LMFDB - Embedded newform 9405.2.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9405.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:	$75.0993031010$
Analytic rank:	$0$
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	$\chi$	$=$	9405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.205229 q^{2} -1.95788 q^{4} +1.00000 q^{5} +2.33231 q^{7} +0.812271 q^{8} +O(q^{10})$	\(q-0.205229 q^{2} -1.95788 q^{4} +1.00000 q^{5} +2.33231 q^{7} +0.812271 q^{8} -0.205229 q^{10} +1.00000 q^{11} -4.30769 q^{13} -0.478657 q^{14} +3.74906 q^{16} -6.71096 q^{17} -1.00000 q^{19} -1.95788 q^{20} -0.205229 q^{22} -8.24907 q^{23} +1.00000 q^{25} +0.884062 q^{26} -4.56638 q^{28} -3.02355 q^{29} -10.2077 q^{31} -2.39396 q^{32} +1.37728 q^{34} +2.33231 q^{35} -5.06393 q^{37} +0.205229 q^{38} +0.812271 q^{40} +6.47966 q^{41} +1.64130 q^{43} -1.95788 q^{44} +1.69295 q^{46} +8.15593 q^{47} -1.56033 q^{49} -0.205229 q^{50} +8.43395 q^{52} +2.35693 q^{53} +1.00000 q^{55} +1.89447 q^{56} +0.620519 q^{58} +4.65880 q^{59} +8.93722 q^{61} +2.09491 q^{62} -7.00681 q^{64} -4.30769 q^{65} +13.2994 q^{67} +13.1393 q^{68} -0.478657 q^{70} +8.76859 q^{71} -2.66639 q^{73} +1.03926 q^{74} +1.95788 q^{76} +2.33231 q^{77} -12.1246 q^{79} +3.74906 q^{80} -1.32981 q^{82} +17.8672 q^{83} -6.71096 q^{85} -0.336843 q^{86} +0.812271 q^{88} +1.27875 q^{89} -10.0469 q^{91} +16.1507 q^{92} -1.67383 q^{94} -1.00000 q^{95} +14.9748 q^{97} +0.320225 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8	$ 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8} + 2 q^{10} + 6 q^{11} - 5 q^{13} + 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 6 q^{25} + 14 q^{26} + 10 q^{28} + 9 q^{29} - 21 q^{31} + q^{32} + 5 q^{35} - 3 q^{37} - 2 q^{38} + 12 q^{40} + 23 q^{41} + 7 q^{43} + 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 13 q^{52} + 17 q^{53} + 6 q^{55} + 2 q^{56} + 23 q^{58} + 29 q^{59} + 17 q^{61} - 2 q^{62} - 18 q^{64} - 5 q^{65} + 8 q^{67} + q^{68} + 8 q^{70} + 12 q^{71} + 2 q^{73} + 37 q^{74} - 4 q^{76} + 5 q^{77} + 3 q^{79} + 4 q^{80} + 24 q^{82} + 11 q^{83} - q^{85} + 12 q^{86} + 12 q^{88} + 22 q^{89} - 18 q^{91} + 15 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{97} + q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 + 2 * q^10 + 6 * q^11 - 5 * q^13 + 8 * q^14 + 4 * q^16 - q^17 - 6 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 6 * q^25 + 14 * q^26 + 10 * q^28 + 9 * q^29 - 21 * q^31 + q^32 + 5 * q^35 - 3 * q^37 - 2 * q^38 + 12 * q^40 + 23 * q^41 + 7 * q^43 + 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 13 * q^52 + 17 * q^53 + 6 * q^55 + 2 * q^56 + 23 * q^58 + 29 * q^59 + 17 * q^61 - 2 * q^62 - 18 * q^64 - 5 * q^65 + 8 * q^67 + q^68 + 8 * q^70 + 12 * q^71 + 2 * q^73 + 37 * q^74 - 4 * q^76 + 5 * q^77 + 3 * q^79 + 4 * q^80 + 24 * q^82 + 11 * q^83 - q^85 + 12 * q^86 + 12 * q^88 + 22 * q^89 - 18 * q^91 + 15 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^97 + q^98

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	$0.523117\pi$
$3$	0	0
$3$	0	0
$4$	−1.95788	−0.978941
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.33231	0.881530	0.440765	−	0.897623i	$-0.354707\pi$
$7$	2.33231	0.881530	0.440765	+	0.897623i	$0.354707\pi$
$8$	0.812271	0.287181
$9$	0	0
$10$	−0.205229	−0.0648990
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.30769	−1.19474	−0.597369	−	0.801966i	$-0.703787\pi$
$13$	−4.30769	−1.19474	−0.597369	+	0.801966i	$0.703787\pi$
$14$	−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	$0.802618\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.95788	−0.437796
$21$	0	0
$22$	−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	$0.829552\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.884062	0.173379
$27$	0	0
$28$	−4.56638	−0.862966
$29$	−3.02355	−0.561459	−0.280730	−	0.959787i	$-0.590576\pi$
$29$	−3.02355	−0.561459	−0.280730	+	0.959787i	$0.590576\pi$
$30$	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
$33$	0	0
$34$	1.37728	0.236202
$35$	2.33231	0.394232
$36$	0	0
$37$	−5.06393	−0.832506	−0.416253	−	0.909249i	$-0.636657\pi$
$37$	−5.06393	−0.832506	−0.416253	+	0.909249i	$0.636657\pi$
$38$	0.205229	0.0332925
$39$	0	0
$40$	0.812271	0.128431
$41$	6.47966	1.01195	0.505976	−	0.862548i	$-0.331132\pi$
$41$	6.47966	1.01195	0.505976	+	0.862548i	$0.331132\pi$
$42$	0	0
$43$	1.64130	0.250296	0.125148	−	0.992138i	$-0.460059\pi$
$43$	1.64130	0.250296	0.125148	+	0.992138i	$0.460059\pi$
$44$	−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	$0.297219\pi$
$48$	0	0
$49$	−1.56033	−0.222905
$50$	−0.205229	−0.0290237
$51$	0	0
$52$	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	$0.448246\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	1.89447	0.253159
$57$	0	0
$58$	0.620519	0.0814782
$59$	4.65880	0.606525	0.303262	−	0.952907i	$-0.401924\pi$
$59$	4.65880	0.606525	0.303262	+	0.952907i	$0.401924\pi$
$60$	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
$63$	0	0
$64$	−7.00681	−0.875852
$65$	−4.30769	−0.534303
$66$	0	0
$67$	13.2994	1.62478	0.812392	−	0.583111i	$-0.198165\pi$
$67$	13.2994	1.62478	0.812392	+	0.583111i	$0.198165\pi$
$68$	13.1393	1.59337
$69$	0	0
$70$	−0.478657	−0.0572104
$71$	8.76859	1.04064	0.520320	−	0.853971i	$-0.325813\pi$
$71$	8.76859	1.04064	0.520320	+	0.853971i	$0.325813\pi$
$72$	0	0
$73$	−2.66639	−0.312077	−0.156038	−	0.987751i	$-0.549872\pi$
$73$	−2.66639	−0.312077	−0.156038	+	0.987751i	$0.549872\pi$
$74$	1.03926	0.120812
$75$	0	0
$76$	1.95788	0.224584
$77$	2.33231	0.265791
$78$	0	0
$79$	−12.1246	−1.36413	−0.682065	−	0.731292i	$-0.738918\pi$
$79$	−12.1246	−1.36413	−0.682065	+	0.731292i	$0.738918\pi$
$80$	3.74906	0.419158
$81$	0	0
$82$	−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	$0.0628220\pi$
$84$	0	0
$85$	−6.71096	−0.727906
$86$	−0.336843	−0.0363227
$87$	0	0
$88$	0.812271	0.0865884
$89$	1.27875	0.135548	0.0677739	−	0.997701i	$-0.478410\pi$
$89$	1.27875	0.135548	0.0677739	+	0.997701i	$0.478410\pi$
$90$	0	0
$91$	−10.0469	−1.05320
$92$	16.1507	1.68383
$93$	0	0
$94$	−1.67383	−0.172642
$95$	−1.00000	−0.102598
$96$	0	0
$97$	14.9748	1.52046	0.760230	−	0.649654i	$-0.225086\pi$
$97$	14.9748	1.52046	0.760230	+	0.649654i	$0.225086\pi$
$98$	0.320225	0.0323476
$99$	0	0
$100$	−1.95788	−0.195788
$101$	−0.465796	−0.0463484	−0.0231742	−	0.999731i	$-0.507377\pi$
$101$	−0.465796	−0.0463484	−0.0231742	+	0.999731i	$0.507377\pi$
$102$	0	0
$103$	12.7767	1.25892	0.629462	−	0.777031i	$-0.283275\pi$
$103$	12.7767	1.25892	0.629462	+	0.777031i	$0.283275\pi$
$104$	−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	$0.559103\pi$
$108$	0	0
$109$	9.14480	0.875913	0.437956	−	0.898996i	$-0.355702\pi$
$109$	9.14480	0.875913	0.437956	+	0.898996i	$0.355702\pi$
$110$	−0.205229	−0.0195678
$111$	0	0
$112$	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	0
$115$	−8.24907	−0.769230
$116$	5.91975	0.549635
$117$	0	0
$118$	−0.956120	−0.0880180
$119$	−15.6520	−1.43482
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.83417	−0.166058
$123$	0	0
$124$	19.9854	1.79475
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.95455	−0.617117	−0.308558	−	0.951205i	$-0.599846\pi$
$127$	−6.95455	−0.617117	−0.308558	+	0.951205i	$0.599846\pi$
$128$	6.22591	0.550298
$129$	0	0
$130$	0.884062	0.0775373
$131$	12.3544	1.07941	0.539703	−	0.841855i	$-0.318537\pi$
$131$	12.3544	1.07941	0.539703	+	0.841855i	$0.318537\pi$
$132$	0	0
$133$	−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	$0.429416\pi$
$138$	0	0
$139$	−6.46045	−0.547968	−0.273984	−	0.961734i	$-0.588342\pi$
$139$	−6.46045	−0.547968	−0.273984	+	0.961734i	$0.588342\pi$
$140$	−4.56638	−0.385930
$141$	0	0
$142$	−1.79957	−0.151016
$143$	−4.30769	−0.360227
$144$	0	0
$145$	−3.02355	−0.251092
$146$	0.547219	0.0452882
$147$	0	0
$148$	9.91458	0.814974
$149$	−1.25732	−0.103004	−0.0515019	−	0.998673i	$-0.516401\pi$
$149$	−1.25732	−0.103004	−0.0515019	+	0.998673i	$0.516401\pi$
$150$	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
$153$	0	0
$154$	−0.478657	−0.0385713
$155$	−10.2077	−0.819901
$156$	0	0
$157$	6.34241	0.506180	0.253090	−	0.967443i	$-0.418553\pi$
$157$	6.34241	0.506180	0.253090	+	0.967443i	$0.418553\pi$
$158$	2.48832	0.197960
$159$	0	0
$160$	−2.39396	−0.189259
$161$	−19.2394	−1.51628
$162$	0	0
$163$	−23.4566	−1.83726	−0.918630	−	0.395119i	$-0.870704\pi$
$163$	−23.4566	−1.83726	−0.918630	+	0.395119i	$0.870704\pi$
$164$	−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	$0.583578\pi$
$168$	0	0
$169$	5.55619	0.427399
$170$	1.37728	0.105633
$171$	0	0
$172$	−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	$0.399255\pi$
$174$	0	0
$175$	2.33231	0.176306
$176$	3.74906	0.282596
$177$	0	0
$178$	−0.262437	−0.0196705
$179$	13.0297	0.973885	0.486943	−	0.873434i	$-0.338112\pi$
$179$	13.0297	0.973885	0.486943	+	0.873434i	$0.338112\pi$
$180$	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
$183$	0	0
$184$	−6.70048	−0.493966
$185$	−5.06393	−0.372308
$186$	0	0
$187$	−6.71096	−0.490754
$188$	−15.9683	−1.16461
$189$	0	0
$190$	0.205229	0.0148889
$191$	−2.87940	−0.208346	−0.104173	−	0.994559i	$-0.533220\pi$
$191$	−2.87940	−0.208346	−0.104173	+	0.994559i	$0.533220\pi$
$192$	0	0
$193$	23.3069	1.67767	0.838834	−	0.544387i	$-0.183238\pi$
$193$	23.3069	1.67767	0.838834	+	0.544387i	$0.183238\pi$
$194$	−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	$0.222450\pi$
$198$	0	0
$199$	−11.8592	−0.840678	−0.420339	−	0.907367i	$-0.638089\pi$
$199$	−11.8592	−0.840678	−0.420339	+	0.907367i	$0.638089\pi$
$200$	0.812271	0.0574362
$201$	0	0
$202$	0.0955946	0.00672601
$203$	−7.05185	−0.494943
$204$	0	0
$205$	6.47966	0.452559
$206$	−2.62214	−0.182693
$207$	0	0
$208$	−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
$213$	0	0
$214$	0.783828	0.0535814
$215$	1.64130	0.111936
$216$	0	0
$217$	−23.8075	−1.61616
$218$	−1.87678	−0.127111
$219$	0	0
$220$	−1.95788	−0.132000
$221$	28.9088	1.94461
$222$	0	0
$223$	18.1532	1.21563	0.607813	−	0.794080i	$-0.292047\pi$
$223$	18.1532	1.21563	0.607813	+	0.794080i	$0.292047\pi$
$224$	−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	$0.283440\pi$
$228$	0	0
$229$	−16.3134	−1.07802	−0.539009	−	0.842300i	$-0.681201\pi$
$229$	−16.3134	−1.07802	−0.539009	+	0.842300i	$0.681201\pi$
$230$	1.69295	0.111630
$231$	0	0
$232$	−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	$0.228650\pi$
$234$	0	0
$235$	8.15593	0.532034
$236$	−9.12138	−0.593751
$237$	0	0
$238$	3.21225	0.208219
$239$	17.2793	1.11771	0.558853	−	0.829267i	$-0.311241\pi$
$239$	17.2793	1.11771	0.558853	+	0.829267i	$0.311241\pi$
$240$	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
$243$	0	0
$244$	−17.4980	−1.12020
$245$	−1.56033	−0.0996860
$246$	0	0
$247$	4.30769	0.274092
$248$	−8.29141	−0.526505
$249$	0	0
$250$	−0.205229	−0.0129798
$251$	−1.20422	−0.0760099	−0.0380050	−	0.999278i	$-0.512100\pi$
$251$	−1.20422	−0.0760099	−0.0380050	+	0.999278i	$0.512100\pi$
$252$	0	0
$253$	−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	$0.320336\pi$
$258$	0	0
$259$	−11.8107	−0.733879
$260$	8.43395	0.523051
$261$	0	0
$262$	−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	$0.565324\pi$
$264$	0	0
$265$	2.35693	0.144785
$266$	0.478657	0.0293483
$267$	0	0
$268$	−26.0387	−1.59057
$269$	2.85724	0.174209	0.0871046	−	0.996199i	$-0.472239\pi$
$269$	2.85724	0.174209	0.0871046	+	0.996199i	$0.472239\pi$
$270$	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
$273$	0	0
$274$	−1.05662	−0.0638329
$275$	1.00000	0.0603023
$276$	0	0
$277$	16.2492	0.976322	0.488161	−	0.872753i	$-0.337668\pi$
$277$	16.2492	0.976322	0.488161	+	0.872753i	$0.337668\pi$
$278$	1.32587	0.0795204
$279$	0	0
$280$	1.89447	0.113216
$281$	29.4723	1.75817	0.879085	−	0.476664i	$-0.158154\pi$
$281$	29.4723	1.75817	0.879085	+	0.476664i	$0.158154\pi$
$282$	0	0
$283$	−32.9008	−1.95575	−0.977874	−	0.209196i	$-0.932915\pi$
$283$	−32.9008	−1.95575	−0.977874	+	0.209196i	$0.932915\pi$
$284$	−17.1679	−1.01872
$285$	0	0
$286$	0.884062	0.0522757
$287$	15.1126	0.892066
$288$	0	0
$289$	28.0370	1.64924
$290$	0.620519	0.0364381
$291$	0	0
$292$	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	0
$295$	4.65880	0.271246
$296$	−4.11329	−0.239080
$297$	0	0
$298$	0.258038	0.0149478
$299$	35.5344	2.05501
$300$	0	0
$301$	3.82803	0.220644
$302$	0.715997	0.0412010
$303$	0	0
$304$	−3.74906	−0.215023
$305$	8.93722	0.511744
$306$	0	0
$307$	−0.783038	−0.0446903	−0.0223452	−	0.999750i	$-0.507113\pi$
$307$	−0.783038	−0.0446903	−0.0223452	+	0.999750i	$0.507113\pi$
$308$	−4.56638	−0.260194
$309$	0	0
$310$	2.09491	0.118983
$311$	10.0442	0.569555	0.284777	−	0.958594i	$-0.408080\pi$
$311$	10.0442	0.569555	0.284777	+	0.958594i	$0.408080\pi$
$312$	0	0
$313$	−1.80765	−0.102174	−0.0510872	−	0.998694i	$-0.516269\pi$
$313$	−1.80765	−0.102174	−0.0510872	+	0.998694i	$0.516269\pi$
$314$	−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	$0.631369\pi$
$318$	0	0
$319$	−3.02355	−0.169286
$320$	−7.00681	−0.391693
$321$	0	0
$322$	3.94847	0.220040
$323$	6.71096	0.373408
$324$	0	0
$325$	−4.30769	−0.238948
$326$	4.81396	0.266621
$327$	0	0
$328$	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
$333$	0	0
$334$	1.37678	0.0753343
$335$	13.2994	0.726626
$336$	0	0
$337$	17.0292	0.927638	0.463819	−	0.885930i	$-0.346479\pi$
$337$	17.0292	0.927638	0.463819	+	0.885930i	$0.346479\pi$
$338$	−1.14029	−0.0620236
$339$	0	0
$340$	13.1393	0.712577
$341$	−10.2077	−0.552777
$342$	0	0
$343$	−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	$0.391189\pi$
$348$	0	0
$349$	−20.9482	−1.12133	−0.560667	−	0.828042i	$-0.689455\pi$
$349$	−20.9482	−1.12133	−0.560667	+	0.828042i	$0.689455\pi$
$350$	−0.478657	−0.0255853
$351$	0	0
$352$	−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	$0.402217\pi$
$354$	0	0
$355$	8.76859	0.465388
$356$	−2.50365	−0.132693
$357$	0	0
$358$	−2.67407	−0.141329
$359$	3.59897	0.189946	0.0949732	−	0.995480i	$-0.469723\pi$
$359$	3.59897	0.189946	0.0949732	+	0.995480i	$0.469723\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.79645	0.146978
$363$	0	0
$364$	19.6706	1.03102
$365$	−2.66639	−0.139565
$366$	0	0
$367$	22.8321	1.19183	0.595914	−	0.803048i	$-0.296790\pi$
$367$	22.8321	1.19183	0.595914	+	0.803048i	$0.296790\pi$
$368$	−30.9263	−1.61214
$369$	0	0
$370$	1.03926	0.0540288
$371$	5.49709	0.285395
$372$	0	0
$373$	−14.5973	−0.755822	−0.377911	−	0.925842i	$-0.623357\pi$
$373$	−14.5973	−0.755822	−0.377911	+	0.925842i	$0.623357\pi$
$374$	1.37728	0.0712176
$375$	0	0
$376$	6.62482	0.341649
$377$	13.0245	0.670797
$378$	0	0
$379$	11.8706	0.609752	0.304876	−	0.952392i	$-0.401385\pi$
$379$	11.8706	0.609752	0.304876	+	0.952392i	$0.401385\pi$
$380$	1.95788	0.100437
$381$	0	0
$382$	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	$0.604400\pi$
$384$	0	0
$385$	2.33231	0.118865
$386$	−4.78325	−0.243461
$387$	0	0
$388$	−29.3189	−1.48844
$389$	−35.3815	−1.79391	−0.896956	−	0.442120i	$-0.854227\pi$
$389$	−35.3815	−1.79391	−0.896956	+	0.442120i	$0.854227\pi$
$390$	0	0
$391$	55.3592	2.79964
$392$	−1.26741	−0.0640140
$393$	0	0
$394$	−4.41057	−0.222201
$395$	−12.1246	−0.610057
$396$	0	0
$397$	−3.65851	−0.183615	−0.0918076	−	0.995777i	$-0.529264\pi$
$397$	−3.65851	−0.183615	−0.0918076	+	0.995777i	$0.529264\pi$
$398$	2.43385	0.121998
$399$	0	0
$400$	3.74906	0.187453
$401$	−7.80707	−0.389867	−0.194933	−	0.980816i	$-0.562449\pi$
$401$	−7.80707	−0.389867	−0.194933	+	0.980816i	$0.562449\pi$
$402$	0	0
$403$	43.9716	2.19038
$404$	0.911972	0.0453723
$405$	0	0
$406$	1.44724	0.0718255
$407$	−5.06393	−0.251010
$408$	0	0
$409$	18.6176	0.920582	0.460291	−	0.887768i	$-0.347745\pi$
$409$	18.6176	0.920582	0.460291	+	0.887768i	$0.347745\pi$
$410$	−1.32981	−0.0656747
$411$	0	0
$412$	−25.0152	−1.23241
$413$	10.8658	0.534670
$414$	0	0
$415$	17.8672	0.877064
$416$	10.3124	0.505608
$417$	0	0
$418$	0.205229	0.0100381
$419$	−4.36059	−0.213029	−0.106514	−	0.994311i	$-0.533969\pi$
$419$	−4.36059	−0.213029	−0.106514	+	0.994311i	$0.533969\pi$
$420$	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
$423$	0	0
$424$	1.91446	0.0929746
$425$	−6.71096	−0.325530
$426$	0	0
$427$	20.8444	1.00873
$428$	7.47771	0.361449
$429$	0	0
$430$	−0.336843	−0.0162440
$431$	22.3219	1.07521	0.537603	−	0.843198i	$-0.319330\pi$
$431$	22.3219	1.07521	0.537603	+	0.843198i	$0.319330\pi$
$432$	0	0
$433$	−32.5287	−1.56323	−0.781614	−	0.623762i	$-0.785603\pi$
$433$	−32.5287	−1.56323	−0.781614	+	0.623762i	$0.785603\pi$
$434$	4.88598	0.234535
$435$	0	0
$436$	−17.9044	−0.857467
$437$	8.24907	0.394607
$438$	0	0
$439$	−11.6856	−0.557723	−0.278862	−	0.960331i	$-0.589957\pi$
$439$	−11.6856	−0.557723	−0.278862	+	0.960331i	$0.589957\pi$
$440$	0.812271	0.0387235
$441$	0	0
$442$	−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	$0.301034\pi$
$444$	0	0
$445$	1.27875	0.0606188
$446$	−3.72555	−0.176410
$447$	0	0
$448$	−16.3421	−0.772090
$449$	0.271967	0.0128349	0.00641746	−	0.999979i	$-0.497957\pi$
$449$	0.271967	0.0128349	0.00641746	+	0.999979i	$0.497957\pi$
$450$	0	0
$451$	6.47966	0.305115
$452$	3.04212	0.143089
$453$	0	0
$454$	−3.89021	−0.182577
$455$	−10.0469	−0.471004
$456$	0	0
$457$	−26.6203	−1.24524	−0.622622	−	0.782523i	$-0.713933\pi$
$457$	−26.6203	−1.24524	−0.622622	+	0.782523i	$0.713933\pi$
$458$	3.34797	0.156440
$459$	0	0
$460$	16.1507	0.753030
$461$	−18.5239	−0.862742	−0.431371	−	0.902175i	$-0.641970\pi$
$461$	−18.5239	−0.862742	−0.431371	+	0.902175i	$0.641970\pi$
$462$	0	0
$463$	14.9154	0.693178	0.346589	−	0.938017i	$-0.387340\pi$
$463$	14.9154	0.693178	0.346589	+	0.938017i	$0.387340\pi$
$464$	−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	$0.691039\pi$
$468$	0	0
$469$	31.0184	1.43230
$470$	−1.67383	−0.0772080
$471$	0	0
$472$	3.78421	0.174182
$473$	1.64130	0.0754672
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	30.6448	1.40460
$477$	0	0
$478$	−3.54621	−0.162200
$479$	−12.0531	−0.550722	−0.275361	−	0.961341i	$-0.588797\pi$
$479$	−12.0531	−0.550722	−0.275361	+	0.961341i	$0.588797\pi$
$480$	0	0
$481$	21.8139	0.994626
$482$	−2.75540	−0.125505
$483$	0	0
$484$	−1.95788	−0.0889946
$485$	14.9748	0.679971
$486$	0	0
$487$	27.5058	1.24641	0.623204	−	0.782059i	$-0.285831\pi$
$487$	27.5058	1.24641	0.623204	+	0.782059i	$0.285831\pi$
$488$	7.25944	0.328620
$489$	0	0
$490$	0.320225	0.0144663
$491$	−33.0788	−1.49283	−0.746413	−	0.665483i	$-0.768226\pi$
$491$	−33.0788	−1.49283	−0.746413	+	0.665483i	$0.768226\pi$
$492$	0	0
$493$	20.2909	0.913858
$494$	−0.884062	−0.0397758
$495$	0	0
$496$	−38.2692	−1.71834
$497$	20.4511	0.917356
$498$	0	0
$499$	9.58033	0.428874	0.214437	−	0.976738i	$-0.431208\pi$
$499$	9.58033	0.428874	0.214437	+	0.976738i	$0.431208\pi$
$500$	−1.95788	−0.0875591
$501$	0	0
$502$	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	$0.616395\pi$
$504$	0	0
$505$	−0.465796	−0.0207276
$506$	1.69295	0.0752606
$507$	0	0
$508$	13.6162	0.604121
$509$	−11.7844	−0.522334	−0.261167	−	0.965294i	$-0.584107\pi$
$509$	−11.7844	−0.522334	−0.261167	+	0.965294i	$0.584107\pi$
$510$	0	0
$511$	−6.21884	−0.275105
$512$	−15.0656	−0.665811
$513$	0	0
$514$	−3.51994	−0.155258
$515$	12.7767	0.563008
$516$	0	0
$517$	8.15593	0.358697
$518$	2.42389	0.106499
$519$	0	0
$520$	−3.49901	−0.153442
$521$	7.41167	0.324711	0.162356	−	0.986732i	$-0.448091\pi$
$521$	7.41167	0.324711	0.162356	+	0.986732i	$0.448091\pi$
$522$	0	0
$523$	5.89974	0.257977	0.128989	−	0.991646i	$-0.458827\pi$
$523$	5.89974	0.257977	0.128989	+	0.991646i	$0.458827\pi$
$524$	−24.1884	−1.05667
$525$	0	0
$526$	1.35648	0.0591455
$527$	68.5034	2.98406
$528$	0	0
$529$	45.0472	1.95857
$530$	−0.483709	−0.0210110
$531$	0	0
$532$	4.56638	0.197978
$533$	−27.9123	−1.20902
$534$	0	0
$535$	−3.81929	−0.165122
$536$	10.8027	0.466608
$537$	0	0
$538$	−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
$543$	0	0
$544$	16.0658	0.688814
$545$	9.14480	0.391720
$546$	0	0
$547$	1.63464	0.0698923	0.0349461	−	0.999389i	$-0.488874\pi$
$547$	1.63464	0.0698923	0.0349461	+	0.999389i	$0.488874\pi$
$548$	−10.0802	−0.430604
$549$	0	0
$550$	−0.205229	−0.00875098
$551$	3.02355	0.128808
$552$	0	0
$553$	−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	$0.436821\pi$
$558$	0	0
$559$	−7.07023	−0.299039
$560$	8.74397	0.369500
$561$	0	0
$562$	−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	$0.489197\pi$
$564$	0	0
$565$	−1.55378	−0.0653681
$566$	6.75218	0.283815
$567$	0	0
$568$	7.12247	0.298852
$569$	17.0331	0.714064	0.357032	−	0.934092i	$-0.383789\pi$
$569$	17.0331	0.714064	0.357032	+	0.934092i	$0.383789\pi$
$570$	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
$573$	0	0
$574$	−3.10153	−0.129455
$575$	−8.24907	−0.344010
$576$	0	0
$577$	−12.6976	−0.528608	−0.264304	−	0.964439i	$-0.585142\pi$
$577$	−12.6976	−0.528608	−0.264304	+	0.964439i	$0.585142\pi$
$578$	−5.75401	−0.239335
$579$	0	0
$580$	5.91975	0.245804
$581$	41.6717	1.72883
$582$	0	0
$583$	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	$0.300031\pi$
$588$	0	0
$589$	10.2077	0.420600
$590$	−0.956120	−0.0393628
$591$	0	0
$592$	−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	$0.247010\pi$
$594$	0	0
$595$	−15.6520	−0.641671
$596$	2.46169	0.100835
$597$	0	0
$598$	−7.29269	−0.298220
$599$	−28.1663	−1.15084	−0.575422	−	0.817857i	$-0.695162\pi$
$599$	−28.1663	−1.15084	−0.575422	+	0.817857i	$0.695162\pi$
$600$	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
$603$	0	0
$604$	6.83061	0.277934
$605$	1.00000	0.0406558
$606$	0	0
$607$	−20.4938	−0.831816	−0.415908	−	0.909407i	$-0.636536\pi$
$607$	−20.4938	−0.831816	−0.415908	+	0.909407i	$0.636536\pi$
$608$	2.39396	0.0970878
$609$	0	0
$610$	−1.83417	−0.0742635
$611$	−35.1332	−1.42134
$612$	0	0
$613$	−2.87465	−0.116106	−0.0580530	−	0.998314i	$-0.518489\pi$
$613$	−2.87465	−0.116106	−0.0580530	+	0.998314i	$0.518489\pi$
$614$	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	$0.584517\pi$
$618$	0	0
$619$	−36.8449	−1.48092	−0.740461	−	0.672099i	$-0.765393\pi$
$619$	−36.8449	−1.48092	−0.740461	+	0.672099i	$0.765393\pi$
$620$	19.9854	0.802635
$621$	0	0
$622$	−2.06136	−0.0826530
$623$	2.98245	0.119489
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.370981	0.0148274
$627$	0	0
$628$	−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
$633$	0	0
$634$	2.93117	0.116412
$635$	−6.95455	−0.275983
$636$	0	0
$637$	6.72143	0.266313
$638$	0.620519	0.0245666
$639$	0	0
$640$	6.22591	0.246101
$641$	13.4739	0.532188	0.266094	−	0.963947i	$-0.414267\pi$
$641$	13.4739	0.532188	0.266094	+	0.963947i	$0.414267\pi$
$642$	0	0
$643$	−25.9686	−1.02410	−0.512051	−	0.858955i	$-0.671114\pi$
$643$	−25.9686	−1.02410	−0.512051	+	0.858955i	$0.671114\pi$
$644$	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	$0.0913505\pi$
$648$	0	0
$649$	4.65880	0.182874
$650$	0.884062	0.0346757
$651$	0	0
$652$	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	$0.535058\pi$
$654$	0	0
$655$	12.3544	0.482725
$656$	24.2926	0.948468
$657$	0	0
$658$	−3.90389	−0.152189
$659$	8.87841	0.345854	0.172927	−	0.984935i	$-0.444678\pi$
$659$	8.87841	0.345854	0.172927	+	0.984935i	$0.444678\pi$
$660$	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
$663$	0	0
$664$	14.5130	0.563212
$665$	−2.33231	−0.0904431
$666$	0	0
$667$	24.9415	0.965738
$668$	13.1345	0.508190
$669$	0	0
$670$	−2.72943	−0.105447
$671$	8.93722	0.345018
$672$	0	0
$673$	−14.2191	−0.548106	−0.274053	−	0.961715i	$-0.588364\pi$
$673$	−14.2191	−0.548106	−0.274053	+	0.961715i	$0.588364\pi$
$674$	−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	0
$679$	34.9259	1.34033
$680$	−5.45112	−0.209041
$681$	0	0
$682$	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	$0.256667\pi$
$684$	0	0
$685$	5.14851	0.196715
$686$	4.09746	0.156442
$687$	0	0
$688$	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
$693$	0	0
$694$	−2.56308	−0.0972932
$695$	−6.46045	−0.245059
$696$	0	0
$697$	−43.4847	−1.64710
$698$	4.29918	0.162726
$699$	0	0
$700$	−4.56638	−0.172593
$701$	19.4759	0.735594	0.367797	−	0.929906i	$-0.380112\pi$
$701$	19.4759	0.735594	0.367797	+	0.929906i	$0.380112\pi$
$702$	0	0
$703$	5.06393	0.190990
$704$	−7.00681	−0.264079
$705$	0	0
$706$	−2.33193	−0.0877632
$707$	−1.08638	−0.0408575
$708$	0	0
$709$	34.6294	1.30053	0.650266	−	0.759706i	$-0.274657\pi$
$709$	34.6294	1.30053	0.650266	+	0.759706i	$0.274657\pi$
$710$	−1.79957	−0.0675365
$711$	0	0
$712$	1.03870	0.0389268
$713$	84.2039	3.15346
$714$	0	0
$715$	−4.30769	−0.161098
$716$	−25.5106	−0.953376
$717$	0	0
$718$	−0.738612	−0.0275648
$719$	38.7823	1.44633	0.723167	−	0.690673i	$-0.242686\pi$
$719$	38.7823	1.44633	0.723167	+	0.690673i	$0.242686\pi$
$720$	0	0
$721$	29.7992	1.10978
$722$	−0.205229	−0.00763782
$723$	0	0
$724$	26.6781	0.991485
$725$	−3.02355	−0.112292
$726$	0	0
$727$	−11.1719	−0.414343	−0.207171	−	0.978305i	$-0.566426\pi$
$727$	−11.1719	−0.414343	−0.207171	+	0.978305i	$0.566426\pi$
$728$	−8.16077	−0.302458
$729$	0	0
$730$	0.547219	0.0202535
$731$	−11.0147	−0.407395
$732$	0	0
$733$	−34.8431	−1.28696	−0.643479	−	0.765464i	$-0.722510\pi$
$733$	−34.8431	−1.28696	−0.643479	+	0.765464i	$0.722510\pi$
$734$	−4.68581	−0.172956
$735$	0	0
$736$	19.7479	0.727918
$737$	13.2994	0.489891
$738$	0	0
$739$	32.1054	1.18102	0.590508	−	0.807032i	$-0.298928\pi$
$739$	32.1054	1.18102	0.590508	+	0.807032i	$0.298928\pi$
$740$	9.91458	0.364467
$741$	0	0
$742$	−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	$0.331946\pi$
$744$	0	0
$745$	−1.25732	−0.0460647
$746$	2.99579	0.109684
$747$	0	0
$748$	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
$753$	0	0
$754$	−2.67300	−0.0973451
$755$	−3.48878	−0.126970
$756$	0	0
$757$	7.87306	0.286151	0.143076	−	0.989712i	$-0.454301\pi$
$757$	7.87306	0.286151	0.143076	+	0.989712i	$0.454301\pi$
$758$	−2.43619	−0.0884864
$759$	0	0
$760$	−0.812271	−0.0294642
$761$	−49.9368	−1.81021	−0.905104	−	0.425189i	$-0.860208\pi$
$761$	−49.9368	−1.81021	−0.905104	+	0.425189i	$0.860208\pi$
$762$	0	0
$763$	21.3285	0.772144
$764$	5.63753	0.203959
$765$	0	0
$766$	2.58764	0.0934953
$767$	−20.0687	−0.724638
$768$	0	0
$769$	20.1374	0.726173	0.363086	−	0.931755i	$-0.381723\pi$
$769$	20.1374	0.726173	0.363086	+	0.931755i	$0.381723\pi$
$770$	−0.478657	−0.0172496
$771$	0	0
$772$	−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	$0.540145\pi$
$774$	0	0
$775$	−10.2077	−0.366671
$776$	12.1636	0.436648
$777$	0	0
$778$	7.26129	0.260330
$779$	−6.47966	−0.232158
$780$	0	0
$781$	8.76859	0.313765
$782$	−11.3613	−0.406279
$783$	0	0
$784$	−5.84979	−0.208921
$785$	6.34241	0.226370
$786$	0	0
$787$	49.1498	1.75200	0.876000	−	0.482311i	$-0.160203\pi$
$787$	49.1498	1.75200	0.876000	+	0.482311i	$0.160203\pi$
$788$	−42.0768	−1.49892
$789$	0	0
$790$	2.48832	0.0885306
$791$	−3.62390	−0.128851
$792$	0	0
$793$	−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	$0.676170\pi$
$798$	0	0
$799$	−54.7341	−1.93635
$800$	−2.39396	−0.0846391
$801$	0	0
$802$	1.60224	0.0565769
$803$	−2.66639	−0.0940947
$804$	0	0
$805$	−19.2394	−0.678099
$806$	−9.02422	−0.317865
$807$	0	0
$808$	−0.378352	−0.0133104
$809$	−11.9905	−0.421562	−0.210781	−	0.977533i	$-0.567601\pi$
$809$	−11.9905	−0.421562	−0.210781	+	0.977533i	$0.567601\pi$
$810$	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
$813$	0	0
$814$	1.03926	0.0364262
$815$	−23.4566	−0.821648
$816$	0	0
$817$	−1.64130	−0.0574220
$818$	−3.82087	−0.133594
$819$	0	0
$820$	−12.6864	−0.443028
$821$	4.91990	0.171706	0.0858529	−	0.996308i	$-0.472638\pi$
$821$	4.91990	0.171706	0.0858529	+	0.996308i	$0.472638\pi$
$822$	0	0
$823$	14.9864	0.522392	0.261196	−	0.965286i	$-0.415883\pi$
$823$	14.9864	0.522392	0.261196	+	0.965286i	$0.415883\pi$
$824$	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	$0.601873\pi$
$828$	0	0
$829$	−20.2361	−0.702827	−0.351414	−	0.936220i	$-0.614299\pi$
$829$	−20.2361	−0.702827	−0.351414	+	0.936220i	$0.614299\pi$
$830$	−3.66685	−0.127278
$831$	0	0
$832$	30.1832	1.04641
$833$	10.4713	0.362810
$834$	0	0
$835$	−6.70854	−0.232159
$836$	1.95788	0.0677147
$837$	0	0
$838$	0.894919	0.0309145
$839$	11.4706	0.396008	0.198004	−	0.980201i	$-0.436554\pi$
$839$	11.4706	0.396008	0.198004	+	0.980201i	$0.436554\pi$
$840$	0	0
$841$	−19.8581	−0.684764
$842$	−2.83318	−0.0976377
$843$	0	0
$844$	24.0367	0.827376
$845$	5.55619	0.191139
$846$	0	0
$847$	2.33231	0.0801391
$848$	8.83627	0.303439
$849$	0	0
$850$	1.37728	0.0472404
$851$	41.7727	1.43195
$852$	0	0
$853$	−48.1136	−1.64738	−0.823690	−	0.567040i	$-0.808088\pi$
$853$	−48.1136	−1.64738	−0.823690	+	0.567040i	$0.808088\pi$
$854$	−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	$0.591362\pi$
$858$	0	0
$859$	−8.14570	−0.277928	−0.138964	−	0.990297i	$-0.544377\pi$
$859$	−8.14570	−0.277928	−0.138964	+	0.990297i	$0.544377\pi$
$860$	−3.21348	−0.109579
$861$	0	0
$862$	−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	$0.487637\pi$
$864$	0	0
$865$	8.18749	0.278383
$866$	6.67582	0.226853
$867$	0	0
$868$	46.6122	1.58212
$869$	−12.1246	−0.411300
$870$	0	0
$871$	−57.2899	−1.94119
$872$	7.42805	0.251546
$873$	0	0
$874$	−1.69295	−0.0572647
$875$	2.33231	0.0788464
$876$	0	0
$877$	31.4497	1.06198	0.530991	−	0.847378i	$-0.321820\pi$
$877$	31.4497	1.06198	0.530991	+	0.847378i	$0.321820\pi$
$878$	2.39822	0.0809360
$879$	0	0
$880$	3.74906	0.126381
$881$	29.6609	0.999301	0.499651	−	0.866227i	$-0.333462\pi$
$881$	29.6609	0.999301	0.499651	+	0.866227i	$0.333462\pi$
$882$	0	0
$883$	44.7897	1.50729	0.753647	−	0.657279i	$-0.228293\pi$
$883$	44.7897	1.50729	0.753647	+	0.657279i	$0.228293\pi$
$884$	−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	$0.165996\pi$
$888$	0	0
$889$	−16.2202	−0.544007
$890$	−0.262437	−0.00879692
$891$	0	0
$892$	−35.5417	−1.19003
$893$	−8.15593	−0.272928
$894$	0	0
$895$	13.0297	0.435535
$896$	14.5208	0.485104
$897$	0	0
$898$	−0.0558154	−0.00186258
$899$	30.8635	1.02935
$900$	0	0
$901$	−15.8173	−0.526949
$902$	−1.32981	−0.0442779
$903$	0	0
$904$	−1.26209	−0.0419765
$905$	−13.6260	−0.452945
$906$	0	0
$907$	24.5110	0.813874	0.406937	−	0.913456i	$-0.366597\pi$
$907$	24.5110	0.813874	0.406937	+	0.913456i	$0.366597\pi$
$908$	−37.1126	−1.23163
$909$	0	0
$910$	2.06190	0.0683515
$911$	−50.0214	−1.65728	−0.828641	−	0.559781i	$-0.810885\pi$
$911$	−50.0214	−1.65728	−0.828641	+	0.559781i	$0.810885\pi$
$912$	0	0
$913$	17.8672	0.591316
$914$	5.46324	0.180708
$915$	0	0
$916$	31.9396	1.05532
$917$	28.8142	0.951529
$918$	0	0
$919$	−30.0880	−0.992511	−0.496255	−	0.868177i	$-0.665292\pi$
$919$	−30.0880	−0.992511	−0.496255	+	0.868177i	$0.665292\pi$
$920$	−6.70048	−0.220908
$921$	0	0
$922$	3.80163	0.125200
$923$	−37.7724	−1.24329
$924$	0	0
$925$	−5.06393	−0.166501
$926$	−3.06107	−0.100593
$927$	0	0
$928$	7.23825	0.237607
$929$	−13.6447	−0.447669	−0.223834	−	0.974627i	$-0.571857\pi$
$929$	−13.6447	−0.447669	−0.223834	+	0.974627i	$0.571857\pi$
$930$	0	0
$931$	1.56033	0.0511379
$932$	−45.0026	−1.47411
$933$	0	0
$934$	5.00963	0.163920
$935$	−6.71096	−0.219472
$936$	0	0
$937$	18.1311	0.592318	0.296159	−	0.955139i	$-0.404294\pi$
$937$	18.1311	0.592318	0.296159	+	0.955139i	$0.404294\pi$
$938$	−6.36587	−0.207853
$939$	0	0
$940$	−15.9683	−0.520830
$941$	48.2974	1.57445	0.787224	−	0.616667i	$-0.211517\pi$
$941$	48.2974	1.57445	0.787224	+	0.616667i	$0.211517\pi$
$942$	0	0
$943$	−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	$0.577203\pi$
$948$	0	0
$949$	11.4860	0.372850
$950$	0.205229	0.00665850
$951$	0	0
$952$	−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	$0.411258\pi$
$954$	0	0
$955$	−2.87940	−0.0931752
$956$	−33.8309	−1.09417
$957$	0	0
$958$	2.47365	0.0799200
$959$	12.0079	0.387756
$960$	0	0
$961$	73.1969	2.36119
$962$	−4.47683	−0.144339
$963$	0	0
$964$	−26.2865	−0.846630
$965$	23.3069	0.750276
$966$	0	0
$967$	−27.7887	−0.893626	−0.446813	−	0.894627i	$-0.647441\pi$
$967$	−27.7887	−0.893626	−0.446813	+	0.894627i	$0.647441\pi$
$968$	0.812271	0.0261074
$969$	0	0
$970$	−3.07326	−0.0986764
$971$	−24.0365	−0.771367	−0.385683	−	0.922631i	$-0.626034\pi$
$971$	−24.0365	−0.771367	−0.385683	+	0.922631i	$0.626034\pi$
$972$	0	0
$973$	−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	$0.429980\pi$
$978$	0	0
$979$	1.27875	0.0408692
$980$	3.05495	0.0975867
$981$	0	0
$982$	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	$0.539780\pi$
$984$	0	0
$985$	21.4910	0.684760
$986$	−4.16428	−0.132618
$987$	0	0
$988$	−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
$993$	0	0
$994$	−4.19714	−0.133125
$995$	−11.8592	−0.375963
$996$	0	0
$997$	31.3614	0.993227	0.496613	−	0.867972i	$-0.334577\pi$
$997$	31.3614	0.993227	0.496613	+	0.867972i	$0.334577\pi$
$998$	−1.96616	−0.0622376
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.f.1.4	✓	6	3.2	odd	2
5225.2.a.l.1.3		6	15.14	odd	2
9405.2.a.z.1.3		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q-0.205229 q^{2} -1.95788 q^{4} +1.00000 q^{5} +2.33231 q^{7} +0.812271 q^{8} +O(q^{10})\)	\(q-0.205229 q^{2} -1.95788 q^{4} +1.00000 q^{5} +2.33231 q^{7} +0.812271 q^{8} -0.205229 q^{10} +1.00000 q^{11} -4.30769 q^{13} -0.478657 q^{14} +3.74906 q^{16} -6.71096 q^{17} -1.00000 q^{19} -1.95788 q^{20} -0.205229 q^{22} -8.24907 q^{23} +1.00000 q^{25} +0.884062 q^{26} -4.56638 q^{28} -3.02355 q^{29} -10.2077 q^{31} -2.39396 q^{32} +1.37728 q^{34} +2.33231 q^{35} -5.06393 q^{37} +0.205229 q^{38} +0.812271 q^{40} +6.47966 q^{41} +1.64130 q^{43} -1.95788 q^{44} +1.69295 q^{46} +8.15593 q^{47} -1.56033 q^{49} -0.205229 q^{50} +8.43395 q^{52} +2.35693 q^{53} +1.00000 q^{55} +1.89447 q^{56} +0.620519 q^{58} +4.65880 q^{59} +8.93722 q^{61} +2.09491 q^{62} -7.00681 q^{64} -4.30769 q^{65} +13.2994 q^{67} +13.1393 q^{68} -0.478657 q^{70} +8.76859 q^{71} -2.66639 q^{73} +1.03926 q^{74} +1.95788 q^{76} +2.33231 q^{77} -12.1246 q^{79} +3.74906 q^{80} -1.32981 q^{82} +17.8672 q^{83} -6.71096 q^{85} -0.336843 q^{86} +0.812271 q^{88} +1.27875 q^{89} -10.0469 q^{91} +16.1507 q^{92} -1.67383 q^{94} -1.00000 q^{95} +14.9748 q^{97} +0.320225 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8	\( 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8} + 2 q^{10} + 6 q^{11} - 5 q^{13} + 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 6 q^{25} + 14 q^{26} + 10 q^{28} + 9 q^{29} - 21 q^{31} + q^{32} + 5 q^{35} - 3 q^{37} - 2 q^{38} + 12 q^{40} + 23 q^{41} + 7 q^{43} + 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 13 q^{52} + 17 q^{53} + 6 q^{55} + 2 q^{56} + 23 q^{58} + 29 q^{59} + 17 q^{61} - 2 q^{62} - 18 q^{64} - 5 q^{65} + 8 q^{67} + q^{68} + 8 q^{70} + 12 q^{71} + 2 q^{73} + 37 q^{74} - 4 q^{76} + 5 q^{77} + 3 q^{79} + 4 q^{80} + 24 q^{82} + 11 q^{83} - q^{85} + 12 q^{86} + 12 q^{88} + 22 q^{89} - 18 q^{91} + 15 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{97} + q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 + 2 * q^10 + 6 * q^11 - 5 * q^13 + 8 * q^14 + 4 * q^16 - q^17 - 6 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 6 * q^25 + 14 * q^26 + 10 * q^28 + 9 * q^29 - 21 * q^31 + q^32 + 5 * q^35 - 3 * q^37 - 2 * q^38 + 12 * q^40 + 23 * q^41 + 7 * q^43 + 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 13 * q^52 + 17 * q^53 + 6 * q^55 + 2 * q^56 + 23 * q^58 + 29 * q^59 + 17 * q^61 - 2 * q^62 - 18 * q^64 - 5 * q^65 + 8 * q^67 + q^68 + 8 * q^70 + 12 * q^71 + 2 * q^73 + 37 * q^74 - 4 * q^76 + 5 * q^77 + 3 * q^79 + 4 * q^80 + 24 * q^82 + 11 * q^83 - q^85 + 12 * q^86 + 12 * q^88 + 22 * q^89 - 18 * q^91 + 15 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^97 + q^98

Introduction

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