# Properties

 Label 2025.4.a.f.1.1 Level $2025$ Weight $4$ Character 2025.1 Self dual yes Analytic conductor $119.479$ 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] = [2025,4,Mod(1,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2025.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2025.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: $$119.478867762$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 405) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2025.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{2} +17.0000 q^{4} -9.00000 q^{7} +45.0000 q^{8} +O(q^{10})$$ $$q+5.00000 q^{2} +17.0000 q^{4} -9.00000 q^{7} +45.0000 q^{8} -8.00000 q^{11} -43.0000 q^{13} -45.0000 q^{14} +89.0000 q^{16} +122.000 q^{17} -59.0000 q^{19} -40.0000 q^{22} +213.000 q^{23} -215.000 q^{26} -153.000 q^{28} +224.000 q^{29} -36.0000 q^{31} +85.0000 q^{32} +610.000 q^{34} -206.000 q^{37} -295.000 q^{38} +413.000 q^{41} +392.000 q^{43} -136.000 q^{44} +1065.00 q^{46} +311.000 q^{47} -262.000 q^{49} -731.000 q^{52} +377.000 q^{53} -405.000 q^{56} +1120.00 q^{58} +337.000 q^{59} +40.0000 q^{61} -180.000 q^{62} -287.000 q^{64} -348.000 q^{67} +2074.00 q^{68} +62.0000 q^{71} +1214.00 q^{73} -1030.00 q^{74} -1003.00 q^{76} +72.0000 q^{77} -294.000 q^{79} +2065.00 q^{82} -534.000 q^{83} +1960.00 q^{86} -360.000 q^{88} -810.000 q^{89} +387.000 q^{91} +3621.00 q^{92} +1555.00 q^{94} +928.000 q^{97} -1310.00 q^{98} +O(q^{100})$$

## 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 $$-0.154920\pi$$
0.883883 + 0.467707i $$0.154920\pi$$
$$3$$ 0 0
$$4$$ 17.0000 2.12500
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −9.00000 −0.485954 −0.242977 0.970032i $$-0.578124\pi$$
−0.242977 + 0.970032i $$0.578124\pi$$
$$8$$ 45.0000 1.98874
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −8.00000 −0.219281 −0.109640 0.993971i $$-0.534970\pi$$
−0.109640 + 0.993971i $$0.534970\pi$$
$$12$$ 0 0
$$13$$ −43.0000 −0.917389 −0.458694 0.888594i $$-0.651683\pi$$
−0.458694 + 0.888594i $$0.651683\pi$$
$$14$$ −45.0000 −0.859054
$$15$$ 0 0
$$16$$ 89.0000 1.39062
$$17$$ 122.000 1.74055 0.870275 0.492566i $$-0.163941\pi$$
0.870275 + 0.492566i $$0.163941\pi$$
$$18$$ 0 0
$$19$$ −59.0000 −0.712396 −0.356198 0.934410i $$-0.615927\pi$$
−0.356198 + 0.934410i $$0.615927\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −40.0000 −0.387638
$$23$$ 213.000 1.93102 0.965512 0.260357i $$-0.0838403\pi$$
0.965512 + 0.260357i $$0.0838403\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −215.000 −1.62173
$$27$$ 0 0
$$28$$ −153.000 −1.03265
$$29$$ 224.000 1.43434 0.717168 0.696900i $$-0.245438\pi$$
0.717168 + 0.696900i $$0.245438\pi$$
$$30$$ 0 0
$$31$$ −36.0000 −0.208574 −0.104287 0.994547i $$-0.533256\pi$$
−0.104287 + 0.994547i $$0.533256\pi$$
$$32$$ 85.0000 0.469563
$$33$$ 0 0
$$34$$ 610.000 3.07689
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −206.000 −0.915302 −0.457651 0.889132i $$-0.651309\pi$$
−0.457651 + 0.889132i $$0.651309\pi$$
$$38$$ −295.000 −1.25935
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 413.000 1.57316 0.786582 0.617485i $$-0.211848\pi$$
0.786582 + 0.617485i $$0.211848\pi$$
$$42$$ 0 0
$$43$$ 392.000 1.39022 0.695110 0.718904i $$-0.255356\pi$$
0.695110 + 0.718904i $$0.255356\pi$$
$$44$$ −136.000 −0.465972
$$45$$ 0 0
$$46$$ 1065.00 3.41360
$$47$$ 311.000 0.965192 0.482596 0.875843i $$-0.339694\pi$$
0.482596 + 0.875843i $$0.339694\pi$$
$$48$$ 0 0
$$49$$ −262.000 −0.763848
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −731.000 −1.94945
$$53$$ 377.000 0.977074 0.488537 0.872543i $$-0.337531\pi$$
0.488537 + 0.872543i $$0.337531\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −405.000 −0.966436
$$57$$ 0 0
$$58$$ 1120.00 2.53557
$$59$$ 337.000 0.743621 0.371811 0.928309i $$-0.378737\pi$$
0.371811 + 0.928309i $$0.378737\pi$$
$$60$$ 0 0
$$61$$ 40.0000 0.0839586 0.0419793 0.999118i $$-0.486634\pi$$
0.0419793 + 0.999118i $$0.486634\pi$$
$$62$$ −180.000 −0.368710
$$63$$ 0 0
$$64$$ −287.000 −0.560547
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −348.000 −0.634552 −0.317276 0.948333i $$-0.602768\pi$$
−0.317276 + 0.948333i $$0.602768\pi$$
$$68$$ 2074.00 3.69867
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 62.0000 0.103634 0.0518172 0.998657i $$-0.483499\pi$$
0.0518172 + 0.998657i $$0.483499\pi$$
$$72$$ 0 0
$$73$$ 1214.00 1.94641 0.973205 0.229939i $$-0.0738525\pi$$
0.973205 + 0.229939i $$0.0738525\pi$$
$$74$$ −1030.00 −1.61804
$$75$$ 0 0
$$76$$ −1003.00 −1.51384
$$77$$ 72.0000 0.106561
$$78$$ 0 0
$$79$$ −294.000 −0.418704 −0.209352 0.977840i $$-0.567135\pi$$
−0.209352 + 0.977840i $$0.567135\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 2065.00 2.78099
$$83$$ −534.000 −0.706194 −0.353097 0.935587i $$-0.614871\pi$$
−0.353097 + 0.935587i $$0.614871\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1960.00 2.45758
$$87$$ 0 0
$$88$$ −360.000 −0.436092
$$89$$ −810.000 −0.964717 −0.482359 0.875974i $$-0.660220\pi$$
−0.482359 + 0.875974i $$0.660220\pi$$
$$90$$ 0 0
$$91$$ 387.000 0.445809
$$92$$ 3621.00 4.10343
$$93$$ 0 0
$$94$$ 1555.00 1.70623
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 928.000 0.971383 0.485691 0.874130i $$-0.338568\pi$$
0.485691 + 0.874130i $$0.338568\pi$$
$$98$$ −1310.00 −1.35031
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −996.000 −0.981245 −0.490622 0.871372i $$-0.663231\pi$$
−0.490622 + 0.871372i $$0.663231\pi$$
$$102$$ 0 0
$$103$$ 433.000 0.414221 0.207110 0.978318i $$-0.433594\pi$$
0.207110 + 0.978318i $$0.433594\pi$$
$$104$$ −1935.00 −1.82445
$$105$$ 0 0
$$106$$ 1885.00 1.72724
$$107$$ 1686.00 1.52329 0.761644 0.647996i $$-0.224393\pi$$
0.761644 + 0.647996i $$0.224393\pi$$
$$108$$ 0 0
$$109$$ 656.000 0.576453 0.288227 0.957562i $$-0.406934\pi$$
0.288227 + 0.957562i $$0.406934\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −801.000 −0.675780
$$113$$ −1018.00 −0.847481 −0.423741 0.905784i $$-0.639283\pi$$
−0.423741 + 0.905784i $$0.639283\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 3808.00 3.04796
$$117$$ 0 0
$$118$$ 1685.00 1.31455
$$119$$ −1098.00 −0.845828
$$120$$ 0 0
$$121$$ −1267.00 −0.951916
$$122$$ 200.000 0.148419
$$123$$ 0 0
$$124$$ −612.000 −0.443220
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1361.00 0.950939 0.475469 0.879732i $$-0.342278\pi$$
0.475469 + 0.879732i $$0.342278\pi$$
$$128$$ −2115.00 −1.46048
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1911.00 1.27454 0.637270 0.770640i $$-0.280063\pi$$
0.637270 + 0.770640i $$0.280063\pi$$
$$132$$ 0 0
$$133$$ 531.000 0.346192
$$134$$ −1740.00 −1.12174
$$135$$ 0 0
$$136$$ 5490.00 3.46150
$$137$$ −654.000 −0.407847 −0.203923 0.978987i $$-0.565369\pi$$
−0.203923 + 0.978987i $$0.565369\pi$$
$$138$$ 0 0
$$139$$ −733.000 −0.447282 −0.223641 0.974672i $$-0.571794\pi$$
−0.223641 + 0.974672i $$0.571794\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 310.000 0.183202
$$143$$ 344.000 0.201166
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 6070.00 3.44080
$$147$$ 0 0
$$148$$ −3502.00 −1.94502
$$149$$ −1126.00 −0.619097 −0.309549 0.950884i $$-0.600178\pi$$
−0.309549 + 0.950884i $$0.600178\pi$$
$$150$$ 0 0
$$151$$ −2546.00 −1.37212 −0.686061 0.727544i $$-0.740662\pi$$
−0.686061 + 0.727544i $$0.740662\pi$$
$$152$$ −2655.00 −1.41677
$$153$$ 0 0
$$154$$ 360.000 0.188374
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −1223.00 −0.621694 −0.310847 0.950460i $$-0.600613\pi$$
−0.310847 + 0.950460i $$0.600613\pi$$
$$158$$ −1470.00 −0.740170
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −1917.00 −0.938390
$$162$$ 0 0
$$163$$ −3176.00 −1.52616 −0.763078 0.646306i $$-0.776313\pi$$
−0.763078 + 0.646306i $$0.776313\pi$$
$$164$$ 7021.00 3.34298
$$165$$ 0 0
$$166$$ −2670.00 −1.24839
$$167$$ −132.000 −0.0611645 −0.0305822 0.999532i $$-0.509736\pi$$
−0.0305822 + 0.999532i $$0.509736\pi$$
$$168$$ 0 0
$$169$$ −348.000 −0.158398
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 6664.00 2.95422
$$173$$ 993.000 0.436395 0.218198 0.975905i $$-0.429982\pi$$
0.218198 + 0.975905i $$0.429982\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −712.000 −0.304938
$$177$$ 0 0
$$178$$ −4050.00 −1.70540
$$179$$ −3101.00 −1.29486 −0.647429 0.762126i $$-0.724156\pi$$
−0.647429 + 0.762126i $$0.724156\pi$$
$$180$$ 0 0
$$181$$ 2846.00 1.16874 0.584369 0.811488i $$-0.301342\pi$$
0.584369 + 0.811488i $$0.301342\pi$$
$$182$$ 1935.00 0.788086
$$183$$ 0 0
$$184$$ 9585.00 3.84030
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −976.000 −0.381669
$$188$$ 5287.00 2.05103
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3080.00 1.16681 0.583406 0.812181i $$-0.301720\pi$$
0.583406 + 0.812181i $$0.301720\pi$$
$$192$$ 0 0
$$193$$ −2588.00 −0.965224 −0.482612 0.875834i $$-0.660312\pi$$
−0.482612 + 0.875834i $$0.660312\pi$$
$$194$$ 4640.00 1.71718
$$195$$ 0 0
$$196$$ −4454.00 −1.62318
$$197$$ 1335.00 0.482816 0.241408 0.970424i $$-0.422391\pi$$
0.241408 + 0.970424i $$0.422391\pi$$
$$198$$ 0 0
$$199$$ −5204.00 −1.85378 −0.926889 0.375336i $$-0.877527\pi$$
−0.926889 + 0.375336i $$0.877527\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −4980.00 −1.73461
$$203$$ −2016.00 −0.697022
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 2165.00 0.732246
$$207$$ 0 0
$$208$$ −3827.00 −1.27574
$$209$$ 472.000 0.156215
$$210$$ 0 0
$$211$$ 1637.00 0.534103 0.267051 0.963682i $$-0.413951\pi$$
0.267051 + 0.963682i $$0.413951\pi$$
$$212$$ 6409.00 2.07628
$$213$$ 0 0
$$214$$ 8430.00 2.69282
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 324.000 0.101357
$$218$$ 3280.00 1.01904
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −5246.00 −1.59676
$$222$$ 0 0
$$223$$ 4480.00 1.34530 0.672652 0.739959i $$-0.265155\pi$$
0.672652 + 0.739959i $$0.265155\pi$$
$$224$$ −765.000 −0.228186
$$225$$ 0 0
$$226$$ −5090.00 −1.49815
$$227$$ 3736.00 1.09237 0.546183 0.837666i $$-0.316080\pi$$
0.546183 + 0.837666i $$0.316080\pi$$
$$228$$ 0 0
$$229$$ −1380.00 −0.398223 −0.199111 0.979977i $$-0.563806\pi$$
−0.199111 + 0.979977i $$0.563806\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 10080.0 2.85252
$$233$$ 2904.00 0.816512 0.408256 0.912867i $$-0.366137\pi$$
0.408256 + 0.912867i $$0.366137\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 5729.00 1.58020
$$237$$ 0 0
$$238$$ −5490.00 −1.49523
$$239$$ −5966.00 −1.61468 −0.807340 0.590087i $$-0.799094\pi$$
−0.807340 + 0.590087i $$0.799094\pi$$
$$240$$ 0 0
$$241$$ −3218.00 −0.860123 −0.430061 0.902800i $$-0.641508\pi$$
−0.430061 + 0.902800i $$0.641508\pi$$
$$242$$ −6335.00 −1.68277
$$243$$ 0 0
$$244$$ 680.000 0.178412
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2537.00 0.653544
$$248$$ −1620.00 −0.414799
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 6123.00 1.53976 0.769881 0.638187i $$-0.220315\pi$$
0.769881 + 0.638187i $$0.220315\pi$$
$$252$$ 0 0
$$253$$ −1704.00 −0.423437
$$254$$ 6805.00 1.68104
$$255$$ 0 0
$$256$$ −8279.00 −2.02124
$$257$$ 1398.00 0.339318 0.169659 0.985503i $$-0.445733\pi$$
0.169659 + 0.985503i $$0.445733\pi$$
$$258$$ 0 0
$$259$$ 1854.00 0.444795
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 9555.00 2.25309
$$263$$ 3211.00 0.752847 0.376423 0.926448i $$-0.377154\pi$$
0.376423 + 0.926448i $$0.377154\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 2655.00 0.611987
$$267$$ 0 0
$$268$$ −5916.00 −1.34842
$$269$$ 4018.00 0.910713 0.455356 0.890309i $$-0.349512\pi$$
0.455356 + 0.890309i $$0.349512\pi$$
$$270$$ 0 0
$$271$$ 2314.00 0.518692 0.259346 0.965784i $$-0.416493\pi$$
0.259346 + 0.965784i $$0.416493\pi$$
$$272$$ 10858.0 2.42045
$$273$$ 0 0
$$274$$ −3270.00 −0.720978
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 4347.00 0.942909 0.471455 0.881890i $$-0.343729\pi$$
0.471455 + 0.881890i $$0.343729\pi$$
$$278$$ −3665.00 −0.790691
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1551.00 0.329270 0.164635 0.986355i $$-0.447355\pi$$
0.164635 + 0.986355i $$0.447355\pi$$
$$282$$ 0 0
$$283$$ −4380.00 −0.920014 −0.460007 0.887915i $$-0.652153\pi$$
−0.460007 + 0.887915i $$0.652153\pi$$
$$284$$ 1054.00 0.220223
$$285$$ 0 0
$$286$$ 1720.00 0.355614
$$287$$ −3717.00 −0.764486
$$288$$ 0 0
$$289$$ 9971.00 2.02951
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 20638.0 4.13612
$$293$$ −5049.00 −1.00671 −0.503354 0.864080i $$-0.667901\pi$$
−0.503354 + 0.864080i $$0.667901\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −9270.00 −1.82030
$$297$$ 0 0
$$298$$ −5630.00 −1.09442
$$299$$ −9159.00 −1.77150
$$300$$ 0 0
$$301$$ −3528.00 −0.675583
$$302$$ −12730.0 −2.42559
$$303$$ 0 0
$$304$$ −5251.00 −0.990676
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −5428.00 −1.00910 −0.504548 0.863384i $$-0.668341\pi$$
−0.504548 + 0.863384i $$0.668341\pi$$
$$308$$ 1224.00 0.226441
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −18.0000 −0.00328195 −0.00164097 0.999999i $$-0.500522\pi$$
−0.00164097 + 0.999999i $$0.500522\pi$$
$$312$$ 0 0
$$313$$ 2116.00 0.382119 0.191060 0.981578i $$-0.438808\pi$$
0.191060 + 0.981578i $$0.438808\pi$$
$$314$$ −6115.00 −1.09901
$$315$$ 0 0
$$316$$ −4998.00 −0.889745
$$317$$ −4415.00 −0.782243 −0.391122 0.920339i $$-0.627913\pi$$
−0.391122 + 0.920339i $$0.627913\pi$$
$$318$$ 0 0
$$319$$ −1792.00 −0.314523
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −9585.00 −1.65885
$$323$$ −7198.00 −1.23996
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −15880.0 −2.69789
$$327$$ 0 0
$$328$$ 18585.0 3.12861
$$329$$ −2799.00 −0.469039
$$330$$ 0 0
$$331$$ 3480.00 0.577879 0.288940 0.957347i $$-0.406697\pi$$
0.288940 + 0.957347i $$0.406697\pi$$
$$332$$ −9078.00 −1.50066
$$333$$ 0 0
$$334$$ −660.000 −0.108125
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −6322.00 −1.02190 −0.510951 0.859610i $$-0.670707\pi$$
−0.510951 + 0.859610i $$0.670707\pi$$
$$338$$ −1740.00 −0.280010
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 288.000 0.0457363
$$342$$ 0 0
$$343$$ 5445.00 0.857150
$$344$$ 17640.0 2.76478
$$345$$ 0 0
$$346$$ 4965.00 0.771445
$$347$$ 10034.0 1.55232 0.776158 0.630539i $$-0.217166\pi$$
0.776158 + 0.630539i $$0.217166\pi$$
$$348$$ 0 0
$$349$$ −2510.00 −0.384978 −0.192489 0.981299i $$-0.561656\pi$$
−0.192489 + 0.981299i $$0.561656\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −680.000 −0.102966
$$353$$ −3726.00 −0.561799 −0.280899 0.959737i $$-0.590633\pi$$
−0.280899 + 0.959737i $$0.590633\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −13770.0 −2.05002
$$357$$ 0 0
$$358$$ −15505.0 −2.28901
$$359$$ −10710.0 −1.57452 −0.787259 0.616622i $$-0.788501\pi$$
−0.787259 + 0.616622i $$0.788501\pi$$
$$360$$ 0 0
$$361$$ −3378.00 −0.492492
$$362$$ 14230.0 2.06606
$$363$$ 0 0
$$364$$ 6579.00 0.947344
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 2160.00 0.307224 0.153612 0.988131i $$-0.450909\pi$$
0.153612 + 0.988131i $$0.450909\pi$$
$$368$$ 18957.0 2.68533
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −3393.00 −0.474813
$$372$$ 0 0
$$373$$ −3394.00 −0.471138 −0.235569 0.971858i $$-0.575695\pi$$
−0.235569 + 0.971858i $$0.575695\pi$$
$$374$$ −4880.00 −0.674703
$$375$$ 0 0
$$376$$ 13995.0 1.91951
$$377$$ −9632.00 −1.31584
$$378$$ 0 0
$$379$$ 9031.00 1.22399 0.611994 0.790863i $$-0.290368\pi$$
0.611994 + 0.790863i $$0.290368\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 15400.0 2.06265
$$383$$ 10305.0 1.37483 0.687416 0.726264i $$-0.258745\pi$$
0.687416 + 0.726264i $$0.258745\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −12940.0 −1.70629
$$387$$ 0 0
$$388$$ 15776.0 2.06419
$$389$$ −3480.00 −0.453581 −0.226790 0.973944i $$-0.572823\pi$$
−0.226790 + 0.973944i $$0.572823\pi$$
$$390$$ 0 0
$$391$$ 25986.0 3.36104
$$392$$ −11790.0 −1.51909
$$393$$ 0 0
$$394$$ 6675.00 0.853507
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −3706.00 −0.468511 −0.234255 0.972175i $$-0.575265\pi$$
−0.234255 + 0.972175i $$0.575265\pi$$
$$398$$ −26020.0 −3.27705
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 2679.00 0.333623 0.166812 0.985989i $$-0.446653\pi$$
0.166812 + 0.985989i $$0.446653\pi$$
$$402$$ 0 0
$$403$$ 1548.00 0.191343
$$404$$ −16932.0 −2.08514
$$405$$ 0 0
$$406$$ −10080.0 −1.23217
$$407$$ 1648.00 0.200708
$$408$$ 0 0
$$409$$ −12499.0 −1.51109 −0.755545 0.655097i $$-0.772628\pi$$
−0.755545 + 0.655097i $$0.772628\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 7361.00 0.880220
$$413$$ −3033.00 −0.361366
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −3655.00 −0.430772
$$417$$ 0 0
$$418$$ 2360.00 0.276152
$$419$$ −928.000 −0.108200 −0.0541000 0.998536i $$-0.517229\pi$$
−0.0541000 + 0.998536i $$0.517229\pi$$
$$420$$ 0 0
$$421$$ 7570.00 0.876340 0.438170 0.898892i $$-0.355627\pi$$
0.438170 + 0.898892i $$0.355627\pi$$
$$422$$ 8185.00 0.944170
$$423$$ 0 0
$$424$$ 16965.0 1.94314
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −360.000 −0.0408000
$$428$$ 28662.0 3.23699
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −2460.00 −0.274928 −0.137464 0.990507i $$-0.543895\pi$$
−0.137464 + 0.990507i $$0.543895\pi$$
$$432$$ 0 0
$$433$$ −1648.00 −0.182905 −0.0914525 0.995809i $$-0.529151\pi$$
−0.0914525 + 0.995809i $$0.529151\pi$$
$$434$$ 1620.00 0.179176
$$435$$ 0 0
$$436$$ 11152.0 1.22496
$$437$$ −12567.0 −1.37565
$$438$$ 0 0
$$439$$ 15826.0 1.72058 0.860289 0.509807i $$-0.170283\pi$$
0.860289 + 0.509807i $$0.170283\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −26230.0 −2.82270
$$443$$ −12774.0 −1.37000 −0.685001 0.728542i $$-0.740198\pi$$
−0.685001 + 0.728542i $$0.740198\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 22400.0 2.37819
$$447$$ 0 0
$$448$$ 2583.00 0.272400
$$449$$ 8875.00 0.932822 0.466411 0.884568i $$-0.345547\pi$$
0.466411 + 0.884568i $$0.345547\pi$$
$$450$$ 0 0
$$451$$ −3304.00 −0.344965
$$452$$ −17306.0 −1.80090
$$453$$ 0 0
$$454$$ 18680.0 1.93105
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11524.0 −1.17958 −0.589792 0.807555i $$-0.700790\pi$$
−0.589792 + 0.807555i $$0.700790\pi$$
$$458$$ −6900.00 −0.703965
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −8544.00 −0.863197 −0.431598 0.902066i $$-0.642050\pi$$
−0.431598 + 0.902066i $$0.642050\pi$$
$$462$$ 0 0
$$463$$ −2523.00 −0.253248 −0.126624 0.991951i $$-0.540414\pi$$
−0.126624 + 0.991951i $$0.540414\pi$$
$$464$$ 19936.0 1.99462
$$465$$ 0 0
$$466$$ 14520.0 1.44340
$$467$$ 2902.00 0.287556 0.143778 0.989610i $$-0.454075\pi$$
0.143778 + 0.989610i $$0.454075\pi$$
$$468$$ 0 0
$$469$$ 3132.00 0.308363
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 15165.0 1.47887
$$473$$ −3136.00 −0.304849
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −18666.0 −1.79738
$$477$$ 0 0
$$478$$ −29830.0 −2.85438
$$479$$ 4362.00 0.416085 0.208043 0.978120i $$-0.433291\pi$$
0.208043 + 0.978120i $$0.433291\pi$$
$$480$$ 0 0
$$481$$ 8858.00 0.839688
$$482$$ −16090.0 −1.52050
$$483$$ 0 0
$$484$$ −21539.0 −2.02282
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −5723.00 −0.532513 −0.266257 0.963902i $$-0.585787\pi$$
−0.266257 + 0.963902i $$0.585787\pi$$
$$488$$ 1800.00 0.166972
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −13339.0 −1.22603 −0.613015 0.790071i $$-0.710043\pi$$
−0.613015 + 0.790071i $$0.710043\pi$$
$$492$$ 0 0
$$493$$ 27328.0 2.49653
$$494$$ 12685.0 1.15531
$$495$$ 0 0
$$496$$ −3204.00 −0.290048
$$497$$ −558.000 −0.0503616
$$498$$ 0 0
$$499$$ −19637.0 −1.76167 −0.880835 0.473424i $$-0.843018\pi$$
−0.880835 + 0.473424i $$0.843018\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 30615.0 2.72194
$$503$$ 5416.00 0.480094 0.240047 0.970761i $$-0.422837\pi$$
0.240047 + 0.970761i $$0.422837\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −8520.00 −0.748538
$$507$$ 0 0
$$508$$ 23137.0 2.02074
$$509$$ −6110.00 −0.532065 −0.266032 0.963964i $$-0.585713\pi$$
−0.266032 + 0.963964i $$0.585713\pi$$
$$510$$ 0 0
$$511$$ −10926.0 −0.945867
$$512$$ −24475.0 −2.11260
$$513$$ 0 0
$$514$$ 6990.00 0.599836
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −2488.00 −0.211648
$$518$$ 9270.00 0.786294
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 20375.0 1.71333 0.856665 0.515873i $$-0.172532\pi$$
0.856665 + 0.515873i $$0.172532\pi$$
$$522$$ 0 0
$$523$$ 19010.0 1.58939 0.794693 0.607011i $$-0.207632\pi$$
0.794693 + 0.607011i $$0.207632\pi$$
$$524$$ 32487.0 2.70840
$$525$$ 0 0
$$526$$ 16055.0 1.33086
$$527$$ −4392.00 −0.363033
$$528$$ 0 0
$$529$$ 33202.0 2.72886
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 9027.00 0.735658
$$533$$ −17759.0 −1.44320
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −15660.0 −1.26196
$$537$$ 0 0
$$538$$ 20090.0 1.60993
$$539$$ 2096.00 0.167497
$$540$$ 0 0
$$541$$ 3288.00 0.261298 0.130649 0.991429i $$-0.458294\pi$$
0.130649 + 0.991429i $$0.458294\pi$$
$$542$$ 11570.0 0.916926
$$543$$ 0 0
$$544$$ 10370.0 0.817298
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −3256.00 −0.254509 −0.127255 0.991870i $$-0.540617\pi$$
−0.127255 + 0.991870i $$0.540617\pi$$
$$548$$ −11118.0 −0.866674
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −13216.0 −1.02182
$$552$$ 0 0
$$553$$ 2646.00 0.203471
$$554$$ 21735.0 1.66684
$$555$$ 0 0
$$556$$ −12461.0 −0.950475
$$557$$ −213.000 −0.0162031 −0.00810153 0.999967i $$-0.502579\pi$$
−0.00810153 + 0.999967i $$0.502579\pi$$
$$558$$ 0 0
$$559$$ −16856.0 −1.27537
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 7755.00 0.582073
$$563$$ −17388.0 −1.30163 −0.650814 0.759237i $$-0.725572\pi$$
−0.650814 + 0.759237i $$0.725572\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −21900.0 −1.62637
$$567$$ 0 0
$$568$$ 2790.00 0.206102
$$569$$ −4353.00 −0.320716 −0.160358 0.987059i $$-0.551265\pi$$
−0.160358 + 0.987059i $$0.551265\pi$$
$$570$$ 0 0
$$571$$ −1924.00 −0.141010 −0.0705052 0.997511i $$-0.522461\pi$$
−0.0705052 + 0.997511i $$0.522461\pi$$
$$572$$ 5848.00 0.427478
$$573$$ 0 0
$$574$$ −18585.0 −1.35143
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −16832.0 −1.21443 −0.607214 0.794538i $$-0.707713\pi$$
−0.607214 + 0.794538i $$0.707713\pi$$
$$578$$ 49855.0 3.58771
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 4806.00 0.343178
$$582$$ 0 0
$$583$$ −3016.00 −0.214254
$$584$$ 54630.0 3.87090
$$585$$ 0 0
$$586$$ −25245.0 −1.77963
$$587$$ 2106.00 0.148082 0.0740408 0.997255i $$-0.476410\pi$$
0.0740408 + 0.997255i $$0.476410\pi$$
$$588$$ 0 0
$$589$$ 2124.00 0.148587
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −18334.0 −1.27284
$$593$$ −4694.00 −0.325058 −0.162529 0.986704i $$-0.551965\pi$$
−0.162529 + 0.986704i $$0.551965\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −19142.0 −1.31558
$$597$$ 0 0
$$598$$ −45795.0 −3.13160
$$599$$ 13226.0 0.902170 0.451085 0.892481i $$-0.351037\pi$$
0.451085 + 0.892481i $$0.351037\pi$$
$$600$$ 0 0
$$601$$ −13291.0 −0.902082 −0.451041 0.892503i $$-0.648947\pi$$
−0.451041 + 0.892503i $$0.648947\pi$$
$$602$$ −17640.0 −1.19427
$$603$$ 0 0
$$604$$ −43282.0 −2.91576
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −6040.00 −0.403881 −0.201941 0.979398i $$-0.564725\pi$$
−0.201941 + 0.979398i $$0.564725\pi$$
$$608$$ −5015.00 −0.334515
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −13373.0 −0.885456
$$612$$ 0 0
$$613$$ 23.0000 0.00151543 0.000757717 1.00000i $$-0.499759\pi$$
0.000757717 1.00000i $$0.499759\pi$$
$$614$$ −27140.0 −1.78385
$$615$$ 0 0
$$616$$ 3240.00 0.211921
$$617$$ 3018.00 0.196921 0.0984604 0.995141i $$-0.468608\pi$$
0.0984604 + 0.995141i $$0.468608\pi$$
$$618$$ 0 0
$$619$$ −9439.00 −0.612901 −0.306450 0.951887i $$-0.599141\pi$$
−0.306450 + 0.951887i $$0.599141\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −90.0000 −0.00580172
$$623$$ 7290.00 0.468808
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 10580.0 0.675498
$$627$$ 0 0
$$628$$ −20791.0 −1.32110
$$629$$ −25132.0 −1.59313
$$630$$ 0 0
$$631$$ −9800.00 −0.618275 −0.309138 0.951017i $$-0.600040\pi$$
−0.309138 + 0.951017i $$0.600040\pi$$
$$632$$ −13230.0 −0.832692
$$633$$ 0 0
$$634$$ −22075.0 −1.38282
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 11266.0 0.700746
$$638$$ −8960.00 −0.556003
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 23442.0 1.44447 0.722233 0.691649i $$-0.243116\pi$$
0.722233 + 0.691649i $$0.243116\pi$$
$$642$$ 0 0
$$643$$ −31308.0 −1.92017 −0.960083 0.279715i $$-0.909760\pi$$
−0.960083 + 0.279715i $$0.909760\pi$$
$$644$$ −32589.0 −1.99408
$$645$$ 0 0
$$646$$ −35990.0 −2.19196
$$647$$ −712.000 −0.0432637 −0.0216318 0.999766i $$-0.506886\pi$$
−0.0216318 + 0.999766i $$0.506886\pi$$
$$648$$ 0 0
$$649$$ −2696.00 −0.163062
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −53992.0 −3.24308
$$653$$ −31478.0 −1.88642 −0.943208 0.332203i $$-0.892208\pi$$
−0.943208 + 0.332203i $$0.892208\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 36757.0 2.18768
$$657$$ 0 0
$$658$$ −13995.0 −0.829152
$$659$$ −16121.0 −0.952936 −0.476468 0.879192i $$-0.658083\pi$$
−0.476468 + 0.879192i $$0.658083\pi$$
$$660$$ 0 0
$$661$$ −19160.0 −1.12744 −0.563720 0.825966i $$-0.690630\pi$$
−0.563720 + 0.825966i $$0.690630\pi$$
$$662$$ 17400.0 1.02156
$$663$$ 0 0
$$664$$ −24030.0 −1.40444
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 47712.0 2.76974
$$668$$ −2244.00 −0.129975
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −320.000 −0.0184105
$$672$$ 0 0
$$673$$ 13422.0 0.768767 0.384383 0.923174i $$-0.374414\pi$$
0.384383 + 0.923174i $$0.374414\pi$$
$$674$$ −31610.0 −1.80649
$$675$$ 0 0
$$676$$ −5916.00 −0.336595
$$677$$ −25905.0 −1.47062 −0.735310 0.677731i $$-0.762964\pi$$
−0.735310 + 0.677731i $$0.762964\pi$$
$$678$$ 0 0
$$679$$ −8352.00 −0.472048
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 1440.00 0.0808511
$$683$$ 9246.00 0.517992 0.258996 0.965878i $$-0.416608\pi$$
0.258996 + 0.965878i $$0.416608\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 27225.0 1.51524
$$687$$ 0 0
$$688$$ 34888.0 1.93327
$$689$$ −16211.0 −0.896357
$$690$$ 0 0
$$691$$ 25039.0 1.37848 0.689239 0.724534i $$-0.257945\pi$$
0.689239 + 0.724534i $$0.257945\pi$$
$$692$$ 16881.0 0.927340
$$693$$ 0 0
$$694$$ 50170.0 2.74413
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 50386.0 2.73817
$$698$$ −12550.0 −0.680551
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 32930.0 1.77425 0.887125 0.461530i $$-0.152699\pi$$
0.887125 + 0.461530i $$0.152699\pi$$
$$702$$ 0 0
$$703$$ 12154.0 0.652058
$$704$$ 2296.00 0.122917
$$705$$ 0 0
$$706$$ −18630.0 −0.993129
$$707$$ 8964.00 0.476840
$$708$$ 0 0
$$709$$ 1882.00 0.0996897 0.0498448 0.998757i $$-0.484127\pi$$
0.0498448 + 0.998757i $$0.484127\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −36450.0 −1.91857
$$713$$ −7668.00 −0.402761
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −52717.0 −2.75157
$$717$$ 0 0
$$718$$ −53550.0 −2.78338
$$719$$ −1962.00 −0.101767 −0.0508833 0.998705i $$-0.516204\pi$$
−0.0508833 + 0.998705i $$0.516204\pi$$
$$720$$ 0 0
$$721$$ −3897.00 −0.201292
$$722$$ −16890.0 −0.870610
$$723$$ 0 0
$$724$$ 48382.0 2.48357
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 13741.0 0.700998 0.350499 0.936563i $$-0.386012\pi$$
0.350499 + 0.936563i $$0.386012\pi$$
$$728$$ 17415.0 0.886597
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 47824.0 2.41975
$$732$$ 0 0
$$733$$ 32458.0 1.63556 0.817779 0.575533i $$-0.195205\pi$$
0.817779 + 0.575533i $$0.195205\pi$$
$$734$$ 10800.0 0.543100
$$735$$ 0 0
$$736$$ 18105.0 0.906738
$$737$$ 2784.00 0.139145
$$738$$ 0 0
$$739$$ 19612.0 0.976237 0.488118 0.872777i $$-0.337683\pi$$
0.488118 + 0.872777i $$0.337683\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −16965.0 −0.839359
$$743$$ −36736.0 −1.81388 −0.906940 0.421259i $$-0.861588\pi$$
−0.906940 + 0.421259i $$0.861588\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −16970.0 −0.832863
$$747$$ 0 0
$$748$$ −16592.0 −0.811048
$$749$$ −15174.0 −0.740248
$$750$$ 0 0
$$751$$ −3746.00 −0.182015 −0.0910076 0.995850i $$-0.529009\pi$$
−0.0910076 + 0.995850i $$0.529009\pi$$
$$752$$ 27679.0 1.34222
$$753$$ 0 0
$$754$$ −48160.0 −2.32611
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −5725.00 −0.274873 −0.137436 0.990511i $$-0.543886\pi$$
−0.137436 + 0.990511i $$0.543886\pi$$
$$758$$ 45155.0 2.16372
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 37323.0 1.77787 0.888934 0.458035i $$-0.151447\pi$$
0.888934 + 0.458035i $$0.151447\pi$$
$$762$$ 0 0
$$763$$ −5904.00 −0.280130
$$764$$ 52360.0 2.47947
$$765$$ 0 0
$$766$$ 51525.0 2.43038
$$767$$ −14491.0 −0.682190
$$768$$ 0 0
$$769$$ −24586.0 −1.15292 −0.576459 0.817126i $$-0.695566\pi$$
−0.576459 + 0.817126i $$0.695566\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −43996.0 −2.05110
$$773$$ 3078.00 0.143219 0.0716093 0.997433i $$-0.477187\pi$$
0.0716093 + 0.997433i $$0.477187\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 41760.0 1.93183
$$777$$ 0 0
$$778$$ −17400.0 −0.801825
$$779$$ −24367.0 −1.12072
$$780$$ 0 0
$$781$$ −496.000 −0.0227251
$$782$$ 129930. 5.94154
$$783$$ 0 0
$$784$$ −23318.0 −1.06223
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −41038.0 −1.85876 −0.929382 0.369120i $$-0.879659\pi$$
−0.929382 + 0.369120i $$0.879659\pi$$
$$788$$ 22695.0 1.02598
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 9162.00 0.411837
$$792$$ 0 0
$$793$$ −1720.00 −0.0770227
$$794$$ −18530.0 −0.828218
$$795$$ 0 0
$$796$$ −88468.0 −3.93928
$$797$$ −9362.00 −0.416084 −0.208042 0.978120i $$-0.566709\pi$$
−0.208042 + 0.978120i $$0.566709\pi$$
$$798$$ 0 0
$$799$$ 37942.0 1.67996
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 13395.0 0.589768
$$803$$ −9712.00 −0.426811
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 7740.00 0.338250
$$807$$ 0 0
$$808$$ −44820.0 −1.95144
$$809$$ −45115.0 −1.96064 −0.980321 0.197411i $$-0.936747\pi$$
−0.980321 + 0.197411i $$0.936747\pi$$
$$810$$ 0 0
$$811$$ −13512.0 −0.585044 −0.292522 0.956259i $$-0.594494\pi$$
−0.292522 + 0.956259i $$0.594494\pi$$
$$812$$ −34272.0 −1.48117
$$813$$ 0 0
$$814$$ 8240.00 0.354806
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −23128.0 −0.990387
$$818$$ −62495.0 −2.67125
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −4530.00 −0.192568 −0.0962839 0.995354i $$-0.530696\pi$$
−0.0962839 + 0.995354i $$0.530696\pi$$
$$822$$ 0 0
$$823$$ −30884.0 −1.30808 −0.654039 0.756461i $$-0.726927\pi$$
−0.654039 + 0.756461i $$0.726927\pi$$
$$824$$ 19485.0 0.823777
$$825$$ 0 0
$$826$$ −15165.0 −0.638811
$$827$$ −12088.0 −0.508272 −0.254136 0.967168i $$-0.581791\pi$$
−0.254136 + 0.967168i $$0.581791\pi$$
$$828$$ 0 0
$$829$$ −14112.0 −0.591230 −0.295615 0.955307i $$-0.595525\pi$$
−0.295615 + 0.955307i $$0.595525\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 12341.0 0.514239
$$833$$ −31964.0 −1.32952
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 8024.00 0.331957
$$837$$ 0 0
$$838$$ −4640.00 −0.191272
$$839$$ 3860.00 0.158834 0.0794172 0.996841i $$-0.474694\pi$$
0.0794172 + 0.996841i $$0.474694\pi$$
$$840$$ 0 0
$$841$$ 25787.0 1.05732
$$842$$ 37850.0 1.54917
$$843$$ 0 0
$$844$$ 27829.0 1.13497
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 11403.0 0.462588
$$848$$ 33553.0 1.35874
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −43878.0 −1.76747
$$852$$ 0 0
$$853$$ −2862.00 −0.114880 −0.0574402 0.998349i $$-0.518294\pi$$
−0.0574402 + 0.998349i $$0.518294\pi$$
$$854$$ −1800.00 −0.0721250
$$855$$ 0 0
$$856$$ 75870.0 3.02942
$$857$$ 32534.0 1.29678 0.648390 0.761308i $$-0.275443\pi$$
0.648390 + 0.761308i $$0.275443\pi$$
$$858$$ 0 0
$$859$$ −500.000 −0.0198600 −0.00993002 0.999951i $$-0.503161\pi$$
−0.00993002 + 0.999951i $$0.503161\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −12300.0 −0.486009
$$863$$ 36595.0 1.44346 0.721731 0.692173i $$-0.243347\pi$$
0.721731 + 0.692173i $$0.243347\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −8240.00 −0.323333
$$867$$ 0 0
$$868$$ 5508.00 0.215384
$$869$$ 2352.00 0.0918137
$$870$$ 0 0
$$871$$ 14964.0 0.582131
$$872$$ 29520.0 1.14641
$$873$$ 0 0
$$874$$ −62835.0 −2.43184
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −40143.0 −1.54565 −0.772824 0.634621i $$-0.781156\pi$$
−0.772824 + 0.634621i $$0.781156\pi$$
$$878$$ 79130.0 3.04158
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 16414.0 0.627698 0.313849 0.949473i $$-0.398381\pi$$
0.313849 + 0.949473i $$0.398381\pi$$
$$882$$ 0 0
$$883$$ 33478.0 1.27591 0.637953 0.770076i $$-0.279782\pi$$
0.637953 + 0.770076i $$0.279782\pi$$
$$884$$ −89182.0 −3.39312
$$885$$ 0 0
$$886$$ −63870.0 −2.42184
$$887$$ 3633.00 0.137524 0.0687622 0.997633i $$-0.478095\pi$$
0.0687622 + 0.997633i $$0.478095\pi$$
$$888$$ 0 0
$$889$$ −12249.0 −0.462113
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 76160.0 2.85877
$$893$$ −18349.0 −0.687599
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 19035.0 0.709726
$$897$$ 0 0
$$898$$ 44375.0 1.64901
$$899$$ −8064.00 −0.299165
$$900$$ 0 0
$$901$$ 45994.0 1.70065
$$902$$ −16520.0 −0.609818
$$903$$ 0 0
$$904$$ −45810.0 −1.68542
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −20466.0 −0.749242 −0.374621 0.927178i $$-0.622227\pi$$
−0.374621 + 0.927178i $$0.622227\pi$$
$$908$$ 63512.0 2.32128
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −15074.0 −0.548215 −0.274108 0.961699i $$-0.588382\pi$$
−0.274108 + 0.961699i $$0.588382\pi$$
$$912$$ 0 0
$$913$$ 4272.00 0.154855
$$914$$ −57620.0 −2.08523
$$915$$ 0 0
$$916$$ −23460.0 −0.846223
$$917$$ −17199.0 −0.619369
$$918$$ 0 0
$$919$$ −36848.0 −1.32264 −0.661318 0.750105i $$-0.730003\pi$$
−0.661318 + 0.750105i $$0.730003\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −42720.0 −1.52593
$$923$$ −2666.00 −0.0950731
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −12615.0 −0.447683
$$927$$ 0 0
$$928$$ 19040.0 0.673511
$$929$$ 35174.0 1.24222 0.621110 0.783724i $$-0.286682\pi$$
0.621110 + 0.783724i $$0.286682\pi$$
$$930$$ 0 0
$$931$$ 15458.0 0.544163
$$932$$ 49368.0 1.73509
$$933$$ 0 0
$$934$$ 14510.0 0.508332
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 10092.0 0.351858 0.175929 0.984403i $$-0.443707\pi$$
0.175929 + 0.984403i $$0.443707\pi$$
$$938$$ 15660.0 0.545114
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −12910.0 −0.447241 −0.223621 0.974676i $$-0.571788\pi$$
−0.223621 + 0.974676i $$0.571788\pi$$
$$942$$ 0 0
$$943$$ 87969.0 3.03782
$$944$$ 29993.0 1.03410
$$945$$ 0 0
$$946$$ −15680.0 −0.538901
$$947$$ 48060.0 1.64914 0.824572 0.565756i $$-0.191416\pi$$
0.824572 + 0.565756i $$0.191416\pi$$
$$948$$ 0 0
$$949$$ −52202.0 −1.78561
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −49410.0 −1.68213
$$953$$ −6316.00 −0.214686 −0.107343 0.994222i $$-0.534234\pi$$
−0.107343 + 0.994222i $$0.534234\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −101422. −3.43119
$$957$$ 0 0
$$958$$ 21810.0 0.735542
$$959$$ 5886.00 0.198195
$$960$$ 0 0
$$961$$ −28495.0 −0.956497
$$962$$ 44290.0 1.48437
$$963$$ 0 0
$$964$$ −54706.0 −1.82776
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −7760.00 −0.258061 −0.129030 0.991641i $$-0.541186\pi$$
−0.129030 + 0.991641i $$0.541186\pi$$
$$968$$ −57015.0 −1.89311
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 22437.0 0.741542 0.370771 0.928724i $$-0.379093\pi$$
0.370771 + 0.928724i $$0.379093\pi$$
$$972$$ 0 0
$$973$$ 6597.00 0.217359
$$974$$ −28615.0 −0.941359
$$975$$ 0 0
$$976$$ 3560.00 0.116755
$$977$$ 3506.00 0.114807 0.0574037 0.998351i $$-0.481718\pi$$
0.0574037 + 0.998351i $$0.481718\pi$$
$$978$$ 0 0
$$979$$ 6480.00 0.211544
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −66695.0 −2.16734
$$983$$ 30648.0 0.994425 0.497212 0.867629i $$-0.334357\pi$$
0.497212 + 0.867629i $$0.334357\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 136640. 4.41329
$$987$$ 0 0
$$988$$ 43129.0 1.38878
$$989$$ 83496.0 2.68455
$$990$$ 0 0
$$991$$ 19232.0 0.616473 0.308236 0.951310i $$-0.400261\pi$$
0.308236 + 0.951310i $$0.400261\pi$$
$$992$$ −3060.00 −0.0979386
$$993$$ 0 0
$$994$$ −2790.00 −0.0890276
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −38415.0 −1.22028 −0.610138 0.792295i $$-0.708886\pi$$
−0.610138 + 0.792295i $$0.708886\pi$$
$$998$$ −98185.0 −3.11422
$$999$$ 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 2025.4.a.f.1.1 1
3.2 odd 2 2025.4.a.a.1.1 1
5.4 even 2 405.4.a.a.1.1 1
15.14 odd 2 405.4.a.b.1.1 yes 1
45.4 even 6 405.4.e.m.136.1 2
45.14 odd 6 405.4.e.a.136.1 2
45.29 odd 6 405.4.e.a.271.1 2
45.34 even 6 405.4.e.m.271.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.a.1.1 1 5.4 even 2
405.4.a.b.1.1 yes 1 15.14 odd 2
405.4.e.a.136.1 2 45.14 odd 6
405.4.e.a.271.1 2 45.29 odd 6
405.4.e.m.136.1 2 45.4 even 6
405.4.e.m.271.1 2 45.34 even 6
2025.4.a.a.1.1 1 3.2 odd 2
2025.4.a.f.1.1 1 1.1 even 1 trivial