# Properties

 Label 33.4.a.d.1.1 Level $33$ Weight $4$ Character 33.1 Self dual yes Analytic conductor $1.947$ Analytic rank $0$ 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] = [33,4,Mod(1,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("33.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.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: $$1.94706303019$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 33.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.37228 q^{2} +3.00000 q^{3} -2.37228 q^{4} +19.4891 q^{5} -7.11684 q^{6} +6.74456 q^{7} +24.6060 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-2.37228 q^{2} +3.00000 q^{3} -2.37228 q^{4} +19.4891 q^{5} -7.11684 q^{6} +6.74456 q^{7} +24.6060 q^{8} +9.00000 q^{9} -46.2337 q^{10} +11.0000 q^{11} -7.11684 q^{12} -60.9783 q^{13} -16.0000 q^{14} +58.4674 q^{15} -39.3940 q^{16} -99.1684 q^{17} -21.3505 q^{18} +24.7011 q^{19} -46.2337 q^{20} +20.2337 q^{21} -26.0951 q^{22} +112.000 q^{23} +73.8179 q^{24} +254.826 q^{25} +144.658 q^{26} +27.0000 q^{27} -16.0000 q^{28} -21.1249 q^{29} -138.701 q^{30} -318.717 q^{31} -103.394 q^{32} +33.0000 q^{33} +235.255 q^{34} +131.446 q^{35} -21.3505 q^{36} -150.380 q^{37} -58.5979 q^{38} -182.935 q^{39} +479.549 q^{40} -252.745 q^{41} -48.0000 q^{42} +214.016 q^{43} -26.0951 q^{44} +175.402 q^{45} -265.696 q^{46} +105.870 q^{47} -118.182 q^{48} -297.511 q^{49} -604.519 q^{50} -297.505 q^{51} +144.658 q^{52} +325.652 q^{53} -64.0516 q^{54} +214.380 q^{55} +165.957 q^{56} +74.1032 q^{57} +50.1143 q^{58} +196.000 q^{59} -138.701 q^{60} -402.641 q^{61} +756.087 q^{62} +60.7011 q^{63} +560.432 q^{64} -1188.41 q^{65} -78.2853 q^{66} +27.4132 q^{67} +235.255 q^{68} +336.000 q^{69} -311.826 q^{70} -300.500 q^{71} +221.454 q^{72} +427.815 q^{73} +356.745 q^{74} +764.478 q^{75} -58.5979 q^{76} +74.1902 q^{77} +433.973 q^{78} +97.5488 q^{79} -767.755 q^{80} +81.0000 q^{81} +599.581 q^{82} +1104.62 q^{83} -48.0000 q^{84} -1932.71 q^{85} -507.707 q^{86} -63.3748 q^{87} +270.666 q^{88} +463.022 q^{89} -416.103 q^{90} -411.272 q^{91} -265.696 q^{92} -956.152 q^{93} -251.152 q^{94} +481.402 q^{95} -310.182 q^{96} -1798.36 q^{97} +705.779 q^{98} +99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 6 q^{3} + q^{4} + 16 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + q^2 + 6 * q^3 + q^4 + 16 * q^5 + 3 * q^6 + 2 * q^7 + 9 * q^8 + 18 * q^9 $$2 q + q^{2} + 6 q^{3} + q^{4} + 16 q^{5} + 3 q^{6} + 2 q^{7} + 9 q^{8} + 18 q^{9} - 58 q^{10} + 22 q^{11} + 3 q^{12} - 76 q^{13} - 32 q^{14} + 48 q^{15} - 119 q^{16} - 26 q^{17} + 9 q^{18} - 54 q^{19} - 58 q^{20} + 6 q^{21} + 11 q^{22} + 224 q^{23} + 27 q^{24} + 142 q^{25} + 94 q^{26} + 54 q^{27} - 32 q^{28} + 222 q^{29} - 174 q^{30} - 40 q^{31} - 247 q^{32} + 66 q^{33} + 482 q^{34} + 148 q^{35} + 9 q^{36} - 48 q^{37} - 324 q^{38} - 228 q^{39} + 534 q^{40} - 494 q^{41} - 96 q^{42} - 66 q^{43} + 11 q^{44} + 144 q^{45} + 112 q^{46} - 64 q^{47} - 357 q^{48} - 618 q^{49} - 985 q^{50} - 78 q^{51} + 94 q^{52} - 84 q^{53} + 27 q^{54} + 176 q^{55} + 240 q^{56} - 162 q^{57} + 870 q^{58} + 392 q^{59} - 174 q^{60} - 1104 q^{61} + 1696 q^{62} + 18 q^{63} + 713 q^{64} - 1136 q^{65} + 33 q^{66} + 928 q^{67} + 482 q^{68} + 672 q^{69} - 256 q^{70} + 456 q^{71} + 81 q^{72} - 592 q^{73} + 702 q^{74} + 426 q^{75} - 324 q^{76} + 22 q^{77} + 282 q^{78} - 230 q^{79} - 490 q^{80} + 162 q^{81} - 214 q^{82} + 348 q^{83} - 96 q^{84} - 2188 q^{85} - 1452 q^{86} + 666 q^{87} + 99 q^{88} + 972 q^{89} - 522 q^{90} - 340 q^{91} + 112 q^{92} - 120 q^{93} - 824 q^{94} + 756 q^{95} - 741 q^{96} - 1184 q^{97} - 375 q^{98} + 198 q^{99}+O(q^{100})$$ 2 * q + q^2 + 6 * q^3 + q^4 + 16 * q^5 + 3 * q^6 + 2 * q^7 + 9 * q^8 + 18 * q^9 - 58 * q^10 + 22 * q^11 + 3 * q^12 - 76 * q^13 - 32 * q^14 + 48 * q^15 - 119 * q^16 - 26 * q^17 + 9 * q^18 - 54 * q^19 - 58 * q^20 + 6 * q^21 + 11 * q^22 + 224 * q^23 + 27 * q^24 + 142 * q^25 + 94 * q^26 + 54 * q^27 - 32 * q^28 + 222 * q^29 - 174 * q^30 - 40 * q^31 - 247 * q^32 + 66 * q^33 + 482 * q^34 + 148 * q^35 + 9 * q^36 - 48 * q^37 - 324 * q^38 - 228 * q^39 + 534 * q^40 - 494 * q^41 - 96 * q^42 - 66 * q^43 + 11 * q^44 + 144 * q^45 + 112 * q^46 - 64 * q^47 - 357 * q^48 - 618 * q^49 - 985 * q^50 - 78 * q^51 + 94 * q^52 - 84 * q^53 + 27 * q^54 + 176 * q^55 + 240 * q^56 - 162 * q^57 + 870 * q^58 + 392 * q^59 - 174 * q^60 - 1104 * q^61 + 1696 * q^62 + 18 * q^63 + 713 * q^64 - 1136 * q^65 + 33 * q^66 + 928 * q^67 + 482 * q^68 + 672 * q^69 - 256 * q^70 + 456 * q^71 + 81 * q^72 - 592 * q^73 + 702 * q^74 + 426 * q^75 - 324 * q^76 + 22 * q^77 + 282 * q^78 - 230 * q^79 - 490 * q^80 + 162 * q^81 - 214 * q^82 + 348 * q^83 - 96 * q^84 - 2188 * q^85 - 1452 * q^86 + 666 * q^87 + 99 * q^88 + 972 * q^89 - 522 * q^90 - 340 * q^91 + 112 * q^92 - 120 * q^93 - 824 * q^94 + 756 * q^95 - 741 * q^96 - 1184 * q^97 - 375 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.37228 −0.838728 −0.419364 0.907818i $$-0.637747\pi$$
−0.419364 + 0.907818i $$0.637747\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −2.37228 −0.296535
$$5$$ 19.4891 1.74316 0.871580 0.490253i $$-0.163096\pi$$
0.871580 + 0.490253i $$0.163096\pi$$
$$6$$ −7.11684 −0.484240
$$7$$ 6.74456 0.364172 0.182086 0.983283i $$-0.441715\pi$$
0.182086 + 0.983283i $$0.441715\pi$$
$$8$$ 24.6060 1.08744
$$9$$ 9.00000 0.333333
$$10$$ −46.2337 −1.46204
$$11$$ 11.0000 0.301511
$$12$$ −7.11684 −0.171205
$$13$$ −60.9783 −1.30095 −0.650474 0.759529i $$-0.725430\pi$$
−0.650474 + 0.759529i $$0.725430\pi$$
$$14$$ −16.0000 −0.305441
$$15$$ 58.4674 1.00641
$$16$$ −39.3940 −0.615532
$$17$$ −99.1684 −1.41482 −0.707408 0.706805i $$-0.750136\pi$$
−0.707408 + 0.706805i $$0.750136\pi$$
$$18$$ −21.3505 −0.279576
$$19$$ 24.7011 0.298253 0.149127 0.988818i $$-0.452354\pi$$
0.149127 + 0.988818i $$0.452354\pi$$
$$20$$ −46.2337 −0.516908
$$21$$ 20.2337 0.210255
$$22$$ −26.0951 −0.252886
$$23$$ 112.000 1.01537 0.507687 0.861541i $$-0.330501\pi$$
0.507687 + 0.861541i $$0.330501\pi$$
$$24$$ 73.8179 0.627834
$$25$$ 254.826 2.03861
$$26$$ 144.658 1.09114
$$27$$ 27.0000 0.192450
$$28$$ −16.0000 −0.107990
$$29$$ −21.1249 −0.135269 −0.0676345 0.997710i $$-0.521545\pi$$
−0.0676345 + 0.997710i $$0.521545\pi$$
$$30$$ −138.701 −0.844108
$$31$$ −318.717 −1.84656 −0.923279 0.384130i $$-0.874502\pi$$
−0.923279 + 0.384130i $$0.874502\pi$$
$$32$$ −103.394 −0.571177
$$33$$ 33.0000 0.174078
$$34$$ 235.255 1.18665
$$35$$ 131.446 0.634810
$$36$$ −21.3505 −0.0988451
$$37$$ −150.380 −0.668172 −0.334086 0.942543i $$-0.608428\pi$$
−0.334086 + 0.942543i $$0.608428\pi$$
$$38$$ −58.5979 −0.250153
$$39$$ −182.935 −0.751103
$$40$$ 479.549 1.89558
$$41$$ −252.745 −0.962733 −0.481367 0.876519i $$-0.659859\pi$$
−0.481367 + 0.876519i $$0.659859\pi$$
$$42$$ −48.0000 −0.176347
$$43$$ 214.016 0.759004 0.379502 0.925191i $$-0.376095\pi$$
0.379502 + 0.925191i $$0.376095\pi$$
$$44$$ −26.0951 −0.0894087
$$45$$ 175.402 0.581053
$$46$$ −265.696 −0.851623
$$47$$ 105.870 0.328567 0.164284 0.986413i $$-0.447469\pi$$
0.164284 + 0.986413i $$0.447469\pi$$
$$48$$ −118.182 −0.355377
$$49$$ −297.511 −0.867379
$$50$$ −604.519 −1.70984
$$51$$ −297.505 −0.816845
$$52$$ 144.658 0.385777
$$53$$ 325.652 0.843995 0.421998 0.906597i $$-0.361329\pi$$
0.421998 + 0.906597i $$0.361329\pi$$
$$54$$ −64.0516 −0.161413
$$55$$ 214.380 0.525583
$$56$$ 165.957 0.396016
$$57$$ 74.1032 0.172197
$$58$$ 50.1143 0.113454
$$59$$ 196.000 0.432492 0.216246 0.976339i $$-0.430619\pi$$
0.216246 + 0.976339i $$0.430619\pi$$
$$60$$ −138.701 −0.298437
$$61$$ −402.641 −0.845130 −0.422565 0.906333i $$-0.638870\pi$$
−0.422565 + 0.906333i $$0.638870\pi$$
$$62$$ 756.087 1.54876
$$63$$ 60.7011 0.121391
$$64$$ 560.432 1.09459
$$65$$ −1188.41 −2.26776
$$66$$ −78.2853 −0.146004
$$67$$ 27.4132 0.0499860 0.0249930 0.999688i $$-0.492044\pi$$
0.0249930 + 0.999688i $$0.492044\pi$$
$$68$$ 235.255 0.419543
$$69$$ 336.000 0.586227
$$70$$ −311.826 −0.532433
$$71$$ −300.500 −0.502292 −0.251146 0.967949i $$-0.580807\pi$$
−0.251146 + 0.967949i $$0.580807\pi$$
$$72$$ 221.454 0.362480
$$73$$ 427.815 0.685917 0.342959 0.939351i $$-0.388571\pi$$
0.342959 + 0.939351i $$0.388571\pi$$
$$74$$ 356.745 0.560415
$$75$$ 764.478 1.17699
$$76$$ −58.5979 −0.0884426
$$77$$ 74.1902 0.109802
$$78$$ 433.973 0.629971
$$79$$ 97.5488 0.138925 0.0694627 0.997585i $$-0.477872\pi$$
0.0694627 + 0.997585i $$0.477872\pi$$
$$80$$ −767.755 −1.07297
$$81$$ 81.0000 0.111111
$$82$$ 599.581 0.807472
$$83$$ 1104.62 1.46082 0.730408 0.683011i $$-0.239330\pi$$
0.730408 + 0.683011i $$0.239330\pi$$
$$84$$ −48.0000 −0.0623480
$$85$$ −1932.71 −2.46625
$$86$$ −507.707 −0.636598
$$87$$ −63.3748 −0.0780976
$$88$$ 270.666 0.327876
$$89$$ 463.022 0.551463 0.275732 0.961235i $$-0.411080\pi$$
0.275732 + 0.961235i $$0.411080\pi$$
$$90$$ −416.103 −0.487346
$$91$$ −411.272 −0.473769
$$92$$ −265.696 −0.301094
$$93$$ −956.152 −1.06611
$$94$$ −251.152 −0.275578
$$95$$ 481.402 0.519903
$$96$$ −310.182 −0.329769
$$97$$ −1798.36 −1.88243 −0.941214 0.337810i $$-0.890314\pi$$
−0.941214 + 0.337810i $$0.890314\pi$$
$$98$$ 705.779 0.727495
$$99$$ 99.0000 0.100504
$$100$$ −604.519 −0.604519
$$101$$ 741.918 0.730927 0.365463 0.930826i $$-0.380911\pi$$
0.365463 + 0.930826i $$0.380911\pi$$
$$102$$ 705.766 0.685111
$$103$$ 389.837 0.372930 0.186465 0.982462i $$-0.440297\pi$$
0.186465 + 0.982462i $$0.440297\pi$$
$$104$$ −1500.43 −1.41470
$$105$$ 394.337 0.366508
$$106$$ −772.538 −0.707882
$$107$$ 1538.42 1.38995 0.694977 0.719032i $$-0.255415\pi$$
0.694977 + 0.719032i $$0.255415\pi$$
$$108$$ −64.0516 −0.0570682
$$109$$ 779.783 0.685226 0.342613 0.939477i $$-0.388688\pi$$
0.342613 + 0.939477i $$0.388688\pi$$
$$110$$ −508.571 −0.440821
$$111$$ −451.141 −0.385770
$$112$$ −265.696 −0.224160
$$113$$ −1514.55 −1.26086 −0.630430 0.776246i $$-0.717122\pi$$
−0.630430 + 0.776246i $$0.717122\pi$$
$$114$$ −175.794 −0.144426
$$115$$ 2182.78 1.76996
$$116$$ 50.1143 0.0401120
$$117$$ −548.804 −0.433649
$$118$$ −464.967 −0.362743
$$119$$ −668.848 −0.515237
$$120$$ 1438.65 1.09442
$$121$$ 121.000 0.0909091
$$122$$ 955.179 0.708834
$$123$$ −758.234 −0.555834
$$124$$ 756.087 0.547569
$$125$$ 2530.20 1.81046
$$126$$ −144.000 −0.101814
$$127$$ 2302.74 1.60894 0.804471 0.593992i $$-0.202449\pi$$
0.804471 + 0.593992i $$0.202449\pi$$
$$128$$ −502.350 −0.346890
$$129$$ 642.049 0.438211
$$130$$ 2819.25 1.90203
$$131$$ −2020.59 −1.34763 −0.673815 0.738900i $$-0.735345\pi$$
−0.673815 + 0.738900i $$0.735345\pi$$
$$132$$ −78.2853 −0.0516201
$$133$$ 166.598 0.108616
$$134$$ −65.0319 −0.0419246
$$135$$ 526.206 0.335471
$$136$$ −2440.14 −1.53853
$$137$$ 1475.40 0.920088 0.460044 0.887896i $$-0.347834\pi$$
0.460044 + 0.887896i $$0.347834\pi$$
$$138$$ −797.087 −0.491685
$$139$$ −1623.71 −0.990802 −0.495401 0.868665i $$-0.664979\pi$$
−0.495401 + 0.868665i $$0.664979\pi$$
$$140$$ −311.826 −0.188244
$$141$$ 317.609 0.189698
$$142$$ 712.870 0.421287
$$143$$ −670.761 −0.392251
$$144$$ −354.546 −0.205177
$$145$$ −411.707 −0.235796
$$146$$ −1014.90 −0.575298
$$147$$ −892.533 −0.500781
$$148$$ 356.745 0.198137
$$149$$ −1104.11 −0.607064 −0.303532 0.952821i $$-0.598166\pi$$
−0.303532 + 0.952821i $$0.598166\pi$$
$$150$$ −1813.56 −0.987175
$$151$$ 2980.07 1.60606 0.803029 0.595940i $$-0.203221\pi$$
0.803029 + 0.595940i $$0.203221\pi$$
$$152$$ 607.794 0.324333
$$153$$ −892.516 −0.471605
$$154$$ −176.000 −0.0920941
$$155$$ −6211.52 −3.21885
$$156$$ 433.973 0.222728
$$157$$ 2844.67 1.44605 0.723024 0.690823i $$-0.242751\pi$$
0.723024 + 0.690823i $$0.242751\pi$$
$$158$$ −231.413 −0.116521
$$159$$ 976.956 0.487281
$$160$$ −2015.06 −0.995653
$$161$$ 755.391 0.369771
$$162$$ −192.155 −0.0931920
$$163$$ −1528.51 −0.734492 −0.367246 0.930124i $$-0.619699\pi$$
−0.367246 + 0.930124i $$0.619699\pi$$
$$164$$ 599.581 0.285484
$$165$$ 643.141 0.303445
$$166$$ −2620.47 −1.22523
$$167$$ −383.881 −0.177878 −0.0889388 0.996037i $$-0.528348\pi$$
−0.0889388 + 0.996037i $$0.528348\pi$$
$$168$$ 497.870 0.228640
$$169$$ 1521.35 0.692466
$$170$$ 4584.92 2.06851
$$171$$ 222.310 0.0994178
$$172$$ −507.707 −0.225071
$$173$$ 2702.85 1.18783 0.593914 0.804529i $$-0.297582\pi$$
0.593914 + 0.804529i $$0.297582\pi$$
$$174$$ 150.343 0.0655027
$$175$$ 1718.69 0.742404
$$176$$ −433.334 −0.185590
$$177$$ 588.000 0.249699
$$178$$ −1098.42 −0.462528
$$179$$ −2777.61 −1.15982 −0.579911 0.814680i $$-0.696913\pi$$
−0.579911 + 0.814680i $$0.696913\pi$$
$$180$$ −416.103 −0.172303
$$181$$ −3993.61 −1.64001 −0.820007 0.572354i $$-0.806030\pi$$
−0.820007 + 0.572354i $$0.806030\pi$$
$$182$$ 975.652 0.397363
$$183$$ −1207.92 −0.487936
$$184$$ 2755.87 1.10416
$$185$$ −2930.78 −1.16473
$$186$$ 2268.26 0.894177
$$187$$ −1090.85 −0.426583
$$188$$ −251.152 −0.0974317
$$189$$ 182.103 0.0700850
$$190$$ −1142.02 −0.436058
$$191$$ 895.587 0.339280 0.169640 0.985506i $$-0.445740\pi$$
0.169640 + 0.985506i $$0.445740\pi$$
$$192$$ 1681.30 0.631964
$$193$$ 1328.24 0.495382 0.247691 0.968839i $$-0.420328\pi$$
0.247691 + 0.968839i $$0.420328\pi$$
$$194$$ 4266.21 1.57885
$$195$$ −3565.24 −1.30929
$$196$$ 705.779 0.257208
$$197$$ 154.114 0.0557370 0.0278685 0.999612i $$-0.491128\pi$$
0.0278685 + 0.999612i $$0.491128\pi$$
$$198$$ −234.856 −0.0842953
$$199$$ 1316.15 0.468842 0.234421 0.972135i $$-0.424681\pi$$
0.234421 + 0.972135i $$0.424681\pi$$
$$200$$ 6270.24 2.21686
$$201$$ 82.2397 0.0288594
$$202$$ −1760.04 −0.613049
$$203$$ −142.478 −0.0492612
$$204$$ 705.766 0.242223
$$205$$ −4925.77 −1.67820
$$206$$ −924.803 −0.312787
$$207$$ 1008.00 0.338458
$$208$$ 2402.18 0.800775
$$209$$ 271.712 0.0899268
$$210$$ −935.478 −0.307401
$$211$$ −1735.76 −0.566324 −0.283162 0.959072i $$-0.591383\pi$$
−0.283162 + 0.959072i $$0.591383\pi$$
$$212$$ −772.538 −0.250274
$$213$$ −901.499 −0.289999
$$214$$ −3649.57 −1.16579
$$215$$ 4170.99 1.32307
$$216$$ 664.361 0.209278
$$217$$ −2149.61 −0.672465
$$218$$ −1849.86 −0.574718
$$219$$ 1283.44 0.396014
$$220$$ −508.571 −0.155854
$$221$$ 6047.12 1.84060
$$222$$ 1070.23 0.323556
$$223$$ 1663.71 0.499597 0.249798 0.968298i $$-0.419636\pi$$
0.249798 + 0.968298i $$0.419636\pi$$
$$224$$ −697.348 −0.208007
$$225$$ 2293.43 0.679536
$$226$$ 3592.95 1.05752
$$227$$ −3658.84 −1.06980 −0.534902 0.844914i $$-0.679651\pi$$
−0.534902 + 0.844914i $$0.679651\pi$$
$$228$$ −175.794 −0.0510624
$$229$$ 1927.07 0.556090 0.278045 0.960568i $$-0.410314\pi$$
0.278045 + 0.960568i $$0.410314\pi$$
$$230$$ −5178.17 −1.48452
$$231$$ 222.571 0.0633942
$$232$$ −519.800 −0.147097
$$233$$ −2784.27 −0.782847 −0.391423 0.920211i $$-0.628017\pi$$
−0.391423 + 0.920211i $$0.628017\pi$$
$$234$$ 1301.92 0.363714
$$235$$ 2063.30 0.572745
$$236$$ −464.967 −0.128249
$$237$$ 292.646 0.0802086
$$238$$ 1586.70 0.432144
$$239$$ 1222.60 0.330892 0.165446 0.986219i $$-0.447094\pi$$
0.165446 + 0.986219i $$0.447094\pi$$
$$240$$ −2303.27 −0.619480
$$241$$ −2013.01 −0.538047 −0.269024 0.963134i $$-0.586701\pi$$
−0.269024 + 0.963134i $$0.586701\pi$$
$$242$$ −287.046 −0.0762480
$$243$$ 243.000 0.0641500
$$244$$ 955.179 0.250611
$$245$$ −5798.23 −1.51198
$$246$$ 1798.74 0.466194
$$247$$ −1506.23 −0.388012
$$248$$ −7842.35 −2.00802
$$249$$ 3313.86 0.843402
$$250$$ −6002.33 −1.51848
$$251$$ 2706.63 0.680641 0.340320 0.940310i $$-0.389464\pi$$
0.340320 + 0.940310i $$0.389464\pi$$
$$252$$ −144.000 −0.0359966
$$253$$ 1232.00 0.306147
$$254$$ −5462.76 −1.34946
$$255$$ −5798.12 −1.42389
$$256$$ −3291.74 −0.803647
$$257$$ 225.741 0.0547912 0.0273956 0.999625i $$-0.491279\pi$$
0.0273956 + 0.999625i $$0.491279\pi$$
$$258$$ −1523.12 −0.367540
$$259$$ −1014.25 −0.243330
$$260$$ 2819.25 0.672471
$$261$$ −190.124 −0.0450897
$$262$$ 4793.40 1.13029
$$263$$ 1953.42 0.457997 0.228998 0.973427i $$-0.426455\pi$$
0.228998 + 0.973427i $$0.426455\pi$$
$$264$$ 811.997 0.189299
$$265$$ 6346.67 1.47122
$$266$$ −395.217 −0.0910989
$$267$$ 1389.07 0.318387
$$268$$ −65.0319 −0.0148226
$$269$$ −350.848 −0.0795225 −0.0397613 0.999209i $$-0.512660\pi$$
−0.0397613 + 0.999209i $$0.512660\pi$$
$$270$$ −1248.31 −0.281369
$$271$$ 254.779 0.0571096 0.0285548 0.999592i $$-0.490909\pi$$
0.0285548 + 0.999592i $$0.490909\pi$$
$$272$$ 3906.64 0.870864
$$273$$ −1233.81 −0.273531
$$274$$ −3500.07 −0.771704
$$275$$ 2803.09 0.614663
$$276$$ −797.087 −0.173837
$$277$$ 116.478 0.0252654 0.0126327 0.999920i $$-0.495979\pi$$
0.0126327 + 0.999920i $$0.495979\pi$$
$$278$$ 3851.90 0.831013
$$279$$ −2868.46 −0.615519
$$280$$ 3234.35 0.690319
$$281$$ 8226.47 1.74644 0.873221 0.487325i $$-0.162027\pi$$
0.873221 + 0.487325i $$0.162027\pi$$
$$282$$ −753.457 −0.159105
$$283$$ 1561.52 0.327995 0.163997 0.986461i $$-0.447561\pi$$
0.163997 + 0.986461i $$0.447561\pi$$
$$284$$ 712.870 0.148947
$$285$$ 1444.21 0.300166
$$286$$ 1591.23 0.328992
$$287$$ −1704.65 −0.350601
$$288$$ −930.546 −0.190392
$$289$$ 4921.38 1.00171
$$290$$ 976.684 0.197768
$$291$$ −5395.07 −1.08682
$$292$$ −1014.90 −0.203399
$$293$$ 9486.19 1.89143 0.945715 0.324997i $$-0.105363\pi$$
0.945715 + 0.324997i $$0.105363\pi$$
$$294$$ 2117.34 0.420019
$$295$$ 3819.87 0.753903
$$296$$ −3700.25 −0.726598
$$297$$ 297.000 0.0580259
$$298$$ 2619.27 0.509161
$$299$$ −6829.56 −1.32095
$$300$$ −1813.56 −0.349019
$$301$$ 1443.45 0.276408
$$302$$ −7069.56 −1.34705
$$303$$ 2225.75 0.422001
$$304$$ −973.074 −0.183584
$$305$$ −7847.13 −1.47320
$$306$$ 2117.30 0.395549
$$307$$ −8443.06 −1.56961 −0.784806 0.619742i $$-0.787237\pi$$
−0.784806 + 0.619742i $$0.787237\pi$$
$$308$$ −176.000 −0.0325602
$$309$$ 1169.51 0.215311
$$310$$ 14735.5 2.69974
$$311$$ 4447.17 0.810856 0.405428 0.914127i $$-0.367123\pi$$
0.405428 + 0.914127i $$0.367123\pi$$
$$312$$ −4501.29 −0.816779
$$313$$ 6480.75 1.17033 0.585165 0.810914i $$-0.301030\pi$$
0.585165 + 0.810914i $$0.301030\pi$$
$$314$$ −6748.36 −1.21284
$$315$$ 1183.01 0.211603
$$316$$ −231.413 −0.0411962
$$317$$ −1252.75 −0.221960 −0.110980 0.993823i $$-0.535399\pi$$
−0.110980 + 0.993823i $$0.535399\pi$$
$$318$$ −2317.61 −0.408696
$$319$$ −232.374 −0.0407852
$$320$$ 10922.3 1.90805
$$321$$ 4615.27 0.802490
$$322$$ −1792.00 −0.310137
$$323$$ −2449.57 −0.421974
$$324$$ −192.155 −0.0329484
$$325$$ −15538.8 −2.65212
$$326$$ 3626.06 0.616039
$$327$$ 2339.35 0.395615
$$328$$ −6219.02 −1.04692
$$329$$ 714.043 0.119655
$$330$$ −1525.71 −0.254508
$$331$$ 3801.44 0.631258 0.315629 0.948883i $$-0.397785\pi$$
0.315629 + 0.948883i $$0.397785\pi$$
$$332$$ −2620.47 −0.433183
$$333$$ −1353.42 −0.222724
$$334$$ 910.673 0.149191
$$335$$ 534.260 0.0871336
$$336$$ −797.087 −0.129419
$$337$$ −5456.53 −0.882007 −0.441003 0.897506i $$-0.645377\pi$$
−0.441003 + 0.897506i $$0.645377\pi$$
$$338$$ −3609.06 −0.580790
$$339$$ −4543.66 −0.727958
$$340$$ 4584.92 0.731330
$$341$$ −3505.89 −0.556758
$$342$$ −527.381 −0.0833845
$$343$$ −4319.97 −0.680047
$$344$$ 5266.08 0.825371
$$345$$ 6548.35 1.02189
$$346$$ −6411.93 −0.996264
$$347$$ −4240.53 −0.656033 −0.328017 0.944672i $$-0.606380\pi$$
−0.328017 + 0.944672i $$0.606380\pi$$
$$348$$ 150.343 0.0231587
$$349$$ −8471.53 −1.29934 −0.649672 0.760215i $$-0.725094\pi$$
−0.649672 + 0.760215i $$0.725094\pi$$
$$350$$ −4077.22 −0.622675
$$351$$ −1646.41 −0.250368
$$352$$ −1137.33 −0.172216
$$353$$ 981.282 0.147956 0.0739778 0.997260i $$-0.476431\pi$$
0.0739778 + 0.997260i $$0.476431\pi$$
$$354$$ −1394.90 −0.209430
$$355$$ −5856.48 −0.875576
$$356$$ −1098.42 −0.163528
$$357$$ −2006.54 −0.297472
$$358$$ 6589.27 0.972776
$$359$$ −3020.09 −0.443995 −0.221997 0.975047i $$-0.571258\pi$$
−0.221997 + 0.975047i $$0.571258\pi$$
$$360$$ 4315.94 0.631861
$$361$$ −6248.86 −0.911045
$$362$$ 9473.96 1.37553
$$363$$ 363.000 0.0524864
$$364$$ 975.652 0.140489
$$365$$ 8337.74 1.19566
$$366$$ 2865.54 0.409246
$$367$$ 8689.40 1.23592 0.617960 0.786209i $$-0.287959\pi$$
0.617960 + 0.786209i $$0.287959\pi$$
$$368$$ −4412.13 −0.624995
$$369$$ −2274.70 −0.320911
$$370$$ 6952.64 0.976893
$$371$$ 2196.38 0.307360
$$372$$ 2268.26 0.316139
$$373$$ 5340.53 0.741346 0.370673 0.928763i $$-0.379127\pi$$
0.370673 + 0.928763i $$0.379127\pi$$
$$374$$ 2587.81 0.357787
$$375$$ 7590.59 1.04527
$$376$$ 2605.02 0.357297
$$377$$ 1288.16 0.175978
$$378$$ −432.000 −0.0587822
$$379$$ −1603.49 −0.217324 −0.108662 0.994079i $$-0.534657\pi$$
−0.108662 + 0.994079i $$0.534657\pi$$
$$380$$ −1142.02 −0.154170
$$381$$ 6908.23 0.928923
$$382$$ −2124.58 −0.284563
$$383$$ −830.236 −0.110765 −0.0553826 0.998465i $$-0.517638\pi$$
−0.0553826 + 0.998465i $$0.517638\pi$$
$$384$$ −1507.05 −0.200277
$$385$$ 1445.90 0.191403
$$386$$ −3150.96 −0.415491
$$387$$ 1926.15 0.253001
$$388$$ 4266.21 0.558206
$$389$$ −1746.12 −0.227588 −0.113794 0.993504i $$-0.536300\pi$$
−0.113794 + 0.993504i $$0.536300\pi$$
$$390$$ 8457.75 1.09814
$$391$$ −11106.9 −1.43657
$$392$$ −7320.54 −0.943223
$$393$$ −6061.76 −0.778054
$$394$$ −365.602 −0.0467482
$$395$$ 1901.14 0.242169
$$396$$ −234.856 −0.0298029
$$397$$ −10016.2 −1.26624 −0.633119 0.774054i $$-0.718226\pi$$
−0.633119 + 0.774054i $$0.718226\pi$$
$$398$$ −3122.28 −0.393231
$$399$$ 499.794 0.0627092
$$400$$ −10038.6 −1.25483
$$401$$ 8228.38 1.02470 0.512351 0.858776i $$-0.328775\pi$$
0.512351 + 0.858776i $$0.328775\pi$$
$$402$$ −195.096 −0.0242052
$$403$$ 19434.8 2.40228
$$404$$ −1760.04 −0.216746
$$405$$ 1578.62 0.193684
$$406$$ 337.999 0.0413168
$$407$$ −1654.18 −0.201462
$$408$$ −7320.41 −0.888270
$$409$$ −12311.5 −1.48843 −0.744213 0.667943i $$-0.767175\pi$$
−0.744213 + 0.667943i $$0.767175\pi$$
$$410$$ 11685.3 1.40755
$$411$$ 4426.21 0.531213
$$412$$ −924.803 −0.110587
$$413$$ 1321.93 0.157502
$$414$$ −2391.26 −0.283874
$$415$$ 21528.1 2.54644
$$416$$ 6304.79 0.743071
$$417$$ −4871.14 −0.572040
$$418$$ −644.577 −0.0754241
$$419$$ 13260.4 1.54610 0.773048 0.634347i $$-0.218731\pi$$
0.773048 + 0.634347i $$0.218731\pi$$
$$420$$ −935.478 −0.108683
$$421$$ −6177.74 −0.715165 −0.357583 0.933881i $$-0.616399\pi$$
−0.357583 + 0.933881i $$0.616399\pi$$
$$422$$ 4117.70 0.474992
$$423$$ 952.826 0.109522
$$424$$ 8012.98 0.917795
$$425$$ −25270.7 −2.88426
$$426$$ 2138.61 0.243230
$$427$$ −2715.64 −0.307773
$$428$$ −3649.57 −0.412170
$$429$$ −2012.28 −0.226466
$$430$$ −9894.76 −1.10969
$$431$$ 1668.05 0.186421 0.0932103 0.995646i $$-0.470287\pi$$
0.0932103 + 0.995646i $$0.470287\pi$$
$$432$$ −1063.64 −0.118459
$$433$$ −731.748 −0.0812138 −0.0406069 0.999175i $$-0.512929\pi$$
−0.0406069 + 0.999175i $$0.512929\pi$$
$$434$$ 5099.48 0.564015
$$435$$ −1235.12 −0.136137
$$436$$ −1849.86 −0.203194
$$437$$ 2766.52 0.302839
$$438$$ −3044.69 −0.332148
$$439$$ −14248.0 −1.54902 −0.774508 0.632564i $$-0.782002\pi$$
−0.774508 + 0.632564i $$0.782002\pi$$
$$440$$ 5275.04 0.571540
$$441$$ −2677.60 −0.289126
$$442$$ −14345.5 −1.54377
$$443$$ 174.262 0.0186895 0.00934473 0.999956i $$-0.497025\pi$$
0.00934473 + 0.999956i $$0.497025\pi$$
$$444$$ 1070.23 0.114394
$$445$$ 9023.89 0.961288
$$446$$ −3946.78 −0.419026
$$447$$ −3312.34 −0.350488
$$448$$ 3779.87 0.398621
$$449$$ 7469.97 0.785144 0.392572 0.919721i $$-0.371585\pi$$
0.392572 + 0.919721i $$0.371585\pi$$
$$450$$ −5440.67 −0.569946
$$451$$ −2780.19 −0.290275
$$452$$ 3592.95 0.373890
$$453$$ 8940.21 0.927258
$$454$$ 8679.79 0.897275
$$455$$ −8015.32 −0.825855
$$456$$ 1823.38 0.187254
$$457$$ 8762.68 0.896939 0.448469 0.893798i $$-0.351969\pi$$
0.448469 + 0.893798i $$0.351969\pi$$
$$458$$ −4571.56 −0.466409
$$459$$ −2677.55 −0.272282
$$460$$ −5178.17 −0.524856
$$461$$ 4339.67 0.438435 0.219218 0.975676i $$-0.429650\pi$$
0.219218 + 0.975676i $$0.429650\pi$$
$$462$$ −528.000 −0.0531705
$$463$$ −3932.49 −0.394726 −0.197363 0.980330i $$-0.563238\pi$$
−0.197363 + 0.980330i $$0.563238\pi$$
$$464$$ 832.197 0.0832624
$$465$$ −18634.6 −1.85840
$$466$$ 6605.06 0.656596
$$467$$ −8383.97 −0.830758 −0.415379 0.909648i $$-0.636351\pi$$
−0.415379 + 0.909648i $$0.636351\pi$$
$$468$$ 1301.92 0.128592
$$469$$ 184.890 0.0182035
$$470$$ −4894.74 −0.480377
$$471$$ 8534.02 0.834876
$$472$$ 4822.77 0.470309
$$473$$ 2354.18 0.228848
$$474$$ −694.240 −0.0672732
$$475$$ 6294.47 0.608022
$$476$$ 1586.70 0.152786
$$477$$ 2930.87 0.281332
$$478$$ −2900.35 −0.277529
$$479$$ 9534.10 0.909445 0.454722 0.890633i $$-0.349739\pi$$
0.454722 + 0.890633i $$0.349739\pi$$
$$480$$ −6045.18 −0.574840
$$481$$ 9169.93 0.869258
$$482$$ 4775.43 0.451275
$$483$$ 2266.17 0.213487
$$484$$ −287.046 −0.0269577
$$485$$ −35048.4 −3.28138
$$486$$ −576.464 −0.0538044
$$487$$ 4451.09 0.414164 0.207082 0.978324i $$-0.433603\pi$$
0.207082 + 0.978324i $$0.433603\pi$$
$$488$$ −9907.38 −0.919029
$$489$$ −4585.53 −0.424059
$$490$$ 13755.0 1.26814
$$491$$ −9757.40 −0.896833 −0.448417 0.893825i $$-0.648012\pi$$
−0.448417 + 0.893825i $$0.648012\pi$$
$$492$$ 1798.74 0.164824
$$493$$ 2094.93 0.191381
$$494$$ 3573.20 0.325437
$$495$$ 1929.42 0.175194
$$496$$ 12555.6 1.13662
$$497$$ −2026.74 −0.182921
$$498$$ −7861.40 −0.707385
$$499$$ −7173.69 −0.643564 −0.321782 0.946814i $$-0.604282\pi$$
−0.321782 + 0.946814i $$0.604282\pi$$
$$500$$ −6002.33 −0.536865
$$501$$ −1151.64 −0.102698
$$502$$ −6420.88 −0.570873
$$503$$ −15617.0 −1.38435 −0.692177 0.721728i $$-0.743348\pi$$
−0.692177 + 0.721728i $$0.743348\pi$$
$$504$$ 1493.61 0.132005
$$505$$ 14459.3 1.27412
$$506$$ −2922.65 −0.256774
$$507$$ 4564.04 0.399795
$$508$$ −5462.76 −0.477108
$$509$$ −8789.23 −0.765375 −0.382688 0.923878i $$-0.625001\pi$$
−0.382688 + 0.923878i $$0.625001\pi$$
$$510$$ 13754.8 1.19426
$$511$$ 2885.42 0.249792
$$512$$ 11827.7 1.02093
$$513$$ 666.929 0.0573989
$$514$$ −535.521 −0.0459549
$$515$$ 7597.58 0.650077
$$516$$ −1523.12 −0.129945
$$517$$ 1164.56 0.0990667
$$518$$ 2406.09 0.204088
$$519$$ 8108.56 0.685792
$$520$$ −29242.0 −2.46606
$$521$$ 13099.3 1.10152 0.550760 0.834664i $$-0.314338\pi$$
0.550760 + 0.834664i $$0.314338\pi$$
$$522$$ 451.029 0.0378180
$$523$$ 16824.2 1.40664 0.703318 0.710876i $$-0.251701\pi$$
0.703318 + 0.710876i $$0.251701\pi$$
$$524$$ 4793.40 0.399620
$$525$$ 5156.07 0.428627
$$526$$ −4634.07 −0.384135
$$527$$ 31606.7 2.61254
$$528$$ −1300.00 −0.107150
$$529$$ 377.000 0.0309855
$$530$$ −15056.1 −1.23395
$$531$$ 1764.00 0.144164
$$532$$ −395.217 −0.0322083
$$533$$ 15411.9 1.25247
$$534$$ −3295.25 −0.267040
$$535$$ 29982.5 2.42291
$$536$$ 674.529 0.0543568
$$537$$ −8332.83 −0.669624
$$538$$ 832.310 0.0666978
$$539$$ −3272.62 −0.261525
$$540$$ −1248.31 −0.0994791
$$541$$ 18863.1 1.49905 0.749527 0.661974i $$-0.230281\pi$$
0.749527 + 0.661974i $$0.230281\pi$$
$$542$$ −604.407 −0.0478995
$$543$$ −11980.8 −0.946862
$$544$$ 10253.4 0.808110
$$545$$ 15197.3 1.19446
$$546$$ 2926.96 0.229418
$$547$$ −12283.0 −0.960119 −0.480059 0.877236i $$-0.659385\pi$$
−0.480059 + 0.877236i $$0.659385\pi$$
$$548$$ −3500.07 −0.272839
$$549$$ −3623.77 −0.281710
$$550$$ −6649.71 −0.515536
$$551$$ −521.809 −0.0403444
$$552$$ 8267.61 0.637487
$$553$$ 657.924 0.0505927
$$554$$ −276.320 −0.0211908
$$555$$ −8792.35 −0.672458
$$556$$ 3851.90 0.293808
$$557$$ −9752.05 −0.741845 −0.370923 0.928664i $$-0.620958\pi$$
−0.370923 + 0.928664i $$0.620958\pi$$
$$558$$ 6804.78 0.516253
$$559$$ −13050.3 −0.987424
$$560$$ −5178.17 −0.390746
$$561$$ −3272.56 −0.246288
$$562$$ −19515.5 −1.46479
$$563$$ −3447.50 −0.258072 −0.129036 0.991640i $$-0.541188\pi$$
−0.129036 + 0.991640i $$0.541188\pi$$
$$564$$ −753.457 −0.0562522
$$565$$ −29517.3 −2.19788
$$566$$ −3704.36 −0.275098
$$567$$ 546.310 0.0404636
$$568$$ −7394.09 −0.546213
$$569$$ −3371.02 −0.248366 −0.124183 0.992259i $$-0.539631\pi$$
−0.124183 + 0.992259i $$0.539631\pi$$
$$570$$ −3426.06 −0.251758
$$571$$ −15852.5 −1.16183 −0.580916 0.813964i $$-0.697305\pi$$
−0.580916 + 0.813964i $$0.697305\pi$$
$$572$$ 1591.23 0.116316
$$573$$ 2686.76 0.195883
$$574$$ 4043.91 0.294059
$$575$$ 28540.5 2.06995
$$576$$ 5043.89 0.364865
$$577$$ 1376.35 0.0993036 0.0496518 0.998767i $$-0.484189\pi$$
0.0496518 + 0.998767i $$0.484189\pi$$
$$578$$ −11674.9 −0.840159
$$579$$ 3984.72 0.286009
$$580$$ 976.684 0.0699217
$$581$$ 7450.17 0.531988
$$582$$ 12798.6 0.911547
$$583$$ 3582.17 0.254474
$$584$$ 10526.8 0.745894
$$585$$ −10695.7 −0.755920
$$586$$ −22503.9 −1.58640
$$587$$ −23021.4 −1.61873 −0.809366 0.587304i $$-0.800189\pi$$
−0.809366 + 0.587304i $$0.800189\pi$$
$$588$$ 2117.34 0.148499
$$589$$ −7872.66 −0.550742
$$590$$ −9061.80 −0.632320
$$591$$ 462.343 0.0321798
$$592$$ 5924.09 0.411281
$$593$$ 2818.69 0.195194 0.0975968 0.995226i $$-0.468884\pi$$
0.0975968 + 0.995226i $$0.468884\pi$$
$$594$$ −704.568 −0.0486679
$$595$$ −13035.3 −0.898140
$$596$$ 2619.27 0.180016
$$597$$ 3948.46 0.270686
$$598$$ 16201.6 1.10792
$$599$$ −17691.4 −1.20676 −0.603380 0.797454i $$-0.706180\pi$$
−0.603380 + 0.797454i $$0.706180\pi$$
$$600$$ 18810.7 1.27991
$$601$$ −24516.1 −1.66394 −0.831972 0.554817i $$-0.812788\pi$$
−0.831972 + 0.554817i $$0.812788\pi$$
$$602$$ −3424.26 −0.231831
$$603$$ 246.719 0.0166620
$$604$$ −7069.56 −0.476252
$$605$$ 2358.18 0.158469
$$606$$ −5280.12 −0.353944
$$607$$ 4288.59 0.286768 0.143384 0.989667i $$-0.454202\pi$$
0.143384 + 0.989667i $$0.454202\pi$$
$$608$$ −2553.94 −0.170355
$$609$$ −427.435 −0.0284410
$$610$$ 18615.6 1.23561
$$611$$ −6455.74 −0.427449
$$612$$ 2117.30 0.139848
$$613$$ −8124.07 −0.535283 −0.267641 0.963519i $$-0.586244\pi$$
−0.267641 + 0.963519i $$0.586244\pi$$
$$614$$ 20029.3 1.31648
$$615$$ −14777.3 −0.968908
$$616$$ 1825.52 0.119403
$$617$$ −14923.1 −0.973714 −0.486857 0.873482i $$-0.661857\pi$$
−0.486857 + 0.873482i $$0.661857\pi$$
$$618$$ −2774.41 −0.180588
$$619$$ 3627.68 0.235556 0.117778 0.993040i $$-0.462423\pi$$
0.117778 + 0.993040i $$0.462423\pi$$
$$620$$ 14735.5 0.954501
$$621$$ 3024.00 0.195409
$$622$$ −10549.9 −0.680087
$$623$$ 3122.88 0.200827
$$624$$ 7206.54 0.462328
$$625$$ 17458.0 1.11731
$$626$$ −15374.2 −0.981589
$$627$$ 815.135 0.0519192
$$628$$ −6748.36 −0.428804
$$629$$ 14913.0 0.945341
$$630$$ −2806.43 −0.177478
$$631$$ 12576.5 0.793445 0.396723 0.917939i $$-0.370147\pi$$
0.396723 + 0.917939i $$0.370147\pi$$
$$632$$ 2400.28 0.151073
$$633$$ −5207.27 −0.326967
$$634$$ 2971.87 0.186164
$$635$$ 44878.5 2.80464
$$636$$ −2317.61 −0.144496
$$637$$ 18141.7 1.12841
$$638$$ 551.257 0.0342077
$$639$$ −2704.50 −0.167431
$$640$$ −9790.36 −0.604685
$$641$$ −7292.77 −0.449371 −0.224686 0.974431i $$-0.572136\pi$$
−0.224686 + 0.974431i $$0.572136\pi$$
$$642$$ −10948.7 −0.673071
$$643$$ −12946.3 −0.794016 −0.397008 0.917815i $$-0.629951\pi$$
−0.397008 + 0.917815i $$0.629951\pi$$
$$644$$ −1792.00 −0.109650
$$645$$ 12513.0 0.763872
$$646$$ 5811.06 0.353921
$$647$$ 21973.3 1.33518 0.667589 0.744530i $$-0.267327\pi$$
0.667589 + 0.744530i $$0.267327\pi$$
$$648$$ 1993.08 0.120827
$$649$$ 2156.00 0.130401
$$650$$ 36862.5 2.22441
$$651$$ −6448.83 −0.388248
$$652$$ 3626.06 0.217803
$$653$$ −27495.6 −1.64776 −0.823880 0.566764i $$-0.808195\pi$$
−0.823880 + 0.566764i $$0.808195\pi$$
$$654$$ −5549.59 −0.331814
$$655$$ −39379.5 −2.34913
$$656$$ 9956.63 0.592593
$$657$$ 3850.33 0.228639
$$658$$ −1693.91 −0.100358
$$659$$ 5156.28 0.304796 0.152398 0.988319i $$-0.451301\pi$$
0.152398 + 0.988319i $$0.451301\pi$$
$$660$$ −1525.71 −0.0899822
$$661$$ 9328.59 0.548926 0.274463 0.961598i $$-0.411500\pi$$
0.274463 + 0.961598i $$0.411500\pi$$
$$662$$ −9018.10 −0.529454
$$663$$ 18141.4 1.06267
$$664$$ 27180.2 1.58855
$$665$$ 3246.85 0.189334
$$666$$ 3210.70 0.186805
$$667$$ −2365.99 −0.137349
$$668$$ 910.673 0.0527470
$$669$$ 4991.12 0.288442
$$670$$ −1267.42 −0.0730814
$$671$$ −4429.06 −0.254816
$$672$$ −2092.04 −0.120093
$$673$$ 22182.4 1.27053 0.635267 0.772293i $$-0.280890\pi$$
0.635267 + 0.772293i $$0.280890\pi$$
$$674$$ 12944.4 0.739764
$$675$$ 6880.30 0.392330
$$676$$ −3609.06 −0.205340
$$677$$ −13507.1 −0.766797 −0.383399 0.923583i $$-0.625246\pi$$
−0.383399 + 0.923583i $$0.625246\pi$$
$$678$$ 10778.8 0.610559
$$679$$ −12129.1 −0.685528
$$680$$ −47556.1 −2.68190
$$681$$ −10976.5 −0.617652
$$682$$ 8316.96 0.466969
$$683$$ −19465.6 −1.09053 −0.545264 0.838264i $$-0.683571\pi$$
−0.545264 + 0.838264i $$0.683571\pi$$
$$684$$ −527.381 −0.0294809
$$685$$ 28754.3 1.60386
$$686$$ 10248.2 0.570375
$$687$$ 5781.22 0.321059
$$688$$ −8430.96 −0.467191
$$689$$ −19857.7 −1.09799
$$690$$ −15534.5 −0.857086
$$691$$ −19971.8 −1.09951 −0.549757 0.835324i $$-0.685280\pi$$
−0.549757 + 0.835324i $$0.685280\pi$$
$$692$$ −6411.93 −0.352233
$$693$$ 667.712 0.0366007
$$694$$ 10059.7 0.550234
$$695$$ −31644.7 −1.72713
$$696$$ −1559.40 −0.0849265
$$697$$ 25064.3 1.36209
$$698$$ 20096.9 1.08980
$$699$$ −8352.80 −0.451977
$$700$$ −4077.22 −0.220149
$$701$$ −14180.1 −0.764018 −0.382009 0.924159i $$-0.624768\pi$$
−0.382009 + 0.924159i $$0.624768\pi$$
$$702$$ 3905.75 0.209990
$$703$$ −3714.56 −0.199285
$$704$$ 6164.75 0.330032
$$705$$ 6189.91 0.330675
$$706$$ −2327.88 −0.124095
$$707$$ 5003.91 0.266183
$$708$$ −1394.90 −0.0740446
$$709$$ 15870.6 0.840667 0.420334 0.907370i $$-0.361913\pi$$
0.420334 + 0.907370i $$0.361913\pi$$
$$710$$ 13893.2 0.734370
$$711$$ 877.939 0.0463084
$$712$$ 11393.1 0.599683
$$713$$ −35696.3 −1.87495
$$714$$ 4760.09 0.249498
$$715$$ −13072.5 −0.683756
$$716$$ 6589.27 0.343928
$$717$$ 3667.79 0.191041
$$718$$ 7164.50 0.372391
$$719$$ −6040.05 −0.313290 −0.156645 0.987655i $$-0.550068\pi$$
−0.156645 + 0.987655i $$0.550068\pi$$
$$720$$ −6909.80 −0.357657
$$721$$ 2629.28 0.135811
$$722$$ 14824.0 0.764119
$$723$$ −6039.03 −0.310642
$$724$$ 9473.96 0.486322
$$725$$ −5383.18 −0.275761
$$726$$ −861.138 −0.0440218
$$727$$ 37252.3 1.90043 0.950213 0.311601i $$-0.100865\pi$$
0.950213 + 0.311601i $$0.100865\pi$$
$$728$$ −10119.7 −0.515196
$$729$$ 729.000 0.0370370
$$730$$ −19779.5 −1.00284
$$731$$ −21223.7 −1.07385
$$732$$ 2865.54 0.144690
$$733$$ 6125.52 0.308665 0.154332 0.988019i $$-0.450677\pi$$
0.154332 + 0.988019i $$0.450677\pi$$
$$734$$ −20613.7 −1.03660
$$735$$ −17394.7 −0.872942
$$736$$ −11580.1 −0.579958
$$737$$ 301.546 0.0150713
$$738$$ 5396.23 0.269157
$$739$$ −5225.97 −0.260136 −0.130068 0.991505i $$-0.541520\pi$$
−0.130068 + 0.991505i $$0.541520\pi$$
$$740$$ 6952.64 0.345384
$$741$$ −4518.68 −0.224019
$$742$$ −5210.43 −0.257791
$$743$$ 7062.96 0.348741 0.174371 0.984680i $$-0.444211\pi$$
0.174371 + 0.984680i $$0.444211\pi$$
$$744$$ −23527.0 −1.15933
$$745$$ −21518.2 −1.05821
$$746$$ −12669.2 −0.621788
$$747$$ 9941.57 0.486939
$$748$$ 2587.81 0.126497
$$749$$ 10376.0 0.506182
$$750$$ −18007.0 −0.876697
$$751$$ −20755.4 −1.00849 −0.504244 0.863561i $$-0.668229\pi$$
−0.504244 + 0.863561i $$0.668229\pi$$
$$752$$ −4170.63 −0.202243
$$753$$ 8119.89 0.392968
$$754$$ −3055.88 −0.147598
$$755$$ 58079.0 2.79962
$$756$$ −432.000 −0.0207827
$$757$$ 31182.9 1.49717 0.748587 0.663037i $$-0.230733\pi$$
0.748587 + 0.663037i $$0.230733\pi$$
$$758$$ 3803.93 0.182276
$$759$$ 3696.00 0.176754
$$760$$ 11845.4 0.565364
$$761$$ 32047.8 1.52659 0.763293 0.646052i $$-0.223581\pi$$
0.763293 + 0.646052i $$0.223581\pi$$
$$762$$ −16388.3 −0.779114
$$763$$ 5259.29 0.249540
$$764$$ −2124.58 −0.100608
$$765$$ −17394.4 −0.822084
$$766$$ 1969.55 0.0929019
$$767$$ −11951.7 −0.562650
$$768$$ −9875.22 −0.463986
$$769$$ 2215.88 0.103910 0.0519548 0.998649i $$-0.483455\pi$$
0.0519548 + 0.998649i $$0.483455\pi$$
$$770$$ −3430.09 −0.160535
$$771$$ 677.223 0.0316337
$$772$$ −3150.96 −0.146898
$$773$$ 5300.56 0.246634 0.123317 0.992367i $$-0.460647\pi$$
0.123317 + 0.992367i $$0.460647\pi$$
$$774$$ −4569.36 −0.212199
$$775$$ −81217.4 −3.76441
$$776$$ −44250.3 −2.04703
$$777$$ −3042.75 −0.140487
$$778$$ 4142.29 0.190885
$$779$$ −6243.06 −0.287138
$$780$$ 8457.75 0.388251
$$781$$ −3305.50 −0.151447
$$782$$ 26348.6 1.20489
$$783$$ −570.373 −0.0260325
$$784$$ 11720.2 0.533899
$$785$$ 55440.2 2.52069
$$786$$ 14380.2 0.652576
$$787$$ 34374.1 1.55693 0.778466 0.627687i $$-0.215998\pi$$
0.778466 + 0.627687i $$0.215998\pi$$
$$788$$ −365.602 −0.0165280
$$789$$ 5860.27 0.264425
$$790$$ −4510.04 −0.203114
$$791$$ −10215.0 −0.459170
$$792$$ 2435.99 0.109292
$$793$$ 24552.4 1.09947
$$794$$ 23761.2 1.06203
$$795$$ 19040.0 0.849409
$$796$$ −3122.28 −0.139028
$$797$$ 29867.1 1.32741 0.663707 0.747993i $$-0.268982\pi$$
0.663707 + 0.747993i $$0.268982\pi$$
$$798$$ −1185.65 −0.0525960
$$799$$ −10498.9 −0.464862
$$800$$ −26347.5 −1.16441
$$801$$ 4167.20 0.183821
$$802$$ −19520.0 −0.859447
$$803$$ 4705.96 0.206812
$$804$$ −195.096 −0.00855783
$$805$$ 14721.9 0.644570
$$806$$ −46104.9 −2.01486
$$807$$ −1052.54 −0.0459124
$$808$$ 18255.6 0.794839
$$809$$ −15857.2 −0.689133 −0.344566 0.938762i $$-0.611974\pi$$
−0.344566 + 0.938762i $$0.611974\pi$$
$$810$$ −3744.93 −0.162449
$$811$$ 36122.6 1.56404 0.782020 0.623253i $$-0.214189\pi$$
0.782020 + 0.623253i $$0.214189\pi$$
$$812$$ 337.999 0.0146077
$$813$$ 764.337 0.0329723
$$814$$ 3924.19 0.168971
$$815$$ −29789.3 −1.28034
$$816$$ 11719.9 0.502794
$$817$$ 5286.43 0.226375
$$818$$ 29206.4 1.24838
$$819$$ −3701.44 −0.157923
$$820$$ 11685.3 0.497645
$$821$$ 25779.6 1.09588 0.547938 0.836519i $$-0.315413\pi$$
0.547938 + 0.836519i $$0.315413\pi$$
$$822$$ −10500.2 −0.445544
$$823$$ −22130.2 −0.937315 −0.468658 0.883380i $$-0.655262\pi$$
−0.468658 + 0.883380i $$0.655262\pi$$
$$824$$ 9592.32 0.405539
$$825$$ 8409.26 0.354876
$$826$$ −3136.00 −0.132101
$$827$$ 18288.6 0.768994 0.384497 0.923126i $$-0.374375\pi$$
0.384497 + 0.923126i $$0.374375\pi$$
$$828$$ −2391.26 −0.100365
$$829$$ −34956.6 −1.46453 −0.732263 0.681022i $$-0.761536\pi$$
−0.732263 + 0.681022i $$0.761536\pi$$
$$830$$ −51070.6 −2.13577
$$831$$ 349.435 0.0145870
$$832$$ −34174.2 −1.42401
$$833$$ 29503.7 1.22718
$$834$$ 11555.7 0.479786
$$835$$ −7481.50 −0.310069
$$836$$ −644.577 −0.0266664
$$837$$ −8605.37 −0.355370
$$838$$ −31457.5 −1.29675
$$839$$ −14507.5 −0.596965 −0.298482 0.954415i $$-0.596480\pi$$
−0.298482 + 0.954415i $$0.596480\pi$$
$$840$$ 9703.04 0.398556
$$841$$ −23942.7 −0.981702
$$842$$ 14655.3 0.599829
$$843$$ 24679.4 1.00831
$$844$$ 4117.70 0.167935
$$845$$ 29649.7 1.20708
$$846$$ −2260.37 −0.0918595
$$847$$ 816.092 0.0331066
$$848$$ −12828.7 −0.519506
$$849$$ 4684.55 0.189368
$$850$$ 59949.2 2.41911
$$851$$ −16842.6 −0.678445
$$852$$ 2138.61 0.0859948
$$853$$ −44146.9 −1.77205 −0.886027 0.463634i $$-0.846545\pi$$
−0.886027 + 0.463634i $$0.846545\pi$$
$$854$$ 6442.26 0.258138
$$855$$ 4332.62 0.173301
$$856$$ 37854.4 1.51149
$$857$$ −47679.3 −1.90046 −0.950230 0.311549i $$-0.899152\pi$$
−0.950230 + 0.311549i $$0.899152\pi$$
$$858$$ 4773.70 0.189943
$$859$$ 7525.09 0.298897 0.149449 0.988769i $$-0.452250\pi$$
0.149449 + 0.988769i $$0.452250\pi$$
$$860$$ −9894.76 −0.392335
$$861$$ −5113.95 −0.202419
$$862$$ −3957.09 −0.156356
$$863$$ 45816.1 1.80718 0.903591 0.428396i $$-0.140921\pi$$
0.903591 + 0.428396i $$0.140921\pi$$
$$864$$ −2791.64 −0.109923
$$865$$ 52676.2 2.07057
$$866$$ 1735.91 0.0681163
$$867$$ 14764.1 0.578335
$$868$$ 5099.48 0.199410
$$869$$ 1073.04 0.0418876
$$870$$ 2930.05 0.114182
$$871$$ −1671.61 −0.0650292
$$872$$ 19187.3 0.745142
$$873$$ −16185.2 −0.627476
$$874$$ −6562.96 −0.253999
$$875$$ 17065.1 0.659319
$$876$$ −3044.69 −0.117432
$$877$$ −34168.3 −1.31560 −0.657801 0.753192i $$-0.728513\pi$$
−0.657801 + 0.753192i $$0.728513\pi$$
$$878$$ 33800.2 1.29920
$$879$$ 28458.6 1.09202
$$880$$ −8445.31 −0.323513
$$881$$ −100.796 −0.00385460 −0.00192730 0.999998i $$-0.500613\pi$$
−0.00192730 + 0.999998i $$0.500613\pi$$
$$882$$ 6352.02 0.242498
$$883$$ −12346.1 −0.470531 −0.235266 0.971931i $$-0.575596\pi$$
−0.235266 + 0.971931i $$0.575596\pi$$
$$884$$ −14345.5 −0.545803
$$885$$ 11459.6 0.435266
$$886$$ −413.398 −0.0156754
$$887$$ −37345.1 −1.41367 −0.706834 0.707379i $$-0.749877\pi$$
−0.706834 + 0.707379i $$0.749877\pi$$
$$888$$ −11100.8 −0.419501
$$889$$ 15531.0 0.585932
$$890$$ −21407.2 −0.806260
$$891$$ 891.000 0.0335013
$$892$$ −3946.78 −0.148148
$$893$$ 2615.09 0.0979962
$$894$$ 7857.80 0.293964
$$895$$ −54133.2 −2.02176
$$896$$ −3388.13 −0.126328
$$897$$ −20488.7 −0.762651
$$898$$ −17720.9 −0.658523
$$899$$ 6732.88 0.249782
$$900$$ −5440.67 −0.201506
$$901$$ −32294.4 −1.19410
$$902$$ 6595.39 0.243462
$$903$$ 4330.34 0.159584
$$904$$ −37267.1 −1.37111
$$905$$ −77831.9 −2.85881
$$906$$ −21208.7 −0.777717
$$907$$ 10308.6 0.377390 0.188695 0.982036i $$-0.439574\pi$$
0.188695 + 0.982036i $$0.439574\pi$$
$$908$$ 8679.79 0.317235
$$909$$ 6677.26 0.243642
$$910$$ 19014.6 0.692668
$$911$$ −22590.8 −0.821589 −0.410795 0.911728i $$-0.634749\pi$$
−0.410795 + 0.911728i $$0.634749\pi$$
$$912$$ −2919.22 −0.105992
$$913$$ 12150.8 0.440453
$$914$$ −20787.5 −0.752288
$$915$$ −23541.4 −0.850551
$$916$$ −4571.56 −0.164900
$$917$$ −13628.0 −0.490769
$$918$$ 6351.90 0.228370
$$919$$ 47712.1 1.71260 0.856299 0.516481i $$-0.172758\pi$$
0.856299 + 0.516481i $$0.172758\pi$$
$$920$$ 53709.5 1.92473
$$921$$ −25329.2 −0.906216
$$922$$ −10294.9 −0.367728
$$923$$ 18323.9 0.653456
$$924$$ −528.000 −0.0187986
$$925$$ −38320.8 −1.36214
$$926$$ 9328.96 0.331068
$$927$$ 3508.53 0.124310
$$928$$ 2184.19 0.0772625
$$929$$ 10714.3 0.378391 0.189196 0.981939i $$-0.439412\pi$$
0.189196 + 0.981939i $$0.439412\pi$$
$$930$$ 44206.4 1.55869
$$931$$ −7348.84 −0.258699
$$932$$ 6605.06 0.232142
$$933$$ 13341.5 0.468148
$$934$$ 19889.1 0.696780
$$935$$ −21259.8 −0.743603
$$936$$ −13503.9 −0.471568
$$937$$ −14719.5 −0.513197 −0.256599 0.966518i $$-0.582602\pi$$
−0.256599 + 0.966518i $$0.582602\pi$$
$$938$$ −438.612 −0.0152678
$$939$$ 19442.2 0.675691
$$940$$ −4894.74 −0.169839
$$941$$ 3694.44 0.127987 0.0639933 0.997950i $$-0.479616\pi$$
0.0639933 + 0.997950i $$0.479616\pi$$
$$942$$ −20245.1 −0.700234
$$943$$ −28307.4 −0.977535
$$944$$ −7721.23 −0.266213
$$945$$ 3549.03 0.122169
$$946$$ −5584.77 −0.191941
$$947$$ 36416.7 1.24961 0.624806 0.780780i $$-0.285178\pi$$
0.624806 + 0.780780i $$0.285178\pi$$
$$948$$ −694.240 −0.0237847
$$949$$ −26087.4 −0.892342
$$950$$ −14932.3 −0.509965
$$951$$ −3758.25 −0.128149
$$952$$ −16457.6 −0.560289
$$953$$ 20779.4 0.706306 0.353153 0.935566i $$-0.385110\pi$$
0.353153 + 0.935566i $$0.385110\pi$$
$$954$$ −6952.84 −0.235961
$$955$$ 17454.2 0.591419
$$956$$ −2900.35 −0.0981212
$$957$$ −697.123 −0.0235473
$$958$$ −22617.6 −0.762777
$$959$$ 9950.94 0.335071
$$960$$ 32767.0 1.10161
$$961$$ 71789.7 2.40978
$$962$$ −21753.7 −0.729071
$$963$$ 13845.8 0.463318
$$964$$ 4775.43 0.159550
$$965$$ 25886.2 0.863531
$$966$$ −5376.00 −0.179058
$$967$$ 56812.8 1.88932 0.944662 0.328044i $$-0.106389\pi$$
0.944662 + 0.328044i $$0.106389\pi$$
$$968$$ 2977.32 0.0988582
$$969$$ −7348.70 −0.243627
$$970$$ 83144.7 2.75218
$$971$$ −26459.7 −0.874493 −0.437247 0.899342i $$-0.644046\pi$$
−0.437247 + 0.899342i $$0.644046\pi$$
$$972$$ −576.464 −0.0190227
$$973$$ −10951.2 −0.360822
$$974$$ −10559.2 −0.347371
$$975$$ −46616.5 −1.53120
$$976$$ 15861.7 0.520204
$$977$$ −21009.2 −0.687967 −0.343984 0.938976i $$-0.611776\pi$$
−0.343984 + 0.938976i $$0.611776\pi$$
$$978$$ 10878.2 0.355670
$$979$$ 5093.24 0.166272
$$980$$ 13755.0 0.448355
$$981$$ 7018.04 0.228409
$$982$$ 23147.3 0.752199
$$983$$ −9076.80 −0.294512 −0.147256 0.989098i $$-0.547044\pi$$
−0.147256 + 0.989098i $$0.547044\pi$$
$$984$$ −18657.1 −0.604437
$$985$$ 3003.55 0.0971585
$$986$$ −4969.76 −0.160517
$$987$$ 2142.13 0.0690828
$$988$$ 3573.20 0.115059
$$989$$ 23969.8 0.770673
$$990$$ −4577.14 −0.146940
$$991$$ 2629.24 0.0842791 0.0421395 0.999112i $$-0.486583\pi$$
0.0421395 + 0.999112i $$0.486583\pi$$
$$992$$ 32953.5 1.05471
$$993$$ 11404.3 0.364457
$$994$$ 4808.00 0.153421
$$995$$ 25650.6 0.817267
$$996$$ −7861.40 −0.250098
$$997$$ −50423.9 −1.60175 −0.800874 0.598833i $$-0.795631\pi$$
−0.800874 + 0.598833i $$0.795631\pi$$
$$998$$ 17018.0 0.539775
$$999$$ −4060.27 −0.128590
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 33.4.a.d.1.1 2
3.2 odd 2 99.4.a.e.1.2 2
4.3 odd 2 528.4.a.o.1.2 2
5.2 odd 4 825.4.c.i.199.2 4
5.3 odd 4 825.4.c.i.199.3 4
5.4 even 2 825.4.a.k.1.2 2
7.6 odd 2 1617.4.a.j.1.1 2
8.3 odd 2 2112.4.a.bh.1.1 2
8.5 even 2 2112.4.a.ba.1.1 2
11.10 odd 2 363.4.a.j.1.2 2
12.11 even 2 1584.4.a.x.1.1 2
15.14 odd 2 2475.4.a.o.1.1 2
33.32 even 2 1089.4.a.t.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.1 2 1.1 even 1 trivial
99.4.a.e.1.2 2 3.2 odd 2
363.4.a.j.1.2 2 11.10 odd 2
528.4.a.o.1.2 2 4.3 odd 2
825.4.a.k.1.2 2 5.4 even 2
825.4.c.i.199.2 4 5.2 odd 4
825.4.c.i.199.3 4 5.3 odd 4
1089.4.a.t.1.1 2 33.32 even 2
1584.4.a.x.1.1 2 12.11 even 2
1617.4.a.j.1.1 2 7.6 odd 2
2112.4.a.ba.1.1 2 8.5 even 2
2112.4.a.bh.1.1 2 8.3 odd 2
2475.4.a.o.1.1 2 15.14 odd 2