# Properties

 Label 75.4.b.b.49.2 Level $75$ Weight $4$ Character 75.49 Analytic conductor $4.425$ 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] = [75,4,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, 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("75.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.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: $$4.42514325043$$ 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 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 75.49 Dual form 75.4.b.b.49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} -24.0000i q^{7} +15.0000i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} -24.0000i q^{7} +15.0000i q^{8} -9.00000 q^{9} +52.0000 q^{11} -21.0000i q^{12} -22.0000i q^{13} +24.0000 q^{14} +41.0000 q^{16} -14.0000i q^{17} -9.00000i q^{18} +20.0000 q^{19} -72.0000 q^{21} +52.0000i q^{22} +168.000i q^{23} +45.0000 q^{24} +22.0000 q^{26} +27.0000i q^{27} -168.000i q^{28} -230.000 q^{29} -288.000 q^{31} +161.000i q^{32} -156.000i q^{33} +14.0000 q^{34} -63.0000 q^{36} -34.0000i q^{37} +20.0000i q^{38} -66.0000 q^{39} +122.000 q^{41} -72.0000i q^{42} +188.000i q^{43} +364.000 q^{44} -168.000 q^{46} +256.000i q^{47} -123.000i q^{48} -233.000 q^{49} -42.0000 q^{51} -154.000i q^{52} +338.000i q^{53} -27.0000 q^{54} +360.000 q^{56} -60.0000i q^{57} -230.000i q^{58} -100.000 q^{59} +742.000 q^{61} -288.000i q^{62} +216.000i q^{63} +167.000 q^{64} +156.000 q^{66} -84.0000i q^{67} -98.0000i q^{68} +504.000 q^{69} -328.000 q^{71} -135.000i q^{72} +38.0000i q^{73} +34.0000 q^{74} +140.000 q^{76} -1248.00i q^{77} -66.0000i q^{78} +240.000 q^{79} +81.0000 q^{81} +122.000i q^{82} -1212.00i q^{83} -504.000 q^{84} -188.000 q^{86} +690.000i q^{87} +780.000i q^{88} -330.000 q^{89} -528.000 q^{91} +1176.00i q^{92} +864.000i q^{93} -256.000 q^{94} +483.000 q^{96} +866.000i q^{97} -233.000i q^{98} -468.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{4} + 6 q^{6} - 18 q^{9}+O(q^{10})$$ 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9 $$2 q + 14 q^{4} + 6 q^{6} - 18 q^{9} + 104 q^{11} + 48 q^{14} + 82 q^{16} + 40 q^{19} - 144 q^{21} + 90 q^{24} + 44 q^{26} - 460 q^{29} - 576 q^{31} + 28 q^{34} - 126 q^{36} - 132 q^{39} + 244 q^{41} + 728 q^{44} - 336 q^{46} - 466 q^{49} - 84 q^{51} - 54 q^{54} + 720 q^{56} - 200 q^{59} + 1484 q^{61} + 334 q^{64} + 312 q^{66} + 1008 q^{69} - 656 q^{71} + 68 q^{74} + 280 q^{76} + 480 q^{79} + 162 q^{81} - 1008 q^{84} - 376 q^{86} - 660 q^{89} - 1056 q^{91} - 512 q^{94} + 966 q^{96} - 936 q^{99}+O(q^{100})$$ 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9 + 104 * q^11 + 48 * q^14 + 82 * q^16 + 40 * q^19 - 144 * q^21 + 90 * q^24 + 44 * q^26 - 460 * q^29 - 576 * q^31 + 28 * q^34 - 126 * q^36 - 132 * q^39 + 244 * q^41 + 728 * q^44 - 336 * q^46 - 466 * q^49 - 84 * q^51 - 54 * q^54 + 720 * q^56 - 200 * q^59 + 1484 * q^61 + 334 * q^64 + 312 * q^66 + 1008 * q^69 - 656 * q^71 + 68 * q^74 + 280 * q^76 + 480 * q^79 + 162 * q^81 - 1008 * q^84 - 376 * q^86 - 660 * q^89 - 1056 * q^91 - 512 * q^94 + 966 * q^96 - 936 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$0.0565670\pi$$
−0.984251 + 0.176777i $$0.943433\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ 7.00000 0.875000
$$5$$ 0 0
$$6$$ 3.00000 0.204124
$$7$$ − 24.0000i − 1.29588i −0.761692 0.647939i $$-0.775631\pi$$
0.761692 0.647939i $$-0.224369\pi$$
$$8$$ 15.0000i 0.662913i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 52.0000 1.42533 0.712663 0.701506i $$-0.247489\pi$$
0.712663 + 0.701506i $$0.247489\pi$$
$$12$$ − 21.0000i − 0.505181i
$$13$$ − 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 $$-0.968158\pi$$
0.995001 0.0998676i $$-0.0318419\pi$$
$$18$$ − 9.00000i − 0.117851i
$$19$$ 20.0000 0.241490 0.120745 0.992684i $$-0.461472\pi$$
0.120745 + 0.992684i $$0.461472\pi$$
$$20$$ 0 0
$$21$$ −72.0000 −0.748176
$$22$$ 52.0000i 0.503929i
$$23$$ 168.000i 1.52306i 0.648129 + 0.761531i $$0.275552\pi$$
−0.648129 + 0.761531i $$0.724448\pi$$
$$24$$ 45.0000 0.382733
$$25$$ 0 0
$$26$$ 22.0000 0.165944
$$27$$ 27.0000i 0.192450i
$$28$$ − 168.000i − 1.13389i
$$29$$ −230.000 −1.47276 −0.736378 0.676570i $$-0.763465\pi$$
−0.736378 + 0.676570i $$0.763465\pi$$
$$30$$ 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
$$33$$ − 156.000i − 0.822913i
$$34$$ 14.0000 0.0706171
$$35$$ 0 0
$$36$$ −63.0000 −0.291667
$$37$$ − 34.0000i − 0.151069i −0.997143 0.0755347i $$-0.975934\pi$$
0.997143 0.0755347i $$-0.0240664\pi$$
$$38$$ 20.0000i 0.0853797i
$$39$$ −66.0000 −0.270986
$$40$$ 0 0
$$41$$ 122.000 0.464712 0.232356 0.972631i $$-0.425357\pi$$
0.232356 + 0.972631i $$0.425357\pi$$
$$42$$ − 72.0000i − 0.264520i
$$43$$ 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 $$0.130035\pi$$
−0.917711 + 0.397249i $$0.869965\pi$$
$$48$$ − 123.000i − 0.369865i
$$49$$ −233.000 −0.679300
$$50$$ 0 0
$$51$$ −42.0000 −0.115317
$$52$$ − 154.000i − 0.410691i
$$53$$ 338.000i 0.875998i 0.898976 + 0.437999i $$0.144313\pi$$
−0.898976 + 0.437999i $$0.855687\pi$$
$$54$$ −27.0000 −0.0680414
$$55$$ 0 0
$$56$$ 360.000 0.859054
$$57$$ − 60.0000i − 0.139424i
$$58$$ − 230.000i − 0.520698i
$$59$$ −100.000 −0.220659 −0.110330 0.993895i $$-0.535191\pi$$
−0.110330 + 0.993895i $$0.535191\pi$$
$$60$$ 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
$$63$$ 216.000i 0.431959i
$$64$$ 167.000 0.326172
$$65$$ 0 0
$$66$$ 156.000 0.290944
$$67$$ − 84.0000i − 0.153168i −0.997063 0.0765838i $$-0.975599\pi$$
0.997063 0.0765838i $$-0.0244013\pi$$
$$68$$ − 98.0000i − 0.174768i
$$69$$ 504.000 0.879340
$$70$$ 0 0
$$71$$ −328.000 −0.548260 −0.274130 0.961693i $$-0.588390\pi$$
−0.274130 + 0.961693i $$0.588390\pi$$
$$72$$ − 135.000i − 0.220971i
$$73$$ 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
$$78$$ − 66.0000i − 0.0958081i
$$79$$ 240.000 0.341799 0.170899 0.985288i $$-0.445333\pi$$
0.170899 + 0.985288i $$0.445333\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 122.000i 0.164301i
$$83$$ − 1212.00i − 1.60282i −0.598114 0.801411i $$-0.704083\pi$$
0.598114 0.801411i $$-0.295917\pi$$
$$84$$ −504.000 −0.654654
$$85$$ 0 0
$$86$$ −188.000 −0.235727
$$87$$ 690.000i 0.850296i
$$88$$ 780.000i 0.944867i
$$89$$ −330.000 −0.393033 −0.196516 0.980501i $$-0.562963\pi$$
−0.196516 + 0.980501i $$0.562963\pi$$
$$90$$ 0 0
$$91$$ −528.000 −0.608236
$$92$$ 1176.00i 1.33268i
$$93$$ 864.000i 0.963362i
$$94$$ −256.000 −0.280898
$$95$$ 0 0
$$96$$ 483.000 0.513500
$$97$$ 866.000i 0.906484i 0.891387 + 0.453242i $$0.149733\pi$$
−0.891387 + 0.453242i $$0.850267\pi$$
$$98$$ − 233.000i − 0.240169i
$$99$$ −468.000 −0.475109
$$100$$ 0 0
$$101$$ −1218.00 −1.19996 −0.599978 0.800017i $$-0.704824\pi$$
−0.599978 + 0.800017i $$0.704824\pi$$
$$102$$ − 42.0000i − 0.0407708i
$$103$$ 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 $$0.00517686\pi$$
−0.999868 + 0.0162629i $$0.994823\pi$$
$$108$$ 189.000i 0.168394i
$$109$$ 970.000 0.852378 0.426189 0.904634i $$-0.359856\pi$$
0.426189 + 0.904634i $$0.359856\pi$$
$$110$$ 0 0
$$111$$ −102.000 −0.0872199
$$112$$ − 984.000i − 0.830172i
$$113$$ − 1042.00i − 0.867461i −0.901043 0.433731i $$-0.857197\pi$$
0.901043 0.433731i $$-0.142803\pi$$
$$114$$ 60.0000 0.0492940
$$115$$ 0 0
$$116$$ −1610.00 −1.28866
$$117$$ 198.000i 0.156454i
$$118$$ − 100.000i − 0.0780148i
$$119$$ −336.000 −0.258833
$$120$$ 0 0
$$121$$ 1373.00 1.03156
$$122$$ 742.000i 0.550635i
$$123$$ − 366.000i − 0.268302i
$$124$$ −2016.00 −1.46002
$$125$$ 0 0
$$126$$ −216.000 −0.152721
$$127$$ 1936.00i 1.35269i 0.736583 + 0.676347i $$0.236438\pi$$
−0.736583 + 0.676347i $$0.763562\pi$$
$$128$$ 1455.00i 1.00473i
$$129$$ 564.000 0.384941
$$130$$ 0 0
$$131$$ 732.000 0.488207 0.244104 0.969749i $$-0.421506\pi$$
0.244104 + 0.969749i $$0.421506\pi$$
$$132$$ − 1092.00i − 0.720048i
$$133$$ − 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 $$-0.757458\pi$$
0.723479 0.690346i $$-0.242542\pi$$
$$138$$ 504.000i 0.310894i
$$139$$ −20.0000 −0.0122042 −0.00610208 0.999981i $$-0.501942\pi$$
−0.00610208 + 0.999981i $$0.501942\pi$$
$$140$$ 0 0
$$141$$ 768.000 0.458704
$$142$$ − 328.000i − 0.193839i
$$143$$ − 1144.00i − 0.668994i
$$144$$ −369.000 −0.213542
$$145$$ 0 0
$$146$$ −38.0000 −0.0215404
$$147$$ 699.000i 0.392194i
$$148$$ − 238.000i − 0.132186i
$$149$$ 1330.00 0.731261 0.365630 0.930760i $$-0.380853\pi$$
0.365630 + 0.930760i $$0.380853\pi$$
$$150$$ 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
$$153$$ 126.000i 0.0665784i
$$154$$ 1248.00 0.653031
$$155$$ 0 0
$$156$$ −462.000 −0.237113
$$157$$ − 3514.00i − 1.78629i −0.449768 0.893146i $$-0.648493\pi$$
0.449768 0.893146i $$-0.351507\pi$$
$$158$$ 240.000i 0.120844i
$$159$$ 1014.00 0.505757
$$160$$ 0 0
$$161$$ 4032.00 1.97370
$$162$$ 81.0000i 0.0392837i
$$163$$ 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 $$-0.998230\pi$$
0.999985 0.00556041i $$-0.00176994\pi$$
$$168$$ − 1080.00i − 0.495975i
$$169$$ 1713.00 0.779700
$$170$$ 0 0
$$171$$ −180.000 −0.0804967
$$172$$ 1316.00i 0.583396i
$$173$$ 618.000i 0.271593i 0.990737 + 0.135797i $$0.0433594\pi$$
−0.990737 + 0.135797i $$0.956641\pi$$
$$174$$ −690.000 −0.300625
$$175$$ 0 0
$$176$$ 2132.00 0.913100
$$177$$ 300.000i 0.127398i
$$178$$ − 330.000i − 0.138958i
$$179$$ −3340.00 −1.39466 −0.697328 0.716752i $$-0.745628\pi$$
−0.697328 + 0.716752i $$0.745628\pi$$
$$180$$ 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
$$183$$ − 2226.00i − 0.899184i
$$184$$ −2520.00 −1.00966
$$185$$ 0 0
$$186$$ −864.000 −0.340600
$$187$$ − 728.000i − 0.284688i
$$188$$ 1792.00i 0.695186i
$$189$$ 648.000 0.249392
$$190$$ 0 0
$$191$$ −1888.00 −0.715240 −0.357620 0.933867i $$-0.616412\pi$$
−0.357620 + 0.933867i $$0.616412\pi$$
$$192$$ − 501.000i − 0.188315i
$$193$$ − 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 $$0.150996\pi$$
−0.889581 + 0.456777i $$0.849004\pi$$
$$198$$ − 468.000i − 0.167976i
$$199$$ 1160.00 0.413217 0.206609 0.978424i $$-0.433757\pi$$
0.206609 + 0.978424i $$0.433757\pi$$
$$200$$ 0 0
$$201$$ −252.000 −0.0884314
$$202$$ − 1218.00i − 0.424248i
$$203$$ 5520.00i 1.90851i
$$204$$ −294.000 −0.100903
$$205$$ 0 0
$$206$$ −88.0000 −0.0297634
$$207$$ − 1512.00i − 0.507687i
$$208$$ − 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
$$213$$ 984.000i 0.316538i
$$214$$ −36.0000 −0.0114996
$$215$$ 0 0
$$216$$ −405.000 −0.127578
$$217$$ 6912.00i 2.16229i
$$218$$ 970.000i 0.301361i
$$219$$ 114.000 0.0351754
$$220$$ 0 0
$$221$$ −308.000 −0.0937481
$$222$$ − 102.000i − 0.0308369i
$$223$$ − 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 $$0.125926\pi$$
−0.922763 + 0.385369i $$0.874074\pi$$
$$228$$ − 420.000i − 0.121996i
$$229$$ −4830.00 −1.39378 −0.696889 0.717179i $$-0.745433\pi$$
−0.696889 + 0.717179i $$0.745433\pi$$
$$230$$ 0 0
$$231$$ −3744.00 −1.06639
$$232$$ − 3450.00i − 0.976309i
$$233$$ − 2682.00i − 0.754093i −0.926194 0.377046i $$-0.876940\pi$$
0.926194 0.377046i $$-0.123060\pi$$
$$234$$ −198.000 −0.0553148
$$235$$ 0 0
$$236$$ −700.000 −0.193077
$$237$$ − 720.000i − 0.197338i
$$238$$ − 336.000i − 0.0915111i
$$239$$ −2320.00 −0.627901 −0.313950 0.949439i $$-0.601653\pi$$
−0.313950 + 0.949439i $$0.601653\pi$$
$$240$$ 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
$$243$$ − 243.000i − 0.0641500i
$$244$$ 5194.00 1.36275
$$245$$ 0 0
$$246$$ 366.000 0.0948590
$$247$$ − 440.000i − 0.113346i
$$248$$ − 4320.00i − 1.10613i
$$249$$ −3636.00 −0.925390
$$250$$ 0 0
$$251$$ 132.000 0.0331943 0.0165971 0.999862i $$-0.494717\pi$$
0.0165971 + 0.999862i $$0.494717\pi$$
$$252$$ 1512.00i 0.377964i
$$253$$ 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 $$-0.624880\pi$$
0.382335 0.924024i $$-0.375120\pi$$
$$258$$ 564.000i 0.136097i
$$259$$ −816.000 −0.195767
$$260$$ 0 0
$$261$$ 2070.00 0.490919
$$262$$ 732.000i 0.172607i
$$263$$ 4888.00i 1.14603i 0.819543 + 0.573017i $$0.194227\pi$$
−0.819543 + 0.573017i $$0.805773\pi$$
$$264$$ 2340.00 0.545519
$$265$$ 0 0
$$266$$ 480.000 0.110642
$$267$$ 990.000i 0.226918i
$$268$$ − 588.000i − 0.134022i
$$269$$ −1270.00 −0.287856 −0.143928 0.989588i $$-0.545973\pi$$
−0.143928 + 0.989588i $$0.545973\pi$$
$$270$$ 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
$$273$$ 1584.00i 0.351165i
$$274$$ 2214.00 0.488148
$$275$$ 0 0
$$276$$ 3528.00 0.769423
$$277$$ − 5394.00i − 1.17001i −0.811028 0.585007i $$-0.801092\pi$$
0.811028 0.585007i $$-0.198908\pi$$
$$278$$ − 20.0000i − 0.00431482i
$$279$$ 2592.00 0.556197
$$280$$ 0 0
$$281$$ 2442.00 0.518425 0.259213 0.965820i $$-0.416537\pi$$
0.259213 + 0.965820i $$0.416537\pi$$
$$282$$ 768.000i 0.162176i
$$283$$ − 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
$$288$$ − 1449.00i − 0.296469i
$$289$$ 4717.00 0.960106
$$290$$ 0 0
$$291$$ 2598.00 0.523359
$$292$$ 266.000i 0.0533098i
$$293$$ − 4542.00i − 0.905619i −0.891607 0.452810i $$-0.850422\pi$$
0.891607 0.452810i $$-0.149578\pi$$
$$294$$ −699.000 −0.138662
$$295$$ 0 0
$$296$$ 510.000 0.100146
$$297$$ 1404.00i 0.274304i
$$298$$ 1330.00i 0.258540i
$$299$$ 3696.00 0.714867
$$300$$ 0 0
$$301$$ 4512.00 0.864011
$$302$$ − 1208.00i − 0.230174i
$$303$$ 3654.00i 0.692795i
$$304$$ 820.000 0.154705
$$305$$ 0 0
$$306$$ −126.000 −0.0235390
$$307$$ 5116.00i 0.951093i 0.879691 + 0.475546i $$0.157750\pi$$
−0.879691 + 0.475546i $$0.842250\pi$$
$$308$$ − 8736.00i − 1.61617i
$$309$$ 264.000 0.0486034
$$310$$ 0 0
$$311$$ −2808.00 −0.511984 −0.255992 0.966679i $$-0.582402\pi$$
−0.255992 + 0.966679i $$0.582402\pi$$
$$312$$ − 990.000i − 0.179640i
$$313$$ 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 $$0.0637601\pi$$
−0.980005 + 0.198971i $$0.936240\pi$$
$$318$$ 1014.00i 0.178812i
$$319$$ −11960.0 −2.09916
$$320$$ 0 0
$$321$$ 108.000 0.0187787
$$322$$ 4032.00i 0.697809i
$$323$$ − 280.000i − 0.0482341i
$$324$$ 567.000 0.0972222
$$325$$ 0 0
$$326$$ −2068.00 −0.351337
$$327$$ − 2910.00i − 0.492120i
$$328$$ 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
$$333$$ 306.000i 0.0503564i
$$334$$ 24.0000 0.00393180
$$335$$ 0 0
$$336$$ −2952.00 −0.479300
$$337$$ − 11534.0i − 1.86438i −0.361966 0.932191i $$-0.617894\pi$$
0.361966 0.932191i $$-0.382106\pi$$
$$338$$ 1713.00i 0.275665i
$$339$$ −3126.00 −0.500829
$$340$$ 0 0
$$341$$ −14976.0 −2.37829
$$342$$ − 180.000i − 0.0284599i
$$343$$ − 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 $$0.375793\pi$$
−0.380382 + 0.924830i $$0.624207\pi$$
$$348$$ 4830.00i 0.744009i
$$349$$ −4870.00 −0.746949 −0.373474 0.927640i $$-0.621834\pi$$
−0.373474 + 0.927640i $$0.621834\pi$$
$$350$$ 0 0
$$351$$ 594.000 0.0903287
$$352$$ 8372.00i 1.26770i
$$353$$ − 10722.0i − 1.61664i −0.588742 0.808321i $$-0.700377\pi$$
0.588742 0.808321i $$-0.299623\pi$$
$$354$$ −300.000 −0.0450419
$$355$$ 0 0
$$356$$ −2310.00 −0.343904
$$357$$ 1008.00i 0.149437i
$$358$$ − 3340.00i − 0.493085i
$$359$$ −120.000 −0.0176417 −0.00882083 0.999961i $$-0.502808\pi$$
−0.00882083 + 0.999961i $$0.502808\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ − 178.000i − 0.0258438i
$$363$$ − 4119.00i − 0.595569i
$$364$$ −3696.00 −0.532206
$$365$$ 0 0
$$366$$ 2226.00 0.317910
$$367$$ 3936.00i 0.559830i 0.960025 + 0.279915i $$0.0903063\pi$$
−0.960025 + 0.279915i $$0.909694\pi$$
$$368$$ 6888.00i 0.975711i
$$369$$ −1098.00 −0.154904
$$370$$ 0 0
$$371$$ 8112.00 1.13519
$$372$$ 6048.00i 0.842941i
$$373$$ − 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
$$378$$ 648.000i 0.0881733i
$$379$$ 13340.0 1.80799 0.903997 0.427539i $$-0.140619\pi$$
0.903997 + 0.427539i $$0.140619\pi$$
$$380$$ 0 0
$$381$$ 5808.00 0.780979
$$382$$ − 1888.00i − 0.252876i
$$383$$ 1008.00i 0.134481i 0.997737 + 0.0672407i $$0.0214195\pi$$
−0.997737 + 0.0672407i $$0.978580\pi$$
$$384$$ 4365.00 0.580079
$$385$$ 0 0
$$386$$ 1922.00 0.253438
$$387$$ − 1692.00i − 0.222246i
$$388$$ 6062.00i 0.793174i
$$389$$ −9630.00 −1.25517 −0.627584 0.778549i $$-0.715956\pi$$
−0.627584 + 0.778549i $$0.715956\pi$$
$$390$$ 0 0
$$391$$ 2352.00 0.304209
$$392$$ − 3495.00i − 0.450317i
$$393$$ − 2196.00i − 0.281867i
$$394$$ −2526.00 −0.322990
$$395$$ 0 0
$$396$$ −3276.00 −0.415720
$$397$$ 7126.00i 0.900866i 0.892810 + 0.450433i $$0.148730\pi$$
−0.892810 + 0.450433i $$0.851270\pi$$
$$398$$ 1160.00i 0.146094i
$$399$$ −1440.00 −0.180677
$$400$$ 0 0
$$401$$ −8718.00 −1.08568 −0.542838 0.839837i $$-0.682650\pi$$
−0.542838 + 0.839837i $$0.682650\pi$$
$$402$$ − 252.000i − 0.0312652i
$$403$$ 6336.00i 0.783173i
$$404$$ −8526.00 −1.04996
$$405$$ 0 0
$$406$$ −5520.00 −0.674761
$$407$$ − 1768.00i − 0.215323i
$$408$$ − 630.000i − 0.0764452i
$$409$$ 10870.0 1.31415 0.657074 0.753826i $$-0.271794\pi$$
0.657074 + 0.753826i $$0.271794\pi$$
$$410$$ 0 0
$$411$$ −6642.00 −0.797143
$$412$$ 616.000i 0.0736605i
$$413$$ 2400.00i 0.285947i
$$414$$ 1512.00 0.179495
$$415$$ 0 0
$$416$$ 3542.00 0.417454
$$417$$ 60.0000i 0.00704607i
$$418$$ 1040.00i 0.121694i
$$419$$ 9700.00 1.13097 0.565484 0.824759i $$-0.308689\pi$$
0.565484 + 0.824759i $$0.308689\pi$$
$$420$$ 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
$$423$$ − 2304.00i − 0.264833i
$$424$$ −5070.00 −0.580710
$$425$$ 0 0
$$426$$ −984.000 −0.111913
$$427$$ − 17808.0i − 2.01824i
$$428$$ 252.000i 0.0284600i
$$429$$ −3432.00 −0.386244
$$430$$ 0 0
$$431$$ 15792.0 1.76490 0.882452 0.470402i $$-0.155891\pi$$
0.882452 + 0.470402i $$0.155891\pi$$
$$432$$ 1107.00i 0.123288i
$$433$$ − 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
$$438$$ 114.000i 0.0124364i
$$439$$ 440.000 0.0478361 0.0239181 0.999714i $$-0.492386\pi$$
0.0239181 + 0.999714i $$0.492386\pi$$
$$440$$ 0 0
$$441$$ 2097.00 0.226433
$$442$$ − 308.000i − 0.0331449i
$$443$$ 10188.0i 1.09266i 0.837571 + 0.546328i $$0.183975\pi$$
−0.837571 + 0.546328i $$0.816025\pi$$
$$444$$ −714.000 −0.0763174
$$445$$ 0 0
$$446$$ 6032.00 0.640411
$$447$$ − 3990.00i − 0.422194i
$$448$$ − 4008.00i − 0.422679i
$$449$$ 13310.0 1.39897 0.699485 0.714647i $$-0.253413\pi$$
0.699485 + 0.714647i $$0.253413\pi$$
$$450$$ 0 0
$$451$$ 6344.00 0.662367
$$452$$ − 7294.00i − 0.759029i
$$453$$ 3624.00i 0.375873i
$$454$$ −2636.00 −0.272497
$$455$$ 0 0
$$456$$ 900.000 0.0924262
$$457$$ 3226.00i 0.330210i 0.986276 + 0.165105i $$0.0527963\pi$$
−0.986276 + 0.165105i $$0.947204\pi$$
$$458$$ − 4830.00i − 0.492775i
$$459$$ 378.000 0.0384391
$$460$$ 0 0
$$461$$ 6582.00 0.664977 0.332488 0.943107i $$-0.392112\pi$$
0.332488 + 0.943107i $$0.392112\pi$$
$$462$$ − 3744.00i − 0.377027i
$$463$$ − 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 $$0.00750744\pi$$
−0.999722 + 0.0235831i $$0.992493\pi$$
$$468$$ 1386.00i 0.136897i
$$469$$ −2016.00 −0.198487
$$470$$ 0 0
$$471$$ −10542.0 −1.03132
$$472$$ − 1500.00i − 0.146278i
$$473$$ 9776.00i 0.950319i
$$474$$ 720.000 0.0697694
$$475$$ 0 0
$$476$$ −2352.00 −0.226478
$$477$$ − 3042.00i − 0.291999i
$$478$$ − 2320.00i − 0.221997i
$$479$$ 19680.0 1.87725 0.938624 0.344941i $$-0.112101\pi$$
0.938624 + 0.344941i $$0.112101\pi$$
$$480$$ 0 0
$$481$$ −748.000 −0.0709062
$$482$$ 2002.00i 0.189188i
$$483$$ − 12096.0i − 1.13952i
$$484$$ 9611.00 0.902611
$$485$$ 0 0
$$486$$ 243.000 0.0226805
$$487$$ − 5944.00i − 0.553077i −0.961003 0.276538i $$-0.910813\pi$$
0.961003 0.276538i $$-0.0891873\pi$$
$$488$$ 11130.0i 1.03244i
$$489$$ 6204.00 0.573731
$$490$$ 0 0
$$491$$ 10772.0 0.990089 0.495044 0.868868i $$-0.335152\pi$$
0.495044 + 0.868868i $$0.335152\pi$$
$$492$$ − 2562.00i − 0.234764i
$$493$$ 3220.00i 0.294161i
$$494$$ 440.000 0.0400740
$$495$$ 0 0
$$496$$ −11808.0 −1.06894
$$497$$ 7872.00i 0.710478i
$$498$$ − 3636.00i − 0.327175i
$$499$$ −8140.00 −0.730253 −0.365127 0.930958i $$-0.618974\pi$$
−0.365127 + 0.930958i $$0.618974\pi$$
$$500$$ 0 0
$$501$$ −72.0000 −0.00642060
$$502$$ 132.000i 0.0117360i
$$503$$ 13768.0i 1.22045i 0.792229 + 0.610223i $$0.208920\pi$$
−0.792229 + 0.610223i $$0.791080\pi$$
$$504$$ −3240.00 −0.286351
$$505$$ 0 0
$$506$$ −8736.00 −0.767515
$$507$$ − 5139.00i − 0.450160i
$$508$$ 13552.0i 1.18361i
$$509$$ −22150.0 −1.92884 −0.964422 0.264368i $$-0.914837\pi$$
−0.964422 + 0.264368i $$0.914837\pi$$
$$510$$ 0 0
$$511$$ 912.000 0.0789521
$$512$$ 11521.0i 0.994455i
$$513$$ 540.000i 0.0464748i
$$514$$ 7614.00 0.653384
$$515$$ 0 0
$$516$$ 3948.00 0.336824
$$517$$ 13312.0i 1.13242i
$$518$$ − 816.000i − 0.0692143i
$$519$$ 1854.00 0.156805
$$520$$ 0 0
$$521$$ 1562.00 0.131348 0.0656741 0.997841i $$-0.479080\pi$$
0.0656741 + 0.997841i $$0.479080\pi$$
$$522$$ 2070.00i 0.173566i
$$523$$ 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
$$528$$ − 6396.00i − 0.527178i
$$529$$ −16057.0 −1.31972
$$530$$ 0 0
$$531$$ 900.000 0.0735531
$$532$$ − 3360.00i − 0.273824i
$$533$$ − 2684.00i − 0.218118i
$$534$$ −990.000 −0.0802275
$$535$$ 0 0
$$536$$ 1260.00 0.101537
$$537$$ 10020.0i 0.805205i
$$538$$ − 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
$$543$$ 534.000i 0.0422028i
$$544$$ 2254.00 0.177646
$$545$$ 0 0
$$546$$ −1584.00 −0.124156
$$547$$ − 10484.0i − 0.819494i −0.912199 0.409747i $$-0.865617\pi$$
0.912199 0.409747i $$-0.134383\pi$$
$$548$$ − 15498.0i − 1.20811i
$$549$$ −6678.00 −0.519144
$$550$$ 0 0
$$551$$ −4600.00 −0.355656
$$552$$ 7560.00i 0.582926i
$$553$$ − 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 $$0.0437960\pi$$
−0.990550 + 0.137155i $$0.956204\pi$$
$$558$$ 2592.00i 0.196645i
$$559$$ 4136.00 0.312941
$$560$$ 0 0
$$561$$ −2184.00 −0.164365
$$562$$ 2442.00i 0.183291i
$$563$$ − 12252.0i − 0.917159i −0.888654 0.458579i $$-0.848359\pi$$
0.888654 0.458579i $$-0.151641\pi$$
$$564$$ 5376.00 0.401366
$$565$$ 0 0
$$566$$ 2772.00 0.205858
$$567$$ − 1944.00i − 0.143986i
$$568$$ − 4920.00i − 0.363448i
$$569$$ 14550.0 1.07200 0.536000 0.844218i $$-0.319935\pi$$
0.536000 + 0.844218i $$0.319935\pi$$
$$570$$ 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
$$573$$ 5664.00i 0.412944i
$$574$$ 2928.00 0.212914
$$575$$ 0 0
$$576$$ −1503.00 −0.108724
$$577$$ 12866.0i 0.928282i 0.885761 + 0.464141i $$0.153637\pi$$
−0.885761 + 0.464141i $$0.846363\pi$$
$$578$$ 4717.00i 0.339449i
$$579$$ −5766.00 −0.413863
$$580$$ 0 0
$$581$$ −29088.0 −2.07706
$$582$$ 2598.00i 0.185035i
$$583$$ 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 $$-0.825234\pi$$
0.853024 0.521872i $$-0.174766\pi$$
$$588$$ 4893.00i 0.343170i
$$589$$ −5760.00 −0.402948
$$590$$ 0 0
$$591$$ 7578.00 0.527440
$$592$$ − 1394.00i − 0.0967788i
$$593$$ − 20402.0i − 1.41283i −0.707797 0.706416i $$-0.750311\pi$$
0.707797 0.706416i $$-0.249689\pi$$
$$594$$ −1404.00 −0.0969812
$$595$$ 0 0
$$596$$ 9310.00 0.639853
$$597$$ − 3480.00i − 0.238571i
$$598$$ 3696.00i 0.252744i
$$599$$ −10760.0 −0.733959 −0.366980 0.930229i $$-0.619608\pi$$
−0.366980 + 0.930229i $$0.619608\pi$$
$$600$$ 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
$$603$$ 756.000i 0.0510559i
$$604$$ −8456.00 −0.569652
$$605$$ 0 0
$$606$$ −3654.00 −0.244940
$$607$$ 11056.0i 0.739290i 0.929173 + 0.369645i $$0.120521\pi$$
−0.929173 + 0.369645i $$0.879479\pi$$
$$608$$ 3220.00i 0.214783i
$$609$$ 16560.0 1.10188
$$610$$ 0 0
$$611$$ 5632.00 0.372907
$$612$$ 882.000i 0.0582561i
$$613$$ 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 $$-0.890099\pi$$
0.940986 0.338445i $$-0.109901\pi$$
$$618$$ 264.000i 0.0171839i
$$619$$ 5260.00 0.341546 0.170773 0.985310i $$-0.445373\pi$$
0.170773 + 0.985310i $$0.445373\pi$$
$$620$$ 0 0
$$621$$ −4536.00 −0.293113
$$622$$ − 2808.00i − 0.181014i
$$623$$ 7920.00i 0.509323i
$$624$$ −2706.00 −0.173600
$$625$$ 0 0
$$626$$ −7318.00 −0.467230
$$627$$ − 3120.00i − 0.198725i
$$628$$ − 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
$$633$$ 13404.0i 0.841645i
$$634$$ −2246.00 −0.140694
$$635$$ 0 0
$$636$$ 7098.00 0.442538
$$637$$ 5126.00i 0.318838i
$$638$$ − 11960.0i − 0.742164i
$$639$$ 2952.00 0.182753
$$640$$ 0 0
$$641$$ −29118.0 −1.79422 −0.897108 0.441812i $$-0.854336\pi$$
−0.897108 + 0.441812i $$0.854336\pi$$
$$642$$ 108.000i 0.00663928i
$$643$$ − 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 $$-0.857326\pi$$
0.901218 0.433366i $$-0.142674\pi$$
$$648$$ 1215.00i 0.0736570i
$$649$$ −5200.00 −0.314511
$$650$$ 0 0
$$651$$ 20736.0 1.24840
$$652$$ 14476.0i 0.869515i
$$653$$ − 6902.00i − 0.413623i −0.978381 0.206812i $$-0.933691\pi$$
0.978381 0.206812i $$-0.0663088\pi$$
$$654$$ 2910.00 0.173991
$$655$$ 0 0
$$656$$ 5002.00 0.297706
$$657$$ − 342.000i − 0.0203085i
$$658$$ 6144.00i 0.364009i
$$659$$ −20140.0 −1.19051 −0.595253 0.803539i $$-0.702948\pi$$
−0.595253 + 0.803539i $$0.702948\pi$$
$$660$$ 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
$$663$$ 924.000i 0.0541255i
$$664$$ 18180.0 1.06253
$$665$$ 0 0
$$666$$ −306.000 −0.0178037
$$667$$ − 38640.0i − 2.24310i
$$668$$ − 168.000i − 0.00973071i
$$669$$ −18096.0 −1.04579
$$670$$ 0 0
$$671$$ 38584.0 2.21985
$$672$$ − 11592.0i − 0.665433i
$$673$$ 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 $$-0.828102\pi$$
0.857692 0.514164i $$-0.171898\pi$$
$$678$$ − 3126.00i − 0.177070i
$$679$$ 20784.0 1.17469
$$680$$ 0 0
$$681$$ 7908.00 0.444986
$$682$$ − 14976.0i − 0.840851i
$$683$$ 23868.0i 1.33716i 0.743638 + 0.668582i $$0.233099\pi$$
−0.743638 + 0.668582i $$0.766901\pi$$
$$684$$ −1260.00 −0.0704347
$$685$$ 0 0
$$686$$ 2640.00 0.146932
$$687$$ 14490.0i 0.804699i
$$688$$ 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
$$693$$ 11232.0i 0.615683i
$$694$$ −11956.0 −0.653953
$$695$$ 0 0
$$696$$ −10350.0 −0.563672
$$697$$ − 1708.00i − 0.0928194i
$$698$$ − 4870.00i − 0.264086i
$$699$$ −8046.00 −0.435376
$$700$$ 0 0
$$701$$ −22138.0 −1.19278 −0.596391 0.802694i $$-0.703399\pi$$
−0.596391 + 0.802694i $$0.703399\pi$$
$$702$$ 594.000i 0.0319360i
$$703$$ − 680.000i − 0.0364818i
$$704$$ 8684.00 0.464901
$$705$$ 0 0
$$706$$ 10722.0 0.571569
$$707$$ 29232.0i 1.55500i
$$708$$ 2100.00i 0.111473i
$$709$$ −3070.00 −0.162618 −0.0813091 0.996689i $$-0.525910\pi$$
−0.0813091 + 0.996689i $$0.525910\pi$$
$$710$$ 0 0
$$711$$ −2160.00 −0.113933
$$712$$ − 4950.00i − 0.260546i
$$713$$ − 48384.0i − 2.54137i
$$714$$ −1008.00 −0.0528340
$$715$$ 0 0
$$716$$ −23380.0 −1.22032
$$717$$ 6960.00i 0.362519i
$$718$$ − 120.000i − 0.00623727i
$$719$$ −15600.0 −0.809154 −0.404577 0.914504i $$-0.632581\pi$$
−0.404577 + 0.914504i $$0.632581\pi$$
$$720$$ 0 0
$$721$$ 2112.00 0.109092
$$722$$ − 6459.00i − 0.332935i
$$723$$ − 6006.00i − 0.308943i
$$724$$ −1246.00 −0.0639603
$$725$$ 0 0
$$726$$ 4119.00 0.210565
$$727$$ 20696.0i 1.05581i 0.849304 + 0.527904i $$0.177022\pi$$
−0.849304 + 0.527904i $$0.822978\pi$$
$$728$$ − 7920.00i − 0.403207i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 2632.00 0.133171
$$732$$ − 15582.0i − 0.786786i
$$733$$ 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
$$738$$ − 1098.00i − 0.0547669i
$$739$$ −11740.0 −0.584388 −0.292194 0.956359i $$-0.594385\pi$$
−0.292194 + 0.956359i $$0.594385\pi$$
$$740$$ 0 0
$$741$$ −1320.00 −0.0654405
$$742$$ 8112.00i 0.401349i
$$743$$ − 2632.00i − 0.129958i −0.997887 0.0649789i $$-0.979302\pi$$
0.997887 0.0649789i $$-0.0206980\pi$$
$$744$$ −12960.0 −0.638625
$$745$$ 0 0
$$746$$ 3022.00 0.148315
$$747$$ 10908.0i 0.534274i
$$748$$ − 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
$$753$$ − 396.000i − 0.0191647i
$$754$$ −5060.00 −0.244396
$$755$$ 0 0
$$756$$ 4536.00 0.218218
$$757$$ 21646.0i 1.03928i 0.854384 + 0.519642i $$0.173934\pi$$
−0.854384 + 0.519642i $$0.826066\pi$$
$$758$$ 13340.0i 0.639222i
$$759$$ 26208.0 1.25335
$$760$$ 0 0
$$761$$ 18282.0 0.870857 0.435428 0.900223i $$-0.356597\pi$$
0.435428 + 0.900223i $$0.356597\pi$$
$$762$$ 5808.00i 0.276118i
$$763$$ − 23280.0i − 1.10458i
$$764$$ −13216.0 −0.625835
$$765$$ 0 0
$$766$$ −1008.00 −0.0475464
$$767$$ 2200.00i 0.103569i
$$768$$ 357.000i 0.0167736i
$$769$$ 24190.0 1.13435 0.567174 0.823598i $$-0.308037\pi$$
0.567174 + 0.823598i $$0.308037\pi$$
$$770$$ 0 0
$$771$$ −22842.0 −1.06697
$$772$$ − 13454.0i − 0.627228i
$$773$$ 25698.0i 1.19572i 0.801600 + 0.597861i $$0.203982\pi$$
−0.801600 + 0.597861i $$0.796018\pi$$
$$774$$ 1692.00 0.0785758
$$775$$ 0 0
$$776$$ −12990.0 −0.600920
$$777$$ 2448.00i 0.113026i
$$778$$ − 9630.00i − 0.443769i
$$779$$ 2440.00 0.112223
$$780$$ 0 0
$$781$$ −17056.0 −0.781449
$$782$$ 2352.00i 0.107554i
$$783$$ − 6210.00i − 0.283432i
$$784$$ −9553.00 −0.435177
$$785$$ 0 0
$$786$$ 2196.00 0.0996549
$$787$$ 33436.0i 1.51444i 0.653160 + 0.757220i $$0.273443\pi$$
−0.653160 + 0.757220i $$0.726557\pi$$
$$788$$ 17682.0i 0.799359i
$$789$$ 14664.0 0.661663
$$790$$ 0 0
$$791$$ −25008.0 −1.12412
$$792$$ − 7020.00i − 0.314956i
$$793$$ − 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 $$-0.685229\pi$$
0.549623 0.835413i $$-0.314771\pi$$
$$798$$ − 1440.00i − 0.0638790i
$$799$$ 3584.00 0.158689
$$800$$ 0 0
$$801$$ 2970.00 0.131011
$$802$$ − 8718.00i − 0.383844i
$$803$$ 1976.00i 0.0868388i
$$804$$ −1764.00 −0.0773775
$$805$$ 0 0
$$806$$ −6336.00 −0.276893
$$807$$ 3810.00i 0.166194i
$$808$$ − 18270.0i − 0.795466i
$$809$$ −4730.00 −0.205560 −0.102780 0.994704i $$-0.532774\pi$$
−0.102780 + 0.994704i $$0.532774\pi$$
$$810$$ 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
$$813$$ − 3216.00i − 0.138733i
$$814$$ 1768.00 0.0761282
$$815$$ 0 0
$$816$$ −1722.00 −0.0738751
$$817$$ 3760.00i 0.161011i
$$818$$ 10870.0i 0.464622i
$$819$$ 4752.00 0.202745
$$820$$ 0 0
$$821$$ 44142.0 1.87645 0.938226 0.346024i $$-0.112468\pi$$
0.938226 + 0.346024i $$0.112468\pi$$
$$822$$ − 6642.00i − 0.281833i
$$823$$ − 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 $$-0.901789\pi$$
0.952778 0.303668i $$-0.0982114\pi$$
$$828$$ − 10584.0i − 0.444226i
$$829$$ −42150.0 −1.76590 −0.882949 0.469468i $$-0.844446\pi$$
−0.882949 + 0.469468i $$0.844446\pi$$
$$830$$ 0 0
$$831$$ −16182.0 −0.675508
$$832$$ − 3674.00i − 0.153093i
$$833$$ 3262.00i 0.135680i
$$834$$ −60.0000 −0.00249116
$$835$$ 0 0
$$836$$ 7280.00 0.301177
$$837$$ − 7776.00i − 0.321121i
$$838$$ 9700.00i 0.399858i
$$839$$ −13400.0 −0.551394 −0.275697 0.961245i $$-0.588909\pi$$
−0.275697 + 0.961245i $$0.588909\pi$$
$$840$$ 0 0
$$841$$ 28511.0 1.16901
$$842$$ 862.000i 0.0352809i
$$843$$ − 7326.00i − 0.299313i
$$844$$ −31276.0 −1.27555
$$845$$ 0 0
$$846$$ 2304.00 0.0936326
$$847$$ − 32952.0i − 1.33677i
$$848$$ 13858.0i 0.561186i
$$849$$ −8316.00 −0.336165
$$850$$ 0 0
$$851$$ 5712.00 0.230088
$$852$$ 6888.00i 0.276971i
$$853$$ 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 $$0.325528\pi$$
−0.521084 + 0.853505i $$0.674472\pi$$
$$858$$ − 3432.00i − 0.136558i
$$859$$ 35900.0 1.42595 0.712976 0.701189i $$-0.247347\pi$$
0.712976 + 0.701189i $$0.247347\pi$$
$$860$$ 0 0
$$861$$ −8784.00 −0.347686
$$862$$ 15792.0i 0.623988i
$$863$$ 3088.00i 0.121804i 0.998144 + 0.0609019i $$0.0193977\pi$$
−0.998144 + 0.0609019i $$0.980602\pi$$
$$864$$ −4347.00 −0.171167
$$865$$ 0 0
$$866$$ 11602.0 0.455256
$$867$$ − 14151.0i − 0.554317i
$$868$$ 48384.0i 1.89200i
$$869$$ 12480.0 0.487175
$$870$$ 0 0
$$871$$ −1848.00 −0.0718910
$$872$$ 14550.0i 0.565052i
$$873$$ − 7794.00i − 0.302161i
$$874$$ −3360.00 −0.130039
$$875$$ 0 0
$$876$$ 798.000 0.0307784
$$877$$ − 35274.0i − 1.35817i −0.734058 0.679087i $$-0.762376\pi$$
0.734058 0.679087i $$-0.237624\pi$$
$$878$$ 440.000i 0.0169126i
$$879$$ −13626.0 −0.522860
$$880$$ 0 0
$$881$$ 25042.0 0.957646 0.478823 0.877911i $$-0.341064\pi$$
0.478823 + 0.877911i $$0.341064\pi$$
$$882$$ 2097.00i 0.0800563i
$$883$$ − 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 $$-0.864193\pi$$
0.910358 0.413823i $$-0.135807\pi$$
$$888$$ − 1530.00i − 0.0578192i
$$889$$ 46464.0 1.75293
$$890$$ 0 0
$$891$$ 4212.00 0.158370
$$892$$ − 42224.0i − 1.58494i
$$893$$ 5120.00i 0.191864i
$$894$$ 3990.00 0.149268
$$895$$ 0 0
$$896$$ 34920.0 1.30200
$$897$$ − 11088.0i − 0.412729i
$$898$$ 13310.0i 0.494611i
$$899$$ 66240.0 2.45743
$$900$$ 0 0
$$901$$ 4732.00 0.174968
$$902$$ 6344.00i 0.234182i
$$903$$ − 13536.0i − 0.498837i
$$904$$ 15630.0 0.575051
$$905$$ 0 0
$$906$$ −3624.00 −0.132891
$$907$$ 31236.0i 1.14352i 0.820420 + 0.571761i $$0.193740\pi$$
−0.820420 + 0.571761i $$0.806260\pi$$
$$908$$ 18452.0i 0.674396i
$$909$$ 10962.0 0.399985
$$910$$ 0 0
$$911$$ 8272.00 0.300838 0.150419 0.988622i $$-0.451938\pi$$
0.150419 + 0.988622i $$0.451938\pi$$
$$912$$ − 2460.00i − 0.0893188i
$$913$$ − 63024.0i − 2.28455i
$$914$$ −3226.00 −0.116747
$$915$$ 0 0
$$916$$ −33810.0 −1.21956
$$917$$ − 17568.0i − 0.632657i
$$918$$ 378.000i 0.0135903i
$$919$$ −20200.0 −0.725067 −0.362533 0.931971i $$-0.618088\pi$$
−0.362533 + 0.931971i $$0.618088\pi$$
$$920$$ 0 0
$$921$$ 15348.0 0.549114
$$922$$ 6582.00i 0.235105i
$$923$$ 7216.00i 0.257332i
$$924$$ −26208.0 −0.933095
$$925$$ 0 0
$$926$$ 15072.0 0.534878
$$927$$ − 792.000i − 0.0280612i
$$928$$ − 37030.0i − 1.30988i
$$929$$ −31010.0 −1.09516 −0.547581 0.836753i $$-0.684451\pi$$
−0.547581 + 0.836753i $$0.684451\pi$$
$$930$$ 0 0
$$931$$ −4660.00 −0.164044
$$932$$ − 18774.0i − 0.659831i
$$933$$ 8424.00i 0.295594i
$$934$$ −476.000 −0.0166758
$$935$$ 0 0
$$936$$ −2970.00 −0.103715
$$937$$ − 39174.0i − 1.36580i −0.730510 0.682902i $$-0.760717\pi$$
0.730510 0.682902i $$-0.239283\pi$$
$$938$$ − 2016.00i − 0.0701756i
$$939$$ 21954.0 0.762984
$$940$$ 0 0
$$941$$ −4138.00 −0.143353 −0.0716764 0.997428i $$-0.522835\pi$$
−0.0716764 + 0.997428i $$0.522835\pi$$
$$942$$ − 10542.0i − 0.364625i
$$943$$ 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 $$0.133151\pi$$
−0.913779 + 0.406213i $$0.866849\pi$$
$$948$$ − 5040.00i − 0.172670i
$$949$$ 836.000 0.0285961
$$950$$ 0 0
$$951$$ 6738.00 0.229752
$$952$$ − 5040.00i − 0.171583i
$$953$$ − 18922.0i − 0.643173i −0.946880 0.321586i $$-0.895784\pi$$
0.946880 0.321586i $$-0.104216\pi$$
$$954$$ 3042.00 0.103237
$$955$$ 0 0
$$956$$ −16240.0 −0.549413
$$957$$ 35880.0i 1.21195i
$$958$$ 19680.0i 0.663708i
$$959$$ −53136.0 −1.78921
$$960$$ 0 0
$$961$$ 53153.0 1.78420
$$962$$ − 748.000i − 0.0250691i
$$963$$ − 324.000i − 0.0108419i
$$964$$ 14014.0 0.468216
$$965$$ 0 0
$$966$$ 12096.0 0.402880
$$967$$ 39656.0i 1.31877i 0.751805 + 0.659385i $$0.229183\pi$$
−0.751805 + 0.659385i $$0.770817\pi$$
$$968$$ 20595.0i 0.683831i
$$969$$ −840.000 −0.0278480
$$970$$ 0 0
$$971$$ −33228.0 −1.09818 −0.549092 0.835762i $$-0.685026\pi$$
−0.549092 + 0.835762i $$0.685026\pi$$
$$972$$ − 1701.00i − 0.0561313i
$$973$$ 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 $$-0.994924\pi$$
0.999873 0.0159473i $$-0.00507640\pi$$
$$978$$ 6204.00i 0.202845i
$$979$$ −17160.0 −0.560200
$$980$$ 0 0
$$981$$ −8730.00 −0.284126
$$982$$ 10772.0i 0.350049i
$$983$$ 13608.0i 0.441534i 0.975327 + 0.220767i $$0.0708560\pi$$
−0.975327 + 0.220767i $$0.929144\pi$$
$$984$$ 5490.00 0.177861
$$985$$ 0 0
$$986$$ −3220.00 −0.104002
$$987$$ − 18432.0i − 0.594425i
$$988$$ − 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
$$993$$ − 3996.00i − 0.127703i
$$994$$ −7872.00 −0.251192
$$995$$ 0 0
$$996$$ −25452.0 −0.809716
$$997$$ − 3234.00i − 0.102730i −0.998680 0.0513650i $$-0.983643\pi$$
0.998680 0.0513650i $$-0.0163572\pi$$
$$998$$ − 8140.00i − 0.258184i
$$999$$ 918.000 0.0290733
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 75.4.b.b.49.2 2
3.2 odd 2 225.4.b.e.199.1 2
4.3 odd 2 1200.4.f.b.49.2 2
5.2 odd 4 75.4.a.b.1.1 1
5.3 odd 4 15.4.a.a.1.1 1
5.4 even 2 inner 75.4.b.b.49.1 2
15.2 even 4 225.4.a.f.1.1 1
15.8 even 4 45.4.a.c.1.1 1
15.14 odd 2 225.4.b.e.199.2 2
20.3 even 4 240.4.a.e.1.1 1
20.7 even 4 1200.4.a.t.1.1 1
20.19 odd 2 1200.4.f.b.49.1 2
35.13 even 4 735.4.a.e.1.1 1
40.3 even 4 960.4.a.ba.1.1 1
40.13 odd 4 960.4.a.b.1.1 1
45.13 odd 12 405.4.e.g.136.1 2
45.23 even 12 405.4.e.i.136.1 2
45.38 even 12 405.4.e.i.271.1 2
45.43 odd 12 405.4.e.g.271.1 2
55.43 even 4 1815.4.a.e.1.1 1
60.23 odd 4 720.4.a.n.1.1 1
105.83 odd 4 2205.4.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 5.3 odd 4
45.4.a.c.1.1 1 15.8 even 4
75.4.a.b.1.1 1 5.2 odd 4
75.4.b.b.49.1 2 5.4 even 2 inner
75.4.b.b.49.2 2 1.1 even 1 trivial
225.4.a.f.1.1 1 15.2 even 4
225.4.b.e.199.1 2 3.2 odd 2
225.4.b.e.199.2 2 15.14 odd 2
240.4.a.e.1.1 1 20.3 even 4
405.4.e.g.136.1 2 45.13 odd 12
405.4.e.g.271.1 2 45.43 odd 12
405.4.e.i.136.1 2 45.23 even 12
405.4.e.i.271.1 2 45.38 even 12
720.4.a.n.1.1 1 60.23 odd 4
735.4.a.e.1.1 1 35.13 even 4
960.4.a.b.1.1 1 40.13 odd 4
960.4.a.ba.1.1 1 40.3 even 4
1200.4.a.t.1.1 1 20.7 even 4
1200.4.f.b.49.1 2 20.19 odd 2
1200.4.f.b.49.2 2 4.3 odd 2
1815.4.a.e.1.1 1 55.43 even 4
2205.4.a.l.1.1 1 105.83 odd 4