# Properties

 Label 315.4.a.a.1.1 Level $315$ Weight $4$ Character 315.1 Self dual yes Analytic conductor $18.586$ 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] = [315,4,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, 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("315.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 315.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: $$18.5856016518$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 315.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.00000 q^{2} +17.0000 q^{4} -5.00000 q^{5} +7.00000 q^{7} -45.0000 q^{8} +O(q^{10})$$ $$q-5.00000 q^{2} +17.0000 q^{4} -5.00000 q^{5} +7.00000 q^{7} -45.0000 q^{8} +25.0000 q^{10} -12.0000 q^{11} +30.0000 q^{13} -35.0000 q^{14} +89.0000 q^{16} +134.000 q^{17} -92.0000 q^{19} -85.0000 q^{20} +60.0000 q^{22} -112.000 q^{23} +25.0000 q^{25} -150.000 q^{26} +119.000 q^{28} +58.0000 q^{29} -224.000 q^{31} -85.0000 q^{32} -670.000 q^{34} -35.0000 q^{35} -146.000 q^{37} +460.000 q^{38} +225.000 q^{40} -18.0000 q^{41} +340.000 q^{43} -204.000 q^{44} +560.000 q^{46} -208.000 q^{47} +49.0000 q^{49} -125.000 q^{50} +510.000 q^{52} +754.000 q^{53} +60.0000 q^{55} -315.000 q^{56} -290.000 q^{58} -380.000 q^{59} +718.000 q^{61} +1120.00 q^{62} -287.000 q^{64} -150.000 q^{65} +412.000 q^{67} +2278.00 q^{68} +175.000 q^{70} +960.000 q^{71} +1066.00 q^{73} +730.000 q^{74} -1564.00 q^{76} -84.0000 q^{77} +896.000 q^{79} -445.000 q^{80} +90.0000 q^{82} -436.000 q^{83} -670.000 q^{85} -1700.00 q^{86} +540.000 q^{88} +1038.00 q^{89} +210.000 q^{91} -1904.00 q^{92} +1040.00 q^{94} +460.000 q^{95} -702.000 q^{97} -245.000 q^{98} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −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
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ −45.0000 −1.98874
$$9$$ 0 0
$$10$$ 25.0000 0.790569
$$11$$ −12.0000 −0.328921 −0.164461 0.986384i $$-0.552588\pi$$
−0.164461 + 0.986384i $$0.552588\pi$$
$$12$$ 0 0
$$13$$ 30.0000 0.640039 0.320019 0.947411i $$-0.396311\pi$$
0.320019 + 0.947411i $$0.396311\pi$$
$$14$$ −35.0000 −0.668153
$$15$$ 0 0
$$16$$ 89.0000 1.39062
$$17$$ 134.000 1.91175 0.955876 0.293771i $$-0.0949105\pi$$
0.955876 + 0.293771i $$0.0949105\pi$$
$$18$$ 0 0
$$19$$ −92.0000 −1.11086 −0.555428 0.831565i $$-0.687445\pi$$
−0.555428 + 0.831565i $$0.687445\pi$$
$$20$$ −85.0000 −0.950329
$$21$$ 0 0
$$22$$ 60.0000 0.581456
$$23$$ −112.000 −1.01537 −0.507687 0.861541i $$-0.669499\pi$$
−0.507687 + 0.861541i $$0.669499\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ −150.000 −1.13144
$$27$$ 0 0
$$28$$ 119.000 0.803175
$$29$$ 58.0000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −224.000 −1.29779 −0.648897 0.760877i $$-0.724769\pi$$
−0.648897 + 0.760877i $$0.724769\pi$$
$$32$$ −85.0000 −0.469563
$$33$$ 0 0
$$34$$ −670.000 −3.37953
$$35$$ −35.0000 −0.169031
$$36$$ 0 0
$$37$$ −146.000 −0.648710 −0.324355 0.945936i $$-0.605147\pi$$
−0.324355 + 0.945936i $$0.605147\pi$$
$$38$$ 460.000 1.96373
$$39$$ 0 0
$$40$$ 225.000 0.889391
$$41$$ −18.0000 −0.0685641 −0.0342820 0.999412i $$-0.510914\pi$$
−0.0342820 + 0.999412i $$0.510914\pi$$
$$42$$ 0 0
$$43$$ 340.000 1.20580 0.602901 0.797816i $$-0.294011\pi$$
0.602901 + 0.797816i $$0.294011\pi$$
$$44$$ −204.000 −0.698958
$$45$$ 0 0
$$46$$ 560.000 1.79495
$$47$$ −208.000 −0.645530 −0.322765 0.946479i $$-0.604612\pi$$
−0.322765 + 0.946479i $$0.604612\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ −125.000 −0.353553
$$51$$ 0 0
$$52$$ 510.000 1.36008
$$53$$ 754.000 1.95415 0.977074 0.212899i $$-0.0682905\pi$$
0.977074 + 0.212899i $$0.0682905\pi$$
$$54$$ 0 0
$$55$$ 60.0000 0.147098
$$56$$ −315.000 −0.751672
$$57$$ 0 0
$$58$$ −290.000 −0.656532
$$59$$ −380.000 −0.838505 −0.419252 0.907870i $$-0.637708\pi$$
−0.419252 + 0.907870i $$0.637708\pi$$
$$60$$ 0 0
$$61$$ 718.000 1.50706 0.753529 0.657415i $$-0.228350\pi$$
0.753529 + 0.657415i $$0.228350\pi$$
$$62$$ 1120.00 2.29420
$$63$$ 0 0
$$64$$ −287.000 −0.560547
$$65$$ −150.000 −0.286234
$$66$$ 0 0
$$67$$ 412.000 0.751251 0.375625 0.926772i $$-0.377428\pi$$
0.375625 + 0.926772i $$0.377428\pi$$
$$68$$ 2278.00 4.06247
$$69$$ 0 0
$$70$$ 175.000 0.298807
$$71$$ 960.000 1.60466 0.802331 0.596879i $$-0.203593\pi$$
0.802331 + 0.596879i $$0.203593\pi$$
$$72$$ 0 0
$$73$$ 1066.00 1.70912 0.854561 0.519352i $$-0.173826\pi$$
0.854561 + 0.519352i $$0.173826\pi$$
$$74$$ 730.000 1.14677
$$75$$ 0 0
$$76$$ −1564.00 −2.36057
$$77$$ −84.0000 −0.124321
$$78$$ 0 0
$$79$$ 896.000 1.27605 0.638025 0.770016i $$-0.279752\pi$$
0.638025 + 0.770016i $$0.279752\pi$$
$$80$$ −445.000 −0.621906
$$81$$ 0 0
$$82$$ 90.0000 0.121205
$$83$$ −436.000 −0.576593 −0.288296 0.957541i $$-0.593089\pi$$
−0.288296 + 0.957541i $$0.593089\pi$$
$$84$$ 0 0
$$85$$ −670.000 −0.854961
$$86$$ −1700.00 −2.13158
$$87$$ 0 0
$$88$$ 540.000 0.654139
$$89$$ 1038.00 1.23627 0.618134 0.786073i $$-0.287889\pi$$
0.618134 + 0.786073i $$0.287889\pi$$
$$90$$ 0 0
$$91$$ 210.000 0.241912
$$92$$ −1904.00 −2.15767
$$93$$ 0 0
$$94$$ 1040.00 1.14115
$$95$$ 460.000 0.496790
$$96$$ 0 0
$$97$$ −702.000 −0.734818 −0.367409 0.930060i $$-0.619755\pi$$
−0.367409 + 0.930060i $$0.619755\pi$$
$$98$$ −245.000 −0.252538
$$99$$ 0 0
$$100$$ 425.000 0.425000
$$101$$ −46.0000 −0.0453185 −0.0226593 0.999743i $$-0.507213\pi$$
−0.0226593 + 0.999743i $$0.507213\pi$$
$$102$$ 0 0
$$103$$ 1880.00 1.79847 0.899233 0.437471i $$-0.144126\pi$$
0.899233 + 0.437471i $$0.144126\pi$$
$$104$$ −1350.00 −1.27287
$$105$$ 0 0
$$106$$ −3770.00 −3.45448
$$107$$ −732.000 −0.661356 −0.330678 0.943744i $$-0.607277\pi$$
−0.330678 + 0.943744i $$0.607277\pi$$
$$108$$ 0 0
$$109$$ −378.000 −0.332164 −0.166082 0.986112i $$-0.553112\pi$$
−0.166082 + 0.986112i $$0.553112\pi$$
$$110$$ −300.000 −0.260035
$$111$$ 0 0
$$112$$ 623.000 0.525607
$$113$$ −1458.00 −1.21378 −0.606890 0.794786i $$-0.707583\pi$$
−0.606890 + 0.794786i $$0.707583\pi$$
$$114$$ 0 0
$$115$$ 560.000 0.454089
$$116$$ 986.000 0.789205
$$117$$ 0 0
$$118$$ 1900.00 1.48228
$$119$$ 938.000 0.722574
$$120$$ 0 0
$$121$$ −1187.00 −0.891811
$$122$$ −3590.00 −2.66413
$$123$$ 0 0
$$124$$ −3808.00 −2.75781
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 608.000 0.424813 0.212407 0.977181i $$-0.431870\pi$$
0.212407 + 0.977181i $$0.431870\pi$$
$$128$$ 2115.00 1.46048
$$129$$ 0 0
$$130$$ 750.000 0.505995
$$131$$ 956.000 0.637604 0.318802 0.947821i $$-0.396720\pi$$
0.318802 + 0.947821i $$0.396720\pi$$
$$132$$ 0 0
$$133$$ −644.000 −0.419864
$$134$$ −2060.00 −1.32804
$$135$$ 0 0
$$136$$ −6030.00 −3.80197
$$137$$ 374.000 0.233233 0.116617 0.993177i $$-0.462795\pi$$
0.116617 + 0.993177i $$0.462795\pi$$
$$138$$ 0 0
$$139$$ 396.000 0.241642 0.120821 0.992674i $$-0.461447\pi$$
0.120821 + 0.992674i $$0.461447\pi$$
$$140$$ −595.000 −0.359191
$$141$$ 0 0
$$142$$ −4800.00 −2.83667
$$143$$ −360.000 −0.210522
$$144$$ 0 0
$$145$$ −290.000 −0.166091
$$146$$ −5330.00 −3.02133
$$147$$ 0 0
$$148$$ −2482.00 −1.37851
$$149$$ 1874.00 1.03036 0.515181 0.857081i $$-0.327725\pi$$
0.515181 + 0.857081i $$0.327725\pi$$
$$150$$ 0 0
$$151$$ −1096.00 −0.590670 −0.295335 0.955394i $$-0.595431\pi$$
−0.295335 + 0.955394i $$0.595431\pi$$
$$152$$ 4140.00 2.20920
$$153$$ 0 0
$$154$$ 420.000 0.219770
$$155$$ 1120.00 0.580391
$$156$$ 0 0
$$157$$ 1918.00 0.974988 0.487494 0.873126i $$-0.337911\pi$$
0.487494 + 0.873126i $$0.337911\pi$$
$$158$$ −4480.00 −2.25576
$$159$$ 0 0
$$160$$ 425.000 0.209995
$$161$$ −784.000 −0.383776
$$162$$ 0 0
$$163$$ 2316.00 1.11290 0.556451 0.830880i $$-0.312163\pi$$
0.556451 + 0.830880i $$0.312163\pi$$
$$164$$ −306.000 −0.145699
$$165$$ 0 0
$$166$$ 2180.00 1.01928
$$167$$ 1736.00 0.804405 0.402203 0.915551i $$-0.368245\pi$$
0.402203 + 0.915551i $$0.368245\pi$$
$$168$$ 0 0
$$169$$ −1297.00 −0.590350
$$170$$ 3350.00 1.51137
$$171$$ 0 0
$$172$$ 5780.00 2.56233
$$173$$ 2442.00 1.07319 0.536595 0.843840i $$-0.319710\pi$$
0.536595 + 0.843840i $$0.319710\pi$$
$$174$$ 0 0
$$175$$ 175.000 0.0755929
$$176$$ −1068.00 −0.457406
$$177$$ 0 0
$$178$$ −5190.00 −2.18543
$$179$$ 4092.00 1.70866 0.854331 0.519730i $$-0.173967\pi$$
0.854331 + 0.519730i $$0.173967\pi$$
$$180$$ 0 0
$$181$$ 1270.00 0.521538 0.260769 0.965401i $$-0.416024\pi$$
0.260769 + 0.965401i $$0.416024\pi$$
$$182$$ −1050.00 −0.427644
$$183$$ 0 0
$$184$$ 5040.00 2.01931
$$185$$ 730.000 0.290112
$$186$$ 0 0
$$187$$ −1608.00 −0.628816
$$188$$ −3536.00 −1.37175
$$189$$ 0 0
$$190$$ −2300.00 −0.878208
$$191$$ −4904.00 −1.85781 −0.928903 0.370323i $$-0.879247\pi$$
−0.928903 + 0.370323i $$0.879247\pi$$
$$192$$ 0 0
$$193$$ 2178.00 0.812310 0.406155 0.913804i $$-0.366869\pi$$
0.406155 + 0.913804i $$0.366869\pi$$
$$194$$ 3510.00 1.29899
$$195$$ 0 0
$$196$$ 833.000 0.303571
$$197$$ 2850.00 1.03073 0.515366 0.856970i $$-0.327656\pi$$
0.515366 + 0.856970i $$0.327656\pi$$
$$198$$ 0 0
$$199$$ −1144.00 −0.407518 −0.203759 0.979021i $$-0.565316\pi$$
−0.203759 + 0.979021i $$0.565316\pi$$
$$200$$ −1125.00 −0.397748
$$201$$ 0 0
$$202$$ 230.000 0.0801126
$$203$$ 406.000 0.140372
$$204$$ 0 0
$$205$$ 90.0000 0.0306628
$$206$$ −9400.00 −3.17927
$$207$$ 0 0
$$208$$ 2670.00 0.890054
$$209$$ 1104.00 0.365384
$$210$$ 0 0
$$211$$ 412.000 0.134423 0.0672115 0.997739i $$-0.478590\pi$$
0.0672115 + 0.997739i $$0.478590\pi$$
$$212$$ 12818.0 4.15257
$$213$$ 0 0
$$214$$ 3660.00 1.16912
$$215$$ −1700.00 −0.539251
$$216$$ 0 0
$$217$$ −1568.00 −0.490520
$$218$$ 1890.00 0.587188
$$219$$ 0 0
$$220$$ 1020.00 0.312584
$$221$$ 4020.00 1.22359
$$222$$ 0 0
$$223$$ −1632.00 −0.490075 −0.245038 0.969514i $$-0.578800\pi$$
−0.245038 + 0.969514i $$0.578800\pi$$
$$224$$ −595.000 −0.177478
$$225$$ 0 0
$$226$$ 7290.00 2.14568
$$227$$ −4084.00 −1.19412 −0.597059 0.802198i $$-0.703664\pi$$
−0.597059 + 0.802198i $$0.703664\pi$$
$$228$$ 0 0
$$229$$ −3386.00 −0.977088 −0.488544 0.872539i $$-0.662472\pi$$
−0.488544 + 0.872539i $$0.662472\pi$$
$$230$$ −2800.00 −0.802724
$$231$$ 0 0
$$232$$ −2610.00 −0.738599
$$233$$ −5322.00 −1.49638 −0.748188 0.663486i $$-0.769076\pi$$
−0.748188 + 0.663486i $$0.769076\pi$$
$$234$$ 0 0
$$235$$ 1040.00 0.288690
$$236$$ −6460.00 −1.78182
$$237$$ 0 0
$$238$$ −4690.00 −1.27734
$$239$$ −3736.00 −1.01114 −0.505569 0.862786i $$-0.668717\pi$$
−0.505569 + 0.862786i $$0.668717\pi$$
$$240$$ 0 0
$$241$$ 210.000 0.0561298 0.0280649 0.999606i $$-0.491065\pi$$
0.0280649 + 0.999606i $$0.491065\pi$$
$$242$$ 5935.00 1.57651
$$243$$ 0 0
$$244$$ 12206.0 3.20250
$$245$$ −245.000 −0.0638877
$$246$$ 0 0
$$247$$ −2760.00 −0.710990
$$248$$ 10080.0 2.58097
$$249$$ 0 0
$$250$$ 625.000 0.158114
$$251$$ 4212.00 1.05920 0.529600 0.848248i $$-0.322342\pi$$
0.529600 + 0.848248i $$0.322342\pi$$
$$252$$ 0 0
$$253$$ 1344.00 0.333978
$$254$$ −3040.00 −0.750971
$$255$$ 0 0
$$256$$ −8279.00 −2.02124
$$257$$ −5130.00 −1.24514 −0.622569 0.782565i $$-0.713911\pi$$
−0.622569 + 0.782565i $$0.713911\pi$$
$$258$$ 0 0
$$259$$ −1022.00 −0.245189
$$260$$ −2550.00 −0.608247
$$261$$ 0 0
$$262$$ −4780.00 −1.12714
$$263$$ −848.000 −0.198821 −0.0994105 0.995047i $$-0.531696\pi$$
−0.0994105 + 0.995047i $$0.531696\pi$$
$$264$$ 0 0
$$265$$ −3770.00 −0.873922
$$266$$ 3220.00 0.742221
$$267$$ 0 0
$$268$$ 7004.00 1.59641
$$269$$ 1274.00 0.288763 0.144381 0.989522i $$-0.453881\pi$$
0.144381 + 0.989522i $$0.453881\pi$$
$$270$$ 0 0
$$271$$ 864.000 0.193669 0.0968344 0.995301i $$-0.469128\pi$$
0.0968344 + 0.995301i $$0.469128\pi$$
$$272$$ 11926.0 2.65853
$$273$$ 0 0
$$274$$ −1870.00 −0.412302
$$275$$ −300.000 −0.0657843
$$276$$ 0 0
$$277$$ −8530.00 −1.85025 −0.925123 0.379668i $$-0.876038\pi$$
−0.925123 + 0.379668i $$0.876038\pi$$
$$278$$ −1980.00 −0.427167
$$279$$ 0 0
$$280$$ 1575.00 0.336158
$$281$$ 5382.00 1.14257 0.571287 0.820750i $$-0.306444\pi$$
0.571287 + 0.820750i $$0.306444\pi$$
$$282$$ 0 0
$$283$$ 6236.00 1.30986 0.654932 0.755687i $$-0.272697\pi$$
0.654932 + 0.755687i $$0.272697\pi$$
$$284$$ 16320.0 3.40991
$$285$$ 0 0
$$286$$ 1800.00 0.372155
$$287$$ −126.000 −0.0259148
$$288$$ 0 0
$$289$$ 13043.0 2.65479
$$290$$ 1450.00 0.293610
$$291$$ 0 0
$$292$$ 18122.0 3.63188
$$293$$ 818.000 0.163099 0.0815496 0.996669i $$-0.474013\pi$$
0.0815496 + 0.996669i $$0.474013\pi$$
$$294$$ 0 0
$$295$$ 1900.00 0.374991
$$296$$ 6570.00 1.29011
$$297$$ 0 0
$$298$$ −9370.00 −1.82144
$$299$$ −3360.00 −0.649879
$$300$$ 0 0
$$301$$ 2380.00 0.455751
$$302$$ 5480.00 1.04417
$$303$$ 0 0
$$304$$ −8188.00 −1.54478
$$305$$ −3590.00 −0.673976
$$306$$ 0 0
$$307$$ −2268.00 −0.421634 −0.210817 0.977526i $$-0.567612\pi$$
−0.210817 + 0.977526i $$0.567612\pi$$
$$308$$ −1428.00 −0.264181
$$309$$ 0 0
$$310$$ −5600.00 −1.02600
$$311$$ −6648.00 −1.21213 −0.606067 0.795414i $$-0.707254\pi$$
−0.606067 + 0.795414i $$0.707254\pi$$
$$312$$ 0 0
$$313$$ 9818.00 1.77299 0.886495 0.462737i $$-0.153133\pi$$
0.886495 + 0.462737i $$0.153133\pi$$
$$314$$ −9590.00 −1.72355
$$315$$ 0 0
$$316$$ 15232.0 2.71160
$$317$$ −934.000 −0.165485 −0.0827424 0.996571i $$-0.526368\pi$$
−0.0827424 + 0.996571i $$0.526368\pi$$
$$318$$ 0 0
$$319$$ −696.000 −0.122158
$$320$$ 1435.00 0.250684
$$321$$ 0 0
$$322$$ 3920.00 0.678426
$$323$$ −12328.0 −2.12368
$$324$$ 0 0
$$325$$ 750.000 0.128008
$$326$$ −11580.0 −1.96735
$$327$$ 0 0
$$328$$ 810.000 0.136356
$$329$$ −1456.00 −0.243987
$$330$$ 0 0
$$331$$ 2292.00 0.380603 0.190302 0.981726i $$-0.439053\pi$$
0.190302 + 0.981726i $$0.439053\pi$$
$$332$$ −7412.00 −1.22526
$$333$$ 0 0
$$334$$ −8680.00 −1.42200
$$335$$ −2060.00 −0.335970
$$336$$ 0 0
$$337$$ −6062.00 −0.979876 −0.489938 0.871757i $$-0.662981\pi$$
−0.489938 + 0.871757i $$0.662981\pi$$
$$338$$ 6485.00 1.04360
$$339$$ 0 0
$$340$$ −11390.0 −1.81679
$$341$$ 2688.00 0.426872
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ −15300.0 −2.39803
$$345$$ 0 0
$$346$$ −12210.0 −1.89715
$$347$$ −1484.00 −0.229583 −0.114791 0.993390i $$-0.536620\pi$$
−0.114791 + 0.993390i $$0.536620\pi$$
$$348$$ 0 0
$$349$$ 254.000 0.0389579 0.0194790 0.999810i $$-0.493799\pi$$
0.0194790 + 0.999810i $$0.493799\pi$$
$$350$$ −875.000 −0.133631
$$351$$ 0 0
$$352$$ 1020.00 0.154449
$$353$$ 10950.0 1.65102 0.825509 0.564388i $$-0.190888\pi$$
0.825509 + 0.564388i $$0.190888\pi$$
$$354$$ 0 0
$$355$$ −4800.00 −0.717627
$$356$$ 17646.0 2.62707
$$357$$ 0 0
$$358$$ −20460.0 −3.02052
$$359$$ −11376.0 −1.67243 −0.836215 0.548402i $$-0.815236\pi$$
−0.836215 + 0.548402i $$0.815236\pi$$
$$360$$ 0 0
$$361$$ 1605.00 0.233999
$$362$$ −6350.00 −0.921957
$$363$$ 0 0
$$364$$ 3570.00 0.514063
$$365$$ −5330.00 −0.764342
$$366$$ 0 0
$$367$$ −1136.00 −0.161577 −0.0807884 0.996731i $$-0.525744\pi$$
−0.0807884 + 0.996731i $$0.525744\pi$$
$$368$$ −9968.00 −1.41201
$$369$$ 0 0
$$370$$ −3650.00 −0.512850
$$371$$ 5278.00 0.738599
$$372$$ 0 0
$$373$$ −8242.00 −1.14411 −0.572057 0.820214i $$-0.693854\pi$$
−0.572057 + 0.820214i $$0.693854\pi$$
$$374$$ 8040.00 1.11160
$$375$$ 0 0
$$376$$ 9360.00 1.28379
$$377$$ 1740.00 0.237704
$$378$$ 0 0
$$379$$ 3620.00 0.490625 0.245313 0.969444i $$-0.421109\pi$$
0.245313 + 0.969444i $$0.421109\pi$$
$$380$$ 7820.00 1.05568
$$381$$ 0 0
$$382$$ 24520.0 3.28417
$$383$$ 8464.00 1.12922 0.564609 0.825359i $$-0.309027\pi$$
0.564609 + 0.825359i $$0.309027\pi$$
$$384$$ 0 0
$$385$$ 420.000 0.0555979
$$386$$ −10890.0 −1.43598
$$387$$ 0 0
$$388$$ −11934.0 −1.56149
$$389$$ −3678.00 −0.479388 −0.239694 0.970848i $$-0.577047\pi$$
−0.239694 + 0.970848i $$0.577047\pi$$
$$390$$ 0 0
$$391$$ −15008.0 −1.94114
$$392$$ −2205.00 −0.284105
$$393$$ 0 0
$$394$$ −14250.0 −1.82209
$$395$$ −4480.00 −0.570666
$$396$$ 0 0
$$397$$ 12590.0 1.59162 0.795811 0.605545i $$-0.207045\pi$$
0.795811 + 0.605545i $$0.207045\pi$$
$$398$$ 5720.00 0.720396
$$399$$ 0 0
$$400$$ 2225.00 0.278125
$$401$$ −2850.00 −0.354918 −0.177459 0.984128i $$-0.556788\pi$$
−0.177459 + 0.984128i $$0.556788\pi$$
$$402$$ 0 0
$$403$$ −6720.00 −0.830638
$$404$$ −782.000 −0.0963019
$$405$$ 0 0
$$406$$ −2030.00 −0.248146
$$407$$ 1752.00 0.213374
$$408$$ 0 0
$$409$$ 1226.00 0.148220 0.0741098 0.997250i $$-0.476388\pi$$
0.0741098 + 0.997250i $$0.476388\pi$$
$$410$$ −450.000 −0.0542047
$$411$$ 0 0
$$412$$ 31960.0 3.82174
$$413$$ −2660.00 −0.316925
$$414$$ 0 0
$$415$$ 2180.00 0.257860
$$416$$ −2550.00 −0.300539
$$417$$ 0 0
$$418$$ −5520.00 −0.645914
$$419$$ −612.000 −0.0713560 −0.0356780 0.999363i $$-0.511359\pi$$
−0.0356780 + 0.999363i $$0.511359\pi$$
$$420$$ 0 0
$$421$$ 5182.00 0.599894 0.299947 0.953956i $$-0.403031\pi$$
0.299947 + 0.953956i $$0.403031\pi$$
$$422$$ −2060.00 −0.237629
$$423$$ 0 0
$$424$$ −33930.0 −3.88629
$$425$$ 3350.00 0.382350
$$426$$ 0 0
$$427$$ 5026.00 0.569614
$$428$$ −12444.0 −1.40538
$$429$$ 0 0
$$430$$ 8500.00 0.953271
$$431$$ 4984.00 0.557009 0.278504 0.960435i $$-0.410161\pi$$
0.278504 + 0.960435i $$0.410161\pi$$
$$432$$ 0 0
$$433$$ −1694.00 −0.188010 −0.0940051 0.995572i $$-0.529967\pi$$
−0.0940051 + 0.995572i $$0.529967\pi$$
$$434$$ 7840.00 0.867125
$$435$$ 0 0
$$436$$ −6426.00 −0.705848
$$437$$ 10304.0 1.12793
$$438$$ 0 0
$$439$$ 13864.0 1.50727 0.753636 0.657292i $$-0.228298\pi$$
0.753636 + 0.657292i $$0.228298\pi$$
$$440$$ −2700.00 −0.292540
$$441$$ 0 0
$$442$$ −20100.0 −2.16303
$$443$$ 4644.00 0.498066 0.249033 0.968495i $$-0.419887\pi$$
0.249033 + 0.968495i $$0.419887\pi$$
$$444$$ 0 0
$$445$$ −5190.00 −0.552875
$$446$$ 8160.00 0.866339
$$447$$ 0 0
$$448$$ −2009.00 −0.211867
$$449$$ 4926.00 0.517756 0.258878 0.965910i $$-0.416647\pi$$
0.258878 + 0.965910i $$0.416647\pi$$
$$450$$ 0 0
$$451$$ 216.000 0.0225522
$$452$$ −24786.0 −2.57928
$$453$$ 0 0
$$454$$ 20420.0 2.11092
$$455$$ −1050.00 −0.108186
$$456$$ 0 0
$$457$$ −14694.0 −1.50406 −0.752031 0.659128i $$-0.770926\pi$$
−0.752031 + 0.659128i $$0.770926\pi$$
$$458$$ 16930.0 1.72726
$$459$$ 0 0
$$460$$ 9520.00 0.964940
$$461$$ −2006.00 −0.202665 −0.101333 0.994853i $$-0.532311\pi$$
−0.101333 + 0.994853i $$0.532311\pi$$
$$462$$ 0 0
$$463$$ 4896.00 0.491439 0.245720 0.969341i $$-0.420976\pi$$
0.245720 + 0.969341i $$0.420976\pi$$
$$464$$ 5162.00 0.516465
$$465$$ 0 0
$$466$$ 26610.0 2.64525
$$467$$ −2660.00 −0.263576 −0.131788 0.991278i $$-0.542072\pi$$
−0.131788 + 0.991278i $$0.542072\pi$$
$$468$$ 0 0
$$469$$ 2884.00 0.283946
$$470$$ −5200.00 −0.510336
$$471$$ 0 0
$$472$$ 17100.0 1.66757
$$473$$ −4080.00 −0.396614
$$474$$ 0 0
$$475$$ −2300.00 −0.222171
$$476$$ 15946.0 1.53547
$$477$$ 0 0
$$478$$ 18680.0 1.78745
$$479$$ 5600.00 0.534176 0.267088 0.963672i $$-0.413938\pi$$
0.267088 + 0.963672i $$0.413938\pi$$
$$480$$ 0 0
$$481$$ −4380.00 −0.415199
$$482$$ −1050.00 −0.0992245
$$483$$ 0 0
$$484$$ −20179.0 −1.89510
$$485$$ 3510.00 0.328620
$$486$$ 0 0
$$487$$ −6424.00 −0.597740 −0.298870 0.954294i $$-0.596610\pi$$
−0.298870 + 0.954294i $$0.596610\pi$$
$$488$$ −32310.0 −2.99714
$$489$$ 0 0
$$490$$ 1225.00 0.112938
$$491$$ 18900.0 1.73716 0.868579 0.495550i $$-0.165033\pi$$
0.868579 + 0.495550i $$0.165033\pi$$
$$492$$ 0 0
$$493$$ 7772.00 0.710007
$$494$$ 13800.0 1.25687
$$495$$ 0 0
$$496$$ −19936.0 −1.80474
$$497$$ 6720.00 0.606505
$$498$$ 0 0
$$499$$ −15364.0 −1.37833 −0.689165 0.724604i $$-0.742023\pi$$
−0.689165 + 0.724604i $$0.742023\pi$$
$$500$$ −2125.00 −0.190066
$$501$$ 0 0
$$502$$ −21060.0 −1.87242
$$503$$ −2216.00 −0.196435 −0.0982173 0.995165i $$-0.531314\pi$$
−0.0982173 + 0.995165i $$0.531314\pi$$
$$504$$ 0 0
$$505$$ 230.000 0.0202671
$$506$$ −6720.00 −0.590396
$$507$$ 0 0
$$508$$ 10336.0 0.902728
$$509$$ 3754.00 0.326902 0.163451 0.986551i $$-0.447737\pi$$
0.163451 + 0.986551i $$0.447737\pi$$
$$510$$ 0 0
$$511$$ 7462.00 0.645987
$$512$$ 24475.0 2.11260
$$513$$ 0 0
$$514$$ 25650.0 2.20111
$$515$$ −9400.00 −0.804298
$$516$$ 0 0
$$517$$ 2496.00 0.212329
$$518$$ 5110.00 0.433437
$$519$$ 0 0
$$520$$ 6750.00 0.569244
$$521$$ 4702.00 0.395390 0.197695 0.980264i $$-0.436654\pi$$
0.197695 + 0.980264i $$0.436654\pi$$
$$522$$ 0 0
$$523$$ −22660.0 −1.89456 −0.947278 0.320413i $$-0.896178\pi$$
−0.947278 + 0.320413i $$0.896178\pi$$
$$524$$ 16252.0 1.35491
$$525$$ 0 0
$$526$$ 4240.00 0.351469
$$527$$ −30016.0 −2.48106
$$528$$ 0 0
$$529$$ 377.000 0.0309855
$$530$$ 18850.0 1.54489
$$531$$ 0 0
$$532$$ −10948.0 −0.892211
$$533$$ −540.000 −0.0438837
$$534$$ 0 0
$$535$$ 3660.00 0.295767
$$536$$ −18540.0 −1.49404
$$537$$ 0 0
$$538$$ −6370.00 −0.510465
$$539$$ −588.000 −0.0469888
$$540$$ 0 0
$$541$$ −8634.00 −0.686145 −0.343073 0.939309i $$-0.611468\pi$$
−0.343073 + 0.939309i $$0.611468\pi$$
$$542$$ −4320.00 −0.342361
$$543$$ 0 0
$$544$$ −11390.0 −0.897688
$$545$$ 1890.00 0.148548
$$546$$ 0 0
$$547$$ −19284.0 −1.50736 −0.753679 0.657243i $$-0.771722\pi$$
−0.753679 + 0.657243i $$0.771722\pi$$
$$548$$ 6358.00 0.495621
$$549$$ 0 0
$$550$$ 1500.00 0.116291
$$551$$ −5336.00 −0.412561
$$552$$ 0 0
$$553$$ 6272.00 0.482301
$$554$$ 42650.0 3.27080
$$555$$ 0 0
$$556$$ 6732.00 0.513490
$$557$$ 19658.0 1.49540 0.747699 0.664038i $$-0.231159\pi$$
0.747699 + 0.664038i $$0.231159\pi$$
$$558$$ 0 0
$$559$$ 10200.0 0.771760
$$560$$ −3115.00 −0.235059
$$561$$ 0 0
$$562$$ −26910.0 −2.01980
$$563$$ 25612.0 1.91726 0.958630 0.284656i $$-0.0918793\pi$$
0.958630 + 0.284656i $$0.0918793\pi$$
$$564$$ 0 0
$$565$$ 7290.00 0.542819
$$566$$ −31180.0 −2.31554
$$567$$ 0 0
$$568$$ −43200.0 −3.19125
$$569$$ −7002.00 −0.515886 −0.257943 0.966160i $$-0.583045\pi$$
−0.257943 + 0.966160i $$0.583045\pi$$
$$570$$ 0 0
$$571$$ −4524.00 −0.331565 −0.165782 0.986162i $$-0.553015\pi$$
−0.165782 + 0.986162i $$0.553015\pi$$
$$572$$ −6120.00 −0.447360
$$573$$ 0 0
$$574$$ 630.000 0.0458113
$$575$$ −2800.00 −0.203075
$$576$$ 0 0
$$577$$ −6014.00 −0.433910 −0.216955 0.976182i $$-0.569612\pi$$
−0.216955 + 0.976182i $$0.569612\pi$$
$$578$$ −65215.0 −4.69306
$$579$$ 0 0
$$580$$ −4930.00 −0.352943
$$581$$ −3052.00 −0.217932
$$582$$ 0 0
$$583$$ −9048.00 −0.642761
$$584$$ −47970.0 −3.39899
$$585$$ 0 0
$$586$$ −4090.00 −0.288321
$$587$$ 11748.0 0.826051 0.413025 0.910719i $$-0.364472\pi$$
0.413025 + 0.910719i $$0.364472\pi$$
$$588$$ 0 0
$$589$$ 20608.0 1.44166
$$590$$ −9500.00 −0.662896
$$591$$ 0 0
$$592$$ −12994.0 −0.902112
$$593$$ 9462.00 0.655241 0.327620 0.944809i $$-0.393753\pi$$
0.327620 + 0.944809i $$0.393753\pi$$
$$594$$ 0 0
$$595$$ −4690.00 −0.323145
$$596$$ 31858.0 2.18952
$$597$$ 0 0
$$598$$ 16800.0 1.14883
$$599$$ −2320.00 −0.158251 −0.0791257 0.996865i $$-0.525213\pi$$
−0.0791257 + 0.996865i $$0.525213\pi$$
$$600$$ 0 0
$$601$$ 4650.00 0.315603 0.157802 0.987471i $$-0.449559\pi$$
0.157802 + 0.987471i $$0.449559\pi$$
$$602$$ −11900.0 −0.805661
$$603$$ 0 0
$$604$$ −18632.0 −1.25517
$$605$$ 5935.00 0.398830
$$606$$ 0 0
$$607$$ −14656.0 −0.980014 −0.490007 0.871718i $$-0.663006\pi$$
−0.490007 + 0.871718i $$0.663006\pi$$
$$608$$ 7820.00 0.521617
$$609$$ 0 0
$$610$$ 17950.0 1.19143
$$611$$ −6240.00 −0.413164
$$612$$ 0 0
$$613$$ 29166.0 1.92170 0.960851 0.277065i $$-0.0893616\pi$$
0.960851 + 0.277065i $$0.0893616\pi$$
$$614$$ 11340.0 0.745350
$$615$$ 0 0
$$616$$ 3780.00 0.247241
$$617$$ −28554.0 −1.86311 −0.931557 0.363597i $$-0.881549\pi$$
−0.931557 + 0.363597i $$0.881549\pi$$
$$618$$ 0 0
$$619$$ −3876.00 −0.251679 −0.125840 0.992051i $$-0.540163\pi$$
−0.125840 + 0.992051i $$0.540163\pi$$
$$620$$ 19040.0 1.23333
$$621$$ 0 0
$$622$$ 33240.0 2.14277
$$623$$ 7266.00 0.467265
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ −49090.0 −3.13423
$$627$$ 0 0
$$628$$ 32606.0 2.07185
$$629$$ −19564.0 −1.24017
$$630$$ 0 0
$$631$$ 2904.00 0.183211 0.0916057 0.995795i $$-0.470800\pi$$
0.0916057 + 0.995795i $$0.470800\pi$$
$$632$$ −40320.0 −2.53773
$$633$$ 0 0
$$634$$ 4670.00 0.292538
$$635$$ −3040.00 −0.189982
$$636$$ 0 0
$$637$$ 1470.00 0.0914341
$$638$$ 3480.00 0.215948
$$639$$ 0 0
$$640$$ −10575.0 −0.653146
$$641$$ −9330.00 −0.574903 −0.287452 0.957795i $$-0.592808\pi$$
−0.287452 + 0.957795i $$0.592808\pi$$
$$642$$ 0 0
$$643$$ −18332.0 −1.12433 −0.562164 0.827025i $$-0.690031\pi$$
−0.562164 + 0.827025i $$0.690031\pi$$
$$644$$ −13328.0 −0.815523
$$645$$ 0 0
$$646$$ 61640.0 3.75417
$$647$$ 2088.00 0.126874 0.0634372 0.997986i $$-0.479794\pi$$
0.0634372 + 0.997986i $$0.479794\pi$$
$$648$$ 0 0
$$649$$ 4560.00 0.275802
$$650$$ −3750.00 −0.226288
$$651$$ 0 0
$$652$$ 39372.0 2.36492
$$653$$ −22.0000 −0.00131842 −0.000659209 1.00000i $$-0.500210\pi$$
−0.000659209 1.00000i $$0.500210\pi$$
$$654$$ 0 0
$$655$$ −4780.00 −0.285145
$$656$$ −1602.00 −0.0953469
$$657$$ 0 0
$$658$$ 7280.00 0.431313
$$659$$ −16260.0 −0.961153 −0.480576 0.876953i $$-0.659573\pi$$
−0.480576 + 0.876953i $$0.659573\pi$$
$$660$$ 0 0
$$661$$ −23818.0 −1.40153 −0.700766 0.713391i $$-0.747158\pi$$
−0.700766 + 0.713391i $$0.747158\pi$$
$$662$$ −11460.0 −0.672818
$$663$$ 0 0
$$664$$ 19620.0 1.14669
$$665$$ 3220.00 0.187769
$$666$$ 0 0
$$667$$ −6496.00 −0.377101
$$668$$ 29512.0 1.70936
$$669$$ 0 0
$$670$$ 10300.0 0.593916
$$671$$ −8616.00 −0.495703
$$672$$ 0 0
$$673$$ 31106.0 1.78165 0.890823 0.454350i $$-0.150128\pi$$
0.890823 + 0.454350i $$0.150128\pi$$
$$674$$ 30310.0 1.73219
$$675$$ 0 0
$$676$$ −22049.0 −1.25449
$$677$$ 1090.00 0.0618790 0.0309395 0.999521i $$-0.490150\pi$$
0.0309395 + 0.999521i $$0.490150\pi$$
$$678$$ 0 0
$$679$$ −4914.00 −0.277735
$$680$$ 30150.0 1.70029
$$681$$ 0 0
$$682$$ −13440.0 −0.754610
$$683$$ 12372.0 0.693121 0.346560 0.938028i $$-0.387350\pi$$
0.346560 + 0.938028i $$0.387350\pi$$
$$684$$ 0 0
$$685$$ −1870.00 −0.104305
$$686$$ −1715.00 −0.0954504
$$687$$ 0 0
$$688$$ 30260.0 1.67682
$$689$$ 22620.0 1.25073
$$690$$ 0 0
$$691$$ 3252.00 0.179033 0.0895166 0.995985i $$-0.471468\pi$$
0.0895166 + 0.995985i $$0.471468\pi$$
$$692$$ 41514.0 2.28053
$$693$$ 0 0
$$694$$ 7420.00 0.405849
$$695$$ −1980.00 −0.108066
$$696$$ 0 0
$$697$$ −2412.00 −0.131077
$$698$$ −1270.00 −0.0688685
$$699$$ 0 0
$$700$$ 2975.00 0.160635
$$701$$ 5434.00 0.292781 0.146390 0.989227i $$-0.453234\pi$$
0.146390 + 0.989227i $$0.453234\pi$$
$$702$$ 0 0
$$703$$ 13432.0 0.720622
$$704$$ 3444.00 0.184376
$$705$$ 0 0
$$706$$ −54750.0 −2.91862
$$707$$ −322.000 −0.0171288
$$708$$ 0 0
$$709$$ −5330.00 −0.282331 −0.141165 0.989986i $$-0.545085\pi$$
−0.141165 + 0.989986i $$0.545085\pi$$
$$710$$ 24000.0 1.26860
$$711$$ 0 0
$$712$$ −46710.0 −2.45861
$$713$$ 25088.0 1.31775
$$714$$ 0 0
$$715$$ 1800.00 0.0941485
$$716$$ 69564.0 3.63091
$$717$$ 0 0
$$718$$ 56880.0 2.95647
$$719$$ 7520.00 0.390054 0.195027 0.980798i $$-0.437521\pi$$
0.195027 + 0.980798i $$0.437521\pi$$
$$720$$ 0 0
$$721$$ 13160.0 0.679756
$$722$$ −8025.00 −0.413656
$$723$$ 0 0
$$724$$ 21590.0 1.10827
$$725$$ 1450.00 0.0742781
$$726$$ 0 0
$$727$$ 19336.0 0.986427 0.493214 0.869908i $$-0.335822\pi$$
0.493214 + 0.869908i $$0.335822\pi$$
$$728$$ −9450.00 −0.481099
$$729$$ 0 0
$$730$$ 26650.0 1.35118
$$731$$ 45560.0 2.30519
$$732$$ 0 0
$$733$$ −22498.0 −1.13367 −0.566837 0.823830i $$-0.691833\pi$$
−0.566837 + 0.823830i $$0.691833\pi$$
$$734$$ 5680.00 0.285630
$$735$$ 0 0
$$736$$ 9520.00 0.476782
$$737$$ −4944.00 −0.247103
$$738$$ 0 0
$$739$$ −18292.0 −0.910531 −0.455265 0.890356i $$-0.650456\pi$$
−0.455265 + 0.890356i $$0.650456\pi$$
$$740$$ 12410.0 0.616487
$$741$$ 0 0
$$742$$ −26390.0 −1.30567
$$743$$ −17904.0 −0.884030 −0.442015 0.897008i $$-0.645736\pi$$
−0.442015 + 0.897008i $$0.645736\pi$$
$$744$$ 0 0
$$745$$ −9370.00 −0.460792
$$746$$ 41210.0 2.02253
$$747$$ 0 0
$$748$$ −27336.0 −1.33623
$$749$$ −5124.00 −0.249969
$$750$$ 0 0
$$751$$ 5408.00 0.262771 0.131385 0.991331i $$-0.458057\pi$$
0.131385 + 0.991331i $$0.458057\pi$$
$$752$$ −18512.0 −0.897690
$$753$$ 0 0
$$754$$ −8700.00 −0.420206
$$755$$ 5480.00 0.264156
$$756$$ 0 0
$$757$$ 8318.00 0.399370 0.199685 0.979860i $$-0.436008\pi$$
0.199685 + 0.979860i $$0.436008\pi$$
$$758$$ −18100.0 −0.867311
$$759$$ 0 0
$$760$$ −20700.0 −0.987984
$$761$$ −6690.00 −0.318676 −0.159338 0.987224i $$-0.550936\pi$$
−0.159338 + 0.987224i $$0.550936\pi$$
$$762$$ 0 0
$$763$$ −2646.00 −0.125546
$$764$$ −83368.0 −3.94784
$$765$$ 0 0
$$766$$ −42320.0 −1.99619
$$767$$ −11400.0 −0.536676
$$768$$ 0 0
$$769$$ 9266.00 0.434513 0.217257 0.976115i $$-0.430289\pi$$
0.217257 + 0.976115i $$0.430289\pi$$
$$770$$ −2100.00 −0.0982841
$$771$$ 0 0
$$772$$ 37026.0 1.72616
$$773$$ −9678.00 −0.450315 −0.225157 0.974322i $$-0.572290\pi$$
−0.225157 + 0.974322i $$0.572290\pi$$
$$774$$ 0 0
$$775$$ −5600.00 −0.259559
$$776$$ 31590.0 1.46136
$$777$$ 0 0
$$778$$ 18390.0 0.847447
$$779$$ 1656.00 0.0761648
$$780$$ 0 0
$$781$$ −11520.0 −0.527808
$$782$$ 75040.0 3.43149
$$783$$ 0 0
$$784$$ 4361.00 0.198661
$$785$$ −9590.00 −0.436028
$$786$$ 0 0
$$787$$ −6860.00 −0.310715 −0.155357 0.987858i $$-0.549653\pi$$
−0.155357 + 0.987858i $$0.549653\pi$$
$$788$$ 48450.0 2.19030
$$789$$ 0 0
$$790$$ 22400.0 1.00881
$$791$$ −10206.0 −0.458766
$$792$$ 0 0
$$793$$ 21540.0 0.964575
$$794$$ −62950.0 −2.81362
$$795$$ 0 0
$$796$$ −19448.0 −0.865975
$$797$$ −10950.0 −0.486661 −0.243331 0.969943i $$-0.578240\pi$$
−0.243331 + 0.969943i $$0.578240\pi$$
$$798$$ 0 0
$$799$$ −27872.0 −1.23409
$$800$$ −2125.00 −0.0939126
$$801$$ 0 0
$$802$$ 14250.0 0.627413
$$803$$ −12792.0 −0.562167
$$804$$ 0 0
$$805$$ 3920.00 0.171630
$$806$$ 33600.0 1.46837
$$807$$ 0 0
$$808$$ 2070.00 0.0901267
$$809$$ −26010.0 −1.13036 −0.565181 0.824967i $$-0.691194\pi$$
−0.565181 + 0.824967i $$0.691194\pi$$
$$810$$ 0 0
$$811$$ −14628.0 −0.633364 −0.316682 0.948532i $$-0.602569\pi$$
−0.316682 + 0.948532i $$0.602569\pi$$
$$812$$ 6902.00 0.298292
$$813$$ 0 0
$$814$$ −8760.00 −0.377196
$$815$$ −11580.0 −0.497705
$$816$$ 0 0
$$817$$ −31280.0 −1.33947
$$818$$ −6130.00 −0.262018
$$819$$ 0 0
$$820$$ 1530.00 0.0651584
$$821$$ −8718.00 −0.370597 −0.185299 0.982682i $$-0.559325\pi$$
−0.185299 + 0.982682i $$0.559325\pi$$
$$822$$ 0 0
$$823$$ −7432.00 −0.314779 −0.157390 0.987537i $$-0.550308\pi$$
−0.157390 + 0.987537i $$0.550308\pi$$
$$824$$ −84600.0 −3.57668
$$825$$ 0 0
$$826$$ 13300.0 0.560250
$$827$$ −17388.0 −0.731125 −0.365562 0.930787i $$-0.619123\pi$$
−0.365562 + 0.930787i $$0.619123\pi$$
$$828$$ 0 0
$$829$$ 7902.00 0.331059 0.165529 0.986205i $$-0.447067\pi$$
0.165529 + 0.986205i $$0.447067\pi$$
$$830$$ −10900.0 −0.455837
$$831$$ 0 0
$$832$$ −8610.00 −0.358772
$$833$$ 6566.00 0.273107
$$834$$ 0 0
$$835$$ −8680.00 −0.359741
$$836$$ 18768.0 0.776441
$$837$$ 0 0
$$838$$ 3060.00 0.126141
$$839$$ 31848.0 1.31051 0.655253 0.755409i $$-0.272562\pi$$
0.655253 + 0.755409i $$0.272562\pi$$
$$840$$ 0 0
$$841$$ −21025.0 −0.862069
$$842$$ −25910.0 −1.06047
$$843$$ 0 0
$$844$$ 7004.00 0.285649
$$845$$ 6485.00 0.264013
$$846$$ 0 0
$$847$$ −8309.00 −0.337073
$$848$$ 67106.0 2.71749
$$849$$ 0 0
$$850$$ −16750.0 −0.675906
$$851$$ 16352.0 0.658683
$$852$$ 0 0
$$853$$ 30150.0 1.21022 0.605109 0.796142i $$-0.293129\pi$$
0.605109 + 0.796142i $$0.293129\pi$$
$$854$$ −25130.0 −1.00694
$$855$$ 0 0
$$856$$ 32940.0 1.31526
$$857$$ 4350.00 0.173388 0.0866938 0.996235i $$-0.472370\pi$$
0.0866938 + 0.996235i $$0.472370\pi$$
$$858$$ 0 0
$$859$$ −30676.0 −1.21845 −0.609227 0.792996i $$-0.708520\pi$$
−0.609227 + 0.792996i $$0.708520\pi$$
$$860$$ −28900.0 −1.14591
$$861$$ 0 0
$$862$$ −24920.0 −0.984662
$$863$$ 23688.0 0.934356 0.467178 0.884163i $$-0.345271\pi$$
0.467178 + 0.884163i $$0.345271\pi$$
$$864$$ 0 0
$$865$$ −12210.0 −0.479945
$$866$$ 8470.00 0.332358
$$867$$ 0 0
$$868$$ −26656.0 −1.04235
$$869$$ −10752.0 −0.419720
$$870$$ 0 0
$$871$$ 12360.0 0.480830
$$872$$ 17010.0 0.660586
$$873$$ 0 0
$$874$$ −51520.0 −1.99392
$$875$$ −875.000 −0.0338062
$$876$$ 0 0
$$877$$ 31910.0 1.22865 0.614324 0.789054i $$-0.289429\pi$$
0.614324 + 0.789054i $$0.289429\pi$$
$$878$$ −69320.0 −2.66451
$$879$$ 0 0
$$880$$ 5340.00 0.204558
$$881$$ −50250.0 −1.92164 −0.960820 0.277172i $$-0.910603\pi$$
−0.960820 + 0.277172i $$0.910603\pi$$
$$882$$ 0 0
$$883$$ 5980.00 0.227908 0.113954 0.993486i $$-0.463648\pi$$
0.113954 + 0.993486i $$0.463648\pi$$
$$884$$ 68340.0 2.60014
$$885$$ 0 0
$$886$$ −23220.0 −0.880464
$$887$$ 24568.0 0.930003 0.465002 0.885310i $$-0.346054\pi$$
0.465002 + 0.885310i $$0.346054\pi$$
$$888$$ 0 0
$$889$$ 4256.00 0.160564
$$890$$ 25950.0 0.977355
$$891$$ 0 0
$$892$$ −27744.0 −1.04141
$$893$$ 19136.0 0.717091
$$894$$ 0 0
$$895$$ −20460.0 −0.764137
$$896$$ 14805.0 0.552009
$$897$$ 0 0
$$898$$ −24630.0 −0.915271
$$899$$ −12992.0 −0.481988
$$900$$ 0 0
$$901$$ 101036. 3.73585
$$902$$ −1080.00 −0.0398670
$$903$$ 0 0
$$904$$ 65610.0 2.41389
$$905$$ −6350.00 −0.233239
$$906$$ 0 0
$$907$$ 13252.0 0.485144 0.242572 0.970133i $$-0.422009\pi$$
0.242572 + 0.970133i $$0.422009\pi$$
$$908$$ −69428.0 −2.53750
$$909$$ 0 0
$$910$$ 5250.00 0.191248
$$911$$ 6744.00 0.245267 0.122634 0.992452i $$-0.460866\pi$$
0.122634 + 0.992452i $$0.460866\pi$$
$$912$$ 0 0
$$913$$ 5232.00 0.189654
$$914$$ 73470.0 2.65883
$$915$$ 0 0
$$916$$ −57562.0 −2.07631
$$917$$ 6692.00 0.240992
$$918$$ 0 0
$$919$$ −45336.0 −1.62731 −0.813654 0.581349i $$-0.802525\pi$$
−0.813654 + 0.581349i $$0.802525\pi$$
$$920$$ −25200.0 −0.903065
$$921$$ 0 0
$$922$$ 10030.0 0.358265
$$923$$ 28800.0 1.02705
$$924$$ 0 0
$$925$$ −3650.00 −0.129742
$$926$$ −24480.0 −0.868750
$$927$$ 0 0
$$928$$ −4930.00 −0.174391
$$929$$ −30074.0 −1.06211 −0.531053 0.847339i $$-0.678203\pi$$
−0.531053 + 0.847339i $$0.678203\pi$$
$$930$$ 0 0
$$931$$ −4508.00 −0.158694
$$932$$ −90474.0 −3.17980
$$933$$ 0 0
$$934$$ 13300.0 0.465941
$$935$$ 8040.00 0.281215
$$936$$ 0 0
$$937$$ 21754.0 0.758455 0.379227 0.925303i $$-0.376190\pi$$
0.379227 + 0.925303i $$0.376190\pi$$
$$938$$ −14420.0 −0.501951
$$939$$ 0 0
$$940$$ 17680.0 0.613466
$$941$$ −14550.0 −0.504056 −0.252028 0.967720i $$-0.581097\pi$$
−0.252028 + 0.967720i $$0.581097\pi$$
$$942$$ 0 0
$$943$$ 2016.00 0.0696182
$$944$$ −33820.0 −1.16605
$$945$$ 0 0
$$946$$ 20400.0 0.701122
$$947$$ −46660.0 −1.60110 −0.800552 0.599263i $$-0.795460\pi$$
−0.800552 + 0.599263i $$0.795460\pi$$
$$948$$ 0 0
$$949$$ 31980.0 1.09390
$$950$$ 11500.0 0.392747
$$951$$ 0 0
$$952$$ −42210.0 −1.43701
$$953$$ −20810.0 −0.707347 −0.353674 0.935369i $$-0.615068\pi$$
−0.353674 + 0.935369i $$0.615068\pi$$
$$954$$ 0 0
$$955$$ 24520.0 0.830836
$$956$$ −63512.0 −2.14867
$$957$$ 0 0
$$958$$ −28000.0 −0.944300
$$959$$ 2618.00 0.0881539
$$960$$ 0 0
$$961$$ 20385.0 0.684267
$$962$$ 21900.0 0.733975
$$963$$ 0 0
$$964$$ 3570.00 0.119276
$$965$$ −10890.0 −0.363276
$$966$$ 0 0
$$967$$ 2776.00 0.0923166 0.0461583 0.998934i $$-0.485302\pi$$
0.0461583 + 0.998934i $$0.485302\pi$$
$$968$$ 53415.0 1.77358
$$969$$ 0 0
$$970$$ −17550.0 −0.580924
$$971$$ −27292.0 −0.902000 −0.451000 0.892524i $$-0.648933\pi$$
−0.451000 + 0.892524i $$0.648933\pi$$
$$972$$ 0 0
$$973$$ 2772.00 0.0913322
$$974$$ 32120.0 1.05666
$$975$$ 0 0
$$976$$ 63902.0 2.09575
$$977$$ 62.0000 0.00203025 0.00101513 0.999999i $$-0.499677\pi$$
0.00101513 + 0.999999i $$0.499677\pi$$
$$978$$ 0 0
$$979$$ −12456.0 −0.406635
$$980$$ −4165.00 −0.135761
$$981$$ 0 0
$$982$$ −94500.0 −3.07089
$$983$$ −37912.0 −1.23012 −0.615058 0.788481i $$-0.710868\pi$$
−0.615058 + 0.788481i $$0.710868\pi$$
$$984$$ 0 0
$$985$$ −14250.0 −0.460957
$$986$$ −38860.0 −1.25513
$$987$$ 0 0
$$988$$ −46920.0 −1.51085
$$989$$ −38080.0 −1.22434
$$990$$ 0 0
$$991$$ 10656.0 0.341573 0.170787 0.985308i $$-0.445369\pi$$
0.170787 + 0.985308i $$0.445369\pi$$
$$992$$ 19040.0 0.609396
$$993$$ 0 0
$$994$$ −33600.0 −1.07216
$$995$$ 5720.00 0.182247
$$996$$ 0 0
$$997$$ −29434.0 −0.934989 −0.467495 0.883996i $$-0.654843\pi$$
−0.467495 + 0.883996i $$0.654843\pi$$
$$998$$ 76820.0 2.43657
$$999$$ 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 315.4.a.a.1.1 1
3.2 odd 2 105.4.a.b.1.1 1
5.4 even 2 1575.4.a.l.1.1 1
7.6 odd 2 2205.4.a.b.1.1 1
12.11 even 2 1680.4.a.u.1.1 1
15.2 even 4 525.4.d.a.274.2 2
15.8 even 4 525.4.d.a.274.1 2
15.14 odd 2 525.4.a.a.1.1 1
21.20 even 2 735.4.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.b.1.1 1 3.2 odd 2
315.4.a.a.1.1 1 1.1 even 1 trivial
525.4.a.a.1.1 1 15.14 odd 2
525.4.d.a.274.1 2 15.8 even 4
525.4.d.a.274.2 2 15.2 even 4
735.4.a.j.1.1 1 21.20 even 2
1575.4.a.l.1.1 1 5.4 even 2
1680.4.a.u.1.1 1 12.11 even 2
2205.4.a.b.1.1 1 7.6 odd 2