# Properties

 Label 825.4.c.j.199.1 Level $825$ Weight $4$ Character 825.199 Analytic conductor $48.677$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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, 1, 0]))

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

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$-2.56155i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.4.c.j.199.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.56155i q^{2} -3.00000i q^{3} +1.43845 q^{4} -7.68466 q^{6} +6.24621i q^{7} -24.1771i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q-2.56155i q^{2} -3.00000i q^{3} +1.43845 q^{4} -7.68466 q^{6} +6.24621i q^{7} -24.1771i q^{8} -9.00000 q^{9} -11.0000 q^{11} -4.31534i q^{12} +49.1231i q^{13} +16.0000 q^{14} -50.4233 q^{16} +82.7083i q^{17} +23.0540i q^{18} +130.354 q^{19} +18.7386 q^{21} +28.1771i q^{22} +185.693i q^{23} -72.5312 q^{24} +125.831 q^{26} +27.0000i q^{27} +8.98485i q^{28} +8.90720 q^{29} +5.26137 q^{31} -64.2547i q^{32} +33.0000i q^{33} +211.862 q^{34} -12.9460 q^{36} -416.894i q^{37} -333.909i q^{38} +147.369 q^{39} -298.479 q^{41} -48.0000i q^{42} +513.633i q^{43} -15.8229 q^{44} +475.663 q^{46} +557.295i q^{47} +151.270i q^{48} +303.985 q^{49} +248.125 q^{51} +70.6610i q^{52} +168.064i q^{53} +69.1619 q^{54} +151.015 q^{56} -391.062i q^{57} -22.8163i q^{58} -618.773 q^{59} +786.405 q^{61} -13.4773i q^{62} -56.2159i q^{63} -567.978 q^{64} +84.5312 q^{66} -339.015i q^{67} +118.972i q^{68} +557.080 q^{69} +1120.71 q^{71} +217.594i q^{72} +123.430i q^{73} -1067.90 q^{74} +187.508 q^{76} -68.7083i q^{77} -377.494i q^{78} +309.835 q^{79} +81.0000 q^{81} +764.570i q^{82} +1021.22i q^{83} +26.9545 q^{84} +1315.70 q^{86} -26.7216i q^{87} +265.948i q^{88} +141.879 q^{89} -306.833 q^{91} +267.110i q^{92} -15.7841i q^{93} +1427.54 q^{94} -192.764 q^{96} +798.345i q^{97} -778.673i q^{98} +99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 14 q^{4} - 6 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q + 14 * q^4 - 6 * q^6 - 36 * q^9 $$4 q + 14 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 64 q^{14} - 78 q^{16} + 340 q^{19} - 24 q^{21} - 18 q^{24} + 124 q^{26} + 316 q^{29} + 120 q^{31} + 732 q^{34} - 126 q^{36} + 540 q^{39} + 76 q^{41} - 154 q^{44} + 1144 q^{46} + 1084 q^{49} - 96 q^{51} + 54 q^{54} + 736 q^{56} - 496 q^{59} + 144 q^{61} - 1538 q^{64} + 66 q^{66} + 744 q^{69} + 4120 q^{71} - 2276 q^{74} + 816 q^{76} - 1284 q^{79} + 324 q^{81} - 288 q^{84} + 2624 q^{86} - 488 q^{89} + 224 q^{91} + 3896 q^{94} + 738 q^{96} + 396 q^{99}+O(q^{100})$$ 4 * q + 14 * q^4 - 6 * q^6 - 36 * q^9 - 44 * q^11 + 64 * q^14 - 78 * q^16 + 340 * q^19 - 24 * q^21 - 18 * q^24 + 124 * q^26 + 316 * q^29 + 120 * q^31 + 732 * q^34 - 126 * q^36 + 540 * q^39 + 76 * q^41 - 154 * q^44 + 1144 * q^46 + 1084 * q^49 - 96 * q^51 + 54 * q^54 + 736 * q^56 - 496 * q^59 + 144 * q^61 - 1538 * q^64 + 66 * q^66 + 744 * q^69 + 4120 * q^71 - 2276 * q^74 + 816 * q^76 - 1284 * q^79 + 324 * q^81 - 288 * q^84 + 2624 * q^86 - 488 * q^89 + 224 * q^91 + 3896 * q^94 + 738 * q^96 + 396 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.56155i − 0.905646i −0.891601 0.452823i $$-0.850417\pi$$
0.891601 0.452823i $$-0.149583\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ 1.43845 0.179806
$$5$$ 0 0
$$6$$ −7.68466 −0.522875
$$7$$ 6.24621i 0.337264i 0.985679 + 0.168632i $$0.0539349\pi$$
−0.985679 + 0.168632i $$0.946065\pi$$
$$8$$ − 24.1771i − 1.06849i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ − 4.31534i − 0.103811i
$$13$$ 49.1231i 1.04802i 0.851711 + 0.524011i $$0.175565\pi$$
−0.851711 + 0.524011i $$0.824435\pi$$
$$14$$ 16.0000 0.305441
$$15$$ 0 0
$$16$$ −50.4233 −0.787864
$$17$$ 82.7083i 1.17998i 0.807409 + 0.589992i $$0.200869\pi$$
−0.807409 + 0.589992i $$0.799131\pi$$
$$18$$ 23.0540i 0.301882i
$$19$$ 130.354 1.57396 0.786981 0.616977i $$-0.211643\pi$$
0.786981 + 0.616977i $$0.211643\pi$$
$$20$$ 0 0
$$21$$ 18.7386 0.194719
$$22$$ 28.1771i 0.273062i
$$23$$ 185.693i 1.68347i 0.539895 + 0.841733i $$0.318464\pi$$
−0.539895 + 0.841733i $$0.681536\pi$$
$$24$$ −72.5312 −0.616891
$$25$$ 0 0
$$26$$ 125.831 0.949137
$$27$$ 27.0000i 0.192450i
$$28$$ 8.98485i 0.0606420i
$$29$$ 8.90720 0.0570354 0.0285177 0.999593i $$-0.490921\pi$$
0.0285177 + 0.999593i $$0.490921\pi$$
$$30$$ 0 0
$$31$$ 5.26137 0.0304829 0.0152414 0.999884i $$-0.495148\pi$$
0.0152414 + 0.999884i $$0.495148\pi$$
$$32$$ − 64.2547i − 0.354961i
$$33$$ 33.0000i 0.174078i
$$34$$ 211.862 1.06865
$$35$$ 0 0
$$36$$ −12.9460 −0.0599353
$$37$$ − 416.894i − 1.85235i −0.377094 0.926175i $$-0.623077\pi$$
0.377094 0.926175i $$-0.376923\pi$$
$$38$$ − 333.909i − 1.42545i
$$39$$ 147.369 0.605076
$$40$$ 0 0
$$41$$ −298.479 −1.13694 −0.568471 0.822703i $$-0.692465\pi$$
−0.568471 + 0.822703i $$0.692465\pi$$
$$42$$ − 48.0000i − 0.176347i
$$43$$ 513.633i 1.82159i 0.412863 + 0.910793i $$0.364529\pi$$
−0.412863 + 0.910793i $$0.635471\pi$$
$$44$$ −15.8229 −0.0542135
$$45$$ 0 0
$$46$$ 475.663 1.52462
$$47$$ 557.295i 1.72957i 0.502140 + 0.864786i $$0.332546\pi$$
−0.502140 + 0.864786i $$0.667454\pi$$
$$48$$ 151.270i 0.454873i
$$49$$ 303.985 0.886253
$$50$$ 0 0
$$51$$ 248.125 0.681264
$$52$$ 70.6610i 0.188441i
$$53$$ 168.064i 0.435574i 0.975996 + 0.217787i $$0.0698838\pi$$
−0.975996 + 0.217787i $$0.930116\pi$$
$$54$$ 69.1619 0.174292
$$55$$ 0 0
$$56$$ 151.015 0.360362
$$57$$ − 391.062i − 0.908728i
$$58$$ − 22.8163i − 0.0516539i
$$59$$ −618.773 −1.36538 −0.682689 0.730709i $$-0.739190\pi$$
−0.682689 + 0.730709i $$0.739190\pi$$
$$60$$ 0 0
$$61$$ 786.405 1.65064 0.825319 0.564667i $$-0.190996\pi$$
0.825319 + 0.564667i $$0.190996\pi$$
$$62$$ − 13.4773i − 0.0276067i
$$63$$ − 56.2159i − 0.112421i
$$64$$ −567.978 −1.10933
$$65$$ 0 0
$$66$$ 84.5312 0.157653
$$67$$ − 339.015i − 0.618169i −0.951035 0.309084i $$-0.899977\pi$$
0.951035 0.309084i $$-0.100023\pi$$
$$68$$ 118.972i 0.212168i
$$69$$ 557.080 0.971949
$$70$$ 0 0
$$71$$ 1120.71 1.87329 0.936645 0.350280i $$-0.113913\pi$$
0.936645 + 0.350280i $$0.113913\pi$$
$$72$$ 217.594i 0.356162i
$$73$$ 123.430i 0.197896i 0.995093 + 0.0989478i $$0.0315477\pi$$
−0.995093 + 0.0989478i $$0.968452\pi$$
$$74$$ −1067.90 −1.67757
$$75$$ 0 0
$$76$$ 187.508 0.283008
$$77$$ − 68.7083i − 0.101689i
$$78$$ − 377.494i − 0.547985i
$$79$$ 309.835 0.441255 0.220628 0.975358i $$-0.429189\pi$$
0.220628 + 0.975358i $$0.429189\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 764.570i 1.02967i
$$83$$ 1021.22i 1.35053i 0.737577 + 0.675263i $$0.235970\pi$$
−0.737577 + 0.675263i $$0.764030\pi$$
$$84$$ 26.9545 0.0350117
$$85$$ 0 0
$$86$$ 1315.70 1.64971
$$87$$ − 26.7216i − 0.0329294i
$$88$$ 265.948i 0.322161i
$$89$$ 141.879 0.168979 0.0844894 0.996424i $$-0.473074\pi$$
0.0844894 + 0.996424i $$0.473074\pi$$
$$90$$ 0 0
$$91$$ −306.833 −0.353460
$$92$$ 267.110i 0.302697i
$$93$$ − 15.7841i − 0.0175993i
$$94$$ 1427.54 1.56638
$$95$$ 0 0
$$96$$ −192.764 −0.204937
$$97$$ 798.345i 0.835666i 0.908524 + 0.417833i $$0.137210\pi$$
−0.908524 + 0.417833i $$0.862790\pi$$
$$98$$ − 778.673i − 0.802631i
$$99$$ 99.0000 0.100504
$$100$$ 0 0
$$101$$ 241.400 0.237823 0.118912 0.992905i $$-0.462059\pi$$
0.118912 + 0.992905i $$0.462059\pi$$
$$102$$ − 635.585i − 0.616983i
$$103$$ − 1168.38i − 1.11771i −0.829267 0.558853i $$-0.811242\pi$$
0.829267 0.558853i $$-0.188758\pi$$
$$104$$ 1187.65 1.11980
$$105$$ 0 0
$$106$$ 430.506 0.394476
$$107$$ − 2106.82i − 1.90350i −0.306882 0.951748i $$-0.599286\pi$$
0.306882 0.951748i $$-0.400714\pi$$
$$108$$ 38.8381i 0.0346037i
$$109$$ −493.792 −0.433914 −0.216957 0.976181i $$-0.569613\pi$$
−0.216957 + 0.976181i $$0.569613\pi$$
$$110$$ 0 0
$$111$$ −1250.68 −1.06945
$$112$$ − 314.955i − 0.265718i
$$113$$ − 170.000i − 0.141524i −0.997493 0.0707622i $$-0.977457\pi$$
0.997493 0.0707622i $$-0.0225431\pi$$
$$114$$ −1001.73 −0.822986
$$115$$ 0 0
$$116$$ 12.8125 0.0102553
$$117$$ − 442.108i − 0.349341i
$$118$$ 1585.02i 1.23655i
$$119$$ −516.614 −0.397966
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ − 2014.42i − 1.49489i
$$123$$ 895.437i 0.656414i
$$124$$ 7.56820 0.00548100
$$125$$ 0 0
$$126$$ −144.000 −0.101814
$$127$$ − 948.182i − 0.662500i −0.943543 0.331250i $$-0.892530\pi$$
0.943543 0.331250i $$-0.107470\pi$$
$$128$$ 940.868i 0.649702i
$$129$$ 1540.90 1.05169
$$130$$ 0 0
$$131$$ −1484.84 −0.990312 −0.495156 0.868804i $$-0.664889\pi$$
−0.495156 + 0.868804i $$0.664889\pi$$
$$132$$ 47.4688i 0.0313002i
$$133$$ 814.220i 0.530841i
$$134$$ −868.405 −0.559842
$$135$$ 0 0
$$136$$ 1999.65 1.26080
$$137$$ 684.928i 0.427134i 0.976928 + 0.213567i $$0.0685082\pi$$
−0.976928 + 0.213567i $$0.931492\pi$$
$$138$$ − 1426.99i − 0.880242i
$$139$$ −830.483 −0.506767 −0.253384 0.967366i $$-0.581543\pi$$
−0.253384 + 0.967366i $$0.581543\pi$$
$$140$$ 0 0
$$141$$ 1671.89 0.998569
$$142$$ − 2870.75i − 1.69654i
$$143$$ − 540.354i − 0.315991i
$$144$$ 453.810 0.262621
$$145$$ 0 0
$$146$$ 316.172 0.179223
$$147$$ − 911.955i − 0.511679i
$$148$$ − 599.680i − 0.333063i
$$149$$ 1213.64 0.667285 0.333642 0.942700i $$-0.391722\pi$$
0.333642 + 0.942700i $$0.391722\pi$$
$$150$$ 0 0
$$151$$ 30.8466 0.0166242 0.00831212 0.999965i $$-0.497354\pi$$
0.00831212 + 0.999965i $$0.497354\pi$$
$$152$$ − 3151.58i − 1.68176i
$$153$$ − 744.375i − 0.393328i
$$154$$ −176.000 −0.0920941
$$155$$ 0 0
$$156$$ 211.983 0.108796
$$157$$ 345.239i 0.175497i 0.996143 + 0.0877485i $$0.0279672\pi$$
−0.996143 + 0.0877485i $$0.972033\pi$$
$$158$$ − 793.659i − 0.399621i
$$159$$ 504.193 0.251479
$$160$$ 0 0
$$161$$ −1159.88 −0.567772
$$162$$ − 207.486i − 0.100627i
$$163$$ − 1921.49i − 0.923331i −0.887054 0.461665i $$-0.847252\pi$$
0.887054 0.461665i $$-0.152748\pi$$
$$164$$ −429.346 −0.204429
$$165$$ 0 0
$$166$$ 2615.91 1.22310
$$167$$ 172.297i 0.0798369i 0.999203 + 0.0399185i $$0.0127098\pi$$
−0.999203 + 0.0399185i $$0.987290\pi$$
$$168$$ − 453.045i − 0.208055i
$$169$$ −216.080 −0.0983521
$$170$$ 0 0
$$171$$ −1173.19 −0.524654
$$172$$ 738.833i 0.327532i
$$173$$ 1025.29i 0.450587i 0.974291 + 0.225293i $$0.0723340\pi$$
−0.974291 + 0.225293i $$0.927666\pi$$
$$174$$ −68.4488 −0.0298224
$$175$$ 0 0
$$176$$ 554.656 0.237550
$$177$$ 1856.32i 0.788302i
$$178$$ − 363.430i − 0.153035i
$$179$$ −1658.29 −0.692437 −0.346219 0.938154i $$-0.612534\pi$$
−0.346219 + 0.938154i $$0.612534\pi$$
$$180$$ 0 0
$$181$$ −2021.81 −0.830275 −0.415137 0.909759i $$-0.636266\pi$$
−0.415137 + 0.909759i $$0.636266\pi$$
$$182$$ 785.970i 0.320110i
$$183$$ − 2359.22i − 0.952996i
$$184$$ 4489.52 1.79876
$$185$$ 0 0
$$186$$ −40.4318 −0.0159387
$$187$$ − 909.792i − 0.355778i
$$188$$ 801.640i 0.310987i
$$189$$ −168.648 −0.0649064
$$190$$ 0 0
$$191$$ 1440.22 0.545607 0.272803 0.962070i $$-0.412049\pi$$
0.272803 + 0.962070i $$0.412049\pi$$
$$192$$ 1703.93i 0.640473i
$$193$$ 2798.05i 1.04356i 0.853079 + 0.521782i $$0.174733\pi$$
−0.853079 + 0.521782i $$0.825267\pi$$
$$194$$ 2045.00 0.756817
$$195$$ 0 0
$$196$$ 437.266 0.159354
$$197$$ 458.943i 0.165982i 0.996550 + 0.0829908i $$0.0264472\pi$$
−0.996550 + 0.0829908i $$0.973553\pi$$
$$198$$ − 253.594i − 0.0910208i
$$199$$ 2371.04 0.844615 0.422308 0.906453i $$-0.361220\pi$$
0.422308 + 0.906453i $$0.361220\pi$$
$$200$$ 0 0
$$201$$ −1017.05 −0.356900
$$202$$ − 618.358i − 0.215384i
$$203$$ 55.6363i 0.0192360i
$$204$$ 356.915 0.122495
$$205$$ 0 0
$$206$$ −2992.86 −1.01225
$$207$$ − 1671.24i − 0.561155i
$$208$$ − 2476.95i − 0.825699i
$$209$$ −1433.90 −0.474568
$$210$$ 0 0
$$211$$ 4319.87 1.40944 0.704721 0.709484i $$-0.251072\pi$$
0.704721 + 0.709484i $$0.251072\pi$$
$$212$$ 241.752i 0.0783187i
$$213$$ − 3362.12i − 1.08154i
$$214$$ −5396.73 −1.72389
$$215$$ 0 0
$$216$$ 652.781 0.205630
$$217$$ 32.8636i 0.0102808i
$$218$$ 1264.87i 0.392973i
$$219$$ 370.290 0.114255
$$220$$ 0 0
$$221$$ −4062.89 −1.23665
$$222$$ 3203.69i 0.968547i
$$223$$ 3837.73i 1.15244i 0.817295 + 0.576219i $$0.195473\pi$$
−0.817295 + 0.576219i $$0.804527\pi$$
$$224$$ 401.349 0.119715
$$225$$ 0 0
$$226$$ −435.464 −0.128171
$$227$$ − 5003.71i − 1.46303i −0.681825 0.731515i $$-0.738813\pi$$
0.681825 0.731515i $$-0.261187\pi$$
$$228$$ − 562.523i − 0.163395i
$$229$$ 277.375 0.0800412 0.0400206 0.999199i $$-0.487258\pi$$
0.0400206 + 0.999199i $$0.487258\pi$$
$$230$$ 0 0
$$231$$ −206.125 −0.0587101
$$232$$ − 215.350i − 0.0609415i
$$233$$ − 2269.91i − 0.638225i −0.947717 0.319113i $$-0.896615\pi$$
0.947717 0.319113i $$-0.103385\pi$$
$$234$$ −1132.48 −0.316379
$$235$$ 0 0
$$236$$ −890.072 −0.245503
$$237$$ − 929.505i − 0.254759i
$$238$$ 1323.33i 0.360416i
$$239$$ 1617.11 0.437665 0.218832 0.975762i $$-0.429775\pi$$
0.218832 + 0.975762i $$0.429775\pi$$
$$240$$ 0 0
$$241$$ 5646.63 1.50926 0.754629 0.656151i $$-0.227817\pi$$
0.754629 + 0.656151i $$0.227817\pi$$
$$242$$ − 309.948i − 0.0823314i
$$243$$ − 243.000i − 0.0641500i
$$244$$ 1131.20 0.296794
$$245$$ 0 0
$$246$$ 2293.71 0.594478
$$247$$ 6403.40i 1.64955i
$$248$$ − 127.204i − 0.0325705i
$$249$$ 3063.66 0.779726
$$250$$ 0 0
$$251$$ −6217.61 −1.56355 −0.781777 0.623558i $$-0.785687\pi$$
−0.781777 + 0.623558i $$0.785687\pi$$
$$252$$ − 80.8636i − 0.0202140i
$$253$$ − 2042.62i − 0.507584i
$$254$$ −2428.82 −0.599991
$$255$$ 0 0
$$256$$ −2133.74 −0.520933
$$257$$ 7712.75i 1.87202i 0.351980 + 0.936008i $$0.385509\pi$$
−0.351980 + 0.936008i $$0.614491\pi$$
$$258$$ − 3947.09i − 0.952462i
$$259$$ 2604.01 0.624730
$$260$$ 0 0
$$261$$ −80.1648 −0.0190118
$$262$$ 3803.49i 0.896871i
$$263$$ − 206.347i − 0.0483798i −0.999707 0.0241899i $$-0.992299\pi$$
0.999707 0.0241899i $$-0.00770063\pi$$
$$264$$ 797.844 0.186000
$$265$$ 0 0
$$266$$ 2085.67 0.480753
$$267$$ − 425.636i − 0.0975600i
$$268$$ − 487.655i − 0.111150i
$$269$$ −1712.47 −0.388146 −0.194073 0.980987i $$-0.562170\pi$$
−0.194073 + 0.980987i $$0.562170\pi$$
$$270$$ 0 0
$$271$$ −477.081 −0.106940 −0.0534698 0.998569i $$-0.517028\pi$$
−0.0534698 + 0.998569i $$0.517028\pi$$
$$272$$ − 4170.43i − 0.929666i
$$273$$ 920.500i 0.204070i
$$274$$ 1754.48 0.386832
$$275$$ 0 0
$$276$$ 801.329 0.174762
$$277$$ 4283.48i 0.929130i 0.885539 + 0.464565i $$0.153789\pi$$
−0.885539 + 0.464565i $$0.846211\pi$$
$$278$$ 2127.33i 0.458952i
$$279$$ −47.3523 −0.0101610
$$280$$ 0 0
$$281$$ 3477.79 0.738319 0.369160 0.929366i $$-0.379646\pi$$
0.369160 + 0.929366i $$0.379646\pi$$
$$282$$ − 4282.62i − 0.904350i
$$283$$ 6568.27i 1.37966i 0.723973 + 0.689829i $$0.242314\pi$$
−0.723973 + 0.689829i $$0.757686\pi$$
$$284$$ 1612.08 0.336829
$$285$$ 0 0
$$286$$ −1384.15 −0.286176
$$287$$ − 1864.36i − 0.383449i
$$288$$ 578.292i 0.118320i
$$289$$ −1927.67 −0.392360
$$290$$ 0 0
$$291$$ 2395.03 0.482472
$$292$$ 177.547i 0.0355828i
$$293$$ − 8352.29i − 1.66534i −0.553766 0.832672i $$-0.686810\pi$$
0.553766 0.832672i $$-0.313190\pi$$
$$294$$ −2336.02 −0.463399
$$295$$ 0 0
$$296$$ −10079.3 −1.97921
$$297$$ − 297.000i − 0.0580259i
$$298$$ − 3108.81i − 0.604324i
$$299$$ −9121.83 −1.76431
$$300$$ 0 0
$$301$$ −3208.26 −0.614355
$$302$$ − 79.0152i − 0.0150557i
$$303$$ − 724.199i − 0.137307i
$$304$$ −6572.89 −1.24007
$$305$$ 0 0
$$306$$ −1906.76 −0.356216
$$307$$ 5383.89i 1.00090i 0.865767 + 0.500448i $$0.166831\pi$$
−0.865767 + 0.500448i $$0.833169\pi$$
$$308$$ − 98.8333i − 0.0182843i
$$309$$ −3505.14 −0.645308
$$310$$ 0 0
$$311$$ −1790.41 −0.326447 −0.163223 0.986589i $$-0.552189\pi$$
−0.163223 + 0.986589i $$0.552189\pi$$
$$312$$ − 3562.96i − 0.646516i
$$313$$ 809.076i 0.146108i 0.997328 + 0.0730538i $$0.0232745\pi$$
−0.997328 + 0.0730538i $$0.976726\pi$$
$$314$$ 884.347 0.158938
$$315$$ 0 0
$$316$$ 445.682 0.0793403
$$317$$ 10744.5i 1.90370i 0.306567 + 0.951849i $$0.400820\pi$$
−0.306567 + 0.951849i $$0.599180\pi$$
$$318$$ − 1291.52i − 0.227751i
$$319$$ −97.9792 −0.0171968
$$320$$ 0 0
$$321$$ −6320.46 −1.09898
$$322$$ 2971.09i 0.514200i
$$323$$ 10781.4i 1.85725i
$$324$$ 116.514 0.0199784
$$325$$ 0 0
$$326$$ −4922.00 −0.836210
$$327$$ 1481.37i 0.250521i
$$328$$ 7216.35i 1.21481i
$$329$$ −3480.98 −0.583322
$$330$$ 0 0
$$331$$ 3399.12 0.564449 0.282224 0.959348i $$-0.408928\pi$$
0.282224 + 0.959348i $$0.408928\pi$$
$$332$$ 1468.97i 0.242832i
$$333$$ 3752.05i 0.617450i
$$334$$ 441.349 0.0723039
$$335$$ 0 0
$$336$$ −944.864 −0.153412
$$337$$ − 11840.0i − 1.91384i −0.290356 0.956919i $$-0.593774\pi$$
0.290356 0.956919i $$-0.406226\pi$$
$$338$$ 553.499i 0.0890721i
$$339$$ −510.000 −0.0817091
$$340$$ 0 0
$$341$$ −57.8750 −0.00919093
$$342$$ 3005.18i 0.475151i
$$343$$ 4041.20i 0.636165i
$$344$$ 12418.1 1.94634
$$345$$ 0 0
$$346$$ 2626.34 0.408072
$$347$$ − 2076.67i − 0.321272i −0.987014 0.160636i $$-0.948645\pi$$
0.987014 0.160636i $$-0.0513545\pi$$
$$348$$ − 38.4376i − 0.00592090i
$$349$$ −5837.37 −0.895322 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$350$$ 0 0
$$351$$ −1326.32 −0.201692
$$352$$ 706.802i 0.107025i
$$353$$ 2423.64i 0.365431i 0.983166 + 0.182715i $$0.0584887\pi$$
−0.983166 + 0.182715i $$0.941511\pi$$
$$354$$ 4755.06 0.713922
$$355$$ 0 0
$$356$$ 204.085 0.0303834
$$357$$ 1549.84i 0.229765i
$$358$$ 4247.79i 0.627103i
$$359$$ 3882.22 0.570740 0.285370 0.958417i $$-0.407884\pi$$
0.285370 + 0.958417i $$0.407884\pi$$
$$360$$ 0 0
$$361$$ 10133.2 1.47736
$$362$$ 5178.96i 0.751935i
$$363$$ − 363.000i − 0.0524864i
$$364$$ −441.363 −0.0635542
$$365$$ 0 0
$$366$$ −6043.26 −0.863077
$$367$$ 5666.65i 0.805986i 0.915203 + 0.402993i $$0.132030\pi$$
−0.915203 + 0.402993i $$0.867970\pi$$
$$368$$ − 9363.26i − 1.32634i
$$369$$ 2686.31 0.378981
$$370$$ 0 0
$$371$$ −1049.77 −0.146903
$$372$$ − 22.7046i − 0.00316446i
$$373$$ − 174.771i − 0.0242608i −0.999926 0.0121304i $$-0.996139\pi$$
0.999926 0.0121304i $$-0.00386132\pi$$
$$374$$ −2330.48 −0.322209
$$375$$ 0 0
$$376$$ 13473.8 1.84802
$$377$$ 437.550i 0.0597744i
$$378$$ 432.000i 0.0587822i
$$379$$ −252.686 −0.0342470 −0.0171235 0.999853i $$-0.505451\pi$$
−0.0171235 + 0.999853i $$0.505451\pi$$
$$380$$ 0 0
$$381$$ −2844.55 −0.382495
$$382$$ − 3689.21i − 0.494126i
$$383$$ 11014.5i 1.46950i 0.678340 + 0.734748i $$0.262700\pi$$
−0.678340 + 0.734748i $$0.737300\pi$$
$$384$$ 2822.61 0.375105
$$385$$ 0 0
$$386$$ 7167.35 0.945099
$$387$$ − 4622.69i − 0.607196i
$$388$$ 1148.38i 0.150258i
$$389$$ −8099.40 −1.05567 −0.527835 0.849347i $$-0.676996\pi$$
−0.527835 + 0.849347i $$0.676996\pi$$
$$390$$ 0 0
$$391$$ −15358.4 −1.98646
$$392$$ − 7349.47i − 0.946949i
$$393$$ 4454.51i 0.571757i
$$394$$ 1175.61 0.150320
$$395$$ 0 0
$$396$$ 142.406 0.0180712
$$397$$ − 424.353i − 0.0536465i −0.999640 0.0268232i $$-0.991461\pi$$
0.999640 0.0268232i $$-0.00853912\pi$$
$$398$$ − 6073.54i − 0.764922i
$$399$$ 2442.66 0.306481
$$400$$ 0 0
$$401$$ −5904.18 −0.735263 −0.367632 0.929972i $$-0.619831\pi$$
−0.367632 + 0.929972i $$0.619831\pi$$
$$402$$ 2605.22i 0.323225i
$$403$$ 258.455i 0.0319468i
$$404$$ 347.241 0.0427620
$$405$$ 0 0
$$406$$ 142.515 0.0174210
$$407$$ 4585.83i 0.558504i
$$408$$ − 5998.94i − 0.727921i
$$409$$ −1370.47 −0.165686 −0.0828430 0.996563i $$-0.526400\pi$$
−0.0828430 + 0.996563i $$0.526400\pi$$
$$410$$ 0 0
$$411$$ 2054.78 0.246606
$$412$$ − 1680.65i − 0.200970i
$$413$$ − 3864.98i − 0.460493i
$$414$$ −4280.97 −0.508208
$$415$$ 0 0
$$416$$ 3156.39 0.372007
$$417$$ 2491.45i 0.292582i
$$418$$ 3673.00i 0.429790i
$$419$$ −1268.73 −0.147927 −0.0739635 0.997261i $$-0.523565\pi$$
−0.0739635 + 0.997261i $$0.523565\pi$$
$$420$$ 0 0
$$421$$ −12241.9 −1.41719 −0.708594 0.705617i $$-0.750670\pi$$
−0.708594 + 0.705617i $$0.750670\pi$$
$$422$$ − 11065.6i − 1.27646i
$$423$$ − 5015.66i − 0.576524i
$$424$$ 4063.31 0.465405
$$425$$ 0 0
$$426$$ −8612.26 −0.979496
$$427$$ 4912.05i 0.556700i
$$428$$ − 3030.55i − 0.342260i
$$429$$ −1621.06 −0.182437
$$430$$ 0 0
$$431$$ 8050.11 0.899675 0.449838 0.893110i $$-0.351482\pi$$
0.449838 + 0.893110i $$0.351482\pi$$
$$432$$ − 1361.43i − 0.151624i
$$433$$ 16565.7i 1.83856i 0.393600 + 0.919282i $$0.371230\pi$$
−0.393600 + 0.919282i $$0.628770\pi$$
$$434$$ 84.1819 0.00931073
$$435$$ 0 0
$$436$$ −710.293 −0.0780203
$$437$$ 24205.9i 2.64971i
$$438$$ − 948.517i − 0.103475i
$$439$$ −4705.80 −0.511607 −0.255804 0.966729i $$-0.582340\pi$$
−0.255804 + 0.966729i $$0.582340\pi$$
$$440$$ 0 0
$$441$$ −2735.86 −0.295418
$$442$$ 10407.3i 1.11997i
$$443$$ − 15094.0i − 1.61882i −0.587246 0.809408i $$-0.699788\pi$$
0.587246 0.809408i $$-0.300212\pi$$
$$444$$ −1799.04 −0.192294
$$445$$ 0 0
$$446$$ 9830.56 1.04370
$$447$$ − 3640.93i − 0.385257i
$$448$$ − 3547.71i − 0.374138i
$$449$$ −973.478 −0.102319 −0.0511595 0.998690i $$-0.516292\pi$$
−0.0511595 + 0.998690i $$0.516292\pi$$
$$450$$ 0 0
$$451$$ 3283.27 0.342801
$$452$$ − 244.536i − 0.0254469i
$$453$$ − 92.5398i − 0.00959801i
$$454$$ −12817.3 −1.32499
$$455$$ 0 0
$$456$$ −9454.75 −0.970963
$$457$$ 62.6577i 0.00641358i 0.999995 + 0.00320679i $$0.00102075\pi$$
−0.999995 + 0.00320679i $$0.998979\pi$$
$$458$$ − 710.510i − 0.0724890i
$$459$$ −2233.12 −0.227088
$$460$$ 0 0
$$461$$ −11866.2 −1.19884 −0.599419 0.800436i $$-0.704602\pi$$
−0.599419 + 0.800436i $$0.704602\pi$$
$$462$$ 528.000i 0.0531705i
$$463$$ 13144.8i 1.31942i 0.751519 + 0.659711i $$0.229321\pi$$
−0.751519 + 0.659711i $$0.770679\pi$$
$$464$$ −449.131 −0.0449361
$$465$$ 0 0
$$466$$ −5814.48 −0.578006
$$467$$ − 10176.8i − 1.00840i −0.863586 0.504201i $$-0.831787\pi$$
0.863586 0.504201i $$-0.168213\pi$$
$$468$$ − 635.949i − 0.0628136i
$$469$$ 2117.56 0.208486
$$470$$ 0 0
$$471$$ 1035.72 0.101323
$$472$$ 14960.1i 1.45889i
$$473$$ − 5649.96i − 0.549229i
$$474$$ −2380.98 −0.230721
$$475$$ 0 0
$$476$$ −743.121 −0.0715565
$$477$$ − 1512.58i − 0.145191i
$$478$$ − 4142.30i − 0.396369i
$$479$$ −3431.25 −0.327302 −0.163651 0.986518i $$-0.552327\pi$$
−0.163651 + 0.986518i $$0.552327\pi$$
$$480$$ 0 0
$$481$$ 20479.1 1.94130
$$482$$ − 14464.1i − 1.36685i
$$483$$ 3479.64i 0.327803i
$$484$$ 174.052 0.0163460
$$485$$ 0 0
$$486$$ −622.457 −0.0580972
$$487$$ − 2833.20i − 0.263624i −0.991275 0.131812i $$-0.957921\pi$$
0.991275 0.131812i $$-0.0420795\pi$$
$$488$$ − 19013.0i − 1.76368i
$$489$$ −5764.48 −0.533085
$$490$$ 0 0
$$491$$ −2667.29 −0.245159 −0.122580 0.992459i $$-0.539117\pi$$
−0.122580 + 0.992459i $$0.539117\pi$$
$$492$$ 1288.04i 0.118027i
$$493$$ 736.700i 0.0673008i
$$494$$ 16402.7 1.49391
$$495$$ 0 0
$$496$$ −265.295 −0.0240164
$$497$$ 7000.18i 0.631793i
$$498$$ − 7847.74i − 0.706156i
$$499$$ 11137.0 0.999120 0.499560 0.866279i $$-0.333495\pi$$
0.499560 + 0.866279i $$0.333495\pi$$
$$500$$ 0 0
$$501$$ 516.892 0.0460939
$$502$$ 15926.7i 1.41603i
$$503$$ − 8780.30i − 0.778319i −0.921170 0.389159i $$-0.872766\pi$$
0.921170 0.389159i $$-0.127234\pi$$
$$504$$ −1359.14 −0.120121
$$505$$ 0 0
$$506$$ −5232.29 −0.459691
$$507$$ 648.239i 0.0567836i
$$508$$ − 1363.91i − 0.119121i
$$509$$ −13597.4 −1.18408 −0.592039 0.805910i $$-0.701677\pi$$
−0.592039 + 0.805910i $$0.701677\pi$$
$$510$$ 0 0
$$511$$ −770.969 −0.0667430
$$512$$ 12992.6i 1.12148i
$$513$$ 3519.56i 0.302909i
$$514$$ 19756.6 1.69538
$$515$$ 0 0
$$516$$ 2216.50 0.189101
$$517$$ − 6130.25i − 0.521486i
$$518$$ − 6670.30i − 0.565784i
$$519$$ 3075.88 0.260146
$$520$$ 0 0
$$521$$ 14001.3 1.17736 0.588682 0.808364i $$-0.299647\pi$$
0.588682 + 0.808364i $$0.299647\pi$$
$$522$$ 205.346i 0.0172180i
$$523$$ 14749.8i 1.23320i 0.787275 + 0.616602i $$0.211491\pi$$
−0.787275 + 0.616602i $$0.788509\pi$$
$$524$$ −2135.86 −0.178064
$$525$$ 0 0
$$526$$ −528.568 −0.0438149
$$527$$ 435.159i 0.0359693i
$$528$$ − 1663.97i − 0.137150i
$$529$$ −22315.0 −1.83406
$$530$$ 0 0
$$531$$ 5568.95 0.455126
$$532$$ 1171.21i 0.0954483i
$$533$$ − 14662.2i − 1.19154i
$$534$$ −1090.29 −0.0883548
$$535$$ 0 0
$$536$$ −8196.40 −0.660505
$$537$$ 4974.86i 0.399779i
$$538$$ 4386.59i 0.351523i
$$539$$ −3343.83 −0.267215
$$540$$ 0 0
$$541$$ 1484.06 0.117939 0.0589694 0.998260i $$-0.481219\pi$$
0.0589694 + 0.998260i $$0.481219\pi$$
$$542$$ 1222.07i 0.0968493i
$$543$$ 6065.42i 0.479359i
$$544$$ 5314.40 0.418847
$$545$$ 0 0
$$546$$ 2357.91 0.184815
$$547$$ − 16562.2i − 1.29460i −0.762234 0.647302i $$-0.775897\pi$$
0.762234 0.647302i $$-0.224103\pi$$
$$548$$ 985.233i 0.0768012i
$$549$$ −7077.65 −0.550212
$$550$$ 0 0
$$551$$ 1161.09 0.0897716
$$552$$ − 13468.6i − 1.03851i
$$553$$ 1935.30i 0.148819i
$$554$$ 10972.3 0.841463
$$555$$ 0 0
$$556$$ −1194.61 −0.0911197
$$557$$ − 8821.52i − 0.671059i −0.942030 0.335529i $$-0.891085\pi$$
0.942030 0.335529i $$-0.108915\pi$$
$$558$$ 121.295i 0.00920223i
$$559$$ −25231.2 −1.90906
$$560$$ 0 0
$$561$$ −2729.37 −0.205409
$$562$$ − 8908.55i − 0.668656i
$$563$$ 5985.53i 0.448064i 0.974582 + 0.224032i $$0.0719220\pi$$
−0.974582 + 0.224032i $$0.928078\pi$$
$$564$$ 2404.92 0.179549
$$565$$ 0 0
$$566$$ 16825.0 1.24948
$$567$$ 505.943i 0.0374737i
$$568$$ − 27095.5i − 2.00158i
$$569$$ 3453.08 0.254413 0.127206 0.991876i $$-0.459399\pi$$
0.127206 + 0.991876i $$0.459399\pi$$
$$570$$ 0 0
$$571$$ −21484.5 −1.57460 −0.787302 0.616568i $$-0.788523\pi$$
−0.787302 + 0.616568i $$0.788523\pi$$
$$572$$ − 777.271i − 0.0568170i
$$573$$ − 4320.67i − 0.315006i
$$574$$ −4775.67 −0.347269
$$575$$ 0 0
$$576$$ 5111.80 0.369777
$$577$$ − 13294.4i − 0.959189i −0.877490 0.479594i $$-0.840784\pi$$
0.877490 0.479594i $$-0.159216\pi$$
$$578$$ 4937.82i 0.355340i
$$579$$ 8394.14 0.602502
$$580$$ 0 0
$$581$$ −6378.77 −0.455483
$$582$$ − 6135.01i − 0.436949i
$$583$$ − 1848.71i − 0.131330i
$$584$$ 2984.18 0.211449
$$585$$ 0 0
$$586$$ −21394.8 −1.50821
$$587$$ 6695.73i 0.470805i 0.971898 + 0.235402i $$0.0756408\pi$$
−0.971898 + 0.235402i $$0.924359\pi$$
$$588$$ − 1311.80i − 0.0920028i
$$589$$ 685.841 0.0479789
$$590$$ 0 0
$$591$$ 1376.83 0.0958295
$$592$$ 21021.2i 1.45940i
$$593$$ 10239.6i 0.709088i 0.935039 + 0.354544i $$0.115364\pi$$
−0.935039 + 0.354544i $$0.884636\pi$$
$$594$$ −760.781 −0.0525509
$$595$$ 0 0
$$596$$ 1745.76 0.119982
$$597$$ − 7113.11i − 0.487639i
$$598$$ 23366.0i 1.59784i
$$599$$ 23890.8 1.62963 0.814817 0.579719i $$-0.196838\pi$$
0.814817 + 0.579719i $$0.196838\pi$$
$$600$$ 0 0
$$601$$ −11343.8 −0.769920 −0.384960 0.922933i $$-0.625785\pi$$
−0.384960 + 0.922933i $$0.625785\pi$$
$$602$$ 8218.12i 0.556388i
$$603$$ 3051.14i 0.206056i
$$604$$ 44.3712 0.00298914
$$605$$ 0 0
$$606$$ −1855.07 −0.124352
$$607$$ − 26032.5i − 1.74074i −0.492399 0.870369i $$-0.663880\pi$$
0.492399 0.870369i $$-0.336120\pi$$
$$608$$ − 8375.87i − 0.558695i
$$609$$ 166.909 0.0111059
$$610$$ 0 0
$$611$$ −27376.1 −1.81263
$$612$$ − 1070.74i − 0.0707226i
$$613$$ 4568.13i 0.300987i 0.988611 + 0.150493i $$0.0480863\pi$$
−0.988611 + 0.150493i $$0.951914\pi$$
$$614$$ 13791.1 0.906457
$$615$$ 0 0
$$616$$ −1661.17 −0.108653
$$617$$ − 12755.9i − 0.832308i −0.909294 0.416154i $$-0.863378\pi$$
0.909294 0.416154i $$-0.136622\pi$$
$$618$$ 8978.59i 0.584421i
$$619$$ 1138.94 0.0739545 0.0369772 0.999316i $$-0.488227\pi$$
0.0369772 + 0.999316i $$0.488227\pi$$
$$620$$ 0 0
$$621$$ −5013.72 −0.323983
$$622$$ 4586.24i 0.295645i
$$623$$ 886.205i 0.0569904i
$$624$$ −7430.85 −0.476718
$$625$$ 0 0
$$626$$ 2072.49 0.132322
$$627$$ 4301.69i 0.273992i
$$628$$ 496.607i 0.0315554i
$$629$$ 34480.6 2.18574
$$630$$ 0 0
$$631$$ 7997.36 0.504548 0.252274 0.967656i $$-0.418822\pi$$
0.252274 + 0.967656i $$0.418822\pi$$
$$632$$ − 7490.91i − 0.471475i
$$633$$ − 12959.6i − 0.813742i
$$634$$ 27522.7 1.72408
$$635$$ 0 0
$$636$$ 725.255 0.0452173
$$637$$ 14932.7i 0.928814i
$$638$$ 250.979i 0.0155742i
$$639$$ −10086.4 −0.624430
$$640$$ 0 0
$$641$$ −573.115 −0.0353146 −0.0176573 0.999844i $$-0.505621\pi$$
−0.0176573 + 0.999844i $$0.505621\pi$$
$$642$$ 16190.2i 0.995290i
$$643$$ 16027.8i 0.983009i 0.870875 + 0.491504i $$0.163553\pi$$
−0.870875 + 0.491504i $$0.836447\pi$$
$$644$$ −1668.42 −0.102089
$$645$$ 0 0
$$646$$ 27617.1 1.68201
$$647$$ − 2622.74i − 0.159367i −0.996820 0.0796837i $$-0.974609\pi$$
0.996820 0.0796837i $$-0.0253910\pi$$
$$648$$ − 1958.34i − 0.118721i
$$649$$ 6806.50 0.411677
$$650$$ 0 0
$$651$$ 98.5908 0.00593560
$$652$$ − 2763.97i − 0.166020i
$$653$$ − 3102.00i − 0.185897i −0.995671 0.0929484i $$-0.970371\pi$$
0.995671 0.0929484i $$-0.0296292\pi$$
$$654$$ 3794.62 0.226883
$$655$$ 0 0
$$656$$ 15050.3 0.895755
$$657$$ − 1110.87i − 0.0659652i
$$658$$ 8916.73i 0.528283i
$$659$$ 20840.2 1.23190 0.615948 0.787787i $$-0.288773\pi$$
0.615948 + 0.787787i $$0.288773\pi$$
$$660$$ 0 0
$$661$$ 18242.9 1.07348 0.536738 0.843749i $$-0.319657\pi$$
0.536738 + 0.843749i $$0.319657\pi$$
$$662$$ − 8707.03i − 0.511191i
$$663$$ 12188.7i 0.713980i
$$664$$ 24690.2 1.44302
$$665$$ 0 0
$$666$$ 9611.06 0.559191
$$667$$ 1654.01i 0.0960171i
$$668$$ 247.841i 0.0143551i
$$669$$ 11513.2 0.665361
$$670$$ 0 0
$$671$$ −8650.46 −0.497686
$$672$$ − 1204.05i − 0.0691177i
$$673$$ 12746.6i 0.730084i 0.930991 + 0.365042i $$0.118945\pi$$
−0.930991 + 0.365042i $$0.881055\pi$$
$$674$$ −30328.7 −1.73326
$$675$$ 0 0
$$676$$ −310.819 −0.0176843
$$677$$ − 7683.11i − 0.436168i −0.975930 0.218084i $$-0.930019\pi$$
0.975930 0.218084i $$-0.0699807\pi$$
$$678$$ 1306.39i 0.0739995i
$$679$$ −4986.63 −0.281840
$$680$$ 0 0
$$681$$ −15011.1 −0.844681
$$682$$ 148.250i 0.00832373i
$$683$$ 21397.1i 1.19874i 0.800473 + 0.599368i $$0.204582\pi$$
−0.800473 + 0.599368i $$0.795418\pi$$
$$684$$ −1687.57 −0.0943359
$$685$$ 0 0
$$686$$ 10351.8 0.576140
$$687$$ − 832.124i − 0.0462118i
$$688$$ − 25899.0i − 1.43516i
$$689$$ −8255.84 −0.456491
$$690$$ 0 0
$$691$$ 26137.5 1.43895 0.719477 0.694516i $$-0.244381\pi$$
0.719477 + 0.694516i $$0.244381\pi$$
$$692$$ 1474.83i 0.0810181i
$$693$$ 618.375i 0.0338963i
$$694$$ −5319.50 −0.290959
$$695$$ 0 0
$$696$$ −646.051 −0.0351846
$$697$$ − 24686.7i − 1.34157i
$$698$$ 14952.7i 0.810845i
$$699$$ −6809.72 −0.368479
$$700$$ 0 0
$$701$$ 13382.4 0.721036 0.360518 0.932752i $$-0.382600\pi$$
0.360518 + 0.932752i $$0.382600\pi$$
$$702$$ 3397.45i 0.182662i
$$703$$ − 54343.9i − 2.91553i
$$704$$ 6247.76 0.334476
$$705$$ 0 0
$$706$$ 6208.27 0.330951
$$707$$ 1507.83i 0.0802092i
$$708$$ 2670.22i 0.141741i
$$709$$ −18164.6 −0.962179 −0.481090 0.876671i $$-0.659759\pi$$
−0.481090 + 0.876671i $$0.659759\pi$$
$$710$$ 0 0
$$711$$ −2788.52 −0.147085
$$712$$ − 3430.21i − 0.180552i
$$713$$ 977.000i 0.0513169i
$$714$$ 3970.00 0.208086
$$715$$ 0 0
$$716$$ −2385.36 −0.124504
$$717$$ − 4851.32i − 0.252686i
$$718$$ − 9944.50i − 0.516888i
$$719$$ −9665.62 −0.501344 −0.250672 0.968072i $$-0.580652\pi$$
−0.250672 + 0.968072i $$0.580652\pi$$
$$720$$ 0 0
$$721$$ 7297.94 0.376962
$$722$$ − 25956.7i − 1.33796i
$$723$$ − 16939.9i − 0.871371i
$$724$$ −2908.26 −0.149288
$$725$$ 0 0
$$726$$ −929.844 −0.0475341
$$727$$ − 29779.6i − 1.51921i −0.650385 0.759605i $$-0.725392\pi$$
0.650385 0.759605i $$-0.274608\pi$$
$$728$$ 7418.33i 0.377667i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −42481.7 −2.14944
$$732$$ − 3393.61i − 0.171354i
$$733$$ − 35029.5i − 1.76513i −0.470187 0.882567i $$-0.655814\pi$$
0.470187 0.882567i $$-0.344186\pi$$
$$734$$ 14515.4 0.729937
$$735$$ 0 0
$$736$$ 11931.7 0.597564
$$737$$ 3729.17i 0.186385i
$$738$$ − 6881.13i − 0.343222i
$$739$$ −23297.3 −1.15968 −0.579842 0.814729i $$-0.696886\pi$$
−0.579842 + 0.814729i $$0.696886\pi$$
$$740$$ 0 0
$$741$$ 19210.2 0.952368
$$742$$ 2689.03i 0.133042i
$$743$$ − 21570.4i − 1.06506i −0.846411 0.532530i $$-0.821241\pi$$
0.846411 0.532530i $$-0.178759\pi$$
$$744$$ −381.613 −0.0188046
$$745$$ 0 0
$$746$$ −447.684 −0.0219717
$$747$$ − 9190.99i − 0.450175i
$$748$$ − 1308.69i − 0.0639710i
$$749$$ 13159.6 0.641980
$$750$$ 0 0
$$751$$ 28554.8 1.38746 0.693729 0.720236i $$-0.255966\pi$$
0.693729 + 0.720236i $$0.255966\pi$$
$$752$$ − 28100.7i − 1.36267i
$$753$$ 18652.8i 0.902719i
$$754$$ 1120.81 0.0541344
$$755$$ 0 0
$$756$$ −242.591 −0.0116706
$$757$$ 7812.81i 0.375114i 0.982254 + 0.187557i $$0.0600570\pi$$
−0.982254 + 0.187557i $$0.939943\pi$$
$$758$$ 647.268i 0.0310156i
$$759$$ −6127.87 −0.293054
$$760$$ 0 0
$$761$$ 2875.13 0.136956 0.0684778 0.997653i $$-0.478186\pi$$
0.0684778 + 0.997653i $$0.478186\pi$$
$$762$$ 7286.45i 0.346405i
$$763$$ − 3084.33i − 0.146344i
$$764$$ 2071.69 0.0981033
$$765$$ 0 0
$$766$$ 28214.3 1.33084
$$767$$ − 30396.0i − 1.43095i
$$768$$ 6401.22i 0.300761i
$$769$$ 27657.7 1.29696 0.648479 0.761233i $$-0.275406\pi$$
0.648479 + 0.761233i $$0.275406\pi$$
$$770$$ 0 0
$$771$$ 23138.2 1.08081
$$772$$ 4024.84i 0.187639i
$$773$$ 3929.35i 0.182832i 0.995813 + 0.0914160i $$0.0291393\pi$$
−0.995813 + 0.0914160i $$0.970861\pi$$
$$774$$ −11841.3 −0.549904
$$775$$ 0 0
$$776$$ 19301.6 0.892898
$$777$$ − 7812.02i − 0.360688i
$$778$$ 20747.0i 0.956064i
$$779$$ −38908.0 −1.78950
$$780$$ 0 0
$$781$$ −12327.8 −0.564818
$$782$$ 39341.3i 1.79903i
$$783$$ 240.495i 0.0109765i
$$784$$ −15327.9 −0.698247
$$785$$ 0 0
$$786$$ 11410.5 0.517809
$$787$$ − 21125.7i − 0.956860i −0.878126 0.478430i $$-0.841206\pi$$
0.878126 0.478430i $$-0.158794\pi$$
$$788$$ 660.166i 0.0298445i
$$789$$ −619.040 −0.0279321
$$790$$ 0 0
$$791$$ 1061.86 0.0477310
$$792$$ − 2393.53i − 0.107387i
$$793$$ 38630.7i 1.72991i
$$794$$ −1087.00 −0.0485847
$$795$$ 0 0
$$796$$ 3410.61 0.151867
$$797$$ 11696.3i 0.519828i 0.965632 + 0.259914i $$0.0836942\pi$$
−0.965632 + 0.259914i $$0.916306\pi$$
$$798$$ − 6257.00i − 0.277563i
$$799$$ −46093.0 −2.04087
$$800$$ 0 0
$$801$$ −1276.91 −0.0563263
$$802$$ 15123.9i 0.665888i
$$803$$ − 1357.73i − 0.0596678i
$$804$$ −1462.97 −0.0641727
$$805$$ 0 0
$$806$$ 662.045 0.0289324
$$807$$ 5137.42i 0.224096i
$$808$$ − 5836.34i − 0.254111i
$$809$$ −14310.2 −0.621902 −0.310951 0.950426i $$-0.600648\pi$$
−0.310951 + 0.950426i $$0.600648\pi$$
$$810$$ 0 0
$$811$$ 21697.9 0.939477 0.469739 0.882806i $$-0.344348\pi$$
0.469739 + 0.882806i $$0.344348\pi$$
$$812$$ 80.0299i 0.00345874i
$$813$$ 1431.24i 0.0617416i
$$814$$ 11746.9 0.505807
$$815$$ 0 0
$$816$$ −12511.3 −0.536743
$$817$$ 66954.1i 2.86711i
$$818$$ 3510.54i 0.150053i
$$819$$ 2761.50 0.117820
$$820$$ 0 0
$$821$$ 3613.00 0.153587 0.0767934 0.997047i $$-0.475532\pi$$
0.0767934 + 0.997047i $$0.475532\pi$$
$$822$$ − 5263.44i − 0.223338i
$$823$$ 4763.98i 0.201776i 0.994898 + 0.100888i $$0.0321684\pi$$
−0.994898 + 0.100888i $$0.967832\pi$$
$$824$$ −28248.0 −1.19425
$$825$$ 0 0
$$826$$ −9900.36 −0.417043
$$827$$ − 33571.7i − 1.41161i −0.708405 0.705806i $$-0.750585\pi$$
0.708405 0.705806i $$-0.249415\pi$$
$$828$$ − 2403.99i − 0.100899i
$$829$$ −17980.5 −0.753303 −0.376652 0.926355i $$-0.622925\pi$$
−0.376652 + 0.926355i $$0.622925\pi$$
$$830$$ 0 0
$$831$$ 12850.4 0.536434
$$832$$ − 27900.9i − 1.16261i
$$833$$ 25142.1i 1.04576i
$$834$$ 6381.98 0.264976
$$835$$ 0 0
$$836$$ −2062.58 −0.0853301
$$837$$ 142.057i 0.00586643i
$$838$$ 3249.91i 0.133969i
$$839$$ 40139.5 1.65169 0.825847 0.563895i $$-0.190698\pi$$
0.825847 + 0.563895i $$0.190698\pi$$
$$840$$ 0 0
$$841$$ −24309.7 −0.996747
$$842$$ 31358.4i 1.28347i
$$843$$ − 10433.4i − 0.426269i
$$844$$ 6213.91 0.253426
$$845$$ 0 0
$$846$$ −12847.9 −0.522127
$$847$$ 755.792i 0.0306603i
$$848$$ − 8474.36i − 0.343173i
$$849$$ 19704.8 0.796546
$$850$$ 0 0
$$851$$ 77414.4 3.11837
$$852$$ − 4836.24i − 0.194468i
$$853$$ 15369.2i 0.616919i 0.951237 + 0.308459i $$0.0998134\pi$$
−0.951237 + 0.308459i $$0.900187\pi$$
$$854$$ 12582.5 0.504173
$$855$$ 0 0
$$856$$ −50936.8 −2.03386
$$857$$ 10324.9i 0.411541i 0.978600 + 0.205770i $$0.0659700\pi$$
−0.978600 + 0.205770i $$0.934030\pi$$
$$858$$ 4152.44i 0.165224i
$$859$$ 27112.5 1.07691 0.538455 0.842655i $$-0.319008\pi$$
0.538455 + 0.842655i $$0.319008\pi$$
$$860$$ 0 0
$$861$$ −5593.09 −0.221384
$$862$$ − 20620.8i − 0.814787i
$$863$$ − 30463.6i − 1.20161i −0.799394 0.600807i $$-0.794846\pi$$
0.799394 0.600807i $$-0.205154\pi$$
$$864$$ 1734.88 0.0683122
$$865$$ 0 0
$$866$$ 42434.0 1.66509
$$867$$ 5783.00i 0.226529i
$$868$$ 47.2726i 0.00184854i
$$869$$ −3408.19 −0.133044
$$870$$ 0 0
$$871$$ 16653.5 0.647855
$$872$$ 11938.4i 0.463631i
$$873$$ − 7185.10i − 0.278555i
$$874$$ 62004.6 2.39970
$$875$$ 0 0
$$876$$ 532.642 0.0205437
$$877$$ − 5086.12i − 0.195833i −0.995195 0.0979167i $$-0.968782\pi$$
0.995195 0.0979167i $$-0.0312179\pi$$
$$878$$ 12054.2i 0.463335i
$$879$$ −25056.9 −0.961487
$$880$$ 0 0
$$881$$ −10625.5 −0.406338 −0.203169 0.979144i $$-0.565124\pi$$
−0.203169 + 0.979144i $$0.565124\pi$$
$$882$$ 7008.06i 0.267544i
$$883$$ − 13112.2i − 0.499728i −0.968281 0.249864i $$-0.919614\pi$$
0.968281 0.249864i $$-0.0803859\pi$$
$$884$$ −5844.25 −0.222357
$$885$$ 0 0
$$886$$ −38664.0 −1.46607
$$887$$ − 14442.8i − 0.546719i −0.961912 0.273360i $$-0.911865\pi$$
0.961912 0.273360i $$-0.0881349\pi$$
$$888$$ 30237.8i 1.14270i
$$889$$ 5922.54 0.223437
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ 5520.38i 0.207215i
$$893$$ 72645.8i 2.72228i
$$894$$ −9326.43 −0.348906
$$895$$ 0 0
$$896$$ −5876.86 −0.219121
$$897$$ 27365.5i 1.01863i
$$898$$ 2493.61i 0.0926648i
$$899$$ 46.8641 0.00173860
$$900$$ 0 0
$$901$$ −13900.3 −0.513970
$$902$$ − 8410.27i − 0.310456i
$$903$$ 9624.77i 0.354698i
$$904$$ −4110.10 −0.151217
$$905$$ 0 0
$$906$$ −237.045 −0.00869239
$$907$$ − 44981.9i − 1.64675i −0.567499 0.823374i $$-0.692089\pi$$
0.567499 0.823374i $$-0.307911\pi$$
$$908$$ − 7197.57i − 0.263062i
$$909$$ −2172.60 −0.0792745
$$910$$ 0 0
$$911$$ 6841.96 0.248830 0.124415 0.992230i $$-0.460295\pi$$
0.124415 + 0.992230i $$0.460295\pi$$
$$912$$ 19718.7i 0.715954i
$$913$$ − 11233.4i − 0.407199i
$$914$$ 160.501 0.00580843
$$915$$ 0 0
$$916$$ 398.989 0.0143919
$$917$$ − 9274.61i − 0.333996i
$$918$$ 5720.27i 0.205661i
$$919$$ 4753.54 0.170625 0.0853127 0.996354i $$-0.472811\pi$$
0.0853127 + 0.996354i $$0.472811\pi$$
$$920$$ 0 0
$$921$$ 16151.7 0.577868
$$922$$ 30395.9i 1.08572i
$$923$$ 55052.7i 1.96325i
$$924$$ −296.500 −0.0105564
$$925$$ 0 0
$$926$$ 33671.2 1.19493
$$927$$ 10515.4i 0.372569i
$$928$$ − 572.330i − 0.0202453i
$$929$$ 7507.93 0.265153 0.132576 0.991173i $$-0.457675\pi$$
0.132576 + 0.991173i $$0.457675\pi$$
$$930$$ 0 0
$$931$$ 39625.7 1.39493
$$932$$ − 3265.14i − 0.114757i
$$933$$ 5371.24i 0.188474i
$$934$$ −26068.3 −0.913256
$$935$$ 0 0
$$936$$ −10688.9 −0.373266
$$937$$ 8540.47i 0.297764i 0.988855 + 0.148882i $$0.0475675\pi$$
−0.988855 + 0.148882i $$0.952433\pi$$
$$938$$ − 5424.24i − 0.188814i
$$939$$ 2427.23 0.0843552
$$940$$ 0 0
$$941$$ 9101.13 0.315290 0.157645 0.987496i $$-0.449610\pi$$
0.157645 + 0.987496i $$0.449610\pi$$
$$942$$ − 2653.04i − 0.0917630i
$$943$$ − 55425.5i − 1.91400i
$$944$$ 31200.6 1.07573
$$945$$ 0 0
$$946$$ −14472.7 −0.497407
$$947$$ 47540.0i 1.63130i 0.578546 + 0.815650i $$0.303620\pi$$
−0.578546 + 0.815650i $$0.696380\pi$$
$$948$$ − 1337.04i − 0.0458072i
$$949$$ −6063.26 −0.207399
$$950$$ 0 0
$$951$$ 32233.6 1.09910
$$952$$ 12490.2i 0.425221i
$$953$$ − 47370.7i − 1.61016i −0.593164 0.805082i $$-0.702121\pi$$
0.593164 0.805082i $$-0.297879\pi$$
$$954$$ −3874.55 −0.131492
$$955$$ 0 0
$$956$$ 2326.12 0.0786947
$$957$$ 293.938i 0.00992859i
$$958$$ 8789.33i 0.296420i
$$959$$ −4278.20 −0.144057
$$960$$ 0 0
$$961$$ −29763.3 −0.999071
$$962$$ − 52458.4i − 1.75813i
$$963$$ 18961.4i 0.634498i
$$964$$ 8122.38 0.271374
$$965$$ 0 0
$$966$$ 8913.27 0.296874
$$967$$ − 36171.6i − 1.20290i −0.798912 0.601448i $$-0.794591\pi$$
0.798912 0.601448i $$-0.205409\pi$$
$$968$$ − 2925.43i − 0.0971351i
$$969$$ 32344.1 1.07228
$$970$$ 0 0
$$971$$ −31713.2 −1.04812 −0.524060 0.851681i $$-0.675583\pi$$
−0.524060 + 0.851681i $$0.675583\pi$$
$$972$$ − 349.543i − 0.0115346i
$$973$$ − 5187.37i − 0.170914i
$$974$$ −7257.40 −0.238750
$$975$$ 0 0
$$976$$ −39653.1 −1.30048
$$977$$ 22800.5i 0.746626i 0.927706 + 0.373313i $$0.121778\pi$$
−0.927706 + 0.373313i $$0.878222\pi$$
$$978$$ 14766.0i 0.482786i
$$979$$ −1560.67 −0.0509490
$$980$$ 0 0
$$981$$ 4444.12 0.144638
$$982$$ 6832.41i 0.222027i
$$983$$ 44597.4i 1.44704i 0.690305 + 0.723518i $$0.257476\pi$$
−0.690305 + 0.723518i $$0.742524\pi$$
$$984$$ 21649.1 0.701369
$$985$$ 0 0
$$986$$ 1887.10 0.0609507
$$987$$ 10443.0i 0.336781i
$$988$$ 9210.95i 0.296599i
$$989$$ −95378.1 −3.06658
$$990$$ 0 0
$$991$$ −34788.1 −1.11512 −0.557558 0.830138i $$-0.688261\pi$$
−0.557558 + 0.830138i $$0.688261\pi$$
$$992$$ − 338.068i − 0.0108202i
$$993$$ − 10197.4i − 0.325885i
$$994$$ 17931.3 0.572180
$$995$$ 0 0
$$996$$ 4406.92 0.140199
$$997$$ − 20360.5i − 0.646765i −0.946268 0.323383i $$-0.895180\pi$$
0.946268 0.323383i $$-0.104820\pi$$
$$998$$ − 28528.0i − 0.904849i
$$999$$ 11256.1 0.356485
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.c.j.199.1 4
5.2 odd 4 825.4.a.m.1.2 2
5.3 odd 4 165.4.a.c.1.1 2
5.4 even 2 inner 825.4.c.j.199.4 4
15.2 even 4 2475.4.a.n.1.1 2
15.8 even 4 495.4.a.d.1.2 2
55.43 even 4 1815.4.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.1 2 5.3 odd 4
495.4.a.d.1.2 2 15.8 even 4
825.4.a.m.1.2 2 5.2 odd 4
825.4.c.j.199.1 4 1.1 even 1 trivial
825.4.c.j.199.4 4 5.4 even 2 inner
1815.4.a.n.1.2 2 55.43 even 4
2475.4.a.n.1.1 2 15.2 even 4