# Properties

 Label 825.4.c.h.199.3 Level $825$ Weight $4$ Character 825.199 Analytic conductor $48.677$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 49x^{2} + 576$$ x^4 + 49*x^2 + 576 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.3 Root $$4.42443i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.4.c.h.199.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.42443i q^{2} -3.00000i q^{3} -11.5756 q^{4} +13.2733 q^{6} -31.6977i q^{7} -15.8199i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+4.42443i q^{2} -3.00000i q^{3} -11.5756 q^{4} +13.2733 q^{6} -31.6977i q^{7} -15.8199i q^{8} -9.00000 q^{9} -11.0000 q^{11} +34.7267i q^{12} +5.15114i q^{13} +140.244 q^{14} -22.6107 q^{16} -121.942i q^{17} -39.8199i q^{18} -34.8489 q^{19} -95.0931 q^{21} -48.6687i q^{22} +116.244i q^{23} -47.4596 q^{24} -22.7909 q^{26} +27.0000i q^{27} +366.919i q^{28} +69.4534 q^{29} +140.605 q^{31} -226.598i q^{32} +33.0000i q^{33} +539.524 q^{34} +104.180 q^{36} +420.070i q^{37} -154.186i q^{38} +15.4534 q^{39} -322.058 q^{41} -420.733i q^{42} +321.035i q^{43} +127.331 q^{44} -514.315 q^{46} +231.408i q^{47} +67.8322i q^{48} -661.745 q^{49} -365.826 q^{51} -59.6274i q^{52} +4.91916i q^{53} -119.460 q^{54} -501.453 q^{56} +104.547i q^{57} +307.292i q^{58} -406.443 q^{59} -556.431 q^{61} +622.095i q^{62} +285.279i q^{63} +821.683 q^{64} -146.006 q^{66} -84.7452i q^{67} +1411.55i q^{68} +348.733 q^{69} +49.0808 q^{71} +142.379i q^{72} +785.884i q^{73} -1858.57 q^{74} +403.395 q^{76} +348.675i q^{77} +68.3726i q^{78} +383.118 q^{79} +81.0000 q^{81} -1424.92i q^{82} -930.211i q^{83} +1100.76 q^{84} -1420.40 q^{86} -208.360i q^{87} +174.018i q^{88} +732.559 q^{89} +163.279 q^{91} -1345.59i q^{92} -421.814i q^{93} -1023.85 q^{94} -679.795 q^{96} +1171.49i q^{97} -2927.84i q^{98} +99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 66 q^{4} - 6 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q - 66 * q^4 - 6 * q^6 - 36 * q^9 $$4 q - 66 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 364 q^{14} + 402 q^{16} - 100 q^{19} - 144 q^{21} + 342 q^{24} + 224 q^{26} + 396 q^{29} + 720 q^{31} + 1252 q^{34} + 594 q^{36} + 180 q^{39} - 1564 q^{41} + 726 q^{44} - 836 q^{46} - 756 q^{49} - 636 q^{51} + 54 q^{54} - 2124 q^{56} + 344 q^{59} - 1556 q^{61} - 1618 q^{64} + 66 q^{66} + 804 q^{69} + 1260 q^{71} - 4716 q^{74} + 1456 q^{76} - 1304 q^{79} + 324 q^{81} + 1212 q^{84} - 2136 q^{86} + 1512 q^{89} - 56 q^{91} - 7444 q^{94} - 5142 q^{96} + 396 q^{99}+O(q^{100})$$ 4 * q - 66 * q^4 - 6 * q^6 - 36 * q^9 - 44 * q^11 + 364 * q^14 + 402 * q^16 - 100 * q^19 - 144 * q^21 + 342 * q^24 + 224 * q^26 + 396 * q^29 + 720 * q^31 + 1252 * q^34 + 594 * q^36 + 180 * q^39 - 1564 * q^41 + 726 * q^44 - 836 * q^46 - 756 * q^49 - 636 * q^51 + 54 * q^54 - 2124 * q^56 + 344 * q^59 - 1556 * q^61 - 1618 * q^64 + 66 * q^66 + 804 * q^69 + 1260 * q^71 - 4716 * q^74 + 1456 * q^76 - 1304 * q^79 + 324 * q^81 + 1212 * q^84 - 2136 * q^86 + 1512 * q^89 - 56 * q^91 - 7444 * q^94 - 5142 * q^96 + 396 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.42443i 1.56427i 0.623108 + 0.782136i $$0.285870\pi$$
−0.623108 + 0.782136i $$0.714130\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ −11.5756 −1.44695
$$5$$ 0 0
$$6$$ 13.2733 0.903133
$$7$$ − 31.6977i − 1.71152i −0.517377 0.855758i $$-0.673091\pi$$
0.517377 0.855758i $$-0.326909\pi$$
$$8$$ − 15.8199i − 0.699146i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 34.7267i 0.835395i
$$13$$ 5.15114i 0.109898i 0.998489 + 0.0549488i $$0.0174996\pi$$
−0.998489 + 0.0549488i $$0.982500\pi$$
$$14$$ 140.244 2.67728
$$15$$ 0 0
$$16$$ −22.6107 −0.353293
$$17$$ − 121.942i − 1.73972i −0.493297 0.869861i $$-0.664208\pi$$
0.493297 0.869861i $$-0.335792\pi$$
$$18$$ − 39.8199i − 0.521424i
$$19$$ −34.8489 −0.420783 −0.210391 0.977617i $$-0.567474\pi$$
−0.210391 + 0.977617i $$0.567474\pi$$
$$20$$ 0 0
$$21$$ −95.0931 −0.988144
$$22$$ − 48.6687i − 0.471646i
$$23$$ 116.244i 1.05385i 0.849911 + 0.526926i $$0.176656\pi$$
−0.849911 + 0.526926i $$0.823344\pi$$
$$24$$ −47.4596 −0.403652
$$25$$ 0 0
$$26$$ −22.7909 −0.171910
$$27$$ 27.0000i 0.192450i
$$28$$ 366.919i 2.47647i
$$29$$ 69.4534 0.444730 0.222365 0.974963i $$-0.428622\pi$$
0.222365 + 0.974963i $$0.428622\pi$$
$$30$$ 0 0
$$31$$ 140.605 0.814623 0.407312 0.913289i $$-0.366466\pi$$
0.407312 + 0.913289i $$0.366466\pi$$
$$32$$ − 226.598i − 1.25179i
$$33$$ 33.0000i 0.174078i
$$34$$ 539.524 2.72140
$$35$$ 0 0
$$36$$ 104.180 0.482315
$$37$$ 420.070i 1.86646i 0.359276 + 0.933232i $$0.383024\pi$$
−0.359276 + 0.933232i $$0.616976\pi$$
$$38$$ − 154.186i − 0.658219i
$$39$$ 15.4534 0.0634495
$$40$$ 0 0
$$41$$ −322.058 −1.22676 −0.613378 0.789789i $$-0.710190\pi$$
−0.613378 + 0.789789i $$0.710190\pi$$
$$42$$ − 420.733i − 1.54573i
$$43$$ 321.035i 1.13854i 0.822149 + 0.569272i $$0.192775\pi$$
−0.822149 + 0.569272i $$0.807225\pi$$
$$44$$ 127.331 0.436271
$$45$$ 0 0
$$46$$ −514.315 −1.64851
$$47$$ 231.408i 0.718176i 0.933304 + 0.359088i $$0.116912\pi$$
−0.933304 + 0.359088i $$0.883088\pi$$
$$48$$ 67.8322i 0.203974i
$$49$$ −661.745 −1.92929
$$50$$ 0 0
$$51$$ −365.826 −1.00443
$$52$$ − 59.6274i − 0.159016i
$$53$$ 4.91916i 0.0127490i 0.999980 + 0.00637452i $$0.00202909\pi$$
−0.999980 + 0.00637452i $$0.997971\pi$$
$$54$$ −119.460 −0.301044
$$55$$ 0 0
$$56$$ −501.453 −1.19660
$$57$$ 104.547i 0.242939i
$$58$$ 307.292i 0.695679i
$$59$$ −406.443 −0.896854 −0.448427 0.893820i $$-0.648016\pi$$
−0.448427 + 0.893820i $$0.648016\pi$$
$$60$$ 0 0
$$61$$ −556.431 −1.16793 −0.583964 0.811779i $$-0.698499\pi$$
−0.583964 + 0.811779i $$0.698499\pi$$
$$62$$ 622.095i 1.27429i
$$63$$ 285.279i 0.570505i
$$64$$ 821.683 1.60485
$$65$$ 0 0
$$66$$ −146.006 −0.272305
$$67$$ − 84.7452i − 0.154526i −0.997011 0.0772632i $$-0.975382\pi$$
0.997011 0.0772632i $$-0.0246182\pi$$
$$68$$ 1411.55i 2.51728i
$$69$$ 348.733 0.608442
$$70$$ 0 0
$$71$$ 49.0808 0.0820398 0.0410199 0.999158i $$-0.486939\pi$$
0.0410199 + 0.999158i $$0.486939\pi$$
$$72$$ 142.379i 0.233049i
$$73$$ 785.884i 1.26001i 0.776591 + 0.630005i $$0.216947\pi$$
−0.776591 + 0.630005i $$0.783053\pi$$
$$74$$ −1858.57 −2.91966
$$75$$ 0 0
$$76$$ 403.395 0.608850
$$77$$ 348.675i 0.516041i
$$78$$ 68.3726i 0.0992522i
$$79$$ 383.118 0.545622 0.272811 0.962068i $$-0.412047\pi$$
0.272811 + 0.962068i $$0.412047\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ − 1424.92i − 1.91898i
$$83$$ − 930.211i − 1.23017i −0.788462 0.615084i $$-0.789122\pi$$
0.788462 0.615084i $$-0.210878\pi$$
$$84$$ 1100.76 1.42979
$$85$$ 0 0
$$86$$ −1420.40 −1.78099
$$87$$ − 208.360i − 0.256765i
$$88$$ 174.018i 0.210800i
$$89$$ 732.559 0.872484 0.436242 0.899829i $$-0.356309\pi$$
0.436242 + 0.899829i $$0.356309\pi$$
$$90$$ 0 0
$$91$$ 163.279 0.188092
$$92$$ − 1345.59i − 1.52487i
$$93$$ − 421.814i − 0.470323i
$$94$$ −1023.85 −1.12342
$$95$$ 0 0
$$96$$ −679.795 −0.722722
$$97$$ 1171.49i 1.22626i 0.789984 + 0.613128i $$0.210089\pi$$
−0.789984 + 0.613128i $$0.789911\pi$$
$$98$$ − 2927.84i − 3.01793i
$$99$$ 99.0000 0.100504
$$100$$ 0 0
$$101$$ −1221.27 −1.20318 −0.601589 0.798806i $$-0.705465\pi$$
−0.601589 + 0.798806i $$0.705465\pi$$
$$102$$ − 1618.57i − 1.57120i
$$103$$ 516.745i 0.494334i 0.968973 + 0.247167i $$0.0794996\pi$$
−0.968973 + 0.247167i $$0.920500\pi$$
$$104$$ 81.4903 0.0768345
$$105$$ 0 0
$$106$$ −21.7645 −0.0199430
$$107$$ 152.025i 0.137353i 0.997639 + 0.0686765i $$0.0218776\pi$$
−0.997639 + 0.0686765i $$0.978122\pi$$
$$108$$ − 312.540i − 0.278465i
$$109$$ −2170.32 −1.90714 −0.953572 0.301164i $$-0.902625\pi$$
−0.953572 + 0.301164i $$0.902625\pi$$
$$110$$ 0 0
$$111$$ 1260.21 1.07760
$$112$$ 716.708i 0.604666i
$$113$$ − 646.397i − 0.538123i −0.963123 0.269062i $$-0.913286\pi$$
0.963123 0.269062i $$-0.0867135\pi$$
$$114$$ −462.559 −0.380023
$$115$$ 0 0
$$116$$ −803.963 −0.643501
$$117$$ − 46.3603i − 0.0366326i
$$118$$ − 1798.28i − 1.40292i
$$119$$ −3865.28 −2.97756
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ − 2461.89i − 1.82696i
$$123$$ 966.174i 0.708268i
$$124$$ −1627.58 −1.17872
$$125$$ 0 0
$$126$$ −1262.20 −0.892425
$$127$$ 993.304i 0.694027i 0.937860 + 0.347014i $$0.112804\pi$$
−0.937860 + 0.347014i $$0.887196\pi$$
$$128$$ 1822.69i 1.25863i
$$129$$ 963.105 0.657339
$$130$$ 0 0
$$131$$ 385.814 0.257318 0.128659 0.991689i $$-0.458933\pi$$
0.128659 + 0.991689i $$0.458933\pi$$
$$132$$ − 381.994i − 0.251881i
$$133$$ 1104.63i 0.720177i
$$134$$ 374.949 0.241721
$$135$$ 0 0
$$136$$ −1929.11 −1.21632
$$137$$ − 884.840i − 0.551803i −0.961186 0.275901i $$-0.911024\pi$$
0.961186 0.275901i $$-0.0889763\pi$$
$$138$$ 1542.94i 0.951769i
$$139$$ 1091.94 0.666312 0.333156 0.942872i $$-0.391886\pi$$
0.333156 + 0.942872i $$0.391886\pi$$
$$140$$ 0 0
$$141$$ 694.223 0.414639
$$142$$ 217.155i 0.128333i
$$143$$ − 56.6626i − 0.0331354i
$$144$$ 203.497 0.117764
$$145$$ 0 0
$$146$$ −3477.09 −1.97100
$$147$$ 1985.24i 1.11387i
$$148$$ − 4862.55i − 2.70067i
$$149$$ −297.014 −0.163304 −0.0816522 0.996661i $$-0.526020\pi$$
−0.0816522 + 0.996661i $$0.526020\pi$$
$$150$$ 0 0
$$151$$ −1887.86 −1.01743 −0.508716 0.860935i $$-0.669880\pi$$
−0.508716 + 0.860935i $$0.669880\pi$$
$$152$$ 551.304i 0.294189i
$$153$$ 1097.48i 0.579907i
$$154$$ −1542.69 −0.807229
$$155$$ 0 0
$$156$$ −178.882 −0.0918080
$$157$$ 56.5343i 0.0287384i 0.999897 + 0.0143692i $$0.00457401\pi$$
−0.999897 + 0.0143692i $$0.995426\pi$$
$$158$$ 1695.08i 0.853501i
$$159$$ 14.7575 0.00736066
$$160$$ 0 0
$$161$$ 3684.68 1.80369
$$162$$ 358.379i 0.173808i
$$163$$ − 49.2338i − 0.0236582i −0.999930 0.0118291i $$-0.996235\pi$$
0.999930 0.0118291i $$-0.00376541\pi$$
$$164$$ 3728.01 1.77505
$$165$$ 0 0
$$166$$ 4115.65 1.92432
$$167$$ − 2068.75i − 0.958589i −0.877654 0.479294i $$-0.840893\pi$$
0.877654 0.479294i $$-0.159107\pi$$
$$168$$ 1504.36i 0.690857i
$$169$$ 2170.47 0.987923
$$170$$ 0 0
$$171$$ 313.640 0.140261
$$172$$ − 3716.17i − 1.64741i
$$173$$ − 604.012i − 0.265446i −0.991153 0.132723i $$-0.957628\pi$$
0.991153 0.132723i $$-0.0423721\pi$$
$$174$$ 921.875 0.401650
$$175$$ 0 0
$$176$$ 248.718 0.106522
$$177$$ 1219.33i 0.517799i
$$178$$ 3241.15i 1.36480i
$$179$$ 2132.02 0.890251 0.445126 0.895468i $$-0.353159\pi$$
0.445126 + 0.895468i $$0.353159\pi$$
$$180$$ 0 0
$$181$$ −589.371 −0.242031 −0.121015 0.992651i $$-0.538615\pi$$
−0.121015 + 0.992651i $$0.538615\pi$$
$$182$$ 722.418i 0.294226i
$$183$$ 1669.29i 0.674304i
$$184$$ 1838.97 0.736796
$$185$$ 0 0
$$186$$ 1866.28 0.735713
$$187$$ 1341.36i 0.524546i
$$188$$ − 2678.68i − 1.03916i
$$189$$ 855.838 0.329381
$$190$$ 0 0
$$191$$ −2160.90 −0.818624 −0.409312 0.912395i $$-0.634231\pi$$
−0.409312 + 0.912395i $$0.634231\pi$$
$$192$$ − 2465.05i − 0.926560i
$$193$$ − 1490.91i − 0.556052i −0.960574 0.278026i $$-0.910320\pi$$
0.960574 0.278026i $$-0.0896802\pi$$
$$194$$ −5183.18 −1.91820
$$195$$ 0 0
$$196$$ 7660.08 2.79157
$$197$$ 230.529i 0.0833732i 0.999131 + 0.0416866i $$0.0132731\pi$$
−0.999131 + 0.0416866i $$0.986727\pi$$
$$198$$ 438.018i 0.157215i
$$199$$ −22.4007 −0.00797963 −0.00398982 0.999992i $$-0.501270\pi$$
−0.00398982 + 0.999992i $$0.501270\pi$$
$$200$$ 0 0
$$201$$ −254.236 −0.0892159
$$202$$ − 5403.43i − 1.88210i
$$203$$ − 2201.51i − 0.761163i
$$204$$ 4234.65 1.45336
$$205$$ 0 0
$$206$$ −2286.30 −0.773273
$$207$$ − 1046.20i − 0.351284i
$$208$$ − 116.471i − 0.0388260i
$$209$$ 383.337 0.126871
$$210$$ 0 0
$$211$$ −1051.64 −0.343117 −0.171558 0.985174i $$-0.554880\pi$$
−0.171558 + 0.985174i $$0.554880\pi$$
$$212$$ − 56.9421i − 0.0184472i
$$213$$ − 147.243i − 0.0473657i
$$214$$ −672.622 −0.214857
$$215$$ 0 0
$$216$$ 427.136 0.134551
$$217$$ − 4456.84i − 1.39424i
$$218$$ − 9602.42i − 2.98329i
$$219$$ 2357.65 0.727467
$$220$$ 0 0
$$221$$ 628.141 0.191191
$$222$$ 5575.71i 1.68566i
$$223$$ 3861.80i 1.15966i 0.814736 + 0.579832i $$0.196882\pi$$
−0.814736 + 0.579832i $$0.803118\pi$$
$$224$$ −7182.65 −2.14246
$$225$$ 0 0
$$226$$ 2859.94 0.841771
$$227$$ 872.721i 0.255174i 0.991827 + 0.127587i $$0.0407232\pi$$
−0.991827 + 0.127587i $$0.959277\pi$$
$$228$$ − 1210.19i − 0.351520i
$$229$$ −1841.72 −0.531459 −0.265730 0.964048i $$-0.585613\pi$$
−0.265730 + 0.964048i $$0.585613\pi$$
$$230$$ 0 0
$$231$$ 1046.02 0.297937
$$232$$ − 1098.74i − 0.310931i
$$233$$ 3932.14i 1.10559i 0.833317 + 0.552796i $$0.186439\pi$$
−0.833317 + 0.552796i $$0.813561\pi$$
$$234$$ 205.118 0.0573033
$$235$$ 0 0
$$236$$ 4704.81 1.29770
$$237$$ − 1149.35i − 0.315015i
$$238$$ − 17101.7i − 4.65772i
$$239$$ −4772.10 −1.29155 −0.645777 0.763526i $$-0.723466\pi$$
−0.645777 + 0.763526i $$0.723466\pi$$
$$240$$ 0 0
$$241$$ 3988.84 1.06616 0.533078 0.846066i $$-0.321035\pi$$
0.533078 + 0.846066i $$0.321035\pi$$
$$242$$ 535.356i 0.142207i
$$243$$ − 243.000i − 0.0641500i
$$244$$ 6441.00 1.68993
$$245$$ 0 0
$$246$$ −4274.77 −1.10792
$$247$$ − 179.511i − 0.0462431i
$$248$$ − 2224.34i − 0.569540i
$$249$$ −2790.63 −0.710238
$$250$$ 0 0
$$251$$ −5474.22 −1.37661 −0.688306 0.725421i $$-0.741645\pi$$
−0.688306 + 0.725421i $$0.741645\pi$$
$$252$$ − 3302.27i − 0.825491i
$$253$$ − 1278.69i − 0.317749i
$$254$$ −4394.80 −1.08565
$$255$$ 0 0
$$256$$ −1490.90 −0.363989
$$257$$ 6434.01i 1.56164i 0.624754 + 0.780822i $$0.285199\pi$$
−0.624754 + 0.780822i $$0.714801\pi$$
$$258$$ 4261.19i 1.02826i
$$259$$ 13315.3 3.19448
$$260$$ 0 0
$$261$$ −625.081 −0.148243
$$262$$ 1707.01i 0.402516i
$$263$$ 7589.00i 1.77931i 0.456636 + 0.889654i $$0.349054\pi$$
−0.456636 + 0.889654i $$0.650946\pi$$
$$264$$ 522.055 0.121706
$$265$$ 0 0
$$266$$ −4887.35 −1.12655
$$267$$ − 2197.68i − 0.503729i
$$268$$ 980.974i 0.223591i
$$269$$ −478.178 −0.108383 −0.0541914 0.998531i $$-0.517258\pi$$
−0.0541914 + 0.998531i $$0.517258\pi$$
$$270$$ 0 0
$$271$$ −122.323 −0.0274192 −0.0137096 0.999906i $$-0.504364\pi$$
−0.0137096 + 0.999906i $$0.504364\pi$$
$$272$$ 2757.20i 0.614631i
$$273$$ − 489.838i − 0.108595i
$$274$$ 3914.91 0.863170
$$275$$ 0 0
$$276$$ −4036.78 −0.880383
$$277$$ − 8199.41i − 1.77854i −0.457385 0.889269i $$-0.651214\pi$$
0.457385 0.889269i $$-0.348786\pi$$
$$278$$ 4831.22i 1.04229i
$$279$$ −1265.44 −0.271541
$$280$$ 0 0
$$281$$ 6943.79 1.47413 0.737067 0.675820i $$-0.236210\pi$$
0.737067 + 0.675820i $$0.236210\pi$$
$$282$$ 3071.54i 0.648609i
$$283$$ 1035.14i 0.217429i 0.994073 + 0.108715i $$0.0346735\pi$$
−0.994073 + 0.108715i $$0.965327\pi$$
$$284$$ −568.139 −0.118707
$$285$$ 0 0
$$286$$ 250.699 0.0518328
$$287$$ 10208.5i 2.09961i
$$288$$ 2039.39i 0.417264i
$$289$$ −9956.85 −2.02663
$$290$$ 0 0
$$291$$ 3514.47 0.707979
$$292$$ − 9097.06i − 1.82317i
$$293$$ − 6144.81i − 1.22520i −0.790393 0.612600i $$-0.790124\pi$$
0.790393 0.612600i $$-0.209876\pi$$
$$294$$ −8783.53 −1.74240
$$295$$ 0 0
$$296$$ 6645.45 1.30493
$$297$$ − 297.000i − 0.0580259i
$$298$$ − 1314.12i − 0.255452i
$$299$$ −598.791 −0.115816
$$300$$ 0 0
$$301$$ 10176.1 1.94864
$$302$$ − 8352.72i − 1.59154i
$$303$$ 3663.81i 0.694655i
$$304$$ 787.958 0.148659
$$305$$ 0 0
$$306$$ −4855.71 −0.907133
$$307$$ 2186.09i 0.406406i 0.979137 + 0.203203i $$0.0651351\pi$$
−0.979137 + 0.203203i $$0.934865\pi$$
$$308$$ − 4036.11i − 0.746684i
$$309$$ 1550.24 0.285404
$$310$$ 0 0
$$311$$ −7484.83 −1.36471 −0.682357 0.731019i $$-0.739045\pi$$
−0.682357 + 0.731019i $$0.739045\pi$$
$$312$$ − 244.471i − 0.0443604i
$$313$$ − 6833.33i − 1.23400i −0.786962 0.617001i $$-0.788347\pi$$
0.786962 0.617001i $$-0.211653\pi$$
$$314$$ −250.132 −0.0449546
$$315$$ 0 0
$$316$$ −4434.81 −0.789485
$$317$$ − 924.265i − 0.163760i −0.996642 0.0818800i $$-0.973908\pi$$
0.996642 0.0818800i $$-0.0260924\pi$$
$$318$$ 65.2934i 0.0115141i
$$319$$ −763.988 −0.134091
$$320$$ 0 0
$$321$$ 456.074 0.0793008
$$322$$ 16302.6i 2.82145i
$$323$$ 4249.54i 0.732046i
$$324$$ −937.621 −0.160772
$$325$$ 0 0
$$326$$ 217.831 0.0370078
$$327$$ 6510.95i 1.10109i
$$328$$ 5094.91i 0.857681i
$$329$$ 7335.10 1.22917
$$330$$ 0 0
$$331$$ −9820.46 −1.63076 −0.815380 0.578927i $$-0.803472\pi$$
−0.815380 + 0.578927i $$0.803472\pi$$
$$332$$ 10767.7i 1.77999i
$$333$$ − 3780.63i − 0.622154i
$$334$$ 9153.02 1.49949
$$335$$ 0 0
$$336$$ 2150.12 0.349104
$$337$$ − 600.808i − 0.0971161i −0.998820 0.0485580i $$-0.984537\pi$$
0.998820 0.0485580i $$-0.0154626\pi$$
$$338$$ 9603.07i 1.54538i
$$339$$ −1939.19 −0.310686
$$340$$ 0 0
$$341$$ −1546.65 −0.245618
$$342$$ 1387.68i 0.219406i
$$343$$ 10103.5i 1.59049i
$$344$$ 5078.73 0.796008
$$345$$ 0 0
$$346$$ 2672.41 0.415230
$$347$$ 3143.41i 0.486303i 0.969988 + 0.243152i $$0.0781813\pi$$
−0.969988 + 0.243152i $$0.921819\pi$$
$$348$$ 2411.89i 0.371525i
$$349$$ −720.663 −0.110533 −0.0552667 0.998472i $$-0.517601\pi$$
−0.0552667 + 0.998472i $$0.517601\pi$$
$$350$$ 0 0
$$351$$ −139.081 −0.0211498
$$352$$ 2492.58i 0.377429i
$$353$$ 1207.12i 0.182007i 0.995851 + 0.0910034i $$0.0290074\pi$$
−0.995851 + 0.0910034i $$0.970993\pi$$
$$354$$ −5394.83 −0.809978
$$355$$ 0 0
$$356$$ −8479.79 −1.26244
$$357$$ 11595.8i 1.71910i
$$358$$ 9432.99i 1.39260i
$$359$$ −8748.31 −1.28612 −0.643062 0.765814i $$-0.722336\pi$$
−0.643062 + 0.765814i $$0.722336\pi$$
$$360$$ 0 0
$$361$$ −5644.56 −0.822942
$$362$$ − 2607.63i − 0.378602i
$$363$$ − 363.000i − 0.0524864i
$$364$$ −1890.05 −0.272158
$$365$$ 0 0
$$366$$ −7385.66 −1.05479
$$367$$ 6730.45i 0.957293i 0.878008 + 0.478647i $$0.158872\pi$$
−0.878008 + 0.478647i $$0.841128\pi$$
$$368$$ − 2628.37i − 0.372318i
$$369$$ 2898.52 0.408919
$$370$$ 0 0
$$371$$ 155.926 0.0218202
$$372$$ 4882.73i 0.680532i
$$373$$ − 227.394i − 0.0315657i −0.999875 0.0157828i $$-0.994976\pi$$
0.999875 0.0157828i $$-0.00502404\pi$$
$$374$$ −5934.76 −0.820533
$$375$$ 0 0
$$376$$ 3660.84 0.502110
$$377$$ 357.764i 0.0488748i
$$378$$ 3786.60i 0.515242i
$$379$$ −11356.2 −1.53913 −0.769565 0.638568i $$-0.779527\pi$$
−0.769565 + 0.638568i $$0.779527\pi$$
$$380$$ 0 0
$$381$$ 2979.91 0.400697
$$382$$ − 9560.74i − 1.28055i
$$383$$ 10753.6i 1.43468i 0.696725 + 0.717338i $$0.254640\pi$$
−0.696725 + 0.717338i $$0.745360\pi$$
$$384$$ 5468.07 0.726670
$$385$$ 0 0
$$386$$ 6596.43 0.869817
$$387$$ − 2889.32i − 0.379515i
$$388$$ − 13560.7i − 1.77433i
$$389$$ 11727.1 1.52850 0.764252 0.644918i $$-0.223109\pi$$
0.764252 + 0.644918i $$0.223109\pi$$
$$390$$ 0 0
$$391$$ 14175.1 1.83341
$$392$$ 10468.7i 1.34885i
$$393$$ − 1157.44i − 0.148563i
$$394$$ −1019.96 −0.130418
$$395$$ 0 0
$$396$$ −1145.98 −0.145424
$$397$$ 359.905i 0.0454990i 0.999741 + 0.0227495i $$0.00724202\pi$$
−0.999741 + 0.0227495i $$0.992758\pi$$
$$398$$ − 99.1105i − 0.0124823i
$$399$$ 3313.89 0.415794
$$400$$ 0 0
$$401$$ −4066.71 −0.506438 −0.253219 0.967409i $$-0.581489\pi$$
−0.253219 + 0.967409i $$0.581489\pi$$
$$402$$ − 1124.85i − 0.139558i
$$403$$ 724.274i 0.0895252i
$$404$$ 14136.9 1.74093
$$405$$ 0 0
$$406$$ 9740.45 1.19067
$$407$$ − 4620.77i − 0.562760i
$$408$$ 5787.32i 0.702242i
$$409$$ 13488.8 1.63076 0.815379 0.578927i $$-0.196528\pi$$
0.815379 + 0.578927i $$0.196528\pi$$
$$410$$ 0 0
$$411$$ −2654.52 −0.318584
$$412$$ − 5981.62i − 0.715275i
$$413$$ 12883.3i 1.53498i
$$414$$ 4628.83 0.549504
$$415$$ 0 0
$$416$$ 1167.24 0.137569
$$417$$ − 3275.83i − 0.384695i
$$418$$ 1696.05i 0.198460i
$$419$$ 7040.12 0.820841 0.410420 0.911896i $$-0.365382\pi$$
0.410420 + 0.911896i $$0.365382\pi$$
$$420$$ 0 0
$$421$$ 9171.74 1.06177 0.530883 0.847445i $$-0.321860\pi$$
0.530883 + 0.847445i $$0.321860\pi$$
$$422$$ − 4652.89i − 0.536728i
$$423$$ − 2082.67i − 0.239392i
$$424$$ 77.8204 0.00891343
$$425$$ 0 0
$$426$$ 651.464 0.0740928
$$427$$ 17637.6i 1.99893i
$$428$$ − 1759.77i − 0.198742i
$$429$$ −169.988 −0.0191307
$$430$$ 0 0
$$431$$ 992.995 0.110976 0.0554882 0.998459i $$-0.482328\pi$$
0.0554882 + 0.998459i $$0.482328\pi$$
$$432$$ − 610.490i − 0.0679912i
$$433$$ 3790.21i 0.420660i 0.977630 + 0.210330i $$0.0674538\pi$$
−0.977630 + 0.210330i $$0.932546\pi$$
$$434$$ 19719.0 2.18097
$$435$$ 0 0
$$436$$ 25122.7 2.75954
$$437$$ − 4050.98i − 0.443443i
$$438$$ 10431.3i 1.13796i
$$439$$ 5136.97 0.558483 0.279242 0.960221i $$-0.409917\pi$$
0.279242 + 0.960221i $$0.409917\pi$$
$$440$$ 0 0
$$441$$ 5955.71 0.643095
$$442$$ 2779.16i 0.299075i
$$443$$ 10676.8i 1.14508i 0.819876 + 0.572541i $$0.194042\pi$$
−0.819876 + 0.572541i $$0.805958\pi$$
$$444$$ −14587.7 −1.55923
$$445$$ 0 0
$$446$$ −17086.2 −1.81403
$$447$$ 891.042i 0.0942838i
$$448$$ − 26045.5i − 2.74672i
$$449$$ −10529.9 −1.10676 −0.553379 0.832929i $$-0.686662\pi$$
−0.553379 + 0.832929i $$0.686662\pi$$
$$450$$ 0 0
$$451$$ 3542.64 0.369881
$$452$$ 7482.42i 0.778636i
$$453$$ 5663.59i 0.587414i
$$454$$ −3861.29 −0.399162
$$455$$ 0 0
$$456$$ 1653.91 0.169850
$$457$$ 14072.5i 1.44045i 0.693741 + 0.720225i $$0.255961\pi$$
−0.693741 + 0.720225i $$0.744039\pi$$
$$458$$ − 8148.55i − 0.831347i
$$459$$ 3292.43 0.334810
$$460$$ 0 0
$$461$$ −30.8173 −0.00311346 −0.00155673 0.999999i $$-0.500496\pi$$
−0.00155673 + 0.999999i $$0.500496\pi$$
$$462$$ 4628.06i 0.466054i
$$463$$ 17591.3i 1.76573i 0.469622 + 0.882867i $$0.344390\pi$$
−0.469622 + 0.882867i $$0.655610\pi$$
$$464$$ −1570.39 −0.157120
$$465$$ 0 0
$$466$$ −17397.5 −1.72945
$$467$$ − 13273.1i − 1.31522i −0.753360 0.657609i $$-0.771568\pi$$
0.753360 0.657609i $$-0.228432\pi$$
$$468$$ 536.647i 0.0530053i
$$469$$ −2686.23 −0.264474
$$470$$ 0 0
$$471$$ 169.603 0.0165921
$$472$$ 6429.87i 0.627031i
$$473$$ − 3531.39i − 0.343284i
$$474$$ 5085.23 0.492769
$$475$$ 0 0
$$476$$ 44742.9 4.30837
$$477$$ − 44.2724i − 0.00424968i
$$478$$ − 21113.8i − 2.02034i
$$479$$ −2496.68 −0.238155 −0.119077 0.992885i $$-0.537994\pi$$
−0.119077 + 0.992885i $$0.537994\pi$$
$$480$$ 0 0
$$481$$ −2163.84 −0.205120
$$482$$ 17648.3i 1.66776i
$$483$$ − 11054.0i − 1.04136i
$$484$$ −1400.64 −0.131541
$$485$$ 0 0
$$486$$ 1075.14 0.100348
$$487$$ 3464.42i 0.322357i 0.986925 + 0.161178i $$0.0515295\pi$$
−0.986925 + 0.161178i $$0.948471\pi$$
$$488$$ 8802.65i 0.816552i
$$489$$ −147.701 −0.0136591
$$490$$ 0 0
$$491$$ −16224.6 −1.49125 −0.745625 0.666366i $$-0.767849\pi$$
−0.745625 + 0.666366i $$0.767849\pi$$
$$492$$ − 11184.0i − 1.02483i
$$493$$ − 8469.29i − 0.773707i
$$494$$ 794.236 0.0723367
$$495$$ 0 0
$$496$$ −3179.17 −0.287800
$$497$$ − 1555.75i − 0.140412i
$$498$$ − 12347.0i − 1.11100i
$$499$$ −9993.81 −0.896562 −0.448281 0.893893i $$-0.647964\pi$$
−0.448281 + 0.893893i $$0.647964\pi$$
$$500$$ 0 0
$$501$$ −6206.24 −0.553441
$$502$$ − 24220.3i − 2.15340i
$$503$$ − 15334.8i − 1.35933i −0.733520 0.679667i $$-0.762124\pi$$
0.733520 0.679667i $$-0.237876\pi$$
$$504$$ 4513.08 0.398866
$$505$$ 0 0
$$506$$ 5657.46 0.497045
$$507$$ − 6511.40i − 0.570377i
$$508$$ − 11498.1i − 1.00422i
$$509$$ 7291.23 0.634927 0.317464 0.948270i $$-0.397169\pi$$
0.317464 + 0.948270i $$0.397169\pi$$
$$510$$ 0 0
$$511$$ 24910.7 2.15653
$$512$$ 7985.14i 0.689251i
$$513$$ − 940.919i − 0.0809797i
$$514$$ −28466.8 −2.44283
$$515$$ 0 0
$$516$$ −11148.5 −0.951134
$$517$$ − 2545.49i − 0.216538i
$$518$$ 58912.5i 4.99704i
$$519$$ −1812.04 −0.153255
$$520$$ 0 0
$$521$$ 16794.3 1.41223 0.706114 0.708098i $$-0.250447\pi$$
0.706114 + 0.708098i $$0.250447\pi$$
$$522$$ − 2765.63i − 0.231893i
$$523$$ − 21009.4i − 1.75655i −0.478157 0.878275i $$-0.658695\pi$$
0.478157 0.878275i $$-0.341305\pi$$
$$524$$ −4466.01 −0.372326
$$525$$ 0 0
$$526$$ −33577.0 −2.78332
$$527$$ − 17145.6i − 1.41722i
$$528$$ − 746.154i − 0.0615003i
$$529$$ −1345.73 −0.110605
$$530$$ 0 0
$$531$$ 3657.99 0.298951
$$532$$ − 12786.7i − 1.04206i
$$533$$ − 1658.97i − 0.134818i
$$534$$ 9723.46 0.787969
$$535$$ 0 0
$$536$$ −1340.66 −0.108036
$$537$$ − 6396.07i − 0.513987i
$$538$$ − 2115.66i − 0.169540i
$$539$$ 7279.20 0.581702
$$540$$ 0 0
$$541$$ −16802.8 −1.33532 −0.667662 0.744464i $$-0.732705\pi$$
−0.667662 + 0.744464i $$0.732705\pi$$
$$542$$ − 541.211i − 0.0428911i
$$543$$ 1768.11i 0.139737i
$$544$$ −27631.9 −2.17777
$$545$$ 0 0
$$546$$ 2167.25 0.169872
$$547$$ − 16784.5i − 1.31198i −0.754770 0.655990i $$-0.772251\pi$$
0.754770 0.655990i $$-0.227749\pi$$
$$548$$ 10242.5i 0.798429i
$$549$$ 5007.88 0.389309
$$550$$ 0 0
$$551$$ −2420.37 −0.187135
$$552$$ − 5516.91i − 0.425390i
$$553$$ − 12144.0i − 0.933840i
$$554$$ 36277.7 2.78212
$$555$$ 0 0
$$556$$ −12639.9 −0.964117
$$557$$ − 18127.0i − 1.37893i −0.724317 0.689467i $$-0.757845\pi$$
0.724317 0.689467i $$-0.242155\pi$$
$$558$$ − 5598.85i − 0.424764i
$$559$$ −1653.70 −0.125123
$$560$$ 0 0
$$561$$ 4024.09 0.302847
$$562$$ 30722.3i 2.30595i
$$563$$ 2090.88i 0.156518i 0.996933 + 0.0782592i $$0.0249362\pi$$
−0.996933 + 0.0782592i $$0.975064\pi$$
$$564$$ −8036.03 −0.599961
$$565$$ 0 0
$$566$$ −4579.89 −0.340119
$$567$$ − 2567.51i − 0.190168i
$$568$$ − 776.452i − 0.0573578i
$$569$$ −6249.23 −0.460424 −0.230212 0.973140i $$-0.573942\pi$$
−0.230212 + 0.973140i $$0.573942\pi$$
$$570$$ 0 0
$$571$$ 6048.79 0.443317 0.221659 0.975124i $$-0.428853\pi$$
0.221659 + 0.975124i $$0.428853\pi$$
$$572$$ 655.902i 0.0479451i
$$573$$ 6482.69i 0.472633i
$$574$$ −45166.8 −3.28437
$$575$$ 0 0
$$576$$ −7395.15 −0.534950
$$577$$ 15729.1i 1.13486i 0.823423 + 0.567429i $$0.192062\pi$$
−0.823423 + 0.567429i $$0.807938\pi$$
$$578$$ − 44053.4i − 3.17021i
$$579$$ −4472.73 −0.321037
$$580$$ 0 0
$$581$$ −29485.6 −2.10545
$$582$$ 15549.5i 1.10747i
$$583$$ − 54.1108i − 0.00384398i
$$584$$ 12432.6 0.880931
$$585$$ 0 0
$$586$$ 27187.3 1.91655
$$587$$ − 15620.5i − 1.09835i −0.835709 0.549173i $$-0.814943\pi$$
0.835709 0.549173i $$-0.185057\pi$$
$$588$$ − 22980.2i − 1.61172i
$$589$$ −4899.91 −0.342780
$$590$$ 0 0
$$591$$ 691.587 0.0481355
$$592$$ − 9498.09i − 0.659407i
$$593$$ − 493.541i − 0.0341776i −0.999854 0.0170888i $$-0.994560\pi$$
0.999854 0.0170888i $$-0.00543980\pi$$
$$594$$ 1314.06 0.0907683
$$595$$ 0 0
$$596$$ 3438.11 0.236293
$$597$$ 67.2022i 0.00460704i
$$598$$ − 2649.31i − 0.181168i
$$599$$ 12455.1 0.849585 0.424793 0.905291i $$-0.360347\pi$$
0.424793 + 0.905291i $$0.360347\pi$$
$$600$$ 0 0
$$601$$ 12454.8 0.845329 0.422664 0.906286i $$-0.361095\pi$$
0.422664 + 0.906286i $$0.361095\pi$$
$$602$$ 45023.3i 3.04820i
$$603$$ 762.707i 0.0515088i
$$604$$ 21853.1 1.47217
$$605$$ 0 0
$$606$$ −16210.3 −1.08663
$$607$$ 4243.19i 0.283733i 0.989886 + 0.141867i $$0.0453104\pi$$
−0.989886 + 0.141867i $$0.954690\pi$$
$$608$$ 7896.70i 0.526732i
$$609$$ −6604.54 −0.439458
$$610$$ 0 0
$$611$$ −1192.01 −0.0789259
$$612$$ − 12703.9i − 0.839095i
$$613$$ 5733.14i 0.377748i 0.982001 + 0.188874i $$0.0604838\pi$$
−0.982001 + 0.188874i $$0.939516\pi$$
$$614$$ −9672.18 −0.635729
$$615$$ 0 0
$$616$$ 5515.99 0.360788
$$617$$ − 15642.1i − 1.02063i −0.859988 0.510314i $$-0.829529\pi$$
0.859988 0.510314i $$-0.170471\pi$$
$$618$$ 6858.91i 0.446449i
$$619$$ 7467.40 0.484879 0.242440 0.970167i $$-0.422052\pi$$
0.242440 + 0.970167i $$0.422052\pi$$
$$620$$ 0 0
$$621$$ −3138.60 −0.202814
$$622$$ − 33116.1i − 2.13478i
$$623$$ − 23220.4i − 1.49327i
$$624$$ −349.413 −0.0224162
$$625$$ 0 0
$$626$$ 30233.6 1.93031
$$627$$ − 1150.01i − 0.0732489i
$$628$$ − 654.416i − 0.0415829i
$$629$$ 51224.2 3.24713
$$630$$ 0 0
$$631$$ −1486.38 −0.0937745 −0.0468872 0.998900i $$-0.514930\pi$$
−0.0468872 + 0.998900i $$0.514930\pi$$
$$632$$ − 6060.87i − 0.381469i
$$633$$ 3154.91i 0.198099i
$$634$$ 4089.35 0.256165
$$635$$ 0 0
$$636$$ −170.826 −0.0106505
$$637$$ − 3408.74i − 0.212024i
$$638$$ − 3380.21i − 0.209755i
$$639$$ −441.728 −0.0273466
$$640$$ 0 0
$$641$$ 12386.0 0.763211 0.381606 0.924325i $$-0.375371\pi$$
0.381606 + 0.924325i $$0.375371\pi$$
$$642$$ 2017.87i 0.124048i
$$643$$ − 14458.1i − 0.886737i −0.896339 0.443369i $$-0.853783\pi$$
0.896339 0.443369i $$-0.146217\pi$$
$$644$$ −42652.3 −2.60984
$$645$$ 0 0
$$646$$ −18801.8 −1.14512
$$647$$ − 15792.8i − 0.959625i −0.877371 0.479813i $$-0.840705\pi$$
0.877371 0.479813i $$-0.159295\pi$$
$$648$$ − 1281.41i − 0.0776828i
$$649$$ 4470.87 0.270412
$$650$$ 0 0
$$651$$ −13370.5 −0.804965
$$652$$ 569.909i 0.0342321i
$$653$$ − 3179.93i − 0.190567i −0.995450 0.0952837i $$-0.969624\pi$$
0.995450 0.0952837i $$-0.0303758\pi$$
$$654$$ −28807.3 −1.72240
$$655$$ 0 0
$$656$$ 7281.96 0.433404
$$657$$ − 7072.96i − 0.420003i
$$658$$ 32453.6i 1.92276i
$$659$$ −11593.5 −0.685308 −0.342654 0.939462i $$-0.611326\pi$$
−0.342654 + 0.939462i $$0.611326\pi$$
$$660$$ 0 0
$$661$$ 3233.88 0.190293 0.0951464 0.995463i $$-0.469668\pi$$
0.0951464 + 0.995463i $$0.469668\pi$$
$$662$$ − 43449.9i − 2.55095i
$$663$$ − 1884.42i − 0.110384i
$$664$$ −14715.8 −0.860066
$$665$$ 0 0
$$666$$ 16727.1 0.973219
$$667$$ 8073.56i 0.468680i
$$668$$ 23946.9i 1.38703i
$$669$$ 11585.4 0.669532
$$670$$ 0 0
$$671$$ 6120.74 0.352144
$$672$$ 21548.0i 1.23695i
$$673$$ − 5495.72i − 0.314776i −0.987537 0.157388i $$-0.949693\pi$$
0.987537 0.157388i $$-0.0503074\pi$$
$$674$$ 2658.23 0.151916
$$675$$ 0 0
$$676$$ −25124.4 −1.42947
$$677$$ − 33836.7i − 1.92090i −0.278448 0.960451i $$-0.589820\pi$$
0.278448 0.960451i $$-0.410180\pi$$
$$678$$ − 8579.82i − 0.485997i
$$679$$ 37133.6 2.09876
$$680$$ 0 0
$$681$$ 2618.16 0.147325
$$682$$ − 6843.04i − 0.384214i
$$683$$ − 21080.3i − 1.18099i −0.807043 0.590493i $$-0.798933\pi$$
0.807043 0.590493i $$-0.201067\pi$$
$$684$$ −3630.56 −0.202950
$$685$$ 0 0
$$686$$ −44702.2 −2.48796
$$687$$ 5525.16i 0.306838i
$$688$$ − 7258.84i − 0.402239i
$$689$$ −25.3393 −0.00140109
$$690$$ 0 0
$$691$$ 11811.3 0.650253 0.325127 0.945671i $$-0.394593\pi$$
0.325127 + 0.945671i $$0.394593\pi$$
$$692$$ 6991.79i 0.384087i
$$693$$ − 3138.07i − 0.172014i
$$694$$ −13907.8 −0.760711
$$695$$ 0 0
$$696$$ −3296.23 −0.179516
$$697$$ 39272.4i 2.13422i
$$698$$ − 3188.52i − 0.172904i
$$699$$ 11796.4 0.638313
$$700$$ 0 0
$$701$$ 4244.99 0.228718 0.114359 0.993440i $$-0.463519\pi$$
0.114359 + 0.993440i $$0.463519\pi$$
$$702$$ − 615.353i − 0.0330841i
$$703$$ − 14639.0i − 0.785376i
$$704$$ −9038.51 −0.483880
$$705$$ 0 0
$$706$$ −5340.81 −0.284708
$$707$$ 38711.5i 2.05926i
$$708$$ − 14114.4i − 0.749227i
$$709$$ 898.822 0.0476107 0.0238053 0.999717i $$-0.492422\pi$$
0.0238053 + 0.999717i $$0.492422\pi$$
$$710$$ 0 0
$$711$$ −3448.06 −0.181874
$$712$$ − 11589.0i − 0.609993i
$$713$$ 16344.5i 0.858493i
$$714$$ −51305.0 −2.68913
$$715$$ 0 0
$$716$$ −24679.4 −1.28815
$$717$$ 14316.3i 0.745679i
$$718$$ − 38706.3i − 2.01185i
$$719$$ −10741.8 −0.557165 −0.278582 0.960412i $$-0.589865\pi$$
−0.278582 + 0.960412i $$0.589865\pi$$
$$720$$ 0 0
$$721$$ 16379.6 0.846061
$$722$$ − 24973.9i − 1.28730i
$$723$$ − 11966.5i − 0.615546i
$$724$$ 6822.30 0.350206
$$725$$ 0 0
$$726$$ 1606.07 0.0821030
$$727$$ − 16794.2i − 0.856758i −0.903599 0.428379i $$-0.859085\pi$$
0.903599 0.428379i $$-0.140915\pi$$
$$728$$ − 2583.06i − 0.131503i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 39147.7 1.98075
$$732$$ − 19323.0i − 0.975681i
$$733$$ 8659.40i 0.436347i 0.975910 + 0.218173i $$0.0700098\pi$$
−0.975910 + 0.218173i $$0.929990\pi$$
$$734$$ −29778.4 −1.49747
$$735$$ 0 0
$$736$$ 26340.8 1.31920
$$737$$ 932.197i 0.0465915i
$$738$$ 12824.3i 0.639660i
$$739$$ −16705.7 −0.831567 −0.415783 0.909464i $$-0.636493\pi$$
−0.415783 + 0.909464i $$0.636493\pi$$
$$740$$ 0 0
$$741$$ −538.534 −0.0266984
$$742$$ 689.884i 0.0341327i
$$743$$ 1292.12i 0.0637996i 0.999491 + 0.0318998i $$0.0101558\pi$$
−0.999491 + 0.0318998i $$0.989844\pi$$
$$744$$ −6673.03 −0.328824
$$745$$ 0 0
$$746$$ 1006.09 0.0493773
$$747$$ 8371.90i 0.410056i
$$748$$ − 15527.0i − 0.758990i
$$749$$ 4818.83 0.235082
$$750$$ 0 0
$$751$$ −14980.4 −0.727886 −0.363943 0.931421i $$-0.618570\pi$$
−0.363943 + 0.931421i $$0.618570\pi$$
$$752$$ − 5232.30i − 0.253726i
$$753$$ 16422.7i 0.794787i
$$754$$ −1582.90 −0.0764535
$$755$$ 0 0
$$756$$ −9906.82 −0.476597
$$757$$ − 3003.41i − 0.144202i −0.997397 0.0721010i $$-0.977030\pi$$
0.997397 0.0721010i $$-0.0229704\pi$$
$$758$$ − 50244.8i − 2.40762i
$$759$$ −3836.06 −0.183452
$$760$$ 0 0
$$761$$ −20375.0 −0.970555 −0.485277 0.874360i $$-0.661281\pi$$
−0.485277 + 0.874360i $$0.661281\pi$$
$$762$$ 13184.4i 0.626799i
$$763$$ 68794.1i 3.26411i
$$764$$ 25013.6 1.18450
$$765$$ 0 0
$$766$$ −47578.4 −2.24422
$$767$$ − 2093.65i − 0.0985621i
$$768$$ 4472.70i 0.210149i
$$769$$ 12372.4 0.580184 0.290092 0.956999i $$-0.406314\pi$$
0.290092 + 0.956999i $$0.406314\pi$$
$$770$$ 0 0
$$771$$ 19302.0 0.901615
$$772$$ 17258.1i 0.804578i
$$773$$ 21023.6i 0.978225i 0.872221 + 0.489113i $$0.162679\pi$$
−0.872221 + 0.489113i $$0.837321\pi$$
$$774$$ 12783.6 0.593664
$$775$$ 0 0
$$776$$ 18532.8 0.857331
$$777$$ − 39945.8i − 1.84433i
$$778$$ 51885.7i 2.39099i
$$779$$ 11223.4 0.516198
$$780$$ 0 0
$$781$$ −539.889 −0.0247359
$$782$$ 62716.6i 2.86795i
$$783$$ 1875.24i 0.0855884i
$$784$$ 14962.5 0.681602
$$785$$ 0 0
$$786$$ 5121.02 0.232393
$$787$$ − 30286.2i − 1.37177i −0.727709 0.685886i $$-0.759415\pi$$
0.727709 0.685886i $$-0.240585\pi$$
$$788$$ − 2668.51i − 0.120637i
$$789$$ 22767.0 1.02728
$$790$$ 0 0
$$791$$ −20489.3 −0.921007
$$792$$ − 1566.17i − 0.0702668i
$$793$$ − 2866.25i − 0.128353i
$$794$$ −1592.37 −0.0711729
$$795$$ 0 0
$$796$$ 259.301 0.0115461
$$797$$ − 32337.8i − 1.43722i −0.695413 0.718610i $$-0.744779\pi$$
0.695413 0.718610i $$-0.255221\pi$$
$$798$$ 14662.1i 0.650415i
$$799$$ 28218.3 1.24943
$$800$$ 0 0
$$801$$ −6593.03 −0.290828
$$802$$ − 17992.9i − 0.792207i
$$803$$ − 8644.72i − 0.379907i
$$804$$ 2942.92 0.129091
$$805$$ 0 0
$$806$$ −3204.50 −0.140042
$$807$$ 1434.53i 0.0625749i
$$808$$ 19320.3i 0.841197i
$$809$$ 891.707 0.0387525 0.0193762 0.999812i $$-0.493832\pi$$
0.0193762 + 0.999812i $$0.493832\pi$$
$$810$$ 0 0
$$811$$ −10114.9 −0.437957 −0.218978 0.975730i $$-0.570272\pi$$
−0.218978 + 0.975730i $$0.570272\pi$$
$$812$$ 25483.8i 1.10136i
$$813$$ 366.970i 0.0158305i
$$814$$ 20444.3 0.880309
$$815$$ 0 0
$$816$$ 8271.59 0.354857
$$817$$ − 11187.7i − 0.479080i
$$818$$ 59680.4i 2.55095i
$$819$$ −1469.51 −0.0626972
$$820$$ 0 0
$$821$$ 10833.5 0.460525 0.230262 0.973129i $$-0.426042\pi$$
0.230262 + 0.973129i $$0.426042\pi$$
$$822$$ − 11744.7i − 0.498351i
$$823$$ 31958.5i 1.35359i 0.736173 + 0.676794i $$0.236631\pi$$
−0.736173 + 0.676794i $$0.763369\pi$$
$$824$$ 8174.84 0.345612
$$825$$ 0 0
$$826$$ −57001.3 −2.40112
$$827$$ − 34847.3i − 1.46525i −0.680634 0.732624i $$-0.738296\pi$$
0.680634 0.732624i $$-0.261704\pi$$
$$828$$ 12110.3i 0.508289i
$$829$$ −6537.91 −0.273910 −0.136955 0.990577i $$-0.543732\pi$$
−0.136955 + 0.990577i $$0.543732\pi$$
$$830$$ 0 0
$$831$$ −24598.2 −1.02684
$$832$$ 4232.60i 0.176369i
$$833$$ 80694.5i 3.35642i
$$834$$ 14493.7 0.601768
$$835$$ 0 0
$$836$$ −4437.35 −0.183575
$$837$$ 3796.32i 0.156774i
$$838$$ 31148.5i 1.28402i
$$839$$ −2710.34 −0.111527 −0.0557635 0.998444i $$-0.517759\pi$$
−0.0557635 + 0.998444i $$0.517759\pi$$
$$840$$ 0 0
$$841$$ −19565.2 −0.802215
$$842$$ 40579.7i 1.66089i
$$843$$ − 20831.4i − 0.851092i
$$844$$ 12173.3 0.496471
$$845$$ 0 0
$$846$$ 9214.62 0.374474
$$847$$ − 3835.42i − 0.155592i
$$848$$ − 111.226i − 0.00450414i
$$849$$ 3105.41 0.125533
$$850$$ 0 0
$$851$$ −48830.8 −1.96698
$$852$$ 1704.42i 0.0685356i
$$853$$ 9759.32i 0.391738i 0.980630 + 0.195869i $$0.0627528\pi$$
−0.980630 + 0.195869i $$0.937247\pi$$
$$854$$ −78036.2 −3.12687
$$855$$ 0 0
$$856$$ 2405.01 0.0960298
$$857$$ 13649.8i 0.544072i 0.962287 + 0.272036i $$0.0876970\pi$$
−0.962287 + 0.272036i $$0.912303\pi$$
$$858$$ − 752.098i − 0.0299257i
$$859$$ −7796.42 −0.309674 −0.154837 0.987940i $$-0.549485\pi$$
−0.154837 + 0.987940i $$0.549485\pi$$
$$860$$ 0 0
$$861$$ 30625.5 1.21221
$$862$$ 4393.43i 0.173597i
$$863$$ 7183.57i 0.283350i 0.989913 + 0.141675i $$0.0452489\pi$$
−0.989913 + 0.141675i $$0.954751\pi$$
$$864$$ 6118.16 0.240907
$$865$$ 0 0
$$866$$ −16769.5 −0.658026
$$867$$ 29870.6i 1.17008i
$$868$$ 51590.5i 2.01739i
$$869$$ −4214.30 −0.164511
$$870$$ 0 0
$$871$$ 436.534 0.0169821
$$872$$ 34334.1i 1.33337i
$$873$$ − 10543.4i − 0.408752i
$$874$$ 17923.3 0.693666
$$875$$ 0 0
$$876$$ −27291.2 −1.05261
$$877$$ − 17063.1i − 0.656991i −0.944506 0.328495i $$-0.893458\pi$$
0.944506 0.328495i $$-0.106542\pi$$
$$878$$ 22728.2i 0.873620i
$$879$$ −18434.4 −0.707369
$$880$$ 0 0
$$881$$ −32174.9 −1.23042 −0.615210 0.788363i $$-0.710929\pi$$
−0.615210 + 0.788363i $$0.710929\pi$$
$$882$$ 26350.6i 1.00598i
$$883$$ 2843.68i 0.108378i 0.998531 + 0.0541889i $$0.0172573\pi$$
−0.998531 + 0.0541889i $$0.982743\pi$$
$$884$$ −7271.09 −0.276644
$$885$$ 0 0
$$886$$ −47238.9 −1.79122
$$887$$ 31417.8i 1.18930i 0.803985 + 0.594649i $$0.202709\pi$$
−0.803985 + 0.594649i $$0.797291\pi$$
$$888$$ − 19936.4i − 0.753401i
$$889$$ 31485.5 1.18784
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ − 44702.5i − 1.67797i
$$893$$ − 8064.30i − 0.302196i
$$894$$ −3942.35 −0.147485
$$895$$ 0 0
$$896$$ 57775.1 2.15416
$$897$$ 1796.37i 0.0668664i
$$898$$ − 46588.6i − 1.73127i
$$899$$ 9765.47 0.362288
$$900$$ 0 0
$$901$$ 599.852 0.0221798
$$902$$ 15674.1i 0.578594i
$$903$$ − 30528.2i − 1.12505i
$$904$$ −10225.9 −0.376227
$$905$$ 0 0
$$906$$ −25058.1 −0.918875
$$907$$ − 12253.1i − 0.448573i −0.974523 0.224287i $$-0.927995\pi$$
0.974523 0.224287i $$-0.0720052\pi$$
$$908$$ − 10102.2i − 0.369223i
$$909$$ 10991.4 0.401059
$$910$$ 0 0
$$911$$ −48422.4 −1.76104 −0.880518 0.474012i $$-0.842805\pi$$
−0.880518 + 0.474012i $$0.842805\pi$$
$$912$$ − 2363.87i − 0.0858286i
$$913$$ 10232.3i 0.370909i
$$914$$ −62262.9 −2.25326
$$915$$ 0 0
$$916$$ 21318.9 0.768993
$$917$$ − 12229.4i − 0.440404i
$$918$$ 14567.1i 0.523733i
$$919$$ −5546.18 −0.199077 −0.0995385 0.995034i $$-0.531737\pi$$
−0.0995385 + 0.995034i $$0.531737\pi$$
$$920$$ 0 0
$$921$$ 6558.26 0.234638
$$922$$ − 136.349i − 0.00487030i
$$923$$ 252.822i 0.00901598i
$$924$$ −12108.3 −0.431098
$$925$$ 0 0
$$926$$ −77831.3 −2.76209
$$927$$ − 4650.71i − 0.164778i
$$928$$ − 15738.0i − 0.556709i
$$929$$ 35684.5 1.26025 0.630125 0.776494i $$-0.283004\pi$$
0.630125 + 0.776494i $$0.283004\pi$$
$$930$$ 0 0
$$931$$ 23061.1 0.811811
$$932$$ − 45516.7i − 1.59973i
$$933$$ 22454.5i 0.787918i
$$934$$ 58726.0 2.05736
$$935$$ 0 0
$$936$$ −733.413 −0.0256115
$$937$$ 48903.6i 1.70503i 0.522705 + 0.852514i $$0.324923\pi$$
−0.522705 + 0.852514i $$0.675077\pi$$
$$938$$ − 11885.0i − 0.413710i
$$939$$ −20500.0 −0.712451
$$940$$ 0 0
$$941$$ −23741.9 −0.822490 −0.411245 0.911525i $$-0.634906\pi$$
−0.411245 + 0.911525i $$0.634906\pi$$
$$942$$ 750.396i 0.0259546i
$$943$$ − 37437.4i − 1.29282i
$$944$$ 9189.97 0.316852
$$945$$ 0 0
$$946$$ 15624.4 0.536989
$$947$$ − 37612.4i − 1.29064i −0.763911 0.645321i $$-0.776724\pi$$
0.763911 0.645321i $$-0.223276\pi$$
$$948$$ 13304.4i 0.455810i
$$949$$ −4048.20 −0.138472
$$950$$ 0 0
$$951$$ −2772.80 −0.0945469
$$952$$ 61148.2i 2.08175i
$$953$$ − 48294.3i − 1.64156i −0.571246 0.820779i $$-0.693540\pi$$
0.571246 0.820779i $$-0.306460\pi$$
$$954$$ 195.880 0.00664765
$$955$$ 0 0
$$956$$ 55239.8 1.86881
$$957$$ 2291.96i 0.0774176i
$$958$$ − 11046.4i − 0.372539i
$$959$$ −28047.4 −0.944419
$$960$$ 0 0
$$961$$ −10021.4 −0.336389
$$962$$ − 9573.76i − 0.320863i
$$963$$ − 1368.22i − 0.0457843i
$$964$$ −46173.1 −1.54267
$$965$$ 0 0
$$966$$ 48907.8 1.62897
$$967$$ − 1840.92i − 0.0612204i −0.999531 0.0306102i $$-0.990255\pi$$
0.999531 0.0306102i $$-0.00974505\pi$$
$$968$$ − 1914.20i − 0.0635587i
$$969$$ 12748.6 0.422647
$$970$$ 0 0
$$971$$ 31461.8 1.03981 0.519906 0.854223i $$-0.325967\pi$$
0.519906 + 0.854223i $$0.325967\pi$$
$$972$$ 2812.86i 0.0928217i
$$973$$ − 34612.1i − 1.14040i
$$974$$ −15328.1 −0.504254
$$975$$ 0 0
$$976$$ 12581.3 0.412620
$$977$$ 7040.11i 0.230535i 0.993334 + 0.115268i $$0.0367726\pi$$
−0.993334 + 0.115268i $$0.963227\pi$$
$$978$$ − 653.494i − 0.0213665i
$$979$$ −8058.15 −0.263064
$$980$$ 0 0
$$981$$ 19532.9 0.635715
$$982$$ − 71784.4i − 2.33272i
$$983$$ − 24610.9i − 0.798541i −0.916833 0.399270i $$-0.869263\pi$$
0.916833 0.399270i $$-0.130737\pi$$
$$984$$ 15284.7 0.495183
$$985$$ 0 0
$$986$$ 37471.8 1.21029
$$987$$ − 22005.3i − 0.709662i
$$988$$ 2077.95i 0.0669112i
$$989$$ −37318.5 −1.19986
$$990$$ 0 0
$$991$$ −40003.3 −1.28229 −0.641144 0.767421i $$-0.721540\pi$$
−0.641144 + 0.767421i $$0.721540\pi$$
$$992$$ − 31860.8i − 1.01974i
$$993$$ 29461.4i 0.941519i
$$994$$ 6883.31 0.219643
$$995$$ 0 0
$$996$$ 32303.2 1.02768
$$997$$ 7342.61i 0.233242i 0.993176 + 0.116621i $$0.0372063\pi$$
−0.993176 + 0.116621i $$0.962794\pi$$
$$998$$ − 44216.9i − 1.40247i
$$999$$ −11341.9 −0.359201
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.h.199.3 4
5.2 odd 4 33.4.a.c.1.1 2
5.3 odd 4 825.4.a.l.1.2 2
5.4 even 2 inner 825.4.c.h.199.2 4
15.2 even 4 99.4.a.f.1.2 2
15.8 even 4 2475.4.a.p.1.1 2
20.7 even 4 528.4.a.p.1.2 2
35.27 even 4 1617.4.a.k.1.1 2
40.27 even 4 2112.4.a.bg.1.1 2
40.37 odd 4 2112.4.a.bn.1.1 2
55.32 even 4 363.4.a.i.1.2 2
60.47 odd 4 1584.4.a.bj.1.1 2
165.32 odd 4 1089.4.a.u.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 5.2 odd 4
99.4.a.f.1.2 2 15.2 even 4
363.4.a.i.1.2 2 55.32 even 4
528.4.a.p.1.2 2 20.7 even 4
825.4.a.l.1.2 2 5.3 odd 4
825.4.c.h.199.2 4 5.4 even 2 inner
825.4.c.h.199.3 4 1.1 even 1 trivial
1089.4.a.u.1.1 2 165.32 odd 4
1584.4.a.bj.1.1 2 60.47 odd 4
1617.4.a.k.1.1 2 35.27 even 4
2112.4.a.bg.1.1 2 40.27 even 4
2112.4.a.bn.1.1 2 40.37 odd 4
2475.4.a.p.1.1 2 15.8 even 4