# Properties

 Label 825.4.a.a.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 $$-0.845080\pi$$
−0.883883 + 0.467707i $$0.845080\pi$$
$$3$$ −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 $$-0.525809\pi$$
−0.0809924 + 0.996715i $$0.525809\pi$$
$$8$$ −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 $$-0.610885\pi$$
−0.341354 + 0.939935i $$0.610885\pi$$
$$14$$ 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 $$0.575641\pi$$
$$18$$ −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 $$-0.671177\pi$$
−0.512220 + 0.858854i $$0.671177\pi$$
$$24$$ 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 $$-0.513440\pi$$
−0.0422106 + 0.999109i $$0.513440\pi$$
$$38$$ −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 $$-0.316090\pi$$
0.546158 + 0.837682i $$0.316090\pi$$
$$44$$ −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 $$0.597852\pi$$
$$48$$ −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 $$-0.563110\pi$$
−0.196970 + 0.980410i $$0.563110\pi$$
$$54$$ 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 $$-0.558368\pi$$
−0.182342 + 0.983235i $$0.558368\pi$$
$$68$$ −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 $$-0.383070\pi$$
0.359140 + 0.933284i $$0.383070\pi$$
$$74$$ 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 $$-0.529932\pi$$
−0.0938947 + 0.995582i $$0.529932\pi$$
$$84$$ 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 $$-0.513165\pi$$
−0.0413466 + 0.999145i $$0.513165\pi$$
$$98$$ 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 $$-0.285230\pi$$
0.624680 + 0.780881i $$0.285230\pi$$
$$104$$ 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 $$0.839457\pi$$
$$108$$ −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 $$-0.348015\pi$$
0.459538 + 0.888158i $$0.348015\pi$$
$$114$$ 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 $$-0.292163\pi$$
0.607525 + 0.794301i $$0.292163\pi$$
$$128$$ 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 $$-0.587445\pi$$
−0.271274 + 0.962502i $$0.587445\pi$$
$$138$$ −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 $$-0.341974\pi$$
0.476310 + 0.879277i $$0.341974\pi$$
$$158$$ 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 $$-0.348758\pi$$
0.457463 + 0.889229i $$0.348758\pi$$
$$164$$ 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 $$0.411856\pi$$
$$168$$ −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 $$-0.254029\pi$$
0.698101 + 0.716000i $$0.254029\pi$$
$$174$$ −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 $$-0.786806\pi$$
−0.783965 + 0.620805i $$0.786806\pi$$
$$194$$ 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 $$0.195741\pi$$
$$198$$ 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 $$-0.679507\pi$$
−0.534518 + 0.845157i $$0.679507\pi$$
$$224$$ 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 $$0.260283\pi$$
$$228$$ −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 $$-0.518402\pi$$
−0.0577801 + 0.998329i $$0.518402\pi$$
$$234$$ 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 $$-0.728395\pi$$
−0.657521 + 0.753437i $$0.728395\pi$$
$$258$$ 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 $$-0.371372\pi$$
0.393187 + 0.919458i $$0.371372\pi$$
$$264$$ −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 $$-0.500138\pi$$
−0.000433821 1.00000i $$0.500138\pi$$
$$278$$ 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 $$-0.351488\pi$$
0.449820 + 0.893119i $$0.351488\pi$$
$$284$$ −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 $$-0.262407\pi$$
0.679015 + 0.734124i $$0.262407\pi$$
$$294$$ −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 $$-0.486385\pi$$
0.0427583 + 0.999085i $$0.486385\pi$$
$$308$$ 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 $$-0.320181\pi$$
0.535346 + 0.844633i $$0.320181\pi$$
$$314$$ −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 $$0.647326\pi$$
$$318$$ −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 $$-0.301164\pi$$
0.584822 + 0.811162i $$0.301164\pi$$
$$338$$ 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 $$-0.536223\pi$$
−0.113554 + 0.993532i $$0.536223\pi$$
$$348$$ 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 $$-0.632830\pi$$
−0.405291 + 0.914188i $$0.632830\pi$$
$$354$$ −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 $$-0.239216\pi$$
0.730652 + 0.682750i $$0.239216\pi$$
$$368$$ −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 $$-0.897144\pi$$
−0.948246 + 0.317537i $$0.897144\pi$$
$$374$$ −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 $$-0.432221\pi$$
0.211328 + 0.977415i $$0.432221\pi$$
$$384$$ −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 $$-0.616677\pi$$
−0.358399 + 0.933568i $$0.616677\pi$$
$$398$$ −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 $$-0.656316\pi$$
−0.471579 + 0.881824i $$0.656316\pi$$
$$434$$ 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 $$-0.304536\pi$$
0.576197 + 0.817311i $$0.304536\pi$$
$$444$$ 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 $$-0.382359\pi$$
0.361225 + 0.932479i $$0.382359\pi$$
$$458$$ 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 $$-0.351143\pi$$
0.450787 + 0.892631i $$0.351143\pi$$
$$464$$ −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 $$0.267296\pi$$
$$468$$ −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 $$-0.409548\pi$$
0.280353 + 0.959897i $$0.409548\pi$$
$$488$$ −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 $$-0.486848\pi$$
0.0413080 + 0.999146i $$0.486848\pi$$
$$504$$ 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 $$-0.366384\pi$$
0.407547 + 0.913184i $$0.366384\pi$$
$$524$$ 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 $$-0.630236\pi$$
−0.397827 + 0.917461i $$0.630236\pi$$
$$548$$ −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 $$-0.471948\pi$$
0.0880138 + 0.996119i $$0.471948\pi$$
$$558$$ −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 $$-0.864415\pi$$
−0.910646 + 0.413188i $$0.864415\pi$$
$$564$$ 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 $$-0.284775\pi$$
0.625793 + 0.779989i $$0.284775\pi$$
$$578$$ 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 $$-0.595180\pi$$
−0.294581 + 0.955626i $$0.595180\pi$$
$$588$$ 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 $$-0.521597\pi$$
−0.0677955 + 0.997699i $$0.521597\pi$$
$$594$$ −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 $$-0.501703\pi$$
−0.00534942 + 0.999986i $$0.501703\pi$$
$$608$$ −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 $$-0.437220\pi$$
0.195952 + 0.980613i $$0.437220\pi$$
$$614$$ −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 $$0.503469\pi$$
$$618$$ 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 $$-0.308527\pi$$
0.565906 + 0.824470i $$0.308527\pi$$
$$644$$ 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 $$0.680121\pi$$
$$648$$ −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 $$-0.783061\pi$$
−0.776608 + 0.629984i $$0.783061\pi$$
$$654$$ −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 $$-0.649299\pi$$
−0.452027 + 0.892004i $$0.649299\pi$$
$$674$$ −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 $$0.235097\pi$$
$$678$$ 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 $$-0.361035\pi$$
0.422836 + 0.906206i $$0.361035\pi$$
$$684$$ 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 $$-0.615889\pi$$
−0.356085 + 0.934454i $$0.615889\pi$$
$$728$$ −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 $$-0.574888\pi$$
−0.233104 + 0.972452i $$0.574888\pi$$
$$734$$ −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 $$-0.504024\pi$$
−0.0126403 + 0.999920i $$0.504024\pi$$
$$744$$ 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 $$-0.561966\pi$$
−0.193443 + 0.981111i $$0.561966\pi$$
$$758$$ 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 $$-0.485895\pi$$
0.0442963 + 0.999018i $$0.485895\pi$$
$$774$$ −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 $$-0.554731\pi$$
−0.171097 + 0.985254i $$0.554731\pi$$
$$788$$ 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 $$-0.687067\pi$$
−0.554438 + 0.832225i $$0.687067\pi$$
$$798$$ −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 $$-0.762669\pi$$
−0.734682 + 0.678411i $$0.762669\pi$$
$$824$$ −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 $$0.836457\pi$$
$$828$$ −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 $$-0.836006\pi$$
−0.870193 + 0.492711i $$0.836006\pi$$
$$854$$ 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 $$0.600531\pi$$
$$858$$ 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 $$-0.686701\pi$$
−0.553482 + 0.832861i $$0.686701\pi$$
$$864$$ 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 $$-0.356349\pi$$
0.436130 + 0.899884i $$0.356349\pi$$
$$878$$ −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 $$-0.733822\pi$$
−0.670271 + 0.742116i $$0.733822\pi$$
$$884$$ 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 $$0.698608\pi$$
$$888$$ −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 $$-0.559010\pi$$
−0.184327 + 0.982865i $$0.559010\pi$$
$$908$$ 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 $$-0.507359\pi$$
−0.0231155 + 0.999733i $$0.507359\pi$$
$$938$$ −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 $$-0.679434\pi$$
−0.534325 + 0.845279i $$0.679434\pi$$
$$948$$ 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 $$-0.495245\pi$$
0.0149389 + 0.999888i $$0.495245\pi$$
$$954$$ 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 $$-0.420723\pi$$
0.246488 + 0.969146i $$0.420723\pi$$
$$968$$ −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 $$-0.684463\pi$$
−0.547611 + 0.836733i $$0.684463\pi$$
$$978$$ 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 $$-0.180184\pi$$
0.844018 + 0.536315i $$0.180184\pi$$
$$984$$ 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 $$-0.499039\pi$$
0.00301773 + 0.999995i $$0.499039\pi$$
$$998$$ 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.a.1.1 1
3.2 odd 2 2475.4.a.k.1.1 1
5.2 odd 4 825.4.c.b.199.1 2
5.3 odd 4 825.4.c.b.199.2 2
5.4 even 2 825.4.a.j.1.1 yes 1
15.14 odd 2 2475.4.a.a.1.1 1

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