# Properties

 Label 2475.4.a.z.1.3 Level $2475$ Weight $4$ Character 2475.1 Self dual yes Analytic conductor $146.030$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$146.029727264$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1957.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 9x + 10$$ x^3 - x^2 - 9*x + 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.12946$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.59486 q^{2} +13.1127 q^{4} -20.6383 q^{7} +23.4921 q^{8} +O(q^{10})$$ $$q+4.59486 q^{2} +13.1127 q^{4} -20.6383 q^{7} +23.4921 q^{8} -11.0000 q^{11} +15.6584 q^{13} -94.8302 q^{14} +3.04132 q^{16} +72.9507 q^{17} +61.0513 q^{19} -50.5434 q^{22} -13.6605 q^{23} +71.9483 q^{26} -270.624 q^{28} +31.4663 q^{29} -243.008 q^{31} -173.963 q^{32} +335.198 q^{34} +65.4018 q^{37} +280.522 q^{38} +109.087 q^{41} +121.750 q^{43} -144.240 q^{44} -62.7678 q^{46} -519.530 q^{47} +82.9413 q^{49} +205.324 q^{52} -542.673 q^{53} -484.839 q^{56} +144.583 q^{58} -109.478 q^{59} -89.6156 q^{61} -1116.59 q^{62} -823.664 q^{64} -488.446 q^{67} +956.581 q^{68} -837.423 q^{71} -351.216 q^{73} +300.512 q^{74} +800.547 q^{76} +227.022 q^{77} -831.205 q^{79} +501.238 q^{82} +1389.13 q^{83} +559.423 q^{86} -258.413 q^{88} -1523.70 q^{89} -323.164 q^{91} -179.125 q^{92} -2387.17 q^{94} +426.612 q^{97} +381.103 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 17 q^{4} - 6 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q + q^2 + 17 * q^4 - 6 * q^7 - 3 * q^8 $$3 q + q^{2} + 17 q^{4} - 6 q^{7} - 3 q^{8} - 33 q^{11} + 20 q^{13} - 144 q^{14} + 25 q^{16} + 32 q^{17} + 116 q^{19} - 11 q^{22} + 240 q^{23} - 302 q^{26} - 160 q^{28} - 238 q^{29} + 92 q^{31} + 197 q^{32} + 354 q^{34} + 90 q^{37} + 324 q^{38} + 46 q^{41} + 134 q^{43} - 187 q^{44} - 240 q^{46} - 220 q^{47} - 457 q^{49} + 1530 q^{52} - 798 q^{53} - 688 q^{56} + 978 q^{58} - 1236 q^{59} + 342 q^{61} - 1792 q^{62} - 1919 q^{64} - 764 q^{67} + 1074 q^{68} - 1816 q^{71} - 100 q^{73} + 1874 q^{74} + 396 q^{76} + 66 q^{77} - 96 q^{79} + 910 q^{82} + 858 q^{83} - 188 q^{86} + 33 q^{88} - 838 q^{89} + 332 q^{91} - 688 q^{92} - 3112 q^{94} + 1322 q^{97} + 1017 q^{98}+O(q^{100})$$ 3 * q + q^2 + 17 * q^4 - 6 * q^7 - 3 * q^8 - 33 * q^11 + 20 * q^13 - 144 * q^14 + 25 * q^16 + 32 * q^17 + 116 * q^19 - 11 * q^22 + 240 * q^23 - 302 * q^26 - 160 * q^28 - 238 * q^29 + 92 * q^31 + 197 * q^32 + 354 * q^34 + 90 * q^37 + 324 * q^38 + 46 * q^41 + 134 * q^43 - 187 * q^44 - 240 * q^46 - 220 * q^47 - 457 * q^49 + 1530 * q^52 - 798 * q^53 - 688 * q^56 + 978 * q^58 - 1236 * q^59 + 342 * q^61 - 1792 * q^62 - 1919 * q^64 - 764 * q^67 + 1074 * q^68 - 1816 * q^71 - 100 * q^73 + 1874 * q^74 + 396 * q^76 + 66 * q^77 - 96 * q^79 + 910 * q^82 + 858 * q^83 - 188 * q^86 + 33 * q^88 - 838 * q^89 + 332 * q^91 - 688 * q^92 - 3112 * q^94 + 1322 * q^97 + 1017 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.59486 1.62453 0.812263 0.583291i $$-0.198235\pi$$
0.812263 + 0.583291i $$0.198235\pi$$
$$3$$ 0 0
$$4$$ 13.1127 1.63909
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −20.6383 −1.11437 −0.557183 0.830390i $$-0.688118\pi$$
−0.557183 + 0.830390i $$0.688118\pi$$
$$8$$ 23.4921 1.03822
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ 15.6584 0.334067 0.167033 0.985951i $$-0.446581\pi$$
0.167033 + 0.985951i $$0.446581\pi$$
$$14$$ −94.8302 −1.81032
$$15$$ 0 0
$$16$$ 3.04132 0.0475206
$$17$$ 72.9507 1.04077 0.520387 0.853931i $$-0.325788\pi$$
0.520387 + 0.853931i $$0.325788\pi$$
$$18$$ 0 0
$$19$$ 61.0513 0.737165 0.368582 0.929595i $$-0.379843\pi$$
0.368582 + 0.929595i $$0.379843\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −50.5434 −0.489813
$$23$$ −13.6605 −0.123844 −0.0619218 0.998081i $$-0.519723\pi$$
−0.0619218 + 0.998081i $$0.519723\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 71.9483 0.542701
$$27$$ 0 0
$$28$$ −270.624 −1.82654
$$29$$ 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 $$0.748587\pi$$
$$32$$ −173.963 −0.961016
$$33$$ 0 0
$$34$$ 335.198 1.69076
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 65.4018 0.290594 0.145297 0.989388i $$-0.453586\pi$$
0.145297 + 0.989388i $$0.453586\pi$$
$$38$$ 280.522 1.19754
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 109.087 0.415524 0.207762 0.978179i $$-0.433382\pi$$
0.207762 + 0.978179i $$0.433382\pi$$
$$42$$ 0 0
$$43$$ 121.750 0.431783 0.215891 0.976417i $$-0.430734\pi$$
0.215891 + 0.976417i $$0.430734\pi$$
$$44$$ −144.240 −0.494204
$$45$$ 0 0
$$46$$ −62.7678 −0.201187
$$47$$ −519.530 −1.61237 −0.806184 0.591665i $$-0.798471\pi$$
−0.806184 + 0.591665i $$0.798471\pi$$
$$48$$ 0 0
$$49$$ 82.9413 0.241811
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 205.324 0.547565
$$53$$ −542.673 −1.40645 −0.703226 0.710967i $$-0.748258\pi$$
−0.703226 + 0.710967i $$0.748258\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −484.839 −1.15695
$$57$$ 0 0
$$58$$ 144.583 0.327322
$$59$$ −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 $$0.529981\pi$$
$$62$$ −1116.59 −2.28721
$$63$$ 0 0
$$64$$ −823.664 −1.60872
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −488.446 −0.890644 −0.445322 0.895371i $$-0.646911\pi$$
−0.445322 + 0.895371i $$0.646911\pi$$
$$68$$ 956.581 1.70592
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −837.423 −1.39977 −0.699887 0.714254i $$-0.746766\pi$$
−0.699887 + 0.714254i $$0.746766\pi$$
$$72$$ 0 0
$$73$$ −351.216 −0.563105 −0.281553 0.959546i $$-0.590849\pi$$
−0.281553 + 0.959546i $$0.590849\pi$$
$$74$$ 300.512 0.472078
$$75$$ 0 0
$$76$$ 800.547 1.20828
$$77$$ 227.022 0.335994
$$78$$ 0 0
$$79$$ −831.205 −1.18377 −0.591885 0.806022i $$-0.701616\pi$$
−0.591885 + 0.806022i $$0.701616\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 501.238 0.675031
$$83$$ 1389.13 1.83707 0.918537 0.395335i $$-0.129371\pi$$
0.918537 + 0.395335i $$0.129371\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 559.423 0.701443
$$87$$ 0 0
$$88$$ −258.413 −0.313034
$$89$$ −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
$$92$$ −179.125 −0.202990
$$93$$ 0 0
$$94$$ −2387.17 −2.61933
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 426.612 0.446555 0.223278 0.974755i $$-0.428324\pi$$
0.223278 + 0.974755i $$0.428324\pi$$
$$98$$ 381.103 0.392829
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −74.1387 −0.0730403 −0.0365202 0.999333i $$-0.511627\pi$$
−0.0365202 + 0.999333i $$0.511627\pi$$
$$102$$ 0 0
$$103$$ 69.3916 0.0663821 0.0331911 0.999449i $$-0.489433\pi$$
0.0331911 + 0.999449i $$0.489433\pi$$
$$104$$ 367.850 0.346833
$$105$$ 0 0
$$106$$ −2493.51 −2.28482
$$107$$ 1141.71 1.03152 0.515761 0.856733i $$-0.327509\pi$$
0.515761 + 0.856733i $$0.327509\pi$$
$$108$$ 0 0
$$109$$ −2226.85 −1.95682 −0.978409 0.206680i $$-0.933734\pi$$
−0.978409 + 0.206680i $$0.933734\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −62.7678 −0.0529554
$$113$$ −1719.76 −1.43169 −0.715847 0.698257i $$-0.753959\pi$$
−0.715847 + 0.698257i $$0.753959\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 412.608 0.330256
$$117$$ 0 0
$$118$$ −503.038 −0.392444
$$119$$ −1505.58 −1.15980
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ −411.771 −0.305574
$$123$$ 0 0
$$124$$ −3186.49 −2.30771
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1601.63 1.11907 0.559534 0.828807i $$-0.310980\pi$$
0.559534 + 0.828807i $$0.310980\pi$$
$$128$$ −2392.91 −1.65239
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2004.13 −1.33665 −0.668327 0.743868i $$-0.732989\pi$$
−0.668327 + 0.743868i $$0.732989\pi$$
$$132$$ 0 0
$$133$$ −1260.00 −0.821471
$$134$$ −2244.34 −1.44687
$$135$$ 0 0
$$136$$ 1713.77 1.08055
$$137$$ −1672.85 −1.04322 −0.521610 0.853184i $$-0.674669\pi$$
−0.521610 + 0.853184i $$0.674669\pi$$
$$138$$ 0 0
$$139$$ 2540.38 1.55016 0.775080 0.631863i $$-0.217709\pi$$
0.775080 + 0.631863i $$0.217709\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −3847.84 −2.27397
$$143$$ −172.243 −0.100725
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −1613.79 −0.914780
$$147$$ 0 0
$$148$$ 857.594 0.476310
$$149$$ −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 $$0.380681\pi$$
$$152$$ 1434.22 0.765335
$$153$$ 0 0
$$154$$ 1043.13 0.545831
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1011.95 0.514411 0.257205 0.966357i $$-0.417198\pi$$
0.257205 + 0.966357i $$0.417198\pi$$
$$158$$ −3819.27 −1.92307
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 281.929 0.138007
$$162$$ 0 0
$$163$$ 2816.37 1.35334 0.676672 0.736285i $$-0.263422\pi$$
0.676672 + 0.736285i $$0.263422\pi$$
$$164$$ 1430.42 0.681081
$$165$$ 0 0
$$166$$ 6382.87 2.98438
$$167$$ 3448.89 1.59810 0.799052 0.601262i $$-0.205335\pi$$
0.799052 + 0.601262i $$0.205335\pi$$
$$168$$ 0 0
$$169$$ −1951.81 −0.888399
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 1596.47 0.707730
$$173$$ −2287.85 −1.00545 −0.502723 0.864448i $$-0.667668\pi$$
−0.502723 + 0.864448i $$0.667668\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −33.4545 −0.0143280
$$177$$ 0 0
$$178$$ −7001.17 −2.94809
$$179$$ −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 $$0.422745\pi$$
$$182$$ −1484.89 −0.604767
$$183$$ 0 0
$$184$$ −320.913 −0.128576
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −802.458 −0.313805
$$188$$ −6812.44 −2.64281
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 2760.35 1.04572 0.522859 0.852419i $$-0.324866\pi$$
0.522859 + 0.852419i $$0.324866\pi$$
$$192$$ 0 0
$$193$$ 1250.61 0.466430 0.233215 0.972425i $$-0.425075\pi$$
0.233215 + 0.972425i $$0.425075\pi$$
$$194$$ 1960.22 0.725441
$$195$$ 0 0
$$196$$ 1087.58 0.396350
$$197$$ −143.991 −0.0520756 −0.0260378 0.999661i $$-0.508289\pi$$
−0.0260378 + 0.999661i $$0.508289\pi$$
$$198$$ 0 0
$$199$$ 761.249 0.271174 0.135587 0.990765i $$-0.456708\pi$$
0.135587 + 0.990765i $$0.456708\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −340.657 −0.118656
$$203$$ −649.411 −0.224531
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 318.844 0.107840
$$207$$ 0 0
$$208$$ 47.6223 0.0158751
$$209$$ −671.564 −0.222263
$$210$$ 0 0
$$211$$ 3976.58 1.29743 0.648717 0.761029i $$-0.275306\pi$$
0.648717 + 0.761029i $$0.275306\pi$$
$$212$$ −7115.91 −2.30530
$$213$$ 0 0
$$214$$ 5245.97 1.67573
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 5015.29 1.56894
$$218$$ −10232.0 −3.17890
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1142.29 0.347688
$$222$$ 0 0
$$223$$ −908.084 −0.272690 −0.136345 0.990661i $$-0.543536\pi$$
−0.136345 + 0.990661i $$0.543536\pi$$
$$224$$ 3590.30 1.07092
$$225$$ 0 0
$$226$$ −7902.05 −2.32583
$$227$$ −2062.15 −0.602951 −0.301475 0.953474i $$-0.597479\pi$$
−0.301475 + 0.953474i $$0.597479\pi$$
$$228$$ 0 0
$$229$$ 4077.47 1.17662 0.588312 0.808634i $$-0.299793\pi$$
0.588312 + 0.808634i $$0.299793\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 739.209 0.209187
$$233$$ 1682.76 0.473138 0.236569 0.971615i $$-0.423977\pi$$
0.236569 + 0.971615i $$0.423977\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −1435.56 −0.395961
$$237$$ 0 0
$$238$$ −6917.93 −1.88413
$$239$$ −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 $$0.621361\pi$$
$$242$$ 555.978 0.147684
$$243$$ 0 0
$$244$$ −1175.10 −0.308313
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 955.968 0.246262
$$248$$ −5708.78 −1.46173
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1827.60 0.459591 0.229796 0.973239i $$-0.426194\pi$$
0.229796 + 0.973239i $$0.426194\pi$$
$$252$$ 0 0
$$253$$ 150.265 0.0373402
$$254$$ 7359.26 1.81796
$$255$$ 0 0
$$256$$ −4405.79 −1.07563
$$257$$ 585.171 0.142031 0.0710155 0.997475i $$-0.477376\pi$$
0.0710155 + 0.997475i $$0.477376\pi$$
$$258$$ 0 0
$$259$$ −1349.78 −0.323828
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −9208.69 −2.17143
$$263$$ −238.098 −0.0558241 −0.0279120 0.999610i $$-0.508886\pi$$
−0.0279120 + 0.999610i $$0.508886\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −5789.51 −1.33450
$$267$$ 0 0
$$268$$ −6404.84 −1.45984
$$269$$ −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 $$0.505117\pi$$
$$272$$ 221.867 0.0494582
$$273$$ 0 0
$$274$$ −7686.51 −1.69474
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 8602.51 1.86597 0.932987 0.359911i $$-0.117193\pi$$
0.932987 + 0.359911i $$0.117193\pi$$
$$278$$ 11672.7 2.51828
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2992.81 0.635360 0.317680 0.948198i $$-0.397096\pi$$
0.317680 + 0.948198i $$0.397096\pi$$
$$282$$ 0 0
$$283$$ 6858.89 1.44070 0.720351 0.693610i $$-0.243981\pi$$
0.720351 + 0.693610i $$0.243981\pi$$
$$284$$ −10980.9 −2.29435
$$285$$ 0 0
$$286$$ −791.431 −0.163630
$$287$$ −2251.37 −0.463046
$$288$$ 0 0
$$289$$ 408.809 0.0832096
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −4605.39 −0.922979
$$293$$ 4049.70 0.807461 0.403731 0.914878i $$-0.367713\pi$$
0.403731 + 0.914878i $$0.367713\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 1536.43 0.301699
$$297$$ 0 0
$$298$$ −14201.2 −2.76059
$$299$$ −213.901 −0.0413720
$$300$$ 0 0
$$301$$ −2512.71 −0.481164
$$302$$ 6243.20 1.18959
$$303$$ 0 0
$$304$$ 185.677 0.0350305
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 9572.69 1.77962 0.889808 0.456335i $$-0.150838\pi$$
0.889808 + 0.456335i $$0.150838\pi$$
$$308$$ 2976.87 0.550724
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −5396.42 −0.983932 −0.491966 0.870614i $$-0.663722\pi$$
−0.491966 + 0.870614i $$0.663722\pi$$
$$312$$ 0 0
$$313$$ −9755.04 −1.76162 −0.880811 0.473469i $$-0.843002\pi$$
−0.880811 + 0.473469i $$0.843002\pi$$
$$314$$ 4649.77 0.835674
$$315$$ 0 0
$$316$$ −10899.3 −1.94030
$$317$$ −4353.75 −0.771391 −0.385695 0.922626i $$-0.626038\pi$$
−0.385695 + 0.922626i $$0.626038\pi$$
$$318$$ 0 0
$$319$$ −346.129 −0.0607508
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 1295.42 0.224196
$$323$$ 4453.74 0.767221
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 12940.8 2.19854
$$327$$ 0 0
$$328$$ 2562.68 0.431404
$$329$$ 10722.2 1.79677
$$330$$ 0 0
$$331$$ 5387.64 0.894656 0.447328 0.894370i $$-0.352376\pi$$
0.447328 + 0.894370i $$0.352376\pi$$
$$332$$ 18215.3 3.01113
$$333$$ 0 0
$$334$$ 15847.2 2.59616
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 4500.27 0.727434 0.363717 0.931509i $$-0.381508\pi$$
0.363717 + 0.931509i $$0.381508\pi$$
$$338$$ −8968.30 −1.44323
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2673.09 0.424504
$$342$$ 0 0
$$343$$ 5367.18 0.844899
$$344$$ 2860.16 0.448283
$$345$$ 0 0
$$346$$ −10512.4 −1.63337
$$347$$ 5906.32 0.913740 0.456870 0.889533i $$-0.348970\pi$$
0.456870 + 0.889533i $$0.348970\pi$$
$$348$$ 0 0
$$349$$ 3636.26 0.557721 0.278860 0.960332i $$-0.410043\pi$$
0.278860 + 0.960332i $$0.410043\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1913.59 0.289757
$$353$$ 210.408 0.0317248 0.0158624 0.999874i $$-0.494951\pi$$
0.0158624 + 0.999874i $$0.494951\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −19979.8 −2.97451
$$357$$ 0 0
$$358$$ −14928.9 −2.20396
$$359$$ −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
$$362$$ 5378.03 0.780837
$$363$$ 0 0
$$364$$ −4237.56 −0.610188
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −5748.70 −0.817656 −0.408828 0.912612i $$-0.634062\pi$$
−0.408828 + 0.912612i $$0.634062\pi$$
$$368$$ −41.5458 −0.00588512
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 11199.9 1.56730
$$372$$ 0 0
$$373$$ −4467.78 −0.620196 −0.310098 0.950705i $$-0.600362\pi$$
−0.310098 + 0.950705i $$0.600362\pi$$
$$374$$ −3687.18 −0.509785
$$375$$ 0 0
$$376$$ −12204.9 −1.67398
$$377$$ 492.712 0.0673103
$$378$$ 0 0
$$379$$ 7804.08 1.05770 0.528851 0.848715i $$-0.322623\pi$$
0.528851 + 0.848715i $$0.322623\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 12683.4 1.69880
$$383$$ −11161.1 −1.48904 −0.744522 0.667597i $$-0.767323\pi$$
−0.744522 + 0.667597i $$0.767323\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 5746.38 0.757728
$$387$$ 0 0
$$388$$ 5594.04 0.731944
$$389$$ −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
$$392$$ 1948.47 0.251052
$$393$$ 0 0
$$394$$ −661.616 −0.0845983
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −6019.74 −0.761013 −0.380507 0.924778i $$-0.624250\pi$$
−0.380507 + 0.924778i $$0.624250\pi$$
$$398$$ 3497.83 0.440529
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10398.8 1.29499 0.647495 0.762069i $$-0.275816\pi$$
0.647495 + 0.762069i $$0.275816\pi$$
$$402$$ 0 0
$$403$$ −3805.13 −0.470340
$$404$$ −972.158 −0.119720
$$405$$ 0 0
$$406$$ −2983.95 −0.364756
$$407$$ −719.420 −0.0876175
$$408$$ 0 0
$$409$$ −4733.68 −0.572287 −0.286144 0.958187i $$-0.592373\pi$$
−0.286144 + 0.958187i $$0.592373\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 909.911 0.108806
$$413$$ 2259.45 0.269202
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2723.98 −0.321044
$$417$$ 0 0
$$418$$ −3085.74 −0.361073
$$419$$ 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 $$0.684981\pi$$
$$422$$ 18271.8 2.10772
$$423$$ 0 0
$$424$$ −12748.5 −1.46020
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1849.52 0.209612
$$428$$ 14970.8 1.69075
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −9335.16 −1.04329 −0.521646 0.853162i $$-0.674682\pi$$
−0.521646 + 0.853162i $$0.674682\pi$$
$$432$$ 0 0
$$433$$ 2983.02 0.331074 0.165537 0.986204i $$-0.447064\pi$$
0.165537 + 0.986204i $$0.447064\pi$$
$$434$$ 23044.5 2.54878
$$435$$ 0 0
$$436$$ −29200.0 −3.20739
$$437$$ −833.988 −0.0912931
$$438$$ 0 0
$$439$$ −5232.32 −0.568850 −0.284425 0.958698i $$-0.591803\pi$$
−0.284425 + 0.958698i $$0.591803\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 5248.68 0.564828
$$443$$ 7517.71 0.806269 0.403135 0.915141i $$-0.367921\pi$$
0.403135 + 0.915141i $$0.367921\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −4172.52 −0.442992
$$447$$ 0 0
$$448$$ 16999.1 1.79270
$$449$$ −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
$$452$$ −22550.7 −2.34667
$$453$$ 0 0
$$454$$ −9475.29 −0.979510
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −9718.51 −0.994776 −0.497388 0.867528i $$-0.665707\pi$$
−0.497388 + 0.867528i $$0.665707\pi$$
$$458$$ 18735.4 1.91146
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 14538.0 1.46877 0.734385 0.678733i $$-0.237471\pi$$
0.734385 + 0.678733i $$0.237471\pi$$
$$462$$ 0 0
$$463$$ 9978.17 1.00157 0.500783 0.865573i $$-0.333045\pi$$
0.500783 + 0.865573i $$0.333045\pi$$
$$464$$ 95.6990 0.00957481
$$465$$ 0 0
$$466$$ 7732.03 0.768625
$$467$$ 15188.2 1.50498 0.752489 0.658605i $$-0.228853\pi$$
0.752489 + 0.658605i $$0.228853\pi$$
$$468$$ 0 0
$$469$$ 10080.7 0.992503
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −2571.88 −0.250806
$$473$$ −1339.25 −0.130187
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −19742.3 −1.90102
$$477$$ 0 0
$$478$$ −18494.1 −1.76967
$$479$$ −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
$$482$$ −12793.3 −1.20896
$$483$$ 0 0
$$484$$ 1586.64 0.149008
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −19086.9 −1.77599 −0.887997 0.459850i $$-0.847903\pi$$
−0.887997 + 0.459850i $$0.847903\pi$$
$$488$$ −2105.26 −0.195288
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 8112.85 0.745677 0.372839 0.927896i $$-0.378384\pi$$
0.372839 + 0.927896i $$0.378384\pi$$
$$492$$ 0 0
$$493$$ 2295.49 0.209703
$$494$$ 4392.53 0.400060
$$495$$ 0 0
$$496$$ −739.066 −0.0669053
$$497$$ 17283.0 1.55986
$$498$$ 0 0
$$499$$ 18329.1 1.64433 0.822167 0.569246i $$-0.192765\pi$$
0.822167 + 0.569246i $$0.192765\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 8397.58 0.746618
$$503$$ 7739.57 0.686064 0.343032 0.939324i $$-0.388546\pi$$
0.343032 + 0.939324i $$0.388546\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 690.446 0.0606602
$$507$$ 0 0
$$508$$ 21001.7 1.83425
$$509$$ −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
$$512$$ −1100.65 −0.0950048
$$513$$ 0 0
$$514$$ 2688.78 0.230733
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 5714.83 0.486147
$$518$$ −6202.07 −0.526068
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −2274.50 −0.191262 −0.0956312 0.995417i $$-0.530487\pi$$
−0.0956312 + 0.995417i $$0.530487\pi$$
$$522$$ 0 0
$$523$$ −10971.1 −0.917274 −0.458637 0.888624i $$-0.651662\pi$$
−0.458637 + 0.888624i $$0.651662\pi$$
$$524$$ −26279.6 −2.19089
$$525$$ 0 0
$$526$$ −1094.02 −0.0906877
$$527$$ −17727.6 −1.46533
$$528$$ 0 0
$$529$$ −11980.4 −0.984663
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −16522.0 −1.34646
$$533$$ 1708.13 0.138813
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −11474.6 −0.924680
$$537$$ 0 0
$$538$$ −21221.2 −1.70058
$$539$$ −912.355 −0.0729089
$$540$$ 0 0
$$541$$ 5313.05 0.422229 0.211115 0.977461i $$-0.432291\pi$$
0.211115 + 0.977461i $$0.432291\pi$$
$$542$$ −659.084 −0.0522326
$$543$$ 0 0
$$544$$ −12690.7 −1.00020
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −20685.1 −1.61688 −0.808439 0.588581i $$-0.799687\pi$$
−0.808439 + 0.588581i $$0.799687\pi$$
$$548$$ −21935.6 −1.70993
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1921.06 0.148529
$$552$$ 0 0
$$553$$ 17154.7 1.31915
$$554$$ 39527.3 3.03132
$$555$$ 0 0
$$556$$ 33311.2 2.54085
$$557$$ −10853.8 −0.825659 −0.412830 0.910808i $$-0.635460\pi$$
−0.412830 + 0.910808i $$0.635460\pi$$
$$558$$ 0 0
$$559$$ 1906.41 0.144244
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 13751.5 1.03216
$$563$$ −15381.2 −1.15141 −0.575704 0.817658i $$-0.695272\pi$$
−0.575704 + 0.817658i $$0.695272\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 31515.6 2.34046
$$567$$ 0 0
$$568$$ −19672.9 −1.45327
$$569$$ 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 $$0.459491\pi$$
$$572$$ −2258.57 −0.165097
$$573$$ 0 0
$$574$$ −10344.7 −0.752231
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −12052.6 −0.869598 −0.434799 0.900528i $$-0.643181\pi$$
−0.434799 + 0.900528i $$0.643181\pi$$
$$578$$ 1878.42 0.135176
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −28669.4 −2.04717
$$582$$ 0 0
$$583$$ 5969.41 0.424061
$$584$$ −8250.80 −0.584624
$$585$$ 0 0
$$586$$ 18607.8 1.31174
$$587$$ −11133.1 −0.782813 −0.391407 0.920218i $$-0.628011\pi$$
−0.391407 + 0.920218i $$0.628011\pi$$
$$588$$ 0 0
$$589$$ −14836.0 −1.03787
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 198.908 0.0138092
$$593$$ −7939.69 −0.549821 −0.274911 0.961470i $$-0.588648\pi$$
−0.274911 + 0.961470i $$0.588648\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −40527.1 −2.78533
$$597$$ 0 0
$$598$$ −982.846 −0.0672100
$$599$$ 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 $$0.736671\pi$$
$$602$$ −11545.6 −0.781664
$$603$$ 0 0
$$604$$ 17816.7 1.20025
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −1427.44 −0.0954496 −0.0477248 0.998861i $$-0.515197\pi$$
−0.0477248 + 0.998861i $$0.515197\pi$$
$$608$$ −10620.6 −0.708427
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −8135.03 −0.538638
$$612$$ 0 0
$$613$$ 8029.40 0.529045 0.264522 0.964380i $$-0.414786\pi$$
0.264522 + 0.964380i $$0.414786\pi$$
$$614$$ 43985.1 2.89103
$$615$$ 0 0
$$616$$ 5333.22 0.348834
$$617$$ 20795.5 1.35688 0.678440 0.734655i $$-0.262656\pi$$
0.678440 + 0.734655i $$0.262656\pi$$
$$618$$ 0 0
$$619$$ 1677.43 0.108920 0.0544602 0.998516i $$-0.482656\pi$$
0.0544602 + 0.998516i $$0.482656\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −24795.8 −1.59842
$$623$$ 31446.6 2.02228
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −44823.0 −2.86180
$$627$$ 0 0
$$628$$ 13269.4 0.843165
$$629$$ 4771.11 0.302443
$$630$$ 0 0
$$631$$ −25225.2 −1.59144 −0.795719 0.605666i $$-0.792907\pi$$
−0.795719 + 0.605666i $$0.792907\pi$$
$$632$$ −19526.8 −1.22901
$$633$$ 0 0
$$634$$ −20004.8 −1.25315
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1298.73 0.0807812
$$638$$ −1590.41 −0.0986912
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −15165.3 −0.934468 −0.467234 0.884134i $$-0.654749\pi$$
−0.467234 + 0.884134i $$0.654749\pi$$
$$642$$ 0 0
$$643$$ −27156.1 −1.66553 −0.832763 0.553630i $$-0.813242\pi$$
−0.832763 + 0.553630i $$0.813242\pi$$
$$644$$ 3696.85 0.226206
$$645$$ 0 0
$$646$$ 20464.3 1.24637
$$647$$ 29154.9 1.77156 0.885778 0.464110i $$-0.153626\pi$$
0.885778 + 0.464110i $$0.153626\pi$$
$$648$$ 0 0
$$649$$ 1204.26 0.0728374
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 36930.2 2.21825
$$653$$ −19141.7 −1.14713 −0.573564 0.819161i $$-0.694440\pi$$
−0.573564 + 0.819161i $$0.694440\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 331.768 0.0197460
$$657$$ 0 0
$$658$$ 49267.2 2.91890
$$659$$ 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 $$0.268239\pi$$
$$662$$ 24755.4 1.45339
$$663$$ 0 0
$$664$$ 32633.7 1.90728
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −429.843 −0.0249529
$$668$$ 45224.3 2.61943
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 985.772 0.0567143
$$672$$ 0 0
$$673$$ 13855.8 0.793615 0.396807 0.917902i $$-0.370118\pi$$
0.396807 + 0.917902i $$0.370118\pi$$
$$674$$ 20678.1 1.18174
$$675$$ 0 0
$$676$$ −25593.5 −1.45616
$$677$$ −24992.8 −1.41884 −0.709419 0.704787i $$-0.751043\pi$$
−0.709419 + 0.704787i $$0.751043\pi$$
$$678$$ 0 0
$$679$$ −8804.57 −0.497626
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 12282.5 0.689619
$$683$$ −14420.5 −0.807887 −0.403943 0.914784i $$-0.632361\pi$$
−0.403943 + 0.914784i $$0.632361\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 24661.4 1.37256
$$687$$ 0 0
$$688$$ 370.280 0.0205186
$$689$$ −8497.42 −0.469849
$$690$$ 0 0
$$691$$ 30552.4 1.68201 0.841005 0.541027i $$-0.181964\pi$$
0.841005 + 0.541027i $$0.181964\pi$$
$$692$$ −29999.9 −1.64801
$$693$$ 0 0
$$694$$ 27138.7 1.48440
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 7957.96 0.432467
$$698$$ 16708.1 0.906032
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 9151.47 0.493076 0.246538 0.969133i $$-0.420707\pi$$
0.246538 + 0.969133i $$0.420707\pi$$
$$702$$ 0 0
$$703$$ 3992.86 0.214216
$$704$$ 9060.30 0.485047
$$705$$ 0 0
$$706$$ 966.793 0.0515378
$$707$$ 1530.10 0.0813937
$$708$$ 0 0
$$709$$ −6261.96 −0.331697 −0.165848 0.986151i $$-0.553036\pi$$
−0.165848 + 0.986151i $$0.553036\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −35794.9 −1.88409
$$713$$ 3319.60 0.174362
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −42603.9 −2.22372
$$717$$ 0 0
$$718$$ −11485.7 −0.596995
$$719$$ 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
$$722$$ −14389.9 −0.741740
$$723$$ 0 0
$$724$$ 15347.7 0.787836
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 7233.66 0.369026 0.184513 0.982830i $$-0.440929\pi$$
0.184513 + 0.982830i $$0.440929\pi$$
$$728$$ −7591.81 −0.386499
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 8881.73 0.449388
$$732$$ 0 0
$$733$$ 13444.8 0.677485 0.338743 0.940879i $$-0.389998\pi$$
0.338743 + 0.940879i $$0.389998\pi$$
$$734$$ −26414.4 −1.32830
$$735$$ 0 0
$$736$$ 2376.41 0.119016
$$737$$ 5372.90 0.268539
$$738$$ 0 0
$$739$$ 18490.9 0.920432 0.460216 0.887807i $$-0.347772\pi$$
0.460216 + 0.887807i $$0.347772\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 51461.8 2.54612
$$743$$ −25160.9 −1.24235 −0.621173 0.783674i $$-0.713343\pi$$
−0.621173 + 0.783674i $$0.713343\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −20528.8 −1.00752
$$747$$ 0 0
$$748$$ −10522.4 −0.514354
$$749$$ −23562.9 −1.14949
$$750$$ 0 0
$$751$$ −13419.5 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$752$$ −1580.06 −0.0766207
$$753$$ 0 0
$$754$$ 2263.94 0.109347
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 7014.90 0.336804 0.168402 0.985718i $$-0.446139\pi$$
0.168402 + 0.985718i $$0.446139\pi$$
$$758$$ 35858.6 1.71826
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30156.9 1.43651 0.718256 0.695779i $$-0.244941\pi$$
0.718256 + 0.695779i $$0.244941\pi$$
$$762$$ 0 0
$$763$$ 45958.4 2.18061
$$764$$ 36195.7 1.71402
$$765$$ 0 0
$$766$$ −51283.5 −2.41899
$$767$$ −1714.26 −0.0807019
$$768$$ 0 0
$$769$$ 11292.2 0.529530 0.264765 0.964313i $$-0.414706\pi$$
0.264765 + 0.964313i $$0.414706\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 16398.9 0.764519
$$773$$ 8524.10 0.396624 0.198312 0.980139i $$-0.436454\pi$$
0.198312 + 0.980139i $$0.436454\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 10022.0 0.463621
$$777$$ 0 0
$$778$$ −39011.4 −1.79772
$$779$$ 6659.89 0.306310
$$780$$ 0 0
$$781$$ 9211.66 0.422047
$$782$$ −4578.96 −0.209390
$$783$$ 0 0
$$784$$ 252.251 0.0114910
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 14983.9 0.678676 0.339338 0.940665i $$-0.389797\pi$$
0.339338 + 0.940665i $$0.389797\pi$$
$$788$$ −1888.10 −0.0853565
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 35493.0 1.59543
$$792$$ 0 0
$$793$$ −1403.24 −0.0628380
$$794$$ −27659.9 −1.23629
$$795$$ 0 0
$$796$$ 9982.04 0.444477
$$797$$ 37172.3 1.65208 0.826041 0.563610i $$-0.190588\pi$$
0.826041 + 0.563610i $$0.190588\pi$$
$$798$$ 0 0
$$799$$ −37900.1 −1.67811
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 47781.0 2.10375
$$803$$ 3863.37 0.169783
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −17484.0 −0.764080
$$807$$ 0 0
$$808$$ −1741.68 −0.0758316
$$809$$ −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 $$0.437663\pi$$
$$812$$ −8515.54 −0.368026
$$813$$ 0 0
$$814$$ −3305.63 −0.142337
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7432.98 0.318295
$$818$$ −21750.6 −0.929696
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 25156.8 1.06940 0.534702 0.845041i $$-0.320424\pi$$
0.534702 + 0.845041i $$0.320424\pi$$
$$822$$ 0 0
$$823$$ −1318.51 −0.0558447 −0.0279224 0.999610i $$-0.508889\pi$$
−0.0279224 + 0.999610i $$0.508889\pi$$
$$824$$ 1630.16 0.0689189
$$825$$ 0 0
$$826$$ 10381.9 0.437326
$$827$$ 124.982 0.00525519 0.00262760 0.999997i $$-0.499164\pi$$
0.00262760 + 0.999997i $$0.499164\pi$$
$$828$$ 0 0
$$829$$ −8886.80 −0.372318 −0.186159 0.982520i $$-0.559604\pi$$
−0.186159 + 0.982520i $$0.559604\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −12897.3 −0.537419
$$833$$ 6050.63 0.251671
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −8806.02 −0.364309
$$837$$ 0 0
$$838$$ 37299.1 1.53756
$$839$$ −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
$$842$$ −43578.8 −1.78364
$$843$$ 0 0
$$844$$ 52143.6 2.12661
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2497.24 −0.101306
$$848$$ −1650.44 −0.0668355
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −893.418 −0.0359882
$$852$$ 0 0
$$853$$ −18130.5 −0.727757 −0.363878 0.931446i $$-0.618548\pi$$
−0.363878 + 0.931446i $$0.618548\pi$$
$$854$$ 8498.27 0.340521
$$855$$ 0 0
$$856$$ 26821.1 1.07094
$$857$$ 26394.1 1.05205 0.526024 0.850470i $$-0.323682\pi$$
0.526024 + 0.850470i $$0.323682\pi$$
$$858$$ 0 0
$$859$$ −29456.2 −1.17000 −0.585002 0.811032i $$-0.698906\pi$$
−0.585002 + 0.811032i $$0.698906\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −42893.7 −1.69486
$$863$$ −762.616 −0.0300808 −0.0150404 0.999887i $$-0.504788\pi$$
−0.0150404 + 0.999887i $$0.504788\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 13706.6 0.537838
$$867$$ 0 0
$$868$$ 65764.0 2.57163
$$869$$ 9143.26 0.356920
$$870$$ 0 0
$$871$$ −7648.29 −0.297535
$$872$$ −52313.3 −2.03160
$$873$$ 0 0
$$874$$ −3832.06 −0.148308
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −44767.2 −1.72369 −0.861847 0.507168i $$-0.830692\pi$$
−0.861847 + 0.507168i $$0.830692\pi$$
$$878$$ −24041.8 −0.924112
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 32057.9 1.22595 0.612973 0.790104i $$-0.289973\pi$$
0.612973 + 0.790104i $$0.289973\pi$$
$$882$$ 0 0
$$883$$ −7078.95 −0.269791 −0.134896 0.990860i $$-0.543070\pi$$
−0.134896 + 0.990860i $$0.543070\pi$$
$$884$$ 14978.6 0.569891
$$885$$ 0 0
$$886$$ 34542.8 1.30981
$$887$$ −25148.1 −0.951964 −0.475982 0.879455i $$-0.657907\pi$$
−0.475982 + 0.879455i $$0.657907\pi$$
$$888$$ 0 0
$$889$$ −33055.0 −1.24705
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −11907.4 −0.446963
$$893$$ −31718.0 −1.18858
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 49385.8 1.84137
$$897$$ 0 0
$$898$$ −73843.6 −2.74409
$$899$$ −7646.56 −0.283679
$$900$$ 0 0
$$901$$ −39588.4 −1.46380
$$902$$ −5513.62 −0.203529
$$903$$ 0 0
$$904$$ −40400.8 −1.48641
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −1269.76 −0.0464848 −0.0232424 0.999730i $$-0.507399\pi$$
−0.0232424 + 0.999730i $$0.507399\pi$$
$$908$$ −27040.4 −0.988289
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 33783.1 1.22863 0.614316 0.789060i $$-0.289432\pi$$
0.614316 + 0.789060i $$0.289432\pi$$
$$912$$ 0 0
$$913$$ −15280.5 −0.553899
$$914$$ −44655.1 −1.61604
$$915$$ 0 0
$$916$$ 53466.6 1.92859
$$917$$ 41361.9 1.48952
$$918$$ 0 0
$$919$$ 39262.5 1.40930 0.704652 0.709553i $$-0.251103\pi$$
0.704652 + 0.709553i $$0.251103\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 66800.1 2.38606
$$923$$ −13112.7 −0.467618
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 45848.3 1.62707
$$927$$ 0 0
$$928$$ −5473.95 −0.193633
$$929$$ 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
$$932$$ 22065.5 0.775515
$$933$$ 0 0
$$934$$ 69787.5 2.44488
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 5135.11 0.179036 0.0895180 0.995985i $$-0.471467\pi$$
0.0895180 + 0.995985i $$0.471467\pi$$
$$938$$ 46319.4 1.61235
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 9702.77 0.336133 0.168067 0.985776i $$-0.446248\pi$$
0.168067 + 0.985776i $$0.446248\pi$$
$$942$$ 0 0
$$943$$ −1490.18 −0.0514600
$$944$$ −332.959 −0.0114798
$$945$$ 0 0
$$946$$ −6153.65 −0.211493
$$947$$ −699.579 −0.0240055 −0.0120028 0.999928i $$-0.503821\pi$$
−0.0120028 + 0.999928i $$0.503821\pi$$
$$948$$ 0 0
$$949$$ −5499.49 −0.188115
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −35369.3 −1.20412
$$953$$ 42039.3 1.42895 0.714473 0.699663i $$-0.246667\pi$$
0.714473 + 0.699663i $$0.246667\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −52778.1 −1.78553
$$957$$ 0 0
$$958$$ −52063.4 −1.75584
$$959$$ 34524.9 1.16253
$$960$$ 0 0
$$961$$ 29262.0 0.982243
$$962$$ 4705.55 0.157706
$$963$$ 0 0
$$964$$ −36509.3 −1.21980
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 32794.8 1.09060 0.545299 0.838242i $$-0.316416\pi$$
0.545299 + 0.838242i $$0.316416\pi$$
$$968$$ 2842.55 0.0943832
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 3322.53 0.109810 0.0549048 0.998492i $$-0.482514\pi$$
0.0549048 + 0.998492i $$0.482514\pi$$
$$972$$ 0 0
$$973$$ −52429.3 −1.72745
$$974$$ −87701.4 −2.88515
$$975$$ 0 0
$$976$$ −272.550 −0.00893864
$$977$$ 22192.5 0.726716 0.363358 0.931650i $$-0.381630\pi$$
0.363358 + 0.931650i $$0.381630\pi$$
$$978$$ 0 0
$$979$$ 16760.7 0.547164
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 37277.4 1.21137
$$983$$ 7383.09 0.239556 0.119778 0.992801i $$-0.461782\pi$$
0.119778 + 0.992801i $$0.461782\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 10547.4 0.340668
$$987$$ 0 0
$$988$$ 12535.3 0.403645
$$989$$ −1663.16 −0.0534735
$$990$$ 0 0
$$991$$ 46260.8 1.48287 0.741434 0.671026i $$-0.234146\pi$$
0.741434 + 0.671026i $$0.234146\pi$$
$$992$$ 42274.3 1.35304
$$993$$ 0 0
$$994$$ 79413.1 2.53403
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −41196.8 −1.30864 −0.654320 0.756217i $$-0.727045\pi$$
−0.654320 + 0.756217i $$0.727045\pi$$
$$998$$ 84219.5 2.67127
$$999$$ 0 0
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 2475.4.a.z.1.3 3
3.2 odd 2 825.4.a.p.1.1 3
5.4 even 2 495.4.a.i.1.1 3
15.2 even 4 825.4.c.m.199.1 6
15.8 even 4 825.4.c.m.199.6 6
15.14 odd 2 165.4.a.g.1.3 3
165.164 even 2 1815.4.a.q.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.g.1.3 3 15.14 odd 2
495.4.a.i.1.1 3 5.4 even 2
825.4.a.p.1.1 3 3.2 odd 2
825.4.c.m.199.1 6 15.2 even 4
825.4.c.m.199.6 6 15.8 even 4
1815.4.a.q.1.1 3 165.164 even 2
2475.4.a.z.1.3 3 1.1 even 1 trivial