# Properties

 Label 1815.4.a.bb.1.1 Level $1815$ Weight $4$ Character 1815.1 Self dual yes Analytic conductor $107.088$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, 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("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.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: $$107.088466660$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 44x^{4} + 495x^{2} - 1200$$ x^6 - 44*x^4 + 495*x^2 - 1200 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 5$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-5.26184$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.26184 q^{2} +3.00000 q^{3} +19.6870 q^{4} +5.00000 q^{5} -15.7855 q^{6} +5.37309 q^{7} -61.4950 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-5.26184 q^{2} +3.00000 q^{3} +19.6870 q^{4} +5.00000 q^{5} -15.7855 q^{6} +5.37309 q^{7} -61.4950 q^{8} +9.00000 q^{9} -26.3092 q^{10} +59.0609 q^{12} -5.15059 q^{13} -28.2724 q^{14} +15.0000 q^{15} +166.081 q^{16} +107.538 q^{17} -47.3566 q^{18} -70.1491 q^{19} +98.4349 q^{20} +16.1193 q^{21} +104.545 q^{23} -184.485 q^{24} +25.0000 q^{25} +27.1016 q^{26} +27.0000 q^{27} +105.780 q^{28} -122.545 q^{29} -78.9276 q^{30} +252.244 q^{31} -381.933 q^{32} -565.849 q^{34} +26.8655 q^{35} +177.183 q^{36} +150.545 q^{37} +369.113 q^{38} -15.4518 q^{39} -307.475 q^{40} -17.3082 q^{41} -84.8171 q^{42} -459.004 q^{43} +45.0000 q^{45} -550.098 q^{46} +310.585 q^{47} +498.244 q^{48} -314.130 q^{49} -131.546 q^{50} +322.615 q^{51} -101.400 q^{52} -322.186 q^{53} -142.070 q^{54} -330.418 q^{56} -210.447 q^{57} +644.812 q^{58} +425.601 q^{59} +295.305 q^{60} +75.2233 q^{61} -1327.27 q^{62} +48.3578 q^{63} +681.020 q^{64} -25.7530 q^{65} -898.576 q^{67} +2117.10 q^{68} +313.634 q^{69} -141.362 q^{70} +1192.10 q^{71} -553.455 q^{72} +277.379 q^{73} -792.142 q^{74} +75.0000 q^{75} -1381.02 q^{76} +81.3048 q^{78} +1257.17 q^{79} +830.406 q^{80} +81.0000 q^{81} +91.0729 q^{82} -156.497 q^{83} +317.340 q^{84} +537.691 q^{85} +2415.21 q^{86} -367.635 q^{87} -96.0978 q^{89} -236.783 q^{90} -27.6746 q^{91} +2058.17 q^{92} +756.731 q^{93} -1634.25 q^{94} -350.746 q^{95} -1145.80 q^{96} -840.690 q^{97} +1652.90 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10})$$ 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9 $$6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9} + 120 q^{12} + 116 q^{14} + 90 q^{15} + 164 q^{16} + 200 q^{20} + 56 q^{23} + 150 q^{25} + 292 q^{26} + 162 q^{27} + 576 q^{31} - 92 q^{34} + 360 q^{36} + 332 q^{37} + 496 q^{38} + 348 q^{42} + 270 q^{45} + 96 q^{47} + 492 q^{48} + 454 q^{49} + 308 q^{53} - 652 q^{56} + 1784 q^{58} + 2080 q^{59} + 600 q^{60} + 1928 q^{64} - 1168 q^{67} + 168 q^{69} + 580 q^{70} + 1064 q^{71} + 450 q^{75} + 876 q^{78} + 820 q^{80} + 486 q^{81} + 24 q^{82} + 5412 q^{86} - 684 q^{89} + 2744 q^{91} + 1368 q^{92} + 1728 q^{93} + 2812 q^{97}+O(q^{100})$$ 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9 + 120 * q^12 + 116 * q^14 + 90 * q^15 + 164 * q^16 + 200 * q^20 + 56 * q^23 + 150 * q^25 + 292 * q^26 + 162 * q^27 + 576 * q^31 - 92 * q^34 + 360 * q^36 + 332 * q^37 + 496 * q^38 + 348 * q^42 + 270 * q^45 + 96 * q^47 + 492 * q^48 + 454 * q^49 + 308 * q^53 - 652 * q^56 + 1784 * q^58 + 2080 * q^59 + 600 * q^60 + 1928 * q^64 - 1168 * q^67 + 168 * q^69 + 580 * q^70 + 1064 * q^71 + 450 * q^75 + 876 * q^78 + 820 * q^80 + 486 * q^81 + 24 * q^82 + 5412 * q^86 - 684 * q^89 + 2744 * q^91 + 1368 * q^92 + 1728 * q^93 + 2812 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.26184 −1.86034 −0.930171 0.367127i $$-0.880342\pi$$
−0.930171 + 0.367127i $$0.880342\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 19.6870 2.46087
$$5$$ 5.00000 0.447214
$$6$$ −15.7855 −1.07407
$$7$$ 5.37309 0.290120 0.145060 0.989423i $$-0.453663\pi$$
0.145060 + 0.989423i $$0.453663\pi$$
$$8$$ −61.4950 −2.71772
$$9$$ 9.00000 0.333333
$$10$$ −26.3092 −0.831970
$$11$$ 0 0
$$12$$ 59.0609 1.42079
$$13$$ −5.15059 −0.109886 −0.0549430 0.998489i $$-0.517498\pi$$
−0.0549430 + 0.998489i $$0.517498\pi$$
$$14$$ −28.2724 −0.539722
$$15$$ 15.0000 0.258199
$$16$$ 166.081 2.59502
$$17$$ 107.538 1.53423 0.767113 0.641512i $$-0.221692\pi$$
0.767113 + 0.641512i $$0.221692\pi$$
$$18$$ −47.3566 −0.620114
$$19$$ −70.1491 −0.847016 −0.423508 0.905892i $$-0.639201\pi$$
−0.423508 + 0.905892i $$0.639201\pi$$
$$20$$ 98.4349 1.10054
$$21$$ 16.1193 0.167501
$$22$$ 0 0
$$23$$ 104.545 0.947786 0.473893 0.880582i $$-0.342848\pi$$
0.473893 + 0.880582i $$0.342848\pi$$
$$24$$ −184.485 −1.56908
$$25$$ 25.0000 0.200000
$$26$$ 27.1016 0.204425
$$27$$ 27.0000 0.192450
$$28$$ 105.780 0.713947
$$29$$ −122.545 −0.784691 −0.392345 0.919818i $$-0.628336\pi$$
−0.392345 + 0.919818i $$0.628336\pi$$
$$30$$ −78.9276 −0.480338
$$31$$ 252.244 1.46143 0.730715 0.682683i $$-0.239187\pi$$
0.730715 + 0.682683i $$0.239187\pi$$
$$32$$ −381.933 −2.10990
$$33$$ 0 0
$$34$$ −565.849 −2.85419
$$35$$ 26.8655 0.129745
$$36$$ 177.183 0.820291
$$37$$ 150.545 0.668903 0.334451 0.942413i $$-0.391449\pi$$
0.334451 + 0.942413i $$0.391449\pi$$
$$38$$ 369.113 1.57574
$$39$$ −15.4518 −0.0634427
$$40$$ −307.475 −1.21540
$$41$$ −17.3082 −0.0659289 −0.0329645 0.999457i $$-0.510495\pi$$
−0.0329645 + 0.999457i $$0.510495\pi$$
$$42$$ −84.8171 −0.311608
$$43$$ −459.004 −1.62785 −0.813924 0.580972i $$-0.802673\pi$$
−0.813924 + 0.580972i $$0.802673\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ −550.098 −1.76321
$$47$$ 310.585 0.963904 0.481952 0.876198i $$-0.339928\pi$$
0.481952 + 0.876198i $$0.339928\pi$$
$$48$$ 498.244 1.49823
$$49$$ −314.130 −0.915831
$$50$$ −131.546 −0.372068
$$51$$ 322.615 0.885786
$$52$$ −101.400 −0.270415
$$53$$ −322.186 −0.835013 −0.417507 0.908674i $$-0.637096\pi$$
−0.417507 + 0.908674i $$0.637096\pi$$
$$54$$ −142.070 −0.358023
$$55$$ 0 0
$$56$$ −330.418 −0.788464
$$57$$ −210.447 −0.489025
$$58$$ 644.812 1.45979
$$59$$ 425.601 0.939128 0.469564 0.882898i $$-0.344411\pi$$
0.469564 + 0.882898i $$0.344411\pi$$
$$60$$ 295.305 0.635394
$$61$$ 75.2233 0.157891 0.0789455 0.996879i $$-0.474845\pi$$
0.0789455 + 0.996879i $$0.474845\pi$$
$$62$$ −1327.27 −2.71876
$$63$$ 48.3578 0.0967065
$$64$$ 681.020 1.33012
$$65$$ −25.7530 −0.0491425
$$66$$ 0 0
$$67$$ −898.576 −1.63849 −0.819243 0.573447i $$-0.805606\pi$$
−0.819243 + 0.573447i $$0.805606\pi$$
$$68$$ 2117.10 3.77554
$$69$$ 313.634 0.547204
$$70$$ −141.362 −0.241371
$$71$$ 1192.10 1.99263 0.996315 0.0857641i $$-0.0273331\pi$$
0.996315 + 0.0857641i $$0.0273331\pi$$
$$72$$ −553.455 −0.905907
$$73$$ 277.379 0.444722 0.222361 0.974964i $$-0.428624\pi$$
0.222361 + 0.974964i $$0.428624\pi$$
$$74$$ −792.142 −1.24439
$$75$$ 75.0000 0.115470
$$76$$ −1381.02 −2.08440
$$77$$ 0 0
$$78$$ 81.3048 0.118025
$$79$$ 1257.17 1.79041 0.895207 0.445651i $$-0.147028\pi$$
0.895207 + 0.445651i $$0.147028\pi$$
$$80$$ 830.406 1.16053
$$81$$ 81.0000 0.111111
$$82$$ 91.0729 0.122650
$$83$$ −156.497 −0.206961 −0.103480 0.994632i $$-0.532998\pi$$
−0.103480 + 0.994632i $$0.532998\pi$$
$$84$$ 317.340 0.412198
$$85$$ 537.691 0.686127
$$86$$ 2415.21 3.02835
$$87$$ −367.635 −0.453042
$$88$$ 0 0
$$89$$ −96.0978 −0.114453 −0.0572267 0.998361i $$-0.518226\pi$$
−0.0572267 + 0.998361i $$0.518226\pi$$
$$90$$ −236.783 −0.277323
$$91$$ −27.6746 −0.0318801
$$92$$ 2058.17 2.33238
$$93$$ 756.731 0.843757
$$94$$ −1634.25 −1.79319
$$95$$ −350.746 −0.378797
$$96$$ −1145.80 −1.21815
$$97$$ −840.690 −0.879991 −0.439995 0.898000i $$-0.645020\pi$$
−0.439995 + 0.898000i $$0.645020\pi$$
$$98$$ 1652.90 1.70376
$$99$$ 0 0
$$100$$ 492.174 0.492174
$$101$$ 1906.92 1.87867 0.939333 0.343008i $$-0.111446\pi$$
0.939333 + 0.343008i $$0.111446\pi$$
$$102$$ −1697.55 −1.64787
$$103$$ 142.967 0.136767 0.0683834 0.997659i $$-0.478216\pi$$
0.0683834 + 0.997659i $$0.478216\pi$$
$$104$$ 316.736 0.298639
$$105$$ 80.5964 0.0749086
$$106$$ 1695.29 1.55341
$$107$$ 1908.21 1.72405 0.862027 0.506862i $$-0.169195\pi$$
0.862027 + 0.506862i $$0.169195\pi$$
$$108$$ 531.548 0.473595
$$109$$ 672.607 0.591046 0.295523 0.955336i $$-0.404506\pi$$
0.295523 + 0.955336i $$0.404506\pi$$
$$110$$ 0 0
$$111$$ 451.634 0.386191
$$112$$ 892.369 0.752866
$$113$$ −1635.11 −1.36123 −0.680613 0.732644i $$-0.738286\pi$$
−0.680613 + 0.732644i $$0.738286\pi$$
$$114$$ 1107.34 0.909754
$$115$$ 522.724 0.423863
$$116$$ −2412.54 −1.93102
$$117$$ −46.3553 −0.0366286
$$118$$ −2239.45 −1.74710
$$119$$ 577.813 0.445109
$$120$$ −922.425 −0.701713
$$121$$ 0 0
$$122$$ −395.813 −0.293731
$$123$$ −51.9246 −0.0380641
$$124$$ 4965.92 3.59639
$$125$$ 125.000 0.0894427
$$126$$ −254.451 −0.179907
$$127$$ −941.002 −0.657484 −0.328742 0.944420i $$-0.606625\pi$$
−0.328742 + 0.944420i $$0.606625\pi$$
$$128$$ −527.958 −0.364573
$$129$$ −1377.01 −0.939838
$$130$$ 135.508 0.0914218
$$131$$ −1243.83 −0.829574 −0.414787 0.909919i $$-0.636144\pi$$
−0.414787 + 0.909919i $$0.636144\pi$$
$$132$$ 0 0
$$133$$ −376.917 −0.245736
$$134$$ 4728.16 3.04814
$$135$$ 135.000 0.0860663
$$136$$ −6613.06 −4.16960
$$137$$ −1127.14 −0.702904 −0.351452 0.936206i $$-0.614312\pi$$
−0.351452 + 0.936206i $$0.614312\pi$$
$$138$$ −1650.29 −1.01799
$$139$$ 2537.94 1.54867 0.774335 0.632776i $$-0.218084\pi$$
0.774335 + 0.632776i $$0.218084\pi$$
$$140$$ 528.899 0.319287
$$141$$ 931.756 0.556510
$$142$$ −6272.67 −3.70697
$$143$$ 0 0
$$144$$ 1494.73 0.865006
$$145$$ −612.725 −0.350924
$$146$$ −1459.52 −0.827335
$$147$$ −942.390 −0.528755
$$148$$ 2963.77 1.64608
$$149$$ 2635.22 1.44890 0.724448 0.689329i $$-0.242095\pi$$
0.724448 + 0.689329i $$0.242095\pi$$
$$150$$ −394.638 −0.214814
$$151$$ −2665.26 −1.43639 −0.718197 0.695839i $$-0.755032\pi$$
−0.718197 + 0.695839i $$0.755032\pi$$
$$152$$ 4313.82 2.30195
$$153$$ 967.844 0.511409
$$154$$ 0 0
$$155$$ 1261.22 0.653571
$$156$$ −304.199 −0.156124
$$157$$ 2341.32 1.19017 0.595087 0.803661i $$-0.297118\pi$$
0.595087 + 0.803661i $$0.297118\pi$$
$$158$$ −6615.03 −3.33078
$$159$$ −966.559 −0.482095
$$160$$ −1909.66 −0.943576
$$161$$ 561.728 0.274971
$$162$$ −426.209 −0.206705
$$163$$ −3408.54 −1.63790 −0.818949 0.573866i $$-0.805443\pi$$
−0.818949 + 0.573866i $$0.805443\pi$$
$$164$$ −340.746 −0.162243
$$165$$ 0 0
$$166$$ 823.460 0.385017
$$167$$ 769.717 0.356662 0.178331 0.983971i $$-0.442930\pi$$
0.178331 + 0.983971i $$0.442930\pi$$
$$168$$ −991.255 −0.455220
$$169$$ −2170.47 −0.987925
$$170$$ −2829.25 −1.27643
$$171$$ −631.342 −0.282339
$$172$$ −9036.40 −4.00592
$$173$$ −2110.51 −0.927508 −0.463754 0.885964i $$-0.653498\pi$$
−0.463754 + 0.885964i $$0.653498\pi$$
$$174$$ 1934.44 0.842812
$$175$$ 134.327 0.0580239
$$176$$ 0 0
$$177$$ 1276.80 0.542206
$$178$$ 505.651 0.212922
$$179$$ 1473.25 0.615173 0.307587 0.951520i $$-0.400479\pi$$
0.307587 + 0.951520i $$0.400479\pi$$
$$180$$ 885.914 0.366845
$$181$$ −2290.00 −0.940413 −0.470206 0.882557i $$-0.655821\pi$$
−0.470206 + 0.882557i $$0.655821\pi$$
$$182$$ 145.619 0.0593078
$$183$$ 225.670 0.0911584
$$184$$ −6428.98 −2.57582
$$185$$ 752.724 0.299142
$$186$$ −3981.80 −1.56968
$$187$$ 0 0
$$188$$ 6114.48 2.37205
$$189$$ 145.073 0.0558335
$$190$$ 1845.57 0.704692
$$191$$ 3880.34 1.47001 0.735003 0.678063i $$-0.237180\pi$$
0.735003 + 0.678063i $$0.237180\pi$$
$$192$$ 2043.06 0.767944
$$193$$ 852.978 0.318128 0.159064 0.987268i $$-0.449152\pi$$
0.159064 + 0.987268i $$0.449152\pi$$
$$194$$ 4423.58 1.63708
$$195$$ −77.2589 −0.0283724
$$196$$ −6184.27 −2.25374
$$197$$ −766.748 −0.277302 −0.138651 0.990341i $$-0.544277\pi$$
−0.138651 + 0.990341i $$0.544277\pi$$
$$198$$ 0 0
$$199$$ 1242.15 0.442479 0.221240 0.975219i $$-0.428990\pi$$
0.221240 + 0.975219i $$0.428990\pi$$
$$200$$ −1537.38 −0.543544
$$201$$ −2695.73 −0.945980
$$202$$ −10033.9 −3.49496
$$203$$ −658.445 −0.227654
$$204$$ 6351.31 2.17981
$$205$$ −86.5409 −0.0294843
$$206$$ −752.271 −0.254433
$$207$$ 940.902 0.315929
$$208$$ −855.416 −0.285156
$$209$$ 0 0
$$210$$ −424.085 −0.139356
$$211$$ −4347.95 −1.41860 −0.709301 0.704906i $$-0.750989\pi$$
−0.709301 + 0.704906i $$0.750989\pi$$
$$212$$ −6342.88 −2.05486
$$213$$ 3576.31 1.15045
$$214$$ −10040.7 −3.20733
$$215$$ −2295.02 −0.727996
$$216$$ −1660.37 −0.523026
$$217$$ 1355.33 0.423989
$$218$$ −3539.15 −1.09955
$$219$$ 832.136 0.256760
$$220$$ 0 0
$$221$$ −553.886 −0.168590
$$222$$ −2376.43 −0.718447
$$223$$ −1123.30 −0.337317 −0.168658 0.985675i $$-0.553943\pi$$
−0.168658 + 0.985675i $$0.553943\pi$$
$$224$$ −2052.16 −0.612123
$$225$$ 225.000 0.0666667
$$226$$ 8603.70 2.53234
$$227$$ −435.002 −0.127190 −0.0635949 0.997976i $$-0.520257\pi$$
−0.0635949 + 0.997976i $$0.520257\pi$$
$$228$$ −4143.07 −1.20343
$$229$$ 4207.36 1.21411 0.607053 0.794661i $$-0.292351\pi$$
0.607053 + 0.794661i $$0.292351\pi$$
$$230$$ −2750.49 −0.788530
$$231$$ 0 0
$$232$$ 7535.91 2.13257
$$233$$ 235.268 0.0661499 0.0330749 0.999453i $$-0.489470\pi$$
0.0330749 + 0.999453i $$0.489470\pi$$
$$234$$ 243.914 0.0681418
$$235$$ 1552.93 0.431071
$$236$$ 8378.80 2.31107
$$237$$ 3771.51 1.03370
$$238$$ −3040.36 −0.828055
$$239$$ −647.975 −0.175373 −0.0876863 0.996148i $$-0.527947\pi$$
−0.0876863 + 0.996148i $$0.527947\pi$$
$$240$$ 2491.22 0.670031
$$241$$ 2281.64 0.609848 0.304924 0.952377i $$-0.401369\pi$$
0.304924 + 0.952377i $$0.401369\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 1480.92 0.388550
$$245$$ −1570.65 −0.409572
$$246$$ 273.219 0.0708122
$$247$$ 361.309 0.0930752
$$248$$ −15511.7 −3.97176
$$249$$ −469.490 −0.119489
$$250$$ −657.730 −0.166394
$$251$$ −4869.34 −1.22450 −0.612251 0.790664i $$-0.709736\pi$$
−0.612251 + 0.790664i $$0.709736\pi$$
$$252$$ 952.019 0.237982
$$253$$ 0 0
$$254$$ 4951.41 1.22314
$$255$$ 1613.07 0.396136
$$256$$ −2670.13 −0.651887
$$257$$ −1381.17 −0.335234 −0.167617 0.985852i $$-0.553607\pi$$
−0.167617 + 0.985852i $$0.553607\pi$$
$$258$$ 7245.62 1.74842
$$259$$ 808.890 0.194062
$$260$$ −506.998 −0.120933
$$261$$ −1102.91 −0.261564
$$262$$ 6544.85 1.54329
$$263$$ 1121.05 0.262841 0.131420 0.991327i $$-0.458046\pi$$
0.131420 + 0.991327i $$0.458046\pi$$
$$264$$ 0 0
$$265$$ −1610.93 −0.373429
$$266$$ 1983.28 0.457153
$$267$$ −288.293 −0.0660797
$$268$$ −17690.2 −4.03210
$$269$$ −1161.36 −0.263232 −0.131616 0.991301i $$-0.542017\pi$$
−0.131616 + 0.991301i $$0.542017\pi$$
$$270$$ −710.349 −0.160113
$$271$$ 3568.57 0.799908 0.399954 0.916535i $$-0.369026\pi$$
0.399954 + 0.916535i $$0.369026\pi$$
$$272$$ 17860.1 3.98135
$$273$$ −83.0238 −0.0184060
$$274$$ 5930.81 1.30764
$$275$$ 0 0
$$276$$ 6174.51 1.34660
$$277$$ 8346.58 1.81046 0.905230 0.424922i $$-0.139698\pi$$
0.905230 + 0.424922i $$0.139698\pi$$
$$278$$ −13354.2 −2.88106
$$279$$ 2270.19 0.487143
$$280$$ −1652.09 −0.352612
$$281$$ −1723.69 −0.365931 −0.182966 0.983119i $$-0.558570\pi$$
−0.182966 + 0.983119i $$0.558570\pi$$
$$282$$ −4902.75 −1.03530
$$283$$ −2086.76 −0.438321 −0.219160 0.975689i $$-0.570332\pi$$
−0.219160 + 0.975689i $$0.570332\pi$$
$$284$$ 23468.9 4.90361
$$285$$ −1052.24 −0.218699
$$286$$ 0 0
$$287$$ −92.9985 −0.0191273
$$288$$ −3437.39 −0.703300
$$289$$ 6651.47 1.35385
$$290$$ 3224.06 0.652839
$$291$$ −2522.07 −0.508063
$$292$$ 5460.75 1.09440
$$293$$ 3127.52 0.623588 0.311794 0.950150i $$-0.399070\pi$$
0.311794 + 0.950150i $$0.399070\pi$$
$$294$$ 4958.71 0.983665
$$295$$ 2128.01 0.419991
$$296$$ −9257.75 −1.81789
$$297$$ 0 0
$$298$$ −13866.1 −2.69544
$$299$$ −538.467 −0.104148
$$300$$ 1476.52 0.284157
$$301$$ −2466.27 −0.472271
$$302$$ 14024.2 2.67219
$$303$$ 5720.75 1.08465
$$304$$ −11650.4 −2.19802
$$305$$ 376.116 0.0706110
$$306$$ −5092.64 −0.951395
$$307$$ −1397.12 −0.259732 −0.129866 0.991532i $$-0.541455\pi$$
−0.129866 + 0.991532i $$0.541455\pi$$
$$308$$ 0 0
$$309$$ 428.902 0.0789623
$$310$$ −6636.33 −1.21587
$$311$$ 5405.45 0.985579 0.492790 0.870148i $$-0.335977\pi$$
0.492790 + 0.870148i $$0.335977\pi$$
$$312$$ 950.207 0.172419
$$313$$ 6240.77 1.12699 0.563497 0.826118i $$-0.309456\pi$$
0.563497 + 0.826118i $$0.309456\pi$$
$$314$$ −12319.6 −2.21413
$$315$$ 241.789 0.0432485
$$316$$ 24749.9 4.40598
$$317$$ −6919.66 −1.22601 −0.613007 0.790077i $$-0.710040\pi$$
−0.613007 + 0.790077i $$0.710040\pi$$
$$318$$ 5085.88 0.896862
$$319$$ 0 0
$$320$$ 3405.10 0.594847
$$321$$ 5724.64 0.995383
$$322$$ −2955.72 −0.511541
$$323$$ −7543.71 −1.29951
$$324$$ 1594.64 0.273430
$$325$$ −128.765 −0.0219772
$$326$$ 17935.2 3.04705
$$327$$ 2017.82 0.341241
$$328$$ 1064.37 0.179176
$$329$$ 1668.80 0.279648
$$330$$ 0 0
$$331$$ −3134.58 −0.520519 −0.260260 0.965539i $$-0.583808\pi$$
−0.260260 + 0.965539i $$0.583808\pi$$
$$332$$ −3080.94 −0.509303
$$333$$ 1354.90 0.222968
$$334$$ −4050.13 −0.663513
$$335$$ −4492.88 −0.732753
$$336$$ 2677.11 0.434667
$$337$$ 6484.89 1.04823 0.524117 0.851647i $$-0.324396\pi$$
0.524117 + 0.851647i $$0.324396\pi$$
$$338$$ 11420.7 1.83788
$$339$$ −4905.34 −0.785904
$$340$$ 10585.5 1.68847
$$341$$ 0 0
$$342$$ 3322.02 0.525247
$$343$$ −3530.82 −0.555820
$$344$$ 28226.5 4.42404
$$345$$ 1568.17 0.244717
$$346$$ 11105.1 1.72548
$$347$$ −2535.12 −0.392198 −0.196099 0.980584i $$-0.562827\pi$$
−0.196099 + 0.980584i $$0.562827\pi$$
$$348$$ −7237.62 −1.11488
$$349$$ −9319.24 −1.42936 −0.714681 0.699451i $$-0.753428\pi$$
−0.714681 + 0.699451i $$0.753428\pi$$
$$350$$ −706.809 −0.107944
$$351$$ −139.066 −0.0211476
$$352$$ 0 0
$$353$$ −3958.44 −0.596846 −0.298423 0.954434i $$-0.596461\pi$$
−0.298423 + 0.954434i $$0.596461\pi$$
$$354$$ −6718.34 −1.00869
$$355$$ 5960.52 0.891132
$$356$$ −1891.88 −0.281655
$$357$$ 1733.44 0.256984
$$358$$ −7752.02 −1.14443
$$359$$ 10876.2 1.59896 0.799480 0.600693i $$-0.205109\pi$$
0.799480 + 0.600693i $$0.205109\pi$$
$$360$$ −2767.28 −0.405134
$$361$$ −1938.10 −0.282564
$$362$$ 12049.6 1.74949
$$363$$ 0 0
$$364$$ −544.829 −0.0784528
$$365$$ 1386.89 0.198886
$$366$$ −1187.44 −0.169586
$$367$$ 4192.86 0.596364 0.298182 0.954509i $$-0.403620\pi$$
0.298182 + 0.954509i $$0.403620\pi$$
$$368$$ 17362.9 2.45952
$$369$$ −155.774 −0.0219763
$$370$$ −3960.71 −0.556507
$$371$$ −1731.14 −0.242254
$$372$$ 14897.7 2.07638
$$373$$ 11108.9 1.54209 0.771044 0.636782i $$-0.219735\pi$$
0.771044 + 0.636782i $$0.219735\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ −19099.4 −2.61962
$$377$$ 631.179 0.0862265
$$378$$ −763.353 −0.103869
$$379$$ −4716.38 −0.639219 −0.319610 0.947549i $$-0.603552\pi$$
−0.319610 + 0.947549i $$0.603552\pi$$
$$380$$ −6905.12 −0.932171
$$381$$ −2823.01 −0.379599
$$382$$ −20417.7 −2.73472
$$383$$ 10117.1 1.34977 0.674883 0.737925i $$-0.264194\pi$$
0.674883 + 0.737925i $$0.264194\pi$$
$$384$$ −1583.87 −0.210486
$$385$$ 0 0
$$386$$ −4488.23 −0.591827
$$387$$ −4131.04 −0.542616
$$388$$ −16550.6 −2.16554
$$389$$ 11322.0 1.47571 0.737853 0.674961i $$-0.235840\pi$$
0.737853 + 0.674961i $$0.235840\pi$$
$$390$$ 406.524 0.0527824
$$391$$ 11242.6 1.45412
$$392$$ 19317.4 2.48897
$$393$$ −3731.50 −0.478955
$$394$$ 4034.51 0.515877
$$395$$ 6285.85 0.800697
$$396$$ 0 0
$$397$$ 5400.42 0.682718 0.341359 0.939933i $$-0.389113\pi$$
0.341359 + 0.939933i $$0.389113\pi$$
$$398$$ −6535.97 −0.823163
$$399$$ −1130.75 −0.141876
$$400$$ 4152.03 0.519004
$$401$$ 13782.4 1.71637 0.858183 0.513344i $$-0.171594\pi$$
0.858183 + 0.513344i $$0.171594\pi$$
$$402$$ 14184.5 1.75985
$$403$$ −1299.20 −0.160591
$$404$$ 37541.4 4.62315
$$405$$ 405.000 0.0496904
$$406$$ 3464.64 0.423515
$$407$$ 0 0
$$408$$ −19839.2 −2.40732
$$409$$ −113.092 −0.0136725 −0.00683626 0.999977i $$-0.502176\pi$$
−0.00683626 + 0.999977i $$0.502176\pi$$
$$410$$ 455.365 0.0548509
$$411$$ −3381.41 −0.405822
$$412$$ 2814.59 0.336565
$$413$$ 2286.79 0.272459
$$414$$ −4950.88 −0.587735
$$415$$ −782.483 −0.0925556
$$416$$ 1967.18 0.231848
$$417$$ 7613.82 0.894125
$$418$$ 0 0
$$419$$ 7352.48 0.857261 0.428630 0.903480i $$-0.358996\pi$$
0.428630 + 0.903480i $$0.358996\pi$$
$$420$$ 1586.70 0.184340
$$421$$ 3306.97 0.382831 0.191416 0.981509i $$-0.438692\pi$$
0.191416 + 0.981509i $$0.438692\pi$$
$$422$$ 22878.2 2.63908
$$423$$ 2795.27 0.321301
$$424$$ 19812.9 2.26933
$$425$$ 2688.46 0.306845
$$426$$ −18818.0 −2.14022
$$427$$ 404.181 0.0458073
$$428$$ 37566.9 4.24268
$$429$$ 0 0
$$430$$ 12076.0 1.35432
$$431$$ 9959.87 1.11311 0.556555 0.830811i $$-0.312123\pi$$
0.556555 + 0.830811i $$0.312123\pi$$
$$432$$ 4484.19 0.499412
$$433$$ 10913.1 1.21120 0.605598 0.795771i $$-0.292934\pi$$
0.605598 + 0.795771i $$0.292934\pi$$
$$434$$ −7131.52 −0.788765
$$435$$ −1838.18 −0.202606
$$436$$ 13241.6 1.45449
$$437$$ −7333.72 −0.802790
$$438$$ −4378.57 −0.477662
$$439$$ −6589.60 −0.716411 −0.358205 0.933643i $$-0.616611\pi$$
−0.358205 + 0.933643i $$0.616611\pi$$
$$440$$ 0 0
$$441$$ −2827.17 −0.305277
$$442$$ 2914.46 0.313635
$$443$$ −2597.43 −0.278572 −0.139286 0.990252i $$-0.544481\pi$$
−0.139286 + 0.990252i $$0.544481\pi$$
$$444$$ 8891.31 0.950367
$$445$$ −480.489 −0.0511851
$$446$$ 5910.62 0.627524
$$447$$ 7905.66 0.836521
$$448$$ 3659.18 0.385893
$$449$$ 6460.30 0.679021 0.339511 0.940602i $$-0.389739\pi$$
0.339511 + 0.940602i $$0.389739\pi$$
$$450$$ −1183.91 −0.124023
$$451$$ 0 0
$$452$$ −32190.4 −3.34980
$$453$$ −7995.77 −0.829303
$$454$$ 2288.91 0.236617
$$455$$ −138.373 −0.0142572
$$456$$ 12941.5 1.32903
$$457$$ 7676.12 0.785720 0.392860 0.919598i $$-0.371486\pi$$
0.392860 + 0.919598i $$0.371486\pi$$
$$458$$ −22138.5 −2.25865
$$459$$ 2903.53 0.295262
$$460$$ 10290.8 1.04307
$$461$$ −7555.51 −0.763330 −0.381665 0.924301i $$-0.624649\pi$$
−0.381665 + 0.924301i $$0.624649\pi$$
$$462$$ 0 0
$$463$$ 5480.50 0.550109 0.275055 0.961429i $$-0.411304\pi$$
0.275055 + 0.961429i $$0.411304\pi$$
$$464$$ −20352.4 −2.03629
$$465$$ 3783.66 0.377339
$$466$$ −1237.94 −0.123061
$$467$$ −10339.7 −1.02455 −0.512275 0.858822i $$-0.671197\pi$$
−0.512275 + 0.858822i $$0.671197\pi$$
$$468$$ −912.596 −0.0901384
$$469$$ −4828.13 −0.475357
$$470$$ −8171.25 −0.801940
$$471$$ 7023.95 0.687147
$$472$$ −26172.3 −2.55229
$$473$$ 0 0
$$474$$ −19845.1 −1.92303
$$475$$ −1753.73 −0.169403
$$476$$ 11375.4 1.09536
$$477$$ −2899.68 −0.278338
$$478$$ 3409.54 0.326253
$$479$$ −4320.65 −0.412141 −0.206070 0.978537i $$-0.566068\pi$$
−0.206070 + 0.978537i $$0.566068\pi$$
$$480$$ −5728.99 −0.544774
$$481$$ −775.394 −0.0735030
$$482$$ −12005.6 −1.13453
$$483$$ 1685.18 0.158755
$$484$$ 0 0
$$485$$ −4203.45 −0.393544
$$486$$ −1278.63 −0.119341
$$487$$ −9426.09 −0.877078 −0.438539 0.898712i $$-0.644504\pi$$
−0.438539 + 0.898712i $$0.644504\pi$$
$$488$$ −4625.86 −0.429104
$$489$$ −10225.6 −0.945641
$$490$$ 8264.51 0.761944
$$491$$ −569.771 −0.0523694 −0.0261847 0.999657i $$-0.508336\pi$$
−0.0261847 + 0.999657i $$0.508336\pi$$
$$492$$ −1022.24 −0.0936708
$$493$$ −13178.3 −1.20389
$$494$$ −1901.15 −0.173152
$$495$$ 0 0
$$496$$ 41892.9 3.79244
$$497$$ 6405.29 0.578101
$$498$$ 2470.38 0.222290
$$499$$ 17072.5 1.53161 0.765804 0.643074i $$-0.222341\pi$$
0.765804 + 0.643074i $$0.222341\pi$$
$$500$$ 2460.87 0.220107
$$501$$ 2309.15 0.205919
$$502$$ 25621.7 2.27799
$$503$$ −18622.7 −1.65079 −0.825395 0.564556i $$-0.809047\pi$$
−0.825395 + 0.564556i $$0.809047\pi$$
$$504$$ −2973.76 −0.262821
$$505$$ 9534.58 0.840165
$$506$$ 0 0
$$507$$ −6511.41 −0.570379
$$508$$ −18525.5 −1.61798
$$509$$ 16294.6 1.41895 0.709473 0.704733i $$-0.248933\pi$$
0.709473 + 0.704733i $$0.248933\pi$$
$$510$$ −8487.74 −0.736948
$$511$$ 1490.38 0.129023
$$512$$ 18273.5 1.57731
$$513$$ −1894.03 −0.163008
$$514$$ 7267.50 0.623649
$$515$$ 714.836 0.0611640
$$516$$ −27109.2 −2.31282
$$517$$ 0 0
$$518$$ −4256.25 −0.361021
$$519$$ −6331.52 −0.535497
$$520$$ 1583.68 0.133556
$$521$$ −20991.0 −1.76513 −0.882565 0.470189i $$-0.844186\pi$$
−0.882565 + 0.470189i $$0.844186\pi$$
$$522$$ 5803.31 0.486598
$$523$$ −11326.3 −0.946966 −0.473483 0.880803i $$-0.657003\pi$$
−0.473483 + 0.880803i $$0.657003\pi$$
$$524$$ −24487.3 −2.04147
$$525$$ 402.982 0.0335001
$$526$$ −5898.80 −0.488973
$$527$$ 27125.8 2.24216
$$528$$ 0 0
$$529$$ −1237.41 −0.101702
$$530$$ 8476.47 0.694706
$$531$$ 3830.41 0.313043
$$532$$ −7420.36 −0.604725
$$533$$ 89.1474 0.00724466
$$534$$ 1516.95 0.122931
$$535$$ 9541.06 0.771021
$$536$$ 55257.9 4.45295
$$537$$ 4419.76 0.355171
$$538$$ 6110.90 0.489702
$$539$$ 0 0
$$540$$ 2657.74 0.211798
$$541$$ −23994.8 −1.90687 −0.953433 0.301603i $$-0.902478\pi$$
−0.953433 + 0.301603i $$0.902478\pi$$
$$542$$ −18777.2 −1.48810
$$543$$ −6870.01 −0.542948
$$544$$ −41072.4 −3.23707
$$545$$ 3363.03 0.264324
$$546$$ 436.858 0.0342414
$$547$$ −14347.6 −1.12150 −0.560748 0.827986i $$-0.689487\pi$$
−0.560748 + 0.827986i $$0.689487\pi$$
$$548$$ −22189.9 −1.72976
$$549$$ 677.009 0.0526303
$$550$$ 0 0
$$551$$ 8596.42 0.664646
$$552$$ −19286.9 −1.48715
$$553$$ 6754.89 0.519434
$$554$$ −43918.4 −3.36807
$$555$$ 2258.17 0.172710
$$556$$ 49964.3 3.81108
$$557$$ 19041.9 1.44853 0.724265 0.689521i $$-0.242179\pi$$
0.724265 + 0.689521i $$0.242179\pi$$
$$558$$ −11945.4 −0.906253
$$559$$ 2364.14 0.178878
$$560$$ 4461.85 0.336692
$$561$$ 0 0
$$562$$ 9069.77 0.680757
$$563$$ 15082.0 1.12900 0.564502 0.825432i $$-0.309068\pi$$
0.564502 + 0.825432i $$0.309068\pi$$
$$564$$ 18343.4 1.36950
$$565$$ −8175.56 −0.608758
$$566$$ 10980.2 0.815426
$$567$$ 435.220 0.0322355
$$568$$ −73308.5 −5.41541
$$569$$ 13321.8 0.981513 0.490756 0.871297i $$-0.336720\pi$$
0.490756 + 0.871297i $$0.336720\pi$$
$$570$$ 5536.70 0.406854
$$571$$ 24112.7 1.76722 0.883611 0.468223i $$-0.155105\pi$$
0.883611 + 0.468223i $$0.155105\pi$$
$$572$$ 0 0
$$573$$ 11641.0 0.848709
$$574$$ 489.343 0.0355833
$$575$$ 2613.62 0.189557
$$576$$ 6129.18 0.443372
$$577$$ −4355.07 −0.314218 −0.157109 0.987581i $$-0.550217\pi$$
−0.157109 + 0.987581i $$0.550217\pi$$
$$578$$ −34999.0 −2.51863
$$579$$ 2558.93 0.183671
$$580$$ −12062.7 −0.863580
$$581$$ −840.870 −0.0600433
$$582$$ 13270.7 0.945171
$$583$$ 0 0
$$584$$ −17057.4 −1.20863
$$585$$ −231.777 −0.0163808
$$586$$ −16456.5 −1.16009
$$587$$ 14765.2 1.03820 0.519101 0.854713i $$-0.326267\pi$$
0.519101 + 0.854713i $$0.326267\pi$$
$$588$$ −18552.8 −1.30120
$$589$$ −17694.7 −1.23785
$$590$$ −11197.2 −0.781327
$$591$$ −2300.24 −0.160100
$$592$$ 25002.6 1.73581
$$593$$ 2990.56 0.207095 0.103548 0.994625i $$-0.466981\pi$$
0.103548 + 0.994625i $$0.466981\pi$$
$$594$$ 0 0
$$595$$ 2889.06 0.199059
$$596$$ 51879.5 3.56555
$$597$$ 3726.44 0.255466
$$598$$ 2833.33 0.193752
$$599$$ −1019.09 −0.0695138 −0.0347569 0.999396i $$-0.511066\pi$$
−0.0347569 + 0.999396i $$0.511066\pi$$
$$600$$ −4612.13 −0.313815
$$601$$ 18514.0 1.25658 0.628289 0.777980i $$-0.283756\pi$$
0.628289 + 0.777980i $$0.283756\pi$$
$$602$$ 12977.1 0.878585
$$603$$ −8087.18 −0.546162
$$604$$ −52470.9 −3.53478
$$605$$ 0 0
$$606$$ −30101.7 −2.01782
$$607$$ 15129.0 1.01164 0.505821 0.862638i $$-0.331190\pi$$
0.505821 + 0.862638i $$0.331190\pi$$
$$608$$ 26792.2 1.78712
$$609$$ −1975.34 −0.131436
$$610$$ −1979.06 −0.131361
$$611$$ −1599.70 −0.105920
$$612$$ 19053.9 1.25851
$$613$$ −10775.7 −0.709992 −0.354996 0.934868i $$-0.615518\pi$$
−0.354996 + 0.934868i $$0.615518\pi$$
$$614$$ 7351.42 0.483191
$$615$$ −259.623 −0.0170228
$$616$$ 0 0
$$617$$ 13290.2 0.867170 0.433585 0.901113i $$-0.357248\pi$$
0.433585 + 0.901113i $$0.357248\pi$$
$$618$$ −2256.81 −0.146897
$$619$$ −21299.7 −1.38305 −0.691526 0.722352i $$-0.743061\pi$$
−0.691526 + 0.722352i $$0.743061\pi$$
$$620$$ 24829.6 1.60835
$$621$$ 2822.71 0.182401
$$622$$ −28442.6 −1.83351
$$623$$ −516.342 −0.0332052
$$624$$ −2566.25 −0.164635
$$625$$ 625.000 0.0400000
$$626$$ −32837.9 −2.09659
$$627$$ 0 0
$$628$$ 46093.4 2.92887
$$629$$ 16189.3 1.02625
$$630$$ −1272.26 −0.0804570
$$631$$ 13486.2 0.850834 0.425417 0.904997i $$-0.360127\pi$$
0.425417 + 0.904997i $$0.360127\pi$$
$$632$$ −77309.7 −4.86584
$$633$$ −13043.8 −0.819030
$$634$$ 36410.1 2.28081
$$635$$ −4705.01 −0.294036
$$636$$ −19028.6 −1.18637
$$637$$ 1617.95 0.100637
$$638$$ 0 0
$$639$$ 10728.9 0.664210
$$640$$ −2639.79 −0.163042
$$641$$ 3234.75 0.199321 0.0996605 0.995021i $$-0.468224\pi$$
0.0996605 + 0.995021i $$0.468224\pi$$
$$642$$ −30122.1 −1.85175
$$643$$ −19623.4 −1.20353 −0.601766 0.798672i $$-0.705536\pi$$
−0.601766 + 0.798672i $$0.705536\pi$$
$$644$$ 11058.7 0.676669
$$645$$ −6885.06 −0.420308
$$646$$ 39693.8 2.41754
$$647$$ −14293.8 −0.868541 −0.434271 0.900782i $$-0.642994\pi$$
−0.434271 + 0.900782i $$0.642994\pi$$
$$648$$ −4981.10 −0.301969
$$649$$ 0 0
$$650$$ 677.540 0.0408851
$$651$$ 4065.98 0.244790
$$652$$ −67103.9 −4.03066
$$653$$ −3963.73 −0.237539 −0.118769 0.992922i $$-0.537895\pi$$
−0.118769 + 0.992922i $$0.537895\pi$$
$$654$$ −10617.5 −0.634825
$$655$$ −6219.16 −0.370997
$$656$$ −2874.56 −0.171087
$$657$$ 2496.41 0.148241
$$658$$ −8780.97 −0.520240
$$659$$ 1516.65 0.0896515 0.0448258 0.998995i $$-0.485727\pi$$
0.0448258 + 0.998995i $$0.485727\pi$$
$$660$$ 0 0
$$661$$ 106.966 0.00629425 0.00314713 0.999995i $$-0.498998\pi$$
0.00314713 + 0.999995i $$0.498998\pi$$
$$662$$ 16493.6 0.968344
$$663$$ −1661.66 −0.0973354
$$664$$ 9623.76 0.562461
$$665$$ −1884.59 −0.109896
$$666$$ −7129.28 −0.414796
$$667$$ −12811.4 −0.743719
$$668$$ 15153.4 0.877699
$$669$$ −3369.90 −0.194750
$$670$$ 23640.8 1.36317
$$671$$ 0 0
$$672$$ −6156.48 −0.353410
$$673$$ −15728.1 −0.900852 −0.450426 0.892814i $$-0.648728\pi$$
−0.450426 + 0.892814i $$0.648728\pi$$
$$674$$ −34122.5 −1.95007
$$675$$ 675.000 0.0384900
$$676$$ −42730.0 −2.43116
$$677$$ 316.806 0.0179850 0.00899249 0.999960i $$-0.497138\pi$$
0.00899249 + 0.999960i $$0.497138\pi$$
$$678$$ 25811.1 1.46205
$$679$$ −4517.10 −0.255303
$$680$$ −33065.3 −1.86470
$$681$$ −1305.01 −0.0734331
$$682$$ 0 0
$$683$$ −11621.2 −0.651057 −0.325528 0.945532i $$-0.605542\pi$$
−0.325528 + 0.945532i $$0.605542\pi$$
$$684$$ −12429.2 −0.694799
$$685$$ −5635.68 −0.314348
$$686$$ 18578.6 1.03402
$$687$$ 12622.1 0.700965
$$688$$ −76231.9 −4.22429
$$689$$ 1659.45 0.0917562
$$690$$ −8251.46 −0.455258
$$691$$ 22281.8 1.22669 0.613344 0.789816i $$-0.289824\pi$$
0.613344 + 0.789816i $$0.289824\pi$$
$$692$$ −41549.5 −2.28248
$$693$$ 0 0
$$694$$ 13339.4 0.729622
$$695$$ 12689.7 0.692586
$$696$$ 22607.7 1.23124
$$697$$ −1861.29 −0.101150
$$698$$ 49036.3 2.65910
$$699$$ 705.804 0.0381917
$$700$$ 2644.50 0.142789
$$701$$ −31634.6 −1.70445 −0.852227 0.523172i $$-0.824749\pi$$
−0.852227 + 0.523172i $$0.824749\pi$$
$$702$$ 731.743 0.0393417
$$703$$ −10560.6 −0.566571
$$704$$ 0 0
$$705$$ 4658.78 0.248879
$$706$$ 20828.7 1.11034
$$707$$ 10246.0 0.545038
$$708$$ 25136.4 1.33430
$$709$$ −7245.11 −0.383774 −0.191887 0.981417i $$-0.561461\pi$$
−0.191887 + 0.981417i $$0.561461\pi$$
$$710$$ −31363.3 −1.65781
$$711$$ 11314.5 0.596804
$$712$$ 5909.54 0.311052
$$713$$ 26370.7 1.38512
$$714$$ −9121.08 −0.478078
$$715$$ 0 0
$$716$$ 29003.9 1.51386
$$717$$ −1943.93 −0.101251
$$718$$ −57229.1 −2.97461
$$719$$ −33452.4 −1.73514 −0.867569 0.497317i $$-0.834319\pi$$
−0.867569 + 0.497317i $$0.834319\pi$$
$$720$$ 7473.65 0.386842
$$721$$ 768.176 0.0396787
$$722$$ 10198.0 0.525665
$$723$$ 6844.92 0.352096
$$724$$ −45083.3 −2.31424
$$725$$ −3063.63 −0.156938
$$726$$ 0 0
$$727$$ 7395.29 0.377271 0.188636 0.982047i $$-0.439594\pi$$
0.188636 + 0.982047i $$0.439594\pi$$
$$728$$ 1701.85 0.0866411
$$729$$ 729.000 0.0370370
$$730$$ −7297.61 −0.369996
$$731$$ −49360.5 −2.49749
$$732$$ 4442.76 0.224329
$$733$$ −666.961 −0.0336082 −0.0168041 0.999859i $$-0.505349\pi$$
−0.0168041 + 0.999859i $$0.505349\pi$$
$$734$$ −22062.2 −1.10944
$$735$$ −4711.95 −0.236466
$$736$$ −39929.0 −1.99973
$$737$$ 0 0
$$738$$ 819.656 0.0408834
$$739$$ 25530.9 1.27087 0.635433 0.772156i $$-0.280822\pi$$
0.635433 + 0.772156i $$0.280822\pi$$
$$740$$ 14818.8 0.736151
$$741$$ 1083.93 0.0537370
$$742$$ 9108.97 0.450675
$$743$$ −11909.0 −0.588019 −0.294009 0.955803i $$-0.594990\pi$$
−0.294009 + 0.955803i $$0.594990\pi$$
$$744$$ −46535.2 −2.29309
$$745$$ 13176.1 0.647966
$$746$$ −58453.4 −2.86881
$$747$$ −1408.47 −0.0689869
$$748$$ 0 0
$$749$$ 10253.0 0.500182
$$750$$ −1973.19 −0.0960676
$$751$$ −16437.5 −0.798685 −0.399343 0.916802i $$-0.630762\pi$$
−0.399343 + 0.916802i $$0.630762\pi$$
$$752$$ 51582.4 2.50135
$$753$$ −14608.0 −0.706966
$$754$$ −3321.17 −0.160411
$$755$$ −13326.3 −0.642375
$$756$$ 2856.06 0.137399
$$757$$ 38642.1 1.85531 0.927657 0.373434i $$-0.121820\pi$$
0.927657 + 0.373434i $$0.121820\pi$$
$$758$$ 24816.8 1.18917
$$759$$ 0 0
$$760$$ 21569.1 1.02946
$$761$$ 32325.7 1.53982 0.769912 0.638150i $$-0.220300\pi$$
0.769912 + 0.638150i $$0.220300\pi$$
$$762$$ 14854.2 0.706183
$$763$$ 3613.98 0.171474
$$764$$ 76392.1 3.61750
$$765$$ 4839.22 0.228709
$$766$$ −53234.6 −2.51102
$$767$$ −2192.10 −0.103197
$$768$$ −8010.39 −0.376367
$$769$$ 29461.5 1.38155 0.690773 0.723072i $$-0.257270\pi$$
0.690773 + 0.723072i $$0.257270\pi$$
$$770$$ 0 0
$$771$$ −4143.51 −0.193547
$$772$$ 16792.6 0.782872
$$773$$ 28275.2 1.31564 0.657820 0.753175i $$-0.271479\pi$$
0.657820 + 0.753175i $$0.271479\pi$$
$$774$$ 21736.9 1.00945
$$775$$ 6306.09 0.292286
$$776$$ 51698.2 2.39157
$$777$$ 2426.67 0.112042
$$778$$ −59574.8 −2.74532
$$779$$ 1214.15 0.0558428
$$780$$ −1520.99 −0.0698209
$$781$$ 0 0
$$782$$ −59156.5 −2.70516
$$783$$ −3308.72 −0.151014
$$784$$ −52171.1 −2.37660
$$785$$ 11706.6 0.532262
$$786$$ 19634.6 0.891019
$$787$$ 27311.6 1.23705 0.618523 0.785767i $$-0.287731\pi$$
0.618523 + 0.785767i $$0.287731\pi$$
$$788$$ −15094.9 −0.682405
$$789$$ 3363.16 0.151751
$$790$$ −33075.1 −1.48957
$$791$$ −8785.61 −0.394918
$$792$$ 0 0
$$793$$ −387.444 −0.0173500
$$794$$ −28416.1 −1.27009
$$795$$ −4832.80 −0.215600
$$796$$ 24454.1 1.08888
$$797$$ −35217.6 −1.56521 −0.782605 0.622518i $$-0.786110\pi$$
−0.782605 + 0.622518i $$0.786110\pi$$
$$798$$ 5949.84 0.263937
$$799$$ 33399.8 1.47885
$$800$$ −9548.32 −0.421980
$$801$$ −864.880 −0.0381511
$$802$$ −72521.1 −3.19303
$$803$$ 0 0
$$804$$ −53070.7 −2.32794
$$805$$ 2808.64 0.122971
$$806$$ 6836.21 0.298753
$$807$$ −3484.08 −0.151977
$$808$$ −117266. −5.10569
$$809$$ −26775.1 −1.16361 −0.581806 0.813327i $$-0.697654\pi$$
−0.581806 + 0.813327i $$0.697654\pi$$
$$810$$ −2131.05 −0.0924411
$$811$$ 22851.7 0.989434 0.494717 0.869054i $$-0.335272\pi$$
0.494717 + 0.869054i $$0.335272\pi$$
$$812$$ −12962.8 −0.560228
$$813$$ 10705.7 0.461827
$$814$$ 0 0
$$815$$ −17042.7 −0.732491
$$816$$ 53580.2 2.29863
$$817$$ 32198.7 1.37881
$$818$$ 595.074 0.0254356
$$819$$ −249.071 −0.0106267
$$820$$ −1703.73 −0.0725571
$$821$$ 11277.7 0.479408 0.239704 0.970846i $$-0.422950\pi$$
0.239704 + 0.970846i $$0.422950\pi$$
$$822$$ 17792.4 0.754967
$$823$$ 3325.42 0.140847 0.0704234 0.997517i $$-0.477565\pi$$
0.0704234 + 0.997517i $$0.477565\pi$$
$$824$$ −8791.77 −0.371694
$$825$$ 0 0
$$826$$ −12032.7 −0.506868
$$827$$ 9271.96 0.389864 0.194932 0.980817i $$-0.437551\pi$$
0.194932 + 0.980817i $$0.437551\pi$$
$$828$$ 18523.5 0.777460
$$829$$ 7773.46 0.325674 0.162837 0.986653i $$-0.447936\pi$$
0.162837 + 0.986653i $$0.447936\pi$$
$$830$$ 4117.30 0.172185
$$831$$ 25039.7 1.04527
$$832$$ −3507.66 −0.146161
$$833$$ −33781.0 −1.40509
$$834$$ −40062.7 −1.66338
$$835$$ 3848.59 0.159504
$$836$$ 0 0
$$837$$ 6810.58 0.281252
$$838$$ −38687.6 −1.59480
$$839$$ 8879.30 0.365372 0.182686 0.983171i $$-0.441521\pi$$
0.182686 + 0.983171i $$0.441521\pi$$
$$840$$ −4956.27 −0.203581
$$841$$ −9371.72 −0.384260
$$842$$ −17400.8 −0.712197
$$843$$ −5171.07 −0.211270
$$844$$ −85597.9 −3.49100
$$845$$ −10852.4 −0.441814
$$846$$ −14708.3 −0.597731
$$847$$ 0 0
$$848$$ −53509.1 −2.16687
$$849$$ −6260.27 −0.253065
$$850$$ −14146.2 −0.570837
$$851$$ 15738.7 0.633976
$$852$$ 70406.8 2.83110
$$853$$ 20374.6 0.817835 0.408917 0.912571i $$-0.365906\pi$$
0.408917 + 0.912571i $$0.365906\pi$$
$$854$$ −2126.74 −0.0852172
$$855$$ −3156.71 −0.126266
$$856$$ −117346. −4.68550
$$857$$ −47483.4 −1.89265 −0.946326 0.323213i $$-0.895237\pi$$
−0.946326 + 0.323213i $$0.895237\pi$$
$$858$$ 0 0
$$859$$ −10722.1 −0.425882 −0.212941 0.977065i $$-0.568304\pi$$
−0.212941 + 0.977065i $$0.568304\pi$$
$$860$$ −45182.0 −1.79150
$$861$$ −278.995 −0.0110431
$$862$$ −52407.3 −2.07076
$$863$$ 18040.1 0.711578 0.355789 0.934566i $$-0.384212\pi$$
0.355789 + 0.934566i $$0.384212\pi$$
$$864$$ −10312.2 −0.406050
$$865$$ −10552.5 −0.414794
$$866$$ −57422.8 −2.25324
$$867$$ 19954.4 0.781647
$$868$$ 26682.3 1.04338
$$869$$ 0 0
$$870$$ 9672.19 0.376917
$$871$$ 4628.20 0.180047
$$872$$ −41362.0 −1.60630
$$873$$ −7566.21 −0.293330
$$874$$ 38588.9 1.49346
$$875$$ 671.636 0.0259491
$$876$$ 16382.2 0.631855
$$877$$ −26506.3 −1.02059 −0.510294 0.860000i $$-0.670463\pi$$
−0.510294 + 0.860000i $$0.670463\pi$$
$$878$$ 34673.4 1.33277
$$879$$ 9382.55 0.360029
$$880$$ 0 0
$$881$$ 14870.8 0.568685 0.284342 0.958723i $$-0.408225\pi$$
0.284342 + 0.958723i $$0.408225\pi$$
$$882$$ 14876.1 0.567919
$$883$$ 42210.9 1.60873 0.804366 0.594134i $$-0.202505\pi$$
0.804366 + 0.594134i $$0.202505\pi$$
$$884$$ −10904.3 −0.414878
$$885$$ 6384.02 0.242482
$$886$$ 13667.3 0.518240
$$887$$ −20743.2 −0.785220 −0.392610 0.919705i $$-0.628428\pi$$
−0.392610 + 0.919705i $$0.628428\pi$$
$$888$$ −27773.2 −1.04956
$$889$$ −5056.09 −0.190749
$$890$$ 2528.26 0.0952218
$$891$$ 0 0
$$892$$ −22114.3 −0.830093
$$893$$ −21787.3 −0.816443
$$894$$ −41598.3 −1.55622
$$895$$ 7366.26 0.275114
$$896$$ −2836.76 −0.105770
$$897$$ −1615.40 −0.0601301
$$898$$ −33993.1 −1.26321
$$899$$ −30911.2 −1.14677
$$900$$ 4429.57 0.164058
$$901$$ −34647.4 −1.28110
$$902$$ 0 0
$$903$$ −7398.81 −0.272666
$$904$$ 100551. 3.69943
$$905$$ −11450.0 −0.420565
$$906$$ 42072.5 1.54279
$$907$$ 34322.7 1.25652 0.628261 0.778002i $$-0.283767\pi$$
0.628261 + 0.778002i $$0.283767\pi$$
$$908$$ −8563.87 −0.312998
$$909$$ 17162.2 0.626222
$$910$$ 728.097 0.0265233
$$911$$ 46372.6 1.68649 0.843245 0.537529i $$-0.180642\pi$$
0.843245 + 0.537529i $$0.180642\pi$$
$$912$$ −34951.3 −1.26903
$$913$$ 0 0
$$914$$ −40390.5 −1.46171
$$915$$ 1128.35 0.0407673
$$916$$ 82830.3 2.98776
$$917$$ −6683.23 −0.240676
$$918$$ −15277.9 −0.549288
$$919$$ 15809.4 0.567469 0.283734 0.958903i $$-0.408427\pi$$
0.283734 + 0.958903i $$0.408427\pi$$
$$920$$ −32144.9 −1.15194
$$921$$ −4191.36 −0.149957
$$922$$ 39755.9 1.42005
$$923$$ −6140.05 −0.218962
$$924$$ 0 0
$$925$$ 3763.62 0.133781
$$926$$ −28837.5 −1.02339
$$927$$ 1286.70 0.0455889
$$928$$ 46804.0 1.65562
$$929$$ 37806.1 1.33517 0.667587 0.744532i $$-0.267327\pi$$
0.667587 + 0.744532i $$0.267327\pi$$
$$930$$ −19909.0 −0.701980
$$931$$ 22035.9 0.775723
$$932$$ 4631.72 0.162786
$$933$$ 16216.4 0.569025
$$934$$ 54405.9 1.90601
$$935$$ 0 0
$$936$$ 2850.62 0.0995464
$$937$$ −2703.76 −0.0942669 −0.0471334 0.998889i $$-0.515009\pi$$
−0.0471334 + 0.998889i $$0.515009\pi$$
$$938$$ 25404.9 0.884326
$$939$$ 18722.3 0.650670
$$940$$ 30572.4 1.06081
$$941$$ 42365.9 1.46768 0.733841 0.679322i $$-0.237726\pi$$
0.733841 + 0.679322i $$0.237726\pi$$
$$942$$ −36958.9 −1.27833
$$943$$ −1809.48 −0.0624865
$$944$$ 70684.3 2.43705
$$945$$ 725.367 0.0249695
$$946$$ 0 0
$$947$$ −7085.97 −0.243150 −0.121575 0.992582i $$-0.538794\pi$$
−0.121575 + 0.992582i $$0.538794\pi$$
$$948$$ 74249.6 2.54379
$$949$$ −1428.66 −0.0488687
$$950$$ 9227.84 0.315148
$$951$$ −20759.0 −0.707840
$$952$$ −35532.6 −1.20968
$$953$$ 26409.0 0.897662 0.448831 0.893617i $$-0.351840\pi$$
0.448831 + 0.893617i $$0.351840\pi$$
$$954$$ 15257.6 0.517803
$$955$$ 19401.7 0.657407
$$956$$ −12756.7 −0.431569
$$957$$ 0 0
$$958$$ 22734.6 0.766723
$$959$$ −6056.21 −0.203926
$$960$$ 10215.3 0.343435
$$961$$ 33835.9 1.13578
$$962$$ 4080.00 0.136741
$$963$$ 17173.9 0.574685
$$964$$ 44918.6 1.50076
$$965$$ 4264.89 0.142271
$$966$$ −8867.17 −0.295338
$$967$$ 31278.1 1.04016 0.520081 0.854117i $$-0.325902\pi$$
0.520081 + 0.854117i $$0.325902\pi$$
$$968$$ 0 0
$$969$$ −22631.1 −0.750275
$$970$$ 22117.9 0.732126
$$971$$ −10461.4 −0.345750 −0.172875 0.984944i $$-0.555306\pi$$
−0.172875 + 0.984944i $$0.555306\pi$$
$$972$$ 4783.93 0.157865
$$973$$ 13636.6 0.449300
$$974$$ 49598.6 1.63167
$$975$$ −386.294 −0.0126885
$$976$$ 12493.2 0.409730
$$977$$ −38539.6 −1.26202 −0.631008 0.775776i $$-0.717359\pi$$
−0.631008 + 0.775776i $$0.717359\pi$$
$$978$$ 53805.6 1.75922
$$979$$ 0 0
$$980$$ −30921.3 −1.00790
$$981$$ 6053.46 0.197015
$$982$$ 2998.04 0.0974250
$$983$$ 48220.9 1.56461 0.782304 0.622897i $$-0.214045\pi$$
0.782304 + 0.622897i $$0.214045\pi$$
$$984$$ 3193.10 0.103448
$$985$$ −3833.74 −0.124013
$$986$$ 69342.0 2.23965
$$987$$ 5006.41 0.161455
$$988$$ 7113.09 0.229046
$$989$$ −47986.4 −1.54285
$$990$$ 0 0
$$991$$ 19959.7 0.639798 0.319899 0.947452i $$-0.396351\pi$$
0.319899 + 0.947452i $$0.396351\pi$$
$$992$$ −96340.1 −3.08347
$$993$$ −9403.73 −0.300522
$$994$$ −33703.6 −1.07547
$$995$$ 6210.73 0.197883
$$996$$ −9242.83 −0.294046
$$997$$ −59491.3 −1.88978 −0.944889 0.327392i $$-0.893830\pi$$
−0.944889 + 0.327392i $$0.893830\pi$$
$$998$$ −89833.0 −2.84931
$$999$$ 4064.71 0.128730
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 1815.4.a.bb.1.1 6
11.10 odd 2 inner 1815.4.a.bb.1.6 yes 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1815.4.a.bb.1.1 6 1.1 even 1 trivial
1815.4.a.bb.1.6 yes 6 11.10 odd 2 inner