# Properties

 Label 2475.4.a.a.1.1 Level $2475$ Weight $4$ Character 2475.1 Self dual yes Analytic conductor $146.030$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$146.029727264$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 825) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2475.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.00000 q^{2} +17.0000 q^{4} +3.00000 q^{7} -45.0000 q^{8} +O(q^{10})$$ $$q-5.00000 q^{2} +17.0000 q^{4} +3.00000 q^{7} -45.0000 q^{8} +11.0000 q^{11} +32.0000 q^{13} -15.0000 q^{14} +89.0000 q^{16} -33.0000 q^{17} +47.0000 q^{19} -55.0000 q^{22} -113.000 q^{23} -160.000 q^{26} +51.0000 q^{28} +54.0000 q^{29} +178.000 q^{31} -85.0000 q^{32} +165.000 q^{34} +19.0000 q^{37} -235.000 q^{38} -139.000 q^{41} -308.000 q^{43} +187.000 q^{44} +565.000 q^{46} -195.000 q^{47} -334.000 q^{49} +544.000 q^{52} -152.000 q^{53} -135.000 q^{56} -270.000 q^{58} +625.000 q^{59} +320.000 q^{61} -890.000 q^{62} -287.000 q^{64} +200.000 q^{67} -561.000 q^{68} +947.000 q^{71} -448.000 q^{73} -95.0000 q^{74} +799.000 q^{76} +33.0000 q^{77} -721.000 q^{79} +695.000 q^{82} -142.000 q^{83} +1540.00 q^{86} -495.000 q^{88} -404.000 q^{89} +96.0000 q^{91} -1921.00 q^{92} +975.000 q^{94} +79.0000 q^{97} +1670.00 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.00000 −1.76777 −0.883883 0.467707i $$-0.845080\pi$$
−0.883883 + 0.467707i $$0.845080\pi$$
$$3$$ 0 0
$$4$$ 17.0000 2.12500
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000 0.161985 0.0809924 0.996715i $$-0.474191\pi$$
0.0809924 + 0.996715i $$0.474191\pi$$
$$8$$ −45.0000 −1.98874
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ 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 $$-0.575641\pi$$
−0.235402 + 0.971898i $$0.575641\pi$$
$$18$$ 0 0
$$19$$ 47.0000 0.567502 0.283751 0.958898i $$-0.408421\pi$$
0.283751 + 0.958898i $$0.408421\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −55.0000 −0.533002
$$23$$ −113.000 −1.02444 −0.512220 0.858854i $$-0.671177\pi$$
−0.512220 + 0.858854i $$0.671177\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −160.000 −1.20687
$$27$$ 0 0
$$28$$ 51.0000 0.344218
$$29$$ 54.0000 0.345778 0.172889 0.984941i $$-0.444690\pi$$
0.172889 + 0.984941i $$0.444690\pi$$
$$30$$ 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
$$33$$ 0 0
$$34$$ 165.000 0.832273
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 19.0000 0.0844211 0.0422106 0.999109i $$-0.486560\pi$$
0.0422106 + 0.999109i $$0.486560\pi$$
$$38$$ −235.000 −1.00321
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −139.000 −0.529467 −0.264734 0.964322i $$-0.585284\pi$$
−0.264734 + 0.964322i $$0.585284\pi$$
$$42$$ 0 0
$$43$$ −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 $$-0.597852\pi$$
−0.302592 + 0.953120i $$0.597852\pi$$
$$48$$ 0 0
$$49$$ −334.000 −0.973761
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 544.000 1.45075
$$53$$ −152.000 −0.393940 −0.196970 0.980410i $$-0.563110\pi$$
−0.196970 + 0.980410i $$0.563110\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −135.000 −0.322145
$$57$$ 0 0
$$58$$ −270.000 −0.611254
$$59$$ 625.000 1.37912 0.689560 0.724229i $$-0.257804\pi$$
0.689560 + 0.724229i $$0.257804\pi$$
$$60$$ 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
$$63$$ 0 0
$$64$$ −287.000 −0.560547
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 200.000 0.364685 0.182342 0.983235i $$-0.441632\pi$$
0.182342 + 0.983235i $$0.441632\pi$$
$$68$$ −561.000 −1.00046
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 947.000 1.58293 0.791466 0.611213i $$-0.209318\pi$$
0.791466 + 0.611213i $$0.209318\pi$$
$$72$$ 0 0
$$73$$ −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
$$78$$ 0 0
$$79$$ −721.000 −1.02682 −0.513410 0.858143i $$-0.671618\pi$$
−0.513410 + 0.858143i $$0.671618\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 695.000 0.935975
$$83$$ −142.000 −0.187789 −0.0938947 0.995582i $$-0.529932\pi$$
−0.0938947 + 0.995582i $$0.529932\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1540.00 1.93096
$$87$$ 0 0
$$88$$ −495.000 −0.599627
$$89$$ −404.000 −0.481168 −0.240584 0.970628i $$-0.577339\pi$$
−0.240584 + 0.970628i $$0.577339\pi$$
$$90$$ 0 0
$$91$$ 96.0000 0.110588
$$92$$ −1921.00 −2.17694
$$93$$ 0 0
$$94$$ 975.000 1.06983
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 79.0000 0.0826931 0.0413466 0.999145i $$-0.486835\pi$$
0.0413466 + 0.999145i $$0.486835\pi$$
$$98$$ 1670.00 1.72138
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 545.000 0.536926 0.268463 0.963290i $$-0.413484\pi$$
0.268463 + 0.963290i $$0.413484\pi$$
$$102$$ 0 0
$$103$$ −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 $$-0.839457\pi$$
−0.875484 + 0.483247i $$0.839457\pi$$
$$108$$ 0 0
$$109$$ −576.000 −0.506154 −0.253077 0.967446i $$-0.581443\pi$$
−0.253077 + 0.967446i $$0.581443\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 267.000 0.225260
$$113$$ 1104.00 0.919076 0.459538 0.888158i $$-0.348015\pi$$
0.459538 + 0.888158i $$0.348015\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 918.000 0.734777
$$117$$ 0 0
$$118$$ −3125.00 −2.43796
$$119$$ −99.0000 −0.0762632
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ −1600.00 −1.18735
$$123$$ 0 0
$$124$$ 3026.00 2.19147
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1739.00 −1.21505 −0.607525 0.794301i $$-0.707837\pi$$
−0.607525 + 0.794301i $$0.707837\pi$$
$$128$$ 2115.00 1.46048
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1818.00 −1.21251 −0.606257 0.795269i $$-0.707330\pi$$
−0.606257 + 0.795269i $$0.707330\pi$$
$$132$$ 0 0
$$133$$ 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 $$-0.587445\pi$$
−0.271274 + 0.962502i $$0.587445\pi$$
$$138$$ 0 0
$$139$$ −636.000 −0.388092 −0.194046 0.980992i $$-0.562161\pi$$
−0.194046 + 0.980992i $$0.562161\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −4735.00 −2.79826
$$143$$ 352.000 0.205844
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 2240.00 1.26975
$$147$$ 0 0
$$148$$ 323.000 0.179395
$$149$$ 239.000 0.131407 0.0657035 0.997839i $$-0.479071\pi$$
0.0657035 + 0.997839i $$0.479071\pi$$
$$150$$ 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
$$153$$ 0 0
$$154$$ −165.000 −0.0863382
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −1874.00 −0.952621 −0.476310 0.879277i $$-0.658026\pi$$
−0.476310 + 0.879277i $$0.658026\pi$$
$$158$$ 3605.00 1.81518
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −339.000 −0.165944
$$162$$ 0 0
$$163$$ −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 $$-0.411856\pi$$
0.273387 + 0.961904i $$0.411856\pi$$
$$168$$ 0 0
$$169$$ −1173.00 −0.533910
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −5236.00 −2.32117
$$173$$ 3177.00 1.39620 0.698101 0.716000i $$-0.254029\pi$$
0.698101 + 0.716000i $$0.254029\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 979.000 0.419289
$$177$$ 0 0
$$178$$ 2020.00 0.850592
$$179$$ −1787.00 −0.746182 −0.373091 0.927795i $$-0.621702\pi$$
−0.373091 + 0.927795i $$0.621702\pi$$
$$180$$ 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
$$183$$ 0 0
$$184$$ 5085.00 2.03734
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −363.000 −0.141953
$$188$$ −3315.00 −1.28602
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3613.00 −1.36873 −0.684365 0.729139i $$-0.739921\pi$$
−0.684365 + 0.729139i $$0.739921\pi$$
$$192$$ 0 0
$$193$$ 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 $$-0.195741\pi$$
0.816809 + 0.576908i $$0.195741\pi$$
$$198$$ 0 0
$$199$$ 4164.00 1.48331 0.741654 0.670783i $$-0.234042\pi$$
0.741654 + 0.670783i $$0.234042\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −2725.00 −0.949160
$$203$$ 162.000 0.0560107
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 6530.00 2.20858
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ 9690.00 3.09530
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 534.000 0.167052
$$218$$ 2880.00 0.894762
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1056.00 −0.321422
$$222$$ 0 0
$$223$$ 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 $$-0.260283\pi$$
0.683898 + 0.729577i $$0.260283\pi$$
$$228$$ 0 0
$$229$$ −4447.00 −1.28326 −0.641629 0.767015i $$-0.721741\pi$$
−0.641629 + 0.767015i $$0.721741\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −2430.00 −0.687661
$$233$$ −411.000 −0.115560 −0.0577801 0.998329i $$-0.518402\pi$$
−0.0577801 + 0.998329i $$0.518402\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 10625.0 2.93063
$$237$$ 0 0
$$238$$ 495.000 0.134815
$$239$$ 6380.00 1.72673 0.863364 0.504582i $$-0.168353\pi$$
0.863364 + 0.504582i $$0.168353\pi$$
$$240$$ 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
$$243$$ 0 0
$$244$$ 5440.00 1.42730
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1504.00 0.387438
$$248$$ −8010.00 −2.05095
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 4728.00 1.18896 0.594480 0.804111i $$-0.297358\pi$$
0.594480 + 0.804111i $$0.297358\pi$$
$$252$$ 0 0
$$253$$ −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 $$-0.728395\pi$$
−0.657521 + 0.753437i $$0.728395\pi$$
$$258$$ 0 0
$$259$$ 57.0000 0.0136749
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 9090.00 2.14344
$$263$$ 3354.00 0.786375 0.393187 0.919458i $$-0.371372\pi$$
0.393187 + 0.919458i $$0.371372\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −705.000 −0.162505
$$267$$ 0 0
$$268$$ 3400.00 0.774955
$$269$$ −1062.00 −0.240711 −0.120356 0.992731i $$-0.538403\pi$$
−0.120356 + 0.992731i $$0.538403\pi$$
$$270$$ 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
$$273$$ 0 0
$$274$$ 4350.00 0.959099
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 4.00000 0.000867642 0 0.000433821 1.00000i $$-0.499862\pi$$
0.000433821 1.00000i $$0.499862\pi$$
$$278$$ 3180.00 0.686057
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4647.00 −0.986537 −0.493268 0.869877i $$-0.664198\pi$$
−0.493268 + 0.869877i $$0.664198\pi$$
$$282$$ 0 0
$$283$$ −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
$$288$$ 0 0
$$289$$ −3824.00 −0.778343
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −7616.00 −1.52634
$$293$$ 6811.00 1.35803 0.679015 0.734124i $$-0.262407\pi$$
0.679015 + 0.734124i $$0.262407\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −855.000 −0.167891
$$297$$ 0 0
$$298$$ −1195.00 −0.232297
$$299$$ −3616.00 −0.699394
$$300$$ 0 0
$$301$$ −924.000 −0.176938
$$302$$ −6040.00 −1.15087
$$303$$ 0 0
$$304$$ 4183.00 0.789183
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −460.000 −0.0855166 −0.0427583 0.999085i $$-0.513615\pi$$
−0.0427583 + 0.999085i $$0.513615\pi$$
$$308$$ 561.000 0.103786
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −8328.00 −1.51845 −0.759224 0.650829i $$-0.774421\pi$$
−0.759224 + 0.650829i $$0.774421\pi$$
$$312$$ 0 0
$$313$$ −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 $$-0.647326\pi$$
−0.446490 + 0.894789i $$0.647326\pi$$
$$318$$ 0 0
$$319$$ 594.000 0.104256
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 1695.00 0.293350
$$323$$ −1551.00 −0.267183
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 9520.00 1.61737
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ −5900.00 −0.966568
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −7236.00 −1.16964 −0.584822 0.811162i $$-0.698836\pi$$
−0.584822 + 0.811162i $$0.698836\pi$$
$$338$$ 5865.00 0.943828
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1958.00 0.310943
$$342$$ 0 0
$$343$$ −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 $$-0.536223\pi$$
−0.113554 + 0.993532i $$0.536223\pi$$
$$348$$ 0 0
$$349$$ 5690.00 0.872718 0.436359 0.899773i $$-0.356268\pi$$
0.436359 + 0.899773i $$0.356268\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −935.000 −0.141579
$$353$$ −5376.00 −0.810582 −0.405291 0.914188i $$-0.632830\pi$$
−0.405291 + 0.914188i $$0.632830\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6868.00 −1.02248
$$357$$ 0 0
$$358$$ 8935.00 1.31908
$$359$$ −3734.00 −0.548950 −0.274475 0.961594i $$-0.588504\pi$$
−0.274475 + 0.961594i $$0.588504\pi$$
$$360$$ 0 0
$$361$$ −4650.00 −0.677941
$$362$$ 4175.00 0.606169
$$363$$ 0 0
$$364$$ 1632.00 0.235000
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −10274.0 −1.46130 −0.730652 0.682750i $$-0.760784\pi$$
−0.730652 + 0.682750i $$0.760784\pi$$
$$368$$ −10057.0 −1.42461
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −456.000 −0.0638122
$$372$$ 0 0
$$373$$ 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
$$378$$ 0 0
$$379$$ −7906.00 −1.07151 −0.535757 0.844372i $$-0.679974\pi$$
−0.535757 + 0.844372i $$0.679974\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 18065.0 2.41960
$$383$$ 3168.00 0.422656 0.211328 0.977415i $$-0.432221\pi$$
0.211328 + 0.977415i $$0.432221\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −21020.0 −2.77174
$$387$$ 0 0
$$388$$ 1343.00 0.175723
$$389$$ −10770.0 −1.40375 −0.701877 0.712298i $$-0.747655\pi$$
−0.701877 + 0.712298i $$0.747655\pi$$
$$390$$ 0 0
$$391$$ 3729.00 0.482311
$$392$$ 15030.0 1.93656
$$393$$ 0 0
$$394$$ −22585.0 −2.88786
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 5670.00 0.716799 0.358399 0.933568i $$-0.383323\pi$$
0.358399 + 0.933568i $$0.383323\pi$$
$$398$$ −20820.0 −2.62214
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −832.000 −0.103611 −0.0518056 0.998657i $$-0.516498\pi$$
−0.0518056 + 0.998657i $$0.516498\pi$$
$$402$$ 0 0
$$403$$ 5696.00 0.704064
$$404$$ 9265.00 1.14097
$$405$$ 0 0
$$406$$ −810.000 −0.0990139
$$407$$ 209.000 0.0254539
$$408$$ 0 0
$$409$$ −5712.00 −0.690563 −0.345281 0.938499i $$-0.612217\pi$$
−0.345281 + 0.938499i $$0.612217\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −22202.0 −2.65489
$$413$$ 1875.00 0.223396
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2720.00 −0.320574
$$417$$ 0 0
$$418$$ −2585.00 −0.302480
$$419$$ 4559.00 0.531555 0.265778 0.964034i $$-0.414371\pi$$
0.265778 + 0.964034i $$0.414371\pi$$
$$420$$ 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
$$423$$ 0 0
$$424$$ 6840.00 0.783443
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 960.000 0.108800
$$428$$ −32946.0 −3.72081
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −10770.0 −1.20365 −0.601824 0.798628i $$-0.705559\pi$$
−0.601824 + 0.798628i $$0.705559\pi$$
$$432$$ 0 0
$$433$$ 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
$$438$$ 0 0
$$439$$ 9835.00 1.06925 0.534623 0.845091i $$-0.320454\pi$$
0.534623 + 0.845091i $$0.320454\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 5280.00 0.568199
$$443$$ 10745.0 1.15239 0.576197 0.817311i $$-0.304536\pi$$
0.576197 + 0.817311i $$0.304536\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −17800.0 −1.88981
$$447$$ 0 0
$$448$$ −861.000 −0.0908001
$$449$$ −8356.00 −0.878272 −0.439136 0.898421i $$-0.644715\pi$$
−0.439136 + 0.898421i $$0.644715\pi$$
$$450$$ 0 0
$$451$$ −1529.00 −0.159640
$$452$$ 18768.0 1.95304
$$453$$ 0 0
$$454$$ −23390.0 −2.41795
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −7058.00 −0.722449 −0.361225 0.932479i $$-0.617641\pi$$
−0.361225 + 0.932479i $$0.617641\pi$$
$$458$$ 22235.0 2.26850
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −646.000 −0.0652651 −0.0326326 0.999467i $$-0.510389\pi$$
−0.0326326 + 0.999467i $$0.510389\pi$$
$$462$$ 0 0
$$463$$ −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 $$-0.267296\pi$$
0.667661 + 0.744466i $$0.267296\pi$$
$$468$$ 0 0
$$469$$ 600.000 0.0590734
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −28125.0 −2.74271
$$473$$ −3388.00 −0.329345
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1683.00 −0.162059
$$477$$ 0 0
$$478$$ −31900.0 −3.05245
$$479$$ −12996.0 −1.23967 −0.619835 0.784732i $$-0.712801\pi$$
−0.619835 + 0.784732i $$0.712801\pi$$
$$480$$ 0 0
$$481$$ 608.000 0.0576350
$$482$$ −36410.0 −3.44073
$$483$$ 0 0
$$484$$ 2057.00 0.193182
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −6026.00 −0.560707 −0.280353 0.959897i $$-0.590452\pi$$
−0.280353 + 0.959897i $$0.590452\pi$$
$$488$$ −14400.0 −1.33577
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −11698.0 −1.07520 −0.537600 0.843200i $$-0.680669\pi$$
−0.537600 + 0.843200i $$0.680669\pi$$
$$492$$ 0 0
$$493$$ −1782.00 −0.162794
$$494$$ −7520.00 −0.684900
$$495$$ 0 0
$$496$$ 15842.0 1.43413
$$497$$ 2841.00 0.256411
$$498$$ 0 0
$$499$$ −17052.0 −1.52976 −0.764882 0.644170i $$-0.777203\pi$$
−0.764882 + 0.644170i $$0.777203\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −23640.0 −2.10180
$$503$$ 932.000 0.0826160 0.0413080 0.999146i $$-0.486848\pi$$
0.0413080 + 0.999146i $$0.486848\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 6215.00 0.546029
$$507$$ 0 0
$$508$$ −29563.0 −2.58198
$$509$$ −4384.00 −0.381763 −0.190882 0.981613i $$-0.561135\pi$$
−0.190882 + 0.981613i $$0.561135\pi$$
$$510$$ 0 0
$$511$$ −1344.00 −0.116350
$$512$$ 24475.0 2.11260
$$513$$ 0 0
$$514$$ 27090.0 2.32469
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −2145.00 −0.182470
$$518$$ −285.000 −0.0241741
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 2322.00 0.195257 0.0976283 0.995223i $$-0.468874\pi$$
0.0976283 + 0.995223i $$0.468874\pi$$
$$522$$ 0 0
$$523$$ −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
$$528$$ 0 0
$$529$$ 602.000 0.0494781
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2397.00 0.195344
$$533$$ −4448.00 −0.361471
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −9000.00 −0.725263
$$537$$ 0 0
$$538$$ 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
$$543$$ 0 0
$$544$$ 2805.00 0.221072
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 10179.0 0.795654 0.397827 0.917461i $$-0.369764\pi$$
0.397827 + 0.917461i $$0.369764\pi$$
$$548$$ −14790.0 −1.15292
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2538.00 0.196229
$$552$$ 0 0
$$553$$ −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 $$-0.471948\pi$$
0.0880138 + 0.996119i $$0.471948\pi$$
$$558$$ 0 0
$$559$$ −9856.00 −0.745732
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 23235.0 1.74397
$$563$$ −24330.0 −1.82129 −0.910646 0.413188i $$-0.864415\pi$$
−0.910646 + 0.413188i $$0.864415\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 21415.0 1.59035
$$567$$ 0 0
$$568$$ −42615.0 −3.14804
$$569$$ −3445.00 −0.253817 −0.126909 0.991914i $$-0.540505\pi$$
−0.126909 + 0.991914i $$0.540505\pi$$
$$570$$ 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
$$573$$ 0 0
$$574$$ 2085.00 0.151614
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −17347.0 −1.25159 −0.625793 0.779989i $$-0.715225\pi$$
−0.625793 + 0.779989i $$0.715225\pi$$
$$578$$ 19120.0 1.37593
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −426.000 −0.0304190
$$582$$ 0 0
$$583$$ −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 $$-0.595180\pi$$
−0.294581 + 0.955626i $$0.595180\pi$$
$$588$$ 0 0
$$589$$ 8366.00 0.585255
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1691.00 0.117398
$$593$$ −1958.00 −0.135591 −0.0677955 0.997699i $$-0.521597\pi$$
−0.0677955 + 0.997699i $$0.521597\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4063.00 0.279240
$$597$$ 0 0
$$598$$ 18080.0 1.23636
$$599$$ −23583.0 −1.60864 −0.804320 0.594196i $$-0.797470\pi$$
−0.804320 + 0.594196i $$0.797470\pi$$
$$600$$ 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
$$603$$ 0 0
$$604$$ 20536.0 1.38344
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 160.000 0.0106988 0.00534942 0.999986i $$-0.498297\pi$$
0.00534942 + 0.999986i $$0.498297\pi$$
$$608$$ −3995.00 −0.266478
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6240.00 −0.413164
$$612$$ 0 0
$$613$$ −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 $$-0.503469\pi$$
−0.0108965 + 0.999941i $$0.503469\pi$$
$$618$$ 0 0
$$619$$ −7202.00 −0.467646 −0.233823 0.972279i $$-0.575124\pi$$
−0.233823 + 0.972279i $$0.575124\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 41640.0 2.68426
$$623$$ −1212.00 −0.0779418
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 29645.0 1.89274
$$627$$ 0 0
$$628$$ −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
$$633$$ 0 0
$$634$$ 25200.0 1.57858
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −10688.0 −0.664794
$$638$$ −2970.00 −0.184300
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1228.00 0.0756678 0.0378339 0.999284i $$-0.487954\pi$$
0.0378339 + 0.999284i $$0.487954\pi$$
$$642$$ 0 0
$$643$$ −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 $$-0.680121\pi$$
−0.536148 + 0.844124i $$0.680121\pi$$
$$648$$ 0 0
$$649$$ 6875.00 0.415820
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −32368.0 −1.94422
$$653$$ −25918.0 −1.55322 −0.776608 0.629984i $$-0.783061\pi$$
−0.776608 + 0.629984i $$0.783061\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −12371.0 −0.736290
$$657$$ 0 0
$$658$$ 2925.00 0.173295
$$659$$ −12864.0 −0.760410 −0.380205 0.924902i $$-0.624147\pi$$
−0.380205 + 0.924902i $$0.624147\pi$$
$$660$$ 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
$$663$$ 0 0
$$664$$ 6390.00 0.373464
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6102.00 −0.354228
$$668$$ 20060.0 1.16189
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 3520.00 0.202516
$$672$$ 0 0
$$673$$ 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 $$-0.235097\pi$$
0.739426 + 0.673238i $$0.235097\pi$$
$$678$$ 0 0
$$679$$ 237.000 0.0133950
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −9790.00 −0.549675
$$683$$ 15095.0 0.845672 0.422836 0.906206i $$-0.361035\pi$$
0.422836 + 0.906206i $$0.361035\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 10155.0 0.565189
$$687$$ 0 0
$$688$$ −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
$$693$$ 0 0
$$694$$ 7340.00 0.401473
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 4587.00 0.249275
$$698$$ −28450.0 −1.54276
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −10529.0 −0.567296 −0.283648 0.958928i $$-0.591545\pi$$
−0.283648 + 0.958928i $$0.591545\pi$$
$$702$$ 0 0
$$703$$ 893.000 0.0479092
$$704$$ −3157.00 −0.169011
$$705$$ 0 0
$$706$$ 26880.0 1.43292
$$707$$ 1635.00 0.0869738
$$708$$ 0 0
$$709$$ −16087.0 −0.852130 −0.426065 0.904693i $$-0.640100\pi$$
−0.426065 + 0.904693i $$0.640100\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 18180.0 0.956916
$$713$$ −20114.0 −1.05649
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −30379.0 −1.58564
$$717$$ 0 0
$$718$$ 18670.0 0.970415
$$719$$ −24336.0 −1.26228 −0.631140 0.775669i $$-0.717413\pi$$
−0.631140 + 0.775669i $$0.717413\pi$$
$$720$$ 0 0
$$721$$ −3918.00 −0.202377
$$722$$ 23250.0 1.19844
$$723$$ 0 0
$$724$$ −14195.0 −0.728664
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 13960.0 0.712170 0.356085 0.934454i $$-0.384111\pi$$
0.356085 + 0.934454i $$0.384111\pi$$
$$728$$ −4320.00 −0.219931
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 10164.0 0.514267
$$732$$ 0 0
$$733$$ 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
$$738$$ 0 0
$$739$$ 28453.0 1.41632 0.708160 0.706052i $$-0.249526\pi$$
0.708160 + 0.706052i $$0.249526\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 2280.00 0.112805
$$743$$ −512.000 −0.0252806 −0.0126403 0.999920i $$-0.504024\pi$$
−0.0126403 + 0.999920i $$0.504024\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −68310.0 −3.35256
$$747$$ 0 0
$$748$$ −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
$$753$$ 0 0
$$754$$ −8640.00 −0.417308
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 8058.00 0.386886 0.193443 0.981111i $$-0.438034\pi$$
0.193443 + 0.981111i $$0.438034\pi$$
$$758$$ 39530.0 1.89419
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18650.0 −0.888386 −0.444193 0.895931i $$-0.646510\pi$$
−0.444193 + 0.895931i $$0.646510\pi$$
$$762$$ 0 0
$$763$$ −1728.00 −0.0819893
$$764$$ −61421.0 −2.90855
$$765$$ 0 0
$$766$$ −15840.0 −0.747157
$$767$$ 20000.0 0.941536
$$768$$ 0 0
$$769$$ −7144.00 −0.335005 −0.167503 0.985872i $$-0.553570\pi$$
−0.167503 + 0.985872i $$0.553570\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 71468.0 3.33185
$$773$$ 1904.00 0.0885927 0.0442963 0.999018i $$-0.485895\pi$$
0.0442963 + 0.999018i $$0.485895\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −3555.00 −0.164455
$$777$$ 0 0
$$778$$ 53850.0 2.48151
$$779$$ −6533.00 −0.300474
$$780$$ 0 0
$$781$$ 10417.0 0.477272
$$782$$ −18645.0 −0.852614
$$783$$ 0 0
$$784$$ −29726.0 −1.35414
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 7555.00 0.342194 0.171097 0.985254i $$-0.445269\pi$$
0.171097 + 0.985254i $$0.445269\pi$$
$$788$$ 76789.0 3.47144
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 3312.00 0.148876
$$792$$ 0 0
$$793$$ 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 $$-0.687067\pi$$
−0.554438 + 0.832225i $$0.687067\pi$$
$$798$$ 0 0
$$799$$ 6435.00 0.284924
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 4160.00 0.183160
$$803$$ −4928.00 −0.216570
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −28480.0 −1.24462
$$807$$ 0 0
$$808$$ −24525.0 −1.06781
$$809$$ −19893.0 −0.864525 −0.432262 0.901748i $$-0.642285\pi$$
−0.432262 + 0.901748i $$0.642285\pi$$
$$810$$ 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
$$813$$ 0 0
$$814$$ −1045.00 −0.0449966
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −14476.0 −0.619891
$$818$$ 28560.0 1.22075
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −16890.0 −0.717984 −0.358992 0.933341i $$-0.616880\pi$$
−0.358992 + 0.933341i $$0.616880\pi$$
$$822$$ 0 0
$$823$$ 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 $$-0.836457\pi$$
−0.870891 + 0.491476i $$0.836457\pi$$
$$828$$ 0 0
$$829$$ −18494.0 −0.774817 −0.387408 0.921908i $$-0.626630\pi$$
−0.387408 + 0.921908i $$0.626630\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −9184.00 −0.382690
$$833$$ 11022.0 0.458451
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 8789.00 0.363605
$$837$$ 0 0
$$838$$ −22795.0 −0.939666
$$839$$ −6680.00 −0.274874 −0.137437 0.990511i $$-0.543886\pi$$
−0.137437 + 0.990511i $$0.543886\pi$$
$$840$$ 0 0
$$841$$ −21473.0 −0.880438
$$842$$ −34275.0 −1.40284
$$843$$ 0 0
$$844$$ 79220.0 3.23088
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 363.000 0.0147259
$$848$$ −13528.0 −0.547822
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −2147.00 −0.0864844
$$852$$ 0 0
$$853$$ 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 $$-0.600531\pi$$
−0.310603 + 0.950540i $$0.600531\pi$$
$$858$$ 0 0
$$859$$ −17036.0 −0.676672 −0.338336 0.941025i $$-0.609864\pi$$
−0.338336 + 0.941025i $$0.609864\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 53850.0 2.12777
$$863$$ −28064.0 −1.10696 −0.553482 0.832861i $$-0.686701\pi$$
−0.553482 + 0.832861i $$0.686701\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −42490.0 −1.66729
$$867$$ 0 0
$$868$$ 9078.00 0.354985
$$869$$ −7931.00 −0.309598
$$870$$ 0 0
$$871$$ 6400.00 0.248973
$$872$$ 25920.0 1.00661
$$873$$ 0 0
$$874$$ 26555.0 1.02773
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −22654.0 −0.872259 −0.436130 0.899884i $$-0.643651\pi$$
−0.436130 + 0.899884i $$0.643651\pi$$
$$878$$ −49175.0 −1.89018
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 22380.0 0.855847 0.427924 0.903815i $$-0.359245\pi$$
0.427924 + 0.903815i $$0.359245\pi$$
$$882$$ 0 0
$$883$$ 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 $$-0.698608\pi$$
−0.584242 + 0.811579i $$0.698608\pi$$
$$888$$ 0 0
$$889$$ −5217.00 −0.196820
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 60520.0 2.27170
$$893$$ −9165.00 −0.343443
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 6345.00 0.236575
$$897$$ 0 0
$$898$$ 41780.0 1.55258
$$899$$ 9612.00 0.356594
$$900$$ 0 0
$$901$$ 5016.00 0.185469
$$902$$ 7645.00 0.282207
$$903$$ 0 0
$$904$$ −49680.0 −1.82780
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 10070.0 0.368654 0.184327 0.982865i $$-0.440990\pi$$
0.184327 + 0.982865i $$0.440990\pi$$
$$908$$ 79526.0 2.90657
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −1885.00 −0.0685542 −0.0342771 0.999412i $$-0.510913\pi$$
−0.0342771 + 0.999412i $$0.510913\pi$$
$$912$$ 0 0
$$913$$ −1562.00 −0.0566207
$$914$$ 35290.0 1.27712
$$915$$ 0 0
$$916$$ −75599.0 −2.72692
$$917$$ −5454.00 −0.196409
$$918$$ 0 0
$$919$$ 23703.0 0.850805 0.425403 0.905004i $$-0.360133\pi$$
0.425403 + 0.905004i $$0.360133\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 3230.00 0.115374
$$923$$ 30304.0 1.08068
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 44910.0 1.59377
$$927$$ 0 0
$$928$$ −4590.00 −0.162364
$$929$$ −53804.0 −1.90016 −0.950082 0.312001i $$-0.899001\pi$$
−0.950082 + 0.312001i $$0.899001\pi$$
$$930$$ 0 0
$$931$$ −15698.0 −0.552611
$$932$$ −6987.00 −0.245565
$$933$$ 0 0
$$934$$ −67380.0 −2.36054
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 1326.00 0.0462311 0.0231155 0.999733i $$-0.492641\pi$$
0.0231155 + 0.999733i $$0.492641\pi$$
$$938$$ −3000.00 −0.104428
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 27109.0 0.939137 0.469569 0.882896i $$-0.344409\pi$$
0.469569 + 0.882896i $$0.344409\pi$$
$$942$$ 0 0
$$943$$ 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 $$-0.679434\pi$$
−0.534325 + 0.845279i $$0.679434\pi$$
$$948$$ 0 0
$$949$$ −14336.0 −0.490375
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 4455.00 0.151667
$$953$$ 879.000 0.0298779 0.0149389 0.999888i $$-0.495245\pi$$
0.0149389 + 0.999888i $$0.495245\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 108460. 3.66930
$$957$$ 0 0
$$958$$ 64980.0 2.19145
$$959$$ −2610.00 −0.0878846
$$960$$ 0 0
$$961$$ 1893.00 0.0635427
$$962$$ −3040.00 −0.101885
$$963$$ 0 0
$$964$$ 123794. 4.13603
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −14824.0 −0.492976 −0.246488 0.969146i $$-0.579277\pi$$
−0.246488 + 0.969146i $$0.579277\pi$$
$$968$$ −5445.00 −0.180794
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −34089.0 −1.12664 −0.563320 0.826239i $$-0.690476\pi$$
−0.563320 + 0.826239i $$0.690476\pi$$
$$972$$ 0 0
$$973$$ −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 $$-0.684463\pi$$
−0.547611 + 0.836733i $$0.684463\pi$$
$$978$$ 0 0
$$979$$ −4444.00 −0.145077
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 58490.0 1.90070
$$983$$ 52025.0 1.68804 0.844018 0.536315i $$-0.180184\pi$$
0.844018 + 0.536315i $$0.180184\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 8910.00 0.287781
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ −14205.0 −0.453275
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −190.000 −0.00603547 −0.00301773 0.999995i $$-0.500961\pi$$
−0.00301773 + 0.999995i $$0.500961\pi$$
$$998$$ 85260.0 2.70427
$$999$$ 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 2475.4.a.a.1.1 1
3.2 odd 2 825.4.a.j.1.1 yes 1
5.4 even 2 2475.4.a.k.1.1 1
15.2 even 4 825.4.c.b.199.2 2
15.8 even 4 825.4.c.b.199.1 2
15.14 odd 2 825.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.a.1.1 1 15.14 odd 2
825.4.a.j.1.1 yes 1 3.2 odd 2
825.4.c.b.199.1 2 15.8 even 4
825.4.c.b.199.2 2 15.2 even 4
2475.4.a.a.1.1 1 1.1 even 1 trivial
2475.4.a.k.1.1 1 5.4 even 2