# Properties

 Label 1216.4.a.n.1.1 Level $1216$ Weight $4$ Character 1216.1 Self dual yes Analytic conductor $71.746$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$71.7463225670$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{93})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 23$$ x^2 - x - 23 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-4.32183$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -17.2873 q^{5} +12.3563 q^{7} -26.0000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -17.2873 q^{5} +12.3563 q^{7} -26.0000 q^{9} -39.2873 q^{11} -18.9310 q^{13} -17.2873 q^{15} +100.149 q^{17} -19.0000 q^{19} +12.3563 q^{21} -158.218 q^{23} +173.851 q^{25} -53.0000 q^{27} -141.229 q^{29} -215.586 q^{31} -39.2873 q^{33} -213.608 q^{35} -209.862 q^{37} -18.9310 q^{39} +276.873 q^{41} +166.713 q^{43} +449.470 q^{45} +60.5524 q^{47} -190.321 q^{49} +100.149 q^{51} +76.2405 q^{53} +679.171 q^{55} -19.0000 q^{57} +575.575 q^{59} +91.0111 q^{61} -321.265 q^{63} +327.265 q^{65} -444.978 q^{67} -158.218 q^{69} +943.287 q^{71} -569.851 q^{73} +173.851 q^{75} -485.448 q^{77} -336.205 q^{79} +649.000 q^{81} +603.929 q^{83} -1731.31 q^{85} -141.229 q^{87} +1635.63 q^{89} -233.917 q^{91} -215.586 q^{93} +328.459 q^{95} +319.470 q^{97} +1021.47 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 4 q^{5} + 44 q^{7} - 52 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 4 * q^5 + 44 * q^7 - 52 * q^9 $$2 q + 2 q^{3} + 4 q^{5} + 44 q^{7} - 52 q^{9} - 40 q^{11} + 20 q^{13} + 4 q^{15} + 46 q^{17} - 38 q^{19} + 44 q^{21} - 220 q^{23} + 502 q^{25} - 106 q^{27} + 84 q^{29} - 84 q^{31} - 40 q^{33} + 460 q^{35} - 304 q^{37} + 20 q^{39} + 168 q^{41} + 372 q^{43} - 104 q^{45} + 584 q^{47} + 468 q^{49} + 46 q^{51} - 484 q^{53} + 664 q^{55} - 38 q^{57} + 1074 q^{59} - 88 q^{61} - 1144 q^{63} + 1156 q^{65} - 1430 q^{67} - 220 q^{69} + 1848 q^{71} - 1294 q^{73} + 502 q^{75} - 508 q^{77} + 832 q^{79} + 1298 q^{81} - 528 q^{83} - 2884 q^{85} + 84 q^{87} + 1844 q^{89} + 998 q^{91} - 84 q^{93} - 76 q^{95} - 364 q^{97} + 1040 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 4 * q^5 + 44 * q^7 - 52 * q^9 - 40 * q^11 + 20 * q^13 + 4 * q^15 + 46 * q^17 - 38 * q^19 + 44 * q^21 - 220 * q^23 + 502 * q^25 - 106 * q^27 + 84 * q^29 - 84 * q^31 - 40 * q^33 + 460 * q^35 - 304 * q^37 + 20 * q^39 + 168 * q^41 + 372 * q^43 - 104 * q^45 + 584 * q^47 + 468 * q^49 + 46 * q^51 - 484 * q^53 + 664 * q^55 - 38 * q^57 + 1074 * q^59 - 88 * q^61 - 1144 * q^63 + 1156 * q^65 - 1430 * q^67 - 220 * q^69 + 1848 * q^71 - 1294 * q^73 + 502 * q^75 - 508 * q^77 + 832 * q^79 + 1298 * q^81 - 528 * q^83 - 2884 * q^85 + 84 * q^87 + 1844 * q^89 + 998 * q^91 - 84 * q^93 - 76 * q^95 - 364 * q^97 + 1040 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.192450 0.0962250 0.995360i $$-0.469323\pi$$
0.0962250 + 0.995360i $$0.469323\pi$$
$$4$$ 0 0
$$5$$ −17.2873 −1.54622 −0.773112 0.634270i $$-0.781301\pi$$
−0.773112 + 0.634270i $$0.781301\pi$$
$$6$$ 0 0
$$7$$ 12.3563 0.667180 0.333590 0.942718i $$-0.391740\pi$$
0.333590 + 0.942718i $$0.391740\pi$$
$$8$$ 0 0
$$9$$ −26.0000 −0.962963
$$10$$ 0 0
$$11$$ −39.2873 −1.07687 −0.538435 0.842667i $$-0.680984\pi$$
−0.538435 + 0.842667i $$0.680984\pi$$
$$12$$ 0 0
$$13$$ −18.9310 −0.403885 −0.201942 0.979397i $$-0.564725\pi$$
−0.201942 + 0.979397i $$0.564725\pi$$
$$14$$ 0 0
$$15$$ −17.2873 −0.297571
$$16$$ 0 0
$$17$$ 100.149 1.42881 0.714404 0.699733i $$-0.246698\pi$$
0.714404 + 0.699733i $$0.246698\pi$$
$$18$$ 0 0
$$19$$ −19.0000 −0.229416
$$20$$ 0 0
$$21$$ 12.3563 0.128399
$$22$$ 0 0
$$23$$ −158.218 −1.43438 −0.717191 0.696877i $$-0.754572\pi$$
−0.717191 + 0.696877i $$0.754572\pi$$
$$24$$ 0 0
$$25$$ 173.851 1.39081
$$26$$ 0 0
$$27$$ −53.0000 −0.377772
$$28$$ 0 0
$$29$$ −141.229 −0.904332 −0.452166 0.891934i $$-0.649349\pi$$
−0.452166 + 0.891934i $$0.649349\pi$$
$$30$$ 0 0
$$31$$ −215.586 −1.24904 −0.624522 0.781008i $$-0.714706\pi$$
−0.624522 + 0.781008i $$0.714706\pi$$
$$32$$ 0 0
$$33$$ −39.2873 −0.207244
$$34$$ 0 0
$$35$$ −213.608 −1.03161
$$36$$ 0 0
$$37$$ −209.862 −0.932462 −0.466231 0.884663i $$-0.654388\pi$$
−0.466231 + 0.884663i $$0.654388\pi$$
$$38$$ 0 0
$$39$$ −18.9310 −0.0777277
$$40$$ 0 0
$$41$$ 276.873 1.05464 0.527321 0.849666i $$-0.323197\pi$$
0.527321 + 0.849666i $$0.323197\pi$$
$$42$$ 0 0
$$43$$ 166.713 0.591243 0.295621 0.955305i $$-0.404473\pi$$
0.295621 + 0.955305i $$0.404473\pi$$
$$44$$ 0 0
$$45$$ 449.470 1.48896
$$46$$ 0 0
$$47$$ 60.5524 0.187925 0.0939625 0.995576i $$-0.470047\pi$$
0.0939625 + 0.995576i $$0.470047\pi$$
$$48$$ 0 0
$$49$$ −190.321 −0.554871
$$50$$ 0 0
$$51$$ 100.149 0.274974
$$52$$ 0 0
$$53$$ 76.2405 0.197593 0.0987966 0.995108i $$-0.468501\pi$$
0.0987966 + 0.995108i $$0.468501\pi$$
$$54$$ 0 0
$$55$$ 679.171 1.66508
$$56$$ 0 0
$$57$$ −19.0000 −0.0441511
$$58$$ 0 0
$$59$$ 575.575 1.27006 0.635029 0.772488i $$-0.280988\pi$$
0.635029 + 0.772488i $$0.280988\pi$$
$$60$$ 0 0
$$61$$ 91.0111 0.191029 0.0955146 0.995428i $$-0.469550\pi$$
0.0955146 + 0.995428i $$0.469550\pi$$
$$62$$ 0 0
$$63$$ −321.265 −0.642470
$$64$$ 0 0
$$65$$ 327.265 0.624496
$$66$$ 0 0
$$67$$ −444.978 −0.811383 −0.405692 0.914010i $$-0.632969\pi$$
−0.405692 + 0.914010i $$0.632969\pi$$
$$68$$ 0 0
$$69$$ −158.218 −0.276047
$$70$$ 0 0
$$71$$ 943.287 1.57673 0.788363 0.615210i $$-0.210929\pi$$
0.788363 + 0.615210i $$0.210929\pi$$
$$72$$ 0 0
$$73$$ −569.851 −0.913644 −0.456822 0.889558i $$-0.651012\pi$$
−0.456822 + 0.889558i $$0.651012\pi$$
$$74$$ 0 0
$$75$$ 173.851 0.267661
$$76$$ 0 0
$$77$$ −485.448 −0.718466
$$78$$ 0 0
$$79$$ −336.205 −0.478810 −0.239405 0.970920i $$-0.576952\pi$$
−0.239405 + 0.970920i $$0.576952\pi$$
$$80$$ 0 0
$$81$$ 649.000 0.890261
$$82$$ 0 0
$$83$$ 603.929 0.798672 0.399336 0.916805i $$-0.369241\pi$$
0.399336 + 0.916805i $$0.369241\pi$$
$$84$$ 0 0
$$85$$ −1731.31 −2.20926
$$86$$ 0 0
$$87$$ −141.229 −0.174039
$$88$$ 0 0
$$89$$ 1635.63 1.94805 0.974025 0.226441i $$-0.0727089\pi$$
0.974025 + 0.226441i $$0.0727089\pi$$
$$90$$ 0 0
$$91$$ −233.917 −0.269464
$$92$$ 0 0
$$93$$ −215.586 −0.240378
$$94$$ 0 0
$$95$$ 328.459 0.354728
$$96$$ 0 0
$$97$$ 319.470 0.334405 0.167202 0.985923i $$-0.446527\pi$$
0.167202 + 0.985923i $$0.446527\pi$$
$$98$$ 0 0
$$99$$ 1021.47 1.03699
$$100$$ 0 0
$$101$$ −1595.19 −1.57156 −0.785781 0.618505i $$-0.787739\pi$$
−0.785781 + 0.618505i $$0.787739\pi$$
$$102$$ 0 0
$$103$$ 1927.51 1.84392 0.921959 0.387287i $$-0.126588\pi$$
0.921959 + 0.387287i $$0.126588\pi$$
$$104$$ 0 0
$$105$$ −213.608 −0.198533
$$106$$ 0 0
$$107$$ −717.089 −0.647884 −0.323942 0.946077i $$-0.605008\pi$$
−0.323942 + 0.946077i $$0.605008\pi$$
$$108$$ 0 0
$$109$$ −383.047 −0.336598 −0.168299 0.985736i $$-0.553827\pi$$
−0.168299 + 0.985736i $$0.553827\pi$$
$$110$$ 0 0
$$111$$ −209.862 −0.179452
$$112$$ 0 0
$$113$$ 622.917 0.518577 0.259288 0.965800i $$-0.416512\pi$$
0.259288 + 0.965800i $$0.416512\pi$$
$$114$$ 0 0
$$115$$ 2735.17 2.21787
$$116$$ 0 0
$$117$$ 492.205 0.388926
$$118$$ 0 0
$$119$$ 1237.48 0.953273
$$120$$ 0 0
$$121$$ 212.492 0.159648
$$122$$ 0 0
$$123$$ 276.873 0.202966
$$124$$ 0 0
$$125$$ −844.498 −0.604274
$$126$$ 0 0
$$127$$ 89.3095 0.0624011 0.0312005 0.999513i $$-0.490067\pi$$
0.0312005 + 0.999513i $$0.490067\pi$$
$$128$$ 0 0
$$129$$ 166.713 0.113785
$$130$$ 0 0
$$131$$ −348.690 −0.232559 −0.116279 0.993217i $$-0.537097\pi$$
−0.116279 + 0.993217i $$0.537097\pi$$
$$132$$ 0 0
$$133$$ −234.771 −0.153062
$$134$$ 0 0
$$135$$ 916.227 0.584120
$$136$$ 0 0
$$137$$ 1356.98 0.846237 0.423118 0.906074i $$-0.360935\pi$$
0.423118 + 0.906074i $$0.360935\pi$$
$$138$$ 0 0
$$139$$ −629.287 −0.383996 −0.191998 0.981395i $$-0.561497\pi$$
−0.191998 + 0.981395i $$0.561497\pi$$
$$140$$ 0 0
$$141$$ 60.5524 0.0361662
$$142$$ 0 0
$$143$$ 743.746 0.434931
$$144$$ 0 0
$$145$$ 2441.47 1.39830
$$146$$ 0 0
$$147$$ −190.321 −0.106785
$$148$$ 0 0
$$149$$ 872.641 0.479796 0.239898 0.970798i $$-0.422886\pi$$
0.239898 + 0.970798i $$0.422886\pi$$
$$150$$ 0 0
$$151$$ −1892.71 −1.02004 −0.510021 0.860162i $$-0.670362\pi$$
−0.510021 + 0.860162i $$0.670362\pi$$
$$152$$ 0 0
$$153$$ −2603.88 −1.37589
$$154$$ 0 0
$$155$$ 3726.90 1.93130
$$156$$ 0 0
$$157$$ 2939.51 1.49426 0.747129 0.664679i $$-0.231432\pi$$
0.747129 + 0.664679i $$0.231432\pi$$
$$158$$ 0 0
$$159$$ 76.2405 0.0380268
$$160$$ 0 0
$$161$$ −1955.00 −0.956991
$$162$$ 0 0
$$163$$ 3301.38 1.58640 0.793202 0.608959i $$-0.208413\pi$$
0.793202 + 0.608959i $$0.208413\pi$$
$$164$$ 0 0
$$165$$ 679.171 0.320445
$$166$$ 0 0
$$167$$ 2630.39 1.21884 0.609418 0.792849i $$-0.291403\pi$$
0.609418 + 0.792849i $$0.291403\pi$$
$$168$$ 0 0
$$169$$ −1838.62 −0.836877
$$170$$ 0 0
$$171$$ 494.000 0.220919
$$172$$ 0 0
$$173$$ 1482.41 0.651479 0.325740 0.945460i $$-0.394387\pi$$
0.325740 + 0.945460i $$0.394387\pi$$
$$174$$ 0 0
$$175$$ 2148.16 0.927918
$$176$$ 0 0
$$177$$ 575.575 0.244423
$$178$$ 0 0
$$179$$ 1048.00 0.437604 0.218802 0.975769i $$-0.429785\pi$$
0.218802 + 0.975769i $$0.429785\pi$$
$$180$$ 0 0
$$181$$ 761.354 0.312657 0.156329 0.987705i $$-0.450034\pi$$
0.156329 + 0.987705i $$0.450034\pi$$
$$182$$ 0 0
$$183$$ 91.0111 0.0367636
$$184$$ 0 0
$$185$$ 3627.95 1.44179
$$186$$ 0 0
$$187$$ −3934.59 −1.53864
$$188$$ 0 0
$$189$$ −654.887 −0.252042
$$190$$ 0 0
$$191$$ −4005.98 −1.51760 −0.758802 0.651322i $$-0.774215\pi$$
−0.758802 + 0.651322i $$0.774215\pi$$
$$192$$ 0 0
$$193$$ −3855.87 −1.43809 −0.719046 0.694962i $$-0.755421\pi$$
−0.719046 + 0.694962i $$0.755421\pi$$
$$194$$ 0 0
$$195$$ 327.265 0.120184
$$196$$ 0 0
$$197$$ 2719.72 0.983616 0.491808 0.870704i $$-0.336336\pi$$
0.491808 + 0.870704i $$0.336336\pi$$
$$198$$ 0 0
$$199$$ 4454.56 1.58681 0.793405 0.608694i $$-0.208306\pi$$
0.793405 + 0.608694i $$0.208306\pi$$
$$200$$ 0 0
$$201$$ −444.978 −0.156151
$$202$$ 0 0
$$203$$ −1745.08 −0.603353
$$204$$ 0 0
$$205$$ −4786.39 −1.63071
$$206$$ 0 0
$$207$$ 4113.67 1.38126
$$208$$ 0 0
$$209$$ 746.459 0.247051
$$210$$ 0 0
$$211$$ −3481.06 −1.13576 −0.567881 0.823111i $$-0.692237\pi$$
−0.567881 + 0.823111i $$0.692237\pi$$
$$212$$ 0 0
$$213$$ 943.287 0.303441
$$214$$ 0 0
$$215$$ −2882.01 −0.914194
$$216$$ 0 0
$$217$$ −2663.85 −0.833337
$$218$$ 0 0
$$219$$ −569.851 −0.175831
$$220$$ 0 0
$$221$$ −1895.92 −0.577074
$$222$$ 0 0
$$223$$ −3193.81 −0.959074 −0.479537 0.877522i $$-0.659195\pi$$
−0.479537 + 0.877522i $$0.659195\pi$$
$$224$$ 0 0
$$225$$ −4520.12 −1.33930
$$226$$ 0 0
$$227$$ 5139.70 1.50279 0.751396 0.659852i $$-0.229381\pi$$
0.751396 + 0.659852i $$0.229381\pi$$
$$228$$ 0 0
$$229$$ −4310.87 −1.24397 −0.621987 0.783027i $$-0.713674\pi$$
−0.621987 + 0.783027i $$0.713674\pi$$
$$230$$ 0 0
$$231$$ −485.448 −0.138269
$$232$$ 0 0
$$233$$ 175.879 0.0494517 0.0247258 0.999694i $$-0.492129\pi$$
0.0247258 + 0.999694i $$0.492129\pi$$
$$234$$ 0 0
$$235$$ −1046.79 −0.290574
$$236$$ 0 0
$$237$$ −336.205 −0.0921470
$$238$$ 0 0
$$239$$ −3811.82 −1.03166 −0.515829 0.856692i $$-0.672516\pi$$
−0.515829 + 0.856692i $$0.672516\pi$$
$$240$$ 0 0
$$241$$ −1356.97 −0.362698 −0.181349 0.983419i $$-0.558046\pi$$
−0.181349 + 0.983419i $$0.558046\pi$$
$$242$$ 0 0
$$243$$ 2080.00 0.549103
$$244$$ 0 0
$$245$$ 3290.13 0.857954
$$246$$ 0 0
$$247$$ 359.688 0.0926575
$$248$$ 0 0
$$249$$ 603.929 0.153704
$$250$$ 0 0
$$251$$ 5465.11 1.37432 0.687160 0.726506i $$-0.258857\pi$$
0.687160 + 0.726506i $$0.258857\pi$$
$$252$$ 0 0
$$253$$ 6215.97 1.54464
$$254$$ 0 0
$$255$$ −1731.31 −0.425172
$$256$$ 0 0
$$257$$ 1249.60 0.303299 0.151649 0.988434i $$-0.451542\pi$$
0.151649 + 0.988434i $$0.451542\pi$$
$$258$$ 0 0
$$259$$ −2593.13 −0.622120
$$260$$ 0 0
$$261$$ 3671.96 0.870838
$$262$$ 0 0
$$263$$ 8043.06 1.88577 0.942883 0.333124i $$-0.108103\pi$$
0.942883 + 0.333124i $$0.108103\pi$$
$$264$$ 0 0
$$265$$ −1317.99 −0.305523
$$266$$ 0 0
$$267$$ 1635.63 0.374902
$$268$$ 0 0
$$269$$ −3051.13 −0.691565 −0.345782 0.938315i $$-0.612386\pi$$
−0.345782 + 0.938315i $$0.612386\pi$$
$$270$$ 0 0
$$271$$ −1967.96 −0.441127 −0.220563 0.975373i $$-0.570790\pi$$
−0.220563 + 0.975373i $$0.570790\pi$$
$$272$$ 0 0
$$273$$ −233.917 −0.0518583
$$274$$ 0 0
$$275$$ −6830.13 −1.49772
$$276$$ 0 0
$$277$$ 4077.51 0.884454 0.442227 0.896903i $$-0.354189\pi$$
0.442227 + 0.896903i $$0.354189\pi$$
$$278$$ 0 0
$$279$$ 5605.23 1.20278
$$280$$ 0 0
$$281$$ −7982.51 −1.69465 −0.847325 0.531075i $$-0.821788\pi$$
−0.847325 + 0.531075i $$0.821788\pi$$
$$282$$ 0 0
$$283$$ 4137.36 0.869049 0.434524 0.900660i $$-0.356917\pi$$
0.434524 + 0.900660i $$0.356917\pi$$
$$284$$ 0 0
$$285$$ 328.459 0.0682674
$$286$$ 0 0
$$287$$ 3421.14 0.703636
$$288$$ 0 0
$$289$$ 5116.86 1.04149
$$290$$ 0 0
$$291$$ 319.470 0.0643562
$$292$$ 0 0
$$293$$ −6807.50 −1.35733 −0.678666 0.734447i $$-0.737442\pi$$
−0.678666 + 0.734447i $$0.737442\pi$$
$$294$$ 0 0
$$295$$ −9950.13 −1.96379
$$296$$ 0 0
$$297$$ 2082.23 0.406812
$$298$$ 0 0
$$299$$ 2995.22 0.579325
$$300$$ 0 0
$$301$$ 2059.96 0.394466
$$302$$ 0 0
$$303$$ −1595.19 −0.302447
$$304$$ 0 0
$$305$$ −1573.34 −0.295374
$$306$$ 0 0
$$307$$ 2822.75 0.524765 0.262383 0.964964i $$-0.415492\pi$$
0.262383 + 0.964964i $$0.415492\pi$$
$$308$$ 0 0
$$309$$ 1927.51 0.354862
$$310$$ 0 0
$$311$$ 8221.36 1.49901 0.749503 0.662001i $$-0.230293\pi$$
0.749503 + 0.662001i $$0.230293\pi$$
$$312$$ 0 0
$$313$$ −1896.42 −0.342466 −0.171233 0.985231i $$-0.554775\pi$$
−0.171233 + 0.985231i $$0.554775\pi$$
$$314$$ 0 0
$$315$$ 5553.81 0.993402
$$316$$ 0 0
$$317$$ 2875.66 0.509505 0.254753 0.967006i $$-0.418006\pi$$
0.254753 + 0.967006i $$0.418006\pi$$
$$318$$ 0 0
$$319$$ 5548.52 0.973848
$$320$$ 0 0
$$321$$ −717.089 −0.124685
$$322$$ 0 0
$$323$$ −1902.83 −0.327791
$$324$$ 0 0
$$325$$ −3291.16 −0.561725
$$326$$ 0 0
$$327$$ −383.047 −0.0647784
$$328$$ 0 0
$$329$$ 748.206 0.125380
$$330$$ 0 0
$$331$$ −9263.48 −1.53827 −0.769134 0.639088i $$-0.779312\pi$$
−0.769134 + 0.639088i $$0.779312\pi$$
$$332$$ 0 0
$$333$$ 5456.41 0.897926
$$334$$ 0 0
$$335$$ 7692.47 1.25458
$$336$$ 0 0
$$337$$ −2071.30 −0.334810 −0.167405 0.985888i $$-0.553539\pi$$
−0.167405 + 0.985888i $$0.553539\pi$$
$$338$$ 0 0
$$339$$ 622.917 0.0998001
$$340$$ 0 0
$$341$$ 8469.78 1.34506
$$342$$ 0 0
$$343$$ −6589.90 −1.03738
$$344$$ 0 0
$$345$$ 2735.17 0.426830
$$346$$ 0 0
$$347$$ 8255.77 1.27721 0.638607 0.769533i $$-0.279511\pi$$
0.638607 + 0.769533i $$0.279511\pi$$
$$348$$ 0 0
$$349$$ 6600.97 1.01244 0.506220 0.862404i $$-0.331042\pi$$
0.506220 + 0.862404i $$0.331042\pi$$
$$350$$ 0 0
$$351$$ 1003.34 0.152577
$$352$$ 0 0
$$353$$ −6136.89 −0.925308 −0.462654 0.886539i $$-0.653103\pi$$
−0.462654 + 0.886539i $$0.653103\pi$$
$$354$$ 0 0
$$355$$ −16306.9 −2.43797
$$356$$ 0 0
$$357$$ 1237.48 0.183457
$$358$$ 0 0
$$359$$ 4314.86 0.634344 0.317172 0.948368i $$-0.397267\pi$$
0.317172 + 0.948368i $$0.397267\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ 0 0
$$363$$ 212.492 0.0307244
$$364$$ 0 0
$$365$$ 9851.18 1.41270
$$366$$ 0 0
$$367$$ −2643.59 −0.376006 −0.188003 0.982168i $$-0.560202\pi$$
−0.188003 + 0.982168i $$0.560202\pi$$
$$368$$ 0 0
$$369$$ −7198.70 −1.01558
$$370$$ 0 0
$$371$$ 942.054 0.131830
$$372$$ 0 0
$$373$$ −3643.51 −0.505774 −0.252887 0.967496i $$-0.581380\pi$$
−0.252887 + 0.967496i $$0.581380\pi$$
$$374$$ 0 0
$$375$$ −844.498 −0.116293
$$376$$ 0 0
$$377$$ 2673.61 0.365246
$$378$$ 0 0
$$379$$ 9731.09 1.31887 0.659436 0.751761i $$-0.270795\pi$$
0.659436 + 0.751761i $$0.270795\pi$$
$$380$$ 0 0
$$381$$ 89.3095 0.0120091
$$382$$ 0 0
$$383$$ 4167.84 0.556049 0.278025 0.960574i $$-0.410320\pi$$
0.278025 + 0.960574i $$0.410320\pi$$
$$384$$ 0 0
$$385$$ 8392.08 1.11091
$$386$$ 0 0
$$387$$ −4334.53 −0.569345
$$388$$ 0 0
$$389$$ 4267.52 0.556226 0.278113 0.960548i $$-0.410291\pi$$
0.278113 + 0.960548i $$0.410291\pi$$
$$390$$ 0 0
$$391$$ −15845.4 −2.04946
$$392$$ 0 0
$$393$$ −348.690 −0.0447560
$$394$$ 0 0
$$395$$ 5812.07 0.740347
$$396$$ 0 0
$$397$$ −7610.34 −0.962096 −0.481048 0.876694i $$-0.659744\pi$$
−0.481048 + 0.876694i $$0.659744\pi$$
$$398$$ 0 0
$$399$$ −234.771 −0.0294567
$$400$$ 0 0
$$401$$ −15827.1 −1.97099 −0.985495 0.169704i $$-0.945719\pi$$
−0.985495 + 0.169704i $$0.945719\pi$$
$$402$$ 0 0
$$403$$ 4081.24 0.504469
$$404$$ 0 0
$$405$$ −11219.5 −1.37654
$$406$$ 0 0
$$407$$ 8244.91 1.00414
$$408$$ 0 0
$$409$$ 5134.26 0.620716 0.310358 0.950620i $$-0.399551\pi$$
0.310358 + 0.950620i $$0.399551\pi$$
$$410$$ 0 0
$$411$$ 1356.98 0.162858
$$412$$ 0 0
$$413$$ 7112.00 0.847358
$$414$$ 0 0
$$415$$ −10440.3 −1.23493
$$416$$ 0 0
$$417$$ −629.287 −0.0739001
$$418$$ 0 0
$$419$$ 5437.64 0.634001 0.317000 0.948425i $$-0.397324\pi$$
0.317000 + 0.948425i $$0.397324\pi$$
$$420$$ 0 0
$$421$$ 1552.15 0.179685 0.0898423 0.995956i $$-0.471364\pi$$
0.0898423 + 0.995956i $$0.471364\pi$$
$$422$$ 0 0
$$423$$ −1574.36 −0.180965
$$424$$ 0 0
$$425$$ 17411.0 1.98720
$$426$$ 0 0
$$427$$ 1124.57 0.127451
$$428$$ 0 0
$$429$$ 743.746 0.0837026
$$430$$ 0 0
$$431$$ 5106.26 0.570673 0.285336 0.958427i $$-0.407895\pi$$
0.285336 + 0.958427i $$0.407895\pi$$
$$432$$ 0 0
$$433$$ 17422.7 1.93367 0.966835 0.255401i $$-0.0822076\pi$$
0.966835 + 0.255401i $$0.0822076\pi$$
$$434$$ 0 0
$$435$$ 2441.47 0.269103
$$436$$ 0 0
$$437$$ 3006.15 0.329070
$$438$$ 0 0
$$439$$ −2682.09 −0.291592 −0.145796 0.989315i $$-0.546574\pi$$
−0.145796 + 0.989315i $$0.546574\pi$$
$$440$$ 0 0
$$441$$ 4948.34 0.534320
$$442$$ 0 0
$$443$$ 7407.64 0.794464 0.397232 0.917718i $$-0.369971\pi$$
0.397232 + 0.917718i $$0.369971\pi$$
$$444$$ 0 0
$$445$$ −28275.6 −3.01212
$$446$$ 0 0
$$447$$ 872.641 0.0923367
$$448$$ 0 0
$$449$$ −15930.7 −1.67442 −0.837212 0.546878i $$-0.815816\pi$$
−0.837212 + 0.546878i $$0.815816\pi$$
$$450$$ 0 0
$$451$$ −10877.6 −1.13571
$$452$$ 0 0
$$453$$ −1892.71 −0.196307
$$454$$ 0 0
$$455$$ 4043.80 0.416651
$$456$$ 0 0
$$457$$ −6952.91 −0.711693 −0.355846 0.934544i $$-0.615807\pi$$
−0.355846 + 0.934544i $$0.615807\pi$$
$$458$$ 0 0
$$459$$ −5307.91 −0.539765
$$460$$ 0 0
$$461$$ −13281.0 −1.34177 −0.670886 0.741561i $$-0.734086\pi$$
−0.670886 + 0.741561i $$0.734086\pi$$
$$462$$ 0 0
$$463$$ 6220.21 0.624358 0.312179 0.950023i $$-0.398941\pi$$
0.312179 + 0.950023i $$0.398941\pi$$
$$464$$ 0 0
$$465$$ 3726.90 0.371679
$$466$$ 0 0
$$467$$ −9875.06 −0.978508 −0.489254 0.872141i $$-0.662731\pi$$
−0.489254 + 0.872141i $$0.662731\pi$$
$$468$$ 0 0
$$469$$ −5498.30 −0.541339
$$470$$ 0 0
$$471$$ 2939.51 0.287570
$$472$$ 0 0
$$473$$ −6549.69 −0.636692
$$474$$ 0 0
$$475$$ −3303.17 −0.319073
$$476$$ 0 0
$$477$$ −1982.25 −0.190275
$$478$$ 0 0
$$479$$ −19014.0 −1.81372 −0.906861 0.421429i $$-0.861529\pi$$
−0.906861 + 0.421429i $$0.861529\pi$$
$$480$$ 0 0
$$481$$ 3972.89 0.376607
$$482$$ 0 0
$$483$$ −1955.00 −0.184173
$$484$$ 0 0
$$485$$ −5522.77 −0.517064
$$486$$ 0 0
$$487$$ −8907.07 −0.828784 −0.414392 0.910099i $$-0.636006\pi$$
−0.414392 + 0.910099i $$0.636006\pi$$
$$488$$ 0 0
$$489$$ 3301.38 0.305303
$$490$$ 0 0
$$491$$ −18451.8 −1.69596 −0.847981 0.530027i $$-0.822182\pi$$
−0.847981 + 0.530027i $$0.822182\pi$$
$$492$$ 0 0
$$493$$ −14144.0 −1.29212
$$494$$ 0 0
$$495$$ −17658.5 −1.60341
$$496$$ 0 0
$$497$$ 11655.6 1.05196
$$498$$ 0 0
$$499$$ −14771.6 −1.32519 −0.662594 0.748979i $$-0.730544\pi$$
−0.662594 + 0.748979i $$0.730544\pi$$
$$500$$ 0 0
$$501$$ 2630.39 0.234565
$$502$$ 0 0
$$503$$ −18175.4 −1.61114 −0.805570 0.592501i $$-0.798141\pi$$
−0.805570 + 0.592501i $$0.798141\pi$$
$$504$$ 0 0
$$505$$ 27576.6 2.42998
$$506$$ 0 0
$$507$$ −1838.62 −0.161057
$$508$$ 0 0
$$509$$ 5172.14 0.450395 0.225198 0.974313i $$-0.427697\pi$$
0.225198 + 0.974313i $$0.427697\pi$$
$$510$$ 0 0
$$511$$ −7041.28 −0.609565
$$512$$ 0 0
$$513$$ 1007.00 0.0866669
$$514$$ 0 0
$$515$$ −33321.5 −2.85111
$$516$$ 0 0
$$517$$ −2378.94 −0.202371
$$518$$ 0 0
$$519$$ 1482.41 0.125377
$$520$$ 0 0
$$521$$ −12920.1 −1.08645 −0.543225 0.839587i $$-0.682797\pi$$
−0.543225 + 0.839587i $$0.682797\pi$$
$$522$$ 0 0
$$523$$ 2486.41 0.207884 0.103942 0.994583i $$-0.466854\pi$$
0.103942 + 0.994583i $$0.466854\pi$$
$$524$$ 0 0
$$525$$ 2148.16 0.178578
$$526$$ 0 0
$$527$$ −21590.7 −1.78464
$$528$$ 0 0
$$529$$ 12866.0 1.05745
$$530$$ 0 0
$$531$$ −14964.9 −1.22302
$$532$$ 0 0
$$533$$ −5241.47 −0.425954
$$534$$ 0 0
$$535$$ 12396.5 1.00177
$$536$$ 0 0
$$537$$ 1048.00 0.0842170
$$538$$ 0 0
$$539$$ 7477.18 0.597523
$$540$$ 0 0
$$541$$ −4553.45 −0.361863 −0.180932 0.983496i $$-0.557911\pi$$
−0.180932 + 0.983496i $$0.557911\pi$$
$$542$$ 0 0
$$543$$ 761.354 0.0601710
$$544$$ 0 0
$$545$$ 6621.85 0.520456
$$546$$ 0 0
$$547$$ −18289.1 −1.42959 −0.714796 0.699333i $$-0.753480\pi$$
−0.714796 + 0.699333i $$0.753480\pi$$
$$548$$ 0 0
$$549$$ −2366.29 −0.183954
$$550$$ 0 0
$$551$$ 2683.36 0.207468
$$552$$ 0 0
$$553$$ −4154.26 −0.319453
$$554$$ 0 0
$$555$$ 3627.95 0.277473
$$556$$ 0 0
$$557$$ 21732.8 1.65323 0.826613 0.562771i $$-0.190265\pi$$
0.826613 + 0.562771i $$0.190265\pi$$
$$558$$ 0 0
$$559$$ −3156.03 −0.238794
$$560$$ 0 0
$$561$$ −3934.59 −0.296112
$$562$$ 0 0
$$563$$ 6231.21 0.466455 0.233227 0.972422i $$-0.425071\pi$$
0.233227 + 0.972422i $$0.425071\pi$$
$$564$$ 0 0
$$565$$ −10768.6 −0.801835
$$566$$ 0 0
$$567$$ 8019.27 0.593964
$$568$$ 0 0
$$569$$ −24670.7 −1.81766 −0.908829 0.417168i $$-0.863023\pi$$
−0.908829 + 0.417168i $$0.863023\pi$$
$$570$$ 0 0
$$571$$ 20912.5 1.53268 0.766342 0.642433i $$-0.222075\pi$$
0.766342 + 0.642433i $$0.222075\pi$$
$$572$$ 0 0
$$573$$ −4005.98 −0.292063
$$574$$ 0 0
$$575$$ −27506.4 −1.99495
$$576$$ 0 0
$$577$$ 20604.1 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$578$$ 0 0
$$579$$ −3855.87 −0.276761
$$580$$ 0 0
$$581$$ 7462.35 0.532858
$$582$$ 0 0
$$583$$ −2995.28 −0.212782
$$584$$ 0 0
$$585$$ −8508.89 −0.601366
$$586$$ 0 0
$$587$$ 5511.88 0.387563 0.193781 0.981045i $$-0.437925\pi$$
0.193781 + 0.981045i $$0.437925\pi$$
$$588$$ 0 0
$$589$$ 4096.13 0.286550
$$590$$ 0 0
$$591$$ 2719.72 0.189297
$$592$$ 0 0
$$593$$ 11836.1 0.819648 0.409824 0.912165i $$-0.365590\pi$$
0.409824 + 0.912165i $$0.365590\pi$$
$$594$$ 0 0
$$595$$ −21392.7 −1.47397
$$596$$ 0 0
$$597$$ 4454.56 0.305382
$$598$$ 0 0
$$599$$ 8407.13 0.573466 0.286733 0.958011i $$-0.407431\pi$$
0.286733 + 0.958011i $$0.407431\pi$$
$$600$$ 0 0
$$601$$ 11313.6 0.767872 0.383936 0.923360i $$-0.374568\pi$$
0.383936 + 0.923360i $$0.374568\pi$$
$$602$$ 0 0
$$603$$ 11569.4 0.781332
$$604$$ 0 0
$$605$$ −3673.41 −0.246852
$$606$$ 0 0
$$607$$ −1581.43 −0.105747 −0.0528733 0.998601i $$-0.516838\pi$$
−0.0528733 + 0.998601i $$0.516838\pi$$
$$608$$ 0 0
$$609$$ −1745.08 −0.116115
$$610$$ 0 0
$$611$$ −1146.31 −0.0759000
$$612$$ 0 0
$$613$$ 2299.35 0.151500 0.0757502 0.997127i $$-0.475865\pi$$
0.0757502 + 0.997127i $$0.475865\pi$$
$$614$$ 0 0
$$615$$ −4786.39 −0.313831
$$616$$ 0 0
$$617$$ −16144.5 −1.05341 −0.526703 0.850049i $$-0.676572\pi$$
−0.526703 + 0.850049i $$0.676572\pi$$
$$618$$ 0 0
$$619$$ −2734.58 −0.177564 −0.0887819 0.996051i $$-0.528297\pi$$
−0.0887819 + 0.996051i $$0.528297\pi$$
$$620$$ 0 0
$$621$$ 8385.57 0.541870
$$622$$ 0 0
$$623$$ 20210.4 1.29970
$$624$$ 0 0
$$625$$ −7132.25 −0.456464
$$626$$ 0 0
$$627$$ 746.459 0.0475450
$$628$$ 0 0
$$629$$ −21017.5 −1.33231
$$630$$ 0 0
$$631$$ 15419.2 0.972787 0.486393 0.873740i $$-0.338312\pi$$
0.486393 + 0.873740i $$0.338312\pi$$
$$632$$ 0 0
$$633$$ −3481.06 −0.218578
$$634$$ 0 0
$$635$$ −1543.92 −0.0964860
$$636$$ 0 0
$$637$$ 3602.95 0.224104
$$638$$ 0 0
$$639$$ −24525.5 −1.51833
$$640$$ 0 0
$$641$$ −11905.5 −0.733600 −0.366800 0.930300i $$-0.619547\pi$$
−0.366800 + 0.930300i $$0.619547\pi$$
$$642$$ 0 0
$$643$$ 3525.47 0.216223 0.108111 0.994139i $$-0.465520\pi$$
0.108111 + 0.994139i $$0.465520\pi$$
$$644$$ 0 0
$$645$$ −2882.01 −0.175937
$$646$$ 0 0
$$647$$ 7502.61 0.455886 0.227943 0.973675i $$-0.426800\pi$$
0.227943 + 0.973675i $$0.426800\pi$$
$$648$$ 0 0
$$649$$ −22612.8 −1.36769
$$650$$ 0 0
$$651$$ −2663.85 −0.160376
$$652$$ 0 0
$$653$$ 6539.35 0.391891 0.195945 0.980615i $$-0.437222\pi$$
0.195945 + 0.980615i $$0.437222\pi$$
$$654$$ 0 0
$$655$$ 6027.92 0.359588
$$656$$ 0 0
$$657$$ 14816.1 0.879805
$$658$$ 0 0
$$659$$ 22229.4 1.31401 0.657007 0.753884i $$-0.271822\pi$$
0.657007 + 0.753884i $$0.271822\pi$$
$$660$$ 0 0
$$661$$ 19497.3 1.14729 0.573643 0.819105i $$-0.305530\pi$$
0.573643 + 0.819105i $$0.305530\pi$$
$$662$$ 0 0
$$663$$ −1895.92 −0.111058
$$664$$ 0 0
$$665$$ 4058.55 0.236667
$$666$$ 0 0
$$667$$ 22345.1 1.29716
$$668$$ 0 0
$$669$$ −3193.81 −0.184574
$$670$$ 0 0
$$671$$ −3575.58 −0.205714
$$672$$ 0 0
$$673$$ −6952.91 −0.398239 −0.199120 0.979975i $$-0.563808\pi$$
−0.199120 + 0.979975i $$0.563808\pi$$
$$674$$ 0 0
$$675$$ −9214.09 −0.525408
$$676$$ 0 0
$$677$$ −2010.38 −0.114129 −0.0570643 0.998371i $$-0.518174\pi$$
−0.0570643 + 0.998371i $$0.518174\pi$$
$$678$$ 0 0
$$679$$ 3947.48 0.223108
$$680$$ 0 0
$$681$$ 5139.70 0.289212
$$682$$ 0 0
$$683$$ −16401.7 −0.918877 −0.459439 0.888209i $$-0.651949\pi$$
−0.459439 + 0.888209i $$0.651949\pi$$
$$684$$ 0 0
$$685$$ −23458.5 −1.30847
$$686$$ 0 0
$$687$$ −4310.87 −0.239403
$$688$$ 0 0
$$689$$ −1443.30 −0.0798048
$$690$$ 0 0
$$691$$ 15967.5 0.879064 0.439532 0.898227i $$-0.355144\pi$$
0.439532 + 0.898227i $$0.355144\pi$$
$$692$$ 0 0
$$693$$ 12621.6 0.691856
$$694$$ 0 0
$$695$$ 10878.7 0.593744
$$696$$ 0 0
$$697$$ 27728.6 1.50688
$$698$$ 0 0
$$699$$ 175.879 0.00951698
$$700$$ 0 0
$$701$$ −22503.7 −1.21248 −0.606242 0.795280i $$-0.707324\pi$$
−0.606242 + 0.795280i $$0.707324\pi$$
$$702$$ 0 0
$$703$$ 3987.38 0.213921
$$704$$ 0 0
$$705$$ −1046.79 −0.0559210
$$706$$ 0 0
$$707$$ −19710.8 −1.04851
$$708$$ 0 0
$$709$$ −14922.1 −0.790426 −0.395213 0.918590i $$-0.629329\pi$$
−0.395213 + 0.918590i $$0.629329\pi$$
$$710$$ 0 0
$$711$$ 8741.32 0.461076
$$712$$ 0 0
$$713$$ 34109.6 1.79161
$$714$$ 0 0
$$715$$ −12857.4 −0.672501
$$716$$ 0 0
$$717$$ −3811.82 −0.198543
$$718$$ 0 0
$$719$$ −3483.14 −0.180667 −0.0903333 0.995912i $$-0.528793\pi$$
−0.0903333 + 0.995912i $$0.528793\pi$$
$$720$$ 0 0
$$721$$ 23817.0 1.23023
$$722$$ 0 0
$$723$$ −1356.97 −0.0698013
$$724$$ 0 0
$$725$$ −24552.8 −1.25775
$$726$$ 0 0
$$727$$ 6572.23 0.335283 0.167641 0.985848i $$-0.446385\pi$$
0.167641 + 0.985848i $$0.446385\pi$$
$$728$$ 0 0
$$729$$ −15443.0 −0.784586
$$730$$ 0 0
$$731$$ 16696.1 0.844773
$$732$$ 0 0
$$733$$ 27149.4 1.36806 0.684030 0.729454i $$-0.260226\pi$$
0.684030 + 0.729454i $$0.260226\pi$$
$$734$$ 0 0
$$735$$ 3290.13 0.165113
$$736$$ 0 0
$$737$$ 17482.0 0.873754
$$738$$ 0 0
$$739$$ 26251.0 1.30671 0.653355 0.757051i $$-0.273361\pi$$
0.653355 + 0.757051i $$0.273361\pi$$
$$740$$ 0 0
$$741$$ 359.688 0.0178319
$$742$$ 0 0
$$743$$ 10583.0 0.522545 0.261273 0.965265i $$-0.415858\pi$$
0.261273 + 0.965265i $$0.415858\pi$$
$$744$$ 0 0
$$745$$ −15085.6 −0.741871
$$746$$ 0 0
$$747$$ −15702.1 −0.769092
$$748$$ 0 0
$$749$$ −8860.60 −0.432255
$$750$$ 0 0
$$751$$ 9812.03 0.476759 0.238380 0.971172i $$-0.423384\pi$$
0.238380 + 0.971172i $$0.423384\pi$$
$$752$$ 0 0
$$753$$ 5465.11 0.264488
$$754$$ 0 0
$$755$$ 32719.8 1.57721
$$756$$ 0 0
$$757$$ −10954.1 −0.525938 −0.262969 0.964804i $$-0.584702\pi$$
−0.262969 + 0.964804i $$0.584702\pi$$
$$758$$ 0 0
$$759$$ 6215.97 0.297267
$$760$$ 0 0
$$761$$ −19350.2 −0.921739 −0.460869 0.887468i $$-0.652462\pi$$
−0.460869 + 0.887468i $$0.652462\pi$$
$$762$$ 0 0
$$763$$ −4733.06 −0.224572
$$764$$ 0 0
$$765$$ 45014.0 2.12743
$$766$$ 0 0
$$767$$ −10896.2 −0.512957
$$768$$ 0 0
$$769$$ 17844.5 0.836786 0.418393 0.908266i $$-0.362593\pi$$
0.418393 + 0.908266i $$0.362593\pi$$
$$770$$ 0 0
$$771$$ 1249.60 0.0583699
$$772$$ 0 0
$$773$$ 38019.3 1.76903 0.884514 0.466513i $$-0.154490\pi$$
0.884514 + 0.466513i $$0.154490\pi$$
$$774$$ 0 0
$$775$$ −37479.7 −1.73718
$$776$$ 0 0
$$777$$ −2593.13 −0.119727
$$778$$ 0 0
$$779$$ −5260.59 −0.241951
$$780$$ 0 0
$$781$$ −37059.2 −1.69793
$$782$$ 0 0
$$783$$ 7485.16 0.341632
$$784$$ 0 0
$$785$$ −50816.2 −2.31046
$$786$$ 0 0
$$787$$ −32884.2 −1.48945 −0.744724 0.667373i $$-0.767419\pi$$
−0.744724 + 0.667373i $$0.767419\pi$$
$$788$$ 0 0
$$789$$ 8043.06 0.362916
$$790$$ 0 0
$$791$$ 7696.99 0.345984
$$792$$ 0 0
$$793$$ −1722.93 −0.0771538
$$794$$ 0 0
$$795$$ −1317.99 −0.0587979
$$796$$ 0 0
$$797$$ −4141.59 −0.184069 −0.0920344 0.995756i $$-0.529337\pi$$
−0.0920344 + 0.995756i $$0.529337\pi$$
$$798$$ 0 0
$$799$$ 6064.27 0.268509
$$800$$ 0 0
$$801$$ −42526.4 −1.87590
$$802$$ 0 0
$$803$$ 22387.9 0.983875
$$804$$ 0 0
$$805$$ 33796.7 1.47972
$$806$$ 0 0
$$807$$ −3051.13 −0.133092
$$808$$ 0 0
$$809$$ 17007.2 0.739111 0.369555 0.929209i $$-0.379510\pi$$
0.369555 + 0.929209i $$0.379510\pi$$
$$810$$ 0 0
$$811$$ 18814.0 0.814611 0.407306 0.913292i $$-0.366468\pi$$
0.407306 + 0.913292i $$0.366468\pi$$
$$812$$ 0 0
$$813$$ −1967.96 −0.0848949
$$814$$ 0 0
$$815$$ −57071.9 −2.45293
$$816$$ 0 0
$$817$$ −3167.54 −0.135640
$$818$$ 0 0
$$819$$ 6081.85 0.259484
$$820$$ 0 0
$$821$$ 30735.2 1.30653 0.653267 0.757128i $$-0.273398\pi$$
0.653267 + 0.757128i $$0.273398\pi$$
$$822$$ 0 0
$$823$$ 11638.0 0.492923 0.246461 0.969153i $$-0.420732\pi$$
0.246461 + 0.969153i $$0.420732\pi$$
$$824$$ 0 0
$$825$$ −6830.13 −0.288236
$$826$$ 0 0
$$827$$ −4691.30 −0.197258 −0.0986291 0.995124i $$-0.531446\pi$$
−0.0986291 + 0.995124i $$0.531446\pi$$
$$828$$ 0 0
$$829$$ 12990.2 0.544232 0.272116 0.962264i $$-0.412276\pi$$
0.272116 + 0.962264i $$0.412276\pi$$
$$830$$ 0 0
$$831$$ 4077.51 0.170213
$$832$$ 0 0
$$833$$ −19060.5 −0.792804
$$834$$ 0 0
$$835$$ −45472.3 −1.88459
$$836$$ 0 0
$$837$$ 11426.0 0.471854
$$838$$ 0 0
$$839$$ 22152.7 0.911558 0.455779 0.890093i $$-0.349361\pi$$
0.455779 + 0.890093i $$0.349361\pi$$
$$840$$ 0 0
$$841$$ −4443.27 −0.182183
$$842$$ 0 0
$$843$$ −7982.51 −0.326136
$$844$$ 0 0
$$845$$ 31784.8 1.29400
$$846$$ 0 0
$$847$$ 2625.63 0.106514
$$848$$ 0 0
$$849$$ 4137.36 0.167248
$$850$$ 0 0
$$851$$ 33204.0 1.33751
$$852$$ 0 0
$$853$$ −27556.6 −1.10612 −0.553060 0.833141i $$-0.686540\pi$$
−0.553060 + 0.833141i $$0.686540\pi$$
$$854$$ 0 0
$$855$$ −8539.93 −0.341590
$$856$$ 0 0
$$857$$ −2728.49 −0.108755 −0.0543777 0.998520i $$-0.517318\pi$$
−0.0543777 + 0.998520i $$0.517318\pi$$
$$858$$ 0 0
$$859$$ −24012.7 −0.953786 −0.476893 0.878961i $$-0.658237\pi$$
−0.476893 + 0.878961i $$0.658237\pi$$
$$860$$ 0 0
$$861$$ 3421.14 0.135415
$$862$$ 0 0
$$863$$ 23821.0 0.939603 0.469802 0.882772i $$-0.344325\pi$$
0.469802 + 0.882772i $$0.344325\pi$$
$$864$$ 0 0
$$865$$ −25626.9 −1.00733
$$866$$ 0 0
$$867$$ 5116.86 0.200436
$$868$$ 0 0
$$869$$ 13208.6 0.515616
$$870$$ 0 0
$$871$$ 8423.85 0.327705
$$872$$ 0 0
$$873$$ −8306.22 −0.322019
$$874$$ 0 0
$$875$$ −10434.9 −0.403160
$$876$$ 0 0
$$877$$ −19.8882 −0.000765765 0 −0.000382882 1.00000i $$-0.500122\pi$$
−0.000382882 1.00000i $$0.500122\pi$$
$$878$$ 0 0
$$879$$ −6807.50 −0.261219
$$880$$ 0 0
$$881$$ −846.013 −0.0323529 −0.0161764 0.999869i $$-0.505149\pi$$
−0.0161764 + 0.999869i $$0.505149\pi$$
$$882$$ 0 0
$$883$$ −14028.9 −0.534666 −0.267333 0.963604i $$-0.586142\pi$$
−0.267333 + 0.963604i $$0.586142\pi$$
$$884$$ 0 0
$$885$$ −9950.13 −0.377932
$$886$$ 0 0
$$887$$ −6993.33 −0.264727 −0.132364 0.991201i $$-0.542257\pi$$
−0.132364 + 0.991201i $$0.542257\pi$$
$$888$$ 0 0
$$889$$ 1103.54 0.0416328
$$890$$ 0 0
$$891$$ −25497.5 −0.958695
$$892$$ 0 0
$$893$$ −1150.50 −0.0431129
$$894$$ 0 0
$$895$$ −18117.1 −0.676634
$$896$$ 0 0
$$897$$ 2995.22 0.111491
$$898$$ 0 0
$$899$$ 30447.0 1.12955
$$900$$ 0 0
$$901$$ 7635.42 0.282323
$$902$$ 0 0
$$903$$ 2059.96 0.0759149
$$904$$ 0 0
$$905$$ −13161.8 −0.483438
$$906$$ 0 0
$$907$$ 26569.7 0.972693 0.486347 0.873766i $$-0.338329\pi$$
0.486347 + 0.873766i $$0.338329\pi$$
$$908$$ 0 0
$$909$$ 41475.0 1.51336
$$910$$ 0 0
$$911$$ 3037.05 0.110452 0.0552260 0.998474i $$-0.482412\pi$$
0.0552260 + 0.998474i $$0.482412\pi$$
$$912$$ 0 0
$$913$$ −23726.7 −0.860066
$$914$$ 0 0
$$915$$ −1573.34 −0.0568447
$$916$$ 0 0
$$917$$ −4308.54 −0.155159
$$918$$ 0 0
$$919$$ 30480.3 1.09407 0.547036 0.837109i $$-0.315756\pi$$
0.547036 + 0.837109i $$0.315756\pi$$
$$920$$ 0 0
$$921$$ 2822.75 0.100991
$$922$$ 0 0
$$923$$ −17857.3 −0.636816
$$924$$ 0 0
$$925$$ −36484.7 −1.29687
$$926$$ 0 0
$$927$$ −50115.4 −1.77563
$$928$$ 0 0
$$929$$ −12218.1 −0.431498 −0.215749 0.976449i $$-0.569219\pi$$
−0.215749 + 0.976449i $$0.569219\pi$$
$$930$$ 0 0
$$931$$ 3616.09 0.127296
$$932$$ 0 0
$$933$$ 8221.36 0.288484
$$934$$ 0 0
$$935$$ 68018.5 2.37908
$$936$$ 0 0
$$937$$ 29036.4 1.01236 0.506178 0.862429i $$-0.331058\pi$$
0.506178 + 0.862429i $$0.331058\pi$$
$$938$$ 0 0
$$939$$ −1896.42 −0.0659076
$$940$$ 0 0
$$941$$ 15188.6 0.526179 0.263090 0.964771i $$-0.415259\pi$$
0.263090 + 0.964771i $$0.415259\pi$$
$$942$$ 0 0
$$943$$ −43806.4 −1.51276
$$944$$ 0 0
$$945$$ 11321.2 0.389714
$$946$$ 0 0
$$947$$ 38267.7 1.31313 0.656565 0.754270i $$-0.272009\pi$$
0.656565 + 0.754270i $$0.272009\pi$$
$$948$$ 0 0
$$949$$ 10787.8 0.369007
$$950$$ 0 0
$$951$$ 2875.66 0.0980544
$$952$$ 0 0
$$953$$ −30092.6 −1.02287 −0.511434 0.859322i $$-0.670886\pi$$
−0.511434 + 0.859322i $$0.670886\pi$$
$$954$$ 0 0
$$955$$ 69252.5 2.34655
$$956$$ 0 0
$$957$$ 5548.52 0.187417
$$958$$ 0 0
$$959$$ 16767.3 0.564592
$$960$$ 0 0
$$961$$ 16686.2 0.560109
$$962$$ 0 0
$$963$$ 18644.3 0.623888
$$964$$ 0 0
$$965$$ 66657.7 2.22361
$$966$$ 0 0
$$967$$ −22521.1 −0.748943 −0.374472 0.927238i $$-0.622176\pi$$
−0.374472 + 0.927238i $$0.622176\pi$$
$$968$$ 0 0
$$969$$ −1902.83 −0.0630835
$$970$$ 0 0
$$971$$ 16791.8 0.554969 0.277485 0.960730i $$-0.410499\pi$$
0.277485 + 0.960730i $$0.410499\pi$$
$$972$$ 0 0
$$973$$ −7775.69 −0.256195
$$974$$ 0 0
$$975$$ −3291.16 −0.108104
$$976$$ 0 0
$$977$$ −28654.3 −0.938312 −0.469156 0.883115i $$-0.655442\pi$$
−0.469156 + 0.883115i $$0.655442\pi$$
$$978$$ 0 0
$$979$$ −64259.5 −2.09780
$$980$$ 0 0
$$981$$ 9959.22 0.324132
$$982$$ 0 0
$$983$$ 43981.3 1.42704 0.713522 0.700633i $$-0.247099\pi$$
0.713522 + 0.700633i $$0.247099\pi$$
$$984$$ 0 0
$$985$$ −47016.7 −1.52089
$$986$$ 0 0
$$987$$ 748.206 0.0241294
$$988$$ 0 0
$$989$$ −26377.0 −0.848068
$$990$$ 0 0
$$991$$ 10308.7 0.330442 0.165221 0.986257i $$-0.447166\pi$$
0.165221 + 0.986257i $$0.447166\pi$$
$$992$$ 0 0
$$993$$ −9263.48 −0.296040
$$994$$ 0 0
$$995$$ −77007.3 −2.45356
$$996$$ 0 0
$$997$$ −56141.7 −1.78337 −0.891687 0.452652i $$-0.850478\pi$$
−0.891687 + 0.452652i $$0.850478\pi$$
$$998$$ 0 0
$$999$$ 11122.7 0.352258
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 1216.4.a.n.1.1 2
4.3 odd 2 1216.4.a.i.1.1 2
8.3 odd 2 608.4.a.d.1.2 yes 2
8.5 even 2 608.4.a.c.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.c.1.2 2 8.5 even 2
608.4.a.d.1.2 yes 2 8.3 odd 2
1216.4.a.i.1.1 2 4.3 odd 2
1216.4.a.n.1.1 2 1.1 even 1 trivial