# Properties

 Label 825.4.a.m.1.2 Level $825$ Weight $4$ Character 825.1 Self dual yes Analytic conductor $48.677$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial 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)

 $$f(q)$$ $$=$$ $$q+2.56155 q^{2} -3.00000 q^{3} -1.43845 q^{4} -7.68466 q^{6} -6.24621 q^{7} -24.1771 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+2.56155 q^{2} -3.00000 q^{3} -1.43845 q^{4} -7.68466 q^{6} -6.24621 q^{7} -24.1771 q^{8} +9.00000 q^{9} -11.0000 q^{11} +4.31534 q^{12} +49.1231 q^{13} -16.0000 q^{14} -50.4233 q^{16} -82.7083 q^{17} +23.0540 q^{18} -130.354 q^{19} +18.7386 q^{21} -28.1771 q^{22} +185.693 q^{23} +72.5312 q^{24} +125.831 q^{26} -27.0000 q^{27} +8.98485 q^{28} -8.90720 q^{29} +5.26137 q^{31} +64.2547 q^{32} +33.0000 q^{33} -211.862 q^{34} -12.9460 q^{36} +416.894 q^{37} -333.909 q^{38} -147.369 q^{39} -298.479 q^{41} +48.0000 q^{42} +513.633 q^{43} +15.8229 q^{44} +475.663 q^{46} -557.295 q^{47} +151.270 q^{48} -303.985 q^{49} +248.125 q^{51} -70.6610 q^{52} +168.064 q^{53} -69.1619 q^{54} +151.015 q^{56} +391.062 q^{57} -22.8163 q^{58} +618.773 q^{59} +786.405 q^{61} +13.4773 q^{62} -56.2159 q^{63} +567.978 q^{64} +84.5312 q^{66} +339.015 q^{67} +118.972 q^{68} -557.080 q^{69} +1120.71 q^{71} -217.594 q^{72} +123.430 q^{73} +1067.90 q^{74} +187.508 q^{76} +68.7083 q^{77} -377.494 q^{78} -309.835 q^{79} +81.0000 q^{81} -764.570 q^{82} +1021.22 q^{83} -26.9545 q^{84} +1315.70 q^{86} +26.7216 q^{87} +265.948 q^{88} -141.879 q^{89} -306.833 q^{91} -267.110 q^{92} -15.7841 q^{93} -1427.54 q^{94} -192.764 q^{96} -798.345 q^{97} -778.673 q^{98} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 6 q^{3} - 7 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + q^2 - 6 * q^3 - 7 * q^4 - 3 * q^6 + 4 * q^7 - 3 * q^8 + 18 * q^9 $$2 q + q^{2} - 6 q^{3} - 7 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9} - 22 q^{11} + 21 q^{12} + 90 q^{13} - 32 q^{14} - 39 q^{16} + 16 q^{17} + 9 q^{18} - 170 q^{19} - 12 q^{21} - 11 q^{22} + 124 q^{23} + 9 q^{24} + 62 q^{26} - 54 q^{27} - 48 q^{28} - 158 q^{29} + 60 q^{31} - 123 q^{32} + 66 q^{33} - 366 q^{34} - 63 q^{36} + 372 q^{37} - 272 q^{38} - 270 q^{39} + 38 q^{41} + 96 q^{42} + 516 q^{43} + 77 q^{44} + 572 q^{46} - 224 q^{47} + 117 q^{48} - 542 q^{49} - 48 q^{51} - 298 q^{52} - 472 q^{53} - 27 q^{54} + 368 q^{56} + 510 q^{57} + 210 q^{58} + 248 q^{59} + 72 q^{61} - 72 q^{62} + 36 q^{63} + 769 q^{64} + 33 q^{66} + 744 q^{67} - 430 q^{68} - 372 q^{69} + 2060 q^{71} - 27 q^{72} + 486 q^{73} + 1138 q^{74} + 408 q^{76} - 44 q^{77} - 186 q^{78} + 642 q^{79} + 162 q^{81} - 1290 q^{82} + 286 q^{83} + 144 q^{84} + 1312 q^{86} + 474 q^{87} + 33 q^{88} + 244 q^{89} + 112 q^{91} + 76 q^{92} - 180 q^{93} - 1948 q^{94} + 369 q^{96} + 168 q^{97} - 407 q^{98} - 198 q^{99}+O(q^{100})$$ 2 * q + q^2 - 6 * q^3 - 7 * q^4 - 3 * q^6 + 4 * q^7 - 3 * q^8 + 18 * q^9 - 22 * q^11 + 21 * q^12 + 90 * q^13 - 32 * q^14 - 39 * q^16 + 16 * q^17 + 9 * q^18 - 170 * q^19 - 12 * q^21 - 11 * q^22 + 124 * q^23 + 9 * q^24 + 62 * q^26 - 54 * q^27 - 48 * q^28 - 158 * q^29 + 60 * q^31 - 123 * q^32 + 66 * q^33 - 366 * q^34 - 63 * q^36 + 372 * q^37 - 272 * q^38 - 270 * q^39 + 38 * q^41 + 96 * q^42 + 516 * q^43 + 77 * q^44 + 572 * q^46 - 224 * q^47 + 117 * q^48 - 542 * q^49 - 48 * q^51 - 298 * q^52 - 472 * q^53 - 27 * q^54 + 368 * q^56 + 510 * q^57 + 210 * q^58 + 248 * q^59 + 72 * q^61 - 72 * q^62 + 36 * q^63 + 769 * q^64 + 33 * q^66 + 744 * q^67 - 430 * q^68 - 372 * q^69 + 2060 * q^71 - 27 * q^72 + 486 * q^73 + 1138 * q^74 + 408 * q^76 - 44 * q^77 - 186 * q^78 + 642 * q^79 + 162 * q^81 - 1290 * q^82 + 286 * q^83 + 144 * q^84 + 1312 * q^86 + 474 * q^87 + 33 * q^88 + 244 * q^89 + 112 * q^91 + 76 * q^92 - 180 * q^93 - 1948 * q^94 + 369 * q^96 + 168 * q^97 - 407 * q^98 - 198 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.56155 0.905646 0.452823 0.891601i $$-0.350417\pi$$
0.452823 + 0.891601i $$0.350417\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ −1.43845 −0.179806
$$5$$ 0 0
$$6$$ −7.68466 −0.522875
$$7$$ −6.24621 −0.337264 −0.168632 0.985679i $$-0.553935\pi$$
−0.168632 + 0.985679i $$0.553935\pi$$
$$8$$ −24.1771 −1.06849
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 4.31534 0.103811
$$13$$ 49.1231 1.04802 0.524011 0.851711i $$-0.324435\pi$$
0.524011 + 0.851711i $$0.324435\pi$$
$$14$$ −16.0000 −0.305441
$$15$$ 0 0
$$16$$ −50.4233 −0.787864
$$17$$ −82.7083 −1.17998 −0.589992 0.807409i $$-0.700869\pi$$
−0.589992 + 0.807409i $$0.700869\pi$$
$$18$$ 23.0540 0.301882
$$19$$ −130.354 −1.57396 −0.786981 0.616977i $$-0.788357\pi$$
−0.786981 + 0.616977i $$0.788357\pi$$
$$20$$ 0 0
$$21$$ 18.7386 0.194719
$$22$$ −28.1771 −0.273062
$$23$$ 185.693 1.68347 0.841733 0.539895i $$-0.181536\pi$$
0.841733 + 0.539895i $$0.181536\pi$$
$$24$$ 72.5312 0.616891
$$25$$ 0 0
$$26$$ 125.831 0.949137
$$27$$ −27.0000 −0.192450
$$28$$ 8.98485 0.0606420
$$29$$ −8.90720 −0.0570354 −0.0285177 0.999593i $$-0.509079\pi$$
−0.0285177 + 0.999593i $$0.509079\pi$$
$$30$$ 0 0
$$31$$ 5.26137 0.0304829 0.0152414 0.999884i $$-0.495148\pi$$
0.0152414 + 0.999884i $$0.495148\pi$$
$$32$$ 64.2547 0.354961
$$33$$ 33.0000 0.174078
$$34$$ −211.862 −1.06865
$$35$$ 0 0
$$36$$ −12.9460 −0.0599353
$$37$$ 416.894 1.85235 0.926175 0.377094i $$-0.123077\pi$$
0.926175 + 0.377094i $$0.123077\pi$$
$$38$$ −333.909 −1.42545
$$39$$ −147.369 −0.605076
$$40$$ 0 0
$$41$$ −298.479 −1.13694 −0.568471 0.822703i $$-0.692465\pi$$
−0.568471 + 0.822703i $$0.692465\pi$$
$$42$$ 48.0000 0.176347
$$43$$ 513.633 1.82159 0.910793 0.412863i $$-0.135471\pi$$
0.910793 + 0.412863i $$0.135471\pi$$
$$44$$ 15.8229 0.0542135
$$45$$ 0 0
$$46$$ 475.663 1.52462
$$47$$ −557.295 −1.72957 −0.864786 0.502140i $$-0.832546\pi$$
−0.864786 + 0.502140i $$0.832546\pi$$
$$48$$ 151.270 0.454873
$$49$$ −303.985 −0.886253
$$50$$ 0 0
$$51$$ 248.125 0.681264
$$52$$ −70.6610 −0.188441
$$53$$ 168.064 0.435574 0.217787 0.975996i $$-0.430116\pi$$
0.217787 + 0.975996i $$0.430116\pi$$
$$54$$ −69.1619 −0.174292
$$55$$ 0 0
$$56$$ 151.015 0.360362
$$57$$ 391.062 0.908728
$$58$$ −22.8163 −0.0516539
$$59$$ 618.773 1.36538 0.682689 0.730709i $$-0.260810\pi$$
0.682689 + 0.730709i $$0.260810\pi$$
$$60$$ 0 0
$$61$$ 786.405 1.65064 0.825319 0.564667i $$-0.190996\pi$$
0.825319 + 0.564667i $$0.190996\pi$$
$$62$$ 13.4773 0.0276067
$$63$$ −56.2159 −0.112421
$$64$$ 567.978 1.10933
$$65$$ 0 0
$$66$$ 84.5312 0.157653
$$67$$ 339.015 0.618169 0.309084 0.951035i $$-0.399977\pi$$
0.309084 + 0.951035i $$0.399977\pi$$
$$68$$ 118.972 0.212168
$$69$$ −557.080 −0.971949
$$70$$ 0 0
$$71$$ 1120.71 1.87329 0.936645 0.350280i $$-0.113913\pi$$
0.936645 + 0.350280i $$0.113913\pi$$
$$72$$ −217.594 −0.356162
$$73$$ 123.430 0.197896 0.0989478 0.995093i $$-0.468452\pi$$
0.0989478 + 0.995093i $$0.468452\pi$$
$$74$$ 1067.90 1.67757
$$75$$ 0 0
$$76$$ 187.508 0.283008
$$77$$ 68.7083 0.101689
$$78$$ −377.494 −0.547985
$$79$$ −309.835 −0.441255 −0.220628 0.975358i $$-0.570811\pi$$
−0.220628 + 0.975358i $$0.570811\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ −764.570 −1.02967
$$83$$ 1021.22 1.35053 0.675263 0.737577i $$-0.264030\pi$$
0.675263 + 0.737577i $$0.264030\pi$$
$$84$$ −26.9545 −0.0350117
$$85$$ 0 0
$$86$$ 1315.70 1.64971
$$87$$ 26.7216 0.0329294
$$88$$ 265.948 0.322161
$$89$$ −141.879 −0.168979 −0.0844894 0.996424i $$-0.526926\pi$$
−0.0844894 + 0.996424i $$0.526926\pi$$
$$90$$ 0 0
$$91$$ −306.833 −0.353460
$$92$$ −267.110 −0.302697
$$93$$ −15.7841 −0.0175993
$$94$$ −1427.54 −1.56638
$$95$$ 0 0
$$96$$ −192.764 −0.204937
$$97$$ −798.345 −0.835666 −0.417833 0.908524i $$-0.637210\pi$$
−0.417833 + 0.908524i $$0.637210\pi$$
$$98$$ −778.673 −0.802631
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ 241.400 0.237823 0.118912 0.992905i $$-0.462059\pi$$
0.118912 + 0.992905i $$0.462059\pi$$
$$102$$ 635.585 0.616983
$$103$$ −1168.38 −1.11771 −0.558853 0.829267i $$-0.688758\pi$$
−0.558853 + 0.829267i $$0.688758\pi$$
$$104$$ −1187.65 −1.11980
$$105$$ 0 0
$$106$$ 430.506 0.394476
$$107$$ 2106.82 1.90350 0.951748 0.306882i $$-0.0992857\pi$$
0.951748 + 0.306882i $$0.0992857\pi$$
$$108$$ 38.8381 0.0346037
$$109$$ 493.792 0.433914 0.216957 0.976181i $$-0.430387\pi$$
0.216957 + 0.976181i $$0.430387\pi$$
$$110$$ 0 0
$$111$$ −1250.68 −1.06945
$$112$$ 314.955 0.265718
$$113$$ −170.000 −0.141524 −0.0707622 0.997493i $$-0.522543\pi$$
−0.0707622 + 0.997493i $$0.522543\pi$$
$$114$$ 1001.73 0.822986
$$115$$ 0 0
$$116$$ 12.8125 0.0102553
$$117$$ 442.108 0.349341
$$118$$ 1585.02 1.23655
$$119$$ 516.614 0.397966
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 2014.42 1.49489
$$123$$ 895.437 0.656414
$$124$$ −7.56820 −0.00548100
$$125$$ 0 0
$$126$$ −144.000 −0.101814
$$127$$ 948.182 0.662500 0.331250 0.943543i $$-0.392530\pi$$
0.331250 + 0.943543i $$0.392530\pi$$
$$128$$ 940.868 0.649702
$$129$$ −1540.90 −1.05169
$$130$$ 0 0
$$131$$ −1484.84 −0.990312 −0.495156 0.868804i $$-0.664889\pi$$
−0.495156 + 0.868804i $$0.664889\pi$$
$$132$$ −47.4688 −0.0313002
$$133$$ 814.220 0.530841
$$134$$ 868.405 0.559842
$$135$$ 0 0
$$136$$ 1999.65 1.26080
$$137$$ −684.928 −0.427134 −0.213567 0.976928i $$-0.568508\pi$$
−0.213567 + 0.976928i $$0.568508\pi$$
$$138$$ −1426.99 −0.880242
$$139$$ 830.483 0.506767 0.253384 0.967366i $$-0.418457\pi$$
0.253384 + 0.967366i $$0.418457\pi$$
$$140$$ 0 0
$$141$$ 1671.89 0.998569
$$142$$ 2870.75 1.69654
$$143$$ −540.354 −0.315991
$$144$$ −453.810 −0.262621
$$145$$ 0 0
$$146$$ 316.172 0.179223
$$147$$ 911.955 0.511679
$$148$$ −599.680 −0.333063
$$149$$ −1213.64 −0.667285 −0.333642 0.942700i $$-0.608278\pi$$
−0.333642 + 0.942700i $$0.608278\pi$$
$$150$$ 0 0
$$151$$ 30.8466 0.0166242 0.00831212 0.999965i $$-0.497354\pi$$
0.00831212 + 0.999965i $$0.497354\pi$$
$$152$$ 3151.58 1.68176
$$153$$ −744.375 −0.393328
$$154$$ 176.000 0.0920941
$$155$$ 0 0
$$156$$ 211.983 0.108796
$$157$$ −345.239 −0.175497 −0.0877485 0.996143i $$-0.527967\pi$$
−0.0877485 + 0.996143i $$0.527967\pi$$
$$158$$ −793.659 −0.399621
$$159$$ −504.193 −0.251479
$$160$$ 0 0
$$161$$ −1159.88 −0.567772
$$162$$ 207.486 0.100627
$$163$$ −1921.49 −0.923331 −0.461665 0.887054i $$-0.652748\pi$$
−0.461665 + 0.887054i $$0.652748\pi$$
$$164$$ 429.346 0.204429
$$165$$ 0 0
$$166$$ 2615.91 1.22310
$$167$$ −172.297 −0.0798369 −0.0399185 0.999203i $$-0.512710\pi$$
−0.0399185 + 0.999203i $$0.512710\pi$$
$$168$$ −453.045 −0.208055
$$169$$ 216.080 0.0983521
$$170$$ 0 0
$$171$$ −1173.19 −0.524654
$$172$$ −738.833 −0.327532
$$173$$ 1025.29 0.450587 0.225293 0.974291i $$-0.427666\pi$$
0.225293 + 0.974291i $$0.427666\pi$$
$$174$$ 68.4488 0.0298224
$$175$$ 0 0
$$176$$ 554.656 0.237550
$$177$$ −1856.32 −0.788302
$$178$$ −363.430 −0.153035
$$179$$ 1658.29 0.692437 0.346219 0.938154i $$-0.387466\pi$$
0.346219 + 0.938154i $$0.387466\pi$$
$$180$$ 0 0
$$181$$ −2021.81 −0.830275 −0.415137 0.909759i $$-0.636266\pi$$
−0.415137 + 0.909759i $$0.636266\pi$$
$$182$$ −785.970 −0.320110
$$183$$ −2359.22 −0.952996
$$184$$ −4489.52 −1.79876
$$185$$ 0 0
$$186$$ −40.4318 −0.0159387
$$187$$ 909.792 0.355778
$$188$$ 801.640 0.310987
$$189$$ 168.648 0.0649064
$$190$$ 0 0
$$191$$ 1440.22 0.545607 0.272803 0.962070i $$-0.412049\pi$$
0.272803 + 0.962070i $$0.412049\pi$$
$$192$$ −1703.93 −0.640473
$$193$$ 2798.05 1.04356 0.521782 0.853079i $$-0.325267\pi$$
0.521782 + 0.853079i $$0.325267\pi$$
$$194$$ −2045.00 −0.756817
$$195$$ 0 0
$$196$$ 437.266 0.159354
$$197$$ −458.943 −0.165982 −0.0829908 0.996550i $$-0.526447\pi$$
−0.0829908 + 0.996550i $$0.526447\pi$$
$$198$$ −253.594 −0.0910208
$$199$$ −2371.04 −0.844615 −0.422308 0.906453i $$-0.638780\pi$$
−0.422308 + 0.906453i $$0.638780\pi$$
$$200$$ 0 0
$$201$$ −1017.05 −0.356900
$$202$$ 618.358 0.215384
$$203$$ 55.6363 0.0192360
$$204$$ −356.915 −0.122495
$$205$$ 0 0
$$206$$ −2992.86 −1.01225
$$207$$ 1671.24 0.561155
$$208$$ −2476.95 −0.825699
$$209$$ 1433.90 0.474568
$$210$$ 0 0
$$211$$ 4319.87 1.40944 0.704721 0.709484i $$-0.251072\pi$$
0.704721 + 0.709484i $$0.251072\pi$$
$$212$$ −241.752 −0.0783187
$$213$$ −3362.12 −1.08154
$$214$$ 5396.73 1.72389
$$215$$ 0 0
$$216$$ 652.781 0.205630
$$217$$ −32.8636 −0.0102808
$$218$$ 1264.87 0.392973
$$219$$ −370.290 −0.114255
$$220$$ 0 0
$$221$$ −4062.89 −1.23665
$$222$$ −3203.69 −0.968547
$$223$$ 3837.73 1.15244 0.576219 0.817295i $$-0.304527\pi$$
0.576219 + 0.817295i $$0.304527\pi$$
$$224$$ −401.349 −0.119715
$$225$$ 0 0
$$226$$ −435.464 −0.128171
$$227$$ 5003.71 1.46303 0.731515 0.681825i $$-0.238813\pi$$
0.731515 + 0.681825i $$0.238813\pi$$
$$228$$ −562.523 −0.163395
$$229$$ −277.375 −0.0800412 −0.0400206 0.999199i $$-0.512742\pi$$
−0.0400206 + 0.999199i $$0.512742\pi$$
$$230$$ 0 0
$$231$$ −206.125 −0.0587101
$$232$$ 215.350 0.0609415
$$233$$ −2269.91 −0.638225 −0.319113 0.947717i $$-0.603385\pi$$
−0.319113 + 0.947717i $$0.603385\pi$$
$$234$$ 1132.48 0.316379
$$235$$ 0 0
$$236$$ −890.072 −0.245503
$$237$$ 929.505 0.254759
$$238$$ 1323.33 0.360416
$$239$$ −1617.11 −0.437665 −0.218832 0.975762i $$-0.570225\pi$$
−0.218832 + 0.975762i $$0.570225\pi$$
$$240$$ 0 0
$$241$$ 5646.63 1.50926 0.754629 0.656151i $$-0.227817\pi$$
0.754629 + 0.656151i $$0.227817\pi$$
$$242$$ 309.948 0.0823314
$$243$$ −243.000 −0.0641500
$$244$$ −1131.20 −0.296794
$$245$$ 0 0
$$246$$ 2293.71 0.594478
$$247$$ −6403.40 −1.64955
$$248$$ −127.204 −0.0325705
$$249$$ −3063.66 −0.779726
$$250$$ 0 0
$$251$$ −6217.61 −1.56355 −0.781777 0.623558i $$-0.785687\pi$$
−0.781777 + 0.623558i $$0.785687\pi$$
$$252$$ 80.8636 0.0202140
$$253$$ −2042.62 −0.507584
$$254$$ 2428.82 0.599991
$$255$$ 0 0
$$256$$ −2133.74 −0.520933
$$257$$ −7712.75 −1.87202 −0.936008 0.351980i $$-0.885509\pi$$
−0.936008 + 0.351980i $$0.885509\pi$$
$$258$$ −3947.09 −0.952462
$$259$$ −2604.01 −0.624730
$$260$$ 0 0
$$261$$ −80.1648 −0.0190118
$$262$$ −3803.49 −0.896871
$$263$$ −206.347 −0.0483798 −0.0241899 0.999707i $$-0.507701\pi$$
−0.0241899 + 0.999707i $$0.507701\pi$$
$$264$$ −797.844 −0.186000
$$265$$ 0 0
$$266$$ 2085.67 0.480753
$$267$$ 425.636 0.0975600
$$268$$ −487.655 −0.111150
$$269$$ 1712.47 0.388146 0.194073 0.980987i $$-0.437830\pi$$
0.194073 + 0.980987i $$0.437830\pi$$
$$270$$ 0 0
$$271$$ −477.081 −0.106940 −0.0534698 0.998569i $$-0.517028\pi$$
−0.0534698 + 0.998569i $$0.517028\pi$$
$$272$$ 4170.43 0.929666
$$273$$ 920.500 0.204070
$$274$$ −1754.48 −0.386832
$$275$$ 0 0
$$276$$ 801.329 0.174762
$$277$$ −4283.48 −0.929130 −0.464565 0.885539i $$-0.653789\pi$$
−0.464565 + 0.885539i $$0.653789\pi$$
$$278$$ 2127.33 0.458952
$$279$$ 47.3523 0.0101610
$$280$$ 0 0
$$281$$ 3477.79 0.738319 0.369160 0.929366i $$-0.379646\pi$$
0.369160 + 0.929366i $$0.379646\pi$$
$$282$$ 4282.62 0.904350
$$283$$ 6568.27 1.37966 0.689829 0.723973i $$-0.257686\pi$$
0.689829 + 0.723973i $$0.257686\pi$$
$$284$$ −1612.08 −0.336829
$$285$$ 0 0
$$286$$ −1384.15 −0.286176
$$287$$ 1864.36 0.383449
$$288$$ 578.292 0.118320
$$289$$ 1927.67 0.392360
$$290$$ 0 0
$$291$$ 2395.03 0.482472
$$292$$ −177.547 −0.0355828
$$293$$ −8352.29 −1.66534 −0.832672 0.553766i $$-0.813190\pi$$
−0.832672 + 0.553766i $$0.813190\pi$$
$$294$$ 2336.02 0.463399
$$295$$ 0 0
$$296$$ −10079.3 −1.97921
$$297$$ 297.000 0.0580259
$$298$$ −3108.81 −0.604324
$$299$$ 9121.83 1.76431
$$300$$ 0 0
$$301$$ −3208.26 −0.614355
$$302$$ 79.0152 0.0150557
$$303$$ −724.199 −0.137307
$$304$$ 6572.89 1.24007
$$305$$ 0 0
$$306$$ −1906.76 −0.356216
$$307$$ −5383.89 −1.00090 −0.500448 0.865767i $$-0.666831\pi$$
−0.500448 + 0.865767i $$0.666831\pi$$
$$308$$ −98.8333 −0.0182843
$$309$$ 3505.14 0.645308
$$310$$ 0 0
$$311$$ −1790.41 −0.326447 −0.163223 0.986589i $$-0.552189\pi$$
−0.163223 + 0.986589i $$0.552189\pi$$
$$312$$ 3562.96 0.646516
$$313$$ 809.076 0.146108 0.0730538 0.997328i $$-0.476726\pi$$
0.0730538 + 0.997328i $$0.476726\pi$$
$$314$$ −884.347 −0.158938
$$315$$ 0 0
$$316$$ 445.682 0.0793403
$$317$$ −10744.5 −1.90370 −0.951849 0.306567i $$-0.900820\pi$$
−0.951849 + 0.306567i $$0.900820\pi$$
$$318$$ −1291.52 −0.227751
$$319$$ 97.9792 0.0171968
$$320$$ 0 0
$$321$$ −6320.46 −1.09898
$$322$$ −2971.09 −0.514200
$$323$$ 10781.4 1.85725
$$324$$ −116.514 −0.0199784
$$325$$ 0 0
$$326$$ −4922.00 −0.836210
$$327$$ −1481.37 −0.250521
$$328$$ 7216.35 1.21481
$$329$$ 3480.98 0.583322
$$330$$ 0 0
$$331$$ 3399.12 0.564449 0.282224 0.959348i $$-0.408928\pi$$
0.282224 + 0.959348i $$0.408928\pi$$
$$332$$ −1468.97 −0.242832
$$333$$ 3752.05 0.617450
$$334$$ −441.349 −0.0723039
$$335$$ 0 0
$$336$$ −944.864 −0.153412
$$337$$ 11840.0 1.91384 0.956919 0.290356i $$-0.0937736\pi$$
0.956919 + 0.290356i $$0.0937736\pi$$
$$338$$ 553.499 0.0890721
$$339$$ 510.000 0.0817091
$$340$$ 0 0
$$341$$ −57.8750 −0.00919093
$$342$$ −3005.18 −0.475151
$$343$$ 4041.20 0.636165
$$344$$ −12418.1 −1.94634
$$345$$ 0 0
$$346$$ 2626.34 0.408072
$$347$$ 2076.67 0.321272 0.160636 0.987014i $$-0.448645\pi$$
0.160636 + 0.987014i $$0.448645\pi$$
$$348$$ −38.4376 −0.00592090
$$349$$ 5837.37 0.895322 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$350$$ 0 0
$$351$$ −1326.32 −0.201692
$$352$$ −706.802 −0.107025
$$353$$ 2423.64 0.365431 0.182715 0.983166i $$-0.441511\pi$$
0.182715 + 0.983166i $$0.441511\pi$$
$$354$$ −4755.06 −0.713922
$$355$$ 0 0
$$356$$ 204.085 0.0303834
$$357$$ −1549.84 −0.229765
$$358$$ 4247.79 0.627103
$$359$$ −3882.22 −0.570740 −0.285370 0.958417i $$-0.592116\pi$$
−0.285370 + 0.958417i $$0.592116\pi$$
$$360$$ 0 0
$$361$$ 10133.2 1.47736
$$362$$ −5178.96 −0.751935
$$363$$ −363.000 −0.0524864
$$364$$ 441.363 0.0635542
$$365$$ 0 0
$$366$$ −6043.26 −0.863077
$$367$$ −5666.65 −0.805986 −0.402993 0.915203i $$-0.632030\pi$$
−0.402993 + 0.915203i $$0.632030\pi$$
$$368$$ −9363.26 −1.32634
$$369$$ −2686.31 −0.378981
$$370$$ 0 0
$$371$$ −1049.77 −0.146903
$$372$$ 22.7046 0.00316446
$$373$$ −174.771 −0.0242608 −0.0121304 0.999926i $$-0.503861\pi$$
−0.0121304 + 0.999926i $$0.503861\pi$$
$$374$$ 2330.48 0.322209
$$375$$ 0 0
$$376$$ 13473.8 1.84802
$$377$$ −437.550 −0.0597744
$$378$$ 432.000 0.0587822
$$379$$ 252.686 0.0342470 0.0171235 0.999853i $$-0.494549\pi$$
0.0171235 + 0.999853i $$0.494549\pi$$
$$380$$ 0 0
$$381$$ −2844.55 −0.382495
$$382$$ 3689.21 0.494126
$$383$$ 11014.5 1.46950 0.734748 0.678340i $$-0.237300\pi$$
0.734748 + 0.678340i $$0.237300\pi$$
$$384$$ −2822.61 −0.375105
$$385$$ 0 0
$$386$$ 7167.35 0.945099
$$387$$ 4622.69 0.607196
$$388$$ 1148.38 0.150258
$$389$$ 8099.40 1.05567 0.527835 0.849347i $$-0.323004\pi$$
0.527835 + 0.849347i $$0.323004\pi$$
$$390$$ 0 0
$$391$$ −15358.4 −1.98646
$$392$$ 7349.47 0.946949
$$393$$ 4454.51 0.571757
$$394$$ −1175.61 −0.150320
$$395$$ 0 0
$$396$$ 142.406 0.0180712
$$397$$ 424.353 0.0536465 0.0268232 0.999640i $$-0.491461\pi$$
0.0268232 + 0.999640i $$0.491461\pi$$
$$398$$ −6073.54 −0.764922
$$399$$ −2442.66 −0.306481
$$400$$ 0 0
$$401$$ −5904.18 −0.735263 −0.367632 0.929972i $$-0.619831\pi$$
−0.367632 + 0.929972i $$0.619831\pi$$
$$402$$ −2605.22 −0.323225
$$403$$ 258.455 0.0319468
$$404$$ −347.241 −0.0427620
$$405$$ 0 0
$$406$$ 142.515 0.0174210
$$407$$ −4585.83 −0.558504
$$408$$ −5998.94 −0.727921
$$409$$ 1370.47 0.165686 0.0828430 0.996563i $$-0.473600\pi$$
0.0828430 + 0.996563i $$0.473600\pi$$
$$410$$ 0 0
$$411$$ 2054.78 0.246606
$$412$$ 1680.65 0.200970
$$413$$ −3864.98 −0.460493
$$414$$ 4280.97 0.508208
$$415$$ 0 0
$$416$$ 3156.39 0.372007
$$417$$ −2491.45 −0.292582
$$418$$ 3673.00 0.429790
$$419$$ 1268.73 0.147927 0.0739635 0.997261i $$-0.476435\pi$$
0.0739635 + 0.997261i $$0.476435\pi$$
$$420$$ 0 0
$$421$$ −12241.9 −1.41719 −0.708594 0.705617i $$-0.750670\pi$$
−0.708594 + 0.705617i $$0.750670\pi$$
$$422$$ 11065.6 1.27646
$$423$$ −5015.66 −0.576524
$$424$$ −4063.31 −0.465405
$$425$$ 0 0
$$426$$ −8612.26 −0.979496
$$427$$ −4912.05 −0.556700
$$428$$ −3030.55 −0.342260
$$429$$ 1621.06 0.182437
$$430$$ 0 0
$$431$$ 8050.11 0.899675 0.449838 0.893110i $$-0.351482\pi$$
0.449838 + 0.893110i $$0.351482\pi$$
$$432$$ 1361.43 0.151624
$$433$$ 16565.7 1.83856 0.919282 0.393600i $$-0.128770\pi$$
0.919282 + 0.393600i $$0.128770\pi$$
$$434$$ −84.1819 −0.00931073
$$435$$ 0 0
$$436$$ −710.293 −0.0780203
$$437$$ −24205.9 −2.64971
$$438$$ −948.517 −0.103475
$$439$$ 4705.80 0.511607 0.255804 0.966729i $$-0.417660\pi$$
0.255804 + 0.966729i $$0.417660\pi$$
$$440$$ 0 0
$$441$$ −2735.86 −0.295418
$$442$$ −10407.3 −1.11997
$$443$$ −15094.0 −1.61882 −0.809408 0.587246i $$-0.800212\pi$$
−0.809408 + 0.587246i $$0.800212\pi$$
$$444$$ 1799.04 0.192294
$$445$$ 0 0
$$446$$ 9830.56 1.04370
$$447$$ 3640.93 0.385257
$$448$$ −3547.71 −0.374138
$$449$$ 973.478 0.102319 0.0511595 0.998690i $$-0.483708\pi$$
0.0511595 + 0.998690i $$0.483708\pi$$
$$450$$ 0 0
$$451$$ 3283.27 0.342801
$$452$$ 244.536 0.0254469
$$453$$ −92.5398 −0.00959801
$$454$$ 12817.3 1.32499
$$455$$ 0 0
$$456$$ −9454.75 −0.970963
$$457$$ −62.6577 −0.00641358 −0.00320679 0.999995i $$-0.501021\pi$$
−0.00320679 + 0.999995i $$0.501021\pi$$
$$458$$ −710.510 −0.0724890
$$459$$ 2233.12 0.227088
$$460$$ 0 0
$$461$$ −11866.2 −1.19884 −0.599419 0.800436i $$-0.704602\pi$$
−0.599419 + 0.800436i $$0.704602\pi$$
$$462$$ −528.000 −0.0531705
$$463$$ 13144.8 1.31942 0.659711 0.751519i $$-0.270679\pi$$
0.659711 + 0.751519i $$0.270679\pi$$
$$464$$ 449.131 0.0449361
$$465$$ 0 0
$$466$$ −5814.48 −0.578006
$$467$$ 10176.8 1.00840 0.504201 0.863586i $$-0.331787\pi$$
0.504201 + 0.863586i $$0.331787\pi$$
$$468$$ −635.949 −0.0628136
$$469$$ −2117.56 −0.208486
$$470$$ 0 0
$$471$$ 1035.72 0.101323
$$472$$ −14960.1 −1.45889
$$473$$ −5649.96 −0.549229
$$474$$ 2380.98 0.230721
$$475$$ 0 0
$$476$$ −743.121 −0.0715565
$$477$$ 1512.58 0.145191
$$478$$ −4142.30 −0.396369
$$479$$ 3431.25 0.327302 0.163651 0.986518i $$-0.447673\pi$$
0.163651 + 0.986518i $$0.447673\pi$$
$$480$$ 0 0
$$481$$ 20479.1 1.94130
$$482$$ 14464.1 1.36685
$$483$$ 3479.64 0.327803
$$484$$ −174.052 −0.0163460
$$485$$ 0 0
$$486$$ −622.457 −0.0580972
$$487$$ 2833.20 0.263624 0.131812 0.991275i $$-0.457921\pi$$
0.131812 + 0.991275i $$0.457921\pi$$
$$488$$ −19013.0 −1.76368
$$489$$ 5764.48 0.533085
$$490$$ 0 0
$$491$$ −2667.29 −0.245159 −0.122580 0.992459i $$-0.539117\pi$$
−0.122580 + 0.992459i $$0.539117\pi$$
$$492$$ −1288.04 −0.118027
$$493$$ 736.700 0.0673008
$$494$$ −16402.7 −1.49391
$$495$$ 0 0
$$496$$ −265.295 −0.0240164
$$497$$ −7000.18 −0.631793
$$498$$ −7847.74 −0.706156
$$499$$ −11137.0 −0.999120 −0.499560 0.866279i $$-0.666505\pi$$
−0.499560 + 0.866279i $$0.666505\pi$$
$$500$$ 0 0
$$501$$ 516.892 0.0460939
$$502$$ −15926.7 −1.41603
$$503$$ −8780.30 −0.778319 −0.389159 0.921170i $$-0.627234\pi$$
−0.389159 + 0.921170i $$0.627234\pi$$
$$504$$ 1359.14 0.120121
$$505$$ 0 0
$$506$$ −5232.29 −0.459691
$$507$$ −648.239 −0.0567836
$$508$$ −1363.91 −0.119121
$$509$$ 13597.4 1.18408 0.592039 0.805910i $$-0.298323\pi$$
0.592039 + 0.805910i $$0.298323\pi$$
$$510$$ 0 0
$$511$$ −770.969 −0.0667430
$$512$$ −12992.6 −1.12148
$$513$$ 3519.56 0.302909
$$514$$ −19756.6 −1.69538
$$515$$ 0 0
$$516$$ 2216.50 0.189101
$$517$$ 6130.25 0.521486
$$518$$ −6670.30 −0.565784
$$519$$ −3075.88 −0.260146
$$520$$ 0 0
$$521$$ 14001.3 1.17736 0.588682 0.808364i $$-0.299647\pi$$
0.588682 + 0.808364i $$0.299647\pi$$
$$522$$ −205.346 −0.0172180
$$523$$ 14749.8 1.23320 0.616602 0.787275i $$-0.288509\pi$$
0.616602 + 0.787275i $$0.288509\pi$$
$$524$$ 2135.86 0.178064
$$525$$ 0 0
$$526$$ −528.568 −0.0438149
$$527$$ −435.159 −0.0359693
$$528$$ −1663.97 −0.137150
$$529$$ 22315.0 1.83406
$$530$$ 0 0
$$531$$ 5568.95 0.455126
$$532$$ −1171.21 −0.0954483
$$533$$ −14662.2 −1.19154
$$534$$ 1090.29 0.0883548
$$535$$ 0 0
$$536$$ −8196.40 −0.660505
$$537$$ −4974.86 −0.399779
$$538$$ 4386.59 0.351523
$$539$$ 3343.83 0.267215
$$540$$ 0 0
$$541$$ 1484.06 0.117939 0.0589694 0.998260i $$-0.481219\pi$$
0.0589694 + 0.998260i $$0.481219\pi$$
$$542$$ −1222.07 −0.0968493
$$543$$ 6065.42 0.479359
$$544$$ −5314.40 −0.418847
$$545$$ 0 0
$$546$$ 2357.91 0.184815
$$547$$ 16562.2 1.29460 0.647302 0.762234i $$-0.275897\pi$$
0.647302 + 0.762234i $$0.275897\pi$$
$$548$$ 985.233 0.0768012
$$549$$ 7077.65 0.550212
$$550$$ 0 0
$$551$$ 1161.09 0.0897716
$$552$$ 13468.6 1.03851
$$553$$ 1935.30 0.148819
$$554$$ −10972.3 −0.841463
$$555$$ 0 0
$$556$$ −1194.61 −0.0911197
$$557$$ 8821.52 0.671059 0.335529 0.942030i $$-0.391085\pi$$
0.335529 + 0.942030i $$0.391085\pi$$
$$558$$ 121.295 0.00920223
$$559$$ 25231.2 1.90906
$$560$$ 0 0
$$561$$ −2729.37 −0.205409
$$562$$ 8908.55 0.668656
$$563$$ 5985.53 0.448064 0.224032 0.974582i $$-0.428078\pi$$
0.224032 + 0.974582i $$0.428078\pi$$
$$564$$ −2404.92 −0.179549
$$565$$ 0 0
$$566$$ 16825.0 1.24948
$$567$$ −505.943 −0.0374737
$$568$$ −27095.5 −2.00158
$$569$$ −3453.08 −0.254413 −0.127206 0.991876i $$-0.540601\pi$$
−0.127206 + 0.991876i $$0.540601\pi$$
$$570$$ 0 0
$$571$$ −21484.5 −1.57460 −0.787302 0.616568i $$-0.788523\pi$$
−0.787302 + 0.616568i $$0.788523\pi$$
$$572$$ 777.271 0.0568170
$$573$$ −4320.67 −0.315006
$$574$$ 4775.67 0.347269
$$575$$ 0 0
$$576$$ 5111.80 0.369777
$$577$$ 13294.4 0.959189 0.479594 0.877490i $$-0.340784\pi$$
0.479594 + 0.877490i $$0.340784\pi$$
$$578$$ 4937.82 0.355340
$$579$$ −8394.14 −0.602502
$$580$$ 0 0
$$581$$ −6378.77 −0.455483
$$582$$ 6135.01 0.436949
$$583$$ −1848.71 −0.131330
$$584$$ −2984.18 −0.211449
$$585$$ 0 0
$$586$$ −21394.8 −1.50821
$$587$$ −6695.73 −0.470805 −0.235402 0.971898i $$-0.575641\pi$$
−0.235402 + 0.971898i $$0.575641\pi$$
$$588$$ −1311.80 −0.0920028
$$589$$ −685.841 −0.0479789
$$590$$ 0 0
$$591$$ 1376.83 0.0958295
$$592$$ −21021.2 −1.45940
$$593$$ 10239.6 0.709088 0.354544 0.935039i $$-0.384636\pi$$
0.354544 + 0.935039i $$0.384636\pi$$
$$594$$ 760.781 0.0525509
$$595$$ 0 0
$$596$$ 1745.76 0.119982
$$597$$ 7113.11 0.487639
$$598$$ 23366.0 1.59784
$$599$$ −23890.8 −1.62963 −0.814817 0.579719i $$-0.803162\pi$$
−0.814817 + 0.579719i $$0.803162\pi$$
$$600$$ 0 0
$$601$$ −11343.8 −0.769920 −0.384960 0.922933i $$-0.625785\pi$$
−0.384960 + 0.922933i $$0.625785\pi$$
$$602$$ −8218.12 −0.556388
$$603$$ 3051.14 0.206056
$$604$$ −44.3712 −0.00298914
$$605$$ 0 0
$$606$$ −1855.07 −0.124352
$$607$$ 26032.5 1.74074 0.870369 0.492399i $$-0.163880\pi$$
0.870369 + 0.492399i $$0.163880\pi$$
$$608$$ −8375.87 −0.558695
$$609$$ −166.909 −0.0111059
$$610$$ 0 0
$$611$$ −27376.1 −1.81263
$$612$$ 1070.74 0.0707226
$$613$$ 4568.13 0.300987 0.150493 0.988611i $$-0.451914\pi$$
0.150493 + 0.988611i $$0.451914\pi$$
$$614$$ −13791.1 −0.906457
$$615$$ 0 0
$$616$$ −1661.17 −0.108653
$$617$$ 12755.9 0.832308 0.416154 0.909294i $$-0.363378\pi$$
0.416154 + 0.909294i $$0.363378\pi$$
$$618$$ 8978.59 0.584421
$$619$$ −1138.94 −0.0739545 −0.0369772 0.999316i $$-0.511773\pi$$
−0.0369772 + 0.999316i $$0.511773\pi$$
$$620$$ 0 0
$$621$$ −5013.72 −0.323983
$$622$$ −4586.24 −0.295645
$$623$$ 886.205 0.0569904
$$624$$ 7430.85 0.476718
$$625$$ 0 0
$$626$$ 2072.49 0.132322
$$627$$ −4301.69 −0.273992
$$628$$ 496.607 0.0315554
$$629$$ −34480.6 −2.18574
$$630$$ 0 0
$$631$$ 7997.36 0.504548 0.252274 0.967656i $$-0.418822\pi$$
0.252274 + 0.967656i $$0.418822\pi$$
$$632$$ 7490.91 0.471475
$$633$$ −12959.6 −0.813742
$$634$$ −27522.7 −1.72408
$$635$$ 0 0
$$636$$ 725.255 0.0452173
$$637$$ −14932.7 −0.928814
$$638$$ 250.979 0.0155742
$$639$$ 10086.4 0.624430
$$640$$ 0 0
$$641$$ −573.115 −0.0353146 −0.0176573 0.999844i $$-0.505621\pi$$
−0.0176573 + 0.999844i $$0.505621\pi$$
$$642$$ −16190.2 −0.995290
$$643$$ 16027.8 0.983009 0.491504 0.870875i $$-0.336447\pi$$
0.491504 + 0.870875i $$0.336447\pi$$
$$644$$ 1668.42 0.102089
$$645$$ 0 0
$$646$$ 27617.1 1.68201
$$647$$ 2622.74 0.159367 0.0796837 0.996820i $$-0.474609\pi$$
0.0796837 + 0.996820i $$0.474609\pi$$
$$648$$ −1958.34 −0.118721
$$649$$ −6806.50 −0.411677
$$650$$ 0 0
$$651$$ 98.5908 0.00593560
$$652$$ 2763.97 0.166020
$$653$$ −3102.00 −0.185897 −0.0929484 0.995671i $$-0.529629\pi$$
−0.0929484 + 0.995671i $$0.529629\pi$$
$$654$$ −3794.62 −0.226883
$$655$$ 0 0
$$656$$ 15050.3 0.895755
$$657$$ 1110.87 0.0659652
$$658$$ 8916.73 0.528283
$$659$$ −20840.2 −1.23190 −0.615948 0.787787i $$-0.711227\pi$$
−0.615948 + 0.787787i $$0.711227\pi$$
$$660$$ 0 0
$$661$$ 18242.9 1.07348 0.536738 0.843749i $$-0.319657\pi$$
0.536738 + 0.843749i $$0.319657\pi$$
$$662$$ 8707.03 0.511191
$$663$$ 12188.7 0.713980
$$664$$ −24690.2 −1.44302
$$665$$ 0 0
$$666$$ 9611.06 0.559191
$$667$$ −1654.01 −0.0960171
$$668$$ 247.841 0.0143551
$$669$$ −11513.2 −0.665361
$$670$$ 0 0
$$671$$ −8650.46 −0.497686
$$672$$ 1204.05 0.0691177
$$673$$ 12746.6 0.730084 0.365042 0.930991i $$-0.381055\pi$$
0.365042 + 0.930991i $$0.381055\pi$$
$$674$$ 30328.7 1.73326
$$675$$ 0 0
$$676$$ −310.819 −0.0176843
$$677$$ 7683.11 0.436168 0.218084 0.975930i $$-0.430019\pi$$
0.218084 + 0.975930i $$0.430019\pi$$
$$678$$ 1306.39 0.0739995
$$679$$ 4986.63 0.281840
$$680$$ 0 0
$$681$$ −15011.1 −0.844681
$$682$$ −148.250 −0.00832373
$$683$$ 21397.1 1.19874 0.599368 0.800473i $$-0.295418\pi$$
0.599368 + 0.800473i $$0.295418\pi$$
$$684$$ 1687.57 0.0943359
$$685$$ 0 0
$$686$$ 10351.8 0.576140
$$687$$ 832.124 0.0462118
$$688$$ −25899.0 −1.43516
$$689$$ 8255.84 0.456491
$$690$$ 0 0
$$691$$ 26137.5 1.43895 0.719477 0.694516i $$-0.244381\pi$$
0.719477 + 0.694516i $$0.244381\pi$$
$$692$$ −1474.83 −0.0810181
$$693$$ 618.375 0.0338963
$$694$$ 5319.50 0.290959
$$695$$ 0 0
$$696$$ −646.051 −0.0351846
$$697$$ 24686.7 1.34157
$$698$$ 14952.7 0.810845
$$699$$ 6809.72 0.368479
$$700$$ 0 0
$$701$$ 13382.4 0.721036 0.360518 0.932752i $$-0.382600\pi$$
0.360518 + 0.932752i $$0.382600\pi$$
$$702$$ −3397.45 −0.182662
$$703$$ −54343.9 −2.91553
$$704$$ −6247.76 −0.334476
$$705$$ 0 0
$$706$$ 6208.27 0.330951
$$707$$ −1507.83 −0.0802092
$$708$$ 2670.22 0.141741
$$709$$ 18164.6 0.962179 0.481090 0.876671i $$-0.340241\pi$$
0.481090 + 0.876671i $$0.340241\pi$$
$$710$$ 0 0
$$711$$ −2788.52 −0.147085
$$712$$ 3430.21 0.180552
$$713$$ 977.000 0.0513169
$$714$$ −3970.00 −0.208086
$$715$$ 0 0
$$716$$ −2385.36 −0.124504
$$717$$ 4851.32 0.252686
$$718$$ −9944.50 −0.516888
$$719$$ 9665.62 0.501344 0.250672 0.968072i $$-0.419348\pi$$
0.250672 + 0.968072i $$0.419348\pi$$
$$720$$ 0 0
$$721$$ 7297.94 0.376962
$$722$$ 25956.7 1.33796
$$723$$ −16939.9 −0.871371
$$724$$ 2908.26 0.149288
$$725$$ 0 0
$$726$$ −929.844 −0.0475341
$$727$$ 29779.6 1.51921 0.759605 0.650385i $$-0.225392\pi$$
0.759605 + 0.650385i $$0.225392\pi$$
$$728$$ 7418.33 0.377667
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −42481.7 −2.14944
$$732$$ 3393.61 0.171354
$$733$$ −35029.5 −1.76513 −0.882567 0.470187i $$-0.844186\pi$$
−0.882567 + 0.470187i $$0.844186\pi$$
$$734$$ −14515.4 −0.729937
$$735$$ 0 0
$$736$$ 11931.7 0.597564
$$737$$ −3729.17 −0.186385
$$738$$ −6881.13 −0.343222
$$739$$ 23297.3 1.15968 0.579842 0.814729i $$-0.303114\pi$$
0.579842 + 0.814729i $$0.303114\pi$$
$$740$$ 0 0
$$741$$ 19210.2 0.952368
$$742$$ −2689.03 −0.133042
$$743$$ −21570.4 −1.06506 −0.532530 0.846411i $$-0.678759\pi$$
−0.532530 + 0.846411i $$0.678759\pi$$
$$744$$ 381.613 0.0188046
$$745$$ 0 0
$$746$$ −447.684 −0.0219717
$$747$$ 9190.99 0.450175
$$748$$ −1308.69 −0.0639710
$$749$$ −13159.6 −0.641980
$$750$$ 0 0
$$751$$ 28554.8 1.38746 0.693729 0.720236i $$-0.255966\pi$$
0.693729 + 0.720236i $$0.255966\pi$$
$$752$$ 28100.7 1.36267
$$753$$ 18652.8 0.902719
$$754$$ −1120.81 −0.0541344
$$755$$ 0 0
$$756$$ −242.591 −0.0116706
$$757$$ −7812.81 −0.375114 −0.187557 0.982254i $$-0.560057\pi$$
−0.187557 + 0.982254i $$0.560057\pi$$
$$758$$ 647.268 0.0310156
$$759$$ 6127.87 0.293054
$$760$$ 0 0
$$761$$ 2875.13 0.136956 0.0684778 0.997653i $$-0.478186\pi$$
0.0684778 + 0.997653i $$0.478186\pi$$
$$762$$ −7286.45 −0.346405
$$763$$ −3084.33 −0.146344
$$764$$ −2071.69 −0.0981033
$$765$$ 0 0
$$766$$ 28214.3 1.33084
$$767$$ 30396.0 1.43095
$$768$$ 6401.22 0.300761
$$769$$ −27657.7 −1.29696 −0.648479 0.761233i $$-0.724594\pi$$
−0.648479 + 0.761233i $$0.724594\pi$$
$$770$$ 0 0
$$771$$ 23138.2 1.08081
$$772$$ −4024.84 −0.187639
$$773$$ 3929.35 0.182832 0.0914160 0.995813i $$-0.470861\pi$$
0.0914160 + 0.995813i $$0.470861\pi$$
$$774$$ 11841.3 0.549904
$$775$$ 0 0
$$776$$ 19301.6 0.892898
$$777$$ 7812.02 0.360688
$$778$$ 20747.0 0.956064
$$779$$ 38908.0 1.78950
$$780$$ 0 0
$$781$$ −12327.8 −0.564818
$$782$$ −39341.3 −1.79903
$$783$$ 240.495 0.0109765
$$784$$ 15327.9 0.698247
$$785$$ 0 0
$$786$$ 11410.5 0.517809
$$787$$ 21125.7 0.956860 0.478430 0.878126i $$-0.341206\pi$$
0.478430 + 0.878126i $$0.341206\pi$$
$$788$$ 660.166 0.0298445
$$789$$ 619.040 0.0279321
$$790$$ 0 0
$$791$$ 1061.86 0.0477310
$$792$$ 2393.53 0.107387
$$793$$ 38630.7 1.72991
$$794$$ 1087.00 0.0485847
$$795$$ 0 0
$$796$$ 3410.61 0.151867
$$797$$ −11696.3 −0.519828 −0.259914 0.965632i $$-0.583694\pi$$
−0.259914 + 0.965632i $$0.583694\pi$$
$$798$$ −6257.00 −0.277563
$$799$$ 46093.0 2.04087
$$800$$ 0 0
$$801$$ −1276.91 −0.0563263
$$802$$ −15123.9 −0.665888
$$803$$ −1357.73 −0.0596678
$$804$$ 1462.97 0.0641727
$$805$$ 0 0
$$806$$ 662.045 0.0289324
$$807$$ −5137.42 −0.224096
$$808$$ −5836.34 −0.254111
$$809$$ 14310.2 0.621902 0.310951 0.950426i $$-0.399352\pi$$
0.310951 + 0.950426i $$0.399352\pi$$
$$810$$ 0 0
$$811$$ 21697.9 0.939477 0.469739 0.882806i $$-0.344348\pi$$
0.469739 + 0.882806i $$0.344348\pi$$
$$812$$ −80.0299 −0.00345874
$$813$$ 1431.24 0.0617416
$$814$$ −11746.9 −0.505807
$$815$$ 0 0
$$816$$ −12511.3 −0.536743
$$817$$ −66954.1 −2.86711
$$818$$ 3510.54 0.150053
$$819$$ −2761.50 −0.117820
$$820$$ 0 0
$$821$$ 3613.00 0.153587 0.0767934 0.997047i $$-0.475532\pi$$
0.0767934 + 0.997047i $$0.475532\pi$$
$$822$$ 5263.44 0.223338
$$823$$ 4763.98 0.201776 0.100888 0.994898i $$-0.467832\pi$$
0.100888 + 0.994898i $$0.467832\pi$$
$$824$$ 28248.0 1.19425
$$825$$ 0 0
$$826$$ −9900.36 −0.417043
$$827$$ 33571.7 1.41161 0.705806 0.708405i $$-0.250585\pi$$
0.705806 + 0.708405i $$0.250585\pi$$
$$828$$ −2403.99 −0.100899
$$829$$ 17980.5 0.753303 0.376652 0.926355i $$-0.377075\pi$$
0.376652 + 0.926355i $$0.377075\pi$$
$$830$$ 0 0
$$831$$ 12850.4 0.536434
$$832$$ 27900.9 1.16261
$$833$$ 25142.1 1.04576
$$834$$ −6381.98 −0.264976
$$835$$ 0 0
$$836$$ −2062.58 −0.0853301
$$837$$ −142.057 −0.00586643
$$838$$ 3249.91 0.133969
$$839$$ −40139.5 −1.65169 −0.825847 0.563895i $$-0.809302\pi$$
−0.825847 + 0.563895i $$0.809302\pi$$
$$840$$ 0 0
$$841$$ −24309.7 −0.996747
$$842$$ −31358.4 −1.28347
$$843$$ −10433.4 −0.426269
$$844$$ −6213.91 −0.253426
$$845$$ 0 0
$$846$$ −12847.9 −0.522127
$$847$$ −755.792 −0.0306603
$$848$$ −8474.36 −0.343173
$$849$$ −19704.8 −0.796546
$$850$$ 0 0
$$851$$ 77414.4 3.11837
$$852$$ 4836.24 0.194468
$$853$$ 15369.2 0.616919 0.308459 0.951237i $$-0.400187\pi$$
0.308459 + 0.951237i $$0.400187\pi$$
$$854$$ −12582.5 −0.504173
$$855$$ 0 0
$$856$$ −50936.8 −2.03386
$$857$$ −10324.9 −0.411541 −0.205770 0.978600i $$-0.565970\pi$$
−0.205770 + 0.978600i $$0.565970\pi$$
$$858$$ 4152.44 0.165224
$$859$$ −27112.5 −1.07691 −0.538455 0.842655i $$-0.680992\pi$$
−0.538455 + 0.842655i $$0.680992\pi$$
$$860$$ 0 0
$$861$$ −5593.09 −0.221384
$$862$$ 20620.8 0.814787
$$863$$ −30463.6 −1.20161 −0.600807 0.799394i $$-0.705154\pi$$
−0.600807 + 0.799394i $$0.705154\pi$$
$$864$$ −1734.88 −0.0683122
$$865$$ 0 0
$$866$$ 42434.0 1.66509
$$867$$ −5783.00 −0.226529
$$868$$ 47.2726 0.00184854
$$869$$ 3408.19 0.133044
$$870$$ 0 0
$$871$$ 16653.5 0.647855
$$872$$ −11938.4 −0.463631
$$873$$ −7185.10 −0.278555
$$874$$ −62004.6 −2.39970
$$875$$ 0 0
$$876$$ 532.642 0.0205437
$$877$$ 5086.12 0.195833 0.0979167 0.995195i $$-0.468782\pi$$
0.0979167 + 0.995195i $$0.468782\pi$$
$$878$$ 12054.2 0.463335
$$879$$ 25056.9 0.961487
$$880$$ 0 0
$$881$$ −10625.5 −0.406338 −0.203169 0.979144i $$-0.565124\pi$$
−0.203169 + 0.979144i $$0.565124\pi$$
$$882$$ −7008.06 −0.267544
$$883$$ −13112.2 −0.499728 −0.249864 0.968281i $$-0.580386\pi$$
−0.249864 + 0.968281i $$0.580386\pi$$
$$884$$ 5844.25 0.222357
$$885$$ 0 0
$$886$$ −38664.0 −1.46607
$$887$$ 14442.8 0.546719 0.273360 0.961912i $$-0.411865\pi$$
0.273360 + 0.961912i $$0.411865\pi$$
$$888$$ 30237.8 1.14270
$$889$$ −5922.54 −0.223437
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ −5520.38 −0.207215
$$893$$ 72645.8 2.72228
$$894$$ 9326.43 0.348906
$$895$$ 0 0
$$896$$ −5876.86 −0.219121
$$897$$ −27365.5 −1.01863
$$898$$ 2493.61 0.0926648
$$899$$ −46.8641 −0.00173860
$$900$$ 0 0
$$901$$ −13900.3 −0.513970
$$902$$ 8410.27 0.310456
$$903$$ 9624.77 0.354698
$$904$$ 4110.10 0.151217
$$905$$ 0 0
$$906$$ −237.045 −0.00869239
$$907$$ 44981.9 1.64675 0.823374 0.567499i $$-0.192089\pi$$
0.823374 + 0.567499i $$0.192089\pi$$
$$908$$ −7197.57 −0.263062
$$909$$ 2172.60 0.0792745
$$910$$ 0 0
$$911$$ 6841.96 0.248830 0.124415 0.992230i $$-0.460295\pi$$
0.124415 + 0.992230i $$0.460295\pi$$
$$912$$ −19718.7 −0.715954
$$913$$ −11233.4 −0.407199
$$914$$ −160.501 −0.00580843
$$915$$ 0 0
$$916$$ 398.989 0.0143919
$$917$$ 9274.61 0.333996
$$918$$ 5720.27 0.205661
$$919$$ −4753.54 −0.170625 −0.0853127 0.996354i $$-0.527189\pi$$
−0.0853127 + 0.996354i $$0.527189\pi$$
$$920$$ 0 0
$$921$$ 16151.7 0.577868
$$922$$ −30395.9 −1.08572
$$923$$ 55052.7 1.96325
$$924$$ 296.500 0.0105564
$$925$$ 0 0
$$926$$ 33671.2 1.19493
$$927$$ −10515.4 −0.372569
$$928$$ −572.330 −0.0202453
$$929$$ −7507.93 −0.265153 −0.132576 0.991173i $$-0.542325\pi$$
−0.132576 + 0.991173i $$0.542325\pi$$
$$930$$ 0 0
$$931$$ 39625.7 1.39493
$$932$$ 3265.14 0.114757
$$933$$ 5371.24 0.188474
$$934$$ 26068.3 0.913256
$$935$$ 0 0
$$936$$ −10688.9 −0.373266
$$937$$ −8540.47 −0.297764 −0.148882 0.988855i $$-0.547567\pi$$
−0.148882 + 0.988855i $$0.547567\pi$$
$$938$$ −5424.24 −0.188814
$$939$$ −2427.23 −0.0843552
$$940$$ 0 0
$$941$$ 9101.13 0.315290 0.157645 0.987496i $$-0.449610\pi$$
0.157645 + 0.987496i $$0.449610\pi$$
$$942$$ 2653.04 0.0917630
$$943$$ −55425.5 −1.91400
$$944$$ −31200.6 −1.07573
$$945$$ 0 0
$$946$$ −14472.7 −0.497407
$$947$$ −47540.0 −1.63130 −0.815650 0.578546i $$-0.803620\pi$$
−0.815650 + 0.578546i $$0.803620\pi$$
$$948$$ −1337.04 −0.0458072
$$949$$ 6063.26 0.207399
$$950$$ 0 0
$$951$$ 32233.6 1.09910
$$952$$ −12490.2 −0.425221
$$953$$ −47370.7 −1.61016 −0.805082 0.593164i $$-0.797879\pi$$
−0.805082 + 0.593164i $$0.797879\pi$$
$$954$$ 3874.55 0.131492
$$955$$ 0 0
$$956$$ 2326.12 0.0786947
$$957$$ −293.938 −0.00992859
$$958$$ 8789.33 0.296420
$$959$$ 4278.20 0.144057
$$960$$ 0 0
$$961$$ −29763.3 −0.999071
$$962$$ 52458.4 1.75813
$$963$$ 18961.4 0.634498
$$964$$ −8122.38 −0.271374
$$965$$ 0 0
$$966$$ 8913.27 0.296874
$$967$$ 36171.6 1.20290 0.601448 0.798912i $$-0.294591\pi$$
0.601448 + 0.798912i $$0.294591\pi$$
$$968$$ −2925.43 −0.0971351
$$969$$ −32344.1 −1.07228
$$970$$ 0 0
$$971$$ −31713.2 −1.04812 −0.524060 0.851681i $$-0.675583\pi$$
−0.524060 + 0.851681i $$0.675583\pi$$
$$972$$ 349.543 0.0115346
$$973$$ −5187.37 −0.170914
$$974$$ 7257.40 0.238750
$$975$$ 0 0
$$976$$ −39653.1 −1.30048
$$977$$ −22800.5 −0.746626 −0.373313 0.927706i $$-0.621778\pi$$
−0.373313 + 0.927706i $$0.621778\pi$$
$$978$$ 14766.0 0.482786
$$979$$ 1560.67 0.0509490
$$980$$ 0 0
$$981$$ 4444.12 0.144638
$$982$$ −6832.41 −0.222027
$$983$$ 44597.4 1.44704 0.723518 0.690305i $$-0.242524\pi$$
0.723518 + 0.690305i $$0.242524\pi$$
$$984$$ −21649.1 −0.701369
$$985$$ 0 0
$$986$$ 1887.10 0.0609507
$$987$$ −10443.0 −0.336781
$$988$$ 9210.95 0.296599
$$989$$ 95378.1 3.06658
$$990$$ 0 0
$$991$$ −34788.1 −1.11512 −0.557558 0.830138i $$-0.688261\pi$$
−0.557558 + 0.830138i $$0.688261\pi$$
$$992$$ 338.068 0.0108202
$$993$$ −10197.4 −0.325885
$$994$$ −17931.3 −0.572180
$$995$$ 0 0
$$996$$ 4406.92 0.140199
$$997$$ 20360.5 0.646765 0.323383 0.946268i $$-0.395180\pi$$
0.323383 + 0.946268i $$0.395180\pi$$
$$998$$ −28528.0 −0.904849
$$999$$ −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.a.m.1.2 2
3.2 odd 2 2475.4.a.n.1.1 2
5.2 odd 4 825.4.c.j.199.4 4
5.3 odd 4 825.4.c.j.199.1 4
5.4 even 2 165.4.a.c.1.1 2
15.14 odd 2 495.4.a.d.1.2 2
55.54 odd 2 1815.4.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.1 2 5.4 even 2
495.4.a.d.1.2 2 15.14 odd 2
825.4.a.m.1.2 2 1.1 even 1 trivial
825.4.c.j.199.1 4 5.3 odd 4
825.4.c.j.199.4 4 5.2 odd 4
1815.4.a.n.1.2 2 55.54 odd 2
2475.4.a.n.1.1 2 3.2 odd 2