# Properties

 Label 825.4.c.m.199.6 Level $825$ Weight $4$ Character 825.199 Analytic conductor $48.677$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.245110336.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 19x^{4} + 101x^{2} + 100$$ x^6 + 19*x^4 + 101*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.6 Root $$-1.12946i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.4.c.m.199.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.59486i q^{2} +3.00000i q^{3} -13.1127 q^{4} -13.7846 q^{6} +20.6383i q^{7} -23.4921i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+4.59486i q^{2} +3.00000i q^{3} -13.1127 q^{4} -13.7846 q^{6} +20.6383i q^{7} -23.4921i q^{8} -9.00000 q^{9} +11.0000 q^{11} -39.3381i q^{12} +15.6584i q^{13} -94.8302 q^{14} +3.04132 q^{16} +72.9507i q^{17} -41.3537i q^{18} -61.0513 q^{19} -61.9150 q^{21} +50.5434i q^{22} +13.6605i q^{23} +70.4764 q^{24} -71.9483 q^{26} -27.0000i q^{27} -270.624i q^{28} +31.4663 q^{29} -243.008 q^{31} -173.963i q^{32} +33.0000i q^{33} -335.198 q^{34} +118.014 q^{36} -65.4018i q^{37} -280.522i q^{38} -46.9753 q^{39} -109.087 q^{41} -284.491i q^{42} +121.750i q^{43} -144.240 q^{44} -62.7678 q^{46} -519.530i q^{47} +9.12396i q^{48} -82.9413 q^{49} -218.852 q^{51} -205.324i q^{52} +542.673i q^{53} +124.061 q^{54} +484.839 q^{56} -183.154i q^{57} +144.583i q^{58} -109.478 q^{59} -89.6156 q^{61} -1116.59i q^{62} -185.745i q^{63} +823.664 q^{64} -151.630 q^{66} +488.446i q^{67} -956.581i q^{68} -40.9814 q^{69} +837.423 q^{71} +211.429i q^{72} -351.216i q^{73} +300.512 q^{74} +800.547 q^{76} +227.022i q^{77} -215.845i q^{78} +831.205 q^{79} +81.0000 q^{81} -501.238i q^{82} -1389.13i q^{83} +811.873 q^{84} -559.423 q^{86} +94.3988i q^{87} -258.413i q^{88} -1523.70 q^{89} -323.164 q^{91} -179.125i q^{92} -729.025i q^{93} +2387.17 q^{94} +521.888 q^{96} -426.612i q^{97} -381.103i q^{98} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 34 q^{4} - 6 q^{6} - 54 q^{9}+O(q^{10})$$ 6 * q - 34 * q^4 - 6 * q^6 - 54 * q^9 $$6 q - 34 q^{4} - 6 q^{6} - 54 q^{9} + 66 q^{11} - 288 q^{14} + 50 q^{16} - 232 q^{19} - 36 q^{21} - 18 q^{24} + 604 q^{26} - 476 q^{29} + 184 q^{31} - 708 q^{34} + 306 q^{36} - 120 q^{39} - 92 q^{41} - 374 q^{44} - 480 q^{46} + 914 q^{49} - 192 q^{51} + 54 q^{54} + 1376 q^{56} - 2472 q^{59} + 684 q^{61} + 3838 q^{64} - 66 q^{66} + 1440 q^{69} + 3632 q^{71} + 3748 q^{74} + 792 q^{76} + 192 q^{79} + 486 q^{81} + 960 q^{84} + 376 q^{86} - 1676 q^{89} + 664 q^{91} + 6224 q^{94} - 1182 q^{96} - 594 q^{99}+O(q^{100})$$ 6 * q - 34 * q^4 - 6 * q^6 - 54 * q^9 + 66 * q^11 - 288 * q^14 + 50 * q^16 - 232 * q^19 - 36 * q^21 - 18 * q^24 + 604 * q^26 - 476 * q^29 + 184 * q^31 - 708 * q^34 + 306 * q^36 - 120 * q^39 - 92 * q^41 - 374 * q^44 - 480 * q^46 + 914 * q^49 - 192 * q^51 + 54 * q^54 + 1376 * q^56 - 2472 * q^59 + 684 * q^61 + 3838 * q^64 - 66 * q^66 + 1440 * q^69 + 3632 * q^71 + 3748 * q^74 + 792 * q^76 + 192 * q^79 + 486 * q^81 + 960 * q^84 + 376 * q^86 - 1676 * q^89 + 664 * q^91 + 6224 * q^94 - 1182 * q^96 - 594 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.59486i 1.62453i 0.583291 + 0.812263i $$0.301765\pi$$
−0.583291 + 0.812263i $$0.698235\pi$$
$$3$$ 3.00000i 0.577350i
$$4$$ −13.1127 −1.63909
$$5$$ 0 0
$$6$$ −13.7846 −0.937921
$$7$$ 20.6383i 1.11437i 0.830390 + 0.557183i $$0.188118\pi$$
−0.830390 + 0.557183i $$0.811882\pi$$
$$8$$ − 23.4921i − 1.03822i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ − 39.3381i − 0.946328i
$$13$$ 15.6584i 0.334067i 0.985951 + 0.167033i $$0.0534188\pi$$
−0.985951 + 0.167033i $$0.946581\pi$$
$$14$$ −94.8302 −1.81032
$$15$$ 0 0
$$16$$ 3.04132 0.0475206
$$17$$ 72.9507i 1.04077i 0.853931 + 0.520387i $$0.174212\pi$$
−0.853931 + 0.520387i $$0.825788\pi$$
$$18$$ − 41.3537i − 0.541509i
$$19$$ −61.0513 −0.737165 −0.368582 0.929595i $$-0.620157\pi$$
−0.368582 + 0.929595i $$0.620157\pi$$
$$20$$ 0 0
$$21$$ −61.9150 −0.643379
$$22$$ 50.5434i 0.489813i
$$23$$ 13.6605i 0.123844i 0.998081 + 0.0619218i $$0.0197229\pi$$
−0.998081 + 0.0619218i $$0.980277\pi$$
$$24$$ 70.4764 0.599414
$$25$$ 0 0
$$26$$ −71.9483 −0.542701
$$27$$ − 27.0000i − 0.192450i
$$28$$ − 270.624i − 1.82654i
$$29$$ 31.4663 0.201487 0.100744 0.994912i $$-0.467878\pi$$
0.100744 + 0.994912i $$0.467878\pi$$
$$30$$ 0 0
$$31$$ −243.008 −1.40792 −0.703961 0.710239i $$-0.748587\pi$$
−0.703961 + 0.710239i $$0.748587\pi$$
$$32$$ − 173.963i − 0.961016i
$$33$$ 33.0000i 0.174078i
$$34$$ −335.198 −1.69076
$$35$$ 0 0
$$36$$ 118.014 0.546363
$$37$$ − 65.4018i − 0.290594i −0.989388 0.145297i $$-0.953586\pi$$
0.989388 0.145297i $$-0.0464138\pi$$
$$38$$ − 280.522i − 1.19754i
$$39$$ −46.9753 −0.192874
$$40$$ 0 0
$$41$$ −109.087 −0.415524 −0.207762 0.978179i $$-0.566618\pi$$
−0.207762 + 0.978179i $$0.566618\pi$$
$$42$$ − 284.491i − 1.04519i
$$43$$ 121.750i 0.431783i 0.976417 + 0.215891i $$0.0692657\pi$$
−0.976417 + 0.215891i $$0.930734\pi$$
$$44$$ −144.240 −0.494204
$$45$$ 0 0
$$46$$ −62.7678 −0.201187
$$47$$ − 519.530i − 1.61237i −0.591665 0.806184i $$-0.701529\pi$$
0.591665 0.806184i $$-0.298471\pi$$
$$48$$ 9.12396i 0.0274361i
$$49$$ −82.9413 −0.241811
$$50$$ 0 0
$$51$$ −218.852 −0.600891
$$52$$ − 205.324i − 0.547565i
$$53$$ 542.673i 1.40645i 0.710967 + 0.703226i $$0.248258\pi$$
−0.710967 + 0.703226i $$0.751742\pi$$
$$54$$ 124.061 0.312640
$$55$$ 0 0
$$56$$ 484.839 1.15695
$$57$$ − 183.154i − 0.425602i
$$58$$ 144.583i 0.327322i
$$59$$ −109.478 −0.241574 −0.120787 0.992678i $$-0.538542\pi$$
−0.120787 + 0.992678i $$0.538542\pi$$
$$60$$ 0 0
$$61$$ −89.6156 −0.188100 −0.0940501 0.995567i $$-0.529981\pi$$
−0.0940501 + 0.995567i $$0.529981\pi$$
$$62$$ − 1116.59i − 2.28721i
$$63$$ − 185.745i − 0.371455i
$$64$$ 823.664 1.60872
$$65$$ 0 0
$$66$$ −151.630 −0.282794
$$67$$ 488.446i 0.890644i 0.895371 + 0.445322i $$0.146911\pi$$
−0.895371 + 0.445322i $$0.853089\pi$$
$$68$$ − 956.581i − 1.70592i
$$69$$ −40.9814 −0.0715011
$$70$$ 0 0
$$71$$ 837.423 1.39977 0.699887 0.714254i $$-0.253234\pi$$
0.699887 + 0.714254i $$0.253234\pi$$
$$72$$ 211.429i 0.346072i
$$73$$ − 351.216i − 0.563105i −0.959546 0.281553i $$-0.909151\pi$$
0.959546 0.281553i $$-0.0908494\pi$$
$$74$$ 300.512 0.472078
$$75$$ 0 0
$$76$$ 800.547 1.20828
$$77$$ 227.022i 0.335994i
$$78$$ − 215.845i − 0.313328i
$$79$$ 831.205 1.18377 0.591885 0.806022i $$-0.298384\pi$$
0.591885 + 0.806022i $$0.298384\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ − 501.238i − 0.675031i
$$83$$ − 1389.13i − 1.83707i −0.395335 0.918537i $$-0.629371\pi$$
0.395335 0.918537i $$-0.370629\pi$$
$$84$$ 811.873 1.05456
$$85$$ 0 0
$$86$$ −559.423 −0.701443
$$87$$ 94.3988i 0.116329i
$$88$$ − 258.413i − 0.313034i
$$89$$ −1523.70 −1.81474 −0.907369 0.420335i $$-0.861912\pi$$
−0.907369 + 0.420335i $$0.861912\pi$$
$$90$$ 0 0
$$91$$ −323.164 −0.372273
$$92$$ − 179.125i − 0.202990i
$$93$$ − 729.025i − 0.812864i
$$94$$ 2387.17 2.61933
$$95$$ 0 0
$$96$$ 521.888 0.554843
$$97$$ − 426.612i − 0.446555i −0.974755 0.223278i $$-0.928324\pi$$
0.974755 0.223278i $$-0.0716757\pi$$
$$98$$ − 381.103i − 0.392829i
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ 74.1387 0.0730403 0.0365202 0.999333i $$-0.488373\pi$$
0.0365202 + 0.999333i $$0.488373\pi$$
$$102$$ − 1005.59i − 0.976163i
$$103$$ 69.3916i 0.0663821i 0.999449 + 0.0331911i $$0.0105670\pi$$
−0.999449 + 0.0331911i $$0.989433\pi$$
$$104$$ 367.850 0.346833
$$105$$ 0 0
$$106$$ −2493.51 −2.28482
$$107$$ 1141.71i 1.03152i 0.856733 + 0.515761i $$0.172491\pi$$
−0.856733 + 0.515761i $$0.827509\pi$$
$$108$$ 354.043i 0.315443i
$$109$$ 2226.85 1.95682 0.978409 0.206680i $$-0.0662659\pi$$
0.978409 + 0.206680i $$0.0662659\pi$$
$$110$$ 0 0
$$111$$ 196.205 0.167775
$$112$$ 62.7678i 0.0529554i
$$113$$ 1719.76i 1.43169i 0.698257 + 0.715847i $$0.253959\pi$$
−0.698257 + 0.715847i $$0.746041\pi$$
$$114$$ 841.566 0.691402
$$115$$ 0 0
$$116$$ −412.608 −0.330256
$$117$$ − 140.926i − 0.111356i
$$118$$ − 503.038i − 0.392444i
$$119$$ −1505.58 −1.15980
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ − 411.771i − 0.305574i
$$123$$ − 327.260i − 0.239903i
$$124$$ 3186.49 2.30771
$$125$$ 0 0
$$126$$ 853.472 0.603439
$$127$$ − 1601.63i − 1.11907i −0.828807 0.559534i $$-0.810980\pi$$
0.828807 0.559534i $$-0.189020\pi$$
$$128$$ 2392.91i 1.65239i
$$129$$ −365.249 −0.249290
$$130$$ 0 0
$$131$$ 2004.13 1.33665 0.668327 0.743868i $$-0.267011\pi$$
0.668327 + 0.743868i $$0.267011\pi$$
$$132$$ − 432.719i − 0.285329i
$$133$$ − 1260.00i − 0.821471i
$$134$$ −2244.34 −1.44687
$$135$$ 0 0
$$136$$ 1713.77 1.08055
$$137$$ − 1672.85i − 1.04322i −0.853184 0.521610i $$-0.825331\pi$$
0.853184 0.521610i $$-0.174669\pi$$
$$138$$ − 188.303i − 0.116155i
$$139$$ −2540.38 −1.55016 −0.775080 0.631863i $$-0.782291\pi$$
−0.775080 + 0.631863i $$0.782291\pi$$
$$140$$ 0 0
$$141$$ 1558.59 0.930901
$$142$$ 3847.84i 2.27397i
$$143$$ 172.243i 0.100725i
$$144$$ −27.3719 −0.0158402
$$145$$ 0 0
$$146$$ 1613.79 0.914780
$$147$$ − 248.824i − 0.139610i
$$148$$ 857.594i 0.476310i
$$149$$ −3090.68 −1.69932 −0.849658 0.527334i $$-0.823192\pi$$
−0.849658 + 0.527334i $$0.823192\pi$$
$$150$$ 0 0
$$151$$ 1358.74 0.732267 0.366134 0.930562i $$-0.380681\pi$$
0.366134 + 0.930562i $$0.380681\pi$$
$$152$$ 1434.22i 0.765335i
$$153$$ − 656.557i − 0.346925i
$$154$$ −1043.13 −0.545831
$$155$$ 0 0
$$156$$ 615.973 0.316137
$$157$$ − 1011.95i − 0.514411i −0.966357 0.257205i $$-0.917198\pi$$
0.966357 0.257205i $$-0.0828017\pi$$
$$158$$ 3819.27i 1.92307i
$$159$$ −1628.02 −0.812015
$$160$$ 0 0
$$161$$ −281.929 −0.138007
$$162$$ 372.183i 0.180503i
$$163$$ 2816.37i 1.35334i 0.736285 + 0.676672i $$0.236578\pi$$
−0.736285 + 0.676672i $$0.763422\pi$$
$$164$$ 1430.42 0.681081
$$165$$ 0 0
$$166$$ 6382.87 2.98438
$$167$$ 3448.89i 1.59810i 0.601262 + 0.799052i $$0.294665\pi$$
−0.601262 + 0.799052i $$0.705335\pi$$
$$168$$ 1454.52i 0.667966i
$$169$$ 1951.81 0.888399
$$170$$ 0 0
$$171$$ 549.462 0.245722
$$172$$ − 1596.47i − 0.707730i
$$173$$ 2287.85i 1.00545i 0.864448 + 0.502723i $$0.167668\pi$$
−0.864448 + 0.502723i $$0.832332\pi$$
$$174$$ −433.749 −0.188979
$$175$$ 0 0
$$176$$ 33.4545 0.0143280
$$177$$ − 328.435i − 0.139473i
$$178$$ − 7001.17i − 2.94809i
$$179$$ −3249.06 −1.35668 −0.678340 0.734748i $$-0.737300\pi$$
−0.678340 + 0.734748i $$0.737300\pi$$
$$180$$ 0 0
$$181$$ 1170.45 0.480655 0.240328 0.970692i $$-0.422745\pi$$
0.240328 + 0.970692i $$0.422745\pi$$
$$182$$ − 1484.89i − 0.604767i
$$183$$ − 268.847i − 0.108600i
$$184$$ 320.913 0.128576
$$185$$ 0 0
$$186$$ 3349.76 1.32052
$$187$$ 802.458i 0.313805i
$$188$$ 6812.44i 2.64281i
$$189$$ 557.235 0.214460
$$190$$ 0 0
$$191$$ −2760.35 −1.04572 −0.522859 0.852419i $$-0.675134\pi$$
−0.522859 + 0.852419i $$0.675134\pi$$
$$192$$ 2470.99i 0.928794i
$$193$$ 1250.61i 0.466430i 0.972425 + 0.233215i $$0.0749245\pi$$
−0.972425 + 0.233215i $$0.925075\pi$$
$$194$$ 1960.22 0.725441
$$195$$ 0 0
$$196$$ 1087.58 0.396350
$$197$$ − 143.991i − 0.0520756i −0.999661 0.0260378i $$-0.991711\pi$$
0.999661 0.0260378i $$-0.00828903\pi$$
$$198$$ − 454.891i − 0.163271i
$$199$$ −761.249 −0.271174 −0.135587 0.990765i $$-0.543292\pi$$
−0.135587 + 0.990765i $$0.543292\pi$$
$$200$$ 0 0
$$201$$ −1465.34 −0.514213
$$202$$ 340.657i 0.118656i
$$203$$ 649.411i 0.224531i
$$204$$ 2869.74 0.984913
$$205$$ 0 0
$$206$$ −318.844 −0.107840
$$207$$ − 122.944i − 0.0412812i
$$208$$ 47.6223i 0.0158751i
$$209$$ −671.564 −0.222263
$$210$$ 0 0
$$211$$ 3976.58 1.29743 0.648717 0.761029i $$-0.275306\pi$$
0.648717 + 0.761029i $$0.275306\pi$$
$$212$$ − 7115.91i − 2.30530i
$$213$$ 2512.27i 0.808159i
$$214$$ −5245.97 −1.67573
$$215$$ 0 0
$$216$$ −634.287 −0.199805
$$217$$ − 5015.29i − 1.56894i
$$218$$ 10232.0i 3.17890i
$$219$$ 1053.65 0.325109
$$220$$ 0 0
$$221$$ −1142.29 −0.347688
$$222$$ 901.535i 0.272555i
$$223$$ − 908.084i − 0.272690i −0.990661 0.136345i $$-0.956464\pi$$
0.990661 0.136345i $$-0.0435355\pi$$
$$224$$ 3590.30 1.07092
$$225$$ 0 0
$$226$$ −7902.05 −2.32583
$$227$$ − 2062.15i − 0.602951i −0.953474 0.301475i $$-0.902521\pi$$
0.953474 0.301475i $$-0.0974791\pi$$
$$228$$ 2401.64i 0.697599i
$$229$$ −4077.47 −1.17662 −0.588312 0.808634i $$-0.700207\pi$$
−0.588312 + 0.808634i $$0.700207\pi$$
$$230$$ 0 0
$$231$$ −681.065 −0.193986
$$232$$ − 739.209i − 0.209187i
$$233$$ − 1682.76i − 0.473138i −0.971615 0.236569i $$-0.923977\pi$$
0.971615 0.236569i $$-0.0760229\pi$$
$$234$$ 647.534 0.180900
$$235$$ 0 0
$$236$$ 1435.56 0.395961
$$237$$ 2493.62i 0.683450i
$$238$$ − 6917.93i − 1.88413i
$$239$$ −4024.96 −1.08934 −0.544672 0.838649i $$-0.683346\pi$$
−0.544672 + 0.838649i $$0.683346\pi$$
$$240$$ 0 0
$$241$$ −2784.27 −0.744194 −0.372097 0.928194i $$-0.621361\pi$$
−0.372097 + 0.928194i $$0.621361\pi$$
$$242$$ 555.978i 0.147684i
$$243$$ 243.000i 0.0641500i
$$244$$ 1175.10 0.308313
$$245$$ 0 0
$$246$$ 1503.71 0.389729
$$247$$ − 955.968i − 0.246262i
$$248$$ 5708.78i 1.46173i
$$249$$ 4167.40 1.06064
$$250$$ 0 0
$$251$$ −1827.60 −0.459591 −0.229796 0.973239i $$-0.573806\pi$$
−0.229796 + 0.973239i $$0.573806\pi$$
$$252$$ 2435.62i 0.608848i
$$253$$ 150.265i 0.0373402i
$$254$$ 7359.26 1.81796
$$255$$ 0 0
$$256$$ −4405.79 −1.07563
$$257$$ 585.171i 0.142031i 0.997475 + 0.0710155i $$0.0226240\pi$$
−0.997475 + 0.0710155i $$0.977376\pi$$
$$258$$ − 1678.27i − 0.404978i
$$259$$ 1349.78 0.323828
$$260$$ 0 0
$$261$$ −283.196 −0.0671625
$$262$$ 9208.69i 2.17143i
$$263$$ 238.098i 0.0558241i 0.999610 + 0.0279120i $$0.00888583\pi$$
−0.999610 + 0.0279120i $$0.991114\pi$$
$$264$$ 775.240 0.180730
$$265$$ 0 0
$$266$$ 5789.51 1.33450
$$267$$ − 4571.09i − 1.04774i
$$268$$ − 6404.84i − 1.45984i
$$269$$ −4618.46 −1.04681 −0.523406 0.852083i $$-0.675339\pi$$
−0.523406 + 0.852083i $$0.675339\pi$$
$$270$$ 0 0
$$271$$ −143.439 −0.0321525 −0.0160762 0.999871i $$-0.505117\pi$$
−0.0160762 + 0.999871i $$0.505117\pi$$
$$272$$ 221.867i 0.0494582i
$$273$$ − 969.493i − 0.214932i
$$274$$ 7686.51 1.69474
$$275$$ 0 0
$$276$$ 537.376 0.117197
$$277$$ − 8602.51i − 1.86597i −0.359911 0.932987i $$-0.617193\pi$$
0.359911 0.932987i $$-0.382807\pi$$
$$278$$ − 11672.7i − 2.51828i
$$279$$ 2187.07 0.469307
$$280$$ 0 0
$$281$$ −2992.81 −0.635360 −0.317680 0.948198i $$-0.602904\pi$$
−0.317680 + 0.948198i $$0.602904\pi$$
$$282$$ 7161.50i 1.51227i
$$283$$ 6858.89i 1.44070i 0.693610 + 0.720351i $$0.256019\pi$$
−0.693610 + 0.720351i $$0.743981\pi$$
$$284$$ −10980.9 −2.29435
$$285$$ 0 0
$$286$$ −791.431 −0.163630
$$287$$ − 2251.37i − 0.463046i
$$288$$ 1565.66i 0.320339i
$$289$$ −408.809 −0.0832096
$$290$$ 0 0
$$291$$ 1279.84 0.257819
$$292$$ 4605.39i 0.922979i
$$293$$ − 4049.70i − 0.807461i −0.914878 0.403731i $$-0.867713\pi$$
0.914878 0.403731i $$-0.132287\pi$$
$$294$$ 1143.31 0.226800
$$295$$ 0 0
$$296$$ −1536.43 −0.301699
$$297$$ − 297.000i − 0.0580259i
$$298$$ − 14201.2i − 2.76059i
$$299$$ −213.901 −0.0413720
$$300$$ 0 0
$$301$$ −2512.71 −0.481164
$$302$$ 6243.20i 1.18959i
$$303$$ 222.416i 0.0421699i
$$304$$ −185.677 −0.0350305
$$305$$ 0 0
$$306$$ 3016.78 0.563588
$$307$$ − 9572.69i − 1.77962i −0.456335 0.889808i $$-0.650838\pi$$
0.456335 0.889808i $$-0.349162\pi$$
$$308$$ − 2976.87i − 0.550724i
$$309$$ −208.175 −0.0383257
$$310$$ 0 0
$$311$$ 5396.42 0.983932 0.491966 0.870614i $$-0.336278\pi$$
0.491966 + 0.870614i $$0.336278\pi$$
$$312$$ 1103.55i 0.200244i
$$313$$ − 9755.04i − 1.76162i −0.473469 0.880811i $$-0.656998\pi$$
0.473469 0.880811i $$-0.343002\pi$$
$$314$$ 4649.77 0.835674
$$315$$ 0 0
$$316$$ −10899.3 −1.94030
$$317$$ − 4353.75i − 0.771391i −0.922626 0.385695i $$-0.873962\pi$$
0.922626 0.385695i $$-0.126038\pi$$
$$318$$ − 7480.52i − 1.31914i
$$319$$ 346.129 0.0607508
$$320$$ 0 0
$$321$$ −3425.12 −0.595549
$$322$$ − 1295.42i − 0.224196i
$$323$$ − 4453.74i − 0.767221i
$$324$$ −1062.13 −0.182121
$$325$$ 0 0
$$326$$ −12940.8 −2.19854
$$327$$ 6680.54i 1.12977i
$$328$$ 2562.68i 0.431404i
$$329$$ 10722.2 1.79677
$$330$$ 0 0
$$331$$ 5387.64 0.894656 0.447328 0.894370i $$-0.352376\pi$$
0.447328 + 0.894370i $$0.352376\pi$$
$$332$$ 18215.3i 3.01113i
$$333$$ 588.616i 0.0968648i
$$334$$ −15847.2 −2.59616
$$335$$ 0 0
$$336$$ −188.303 −0.0305738
$$337$$ − 4500.27i − 0.727434i −0.931509 0.363717i $$-0.881508\pi$$
0.931509 0.363717i $$-0.118492\pi$$
$$338$$ 8968.30i 1.44323i
$$339$$ −5159.28 −0.826589
$$340$$ 0 0
$$341$$ −2673.09 −0.424504
$$342$$ 2524.70i 0.399181i
$$343$$ 5367.18i 0.844899i
$$344$$ 2860.16 0.448283
$$345$$ 0 0
$$346$$ −10512.4 −1.63337
$$347$$ 5906.32i 0.913740i 0.889533 + 0.456870i $$0.151030\pi$$
−0.889533 + 0.456870i $$0.848970\pi$$
$$348$$ − 1237.82i − 0.190673i
$$349$$ −3636.26 −0.557721 −0.278860 0.960332i $$-0.589957\pi$$
−0.278860 + 0.960332i $$0.589957\pi$$
$$350$$ 0 0
$$351$$ 422.778 0.0642912
$$352$$ − 1913.59i − 0.289757i
$$353$$ − 210.408i − 0.0317248i −0.999874 0.0158624i $$-0.994951\pi$$
0.999874 0.0158624i $$-0.00504938\pi$$
$$354$$ 1509.11 0.226577
$$355$$ 0 0
$$356$$ 19979.8 2.97451
$$357$$ − 4516.75i − 0.669612i
$$358$$ − 14928.9i − 2.20396i
$$359$$ −2499.68 −0.367488 −0.183744 0.982974i $$-0.558822\pi$$
−0.183744 + 0.982974i $$0.558822\pi$$
$$360$$ 0 0
$$361$$ −3131.74 −0.456588
$$362$$ 5378.03i 0.780837i
$$363$$ 363.000i 0.0524864i
$$364$$ 4237.56 0.610188
$$365$$ 0 0
$$366$$ 1235.31 0.176423
$$367$$ 5748.70i 0.817656i 0.912612 + 0.408828i $$0.134062\pi$$
−0.912612 + 0.408828i $$0.865938\pi$$
$$368$$ 41.5458i 0.00588512i
$$369$$ 981.781 0.138508
$$370$$ 0 0
$$371$$ −11199.9 −1.56730
$$372$$ 9559.48i 1.33236i
$$373$$ − 4467.78i − 0.620196i −0.950705 0.310098i $$-0.899638\pi$$
0.950705 0.310098i $$-0.100362\pi$$
$$374$$ −3687.18 −0.509785
$$375$$ 0 0
$$376$$ −12204.9 −1.67398
$$377$$ 492.712i 0.0673103i
$$378$$ 2560.42i 0.348396i
$$379$$ −7804.08 −1.05770 −0.528851 0.848715i $$-0.677377\pi$$
−0.528851 + 0.848715i $$0.677377\pi$$
$$380$$ 0 0
$$381$$ 4804.89 0.646094
$$382$$ − 12683.4i − 1.69880i
$$383$$ 11161.1i 1.48904i 0.667597 + 0.744522i $$0.267323\pi$$
−0.667597 + 0.744522i $$0.732677\pi$$
$$384$$ −7178.74 −0.954007
$$385$$ 0 0
$$386$$ −5746.38 −0.757728
$$387$$ − 1095.75i − 0.143928i
$$388$$ 5594.04i 0.731944i
$$389$$ −8490.24 −1.10661 −0.553306 0.832978i $$-0.686634\pi$$
−0.553306 + 0.832978i $$0.686634\pi$$
$$390$$ 0 0
$$391$$ −996.540 −0.128893
$$392$$ 1948.47i 0.251052i
$$393$$ 6012.39i 0.771717i
$$394$$ 661.616 0.0845983
$$395$$ 0 0
$$396$$ 1298.16 0.164735
$$397$$ 6019.74i 0.761013i 0.924778 + 0.380507i $$0.124250\pi$$
−0.924778 + 0.380507i $$0.875750\pi$$
$$398$$ − 3497.83i − 0.440529i
$$399$$ 3779.99 0.474277
$$400$$ 0 0
$$401$$ −10398.8 −1.29499 −0.647495 0.762069i $$-0.724184\pi$$
−0.647495 + 0.762069i $$0.724184\pi$$
$$402$$ − 6733.01i − 0.835354i
$$403$$ − 3805.13i − 0.470340i
$$404$$ −972.158 −0.119720
$$405$$ 0 0
$$406$$ −2983.95 −0.364756
$$407$$ − 719.420i − 0.0876175i
$$408$$ 5141.30i 0.623854i
$$409$$ 4733.68 0.572287 0.286144 0.958187i $$-0.407627\pi$$
0.286144 + 0.958187i $$0.407627\pi$$
$$410$$ 0 0
$$411$$ 5018.55 0.602304
$$412$$ − 909.911i − 0.108806i
$$413$$ − 2259.45i − 0.269202i
$$414$$ 564.910 0.0670624
$$415$$ 0 0
$$416$$ 2723.98 0.321044
$$417$$ − 7621.14i − 0.894985i
$$418$$ − 3085.74i − 0.361073i
$$419$$ 8117.57 0.946466 0.473233 0.880937i $$-0.343087\pi$$
0.473233 + 0.880937i $$0.343087\pi$$
$$420$$ 0 0
$$421$$ −9484.27 −1.09795 −0.548973 0.835840i $$-0.684981\pi$$
−0.548973 + 0.835840i $$0.684981\pi$$
$$422$$ 18271.8i 2.10772i
$$423$$ 4675.77i 0.537456i
$$424$$ 12748.5 1.46020
$$425$$ 0 0
$$426$$ −11543.5 −1.31288
$$427$$ − 1849.52i − 0.209612i
$$428$$ − 14970.8i − 1.69075i
$$429$$ −516.728 −0.0581536
$$430$$ 0 0
$$431$$ 9335.16 1.04329 0.521646 0.853162i $$-0.325318\pi$$
0.521646 + 0.853162i $$0.325318\pi$$
$$432$$ − 82.1157i − 0.00914535i
$$433$$ 2983.02i 0.331074i 0.986204 + 0.165537i $$0.0529357\pi$$
−0.986204 + 0.165537i $$0.947064\pi$$
$$434$$ 23044.5 2.54878
$$435$$ 0 0
$$436$$ −29200.0 −3.20739
$$437$$ − 833.988i − 0.0912931i
$$438$$ 4841.36i 0.528148i
$$439$$ 5232.32 0.568850 0.284425 0.958698i $$-0.408197\pi$$
0.284425 + 0.958698i $$0.408197\pi$$
$$440$$ 0 0
$$441$$ 746.472 0.0806038
$$442$$ − 5248.68i − 0.564828i
$$443$$ − 7517.71i − 0.806269i −0.915141 0.403135i $$-0.867921\pi$$
0.915141 0.403135i $$-0.132079\pi$$
$$444$$ −2572.78 −0.274997
$$445$$ 0 0
$$446$$ 4172.52 0.442992
$$447$$ − 9272.03i − 0.981101i
$$448$$ 16999.1i 1.79270i
$$449$$ −16070.9 −1.68916 −0.844581 0.535428i $$-0.820150\pi$$
−0.844581 + 0.535428i $$0.820150\pi$$
$$450$$ 0 0
$$451$$ −1199.96 −0.125285
$$452$$ − 22550.7i − 2.34667i
$$453$$ 4076.21i 0.422775i
$$454$$ 9475.29 0.979510
$$455$$ 0 0
$$456$$ −4302.67 −0.441867
$$457$$ 9718.51i 0.994776i 0.867528 + 0.497388i $$0.165707\pi$$
−0.867528 + 0.497388i $$0.834293\pi$$
$$458$$ − 18735.4i − 1.91146i
$$459$$ 1969.67 0.200297
$$460$$ 0 0
$$461$$ −14538.0 −1.46877 −0.734385 0.678733i $$-0.762529\pi$$
−0.734385 + 0.678733i $$0.762529\pi$$
$$462$$ − 3129.40i − 0.315136i
$$463$$ 9978.17i 1.00157i 0.865573 + 0.500783i $$0.166955\pi$$
−0.865573 + 0.500783i $$0.833045\pi$$
$$464$$ 95.6990 0.00957481
$$465$$ 0 0
$$466$$ 7732.03 0.768625
$$467$$ 15188.2i 1.50498i 0.658605 + 0.752489i $$0.271147\pi$$
−0.658605 + 0.752489i $$0.728853\pi$$
$$468$$ 1847.92i 0.182522i
$$469$$ −10080.7 −0.992503
$$470$$ 0 0
$$471$$ 3035.85 0.296995
$$472$$ 2571.88i 0.250806i
$$473$$ 1339.25i 0.130187i
$$474$$ −11457.8 −1.11028
$$475$$ 0 0
$$476$$ 19742.3 1.90102
$$477$$ − 4884.06i − 0.468817i
$$478$$ − 18494.1i − 1.76967i
$$479$$ −11330.8 −1.08083 −0.540415 0.841399i $$-0.681733\pi$$
−0.540415 + 0.841399i $$0.681733\pi$$
$$480$$ 0 0
$$481$$ 1024.09 0.0970779
$$482$$ − 12793.3i − 1.20896i
$$483$$ − 845.788i − 0.0796784i
$$484$$ −1586.64 −0.149008
$$485$$ 0 0
$$486$$ −1116.55 −0.104213
$$487$$ 19086.9i 1.77599i 0.459850 + 0.887997i $$0.347903\pi$$
−0.459850 + 0.887997i $$0.652097\pi$$
$$488$$ 2105.26i 0.195288i
$$489$$ −8449.11 −0.781353
$$490$$ 0 0
$$491$$ −8112.85 −0.745677 −0.372839 0.927896i $$-0.621616\pi$$
−0.372839 + 0.927896i $$0.621616\pi$$
$$492$$ 4291.27i 0.393222i
$$493$$ 2295.49i 0.209703i
$$494$$ 4392.53 0.400060
$$495$$ 0 0
$$496$$ −739.066 −0.0669053
$$497$$ 17283.0i 1.55986i
$$498$$ 19148.6i 1.72303i
$$499$$ −18329.1 −1.64433 −0.822167 0.569246i $$-0.807235\pi$$
−0.822167 + 0.569246i $$0.807235\pi$$
$$500$$ 0 0
$$501$$ −10346.7 −0.922666
$$502$$ − 8397.58i − 0.746618i
$$503$$ − 7739.57i − 0.686064i −0.939324 0.343032i $$-0.888546\pi$$
0.939324 0.343032i $$-0.111454\pi$$
$$504$$ −4363.55 −0.385651
$$505$$ 0 0
$$506$$ −690.446 −0.0606602
$$507$$ 5855.44i 0.512918i
$$508$$ 21001.7i 1.83425i
$$509$$ −15914.9 −1.38589 −0.692943 0.720993i $$-0.743686\pi$$
−0.692943 + 0.720993i $$0.743686\pi$$
$$510$$ 0 0
$$511$$ 7248.51 0.627505
$$512$$ − 1100.65i − 0.0950048i
$$513$$ 1648.38i 0.141867i
$$514$$ −2688.78 −0.230733
$$515$$ 0 0
$$516$$ 4789.40 0.408608
$$517$$ − 5714.83i − 0.486147i
$$518$$ 6202.07i 0.526068i
$$519$$ −6863.56 −0.580495
$$520$$ 0 0
$$521$$ 2274.50 0.191262 0.0956312 0.995417i $$-0.469513\pi$$
0.0956312 + 0.995417i $$0.469513\pi$$
$$522$$ − 1301.25i − 0.109107i
$$523$$ − 10971.1i − 0.917274i −0.888624 0.458637i $$-0.848338\pi$$
0.888624 0.458637i $$-0.151662\pi$$
$$524$$ −26279.6 −2.19089
$$525$$ 0 0
$$526$$ −1094.02 −0.0906877
$$527$$ − 17727.6i − 1.46533i
$$528$$ 100.364i 0.00827228i
$$529$$ 11980.4 0.984663
$$530$$ 0 0
$$531$$ 985.306 0.0805247
$$532$$ 16522.0i 1.34646i
$$533$$ − 1708.13i − 0.138813i
$$534$$ 21003.5 1.70208
$$535$$ 0 0
$$536$$ 11474.6 0.924680
$$537$$ − 9747.17i − 0.783280i
$$538$$ − 21221.2i − 1.70058i
$$539$$ −912.355 −0.0729089
$$540$$ 0 0
$$541$$ 5313.05 0.422229 0.211115 0.977461i $$-0.432291\pi$$
0.211115 + 0.977461i $$0.432291\pi$$
$$542$$ − 659.084i − 0.0522326i
$$543$$ 3511.34i 0.277506i
$$544$$ 12690.7 1.00020
$$545$$ 0 0
$$546$$ 4454.68 0.349162
$$547$$ 20685.1i 1.61688i 0.588581 + 0.808439i $$0.299687\pi$$
−0.588581 + 0.808439i $$0.700313\pi$$
$$548$$ 21935.6i 1.70993i
$$549$$ 806.541 0.0627000
$$550$$ 0 0
$$551$$ −1921.06 −0.148529
$$552$$ 962.739i 0.0742335i
$$553$$ 17154.7i 1.31915i
$$554$$ 39527.3 3.03132
$$555$$ 0 0
$$556$$ 33311.2 2.54085
$$557$$ − 10853.8i − 0.825659i −0.910808 0.412830i $$-0.864540\pi$$
0.910808 0.412830i $$-0.135460\pi$$
$$558$$ 10049.3i 0.762402i
$$559$$ −1906.41 −0.144244
$$560$$ 0 0
$$561$$ −2407.37 −0.181175
$$562$$ − 13751.5i − 1.03216i
$$563$$ 15381.2i 1.15141i 0.817658 + 0.575704i $$0.195272\pi$$
−0.817658 + 0.575704i $$0.804728\pi$$
$$564$$ −20437.3 −1.52583
$$565$$ 0 0
$$566$$ −31515.6 −2.34046
$$567$$ 1671.71i 0.123818i
$$568$$ − 19672.9i − 1.45327i
$$569$$ 1348.88 0.0993814 0.0496907 0.998765i $$-0.484176\pi$$
0.0496907 + 0.998765i $$0.484176\pi$$
$$570$$ 0 0
$$571$$ 3463.51 0.253841 0.126920 0.991913i $$-0.459491\pi$$
0.126920 + 0.991913i $$0.459491\pi$$
$$572$$ − 2258.57i − 0.165097i
$$573$$ − 8281.05i − 0.603745i
$$574$$ 10344.7 0.752231
$$575$$ 0 0
$$576$$ −7412.97 −0.536239
$$577$$ 12052.6i 0.869598i 0.900528 + 0.434799i $$0.143181\pi$$
−0.900528 + 0.434799i $$0.856819\pi$$
$$578$$ − 1878.42i − 0.135176i
$$579$$ −3751.83 −0.269293
$$580$$ 0 0
$$581$$ 28669.4 2.04717
$$582$$ 5880.66i 0.418834i
$$583$$ 5969.41i 0.424061i
$$584$$ −8250.80 −0.584624
$$585$$ 0 0
$$586$$ 18607.8 1.31174
$$587$$ − 11133.1i − 0.782813i −0.920218 0.391407i $$-0.871989\pi$$
0.920218 0.391407i $$-0.128011\pi$$
$$588$$ 3262.75i 0.228833i
$$589$$ 14836.0 1.03787
$$590$$ 0 0
$$591$$ 431.972 0.0300659
$$592$$ − 198.908i − 0.0138092i
$$593$$ 7939.69i 0.549821i 0.961470 + 0.274911i $$0.0886482\pi$$
−0.961470 + 0.274911i $$0.911352\pi$$
$$594$$ 1364.67 0.0942646
$$595$$ 0 0
$$596$$ 40527.1 2.78533
$$597$$ − 2283.75i − 0.156562i
$$598$$ − 982.846i − 0.0672100i
$$599$$ 19474.7 1.32840 0.664202 0.747553i $$-0.268771\pi$$
0.664202 + 0.747553i $$0.268771\pi$$
$$600$$ 0 0
$$601$$ −19946.1 −1.35377 −0.676887 0.736087i $$-0.736671\pi$$
−0.676887 + 0.736087i $$0.736671\pi$$
$$602$$ − 11545.6i − 0.781664i
$$603$$ − 4396.01i − 0.296881i
$$604$$ −17816.7 −1.20025
$$605$$ 0 0
$$606$$ −1021.97 −0.0685061
$$607$$ 1427.44i 0.0954496i 0.998861 + 0.0477248i $$0.0151970\pi$$
−0.998861 + 0.0477248i $$0.984803\pi$$
$$608$$ 10620.6i 0.708427i
$$609$$ −1948.23 −0.129633
$$610$$ 0 0
$$611$$ 8135.03 0.538638
$$612$$ 8609.23i 0.568640i
$$613$$ 8029.40i 0.529045i 0.964380 + 0.264522i $$0.0852143\pi$$
−0.964380 + 0.264522i $$0.914786\pi$$
$$614$$ 43985.1 2.89103
$$615$$ 0 0
$$616$$ 5333.22 0.348834
$$617$$ 20795.5i 1.35688i 0.734655 + 0.678440i $$0.237344\pi$$
−0.734655 + 0.678440i $$0.762656\pi$$
$$618$$ − 956.533i − 0.0622612i
$$619$$ −1677.43 −0.108920 −0.0544602 0.998516i $$-0.517344\pi$$
−0.0544602 + 0.998516i $$0.517344\pi$$
$$620$$ 0 0
$$621$$ 368.832 0.0238337
$$622$$ 24795.8i 1.59842i
$$623$$ − 31446.6i − 2.02228i
$$624$$ −142.867 −0.00916547
$$625$$ 0 0
$$626$$ 44823.0 2.86180
$$627$$ − 2014.69i − 0.128324i
$$628$$ 13269.4i 0.843165i
$$629$$ 4771.11 0.302443
$$630$$ 0 0
$$631$$ −25225.2 −1.59144 −0.795719 0.605666i $$-0.792907\pi$$
−0.795719 + 0.605666i $$0.792907\pi$$
$$632$$ − 19526.8i − 1.22901i
$$633$$ 11929.7i 0.749074i
$$634$$ 20004.8 1.25315
$$635$$ 0 0
$$636$$ 21347.7 1.33096
$$637$$ − 1298.73i − 0.0807812i
$$638$$ 1590.41i 0.0986912i
$$639$$ −7536.81 −0.466591
$$640$$ 0 0
$$641$$ 15165.3 0.934468 0.467234 0.884134i $$-0.345251\pi$$
0.467234 + 0.884134i $$0.345251\pi$$
$$642$$ − 15737.9i − 0.967486i
$$643$$ − 27156.1i − 1.66553i −0.553630 0.832763i $$-0.686758\pi$$
0.553630 0.832763i $$-0.313242\pi$$
$$644$$ 3696.85 0.226206
$$645$$ 0 0
$$646$$ 20464.3 1.24637
$$647$$ 29154.9i 1.77156i 0.464110 + 0.885778i $$0.346374\pi$$
−0.464110 + 0.885778i $$0.653626\pi$$
$$648$$ − 1902.86i − 0.115357i
$$649$$ −1204.26 −0.0728374
$$650$$ 0 0
$$651$$ 15045.9 0.905828
$$652$$ − 36930.2i − 2.21825i
$$653$$ 19141.7i 1.14713i 0.819161 + 0.573564i $$0.194440\pi$$
−0.819161 + 0.573564i $$0.805560\pi$$
$$654$$ −30696.1 −1.83534
$$655$$ 0 0
$$656$$ −331.768 −0.0197460
$$657$$ 3160.94i 0.187702i
$$658$$ 49267.2i 2.91890i
$$659$$ 24939.6 1.47422 0.737110 0.675773i $$-0.236190\pi$$
0.737110 + 0.675773i $$0.236190\pi$$
$$660$$ 0 0
$$661$$ 22617.7 1.33090 0.665452 0.746440i $$-0.268239\pi$$
0.665452 + 0.746440i $$0.268239\pi$$
$$662$$ 24755.4i 1.45339i
$$663$$ − 3426.88i − 0.200738i
$$664$$ −32633.7 −1.90728
$$665$$ 0 0
$$666$$ −2704.61 −0.157359
$$667$$ 429.843i 0.0249529i
$$668$$ − 45224.3i − 2.61943i
$$669$$ 2724.25 0.157438
$$670$$ 0 0
$$671$$ −985.772 −0.0567143
$$672$$ 10770.9i 0.618298i
$$673$$ 13855.8i 0.793615i 0.917902 + 0.396807i $$0.129882\pi$$
−0.917902 + 0.396807i $$0.870118\pi$$
$$674$$ 20678.1 1.18174
$$675$$ 0 0
$$676$$ −25593.5 −1.45616
$$677$$ − 24992.8i − 1.41884i −0.704787 0.709419i $$-0.748957\pi$$
0.704787 0.709419i $$-0.251043\pi$$
$$678$$ − 23706.1i − 1.34282i
$$679$$ 8804.57 0.497626
$$680$$ 0 0
$$681$$ 6186.46 0.348114
$$682$$ − 12282.5i − 0.689619i
$$683$$ 14420.5i 0.807887i 0.914784 + 0.403943i $$0.132361\pi$$
−0.914784 + 0.403943i $$0.867639\pi$$
$$684$$ −7204.93 −0.402759
$$685$$ 0 0
$$686$$ −24661.4 −1.37256
$$687$$ − 12232.4i − 0.679324i
$$688$$ 370.280i 0.0205186i
$$689$$ −8497.42 −0.469849
$$690$$ 0 0
$$691$$ 30552.4 1.68201 0.841005 0.541027i $$-0.181964\pi$$
0.841005 + 0.541027i $$0.181964\pi$$
$$692$$ − 29999.9i − 1.64801i
$$693$$ − 2043.20i − 0.111998i
$$694$$ −27138.7 −1.48440
$$695$$ 0 0
$$696$$ 2217.63 0.120774
$$697$$ − 7957.96i − 0.432467i
$$698$$ − 16708.1i − 0.906032i
$$699$$ 5048.27 0.273166
$$700$$ 0 0
$$701$$ −9151.47 −0.493076 −0.246538 0.969133i $$-0.579293\pi$$
−0.246538 + 0.969133i $$0.579293\pi$$
$$702$$ 1942.60i 0.104443i
$$703$$ 3992.86i 0.214216i
$$704$$ 9060.30 0.485047
$$705$$ 0 0
$$706$$ 966.793 0.0515378
$$707$$ 1530.10i 0.0813937i
$$708$$ 4306.67i 0.228608i
$$709$$ 6261.96 0.331697 0.165848 0.986151i $$-0.446964\pi$$
0.165848 + 0.986151i $$0.446964\pi$$
$$710$$ 0 0
$$711$$ −7480.85 −0.394590
$$712$$ 35794.9i 1.88409i
$$713$$ − 3319.60i − 0.174362i
$$714$$ 20753.8 1.08780
$$715$$ 0 0
$$716$$ 42603.9 2.22372
$$717$$ − 12074.9i − 0.628933i
$$718$$ − 11485.7i − 0.596995i
$$719$$ 18228.7 0.945500 0.472750 0.881197i $$-0.343261\pi$$
0.472750 + 0.881197i $$0.343261\pi$$
$$720$$ 0 0
$$721$$ −1432.13 −0.0739740
$$722$$ − 14389.9i − 0.741740i
$$723$$ − 8352.81i − 0.429660i
$$724$$ −15347.7 −0.787836
$$725$$ 0 0
$$726$$ −1667.93 −0.0852655
$$727$$ − 7233.66i − 0.369026i −0.982830 0.184513i $$-0.940929\pi$$
0.982830 0.184513i $$-0.0590707\pi$$
$$728$$ 7591.81i 0.386499i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −8881.73 −0.449388
$$732$$ 3525.31i 0.178004i
$$733$$ 13444.8i 0.677485i 0.940879 + 0.338743i $$0.110002\pi$$
−0.940879 + 0.338743i $$0.889998\pi$$
$$734$$ −26414.4 −1.32830
$$735$$ 0 0
$$736$$ 2376.41 0.119016
$$737$$ 5372.90i 0.268539i
$$738$$ 4511.14i 0.225010i
$$739$$ −18490.9 −0.920432 −0.460216 0.887807i $$-0.652228\pi$$
−0.460216 + 0.887807i $$0.652228\pi$$
$$740$$ 0 0
$$741$$ 2867.90 0.142180
$$742$$ − 51461.8i − 2.54612i
$$743$$ 25160.9i 1.24235i 0.783674 + 0.621173i $$0.213343\pi$$
−0.783674 + 0.621173i $$0.786657\pi$$
$$744$$ −17126.3 −0.843927
$$745$$ 0 0
$$746$$ 20528.8 1.00752
$$747$$ 12502.2i 0.612358i
$$748$$ − 10522.4i − 0.514354i
$$749$$ −23562.9 −1.14949
$$750$$ 0 0
$$751$$ −13419.5 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$752$$ − 1580.06i − 0.0766207i
$$753$$ − 5482.81i − 0.265345i
$$754$$ −2263.94 −0.109347
$$755$$ 0 0
$$756$$ −7306.86 −0.351518
$$757$$ − 7014.90i − 0.336804i −0.985718 0.168402i $$-0.946139\pi$$
0.985718 0.168402i $$-0.0538607\pi$$
$$758$$ − 35858.6i − 1.71826i
$$759$$ −450.795 −0.0215584
$$760$$ 0 0
$$761$$ −30156.9 −1.43651 −0.718256 0.695779i $$-0.755059\pi$$
−0.718256 + 0.695779i $$0.755059\pi$$
$$762$$ 22077.8i 1.04960i
$$763$$ 45958.4i 2.18061i
$$764$$ 36195.7 1.71402
$$765$$ 0 0
$$766$$ −51283.5 −2.41899
$$767$$ − 1714.26i − 0.0807019i
$$768$$ − 13217.4i − 0.621017i
$$769$$ −11292.2 −0.529530 −0.264765 0.964313i $$-0.585294\pi$$
−0.264765 + 0.964313i $$0.585294\pi$$
$$770$$ 0 0
$$771$$ −1755.51 −0.0820016
$$772$$ − 16398.9i − 0.764519i
$$773$$ − 8524.10i − 0.396624i −0.980139 0.198312i $$-0.936454\pi$$
0.980139 0.198312i $$-0.0635460\pi$$
$$774$$ 5034.80 0.233814
$$775$$ 0 0
$$776$$ −10022.0 −0.463621
$$777$$ 4049.35i 0.186962i
$$778$$ − 39011.4i − 1.79772i
$$779$$ 6659.89 0.306310
$$780$$ 0 0
$$781$$ 9211.66 0.422047
$$782$$ − 4578.96i − 0.209390i
$$783$$ − 849.589i − 0.0387763i
$$784$$ −252.251 −0.0114910
$$785$$ 0 0
$$786$$ −27626.1 −1.25368
$$787$$ − 14983.9i − 0.678676i −0.940665 0.339338i $$-0.889797\pi$$
0.940665 0.339338i $$-0.110203\pi$$
$$788$$ 1888.10i 0.0853565i
$$789$$ −714.293 −0.0322300
$$790$$ 0 0
$$791$$ −35493.0 −1.59543
$$792$$ 2325.72i 0.104345i
$$793$$ − 1403.24i − 0.0628380i
$$794$$ −27659.9 −1.23629
$$795$$ 0 0
$$796$$ 9982.04 0.444477
$$797$$ 37172.3i 1.65208i 0.563610 + 0.826041i $$0.309412\pi$$
−0.563610 + 0.826041i $$0.690588\pi$$
$$798$$ 17368.5i 0.770475i
$$799$$ 37900.1 1.67811
$$800$$ 0 0
$$801$$ 13713.3 0.604913
$$802$$ − 47781.0i − 2.10375i
$$803$$ − 3863.37i − 0.169783i
$$804$$ 19214.5 0.842841
$$805$$ 0 0
$$806$$ 17484.0 0.764080
$$807$$ − 13855.4i − 0.604378i
$$808$$ − 1741.68i − 0.0758316i
$$809$$ −23797.1 −1.03419 −0.517096 0.855928i $$-0.672987\pi$$
−0.517096 + 0.855928i $$0.672987\pi$$
$$810$$ 0 0
$$811$$ 8988.35 0.389178 0.194589 0.980885i $$-0.437663\pi$$
0.194589 + 0.980885i $$0.437663\pi$$
$$812$$ − 8515.54i − 0.368026i
$$813$$ − 430.318i − 0.0185633i
$$814$$ 3305.63 0.142337
$$815$$ 0 0
$$816$$ −665.600 −0.0285547
$$817$$ − 7432.98i − 0.318295i
$$818$$ 21750.6i 0.929696i
$$819$$ 2908.48 0.124091
$$820$$ 0 0
$$821$$ −25156.8 −1.06940 −0.534702 0.845041i $$-0.679576\pi$$
−0.534702 + 0.845041i $$0.679576\pi$$
$$822$$ 23059.5i 0.978459i
$$823$$ − 1318.51i − 0.0558447i −0.999610 0.0279224i $$-0.991111\pi$$
0.999610 0.0279224i $$-0.00888912\pi$$
$$824$$ 1630.16 0.0689189
$$825$$ 0 0
$$826$$ 10381.9 0.437326
$$827$$ 124.982i 0.00525519i 0.999997 + 0.00262760i $$0.000836391\pi$$
−0.999997 + 0.00262760i $$0.999164\pi$$
$$828$$ 1612.13i 0.0676635i
$$829$$ 8886.80 0.372318 0.186159 0.982520i $$-0.440396\pi$$
0.186159 + 0.982520i $$0.440396\pi$$
$$830$$ 0 0
$$831$$ 25807.5 1.07732
$$832$$ 12897.3i 0.537419i
$$833$$ − 6050.63i − 0.251671i
$$834$$ 35018.0 1.45393
$$835$$ 0 0
$$836$$ 8806.02 0.364309
$$837$$ 6561.22i 0.270955i
$$838$$ 37299.1i 1.53756i
$$839$$ −2995.21 −0.123249 −0.0616247 0.998099i $$-0.519628\pi$$
−0.0616247 + 0.998099i $$0.519628\pi$$
$$840$$ 0 0
$$841$$ −23398.9 −0.959403
$$842$$ − 43578.8i − 1.78364i
$$843$$ − 8978.43i − 0.366825i
$$844$$ −52143.6 −2.12661
$$845$$ 0 0
$$846$$ −21484.5 −0.873111
$$847$$ 2497.24i 0.101306i
$$848$$ 1650.44i 0.0668355i
$$849$$ −20576.7 −0.831789
$$850$$ 0 0
$$851$$ 893.418 0.0359882
$$852$$ − 32942.7i − 1.32464i
$$853$$ − 18130.5i − 0.727757i −0.931446 0.363878i $$-0.881452\pi$$
0.931446 0.363878i $$-0.118548\pi$$
$$854$$ 8498.27 0.340521
$$855$$ 0 0
$$856$$ 26821.1 1.07094
$$857$$ 26394.1i 1.05205i 0.850470 + 0.526024i $$0.176318\pi$$
−0.850470 + 0.526024i $$0.823682\pi$$
$$858$$ − 2374.29i − 0.0944720i
$$859$$ 29456.2 1.17000 0.585002 0.811032i $$-0.301094\pi$$
0.585002 + 0.811032i $$0.301094\pi$$
$$860$$ 0 0
$$861$$ 6754.12 0.267340
$$862$$ 42893.7i 1.69486i
$$863$$ 762.616i 0.0300808i 0.999887 + 0.0150404i $$0.00478769\pi$$
−0.999887 + 0.0150404i $$0.995212\pi$$
$$864$$ −4696.99 −0.184948
$$865$$ 0 0
$$866$$ −13706.6 −0.537838
$$867$$ − 1226.43i − 0.0480411i
$$868$$ 65764.0i 2.57163i
$$869$$ 9143.26 0.356920
$$870$$ 0 0
$$871$$ −7648.29 −0.297535
$$872$$ − 52313.3i − 2.03160i
$$873$$ 3839.51i 0.148852i
$$874$$ 3832.06 0.148308
$$875$$ 0 0
$$876$$ −13816.2 −0.532882
$$877$$ 44767.2i 1.72369i 0.507168 + 0.861847i $$0.330692\pi$$
−0.507168 + 0.861847i $$0.669308\pi$$
$$878$$ 24041.8i 0.924112i
$$879$$ 12149.1 0.466188
$$880$$ 0 0
$$881$$ −32057.9 −1.22595 −0.612973 0.790104i $$-0.710027\pi$$
−0.612973 + 0.790104i $$0.710027\pi$$
$$882$$ 3429.93i 0.130943i
$$883$$ − 7078.95i − 0.269791i −0.990860 0.134896i $$-0.956930\pi$$
0.990860 0.134896i $$-0.0430699\pi$$
$$884$$ 14978.6 0.569891
$$885$$ 0 0
$$886$$ 34542.8 1.30981
$$887$$ − 25148.1i − 0.951964i −0.879455 0.475982i $$-0.842093\pi$$
0.879455 0.475982i $$-0.157907\pi$$
$$888$$ − 4609.28i − 0.174186i
$$889$$ 33055.0 1.24705
$$890$$ 0 0
$$891$$ 891.000 0.0335013
$$892$$ 11907.4i 0.446963i
$$893$$ 31718.0i 1.18858i
$$894$$ 42603.6 1.59382
$$895$$ 0 0
$$896$$ −49385.8 −1.84137
$$897$$ − 641.704i − 0.0238862i
$$898$$ − 73843.6i − 2.74409i
$$899$$ −7646.56 −0.283679
$$900$$ 0 0
$$901$$ −39588.4 −1.46380
$$902$$ − 5513.62i − 0.203529i
$$903$$ − 7538.14i − 0.277800i
$$904$$ 40400.8 1.48641
$$905$$ 0 0
$$906$$ −18729.6 −0.686809
$$907$$ 1269.76i 0.0464848i 0.999730 + 0.0232424i $$0.00739895\pi$$
−0.999730 + 0.0232424i $$0.992601\pi$$
$$908$$ 27040.4i 0.988289i
$$909$$ −667.248 −0.0243468
$$910$$ 0 0
$$911$$ −33783.1 −1.22863 −0.614316 0.789060i $$-0.710568\pi$$
−0.614316 + 0.789060i $$0.710568\pi$$
$$912$$ − 557.030i − 0.0202249i
$$913$$ − 15280.5i − 0.553899i
$$914$$ −44655.1 −1.61604
$$915$$ 0 0
$$916$$ 53466.6 1.92859
$$917$$ 41361.9i 1.48952i
$$918$$ 9050.35i 0.325388i
$$919$$ −39262.5 −1.40930 −0.704652 0.709553i $$-0.748897\pi$$
−0.704652 + 0.709553i $$0.748897\pi$$
$$920$$ 0 0
$$921$$ 28718.1 1.02746
$$922$$ − 66800.1i − 2.38606i
$$923$$ 13112.7i 0.467618i
$$924$$ 8930.61 0.317960
$$925$$ 0 0
$$926$$ −45848.3 −1.62707
$$927$$ − 624.525i − 0.0221274i
$$928$$ − 5473.95i − 0.193633i
$$929$$ 21175.0 0.747825 0.373913 0.927464i $$-0.378016\pi$$
0.373913 + 0.927464i $$0.378016\pi$$
$$930$$ 0 0
$$931$$ 5063.68 0.178255
$$932$$ 22065.5i 0.775515i
$$933$$ 16189.3i 0.568073i
$$934$$ −69787.5 −2.44488
$$935$$ 0 0
$$936$$ −3310.65 −0.115611
$$937$$ − 5135.11i − 0.179036i −0.995985 0.0895180i $$-0.971467\pi$$
0.995985 0.0895180i $$-0.0285327\pi$$
$$938$$ − 46319.4i − 1.61235i
$$939$$ 29265.1 1.01707
$$940$$ 0 0
$$941$$ −9702.77 −0.336133 −0.168067 0.985776i $$-0.553752\pi$$
−0.168067 + 0.985776i $$0.553752\pi$$
$$942$$ 13949.3i 0.482477i
$$943$$ − 1490.18i − 0.0514600i
$$944$$ −332.959 −0.0114798
$$945$$ 0 0
$$946$$ −6153.65 −0.211493
$$947$$ − 699.579i − 0.0240055i −0.999928 0.0120028i $$-0.996179\pi$$
0.999928 0.0120028i $$-0.00382069\pi$$
$$948$$ − 32698.0i − 1.12024i
$$949$$ 5499.49 0.188115
$$950$$ 0 0
$$951$$ 13061.2 0.445363
$$952$$ 35369.3i 1.20412i
$$953$$ − 42039.3i − 1.42895i −0.699663 0.714473i $$-0.746667\pi$$
0.699663 0.714473i $$-0.253333\pi$$
$$954$$ 22441.6 0.761606
$$955$$ 0 0
$$956$$ 52778.1 1.78553
$$957$$ 1038.39i 0.0350745i
$$958$$ − 52063.4i − 1.75584i
$$959$$ 34524.9 1.16253
$$960$$ 0 0
$$961$$ 29262.0 0.982243
$$962$$ 4705.55i 0.157706i
$$963$$ − 10275.3i − 0.343841i
$$964$$ 36509.3 1.21980
$$965$$ 0 0
$$966$$ 3886.27 0.129440
$$967$$ − 32794.8i − 1.09060i −0.838242 0.545299i $$-0.816416\pi$$
0.838242 0.545299i $$-0.183584\pi$$
$$968$$ − 2842.55i − 0.0943832i
$$969$$ 13361.2 0.442955
$$970$$ 0 0
$$971$$ −3322.53 −0.109810 −0.0549048 0.998492i $$-0.517486\pi$$
−0.0549048 + 0.998492i $$0.517486\pi$$
$$972$$ − 3186.39i − 0.105148i
$$973$$ − 52429.3i − 1.72745i
$$974$$ −87701.4 −2.88515
$$975$$ 0 0
$$976$$ −272.550 −0.00893864
$$977$$ 22192.5i 0.726716i 0.931650 + 0.363358i $$0.118370\pi$$
−0.931650 + 0.363358i $$0.881630\pi$$
$$978$$ − 38822.4i − 1.26933i
$$979$$ −16760.7 −0.547164
$$980$$ 0 0
$$981$$ −20041.6 −0.652272
$$982$$ − 37277.4i − 1.21137i
$$983$$ − 7383.09i − 0.239556i −0.992801 0.119778i $$-0.961782\pi$$
0.992801 0.119778i $$-0.0382183\pi$$
$$984$$ −7688.04 −0.249071
$$985$$ 0 0
$$986$$ −10547.4 −0.340668
$$987$$ 32166.7i 1.03736i
$$988$$ 12535.3i 0.403645i
$$989$$ −1663.16 −0.0534735
$$990$$ 0 0
$$991$$ 46260.8 1.48287 0.741434 0.671026i $$-0.234146\pi$$
0.741434 + 0.671026i $$0.234146\pi$$
$$992$$ 42274.3i 1.35304i
$$993$$ 16162.9i 0.516530i
$$994$$ −79413.1 −2.53403
$$995$$ 0 0
$$996$$ −54645.9 −1.73847
$$997$$ 41196.8i 1.30864i 0.756217 + 0.654320i $$0.227045\pi$$
−0.756217 + 0.654320i $$0.772955\pi$$
$$998$$ − 84219.5i − 2.67127i
$$999$$ −1765.85 −0.0559249
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.m.199.6 6
5.2 odd 4 825.4.a.p.1.1 3
5.3 odd 4 165.4.a.g.1.3 3
5.4 even 2 inner 825.4.c.m.199.1 6
15.2 even 4 2475.4.a.z.1.3 3
15.8 even 4 495.4.a.i.1.1 3
55.43 even 4 1815.4.a.q.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.g.1.3 3 5.3 odd 4
495.4.a.i.1.1 3 15.8 even 4
825.4.a.p.1.1 3 5.2 odd 4
825.4.c.m.199.1 6 5.4 even 2 inner
825.4.c.m.199.6 6 1.1 even 1 trivial
1815.4.a.q.1.1 3 55.43 even 4
2475.4.a.z.1.3 3 15.2 even 4