# Properties

 Label 825.4.c.f.199.2 Level $825$ Weight $4$ Character 825.199 Analytic conductor $48.677$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (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: $$48.6765757547$$ 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 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.4.c.f.199.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +26.0000i q^{7} +15.0000i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +26.0000i q^{7} +15.0000i q^{8} -9.00000 q^{9} +11.0000 q^{11} -21.0000i q^{12} -32.0000i q^{13} -26.0000 q^{14} +41.0000 q^{16} -74.0000i q^{17} -9.00000i q^{18} +60.0000 q^{19} +78.0000 q^{21} +11.0000i q^{22} -182.000i q^{23} +45.0000 q^{24} +32.0000 q^{26} +27.0000i q^{27} +182.000i q^{28} +90.0000 q^{29} -8.00000 q^{31} +161.000i q^{32} -33.0000i q^{33} +74.0000 q^{34} -63.0000 q^{36} +66.0000i q^{37} +60.0000i q^{38} -96.0000 q^{39} +422.000 q^{41} +78.0000i q^{42} +408.000i q^{43} +77.0000 q^{44} +182.000 q^{46} +506.000i q^{47} -123.000i q^{48} -333.000 q^{49} -222.000 q^{51} -224.000i q^{52} +348.000i q^{53} -27.0000 q^{54} -390.000 q^{56} -180.000i q^{57} +90.0000i q^{58} +200.000 q^{59} +132.000 q^{61} -8.00000i q^{62} -234.000i q^{63} +167.000 q^{64} +33.0000 q^{66} +1036.00i q^{67} -518.000i q^{68} -546.000 q^{69} +762.000 q^{71} -135.000i q^{72} -542.000i q^{73} -66.0000 q^{74} +420.000 q^{76} +286.000i q^{77} -96.0000i q^{78} +550.000 q^{79} +81.0000 q^{81} +422.000i q^{82} -132.000i q^{83} +546.000 q^{84} -408.000 q^{86} -270.000i q^{87} +165.000i q^{88} -570.000 q^{89} +832.000 q^{91} -1274.00i q^{92} +24.0000i q^{93} -506.000 q^{94} +483.000 q^{96} -14.0000i q^{97} -333.000i q^{98} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{4} + 6 q^{6} - 18 q^{9}+O(q^{10})$$ 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9 $$2 q + 14 q^{4} + 6 q^{6} - 18 q^{9} + 22 q^{11} - 52 q^{14} + 82 q^{16} + 120 q^{19} + 156 q^{21} + 90 q^{24} + 64 q^{26} + 180 q^{29} - 16 q^{31} + 148 q^{34} - 126 q^{36} - 192 q^{39} + 844 q^{41} + 154 q^{44} + 364 q^{46} - 666 q^{49} - 444 q^{51} - 54 q^{54} - 780 q^{56} + 400 q^{59} + 264 q^{61} + 334 q^{64} + 66 q^{66} - 1092 q^{69} + 1524 q^{71} - 132 q^{74} + 840 q^{76} + 1100 q^{79} + 162 q^{81} + 1092 q^{84} - 816 q^{86} - 1140 q^{89} + 1664 q^{91} - 1012 q^{94} + 966 q^{96} - 198 q^{99}+O(q^{100})$$ 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9 + 22 * q^11 - 52 * q^14 + 82 * q^16 + 120 * q^19 + 156 * q^21 + 90 * q^24 + 64 * q^26 + 180 * q^29 - 16 * q^31 + 148 * q^34 - 126 * q^36 - 192 * q^39 + 844 * q^41 + 154 * q^44 + 364 * q^46 - 666 * q^49 - 444 * q^51 - 54 * q^54 - 780 * q^56 + 400 * q^59 + 264 * q^61 + 334 * q^64 + 66 * q^66 - 1092 * q^69 + 1524 * q^71 - 132 * q^74 + 840 * q^76 + 1100 * q^79 + 162 * q^81 + 1092 * q^84 - 816 * q^86 - 1140 * q^89 + 1664 * q^91 - 1012 * q^94 + 966 * q^96 - 198 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$7$$ 26.0000i 1.40387i 0.712242 + 0.701934i $$0.247680\pi$$
−0.712242 + 0.701934i $$0.752320\pi$$
$$8$$ 15.0000i 0.662913i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ − 21.0000i − 0.505181i
$$13$$ − 32.0000i − 0.682708i −0.939935 0.341354i $$-0.889115\pi$$
0.939935 0.341354i $$-0.110885\pi$$
$$14$$ −26.0000 −0.496342
$$15$$ 0 0
$$16$$ 41.0000 0.640625
$$17$$ − 74.0000i − 1.05574i −0.849324 0.527872i $$-0.822990\pi$$
0.849324 0.527872i $$-0.177010\pi$$
$$18$$ − 9.00000i − 0.117851i
$$19$$ 60.0000 0.724471 0.362235 0.932087i $$-0.382014\pi$$
0.362235 + 0.932087i $$0.382014\pi$$
$$20$$ 0 0
$$21$$ 78.0000 0.810524
$$22$$ 11.0000i 0.106600i
$$23$$ − 182.000i − 1.64998i −0.565145 0.824992i $$-0.691180\pi$$
0.565145 0.824992i $$-0.308820\pi$$
$$24$$ 45.0000 0.382733
$$25$$ 0 0
$$26$$ 32.0000 0.241374
$$27$$ 27.0000i 0.192450i
$$28$$ 182.000i 1.22838i
$$29$$ 90.0000 0.576296 0.288148 0.957586i $$-0.406961\pi$$
0.288148 + 0.957586i $$0.406961\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −0.0463498 −0.0231749 0.999731i $$-0.507377\pi$$
−0.0231749 + 0.999731i $$0.507377\pi$$
$$32$$ 161.000i 0.889408i
$$33$$ − 33.0000i − 0.174078i
$$34$$ 74.0000 0.373262
$$35$$ 0 0
$$36$$ −63.0000 −0.291667
$$37$$ 66.0000i 0.293252i 0.989192 + 0.146626i $$0.0468414\pi$$
−0.989192 + 0.146626i $$0.953159\pi$$
$$38$$ 60.0000i 0.256139i
$$39$$ −96.0000 −0.394162
$$40$$ 0 0
$$41$$ 422.000 1.60745 0.803724 0.595003i $$-0.202849\pi$$
0.803724 + 0.595003i $$0.202849\pi$$
$$42$$ 78.0000i 0.286563i
$$43$$ 408.000i 1.44696i 0.690344 + 0.723482i $$0.257459\pi$$
−0.690344 + 0.723482i $$0.742541\pi$$
$$44$$ 77.0000 0.263822
$$45$$ 0 0
$$46$$ 182.000 0.583357
$$47$$ 506.000i 1.57038i 0.619257 + 0.785188i $$0.287434\pi$$
−0.619257 + 0.785188i $$0.712566\pi$$
$$48$$ − 123.000i − 0.369865i
$$49$$ −333.000 −0.970845
$$50$$ 0 0
$$51$$ −222.000 −0.609534
$$52$$ − 224.000i − 0.597369i
$$53$$ 348.000i 0.901915i 0.892546 + 0.450957i $$0.148917\pi$$
−0.892546 + 0.450957i $$0.851083\pi$$
$$54$$ −27.0000 −0.0680414
$$55$$ 0 0
$$56$$ −390.000 −0.930642
$$57$$ − 180.000i − 0.418273i
$$58$$ 90.0000i 0.203751i
$$59$$ 200.000 0.441318 0.220659 0.975351i $$-0.429179\pi$$
0.220659 + 0.975351i $$0.429179\pi$$
$$60$$ 0 0
$$61$$ 132.000 0.277063 0.138532 0.990358i $$-0.455762\pi$$
0.138532 + 0.990358i $$0.455762\pi$$
$$62$$ − 8.00000i − 0.0163871i
$$63$$ − 234.000i − 0.467956i
$$64$$ 167.000 0.326172
$$65$$ 0 0
$$66$$ 33.0000 0.0615457
$$67$$ 1036.00i 1.88907i 0.328414 + 0.944534i $$0.393486\pi$$
−0.328414 + 0.944534i $$0.606514\pi$$
$$68$$ − 518.000i − 0.923775i
$$69$$ −546.000 −0.952618
$$70$$ 0 0
$$71$$ 762.000 1.27370 0.636850 0.770987i $$-0.280237\pi$$
0.636850 + 0.770987i $$0.280237\pi$$
$$72$$ − 135.000i − 0.220971i
$$73$$ − 542.000i − 0.868990i −0.900674 0.434495i $$-0.856927\pi$$
0.900674 0.434495i $$-0.143073\pi$$
$$74$$ −66.0000 −0.103680
$$75$$ 0 0
$$76$$ 420.000 0.633912
$$77$$ 286.000i 0.423282i
$$78$$ − 96.0000i − 0.139357i
$$79$$ 550.000 0.783289 0.391645 0.920117i $$-0.371906\pi$$
0.391645 + 0.920117i $$0.371906\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 422.000i 0.568318i
$$83$$ − 132.000i − 0.174565i −0.996184 0.0872824i $$-0.972182\pi$$
0.996184 0.0872824i $$-0.0278183\pi$$
$$84$$ 546.000 0.709208
$$85$$ 0 0
$$86$$ −408.000 −0.511579
$$87$$ − 270.000i − 0.332725i
$$88$$ 165.000i 0.199876i
$$89$$ −570.000 −0.678875 −0.339438 0.940629i $$-0.610237\pi$$
−0.339438 + 0.940629i $$0.610237\pi$$
$$90$$ 0 0
$$91$$ 832.000 0.958432
$$92$$ − 1274.00i − 1.44374i
$$93$$ 24.0000i 0.0267600i
$$94$$ −506.000 −0.555212
$$95$$ 0 0
$$96$$ 483.000 0.513500
$$97$$ − 14.0000i − 0.0146545i −0.999973 0.00732724i $$-0.997668\pi$$
0.999973 0.00732724i $$-0.00233235\pi$$
$$98$$ − 333.000i − 0.343246i
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ 1702.00 1.67679 0.838393 0.545067i $$-0.183496\pi$$
0.838393 + 0.545067i $$0.183496\pi$$
$$102$$ − 222.000i − 0.215503i
$$103$$ − 1132.00i − 1.08291i −0.840731 0.541453i $$-0.817874\pi$$
0.840731 0.541453i $$-0.182126\pi$$
$$104$$ 480.000 0.452576
$$105$$ 0 0
$$106$$ −348.000 −0.318875
$$107$$ − 564.000i − 0.509570i −0.966998 0.254785i $$-0.917995\pi$$
0.966998 0.254785i $$-0.0820046\pi$$
$$108$$ 189.000i 0.168394i
$$109$$ 320.000 0.281197 0.140598 0.990067i $$-0.455097\pi$$
0.140598 + 0.990067i $$0.455097\pi$$
$$110$$ 0 0
$$111$$ 198.000 0.169309
$$112$$ 1066.00i 0.899353i
$$113$$ − 2142.00i − 1.78321i −0.452817 0.891604i $$-0.649581\pi$$
0.452817 0.891604i $$-0.350419\pi$$
$$114$$ 180.000 0.147882
$$115$$ 0 0
$$116$$ 630.000 0.504259
$$117$$ 288.000i 0.227569i
$$118$$ 200.000i 0.156030i
$$119$$ 1924.00 1.48212
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 132.000i 0.0979567i
$$123$$ − 1266.00i − 0.928060i
$$124$$ −56.0000 −0.0405560
$$125$$ 0 0
$$126$$ 234.000 0.165447
$$127$$ 1606.00i 1.12212i 0.827775 + 0.561061i $$0.189607\pi$$
−0.827775 + 0.561061i $$0.810393\pi$$
$$128$$ 1455.00i 1.00473i
$$129$$ 1224.00 0.835405
$$130$$ 0 0
$$131$$ −1908.00 −1.27254 −0.636270 0.771466i $$-0.719524\pi$$
−0.636270 + 0.771466i $$0.719524\pi$$
$$132$$ − 231.000i − 0.152318i
$$133$$ 1560.00i 1.01706i
$$134$$ −1036.00 −0.667886
$$135$$ 0 0
$$136$$ 1110.00 0.699866
$$137$$ 2186.00i 1.36323i 0.731711 + 0.681615i $$0.238722\pi$$
−0.731711 + 0.681615i $$0.761278\pi$$
$$138$$ − 546.000i − 0.336801i
$$139$$ −2740.00 −1.67197 −0.835985 0.548753i $$-0.815103\pi$$
−0.835985 + 0.548753i $$0.815103\pi$$
$$140$$ 0 0
$$141$$ 1518.00 0.906657
$$142$$ 762.000i 0.450321i
$$143$$ − 352.000i − 0.205844i
$$144$$ −369.000 −0.213542
$$145$$ 0 0
$$146$$ 542.000 0.307235
$$147$$ 999.000i 0.560518i
$$148$$ 462.000i 0.256596i
$$149$$ 1310.00 0.720264 0.360132 0.932901i $$-0.382732\pi$$
0.360132 + 0.932901i $$0.382732\pi$$
$$150$$ 0 0
$$151$$ −1198.00 −0.645641 −0.322821 0.946460i $$-0.604631\pi$$
−0.322821 + 0.946460i $$0.604631\pi$$
$$152$$ 900.000i 0.480261i
$$153$$ 666.000i 0.351914i
$$154$$ −286.000 −0.149653
$$155$$ 0 0
$$156$$ −672.000 −0.344891
$$157$$ − 2114.00i − 1.07462i −0.843384 0.537311i $$-0.819440\pi$$
0.843384 0.537311i $$-0.180560\pi$$
$$158$$ 550.000i 0.276934i
$$159$$ 1044.00 0.520721
$$160$$ 0 0
$$161$$ 4732.00 2.31636
$$162$$ 81.0000i 0.0392837i
$$163$$ 3868.00i 1.85868i 0.369223 + 0.929341i $$0.379624\pi$$
−0.369223 + 0.929341i $$0.620376\pi$$
$$164$$ 2954.00 1.40652
$$165$$ 0 0
$$166$$ 132.000 0.0617180
$$167$$ − 2004.00i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 1170.00i 0.537306i
$$169$$ 1173.00 0.533910
$$170$$ 0 0
$$171$$ −540.000 −0.241490
$$172$$ 2856.00i 1.26609i
$$173$$ 678.000i 0.297962i 0.988840 + 0.148981i $$0.0475993\pi$$
−0.988840 + 0.148981i $$0.952401\pi$$
$$174$$ 270.000 0.117636
$$175$$ 0 0
$$176$$ 451.000 0.193156
$$177$$ − 600.000i − 0.254795i
$$178$$ − 570.000i − 0.240019i
$$179$$ 1680.00 0.701503 0.350752 0.936469i $$-0.385926\pi$$
0.350752 + 0.936469i $$0.385926\pi$$
$$180$$ 0 0
$$181$$ −4358.00 −1.78966 −0.894828 0.446412i $$-0.852702\pi$$
−0.894828 + 0.446412i $$0.852702\pi$$
$$182$$ 832.000i 0.338857i
$$183$$ − 396.000i − 0.159963i
$$184$$ 2730.00 1.09379
$$185$$ 0 0
$$186$$ −24.0000 −0.00946110
$$187$$ − 814.000i − 0.318319i
$$188$$ 3542.00i 1.37408i
$$189$$ −702.000 −0.270175
$$190$$ 0 0
$$191$$ −1778.00 −0.673568 −0.336784 0.941582i $$-0.609339\pi$$
−0.336784 + 0.941582i $$0.609339\pi$$
$$192$$ − 501.000i − 0.188315i
$$193$$ − 3962.00i − 1.47767i −0.673884 0.738837i $$-0.735375\pi$$
0.673884 0.738837i $$-0.264625\pi$$
$$194$$ 14.0000 0.00518114
$$195$$ 0 0
$$196$$ −2331.00 −0.849490
$$197$$ − 374.000i − 0.135261i −0.997710 0.0676304i $$-0.978456\pi$$
0.997710 0.0676304i $$-0.0215439\pi$$
$$198$$ − 99.0000i − 0.0355335i
$$199$$ −2100.00 −0.748066 −0.374033 0.927415i $$-0.622025\pi$$
−0.374033 + 0.927415i $$0.622025\pi$$
$$200$$ 0 0
$$201$$ 3108.00 1.09065
$$202$$ 1702.00i 0.592833i
$$203$$ 2340.00i 0.809043i
$$204$$ −1554.00 −0.533342
$$205$$ 0 0
$$206$$ 1132.00 0.382865
$$207$$ 1638.00i 0.549995i
$$208$$ − 1312.00i − 0.437360i
$$209$$ 660.000 0.218436
$$210$$ 0 0
$$211$$ 2232.00 0.728233 0.364117 0.931353i $$-0.381371\pi$$
0.364117 + 0.931353i $$0.381371\pi$$
$$212$$ 2436.00i 0.789175i
$$213$$ − 2286.00i − 0.735372i
$$214$$ 564.000 0.180160
$$215$$ 0 0
$$216$$ −405.000 −0.127578
$$217$$ − 208.000i − 0.0650689i
$$218$$ 320.000i 0.0994180i
$$219$$ −1626.00 −0.501712
$$220$$ 0 0
$$221$$ −2368.00 −0.720764
$$222$$ 198.000i 0.0598599i
$$223$$ 2128.00i 0.639020i 0.947583 + 0.319510i $$0.103518\pi$$
−0.947583 + 0.319510i $$0.896482\pi$$
$$224$$ −4186.00 −1.24861
$$225$$ 0 0
$$226$$ 2142.00 0.630459
$$227$$ − 2964.00i − 0.866641i −0.901240 0.433321i $$-0.857342\pi$$
0.901240 0.433321i $$-0.142658\pi$$
$$228$$ − 1260.00i − 0.365989i
$$229$$ 2550.00 0.735846 0.367923 0.929856i $$-0.380069\pi$$
0.367923 + 0.929856i $$0.380069\pi$$
$$230$$ 0 0
$$231$$ 858.000 0.244382
$$232$$ 1350.00i 0.382034i
$$233$$ − 3042.00i − 0.855314i −0.903941 0.427657i $$-0.859339\pi$$
0.903941 0.427657i $$-0.140661\pi$$
$$234$$ −288.000 −0.0804579
$$235$$ 0 0
$$236$$ 1400.00 0.386154
$$237$$ − 1650.00i − 0.452232i
$$238$$ 1924.00i 0.524010i
$$239$$ −2700.00 −0.730747 −0.365373 0.930861i $$-0.619059\pi$$
−0.365373 + 0.930861i $$0.619059\pi$$
$$240$$ 0 0
$$241$$ −578.000 −0.154491 −0.0772453 0.997012i $$-0.524612\pi$$
−0.0772453 + 0.997012i $$0.524612\pi$$
$$242$$ 121.000i 0.0321412i
$$243$$ − 243.000i − 0.0641500i
$$244$$ 924.000 0.242430
$$245$$ 0 0
$$246$$ 1266.00 0.328119
$$247$$ − 1920.00i − 0.494602i
$$248$$ − 120.000i − 0.0307258i
$$249$$ −396.000 −0.100785
$$250$$ 0 0
$$251$$ 3752.00 0.943522 0.471761 0.881726i $$-0.343618\pi$$
0.471761 + 0.881726i $$0.343618\pi$$
$$252$$ − 1638.00i − 0.409462i
$$253$$ − 2002.00i − 0.497489i
$$254$$ −1606.00 −0.396730
$$255$$ 0 0
$$256$$ −119.000 −0.0290527
$$257$$ − 674.000i − 0.163591i −0.996649 0.0817957i $$-0.973935\pi$$
0.996649 0.0817957i $$-0.0260655\pi$$
$$258$$ 1224.00i 0.295360i
$$259$$ −1716.00 −0.411687
$$260$$ 0 0
$$261$$ −810.000 −0.192099
$$262$$ − 1908.00i − 0.449911i
$$263$$ − 4352.00i − 1.02036i −0.860066 0.510182i $$-0.829578\pi$$
0.860066 0.510182i $$-0.170422\pi$$
$$264$$ 495.000 0.115398
$$265$$ 0 0
$$266$$ −1560.00 −0.359585
$$267$$ 1710.00i 0.391949i
$$268$$ 7252.00i 1.65293i
$$269$$ −500.000 −0.113329 −0.0566646 0.998393i $$-0.518047\pi$$
−0.0566646 + 0.998393i $$0.518047\pi$$
$$270$$ 0 0
$$271$$ −6538.00 −1.46552 −0.732759 0.680489i $$-0.761768\pi$$
−0.732759 + 0.680489i $$0.761768\pi$$
$$272$$ − 3034.00i − 0.676336i
$$273$$ − 2496.00i − 0.553351i
$$274$$ −2186.00 −0.481975
$$275$$ 0 0
$$276$$ −3822.00 −0.833541
$$277$$ − 124.000i − 0.0268969i −0.999910 0.0134484i $$-0.995719\pi$$
0.999910 0.0134484i $$-0.00428090\pi$$
$$278$$ − 2740.00i − 0.591131i
$$279$$ 72.0000 0.0154499
$$280$$ 0 0
$$281$$ 3642.00 0.773180 0.386590 0.922252i $$-0.373653\pi$$
0.386590 + 0.922252i $$0.373653\pi$$
$$282$$ 1518.00i 0.320552i
$$283$$ 4648.00i 0.976307i 0.872758 + 0.488154i $$0.162329\pi$$
−0.872758 + 0.488154i $$0.837671\pi$$
$$284$$ 5334.00 1.11449
$$285$$ 0 0
$$286$$ 352.000 0.0727769
$$287$$ 10972.0i 2.25664i
$$288$$ − 1449.00i − 0.296469i
$$289$$ −563.000 −0.114594
$$290$$ 0 0
$$291$$ −42.0000 −0.00846077
$$292$$ − 3794.00i − 0.760367i
$$293$$ − 3102.00i − 0.618501i −0.950981 0.309250i $$-0.899922\pi$$
0.950981 0.309250i $$-0.100078\pi$$
$$294$$ −999.000 −0.198173
$$295$$ 0 0
$$296$$ −990.000 −0.194401
$$297$$ 297.000i 0.0580259i
$$298$$ 1310.00i 0.254652i
$$299$$ −5824.00 −1.12646
$$300$$ 0 0
$$301$$ −10608.0 −2.03135
$$302$$ − 1198.00i − 0.228269i
$$303$$ − 5106.00i − 0.968093i
$$304$$ 2460.00 0.464114
$$305$$ 0 0
$$306$$ −666.000 −0.124421
$$307$$ − 1244.00i − 0.231267i −0.993292 0.115633i $$-0.963110\pi$$
0.993292 0.115633i $$-0.0368897\pi$$
$$308$$ 2002.00i 0.370372i
$$309$$ −3396.00 −0.625216
$$310$$ 0 0
$$311$$ 2082.00 0.379612 0.189806 0.981822i $$-0.439214\pi$$
0.189806 + 0.981822i $$0.439214\pi$$
$$312$$ − 1440.00i − 0.261295i
$$313$$ 2378.00i 0.429433i 0.976676 + 0.214716i $$0.0688827\pi$$
−0.976676 + 0.214716i $$0.931117\pi$$
$$314$$ 2114.00 0.379936
$$315$$ 0 0
$$316$$ 3850.00 0.685378
$$317$$ 496.000i 0.0878806i 0.999034 + 0.0439403i $$0.0139911\pi$$
−0.999034 + 0.0439403i $$0.986009\pi$$
$$318$$ 1044.00i 0.184103i
$$319$$ 990.000 0.173760
$$320$$ 0 0
$$321$$ −1692.00 −0.294200
$$322$$ 4732.00i 0.818957i
$$323$$ − 4440.00i − 0.764855i
$$324$$ 567.000 0.0972222
$$325$$ 0 0
$$326$$ −3868.00 −0.657143
$$327$$ − 960.000i − 0.162349i
$$328$$ 6330.00i 1.06560i
$$329$$ −13156.0 −2.20460
$$330$$ 0 0
$$331$$ −2708.00 −0.449683 −0.224842 0.974395i $$-0.572186\pi$$
−0.224842 + 0.974395i $$0.572186\pi$$
$$332$$ − 924.000i − 0.152744i
$$333$$ − 594.000i − 0.0977507i
$$334$$ 2004.00 0.328305
$$335$$ 0 0
$$336$$ 3198.00 0.519242
$$337$$ − 4034.00i − 0.652065i −0.945359 0.326033i $$-0.894288\pi$$
0.945359 0.326033i $$-0.105712\pi$$
$$338$$ 1173.00i 0.188766i
$$339$$ −6426.00 −1.02954
$$340$$ 0 0
$$341$$ −88.0000 −0.0139750
$$342$$ − 540.000i − 0.0853797i
$$343$$ 260.000i 0.0409291i
$$344$$ −6120.00 −0.959210
$$345$$ 0 0
$$346$$ −678.000 −0.105345
$$347$$ − 11084.0i − 1.71476i −0.514687 0.857378i $$-0.672092\pi$$
0.514687 0.857378i $$-0.327908\pi$$
$$348$$ − 1890.00i − 0.291134i
$$349$$ 3120.00 0.478538 0.239269 0.970953i $$-0.423092\pi$$
0.239269 + 0.970953i $$0.423092\pi$$
$$350$$ 0 0
$$351$$ 864.000 0.131387
$$352$$ 1771.00i 0.268167i
$$353$$ − 5622.00i − 0.847674i −0.905739 0.423837i $$-0.860683\pi$$
0.905739 0.423837i $$-0.139317\pi$$
$$354$$ 600.000 0.0900837
$$355$$ 0 0
$$356$$ −3990.00 −0.594016
$$357$$ − 5772.00i − 0.855705i
$$358$$ 1680.00i 0.248019i
$$359$$ 8500.00 1.24962 0.624809 0.780778i $$-0.285177\pi$$
0.624809 + 0.780778i $$0.285177\pi$$
$$360$$ 0 0
$$361$$ −3259.00 −0.475142
$$362$$ − 4358.00i − 0.632739i
$$363$$ − 363.000i − 0.0524864i
$$364$$ 5824.00 0.838628
$$365$$ 0 0
$$366$$ 396.000 0.0565553
$$367$$ − 7144.00i − 1.01611i −0.861324 0.508057i $$-0.830364\pi$$
0.861324 0.508057i $$-0.169636\pi$$
$$368$$ − 7462.00i − 1.05702i
$$369$$ −3798.00 −0.535816
$$370$$ 0 0
$$371$$ −9048.00 −1.26617
$$372$$ 168.000i 0.0234150i
$$373$$ − 632.000i − 0.0877312i −0.999037 0.0438656i $$-0.986033\pi$$
0.999037 0.0438656i $$-0.0139673\pi$$
$$374$$ 814.000 0.112543
$$375$$ 0 0
$$376$$ −7590.00 −1.04102
$$377$$ − 2880.00i − 0.393442i
$$378$$ − 702.000i − 0.0955211i
$$379$$ 4220.00 0.571944 0.285972 0.958238i $$-0.407684\pi$$
0.285972 + 0.958238i $$0.407684\pi$$
$$380$$ 0 0
$$381$$ 4818.00 0.647857
$$382$$ − 1778.00i − 0.238142i
$$383$$ 8458.00i 1.12842i 0.825632 + 0.564208i $$0.190819\pi$$
−0.825632 + 0.564208i $$0.809181\pi$$
$$384$$ 4365.00 0.580079
$$385$$ 0 0
$$386$$ 3962.00 0.522437
$$387$$ − 3672.00i − 0.482321i
$$388$$ − 98.0000i − 0.0128227i
$$389$$ −1740.00 −0.226790 −0.113395 0.993550i $$-0.536173\pi$$
−0.113395 + 0.993550i $$0.536173\pi$$
$$390$$ 0 0
$$391$$ −13468.0 −1.74196
$$392$$ − 4995.00i − 0.643586i
$$393$$ 5724.00i 0.734701i
$$394$$ 374.000 0.0478219
$$395$$ 0 0
$$396$$ −693.000 −0.0879408
$$397$$ 5126.00i 0.648027i 0.946053 + 0.324013i $$0.105032\pi$$
−0.946053 + 0.324013i $$0.894968\pi$$
$$398$$ − 2100.00i − 0.264481i
$$399$$ 4680.00 0.587201
$$400$$ 0 0
$$401$$ −3098.00 −0.385802 −0.192901 0.981218i $$-0.561790\pi$$
−0.192901 + 0.981218i $$0.561790\pi$$
$$402$$ 3108.00i 0.385604i
$$403$$ 256.000i 0.0316433i
$$404$$ 11914.0 1.46719
$$405$$ 0 0
$$406$$ −2340.00 −0.286040
$$407$$ 726.000i 0.0884189i
$$408$$ − 3330.00i − 0.404068i
$$409$$ −6390.00 −0.772531 −0.386265 0.922388i $$-0.626235\pi$$
−0.386265 + 0.922388i $$0.626235\pi$$
$$410$$ 0 0
$$411$$ 6558.00 0.787062
$$412$$ − 7924.00i − 0.947542i
$$413$$ 5200.00i 0.619553i
$$414$$ −1638.00 −0.194452
$$415$$ 0 0
$$416$$ 5152.00 0.607206
$$417$$ 8220.00i 0.965312i
$$418$$ 660.000i 0.0772288i
$$419$$ −9760.00 −1.13796 −0.568982 0.822350i $$-0.692663\pi$$
−0.568982 + 0.822350i $$0.692663\pi$$
$$420$$ 0 0
$$421$$ −5138.00 −0.594800 −0.297400 0.954753i $$-0.596119\pi$$
−0.297400 + 0.954753i $$0.596119\pi$$
$$422$$ 2232.00i 0.257469i
$$423$$ − 4554.00i − 0.523459i
$$424$$ −5220.00 −0.597891
$$425$$ 0 0
$$426$$ 2286.00 0.259993
$$427$$ 3432.00i 0.388960i
$$428$$ − 3948.00i − 0.445873i
$$429$$ −1056.00 −0.118844
$$430$$ 0 0
$$431$$ −7008.00 −0.783210 −0.391605 0.920133i $$-0.628080\pi$$
−0.391605 + 0.920133i $$0.628080\pi$$
$$432$$ 1107.00i 0.123288i
$$433$$ 5578.00i 0.619080i 0.950886 + 0.309540i $$0.100175\pi$$
−0.950886 + 0.309540i $$0.899825\pi$$
$$434$$ 208.000 0.0230053
$$435$$ 0 0
$$436$$ 2240.00 0.246047
$$437$$ − 10920.0i − 1.19536i
$$438$$ − 1626.00i − 0.177382i
$$439$$ 10430.0 1.13393 0.566967 0.823741i $$-0.308117\pi$$
0.566967 + 0.823741i $$0.308117\pi$$
$$440$$ 0 0
$$441$$ 2997.00 0.323615
$$442$$ − 2368.00i − 0.254829i
$$443$$ − 4432.00i − 0.475329i −0.971347 0.237664i $$-0.923618\pi$$
0.971347 0.237664i $$-0.0763819\pi$$
$$444$$ 1386.00 0.148146
$$445$$ 0 0
$$446$$ −2128.00 −0.225928
$$447$$ − 3930.00i − 0.415845i
$$448$$ 4342.00i 0.457902i
$$449$$ 6290.00 0.661121 0.330561 0.943785i $$-0.392762\pi$$
0.330561 + 0.943785i $$0.392762\pi$$
$$450$$ 0 0
$$451$$ 4642.00 0.484664
$$452$$ − 14994.0i − 1.56031i
$$453$$ 3594.00i 0.372761i
$$454$$ 2964.00 0.306404
$$455$$ 0 0
$$456$$ 2700.00 0.277279
$$457$$ − 3054.00i − 0.312604i −0.987709 0.156302i $$-0.950043\pi$$
0.987709 0.156302i $$-0.0499573\pi$$
$$458$$ 2550.00i 0.260161i
$$459$$ 1998.00 0.203178
$$460$$ 0 0
$$461$$ 12882.0 1.30146 0.650732 0.759308i $$-0.274462\pi$$
0.650732 + 0.759308i $$0.274462\pi$$
$$462$$ 858.000i 0.0864021i
$$463$$ 6148.00i 0.617110i 0.951207 + 0.308555i $$0.0998453\pi$$
−0.951207 + 0.308555i $$0.900155\pi$$
$$464$$ 3690.00 0.369190
$$465$$ 0 0
$$466$$ 3042.00 0.302399
$$467$$ − 5124.00i − 0.507731i −0.967240 0.253866i $$-0.918298\pi$$
0.967240 0.253866i $$-0.0817021\pi$$
$$468$$ 2016.00i 0.199123i
$$469$$ −26936.0 −2.65200
$$470$$ 0 0
$$471$$ −6342.00 −0.620433
$$472$$ 3000.00i 0.292555i
$$473$$ 4488.00i 0.436276i
$$474$$ 1650.00 0.159888
$$475$$ 0 0
$$476$$ 13468.0 1.29686
$$477$$ − 3132.00i − 0.300638i
$$478$$ − 2700.00i − 0.258358i
$$479$$ 16520.0 1.57582 0.787910 0.615790i $$-0.211163\pi$$
0.787910 + 0.615790i $$0.211163\pi$$
$$480$$ 0 0
$$481$$ 2112.00 0.200206
$$482$$ − 578.000i − 0.0546207i
$$483$$ − 14196.0i − 1.33735i
$$484$$ 847.000 0.0795455
$$485$$ 0 0
$$486$$ 243.000 0.0226805
$$487$$ − 524.000i − 0.0487571i −0.999703 0.0243785i $$-0.992239\pi$$
0.999703 0.0243785i $$-0.00776070\pi$$
$$488$$ 1980.00i 0.183669i
$$489$$ 11604.0 1.07311
$$490$$ 0 0
$$491$$ −15028.0 −1.38127 −0.690636 0.723203i $$-0.742669\pi$$
−0.690636 + 0.723203i $$0.742669\pi$$
$$492$$ − 8862.00i − 0.812052i
$$493$$ − 6660.00i − 0.608421i
$$494$$ 1920.00 0.174868
$$495$$ 0 0
$$496$$ −328.000 −0.0296928
$$497$$ 19812.0i 1.78811i
$$498$$ − 396.000i − 0.0356329i
$$499$$ −9020.00 −0.809200 −0.404600 0.914494i $$-0.632589\pi$$
−0.404600 + 0.914494i $$0.632589\pi$$
$$500$$ 0 0
$$501$$ −6012.00 −0.536120
$$502$$ 3752.00i 0.333586i
$$503$$ − 14812.0i − 1.31299i −0.754330 0.656495i $$-0.772038\pi$$
0.754330 0.656495i $$-0.227962\pi$$
$$504$$ 3510.00 0.310214
$$505$$ 0 0
$$506$$ 2002.00 0.175889
$$507$$ − 3519.00i − 0.308253i
$$508$$ 11242.0i 0.981856i
$$509$$ −12660.0 −1.10245 −0.551223 0.834358i $$-0.685839\pi$$
−0.551223 + 0.834358i $$0.685839\pi$$
$$510$$ 0 0
$$511$$ 14092.0 1.21995
$$512$$ 11521.0i 0.994455i
$$513$$ 1620.00i 0.139424i
$$514$$ 674.000 0.0578383
$$515$$ 0 0
$$516$$ 8568.00 0.730979
$$517$$ 5566.00i 0.473486i
$$518$$ − 1716.00i − 0.145553i
$$519$$ 2034.00 0.172028
$$520$$ 0 0
$$521$$ −3738.00 −0.314328 −0.157164 0.987573i $$-0.550235\pi$$
−0.157164 + 0.987573i $$0.550235\pi$$
$$522$$ − 810.000i − 0.0679171i
$$523$$ − 6352.00i − 0.531078i −0.964100 0.265539i $$-0.914450\pi$$
0.964100 0.265539i $$-0.0855498\pi$$
$$524$$ −13356.0 −1.11347
$$525$$ 0 0
$$526$$ 4352.00 0.360753
$$527$$ 592.000i 0.0489334i
$$528$$ − 1353.00i − 0.111518i
$$529$$ −20957.0 −1.72245
$$530$$ 0 0
$$531$$ −1800.00 −0.147106
$$532$$ 10920.0i 0.889929i
$$533$$ − 13504.0i − 1.09742i
$$534$$ −1710.00 −0.138575
$$535$$ 0 0
$$536$$ −15540.0 −1.25229
$$537$$ − 5040.00i − 0.405013i
$$538$$ − 500.000i − 0.0400679i
$$539$$ −3663.00 −0.292721
$$540$$ 0 0
$$541$$ −24728.0 −1.96514 −0.982569 0.185898i $$-0.940481\pi$$
−0.982569 + 0.185898i $$0.940481\pi$$
$$542$$ − 6538.00i − 0.518139i
$$543$$ 13074.0i 1.03326i
$$544$$ 11914.0 0.938986
$$545$$ 0 0
$$546$$ 2496.00 0.195639
$$547$$ 22756.0i 1.77875i 0.457178 + 0.889375i $$0.348860\pi$$
−0.457178 + 0.889375i $$0.651140\pi$$
$$548$$ 15302.0i 1.19283i
$$549$$ −1188.00 −0.0923545
$$550$$ 0 0
$$551$$ 5400.00 0.417509
$$552$$ − 8190.00i − 0.631503i
$$553$$ 14300.0i 1.09963i
$$554$$ 124.000 0.00950949
$$555$$ 0 0
$$556$$ −19180.0 −1.46297
$$557$$ 9526.00i 0.724649i 0.932052 + 0.362325i $$0.118017\pi$$
−0.932052 + 0.362325i $$0.881983\pi$$
$$558$$ 72.0000i 0.00546237i
$$559$$ 13056.0 0.987853
$$560$$ 0 0
$$561$$ −2442.00 −0.183781
$$562$$ 3642.00i 0.273360i
$$563$$ 12068.0i 0.903385i 0.892174 + 0.451692i $$0.149180\pi$$
−0.892174 + 0.451692i $$0.850820\pi$$
$$564$$ 10626.0 0.793325
$$565$$ 0 0
$$566$$ −4648.00 −0.345177
$$567$$ 2106.00i 0.155985i
$$568$$ 11430.0i 0.844352i
$$569$$ −15090.0 −1.11179 −0.555893 0.831254i $$-0.687623\pi$$
−0.555893 + 0.831254i $$0.687623\pi$$
$$570$$ 0 0
$$571$$ 4412.00 0.323356 0.161678 0.986844i $$-0.448309\pi$$
0.161678 + 0.986844i $$0.448309\pi$$
$$572$$ − 2464.00i − 0.180114i
$$573$$ 5334.00i 0.388885i
$$574$$ −10972.0 −0.797844
$$575$$ 0 0
$$576$$ −1503.00 −0.108724
$$577$$ 3906.00i 0.281818i 0.990023 + 0.140909i $$0.0450025\pi$$
−0.990023 + 0.140909i $$0.954998\pi$$
$$578$$ − 563.000i − 0.0405151i
$$579$$ −11886.0 −0.853135
$$580$$ 0 0
$$581$$ 3432.00 0.245066
$$582$$ − 42.0000i − 0.00299133i
$$583$$ 3828.00i 0.271937i
$$584$$ 8130.00 0.576065
$$585$$ 0 0
$$586$$ 3102.00 0.218673
$$587$$ 12016.0i 0.844895i 0.906387 + 0.422448i $$0.138829\pi$$
−0.906387 + 0.422448i $$0.861171\pi$$
$$588$$ 6993.00i 0.490453i
$$589$$ −480.000 −0.0335790
$$590$$ 0 0
$$591$$ −1122.00 −0.0780929
$$592$$ 2706.00i 0.187865i
$$593$$ − 11342.0i − 0.785430i −0.919660 0.392715i $$-0.871536\pi$$
0.919660 0.392715i $$-0.128464\pi$$
$$594$$ −297.000 −0.0205152
$$595$$ 0 0
$$596$$ 9170.00 0.630231
$$597$$ 6300.00i 0.431896i
$$598$$ − 5824.00i − 0.398263i
$$599$$ −20690.0 −1.41130 −0.705651 0.708559i $$-0.749346\pi$$
−0.705651 + 0.708559i $$0.749346\pi$$
$$600$$ 0 0
$$601$$ −598.000 −0.0405872 −0.0202936 0.999794i $$-0.506460\pi$$
−0.0202936 + 0.999794i $$0.506460\pi$$
$$602$$ − 10608.0i − 0.718189i
$$603$$ − 9324.00i − 0.629689i
$$604$$ −8386.00 −0.564936
$$605$$ 0 0
$$606$$ 5106.00 0.342272
$$607$$ 166.000i 0.0111001i 0.999985 + 0.00555003i $$0.00176664\pi$$
−0.999985 + 0.00555003i $$0.998233\pi$$
$$608$$ 9660.00i 0.644350i
$$609$$ 7020.00 0.467101
$$610$$ 0 0
$$611$$ 16192.0 1.07211
$$612$$ 4662.00i 0.307925i
$$613$$ 20108.0i 1.32488i 0.749113 + 0.662442i $$0.230480\pi$$
−0.749113 + 0.662442i $$0.769520\pi$$
$$614$$ 1244.00 0.0817651
$$615$$ 0 0
$$616$$ −4290.00 −0.280599
$$617$$ 2286.00i 0.149159i 0.997215 + 0.0745793i $$0.0237614\pi$$
−0.997215 + 0.0745793i $$0.976239\pi$$
$$618$$ − 3396.00i − 0.221047i
$$619$$ 25660.0 1.66618 0.833088 0.553141i $$-0.186571\pi$$
0.833088 + 0.553141i $$0.186571\pi$$
$$620$$ 0 0
$$621$$ 4914.00 0.317539
$$622$$ 2082.00i 0.134213i
$$623$$ − 14820.0i − 0.953051i
$$624$$ −3936.00 −0.252510
$$625$$ 0 0
$$626$$ −2378.00 −0.151827
$$627$$ − 1980.00i − 0.126114i
$$628$$ − 14798.0i − 0.940294i
$$629$$ 4884.00 0.309599
$$630$$ 0 0
$$631$$ −11408.0 −0.719723 −0.359862 0.933006i $$-0.617176\pi$$
−0.359862 + 0.933006i $$0.617176\pi$$
$$632$$ 8250.00i 0.519252i
$$633$$ − 6696.00i − 0.420446i
$$634$$ −496.000 −0.0310705
$$635$$ 0 0
$$636$$ 7308.00 0.455631
$$637$$ 10656.0i 0.662804i
$$638$$ 990.000i 0.0614333i
$$639$$ −6858.00 −0.424567
$$640$$ 0 0
$$641$$ −3378.00 −0.208148 −0.104074 0.994570i $$-0.533188\pi$$
−0.104074 + 0.994570i $$0.533188\pi$$
$$642$$ − 1692.00i − 0.104015i
$$643$$ − 11212.0i − 0.687649i −0.939034 0.343824i $$-0.888278\pi$$
0.939034 0.343824i $$-0.111722\pi$$
$$644$$ 33124.0 2.02681
$$645$$ 0 0
$$646$$ 4440.00 0.270417
$$647$$ 86.0000i 0.00522567i 0.999997 + 0.00261284i $$0.000831692\pi$$
−0.999997 + 0.00261284i $$0.999168\pi$$
$$648$$ 1215.00i 0.0736570i
$$649$$ 2200.00 0.133062
$$650$$ 0 0
$$651$$ −624.000 −0.0375676
$$652$$ 27076.0i 1.62635i
$$653$$ − 4432.00i − 0.265601i −0.991143 0.132801i $$-0.957603\pi$$
0.991143 0.132801i $$-0.0423970\pi$$
$$654$$ 960.000 0.0573990
$$655$$ 0 0
$$656$$ 17302.0 1.02977
$$657$$ 4878.00i 0.289663i
$$658$$ − 13156.0i − 0.779444i
$$659$$ −4580.00 −0.270731 −0.135365 0.990796i $$-0.543221\pi$$
−0.135365 + 0.990796i $$0.543221\pi$$
$$660$$ 0 0
$$661$$ 4282.00 0.251967 0.125984 0.992032i $$-0.459791\pi$$
0.125984 + 0.992032i $$0.459791\pi$$
$$662$$ − 2708.00i − 0.158987i
$$663$$ 7104.00i 0.416133i
$$664$$ 1980.00 0.115721
$$665$$ 0 0
$$666$$ 594.000 0.0345601
$$667$$ − 16380.0i − 0.950879i
$$668$$ − 14028.0i − 0.812514i
$$669$$ 6384.00 0.368938
$$670$$ 0 0
$$671$$ 1452.00 0.0835378
$$672$$ 12558.0i 0.720886i
$$673$$ 8438.00i 0.483300i 0.970363 + 0.241650i $$0.0776886\pi$$
−0.970363 + 0.241650i $$0.922311\pi$$
$$674$$ 4034.00 0.230540
$$675$$ 0 0
$$676$$ 8211.00 0.467171
$$677$$ − 34494.0i − 1.95822i −0.203341 0.979108i $$-0.565180\pi$$
0.203341 0.979108i $$-0.434820\pi$$
$$678$$ − 6426.00i − 0.363996i
$$679$$ 364.000 0.0205730
$$680$$ 0 0
$$681$$ −8892.00 −0.500356
$$682$$ − 88.0000i − 0.00494090i
$$683$$ − 13712.0i − 0.768192i −0.923293 0.384096i $$-0.874513\pi$$
0.923293 0.384096i $$-0.125487\pi$$
$$684$$ −3780.00 −0.211304
$$685$$ 0 0
$$686$$ −260.000 −0.0144706
$$687$$ − 7650.00i − 0.424841i
$$688$$ 16728.0i 0.926961i
$$689$$ 11136.0 0.615744
$$690$$ 0 0
$$691$$ 11372.0 0.626066 0.313033 0.949742i $$-0.398655\pi$$
0.313033 + 0.949742i $$0.398655\pi$$
$$692$$ 4746.00i 0.260717i
$$693$$ − 2574.00i − 0.141094i
$$694$$ 11084.0 0.606258
$$695$$ 0 0
$$696$$ 4050.00 0.220567
$$697$$ − 31228.0i − 1.69705i
$$698$$ 3120.00i 0.169189i
$$699$$ −9126.00 −0.493815
$$700$$ 0 0
$$701$$ −6398.00 −0.344721 −0.172360 0.985034i $$-0.555139\pi$$
−0.172360 + 0.985034i $$0.555139\pi$$
$$702$$ 864.000i 0.0464524i
$$703$$ 3960.00i 0.212453i
$$704$$ 1837.00 0.0983445
$$705$$ 0 0
$$706$$ 5622.00 0.299698
$$707$$ 44252.0i 2.35399i
$$708$$ − 4200.00i − 0.222946i
$$709$$ 5830.00 0.308816 0.154408 0.988007i $$-0.450653\pi$$
0.154408 + 0.988007i $$0.450653\pi$$
$$710$$ 0 0
$$711$$ −4950.00 −0.261096
$$712$$ − 8550.00i − 0.450035i
$$713$$ 1456.00i 0.0764763i
$$714$$ 5772.00 0.302537
$$715$$ 0 0
$$716$$ 11760.0 0.613815
$$717$$ 8100.00i 0.421897i
$$718$$ 8500.00i 0.441807i
$$719$$ −34530.0 −1.79103 −0.895516 0.445030i $$-0.853193\pi$$
−0.895516 + 0.445030i $$0.853193\pi$$
$$720$$ 0 0
$$721$$ 29432.0 1.52026
$$722$$ − 3259.00i − 0.167988i
$$723$$ 1734.00i 0.0891952i
$$724$$ −30506.0 −1.56595
$$725$$ 0 0
$$726$$ 363.000 0.0185567
$$727$$ 17316.0i 0.883377i 0.897169 + 0.441688i $$0.145620\pi$$
−0.897169 + 0.441688i $$0.854380\pi$$
$$728$$ 12480.0i 0.635357i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 30192.0 1.52762
$$732$$ − 2772.00i − 0.139967i
$$733$$ − 27072.0i − 1.36416i −0.731279 0.682079i $$-0.761076\pi$$
0.731279 0.682079i $$-0.238924\pi$$
$$734$$ 7144.00 0.359250
$$735$$ 0 0
$$736$$ 29302.0 1.46751
$$737$$ 11396.0i 0.569575i
$$738$$ − 3798.00i − 0.189439i
$$739$$ 17320.0 0.862147 0.431073 0.902317i $$-0.358135\pi$$
0.431073 + 0.902317i $$0.358135\pi$$
$$740$$ 0 0
$$741$$ −5760.00 −0.285559
$$742$$ − 9048.00i − 0.447658i
$$743$$ 14588.0i 0.720299i 0.932895 + 0.360149i $$0.117274\pi$$
−0.932895 + 0.360149i $$0.882726\pi$$
$$744$$ −360.000 −0.0177396
$$745$$ 0 0
$$746$$ 632.000 0.0310176
$$747$$ 1188.00i 0.0581883i
$$748$$ − 5698.00i − 0.278529i
$$749$$ 14664.0 0.715368
$$750$$ 0 0
$$751$$ 26152.0 1.27071 0.635353 0.772222i $$-0.280855\pi$$
0.635353 + 0.772222i $$0.280855\pi$$
$$752$$ 20746.0i 1.00602i
$$753$$ − 11256.0i − 0.544743i
$$754$$ 2880.00 0.139103
$$755$$ 0 0
$$756$$ −4914.00 −0.236403
$$757$$ 1066.00i 0.0511815i 0.999673 + 0.0255908i $$0.00814669\pi$$
−0.999673 + 0.0255908i $$0.991853\pi$$
$$758$$ 4220.00i 0.202213i
$$759$$ −6006.00 −0.287225
$$760$$ 0 0
$$761$$ −37518.0 −1.78716 −0.893578 0.448907i $$-0.851813\pi$$
−0.893578 + 0.448907i $$0.851813\pi$$
$$762$$ 4818.00i 0.229052i
$$763$$ 8320.00i 0.394763i
$$764$$ −12446.0 −0.589372
$$765$$ 0 0
$$766$$ −8458.00 −0.398956
$$767$$ − 6400.00i − 0.301292i
$$768$$ 357.000i 0.0167736i
$$769$$ 17290.0 0.810785 0.405392 0.914143i $$-0.367135\pi$$
0.405392 + 0.914143i $$0.367135\pi$$
$$770$$ 0 0
$$771$$ −2022.00 −0.0944495
$$772$$ − 27734.0i − 1.29296i
$$773$$ − 17172.0i − 0.799009i −0.916731 0.399504i $$-0.869182\pi$$
0.916731 0.399504i $$-0.130818\pi$$
$$774$$ 3672.00 0.170526
$$775$$ 0 0
$$776$$ 210.000 0.00971464
$$777$$ 5148.00i 0.237688i
$$778$$ − 1740.00i − 0.0801825i
$$779$$ 25320.0 1.16455
$$780$$ 0 0
$$781$$ 8382.00 0.384035
$$782$$ − 13468.0i − 0.615876i
$$783$$ 2430.00i 0.110908i
$$784$$ −13653.0 −0.621948
$$785$$ 0 0
$$786$$ −5724.00 −0.259756
$$787$$ 9536.00i 0.431921i 0.976402 + 0.215960i $$0.0692882\pi$$
−0.976402 + 0.215960i $$0.930712\pi$$
$$788$$ − 2618.00i − 0.118353i
$$789$$ −13056.0 −0.589108
$$790$$ 0 0
$$791$$ 55692.0 2.50339
$$792$$ − 1485.00i − 0.0666252i
$$793$$ − 4224.00i − 0.189153i
$$794$$ −5126.00 −0.229112
$$795$$ 0 0
$$796$$ −14700.0 −0.654557
$$797$$ 20516.0i 0.911812i 0.890028 + 0.455906i $$0.150685\pi$$
−0.890028 + 0.455906i $$0.849315\pi$$
$$798$$ 4680.00i 0.207607i
$$799$$ 37444.0 1.65791
$$800$$ 0 0
$$801$$ 5130.00 0.226292
$$802$$ − 3098.00i − 0.136402i
$$803$$ − 5962.00i − 0.262010i
$$804$$ 21756.0 0.954322
$$805$$ 0 0
$$806$$ −256.000 −0.0111876
$$807$$ 1500.00i 0.0654306i
$$808$$ 25530.0i 1.11156i
$$809$$ −22470.0 −0.976518 −0.488259 0.872699i $$-0.662368\pi$$
−0.488259 + 0.872699i $$0.662368\pi$$
$$810$$ 0 0
$$811$$ −3368.00 −0.145828 −0.0729140 0.997338i $$-0.523230\pi$$
−0.0729140 + 0.997338i $$0.523230\pi$$
$$812$$ 16380.0i 0.707913i
$$813$$ 19614.0i 0.846117i
$$814$$ −726.000 −0.0312608
$$815$$ 0 0
$$816$$ −9102.00 −0.390483
$$817$$ 24480.0i 1.04828i
$$818$$ − 6390.00i − 0.273131i
$$819$$ −7488.00 −0.319477
$$820$$ 0 0
$$821$$ −10738.0 −0.456466 −0.228233 0.973607i $$-0.573295\pi$$
−0.228233 + 0.973607i $$0.573295\pi$$
$$822$$ 6558.00i 0.278268i
$$823$$ − 15912.0i − 0.673946i −0.941514 0.336973i $$-0.890597\pi$$
0.941514 0.336973i $$-0.109403\pi$$
$$824$$ 16980.0 0.717872
$$825$$ 0 0
$$826$$ −5200.00 −0.219045
$$827$$ − 22924.0i − 0.963900i −0.876199 0.481950i $$-0.839929\pi$$
0.876199 0.481950i $$-0.160071\pi$$
$$828$$ 11466.0i 0.481245i
$$829$$ 41690.0 1.74663 0.873313 0.487159i $$-0.161967\pi$$
0.873313 + 0.487159i $$0.161967\pi$$
$$830$$ 0 0
$$831$$ −372.000 −0.0155289
$$832$$ − 5344.00i − 0.222680i
$$833$$ 24642.0i 1.02496i
$$834$$ −8220.00 −0.341289
$$835$$ 0 0
$$836$$ 4620.00 0.191132
$$837$$ − 216.000i − 0.00892001i
$$838$$ − 9760.00i − 0.402331i
$$839$$ 16450.0 0.676898 0.338449 0.940985i $$-0.390098\pi$$
0.338449 + 0.940985i $$0.390098\pi$$
$$840$$ 0 0
$$841$$ −16289.0 −0.667883
$$842$$ − 5138.00i − 0.210294i
$$843$$ − 10926.0i − 0.446396i
$$844$$ 15624.0 0.637204
$$845$$ 0 0
$$846$$ 4554.00 0.185071
$$847$$ 3146.00i 0.127624i
$$848$$ 14268.0i 0.577789i
$$849$$ 13944.0 0.563671
$$850$$ 0 0
$$851$$ 12012.0 0.483861
$$852$$ − 16002.0i − 0.643450i
$$853$$ − 30892.0i − 1.24000i −0.784601 0.620001i $$-0.787132\pi$$
0.784601 0.620001i $$-0.212868\pi$$
$$854$$ −3432.00 −0.137518
$$855$$ 0 0
$$856$$ 8460.00 0.337800
$$857$$ 38906.0i 1.55076i 0.631493 + 0.775381i $$0.282442\pi$$
−0.631493 + 0.775381i $$0.717558\pi$$
$$858$$ − 1056.00i − 0.0420178i
$$859$$ 1020.00 0.0405145 0.0202572 0.999795i $$-0.493551\pi$$
0.0202572 + 0.999795i $$0.493551\pi$$
$$860$$ 0 0
$$861$$ 32916.0 1.30287
$$862$$ − 7008.00i − 0.276907i
$$863$$ 15078.0i 0.594741i 0.954762 + 0.297370i $$0.0961096\pi$$
−0.954762 + 0.297370i $$0.903890\pi$$
$$864$$ −4347.00 −0.171167
$$865$$ 0 0
$$866$$ −5578.00 −0.218878
$$867$$ 1689.00i 0.0661608i
$$868$$ − 1456.00i − 0.0569353i
$$869$$ 6050.00 0.236171
$$870$$ 0 0
$$871$$ 33152.0 1.28968
$$872$$ 4800.00i 0.186409i
$$873$$ 126.000i 0.00488483i
$$874$$ 10920.0 0.422625
$$875$$ 0 0
$$876$$ −11382.0 −0.438998
$$877$$ − 22704.0i − 0.874184i −0.899417 0.437092i $$-0.856008\pi$$
0.899417 0.437092i $$-0.143992\pi$$
$$878$$ 10430.0i 0.400906i
$$879$$ −9306.00 −0.357092
$$880$$ 0 0
$$881$$ −19358.0 −0.740281 −0.370141 0.928976i $$-0.620690\pi$$
−0.370141 + 0.928976i $$0.620690\pi$$
$$882$$ 2997.00i 0.114415i
$$883$$ − 11252.0i − 0.428833i −0.976742 0.214417i $$-0.931215\pi$$
0.976742 0.214417i $$-0.0687851\pi$$
$$884$$ −16576.0 −0.630669
$$885$$ 0 0
$$886$$ 4432.00 0.168054
$$887$$ − 43684.0i − 1.65362i −0.562478 0.826812i $$-0.690152\pi$$
0.562478 0.826812i $$-0.309848\pi$$
$$888$$ 2970.00i 0.112237i
$$889$$ −41756.0 −1.57531
$$890$$ 0 0
$$891$$ 891.000 0.0335013
$$892$$ 14896.0i 0.559142i
$$893$$ 30360.0i 1.13769i
$$894$$ 3930.00 0.147023
$$895$$ 0 0
$$896$$ −37830.0 −1.41050
$$897$$ 17472.0i 0.650360i
$$898$$ 6290.00i 0.233742i
$$899$$ −720.000 −0.0267112
$$900$$ 0 0
$$901$$ 25752.0 0.952190
$$902$$ 4642.00i 0.171354i
$$903$$ 31824.0i 1.17280i
$$904$$ 32130.0 1.18211
$$905$$ 0 0
$$906$$ −3594.00 −0.131791
$$907$$ − 45804.0i − 1.67684i −0.545022 0.838422i $$-0.683479\pi$$
0.545022 0.838422i $$-0.316521\pi$$
$$908$$ − 20748.0i − 0.758311i
$$909$$ −15318.0 −0.558928
$$910$$ 0 0
$$911$$ −15318.0 −0.557089 −0.278544 0.960423i $$-0.589852\pi$$
−0.278544 + 0.960423i $$0.589852\pi$$
$$912$$ − 7380.00i − 0.267956i
$$913$$ − 1452.00i − 0.0526333i
$$914$$ 3054.00 0.110522
$$915$$ 0 0
$$916$$ 17850.0 0.643865
$$917$$ − 49608.0i − 1.78648i
$$918$$ 1998.00i 0.0718342i
$$919$$ −11350.0 −0.407401 −0.203701 0.979033i $$-0.565297\pi$$
−0.203701 + 0.979033i $$0.565297\pi$$
$$920$$ 0 0
$$921$$ −3732.00 −0.133522
$$922$$ 12882.0i 0.460137i
$$923$$ − 24384.0i − 0.869566i
$$924$$ 6006.00 0.213834
$$925$$ 0 0
$$926$$ −6148.00 −0.218181
$$927$$ 10188.0i 0.360969i
$$928$$ 14490.0i 0.512562i
$$929$$ −33030.0 −1.16650 −0.583250 0.812292i $$-0.698219\pi$$
−0.583250 + 0.812292i $$0.698219\pi$$
$$930$$ 0 0
$$931$$ −19980.0 −0.703349
$$932$$ − 21294.0i − 0.748399i
$$933$$ − 6246.00i − 0.219169i
$$934$$ 5124.00 0.179510
$$935$$ 0 0
$$936$$ −4320.00 −0.150859
$$937$$ 10006.0i 0.348860i 0.984670 + 0.174430i $$0.0558083\pi$$
−0.984670 + 0.174430i $$0.944192\pi$$
$$938$$ − 26936.0i − 0.937624i
$$939$$ 7134.00 0.247933
$$940$$ 0 0
$$941$$ 2622.00 0.0908340 0.0454170 0.998968i $$-0.485538\pi$$
0.0454170 + 0.998968i $$0.485538\pi$$
$$942$$ − 6342.00i − 0.219356i
$$943$$ − 76804.0i − 2.65226i
$$944$$ 8200.00 0.282720
$$945$$ 0 0
$$946$$ −4488.00 −0.154247
$$947$$ 39876.0i 1.36832i 0.729334 + 0.684158i $$0.239830\pi$$
−0.729334 + 0.684158i $$0.760170\pi$$
$$948$$ − 11550.0i − 0.395703i
$$949$$ −17344.0 −0.593267
$$950$$ 0 0
$$951$$ 1488.00 0.0507379
$$952$$ 28860.0i 0.982519i
$$953$$ 38918.0i 1.32285i 0.750011 + 0.661426i $$0.230048\pi$$
−0.750011 + 0.661426i $$0.769952\pi$$
$$954$$ 3132.00 0.106292
$$955$$ 0 0
$$956$$ −18900.0 −0.639403
$$957$$ − 2970.00i − 0.100320i
$$958$$ 16520.0i 0.557137i
$$959$$ −56836.0 −1.91380
$$960$$ 0 0
$$961$$ −29727.0 −0.997852
$$962$$ 2112.00i 0.0707834i
$$963$$ 5076.00i 0.169857i
$$964$$ −4046.00 −0.135179
$$965$$ 0 0
$$966$$ 14196.0 0.472825
$$967$$ − 1114.00i − 0.0370464i −0.999828 0.0185232i $$-0.994104\pi$$
0.999828 0.0185232i $$-0.00589645\pi$$
$$968$$ 1815.00i 0.0602648i
$$969$$ −13320.0 −0.441589
$$970$$ 0 0
$$971$$ −1688.00 −0.0557884 −0.0278942 0.999611i $$-0.508880\pi$$
−0.0278942 + 0.999611i $$0.508880\pi$$
$$972$$ − 1701.00i − 0.0561313i
$$973$$ − 71240.0i − 2.34722i
$$974$$ 524.000 0.0172382
$$975$$ 0 0
$$976$$ 5412.00 0.177494
$$977$$ 41826.0i 1.36963i 0.728715 + 0.684817i $$0.240118\pi$$
−0.728715 + 0.684817i $$0.759882\pi$$
$$978$$ 11604.0i 0.379402i
$$979$$ −6270.00 −0.204689
$$980$$ 0 0
$$981$$ −2880.00 −0.0937322
$$982$$ − 15028.0i − 0.488353i
$$983$$ 978.000i 0.0317328i 0.999874 + 0.0158664i $$0.00505065\pi$$
−0.999874 + 0.0158664i $$0.994949\pi$$
$$984$$ 18990.0 0.615223
$$985$$ 0 0
$$986$$ 6660.00 0.215109
$$987$$ 39468.0i 1.27283i
$$988$$ − 13440.0i − 0.432777i
$$989$$ 74256.0 2.38747
$$990$$ 0 0
$$991$$ 47272.0 1.51528 0.757641 0.652671i $$-0.226352\pi$$
0.757641 + 0.652671i $$0.226352\pi$$
$$992$$ − 1288.00i − 0.0412238i
$$993$$ 8124.00i 0.259625i
$$994$$ −19812.0 −0.632192
$$995$$ 0 0
$$996$$ −2772.00 −0.0881869
$$997$$ − 51104.0i − 1.62335i −0.584109 0.811675i $$-0.698556\pi$$
0.584109 0.811675i $$-0.301444\pi$$
$$998$$ − 9020.00i − 0.286095i
$$999$$ −1782.00 −0.0564364
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 825.4.c.f.199.2 2
5.2 odd 4 33.4.a.b.1.1 1
5.3 odd 4 825.4.a.f.1.1 1
5.4 even 2 inner 825.4.c.f.199.1 2
15.2 even 4 99.4.a.a.1.1 1
15.8 even 4 2475.4.a.e.1.1 1
20.7 even 4 528.4.a.h.1.1 1
35.27 even 4 1617.4.a.d.1.1 1
40.27 even 4 2112.4.a.h.1.1 1
40.37 odd 4 2112.4.a.u.1.1 1
55.32 even 4 363.4.a.d.1.1 1
60.47 odd 4 1584.4.a.l.1.1 1
165.32 odd 4 1089.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.b.1.1 1 5.2 odd 4
99.4.a.a.1.1 1 15.2 even 4
363.4.a.d.1.1 1 55.32 even 4
528.4.a.h.1.1 1 20.7 even 4
825.4.a.f.1.1 1 5.3 odd 4
825.4.c.f.199.1 2 5.4 even 2 inner
825.4.c.f.199.2 2 1.1 even 1 trivial
1089.4.a.e.1.1 1 165.32 odd 4
1584.4.a.l.1.1 1 60.47 odd 4
1617.4.a.d.1.1 1 35.27 even 4
2112.4.a.h.1.1 1 40.27 even 4
2112.4.a.u.1.1 1 40.37 odd 4
2475.4.a.e.1.1 1 15.8 even 4