# Properties

 Label 1225.4.a.m.1.1 Level $1225$ Weight $4$ Character 1225.1 Self dual yes Analytic conductor $72.277$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1225.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: $$72.2773397570$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.41421 q^{2} -4.65685 q^{3} +21.3137 q^{4} +25.2132 q^{6} -72.0833 q^{8} -5.31371 q^{9} +O(q^{10})$$ $$q-5.41421 q^{2} -4.65685 q^{3} +21.3137 q^{4} +25.2132 q^{6} -72.0833 q^{8} -5.31371 q^{9} -52.2548 q^{11} -99.2548 q^{12} +30.6569 q^{13} +219.765 q^{16} +37.2254 q^{17} +28.7696 q^{18} -80.2254 q^{19} +282.919 q^{22} -25.8335 q^{23} +335.681 q^{24} -165.983 q^{26} +150.480 q^{27} +20.9411 q^{29} +314.558 q^{31} -613.186 q^{32} +243.343 q^{33} -201.546 q^{34} -113.255 q^{36} -197.147 q^{37} +434.357 q^{38} -142.765 q^{39} -11.3625 q^{41} +33.8335 q^{43} -1113.74 q^{44} +139.868 q^{46} -361.676 q^{47} -1023.41 q^{48} -173.353 q^{51} +653.411 q^{52} -153.019 q^{53} -814.732 q^{54} +373.598 q^{57} -113.380 q^{58} +616.000 q^{59} -15.2649 q^{61} -1703.09 q^{62} +1561.80 q^{64} -1317.51 q^{66} +166.510 q^{67} +793.411 q^{68} +120.303 q^{69} -952.000 q^{71} +383.029 q^{72} -148.489 q^{73} +1067.40 q^{74} -1709.90 q^{76} +772.958 q^{78} +857.725 q^{79} -557.294 q^{81} +61.5189 q^{82} +660.528 q^{83} -183.182 q^{86} -97.5198 q^{87} +3766.70 q^{88} +45.7746 q^{89} -550.607 q^{92} -1464.85 q^{93} +1958.19 q^{94} +2855.52 q^{96} +1682.13 q^{97} +277.667 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9}+O(q^{10})$$ 2 * q - 8 * q^2 + 2 * q^3 + 20 * q^4 + 8 * q^6 - 48 * q^8 + 12 * q^9 $$2 q - 8 q^{2} + 2 q^{3} + 20 q^{4} + 8 q^{6} - 48 q^{8} + 12 q^{9} - 14 q^{11} - 108 q^{12} + 50 q^{13} + 168 q^{16} - 50 q^{17} - 16 q^{18} - 36 q^{19} + 184 q^{22} - 244 q^{23} + 496 q^{24} - 216 q^{26} + 86 q^{27} - 26 q^{29} + 120 q^{31} - 672 q^{32} + 498 q^{33} + 24 q^{34} - 136 q^{36} - 564 q^{37} + 320 q^{38} - 14 q^{39} + 328 q^{41} + 260 q^{43} - 1164 q^{44} + 704 q^{46} - 350 q^{47} - 1368 q^{48} - 754 q^{51} + 628 q^{52} + 56 q^{53} - 648 q^{54} + 668 q^{57} + 8 q^{58} + 1232 q^{59} - 336 q^{61} - 1200 q^{62} + 2128 q^{64} - 1976 q^{66} + 152 q^{67} + 908 q^{68} - 1332 q^{69} - 1904 q^{71} + 800 q^{72} + 676 q^{73} + 2016 q^{74} - 1768 q^{76} + 440 q^{78} + 1014 q^{79} - 1454 q^{81} - 816 q^{82} - 376 q^{83} - 768 q^{86} - 410 q^{87} + 4688 q^{88} + 216 q^{89} - 264 q^{92} - 2760 q^{93} + 1928 q^{94} + 2464 q^{96} + 2742 q^{97} + 940 q^{99}+O(q^{100})$$ 2 * q - 8 * q^2 + 2 * q^3 + 20 * q^4 + 8 * q^6 - 48 * q^8 + 12 * q^9 - 14 * q^11 - 108 * q^12 + 50 * q^13 + 168 * q^16 - 50 * q^17 - 16 * q^18 - 36 * q^19 + 184 * q^22 - 244 * q^23 + 496 * q^24 - 216 * q^26 + 86 * q^27 - 26 * q^29 + 120 * q^31 - 672 * q^32 + 498 * q^33 + 24 * q^34 - 136 * q^36 - 564 * q^37 + 320 * q^38 - 14 * q^39 + 328 * q^41 + 260 * q^43 - 1164 * q^44 + 704 * q^46 - 350 * q^47 - 1368 * q^48 - 754 * q^51 + 628 * q^52 + 56 * q^53 - 648 * q^54 + 668 * q^57 + 8 * q^58 + 1232 * q^59 - 336 * q^61 - 1200 * q^62 + 2128 * q^64 - 1976 * q^66 + 152 * q^67 + 908 * q^68 - 1332 * q^69 - 1904 * q^71 + 800 * q^72 + 676 * q^73 + 2016 * q^74 - 1768 * q^76 + 440 * q^78 + 1014 * q^79 - 1454 * q^81 - 816 * q^82 - 376 * q^83 - 768 * q^86 - 410 * q^87 + 4688 * q^88 + 216 * q^89 - 264 * q^92 - 2760 * q^93 + 1928 * q^94 + 2464 * q^96 + 2742 * q^97 + 940 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.41421 −1.91421 −0.957107 0.289735i $$-0.906433\pi$$
−0.957107 + 0.289735i $$0.906433\pi$$
$$3$$ −4.65685 −0.896212 −0.448106 0.893980i $$-0.647901\pi$$
−0.448106 + 0.893980i $$0.647901\pi$$
$$4$$ 21.3137 2.66421
$$5$$ 0 0
$$6$$ 25.2132 1.71554
$$7$$ 0 0
$$8$$ −72.0833 −3.18566
$$9$$ −5.31371 −0.196804
$$10$$ 0 0
$$11$$ −52.2548 −1.43231 −0.716156 0.697941i $$-0.754100\pi$$
−0.716156 + 0.697941i $$0.754100\pi$$
$$12$$ −99.2548 −2.38770
$$13$$ 30.6569 0.654052 0.327026 0.945015i $$-0.393953\pi$$
0.327026 + 0.945015i $$0.393953\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 219.765 3.43382
$$17$$ 37.2254 0.531087 0.265544 0.964099i $$-0.414449\pi$$
0.265544 + 0.964099i $$0.414449\pi$$
$$18$$ 28.7696 0.376725
$$19$$ −80.2254 −0.968683 −0.484341 0.874879i $$-0.660941\pi$$
−0.484341 + 0.874879i $$0.660941\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 282.919 2.74175
$$23$$ −25.8335 −0.234202 −0.117101 0.993120i $$-0.537360\pi$$
−0.117101 + 0.993120i $$0.537360\pi$$
$$24$$ 335.681 2.85503
$$25$$ 0 0
$$26$$ −165.983 −1.25200
$$27$$ 150.480 1.07259
$$28$$ 0 0
$$29$$ 20.9411 0.134092 0.0670460 0.997750i $$-0.478643\pi$$
0.0670460 + 0.997750i $$0.478643\pi$$
$$30$$ 0 0
$$31$$ 314.558 1.82246 0.911232 0.411894i $$-0.135133\pi$$
0.911232 + 0.411894i $$0.135133\pi$$
$$32$$ −613.186 −3.38741
$$33$$ 243.343 1.28365
$$34$$ −201.546 −1.01661
$$35$$ 0 0
$$36$$ −113.255 −0.524328
$$37$$ −197.147 −0.875968 −0.437984 0.898983i $$-0.644307\pi$$
−0.437984 + 0.898983i $$0.644307\pi$$
$$38$$ 434.357 1.85427
$$39$$ −142.765 −0.586170
$$40$$ 0 0
$$41$$ −11.3625 −0.0432810 −0.0216405 0.999766i $$-0.506889\pi$$
−0.0216405 + 0.999766i $$0.506889\pi$$
$$42$$ 0 0
$$43$$ 33.8335 0.119990 0.0599948 0.998199i $$-0.480892\pi$$
0.0599948 + 0.998199i $$0.480892\pi$$
$$44$$ −1113.74 −3.81598
$$45$$ 0 0
$$46$$ 139.868 0.448313
$$47$$ −361.676 −1.12247 −0.561233 0.827658i $$-0.689673\pi$$
−0.561233 + 0.827658i $$0.689673\pi$$
$$48$$ −1023.41 −3.07743
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −173.353 −0.475967
$$52$$ 653.411 1.74254
$$53$$ −153.019 −0.396582 −0.198291 0.980143i $$-0.563539\pi$$
−0.198291 + 0.980143i $$0.563539\pi$$
$$54$$ −814.732 −2.05317
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 373.598 0.868145
$$58$$ −113.380 −0.256681
$$59$$ 616.000 1.35926 0.679630 0.733555i $$-0.262140\pi$$
0.679630 + 0.733555i $$0.262140\pi$$
$$60$$ 0 0
$$61$$ −15.2649 −0.0320406 −0.0160203 0.999872i $$-0.505100\pi$$
−0.0160203 + 0.999872i $$0.505100\pi$$
$$62$$ −1703.09 −3.48858
$$63$$ 0 0
$$64$$ 1561.80 3.05040
$$65$$ 0 0
$$66$$ −1317.51 −2.45719
$$67$$ 166.510 0.303618 0.151809 0.988410i $$-0.451490\pi$$
0.151809 + 0.988410i $$0.451490\pi$$
$$68$$ 793.411 1.41493
$$69$$ 120.303 0.209895
$$70$$ 0 0
$$71$$ −952.000 −1.59129 −0.795645 0.605763i $$-0.792868\pi$$
−0.795645 + 0.605763i $$0.792868\pi$$
$$72$$ 383.029 0.626951
$$73$$ −148.489 −0.238074 −0.119037 0.992890i $$-0.537981\pi$$
−0.119037 + 0.992890i $$0.537981\pi$$
$$74$$ 1067.40 1.67679
$$75$$ 0 0
$$76$$ −1709.90 −2.58078
$$77$$ 0 0
$$78$$ 772.958 1.12205
$$79$$ 857.725 1.22154 0.610770 0.791808i $$-0.290860\pi$$
0.610770 + 0.791808i $$0.290860\pi$$
$$80$$ 0 0
$$81$$ −557.294 −0.764464
$$82$$ 61.5189 0.0828491
$$83$$ 660.528 0.873523 0.436761 0.899577i $$-0.356125\pi$$
0.436761 + 0.899577i $$0.356125\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −183.182 −0.229686
$$87$$ −97.5198 −0.120175
$$88$$ 3766.70 4.56286
$$89$$ 45.7746 0.0545180 0.0272590 0.999628i $$-0.491322\pi$$
0.0272590 + 0.999628i $$0.491322\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −550.607 −0.623965
$$93$$ −1464.85 −1.63331
$$94$$ 1958.19 2.14864
$$95$$ 0 0
$$96$$ 2855.52 3.03583
$$97$$ 1682.13 1.76076 0.880382 0.474265i $$-0.157286\pi$$
0.880382 + 0.474265i $$0.157286\pi$$
$$98$$ 0 0
$$99$$ 277.667 0.281885
$$100$$ 0 0
$$101$$ 434.167 0.427734 0.213867 0.976863i $$-0.431394\pi$$
0.213867 + 0.976863i $$0.431394\pi$$
$$102$$ 938.572 0.911102
$$103$$ 345.577 0.330589 0.165295 0.986244i $$-0.447142\pi$$
0.165295 + 0.986244i $$0.447142\pi$$
$$104$$ −2209.85 −2.08359
$$105$$ 0 0
$$106$$ 828.479 0.759142
$$107$$ −217.119 −0.196165 −0.0980825 0.995178i $$-0.531271\pi$$
−0.0980825 + 0.995178i $$0.531271\pi$$
$$108$$ 3207.29 2.85761
$$109$$ 1734.41 1.52409 0.762047 0.647521i $$-0.224194\pi$$
0.762047 + 0.647521i $$0.224194\pi$$
$$110$$ 0 0
$$111$$ 918.086 0.785053
$$112$$ 0 0
$$113$$ 1854.20 1.54362 0.771809 0.635855i $$-0.219352\pi$$
0.771809 + 0.635855i $$0.219352\pi$$
$$114$$ −2022.74 −1.66181
$$115$$ 0 0
$$116$$ 446.333 0.357250
$$117$$ −162.902 −0.128720
$$118$$ −3335.16 −2.60191
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1399.57 1.05152
$$122$$ 82.6476 0.0613325
$$123$$ 52.9134 0.0387890
$$124$$ 6704.41 4.85543
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1394.51 −0.974352 −0.487176 0.873304i $$-0.661973\pi$$
−0.487176 + 0.873304i $$0.661973\pi$$
$$128$$ −3550.45 −2.45171
$$129$$ −157.558 −0.107536
$$130$$ 0 0
$$131$$ −1762.42 −1.17544 −0.587722 0.809063i $$-0.699975\pi$$
−0.587722 + 0.809063i $$0.699975\pi$$
$$132$$ 5186.54 3.41993
$$133$$ 0 0
$$134$$ −901.519 −0.581189
$$135$$ 0 0
$$136$$ −2683.33 −1.69186
$$137$$ 922.949 0.575568 0.287784 0.957695i $$-0.407081\pi$$
0.287784 + 0.957695i $$0.407081\pi$$
$$138$$ −651.345 −0.401784
$$139$$ 196.039 0.119624 0.0598122 0.998210i $$-0.480950\pi$$
0.0598122 + 0.998210i $$0.480950\pi$$
$$140$$ 0 0
$$141$$ 1684.27 1.00597
$$142$$ 5154.33 3.04607
$$143$$ −1601.97 −0.936807
$$144$$ −1167.76 −0.675790
$$145$$ 0 0
$$146$$ 803.954 0.455724
$$147$$ 0 0
$$148$$ −4201.94 −2.33376
$$149$$ 780.372 0.429064 0.214532 0.976717i $$-0.431177\pi$$
0.214532 + 0.976717i $$0.431177\pi$$
$$150$$ 0 0
$$151$$ −2319.43 −1.25002 −0.625008 0.780618i $$-0.714904\pi$$
−0.625008 + 0.780618i $$0.714904\pi$$
$$152$$ 5782.91 3.08589
$$153$$ −197.805 −0.104520
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −3042.84 −1.56168
$$157$$ 1022.90 0.519977 0.259989 0.965612i $$-0.416281\pi$$
0.259989 + 0.965612i $$0.416281\pi$$
$$158$$ −4643.91 −2.33829
$$159$$ 712.589 0.355421
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 3017.31 1.46335
$$163$$ 1350.63 0.649013 0.324507 0.945883i $$-0.394802\pi$$
0.324507 + 0.945883i $$0.394802\pi$$
$$164$$ −242.177 −0.115310
$$165$$ 0 0
$$166$$ −3576.24 −1.67211
$$167$$ −1230.58 −0.570209 −0.285105 0.958496i $$-0.592028\pi$$
−0.285105 + 0.958496i $$0.592028\pi$$
$$168$$ 0 0
$$169$$ −1257.16 −0.572215
$$170$$ 0 0
$$171$$ 426.294 0.190641
$$172$$ 721.117 0.319678
$$173$$ −2487.65 −1.09325 −0.546626 0.837377i $$-0.684088\pi$$
−0.546626 + 0.837377i $$0.684088\pi$$
$$174$$ 527.993 0.230040
$$175$$ 0 0
$$176$$ −11483.8 −4.91830
$$177$$ −2868.62 −1.21819
$$178$$ −247.833 −0.104359
$$179$$ 1621.18 0.676941 0.338471 0.940977i $$-0.390090\pi$$
0.338471 + 0.940977i $$0.390090\pi$$
$$180$$ 0 0
$$181$$ −2593.69 −1.06512 −0.532561 0.846392i $$-0.678770\pi$$
−0.532561 + 0.846392i $$0.678770\pi$$
$$182$$ 0 0
$$183$$ 71.0866 0.0287151
$$184$$ 1862.16 0.746089
$$185$$ 0 0
$$186$$ 7931.03 3.12651
$$187$$ −1945.21 −0.760682
$$188$$ −7708.66 −2.99049
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1823.08 −0.690645 −0.345323 0.938484i $$-0.612231\pi$$
−0.345323 + 0.938484i $$0.612231\pi$$
$$192$$ −7273.09 −2.73380
$$193$$ 1541.03 0.574744 0.287372 0.957819i $$-0.407218\pi$$
0.287372 + 0.957819i $$0.407218\pi$$
$$194$$ −9107.39 −3.37048
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −701.243 −0.253612 −0.126806 0.991928i $$-0.540473\pi$$
−0.126806 + 0.991928i $$0.540473\pi$$
$$198$$ −1503.35 −0.539587
$$199$$ −3294.96 −1.17374 −0.586868 0.809682i $$-0.699639\pi$$
−0.586868 + 0.809682i $$0.699639\pi$$
$$200$$ 0 0
$$201$$ −775.411 −0.272106
$$202$$ −2350.67 −0.818775
$$203$$ 0 0
$$204$$ −3694.80 −1.26808
$$205$$ 0 0
$$206$$ −1871.03 −0.632819
$$207$$ 137.272 0.0460920
$$208$$ 6737.29 2.24590
$$209$$ 4192.16 1.38746
$$210$$ 0 0
$$211$$ 4082.35 1.33195 0.665974 0.745975i $$-0.268016\pi$$
0.665974 + 0.745975i $$0.268016\pi$$
$$212$$ −3261.41 −1.05658
$$213$$ 4433.33 1.42613
$$214$$ 1175.53 0.375502
$$215$$ 0 0
$$216$$ −10847.1 −3.41691
$$217$$ 0 0
$$218$$ −9390.46 −2.91744
$$219$$ 691.494 0.213364
$$220$$ 0 0
$$221$$ 1141.21 0.347359
$$222$$ −4970.71 −1.50276
$$223$$ 747.161 0.224366 0.112183 0.993688i $$-0.464216\pi$$
0.112183 + 0.993688i $$0.464216\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −10039.1 −2.95481
$$227$$ 1665.67 0.487025 0.243513 0.969898i $$-0.421700\pi$$
0.243513 + 0.969898i $$0.421700\pi$$
$$228$$ 7962.76 2.31292
$$229$$ 6628.35 1.91272 0.956362 0.292183i $$-0.0943816\pi$$
0.956362 + 0.292183i $$0.0943816\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −1509.50 −0.427172
$$233$$ 432.431 0.121586 0.0607929 0.998150i $$-0.480637\pi$$
0.0607929 + 0.998150i $$0.480637\pi$$
$$234$$ 881.984 0.246398
$$235$$ 0 0
$$236$$ 13129.2 3.62136
$$237$$ −3994.30 −1.09476
$$238$$ 0 0
$$239$$ 5580.44 1.51033 0.755165 0.655535i $$-0.227557\pi$$
0.755165 + 0.655535i $$0.227557\pi$$
$$240$$ 0 0
$$241$$ 6296.87 1.68306 0.841529 0.540212i $$-0.181656\pi$$
0.841529 + 0.540212i $$0.181656\pi$$
$$242$$ −7577.56 −2.01283
$$243$$ −1467.73 −0.387468
$$244$$ −325.352 −0.0853629
$$245$$ 0 0
$$246$$ −286.485 −0.0742504
$$247$$ −2459.46 −0.633569
$$248$$ −22674.4 −5.80575
$$249$$ −3075.98 −0.782862
$$250$$ 0 0
$$251$$ −311.921 −0.0784393 −0.0392197 0.999231i $$-0.512487\pi$$
−0.0392197 + 0.999231i $$0.512487\pi$$
$$252$$ 0 0
$$253$$ 1349.92 0.335451
$$254$$ 7550.17 1.86512
$$255$$ 0 0
$$256$$ 6728.46 1.64269
$$257$$ −7861.39 −1.90809 −0.954046 0.299659i $$-0.903127\pi$$
−0.954046 + 0.299659i $$0.903127\pi$$
$$258$$ 853.050 0.205847
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −111.275 −0.0263899
$$262$$ 9542.11 2.25005
$$263$$ −5227.09 −1.22554 −0.612769 0.790262i $$-0.709944\pi$$
−0.612769 + 0.790262i $$0.709944\pi$$
$$264$$ −17541.0 −4.08929
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −213.166 −0.0488596
$$268$$ 3548.94 0.808903
$$269$$ −1281.71 −0.290510 −0.145255 0.989394i $$-0.546400\pi$$
−0.145255 + 0.989394i $$0.546400\pi$$
$$270$$ 0 0
$$271$$ −4704.14 −1.05445 −0.527226 0.849725i $$-0.676768\pi$$
−0.527226 + 0.849725i $$0.676768\pi$$
$$272$$ 8180.82 1.82366
$$273$$ 0 0
$$274$$ −4997.04 −1.10176
$$275$$ 0 0
$$276$$ 2564.10 0.559205
$$277$$ −8958.56 −1.94321 −0.971603 0.236619i $$-0.923961\pi$$
−0.971603 + 0.236619i $$0.923961\pi$$
$$278$$ −1061.40 −0.228987
$$279$$ −1671.47 −0.358668
$$280$$ 0 0
$$281$$ −370.904 −0.0787412 −0.0393706 0.999225i $$-0.512535\pi$$
−0.0393706 + 0.999225i $$0.512535\pi$$
$$282$$ −9119.02 −1.92564
$$283$$ −5822.26 −1.22296 −0.611479 0.791261i $$-0.709425\pi$$
−0.611479 + 0.791261i $$0.709425\pi$$
$$284$$ −20290.7 −4.23954
$$285$$ 0 0
$$286$$ 8673.40 1.79325
$$287$$ 0 0
$$288$$ 3258.29 0.666655
$$289$$ −3527.27 −0.717946
$$290$$ 0 0
$$291$$ −7833.42 −1.57802
$$292$$ −3164.86 −0.634279
$$293$$ 7443.79 1.48420 0.742100 0.670289i $$-0.233830\pi$$
0.742100 + 0.670289i $$0.233830\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 14211.0 2.79053
$$297$$ −7863.32 −1.53628
$$298$$ −4225.10 −0.821320
$$299$$ −791.973 −0.153181
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 12557.9 2.39280
$$303$$ −2021.85 −0.383341
$$304$$ −17630.7 −3.32628
$$305$$ 0 0
$$306$$ 1070.96 0.200074
$$307$$ −761.674 −0.141600 −0.0707998 0.997491i $$-0.522555\pi$$
−0.0707998 + 0.997491i $$0.522555\pi$$
$$308$$ 0 0
$$309$$ −1609.30 −0.296278
$$310$$ 0 0
$$311$$ −7718.69 −1.40735 −0.703677 0.710520i $$-0.748460\pi$$
−0.703677 + 0.710520i $$0.748460\pi$$
$$312$$ 10290.9 1.86734
$$313$$ 8556.00 1.54509 0.772546 0.634959i $$-0.218983\pi$$
0.772546 + 0.634959i $$0.218983\pi$$
$$314$$ −5538.21 −0.995348
$$315$$ 0 0
$$316$$ 18281.3 3.25444
$$317$$ 7780.95 1.37862 0.689309 0.724468i $$-0.257914\pi$$
0.689309 + 0.724468i $$0.257914\pi$$
$$318$$ −3858.11 −0.680352
$$319$$ −1094.28 −0.192062
$$320$$ 0 0
$$321$$ 1011.09 0.175805
$$322$$ 0 0
$$323$$ −2986.42 −0.514455
$$324$$ −11878.0 −2.03670
$$325$$ 0 0
$$326$$ −7312.58 −1.24235
$$327$$ −8076.89 −1.36591
$$328$$ 819.045 0.137879
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4932.12 −0.819015 −0.409507 0.912307i $$-0.634299\pi$$
−0.409507 + 0.912307i $$0.634299\pi$$
$$332$$ 14078.3 2.32725
$$333$$ 1047.58 0.172394
$$334$$ 6662.61 1.09150
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 7121.13 1.15108 0.575538 0.817775i $$-0.304793\pi$$
0.575538 + 0.817775i $$0.304793\pi$$
$$338$$ 6806.52 1.09534
$$339$$ −8634.76 −1.38341
$$340$$ 0 0
$$341$$ −16437.2 −2.61034
$$342$$ −2308.05 −0.364927
$$343$$ 0 0
$$344$$ −2438.83 −0.382246
$$345$$ 0 0
$$346$$ 13468.7 2.09272
$$347$$ −9540.58 −1.47598 −0.737991 0.674811i $$-0.764225\pi$$
−0.737991 + 0.674811i $$0.764225\pi$$
$$348$$ −2078.51 −0.320172
$$349$$ −1281.65 −0.196576 −0.0982880 0.995158i $$-0.531337\pi$$
−0.0982880 + 0.995158i $$0.531337\pi$$
$$350$$ 0 0
$$351$$ 4613.25 0.701530
$$352$$ 32041.9 4.85182
$$353$$ 5798.07 0.874221 0.437110 0.899408i $$-0.356002\pi$$
0.437110 + 0.899408i $$0.356002\pi$$
$$354$$ 15531.3 2.33187
$$355$$ 0 0
$$356$$ 975.627 0.145247
$$357$$ 0 0
$$358$$ −8777.40 −1.29581
$$359$$ 2267.29 0.333323 0.166662 0.986014i $$-0.446701\pi$$
0.166662 + 0.986014i $$0.446701\pi$$
$$360$$ 0 0
$$361$$ −422.886 −0.0616541
$$362$$ 14042.8 2.03887
$$363$$ −6517.58 −0.942381
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −384.878 −0.0549669
$$367$$ −7372.85 −1.04866 −0.524332 0.851514i $$-0.675685\pi$$
−0.524332 + 0.851514i $$0.675685\pi$$
$$368$$ −5677.28 −0.804209
$$369$$ 60.3769 0.00851788
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −31221.4 −4.35150
$$373$$ −6447.14 −0.894961 −0.447480 0.894294i $$-0.647679\pi$$
−0.447480 + 0.894294i $$0.647679\pi$$
$$374$$ 10531.8 1.45611
$$375$$ 0 0
$$376$$ 26070.8 3.57579
$$377$$ 641.989 0.0877032
$$378$$ 0 0
$$379$$ −4247.57 −0.575680 −0.287840 0.957678i $$-0.592937\pi$$
−0.287840 + 0.957678i $$0.592937\pi$$
$$380$$ 0 0
$$381$$ 6494.03 0.873226
$$382$$ 9870.53 1.32204
$$383$$ −6681.86 −0.891454 −0.445727 0.895169i $$-0.647055\pi$$
−0.445727 + 0.895169i $$0.647055\pi$$
$$384$$ 16533.9 2.19725
$$385$$ 0 0
$$386$$ −8343.45 −1.10018
$$387$$ −179.781 −0.0236145
$$388$$ 35852.4 4.69105
$$389$$ −6371.78 −0.830494 −0.415247 0.909709i $$-0.636305\pi$$
−0.415247 + 0.909709i $$0.636305\pi$$
$$390$$ 0 0
$$391$$ −961.661 −0.124382
$$392$$ 0 0
$$393$$ 8207.33 1.05345
$$394$$ 3796.68 0.485467
$$395$$ 0 0
$$396$$ 5918.11 0.751001
$$397$$ 4247.93 0.537021 0.268510 0.963277i $$-0.413469\pi$$
0.268510 + 0.963277i $$0.413469\pi$$
$$398$$ 17839.6 2.24678
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −8833.62 −1.10008 −0.550038 0.835140i $$-0.685387\pi$$
−0.550038 + 0.835140i $$0.685387\pi$$
$$402$$ 4198.24 0.520869
$$403$$ 9643.37 1.19199
$$404$$ 9253.70 1.13958
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 10301.9 1.25466
$$408$$ 12495.9 1.51627
$$409$$ 319.205 0.0385908 0.0192954 0.999814i $$-0.493858\pi$$
0.0192954 + 0.999814i $$0.493858\pi$$
$$410$$ 0 0
$$411$$ −4298.04 −0.515831
$$412$$ 7365.53 0.880761
$$413$$ 0 0
$$414$$ −743.218 −0.0882298
$$415$$ 0 0
$$416$$ −18798.3 −2.21554
$$417$$ −912.924 −0.107209
$$418$$ −22697.3 −2.65589
$$419$$ 12789.2 1.49115 0.745577 0.666420i $$-0.232174\pi$$
0.745577 + 0.666420i $$0.232174\pi$$
$$420$$ 0 0
$$421$$ −6747.40 −0.781112 −0.390556 0.920579i $$-0.627717\pi$$
−0.390556 + 0.920579i $$0.627717\pi$$
$$422$$ −22102.7 −2.54963
$$423$$ 1921.84 0.220906
$$424$$ 11030.1 1.26337
$$425$$ 0 0
$$426$$ −24003.0 −2.72992
$$427$$ 0 0
$$428$$ −4627.60 −0.522625
$$429$$ 7460.14 0.839577
$$430$$ 0 0
$$431$$ −5184.75 −0.579444 −0.289722 0.957111i $$-0.593563\pi$$
−0.289722 + 0.957111i $$0.593563\pi$$
$$432$$ 33070.2 3.68308
$$433$$ −4242.03 −0.470806 −0.235403 0.971898i $$-0.575641\pi$$
−0.235403 + 0.971898i $$0.575641\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 36966.7 4.06051
$$437$$ 2072.50 0.226868
$$438$$ −3743.90 −0.408425
$$439$$ 5434.12 0.590789 0.295394 0.955375i $$-0.404549\pi$$
0.295394 + 0.955375i $$0.404549\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −6178.77 −0.664919
$$443$$ 11493.8 1.23270 0.616350 0.787472i $$-0.288611\pi$$
0.616350 + 0.787472i $$0.288611\pi$$
$$444$$ 19567.8 2.09155
$$445$$ 0 0
$$446$$ −4045.29 −0.429484
$$447$$ −3634.08 −0.384532
$$448$$ 0 0
$$449$$ −16849.3 −1.77098 −0.885489 0.464661i $$-0.846176\pi$$
−0.885489 + 0.464661i $$0.846176\pi$$
$$450$$ 0 0
$$451$$ 593.745 0.0619919
$$452$$ 39520.0 4.11253
$$453$$ 10801.2 1.12028
$$454$$ −9018.32 −0.932270
$$455$$ 0 0
$$456$$ −26930.2 −2.76561
$$457$$ −15348.5 −1.57106 −0.785528 0.618826i $$-0.787609\pi$$
−0.785528 + 0.618826i $$0.787609\pi$$
$$458$$ −35887.3 −3.66136
$$459$$ 5601.69 0.569639
$$460$$ 0 0
$$461$$ −14038.4 −1.41830 −0.709148 0.705059i $$-0.750920\pi$$
−0.709148 + 0.705059i $$0.750920\pi$$
$$462$$ 0 0
$$463$$ 8661.23 0.869377 0.434689 0.900581i $$-0.356858\pi$$
0.434689 + 0.900581i $$0.356858\pi$$
$$464$$ 4602.12 0.460448
$$465$$ 0 0
$$466$$ −2341.27 −0.232741
$$467$$ 7014.71 0.695079 0.347539 0.937665i $$-0.387017\pi$$
0.347539 + 0.937665i $$0.387017\pi$$
$$468$$ −3472.04 −0.342938
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −4763.50 −0.466010
$$472$$ −44403.3 −4.33014
$$473$$ −1767.96 −0.171863
$$474$$ 21626.0 2.09560
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 813.100 0.0780488
$$478$$ −30213.7 −2.89109
$$479$$ −18134.7 −1.72984 −0.864922 0.501907i $$-0.832632\pi$$
−0.864922 + 0.501907i $$0.832632\pi$$
$$480$$ 0 0
$$481$$ −6043.91 −0.572929
$$482$$ −34092.6 −3.22173
$$483$$ 0 0
$$484$$ 29830.0 2.80146
$$485$$ 0 0
$$486$$ 7946.59 0.741697
$$487$$ −16537.8 −1.53881 −0.769405 0.638761i $$-0.779447\pi$$
−0.769405 + 0.638761i $$0.779447\pi$$
$$488$$ 1100.35 0.102070
$$489$$ −6289.67 −0.581654
$$490$$ 0 0
$$491$$ 220.608 0.0202768 0.0101384 0.999949i $$-0.496773\pi$$
0.0101384 + 0.999949i $$0.496773\pi$$
$$492$$ 1127.78 0.103342
$$493$$ 779.542 0.0712146
$$494$$ 13316.0 1.21279
$$495$$ 0 0
$$496$$ 69128.8 6.25801
$$497$$ 0 0
$$498$$ 16654.0 1.49856
$$499$$ 5939.04 0.532801 0.266401 0.963862i $$-0.414166\pi$$
0.266401 + 0.963862i $$0.414166\pi$$
$$500$$ 0 0
$$501$$ 5730.62 0.511029
$$502$$ 1688.81 0.150150
$$503$$ −11604.8 −1.02869 −0.514345 0.857584i $$-0.671965\pi$$
−0.514345 + 0.857584i $$0.671965\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −7308.78 −0.642124
$$507$$ 5854.40 0.512826
$$508$$ −29722.2 −2.59588
$$509$$ 1867.67 0.162639 0.0813193 0.996688i $$-0.474087\pi$$
0.0813193 + 0.996688i $$0.474087\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −8025.75 −0.692757
$$513$$ −12072.3 −1.03900
$$514$$ 42563.2 3.65250
$$515$$ 0 0
$$516$$ −3358.14 −0.286499
$$517$$ 18899.3 1.60772
$$518$$ 0 0
$$519$$ 11584.6 0.979786
$$520$$ 0 0
$$521$$ −6117.21 −0.514395 −0.257197 0.966359i $$-0.582799\pi$$
−0.257197 + 0.966359i $$0.582799\pi$$
$$522$$ 602.467 0.0505158
$$523$$ −16685.6 −1.39505 −0.697524 0.716561i $$-0.745715\pi$$
−0.697524 + 0.716561i $$0.745715\pi$$
$$524$$ −37563.7 −3.13164
$$525$$ 0 0
$$526$$ 28300.6 2.34594
$$527$$ 11709.6 0.967887
$$528$$ 53478.2 4.40784
$$529$$ −11499.6 −0.945149
$$530$$ 0 0
$$531$$ −3273.24 −0.267508
$$532$$ 0 0
$$533$$ −348.338 −0.0283081
$$534$$ 1154.12 0.0935278
$$535$$ 0 0
$$536$$ −12002.6 −0.967223
$$537$$ −7549.59 −0.606683
$$538$$ 6939.44 0.556097
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 9309.03 0.739790 0.369895 0.929074i $$-0.379394\pi$$
0.369895 + 0.929074i $$0.379394\pi$$
$$542$$ 25469.2 2.01845
$$543$$ 12078.4 0.954575
$$544$$ −22826.1 −1.79901
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −10894.7 −0.851598 −0.425799 0.904818i $$-0.640007\pi$$
−0.425799 + 0.904818i $$0.640007\pi$$
$$548$$ 19671.5 1.53344
$$549$$ 81.1134 0.00630571
$$550$$ 0 0
$$551$$ −1680.01 −0.129893
$$552$$ −8671.81 −0.668654
$$553$$ 0 0
$$554$$ 48503.6 3.71971
$$555$$ 0 0
$$556$$ 4178.31 0.318705
$$557$$ 7873.90 0.598973 0.299486 0.954101i $$-0.403185\pi$$
0.299486 + 0.954101i $$0.403185\pi$$
$$558$$ 9049.71 0.686567
$$559$$ 1037.23 0.0784796
$$560$$ 0 0
$$561$$ 9058.55 0.681733
$$562$$ 2008.15 0.150728
$$563$$ 21770.7 1.62971 0.814854 0.579666i $$-0.196817\pi$$
0.814854 + 0.579666i $$0.196817\pi$$
$$564$$ 35898.1 2.68011
$$565$$ 0 0
$$566$$ 31522.9 2.34100
$$567$$ 0 0
$$568$$ 68623.3 5.06931
$$569$$ −12381.3 −0.912213 −0.456106 0.889925i $$-0.650756\pi$$
−0.456106 + 0.889925i $$0.650756\pi$$
$$570$$ 0 0
$$571$$ −5768.38 −0.422765 −0.211383 0.977403i $$-0.567797\pi$$
−0.211383 + 0.977403i $$0.567797\pi$$
$$572$$ −34143.9 −2.49585
$$573$$ 8489.81 0.618965
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −8298.97 −0.600330
$$577$$ 4733.38 0.341513 0.170757 0.985313i $$-0.445379\pi$$
0.170757 + 0.985313i $$0.445379\pi$$
$$578$$ 19097.4 1.37430
$$579$$ −7176.34 −0.515093
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 42411.8 3.02066
$$583$$ 7996.00 0.568028
$$584$$ 10703.6 0.758422
$$585$$ 0 0
$$586$$ −40302.3 −2.84108
$$587$$ −8441.67 −0.593569 −0.296785 0.954944i $$-0.595914\pi$$
−0.296785 + 0.954944i $$0.595914\pi$$
$$588$$ 0 0
$$589$$ −25235.6 −1.76539
$$590$$ 0 0
$$591$$ 3265.59 0.227290
$$592$$ −43326.0 −3.00792
$$593$$ 18939.9 1.31158 0.655791 0.754943i $$-0.272335\pi$$
0.655791 + 0.754943i $$0.272335\pi$$
$$594$$ 42573.7 2.94077
$$595$$ 0 0
$$596$$ 16632.6 1.14312
$$597$$ 15344.2 1.05192
$$598$$ 4287.91 0.293220
$$599$$ 22655.3 1.54536 0.772681 0.634794i $$-0.218915\pi$$
0.772681 + 0.634794i $$0.218915\pi$$
$$600$$ 0 0
$$601$$ 15947.4 1.08237 0.541187 0.840902i $$-0.317975\pi$$
0.541187 + 0.840902i $$0.317975\pi$$
$$602$$ 0 0
$$603$$ −884.784 −0.0597532
$$604$$ −49435.6 −3.33031
$$605$$ 0 0
$$606$$ 10946.7 0.733796
$$607$$ −25993.2 −1.73811 −0.869053 0.494719i $$-0.835271\pi$$
−0.869053 + 0.494719i $$0.835271\pi$$
$$608$$ 49193.1 3.28132
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −11087.9 −0.734152
$$612$$ −4215.96 −0.278464
$$613$$ −665.408 −0.0438427 −0.0219213 0.999760i $$-0.506978\pi$$
−0.0219213 + 0.999760i $$0.506978\pi$$
$$614$$ 4123.87 0.271052
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18401.3 −1.20066 −0.600330 0.799752i $$-0.704964\pi$$
−0.600330 + 0.799752i $$0.704964\pi$$
$$618$$ 8713.10 0.567140
$$619$$ 11150.6 0.724040 0.362020 0.932170i $$-0.382087\pi$$
0.362020 + 0.932170i $$0.382087\pi$$
$$620$$ 0 0
$$621$$ −3887.43 −0.251203
$$622$$ 41790.6 2.69397
$$623$$ 0 0
$$624$$ −31374.6 −2.01280
$$625$$ 0 0
$$626$$ −46324.0 −2.95764
$$627$$ −19522.3 −1.24345
$$628$$ 21801.8 1.38533
$$629$$ −7338.88 −0.465215
$$630$$ 0 0
$$631$$ 5381.79 0.339534 0.169767 0.985484i $$-0.445699\pi$$
0.169767 + 0.985484i $$0.445699\pi$$
$$632$$ −61827.6 −3.89141
$$633$$ −19010.9 −1.19371
$$634$$ −42127.7 −2.63897
$$635$$ 0 0
$$636$$ 15187.9 0.946918
$$637$$ 0 0
$$638$$ 5924.64 0.367647
$$639$$ 5058.65 0.313172
$$640$$ 0 0
$$641$$ −19455.1 −1.19880 −0.599398 0.800451i $$-0.704593\pi$$
−0.599398 + 0.800451i $$0.704593\pi$$
$$642$$ −5474.26 −0.336529
$$643$$ −14695.8 −0.901317 −0.450658 0.892696i $$-0.648811\pi$$
−0.450658 + 0.892696i $$0.648811\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 16169.1 0.984777
$$647$$ −12694.8 −0.771383 −0.385691 0.922628i $$-0.626037\pi$$
−0.385691 + 0.922628i $$0.626037\pi$$
$$648$$ 40171.6 2.43532
$$649$$ −32189.0 −1.94688
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 28786.8 1.72911
$$653$$ 12385.6 0.742247 0.371124 0.928583i $$-0.378973\pi$$
0.371124 + 0.928583i $$0.378973\pi$$
$$654$$ 43730.0 2.61465
$$655$$ 0 0
$$656$$ −2497.07 −0.148619
$$657$$ 789.030 0.0468539
$$658$$ 0 0
$$659$$ −2072.18 −0.122489 −0.0612447 0.998123i $$-0.519507\pi$$
−0.0612447 + 0.998123i $$0.519507\pi$$
$$660$$ 0 0
$$661$$ −1074.36 −0.0632193 −0.0316096 0.999500i $$-0.510063\pi$$
−0.0316096 + 0.999500i $$0.510063\pi$$
$$662$$ 26703.6 1.56777
$$663$$ −5314.47 −0.311307
$$664$$ −47613.0 −2.78275
$$665$$ 0 0
$$666$$ −5671.84 −0.329999
$$667$$ −540.982 −0.0314047
$$668$$ −26228.2 −1.51916
$$669$$ −3479.42 −0.201080
$$670$$ 0 0
$$671$$ 797.667 0.0458921
$$672$$ 0 0
$$673$$ −26195.2 −1.50037 −0.750186 0.661226i $$-0.770036\pi$$
−0.750186 + 0.661226i $$0.770036\pi$$
$$674$$ −38555.3 −2.20341
$$675$$ 0 0
$$676$$ −26794.7 −1.52450
$$677$$ −4228.44 −0.240047 −0.120024 0.992771i $$-0.538297\pi$$
−0.120024 + 0.992771i $$0.538297\pi$$
$$678$$ 46750.4 2.64814
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −7756.80 −0.436478
$$682$$ 88994.5 4.99674
$$683$$ −27525.5 −1.54207 −0.771036 0.636792i $$-0.780261\pi$$
−0.771036 + 0.636792i $$0.780261\pi$$
$$684$$ 9085.91 0.507907
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −30867.3 −1.71421
$$688$$ 7435.40 0.412023
$$689$$ −4691.09 −0.259385
$$690$$ 0 0
$$691$$ 33324.4 1.83462 0.917309 0.398177i $$-0.130357\pi$$
0.917309 + 0.398177i $$0.130357\pi$$
$$692$$ −53021.1 −2.91266
$$693$$ 0 0
$$694$$ 51654.8 2.82534
$$695$$ 0 0
$$696$$ 7029.54 0.382836
$$697$$ −422.973 −0.0229860
$$698$$ 6939.12 0.376289
$$699$$ −2013.77 −0.108967
$$700$$ 0 0
$$701$$ −33262.9 −1.79219 −0.896094 0.443864i $$-0.853607\pi$$
−0.896094 + 0.443864i $$0.853607\pi$$
$$702$$ −24977.1 −1.34288
$$703$$ 15816.2 0.848534
$$704$$ −81611.8 −4.36912
$$705$$ 0 0
$$706$$ −31392.0 −1.67345
$$707$$ 0 0
$$708$$ −61141.0 −3.24551
$$709$$ 13703.0 0.725851 0.362926 0.931818i $$-0.381778\pi$$
0.362926 + 0.931818i $$0.381778\pi$$
$$710$$ 0 0
$$711$$ −4557.70 −0.240404
$$712$$ −3299.58 −0.173676
$$713$$ −8126.14 −0.426825
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 34553.3 1.80352
$$717$$ −25987.3 −1.35358
$$718$$ −12275.6 −0.638052
$$719$$ 8074.93 0.418838 0.209419 0.977826i $$-0.432843\pi$$
0.209419 + 0.977826i $$0.432843\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 2289.59 0.118019
$$723$$ −29323.6 −1.50838
$$724$$ −55281.1 −2.83771
$$725$$ 0 0
$$726$$ 35287.6 1.80392
$$727$$ −3668.70 −0.187159 −0.0935794 0.995612i $$-0.529831\pi$$
−0.0935794 + 0.995612i $$0.529831\pi$$
$$728$$ 0 0
$$729$$ 21881.9 1.11172
$$730$$ 0 0
$$731$$ 1259.46 0.0637250
$$732$$ 1515.12 0.0765033
$$733$$ −14980.3 −0.754857 −0.377428 0.926039i $$-0.623192\pi$$
−0.377428 + 0.926039i $$0.623192\pi$$
$$734$$ 39918.2 2.00737
$$735$$ 0 0
$$736$$ 15840.7 0.793338
$$737$$ −8700.94 −0.434875
$$738$$ −326.894 −0.0163050
$$739$$ 6530.59 0.325077 0.162538 0.986702i $$-0.448032\pi$$
0.162538 + 0.986702i $$0.448032\pi$$
$$740$$ 0 0
$$741$$ 11453.3 0.567812
$$742$$ 0 0
$$743$$ −25952.0 −1.28141 −0.640704 0.767788i $$-0.721357\pi$$
−0.640704 + 0.767788i $$0.721357\pi$$
$$744$$ 105591. 5.20318
$$745$$ 0 0
$$746$$ 34906.2 1.71315
$$747$$ −3509.85 −0.171913
$$748$$ −41459.6 −2.02662
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −14093.9 −0.684813 −0.342407 0.939552i $$-0.611242\pi$$
−0.342407 + 0.939552i $$0.611242\pi$$
$$752$$ −79483.6 −3.85435
$$753$$ 1452.57 0.0702983
$$754$$ −3475.87 −0.167883
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2554.41 0.122644 0.0613220 0.998118i $$-0.480468\pi$$
0.0613220 + 0.998118i $$0.480468\pi$$
$$758$$ 22997.2 1.10197
$$759$$ −6286.40 −0.300635
$$760$$ 0 0
$$761$$ −2219.08 −0.105705 −0.0528527 0.998602i $$-0.516831\pi$$
−0.0528527 + 0.998602i $$0.516831\pi$$
$$762$$ −35160.1 −1.67154
$$763$$ 0 0
$$764$$ −38856.5 −1.84003
$$765$$ 0 0
$$766$$ 36177.0 1.70643
$$767$$ 18884.6 0.889028
$$768$$ −31333.5 −1.47220
$$769$$ 22466.2 1.05352 0.526758 0.850015i $$-0.323408\pi$$
0.526758 + 0.850015i $$0.323408\pi$$
$$770$$ 0 0
$$771$$ 36609.3 1.71006
$$772$$ 32845.0 1.53124
$$773$$ 9674.79 0.450165 0.225083 0.974340i $$-0.427735\pi$$
0.225083 + 0.974340i $$0.427735\pi$$
$$774$$ 973.374 0.0452031
$$775$$ 0 0
$$776$$ −121253. −5.60920
$$777$$ 0 0
$$778$$ 34498.2 1.58974
$$779$$ 911.560 0.0419256
$$780$$ 0 0
$$781$$ 49746.6 2.27922
$$782$$ 5206.64 0.238093
$$783$$ 3151.23 0.143826
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −44436.2 −2.01652
$$787$$ −20942.8 −0.948577 −0.474288 0.880370i $$-0.657295\pi$$
−0.474288 + 0.880370i $$0.657295\pi$$
$$788$$ −14946.1 −0.675676
$$789$$ 24341.8 1.09834
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −20015.1 −0.897989
$$793$$ −467.975 −0.0209562
$$794$$ −22999.2 −1.02797
$$795$$ 0 0
$$796$$ −70227.8 −3.12708
$$797$$ 23526.6 1.04561 0.522807 0.852451i $$-0.324885\pi$$
0.522807 + 0.852451i $$0.324885\pi$$
$$798$$ 0 0
$$799$$ −13463.5 −0.596127
$$800$$ 0 0
$$801$$ −243.233 −0.0107294
$$802$$ 47827.1 2.10578
$$803$$ 7759.29 0.340996
$$804$$ −16526.9 −0.724948
$$805$$ 0 0
$$806$$ −52211.3 −2.28172
$$807$$ 5968.72 0.260358
$$808$$ −31296.1 −1.36262
$$809$$ −18202.2 −0.791047 −0.395523 0.918456i $$-0.629437\pi$$
−0.395523 + 0.918456i $$0.629437\pi$$
$$810$$ 0 0
$$811$$ 2510.24 0.108689 0.0543443 0.998522i $$-0.482693\pi$$
0.0543443 + 0.998522i $$0.482693\pi$$
$$812$$ 0 0
$$813$$ 21906.5 0.945012
$$814$$ −55776.7 −2.40168
$$815$$ 0 0
$$816$$ −38096.9 −1.63438
$$817$$ −2714.30 −0.116232
$$818$$ −1728.24 −0.0738711
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 17899.6 0.760903 0.380451 0.924801i $$-0.375769\pi$$
0.380451 + 0.924801i $$0.375769\pi$$
$$822$$ 23270.5 0.987411
$$823$$ −14039.5 −0.594637 −0.297318 0.954778i $$-0.596092\pi$$
−0.297318 + 0.954778i $$0.596092\pi$$
$$824$$ −24910.3 −1.05315
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −15127.4 −0.636073 −0.318036 0.948079i $$-0.603023\pi$$
−0.318036 + 0.948079i $$0.603023\pi$$
$$828$$ 2925.77 0.122799
$$829$$ −21986.5 −0.921136 −0.460568 0.887624i $$-0.652354\pi$$
−0.460568 + 0.887624i $$0.652354\pi$$
$$830$$ 0 0
$$831$$ 41718.7 1.74152
$$832$$ 47880.0 1.99512
$$833$$ 0 0
$$834$$ 4942.76 0.205220
$$835$$ 0 0
$$836$$ 89350.6 3.69648
$$837$$ 47334.8 1.95476
$$838$$ −69243.5 −2.85439
$$839$$ 2276.89 0.0936914 0.0468457 0.998902i $$-0.485083\pi$$
0.0468457 + 0.998902i $$0.485083\pi$$
$$840$$ 0 0
$$841$$ −23950.5 −0.982019
$$842$$ 36531.8 1.49521
$$843$$ 1727.25 0.0705688
$$844$$ 87010.1 3.54859
$$845$$ 0 0
$$846$$ −10405.3 −0.422861
$$847$$ 0 0
$$848$$ −33628.2 −1.36179
$$849$$ 27113.4 1.09603
$$850$$ 0 0
$$851$$ 5093.00 0.205154
$$852$$ 94490.6 3.79952
$$853$$ 13342.6 0.535570 0.267785 0.963479i $$-0.413708\pi$$
0.267785 + 0.963479i $$0.413708\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 15650.6 0.624915
$$857$$ 18690.9 0.745003 0.372502 0.928032i $$-0.378500\pi$$
0.372502 + 0.928032i $$0.378500\pi$$
$$858$$ −40390.8 −1.60713
$$859$$ −18318.9 −0.727628 −0.363814 0.931472i $$-0.618526\pi$$
−0.363814 + 0.931472i $$0.618526\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 28071.3 1.10918
$$863$$ −38133.1 −1.50413 −0.752067 0.659087i $$-0.770943\pi$$
−0.752067 + 0.659087i $$0.770943\pi$$
$$864$$ −92272.3 −3.63330
$$865$$ 0 0
$$866$$ 22967.3 0.901223
$$867$$ 16426.0 0.643432
$$868$$ 0 0
$$869$$ −44820.3 −1.74962
$$870$$ 0 0
$$871$$ 5104.66 0.198582
$$872$$ −125022. −4.85525
$$873$$ −8938.33 −0.346525
$$874$$ −11221.0 −0.434273
$$875$$ 0 0
$$876$$ 14738.3 0.568449
$$877$$ 19707.5 0.758807 0.379404 0.925231i $$-0.376129\pi$$
0.379404 + 0.925231i $$0.376129\pi$$
$$878$$ −29421.5 −1.13090
$$879$$ −34664.6 −1.33016
$$880$$ 0 0
$$881$$ 14091.5 0.538883 0.269441 0.963017i $$-0.413161\pi$$
0.269441 + 0.963017i $$0.413161\pi$$
$$882$$ 0 0
$$883$$ −3115.87 −0.118751 −0.0593757 0.998236i $$-0.518911\pi$$
−0.0593757 + 0.998236i $$0.518911\pi$$
$$884$$ 24323.5 0.925438
$$885$$ 0 0
$$886$$ −62229.8 −2.35965
$$887$$ 38734.6 1.46627 0.733134 0.680084i $$-0.238057\pi$$
0.733134 + 0.680084i $$0.238057\pi$$
$$888$$ −66178.6 −2.50091
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 29121.3 1.09495
$$892$$ 15924.8 0.597759
$$893$$ 29015.6 1.08731
$$894$$ 19675.7 0.736077
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 3688.10 0.137282
$$898$$ 91225.8 3.39003
$$899$$ 6587.21 0.244378
$$900$$ 0 0
$$901$$ −5696.21 −0.210619
$$902$$ −3214.66 −0.118666
$$903$$ 0 0
$$904$$ −133657. −4.91744
$$905$$ 0 0
$$906$$ −58480.2 −2.14445
$$907$$ −19242.9 −0.704464 −0.352232 0.935913i $$-0.614577\pi$$
−0.352232 + 0.935913i $$0.614577\pi$$
$$908$$ 35501.7 1.29754
$$909$$ −2307.03 −0.0841799
$$910$$ 0 0
$$911$$ 34613.3 1.25882 0.629412 0.777072i $$-0.283296\pi$$
0.629412 + 0.777072i $$0.283296\pi$$
$$912$$ 82103.6 2.98105
$$913$$ −34515.8 −1.25116
$$914$$ 83100.1 3.00734
$$915$$ 0 0
$$916$$ 141275. 5.09591
$$917$$ 0 0
$$918$$ −30328.7 −1.09041
$$919$$ 25826.4 0.927022 0.463511 0.886091i $$-0.346589\pi$$
0.463511 + 0.886091i $$0.346589\pi$$
$$920$$ 0 0
$$921$$ 3547.01 0.126903
$$922$$ 76007.0 2.71492
$$923$$ −29185.3 −1.04079
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −46893.8 −1.66417
$$927$$ −1836.29 −0.0650613
$$928$$ −12840.8 −0.454224
$$929$$ −19451.6 −0.686960 −0.343480 0.939160i $$-0.611606\pi$$
−0.343480 + 0.939160i $$0.611606\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 9216.70 0.323930
$$933$$ 35944.8 1.26129
$$934$$ −37979.1 −1.33053
$$935$$ 0 0
$$936$$ 11742.5 0.410059
$$937$$ 34469.1 1.20177 0.600884 0.799336i $$-0.294815\pi$$
0.600884 + 0.799336i $$0.294815\pi$$
$$938$$ 0 0
$$939$$ −39844.1 −1.38473
$$940$$ 0 0
$$941$$ −14156.4 −0.490419 −0.245209 0.969470i $$-0.578857\pi$$
−0.245209 + 0.969470i $$0.578857\pi$$
$$942$$ 25790.6 0.892042
$$943$$ 293.532 0.0101365
$$944$$ 135375. 4.66746
$$945$$ 0 0
$$946$$ 9572.13 0.328982
$$947$$ 38092.4 1.30711 0.653557 0.756877i $$-0.273276\pi$$
0.653557 + 0.756877i $$0.273276\pi$$
$$948$$ −85133.3 −2.91667
$$949$$ −4552.22 −0.155713
$$950$$ 0 0
$$951$$ −36234.8 −1.23553
$$952$$ 0 0
$$953$$ −5037.40 −0.171225 −0.0856126 0.996329i $$-0.527285\pi$$
−0.0856126 + 0.996329i $$0.527285\pi$$
$$954$$ −4402.30 −0.149402
$$955$$ 0 0
$$956$$ 118940. 4.02384
$$957$$ 5095.88 0.172128
$$958$$ 98185.1 3.31129
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 69156.0 2.32137
$$962$$ 32723.0 1.09671
$$963$$ 1153.71 0.0386060
$$964$$ 134210. 4.48403
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −11495.3 −0.382278 −0.191139 0.981563i $$-0.561218\pi$$
−0.191139 + 0.981563i $$0.561218\pi$$
$$968$$ −100885. −3.34977
$$969$$ 13907.3 0.461061
$$970$$ 0 0
$$971$$ −22352.7 −0.738757 −0.369379 0.929279i $$-0.620429\pi$$
−0.369379 + 0.929279i $$0.620429\pi$$
$$972$$ −31282.7 −1.03230
$$973$$ 0 0
$$974$$ 89539.4 2.94561
$$975$$ 0 0
$$976$$ −3354.69 −0.110022
$$977$$ −14345.7 −0.469765 −0.234882 0.972024i $$-0.575470\pi$$
−0.234882 + 0.972024i $$0.575470\pi$$
$$978$$ 34053.6 1.11341
$$979$$ −2391.94 −0.0780867
$$980$$ 0 0
$$981$$ −9216.15 −0.299948
$$982$$ −1194.42 −0.0388141
$$983$$ 34460.9 1.11814 0.559070 0.829120i $$-0.311158\pi$$
0.559070 + 0.829120i $$0.311158\pi$$
$$984$$ −3814.17 −0.123568
$$985$$ 0 0
$$986$$ −4220.61 −0.136320
$$987$$ 0 0
$$988$$ −52420.2 −1.68796
$$989$$ −874.036 −0.0281019
$$990$$ 0 0
$$991$$ −35189.6 −1.12799 −0.563993 0.825780i $$-0.690735\pi$$
−0.563993 + 0.825780i $$0.690735\pi$$
$$992$$ −192883. −6.17342
$$993$$ 22968.2 0.734011
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −65560.6 −2.08571
$$997$$ −50730.0 −1.61147 −0.805734 0.592277i $$-0.798229\pi$$
−0.805734 + 0.592277i $$0.798229\pi$$
$$998$$ −32155.2 −1.01990
$$999$$ −29666.8 −0.939554
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 1225.4.a.m.1.1 2
5.4 even 2 245.4.a.k.1.2 2
7.6 odd 2 175.4.a.c.1.1 2
15.14 odd 2 2205.4.a.u.1.1 2
21.20 even 2 1575.4.a.z.1.2 2
35.4 even 6 245.4.e.i.226.1 4
35.9 even 6 245.4.e.i.116.1 4
35.13 even 4 175.4.b.c.99.4 4
35.19 odd 6 245.4.e.h.116.1 4
35.24 odd 6 245.4.e.h.226.1 4
35.27 even 4 175.4.b.c.99.1 4
35.34 odd 2 35.4.a.b.1.2 2
105.104 even 2 315.4.a.f.1.1 2
140.139 even 2 560.4.a.r.1.2 2
280.69 odd 2 2240.4.a.bn.1.2 2
280.139 even 2 2240.4.a.bo.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 2 35.34 odd 2
175.4.a.c.1.1 2 7.6 odd 2
175.4.b.c.99.1 4 35.27 even 4
175.4.b.c.99.4 4 35.13 even 4
245.4.a.k.1.2 2 5.4 even 2
245.4.e.h.116.1 4 35.19 odd 6
245.4.e.h.226.1 4 35.24 odd 6
245.4.e.i.116.1 4 35.9 even 6
245.4.e.i.226.1 4 35.4 even 6
315.4.a.f.1.1 2 105.104 even 2
560.4.a.r.1.2 2 140.139 even 2
1225.4.a.m.1.1 2 1.1 even 1 trivial
1575.4.a.z.1.2 2 21.20 even 2
2205.4.a.u.1.1 2 15.14 odd 2
2240.4.a.bn.1.2 2 280.69 odd 2
2240.4.a.bo.1.1 2 280.139 even 2