# Properties

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

## $q$-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 $$-0.698235\pi$$
0.583291 0.812263i $$-0.301765\pi$$
$$3$$ − 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 $$-0.811882\pi$$
0.830390 0.557183i $$-0.188118\pi$$
$$8$$ 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 $$-0.946581\pi$$
0.985951 0.167033i $$-0.0534188\pi$$
$$14$$ −94.8302 −1.81032
$$15$$ 0 0
$$16$$ 3.04132 0.0475206
$$17$$ − 72.9507i − 1.04077i −0.853931 0.520387i $$-0.825788\pi$$
0.853931 0.520387i $$-0.174212\pi$$
$$18$$ 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 $$-0.980277\pi$$
0.998081 0.0619218i $$-0.0197229\pi$$
$$24$$ 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 $$0.0464138\pi$$
−0.989388 + 0.145297i $$0.953586\pi$$
$$38$$ 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 $$-0.930734\pi$$
0.976417 0.215891i $$-0.0692657\pi$$
$$44$$ −144.240 −0.494204
$$45$$ 0 0
$$46$$ −62.7678 −0.201187
$$47$$ 519.530i 1.61237i 0.591665 + 0.806184i $$0.298471\pi$$
−0.591665 + 0.806184i $$0.701529\pi$$
$$48$$ − 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 $$-0.751742\pi$$
0.710967 0.703226i $$-0.248258\pi$$
$$54$$ 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 $$-0.853089\pi$$
0.895371 0.445322i $$-0.146911\pi$$
$$68$$ 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 $$0.0908494\pi$$
−0.959546 + 0.281553i $$0.909151\pi$$
$$74$$ 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 $$0.370629\pi$$
−0.395335 + 0.918537i $$0.629371\pi$$
$$84$$ 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 $$0.0716757\pi$$
−0.974755 + 0.223278i $$0.928324\pi$$
$$98$$ 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 $$-0.989433\pi$$
0.999449 0.0331911i $$-0.0105670\pi$$
$$104$$ 367.850 0.346833
$$105$$ 0 0
$$106$$ −2493.51 −2.28482
$$107$$ − 1141.71i − 1.03152i −0.856733 0.515761i $$-0.827509\pi$$
0.856733 0.515761i $$-0.172491\pi$$
$$108$$ − 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 $$-0.746041\pi$$
0.698257 0.715847i $$-0.253959\pi$$
$$114$$ 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 $$0.189020\pi$$
−0.828807 + 0.559534i $$0.810980\pi$$
$$128$$ − 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 $$0.174669\pi$$
−0.853184 + 0.521610i $$0.825331\pi$$
$$138$$ 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 $$0.0828017\pi$$
−0.966357 + 0.257205i $$0.917198\pi$$
$$158$$ − 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 $$-0.763422\pi$$
0.736285 0.676672i $$-0.236578\pi$$
$$164$$ 1430.42 0.681081
$$165$$ 0 0
$$166$$ 6382.87 2.98438
$$167$$ − 3448.89i − 1.59810i −0.601262 0.799052i $$-0.705335\pi$$
0.601262 0.799052i $$-0.294665\pi$$
$$168$$ − 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 $$-0.832332\pi$$
0.864448 0.502723i $$-0.167668\pi$$
$$174$$ −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 $$-0.925075\pi$$
0.972425 0.233215i $$-0.0749245\pi$$
$$194$$ 1960.22 0.725441
$$195$$ 0 0
$$196$$ 1087.58 0.396350
$$197$$ 143.991i 0.0520756i 0.999661 + 0.0260378i $$0.00828903\pi$$
−0.999661 + 0.0260378i $$0.991711\pi$$
$$198$$ 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 $$0.0435355\pi$$
−0.990661 + 0.136345i $$0.956464\pi$$
$$224$$ 3590.30 1.07092
$$225$$ 0 0
$$226$$ −7902.05 −2.32583
$$227$$ 2062.15i 0.602951i 0.953474 + 0.301475i $$0.0974791\pi$$
−0.953474 + 0.301475i $$0.902521\pi$$
$$228$$ − 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 $$0.0760229\pi$$
−0.971615 + 0.236569i $$0.923977\pi$$
$$234$$ 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 $$-0.977376\pi$$
0.997475 0.0710155i $$-0.0226240\pi$$
$$258$$ 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 $$-0.991114\pi$$
0.999610 0.0279120i $$-0.00888583\pi$$
$$264$$ 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 $$0.382807\pi$$
−0.359911 + 0.932987i $$0.617193\pi$$
$$278$$ 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 $$-0.743981\pi$$
0.693610 0.720351i $$-0.256019\pi$$
$$284$$ −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 $$0.132287\pi$$
−0.914878 + 0.403731i $$0.867713\pi$$
$$294$$ 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 $$0.349162\pi$$
−0.456335 + 0.889808i $$0.650838\pi$$
$$308$$ 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 $$0.343002\pi$$
−0.473469 + 0.880811i $$0.656998\pi$$
$$314$$ 4649.77 0.835674
$$315$$ 0 0
$$316$$ −10899.3 −1.94030
$$317$$ 4353.75i 0.771391i 0.922626 + 0.385695i $$0.126038\pi$$
−0.922626 + 0.385695i $$0.873962\pi$$
$$318$$ 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 $$0.118492\pi$$
−0.931509 + 0.363717i $$0.881508\pi$$
$$338$$ − 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 $$-0.848970\pi$$
0.889533 0.456870i $$-0.151030\pi$$
$$348$$ 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 $$0.00504938\pi$$
−0.999874 + 0.0158624i $$0.994951\pi$$
$$354$$ 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 $$-0.865938\pi$$
0.912612 0.408828i $$-0.134062\pi$$
$$368$$ − 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 $$0.100362\pi$$
−0.950705 + 0.310098i $$0.899638\pi$$
$$374$$ −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 $$-0.732677\pi$$
0.667597 0.744522i $$-0.267323\pi$$
$$384$$ −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 $$-0.875750\pi$$
0.924778 0.380507i $$-0.124250\pi$$
$$398$$ 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 $$-0.947064\pi$$
0.986204 0.165537i $$-0.0529357\pi$$
$$434$$ 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 $$0.132079\pi$$
−0.915141 + 0.403135i $$0.867921\pi$$
$$444$$ −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 $$-0.834293\pi$$
0.867528 0.497388i $$-0.165707\pi$$
$$458$$ 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 $$-0.833045\pi$$
0.865573 0.500783i $$-0.166955\pi$$
$$464$$ 95.6990 0.00957481
$$465$$ 0 0
$$466$$ 7732.03 0.768625
$$467$$ − 15188.2i − 1.50498i −0.658605 0.752489i $$-0.728853\pi$$
0.658605 0.752489i $$-0.271147\pi$$
$$468$$ − 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 $$-0.652097\pi$$
0.459850 0.887997i $$-0.347903\pi$$
$$488$$ − 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 $$0.111454\pi$$
−0.939324 + 0.343032i $$0.888546\pi$$
$$504$$ −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 $$0.151662\pi$$
−0.888624 + 0.458637i $$0.848338\pi$$
$$524$$ −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 $$-0.700313\pi$$
0.588581 0.808439i $$-0.299687\pi$$
$$548$$ − 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 $$0.135460\pi$$
−0.910808 + 0.412830i $$0.864540\pi$$
$$558$$ − 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 $$-0.804728\pi$$
0.817658 0.575704i $$-0.195272\pi$$
$$564$$ −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 $$-0.856819\pi$$
0.900528 0.434799i $$-0.143181\pi$$
$$578$$ 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 $$0.128011\pi$$
−0.920218 + 0.391407i $$0.871989\pi$$
$$588$$ − 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 $$-0.911352\pi$$
0.961470 0.274911i $$-0.0886482\pi$$
$$594$$ 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 $$-0.984803\pi$$
0.998861 0.0477248i $$-0.0151970\pi$$
$$608$$ − 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 $$-0.914786\pi$$
0.964380 0.264522i $$-0.0852143\pi$$
$$614$$ 43985.1 2.89103
$$615$$ 0 0
$$616$$ 5333.22 0.348834
$$617$$ − 20795.5i − 1.35688i −0.734655 0.678440i $$-0.762656\pi$$
0.734655 0.678440i $$-0.237344\pi$$
$$618$$ 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 $$0.313242\pi$$
−0.553630 + 0.832763i $$0.686758\pi$$
$$644$$ 3696.85 0.226206
$$645$$ 0 0
$$646$$ 20464.3 1.24637
$$647$$ − 29154.9i − 1.77156i −0.464110 0.885778i $$-0.653626\pi$$
0.464110 0.885778i $$-0.346374\pi$$
$$648$$ 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 $$-0.805560\pi$$
0.819161 0.573564i $$-0.194440\pi$$
$$654$$ −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 $$-0.870118\pi$$
0.917902 0.396807i $$-0.129882\pi$$
$$674$$ 20678.1 1.18174
$$675$$ 0 0
$$676$$ −25593.5 −1.45616
$$677$$ 24992.8i 1.41884i 0.704787 + 0.709419i $$0.251043\pi$$
−0.704787 + 0.709419i $$0.748957\pi$$
$$678$$ 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 $$-0.867639\pi$$
0.914784 0.403943i $$-0.132361\pi$$
$$684$$ −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 $$0.0590707\pi$$
−0.982830 + 0.184513i $$0.940929\pi$$
$$728$$ − 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 $$-0.889998\pi$$
0.940879 0.338743i $$-0.110002\pi$$
$$734$$ −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 $$-0.786657\pi$$
0.783674 0.621173i $$-0.213343\pi$$
$$744$$ −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 $$0.0538607\pi$$
−0.985718 + 0.168402i $$0.946139\pi$$
$$758$$ 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 $$0.0635460\pi$$
−0.980139 + 0.198312i $$0.936454\pi$$
$$774$$ 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 $$0.110203\pi$$
−0.940665 + 0.339338i $$0.889797\pi$$
$$788$$ − 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 $$-0.690588\pi$$
0.563610 0.826041i $$-0.309412\pi$$
$$798$$ − 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 $$0.00888912\pi$$
−0.999610 + 0.0279224i $$0.991111\pi$$
$$824$$ 1630.16 0.0689189
$$825$$ 0 0
$$826$$ 10381.9 0.437326
$$827$$ − 124.982i − 0.00525519i −0.999997 0.00262760i $$-0.999164\pi$$
0.999997 0.00262760i $$-0.000836391\pi$$
$$828$$ − 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 $$0.118548\pi$$
−0.931446 + 0.363878i $$0.881452\pi$$
$$854$$ 8498.27 0.340521
$$855$$ 0 0
$$856$$ 26821.1 1.07094
$$857$$ − 26394.1i − 1.05205i −0.850470 0.526024i $$-0.823682\pi$$
0.850470 0.526024i $$-0.176318\pi$$
$$858$$ 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 $$-0.995212\pi$$
0.999887 0.0150404i $$-0.00478769\pi$$
$$864$$ −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 $$-0.669308\pi$$
0.507168 0.861847i $$-0.330692\pi$$
$$878$$ − 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 $$0.0430699\pi$$
−0.990860 + 0.134896i $$0.956930\pi$$
$$884$$ 14978.6 0.569891
$$885$$ 0 0
$$886$$ 34542.8 1.30981
$$887$$ 25148.1i 0.951964i 0.879455 + 0.475982i $$0.157907\pi$$
−0.879455 + 0.475982i $$0.842093\pi$$
$$888$$ 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 $$-0.992601\pi$$
0.999730 0.0232424i $$-0.00739895\pi$$
$$908$$ − 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 $$0.0285327\pi$$
−0.995985 + 0.0895180i $$0.971467\pi$$
$$938$$ 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 $$0.00382069\pi$$
−0.999928 + 0.0120028i $$0.996179\pi$$
$$948$$ 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 $$0.253333\pi$$
−0.699663 + 0.714473i $$0.746667\pi$$
$$954$$ 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 $$0.183584\pi$$
−0.838242 + 0.545299i $$0.816416\pi$$
$$968$$ 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 $$-0.881630\pi$$
0.931650 0.363358i $$-0.118370\pi$$
$$978$$ 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 $$0.0382183\pi$$
−0.992801 + 0.119778i $$0.961782\pi$$
$$984$$ −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 $$-0.772955\pi$$
0.756217 0.654320i $$-0.227045\pi$$
$$998$$ 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.199.1 6
5.2 odd 4 165.4.a.g.1.3 3
5.3 odd 4 825.4.a.p.1.1 3
5.4 even 2 inner 825.4.c.m.199.6 6
15.2 even 4 495.4.a.i.1.1 3
15.8 even 4 2475.4.a.z.1.3 3
55.32 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 5.2 odd 4
495.4.a.i.1.1 3 15.2 even 4
825.4.a.p.1.1 3 5.3 odd 4
825.4.c.m.199.1 6 1.1 even 1 trivial
825.4.c.m.199.6 6 5.4 even 2 inner
1815.4.a.q.1.1 3 55.32 even 4
2475.4.a.z.1.3 3 15.8 even 4