LMFDB - Embedded newform 5225.2.a.l.1.3

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.7281497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 $ x^6 - 2x^5 - 5x^4 + 7x^3 + 6x^2 - 2*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.77015$ of defining polynomial
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)

$f(q)$	$=$	$q-0.205229 q^{2} +3.10246 q^{3} -1.95788 q^{4} -0.636714 q^{6} -2.33231 q^{7} +0.812271 q^{8} +6.62527 q^{9} +O(q^{10})$	\(q-0.205229 q^{2} +3.10246 q^{3} -1.95788 q^{4} -0.636714 q^{6} -2.33231 q^{7} +0.812271 q^{8} +6.62527 q^{9} -1.00000 q^{11} -6.07425 q^{12} +4.30769 q^{13} +0.478657 q^{14} +3.74906 q^{16} -6.71096 q^{17} -1.35969 q^{18} -1.00000 q^{19} -7.23590 q^{21} +0.205229 q^{22} -8.24907 q^{23} +2.52004 q^{24} -0.884062 q^{26} +11.2472 q^{27} +4.56638 q^{28} +3.02355 q^{29} -10.2077 q^{31} -2.39396 q^{32} -3.10246 q^{33} +1.37728 q^{34} -12.9715 q^{36} +5.06393 q^{37} +0.205229 q^{38} +13.3644 q^{39} -6.47966 q^{41} +1.48501 q^{42} -1.64130 q^{43} +1.95788 q^{44} +1.69295 q^{46} +8.15593 q^{47} +11.6313 q^{48} -1.56033 q^{49} -20.8205 q^{51} -8.43395 q^{52} +2.35693 q^{53} -2.30826 q^{54} -1.89447 q^{56} -3.10246 q^{57} -0.620519 q^{58} -4.65880 q^{59} +8.93722 q^{61} +2.09491 q^{62} -15.4522 q^{63} -7.00681 q^{64} +0.636714 q^{66} -13.2994 q^{67} +13.1393 q^{68} -25.5924 q^{69} -8.76859 q^{71} +5.38151 q^{72} +2.66639 q^{73} -1.03926 q^{74} +1.95788 q^{76} +2.33231 q^{77} -2.74277 q^{78} -12.1246 q^{79} +15.0183 q^{81} +1.32981 q^{82} +17.8672 q^{83} +14.1670 q^{84} +0.336843 q^{86} +9.38045 q^{87} -0.812271 q^{88} -1.27875 q^{89} -10.0469 q^{91} +16.1507 q^{92} -31.6690 q^{93} -1.67383 q^{94} -7.42716 q^{96} -14.9748 q^{97} +0.320225 q^{98} -6.62527 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9}+O(q^{10}) $ 6 * q + 2 * q^2 + q^3 + 4 * q^4 - 5 * q^7 + 12 * q^8 + q^9	$ 6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9} - 6 q^{11} - q^{12} + 5 q^{13} - 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{18} - 6 q^{19} - 21 q^{21} - 2 q^{22} - 4 q^{23} - q^{24} - 14 q^{26} + 16 q^{27} - 10 q^{28} - 9 q^{29} - 21 q^{31} + q^{32} - q^{33} - 28 q^{36} + 3 q^{37} - 2 q^{38} + 20 q^{39} - 23 q^{41} - q^{42} - 7 q^{43} - 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} - 16 q^{51} - 13 q^{52} + 17 q^{53} + q^{54} - 2 q^{56} - q^{57} - 23 q^{58} - 29 q^{59} + 17 q^{61} - 2 q^{62} - 6 q^{63} - 18 q^{64} - 8 q^{67} + q^{68} - 38 q^{69} - 12 q^{71} - 13 q^{72} - 2 q^{73} - 37 q^{74} - 4 q^{76} + 5 q^{77} - q^{78} + 3 q^{79} - 2 q^{81} - 24 q^{82} + 11 q^{83} - 3 q^{84} - 12 q^{86} + 12 q^{87} - 12 q^{88} - 22 q^{89} - 18 q^{91} + 15 q^{92} - 18 q^{93} + 22 q^{94} - 17 q^{96} + 2 q^{97} + q^{98} - q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 + q^3 + 4 * q^4 - 5 * q^7 + 12 * q^8 + q^9 - 6 * q^11 - q^12 + 5 * q^13 - 8 * q^14 + 4 * q^16 - q^17 - 6 * q^18 - 6 * q^19 - 21 * q^21 - 2 * q^22 - 4 * q^23 - q^24 - 14 * q^26 + 16 * q^27 - 10 * q^28 - 9 * q^29 - 21 * q^31 + q^32 - q^33 - 28 * q^36 + 3 * q^37 - 2 * q^38 + 20 * q^39 - 23 * q^41 - q^42 - 7 * q^43 - 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 - 16 * q^51 - 13 * q^52 + 17 * q^53 + q^54 - 2 * q^56 - q^57 - 23 * q^58 - 29 * q^59 + 17 * q^61 - 2 * q^62 - 6 * q^63 - 18 * q^64 - 8 * q^67 + q^68 - 38 * q^69 - 12 * q^71 - 13 * q^72 - 2 * q^73 - 37 * q^74 - 4 * q^76 + 5 * q^77 - q^78 + 3 * q^79 - 2 * q^81 - 24 * q^82 + 11 * q^83 - 3 * q^84 - 12 * q^86 + 12 * q^87 - 12 * q^88 - 22 * q^89 - 18 * q^91 + 15 * q^92 - 18 * q^93 + 22 * q^94 - 17 * q^96 + 2 * q^97 + q^98 - q^99

Display 100 coefficients 10 coefficients

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.205229	−0.145119	−0.0725593	−	0.997364i	$-0.523117\pi$
$2$	−0.205229	−0.145119	−0.0725593	+	0.997364i	$0.523117\pi$
$3$	3.10246	1.79121	0.895603	−	0.444853i	$-0.146744\pi$
$3$	3.10246	1.79121	0.895603	+	0.444853i	$0.146744\pi$
$4$	−1.95788	−0.978941
$5$	0	0
$5$	0	0
$6$	−0.636714	−0.259937
$7$	−2.33231	−0.881530	−0.440765	−	0.897623i	$-0.645293\pi$
$7$	−2.33231	−0.881530	−0.440765	+	0.897623i	$0.645293\pi$
$8$	0.812271	0.287181
$9$	6.62527	2.20842
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−6.07425	−1.75349
$13$	4.30769	1.19474	0.597369	−	0.801966i	$-0.296213\pi$
$13$	4.30769	1.19474	0.597369	+	0.801966i	$0.296213\pi$
$14$	0.478657	0.127926
$15$	0	0
$16$	3.74906	0.937265
$17$	−6.71096	−1.62765	−0.813824	−	0.581111i	$-0.802618\pi$
$17$	−6.71096	−1.62765	−0.813824	+	0.581111i	$0.802618\pi$
$18$	−1.35969	−0.320483
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−7.23590	−1.57900
$22$	0.205229	0.0437549
$23$	−8.24907	−1.72005	−0.860025	−	0.510252i	$-0.829552\pi$
$23$	−8.24907	−1.72005	−0.860025	+	0.510252i	$0.829552\pi$
$24$	2.52004	0.514401
$25$	0	0
$26$	−0.884062	−0.173379
$27$	11.2472	2.16453
$28$	4.56638	0.862966
$29$	3.02355	0.561459	0.280730	−	0.959787i	$-0.409424\pi$
$29$	3.02355	0.561459	0.280730	+	0.959787i	$0.409424\pi$
$30$	0	0
$31$	−10.2077	−1.83335	−0.916677	−	0.399628i	$-0.869139\pi$
$31$	−10.2077	−1.83335	−0.916677	+	0.399628i	$0.869139\pi$
$32$	−2.39396	−0.423196
$33$	−3.10246	−0.540069
$34$	1.37728	0.236202
$35$	0	0
$36$	−12.9715	−2.16191
$37$	5.06393	0.832506	0.416253	−	0.909249i	$-0.363343\pi$
$37$	5.06393	0.832506	0.416253	+	0.909249i	$0.363343\pi$
$38$	0.205229	0.0332925
$39$	13.3644	2.14002
$40$	0	0
$41$	−6.47966	−1.01195	−0.505976	−	0.862548i	$-0.668868\pi$
$41$	−6.47966	−1.01195	−0.505976	+	0.862548i	$0.668868\pi$
$42$	1.48501	0.229143
$43$	−1.64130	−0.250296	−0.125148	−	0.992138i	$-0.539941\pi$
$43$	−1.64130	−0.250296	−0.125148	+	0.992138i	$0.539941\pi$
$44$	1.95788	0.295162
$45$	0	0
$46$	1.69295	0.249611
$47$	8.15593	1.18966	0.594832	−	0.803850i	$-0.297219\pi$
$47$	8.15593	1.18966	0.594832	+	0.803850i	$0.297219\pi$
$48$	11.6313	1.67884
$49$	−1.56033	−0.222905
$50$	0	0
$51$	−20.8205	−2.91545
$52$	−8.43395	−1.16958
$53$	2.35693	0.323749	0.161875	−	0.986811i	$-0.448246\pi$
$53$	2.35693	0.323749	0.161875	+	0.986811i	$0.448246\pi$
$54$	−2.30826	−0.314114
$55$	0	0
$56$	−1.89447	−0.253159
$57$	−3.10246	−0.410931
$58$	−0.620519	−0.0814782
$59$	−4.65880	−0.606525	−0.303262	−	0.952907i	$-0.598076\pi$
$59$	−4.65880	−0.606525	−0.303262	+	0.952907i	$0.598076\pi$
$60$	0	0
$61$	8.93722	1.14429	0.572147	−	0.820151i	$-0.306111\pi$
$61$	8.93722	1.14429	0.572147	+	0.820151i	$0.306111\pi$
$62$	2.09491	0.266054
$63$	−15.4522	−1.94679
$64$	−7.00681	−0.875852
$65$	0	0
$66$	0.636714	0.0783741
$67$	−13.2994	−1.62478	−0.812392	−	0.583111i	$-0.801835\pi$
$67$	−13.2994	−1.62478	−0.812392	+	0.583111i	$0.801835\pi$
$68$	13.1393	1.59337
$69$	−25.5924	−3.08097
$70$	0	0
$71$	−8.76859	−1.04064	−0.520320	−	0.853971i	$-0.674187\pi$
$71$	−8.76859	−1.04064	−0.520320	+	0.853971i	$0.674187\pi$
$72$	5.38151	0.634217
$73$	2.66639	0.312077	0.156038	−	0.987751i	$-0.450128\pi$
$73$	2.66639	0.312077	0.156038	+	0.987751i	$0.450128\pi$
$74$	−1.03926	−0.120812
$75$	0	0
$76$	1.95788	0.224584
$77$	2.33231	0.265791
$78$	−2.74277	−0.310557
$79$	−12.1246	−1.36413	−0.682065	−	0.731292i	$-0.738918\pi$
$79$	−12.1246	−1.36413	−0.682065	+	0.731292i	$0.738918\pi$
$80$	0	0
$81$	15.0183	1.66871
$82$	1.32981	0.146853
$83$	17.8672	1.96117	0.980587	−	0.196082i	$-0.0628220\pi$
$83$	17.8672	1.96117	0.980587	+	0.196082i	$0.0628220\pi$
$84$	14.1670	1.54575
$85$	0	0
$86$	0.336843	0.0363227
$87$	9.38045	1.00569
$88$	−0.812271	−0.0865884
$89$	−1.27875	−0.135548	−0.0677739	−	0.997701i	$-0.521590\pi$
$89$	−1.27875	−0.135548	−0.0677739	+	0.997701i	$0.521590\pi$
$90$	0	0
$91$	−10.0469	−1.05320
$92$	16.1507	1.68383
$93$	−31.6690	−3.28392
$94$	−1.67383	−0.172642
$95$	0	0
$96$	−7.42716	−0.758031
$97$	−14.9748	−1.52046	−0.760230	−	0.649654i	$-0.774914\pi$
$97$	−14.9748	−1.52046	−0.760230	+	0.649654i	$0.774914\pi$
$98$	0.320225	0.0323476
$99$	−6.62527	−0.665864
$100$	0	0
$101$	0.465796	0.0463484	0.0231742	−	0.999731i	$-0.492623\pi$
$101$	0.465796	0.0463484	0.0231742	+	0.999731i	$0.492623\pi$
$102$	4.27297	0.423087
$103$	−12.7767	−1.25892	−0.629462	−	0.777031i	$-0.716725\pi$
$103$	−12.7767	−1.25892	−0.629462	+	0.777031i	$0.716725\pi$
$104$	3.49901	0.343106
$105$	0	0
$106$	−0.483709	−0.0469820
$107$	−3.81929	−0.369225	−0.184612	−	0.982811i	$-0.559103\pi$
$107$	−3.81929	−0.369225	−0.184612	+	0.982811i	$0.559103\pi$
$108$	−22.0208	−2.11895
$109$	9.14480	0.875913	0.437956	−	0.898996i	$-0.355702\pi$
$109$	9.14480	0.875913	0.437956	+	0.898996i	$0.355702\pi$
$110$	0	0
$111$	15.7107	1.49119
$112$	−8.74397	−0.826228
$113$	−1.55378	−0.146167	−0.0730837	−	0.997326i	$-0.523284\pi$
$113$	−1.55378	−0.146167	−0.0730837	+	0.997326i	$0.523284\pi$
$114$	0.636714	0.0596337
$115$	0	0
$116$	−5.91975	−0.549635
$117$	28.5396	2.63849
$118$	0.956120	0.0880180
$119$	15.6520	1.43482
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.83417	−0.166058
$123$	−20.1029	−1.81262
$124$	19.9854	1.79475
$125$	0	0
$126$	3.17123	0.282515
$127$	6.95455	0.617117	0.308558	−	0.951205i	$-0.400154\pi$
$127$	6.95455	0.617117	0.308558	+	0.951205i	$0.400154\pi$
$128$	6.22591	0.550298
$129$	−5.09208	−0.448333
$130$	0	0
$131$	−12.3544	−1.07941	−0.539703	−	0.841855i	$-0.681463\pi$
$131$	−12.3544	−1.07941	−0.539703	+	0.841855i	$0.681463\pi$
$132$	6.07425	0.528696
$133$	2.33231	0.202237
$134$	2.72943	0.235787
$135$	0	0
$136$	−5.45112	−0.467430
$137$	5.14851	0.439867	0.219934	−	0.975515i	$-0.429416\pi$
$137$	5.14851	0.439867	0.219934	+	0.975515i	$0.429416\pi$
$138$	5.25230	0.447105
$139$	−6.46045	−0.547968	−0.273984	−	0.961734i	$-0.588342\pi$
$139$	−6.46045	−0.547968	−0.273984	+	0.961734i	$0.588342\pi$
$140$	0	0
$141$	25.3034	2.13093
$142$	1.79957	0.151016
$143$	−4.30769	−0.360227
$144$	24.8385	2.06988
$145$	0	0
$146$	−0.547219	−0.0452882
$147$	−4.84087	−0.399269
$148$	−9.91458	−0.814974
$149$	1.25732	0.103004	0.0515019	−	0.998673i	$-0.483599\pi$
$149$	1.25732	0.103004	0.0515019	+	0.998673i	$0.483599\pi$
$150$	0	0
$151$	−3.48878	−0.283913	−0.141956	−	0.989873i	$-0.545339\pi$
$151$	−3.48878	−0.283913	−0.141956	+	0.989873i	$0.545339\pi$
$152$	−0.812271	−0.0658839
$153$	−44.4619	−3.59453
$154$	−0.478657	−0.0385713
$155$	0	0
$156$	−26.1660	−2.09496
$157$	−6.34241	−0.506180	−0.253090	−	0.967443i	$-0.581447\pi$
$157$	−6.34241	−0.506180	−0.253090	+	0.967443i	$0.581447\pi$
$158$	2.48832	0.197960
$159$	7.31228	0.579902
$160$	0	0
$161$	19.2394	1.51628
$162$	−3.08220	−0.242160
$163$	23.4566	1.83726	0.918630	−	0.395119i	$-0.129296\pi$
$163$	23.4566	1.83726	0.918630	+	0.395119i	$0.129296\pi$
$164$	12.6864	0.990641
$165$	0	0
$166$	−3.66685	−0.284603
$167$	−6.70854	−0.519122	−0.259561	−	0.965727i	$-0.583578\pi$
$167$	−6.70854	−0.519122	−0.259561	+	0.965727i	$0.583578\pi$
$168$	−5.87751	−0.453460
$169$	5.55619	0.427399
$170$	0	0
$171$	−6.62527	−0.506647
$172$	3.21348	0.245025
$173$	8.18749	0.622483	0.311242	−	0.950331i	$-0.399255\pi$
$173$	8.18749	0.622483	0.311242	+	0.950331i	$0.399255\pi$
$174$	−1.92514	−0.145944
$175$	0	0
$176$	−3.74906	−0.282596
$177$	−14.4538	−1.08641
$178$	0.262437	0.0196705
$179$	−13.0297	−0.973885	−0.486943	−	0.873434i	$-0.661888\pi$
$179$	−13.0297	−0.973885	−0.486943	+	0.873434i	$0.661888\pi$
$180$	0	0
$181$	−13.6260	−1.01281	−0.506407	−	0.862294i	$-0.669027\pi$
$181$	−13.6260	−1.01281	−0.506407	+	0.862294i	$0.669027\pi$
$182$	2.06190	0.152839
$183$	27.7274	2.04967
$184$	−6.70048	−0.493966
$185$	0	0
$186$	6.49938	0.476558
$187$	6.71096	0.490754
$188$	−15.9683	−1.16461
$189$	−26.2321	−1.90810
$190$	0	0
$191$	2.87940	0.208346	0.104173	−	0.994559i	$-0.466780\pi$
$191$	2.87940	0.208346	0.104173	+	0.994559i	$0.466780\pi$
$192$	−21.7384	−1.56883
$193$	−23.3069	−1.67767	−0.838834	−	0.544387i	$-0.816762\pi$
$193$	−23.3069	−1.67767	−0.838834	+	0.544387i	$0.816762\pi$
$194$	3.07326	0.220647
$195$	0	0
$196$	3.05495	0.218211
$197$	21.4910	1.53117	0.765585	−	0.643335i	$-0.222450\pi$
$197$	21.4910	1.53117	0.765585	+	0.643335i	$0.222450\pi$
$198$	1.35969	0.0966293
$199$	−11.8592	−0.840678	−0.420339	−	0.907367i	$-0.638089\pi$
$199$	−11.8592	−0.840678	−0.420339	+	0.907367i	$0.638089\pi$
$200$	0	0
$201$	−41.2610	−2.91033
$202$	−0.0955946	−0.00672601
$203$	−7.05185	−0.494943
$204$	40.7641	2.85406
$205$	0	0
$206$	2.62214	0.182693
$207$	−54.6523	−3.79860
$208$	16.1498	1.11979
$209$	1.00000	0.0691714
$210$	0	0
$211$	−12.2769	−0.845175	−0.422587	−	0.906322i	$-0.638878\pi$
$211$	−12.2769	−0.845175	−0.422587	+	0.906322i	$0.638878\pi$
$212$	−4.61459	−0.316931
$213$	−27.2042	−1.86400
$214$	0.783828	0.0535814
$215$	0	0
$216$	9.13581	0.621613
$217$	23.8075	1.61616
$218$	−1.87678	−0.127111
$219$	8.27236	0.558994
$220$	0	0
$221$	−28.9088	−1.94461
$222$	−3.22428	−0.216399
$223$	−18.1532	−1.21563	−0.607813	−	0.794080i	$-0.707953\pi$
$223$	−18.1532	−1.21563	−0.607813	+	0.794080i	$0.707953\pi$
$224$	5.58345	0.373060
$225$	0	0
$226$	0.318880	0.0212116
$227$	18.9555	1.25812	0.629061	−	0.777356i	$-0.283440\pi$
$227$	18.9555	1.25812	0.629061	+	0.777356i	$0.283440\pi$
$228$	6.07425	0.402277
$229$	−16.3134	−1.07802	−0.539009	−	0.842300i	$-0.681201\pi$
$229$	−16.3134	−1.07802	−0.539009	+	0.842300i	$0.681201\pi$
$230$	0	0
$231$	7.23590	0.476087
$232$	2.45594	0.161240
$233$	22.9853	1.50582	0.752910	−	0.658124i	$-0.228650\pi$
$233$	22.9853	1.50582	0.752910	+	0.658124i	$0.228650\pi$
$234$	−5.85714	−0.382893
$235$	0	0
$236$	9.12138	0.593751
$237$	−37.6162	−2.44344
$238$	−3.21225	−0.208219
$239$	−17.2793	−1.11771	−0.558853	−	0.829267i	$-0.688759\pi$
$239$	−17.2793	−1.11771	−0.558853	+	0.829267i	$0.688759\pi$
$240$	0	0
$241$	13.4260	0.864843	0.432422	−	0.901672i	$-0.357659\pi$
$241$	13.4260	0.864843	0.432422	+	0.901672i	$0.357659\pi$
$242$	−0.205229	−0.0131926
$243$	12.8521	0.824463
$244$	−17.4980	−1.12020
$245$	0	0
$246$	4.12569	0.263044
$247$	−4.30769	−0.274092
$248$	−8.29141	−0.526505
$249$	55.4321	3.51287
$250$	0	0
$251$	1.20422	0.0760099	0.0380050	−	0.999278i	$-0.487900\pi$
$251$	1.20422	0.0760099	0.0380050	+	0.999278i	$0.487900\pi$
$252$	30.2535	1.90579
$253$	8.24907	0.518615
$254$	−1.42727	−0.0895551
$255$	0	0
$256$	12.7359	0.795993
$257$	17.1513	1.06987	0.534935	−	0.844893i	$-0.320336\pi$
$257$	17.1513	1.06987	0.534935	+	0.844893i	$0.320336\pi$
$258$	1.04504	0.0650614
$259$	−11.8107	−0.733879
$260$	0	0
$261$	20.0318	1.23994
$262$	2.53547	0.156642
$263$	−6.60962	−0.407567	−0.203783	−	0.979016i	$-0.565324\pi$
$263$	−6.60962	−0.407567	−0.203783	+	0.979016i	$0.565324\pi$
$264$	−2.52004	−0.155098
$265$	0	0
$266$	−0.478657	−0.0293483
$267$	−3.96729	−0.242794
$268$	26.0387	1.59057
$269$	−2.85724	−0.174209	−0.0871046	−	0.996199i	$-0.527761\pi$
$269$	−2.85724	−0.174209	−0.0871046	+	0.996199i	$0.527761\pi$
$270$	0	0
$271$	−1.28250	−0.0779062	−0.0389531	−	0.999241i	$-0.512402\pi$
$271$	−1.28250	−0.0779062	−0.0389531	+	0.999241i	$0.512402\pi$
$272$	−25.1598	−1.52554
$273$	−31.1700	−1.88649
$274$	−1.05662	−0.0638329
$275$	0	0
$276$	50.1069	3.01608
$277$	−16.2492	−0.976322	−0.488161	−	0.872753i	$-0.662332\pi$
$277$	−16.2492	−0.976322	−0.488161	+	0.872753i	$0.662332\pi$
$278$	1.32587	0.0795204
$279$	−67.6286	−4.04882
$280$	0	0
$281$	−29.4723	−1.75817	−0.879085	−	0.476664i	$-0.841846\pi$
$281$	−29.4723	−1.75817	−0.879085	+	0.476664i	$0.841846\pi$
$282$	−5.19299	−0.309238
$283$	32.9008	1.95575	0.977874	−	0.209196i	$-0.0670845\pi$
$283$	32.9008	1.95575	0.977874	+	0.209196i	$0.0670845\pi$
$284$	17.1679	1.01872
$285$	0	0
$286$	0.884062	0.0522757
$287$	15.1126	0.892066
$288$	−15.8606	−0.934595
$289$	28.0370	1.64924
$290$	0	0
$291$	−46.4587	−2.72346
$292$	−5.22047	−0.305505
$293$	−27.9704	−1.63405	−0.817023	−	0.576605i	$-0.804377\pi$
$293$	−27.9704	−1.63405	−0.817023	+	0.576605i	$0.804377\pi$
$294$	0.993486	0.0579413
$295$	0	0
$296$	4.11329	0.239080
$297$	−11.2472	−0.652631
$298$	−0.258038	−0.0149478
$299$	−35.5344	−2.05501
$300$	0	0
$301$	3.82803	0.220644
$302$	0.715997	0.0412010
$303$	1.44511	0.0830196
$304$	−3.74906	−0.215023
$305$	0	0
$306$	9.12486	0.521634
$307$	0.783038	0.0446903	0.0223452	−	0.999750i	$-0.492887\pi$
$307$	0.783038	0.0446903	0.0223452	+	0.999750i	$0.492887\pi$
$308$	−4.56638	−0.260194
$309$	−39.6392	−2.25499
$310$	0	0
$311$	−10.0442	−0.569555	−0.284777	−	0.958594i	$-0.591920\pi$
$311$	−10.0442	−0.569555	−0.284777	+	0.958594i	$0.591920\pi$
$312$	10.8555	0.614574
$313$	1.80765	0.102174	0.0510872	−	0.998694i	$-0.483731\pi$
$313$	1.80765	0.102174	0.0510872	+	0.998694i	$0.483731\pi$
$314$	1.30165	0.0734561
$315$	0	0
$316$	23.7386	1.33540
$317$	−14.2824	−0.802182	−0.401091	−	0.916038i	$-0.631369\pi$
$317$	−14.2824	−0.802182	−0.401091	+	0.916038i	$0.631369\pi$
$318$	−1.50069	−0.0841545
$319$	−3.02355	−0.169286
$320$	0	0
$321$	−11.8492	−0.661358
$322$	−3.94847	−0.220040
$323$	6.71096	0.373408
$324$	−29.4041	−1.63356
$325$	0	0
$326$	−4.81396	−0.266621
$327$	28.3714	1.56894
$328$	−5.26323	−0.290614
$329$	−19.0221	−1.04872
$330$	0	0
$331$	−14.4996	−0.796969	−0.398484	−	0.917175i	$-0.630464\pi$
$331$	−14.4996	−0.796969	−0.398484	+	0.917175i	$0.630464\pi$
$332$	−34.9818	−1.91987
$333$	33.5499	1.83852
$334$	1.37678	0.0753343
$335$	0	0
$336$	−27.1278	−1.47994
$337$	−17.0292	−0.927638	−0.463819	−	0.885930i	$-0.653521\pi$
$337$	−17.0292	−0.927638	−0.463819	+	0.885930i	$0.653521\pi$
$338$	−1.14029	−0.0620236
$339$	−4.82055	−0.261816
$340$	0	0
$341$	10.2077	0.552777
$342$	1.35969	0.0735239
$343$	19.9653	1.07803
$344$	−1.33318	−0.0718804
$345$	0	0
$346$	−1.68031	−0.0903339
$347$	12.4889	0.670439	0.335220	−	0.942140i	$-0.391189\pi$
$347$	12.4889	0.670439	0.335220	+	0.942140i	$0.391189\pi$
$348$	−18.3658	−0.984510
$349$	−20.9482	−1.12133	−0.560667	−	0.828042i	$-0.689455\pi$
$349$	−20.9482	−1.12133	−0.560667	+	0.828042i	$0.689455\pi$
$350$	0	0
$351$	48.4496	2.58605
$352$	2.39396	0.127598
$353$	11.3626	0.604769	0.302384	−	0.953186i	$-0.402217\pi$
$353$	11.3626	0.604769	0.302384	+	0.953186i	$0.402217\pi$
$354$	2.96633	0.157658
$355$	0	0
$356$	2.50365	0.132693
$357$	48.5599	2.57006
$358$	2.67407	0.141329
$359$	−3.59897	−0.189946	−0.0949732	−	0.995480i	$-0.530277\pi$
$359$	−3.59897	−0.189946	−0.0949732	+	0.995480i	$0.530277\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.79645	0.146978
$363$	3.10246	0.162837
$364$	19.6706	1.03102
$365$	0	0
$366$	−5.69045	−0.297445
$367$	−22.8321	−1.19183	−0.595914	−	0.803048i	$-0.703210\pi$
$367$	−22.8321	−1.19183	−0.595914	+	0.803048i	$0.703210\pi$
$368$	−30.9263	−1.61214
$369$	−42.9294	−2.23482
$370$	0	0
$371$	−5.49709	−0.285395
$372$	62.0041	3.21476
$373$	14.5973	0.755822	0.377911	−	0.925842i	$-0.376643\pi$
$373$	14.5973	0.755822	0.377911	+	0.925842i	$0.376643\pi$
$374$	−1.37728	−0.0712176
$375$	0	0
$376$	6.62482	0.341649
$377$	13.0245	0.670797
$378$	5.38357	0.276901
$379$	11.8706	0.609752	0.304876	−	0.952392i	$-0.401385\pi$
$379$	11.8706	0.609752	0.304876	+	0.952392i	$0.401385\pi$
$380$	0	0
$381$	21.5762	1.10538
$382$	−0.590936	−0.0302349
$383$	−12.6086	−0.644268	−0.322134	−	0.946694i	$-0.604400\pi$
$383$	−12.6086	−0.644268	−0.322134	+	0.946694i	$0.604400\pi$
$384$	19.3157	0.985698
$385$	0	0
$386$	4.78325	0.243461
$387$	−10.8741	−0.552760
$388$	29.3189	1.48844
$389$	35.3815	1.79391	0.896956	−	0.442120i	$-0.145773\pi$
$389$	35.3815	1.79391	0.896956	+	0.442120i	$0.145773\pi$
$390$	0	0
$391$	55.3592	2.79964
$392$	−1.26741	−0.0640140
$393$	−38.3289	−1.93344
$394$	−4.41057	−0.222201
$395$	0	0
$396$	12.9715	0.651842
$397$	3.65851	0.183615	0.0918076	−	0.995777i	$-0.470736\pi$
$397$	3.65851	0.183615	0.0918076	+	0.995777i	$0.470736\pi$
$398$	2.43385	0.121998
$399$	7.23590	0.362248
$400$	0	0
$401$	7.80707	0.389867	0.194933	−	0.980816i	$-0.437551\pi$
$401$	7.80707	0.389867	0.194933	+	0.980816i	$0.437551\pi$
$402$	8.46794	0.422342
$403$	−43.9716	−2.19038
$404$	−0.911972	−0.0453723
$405$	0	0
$406$	1.44724	0.0718255
$407$	−5.06393	−0.251010
$408$	−16.9119	−0.837263
$409$	18.6176	0.920582	0.460291	−	0.887768i	$-0.347745\pi$
$409$	18.6176	0.920582	0.460291	+	0.887768i	$0.347745\pi$
$410$	0	0
$411$	15.9731	0.787893
$412$	25.0152	1.23241
$413$	10.8658	0.534670
$414$	11.2162	0.551247
$415$	0	0
$416$	−10.3124	−0.505608
$417$	−20.0433	−0.981525
$418$	−0.205229	−0.0100381
$419$	4.36059	0.213029	0.106514	−	0.994311i	$-0.466031\pi$
$419$	4.36059	0.213029	0.106514	+	0.994311i	$0.466031\pi$
$420$	0	0
$421$	13.8050	0.672813	0.336407	−	0.941717i	$-0.390788\pi$
$421$	13.8050	0.672813	0.336407	+	0.941717i	$0.390788\pi$
$422$	2.51957	0.122651
$423$	54.0352	2.62728
$424$	1.91446	0.0929746
$425$	0	0
$426$	5.58308	0.270501
$427$	−20.8444	−1.00873
$428$	7.47771	0.361449
$429$	−13.3644	−0.645241
$430$	0	0
$431$	−22.3219	−1.07521	−0.537603	−	0.843198i	$-0.680670\pi$
$431$	−22.3219	−1.07521	−0.537603	+	0.843198i	$0.680670\pi$
$432$	42.1666	2.02874
$433$	32.5287	1.56323	0.781614	−	0.623762i	$-0.214397\pi$
$433$	32.5287	1.56323	0.781614	+	0.623762i	$0.214397\pi$
$434$	−4.88598	−0.234535
$435$	0	0
$436$	−17.9044	−0.857467
$437$	8.24907	0.394607
$438$	−1.69773	−0.0811205
$439$	−11.6856	−0.557723	−0.278862	−	0.960331i	$-0.589957\pi$
$439$	−11.6856	−0.557723	−0.278862	+	0.960331i	$0.589957\pi$
$440$	0	0
$441$	−10.3376	−0.492268
$442$	5.93291	0.282200
$443$	24.6321	1.17031	0.585154	−	0.810923i	$-0.301034\pi$
$443$	24.6321	1.17031	0.585154	+	0.810923i	$0.301034\pi$
$444$	−30.7596	−1.45979
$445$	0	0
$446$	3.72555	0.176410
$447$	3.90079	0.184501
$448$	16.3421	0.772090
$449$	−0.271967	−0.0128349	−0.00641746	−	0.999979i	$-0.502043\pi$
$449$	−0.271967	−0.0128349	−0.00641746	+	0.999979i	$0.502043\pi$
$450$	0	0
$451$	6.47966	0.305115
$452$	3.04212	0.143089
$453$	−10.8238	−0.508546
$454$	−3.89021	−0.182577
$455$	0	0
$456$	−2.52004	−0.118012
$457$	26.6203	1.24524	0.622622	−	0.782523i	$-0.286067\pi$
$457$	26.6203	1.24524	0.622622	+	0.782523i	$0.286067\pi$
$458$	3.34797	0.156440
$459$	−75.4799	−3.52310
$460$	0	0
$461$	18.5239	0.862742	0.431371	−	0.902175i	$-0.358030\pi$
$461$	18.5239	0.862742	0.431371	+	0.902175i	$0.358030\pi$
$462$	−1.48501	−0.0690891
$463$	−14.9154	−0.693178	−0.346589	−	0.938017i	$-0.612660\pi$
$463$	−14.9154	−0.693178	−0.346589	+	0.938017i	$0.612660\pi$
$464$	11.3355	0.526236
$465$	0	0
$466$	−4.71725	−0.218522
$467$	−24.4100	−1.12956	−0.564780	−	0.825242i	$-0.691039\pi$
$467$	−24.4100	−1.12956	−0.564780	+	0.825242i	$0.691039\pi$
$468$	−55.8771	−2.58292
$469$	31.0184	1.43230
$470$	0	0
$471$	−19.6771	−0.906673
$472$	−3.78421	−0.174182
$473$	1.64130	0.0754672
$474$	7.71993	0.354588
$475$	0	0
$476$	−30.6448	−1.40460
$477$	15.6153	0.714974
$478$	3.54621	0.162200
$479$	12.0531	0.550722	0.275361	−	0.961341i	$-0.411203\pi$
$479$	12.0531	0.550722	0.275361	+	0.961341i	$0.411203\pi$
$480$	0	0
$481$	21.8139	0.994626
$482$	−2.75540	−0.125505
$483$	59.6894	2.71596
$484$	−1.95788	−0.0889946
$485$	0	0
$486$	−2.63762	−0.119645
$487$	−27.5058	−1.24641	−0.623204	−	0.782059i	$-0.714169\pi$
$487$	−27.5058	−1.24641	−0.623204	+	0.782059i	$0.714169\pi$
$488$	7.25944	0.328620
$489$	72.7731	3.29091
$490$	0	0
$491$	33.0788	1.49283	0.746413	−	0.665483i	$-0.231774\pi$
$491$	33.0788	1.49283	0.746413	+	0.665483i	$0.231774\pi$
$492$	39.3590	1.77444
$493$	−20.2909	−0.913858
$494$	0.884062	0.0397758
$495$	0	0
$496$	−38.2692	−1.71834
$497$	20.4511	0.917356
$498$	−11.3763	−0.509783
$499$	9.58033	0.428874	0.214437	−	0.976738i	$-0.431208\pi$
$499$	9.58033	0.428874	0.214437	+	0.976738i	$0.431208\pi$
$500$	0	0
$501$	−20.8130	−0.929856
$502$	−0.247141	−0.0110305
$503$	−16.0390	−0.715142	−0.357571	−	0.933886i	$-0.616395\pi$
$503$	−16.0390	−0.715142	−0.357571	+	0.933886i	$0.616395\pi$
$504$	−12.5513	−0.559081
$505$	0	0
$506$	−1.69295	−0.0752606
$507$	17.2379	0.765561
$508$	−13.6162	−0.604121
$509$	11.7844	0.522334	0.261167	−	0.965294i	$-0.415893\pi$
$509$	11.7844	0.522334	0.261167	+	0.965294i	$0.415893\pi$
$510$	0	0
$511$	−6.21884	−0.275105
$512$	−15.0656	−0.665811
$513$	−11.2472	−0.496578
$514$	−3.51994	−0.155258
$515$	0	0
$516$	9.96969	0.438891
$517$	−8.15593	−0.358697
$518$	2.42389	0.106499
$519$	25.4014	1.11500
$520$	0	0
$521$	−7.41167	−0.324711	−0.162356	−	0.986732i	$-0.551909\pi$
$521$	−7.41167	−0.324711	−0.162356	+	0.986732i	$0.551909\pi$
$522$	−4.11111	−0.179938
$523$	−5.89974	−0.257977	−0.128989	−	0.991646i	$-0.541173\pi$
$523$	−5.89974	−0.257977	−0.128989	+	0.991646i	$0.541173\pi$
$524$	24.1884	1.05667
$525$	0	0
$526$	1.35648	0.0591455
$527$	68.5034	2.98406
$528$	−11.6313	−0.506188
$529$	45.0472	1.95857
$530$	0	0
$531$	−30.8658	−1.33946
$532$	−4.56638	−0.197978
$533$	−27.9123	−1.20902
$534$	0.814201	0.0352339
$535$	0	0
$536$	−10.8027	−0.466608
$537$	−40.4241	−1.74443
$538$	0.586388	0.0252810
$539$	1.56033	0.0672083
$540$	0	0
$541$	23.5417	1.01214	0.506068	−	0.862493i	$-0.331098\pi$
$541$	23.5417	1.01214	0.506068	+	0.862493i	$0.331098\pi$
$542$	0.263205	0.0113056
$543$	−42.2742	−1.81416
$544$	16.0658	0.688814
$545$	0	0
$546$	6.39698	0.273765
$547$	−1.63464	−0.0698923	−0.0349461	−	0.999389i	$-0.511126\pi$
$547$	−1.63464	−0.0698923	−0.0349461	+	0.999389i	$0.511126\pi$
$548$	−10.0802	−0.430604
$549$	59.2115	2.52708
$550$	0	0
$551$	−3.02355	−0.128808
$552$	−20.7880	−0.884795
$553$	28.2784	1.20252
$554$	3.33481	0.141683
$555$	0	0
$556$	12.6488	0.536428
$557$	9.30739	0.394367	0.197183	−	0.980367i	$-0.436821\pi$
$557$	9.30739	0.394367	0.197183	+	0.980367i	$0.436821\pi$
$558$	13.8793	0.587559
$559$	−7.07023	−0.299039
$560$	0	0
$561$	20.8205	0.879043
$562$	6.04856	0.255143
$563$	1.61027	0.0678646	0.0339323	−	0.999424i	$-0.489197\pi$
$563$	1.61027	0.0678646	0.0339323	+	0.999424i	$0.489197\pi$
$564$	−49.5411	−2.08606
$565$	0	0
$566$	−6.75218	−0.283815
$567$	−35.0274	−1.47101
$568$	−7.12247	−0.298852
$569$	−17.0331	−0.714064	−0.357032	−	0.934092i	$-0.616211\pi$
$569$	−17.0331	−0.714064	−0.357032	+	0.934092i	$0.616211\pi$
$570$	0	0
$571$	−5.32472	−0.222833	−0.111416	−	0.993774i	$-0.535539\pi$
$571$	−5.32472	−0.222833	−0.111416	+	0.993774i	$0.535539\pi$
$572$	8.43395	0.352641
$573$	8.93323	0.373191
$574$	−3.10153	−0.129455
$575$	0	0
$576$	−46.4220	−1.93425
$577$	12.6976	0.528608	0.264304	−	0.964439i	$-0.414858\pi$
$577$	12.6976	0.528608	0.264304	+	0.964439i	$0.414858\pi$
$578$	−5.75401	−0.239335
$579$	−72.3088	−3.00505
$580$	0	0
$581$	−41.6717	−1.72883
$582$	9.53467	0.395225
$583$	−2.35693	−0.0976140
$584$	2.16583	0.0896226
$585$	0	0
$586$	5.74032	0.237131
$587$	28.4780	1.17541	0.587707	−	0.809074i	$-0.300031\pi$
$587$	28.4780	1.17541	0.587707	+	0.809074i	$0.300031\pi$
$588$	9.47786	0.390860
$589$	10.2077	0.420600
$590$	0	0
$591$	66.6750	2.74264
$592$	18.9850	0.780279
$593$	34.7604	1.42744	0.713719	−	0.700433i	$-0.247010\pi$
$593$	34.7604	1.42744	0.713719	+	0.700433i	$0.247010\pi$
$594$	2.30826	0.0947090
$595$	0	0
$596$	−2.46169	−0.100835
$597$	−36.7928	−1.50583
$598$	7.29269	0.298220
$599$	28.1663	1.15084	0.575422	−	0.817857i	$-0.304838\pi$
$599$	28.1663	1.15084	0.575422	+	0.817857i	$0.304838\pi$
$600$	0	0
$601$	35.3257	1.44097	0.720484	−	0.693472i	$-0.243920\pi$
$601$	35.3257	1.44097	0.720484	+	0.693472i	$0.243920\pi$
$602$	−0.785621	−0.0320195
$603$	−88.1123	−3.58821
$604$	6.83061	0.277934
$605$	0	0
$606$	−0.296579	−0.0120477
$607$	20.4938	0.831816	0.415908	−	0.909407i	$-0.363464\pi$
$607$	20.4938	0.831816	0.415908	+	0.909407i	$0.363464\pi$
$608$	2.39396	0.0970878
$609$	−21.8781	−0.886546
$610$	0	0
$611$	35.1332	1.42134
$612$	87.0512	3.51883
$613$	2.87465	0.116106	0.0580530	−	0.998314i	$-0.481511\pi$
$613$	2.87465	0.116106	0.0580530	+	0.998314i	$0.481511\pi$
$614$	−0.160702	−0.00648540
$615$	0	0
$616$	1.89447	0.0763302
$617$	−13.0363	−0.524820	−0.262410	−	0.964956i	$-0.584517\pi$
$617$	−13.0363	−0.524820	−0.262410	+	0.964956i	$0.584517\pi$
$618$	8.13510	0.327242
$619$	−36.8449	−1.48092	−0.740461	−	0.672099i	$-0.765393\pi$
$619$	−36.8449	−1.48092	−0.740461	+	0.672099i	$0.765393\pi$
$620$	0	0
$621$	−92.7793	−3.72311
$622$	2.06136	0.0826530
$623$	2.98245	0.119489
$624$	50.1041	2.00577
$625$	0	0
$626$	−0.370981	−0.0148274
$627$	3.10246	0.123900
$628$	12.4177	0.495520
$629$	−33.9839	−1.35503
$630$	0	0
$631$	−31.5517	−1.25605	−0.628026	−	0.778192i	$-0.716137\pi$
$631$	−31.5517	−1.25605	−0.628026	+	0.778192i	$0.716137\pi$
$632$	−9.84849	−0.391752
$633$	−38.0885	−1.51388
$634$	2.93117	0.116412
$635$	0	0
$636$	−14.3166	−0.567689
$637$	−6.72143	−0.266313
$638$	0.620519	0.0245666
$639$	−58.0942	−2.29817
$640$	0	0
$641$	−13.4739	−0.532188	−0.266094	−	0.963947i	$-0.585733\pi$
$641$	−13.4739	−0.532188	−0.266094	+	0.963947i	$0.585733\pi$
$642$	2.43180	0.0959753
$643$	25.9686	1.02410	0.512051	−	0.858955i	$-0.328886\pi$
$643$	25.9686	1.02410	0.512051	+	0.858955i	$0.328886\pi$
$644$	−37.6684	−1.48434
$645$	0	0
$646$	−1.37728	−0.0541885
$647$	48.7918	1.91820	0.959101	−	0.283063i	$-0.0913505\pi$
$647$	48.7918	1.91820	0.959101	+	0.283063i	$0.0913505\pi$
$648$	12.1990	0.479221
$649$	4.65880	0.182874
$650$	0	0
$651$	73.8618	2.89487
$652$	−45.9252	−1.79857
$653$	−5.61760	−0.219834	−0.109917	−	0.993941i	$-0.535058\pi$
$653$	−5.61760	−0.219834	−0.109917	+	0.993941i	$0.535058\pi$
$654$	−5.82262	−0.227683
$655$	0	0
$656$	−24.2926	−0.948468
$657$	17.6655	0.689197
$658$	3.90389	0.152189
$659$	−8.87841	−0.345854	−0.172927	−	0.984935i	$-0.555322\pi$
$659$	−8.87841	−0.345854	−0.172927	+	0.984935i	$0.555322\pi$
$660$	0	0
$661$	−19.6584	−0.764625	−0.382312	−	0.924033i	$-0.624872\pi$
$661$	−19.6584	−0.764625	−0.382312	+	0.924033i	$0.624872\pi$
$662$	2.97573	0.115655
$663$	−89.6883	−3.48320
$664$	14.5130	0.563212
$665$	0	0
$666$	−6.88540	−0.266804
$667$	−24.9415	−0.965738
$668$	13.1345	0.508190
$669$	−56.3195	−2.17744
$670$	0	0
$671$	−8.93722	−0.345018
$672$	17.3224	0.668227
$673$	14.2191	0.548106	0.274053	−	0.961715i	$-0.411636\pi$
$673$	14.2191	0.548106	0.274053	+	0.961715i	$0.411636\pi$
$674$	3.49487	0.134618
$675$	0	0
$676$	−10.8784	−0.418399
$677$	−51.0566	−1.96226	−0.981132	−	0.193339i	$-0.938068\pi$
$677$	−51.0566	−1.96226	−0.981132	+	0.193339i	$0.938068\pi$
$678$	0.989314	0.0379944
$679$	34.9259	1.34033
$680$	0	0
$681$	58.8087	2.25356
$682$	−2.09491	−0.0802183
$683$	36.1772	1.38428	0.692142	−	0.721762i	$-0.256667\pi$
$683$	36.1772	1.38428	0.692142	+	0.721762i	$0.256667\pi$
$684$	12.9715	0.495977
$685$	0	0
$686$	−4.09746	−0.156442
$687$	−50.6116	−1.93095
$688$	−6.15335	−0.234594
$689$	10.1529	0.386795
$690$	0	0
$691$	44.3403	1.68678	0.843392	−	0.537299i	$-0.180555\pi$
$691$	44.3403	1.68678	0.843392	+	0.537299i	$0.180555\pi$
$692$	−16.0301	−0.609374
$693$	15.4522	0.586979
$694$	−2.56308	−0.0972932
$695$	0	0
$696$	7.61946	0.288815
$697$	43.4847	1.64710
$698$	4.29918	0.162726
$699$	71.3111	2.69723
$700$	0	0
$701$	−19.4759	−0.735594	−0.367797	−	0.929906i	$-0.619888\pi$
$701$	−19.4759	−0.735594	−0.367797	+	0.929906i	$0.619888\pi$
$702$	−9.94326	−0.375284
$703$	−5.06393	−0.190990
$704$	7.00681	0.264079
$705$	0	0
$706$	−2.33193	−0.0877632
$707$	−1.08638	−0.0408575
$708$	28.2987	1.06353
$709$	34.6294	1.30053	0.650266	−	0.759706i	$-0.274657\pi$
$709$	34.6294	1.30053	0.650266	+	0.759706i	$0.274657\pi$
$710$	0	0
$711$	−80.3290	−3.01257
$712$	−1.03870	−0.0389268
$713$	84.2039	3.15346
$714$	−9.96588	−0.372964
$715$	0	0
$716$	25.5106	0.953376
$717$	−53.6084	−2.00204
$718$	0.738612	0.0275648
$719$	−38.7823	−1.44633	−0.723167	−	0.690673i	$-0.757314\pi$
$719$	−38.7823	−1.44633	−0.723167	+	0.690673i	$0.757314\pi$
$720$	0	0
$721$	29.7992	1.10978
$722$	−0.205229	−0.00763782
$723$	41.6536	1.54911
$724$	26.6781	0.991485
$725$	0	0
$726$	−0.636714	−0.0236307
$727$	11.1719	0.414343	0.207171	−	0.978305i	$-0.433574\pi$
$727$	11.1719	0.414343	0.207171	+	0.978305i	$0.433574\pi$
$728$	−8.16077	−0.302458
$729$	−5.18189	−0.191922
$730$	0	0
$731$	11.0147	0.407395
$732$	−54.2869	−2.00650
$733$	34.8431	1.28696	0.643479	−	0.765464i	$-0.277490\pi$
$733$	34.8431	1.28696	0.643479	+	0.765464i	$0.277490\pi$
$734$	4.68581	0.172956
$735$	0	0
$736$	19.7479	0.727918
$737$	13.2994	0.489891
$738$	8.81035	0.324314
$739$	32.1054	1.18102	0.590508	−	0.807032i	$-0.298928\pi$
$739$	32.1054	1.18102	0.590508	+	0.807032i	$0.298928\pi$
$740$	0	0
$741$	−13.3644	−0.490955
$742$	1.12816	0.0414161
$743$	27.4635	1.00754	0.503769	−	0.863838i	$-0.331946\pi$
$743$	27.4635	1.00754	0.503769	+	0.863838i	$0.331946\pi$
$744$	−25.7238	−0.943079
$745$	0	0
$746$	−2.99579	−0.109684
$747$	118.375	4.33110
$748$	−13.1393	−0.480419
$749$	8.90776	0.325483
$750$	0	0
$751$	−13.0173	−0.475007	−0.237504	−	0.971387i	$-0.576329\pi$
$751$	−13.0173	−0.475007	−0.237504	+	0.971387i	$0.576329\pi$
$752$	30.5771	1.11503
$753$	3.73606	0.136150
$754$	−2.67300	−0.0973451
$755$	0	0
$756$	51.3593	1.86792
$757$	−7.87306	−0.286151	−0.143076	−	0.989712i	$-0.545699\pi$
$757$	−7.87306	−0.286151	−0.143076	+	0.989712i	$0.545699\pi$
$758$	−2.43619	−0.0884864
$759$	25.5924	0.928946
$760$	0	0
$761$	49.9368	1.81021	0.905104	−	0.425189i	$-0.139792\pi$
$761$	49.9368	1.81021	0.905104	+	0.425189i	$0.139792\pi$
$762$	−4.42806	−0.160412
$763$	−21.3285	−0.772144
$764$	−5.63753	−0.203959
$765$	0	0
$766$	2.58764	0.0934953
$767$	−20.0687	−0.724638
$768$	39.5126	1.42579
$769$	20.1374	0.726173	0.363086	−	0.931755i	$-0.381723\pi$
$769$	20.1374	0.726173	0.363086	+	0.931755i	$0.381723\pi$
$770$	0	0
$771$	53.2113	1.91636
$772$	45.6322	1.64234
$773$	−6.99436	−0.251570	−0.125785	−	0.992058i	$-0.540145\pi$
$773$	−6.99436	−0.251570	−0.125785	+	0.992058i	$0.540145\pi$
$774$	2.23167	0.0802158
$775$	0	0
$776$	−12.1636	−0.436648
$777$	−36.6421	−1.31453
$778$	−7.26129	−0.260330
$779$	6.47966	0.232158
$780$	0	0
$781$	8.76859	0.313765
$782$	−11.3613	−0.406279
$783$	34.0066	1.21530
$784$	−5.84979	−0.208921
$785$	0	0
$786$	7.86620	0.280578
$787$	−49.1498	−1.75200	−0.876000	−	0.482311i	$-0.839797\pi$
$787$	−49.1498	−1.75200	−0.876000	+	0.482311i	$0.839797\pi$
$788$	−42.0768	−1.49892
$789$	−20.5061	−0.730036
$790$	0	0
$791$	3.62390	0.128851
$792$	−5.38151	−0.191224
$793$	38.4988	1.36713
$794$	−0.750831	−0.0266460
$795$	0	0
$796$	23.2190	0.822974
$797$	−29.6783	−1.05126	−0.525630	−	0.850713i	$-0.676170\pi$
$797$	−29.6783	−1.05126	−0.525630	+	0.850713i	$0.676170\pi$
$798$	−1.48501	−0.0525689
$799$	−54.7341	−1.93635
$800$	0	0
$801$	−8.47209	−0.299347
$802$	−1.60224	−0.0565769
$803$	−2.66639	−0.0940947
$804$	80.7841	2.84904
$805$	0	0
$806$	9.02422	0.317865
$807$	−8.86449	−0.312045
$808$	0.378352	0.0133104
$809$	11.9905	0.421562	0.210781	−	0.977533i	$-0.432399\pi$
$809$	11.9905	0.421562	0.210781	+	0.977533i	$0.432399\pi$
$810$	0	0
$811$	−17.0389	−0.598318	−0.299159	−	0.954203i	$-0.596706\pi$
$811$	−17.0389	−0.598318	−0.299159	+	0.954203i	$0.596706\pi$
$812$	13.8067	0.484520
$813$	−3.97890	−0.139546
$814$	1.03926	0.0364262
$815$	0	0
$816$	−78.0574	−2.73255
$817$	1.64130	0.0574220
$818$	−3.82087	−0.133594
$819$	−66.5632	−2.32590
$820$	0	0
$821$	−4.91990	−0.171706	−0.0858529	−	0.996308i	$-0.527362\pi$
$821$	−4.91990	−0.171706	−0.0858529	+	0.996308i	$0.527362\pi$
$822$	−3.27813	−0.114338
$823$	−14.9864	−0.522392	−0.261196	−	0.965286i	$-0.584117\pi$
$823$	−14.9864	−0.522392	−0.261196	+	0.965286i	$0.584117\pi$
$824$	−10.3781	−0.361539
$825$	0	0
$826$	−2.22997	−0.0775905
$827$	−18.0947	−0.629213	−0.314607	−	0.949222i	$-0.601873\pi$
$827$	−18.0947	−0.629213	−0.314607	+	0.949222i	$0.601873\pi$
$828$	107.003	3.71860
$829$	−20.2361	−0.702827	−0.351414	−	0.936220i	$-0.614299\pi$
$829$	−20.2361	−0.702827	−0.351414	+	0.936220i	$0.614299\pi$
$830$	0	0
$831$	−50.4126	−1.74880
$832$	−30.1832	−1.04641
$833$	10.4713	0.362810
$834$	4.11346	0.142437
$835$	0	0
$836$	−1.95788	−0.0677147
$837$	−114.808	−3.96836
$838$	−0.894919	−0.0309145
$839$	−11.4706	−0.396008	−0.198004	−	0.980201i	$-0.563446\pi$
$839$	−11.4706	−0.396008	−0.198004	+	0.980201i	$0.563446\pi$
$840$	0	0
$841$	−19.8581	−0.684764
$842$	−2.83318	−0.0976377
$843$	−91.4367	−3.14925
$844$	24.0367	0.827376
$845$	0	0
$846$	−11.0896	−0.381267
$847$	−2.33231	−0.0801391
$848$	8.83627	0.303439
$849$	102.073	3.50315
$850$	0	0
$851$	−41.7727	−1.43195
$852$	53.2626	1.82475
$853$	48.1136	1.64738	0.823690	−	0.567040i	$-0.191912\pi$
$853$	48.1136	1.64738	0.823690	+	0.567040i	$0.191912\pi$
$854$	4.27786	0.146385
$855$	0	0
$856$	−3.10230	−0.106034
$857$	−16.5752	−0.566197	−0.283099	−	0.959091i	$-0.591362\pi$
$857$	−16.5752	−0.566197	−0.283099	+	0.959091i	$0.591362\pi$
$858$	2.74277	0.0936365
$859$	−8.14570	−0.277928	−0.138964	−	0.990297i	$-0.544377\pi$
$859$	−8.14570	−0.277928	−0.138964	+	0.990297i	$0.544377\pi$
$860$	0	0
$861$	46.8861	1.59788
$862$	4.58109	0.156032
$863$	2.28140	0.0776597	0.0388298	−	0.999246i	$-0.487637\pi$
$863$	2.28140	0.0776597	0.0388298	+	0.999246i	$0.487637\pi$
$864$	−26.9254	−0.916021
$865$	0	0
$866$	−6.67582	−0.226853
$867$	86.9838	2.95413
$868$	−46.6122	−1.58212
$869$	12.1246	0.411300
$870$	0	0
$871$	−57.2899	−1.94119
$872$	7.42805	0.251546
$873$	−99.2120	−3.35782
$874$	−1.69295	−0.0572647
$875$	0	0
$876$	−16.1963	−0.547222
$877$	−31.4497	−1.06198	−0.530991	−	0.847378i	$-0.678180\pi$
$877$	−31.4497	−1.06198	−0.530991	+	0.847378i	$0.678180\pi$
$878$	2.39822	0.0809360
$879$	−86.7770	−2.92692
$880$	0	0
$881$	−29.6609	−0.999301	−0.499651	−	0.866227i	$-0.666538\pi$
$881$	−29.6609	−0.999301	−0.499651	+	0.866227i	$0.666538\pi$
$882$	2.12158	0.0714372
$883$	−44.7897	−1.50729	−0.753647	−	0.657279i	$-0.771707\pi$
$883$	−44.7897	−1.50729	−0.753647	+	0.657279i	$0.771707\pi$
$884$	56.5999	1.90366
$885$	0	0
$886$	−5.05522	−0.169833
$887$	51.6475	1.73415	0.867077	−	0.498175i	$-0.165996\pi$
$887$	51.6475	1.73415	0.867077	+	0.498175i	$0.165996\pi$
$888$	12.7613	0.428241
$889$	−16.2202	−0.544007
$890$	0	0
$891$	−15.0183	−0.503134
$892$	35.5417	1.19003
$893$	−8.15593	−0.272928
$894$	−0.800554	−0.0267745
$895$	0	0
$896$	−14.5208	−0.485104
$897$	−110.244	−3.68095
$898$	0.0558154	0.00186258
$899$	−30.8635	−1.02935
$900$	0	0
$901$	−15.8173	−0.526949
$902$	−1.32981	−0.0442779
$903$	11.8763	0.395219
$904$	−1.26209	−0.0419765
$905$	0	0
$906$	2.22135	0.0737996
$907$	−24.5110	−0.813874	−0.406937	−	0.913456i	$-0.633403\pi$
$907$	−24.5110	−0.813874	−0.406937	+	0.913456i	$0.633403\pi$
$908$	−37.1126	−1.23163
$909$	3.08602	0.102357
$910$	0	0
$911$	50.0214	1.65728	0.828641	−	0.559781i	$-0.189115\pi$
$911$	50.0214	1.65728	0.828641	+	0.559781i	$0.189115\pi$
$912$	−11.6313	−0.385151
$913$	−17.8672	−0.591316
$914$	−5.46324	−0.180708
$915$	0	0
$916$	31.9396	1.05532
$917$	28.8142	0.951529
$918$	15.4906	0.511267
$919$	−30.0880	−0.992511	−0.496255	−	0.868177i	$-0.665292\pi$
$919$	−30.0880	−0.992511	−0.496255	+	0.868177i	$0.665292\pi$
$920$	0	0
$921$	2.42934	0.0800496
$922$	−3.80163	−0.125200
$923$	−37.7724	−1.24329
$924$	−14.1670	−0.466061
$925$	0	0
$926$	3.06107	0.100593
$927$	−84.6490	−2.78024
$928$	−7.23825	−0.237607
$929$	13.6447	0.447669	0.223834	−	0.974627i	$-0.428143\pi$
$929$	13.6447	0.447669	0.223834	+	0.974627i	$0.428143\pi$
$930$	0	0
$931$	1.56033	0.0511379
$932$	−45.0026	−1.47411
$933$	−31.1618	−1.02019
$934$	5.00963	0.163920
$935$	0	0
$936$	23.1819	0.757723
$937$	−18.1311	−0.592318	−0.296159	−	0.955139i	$-0.595706\pi$
$937$	−18.1311	−0.592318	−0.296159	+	0.955139i	$0.595706\pi$
$938$	−6.36587	−0.207853
$939$	5.60816	0.183015
$940$	0	0
$941$	−48.2974	−1.57445	−0.787224	−	0.616667i	$-0.788483\pi$
$941$	−48.2974	−1.57445	−0.787224	+	0.616667i	$0.788483\pi$
$942$	4.03830	0.131575
$943$	53.4511	1.74061
$944$	−17.4661	−0.568474
$945$	0	0
$946$	−0.336843	−0.0109517
$947$	−14.7816	−0.480337	−0.240169	−	0.970731i	$-0.577203\pi$
$947$	−14.7816	−0.480337	−0.240169	+	0.970731i	$0.577203\pi$
$948$	73.6481	2.39198
$949$	11.4860	0.372850
$950$	0	0
$951$	−44.3107	−1.43687
$952$	12.7137	0.412053
$953$	16.9908	0.550387	0.275193	−	0.961389i	$-0.411258\pi$
$953$	16.9908	0.550387	0.275193	+	0.961389i	$0.411258\pi$
$954$	−3.20470	−0.103756
$955$	0	0
$956$	33.8309	1.09417
$957$	−9.38045	−0.303227
$958$	−2.47365	−0.0799200
$959$	−12.0079	−0.387756
$960$	0	0
$961$	73.1969	2.36119
$962$	−4.47683	−0.144339
$963$	−25.3038	−0.815404
$964$	−26.2865	−0.846630
$965$	0	0
$966$	−12.2500	−0.394137
$967$	27.7887	0.893626	0.446813	−	0.894627i	$-0.352559\pi$
$967$	27.7887	0.893626	0.446813	+	0.894627i	$0.352559\pi$
$968$	0.812271	0.0261074
$969$	20.8205	0.668851
$970$	0	0
$971$	24.0365	0.771367	0.385683	−	0.922631i	$-0.373966\pi$
$971$	24.0365	0.771367	0.385683	+	0.922631i	$0.373966\pi$
$972$	−25.1629	−0.807100
$973$	15.0678	0.483051
$974$	5.64499	0.180877
$975$	0	0
$976$	33.5062	1.07251
$977$	13.6409	0.436412	0.218206	−	0.975903i	$-0.429980\pi$
$977$	13.6409	0.436412	0.218206	+	0.975903i	$0.429980\pi$
$978$	−14.9351	−0.477573
$979$	1.27875	0.0408692
$980$	0	0
$981$	60.5867	1.93439
$982$	−6.78872	−0.216637
$983$	−7.81613	−0.249296	−0.124648	−	0.992201i	$-0.539780\pi$
$983$	−7.81613	−0.249296	−0.124648	+	0.992201i	$0.539780\pi$
$984$	−16.3290	−0.520549
$985$	0	0
$986$	4.16428	0.132618
$987$	−59.0155	−1.87848
$988$	8.43395	0.268320
$989$	13.5392	0.430522
$990$	0	0
$991$	−44.7494	−1.42151	−0.710756	−	0.703439i	$-0.751647\pi$
$991$	−44.7494	−1.42151	−0.710756	+	0.703439i	$0.751647\pi$
$992$	24.4368	0.775868
$993$	−44.9844	−1.42754
$994$	−4.19714	−0.133125
$995$	0	0
$996$	−108.530	−3.43889
$997$	−31.3614	−0.993227	−0.496613	−	0.867972i	$-0.665423\pi$
$997$	−31.3614	−0.993227	−0.496613	+	0.867972i	$0.665423\pi$
$998$	−1.96616	−0.0622376
$999$	56.9553	1.80199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.l.1.3		6
5.4	even	2		1045.2.a.f.1.4	✓	6
15.14	odd	2		9405.2.a.z.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.f.1.4	✓	6	5.4	even	2
5225.2.a.l.1.3		6	1.1	even	1	trivial
9405.2.a.z.1.3		6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.205229 q^{2} +3.10246 q^{3} -1.95788 q^{4} -0.636714 q^{6} -2.33231 q^{7} +0.812271 q^{8} +6.62527 q^{9} +O(q^{10})\)	\(q-0.205229 q^{2} +3.10246 q^{3} -1.95788 q^{4} -0.636714 q^{6} -2.33231 q^{7} +0.812271 q^{8} +6.62527 q^{9} -1.00000 q^{11} -6.07425 q^{12} +4.30769 q^{13} +0.478657 q^{14} +3.74906 q^{16} -6.71096 q^{17} -1.35969 q^{18} -1.00000 q^{19} -7.23590 q^{21} +0.205229 q^{22} -8.24907 q^{23} +2.52004 q^{24} -0.884062 q^{26} +11.2472 q^{27} +4.56638 q^{28} +3.02355 q^{29} -10.2077 q^{31} -2.39396 q^{32} -3.10246 q^{33} +1.37728 q^{34} -12.9715 q^{36} +5.06393 q^{37} +0.205229 q^{38} +13.3644 q^{39} -6.47966 q^{41} +1.48501 q^{42} -1.64130 q^{43} +1.95788 q^{44} +1.69295 q^{46} +8.15593 q^{47} +11.6313 q^{48} -1.56033 q^{49} -20.8205 q^{51} -8.43395 q^{52} +2.35693 q^{53} -2.30826 q^{54} -1.89447 q^{56} -3.10246 q^{57} -0.620519 q^{58} -4.65880 q^{59} +8.93722 q^{61} +2.09491 q^{62} -15.4522 q^{63} -7.00681 q^{64} +0.636714 q^{66} -13.2994 q^{67} +13.1393 q^{68} -25.5924 q^{69} -8.76859 q^{71} +5.38151 q^{72} +2.66639 q^{73} -1.03926 q^{74} +1.95788 q^{76} +2.33231 q^{77} -2.74277 q^{78} -12.1246 q^{79} +15.0183 q^{81} +1.32981 q^{82} +17.8672 q^{83} +14.1670 q^{84} +0.336843 q^{86} +9.38045 q^{87} -0.812271 q^{88} -1.27875 q^{89} -10.0469 q^{91} +16.1507 q^{92} -31.6690 q^{93} -1.67383 q^{94} -7.42716 q^{96} -14.9748 q^{97} +0.320225 q^{98} -6.62527 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9}+O(q^{10}) \) 6 * q + 2 * q^2 + q^3 + 4 * q^4 - 5 * q^7 + 12 * q^8 + q^9	\( 6 q + 2 q^{2} + q^{3} + 4 q^{4} - 5 q^{7} + 12 q^{8} + q^{9} - 6 q^{11} - q^{12} + 5 q^{13} - 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{18} - 6 q^{19} - 21 q^{21} - 2 q^{22} - 4 q^{23} - q^{24} - 14 q^{26} + 16 q^{27} - 10 q^{28} - 9 q^{29} - 21 q^{31} + q^{32} - q^{33} - 28 q^{36} + 3 q^{37} - 2 q^{38} + 20 q^{39} - 23 q^{41} - q^{42} - 7 q^{43} - 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} - 16 q^{51} - 13 q^{52} + 17 q^{53} + q^{54} - 2 q^{56} - q^{57} - 23 q^{58} - 29 q^{59} + 17 q^{61} - 2 q^{62} - 6 q^{63} - 18 q^{64} - 8 q^{67} + q^{68} - 38 q^{69} - 12 q^{71} - 13 q^{72} - 2 q^{73} - 37 q^{74} - 4 q^{76} + 5 q^{77} - q^{78} + 3 q^{79} - 2 q^{81} - 24 q^{82} + 11 q^{83} - 3 q^{84} - 12 q^{86} + 12 q^{87} - 12 q^{88} - 22 q^{89} - 18 q^{91} + 15 q^{92} - 18 q^{93} + 22 q^{94} - 17 q^{96} + 2 q^{97} + q^{98} - q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 + q^3 + 4 * q^4 - 5 * q^7 + 12 * q^8 + q^9 - 6 * q^11 - q^12 + 5 * q^13 - 8 * q^14 + 4 * q^16 - q^17 - 6 * q^18 - 6 * q^19 - 21 * q^21 - 2 * q^22 - 4 * q^23 - q^24 - 14 * q^26 + 16 * q^27 - 10 * q^28 - 9 * q^29 - 21 * q^31 + q^32 - q^33 - 28 * q^36 + 3 * q^37 - 2 * q^38 + 20 * q^39 - 23 * q^41 - q^42 - 7 * q^43 - 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 - 16 * q^51 - 13 * q^52 + 17 * q^53 + q^54 - 2 * q^56 - q^57 - 23 * q^58 - 29 * q^59 + 17 * q^61 - 2 * q^62 - 6 * q^63 - 18 * q^64 - 8 * q^67 + q^68 - 38 * q^69 - 12 * q^71 - 13 * q^72 - 2 * q^73 - 37 * q^74 - 4 * q^76 + 5 * q^77 - q^78 + 3 * q^79 - 2 * q^81 - 24 * q^82 + 11 * q^83 - 3 * q^84 - 12 * q^86 + 12 * q^87 - 12 * q^88 - 22 * q^89 - 18 * q^91 + 15 * q^92 - 18 * q^93 + 22 * q^94 - 17 * q^96 + 2 * q^97 + q^98 - q^99

Introduction

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