# Properties

 Label 825.4.a.i.1.1 Level $825$ Weight $4$ Character 825.1 Self dual yes Analytic conductor $48.677$ Analytic rank $0$ 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] = [825,4,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.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: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 825.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{2} -3.00000 q^{3} +17.0000 q^{4} -15.0000 q^{6} +32.0000 q^{7} +45.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+5.00000 q^{2} -3.00000 q^{3} +17.0000 q^{4} -15.0000 q^{6} +32.0000 q^{7} +45.0000 q^{8} +9.00000 q^{9} -11.0000 q^{11} -51.0000 q^{12} +38.0000 q^{13} +160.000 q^{14} +89.0000 q^{16} +2.00000 q^{17} +45.0000 q^{18} +72.0000 q^{19} -96.0000 q^{21} -55.0000 q^{22} -68.0000 q^{23} -135.000 q^{24} +190.000 q^{26} -27.0000 q^{27} +544.000 q^{28} -54.0000 q^{29} -152.000 q^{31} +85.0000 q^{32} +33.0000 q^{33} +10.0000 q^{34} +153.000 q^{36} -174.000 q^{37} +360.000 q^{38} -114.000 q^{39} +94.0000 q^{41} -480.000 q^{42} +528.000 q^{43} -187.000 q^{44} -340.000 q^{46} +340.000 q^{47} -267.000 q^{48} +681.000 q^{49} -6.00000 q^{51} +646.000 q^{52} +438.000 q^{53} -135.000 q^{54} +1440.00 q^{56} -216.000 q^{57} -270.000 q^{58} +20.0000 q^{59} +570.000 q^{61} -760.000 q^{62} +288.000 q^{63} -287.000 q^{64} +165.000 q^{66} +460.000 q^{67} +34.0000 q^{68} +204.000 q^{69} -1092.00 q^{71} +405.000 q^{72} -562.000 q^{73} -870.000 q^{74} +1224.00 q^{76} -352.000 q^{77} -570.000 q^{78} -16.0000 q^{79} +81.0000 q^{81} +470.000 q^{82} -372.000 q^{83} -1632.00 q^{84} +2640.00 q^{86} +162.000 q^{87} -495.000 q^{88} -966.000 q^{89} +1216.00 q^{91} -1156.00 q^{92} +456.000 q^{93} +1700.00 q^{94} -255.000 q^{96} +526.000 q^{97} +3405.00 q^{98} -99.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.00000 1.76777 0.883883 0.467707i $$-0.154920\pi$$
0.883883 + 0.467707i $$0.154920\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 17.0000 2.12500
$$5$$ 0 0
$$6$$ −15.0000 −1.02062
$$7$$ 32.0000 1.72784 0.863919 0.503631i $$-0.168003\pi$$
0.863919 + 0.503631i $$0.168003\pi$$
$$8$$ 45.0000 1.98874
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ −51.0000 −1.22687
$$13$$ 38.0000 0.810716 0.405358 0.914158i $$-0.367147\pi$$
0.405358 + 0.914158i $$0.367147\pi$$
$$14$$ 160.000 3.05441
$$15$$ 0 0
$$16$$ 89.0000 1.39062
$$17$$ 2.00000 0.0285336 0.0142668 0.999898i $$-0.495459\pi$$
0.0142668 + 0.999898i $$0.495459\pi$$
$$18$$ 45.0000 0.589256
$$19$$ 72.0000 0.869365 0.434682 0.900584i $$-0.356861\pi$$
0.434682 + 0.900584i $$0.356861\pi$$
$$20$$ 0 0
$$21$$ −96.0000 −0.997567
$$22$$ −55.0000 −0.533002
$$23$$ −68.0000 −0.616477 −0.308239 0.951309i $$-0.599740\pi$$
−0.308239 + 0.951309i $$0.599740\pi$$
$$24$$ −135.000 −1.14820
$$25$$ 0 0
$$26$$ 190.000 1.43316
$$27$$ −27.0000 −0.192450
$$28$$ 544.000 3.67165
$$29$$ −54.0000 −0.345778 −0.172889 0.984941i $$-0.555310\pi$$
−0.172889 + 0.984941i $$0.555310\pi$$
$$30$$ 0 0
$$31$$ −152.000 −0.880645 −0.440323 0.897840i $$-0.645136\pi$$
−0.440323 + 0.897840i $$0.645136\pi$$
$$32$$ 85.0000 0.469563
$$33$$ 33.0000 0.174078
$$34$$ 10.0000 0.0504408
$$35$$ 0 0
$$36$$ 153.000 0.708333
$$37$$ −174.000 −0.773120 −0.386560 0.922264i $$-0.626337\pi$$
−0.386560 + 0.922264i $$0.626337\pi$$
$$38$$ 360.000 1.53683
$$39$$ −114.000 −0.468067
$$40$$ 0 0
$$41$$ 94.0000 0.358057 0.179028 0.983844i $$-0.442705\pi$$
0.179028 + 0.983844i $$0.442705\pi$$
$$42$$ −480.000 −1.76347
$$43$$ 528.000 1.87254 0.936270 0.351280i $$-0.114254\pi$$
0.936270 + 0.351280i $$0.114254\pi$$
$$44$$ −187.000 −0.640712
$$45$$ 0 0
$$46$$ −340.000 −1.08979
$$47$$ 340.000 1.05519 0.527597 0.849495i $$-0.323093\pi$$
0.527597 + 0.849495i $$0.323093\pi$$
$$48$$ −267.000 −0.802878
$$49$$ 681.000 1.98542
$$50$$ 0 0
$$51$$ −6.00000 −0.0164739
$$52$$ 646.000 1.72277
$$53$$ 438.000 1.13517 0.567584 0.823315i $$-0.307878\pi$$
0.567584 + 0.823315i $$0.307878\pi$$
$$54$$ −135.000 −0.340207
$$55$$ 0 0
$$56$$ 1440.00 3.43622
$$57$$ −216.000 −0.501928
$$58$$ −270.000 −0.611254
$$59$$ 20.0000 0.0441318 0.0220659 0.999757i $$-0.492976\pi$$
0.0220659 + 0.999757i $$0.492976\pi$$
$$60$$ 0 0
$$61$$ 570.000 1.19641 0.598205 0.801343i $$-0.295881\pi$$
0.598205 + 0.801343i $$0.295881\pi$$
$$62$$ −760.000 −1.55678
$$63$$ 288.000 0.575946
$$64$$ −287.000 −0.560547
$$65$$ 0 0
$$66$$ 165.000 0.307729
$$67$$ 460.000 0.838775 0.419388 0.907807i $$-0.362245\pi$$
0.419388 + 0.907807i $$0.362245\pi$$
$$68$$ 34.0000 0.0606339
$$69$$ 204.000 0.355923
$$70$$ 0 0
$$71$$ −1092.00 −1.82530 −0.912652 0.408738i $$-0.865969\pi$$
−0.912652 + 0.408738i $$0.865969\pi$$
$$72$$ 405.000 0.662913
$$73$$ −562.000 −0.901057 −0.450528 0.892762i $$-0.648764\pi$$
−0.450528 + 0.892762i $$0.648764\pi$$
$$74$$ −870.000 −1.36670
$$75$$ 0 0
$$76$$ 1224.00 1.84740
$$77$$ −352.000 −0.520963
$$78$$ −570.000 −0.827433
$$79$$ −16.0000 −0.0227866 −0.0113933 0.999935i $$-0.503627\pi$$
−0.0113933 + 0.999935i $$0.503627\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 470.000 0.632961
$$83$$ −372.000 −0.491955 −0.245978 0.969275i $$-0.579109\pi$$
−0.245978 + 0.969275i $$0.579109\pi$$
$$84$$ −1632.00 −2.11983
$$85$$ 0 0
$$86$$ 2640.00 3.31022
$$87$$ 162.000 0.199635
$$88$$ −495.000 −0.599627
$$89$$ −966.000 −1.15051 −0.575257 0.817973i $$-0.695098\pi$$
−0.575257 + 0.817973i $$0.695098\pi$$
$$90$$ 0 0
$$91$$ 1216.00 1.40079
$$92$$ −1156.00 −1.31001
$$93$$ 456.000 0.508441
$$94$$ 1700.00 1.86534
$$95$$ 0 0
$$96$$ −255.000 −0.271102
$$97$$ 526.000 0.550590 0.275295 0.961360i $$-0.411225\pi$$
0.275295 + 0.961360i $$0.411225\pi$$
$$98$$ 3405.00 3.50976
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ 50.0000 0.0492593 0.0246296 0.999697i $$-0.492159\pi$$
0.0246296 + 0.999697i $$0.492159\pi$$
$$102$$ −30.0000 −0.0291220
$$103$$ −944.000 −0.903059 −0.451530 0.892256i $$-0.649121\pi$$
−0.451530 + 0.892256i $$0.649121\pi$$
$$104$$ 1710.00 1.61230
$$105$$ 0 0
$$106$$ 2190.00 2.00671
$$107$$ −468.000 −0.422834 −0.211417 0.977396i $$-0.567808\pi$$
−0.211417 + 0.977396i $$0.567808\pi$$
$$108$$ −459.000 −0.408956
$$109$$ 154.000 0.135326 0.0676630 0.997708i $$-0.478446\pi$$
0.0676630 + 0.997708i $$0.478446\pi$$
$$110$$ 0 0
$$111$$ 522.000 0.446361
$$112$$ 2848.00 2.40277
$$113$$ 54.0000 0.0449548 0.0224774 0.999747i $$-0.492845\pi$$
0.0224774 + 0.999747i $$0.492845\pi$$
$$114$$ −1080.00 −0.887292
$$115$$ 0 0
$$116$$ −918.000 −0.734777
$$117$$ 342.000 0.270239
$$118$$ 100.000 0.0780148
$$119$$ 64.0000 0.0493014
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 2850.00 2.11497
$$123$$ −282.000 −0.206724
$$124$$ −2584.00 −1.87137
$$125$$ 0 0
$$126$$ 1440.00 1.01814
$$127$$ 2224.00 1.55392 0.776961 0.629549i $$-0.216760\pi$$
0.776961 + 0.629549i $$0.216760\pi$$
$$128$$ −2115.00 −1.46048
$$129$$ −1584.00 −1.08111
$$130$$ 0 0
$$131$$ −2772.00 −1.84878 −0.924392 0.381443i $$-0.875427\pi$$
−0.924392 + 0.381443i $$0.875427\pi$$
$$132$$ 561.000 0.369915
$$133$$ 2304.00 1.50212
$$134$$ 2300.00 1.48276
$$135$$ 0 0
$$136$$ 90.0000 0.0567459
$$137$$ −1130.00 −0.704689 −0.352345 0.935870i $$-0.614615\pi$$
−0.352345 + 0.935870i $$0.614615\pi$$
$$138$$ 1020.00 0.629190
$$139$$ −1616.00 −0.986096 −0.493048 0.870002i $$-0.664117\pi$$
−0.493048 + 0.870002i $$0.664117\pi$$
$$140$$ 0 0
$$141$$ −1020.00 −0.609216
$$142$$ −5460.00 −3.22671
$$143$$ −418.000 −0.244440
$$144$$ 801.000 0.463542
$$145$$ 0 0
$$146$$ −2810.00 −1.59286
$$147$$ −2043.00 −1.14628
$$148$$ −2958.00 −1.64288
$$149$$ 2066.00 1.13593 0.567964 0.823053i $$-0.307731\pi$$
0.567964 + 0.823053i $$0.307731\pi$$
$$150$$ 0 0
$$151$$ 248.000 0.133655 0.0668277 0.997765i $$-0.478712\pi$$
0.0668277 + 0.997765i $$0.478712\pi$$
$$152$$ 3240.00 1.72894
$$153$$ 18.0000 0.00951120
$$154$$ −1760.00 −0.920941
$$155$$ 0 0
$$156$$ −1938.00 −0.994642
$$157$$ −2366.00 −1.20272 −0.601361 0.798977i $$-0.705375\pi$$
−0.601361 + 0.798977i $$0.705375\pi$$
$$158$$ −80.0000 −0.0402814
$$159$$ −1314.00 −0.655390
$$160$$ 0 0
$$161$$ −2176.00 −1.06517
$$162$$ 405.000 0.196419
$$163$$ 284.000 0.136470 0.0682350 0.997669i $$-0.478263\pi$$
0.0682350 + 0.997669i $$0.478263\pi$$
$$164$$ 1598.00 0.760871
$$165$$ 0 0
$$166$$ −1860.00 −0.869663
$$167$$ −600.000 −0.278020 −0.139010 0.990291i $$-0.544392\pi$$
−0.139010 + 0.990291i $$0.544392\pi$$
$$168$$ −4320.00 −1.98390
$$169$$ −753.000 −0.342740
$$170$$ 0 0
$$171$$ 648.000 0.289788
$$172$$ 8976.00 3.97915
$$173$$ −138.000 −0.0606471 −0.0303235 0.999540i $$-0.509654\pi$$
−0.0303235 + 0.999540i $$0.509654\pi$$
$$174$$ 810.000 0.352908
$$175$$ 0 0
$$176$$ −979.000 −0.419289
$$177$$ −60.0000 −0.0254795
$$178$$ −4830.00 −2.03384
$$179$$ 3972.00 1.65855 0.829277 0.558838i $$-0.188752\pi$$
0.829277 + 0.558838i $$0.188752\pi$$
$$180$$ 0 0
$$181$$ 2230.00 0.915771 0.457886 0.889011i $$-0.348607\pi$$
0.457886 + 0.889011i $$0.348607\pi$$
$$182$$ 6080.00 2.47626
$$183$$ −1710.00 −0.690748
$$184$$ −3060.00 −1.22601
$$185$$ 0 0
$$186$$ 2280.00 0.898805
$$187$$ −22.0000 −0.00860320
$$188$$ 5780.00 2.24229
$$189$$ −864.000 −0.332522
$$190$$ 0 0
$$191$$ −772.000 −0.292461 −0.146230 0.989251i $$-0.546714\pi$$
−0.146230 + 0.989251i $$0.546714\pi$$
$$192$$ 861.000 0.323632
$$193$$ −394.000 −0.146947 −0.0734734 0.997297i $$-0.523408\pi$$
−0.0734734 + 0.997297i $$0.523408\pi$$
$$194$$ 2630.00 0.973314
$$195$$ 0 0
$$196$$ 11577.0 4.21902
$$197$$ −3058.00 −1.10596 −0.552978 0.833196i $$-0.686509\pi$$
−0.552978 + 0.833196i $$0.686509\pi$$
$$198$$ −495.000 −0.177667
$$199$$ 2664.00 0.948975 0.474487 0.880262i $$-0.342633\pi$$
0.474487 + 0.880262i $$0.342633\pi$$
$$200$$ 0 0
$$201$$ −1380.00 −0.484267
$$202$$ 250.000 0.0870789
$$203$$ −1728.00 −0.597447
$$204$$ −102.000 −0.0350070
$$205$$ 0 0
$$206$$ −4720.00 −1.59640
$$207$$ −612.000 −0.205492
$$208$$ 3382.00 1.12740
$$209$$ −792.000 −0.262123
$$210$$ 0 0
$$211$$ −6000.00 −1.95762 −0.978808 0.204779i $$-0.934352\pi$$
−0.978808 + 0.204779i $$0.934352\pi$$
$$212$$ 7446.00 2.41223
$$213$$ 3276.00 1.05384
$$214$$ −2340.00 −0.747472
$$215$$ 0 0
$$216$$ −1215.00 −0.382733
$$217$$ −4864.00 −1.52161
$$218$$ 770.000 0.239225
$$219$$ 1686.00 0.520225
$$220$$ 0 0
$$221$$ 76.0000 0.0231326
$$222$$ 2610.00 0.789062
$$223$$ 560.000 0.168163 0.0840816 0.996459i $$-0.473204\pi$$
0.0840816 + 0.996459i $$0.473204\pi$$
$$224$$ 2720.00 0.811329
$$225$$ 0 0
$$226$$ 270.000 0.0794696
$$227$$ −5292.00 −1.54732 −0.773662 0.633599i $$-0.781577\pi$$
−0.773662 + 0.633599i $$0.781577\pi$$
$$228$$ −3672.00 −1.06660
$$229$$ −5322.00 −1.53575 −0.767877 0.640597i $$-0.778687\pi$$
−0.767877 + 0.640597i $$0.778687\pi$$
$$230$$ 0 0
$$231$$ 1056.00 0.300778
$$232$$ −2430.00 −0.687661
$$233$$ 3954.00 1.11174 0.555869 0.831270i $$-0.312385\pi$$
0.555869 + 0.831270i $$0.312385\pi$$
$$234$$ 1710.00 0.477719
$$235$$ 0 0
$$236$$ 340.000 0.0937801
$$237$$ 48.0000 0.0131558
$$238$$ 320.000 0.0871534
$$239$$ −3360.00 −0.909374 −0.454687 0.890651i $$-0.650249\pi$$
−0.454687 + 0.890651i $$0.650249\pi$$
$$240$$ 0 0
$$241$$ −3278.00 −0.876160 −0.438080 0.898936i $$-0.644341\pi$$
−0.438080 + 0.898936i $$0.644341\pi$$
$$242$$ 605.000 0.160706
$$243$$ −243.000 −0.0641500
$$244$$ 9690.00 2.54237
$$245$$ 0 0
$$246$$ −1410.00 −0.365440
$$247$$ 2736.00 0.704808
$$248$$ −6840.00 −1.75137
$$249$$ 1116.00 0.284031
$$250$$ 0 0
$$251$$ 2092.00 0.526079 0.263040 0.964785i $$-0.415275\pi$$
0.263040 + 0.964785i $$0.415275\pi$$
$$252$$ 4896.00 1.22388
$$253$$ 748.000 0.185875
$$254$$ 11120.0 2.74697
$$255$$ 0 0
$$256$$ −8279.00 −2.02124
$$257$$ −658.000 −0.159708 −0.0798539 0.996807i $$-0.525445\pi$$
−0.0798539 + 0.996807i $$0.525445\pi$$
$$258$$ −7920.00 −1.91115
$$259$$ −5568.00 −1.33583
$$260$$ 0 0
$$261$$ −486.000 −0.115259
$$262$$ −13860.0 −3.26822
$$263$$ 5104.00 1.19668 0.598339 0.801243i $$-0.295828\pi$$
0.598339 + 0.801243i $$0.295828\pi$$
$$264$$ 1485.00 0.346195
$$265$$ 0 0
$$266$$ 11520.0 2.65540
$$267$$ 2898.00 0.664250
$$268$$ 7820.00 1.78240
$$269$$ −4238.00 −0.960578 −0.480289 0.877110i $$-0.659468\pi$$
−0.480289 + 0.877110i $$0.659468\pi$$
$$270$$ 0 0
$$271$$ −3376.00 −0.756743 −0.378372 0.925654i $$-0.623516\pi$$
−0.378372 + 0.925654i $$0.623516\pi$$
$$272$$ 178.000 0.0396795
$$273$$ −3648.00 −0.808744
$$274$$ −5650.00 −1.24573
$$275$$ 0 0
$$276$$ 3468.00 0.756337
$$277$$ −2074.00 −0.449872 −0.224936 0.974374i $$-0.572217\pi$$
−0.224936 + 0.974374i $$0.572217\pi$$
$$278$$ −8080.00 −1.74319
$$279$$ −1368.00 −0.293548
$$280$$ 0 0
$$281$$ 702.000 0.149031 0.0745157 0.997220i $$-0.476259\pi$$
0.0745157 + 0.997220i $$0.476259\pi$$
$$282$$ −5100.00 −1.07695
$$283$$ −4912.00 −1.03176 −0.515880 0.856661i $$-0.672535\pi$$
−0.515880 + 0.856661i $$0.672535\pi$$
$$284$$ −18564.0 −3.87877
$$285$$ 0 0
$$286$$ −2090.00 −0.432113
$$287$$ 3008.00 0.618664
$$288$$ 765.000 0.156521
$$289$$ −4909.00 −0.999186
$$290$$ 0 0
$$291$$ −1578.00 −0.317883
$$292$$ −9554.00 −1.91475
$$293$$ 3486.00 0.695066 0.347533 0.937668i $$-0.387019\pi$$
0.347533 + 0.937668i $$0.387019\pi$$
$$294$$ −10215.0 −2.02636
$$295$$ 0 0
$$296$$ −7830.00 −1.53753
$$297$$ 297.000 0.0580259
$$298$$ 10330.0 2.00806
$$299$$ −2584.00 −0.499788
$$300$$ 0 0
$$301$$ 16896.0 3.23545
$$302$$ 1240.00 0.236271
$$303$$ −150.000 −0.0284399
$$304$$ 6408.00 1.20896
$$305$$ 0 0
$$306$$ 90.0000 0.0168136
$$307$$ −8360.00 −1.55417 −0.777085 0.629395i $$-0.783303\pi$$
−0.777085 + 0.629395i $$0.783303\pi$$
$$308$$ −5984.00 −1.10705
$$309$$ 2832.00 0.521381
$$310$$ 0 0
$$311$$ −5532.00 −1.00865 −0.504326 0.863513i $$-0.668259\pi$$
−0.504326 + 0.863513i $$0.668259\pi$$
$$312$$ −5130.00 −0.930862
$$313$$ −4826.00 −0.871507 −0.435753 0.900066i $$-0.643518\pi$$
−0.435753 + 0.900066i $$0.643518\pi$$
$$314$$ −11830.0 −2.12613
$$315$$ 0 0
$$316$$ −272.000 −0.0484215
$$317$$ −7570.00 −1.34124 −0.670621 0.741800i $$-0.733972\pi$$
−0.670621 + 0.741800i $$0.733972\pi$$
$$318$$ −6570.00 −1.15858
$$319$$ 594.000 0.104256
$$320$$ 0 0
$$321$$ 1404.00 0.244123
$$322$$ −10880.0 −1.88298
$$323$$ 144.000 0.0248061
$$324$$ 1377.00 0.236111
$$325$$ 0 0
$$326$$ 1420.00 0.241247
$$327$$ −462.000 −0.0781305
$$328$$ 4230.00 0.712081
$$329$$ 10880.0 1.82320
$$330$$ 0 0
$$331$$ 3676.00 0.610427 0.305213 0.952284i $$-0.401272\pi$$
0.305213 + 0.952284i $$0.401272\pi$$
$$332$$ −6324.00 −1.04541
$$333$$ −1566.00 −0.257707
$$334$$ −3000.00 −0.491475
$$335$$ 0 0
$$336$$ −8544.00 −1.38724
$$337$$ 5686.00 0.919098 0.459549 0.888152i $$-0.348011\pi$$
0.459549 + 0.888152i $$0.348011\pi$$
$$338$$ −3765.00 −0.605885
$$339$$ −162.000 −0.0259547
$$340$$ 0 0
$$341$$ 1672.00 0.265525
$$342$$ 3240.00 0.512278
$$343$$ 10816.0 1.70265
$$344$$ 23760.0 3.72399
$$345$$ 0 0
$$346$$ −690.000 −0.107210
$$347$$ 1652.00 0.255574 0.127787 0.991802i $$-0.459213\pi$$
0.127787 + 0.991802i $$0.459213\pi$$
$$348$$ 2754.00 0.424224
$$349$$ −6990.00 −1.07211 −0.536055 0.844183i $$-0.680086\pi$$
−0.536055 + 0.844183i $$0.680086\pi$$
$$350$$ 0 0
$$351$$ −1026.00 −0.156022
$$352$$ −935.000 −0.141579
$$353$$ 8094.00 1.22040 0.610199 0.792249i $$-0.291090\pi$$
0.610199 + 0.792249i $$0.291090\pi$$
$$354$$ −300.000 −0.0450419
$$355$$ 0 0
$$356$$ −16422.0 −2.44484
$$357$$ −192.000 −0.0284642
$$358$$ 19860.0 2.93194
$$359$$ 1024.00 0.150542 0.0752711 0.997163i $$-0.476018\pi$$
0.0752711 + 0.997163i $$0.476018\pi$$
$$360$$ 0 0
$$361$$ −1675.00 −0.244205
$$362$$ 11150.0 1.61887
$$363$$ −363.000 −0.0524864
$$364$$ 20672.0 2.97667
$$365$$ 0 0
$$366$$ −8550.00 −1.22108
$$367$$ 13664.0 1.94347 0.971737 0.236066i $$-0.0758581\pi$$
0.971737 + 0.236066i $$0.0758581\pi$$
$$368$$ −6052.00 −0.857289
$$369$$ 846.000 0.119352
$$370$$ 0 0
$$371$$ 14016.0 1.96139
$$372$$ 7752.00 1.08044
$$373$$ 1958.00 0.271800 0.135900 0.990723i $$-0.456607\pi$$
0.135900 + 0.990723i $$0.456607\pi$$
$$374$$ −110.000 −0.0152085
$$375$$ 0 0
$$376$$ 15300.0 2.09850
$$377$$ −2052.00 −0.280327
$$378$$ −4320.00 −0.587822
$$379$$ 6124.00 0.829997 0.414998 0.909822i $$-0.363782\pi$$
0.414998 + 0.909822i $$0.363782\pi$$
$$380$$ 0 0
$$381$$ −6672.00 −0.897157
$$382$$ −3860.00 −0.517002
$$383$$ −5612.00 −0.748720 −0.374360 0.927283i $$-0.622138\pi$$
−0.374360 + 0.927283i $$0.622138\pi$$
$$384$$ 6345.00 0.843208
$$385$$ 0 0
$$386$$ −1970.00 −0.259768
$$387$$ 4752.00 0.624180
$$388$$ 8942.00 1.17000
$$389$$ 12450.0 1.62273 0.811363 0.584543i $$-0.198726\pi$$
0.811363 + 0.584543i $$0.198726\pi$$
$$390$$ 0 0
$$391$$ −136.000 −0.0175903
$$392$$ 30645.0 3.94849
$$393$$ 8316.00 1.06740
$$394$$ −15290.0 −1.95507
$$395$$ 0 0
$$396$$ −1683.00 −0.213571
$$397$$ −14830.0 −1.87480 −0.937401 0.348252i $$-0.886775\pi$$
−0.937401 + 0.348252i $$0.886775\pi$$
$$398$$ 13320.0 1.67757
$$399$$ −6912.00 −0.867250
$$400$$ 0 0
$$401$$ −3358.00 −0.418181 −0.209090 0.977896i $$-0.567050\pi$$
−0.209090 + 0.977896i $$0.567050\pi$$
$$402$$ −6900.00 −0.856071
$$403$$ −5776.00 −0.713953
$$404$$ 850.000 0.104676
$$405$$ 0 0
$$406$$ −8640.00 −1.05615
$$407$$ 1914.00 0.233104
$$408$$ −270.000 −0.0327622
$$409$$ 10698.0 1.29335 0.646677 0.762764i $$-0.276158\pi$$
0.646677 + 0.762764i $$0.276158\pi$$
$$410$$ 0 0
$$411$$ 3390.00 0.406852
$$412$$ −16048.0 −1.91900
$$413$$ 640.000 0.0762526
$$414$$ −3060.00 −0.363263
$$415$$ 0 0
$$416$$ 3230.00 0.380682
$$417$$ 4848.00 0.569323
$$418$$ −3960.00 −0.463373
$$419$$ −2044.00 −0.238320 −0.119160 0.992875i $$-0.538020\pi$$
−0.119160 + 0.992875i $$0.538020\pi$$
$$420$$ 0 0
$$421$$ 3070.00 0.355398 0.177699 0.984085i $$-0.443135\pi$$
0.177699 + 0.984085i $$0.443135\pi$$
$$422$$ −30000.0 −3.46061
$$423$$ 3060.00 0.351731
$$424$$ 19710.0 2.25755
$$425$$ 0 0
$$426$$ 16380.0 1.86294
$$427$$ 18240.0 2.06720
$$428$$ −7956.00 −0.898523
$$429$$ 1254.00 0.141127
$$430$$ 0 0
$$431$$ −12600.0 −1.40817 −0.704084 0.710116i $$-0.748642\pi$$
−0.704084 + 0.710116i $$0.748642\pi$$
$$432$$ −2403.00 −0.267626
$$433$$ 9902.00 1.09898 0.549492 0.835499i $$-0.314821\pi$$
0.549492 + 0.835499i $$0.314821\pi$$
$$434$$ −24320.0 −2.68986
$$435$$ 0 0
$$436$$ 2618.00 0.287568
$$437$$ −4896.00 −0.535944
$$438$$ 8430.00 0.919637
$$439$$ 11440.0 1.24374 0.621869 0.783121i $$-0.286373\pi$$
0.621869 + 0.783121i $$0.286373\pi$$
$$440$$ 0 0
$$441$$ 6129.00 0.661808
$$442$$ 380.000 0.0408931
$$443$$ 5180.00 0.555551 0.277776 0.960646i $$-0.410403\pi$$
0.277776 + 0.960646i $$0.410403\pi$$
$$444$$ 8874.00 0.948517
$$445$$ 0 0
$$446$$ 2800.00 0.297273
$$447$$ −6198.00 −0.655829
$$448$$ −9184.00 −0.968534
$$449$$ 10826.0 1.13789 0.568943 0.822377i $$-0.307353\pi$$
0.568943 + 0.822377i $$0.307353\pi$$
$$450$$ 0 0
$$451$$ −1034.00 −0.107958
$$452$$ 918.000 0.0955290
$$453$$ −744.000 −0.0771659
$$454$$ −26460.0 −2.73531
$$455$$ 0 0
$$456$$ −9720.00 −0.998203
$$457$$ 15798.0 1.61707 0.808533 0.588451i $$-0.200262\pi$$
0.808533 + 0.588451i $$0.200262\pi$$
$$458$$ −26610.0 −2.71486
$$459$$ −54.0000 −0.00549129
$$460$$ 0 0
$$461$$ −3894.00 −0.393409 −0.196705 0.980463i $$-0.563024\pi$$
−0.196705 + 0.980463i $$0.563024\pi$$
$$462$$ 5280.00 0.531705
$$463$$ 15992.0 1.60521 0.802604 0.596512i $$-0.203447\pi$$
0.802604 + 0.596512i $$0.203447\pi$$
$$464$$ −4806.00 −0.480847
$$465$$ 0 0
$$466$$ 19770.0 1.96530
$$467$$ −11844.0 −1.17361 −0.586804 0.809729i $$-0.699614\pi$$
−0.586804 + 0.809729i $$0.699614\pi$$
$$468$$ 5814.00 0.574257
$$469$$ 14720.0 1.44927
$$470$$ 0 0
$$471$$ 7098.00 0.694392
$$472$$ 900.000 0.0877666
$$473$$ −5808.00 −0.564592
$$474$$ 240.000 0.0232565
$$475$$ 0 0
$$476$$ 1088.00 0.104766
$$477$$ 3942.00 0.378389
$$478$$ −16800.0 −1.60756
$$479$$ 14936.0 1.42472 0.712362 0.701812i $$-0.247625\pi$$
0.712362 + 0.701812i $$0.247625\pi$$
$$480$$ 0 0
$$481$$ −6612.00 −0.626780
$$482$$ −16390.0 −1.54885
$$483$$ 6528.00 0.614978
$$484$$ 2057.00 0.193182
$$485$$ 0 0
$$486$$ −1215.00 −0.113402
$$487$$ 2056.00 0.191306 0.0956532 0.995415i $$-0.469506\pi$$
0.0956532 + 0.995415i $$0.469506\pi$$
$$488$$ 25650.0 2.37935
$$489$$ −852.000 −0.0787909
$$490$$ 0 0
$$491$$ −17852.0 −1.64083 −0.820417 0.571766i $$-0.806259\pi$$
−0.820417 + 0.571766i $$0.806259\pi$$
$$492$$ −4794.00 −0.439289
$$493$$ −108.000 −0.00986628
$$494$$ 13680.0 1.24594
$$495$$ 0 0
$$496$$ −13528.0 −1.22465
$$497$$ −34944.0 −3.15383
$$498$$ 5580.00 0.502100
$$499$$ 4508.00 0.404420 0.202210 0.979342i $$-0.435188\pi$$
0.202210 + 0.979342i $$0.435188\pi$$
$$500$$ 0 0
$$501$$ 1800.00 0.160515
$$502$$ 10460.0 0.929985
$$503$$ 5912.00 0.524062 0.262031 0.965059i $$-0.415608\pi$$
0.262031 + 0.965059i $$0.415608\pi$$
$$504$$ 12960.0 1.14541
$$505$$ 0 0
$$506$$ 3740.00 0.328584
$$507$$ 2259.00 0.197881
$$508$$ 37808.0 3.30208
$$509$$ −11406.0 −0.993246 −0.496623 0.867966i $$-0.665427\pi$$
−0.496623 + 0.867966i $$0.665427\pi$$
$$510$$ 0 0
$$511$$ −17984.0 −1.55688
$$512$$ −24475.0 −2.11260
$$513$$ −1944.00 −0.167309
$$514$$ −3290.00 −0.282326
$$515$$ 0 0
$$516$$ −26928.0 −2.29736
$$517$$ −3740.00 −0.318153
$$518$$ −27840.0 −2.36143
$$519$$ 414.000 0.0350146
$$520$$ 0 0
$$521$$ −1542.00 −0.129667 −0.0648333 0.997896i $$-0.520652\pi$$
−0.0648333 + 0.997896i $$0.520652\pi$$
$$522$$ −2430.00 −0.203751
$$523$$ 7504.00 0.627394 0.313697 0.949523i $$-0.398432\pi$$
0.313697 + 0.949523i $$0.398432\pi$$
$$524$$ −47124.0 −3.92867
$$525$$ 0 0
$$526$$ 25520.0 2.11545
$$527$$ −304.000 −0.0251280
$$528$$ 2937.00 0.242077
$$529$$ −7543.00 −0.619956
$$530$$ 0 0
$$531$$ 180.000 0.0147106
$$532$$ 39168.0 3.19201
$$533$$ 3572.00 0.290282
$$534$$ 14490.0 1.17424
$$535$$ 0 0
$$536$$ 20700.0 1.66810
$$537$$ −11916.0 −0.957567
$$538$$ −21190.0 −1.69808
$$539$$ −7491.00 −0.598627
$$540$$ 0 0
$$541$$ 1018.00 0.0809006 0.0404503 0.999182i $$-0.487121\pi$$
0.0404503 + 0.999182i $$0.487121\pi$$
$$542$$ −16880.0 −1.33775
$$543$$ −6690.00 −0.528721
$$544$$ 170.000 0.0133983
$$545$$ 0 0
$$546$$ −18240.0 −1.42967
$$547$$ −7904.00 −0.617826 −0.308913 0.951090i $$-0.599965\pi$$
−0.308913 + 0.951090i $$0.599965\pi$$
$$548$$ −19210.0 −1.49746
$$549$$ 5130.00 0.398803
$$550$$ 0 0
$$551$$ −3888.00 −0.300607
$$552$$ 9180.00 0.707838
$$553$$ −512.000 −0.0393715
$$554$$ −10370.0 −0.795269
$$555$$ 0 0
$$556$$ −27472.0 −2.09545
$$557$$ 22934.0 1.74460 0.872302 0.488967i $$-0.162626\pi$$
0.872302 + 0.488967i $$0.162626\pi$$
$$558$$ −6840.00 −0.518925
$$559$$ 20064.0 1.51810
$$560$$ 0 0
$$561$$ 66.0000 0.00496706
$$562$$ 3510.00 0.263453
$$563$$ −14020.0 −1.04951 −0.524754 0.851254i $$-0.675843\pi$$
−0.524754 + 0.851254i $$0.675843\pi$$
$$564$$ −17340.0 −1.29458
$$565$$ 0 0
$$566$$ −24560.0 −1.82391
$$567$$ 2592.00 0.191982
$$568$$ −49140.0 −3.63005
$$569$$ 4230.00 0.311653 0.155827 0.987784i $$-0.450196\pi$$
0.155827 + 0.987784i $$0.450196\pi$$
$$570$$ 0 0
$$571$$ −8536.00 −0.625605 −0.312803 0.949818i $$-0.601268\pi$$
−0.312803 + 0.949818i $$0.601268\pi$$
$$572$$ −7106.00 −0.519435
$$573$$ 2316.00 0.168852
$$574$$ 15040.0 1.09365
$$575$$ 0 0
$$576$$ −2583.00 −0.186849
$$577$$ 11982.0 0.864501 0.432251 0.901754i $$-0.357720\pi$$
0.432251 + 0.901754i $$0.357720\pi$$
$$578$$ −24545.0 −1.76633
$$579$$ 1182.00 0.0848398
$$580$$ 0 0
$$581$$ −11904.0 −0.850019
$$582$$ −7890.00 −0.561943
$$583$$ −4818.00 −0.342266
$$584$$ −25290.0 −1.79197
$$585$$ 0 0
$$586$$ 17430.0 1.22871
$$587$$ 20396.0 1.43413 0.717064 0.697007i $$-0.245486\pi$$
0.717064 + 0.697007i $$0.245486\pi$$
$$588$$ −34731.0 −2.43585
$$589$$ −10944.0 −0.765602
$$590$$ 0 0
$$591$$ 9174.00 0.638524
$$592$$ −15486.0 −1.07512
$$593$$ −12518.0 −0.866868 −0.433434 0.901185i $$-0.642698\pi$$
−0.433434 + 0.901185i $$0.642698\pi$$
$$594$$ 1485.00 0.102576
$$595$$ 0 0
$$596$$ 35122.0 2.41385
$$597$$ −7992.00 −0.547891
$$598$$ −12920.0 −0.883509
$$599$$ −25292.0 −1.72521 −0.862607 0.505875i $$-0.831170\pi$$
−0.862607 + 0.505875i $$0.831170\pi$$
$$600$$ 0 0
$$601$$ 15962.0 1.08337 0.541683 0.840583i $$-0.317787\pi$$
0.541683 + 0.840583i $$0.317787\pi$$
$$602$$ 84480.0 5.71951
$$603$$ 4140.00 0.279592
$$604$$ 4216.00 0.284018
$$605$$ 0 0
$$606$$ −750.000 −0.0502750
$$607$$ 1600.00 0.106988 0.0534942 0.998568i $$-0.482964\pi$$
0.0534942 + 0.998568i $$0.482964\pi$$
$$608$$ 6120.00 0.408222
$$609$$ 5184.00 0.344936
$$610$$ 0 0
$$611$$ 12920.0 0.855462
$$612$$ 306.000 0.0202113
$$613$$ −2162.00 −0.142451 −0.0712254 0.997460i $$-0.522691\pi$$
−0.0712254 + 0.997460i $$0.522691\pi$$
$$614$$ −41800.0 −2.74741
$$615$$ 0 0
$$616$$ −15840.0 −1.03606
$$617$$ 18126.0 1.18270 0.591350 0.806415i $$-0.298595\pi$$
0.591350 + 0.806415i $$0.298595\pi$$
$$618$$ 14160.0 0.921681
$$619$$ 17348.0 1.12645 0.563227 0.826302i $$-0.309560\pi$$
0.563227 + 0.826302i $$0.309560\pi$$
$$620$$ 0 0
$$621$$ 1836.00 0.118641
$$622$$ −27660.0 −1.78306
$$623$$ −30912.0 −1.98790
$$624$$ −10146.0 −0.650906
$$625$$ 0 0
$$626$$ −24130.0 −1.54062
$$627$$ 2376.00 0.151337
$$628$$ −40222.0 −2.55578
$$629$$ −348.000 −0.0220599
$$630$$ 0 0
$$631$$ 10096.0 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ −720.000 −0.0453166
$$633$$ 18000.0 1.13023
$$634$$ −37850.0 −2.37100
$$635$$ 0 0
$$636$$ −22338.0 −1.39270
$$637$$ 25878.0 1.60961
$$638$$ 2970.00 0.184300
$$639$$ −9828.00 −0.608435
$$640$$ 0 0
$$641$$ 8922.00 0.549763 0.274881 0.961478i $$-0.411361\pi$$
0.274881 + 0.961478i $$0.411361\pi$$
$$642$$ 7020.00 0.431553
$$643$$ 14644.0 0.898138 0.449069 0.893497i $$-0.351756\pi$$
0.449069 + 0.893497i $$0.351756\pi$$
$$644$$ −36992.0 −2.26349
$$645$$ 0 0
$$646$$ 720.000 0.0438514
$$647$$ −6932.00 −0.421213 −0.210607 0.977571i $$-0.567544\pi$$
−0.210607 + 0.977571i $$0.567544\pi$$
$$648$$ 3645.00 0.220971
$$649$$ −220.000 −0.0133062
$$650$$ 0 0
$$651$$ 14592.0 0.878503
$$652$$ 4828.00 0.289999
$$653$$ 5942.00 0.356093 0.178046 0.984022i $$-0.443022\pi$$
0.178046 + 0.984022i $$0.443022\pi$$
$$654$$ −2310.00 −0.138116
$$655$$ 0 0
$$656$$ 8366.00 0.497923
$$657$$ −5058.00 −0.300352
$$658$$ 54400.0 3.22300
$$659$$ 484.000 0.0286100 0.0143050 0.999898i $$-0.495446\pi$$
0.0143050 + 0.999898i $$0.495446\pi$$
$$660$$ 0 0
$$661$$ −17114.0 −1.00705 −0.503523 0.863982i $$-0.667963\pi$$
−0.503523 + 0.863982i $$0.667963\pi$$
$$662$$ 18380.0 1.07909
$$663$$ −228.000 −0.0133556
$$664$$ −16740.0 −0.978370
$$665$$ 0 0
$$666$$ −7830.00 −0.455565
$$667$$ 3672.00 0.213164
$$668$$ −10200.0 −0.590793
$$669$$ −1680.00 −0.0970890
$$670$$ 0 0
$$671$$ −6270.00 −0.360731
$$672$$ −8160.00 −0.468421
$$673$$ −16154.0 −0.925247 −0.462623 0.886555i $$-0.653092\pi$$
−0.462623 + 0.886555i $$0.653092\pi$$
$$674$$ 28430.0 1.62475
$$675$$ 0 0
$$676$$ −12801.0 −0.728323
$$677$$ 3390.00 0.192449 0.0962247 0.995360i $$-0.469323\pi$$
0.0962247 + 0.995360i $$0.469323\pi$$
$$678$$ −810.000 −0.0458818
$$679$$ 16832.0 0.951330
$$680$$ 0 0
$$681$$ 15876.0 0.893347
$$682$$ 8360.00 0.469386
$$683$$ 25540.0 1.43084 0.715418 0.698697i $$-0.246236\pi$$
0.715418 + 0.698697i $$0.246236\pi$$
$$684$$ 11016.0 0.615800
$$685$$ 0 0
$$686$$ 54080.0 3.00989
$$687$$ 15966.0 0.886668
$$688$$ 46992.0 2.60400
$$689$$ 16644.0 0.920299
$$690$$ 0 0
$$691$$ 12476.0 0.686844 0.343422 0.939181i $$-0.388414\pi$$
0.343422 + 0.939181i $$0.388414\pi$$
$$692$$ −2346.00 −0.128875
$$693$$ −3168.00 −0.173654
$$694$$ 8260.00 0.451794
$$695$$ 0 0
$$696$$ 7290.00 0.397021
$$697$$ 188.000 0.0102167
$$698$$ −34950.0 −1.89524
$$699$$ −11862.0 −0.641863
$$700$$ 0 0
$$701$$ −20806.0 −1.12102 −0.560508 0.828149i $$-0.689394\pi$$
−0.560508 + 0.828149i $$0.689394\pi$$
$$702$$ −5130.00 −0.275811
$$703$$ −12528.0 −0.672123
$$704$$ 3157.00 0.169011
$$705$$ 0 0
$$706$$ 40470.0 2.15738
$$707$$ 1600.00 0.0851120
$$708$$ −1020.00 −0.0541440
$$709$$ 14198.0 0.752069 0.376035 0.926606i $$-0.377287\pi$$
0.376035 + 0.926606i $$0.377287\pi$$
$$710$$ 0 0
$$711$$ −144.000 −0.00759553
$$712$$ −43470.0 −2.28807
$$713$$ 10336.0 0.542898
$$714$$ −960.000 −0.0503181
$$715$$ 0 0
$$716$$ 67524.0 3.52443
$$717$$ 10080.0 0.525027
$$718$$ 5120.00 0.266124
$$719$$ 4596.00 0.238389 0.119195 0.992871i $$-0.461969\pi$$
0.119195 + 0.992871i $$0.461969\pi$$
$$720$$ 0 0
$$721$$ −30208.0 −1.56034
$$722$$ −8375.00 −0.431697
$$723$$ 9834.00 0.505851
$$724$$ 37910.0 1.94601
$$725$$ 0 0
$$726$$ −1815.00 −0.0927837
$$727$$ −19560.0 −0.997855 −0.498927 0.866644i $$-0.666273\pi$$
−0.498927 + 0.866644i $$0.666273\pi$$
$$728$$ 54720.0 2.78579
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 1056.00 0.0534303
$$732$$ −29070.0 −1.46784
$$733$$ 1638.00 0.0825388 0.0412694 0.999148i $$-0.486860\pi$$
0.0412694 + 0.999148i $$0.486860\pi$$
$$734$$ 68320.0 3.43561
$$735$$ 0 0
$$736$$ −5780.00 −0.289475
$$737$$ −5060.00 −0.252900
$$738$$ 4230.00 0.210987
$$739$$ −15592.0 −0.776131 −0.388066 0.921632i $$-0.626857\pi$$
−0.388066 + 0.921632i $$0.626857\pi$$
$$740$$ 0 0
$$741$$ −8208.00 −0.406921
$$742$$ 70080.0 3.46727
$$743$$ −592.000 −0.0292307 −0.0146153 0.999893i $$-0.504652\pi$$
−0.0146153 + 0.999893i $$0.504652\pi$$
$$744$$ 20520.0 1.01116
$$745$$ 0 0
$$746$$ 9790.00 0.480479
$$747$$ −3348.00 −0.163985
$$748$$ −374.000 −0.0182818
$$749$$ −14976.0 −0.730589
$$750$$ 0 0
$$751$$ 39832.0 1.93541 0.967703 0.252092i $$-0.0811186\pi$$
0.967703 + 0.252092i $$0.0811186\pi$$
$$752$$ 30260.0 1.46738
$$753$$ −6276.00 −0.303732
$$754$$ −10260.0 −0.495553
$$755$$ 0 0
$$756$$ −14688.0 −0.706610
$$757$$ −10958.0 −0.526123 −0.263062 0.964779i $$-0.584732\pi$$
−0.263062 + 0.964779i $$0.584732\pi$$
$$758$$ 30620.0 1.46724
$$759$$ −2244.00 −0.107315
$$760$$ 0 0
$$761$$ −8970.00 −0.427283 −0.213641 0.976912i $$-0.568532\pi$$
−0.213641 + 0.976912i $$0.568532\pi$$
$$762$$ −33360.0 −1.58596
$$763$$ 4928.00 0.233821
$$764$$ −13124.0 −0.621479
$$765$$ 0 0
$$766$$ −28060.0 −1.32356
$$767$$ 760.000 0.0357784
$$768$$ 24837.0 1.16696
$$769$$ −10054.0 −0.471465 −0.235732 0.971818i $$-0.575749\pi$$
−0.235732 + 0.971818i $$0.575749\pi$$
$$770$$ 0 0
$$771$$ 1974.00 0.0922074
$$772$$ −6698.00 −0.312262
$$773$$ −26346.0 −1.22587 −0.612936 0.790132i $$-0.710012\pi$$
−0.612936 + 0.790132i $$0.710012\pi$$
$$774$$ 23760.0 1.10341
$$775$$ 0 0
$$776$$ 23670.0 1.09498
$$777$$ 16704.0 0.771239
$$778$$ 62250.0 2.86860
$$779$$ 6768.00 0.311282
$$780$$ 0 0
$$781$$ 12012.0 0.550350
$$782$$ −680.000 −0.0310956
$$783$$ 1458.00 0.0665449
$$784$$ 60609.0 2.76098
$$785$$ 0 0
$$786$$ 41580.0 1.88691
$$787$$ 16040.0 0.726511 0.363256 0.931690i $$-0.381665\pi$$
0.363256 + 0.931690i $$0.381665\pi$$
$$788$$ −51986.0 −2.35016
$$789$$ −15312.0 −0.690902
$$790$$ 0 0
$$791$$ 1728.00 0.0776746
$$792$$ −4455.00 −0.199876
$$793$$ 21660.0 0.969948
$$794$$ −74150.0 −3.31421
$$795$$ 0 0
$$796$$ 45288.0 2.01657
$$797$$ −32810.0 −1.45821 −0.729103 0.684404i $$-0.760062\pi$$
−0.729103 + 0.684404i $$0.760062\pi$$
$$798$$ −34560.0 −1.53310
$$799$$ 680.000 0.0301085
$$800$$ 0 0
$$801$$ −8694.00 −0.383505
$$802$$ −16790.0 −0.739246
$$803$$ 6182.00 0.271679
$$804$$ −23460.0 −1.02907
$$805$$ 0 0
$$806$$ −28880.0 −1.26210
$$807$$ 12714.0 0.554590
$$808$$ 2250.00 0.0979638
$$809$$ 18918.0 0.822153 0.411076 0.911601i $$-0.365153\pi$$
0.411076 + 0.911601i $$0.365153\pi$$
$$810$$ 0 0
$$811$$ −8552.00 −0.370285 −0.185143 0.982712i $$-0.559275\pi$$
−0.185143 + 0.982712i $$0.559275\pi$$
$$812$$ −29376.0 −1.26958
$$813$$ 10128.0 0.436906
$$814$$ 9570.00 0.412074
$$815$$ 0 0
$$816$$ −534.000 −0.0229090
$$817$$ 38016.0 1.62792
$$818$$ 53490.0 2.28635
$$819$$ 10944.0 0.466928
$$820$$ 0 0
$$821$$ −46430.0 −1.97371 −0.986856 0.161600i $$-0.948335\pi$$
−0.986856 + 0.161600i $$0.948335\pi$$
$$822$$ 16950.0 0.719220
$$823$$ −16392.0 −0.694276 −0.347138 0.937814i $$-0.612846\pi$$
−0.347138 + 0.937814i $$0.612846\pi$$
$$824$$ −42480.0 −1.79595
$$825$$ 0 0
$$826$$ 3200.00 0.134797
$$827$$ 13876.0 0.583453 0.291727 0.956502i $$-0.405770\pi$$
0.291727 + 0.956502i $$0.405770\pi$$
$$828$$ −10404.0 −0.436671
$$829$$ −24554.0 −1.02870 −0.514352 0.857579i $$-0.671968\pi$$
−0.514352 + 0.857579i $$0.671968\pi$$
$$830$$ 0 0
$$831$$ 6222.00 0.259734
$$832$$ −10906.0 −0.454444
$$833$$ 1362.00 0.0566513
$$834$$ 24240.0 1.00643
$$835$$ 0 0
$$836$$ −13464.0 −0.557012
$$837$$ 4104.00 0.169480
$$838$$ −10220.0 −0.421294
$$839$$ 19900.0 0.818861 0.409430 0.912341i $$-0.365727\pi$$
0.409430 + 0.912341i $$0.365727\pi$$
$$840$$ 0 0
$$841$$ −21473.0 −0.880438
$$842$$ 15350.0 0.628261
$$843$$ −2106.00 −0.0860433
$$844$$ −102000. −4.15993
$$845$$ 0 0
$$846$$ 15300.0 0.621779
$$847$$ 3872.00 0.157076
$$848$$ 38982.0 1.57859
$$849$$ 14736.0 0.595687
$$850$$ 0 0
$$851$$ 11832.0 0.476611
$$852$$ 55692.0 2.23941
$$853$$ −41138.0 −1.65128 −0.825638 0.564200i $$-0.809185\pi$$
−0.825638 + 0.564200i $$0.809185\pi$$
$$854$$ 91200.0 3.65433
$$855$$ 0 0
$$856$$ −21060.0 −0.840907
$$857$$ −19910.0 −0.793597 −0.396799 0.917906i $$-0.629879\pi$$
−0.396799 + 0.917906i $$0.629879\pi$$
$$858$$ 6270.00 0.249481
$$859$$ 42924.0 1.70495 0.852473 0.522772i $$-0.175102\pi$$
0.852473 + 0.522772i $$0.175102\pi$$
$$860$$ 0 0
$$861$$ −9024.00 −0.357186
$$862$$ −63000.0 −2.48931
$$863$$ 46236.0 1.82374 0.911872 0.410474i $$-0.134637\pi$$
0.911872 + 0.410474i $$0.134637\pi$$
$$864$$ −2295.00 −0.0903675
$$865$$ 0 0
$$866$$ 49510.0 1.94275
$$867$$ 14727.0 0.576880
$$868$$ −82688.0 −3.23343
$$869$$ 176.000 0.00687042
$$870$$ 0 0
$$871$$ 17480.0 0.680008
$$872$$ 6930.00 0.269128
$$873$$ 4734.00 0.183530
$$874$$ −24480.0 −0.947424
$$875$$ 0 0
$$876$$ 28662.0 1.10548
$$877$$ −25746.0 −0.991312 −0.495656 0.868519i $$-0.665072\pi$$
−0.495656 + 0.868519i $$0.665072\pi$$
$$878$$ 57200.0 2.19864
$$879$$ −10458.0 −0.401296
$$880$$ 0 0
$$881$$ −24550.0 −0.938831 −0.469416 0.882977i $$-0.655535\pi$$
−0.469416 + 0.882977i $$0.655535\pi$$
$$882$$ 30645.0 1.16992
$$883$$ 19436.0 0.740740 0.370370 0.928884i $$-0.379231\pi$$
0.370370 + 0.928884i $$0.379231\pi$$
$$884$$ 1292.00 0.0491569
$$885$$ 0 0
$$886$$ 25900.0 0.982085
$$887$$ 22912.0 0.867316 0.433658 0.901077i $$-0.357223\pi$$
0.433658 + 0.901077i $$0.357223\pi$$
$$888$$ 23490.0 0.887695
$$889$$ 71168.0 2.68492
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ 9520.00 0.357347
$$893$$ 24480.0 0.917348
$$894$$ −30990.0 −1.15935
$$895$$ 0 0
$$896$$ −67680.0 −2.52347
$$897$$ 7752.00 0.288553
$$898$$ 54130.0 2.01152
$$899$$ 8208.00 0.304507
$$900$$ 0 0
$$901$$ 876.000 0.0323904
$$902$$ −5170.00 −0.190845
$$903$$ −50688.0 −1.86799
$$904$$ 2430.00 0.0894033
$$905$$ 0 0
$$906$$ −3720.00 −0.136411
$$907$$ 39900.0 1.46070 0.730352 0.683071i $$-0.239356\pi$$
0.730352 + 0.683071i $$0.239356\pi$$
$$908$$ −89964.0 −3.28806
$$909$$ 450.000 0.0164198
$$910$$ 0 0
$$911$$ 29460.0 1.07141 0.535704 0.844406i $$-0.320046\pi$$
0.535704 + 0.844406i $$0.320046\pi$$
$$912$$ −19224.0 −0.697994
$$913$$ 4092.00 0.148330
$$914$$ 78990.0 2.85860
$$915$$ 0 0
$$916$$ −90474.0 −3.26348
$$917$$ −88704.0 −3.19440
$$918$$ −270.000 −0.00970733
$$919$$ 29368.0 1.05415 0.527073 0.849820i $$-0.323289\pi$$
0.527073 + 0.849820i $$0.323289\pi$$
$$920$$ 0 0
$$921$$ 25080.0 0.897301
$$922$$ −19470.0 −0.695456
$$923$$ −41496.0 −1.47980
$$924$$ 17952.0 0.639153
$$925$$ 0 0
$$926$$ 79960.0 2.83763
$$927$$ −8496.00 −0.301020
$$928$$ −4590.00 −0.162364
$$929$$ 33954.0 1.19913 0.599567 0.800325i $$-0.295340\pi$$
0.599567 + 0.800325i $$0.295340\pi$$
$$930$$ 0 0
$$931$$ 49032.0 1.72606
$$932$$ 67218.0 2.36245
$$933$$ 16596.0 0.582346
$$934$$ −59220.0 −2.07467
$$935$$ 0 0
$$936$$ 15390.0 0.537434
$$937$$ 2854.00 0.0995049 0.0497525 0.998762i $$-0.484157\pi$$
0.0497525 + 0.998762i $$0.484157\pi$$
$$938$$ 73600.0 2.56197
$$939$$ 14478.0 0.503165
$$940$$ 0 0
$$941$$ −6294.00 −0.218043 −0.109022 0.994039i $$-0.534772\pi$$
−0.109022 + 0.994039i $$0.534772\pi$$
$$942$$ 35490.0 1.22752
$$943$$ −6392.00 −0.220734
$$944$$ 1780.00 0.0613708
$$945$$ 0 0
$$946$$ −29040.0 −0.998067
$$947$$ −2268.00 −0.0778248 −0.0389124 0.999243i $$-0.512389\pi$$
−0.0389124 + 0.999243i $$0.512389\pi$$
$$948$$ 816.000 0.0279562
$$949$$ −21356.0 −0.730501
$$950$$ 0 0
$$951$$ 22710.0 0.774366
$$952$$ 2880.00 0.0980476
$$953$$ −26566.0 −0.902998 −0.451499 0.892272i $$-0.649111\pi$$
−0.451499 + 0.892272i $$0.649111\pi$$
$$954$$ 19710.0 0.668904
$$955$$ 0 0
$$956$$ −57120.0 −1.93242
$$957$$ −1782.00 −0.0601921
$$958$$ 74680.0 2.51858
$$959$$ −36160.0 −1.21759
$$960$$ 0 0
$$961$$ −6687.00 −0.224464
$$962$$ −33060.0 −1.10800
$$963$$ −4212.00 −0.140945
$$964$$ −55726.0 −1.86184
$$965$$ 0 0
$$966$$ 32640.0 1.08714
$$967$$ −11176.0 −0.371661 −0.185830 0.982582i $$-0.559497\pi$$
−0.185830 + 0.982582i $$0.559497\pi$$
$$968$$ 5445.00 0.180794
$$969$$ −432.000 −0.0143218
$$970$$ 0 0
$$971$$ −42316.0 −1.39854 −0.699271 0.714856i $$-0.746492\pi$$
−0.699271 + 0.714856i $$0.746492\pi$$
$$972$$ −4131.00 −0.136319
$$973$$ −51712.0 −1.70381
$$974$$ 10280.0 0.338185
$$975$$ 0 0
$$976$$ 50730.0 1.66376
$$977$$ 45054.0 1.47534 0.737669 0.675163i $$-0.235927\pi$$
0.737669 + 0.675163i $$0.235927\pi$$
$$978$$ −4260.00 −0.139284
$$979$$ 10626.0 0.346893
$$980$$ 0 0
$$981$$ 1386.00 0.0451086
$$982$$ −89260.0 −2.90061
$$983$$ 12300.0 0.399094 0.199547 0.979888i $$-0.436053\pi$$
0.199547 + 0.979888i $$0.436053\pi$$
$$984$$ −12690.0 −0.411120
$$985$$ 0 0
$$986$$ −540.000 −0.0174413
$$987$$ −32640.0 −1.05263
$$988$$ 46512.0 1.49772
$$989$$ −35904.0 −1.15438
$$990$$ 0 0
$$991$$ 36280.0 1.16294 0.581469 0.813568i $$-0.302478\pi$$
0.581469 + 0.813568i $$0.302478\pi$$
$$992$$ −12920.0 −0.413519
$$993$$ −11028.0 −0.352430
$$994$$ −174720. −5.57523
$$995$$ 0 0
$$996$$ 18972.0 0.603565
$$997$$ −3290.00 −0.104509 −0.0522544 0.998634i $$-0.516641\pi$$
−0.0522544 + 0.998634i $$0.516641\pi$$
$$998$$ 22540.0 0.714921
$$999$$ 4698.00 0.148787
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 825.4.a.i.1.1 1
3.2 odd 2 2475.4.a.b.1.1 1
5.2 odd 4 825.4.c.a.199.2 2
5.3 odd 4 825.4.c.a.199.1 2
5.4 even 2 33.4.a.a.1.1 1
15.14 odd 2 99.4.a.b.1.1 1
20.19 odd 2 528.4.a.a.1.1 1
35.34 odd 2 1617.4.a.a.1.1 1
40.19 odd 2 2112.4.a.y.1.1 1
40.29 even 2 2112.4.a.l.1.1 1
55.54 odd 2 363.4.a.h.1.1 1
60.59 even 2 1584.4.a.t.1.1 1
165.164 even 2 1089.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.a.1.1 1 5.4 even 2
99.4.a.b.1.1 1 15.14 odd 2
363.4.a.h.1.1 1 55.54 odd 2
528.4.a.a.1.1 1 20.19 odd 2
825.4.a.i.1.1 1 1.1 even 1 trivial
825.4.c.a.199.1 2 5.3 odd 4
825.4.c.a.199.2 2 5.2 odd 4
1089.4.a.a.1.1 1 165.164 even 2
1584.4.a.t.1.1 1 60.59 even 2
1617.4.a.a.1.1 1 35.34 odd 2
2112.4.a.l.1.1 1 40.29 even 2
2112.4.a.y.1.1 1 40.19 odd 2
2475.4.a.b.1.1 1 3.2 odd 2