# Properties

 Label 5225.2.a.bc.1.19 Level $5225$ Weight $2$ Character 5225.1 Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $30$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ Analytic rank: $$0$$ Dimension: $$30$$ Twist minimal: no (minimal twist has level 1045) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.19 Character $$\chi$$ $$=$$ 5225.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.661223 q^{2} -3.01131 q^{3} -1.56278 q^{4} -1.99115 q^{6} -0.592449 q^{7} -2.35579 q^{8} +6.06799 q^{9} +O(q^{10})$$ $$q+0.661223 q^{2} -3.01131 q^{3} -1.56278 q^{4} -1.99115 q^{6} -0.592449 q^{7} -2.35579 q^{8} +6.06799 q^{9} +1.00000 q^{11} +4.70603 q^{12} -3.68157 q^{13} -0.391741 q^{14} +1.56786 q^{16} -0.251015 q^{17} +4.01230 q^{18} +1.00000 q^{19} +1.78405 q^{21} +0.661223 q^{22} -2.58194 q^{23} +7.09403 q^{24} -2.43434 q^{26} -9.23869 q^{27} +0.925870 q^{28} +2.22782 q^{29} -7.50257 q^{31} +5.74830 q^{32} -3.01131 q^{33} -0.165977 q^{34} -9.48297 q^{36} -9.64498 q^{37} +0.661223 q^{38} +11.0864 q^{39} +5.78336 q^{41} +1.17965 q^{42} +0.283462 q^{43} -1.56278 q^{44} -1.70724 q^{46} +3.78624 q^{47} -4.72132 q^{48} -6.64900 q^{49} +0.755885 q^{51} +5.75350 q^{52} -3.89730 q^{53} -6.10883 q^{54} +1.39569 q^{56} -3.01131 q^{57} +1.47309 q^{58} -2.98649 q^{59} -8.94657 q^{61} -4.96087 q^{62} -3.59498 q^{63} +0.665182 q^{64} -1.99115 q^{66} -3.70414 q^{67} +0.392283 q^{68} +7.77503 q^{69} +16.1865 q^{71} -14.2950 q^{72} -7.91613 q^{73} -6.37748 q^{74} -1.56278 q^{76} -0.592449 q^{77} +7.33056 q^{78} -15.4023 q^{79} +9.61657 q^{81} +3.82409 q^{82} -3.74222 q^{83} -2.78808 q^{84} +0.187432 q^{86} -6.70866 q^{87} -2.35579 q^{88} -3.78573 q^{89} +2.18114 q^{91} +4.03502 q^{92} +22.5926 q^{93} +2.50355 q^{94} -17.3099 q^{96} -9.18878 q^{97} -4.39647 q^{98} +6.06799 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$30 q + 42 q^{4} + 12 q^{6} + 40 q^{9}+O(q^{10})$$ 30 * q + 42 * q^4 + 12 * q^6 + 40 * q^9 $$30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46} + 60 q^{49} + 64 q^{51} + 62 q^{54} - 32 q^{56} - 14 q^{59} + 78 q^{61} + 90 q^{64} + 12 q^{66} - 28 q^{69} + 20 q^{71} + 42 q^{74} + 42 q^{76} + 102 q^{79} + 42 q^{81} + 98 q^{84} - 52 q^{86} - 8 q^{89} + 56 q^{91} + 40 q^{94} - 74 q^{96} + 40 q^{99}+O(q^{100})$$ 30 * q + 42 * q^4 + 12 * q^6 + 40 * q^9 + 30 * q^11 - 4 * q^14 + 66 * q^16 + 30 * q^19 + 14 * q^21 + 22 * q^24 + 30 * q^29 + 26 * q^31 + 12 * q^34 + 78 * q^36 + 64 * q^39 + 22 * q^41 + 42 * q^44 + 28 * q^46 + 60 * q^49 + 64 * q^51 + 62 * q^54 - 32 * q^56 - 14 * q^59 + 78 * q^61 + 90 * q^64 + 12 * q^66 - 28 * q^69 + 20 * q^71 + 42 * q^74 + 42 * q^76 + 102 * q^79 + 42 * q^81 + 98 * q^84 - 52 * q^86 - 8 * q^89 + 56 * q^91 + 40 * q^94 - 74 * q^96 + 40 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.661223 0.467555 0.233778 0.972290i $$-0.424891\pi$$
0.233778 + 0.972290i $$0.424891\pi$$
$$3$$ −3.01131 −1.73858 −0.869291 0.494301i $$-0.835424\pi$$
−0.869291 + 0.494301i $$0.835424\pi$$
$$4$$ −1.56278 −0.781392
$$5$$ 0 0
$$6$$ −1.99115 −0.812883
$$7$$ −0.592449 −0.223925 −0.111962 0.993712i $$-0.535714\pi$$
−0.111962 + 0.993712i $$0.535714\pi$$
$$8$$ −2.35579 −0.832899
$$9$$ 6.06799 2.02266
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 4.70603 1.35851
$$13$$ −3.68157 −1.02108 −0.510542 0.859853i $$-0.670555\pi$$
−0.510542 + 0.859853i $$0.670555\pi$$
$$14$$ −0.391741 −0.104697
$$15$$ 0 0
$$16$$ 1.56786 0.391966
$$17$$ −0.251015 −0.0608801 −0.0304401 0.999537i $$-0.509691\pi$$
−0.0304401 + 0.999537i $$0.509691\pi$$
$$18$$ 4.01230 0.945708
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 1.78405 0.389311
$$22$$ 0.661223 0.140973
$$23$$ −2.58194 −0.538372 −0.269186 0.963088i $$-0.586755\pi$$
−0.269186 + 0.963088i $$0.586755\pi$$
$$24$$ 7.09403 1.44806
$$25$$ 0 0
$$26$$ −2.43434 −0.477414
$$27$$ −9.23869 −1.77799
$$28$$ 0.925870 0.174973
$$29$$ 2.22782 0.413696 0.206848 0.978373i $$-0.433679\pi$$
0.206848 + 0.978373i $$0.433679\pi$$
$$30$$ 0 0
$$31$$ −7.50257 −1.34750 −0.673751 0.738959i $$-0.735318\pi$$
−0.673751 + 0.738959i $$0.735318\pi$$
$$32$$ 5.74830 1.01616
$$33$$ −3.01131 −0.524202
$$34$$ −0.165977 −0.0284648
$$35$$ 0 0
$$36$$ −9.48297 −1.58049
$$37$$ −9.64498 −1.58562 −0.792812 0.609466i $$-0.791384\pi$$
−0.792812 + 0.609466i $$0.791384\pi$$
$$38$$ 0.661223 0.107265
$$39$$ 11.0864 1.77524
$$40$$ 0 0
$$41$$ 5.78336 0.903209 0.451605 0.892218i $$-0.350852\pi$$
0.451605 + 0.892218i $$0.350852\pi$$
$$42$$ 1.17965 0.182025
$$43$$ 0.283462 0.0432275 0.0216138 0.999766i $$-0.493120\pi$$
0.0216138 + 0.999766i $$0.493120\pi$$
$$44$$ −1.56278 −0.235599
$$45$$ 0 0
$$46$$ −1.70724 −0.251719
$$47$$ 3.78624 0.552279 0.276140 0.961118i $$-0.410945\pi$$
0.276140 + 0.961118i $$0.410945\pi$$
$$48$$ −4.72132 −0.681464
$$49$$ −6.64900 −0.949858
$$50$$ 0 0
$$51$$ 0.755885 0.105845
$$52$$ 5.75350 0.797868
$$53$$ −3.89730 −0.535336 −0.267668 0.963511i $$-0.586253\pi$$
−0.267668 + 0.963511i $$0.586253\pi$$
$$54$$ −6.10883 −0.831307
$$55$$ 0 0
$$56$$ 1.39569 0.186507
$$57$$ −3.01131 −0.398858
$$58$$ 1.47309 0.193426
$$59$$ −2.98649 −0.388808 −0.194404 0.980922i $$-0.562277\pi$$
−0.194404 + 0.980922i $$0.562277\pi$$
$$60$$ 0 0
$$61$$ −8.94657 −1.14549 −0.572746 0.819733i $$-0.694122\pi$$
−0.572746 + 0.819733i $$0.694122\pi$$
$$62$$ −4.96087 −0.630031
$$63$$ −3.59498 −0.452925
$$64$$ 0.665182 0.0831477
$$65$$ 0 0
$$66$$ −1.99115 −0.245093
$$67$$ −3.70414 −0.452533 −0.226266 0.974065i $$-0.572652\pi$$
−0.226266 + 0.974065i $$0.572652\pi$$
$$68$$ 0.392283 0.0475713
$$69$$ 7.77503 0.936003
$$70$$ 0 0
$$71$$ 16.1865 1.92099 0.960493 0.278304i $$-0.0897724\pi$$
0.960493 + 0.278304i $$0.0897724\pi$$
$$72$$ −14.2950 −1.68468
$$73$$ −7.91613 −0.926513 −0.463257 0.886224i $$-0.653319\pi$$
−0.463257 + 0.886224i $$0.653319\pi$$
$$74$$ −6.37748 −0.741367
$$75$$ 0 0
$$76$$ −1.56278 −0.179264
$$77$$ −0.592449 −0.0675158
$$78$$ 7.33056 0.830022
$$79$$ −15.4023 −1.73289 −0.866447 0.499270i $$-0.833602\pi$$
−0.866447 + 0.499270i $$0.833602\pi$$
$$80$$ 0 0
$$81$$ 9.61657 1.06851
$$82$$ 3.82409 0.422300
$$83$$ −3.74222 −0.410762 −0.205381 0.978682i $$-0.565843\pi$$
−0.205381 + 0.978682i $$0.565843\pi$$
$$84$$ −2.78808 −0.304205
$$85$$ 0 0
$$86$$ 0.187432 0.0202113
$$87$$ −6.70866 −0.719244
$$88$$ −2.35579 −0.251129
$$89$$ −3.78573 −0.401286 −0.200643 0.979664i $$-0.564303\pi$$
−0.200643 + 0.979664i $$0.564303\pi$$
$$90$$ 0 0
$$91$$ 2.18114 0.228646
$$92$$ 4.03502 0.420680
$$93$$ 22.5926 2.34274
$$94$$ 2.50355 0.258221
$$95$$ 0 0
$$96$$ −17.3099 −1.76669
$$97$$ −9.18878 −0.932979 −0.466489 0.884527i $$-0.654481\pi$$
−0.466489 + 0.884527i $$0.654481\pi$$
$$98$$ −4.39647 −0.444111
$$99$$ 6.06799 0.609856
$$100$$ 0 0
$$101$$ −14.1183 −1.40482 −0.702411 0.711772i $$-0.747893\pi$$
−0.702411 + 0.711772i $$0.747893\pi$$
$$102$$ 0.499809 0.0494884
$$103$$ −5.20319 −0.512686 −0.256343 0.966586i $$-0.582518\pi$$
−0.256343 + 0.966586i $$0.582518\pi$$
$$104$$ 8.67303 0.850461
$$105$$ 0 0
$$106$$ −2.57699 −0.250299
$$107$$ 12.8863 1.24576 0.622881 0.782316i $$-0.285962\pi$$
0.622881 + 0.782316i $$0.285962\pi$$
$$108$$ 14.4381 1.38930
$$109$$ 3.10981 0.297866 0.148933 0.988847i $$-0.452416\pi$$
0.148933 + 0.988847i $$0.452416\pi$$
$$110$$ 0 0
$$111$$ 29.0440 2.75674
$$112$$ −0.928879 −0.0877708
$$113$$ −7.27809 −0.684665 −0.342333 0.939579i $$-0.611217\pi$$
−0.342333 + 0.939579i $$0.611217\pi$$
$$114$$ −1.99115 −0.186488
$$115$$ 0 0
$$116$$ −3.48160 −0.323259
$$117$$ −22.3398 −2.06531
$$118$$ −1.97474 −0.181789
$$119$$ 0.148714 0.0136326
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −5.91568 −0.535581
$$123$$ −17.4155 −1.57030
$$124$$ 11.7249 1.05293
$$125$$ 0 0
$$126$$ −2.37708 −0.211767
$$127$$ −16.8696 −1.49694 −0.748468 0.663171i $$-0.769210\pi$$
−0.748468 + 0.663171i $$0.769210\pi$$
$$128$$ −11.0568 −0.977289
$$129$$ −0.853592 −0.0751546
$$130$$ 0 0
$$131$$ −11.4376 −0.999309 −0.499655 0.866225i $$-0.666540\pi$$
−0.499655 + 0.866225i $$0.666540\pi$$
$$132$$ 4.70603 0.409607
$$133$$ −0.592449 −0.0513718
$$134$$ −2.44926 −0.211584
$$135$$ 0 0
$$136$$ 0.591340 0.0507070
$$137$$ 16.7276 1.42914 0.714569 0.699565i $$-0.246623\pi$$
0.714569 + 0.699565i $$0.246623\pi$$
$$138$$ 5.14103 0.437633
$$139$$ 16.6330 1.41079 0.705397 0.708813i $$-0.250769\pi$$
0.705397 + 0.708813i $$0.250769\pi$$
$$140$$ 0 0
$$141$$ −11.4015 −0.960182
$$142$$ 10.7029 0.898167
$$143$$ −3.68157 −0.307869
$$144$$ 9.51378 0.792815
$$145$$ 0 0
$$146$$ −5.23433 −0.433196
$$147$$ 20.0222 1.65140
$$148$$ 15.0730 1.23899
$$149$$ 14.3567 1.17614 0.588072 0.808808i $$-0.299887\pi$$
0.588072 + 0.808808i $$0.299887\pi$$
$$150$$ 0 0
$$151$$ 16.0364 1.30502 0.652511 0.757779i $$-0.273716\pi$$
0.652511 + 0.757779i $$0.273716\pi$$
$$152$$ −2.35579 −0.191080
$$153$$ −1.52316 −0.123140
$$154$$ −0.391741 −0.0315674
$$155$$ 0 0
$$156$$ −17.3256 −1.38716
$$157$$ 24.1525 1.92758 0.963788 0.266670i $$-0.0859234\pi$$
0.963788 + 0.266670i $$0.0859234\pi$$
$$158$$ −10.1844 −0.810223
$$159$$ 11.7360 0.930725
$$160$$ 0 0
$$161$$ 1.52967 0.120555
$$162$$ 6.35870 0.499587
$$163$$ 16.3016 1.27684 0.638421 0.769687i $$-0.279588\pi$$
0.638421 + 0.769687i $$0.279588\pi$$
$$164$$ −9.03814 −0.705760
$$165$$ 0 0
$$166$$ −2.47444 −0.192054
$$167$$ 16.9851 1.31435 0.657173 0.753740i $$-0.271752\pi$$
0.657173 + 0.753740i $$0.271752\pi$$
$$168$$ −4.20285 −0.324257
$$169$$ 0.553985 0.0426142
$$170$$ 0 0
$$171$$ 6.06799 0.464031
$$172$$ −0.442990 −0.0337776
$$173$$ 6.48231 0.492841 0.246420 0.969163i $$-0.420746\pi$$
0.246420 + 0.969163i $$0.420746\pi$$
$$174$$ −4.43592 −0.336286
$$175$$ 0 0
$$176$$ 1.56786 0.118182
$$177$$ 8.99326 0.675975
$$178$$ −2.50321 −0.187624
$$179$$ 2.50231 0.187031 0.0935157 0.995618i $$-0.470189\pi$$
0.0935157 + 0.995618i $$0.470189\pi$$
$$180$$ 0 0
$$181$$ 3.08136 0.229036 0.114518 0.993421i $$-0.463468\pi$$
0.114518 + 0.993421i $$0.463468\pi$$
$$182$$ 1.44222 0.106905
$$183$$ 26.9409 1.99153
$$184$$ 6.08252 0.448410
$$185$$ 0 0
$$186$$ 14.9387 1.09536
$$187$$ −0.251015 −0.0183561
$$188$$ −5.91707 −0.431547
$$189$$ 5.47345 0.398135
$$190$$ 0 0
$$191$$ 6.02462 0.435926 0.217963 0.975957i $$-0.430059\pi$$
0.217963 + 0.975957i $$0.430059\pi$$
$$192$$ −2.00307 −0.144559
$$193$$ −9.85410 −0.709314 −0.354657 0.934996i $$-0.615402\pi$$
−0.354657 + 0.934996i $$0.615402\pi$$
$$194$$ −6.07583 −0.436219
$$195$$ 0 0
$$196$$ 10.3910 0.742211
$$197$$ −10.3472 −0.737208 −0.368604 0.929587i $$-0.620164\pi$$
−0.368604 + 0.929587i $$0.620164\pi$$
$$198$$ 4.01230 0.285142
$$199$$ 9.67189 0.685622 0.342811 0.939404i $$-0.388621\pi$$
0.342811 + 0.939404i $$0.388621\pi$$
$$200$$ 0 0
$$201$$ 11.1543 0.786765
$$202$$ −9.33533 −0.656832
$$203$$ −1.31987 −0.0926367
$$204$$ −1.18128 −0.0827065
$$205$$ 0 0
$$206$$ −3.44047 −0.239709
$$207$$ −15.6672 −1.08895
$$208$$ −5.77220 −0.400230
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ −5.34846 −0.368203 −0.184102 0.982907i $$-0.558938\pi$$
−0.184102 + 0.982907i $$0.558938\pi$$
$$212$$ 6.09065 0.418307
$$213$$ −48.7426 −3.33979
$$214$$ 8.52070 0.582463
$$215$$ 0 0
$$216$$ 21.7644 1.48088
$$217$$ 4.44489 0.301739
$$218$$ 2.05628 0.139269
$$219$$ 23.8379 1.61082
$$220$$ 0 0
$$221$$ 0.924131 0.0621638
$$222$$ 19.2046 1.28893
$$223$$ −3.76862 −0.252365 −0.126183 0.992007i $$-0.540273\pi$$
−0.126183 + 0.992007i $$0.540273\pi$$
$$224$$ −3.40557 −0.227544
$$225$$ 0 0
$$226$$ −4.81244 −0.320119
$$227$$ 3.70626 0.245993 0.122997 0.992407i $$-0.460750\pi$$
0.122997 + 0.992407i $$0.460750\pi$$
$$228$$ 4.70603 0.311664
$$229$$ −7.17137 −0.473897 −0.236949 0.971522i $$-0.576147\pi$$
−0.236949 + 0.971522i $$0.576147\pi$$
$$230$$ 0 0
$$231$$ 1.78405 0.117382
$$232$$ −5.24829 −0.344567
$$233$$ 6.50149 0.425927 0.212964 0.977060i $$-0.431688\pi$$
0.212964 + 0.977060i $$0.431688\pi$$
$$234$$ −14.7716 −0.965648
$$235$$ 0 0
$$236$$ 4.66724 0.303812
$$237$$ 46.3811 3.01278
$$238$$ 0.0983329 0.00637398
$$239$$ 23.9030 1.54616 0.773079 0.634310i $$-0.218716\pi$$
0.773079 + 0.634310i $$0.218716\pi$$
$$240$$ 0 0
$$241$$ 15.7035 1.01155 0.505777 0.862664i $$-0.331206\pi$$
0.505777 + 0.862664i $$0.331206\pi$$
$$242$$ 0.661223 0.0425050
$$243$$ −1.24243 −0.0797022
$$244$$ 13.9816 0.895078
$$245$$ 0 0
$$246$$ −11.5155 −0.734203
$$247$$ −3.68157 −0.234253
$$248$$ 17.6745 1.12233
$$249$$ 11.2690 0.714144
$$250$$ 0 0
$$251$$ 8.82062 0.556753 0.278376 0.960472i $$-0.410204\pi$$
0.278376 + 0.960472i $$0.410204\pi$$
$$252$$ 5.61817 0.353912
$$253$$ −2.58194 −0.162325
$$254$$ −11.1546 −0.699901
$$255$$ 0 0
$$256$$ −8.64135 −0.540084
$$257$$ −9.32567 −0.581719 −0.290860 0.956766i $$-0.593941\pi$$
−0.290860 + 0.956766i $$0.593941\pi$$
$$258$$ −0.564415 −0.0351389
$$259$$ 5.71416 0.355060
$$260$$ 0 0
$$261$$ 13.5184 0.836768
$$262$$ −7.56282 −0.467232
$$263$$ −23.7768 −1.46614 −0.733070 0.680153i $$-0.761913\pi$$
−0.733070 + 0.680153i $$0.761913\pi$$
$$264$$ 7.09403 0.436607
$$265$$ 0 0
$$266$$ −0.391741 −0.0240192
$$267$$ 11.4000 0.697669
$$268$$ 5.78877 0.353605
$$269$$ −14.8523 −0.905562 −0.452781 0.891622i $$-0.649568\pi$$
−0.452781 + 0.891622i $$0.649568\pi$$
$$270$$ 0 0
$$271$$ 13.9947 0.850117 0.425059 0.905166i $$-0.360253\pi$$
0.425059 + 0.905166i $$0.360253\pi$$
$$272$$ −0.393557 −0.0238629
$$273$$ −6.56811 −0.397520
$$274$$ 11.0607 0.668201
$$275$$ 0 0
$$276$$ −12.1507 −0.731386
$$277$$ 3.84886 0.231256 0.115628 0.993293i $$-0.463112\pi$$
0.115628 + 0.993293i $$0.463112\pi$$
$$278$$ 10.9981 0.659624
$$279$$ −45.5255 −2.72554
$$280$$ 0 0
$$281$$ 17.9323 1.06975 0.534877 0.844930i $$-0.320358\pi$$
0.534877 + 0.844930i $$0.320358\pi$$
$$282$$ −7.53896 −0.448938
$$283$$ −15.7342 −0.935304 −0.467652 0.883913i $$-0.654900\pi$$
−0.467652 + 0.883913i $$0.654900\pi$$
$$284$$ −25.2960 −1.50104
$$285$$ 0 0
$$286$$ −2.43434 −0.143946
$$287$$ −3.42635 −0.202251
$$288$$ 34.8806 2.05536
$$289$$ −16.9370 −0.996294
$$290$$ 0 0
$$291$$ 27.6703 1.62206
$$292$$ 12.3712 0.723970
$$293$$ −20.1233 −1.17562 −0.587808 0.809000i $$-0.700009\pi$$
−0.587808 + 0.809000i $$0.700009\pi$$
$$294$$ 13.2392 0.772123
$$295$$ 0 0
$$296$$ 22.7216 1.32067
$$297$$ −9.23869 −0.536083
$$298$$ 9.49297 0.549913
$$299$$ 9.50561 0.549723
$$300$$ 0 0
$$301$$ −0.167937 −0.00967971
$$302$$ 10.6036 0.610170
$$303$$ 42.5145 2.44240
$$304$$ 1.56786 0.0899231
$$305$$ 0 0
$$306$$ −1.00715 −0.0575748
$$307$$ −13.0048 −0.742226 −0.371113 0.928588i $$-0.621024\pi$$
−0.371113 + 0.928588i $$0.621024\pi$$
$$308$$ 0.925870 0.0527563
$$309$$ 15.6684 0.891346
$$310$$ 0 0
$$311$$ 23.0064 1.30457 0.652287 0.757972i $$-0.273810\pi$$
0.652287 + 0.757972i $$0.273810\pi$$
$$312$$ −26.1172 −1.47860
$$313$$ 10.8793 0.614936 0.307468 0.951558i $$-0.400518\pi$$
0.307468 + 0.951558i $$0.400518\pi$$
$$314$$ 15.9702 0.901248
$$315$$ 0 0
$$316$$ 24.0705 1.35407
$$317$$ 13.6489 0.766599 0.383300 0.923624i $$-0.374788\pi$$
0.383300 + 0.923624i $$0.374788\pi$$
$$318$$ 7.76011 0.435165
$$319$$ 2.22782 0.124734
$$320$$ 0 0
$$321$$ −38.8046 −2.16586
$$322$$ 1.01145 0.0563660
$$323$$ −0.251015 −0.0139669
$$324$$ −15.0286 −0.834924
$$325$$ 0 0
$$326$$ 10.7790 0.596994
$$327$$ −9.36461 −0.517864
$$328$$ −13.6244 −0.752282
$$329$$ −2.24315 −0.123669
$$330$$ 0 0
$$331$$ 8.45886 0.464941 0.232471 0.972603i $$-0.425319\pi$$
0.232471 + 0.972603i $$0.425319\pi$$
$$332$$ 5.84829 0.320966
$$333$$ −58.5257 −3.20719
$$334$$ 11.2309 0.614529
$$335$$ 0 0
$$336$$ 2.79714 0.152597
$$337$$ −5.76453 −0.314014 −0.157007 0.987598i $$-0.550184\pi$$
−0.157007 + 0.987598i $$0.550184\pi$$
$$338$$ 0.366308 0.0199245
$$339$$ 21.9166 1.19035
$$340$$ 0 0
$$341$$ −7.50257 −0.406287
$$342$$ 4.01230 0.216960
$$343$$ 8.08634 0.436621
$$344$$ −0.667778 −0.0360042
$$345$$ 0 0
$$346$$ 4.28625 0.230430
$$347$$ −20.2740 −1.08837 −0.544183 0.838966i $$-0.683160\pi$$
−0.544183 + 0.838966i $$0.683160\pi$$
$$348$$ 10.4842 0.562011
$$349$$ −4.09845 −0.219385 −0.109693 0.993966i $$-0.534987\pi$$
−0.109693 + 0.993966i $$0.534987\pi$$
$$350$$ 0 0
$$351$$ 34.0129 1.81547
$$352$$ 5.74830 0.306385
$$353$$ 26.6199 1.41684 0.708418 0.705793i $$-0.249409\pi$$
0.708418 + 0.705793i $$0.249409\pi$$
$$354$$ 5.94655 0.316056
$$355$$ 0 0
$$356$$ 5.91628 0.313562
$$357$$ −0.447823 −0.0237013
$$358$$ 1.65459 0.0874476
$$359$$ 22.2264 1.17307 0.586534 0.809925i $$-0.300492\pi$$
0.586534 + 0.809925i $$0.300492\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 2.03747 0.107087
$$363$$ −3.01131 −0.158053
$$364$$ −3.40866 −0.178662
$$365$$ 0 0
$$366$$ 17.8140 0.931150
$$367$$ 24.0546 1.25564 0.627820 0.778358i $$-0.283947\pi$$
0.627820 + 0.778358i $$0.283947\pi$$
$$368$$ −4.04813 −0.211023
$$369$$ 35.0934 1.82689
$$370$$ 0 0
$$371$$ 2.30895 0.119875
$$372$$ −35.3073 −1.83060
$$373$$ 0.342698 0.0177442 0.00887211 0.999961i $$-0.497176\pi$$
0.00887211 + 0.999961i $$0.497176\pi$$
$$374$$ −0.165977 −0.00858247
$$375$$ 0 0
$$376$$ −8.91959 −0.459993
$$377$$ −8.20188 −0.422418
$$378$$ 3.61917 0.186150
$$379$$ −30.4031 −1.56170 −0.780851 0.624717i $$-0.785214\pi$$
−0.780851 + 0.624717i $$0.785214\pi$$
$$380$$ 0 0
$$381$$ 50.7997 2.60255
$$382$$ 3.98361 0.203819
$$383$$ 26.5955 1.35897 0.679483 0.733692i $$-0.262204\pi$$
0.679483 + 0.733692i $$0.262204\pi$$
$$384$$ 33.2953 1.69910
$$385$$ 0 0
$$386$$ −6.51576 −0.331643
$$387$$ 1.72005 0.0874348
$$388$$ 14.3601 0.729022
$$389$$ 4.46007 0.226134 0.113067 0.993587i $$-0.463932\pi$$
0.113067 + 0.993587i $$0.463932\pi$$
$$390$$ 0 0
$$391$$ 0.648107 0.0327762
$$392$$ 15.6637 0.791136
$$393$$ 34.4422 1.73738
$$394$$ −6.84181 −0.344686
$$395$$ 0 0
$$396$$ −9.48297 −0.476537
$$397$$ −6.23802 −0.313077 −0.156539 0.987672i $$-0.550034\pi$$
−0.156539 + 0.987672i $$0.550034\pi$$
$$398$$ 6.39528 0.320566
$$399$$ 1.78405 0.0893141
$$400$$ 0 0
$$401$$ 18.8302 0.940336 0.470168 0.882577i $$-0.344193\pi$$
0.470168 + 0.882577i $$0.344193\pi$$
$$402$$ 7.37549 0.367856
$$403$$ 27.6213 1.37591
$$404$$ 22.0638 1.09772
$$405$$ 0 0
$$406$$ −0.872728 −0.0433128
$$407$$ −9.64498 −0.478084
$$408$$ −1.78071 −0.0881583
$$409$$ −29.9488 −1.48087 −0.740437 0.672126i $$-0.765381\pi$$
−0.740437 + 0.672126i $$0.765381\pi$$
$$410$$ 0 0
$$411$$ −50.3721 −2.48467
$$412$$ 8.13146 0.400609
$$413$$ 1.76935 0.0870638
$$414$$ −10.3595 −0.509142
$$415$$ 0 0
$$416$$ −21.1628 −1.03759
$$417$$ −50.0872 −2.45278
$$418$$ 0.661223 0.0323415
$$419$$ −27.8201 −1.35910 −0.679551 0.733628i $$-0.737826\pi$$
−0.679551 + 0.733628i $$0.737826\pi$$
$$420$$ 0 0
$$421$$ −28.7544 −1.40140 −0.700701 0.713455i $$-0.747129\pi$$
−0.700701 + 0.713455i $$0.747129\pi$$
$$422$$ −3.53652 −0.172155
$$423$$ 22.9749 1.11708
$$424$$ 9.18125 0.445881
$$425$$ 0 0
$$426$$ −32.2298 −1.56154
$$427$$ 5.30039 0.256504
$$428$$ −20.1385 −0.973429
$$429$$ 11.0864 0.535255
$$430$$ 0 0
$$431$$ 6.72669 0.324013 0.162007 0.986790i $$-0.448203\pi$$
0.162007 + 0.986790i $$0.448203\pi$$
$$432$$ −14.4850 −0.696909
$$433$$ −32.1724 −1.54611 −0.773053 0.634341i $$-0.781272\pi$$
−0.773053 + 0.634341i $$0.781272\pi$$
$$434$$ 2.93906 0.141080
$$435$$ 0 0
$$436$$ −4.85997 −0.232750
$$437$$ −2.58194 −0.123511
$$438$$ 15.7622 0.753147
$$439$$ 31.4984 1.50334 0.751668 0.659541i $$-0.229249\pi$$
0.751668 + 0.659541i $$0.229249\pi$$
$$440$$ 0 0
$$441$$ −40.3461 −1.92124
$$442$$ 0.611057 0.0290650
$$443$$ 18.9443 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$444$$ −45.3895 −2.15409
$$445$$ 0 0
$$446$$ −2.49190 −0.117995
$$447$$ −43.2324 −2.04482
$$448$$ −0.394086 −0.0186188
$$449$$ 31.1737 1.47118 0.735590 0.677427i $$-0.236905\pi$$
0.735590 + 0.677427i $$0.236905\pi$$
$$450$$ 0 0
$$451$$ 5.78336 0.272328
$$452$$ 11.3741 0.534992
$$453$$ −48.2905 −2.26889
$$454$$ 2.45067 0.115016
$$455$$ 0 0
$$456$$ 7.09403 0.332208
$$457$$ 34.4215 1.61017 0.805085 0.593159i $$-0.202120\pi$$
0.805085 + 0.593159i $$0.202120\pi$$
$$458$$ −4.74187 −0.221573
$$459$$ 2.31905 0.108244
$$460$$ 0 0
$$461$$ −24.8901 −1.15925 −0.579624 0.814884i $$-0.696801\pi$$
−0.579624 + 0.814884i $$0.696801\pi$$
$$462$$ 1.17965 0.0548825
$$463$$ 26.3939 1.22663 0.613314 0.789839i $$-0.289836\pi$$
0.613314 + 0.789839i $$0.289836\pi$$
$$464$$ 3.49291 0.162154
$$465$$ 0 0
$$466$$ 4.29894 0.199144
$$467$$ −26.6487 −1.23315 −0.616577 0.787295i $$-0.711481\pi$$
−0.616577 + 0.787295i $$0.711481\pi$$
$$468$$ 34.9122 1.61382
$$469$$ 2.19451 0.101333
$$470$$ 0 0
$$471$$ −72.7306 −3.35125
$$472$$ 7.03557 0.323838
$$473$$ 0.283462 0.0130336
$$474$$ 30.6682 1.40864
$$475$$ 0 0
$$476$$ −0.232407 −0.0106524
$$477$$ −23.6488 −1.08281
$$478$$ 15.8052 0.722914
$$479$$ −24.1076 −1.10150 −0.550751 0.834669i $$-0.685659\pi$$
−0.550751 + 0.834669i $$0.685659\pi$$
$$480$$ 0 0
$$481$$ 35.5087 1.61906
$$482$$ 10.3835 0.472958
$$483$$ −4.60631 −0.209594
$$484$$ −1.56278 −0.0710356
$$485$$ 0 0
$$486$$ −0.821527 −0.0372652
$$487$$ 11.9390 0.541006 0.270503 0.962719i $$-0.412810\pi$$
0.270503 + 0.962719i $$0.412810\pi$$
$$488$$ 21.0763 0.954079
$$489$$ −49.0893 −2.21989
$$490$$ 0 0
$$491$$ 10.8304 0.488768 0.244384 0.969678i $$-0.421414\pi$$
0.244384 + 0.969678i $$0.421414\pi$$
$$492$$ 27.2167 1.22702
$$493$$ −0.559217 −0.0251858
$$494$$ −2.43434 −0.109526
$$495$$ 0 0
$$496$$ −11.7630 −0.528174
$$497$$ −9.58969 −0.430156
$$498$$ 7.45132 0.333902
$$499$$ 14.6568 0.656130 0.328065 0.944655i $$-0.393604\pi$$
0.328065 + 0.944655i $$0.393604\pi$$
$$500$$ 0 0
$$501$$ −51.1474 −2.28510
$$502$$ 5.83240 0.260313
$$503$$ 21.7295 0.968872 0.484436 0.874827i $$-0.339025\pi$$
0.484436 + 0.874827i $$0.339025\pi$$
$$504$$ 8.46903 0.377241
$$505$$ 0 0
$$506$$ −1.70724 −0.0758960
$$507$$ −1.66822 −0.0740883
$$508$$ 26.3636 1.16969
$$509$$ −22.4914 −0.996914 −0.498457 0.866914i $$-0.666100\pi$$
−0.498457 + 0.866914i $$0.666100\pi$$
$$510$$ 0 0
$$511$$ 4.68990 0.207469
$$512$$ 16.3997 0.724769
$$513$$ −9.23869 −0.407898
$$514$$ −6.16635 −0.271986
$$515$$ 0 0
$$516$$ 1.33398 0.0587252
$$517$$ 3.78624 0.166518
$$518$$ 3.77833 0.166010
$$519$$ −19.5202 −0.856843
$$520$$ 0 0
$$521$$ −14.1999 −0.622109 −0.311055 0.950392i $$-0.600682\pi$$
−0.311055 + 0.950392i $$0.600682\pi$$
$$522$$ 8.93868 0.391235
$$523$$ 12.2976 0.537735 0.268867 0.963177i $$-0.413351\pi$$
0.268867 + 0.963177i $$0.413351\pi$$
$$524$$ 17.8745 0.780852
$$525$$ 0 0
$$526$$ −15.7218 −0.685502
$$527$$ 1.88326 0.0820360
$$528$$ −4.72132 −0.205469
$$529$$ −16.3336 −0.710156
$$530$$ 0 0
$$531$$ −18.1220 −0.786429
$$532$$ 0.925870 0.0401416
$$533$$ −21.2919 −0.922253
$$534$$ 7.53795 0.326199
$$535$$ 0 0
$$536$$ 8.72619 0.376914
$$537$$ −7.53524 −0.325169
$$538$$ −9.82069 −0.423400
$$539$$ −6.64900 −0.286393
$$540$$ 0 0
$$541$$ 31.2468 1.34341 0.671703 0.740821i $$-0.265563\pi$$
0.671703 + 0.740821i $$0.265563\pi$$
$$542$$ 9.25362 0.397477
$$543$$ −9.27894 −0.398197
$$544$$ −1.44291 −0.0618643
$$545$$ 0 0
$$546$$ −4.34298 −0.185863
$$547$$ −39.7365 −1.69901 −0.849505 0.527581i $$-0.823099\pi$$
−0.849505 + 0.527581i $$0.823099\pi$$
$$548$$ −26.1417 −1.11672
$$549$$ −54.2878 −2.31695
$$550$$ 0 0
$$551$$ 2.22782 0.0949083
$$552$$ −18.3164 −0.779597
$$553$$ 9.12507 0.388038
$$554$$ 2.54496 0.108125
$$555$$ 0 0
$$556$$ −25.9938 −1.10238
$$557$$ 14.7208 0.623740 0.311870 0.950125i $$-0.399045\pi$$
0.311870 + 0.950125i $$0.399045\pi$$
$$558$$ −30.1025 −1.27434
$$559$$ −1.04359 −0.0441390
$$560$$ 0 0
$$561$$ 0.755885 0.0319135
$$562$$ 11.8573 0.500169
$$563$$ 29.4737 1.24217 0.621083 0.783745i $$-0.286693\pi$$
0.621083 + 0.783745i $$0.286693\pi$$
$$564$$ 17.8181 0.750279
$$565$$ 0 0
$$566$$ −10.4038 −0.437306
$$567$$ −5.69733 −0.239265
$$568$$ −38.1321 −1.59999
$$569$$ −42.0413 −1.76246 −0.881231 0.472685i $$-0.843285\pi$$
−0.881231 + 0.472685i $$0.843285\pi$$
$$570$$ 0 0
$$571$$ 13.1875 0.551878 0.275939 0.961175i $$-0.411011\pi$$
0.275939 + 0.961175i $$0.411011\pi$$
$$572$$ 5.75350 0.240566
$$573$$ −18.1420 −0.757893
$$574$$ −2.26558 −0.0945634
$$575$$ 0 0
$$576$$ 4.03632 0.168180
$$577$$ 28.1175 1.17055 0.585273 0.810836i $$-0.300987\pi$$
0.585273 + 0.810836i $$0.300987\pi$$
$$578$$ −11.1991 −0.465822
$$579$$ 29.6738 1.23320
$$580$$ 0 0
$$581$$ 2.21708 0.0919798
$$582$$ 18.2962 0.758403
$$583$$ −3.89730 −0.161410
$$584$$ 18.6488 0.771692
$$585$$ 0 0
$$586$$ −13.3060 −0.549665
$$587$$ 0.510507 0.0210709 0.0105354 0.999945i $$-0.496646\pi$$
0.0105354 + 0.999945i $$0.496646\pi$$
$$588$$ −31.2904 −1.29039
$$589$$ −7.50257 −0.309138
$$590$$ 0 0
$$591$$ 31.1587 1.28170
$$592$$ −15.1220 −0.621510
$$593$$ 17.7779 0.730052 0.365026 0.930997i $$-0.381060\pi$$
0.365026 + 0.930997i $$0.381060\pi$$
$$594$$ −6.10883 −0.250648
$$595$$ 0 0
$$596$$ −22.4364 −0.919030
$$597$$ −29.1251 −1.19201
$$598$$ 6.28533 0.257026
$$599$$ 31.6622 1.29368 0.646842 0.762624i $$-0.276089\pi$$
0.646842 + 0.762624i $$0.276089\pi$$
$$600$$ 0 0
$$601$$ −35.7075 −1.45654 −0.728270 0.685290i $$-0.759675\pi$$
−0.728270 + 0.685290i $$0.759675\pi$$
$$602$$ −0.111044 −0.00452580
$$603$$ −22.4767 −0.915322
$$604$$ −25.0614 −1.01973
$$605$$ 0 0
$$606$$ 28.1116 1.14196
$$607$$ −26.6864 −1.08317 −0.541584 0.840647i $$-0.682175\pi$$
−0.541584 + 0.840647i $$0.682175\pi$$
$$608$$ 5.74830 0.233124
$$609$$ 3.97454 0.161056
$$610$$ 0 0
$$611$$ −13.9393 −0.563924
$$612$$ 2.38037 0.0962207
$$613$$ −37.6483 −1.52060 −0.760301 0.649571i $$-0.774949\pi$$
−0.760301 + 0.649571i $$0.774949\pi$$
$$614$$ −8.59910 −0.347032
$$615$$ 0 0
$$616$$ 1.39569 0.0562339
$$617$$ 38.9725 1.56897 0.784486 0.620146i $$-0.212927\pi$$
0.784486 + 0.620146i $$0.212927\pi$$
$$618$$ 10.3603 0.416753
$$619$$ −18.4615 −0.742029 −0.371015 0.928627i $$-0.620990\pi$$
−0.371015 + 0.928627i $$0.620990\pi$$
$$620$$ 0 0
$$621$$ 23.8537 0.957218
$$622$$ 15.2124 0.609960
$$623$$ 2.24285 0.0898579
$$624$$ 17.3819 0.695833
$$625$$ 0 0
$$626$$ 7.19366 0.287517
$$627$$ −3.01131 −0.120260
$$628$$ −37.7451 −1.50619
$$629$$ 2.42104 0.0965330
$$630$$ 0 0
$$631$$ −13.9700 −0.556137 −0.278069 0.960561i $$-0.589694\pi$$
−0.278069 + 0.960561i $$0.589694\pi$$
$$632$$ 36.2846 1.44333
$$633$$ 16.1059 0.640151
$$634$$ 9.02497 0.358427
$$635$$ 0 0
$$636$$ −18.3408 −0.727261
$$637$$ 24.4788 0.969885
$$638$$ 1.47309 0.0583200
$$639$$ 98.2197 3.88551
$$640$$ 0 0
$$641$$ −15.9666 −0.630642 −0.315321 0.948985i $$-0.602112\pi$$
−0.315321 + 0.948985i $$0.602112\pi$$
$$642$$ −25.6585 −1.01266
$$643$$ 24.3556 0.960492 0.480246 0.877134i $$-0.340547\pi$$
0.480246 + 0.877134i $$0.340547\pi$$
$$644$$ −2.39054 −0.0942006
$$645$$ 0 0
$$646$$ −0.165977 −0.00653028
$$647$$ −0.437751 −0.0172098 −0.00860488 0.999963i $$-0.502739\pi$$
−0.00860488 + 0.999963i $$0.502739\pi$$
$$648$$ −22.6547 −0.889960
$$649$$ −2.98649 −0.117230
$$650$$ 0 0
$$651$$ −13.3849 −0.524597
$$652$$ −25.4759 −0.997714
$$653$$ 33.2515 1.30123 0.650616 0.759407i $$-0.274511\pi$$
0.650616 + 0.759407i $$0.274511\pi$$
$$654$$ −6.19210 −0.242130
$$655$$ 0 0
$$656$$ 9.06751 0.354027
$$657$$ −48.0350 −1.87403
$$658$$ −1.48322 −0.0578221
$$659$$ 2.76267 0.107618 0.0538092 0.998551i $$-0.482864\pi$$
0.0538092 + 0.998551i $$0.482864\pi$$
$$660$$ 0 0
$$661$$ 20.4256 0.794466 0.397233 0.917718i $$-0.369971\pi$$
0.397233 + 0.917718i $$0.369971\pi$$
$$662$$ 5.59319 0.217386
$$663$$ −2.78285 −0.108077
$$664$$ 8.81591 0.342124
$$665$$ 0 0
$$666$$ −38.6985 −1.49954
$$667$$ −5.75210 −0.222722
$$668$$ −26.5440 −1.02702
$$669$$ 11.3485 0.438758
$$670$$ 0 0
$$671$$ −8.94657 −0.345379
$$672$$ 10.2552 0.395604
$$673$$ 40.3538 1.55553 0.777763 0.628558i $$-0.216354\pi$$
0.777763 + 0.628558i $$0.216354\pi$$
$$674$$ −3.81164 −0.146819
$$675$$ 0 0
$$676$$ −0.865759 −0.0332984
$$677$$ 0.670468 0.0257682 0.0128841 0.999917i $$-0.495899\pi$$
0.0128841 + 0.999917i $$0.495899\pi$$
$$678$$ 14.4918 0.556553
$$679$$ 5.44388 0.208917
$$680$$ 0 0
$$681$$ −11.1607 −0.427680
$$682$$ −4.96087 −0.189962
$$683$$ 23.2360 0.889101 0.444551 0.895754i $$-0.353363\pi$$
0.444551 + 0.895754i $$0.353363\pi$$
$$684$$ −9.48297 −0.362590
$$685$$ 0 0
$$686$$ 5.34687 0.204145
$$687$$ 21.5952 0.823909
$$688$$ 0.444429 0.0169437
$$689$$ 14.3482 0.546623
$$690$$ 0 0
$$691$$ 38.1311 1.45058 0.725288 0.688446i $$-0.241707\pi$$
0.725288 + 0.688446i $$0.241707\pi$$
$$692$$ −10.1304 −0.385102
$$693$$ −3.59498 −0.136562
$$694$$ −13.4056 −0.508871
$$695$$ 0 0
$$696$$ 15.8042 0.599057
$$697$$ −1.45171 −0.0549875
$$698$$ −2.70999 −0.102575
$$699$$ −19.5780 −0.740509
$$700$$ 0 0
$$701$$ −10.9408 −0.413228 −0.206614 0.978423i $$-0.566244\pi$$
−0.206614 + 0.978423i $$0.566244\pi$$
$$702$$ 22.4901 0.848835
$$703$$ −9.64498 −0.363767
$$704$$ 0.665182 0.0250700
$$705$$ 0 0
$$706$$ 17.6017 0.662449
$$707$$ 8.36436 0.314574
$$708$$ −14.0545 −0.528201
$$709$$ 47.7317 1.79260 0.896300 0.443448i $$-0.146245\pi$$
0.896300 + 0.443448i $$0.146245\pi$$
$$710$$ 0 0
$$711$$ −93.4610 −3.50506
$$712$$ 8.91840 0.334231
$$713$$ 19.3712 0.725457
$$714$$ −0.296111 −0.0110817
$$715$$ 0 0
$$716$$ −3.91057 −0.146145
$$717$$ −71.9794 −2.68812
$$718$$ 14.6966 0.548474
$$719$$ −43.5137 −1.62279 −0.811394 0.584499i $$-0.801291\pi$$
−0.811394 + 0.584499i $$0.801291\pi$$
$$720$$ 0 0
$$721$$ 3.08263 0.114803
$$722$$ 0.661223 0.0246082
$$723$$ −47.2883 −1.75867
$$724$$ −4.81550 −0.178967
$$725$$ 0 0
$$726$$ −1.99115 −0.0738984
$$727$$ −6.09893 −0.226197 −0.113098 0.993584i $$-0.536078\pi$$
−0.113098 + 0.993584i $$0.536078\pi$$
$$728$$ −5.13833 −0.190439
$$729$$ −25.1084 −0.929939
$$730$$ 0 0
$$731$$ −0.0711532 −0.00263170
$$732$$ −42.1028 −1.55617
$$733$$ −18.9127 −0.698557 −0.349279 0.937019i $$-0.613573\pi$$
−0.349279 + 0.937019i $$0.613573\pi$$
$$734$$ 15.9055 0.587081
$$735$$ 0 0
$$736$$ −14.8418 −0.547075
$$737$$ −3.70414 −0.136444
$$738$$ 23.2046 0.854172
$$739$$ −37.3950 −1.37560 −0.687799 0.725901i $$-0.741423\pi$$
−0.687799 + 0.725901i $$0.741423\pi$$
$$740$$ 0 0
$$741$$ 11.0864 0.407268
$$742$$ 1.52673 0.0560482
$$743$$ 47.1521 1.72984 0.864921 0.501908i $$-0.167369\pi$$
0.864921 + 0.501908i $$0.167369\pi$$
$$744$$ −53.2235 −1.95127
$$745$$ 0 0
$$746$$ 0.226600 0.00829640
$$747$$ −22.7078 −0.830834
$$748$$ 0.392283 0.0143433
$$749$$ −7.63446 −0.278957
$$750$$ 0 0
$$751$$ 41.7913 1.52499 0.762493 0.646996i $$-0.223975\pi$$
0.762493 + 0.646996i $$0.223975\pi$$
$$752$$ 5.93630 0.216474
$$753$$ −26.5616 −0.967960
$$754$$ −5.42327 −0.197504
$$755$$ 0 0
$$756$$ −8.55382 −0.311099
$$757$$ −46.1909 −1.67884 −0.839419 0.543485i $$-0.817105\pi$$
−0.839419 + 0.543485i $$0.817105\pi$$
$$758$$ −20.1032 −0.730182
$$759$$ 7.77503 0.282216
$$760$$ 0 0
$$761$$ 29.8786 1.08310 0.541550 0.840669i $$-0.317838\pi$$
0.541550 + 0.840669i $$0.317838\pi$$
$$762$$ 33.5899 1.21683
$$763$$ −1.84241 −0.0666996
$$764$$ −9.41517 −0.340629
$$765$$ 0 0
$$766$$ 17.5855 0.635391
$$767$$ 10.9950 0.397006
$$768$$ 26.0218 0.938980
$$769$$ −26.3059 −0.948615 −0.474307 0.880359i $$-0.657301\pi$$
−0.474307 + 0.880359i $$0.657301\pi$$
$$770$$ 0 0
$$771$$ 28.0825 1.01137
$$772$$ 15.3998 0.554252
$$773$$ 11.6861 0.420318 0.210159 0.977667i $$-0.432602\pi$$
0.210159 + 0.977667i $$0.432602\pi$$
$$774$$ 1.13733 0.0408806
$$775$$ 0 0
$$776$$ 21.6469 0.777077
$$777$$ −17.2071 −0.617301
$$778$$ 2.94910 0.105730
$$779$$ 5.78336 0.207210
$$780$$ 0 0
$$781$$ 16.1865 0.579199
$$782$$ 0.428543 0.0153247
$$783$$ −20.5821 −0.735545
$$784$$ −10.4247 −0.372312
$$785$$ 0 0
$$786$$ 22.7740 0.812321
$$787$$ −26.6657 −0.950529 −0.475265 0.879843i $$-0.657648\pi$$
−0.475265 + 0.879843i $$0.657648\pi$$
$$788$$ 16.1705 0.576049
$$789$$ 71.5994 2.54900
$$790$$ 0 0
$$791$$ 4.31190 0.153313
$$792$$ −14.2950 −0.507949
$$793$$ 32.9375 1.16964
$$794$$ −4.12472 −0.146381
$$795$$ 0 0
$$796$$ −15.1151 −0.535740
$$797$$ 4.54909 0.161137 0.0805686 0.996749i $$-0.474326\pi$$
0.0805686 + 0.996749i $$0.474326\pi$$
$$798$$ 1.17965 0.0417593
$$799$$ −0.950403 −0.0336228
$$800$$ 0 0
$$801$$ −22.9718 −0.811668
$$802$$ 12.4510 0.439659
$$803$$ −7.91613 −0.279354
$$804$$ −17.4318 −0.614772
$$805$$ 0 0
$$806$$ 18.2638 0.643315
$$807$$ 44.7249 1.57439
$$808$$ 33.2598 1.17007
$$809$$ −0.704828 −0.0247804 −0.0123902 0.999923i $$-0.503944\pi$$
−0.0123902 + 0.999923i $$0.503944\pi$$
$$810$$ 0 0
$$811$$ −1.69947 −0.0596763 −0.0298382 0.999555i $$-0.509499\pi$$
−0.0298382 + 0.999555i $$0.509499\pi$$
$$812$$ 2.06267 0.0723856
$$813$$ −42.1424 −1.47800
$$814$$ −6.37748 −0.223531
$$815$$ 0 0
$$816$$ 1.18512 0.0414876
$$817$$ 0.283462 0.00991708
$$818$$ −19.8028 −0.692390
$$819$$ 13.2352 0.462474
$$820$$ 0 0
$$821$$ 0.312402 0.0109029 0.00545146 0.999985i $$-0.498265\pi$$
0.00545146 + 0.999985i $$0.498265\pi$$
$$822$$ −33.3072 −1.16172
$$823$$ −37.7601 −1.31623 −0.658116 0.752916i $$-0.728647\pi$$
−0.658116 + 0.752916i $$0.728647\pi$$
$$824$$ 12.2577 0.427016
$$825$$ 0 0
$$826$$ 1.16993 0.0407071
$$827$$ −47.5469 −1.65337 −0.826684 0.562667i $$-0.809775\pi$$
−0.826684 + 0.562667i $$0.809775\pi$$
$$828$$ 24.4845 0.850894
$$829$$ −19.6186 −0.681381 −0.340690 0.940176i $$-0.610661\pi$$
−0.340690 + 0.940176i $$0.610661\pi$$
$$830$$ 0 0
$$831$$ −11.5901 −0.402057
$$832$$ −2.44891 −0.0849008
$$833$$ 1.66900 0.0578275
$$834$$ −33.1188 −1.14681
$$835$$ 0 0
$$836$$ −1.56278 −0.0540500
$$837$$ 69.3139 2.39584
$$838$$ −18.3953 −0.635456
$$839$$ −31.4596 −1.08611 −0.543053 0.839698i $$-0.682732\pi$$
−0.543053 + 0.839698i $$0.682732\pi$$
$$840$$ 0 0
$$841$$ −24.0368 −0.828856
$$842$$ −19.0130 −0.655233
$$843$$ −53.9999 −1.85985
$$844$$ 8.35849 0.287711
$$845$$ 0 0
$$846$$ 15.1915 0.522295
$$847$$ −0.592449 −0.0203568
$$848$$ −6.11044 −0.209833
$$849$$ 47.3807 1.62610
$$850$$ 0 0
$$851$$ 24.9028 0.853656
$$852$$ 76.1742 2.60969
$$853$$ 51.7295 1.77119 0.885593 0.464463i $$-0.153753\pi$$
0.885593 + 0.464463i $$0.153753\pi$$
$$854$$ 3.50474 0.119930
$$855$$ 0 0
$$856$$ −30.3574 −1.03759
$$857$$ −23.5776 −0.805394 −0.402697 0.915333i $$-0.631927\pi$$
−0.402697 + 0.915333i $$0.631927\pi$$
$$858$$ 7.33056 0.250261
$$859$$ 7.75928 0.264743 0.132372 0.991200i $$-0.457741\pi$$
0.132372 + 0.991200i $$0.457741\pi$$
$$860$$ 0 0
$$861$$ 10.3178 0.351629
$$862$$ 4.44784 0.151494
$$863$$ 46.2297 1.57368 0.786839 0.617158i $$-0.211716\pi$$
0.786839 + 0.617158i $$0.211716\pi$$
$$864$$ −53.1067 −1.80673
$$865$$ 0 0
$$866$$ −21.2731 −0.722890
$$867$$ 51.0025 1.73214
$$868$$ −6.94640 −0.235776
$$869$$ −15.4023 −0.522487
$$870$$ 0 0
$$871$$ 13.6371 0.462074
$$872$$ −7.32608 −0.248092
$$873$$ −55.7574 −1.88710
$$874$$ −1.70724 −0.0577482
$$875$$ 0 0
$$876$$ −37.2535 −1.25868
$$877$$ 39.3550 1.32892 0.664461 0.747323i $$-0.268661\pi$$
0.664461 + 0.747323i $$0.268661\pi$$
$$878$$ 20.8275 0.702893
$$879$$ 60.5975 2.04390
$$880$$ 0 0
$$881$$ −0.350505 −0.0118088 −0.00590441 0.999983i $$-0.501879\pi$$
−0.00590441 + 0.999983i $$0.501879\pi$$
$$882$$ −26.6778 −0.898288
$$883$$ −15.3175 −0.515476 −0.257738 0.966215i $$-0.582977\pi$$
−0.257738 + 0.966215i $$0.582977\pi$$
$$884$$ −1.44422 −0.0485743
$$885$$ 0 0
$$886$$ 12.5264 0.420833
$$887$$ 55.8068 1.87381 0.936904 0.349586i $$-0.113678\pi$$
0.936904 + 0.349586i $$0.113678\pi$$
$$888$$ −68.4218 −2.29608
$$889$$ 9.99439 0.335201
$$890$$ 0 0
$$891$$ 9.61657 0.322167
$$892$$ 5.88954 0.197196
$$893$$ 3.78624 0.126702
$$894$$ −28.5863 −0.956068
$$895$$ 0 0
$$896$$ 6.55057 0.218839
$$897$$ −28.6243 −0.955739
$$898$$ 20.6128 0.687858
$$899$$ −16.7144 −0.557455
$$900$$ 0 0
$$901$$ 0.978283 0.0325913
$$902$$ 3.82409 0.127328
$$903$$ 0.505710 0.0168290
$$904$$ 17.1457 0.570257
$$905$$ 0 0
$$906$$ −31.9308 −1.06083
$$907$$ −35.6759 −1.18460 −0.592300 0.805718i $$-0.701780\pi$$
−0.592300 + 0.805718i $$0.701780\pi$$
$$908$$ −5.79209 −0.192217
$$909$$ −85.6696 −2.84148
$$910$$ 0 0
$$911$$ 48.6660 1.61238 0.806189 0.591658i $$-0.201526\pi$$
0.806189 + 0.591658i $$0.201526\pi$$
$$912$$ −4.72132 −0.156339
$$913$$ −3.74222 −0.123849
$$914$$ 22.7603 0.752844
$$915$$ 0 0
$$916$$ 11.2073 0.370300
$$917$$ 6.77621 0.223770
$$918$$ 1.53341 0.0506101
$$919$$ 44.9545 1.48291 0.741456 0.671002i $$-0.234136\pi$$
0.741456 + 0.671002i $$0.234136\pi$$
$$920$$ 0 0
$$921$$ 39.1616 1.29042
$$922$$ −16.4579 −0.542013
$$923$$ −59.5918 −1.96149
$$924$$ −2.78808 −0.0917212
$$925$$ 0 0
$$926$$ 17.4523 0.573517
$$927$$ −31.5729 −1.03699
$$928$$ 12.8062 0.420383
$$929$$ −32.8433 −1.07756 −0.538778 0.842448i $$-0.681114\pi$$
−0.538778 + 0.842448i $$0.681114\pi$$
$$930$$ 0 0
$$931$$ −6.64900 −0.217912
$$932$$ −10.1604 −0.332816
$$933$$ −69.2794 −2.26811
$$934$$ −17.6207 −0.576567
$$935$$ 0 0
$$936$$ 52.6279 1.72020
$$937$$ 32.8654 1.07367 0.536833 0.843688i $$-0.319620\pi$$
0.536833 + 0.843688i $$0.319620\pi$$
$$938$$ 1.45106 0.0473789
$$939$$ −32.7610 −1.06912
$$940$$ 0 0
$$941$$ 42.2849 1.37845 0.689225 0.724548i $$-0.257951\pi$$
0.689225 + 0.724548i $$0.257951\pi$$
$$942$$ −48.0911 −1.56689
$$943$$ −14.9323 −0.486262
$$944$$ −4.68241 −0.152399
$$945$$ 0 0
$$946$$ 0.187432 0.00609392
$$947$$ −0.322414 −0.0104770 −0.00523852 0.999986i $$-0.501667\pi$$
−0.00523852 + 0.999986i $$0.501667\pi$$
$$948$$ −72.4836 −2.35416
$$949$$ 29.1438 0.946048
$$950$$ 0 0
$$951$$ −41.1011 −1.33279
$$952$$ −0.350339 −0.0113546
$$953$$ −36.2919 −1.17561 −0.587806 0.809002i $$-0.700008\pi$$
−0.587806 + 0.809002i $$0.700008\pi$$
$$954$$ −15.6371 −0.506271
$$955$$ 0 0
$$956$$ −37.3552 −1.20816
$$957$$ −6.70866 −0.216860
$$958$$ −15.9405 −0.515013
$$959$$ −9.91027 −0.320019
$$960$$ 0 0
$$961$$ 25.2885 0.815759
$$962$$ 23.4792 0.756999
$$963$$ 78.1938 2.51976
$$964$$ −24.5413 −0.790421
$$965$$ 0 0
$$966$$ −3.04580 −0.0979969
$$967$$ −3.37752 −0.108614 −0.0543069 0.998524i $$-0.517295\pi$$
−0.0543069 + 0.998524i $$0.517295\pi$$
$$968$$ −2.35579 −0.0757181
$$969$$ 0.755885 0.0242825
$$970$$ 0 0
$$971$$ −37.0852 −1.19012 −0.595061 0.803681i $$-0.702872\pi$$
−0.595061 + 0.803681i $$0.702872\pi$$
$$972$$ 1.94166 0.0622787
$$973$$ −9.85421 −0.315911
$$974$$ 7.89432 0.252950
$$975$$ 0 0
$$976$$ −14.0270 −0.448993
$$977$$ −15.5780 −0.498385 −0.249192 0.968454i $$-0.580165\pi$$
−0.249192 + 0.968454i $$0.580165\pi$$
$$978$$ −32.4590 −1.03792
$$979$$ −3.78573 −0.120992
$$980$$ 0 0
$$981$$ 18.8703 0.602483
$$982$$ 7.16130 0.228526
$$983$$ −17.9951 −0.573956 −0.286978 0.957937i $$-0.592651\pi$$
−0.286978 + 0.957937i $$0.592651\pi$$
$$984$$ 41.0273 1.30790
$$985$$ 0 0
$$986$$ −0.369767 −0.0117758
$$987$$ 6.75483 0.215009
$$988$$ 5.75350 0.183043
$$989$$ −0.731882 −0.0232725
$$990$$ 0 0
$$991$$ 5.95229 0.189081 0.0945403 0.995521i $$-0.469862\pi$$
0.0945403 + 0.995521i $$0.469862\pi$$
$$992$$ −43.1270 −1.36928
$$993$$ −25.4723 −0.808338
$$994$$ −6.34092 −0.201122
$$995$$ 0 0
$$996$$ −17.6110 −0.558026
$$997$$ 4.21169 0.133386 0.0666928 0.997774i $$-0.478755\pi$$
0.0666928 + 0.997774i $$0.478755\pi$$
$$998$$ 9.69143 0.306777
$$999$$ 89.1069 2.81922
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 5225.2.a.bc.1.19 30
5.2 odd 4 1045.2.b.e.419.19 yes 30
5.3 odd 4 1045.2.b.e.419.12 30
5.4 even 2 inner 5225.2.a.bc.1.12 30

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.b.e.419.12 30 5.3 odd 4
1045.2.b.e.419.19 yes 30 5.2 odd 4
5225.2.a.bc.1.12 30 5.4 even 2 inner
5225.2.a.bc.1.19 30 1.1 even 1 trivial