# Properties

 Label 225.4.b.e.199.1 Level $225$ Weight $4$ Character 225.199 Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 225.199 Dual form 225.4.b.e.199.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +7.00000 q^{4} -24.0000i q^{7} -15.0000i q^{8} +O(q^{10})$$ $$q-1.00000i q^{2} +7.00000 q^{4} -24.0000i q^{7} -15.0000i q^{8} -52.0000 q^{11} -22.0000i q^{13} -24.0000 q^{14} +41.0000 q^{16} +14.0000i q^{17} +20.0000 q^{19} +52.0000i q^{22} -168.000i q^{23} -22.0000 q^{26} -168.000i q^{28} +230.000 q^{29} -288.000 q^{31} -161.000i q^{32} +14.0000 q^{34} -34.0000i q^{37} -20.0000i q^{38} -122.000 q^{41} +188.000i q^{43} -364.000 q^{44} -168.000 q^{46} -256.000i q^{47} -233.000 q^{49} -154.000i q^{52} -338.000i q^{53} -360.000 q^{56} -230.000i q^{58} +100.000 q^{59} +742.000 q^{61} +288.000i q^{62} +167.000 q^{64} -84.0000i q^{67} +98.0000i q^{68} +328.000 q^{71} +38.0000i q^{73} -34.0000 q^{74} +140.000 q^{76} +1248.00i q^{77} +240.000 q^{79} +122.000i q^{82} +1212.00i q^{83} +188.000 q^{86} +780.000i q^{88} +330.000 q^{89} -528.000 q^{91} -1176.00i q^{92} -256.000 q^{94} +866.000i q^{97} +233.000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{4}+O(q^{10})$$ 2 * q + 14 * q^4 $$2 q + 14 q^{4} - 104 q^{11} - 48 q^{14} + 82 q^{16} + 40 q^{19} - 44 q^{26} + 460 q^{29} - 576 q^{31} + 28 q^{34} - 244 q^{41} - 728 q^{44} - 336 q^{46} - 466 q^{49} - 720 q^{56} + 200 q^{59} + 1484 q^{61} + 334 q^{64} + 656 q^{71} - 68 q^{74} + 280 q^{76} + 480 q^{79} + 376 q^{86} + 660 q^{89} - 1056 q^{91} - 512 q^{94}+O(q^{100})$$ 2 * q + 14 * q^4 - 104 * q^11 - 48 * q^14 + 82 * q^16 + 40 * q^19 - 44 * q^26 + 460 * q^29 - 576 * q^31 + 28 * q^34 - 244 * q^41 - 728 * q^44 - 336 * q^46 - 466 * q^49 - 720 * q^56 + 200 * q^59 + 1484 * q^61 + 334 * q^64 + 656 * q^71 - 68 * q^74 + 280 * q^76 + 480 * q^79 + 376 * q^86 + 660 * q^89 - 1056 * q^91 - 512 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.353553i −0.984251 0.176777i $$-0.943433\pi$$
0.984251 0.176777i $$-0.0565670\pi$$
$$3$$ 0 0
$$4$$ 7.00000 0.875000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 24.0000i − 1.29588i −0.761692 0.647939i $$-0.775631\pi$$
0.761692 0.647939i $$-0.224369\pi$$
$$8$$ − 15.0000i − 0.662913i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −52.0000 −1.42533 −0.712663 0.701506i $$-0.752511\pi$$
−0.712663 + 0.701506i $$0.752511\pi$$
$$12$$ 0 0
$$13$$ − 22.0000i − 0.469362i −0.972072 0.234681i $$-0.924595\pi$$
0.972072 0.234681i $$-0.0754045\pi$$
$$14$$ −24.0000 −0.458162
$$15$$ 0 0
$$16$$ 41.0000 0.640625
$$17$$ 14.0000i 0.199735i 0.995001 + 0.0998676i $$0.0318419\pi$$
−0.995001 + 0.0998676i $$0.968158\pi$$
$$18$$ 0 0
$$19$$ 20.0000 0.241490 0.120745 0.992684i $$-0.461472\pi$$
0.120745 + 0.992684i $$0.461472\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 52.0000i 0.503929i
$$23$$ − 168.000i − 1.52306i −0.648129 0.761531i $$-0.724448\pi$$
0.648129 0.761531i $$-0.275552\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −22.0000 −0.165944
$$27$$ 0 0
$$28$$ − 168.000i − 1.13389i
$$29$$ 230.000 1.47276 0.736378 0.676570i $$-0.236535\pi$$
0.736378 + 0.676570i $$0.236535\pi$$
$$30$$ 0 0
$$31$$ −288.000 −1.66859 −0.834296 0.551317i $$-0.814125\pi$$
−0.834296 + 0.551317i $$0.814125\pi$$
$$32$$ − 161.000i − 0.889408i
$$33$$ 0 0
$$34$$ 14.0000 0.0706171
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 34.0000i − 0.151069i −0.997143 0.0755347i $$-0.975934\pi$$
0.997143 0.0755347i $$-0.0240664\pi$$
$$38$$ − 20.0000i − 0.0853797i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −122.000 −0.464712 −0.232356 0.972631i $$-0.574643\pi$$
−0.232356 + 0.972631i $$0.574643\pi$$
$$42$$ 0 0
$$43$$ 188.000i 0.666738i 0.942796 + 0.333369i $$0.108185\pi$$
−0.942796 + 0.333369i $$0.891815\pi$$
$$44$$ −364.000 −1.24716
$$45$$ 0 0
$$46$$ −168.000 −0.538484
$$47$$ − 256.000i − 0.794499i −0.917711 0.397249i $$-0.869965\pi$$
0.917711 0.397249i $$-0.130035\pi$$
$$48$$ 0 0
$$49$$ −233.000 −0.679300
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 154.000i − 0.410691i
$$53$$ − 338.000i − 0.875998i −0.898976 0.437999i $$-0.855687\pi$$
0.898976 0.437999i $$-0.144313\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −360.000 −0.859054
$$57$$ 0 0
$$58$$ − 230.000i − 0.520698i
$$59$$ 100.000 0.220659 0.110330 0.993895i $$-0.464809\pi$$
0.110330 + 0.993895i $$0.464809\pi$$
$$60$$ 0 0
$$61$$ 742.000 1.55743 0.778716 0.627376i $$-0.215871\pi$$
0.778716 + 0.627376i $$0.215871\pi$$
$$62$$ 288.000i 0.589936i
$$63$$ 0 0
$$64$$ 167.000 0.326172
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 84.0000i − 0.153168i −0.997063 0.0765838i $$-0.975599\pi$$
0.997063 0.0765838i $$-0.0244013\pi$$
$$68$$ 98.0000i 0.174768i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 328.000 0.548260 0.274130 0.961693i $$-0.411610\pi$$
0.274130 + 0.961693i $$0.411610\pi$$
$$72$$ 0 0
$$73$$ 38.0000i 0.0609255i 0.999536 + 0.0304628i $$0.00969810\pi$$
−0.999536 + 0.0304628i $$0.990302\pi$$
$$74$$ −34.0000 −0.0534111
$$75$$ 0 0
$$76$$ 140.000 0.211304
$$77$$ 1248.00i 1.84705i
$$78$$ 0 0
$$79$$ 240.000 0.341799 0.170899 0.985288i $$-0.445333\pi$$
0.170899 + 0.985288i $$0.445333\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 122.000i 0.164301i
$$83$$ 1212.00i 1.60282i 0.598114 + 0.801411i $$0.295917\pi$$
−0.598114 + 0.801411i $$0.704083\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 188.000 0.235727
$$87$$ 0 0
$$88$$ 780.000i 0.944867i
$$89$$ 330.000 0.393033 0.196516 0.980501i $$-0.437037\pi$$
0.196516 + 0.980501i $$0.437037\pi$$
$$90$$ 0 0
$$91$$ −528.000 −0.608236
$$92$$ − 1176.00i − 1.33268i
$$93$$ 0 0
$$94$$ −256.000 −0.280898
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 866.000i 0.906484i 0.891387 + 0.453242i $$0.149733\pi$$
−0.891387 + 0.453242i $$0.850267\pi$$
$$98$$ 233.000i 0.240169i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1218.00 1.19996 0.599978 0.800017i $$-0.295176\pi$$
0.599978 + 0.800017i $$0.295176\pi$$
$$102$$ 0 0
$$103$$ 88.0000i 0.0841835i 0.999114 + 0.0420917i $$0.0134022\pi$$
−0.999114 + 0.0420917i $$0.986598\pi$$
$$104$$ −330.000 −0.311146
$$105$$ 0 0
$$106$$ −338.000 −0.309712
$$107$$ − 36.0000i − 0.0325257i −0.999868 0.0162629i $$-0.994823\pi$$
0.999868 0.0162629i $$-0.00517686\pi$$
$$108$$ 0 0
$$109$$ 970.000 0.852378 0.426189 0.904634i $$-0.359856\pi$$
0.426189 + 0.904634i $$0.359856\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 984.000i − 0.830172i
$$113$$ 1042.00i 0.867461i 0.901043 + 0.433731i $$0.142803\pi$$
−0.901043 + 0.433731i $$0.857197\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1610.00 1.28866
$$117$$ 0 0
$$118$$ − 100.000i − 0.0780148i
$$119$$ 336.000 0.258833
$$120$$ 0 0
$$121$$ 1373.00 1.03156
$$122$$ − 742.000i − 0.550635i
$$123$$ 0 0
$$124$$ −2016.00 −1.46002
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1936.00i 1.35269i 0.736583 + 0.676347i $$0.236438\pi$$
−0.736583 + 0.676347i $$0.763562\pi$$
$$128$$ − 1455.00i − 1.00473i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −732.000 −0.488207 −0.244104 0.969749i $$-0.578494\pi$$
−0.244104 + 0.969749i $$0.578494\pi$$
$$132$$ 0 0
$$133$$ − 480.000i − 0.312942i
$$134$$ −84.0000 −0.0541529
$$135$$ 0 0
$$136$$ 210.000 0.132407
$$137$$ 2214.00i 1.38069i 0.723479 + 0.690346i $$0.242542\pi$$
−0.723479 + 0.690346i $$0.757458\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −0.0122042 −0.00610208 0.999981i $$-0.501942\pi$$
−0.00610208 + 0.999981i $$0.501942\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 328.000i − 0.193839i
$$143$$ 1144.00i 0.668994i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 38.0000 0.0215404
$$147$$ 0 0
$$148$$ − 238.000i − 0.132186i
$$149$$ −1330.00 −0.731261 −0.365630 0.930760i $$-0.619147\pi$$
−0.365630 + 0.930760i $$0.619147\pi$$
$$150$$ 0 0
$$151$$ −1208.00 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ − 300.000i − 0.160087i
$$153$$ 0 0
$$154$$ 1248.00 0.653031
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 3514.00i − 1.78629i −0.449768 0.893146i $$-0.648493\pi$$
0.449768 0.893146i $$-0.351507\pi$$
$$158$$ − 240.000i − 0.120844i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4032.00 −1.97370
$$162$$ 0 0
$$163$$ 2068.00i 0.993732i 0.867827 + 0.496866i $$0.165516\pi$$
−0.867827 + 0.496866i $$0.834484\pi$$
$$164$$ −854.000 −0.406623
$$165$$ 0 0
$$166$$ 1212.00 0.566683
$$167$$ 24.0000i 0.0111208i 0.999985 + 0.00556041i $$0.00176994\pi$$
−0.999985 + 0.00556041i $$0.998230\pi$$
$$168$$ 0 0
$$169$$ 1713.00 0.779700
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 1316.00i 0.583396i
$$173$$ − 618.000i − 0.271593i −0.990737 0.135797i $$-0.956641\pi$$
0.990737 0.135797i $$-0.0433594\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2132.00 −0.913100
$$177$$ 0 0
$$178$$ − 330.000i − 0.138958i
$$179$$ 3340.00 1.39466 0.697328 0.716752i $$-0.254372\pi$$
0.697328 + 0.716752i $$0.254372\pi$$
$$180$$ 0 0
$$181$$ −178.000 −0.0730974 −0.0365487 0.999332i $$-0.511636\pi$$
−0.0365487 + 0.999332i $$0.511636\pi$$
$$182$$ 528.000i 0.215044i
$$183$$ 0 0
$$184$$ −2520.00 −1.00966
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 728.000i − 0.284688i
$$188$$ − 1792.00i − 0.695186i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1888.00 0.715240 0.357620 0.933867i $$-0.383588\pi$$
0.357620 + 0.933867i $$0.383588\pi$$
$$192$$ 0 0
$$193$$ − 1922.00i − 0.716832i −0.933562 0.358416i $$-0.883317\pi$$
0.933562 0.358416i $$-0.116683\pi$$
$$194$$ 866.000 0.320491
$$195$$ 0 0
$$196$$ −1631.00 −0.594388
$$197$$ − 2526.00i − 0.913554i −0.889581 0.456777i $$-0.849004\pi$$
0.889581 0.456777i $$-0.150996\pi$$
$$198$$ 0 0
$$199$$ 1160.00 0.413217 0.206609 0.978424i $$-0.433757\pi$$
0.206609 + 0.978424i $$0.433757\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 1218.00i − 0.424248i
$$203$$ − 5520.00i − 1.90851i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 88.0000 0.0297634
$$207$$ 0 0
$$208$$ − 902.000i − 0.300685i
$$209$$ −1040.00 −0.344202
$$210$$ 0 0
$$211$$ −4468.00 −1.45777 −0.728886 0.684635i $$-0.759961\pi$$
−0.728886 + 0.684635i $$0.759961\pi$$
$$212$$ − 2366.00i − 0.766498i
$$213$$ 0 0
$$214$$ −36.0000 −0.0114996
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 6912.00i 2.16229i
$$218$$ − 970.000i − 0.301361i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 308.000 0.0937481
$$222$$ 0 0
$$223$$ − 6032.00i − 1.81136i −0.423965 0.905678i $$-0.639362\pi$$
0.423965 0.905678i $$-0.360638\pi$$
$$224$$ −3864.00 −1.15256
$$225$$ 0 0
$$226$$ 1042.00 0.306694
$$227$$ − 2636.00i − 0.770738i −0.922763 0.385369i $$-0.874074\pi$$
0.922763 0.385369i $$-0.125926\pi$$
$$228$$ 0 0
$$229$$ −4830.00 −1.39378 −0.696889 0.717179i $$-0.745433\pi$$
−0.696889 + 0.717179i $$0.745433\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 3450.00i − 0.976309i
$$233$$ 2682.00i 0.754093i 0.926194 + 0.377046i $$0.123060\pi$$
−0.926194 + 0.377046i $$0.876940\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 700.000 0.193077
$$237$$ 0 0
$$238$$ − 336.000i − 0.0915111i
$$239$$ 2320.00 0.627901 0.313950 0.949439i $$-0.398347\pi$$
0.313950 + 0.949439i $$0.398347\pi$$
$$240$$ 0 0
$$241$$ 2002.00 0.535104 0.267552 0.963543i $$-0.413785\pi$$
0.267552 + 0.963543i $$0.413785\pi$$
$$242$$ − 1373.00i − 0.364710i
$$243$$ 0 0
$$244$$ 5194.00 1.36275
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 440.000i − 0.113346i
$$248$$ 4320.00i 1.10613i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −132.000 −0.0331943 −0.0165971 0.999862i $$-0.505283\pi$$
−0.0165971 + 0.999862i $$0.505283\pi$$
$$252$$ 0 0
$$253$$ 8736.00i 2.17086i
$$254$$ 1936.00 0.478250
$$255$$ 0 0
$$256$$ −119.000 −0.0290527
$$257$$ 7614.00i 1.84805i 0.382335 + 0.924024i $$0.375120\pi$$
−0.382335 + 0.924024i $$0.624880\pi$$
$$258$$ 0 0
$$259$$ −816.000 −0.195767
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 732.000i 0.172607i
$$263$$ − 4888.00i − 1.14603i −0.819543 0.573017i $$-0.805773\pi$$
0.819543 0.573017i $$-0.194227\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −480.000 −0.110642
$$267$$ 0 0
$$268$$ − 588.000i − 0.134022i
$$269$$ 1270.00 0.287856 0.143928 0.989588i $$-0.454027\pi$$
0.143928 + 0.989588i $$0.454027\pi$$
$$270$$ 0 0
$$271$$ 1072.00 0.240293 0.120146 0.992756i $$-0.461664\pi$$
0.120146 + 0.992756i $$0.461664\pi$$
$$272$$ 574.000i 0.127955i
$$273$$ 0 0
$$274$$ 2214.00 0.488148
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 5394.00i − 1.17001i −0.811028 0.585007i $$-0.801092\pi$$
0.811028 0.585007i $$-0.198908\pi$$
$$278$$ 20.0000i 0.00431482i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2442.00 −0.518425 −0.259213 0.965820i $$-0.583463\pi$$
−0.259213 + 0.965820i $$0.583463\pi$$
$$282$$ 0 0
$$283$$ − 2772.00i − 0.582255i −0.956684 0.291128i $$-0.905970\pi$$
0.956684 0.291128i $$-0.0940305\pi$$
$$284$$ 2296.00 0.479727
$$285$$ 0 0
$$286$$ 1144.00 0.236525
$$287$$ 2928.00i 0.602210i
$$288$$ 0 0
$$289$$ 4717.00 0.960106
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 266.000i 0.0533098i
$$293$$ 4542.00i 0.905619i 0.891607 + 0.452810i $$0.149578\pi$$
−0.891607 + 0.452810i $$0.850422\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −510.000 −0.100146
$$297$$ 0 0
$$298$$ 1330.00i 0.258540i
$$299$$ −3696.00 −0.714867
$$300$$ 0 0
$$301$$ 4512.00 0.864011
$$302$$ 1208.00i 0.230174i
$$303$$ 0 0
$$304$$ 820.000 0.154705
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5116.00i 0.951093i 0.879691 + 0.475546i $$0.157750\pi$$
−0.879691 + 0.475546i $$0.842250\pi$$
$$308$$ 8736.00i 1.61617i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2808.00 0.511984 0.255992 0.966679i $$-0.417598\pi$$
0.255992 + 0.966679i $$0.417598\pi$$
$$312$$ 0 0
$$313$$ 7318.00i 1.32153i 0.750594 + 0.660763i $$0.229767\pi$$
−0.750594 + 0.660763i $$0.770233\pi$$
$$314$$ −3514.00 −0.631549
$$315$$ 0 0
$$316$$ 1680.00 0.299074
$$317$$ − 2246.00i − 0.397943i −0.980005 0.198971i $$-0.936240\pi$$
0.980005 0.198971i $$-0.0637601\pi$$
$$318$$ 0 0
$$319$$ −11960.0 −2.09916
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 4032.00i 0.697809i
$$323$$ 280.000i 0.0482341i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 2068.00 0.351337
$$327$$ 0 0
$$328$$ 1830.00i 0.308064i
$$329$$ −6144.00 −1.02957
$$330$$ 0 0
$$331$$ 1332.00 0.221188 0.110594 0.993866i $$-0.464725\pi$$
0.110594 + 0.993866i $$0.464725\pi$$
$$332$$ 8484.00i 1.40247i
$$333$$ 0 0
$$334$$ 24.0000 0.00393180
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 11534.0i − 1.86438i −0.361966 0.932191i $$-0.617894\pi$$
0.361966 0.932191i $$-0.382106\pi$$
$$338$$ − 1713.00i − 0.275665i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 14976.0 2.37829
$$342$$ 0 0
$$343$$ − 2640.00i − 0.415588i
$$344$$ 2820.00 0.441989
$$345$$ 0 0
$$346$$ −618.000 −0.0960228
$$347$$ − 11956.0i − 1.84966i −0.380382 0.924830i $$-0.624207\pi$$
0.380382 0.924830i $$-0.375793\pi$$
$$348$$ 0 0
$$349$$ −4870.00 −0.746949 −0.373474 0.927640i $$-0.621834\pi$$
−0.373474 + 0.927640i $$0.621834\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 8372.00i 1.26770i
$$353$$ 10722.0i 1.61664i 0.588742 + 0.808321i $$0.299623\pi$$
−0.588742 + 0.808321i $$0.700377\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 2310.00 0.343904
$$357$$ 0 0
$$358$$ − 3340.00i − 0.493085i
$$359$$ 120.000 0.0176417 0.00882083 0.999961i $$-0.497192\pi$$
0.00882083 + 0.999961i $$0.497192\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ 178.000i 0.0258438i
$$363$$ 0 0
$$364$$ −3696.00 −0.532206
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3936.00i 0.559830i 0.960025 + 0.279915i $$0.0903063\pi$$
−0.960025 + 0.279915i $$0.909694\pi$$
$$368$$ − 6888.00i − 0.975711i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −8112.00 −1.13519
$$372$$ 0 0
$$373$$ − 3022.00i − 0.419499i −0.977755 0.209750i $$-0.932735\pi$$
0.977755 0.209750i $$-0.0672649\pi$$
$$374$$ −728.000 −0.100652
$$375$$ 0 0
$$376$$ −3840.00 −0.526683
$$377$$ − 5060.00i − 0.691255i
$$378$$ 0 0
$$379$$ 13340.0 1.80799 0.903997 0.427539i $$-0.140619\pi$$
0.903997 + 0.427539i $$0.140619\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 1888.00i − 0.252876i
$$383$$ − 1008.00i − 0.134481i −0.997737 0.0672407i $$-0.978580\pi$$
0.997737 0.0672407i $$-0.0214195\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1922.00 −0.253438
$$387$$ 0 0
$$388$$ 6062.00i 0.793174i
$$389$$ 9630.00 1.25517 0.627584 0.778549i $$-0.284044\pi$$
0.627584 + 0.778549i $$0.284044\pi$$
$$390$$ 0 0
$$391$$ 2352.00 0.304209
$$392$$ 3495.00i 0.450317i
$$393$$ 0 0
$$394$$ −2526.00 −0.322990
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 7126.00i 0.900866i 0.892810 + 0.450433i $$0.148730\pi$$
−0.892810 + 0.450433i $$0.851270\pi$$
$$398$$ − 1160.00i − 0.146094i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 8718.00 1.08568 0.542838 0.839837i $$-0.317350\pi$$
0.542838 + 0.839837i $$0.317350\pi$$
$$402$$ 0 0
$$403$$ 6336.00i 0.783173i
$$404$$ 8526.00 1.04996
$$405$$ 0 0
$$406$$ −5520.00 −0.674761
$$407$$ 1768.00i 0.215323i
$$408$$ 0 0
$$409$$ 10870.0 1.31415 0.657074 0.753826i $$-0.271794\pi$$
0.657074 + 0.753826i $$0.271794\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 616.000i 0.0736605i
$$413$$ − 2400.00i − 0.285947i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −3542.00 −0.417454
$$417$$ 0 0
$$418$$ 1040.00i 0.121694i
$$419$$ −9700.00 −1.13097 −0.565484 0.824759i $$-0.691311\pi$$
−0.565484 + 0.824759i $$0.691311\pi$$
$$420$$ 0 0
$$421$$ 862.000 0.0997893 0.0498947 0.998754i $$-0.484111\pi$$
0.0498947 + 0.998754i $$0.484111\pi$$
$$422$$ 4468.00i 0.515400i
$$423$$ 0 0
$$424$$ −5070.00 −0.580710
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 17808.0i − 2.01824i
$$428$$ − 252.000i − 0.0284600i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −15792.0 −1.76490 −0.882452 0.470402i $$-0.844109\pi$$
−0.882452 + 0.470402i $$0.844109\pi$$
$$432$$ 0 0
$$433$$ − 11602.0i − 1.28766i −0.765169 0.643830i $$-0.777345\pi$$
0.765169 0.643830i $$-0.222655\pi$$
$$434$$ 6912.00 0.764485
$$435$$ 0 0
$$436$$ 6790.00 0.745830
$$437$$ − 3360.00i − 0.367805i
$$438$$ 0 0
$$439$$ 440.000 0.0478361 0.0239181 0.999714i $$-0.492386\pi$$
0.0239181 + 0.999714i $$0.492386\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 308.000i − 0.0331449i
$$443$$ − 10188.0i − 1.09266i −0.837571 0.546328i $$-0.816025\pi$$
0.837571 0.546328i $$-0.183975\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −6032.00 −0.640411
$$447$$ 0 0
$$448$$ − 4008.00i − 0.422679i
$$449$$ −13310.0 −1.39897 −0.699485 0.714647i $$-0.746587\pi$$
−0.699485 + 0.714647i $$0.746587\pi$$
$$450$$ 0 0
$$451$$ 6344.00 0.662367
$$452$$ 7294.00i 0.759029i
$$453$$ 0 0
$$454$$ −2636.00 −0.272497
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3226.00i 0.330210i 0.986276 + 0.165105i $$0.0527963\pi$$
−0.986276 + 0.165105i $$0.947204\pi$$
$$458$$ 4830.00i 0.492775i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6582.00 −0.664977 −0.332488 0.943107i $$-0.607888\pi$$
−0.332488 + 0.943107i $$0.607888\pi$$
$$462$$ 0 0
$$463$$ − 15072.0i − 1.51286i −0.654073 0.756431i $$-0.726941\pi$$
0.654073 0.756431i $$-0.273059\pi$$
$$464$$ 9430.00 0.943484
$$465$$ 0 0
$$466$$ 2682.00 0.266612
$$467$$ − 476.000i − 0.0471663i −0.999722 0.0235831i $$-0.992493\pi$$
0.999722 0.0235831i $$-0.00750744\pi$$
$$468$$ 0 0
$$469$$ −2016.00 −0.198487
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 1500.00i − 0.146278i
$$473$$ − 9776.00i − 0.950319i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2352.00 0.226478
$$477$$ 0 0
$$478$$ − 2320.00i − 0.221997i
$$479$$ −19680.0 −1.87725 −0.938624 0.344941i $$-0.887899\pi$$
−0.938624 + 0.344941i $$0.887899\pi$$
$$480$$ 0 0
$$481$$ −748.000 −0.0709062
$$482$$ − 2002.00i − 0.189188i
$$483$$ 0 0
$$484$$ 9611.00 0.902611
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 5944.00i − 0.553077i −0.961003 0.276538i $$-0.910813\pi$$
0.961003 0.276538i $$-0.0891873\pi$$
$$488$$ − 11130.0i − 1.03244i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −10772.0 −0.990089 −0.495044 0.868868i $$-0.664848\pi$$
−0.495044 + 0.868868i $$0.664848\pi$$
$$492$$ 0 0
$$493$$ 3220.00i 0.294161i
$$494$$ −440.000 −0.0400740
$$495$$ 0 0
$$496$$ −11808.0 −1.06894
$$497$$ − 7872.00i − 0.710478i
$$498$$ 0 0
$$499$$ −8140.00 −0.730253 −0.365127 0.930958i $$-0.618974\pi$$
−0.365127 + 0.930958i $$0.618974\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 132.000i 0.0117360i
$$503$$ − 13768.0i − 1.22045i −0.792229 0.610223i $$-0.791080\pi$$
0.792229 0.610223i $$-0.208920\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 8736.00 0.767515
$$507$$ 0 0
$$508$$ 13552.0i 1.18361i
$$509$$ 22150.0 1.92884 0.964422 0.264368i $$-0.0851633\pi$$
0.964422 + 0.264368i $$0.0851633\pi$$
$$510$$ 0 0
$$511$$ 912.000 0.0789521
$$512$$ − 11521.0i − 0.994455i
$$513$$ 0 0
$$514$$ 7614.00 0.653384
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 13312.0i 1.13242i
$$518$$ 816.000i 0.0692143i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −1562.00 −0.131348 −0.0656741 0.997841i $$-0.520920\pi$$
−0.0656741 + 0.997841i $$0.520920\pi$$
$$522$$ 0 0
$$523$$ 668.000i 0.0558501i 0.999610 + 0.0279250i $$0.00888997\pi$$
−0.999610 + 0.0279250i $$0.991110\pi$$
$$524$$ −5124.00 −0.427181
$$525$$ 0 0
$$526$$ −4888.00 −0.405184
$$527$$ − 4032.00i − 0.333276i
$$528$$ 0 0
$$529$$ −16057.0 −1.31972
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 3360.00i − 0.273824i
$$533$$ 2684.00i 0.218118i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −1260.00 −0.101537
$$537$$ 0 0
$$538$$ − 1270.00i − 0.101772i
$$539$$ 12116.0 0.968225
$$540$$ 0 0
$$541$$ −6138.00 −0.487788 −0.243894 0.969802i $$-0.578425\pi$$
−0.243894 + 0.969802i $$0.578425\pi$$
$$542$$ − 1072.00i − 0.0849564i
$$543$$ 0 0
$$544$$ 2254.00 0.177646
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 10484.0i − 0.819494i −0.912199 0.409747i $$-0.865617\pi$$
0.912199 0.409747i $$-0.134383\pi$$
$$548$$ 15498.0i 1.20811i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 4600.00 0.355656
$$552$$ 0 0
$$553$$ − 5760.00i − 0.442930i
$$554$$ −5394.00 −0.413663
$$555$$ 0 0
$$556$$ −140.000 −0.0106786
$$557$$ − 3606.00i − 0.274311i −0.990550 0.137155i $$-0.956204\pi$$
0.990550 0.137155i $$-0.0437960\pi$$
$$558$$ 0 0
$$559$$ 4136.00 0.312941
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2442.00i 0.183291i
$$563$$ 12252.0i 0.917159i 0.888654 + 0.458579i $$0.151641\pi$$
−0.888654 + 0.458579i $$0.848359\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −2772.00 −0.205858
$$567$$ 0 0
$$568$$ − 4920.00i − 0.363448i
$$569$$ −14550.0 −1.07200 −0.536000 0.844218i $$-0.680065\pi$$
−0.536000 + 0.844218i $$0.680065\pi$$
$$570$$ 0 0
$$571$$ −25468.0 −1.86655 −0.933277 0.359157i $$-0.883064\pi$$
−0.933277 + 0.359157i $$0.883064\pi$$
$$572$$ 8008.00i 0.585369i
$$573$$ 0 0
$$574$$ 2928.00 0.212914
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 12866.0i 0.928282i 0.885761 + 0.464141i $$0.153637\pi$$
−0.885761 + 0.464141i $$0.846363\pi$$
$$578$$ − 4717.00i − 0.339449i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 29088.0 2.07706
$$582$$ 0 0
$$583$$ 17576.0i 1.24858i
$$584$$ 570.000 0.0403883
$$585$$ 0 0
$$586$$ 4542.00 0.320185
$$587$$ 14844.0i 1.04374i 0.853024 + 0.521872i $$0.174766\pi$$
−0.853024 + 0.521872i $$0.825234\pi$$
$$588$$ 0 0
$$589$$ −5760.00 −0.402948
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 1394.00i − 0.0967788i
$$593$$ 20402.0i 1.41283i 0.707797 + 0.706416i $$0.249689\pi$$
−0.707797 + 0.706416i $$0.750311\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −9310.00 −0.639853
$$597$$ 0 0
$$598$$ 3696.00i 0.252744i
$$599$$ 10760.0 0.733959 0.366980 0.930229i $$-0.380392\pi$$
0.366980 + 0.930229i $$0.380392\pi$$
$$600$$ 0 0
$$601$$ 14282.0 0.969343 0.484671 0.874696i $$-0.338939\pi$$
0.484671 + 0.874696i $$0.338939\pi$$
$$602$$ − 4512.00i − 0.305474i
$$603$$ 0 0
$$604$$ −8456.00 −0.569652
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 11056.0i 0.739290i 0.929173 + 0.369645i $$0.120521\pi$$
−0.929173 + 0.369645i $$0.879479\pi$$
$$608$$ − 3220.00i − 0.214783i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −5632.00 −0.372907
$$612$$ 0 0
$$613$$ 16418.0i 1.08176i 0.841101 + 0.540878i $$0.181908\pi$$
−0.841101 + 0.540878i $$0.818092\pi$$
$$614$$ 5116.00 0.336262
$$615$$ 0 0
$$616$$ 18720.0 1.22443
$$617$$ 10374.0i 0.676891i 0.940986 + 0.338445i $$0.109901\pi$$
−0.940986 + 0.338445i $$0.890099\pi$$
$$618$$ 0 0
$$619$$ 5260.00 0.341546 0.170773 0.985310i $$-0.445373\pi$$
0.170773 + 0.985310i $$0.445373\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 2808.00i − 0.181014i
$$623$$ − 7920.00i − 0.509323i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 7318.00 0.467230
$$627$$ 0 0
$$628$$ − 24598.0i − 1.56300i
$$629$$ 476.000 0.0301739
$$630$$ 0 0
$$631$$ 21352.0 1.34708 0.673542 0.739149i $$-0.264772\pi$$
0.673542 + 0.739149i $$0.264772\pi$$
$$632$$ − 3600.00i − 0.226583i
$$633$$ 0 0
$$634$$ −2246.00 −0.140694
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 5126.00i 0.318838i
$$638$$ 11960.0i 0.742164i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 29118.0 1.79422 0.897108 0.441812i $$-0.145664\pi$$
0.897108 + 0.441812i $$0.145664\pi$$
$$642$$ 0 0
$$643$$ − 5772.00i − 0.354005i −0.984210 0.177003i $$-0.943360\pi$$
0.984210 0.177003i $$-0.0566401\pi$$
$$644$$ −28224.0 −1.72699
$$645$$ 0 0
$$646$$ 280.000 0.0170533
$$647$$ 14264.0i 0.866732i 0.901218 + 0.433366i $$0.142674\pi$$
−0.901218 + 0.433366i $$0.857326\pi$$
$$648$$ 0 0
$$649$$ −5200.00 −0.314511
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 14476.0i 0.869515i
$$653$$ 6902.00i 0.413623i 0.978381 + 0.206812i $$0.0663088\pi$$
−0.978381 + 0.206812i $$0.933691\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −5002.00 −0.297706
$$657$$ 0 0
$$658$$ 6144.00i 0.364009i
$$659$$ 20140.0 1.19051 0.595253 0.803539i $$-0.297052\pi$$
0.595253 + 0.803539i $$0.297052\pi$$
$$660$$ 0 0
$$661$$ −3218.00 −0.189358 −0.0946790 0.995508i $$-0.530182\pi$$
−0.0946790 + 0.995508i $$0.530182\pi$$
$$662$$ − 1332.00i − 0.0782019i
$$663$$ 0 0
$$664$$ 18180.0 1.06253
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 38640.0i − 2.24310i
$$668$$ 168.000i 0.00973071i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −38584.0 −2.21985
$$672$$ 0 0
$$673$$ 7518.00i 0.430606i 0.976547 + 0.215303i $$0.0690739\pi$$
−0.976547 + 0.215303i $$0.930926\pi$$
$$674$$ −11534.0 −0.659159
$$675$$ 0 0
$$676$$ 11991.0 0.682237
$$677$$ 18114.0i 1.02833i 0.857692 + 0.514164i $$0.171898\pi$$
−0.857692 + 0.514164i $$0.828102\pi$$
$$678$$ 0 0
$$679$$ 20784.0 1.17469
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 14976.0i − 0.840851i
$$683$$ − 23868.0i − 1.33716i −0.743638 0.668582i $$-0.766901\pi$$
0.743638 0.668582i $$-0.233099\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −2640.00 −0.146932
$$687$$ 0 0
$$688$$ 7708.00i 0.427129i
$$689$$ −7436.00 −0.411160
$$690$$ 0 0
$$691$$ 172.000 0.00946916 0.00473458 0.999989i $$-0.498493\pi$$
0.00473458 + 0.999989i $$0.498493\pi$$
$$692$$ − 4326.00i − 0.237644i
$$693$$ 0 0
$$694$$ −11956.0 −0.653953
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 1708.00i − 0.0928194i
$$698$$ 4870.00i 0.264086i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22138.0 1.19278 0.596391 0.802694i $$-0.296601\pi$$
0.596391 + 0.802694i $$0.296601\pi$$
$$702$$ 0 0
$$703$$ − 680.000i − 0.0364818i
$$704$$ −8684.00 −0.464901
$$705$$ 0 0
$$706$$ 10722.0 0.571569
$$707$$ − 29232.0i − 1.55500i
$$708$$ 0 0
$$709$$ −3070.00 −0.162618 −0.0813091 0.996689i $$-0.525910\pi$$
−0.0813091 + 0.996689i $$0.525910\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 4950.00i − 0.260546i
$$713$$ 48384.0i 2.54137i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 23380.0 1.22032
$$717$$ 0 0
$$718$$ − 120.000i − 0.00623727i
$$719$$ 15600.0 0.809154 0.404577 0.914504i $$-0.367419\pi$$
0.404577 + 0.914504i $$0.367419\pi$$
$$720$$ 0 0
$$721$$ 2112.00 0.109092
$$722$$ 6459.00i 0.332935i
$$723$$ 0 0
$$724$$ −1246.00 −0.0639603
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 20696.0i 1.05581i 0.849304 + 0.527904i $$0.177022\pi$$
−0.849304 + 0.527904i $$0.822978\pi$$
$$728$$ 7920.00i 0.403207i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2632.00 −0.133171
$$732$$ 0 0
$$733$$ 30778.0i 1.55090i 0.631408 + 0.775451i $$0.282478\pi$$
−0.631408 + 0.775451i $$0.717522\pi$$
$$734$$ 3936.00 0.197930
$$735$$ 0 0
$$736$$ −27048.0 −1.35462
$$737$$ 4368.00i 0.218314i
$$738$$ 0 0
$$739$$ −11740.0 −0.584388 −0.292194 0.956359i $$-0.594385\pi$$
−0.292194 + 0.956359i $$0.594385\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 8112.00i 0.401349i
$$743$$ 2632.00i 0.129958i 0.997887 + 0.0649789i $$0.0206980\pi$$
−0.997887 + 0.0649789i $$0.979302\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −3022.00 −0.148315
$$747$$ 0 0
$$748$$ − 5096.00i − 0.249102i
$$749$$ −864.000 −0.0421494
$$750$$ 0 0
$$751$$ −20528.0 −0.997440 −0.498720 0.866763i $$-0.666196\pi$$
−0.498720 + 0.866763i $$0.666196\pi$$
$$752$$ − 10496.0i − 0.508976i
$$753$$ 0 0
$$754$$ −5060.00 −0.244396
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 21646.0i 1.03928i 0.854384 + 0.519642i $$0.173934\pi$$
−0.854384 + 0.519642i $$0.826066\pi$$
$$758$$ − 13340.0i − 0.639222i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18282.0 −0.870857 −0.435428 0.900223i $$-0.643403\pi$$
−0.435428 + 0.900223i $$0.643403\pi$$
$$762$$ 0 0
$$763$$ − 23280.0i − 1.10458i
$$764$$ 13216.0 0.625835
$$765$$ 0 0
$$766$$ −1008.00 −0.0475464
$$767$$ − 2200.00i − 0.103569i
$$768$$ 0 0
$$769$$ 24190.0 1.13435 0.567174 0.823598i $$-0.308037\pi$$
0.567174 + 0.823598i $$0.308037\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 13454.0i − 0.627228i
$$773$$ − 25698.0i − 1.19572i −0.801600 0.597861i $$-0.796018\pi$$
0.801600 0.597861i $$-0.203982\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 12990.0 0.600920
$$777$$ 0 0
$$778$$ − 9630.00i − 0.443769i
$$779$$ −2440.00 −0.112223
$$780$$ 0 0
$$781$$ −17056.0 −0.781449
$$782$$ − 2352.00i − 0.107554i
$$783$$ 0 0
$$784$$ −9553.00 −0.435177
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 33436.0i 1.51444i 0.653160 + 0.757220i $$0.273443\pi$$
−0.653160 + 0.757220i $$0.726557\pi$$
$$788$$ − 17682.0i − 0.799359i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 25008.0 1.12412
$$792$$ 0 0
$$793$$ − 16324.0i − 0.730999i
$$794$$ 7126.00 0.318504
$$795$$ 0 0
$$796$$ 8120.00 0.361565
$$797$$ 37594.0i 1.67083i 0.549623 + 0.835413i $$0.314771\pi$$
−0.549623 + 0.835413i $$0.685229\pi$$
$$798$$ 0 0
$$799$$ 3584.00 0.158689
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 8718.00i − 0.383844i
$$803$$ − 1976.00i − 0.0868388i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 6336.00 0.276893
$$807$$ 0 0
$$808$$ − 18270.0i − 0.795466i
$$809$$ 4730.00 0.205560 0.102780 0.994704i $$-0.467226\pi$$
0.102780 + 0.994704i $$0.467226\pi$$
$$810$$ 0 0
$$811$$ −8748.00 −0.378772 −0.189386 0.981903i $$-0.560650\pi$$
−0.189386 + 0.981903i $$0.560650\pi$$
$$812$$ − 38640.0i − 1.66995i
$$813$$ 0 0
$$814$$ 1768.00 0.0761282
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 3760.00i 0.161011i
$$818$$ − 10870.0i − 0.464622i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −44142.0 −1.87645 −0.938226 0.346024i $$-0.887532\pi$$
−0.938226 + 0.346024i $$0.887532\pi$$
$$822$$ 0 0
$$823$$ − 3992.00i − 0.169079i −0.996420 0.0845397i $$-0.973058\pi$$
0.996420 0.0845397i $$-0.0269420\pi$$
$$824$$ 1320.00 0.0558063
$$825$$ 0 0
$$826$$ −2400.00 −0.101098
$$827$$ 14444.0i 0.607336i 0.952778 + 0.303668i $$0.0982114\pi$$
−0.952778 + 0.303668i $$0.901789\pi$$
$$828$$ 0 0
$$829$$ −42150.0 −1.76590 −0.882949 0.469468i $$-0.844446\pi$$
−0.882949 + 0.469468i $$0.844446\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 3674.00i − 0.153093i
$$833$$ − 3262.00i − 0.135680i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −7280.00 −0.301177
$$837$$ 0 0
$$838$$ 9700.00i 0.399858i
$$839$$ 13400.0 0.551394 0.275697 0.961245i $$-0.411091\pi$$
0.275697 + 0.961245i $$0.411091\pi$$
$$840$$ 0 0
$$841$$ 28511.0 1.16901
$$842$$ − 862.000i − 0.0352809i
$$843$$ 0 0
$$844$$ −31276.0 −1.27555
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 32952.0i − 1.33677i
$$848$$ − 13858.0i − 0.561186i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −5712.00 −0.230088
$$852$$ 0 0
$$853$$ 8658.00i 0.347531i 0.984787 + 0.173766i $$0.0555935\pi$$
−0.984787 + 0.173766i $$0.944406\pi$$
$$854$$ −17808.0 −0.713556
$$855$$ 0 0
$$856$$ −540.000 −0.0215617
$$857$$ − 42826.0i − 1.70701i −0.521084 0.853505i $$-0.674472\pi$$
0.521084 0.853505i $$-0.325528\pi$$
$$858$$ 0 0
$$859$$ 35900.0 1.42595 0.712976 0.701189i $$-0.247347\pi$$
0.712976 + 0.701189i $$0.247347\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 15792.0i 0.623988i
$$863$$ − 3088.00i − 0.121804i −0.998144 0.0609019i $$-0.980602\pi$$
0.998144 0.0609019i $$-0.0193977\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −11602.0 −0.455256
$$867$$ 0 0
$$868$$ 48384.0i 1.89200i
$$869$$ −12480.0 −0.487175
$$870$$ 0 0
$$871$$ −1848.00 −0.0718910
$$872$$ − 14550.0i − 0.565052i
$$873$$ 0 0
$$874$$ −3360.00 −0.130039
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 35274.0i − 1.35817i −0.734058 0.679087i $$-0.762376\pi$$
0.734058 0.679087i $$-0.237624\pi$$
$$878$$ − 440.000i − 0.0169126i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −25042.0 −0.957646 −0.478823 0.877911i $$-0.658936\pi$$
−0.478823 + 0.877911i $$0.658936\pi$$
$$882$$ 0 0
$$883$$ − 12572.0i − 0.479141i −0.970879 0.239570i $$-0.922993\pi$$
0.970879 0.239570i $$-0.0770066\pi$$
$$884$$ 2156.00 0.0820296
$$885$$ 0 0
$$886$$ −10188.0 −0.386312
$$887$$ 21864.0i 0.827645i 0.910358 + 0.413823i $$0.135807\pi$$
−0.910358 + 0.413823i $$0.864193\pi$$
$$888$$ 0 0
$$889$$ 46464.0 1.75293
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 42224.0i − 1.58494i
$$893$$ − 5120.00i − 0.191864i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −34920.0 −1.30200
$$897$$ 0 0
$$898$$ 13310.0i 0.494611i
$$899$$ −66240.0 −2.45743
$$900$$ 0 0
$$901$$ 4732.00 0.174968
$$902$$ − 6344.00i − 0.234182i
$$903$$ 0 0
$$904$$ 15630.0 0.575051
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 31236.0i 1.14352i 0.820420 + 0.571761i $$0.193740\pi$$
−0.820420 + 0.571761i $$0.806260\pi$$
$$908$$ − 18452.0i − 0.674396i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −8272.00 −0.300838 −0.150419 0.988622i $$-0.548062\pi$$
−0.150419 + 0.988622i $$0.548062\pi$$
$$912$$ 0 0
$$913$$ − 63024.0i − 2.28455i
$$914$$ 3226.00 0.116747
$$915$$ 0 0
$$916$$ −33810.0 −1.21956
$$917$$ 17568.0i 0.632657i
$$918$$ 0 0
$$919$$ −20200.0 −0.725067 −0.362533 0.931971i $$-0.618088\pi$$
−0.362533 + 0.931971i $$0.618088\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 6582.00i 0.235105i
$$923$$ − 7216.00i − 0.257332i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −15072.0 −0.534878
$$927$$ 0 0
$$928$$ − 37030.0i − 1.30988i
$$929$$ 31010.0 1.09516 0.547581 0.836753i $$-0.315549\pi$$
0.547581 + 0.836753i $$0.315549\pi$$
$$930$$ 0 0
$$931$$ −4660.00 −0.164044
$$932$$ 18774.0i 0.659831i
$$933$$ 0 0
$$934$$ −476.000 −0.0166758
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 39174.0i − 1.36580i −0.730510 0.682902i $$-0.760717\pi$$
0.730510 0.682902i $$-0.239283\pi$$
$$938$$ 2016.00i 0.0701756i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 4138.00 0.143353 0.0716764 0.997428i $$-0.477165\pi$$
0.0716764 + 0.997428i $$0.477165\pi$$
$$942$$ 0 0
$$943$$ 20496.0i 0.707785i
$$944$$ 4100.00 0.141360
$$945$$ 0 0
$$946$$ −9776.00 −0.335989
$$947$$ − 23676.0i − 0.812425i −0.913779 0.406213i $$-0.866849\pi$$
0.913779 0.406213i $$-0.133151\pi$$
$$948$$ 0 0
$$949$$ 836.000 0.0285961
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 5040.00i − 0.171583i
$$953$$ 18922.0i 0.643173i 0.946880 + 0.321586i $$0.104216\pi$$
−0.946880 + 0.321586i $$0.895784\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 16240.0 0.549413
$$957$$ 0 0
$$958$$ 19680.0i 0.663708i
$$959$$ 53136.0 1.78921
$$960$$ 0 0
$$961$$ 53153.0 1.78420
$$962$$ 748.000i 0.0250691i
$$963$$ 0 0
$$964$$ 14014.0 0.468216
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 39656.0i 1.31877i 0.751805 + 0.659385i $$0.229183\pi$$
−0.751805 + 0.659385i $$0.770817\pi$$
$$968$$ − 20595.0i − 0.683831i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 33228.0 1.09818 0.549092 0.835762i $$-0.314974\pi$$
0.549092 + 0.835762i $$0.314974\pi$$
$$972$$ 0 0
$$973$$ 480.000i 0.0158151i
$$974$$ −5944.00 −0.195542
$$975$$ 0 0
$$976$$ 30422.0 0.997730
$$977$$ 974.000i 0.0318946i 0.999873 + 0.0159473i $$0.00507640\pi$$
−0.999873 + 0.0159473i $$0.994924\pi$$
$$978$$ 0 0
$$979$$ −17160.0 −0.560200
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 10772.0i 0.350049i
$$983$$ − 13608.0i − 0.441534i −0.975327 0.220767i $$-0.929144\pi$$
0.975327 0.220767i $$-0.0708560\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 3220.00 0.104002
$$987$$ 0 0
$$988$$ − 3080.00i − 0.0991780i
$$989$$ 31584.0 1.01548
$$990$$ 0 0
$$991$$ 13472.0 0.431839 0.215919 0.976411i $$-0.430725\pi$$
0.215919 + 0.976411i $$0.430725\pi$$
$$992$$ 46368.0i 1.48406i
$$993$$ 0 0
$$994$$ −7872.00 −0.251192
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 3234.00i − 0.102730i −0.998680 0.0513650i $$-0.983643\pi$$
0.998680 0.0513650i $$-0.0163572\pi$$
$$998$$ 8140.00i 0.258184i
$$999$$ 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 225.4.b.e.199.1 2
3.2 odd 2 75.4.b.b.49.2 2
5.2 odd 4 225.4.a.f.1.1 1
5.3 odd 4 45.4.a.c.1.1 1
5.4 even 2 inner 225.4.b.e.199.2 2
12.11 even 2 1200.4.f.b.49.2 2
15.2 even 4 75.4.a.b.1.1 1
15.8 even 4 15.4.a.a.1.1 1
15.14 odd 2 75.4.b.b.49.1 2
20.3 even 4 720.4.a.n.1.1 1
35.13 even 4 2205.4.a.l.1.1 1
45.13 odd 12 405.4.e.i.136.1 2
45.23 even 12 405.4.e.g.136.1 2
45.38 even 12 405.4.e.g.271.1 2
45.43 odd 12 405.4.e.i.271.1 2
60.23 odd 4 240.4.a.e.1.1 1
60.47 odd 4 1200.4.a.t.1.1 1
60.59 even 2 1200.4.f.b.49.1 2
105.83 odd 4 735.4.a.e.1.1 1
120.53 even 4 960.4.a.b.1.1 1
120.83 odd 4 960.4.a.ba.1.1 1
165.98 odd 4 1815.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 15.8 even 4
45.4.a.c.1.1 1 5.3 odd 4
75.4.a.b.1.1 1 15.2 even 4
75.4.b.b.49.1 2 15.14 odd 2
75.4.b.b.49.2 2 3.2 odd 2
225.4.a.f.1.1 1 5.2 odd 4
225.4.b.e.199.1 2 1.1 even 1 trivial
225.4.b.e.199.2 2 5.4 even 2 inner
240.4.a.e.1.1 1 60.23 odd 4
405.4.e.g.136.1 2 45.23 even 12
405.4.e.g.271.1 2 45.38 even 12
405.4.e.i.136.1 2 45.13 odd 12
405.4.e.i.271.1 2 45.43 odd 12
720.4.a.n.1.1 1 20.3 even 4
735.4.a.e.1.1 1 105.83 odd 4
960.4.a.b.1.1 1 120.53 even 4
960.4.a.ba.1.1 1 120.83 odd 4
1200.4.a.t.1.1 1 60.47 odd 4
1200.4.f.b.49.1 2 60.59 even 2
1200.4.f.b.49.2 2 12.11 even 2
1815.4.a.e.1.1 1 165.98 odd 4
2205.4.a.l.1.1 1 35.13 even 4