# Properties

 Label 1840.4.a.h.1.1 Level $1840$ Weight $4$ Character 1840.1 Self dual yes Analytic conductor $108.564$ 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] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$108.563514411$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{109})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 27$$ x^2 - x - 27 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 115) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-4.72015$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.72015 q^{3} +5.00000 q^{5} +25.6008 q^{7} -13.1605 q^{9} +O(q^{10})$$ $$q-3.72015 q^{3} +5.00000 q^{5} +25.6008 q^{7} -13.1605 q^{9} -12.6008 q^{11} +8.16046 q^{13} -18.6008 q^{15} -76.0411 q^{17} +103.362 q^{19} -95.2388 q^{21} +23.0000 q^{23} +25.0000 q^{25} +149.403 q^{27} -267.202 q^{29} +63.7574 q^{31} +46.8768 q^{33} +128.004 q^{35} +112.164 q^{37} -30.3582 q^{39} -239.078 q^{41} -282.239 q^{43} -65.8023 q^{45} -577.291 q^{47} +312.399 q^{49} +282.884 q^{51} -2.31326 q^{53} -63.0038 q^{55} -384.522 q^{57} -272.888 q^{59} +294.049 q^{61} -336.918 q^{63} +40.8023 q^{65} -426.732 q^{67} -85.5635 q^{69} +1020.85 q^{71} -286.650 q^{73} -93.0038 q^{75} -322.589 q^{77} +551.866 q^{79} -200.470 q^{81} -21.7021 q^{83} -380.205 q^{85} +994.031 q^{87} -1049.63 q^{89} +208.914 q^{91} -237.187 q^{93} +516.810 q^{95} +1729.19 q^{97} +165.832 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 10 q^{5} - q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 10 * q^5 - q^7 + 5 * q^9 $$2 q + 3 q^{3} + 10 q^{5} - q^{7} + 5 q^{9} + 27 q^{11} - 15 q^{13} + 15 q^{15} - 79 q^{17} + 71 q^{19} - 274 q^{21} + 46 q^{23} + 50 q^{25} + 90 q^{27} - 430 q^{29} + 305 q^{31} + 313 q^{33} - 5 q^{35} - 68 q^{37} - 186 q^{39} - 593 q^{41} - 648 q^{43} + 25 q^{45} - 382 q^{47} + 677 q^{49} + 263 q^{51} - 464 q^{53} + 135 q^{55} - 602 q^{57} + 18 q^{59} - 7 q^{61} - 820 q^{63} - 75 q^{65} - 60 q^{67} + 69 q^{69} + 1029 q^{71} + 74 q^{73} + 75 q^{75} - 1376 q^{77} - 692 q^{79} - 1090 q^{81} + 1460 q^{83} - 395 q^{85} - 100 q^{87} - 220 q^{89} + 825 q^{91} + 1384 q^{93} + 355 q^{95} + 1339 q^{97} + 885 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + 10 * q^5 - q^7 + 5 * q^9 + 27 * q^11 - 15 * q^13 + 15 * q^15 - 79 * q^17 + 71 * q^19 - 274 * q^21 + 46 * q^23 + 50 * q^25 + 90 * q^27 - 430 * q^29 + 305 * q^31 + 313 * q^33 - 5 * q^35 - 68 * q^37 - 186 * q^39 - 593 * q^41 - 648 * q^43 + 25 * q^45 - 382 * q^47 + 677 * q^49 + 263 * q^51 - 464 * q^53 + 135 * q^55 - 602 * q^57 + 18 * q^59 - 7 * q^61 - 820 * q^63 - 75 * q^65 - 60 * q^67 + 69 * q^69 + 1029 * q^71 + 74 * q^73 + 75 * q^75 - 1376 * q^77 - 692 * q^79 - 1090 * q^81 + 1460 * q^83 - 395 * q^85 - 100 * q^87 - 220 * q^89 + 825 * q^91 + 1384 * q^93 + 355 * q^95 + 1339 * q^97 + 885 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −3.72015 −0.715944 −0.357972 0.933732i $$-0.616532\pi$$
−0.357972 + 0.933732i $$0.616532\pi$$
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ 25.6008 1.38231 0.691156 0.722706i $$-0.257102\pi$$
0.691156 + 0.722706i $$0.257102\pi$$
$$8$$ 0 0
$$9$$ −13.1605 −0.487424
$$10$$ 0 0
$$11$$ −12.6008 −0.345389 −0.172694 0.984975i $$-0.555247\pi$$
−0.172694 + 0.984975i $$0.555247\pi$$
$$12$$ 0 0
$$13$$ 8.16046 0.174100 0.0870502 0.996204i $$-0.472256\pi$$
0.0870502 + 0.996204i $$0.472256\pi$$
$$14$$ 0 0
$$15$$ −18.6008 −0.320180
$$16$$ 0 0
$$17$$ −76.0411 −1.08486 −0.542431 0.840100i $$-0.682496\pi$$
−0.542431 + 0.840100i $$0.682496\pi$$
$$18$$ 0 0
$$19$$ 103.362 1.24805 0.624023 0.781406i $$-0.285497\pi$$
0.624023 + 0.781406i $$0.285497\pi$$
$$20$$ 0 0
$$21$$ −95.2388 −0.989657
$$22$$ 0 0
$$23$$ 23.0000 0.208514
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 149.403 1.06491
$$28$$ 0 0
$$29$$ −267.202 −1.71097 −0.855484 0.517829i $$-0.826740\pi$$
−0.855484 + 0.517829i $$0.826740\pi$$
$$30$$ 0 0
$$31$$ 63.7574 0.369392 0.184696 0.982796i $$-0.440870\pi$$
0.184696 + 0.982796i $$0.440870\pi$$
$$32$$ 0 0
$$33$$ 46.8768 0.247279
$$34$$ 0 0
$$35$$ 128.004 0.618188
$$36$$ 0 0
$$37$$ 112.164 0.498370 0.249185 0.968456i $$-0.419837\pi$$
0.249185 + 0.968456i $$0.419837\pi$$
$$38$$ 0 0
$$39$$ −30.3582 −0.124646
$$40$$ 0 0
$$41$$ −239.078 −0.910677 −0.455339 0.890318i $$-0.650482\pi$$
−0.455339 + 0.890318i $$0.650482\pi$$
$$42$$ 0 0
$$43$$ −282.239 −1.00095 −0.500477 0.865750i $$-0.666842\pi$$
−0.500477 + 0.865750i $$0.666842\pi$$
$$44$$ 0 0
$$45$$ −65.8023 −0.217983
$$46$$ 0 0
$$47$$ −577.291 −1.79163 −0.895815 0.444427i $$-0.853407\pi$$
−0.895815 + 0.444427i $$0.853407\pi$$
$$48$$ 0 0
$$49$$ 312.399 0.910785
$$50$$ 0 0
$$51$$ 282.884 0.776701
$$52$$ 0 0
$$53$$ −2.31326 −0.00599529 −0.00299764 0.999996i $$-0.500954\pi$$
−0.00299764 + 0.999996i $$0.500954\pi$$
$$54$$ 0 0
$$55$$ −63.0038 −0.154462
$$56$$ 0 0
$$57$$ −384.522 −0.893531
$$58$$ 0 0
$$59$$ −272.888 −0.602153 −0.301077 0.953600i $$-0.597346\pi$$
−0.301077 + 0.953600i $$0.597346\pi$$
$$60$$ 0 0
$$61$$ 294.049 0.617198 0.308599 0.951192i $$-0.400140\pi$$
0.308599 + 0.951192i $$0.400140\pi$$
$$62$$ 0 0
$$63$$ −336.918 −0.673772
$$64$$ 0 0
$$65$$ 40.8023 0.0778600
$$66$$ 0 0
$$67$$ −426.732 −0.778113 −0.389056 0.921214i $$-0.627199\pi$$
−0.389056 + 0.921214i $$0.627199\pi$$
$$68$$ 0 0
$$69$$ −85.5635 −0.149285
$$70$$ 0 0
$$71$$ 1020.85 1.70638 0.853191 0.521598i $$-0.174664\pi$$
0.853191 + 0.521598i $$0.174664\pi$$
$$72$$ 0 0
$$73$$ −286.650 −0.459586 −0.229793 0.973240i $$-0.573805\pi$$
−0.229793 + 0.973240i $$0.573805\pi$$
$$74$$ 0 0
$$75$$ −93.0038 −0.143189
$$76$$ 0 0
$$77$$ −322.589 −0.477435
$$78$$ 0 0
$$79$$ 551.866 0.785947 0.392974 0.919550i $$-0.371446\pi$$
0.392974 + 0.919550i $$0.371446\pi$$
$$80$$ 0 0
$$81$$ −200.470 −0.274993
$$82$$ 0 0
$$83$$ −21.7021 −0.0287001 −0.0143501 0.999897i $$-0.504568\pi$$
−0.0143501 + 0.999897i $$0.504568\pi$$
$$84$$ 0 0
$$85$$ −380.205 −0.485165
$$86$$ 0 0
$$87$$ 994.031 1.22496
$$88$$ 0 0
$$89$$ −1049.63 −1.25012 −0.625058 0.780578i $$-0.714925\pi$$
−0.625058 + 0.780578i $$0.714925\pi$$
$$90$$ 0 0
$$91$$ 208.914 0.240661
$$92$$ 0 0
$$93$$ −237.187 −0.264464
$$94$$ 0 0
$$95$$ 516.810 0.558143
$$96$$ 0 0
$$97$$ 1729.19 1.81003 0.905014 0.425381i $$-0.139860\pi$$
0.905014 + 0.425381i $$0.139860\pi$$
$$98$$ 0 0
$$99$$ 165.832 0.168351
$$100$$ 0 0
$$101$$ 855.874 0.843195 0.421597 0.906783i $$-0.361470\pi$$
0.421597 + 0.906783i $$0.361470\pi$$
$$102$$ 0 0
$$103$$ 632.534 0.605101 0.302551 0.953133i $$-0.402162\pi$$
0.302551 + 0.953133i $$0.402162\pi$$
$$104$$ 0 0
$$105$$ −476.194 −0.442588
$$106$$ 0 0
$$107$$ −1309.62 −1.18323 −0.591616 0.806220i $$-0.701510\pi$$
−0.591616 + 0.806220i $$0.701510\pi$$
$$108$$ 0 0
$$109$$ 1726.21 1.51689 0.758443 0.651739i $$-0.225960\pi$$
0.758443 + 0.651739i $$0.225960\pi$$
$$110$$ 0 0
$$111$$ −417.268 −0.356805
$$112$$ 0 0
$$113$$ −1492.24 −1.24228 −0.621142 0.783698i $$-0.713331\pi$$
−0.621142 + 0.783698i $$0.713331\pi$$
$$114$$ 0 0
$$115$$ 115.000 0.0932505
$$116$$ 0 0
$$117$$ −107.395 −0.0848608
$$118$$ 0 0
$$119$$ −1946.71 −1.49962
$$120$$ 0 0
$$121$$ −1172.22 −0.880707
$$122$$ 0 0
$$123$$ 889.408 0.651994
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 1368.29 0.956034 0.478017 0.878351i $$-0.341356\pi$$
0.478017 + 0.878351i $$0.341356\pi$$
$$128$$ 0 0
$$129$$ 1049.97 0.716627
$$130$$ 0 0
$$131$$ −1984.24 −1.32339 −0.661694 0.749774i $$-0.730162\pi$$
−0.661694 + 0.749774i $$0.730162\pi$$
$$132$$ 0 0
$$133$$ 2646.15 1.72519
$$134$$ 0 0
$$135$$ 747.015 0.476243
$$136$$ 0 0
$$137$$ 273.331 0.170455 0.0852273 0.996362i $$-0.472838\pi$$
0.0852273 + 0.996362i $$0.472838\pi$$
$$138$$ 0 0
$$139$$ 557.889 0.340428 0.170214 0.985407i $$-0.445554\pi$$
0.170214 + 0.985407i $$0.445554\pi$$
$$140$$ 0 0
$$141$$ 2147.61 1.28271
$$142$$ 0 0
$$143$$ −102.828 −0.0601323
$$144$$ 0 0
$$145$$ −1336.01 −0.765168
$$146$$ 0 0
$$147$$ −1162.17 −0.652071
$$148$$ 0 0
$$149$$ −1354.16 −0.744545 −0.372272 0.928124i $$-0.621421\pi$$
−0.372272 + 0.928124i $$0.621421\pi$$
$$150$$ 0 0
$$151$$ −162.652 −0.0876586 −0.0438293 0.999039i $$-0.513956\pi$$
−0.0438293 + 0.999039i $$0.513956\pi$$
$$152$$ 0 0
$$153$$ 1000.74 0.528789
$$154$$ 0 0
$$155$$ 318.787 0.165197
$$156$$ 0 0
$$157$$ −2364.83 −1.20213 −0.601063 0.799201i $$-0.705256\pi$$
−0.601063 + 0.799201i $$0.705256\pi$$
$$158$$ 0 0
$$159$$ 8.60567 0.00429229
$$160$$ 0 0
$$161$$ 588.818 0.288232
$$162$$ 0 0
$$163$$ −471.162 −0.226406 −0.113203 0.993572i $$-0.536111\pi$$
−0.113203 + 0.993572i $$0.536111\pi$$
$$164$$ 0 0
$$165$$ 234.384 0.110586
$$166$$ 0 0
$$167$$ 3735.73 1.73102 0.865508 0.500895i $$-0.166996\pi$$
0.865508 + 0.500895i $$0.166996\pi$$
$$168$$ 0 0
$$169$$ −2130.41 −0.969689
$$170$$ 0 0
$$171$$ −1360.29 −0.608328
$$172$$ 0 0
$$173$$ −2709.33 −1.19067 −0.595336 0.803477i $$-0.702981\pi$$
−0.595336 + 0.803477i $$0.702981\pi$$
$$174$$ 0 0
$$175$$ 640.019 0.276462
$$176$$ 0 0
$$177$$ 1015.19 0.431108
$$178$$ 0 0
$$179$$ −3300.43 −1.37813 −0.689066 0.724699i $$-0.741979\pi$$
−0.689066 + 0.724699i $$0.741979\pi$$
$$180$$ 0 0
$$181$$ −583.868 −0.239771 −0.119886 0.992788i $$-0.538253\pi$$
−0.119886 + 0.992788i $$0.538253\pi$$
$$182$$ 0 0
$$183$$ −1093.91 −0.441879
$$184$$ 0 0
$$185$$ 560.821 0.222878
$$186$$ 0 0
$$187$$ 958.176 0.374699
$$188$$ 0 0
$$189$$ 3824.83 1.47204
$$190$$ 0 0
$$191$$ 3058.98 1.15885 0.579424 0.815026i $$-0.303278\pi$$
0.579424 + 0.815026i $$0.303278\pi$$
$$192$$ 0 0
$$193$$ −1336.93 −0.498625 −0.249313 0.968423i $$-0.580205\pi$$
−0.249313 + 0.968423i $$0.580205\pi$$
$$194$$ 0 0
$$195$$ −151.791 −0.0557434
$$196$$ 0 0
$$197$$ −2449.21 −0.885783 −0.442892 0.896575i $$-0.646047\pi$$
−0.442892 + 0.896575i $$0.646047\pi$$
$$198$$ 0 0
$$199$$ −1138.66 −0.405614 −0.202807 0.979219i $$-0.565006\pi$$
−0.202807 + 0.979219i $$0.565006\pi$$
$$200$$ 0 0
$$201$$ 1587.51 0.557085
$$202$$ 0 0
$$203$$ −6840.56 −2.36509
$$204$$ 0 0
$$205$$ −1195.39 −0.407267
$$206$$ 0 0
$$207$$ −302.691 −0.101635
$$208$$ 0 0
$$209$$ −1302.44 −0.431061
$$210$$ 0 0
$$211$$ −5596.81 −1.82607 −0.913034 0.407884i $$-0.866267\pi$$
−0.913034 + 0.407884i $$0.866267\pi$$
$$212$$ 0 0
$$213$$ −3797.74 −1.22167
$$214$$ 0 0
$$215$$ −1411.19 −0.447640
$$216$$ 0 0
$$217$$ 1632.24 0.510615
$$218$$ 0 0
$$219$$ 1066.38 0.329038
$$220$$ 0 0
$$221$$ −620.530 −0.188875
$$222$$ 0 0
$$223$$ −5580.60 −1.67581 −0.837903 0.545820i $$-0.816218\pi$$
−0.837903 + 0.545820i $$0.816218\pi$$
$$224$$ 0 0
$$225$$ −329.011 −0.0974849
$$226$$ 0 0
$$227$$ 566.323 0.165587 0.0827934 0.996567i $$-0.473616\pi$$
0.0827934 + 0.996567i $$0.473616\pi$$
$$228$$ 0 0
$$229$$ −693.702 −0.200180 −0.100090 0.994978i $$-0.531913\pi$$
−0.100090 + 0.994978i $$0.531913\pi$$
$$230$$ 0 0
$$231$$ 1200.08 0.341816
$$232$$ 0 0
$$233$$ −2208.68 −0.621011 −0.310506 0.950572i $$-0.600498\pi$$
−0.310506 + 0.950572i $$0.600498\pi$$
$$234$$ 0 0
$$235$$ −2886.46 −0.801241
$$236$$ 0 0
$$237$$ −2053.03 −0.562694
$$238$$ 0 0
$$239$$ 3876.79 1.04924 0.524621 0.851336i $$-0.324207\pi$$
0.524621 + 0.851336i $$0.324207\pi$$
$$240$$ 0 0
$$241$$ −2024.57 −0.541136 −0.270568 0.962701i $$-0.587211\pi$$
−0.270568 + 0.962701i $$0.587211\pi$$
$$242$$ 0 0
$$243$$ −3288.10 −0.868033
$$244$$ 0 0
$$245$$ 1562.00 0.407315
$$246$$ 0 0
$$247$$ 843.481 0.217285
$$248$$ 0 0
$$249$$ 80.7350 0.0205477
$$250$$ 0 0
$$251$$ 733.444 0.184441 0.0922203 0.995739i $$-0.470604\pi$$
0.0922203 + 0.995739i $$0.470604\pi$$
$$252$$ 0 0
$$253$$ −289.818 −0.0720185
$$254$$ 0 0
$$255$$ 1414.42 0.347351
$$256$$ 0 0
$$257$$ 2367.83 0.574713 0.287356 0.957824i $$-0.407224\pi$$
0.287356 + 0.957824i $$0.407224\pi$$
$$258$$ 0 0
$$259$$ 2871.49 0.688903
$$260$$ 0 0
$$261$$ 3516.50 0.833968
$$262$$ 0 0
$$263$$ −6632.76 −1.55511 −0.777554 0.628816i $$-0.783540\pi$$
−0.777554 + 0.628816i $$0.783540\pi$$
$$264$$ 0 0
$$265$$ −11.5663 −0.00268117
$$266$$ 0 0
$$267$$ 3904.78 0.895013
$$268$$ 0 0
$$269$$ −6092.97 −1.38102 −0.690511 0.723322i $$-0.742614\pi$$
−0.690511 + 0.723322i $$0.742614\pi$$
$$270$$ 0 0
$$271$$ 1981.30 0.444115 0.222058 0.975034i $$-0.428723\pi$$
0.222058 + 0.975034i $$0.428723\pi$$
$$272$$ 0 0
$$273$$ −777.192 −0.172300
$$274$$ 0 0
$$275$$ −315.019 −0.0690777
$$276$$ 0 0
$$277$$ −4097.42 −0.888773 −0.444386 0.895835i $$-0.646578\pi$$
−0.444386 + 0.895835i $$0.646578\pi$$
$$278$$ 0 0
$$279$$ −839.077 −0.180051
$$280$$ 0 0
$$281$$ −3866.40 −0.820819 −0.410409 0.911901i $$-0.634614\pi$$
−0.410409 + 0.911901i $$0.634614\pi$$
$$282$$ 0 0
$$283$$ −3982.00 −0.836415 −0.418208 0.908351i $$-0.637342\pi$$
−0.418208 + 0.908351i $$0.637342\pi$$
$$284$$ 0 0
$$285$$ −1922.61 −0.399599
$$286$$ 0 0
$$287$$ −6120.59 −1.25884
$$288$$ 0 0
$$289$$ 869.245 0.176927
$$290$$ 0 0
$$291$$ −6432.86 −1.29588
$$292$$ 0 0
$$293$$ −7491.70 −1.49375 −0.746877 0.664962i $$-0.768448\pi$$
−0.746877 + 0.664962i $$0.768448\pi$$
$$294$$ 0 0
$$295$$ −1364.44 −0.269291
$$296$$ 0 0
$$297$$ −1882.59 −0.367809
$$298$$ 0 0
$$299$$ 187.691 0.0363024
$$300$$ 0 0
$$301$$ −7225.53 −1.38363
$$302$$ 0 0
$$303$$ −3183.98 −0.603680
$$304$$ 0 0
$$305$$ 1470.24 0.276019
$$306$$ 0 0
$$307$$ 5813.01 1.08067 0.540335 0.841450i $$-0.318297\pi$$
0.540335 + 0.841450i $$0.318297\pi$$
$$308$$ 0 0
$$309$$ −2353.12 −0.433218
$$310$$ 0 0
$$311$$ −4769.43 −0.869613 −0.434807 0.900524i $$-0.643183\pi$$
−0.434807 + 0.900524i $$0.643183\pi$$
$$312$$ 0 0
$$313$$ 7053.43 1.27375 0.636875 0.770967i $$-0.280227\pi$$
0.636875 + 0.770967i $$0.280227\pi$$
$$314$$ 0 0
$$315$$ −1684.59 −0.301320
$$316$$ 0 0
$$317$$ 8845.36 1.56721 0.783604 0.621261i $$-0.213379\pi$$
0.783604 + 0.621261i $$0.213379\pi$$
$$318$$ 0 0
$$319$$ 3366.94 0.590949
$$320$$ 0 0
$$321$$ 4871.99 0.847127
$$322$$ 0 0
$$323$$ −7859.76 −1.35396
$$324$$ 0 0
$$325$$ 204.011 0.0348201
$$326$$ 0 0
$$327$$ −6421.75 −1.08601
$$328$$ 0 0
$$329$$ −14779.1 −2.47659
$$330$$ 0 0
$$331$$ −4264.07 −0.708079 −0.354040 0.935230i $$-0.615192\pi$$
−0.354040 + 0.935230i $$0.615192\pi$$
$$332$$ 0 0
$$333$$ −1476.13 −0.242918
$$334$$ 0 0
$$335$$ −2133.66 −0.347983
$$336$$ 0 0
$$337$$ −1290.58 −0.208613 −0.104306 0.994545i $$-0.533262\pi$$
−0.104306 + 0.994545i $$0.533262\pi$$
$$338$$ 0 0
$$339$$ 5551.36 0.889405
$$340$$ 0 0
$$341$$ −803.392 −0.127584
$$342$$ 0 0
$$343$$ −783.403 −0.123323
$$344$$ 0 0
$$345$$ −427.818 −0.0667621
$$346$$ 0 0
$$347$$ 1805.09 0.279257 0.139629 0.990204i $$-0.455409\pi$$
0.139629 + 0.990204i $$0.455409\pi$$
$$348$$ 0 0
$$349$$ −3586.81 −0.550136 −0.275068 0.961425i $$-0.588700\pi$$
−0.275068 + 0.961425i $$0.588700\pi$$
$$350$$ 0 0
$$351$$ 1219.20 0.185402
$$352$$ 0 0
$$353$$ −2292.90 −0.345719 −0.172859 0.984947i $$-0.555301\pi$$
−0.172859 + 0.984947i $$0.555301\pi$$
$$354$$ 0 0
$$355$$ 5104.27 0.763118
$$356$$ 0 0
$$357$$ 7242.06 1.07364
$$358$$ 0 0
$$359$$ 3077.18 0.452388 0.226194 0.974082i $$-0.427372\pi$$
0.226194 + 0.974082i $$0.427372\pi$$
$$360$$ 0 0
$$361$$ 3824.70 0.557618
$$362$$ 0 0
$$363$$ 4360.84 0.630537
$$364$$ 0 0
$$365$$ −1433.25 −0.205533
$$366$$ 0 0
$$367$$ 1378.56 0.196076 0.0980382 0.995183i $$-0.468743\pi$$
0.0980382 + 0.995183i $$0.468743\pi$$
$$368$$ 0 0
$$369$$ 3146.38 0.443886
$$370$$ 0 0
$$371$$ −59.2211 −0.00828736
$$372$$ 0 0
$$373$$ 11182.3 1.55227 0.776135 0.630567i $$-0.217178\pi$$
0.776135 + 0.630567i $$0.217178\pi$$
$$374$$ 0 0
$$375$$ −465.019 −0.0640360
$$376$$ 0 0
$$377$$ −2180.49 −0.297880
$$378$$ 0 0
$$379$$ −12021.9 −1.62935 −0.814677 0.579915i $$-0.803086\pi$$
−0.814677 + 0.579915i $$0.803086\pi$$
$$380$$ 0 0
$$381$$ −5090.26 −0.684467
$$382$$ 0 0
$$383$$ 13274.6 1.77102 0.885512 0.464617i $$-0.153808\pi$$
0.885512 + 0.464617i $$0.153808\pi$$
$$384$$ 0 0
$$385$$ −1612.95 −0.213515
$$386$$ 0 0
$$387$$ 3714.39 0.487889
$$388$$ 0 0
$$389$$ 11016.4 1.43587 0.717934 0.696112i $$-0.245088\pi$$
0.717934 + 0.696112i $$0.245088\pi$$
$$390$$ 0 0
$$391$$ −1748.94 −0.226210
$$392$$ 0 0
$$393$$ 7381.67 0.947471
$$394$$ 0 0
$$395$$ 2759.33 0.351486
$$396$$ 0 0
$$397$$ −1893.18 −0.239335 −0.119668 0.992814i $$-0.538183\pi$$
−0.119668 + 0.992814i $$0.538183\pi$$
$$398$$ 0 0
$$399$$ −9844.07 −1.23514
$$400$$ 0 0
$$401$$ 11281.8 1.40495 0.702475 0.711709i $$-0.252078\pi$$
0.702475 + 0.711709i $$0.252078\pi$$
$$402$$ 0 0
$$403$$ 520.290 0.0643113
$$404$$ 0 0
$$405$$ −1002.35 −0.122981
$$406$$ 0 0
$$407$$ −1413.36 −0.172131
$$408$$ 0 0
$$409$$ −14677.6 −1.77447 −0.887236 0.461316i $$-0.847377\pi$$
−0.887236 + 0.461316i $$0.847377\pi$$
$$410$$ 0 0
$$411$$ −1016.83 −0.122036
$$412$$ 0 0
$$413$$ −6986.15 −0.832363
$$414$$ 0 0
$$415$$ −108.510 −0.0128351
$$416$$ 0 0
$$417$$ −2075.43 −0.243728
$$418$$ 0 0
$$419$$ −2196.50 −0.256100 −0.128050 0.991768i $$-0.540872\pi$$
−0.128050 + 0.991768i $$0.540872\pi$$
$$420$$ 0 0
$$421$$ −6571.28 −0.760723 −0.380362 0.924838i $$-0.624200\pi$$
−0.380362 + 0.924838i $$0.624200\pi$$
$$422$$ 0 0
$$423$$ 7597.42 0.873284
$$424$$ 0 0
$$425$$ −1901.03 −0.216973
$$426$$ 0 0
$$427$$ 7527.87 0.853160
$$428$$ 0 0
$$429$$ 382.536 0.0430513
$$430$$ 0 0
$$431$$ 583.607 0.0652235 0.0326118 0.999468i $$-0.489618\pi$$
0.0326118 + 0.999468i $$0.489618\pi$$
$$432$$ 0 0
$$433$$ −2677.60 −0.297176 −0.148588 0.988899i $$-0.547473\pi$$
−0.148588 + 0.988899i $$0.547473\pi$$
$$434$$ 0 0
$$435$$ 4970.15 0.547817
$$436$$ 0 0
$$437$$ 2377.33 0.260236
$$438$$ 0 0
$$439$$ 5509.18 0.598949 0.299475 0.954104i $$-0.403189\pi$$
0.299475 + 0.954104i $$0.403189\pi$$
$$440$$ 0 0
$$441$$ −4111.32 −0.443939
$$442$$ 0 0
$$443$$ 13054.6 1.40009 0.700047 0.714097i $$-0.253162\pi$$
0.700047 + 0.714097i $$0.253162\pi$$
$$444$$ 0 0
$$445$$ −5248.14 −0.559069
$$446$$ 0 0
$$447$$ 5037.68 0.533052
$$448$$ 0 0
$$449$$ −11819.1 −1.24227 −0.621135 0.783704i $$-0.713328\pi$$
−0.621135 + 0.783704i $$0.713328\pi$$
$$450$$ 0 0
$$451$$ 3012.57 0.314537
$$452$$ 0 0
$$453$$ 605.091 0.0627587
$$454$$ 0 0
$$455$$ 1044.57 0.107627
$$456$$ 0 0
$$457$$ −4144.87 −0.424264 −0.212132 0.977241i $$-0.568041\pi$$
−0.212132 + 0.977241i $$0.568041\pi$$
$$458$$ 0 0
$$459$$ −11360.8 −1.15528
$$460$$ 0 0
$$461$$ −6061.38 −0.612379 −0.306189 0.951971i $$-0.599054\pi$$
−0.306189 + 0.951971i $$0.599054\pi$$
$$462$$ 0 0
$$463$$ 5729.13 0.575066 0.287533 0.957771i $$-0.407165\pi$$
0.287533 + 0.957771i $$0.407165\pi$$
$$464$$ 0 0
$$465$$ −1185.94 −0.118272
$$466$$ 0 0
$$467$$ −8958.05 −0.887643 −0.443821 0.896115i $$-0.646378\pi$$
−0.443821 + 0.896115i $$0.646378\pi$$
$$468$$ 0 0
$$469$$ −10924.7 −1.07559
$$470$$ 0 0
$$471$$ 8797.53 0.860655
$$472$$ 0 0
$$473$$ 3556.42 0.345718
$$474$$ 0 0
$$475$$ 2584.05 0.249609
$$476$$ 0 0
$$477$$ 30.4435 0.00292225
$$478$$ 0 0
$$479$$ −6356.60 −0.606347 −0.303174 0.952935i $$-0.598046\pi$$
−0.303174 + 0.952935i $$0.598046\pi$$
$$480$$ 0 0
$$481$$ 915.312 0.0867664
$$482$$ 0 0
$$483$$ −2190.49 −0.206358
$$484$$ 0 0
$$485$$ 8645.96 0.809469
$$486$$ 0 0
$$487$$ −13991.2 −1.30185 −0.650926 0.759141i $$-0.725619\pi$$
−0.650926 + 0.759141i $$0.725619\pi$$
$$488$$ 0 0
$$489$$ 1752.79 0.162094
$$490$$ 0 0
$$491$$ 5301.96 0.487320 0.243660 0.969861i $$-0.421652\pi$$
0.243660 + 0.969861i $$0.421652\pi$$
$$492$$ 0 0
$$493$$ 20318.3 1.85617
$$494$$ 0 0
$$495$$ 829.159 0.0752888
$$496$$ 0 0
$$497$$ 26134.7 2.35875
$$498$$ 0 0
$$499$$ 8013.45 0.718900 0.359450 0.933164i $$-0.382964\pi$$
0.359450 + 0.933164i $$0.382964\pi$$
$$500$$ 0 0
$$501$$ −13897.5 −1.23931
$$502$$ 0 0
$$503$$ 11881.9 1.05325 0.526627 0.850097i $$-0.323456\pi$$
0.526627 + 0.850097i $$0.323456\pi$$
$$504$$ 0 0
$$505$$ 4279.37 0.377088
$$506$$ 0 0
$$507$$ 7925.44 0.694243
$$508$$ 0 0
$$509$$ −12113.8 −1.05488 −0.527441 0.849592i $$-0.676848\pi$$
−0.527441 + 0.849592i $$0.676848\pi$$
$$510$$ 0 0
$$511$$ −7338.45 −0.635291
$$512$$ 0 0
$$513$$ 15442.6 1.32906
$$514$$ 0 0
$$515$$ 3162.67 0.270609
$$516$$ 0 0
$$517$$ 7274.31 0.618808
$$518$$ 0 0
$$519$$ 10079.1 0.852454
$$520$$ 0 0
$$521$$ −15327.6 −1.28890 −0.644449 0.764647i $$-0.722913\pi$$
−0.644449 + 0.764647i $$0.722913\pi$$
$$522$$ 0 0
$$523$$ −14545.8 −1.21614 −0.608070 0.793883i $$-0.708056\pi$$
−0.608070 + 0.793883i $$0.708056\pi$$
$$524$$ 0 0
$$525$$ −2380.97 −0.197931
$$526$$ 0 0
$$527$$ −4848.18 −0.400740
$$528$$ 0 0
$$529$$ 529.000 0.0434783
$$530$$ 0 0
$$531$$ 3591.34 0.293504
$$532$$ 0 0
$$533$$ −1950.99 −0.158549
$$534$$ 0 0
$$535$$ −6548.10 −0.529157
$$536$$ 0 0
$$537$$ 12278.1 0.986665
$$538$$ 0 0
$$539$$ −3936.47 −0.314575
$$540$$ 0 0
$$541$$ −5468.93 −0.434617 −0.217308 0.976103i $$-0.569728\pi$$
−0.217308 + 0.976103i $$0.569728\pi$$
$$542$$ 0 0
$$543$$ 2172.08 0.171663
$$544$$ 0 0
$$545$$ 8631.03 0.678372
$$546$$ 0 0
$$547$$ 19566.3 1.52943 0.764713 0.644371i $$-0.222881\pi$$
0.764713 + 0.644371i $$0.222881\pi$$
$$548$$ 0 0
$$549$$ −3869.82 −0.300837
$$550$$ 0 0
$$551$$ −27618.5 −2.13537
$$552$$ 0 0
$$553$$ 14128.2 1.08642
$$554$$ 0 0
$$555$$ −2086.34 −0.159568
$$556$$ 0 0
$$557$$ −9803.11 −0.745729 −0.372865 0.927886i $$-0.621624\pi$$
−0.372865 + 0.927886i $$0.621624\pi$$
$$558$$ 0 0
$$559$$ −2303.20 −0.174266
$$560$$ 0 0
$$561$$ −3564.56 −0.268264
$$562$$ 0 0
$$563$$ 4909.73 0.367532 0.183766 0.982970i $$-0.441171\pi$$
0.183766 + 0.982970i $$0.441171\pi$$
$$564$$ 0 0
$$565$$ −7461.19 −0.555566
$$566$$ 0 0
$$567$$ −5132.18 −0.380126
$$568$$ 0 0
$$569$$ 11498.2 0.847155 0.423578 0.905860i $$-0.360774\pi$$
0.423578 + 0.905860i $$0.360774\pi$$
$$570$$ 0 0
$$571$$ −14766.6 −1.08225 −0.541124 0.840943i $$-0.682001\pi$$
−0.541124 + 0.840943i $$0.682001\pi$$
$$572$$ 0 0
$$573$$ −11379.9 −0.829670
$$574$$ 0 0
$$575$$ 575.000 0.0417029
$$576$$ 0 0
$$577$$ 13364.0 0.964212 0.482106 0.876113i $$-0.339872\pi$$
0.482106 + 0.876113i $$0.339872\pi$$
$$578$$ 0 0
$$579$$ 4973.60 0.356988
$$580$$ 0 0
$$581$$ −555.590 −0.0396725
$$582$$ 0 0
$$583$$ 29.1488 0.00207070
$$584$$ 0 0
$$585$$ −536.977 −0.0379509
$$586$$ 0 0
$$587$$ −12524.7 −0.880664 −0.440332 0.897835i $$-0.645139\pi$$
−0.440332 + 0.897835i $$0.645139\pi$$
$$588$$ 0 0
$$589$$ 6590.09 0.461019
$$590$$ 0 0
$$591$$ 9111.45 0.634171
$$592$$ 0 0
$$593$$ −17938.6 −1.24224 −0.621121 0.783714i $$-0.713323\pi$$
−0.621121 + 0.783714i $$0.713323\pi$$
$$594$$ 0 0
$$595$$ −9733.55 −0.670650
$$596$$ 0 0
$$597$$ 4235.98 0.290397
$$598$$ 0 0
$$599$$ 26735.8 1.82370 0.911848 0.410528i $$-0.134656\pi$$
0.911848 + 0.410528i $$0.134656\pi$$
$$600$$ 0 0
$$601$$ −21043.6 −1.42826 −0.714131 0.700012i $$-0.753178\pi$$
−0.714131 + 0.700012i $$0.753178\pi$$
$$602$$ 0 0
$$603$$ 5615.98 0.379271
$$604$$ 0 0
$$605$$ −5861.10 −0.393864
$$606$$ 0 0
$$607$$ −5098.83 −0.340947 −0.170474 0.985362i $$-0.554530\pi$$
−0.170474 + 0.985362i $$0.554530\pi$$
$$608$$ 0 0
$$609$$ 25447.9 1.69327
$$610$$ 0 0
$$611$$ −4710.96 −0.311923
$$612$$ 0 0
$$613$$ −12784.8 −0.842369 −0.421185 0.906975i $$-0.638386\pi$$
−0.421185 + 0.906975i $$0.638386\pi$$
$$614$$ 0 0
$$615$$ 4447.04 0.291580
$$616$$ 0 0
$$617$$ −8940.73 −0.583372 −0.291686 0.956514i $$-0.594216\pi$$
−0.291686 + 0.956514i $$0.594216\pi$$
$$618$$ 0 0
$$619$$ −1333.26 −0.0865725 −0.0432863 0.999063i $$-0.513783\pi$$
−0.0432863 + 0.999063i $$0.513783\pi$$
$$620$$ 0 0
$$621$$ 3436.27 0.222050
$$622$$ 0 0
$$623$$ −26871.3 −1.72805
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 4845.28 0.308615
$$628$$ 0 0
$$629$$ −8529.09 −0.540663
$$630$$ 0 0
$$631$$ 16667.9 1.05157 0.525784 0.850618i $$-0.323772\pi$$
0.525784 + 0.850618i $$0.323772\pi$$
$$632$$ 0 0
$$633$$ 20821.0 1.30736
$$634$$ 0 0
$$635$$ 6841.46 0.427551
$$636$$ 0 0
$$637$$ 2549.32 0.158568
$$638$$ 0 0
$$639$$ −13434.9 −0.831733
$$640$$ 0 0
$$641$$ 27431.9 1.69032 0.845158 0.534516i $$-0.179506\pi$$
0.845158 + 0.534516i $$0.179506\pi$$
$$642$$ 0 0
$$643$$ −9618.00 −0.589886 −0.294943 0.955515i $$-0.595301\pi$$
−0.294943 + 0.955515i $$0.595301\pi$$
$$644$$ 0 0
$$645$$ 5249.86 0.320485
$$646$$ 0 0
$$647$$ 12548.6 0.762496 0.381248 0.924473i $$-0.375494\pi$$
0.381248 + 0.924473i $$0.375494\pi$$
$$648$$ 0 0
$$649$$ 3438.60 0.207977
$$650$$ 0 0
$$651$$ −6072.18 −0.365572
$$652$$ 0 0
$$653$$ −8829.79 −0.529152 −0.264576 0.964365i $$-0.585232\pi$$
−0.264576 + 0.964365i $$0.585232\pi$$
$$654$$ 0 0
$$655$$ −9921.19 −0.591837
$$656$$ 0 0
$$657$$ 3772.44 0.224014
$$658$$ 0 0
$$659$$ −5935.56 −0.350860 −0.175430 0.984492i $$-0.556132\pi$$
−0.175430 + 0.984492i $$0.556132\pi$$
$$660$$ 0 0
$$661$$ −1182.03 −0.0695549 −0.0347775 0.999395i $$-0.511072\pi$$
−0.0347775 + 0.999395i $$0.511072\pi$$
$$662$$ 0 0
$$663$$ 2308.47 0.135224
$$664$$ 0 0
$$665$$ 13230.7 0.771527
$$666$$ 0 0
$$667$$ −6145.64 −0.356762
$$668$$ 0 0
$$669$$ 20760.7 1.19978
$$670$$ 0 0
$$671$$ −3705.24 −0.213173
$$672$$ 0 0
$$673$$ −6053.25 −0.346710 −0.173355 0.984859i $$-0.555461\pi$$
−0.173355 + 0.984859i $$0.555461\pi$$
$$674$$ 0 0
$$675$$ 3735.08 0.212982
$$676$$ 0 0
$$677$$ 16694.2 0.947728 0.473864 0.880598i $$-0.342859\pi$$
0.473864 + 0.880598i $$0.342859\pi$$
$$678$$ 0 0
$$679$$ 44268.6 2.50202
$$680$$ 0 0
$$681$$ −2106.81 −0.118551
$$682$$ 0 0
$$683$$ 4150.03 0.232499 0.116249 0.993220i $$-0.462913\pi$$
0.116249 + 0.993220i $$0.462913\pi$$
$$684$$ 0 0
$$685$$ 1366.66 0.0762296
$$686$$ 0 0
$$687$$ 2580.68 0.143317
$$688$$ 0 0
$$689$$ −18.8772 −0.00104378
$$690$$ 0 0
$$691$$ −33382.4 −1.83781 −0.918904 0.394481i $$-0.870924\pi$$
−0.918904 + 0.394481i $$0.870924\pi$$
$$692$$ 0 0
$$693$$ 4245.42 0.232713
$$694$$ 0 0
$$695$$ 2789.45 0.152244
$$696$$ 0 0
$$697$$ 18179.8 0.987960
$$698$$ 0 0
$$699$$ 8216.64 0.444609
$$700$$ 0 0
$$701$$ −29528.7 −1.59099 −0.795494 0.605961i $$-0.792789\pi$$
−0.795494 + 0.605961i $$0.792789\pi$$
$$702$$ 0 0
$$703$$ 11593.5 0.621989
$$704$$ 0 0
$$705$$ 10738.1 0.573644
$$706$$ 0 0
$$707$$ 21911.0 1.16556
$$708$$ 0 0
$$709$$ 27016.3 1.43106 0.715529 0.698583i $$-0.246186\pi$$
0.715529 + 0.698583i $$0.246186\pi$$
$$710$$ 0 0
$$711$$ −7262.82 −0.383090
$$712$$ 0 0
$$713$$ 1466.42 0.0770237
$$714$$ 0 0
$$715$$ −514.140 −0.0268920
$$716$$ 0 0
$$717$$ −14422.3 −0.751198
$$718$$ 0 0
$$719$$ −24752.9 −1.28390 −0.641952 0.766745i $$-0.721875\pi$$
−0.641952 + 0.766745i $$0.721875\pi$$
$$720$$ 0 0
$$721$$ 16193.4 0.836438
$$722$$ 0 0
$$723$$ 7531.69 0.387423
$$724$$ 0 0
$$725$$ −6680.04 −0.342194
$$726$$ 0 0
$$727$$ 17574.5 0.896565 0.448283 0.893892i $$-0.352036\pi$$
0.448283 + 0.893892i $$0.352036\pi$$
$$728$$ 0 0
$$729$$ 17644.9 0.896456
$$730$$ 0 0
$$731$$ 21461.7 1.08590
$$732$$ 0 0
$$733$$ 27739.6 1.39780 0.698898 0.715221i $$-0.253674\pi$$
0.698898 + 0.715221i $$0.253674\pi$$
$$734$$ 0 0
$$735$$ −5810.87 −0.291615
$$736$$ 0 0
$$737$$ 5377.15 0.268751
$$738$$ 0 0
$$739$$ 3140.78 0.156340 0.0781701 0.996940i $$-0.475092\pi$$
0.0781701 + 0.996940i $$0.475092\pi$$
$$740$$ 0 0
$$741$$ −3137.88 −0.155564
$$742$$ 0 0
$$743$$ 35881.5 1.77169 0.885844 0.463984i $$-0.153580\pi$$
0.885844 + 0.463984i $$0.153580\pi$$
$$744$$ 0 0
$$745$$ −6770.80 −0.332970
$$746$$ 0 0
$$747$$ 285.609 0.0139891
$$748$$ 0 0
$$749$$ −33527.3 −1.63559
$$750$$ 0 0
$$751$$ 7009.03 0.340563 0.170282 0.985395i $$-0.445532\pi$$
0.170282 + 0.985395i $$0.445532\pi$$
$$752$$ 0 0
$$753$$ −2728.52 −0.132049
$$754$$ 0 0
$$755$$ −813.261 −0.0392021
$$756$$ 0 0
$$757$$ 29998.3 1.44030 0.720150 0.693818i $$-0.244073\pi$$
0.720150 + 0.693818i $$0.244073\pi$$
$$758$$ 0 0
$$759$$ 1078.17 0.0515612
$$760$$ 0 0
$$761$$ 5283.49 0.251677 0.125839 0.992051i $$-0.459838\pi$$
0.125839 + 0.992051i $$0.459838\pi$$
$$762$$ 0 0
$$763$$ 44192.2 2.09681
$$764$$ 0 0
$$765$$ 5003.68 0.236481
$$766$$ 0 0
$$767$$ −2226.89 −0.104835
$$768$$ 0 0
$$769$$ −15216.2 −0.713538 −0.356769 0.934193i $$-0.616122\pi$$
−0.356769 + 0.934193i $$0.616122\pi$$
$$770$$ 0 0
$$771$$ −8808.69 −0.411462
$$772$$ 0 0
$$773$$ 15651.3 0.728250 0.364125 0.931350i $$-0.381368\pi$$
0.364125 + 0.931350i $$0.381368\pi$$
$$774$$ 0 0
$$775$$ 1593.93 0.0738785
$$776$$ 0 0
$$777$$ −10682.4 −0.493216
$$778$$ 0 0
$$779$$ −24711.6 −1.13657
$$780$$ 0 0
$$781$$ −12863.6 −0.589365
$$782$$ 0 0
$$783$$ −39920.7 −1.82203
$$784$$ 0 0
$$785$$ −11824.1 −0.537607
$$786$$ 0 0
$$787$$ −9401.09 −0.425810 −0.212905 0.977073i $$-0.568293\pi$$
−0.212905 + 0.977073i $$0.568293\pi$$
$$788$$ 0 0
$$789$$ 24674.9 1.11337
$$790$$ 0 0
$$791$$ −38202.5 −1.71722
$$792$$ 0 0
$$793$$ 2399.57 0.107454
$$794$$ 0 0
$$795$$ 43.0283 0.00191957
$$796$$ 0 0
$$797$$ 37388.5 1.66169 0.830845 0.556504i $$-0.187857\pi$$
0.830845 + 0.556504i $$0.187857\pi$$
$$798$$ 0 0
$$799$$ 43897.9 1.94367
$$800$$ 0 0
$$801$$ 13813.6 0.609337
$$802$$ 0 0
$$803$$ 3612.00 0.158736
$$804$$ 0 0
$$805$$ 2944.09 0.128901
$$806$$ 0 0
$$807$$ 22666.8 0.988734
$$808$$ 0 0
$$809$$ 24390.6 1.05999 0.529993 0.848002i $$-0.322195\pi$$
0.529993 + 0.848002i $$0.322195\pi$$
$$810$$ 0 0
$$811$$ −23750.8 −1.02836 −0.514181 0.857682i $$-0.671904\pi$$
−0.514181 + 0.857682i $$0.671904\pi$$
$$812$$ 0 0
$$813$$ −7370.73 −0.317962
$$814$$ 0 0
$$815$$ −2355.81 −0.101252
$$816$$ 0 0
$$817$$ −29172.8 −1.24924
$$818$$ 0 0
$$819$$ −2749.40 −0.117304
$$820$$ 0 0
$$821$$ 3041.59 0.129296 0.0646481 0.997908i $$-0.479408\pi$$
0.0646481 + 0.997908i $$0.479408\pi$$
$$822$$ 0 0
$$823$$ −28411.6 −1.20336 −0.601681 0.798737i $$-0.705502\pi$$
−0.601681 + 0.798737i $$0.705502\pi$$
$$824$$ 0 0
$$825$$ 1171.92 0.0494558
$$826$$ 0 0
$$827$$ 24351.5 1.02392 0.511962 0.859008i $$-0.328919\pi$$
0.511962 + 0.859008i $$0.328919\pi$$
$$828$$ 0 0
$$829$$ −47240.8 −1.97918 −0.989590 0.143914i $$-0.954031\pi$$
−0.989590 + 0.143914i $$0.954031\pi$$
$$830$$ 0 0
$$831$$ 15243.0 0.636311
$$832$$ 0 0
$$833$$ −23755.2 −0.988077
$$834$$ 0 0
$$835$$ 18678.7 0.774134
$$836$$ 0 0
$$837$$ 9525.55 0.393371
$$838$$ 0 0
$$839$$ −2646.33 −0.108893 −0.0544466 0.998517i $$-0.517339\pi$$
−0.0544466 + 0.998517i $$0.517339\pi$$
$$840$$ 0 0
$$841$$ 47007.7 1.92741
$$842$$ 0 0
$$843$$ 14383.6 0.587660
$$844$$ 0 0
$$845$$ −10652.0 −0.433658
$$846$$ 0 0
$$847$$ −30009.7 −1.21741
$$848$$ 0 0
$$849$$ 14813.7 0.598827
$$850$$ 0 0
$$851$$ 2579.78 0.103917
$$852$$ 0 0
$$853$$ −40826.4 −1.63877 −0.819385 0.573244i $$-0.805685\pi$$
−0.819385 + 0.573244i $$0.805685\pi$$
$$854$$ 0 0
$$855$$ −6801.46 −0.272053
$$856$$ 0 0
$$857$$ 1516.61 0.0604510 0.0302255 0.999543i $$-0.490377\pi$$
0.0302255 + 0.999543i $$0.490377\pi$$
$$858$$ 0 0
$$859$$ −6514.74 −0.258766 −0.129383 0.991595i $$-0.541300\pi$$
−0.129383 + 0.991595i $$0.541300\pi$$
$$860$$ 0 0
$$861$$ 22769.5 0.901258
$$862$$ 0 0
$$863$$ 7120.18 0.280850 0.140425 0.990091i $$-0.455153\pi$$
0.140425 + 0.990091i $$0.455153\pi$$
$$864$$ 0 0
$$865$$ −13546.6 −0.532485
$$866$$ 0 0
$$867$$ −3233.72 −0.126670
$$868$$ 0 0
$$869$$ −6953.94 −0.271457
$$870$$ 0 0
$$871$$ −3482.33 −0.135470
$$872$$ 0 0
$$873$$ −22757.0 −0.882252
$$874$$ 0 0
$$875$$ 3200.10 0.123638
$$876$$ 0 0
$$877$$ −34013.4 −1.30964 −0.654818 0.755787i $$-0.727255\pi$$
−0.654818 + 0.755787i $$0.727255\pi$$
$$878$$ 0 0
$$879$$ 27870.3 1.06944
$$880$$ 0 0
$$881$$ 6704.11 0.256376 0.128188 0.991750i $$-0.459084\pi$$
0.128188 + 0.991750i $$0.459084\pi$$
$$882$$ 0 0
$$883$$ −47808.5 −1.82207 −0.911033 0.412333i $$-0.864714\pi$$
−0.911033 + 0.412333i $$0.864714\pi$$
$$884$$ 0 0
$$885$$ 5075.93 0.192797
$$886$$ 0 0
$$887$$ 38955.7 1.47464 0.737320 0.675544i $$-0.236091\pi$$
0.737320 + 0.675544i $$0.236091\pi$$
$$888$$ 0 0
$$889$$ 35029.3 1.32154
$$890$$ 0 0
$$891$$ 2526.07 0.0949794
$$892$$ 0 0
$$893$$ −59670.0 −2.23604
$$894$$ 0 0
$$895$$ −16502.1 −0.616319
$$896$$ 0 0
$$897$$ −698.238 −0.0259905
$$898$$ 0 0
$$899$$ −17036.1 −0.632019
$$900$$ 0 0
$$901$$ 175.903 0.00650407
$$902$$ 0 0
$$903$$ 26880.1 0.990601
$$904$$ 0 0
$$905$$ −2919.34 −0.107229
$$906$$ 0 0
$$907$$ −7725.66 −0.282830 −0.141415 0.989950i $$-0.545165\pi$$
−0.141415 + 0.989950i $$0.545165\pi$$
$$908$$ 0 0
$$909$$ −11263.7 −0.410994
$$910$$ 0 0
$$911$$ 13150.7 0.478269 0.239135 0.970986i $$-0.423136\pi$$
0.239135 + 0.970986i $$0.423136\pi$$
$$912$$ 0 0
$$913$$ 273.463 0.00991270
$$914$$ 0 0
$$915$$ −5469.53 −0.197614
$$916$$ 0 0
$$917$$ −50798.0 −1.82933
$$918$$ 0 0
$$919$$ −33083.7 −1.18752 −0.593759 0.804643i $$-0.702357\pi$$
−0.593759 + 0.804643i $$0.702357\pi$$
$$920$$ 0 0
$$921$$ −21625.3 −0.773699
$$922$$ 0 0
$$923$$ 8330.65 0.297082
$$924$$ 0 0
$$925$$ 2804.11 0.0996740
$$926$$ 0 0
$$927$$ −8324.44 −0.294941
$$928$$ 0 0
$$929$$ −1310.12 −0.0462686 −0.0231343 0.999732i $$-0.507365\pi$$
−0.0231343 + 0.999732i $$0.507365\pi$$
$$930$$ 0 0
$$931$$ 32290.2 1.13670
$$932$$ 0 0
$$933$$ 17743.0 0.622594
$$934$$ 0 0
$$935$$ 4790.88 0.167571
$$936$$ 0 0
$$937$$ 40623.1 1.41633 0.708163 0.706049i $$-0.249524\pi$$
0.708163 + 0.706049i $$0.249524\pi$$
$$938$$ 0 0
$$939$$ −26239.8 −0.911933
$$940$$ 0 0
$$941$$ −10434.0 −0.361466 −0.180733 0.983532i $$-0.557847\pi$$
−0.180733 + 0.983532i $$0.557847\pi$$
$$942$$ 0 0
$$943$$ −5498.80 −0.189889
$$944$$ 0 0
$$945$$ 19124.2 0.658317
$$946$$ 0 0
$$947$$ 51333.4 1.76147 0.880734 0.473612i $$-0.157050\pi$$
0.880734 + 0.473612i $$0.157050\pi$$
$$948$$ 0 0
$$949$$ −2339.19 −0.0800141
$$950$$ 0 0
$$951$$ −32906.1 −1.12203
$$952$$ 0 0
$$953$$ 40263.6 1.36859 0.684294 0.729206i $$-0.260110\pi$$
0.684294 + 0.729206i $$0.260110\pi$$
$$954$$ 0 0
$$955$$ 15294.9 0.518252
$$956$$ 0 0
$$957$$ −12525.5 −0.423086
$$958$$ 0 0
$$959$$ 6997.49 0.235621
$$960$$ 0 0
$$961$$ −25726.0 −0.863549
$$962$$ 0 0
$$963$$ 17235.2 0.576736
$$964$$ 0 0
$$965$$ −6684.67 −0.222992
$$966$$ 0 0
$$967$$ 8353.12 0.277785 0.138893 0.990307i $$-0.455646\pi$$
0.138893 + 0.990307i $$0.455646\pi$$
$$968$$ 0 0
$$969$$ 29239.5 0.969358
$$970$$ 0 0
$$971$$ 28020.1 0.926063 0.463032 0.886342i $$-0.346762\pi$$
0.463032 + 0.886342i $$0.346762\pi$$
$$972$$ 0 0
$$973$$ 14282.4 0.470578
$$974$$ 0 0
$$975$$ −758.954 −0.0249292
$$976$$ 0 0
$$977$$ −50115.3 −1.64107 −0.820537 0.571593i $$-0.806326\pi$$
−0.820537 + 0.571593i $$0.806326\pi$$
$$978$$ 0 0
$$979$$ 13226.1 0.431776
$$980$$ 0 0
$$981$$ −22717.7 −0.739367
$$982$$ 0 0
$$983$$ 38113.0 1.23664 0.618320 0.785926i $$-0.287814\pi$$
0.618320 + 0.785926i $$0.287814\pi$$
$$984$$ 0 0
$$985$$ −12246.1 −0.396134
$$986$$ 0 0
$$987$$ 54980.5 1.77310
$$988$$ 0 0
$$989$$ −6491.49 −0.208713
$$990$$ 0 0
$$991$$ 5245.02 0.168127 0.0840634 0.996460i $$-0.473210\pi$$
0.0840634 + 0.996460i $$0.473210\pi$$
$$992$$ 0 0
$$993$$ 15863.0 0.506945
$$994$$ 0 0
$$995$$ −5693.29 −0.181396
$$996$$ 0 0
$$997$$ 21896.1 0.695544 0.347772 0.937579i $$-0.386938\pi$$
0.347772 + 0.937579i $$0.386938\pi$$
$$998$$ 0 0
$$999$$ 16757.7 0.530721
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 1840.4.a.h.1.1 2
4.3 odd 2 115.4.a.c.1.2 2
12.11 even 2 1035.4.a.g.1.1 2
20.3 even 4 575.4.b.f.24.4 4
20.7 even 4 575.4.b.f.24.1 4
20.19 odd 2 575.4.a.h.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.c.1.2 2 4.3 odd 2
575.4.a.h.1.1 2 20.19 odd 2
575.4.b.f.24.1 4 20.7 even 4
575.4.b.f.24.4 4 20.3 even 4
1035.4.a.g.1.1 2 12.11 even 2
1840.4.a.h.1.1 2 1.1 even 1 trivial