.0471663 −0.0235831 0.999722i $$-0.507507\pi$$
−0.0235831 + 0.999722i $$0.507507\pi$$
$$468$$ 0 0
$$469$$ 2016.00 0.198487
$$470$$ 0 0
$$471$$ 10542.0 1.03132
$$472$$ 0 0
$$473$$ −9776.00 −0.950319
$$474$$ 0 0
$$475$$ 500.000 0.0482980
$$476$$ 0 0
$$477$$ −3042.00 −0.291999
$$478$$ 0 0
$$479$$ 19680.0 1.87725 0.938624 0.344941i $$-0.112101\pi$$
0.938624 + 0.344941i $$0.112101\pi$$
$$480$$ 0 0
$$481$$ −748.000 −0.0709062
$$482$$ 0 0
$$483$$ −12096.0 −1.13952
$$484$$ 0 0
$$485$$ 4330.00 0.405392
$$486$$ 0 0
$$487$$ 5944.00 0.553077 0.276538 0.961003i $$-0.410813\pi$$
0.276538 + 0.961003i $$0.410813\pi$$
$$488$$ 0 0
$$489$$ −6204.00 −0.573731
$$490$$ 0 0
$$491$$ −10772.0 −0.990089 −0.495044 0.868868i $$-0.664848\pi$$
−0.495044 + 0.868868i $$0.664848\pi$$
$$492$$ 0 0
$$493$$ −3220.00 −0.294161
$$494$$ 0 0
$$495$$ −2340.00 −0.212475
$$496$$ 0 0
$$497$$ 7872.00 0.710478
$$498$$ 0 0
$$499$$ −8140.00 −0.730253 −0.365127 0.930958i $$-0.618974\pi$$
−0.365127 + 0.930958i $$0.618974\pi$$
$$500$$ 0 0
$$501$$ −72.0000 −0.00642060
$$502$$ 0 0
$$503$$ 13768.0 1.22045 0.610223 0.792229i $$-0.291080\pi$$
0.610223 + 0.792229i $$0.291080\pi$$
$$504$$ 0 0
$$505$$ −6090.00 −0.536637
$$506$$ 0 0
$$507$$ 5139.00 0.450160
$$508$$ 0 0
$$509$$ 22150.0 1.92884 0.964422 0.264368i $$-0.0851633\pi$$
0.964422 + 0.264368i $$0.0851633\pi$$
$$510$$ 0 0
$$511$$ −912.000 −0.0789521
$$512$$ 0 0
$$513$$ −540.000 −0.0464748
$$514$$ 0 0
$$515$$ 440.000 0.0376480
$$516$$ 0 0
$$517$$ 13312.0 1.13242
$$518$$ 0 0
$$519$$ 1854.00 0.156805
$$520$$ 0 0
$$521$$ 1562.00 0.131348 0.0656741 0.997841i $$-0.479080\pi$$
0.0656741 + 0.997841i $$0.479080\pi$$
$$522$$ 0 0
$$523$$ 668.000 0.0558501 0.0279250 0.999610i $$-0.491110\pi$$
0.0279250 + 0.999610i $$0.491110\pi$$
$$524$$ 0 0
$$525$$ −1800.00 −0.149635
$$526$$ 0 0
$$527$$ −4032.00 −0.333276
$$528$$ 0 0
$$529$$ 16057.0 1.31972
$$530$$ 0 0
$$531$$ −900.000 −0.0735531
$$532$$ 0 0
$$533$$ 2684.00 0.218118
$$534$$ 0 0
$$535$$ −180.000 −0.0145459
$$536$$ 0 0
$$537$$ 10020.0 0.805205
$$538$$ 0 0
$$539$$ −12116.0 −0.968225
$$540$$ 0 0
$$541$$ −6138.00 −0.487788 −0.243894 0.969802i $$-0.578425\pi$$
−0.243894 + 0.969802i $$0.578425\pi$$
$$542$$ 0 0
$$543$$ 534.000 0.0422028
$$544$$ 0 0
$$545$$ −4850.00 −0.381195
$$546$$ 0 0
$$547$$ 10484.0 0.819494 0.409747 0.912199i $$-0.365617\pi$$
0.409747 + 0.912199i $$0.365617\pi$$
$$548$$ 0 0
$$549$$ 6678.00 0.519144
$$550$$ 0 0
$$551$$ 4600.00 0.355656
$$552$$ 0 0
$$553$$ 5760.00 0.442930
$$554$$ 0 0
$$555$$ 510.000 0.0390059
$$556$$ 0 0
$$557$$ 3606.00 0.274311 0.137155 0.990550i $$-0.456204\pi$$
0.137155 + 0.990550i $$0.456204\pi$$
$$558$$ 0 0
$$559$$ 4136.00 0.312941
$$560$$ 0 0
$$561$$ −2184.00 −0.164365
$$562$$ 0 0
$$563$$ −12252.0 −0.917159 −0.458579 0.888654i $$-0.651641\pi$$
−0.458579 + 0.888654i $$0.651641\pi$$
$$564$$ 0 0
$$565$$ 5210.00 0.387940
$$566$$ 0 0
$$567$$ 1944.00 0.143986
$$568$$ 0 0
$$569$$ −14550.0 −1.07200 −0.536000 0.844218i $$-0.680065\pi$$
−0.536000 + 0.844218i $$0.680065\pi$$
$$570$$ 0 0
$$571$$ 25468.0 1.86655 0.933277 0.359157i $$-0.116936\pi$$
0.933277 + 0.359157i $$0.116936\pi$$
$$572$$ 0 0
$$573$$ −5664.00 −0.412944
$$574$$ 0 0
$$575$$ 4200.00 0.304612
$$576$$ 0 0
$$577$$ 12866.0 0.928282 0.464141 0.885761i $$-0.346363\pi$$
0.464141 + 0.885761i $$0.346363\pi$$
$$578$$ 0 0
$$579$$ −5766.00 −0.413863
$$580$$ 0 0
$$581$$ −29088.0 −2.07706
$$582$$ 0 0
$$583$$ 17576.0 1.24858
$$584$$ 0 0
$$585$$ 990.000 0.0699683
$$586$$ 0 0
$$587$$ 14844.0 1.04374 0.521872 0.853024i $$-0.325234\pi$$
0.521872 + 0.853024i $$0.325234\pi$$
$$588$$ 0 0
$$589$$ 5760.00 0.402948
$$590$$ 0 0
$$591$$ −7578.00 −0.527440
$$592$$ 0 0
$$593$$ 20402.0 1.41283 0.706416 0.707797i $$-0.250311\pi$$
0.706416 + 0.707797i $$0.250311\pi$$
$$594$$ 0 0
$$595$$ −1680.00 −0.115753
$$596$$ 0 0
$$597$$ −3480.00 −0.238571
$$598$$ 0 0
$$599$$ −10760.0 −0.733959 −0.366980 0.930229i $$-0.619608\pi$$
−0.366980 + 0.930229i $$0.619608\pi$$
$$600$$ 0 0
$$601$$ 14282.0 0.969343 0.484671 0.874696i $$-0.338939\pi$$
0.484671 + 0.874696i $$0.338939\pi$$
$$602$$ 0 0
$$603$$ 756.000 0.0510559
$$604$$ 0 0
$$605$$ 6865.00 0.461326
$$606$$ 0 0
$$607$$ −11056.0 −0.739290 −0.369645 0.929173i $$-0.620521\pi$$
−0.369645 + 0.929173i $$0.620521\pi$$
$$608$$ 0 0
$$609$$ −16560.0 −1.10188
$$610$$ 0 0
$$611$$ −5632.00 −0.372907
$$612$$ 0 0
$$613$$ −16418.0 −1.08176 −0.540878 0.841101i $$-0.681908\pi$$
−0.540878 + 0.841101i $$0.681908\pi$$
$$614$$ 0 0
$$615$$ −1830.00 −0.119988
$$616$$ 0 0
$$617$$ −10374.0 −0.676891 −0.338445 0.940986i $$-0.609901\pi$$
−0.338445 + 0.940986i $$0.609901\pi$$
$$618$$ 0 0
$$619$$ 5260.00 0.341546 0.170773 0.985310i $$-0.445373\pi$$
0.170773 + 0.985310i $$0.445373\pi$$
$$620$$ 0 0
$$621$$ −4536.00 −0.293113
$$622$$ 0 0
$$623$$ 7920.00 0.509323
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 3120.00 0.198725
$$628$$ 0 0
$$629$$ 476.000 0.0301739
$$630$$ 0 0
$$631$$ −21352.0 −1.34708 −0.673542 0.739149i $$-0.735228\pi$$
−0.673542 + 0.739149i $$0.735228\pi$$
$$632$$ 0 0
$$633$$ −13404.0 −0.841645
$$634$$ 0 0
$$635$$ −9680.00 −0.604943
$$636$$ 0 0
$$637$$ 5126.00 0.318838
$$638$$ 0 0
$$639$$ 2952.00 0.182753
$$640$$ 0 0
$$641$$ −29118.0 −1.79422 −0.897108 0.441812i $$-0.854336\pi$$
−0.897108 + 0.441812i $$0.854336\pi$$
$$642$$ 0 0
$$643$$ −5772.00 −0.354005 −0.177003 0.984210i $$-0.556640\pi$$
−0.177003 + 0.984210i $$0.556640\pi$$
$$644$$ 0 0
$$645$$ −2820.00 −0.172151
$$646$$ 0 0
$$647$$ 14264.0 0.866732 0.433366 0.901218i $$-0.357326\pi$$
0.433366 + 0.901218i $$0.357326\pi$$
$$648$$ 0 0
$$649$$ 5200.00 0.314511
$$650$$ 0 0
$$651$$ −20736.0 −1.24840
$$652$$ 0 0
$$653$$ 6902.00 0.413623 0.206812 0.978381i $$-0.433691\pi$$
0.206812 + 0.978381i $$0.433691\pi$$
$$654$$ 0 0
$$655$$ −3660.00 −0.218333
$$656$$ 0 0
$$657$$ −342.000 −0.0203085
$$658$$ 0 0
$$659$$ −20140.0 −1.19051 −0.595253 0.803539i $$-0.702948\pi$$
−0.595253 + 0.803539i $$0.702948\pi$$
$$660$$ 0 0
$$661$$ −3218.00 −0.189358 −0.0946790 0.995508i $$-0.530182\pi$$
−0.0946790 + 0.995508i $$0.530182\pi$$
$$662$$ 0 0
$$663$$ 924.000 0.0541255
$$664$$ 0 0
$$665$$ 2400.00 0.139952
$$666$$ 0 0
$$667$$ 38640.0 2.24310
$$668$$ 0 0
$$669$$ 18096.0 1.04579
$$670$$ 0 0
$$671$$ −38584.0 −2.21985
$$672$$ 0 0
$$673$$ −7518.00 −0.430606 −0.215303 0.976547i $$-0.569074\pi$$
−0.215303 + 0.976547i $$0.569074\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ −18114.0 −1.02833 −0.514164 0.857692i $$-0.671898\pi$$
−0.514164 + 0.857692i $$0.671898\pi$$
$$678$$ 0 0
$$679$$ 20784.0 1.17469
$$680$$ 0 0
$$681$$ 7908.00 0.444986
$$682$$ 0 0
$$683$$ 23868.0 1.33716 0.668582 0.743638i $$-0.266901\pi$$
0.668582 + 0.743638i $$0.266901\pi$$
$$684$$ 0 0
$$685$$ −11070.0 −0.617464
$$686$$ 0 0
$$687$$ −14490.0 −0.804699
$$688$$ 0 0
$$689$$ −7436.00 −0.411160
$$690$$ 0 0
$$691$$ −172.000 −0.00946916 −0.00473458 0.999989i $$-0.501507\pi$$
−0.00473458 + 0.999989i $$0.501507\pi$$
$$692$$ 0 0
$$693$$ −11232.0 −0.615683
$$694$$ 0 0
$$695$$ −100.000 −0.00545787
$$696$$ 0 0
$$697$$ −1708.00 −0.0928194
$$698$$ 0 0
$$699$$ −8046.00 −0.435376
$$700$$ 0 0
$$701$$ −22138.0 −1.19278 −0.596391 0.802694i $$-0.703399\pi$$
−0.596391 + 0.802694i $$0.703399\pi$$
$$702$$ 0 0
$$703$$ −680.000 −0.0364818
$$704$$ 0 0
$$705$$ 3840.00 0.205139
$$706$$ 0 0
$$707$$ −29232.0 −1.55500
$$708$$ 0 0
$$709$$ 3070.00 0.162618 0.0813091 0.996689i $$-0.474090\pi$$
0.0813091 + 0.996689i $$0.474090\pi$$
$$710$$ 0 0
$$711$$ 2160.00 0.113933
$$712$$ 0 0
$$713$$ 48384.0 2.54137
$$714$$ 0 0
$$715$$ −5720.00 −0.299183
$$716$$ 0 0
$$717$$ 6960.00 0.362519
$$718$$ 0 0
$$719$$ −15600.0 −0.809154 −0.404577 0.914504i $$-0.632581\pi$$
−0.404577 + 0.914504i $$0.632581\pi$$
$$720$$ 0 0
$$721$$ 2112.00 0.109092
$$722$$ 0 0
$$723$$ −6006.00 −0.308943
$$724$$ 0 0
$$725$$ 5750.00 0.294551
$$726$$ 0 0
$$727$$ −20696.0 −1.05581 −0.527904 0.849304i $$-0.677022\pi$$
−0.527904 + 0.849304i $$0.677022\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −2632.00 −0.133171
$$732$$ 0 0
$$733$$ −30778.0 −1.55090 −0.775451 0.631408i $$-0.782478\pi$$
−0.775451 + 0.631408i $$0.782478\pi$$
$$734$$ 0 0
$$735$$ −3495.00 −0.175395
$$736$$ 0 0
$$737$$ −4368.00 −0.218314
$$738$$ 0 0
$$739$$ −11740.0 −0.584388 −0.292194 0.956359i $$-0.594385\pi$$
−0.292194 + 0.956359i $$0.594385\pi$$
$$740$$ 0 0
$$741$$ −1320.00 −0.0654405
$$742$$ 0 0
$$743$$ −2632.00 −0.129958 −0.0649789 0.997887i $$-0.520698\pi$$
−0.0649789 + 0.997887i $$0.520698\pi$$
$$744$$ 0 0
$$745$$ −6650.00 −0.327030
$$746$$ 0 0
$$747$$ −10908.0 −0.534274
$$748$$ 0 0
$$749$$ −864.000 −0.0421494
$$750$$ 0 0
$$751$$ 20528.0 0.997440 0.498720 0.866763i $$-0.333804\pi$$
0.498720 + 0.866763i $$0.333804\pi$$
$$752$$ 0 0
$$753$$ 396.000 0.0191647
$$754$$ 0 0
$$755$$ 6040.00 0.291150
$$756$$ 0 0
$$757$$ 21646.0 1.03928 0.519642 0.854384i $$-0.326066\pi$$
0.519642 + 0.854384i $$0.326066\pi$$
$$758$$ 0 0
$$759$$ 26208.0 1.25335
$$760$$ 0 0
$$761$$ 18282.0 0.870857 0.435428 0.900223i $$-0.356597\pi$$
0.435428 + 0.900223i $$0.356597\pi$$
$$762$$ 0 0
$$763$$ −23280.0 −1.10458
$$764$$ 0 0
$$765$$ −630.000 −0.0297748
$$766$$ 0 0
$$767$$ −2200.00 −0.103569
$$768$$ 0 0
$$769$$ −24190.0 −1.13435 −0.567174 0.823598i $$-0.691963\pi$$
−0.567174 + 0.823598i $$0.691963\pi$$
$$770$$ 0 0
$$771$$ 22842.0 1.06697
$$772$$ 0 0
$$773$$ −25698.0 −1.19572 −0.597861 0.801600i $$-0.703982\pi$$
−0.597861 + 0.801600i $$0.703982\pi$$
$$774$$ 0 0
$$775$$ 7200.00 0.333718
$$776$$ 0 0
$$777$$ 2448.00 0.113026
$$778$$ 0 0
$$779$$ 2440.00 0.112223
$$780$$ 0 0
$$781$$ −17056.0 −0.781449
$$782$$ 0 0
$$783$$ −6210.00 −0.283432
$$784$$ 0 0
$$785$$ −17570.0 −0.798854
$$786$$ 0 0
$$787$$ −33436.0 −1.51444 −0.757220 0.653160i $$-0.773443\pi$$
−0.757220 + 0.653160i $$0.773443\pi$$
$$788$$ 0 0
$$789$$ −14664.0 −0.661663
$$790$$ 0 0
$$791$$ 25008.0 1.12412
$$792$$ 0 0
$$793$$ 16324.0 0.730999
$$794$$ 0 0
$$795$$ 5070.00 0.226182
$$796$$ 0 0
$$797$$ −37594.0 −1.67083 −0.835413 0.549623i $$-0.814771\pi$$
−0.835413 + 0.549623i $$0.814771\pi$$
$$798$$ 0 0
$$799$$ 3584.00 0.158689
$$800$$ 0 0
$$801$$ 2970.00 0.131011
$$802$$ 0 0
$$803$$ 1976.00 0.0868388
$$804$$ 0 0
$$805$$ 20160.0 0.882667
$$806$$ 0 0
$$807$$ −3810.00 −0.166194
$$808$$ 0 0
$$809$$ 4730.00 0.205560 0.102780 0.994704i $$-0.467226\pi$$
0.102780 + 0.994704i $$0.467226\pi$$
$$810$$ 0 0
$$811$$ 8748.00 0.378772 0.189386 0.981903i $$-0.439350\pi$$
0.189386 + 0.981903i $$0.439350\pi$$
$$812$$ 0 0
$$813$$ 3216.00 0.138733
$$814$$ 0 0
$$815$$ 10340.0 0.444410
$$816$$ 0 0
$$817$$ 3760.00 0.161011
$$818$$ 0 0
$$819$$ 4752.00 0.202745
$$820$$ 0 0
$$821$$ 44142.0 1.87645 0.938226 0.346024i $$-0.112468\pi$$
0.938226 + 0.346024i $$0.112468\pi$$
$$822$$ 0 0
$$823$$ −3992.00 −0.169079 −0.0845397 0.996420i $$-0.526942\pi$$
−0.0845397 + 0.996420i $$0.526942\pi$$
$$824$$ 0 0
$$825$$ 3900.00 0.164583
$$826$$ 0 0
$$827$$ 14444.0 0.607336 0.303668 0.952778i $$-0.401789\pi$$
0.303668 + 0.952778i $$0.401789\pi$$
$$828$$ 0 0
$$829$$ 42150.0 1.76590 0.882949 0.469468i $$-0.155554\pi$$
0.882949 + 0.469468i $$0.155554\pi$$
$$830$$ 0 0
$$831$$ 16182.0 0.675508
$$832$$ 0 0
$$833$$ −3262.00 −0.135680
$$834$$ 0 0
$$835$$ 120.000 0.00497338
$$836$$ 0 0
$$837$$ −7776.00 −0.321121
$$838$$ 0 0
$$839$$ −13400.0 −0.551394 −0.275697 0.961245i $$-0.588909\pi$$
−0.275697 + 0.961245i $$0.588909\pi$$
$$840$$ 0 0
$$841$$ 28511.0 1.16901
$$842$$ 0 0
$$843$$ −7326.00 −0.299313
$$844$$ 0 0
$$845$$ −8565.00 −0.348692
$$846$$ 0 0
$$847$$ 32952.0 1.33677
$$848$$ 0 0
$$849$$ 8316.00 0.336165
$$850$$ 0 0
$$851$$ −5712.00 −0.230088
$$852$$ 0 0
$$853$$ −8658.00 −0.347531 −0.173766 0.984787i $$-0.555594\pi$$
−0.173766 + 0.984787i $$0.555594\pi$$
$$854$$ 0 0
$$855$$ 900.000 0.0359992
$$856$$ 0 0
$$857$$ 42826.0 1.70701 0.853505 0.521084i $$-0.174472\pi$$
0.853505 + 0.521084i $$0.174472\pi$$
$$858$$ 0 0
$$859$$ 35900.0 1.42595 0.712976 0.701189i $$-0.247347\pi$$
0.712976 + 0.701189i $$0.247347\pi$$
$$860$$ 0 0
$$861$$ −8784.00 −0.347686
$$862$$ 0 0
$$863$$ 3088.00 0.121804 0.0609019 0.998144i $$-0.480602\pi$$
0.0609019 + 0.998144i $$0.480602\pi$$
$$864$$ 0 0
$$865$$ −3090.00 −0.121460
$$866$$ 0 0
$$867$$ 14151.0 0.554317
$$868$$ 0 0
$$869$$ −12480.0 −0.487175
$$870$$ 0 0
$$871$$ 1848.00 0.0718910
$$872$$ 0 0
$$873$$ 7794.00 0.302161
$$874$$ 0 0
$$875$$ 3000.00 0.115907
$$876$$ 0 0
$$877$$ −35274.0 −1.35817 −0.679087 0.734058i $$-0.737624\pi$$
−0.679087 + 0.734058i $$0.737624\pi$$
$$878$$ 0 0
$$879$$ −13626.0 −0.522860
$$880$$ 0 0
$$881$$ 25042.0 0.957646 0.478823 0.877911i $$-0.341064\pi$$
0.478823 + 0.877911i $$0.341064\pi$$
$$882$$ 0 0
$$883$$ −12572.0 −0.479141 −0.239570 0.970879i $$-0.577007\pi$$
−0.239570 + 0.970879i $$0.577007\pi$$
$$884$$ 0 0
$$885$$ 1500.00 0.0569740
$$886$$ 0 0
$$887$$ 21864.0 0.827645 0.413823 0.910358i $$-0.364193\pi$$
0.413823 + 0.910358i $$0.364193\pi$$
$$888$$ 0 0
$$889$$ −46464.0 −1.75293
$$890$$ 0 0
$$891$$ −4212.00 −0.158370
$$892$$ 0 0
$$893$$ −5120.00 −0.191864
$$894$$ 0 0
$$895$$ −16700.0 −0.623709
$$896$$ 0 0
$$897$$ −11088.0 −0.412729
$$898$$ 0 0
$$899$$ 66240.0 2.45743
$$900$$ 0 0
$$901$$ 4732.00 0.174968
$$902$$ 0 0
$$903$$ −13536.0 −0.498837
$$904$$ 0 0
$$905$$ −890.000 −0.0326902
$$906$$ 0 0
$$907$$ −31236.0 −1.14352 −0.571761 0.820420i $$-0.693740\pi$$
−0.571761 + 0.820420i $$0.693740\pi$$
$$908$$ 0 0
$$909$$ −10962.0 −0.399985
$$910$$ 0 0
$$911$$ −8272.00 −0.300838 −0.150419 0.988622i $$-0.548062\pi$$
−0.150419 + 0.988622i $$0.548062\pi$$
$$912$$ 0 0
$$913$$ 63024.0 2.28455
$$914$$ 0 0
$$915$$ −11130.0 −0.402127
$$916$$ 0 0
$$917$$ −17568.0 −0.632657
$$918$$ 0 0
$$919$$ −20200.0 −0.725067 −0.362533 0.931971i $$-0.618088\pi$$
−0.362533 + 0.931971i $$0.618088\pi$$
$$920$$ 0 0
$$921$$ 15348.0 0.549114
$$922$$ 0 0
$$923$$ 7216.00 0.257332
$$924$$ 0 0
$$925$$ −850.000 −0.0302139
$$926$$ 0 0
$$927$$ 792.000 0.0280612
$$928$$ 0 0
$$929$$ 31010.0 1.09516 0.547581 0.836753i $$-0.315549\pi$$
0.547581 + 0.836753i $$0.315549\pi$$
$$930$$ 0 0
$$931$$ 4660.00 0.164044
$$932$$ 0 0
$$933$$ −8424.00 −0.295594
$$934$$ 0 0
$$935$$ 3640.00 0.127316
$$936$$ 0 0
$$937$$ −39174.0 −1.36580 −0.682902 0.730510i $$-0.739283\pi$$
−0.682902 + 0.730510i $$0.739283\pi$$
$$938$$ 0 0
$$939$$ 21954.0 0.762984
$$940$$ 0 0
$$941$$ −4138.00 −0.143353 −0.0716764 0.997428i $$-0.522835\pi$$
−0.0716764 + 0.997428i $$0.522835\pi$$
$$942$$ 0 0
$$943$$ 20496.0 0.707785
$$944$$ 0 0
$$945$$ −3240.00 −0.111531
$$946$$ 0 0
$$947$$ −23676.0 −0.812425 −0.406213 0.913779i $$-0.633151\pi$$
−0.406213 + 0.913779i $$0.633151\pi$$
$$948$$ 0 0
$$949$$ −836.000 −0.0285961
$$950$$ 0 0
$$951$$ −6738.00 −0.229752
$$952$$ 0 0
$$953$$ 18922.0 0.643173 0.321586 0.946880i $$-0.395784\pi$$
0.321586 + 0.946880i $$0.395784\pi$$
$$954$$ 0 0
$$955$$ 9440.00 0.319865
$$956$$ 0 0
$$957$$ 35880.0 1.21195
$$958$$ 0 0
$$959$$ −53136.0 −1.78921
$$960$$ 0 0
$$961$$ 53153.0 1.78420
$$962$$ 0 0
$$963$$ −324.000 −0.0108419
$$964$$ 0 0
$$965$$ 9610.00 0.320577
$$966$$ 0 0
$$967$$ −39656.0 −1.31877 −0.659385 0.751805i $$-0.729183\pi$$
−0.659385 + 0.751805i $$0.729183\pi$$
$$968$$ 0 0
$$969$$ 840.000 0.0278480
$$970$$ 0 0
$$971$$ 33228.0 1.09818 0.549092 0.835762i $$-0.314974\pi$$
0.549092 + 0.835762i $$0.314974\pi$$
$$972$$ 0 0
$$973$$ −480.000 −0.0158151
$$974$$ 0 0
$$975$$ −1650.00 −0.0541972
$$976$$ 0 0
$$977$$ −974.000 −0.0318946 −0.0159473 0.999873i $$-0.505076\pi$$
−0.0159473 + 0.999873i $$0.505076\pi$$
$$978$$ 0 0
$$979$$ −17160.0 −0.560200
$$980$$ 0 0
$$981$$ −8730.00 −0.284126
$$982$$ 0 0
$$983$$ 13608.0 0.441534 0.220767 0.975327i $$-0.429144\pi$$
0.220767 + 0.975327i $$0.429144\pi$$
$$984$$ 0 0
$$985$$ 12630.0 0.408554
$$986$$ 0 0
$$987$$ 18432.0 0.594425
$$988$$ 0 0
$$989$$ 31584.0 1.01548
$$990$$ 0 0
$$991$$ −13472.0 −0.431839 −0.215919 0.976411i $$-0.569275\pi$$
−0.215919 + 0.976411i $$0.569275\pi$$
$$992$$ 0 0
$$993$$ 3996.00 0.127703
$$994$$ 0 0
$$995$$ 5800.00 0.184796
$$996$$ 0 0
$$997$$ −3234.00 −0.102730 −0.0513650 0.998680i $$-0.516357\pi$$
−0.0513650 + 0.998680i $$0.516357\pi$$
$$998$$ 0 0
$$999$$ 918.000 0.0290733
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 240.4.a.e.1.1 1
3.2 odd 2 720.4.a.n.1.1 1
4.3 odd 2 15.4.a.a.1.1 1
5.2 odd 4 1200.4.f.b.49.2 2
5.3 odd 4 1200.4.f.b.49.1 2
5.4 even 2 1200.4.a.t.1.1 1
8.3 odd 2 960.4.a.b.1.1 1
8.5 even 2 960.4.a.ba.1.1 1
12.11 even 2 45.4.a.c.1.1 1
20.3 even 4 75.4.b.b.49.1 2
20.7 even 4 75.4.b.b.49.2 2
20.19 odd 2 75.4.a.b.1.1 1
28.27 even 2 735.4.a.e.1.1 1
36.7 odd 6 405.4.e.g.271.1 2
36.11 even 6 405.4.e.i.271.1 2
36.23 even 6 405.4.e.i.136.1 2
36.31 odd 6 405.4.e.g.136.1 2
44.43 even 2 1815.4.a.e.1.1 1
60.23 odd 4 225.4.b.e.199.2 2
60.47 odd 4 225.4.b.e.199.1 2
60.59 even 2 225.4.a.f.1.1 1
84.83 odd 2 2205.4.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 4.3 odd 2
45.4.a.c.1.1 1 12.11 even 2
75.4.a.b.1.1 1 20.19 odd 2
75.4.b.b.49.1 2 20.3 even 4
75.4.b.b.49.2 2 20.7 even 4
225.4.a.f.1.1 1 60.59 even 2
225.4.b.e.199.1 2 60.47 odd 4
225.4.b.e.199.2 2 60.23 odd 4
240.4.a.e.1.1 1 1.1 even 1 trivial
405.4.e.g.136.1 2 36.31 odd 6
405.4.e.g.271.1 2 36.7 odd 6
405.4.e.i.136.1 2 36.23 even 6
405.4.e.i.271.1 2 36.11 even 6
720.4.a.n.1.1 1 3.2 odd 2
735.4.a.e.1.1 1 28.27 even 2
960.4.a.b.1.1 1 8.3 odd 2
960.4.a.ba.1.1 1 8.5 even 2
1200.4.a.t.1.1 1 5.4 even 2
1200.4.f.b.49.1 2 5.3 odd 4
1200.4.f.b.49.2 2 5.2 odd 4
1815.4.a.e.1.1 1 44.43 even 2
2205.4.a.l.1.1 1 84.83 odd 2