# Properties

 Label 1452.4.a.t.1.3 Level $1452$ Weight $4$ Character 1452.1 Self dual yes Analytic conductor $85.671$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1452,4,Mod(1,1452)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1452, 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("1452.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1452 = 2^{2} \cdot 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1452.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: $$85.6707733283$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 174x^{4} + 63x^{3} + 7614x^{2} + 1579x - 12611$$ x^6 - x^5 - 174*x^4 + 63*x^3 + 7614*x^2 + 1579*x - 12611 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 11$$ Twist minimal: no (minimal twist has level 132) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-9.47700$$ of defining polynomial Character $$\chi$$ $$=$$ 1452.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -0.981524 q^{5} +17.7745 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -0.981524 q^{5} +17.7745 q^{7} +9.00000 q^{9} +61.8967 q^{13} -2.94457 q^{15} -130.198 q^{17} -116.281 q^{19} +53.3235 q^{21} -26.4920 q^{23} -124.037 q^{25} +27.0000 q^{27} -179.702 q^{29} -319.935 q^{31} -17.4461 q^{35} -177.998 q^{37} +185.690 q^{39} +338.949 q^{41} -159.872 q^{43} -8.83372 q^{45} +98.9479 q^{47} -27.0675 q^{49} -390.593 q^{51} +562.690 q^{53} -348.844 q^{57} +443.734 q^{59} -551.701 q^{61} +159.970 q^{63} -60.7531 q^{65} +454.264 q^{67} -79.4761 q^{69} -214.257 q^{71} -162.610 q^{73} -372.110 q^{75} -167.620 q^{79} +81.0000 q^{81} -357.601 q^{83} +127.792 q^{85} -539.105 q^{87} +1215.19 q^{89} +1100.18 q^{91} -959.805 q^{93} +114.133 q^{95} -1117.50 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9}+O(q^{10})$$ 6 * q + 18 * q^3 + 13 * q^5 - 23 * q^7 + 54 * q^9 $$6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9} - 66 q^{13} + 39 q^{15} - 44 q^{17} - 270 q^{19} - 69 q^{21} - 124 q^{23} + 131 q^{25} + 162 q^{27} - 141 q^{29} - 253 q^{31} - 884 q^{35} + 288 q^{37} - 198 q^{39} - 428 q^{41} - 1006 q^{43} + 117 q^{45} - 674 q^{47} + 181 q^{49} - 132 q^{51} - 773 q^{53} - 810 q^{57} - 17 q^{59} - 1016 q^{61} - 207 q^{63} - 1220 q^{65} + 1836 q^{67} - 372 q^{69} - 208 q^{71} - 1521 q^{73} + 393 q^{75} - 1425 q^{79} + 486 q^{81} - 3065 q^{83} - 1304 q^{85} - 423 q^{87} + 1444 q^{89} + 1328 q^{91} - 759 q^{93} - 1760 q^{95} - 3887 q^{97}+O(q^{100})$$ 6 * q + 18 * q^3 + 13 * q^5 - 23 * q^7 + 54 * q^9 - 66 * q^13 + 39 * q^15 - 44 * q^17 - 270 * q^19 - 69 * q^21 - 124 * q^23 + 131 * q^25 + 162 * q^27 - 141 * q^29 - 253 * q^31 - 884 * q^35 + 288 * q^37 - 198 * q^39 - 428 * q^41 - 1006 * q^43 + 117 * q^45 - 674 * q^47 + 181 * q^49 - 132 * q^51 - 773 * q^53 - 810 * q^57 - 17 * q^59 - 1016 * q^61 - 207 * q^63 - 1220 * q^65 + 1836 * q^67 - 372 * q^69 - 208 * q^71 - 1521 * q^73 + 393 * q^75 - 1425 * q^79 + 486 * q^81 - 3065 * q^83 - 1304 * q^85 - 423 * q^87 + 1444 * q^89 + 1328 * q^91 - 759 * q^93 - 1760 * q^95 - 3887 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −0.981524 −0.0877902 −0.0438951 0.999036i $$-0.513977\pi$$
−0.0438951 + 0.999036i $$0.513977\pi$$
$$6$$ 0 0
$$7$$ 17.7745 0.959732 0.479866 0.877342i $$-0.340685\pi$$
0.479866 + 0.877342i $$0.340685\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 61.8967 1.32054 0.660272 0.751027i $$-0.270441\pi$$
0.660272 + 0.751027i $$0.270441\pi$$
$$14$$ 0 0
$$15$$ −2.94457 −0.0506857
$$16$$ 0 0
$$17$$ −130.198 −1.85751 −0.928753 0.370700i $$-0.879118\pi$$
−0.928753 + 0.370700i $$0.879118\pi$$
$$18$$ 0 0
$$19$$ −116.281 −1.40404 −0.702019 0.712158i $$-0.747718\pi$$
−0.702019 + 0.712158i $$0.747718\pi$$
$$20$$ 0 0
$$21$$ 53.3235 0.554102
$$22$$ 0 0
$$23$$ −26.4920 −0.240173 −0.120086 0.992763i $$-0.538317\pi$$
−0.120086 + 0.992763i $$0.538317\pi$$
$$24$$ 0 0
$$25$$ −124.037 −0.992293
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −179.702 −1.15068 −0.575341 0.817914i $$-0.695131\pi$$
−0.575341 + 0.817914i $$0.695131\pi$$
$$30$$ 0 0
$$31$$ −319.935 −1.85361 −0.926807 0.375538i $$-0.877458\pi$$
−0.926807 + 0.375538i $$0.877458\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −17.4461 −0.0842551
$$36$$ 0 0
$$37$$ −177.998 −0.790885 −0.395442 0.918491i $$-0.629409\pi$$
−0.395442 + 0.918491i $$0.629409\pi$$
$$38$$ 0 0
$$39$$ 185.690 0.762416
$$40$$ 0 0
$$41$$ 338.949 1.29110 0.645548 0.763720i $$-0.276629\pi$$
0.645548 + 0.763720i $$0.276629\pi$$
$$42$$ 0 0
$$43$$ −159.872 −0.566984 −0.283492 0.958975i $$-0.591493\pi$$
−0.283492 + 0.958975i $$0.591493\pi$$
$$44$$ 0 0
$$45$$ −8.83372 −0.0292634
$$46$$ 0 0
$$47$$ 98.9479 0.307086 0.153543 0.988142i $$-0.450932\pi$$
0.153543 + 0.988142i $$0.450932\pi$$
$$48$$ 0 0
$$49$$ −27.0675 −0.0789141
$$50$$ 0 0
$$51$$ −390.593 −1.07243
$$52$$ 0 0
$$53$$ 562.690 1.45833 0.729164 0.684339i $$-0.239909\pi$$
0.729164 + 0.684339i $$0.239909\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −348.844 −0.810622
$$58$$ 0 0
$$59$$ 443.734 0.979140 0.489570 0.871964i $$-0.337154\pi$$
0.489570 + 0.871964i $$0.337154\pi$$
$$60$$ 0 0
$$61$$ −551.701 −1.15800 −0.579001 0.815327i $$-0.696557\pi$$
−0.579001 + 0.815327i $$0.696557\pi$$
$$62$$ 0 0
$$63$$ 159.970 0.319911
$$64$$ 0 0
$$65$$ −60.7531 −0.115931
$$66$$ 0 0
$$67$$ 454.264 0.828316 0.414158 0.910205i $$-0.364076\pi$$
0.414158 + 0.910205i $$0.364076\pi$$
$$68$$ 0 0
$$69$$ −79.4761 −0.138664
$$70$$ 0 0
$$71$$ −214.257 −0.358136 −0.179068 0.983837i $$-0.557308\pi$$
−0.179068 + 0.983837i $$0.557308\pi$$
$$72$$ 0 0
$$73$$ −162.610 −0.260713 −0.130357 0.991467i $$-0.541612\pi$$
−0.130357 + 0.991467i $$0.541612\pi$$
$$74$$ 0 0
$$75$$ −372.110 −0.572901
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −167.620 −0.238719 −0.119359 0.992851i $$-0.538084\pi$$
−0.119359 + 0.992851i $$0.538084\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −357.601 −0.472913 −0.236457 0.971642i $$-0.575986\pi$$
−0.236457 + 0.971642i $$0.575986\pi$$
$$84$$ 0 0
$$85$$ 127.792 0.163071
$$86$$ 0 0
$$87$$ −539.105 −0.664347
$$88$$ 0 0
$$89$$ 1215.19 1.44730 0.723648 0.690169i $$-0.242464\pi$$
0.723648 + 0.690169i $$0.242464\pi$$
$$90$$ 0 0
$$91$$ 1100.18 1.26737
$$92$$ 0 0
$$93$$ −959.805 −1.07018
$$94$$ 0 0
$$95$$ 114.133 0.123261
$$96$$ 0 0
$$97$$ −1117.50 −1.16974 −0.584868 0.811128i $$-0.698854\pi$$
−0.584868 + 0.811128i $$0.698854\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −405.605 −0.399596 −0.199798 0.979837i $$-0.564029\pi$$
−0.199798 + 0.979837i $$0.564029\pi$$
$$102$$ 0 0
$$103$$ 444.313 0.425044 0.212522 0.977156i $$-0.431832\pi$$
0.212522 + 0.977156i $$0.431832\pi$$
$$104$$ 0 0
$$105$$ −52.3383 −0.0486447
$$106$$ 0 0
$$107$$ −1247.30 −1.12693 −0.563465 0.826140i $$-0.690532\pi$$
−0.563465 + 0.826140i $$0.690532\pi$$
$$108$$ 0 0
$$109$$ −1343.63 −1.18070 −0.590349 0.807148i $$-0.701010\pi$$
−0.590349 + 0.807148i $$0.701010\pi$$
$$110$$ 0 0
$$111$$ −533.995 −0.456618
$$112$$ 0 0
$$113$$ 1193.63 0.993693 0.496846 0.867838i $$-0.334491\pi$$
0.496846 + 0.867838i $$0.334491\pi$$
$$114$$ 0 0
$$115$$ 26.0026 0.0210848
$$116$$ 0 0
$$117$$ 557.071 0.440181
$$118$$ 0 0
$$119$$ −2314.20 −1.78271
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 1016.85 0.745414
$$124$$ 0 0
$$125$$ 244.435 0.174904
$$126$$ 0 0
$$127$$ −640.650 −0.447626 −0.223813 0.974632i $$-0.571851\pi$$
−0.223813 + 0.974632i $$0.571851\pi$$
$$128$$ 0 0
$$129$$ −479.617 −0.327348
$$130$$ 0 0
$$131$$ −379.349 −0.253007 −0.126503 0.991966i $$-0.540375\pi$$
−0.126503 + 0.991966i $$0.540375\pi$$
$$132$$ 0 0
$$133$$ −2066.84 −1.34750
$$134$$ 0 0
$$135$$ −26.5011 −0.0168952
$$136$$ 0 0
$$137$$ −1193.17 −0.744083 −0.372041 0.928216i $$-0.621342\pi$$
−0.372041 + 0.928216i $$0.621342\pi$$
$$138$$ 0 0
$$139$$ −2945.39 −1.79730 −0.898650 0.438666i $$-0.855451\pi$$
−0.898650 + 0.438666i $$0.855451\pi$$
$$140$$ 0 0
$$141$$ 296.844 0.177296
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 176.382 0.101019
$$146$$ 0 0
$$147$$ −81.2026 −0.0455611
$$148$$ 0 0
$$149$$ −3305.66 −1.81752 −0.908760 0.417320i $$-0.862969\pi$$
−0.908760 + 0.417320i $$0.862969\pi$$
$$150$$ 0 0
$$151$$ 1670.75 0.900421 0.450210 0.892923i $$-0.351349\pi$$
0.450210 + 0.892923i $$0.351349\pi$$
$$152$$ 0 0
$$153$$ −1171.78 −0.619168
$$154$$ 0 0
$$155$$ 314.024 0.162729
$$156$$ 0 0
$$157$$ 561.400 0.285379 0.142690 0.989767i $$-0.454425\pi$$
0.142690 + 0.989767i $$0.454425\pi$$
$$158$$ 0 0
$$159$$ 1688.07 0.841966
$$160$$ 0 0
$$161$$ −470.883 −0.230502
$$162$$ 0 0
$$163$$ 3799.01 1.82553 0.912765 0.408485i $$-0.133943\pi$$
0.912765 + 0.408485i $$0.133943\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 693.606 0.321394 0.160697 0.987004i $$-0.448626\pi$$
0.160697 + 0.987004i $$0.448626\pi$$
$$168$$ 0 0
$$169$$ 1634.21 0.743835
$$170$$ 0 0
$$171$$ −1046.53 −0.468013
$$172$$ 0 0
$$173$$ 2766.95 1.21600 0.607998 0.793939i $$-0.291973\pi$$
0.607998 + 0.793939i $$0.291973\pi$$
$$174$$ 0 0
$$175$$ −2204.69 −0.952335
$$176$$ 0 0
$$177$$ 1331.20 0.565307
$$178$$ 0 0
$$179$$ 2640.76 1.10268 0.551340 0.834281i $$-0.314117\pi$$
0.551340 + 0.834281i $$0.314117\pi$$
$$180$$ 0 0
$$181$$ −4399.56 −1.80672 −0.903361 0.428882i $$-0.858908\pi$$
−0.903361 + 0.428882i $$0.858908\pi$$
$$182$$ 0 0
$$183$$ −1655.10 −0.668572
$$184$$ 0 0
$$185$$ 174.710 0.0694319
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 479.911 0.184701
$$190$$ 0 0
$$191$$ −1575.94 −0.597019 −0.298510 0.954407i $$-0.596490\pi$$
−0.298510 + 0.954407i $$0.596490\pi$$
$$192$$ 0 0
$$193$$ 187.213 0.0698232 0.0349116 0.999390i $$-0.488885\pi$$
0.0349116 + 0.999390i $$0.488885\pi$$
$$194$$ 0 0
$$195$$ −182.259 −0.0669326
$$196$$ 0 0
$$197$$ −1465.07 −0.529856 −0.264928 0.964268i $$-0.585348\pi$$
−0.264928 + 0.964268i $$0.585348\pi$$
$$198$$ 0 0
$$199$$ 3519.83 1.25384 0.626920 0.779083i $$-0.284315\pi$$
0.626920 + 0.779083i $$0.284315\pi$$
$$200$$ 0 0
$$201$$ 1362.79 0.478228
$$202$$ 0 0
$$203$$ −3194.11 −1.10435
$$204$$ 0 0
$$205$$ −332.686 −0.113345
$$206$$ 0 0
$$207$$ −238.428 −0.0800576
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 504.847 0.164716 0.0823580 0.996603i $$-0.473755\pi$$
0.0823580 + 0.996603i $$0.473755\pi$$
$$212$$ 0 0
$$213$$ −642.772 −0.206770
$$214$$ 0 0
$$215$$ 156.919 0.0497756
$$216$$ 0 0
$$217$$ −5686.68 −1.77897
$$218$$ 0 0
$$219$$ −487.830 −0.150523
$$220$$ 0 0
$$221$$ −8058.81 −2.45292
$$222$$ 0 0
$$223$$ −599.729 −0.180093 −0.0900467 0.995938i $$-0.528702\pi$$
−0.0900467 + 0.995938i $$0.528702\pi$$
$$224$$ 0 0
$$225$$ −1116.33 −0.330764
$$226$$ 0 0
$$227$$ 2340.46 0.684326 0.342163 0.939641i $$-0.388840\pi$$
0.342163 + 0.939641i $$0.388840\pi$$
$$228$$ 0 0
$$229$$ −3465.87 −1.00014 −0.500069 0.865986i $$-0.666692\pi$$
−0.500069 + 0.865986i $$0.666692\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2976.18 0.836807 0.418404 0.908261i $$-0.362590\pi$$
0.418404 + 0.908261i $$0.362590\pi$$
$$234$$ 0 0
$$235$$ −97.1197 −0.0269591
$$236$$ 0 0
$$237$$ −502.861 −0.137824
$$238$$ 0 0
$$239$$ 4379.37 1.18526 0.592632 0.805473i $$-0.298089\pi$$
0.592632 + 0.805473i $$0.298089\pi$$
$$240$$ 0 0
$$241$$ −4045.04 −1.08118 −0.540590 0.841286i $$-0.681799\pi$$
−0.540590 + 0.841286i $$0.681799\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 26.5674 0.00692788
$$246$$ 0 0
$$247$$ −7197.42 −1.85409
$$248$$ 0 0
$$249$$ −1072.80 −0.273037
$$250$$ 0 0
$$251$$ −6862.99 −1.72585 −0.862925 0.505333i $$-0.831370\pi$$
−0.862925 + 0.505333i $$0.831370\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 383.377 0.0941489
$$256$$ 0 0
$$257$$ 6424.73 1.55939 0.779696 0.626158i $$-0.215373\pi$$
0.779696 + 0.626158i $$0.215373\pi$$
$$258$$ 0 0
$$259$$ −3163.83 −0.759038
$$260$$ 0 0
$$261$$ −1617.32 −0.383561
$$262$$ 0 0
$$263$$ −6510.99 −1.52656 −0.763280 0.646068i $$-0.776412\pi$$
−0.763280 + 0.646068i $$0.776412\pi$$
$$264$$ 0 0
$$265$$ −552.294 −0.128027
$$266$$ 0 0
$$267$$ 3645.56 0.835597
$$268$$ 0 0
$$269$$ 5029.52 1.13998 0.569992 0.821650i $$-0.306946\pi$$
0.569992 + 0.821650i $$0.306946\pi$$
$$270$$ 0 0
$$271$$ 900.678 0.201890 0.100945 0.994892i $$-0.467813\pi$$
0.100945 + 0.994892i $$0.467813\pi$$
$$272$$ 0 0
$$273$$ 3300.55 0.731715
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 4634.96 1.00537 0.502686 0.864469i $$-0.332345\pi$$
0.502686 + 0.864469i $$0.332345\pi$$
$$278$$ 0 0
$$279$$ −2879.42 −0.617871
$$280$$ 0 0
$$281$$ −934.894 −0.198474 −0.0992368 0.995064i $$-0.531640\pi$$
−0.0992368 + 0.995064i $$0.531640\pi$$
$$282$$ 0 0
$$283$$ 1335.92 0.280608 0.140304 0.990108i $$-0.455192\pi$$
0.140304 + 0.990108i $$0.455192\pi$$
$$284$$ 0 0
$$285$$ 342.398 0.0711646
$$286$$ 0 0
$$287$$ 6024.64 1.23911
$$288$$ 0 0
$$289$$ 12038.5 2.45033
$$290$$ 0 0
$$291$$ −3352.49 −0.675348
$$292$$ 0 0
$$293$$ −2972.82 −0.592745 −0.296372 0.955072i $$-0.595777\pi$$
−0.296372 + 0.955072i $$0.595777\pi$$
$$294$$ 0 0
$$295$$ −435.536 −0.0859589
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1639.77 −0.317159
$$300$$ 0 0
$$301$$ −2841.65 −0.544153
$$302$$ 0 0
$$303$$ −1216.82 −0.230707
$$304$$ 0 0
$$305$$ 541.508 0.101661
$$306$$ 0 0
$$307$$ −644.628 −0.119840 −0.0599200 0.998203i $$-0.519085\pi$$
−0.0599200 + 0.998203i $$0.519085\pi$$
$$308$$ 0 0
$$309$$ 1332.94 0.245399
$$310$$ 0 0
$$311$$ 4531.91 0.826305 0.413152 0.910662i $$-0.364428\pi$$
0.413152 + 0.910662i $$0.364428\pi$$
$$312$$ 0 0
$$313$$ 4647.06 0.839193 0.419597 0.907711i $$-0.362172\pi$$
0.419597 + 0.907711i $$0.362172\pi$$
$$314$$ 0 0
$$315$$ −157.015 −0.0280850
$$316$$ 0 0
$$317$$ −628.109 −0.111287 −0.0556437 0.998451i $$-0.517721\pi$$
−0.0556437 + 0.998451i $$0.517721\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −3741.91 −0.650633
$$322$$ 0 0
$$323$$ 15139.5 2.60801
$$324$$ 0 0
$$325$$ −7677.46 −1.31037
$$326$$ 0 0
$$327$$ −4030.88 −0.681676
$$328$$ 0 0
$$329$$ 1758.75 0.294720
$$330$$ 0 0
$$331$$ −1614.45 −0.268091 −0.134045 0.990975i $$-0.542797\pi$$
−0.134045 + 0.990975i $$0.542797\pi$$
$$332$$ 0 0
$$333$$ −1601.98 −0.263628
$$334$$ 0 0
$$335$$ −445.871 −0.0727180
$$336$$ 0 0
$$337$$ 359.175 0.0580579 0.0290290 0.999579i $$-0.490758\pi$$
0.0290290 + 0.999579i $$0.490758\pi$$
$$338$$ 0 0
$$339$$ 3580.89 0.573709
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −6577.76 −1.03547
$$344$$ 0 0
$$345$$ 78.0077 0.0121733
$$346$$ 0 0
$$347$$ −1980.58 −0.306407 −0.153203 0.988195i $$-0.548959\pi$$
−0.153203 + 0.988195i $$0.548959\pi$$
$$348$$ 0 0
$$349$$ −1550.65 −0.237835 −0.118918 0.992904i $$-0.537942\pi$$
−0.118918 + 0.992904i $$0.537942\pi$$
$$350$$ 0 0
$$351$$ 1671.21 0.254139
$$352$$ 0 0
$$353$$ 261.089 0.0393664 0.0196832 0.999806i $$-0.493734\pi$$
0.0196832 + 0.999806i $$0.493734\pi$$
$$354$$ 0 0
$$355$$ 210.299 0.0314408
$$356$$ 0 0
$$357$$ −6942.59 −1.02925
$$358$$ 0 0
$$359$$ −3.02135 −0.000444181 0 −0.000222090 1.00000i $$-0.500071\pi$$
−0.000222090 1.00000i $$0.500071\pi$$
$$360$$ 0 0
$$361$$ 6662.31 0.971324
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 159.606 0.0228881
$$366$$ 0 0
$$367$$ −10582.1 −1.50513 −0.752563 0.658520i $$-0.771183\pi$$
−0.752563 + 0.658520i $$0.771183\pi$$
$$368$$ 0 0
$$369$$ 3050.54 0.430365
$$370$$ 0 0
$$371$$ 10001.5 1.39960
$$372$$ 0 0
$$373$$ 0.122254 1.69707e−5 0 8.48537e−6 1.00000i $$-0.499997\pi$$
8.48537e−6 1.00000i $$0.499997\pi$$
$$374$$ 0 0
$$375$$ 733.306 0.100981
$$376$$ 0 0
$$377$$ −11123.0 −1.51953
$$378$$ 0 0
$$379$$ 6236.92 0.845300 0.422650 0.906293i $$-0.361100\pi$$
0.422650 + 0.906293i $$0.361100\pi$$
$$380$$ 0 0
$$381$$ −1921.95 −0.258437
$$382$$ 0 0
$$383$$ −8554.88 −1.14134 −0.570671 0.821179i $$-0.693317\pi$$
−0.570671 + 0.821179i $$0.693317\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1438.85 −0.188995
$$388$$ 0 0
$$389$$ −3604.00 −0.469743 −0.234871 0.972026i $$-0.575467\pi$$
−0.234871 + 0.972026i $$0.575467\pi$$
$$390$$ 0 0
$$391$$ 3449.20 0.446122
$$392$$ 0 0
$$393$$ −1138.05 −0.146074
$$394$$ 0 0
$$395$$ 164.523 0.0209572
$$396$$ 0 0
$$397$$ 10515.4 1.32935 0.664674 0.747133i $$-0.268570\pi$$
0.664674 + 0.747133i $$0.268570\pi$$
$$398$$ 0 0
$$399$$ −6200.52 −0.777980
$$400$$ 0 0
$$401$$ −10070.2 −1.25407 −0.627035 0.778991i $$-0.715732\pi$$
−0.627035 + 0.778991i $$0.715732\pi$$
$$402$$ 0 0
$$403$$ −19802.9 −2.44778
$$404$$ 0 0
$$405$$ −79.5034 −0.00975446
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 146.224 0.0176780 0.00883900 0.999961i $$-0.497186\pi$$
0.00883900 + 0.999961i $$0.497186\pi$$
$$410$$ 0 0
$$411$$ −3579.51 −0.429596
$$412$$ 0 0
$$413$$ 7887.15 0.939712
$$414$$ 0 0
$$415$$ 350.994 0.0415171
$$416$$ 0 0
$$417$$ −8836.17 −1.03767
$$418$$ 0 0
$$419$$ −12686.3 −1.47915 −0.739577 0.673072i $$-0.764974\pi$$
−0.739577 + 0.673072i $$0.764974\pi$$
$$420$$ 0 0
$$421$$ −8158.85 −0.944508 −0.472254 0.881462i $$-0.656560\pi$$
−0.472254 + 0.881462i $$0.656560\pi$$
$$422$$ 0 0
$$423$$ 890.531 0.102362
$$424$$ 0 0
$$425$$ 16149.3 1.84319
$$426$$ 0 0
$$427$$ −9806.20 −1.11137
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −6289.60 −0.702922 −0.351461 0.936202i $$-0.614315\pi$$
−0.351461 + 0.936202i $$0.614315\pi$$
$$432$$ 0 0
$$433$$ 5741.35 0.637210 0.318605 0.947888i $$-0.396786\pi$$
0.318605 + 0.947888i $$0.396786\pi$$
$$434$$ 0 0
$$435$$ 529.145 0.0583231
$$436$$ 0 0
$$437$$ 3080.53 0.337212
$$438$$ 0 0
$$439$$ −3601.82 −0.391584 −0.195792 0.980645i $$-0.562728\pi$$
−0.195792 + 0.980645i $$0.562728\pi$$
$$440$$ 0 0
$$441$$ −243.608 −0.0263047
$$442$$ 0 0
$$443$$ −3487.54 −0.374036 −0.187018 0.982356i $$-0.559882\pi$$
−0.187018 + 0.982356i $$0.559882\pi$$
$$444$$ 0 0
$$445$$ −1192.73 −0.127058
$$446$$ 0 0
$$447$$ −9916.99 −1.04935
$$448$$ 0 0
$$449$$ −9135.55 −0.960208 −0.480104 0.877212i $$-0.659401\pi$$
−0.480104 + 0.877212i $$0.659401\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 5012.24 0.519858
$$454$$ 0 0
$$455$$ −1079.86 −0.111262
$$456$$ 0 0
$$457$$ −858.984 −0.0879247 −0.0439623 0.999033i $$-0.513998\pi$$
−0.0439623 + 0.999033i $$0.513998\pi$$
$$458$$ 0 0
$$459$$ −3515.34 −0.357477
$$460$$ 0 0
$$461$$ 15172.7 1.53289 0.766447 0.642308i $$-0.222023\pi$$
0.766447 + 0.642308i $$0.222023\pi$$
$$462$$ 0 0
$$463$$ 808.597 0.0811634 0.0405817 0.999176i $$-0.487079\pi$$
0.0405817 + 0.999176i $$0.487079\pi$$
$$464$$ 0 0
$$465$$ 942.072 0.0939517
$$466$$ 0 0
$$467$$ 16387.7 1.62384 0.811921 0.583767i $$-0.198422\pi$$
0.811921 + 0.583767i $$0.198422\pi$$
$$468$$ 0 0
$$469$$ 8074.31 0.794962
$$470$$ 0 0
$$471$$ 1684.20 0.164764
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 14423.1 1.39322
$$476$$ 0 0
$$477$$ 5064.21 0.486110
$$478$$ 0 0
$$479$$ −18147.3 −1.73105 −0.865523 0.500870i $$-0.833014\pi$$
−0.865523 + 0.500870i $$0.833014\pi$$
$$480$$ 0 0
$$481$$ −11017.5 −1.04440
$$482$$ 0 0
$$483$$ −1412.65 −0.133080
$$484$$ 0 0
$$485$$ 1096.85 0.102691
$$486$$ 0 0
$$487$$ −6192.06 −0.576158 −0.288079 0.957607i $$-0.593017\pi$$
−0.288079 + 0.957607i $$0.593017\pi$$
$$488$$ 0 0
$$489$$ 11397.0 1.05397
$$490$$ 0 0
$$491$$ 691.676 0.0635741 0.0317871 0.999495i $$-0.489880\pi$$
0.0317871 + 0.999495i $$0.489880\pi$$
$$492$$ 0 0
$$493$$ 23396.8 2.13740
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −3808.31 −0.343715
$$498$$ 0 0
$$499$$ −6553.93 −0.587964 −0.293982 0.955811i $$-0.594981\pi$$
−0.293982 + 0.955811i $$0.594981\pi$$
$$500$$ 0 0
$$501$$ 2080.82 0.185557
$$502$$ 0 0
$$503$$ 7930.22 0.702964 0.351482 0.936195i $$-0.385678\pi$$
0.351482 + 0.936195i $$0.385678\pi$$
$$504$$ 0 0
$$505$$ 398.111 0.0350806
$$506$$ 0 0
$$507$$ 4902.62 0.429453
$$508$$ 0 0
$$509$$ 8186.17 0.712860 0.356430 0.934322i $$-0.383994\pi$$
0.356430 + 0.934322i $$0.383994\pi$$
$$510$$ 0 0
$$511$$ −2890.31 −0.250215
$$512$$ 0 0
$$513$$ −3139.59 −0.270207
$$514$$ 0 0
$$515$$ −436.104 −0.0373147
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 8300.85 0.702056
$$520$$ 0 0
$$521$$ −6788.92 −0.570879 −0.285440 0.958397i $$-0.592140\pi$$
−0.285440 + 0.958397i $$0.592140\pi$$
$$522$$ 0 0
$$523$$ 12912.6 1.07959 0.539797 0.841795i $$-0.318501\pi$$
0.539797 + 0.841795i $$0.318501\pi$$
$$524$$ 0 0
$$525$$ −6614.06 −0.549831
$$526$$ 0 0
$$527$$ 41654.8 3.44310
$$528$$ 0 0
$$529$$ −11465.2 −0.942317
$$530$$ 0 0
$$531$$ 3993.61 0.326380
$$532$$ 0 0
$$533$$ 20979.8 1.70495
$$534$$ 0 0
$$535$$ 1224.26 0.0989333
$$536$$ 0 0
$$537$$ 7922.28 0.636633
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −8014.83 −0.636940 −0.318470 0.947933i $$-0.603169\pi$$
−0.318470 + 0.947933i $$0.603169\pi$$
$$542$$ 0 0
$$543$$ −13198.7 −1.04311
$$544$$ 0 0
$$545$$ 1318.80 0.103654
$$546$$ 0 0
$$547$$ −9922.03 −0.775567 −0.387784 0.921750i $$-0.626759\pi$$
−0.387784 + 0.921750i $$0.626759\pi$$
$$548$$ 0 0
$$549$$ −4965.31 −0.386000
$$550$$ 0 0
$$551$$ 20895.9 1.61560
$$552$$ 0 0
$$553$$ −2979.37 −0.229106
$$554$$ 0 0
$$555$$ 524.129 0.0400865
$$556$$ 0 0
$$557$$ 22180.3 1.68727 0.843636 0.536916i $$-0.180411\pi$$
0.843636 + 0.536916i $$0.180411\pi$$
$$558$$ 0 0
$$559$$ −9895.58 −0.748727
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −16205.7 −1.21312 −0.606562 0.795036i $$-0.707452\pi$$
−0.606562 + 0.795036i $$0.707452\pi$$
$$564$$ 0 0
$$565$$ −1171.58 −0.0872365
$$566$$ 0 0
$$567$$ 1439.73 0.106637
$$568$$ 0 0
$$569$$ 17252.2 1.27109 0.635543 0.772065i $$-0.280776\pi$$
0.635543 + 0.772065i $$0.280776\pi$$
$$570$$ 0 0
$$571$$ 3002.90 0.220083 0.110042 0.993927i $$-0.464902\pi$$
0.110042 + 0.993927i $$0.464902\pi$$
$$572$$ 0 0
$$573$$ −4727.81 −0.344689
$$574$$ 0 0
$$575$$ 3285.98 0.238322
$$576$$ 0 0
$$577$$ −11249.6 −0.811659 −0.405830 0.913949i $$-0.633017\pi$$
−0.405830 + 0.913949i $$0.633017\pi$$
$$578$$ 0 0
$$579$$ 561.639 0.0403124
$$580$$ 0 0
$$581$$ −6356.17 −0.453870
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −546.778 −0.0386436
$$586$$ 0 0
$$587$$ 12915.4 0.908139 0.454070 0.890966i $$-0.349972\pi$$
0.454070 + 0.890966i $$0.349972\pi$$
$$588$$ 0 0
$$589$$ 37202.4 2.60255
$$590$$ 0 0
$$591$$ −4395.20 −0.305913
$$592$$ 0 0
$$593$$ 8524.23 0.590300 0.295150 0.955451i $$-0.404630\pi$$
0.295150 + 0.955451i $$0.404630\pi$$
$$594$$ 0 0
$$595$$ 2271.44 0.156504
$$596$$ 0 0
$$597$$ 10559.5 0.723905
$$598$$ 0 0
$$599$$ 9761.84 0.665873 0.332936 0.942949i $$-0.391961\pi$$
0.332936 + 0.942949i $$0.391961\pi$$
$$600$$ 0 0
$$601$$ −3962.71 −0.268956 −0.134478 0.990917i $$-0.542936\pi$$
−0.134478 + 0.990917i $$0.542936\pi$$
$$602$$ 0 0
$$603$$ 4088.38 0.276105
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −27291.7 −1.82493 −0.912467 0.409151i $$-0.865825\pi$$
−0.912467 + 0.409151i $$0.865825\pi$$
$$608$$ 0 0
$$609$$ −9582.32 −0.637595
$$610$$ 0 0
$$611$$ 6124.55 0.405520
$$612$$ 0 0
$$613$$ 9202.01 0.606306 0.303153 0.952942i $$-0.401961\pi$$
0.303153 + 0.952942i $$0.401961\pi$$
$$614$$ 0 0
$$615$$ −998.059 −0.0654400
$$616$$ 0 0
$$617$$ −21019.6 −1.37150 −0.685752 0.727835i $$-0.740526\pi$$
−0.685752 + 0.727835i $$0.740526\pi$$
$$618$$ 0 0
$$619$$ −28268.6 −1.83556 −0.917778 0.397094i $$-0.870019\pi$$
−0.917778 + 0.397094i $$0.870019\pi$$
$$620$$ 0 0
$$621$$ −715.285 −0.0462213
$$622$$ 0 0
$$623$$ 21599.3 1.38902
$$624$$ 0 0
$$625$$ 15264.7 0.976938
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 23175.0 1.46907
$$630$$ 0 0
$$631$$ 17270.9 1.08961 0.544806 0.838562i $$-0.316603\pi$$
0.544806 + 0.838562i $$0.316603\pi$$
$$632$$ 0 0
$$633$$ 1514.54 0.0950988
$$634$$ 0 0
$$635$$ 628.814 0.0392972
$$636$$ 0 0
$$637$$ −1675.39 −0.104210
$$638$$ 0 0
$$639$$ −1928.31 −0.119379
$$640$$ 0 0
$$641$$ −23785.2 −1.46562 −0.732809 0.680435i $$-0.761791\pi$$
−0.732809 + 0.680435i $$0.761791\pi$$
$$642$$ 0 0
$$643$$ 5723.75 0.351046 0.175523 0.984475i $$-0.443838\pi$$
0.175523 + 0.984475i $$0.443838\pi$$
$$644$$ 0 0
$$645$$ 470.756 0.0287380
$$646$$ 0 0
$$647$$ 11170.0 0.678731 0.339366 0.940655i $$-0.389788\pi$$
0.339366 + 0.940655i $$0.389788\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −17060.0 −1.02709
$$652$$ 0 0
$$653$$ 31473.7 1.88615 0.943077 0.332573i $$-0.107917\pi$$
0.943077 + 0.332573i $$0.107917\pi$$
$$654$$ 0 0
$$655$$ 372.341 0.0222115
$$656$$ 0 0
$$657$$ −1463.49 −0.0869044
$$658$$ 0 0
$$659$$ −20262.0 −1.19772 −0.598858 0.800855i $$-0.704379\pi$$
−0.598858 + 0.800855i $$0.704379\pi$$
$$660$$ 0 0
$$661$$ −6117.04 −0.359948 −0.179974 0.983671i $$-0.557601\pi$$
−0.179974 + 0.983671i $$0.557601\pi$$
$$662$$ 0 0
$$663$$ −24176.4 −1.41619
$$664$$ 0 0
$$665$$ 2028.65 0.118297
$$666$$ 0 0
$$667$$ 4760.67 0.276362
$$668$$ 0 0
$$669$$ −1799.19 −0.103977
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 15872.7 0.909137 0.454569 0.890712i $$-0.349793\pi$$
0.454569 + 0.890712i $$0.349793\pi$$
$$674$$ 0 0
$$675$$ −3348.99 −0.190967
$$676$$ 0 0
$$677$$ 18848.6 1.07003 0.535016 0.844842i $$-0.320305\pi$$
0.535016 + 0.844842i $$0.320305\pi$$
$$678$$ 0 0
$$679$$ −19862.9 −1.12263
$$680$$ 0 0
$$681$$ 7021.39 0.395096
$$682$$ 0 0
$$683$$ 2430.48 0.136164 0.0680819 0.997680i $$-0.478312\pi$$
0.0680819 + 0.997680i $$0.478312\pi$$
$$684$$ 0 0
$$685$$ 1171.12 0.0653232
$$686$$ 0 0
$$687$$ −10397.6 −0.577430
$$688$$ 0 0
$$689$$ 34828.7 1.92579
$$690$$ 0 0
$$691$$ 28647.5 1.57714 0.788568 0.614947i $$-0.210823\pi$$
0.788568 + 0.614947i $$0.210823\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 2890.97 0.157785
$$696$$ 0 0
$$697$$ −44130.4 −2.39822
$$698$$ 0 0
$$699$$ 8928.54 0.483131
$$700$$ 0 0
$$701$$ 3160.02 0.170260 0.0851300 0.996370i $$-0.472869\pi$$
0.0851300 + 0.996370i $$0.472869\pi$$
$$702$$ 0 0
$$703$$ 20697.9 1.11043
$$704$$ 0 0
$$705$$ −291.359 −0.0155649
$$706$$ 0 0
$$707$$ −7209.43 −0.383506
$$708$$ 0 0
$$709$$ 1450.00 0.0768066 0.0384033 0.999262i $$-0.487773\pi$$
0.0384033 + 0.999262i $$0.487773\pi$$
$$710$$ 0 0
$$711$$ −1508.58 −0.0795729
$$712$$ 0 0
$$713$$ 8475.74 0.445188
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 13138.1 0.684313
$$718$$ 0 0
$$719$$ 8831.15 0.458062 0.229031 0.973419i $$-0.426444\pi$$
0.229031 + 0.973419i $$0.426444\pi$$
$$720$$ 0 0
$$721$$ 7897.44 0.407928
$$722$$ 0 0
$$723$$ −12135.1 −0.624219
$$724$$ 0 0
$$725$$ 22289.6 1.14181
$$726$$ 0 0
$$727$$ 20055.4 1.02313 0.511565 0.859245i $$-0.329066\pi$$
0.511565 + 0.859245i $$0.329066\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 20815.0 1.05318
$$732$$ 0 0
$$733$$ 8074.12 0.406854 0.203427 0.979090i $$-0.434792\pi$$
0.203427 + 0.979090i $$0.434792\pi$$
$$734$$ 0 0
$$735$$ 79.7023 0.00399982
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 3891.01 0.193685 0.0968425 0.995300i $$-0.469126\pi$$
0.0968425 + 0.995300i $$0.469126\pi$$
$$740$$ 0 0
$$741$$ −21592.3 −1.07046
$$742$$ 0 0
$$743$$ 20048.6 0.989921 0.494960 0.868916i $$-0.335183\pi$$
0.494960 + 0.868916i $$0.335183\pi$$
$$744$$ 0 0
$$745$$ 3244.59 0.159560
$$746$$ 0 0
$$747$$ −3218.41 −0.157638
$$748$$ 0 0
$$749$$ −22170.2 −1.08155
$$750$$ 0 0
$$751$$ 23590.4 1.14624 0.573120 0.819472i $$-0.305733\pi$$
0.573120 + 0.819472i $$0.305733\pi$$
$$752$$ 0 0
$$753$$ −20589.0 −0.996419
$$754$$ 0 0
$$755$$ −1639.88 −0.0790481
$$756$$ 0 0
$$757$$ 13443.2 0.645446 0.322723 0.946493i $$-0.395402\pi$$
0.322723 + 0.946493i $$0.395402\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 22458.3 1.06980 0.534898 0.844917i $$-0.320350\pi$$
0.534898 + 0.844917i $$0.320350\pi$$
$$762$$ 0 0
$$763$$ −23882.3 −1.13315
$$764$$ 0 0
$$765$$ 1150.13 0.0543569
$$766$$ 0 0
$$767$$ 27465.7 1.29300
$$768$$ 0 0
$$769$$ 32749.5 1.53573 0.767865 0.640612i $$-0.221319\pi$$
0.767865 + 0.640612i $$0.221319\pi$$
$$770$$ 0 0
$$771$$ 19274.2 0.900316
$$772$$ 0 0
$$773$$ −1660.59 −0.0772671 −0.0386335 0.999253i $$-0.512300\pi$$
−0.0386335 + 0.999253i $$0.512300\pi$$
$$774$$ 0 0
$$775$$ 39683.7 1.83933
$$776$$ 0 0
$$777$$ −9491.49 −0.438231
$$778$$ 0 0
$$779$$ −39413.4 −1.81275
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −4851.95 −0.221449
$$784$$ 0 0
$$785$$ −551.027 −0.0250535
$$786$$ 0 0
$$787$$ −41436.3 −1.87680 −0.938402 0.345545i $$-0.887694\pi$$
−0.938402 + 0.345545i $$0.887694\pi$$
$$788$$ 0 0
$$789$$ −19533.0 −0.881359
$$790$$ 0 0
$$791$$ 21216.2 0.953679
$$792$$ 0 0
$$793$$ −34148.5 −1.52919
$$794$$ 0 0
$$795$$ −1656.88 −0.0739164
$$796$$ 0 0
$$797$$ 23056.3 1.02471 0.512356 0.858773i $$-0.328773\pi$$
0.512356 + 0.858773i $$0.328773\pi$$
$$798$$ 0 0
$$799$$ −12882.8 −0.570414
$$800$$ 0 0
$$801$$ 10936.7 0.482432
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 462.183 0.0202358
$$806$$ 0 0
$$807$$ 15088.6 0.658170
$$808$$ 0 0
$$809$$ −15626.1 −0.679091 −0.339546 0.940590i $$-0.610273\pi$$
−0.339546 + 0.940590i $$0.610273\pi$$
$$810$$ 0 0
$$811$$ −19198.1 −0.831241 −0.415620 0.909538i $$-0.636436\pi$$
−0.415620 + 0.909538i $$0.636436\pi$$
$$812$$ 0 0
$$813$$ 2702.03 0.116561
$$814$$ 0 0
$$815$$ −3728.82 −0.160264
$$816$$ 0 0
$$817$$ 18590.2 0.796068
$$818$$ 0 0
$$819$$ 9901.64 0.422456
$$820$$ 0 0
$$821$$ 16425.9 0.698257 0.349129 0.937075i $$-0.386478\pi$$
0.349129 + 0.937075i $$0.386478\pi$$
$$822$$ 0 0
$$823$$ −11524.3 −0.488107 −0.244054 0.969762i $$-0.578477\pi$$
−0.244054 + 0.969762i $$0.578477\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −32655.7 −1.37310 −0.686548 0.727084i $$-0.740875\pi$$
−0.686548 + 0.727084i $$0.740875\pi$$
$$828$$ 0 0
$$829$$ −15368.2 −0.643861 −0.321930 0.946763i $$-0.604332\pi$$
−0.321930 + 0.946763i $$0.604332\pi$$
$$830$$ 0 0
$$831$$ 13904.9 0.580451
$$832$$ 0 0
$$833$$ 3524.13 0.146583
$$834$$ 0 0
$$835$$ −680.791 −0.0282153
$$836$$ 0 0
$$837$$ −8638.25 −0.356728
$$838$$ 0 0
$$839$$ −40339.9 −1.65994 −0.829969 0.557809i $$-0.811642\pi$$
−0.829969 + 0.557809i $$0.811642\pi$$
$$840$$ 0 0
$$841$$ 7903.72 0.324069
$$842$$ 0 0
$$843$$ −2804.68 −0.114589
$$844$$ 0 0
$$845$$ −1604.01 −0.0653014
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 4007.75 0.162009
$$850$$ 0 0
$$851$$ 4715.54 0.189949
$$852$$ 0 0
$$853$$ 12664.0 0.508331 0.254166 0.967161i $$-0.418199\pi$$
0.254166 + 0.967161i $$0.418199\pi$$
$$854$$ 0 0
$$855$$ 1027.19 0.0410869
$$856$$ 0 0
$$857$$ 16651.8 0.663729 0.331865 0.943327i $$-0.392322\pi$$
0.331865 + 0.943327i $$0.392322\pi$$
$$858$$ 0 0
$$859$$ 23620.4 0.938205 0.469103 0.883144i $$-0.344577\pi$$
0.469103 + 0.883144i $$0.344577\pi$$
$$860$$ 0 0
$$861$$ 18073.9 0.715398
$$862$$ 0 0
$$863$$ −19172.7 −0.756252 −0.378126 0.925754i $$-0.623431\pi$$
−0.378126 + 0.925754i $$0.623431\pi$$
$$864$$ 0 0
$$865$$ −2715.83 −0.106752
$$866$$ 0 0
$$867$$ 36115.4 1.41470
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 28117.5 1.09383
$$872$$ 0 0
$$873$$ −10057.5 −0.389912
$$874$$ 0 0
$$875$$ 4344.71 0.167861
$$876$$ 0 0
$$877$$ −6727.62 −0.259037 −0.129519 0.991577i $$-0.541343\pi$$
−0.129519 + 0.991577i $$0.541343\pi$$
$$878$$ 0 0
$$879$$ −8918.47 −0.342221
$$880$$ 0 0
$$881$$ −38006.5 −1.45343 −0.726714 0.686940i $$-0.758953\pi$$
−0.726714 + 0.686940i $$0.758953\pi$$
$$882$$ 0 0
$$883$$ −34461.2 −1.31338 −0.656688 0.754163i $$-0.728043\pi$$
−0.656688 + 0.754163i $$0.728043\pi$$
$$884$$ 0 0
$$885$$ −1306.61 −0.0496284
$$886$$ 0 0
$$887$$ −48302.1 −1.82844 −0.914220 0.405218i $$-0.867196\pi$$
−0.914220 + 0.405218i $$0.867196\pi$$
$$888$$ 0 0
$$889$$ −11387.2 −0.429601
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −11505.8 −0.431160
$$894$$ 0 0
$$895$$ −2591.97 −0.0968045
$$896$$ 0 0
$$897$$ −4919.31 −0.183112
$$898$$ 0 0
$$899$$ 57492.9 2.13292
$$900$$ 0 0
$$901$$ −73261.0 −2.70885
$$902$$ 0 0
$$903$$ −8524.95 −0.314167
$$904$$ 0 0
$$905$$ 4318.27 0.158612
$$906$$ 0 0
$$907$$ 14658.4 0.536631 0.268316 0.963331i $$-0.413533\pi$$
0.268316 + 0.963331i $$0.413533\pi$$
$$908$$ 0 0
$$909$$ −3650.45 −0.133199
$$910$$ 0 0
$$911$$ −47552.5 −1.72940 −0.864701 0.502286i $$-0.832492\pi$$
−0.864701 + 0.502286i $$0.832492\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 1624.52 0.0586941
$$916$$ 0 0
$$917$$ −6742.74 −0.242819
$$918$$ 0 0
$$919$$ 8911.32 0.319867 0.159933 0.987128i $$-0.448872\pi$$
0.159933 + 0.987128i $$0.448872\pi$$
$$920$$ 0 0
$$921$$ −1933.88 −0.0691897
$$922$$ 0 0
$$923$$ −13261.8 −0.472934
$$924$$ 0 0
$$925$$ 22078.3 0.784789
$$926$$ 0 0
$$927$$ 3998.82 0.141681
$$928$$ 0 0
$$929$$ −8023.11 −0.283347 −0.141674 0.989913i $$-0.545248\pi$$
−0.141674 + 0.989913i $$0.545248\pi$$
$$930$$ 0 0
$$931$$ 3147.45 0.110798
$$932$$ 0 0
$$933$$ 13595.7 0.477067
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 20712.2 0.722131 0.361065 0.932541i $$-0.382413\pi$$
0.361065 + 0.932541i $$0.382413\pi$$
$$938$$ 0 0
$$939$$ 13941.2 0.484509
$$940$$ 0 0
$$941$$ 39715.8 1.37588 0.687938 0.725770i $$-0.258516\pi$$
0.687938 + 0.725770i $$0.258516\pi$$
$$942$$ 0 0
$$943$$ −8979.45 −0.310086
$$944$$ 0 0
$$945$$ −471.044 −0.0162149
$$946$$ 0 0
$$947$$ −5331.12 −0.182934 −0.0914668 0.995808i $$-0.529156\pi$$
−0.0914668 + 0.995808i $$0.529156\pi$$
$$948$$ 0 0
$$949$$ −10065.0 −0.344283
$$950$$ 0 0
$$951$$ −1884.33 −0.0642518
$$952$$ 0 0
$$953$$ 44970.8 1.52859 0.764295 0.644866i $$-0.223087\pi$$
0.764295 + 0.644866i $$0.223087\pi$$
$$954$$ 0 0
$$955$$ 1546.82 0.0524124
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −21208.0 −0.714120
$$960$$ 0 0
$$961$$ 72567.5 2.43589
$$962$$ 0 0
$$963$$ −11225.7 −0.375643
$$964$$ 0 0
$$965$$ −183.754 −0.00612979
$$966$$ 0 0
$$967$$ 15679.1 0.521412 0.260706 0.965418i $$-0.416045\pi$$
0.260706 + 0.965418i $$0.416045\pi$$
$$968$$ 0 0
$$969$$ 45418.6 1.50573
$$970$$ 0 0
$$971$$ −2532.55 −0.0837008 −0.0418504 0.999124i $$-0.513325\pi$$
−0.0418504 + 0.999124i $$0.513325\pi$$
$$972$$ 0 0
$$973$$ −52352.8 −1.72493
$$974$$ 0 0
$$975$$ −23032.4 −0.756540
$$976$$ 0 0
$$977$$ 30233.1 0.990014 0.495007 0.868889i $$-0.335165\pi$$
0.495007 + 0.868889i $$0.335165\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −12092.6 −0.393566
$$982$$ 0 0
$$983$$ 16703.1 0.541960 0.270980 0.962585i $$-0.412652\pi$$
0.270980 + 0.962585i $$0.412652\pi$$
$$984$$ 0 0
$$985$$ 1438.00 0.0465162
$$986$$ 0 0
$$987$$ 5276.25 0.170157
$$988$$ 0 0
$$989$$ 4235.35 0.136174
$$990$$ 0 0
$$991$$ 13840.2 0.443640 0.221820 0.975088i $$-0.428800\pi$$
0.221820 + 0.975088i $$0.428800\pi$$
$$992$$ 0 0
$$993$$ −4843.34 −0.154782
$$994$$ 0 0
$$995$$ −3454.80 −0.110075
$$996$$ 0 0
$$997$$ 41518.7 1.31887 0.659434 0.751763i $$-0.270796\pi$$
0.659434 + 0.751763i $$0.270796\pi$$
$$998$$ 0 0
$$999$$ −4805.95 −0.152206
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 1452.4.a.t.1.3 6
11.7 odd 10 132.4.i.c.49.2 12
11.8 odd 10 132.4.i.c.97.2 yes 12
11.10 odd 2 1452.4.a.u.1.3 6
33.8 even 10 396.4.j.c.361.2 12
33.29 even 10 396.4.j.c.181.2 12

By twisted newform
Twist Min Dim Char Parity Ord Type
132.4.i.c.49.2 12 11.7 odd 10
132.4.i.c.97.2 yes 12 11.8 odd 10
396.4.j.c.181.2 12 33.29 even 10
396.4.j.c.361.2 12 33.8 even 10
1452.4.a.t.1.3 6 1.1 even 1 trivial
1452.4.a.u.1.3 6 11.10 odd 2