LMFDB - Embedded newform 3645.2.a.g.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3645,2,Mod(1,3645)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3645, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3645.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3645 = 3^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3645.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:	$29.1054715368$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 15 x^{13} + 47 x^{12} + 84 x^{11} - 279 x^{10} - 219 x^{9} + 783 x^{8} + 279 x^{7} + \cdots + 3 $ x^15 - 3x^14 - 15x^13 + 47x^12 + 84x^11 - 279x^10 - 219x^9 + 783x^8 + 279x^7 - 1054x^6 - 195x^5 + 609x^4 + 88x^3 - 102*x^2 + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-0.694378$ of defining polynomial
Character	$\chi$	$=$	3645.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.694378 q^{2} -1.51784 q^{4} +1.00000 q^{5} -0.291597 q^{7} -2.44271 q^{8} +O(q^{10})$	\(q+0.694378 q^{2} -1.51784 q^{4} +1.00000 q^{5} -0.291597 q^{7} -2.44271 q^{8} +0.694378 q^{10} -1.78834 q^{11} +2.19103 q^{13} -0.202478 q^{14} +1.33951 q^{16} +5.46565 q^{17} -6.84486 q^{19} -1.51784 q^{20} -1.24178 q^{22} -5.34257 q^{23} +1.00000 q^{25} +1.52140 q^{26} +0.442597 q^{28} +2.95275 q^{29} +6.22487 q^{31} +5.81555 q^{32} +3.79523 q^{34} -0.291597 q^{35} -8.54729 q^{37} -4.75292 q^{38} -2.44271 q^{40} -4.55506 q^{41} +9.37931 q^{43} +2.71440 q^{44} -3.70976 q^{46} -5.46220 q^{47} -6.91497 q^{49} +0.694378 q^{50} -3.32563 q^{52} +2.24096 q^{53} -1.78834 q^{55} +0.712286 q^{56} +2.05033 q^{58} -10.4333 q^{59} +0.296654 q^{61} +4.32241 q^{62} +1.35917 q^{64} +2.19103 q^{65} -4.45758 q^{67} -8.29598 q^{68} -0.202478 q^{70} -9.49280 q^{71} -10.4842 q^{73} -5.93505 q^{74} +10.3894 q^{76} +0.521473 q^{77} -7.12060 q^{79} +1.33951 q^{80} -3.16293 q^{82} +6.21102 q^{83} +5.46565 q^{85} +6.51279 q^{86} +4.36839 q^{88} -12.3795 q^{89} -0.638896 q^{91} +8.10916 q^{92} -3.79283 q^{94} -6.84486 q^{95} +14.0798 q^{97} -4.80161 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 3 q^{2} + 9 q^{4} + 15 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) $ 15 * q - 3 * q^2 + 9 * q^4 + 15 * q^5 - 12 * q^7 - 9 * q^8	$ 15 q - 3 q^{2} + 9 q^{4} + 15 q^{5} - 12 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} - 3 q^{16} - 12 q^{17} - 24 q^{19} + 9 q^{20} - 18 q^{22} - 18 q^{23} + 15 q^{25} + 9 q^{26} - 30 q^{28} + 3 q^{29} - 24 q^{31} - 6 q^{32} - 18 q^{34} - 12 q^{35} - 24 q^{37} - 18 q^{38} - 9 q^{40} + 9 q^{41} - 42 q^{43} + 12 q^{44} - 30 q^{46} - 21 q^{47} - 3 q^{49} - 3 q^{50} - 36 q^{52} - 18 q^{53} + 30 q^{56} - 30 q^{58} + 6 q^{59} - 15 q^{61} + 36 q^{62} - 27 q^{64} - 12 q^{65} - 45 q^{67} - 36 q^{68} + 12 q^{71} - 21 q^{73} + 21 q^{74} - 48 q^{76} - 9 q^{77} - 48 q^{79} - 3 q^{80} - 24 q^{82} - 33 q^{83} - 12 q^{85} + 15 q^{86} - 54 q^{88} + 9 q^{89} - 51 q^{91} + 33 q^{92} - 30 q^{94} - 24 q^{95} - 30 q^{97} - 15 q^{98}+O(q^{100}) $ 15 * q - 3 * q^2 + 9 * q^4 + 15 * q^5 - 12 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 - 3 * q^16 - 12 * q^17 - 24 * q^19 + 9 * q^20 - 18 * q^22 - 18 * q^23 + 15 * q^25 + 9 * q^26 - 30 * q^28 + 3 * q^29 - 24 * q^31 - 6 * q^32 - 18 * q^34 - 12 * q^35 - 24 * q^37 - 18 * q^38 - 9 * q^40 + 9 * q^41 - 42 * q^43 + 12 * q^44 - 30 * q^46 - 21 * q^47 - 3 * q^49 - 3 * q^50 - 36 * q^52 - 18 * q^53 + 30 * q^56 - 30 * q^58 + 6 * q^59 - 15 * q^61 + 36 * q^62 - 27 * q^64 - 12 * q^65 - 45 * q^67 - 36 * q^68 + 12 * q^71 - 21 * q^73 + 21 * q^74 - 48 * q^76 - 9 * q^77 - 48 * q^79 - 3 * q^80 - 24 * q^82 - 33 * q^83 - 12 * q^85 + 15 * q^86 - 54 * q^88 + 9 * q^89 - 51 * q^91 + 33 * q^92 - 30 * q^94 - 24 * q^95 - 30 * q^97 - 15 * 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 $

$2$	0.694378	0.491000	0.245500	−	0.969397i	$-0.421048\pi$
$2$	0.694378	0.491000	0.245500	+	0.969397i	$0.421048\pi$
$3$	0	0
$3$	0	0
$4$	−1.51784	−0.758919
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.291597	−0.110213	−0.0551066	−	0.998480i	$-0.517550\pi$
$7$	−0.291597	−0.110213	−0.0551066	+	0.998480i	$0.517550\pi$
$8$	−2.44271	−0.863629
$9$	0	0
$10$	0.694378	0.219582
$11$	−1.78834	−0.539203	−0.269602	−	0.962972i	$-0.586892\pi$
$11$	−1.78834	−0.539203	−0.269602	+	0.962972i	$0.586892\pi$
$12$	0	0
$13$	2.19103	0.607682	0.303841	−	0.952723i	$-0.401731\pi$
$13$	2.19103	0.607682	0.303841	+	0.952723i	$0.401731\pi$
$14$	−0.202478	−0.0541146
$15$	0	0
$16$	1.33951	0.334878
$17$	5.46565	1.32562	0.662808	−	0.748790i	$-0.269365\pi$
$17$	5.46565	1.32562	0.662808	+	0.748790i	$0.269365\pi$
$18$	0	0
$19$	−6.84486	−1.57032	−0.785159	−	0.619295i	$-0.787419\pi$
$19$	−6.84486	−1.57032	−0.785159	+	0.619295i	$0.787419\pi$
$20$	−1.51784	−0.339399
$21$	0	0
$22$	−1.24178	−0.264749
$23$	−5.34257	−1.11400	−0.557001	−	0.830512i	$-0.688048\pi$
$23$	−5.34257	−1.11400	−0.557001	+	0.830512i	$0.688048\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.52140	0.298372
$27$	0	0
$28$	0.442597	0.0836429
$29$	2.95275	0.548312	0.274156	−	0.961685i	$-0.411602\pi$
$29$	2.95275	0.548312	0.274156	+	0.961685i	$0.411602\pi$
$30$	0	0
$31$	6.22487	1.11802	0.559010	−	0.829161i	$-0.311181\pi$
$31$	6.22487	1.11802	0.559010	+	0.829161i	$0.311181\pi$
$32$	5.81555	1.02805
$33$	0	0
$34$	3.79523	0.650877
$35$	−0.291597	−0.0492888
$36$	0	0
$37$	−8.54729	−1.40517	−0.702583	−	0.711602i	$-0.747970\pi$
$37$	−8.54729	−1.40517	−0.702583	+	0.711602i	$0.747970\pi$
$38$	−4.75292	−0.771025
$39$	0	0
$40$	−2.44271	−0.386226
$41$	−4.55506	−0.711381	−0.355690	−	0.934604i	$-0.615754\pi$
$41$	−4.55506	−0.711381	−0.355690	+	0.934604i	$0.615754\pi$
$42$	0	0
$43$	9.37931	1.43033	0.715166	−	0.698955i	$-0.246351\pi$
$43$	9.37931	1.43033	0.715166	+	0.698955i	$0.246351\pi$
$44$	2.71440	0.409212
$45$	0	0
$46$	−3.70976	−0.546975
$47$	−5.46220	−0.796744	−0.398372	−	0.917224i	$-0.630425\pi$
$47$	−5.46220	−0.796744	−0.398372	+	0.917224i	$0.630425\pi$
$48$	0	0
$49$	−6.91497	−0.987853
$50$	0.694378	0.0981999
$51$	0	0
$52$	−3.32563	−0.461182
$53$	2.24096	0.307820	0.153910	−	0.988085i	$-0.450813\pi$
$53$	2.24096	0.307820	0.153910	+	0.988085i	$0.450813\pi$
$54$	0	0
$55$	−1.78834	−0.241139
$56$	0.712286	0.0951833
$57$	0	0
$58$	2.05033	0.269221
$59$	−10.4333	−1.35831	−0.679153	−	0.733997i	$-0.737653\pi$
$59$	−10.4333	−1.35831	−0.679153	+	0.733997i	$0.737653\pi$
$60$	0	0
$61$	0.296654	0.0379827	0.0189913	−	0.999820i	$-0.493955\pi$
$61$	0.296654	0.0379827	0.0189913	+	0.999820i	$0.493955\pi$
$62$	4.32241	0.548947
$63$	0	0
$64$	1.35917	0.169896
$65$	2.19103	0.271764
$66$	0	0
$67$	−4.45758	−0.544580	−0.272290	−	0.962215i	$-0.587781\pi$
$67$	−4.45758	−0.544580	−0.272290	+	0.962215i	$0.587781\pi$
$68$	−8.29598	−1.00604
$69$	0	0
$70$	−0.202478	−0.0242008
$71$	−9.49280	−1.12659	−0.563294	−	0.826256i	$-0.690466\pi$
$71$	−9.49280	−1.12659	−0.563294	+	0.826256i	$0.690466\pi$
$72$	0	0
$73$	−10.4842	−1.22709	−0.613544	−	0.789661i	$-0.710257\pi$
$73$	−10.4842	−1.22709	−0.613544	+	0.789661i	$0.710257\pi$
$74$	−5.93505	−0.689936
$75$	0	0
$76$	10.3894	1.19174
$77$	0.521473	0.0594273
$78$	0	0
$79$	−7.12060	−0.801131	−0.400565	−	0.916268i	$-0.631186\pi$
$79$	−7.12060	−0.801131	−0.400565	+	0.916268i	$0.631186\pi$
$80$	1.33951	0.149762
$81$	0	0
$82$	−3.16293	−0.349288
$83$	6.21102	0.681748	0.340874	−	0.940109i	$-0.389277\pi$
$83$	6.21102	0.681748	0.340874	+	0.940109i	$0.389277\pi$
$84$	0	0
$85$	5.46565	0.592833
$86$	6.51279	0.702292
$87$	0	0
$88$	4.36839	0.465671
$89$	−12.3795	−1.31223	−0.656113	−	0.754662i	$-0.727801\pi$
$89$	−12.3795	−1.31223	−0.656113	+	0.754662i	$0.727801\pi$
$90$	0	0
$91$	−0.638896	−0.0669745
$92$	8.10916	0.845438
$93$	0	0
$94$	−3.79283	−0.391201
$95$	−6.84486	−0.702267
$96$	0	0
$97$	14.0798	1.42958	0.714792	−	0.699337i	$-0.246521\pi$
$97$	14.0798	1.42958	0.714792	+	0.699337i	$0.246521\pi$
$98$	−4.80161	−0.485035
$99$	0	0
$100$	−1.51784	−0.151784
$101$	12.6104	1.25478	0.627389	−	0.778706i	$-0.284124\pi$
$101$	12.6104	1.25478	0.627389	+	0.778706i	$0.284124\pi$
$102$	0	0
$103$	−5.89515	−0.580867	−0.290433	−	0.956895i	$-0.593799\pi$
$103$	−5.89515	−0.580867	−0.290433	+	0.956895i	$0.593799\pi$
$104$	−5.35205	−0.524811
$105$	0	0
$106$	1.55608	0.151140
$107$	−6.13206	−0.592808	−0.296404	−	0.955063i	$-0.595788\pi$
$107$	−6.13206	−0.592808	−0.296404	+	0.955063i	$0.595788\pi$
$108$	0	0
$109$	−5.00653	−0.479538	−0.239769	−	0.970830i	$-0.577072\pi$
$109$	−5.00653	−0.479538	−0.239769	+	0.970830i	$0.577072\pi$
$110$	−1.24178	−0.118399
$111$	0	0
$112$	−0.390597	−0.0369080
$113$	−2.24677	−0.211358	−0.105679	−	0.994400i	$-0.533702\pi$
$113$	−2.24677	−0.211358	−0.105679	+	0.994400i	$0.533702\pi$
$114$	0	0
$115$	−5.34257	−0.498197
$116$	−4.48180	−0.416125
$117$	0	0
$118$	−7.24469	−0.666928
$119$	−1.59377	−0.146100
$120$	0	0
$121$	−7.80186	−0.709260
$122$	0.205990	0.0186495
$123$	0	0
$124$	−9.44835	−0.848487
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.01072	0.355894	0.177947	−	0.984040i	$-0.443054\pi$
$127$	4.01072	0.355894	0.177947	+	0.984040i	$0.443054\pi$
$128$	−10.6873	−0.944635
$129$	0	0
$130$	1.52140	0.133436
$131$	14.2980	1.24922	0.624610	−	0.780937i	$-0.285258\pi$
$131$	14.2980	1.24922	0.624610	+	0.780937i	$0.285258\pi$
$132$	0	0
$133$	1.99594	0.173070
$134$	−3.09524	−0.267388
$135$	0	0
$136$	−13.3510	−1.14484
$137$	−1.49411	−0.127650	−0.0638250	−	0.997961i	$-0.520330\pi$
$137$	−1.49411	−0.127650	−0.0638250	+	0.997961i	$0.520330\pi$
$138$	0	0
$139$	−4.34934	−0.368906	−0.184453	−	0.982841i	$-0.559051\pi$
$139$	−4.34934	−0.368906	−0.184453	+	0.982841i	$0.559051\pi$
$140$	0.442597	0.0374062
$141$	0	0
$142$	−6.59160	−0.553155
$143$	−3.91829	−0.327664
$144$	0	0
$145$	2.95275	0.245213
$146$	−7.28003	−0.602499
$147$	0	0
$148$	12.9734	1.06641
$149$	11.8839	0.973566	0.486783	−	0.873523i	$-0.338170\pi$
$149$	11.8839	0.973566	0.486783	+	0.873523i	$0.338170\pi$
$150$	0	0
$151$	−21.7802	−1.77245	−0.886225	−	0.463254i	$-0.846682\pi$
$151$	−21.7802	−1.77245	−0.886225	+	0.463254i	$0.846682\pi$
$152$	16.7200	1.35617
$153$	0	0
$154$	0.362099	0.0291788
$155$	6.22487	0.499993
$156$	0	0
$157$	−12.8666	−1.02687	−0.513435	−	0.858129i	$-0.671627\pi$
$157$	−12.8666	−1.02687	−0.513435	+	0.858129i	$0.671627\pi$
$158$	−4.94439	−0.393355
$159$	0	0
$160$	5.81555	0.459760
$161$	1.55788	0.122778
$162$	0	0
$163$	−16.1843	−1.26765	−0.633826	−	0.773476i	$-0.718516\pi$
$163$	−16.1843	−1.26765	−0.633826	+	0.773476i	$0.718516\pi$
$164$	6.91385	0.539881
$165$	0	0
$166$	4.31280	0.334738
$167$	−16.4702	−1.27450	−0.637250	−	0.770658i	$-0.719928\pi$
$167$	−16.4702	−1.27450	−0.637250	+	0.770658i	$0.719928\pi$
$168$	0	0
$169$	−8.19940	−0.630723
$170$	3.79523	0.291081
$171$	0	0
$172$	−14.2363	−1.08551
$173$	10.2644	0.780389	0.390195	−	0.920732i	$-0.372408\pi$
$173$	10.2644	0.780389	0.390195	+	0.920732i	$0.372408\pi$
$174$	0	0
$175$	−0.291597	−0.0220426
$176$	−2.39550	−0.180567
$177$	0	0
$178$	−8.59607	−0.644303
$179$	−6.81389	−0.509294	−0.254647	−	0.967034i	$-0.581959\pi$
$179$	−6.81389	−0.509294	−0.254647	+	0.967034i	$0.581959\pi$
$180$	0	0
$181$	−25.9084	−1.92575	−0.962876	−	0.269943i	$-0.912995\pi$
$181$	−25.9084	−1.92575	−0.962876	+	0.269943i	$0.912995\pi$
$182$	−0.443636	−0.0328845
$183$	0	0
$184$	13.0504	0.962085
$185$	−8.54729	−0.628409
$186$	0	0
$187$	−9.77442	−0.714776
$188$	8.29074	0.604665
$189$	0	0
$190$	−4.75292	−0.344813
$191$	−19.5004	−1.41100	−0.705500	−	0.708710i	$-0.749278\pi$
$191$	−19.5004	−1.41100	−0.705500	+	0.708710i	$0.749278\pi$
$192$	0	0
$193$	−17.1259	−1.23275	−0.616374	−	0.787454i	$-0.711399\pi$
$193$	−17.1259	−1.23275	−0.616374	+	0.787454i	$0.711399\pi$
$194$	9.77669	0.701925
$195$	0	0
$196$	10.4958	0.749701
$197$	19.5552	1.39325	0.696624	−	0.717436i	$-0.254684\pi$
$197$	19.5552	1.39325	0.696624	+	0.717436i	$0.254684\pi$
$198$	0	0
$199$	−21.3564	−1.51391	−0.756956	−	0.653466i	$-0.773314\pi$
$199$	−21.3564	−1.51391	−0.756956	+	0.653466i	$0.773314\pi$
$200$	−2.44271	−0.172726
$201$	0	0
$202$	8.75636	0.616095
$203$	−0.861012	−0.0604312
$204$	0	0
$205$	−4.55506	−0.318139
$206$	−4.09347	−0.285205
$207$	0	0
$208$	2.93491	0.203499
$209$	12.2409	0.846720
$210$	0	0
$211$	1.84964	0.127335	0.0636673	−	0.997971i	$-0.479720\pi$
$211$	1.84964	0.127335	0.0636673	+	0.997971i	$0.479720\pi$
$212$	−3.40142	−0.233611
$213$	0	0
$214$	−4.25797	−0.291069
$215$	9.37931	0.639664
$216$	0	0
$217$	−1.81515	−0.123220
$218$	−3.47642	−0.235453
$219$	0	0
$220$	2.71440	0.183005
$221$	11.9754	0.805552
$222$	0	0
$223$	−2.66619	−0.178542	−0.0892708	−	0.996007i	$-0.528454\pi$
$223$	−2.66619	−0.178542	−0.0892708	+	0.996007i	$0.528454\pi$
$224$	−1.69579	−0.113305
$225$	0	0
$226$	−1.56011	−0.103777
$227$	4.48110	0.297421	0.148711	−	0.988881i	$-0.452488\pi$
$227$	4.48110	0.297421	0.148711	+	0.988881i	$0.452488\pi$
$228$	0	0
$229$	24.3717	1.61053	0.805265	−	0.592915i	$-0.202023\pi$
$229$	24.3717	1.61053	0.805265	+	0.592915i	$0.202023\pi$
$230$	−3.70976	−0.244615
$231$	0	0
$232$	−7.21272	−0.473538
$233$	4.80472	0.314767	0.157384	−	0.987538i	$-0.449694\pi$
$233$	4.80472	0.314767	0.157384	+	0.987538i	$0.449694\pi$
$234$	0	0
$235$	−5.46220	−0.356315
$236$	15.8361	1.03084
$237$	0	0
$238$	−1.10668	−0.0717352
$239$	16.5595	1.07115	0.535573	−	0.844489i	$-0.320096\pi$
$239$	16.5595	1.07115	0.535573	+	0.844489i	$0.320096\pi$
$240$	0	0
$241$	27.9515	1.80052	0.900258	−	0.435356i	$-0.143378\pi$
$241$	27.9515	1.80052	0.900258	+	0.435356i	$0.143378\pi$
$242$	−5.41744	−0.348246
$243$	0	0
$244$	−0.450273	−0.0288258
$245$	−6.91497	−0.441781
$246$	0	0
$247$	−14.9973	−0.954253
$248$	−15.2056	−0.965554
$249$	0	0
$250$	0.694378	0.0439163
$251$	23.1222	1.45946	0.729730	−	0.683736i	$-0.239646\pi$
$251$	23.1222	1.45946	0.729730	+	0.683736i	$0.239646\pi$
$252$	0	0
$253$	9.55431	0.600674
$254$	2.78496	0.174744
$255$	0	0
$256$	−10.1394	−0.633711
$257$	27.8963	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$257$	27.8963	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$258$	0	0
$259$	2.49236	0.154868
$260$	−3.32563	−0.206247
$261$	0	0
$262$	9.92820	0.613367
$263$	15.4147	0.950514	0.475257	−	0.879847i	$-0.342355\pi$
$263$	15.4147	0.950514	0.475257	+	0.879847i	$0.342355\pi$
$264$	0	0
$265$	2.24096	0.137661
$266$	1.38594	0.0849771
$267$	0	0
$268$	6.76588	0.413292
$269$	−16.5120	−1.00675	−0.503377	−	0.864067i	$-0.667909\pi$
$269$	−16.5120	−1.00675	−0.503377	+	0.864067i	$0.667909\pi$
$270$	0	0
$271$	−2.72926	−0.165791	−0.0828953	−	0.996558i	$-0.526417\pi$
$271$	−2.72926	−0.165791	−0.0828953	+	0.996558i	$0.526417\pi$
$272$	7.32131	0.443920
$273$	0	0
$274$	−1.03747	−0.0626761
$275$	−1.78834	−0.107841
$276$	0	0
$277$	9.58747	0.576055	0.288028	−	0.957622i	$-0.407001\pi$
$277$	9.58747	0.576055	0.288028	+	0.957622i	$0.407001\pi$
$278$	−3.02009	−0.181133
$279$	0	0
$280$	0.712286	0.0425672
$281$	−3.21598	−0.191849	−0.0959245	−	0.995389i	$-0.530581\pi$
$281$	−3.21598	−0.191849	−0.0959245	+	0.995389i	$0.530581\pi$
$282$	0	0
$283$	−25.3104	−1.50455	−0.752273	−	0.658852i	$-0.771042\pi$
$283$	−25.3104	−1.50455	−0.752273	+	0.658852i	$0.771042\pi$
$284$	14.4085	0.854990
$285$	0	0
$286$	−2.72078	−0.160883
$287$	1.32824	0.0784035
$288$	0	0
$289$	12.8734	0.757256
$290$	2.05033	0.120399
$291$	0	0
$292$	15.9134	0.931260
$293$	−23.8942	−1.39591	−0.697957	−	0.716140i	$-0.745907\pi$
$293$	−23.8942	−1.39591	−0.697957	+	0.716140i	$0.745907\pi$
$294$	0	0
$295$	−10.4333	−0.607453
$296$	20.8785	1.21354
$297$	0	0
$298$	8.25191	0.478020
$299$	−11.7057	−0.676959
$300$	0	0
$301$	−2.73497	−0.157641
$302$	−15.1237	−0.870273
$303$	0	0
$304$	−9.16877	−0.525865
$305$	0.296654	0.0169864
$306$	0	0
$307$	−16.0776	−0.917597	−0.458799	−	0.888540i	$-0.651720\pi$
$307$	−16.0776	−0.917597	−0.458799	+	0.888540i	$0.651720\pi$
$308$	−0.791511	−0.0451005
$309$	0	0
$310$	4.32241	0.245497
$311$	17.3187	0.982053	0.491027	−	0.871145i	$-0.336622\pi$
$311$	17.3187	0.982053	0.491027	+	0.871145i	$0.336622\pi$
$312$	0	0
$313$	13.4037	0.757621	0.378811	−	0.925474i	$-0.376333\pi$
$313$	13.4037	0.757621	0.378811	+	0.925474i	$0.376333\pi$
$314$	−8.93431	−0.504192
$315$	0	0
$316$	10.8079	0.607994
$317$	−21.4215	−1.20315	−0.601576	−	0.798815i	$-0.705460\pi$
$317$	−21.4215	−1.20315	−0.601576	+	0.798815i	$0.705460\pi$
$318$	0	0
$319$	−5.28051	−0.295652
$320$	1.35917	0.0759797
$321$	0	0
$322$	1.08175	0.0602838
$323$	−37.4116	−2.08164
$324$	0	0
$325$	2.19103	0.121536
$326$	−11.2380	−0.622416
$327$	0	0
$328$	11.1267	0.614369
$329$	1.59276	0.0878117
$330$	0	0
$331$	−8.04681	−0.442293	−0.221146	−	0.975241i	$-0.570980\pi$
$331$	−8.04681	−0.442293	−0.221146	+	0.975241i	$0.570980\pi$
$332$	−9.42732	−0.517392
$333$	0	0
$334$	−11.4365	−0.625779
$335$	−4.45758	−0.243543
$336$	0	0
$337$	20.6757	1.12628	0.563139	−	0.826362i	$-0.309594\pi$
$337$	20.6757	1.12628	0.563139	+	0.826362i	$0.309594\pi$
$338$	−5.69348	−0.309685
$339$	0	0
$340$	−8.29598	−0.449913
$341$	−11.1322	−0.602840
$342$	0	0
$343$	4.05756	0.219088
$344$	−22.9109	−1.23528
$345$	0	0
$346$	7.12739	0.383171
$347$	9.17856	0.492731	0.246366	−	0.969177i	$-0.420764\pi$
$347$	9.17856	0.492731	0.246366	+	0.969177i	$0.420764\pi$
$348$	0	0
$349$	−20.3345	−1.08848	−0.544240	−	0.838930i	$-0.683182\pi$
$349$	−20.3345	−1.08848	−0.544240	+	0.838930i	$0.683182\pi$
$350$	−0.202478	−0.0108229
$351$	0	0
$352$	−10.4002	−0.554330
$353$	−5.72152	−0.304526	−0.152263	−	0.988340i	$-0.548656\pi$
$353$	−5.72152	−0.304526	−0.152263	+	0.988340i	$0.548656\pi$
$354$	0	0
$355$	−9.49280	−0.503826
$356$	18.7901	0.995874
$357$	0	0
$358$	−4.73141	−0.250063
$359$	32.4662	1.71350	0.856749	−	0.515733i	$-0.172480\pi$
$359$	32.4662	1.71350	0.856749	+	0.515733i	$0.172480\pi$
$360$	0	0
$361$	27.8520	1.46590
$362$	−17.9902	−0.945544
$363$	0	0
$364$	0.969742	0.0508283
$365$	−10.4842	−0.548770
$366$	0	0
$367$	4.18664	0.218541	0.109271	−	0.994012i	$-0.465149\pi$
$367$	4.18664	0.218541	0.109271	+	0.994012i	$0.465149\pi$
$368$	−7.15644	−0.373055
$369$	0	0
$370$	−5.93505	−0.308549
$371$	−0.653458	−0.0339258
$372$	0	0
$373$	6.90859	0.357713	0.178857	−	0.983875i	$-0.442760\pi$
$373$	6.90859	0.357713	0.178857	+	0.983875i	$0.442760\pi$
$374$	−6.78714	−0.350955
$375$	0	0
$376$	13.3426	0.688091
$377$	6.46956	0.333199
$378$	0	0
$379$	−9.83488	−0.505184	−0.252592	−	0.967573i	$-0.581283\pi$
$379$	−9.83488	−0.505184	−0.252592	+	0.967573i	$0.581283\pi$
$380$	10.3894	0.532964
$381$	0	0
$382$	−13.5407	−0.692800
$383$	−29.9760	−1.53170	−0.765851	−	0.643018i	$-0.777682\pi$
$383$	−29.9760	−1.53170	−0.765851	+	0.643018i	$0.777682\pi$
$384$	0	0
$385$	0.521473	0.0265767
$386$	−11.8918	−0.605279
$387$	0	0
$388$	−21.3708	−1.08494
$389$	11.9460	0.605685	0.302843	−	0.953041i	$-0.402064\pi$
$389$	11.9460	0.605685	0.302843	+	0.953041i	$0.402064\pi$
$390$	0	0
$391$	−29.2006	−1.47674
$392$	16.8913	0.853138
$393$	0	0
$394$	13.5787	0.684085
$395$	−7.12060	−0.358277
$396$	0	0
$397$	−11.9255	−0.598526	−0.299263	−	0.954171i	$-0.596741\pi$
$397$	−11.9255	−0.598526	−0.299263	+	0.954171i	$0.596741\pi$
$398$	−14.8294	−0.743330
$399$	0	0
$400$	1.33951	0.0669756
$401$	34.9016	1.74290	0.871452	−	0.490481i	$-0.163179\pi$
$401$	34.9016	1.74290	0.871452	+	0.490481i	$0.163179\pi$
$402$	0	0
$403$	13.6389	0.679400
$404$	−19.1405	−0.952275
$405$	0	0
$406$	−0.597868	−0.0296717
$407$	15.2854	0.757670
$408$	0	0
$409$	−11.4755	−0.567427	−0.283713	−	0.958909i	$-0.591566\pi$
$409$	−11.4755	−0.567427	−0.283713	+	0.958909i	$0.591566\pi$
$410$	−3.16293	−0.156206
$411$	0	0
$412$	8.94789	0.440831
$413$	3.04233	0.149703
$414$	0	0
$415$	6.21102	0.304887
$416$	12.7420	0.624730
$417$	0	0
$418$	8.49981	0.415739
$419$	−30.9446	−1.51174	−0.755872	−	0.654719i	$-0.772787\pi$
$419$	−30.9446	−1.51174	−0.755872	+	0.654719i	$0.772787\pi$
$420$	0	0
$421$	9.46162	0.461131	0.230566	−	0.973057i	$-0.425942\pi$
$421$	9.46162	0.461131	0.230566	+	0.973057i	$0.425942\pi$
$422$	1.28435	0.0625213
$423$	0	0
$424$	−5.47403	−0.265842
$425$	5.46565	0.265123
$426$	0	0
$427$	−0.0865033	−0.00418619
$428$	9.30747	0.449894
$429$	0	0
$430$	6.51279	0.314075
$431$	2.72680	0.131345	0.0656727	−	0.997841i	$-0.479081\pi$
$431$	2.72680	0.131345	0.0656727	+	0.997841i	$0.479081\pi$
$432$	0	0
$433$	−18.8159	−0.904237	−0.452118	−	0.891958i	$-0.649332\pi$
$433$	−18.8159	−0.904237	−0.452118	+	0.891958i	$0.649332\pi$
$434$	−1.26040	−0.0605012
$435$	0	0
$436$	7.59910	0.363931
$437$	36.5691	1.74934
$438$	0	0
$439$	−14.2550	−0.680356	−0.340178	−	0.940361i	$-0.610487\pi$
$439$	−14.2550	−0.680356	−0.340178	+	0.940361i	$0.610487\pi$
$440$	4.36839	0.208255
$441$	0	0
$442$	8.31545	0.395526
$443$	4.24329	0.201605	0.100802	−	0.994906i	$-0.467859\pi$
$443$	4.24329	0.201605	0.100802	+	0.994906i	$0.467859\pi$
$444$	0	0
$445$	−12.3795	−0.586846
$446$	−1.85135	−0.0876638
$447$	0	0
$448$	−0.396328	−0.0187248
$449$	1.09920	0.0518742	0.0259371	−	0.999664i	$-0.491743\pi$
$449$	1.09920	0.0518742	0.0259371	+	0.999664i	$0.491743\pi$
$450$	0	0
$451$	8.14597	0.383579
$452$	3.41023	0.160404
$453$	0	0
$454$	3.11158	0.146034
$455$	−0.638896	−0.0299519
$456$	0	0
$457$	−3.53135	−0.165189	−0.0825947	−	0.996583i	$-0.526321\pi$
$457$	−3.53135	−0.165189	−0.0825947	+	0.996583i	$0.526321\pi$
$458$	16.9232	0.790770
$459$	0	0
$460$	8.10916	0.378092
$461$	21.4747	1.00017	0.500087	−	0.865975i	$-0.333301\pi$
$461$	21.4747	1.00017	0.500087	+	0.865975i	$0.333301\pi$
$462$	0	0
$463$	23.4028	1.08762	0.543810	−	0.839208i	$-0.316981\pi$
$463$	23.4028	1.08762	0.543810	+	0.839208i	$0.316981\pi$
$464$	3.95525	0.183618
$465$	0	0
$466$	3.33629	0.154551
$467$	−29.7095	−1.37479	−0.687396	−	0.726283i	$-0.741246\pi$
$467$	−29.7095	−1.37479	−0.687396	+	0.726283i	$0.741246\pi$
$468$	0	0
$469$	1.29981	0.0600198
$470$	−3.79283	−0.174950
$471$	0	0
$472$	25.4857	1.17307
$473$	−16.7733	−0.771239
$474$	0	0
$475$	−6.84486	−0.314064
$476$	2.41908	0.110878
$477$	0	0
$478$	11.4986	0.525932
$479$	10.7927	0.493131	0.246566	−	0.969126i	$-0.420698\pi$
$479$	10.7927	0.493131	0.246566	+	0.969126i	$0.420698\pi$
$480$	0	0
$481$	−18.7273	−0.853893
$482$	19.4089	0.884053
$483$	0	0
$484$	11.8420	0.538271
$485$	14.0798	0.639329
$486$	0	0
$487$	27.9786	1.26783	0.633916	−	0.773402i	$-0.281446\pi$
$487$	27.9786	1.26783	0.633916	+	0.773402i	$0.281446\pi$
$488$	−0.724640	−0.0328029
$489$	0	0
$490$	−4.80161	−0.216914
$491$	32.9123	1.48531	0.742655	−	0.669674i	$-0.233566\pi$
$491$	32.9123	1.48531	0.742655	+	0.669674i	$0.233566\pi$
$492$	0	0
$493$	16.1387	0.726851
$494$	−10.4138	−0.468538
$495$	0	0
$496$	8.33829	0.374400
$497$	2.76807	0.124165
$498$	0	0
$499$	−15.7063	−0.703112	−0.351556	−	0.936167i	$-0.614347\pi$
$499$	−15.7063	−0.703112	−0.351556	+	0.936167i	$0.614347\pi$
$500$	−1.51784	−0.0678798
$501$	0	0
$502$	16.0555	0.716594
$503$	19.0851	0.850964	0.425482	−	0.904967i	$-0.360104\pi$
$503$	19.0851	0.850964	0.425482	+	0.904967i	$0.360104\pi$
$504$	0	0
$505$	12.6104	0.561154
$506$	6.63430	0.294931
$507$	0	0
$508$	−6.08763	−0.270095
$509$	−13.7678	−0.610245	−0.305122	−	0.952313i	$-0.598697\pi$
$509$	−13.7678	−0.610245	−0.305122	+	0.952313i	$0.598697\pi$
$510$	0	0
$511$	3.05717	0.135241
$512$	14.3341	0.633483
$513$	0	0
$514$	19.3706	0.854401
$515$	−5.89515	−0.259772
$516$	0	0
$517$	9.76825	0.429607
$518$	1.73064	0.0760400
$519$	0	0
$520$	−5.35205	−0.234703
$521$	−15.6068	−0.683748	−0.341874	−	0.939746i	$-0.611062\pi$
$521$	−15.6068	−0.683748	−0.341874	+	0.939746i	$0.611062\pi$
$522$	0	0
$523$	14.5475	0.636117	0.318058	−	0.948071i	$-0.396969\pi$
$523$	14.5475	0.636117	0.318058	+	0.948071i	$0.396969\pi$
$524$	−21.7020	−0.948057
$525$	0	0
$526$	10.7037	0.466702
$527$	34.0230	1.48206
$528$	0	0
$529$	5.54305	0.241002
$530$	1.55608	0.0675917
$531$	0	0
$532$	−3.02951	−0.131346
$533$	−9.98026	−0.432293
$534$	0	0
$535$	−6.13206	−0.265112
$536$	10.8886	0.470315
$537$	0	0
$538$	−11.4656	−0.494316
$539$	12.3663	0.532654
$540$	0	0
$541$	−1.02903	−0.0442416	−0.0221208	−	0.999755i	$-0.507042\pi$
$541$	−1.02903	−0.0442416	−0.0221208	+	0.999755i	$0.507042\pi$
$542$	−1.89514	−0.0814032
$543$	0	0
$544$	31.7858	1.36280
$545$	−5.00653	−0.214456
$546$	0	0
$547$	27.3080	1.16761	0.583803	−	0.811896i	$-0.301564\pi$
$547$	27.3080	1.16761	0.583803	+	0.811896i	$0.301564\pi$
$548$	2.26781	0.0968761
$549$	0	0
$550$	−1.24178	−0.0529497
$551$	−20.2112	−0.861024
$552$	0	0
$553$	2.07634	0.0882951
$554$	6.65733	0.282843
$555$	0	0
$556$	6.60159	0.279970
$557$	14.3369	0.607476	0.303738	−	0.952756i	$-0.401765\pi$
$557$	14.3369	0.607476	0.303738	+	0.952756i	$0.401765\pi$
$558$	0	0
$559$	20.5503	0.869186
$560$	−0.390597	−0.0165057
$561$	0	0
$562$	−2.23310	−0.0941978
$563$	−29.0423	−1.22399	−0.611994	−	0.790863i	$-0.709632\pi$
$563$	−29.0423	−1.22399	−0.611994	+	0.790863i	$0.709632\pi$
$564$	0	0
$565$	−2.24677	−0.0945221
$566$	−17.5750	−0.738731
$567$	0	0
$568$	23.1882	0.972954
$569$	3.69642	0.154962	0.0774810	−	0.996994i	$-0.475312\pi$
$569$	3.69642	0.154962	0.0774810	+	0.996994i	$0.475312\pi$
$570$	0	0
$571$	5.33907	0.223433	0.111717	−	0.993740i	$-0.464365\pi$
$571$	5.33907	0.223433	0.111717	+	0.993740i	$0.464365\pi$
$572$	5.94734	0.248671
$573$	0	0
$574$	0.922301	0.0384961
$575$	−5.34257	−0.222801
$576$	0	0
$577$	21.4615	0.893455	0.446727	−	0.894670i	$-0.352589\pi$
$577$	21.4615	0.893455	0.446727	+	0.894670i	$0.352589\pi$
$578$	8.93898	0.371812
$579$	0	0
$580$	−4.48180	−0.186097
$581$	−1.81111	−0.0751376
$582$	0	0
$583$	−4.00760	−0.165978
$584$	25.6100	1.05975
$585$	0	0
$586$	−16.5916	−0.685393
$587$	6.08738	0.251253	0.125627	−	0.992078i	$-0.459906\pi$
$587$	6.08738	0.251253	0.125627	+	0.992078i	$0.459906\pi$
$588$	0	0
$589$	−42.6083	−1.75565
$590$	−7.24469	−0.298259
$591$	0	0
$592$	−11.4492	−0.470559
$593$	−0.198543	−0.00815319	−0.00407660	−	0.999992i	$-0.501298\pi$
$593$	−0.198543	−0.00815319	−0.00407660	+	0.999992i	$0.501298\pi$
$594$	0	0
$595$	−1.59377	−0.0653380
$596$	−18.0378	−0.738858
$597$	0	0
$598$	−8.12820	−0.332387
$599$	8.54109	0.348979	0.174490	−	0.984659i	$-0.444172\pi$
$599$	8.54109	0.348979	0.174490	+	0.984659i	$0.444172\pi$
$600$	0	0
$601$	13.3212	0.543383	0.271691	−	0.962384i	$-0.412417\pi$
$601$	13.3212	0.543383	0.271691	+	0.962384i	$0.412417\pi$
$602$	−1.89911	−0.0774018
$603$	0	0
$604$	33.0589	1.34515
$605$	−7.80186	−0.317191
$606$	0	0
$607$	−1.25384	−0.0508917	−0.0254458	−	0.999676i	$-0.508101\pi$
$607$	−1.25384	−0.0508917	−0.0254458	+	0.999676i	$0.508101\pi$
$608$	−39.8066	−1.61437
$609$	0	0
$610$	0.205990	0.00834030
$611$	−11.9678	−0.484167
$612$	0	0
$613$	−14.7765	−0.596818	−0.298409	−	0.954438i	$-0.596456\pi$
$613$	−14.7765	−0.596818	−0.298409	+	0.954438i	$0.596456\pi$
$614$	−11.1639	−0.450540
$615$	0	0
$616$	−1.27381	−0.0513231
$617$	−2.12796	−0.0856684	−0.0428342	−	0.999082i	$-0.513639\pi$
$617$	−2.12796	−0.0856684	−0.0428342	+	0.999082i	$0.513639\pi$
$618$	0	0
$619$	−17.7002	−0.711430	−0.355715	−	0.934594i	$-0.615763\pi$
$619$	−17.7002	−0.711430	−0.355715	+	0.934594i	$0.615763\pi$
$620$	−9.44835	−0.379455
$621$	0	0
$622$	12.0257	0.482188
$623$	3.60983	0.144625
$624$	0	0
$625$	1.00000	0.0400000
$626$	9.30723	0.371992
$627$	0	0
$628$	19.5295	0.779311
$629$	−46.7165	−1.86271
$630$	0	0
$631$	12.0809	0.480933	0.240466	−	0.970657i	$-0.422700\pi$
$631$	12.0809	0.480933	0.240466	+	0.970657i	$0.422700\pi$
$632$	17.3936	0.691879
$633$	0	0
$634$	−14.8746	−0.590747
$635$	4.01072	0.159161
$636$	0	0
$637$	−15.1509	−0.600300
$638$	−3.66667	−0.145165
$639$	0	0
$640$	−10.6873	−0.422454
$641$	−14.1528	−0.559004	−0.279502	−	0.960145i	$-0.590169\pi$
$641$	−14.1528	−0.559004	−0.279502	+	0.960145i	$0.590169\pi$
$642$	0	0
$643$	23.5908	0.930329	0.465164	−	0.885224i	$-0.345995\pi$
$643$	23.5908	0.930329	0.465164	+	0.885224i	$0.345995\pi$
$644$	−2.36460	−0.0931784
$645$	0	0
$646$	−25.9778	−1.02208
$647$	−0.0994615	−0.00391023	−0.00195512	−	0.999998i	$-0.500622\pi$
$647$	−0.0994615	−0.00391023	−0.00195512	+	0.999998i	$0.500622\pi$
$648$	0	0
$649$	18.6583	0.732403
$650$	1.52140	0.0596743
$651$	0	0
$652$	24.5652	0.962046
$653$	−16.4226	−0.642666	−0.321333	−	0.946966i	$-0.604131\pi$
$653$	−16.4226	−0.642666	−0.321333	+	0.946966i	$0.604131\pi$
$654$	0	0
$655$	14.2980	0.558668
$656$	−6.10156	−0.238226
$657$	0	0
$658$	1.10598	0.0431155
$659$	37.9333	1.47767	0.738836	−	0.673885i	$-0.235376\pi$
$659$	37.9333	1.47767	0.738836	+	0.673885i	$0.235376\pi$
$660$	0	0
$661$	−13.0649	−0.508166	−0.254083	−	0.967182i	$-0.581774\pi$
$661$	−13.0649	−0.508166	−0.254083	+	0.967182i	$0.581774\pi$
$662$	−5.58753	−0.217165
$663$	0	0
$664$	−15.1717	−0.588777
$665$	1.99594	0.0773991
$666$	0	0
$667$	−15.7753	−0.610821
$668$	24.9990	0.967242
$669$	0	0
$670$	−3.09524	−0.119580
$671$	−0.530517	−0.0204804
$672$	0	0
$673$	4.04776	0.156030	0.0780148	−	0.996952i	$-0.475142\pi$
$673$	4.04776	0.156030	0.0780148	+	0.996952i	$0.475142\pi$
$674$	14.3568	0.553002
$675$	0	0
$676$	12.4454	0.478668
$677$	14.9229	0.573535	0.286768	−	0.958000i	$-0.407419\pi$
$677$	14.9229	0.573535	0.286768	+	0.958000i	$0.407419\pi$
$678$	0	0
$679$	−4.10561	−0.157559
$680$	−13.3510	−0.511988
$681$	0	0
$682$	−7.72992	−0.295994
$683$	23.3698	0.894220	0.447110	−	0.894479i	$-0.352453\pi$
$683$	23.3698	0.894220	0.447110	+	0.894479i	$0.352453\pi$
$684$	0	0
$685$	−1.49411	−0.0570868
$686$	2.81748	0.107572
$687$	0	0
$688$	12.5637	0.478987
$689$	4.91002	0.187057
$690$	0	0
$691$	12.5671	0.478077	0.239038	−	0.971010i	$-0.423168\pi$
$691$	12.5671	0.478077	0.239038	+	0.971010i	$0.423168\pi$
$692$	−15.5797	−0.592252
$693$	0	0
$694$	6.37340	0.241931
$695$	−4.34934	−0.164980
$696$	0	0
$697$	−24.8964	−0.943017
$698$	−14.1198	−0.534443
$699$	0	0
$700$	0.442597	0.0167286
$701$	38.3903	1.44998	0.724991	−	0.688758i	$-0.241844\pi$
$701$	38.3903	1.44998	0.724991	+	0.688758i	$0.241844\pi$
$702$	0	0
$703$	58.5049	2.20656
$704$	−2.43065	−0.0916084
$705$	0	0
$706$	−3.97290	−0.149522
$707$	−3.67714	−0.138293
$708$	0	0
$709$	−7.80062	−0.292959	−0.146479	−	0.989214i	$-0.546794\pi$
$709$	−7.80062	−0.292959	−0.146479	+	0.989214i	$0.546794\pi$
$710$	−6.59160	−0.247378
$711$	0	0
$712$	30.2396	1.13328
$713$	−33.2568	−1.24548
$714$	0	0
$715$	−3.91829	−0.146536
$716$	10.3424	0.386513
$717$	0	0
$718$	22.5438	0.841327
$719$	29.6952	1.10744	0.553722	−	0.832702i	$-0.313207\pi$
$719$	29.6952	1.10744	0.553722	+	0.832702i	$0.313207\pi$
$720$	0	0
$721$	1.71901	0.0640192
$722$	19.3399	0.719755
$723$	0	0
$724$	39.3247	1.46149
$725$	2.95275	0.109662
$726$	0	0
$727$	0.771126	0.0285995	0.0142997	−	0.999898i	$-0.495448\pi$
$727$	0.771126	0.0285995	0.0142997	+	0.999898i	$0.495448\pi$
$728$	1.56064	0.0578411
$729$	0	0
$730$	−7.28003	−0.269446
$731$	51.2640	1.89607
$732$	0	0
$733$	−2.04505	−0.0755358	−0.0377679	−	0.999287i	$-0.512025\pi$
$733$	−2.04505	−0.0755358	−0.0377679	+	0.999287i	$0.512025\pi$
$734$	2.90711	0.107304
$735$	0	0
$736$	−31.0700	−1.14525
$737$	7.97164	0.293639
$738$	0	0
$739$	−8.56527	−0.315079	−0.157539	−	0.987513i	$-0.550356\pi$
$739$	−8.56527	−0.315079	−0.157539	+	0.987513i	$0.550356\pi$
$740$	12.9734	0.476912
$741$	0	0
$742$	−0.453747	−0.0166576
$743$	−33.5742	−1.23172	−0.615860	−	0.787856i	$-0.711191\pi$
$743$	−33.5742	−1.23172	−0.615860	+	0.787856i	$0.711191\pi$
$744$	0	0
$745$	11.8839	0.435392
$746$	4.79717	0.175637
$747$	0	0
$748$	14.8360	0.542458
$749$	1.78809	0.0653353
$750$	0	0
$751$	−23.4434	−0.855463	−0.427731	−	0.903906i	$-0.640687\pi$
$751$	−23.4434	−0.855463	−0.427731	+	0.903906i	$0.640687\pi$
$752$	−7.31669	−0.266812
$753$	0	0
$754$	4.49232	0.163601
$755$	−21.7802	−0.792664
$756$	0	0
$757$	2.94896	0.107182	0.0535909	−	0.998563i	$-0.482933\pi$
$757$	2.94896	0.107182	0.0535909	+	0.998563i	$0.482933\pi$
$758$	−6.82913	−0.248045
$759$	0	0
$760$	16.7200	0.606498
$761$	29.7536	1.07857	0.539284	−	0.842124i	$-0.318695\pi$
$761$	29.7536	1.07857	0.539284	+	0.842124i	$0.318695\pi$
$762$	0	0
$763$	1.45989	0.0528515
$764$	29.5985	1.07084
$765$	0	0
$766$	−20.8147	−0.752065
$767$	−22.8598	−0.825418
$768$	0	0
$769$	−7.98003	−0.287767	−0.143884	−	0.989595i	$-0.545959\pi$
$769$	−7.98003	−0.287767	−0.143884	+	0.989595i	$0.545959\pi$
$770$	0.362099	0.0130491
$771$	0	0
$772$	25.9943	0.935556
$773$	−0.375937	−0.0135215	−0.00676076	−	0.999977i	$-0.502152\pi$
$773$	−0.375937	−0.0135215	−0.00676076	+	0.999977i	$0.502152\pi$
$774$	0	0
$775$	6.22487	0.223604
$776$	−34.3928	−1.23463
$777$	0	0
$778$	8.29503	0.297391
$779$	31.1787	1.11709
$780$	0	0
$781$	16.9763	0.607460
$782$	−20.2763	−0.725078
$783$	0	0
$784$	−9.26269	−0.330810
$785$	−12.8666	−0.459230
$786$	0	0
$787$	−5.90797	−0.210596	−0.105298	−	0.994441i	$-0.533580\pi$
$787$	−5.90797	−0.210596	−0.105298	+	0.994441i	$0.533580\pi$
$788$	−29.6816	−1.05736
$789$	0	0
$790$	−4.94439	−0.175914
$791$	0.655149	0.0232944
$792$	0	0
$793$	0.649977	0.0230814
$794$	−8.28083	−0.293876
$795$	0	0
$796$	32.4155	1.14894
$797$	−51.5520	−1.82607	−0.913033	−	0.407886i	$-0.866266\pi$
$797$	−51.5520	−1.82607	−0.913033	+	0.407886i	$0.866266\pi$
$798$	0	0
$799$	−29.8545	−1.05618
$800$	5.81555	0.205611
$801$	0	0
$802$	24.2349	0.855765
$803$	18.7493	0.661649
$804$	0	0
$805$	1.55788	0.0549079
$806$	9.47053	0.333585
$807$	0	0
$808$	−30.8035	−1.08366
$809$	37.4718	1.31744	0.658719	−	0.752389i	$-0.271099\pi$
$809$	37.4718	1.31744	0.658719	+	0.752389i	$0.271099\pi$
$810$	0	0
$811$	27.5789	0.968426	0.484213	−	0.874950i	$-0.339106\pi$
$811$	27.5789	0.968426	0.484213	+	0.874950i	$0.339106\pi$
$812$	1.30688	0.0458624
$813$	0	0
$814$	10.6139	0.372016
$815$	−16.1843	−0.566911
$816$	0	0
$817$	−64.2000	−2.24607
$818$	−7.96834	−0.278606
$819$	0	0
$820$	6.91385	0.241442
$821$	44.5274	1.55402	0.777008	−	0.629491i	$-0.216737\pi$
$821$	44.5274	1.55402	0.777008	+	0.629491i	$0.216737\pi$
$822$	0	0
$823$	−25.2956	−0.881748	−0.440874	−	0.897569i	$-0.645331\pi$
$823$	−25.2956	−0.881748	−0.440874	+	0.897569i	$0.645331\pi$
$824$	14.4002	0.501653
$825$	0	0
$826$	2.11253	0.0735042
$827$	−18.9379	−0.658534	−0.329267	−	0.944237i	$-0.606802\pi$
$827$	−18.9379	−0.658534	−0.329267	+	0.944237i	$0.606802\pi$
$828$	0	0
$829$	3.45210	0.119896	0.0599482	−	0.998201i	$-0.480906\pi$
$829$	3.45210	0.119896	0.0599482	+	0.998201i	$0.480906\pi$
$830$	4.31280	0.149699
$831$	0	0
$832$	2.97797	0.103243
$833$	−37.7948	−1.30951
$834$	0	0
$835$	−16.4702	−0.569973
$836$	−18.5797	−0.642593
$837$	0	0
$838$	−21.4873	−0.742266
$839$	19.1777	0.662088	0.331044	−	0.943615i	$-0.392599\pi$
$839$	19.1777	0.662088	0.331044	+	0.943615i	$0.392599\pi$
$840$	0	0
$841$	−20.2813	−0.699354
$842$	6.56994	0.226415
$843$	0	0
$844$	−2.80746	−0.0966367
$845$	−8.19940	−0.282068
$846$	0	0
$847$	2.27500	0.0781698
$848$	3.00180	0.103082
$849$	0	0
$850$	3.79523	0.130175
$851$	45.6645	1.56536
$852$	0	0
$853$	−5.84922	−0.200273	−0.100137	−	0.994974i	$-0.531928\pi$
$853$	−5.84922	−0.200273	−0.100137	+	0.994974i	$0.531928\pi$
$854$	−0.0600660	−0.00205542
$855$	0	0
$856$	14.9788	0.511966
$857$	−49.4113	−1.68786	−0.843930	−	0.536454i	$-0.819764\pi$
$857$	−49.4113	−1.68786	−0.843930	+	0.536454i	$0.819764\pi$
$858$	0	0
$859$	7.77965	0.265438	0.132719	−	0.991154i	$-0.457629\pi$
$859$	7.77965	0.265438	0.132719	+	0.991154i	$0.457629\pi$
$860$	−14.2363	−0.485453
$861$	0	0
$862$	1.89343	0.0644905
$863$	11.8283	0.402639	0.201320	−	0.979526i	$-0.435477\pi$
$863$	11.8283	0.402639	0.201320	+	0.979526i	$0.435477\pi$
$864$	0	0
$865$	10.2644	0.349001
$866$	−13.0654	−0.443980
$867$	0	0
$868$	2.75511	0.0935144
$869$	12.7340	0.431972
$870$	0	0
$871$	−9.76667	−0.330931
$872$	12.2295	0.414143
$873$	0	0
$874$	25.3928	0.858924
$875$	−0.291597	−0.00985777
$876$	0	0
$877$	−47.5832	−1.60677	−0.803386	−	0.595459i	$-0.796970\pi$
$877$	−47.5832	−1.60677	−0.803386	+	0.595459i	$0.796970\pi$
$878$	−9.89839	−0.334054
$879$	0	0
$880$	−2.39550	−0.0807522
$881$	25.7202	0.866535	0.433268	−	0.901265i	$-0.357361\pi$
$881$	25.7202	0.866535	0.433268	+	0.901265i	$0.357361\pi$
$882$	0	0
$883$	44.9567	1.51291	0.756456	−	0.654044i	$-0.226929\pi$
$883$	44.9567	1.51291	0.756456	+	0.654044i	$0.226929\pi$
$884$	−18.1767	−0.611349
$885$	0	0
$886$	2.94645	0.0989879
$887$	14.6098	0.490548	0.245274	−	0.969454i	$-0.421122\pi$
$887$	14.6098	0.490548	0.245274	+	0.969454i	$0.421122\pi$
$888$	0	0
$889$	−1.16951	−0.0392242
$890$	−8.59607	−0.288141
$891$	0	0
$892$	4.04685	0.135499
$893$	37.3880	1.25114
$894$	0	0
$895$	−6.81389	−0.227763
$896$	3.11639	0.104111
$897$	0	0
$898$	0.763258	0.0254702
$899$	18.3805	0.613023
$900$	0	0
$901$	12.2483	0.408051
$902$	5.65639	0.188337
$903$	0	0
$904$	5.48820	0.182535
$905$	−25.9084	−0.861223
$906$	0	0
$907$	−32.3136	−1.07296	−0.536479	−	0.843914i	$-0.680246\pi$
$907$	−32.3136	−1.07296	−0.536479	+	0.843914i	$0.680246\pi$
$908$	−6.80159	−0.225719
$909$	0	0
$910$	−0.443636	−0.0147064
$911$	10.8366	0.359033	0.179517	−	0.983755i	$-0.442547\pi$
$911$	10.8366	0.359033	0.179517	+	0.983755i	$0.442547\pi$
$912$	0	0
$913$	−11.1074	−0.367601
$914$	−2.45209	−0.0811080
$915$	0	0
$916$	−36.9924	−1.22226
$917$	−4.16924	−0.137681
$918$	0	0
$919$	−7.87445	−0.259754	−0.129877	−	0.991530i	$-0.541458\pi$
$919$	−7.87445	−0.259754	−0.129877	+	0.991530i	$0.541458\pi$
$920$	13.0504	0.430257
$921$	0	0
$922$	14.9115	0.491085
$923$	−20.7990	−0.684607
$924$	0	0
$925$	−8.54729	−0.281033
$926$	16.2504	0.534021
$927$	0	0
$928$	17.1719	0.563694
$929$	−32.3988	−1.06297	−0.531485	−	0.847068i	$-0.678366\pi$
$929$	−32.3988	−1.06297	−0.531485	+	0.847068i	$0.678366\pi$
$930$	0	0
$931$	47.3320	1.55124
$932$	−7.29279	−0.238883
$933$	0	0
$934$	−20.6296	−0.675022
$935$	−9.77442	−0.319658
$936$	0	0
$937$	60.9943	1.99260	0.996298	−	0.0859685i	$-0.0273984\pi$
$937$	60.9943	1.99260	0.996298	+	0.0859685i	$0.0273984\pi$
$938$	0.902563	0.0294697
$939$	0	0
$940$	8.29074	0.270414
$941$	−15.3190	−0.499387	−0.249693	−	0.968325i	$-0.580330\pi$
$941$	−15.3190	−0.499387	−0.249693	+	0.968325i	$0.580330\pi$
$942$	0	0
$943$	24.3357	0.792480
$944$	−13.9756	−0.454867
$945$	0	0
$946$	−11.6470	−0.378678
$947$	−55.4559	−1.80207	−0.901037	−	0.433742i	$-0.857193\pi$
$947$	−55.4559	−1.80207	−0.901037	+	0.433742i	$0.857193\pi$
$948$	0	0
$949$	−22.9713	−0.745679
$950$	−4.75292	−0.154205
$951$	0	0
$952$	3.89311	0.126176
$953$	−2.86574	−0.0928304	−0.0464152	−	0.998922i	$-0.514780\pi$
$953$	−2.86574	−0.0928304	−0.0464152	+	0.998922i	$0.514780\pi$
$954$	0	0
$955$	−19.5004	−0.631018
$956$	−25.1347	−0.812914
$957$	0	0
$958$	7.49422	0.242127
$959$	0.435676	0.0140687
$960$	0	0
$961$	7.74898	0.249967
$962$	−13.0039	−0.419261
$963$	0	0
$964$	−42.4259	−1.36645
$965$	−17.1259	−0.551301
$966$	0	0
$967$	46.2225	1.48642	0.743208	−	0.669061i	$-0.233303\pi$
$967$	46.2225	1.48642	0.743208	+	0.669061i	$0.233303\pi$
$968$	19.0577	0.612537
$969$	0	0
$970$	9.77669	0.313910
$971$	−35.1848	−1.12914	−0.564568	−	0.825387i	$-0.690957\pi$
$971$	−35.1848	−1.12914	−0.564568	+	0.825387i	$0.690957\pi$
$972$	0	0
$973$	1.26825	0.0406583
$974$	19.4277	0.622505
$975$	0	0
$976$	0.397372	0.0127196
$977$	25.8171	0.825962	0.412981	−	0.910740i	$-0.364488\pi$
$977$	25.8171	0.825962	0.412981	+	0.910740i	$0.364488\pi$
$978$	0	0
$979$	22.1387	0.707557
$980$	10.4958	0.335276
$981$	0	0
$982$	22.8536	0.729287
$983$	−10.6669	−0.340221	−0.170111	−	0.985425i	$-0.554412\pi$
$983$	−10.6669	−0.340221	−0.170111	+	0.985425i	$0.554412\pi$
$984$	0	0
$985$	19.5552	0.623080
$986$	11.2064	0.356883
$987$	0	0
$988$	22.7634	0.724201
$989$	−50.1096	−1.59339
$990$	0	0
$991$	59.3676	1.88587	0.942937	−	0.332972i	$-0.108051\pi$
$991$	59.3676	1.88587	0.942937	+	0.332972i	$0.108051\pi$
$992$	36.2010	1.14938
$993$	0	0
$994$	1.92209	0.0609649
$995$	−21.3564	−0.677042
$996$	0	0
$997$	27.8247	0.881216	0.440608	−	0.897699i	$-0.354763\pi$
$997$	27.8247	0.881216	0.440608	+	0.897699i	$0.354763\pi$
$998$	−10.9061	−0.345228
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3645.2.a.g.1.10		15
3.2	odd	2		3645.2.a.h.1.6		15
27.4	even	9		405.2.k.a.46.4		30
27.7	even	9		405.2.k.a.361.4		30
27.20	odd	18		135.2.k.a.76.2	yes	30
27.23	odd	18		135.2.k.a.16.2	✓	30
135.23	even	36		675.2.u.c.124.4		60
135.47	even	36		675.2.u.c.49.4		60
135.74	odd	18		675.2.l.d.76.4		30
135.77	even	36		675.2.u.c.124.7		60
135.104	odd	18		675.2.l.d.151.4		30
135.128	even	36		675.2.u.c.49.7		60

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.k.a.16.2	✓	30	27.23	odd	18
135.2.k.a.76.2	yes	30	27.20	odd	18
405.2.k.a.46.4		30	27.4	even	9
405.2.k.a.361.4		30	27.7	even	9
675.2.l.d.76.4		30	135.74	odd	18
675.2.l.d.151.4		30	135.104	odd	18
675.2.u.c.49.4		60	135.47	even	36
675.2.u.c.49.7		60	135.128	even	36
675.2.u.c.124.4		60	135.23	even	36
675.2.u.c.124.7		60	135.77	even	36
3645.2.a.g.1.10		15	1.1	even	1	trivial
3645.2.a.h.1.6		15	3.2	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:	\( N \)	\(=\)	\( 3645 = 3^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3645.a (trivial)

\(f(q)\)	\(=\)	\(q+0.694378 q^{2} -1.51784 q^{4} +1.00000 q^{5} -0.291597 q^{7} -2.44271 q^{8} +O(q^{10})\)	\(q+0.694378 q^{2} -1.51784 q^{4} +1.00000 q^{5} -0.291597 q^{7} -2.44271 q^{8} +0.694378 q^{10} -1.78834 q^{11} +2.19103 q^{13} -0.202478 q^{14} +1.33951 q^{16} +5.46565 q^{17} -6.84486 q^{19} -1.51784 q^{20} -1.24178 q^{22} -5.34257 q^{23} +1.00000 q^{25} +1.52140 q^{26} +0.442597 q^{28} +2.95275 q^{29} +6.22487 q^{31} +5.81555 q^{32} +3.79523 q^{34} -0.291597 q^{35} -8.54729 q^{37} -4.75292 q^{38} -2.44271 q^{40} -4.55506 q^{41} +9.37931 q^{43} +2.71440 q^{44} -3.70976 q^{46} -5.46220 q^{47} -6.91497 q^{49} +0.694378 q^{50} -3.32563 q^{52} +2.24096 q^{53} -1.78834 q^{55} +0.712286 q^{56} +2.05033 q^{58} -10.4333 q^{59} +0.296654 q^{61} +4.32241 q^{62} +1.35917 q^{64} +2.19103 q^{65} -4.45758 q^{67} -8.29598 q^{68} -0.202478 q^{70} -9.49280 q^{71} -10.4842 q^{73} -5.93505 q^{74} +10.3894 q^{76} +0.521473 q^{77} -7.12060 q^{79} +1.33951 q^{80} -3.16293 q^{82} +6.21102 q^{83} +5.46565 q^{85} +6.51279 q^{86} +4.36839 q^{88} -12.3795 q^{89} -0.638896 q^{91} +8.10916 q^{92} -3.79283 q^{94} -6.84486 q^{95} +14.0798 q^{97} -4.80161 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{2} + 9 q^{4} + 15 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) \) 15 * q - 3 * q^2 + 9 * q^4 + 15 * q^5 - 12 * q^7 - 9 * q^8	\( 15 q - 3 q^{2} + 9 q^{4} + 15 q^{5} - 12 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} - 3 q^{16} - 12 q^{17} - 24 q^{19} + 9 q^{20} - 18 q^{22} - 18 q^{23} + 15 q^{25} + 9 q^{26} - 30 q^{28} + 3 q^{29} - 24 q^{31} - 6 q^{32} - 18 q^{34} - 12 q^{35} - 24 q^{37} - 18 q^{38} - 9 q^{40} + 9 q^{41} - 42 q^{43} + 12 q^{44} - 30 q^{46} - 21 q^{47} - 3 q^{49} - 3 q^{50} - 36 q^{52} - 18 q^{53} + 30 q^{56} - 30 q^{58} + 6 q^{59} - 15 q^{61} + 36 q^{62} - 27 q^{64} - 12 q^{65} - 45 q^{67} - 36 q^{68} + 12 q^{71} - 21 q^{73} + 21 q^{74} - 48 q^{76} - 9 q^{77} - 48 q^{79} - 3 q^{80} - 24 q^{82} - 33 q^{83} - 12 q^{85} + 15 q^{86} - 54 q^{88} + 9 q^{89} - 51 q^{91} + 33 q^{92} - 30 q^{94} - 24 q^{95} - 30 q^{97} - 15 q^{98}+O(q^{100}) \) 15 * q - 3 * q^2 + 9 * q^4 + 15 * q^5 - 12 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 - 3 * q^16 - 12 * q^17 - 24 * q^19 + 9 * q^20 - 18 * q^22 - 18 * q^23 + 15 * q^25 + 9 * q^26 - 30 * q^28 + 3 * q^29 - 24 * q^31 - 6 * q^32 - 18 * q^34 - 12 * q^35 - 24 * q^37 - 18 * q^38 - 9 * q^40 + 9 * q^41 - 42 * q^43 + 12 * q^44 - 30 * q^46 - 21 * q^47 - 3 * q^49 - 3 * q^50 - 36 * q^52 - 18 * q^53 + 30 * q^56 - 30 * q^58 + 6 * q^59 - 15 * q^61 + 36 * q^62 - 27 * q^64 - 12 * q^65 - 45 * q^67 - 36 * q^68 + 12 * q^71 - 21 * q^73 + 21 * q^74 - 48 * q^76 - 9 * q^77 - 48 * q^79 - 3 * q^80 - 24 * q^82 - 33 * q^83 - 12 * q^85 + 15 * q^86 - 54 * q^88 + 9 * q^89 - 51 * q^91 + 33 * q^92 - 30 * q^94 - 24 * q^95 - 30 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more