# Properties

 Label 225.4.a.f.1.1 Level $225$ Weight $4$ Character 225.1 Self dual yes Analytic conductor $13.275$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.2754297513$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 225.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -7.00000 q^{4} +24.0000 q^{7} -15.0000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} -7.00000 q^{4} +24.0000 q^{7} -15.0000 q^{8} -52.0000 q^{11} -22.0000 q^{13} +24.0000 q^{14} +41.0000 q^{16} -14.0000 q^{17} -20.0000 q^{19} -52.0000 q^{22} -168.000 q^{23} -22.0000 q^{26} -168.000 q^{28} -230.000 q^{29} -288.000 q^{31} +161.000 q^{32} -14.0000 q^{34} +34.0000 q^{37} -20.0000 q^{38} -122.000 q^{41} +188.000 q^{43} +364.000 q^{44} -168.000 q^{46} +256.000 q^{47} +233.000 q^{49} +154.000 q^{52} -338.000 q^{53} -360.000 q^{56} -230.000 q^{58} -100.000 q^{59} +742.000 q^{61} -288.000 q^{62} -167.000 q^{64} +84.0000 q^{67} +98.0000 q^{68} +328.000 q^{71} +38.0000 q^{73} +34.0000 q^{74} +140.000 q^{76} -1248.00 q^{77} -240.000 q^{79} -122.000 q^{82} +1212.00 q^{83} +188.000 q^{86} +780.000 q^{88} -330.000 q^{89} -528.000 q^{91} +1176.00 q^{92} +256.000 q^{94} -866.000 q^{97} +233.000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.353553 0.176777 0.984251i $$-0.443433\pi$$
0.176777 + 0.984251i $$0.443433\pi$$
$$3$$ 0 0
$$4$$ −7.00000 −0.875000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 24.0000 1.29588 0.647939 0.761692i $$-0.275631\pi$$
0.647939 + 0.761692i $$0.275631\pi$$
$$8$$ −15.0000 −0.662913
$$9$$ 0 0
$$10$$ 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 $$-0.575405\pi$$
−0.234681 + 0.972072i $$0.575405\pi$$
$$14$$ 24.0000 0.458162
$$15$$ 0 0
$$16$$ 41.0000 0.640625
$$17$$ −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 $$-0.538528\pi$$
−0.120745 + 0.992684i $$0.538528\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −52.0000 −0.503929
$$23$$ −168.000 −1.52306 −0.761531 0.648129i $$-0.775552\pi$$
−0.761531 + 0.648129i $$0.775552\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −22.0000 −0.165944
$$27$$ 0 0
$$28$$ −168.000 −1.13389
$$29$$ −230.000 −1.47276 −0.736378 0.676570i $$-0.763465\pi$$
−0.736378 + 0.676570i $$0.763465\pi$$
$$30$$ 0 0
$$31$$ −288.000 −1.66859 −0.834296 0.551317i $$-0.814125\pi$$
−0.834296 + 0.551317i $$0.814125\pi$$
$$32$$ 161.000 0.889408
$$33$$ 0 0
$$34$$ −14.0000 −0.0706171
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 34.0000 0.151069 0.0755347 0.997143i $$-0.475934\pi$$
0.0755347 + 0.997143i $$0.475934\pi$$
$$38$$ −20.0000 −0.0853797
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −122.000 −0.464712 −0.232356 0.972631i $$-0.574643\pi$$
−0.232356 + 0.972631i $$0.574643\pi$$
$$42$$ 0 0
$$43$$ 188.000 0.666738 0.333369 0.942796i $$-0.391815\pi$$
0.333369 + 0.942796i $$0.391815\pi$$
$$44$$ 364.000 1.24716
$$45$$ 0 0
$$46$$ −168.000 −0.538484
$$47$$ 256.000 0.794499 0.397249 0.917711i $$-0.369965\pi$$
0.397249 + 0.917711i $$0.369965\pi$$
$$48$$ 0 0
$$49$$ 233.000 0.679300
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 154.000 0.410691
$$53$$ −338.000 −0.875998 −0.437999 0.898976i $$-0.644313\pi$$
−0.437999 + 0.898976i $$0.644313\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −360.000 −0.859054
$$57$$ 0 0
$$58$$ −230.000 −0.520698
$$59$$ −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 $$0.215871\pi$$
$$62$$ −288.000 −0.589936
$$63$$ 0 0
$$64$$ −167.000 −0.326172
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 84.0000 0.153168 0.0765838 0.997063i $$-0.475599\pi$$
0.0765838 + 0.997063i $$0.475599\pi$$
$$68$$ 98.0000 0.174768
$$69$$ 0 0
$$70$$ 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 $$-0.490302\pi$$
0.0304628 + 0.999536i $$0.490302\pi$$
$$74$$ 34.0000 0.0534111
$$75$$ 0 0
$$76$$ 140.000 0.211304
$$77$$ −1248.00 −1.84705
$$78$$ 0 0
$$79$$ −240.000 −0.341799 −0.170899 0.985288i $$-0.554667\pi$$
−0.170899 + 0.985288i $$0.554667\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −122.000 −0.164301
$$83$$ 1212.00 1.60282 0.801411 0.598114i $$-0.204083\pi$$
0.801411 + 0.598114i $$0.204083\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 188.000 0.235727
$$87$$ 0 0
$$88$$ 780.000 0.944867
$$89$$ −330.000 −0.393033 −0.196516 0.980501i $$-0.562963\pi$$
−0.196516 + 0.980501i $$0.562963\pi$$
$$90$$ 0 0
$$91$$ −528.000 −0.608236
$$92$$ 1176.00 1.33268
$$93$$ 0 0
$$94$$ 256.000 0.280898
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −866.000 −0.906484 −0.453242 0.891387i $$-0.649733\pi$$
−0.453242 + 0.891387i $$0.649733\pi$$
$$98$$ 233.000 0.240169
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1218.00 1.19996 0.599978 0.800017i $$-0.295176\pi$$
0.599978 + 0.800017i $$0.295176\pi$$
$$102$$ 0 0
$$103$$ 88.0000 0.0841835 0.0420917 0.999114i $$-0.486598\pi$$
0.0420917 + 0.999114i $$0.486598\pi$$
$$104$$ 330.000 0.311146
$$105$$ 0 0
$$106$$ −338.000 −0.309712
$$107$$ 36.0000 0.0325257 0.0162629 0.999868i $$-0.494823\pi$$
0.0162629 + 0.999868i $$0.494823\pi$$
$$108$$ 0 0
$$109$$ −970.000 −0.852378 −0.426189 0.904634i $$-0.640144\pi$$
−0.426189 + 0.904634i $$0.640144\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 984.000 0.830172
$$113$$ 1042.00 0.867461 0.433731 0.901043i $$-0.357197\pi$$
0.433731 + 0.901043i $$0.357197\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1610.00 1.28866
$$117$$ 0 0
$$118$$ −100.000 −0.0780148
$$119$$ −336.000 −0.258833
$$120$$ 0 0
$$121$$ 1373.00 1.03156
$$122$$ 742.000 0.550635
$$123$$ 0 0
$$124$$ 2016.00 1.46002
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1936.00 −1.35269 −0.676347 0.736583i $$-0.736438\pi$$
−0.676347 + 0.736583i $$0.736438\pi$$
$$128$$ −1455.00 −1.00473
$$129$$ 0 0
$$130$$ 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
$$134$$ 84.0000 0.0541529
$$135$$ 0 0
$$136$$ 210.000 0.132407
$$137$$ −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 $$-0.498058\pi$$
0.00610208 + 0.999981i $$0.498058\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 328.000 0.193839
$$143$$ 1144.00 0.668994
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 38.0000 0.0215404
$$147$$ 0 0
$$148$$ −238.000 −0.132186
$$149$$ 1330.00 0.731261 0.365630 0.930760i $$-0.380853\pi$$
0.365630 + 0.930760i $$0.380853\pi$$
$$150$$ 0 0
$$151$$ −1208.00 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 300.000 0.160087
$$153$$ 0 0
$$154$$ −1248.00 −0.653031
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3514.00 1.78629 0.893146 0.449768i $$-0.148493\pi$$
0.893146 + 0.449768i $$0.148493\pi$$
$$158$$ −240.000 −0.120844
$$159$$ 0 0
$$160$$ 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 $$0.334484\pi$$
$$164$$ 854.000 0.406623
$$165$$ 0 0
$$166$$ 1212.00 0.566683
$$167$$ −24.0000 −0.0111208 −0.00556041 0.999985i $$-0.501770\pi$$
−0.00556041 + 0.999985i $$0.501770\pi$$
$$168$$ 0 0
$$169$$ −1713.00 −0.779700
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1316.00 −0.583396
$$173$$ −618.000 −0.271593 −0.135797 0.990737i $$-0.543359\pi$$
−0.135797 + 0.990737i $$0.543359\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2132.00 −0.913100
$$177$$ 0 0
$$178$$ −330.000 −0.138958
$$179$$ −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 $$0.511636\pi$$
$$182$$ −528.000 −0.215044
$$183$$ 0 0
$$184$$ 2520.00 1.00966
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 728.000 0.284688
$$188$$ −1792.00 −0.695186
$$189$$ 0 0
$$190$$ 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 $$-0.616683\pi$$
−0.358416 + 0.933562i $$0.616683\pi$$
$$194$$ −866.000 −0.320491
$$195$$ 0 0
$$196$$ −1631.00 −0.594388
$$197$$ 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 $$-0.566243\pi$$
−0.206609 + 0.978424i $$0.566243\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 1218.00 0.424248
$$203$$ −5520.00 −1.90851
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 88.0000 0.0297634
$$207$$ 0 0
$$208$$ −902.000 −0.300685
$$209$$ 1040.00 0.344202
$$210$$ 0 0
$$211$$ −4468.00 −1.45777 −0.728886 0.684635i $$-0.759961\pi$$
−0.728886 + 0.684635i $$0.759961\pi$$
$$212$$ 2366.00 0.766498
$$213$$ 0 0
$$214$$ 36.0000 0.0114996
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −6912.00 −2.16229
$$218$$ −970.000 −0.301361
$$219$$ 0 0
$$220$$ 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 $$0.860638\pi$$
$$224$$ 3864.00 1.15256
$$225$$ 0 0
$$226$$ 1042.00 0.306694
$$227$$ 2636.00 0.770738 0.385369 0.922763i $$-0.374074\pi$$
0.385369 + 0.922763i $$0.374074\pi$$
$$228$$ 0 0
$$229$$ 4830.00 1.39378 0.696889 0.717179i $$-0.254567\pi$$
0.696889 + 0.717179i $$0.254567\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3450.00 0.976309
$$233$$ 2682.00 0.754093 0.377046 0.926194i $$-0.376940\pi$$
0.377046 + 0.926194i $$0.376940\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 700.000 0.193077
$$237$$ 0 0
$$238$$ −336.000 −0.0915111
$$239$$ −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 $$0.413785\pi$$
$$242$$ 1373.00 0.364710
$$243$$ 0 0
$$244$$ −5194.00 −1.36275
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 440.000 0.113346
$$248$$ 4320.00 1.10613
$$249$$ 0 0
$$250$$ 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
$$254$$ −1936.00 −0.478250
$$255$$ 0 0
$$256$$ −119.000 −0.0290527
$$257$$ −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
$$261$$ 0 0
$$262$$ −732.000 −0.172607
$$263$$ −4888.00 −1.14603 −0.573017 0.819543i $$-0.694227\pi$$
−0.573017 + 0.819543i $$0.694227\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −480.000 −0.110642
$$267$$ 0 0
$$268$$ −588.000 −0.134022
$$269$$ −1270.00 −0.287856 −0.143928 0.989588i $$-0.545973\pi$$
−0.143928 + 0.989588i $$0.545973\pi$$
$$270$$ 0 0
$$271$$ 1072.00 0.240293 0.120146 0.992756i $$-0.461664\pi$$
0.120146 + 0.992756i $$0.461664\pi$$
$$272$$ −574.000 −0.127955
$$273$$ 0 0
$$274$$ −2214.00 −0.488148
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 5394.00 1.17001 0.585007 0.811028i $$-0.301092\pi$$
0.585007 + 0.811028i $$0.301092\pi$$
$$278$$ 20.0000 0.00431482
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2442.00 −0.518425 −0.259213 0.965820i $$-0.583463\pi$$
−0.259213 + 0.965820i $$0.583463\pi$$
$$282$$ 0 0
$$283$$ −2772.00 −0.582255 −0.291128 0.956684i $$-0.594030\pi$$
−0.291128 + 0.956684i $$0.594030\pi$$
$$284$$ −2296.00 −0.479727
$$285$$ 0 0
$$286$$ 1144.00 0.236525
$$287$$ −2928.00 −0.602210
$$288$$ 0 0
$$289$$ −4717.00 −0.960106
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −266.000 −0.0533098
$$293$$ 4542.00 0.905619 0.452810 0.891607i $$-0.350422\pi$$
0.452810 + 0.891607i $$0.350422\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −510.000 −0.100146
$$297$$ 0 0
$$298$$ 1330.00 0.258540
$$299$$ 3696.00 0.714867
$$300$$ 0 0
$$301$$ 4512.00 0.864011
$$302$$ −1208.00 −0.230174
$$303$$ 0 0
$$304$$ −820.000 −0.154705
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −5116.00 −0.951093 −0.475546 0.879691i $$-0.657750\pi$$
−0.475546 + 0.879691i $$0.657750\pi$$
$$308$$ 8736.00 1.61617
$$309$$ 0 0
$$310$$ 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 $$-0.270233\pi$$
0.660763 + 0.750594i $$0.270233\pi$$
$$314$$ 3514.00 0.631549
$$315$$ 0 0
$$316$$ 1680.00 0.299074
$$317$$ 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
$$321$$ 0 0
$$322$$ −4032.00 −0.697809
$$323$$ 280.000 0.0482341
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 2068.00 0.351337
$$327$$ 0 0
$$328$$ 1830.00 0.308064
$$329$$ 6144.00 1.02957
$$330$$ 0 0
$$331$$ 1332.00 0.221188 0.110594 0.993866i $$-0.464725\pi$$
0.110594 + 0.993866i $$0.464725\pi$$
$$332$$ −8484.00 −1.40247
$$333$$ 0 0
$$334$$ −24.0000 −0.00393180
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 11534.0 1.86438 0.932191 0.361966i $$-0.117894\pi$$
0.932191 + 0.361966i $$0.117894\pi$$
$$338$$ −1713.00 −0.275665
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 14976.0 2.37829
$$342$$ 0 0
$$343$$ −2640.00 −0.415588
$$344$$ −2820.00 −0.441989
$$345$$ 0 0
$$346$$ −618.000 −0.0960228
$$347$$ 11956.0 1.84966 0.924830 0.380382i $$-0.124207\pi$$
0.924830 + 0.380382i $$0.124207\pi$$
$$348$$ 0 0
$$349$$ 4870.00 0.746949 0.373474 0.927640i $$-0.378166\pi$$
0.373474 + 0.927640i $$0.378166\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −8372.00 −1.26770
$$353$$ 10722.0 1.61664 0.808321 0.588742i $$-0.200377\pi$$
0.808321 + 0.588742i $$0.200377\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 2310.00 0.343904
$$357$$ 0 0
$$358$$ −3340.00 −0.493085
$$359$$ −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
$$362$$ −178.000 −0.0258438
$$363$$ 0 0
$$364$$ 3696.00 0.532206
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −3936.00 −0.559830 −0.279915 0.960025i $$-0.590306\pi$$
−0.279915 + 0.960025i $$0.590306\pi$$
$$368$$ −6888.00 −0.975711
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −8112.00 −1.13519
$$372$$ 0 0
$$373$$ −3022.00 −0.419499 −0.209750 0.977755i $$-0.567265\pi$$
−0.209750 + 0.977755i $$0.567265\pi$$
$$374$$ 728.000 0.100652
$$375$$ 0 0
$$376$$ −3840.00 −0.526683
$$377$$ 5060.00 0.691255
$$378$$ 0 0
$$379$$ −13340.0 −1.80799 −0.903997 0.427539i $$-0.859381\pi$$
−0.903997 + 0.427539i $$0.859381\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 1888.00 0.252876
$$383$$ −1008.00 −0.134481 −0.0672407 0.997737i $$-0.521420\pi$$
−0.0672407 + 0.997737i $$0.521420\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1922.00 −0.253438
$$387$$ 0 0
$$388$$ 6062.00 0.793174
$$389$$ −9630.00 −1.25517 −0.627584 0.778549i $$-0.715956\pi$$
−0.627584 + 0.778549i $$0.715956\pi$$
$$390$$ 0 0
$$391$$ 2352.00 0.304209
$$392$$ −3495.00 −0.450317
$$393$$ 0 0
$$394$$ 2526.00 0.322990
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −7126.00 −0.900866 −0.450433 0.892810i $$-0.648730\pi$$
−0.450433 + 0.892810i $$0.648730\pi$$
$$398$$ −1160.00 −0.146094
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 8718.00 1.08568 0.542838 0.839837i $$-0.317350\pi$$
0.542838 + 0.839837i $$0.317350\pi$$
$$402$$ 0 0
$$403$$ 6336.00 0.783173
$$404$$ −8526.00 −1.04996
$$405$$ 0 0
$$406$$ −5520.00 −0.674761
$$407$$ −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
$$411$$ 0 0
$$412$$ −616.000 −0.0736605
$$413$$ −2400.00 −0.285947
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −3542.00 −0.417454
$$417$$ 0 0
$$418$$ 1040.00 0.121694
$$419$$ 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 $$0.484111\pi$$
$$422$$ −4468.00 −0.515400
$$423$$ 0 0
$$424$$ 5070.00 0.580710
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 17808.0 2.01824
$$428$$ −252.000 −0.0284600
$$429$$ 0 0
$$430$$ 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 $$-0.722655\pi$$
−0.643830 + 0.765169i $$0.722655\pi$$
$$434$$ −6912.00 −0.764485
$$435$$ 0 0
$$436$$ 6790.00 0.745830
$$437$$ 3360.00 0.367805
$$438$$ 0 0
$$439$$ −440.000 −0.0478361 −0.0239181 0.999714i $$-0.507614\pi$$
−0.0239181 + 0.999714i $$0.507614\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 308.000 0.0331449
$$443$$ −10188.0 −1.09266 −0.546328 0.837571i $$-0.683975\pi$$
−0.546328 + 0.837571i $$0.683975\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −6032.00 −0.640411
$$447$$ 0 0
$$448$$ −4008.00 −0.422679
$$449$$ 13310.0 1.39897 0.699485 0.714647i $$-0.253413\pi$$
0.699485 + 0.714647i $$0.253413\pi$$
$$450$$ 0 0
$$451$$ 6344.00 0.662367
$$452$$ −7294.00 −0.759029
$$453$$ 0 0
$$454$$ 2636.00 0.272497
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3226.00 −0.330210 −0.165105 0.986276i $$-0.552796\pi$$
−0.165105 + 0.986276i $$0.552796\pi$$
$$458$$ 4830.00 0.492775
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6582.00 −0.664977 −0.332488 0.943107i $$-0.607888\pi$$
−0.332488 + 0.943107i $$0.607888\pi$$
$$462$$ 0 0
$$463$$ −15072.0 −1.51286 −0.756431 0.654073i $$-0.773059\pi$$
−0.756431 + 0.654073i $$0.773059\pi$$
$$464$$ −9430.00 −0.943484
$$465$$ 0 0
$$466$$ 2682.00 0.266612
$$467$$ 476.000 0.0471663 0.0235831 0.999722i $$-0.492493\pi$$
0.0235831 + 0.999722i $$0.492493\pi$$
$$468$$ 0 0
$$469$$ 2016.00 0.198487
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 1500.00 0.146278
$$473$$ −9776.00 −0.950319
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2352.00 0.226478
$$477$$ 0 0
$$478$$ −2320.00 −0.221997
$$479$$ 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
$$482$$ 2002.00 0.189188
$$483$$ 0 0
$$484$$ −9611.00 −0.902611
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 5944.00 0.553077 0.276538 0.961003i $$-0.410813\pi$$
0.276538 + 0.961003i $$0.410813\pi$$
$$488$$ −11130.0 −1.03244
$$489$$ 0 0
$$490$$ 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
$$494$$ 440.000 0.0400740
$$495$$ 0 0
$$496$$ −11808.0 −1.06894
$$497$$ 7872.00 0.710478
$$498$$ 0 0
$$499$$ 8140.00 0.730253 0.365127 0.930958i $$-0.381026\pi$$
0.365127 + 0.930958i $$0.381026\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −132.000 −0.0117360
$$503$$ −13768.0 −1.22045 −0.610223 0.792229i $$-0.708920\pi$$
−0.610223 + 0.792229i $$0.708920\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 8736.00 0.767515
$$507$$ 0 0
$$508$$ 13552.0 1.18361
$$509$$ −22150.0 −1.92884 −0.964422 0.264368i $$-0.914837\pi$$
−0.964422 + 0.264368i $$0.914837\pi$$
$$510$$ 0 0
$$511$$ 912.000 0.0789521
$$512$$ 11521.0 0.994455
$$513$$ 0 0
$$514$$ −7614.00 −0.653384
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −13312.0 −1.13242
$$518$$ 816.000 0.0692143
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −1562.00 −0.131348 −0.0656741 0.997841i $$-0.520920\pi$$
−0.0656741 + 0.997841i $$0.520920\pi$$
$$522$$ 0 0
$$523$$ 668.000 0.0558501 0.0279250 0.999610i $$-0.491110\pi$$
0.0279250 + 0.999610i $$0.491110\pi$$
$$524$$ 5124.00 0.427181
$$525$$ 0 0
$$526$$ −4888.00 −0.405184
$$527$$ 4032.00 0.333276
$$528$$ 0 0
$$529$$ 16057.0 1.31972
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 3360.00 0.273824
$$533$$ 2684.00 0.218118
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −1260.00 −0.101537
$$537$$ 0 0
$$538$$ −1270.00 −0.101772
$$539$$ −12116.0 −0.968225
$$540$$ 0 0
$$541$$ −6138.00 −0.487788 −0.243894 0.969802i $$-0.578425\pi$$
−0.243894 + 0.969802i $$0.578425\pi$$
$$542$$ 1072.00 0.0849564
$$543$$ 0 0
$$544$$ −2254.00 −0.177646
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 10484.0 0.819494 0.409747 0.912199i $$-0.365617\pi$$
0.409747 + 0.912199i $$0.365617\pi$$
$$548$$ 15498.0 1.20811
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 4600.00 0.355656
$$552$$ 0 0
$$553$$ −5760.00 −0.442930
$$554$$ 5394.00 0.413663
$$555$$ 0 0
$$556$$ −140.000 −0.0106786
$$557$$ 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
$$561$$ 0 0
$$562$$ −2442.00 −0.183291
$$563$$ 12252.0 0.917159 0.458579 0.888654i $$-0.348359\pi$$
0.458579 + 0.888654i $$0.348359\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −2772.00 −0.205858
$$567$$ 0 0
$$568$$ −4920.00 −0.363448
$$569$$ 14550.0 1.07200 0.536000 0.844218i $$-0.319935\pi$$
0.536000 + 0.844218i $$0.319935\pi$$
$$570$$ 0 0
$$571$$ −25468.0 −1.86655 −0.933277 0.359157i $$-0.883064\pi$$
−0.933277 + 0.359157i $$0.883064\pi$$
$$572$$ −8008.00 −0.585369
$$573$$ 0 0
$$574$$ −2928.00 −0.212914
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −12866.0 −0.928282 −0.464141 0.885761i $$-0.653637\pi$$
−0.464141 + 0.885761i $$0.653637\pi$$
$$578$$ −4717.00 −0.339449
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 29088.0 2.07706
$$582$$ 0 0
$$583$$ 17576.0 1.24858
$$584$$ −570.000 −0.0403883
$$585$$ 0 0
$$586$$ 4542.00 0.320185
$$587$$ −14844.0 −1.04374 −0.521872 0.853024i $$-0.674766\pi$$
−0.521872 + 0.853024i $$0.674766\pi$$
$$588$$ 0 0
$$589$$ 5760.00 0.402948
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1394.00 0.0967788
$$593$$ 20402.0 1.41283 0.706416 0.707797i $$-0.250311\pi$$
0.706416 + 0.707797i $$0.250311\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −9310.00 −0.639853
$$597$$ 0 0
$$598$$ 3696.00 0.252744
$$599$$ −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 $$0.338939\pi$$
$$602$$ 4512.00 0.305474
$$603$$ 0 0
$$604$$ 8456.00 0.569652
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −11056.0 −0.739290 −0.369645 0.929173i $$-0.620521\pi$$
−0.369645 + 0.929173i $$0.620521\pi$$
$$608$$ −3220.00 −0.214783
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −5632.00 −0.372907
$$612$$ 0 0
$$613$$ 16418.0 1.08176 0.540878 0.841101i $$-0.318092\pi$$
0.540878 + 0.841101i $$0.318092\pi$$
$$614$$ −5116.00 −0.336262
$$615$$ 0 0
$$616$$ 18720.0 1.22443
$$617$$ −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 $$-0.554627\pi$$
−0.170773 + 0.985310i $$0.554627\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 2808.00 0.181014
$$623$$ −7920.00 −0.509323
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 7318.00 0.467230
$$627$$ 0 0
$$628$$ −24598.0 −1.56300
$$629$$ −476.000 −0.0301739
$$630$$ 0 0
$$631$$ 21352.0 1.34708 0.673542 0.739149i $$-0.264772\pi$$
0.673542 + 0.739149i $$0.264772\pi$$
$$632$$ 3600.00 0.226583
$$633$$ 0 0
$$634$$ 2246.00 0.140694
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −5126.00 −0.318838
$$638$$ 11960.0 0.742164
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 29118.0 1.79422 0.897108 0.441812i $$-0.145664\pi$$
0.897108 + 0.441812i $$0.145664\pi$$
$$642$$ 0 0
$$643$$ −5772.00 −0.354005 −0.177003 0.984210i $$-0.556640\pi$$
−0.177003 + 0.984210i $$0.556640\pi$$
$$644$$ 28224.0 1.72699
$$645$$ 0 0
$$646$$ 280.000 0.0170533
$$647$$ −14264.0 −0.866732 −0.433366 0.901218i $$-0.642674\pi$$
−0.433366 + 0.901218i $$0.642674\pi$$
$$648$$ 0 0
$$649$$ 5200.00 0.314511
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −14476.0 −0.869515
$$653$$ 6902.00 0.413623 0.206812 0.978381i $$-0.433691\pi$$
0.206812 + 0.978381i $$0.433691\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −5002.00 −0.297706
$$657$$ 0 0
$$658$$ 6144.00 0.364009
$$659$$ −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 $$0.530182\pi$$
$$662$$ 1332.00 0.0782019
$$663$$ 0 0
$$664$$ −18180.0 −1.06253
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 38640.0 2.24310
$$668$$ 168.000 0.00973071
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −38584.0 −2.21985
$$672$$ 0 0
$$673$$ 7518.00 0.430606 0.215303 0.976547i $$-0.430926\pi$$
0.215303 + 0.976547i $$0.430926\pi$$
$$674$$ 11534.0 0.659159
$$675$$ 0 0
$$676$$ 11991.0 0.682237
$$677$$ −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
$$681$$ 0 0
$$682$$ 14976.0 0.840851
$$683$$ −23868.0 −1.33716 −0.668582 0.743638i $$-0.733099\pi$$
−0.668582 + 0.743638i $$0.733099\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −2640.00 −0.146932
$$687$$ 0 0
$$688$$ 7708.00 0.427129
$$689$$ 7436.00 0.411160
$$690$$ 0 0
$$691$$ 172.000 0.00946916 0.00473458 0.999989i $$-0.498493\pi$$
0.00473458 + 0.999989i $$0.498493\pi$$
$$692$$ 4326.00 0.237644
$$693$$ 0 0
$$694$$ 11956.0 0.653953
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1708.00 0.0928194
$$698$$ 4870.00 0.264086
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22138.0 1.19278 0.596391 0.802694i $$-0.296601\pi$$
0.596391 + 0.802694i $$0.296601\pi$$
$$702$$ 0 0
$$703$$ −680.000 −0.0364818
$$704$$ 8684.00 0.464901
$$705$$ 0 0
$$706$$ 10722.0 0.571569
$$707$$ 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
$$711$$ 0 0
$$712$$ 4950.00 0.260546
$$713$$ 48384.0 2.54137
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 23380.0 1.22032
$$717$$ 0 0
$$718$$ −120.000 −0.00623727
$$719$$ −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
$$722$$ −6459.00 −0.332935
$$723$$ 0 0
$$724$$ 1246.00 0.0639603
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −20696.0 −1.05581 −0.527904 0.849304i $$-0.677022\pi$$
−0.527904 + 0.849304i $$0.677022\pi$$
$$728$$ 7920.00 0.403207
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2632.00 −0.133171
$$732$$ 0 0
$$733$$ 30778.0 1.55090 0.775451 0.631408i $$-0.217522\pi$$
0.775451 + 0.631408i $$0.217522\pi$$
$$734$$ −3936.00 −0.197930
$$735$$ 0 0
$$736$$ −27048.0 −1.35462
$$737$$ −4368.00 −0.218314
$$738$$ 0 0
$$739$$ 11740.0 0.584388 0.292194 0.956359i $$-0.405615\pi$$
0.292194 + 0.956359i $$0.405615\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −8112.00 −0.401349
$$743$$ 2632.00 0.129958 0.0649789 0.997887i $$-0.479302\pi$$
0.0649789 + 0.997887i $$0.479302\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −3022.00 −0.148315
$$747$$ 0 0
$$748$$ −5096.00 −0.249102
$$749$$ 864.000 0.0421494
$$750$$ 0 0
$$751$$ −20528.0 −0.997440 −0.498720 0.866763i $$-0.666196\pi$$
−0.498720 + 0.866763i $$0.666196\pi$$
$$752$$ 10496.0 0.508976
$$753$$ 0 0
$$754$$ 5060.00 0.244396
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −21646.0 −1.03928 −0.519642 0.854384i $$-0.673934\pi$$
−0.519642 + 0.854384i $$0.673934\pi$$
$$758$$ −13340.0 −0.639222
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18282.0 −0.870857 −0.435428 0.900223i $$-0.643403\pi$$
−0.435428 + 0.900223i $$0.643403\pi$$
$$762$$ 0 0
$$763$$ −23280.0 −1.10458
$$764$$ −13216.0 −0.625835
$$765$$ 0 0
$$766$$ −1008.00 −0.0475464
$$767$$ 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
$$771$$ 0 0
$$772$$ 13454.0 0.627228
$$773$$ −25698.0 −1.19572 −0.597861 0.801600i $$-0.703982\pi$$
−0.597861 + 0.801600i $$0.703982\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 12990.0 0.600920
$$777$$ 0 0
$$778$$ −9630.00 −0.443769
$$779$$ 2440.00 0.112223
$$780$$ 0 0
$$781$$ −17056.0 −0.781449
$$782$$ 2352.00 0.107554
$$783$$ 0 0
$$784$$ 9553.00 0.435177
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −33436.0 −1.51444 −0.757220 0.653160i $$-0.773443\pi$$
−0.757220 + 0.653160i $$0.773443\pi$$
$$788$$ −17682.0 −0.799359
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 25008.0 1.12412
$$792$$ 0 0
$$793$$ −16324.0 −0.730999
$$794$$ −7126.00 −0.318504
$$795$$ 0 0
$$796$$ 8120.00 0.361565
$$797$$ −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
$$801$$ 0 0
$$802$$ 8718.00 0.383844
$$803$$ −1976.00 −0.0868388
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 6336.00 0.276893
$$807$$ 0 0
$$808$$ −18270.0 −0.795466
$$809$$ −4730.00 −0.205560 −0.102780 0.994704i $$-0.532774\pi$$
−0.102780 + 0.994704i $$0.532774\pi$$
$$810$$ 0 0
$$811$$ −8748.00 −0.378772 −0.189386 0.981903i $$-0.560650\pi$$
−0.189386 + 0.981903i $$0.560650\pi$$
$$812$$ 38640.0 1.66995
$$813$$ 0 0
$$814$$ −1768.00 −0.0761282
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −3760.00 −0.161011
$$818$$ −10870.0 −0.464622
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −44142.0 −1.87645 −0.938226 0.346024i $$-0.887532\pi$$
−0.938226 + 0.346024i $$0.887532\pi$$
$$822$$ 0 0
$$823$$ −3992.00 −0.169079 −0.0845397 0.996420i $$-0.526942\pi$$
−0.0845397 + 0.996420i $$0.526942\pi$$
$$824$$ −1320.00 −0.0558063
$$825$$ 0 0
$$826$$ −2400.00 −0.101098
$$827$$ −14444.0 −0.607336 −0.303668 0.952778i $$-0.598211\pi$$
−0.303668 + 0.952778i $$0.598211\pi$$
$$828$$ 0 0
$$829$$ 42150.0 1.76590 0.882949 0.469468i $$-0.155554\pi$$
0.882949 + 0.469468i $$0.155554\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 3674.00 0.153093
$$833$$ −3262.00 −0.135680
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −7280.00 −0.301177
$$837$$ 0 0
$$838$$ 9700.00 0.399858
$$839$$ −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
$$842$$ 862.000 0.0352809
$$843$$ 0 0
$$844$$ 31276.0 1.27555
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 32952.0 1.33677
$$848$$ −13858.0 −0.561186
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −5712.00 −0.230088
$$852$$ 0 0
$$853$$ 8658.00 0.347531 0.173766 0.984787i $$-0.444406\pi$$
0.173766 + 0.984787i $$0.444406\pi$$
$$854$$ 17808.0 0.713556
$$855$$ 0 0
$$856$$ −540.000 −0.0215617
$$857$$ 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 $$-0.752653\pi$$
−0.712976 + 0.701189i $$0.752653\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −15792.0 −0.623988
$$863$$ −3088.00 −0.121804 −0.0609019 0.998144i $$-0.519398\pi$$
−0.0609019 + 0.998144i $$0.519398\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −11602.0 −0.455256
$$867$$ 0 0
$$868$$ 48384.0 1.89200
$$869$$ 12480.0 0.487175
$$870$$ 0 0
$$871$$ −1848.00 −0.0718910
$$872$$ 14550.0 0.565052
$$873$$ 0 0
$$874$$ 3360.00 0.130039
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 35274.0 1.35817 0.679087 0.734058i $$-0.262376\pi$$
0.679087 + 0.734058i $$0.262376\pi$$
$$878$$ −440.000 −0.0169126
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −25042.0 −0.957646 −0.478823 0.877911i $$-0.658936\pi$$
−0.478823 + 0.877911i $$0.658936\pi$$
$$882$$ 0 0
$$883$$ −12572.0 −0.479141 −0.239570 0.970879i $$-0.577007\pi$$
−0.239570 + 0.970879i $$0.577007\pi$$
$$884$$ −2156.00 −0.0820296
$$885$$ 0 0
$$886$$ −10188.0 −0.386312
$$887$$ −21864.0 −0.827645 −0.413823 0.910358i $$-0.635807\pi$$
−0.413823 + 0.910358i $$0.635807\pi$$
$$888$$ 0 0
$$889$$ −46464.0 −1.75293
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 42224.0 1.58494
$$893$$ −5120.00 −0.191864
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −34920.0 −1.30200
$$897$$ 0 0
$$898$$ 13310.0 0.494611
$$899$$ 66240.0 2.45743
$$900$$ 0 0
$$901$$ 4732.00 0.174968
$$902$$ 6344.00 0.234182
$$903$$ 0 0
$$904$$ −15630.0 −0.575051
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −31236.0 −1.14352 −0.571761 0.820420i $$-0.693740\pi$$
−0.571761 + 0.820420i $$0.693740\pi$$
$$908$$ −18452.0 −0.674396
$$909$$ 0 0
$$910$$ 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
$$914$$ −3226.00 −0.116747
$$915$$ 0 0
$$916$$ −33810.0 −1.21956
$$917$$ −17568.0 −0.632657
$$918$$ 0 0
$$919$$ 20200.0 0.725067 0.362533 0.931971i $$-0.381912\pi$$
0.362533 + 0.931971i $$0.381912\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −6582.00 −0.235105
$$923$$ −7216.00 −0.257332
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −15072.0 −0.534878
$$927$$ 0 0
$$928$$ −37030.0 −1.30988
$$929$$ −31010.0 −1.09516 −0.547581 0.836753i $$-0.684451\pi$$
−0.547581 + 0.836753i $$0.684451\pi$$
$$930$$ 0 0
$$931$$ −4660.00 −0.164044
$$932$$ −18774.0 −0.659831
$$933$$ 0 0
$$934$$ 476.000 0.0166758
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 39174.0 1.36580 0.682902 0.730510i $$-0.260717\pi$$
0.682902 + 0.730510i $$0.260717\pi$$
$$938$$ 2016.00 0.0701756
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 4138.00 0.143353 0.0716764 0.997428i $$-0.477165\pi$$
0.0716764 + 0.997428i $$0.477165\pi$$
$$942$$ 0 0
$$943$$ 20496.0 0.707785
$$944$$ −4100.00 −0.141360
$$945$$ 0 0
$$946$$ −9776.00 −0.335989
$$947$$ 23676.0 0.812425 0.406213 0.913779i $$-0.366849\pi$$
0.406213 + 0.913779i $$0.366849\pi$$
$$948$$ 0 0
$$949$$ −836.000 −0.0285961
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 5040.00 0.171583
$$953$$ 18922.0 0.643173 0.321586 0.946880i $$-0.395784\pi$$
0.321586 + 0.946880i $$0.395784\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 16240.0 0.549413
$$957$$ 0 0
$$958$$ 19680.0 0.663708
$$959$$ −53136.0 −1.78921
$$960$$ 0 0
$$961$$ 53153.0 1.78420
$$962$$ −748.000 −0.0250691
$$963$$ 0 0
$$964$$ −14014.0 −0.468216
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −39656.0 −1.31877 −0.659385 0.751805i $$-0.729183\pi$$
−0.659385 + 0.751805i $$0.729183\pi$$
$$968$$ −20595.0 −0.683831
$$969$$ 0 0
$$970$$ 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
$$974$$ 5944.00 0.195542
$$975$$ 0 0
$$976$$ 30422.0 0.997730
$$977$$ −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
$$981$$ 0 0
$$982$$ −10772.0 −0.350049
$$983$$ −13608.0 −0.441534 −0.220767 0.975327i $$-0.570856\pi$$
−0.220767 + 0.975327i $$0.570856\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 3220.00 0.104002
$$987$$ 0 0
$$988$$ −3080.00 −0.0991780
$$989$$ −31584.0 −1.01548
$$990$$ 0 0
$$991$$ 13472.0 0.431839 0.215919 0.976411i $$-0.430725\pi$$
0.215919 + 0.976411i $$0.430725\pi$$
$$992$$ −46368.0 −1.48406
$$993$$ 0 0
$$994$$ 7872.00 0.251192
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 3234.00 0.102730 0.0513650 0.998680i $$-0.483643\pi$$
0.0513650 + 0.998680i $$0.483643\pi$$
$$998$$ 8140.00 0.258184
$$999$$ 0 0
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 225.4.a.f.1.1 1
3.2 odd 2 75.4.a.b.1.1 1
5.2 odd 4 225.4.b.e.199.2 2
5.3 odd 4 225.4.b.e.199.1 2
5.4 even 2 45.4.a.c.1.1 1
12.11 even 2 1200.4.a.t.1.1 1
15.2 even 4 75.4.b.b.49.1 2
15.8 even 4 75.4.b.b.49.2 2
15.14 odd 2 15.4.a.a.1.1 1
20.19 odd 2 720.4.a.n.1.1 1
35.34 odd 2 2205.4.a.l.1.1 1
45.4 even 6 405.4.e.i.136.1 2
45.14 odd 6 405.4.e.g.136.1 2
45.29 odd 6 405.4.e.g.271.1 2
45.34 even 6 405.4.e.i.271.1 2
60.23 odd 4 1200.4.f.b.49.2 2
60.47 odd 4 1200.4.f.b.49.1 2
60.59 even 2 240.4.a.e.1.1 1
105.104 even 2 735.4.a.e.1.1 1
120.29 odd 2 960.4.a.b.1.1 1
120.59 even 2 960.4.a.ba.1.1 1
165.164 even 2 1815.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 15.14 odd 2
45.4.a.c.1.1 1 5.4 even 2
75.4.a.b.1.1 1 3.2 odd 2
75.4.b.b.49.1 2 15.2 even 4
75.4.b.b.49.2 2 15.8 even 4
225.4.a.f.1.1 1 1.1 even 1 trivial
225.4.b.e.199.1 2 5.3 odd 4
225.4.b.e.199.2 2 5.2 odd 4
240.4.a.e.1.1 1 60.59 even 2
405.4.e.g.136.1 2 45.14 odd 6
405.4.e.g.271.1 2 45.29 odd 6
405.4.e.i.136.1 2 45.4 even 6
405.4.e.i.271.1 2 45.34 even 6
720.4.a.n.1.1 1 20.19 odd 2
735.4.a.e.1.1 1 105.104 even 2
960.4.a.b.1.1 1 120.29 odd 2
960.4.a.ba.1.1 1 120.59 even 2
1200.4.a.t.1.1 1 12.11 even 2
1200.4.f.b.49.1 2 60.47 odd 4
1200.4.f.b.49.2 2 60.23 odd 4
1815.4.a.e.1.1 1 165.164 even 2
2205.4.a.l.1.1 1 35.34 odd 2