# Properties

 Label 2475.4.a.p.1.2 Level $2475$ Weight $4$ Character 2475.1 Self dual yes Analytic conductor $146.030$ 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] = [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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$5.42443$$ 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)

 $$f(q)$$ $$=$$ $$q+5.42443 q^{2} +21.4244 q^{4} +7.69772 q^{7} +72.8199 q^{8} +O(q^{10})$$ $$q+5.42443 q^{2} +21.4244 q^{4} +7.69772 q^{7} +72.8199 q^{8} +11.0000 q^{11} -24.8489 q^{13} +41.7557 q^{14} +223.611 q^{16} -15.9420 q^{17} +15.1511 q^{19} +59.6687 q^{22} +17.7557 q^{23} -134.791 q^{26} +164.919 q^{28} +128.547 q^{29} +219.395 q^{31} +630.402 q^{32} -86.4763 q^{34} -92.0703 q^{37} +82.1863 q^{38} +459.942 q^{41} -64.9648 q^{43} +235.669 q^{44} +96.3146 q^{46} +497.408 q^{47} -283.745 q^{49} -532.373 q^{52} -526.919 q^{53} +560.547 q^{56} +697.292 q^{58} +578.443 q^{59} -221.569 q^{61} +1190.09 q^{62} +1630.68 q^{64} +860.745 q^{67} -341.548 q^{68} -580.919 q^{71} -510.116 q^{73} -499.429 q^{74} +324.605 q^{76} +84.6749 q^{77} +1035.12 q^{79} +2494.92 q^{82} +606.211 q^{83} -352.397 q^{86} +801.018 q^{88} +23.4411 q^{89} -191.279 q^{91} +380.406 q^{92} +2698.15 q^{94} -719.490 q^{97} -1539.16 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 33 q^{4} - 24 q^{7} + 57 q^{8}+O(q^{10})$$ 2 * q + q^2 + 33 * q^4 - 24 * q^7 + 57 * q^8 $$2 q + q^{2} + 33 q^{4} - 24 q^{7} + 57 q^{8} + 22 q^{11} - 30 q^{13} + 182 q^{14} + 201 q^{16} + 106 q^{17} + 50 q^{19} + 11 q^{22} + 134 q^{23} - 112 q^{26} - 202 q^{28} + 198 q^{29} + 360 q^{31} + 857 q^{32} - 626 q^{34} + 328 q^{37} - 72 q^{38} + 782 q^{41} - 386 q^{43} + 363 q^{44} - 418 q^{46} + 266 q^{47} + 378 q^{49} - 592 q^{52} - 522 q^{53} + 1062 q^{56} + 390 q^{58} + 172 q^{59} - 778 q^{61} + 568 q^{62} + 809 q^{64} + 776 q^{67} + 1070 q^{68} - 630 q^{71} - 1296 q^{73} - 2358 q^{74} + 728 q^{76} - 264 q^{77} + 652 q^{79} + 1070 q^{82} - 324 q^{83} + 1068 q^{86} + 627 q^{88} + 756 q^{89} - 28 q^{91} + 1726 q^{92} + 3722 q^{94} + 452 q^{97} - 4467 q^{98}+O(q^{100})$$ 2 * q + q^2 + 33 * q^4 - 24 * q^7 + 57 * q^8 + 22 * q^11 - 30 * q^13 + 182 * q^14 + 201 * q^16 + 106 * q^17 + 50 * q^19 + 11 * q^22 + 134 * q^23 - 112 * q^26 - 202 * q^28 + 198 * q^29 + 360 * q^31 + 857 * q^32 - 626 * q^34 + 328 * q^37 - 72 * q^38 + 782 * q^41 - 386 * q^43 + 363 * q^44 - 418 * q^46 + 266 * q^47 + 378 * q^49 - 592 * q^52 - 522 * q^53 + 1062 * q^56 + 390 * q^58 + 172 * q^59 - 778 * q^61 + 568 * q^62 + 809 * q^64 + 776 * q^67 + 1070 * q^68 - 630 * q^71 - 1296 * q^73 - 2358 * q^74 + 728 * q^76 - 264 * q^77 + 652 * q^79 + 1070 * q^82 - 324 * q^83 + 1068 * q^86 + 627 * q^88 + 756 * q^89 - 28 * q^91 + 1726 * q^92 + 3722 * q^94 + 452 * q^97 - 4467 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.42443 1.91783 0.958913 0.283702i $$-0.0915625\pi$$
0.958913 + 0.283702i $$0.0915625\pi$$
$$3$$ 0 0
$$4$$ 21.4244 2.67805
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 7.69772 0.415638 0.207819 0.978167i $$-0.433364\pi$$
0.207819 + 0.978167i $$0.433364\pi$$
$$8$$ 72.8199 3.21821
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ −24.8489 −0.530141 −0.265071 0.964229i $$-0.585395\pi$$
−0.265071 + 0.964229i $$0.585395\pi$$
$$14$$ 41.7557 0.797120
$$15$$ 0 0
$$16$$ 223.611 3.49392
$$17$$ −15.9420 −0.227441 −0.113721 0.993513i $$-0.536277\pi$$
−0.113721 + 0.993513i $$0.536277\pi$$
$$18$$ 0 0
$$19$$ 15.1511 0.182943 0.0914713 0.995808i $$-0.470843\pi$$
0.0914713 + 0.995808i $$0.470843\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 59.6687 0.578246
$$23$$ 17.7557 0.160971 0.0804853 0.996756i $$-0.474353\pi$$
0.0804853 + 0.996756i $$0.474353\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −134.791 −1.01672
$$27$$ 0 0
$$28$$ 164.919 1.11310
$$29$$ 128.547 0.823121 0.411560 0.911383i $$-0.364984\pi$$
0.411560 + 0.911383i $$0.364984\pi$$
$$30$$ 0 0
$$31$$ 219.395 1.27112 0.635558 0.772053i $$-0.280770\pi$$
0.635558 + 0.772053i $$0.280770\pi$$
$$32$$ 630.402 3.48251
$$33$$ 0 0
$$34$$ −86.4763 −0.436193
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −92.0703 −0.409088 −0.204544 0.978857i $$-0.565571\pi$$
−0.204544 + 0.978857i $$0.565571\pi$$
$$38$$ 82.1863 0.350852
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 459.942 1.75197 0.875986 0.482336i $$-0.160212\pi$$
0.875986 + 0.482336i $$0.160212\pi$$
$$42$$ 0 0
$$43$$ −64.9648 −0.230396 −0.115198 0.993343i $$-0.536750\pi$$
−0.115198 + 0.993343i $$0.536750\pi$$
$$44$$ 235.669 0.807464
$$45$$ 0 0
$$46$$ 96.3146 0.308713
$$47$$ 497.408 1.54371 0.771855 0.635799i $$-0.219329\pi$$
0.771855 + 0.635799i $$0.219329\pi$$
$$48$$ 0 0
$$49$$ −283.745 −0.827245
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −532.373 −1.41975
$$53$$ −526.919 −1.36562 −0.682811 0.730596i $$-0.739243\pi$$
−0.682811 + 0.730596i $$0.739243\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 560.547 1.33761
$$57$$ 0 0
$$58$$ 697.292 1.57860
$$59$$ 578.443 1.27639 0.638194 0.769876i $$-0.279682\pi$$
0.638194 + 0.769876i $$0.279682\pi$$
$$60$$ 0 0
$$61$$ −221.569 −0.465067 −0.232533 0.972588i $$-0.574701\pi$$
−0.232533 + 0.972588i $$0.574701\pi$$
$$62$$ 1190.09 2.43778
$$63$$ 0 0
$$64$$ 1630.68 3.18493
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 860.745 1.56950 0.784752 0.619810i $$-0.212790\pi$$
0.784752 + 0.619810i $$0.212790\pi$$
$$68$$ −341.548 −0.609100
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −580.919 −0.971020 −0.485510 0.874231i $$-0.661366\pi$$
−0.485510 + 0.874231i $$0.661366\pi$$
$$72$$ 0 0
$$73$$ −510.116 −0.817871 −0.408935 0.912563i $$-0.634100\pi$$
−0.408935 + 0.912563i $$0.634100\pi$$
$$74$$ −499.429 −0.784560
$$75$$ 0 0
$$76$$ 324.605 0.489930
$$77$$ 84.6749 0.125319
$$78$$ 0 0
$$79$$ 1035.12 1.47418 0.737088 0.675797i $$-0.236200\pi$$
0.737088 + 0.675797i $$0.236200\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 2494.92 3.35998
$$83$$ 606.211 0.801690 0.400845 0.916146i $$-0.368717\pi$$
0.400845 + 0.916146i $$0.368717\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −352.397 −0.441860
$$87$$ 0 0
$$88$$ 801.018 0.970328
$$89$$ 23.4411 0.0279186 0.0139593 0.999903i $$-0.495556\pi$$
0.0139593 + 0.999903i $$0.495556\pi$$
$$90$$ 0 0
$$91$$ −191.279 −0.220347
$$92$$ 380.406 0.431088
$$93$$ 0 0
$$94$$ 2698.15 2.96057
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −719.490 −0.753126 −0.376563 0.926391i $$-0.622894\pi$$
−0.376563 + 0.926391i $$0.622894\pi$$
$$98$$ −1539.16 −1.58651
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1871.27 −1.84355 −0.921774 0.387727i $$-0.873260\pi$$
−0.921774 + 0.387727i $$0.873260\pi$$
$$102$$ 0 0
$$103$$ 428.745 0.410151 0.205075 0.978746i $$-0.434256\pi$$
0.205075 + 0.978746i $$0.434256\pi$$
$$104$$ −1809.49 −1.70611
$$105$$ 0 0
$$106$$ −2858.24 −2.61902
$$107$$ 1148.02 1.03723 0.518616 0.855008i $$-0.326448\pi$$
0.518616 + 0.855008i $$0.326448\pi$$
$$108$$ 0 0
$$109$$ −1828.32 −1.60662 −0.803308 0.595564i $$-0.796929\pi$$
−0.803308 + 0.595564i $$0.796929\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1721.29 1.45220
$$113$$ 1126.40 0.937722 0.468861 0.883272i $$-0.344665\pi$$
0.468861 + 0.883272i $$0.344665\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2754.04 2.20436
$$117$$ 0 0
$$118$$ 3137.72 2.44789
$$119$$ −122.717 −0.0945332
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ −1201.89 −0.891916
$$123$$ 0 0
$$124$$ 4700.42 3.40412
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −661.304 −0.462057 −0.231029 0.972947i $$-0.574209\pi$$
−0.231029 + 0.972947i $$0.574209\pi$$
$$128$$ 3802.31 2.62562
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −622.186 −0.414967 −0.207483 0.978239i $$-0.566527\pi$$
−0.207483 + 0.978239i $$0.566527\pi$$
$$132$$ 0 0
$$133$$ 116.629 0.0760378
$$134$$ 4669.05 3.01003
$$135$$ 0 0
$$136$$ −1160.89 −0.731955
$$137$$ −1872.84 −1.16794 −0.583969 0.811776i $$-0.698501\pi$$
−0.583969 + 0.811776i $$0.698501\pi$$
$$138$$ 0 0
$$139$$ −954.058 −0.582174 −0.291087 0.956697i $$-0.594017\pi$$
−0.291087 + 0.956697i $$0.594017\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −3151.15 −1.86225
$$143$$ −273.337 −0.159844
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2767.09 −1.56853
$$147$$ 0 0
$$148$$ −1972.55 −1.09556
$$149$$ 2047.01 1.12549 0.562745 0.826631i $$-0.309745\pi$$
0.562745 + 0.826631i $$0.309745\pi$$
$$150$$ 0 0
$$151$$ 475.863 0.256458 0.128229 0.991745i $$-0.459071\pi$$
0.128229 + 0.991745i $$0.459071\pi$$
$$152$$ 1103.30 0.588749
$$153$$ 0 0
$$154$$ 459.313 0.240341
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 647.466 0.329130 0.164565 0.986366i $$-0.447378\pi$$
0.164565 + 0.986366i $$0.447378\pi$$
$$158$$ 5614.92 2.82721
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 136.678 0.0669054
$$162$$ 0 0
$$163$$ −1093.23 −0.525329 −0.262665 0.964887i $$-0.584601\pi$$
−0.262665 + 0.964887i $$0.584601\pi$$
$$164$$ 9853.99 4.69188
$$165$$ 0 0
$$166$$ 3288.35 1.53750
$$167$$ 1123.25 0.520479 0.260240 0.965544i $$-0.416198\pi$$
0.260240 + 0.965544i $$0.416198\pi$$
$$168$$ 0 0
$$169$$ −1579.53 −0.718951
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −1391.83 −0.617014
$$173$$ 46.0123 0.0202211 0.0101106 0.999949i $$-0.496782\pi$$
0.0101106 + 0.999949i $$0.496782\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2459.72 1.05346
$$177$$ 0 0
$$178$$ 127.155 0.0535430
$$179$$ 831.975 0.347401 0.173700 0.984799i $$-0.444428\pi$$
0.173700 + 0.984799i $$0.444428\pi$$
$$180$$ 0 0
$$181$$ −1810.63 −0.743553 −0.371776 0.928322i $$-0.621251\pi$$
−0.371776 + 0.928322i $$0.621251\pi$$
$$182$$ −1037.58 −0.422586
$$183$$ 0 0
$$184$$ 1292.97 0.518037
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −175.362 −0.0685762
$$188$$ 10656.7 4.13414
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −458.898 −0.173847 −0.0869233 0.996215i $$-0.527704\pi$$
−0.0869233 + 0.996215i $$0.527704\pi$$
$$192$$ 0 0
$$193$$ −1778.91 −0.663465 −0.331733 0.943373i $$-0.607633\pi$$
−0.331733 + 0.943373i $$0.607633\pi$$
$$194$$ −3902.82 −1.44436
$$195$$ 0 0
$$196$$ −6079.08 −2.21541
$$197$$ 5304.53 1.91844 0.959218 0.282666i $$-0.0912188\pi$$
0.959218 + 0.282666i $$0.0912188\pi$$
$$198$$ 0 0
$$199$$ −5138.40 −1.83041 −0.915205 0.402989i $$-0.867971\pi$$
−0.915205 + 0.402989i $$0.867971\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −10150.6 −3.53560
$$203$$ 989.515 0.342120
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 2325.70 0.786597
$$207$$ 0 0
$$208$$ −5556.47 −1.85227
$$209$$ 166.663 0.0551593
$$210$$ 0 0
$$211$$ −4262.36 −1.39068 −0.695339 0.718682i $$-0.744746\pi$$
−0.695339 + 0.718682i $$0.744746\pi$$
$$212$$ −11288.9 −3.65721
$$213$$ 0 0
$$214$$ 6227.38 1.98923
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1688.84 0.528323
$$218$$ −9917.58 −3.08121
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 396.141 0.120576
$$222$$ 0 0
$$223$$ 1377.80 0.413740 0.206870 0.978368i $$-0.433672\pi$$
0.206870 + 0.978368i $$0.433672\pi$$
$$224$$ 4852.65 1.44746
$$225$$ 0 0
$$226$$ 6110.06 1.79839
$$227$$ −1227.28 −0.358843 −0.179422 0.983772i $$-0.557423\pi$$
−0.179422 + 0.983772i $$0.557423\pi$$
$$228$$ 0 0
$$229$$ 3890.28 1.12261 0.561304 0.827610i $$-0.310300\pi$$
0.561304 + 0.827610i $$0.310300\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 9360.74 2.64898
$$233$$ −3218.14 −0.904837 −0.452419 0.891806i $$-0.649439\pi$$
−0.452419 + 0.891806i $$0.649439\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 12392.8 3.41823
$$237$$ 0 0
$$238$$ −665.670 −0.181298
$$239$$ 428.098 0.115864 0.0579318 0.998321i $$-0.481549\pi$$
0.0579318 + 0.998321i $$0.481549\pi$$
$$240$$ 0 0
$$241$$ 1231.16 0.329070 0.164535 0.986371i $$-0.447388\pi$$
0.164535 + 0.986371i $$0.447388\pi$$
$$242$$ 656.356 0.174348
$$243$$ 0 0
$$244$$ −4747.00 −1.24547
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −376.489 −0.0969854
$$248$$ 15976.3 4.09072
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −2838.22 −0.713732 −0.356866 0.934156i $$-0.616155\pi$$
−0.356866 + 0.934156i $$0.616155\pi$$
$$252$$ 0 0
$$253$$ 195.313 0.0485344
$$254$$ −3587.20 −0.886145
$$255$$ 0 0
$$256$$ 7579.90 1.85056
$$257$$ 342.007 0.0830110 0.0415055 0.999138i $$-0.486785\pi$$
0.0415055 + 0.999138i $$0.486785\pi$$
$$258$$ 0 0
$$259$$ −708.731 −0.170032
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −3375.01 −0.795834
$$263$$ 5895.00 1.38213 0.691067 0.722791i $$-0.257141\pi$$
0.691067 + 0.722791i $$0.257141\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 632.647 0.145827
$$267$$ 0 0
$$268$$ 18441.0 4.20322
$$269$$ 2496.18 0.565779 0.282890 0.959152i $$-0.408707\pi$$
0.282890 + 0.959152i $$0.408707\pi$$
$$270$$ 0 0
$$271$$ −2249.68 −0.504274 −0.252137 0.967692i $$-0.581133\pi$$
−0.252137 + 0.967692i $$0.581133\pi$$
$$272$$ −3564.80 −0.794662
$$273$$ 0 0
$$274$$ −10159.1 −2.23990
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −4082.59 −0.885556 −0.442778 0.896631i $$-0.646007\pi$$
−0.442778 + 0.896631i $$0.646007\pi$$
$$278$$ −5175.22 −1.11651
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1033.79 0.219468 0.109734 0.993961i $$-0.465000\pi$$
0.109734 + 0.993961i $$0.465000\pi$$
$$282$$ 0 0
$$283$$ 7809.14 1.64030 0.820150 0.572148i $$-0.193890\pi$$
0.820150 + 0.572148i $$0.193890\pi$$
$$284$$ −12445.9 −2.60044
$$285$$ 0 0
$$286$$ −1482.70 −0.306552
$$287$$ 3540.50 0.728186
$$288$$ 0 0
$$289$$ −4658.85 −0.948270
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −10928.9 −2.19030
$$293$$ −1949.19 −0.388645 −0.194323 0.980938i $$-0.562251\pi$$
−0.194323 + 0.980938i $$0.562251\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −6704.55 −1.31653
$$297$$ 0 0
$$298$$ 11103.9 2.15849
$$299$$ −441.209 −0.0853371
$$300$$ 0 0
$$301$$ −500.081 −0.0957614
$$302$$ 2581.28 0.491842
$$303$$ 0 0
$$304$$ 3387.96 0.639187
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2364.09 −0.439497 −0.219748 0.975557i $$-0.570524\pi$$
−0.219748 + 0.975557i $$0.570524\pi$$
$$308$$ 1814.11 0.335612
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 1989.17 0.362686 0.181343 0.983420i $$-0.441956\pi$$
0.181343 + 0.983420i $$0.441956\pi$$
$$312$$ 0 0
$$313$$ 3878.67 0.700433 0.350216 0.936669i $$-0.386108\pi$$
0.350216 + 0.936669i $$0.386108\pi$$
$$314$$ 3512.13 0.631214
$$315$$ 0 0
$$316$$ 22176.8 3.94792
$$317$$ 2913.73 0.516251 0.258126 0.966111i $$-0.416895\pi$$
0.258126 + 0.966111i $$0.416895\pi$$
$$318$$ 0 0
$$319$$ 1414.01 0.248180
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 741.402 0.128313
$$323$$ −241.540 −0.0416087
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −5930.17 −1.00749
$$327$$ 0 0
$$328$$ 33492.9 5.63822
$$329$$ 3828.90 0.641624
$$330$$ 0 0
$$331$$ 8104.46 1.34580 0.672902 0.739731i $$-0.265047\pi$$
0.672902 + 0.739731i $$0.265047\pi$$
$$332$$ 12987.7 2.14697
$$333$$ 0 0
$$334$$ 6093.02 0.998189
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −5919.19 −0.956792 −0.478396 0.878144i $$-0.658782\pi$$
−0.478396 + 0.878144i $$0.658782\pi$$
$$338$$ −8568.07 −1.37882
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2413.35 0.383256
$$342$$ 0 0
$$343$$ −4824.51 −0.759472
$$344$$ −4730.73 −0.741465
$$345$$ 0 0
$$346$$ 249.590 0.0387805
$$347$$ −8540.59 −1.32128 −0.660638 0.750705i $$-0.729714\pi$$
−0.660638 + 0.750705i $$0.729714\pi$$
$$348$$ 0 0
$$349$$ 937.337 0.143767 0.0718833 0.997413i $$-0.477099\pi$$
0.0718833 + 0.997413i $$0.477099\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 6934.42 1.05002
$$353$$ −211.118 −0.0318319 −0.0159160 0.999873i $$-0.505066\pi$$
−0.0159160 + 0.999873i $$0.505066\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 502.213 0.0747675
$$357$$ 0 0
$$358$$ 4512.99 0.666254
$$359$$ 1376.31 0.202337 0.101169 0.994869i $$-0.467742\pi$$
0.101169 + 0.994869i $$0.467742\pi$$
$$360$$ 0 0
$$361$$ −6629.44 −0.966532
$$362$$ −9821.63 −1.42600
$$363$$ 0 0
$$364$$ −4098.05 −0.590100
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −1030.45 −0.146564 −0.0732821 0.997311i $$-0.523347\pi$$
−0.0732821 + 0.997311i $$0.523347\pi$$
$$368$$ 3970.37 0.562418
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −4056.07 −0.567603
$$372$$ 0 0
$$373$$ −9365.39 −1.30006 −0.650029 0.759909i $$-0.725243\pi$$
−0.650029 + 0.759909i $$0.725243\pi$$
$$374$$ −951.239 −0.131517
$$375$$ 0 0
$$376$$ 36221.2 4.96799
$$377$$ −3194.24 −0.436370
$$378$$ 0 0
$$379$$ −7120.23 −0.965017 −0.482509 0.875891i $$-0.660274\pi$$
−0.482509 + 0.875891i $$0.660274\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −2489.26 −0.333407
$$383$$ −1163.56 −0.155235 −0.0776176 0.996983i $$-0.524731\pi$$
−0.0776176 + 0.996983i $$0.524731\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −9649.57 −1.27241
$$387$$ 0 0
$$388$$ −15414.7 −2.01691
$$389$$ 10958.9 1.42838 0.714188 0.699954i $$-0.246796\pi$$
0.714188 + 0.699954i $$0.246796\pi$$
$$390$$ 0 0
$$391$$ −283.062 −0.0366114
$$392$$ −20662.3 −2.66225
$$393$$ 0 0
$$394$$ 28774.0 3.67923
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2172.09 0.274595 0.137298 0.990530i $$-0.456158\pi$$
0.137298 + 0.990530i $$0.456158\pi$$
$$398$$ −27872.9 −3.51041
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −7830.71 −0.975180 −0.487590 0.873073i $$-0.662124\pi$$
−0.487590 + 0.873073i $$0.662124\pi$$
$$402$$ 0 0
$$403$$ −5451.73 −0.673870
$$404$$ −40090.9 −4.93712
$$405$$ 0 0
$$406$$ 5367.55 0.656126
$$407$$ −1012.77 −0.123345
$$408$$ 0 0
$$409$$ −10731.2 −1.29736 −0.648682 0.761060i $$-0.724679\pi$$
−0.648682 + 0.761060i $$0.724679\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 9185.62 1.09841
$$413$$ 4452.69 0.530515
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −15664.8 −1.84622
$$417$$ 0 0
$$418$$ 904.049 0.105786
$$419$$ 7315.88 0.852994 0.426497 0.904489i $$-0.359748\pi$$
0.426497 + 0.904489i $$0.359748\pi$$
$$420$$ 0 0
$$421$$ −12495.7 −1.44657 −0.723284 0.690551i $$-0.757368\pi$$
−0.723284 + 0.690551i $$0.757368\pi$$
$$422$$ −23120.9 −2.66708
$$423$$ 0 0
$$424$$ −38370.2 −4.39486
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1705.58 −0.193299
$$428$$ 24595.8 2.77776
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −6075.01 −0.678939 −0.339470 0.940617i $$-0.610248\pi$$
−0.339470 + 0.940617i $$0.610248\pi$$
$$432$$ 0 0
$$433$$ −5641.79 −0.626160 −0.313080 0.949727i $$-0.601361\pi$$
−0.313080 + 0.949727i $$0.601361\pi$$
$$434$$ 9161.01 1.01323
$$435$$ 0 0
$$436$$ −39170.7 −4.30260
$$437$$ 269.019 0.0294484
$$438$$ 0 0
$$439$$ 10897.0 1.18470 0.592351 0.805680i $$-0.298200\pi$$
0.592351 + 0.805680i $$0.298200\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 2148.84 0.231244
$$443$$ −7720.83 −0.828054 −0.414027 0.910265i $$-0.635878\pi$$
−0.414027 + 0.910265i $$0.635878\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 7473.76 0.793481
$$447$$ 0 0
$$448$$ 12552.5 1.32378
$$449$$ 7473.86 0.785553 0.392776 0.919634i $$-0.371515\pi$$
0.392776 + 0.919634i $$0.371515\pi$$
$$450$$ 0 0
$$451$$ 5059.36 0.528240
$$452$$ 24132.4 2.51127
$$453$$ 0 0
$$454$$ −6657.29 −0.688198
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11140.5 −1.14033 −0.570167 0.821529i $$-0.693121\pi$$
−0.570167 + 0.821529i $$0.693121\pi$$
$$458$$ 21102.6 2.15296
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14328.8 −1.44763 −0.723817 0.689992i $$-0.757614\pi$$
−0.723817 + 0.689992i $$0.757614\pi$$
$$462$$ 0 0
$$463$$ −11760.7 −1.18049 −0.590246 0.807223i $$-0.700969\pi$$
−0.590246 + 0.807223i $$0.700969\pi$$
$$464$$ 28744.4 2.87592
$$465$$ 0 0
$$466$$ −17456.5 −1.73532
$$467$$ 11854.9 1.17469 0.587343 0.809338i $$-0.300174\pi$$
0.587343 + 0.809338i $$0.300174\pi$$
$$468$$ 0 0
$$469$$ 6625.77 0.652345
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 42122.1 4.10769
$$473$$ −714.613 −0.0694671
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2629.14 −0.253165
$$477$$ 0 0
$$478$$ 2322.19 0.222206
$$479$$ 1324.68 0.126359 0.0631796 0.998002i $$-0.479876\pi$$
0.0631796 + 0.998002i $$0.479876\pi$$
$$480$$ 0 0
$$481$$ 2287.84 0.216874
$$482$$ 6678.34 0.631100
$$483$$ 0 0
$$484$$ 2592.36 0.243459
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −18636.4 −1.73408 −0.867040 0.498239i $$-0.833980\pi$$
−0.867040 + 0.498239i $$0.833980\pi$$
$$488$$ −16134.7 −1.49668
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −124.552 −0.0114480 −0.00572398 0.999984i $$-0.501822\pi$$
−0.00572398 + 0.999984i $$0.501822\pi$$
$$492$$ 0 0
$$493$$ −2049.29 −0.187212
$$494$$ −2042.24 −0.186001
$$495$$ 0 0
$$496$$ 49059.2 4.44117
$$497$$ −4471.75 −0.403592
$$498$$ 0 0
$$499$$ 10230.2 0.917768 0.458884 0.888496i $$-0.348249\pi$$
0.458884 + 0.888496i $$0.348249\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −15395.7 −1.36881
$$503$$ 5150.81 0.456587 0.228294 0.973592i $$-0.426685\pi$$
0.228294 + 0.973592i $$0.426685\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1059.46 0.0930806
$$507$$ 0 0
$$508$$ −14168.1 −1.23741
$$509$$ 22.7715 0.00198296 0.000991481 1.00000i $$-0.499684\pi$$
0.000991481 1.00000i $$0.499684\pi$$
$$510$$ 0 0
$$511$$ −3926.73 −0.339938
$$512$$ 10698.1 0.923429
$$513$$ 0 0
$$514$$ 1855.19 0.159201
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 5471.49 0.465446
$$518$$ −3844.46 −0.326093
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −21521.7 −1.80976 −0.904879 0.425669i $$-0.860039\pi$$
−0.904879 + 0.425669i $$0.860039\pi$$
$$522$$ 0 0
$$523$$ −2923.36 −0.244416 −0.122208 0.992504i $$-0.538998\pi$$
−0.122208 + 0.992504i $$0.538998\pi$$
$$524$$ −13330.0 −1.11130
$$525$$ 0 0
$$526$$ 31977.0 2.65069
$$527$$ −3497.60 −0.289104
$$528$$ 0 0
$$529$$ −11851.7 −0.974088
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2498.71 0.203633
$$533$$ −11429.0 −0.928792
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 62679.3 5.05100
$$537$$ 0 0
$$538$$ 13540.3 1.08507
$$539$$ −3121.20 −0.249424
$$540$$ 0 0
$$541$$ 21272.8 1.69056 0.845278 0.534327i $$-0.179435\pi$$
0.845278 + 0.534327i $$0.179435\pi$$
$$542$$ −12203.2 −0.967108
$$543$$ 0 0
$$544$$ −10049.9 −0.792067
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 18730.5 1.46409 0.732046 0.681256i $$-0.238566\pi$$
0.732046 + 0.681256i $$0.238566\pi$$
$$548$$ −40124.5 −3.12780
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1947.63 0.150584
$$552$$ 0 0
$$553$$ 7968.04 0.612723
$$554$$ −22145.7 −1.69834
$$555$$ 0 0
$$556$$ −20440.1 −1.55909
$$557$$ −18885.0 −1.43659 −0.718297 0.695736i $$-0.755078\pi$$
−0.718297 + 0.695736i $$0.755078\pi$$
$$558$$ 0 0
$$559$$ 1614.30 0.122143
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 5607.71 0.420902
$$563$$ 10285.1 0.769922 0.384961 0.922933i $$-0.374215\pi$$
0.384961 + 0.922933i $$0.374215\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 42360.1 3.14581
$$567$$ 0 0
$$568$$ −42302.5 −3.12495
$$569$$ −18008.8 −1.32683 −0.663415 0.748251i $$-0.730894\pi$$
−0.663415 + 0.748251i $$0.730894\pi$$
$$570$$ 0 0
$$571$$ −7010.79 −0.513822 −0.256911 0.966435i $$-0.582705\pi$$
−0.256911 + 0.966435i $$0.582705\pi$$
$$572$$ −5856.10 −0.428070
$$573$$ 0 0
$$574$$ 19205.2 1.39653
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 16398.9 1.18318 0.591589 0.806240i $$-0.298501\pi$$
0.591589 + 0.806240i $$0.298501\pi$$
$$578$$ −25271.6 −1.81862
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 4666.44 0.333213
$$582$$ 0 0
$$583$$ −5796.11 −0.411750
$$584$$ −37146.6 −2.63208
$$585$$ 0 0
$$586$$ −10573.3 −0.745354
$$587$$ 12823.5 0.901671 0.450836 0.892607i $$-0.351126\pi$$
0.450836 + 0.892607i $$0.351126\pi$$
$$588$$ 0 0
$$589$$ 3324.09 0.232541
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −20587.9 −1.42932
$$593$$ 16899.5 1.17029 0.585144 0.810929i $$-0.301038\pi$$
0.585144 + 0.810929i $$0.301038\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 43856.1 3.01412
$$597$$ 0 0
$$598$$ −2393.31 −0.163662
$$599$$ 15074.9 1.02829 0.514143 0.857704i $$-0.328110\pi$$
0.514143 + 0.857704i $$0.328110\pi$$
$$600$$ 0 0
$$601$$ −11418.8 −0.775014 −0.387507 0.921867i $$-0.626664\pi$$
−0.387507 + 0.921867i $$0.626664\pi$$
$$602$$ −2712.65 −0.183654
$$603$$ 0 0
$$604$$ 10195.1 0.686809
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 17952.8 1.20046 0.600232 0.799826i $$-0.295075\pi$$
0.600232 + 0.799826i $$0.295075\pi$$
$$608$$ 9551.30 0.637100
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −12360.0 −0.818384
$$612$$ 0 0
$$613$$ −12528.9 −0.825507 −0.412753 0.910843i $$-0.635433\pi$$
−0.412753 + 0.910843i $$0.635433\pi$$
$$614$$ −12823.8 −0.842878
$$615$$ 0 0
$$616$$ 6166.01 0.403305
$$617$$ −8586.10 −0.560232 −0.280116 0.959966i $$-0.590373\pi$$
−0.280116 + 0.959966i $$0.590373\pi$$
$$618$$ 0 0
$$619$$ 18415.4 1.19576 0.597882 0.801584i $$-0.296009\pi$$
0.597882 + 0.801584i $$0.296009\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 10790.1 0.695569
$$623$$ 180.443 0.0116040
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 21039.6 1.34331
$$627$$ 0 0
$$628$$ 13871.6 0.881427
$$629$$ 1467.79 0.0930436
$$630$$ 0 0
$$631$$ 2374.38 0.149798 0.0748989 0.997191i $$-0.476137\pi$$
0.0748989 + 0.997191i $$0.476137\pi$$
$$632$$ 75377.1 4.74421
$$633$$ 0 0
$$634$$ 15805.3 0.990080
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 7050.74 0.438557
$$638$$ 7670.21 0.475966
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −11086.0 −0.683104 −0.341552 0.939863i $$-0.610953\pi$$
−0.341552 + 0.939863i $$0.610953\pi$$
$$642$$ 0 0
$$643$$ −19934.1 −1.22259 −0.611294 0.791403i $$-0.709351\pi$$
−0.611294 + 0.791403i $$0.709351\pi$$
$$644$$ 2928.26 0.179176
$$645$$ 0 0
$$646$$ −1310.21 −0.0797983
$$647$$ −30634.8 −1.86148 −0.930739 0.365684i $$-0.880835\pi$$
−0.930739 + 0.365684i $$0.880835\pi$$
$$648$$ 0 0
$$649$$ 6362.87 0.384845
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −23421.9 −1.40686
$$653$$ −9818.07 −0.588378 −0.294189 0.955747i $$-0.595049\pi$$
−0.294189 + 0.955747i $$0.595049\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 102848. 6.12125
$$657$$ 0 0
$$658$$ 20769.6 1.23052
$$659$$ −16478.5 −0.974070 −0.487035 0.873383i $$-0.661922\pi$$
−0.487035 + 0.873383i $$0.661922\pi$$
$$660$$ 0 0
$$661$$ 2958.12 0.174066 0.0870328 0.996205i $$-0.472262\pi$$
0.0870328 + 0.996205i $$0.472262\pi$$
$$662$$ 43962.1 2.58102
$$663$$ 0 0
$$664$$ 44144.2 2.58001
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2282.44 0.132498
$$668$$ 24065.1 1.39387
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −2437.26 −0.140223
$$672$$ 0 0
$$673$$ 29960.3 1.71602 0.858012 0.513630i $$-0.171700\pi$$
0.858012 + 0.513630i $$0.171700\pi$$
$$674$$ −32108.2 −1.83496
$$675$$ 0 0
$$676$$ −33840.6 −1.92539
$$677$$ −4514.73 −0.256300 −0.128150 0.991755i $$-0.540904\pi$$
−0.128150 + 0.991755i $$0.540904\pi$$
$$678$$ 0 0
$$679$$ −5538.43 −0.313027
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 13091.0 0.735018
$$683$$ −13555.7 −0.759438 −0.379719 0.925102i $$-0.623979\pi$$
−0.379719 + 0.925102i $$0.623979\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −26170.2 −1.45653
$$687$$ 0 0
$$688$$ −14526.8 −0.804986
$$689$$ 13093.3 0.723972
$$690$$ 0 0
$$691$$ −11471.3 −0.631535 −0.315768 0.948837i $$-0.602262\pi$$
−0.315768 + 0.948837i $$0.602262\pi$$
$$692$$ 985.787 0.0541532
$$693$$ 0 0
$$694$$ −46327.8 −2.53398
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −7332.40 −0.398471
$$698$$ 5084.52 0.275719
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −22229.0 −1.19769 −0.598843 0.800866i $$-0.704373\pi$$
−0.598843 + 0.800866i $$0.704373\pi$$
$$702$$ 0 0
$$703$$ −1394.97 −0.0748397
$$704$$ 17937.5 0.960292
$$705$$ 0 0
$$706$$ −1145.19 −0.0610480
$$707$$ −14404.5 −0.766248
$$708$$ 0 0
$$709$$ −15081.2 −0.798851 −0.399426 0.916766i $$-0.630790\pi$$
−0.399426 + 0.916766i $$0.630790\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 1706.98 0.0898480
$$713$$ 3895.52 0.204612
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 17824.6 0.930358
$$717$$ 0 0
$$718$$ 7465.71 0.388047
$$719$$ 7399.80 0.383819 0.191910 0.981413i $$-0.438532\pi$$
0.191910 + 0.981413i $$0.438532\pi$$
$$720$$ 0 0
$$721$$ 3300.36 0.170474
$$722$$ −35960.9 −1.85364
$$723$$ 0 0
$$724$$ −38791.7 −1.99127
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −1705.77 −0.0870202 −0.0435101 0.999053i $$-0.513854\pi$$
−0.0435101 + 0.999053i $$0.513854\pi$$
$$728$$ −13928.9 −0.709122
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1035.67 0.0524017
$$732$$ 0 0
$$733$$ −37122.6 −1.87061 −0.935303 0.353847i $$-0.884873\pi$$
−0.935303 + 0.353847i $$0.884873\pi$$
$$734$$ −5589.60 −0.281084
$$735$$ 0 0
$$736$$ 11193.2 0.560581
$$737$$ 9468.20 0.473223
$$738$$ 0 0
$$739$$ 34256.3 1.70520 0.852598 0.522568i $$-0.175026\pi$$
0.852598 + 0.522568i $$0.175026\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −22001.9 −1.08856
$$743$$ 1567.88 0.0774160 0.0387080 0.999251i $$-0.487676\pi$$
0.0387080 + 0.999251i $$0.487676\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −50801.9 −2.49328
$$747$$ 0 0
$$748$$ −3757.03 −0.183651
$$749$$ 8837.17 0.431112
$$750$$ 0 0
$$751$$ −955.613 −0.0464325 −0.0232163 0.999730i $$-0.507391\pi$$
−0.0232163 + 0.999730i $$0.507391\pi$$
$$752$$ 111226. 5.39360
$$753$$ 0 0
$$754$$ −17326.9 −0.836881
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 14015.4 0.672918 0.336459 0.941698i $$-0.390771\pi$$
0.336459 + 0.941698i $$0.390771\pi$$
$$758$$ −38623.2 −1.85073
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 36271.0 1.72776 0.863879 0.503699i $$-0.168028\pi$$
0.863879 + 0.503699i $$0.168028\pi$$
$$762$$ 0 0
$$763$$ −14073.9 −0.667770
$$764$$ −9831.63 −0.465571
$$765$$ 0 0
$$766$$ −6311.64 −0.297714
$$767$$ −14373.6 −0.676665
$$768$$ 0 0
$$769$$ −18163.6 −0.851749 −0.425874 0.904782i $$-0.640033\pi$$
−0.425874 + 0.904782i $$0.640033\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −38112.1 −1.77680
$$773$$ −8345.65 −0.388321 −0.194160 0.980970i $$-0.562198\pi$$
−0.194160 + 0.980970i $$0.562198\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −52393.2 −2.42372
$$777$$ 0 0
$$778$$ 59445.7 2.73937
$$779$$ 6968.65 0.320510
$$780$$ 0 0
$$781$$ −6390.11 −0.292774
$$782$$ −1535.45 −0.0702142
$$783$$ 0 0
$$784$$ −63448.5 −2.89033
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 22996.2 1.04158 0.520791 0.853684i $$-0.325637\pi$$
0.520791 + 0.853684i $$0.325637\pi$$
$$788$$ 113647. 5.13768
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 8670.69 0.389752
$$792$$ 0 0
$$793$$ 5505.75 0.246551
$$794$$ 11782.4 0.526626
$$795$$ 0 0
$$796$$ −110087. −4.90194
$$797$$ −2743.82 −0.121946 −0.0609730 0.998139i $$-0.519420\pi$$
−0.0609730 + 0.998139i $$0.519420\pi$$
$$798$$ 0 0
$$799$$ −7929.68 −0.351104
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −42477.1 −1.87022
$$803$$ −5611.28 −0.246597
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −29572.5 −1.29237
$$807$$ 0 0
$$808$$ −136266. −5.93293
$$809$$ −41241.7 −1.79231 −0.896156 0.443738i $$-0.853652\pi$$
−0.896156 + 0.443738i $$0.853652\pi$$
$$810$$ 0 0
$$811$$ 12832.9 0.555641 0.277820 0.960633i $$-0.410388\pi$$
0.277820 + 0.960633i $$0.410388\pi$$
$$812$$ 21199.8 0.916215
$$813$$ 0 0
$$814$$ −5493.72 −0.236554
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −984.292 −0.0421493
$$818$$ −58210.4 −2.48812
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −16368.5 −0.695817 −0.347908 0.937529i $$-0.613108\pi$$
−0.347908 + 0.937529i $$0.613108\pi$$
$$822$$ 0 0
$$823$$ −3869.53 −0.163892 −0.0819461 0.996637i $$-0.526114\pi$$
−0.0819461 + 0.996637i $$0.526114\pi$$
$$824$$ 31221.2 1.31995
$$825$$ 0 0
$$826$$ 24153.3 1.01743
$$827$$ 7388.69 0.310677 0.155339 0.987861i $$-0.450353\pi$$
0.155339 + 0.987861i $$0.450353\pi$$
$$828$$ 0 0
$$829$$ 23990.1 1.00508 0.502539 0.864554i $$-0.332399\pi$$
0.502539 + 0.864554i $$0.332399\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −40520.6 −1.68846
$$833$$ 4523.47 0.188150
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 3570.65 0.147720
$$837$$ 0 0
$$838$$ 39684.5 1.63589
$$839$$ 18228.3 0.750074 0.375037 0.927010i $$-0.377630\pi$$
0.375037 + 0.927010i $$0.377630\pi$$
$$840$$ 0 0
$$841$$ −7864.78 −0.322472
$$842$$ −67782.3 −2.77427
$$843$$ 0 0
$$844$$ −91318.7 −3.72431
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 931.424 0.0377852
$$848$$ −117825. −4.77137
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1634.77 −0.0658511
$$852$$ 0 0
$$853$$ 21737.3 0.872534 0.436267 0.899817i $$-0.356300\pi$$
0.436267 + 0.899817i $$0.356300\pi$$
$$854$$ −9251.79 −0.370714
$$855$$ 0 0
$$856$$ 83599.0 3.33803
$$857$$ −18712.2 −0.745852 −0.372926 0.927861i $$-0.621645\pi$$
−0.372926 + 0.927861i $$0.621645\pi$$
$$858$$ 0 0
$$859$$ 30527.6 1.21256 0.606279 0.795252i $$-0.292661\pi$$
0.606279 + 0.795252i $$0.292661\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −32953.4 −1.30209
$$863$$ 10906.4 0.430196 0.215098 0.976592i $$-0.430993\pi$$
0.215098 + 0.976592i $$0.430993\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −30603.5 −1.20087
$$867$$ 0 0
$$868$$ 36182.5 1.41488
$$869$$ 11386.3 0.444481
$$870$$ 0 0
$$871$$ −21388.5 −0.832058
$$872$$ −133138. −5.17043
$$873$$ 0 0
$$874$$ 1459.28 0.0564768
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −21770.9 −0.838256 −0.419128 0.907927i $$-0.637664\pi$$
−0.419128 + 0.907927i $$0.637664\pi$$
$$878$$ 59109.8 2.27205
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −47206.9 −1.80527 −0.902634 0.430409i $$-0.858369\pi$$
−0.902634 + 0.430409i $$0.858369\pi$$
$$882$$ 0 0
$$883$$ 6059.68 0.230945 0.115473 0.993311i $$-0.463162\pi$$
0.115473 + 0.993311i $$0.463162\pi$$
$$884$$ 8487.09 0.322909
$$885$$ 0 0
$$886$$ −41881.1 −1.58806
$$887$$ −37130.2 −1.40553 −0.702767 0.711420i $$-0.748052\pi$$
−0.702767 + 0.711420i $$0.748052\pi$$
$$888$$ 0 0
$$889$$ −5090.53 −0.192048
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 29518.5 1.10802
$$893$$ 7536.30 0.282410
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 29269.1 1.09131
$$897$$ 0 0
$$898$$ 40541.4 1.50655
$$899$$ 28202.5 1.04628
$$900$$ 0 0
$$901$$ 8400.15 0.310599
$$902$$ 27444.1 1.01307
$$903$$ 0 0
$$904$$ 82024.1 3.01779
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −1182.94 −0.0433064 −0.0216532 0.999766i $$-0.506893\pi$$
−0.0216532 + 0.999766i $$0.506893\pi$$
$$908$$ −26293.8 −0.961001
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −37676.4 −1.37022 −0.685112 0.728438i $$-0.740247\pi$$
−0.685112 + 0.728438i $$0.740247\pi$$
$$912$$ 0 0
$$913$$ 6668.32 0.241719
$$914$$ −60431.1 −2.18696
$$915$$ 0 0
$$916$$ 83347.1 3.00640
$$917$$ −4789.41 −0.172476
$$918$$ 0 0
$$919$$ 8697.82 0.312203 0.156101 0.987741i $$-0.450107\pi$$
0.156101 + 0.987741i $$0.450107\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −77725.7 −2.77631
$$923$$ 14435.2 0.514778
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −63795.3 −2.26398
$$927$$ 0 0
$$928$$ 81036.0 2.86653
$$929$$ 17247.5 0.609119 0.304559 0.952493i $$-0.401491\pi$$
0.304559 + 0.952493i $$0.401491\pi$$
$$930$$ 0 0
$$931$$ −4299.06 −0.151338
$$932$$ −68946.7 −2.42320
$$933$$ 0 0
$$934$$ 64306.0 2.25284
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 41812.4 1.45779 0.728896 0.684624i $$-0.240034\pi$$
0.728896 + 0.684624i $$0.240034\pi$$
$$938$$ 35941.0 1.25108
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −37655.9 −1.30451 −0.652257 0.757998i $$-0.726178\pi$$
−0.652257 + 0.757998i $$0.726178\pi$$
$$942$$ 0 0
$$943$$ 8166.60 0.282016
$$944$$ 129346. 4.45959
$$945$$ 0 0
$$946$$ −3876.37 −0.133226
$$947$$ −21244.4 −0.728986 −0.364493 0.931206i $$-0.618758\pi$$
−0.364493 + 0.931206i $$0.618758\pi$$
$$948$$ 0 0
$$949$$ 12675.8 0.433587
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −8936.24 −0.304228
$$953$$ 1324.27 0.0450130 0.0225065 0.999747i $$-0.492835\pi$$
0.0225065 + 0.999747i $$0.492835\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 9171.77 0.310289
$$957$$ 0 0
$$958$$ 7185.62 0.242335
$$959$$ −14416.6 −0.485439
$$960$$ 0 0
$$961$$ 18343.4 0.615735
$$962$$ 12410.2 0.415927
$$963$$ 0 0
$$964$$ 26376.9 0.881268
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −52267.1 −1.73815 −0.869077 0.494676i $$-0.835287\pi$$
−0.869077 + 0.494676i $$0.835287\pi$$
$$968$$ 8811.20 0.292565
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 52489.8 1.73479 0.867394 0.497622i $$-0.165793\pi$$
0.867394 + 0.497622i $$0.165793\pi$$
$$972$$ 0 0
$$973$$ −7344.07 −0.241973
$$974$$ −101092. −3.32566
$$975$$ 0 0
$$976$$ −49545.3 −1.62490
$$977$$ 8324.11 0.272581 0.136291 0.990669i $$-0.456482\pi$$
0.136291 + 0.990669i $$0.456482\pi$$
$$978$$ 0 0
$$979$$ 257.852 0.00841777
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −675.623 −0.0219552
$$983$$ −44407.1 −1.44086 −0.720431 0.693527i $$-0.756056\pi$$
−0.720431 + 0.693527i $$0.756056\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −11116.2 −0.359039
$$987$$ 0 0
$$988$$ −8066.05 −0.259732
$$989$$ −1153.50 −0.0370870
$$990$$ 0 0
$$991$$ −45124.7 −1.44645 −0.723226 0.690612i $$-0.757341\pi$$
−0.723226 + 0.690612i $$0.757341\pi$$
$$992$$ 138307. 4.42667
$$993$$ 0 0
$$994$$ −24256.7 −0.774020
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −5480.61 −0.174095 −0.0870474 0.996204i $$-0.527743\pi$$
−0.0870474 + 0.996204i $$0.527743\pi$$
$$998$$ 55492.9 1.76012
$$999$$ 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.p.1.2 2
3.2 odd 2 825.4.a.l.1.1 2
5.4 even 2 99.4.a.f.1.1 2
15.2 even 4 825.4.c.h.199.1 4
15.8 even 4 825.4.c.h.199.4 4
15.14 odd 2 33.4.a.c.1.2 2
20.19 odd 2 1584.4.a.bj.1.2 2
55.54 odd 2 1089.4.a.u.1.2 2
60.59 even 2 528.4.a.p.1.1 2
105.104 even 2 1617.4.a.k.1.2 2
120.29 odd 2 2112.4.a.bn.1.2 2
120.59 even 2 2112.4.a.bg.1.2 2
165.164 even 2 363.4.a.i.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.2 2 15.14 odd 2
99.4.a.f.1.1 2 5.4 even 2
363.4.a.i.1.1 2 165.164 even 2
528.4.a.p.1.1 2 60.59 even 2
825.4.a.l.1.1 2 3.2 odd 2
825.4.c.h.199.1 4 15.2 even 4
825.4.c.h.199.4 4 15.8 even 4
1089.4.a.u.1.2 2 55.54 odd 2
1584.4.a.bj.1.2 2 20.19 odd 2
1617.4.a.k.1.2 2 105.104 even 2
2112.4.a.bg.1.2 2 120.59 even 2
2112.4.a.bn.1.2 2 120.29 odd 2
2475.4.a.p.1.2 2 1.1 even 1 trivial