LMFDB - Embedded newform 3040.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,2,Mod(1,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, 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("3040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3040.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:	$24.2745222145$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.08613$ of defining polynomial
Character	$\chi$	$=$	3040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.08613 q^{3} -1.00000 q^{5} -0.648061 q^{7} +6.52420 q^{9} +1.35194 q^{11} +4.43807 q^{13} -3.08613 q^{15} +2.00000 q^{17} -1.00000 q^{19} -2.00000 q^{21} -3.35194 q^{23} +1.00000 q^{25} +10.8761 q^{27} -4.17226 q^{29} +4.17226 q^{31} +4.17226 q^{33} +0.648061 q^{35} -2.43807 q^{37} +13.6965 q^{39} +10.1723 q^{41} +6.82032 q^{43} -6.52420 q^{45} -0.648061 q^{47} -6.58002 q^{49} +6.17226 q^{51} +6.43807 q^{53} -1.35194 q^{55} -3.08613 q^{57} +10.3445 q^{59} -2.11644 q^{61} -4.22808 q^{63} -4.43807 q^{65} -12.1345 q^{67} -10.3445 q^{69} -3.82774 q^{71} -4.17226 q^{73} +3.08613 q^{75} -0.876139 q^{77} +2.00000 q^{79} +13.9926 q^{81} +9.52420 q^{83} -2.00000 q^{85} -12.8761 q^{87} +3.82774 q^{89} -2.87614 q^{91} +12.8761 q^{93} +1.00000 q^{95} +7.73419 q^{97} +8.82032 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 2 q^{15} + 6 q^{17} - 3 q^{19} - 6 q^{21} - 8 q^{23} + 3 q^{25} + 14 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{35} + 2 q^{37} + 10 q^{39}+ \cdots + 14 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - 2 * q^15 + 6 * q^17 - 3 * q^19 - 6 * q^21 - 8 * q^23 + 3 * q^25 + 14 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^33 + 4 * q^35 + 2 * q^37 + 10 * q^39 + 16 * q^41 + 8 * q^43 - 3 * q^45 - 4 * q^47 + 3 * q^49 + 4 * q^51 + 10 * q^53 - 2 * q^55 - 2 * q^57 + 2 * q^59 + 2 * q^61 + 8 * q^63 - 4 * q^65 + 4 * q^67 - 2 * q^69 - 26 * q^71 + 2 * q^73 + 2 * q^75 + 16 * q^77 + 6 * q^79 + 15 * q^81 + 12 * q^83 - 6 * q^85 - 20 * q^87 + 26 * q^89 + 10 * q^91 + 20 * q^93 + 3 * q^95 + 18 * q^97 + 14 * q^99

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
$3$	3.08613	1.78178	0.890889	−	0.454221i	$-0.150082\pi$
$3$	3.08613	1.78178	0.890889	+	0.454221i	$0.150082\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.648061	−0.244944	−0.122472	−	0.992472i	$-0.539082\pi$
$7$	−0.648061	−0.244944	−0.122472	+	0.992472i	$0.539082\pi$
$8$	0	0
$9$	6.52420	2.17473
$10$	0	0
$11$	1.35194	0.407625	0.203813	−	0.979010i	$-0.434667\pi$
$11$	1.35194	0.407625	0.203813	+	0.979010i	$0.434667\pi$
$12$	0	0
$13$	4.43807	1.23090	0.615449	−	0.788176i	$-0.288975\pi$
$13$	4.43807	1.23090	0.615449	+	0.788176i	$0.288975\pi$
$14$	0	0
$15$	−3.08613	−0.796835
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−3.35194	−0.698928	−0.349464	−	0.936950i	$-0.613636\pi$
$23$	−3.35194	−0.698928	−0.349464	+	0.936950i	$0.613636\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	10.8761	2.09311
$28$	0	0
$29$	−4.17226	−0.774769	−0.387385	−	0.921918i	$-0.626621\pi$
$29$	−4.17226	−0.774769	−0.387385	+	0.921918i	$0.626621\pi$
$30$	0	0
$31$	4.17226	0.749360	0.374680	−	0.927154i	$-0.377753\pi$
$31$	4.17226	0.749360	0.374680	+	0.927154i	$0.377753\pi$
$32$	0	0
$33$	4.17226	0.726297
$34$	0	0
$35$	0.648061	0.109542
$36$	0	0
$37$	−2.43807	−0.400816	−0.200408	−	0.979713i	$-0.564227\pi$
$37$	−2.43807	−0.400816	−0.200408	+	0.979713i	$0.564227\pi$
$38$	0	0
$39$	13.6965	2.19319
$40$	0	0
$41$	10.1723	1.58864	0.794320	−	0.607499i	$-0.207827\pi$
$41$	10.1723	1.58864	0.794320	+	0.607499i	$0.207827\pi$
$42$	0	0
$43$	6.82032	1.04009	0.520045	−	0.854139i	$-0.325915\pi$
$43$	6.82032	1.04009	0.520045	+	0.854139i	$0.325915\pi$
$44$	0	0
$45$	−6.52420	−0.972570
$46$	0	0
$47$	−0.648061	−0.0945294	−0.0472647	−	0.998882i	$-0.515050\pi$
$47$	−0.648061	−0.0945294	−0.0472647	+	0.998882i	$0.515050\pi$
$48$	0	0
$49$	−6.58002	−0.940002
$50$	0	0
$51$	6.17226	0.864289
$52$	0	0
$53$	6.43807	0.884337	0.442168	−	0.896932i	$-0.354209\pi$
$53$	6.43807	0.884337	0.442168	+	0.896932i	$0.354209\pi$
$54$	0	0
$55$	−1.35194	−0.182295
$56$	0	0
$57$	−3.08613	−0.408768
$58$	0	0
$59$	10.3445	1.34674	0.673371	−	0.739305i	$-0.264846\pi$
$59$	10.3445	1.34674	0.673371	+	0.739305i	$0.264846\pi$
$60$	0	0
$61$	−2.11644	−0.270983	−0.135491	−	0.990779i	$-0.543261\pi$
$61$	−2.11644	−0.270983	−0.135491	+	0.990779i	$0.543261\pi$
$62$	0	0
$63$	−4.22808	−0.532688
$64$	0	0
$65$	−4.43807	−0.550475
$66$	0	0
$67$	−12.1345	−1.48247	−0.741234	−	0.671246i	$-0.765759\pi$
$67$	−12.1345	−1.48247	−0.741234	+	0.671246i	$0.765759\pi$
$68$	0	0
$69$	−10.3445	−1.24533
$70$	0	0
$71$	−3.82774	−0.454269	−0.227135	−	0.973863i	$-0.572936\pi$
$71$	−3.82774	−0.454269	−0.227135	+	0.973863i	$0.572936\pi$
$72$	0	0
$73$	−4.17226	−0.488326	−0.244163	−	0.969734i	$-0.578513\pi$
$73$	−4.17226	−0.488326	−0.244163	+	0.969734i	$0.578513\pi$
$74$	0	0
$75$	3.08613	0.356356
$76$	0	0
$77$	−0.876139	−0.0998453
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	13.9926	1.55473
$82$	0	0
$83$	9.52420	1.04542	0.522708	−	0.852512i	$-0.324922\pi$
$83$	9.52420	1.04542	0.522708	+	0.852512i	$0.324922\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−12.8761	−1.38047
$88$	0	0
$89$	3.82774	0.405740	0.202870	−	0.979206i	$-0.434973\pi$
$89$	3.82774	0.405740	0.202870	+	0.979206i	$0.434973\pi$
$90$	0	0
$91$	−2.87614	−0.301501
$92$	0	0
$93$	12.8761	1.33519
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	7.73419	0.785288	0.392644	−	0.919691i	$-0.371560\pi$
$97$	7.73419	0.785288	0.392644	+	0.919691i	$0.371560\pi$
$98$	0	0
$99$	8.82032	0.886476
$100$	0	0
$101$	−1.35194	−0.134523	−0.0672615	−	0.997735i	$-0.521426\pi$
$101$	−1.35194	−0.134523	−0.0672615	+	0.997735i	$0.521426\pi$
$102$	0	0
$103$	−3.79001	−0.373441	−0.186720	−	0.982413i	$-0.559786\pi$
$103$	−3.79001	−0.373441	−0.186720	+	0.982413i	$0.559786\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−8.55451	−0.826996	−0.413498	−	0.910505i	$-0.635693\pi$
$107$	−8.55451	−0.826996	−0.413498	+	0.910505i	$0.635693\pi$
$108$	0	0
$109$	−7.04840	−0.675114	−0.337557	−	0.941305i	$-0.609601\pi$
$109$	−7.04840	−0.675114	−0.337557	+	0.941305i	$0.609601\pi$
$110$	0	0
$111$	−7.52420	−0.714165
$112$	0	0
$113$	18.6103	1.75071	0.875356	−	0.483478i	$-0.160627\pi$
$113$	18.6103	1.75071	0.875356	+	0.483478i	$0.160627\pi$
$114$	0	0
$115$	3.35194	0.312570
$116$	0	0
$117$	28.9549	2.67688
$118$	0	0
$119$	−1.29612	−0.118815
$120$	0	0
$121$	−9.17226	−0.833842
$122$	0	0
$123$	31.3929	2.83060
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.79001	0.158838	0.0794188	−	0.996841i	$-0.474694\pi$
$127$	1.79001	0.158838	0.0794188	+	0.996841i	$0.474694\pi$
$128$	0	0
$129$	21.0484	1.85321
$130$	0	0
$131$	−17.2207	−1.50458	−0.752288	−	0.658834i	$-0.771050\pi$
$131$	−17.2207	−1.50458	−0.752288	+	0.658834i	$0.771050\pi$
$132$	0	0
$133$	0.648061	0.0561940
$134$	0	0
$135$	−10.8761	−0.936069
$136$	0	0
$137$	10.1116	0.863895	0.431948	−	0.901899i	$-0.357827\pi$
$137$	10.1116	0.863895	0.431948	+	0.901899i	$0.357827\pi$
$138$	0	0
$139$	−9.69646	−0.822443	−0.411222	−	0.911535i	$-0.634898\pi$
$139$	−9.69646	−0.822443	−0.411222	+	0.911535i	$0.634898\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	4.17226	0.346487
$146$	0	0
$147$	−20.3068	−1.67488
$148$	0	0
$149$	5.88356	0.482000	0.241000	−	0.970525i	$-0.422525\pi$
$149$	5.88356	0.482000	0.241000	+	0.970525i	$0.422525\pi$
$150$	0	0
$151$	18.6284	1.51596	0.757980	−	0.652278i	$-0.226187\pi$
$151$	18.6284	1.51596	0.757980	+	0.652278i	$0.226187\pi$
$152$	0	0
$153$	13.0484	1.05490
$154$	0	0
$155$	−4.17226	−0.335124
$156$	0	0
$157$	−17.3929	−1.38811	−0.694053	−	0.719924i	$-0.744177\pi$
$157$	−17.3929	−1.38811	−0.694053	+	0.719924i	$0.744177\pi$
$158$	0	0
$159$	19.8687	1.57569
$160$	0	0
$161$	2.17226	0.171198
$162$	0	0
$163$	1.17968	0.0923996	0.0461998	−	0.998932i	$-0.485289\pi$
$163$	1.17968	0.0923996	0.0461998	+	0.998932i	$0.485289\pi$
$164$	0	0
$165$	−4.17226	−0.324810
$166$	0	0
$167$	−24.7268	−1.91342	−0.956708	−	0.291051i	$-0.905995\pi$
$167$	−24.7268	−1.91342	−0.956708	+	0.291051i	$0.905995\pi$
$168$	0	0
$169$	6.69646	0.515112
$170$	0	0
$171$	−6.52420	−0.498918
$172$	0	0
$173$	17.2026	1.30789	0.653944	−	0.756543i	$-0.273113\pi$
$173$	17.2026	1.30789	0.653944	+	0.756543i	$0.273113\pi$
$174$	0	0
$175$	−0.648061	−0.0489888
$176$	0	0
$177$	31.9245	2.39960
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	16.0968	1.19647	0.598233	−	0.801322i	$-0.295870\pi$
$181$	16.0968	1.19647	0.598233	+	0.801322i	$0.295870\pi$
$182$	0	0
$183$	−6.53162	−0.482831
$184$	0	0
$185$	2.43807	0.179250
$186$	0	0
$187$	2.70388	0.197727
$188$	0	0
$189$	−7.04840	−0.512696
$190$	0	0
$191$	0.111635	0.00807764	0.00403882	−	0.999992i	$-0.498714\pi$
$191$	0.111635	0.00807764	0.00403882	+	0.999992i	$0.498714\pi$
$192$	0	0
$193$	−6.01809	−0.433191	−0.216596	−	0.976261i	$-0.569495\pi$
$193$	−6.01809	−0.433191	−0.216596	+	0.976261i	$0.569495\pi$
$194$	0	0
$195$	−13.6965	−0.980824
$196$	0	0
$197$	14.7645	1.05193	0.525964	−	0.850507i	$-0.323705\pi$
$197$	14.7645	1.05193	0.525964	+	0.850507i	$0.323705\pi$
$198$	0	0
$199$	9.75228	0.691321	0.345660	−	0.938360i	$-0.387655\pi$
$199$	9.75228	0.691321	0.345660	+	0.938360i	$0.387655\pi$
$200$	0	0
$201$	−37.4487	−2.64143
$202$	0	0
$203$	2.70388	0.189775
$204$	0	0
$205$	−10.1723	−0.710461
$206$	0	0
$207$	−21.8687	−1.51998
$208$	0	0
$209$	−1.35194	−0.0935156
$210$	0	0
$211$	4.87614	0.335687	0.167844	−	0.985814i	$-0.446320\pi$
$211$	4.87614	0.335687	0.167844	+	0.985814i	$0.446320\pi$
$212$	0	0
$213$	−11.8129	−0.809407
$214$	0	0
$215$	−6.82032	−0.465142
$216$	0	0
$217$	−2.70388	−0.183551
$218$	0	0
$219$	−12.8761	−0.870089
$220$	0	0
$221$	8.87614	0.597074
$222$	0	0
$223$	−20.8384	−1.39544	−0.697721	−	0.716369i	$-0.745803\pi$
$223$	−20.8384	−1.39544	−0.697721	+	0.716369i	$0.745803\pi$
$224$	0	0
$225$	6.52420	0.434947
$226$	0	0
$227$	−25.3552	−1.68288	−0.841441	−	0.540348i	$-0.818293\pi$
$227$	−25.3552	−1.68288	−0.841441	+	0.540348i	$0.818293\pi$
$228$	0	0
$229$	26.0410	1.72084	0.860418	−	0.509589i	$-0.170202\pi$
$229$	26.0410	1.72084	0.860418	+	0.509589i	$0.170202\pi$
$230$	0	0
$231$	−2.70388	−0.177902
$232$	0	0
$233$	−15.1090	−0.989825	−0.494913	−	0.868943i	$-0.664800\pi$
$233$	−15.1090	−0.989825	−0.494913	+	0.868943i	$0.664800\pi$
$234$	0	0
$235$	0.648061	0.0422748
$236$	0	0
$237$	6.17226	0.400931
$238$	0	0
$239$	−20.9878	−1.35759	−0.678793	−	0.734330i	$-0.737497\pi$
$239$	−20.9878	−1.35759	−0.678793	+	0.734330i	$0.737497\pi$
$240$	0	0
$241$	−8.87614	−0.571762	−0.285881	−	0.958265i	$-0.592286\pi$
$241$	−8.87614	−0.571762	−0.285881	+	0.958265i	$0.592286\pi$
$242$	0	0
$243$	10.5545	0.677072
$244$	0	0
$245$	6.58002	0.420382
$246$	0	0
$247$	−4.43807	−0.282388
$248$	0	0
$249$	29.3929	1.86270
$250$	0	0
$251$	4.11164	0.259524	0.129762	−	0.991545i	$-0.458579\pi$
$251$	4.11164	0.259524	0.129762	+	0.991545i	$0.458579\pi$
$252$	0	0
$253$	−4.53162	−0.284900
$254$	0	0
$255$	−6.17226	−0.386522
$256$	0	0
$257$	−5.37483	−0.335273	−0.167636	−	0.985849i	$-0.553613\pi$
$257$	−5.37483	−0.335273	−0.167636	+	0.985849i	$0.553613\pi$
$258$	0	0
$259$	1.58002	0.0981775
$260$	0	0
$261$	−27.2207	−1.68492
$262$	0	0
$263$	−7.58482	−0.467700	−0.233850	−	0.972273i	$-0.575133\pi$
$263$	−7.58482	−0.467700	−0.233850	+	0.972273i	$0.575133\pi$
$264$	0	0
$265$	−6.43807	−0.395487
$266$	0	0
$267$	11.8129	0.722938
$268$	0	0
$269$	−2.81551	−0.171665	−0.0858324	−	0.996310i	$-0.527355\pi$
$269$	−2.81551	−0.171665	−0.0858324	+	0.996310i	$0.527355\pi$
$270$	0	0
$271$	−16.7449	−1.01718	−0.508589	−	0.861009i	$-0.669833\pi$
$271$	−16.7449	−1.01718	−0.508589	+	0.861009i	$0.669833\pi$
$272$	0	0
$273$	−8.87614	−0.537208
$274$	0	0
$275$	1.35194	0.0815250
$276$	0	0
$277$	25.0484	1.50501	0.752506	−	0.658585i	$-0.228845\pi$
$277$	25.0484	1.50501	0.752506	+	0.658585i	$0.228845\pi$
$278$	0	0
$279$	27.2207	1.62966
$280$	0	0
$281$	−26.8613	−1.60241	−0.801205	−	0.598389i	$-0.795808\pi$
$281$	−26.8613	−1.60241	−0.801205	+	0.598389i	$0.795808\pi$
$282$	0	0
$283$	−21.5242	−1.27948	−0.639740	−	0.768591i	$-0.720958\pi$
$283$	−21.5242	−1.27948	−0.639740	+	0.768591i	$0.720958\pi$
$284$	0	0
$285$	3.08613	0.182807
$286$	0	0
$287$	−6.59224	−0.389128
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	23.8687	1.39921
$292$	0	0
$293$	18.6710	1.09077	0.545384	−	0.838186i	$-0.316384\pi$
$293$	18.6710	1.09077	0.545384	+	0.838186i	$0.316384\pi$
$294$	0	0
$295$	−10.3445	−0.602281
$296$	0	0
$297$	14.7039	0.853206
$298$	0	0
$299$	−14.8761	−0.860309
$300$	0	0
$301$	−4.41998	−0.254764
$302$	0	0
$303$	−4.17226	−0.239690
$304$	0	0
$305$	2.11644	0.121187
$306$	0	0
$307$	−10.3068	−0.588240	−0.294120	−	0.955769i	$-0.595026\pi$
$307$	−10.3068	−0.588240	−0.294120	+	0.955769i	$0.595026\pi$
$308$	0	0
$309$	−11.6965	−0.665388
$310$	0	0
$311$	−12.4003	−0.703159	−0.351579	−	0.936158i	$-0.614355\pi$
$311$	−12.4003	−0.703159	−0.351579	+	0.936158i	$0.614355\pi$
$312$	0	0
$313$	−27.4535	−1.55177	−0.775883	−	0.630877i	$-0.782695\pi$
$313$	−27.4535	−1.55177	−0.775883	+	0.630877i	$0.782695\pi$
$314$	0	0
$315$	4.22808	0.238225
$316$	0	0
$317$	−34.1149	−1.91608	−0.958041	−	0.286630i	$-0.907465\pi$
$317$	−34.1149	−1.91608	−0.958041	+	0.286630i	$0.907465\pi$
$318$	0	0
$319$	−5.64064	−0.315815
$320$	0	0
$321$	−26.4003	−1.47352
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	4.43807	0.246180
$326$	0	0
$327$	−21.7523	−1.20290
$328$	0	0
$329$	0.419983	0.0231544
$330$	0	0
$331$	−16.2691	−0.894228	−0.447114	−	0.894477i	$-0.647548\pi$
$331$	−16.2691	−0.894228	−0.447114	+	0.894477i	$0.647548\pi$
$332$	0	0
$333$	−15.9065	−0.871668
$334$	0	0
$335$	12.1345	0.662980
$336$	0	0
$337$	23.0665	1.25651	0.628256	−	0.778007i	$-0.283769\pi$
$337$	23.0665	1.25651	0.628256	+	0.778007i	$0.283769\pi$
$338$	0	0
$339$	57.4339	3.11938
$340$	0	0
$341$	5.64064	0.305458
$342$	0	0
$343$	8.80068	0.475192
$344$	0	0
$345$	10.3445	0.556930
$346$	0	0
$347$	26.4003	1.41724	0.708622	−	0.705588i	$-0.249317\pi$
$347$	26.4003	1.41724	0.708622	+	0.705588i	$0.249317\pi$
$348$	0	0
$349$	−35.4535	−1.89778	−0.948892	−	0.315600i	$-0.897794\pi$
$349$	−35.4535	−1.89778	−0.948892	+	0.315600i	$0.897794\pi$
$350$	0	0
$351$	48.2691	2.57641
$352$	0	0
$353$	−19.0336	−1.01305	−0.506527	−	0.862224i	$-0.669071\pi$
$353$	−19.0336	−1.01305	−0.506527	+	0.862224i	$0.669071\pi$
$354$	0	0
$355$	3.82774	0.203155
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	−4.16745	−0.219950	−0.109975	−	0.993934i	$-0.535077\pi$
$359$	−4.16745	−0.219950	−0.109975	+	0.993934i	$0.535077\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−28.3068	−1.48572
$364$	0	0
$365$	4.17226	0.218386
$366$	0	0
$367$	−34.3249	−1.79174	−0.895872	−	0.444312i	$-0.853448\pi$
$367$	−34.3249	−1.79174	−0.895872	+	0.444312i	$0.853448\pi$
$368$	0	0
$369$	66.3659	3.45487
$370$	0	0
$371$	−4.17226	−0.216613
$372$	0	0
$373$	−1.32905	−0.0688153	−0.0344077	−	0.999408i	$-0.510954\pi$
$373$	−1.32905	−0.0688153	−0.0344077	+	0.999408i	$0.510954\pi$
$374$	0	0
$375$	−3.08613	−0.159367
$376$	0	0
$377$	−18.5168	−0.953663
$378$	0	0
$379$	10.5922	0.544087	0.272043	−	0.962285i	$-0.412301\pi$
$379$	10.5922	0.544087	0.272043	+	0.962285i	$0.412301\pi$
$380$	0	0
$381$	5.52420	0.283013
$382$	0	0
$383$	2.38225	0.121727	0.0608637	−	0.998146i	$-0.480615\pi$
$383$	2.38225	0.121727	0.0608637	+	0.998146i	$0.480615\pi$
$384$	0	0
$385$	0.876139	0.0446522
$386$	0	0
$387$	44.4971	2.26192
$388$	0	0
$389$	23.3323	1.18299	0.591497	−	0.806307i	$-0.298537\pi$
$389$	23.3323	1.18299	0.591497	+	0.806307i	$0.298537\pi$
$390$	0	0
$391$	−6.70388	−0.339030
$392$	0	0
$393$	−53.1452	−2.68082
$394$	0	0
$395$	−2.00000	−0.100631
$396$	0	0
$397$	34.3807	1.72552	0.862759	−	0.505616i	$-0.168735\pi$
$397$	34.3807	1.72552	0.862759	+	0.505616i	$0.168735\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	24.3297	1.21497	0.607483	−	0.794333i	$-0.292179\pi$
$401$	24.3297	1.21497	0.607483	+	0.794333i	$0.292179\pi$
$402$	0	0
$403$	18.5168	0.922387
$404$	0	0
$405$	−13.9926	−0.695297
$406$	0	0
$407$	−3.29612	−0.163383
$408$	0	0
$409$	−9.75228	−0.482219	−0.241110	−	0.970498i	$-0.577511\pi$
$409$	−9.75228	−0.482219	−0.241110	+	0.970498i	$0.577511\pi$
$410$	0	0
$411$	31.2058	1.53927
$412$	0	0
$413$	−6.70388	−0.329876
$414$	0	0
$415$	−9.52420	−0.467525
$416$	0	0
$417$	−29.9245	−1.46541
$418$	0	0
$419$	6.51678	0.318366	0.159183	−	0.987249i	$-0.449114\pi$
$419$	6.51678	0.318366	0.159183	+	0.987249i	$0.449114\pi$
$420$	0	0
$421$	2.40515	0.117220	0.0586098	−	0.998281i	$-0.481333\pi$
$421$	2.40515	0.117220	0.0586098	+	0.998281i	$0.481333\pi$
$422$	0	0
$423$	−4.22808	−0.205576
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	1.37158	0.0663756
$428$	0	0
$429$	18.5168	0.893999
$430$	0	0
$431$	13.1090	0.631439	0.315720	−	0.948852i	$-0.397754\pi$
$431$	13.1090	0.631439	0.315720	+	0.948852i	$0.397754\pi$
$432$	0	0
$433$	−20.8942	−1.00411	−0.502056	−	0.864835i	$-0.667423\pi$
$433$	−20.8942	−1.00411	−0.502056	+	0.864835i	$0.667423\pi$
$434$	0	0
$435$	12.8761	0.617364
$436$	0	0
$437$	3.35194	0.160345
$438$	0	0
$439$	−41.3929	−1.97558	−0.987788	−	0.155803i	$-0.950203\pi$
$439$	−41.3929	−1.97558	−0.987788	+	0.155803i	$0.950203\pi$
$440$	0	0
$441$	−42.9293	−2.04425
$442$	0	0
$443$	10.0558	0.477766	0.238883	−	0.971048i	$-0.423219\pi$
$443$	10.0558	0.477766	0.238883	+	0.971048i	$0.423219\pi$
$444$	0	0
$445$	−3.82774	−0.181452
$446$	0	0
$447$	18.1574	0.858817
$448$	0	0
$449$	12.1116	0.571583	0.285792	−	0.958292i	$-0.407743\pi$
$449$	12.1116	0.571583	0.285792	+	0.958292i	$0.407743\pi$
$450$	0	0
$451$	13.7523	0.647569
$452$	0	0
$453$	57.4897	2.70110
$454$	0	0
$455$	2.87614	0.134835
$456$	0	0
$457$	−30.9123	−1.44602	−0.723008	−	0.690839i	$-0.757241\pi$
$457$	−30.9123	−1.44602	−0.723008	+	0.690839i	$0.757241\pi$
$458$	0	0
$459$	21.7523	1.01531
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	31.0894	1.44485	0.722423	−	0.691451i	$-0.243028\pi$
$463$	31.0894	1.44485	0.722423	+	0.691451i	$0.243028\pi$
$464$	0	0
$465$	−12.8761	−0.597117
$466$	0	0
$467$	−12.0410	−0.557190	−0.278595	−	0.960409i	$-0.589869\pi$
$467$	−12.0410	−0.557190	−0.278595	+	0.960409i	$0.589869\pi$
$468$	0	0
$469$	7.86391	0.363122
$470$	0	0
$471$	−53.6768	−2.47330
$472$	0	0
$473$	9.22066	0.423966
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	42.0032	1.92320
$478$	0	0
$479$	10.3035	0.470781	0.235390	−	0.971901i	$-0.424363\pi$
$479$	10.3035	0.470781	0.235390	+	0.971901i	$0.424363\pi$
$480$	0	0
$481$	−10.8203	−0.493364
$482$	0	0
$483$	6.70388	0.305037
$484$	0	0
$485$	−7.73419	−0.351192
$486$	0	0
$487$	−22.4939	−1.01930	−0.509648	−	0.860383i	$-0.670224\pi$
$487$	−22.4939	−1.01930	−0.509648	+	0.860383i	$0.670224\pi$
$488$	0	0
$489$	3.64064	0.164636
$490$	0	0
$491$	−35.6162	−1.60734	−0.803668	−	0.595078i	$-0.797121\pi$
$491$	−35.6162	−1.60734	−0.803668	+	0.595078i	$0.797121\pi$
$492$	0	0
$493$	−8.34452	−0.375818
$494$	0	0
$495$	−8.82032	−0.396444
$496$	0	0
$497$	2.48061	0.111270
$498$	0	0
$499$	−0.0558176	−0.00249874	−0.00124937	−	0.999999i	$-0.500398\pi$
$499$	−0.0558176	−0.00249874	−0.00124937	+	0.999999i	$0.500398\pi$
$500$	0	0
$501$	−76.3100	−3.40928
$502$	0	0
$503$	23.8081	1.06155	0.530775	−	0.847513i	$-0.321901\pi$
$503$	23.8081	1.06155	0.530775	+	0.847513i	$0.321901\pi$
$504$	0	0
$505$	1.35194	0.0601605
$506$	0	0
$507$	20.6661	0.917816
$508$	0	0
$509$	10.5316	0.466806	0.233403	−	0.972380i	$-0.425014\pi$
$509$	10.5316	0.466806	0.233403	+	0.972380i	$0.425014\pi$
$510$	0	0
$511$	2.70388	0.119613
$512$	0	0
$513$	−10.8761	−0.480193
$514$	0	0
$515$	3.79001	0.167008
$516$	0	0
$517$	−0.876139	−0.0385325
$518$	0	0
$519$	53.0894	2.33037
$520$	0	0
$521$	−11.2961	−0.494892	−0.247446	−	0.968902i	$-0.579591\pi$
$521$	−11.2961	−0.494892	−0.247446	+	0.968902i	$0.579591\pi$
$522$	0	0
$523$	−11.0861	−0.484763	−0.242381	−	0.970181i	$-0.577929\pi$
$523$	−11.0861	−0.484763	−0.242381	+	0.970181i	$0.577929\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	8.34452	0.363493
$528$	0	0
$529$	−11.7645	−0.511500
$530$	0	0
$531$	67.4897	2.92880
$532$	0	0
$533$	45.1452	1.95546
$534$	0	0
$535$	8.55451	0.369844
$536$	0	0
$537$	18.5168	0.799058
$538$	0	0
$539$	−8.89578	−0.383169
$540$	0	0
$541$	16.8565	0.724717	0.362359	−	0.932039i	$-0.381972\pi$
$541$	16.8565	0.724717	0.362359	+	0.932039i	$0.381972\pi$
$542$	0	0
$543$	49.6768	2.13184
$544$	0	0
$545$	7.04840	0.301920
$546$	0	0
$547$	16.6151	0.710412	0.355206	−	0.934788i	$-0.384411\pi$
$547$	16.6151	0.710412	0.355206	+	0.934788i	$0.384411\pi$
$548$	0	0
$549$	−13.8081	−0.589315
$550$	0	0
$551$	4.17226	0.177744
$552$	0	0
$553$	−1.29612	−0.0551167
$554$	0	0
$555$	7.52420	0.319384
$556$	0	0
$557$	38.4562	1.62944	0.814720	−	0.579855i	$-0.196891\pi$
$557$	38.4562	1.62944	0.814720	+	0.579855i	$0.196891\pi$
$558$	0	0
$559$	30.2691	1.28024
$560$	0	0
$561$	8.34452	0.352306
$562$	0	0
$563$	40.4184	1.70343	0.851717	−	0.524002i	$-0.175562\pi$
$563$	40.4184	1.70343	0.851717	+	0.524002i	$0.175562\pi$
$564$	0	0
$565$	−18.6103	−0.782942
$566$	0	0
$567$	−9.06804	−0.380822
$568$	0	0
$569$	26.9368	1.12925	0.564624	−	0.825348i	$-0.309021\pi$
$569$	26.9368	1.12925	0.564624	+	0.825348i	$0.309021\pi$
$570$	0	0
$571$	−21.4636	−0.898223	−0.449111	−	0.893476i	$-0.648259\pi$
$571$	−21.4636	−0.898223	−0.449111	+	0.893476i	$0.648259\pi$
$572$	0	0
$573$	0.344521	0.0143926
$574$	0	0
$575$	−3.35194	−0.139786
$576$	0	0
$577$	−12.8613	−0.535423	−0.267712	−	0.963499i	$-0.586267\pi$
$577$	−12.8613	−0.535423	−0.267712	+	0.963499i	$0.586267\pi$
$578$	0	0
$579$	−18.5726	−0.771851
$580$	0	0
$581$	−6.17226	−0.256069
$582$	0	0
$583$	8.70388	0.360478
$584$	0	0
$585$	−28.9549	−1.19714
$586$	0	0
$587$	−13.8687	−0.572423	−0.286212	−	0.958166i	$-0.592396\pi$
$587$	−13.8687	−0.572423	−0.286212	+	0.958166i	$0.592396\pi$
$588$	0	0
$589$	−4.17226	−0.171915
$590$	0	0
$591$	45.5652	1.87430
$592$	0	0
$593$	−20.9368	−0.859770	−0.429885	−	0.902884i	$-0.641446\pi$
$593$	−20.9368	−0.859770	−0.429885	+	0.902884i	$0.641446\pi$
$594$	0	0
$595$	1.29612	0.0531358
$596$	0	0
$597$	30.0968	1.23178
$598$	0	0
$599$	−0.703878	−0.0287597	−0.0143798	−	0.999897i	$-0.504577\pi$
$599$	−0.703878	−0.0287597	−0.0143798	+	0.999897i	$0.504577\pi$
$600$	0	0
$601$	5.22066	0.212955	0.106478	−	0.994315i	$-0.466043\pi$
$601$	5.22066	0.212955	0.106478	+	0.994315i	$0.466043\pi$
$602$	0	0
$603$	−79.1681	−3.22397
$604$	0	0
$605$	9.17226	0.372905
$606$	0	0
$607$	−16.1345	−0.654880	−0.327440	−	0.944872i	$-0.606186\pi$
$607$	−16.1345	−0.654880	−0.327440	+	0.944872i	$0.606186\pi$
$608$	0	0
$609$	8.34452	0.338137
$610$	0	0
$611$	−2.87614	−0.116356
$612$	0	0
$613$	−19.3323	−0.780824	−0.390412	−	0.920640i	$-0.627667\pi$
$613$	−19.3323	−0.780824	−0.390412	+	0.920640i	$0.627667\pi$
$614$	0	0
$615$	−31.3929	−1.26588
$616$	0	0
$617$	1.01223	0.0407507	0.0203753	−	0.999792i	$-0.493514\pi$
$617$	1.01223	0.0407507	0.0203753	+	0.999792i	$0.493514\pi$
$618$	0	0
$619$	−18.8809	−0.758889	−0.379445	−	0.925214i	$-0.623885\pi$
$619$	−18.8809	−0.758889	−0.379445	+	0.925214i	$0.623885\pi$
$620$	0	0
$621$	−36.4562	−1.46294
$622$	0	0
$623$	−2.48061	−0.0993835
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.17226	−0.166624
$628$	0	0
$629$	−4.87614	−0.194424
$630$	0	0
$631$	−17.4636	−0.695214	−0.347607	−	0.937640i	$-0.613006\pi$
$631$	−17.4636	−0.695214	−0.347607	+	0.937640i	$0.613006\pi$
$632$	0	0
$633$	15.0484	0.598120
$634$	0	0
$635$	−1.79001	−0.0710343
$636$	0	0
$637$	−29.2026	−1.15705
$638$	0	0
$639$	−24.9729	−0.987914
$640$	0	0
$641$	−22.4051	−0.884950	−0.442475	−	0.896781i	$-0.645899\pi$
$641$	−22.4051	−0.884950	−0.442475	+	0.896781i	$0.645899\pi$
$642$	0	0
$643$	22.6332	0.892567	0.446284	−	0.894892i	$-0.352747\pi$
$643$	22.6332	0.892567	0.446284	+	0.894892i	$0.352747\pi$
$644$	0	0
$645$	−21.0484	−0.828780
$646$	0	0
$647$	−34.8203	−1.36893	−0.684464	−	0.729047i	$-0.739964\pi$
$647$	−34.8203	−1.36893	−0.684464	+	0.729047i	$0.739964\pi$
$648$	0	0
$649$	13.9852	0.548966
$650$	0	0
$651$	−8.34452	−0.327048
$652$	0	0
$653$	−23.1090	−0.904326	−0.452163	−	0.891935i	$-0.649347\pi$
$653$	−23.1090	−0.904326	−0.452163	+	0.891935i	$0.649347\pi$
$654$	0	0
$655$	17.2207	0.672867
$656$	0	0
$657$	−27.2207	−1.06198
$658$	0	0
$659$	4.24772	0.165468	0.0827339	−	0.996572i	$-0.473635\pi$
$659$	4.24772	0.165468	0.0827339	+	0.996572i	$0.473635\pi$
$660$	0	0
$661$	8.15742	0.317287	0.158644	−	0.987336i	$-0.449288\pi$
$661$	8.15742	0.317287	0.158644	+	0.987336i	$0.449288\pi$
$662$	0	0
$663$	27.3929	1.06385
$664$	0	0
$665$	−0.648061	−0.0251307
$666$	0	0
$667$	13.9852	0.541508
$668$	0	0
$669$	−64.3100	−2.48637
$670$	0	0
$671$	−2.86130	−0.110459
$672$	0	0
$673$	39.2239	1.51197	0.755985	−	0.654589i	$-0.227158\pi$
$673$	39.2239	1.51197	0.755985	+	0.654589i	$0.227158\pi$
$674$	0	0
$675$	10.8761	0.418623
$676$	0	0
$677$	32.4232	1.24613	0.623063	−	0.782172i	$-0.285888\pi$
$677$	32.4232	1.24613	0.623063	+	0.782172i	$0.285888\pi$
$678$	0	0
$679$	−5.01223	−0.192352
$680$	0	0
$681$	−78.2494	−2.99852
$682$	0	0
$683$	−35.6539	−1.36426	−0.682130	−	0.731231i	$-0.738946\pi$
$683$	−35.6539	−1.36426	−0.682130	+	0.731231i	$0.738946\pi$
$684$	0	0
$685$	−10.1116	−0.386346
$686$	0	0
$687$	80.3659	3.06615
$688$	0	0
$689$	28.5726	1.08853
$690$	0	0
$691$	−44.0261	−1.67483	−0.837417	−	0.546565i	$-0.815935\pi$
$691$	−44.0261	−1.67483	−0.837417	+	0.546565i	$0.815935\pi$
$692$	0	0
$693$	−5.71610	−0.217137
$694$	0	0
$695$	9.69646	0.367808
$696$	0	0
$697$	20.3445	0.770604
$698$	0	0
$699$	−46.6284	−1.76365
$700$	0	0
$701$	28.9023	1.09162	0.545812	−	0.837908i	$-0.316221\pi$
$701$	28.9023	1.09162	0.545812	+	0.837908i	$0.316221\pi$
$702$	0	0
$703$	2.43807	0.0919535
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	0.876139	0.0329506
$708$	0	0
$709$	−10.6433	−0.399716	−0.199858	−	0.979825i	$-0.564048\pi$
$709$	−10.6433	−0.399716	−0.199858	+	0.979825i	$0.564048\pi$
$710$	0	0
$711$	13.0484	0.489353
$712$	0	0
$713$	−13.9852	−0.523748
$714$	0	0
$715$	−6.00000	−0.224387
$716$	0	0
$717$	−64.7710	−2.41892
$718$	0	0
$719$	−41.4339	−1.54522	−0.772612	−	0.634879i	$-0.781050\pi$
$719$	−41.4339	−1.54522	−0.772612	+	0.634879i	$0.781050\pi$
$720$	0	0
$721$	2.45616	0.0914720
$722$	0	0
$723$	−27.3929	−1.01875
$724$	0	0
$725$	−4.17226	−0.154954
$726$	0	0
$727$	−4.72352	−0.175186	−0.0875929	−	0.996156i	$-0.527917\pi$
$727$	−4.72352	−0.175186	−0.0875929	+	0.996156i	$0.527917\pi$
$728$	0	0
$729$	−9.40515	−0.348339
$730$	0	0
$731$	13.6406	0.504517
$732$	0	0
$733$	17.6917	0.653456	0.326728	−	0.945118i	$-0.394054\pi$
$733$	17.6917	0.653456	0.326728	+	0.945118i	$0.394054\pi$
$734$	0	0
$735$	20.3068	0.749027
$736$	0	0
$737$	−16.4051	−0.604291
$738$	0	0
$739$	12.2233	0.449640	0.224820	−	0.974400i	$-0.427821\pi$
$739$	12.2233	0.449640	0.224820	+	0.974400i	$0.427821\pi$
$740$	0	0
$741$	−13.6965	−0.503152
$742$	0	0
$743$	19.6635	0.721385	0.360693	−	0.932685i	$-0.382540\pi$
$743$	19.6635	0.721385	0.360693	+	0.932685i	$0.382540\pi$
$744$	0	0
$745$	−5.88356	−0.215557
$746$	0	0
$747$	62.1378	2.27350
$748$	0	0
$749$	5.54384	0.202568
$750$	0	0
$751$	−41.6768	−1.52081	−0.760404	−	0.649450i	$-0.774999\pi$
$751$	−41.6768	−1.52081	−0.760404	+	0.649450i	$0.774999\pi$
$752$	0	0
$753$	12.6890	0.462414
$754$	0	0
$755$	−18.6284	−0.677957
$756$	0	0
$757$	−42.6136	−1.54882	−0.774408	−	0.632686i	$-0.781952\pi$
$757$	−42.6136	−1.54882	−0.774408	+	0.632686i	$0.781952\pi$
$758$	0	0
$759$	−13.9852	−0.507629
$760$	0	0
$761$	−30.8809	−1.11943	−0.559717	−	0.828684i	$-0.689090\pi$
$761$	−30.8809	−1.11943	−0.559717	+	0.828684i	$0.689090\pi$
$762$	0	0
$763$	4.56779	0.165365
$764$	0	0
$765$	−13.0484	−0.471766
$766$	0	0
$767$	45.9097	1.65770
$768$	0	0
$769$	−5.12867	−0.184945	−0.0924723	−	0.995715i	$-0.529477\pi$
$769$	−5.12867	−0.184945	−0.0924723	+	0.995715i	$0.529477\pi$
$770$	0	0
$771$	−16.5874	−0.597382
$772$	0	0
$773$	42.0787	1.51347	0.756733	−	0.653724i	$-0.226794\pi$
$773$	42.0787	1.51347	0.756733	+	0.653724i	$0.226794\pi$
$774$	0	0
$775$	4.17226	0.149872
$776$	0	0
$777$	4.87614	0.174931
$778$	0	0
$779$	−10.1723	−0.364459
$780$	0	0
$781$	−5.17487	−0.185171
$782$	0	0
$783$	−45.3781	−1.62168
$784$	0	0
$785$	17.3929	0.620780
$786$	0	0
$787$	43.4307	1.54814	0.774068	−	0.633103i	$-0.218219\pi$
$787$	43.4307	1.54814	0.774068	+	0.633103i	$0.218219\pi$
$788$	0	0
$789$	−23.4078	−0.833338
$790$	0	0
$791$	−12.0606	−0.428826
$792$	0	0
$793$	−9.39292	−0.333552
$794$	0	0
$795$	−19.8687	−0.704671
$796$	0	0
$797$	−40.0032	−1.41699	−0.708494	−	0.705717i	$-0.750625\pi$
$797$	−40.0032	−1.41699	−0.708494	+	0.705717i	$0.750625\pi$
$798$	0	0
$799$	−1.29612	−0.0458535
$800$	0	0
$801$	24.9729	0.882375
$802$	0	0
$803$	−5.64064	−0.199054
$804$	0	0
$805$	−2.17226	−0.0765621
$806$	0	0
$807$	−8.68904	−0.305869
$808$	0	0
$809$	−28.7401	−1.01045	−0.505223	−	0.862989i	$-0.668590\pi$
$809$	−28.7401	−1.01045	−0.505223	+	0.862989i	$0.668590\pi$
$810$	0	0
$811$	16.0510	0.563627	0.281814	−	0.959469i	$-0.409064\pi$
$811$	16.0510	0.563627	0.281814	+	0.959469i	$0.409064\pi$
$812$	0	0
$813$	−51.6768	−1.81239
$814$	0	0
$815$	−1.17968	−0.0413223
$816$	0	0
$817$	−6.82032	−0.238613
$818$	0	0
$819$	−18.7645	−0.655685
$820$	0	0
$821$	7.82774	0.273190	0.136595	−	0.990627i	$-0.456384\pi$
$821$	7.82774	0.273190	0.136595	+	0.990627i	$0.456384\pi$
$822$	0	0
$823$	9.78676	0.341145	0.170572	−	0.985345i	$-0.445438\pi$
$823$	9.78676	0.341145	0.170572	+	0.985345i	$0.445438\pi$
$824$	0	0
$825$	4.17226	0.145259
$826$	0	0
$827$	50.1707	1.74461	0.872303	−	0.488965i	$-0.162626\pi$
$827$	50.1707	1.74461	0.872303	+	0.488965i	$0.162626\pi$
$828$	0	0
$829$	−21.2961	−0.739645	−0.369822	−	0.929102i	$-0.620581\pi$
$829$	−21.2961	−0.739645	−0.369822	+	0.929102i	$0.620581\pi$
$830$	0	0
$831$	77.3026	2.68160
$832$	0	0
$833$	−13.1600	−0.455968
$834$	0	0
$835$	24.7268	0.855705
$836$	0	0
$837$	45.3781	1.56850
$838$	0	0
$839$	30.9516	1.06857	0.534284	−	0.845305i	$-0.320581\pi$
$839$	30.9516	1.06857	0.534284	+	0.845305i	$0.320581\pi$
$840$	0	0
$841$	−11.5922	−0.399733
$842$	0	0
$843$	−82.8975	−2.85514
$844$	0	0
$845$	−6.69646	−0.230365
$846$	0	0
$847$	5.94418	0.204245
$848$	0	0
$849$	−66.4265	−2.27975
$850$	0	0
$851$	8.17226	0.280141
$852$	0	0
$853$	38.9219	1.33266	0.666331	−	0.745656i	$-0.267864\pi$
$853$	38.9219	1.33266	0.666331	+	0.745656i	$0.267864\pi$
$854$	0	0
$855$	6.52420	0.223123
$856$	0	0
$857$	52.1245	1.78054	0.890270	−	0.455434i	$-0.150516\pi$
$857$	52.1245	1.78054	0.890270	+	0.455434i	$0.150516\pi$
$858$	0	0
$859$	−4.41998	−0.150808	−0.0754039	−	0.997153i	$-0.524025\pi$
$859$	−4.41998	−0.150808	−0.0754039	+	0.997153i	$0.524025\pi$
$860$	0	0
$861$	−20.3445	−0.693339
$862$	0	0
$863$	−12.4184	−0.422728	−0.211364	−	0.977407i	$-0.567791\pi$
$863$	−12.4184	−0.422728	−0.211364	+	0.977407i	$0.567791\pi$
$864$	0	0
$865$	−17.2026	−0.584905
$866$	0	0
$867$	−40.1197	−1.36254
$868$	0	0
$869$	2.70388	0.0917228
$870$	0	0
$871$	−53.8539	−1.82477
$872$	0	0
$873$	50.4594	1.70779
$874$	0	0
$875$	0.648061	0.0219085
$876$	0	0
$877$	40.3020	1.36090	0.680451	−	0.732794i	$-0.261784\pi$
$877$	40.3020	1.36090	0.680451	+	0.732794i	$0.261784\pi$
$878$	0	0
$879$	57.6210	1.94351
$880$	0	0
$881$	−4.47580	−0.150794	−0.0753968	−	0.997154i	$-0.524022\pi$
$881$	−4.47580	−0.150794	−0.0753968	+	0.997154i	$0.524022\pi$
$882$	0	0
$883$	−27.5455	−0.926981	−0.463491	−	0.886102i	$-0.653403\pi$
$883$	−27.5455	−0.926981	−0.463491	+	0.886102i	$0.653403\pi$
$884$	0	0
$885$	−31.9245	−1.07313
$886$	0	0
$887$	−48.5152	−1.62898	−0.814491	−	0.580176i	$-0.802984\pi$
$887$	−48.5152	−1.62898	−0.814491	+	0.580176i	$0.802984\pi$
$888$	0	0
$889$	−1.16003	−0.0389063
$890$	0	0
$891$	18.9171	0.633747
$892$	0	0
$893$	0.648061	0.0216865
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	−45.9097	−1.53288
$898$	0	0
$899$	−17.4078	−0.580581
$900$	0	0
$901$	12.8761	0.428966
$902$	0	0
$903$	−13.6406	−0.453932
$904$	0	0
$905$	−16.0968	−0.535076
$906$	0	0
$907$	−4.55451	−0.151230	−0.0756150	−	0.997137i	$-0.524092\pi$
$907$	−4.55451	−0.151230	−0.0756150	+	0.997137i	$0.524092\pi$
$908$	0	0
$909$	−8.82032	−0.292552
$910$	0	0
$911$	0.298732	0.00989745	0.00494872	−	0.999988i	$-0.498425\pi$
$911$	0.298732	0.00989745	0.00494872	+	0.999988i	$0.498425\pi$
$912$	0	0
$913$	12.8761	0.426138
$914$	0	0
$915$	6.53162	0.215929
$916$	0	0
$917$	11.1600	0.368537
$918$	0	0
$919$	−0.0754620	−0.00248926	−0.00124463	−	0.999999i	$-0.500396\pi$
$919$	−0.0754620	−0.00248926	−0.00124463	+	0.999999i	$0.500396\pi$
$920$	0	0
$921$	−31.8081	−1.04811
$922$	0	0
$923$	−16.9878	−0.559159
$924$	0	0
$925$	−2.43807	−0.0801632
$926$	0	0
$927$	−24.7268	−0.812134
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	6.58002	0.215651
$932$	0	0
$933$	−38.2691	−1.25287
$934$	0	0
$935$	−2.70388	−0.0884263
$936$	0	0
$937$	42.9581	1.40338	0.701690	−	0.712482i	$-0.252429\pi$
$937$	42.9581	1.40338	0.701690	+	0.712482i	$0.252429\pi$
$938$	0	0
$939$	−84.7252	−2.76490
$940$	0	0
$941$	20.3297	0.662729	0.331364	−	0.943503i	$-0.392491\pi$
$941$	20.3297	0.662729	0.331364	+	0.943503i	$0.392491\pi$
$942$	0	0
$943$	−34.0968	−1.11034
$944$	0	0
$945$	7.04840	0.229284
$946$	0	0
$947$	57.0745	1.85467	0.927337	−	0.374228i	$-0.122092\pi$
$947$	57.0745	1.85467	0.927337	+	0.374228i	$0.122092\pi$
$948$	0	0
$949$	−18.5168	−0.601080
$950$	0	0
$951$	−105.283	−3.41403
$952$	0	0
$953$	44.4594	1.44018	0.720091	−	0.693880i	$-0.244100\pi$
$953$	44.4594	1.44018	0.720091	+	0.693880i	$0.244100\pi$
$954$	0	0
$955$	−0.111635	−0.00361243
$956$	0	0
$957$	−17.4078	−0.562713
$958$	0	0
$959$	−6.55295	−0.211606
$960$	0	0
$961$	−13.5922	−0.438459
$962$	0	0
$963$	−55.8113	−1.79850
$964$	0	0
$965$	6.01809	0.193729
$966$	0	0
$967$	38.1016	1.22527	0.612633	−	0.790368i	$-0.290111\pi$
$967$	38.1016	1.22527	0.612633	+	0.790368i	$0.290111\pi$
$968$	0	0
$969$	−6.17226	−0.198282
$970$	0	0
$971$	−15.3716	−0.493298	−0.246649	−	0.969105i	$-0.579329\pi$
$971$	−15.3716	−0.493298	−0.246649	+	0.969105i	$0.579329\pi$
$972$	0	0
$973$	6.28390	0.201452
$974$	0	0
$975$	13.6965	0.438638
$976$	0	0
$977$	25.1271	0.803888	0.401944	−	0.915664i	$-0.368335\pi$
$977$	25.1271	0.803888	0.401944	+	0.915664i	$0.368335\pi$
$978$	0	0
$979$	5.17487	0.165390
$980$	0	0
$981$	−45.9852	−1.46819
$982$	0	0
$983$	27.3042	0.870868	0.435434	−	0.900221i	$-0.356595\pi$
$983$	27.3042	0.870868	0.435434	+	0.900221i	$0.356595\pi$
$984$	0	0
$985$	−14.7645	−0.470436
$986$	0	0
$987$	1.29612	0.0412560
$988$	0	0
$989$	−22.8613	−0.726947
$990$	0	0
$991$	−57.0697	−1.81288	−0.906440	−	0.422335i	$-0.861211\pi$
$991$	−57.0697	−1.81288	−0.906440	+	0.422335i	$0.861211\pi$
$992$	0	0
$993$	−50.2084	−1.59332
$994$	0	0
$995$	−9.75228	−0.309168
$996$	0	0
$997$	18.1213	0.573906	0.286953	−	0.957945i	$-0.407358\pi$
$997$	18.1213	0.573906	0.286953	+	0.957945i	$0.407358\pi$
$998$	0	0
$999$	−26.5168	−0.838954

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.a.n.1.3	yes	3
4.3	odd	2		3040.2.a.k.1.1	✓	3
8.3	odd	2		6080.2.a.bz.1.3		3
8.5	even	2		6080.2.a.bp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.k.1.1	✓	3	4.3	odd	2
3040.2.a.n.1.3	yes	3	1.1	even	1	trivial
6080.2.a.bp.1.1		3	8.5	even	2
6080.2.a.bz.1.3		3	8.3	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:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.a (trivial)

\(f(q)\)	\(=\)	\(q+3.08613 q^{3} -1.00000 q^{5} -0.648061 q^{7} +6.52420 q^{9} +1.35194 q^{11} +4.43807 q^{13} -3.08613 q^{15} +2.00000 q^{17} -1.00000 q^{19} -2.00000 q^{21} -3.35194 q^{23} +1.00000 q^{25} +10.8761 q^{27} -4.17226 q^{29} +4.17226 q^{31} +4.17226 q^{33} +0.648061 q^{35} -2.43807 q^{37} +13.6965 q^{39} +10.1723 q^{41} +6.82032 q^{43} -6.52420 q^{45} -0.648061 q^{47} -6.58002 q^{49} +6.17226 q^{51} +6.43807 q^{53} -1.35194 q^{55} -3.08613 q^{57} +10.3445 q^{59} -2.11644 q^{61} -4.22808 q^{63} -4.43807 q^{65} -12.1345 q^{67} -10.3445 q^{69} -3.82774 q^{71} -4.17226 q^{73} +3.08613 q^{75} -0.876139 q^{77} +2.00000 q^{79} +13.9926 q^{81} +9.52420 q^{83} -2.00000 q^{85} -12.8761 q^{87} +3.82774 q^{89} -2.87614 q^{91} +12.8761 q^{93} +1.00000 q^{95} +7.73419 q^{97} +8.82032 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 2 q^{15} + 6 q^{17} - 3 q^{19} - 6 q^{21} - 8 q^{23} + 3 q^{25} + 14 q^{27} + 2 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{35} + 2 q^{37} + 10 q^{39}+ \cdots + 14 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 3 * q^5 - 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - 2 * q^15 + 6 * q^17 - 3 * q^19 - 6 * q^21 - 8 * q^23 + 3 * q^25 + 14 * q^27 + 2 * q^29 - 2 * q^31 - 2 * q^33 + 4 * q^35 + 2 * q^37 + 10 * q^39 + 16 * q^41 + 8 * q^43 - 3 * q^45 - 4 * q^47 + 3 * q^49 + 4 * q^51 + 10 * q^53 - 2 * q^55 - 2 * q^57 + 2 * q^59 + 2 * q^61 + 8 * q^63 - 4 * q^65 + 4 * q^67 - 2 * q^69 - 26 * q^71 + 2 * q^73 + 2 * q^75 + 16 * q^77 + 6 * q^79 + 15 * q^81 + 12 * q^83 - 6 * q^85 - 20 * q^87 + 26 * q^89 + 10 * q^91 + 20 * q^93 + 3 * q^95 + 18 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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