# Properties

 Label 825.4.a.j.1.1 Level $825$ Weight $4$ Character 825.1 Self dual yes Analytic conductor $48.677$ 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] = [825,4,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.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: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{2} +3.00000 q^{3} +17.0000 q^{4} +15.0000 q^{6} +3.00000 q^{7} +45.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+5.00000 q^{2} +3.00000 q^{3} +17.0000 q^{4} +15.0000 q^{6} +3.00000 q^{7} +45.0000 q^{8} +9.00000 q^{9} -11.0000 q^{11} +51.0000 q^{12} +32.0000 q^{13} +15.0000 q^{14} +89.0000 q^{16} +33.0000 q^{17} +45.0000 q^{18} +47.0000 q^{19} +9.00000 q^{21} -55.0000 q^{22} +113.000 q^{23} +135.000 q^{24} +160.000 q^{26} +27.0000 q^{27} +51.0000 q^{28} -54.0000 q^{29} +178.000 q^{31} +85.0000 q^{32} -33.0000 q^{33} +165.000 q^{34} +153.000 q^{36} +19.0000 q^{37} +235.000 q^{38} +96.0000 q^{39} +139.000 q^{41} +45.0000 q^{42} -308.000 q^{43} -187.000 q^{44} +565.000 q^{46} +195.000 q^{47} +267.000 q^{48} -334.000 q^{49} +99.0000 q^{51} +544.000 q^{52} +152.000 q^{53} +135.000 q^{54} +135.000 q^{56} +141.000 q^{57} -270.000 q^{58} -625.000 q^{59} +320.000 q^{61} +890.000 q^{62} +27.0000 q^{63} -287.000 q^{64} -165.000 q^{66} +200.000 q^{67} +561.000 q^{68} +339.000 q^{69} -947.000 q^{71} +405.000 q^{72} -448.000 q^{73} +95.0000 q^{74} +799.000 q^{76} -33.0000 q^{77} +480.000 q^{78} -721.000 q^{79} +81.0000 q^{81} +695.000 q^{82} +142.000 q^{83} +153.000 q^{84} -1540.00 q^{86} -162.000 q^{87} -495.000 q^{88} +404.000 q^{89} +96.0000 q^{91} +1921.00 q^{92} +534.000 q^{93} +975.000 q^{94} +255.000 q^{96} +79.0000 q^{97} -1670.00 q^{98} -99.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.00000 1.76777 0.883883 0.467707i $$-0.154920\pi$$
0.883883 + 0.467707i $$0.154920\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 17.0000 2.12500
$$5$$ 0 0
$$6$$ 15.0000 1.02062
$$7$$ 3.00000 0.161985 0.0809924 0.996715i $$-0.474191\pi$$
0.0809924 + 0.996715i $$0.474191\pi$$
$$8$$ 45.0000 1.98874
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 51.0000 1.22687
$$13$$ 32.0000 0.682708 0.341354 0.939935i $$-0.389115\pi$$
0.341354 + 0.939935i $$0.389115\pi$$
$$14$$ 15.0000 0.286351
$$15$$ 0 0
$$16$$ 89.0000 1.39062
$$17$$ 33.0000 0.470804 0.235402 0.971898i $$-0.424359\pi$$
0.235402 + 0.971898i $$0.424359\pi$$
$$18$$ 45.0000 0.589256
$$19$$ 47.0000 0.567502 0.283751 0.958898i $$-0.408421\pi$$
0.283751 + 0.958898i $$0.408421\pi$$
$$20$$ 0 0
$$21$$ 9.00000 0.0935220
$$22$$ −55.0000 −0.533002
$$23$$ 113.000 1.02444 0.512220 0.858854i $$-0.328823\pi$$
0.512220 + 0.858854i $$0.328823\pi$$
$$24$$ 135.000 1.14820
$$25$$ 0 0
$$26$$ 160.000 1.20687
$$27$$ 27.0000 0.192450
$$28$$ 51.0000 0.344218
$$29$$ −54.0000 −0.345778 −0.172889 0.984941i $$-0.555310\pi$$
−0.172889 + 0.984941i $$0.555310\pi$$
$$30$$ 0 0
$$31$$ 178.000 1.03128 0.515641 0.856805i $$-0.327554\pi$$
0.515641 + 0.856805i $$0.327554\pi$$
$$32$$ 85.0000 0.469563
$$33$$ −33.0000 −0.174078
$$34$$ 165.000 0.832273
$$35$$ 0 0
$$36$$ 153.000 0.708333
$$37$$ 19.0000 0.0844211 0.0422106 0.999109i $$-0.486560\pi$$
0.0422106 + 0.999109i $$0.486560\pi$$
$$38$$ 235.000 1.00321
$$39$$ 96.0000 0.394162
$$40$$ 0 0
$$41$$ 139.000 0.529467 0.264734 0.964322i $$-0.414716\pi$$
0.264734 + 0.964322i $$0.414716\pi$$
$$42$$ 45.0000 0.165325
$$43$$ −308.000 −1.09232 −0.546158 0.837682i $$-0.683910\pi$$
−0.546158 + 0.837682i $$0.683910\pi$$
$$44$$ −187.000 −0.640712
$$45$$ 0 0
$$46$$ 565.000 1.81097
$$47$$ 195.000 0.605185 0.302592 0.953120i $$-0.402148\pi$$
0.302592 + 0.953120i $$0.402148\pi$$
$$48$$ 267.000 0.802878
$$49$$ −334.000 −0.973761
$$50$$ 0 0
$$51$$ 99.0000 0.271819
$$52$$ 544.000 1.45075
$$53$$ 152.000 0.393940 0.196970 0.980410i $$-0.436890\pi$$
0.196970 + 0.980410i $$0.436890\pi$$
$$54$$ 135.000 0.340207
$$55$$ 0 0
$$56$$ 135.000 0.322145
$$57$$ 141.000 0.327647
$$58$$ −270.000 −0.611254
$$59$$ −625.000 −1.37912 −0.689560 0.724229i $$-0.742196\pi$$
−0.689560 + 0.724229i $$0.742196\pi$$
$$60$$ 0 0
$$61$$ 320.000 0.671669 0.335834 0.941921i $$-0.390982\pi$$
0.335834 + 0.941921i $$0.390982\pi$$
$$62$$ 890.000 1.82307
$$63$$ 27.0000 0.0539949
$$64$$ −287.000 −0.560547
$$65$$ 0 0
$$66$$ −165.000 −0.307729
$$67$$ 200.000 0.364685 0.182342 0.983235i $$-0.441632\pi$$
0.182342 + 0.983235i $$0.441632\pi$$
$$68$$ 561.000 1.00046
$$69$$ 339.000 0.591461
$$70$$ 0 0
$$71$$ −947.000 −1.58293 −0.791466 0.611213i $$-0.790682\pi$$
−0.791466 + 0.611213i $$0.790682\pi$$
$$72$$ 405.000 0.662913
$$73$$ −448.000 −0.718280 −0.359140 0.933284i $$-0.616930\pi$$
−0.359140 + 0.933284i $$0.616930\pi$$
$$74$$ 95.0000 0.149237
$$75$$ 0 0
$$76$$ 799.000 1.20594
$$77$$ −33.0000 −0.0488402
$$78$$ 480.000 0.696786
$$79$$ −721.000 −1.02682 −0.513410 0.858143i $$-0.671618\pi$$
−0.513410 + 0.858143i $$0.671618\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 695.000 0.935975
$$83$$ 142.000 0.187789 0.0938947 0.995582i $$-0.470068\pi$$
0.0938947 + 0.995582i $$0.470068\pi$$
$$84$$ 153.000 0.198734
$$85$$ 0 0
$$86$$ −1540.00 −1.93096
$$87$$ −162.000 −0.199635
$$88$$ −495.000 −0.599627
$$89$$ 404.000 0.481168 0.240584 0.970628i $$-0.422661\pi$$
0.240584 + 0.970628i $$0.422661\pi$$
$$90$$ 0 0
$$91$$ 96.0000 0.110588
$$92$$ 1921.00 2.17694
$$93$$ 534.000 0.595411
$$94$$ 975.000 1.06983
$$95$$ 0 0
$$96$$ 255.000 0.271102
$$97$$ 79.0000 0.0826931 0.0413466 0.999145i $$-0.486835\pi$$
0.0413466 + 0.999145i $$0.486835\pi$$
$$98$$ −1670.00 −1.72138
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ −545.000 −0.536926 −0.268463 0.963290i $$-0.586516\pi$$
−0.268463 + 0.963290i $$0.586516\pi$$
$$102$$ 495.000 0.480513
$$103$$ −1306.00 −1.24936 −0.624680 0.780881i $$-0.714770\pi$$
−0.624680 + 0.780881i $$0.714770\pi$$
$$104$$ 1440.00 1.35773
$$105$$ 0 0
$$106$$ 760.000 0.696394
$$107$$ 1938.00 1.75097 0.875484 0.483247i $$-0.160543\pi$$
0.875484 + 0.483247i $$0.160543\pi$$
$$108$$ 459.000 0.408956
$$109$$ −576.000 −0.506154 −0.253077 0.967446i $$-0.581443\pi$$
−0.253077 + 0.967446i $$0.581443\pi$$
$$110$$ 0 0
$$111$$ 57.0000 0.0487405
$$112$$ 267.000 0.225260
$$113$$ −1104.00 −0.919076 −0.459538 0.888158i $$-0.651985\pi$$
−0.459538 + 0.888158i $$0.651985\pi$$
$$114$$ 705.000 0.579204
$$115$$ 0 0
$$116$$ −918.000 −0.734777
$$117$$ 288.000 0.227569
$$118$$ −3125.00 −2.43796
$$119$$ 99.0000 0.0762632
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 1600.00 1.18735
$$123$$ 417.000 0.305688
$$124$$ 3026.00 2.19147
$$125$$ 0 0
$$126$$ 135.000 0.0954504
$$127$$ −1739.00 −1.21505 −0.607525 0.794301i $$-0.707837\pi$$
−0.607525 + 0.794301i $$0.707837\pi$$
$$128$$ −2115.00 −1.46048
$$129$$ −924.000 −0.630649
$$130$$ 0 0
$$131$$ 1818.00 1.21251 0.606257 0.795269i $$-0.292670\pi$$
0.606257 + 0.795269i $$0.292670\pi$$
$$132$$ −561.000 −0.369915
$$133$$ 141.000 0.0919267
$$134$$ 1000.00 0.644678
$$135$$ 0 0
$$136$$ 1485.00 0.936307
$$137$$ 870.000 0.542548 0.271274 0.962502i $$-0.412555\pi$$
0.271274 + 0.962502i $$0.412555\pi$$
$$138$$ 1695.00 1.04557
$$139$$ −636.000 −0.388092 −0.194046 0.980992i $$-0.562161\pi$$
−0.194046 + 0.980992i $$0.562161\pi$$
$$140$$ 0 0
$$141$$ 585.000 0.349403
$$142$$ −4735.00 −2.79826
$$143$$ −352.000 −0.205844
$$144$$ 801.000 0.463542
$$145$$ 0 0
$$146$$ −2240.00 −1.26975
$$147$$ −1002.00 −0.562201
$$148$$ 323.000 0.179395
$$149$$ −239.000 −0.131407 −0.0657035 0.997839i $$-0.520929\pi$$
−0.0657035 + 0.997839i $$0.520929\pi$$
$$150$$ 0 0
$$151$$ 1208.00 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 2115.00 1.12861
$$153$$ 297.000 0.156935
$$154$$ −165.000 −0.0863382
$$155$$ 0 0
$$156$$ 1632.00 0.837593
$$157$$ −1874.00 −0.952621 −0.476310 0.879277i $$-0.658026\pi$$
−0.476310 + 0.879277i $$0.658026\pi$$
$$158$$ −3605.00 −1.81518
$$159$$ 456.000 0.227441
$$160$$ 0 0
$$161$$ 339.000 0.165944
$$162$$ 405.000 0.196419
$$163$$ −1904.00 −0.914925 −0.457463 0.889229i $$-0.651242\pi$$
−0.457463 + 0.889229i $$0.651242\pi$$
$$164$$ 2363.00 1.12512
$$165$$ 0 0
$$166$$ 710.000 0.331968
$$167$$ −1180.00 −0.546773 −0.273387 0.961904i $$-0.588144\pi$$
−0.273387 + 0.961904i $$0.588144\pi$$
$$168$$ 405.000 0.185991
$$169$$ −1173.00 −0.533910
$$170$$ 0 0
$$171$$ 423.000 0.189167
$$172$$ −5236.00 −2.32117
$$173$$ −3177.00 −1.39620 −0.698101 0.716000i $$-0.745971\pi$$
−0.698101 + 0.716000i $$0.745971\pi$$
$$174$$ −810.000 −0.352908
$$175$$ 0 0
$$176$$ −979.000 −0.419289
$$177$$ −1875.00 −0.796235
$$178$$ 2020.00 0.850592
$$179$$ 1787.00 0.746182 0.373091 0.927795i $$-0.378298\pi$$
0.373091 + 0.927795i $$0.378298\pi$$
$$180$$ 0 0
$$181$$ −835.000 −0.342901 −0.171450 0.985193i $$-0.554845\pi$$
−0.171450 + 0.985193i $$0.554845\pi$$
$$182$$ 480.000 0.195494
$$183$$ 960.000 0.387788
$$184$$ 5085.00 2.03734
$$185$$ 0 0
$$186$$ 2670.00 1.05255
$$187$$ −363.000 −0.141953
$$188$$ 3315.00 1.28602
$$189$$ 81.0000 0.0311740
$$190$$ 0 0
$$191$$ 3613.00 1.36873 0.684365 0.729139i $$-0.260079\pi$$
0.684365 + 0.729139i $$0.260079\pi$$
$$192$$ −861.000 −0.323632
$$193$$ 4204.00 1.56793 0.783965 0.620805i $$-0.213194\pi$$
0.783965 + 0.620805i $$0.213194\pi$$
$$194$$ 395.000 0.146182
$$195$$ 0 0
$$196$$ −5678.00 −2.06924
$$197$$ −4517.00 −1.63362 −0.816809 0.576908i $$-0.804259\pi$$
−0.816809 + 0.576908i $$0.804259\pi$$
$$198$$ −495.000 −0.177667
$$199$$ 4164.00 1.48331 0.741654 0.670783i $$-0.234042\pi$$
0.741654 + 0.670783i $$0.234042\pi$$
$$200$$ 0 0
$$201$$ 600.000 0.210551
$$202$$ −2725.00 −0.949160
$$203$$ −162.000 −0.0560107
$$204$$ 1683.00 0.577616
$$205$$ 0 0
$$206$$ −6530.00 −2.20858
$$207$$ 1017.00 0.341480
$$208$$ 2848.00 0.949391
$$209$$ −517.000 −0.171108
$$210$$ 0 0
$$211$$ 4660.00 1.52042 0.760208 0.649680i $$-0.225097\pi$$
0.760208 + 0.649680i $$0.225097\pi$$
$$212$$ 2584.00 0.837122
$$213$$ −2841.00 −0.913907
$$214$$ 9690.00 3.09530
$$215$$ 0 0
$$216$$ 1215.00 0.382733
$$217$$ 534.000 0.167052
$$218$$ −2880.00 −0.894762
$$219$$ −1344.00 −0.414699
$$220$$ 0 0
$$221$$ 1056.00 0.321422
$$222$$ 285.000 0.0861619
$$223$$ 3560.00 1.06904 0.534518 0.845157i $$-0.320493\pi$$
0.534518 + 0.845157i $$0.320493\pi$$
$$224$$ 255.000 0.0760621
$$225$$ 0 0
$$226$$ −5520.00 −1.62471
$$227$$ −4678.00 −1.36780 −0.683898 0.729577i $$-0.739717\pi$$
−0.683898 + 0.729577i $$0.739717\pi$$
$$228$$ 2397.00 0.696251
$$229$$ −4447.00 −1.28326 −0.641629 0.767015i $$-0.721741\pi$$
−0.641629 + 0.767015i $$0.721741\pi$$
$$230$$ 0 0
$$231$$ −99.0000 −0.0281979
$$232$$ −2430.00 −0.687661
$$233$$ 411.000 0.115560 0.0577801 0.998329i $$-0.481598\pi$$
0.0577801 + 0.998329i $$0.481598\pi$$
$$234$$ 1440.00 0.402290
$$235$$ 0 0
$$236$$ −10625.0 −2.93063
$$237$$ −2163.00 −0.592835
$$238$$ 495.000 0.134815
$$239$$ −6380.00 −1.72673 −0.863364 0.504582i $$-0.831647\pi$$
−0.863364 + 0.504582i $$0.831647\pi$$
$$240$$ 0 0
$$241$$ 7282.00 1.94637 0.973184 0.230027i $$-0.0738813\pi$$
0.973184 + 0.230027i $$0.0738813\pi$$
$$242$$ 605.000 0.160706
$$243$$ 243.000 0.0641500
$$244$$ 5440.00 1.42730
$$245$$ 0 0
$$246$$ 2085.00 0.540385
$$247$$ 1504.00 0.387438
$$248$$ 8010.00 2.05095
$$249$$ 426.000 0.108420
$$250$$ 0 0
$$251$$ −4728.00 −1.18896 −0.594480 0.804111i $$-0.702642\pi$$
−0.594480 + 0.804111i $$0.702642\pi$$
$$252$$ 459.000 0.114739
$$253$$ −1243.00 −0.308880
$$254$$ −8695.00 −2.14792
$$255$$ 0 0
$$256$$ −8279.00 −2.02124
$$257$$ 5418.00 1.31504 0.657521 0.753437i $$-0.271605\pi$$
0.657521 + 0.753437i $$0.271605\pi$$
$$258$$ −4620.00 −1.11484
$$259$$ 57.0000 0.0136749
$$260$$ 0 0
$$261$$ −486.000 −0.115259
$$262$$ 9090.00 2.14344
$$263$$ −3354.00 −0.786375 −0.393187 0.919458i $$-0.628628\pi$$
−0.393187 + 0.919458i $$0.628628\pi$$
$$264$$ −1485.00 −0.346195
$$265$$ 0 0
$$266$$ 705.000 0.162505
$$267$$ 1212.00 0.277802
$$268$$ 3400.00 0.774955
$$269$$ 1062.00 0.240711 0.120356 0.992731i $$-0.461597\pi$$
0.120356 + 0.992731i $$0.461597\pi$$
$$270$$ 0 0
$$271$$ −4821.00 −1.08065 −0.540323 0.841458i $$-0.681698\pi$$
−0.540323 + 0.841458i $$0.681698\pi$$
$$272$$ 2937.00 0.654712
$$273$$ 288.000 0.0638482
$$274$$ 4350.00 0.959099
$$275$$ 0 0
$$276$$ 5763.00 1.25685
$$277$$ 4.00000 0.000867642 0 0.000433821 1.00000i $$-0.499862\pi$$
0.000433821 1.00000i $$0.499862\pi$$
$$278$$ −3180.00 −0.686057
$$279$$ 1602.00 0.343761
$$280$$ 0 0
$$281$$ 4647.00 0.986537 0.493268 0.869877i $$-0.335802\pi$$
0.493268 + 0.869877i $$0.335802\pi$$
$$282$$ 2925.00 0.617664
$$283$$ −4283.00 −0.899639 −0.449820 0.893119i $$-0.648512\pi$$
−0.449820 + 0.893119i $$0.648512\pi$$
$$284$$ −16099.0 −3.36373
$$285$$ 0 0
$$286$$ −1760.00 −0.363885
$$287$$ 417.000 0.0857656
$$288$$ 765.000 0.156521
$$289$$ −3824.00 −0.778343
$$290$$ 0 0
$$291$$ 237.000 0.0477429
$$292$$ −7616.00 −1.52634
$$293$$ −6811.00 −1.35803 −0.679015 0.734124i $$-0.737593\pi$$
−0.679015 + 0.734124i $$0.737593\pi$$
$$294$$ −5010.00 −0.993841
$$295$$ 0 0
$$296$$ 855.000 0.167891
$$297$$ −297.000 −0.0580259
$$298$$ −1195.00 −0.232297
$$299$$ 3616.00 0.699394
$$300$$ 0 0
$$301$$ −924.000 −0.176938
$$302$$ 6040.00 1.15087
$$303$$ −1635.00 −0.309994
$$304$$ 4183.00 0.789183
$$305$$ 0 0
$$306$$ 1485.00 0.277424
$$307$$ −460.000 −0.0855166 −0.0427583 0.999085i $$-0.513615\pi$$
−0.0427583 + 0.999085i $$0.513615\pi$$
$$308$$ −561.000 −0.103786
$$309$$ −3918.00 −0.721318
$$310$$ 0 0
$$311$$ 8328.00 1.51845 0.759224 0.650829i $$-0.225579\pi$$
0.759224 + 0.650829i $$0.225579\pi$$
$$312$$ 4320.00 0.783884
$$313$$ −5929.00 −1.07069 −0.535346 0.844633i $$-0.679819\pi$$
−0.535346 + 0.844633i $$0.679819\pi$$
$$314$$ −9370.00 −1.68401
$$315$$ 0 0
$$316$$ −12257.0 −2.18199
$$317$$ 5040.00 0.892980 0.446490 0.894789i $$-0.352674\pi$$
0.446490 + 0.894789i $$0.352674\pi$$
$$318$$ 2280.00 0.402063
$$319$$ 594.000 0.104256
$$320$$ 0 0
$$321$$ 5814.00 1.01092
$$322$$ 1695.00 0.293350
$$323$$ 1551.00 0.267183
$$324$$ 1377.00 0.236111
$$325$$ 0 0
$$326$$ −9520.00 −1.61737
$$327$$ −1728.00 −0.292228
$$328$$ 6255.00 1.05297
$$329$$ 585.000 0.0980307
$$330$$ 0 0
$$331$$ 10396.0 1.72633 0.863166 0.504920i $$-0.168478\pi$$
0.863166 + 0.504920i $$0.168478\pi$$
$$332$$ 2414.00 0.399053
$$333$$ 171.000 0.0281404
$$334$$ −5900.00 −0.966568
$$335$$ 0 0
$$336$$ 801.000 0.130054
$$337$$ −7236.00 −1.16964 −0.584822 0.811162i $$-0.698836\pi$$
−0.584822 + 0.811162i $$0.698836\pi$$
$$338$$ −5865.00 −0.943828
$$339$$ −3312.00 −0.530629
$$340$$ 0 0
$$341$$ −1958.00 −0.310943
$$342$$ 2115.00 0.334404
$$343$$ −2031.00 −0.319719
$$344$$ −13860.0 −2.17233
$$345$$ 0 0
$$346$$ −15885.0 −2.46816
$$347$$ 1468.00 0.227108 0.113554 0.993532i $$-0.463777\pi$$
0.113554 + 0.993532i $$0.463777\pi$$
$$348$$ −2754.00 −0.424224
$$349$$ 5690.00 0.872718 0.436359 0.899773i $$-0.356268\pi$$
0.436359 + 0.899773i $$0.356268\pi$$
$$350$$ 0 0
$$351$$ 864.000 0.131387
$$352$$ −935.000 −0.141579
$$353$$ 5376.00 0.810582 0.405291 0.914188i $$-0.367170\pi$$
0.405291 + 0.914188i $$0.367170\pi$$
$$354$$ −9375.00 −1.40756
$$355$$ 0 0
$$356$$ 6868.00 1.02248
$$357$$ 297.000 0.0440306
$$358$$ 8935.00 1.31908
$$359$$ 3734.00 0.548950 0.274475 0.961594i $$-0.411496\pi$$
0.274475 + 0.961594i $$0.411496\pi$$
$$360$$ 0 0
$$361$$ −4650.00 −0.677941
$$362$$ −4175.00 −0.606169
$$363$$ 363.000 0.0524864
$$364$$ 1632.00 0.235000
$$365$$ 0 0
$$366$$ 4800.00 0.685519
$$367$$ −10274.0 −1.46130 −0.730652 0.682750i $$-0.760784\pi$$
−0.730652 + 0.682750i $$0.760784\pi$$
$$368$$ 10057.0 1.42461
$$369$$ 1251.00 0.176489
$$370$$ 0 0
$$371$$ 456.000 0.0638122
$$372$$ 9078.00 1.26525
$$373$$ 13662.0 1.89649 0.948246 0.317537i $$-0.102856\pi$$
0.948246 + 0.317537i $$0.102856\pi$$
$$374$$ −1815.00 −0.250940
$$375$$ 0 0
$$376$$ 8775.00 1.20355
$$377$$ −1728.00 −0.236065
$$378$$ 405.000 0.0551083
$$379$$ −7906.00 −1.07151 −0.535757 0.844372i $$-0.679974\pi$$
−0.535757 + 0.844372i $$0.679974\pi$$
$$380$$ 0 0
$$381$$ −5217.00 −0.701509
$$382$$ 18065.0 2.41960
$$383$$ −3168.00 −0.422656 −0.211328 0.977415i $$-0.567779\pi$$
−0.211328 + 0.977415i $$0.567779\pi$$
$$384$$ −6345.00 −0.843208
$$385$$ 0 0
$$386$$ 21020.0 2.77174
$$387$$ −2772.00 −0.364105
$$388$$ 1343.00 0.175723
$$389$$ 10770.0 1.40375 0.701877 0.712298i $$-0.252345\pi$$
0.701877 + 0.712298i $$0.252345\pi$$
$$390$$ 0 0
$$391$$ 3729.00 0.482311
$$392$$ −15030.0 −1.93656
$$393$$ 5454.00 0.700046
$$394$$ −22585.0 −2.88786
$$395$$ 0 0
$$396$$ −1683.00 −0.213571
$$397$$ 5670.00 0.716799 0.358399 0.933568i $$-0.383323\pi$$
0.358399 + 0.933568i $$0.383323\pi$$
$$398$$ 20820.0 2.62214
$$399$$ 423.000 0.0530739
$$400$$ 0 0
$$401$$ 832.000 0.103611 0.0518056 0.998657i $$-0.483502\pi$$
0.0518056 + 0.998657i $$0.483502\pi$$
$$402$$ 3000.00 0.372205
$$403$$ 5696.00 0.704064
$$404$$ −9265.00 −1.14097
$$405$$ 0 0
$$406$$ −810.000 −0.0990139
$$407$$ −209.000 −0.0254539
$$408$$ 4455.00 0.540577
$$409$$ −5712.00 −0.690563 −0.345281 0.938499i $$-0.612217\pi$$
−0.345281 + 0.938499i $$0.612217\pi$$
$$410$$ 0 0
$$411$$ 2610.00 0.313240
$$412$$ −22202.0 −2.65489
$$413$$ −1875.00 −0.223396
$$414$$ 5085.00 0.603657
$$415$$ 0 0
$$416$$ 2720.00 0.320574
$$417$$ −1908.00 −0.224065
$$418$$ −2585.00 −0.302480
$$419$$ −4559.00 −0.531555 −0.265778 0.964034i $$-0.585629\pi$$
−0.265778 + 0.964034i $$0.585629\pi$$
$$420$$ 0 0
$$421$$ 6855.00 0.793568 0.396784 0.917912i $$-0.370126\pi$$
0.396784 + 0.917912i $$0.370126\pi$$
$$422$$ 23300.0 2.68774
$$423$$ 1755.00 0.201728
$$424$$ 6840.00 0.783443
$$425$$ 0 0
$$426$$ −14205.0 −1.61557
$$427$$ 960.000 0.108800
$$428$$ 32946.0 3.72081
$$429$$ −1056.00 −0.118844
$$430$$ 0 0
$$431$$ 10770.0 1.20365 0.601824 0.798628i $$-0.294441\pi$$
0.601824 + 0.798628i $$0.294441\pi$$
$$432$$ 2403.00 0.267626
$$433$$ 8498.00 0.943159 0.471579 0.881824i $$-0.343684\pi$$
0.471579 + 0.881824i $$0.343684\pi$$
$$434$$ 2670.00 0.295309
$$435$$ 0 0
$$436$$ −9792.00 −1.07558
$$437$$ 5311.00 0.581372
$$438$$ −6720.00 −0.733091
$$439$$ 9835.00 1.06925 0.534623 0.845091i $$-0.320454\pi$$
0.534623 + 0.845091i $$0.320454\pi$$
$$440$$ 0 0
$$441$$ −3006.00 −0.324587
$$442$$ 5280.00 0.568199
$$443$$ −10745.0 −1.15239 −0.576197 0.817311i $$-0.695464\pi$$
−0.576197 + 0.817311i $$0.695464\pi$$
$$444$$ 969.000 0.103574
$$445$$ 0 0
$$446$$ 17800.0 1.88981
$$447$$ −717.000 −0.0758679
$$448$$ −861.000 −0.0908001
$$449$$ 8356.00 0.878272 0.439136 0.898421i $$-0.355285\pi$$
0.439136 + 0.898421i $$0.355285\pi$$
$$450$$ 0 0
$$451$$ −1529.00 −0.159640
$$452$$ −18768.0 −1.95304
$$453$$ 3624.00 0.375873
$$454$$ −23390.0 −2.41795
$$455$$ 0 0
$$456$$ 6345.00 0.651605
$$457$$ −7058.00 −0.722449 −0.361225 0.932479i $$-0.617641\pi$$
−0.361225 + 0.932479i $$0.617641\pi$$
$$458$$ −22235.0 −2.26850
$$459$$ 891.000 0.0906064
$$460$$ 0 0
$$461$$ 646.000 0.0652651 0.0326326 0.999467i $$-0.489611\pi$$
0.0326326 + 0.999467i $$0.489611\pi$$
$$462$$ −495.000 −0.0498474
$$463$$ −8982.00 −0.901574 −0.450787 0.892631i $$-0.648857\pi$$
−0.450787 + 0.892631i $$0.648857\pi$$
$$464$$ −4806.00 −0.480847
$$465$$ 0 0
$$466$$ 2055.00 0.204283
$$467$$ −13476.0 −1.33532 −0.667661 0.744466i $$-0.732704\pi$$
−0.667661 + 0.744466i $$0.732704\pi$$
$$468$$ 4896.00 0.483585
$$469$$ 600.000 0.0590734
$$470$$ 0 0
$$471$$ −5622.00 −0.549996
$$472$$ −28125.0 −2.74271
$$473$$ 3388.00 0.329345
$$474$$ −10815.0 −1.04799
$$475$$ 0 0
$$476$$ 1683.00 0.162059
$$477$$ 1368.00 0.131313
$$478$$ −31900.0 −3.05245
$$479$$ 12996.0 1.23967 0.619835 0.784732i $$-0.287199\pi$$
0.619835 + 0.784732i $$0.287199\pi$$
$$480$$ 0 0
$$481$$ 608.000 0.0576350
$$482$$ 36410.0 3.44073
$$483$$ 1017.00 0.0958077
$$484$$ 2057.00 0.193182
$$485$$ 0 0
$$486$$ 1215.00 0.113402
$$487$$ −6026.00 −0.560707 −0.280353 0.959897i $$-0.590452\pi$$
−0.280353 + 0.959897i $$0.590452\pi$$
$$488$$ 14400.0 1.33577
$$489$$ −5712.00 −0.528232
$$490$$ 0 0
$$491$$ 11698.0 1.07520 0.537600 0.843200i $$-0.319331\pi$$
0.537600 + 0.843200i $$0.319331\pi$$
$$492$$ 7089.00 0.649587
$$493$$ −1782.00 −0.162794
$$494$$ 7520.00 0.684900
$$495$$ 0 0
$$496$$ 15842.0 1.43413
$$497$$ −2841.00 −0.256411
$$498$$ 2130.00 0.191662
$$499$$ −17052.0 −1.52976 −0.764882 0.644170i $$-0.777203\pi$$
−0.764882 + 0.644170i $$0.777203\pi$$
$$500$$ 0 0
$$501$$ −3540.00 −0.315680
$$502$$ −23640.0 −2.10180
$$503$$ −932.000 −0.0826160 −0.0413080 0.999146i $$-0.513152\pi$$
−0.0413080 + 0.999146i $$0.513152\pi$$
$$504$$ 1215.00 0.107382
$$505$$ 0 0
$$506$$ −6215.00 −0.546029
$$507$$ −3519.00 −0.308253
$$508$$ −29563.0 −2.58198
$$509$$ 4384.00 0.381763 0.190882 0.981613i $$-0.438865\pi$$
0.190882 + 0.981613i $$0.438865\pi$$
$$510$$ 0 0
$$511$$ −1344.00 −0.116350
$$512$$ −24475.0 −2.11260
$$513$$ 1269.00 0.109216
$$514$$ 27090.0 2.32469
$$515$$ 0 0
$$516$$ −15708.0 −1.34013
$$517$$ −2145.00 −0.182470
$$518$$ 285.000 0.0241741
$$519$$ −9531.00 −0.806097
$$520$$ 0 0
$$521$$ −2322.00 −0.195257 −0.0976283 0.995223i $$-0.531126\pi$$
−0.0976283 + 0.995223i $$0.531126\pi$$
$$522$$ −2430.00 −0.203751
$$523$$ −9749.00 −0.815094 −0.407547 0.913184i $$-0.633616\pi$$
−0.407547 + 0.913184i $$0.633616\pi$$
$$524$$ 30906.0 2.57659
$$525$$ 0 0
$$526$$ −16770.0 −1.39013
$$527$$ 5874.00 0.485532
$$528$$ −2937.00 −0.242077
$$529$$ 602.000 0.0494781
$$530$$ 0 0
$$531$$ −5625.00 −0.459707
$$532$$ 2397.00 0.195344
$$533$$ 4448.00 0.361471
$$534$$ 6060.00 0.491090
$$535$$ 0 0
$$536$$ 9000.00 0.725263
$$537$$ 5361.00 0.430809
$$538$$ 5310.00 0.425521
$$539$$ 3674.00 0.293600
$$540$$ 0 0
$$541$$ 4208.00 0.334410 0.167205 0.985922i $$-0.446526\pi$$
0.167205 + 0.985922i $$0.446526\pi$$
$$542$$ −24105.0 −1.91033
$$543$$ −2505.00 −0.197974
$$544$$ 2805.00 0.221072
$$545$$ 0 0
$$546$$ 1440.00 0.112869
$$547$$ 10179.0 0.795654 0.397827 0.917461i $$-0.369764\pi$$
0.397827 + 0.917461i $$0.369764\pi$$
$$548$$ 14790.0 1.15292
$$549$$ 2880.00 0.223890
$$550$$ 0 0
$$551$$ −2538.00 −0.196229
$$552$$ 15255.0 1.17626
$$553$$ −2163.00 −0.166329
$$554$$ 20.0000 0.00153379
$$555$$ 0 0
$$556$$ −10812.0 −0.824696
$$557$$ −2314.00 −0.176028 −0.0880138 0.996119i $$-0.528052\pi$$
−0.0880138 + 0.996119i $$0.528052\pi$$
$$558$$ 8010.00 0.607689
$$559$$ −9856.00 −0.745732
$$560$$ 0 0
$$561$$ −1089.00 −0.0819565
$$562$$ 23235.0 1.74397
$$563$$ 24330.0 1.82129 0.910646 0.413188i $$-0.135585\pi$$
0.910646 + 0.413188i $$0.135585\pi$$
$$564$$ 9945.00 0.742482
$$565$$ 0 0
$$566$$ −21415.0 −1.59035
$$567$$ 243.000 0.0179983
$$568$$ −42615.0 −3.14804
$$569$$ 3445.00 0.253817 0.126909 0.991914i $$-0.459495\pi$$
0.126909 + 0.991914i $$0.459495\pi$$
$$570$$ 0 0
$$571$$ −13056.0 −0.956877 −0.478438 0.878121i $$-0.658797\pi$$
−0.478438 + 0.878121i $$0.658797\pi$$
$$572$$ −5984.00 −0.437419
$$573$$ 10839.0 0.790237
$$574$$ 2085.00 0.151614
$$575$$ 0 0
$$576$$ −2583.00 −0.186849
$$577$$ −17347.0 −1.25159 −0.625793 0.779989i $$-0.715225\pi$$
−0.625793 + 0.779989i $$0.715225\pi$$
$$578$$ −19120.0 −1.37593
$$579$$ 12612.0 0.905245
$$580$$ 0 0
$$581$$ 426.000 0.0304190
$$582$$ 1185.00 0.0843983
$$583$$ −1672.00 −0.118777
$$584$$ −20160.0 −1.42847
$$585$$ 0 0
$$586$$ −34055.0 −2.40068
$$587$$ 8379.00 0.589162 0.294581 0.955626i $$-0.404820\pi$$
0.294581 + 0.955626i $$0.404820\pi$$
$$588$$ −17034.0 −1.19468
$$589$$ 8366.00 0.585255
$$590$$ 0 0
$$591$$ −13551.0 −0.943170
$$592$$ 1691.00 0.117398
$$593$$ 1958.00 0.135591 0.0677955 0.997699i $$-0.478403\pi$$
0.0677955 + 0.997699i $$0.478403\pi$$
$$594$$ −1485.00 −0.102576
$$595$$ 0 0
$$596$$ −4063.00 −0.279240
$$597$$ 12492.0 0.856388
$$598$$ 18080.0 1.23636
$$599$$ 23583.0 1.60864 0.804320 0.594196i $$-0.202530\pi$$
0.804320 + 0.594196i $$0.202530\pi$$
$$600$$ 0 0
$$601$$ −15328.0 −1.04034 −0.520168 0.854064i $$-0.674131\pi$$
−0.520168 + 0.854064i $$0.674131\pi$$
$$602$$ −4620.00 −0.312786
$$603$$ 1800.00 0.121562
$$604$$ 20536.0 1.38344
$$605$$ 0 0
$$606$$ −8175.00 −0.547998
$$607$$ 160.000 0.0106988 0.00534942 0.999986i $$-0.498297\pi$$
0.00534942 + 0.999986i $$0.498297\pi$$
$$608$$ 3995.00 0.266478
$$609$$ −486.000 −0.0323378
$$610$$ 0 0
$$611$$ 6240.00 0.413164
$$612$$ 5049.00 0.333486
$$613$$ −5948.00 −0.391904 −0.195952 0.980613i $$-0.562780\pi$$
−0.195952 + 0.980613i $$0.562780\pi$$
$$614$$ −2300.00 −0.151173
$$615$$ 0 0
$$616$$ −1485.00 −0.0971304
$$617$$ 334.000 0.0217931 0.0108965 0.999941i $$-0.496531\pi$$
0.0108965 + 0.999941i $$0.496531\pi$$
$$618$$ −19590.0 −1.27512
$$619$$ −7202.00 −0.467646 −0.233823 0.972279i $$-0.575124\pi$$
−0.233823 + 0.972279i $$0.575124\pi$$
$$620$$ 0 0
$$621$$ 3051.00 0.197154
$$622$$ 41640.0 2.68426
$$623$$ 1212.00 0.0779418
$$624$$ 8544.00 0.548131
$$625$$ 0 0
$$626$$ −29645.0 −1.89274
$$627$$ −1551.00 −0.0987894
$$628$$ −31858.0 −2.02432
$$629$$ 627.000 0.0397458
$$630$$ 0 0
$$631$$ 10306.0 0.650199 0.325099 0.945680i $$-0.394602\pi$$
0.325099 + 0.945680i $$0.394602\pi$$
$$632$$ −32445.0 −2.04208
$$633$$ 13980.0 0.877812
$$634$$ 25200.0 1.57858
$$635$$ 0 0
$$636$$ 7752.00 0.483313
$$637$$ −10688.0 −0.664794
$$638$$ 2970.00 0.184300
$$639$$ −8523.00 −0.527644
$$640$$ 0 0
$$641$$ −1228.00 −0.0756678 −0.0378339 0.999284i $$-0.512046\pi$$
−0.0378339 + 0.999284i $$0.512046\pi$$
$$642$$ 29070.0 1.78707
$$643$$ −18454.0 −1.13181 −0.565906 0.824470i $$-0.691473\pi$$
−0.565906 + 0.824470i $$0.691473\pi$$
$$644$$ 5763.00 0.352630
$$645$$ 0 0
$$646$$ 7755.00 0.472316
$$647$$ 17647.0 1.07230 0.536148 0.844124i $$-0.319879\pi$$
0.536148 + 0.844124i $$0.319879\pi$$
$$648$$ 3645.00 0.220971
$$649$$ 6875.00 0.415820
$$650$$ 0 0
$$651$$ 1602.00 0.0964475
$$652$$ −32368.0 −1.94422
$$653$$ 25918.0 1.55322 0.776608 0.629984i $$-0.216939\pi$$
0.776608 + 0.629984i $$0.216939\pi$$
$$654$$ −8640.00 −0.516591
$$655$$ 0 0
$$656$$ 12371.0 0.736290
$$657$$ −4032.00 −0.239427
$$658$$ 2925.00 0.173295
$$659$$ 12864.0 0.760410 0.380205 0.924902i $$-0.375853\pi$$
0.380205 + 0.924902i $$0.375853\pi$$
$$660$$ 0 0
$$661$$ −11419.0 −0.671933 −0.335966 0.941874i $$-0.609063\pi$$
−0.335966 + 0.941874i $$0.609063\pi$$
$$662$$ 51980.0 3.05175
$$663$$ 3168.00 0.185573
$$664$$ 6390.00 0.373464
$$665$$ 0 0
$$666$$ 855.000 0.0497456
$$667$$ −6102.00 −0.354228
$$668$$ −20060.0 −1.16189
$$669$$ 10680.0 0.617209
$$670$$ 0 0
$$671$$ −3520.00 −0.202516
$$672$$ 765.000 0.0439145
$$673$$ 15784.0 0.904054 0.452027 0.892004i $$-0.350701\pi$$
0.452027 + 0.892004i $$0.350701\pi$$
$$674$$ −36180.0 −2.06766
$$675$$ 0 0
$$676$$ −19941.0 −1.13456
$$677$$ −26050.0 −1.47885 −0.739426 0.673238i $$-0.764903\pi$$
−0.739426 + 0.673238i $$0.764903\pi$$
$$678$$ −16560.0 −0.938028
$$679$$ 237.000 0.0133950
$$680$$ 0 0
$$681$$ −14034.0 −0.789698
$$682$$ −9790.00 −0.549675
$$683$$ −15095.0 −0.845672 −0.422836 0.906206i $$-0.638965\pi$$
−0.422836 + 0.906206i $$0.638965\pi$$
$$684$$ 7191.00 0.401981
$$685$$ 0 0
$$686$$ −10155.0 −0.565189
$$687$$ −13341.0 −0.740889
$$688$$ −27412.0 −1.51900
$$689$$ 4864.00 0.268946
$$690$$ 0 0
$$691$$ 15896.0 0.875126 0.437563 0.899188i $$-0.355842\pi$$
0.437563 + 0.899188i $$0.355842\pi$$
$$692$$ −54009.0 −2.96693
$$693$$ −297.000 −0.0162801
$$694$$ 7340.00 0.401473
$$695$$ 0 0
$$696$$ −7290.00 −0.397021
$$697$$ 4587.00 0.249275
$$698$$ 28450.0 1.54276
$$699$$ 1233.00 0.0667187
$$700$$ 0 0
$$701$$ 10529.0 0.567296 0.283648 0.958928i $$-0.408455\pi$$
0.283648 + 0.958928i $$0.408455\pi$$
$$702$$ 4320.00 0.232262
$$703$$ 893.000 0.0479092
$$704$$ 3157.00 0.169011
$$705$$ 0 0
$$706$$ 26880.0 1.43292
$$707$$ −1635.00 −0.0869738
$$708$$ −31875.0 −1.69200
$$709$$ −16087.0 −0.852130 −0.426065 0.904693i $$-0.640100\pi$$
−0.426065 + 0.904693i $$0.640100\pi$$
$$710$$ 0 0
$$711$$ −6489.00 −0.342274
$$712$$ 18180.0 0.956916
$$713$$ 20114.0 1.05649
$$714$$ 1485.00 0.0778358
$$715$$ 0 0
$$716$$ 30379.0 1.58564
$$717$$ −19140.0 −0.996927
$$718$$ 18670.0 0.970415
$$719$$ 24336.0 1.26228 0.631140 0.775669i $$-0.282587\pi$$
0.631140 + 0.775669i $$0.282587\pi$$
$$720$$ 0 0
$$721$$ −3918.00 −0.202377
$$722$$ −23250.0 −1.19844
$$723$$ 21846.0 1.12374
$$724$$ −14195.0 −0.728664
$$725$$ 0 0
$$726$$ 1815.00 0.0927837
$$727$$ 13960.0 0.712170 0.356085 0.934454i $$-0.384111\pi$$
0.356085 + 0.934454i $$0.384111\pi$$
$$728$$ 4320.00 0.219931
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −10164.0 −0.514267
$$732$$ 16320.0 0.824050
$$733$$ 9252.00 0.466208 0.233104 0.972452i $$-0.425112\pi$$
0.233104 + 0.972452i $$0.425112\pi$$
$$734$$ −51370.0 −2.58324
$$735$$ 0 0
$$736$$ 9605.00 0.481039
$$737$$ −2200.00 −0.109957
$$738$$ 6255.00 0.311992
$$739$$ 28453.0 1.41632 0.708160 0.706052i $$-0.249526\pi$$
0.708160 + 0.706052i $$0.249526\pi$$
$$740$$ 0 0
$$741$$ 4512.00 0.223688
$$742$$ 2280.00 0.112805
$$743$$ 512.000 0.0252806 0.0126403 0.999920i $$-0.495976\pi$$
0.0126403 + 0.999920i $$0.495976\pi$$
$$744$$ 24030.0 1.18412
$$745$$ 0 0
$$746$$ 68310.0 3.35256
$$747$$ 1278.00 0.0625965
$$748$$ −6171.00 −0.301650
$$749$$ 5814.00 0.283630
$$750$$ 0 0
$$751$$ 772.000 0.0375109 0.0187554 0.999824i $$-0.494030\pi$$
0.0187554 + 0.999824i $$0.494030\pi$$
$$752$$ 17355.0 0.841585
$$753$$ −14184.0 −0.686446
$$754$$ −8640.00 −0.417308
$$755$$ 0 0
$$756$$ 1377.00 0.0662447
$$757$$ 8058.00 0.386886 0.193443 0.981111i $$-0.438034\pi$$
0.193443 + 0.981111i $$0.438034\pi$$
$$758$$ −39530.0 −1.89419
$$759$$ −3729.00 −0.178332
$$760$$ 0 0
$$761$$ 18650.0 0.888386 0.444193 0.895931i $$-0.353490\pi$$
0.444193 + 0.895931i $$0.353490\pi$$
$$762$$ −26085.0 −1.24010
$$763$$ −1728.00 −0.0819893
$$764$$ 61421.0 2.90855
$$765$$ 0 0
$$766$$ −15840.0 −0.747157
$$767$$ −20000.0 −0.941536
$$768$$ −24837.0 −1.16696
$$769$$ −7144.00 −0.335005 −0.167503 0.985872i $$-0.553570\pi$$
−0.167503 + 0.985872i $$0.553570\pi$$
$$770$$ 0 0
$$771$$ 16254.0 0.759239
$$772$$ 71468.0 3.33185
$$773$$ −1904.00 −0.0885927 −0.0442963 0.999018i $$-0.514105\pi$$
−0.0442963 + 0.999018i $$0.514105\pi$$
$$774$$ −13860.0 −0.643653
$$775$$ 0 0
$$776$$ 3555.00 0.164455
$$777$$ 171.000 0.00789523
$$778$$ 53850.0 2.48151
$$779$$ 6533.00 0.300474
$$780$$ 0 0
$$781$$ 10417.0 0.477272
$$782$$ 18645.0 0.852614
$$783$$ −1458.00 −0.0665449
$$784$$ −29726.0 −1.35414
$$785$$ 0 0
$$786$$ 27270.0 1.23752
$$787$$ 7555.00 0.342194 0.171097 0.985254i $$-0.445269\pi$$
0.171097 + 0.985254i $$0.445269\pi$$
$$788$$ −76789.0 −3.47144
$$789$$ −10062.0 −0.454014
$$790$$ 0 0
$$791$$ −3312.00 −0.148876
$$792$$ −4455.00 −0.199876
$$793$$ 10240.0 0.458554
$$794$$ 28350.0 1.26713
$$795$$ 0 0
$$796$$ 70788.0 3.15203
$$797$$ 24950.0 1.10888 0.554438 0.832225i $$-0.312933\pi$$
0.554438 + 0.832225i $$0.312933\pi$$
$$798$$ 2115.00 0.0938223
$$799$$ 6435.00 0.284924
$$800$$ 0 0
$$801$$ 3636.00 0.160389
$$802$$ 4160.00 0.183160
$$803$$ 4928.00 0.216570
$$804$$ 10200.0 0.447421
$$805$$ 0 0
$$806$$ 28480.0 1.24462
$$807$$ 3186.00 0.138975
$$808$$ −24525.0 −1.06781
$$809$$ 19893.0 0.864525 0.432262 0.901748i $$-0.357715\pi$$
0.432262 + 0.901748i $$0.357715\pi$$
$$810$$ 0 0
$$811$$ 34503.0 1.49391 0.746957 0.664872i $$-0.231514\pi$$
0.746957 + 0.664872i $$0.231514\pi$$
$$812$$ −2754.00 −0.119023
$$813$$ −14463.0 −0.623911
$$814$$ −1045.00 −0.0449966
$$815$$ 0 0
$$816$$ 8811.00 0.377998
$$817$$ −14476.0 −0.619891
$$818$$ −28560.0 −1.22075
$$819$$ 864.000 0.0368628
$$820$$ 0 0
$$821$$ 16890.0 0.717984 0.358992 0.933341i $$-0.383120\pi$$
0.358992 + 0.933341i $$0.383120\pi$$
$$822$$ 13050.0 0.553736
$$823$$ 34692.0 1.46936 0.734682 0.678411i $$-0.237331\pi$$
0.734682 + 0.678411i $$0.237331\pi$$
$$824$$ −58770.0 −2.48465
$$825$$ 0 0
$$826$$ −9375.00 −0.394913
$$827$$ 41424.0 1.74178 0.870891 0.491476i $$-0.163543\pi$$
0.870891 + 0.491476i $$0.163543\pi$$
$$828$$ 17289.0 0.725645
$$829$$ −18494.0 −0.774817 −0.387408 0.921908i $$-0.626630\pi$$
−0.387408 + 0.921908i $$0.626630\pi$$
$$830$$ 0 0
$$831$$ 12.0000 0.000500933 0
$$832$$ −9184.00 −0.382690
$$833$$ −11022.0 −0.458451
$$834$$ −9540.00 −0.396095
$$835$$ 0 0
$$836$$ −8789.00 −0.363605
$$837$$ 4806.00 0.198470
$$838$$ −22795.0 −0.939666
$$839$$ 6680.00 0.274874 0.137437 0.990511i $$-0.456114\pi$$
0.137437 + 0.990511i $$0.456114\pi$$
$$840$$ 0 0
$$841$$ −21473.0 −0.880438
$$842$$ 34275.0 1.40284
$$843$$ 13941.0 0.569577
$$844$$ 79220.0 3.23088
$$845$$ 0 0
$$846$$ 8775.00 0.356608
$$847$$ 363.000 0.0147259
$$848$$ 13528.0 0.547822
$$849$$ −12849.0 −0.519407
$$850$$ 0 0
$$851$$ 2147.00 0.0864844
$$852$$ −48297.0 −1.94205
$$853$$ 43358.0 1.74039 0.870193 0.492711i $$-0.163994\pi$$
0.870193 + 0.492711i $$0.163994\pi$$
$$854$$ 4800.00 0.192333
$$855$$ 0 0
$$856$$ 87210.0 3.48222
$$857$$ 15585.0 0.621206 0.310603 0.950540i $$-0.399469\pi$$
0.310603 + 0.950540i $$0.399469\pi$$
$$858$$ −5280.00 −0.210089
$$859$$ −17036.0 −0.676672 −0.338336 0.941025i $$-0.609864\pi$$
−0.338336 + 0.941025i $$0.609864\pi$$
$$860$$ 0 0
$$861$$ 1251.00 0.0495168
$$862$$ 53850.0 2.12777
$$863$$ 28064.0 1.10696 0.553482 0.832861i $$-0.313299\pi$$
0.553482 + 0.832861i $$0.313299\pi$$
$$864$$ 2295.00 0.0903675
$$865$$ 0 0
$$866$$ 42490.0 1.66729
$$867$$ −11472.0 −0.449377
$$868$$ 9078.00 0.354985
$$869$$ 7931.00 0.309598
$$870$$ 0 0
$$871$$ 6400.00 0.248973
$$872$$ −25920.0 −1.00661
$$873$$ 711.000 0.0275644
$$874$$ 26555.0 1.02773
$$875$$ 0 0
$$876$$ −22848.0 −0.881236
$$877$$ −22654.0 −0.872259 −0.436130 0.899884i $$-0.643651\pi$$
−0.436130 + 0.899884i $$0.643651\pi$$
$$878$$ 49175.0 1.89018
$$879$$ −20433.0 −0.784059
$$880$$ 0 0
$$881$$ −22380.0 −0.855847 −0.427924 0.903815i $$-0.640755\pi$$
−0.427924 + 0.903815i $$0.640755\pi$$
$$882$$ −15030.0 −0.573794
$$883$$ 35174.0 1.34054 0.670271 0.742116i $$-0.266178\pi$$
0.670271 + 0.742116i $$0.266178\pi$$
$$884$$ 17952.0 0.683022
$$885$$ 0 0
$$886$$ −53725.0 −2.03716
$$887$$ 30868.0 1.16848 0.584242 0.811579i $$-0.301392\pi$$
0.584242 + 0.811579i $$0.301392\pi$$
$$888$$ 2565.00 0.0969322
$$889$$ −5217.00 −0.196820
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ 60520.0 2.27170
$$893$$ 9165.00 0.343443
$$894$$ −3585.00 −0.134117
$$895$$ 0 0
$$896$$ −6345.00 −0.236575
$$897$$ 10848.0 0.403795
$$898$$ 41780.0 1.55258
$$899$$ −9612.00 −0.356594
$$900$$ 0 0
$$901$$ 5016.00 0.185469
$$902$$ −7645.00 −0.282207
$$903$$ −2772.00 −0.102155
$$904$$ −49680.0 −1.82780
$$905$$ 0 0
$$906$$ 18120.0 0.664455
$$907$$ 10070.0 0.368654 0.184327 0.982865i $$-0.440990\pi$$
0.184327 + 0.982865i $$0.440990\pi$$
$$908$$ −79526.0 −2.90657
$$909$$ −4905.00 −0.178975
$$910$$ 0 0
$$911$$ 1885.00 0.0685542 0.0342771 0.999412i $$-0.489087\pi$$
0.0342771 + 0.999412i $$0.489087\pi$$
$$912$$ 12549.0 0.455635
$$913$$ −1562.00 −0.0566207
$$914$$ −35290.0 −1.27712
$$915$$ 0 0
$$916$$ −75599.0 −2.72692
$$917$$ 5454.00 0.196409
$$918$$ 4455.00 0.160171
$$919$$ 23703.0 0.850805 0.425403 0.905004i $$-0.360133\pi$$
0.425403 + 0.905004i $$0.360133\pi$$
$$920$$ 0 0
$$921$$ −1380.00 −0.0493730
$$922$$ 3230.00 0.115374
$$923$$ −30304.0 −1.08068
$$924$$ −1683.00 −0.0599206
$$925$$ 0 0
$$926$$ −44910.0 −1.59377
$$927$$ −11754.0 −0.416453
$$928$$ −4590.00 −0.162364
$$929$$ 53804.0 1.90016 0.950082 0.312001i $$-0.100999\pi$$
0.950082 + 0.312001i $$0.100999\pi$$
$$930$$ 0 0
$$931$$ −15698.0 −0.552611
$$932$$ 6987.00 0.245565
$$933$$ 24984.0 0.876677
$$934$$ −67380.0 −2.36054
$$935$$ 0 0
$$936$$ 12960.0 0.452576
$$937$$ 1326.00 0.0462311 0.0231155 0.999733i $$-0.492641\pi$$
0.0231155 + 0.999733i $$0.492641\pi$$
$$938$$ 3000.00 0.104428
$$939$$ −17787.0 −0.618165
$$940$$ 0 0
$$941$$ −27109.0 −0.939137 −0.469569 0.882896i $$-0.655591\pi$$
−0.469569 + 0.882896i $$0.655591\pi$$
$$942$$ −28110.0 −0.972265
$$943$$ 15707.0 0.542408
$$944$$ −55625.0 −1.91784
$$945$$ 0 0
$$946$$ 16940.0 0.582206
$$947$$ 31143.0 1.06865 0.534325 0.845279i $$-0.320566\pi$$
0.534325 + 0.845279i $$0.320566\pi$$
$$948$$ −36771.0 −1.25977
$$949$$ −14336.0 −0.490375
$$950$$ 0 0
$$951$$ 15120.0 0.515562
$$952$$ 4455.00 0.151667
$$953$$ −879.000 −0.0298779 −0.0149389 0.999888i $$-0.504755\pi$$
−0.0149389 + 0.999888i $$0.504755\pi$$
$$954$$ 6840.00 0.232131
$$955$$ 0 0
$$956$$ −108460. −3.66930
$$957$$ 1782.00 0.0601921
$$958$$ 64980.0 2.19145
$$959$$ 2610.00 0.0878846
$$960$$ 0 0
$$961$$ 1893.00 0.0635427
$$962$$ 3040.00 0.101885
$$963$$ 17442.0 0.583656
$$964$$ 123794. 4.13603
$$965$$ 0 0
$$966$$ 5085.00 0.169366
$$967$$ −14824.0 −0.492976 −0.246488 0.969146i $$-0.579277\pi$$
−0.246488 + 0.969146i $$0.579277\pi$$
$$968$$ 5445.00 0.180794
$$969$$ 4653.00 0.154258
$$970$$ 0 0
$$971$$ 34089.0 1.12664 0.563320 0.826239i $$-0.309524\pi$$
0.563320 + 0.826239i $$0.309524\pi$$
$$972$$ 4131.00 0.136319
$$973$$ −1908.00 −0.0628650
$$974$$ −30130.0 −0.991199
$$975$$ 0 0
$$976$$ 28480.0 0.934040
$$977$$ 33446.0 1.09522 0.547611 0.836733i $$-0.315537\pi$$
0.547611 + 0.836733i $$0.315537\pi$$
$$978$$ −28560.0 −0.933792
$$979$$ −4444.00 −0.145077
$$980$$ 0 0
$$981$$ −5184.00 −0.168718
$$982$$ 58490.0 1.90070
$$983$$ −52025.0 −1.68804 −0.844018 0.536315i $$-0.819816\pi$$
−0.844018 + 0.536315i $$0.819816\pi$$
$$984$$ 18765.0 0.607933
$$985$$ 0 0
$$986$$ −8910.00 −0.287781
$$987$$ 1755.00 0.0565980
$$988$$ 25568.0 0.823306
$$989$$ −34804.0 −1.11901
$$990$$ 0 0
$$991$$ −41260.0 −1.32257 −0.661285 0.750135i $$-0.729989\pi$$
−0.661285 + 0.750135i $$0.729989\pi$$
$$992$$ 15130.0 0.484252
$$993$$ 31188.0 0.996698
$$994$$ −14205.0 −0.453275
$$995$$ 0 0
$$996$$ 7242.00 0.230393
$$997$$ −190.000 −0.00603547 −0.00301773 0.999995i $$-0.500961\pi$$
−0.00301773 + 0.999995i $$0.500961\pi$$
$$998$$ −85260.0 −2.70427
$$999$$ 513.000 0.0162468
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 825.4.a.j.1.1 yes 1
3.2 odd 2 2475.4.a.a.1.1 1
5.2 odd 4 825.4.c.b.199.2 2
5.3 odd 4 825.4.c.b.199.1 2
5.4 even 2 825.4.a.a.1.1 1
15.14 odd 2 2475.4.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.a.1.1 1 5.4 even 2
825.4.a.j.1.1 yes 1 1.1 even 1 trivial
825.4.c.b.199.1 2 5.3 odd 4
825.4.c.b.199.2 2 5.2 odd 4
2475.4.a.a.1.1 1 3.2 odd 2
2475.4.a.k.1.1 1 15.14 odd 2