LMFDB - Embedded newform 6024.2.a.p.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 29 x^{12} + 210 x^{11} + 280 x^{10} - 2796 x^{9} - 863 x^{8} + 17652 x^{7} + \cdots - 3364 $ x^14 - 6x^13 - 29x^12 + 210x^11 + 280x^10 - 2796x^9 - 863x^8 + 17652x^7 - 1300x^6 - 53458x^5 + 10892x^4 + 67298x^3 - 13761x^2 - 22140*x - 3364
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.198997$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.198997 q^{5} +4.17683 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.198997 q^{5} +4.17683 q^{7} +1.00000 q^{9} -3.15027 q^{11} -2.25294 q^{13} +0.198997 q^{15} +4.54891 q^{17} -7.17348 q^{19} -4.17683 q^{21} +4.93189 q^{23} -4.96040 q^{25} -1.00000 q^{27} +2.19608 q^{29} +3.49112 q^{31} +3.15027 q^{33} -0.831177 q^{35} +4.18756 q^{37} +2.25294 q^{39} -3.16436 q^{41} +3.95535 q^{43} -0.198997 q^{45} -7.14201 q^{47} +10.4459 q^{49} -4.54891 q^{51} +0.0629449 q^{53} +0.626895 q^{55} +7.17348 q^{57} +7.16608 q^{59} +0.547121 q^{61} +4.17683 q^{63} +0.448329 q^{65} -0.268787 q^{67} -4.93189 q^{69} +5.33220 q^{71} +8.29308 q^{73} +4.96040 q^{75} -13.1581 q^{77} -0.512972 q^{79} +1.00000 q^{81} +13.8697 q^{83} -0.905220 q^{85} -2.19608 q^{87} +1.21408 q^{89} -9.41015 q^{91} -3.49112 q^{93} +1.42750 q^{95} +16.8801 q^{97} -3.15027 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * 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	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.198997	−0.0889942	−0.0444971	−	0.999010i	$-0.514169\pi$
$5$	−0.198997	−0.0889942	−0.0444971	+	0.999010i	$0.514169\pi$
$6$	0	0
$7$	4.17683	1.57869	0.789346	−	0.613949i	$-0.210420\pi$
$7$	4.17683	1.57869	0.789346	+	0.613949i	$0.210420\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.15027	−0.949842	−0.474921	−	0.880028i	$-0.657523\pi$
$11$	−3.15027	−0.949842	−0.474921	+	0.880028i	$0.657523\pi$
$12$	0	0
$13$	−2.25294	−0.624853	−0.312427	−	0.949942i	$-0.601142\pi$
$13$	−2.25294	−0.624853	−0.312427	+	0.949942i	$0.601142\pi$
$14$	0	0
$15$	0.198997	0.0513808
$16$	0	0
$17$	4.54891	1.10327	0.551637	−	0.834085i	$-0.314004\pi$
$17$	4.54891	1.10327	0.551637	+	0.834085i	$0.314004\pi$
$18$	0	0
$19$	−7.17348	−1.64571	−0.822854	−	0.568253i	$-0.807620\pi$
$19$	−7.17348	−1.64571	−0.822854	+	0.568253i	$0.807620\pi$
$20$	0	0
$21$	−4.17683	−0.911458
$22$	0	0
$23$	4.93189	1.02837	0.514185	−	0.857679i	$-0.328094\pi$
$23$	4.93189	1.02837	0.514185	+	0.857679i	$0.328094\pi$
$24$	0	0
$25$	−4.96040	−0.992080
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.19608	0.407802	0.203901	−	0.978991i	$-0.434638\pi$
$29$	2.19608	0.407802	0.203901	+	0.978991i	$0.434638\pi$
$30$	0	0
$31$	3.49112	0.627023	0.313511	−	0.949584i	$-0.398495\pi$
$31$	3.49112	0.627023	0.313511	+	0.949584i	$0.398495\pi$
$32$	0	0
$33$	3.15027	0.548392
$34$	0	0
$35$	−0.831177	−0.140494
$36$	0	0
$37$	4.18756	0.688431	0.344215	−	0.938891i	$-0.388145\pi$
$37$	4.18756	0.688431	0.344215	+	0.938891i	$0.388145\pi$
$38$	0	0
$39$	2.25294	0.360759
$40$	0	0
$41$	−3.16436	−0.494190	−0.247095	−	0.968991i	$-0.579476\pi$
$41$	−3.16436	−0.494190	−0.247095	+	0.968991i	$0.579476\pi$
$42$	0	0
$43$	3.95535	0.603185	0.301592	−	0.953437i	$-0.402482\pi$
$43$	3.95535	0.603185	0.301592	+	0.953437i	$0.402482\pi$
$44$	0	0
$45$	−0.198997	−0.0296647
$46$	0	0
$47$	−7.14201	−1.04177	−0.520884	−	0.853627i	$-0.674398\pi$
$47$	−7.14201	−1.04177	−0.520884	+	0.853627i	$0.674398\pi$
$48$	0	0
$49$	10.4459	1.49227
$50$	0	0
$51$	−4.54891	−0.636975
$52$	0	0
$53$	0.0629449	0.00864615	0.00432307	−	0.999991i	$-0.498624\pi$
$53$	0.0629449	0.00864615	0.00432307	+	0.999991i	$0.498624\pi$
$54$	0	0
$55$	0.626895	0.0845305
$56$	0	0
$57$	7.17348	0.950150
$58$	0	0
$59$	7.16608	0.932945	0.466472	−	0.884536i	$-0.345525\pi$
$59$	7.16608	0.932945	0.466472	+	0.884536i	$0.345525\pi$
$60$	0	0
$61$	0.547121	0.0700517	0.0350258	−	0.999386i	$-0.488849\pi$
$61$	0.547121	0.0700517	0.0350258	+	0.999386i	$0.488849\pi$
$62$	0	0
$63$	4.17683	0.526231
$64$	0	0
$65$	0.448329	0.0556084
$66$	0	0
$67$	−0.268787	−0.0328375	−0.0164188	−	0.999865i	$-0.505226\pi$
$67$	−0.268787	−0.0328375	−0.0164188	+	0.999865i	$0.505226\pi$
$68$	0	0
$69$	−4.93189	−0.593730
$70$	0	0
$71$	5.33220	0.632816	0.316408	−	0.948623i	$-0.397523\pi$
$71$	5.33220	0.632816	0.316408	+	0.948623i	$0.397523\pi$
$72$	0	0
$73$	8.29308	0.970632	0.485316	−	0.874339i	$-0.338705\pi$
$73$	8.29308	0.970632	0.485316	+	0.874339i	$0.338705\pi$
$74$	0	0
$75$	4.96040	0.572778
$76$	0	0
$77$	−13.1581	−1.49951
$78$	0	0
$79$	−0.512972	−0.0577138	−0.0288569	−	0.999584i	$-0.509187\pi$
$79$	−0.512972	−0.0577138	−0.0288569	+	0.999584i	$0.509187\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.8697	1.52240	0.761200	−	0.648517i	$-0.224610\pi$
$83$	13.8697	1.52240	0.761200	+	0.648517i	$0.224610\pi$
$84$	0	0
$85$	−0.905220	−0.0981849
$86$	0	0
$87$	−2.19608	−0.235445
$88$	0	0
$89$	1.21408	0.128693	0.0643463	−	0.997928i	$-0.479504\pi$
$89$	1.21408	0.128693	0.0643463	+	0.997928i	$0.479504\pi$
$90$	0	0
$91$	−9.41015	−0.986451
$92$	0	0
$93$	−3.49112	−0.362012
$94$	0	0
$95$	1.42750	0.146459
$96$	0	0
$97$	16.8801	1.71391	0.856957	−	0.515389i	$-0.172352\pi$
$97$	16.8801	1.71391	0.856957	+	0.515389i	$0.172352\pi$
$98$	0	0
$99$	−3.15027	−0.316614
$100$	0	0
$101$	1.72270	0.171415	0.0857074	−	0.996320i	$-0.472685\pi$
$101$	1.72270	0.171415	0.0857074	+	0.996320i	$0.472685\pi$
$102$	0	0
$103$	−2.22594	−0.219328	−0.109664	−	0.993969i	$-0.534978\pi$
$103$	−2.22594	−0.219328	−0.109664	+	0.993969i	$0.534978\pi$
$104$	0	0
$105$	0.831177	0.0811145
$106$	0	0
$107$	−4.82892	−0.466829	−0.233415	−	0.972377i	$-0.574990\pi$
$107$	−4.82892	−0.466829	−0.233415	+	0.972377i	$0.574990\pi$
$108$	0	0
$109$	3.05638	0.292748	0.146374	−	0.989229i	$-0.453240\pi$
$109$	3.05638	0.292748	0.146374	+	0.989229i	$0.453240\pi$
$110$	0	0
$111$	−4.18756	−0.397466
$112$	0	0
$113$	13.4355	1.26390	0.631951	−	0.775008i	$-0.282254\pi$
$113$	13.4355	1.26390	0.631951	+	0.775008i	$0.282254\pi$
$114$	0	0
$115$	−0.981432	−0.0915190
$116$	0	0
$117$	−2.25294	−0.208284
$118$	0	0
$119$	19.0000	1.74173
$120$	0	0
$121$	−1.07579	−0.0977993
$122$	0	0
$123$	3.16436	0.285321
$124$	0	0
$125$	1.98209	0.177284
$126$	0	0
$127$	−17.3270	−1.53752	−0.768762	−	0.639534i	$-0.779127\pi$
$127$	−17.3270	−1.53752	−0.768762	+	0.639534i	$0.779127\pi$
$128$	0	0
$129$	−3.95535	−0.348249
$130$	0	0
$131$	−7.94832	−0.694448	−0.347224	−	0.937782i	$-0.612876\pi$
$131$	−7.94832	−0.694448	−0.347224	+	0.937782i	$0.612876\pi$
$132$	0	0
$133$	−29.9624	−2.59807
$134$	0	0
$135$	0.198997	0.0171269
$136$	0	0
$137$	10.2338	0.874328	0.437164	−	0.899382i	$-0.355983\pi$
$137$	10.2338	0.874328	0.437164	+	0.899382i	$0.355983\pi$
$138$	0	0
$139$	1.80111	0.152768	0.0763840	−	0.997078i	$-0.475663\pi$
$139$	1.80111	0.152768	0.0763840	+	0.997078i	$0.475663\pi$
$140$	0	0
$141$	7.14201	0.601465
$142$	0	0
$143$	7.09738	0.593512
$144$	0	0
$145$	−0.437014	−0.0362921
$146$	0	0
$147$	−10.4459	−0.861562
$148$	0	0
$149$	−8.00894	−0.656118	−0.328059	−	0.944657i	$-0.606395\pi$
$149$	−8.00894	−0.656118	−0.328059	+	0.944657i	$0.606395\pi$
$150$	0	0
$151$	−13.8225	−1.12486	−0.562429	−	0.826846i	$-0.690133\pi$
$151$	−13.8225	−1.12486	−0.562429	+	0.826846i	$0.690133\pi$
$152$	0	0
$153$	4.54891	0.367758
$154$	0	0
$155$	−0.694722	−0.0558014
$156$	0	0
$157$	11.0672	0.883255	0.441627	−	0.897199i	$-0.354401\pi$
$157$	11.0672	0.883255	0.441627	+	0.897199i	$0.354401\pi$
$158$	0	0
$159$	−0.0629449	−0.00499186
$160$	0	0
$161$	20.5996	1.62348
$162$	0	0
$163$	10.0025	0.783454	0.391727	−	0.920082i	$-0.371878\pi$
$163$	10.0025	0.783454	0.391727	+	0.920082i	$0.371878\pi$
$164$	0	0
$165$	−0.626895	−0.0488037
$166$	0	0
$167$	22.3266	1.72768	0.863842	−	0.503763i	$-0.168052\pi$
$167$	22.3266	1.72768	0.863842	+	0.503763i	$0.168052\pi$
$168$	0	0
$169$	−7.92426	−0.609558
$170$	0	0
$171$	−7.17348	−0.548569
$172$	0	0
$173$	1.02565	0.0779786	0.0389893	−	0.999240i	$-0.487586\pi$
$173$	1.02565	0.0779786	0.0389893	+	0.999240i	$0.487586\pi$
$174$	0	0
$175$	−20.7187	−1.56619
$176$	0	0
$177$	−7.16608	−0.538636
$178$	0	0
$179$	−5.22905	−0.390837	−0.195419	−	0.980720i	$-0.562607\pi$
$179$	−5.22905	−0.390837	−0.195419	+	0.980720i	$0.562607\pi$
$180$	0	0
$181$	−5.61564	−0.417407	−0.208704	−	0.977979i	$-0.566924\pi$
$181$	−5.61564	−0.417407	−0.208704	+	0.977979i	$0.566924\pi$
$182$	0	0
$183$	−0.547121	−0.0404443
$184$	0	0
$185$	−0.833313	−0.0612664
$186$	0	0
$187$	−14.3303	−1.04794
$188$	0	0
$189$	−4.17683	−0.303819
$190$	0	0
$191$	8.74408	0.632700	0.316350	−	0.948643i	$-0.397543\pi$
$191$	8.74408	0.632700	0.316350	+	0.948643i	$0.397543\pi$
$192$	0	0
$193$	7.81172	0.562300	0.281150	−	0.959664i	$-0.409284\pi$
$193$	7.81172	0.562300	0.281150	+	0.959664i	$0.409284\pi$
$194$	0	0
$195$	−0.448329	−0.0321055
$196$	0	0
$197$	−3.78723	−0.269829	−0.134914	−	0.990857i	$-0.543076\pi$
$197$	−3.78723	−0.269829	−0.134914	+	0.990857i	$0.543076\pi$
$198$	0	0
$199$	−13.7063	−0.971616	−0.485808	−	0.874066i	$-0.661474\pi$
$199$	−13.7063	−0.971616	−0.485808	+	0.874066i	$0.661474\pi$
$200$	0	0
$201$	0.268787	0.0189588
$202$	0	0
$203$	9.17266	0.643794
$204$	0	0
$205$	0.629699	0.0439801
$206$	0	0
$207$	4.93189	0.342790
$208$	0	0
$209$	22.5984	1.56316
$210$	0	0
$211$	−1.35191	−0.0930694	−0.0465347	−	0.998917i	$-0.514818\pi$
$211$	−1.35191	−0.0930694	−0.0465347	+	0.998917i	$0.514818\pi$
$212$	0	0
$213$	−5.33220	−0.365356
$214$	0	0
$215$	−0.787103	−0.0536800
$216$	0	0
$217$	14.5818	0.989876
$218$	0	0
$219$	−8.29308	−0.560395
$220$	0	0
$221$	−10.2484	−0.689384
$222$	0	0
$223$	−2.31466	−0.155001	−0.0775006	−	0.996992i	$-0.524694\pi$
$223$	−2.31466	−0.155001	−0.0775006	+	0.996992i	$0.524694\pi$
$224$	0	0
$225$	−4.96040	−0.330693
$226$	0	0
$227$	−6.62137	−0.439476	−0.219738	−	0.975559i	$-0.570520\pi$
$227$	−6.62137	−0.439476	−0.219738	+	0.975559i	$0.570520\pi$
$228$	0	0
$229$	−25.9218	−1.71296	−0.856480	−	0.516180i	$-0.827354\pi$
$229$	−25.9218	−1.71296	−0.856480	+	0.516180i	$0.827354\pi$
$230$	0	0
$231$	13.1581	0.865742
$232$	0	0
$233$	17.3199	1.13467	0.567333	−	0.823488i	$-0.307975\pi$
$233$	17.3199	1.13467	0.567333	+	0.823488i	$0.307975\pi$
$234$	0	0
$235$	1.42124	0.0927114
$236$	0	0
$237$	0.512972	0.0333211
$238$	0	0
$239$	12.4352	0.804366	0.402183	−	0.915559i	$-0.368251\pi$
$239$	12.4352	0.804366	0.402183	+	0.915559i	$0.368251\pi$
$240$	0	0
$241$	−18.0976	−1.16577	−0.582886	−	0.812554i	$-0.698076\pi$
$241$	−18.0976	−1.16577	−0.582886	+	0.812554i	$0.698076\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.07870	−0.132803
$246$	0	0
$247$	16.1614	1.02833
$248$	0	0
$249$	−13.8697	−0.878958
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−15.5368	−0.976789
$254$	0	0
$255$	0.905220	0.0566871
$256$	0	0
$257$	11.3975	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$257$	11.3975	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$258$	0	0
$259$	17.4907	1.08682
$260$	0	0
$261$	2.19608	0.135934
$262$	0	0
$263$	16.1156	0.993728	0.496864	−	0.867828i	$-0.334485\pi$
$263$	16.1156	0.993728	0.496864	+	0.867828i	$0.334485\pi$
$264$	0	0
$265$	−0.0125259	−0.000769457 0
$266$	0	0
$267$	−1.21408	−0.0743007
$268$	0	0
$269$	13.1491	0.801712	0.400856	−	0.916141i	$-0.368713\pi$
$269$	13.1491	0.801712	0.400856	+	0.916141i	$0.368713\pi$
$270$	0	0
$271$	7.86192	0.477578	0.238789	−	0.971072i	$-0.423250\pi$
$271$	7.86192	0.477578	0.238789	+	0.971072i	$0.423250\pi$
$272$	0	0
$273$	9.41015	0.569528
$274$	0	0
$275$	15.6266	0.942320
$276$	0	0
$277$	12.7694	0.767241	0.383620	−	0.923491i	$-0.374677\pi$
$277$	12.7694	0.767241	0.383620	+	0.923491i	$0.374677\pi$
$278$	0	0
$279$	3.49112	0.209008
$280$	0	0
$281$	21.2331	1.26666	0.633330	−	0.773882i	$-0.281687\pi$
$281$	21.2331	1.26666	0.633330	+	0.773882i	$0.281687\pi$
$282$	0	0
$283$	−3.86846	−0.229956	−0.114978	−	0.993368i	$-0.536680\pi$
$283$	−3.86846	−0.229956	−0.114978	+	0.993368i	$0.536680\pi$
$284$	0	0
$285$	−1.42750	−0.0845579
$286$	0	0
$287$	−13.2170	−0.780174
$288$	0	0
$289$	3.69260	0.217212
$290$	0	0
$291$	−16.8801	−0.989528
$292$	0	0
$293$	29.5898	1.72865	0.864327	−	0.502931i	$-0.167745\pi$
$293$	29.5898	1.72865	0.864327	+	0.502931i	$0.167745\pi$
$294$	0	0
$295$	−1.42603	−0.0830267
$296$	0	0
$297$	3.15027	0.182797
$298$	0	0
$299$	−11.1113	−0.642581
$300$	0	0
$301$	16.5208	0.952243
$302$	0	0
$303$	−1.72270	−0.0989663
$304$	0	0
$305$	−0.108876	−0.00623419
$306$	0	0
$307$	−1.55930	−0.0889942	−0.0444971	−	0.999010i	$-0.514169\pi$
$307$	−1.55930	−0.0889942	−0.0444971	+	0.999010i	$0.514169\pi$
$308$	0	0
$309$	2.22594	0.126629
$310$	0	0
$311$	13.4807	0.764419	0.382210	−	0.924076i	$-0.375163\pi$
$311$	13.4807	0.764419	0.382210	+	0.924076i	$0.375163\pi$
$312$	0	0
$313$	21.4596	1.21297	0.606484	−	0.795096i	$-0.292580\pi$
$313$	21.4596	1.21297	0.606484	+	0.795096i	$0.292580\pi$
$314$	0	0
$315$	−0.831177	−0.0468315
$316$	0	0
$317$	30.7668	1.72804	0.864018	−	0.503461i	$-0.167940\pi$
$317$	30.7668	1.72804	0.864018	+	0.503461i	$0.167940\pi$
$318$	0	0
$319$	−6.91826	−0.387348
$320$	0	0
$321$	4.82892	0.269524
$322$	0	0
$323$	−32.6315	−1.81567
$324$	0	0
$325$	11.1755	0.619905
$326$	0	0
$327$	−3.05638	−0.169018
$328$	0	0
$329$	−29.8309	−1.64463
$330$	0	0
$331$	−8.73735	−0.480248	−0.240124	−	0.970742i	$-0.577188\pi$
$331$	−8.73735	−0.480248	−0.240124	+	0.970742i	$0.577188\pi$
$332$	0	0
$333$	4.18756	0.229477
$334$	0	0
$335$	0.0534878	0.00292235
$336$	0	0
$337$	13.6628	0.744258	0.372129	−	0.928181i	$-0.378628\pi$
$337$	13.6628	0.744258	0.372129	+	0.928181i	$0.378628\pi$
$338$	0	0
$339$	−13.4355	−0.729715
$340$	0	0
$341$	−10.9980	−0.595573
$342$	0	0
$343$	14.3929	0.777141
$344$	0	0
$345$	0.981432	0.0528385
$346$	0	0
$347$	−1.61590	−0.0867462	−0.0433731	−	0.999059i	$-0.513810\pi$
$347$	−1.61590	−0.0867462	−0.0433731	+	0.999059i	$0.513810\pi$
$348$	0	0
$349$	−14.4763	−0.774900	−0.387450	−	0.921891i	$-0.626644\pi$
$349$	−14.4763	−0.774900	−0.387450	+	0.921891i	$0.626644\pi$
$350$	0	0
$351$	2.25294	0.120253
$352$	0	0
$353$	15.5057	0.825284	0.412642	−	0.910893i	$-0.364606\pi$
$353$	15.5057	0.825284	0.412642	+	0.910893i	$0.364606\pi$
$354$	0	0
$355$	−1.06109	−0.0563170
$356$	0	0
$357$	−19.0000	−1.00559
$358$	0	0
$359$	−8.03444	−0.424042	−0.212021	−	0.977265i	$-0.568004\pi$
$359$	−8.03444	−0.424042	−0.212021	+	0.977265i	$0.568004\pi$
$360$	0	0
$361$	32.4588	1.70836
$362$	0	0
$363$	1.07579	0.0564644
$364$	0	0
$365$	−1.65030	−0.0863806
$366$	0	0
$367$	2.92106	0.152478	0.0762391	−	0.997090i	$-0.475709\pi$
$367$	2.92106	0.152478	0.0762391	+	0.997090i	$0.475709\pi$
$368$	0	0
$369$	−3.16436	−0.164730
$370$	0	0
$371$	0.262910	0.0136496
$372$	0	0
$373$	22.8340	1.18230	0.591151	−	0.806561i	$-0.298674\pi$
$373$	22.8340	1.18230	0.591151	+	0.806561i	$0.298674\pi$
$374$	0	0
$375$	−1.98209	−0.102355
$376$	0	0
$377$	−4.94765	−0.254817
$378$	0	0
$379$	−32.1900	−1.65349	−0.826744	−	0.562578i	$-0.809810\pi$
$379$	−32.1900	−1.65349	−0.826744	+	0.562578i	$0.809810\pi$
$380$	0	0
$381$	17.3270	0.887690
$382$	0	0
$383$	−14.4608	−0.738910	−0.369455	−	0.929249i	$-0.620456\pi$
$383$	−14.4608	−0.738910	−0.369455	+	0.929249i	$0.620456\pi$
$384$	0	0
$385$	2.61843	0.133448
$386$	0	0
$387$	3.95535	0.201062
$388$	0	0
$389$	12.7873	0.648340	0.324170	−	0.945999i	$-0.394915\pi$
$389$	12.7873	0.648340	0.324170	+	0.945999i	$0.394915\pi$
$390$	0	0
$391$	22.4347	1.13457
$392$	0	0
$393$	7.94832	0.400940
$394$	0	0
$395$	0.102080	0.00513620
$396$	0	0
$397$	24.3813	1.22366	0.611831	−	0.790989i	$-0.290433\pi$
$397$	24.3813	1.22366	0.611831	+	0.790989i	$0.290433\pi$
$398$	0	0
$399$	29.9624	1.49999
$400$	0	0
$401$	16.7575	0.836830	0.418415	−	0.908256i	$-0.362586\pi$
$401$	16.7575	0.836830	0.418415	+	0.908256i	$0.362586\pi$
$402$	0	0
$403$	−7.86528	−0.391797
$404$	0	0
$405$	−0.198997	−0.00988825
$406$	0	0
$407$	−13.1920	−0.653901
$408$	0	0
$409$	37.7099	1.86464	0.932318	−	0.361639i	$-0.117783\pi$
$409$	37.7099	1.86464	0.932318	+	0.361639i	$0.117783\pi$
$410$	0	0
$411$	−10.2338	−0.504794
$412$	0	0
$413$	29.9315	1.47283
$414$	0	0
$415$	−2.76004	−0.135485
$416$	0	0
$417$	−1.80111	−0.0882007
$418$	0	0
$419$	−16.3657	−0.799516	−0.399758	−	0.916621i	$-0.630906\pi$
$419$	−16.3657	−0.799516	−0.399758	+	0.916621i	$0.630906\pi$
$420$	0	0
$421$	25.0416	1.22046	0.610228	−	0.792226i	$-0.291078\pi$
$421$	25.0416	1.22046	0.610228	+	0.792226i	$0.291078\pi$
$422$	0	0
$423$	−7.14201	−0.347256
$424$	0	0
$425$	−22.5644	−1.09454
$426$	0	0
$427$	2.28523	0.110590
$428$	0	0
$429$	−7.09738	−0.342665
$430$	0	0
$431$	−35.2488	−1.69788	−0.848938	−	0.528493i	$-0.822757\pi$
$431$	−35.2488	−1.69788	−0.848938	+	0.528493i	$0.822757\pi$
$432$	0	0
$433$	30.9824	1.48892	0.744461	−	0.667667i	$-0.232707\pi$
$433$	30.9824	1.48892	0.744461	+	0.667667i	$0.232707\pi$
$434$	0	0
$435$	0.437014	0.0209532
$436$	0	0
$437$	−35.3788	−1.69240
$438$	0	0
$439$	−2.75573	−0.131524	−0.0657619	−	0.997835i	$-0.520948\pi$
$439$	−2.75573	−0.131524	−0.0657619	+	0.997835i	$0.520948\pi$
$440$	0	0
$441$	10.4459	0.497423
$442$	0	0
$443$	38.9175	1.84903	0.924513	−	0.381151i	$-0.124472\pi$
$443$	38.9175	1.84903	0.924513	+	0.381151i	$0.124472\pi$
$444$	0	0
$445$	−0.241599	−0.0114529
$446$	0	0
$447$	8.00894	0.378810
$448$	0	0
$449$	−24.7968	−1.17023	−0.585117	−	0.810949i	$-0.698952\pi$
$449$	−24.7968	−1.17023	−0.585117	+	0.810949i	$0.698952\pi$
$450$	0	0
$451$	9.96860	0.469403
$452$	0	0
$453$	13.8225	0.649437
$454$	0	0
$455$	1.87259	0.0877885
$456$	0	0
$457$	−16.9705	−0.793848	−0.396924	−	0.917852i	$-0.629922\pi$
$457$	−16.9705	−0.793848	−0.396924	+	0.917852i	$0.629922\pi$
$458$	0	0
$459$	−4.54891	−0.212325
$460$	0	0
$461$	15.8774	0.739483	0.369742	−	0.929135i	$-0.379446\pi$
$461$	15.8774	0.739483	0.369742	+	0.929135i	$0.379446\pi$
$462$	0	0
$463$	−30.0369	−1.39593	−0.697967	−	0.716130i	$-0.745912\pi$
$463$	−30.0369	−1.39593	−0.697967	+	0.716130i	$0.745912\pi$
$464$	0	0
$465$	0.694722	0.0322170
$466$	0	0
$467$	20.2569	0.937378	0.468689	−	0.883363i	$-0.344726\pi$
$467$	20.2569	0.937378	0.468689	+	0.883363i	$0.344726\pi$
$468$	0	0
$469$	−1.12268	−0.0518404
$470$	0	0
$471$	−11.0672	−0.509948
$472$	0	0
$473$	−12.4604	−0.572931
$474$	0	0
$475$	35.5833	1.63267
$476$	0	0
$477$	0.0629449	0.00288205
$478$	0	0
$479$	0.255404	0.0116697	0.00583486	−	0.999983i	$-0.498143\pi$
$479$	0.255404	0.0116697	0.00583486	+	0.999983i	$0.498143\pi$
$480$	0	0
$481$	−9.43433	−0.430168
$482$	0	0
$483$	−20.5996	−0.937316
$484$	0	0
$485$	−3.35909	−0.152528
$486$	0	0
$487$	34.2932	1.55397	0.776987	−	0.629517i	$-0.216747\pi$
$487$	34.2932	1.55397	0.776987	+	0.629517i	$0.216747\pi$
$488$	0	0
$489$	−10.0025	−0.452327
$490$	0	0
$491$	0.832284	0.0375605	0.0187802	−	0.999824i	$-0.494022\pi$
$491$	0.832284	0.0375605	0.0187802	+	0.999824i	$0.494022\pi$
$492$	0	0
$493$	9.98979	0.449917
$494$	0	0
$495$	0.626895	0.0281768
$496$	0	0
$497$	22.2717	0.999021
$498$	0	0
$499$	13.8565	0.620304	0.310152	−	0.950687i	$-0.399620\pi$
$499$	13.8565	0.620304	0.310152	+	0.950687i	$0.399620\pi$
$500$	0	0
$501$	−22.3266	−0.997478
$502$	0	0
$503$	10.6072	0.472951	0.236475	−	0.971637i	$-0.424008\pi$
$503$	10.6072	0.472951	0.236475	+	0.971637i	$0.424008\pi$
$504$	0	0
$505$	−0.342812	−0.0152549
$506$	0	0
$507$	7.92426	0.351929
$508$	0	0
$509$	11.5310	0.511104	0.255552	−	0.966795i	$-0.417743\pi$
$509$	11.5310	0.511104	0.255552	+	0.966795i	$0.417743\pi$
$510$	0	0
$511$	34.6388	1.53233
$512$	0	0
$513$	7.17348	0.316717
$514$	0	0
$515$	0.442955	0.0195189
$516$	0	0
$517$	22.4993	0.989516
$518$	0	0
$519$	−1.02565	−0.0450209
$520$	0	0
$521$	37.9279	1.66165	0.830826	−	0.556532i	$-0.187868\pi$
$521$	37.9279	1.66165	0.830826	+	0.556532i	$0.187868\pi$
$522$	0	0
$523$	18.2703	0.798903	0.399452	−	0.916754i	$-0.369201\pi$
$523$	18.2703	0.798903	0.399452	+	0.916754i	$0.369201\pi$
$524$	0	0
$525$	20.7187	0.904240
$526$	0	0
$527$	15.8808	0.691777
$528$	0	0
$529$	1.32353	0.0575448
$530$	0	0
$531$	7.16608	0.310982
$532$	0	0
$533$	7.12912	0.308796
$534$	0	0
$535$	0.960941	0.0415451
$536$	0	0
$537$	5.22905	0.225650
$538$	0	0
$539$	−32.9074	−1.41742
$540$	0	0
$541$	6.10806	0.262606	0.131303	−	0.991342i	$-0.458084\pi$
$541$	6.10806	0.262606	0.131303	+	0.991342i	$0.458084\pi$
$542$	0	0
$543$	5.61564	0.240990
$544$	0	0
$545$	−0.608212	−0.0260529
$546$	0	0
$547$	3.41332	0.145943	0.0729715	−	0.997334i	$-0.476752\pi$
$547$	3.41332	0.145943	0.0729715	+	0.997334i	$0.476752\pi$
$548$	0	0
$549$	0.547121	0.0233506
$550$	0	0
$551$	−15.7536	−0.671124
$552$	0	0
$553$	−2.14259	−0.0911123
$554$	0	0
$555$	0.833313	0.0353722
$556$	0	0
$557$	12.1414	0.514448	0.257224	−	0.966352i	$-0.417192\pi$
$557$	12.1414	0.514448	0.257224	+	0.966352i	$0.417192\pi$
$558$	0	0
$559$	−8.91116	−0.376902
$560$	0	0
$561$	14.3303	0.605026
$562$	0	0
$563$	9.14187	0.385284	0.192642	−	0.981269i	$-0.438294\pi$
$563$	9.14187	0.385284	0.192642	+	0.981269i	$0.438294\pi$
$564$	0	0
$565$	−2.67362	−0.112480
$566$	0	0
$567$	4.17683	0.175410
$568$	0	0
$569$	−16.4144	−0.688129	−0.344065	−	0.938946i	$-0.611804\pi$
$569$	−16.4144	−0.688129	−0.344065	+	0.938946i	$0.611804\pi$
$570$	0	0
$571$	−23.7852	−0.995378	−0.497689	−	0.867356i	$-0.665818\pi$
$571$	−23.7852	−0.995378	−0.497689	+	0.867356i	$0.665818\pi$
$572$	0	0
$573$	−8.74408	−0.365289
$574$	0	0
$575$	−24.4641	−1.02023
$576$	0	0
$577$	−0.983060	−0.0409253	−0.0204627	−	0.999791i	$-0.506514\pi$
$577$	−0.983060	−0.0409253	−0.0204627	+	0.999791i	$0.506514\pi$
$578$	0	0
$579$	−7.81172	−0.324644
$580$	0	0
$581$	57.9315	2.40340
$582$	0	0
$583$	−0.198294	−0.00821248
$584$	0	0
$585$	0.448329	0.0185361
$586$	0	0
$587$	1.21459	0.0501313	0.0250657	−	0.999686i	$-0.492021\pi$
$587$	1.21459	0.0501313	0.0250657	+	0.999686i	$0.492021\pi$
$588$	0	0
$589$	−25.0434	−1.03190
$590$	0	0
$591$	3.78723	0.155786
$592$	0	0
$593$	40.3903	1.65863	0.829315	−	0.558781i	$-0.188731\pi$
$593$	40.3903	1.65863	0.829315	+	0.558781i	$0.188731\pi$
$594$	0	0
$595$	−3.78095	−0.155004
$596$	0	0
$597$	13.7063	0.560962
$598$	0	0
$599$	−11.8088	−0.482494	−0.241247	−	0.970464i	$-0.577556\pi$
$599$	−11.8088	−0.482494	−0.241247	+	0.970464i	$0.577556\pi$
$600$	0	0
$601$	−40.5448	−1.65386	−0.826928	−	0.562307i	$-0.809914\pi$
$601$	−40.5448	−1.65386	−0.826928	+	0.562307i	$0.809914\pi$
$602$	0	0
$603$	−0.268787	−0.0109458
$604$	0	0
$605$	0.214080	0.00870357
$606$	0	0
$607$	−37.0879	−1.50535	−0.752676	−	0.658391i	$-0.771237\pi$
$607$	−37.0879	−1.50535	−0.752676	+	0.658391i	$0.771237\pi$
$608$	0	0
$609$	−9.17266	−0.371695
$610$	0	0
$611$	16.0905	0.650953
$612$	0	0
$613$	−20.3062	−0.820158	−0.410079	−	0.912050i	$-0.634499\pi$
$613$	−20.3062	−0.820158	−0.410079	+	0.912050i	$0.634499\pi$
$614$	0	0
$615$	−0.629699	−0.0253919
$616$	0	0
$617$	−30.1181	−1.21251	−0.606254	−	0.795271i	$-0.707328\pi$
$617$	−30.1181	−1.21251	−0.606254	+	0.795271i	$0.707328\pi$
$618$	0	0
$619$	26.9068	1.08147	0.540737	−	0.841191i	$-0.318145\pi$
$619$	26.9068	1.08147	0.540737	+	0.841191i	$0.318145\pi$
$620$	0	0
$621$	−4.93189	−0.197910
$622$	0	0
$623$	5.07101	0.203166
$624$	0	0
$625$	24.4076	0.976303
$626$	0	0
$627$	−22.5984	−0.902493
$628$	0	0
$629$	19.0488	0.759527
$630$	0	0
$631$	−6.58424	−0.262115	−0.131057	−	0.991375i	$-0.541837\pi$
$631$	−6.58424	−0.262115	−0.131057	+	0.991375i	$0.541837\pi$
$632$	0	0
$633$	1.35191	0.0537336
$634$	0	0
$635$	3.44803	0.136831
$636$	0	0
$637$	−23.5340	−0.932449
$638$	0	0
$639$	5.33220	0.210939
$640$	0	0
$641$	39.5375	1.56164	0.780818	−	0.624758i	$-0.214802\pi$
$641$	39.5375	1.56164	0.780818	+	0.624758i	$0.214802\pi$
$642$	0	0
$643$	−4.05093	−0.159753	−0.0798766	−	0.996805i	$-0.525453\pi$
$643$	−4.05093	−0.159753	−0.0798766	+	0.996805i	$0.525453\pi$
$644$	0	0
$645$	0.787103	0.0309921
$646$	0	0
$647$	−26.2087	−1.03037	−0.515185	−	0.857079i	$-0.672277\pi$
$647$	−26.2087	−1.03037	−0.515185	+	0.857079i	$0.672277\pi$
$648$	0	0
$649$	−22.5751	−0.886150
$650$	0	0
$651$	−14.5818	−0.571505
$652$	0	0
$653$	−39.8685	−1.56017	−0.780087	−	0.625671i	$-0.784825\pi$
$653$	−39.8685	−1.56017	−0.780087	+	0.625671i	$0.784825\pi$
$654$	0	0
$655$	1.58169	0.0618019
$656$	0	0
$657$	8.29308	0.323544
$658$	0	0
$659$	−29.2316	−1.13870	−0.569350	−	0.822095i	$-0.692805\pi$
$659$	−29.2316	−1.13870	−0.569350	+	0.822095i	$0.692805\pi$
$660$	0	0
$661$	18.3041	0.711945	0.355973	−	0.934496i	$-0.384150\pi$
$661$	18.3041	0.711945	0.355973	+	0.934496i	$0.384150\pi$
$662$	0	0
$663$	10.2484	0.398016
$664$	0	0
$665$	5.96243	0.231213
$666$	0	0
$667$	10.8308	0.419372
$668$	0	0
$669$	2.31466	0.0894899
$670$	0	0
$671$	−1.72358	−0.0665380
$672$	0	0
$673$	−38.2487	−1.47438	−0.737189	−	0.675686i	$-0.763847\pi$
$673$	−38.2487	−1.47438	−0.737189	+	0.675686i	$0.763847\pi$
$674$	0	0
$675$	4.96040	0.190926
$676$	0	0
$677$	−22.5806	−0.867842	−0.433921	−	0.900951i	$-0.642870\pi$
$677$	−22.5806	−0.867842	−0.433921	+	0.900951i	$0.642870\pi$
$678$	0	0
$679$	70.5052	2.70574
$680$	0	0
$681$	6.62137	0.253732
$682$	0	0
$683$	−11.3213	−0.433198	−0.216599	−	0.976261i	$-0.569496\pi$
$683$	−11.3213	−0.433198	−0.216599	+	0.976261i	$0.569496\pi$
$684$	0	0
$685$	−2.03649	−0.0778102
$686$	0	0
$687$	25.9218	0.988978
$688$	0	0
$689$	−0.141811	−0.00540258
$690$	0	0
$691$	−2.40088	−0.0913337	−0.0456668	−	0.998957i	$-0.514541\pi$
$691$	−2.40088	−0.0913337	−0.0456668	+	0.998957i	$0.514541\pi$
$692$	0	0
$693$	−13.1581	−0.499836
$694$	0	0
$695$	−0.358415	−0.0135955
$696$	0	0
$697$	−14.3944	−0.545227
$698$	0	0
$699$	−17.3199	−0.655100
$700$	0	0
$701$	−6.83739	−0.258245	−0.129122	−	0.991629i	$-0.541216\pi$
$701$	−6.83739	−0.258245	−0.129122	+	0.991629i	$0.541216\pi$
$702$	0	0
$703$	−30.0394	−1.13296
$704$	0	0
$705$	−1.42124	−0.0535270
$706$	0	0
$707$	7.19540	0.270611
$708$	0	0
$709$	−39.2873	−1.47546	−0.737732	−	0.675094i	$-0.764103\pi$
$709$	−39.2873	−1.47546	−0.737732	+	0.675094i	$0.764103\pi$
$710$	0	0
$711$	−0.512972	−0.0192379
$712$	0	0
$713$	17.2178	0.644811
$714$	0	0
$715$	−1.41236	−0.0528192
$716$	0	0
$717$	−12.4352	−0.464401
$718$	0	0
$719$	−1.23447	−0.0460378	−0.0230189	−	0.999735i	$-0.507328\pi$
$719$	−1.23447	−0.0460378	−0.0230189	+	0.999735i	$0.507328\pi$
$720$	0	0
$721$	−9.29736	−0.346252
$722$	0	0
$723$	18.0976	0.673058
$724$	0	0
$725$	−10.8935	−0.404573
$726$	0	0
$727$	−31.1611	−1.15570	−0.577850	−	0.816143i	$-0.696108\pi$
$727$	−31.1611	−1.15570	−0.577850	+	0.816143i	$0.696108\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.9925	0.665478
$732$	0	0
$733$	−2.63406	−0.0972913	−0.0486456	−	0.998816i	$-0.515490\pi$
$733$	−2.63406	−0.0972913	−0.0486456	+	0.998816i	$0.515490\pi$
$734$	0	0
$735$	2.07870	0.0766740
$736$	0	0
$737$	0.846751	0.0311905
$738$	0	0
$739$	−26.8967	−0.989411	−0.494706	−	0.869061i	$-0.664724\pi$
$739$	−26.8967	−0.989411	−0.494706	+	0.869061i	$0.664724\pi$
$740$	0	0
$741$	−16.1614	−0.593705
$742$	0	0
$743$	−20.8526	−0.765009	−0.382505	−	0.923954i	$-0.624938\pi$
$743$	−20.8526	−0.765009	−0.382505	+	0.923954i	$0.624938\pi$
$744$	0	0
$745$	1.59376	0.0583907
$746$	0	0
$747$	13.8697	0.507467
$748$	0	0
$749$	−20.1696	−0.736980
$750$	0	0
$751$	−37.3712	−1.36369	−0.681847	−	0.731495i	$-0.738823\pi$
$751$	−37.3712	−1.36369	−0.681847	+	0.731495i	$0.738823\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	2.75063	0.100106
$756$	0	0
$757$	29.2784	1.06414	0.532070	−	0.846700i	$-0.321414\pi$
$757$	29.2784	1.06414	0.532070	+	0.846700i	$0.321414\pi$
$758$	0	0
$759$	15.5368	0.563950
$760$	0	0
$761$	−34.2832	−1.24277	−0.621383	−	0.783507i	$-0.713429\pi$
$761$	−34.2832	−1.24277	−0.621383	+	0.783507i	$0.713429\pi$
$762$	0	0
$763$	12.7660	0.462160
$764$	0	0
$765$	−0.905220	−0.0327283
$766$	0	0
$767$	−16.1448	−0.582954
$768$	0	0
$769$	−50.7820	−1.83125	−0.915624	−	0.402036i	$-0.868303\pi$
$769$	−50.7820	−1.83125	−0.915624	+	0.402036i	$0.868303\pi$
$770$	0	0
$771$	−11.3975	−0.410471
$772$	0	0
$773$	38.2862	1.37706	0.688529	−	0.725209i	$-0.258257\pi$
$773$	38.2862	1.37706	0.688529	+	0.725209i	$0.258257\pi$
$774$	0	0
$775$	−17.3173	−0.622057
$776$	0	0
$777$	−17.4907	−0.627476
$778$	0	0
$779$	22.6995	0.813293
$780$	0	0
$781$	−16.7979	−0.601075
$782$	0	0
$783$	−2.19608	−0.0784816
$784$	0	0
$785$	−2.20233	−0.0786046
$786$	0	0
$787$	5.23261	0.186522	0.0932612	−	0.995642i	$-0.470271\pi$
$787$	5.23261	0.186522	0.0932612	+	0.995642i	$0.470271\pi$
$788$	0	0
$789$	−16.1156	−0.573729
$790$	0	0
$791$	56.1176	1.99531
$792$	0	0
$793$	−1.23263	−0.0437720
$794$	0	0
$795$	0.0125259	0.000444246 0
$796$	0	0
$797$	29.5870	1.04802	0.524012	−	0.851711i	$-0.324435\pi$
$797$	29.5870	1.04802	0.524012	+	0.851711i	$0.324435\pi$
$798$	0	0
$799$	−32.4884	−1.14936
$800$	0	0
$801$	1.21408	0.0428975
$802$	0	0
$803$	−26.1255	−0.921947
$804$	0	0
$805$	−4.09927	−0.144480
$806$	0	0
$807$	−13.1491	−0.462869
$808$	0	0
$809$	−33.9588	−1.19393	−0.596965	−	0.802267i	$-0.703627\pi$
$809$	−33.9588	−1.19393	−0.596965	+	0.802267i	$0.703627\pi$
$810$	0	0
$811$	53.0609	1.86322	0.931609	−	0.363461i	$-0.118405\pi$
$811$	53.0609	1.86322	0.931609	+	0.363461i	$0.118405\pi$
$812$	0	0
$813$	−7.86192	−0.275730
$814$	0	0
$815$	−1.99046	−0.0697229
$816$	0	0
$817$	−28.3736	−0.992666
$818$	0	0
$819$	−9.41015	−0.328817
$820$	0	0
$821$	−23.9973	−0.837512	−0.418756	−	0.908099i	$-0.637534\pi$
$821$	−23.9973	−0.837512	−0.418756	+	0.908099i	$0.637534\pi$
$822$	0	0
$823$	−3.25718	−0.113538	−0.0567691	−	0.998387i	$-0.518080\pi$
$823$	−3.25718	−0.113538	−0.0567691	+	0.998387i	$0.518080\pi$
$824$	0	0
$825$	−15.6266	−0.544049
$826$	0	0
$827$	−47.2165	−1.64188	−0.820939	−	0.571016i	$-0.806549\pi$
$827$	−47.2165	−1.64188	−0.820939	+	0.571016i	$0.806549\pi$
$828$	0	0
$829$	−6.54091	−0.227175	−0.113588	−	0.993528i	$-0.536234\pi$
$829$	−6.54091	−0.227175	−0.113588	+	0.993528i	$0.536234\pi$
$830$	0	0
$831$	−12.7694	−0.442967
$832$	0	0
$833$	47.5174	1.64638
$834$	0	0
$835$	−4.44293	−0.153754
$836$	0	0
$837$	−3.49112	−0.120671
$838$	0	0
$839$	−47.3504	−1.63472	−0.817359	−	0.576129i	$-0.804563\pi$
$839$	−47.3504	−1.63472	−0.817359	+	0.576129i	$0.804563\pi$
$840$	0	0
$841$	−24.1772	−0.833697
$842$	0	0
$843$	−21.2331	−0.731307
$844$	0	0
$845$	1.57690	0.0542472
$846$	0	0
$847$	−4.49340	−0.154395
$848$	0	0
$849$	3.86846	0.132765
$850$	0	0
$851$	20.6526	0.707962
$852$	0	0
$853$	−34.0676	−1.16645	−0.583227	−	0.812309i	$-0.698210\pi$
$853$	−34.0676	−1.16645	−0.583227	+	0.812309i	$0.698210\pi$
$854$	0	0
$855$	1.42750	0.0488195
$856$	0	0
$857$	25.1592	0.859421	0.429710	−	0.902967i	$-0.358616\pi$
$857$	25.1592	0.859421	0.429710	+	0.902967i	$0.358616\pi$
$858$	0	0
$859$	29.2762	0.998890	0.499445	−	0.866346i	$-0.333537\pi$
$859$	29.2762	0.998890	0.499445	+	0.866346i	$0.333537\pi$
$860$	0	0
$861$	13.2170	0.450434
$862$	0	0
$863$	−6.32136	−0.215182	−0.107591	−	0.994195i	$-0.534314\pi$
$863$	−6.32136	−0.215182	−0.107591	+	0.994195i	$0.534314\pi$
$864$	0	0
$865$	−0.204101	−0.00693964
$866$	0	0
$867$	−3.69260	−0.125407
$868$	0	0
$869$	1.61600	0.0548190
$870$	0	0
$871$	0.605561	0.0205186
$872$	0	0
$873$	16.8801	0.571304
$874$	0	0
$875$	8.27885	0.279876
$876$	0	0
$877$	−40.7072	−1.37459	−0.687293	−	0.726380i	$-0.741201\pi$
$877$	−40.7072	−1.37459	−0.687293	+	0.726380i	$0.741201\pi$
$878$	0	0
$879$	−29.5898	−0.998039
$880$	0	0
$881$	−19.7440	−0.665192	−0.332596	−	0.943069i	$-0.607925\pi$
$881$	−19.7440	−0.665192	−0.332596	+	0.943069i	$0.607925\pi$
$882$	0	0
$883$	19.8135	0.666778	0.333389	−	0.942789i	$-0.391808\pi$
$883$	19.8135	0.666778	0.333389	+	0.942789i	$0.391808\pi$
$884$	0	0
$885$	1.42603	0.0479355
$886$	0	0
$887$	−4.81369	−0.161628	−0.0808139	−	0.996729i	$-0.525752\pi$
$887$	−4.81369	−0.161628	−0.0808139	+	0.996729i	$0.525752\pi$
$888$	0	0
$889$	−72.3720	−2.42728
$890$	0	0
$891$	−3.15027	−0.105538
$892$	0	0
$893$	51.2330	1.71445
$894$	0	0
$895$	1.04057	0.0347823
$896$	0	0
$897$	11.1113	0.370994
$898$	0	0
$899$	7.66678	0.255701
$900$	0	0
$901$	0.286331	0.00953906
$902$	0	0
$903$	−16.5208	−0.549778
$904$	0	0
$905$	1.11750	0.0371468
$906$	0	0
$907$	−2.68217	−0.0890598	−0.0445299	−	0.999008i	$-0.514179\pi$
$907$	−2.68217	−0.0890598	−0.0445299	+	0.999008i	$0.514179\pi$
$908$	0	0
$909$	1.72270	0.0571382
$910$	0	0
$911$	42.0890	1.39447	0.697235	−	0.716842i	$-0.254413\pi$
$911$	42.0890	1.39447	0.697235	+	0.716842i	$0.254413\pi$
$912$	0	0
$913$	−43.6934	−1.44604
$914$	0	0
$915$	0.108876	0.00359931
$916$	0	0
$917$	−33.1988	−1.09632
$918$	0	0
$919$	40.4880	1.33558	0.667788	−	0.744351i	$-0.267241\pi$
$919$	40.4880	1.33558	0.667788	+	0.744351i	$0.267241\pi$
$920$	0	0
$921$	1.55930	0.0513808
$922$	0	0
$923$	−12.0131	−0.395417
$924$	0	0
$925$	−20.7720	−0.682979
$926$	0	0
$927$	−2.22594	−0.0731094
$928$	0	0
$929$	−16.6248	−0.545443	−0.272722	−	0.962093i	$-0.587924\pi$
$929$	−16.6248	−0.545443	−0.272722	+	0.962093i	$0.587924\pi$
$930$	0	0
$931$	−74.9333	−2.45584
$932$	0	0
$933$	−13.4807	−0.441338
$934$	0	0
$935$	2.85169	0.0932602
$936$	0	0
$937$	54.0954	1.76722	0.883610	−	0.468224i	$-0.155106\pi$
$937$	54.0954	1.76722	0.883610	+	0.468224i	$0.155106\pi$
$938$	0	0
$939$	−21.4596	−0.700307
$940$	0	0
$941$	−22.1565	−0.722281	−0.361141	−	0.932511i	$-0.617613\pi$
$941$	−22.1565	−0.722281	−0.361141	+	0.932511i	$0.617613\pi$
$942$	0	0
$943$	−15.6063	−0.508210
$944$	0	0
$945$	0.831177	0.0270382
$946$	0	0
$947$	−21.4686	−0.697637	−0.348819	−	0.937190i	$-0.613417\pi$
$947$	−21.4686	−0.697637	−0.348819	+	0.937190i	$0.613417\pi$
$948$	0	0
$949$	−18.6838	−0.606503
$950$	0	0
$951$	−30.7668	−0.997682
$952$	0	0
$953$	8.11097	0.262740	0.131370	−	0.991333i	$-0.458062\pi$
$953$	8.11097	0.262740	0.131370	+	0.991333i	$0.458062\pi$
$954$	0	0
$955$	−1.74005	−0.0563066
$956$	0	0
$957$	6.91826	0.223636
$958$	0	0
$959$	42.7446	1.38030
$960$	0	0
$961$	−18.8121	−0.606842
$962$	0	0
$963$	−4.82892	−0.155610
$964$	0	0
$965$	−1.55451	−0.0500415
$966$	0	0
$967$	−18.4764	−0.594162	−0.297081	−	0.954852i	$-0.596013\pi$
$967$	−18.4764	−0.594162	−0.297081	+	0.954852i	$0.596013\pi$
$968$	0	0
$969$	32.6315	1.04828
$970$	0	0
$971$	7.82442	0.251098	0.125549	−	0.992087i	$-0.459931\pi$
$971$	7.82442	0.251098	0.125549	+	0.992087i	$0.459931\pi$
$972$	0	0
$973$	7.52292	0.241174
$974$	0	0
$975$	−11.1755	−0.357902
$976$	0	0
$977$	−14.7563	−0.472097	−0.236049	−	0.971741i	$-0.575852\pi$
$977$	−14.7563	−0.472097	−0.236049	+	0.971741i	$0.575852\pi$
$978$	0	0
$979$	−3.82469	−0.122238
$980$	0	0
$981$	3.05638	0.0975828
$982$	0	0
$983$	14.7794	0.471388	0.235694	−	0.971827i	$-0.424264\pi$
$983$	14.7794	0.471388	0.235694	+	0.971827i	$0.424264\pi$
$984$	0	0
$985$	0.753647	0.0240132
$986$	0	0
$987$	29.8309	0.949529
$988$	0	0
$989$	19.5073	0.620297
$990$	0	0
$991$	26.0166	0.826446	0.413223	−	0.910630i	$-0.364403\pi$
$991$	26.0166	0.826446	0.413223	+	0.910630i	$0.364403\pi$
$992$	0	0
$993$	8.73735	0.277272
$994$	0	0
$995$	2.72752	0.0864682
$996$	0	0
$997$	−26.5337	−0.840330	−0.420165	−	0.907448i	$-0.638028\pi$
$997$	−26.5337	−0.840330	−0.420165	+	0.907448i	$0.638028\pi$
$998$	0	0
$999$	−4.18756	−0.132489

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.p.1.7	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.p.1.7	✓	14	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.198997 q^{5} +4.17683 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.198997 q^{5} +4.17683 q^{7} +1.00000 q^{9} -3.15027 q^{11} -2.25294 q^{13} +0.198997 q^{15} +4.54891 q^{17} -7.17348 q^{19} -4.17683 q^{21} +4.93189 q^{23} -4.96040 q^{25} -1.00000 q^{27} +2.19608 q^{29} +3.49112 q^{31} +3.15027 q^{33} -0.831177 q^{35} +4.18756 q^{37} +2.25294 q^{39} -3.16436 q^{41} +3.95535 q^{43} -0.198997 q^{45} -7.14201 q^{47} +10.4459 q^{49} -4.54891 q^{51} +0.0629449 q^{53} +0.626895 q^{55} +7.17348 q^{57} +7.16608 q^{59} +0.547121 q^{61} +4.17683 q^{63} +0.448329 q^{65} -0.268787 q^{67} -4.93189 q^{69} +5.33220 q^{71} +8.29308 q^{73} +4.96040 q^{75} -13.1581 q^{77} -0.512972 q^{79} +1.00000 q^{81} +13.8697 q^{83} -0.905220 q^{85} -2.19608 q^{87} +1.21408 q^{89} -9.41015 q^{91} -3.49112 q^{93} +1.42750 q^{95} +16.8801 q^{97} -3.15027 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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