# Properties

 Label 75.4.a.c.1.1 Level $75$ Weight $4$ Character 75.1 Self dual yes Analytic conductor $4.425$ 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] = [75,4,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, 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("75.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.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: $$4.42514325043$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 75.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.70156 q^{2} -3.00000 q^{3} +14.1047 q^{4} +14.1047 q^{6} +16.2094 q^{7} -28.7016 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-4.70156 q^{2} -3.00000 q^{3} +14.1047 q^{4} +14.1047 q^{6} +16.2094 q^{7} -28.7016 q^{8} +9.00000 q^{9} -40.2094 q^{11} -42.3141 q^{12} -19.7906 q^{13} -76.2094 q^{14} +22.1047 q^{16} -83.0156 q^{17} -42.3141 q^{18} -48.8375 q^{19} -48.6281 q^{21} +189.047 q^{22} -1.61250 q^{23} +86.1047 q^{24} +93.0469 q^{26} -27.0000 q^{27} +228.628 q^{28} -24.5344 q^{29} -12.4187 q^{31} +125.686 q^{32} +120.628 q^{33} +390.303 q^{34} +126.942 q^{36} -325.884 q^{37} +229.612 q^{38} +59.3719 q^{39} -242.419 q^{41} +228.628 q^{42} -367.350 q^{43} -567.141 q^{44} +7.58125 q^{46} +204.544 q^{47} -66.3141 q^{48} -80.2562 q^{49} +249.047 q^{51} -279.141 q^{52} +61.5281 q^{53} +126.942 q^{54} -465.234 q^{56} +146.512 q^{57} +115.350 q^{58} -112.209 q^{59} +477.350 q^{61} +58.3875 q^{62} +145.884 q^{63} -767.758 q^{64} -567.141 q^{66} -558.094 q^{67} -1170.91 q^{68} +4.83749 q^{69} +558.281 q^{71} -258.314 q^{72} -1011.77 q^{73} +1532.17 q^{74} -688.837 q^{76} -651.769 q^{77} -279.141 q^{78} +1150.47 q^{79} +81.0000 q^{81} +1139.75 q^{82} +1157.92 q^{83} -685.884 q^{84} +1727.12 q^{86} +73.6032 q^{87} +1154.07 q^{88} +96.9751 q^{89} -320.794 q^{91} -22.7438 q^{92} +37.2562 q^{93} -961.675 q^{94} -377.058 q^{96} +1152.37 q^{97} +377.330 q^{98} -361.884 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 6 q^{7} - 51 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 - 6 * q^3 + 9 * q^4 + 9 * q^6 - 6 * q^7 - 51 * q^8 + 18 * q^9 $$2 q - 3 q^{2} - 6 q^{3} + 9 q^{4} + 9 q^{6} - 6 q^{7} - 51 q^{8} + 18 q^{9} - 42 q^{11} - 27 q^{12} - 78 q^{13} - 114 q^{14} + 25 q^{16} - 102 q^{17} - 27 q^{18} + 56 q^{19} + 18 q^{21} + 186 q^{22} + 48 q^{23} + 153 q^{24} - 6 q^{26} - 54 q^{27} + 342 q^{28} - 318 q^{29} + 52 q^{31} + 309 q^{32} + 126 q^{33} + 358 q^{34} + 81 q^{36} - 306 q^{37} + 408 q^{38} + 234 q^{39} - 408 q^{41} + 342 q^{42} - 120 q^{43} - 558 q^{44} + 92 q^{46} - 180 q^{47} - 75 q^{48} + 70 q^{49} + 306 q^{51} + 18 q^{52} - 402 q^{53} + 81 q^{54} + 30 q^{56} - 168 q^{57} - 384 q^{58} - 186 q^{59} + 340 q^{61} + 168 q^{62} - 54 q^{63} - 479 q^{64} - 558 q^{66} - 732 q^{67} - 1074 q^{68} - 144 q^{69} - 36 q^{71} - 459 q^{72} - 1332 q^{73} + 1566 q^{74} - 1224 q^{76} - 612 q^{77} + 18 q^{78} + 380 q^{79} + 162 q^{81} + 858 q^{82} + 984 q^{83} - 1026 q^{84} + 2148 q^{86} + 954 q^{87} + 1194 q^{88} + 1116 q^{89} + 972 q^{91} - 276 q^{92} - 156 q^{93} - 1616 q^{94} - 927 q^{96} + 768 q^{97} + 633 q^{98} - 378 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 - 6 * q^3 + 9 * q^4 + 9 * q^6 - 6 * q^7 - 51 * q^8 + 18 * q^9 - 42 * q^11 - 27 * q^12 - 78 * q^13 - 114 * q^14 + 25 * q^16 - 102 * q^17 - 27 * q^18 + 56 * q^19 + 18 * q^21 + 186 * q^22 + 48 * q^23 + 153 * q^24 - 6 * q^26 - 54 * q^27 + 342 * q^28 - 318 * q^29 + 52 * q^31 + 309 * q^32 + 126 * q^33 + 358 * q^34 + 81 * q^36 - 306 * q^37 + 408 * q^38 + 234 * q^39 - 408 * q^41 + 342 * q^42 - 120 * q^43 - 558 * q^44 + 92 * q^46 - 180 * q^47 - 75 * q^48 + 70 * q^49 + 306 * q^51 + 18 * q^52 - 402 * q^53 + 81 * q^54 + 30 * q^56 - 168 * q^57 - 384 * q^58 - 186 * q^59 + 340 * q^61 + 168 * q^62 - 54 * q^63 - 479 * q^64 - 558 * q^66 - 732 * q^67 - 1074 * q^68 - 144 * q^69 - 36 * q^71 - 459 * q^72 - 1332 * q^73 + 1566 * q^74 - 1224 * q^76 - 612 * q^77 + 18 * q^78 + 380 * q^79 + 162 * q^81 + 858 * q^82 + 984 * q^83 - 1026 * q^84 + 2148 * q^86 + 954 * q^87 + 1194 * q^88 + 1116 * q^89 + 972 * q^91 - 276 * q^92 - 156 * q^93 - 1616 * q^94 - 927 * q^96 + 768 * q^97 + 633 * q^98 - 378 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.70156 −1.66225 −0.831127 0.556083i $$-0.812304\pi$$
−0.831127 + 0.556083i $$0.812304\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 14.1047 1.76309
$$5$$ 0 0
$$6$$ 14.1047 0.959702
$$7$$ 16.2094 0.875224 0.437612 0.899164i $$-0.355824\pi$$
0.437612 + 0.899164i $$0.355824\pi$$
$$8$$ −28.7016 −1.26844
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −40.2094 −1.10214 −0.551072 0.834458i $$-0.685781\pi$$
−0.551072 + 0.834458i $$0.685781\pi$$
$$12$$ −42.3141 −1.01792
$$13$$ −19.7906 −0.422226 −0.211113 0.977462i $$-0.567709\pi$$
−0.211113 + 0.977462i $$0.567709\pi$$
$$14$$ −76.2094 −1.45484
$$15$$ 0 0
$$16$$ 22.1047 0.345386
$$17$$ −83.0156 −1.18437 −0.592184 0.805803i $$-0.701734\pi$$
−0.592184 + 0.805803i $$0.701734\pi$$
$$18$$ −42.3141 −0.554084
$$19$$ −48.8375 −0.589689 −0.294844 0.955545i $$-0.595268\pi$$
−0.294844 + 0.955545i $$0.595268\pi$$
$$20$$ 0 0
$$21$$ −48.6281 −0.505311
$$22$$ 189.047 1.83204
$$23$$ −1.61250 −0.0146186 −0.00730932 0.999973i $$-0.502327\pi$$
−0.00730932 + 0.999973i $$0.502327\pi$$
$$24$$ 86.1047 0.732335
$$25$$ 0 0
$$26$$ 93.0469 0.701846
$$27$$ −27.0000 −0.192450
$$28$$ 228.628 1.54309
$$29$$ −24.5344 −0.157101 −0.0785504 0.996910i $$-0.525029\pi$$
−0.0785504 + 0.996910i $$0.525029\pi$$
$$30$$ 0 0
$$31$$ −12.4187 −0.0719507 −0.0359754 0.999353i $$-0.511454\pi$$
−0.0359754 + 0.999353i $$0.511454\pi$$
$$32$$ 125.686 0.694323
$$33$$ 120.628 0.636323
$$34$$ 390.303 1.96872
$$35$$ 0 0
$$36$$ 126.942 0.587695
$$37$$ −325.884 −1.44797 −0.723987 0.689813i $$-0.757693\pi$$
−0.723987 + 0.689813i $$0.757693\pi$$
$$38$$ 229.612 0.980212
$$39$$ 59.3719 0.243772
$$40$$ 0 0
$$41$$ −242.419 −0.923401 −0.461701 0.887036i $$-0.652761\pi$$
−0.461701 + 0.887036i $$0.652761\pi$$
$$42$$ 228.628 0.839954
$$43$$ −367.350 −1.30280 −0.651399 0.758735i $$-0.725818\pi$$
−0.651399 + 0.758735i $$0.725818\pi$$
$$44$$ −567.141 −1.94317
$$45$$ 0 0
$$46$$ 7.58125 0.0242999
$$47$$ 204.544 0.634804 0.317402 0.948291i $$-0.397190\pi$$
0.317402 + 0.948291i $$0.397190\pi$$
$$48$$ −66.3141 −0.199409
$$49$$ −80.2562 −0.233983
$$50$$ 0 0
$$51$$ 249.047 0.683795
$$52$$ −279.141 −0.744420
$$53$$ 61.5281 0.159463 0.0797314 0.996816i $$-0.474594\pi$$
0.0797314 + 0.996816i $$0.474594\pi$$
$$54$$ 126.942 0.319901
$$55$$ 0 0
$$56$$ −465.234 −1.11017
$$57$$ 146.512 0.340457
$$58$$ 115.350 0.261141
$$59$$ −112.209 −0.247600 −0.123800 0.992307i $$-0.539508\pi$$
−0.123800 + 0.992307i $$0.539508\pi$$
$$60$$ 0 0
$$61$$ 477.350 1.00194 0.500970 0.865464i $$-0.332977\pi$$
0.500970 + 0.865464i $$0.332977\pi$$
$$62$$ 58.3875 0.119600
$$63$$ 145.884 0.291741
$$64$$ −767.758 −1.49953
$$65$$ 0 0
$$66$$ −567.141 −1.05773
$$67$$ −558.094 −1.01764 −0.508821 0.860872i $$-0.669918\pi$$
−0.508821 + 0.860872i $$0.669918\pi$$
$$68$$ −1170.91 −2.08814
$$69$$ 4.83749 0.00844008
$$70$$ 0 0
$$71$$ 558.281 0.933180 0.466590 0.884474i $$-0.345482\pi$$
0.466590 + 0.884474i $$0.345482\pi$$
$$72$$ −258.314 −0.422814
$$73$$ −1011.77 −1.62217 −0.811086 0.584927i $$-0.801123\pi$$
−0.811086 + 0.584927i $$0.801123\pi$$
$$74$$ 1532.17 2.40690
$$75$$ 0 0
$$76$$ −688.837 −1.03967
$$77$$ −651.769 −0.964623
$$78$$ −279.141 −0.405211
$$79$$ 1150.47 1.63845 0.819227 0.573470i $$-0.194403\pi$$
0.819227 + 0.573470i $$0.194403\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 1139.75 1.53493
$$83$$ 1157.92 1.53131 0.765655 0.643251i $$-0.222415\pi$$
0.765655 + 0.643251i $$0.222415\pi$$
$$84$$ −685.884 −0.890906
$$85$$ 0 0
$$86$$ 1727.12 2.16558
$$87$$ 73.6032 0.0907022
$$88$$ 1154.07 1.39801
$$89$$ 96.9751 0.115498 0.0577491 0.998331i $$-0.481608\pi$$
0.0577491 + 0.998331i $$0.481608\pi$$
$$90$$ 0 0
$$91$$ −320.794 −0.369542
$$92$$ −22.7438 −0.0257739
$$93$$ 37.2562 0.0415408
$$94$$ −961.675 −1.05520
$$95$$ 0 0
$$96$$ −377.058 −0.400868
$$97$$ 1152.37 1.20625 0.603123 0.797648i $$-0.293923\pi$$
0.603123 + 0.797648i $$0.293923\pi$$
$$98$$ 377.330 0.388939
$$99$$ −361.884 −0.367381
$$100$$ 0 0
$$101$$ −1156.49 −1.13936 −0.569679 0.821867i $$-0.692932\pi$$
−0.569679 + 0.821867i $$0.692932\pi$$
$$102$$ −1170.91 −1.13664
$$103$$ 1333.70 1.27585 0.637927 0.770096i $$-0.279792\pi$$
0.637927 + 0.770096i $$0.279792\pi$$
$$104$$ 568.022 0.535569
$$105$$ 0 0
$$106$$ −289.278 −0.265068
$$107$$ 798.263 0.721224 0.360612 0.932716i $$-0.382568\pi$$
0.360612 + 0.932716i $$0.382568\pi$$
$$108$$ −380.827 −0.339306
$$109$$ −985.119 −0.865663 −0.432831 0.901475i $$-0.642485\pi$$
−0.432831 + 0.901475i $$0.642485\pi$$
$$110$$ 0 0
$$111$$ 977.653 0.835988
$$112$$ 358.303 0.302290
$$113$$ −1888.25 −1.57196 −0.785979 0.618253i $$-0.787841\pi$$
−0.785979 + 0.618253i $$0.787841\pi$$
$$114$$ −688.837 −0.565926
$$115$$ 0 0
$$116$$ −346.050 −0.276982
$$117$$ −178.116 −0.140742
$$118$$ 527.559 0.411574
$$119$$ −1345.63 −1.03659
$$120$$ 0 0
$$121$$ 285.794 0.214721
$$122$$ −2244.29 −1.66548
$$123$$ 727.256 0.533126
$$124$$ −175.163 −0.126855
$$125$$ 0 0
$$126$$ −685.884 −0.484948
$$127$$ 620.859 0.433798 0.216899 0.976194i $$-0.430406\pi$$
0.216899 + 0.976194i $$0.430406\pi$$
$$128$$ 2604.17 1.79827
$$129$$ 1102.05 0.752171
$$130$$ 0 0
$$131$$ −2588.35 −1.72630 −0.863151 0.504947i $$-0.831512\pi$$
−0.863151 + 0.504947i $$0.831512\pi$$
$$132$$ 1701.42 1.12189
$$133$$ −791.625 −0.516110
$$134$$ 2623.91 1.69158
$$135$$ 0 0
$$136$$ 2382.68 1.50230
$$137$$ 1656.29 1.03289 0.516447 0.856319i $$-0.327254\pi$$
0.516447 + 0.856319i $$0.327254\pi$$
$$138$$ −22.7438 −0.0140295
$$139$$ −153.256 −0.0935182 −0.0467591 0.998906i $$-0.514889\pi$$
−0.0467591 + 0.998906i $$0.514889\pi$$
$$140$$ 0 0
$$141$$ −613.631 −0.366504
$$142$$ −2624.79 −1.55118
$$143$$ 795.769 0.465353
$$144$$ 198.942 0.115129
$$145$$ 0 0
$$146$$ 4756.89 2.69646
$$147$$ 240.769 0.135090
$$148$$ −4596.50 −2.55290
$$149$$ 1483.38 0.815591 0.407795 0.913073i $$-0.366298\pi$$
0.407795 + 0.913073i $$0.366298\pi$$
$$150$$ 0 0
$$151$$ −394.281 −0.212491 −0.106246 0.994340i $$-0.533883\pi$$
−0.106246 + 0.994340i $$0.533883\pi$$
$$152$$ 1401.71 0.747986
$$153$$ −747.141 −0.394789
$$154$$ 3064.33 1.60345
$$155$$ 0 0
$$156$$ 837.422 0.429791
$$157$$ −1727.05 −0.877922 −0.438961 0.898506i $$-0.644653\pi$$
−0.438961 + 0.898506i $$0.644653\pi$$
$$158$$ −5409.00 −2.72352
$$159$$ −184.584 −0.0920659
$$160$$ 0 0
$$161$$ −26.1376 −0.0127946
$$162$$ −380.827 −0.184695
$$163$$ 2034.28 0.977529 0.488764 0.872416i $$-0.337448\pi$$
0.488764 + 0.872416i $$0.337448\pi$$
$$164$$ −3419.24 −1.62804
$$165$$ 0 0
$$166$$ −5444.06 −2.54543
$$167$$ −192.900 −0.0893835 −0.0446918 0.999001i $$-0.514231\pi$$
−0.0446918 + 0.999001i $$0.514231\pi$$
$$168$$ 1395.70 0.640957
$$169$$ −1805.33 −0.821726
$$170$$ 0 0
$$171$$ −439.537 −0.196563
$$172$$ −5181.36 −2.29695
$$173$$ 1239.91 0.544905 0.272452 0.962169i $$-0.412165\pi$$
0.272452 + 0.962169i $$0.412165\pi$$
$$174$$ −346.050 −0.150770
$$175$$ 0 0
$$176$$ −888.816 −0.380665
$$177$$ 336.628 0.142952
$$178$$ −455.934 −0.191987
$$179$$ −2636.86 −1.10105 −0.550525 0.834818i $$-0.685573\pi$$
−0.550525 + 0.834818i $$0.685573\pi$$
$$180$$ 0 0
$$181$$ 3317.58 1.36240 0.681199 0.732099i $$-0.261459\pi$$
0.681199 + 0.732099i $$0.261459\pi$$
$$182$$ 1508.23 0.614272
$$183$$ −1432.05 −0.578471
$$184$$ 46.2812 0.0185429
$$185$$ 0 0
$$186$$ −175.163 −0.0690513
$$187$$ 3338.01 1.30534
$$188$$ 2885.02 1.11921
$$189$$ −437.653 −0.168437
$$190$$ 0 0
$$191$$ −624.506 −0.236585 −0.118292 0.992979i $$-0.537742\pi$$
−0.118292 + 0.992979i $$0.537742\pi$$
$$192$$ 2303.27 0.865752
$$193$$ 436.144 0.162665 0.0813324 0.996687i $$-0.474082\pi$$
0.0813324 + 0.996687i $$0.474082\pi$$
$$194$$ −5417.96 −2.00509
$$195$$ 0 0
$$196$$ −1131.99 −0.412532
$$197$$ −3355.81 −1.21366 −0.606831 0.794831i $$-0.707560\pi$$
−0.606831 + 0.794831i $$0.707560\pi$$
$$198$$ 1701.42 0.610681
$$199$$ 3799.77 1.35356 0.676780 0.736185i $$-0.263375\pi$$
0.676780 + 0.736185i $$0.263375\pi$$
$$200$$ 0 0
$$201$$ 1674.28 0.587536
$$202$$ 5437.31 1.89390
$$203$$ −397.687 −0.137498
$$204$$ 3512.73 1.20559
$$205$$ 0 0
$$206$$ −6270.46 −2.12079
$$207$$ −14.5125 −0.00487288
$$208$$ −437.466 −0.145831
$$209$$ 1963.72 0.649922
$$210$$ 0 0
$$211$$ 2365.27 0.771715 0.385857 0.922558i $$-0.373906\pi$$
0.385857 + 0.922558i $$0.373906\pi$$
$$212$$ 867.834 0.281147
$$213$$ −1674.84 −0.538772
$$214$$ −3753.08 −1.19886
$$215$$ 0 0
$$216$$ 774.942 0.244112
$$217$$ −201.300 −0.0629730
$$218$$ 4631.60 1.43895
$$219$$ 3035.31 0.936562
$$220$$ 0 0
$$221$$ 1642.93 0.500070
$$222$$ −4596.50 −1.38962
$$223$$ 3328.58 0.999545 0.499772 0.866157i $$-0.333417\pi$$
0.499772 + 0.866157i $$0.333417\pi$$
$$224$$ 2037.29 0.607688
$$225$$ 0 0
$$226$$ 8877.71 2.61299
$$227$$ 527.100 0.154118 0.0770592 0.997027i $$-0.475447\pi$$
0.0770592 + 0.997027i $$0.475447\pi$$
$$228$$ 2066.51 0.600255
$$229$$ 2566.06 0.740479 0.370240 0.928936i $$-0.379276\pi$$
0.370240 + 0.928936i $$0.379276\pi$$
$$230$$ 0 0
$$231$$ 1955.31 0.556925
$$232$$ 704.175 0.199273
$$233$$ −5534.99 −1.55626 −0.778132 0.628101i $$-0.783832\pi$$
−0.778132 + 0.628101i $$0.783832\pi$$
$$234$$ 837.422 0.233949
$$235$$ 0 0
$$236$$ −1582.68 −0.436541
$$237$$ −3451.41 −0.945962
$$238$$ 6326.57 1.72307
$$239$$ 1010.01 0.273355 0.136678 0.990616i $$-0.456358\pi$$
0.136678 + 0.990616i $$0.456358\pi$$
$$240$$ 0 0
$$241$$ −4074.29 −1.08900 −0.544498 0.838762i $$-0.683280\pi$$
−0.544498 + 0.838762i $$0.683280\pi$$
$$242$$ −1343.68 −0.356921
$$243$$ −243.000 −0.0641500
$$244$$ 6732.87 1.76651
$$245$$ 0 0
$$246$$ −3419.24 −0.886190
$$247$$ 966.525 0.248982
$$248$$ 356.437 0.0912653
$$249$$ −3473.77 −0.884103
$$250$$ 0 0
$$251$$ 1773.98 0.446107 0.223054 0.974806i $$-0.428398\pi$$
0.223054 + 0.974806i $$0.428398\pi$$
$$252$$ 2057.65 0.514365
$$253$$ 64.8375 0.0161119
$$254$$ −2919.01 −0.721082
$$255$$ 0 0
$$256$$ −6101.62 −1.48965
$$257$$ −662.784 −0.160869 −0.0804345 0.996760i $$-0.525631\pi$$
−0.0804345 + 0.996760i $$0.525631\pi$$
$$258$$ −5181.36 −1.25030
$$259$$ −5282.38 −1.26730
$$260$$ 0 0
$$261$$ −220.810 −0.0523669
$$262$$ 12169.3 2.86955
$$263$$ −712.312 −0.167008 −0.0835039 0.996507i $$-0.526611\pi$$
−0.0835039 + 0.996507i $$0.526611\pi$$
$$264$$ −3462.22 −0.807139
$$265$$ 0 0
$$266$$ 3721.87 0.857905
$$267$$ −290.925 −0.0666829
$$268$$ −7871.74 −1.79419
$$269$$ −3136.41 −0.710894 −0.355447 0.934696i $$-0.615671\pi$$
−0.355447 + 0.934696i $$0.615671\pi$$
$$270$$ 0 0
$$271$$ −2275.69 −0.510105 −0.255053 0.966927i $$-0.582093\pi$$
−0.255053 + 0.966927i $$0.582093\pi$$
$$272$$ −1835.03 −0.409064
$$273$$ 962.381 0.213355
$$274$$ −7787.15 −1.71693
$$275$$ 0 0
$$276$$ 68.2313 0.0148806
$$277$$ −5171.00 −1.12164 −0.560821 0.827937i $$-0.689515\pi$$
−0.560821 + 0.827937i $$0.689515\pi$$
$$278$$ 720.544 0.155451
$$279$$ −111.769 −0.0239836
$$280$$ 0 0
$$281$$ 2240.14 0.475571 0.237785 0.971318i $$-0.423578\pi$$
0.237785 + 0.971318i $$0.423578\pi$$
$$282$$ 2885.02 0.609222
$$283$$ −225.244 −0.0473123 −0.0236561 0.999720i $$-0.507531\pi$$
−0.0236561 + 0.999720i $$0.507531\pi$$
$$284$$ 7874.38 1.64528
$$285$$ 0 0
$$286$$ −3741.36 −0.773535
$$287$$ −3929.46 −0.808183
$$288$$ 1131.17 0.231441
$$289$$ 1978.59 0.402726
$$290$$ 0 0
$$291$$ −3457.12 −0.696427
$$292$$ −14270.7 −2.86003
$$293$$ −1139.86 −0.227274 −0.113637 0.993522i $$-0.536250\pi$$
−0.113637 + 0.993522i $$0.536250\pi$$
$$294$$ −1131.99 −0.224554
$$295$$ 0 0
$$296$$ 9353.39 1.83667
$$297$$ 1085.65 0.212108
$$298$$ −6974.19 −1.35572
$$299$$ 31.9123 0.00617237
$$300$$ 0 0
$$301$$ −5954.51 −1.14024
$$302$$ 1853.74 0.353214
$$303$$ 3469.47 0.657808
$$304$$ −1079.54 −0.203670
$$305$$ 0 0
$$306$$ 3512.73 0.656240
$$307$$ −5244.86 −0.975049 −0.487525 0.873109i $$-0.662100\pi$$
−0.487525 + 0.873109i $$0.662100\pi$$
$$308$$ −9192.99 −1.70071
$$309$$ −4001.09 −0.736615
$$310$$ 0 0
$$311$$ −5188.26 −0.945977 −0.472989 0.881068i $$-0.656825\pi$$
−0.472989 + 0.881068i $$0.656825\pi$$
$$312$$ −1704.07 −0.309211
$$313$$ −486.656 −0.0878832 −0.0439416 0.999034i $$-0.513992\pi$$
−0.0439416 + 0.999034i $$0.513992\pi$$
$$314$$ 8119.85 1.45933
$$315$$ 0 0
$$316$$ 16227.0 2.88873
$$317$$ 4218.87 0.747493 0.373747 0.927531i $$-0.378073\pi$$
0.373747 + 0.927531i $$0.378073\pi$$
$$318$$ 867.834 0.153037
$$319$$ 986.512 0.173148
$$320$$ 0 0
$$321$$ −2394.79 −0.416399
$$322$$ 122.887 0.0212678
$$323$$ 4054.27 0.698408
$$324$$ 1142.48 0.195898
$$325$$ 0 0
$$326$$ −9564.30 −1.62490
$$327$$ 2955.36 0.499791
$$328$$ 6957.80 1.17128
$$329$$ 3315.53 0.555595
$$330$$ 0 0
$$331$$ 7439.94 1.23546 0.617728 0.786392i $$-0.288053\pi$$
0.617728 + 0.786392i $$0.288053\pi$$
$$332$$ 16332.2 2.69983
$$333$$ −2932.96 −0.482658
$$334$$ 906.931 0.148578
$$335$$ 0 0
$$336$$ −1074.91 −0.174527
$$337$$ 6555.39 1.05963 0.529815 0.848113i $$-0.322261\pi$$
0.529815 + 0.848113i $$0.322261\pi$$
$$338$$ 8487.88 1.36592
$$339$$ 5664.74 0.907571
$$340$$ 0 0
$$341$$ 499.350 0.0793000
$$342$$ 2066.51 0.326737
$$343$$ −6860.72 −1.08001
$$344$$ 10543.5 1.65252
$$345$$ 0 0
$$346$$ −5829.51 −0.905770
$$347$$ −1950.56 −0.301763 −0.150881 0.988552i $$-0.548211\pi$$
−0.150881 + 0.988552i $$0.548211\pi$$
$$348$$ 1038.15 0.159916
$$349$$ −1426.74 −0.218830 −0.109415 0.993996i $$-0.534898\pi$$
−0.109415 + 0.993996i $$0.534898\pi$$
$$350$$ 0 0
$$351$$ 534.347 0.0812573
$$352$$ −5053.75 −0.765244
$$353$$ 7078.96 1.06735 0.533676 0.845689i $$-0.320810\pi$$
0.533676 + 0.845689i $$0.320810\pi$$
$$354$$ −1582.68 −0.237623
$$355$$ 0 0
$$356$$ 1367.80 0.203633
$$357$$ 4036.89 0.598474
$$358$$ 12397.4 1.83023
$$359$$ −5409.79 −0.795314 −0.397657 0.917534i $$-0.630177\pi$$
−0.397657 + 0.917534i $$0.630177\pi$$
$$360$$ 0 0
$$361$$ −4473.90 −0.652267
$$362$$ −15597.8 −2.26465
$$363$$ −857.381 −0.123969
$$364$$ −4524.69 −0.651534
$$365$$ 0 0
$$366$$ 6732.87 0.961565
$$367$$ −4940.09 −0.702645 −0.351322 0.936255i $$-0.614268\pi$$
−0.351322 + 0.936255i $$0.614268\pi$$
$$368$$ −35.6437 −0.00504907
$$369$$ −2181.77 −0.307800
$$370$$ 0 0
$$371$$ 997.332 0.139566
$$372$$ 525.488 0.0732399
$$373$$ −12891.9 −1.78959 −0.894797 0.446473i $$-0.852680\pi$$
−0.894797 + 0.446473i $$0.852680\pi$$
$$374$$ −15693.8 −2.16981
$$375$$ 0 0
$$376$$ −5870.72 −0.805211
$$377$$ 485.551 0.0663320
$$378$$ 2057.65 0.279985
$$379$$ −9475.15 −1.28418 −0.642092 0.766627i $$-0.721933\pi$$
−0.642092 + 0.766627i $$0.721933\pi$$
$$380$$ 0 0
$$381$$ −1862.58 −0.250453
$$382$$ 2936.16 0.393264
$$383$$ 5800.97 0.773931 0.386966 0.922094i $$-0.373523\pi$$
0.386966 + 0.922094i $$0.373523\pi$$
$$384$$ −7812.52 −1.03823
$$385$$ 0 0
$$386$$ −2050.56 −0.270390
$$387$$ −3306.15 −0.434266
$$388$$ 16253.9 2.12672
$$389$$ 13779.7 1.79603 0.898016 0.439962i $$-0.145008\pi$$
0.898016 + 0.439962i $$0.145008\pi$$
$$390$$ 0 0
$$391$$ 133.862 0.0173138
$$392$$ 2303.48 0.296794
$$393$$ 7765.06 0.996680
$$394$$ 15777.5 2.01741
$$395$$ 0 0
$$396$$ −5104.27 −0.647725
$$397$$ 2816.46 0.356056 0.178028 0.984025i $$-0.443028\pi$$
0.178028 + 0.984025i $$0.443028\pi$$
$$398$$ −17864.8 −2.24996
$$399$$ 2374.88 0.297976
$$400$$ 0 0
$$401$$ 11986.4 1.49270 0.746352 0.665551i $$-0.231804\pi$$
0.746352 + 0.665551i $$0.231804\pi$$
$$402$$ −7871.74 −0.976633
$$403$$ 245.775 0.0303794
$$404$$ −16311.9 −2.00879
$$405$$ 0 0
$$406$$ 1869.75 0.228557
$$407$$ 13103.6 1.59588
$$408$$ −7148.03 −0.867354
$$409$$ −3339.07 −0.403683 −0.201841 0.979418i $$-0.564693\pi$$
−0.201841 + 0.979418i $$0.564693\pi$$
$$410$$ 0 0
$$411$$ −4968.87 −0.596342
$$412$$ 18811.4 2.24944
$$413$$ −1818.84 −0.216706
$$414$$ 68.2313 0.00809996
$$415$$ 0 0
$$416$$ −2487.40 −0.293161
$$417$$ 459.769 0.0539927
$$418$$ −9232.57 −1.08033
$$419$$ 1688.52 0.196873 0.0984363 0.995143i $$-0.468616\pi$$
0.0984363 + 0.995143i $$0.468616\pi$$
$$420$$ 0 0
$$421$$ −2664.27 −0.308429 −0.154214 0.988037i $$-0.549285\pi$$
−0.154214 + 0.988037i $$0.549285\pi$$
$$422$$ −11120.5 −1.28279
$$423$$ 1840.89 0.211601
$$424$$ −1765.95 −0.202269
$$425$$ 0 0
$$426$$ 7874.38 0.895575
$$427$$ 7737.54 0.876923
$$428$$ 11259.2 1.27158
$$429$$ −2387.31 −0.268672
$$430$$ 0 0
$$431$$ 12266.0 1.37084 0.685420 0.728148i $$-0.259619\pi$$
0.685420 + 0.728148i $$0.259619\pi$$
$$432$$ −596.827 −0.0664695
$$433$$ 15647.3 1.73664 0.868318 0.496008i $$-0.165201\pi$$
0.868318 + 0.496008i $$0.165201\pi$$
$$434$$ 946.425 0.104677
$$435$$ 0 0
$$436$$ −13894.8 −1.52624
$$437$$ 78.7503 0.00862045
$$438$$ −14270.7 −1.55680
$$439$$ 16131.0 1.75373 0.876867 0.480733i $$-0.159629\pi$$
0.876867 + 0.480733i $$0.159629\pi$$
$$440$$ 0 0
$$441$$ −722.306 −0.0779944
$$442$$ −7724.34 −0.831243
$$443$$ −10053.7 −1.07825 −0.539127 0.842225i $$-0.681246\pi$$
−0.539127 + 0.842225i $$0.681246\pi$$
$$444$$ 13789.5 1.47392
$$445$$ 0 0
$$446$$ −15649.5 −1.66150
$$447$$ −4450.13 −0.470882
$$448$$ −12444.9 −1.31242
$$449$$ 7477.71 0.785957 0.392979 0.919548i $$-0.371445\pi$$
0.392979 + 0.919548i $$0.371445\pi$$
$$450$$ 0 0
$$451$$ 9747.51 1.01772
$$452$$ −26633.1 −2.77150
$$453$$ 1182.84 0.122682
$$454$$ −2478.19 −0.256184
$$455$$ 0 0
$$456$$ −4205.14 −0.431850
$$457$$ −1363.46 −0.139562 −0.0697812 0.997562i $$-0.522230\pi$$
−0.0697812 + 0.997562i $$0.522230\pi$$
$$458$$ −12064.5 −1.23086
$$459$$ 2241.42 0.227932
$$460$$ 0 0
$$461$$ 5276.77 0.533109 0.266555 0.963820i $$-0.414115\pi$$
0.266555 + 0.963820i $$0.414115\pi$$
$$462$$ −9192.99 −0.925751
$$463$$ −5740.02 −0.576159 −0.288079 0.957607i $$-0.593017\pi$$
−0.288079 + 0.957607i $$0.593017\pi$$
$$464$$ −542.325 −0.0542604
$$465$$ 0 0
$$466$$ 26023.1 2.58690
$$467$$ −6233.36 −0.617657 −0.308828 0.951118i $$-0.599937\pi$$
−0.308828 + 0.951118i $$0.599937\pi$$
$$468$$ −2512.27 −0.248140
$$469$$ −9046.35 −0.890664
$$470$$ 0 0
$$471$$ 5181.16 0.506869
$$472$$ 3220.58 0.314067
$$473$$ 14770.9 1.43587
$$474$$ 16227.0 1.57243
$$475$$ 0 0
$$476$$ −18979.7 −1.82759
$$477$$ 553.753 0.0531543
$$478$$ −4748.61 −0.454385
$$479$$ −19688.2 −1.87803 −0.939013 0.343881i $$-0.888258\pi$$
−0.939013 + 0.343881i $$0.888258\pi$$
$$480$$ 0 0
$$481$$ 6449.46 0.611372
$$482$$ 19155.5 1.81019
$$483$$ 78.4127 0.00738696
$$484$$ 4031.03 0.378572
$$485$$ 0 0
$$486$$ 1142.48 0.106634
$$487$$ 3955.08 0.368012 0.184006 0.982925i $$-0.441093\pi$$
0.184006 + 0.982925i $$0.441093\pi$$
$$488$$ −13700.7 −1.27090
$$489$$ −6102.84 −0.564377
$$490$$ 0 0
$$491$$ −13893.5 −1.27699 −0.638497 0.769624i $$-0.720443\pi$$
−0.638497 + 0.769624i $$0.720443\pi$$
$$492$$ 10257.7 0.939947
$$493$$ 2036.74 0.186065
$$494$$ −4544.18 −0.413871
$$495$$ 0 0
$$496$$ −274.512 −0.0248508
$$497$$ 9049.39 0.816741
$$498$$ 16332.2 1.46960
$$499$$ 13523.7 1.21324 0.606618 0.794993i $$-0.292526\pi$$
0.606618 + 0.794993i $$0.292526\pi$$
$$500$$ 0 0
$$501$$ 578.700 0.0516056
$$502$$ −8340.50 −0.741543
$$503$$ −13135.4 −1.16437 −0.582184 0.813057i $$-0.697802\pi$$
−0.582184 + 0.813057i $$0.697802\pi$$
$$504$$ −4187.11 −0.370057
$$505$$ 0 0
$$506$$ −304.837 −0.0267820
$$507$$ 5415.99 0.474423
$$508$$ 8757.03 0.764823
$$509$$ −2222.71 −0.193556 −0.0967778 0.995306i $$-0.530854\pi$$
−0.0967778 + 0.995306i $$0.530854\pi$$
$$510$$ 0 0
$$511$$ −16400.1 −1.41976
$$512$$ 7853.76 0.677911
$$513$$ 1318.61 0.113486
$$514$$ 3116.12 0.267405
$$515$$ 0 0
$$516$$ 15544.1 1.32614
$$517$$ −8224.57 −0.699645
$$518$$ 24835.4 2.10658
$$519$$ −3719.73 −0.314601
$$520$$ 0 0
$$521$$ −4916.42 −0.413421 −0.206710 0.978402i $$-0.566276\pi$$
−0.206710 + 0.978402i $$0.566276\pi$$
$$522$$ 1038.15 0.0870471
$$523$$ −17743.4 −1.48349 −0.741746 0.670681i $$-0.766002\pi$$
−0.741746 + 0.670681i $$0.766002\pi$$
$$524$$ −36507.9 −3.04362
$$525$$ 0 0
$$526$$ 3348.98 0.277609
$$527$$ 1030.95 0.0852161
$$528$$ 2666.45 0.219777
$$529$$ −12164.4 −0.999786
$$530$$ 0 0
$$531$$ −1009.88 −0.0825334
$$532$$ −11165.6 −0.909946
$$533$$ 4797.62 0.389884
$$534$$ 1367.80 0.110844
$$535$$ 0 0
$$536$$ 16018.2 1.29082
$$537$$ 7910.58 0.635692
$$538$$ 14746.0 1.18169
$$539$$ 3227.05 0.257883
$$540$$ 0 0
$$541$$ −12671.3 −1.00699 −0.503495 0.863998i $$-0.667953\pi$$
−0.503495 + 0.863998i $$0.667953\pi$$
$$542$$ 10699.3 0.847924
$$543$$ −9952.74 −0.786580
$$544$$ −10433.9 −0.822334
$$545$$ 0 0
$$546$$ −4524.69 −0.354650
$$547$$ −5250.90 −0.410443 −0.205221 0.978716i $$-0.565791\pi$$
−0.205221 + 0.978716i $$0.565791\pi$$
$$548$$ 23361.5 1.82108
$$549$$ 4296.15 0.333980
$$550$$ 0 0
$$551$$ 1198.20 0.0926406
$$552$$ −138.844 −0.0107057
$$553$$ 18648.4 1.43401
$$554$$ 24311.8 1.86445
$$555$$ 0 0
$$556$$ −2161.63 −0.164881
$$557$$ −25830.2 −1.96492 −0.982462 0.186465i $$-0.940297\pi$$
−0.982462 + 0.186465i $$0.940297\pi$$
$$558$$ 525.488 0.0398668
$$559$$ 7270.09 0.550075
$$560$$ 0 0
$$561$$ −10014.0 −0.753640
$$562$$ −10532.1 −0.790519
$$563$$ −2021.14 −0.151298 −0.0756490 0.997135i $$-0.524103\pi$$
−0.0756490 + 0.997135i $$0.524103\pi$$
$$564$$ −8655.07 −0.646178
$$565$$ 0 0
$$566$$ 1059.00 0.0786450
$$567$$ 1312.96 0.0972471
$$568$$ −16023.5 −1.18368
$$569$$ 8706.51 0.641469 0.320734 0.947169i $$-0.396070\pi$$
0.320734 + 0.947169i $$0.396070\pi$$
$$570$$ 0 0
$$571$$ −12194.5 −0.893740 −0.446870 0.894599i $$-0.647461\pi$$
−0.446870 + 0.894599i $$0.647461\pi$$
$$572$$ 11224.1 0.820458
$$573$$ 1873.52 0.136592
$$574$$ 18474.6 1.34340
$$575$$ 0 0
$$576$$ −6909.82 −0.499842
$$577$$ 15264.0 1.10130 0.550649 0.834737i $$-0.314380\pi$$
0.550649 + 0.834737i $$0.314380\pi$$
$$578$$ −9302.48 −0.669433
$$579$$ −1308.43 −0.0939146
$$580$$ 0 0
$$581$$ 18769.2 1.34024
$$582$$ 16253.9 1.15764
$$583$$ −2474.01 −0.175751
$$584$$ 29039.3 2.05763
$$585$$ 0 0
$$586$$ 5359.12 0.377787
$$587$$ 8456.89 0.594639 0.297319 0.954778i $$-0.403907\pi$$
0.297319 + 0.954778i $$0.403907\pi$$
$$588$$ 3395.97 0.238176
$$589$$ 606.500 0.0424285
$$590$$ 0 0
$$591$$ 10067.4 0.700708
$$592$$ −7203.57 −0.500110
$$593$$ 1225.23 0.0848467 0.0424234 0.999100i $$-0.486492\pi$$
0.0424234 + 0.999100i $$0.486492\pi$$
$$594$$ −5104.27 −0.352577
$$595$$ 0 0
$$596$$ 20922.6 1.43796
$$597$$ −11399.3 −0.781478
$$598$$ −150.038 −0.0102600
$$599$$ −16060.0 −1.09548 −0.547741 0.836648i $$-0.684512\pi$$
−0.547741 + 0.836648i $$0.684512\pi$$
$$600$$ 0 0
$$601$$ 9699.93 0.658350 0.329175 0.944269i $$-0.393229\pi$$
0.329175 + 0.944269i $$0.393229\pi$$
$$602$$ 27995.5 1.89537
$$603$$ −5022.84 −0.339214
$$604$$ −5561.21 −0.374640
$$605$$ 0 0
$$606$$ −16311.9 −1.09344
$$607$$ −23661.2 −1.58217 −0.791087 0.611703i $$-0.790485\pi$$
−0.791087 + 0.611703i $$0.790485\pi$$
$$608$$ −6138.19 −0.409435
$$609$$ 1193.06 0.0793847
$$610$$ 0 0
$$611$$ −4048.05 −0.268030
$$612$$ −10538.2 −0.696047
$$613$$ 8085.63 0.532749 0.266375 0.963870i $$-0.414174\pi$$
0.266375 + 0.963870i $$0.414174\pi$$
$$614$$ 24659.0 1.62078
$$615$$ 0 0
$$616$$ 18706.8 1.22357
$$617$$ 11035.1 0.720029 0.360014 0.932947i $$-0.382772\pi$$
0.360014 + 0.932947i $$0.382772\pi$$
$$618$$ 18811.4 1.22444
$$619$$ −16826.3 −1.09258 −0.546290 0.837596i $$-0.683960\pi$$
−0.546290 + 0.837596i $$0.683960\pi$$
$$620$$ 0 0
$$621$$ 43.5374 0.00281336
$$622$$ 24392.9 1.57245
$$623$$ 1571.90 0.101087
$$624$$ 1312.40 0.0841954
$$625$$ 0 0
$$626$$ 2288.04 0.146084
$$627$$ −5891.17 −0.375233
$$628$$ −24359.5 −1.54785
$$629$$ 27053.5 1.71493
$$630$$ 0 0
$$631$$ −3705.91 −0.233803 −0.116902 0.993144i $$-0.537296\pi$$
−0.116902 + 0.993144i $$0.537296\pi$$
$$632$$ −33020.2 −2.07828
$$633$$ −7095.81 −0.445550
$$634$$ −19835.3 −1.24252
$$635$$ 0 0
$$636$$ −2603.50 −0.162320
$$637$$ 1588.32 0.0987937
$$638$$ −4638.15 −0.287815
$$639$$ 5024.53 0.311060
$$640$$ 0 0
$$641$$ −24597.4 −1.51566 −0.757829 0.652453i $$-0.773740\pi$$
−0.757829 + 0.652453i $$0.773740\pi$$
$$642$$ 11259.2 0.692160
$$643$$ 21479.5 1.31737 0.658685 0.752419i $$-0.271113\pi$$
0.658685 + 0.752419i $$0.271113\pi$$
$$644$$ −368.662 −0.0225580
$$645$$ 0 0
$$646$$ −19061.4 −1.16093
$$647$$ −27119.7 −1.64789 −0.823946 0.566668i $$-0.808232\pi$$
−0.823946 + 0.566668i $$0.808232\pi$$
$$648$$ −2324.83 −0.140938
$$649$$ 4511.87 0.272891
$$650$$ 0 0
$$651$$ 603.900 0.0363575
$$652$$ 28692.9 1.72347
$$653$$ −18476.4 −1.10725 −0.553627 0.832765i $$-0.686757\pi$$
−0.553627 + 0.832765i $$0.686757\pi$$
$$654$$ −13894.8 −0.830779
$$655$$ 0 0
$$656$$ −5358.59 −0.318930
$$657$$ −9105.92 −0.540724
$$658$$ −15588.1 −0.923540
$$659$$ −19273.5 −1.13928 −0.569641 0.821894i $$-0.692918\pi$$
−0.569641 + 0.821894i $$0.692918\pi$$
$$660$$ 0 0
$$661$$ 25605.3 1.50670 0.753352 0.657618i $$-0.228436\pi$$
0.753352 + 0.657618i $$0.228436\pi$$
$$662$$ −34979.3 −2.05364
$$663$$ −4928.79 −0.288716
$$664$$ −33234.3 −1.94238
$$665$$ 0 0
$$666$$ 13789.5 0.802300
$$667$$ 39.5616 0.00229660
$$668$$ −2720.79 −0.157591
$$669$$ −9985.75 −0.577087
$$670$$ 0 0
$$671$$ −19193.9 −1.10428
$$672$$ −6111.87 −0.350849
$$673$$ −7855.52 −0.449938 −0.224969 0.974366i $$-0.572228\pi$$
−0.224969 + 0.974366i $$0.572228\pi$$
$$674$$ −30820.6 −1.76137
$$675$$ 0 0
$$676$$ −25463.6 −1.44877
$$677$$ 6763.09 0.383939 0.191970 0.981401i $$-0.438512\pi$$
0.191970 + 0.981401i $$0.438512\pi$$
$$678$$ −26633.1 −1.50861
$$679$$ 18679.3 1.05574
$$680$$ 0 0
$$681$$ −1581.30 −0.0889803
$$682$$ −2347.72 −0.131817
$$683$$ −15608.6 −0.874447 −0.437224 0.899353i $$-0.644038\pi$$
−0.437224 + 0.899353i $$0.644038\pi$$
$$684$$ −6199.54 −0.346557
$$685$$ 0 0
$$686$$ 32256.1 1.79525
$$687$$ −7698.17 −0.427516
$$688$$ −8120.16 −0.449968
$$689$$ −1217.68 −0.0673293
$$690$$ 0 0
$$691$$ 6203.15 0.341504 0.170752 0.985314i $$-0.445380\pi$$
0.170752 + 0.985314i $$0.445380\pi$$
$$692$$ 17488.5 0.960714
$$693$$ −5865.92 −0.321541
$$694$$ 9170.69 0.501606
$$695$$ 0 0
$$696$$ −2112.53 −0.115050
$$697$$ 20124.5 1.09365
$$698$$ 6707.90 0.363750
$$699$$ 16605.0 0.898509
$$700$$ 0 0
$$701$$ −16507.9 −0.889435 −0.444718 0.895671i $$-0.646696\pi$$
−0.444718 + 0.895671i $$0.646696\pi$$
$$702$$ −2512.27 −0.135070
$$703$$ 15915.4 0.853854
$$704$$ 30871.1 1.65269
$$705$$ 0 0
$$706$$ −33282.2 −1.77421
$$707$$ −18746.0 −0.997193
$$708$$ 4748.03 0.252037
$$709$$ −25539.6 −1.35283 −0.676416 0.736520i $$-0.736468\pi$$
−0.676416 + 0.736520i $$0.736468\pi$$
$$710$$ 0 0
$$711$$ 10354.2 0.546151
$$712$$ −2783.34 −0.146503
$$713$$ 20.0252 0.00105182
$$714$$ −18979.7 −0.994815
$$715$$ 0 0
$$716$$ −37192.1 −1.94125
$$717$$ −3030.02 −0.157822
$$718$$ 25434.4 1.32201
$$719$$ −7353.45 −0.381415 −0.190708 0.981647i $$-0.561078\pi$$
−0.190708 + 0.981647i $$0.561078\pi$$
$$720$$ 0 0
$$721$$ 21618.4 1.11666
$$722$$ 21034.3 1.08423
$$723$$ 12222.9 0.628732
$$724$$ 46793.4 2.40202
$$725$$ 0 0
$$726$$ 4031.03 0.206068
$$727$$ 21696.5 1.10685 0.553424 0.832900i $$-0.313321\pi$$
0.553424 + 0.832900i $$0.313321\pi$$
$$728$$ 9207.28 0.468742
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 30495.8 1.54299
$$732$$ −20198.6 −1.01989
$$733$$ −90.2714 −0.00454877 −0.00227439 0.999997i $$-0.500724\pi$$
−0.00227439 + 0.999997i $$0.500724\pi$$
$$734$$ 23226.1 1.16797
$$735$$ 0 0
$$736$$ −202.668 −0.0101501
$$737$$ 22440.6 1.12159
$$738$$ 10257.7 0.511642
$$739$$ 14273.1 0.710479 0.355239 0.934775i $$-0.384399\pi$$
0.355239 + 0.934775i $$0.384399\pi$$
$$740$$ 0 0
$$741$$ −2899.57 −0.143750
$$742$$ −4689.02 −0.231994
$$743$$ −15866.6 −0.783429 −0.391715 0.920087i $$-0.628118\pi$$
−0.391715 + 0.920087i $$0.628118\pi$$
$$744$$ −1069.31 −0.0526921
$$745$$ 0 0
$$746$$ 60612.2 2.97476
$$747$$ 10421.3 0.510437
$$748$$ 47081.5 2.30143
$$749$$ 12939.3 0.631232
$$750$$ 0 0
$$751$$ 26776.9 1.30107 0.650534 0.759477i $$-0.274545\pi$$
0.650534 + 0.759477i $$0.274545\pi$$
$$752$$ 4521.37 0.219252
$$753$$ −5321.95 −0.257560
$$754$$ −2282.85 −0.110261
$$755$$ 0 0
$$756$$ −6172.96 −0.296969
$$757$$ −30478.0 −1.46333 −0.731666 0.681663i $$-0.761257\pi$$
−0.731666 + 0.681663i $$0.761257\pi$$
$$758$$ 44548.0 2.13464
$$759$$ −194.512 −0.00930218
$$760$$ 0 0
$$761$$ 29104.7 1.38639 0.693195 0.720750i $$-0.256202\pi$$
0.693195 + 0.720750i $$0.256202\pi$$
$$762$$ 8757.03 0.416317
$$763$$ −15968.2 −0.757649
$$764$$ −8808.47 −0.417119
$$765$$ 0 0
$$766$$ −27273.6 −1.28647
$$767$$ 2220.69 0.104543
$$768$$ 18304.9 0.860052
$$769$$ 4170.65 0.195575 0.0977876 0.995207i $$-0.468823\pi$$
0.0977876 + 0.995207i $$0.468823\pi$$
$$770$$ 0 0
$$771$$ 1988.35 0.0928778
$$772$$ 6151.67 0.286792
$$773$$ 17738.5 0.825367 0.412684 0.910874i $$-0.364591\pi$$
0.412684 + 0.910874i $$0.364591\pi$$
$$774$$ 15544.1 0.721860
$$775$$ 0 0
$$776$$ −33075.0 −1.53005
$$777$$ 15847.1 0.731677
$$778$$ −64786.0 −2.98546
$$779$$ 11839.1 0.544519
$$780$$ 0 0
$$781$$ −22448.1 −1.02850
$$782$$ −629.363 −0.0287800
$$783$$ 662.429 0.0302341
$$784$$ −1774.04 −0.0808145
$$785$$ 0 0
$$786$$ −36507.9 −1.65674
$$787$$ 3807.92 0.172475 0.0862374 0.996275i $$-0.472516\pi$$
0.0862374 + 0.996275i $$0.472516\pi$$
$$788$$ −47332.6 −2.13979
$$789$$ 2136.94 0.0964220
$$790$$ 0 0
$$791$$ −30607.3 −1.37582
$$792$$ 10386.6 0.466002
$$793$$ −9447.06 −0.423045
$$794$$ −13241.8 −0.591856
$$795$$ 0 0
$$796$$ 53594.5 2.38644
$$797$$ −23840.3 −1.05956 −0.529779 0.848136i $$-0.677725\pi$$
−0.529779 + 0.848136i $$0.677725\pi$$
$$798$$ −11165.6 −0.495312
$$799$$ −16980.3 −0.751841
$$800$$ 0 0
$$801$$ 872.775 0.0384994
$$802$$ −56355.0 −2.48125
$$803$$ 40682.6 1.78787
$$804$$ 23615.2 1.03588
$$805$$ 0 0
$$806$$ −1155.53 −0.0504983
$$807$$ 9409.24 0.410435
$$808$$ 33193.1 1.44521
$$809$$ −1984.22 −0.0862316 −0.0431158 0.999070i $$-0.513728\pi$$
−0.0431158 + 0.999070i $$0.513728\pi$$
$$810$$ 0 0
$$811$$ −9713.78 −0.420588 −0.210294 0.977638i $$-0.567442\pi$$
−0.210294 + 0.977638i $$0.567442\pi$$
$$812$$ −5609.25 −0.242421
$$813$$ 6827.08 0.294509
$$814$$ −61607.4 −2.65275
$$815$$ 0 0
$$816$$ 5505.10 0.236173
$$817$$ 17940.5 0.768246
$$818$$ 15698.8 0.671023
$$819$$ −2887.14 −0.123181
$$820$$ 0 0
$$821$$ −19235.4 −0.817686 −0.408843 0.912605i $$-0.634068\pi$$
−0.408843 + 0.912605i $$0.634068\pi$$
$$822$$ 23361.5 0.991271
$$823$$ 12717.6 0.538650 0.269325 0.963049i $$-0.413199\pi$$
0.269325 + 0.963049i $$0.413199\pi$$
$$824$$ −38279.2 −1.61835
$$825$$ 0 0
$$826$$ 8551.41 0.360220
$$827$$ 6744.75 0.283601 0.141800 0.989895i $$-0.454711\pi$$
0.141800 + 0.989895i $$0.454711\pi$$
$$828$$ −204.694 −0.00859131
$$829$$ 3404.22 0.142622 0.0713108 0.997454i $$-0.477282\pi$$
0.0713108 + 0.997454i $$0.477282\pi$$
$$830$$ 0 0
$$831$$ 15513.0 0.647581
$$832$$ 15194.4 0.633139
$$833$$ 6662.52 0.277122
$$834$$ −2161.63 −0.0897496
$$835$$ 0 0
$$836$$ 27697.7 1.14587
$$837$$ 335.306 0.0138469
$$838$$ −7938.69 −0.327252
$$839$$ −21361.9 −0.879015 −0.439508 0.898239i $$-0.644847\pi$$
−0.439508 + 0.898239i $$0.644847\pi$$
$$840$$ 0 0
$$841$$ −23787.1 −0.975319
$$842$$ 12526.2 0.512687
$$843$$ −6720.41 −0.274571
$$844$$ 33361.4 1.36060
$$845$$ 0 0
$$846$$ −8655.07 −0.351735
$$847$$ 4632.54 0.187929
$$848$$ 1360.06 0.0550762
$$849$$ 675.732 0.0273158
$$850$$ 0 0
$$851$$ 525.488 0.0211674
$$852$$ −23623.1 −0.949901
$$853$$ 10728.9 0.430657 0.215328 0.976542i $$-0.430918\pi$$
0.215328 + 0.976542i $$0.430918\pi$$
$$854$$ −36378.5 −1.45767
$$855$$ 0 0
$$856$$ −22911.4 −0.914830
$$857$$ 42895.2 1.70977 0.854885 0.518817i $$-0.173627\pi$$
0.854885 + 0.518817i $$0.173627\pi$$
$$858$$ 11224.1 0.446601
$$859$$ 35530.5 1.41127 0.705637 0.708574i $$-0.250661\pi$$
0.705637 + 0.708574i $$0.250661\pi$$
$$860$$ 0 0
$$861$$ 11788.4 0.466605
$$862$$ −57669.3 −2.27868
$$863$$ −5704.35 −0.225004 −0.112502 0.993652i $$-0.535886\pi$$
−0.112502 + 0.993652i $$0.535886\pi$$
$$864$$ −3393.52 −0.133623
$$865$$ 0 0
$$866$$ −73567.0 −2.88673
$$867$$ −5935.78 −0.232514
$$868$$ −2839.27 −0.111027
$$869$$ −46259.6 −1.80581
$$870$$ 0 0
$$871$$ 11045.0 0.429674
$$872$$ 28274.4 1.09804
$$873$$ 10371.4 0.402082
$$874$$ −370.249 −0.0143294
$$875$$ 0 0
$$876$$ 42812.0 1.65124
$$877$$ −50249.0 −1.93476 −0.967382 0.253324i $$-0.918476\pi$$
−0.967382 + 0.253324i $$0.918476\pi$$
$$878$$ −75840.7 −2.91515
$$879$$ 3419.58 0.131217
$$880$$ 0 0
$$881$$ −26864.5 −1.02734 −0.513672 0.857987i $$-0.671715\pi$$
−0.513672 + 0.857987i $$0.671715\pi$$
$$882$$ 3395.97 0.129646
$$883$$ −18942.1 −0.721918 −0.360959 0.932582i $$-0.617551\pi$$
−0.360959 + 0.932582i $$0.617551\pi$$
$$884$$ 23173.0 0.881667
$$885$$ 0 0
$$886$$ 47268.2 1.79233
$$887$$ 25344.8 0.959409 0.479705 0.877430i $$-0.340744\pi$$
0.479705 + 0.877430i $$0.340744\pi$$
$$888$$ −28060.2 −1.06040
$$889$$ 10063.7 0.379670
$$890$$ 0 0
$$891$$ −3256.96 −0.122460
$$892$$ 46948.6 1.76228
$$893$$ −9989.40 −0.374337
$$894$$ 20922.6 0.782725
$$895$$ 0 0
$$896$$ 42212.0 1.57389
$$897$$ −95.7370 −0.00356362
$$898$$ −35156.9 −1.30646
$$899$$ 304.686 0.0113035
$$900$$ 0 0
$$901$$ −5107.79 −0.188863
$$902$$ −45828.5 −1.69171
$$903$$ 17863.5 0.658318
$$904$$ 54195.6 1.99394
$$905$$ 0 0
$$906$$ −5561.21 −0.203928
$$907$$ −4800.11 −0.175728 −0.0878639 0.996132i $$-0.528004\pi$$
−0.0878639 + 0.996132i $$0.528004\pi$$
$$908$$ 7434.58 0.271724
$$909$$ −10408.4 −0.379786
$$910$$ 0 0
$$911$$ −25731.7 −0.935819 −0.467909 0.883776i $$-0.654993\pi$$
−0.467909 + 0.883776i $$0.654993\pi$$
$$912$$ 3238.61 0.117589
$$913$$ −46559.4 −1.68772
$$914$$ 6410.40 0.231988
$$915$$ 0 0
$$916$$ 36193.4 1.30553
$$917$$ −41955.6 −1.51090
$$918$$ −10538.2 −0.378880
$$919$$ −12751.9 −0.457722 −0.228861 0.973459i $$-0.573500\pi$$
−0.228861 + 0.973459i $$0.573500\pi$$
$$920$$ 0 0
$$921$$ 15734.6 0.562945
$$922$$ −24809.0 −0.886163
$$923$$ −11048.7 −0.394012
$$924$$ 27579.0 0.981907
$$925$$ 0 0
$$926$$ 26987.1 0.957722
$$927$$ 12003.3 0.425285
$$928$$ −3083.63 −0.109079
$$929$$ 15557.8 0.549444 0.274722 0.961524i $$-0.411414\pi$$
0.274722 + 0.961524i $$0.411414\pi$$
$$930$$ 0 0
$$931$$ 3919.51 0.137977
$$932$$ −78069.3 −2.74383
$$933$$ 15564.8 0.546160
$$934$$ 29306.5 1.02670
$$935$$ 0 0
$$936$$ 5112.20 0.178523
$$937$$ 23858.0 0.831811 0.415905 0.909408i $$-0.363465\pi$$
0.415905 + 0.909408i $$0.363465\pi$$
$$938$$ 42532.0 1.48051
$$939$$ 1459.97 0.0507394
$$940$$ 0 0
$$941$$ 9748.00 0.337700 0.168850 0.985642i $$-0.445995\pi$$
0.168850 + 0.985642i $$0.445995\pi$$
$$942$$ −24359.5 −0.842544
$$943$$ 390.899 0.0134989
$$944$$ −2480.35 −0.0855176
$$945$$ 0 0
$$946$$ −69446.4 −2.38678
$$947$$ 51537.0 1.76845 0.884227 0.467057i $$-0.154686\pi$$
0.884227 + 0.467057i $$0.154686\pi$$
$$948$$ −48681.0 −1.66781
$$949$$ 20023.5 0.684923
$$950$$ 0 0
$$951$$ −12656.6 −0.431566
$$952$$ 38621.7 1.31485
$$953$$ 5631.36 0.191414 0.0957071 0.995410i $$-0.469489\pi$$
0.0957071 + 0.995410i $$0.469489\pi$$
$$954$$ −2603.50 −0.0883559
$$955$$ 0 0
$$956$$ 14245.8 0.481948
$$957$$ −2959.54 −0.0999668
$$958$$ 92565.1 3.12176
$$959$$ 26847.4 0.904013
$$960$$ 0 0
$$961$$ −29636.8 −0.994823
$$962$$ −30322.5 −1.01625
$$963$$ 7184.36 0.240408
$$964$$ −57466.5 −1.91999
$$965$$ 0 0
$$966$$ −368.662 −0.0122790
$$967$$ 43360.9 1.44198 0.720989 0.692946i $$-0.243688\pi$$
0.720989 + 0.692946i $$0.243688\pi$$
$$968$$ −8202.72 −0.272361
$$969$$ −12162.8 −0.403226
$$970$$ 0 0
$$971$$ 12920.0 0.427007 0.213503 0.976942i $$-0.431513\pi$$
0.213503 + 0.976942i $$0.431513\pi$$
$$972$$ −3427.44 −0.113102
$$973$$ −2484.19 −0.0818493
$$974$$ −18595.0 −0.611728
$$975$$ 0 0
$$976$$ 10551.7 0.346056
$$977$$ 10650.4 0.348759 0.174379 0.984679i $$-0.444208\pi$$
0.174379 + 0.984679i $$0.444208\pi$$
$$978$$ 28692.9 0.938137
$$979$$ −3899.31 −0.127296
$$980$$ 0 0
$$981$$ −8866.07 −0.288554
$$982$$ 65321.0 2.12269
$$983$$ 49450.3 1.60450 0.802248 0.596991i $$-0.203637\pi$$
0.802248 + 0.596991i $$0.203637\pi$$
$$984$$ −20873.4 −0.676239
$$985$$ 0 0
$$986$$ −9575.85 −0.309287
$$987$$ −9946.58 −0.320773
$$988$$ 13632.5 0.438976
$$989$$ 592.351 0.0190452
$$990$$ 0 0
$$991$$ 9410.47 0.301648 0.150824 0.988561i $$-0.451807\pi$$
0.150824 + 0.988561i $$0.451807\pi$$
$$992$$ −1560.86 −0.0499571
$$993$$ −22319.8 −0.713291
$$994$$ −42546.3 −1.35763
$$995$$ 0 0
$$996$$ −48996.5 −1.55875
$$997$$ −532.117 −0.0169030 −0.00845151 0.999964i $$-0.502690\pi$$
−0.00845151 + 0.999964i $$0.502690\pi$$
$$998$$ −63582.6 −2.01671
$$999$$ 8798.88 0.278663
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 75.4.a.c.1.1 2
3.2 odd 2 225.4.a.o.1.2 2
4.3 odd 2 1200.4.a.bt.1.1 2
5.2 odd 4 15.4.b.a.4.1 4
5.3 odd 4 15.4.b.a.4.4 yes 4
5.4 even 2 75.4.a.f.1.2 2
15.2 even 4 45.4.b.b.19.4 4
15.8 even 4 45.4.b.b.19.1 4
15.14 odd 2 225.4.a.i.1.1 2
20.3 even 4 240.4.f.f.49.4 4
20.7 even 4 240.4.f.f.49.2 4
20.19 odd 2 1200.4.a.bn.1.2 2
40.3 even 4 960.4.f.p.769.1 4
40.13 odd 4 960.4.f.q.769.3 4
40.27 even 4 960.4.f.p.769.3 4
40.37 odd 4 960.4.f.q.769.1 4
60.23 odd 4 720.4.f.j.289.1 4
60.47 odd 4 720.4.f.j.289.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.b.a.4.1 4 5.2 odd 4
15.4.b.a.4.4 yes 4 5.3 odd 4
45.4.b.b.19.1 4 15.8 even 4
45.4.b.b.19.4 4 15.2 even 4
75.4.a.c.1.1 2 1.1 even 1 trivial
75.4.a.f.1.2 2 5.4 even 2
225.4.a.i.1.1 2 15.14 odd 2
225.4.a.o.1.2 2 3.2 odd 2
240.4.f.f.49.2 4 20.7 even 4
240.4.f.f.49.4 4 20.3 even 4
720.4.f.j.289.1 4 60.23 odd 4
720.4.f.j.289.2 4 60.47 odd 4
960.4.f.p.769.1 4 40.3 even 4
960.4.f.p.769.3 4 40.27 even 4
960.4.f.q.769.1 4 40.37 odd 4
960.4.f.q.769.3 4 40.13 odd 4
1200.4.a.bn.1.2 2 20.19 odd 2
1200.4.a.bt.1.1 2 4.3 odd 2