# Properties

 Label 825.4.a.l.1.2 Level $825$ Weight $4$ Character 825.1 Self dual yes Analytic conductor $48.677$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(1,825)]

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, 0, 0]))

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$48.6765757547$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-4.42443$$ of defining polynomial Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.42443 q^{2} +3.00000 q^{3} +11.5756 q^{4} +13.2733 q^{6} -31.6977 q^{7} +15.8199 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+4.42443 q^{2} +3.00000 q^{3} +11.5756 q^{4} +13.2733 q^{6} -31.6977 q^{7} +15.8199 q^{8} +9.00000 q^{9} -11.0000 q^{11} +34.7267 q^{12} -5.15114 q^{13} -140.244 q^{14} -22.6107 q^{16} -121.942 q^{17} +39.8199 q^{18} +34.8489 q^{19} -95.0931 q^{21} -48.6687 q^{22} -116.244 q^{23} +47.4596 q^{24} -22.7909 q^{26} +27.0000 q^{27} -366.919 q^{28} -69.4534 q^{29} +140.605 q^{31} -226.598 q^{32} -33.0000 q^{33} -539.524 q^{34} +104.180 q^{36} +420.070 q^{37} +154.186 q^{38} -15.4534 q^{39} -322.058 q^{41} -420.733 q^{42} -321.035 q^{43} -127.331 q^{44} -514.315 q^{46} +231.408 q^{47} -67.8322 q^{48} +661.745 q^{49} -365.826 q^{51} -59.6274 q^{52} -4.91916 q^{53} +119.460 q^{54} -501.453 q^{56} +104.547 q^{57} -307.292 q^{58} +406.443 q^{59} -556.431 q^{61} +622.095 q^{62} -285.279 q^{63} -821.683 q^{64} -146.006 q^{66} -84.7452 q^{67} -1411.55 q^{68} -348.733 q^{69} +49.0808 q^{71} +142.379 q^{72} -785.884 q^{73} +1858.57 q^{74} +403.395 q^{76} +348.675 q^{77} -68.3726 q^{78} -383.118 q^{79} +81.0000 q^{81} -1424.92 q^{82} +930.211 q^{83} -1100.76 q^{84} -1420.40 q^{86} -208.360 q^{87} -174.018 q^{88} -732.559 q^{89} +163.279 q^{91} -1345.59 q^{92} +421.814 q^{93} +1023.85 q^{94} -679.795 q^{96} +1171.49 q^{97} +2927.84 q^{98} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 6 q^{3} + 33 q^{4} - 3 q^{6} - 24 q^{7} - 57 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - q^2 + 6 * q^3 + 33 * q^4 - 3 * q^6 - 24 * q^7 - 57 * q^8 + 18 * q^9 $$2 q - q^{2} + 6 q^{3} + 33 q^{4} - 3 q^{6} - 24 q^{7} - 57 q^{8} + 18 q^{9} - 22 q^{11} + 99 q^{12} - 30 q^{13} - 182 q^{14} + 201 q^{16} - 106 q^{17} - 9 q^{18} + 50 q^{19} - 72 q^{21} + 11 q^{22} - 134 q^{23} - 171 q^{24} + 112 q^{26} + 54 q^{27} - 202 q^{28} - 198 q^{29} + 360 q^{31} - 857 q^{32} - 66 q^{33} - 626 q^{34} + 297 q^{36} + 328 q^{37} + 72 q^{38} - 90 q^{39} - 782 q^{41} - 546 q^{42} - 386 q^{43} - 363 q^{44} - 418 q^{46} - 266 q^{47} + 603 q^{48} + 378 q^{49} - 318 q^{51} - 592 q^{52} + 522 q^{53} - 27 q^{54} - 1062 q^{56} + 150 q^{57} + 390 q^{58} - 172 q^{59} - 778 q^{61} - 568 q^{62} - 216 q^{63} + 809 q^{64} + 33 q^{66} + 776 q^{67} - 1070 q^{68} - 402 q^{69} + 630 q^{71} - 513 q^{72} - 1296 q^{73} + 2358 q^{74} + 728 q^{76} + 264 q^{77} + 336 q^{78} + 652 q^{79} + 162 q^{81} + 1070 q^{82} + 324 q^{83} - 606 q^{84} - 1068 q^{86} - 594 q^{87} + 627 q^{88} - 756 q^{89} - 28 q^{91} - 1726 q^{92} + 1080 q^{93} + 3722 q^{94} - 2571 q^{96} + 452 q^{97} + 4467 q^{98} - 198 q^{99}+O(q^{100})$$ 2 * q - q^2 + 6 * q^3 + 33 * q^4 - 3 * q^6 - 24 * q^7 - 57 * q^8 + 18 * q^9 - 22 * q^11 + 99 * q^12 - 30 * q^13 - 182 * q^14 + 201 * q^16 - 106 * q^17 - 9 * q^18 + 50 * q^19 - 72 * q^21 + 11 * q^22 - 134 * q^23 - 171 * q^24 + 112 * q^26 + 54 * q^27 - 202 * q^28 - 198 * q^29 + 360 * q^31 - 857 * q^32 - 66 * q^33 - 626 * q^34 + 297 * q^36 + 328 * q^37 + 72 * q^38 - 90 * q^39 - 782 * q^41 - 546 * q^42 - 386 * q^43 - 363 * q^44 - 418 * q^46 - 266 * q^47 + 603 * q^48 + 378 * q^49 - 318 * q^51 - 592 * q^52 + 522 * q^53 - 27 * q^54 - 1062 * q^56 + 150 * q^57 + 390 * q^58 - 172 * q^59 - 778 * q^61 - 568 * q^62 - 216 * q^63 + 809 * q^64 + 33 * q^66 + 776 * q^67 - 1070 * q^68 - 402 * q^69 + 630 * q^71 - 513 * q^72 - 1296 * q^73 + 2358 * q^74 + 728 * q^76 + 264 * q^77 + 336 * q^78 + 652 * q^79 + 162 * q^81 + 1070 * q^82 + 324 * q^83 - 606 * q^84 - 1068 * q^86 - 594 * q^87 + 627 * q^88 - 756 * q^89 - 28 * q^91 - 1726 * q^92 + 1080 * q^93 + 3722 * q^94 - 2571 * q^96 + 452 * q^97 + 4467 * q^98 - 198 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.42443 1.56427 0.782136 0.623108i $$-0.214130\pi$$
0.782136 + 0.623108i $$0.214130\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 11.5756 1.44695
$$5$$ 0 0
$$6$$ 13.2733 0.903133
$$7$$ −31.6977 −1.71152 −0.855758 0.517377i $$-0.826909\pi$$
−0.855758 + 0.517377i $$0.826909\pi$$
$$8$$ 15.8199 0.699146
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 34.7267 0.835395
$$13$$ −5.15114 −0.109898 −0.0549488 0.998489i $$-0.517500\pi$$
−0.0549488 + 0.998489i $$0.517500\pi$$
$$14$$ −140.244 −2.67728
$$15$$ 0 0
$$16$$ −22.6107 −0.353293
$$17$$ −121.942 −1.73972 −0.869861 0.493297i $$-0.835792\pi$$
−0.869861 + 0.493297i $$0.835792\pi$$
$$18$$ 39.8199 0.521424
$$19$$ 34.8489 0.420783 0.210391 0.977617i $$-0.432526\pi$$
0.210391 + 0.977617i $$0.432526\pi$$
$$20$$ 0 0
$$21$$ −95.0931 −0.988144
$$22$$ −48.6687 −0.471646
$$23$$ −116.244 −1.05385 −0.526926 0.849911i $$-0.676656\pi$$
−0.526926 + 0.849911i $$0.676656\pi$$
$$24$$ 47.4596 0.403652
$$25$$ 0 0
$$26$$ −22.7909 −0.171910
$$27$$ 27.0000 0.192450
$$28$$ −366.919 −2.47647
$$29$$ −69.4534 −0.444730 −0.222365 0.974963i $$-0.571378\pi$$
−0.222365 + 0.974963i $$0.571378\pi$$
$$30$$ 0 0
$$31$$ 140.605 0.814623 0.407312 0.913289i $$-0.366466\pi$$
0.407312 + 0.913289i $$0.366466\pi$$
$$32$$ −226.598 −1.25179
$$33$$ −33.0000 −0.174078
$$34$$ −539.524 −2.72140
$$35$$ 0 0
$$36$$ 104.180 0.482315
$$37$$ 420.070 1.86646 0.933232 0.359276i $$-0.116976\pi$$
0.933232 + 0.359276i $$0.116976\pi$$
$$38$$ 154.186 0.658219
$$39$$ −15.4534 −0.0634495
$$40$$ 0 0
$$41$$ −322.058 −1.22676 −0.613378 0.789789i $$-0.710190\pi$$
−0.613378 + 0.789789i $$0.710190\pi$$
$$42$$ −420.733 −1.54573
$$43$$ −321.035 −1.13854 −0.569272 0.822149i $$-0.692775\pi$$
−0.569272 + 0.822149i $$0.692775\pi$$
$$44$$ −127.331 −0.436271
$$45$$ 0 0
$$46$$ −514.315 −1.64851
$$47$$ 231.408 0.718176 0.359088 0.933304i $$-0.383088\pi$$
0.359088 + 0.933304i $$0.383088\pi$$
$$48$$ −67.8322 −0.203974
$$49$$ 661.745 1.92929
$$50$$ 0 0
$$51$$ −365.826 −1.00443
$$52$$ −59.6274 −0.159016
$$53$$ −4.91916 −0.0127490 −0.00637452 0.999980i $$-0.502029\pi$$
−0.00637452 + 0.999980i $$0.502029\pi$$
$$54$$ 119.460 0.301044
$$55$$ 0 0
$$56$$ −501.453 −1.19660
$$57$$ 104.547 0.242939
$$58$$ −307.292 −0.695679
$$59$$ 406.443 0.896854 0.448427 0.893820i $$-0.351984\pi$$
0.448427 + 0.893820i $$0.351984\pi$$
$$60$$ 0 0
$$61$$ −556.431 −1.16793 −0.583964 0.811779i $$-0.698499\pi$$
−0.583964 + 0.811779i $$0.698499\pi$$
$$62$$ 622.095 1.27429
$$63$$ −285.279 −0.570505
$$64$$ −821.683 −1.60485
$$65$$ 0 0
$$66$$ −146.006 −0.272305
$$67$$ −84.7452 −0.154526 −0.0772632 0.997011i $$-0.524618\pi$$
−0.0772632 + 0.997011i $$0.524618\pi$$
$$68$$ −1411.55 −2.51728
$$69$$ −348.733 −0.608442
$$70$$ 0 0
$$71$$ 49.0808 0.0820398 0.0410199 0.999158i $$-0.486939\pi$$
0.0410199 + 0.999158i $$0.486939\pi$$
$$72$$ 142.379 0.233049
$$73$$ −785.884 −1.26001 −0.630005 0.776591i $$-0.716947\pi$$
−0.630005 + 0.776591i $$0.716947\pi$$
$$74$$ 1858.57 2.91966
$$75$$ 0 0
$$76$$ 403.395 0.608850
$$77$$ 348.675 0.516041
$$78$$ −68.3726 −0.0992522
$$79$$ −383.118 −0.545622 −0.272811 0.962068i $$-0.587953\pi$$
−0.272811 + 0.962068i $$0.587953\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ −1424.92 −1.91898
$$83$$ 930.211 1.23017 0.615084 0.788462i $$-0.289122\pi$$
0.615084 + 0.788462i $$0.289122\pi$$
$$84$$ −1100.76 −1.42979
$$85$$ 0 0
$$86$$ −1420.40 −1.78099
$$87$$ −208.360 −0.256765
$$88$$ −174.018 −0.210800
$$89$$ −732.559 −0.872484 −0.436242 0.899829i $$-0.643691\pi$$
−0.436242 + 0.899829i $$0.643691\pi$$
$$90$$ 0 0
$$91$$ 163.279 0.188092
$$92$$ −1345.59 −1.52487
$$93$$ 421.814 0.470323
$$94$$ 1023.85 1.12342
$$95$$ 0 0
$$96$$ −679.795 −0.722722
$$97$$ 1171.49 1.22626 0.613128 0.789984i $$-0.289911\pi$$
0.613128 + 0.789984i $$0.289911\pi$$
$$98$$ 2927.84 3.01793
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ −1221.27 −1.20318 −0.601589 0.798806i $$-0.705465\pi$$
−0.601589 + 0.798806i $$0.705465\pi$$
$$102$$ −1618.57 −1.57120
$$103$$ −516.745 −0.494334 −0.247167 0.968973i $$-0.579500\pi$$
−0.247167 + 0.968973i $$0.579500\pi$$
$$104$$ −81.4903 −0.0768345
$$105$$ 0 0
$$106$$ −21.7645 −0.0199430
$$107$$ 152.025 0.137353 0.0686765 0.997639i $$-0.478122\pi$$
0.0686765 + 0.997639i $$0.478122\pi$$
$$108$$ 312.540 0.278465
$$109$$ 2170.32 1.90714 0.953572 0.301164i $$-0.0973752\pi$$
0.953572 + 0.301164i $$0.0973752\pi$$
$$110$$ 0 0
$$111$$ 1260.21 1.07760
$$112$$ 716.708 0.604666
$$113$$ 646.397 0.538123 0.269062 0.963123i $$-0.413286\pi$$
0.269062 + 0.963123i $$0.413286\pi$$
$$114$$ 462.559 0.380023
$$115$$ 0 0
$$116$$ −803.963 −0.643501
$$117$$ −46.3603 −0.0366326
$$118$$ 1798.28 1.40292
$$119$$ 3865.28 2.97756
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ −2461.89 −1.82696
$$123$$ −966.174 −0.708268
$$124$$ 1627.58 1.17872
$$125$$ 0 0
$$126$$ −1262.20 −0.892425
$$127$$ 993.304 0.694027 0.347014 0.937860i $$-0.387196\pi$$
0.347014 + 0.937860i $$0.387196\pi$$
$$128$$ −1822.69 −1.25863
$$129$$ −963.105 −0.657339
$$130$$ 0 0
$$131$$ 385.814 0.257318 0.128659 0.991689i $$-0.458933\pi$$
0.128659 + 0.991689i $$0.458933\pi$$
$$132$$ −381.994 −0.251881
$$133$$ −1104.63 −0.720177
$$134$$ −374.949 −0.241721
$$135$$ 0 0
$$136$$ −1929.11 −1.21632
$$137$$ −884.840 −0.551803 −0.275901 0.961186i $$-0.588976\pi$$
−0.275901 + 0.961186i $$0.588976\pi$$
$$138$$ −1542.94 −0.951769
$$139$$ −1091.94 −0.666312 −0.333156 0.942872i $$-0.608114\pi$$
−0.333156 + 0.942872i $$0.608114\pi$$
$$140$$ 0 0
$$141$$ 694.223 0.414639
$$142$$ 217.155 0.128333
$$143$$ 56.6626 0.0331354
$$144$$ −203.497 −0.117764
$$145$$ 0 0
$$146$$ −3477.09 −1.97100
$$147$$ 1985.24 1.11387
$$148$$ 4862.55 2.70067
$$149$$ 297.014 0.163304 0.0816522 0.996661i $$-0.473980\pi$$
0.0816522 + 0.996661i $$0.473980\pi$$
$$150$$ 0 0
$$151$$ −1887.86 −1.01743 −0.508716 0.860935i $$-0.669880\pi$$
−0.508716 + 0.860935i $$0.669880\pi$$
$$152$$ 551.304 0.294189
$$153$$ −1097.48 −0.579907
$$154$$ 1542.69 0.807229
$$155$$ 0 0
$$156$$ −178.882 −0.0918080
$$157$$ 56.5343 0.0287384 0.0143692 0.999897i $$-0.495426\pi$$
0.0143692 + 0.999897i $$0.495426\pi$$
$$158$$ −1695.08 −0.853501
$$159$$ −14.7575 −0.00736066
$$160$$ 0 0
$$161$$ 3684.68 1.80369
$$162$$ 358.379 0.173808
$$163$$ 49.2338 0.0236582 0.0118291 0.999930i $$-0.496235\pi$$
0.0118291 + 0.999930i $$0.496235\pi$$
$$164$$ −3728.01 −1.77505
$$165$$ 0 0
$$166$$ 4115.65 1.92432
$$167$$ −2068.75 −0.958589 −0.479294 0.877654i $$-0.659107\pi$$
−0.479294 + 0.877654i $$0.659107\pi$$
$$168$$ −1504.36 −0.690857
$$169$$ −2170.47 −0.987923
$$170$$ 0 0
$$171$$ 313.640 0.140261
$$172$$ −3716.17 −1.64741
$$173$$ 604.012 0.265446 0.132723 0.991153i $$-0.457628\pi$$
0.132723 + 0.991153i $$0.457628\pi$$
$$174$$ −921.875 −0.401650
$$175$$ 0 0
$$176$$ 248.718 0.106522
$$177$$ 1219.33 0.517799
$$178$$ −3241.15 −1.36480
$$179$$ −2132.02 −0.890251 −0.445126 0.895468i $$-0.646841\pi$$
−0.445126 + 0.895468i $$0.646841\pi$$
$$180$$ 0 0
$$181$$ −589.371 −0.242031 −0.121015 0.992651i $$-0.538615\pi$$
−0.121015 + 0.992651i $$0.538615\pi$$
$$182$$ 722.418 0.294226
$$183$$ −1669.29 −0.674304
$$184$$ −1838.97 −0.736796
$$185$$ 0 0
$$186$$ 1866.28 0.735713
$$187$$ 1341.36 0.524546
$$188$$ 2678.68 1.03916
$$189$$ −855.838 −0.329381
$$190$$ 0 0
$$191$$ −2160.90 −0.818624 −0.409312 0.912395i $$-0.634231\pi$$
−0.409312 + 0.912395i $$0.634231\pi$$
$$192$$ −2465.05 −0.926560
$$193$$ 1490.91 0.556052 0.278026 0.960574i $$-0.410320\pi$$
0.278026 + 0.960574i $$0.410320\pi$$
$$194$$ 5183.18 1.91820
$$195$$ 0 0
$$196$$ 7660.08 2.79157
$$197$$ 230.529 0.0833732 0.0416866 0.999131i $$-0.486727\pi$$
0.0416866 + 0.999131i $$0.486727\pi$$
$$198$$ −438.018 −0.157215
$$199$$ 22.4007 0.00797963 0.00398982 0.999992i $$-0.498730\pi$$
0.00398982 + 0.999992i $$0.498730\pi$$
$$200$$ 0 0
$$201$$ −254.236 −0.0892159
$$202$$ −5403.43 −1.88210
$$203$$ 2201.51 0.761163
$$204$$ −4234.65 −1.45336
$$205$$ 0 0
$$206$$ −2286.30 −0.773273
$$207$$ −1046.20 −0.351284
$$208$$ 116.471 0.0388260
$$209$$ −383.337 −0.126871
$$210$$ 0 0
$$211$$ −1051.64 −0.343117 −0.171558 0.985174i $$-0.554880\pi$$
−0.171558 + 0.985174i $$0.554880\pi$$
$$212$$ −56.9421 −0.0184472
$$213$$ 147.243 0.0473657
$$214$$ 672.622 0.214857
$$215$$ 0 0
$$216$$ 427.136 0.134551
$$217$$ −4456.84 −1.39424
$$218$$ 9602.42 2.98329
$$219$$ −2357.65 −0.727467
$$220$$ 0 0
$$221$$ 628.141 0.191191
$$222$$ 5575.71 1.68566
$$223$$ −3861.80 −1.15966 −0.579832 0.814736i $$-0.696882\pi$$
−0.579832 + 0.814736i $$0.696882\pi$$
$$224$$ 7182.65 2.14246
$$225$$ 0 0
$$226$$ 2859.94 0.841771
$$227$$ 872.721 0.255174 0.127587 0.991827i $$-0.459277\pi$$
0.127587 + 0.991827i $$0.459277\pi$$
$$228$$ 1210.19 0.351520
$$229$$ 1841.72 0.531459 0.265730 0.964048i $$-0.414387\pi$$
0.265730 + 0.964048i $$0.414387\pi$$
$$230$$ 0 0
$$231$$ 1046.02 0.297937
$$232$$ −1098.74 −0.310931
$$233$$ −3932.14 −1.10559 −0.552796 0.833317i $$-0.686439\pi$$
−0.552796 + 0.833317i $$0.686439\pi$$
$$234$$ −205.118 −0.0573033
$$235$$ 0 0
$$236$$ 4704.81 1.29770
$$237$$ −1149.35 −0.315015
$$238$$ 17101.7 4.65772
$$239$$ 4772.10 1.29155 0.645777 0.763526i $$-0.276534\pi$$
0.645777 + 0.763526i $$0.276534\pi$$
$$240$$ 0 0
$$241$$ 3988.84 1.06616 0.533078 0.846066i $$-0.321035\pi$$
0.533078 + 0.846066i $$0.321035\pi$$
$$242$$ 535.356 0.142207
$$243$$ 243.000 0.0641500
$$244$$ −6441.00 −1.68993
$$245$$ 0 0
$$246$$ −4274.77 −1.10792
$$247$$ −179.511 −0.0462431
$$248$$ 2224.34 0.569540
$$249$$ 2790.63 0.710238
$$250$$ 0 0
$$251$$ −5474.22 −1.37661 −0.688306 0.725421i $$-0.741645\pi$$
−0.688306 + 0.725421i $$0.741645\pi$$
$$252$$ −3302.27 −0.825491
$$253$$ 1278.69 0.317749
$$254$$ 4394.80 1.08565
$$255$$ 0 0
$$256$$ −1490.90 −0.363989
$$257$$ 6434.01 1.56164 0.780822 0.624754i $$-0.214801\pi$$
0.780822 + 0.624754i $$0.214801\pi$$
$$258$$ −4261.19 −1.02826
$$259$$ −13315.3 −3.19448
$$260$$ 0 0
$$261$$ −625.081 −0.148243
$$262$$ 1707.01 0.402516
$$263$$ −7589.00 −1.77931 −0.889654 0.456636i $$-0.849054\pi$$
−0.889654 + 0.456636i $$0.849054\pi$$
$$264$$ −522.055 −0.121706
$$265$$ 0 0
$$266$$ −4887.35 −1.12655
$$267$$ −2197.68 −0.503729
$$268$$ −980.974 −0.223591
$$269$$ 478.178 0.108383 0.0541914 0.998531i $$-0.482742\pi$$
0.0541914 + 0.998531i $$0.482742\pi$$
$$270$$ 0 0
$$271$$ −122.323 −0.0274192 −0.0137096 0.999906i $$-0.504364\pi$$
−0.0137096 + 0.999906i $$0.504364\pi$$
$$272$$ 2757.20 0.614631
$$273$$ 489.838 0.108595
$$274$$ −3914.91 −0.863170
$$275$$ 0 0
$$276$$ −4036.78 −0.880383
$$277$$ −8199.41 −1.77854 −0.889269 0.457385i $$-0.848786\pi$$
−0.889269 + 0.457385i $$0.848786\pi$$
$$278$$ −4831.22 −1.04229
$$279$$ 1265.44 0.271541
$$280$$ 0 0
$$281$$ 6943.79 1.47413 0.737067 0.675820i $$-0.236210\pi$$
0.737067 + 0.675820i $$0.236210\pi$$
$$282$$ 3071.54 0.648609
$$283$$ −1035.14 −0.217429 −0.108715 0.994073i $$-0.534673\pi$$
−0.108715 + 0.994073i $$0.534673\pi$$
$$284$$ 568.139 0.118707
$$285$$ 0 0
$$286$$ 250.699 0.0518328
$$287$$ 10208.5 2.09961
$$288$$ −2039.39 −0.417264
$$289$$ 9956.85 2.02663
$$290$$ 0 0
$$291$$ 3514.47 0.707979
$$292$$ −9097.06 −1.82317
$$293$$ 6144.81 1.22520 0.612600 0.790393i $$-0.290124\pi$$
0.612600 + 0.790393i $$0.290124\pi$$
$$294$$ 8783.53 1.74240
$$295$$ 0 0
$$296$$ 6645.45 1.30493
$$297$$ −297.000 −0.0580259
$$298$$ 1314.12 0.255452
$$299$$ 598.791 0.115816
$$300$$ 0 0
$$301$$ 10176.1 1.94864
$$302$$ −8352.72 −1.59154
$$303$$ −3663.81 −0.694655
$$304$$ −787.958 −0.148659
$$305$$ 0 0
$$306$$ −4855.71 −0.907133
$$307$$ 2186.09 0.406406 0.203203 0.979137i $$-0.434865\pi$$
0.203203 + 0.979137i $$0.434865\pi$$
$$308$$ 4036.11 0.746684
$$309$$ −1550.24 −0.285404
$$310$$ 0 0
$$311$$ −7484.83 −1.36471 −0.682357 0.731019i $$-0.739045\pi$$
−0.682357 + 0.731019i $$0.739045\pi$$
$$312$$ −244.471 −0.0443604
$$313$$ 6833.33 1.23400 0.617001 0.786962i $$-0.288347\pi$$
0.617001 + 0.786962i $$0.288347\pi$$
$$314$$ 250.132 0.0449546
$$315$$ 0 0
$$316$$ −4434.81 −0.789485
$$317$$ −924.265 −0.163760 −0.0818800 0.996642i $$-0.526092\pi$$
−0.0818800 + 0.996642i $$0.526092\pi$$
$$318$$ −65.2934 −0.0115141
$$319$$ 763.988 0.134091
$$320$$ 0 0
$$321$$ 456.074 0.0793008
$$322$$ 16302.6 2.82145
$$323$$ −4249.54 −0.732046
$$324$$ 937.621 0.160772
$$325$$ 0 0
$$326$$ 217.831 0.0370078
$$327$$ 6510.95 1.10109
$$328$$ −5094.91 −0.857681
$$329$$ −7335.10 −1.22917
$$330$$ 0 0
$$331$$ −9820.46 −1.63076 −0.815380 0.578927i $$-0.803472\pi$$
−0.815380 + 0.578927i $$0.803472\pi$$
$$332$$ 10767.7 1.77999
$$333$$ 3780.63 0.622154
$$334$$ −9153.02 −1.49949
$$335$$ 0 0
$$336$$ 2150.12 0.349104
$$337$$ −600.808 −0.0971161 −0.0485580 0.998820i $$-0.515463\pi$$
−0.0485580 + 0.998820i $$0.515463\pi$$
$$338$$ −9603.07 −1.54538
$$339$$ 1939.19 0.310686
$$340$$ 0 0
$$341$$ −1546.65 −0.245618
$$342$$ 1387.68 0.219406
$$343$$ −10103.5 −1.59049
$$344$$ −5078.73 −0.796008
$$345$$ 0 0
$$346$$ 2672.41 0.415230
$$347$$ 3143.41 0.486303 0.243152 0.969988i $$-0.421819\pi$$
0.243152 + 0.969988i $$0.421819\pi$$
$$348$$ −2411.89 −0.371525
$$349$$ 720.663 0.110533 0.0552667 0.998472i $$-0.482399\pi$$
0.0552667 + 0.998472i $$0.482399\pi$$
$$350$$ 0 0
$$351$$ −139.081 −0.0211498
$$352$$ 2492.58 0.377429
$$353$$ −1207.12 −0.182007 −0.0910034 0.995851i $$-0.529007\pi$$
−0.0910034 + 0.995851i $$0.529007\pi$$
$$354$$ 5394.83 0.809978
$$355$$ 0 0
$$356$$ −8479.79 −1.26244
$$357$$ 11595.8 1.71910
$$358$$ −9432.99 −1.39260
$$359$$ 8748.31 1.28612 0.643062 0.765814i $$-0.277664\pi$$
0.643062 + 0.765814i $$0.277664\pi$$
$$360$$ 0 0
$$361$$ −5644.56 −0.822942
$$362$$ −2607.63 −0.378602
$$363$$ 363.000 0.0524864
$$364$$ 1890.05 0.272158
$$365$$ 0 0
$$366$$ −7385.66 −1.05479
$$367$$ 6730.45 0.957293 0.478647 0.878008i $$-0.341128\pi$$
0.478647 + 0.878008i $$0.341128\pi$$
$$368$$ 2628.37 0.372318
$$369$$ −2898.52 −0.408919
$$370$$ 0 0
$$371$$ 155.926 0.0218202
$$372$$ 4882.73 0.680532
$$373$$ 227.394 0.0315657 0.0157828 0.999875i $$-0.494976\pi$$
0.0157828 + 0.999875i $$0.494976\pi$$
$$374$$ 5934.76 0.820533
$$375$$ 0 0
$$376$$ 3660.84 0.502110
$$377$$ 357.764 0.0488748
$$378$$ −3786.60 −0.515242
$$379$$ 11356.2 1.53913 0.769565 0.638568i $$-0.220473\pi$$
0.769565 + 0.638568i $$0.220473\pi$$
$$380$$ 0 0
$$381$$ 2979.91 0.400697
$$382$$ −9560.74 −1.28055
$$383$$ −10753.6 −1.43468 −0.717338 0.696725i $$-0.754640\pi$$
−0.717338 + 0.696725i $$0.754640\pi$$
$$384$$ −5468.07 −0.726670
$$385$$ 0 0
$$386$$ 6596.43 0.869817
$$387$$ −2889.32 −0.379515
$$388$$ 13560.7 1.77433
$$389$$ −11727.1 −1.52850 −0.764252 0.644918i $$-0.776891\pi$$
−0.764252 + 0.644918i $$0.776891\pi$$
$$390$$ 0 0
$$391$$ 14175.1 1.83341
$$392$$ 10468.7 1.34885
$$393$$ 1157.44 0.148563
$$394$$ 1019.96 0.130418
$$395$$ 0 0
$$396$$ −1145.98 −0.145424
$$397$$ 359.905 0.0454990 0.0227495 0.999741i $$-0.492758\pi$$
0.0227495 + 0.999741i $$0.492758\pi$$
$$398$$ 99.1105 0.0124823
$$399$$ −3313.89 −0.415794
$$400$$ 0 0
$$401$$ −4066.71 −0.506438 −0.253219 0.967409i $$-0.581489\pi$$
−0.253219 + 0.967409i $$0.581489\pi$$
$$402$$ −1124.85 −0.139558
$$403$$ −724.274 −0.0895252
$$404$$ −14136.9 −1.74093
$$405$$ 0 0
$$406$$ 9740.45 1.19067
$$407$$ −4620.77 −0.562760
$$408$$ −5787.32 −0.702242
$$409$$ −13488.8 −1.63076 −0.815379 0.578927i $$-0.803472\pi$$
−0.815379 + 0.578927i $$0.803472\pi$$
$$410$$ 0 0
$$411$$ −2654.52 −0.318584
$$412$$ −5981.62 −0.715275
$$413$$ −12883.3 −1.53498
$$414$$ −4628.83 −0.549504
$$415$$ 0 0
$$416$$ 1167.24 0.137569
$$417$$ −3275.83 −0.384695
$$418$$ −1696.05 −0.198460
$$419$$ −7040.12 −0.820841 −0.410420 0.911896i $$-0.634618\pi$$
−0.410420 + 0.911896i $$0.634618\pi$$
$$420$$ 0 0
$$421$$ 9171.74 1.06177 0.530883 0.847445i $$-0.321860\pi$$
0.530883 + 0.847445i $$0.321860\pi$$
$$422$$ −4652.89 −0.536728
$$423$$ 2082.67 0.239392
$$424$$ −77.8204 −0.00891343
$$425$$ 0 0
$$426$$ 651.464 0.0740928
$$427$$ 17637.6 1.99893
$$428$$ 1759.77 0.198742
$$429$$ 169.988 0.0191307
$$430$$ 0 0
$$431$$ 992.995 0.110976 0.0554882 0.998459i $$-0.482328\pi$$
0.0554882 + 0.998459i $$0.482328\pi$$
$$432$$ −610.490 −0.0679912
$$433$$ −3790.21 −0.420660 −0.210330 0.977630i $$-0.567454\pi$$
−0.210330 + 0.977630i $$0.567454\pi$$
$$434$$ −19719.0 −2.18097
$$435$$ 0 0
$$436$$ 25122.7 2.75954
$$437$$ −4050.98 −0.443443
$$438$$ −10431.3 −1.13796
$$439$$ −5136.97 −0.558483 −0.279242 0.960221i $$-0.590083\pi$$
−0.279242 + 0.960221i $$0.590083\pi$$
$$440$$ 0 0
$$441$$ 5955.71 0.643095
$$442$$ 2779.16 0.299075
$$443$$ −10676.8 −1.14508 −0.572541 0.819876i $$-0.694042\pi$$
−0.572541 + 0.819876i $$0.694042\pi$$
$$444$$ 14587.7 1.55923
$$445$$ 0 0
$$446$$ −17086.2 −1.81403
$$447$$ 891.042 0.0942838
$$448$$ 26045.5 2.74672
$$449$$ 10529.9 1.10676 0.553379 0.832929i $$-0.313338\pi$$
0.553379 + 0.832929i $$0.313338\pi$$
$$450$$ 0 0
$$451$$ 3542.64 0.369881
$$452$$ 7482.42 0.778636
$$453$$ −5663.59 −0.587414
$$454$$ 3861.29 0.399162
$$455$$ 0 0
$$456$$ 1653.91 0.169850
$$457$$ 14072.5 1.44045 0.720225 0.693741i $$-0.244039\pi$$
0.720225 + 0.693741i $$0.244039\pi$$
$$458$$ 8148.55 0.831347
$$459$$ −3292.43 −0.334810
$$460$$ 0 0
$$461$$ −30.8173 −0.00311346 −0.00155673 0.999999i $$-0.500496\pi$$
−0.00155673 + 0.999999i $$0.500496\pi$$
$$462$$ 4628.06 0.466054
$$463$$ −17591.3 −1.76573 −0.882867 0.469622i $$-0.844390\pi$$
−0.882867 + 0.469622i $$0.844390\pi$$
$$464$$ 1570.39 0.157120
$$465$$ 0 0
$$466$$ −17397.5 −1.72945
$$467$$ −13273.1 −1.31522 −0.657609 0.753360i $$-0.728432\pi$$
−0.657609 + 0.753360i $$0.728432\pi$$
$$468$$ −536.647 −0.0530053
$$469$$ 2686.23 0.264474
$$470$$ 0 0
$$471$$ 169.603 0.0165921
$$472$$ 6429.87 0.627031
$$473$$ 3531.39 0.343284
$$474$$ −5085.23 −0.492769
$$475$$ 0 0
$$476$$ 44742.9 4.30837
$$477$$ −44.2724 −0.00424968
$$478$$ 21113.8 2.02034
$$479$$ 2496.68 0.238155 0.119077 0.992885i $$-0.462006\pi$$
0.119077 + 0.992885i $$0.462006\pi$$
$$480$$ 0 0
$$481$$ −2163.84 −0.205120
$$482$$ 17648.3 1.66776
$$483$$ 11054.0 1.04136
$$484$$ 1400.64 0.131541
$$485$$ 0 0
$$486$$ 1075.14 0.100348
$$487$$ 3464.42 0.322357 0.161178 0.986925i $$-0.448471\pi$$
0.161178 + 0.986925i $$0.448471\pi$$
$$488$$ −8802.65 −0.816552
$$489$$ 147.701 0.0136591
$$490$$ 0 0
$$491$$ −16224.6 −1.49125 −0.745625 0.666366i $$-0.767849\pi$$
−0.745625 + 0.666366i $$0.767849\pi$$
$$492$$ −11184.0 −1.02483
$$493$$ 8469.29 0.773707
$$494$$ −794.236 −0.0723367
$$495$$ 0 0
$$496$$ −3179.17 −0.287800
$$497$$ −1555.75 −0.140412
$$498$$ 12347.0 1.11100
$$499$$ 9993.81 0.896562 0.448281 0.893893i $$-0.352036\pi$$
0.448281 + 0.893893i $$0.352036\pi$$
$$500$$ 0 0
$$501$$ −6206.24 −0.553441
$$502$$ −24220.3 −2.15340
$$503$$ 15334.8 1.35933 0.679667 0.733520i $$-0.262124\pi$$
0.679667 + 0.733520i $$0.262124\pi$$
$$504$$ −4513.08 −0.398866
$$505$$ 0 0
$$506$$ 5657.46 0.497045
$$507$$ −6511.40 −0.570377
$$508$$ 11498.1 1.00422
$$509$$ −7291.23 −0.634927 −0.317464 0.948270i $$-0.602831\pi$$
−0.317464 + 0.948270i $$0.602831\pi$$
$$510$$ 0 0
$$511$$ 24910.7 2.15653
$$512$$ 7985.14 0.689251
$$513$$ 940.919 0.0809797
$$514$$ 28466.8 2.44283
$$515$$ 0 0
$$516$$ −11148.5 −0.951134
$$517$$ −2545.49 −0.216538
$$518$$ −58912.5 −4.99704
$$519$$ 1812.04 0.153255
$$520$$ 0 0
$$521$$ 16794.3 1.41223 0.706114 0.708098i $$-0.250447\pi$$
0.706114 + 0.708098i $$0.250447\pi$$
$$522$$ −2765.63 −0.231893
$$523$$ 21009.4 1.75655 0.878275 0.478157i $$-0.158695\pi$$
0.878275 + 0.478157i $$0.158695\pi$$
$$524$$ 4466.01 0.372326
$$525$$ 0 0
$$526$$ −33577.0 −2.78332
$$527$$ −17145.6 −1.41722
$$528$$ 746.154 0.0615003
$$529$$ 1345.73 0.110605
$$530$$ 0 0
$$531$$ 3657.99 0.298951
$$532$$ −12786.7 −1.04206
$$533$$ 1658.97 0.134818
$$534$$ −9723.46 −0.787969
$$535$$ 0 0
$$536$$ −1340.66 −0.108036
$$537$$ −6396.07 −0.513987
$$538$$ 2115.66 0.169540
$$539$$ −7279.20 −0.581702
$$540$$ 0 0
$$541$$ −16802.8 −1.33532 −0.667662 0.744464i $$-0.732705\pi$$
−0.667662 + 0.744464i $$0.732705\pi$$
$$542$$ −541.211 −0.0428911
$$543$$ −1768.11 −0.139737
$$544$$ 27631.9 2.17777
$$545$$ 0 0
$$546$$ 2167.25 0.169872
$$547$$ −16784.5 −1.31198 −0.655990 0.754770i $$-0.727749\pi$$
−0.655990 + 0.754770i $$0.727749\pi$$
$$548$$ −10242.5 −0.798429
$$549$$ −5007.88 −0.389309
$$550$$ 0 0
$$551$$ −2420.37 −0.187135
$$552$$ −5516.91 −0.425390
$$553$$ 12144.0 0.933840
$$554$$ −36277.7 −2.78212
$$555$$ 0 0
$$556$$ −12639.9 −0.964117
$$557$$ −18127.0 −1.37893 −0.689467 0.724317i $$-0.742155\pi$$
−0.689467 + 0.724317i $$0.742155\pi$$
$$558$$ 5598.85 0.424764
$$559$$ 1653.70 0.125123
$$560$$ 0 0
$$561$$ 4024.09 0.302847
$$562$$ 30722.3 2.30595
$$563$$ −2090.88 −0.156518 −0.0782592 0.996933i $$-0.524936\pi$$
−0.0782592 + 0.996933i $$0.524936\pi$$
$$564$$ 8036.03 0.599961
$$565$$ 0 0
$$566$$ −4579.89 −0.340119
$$567$$ −2567.51 −0.190168
$$568$$ 776.452 0.0573578
$$569$$ 6249.23 0.460424 0.230212 0.973140i $$-0.426058\pi$$
0.230212 + 0.973140i $$0.426058\pi$$
$$570$$ 0 0
$$571$$ 6048.79 0.443317 0.221659 0.975124i $$-0.428853\pi$$
0.221659 + 0.975124i $$0.428853\pi$$
$$572$$ 655.902 0.0479451
$$573$$ −6482.69 −0.472633
$$574$$ 45166.8 3.28437
$$575$$ 0 0
$$576$$ −7395.15 −0.534950
$$577$$ 15729.1 1.13486 0.567429 0.823423i $$-0.307938\pi$$
0.567429 + 0.823423i $$0.307938\pi$$
$$578$$ 44053.4 3.17021
$$579$$ 4472.73 0.321037
$$580$$ 0 0
$$581$$ −29485.6 −2.10545
$$582$$ 15549.5 1.10747
$$583$$ 54.1108 0.00384398
$$584$$ −12432.6 −0.880931
$$585$$ 0 0
$$586$$ 27187.3 1.91655
$$587$$ −15620.5 −1.09835 −0.549173 0.835709i $$-0.685057\pi$$
−0.549173 + 0.835709i $$0.685057\pi$$
$$588$$ 22980.2 1.61172
$$589$$ 4899.91 0.342780
$$590$$ 0 0
$$591$$ 691.587 0.0481355
$$592$$ −9498.09 −0.659407
$$593$$ 493.541 0.0341776 0.0170888 0.999854i $$-0.494560\pi$$
0.0170888 + 0.999854i $$0.494560\pi$$
$$594$$ −1314.06 −0.0907683
$$595$$ 0 0
$$596$$ 3438.11 0.236293
$$597$$ 67.2022 0.00460704
$$598$$ 2649.31 0.181168
$$599$$ −12455.1 −0.849585 −0.424793 0.905291i $$-0.639653\pi$$
−0.424793 + 0.905291i $$0.639653\pi$$
$$600$$ 0 0
$$601$$ 12454.8 0.845329 0.422664 0.906286i $$-0.361095\pi$$
0.422664 + 0.906286i $$0.361095\pi$$
$$602$$ 45023.3 3.04820
$$603$$ −762.707 −0.0515088
$$604$$ −21853.1 −1.47217
$$605$$ 0 0
$$606$$ −16210.3 −1.08663
$$607$$ 4243.19 0.283733 0.141867 0.989886i $$-0.454690\pi$$
0.141867 + 0.989886i $$0.454690\pi$$
$$608$$ −7896.70 −0.526732
$$609$$ 6604.54 0.439458
$$610$$ 0 0
$$611$$ −1192.01 −0.0789259
$$612$$ −12703.9 −0.839095
$$613$$ −5733.14 −0.377748 −0.188874 0.982001i $$-0.560484\pi$$
−0.188874 + 0.982001i $$0.560484\pi$$
$$614$$ 9672.18 0.635729
$$615$$ 0 0
$$616$$ 5515.99 0.360788
$$617$$ −15642.1 −1.02063 −0.510314 0.859988i $$-0.670471\pi$$
−0.510314 + 0.859988i $$0.670471\pi$$
$$618$$ −6858.91 −0.446449
$$619$$ −7467.40 −0.484879 −0.242440 0.970167i $$-0.577948\pi$$
−0.242440 + 0.970167i $$0.577948\pi$$
$$620$$ 0 0
$$621$$ −3138.60 −0.202814
$$622$$ −33116.1 −2.13478
$$623$$ 23220.4 1.49327
$$624$$ 349.413 0.0224162
$$625$$ 0 0
$$626$$ 30233.6 1.93031
$$627$$ −1150.01 −0.0732489
$$628$$ 654.416 0.0415829
$$629$$ −51224.2 −3.24713
$$630$$ 0 0
$$631$$ −1486.38 −0.0937745 −0.0468872 0.998900i $$-0.514930\pi$$
−0.0468872 + 0.998900i $$0.514930\pi$$
$$632$$ −6060.87 −0.381469
$$633$$ −3154.91 −0.198099
$$634$$ −4089.35 −0.256165
$$635$$ 0 0
$$636$$ −170.826 −0.0106505
$$637$$ −3408.74 −0.212024
$$638$$ 3380.21 0.209755
$$639$$ 441.728 0.0273466
$$640$$ 0 0
$$641$$ 12386.0 0.763211 0.381606 0.924325i $$-0.375371\pi$$
0.381606 + 0.924325i $$0.375371\pi$$
$$642$$ 2017.87 0.124048
$$643$$ 14458.1 0.886737 0.443369 0.896339i $$-0.353783\pi$$
0.443369 + 0.896339i $$0.353783\pi$$
$$644$$ 42652.3 2.60984
$$645$$ 0 0
$$646$$ −18801.8 −1.14512
$$647$$ −15792.8 −0.959625 −0.479813 0.877371i $$-0.659295\pi$$
−0.479813 + 0.877371i $$0.659295\pi$$
$$648$$ 1281.41 0.0776828
$$649$$ −4470.87 −0.270412
$$650$$ 0 0
$$651$$ −13370.5 −0.804965
$$652$$ 569.909 0.0342321
$$653$$ 3179.93 0.190567 0.0952837 0.995450i $$-0.469624\pi$$
0.0952837 + 0.995450i $$0.469624\pi$$
$$654$$ 28807.3 1.72240
$$655$$ 0 0
$$656$$ 7281.96 0.433404
$$657$$ −7072.96 −0.420003
$$658$$ −32453.6 −1.92276
$$659$$ 11593.5 0.685308 0.342654 0.939462i $$-0.388674\pi$$
0.342654 + 0.939462i $$0.388674\pi$$
$$660$$ 0 0
$$661$$ 3233.88 0.190293 0.0951464 0.995463i $$-0.469668\pi$$
0.0951464 + 0.995463i $$0.469668\pi$$
$$662$$ −43449.9 −2.55095
$$663$$ 1884.42 0.110384
$$664$$ 14715.8 0.860066
$$665$$ 0 0
$$666$$ 16727.1 0.973219
$$667$$ 8073.56 0.468680
$$668$$ −23946.9 −1.38703
$$669$$ −11585.4 −0.669532
$$670$$ 0 0
$$671$$ 6120.74 0.352144
$$672$$ 21548.0 1.23695
$$673$$ 5495.72 0.314776 0.157388 0.987537i $$-0.449693\pi$$
0.157388 + 0.987537i $$0.449693\pi$$
$$674$$ −2658.23 −0.151916
$$675$$ 0 0
$$676$$ −25124.4 −1.42947
$$677$$ −33836.7 −1.92090 −0.960451 0.278448i $$-0.910180\pi$$
−0.960451 + 0.278448i $$0.910180\pi$$
$$678$$ 8579.82 0.485997
$$679$$ −37133.6 −2.09876
$$680$$ 0 0
$$681$$ 2618.16 0.147325
$$682$$ −6843.04 −0.384214
$$683$$ 21080.3 1.18099 0.590493 0.807043i $$-0.298933\pi$$
0.590493 + 0.807043i $$0.298933\pi$$
$$684$$ 3630.56 0.202950
$$685$$ 0 0
$$686$$ −44702.2 −2.48796
$$687$$ 5525.16 0.306838
$$688$$ 7258.84 0.402239
$$689$$ 25.3393 0.00140109
$$690$$ 0 0
$$691$$ 11811.3 0.650253 0.325127 0.945671i $$-0.394593\pi$$
0.325127 + 0.945671i $$0.394593\pi$$
$$692$$ 6991.79 0.384087
$$693$$ 3138.07 0.172014
$$694$$ 13907.8 0.760711
$$695$$ 0 0
$$696$$ −3296.23 −0.179516
$$697$$ 39272.4 2.13422
$$698$$ 3188.52 0.172904
$$699$$ −11796.4 −0.638313
$$700$$ 0 0
$$701$$ 4244.99 0.228718 0.114359 0.993440i $$-0.463519\pi$$
0.114359 + 0.993440i $$0.463519\pi$$
$$702$$ −615.353 −0.0330841
$$703$$ 14639.0 0.785376
$$704$$ 9038.51 0.483880
$$705$$ 0 0
$$706$$ −5340.81 −0.284708
$$707$$ 38711.5 2.05926
$$708$$ 14114.4 0.749227
$$709$$ −898.822 −0.0476107 −0.0238053 0.999717i $$-0.507578\pi$$
−0.0238053 + 0.999717i $$0.507578\pi$$
$$710$$ 0 0
$$711$$ −3448.06 −0.181874
$$712$$ −11589.0 −0.609993
$$713$$ −16344.5 −0.858493
$$714$$ 51305.0 2.68913
$$715$$ 0 0
$$716$$ −24679.4 −1.28815
$$717$$ 14316.3 0.745679
$$718$$ 38706.3 2.01185
$$719$$ 10741.8 0.557165 0.278582 0.960412i $$-0.410135\pi$$
0.278582 + 0.960412i $$0.410135\pi$$
$$720$$ 0 0
$$721$$ 16379.6 0.846061
$$722$$ −24973.9 −1.28730
$$723$$ 11966.5 0.615546
$$724$$ −6822.30 −0.350206
$$725$$ 0 0
$$726$$ 1606.07 0.0821030
$$727$$ −16794.2 −0.856758 −0.428379 0.903599i $$-0.640915\pi$$
−0.428379 + 0.903599i $$0.640915\pi$$
$$728$$ 2583.06 0.131503
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 39147.7 1.98075
$$732$$ −19323.0 −0.975681
$$733$$ −8659.40 −0.436347 −0.218173 0.975910i $$-0.570010\pi$$
−0.218173 + 0.975910i $$0.570010\pi$$
$$734$$ 29778.4 1.49747
$$735$$ 0 0
$$736$$ 26340.8 1.31920
$$737$$ 932.197 0.0465915
$$738$$ −12824.3 −0.639660
$$739$$ 16705.7 0.831567 0.415783 0.909464i $$-0.363507\pi$$
0.415783 + 0.909464i $$0.363507\pi$$
$$740$$ 0 0
$$741$$ −538.534 −0.0266984
$$742$$ 689.884 0.0341327
$$743$$ −1292.12 −0.0637996 −0.0318998 0.999491i $$-0.510156\pi$$
−0.0318998 + 0.999491i $$0.510156\pi$$
$$744$$ 6673.03 0.328824
$$745$$ 0 0
$$746$$ 1006.09 0.0493773
$$747$$ 8371.90 0.410056
$$748$$ 15527.0 0.758990
$$749$$ −4818.83 −0.235082
$$750$$ 0 0
$$751$$ −14980.4 −0.727886 −0.363943 0.931421i $$-0.618570\pi$$
−0.363943 + 0.931421i $$0.618570\pi$$
$$752$$ −5232.30 −0.253726
$$753$$ −16422.7 −0.794787
$$754$$ 1582.90 0.0764535
$$755$$ 0 0
$$756$$ −9906.82 −0.476597
$$757$$ −3003.41 −0.144202 −0.0721010 0.997397i $$-0.522970\pi$$
−0.0721010 + 0.997397i $$0.522970\pi$$
$$758$$ 50244.8 2.40762
$$759$$ 3836.06 0.183452
$$760$$ 0 0
$$761$$ −20375.0 −0.970555 −0.485277 0.874360i $$-0.661281\pi$$
−0.485277 + 0.874360i $$0.661281\pi$$
$$762$$ 13184.4 0.626799
$$763$$ −68794.1 −3.26411
$$764$$ −25013.6 −1.18450
$$765$$ 0 0
$$766$$ −47578.4 −2.24422
$$767$$ −2093.65 −0.0985621
$$768$$ −4472.70 −0.210149
$$769$$ −12372.4 −0.580184 −0.290092 0.956999i $$-0.593686\pi$$
−0.290092 + 0.956999i $$0.593686\pi$$
$$770$$ 0 0
$$771$$ 19302.0 0.901615
$$772$$ 17258.1 0.804578
$$773$$ −21023.6 −0.978225 −0.489113 0.872221i $$-0.662679\pi$$
−0.489113 + 0.872221i $$0.662679\pi$$
$$774$$ −12783.6 −0.593664
$$775$$ 0 0
$$776$$ 18532.8 0.857331
$$777$$ −39945.8 −1.84433
$$778$$ −51885.7 −2.39099
$$779$$ −11223.4 −0.516198
$$780$$ 0 0
$$781$$ −539.889 −0.0247359
$$782$$ 62716.6 2.86795
$$783$$ −1875.24 −0.0855884
$$784$$ −14962.5 −0.681602
$$785$$ 0 0
$$786$$ 5121.02 0.232393
$$787$$ −30286.2 −1.37177 −0.685886 0.727709i $$-0.740585\pi$$
−0.685886 + 0.727709i $$0.740585\pi$$
$$788$$ 2668.51 0.120637
$$789$$ −22767.0 −1.02728
$$790$$ 0 0
$$791$$ −20489.3 −0.921007
$$792$$ −1566.17 −0.0702668
$$793$$ 2866.25 0.128353
$$794$$ 1592.37 0.0711729
$$795$$ 0 0
$$796$$ 259.301 0.0115461
$$797$$ −32337.8 −1.43722 −0.718610 0.695413i $$-0.755221\pi$$
−0.718610 + 0.695413i $$0.755221\pi$$
$$798$$ −14662.1 −0.650415
$$799$$ −28218.3 −1.24943
$$800$$ 0 0
$$801$$ −6593.03 −0.290828
$$802$$ −17992.9 −0.792207
$$803$$ 8644.72 0.379907
$$804$$ −2942.92 −0.129091
$$805$$ 0 0
$$806$$ −3204.50 −0.140042
$$807$$ 1434.53 0.0625749
$$808$$ −19320.3 −0.841197
$$809$$ −891.707 −0.0387525 −0.0193762 0.999812i $$-0.506168\pi$$
−0.0193762 + 0.999812i $$0.506168\pi$$
$$810$$ 0 0
$$811$$ −10114.9 −0.437957 −0.218978 0.975730i $$-0.570272\pi$$
−0.218978 + 0.975730i $$0.570272\pi$$
$$812$$ 25483.8 1.10136
$$813$$ −366.970 −0.0158305
$$814$$ −20444.3 −0.880309
$$815$$ 0 0
$$816$$ 8271.59 0.354857
$$817$$ −11187.7 −0.479080
$$818$$ −59680.4 −2.55095
$$819$$ 1469.51 0.0626972
$$820$$ 0 0
$$821$$ 10833.5 0.460525 0.230262 0.973129i $$-0.426042\pi$$
0.230262 + 0.973129i $$0.426042\pi$$
$$822$$ −11744.7 −0.498351
$$823$$ −31958.5 −1.35359 −0.676794 0.736173i $$-0.736631\pi$$
−0.676794 + 0.736173i $$0.736631\pi$$
$$824$$ −8174.84 −0.345612
$$825$$ 0 0
$$826$$ −57001.3 −2.40112
$$827$$ −34847.3 −1.46525 −0.732624 0.680634i $$-0.761704\pi$$
−0.732624 + 0.680634i $$0.761704\pi$$
$$828$$ −12110.3 −0.508289
$$829$$ 6537.91 0.273910 0.136955 0.990577i $$-0.456268\pi$$
0.136955 + 0.990577i $$0.456268\pi$$
$$830$$ 0 0
$$831$$ −24598.2 −1.02684
$$832$$ 4232.60 0.176369
$$833$$ −80694.5 −3.35642
$$834$$ −14493.7 −0.601768
$$835$$ 0 0
$$836$$ −4437.35 −0.183575
$$837$$ 3796.32 0.156774
$$838$$ −31148.5 −1.28402
$$839$$ 2710.34 0.111527 0.0557635 0.998444i $$-0.482241\pi$$
0.0557635 + 0.998444i $$0.482241\pi$$
$$840$$ 0 0
$$841$$ −19565.2 −0.802215
$$842$$ 40579.7 1.66089
$$843$$ 20831.4 0.851092
$$844$$ −12173.3 −0.496471
$$845$$ 0 0
$$846$$ 9214.62 0.374474
$$847$$ −3835.42 −0.155592
$$848$$ 111.226 0.00450414
$$849$$ −3105.41 −0.125533
$$850$$ 0 0
$$851$$ −48830.8 −1.96698
$$852$$ 1704.42 0.0685356
$$853$$ −9759.32 −0.391738 −0.195869 0.980630i $$-0.562753\pi$$
−0.195869 + 0.980630i $$0.562753\pi$$
$$854$$ 78036.2 3.12687
$$855$$ 0 0
$$856$$ 2405.01 0.0960298
$$857$$ 13649.8 0.544072 0.272036 0.962287i $$-0.412303\pi$$
0.272036 + 0.962287i $$0.412303\pi$$
$$858$$ 752.098 0.0299257
$$859$$ 7796.42 0.309674 0.154837 0.987940i $$-0.450515\pi$$
0.154837 + 0.987940i $$0.450515\pi$$
$$860$$ 0 0
$$861$$ 30625.5 1.21221
$$862$$ 4393.43 0.173597
$$863$$ −7183.57 −0.283350 −0.141675 0.989913i $$-0.545249\pi$$
−0.141675 + 0.989913i $$0.545249\pi$$
$$864$$ −6118.16 −0.240907
$$865$$ 0 0
$$866$$ −16769.5 −0.658026
$$867$$ 29870.6 1.17008
$$868$$ −51590.5 −2.01739
$$869$$ 4214.30 0.164511
$$870$$ 0 0
$$871$$ 436.534 0.0169821
$$872$$ 34334.1 1.33337
$$873$$ 10543.4 0.408752
$$874$$ −17923.3 −0.693666
$$875$$ 0 0
$$876$$ −27291.2 −1.05261
$$877$$ −17063.1 −0.656991 −0.328495 0.944506i $$-0.606542\pi$$
−0.328495 + 0.944506i $$0.606542\pi$$
$$878$$ −22728.2 −0.873620
$$879$$ 18434.4 0.707369
$$880$$ 0 0
$$881$$ −32174.9 −1.23042 −0.615210 0.788363i $$-0.710929\pi$$
−0.615210 + 0.788363i $$0.710929\pi$$
$$882$$ 26350.6 1.00598
$$883$$ −2843.68 −0.108378 −0.0541889 0.998531i $$-0.517257\pi$$
−0.0541889 + 0.998531i $$0.517257\pi$$
$$884$$ 7271.09 0.276644
$$885$$ 0 0
$$886$$ −47238.9 −1.79122
$$887$$ 31417.8 1.18930 0.594649 0.803985i $$-0.297291\pi$$
0.594649 + 0.803985i $$0.297291\pi$$
$$888$$ 19936.4 0.753401
$$889$$ −31485.5 −1.18784
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ −44702.5 −1.67797
$$893$$ 8064.30 0.302196
$$894$$ 3942.35 0.147485
$$895$$ 0 0
$$896$$ 57775.1 2.15416
$$897$$ 1796.37 0.0668664
$$898$$ 46588.6 1.73127
$$899$$ −9765.47 −0.362288
$$900$$ 0 0
$$901$$ 599.852 0.0221798
$$902$$ 15674.1 0.578594
$$903$$ 30528.2 1.12505
$$904$$ 10225.9 0.376227
$$905$$ 0 0
$$906$$ −25058.1 −0.918875
$$907$$ −12253.1 −0.448573 −0.224287 0.974523i $$-0.572005\pi$$
−0.224287 + 0.974523i $$0.572005\pi$$
$$908$$ 10102.2 0.369223
$$909$$ −10991.4 −0.401059
$$910$$ 0 0
$$911$$ −48422.4 −1.76104 −0.880518 0.474012i $$-0.842805\pi$$
−0.880518 + 0.474012i $$0.842805\pi$$
$$912$$ −2363.87 −0.0858286
$$913$$ −10232.3 −0.370909
$$914$$ 62262.9 2.25326
$$915$$ 0 0
$$916$$ 21318.9 0.768993
$$917$$ −12229.4 −0.440404
$$918$$ −14567.1 −0.523733
$$919$$ 5546.18 0.199077 0.0995385 0.995034i $$-0.468263\pi$$
0.0995385 + 0.995034i $$0.468263\pi$$
$$920$$ 0 0
$$921$$ 6558.26 0.234638
$$922$$ −136.349 −0.00487030
$$923$$ −252.822 −0.00901598
$$924$$ 12108.3 0.431098
$$925$$ 0 0
$$926$$ −77831.3 −2.76209
$$927$$ −4650.71 −0.164778
$$928$$ 15738.0 0.556709
$$929$$ −35684.5 −1.26025 −0.630125 0.776494i $$-0.716996\pi$$
−0.630125 + 0.776494i $$0.716996\pi$$
$$930$$ 0 0
$$931$$ 23061.1 0.811811
$$932$$ −45516.7 −1.59973
$$933$$ −22454.5 −0.787918
$$934$$ −58726.0 −2.05736
$$935$$ 0 0
$$936$$ −733.413 −0.0256115
$$937$$ 48903.6 1.70503 0.852514 0.522705i $$-0.175077\pi$$
0.852514 + 0.522705i $$0.175077\pi$$
$$938$$ 11885.0 0.413710
$$939$$ 20500.0 0.712451
$$940$$ 0 0
$$941$$ −23741.9 −0.822490 −0.411245 0.911525i $$-0.634906\pi$$
−0.411245 + 0.911525i $$0.634906\pi$$
$$942$$ 750.396 0.0259546
$$943$$ 37437.4 1.29282
$$944$$ −9189.97 −0.316852
$$945$$ 0 0
$$946$$ 15624.4 0.536989
$$947$$ −37612.4 −1.29064 −0.645321 0.763911i $$-0.723276\pi$$
−0.645321 + 0.763911i $$0.723276\pi$$
$$948$$ −13304.4 −0.455810
$$949$$ 4048.20 0.138472
$$950$$ 0 0
$$951$$ −2772.80 −0.0945469
$$952$$ 61148.2 2.08175
$$953$$ 48294.3 1.64156 0.820779 0.571246i $$-0.193540\pi$$
0.820779 + 0.571246i $$0.193540\pi$$
$$954$$ −195.880 −0.00664765
$$955$$ 0 0
$$956$$ 55239.8 1.86881
$$957$$ 2291.96 0.0774176
$$958$$ 11046.4 0.372539
$$959$$ 28047.4 0.944419
$$960$$ 0 0
$$961$$ −10021.4 −0.336389
$$962$$ −9573.76 −0.320863
$$963$$ 1368.22 0.0457843
$$964$$ 46173.1 1.54267
$$965$$ 0 0
$$966$$ 48907.8 1.62897
$$967$$ −1840.92 −0.0612204 −0.0306102 0.999531i $$-0.509745\pi$$
−0.0306102 + 0.999531i $$0.509745\pi$$
$$968$$ 1914.20 0.0635587
$$969$$ −12748.6 −0.422647
$$970$$ 0 0
$$971$$ 31461.8 1.03981 0.519906 0.854223i $$-0.325967\pi$$
0.519906 + 0.854223i $$0.325967\pi$$
$$972$$ 2812.86 0.0928217
$$973$$ 34612.1 1.14040
$$974$$ 15328.1 0.504254
$$975$$ 0 0
$$976$$ 12581.3 0.412620
$$977$$ 7040.11 0.230535 0.115268 0.993334i $$-0.463227\pi$$
0.115268 + 0.993334i $$0.463227\pi$$
$$978$$ 653.494 0.0213665
$$979$$ 8058.15 0.263064
$$980$$ 0 0
$$981$$ 19532.9 0.635715
$$982$$ −71784.4 −2.33272
$$983$$ 24610.9 0.798541 0.399270 0.916833i $$-0.369263\pi$$
0.399270 + 0.916833i $$0.369263\pi$$
$$984$$ −15284.7 −0.495183
$$985$$ 0 0
$$986$$ 37471.8 1.21029
$$987$$ −22005.3 −0.709662
$$988$$ −2077.95 −0.0669112
$$989$$ 37318.5 1.19986
$$990$$ 0 0
$$991$$ −40003.3 −1.28229 −0.641144 0.767421i $$-0.721540\pi$$
−0.641144 + 0.767421i $$0.721540\pi$$
$$992$$ −31860.8 −1.01974
$$993$$ −29461.4 −0.941519
$$994$$ −6883.31 −0.219643
$$995$$ 0 0
$$996$$ 32303.2 1.02768
$$997$$ 7342.61 0.233242 0.116621 0.993176i $$-0.462794\pi$$
0.116621 + 0.993176i $$0.462794\pi$$
$$998$$ 44216.9 1.40247
$$999$$ 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.a.l.1.2 2
3.2 odd 2 2475.4.a.p.1.1 2
5.2 odd 4 825.4.c.h.199.3 4
5.3 odd 4 825.4.c.h.199.2 4
5.4 even 2 33.4.a.c.1.1 2
15.14 odd 2 99.4.a.f.1.2 2
20.19 odd 2 528.4.a.p.1.2 2
35.34 odd 2 1617.4.a.k.1.1 2
40.19 odd 2 2112.4.a.bg.1.1 2
40.29 even 2 2112.4.a.bn.1.1 2
55.54 odd 2 363.4.a.i.1.2 2
60.59 even 2 1584.4.a.bj.1.1 2
165.164 even 2 1089.4.a.u.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 5.4 even 2
99.4.a.f.1.2 2 15.14 odd 2
363.4.a.i.1.2 2 55.54 odd 2
528.4.a.p.1.2 2 20.19 odd 2
825.4.a.l.1.2 2 1.1 even 1 trivial
825.4.c.h.199.2 4 5.3 odd 4
825.4.c.h.199.3 4 5.2 odd 4
1089.4.a.u.1.1 2 165.164 even 2
1584.4.a.bj.1.1 2 60.59 even 2
1617.4.a.k.1.1 2 35.34 odd 2
2112.4.a.bg.1.1 2 40.19 odd 2
2112.4.a.bn.1.1 2 40.29 even 2
2475.4.a.p.1.1 2 3.2 odd 2