# Properties

 Label 361.4.a.b.1.1 Level $361$ Weight $4$ Character 361.1 Self dual yes Analytic conductor $21.300$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [361,4,Mod(1,361)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(361, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

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

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

chi := DirichletCharacter("361.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 361.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: $$21.2996895121$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 361.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{2} +5.00000 q^{3} +1.00000 q^{4} -12.0000 q^{5} +15.0000 q^{6} +11.0000 q^{7} -21.0000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{2} +5.00000 q^{3} +1.00000 q^{4} -12.0000 q^{5} +15.0000 q^{6} +11.0000 q^{7} -21.0000 q^{8} -2.00000 q^{9} -36.0000 q^{10} -54.0000 q^{11} +5.00000 q^{12} -11.0000 q^{13} +33.0000 q^{14} -60.0000 q^{15} -71.0000 q^{16} -93.0000 q^{17} -6.00000 q^{18} -12.0000 q^{20} +55.0000 q^{21} -162.000 q^{22} +183.000 q^{23} -105.000 q^{24} +19.0000 q^{25} -33.0000 q^{26} -145.000 q^{27} +11.0000 q^{28} +249.000 q^{29} -180.000 q^{30} -56.0000 q^{31} -45.0000 q^{32} -270.000 q^{33} -279.000 q^{34} -132.000 q^{35} -2.00000 q^{36} +250.000 q^{37} -55.0000 q^{39} +252.000 q^{40} -240.000 q^{41} +165.000 q^{42} -196.000 q^{43} -54.0000 q^{44} +24.0000 q^{45} +549.000 q^{46} -168.000 q^{47} -355.000 q^{48} -222.000 q^{49} +57.0000 q^{50} -465.000 q^{51} -11.0000 q^{52} -435.000 q^{53} -435.000 q^{54} +648.000 q^{55} -231.000 q^{56} +747.000 q^{58} -195.000 q^{59} -60.0000 q^{60} -358.000 q^{61} -168.000 q^{62} -22.0000 q^{63} +433.000 q^{64} +132.000 q^{65} -810.000 q^{66} +961.000 q^{67} -93.0000 q^{68} +915.000 q^{69} -396.000 q^{70} +246.000 q^{71} +42.0000 q^{72} +353.000 q^{73} +750.000 q^{74} +95.0000 q^{75} -594.000 q^{77} -165.000 q^{78} +34.0000 q^{79} +852.000 q^{80} -671.000 q^{81} -720.000 q^{82} +234.000 q^{83} +55.0000 q^{84} +1116.00 q^{85} -588.000 q^{86} +1245.00 q^{87} +1134.00 q^{88} +168.000 q^{89} +72.0000 q^{90} -121.000 q^{91} +183.000 q^{92} -280.000 q^{93} -504.000 q^{94} -225.000 q^{96} -758.000 q^{97} -666.000 q^{98} +108.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.00000 1.06066 0.530330 0.847791i $$-0.322068\pi$$
0.530330 + 0.847791i $$0.322068\pi$$
$$3$$ 5.00000 0.962250 0.481125 0.876652i $$-0.340228\pi$$
0.481125 + 0.876652i $$0.340228\pi$$
$$4$$ 1.00000 0.125000
$$5$$ −12.0000 −1.07331 −0.536656 0.843801i $$-0.680313\pi$$
−0.536656 + 0.843801i $$0.680313\pi$$
$$6$$ 15.0000 1.02062
$$7$$ 11.0000 0.593944 0.296972 0.954886i $$-0.404023\pi$$
0.296972 + 0.954886i $$0.404023\pi$$
$$8$$ −21.0000 −0.928078
$$9$$ −2.00000 −0.0740741
$$10$$ −36.0000 −1.13842
$$11$$ −54.0000 −1.48015 −0.740073 0.672526i $$-0.765209\pi$$
−0.740073 + 0.672526i $$0.765209\pi$$
$$12$$ 5.00000 0.120281
$$13$$ −11.0000 −0.234681 −0.117340 0.993092i $$-0.537437\pi$$
−0.117340 + 0.993092i $$0.537437\pi$$
$$14$$ 33.0000 0.629973
$$15$$ −60.0000 −1.03280
$$16$$ −71.0000 −1.10938
$$17$$ −93.0000 −1.32681 −0.663406 0.748259i $$-0.730890\pi$$
−0.663406 + 0.748259i $$0.730890\pi$$
$$18$$ −6.00000 −0.0785674
$$19$$ 0 0
$$20$$ −12.0000 −0.134164
$$21$$ 55.0000 0.571523
$$22$$ −162.000 −1.56993
$$23$$ 183.000 1.65905 0.829525 0.558470i $$-0.188611\pi$$
0.829525 + 0.558470i $$0.188611\pi$$
$$24$$ −105.000 −0.893043
$$25$$ 19.0000 0.152000
$$26$$ −33.0000 −0.248917
$$27$$ −145.000 −1.03353
$$28$$ 11.0000 0.0742430
$$29$$ 249.000 1.59442 0.797209 0.603703i $$-0.206309\pi$$
0.797209 + 0.603703i $$0.206309\pi$$
$$30$$ −180.000 −1.09545
$$31$$ −56.0000 −0.324448 −0.162224 0.986754i $$-0.551867\pi$$
−0.162224 + 0.986754i $$0.551867\pi$$
$$32$$ −45.0000 −0.248592
$$33$$ −270.000 −1.42427
$$34$$ −279.000 −1.40730
$$35$$ −132.000 −0.637488
$$36$$ −2.00000 −0.00925926
$$37$$ 250.000 1.11080 0.555402 0.831582i $$-0.312564\pi$$
0.555402 + 0.831582i $$0.312564\pi$$
$$38$$ 0 0
$$39$$ −55.0000 −0.225822
$$40$$ 252.000 0.996117
$$41$$ −240.000 −0.914188 −0.457094 0.889418i $$-0.651110\pi$$
−0.457094 + 0.889418i $$0.651110\pi$$
$$42$$ 165.000 0.606192
$$43$$ −196.000 −0.695110 −0.347555 0.937660i $$-0.612988\pi$$
−0.347555 + 0.937660i $$0.612988\pi$$
$$44$$ −54.0000 −0.185018
$$45$$ 24.0000 0.0795046
$$46$$ 549.000 1.75969
$$47$$ −168.000 −0.521390 −0.260695 0.965421i $$-0.583952\pi$$
−0.260695 + 0.965421i $$0.583952\pi$$
$$48$$ −355.000 −1.06750
$$49$$ −222.000 −0.647230
$$50$$ 57.0000 0.161220
$$51$$ −465.000 −1.27673
$$52$$ −11.0000 −0.0293351
$$53$$ −435.000 −1.12739 −0.563697 0.825982i $$-0.690621\pi$$
−0.563697 + 0.825982i $$0.690621\pi$$
$$54$$ −435.000 −1.09622
$$55$$ 648.000 1.58866
$$56$$ −231.000 −0.551226
$$57$$ 0 0
$$58$$ 747.000 1.69114
$$59$$ −195.000 −0.430285 −0.215143 0.976583i $$-0.569022\pi$$
−0.215143 + 0.976583i $$0.569022\pi$$
$$60$$ −60.0000 −0.129099
$$61$$ −358.000 −0.751430 −0.375715 0.926735i $$-0.622603\pi$$
−0.375715 + 0.926735i $$0.622603\pi$$
$$62$$ −168.000 −0.344129
$$63$$ −22.0000 −0.0439959
$$64$$ 433.000 0.845703
$$65$$ 132.000 0.251886
$$66$$ −810.000 −1.51067
$$67$$ 961.000 1.75231 0.876155 0.482029i $$-0.160100\pi$$
0.876155 + 0.482029i $$0.160100\pi$$
$$68$$ −93.0000 −0.165852
$$69$$ 915.000 1.59642
$$70$$ −396.000 −0.676158
$$71$$ 246.000 0.411195 0.205597 0.978637i $$-0.434086\pi$$
0.205597 + 0.978637i $$0.434086\pi$$
$$72$$ 42.0000 0.0687465
$$73$$ 353.000 0.565966 0.282983 0.959125i $$-0.408676\pi$$
0.282983 + 0.959125i $$0.408676\pi$$
$$74$$ 750.000 1.17819
$$75$$ 95.0000 0.146262
$$76$$ 0 0
$$77$$ −594.000 −0.879124
$$78$$ −165.000 −0.239520
$$79$$ 34.0000 0.0484215 0.0242108 0.999707i $$-0.492293\pi$$
0.0242108 + 0.999707i $$0.492293\pi$$
$$80$$ 852.000 1.19071
$$81$$ −671.000 −0.920439
$$82$$ −720.000 −0.969643
$$83$$ 234.000 0.309456 0.154728 0.987957i $$-0.450550\pi$$
0.154728 + 0.987957i $$0.450550\pi$$
$$84$$ 55.0000 0.0714404
$$85$$ 1116.00 1.42408
$$86$$ −588.000 −0.737275
$$87$$ 1245.00 1.53423
$$88$$ 1134.00 1.37369
$$89$$ 168.000 0.200089 0.100045 0.994983i $$-0.468101\pi$$
0.100045 + 0.994983i $$0.468101\pi$$
$$90$$ 72.0000 0.0843274
$$91$$ −121.000 −0.139387
$$92$$ 183.000 0.207381
$$93$$ −280.000 −0.312201
$$94$$ −504.000 −0.553017
$$95$$ 0 0
$$96$$ −225.000 −0.239208
$$97$$ −758.000 −0.793435 −0.396718 0.917941i $$-0.629851\pi$$
−0.396718 + 0.917941i $$0.629851\pi$$
$$98$$ −666.000 −0.686491
$$99$$ 108.000 0.109640
$$100$$ 19.0000 0.0190000
$$101$$ −726.000 −0.715245 −0.357622 0.933866i $$-0.616412\pi$$
−0.357622 + 0.933866i $$0.616412\pi$$
$$102$$ −1395.00 −1.35417
$$103$$ −2.00000 −0.00191326 −0.000956630 1.00000i $$-0.500305\pi$$
−0.000956630 1.00000i $$0.500305\pi$$
$$104$$ 231.000 0.217802
$$105$$ −660.000 −0.613423
$$106$$ −1305.00 −1.19578
$$107$$ −1413.00 −1.27663 −0.638317 0.769773i $$-0.720369\pi$$
−0.638317 + 0.769773i $$0.720369\pi$$
$$108$$ −145.000 −0.129191
$$109$$ −389.000 −0.341830 −0.170915 0.985286i $$-0.554672\pi$$
−0.170915 + 0.985286i $$0.554672\pi$$
$$110$$ 1944.00 1.68503
$$111$$ 1250.00 1.06887
$$112$$ −781.000 −0.658907
$$113$$ −342.000 −0.284714 −0.142357 0.989815i $$-0.545468\pi$$
−0.142357 + 0.989815i $$0.545468\pi$$
$$114$$ 0 0
$$115$$ −2196.00 −1.78068
$$116$$ 249.000 0.199302
$$117$$ 22.0000 0.0173838
$$118$$ −585.000 −0.456387
$$119$$ −1023.00 −0.788053
$$120$$ 1260.00 0.958514
$$121$$ 1585.00 1.19083
$$122$$ −1074.00 −0.797011
$$123$$ −1200.00 −0.879678
$$124$$ −56.0000 −0.0405560
$$125$$ 1272.00 0.910169
$$126$$ −66.0000 −0.0466647
$$127$$ 1150.00 0.803512 0.401756 0.915747i $$-0.368400\pi$$
0.401756 + 0.915747i $$0.368400\pi$$
$$128$$ 1659.00 1.14560
$$129$$ −980.000 −0.668870
$$130$$ 396.000 0.267165
$$131$$ −1452.00 −0.968411 −0.484205 0.874954i $$-0.660891\pi$$
−0.484205 + 0.874954i $$0.660891\pi$$
$$132$$ −270.000 −0.178034
$$133$$ 0 0
$$134$$ 2883.00 1.85861
$$135$$ 1740.00 1.10930
$$136$$ 1953.00 1.23139
$$137$$ −1689.00 −1.05329 −0.526646 0.850085i $$-0.676551\pi$$
−0.526646 + 0.850085i $$0.676551\pi$$
$$138$$ 2745.00 1.69326
$$139$$ 2144.00 1.30829 0.654143 0.756371i $$-0.273030\pi$$
0.654143 + 0.756371i $$0.273030\pi$$
$$140$$ −132.000 −0.0796860
$$141$$ −840.000 −0.501708
$$142$$ 738.000 0.436138
$$143$$ 594.000 0.347362
$$144$$ 142.000 0.0821759
$$145$$ −2988.00 −1.71131
$$146$$ 1059.00 0.600298
$$147$$ −1110.00 −0.622798
$$148$$ 250.000 0.138850
$$149$$ −3000.00 −1.64946 −0.824730 0.565527i $$-0.808673\pi$$
−0.824730 + 0.565527i $$0.808673\pi$$
$$150$$ 285.000 0.155134
$$151$$ 1006.00 0.542166 0.271083 0.962556i $$-0.412618\pi$$
0.271083 + 0.962556i $$0.412618\pi$$
$$152$$ 0 0
$$153$$ 186.000 0.0982824
$$154$$ −1782.00 −0.932452
$$155$$ 672.000 0.348234
$$156$$ −55.0000 −0.0282277
$$157$$ 2846.00 1.44672 0.723362 0.690469i $$-0.242596\pi$$
0.723362 + 0.690469i $$0.242596\pi$$
$$158$$ 102.000 0.0513588
$$159$$ −2175.00 −1.08483
$$160$$ 540.000 0.266817
$$161$$ 2013.00 0.985383
$$162$$ −2013.00 −0.976273
$$163$$ −1600.00 −0.768845 −0.384422 0.923157i $$-0.625599\pi$$
−0.384422 + 0.923157i $$0.625599\pi$$
$$164$$ −240.000 −0.114273
$$165$$ 3240.00 1.52869
$$166$$ 702.000 0.328228
$$167$$ 2004.00 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ −1155.00 −0.530418
$$169$$ −2076.00 −0.944925
$$170$$ 3348.00 1.51047
$$171$$ 0 0
$$172$$ −196.000 −0.0868887
$$173$$ 462.000 0.203036 0.101518 0.994834i $$-0.467630\pi$$
0.101518 + 0.994834i $$0.467630\pi$$
$$174$$ 3735.00 1.62730
$$175$$ 209.000 0.0902795
$$176$$ 3834.00 1.64204
$$177$$ −975.000 −0.414042
$$178$$ 504.000 0.212227
$$179$$ −720.000 −0.300644 −0.150322 0.988637i $$-0.548031\pi$$
−0.150322 + 0.988637i $$0.548031\pi$$
$$180$$ 24.0000 0.00993808
$$181$$ 2338.00 0.960122 0.480061 0.877235i $$-0.340614\pi$$
0.480061 + 0.877235i $$0.340614\pi$$
$$182$$ −363.000 −0.147843
$$183$$ −1790.00 −0.723063
$$184$$ −3843.00 −1.53973
$$185$$ −3000.00 −1.19224
$$186$$ −840.000 −0.331139
$$187$$ 5022.00 1.96388
$$188$$ −168.000 −0.0651737
$$189$$ −1595.00 −0.613858
$$190$$ 0 0
$$191$$ 2871.00 1.08763 0.543817 0.839204i $$-0.316978\pi$$
0.543817 + 0.839204i $$0.316978\pi$$
$$192$$ 2165.00 0.813778
$$193$$ −1658.00 −0.618370 −0.309185 0.951002i $$-0.600056\pi$$
−0.309185 + 0.951002i $$0.600056\pi$$
$$194$$ −2274.00 −0.841565
$$195$$ 660.000 0.242377
$$196$$ −222.000 −0.0809038
$$197$$ −4176.00 −1.51029 −0.755146 0.655556i $$-0.772434\pi$$
−0.755146 + 0.655556i $$0.772434\pi$$
$$198$$ 324.000 0.116291
$$199$$ −241.000 −0.0858494 −0.0429247 0.999078i $$-0.513668\pi$$
−0.0429247 + 0.999078i $$0.513668\pi$$
$$200$$ −399.000 −0.141068
$$201$$ 4805.00 1.68616
$$202$$ −2178.00 −0.758631
$$203$$ 2739.00 0.946996
$$204$$ −465.000 −0.159591
$$205$$ 2880.00 0.981209
$$206$$ −6.00000 −0.00202932
$$207$$ −366.000 −0.122893
$$208$$ 781.000 0.260349
$$209$$ 0 0
$$210$$ −1980.00 −0.650633
$$211$$ 745.000 0.243071 0.121535 0.992587i $$-0.461218\pi$$
0.121535 + 0.992587i $$0.461218\pi$$
$$212$$ −435.000 −0.140924
$$213$$ 1230.00 0.395672
$$214$$ −4239.00 −1.35408
$$215$$ 2352.00 0.746070
$$216$$ 3045.00 0.959194
$$217$$ −616.000 −0.192704
$$218$$ −1167.00 −0.362565
$$219$$ 1765.00 0.544601
$$220$$ 648.000 0.198583
$$221$$ 1023.00 0.311377
$$222$$ 3750.00 1.13371
$$223$$ 1978.00 0.593976 0.296988 0.954881i $$-0.404018\pi$$
0.296988 + 0.954881i $$0.404018\pi$$
$$224$$ −495.000 −0.147650
$$225$$ −38.0000 −0.0112593
$$226$$ −1026.00 −0.301985
$$227$$ −5355.00 −1.56574 −0.782872 0.622183i $$-0.786246\pi$$
−0.782872 + 0.622183i $$0.786246\pi$$
$$228$$ 0 0
$$229$$ −6370.00 −1.83817 −0.919086 0.394057i $$-0.871071\pi$$
−0.919086 + 0.394057i $$0.871071\pi$$
$$230$$ −6588.00 −1.88870
$$231$$ −2970.00 −0.845938
$$232$$ −5229.00 −1.47974
$$233$$ −2838.00 −0.797955 −0.398978 0.916961i $$-0.630635\pi$$
−0.398978 + 0.916961i $$0.630635\pi$$
$$234$$ 66.0000 0.0184383
$$235$$ 2016.00 0.559614
$$236$$ −195.000 −0.0537857
$$237$$ 170.000 0.0465936
$$238$$ −3069.00 −0.835856
$$239$$ −369.000 −0.0998687 −0.0499344 0.998753i $$-0.515901\pi$$
−0.0499344 + 0.998753i $$0.515901\pi$$
$$240$$ 4260.00 1.14576
$$241$$ −6608.00 −1.76622 −0.883109 0.469167i $$-0.844554\pi$$
−0.883109 + 0.469167i $$0.844554\pi$$
$$242$$ 4755.00 1.26307
$$243$$ 560.000 0.147835
$$244$$ −358.000 −0.0939287
$$245$$ 2664.00 0.694680
$$246$$ −3600.00 −0.933039
$$247$$ 0 0
$$248$$ 1176.00 0.301113
$$249$$ 1170.00 0.297774
$$250$$ 3816.00 0.965380
$$251$$ 4674.00 1.17538 0.587690 0.809086i $$-0.300038\pi$$
0.587690 + 0.809086i $$0.300038\pi$$
$$252$$ −22.0000 −0.00549948
$$253$$ −9882.00 −2.45564
$$254$$ 3450.00 0.852253
$$255$$ 5580.00 1.37033
$$256$$ 1513.00 0.369385
$$257$$ −4512.00 −1.09514 −0.547570 0.836760i $$-0.684447\pi$$
−0.547570 + 0.836760i $$0.684447\pi$$
$$258$$ −2940.00 −0.709443
$$259$$ 2750.00 0.659756
$$260$$ 132.000 0.0314857
$$261$$ −498.000 −0.118105
$$262$$ −4356.00 −1.02715
$$263$$ 3768.00 0.883440 0.441720 0.897153i $$-0.354368\pi$$
0.441720 + 0.897153i $$0.354368\pi$$
$$264$$ 5670.00 1.32183
$$265$$ 5220.00 1.21005
$$266$$ 0 0
$$267$$ 840.000 0.192536
$$268$$ 961.000 0.219039
$$269$$ −4758.00 −1.07844 −0.539220 0.842165i $$-0.681281\pi$$
−0.539220 + 0.842165i $$0.681281\pi$$
$$270$$ 5220.00 1.17659
$$271$$ −2041.00 −0.457498 −0.228749 0.973485i $$-0.573463\pi$$
−0.228749 + 0.973485i $$0.573463\pi$$
$$272$$ 6603.00 1.47193
$$273$$ −605.000 −0.134126
$$274$$ −5067.00 −1.11718
$$275$$ −1026.00 −0.224982
$$276$$ 915.000 0.199553
$$277$$ 1964.00 0.426012 0.213006 0.977051i $$-0.431675\pi$$
0.213006 + 0.977051i $$0.431675\pi$$
$$278$$ 6432.00 1.38765
$$279$$ 112.000 0.0240332
$$280$$ 2772.00 0.591638
$$281$$ 5496.00 1.16678 0.583388 0.812194i $$-0.301727\pi$$
0.583388 + 0.812194i $$0.301727\pi$$
$$282$$ −2520.00 −0.532141
$$283$$ 3098.00 0.650731 0.325366 0.945588i $$-0.394513\pi$$
0.325366 + 0.945588i $$0.394513\pi$$
$$284$$ 246.000 0.0513993
$$285$$ 0 0
$$286$$ 1782.00 0.368433
$$287$$ −2640.00 −0.542977
$$288$$ 90.0000 0.0184142
$$289$$ 3736.00 0.760432
$$290$$ −8964.00 −1.81512
$$291$$ −3790.00 −0.763484
$$292$$ 353.000 0.0707458
$$293$$ −117.000 −0.0233284 −0.0116642 0.999932i $$-0.503713\pi$$
−0.0116642 + 0.999932i $$0.503713\pi$$
$$294$$ −3330.00 −0.660577
$$295$$ 2340.00 0.461831
$$296$$ −5250.00 −1.03091
$$297$$ 7830.00 1.52977
$$298$$ −9000.00 −1.74952
$$299$$ −2013.00 −0.389347
$$300$$ 95.0000 0.0182828
$$301$$ −2156.00 −0.412856
$$302$$ 3018.00 0.575054
$$303$$ −3630.00 −0.688244
$$304$$ 0 0
$$305$$ 4296.00 0.806519
$$306$$ 558.000 0.104244
$$307$$ 1420.00 0.263986 0.131993 0.991251i $$-0.457862\pi$$
0.131993 + 0.991251i $$0.457862\pi$$
$$308$$ −594.000 −0.109891
$$309$$ −10.0000 −0.00184104
$$310$$ 2016.00 0.369358
$$311$$ −6561.00 −1.19627 −0.598135 0.801395i $$-0.704091\pi$$
−0.598135 + 0.801395i $$0.704091\pi$$
$$312$$ 1155.00 0.209580
$$313$$ −1483.00 −0.267809 −0.133904 0.990994i $$-0.542751\pi$$
−0.133904 + 0.990994i $$0.542751\pi$$
$$314$$ 8538.00 1.53448
$$315$$ 264.000 0.0472213
$$316$$ 34.0000 0.00605269
$$317$$ 1239.00 0.219524 0.109762 0.993958i $$-0.464991\pi$$
0.109762 + 0.993958i $$0.464991\pi$$
$$318$$ −6525.00 −1.15064
$$319$$ −13446.0 −2.35997
$$320$$ −5196.00 −0.907704
$$321$$ −7065.00 −1.22844
$$322$$ 6039.00 1.04516
$$323$$ 0 0
$$324$$ −671.000 −0.115055
$$325$$ −209.000 −0.0356715
$$326$$ −4800.00 −0.815483
$$327$$ −1945.00 −0.328926
$$328$$ 5040.00 0.848437
$$329$$ −1848.00 −0.309676
$$330$$ 9720.00 1.62142
$$331$$ 8899.00 1.47774 0.738872 0.673846i $$-0.235359\pi$$
0.738872 + 0.673846i $$0.235359\pi$$
$$332$$ 234.000 0.0386820
$$333$$ −500.000 −0.0822818
$$334$$ 6012.00 0.984916
$$335$$ −11532.0 −1.88078
$$336$$ −3905.00 −0.634033
$$337$$ −5816.00 −0.940112 −0.470056 0.882637i $$-0.655766\pi$$
−0.470056 + 0.882637i $$0.655766\pi$$
$$338$$ −6228.00 −1.00224
$$339$$ −1710.00 −0.273966
$$340$$ 1116.00 0.178011
$$341$$ 3024.00 0.480231
$$342$$ 0 0
$$343$$ −6215.00 −0.978363
$$344$$ 4116.00 0.645116
$$345$$ −10980.0 −1.71346
$$346$$ 1386.00 0.215352
$$347$$ −1578.00 −0.244125 −0.122063 0.992522i $$-0.538951\pi$$
−0.122063 + 0.992522i $$0.538951\pi$$
$$348$$ 1245.00 0.191779
$$349$$ 1658.00 0.254300 0.127150 0.991883i $$-0.459417\pi$$
0.127150 + 0.991883i $$0.459417\pi$$
$$350$$ 627.000 0.0957559
$$351$$ 1595.00 0.242549
$$352$$ 2430.00 0.367953
$$353$$ −11367.0 −1.71389 −0.856947 0.515405i $$-0.827641\pi$$
−0.856947 + 0.515405i $$0.827641\pi$$
$$354$$ −2925.00 −0.439158
$$355$$ −2952.00 −0.441341
$$356$$ 168.000 0.0250112
$$357$$ −5115.00 −0.758304
$$358$$ −2160.00 −0.318881
$$359$$ 2553.00 0.375326 0.187663 0.982233i $$-0.439909\pi$$
0.187663 + 0.982233i $$0.439909\pi$$
$$360$$ −504.000 −0.0737865
$$361$$ 0 0
$$362$$ 7014.00 1.01836
$$363$$ 7925.00 1.14588
$$364$$ −121.000 −0.0174234
$$365$$ −4236.00 −0.607459
$$366$$ −5370.00 −0.766925
$$367$$ −196.000 −0.0278777 −0.0139389 0.999903i $$-0.504437\pi$$
−0.0139389 + 0.999903i $$0.504437\pi$$
$$368$$ −12993.0 −1.84051
$$369$$ 480.000 0.0677176
$$370$$ −9000.00 −1.26456
$$371$$ −4785.00 −0.669609
$$372$$ −280.000 −0.0390251
$$373$$ −9353.00 −1.29834 −0.649169 0.760644i $$-0.724883\pi$$
−0.649169 + 0.760644i $$0.724883\pi$$
$$374$$ 15066.0 2.08301
$$375$$ 6360.00 0.875811
$$376$$ 3528.00 0.483890
$$377$$ −2739.00 −0.374180
$$378$$ −4785.00 −0.651095
$$379$$ −3827.00 −0.518680 −0.259340 0.965786i $$-0.583505\pi$$
−0.259340 + 0.965786i $$0.583505\pi$$
$$380$$ 0 0
$$381$$ 5750.00 0.773180
$$382$$ 8613.00 1.15361
$$383$$ −5694.00 −0.759660 −0.379830 0.925056i $$-0.624018\pi$$
−0.379830 + 0.925056i $$0.624018\pi$$
$$384$$ 8295.00 1.10235
$$385$$ 7128.00 0.943575
$$386$$ −4974.00 −0.655881
$$387$$ 392.000 0.0514896
$$388$$ −758.000 −0.0991794
$$389$$ 1290.00 0.168138 0.0840689 0.996460i $$-0.473208\pi$$
0.0840689 + 0.996460i $$0.473208\pi$$
$$390$$ 1980.00 0.257080
$$391$$ −17019.0 −2.20125
$$392$$ 4662.00 0.600680
$$393$$ −7260.00 −0.931854
$$394$$ −12528.0 −1.60191
$$395$$ −408.000 −0.0519714
$$396$$ 108.000 0.0137051
$$397$$ 6536.00 0.826278 0.413139 0.910668i $$-0.364432\pi$$
0.413139 + 0.910668i $$0.364432\pi$$
$$398$$ −723.000 −0.0910571
$$399$$ 0 0
$$400$$ −1349.00 −0.168625
$$401$$ −2328.00 −0.289912 −0.144956 0.989438i $$-0.546304\pi$$
−0.144956 + 0.989438i $$0.546304\pi$$
$$402$$ 14415.0 1.78844
$$403$$ 616.000 0.0761418
$$404$$ −726.000 −0.0894056
$$405$$ 8052.00 0.987919
$$406$$ 8217.00 1.00444
$$407$$ −13500.0 −1.64415
$$408$$ 9765.00 1.18490
$$409$$ 6676.00 0.807107 0.403554 0.914956i $$-0.367775\pi$$
0.403554 + 0.914956i $$0.367775\pi$$
$$410$$ 8640.00 1.04073
$$411$$ −8445.00 −1.01353
$$412$$ −2.00000 −0.000239158 0
$$413$$ −2145.00 −0.255565
$$414$$ −1098.00 −0.130347
$$415$$ −2808.00 −0.332143
$$416$$ 495.000 0.0583398
$$417$$ 10720.0 1.25890
$$418$$ 0 0
$$419$$ −8136.00 −0.948615 −0.474307 0.880359i $$-0.657301\pi$$
−0.474307 + 0.880359i $$0.657301\pi$$
$$420$$ −660.000 −0.0766779
$$421$$ 8665.00 1.00310 0.501551 0.865128i $$-0.332763\pi$$
0.501551 + 0.865128i $$0.332763\pi$$
$$422$$ 2235.00 0.257815
$$423$$ 336.000 0.0386215
$$424$$ 9135.00 1.04631
$$425$$ −1767.00 −0.201676
$$426$$ 3690.00 0.419674
$$427$$ −3938.00 −0.446307
$$428$$ −1413.00 −0.159579
$$429$$ 2970.00 0.334249
$$430$$ 7056.00 0.791327
$$431$$ −750.000 −0.0838196 −0.0419098 0.999121i $$-0.513344\pi$$
−0.0419098 + 0.999121i $$0.513344\pi$$
$$432$$ 10295.0 1.14657
$$433$$ 4858.00 0.539170 0.269585 0.962977i $$-0.413113\pi$$
0.269585 + 0.962977i $$0.413113\pi$$
$$434$$ −1848.00 −0.204394
$$435$$ −14940.0 −1.64671
$$436$$ −389.000 −0.0427287
$$437$$ 0 0
$$438$$ 5295.00 0.577637
$$439$$ −6500.00 −0.706670 −0.353335 0.935497i $$-0.614952\pi$$
−0.353335 + 0.935497i $$0.614952\pi$$
$$440$$ −13608.0 −1.47440
$$441$$ 444.000 0.0479430
$$442$$ 3069.00 0.330266
$$443$$ 3486.00 0.373871 0.186936 0.982372i $$-0.440144\pi$$
0.186936 + 0.982372i $$0.440144\pi$$
$$444$$ 1250.00 0.133609
$$445$$ −2016.00 −0.214759
$$446$$ 5934.00 0.630007
$$447$$ −15000.0 −1.58719
$$448$$ 4763.00 0.502300
$$449$$ 15030.0 1.57975 0.789877 0.613265i $$-0.210144\pi$$
0.789877 + 0.613265i $$0.210144\pi$$
$$450$$ −114.000 −0.0119422
$$451$$ 12960.0 1.35313
$$452$$ −342.000 −0.0355892
$$453$$ 5030.00 0.521700
$$454$$ −16065.0 −1.66072
$$455$$ 1452.00 0.149606
$$456$$ 0 0
$$457$$ −2959.00 −0.302880 −0.151440 0.988466i $$-0.548391\pi$$
−0.151440 + 0.988466i $$0.548391\pi$$
$$458$$ −19110.0 −1.94968
$$459$$ 13485.0 1.37130
$$460$$ −2196.00 −0.222585
$$461$$ −156.000 −0.0157606 −0.00788031 0.999969i $$-0.502508\pi$$
−0.00788031 + 0.999969i $$0.502508\pi$$
$$462$$ −8910.00 −0.897253
$$463$$ 4484.00 0.450085 0.225042 0.974349i $$-0.427748\pi$$
0.225042 + 0.974349i $$0.427748\pi$$
$$464$$ −17679.0 −1.76881
$$465$$ 3360.00 0.335089
$$466$$ −8514.00 −0.846359
$$467$$ 8766.00 0.868613 0.434306 0.900765i $$-0.356994\pi$$
0.434306 + 0.900765i $$0.356994\pi$$
$$468$$ 22.0000 0.00217297
$$469$$ 10571.0 1.04077
$$470$$ 6048.00 0.593561
$$471$$ 14230.0 1.39211
$$472$$ 4095.00 0.399338
$$473$$ 10584.0 1.02886
$$474$$ 510.000 0.0494200
$$475$$ 0 0
$$476$$ −1023.00 −0.0985066
$$477$$ 870.000 0.0835106
$$478$$ −1107.00 −0.105927
$$479$$ −18996.0 −1.81200 −0.906001 0.423275i $$-0.860881\pi$$
−0.906001 + 0.423275i $$0.860881\pi$$
$$480$$ 2700.00 0.256745
$$481$$ −2750.00 −0.260684
$$482$$ −19824.0 −1.87336
$$483$$ 10065.0 0.948185
$$484$$ 1585.00 0.148854
$$485$$ 9096.00 0.851604
$$486$$ 1680.00 0.156803
$$487$$ 7450.00 0.693207 0.346603 0.938012i $$-0.387335\pi$$
0.346603 + 0.938012i $$0.387335\pi$$
$$488$$ 7518.00 0.697385
$$489$$ −8000.00 −0.739821
$$490$$ 7992.00 0.736820
$$491$$ 6180.00 0.568023 0.284012 0.958821i $$-0.408335\pi$$
0.284012 + 0.958821i $$0.408335\pi$$
$$492$$ −1200.00 −0.109960
$$493$$ −23157.0 −2.11549
$$494$$ 0 0
$$495$$ −1296.00 −0.117679
$$496$$ 3976.00 0.359935
$$497$$ 2706.00 0.244227
$$498$$ 3510.00 0.315837
$$499$$ 2576.00 0.231097 0.115549 0.993302i $$-0.463137\pi$$
0.115549 + 0.993302i $$0.463137\pi$$
$$500$$ 1272.00 0.113771
$$501$$ 10020.0 0.893534
$$502$$ 14022.0 1.24668
$$503$$ −10545.0 −0.934748 −0.467374 0.884060i $$-0.654800\pi$$
−0.467374 + 0.884060i $$0.654800\pi$$
$$504$$ 462.000 0.0408316
$$505$$ 8712.00 0.767681
$$506$$ −29646.0 −2.60460
$$507$$ −10380.0 −0.909254
$$508$$ 1150.00 0.100439
$$509$$ 14694.0 1.27957 0.639784 0.768555i $$-0.279024\pi$$
0.639784 + 0.768555i $$0.279024\pi$$
$$510$$ 16740.0 1.45345
$$511$$ 3883.00 0.336152
$$512$$ −8733.00 −0.753804
$$513$$ 0 0
$$514$$ −13536.0 −1.16157
$$515$$ 24.0000 0.00205353
$$516$$ −980.000 −0.0836087
$$517$$ 9072.00 0.771733
$$518$$ 8250.00 0.699776
$$519$$ 2310.00 0.195371
$$520$$ −2772.00 −0.233770
$$521$$ −10332.0 −0.868816 −0.434408 0.900716i $$-0.643042\pi$$
−0.434408 + 0.900716i $$0.643042\pi$$
$$522$$ −1494.00 −0.125269
$$523$$ −10937.0 −0.914420 −0.457210 0.889359i $$-0.651151\pi$$
−0.457210 + 0.889359i $$0.651151\pi$$
$$524$$ −1452.00 −0.121051
$$525$$ 1045.00 0.0868715
$$526$$ 11304.0 0.937030
$$527$$ 5208.00 0.430482
$$528$$ 19170.0 1.58005
$$529$$ 21322.0 1.75245
$$530$$ 15660.0 1.28345
$$531$$ 390.000 0.0318730
$$532$$ 0 0
$$533$$ 2640.00 0.214542
$$534$$ 2520.00 0.204215
$$535$$ 16956.0 1.37023
$$536$$ −20181.0 −1.62628
$$537$$ −3600.00 −0.289295
$$538$$ −14274.0 −1.14386
$$539$$ 11988.0 0.957996
$$540$$ 1740.00 0.138662
$$541$$ 18578.0 1.47640 0.738198 0.674584i $$-0.235677\pi$$
0.738198 + 0.674584i $$0.235677\pi$$
$$542$$ −6123.00 −0.485250
$$543$$ 11690.0 0.923878
$$544$$ 4185.00 0.329835
$$545$$ 4668.00 0.366890
$$546$$ −1815.00 −0.142262
$$547$$ −21404.0 −1.67307 −0.836535 0.547914i $$-0.815422\pi$$
−0.836535 + 0.547914i $$0.815422\pi$$
$$548$$ −1689.00 −0.131662
$$549$$ 716.000 0.0556614
$$550$$ −3078.00 −0.238630
$$551$$ 0 0
$$552$$ −19215.0 −1.48160
$$553$$ 374.000 0.0287597
$$554$$ 5892.00 0.451854
$$555$$ −15000.0 −1.14723
$$556$$ 2144.00 0.163536
$$557$$ −3948.00 −0.300327 −0.150163 0.988661i $$-0.547980\pi$$
−0.150163 + 0.988661i $$0.547980\pi$$
$$558$$ 336.000 0.0254911
$$559$$ 2156.00 0.163129
$$560$$ 9372.00 0.707213
$$561$$ 25110.0 1.88974
$$562$$ 16488.0 1.23755
$$563$$ −5724.00 −0.428486 −0.214243 0.976780i $$-0.568729\pi$$
−0.214243 + 0.976780i $$0.568729\pi$$
$$564$$ −840.000 −0.0627134
$$565$$ 4104.00 0.305587
$$566$$ 9294.00 0.690205
$$567$$ −7381.00 −0.546689
$$568$$ −5166.00 −0.381621
$$569$$ 20592.0 1.51716 0.758578 0.651582i $$-0.225895\pi$$
0.758578 + 0.651582i $$0.225895\pi$$
$$570$$ 0 0
$$571$$ 20684.0 1.51593 0.757967 0.652293i $$-0.226193\pi$$
0.757967 + 0.652293i $$0.226193\pi$$
$$572$$ 594.000 0.0434203
$$573$$ 14355.0 1.04658
$$574$$ −7920.00 −0.575914
$$575$$ 3477.00 0.252176
$$576$$ −866.000 −0.0626447
$$577$$ −19573.0 −1.41219 −0.706096 0.708116i $$-0.749545\pi$$
−0.706096 + 0.708116i $$0.749545\pi$$
$$578$$ 11208.0 0.806559
$$579$$ −8290.00 −0.595027
$$580$$ −2988.00 −0.213914
$$581$$ 2574.00 0.183800
$$582$$ −11370.0 −0.809797
$$583$$ 23490.0 1.66871
$$584$$ −7413.00 −0.525260
$$585$$ −264.000 −0.0186582
$$586$$ −351.000 −0.0247435
$$587$$ 13524.0 0.950929 0.475464 0.879735i $$-0.342280\pi$$
0.475464 + 0.879735i $$0.342280\pi$$
$$588$$ −1110.00 −0.0778497
$$589$$ 0 0
$$590$$ 7020.00 0.489845
$$591$$ −20880.0 −1.45328
$$592$$ −17750.0 −1.23230
$$593$$ 8994.00 0.622832 0.311416 0.950274i $$-0.399197\pi$$
0.311416 + 0.950274i $$0.399197\pi$$
$$594$$ 23490.0 1.62257
$$595$$ 12276.0 0.845827
$$596$$ −3000.00 −0.206183
$$597$$ −1205.00 −0.0826087
$$598$$ −6039.00 −0.412965
$$599$$ −10128.0 −0.690850 −0.345425 0.938446i $$-0.612265\pi$$
−0.345425 + 0.938446i $$0.612265\pi$$
$$600$$ −1995.00 −0.135743
$$601$$ 22696.0 1.54041 0.770207 0.637794i $$-0.220153\pi$$
0.770207 + 0.637794i $$0.220153\pi$$
$$602$$ −6468.00 −0.437900
$$603$$ −1922.00 −0.129801
$$604$$ 1006.00 0.0677708
$$605$$ −19020.0 −1.27814
$$606$$ −10890.0 −0.729993
$$607$$ 5182.00 0.346509 0.173254 0.984877i $$-0.444572\pi$$
0.173254 + 0.984877i $$0.444572\pi$$
$$608$$ 0 0
$$609$$ 13695.0 0.911247
$$610$$ 12888.0 0.855442
$$611$$ 1848.00 0.122360
$$612$$ 186.000 0.0122853
$$613$$ 10082.0 0.664287 0.332144 0.943229i $$-0.392228\pi$$
0.332144 + 0.943229i $$0.392228\pi$$
$$614$$ 4260.00 0.279999
$$615$$ 14400.0 0.944169
$$616$$ 12474.0 0.815896
$$617$$ −12174.0 −0.794338 −0.397169 0.917745i $$-0.630007\pi$$
−0.397169 + 0.917745i $$0.630007\pi$$
$$618$$ −30.0000 −0.00195271
$$619$$ 7490.00 0.486347 0.243173 0.969983i $$-0.421812\pi$$
0.243173 + 0.969983i $$0.421812\pi$$
$$620$$ 672.000 0.0435293
$$621$$ −26535.0 −1.71467
$$622$$ −19683.0 −1.26884
$$623$$ 1848.00 0.118842
$$624$$ 3905.00 0.250521
$$625$$ −17639.0 −1.12890
$$626$$ −4449.00 −0.284054
$$627$$ 0 0
$$628$$ 2846.00 0.180840
$$629$$ −23250.0 −1.47383
$$630$$ 792.000 0.0500858
$$631$$ 11072.0 0.698525 0.349263 0.937025i $$-0.386432\pi$$
0.349263 + 0.937025i $$0.386432\pi$$
$$632$$ −714.000 −0.0449389
$$633$$ 3725.00 0.233895
$$634$$ 3717.00 0.232841
$$635$$ −13800.0 −0.862419
$$636$$ −2175.00 −0.135604
$$637$$ 2442.00 0.151893
$$638$$ −40338.0 −2.50313
$$639$$ −492.000 −0.0304589
$$640$$ −19908.0 −1.22958
$$641$$ 18894.0 1.16422 0.582112 0.813108i $$-0.302226\pi$$
0.582112 + 0.813108i $$0.302226\pi$$
$$642$$ −21195.0 −1.30296
$$643$$ −19834.0 −1.21645 −0.608224 0.793765i $$-0.708118\pi$$
−0.608224 + 0.793765i $$0.708118\pi$$
$$644$$ 2013.00 0.123173
$$645$$ 11760.0 0.717906
$$646$$ 0 0
$$647$$ 3375.00 0.205077 0.102539 0.994729i $$-0.467303\pi$$
0.102539 + 0.994729i $$0.467303\pi$$
$$648$$ 14091.0 0.854239
$$649$$ 10530.0 0.636885
$$650$$ −627.000 −0.0378353
$$651$$ −3080.00 −0.185430
$$652$$ −1600.00 −0.0961056
$$653$$ −24948.0 −1.49509 −0.747543 0.664214i $$-0.768766\pi$$
−0.747543 + 0.664214i $$0.768766\pi$$
$$654$$ −5835.00 −0.348879
$$655$$ 17424.0 1.03941
$$656$$ 17040.0 1.01418
$$657$$ −706.000 −0.0419234
$$658$$ −5544.00 −0.328461
$$659$$ 9879.00 0.583962 0.291981 0.956424i $$-0.405686\pi$$
0.291981 + 0.956424i $$0.405686\pi$$
$$660$$ 3240.00 0.191086
$$661$$ 14155.0 0.832928 0.416464 0.909152i $$-0.363269\pi$$
0.416464 + 0.909152i $$0.363269\pi$$
$$662$$ 26697.0 1.56738
$$663$$ 5115.00 0.299623
$$664$$ −4914.00 −0.287199
$$665$$ 0 0
$$666$$ −1500.00 −0.0872730
$$667$$ 45567.0 2.64522
$$668$$ 2004.00 0.116073
$$669$$ 9890.00 0.571554
$$670$$ −34596.0 −1.99487
$$671$$ 19332.0 1.11223
$$672$$ −2475.00 −0.142076
$$673$$ −8948.00 −0.512511 −0.256256 0.966609i $$-0.582489\pi$$
−0.256256 + 0.966609i $$0.582489\pi$$
$$674$$ −17448.0 −0.997139
$$675$$ −2755.00 −0.157096
$$676$$ −2076.00 −0.118116
$$677$$ 11511.0 0.653477 0.326738 0.945115i $$-0.394050\pi$$
0.326738 + 0.945115i $$0.394050\pi$$
$$678$$ −5130.00 −0.290585
$$679$$ −8338.00 −0.471256
$$680$$ −23436.0 −1.32166
$$681$$ −26775.0 −1.50664
$$682$$ 9072.00 0.509362
$$683$$ 10476.0 0.586900 0.293450 0.955974i $$-0.405197\pi$$
0.293450 + 0.955974i $$0.405197\pi$$
$$684$$ 0 0
$$685$$ 20268.0 1.13051
$$686$$ −18645.0 −1.03771
$$687$$ −31850.0 −1.76878
$$688$$ 13916.0 0.771137
$$689$$ 4785.00 0.264578
$$690$$ −32940.0 −1.81740
$$691$$ 30098.0 1.65699 0.828496 0.559995i $$-0.189197\pi$$
0.828496 + 0.559995i $$0.189197\pi$$
$$692$$ 462.000 0.0253795
$$693$$ 1188.00 0.0651203
$$694$$ −4734.00 −0.258934
$$695$$ −25728.0 −1.40420
$$696$$ −26145.0 −1.42388
$$697$$ 22320.0 1.21296
$$698$$ 4974.00 0.269726
$$699$$ −14190.0 −0.767833
$$700$$ 209.000 0.0112849
$$701$$ −14700.0 −0.792028 −0.396014 0.918245i $$-0.629607\pi$$
−0.396014 + 0.918245i $$0.629607\pi$$
$$702$$ 4785.00 0.257262
$$703$$ 0 0
$$704$$ −23382.0 −1.25176
$$705$$ 10080.0 0.538489
$$706$$ −34101.0 −1.81786
$$707$$ −7986.00 −0.424815
$$708$$ −975.000 −0.0517553
$$709$$ 31178.0 1.65150 0.825751 0.564035i $$-0.190752\pi$$
0.825751 + 0.564035i $$0.190752\pi$$
$$710$$ −8856.00 −0.468112
$$711$$ −68.0000 −0.00358678
$$712$$ −3528.00 −0.185699
$$713$$ −10248.0 −0.538276
$$714$$ −15345.0 −0.804303
$$715$$ −7128.00 −0.372828
$$716$$ −720.000 −0.0375805
$$717$$ −1845.00 −0.0960987
$$718$$ 7659.00 0.398094
$$719$$ −33285.0 −1.72645 −0.863227 0.504815i $$-0.831561\pi$$
−0.863227 + 0.504815i $$0.831561\pi$$
$$720$$ −1704.00 −0.0882005
$$721$$ −22.0000 −0.00113637
$$722$$ 0 0
$$723$$ −33040.0 −1.69954
$$724$$ 2338.00 0.120015
$$725$$ 4731.00 0.242352
$$726$$ 23775.0 1.21539
$$727$$ −34729.0 −1.77170 −0.885851 0.463970i $$-0.846425\pi$$
−0.885851 + 0.463970i $$0.846425\pi$$
$$728$$ 2541.00 0.129362
$$729$$ 20917.0 1.06269
$$730$$ −12708.0 −0.644307
$$731$$ 18228.0 0.922280
$$732$$ −1790.00 −0.0903829
$$733$$ 4196.00 0.211436 0.105718 0.994396i $$-0.466286\pi$$
0.105718 + 0.994396i $$0.466286\pi$$
$$734$$ −588.000 −0.0295688
$$735$$ 13320.0 0.668457
$$736$$ −8235.00 −0.412427
$$737$$ −51894.0 −2.59368
$$738$$ 1440.00 0.0718254
$$739$$ −10744.0 −0.534810 −0.267405 0.963584i $$-0.586166\pi$$
−0.267405 + 0.963584i $$0.586166\pi$$
$$740$$ −3000.00 −0.149030
$$741$$ 0 0
$$742$$ −14355.0 −0.710227
$$743$$ 2208.00 0.109022 0.0545112 0.998513i $$-0.482640\pi$$
0.0545112 + 0.998513i $$0.482640\pi$$
$$744$$ 5880.00 0.289746
$$745$$ 36000.0 1.77039
$$746$$ −28059.0 −1.37710
$$747$$ −468.000 −0.0229227
$$748$$ 5022.00 0.245485
$$749$$ −15543.0 −0.758249
$$750$$ 19080.0 0.928937
$$751$$ −13160.0 −0.639434 −0.319717 0.947513i $$-0.603588\pi$$
−0.319717 + 0.947513i $$0.603588\pi$$
$$752$$ 11928.0 0.578417
$$753$$ 23370.0 1.13101
$$754$$ −8217.00 −0.396877
$$755$$ −12072.0 −0.581914
$$756$$ −1595.00 −0.0767323
$$757$$ 758.000 0.0363936 0.0181968 0.999834i $$-0.494207\pi$$
0.0181968 + 0.999834i $$0.494207\pi$$
$$758$$ −11481.0 −0.550143
$$759$$ −49410.0 −2.36294
$$760$$ 0 0
$$761$$ 4851.00 0.231076 0.115538 0.993303i $$-0.463141\pi$$
0.115538 + 0.993303i $$0.463141\pi$$
$$762$$ 17250.0 0.820081
$$763$$ −4279.00 −0.203028
$$764$$ 2871.00 0.135954
$$765$$ −2232.00 −0.105488
$$766$$ −17082.0 −0.805741
$$767$$ 2145.00 0.100980
$$768$$ 7565.00 0.355441
$$769$$ −33091.0 −1.55175 −0.775873 0.630890i $$-0.782690\pi$$
−0.775873 + 0.630890i $$0.782690\pi$$
$$770$$ 21384.0 1.00081
$$771$$ −22560.0 −1.05380
$$772$$ −1658.00 −0.0772963
$$773$$ −42357.0 −1.97086 −0.985430 0.170079i $$-0.945598\pi$$
−0.985430 + 0.170079i $$0.945598\pi$$
$$774$$ 1176.00 0.0546130
$$775$$ −1064.00 −0.0493161
$$776$$ 15918.0 0.736370
$$777$$ 13750.0 0.634850
$$778$$ 3870.00 0.178337
$$779$$ 0 0
$$780$$ 660.000 0.0302972
$$781$$ −13284.0 −0.608629
$$782$$ −51057.0 −2.33478
$$783$$ −36105.0 −1.64788
$$784$$ 15762.0 0.718021
$$785$$ −34152.0 −1.55279
$$786$$ −21780.0 −0.988380
$$787$$ 39877.0 1.80618 0.903089 0.429454i $$-0.141294\pi$$
0.903089 + 0.429454i $$0.141294\pi$$
$$788$$ −4176.00 −0.188787
$$789$$ 18840.0 0.850091
$$790$$ −1224.00 −0.0551240
$$791$$ −3762.00 −0.169104
$$792$$ −2268.00 −0.101755
$$793$$ 3938.00 0.176346
$$794$$ 19608.0 0.876400
$$795$$ 26100.0 1.16437
$$796$$ −241.000 −0.0107312
$$797$$ 30033.0 1.33478 0.667392 0.744706i $$-0.267410\pi$$
0.667392 + 0.744706i $$0.267410\pi$$
$$798$$ 0 0
$$799$$ 15624.0 0.691786
$$800$$ −855.000 −0.0377860
$$801$$ −336.000 −0.0148214
$$802$$ −6984.00 −0.307498
$$803$$ −19062.0 −0.837713
$$804$$ 4805.00 0.210770
$$805$$ −24156.0 −1.05762
$$806$$ 1848.00 0.0807606
$$807$$ −23790.0 −1.03773
$$808$$ 15246.0 0.663802
$$809$$ 585.000 0.0254234 0.0127117 0.999919i $$-0.495954\pi$$
0.0127117 + 0.999919i $$0.495954\pi$$
$$810$$ 24156.0 1.04785
$$811$$ −28361.0 −1.22798 −0.613989 0.789315i $$-0.710436\pi$$
−0.613989 + 0.789315i $$0.710436\pi$$
$$812$$ 2739.00 0.118374
$$813$$ −10205.0 −0.440228
$$814$$ −40500.0 −1.74389
$$815$$ 19200.0 0.825211
$$816$$ 33015.0 1.41637
$$817$$ 0 0
$$818$$ 20028.0 0.856067
$$819$$ 242.000 0.0103250
$$820$$ 2880.00 0.122651
$$821$$ 25068.0 1.06563 0.532813 0.846233i $$-0.321135\pi$$
0.532813 + 0.846233i $$0.321135\pi$$
$$822$$ −25335.0 −1.07501
$$823$$ 10901.0 0.461707 0.230854 0.972989i $$-0.425848\pi$$
0.230854 + 0.972989i $$0.425848\pi$$
$$824$$ 42.0000 0.00177565
$$825$$ −5130.00 −0.216489
$$826$$ −6435.00 −0.271068
$$827$$ −12027.0 −0.505707 −0.252854 0.967505i $$-0.581369\pi$$
−0.252854 + 0.967505i $$0.581369\pi$$
$$828$$ −366.000 −0.0153616
$$829$$ 19339.0 0.810219 0.405109 0.914268i $$-0.367233\pi$$
0.405109 + 0.914268i $$0.367233\pi$$
$$830$$ −8424.00 −0.352291
$$831$$ 9820.00 0.409930
$$832$$ −4763.00 −0.198470
$$833$$ 20646.0 0.858753
$$834$$ 32160.0 1.33526
$$835$$ −24048.0 −0.996665
$$836$$ 0 0
$$837$$ 8120.00 0.335326
$$838$$ −24408.0 −1.00616
$$839$$ 13188.0 0.542670 0.271335 0.962485i $$-0.412535\pi$$
0.271335 + 0.962485i $$0.412535\pi$$
$$840$$ 13860.0 0.569304
$$841$$ 37612.0 1.54217
$$842$$ 25995.0 1.06395
$$843$$ 27480.0 1.12273
$$844$$ 745.000 0.0303838
$$845$$ 24912.0 1.01420
$$846$$ 1008.00 0.0409642
$$847$$ 17435.0 0.707289
$$848$$ 30885.0 1.25070
$$849$$ 15490.0 0.626167
$$850$$ −5301.00 −0.213909
$$851$$ 45750.0 1.84288
$$852$$ 1230.00 0.0494590
$$853$$ −4678.00 −0.187775 −0.0938873 0.995583i $$-0.529929\pi$$
−0.0938873 + 0.995583i $$0.529929\pi$$
$$854$$ −11814.0 −0.473380
$$855$$ 0 0
$$856$$ 29673.0 1.18482
$$857$$ −15252.0 −0.607933 −0.303966 0.952683i $$-0.598311\pi$$
−0.303966 + 0.952683i $$0.598311\pi$$
$$858$$ 8910.00 0.354525
$$859$$ −610.000 −0.0242293 −0.0121146 0.999927i $$-0.503856\pi$$
−0.0121146 + 0.999927i $$0.503856\pi$$
$$860$$ 2352.00 0.0932588
$$861$$ −13200.0 −0.522479
$$862$$ −2250.00 −0.0889041
$$863$$ −774.000 −0.0305299 −0.0152649 0.999883i $$-0.504859\pi$$
−0.0152649 + 0.999883i $$0.504859\pi$$
$$864$$ 6525.00 0.256927
$$865$$ −5544.00 −0.217921
$$866$$ 14574.0 0.571876
$$867$$ 18680.0 0.731726
$$868$$ −616.000 −0.0240880
$$869$$ −1836.00 −0.0716709
$$870$$ −44820.0 −1.74660
$$871$$ −10571.0 −0.411234
$$872$$ 8169.00 0.317245
$$873$$ 1516.00 0.0587730
$$874$$ 0 0
$$875$$ 13992.0 0.540590
$$876$$ 1765.00 0.0680751
$$877$$ 31039.0 1.19511 0.597556 0.801827i $$-0.296139\pi$$
0.597556 + 0.801827i $$0.296139\pi$$
$$878$$ −19500.0 −0.749537
$$879$$ −585.000 −0.0224477
$$880$$ −46008.0 −1.76242
$$881$$ 33678.0 1.28790 0.643950 0.765067i $$-0.277294\pi$$
0.643950 + 0.765067i $$0.277294\pi$$
$$882$$ 1332.00 0.0508512
$$883$$ −42982.0 −1.63812 −0.819060 0.573708i $$-0.805504\pi$$
−0.819060 + 0.573708i $$0.805504\pi$$
$$884$$ 1023.00 0.0389222
$$885$$ 11700.0 0.444397
$$886$$ 10458.0 0.396550
$$887$$ −4494.00 −0.170117 −0.0850585 0.996376i $$-0.527108\pi$$
−0.0850585 + 0.996376i $$0.527108\pi$$
$$888$$ −26250.0 −0.991996
$$889$$ 12650.0 0.477241
$$890$$ −6048.00 −0.227786
$$891$$ 36234.0 1.36238
$$892$$ 1978.00 0.0742470
$$893$$ 0 0
$$894$$ −45000.0 −1.68347
$$895$$ 8640.00 0.322685
$$896$$ 18249.0 0.680420
$$897$$ −10065.0 −0.374649
$$898$$ 45090.0 1.67558
$$899$$ −13944.0 −0.517306
$$900$$ −38.0000 −0.00140741
$$901$$ 40455.0 1.49584
$$902$$ 38880.0 1.43521
$$903$$ −10780.0 −0.397271
$$904$$ 7182.00 0.264236
$$905$$ −28056.0 −1.03051
$$906$$ 15090.0 0.553346
$$907$$ 23839.0 0.872724 0.436362 0.899771i $$-0.356267\pi$$
0.436362 + 0.899771i $$0.356267\pi$$
$$908$$ −5355.00 −0.195718
$$909$$ 1452.00 0.0529811
$$910$$ 4356.00 0.158681
$$911$$ 10332.0 0.375757 0.187878 0.982192i $$-0.439839\pi$$
0.187878 + 0.982192i $$0.439839\pi$$
$$912$$ 0 0
$$913$$ −12636.0 −0.458040
$$914$$ −8877.00 −0.321253
$$915$$ 21480.0 0.776073
$$916$$ −6370.00 −0.229772
$$917$$ −15972.0 −0.575182
$$918$$ 40455.0 1.45448
$$919$$ −14371.0 −0.515838 −0.257919 0.966166i $$-0.583037\pi$$
−0.257919 + 0.966166i $$0.583037\pi$$
$$920$$ 46116.0 1.65261
$$921$$ 7100.00 0.254021
$$922$$ −468.000 −0.0167167
$$923$$ −2706.00 −0.0964995
$$924$$ −2970.00 −0.105742
$$925$$ 4750.00 0.168842
$$926$$ 13452.0 0.477387
$$927$$ 4.00000 0.000141723 0
$$928$$ −11205.0 −0.396360
$$929$$ 26889.0 0.949623 0.474811 0.880088i $$-0.342516\pi$$
0.474811 + 0.880088i $$0.342516\pi$$
$$930$$ 10080.0 0.355415
$$931$$ 0 0
$$932$$ −2838.00 −0.0997444
$$933$$ −32805.0 −1.15111
$$934$$ 26298.0 0.921303
$$935$$ −60264.0 −2.10785
$$936$$ −462.000 −0.0161335
$$937$$ 785.000 0.0273691 0.0136845 0.999906i $$-0.495644\pi$$
0.0136845 + 0.999906i $$0.495644\pi$$
$$938$$ 31713.0 1.10391
$$939$$ −7415.00 −0.257699
$$940$$ 2016.00 0.0699518
$$941$$ 18141.0 0.628459 0.314229 0.949347i $$-0.398254\pi$$
0.314229 + 0.949347i $$0.398254\pi$$
$$942$$ 42690.0 1.47656
$$943$$ −43920.0 −1.51668
$$944$$ 13845.0 0.477348
$$945$$ 19140.0 0.658862
$$946$$ 31752.0 1.09128
$$947$$ 23100.0 0.792660 0.396330 0.918108i $$-0.370284\pi$$
0.396330 + 0.918108i $$0.370284\pi$$
$$948$$ 170.000 0.00582420
$$949$$ −3883.00 −0.132821
$$950$$ 0 0
$$951$$ 6195.00 0.211237
$$952$$ 21483.0 0.731374
$$953$$ −45690.0 −1.55304 −0.776519 0.630094i $$-0.783016\pi$$
−0.776519 + 0.630094i $$0.783016\pi$$
$$954$$ 2610.00 0.0885764
$$955$$ −34452.0 −1.16737
$$956$$ −369.000 −0.0124836
$$957$$ −67230.0 −2.27089
$$958$$ −56988.0 −1.92192
$$959$$ −18579.0 −0.625597
$$960$$ −25980.0 −0.873438
$$961$$ −26655.0 −0.894733
$$962$$ −8250.00 −0.276498
$$963$$ 2826.00 0.0945655
$$964$$ −6608.00 −0.220777
$$965$$ 19896.0 0.663705
$$966$$ 30195.0 1.00570
$$967$$ 21584.0 0.717781 0.358891 0.933380i $$-0.383155\pi$$
0.358891 + 0.933380i $$0.383155\pi$$
$$968$$ −33285.0 −1.10519
$$969$$ 0 0
$$970$$ 27288.0 0.903263
$$971$$ 50556.0 1.67087 0.835437 0.549586i $$-0.185214\pi$$
0.835437 + 0.549586i $$0.185214\pi$$
$$972$$ 560.000 0.0184794
$$973$$ 23584.0 0.777049
$$974$$ 22350.0 0.735257
$$975$$ −1045.00 −0.0343249
$$976$$ 25418.0 0.833617
$$977$$ −8568.00 −0.280568 −0.140284 0.990111i $$-0.544802\pi$$
−0.140284 + 0.990111i $$0.544802\pi$$
$$978$$ −24000.0 −0.784699
$$979$$ −9072.00 −0.296162
$$980$$ 2664.00 0.0868351
$$981$$ 778.000 0.0253207
$$982$$ 18540.0 0.602480
$$983$$ −29706.0 −0.963860 −0.481930 0.876210i $$-0.660064\pi$$
−0.481930 + 0.876210i $$0.660064\pi$$
$$984$$ 25200.0 0.816409
$$985$$ 50112.0 1.62102
$$986$$ −69471.0 −2.24382
$$987$$ −9240.00 −0.297986
$$988$$ 0 0
$$989$$ −35868.0 −1.15322
$$990$$ −3888.00 −0.124817
$$991$$ −30512.0 −0.978048 −0.489024 0.872270i $$-0.662647\pi$$
−0.489024 + 0.872270i $$0.662647\pi$$
$$992$$ 2520.00 0.0806553
$$993$$ 44495.0 1.42196
$$994$$ 8118.00 0.259042
$$995$$ 2892.00 0.0921433
$$996$$ 1170.00 0.0372218
$$997$$ 47756.0 1.51700 0.758499 0.651674i $$-0.225933\pi$$
0.758499 + 0.651674i $$0.225933\pi$$
$$998$$ 7728.00 0.245116
$$999$$ −36250.0 −1.14805
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 361.4.a.b.1.1 1
19.18 odd 2 19.4.a.a.1.1 1
57.56 even 2 171.4.a.d.1.1 1
76.75 even 2 304.4.a.b.1.1 1
95.18 even 4 475.4.b.c.324.2 2
95.37 even 4 475.4.b.c.324.1 2
95.94 odd 2 475.4.a.e.1.1 1
133.132 even 2 931.4.a.a.1.1 1
152.37 odd 2 1216.4.a.f.1.1 1
152.75 even 2 1216.4.a.a.1.1 1
209.208 even 2 2299.4.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
19.4.a.a.1.1 1 19.18 odd 2
171.4.a.d.1.1 1 57.56 even 2
304.4.a.b.1.1 1 76.75 even 2
361.4.a.b.1.1 1 1.1 even 1 trivial
475.4.a.e.1.1 1 95.94 odd 2
475.4.b.c.324.1 2 95.37 even 4
475.4.b.c.324.2 2 95.18 even 4
931.4.a.a.1.1 1 133.132 even 2
1216.4.a.a.1.1 1 152.75 even 2
1216.4.a.f.1.1 1 152.37 odd 2
2299.4.a.b.1.1 1 209.208 even 2