LMFDB - Embedded newform 6040.2.a.o.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
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} - 5 x^{14} - 19 x^{13} + 119 x^{12} + 106 x^{11} - 1063 x^{10} - 48 x^{9} + 4510 x^{8} + \cdots + 272 $ x^15 - 5x^14 - 19x^13 + 119x^12 + 106x^11 - 1063x^10 - 48x^9 + 4510x^8 - 1130x^7 - 9512x^6 + 2839x^5 + 9383x^4 - 1781x^3 - 3692x^2 - 92x + 272
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.422275$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.422275 q^{3} +1.00000 q^{5} -2.22380 q^{7} -2.82168 q^{9} +O(q^{10})$	$q-0.422275 q^{3} +1.00000 q^{5} -2.22380 q^{7} -2.82168 q^{9} -3.70557 q^{11} -0.141865 q^{13} -0.422275 q^{15} -6.15177 q^{17} +2.38891 q^{19} +0.939056 q^{21} -0.264060 q^{23} +1.00000 q^{25} +2.45835 q^{27} +0.575710 q^{29} +7.70694 q^{31} +1.56477 q^{33} -2.22380 q^{35} -2.83131 q^{37} +0.0599061 q^{39} +2.54575 q^{41} -11.1538 q^{43} -2.82168 q^{45} -5.40902 q^{47} -2.05470 q^{49} +2.59774 q^{51} -6.57501 q^{53} -3.70557 q^{55} -1.00878 q^{57} +3.40647 q^{59} -2.76821 q^{61} +6.27487 q^{63} -0.141865 q^{65} +12.7343 q^{67} +0.111506 q^{69} +8.14546 q^{71} -6.38416 q^{73} -0.422275 q^{75} +8.24047 q^{77} -10.7647 q^{79} +7.42695 q^{81} +6.69742 q^{83} -6.15177 q^{85} -0.243108 q^{87} +5.81483 q^{89} +0.315480 q^{91} -3.25445 q^{93} +2.38891 q^{95} +8.17820 q^{97} +10.4560 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * 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
$3$	−0.422275	−0.243800	−0.121900	−	0.992542i	$-0.538899\pi$
$3$	−0.422275	−0.243800	−0.121900	+	0.992542i	$0.538899\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.22380	−0.840519	−0.420259	−	0.907404i	$-0.638061\pi$
$7$	−2.22380	−0.840519	−0.420259	+	0.907404i	$0.638061\pi$
$8$	0	0
$9$	−2.82168	−0.940561
$10$	0	0
$11$	−3.70557	−1.11727	−0.558636	−	0.829413i	$-0.688675\pi$
$11$	−3.70557	−1.11727	−0.558636	+	0.829413i	$0.688675\pi$
$12$	0	0
$13$	−0.141865	−0.0393463	−0.0196732	−	0.999806i	$-0.506263\pi$
$13$	−0.141865	−0.0393463	−0.0196732	+	0.999806i	$0.506263\pi$
$14$	0	0
$15$	−0.422275	−0.109031
$16$	0	0
$17$	−6.15177	−1.49202	−0.746012	−	0.665932i	$-0.768034\pi$
$17$	−6.15177	−1.49202	−0.746012	+	0.665932i	$0.768034\pi$
$18$	0	0
$19$	2.38891	0.548054	0.274027	−	0.961722i	$-0.411644\pi$
$19$	2.38891	0.548054	0.274027	+	0.961722i	$0.411644\pi$
$20$	0	0
$21$	0.939056	0.204919
$22$	0	0
$23$	−0.264060	−0.0550603	−0.0275301	−	0.999621i	$-0.508764\pi$
$23$	−0.264060	−0.0550603	−0.0275301	+	0.999621i	$0.508764\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.45835	0.473110
$28$	0	0
$29$	0.575710	0.106907	0.0534533	−	0.998570i	$-0.482977\pi$
$29$	0.575710	0.106907	0.0534533	+	0.998570i	$0.482977\pi$
$30$	0	0
$31$	7.70694	1.38421	0.692104	−	0.721798i	$-0.256684\pi$
$31$	7.70694	1.38421	0.692104	+	0.721798i	$0.256684\pi$
$32$	0	0
$33$	1.56477	0.272392
$34$	0	0
$35$	−2.22380	−0.375891
$36$	0	0
$37$	−2.83131	−0.465464	−0.232732	−	0.972541i	$-0.574766\pi$
$37$	−2.83131	−0.465464	−0.232732	+	0.972541i	$0.574766\pi$
$38$	0	0
$39$	0.0599061	0.00959265
$40$	0	0
$41$	2.54575	0.397580	0.198790	−	0.980042i	$-0.436299\pi$
$41$	2.54575	0.397580	0.198790	+	0.980042i	$0.436299\pi$
$42$	0	0
$43$	−11.1538	−1.70094	−0.850471	−	0.526022i	$-0.823683\pi$
$43$	−11.1538	−1.70094	−0.850471	+	0.526022i	$0.823683\pi$
$44$	0	0
$45$	−2.82168	−0.420632
$46$	0	0
$47$	−5.40902	−0.788987	−0.394493	−	0.918899i	$-0.629080\pi$
$47$	−5.40902	−0.788987	−0.394493	+	0.918899i	$0.629080\pi$
$48$	0	0
$49$	−2.05470	−0.293528
$50$	0	0
$51$	2.59774	0.363756
$52$	0	0
$53$	−6.57501	−0.903148	−0.451574	−	0.892234i	$-0.649137\pi$
$53$	−6.57501	−0.903148	−0.451574	+	0.892234i	$0.649137\pi$
$54$	0	0
$55$	−3.70557	−0.499660
$56$	0	0
$57$	−1.00878	−0.133616
$58$	0	0
$59$	3.40647	0.443485	0.221742	−	0.975105i	$-0.428826\pi$
$59$	3.40647	0.443485	0.221742	+	0.975105i	$0.428826\pi$
$60$	0	0
$61$	−2.76821	−0.354433	−0.177217	−	0.984172i	$-0.556709\pi$
$61$	−2.76821	−0.354433	−0.177217	+	0.984172i	$0.556709\pi$
$62$	0	0
$63$	6.27487	0.790560
$64$	0	0
$65$	−0.141865	−0.0175962
$66$	0	0
$67$	12.7343	1.55575	0.777874	−	0.628420i	$-0.216298\pi$
$67$	12.7343	1.55575	0.777874	+	0.628420i	$0.216298\pi$
$68$	0	0
$69$	0.111506	0.0134237
$70$	0	0
$71$	8.14546	0.966688	0.483344	−	0.875431i	$-0.339422\pi$
$71$	8.14546	0.966688	0.483344	+	0.875431i	$0.339422\pi$
$72$	0	0
$73$	−6.38416	−0.747209	−0.373605	−	0.927588i	$-0.621878\pi$
$73$	−6.38416	−0.747209	−0.373605	+	0.927588i	$0.621878\pi$
$74$	0	0
$75$	−0.422275	−0.0487601
$76$	0	0
$77$	8.24047	0.939089
$78$	0	0
$79$	−10.7647	−1.21113	−0.605564	−	0.795796i	$-0.707053\pi$
$79$	−10.7647	−1.21113	−0.605564	+	0.795796i	$0.707053\pi$
$80$	0	0
$81$	7.42695	0.825217
$82$	0	0
$83$	6.69742	0.735137	0.367569	−	0.929996i	$-0.380190\pi$
$83$	6.69742	0.735137	0.367569	+	0.929996i	$0.380190\pi$
$84$	0	0
$85$	−6.15177	−0.667254
$86$	0	0
$87$	−0.243108	−0.0260639
$88$	0	0
$89$	5.81483	0.616371	0.308186	−	0.951326i	$-0.400278\pi$
$89$	5.81483	0.616371	0.308186	+	0.951326i	$0.400278\pi$
$90$	0	0
$91$	0.315480	0.0330713
$92$	0	0
$93$	−3.25445	−0.337470
$94$	0	0
$95$	2.38891	0.245097
$96$	0	0
$97$	8.17820	0.830370	0.415185	−	0.909737i	$-0.363717\pi$
$97$	8.17820	0.830370	0.415185	+	0.909737i	$0.363717\pi$
$98$	0	0
$99$	10.4560	1.05086
$100$	0	0
$101$	0.156304	0.0155529	0.00777643	−	0.999970i	$-0.497525\pi$
$101$	0.156304	0.0155529	0.00777643	+	0.999970i	$0.497525\pi$
$102$	0	0
$103$	2.20630	0.217393	0.108697	−	0.994075i	$-0.465332\pi$
$103$	2.20630	0.217393	0.108697	+	0.994075i	$0.465332\pi$
$104$	0	0
$105$	0.939056	0.0916425
$106$	0	0
$107$	16.3973	1.58519	0.792595	−	0.609749i	$-0.208730\pi$
$107$	16.3973	1.58519	0.792595	+	0.609749i	$0.208730\pi$
$108$	0	0
$109$	−1.37589	−0.131786	−0.0658932	−	0.997827i	$-0.520990\pi$
$109$	−1.37589	−0.131786	−0.0658932	+	0.997827i	$0.520990\pi$
$110$	0	0
$111$	1.19559	0.113480
$112$	0	0
$113$	12.1197	1.14013	0.570063	−	0.821601i	$-0.306919\pi$
$113$	12.1197	1.14013	0.570063	+	0.821601i	$0.306919\pi$
$114$	0	0
$115$	−0.264060	−0.0246237
$116$	0	0
$117$	0.400299	0.0370076
$118$	0	0
$119$	13.6803	1.25407
$120$	0	0
$121$	2.73128	0.248299
$122$	0	0
$123$	−1.07501	−0.0969301
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.4526	1.10499	0.552494	−	0.833517i	$-0.313676\pi$
$127$	12.4526	1.10499	0.552494	+	0.833517i	$0.313676\pi$
$128$	0	0
$129$	4.70998	0.414690
$130$	0	0
$131$	19.8390	1.73335	0.866673	−	0.498877i	$-0.166254\pi$
$131$	19.8390	1.73335	0.866673	+	0.498877i	$0.166254\pi$
$132$	0	0
$133$	−5.31247	−0.460650
$134$	0	0
$135$	2.45835	0.211581
$136$	0	0
$137$	−7.59680	−0.649039	−0.324519	−	0.945879i	$-0.605203\pi$
$137$	−7.59680	−0.649039	−0.324519	+	0.945879i	$0.605203\pi$
$138$	0	0
$139$	12.3886	1.05079	0.525394	−	0.850859i	$-0.323918\pi$
$139$	12.3886	1.05079	0.525394	+	0.850859i	$0.323918\pi$
$140$	0	0
$141$	2.28409	0.192355
$142$	0	0
$143$	0.525692	0.0439606
$144$	0	0
$145$	0.575710	0.0478101
$146$	0	0
$147$	0.867646	0.0715622
$148$	0	0
$149$	−3.48682	−0.285651	−0.142826	−	0.989748i	$-0.545619\pi$
$149$	−3.48682	−0.285651	−0.142826	+	0.989748i	$0.545619\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	17.3584	1.40334
$154$	0	0
$155$	7.70694	0.619037
$156$	0	0
$157$	−1.35694	−0.108296	−0.0541479	−	0.998533i	$-0.517244\pi$
$157$	−1.35694	−0.108296	−0.0541479	+	0.998533i	$0.517244\pi$
$158$	0	0
$159$	2.77646	0.220188
$160$	0	0
$161$	0.587217	0.0462792
$162$	0	0
$163$	−16.5730	−1.29810	−0.649050	−	0.760746i	$-0.724833\pi$
$163$	−16.5730	−1.29810	−0.649050	+	0.760746i	$0.724833\pi$
$164$	0	0
$165$	1.56477	0.121817
$166$	0	0
$167$	11.4432	0.885505	0.442752	−	0.896644i	$-0.354002\pi$
$167$	11.4432	0.885505	0.442752	+	0.896644i	$0.354002\pi$
$168$	0	0
$169$	−12.9799	−0.998452
$170$	0	0
$171$	−6.74075	−0.515478
$172$	0	0
$173$	7.67050	0.583178	0.291589	−	0.956544i	$-0.405816\pi$
$173$	7.67050	0.583178	0.291589	+	0.956544i	$0.405816\pi$
$174$	0	0
$175$	−2.22380	−0.168104
$176$	0	0
$177$	−1.43847	−0.108122
$178$	0	0
$179$	−8.70259	−0.650462	−0.325231	−	0.945635i	$-0.605442\pi$
$179$	−8.70259	−0.650462	−0.325231	+	0.945635i	$0.605442\pi$
$180$	0	0
$181$	−12.5611	−0.933663	−0.466831	−	0.884346i	$-0.654604\pi$
$181$	−12.5611	−0.933663	−0.466831	+	0.884346i	$0.654604\pi$
$182$	0	0
$183$	1.16895	0.0864110
$184$	0	0
$185$	−2.83131	−0.208162
$186$	0	0
$187$	22.7959	1.66700
$188$	0	0
$189$	−5.46689	−0.397658
$190$	0	0
$191$	4.67003	0.337912	0.168956	−	0.985624i	$-0.445960\pi$
$191$	4.67003	0.337912	0.168956	+	0.985624i	$0.445960\pi$
$192$	0	0
$193$	5.58858	0.402275	0.201137	−	0.979563i	$-0.435536\pi$
$193$	5.58858	0.402275	0.201137	+	0.979563i	$0.435536\pi$
$194$	0	0
$195$	0.0599061	0.00428996
$196$	0	0
$197$	−23.3058	−1.66047	−0.830236	−	0.557412i	$-0.811794\pi$
$197$	−23.3058	−1.66047	−0.830236	+	0.557412i	$0.811794\pi$
$198$	0	0
$199$	−5.61798	−0.398248	−0.199124	−	0.979974i	$-0.563810\pi$
$199$	−5.61798	−0.398248	−0.199124	+	0.979974i	$0.563810\pi$
$200$	0	0
$201$	−5.37739	−0.379292
$202$	0	0
$203$	−1.28027	−0.0898571
$204$	0	0
$205$	2.54575	0.177803
$206$	0	0
$207$	0.745093	0.0517876
$208$	0	0
$209$	−8.85229	−0.612325
$210$	0	0
$211$	−4.17626	−0.287505	−0.143753	−	0.989614i	$-0.545917\pi$
$211$	−4.17626	−0.287505	−0.143753	+	0.989614i	$0.545917\pi$
$212$	0	0
$213$	−3.43962	−0.235679
$214$	0	0
$215$	−11.1538	−0.760684
$216$	0	0
$217$	−17.1387	−1.16345
$218$	0	0
$219$	2.69587	0.182170
$220$	0	0
$221$	0.872723	0.0587057
$222$	0	0
$223$	17.3837	1.16410	0.582048	−	0.813154i	$-0.302252\pi$
$223$	17.3837	1.16410	0.582048	+	0.813154i	$0.302252\pi$
$224$	0	0
$225$	−2.82168	−0.188112
$226$	0	0
$227$	4.28541	0.284433	0.142216	−	0.989836i	$-0.454577\pi$
$227$	4.28541	0.284433	0.142216	+	0.989836i	$0.454577\pi$
$228$	0	0
$229$	−2.36604	−0.156352	−0.0781762	−	0.996940i	$-0.524910\pi$
$229$	−2.36604	−0.156352	−0.0781762	+	0.996940i	$0.524910\pi$
$230$	0	0
$231$	−3.47974	−0.228950
$232$	0	0
$233$	16.3544	1.07142	0.535708	−	0.844404i	$-0.320045\pi$
$233$	16.3544	1.07142	0.535708	+	0.844404i	$0.320045\pi$
$234$	0	0
$235$	−5.40902	−0.352846
$236$	0	0
$237$	4.54568	0.295274
$238$	0	0
$239$	2.93260	0.189694	0.0948470	−	0.995492i	$-0.469764\pi$
$239$	2.93260	0.189694	0.0948470	+	0.995492i	$0.469764\pi$
$240$	0	0
$241$	9.99407	0.643775	0.321887	−	0.946778i	$-0.395683\pi$
$241$	9.99407	0.643775	0.321887	+	0.946778i	$0.395683\pi$
$242$	0	0
$243$	−10.5113	−0.674298
$244$	0	0
$245$	−2.05470	−0.131270
$246$	0	0
$247$	−0.338903	−0.0215639
$248$	0	0
$249$	−2.82815	−0.179227
$250$	0	0
$251$	4.41747	0.278828	0.139414	−	0.990234i	$-0.455478\pi$
$251$	4.41747	0.278828	0.139414	+	0.990234i	$0.455478\pi$
$252$	0	0
$253$	0.978493	0.0615173
$254$	0	0
$255$	2.59774	0.162677
$256$	0	0
$257$	20.8241	1.29897	0.649485	−	0.760375i	$-0.274985\pi$
$257$	20.8241	1.29897	0.649485	+	0.760375i	$0.274985\pi$
$258$	0	0
$259$	6.29627	0.391231
$260$	0	0
$261$	−1.62447	−0.100552
$262$	0	0
$263$	20.1137	1.24027	0.620133	−	0.784496i	$-0.287078\pi$
$263$	20.1137	1.24027	0.620133	+	0.784496i	$0.287078\pi$
$264$	0	0
$265$	−6.57501	−0.403900
$266$	0	0
$267$	−2.45546	−0.150272
$268$	0	0
$269$	14.7069	0.896697	0.448349	−	0.893859i	$-0.352012\pi$
$269$	14.7069	0.896697	0.448349	+	0.893859i	$0.352012\pi$
$270$	0	0
$271$	8.56866	0.520509	0.260254	−	0.965540i	$-0.416194\pi$
$271$	8.56866	0.520509	0.260254	+	0.965540i	$0.416194\pi$
$272$	0	0
$273$	−0.133219	−0.00806281
$274$	0	0
$275$	−3.70557	−0.223455
$276$	0	0
$277$	17.7224	1.06484	0.532419	−	0.846481i	$-0.321283\pi$
$277$	17.7224	1.06484	0.532419	+	0.846481i	$0.321283\pi$
$278$	0	0
$279$	−21.7466	−1.30193
$280$	0	0
$281$	16.8108	1.00285	0.501425	−	0.865201i	$-0.332809\pi$
$281$	16.8108	1.00285	0.501425	+	0.865201i	$0.332809\pi$
$282$	0	0
$283$	−20.1333	−1.19680	−0.598401	−	0.801197i	$-0.704197\pi$
$283$	−20.1333	−1.19680	−0.598401	+	0.801197i	$0.704197\pi$
$284$	0	0
$285$	−1.00878	−0.0597548
$286$	0	0
$287$	−5.66126	−0.334173
$288$	0	0
$289$	20.8443	1.22614
$290$	0	0
$291$	−3.45344	−0.202445
$292$	0	0
$293$	1.82170	0.106425	0.0532124	−	0.998583i	$-0.483054\pi$
$293$	1.82170	0.106425	0.0532124	+	0.998583i	$0.483054\pi$
$294$	0	0
$295$	3.40647	0.198332
$296$	0	0
$297$	−9.10960	−0.528593
$298$	0	0
$299$	0.0374609	0.00216642
$300$	0	0
$301$	24.8039	1.42967
$302$	0	0
$303$	−0.0660034	−0.00379180
$304$	0	0
$305$	−2.76821	−0.158507
$306$	0	0
$307$	−13.3002	−0.759085	−0.379543	−	0.925174i	$-0.623919\pi$
$307$	−13.3002	−0.759085	−0.379543	+	0.925174i	$0.623919\pi$
$308$	0	0
$309$	−0.931664	−0.0530005
$310$	0	0
$311$	23.0928	1.30947	0.654736	−	0.755858i	$-0.272780\pi$
$311$	23.0928	1.30947	0.654736	+	0.755858i	$0.272780\pi$
$312$	0	0
$313$	11.7479	0.664029	0.332014	−	0.943274i	$-0.392272\pi$
$313$	11.7479	0.664029	0.332014	+	0.943274i	$0.392272\pi$
$314$	0	0
$315$	6.27487	0.353549
$316$	0	0
$317$	−26.1313	−1.46768	−0.733840	−	0.679322i	$-0.762274\pi$
$317$	−26.1313	−1.46768	−0.733840	+	0.679322i	$0.762274\pi$
$318$	0	0
$319$	−2.13334	−0.119444
$320$	0	0
$321$	−6.92418	−0.386470
$322$	0	0
$323$	−14.6960	−0.817709
$324$	0	0
$325$	−0.141865	−0.00786927
$326$	0	0
$327$	0.581004	0.0321296
$328$	0	0
$329$	12.0286	0.663158
$330$	0	0
$331$	31.4456	1.72841	0.864204	−	0.503142i	$-0.167823\pi$
$331$	31.4456	1.72841	0.864204	+	0.503142i	$0.167823\pi$
$332$	0	0
$333$	7.98905	0.437797
$334$	0	0
$335$	12.7343	0.695752
$336$	0	0
$337$	−14.0181	−0.763615	−0.381808	−	0.924242i	$-0.624698\pi$
$337$	−14.0181	−0.763615	−0.381808	+	0.924242i	$0.624698\pi$
$338$	0	0
$339$	−5.11784	−0.277963
$340$	0	0
$341$	−28.5587	−1.54654
$342$	0	0
$343$	20.1359	1.08723
$344$	0	0
$345$	0.111506	0.00600327
$346$	0	0
$347$	−23.2312	−1.24712	−0.623559	−	0.781776i	$-0.714314\pi$
$347$	−23.2312	−1.24712	−0.623559	+	0.781776i	$0.714314\pi$
$348$	0	0
$349$	31.9998	1.71291	0.856455	−	0.516221i	$-0.172662\pi$
$349$	31.9998	1.71291	0.856455	+	0.516221i	$0.172662\pi$
$350$	0	0
$351$	−0.348754	−0.0186151
$352$	0	0
$353$	−5.16148	−0.274718	−0.137359	−	0.990521i	$-0.543861\pi$
$353$	−5.16148	−0.274718	−0.137359	+	0.990521i	$0.543861\pi$
$354$	0	0
$355$	8.14546	0.432316
$356$	0	0
$357$	−5.77686	−0.305744
$358$	0	0
$359$	−22.1121	−1.16703	−0.583515	−	0.812102i	$-0.698323\pi$
$359$	−22.1121	−1.16703	−0.583515	+	0.812102i	$0.698323\pi$
$360$	0	0
$361$	−13.2931	−0.699637
$362$	0	0
$363$	−1.15335	−0.0605353
$364$	0	0
$365$	−6.38416	−0.334162
$366$	0	0
$367$	−10.9382	−0.570972	−0.285486	−	0.958383i	$-0.592155\pi$
$367$	−10.9382	−0.570972	−0.285486	+	0.958383i	$0.592155\pi$
$368$	0	0
$369$	−7.18331	−0.373948
$370$	0	0
$371$	14.6215	0.759113
$372$	0	0
$373$	−14.4506	−0.748224	−0.374112	−	0.927384i	$-0.622052\pi$
$373$	−14.4506	−0.748224	−0.374112	+	0.927384i	$0.622052\pi$
$374$	0	0
$375$	−0.422275	−0.0218062
$376$	0	0
$377$	−0.0816732	−0.00420639
$378$	0	0
$379$	1.71730	0.0882117	0.0441058	−	0.999027i	$-0.485956\pi$
$379$	1.71730	0.0882117	0.0441058	+	0.999027i	$0.485956\pi$
$380$	0	0
$381$	−5.25841	−0.269396
$382$	0	0
$383$	−5.66580	−0.289509	−0.144754	−	0.989468i	$-0.546239\pi$
$383$	−5.66580	−0.289509	−0.144754	+	0.989468i	$0.546239\pi$
$384$	0	0
$385$	8.24047	0.419973
$386$	0	0
$387$	31.4726	1.59984
$388$	0	0
$389$	−2.59835	−0.131742	−0.0658708	−	0.997828i	$-0.520983\pi$
$389$	−2.59835	−0.131742	−0.0658708	+	0.997828i	$0.520983\pi$
$390$	0	0
$391$	1.62444	0.0821512
$392$	0	0
$393$	−8.37752	−0.422590
$394$	0	0
$395$	−10.7647	−0.541633
$396$	0	0
$397$	−28.4544	−1.42809	−0.714043	−	0.700101i	$-0.753138\pi$
$397$	−28.4544	−1.42809	−0.714043	+	0.700101i	$0.753138\pi$
$398$	0	0
$399$	2.24332	0.112307
$400$	0	0
$401$	−6.69712	−0.334438	−0.167219	−	0.985920i	$-0.553479\pi$
$401$	−6.69712	−0.334438	−0.167219	+	0.985920i	$0.553479\pi$
$402$	0	0
$403$	−1.09335	−0.0544635
$404$	0	0
$405$	7.42695	0.369048
$406$	0	0
$407$	10.4916	0.520050
$408$	0	0
$409$	27.2285	1.34636	0.673182	−	0.739477i	$-0.264927\pi$
$409$	27.2285	1.34636	0.673182	+	0.739477i	$0.264927\pi$
$410$	0	0
$411$	3.20794	0.158236
$412$	0	0
$413$	−7.57532	−0.372757
$414$	0	0
$415$	6.69742	0.328763
$416$	0	0
$417$	−5.23140	−0.256183
$418$	0	0
$419$	−24.1260	−1.17863	−0.589317	−	0.807902i	$-0.700603\pi$
$419$	−24.1260	−1.17863	−0.589317	+	0.807902i	$0.700603\pi$
$420$	0	0
$421$	17.0860	0.832719	0.416360	−	0.909200i	$-0.363306\pi$
$421$	17.0860	0.832719	0.416360	+	0.909200i	$0.363306\pi$
$422$	0	0
$423$	15.2625	0.742090
$424$	0	0
$425$	−6.15177	−0.298405
$426$	0	0
$427$	6.15596	0.297908
$428$	0	0
$429$	−0.221986	−0.0107176
$430$	0	0
$431$	19.1692	0.923346	0.461673	−	0.887050i	$-0.347249\pi$
$431$	19.1692	0.923346	0.461673	+	0.887050i	$0.347249\pi$
$432$	0	0
$433$	−35.7878	−1.71985	−0.859926	−	0.510419i	$-0.829490\pi$
$433$	−35.7878	−1.71985	−0.859926	+	0.510419i	$0.829490\pi$
$434$	0	0
$435$	−0.243108	−0.0116561
$436$	0	0
$437$	−0.630815	−0.0301760
$438$	0	0
$439$	15.2533	0.728000	0.364000	−	0.931399i	$-0.381411\pi$
$439$	15.2533	0.728000	0.364000	+	0.931399i	$0.381411\pi$
$440$	0	0
$441$	5.79770	0.276081
$442$	0	0
$443$	21.0854	1.00180	0.500899	−	0.865506i	$-0.333003\pi$
$443$	21.0854	1.00180	0.500899	+	0.865506i	$0.333003\pi$
$444$	0	0
$445$	5.81483	0.275650
$446$	0	0
$447$	1.47240	0.0696419
$448$	0	0
$449$	6.33590	0.299010	0.149505	−	0.988761i	$-0.452232\pi$
$449$	6.33590	0.299010	0.149505	+	0.988761i	$0.452232\pi$
$450$	0	0
$451$	−9.43348	−0.444205
$452$	0	0
$453$	0.422275	0.0198402
$454$	0	0
$455$	0.315480	0.0147900
$456$	0	0
$457$	−36.7064	−1.71705	−0.858527	−	0.512768i	$-0.828620\pi$
$457$	−36.7064	−1.71705	−0.858527	+	0.512768i	$0.828620\pi$
$458$	0	0
$459$	−15.1232	−0.705891
$460$	0	0
$461$	31.5834	1.47099	0.735493	−	0.677532i	$-0.236951\pi$
$461$	31.5834	1.47099	0.735493	+	0.677532i	$0.236951\pi$
$462$	0	0
$463$	1.71467	0.0796874	0.0398437	−	0.999206i	$-0.487314\pi$
$463$	1.71467	0.0796874	0.0398437	+	0.999206i	$0.487314\pi$
$464$	0	0
$465$	−3.25445	−0.150921
$466$	0	0
$467$	17.3750	0.804020	0.402010	−	0.915635i	$-0.368312\pi$
$467$	17.3750	0.804020	0.402010	+	0.915635i	$0.368312\pi$
$468$	0	0
$469$	−28.3187	−1.30764
$470$	0	0
$471$	0.573003	0.0264026
$472$	0	0
$473$	41.3313	1.90042
$474$	0	0
$475$	2.38891	0.109611
$476$	0	0
$477$	18.5526	0.849466
$478$	0	0
$479$	−8.46515	−0.386783	−0.193391	−	0.981122i	$-0.561949\pi$
$479$	−8.46515	−0.386783	−0.193391	+	0.981122i	$0.561949\pi$
$480$	0	0
$481$	0.401664	0.0183143
$482$	0	0
$483$	−0.247967	−0.0112829
$484$	0	0
$485$	8.17820	0.371353
$486$	0	0
$487$	−30.2266	−1.36970	−0.684849	−	0.728685i	$-0.740132\pi$
$487$	−30.2266	−1.36970	−0.684849	+	0.728685i	$0.740132\pi$
$488$	0	0
$489$	6.99837	0.316477
$490$	0	0
$491$	13.5315	0.610667	0.305334	−	0.952245i	$-0.401232\pi$
$491$	13.5315	0.610667	0.305334	+	0.952245i	$0.401232\pi$
$492$	0	0
$493$	−3.54164	−0.159507
$494$	0	0
$495$	10.4560	0.469961
$496$	0	0
$497$	−18.1139	−0.812520
$498$	0	0
$499$	16.6760	0.746520	0.373260	−	0.927727i	$-0.378240\pi$
$499$	16.6760	0.746520	0.373260	+	0.927727i	$0.378240\pi$
$500$	0	0
$501$	−4.83219	−0.215886
$502$	0	0
$503$	−17.3046	−0.771574	−0.385787	−	0.922588i	$-0.626070\pi$
$503$	−17.3046	−0.771574	−0.385787	+	0.922588i	$0.626070\pi$
$504$	0	0
$505$	0.156304	0.00695545
$506$	0	0
$507$	5.48107	0.243423
$508$	0	0
$509$	−1.25479	−0.0556174	−0.0278087	−	0.999613i	$-0.508853\pi$
$509$	−1.25479	−0.0556174	−0.0278087	+	0.999613i	$0.508853\pi$
$510$	0	0
$511$	14.1971	0.628043
$512$	0	0
$513$	5.87278	0.259289
$514$	0	0
$515$	2.20630	0.0972211
$516$	0	0
$517$	20.0435	0.881513
$518$	0	0
$519$	−3.23906	−0.142179
$520$	0	0
$521$	2.16601	0.0948948	0.0474474	−	0.998874i	$-0.484891\pi$
$521$	2.16601	0.0948948	0.0474474	+	0.998874i	$0.484891\pi$
$522$	0	0
$523$	17.7583	0.776514	0.388257	−	0.921551i	$-0.373077\pi$
$523$	17.7583	0.776514	0.388257	+	0.921551i	$0.373077\pi$
$524$	0	0
$525$	0.939056	0.0409838
$526$	0	0
$527$	−47.4114	−2.06527
$528$	0	0
$529$	−22.9303	−0.996968
$530$	0	0
$531$	−9.61198	−0.417125
$532$	0	0
$533$	−0.361154	−0.0156433
$534$	0	0
$535$	16.3973	0.708918
$536$	0	0
$537$	3.67488	0.158583
$538$	0	0
$539$	7.61383	0.327951
$540$	0	0
$541$	−5.50805	−0.236810	−0.118405	−	0.992965i	$-0.537778\pi$
$541$	−5.50805	−0.236810	−0.118405	+	0.992965i	$0.537778\pi$
$542$	0	0
$543$	5.30425	0.227627
$544$	0	0
$545$	−1.37589	−0.0589367
$546$	0	0
$547$	−2.56303	−0.109587	−0.0547936	−	0.998498i	$-0.517450\pi$
$547$	−2.56303	−0.109587	−0.0547936	+	0.998498i	$0.517450\pi$
$548$	0	0
$549$	7.81102	0.333366
$550$	0	0
$551$	1.37532	0.0585906
$552$	0	0
$553$	23.9387	1.01798
$554$	0	0
$555$	1.19559	0.0507499
$556$	0	0
$557$	9.86710	0.418082	0.209041	−	0.977907i	$-0.432966\pi$
$557$	9.86710	0.418082	0.209041	+	0.977907i	$0.432966\pi$
$558$	0	0
$559$	1.58234	0.0669258
$560$	0	0
$561$	−9.62611	−0.406415
$562$	0	0
$563$	14.1072	0.594547	0.297274	−	0.954792i	$-0.403923\pi$
$563$	14.1072	0.594547	0.297274	+	0.954792i	$0.403923\pi$
$564$	0	0
$565$	12.1197	0.509880
$566$	0	0
$567$	−16.5161	−0.693611
$568$	0	0
$569$	−27.7403	−1.16293	−0.581466	−	0.813571i	$-0.697521\pi$
$569$	−27.7403	−1.16293	−0.581466	+	0.813571i	$0.697521\pi$
$570$	0	0
$571$	40.1795	1.68146	0.840730	−	0.541454i	$-0.182126\pi$
$571$	40.1795	1.68146	0.840730	+	0.541454i	$0.182126\pi$
$572$	0	0
$573$	−1.97204	−0.0823830
$574$	0	0
$575$	−0.264060	−0.0110121
$576$	0	0
$577$	−11.7077	−0.487397	−0.243698	−	0.969851i	$-0.578361\pi$
$577$	−11.7077	−0.487397	−0.243698	+	0.969851i	$0.578361\pi$
$578$	0	0
$579$	−2.35991	−0.0980747
$580$	0	0
$581$	−14.8938	−0.617897
$582$	0	0
$583$	24.3642	1.00906
$584$	0	0
$585$	0.400299	0.0165503
$586$	0	0
$587$	−19.4265	−0.801819	−0.400909	−	0.916118i	$-0.631306\pi$
$587$	−19.4265	−0.801819	−0.400909	+	0.916118i	$0.631306\pi$
$588$	0	0
$589$	18.4112	0.758620
$590$	0	0
$591$	9.84146	0.404824
$592$	0	0
$593$	−4.23253	−0.173809	−0.0869045	−	0.996217i	$-0.527698\pi$
$593$	−4.23253	−0.173809	−0.0869045	+	0.996217i	$0.527698\pi$
$594$	0	0
$595$	13.6803	0.560839
$596$	0	0
$597$	2.37233	0.0970929
$598$	0	0
$599$	−4.33777	−0.177237	−0.0886183	−	0.996066i	$-0.528245\pi$
$599$	−4.33777	−0.177237	−0.0886183	+	0.996066i	$0.528245\pi$
$600$	0	0
$601$	33.4829	1.36580	0.682898	−	0.730514i	$-0.260719\pi$
$601$	33.4829	1.36580	0.682898	+	0.730514i	$0.260719\pi$
$602$	0	0
$603$	−35.9323	−1.46328
$604$	0	0
$605$	2.73128	0.111042
$606$	0	0
$607$	14.3462	0.582293	0.291146	−	0.956678i	$-0.405963\pi$
$607$	14.3462	0.582293	0.291146	+	0.956678i	$0.405963\pi$
$608$	0	0
$609$	0.540624	0.0219072
$610$	0	0
$611$	0.767352	0.0310437
$612$	0	0
$613$	29.3348	1.18482	0.592410	−	0.805637i	$-0.298177\pi$
$613$	29.3348	1.18482	0.592410	+	0.805637i	$0.298177\pi$
$614$	0	0
$615$	−1.07501	−0.0433485
$616$	0	0
$617$	−28.0012	−1.12729	−0.563643	−	0.826019i	$-0.690600\pi$
$617$	−28.0012	−1.12729	−0.563643	+	0.826019i	$0.690600\pi$
$618$	0	0
$619$	9.96709	0.400611	0.200306	−	0.979733i	$-0.435807\pi$
$619$	9.96709	0.400611	0.200306	+	0.979733i	$0.435807\pi$
$620$	0	0
$621$	−0.649151	−0.0260495
$622$	0	0
$623$	−12.9311	−0.518072
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.73810	0.149285
$628$	0	0
$629$	17.4176	0.694483
$630$	0	0
$631$	−3.31168	−0.131836	−0.0659179	−	0.997825i	$-0.520998\pi$
$631$	−3.31168	−0.131836	−0.0659179	+	0.997825i	$0.520998\pi$
$632$	0	0
$633$	1.76353	0.0700939
$634$	0	0
$635$	12.4526	0.494165
$636$	0	0
$637$	0.291490	0.0115492
$638$	0	0
$639$	−22.9839	−0.909229
$640$	0	0
$641$	−26.4625	−1.04521	−0.522603	−	0.852576i	$-0.675039\pi$
$641$	−26.4625	−1.04521	−0.522603	+	0.852576i	$0.675039\pi$
$642$	0	0
$643$	−34.3156	−1.35328	−0.676638	−	0.736316i	$-0.736564\pi$
$643$	−34.3156	−1.35328	−0.676638	+	0.736316i	$0.736564\pi$
$644$	0	0
$645$	4.70998	0.185455
$646$	0	0
$647$	10.5859	0.416175	0.208087	−	0.978110i	$-0.433276\pi$
$647$	10.5859	0.416175	0.208087	+	0.978110i	$0.433276\pi$
$648$	0	0
$649$	−12.6229	−0.495493
$650$	0	0
$651$	7.23725	0.283650
$652$	0	0
$653$	42.3251	1.65631	0.828155	−	0.560500i	$-0.189391\pi$
$653$	42.3251	1.65631	0.828155	+	0.560500i	$0.189391\pi$
$654$	0	0
$655$	19.8390	0.775176
$656$	0	0
$657$	18.0141	0.702796
$658$	0	0
$659$	28.4235	1.10722	0.553611	−	0.832775i	$-0.313249\pi$
$659$	28.4235	1.10722	0.553611	+	0.832775i	$0.313249\pi$
$660$	0	0
$661$	14.2739	0.555192	0.277596	−	0.960698i	$-0.410462\pi$
$661$	14.2739	0.555192	0.277596	+	0.960698i	$0.410462\pi$
$662$	0	0
$663$	−0.368529	−0.0143125
$664$	0	0
$665$	−5.31247	−0.206009
$666$	0	0
$667$	−0.152022	−0.00588631
$668$	0	0
$669$	−7.34068	−0.283807
$670$	0	0
$671$	10.2578	0.395999
$672$	0	0
$673$	−25.4649	−0.981598	−0.490799	−	0.871273i	$-0.663295\pi$
$673$	−25.4649	−0.981598	−0.490799	+	0.871273i	$0.663295\pi$
$674$	0	0
$675$	2.45835	0.0946219
$676$	0	0
$677$	8.23002	0.316305	0.158153	−	0.987415i	$-0.449446\pi$
$677$	8.23002	0.316305	0.158153	+	0.987415i	$0.449446\pi$
$678$	0	0
$679$	−18.1867	−0.697942
$680$	0	0
$681$	−1.80962	−0.0693448
$682$	0	0
$683$	−13.6547	−0.522484	−0.261242	−	0.965273i	$-0.584132\pi$
$683$	−13.6547	−0.522484	−0.261242	+	0.965273i	$0.584132\pi$
$684$	0	0
$685$	−7.59680	−0.290259
$686$	0	0
$687$	0.999119	0.0381188
$688$	0	0
$689$	0.932766	0.0355356
$690$	0	0
$691$	−6.00656	−0.228500	−0.114250	−	0.993452i	$-0.536447\pi$
$691$	−6.00656	−0.228500	−0.114250	+	0.993452i	$0.536447\pi$
$692$	0	0
$693$	−23.2520	−0.883271
$694$	0	0
$695$	12.3886	0.469927
$696$	0	0
$697$	−15.6609	−0.593199
$698$	0	0
$699$	−6.90607	−0.261211
$700$	0	0
$701$	39.5443	1.49357	0.746784	−	0.665067i	$-0.231597\pi$
$701$	39.5443	1.49357	0.746784	+	0.665067i	$0.231597\pi$
$702$	0	0
$703$	−6.76374	−0.255099
$704$	0	0
$705$	2.28409	0.0860239
$706$	0	0
$707$	−0.347590	−0.0130725
$708$	0	0
$709$	29.5768	1.11078	0.555390	−	0.831590i	$-0.312569\pi$
$709$	29.5768	1.11078	0.555390	+	0.831590i	$0.312569\pi$
$710$	0	0
$711$	30.3747	1.13914
$712$	0	0
$713$	−2.03509	−0.0762148
$714$	0	0
$715$	0.525692	0.0196598
$716$	0	0
$717$	−1.23836	−0.0462475
$718$	0	0
$719$	11.3573	0.423557	0.211778	−	0.977318i	$-0.432075\pi$
$719$	11.3573	0.423557	0.211778	+	0.977318i	$0.432075\pi$
$720$	0	0
$721$	−4.90638	−0.182723
$722$	0	0
$723$	−4.22024	−0.156953
$724$	0	0
$725$	0.575710	0.0213813
$726$	0	0
$727$	21.0135	0.779346	0.389673	−	0.920953i	$-0.372588\pi$
$727$	21.0135	0.779346	0.389673	+	0.920953i	$0.372588\pi$
$728$	0	0
$729$	−17.8422	−0.660823
$730$	0	0
$731$	68.6158	2.53785
$732$	0	0
$733$	−33.9168	−1.25274	−0.626372	−	0.779524i	$-0.715461\pi$
$733$	−33.9168	−1.25274	−0.626372	+	0.779524i	$0.715461\pi$
$734$	0	0
$735$	0.867646	0.0320036
$736$	0	0
$737$	−47.1881	−1.73819
$738$	0	0
$739$	29.5365	1.08652	0.543259	−	0.839565i	$-0.317190\pi$
$739$	29.5365	1.08652	0.543259	+	0.839565i	$0.317190\pi$
$740$	0	0
$741$	0.143110	0.00525729
$742$	0	0
$743$	15.8602	0.581855	0.290928	−	0.956745i	$-0.406036\pi$
$743$	15.8602	0.581855	0.290928	+	0.956745i	$0.406036\pi$
$744$	0	0
$745$	−3.48682	−0.127747
$746$	0	0
$747$	−18.8980	−0.691442
$748$	0	0
$749$	−36.4644	−1.33238
$750$	0	0
$751$	−31.8147	−1.16094	−0.580468	−	0.814283i	$-0.697130\pi$
$751$	−31.8147	−1.16094	−0.580468	+	0.814283i	$0.697130\pi$
$752$	0	0
$753$	−1.86539	−0.0679785
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−30.6344	−1.11343	−0.556713	−	0.830705i	$-0.687938\pi$
$757$	−30.6344	−1.11343	−0.556713	+	0.830705i	$0.687938\pi$
$758$	0	0
$759$	−0.413193	−0.0149979
$760$	0	0
$761$	−11.2332	−0.407204	−0.203602	−	0.979054i	$-0.565265\pi$
$761$	−11.2332	−0.407204	−0.203602	+	0.979054i	$0.565265\pi$
$762$	0	0
$763$	3.05971	0.110769
$764$	0	0
$765$	17.3584	0.627593
$766$	0	0
$767$	−0.483260	−0.0174495
$768$	0	0
$769$	14.0067	0.505096	0.252548	−	0.967584i	$-0.418731\pi$
$769$	14.0067	0.505096	0.252548	+	0.967584i	$0.418731\pi$
$770$	0	0
$771$	−8.79347	−0.316689
$772$	0	0
$773$	−3.31025	−0.119061	−0.0595306	−	0.998226i	$-0.518960\pi$
$773$	−3.31025	−0.119061	−0.0595306	+	0.998226i	$0.518960\pi$
$774$	0	0
$775$	7.70694	0.276842
$776$	0	0
$777$	−2.65876	−0.0953823
$778$	0	0
$779$	6.08158	0.217895
$780$	0	0
$781$	−30.1836	−1.08005
$782$	0	0
$783$	1.41530	0.0505786
$784$	0	0
$785$	−1.35694	−0.0484314
$786$	0	0
$787$	−21.5314	−0.767512	−0.383756	−	0.923434i	$-0.625370\pi$
$787$	−21.5314	−0.767512	−0.383756	+	0.923434i	$0.625370\pi$
$788$	0	0
$789$	−8.49353	−0.302377
$790$	0	0
$791$	−26.9518	−0.958297
$792$	0	0
$793$	0.392713	0.0139457
$794$	0	0
$795$	2.77646	0.0984710
$796$	0	0
$797$	39.9322	1.41447	0.707236	−	0.706978i	$-0.249942\pi$
$797$	39.9322	1.41447	0.707236	+	0.706978i	$0.249942\pi$
$798$	0	0
$799$	33.2751	1.17719
$800$	0	0
$801$	−16.4076	−0.579735
$802$	0	0
$803$	23.6570	0.834836
$804$	0	0
$805$	0.587217	0.0206967
$806$	0	0
$807$	−6.21036	−0.218615
$808$	0	0
$809$	−21.5580	−0.757939	−0.378969	−	0.925409i	$-0.623721\pi$
$809$	−21.5580	−0.757939	−0.378969	+	0.925409i	$0.623721\pi$
$810$	0	0
$811$	37.8425	1.32883	0.664415	−	0.747364i	$-0.268681\pi$
$811$	37.8425	1.32883	0.664415	+	0.747364i	$0.268681\pi$
$812$	0	0
$813$	−3.61833	−0.126900
$814$	0	0
$815$	−16.5730	−0.580528
$816$	0	0
$817$	−26.6455	−0.932207
$818$	0	0
$819$	−0.890186	−0.0311056
$820$	0	0
$821$	−48.5043	−1.69281	−0.846405	−	0.532539i	$-0.821238\pi$
$821$	−48.5043	−1.69281	−0.846405	+	0.532539i	$0.821238\pi$
$822$	0	0
$823$	−12.7195	−0.443374	−0.221687	−	0.975118i	$-0.571156\pi$
$823$	−12.7195	−0.443374	−0.221687	+	0.975118i	$0.571156\pi$
$824$	0	0
$825$	1.56477	0.0544783
$826$	0	0
$827$	−20.0613	−0.697601	−0.348800	−	0.937197i	$-0.613411\pi$
$827$	−20.0613	−0.697601	−0.348800	+	0.937197i	$0.613411\pi$
$828$	0	0
$829$	7.10881	0.246899	0.123450	−	0.992351i	$-0.460604\pi$
$829$	7.10881	0.246899	0.123450	+	0.992351i	$0.460604\pi$
$830$	0	0
$831$	−7.48373	−0.259608
$832$	0	0
$833$	12.6400	0.437951
$834$	0	0
$835$	11.4432	0.396010
$836$	0	0
$837$	18.9464	0.654882
$838$	0	0
$839$	1.59059	0.0549132	0.0274566	−	0.999623i	$-0.491259\pi$
$839$	1.59059	0.0549132	0.0274566	+	0.999623i	$0.491259\pi$
$840$	0	0
$841$	−28.6686	−0.988571
$842$	0	0
$843$	−7.09879	−0.244495
$844$	0	0
$845$	−12.9799	−0.446521
$846$	0	0
$847$	−6.07384	−0.208700
$848$	0	0
$849$	8.50179	0.291781
$850$	0	0
$851$	0.747634	0.0256286
$852$	0	0
$853$	41.1692	1.40961	0.704803	−	0.709403i	$-0.251036\pi$
$853$	41.1692	1.40961	0.704803	+	0.709403i	$0.251036\pi$
$854$	0	0
$855$	−6.74075	−0.230529
$856$	0	0
$857$	16.0759	0.549142	0.274571	−	0.961567i	$-0.411464\pi$
$857$	16.0759	0.549142	0.274571	+	0.961567i	$0.411464\pi$
$858$	0	0
$859$	−0.868095	−0.0296190	−0.0148095	−	0.999890i	$-0.504714\pi$
$859$	−0.868095	−0.0296190	−0.0148095	+	0.999890i	$0.504714\pi$
$860$	0	0
$861$	2.39060	0.0814716
$862$	0	0
$863$	−33.8562	−1.15248	−0.576239	−	0.817281i	$-0.695480\pi$
$863$	−33.8562	−1.15248	−0.576239	+	0.817281i	$0.695480\pi$
$864$	0	0
$865$	7.67050	0.260805
$866$	0	0
$867$	−8.80203	−0.298933
$868$	0	0
$869$	39.8896	1.35316
$870$	0	0
$871$	−1.80656	−0.0612130
$872$	0	0
$873$	−23.0763	−0.781014
$874$	0	0
$875$	−2.22380	−0.0751783
$876$	0	0
$877$	5.98119	0.201970	0.100985	−	0.994888i	$-0.467801\pi$
$877$	5.98119	0.201970	0.100985	+	0.994888i	$0.467801\pi$
$878$	0	0
$879$	−0.769257	−0.0259464
$880$	0	0
$881$	44.1437	1.48724	0.743619	−	0.668604i	$-0.233108\pi$
$881$	44.1437	1.48724	0.743619	+	0.668604i	$0.233108\pi$
$882$	0	0
$883$	29.0691	0.978253	0.489127	−	0.872213i	$-0.337316\pi$
$883$	29.0691	0.978253	0.489127	+	0.872213i	$0.337316\pi$
$884$	0	0
$885$	−1.43847	−0.0483535
$886$	0	0
$887$	41.3291	1.38770	0.693848	−	0.720121i	$-0.255914\pi$
$887$	41.3291	1.38770	0.693848	+	0.720121i	$0.255914\pi$
$888$	0	0
$889$	−27.6921	−0.928763
$890$	0	0
$891$	−27.5211	−0.921993
$892$	0	0
$893$	−12.9217	−0.432407
$894$	0	0
$895$	−8.70259	−0.290895
$896$	0	0
$897$	−0.0158188	−0.000528174 0
$898$	0	0
$899$	4.43696	0.147981
$900$	0	0
$901$	40.4480	1.34752
$902$	0	0
$903$	−10.4741	−0.348555
$904$	0	0
$905$	−12.5611	−0.417547
$906$	0	0
$907$	−17.9440	−0.595820	−0.297910	−	0.954594i	$-0.596290\pi$
$907$	−17.9440	−0.595820	−0.297910	+	0.954594i	$0.596290\pi$
$908$	0	0
$909$	−0.441042	−0.0146284
$910$	0	0
$911$	−38.2626	−1.26770	−0.633849	−	0.773457i	$-0.718526\pi$
$911$	−38.2626	−1.26770	−0.633849	+	0.773457i	$0.718526\pi$
$912$	0	0
$913$	−24.8178	−0.821349
$914$	0	0
$915$	1.16895	0.0386442
$916$	0	0
$917$	−44.1181	−1.45691
$918$	0	0
$919$	7.81982	0.257952	0.128976	−	0.991648i	$-0.458831\pi$
$919$	7.81982	0.257952	0.128976	+	0.991648i	$0.458831\pi$
$920$	0	0
$921$	5.61636	0.185065
$922$	0	0
$923$	−1.15556	−0.0380356
$924$	0	0
$925$	−2.83131	−0.0930928
$926$	0	0
$927$	−6.22548	−0.204472
$928$	0	0
$929$	−27.3537	−0.897447	−0.448724	−	0.893671i	$-0.648121\pi$
$929$	−27.3537	−0.897447	−0.448724	+	0.893671i	$0.648121\pi$
$930$	0	0
$931$	−4.90848	−0.160869
$932$	0	0
$933$	−9.75150	−0.319250
$934$	0	0
$935$	22.7959	0.745504
$936$	0	0
$937$	16.8613	0.550835	0.275417	−	0.961325i	$-0.411184\pi$
$937$	16.8613	0.550835	0.275417	+	0.961325i	$0.411184\pi$
$938$	0	0
$939$	−4.96083	−0.161890
$940$	0	0
$941$	−0.602207	−0.0196314	−0.00981569	−	0.999952i	$-0.503124\pi$
$941$	−0.602207	−0.0196314	−0.00981569	+	0.999952i	$0.503124\pi$
$942$	0	0
$943$	−0.672231	−0.0218908
$944$	0	0
$945$	−5.46689	−0.177838
$946$	0	0
$947$	−9.92585	−0.322547	−0.161273	−	0.986910i	$-0.551560\pi$
$947$	−9.92585	−0.322547	−0.161273	+	0.986910i	$0.551560\pi$
$948$	0	0
$949$	0.905690	0.0293999
$950$	0	0
$951$	11.0346	0.357821
$952$	0	0
$953$	10.0333	0.325009	0.162505	−	0.986708i	$-0.448043\pi$
$953$	10.0333	0.325009	0.162505	+	0.986708i	$0.448043\pi$
$954$	0	0
$955$	4.67003	0.151119
$956$	0	0
$957$	0.900854	0.0291205
$958$	0	0
$959$	16.8938	0.545529
$960$	0	0
$961$	28.3970	0.916031
$962$	0	0
$963$	−46.2681	−1.49097
$964$	0	0
$965$	5.58858	0.179903
$966$	0	0
$967$	−6.45188	−0.207478	−0.103739	−	0.994605i	$-0.533081\pi$
$967$	−6.45188	−0.207478	−0.103739	+	0.994605i	$0.533081\pi$
$968$	0	0
$969$	6.20576	0.199358
$970$	0	0
$971$	−27.6752	−0.888139	−0.444069	−	0.895992i	$-0.646466\pi$
$971$	−27.6752	−0.888139	−0.444069	+	0.895992i	$0.646466\pi$
$972$	0	0
$973$	−27.5499	−0.883208
$974$	0	0
$975$	0.0599061	0.00191853
$976$	0	0
$977$	8.36221	0.267531	0.133765	−	0.991013i	$-0.457293\pi$
$977$	8.36221	0.267531	0.133765	+	0.991013i	$0.457293\pi$
$978$	0	0
$979$	−21.5473	−0.688655
$980$	0	0
$981$	3.88233	0.123953
$982$	0	0
$983$	−49.4020	−1.57568	−0.787839	−	0.615881i	$-0.788800\pi$
$983$	−49.4020	−1.57568	−0.787839	+	0.615881i	$0.788800\pi$
$984$	0	0
$985$	−23.3058	−0.742585
$986$	0	0
$987$	−5.07937	−0.161678
$988$	0	0
$989$	2.94527	0.0936543
$990$	0	0
$991$	17.1167	0.543730	0.271865	−	0.962335i	$-0.412360\pi$
$991$	17.1167	0.543730	0.271865	+	0.962335i	$0.412360\pi$
$992$	0	0
$993$	−13.2787	−0.421386
$994$	0	0
$995$	−5.61798	−0.178102
$996$	0	0
$997$	−29.6389	−0.938672	−0.469336	−	0.883020i	$-0.655507\pi$
$997$	−29.6389	−0.938672	−0.469336	+	0.883020i	$0.655507\pi$
$998$	0	0
$999$	−6.96034	−0.220215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.o.1.7	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.o.1.7	✓	15	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-0.422275 q^{3} +1.00000 q^{5} -2.22380 q^{7} -2.82168 q^{9} +O(q^{10})\)	\(q-0.422275 q^{3} +1.00000 q^{5} -2.22380 q^{7} -2.82168 q^{9} -3.70557 q^{11} -0.141865 q^{13} -0.422275 q^{15} -6.15177 q^{17} +2.38891 q^{19} +0.939056 q^{21} -0.264060 q^{23} +1.00000 q^{25} +2.45835 q^{27} +0.575710 q^{29} +7.70694 q^{31} +1.56477 q^{33} -2.22380 q^{35} -2.83131 q^{37} +0.0599061 q^{39} +2.54575 q^{41} -11.1538 q^{43} -2.82168 q^{45} -5.40902 q^{47} -2.05470 q^{49} +2.59774 q^{51} -6.57501 q^{53} -3.70557 q^{55} -1.00878 q^{57} +3.40647 q^{59} -2.76821 q^{61} +6.27487 q^{63} -0.141865 q^{65} +12.7343 q^{67} +0.111506 q^{69} +8.14546 q^{71} -6.38416 q^{73} -0.422275 q^{75} +8.24047 q^{77} -10.7647 q^{79} +7.42695 q^{81} +6.69742 q^{83} -6.15177 q^{85} -0.243108 q^{87} +5.81483 q^{89} +0.315480 q^{91} -3.25445 q^{93} +2.38891 q^{95} +8.17820 q^{97} +10.4560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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