LMFDB - Embedded newform 3640.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.0655463357$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
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			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47283 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.11491 q^{9} +O(q^{10})$	$q-2.47283 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.11491 q^{9} +3.58774 q^{11} +1.00000 q^{13} +2.47283 q^{15} -7.70265 q^{17} -1.47283 q^{19} +2.47283 q^{21} -2.87189 q^{23} +1.00000 q^{25} -0.284147 q^{27} +8.29039 q^{29} +6.47283 q^{31} -8.87189 q^{33} +1.00000 q^{35} -7.75698 q^{37} -2.47283 q^{39} -4.34472 q^{41} +0.945668 q^{43} -3.11491 q^{45} +8.15604 q^{47} +1.00000 q^{49} +19.0474 q^{51} +5.17548 q^{53} -3.58774 q^{55} +3.64207 q^{57} -1.45339 q^{59} -9.58774 q^{61} -3.11491 q^{63} -1.00000 q^{65} +6.11491 q^{67} +7.10170 q^{69} +12.1212 q^{71} +13.3315 q^{73} -2.47283 q^{75} -3.58774 q^{77} +9.06058 q^{79} -8.64207 q^{81} -14.4596 q^{83} +7.70265 q^{85} -20.5008 q^{87} -1.60095 q^{89} -1.00000 q^{91} -16.0062 q^{93} +1.47283 q^{95} -9.24926 q^{97} +11.1755 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} + 2 q^{15} - 5 q^{17} + q^{19} + 2 q^{21} + 5 q^{23} + 3 q^{25} + q^{27} - 5 q^{29} + 14 q^{31} - 13 q^{33} + 3 q^{35} - 16 q^{37} - 2 q^{39} + 6 q^{41} - 8 q^{43} - 3 q^{45} + 9 q^{47} + 3 q^{49} + 20 q^{51} - 8 q^{53} + q^{55} + 10 q^{57} - 7 q^{59} - 17 q^{61} - 3 q^{63} - 3 q^{65} + 12 q^{67} - 5 q^{69} + 2 q^{71} + q^{73} - 2 q^{75} + q^{77} + 10 q^{79} - 25 q^{81} - 18 q^{83} + 5 q^{85} - 27 q^{87} - 13 q^{89} - 3 q^{91} - 20 q^{93} - q^{95} - 7 q^{97} + 10 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 - q^11 + 3 * q^13 + 2 * q^15 - 5 * q^17 + q^19 + 2 * q^21 + 5 * q^23 + 3 * q^25 + q^27 - 5 * q^29 + 14 * q^31 - 13 * q^33 + 3 * q^35 - 16 * q^37 - 2 * q^39 + 6 * q^41 - 8 * q^43 - 3 * q^45 + 9 * q^47 + 3 * q^49 + 20 * q^51 - 8 * q^53 + q^55 + 10 * q^57 - 7 * q^59 - 17 * q^61 - 3 * q^63 - 3 * q^65 + 12 * q^67 - 5 * q^69 + 2 * q^71 + q^73 - 2 * q^75 + q^77 + 10 * q^79 - 25 * q^81 - 18 * q^83 + 5 * q^85 - 27 * q^87 - 13 * q^89 - 3 * q^91 - 20 * q^93 - q^95 - 7 * q^97 + 10 * 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
$3$	−2.47283	−1.42769	−0.713846	−	0.700303i	$-0.753048\pi$
$3$	−2.47283	−1.42769	−0.713846	+	0.700303i	$0.753048\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.11491	1.03830
$10$	0	0
$11$	3.58774	1.08174	0.540872	−	0.841105i	$-0.318094\pi$
$11$	3.58774	1.08174	0.540872	+	0.841105i	$0.318094\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.47283	0.638483
$16$	0	0
$17$	−7.70265	−1.86817	−0.934083	−	0.357055i	$-0.883781\pi$
$17$	−7.70265	−1.86817	−0.934083	+	0.357055i	$0.883781\pi$
$18$	0	0
$19$	−1.47283	−0.337891	−0.168946	−	0.985625i	$-0.554036\pi$
$19$	−1.47283	−0.337891	−0.168946	+	0.985625i	$0.554036\pi$
$20$	0	0
$21$	2.47283	0.539617
$22$	0	0
$23$	−2.87189	−0.598830	−0.299415	−	0.954123i	$-0.596792\pi$
$23$	−2.87189	−0.598830	−0.299415	+	0.954123i	$0.596792\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.284147	−0.0546842
$28$	0	0
$29$	8.29039	1.53949	0.769743	−	0.638353i	$-0.220384\pi$
$29$	8.29039	1.53949	0.769743	+	0.638353i	$0.220384\pi$
$30$	0	0
$31$	6.47283	1.16256	0.581278	−	0.813705i	$-0.302553\pi$
$31$	6.47283	1.16256	0.581278	+	0.813705i	$0.302553\pi$
$32$	0	0
$33$	−8.87189	−1.54440
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−7.75698	−1.27524	−0.637620	−	0.770351i	$-0.720081\pi$
$37$	−7.75698	−1.27524	−0.637620	+	0.770351i	$0.720081\pi$
$38$	0	0
$39$	−2.47283	−0.395970
$40$	0	0
$41$	−4.34472	−0.678532	−0.339266	−	0.940691i	$-0.610179\pi$
$41$	−4.34472	−0.678532	−0.339266	+	0.940691i	$0.610179\pi$
$42$	0	0
$43$	0.945668	0.144213	0.0721065	−	0.997397i	$-0.477028\pi$
$43$	0.945668	0.144213	0.0721065	+	0.997397i	$0.477028\pi$
$44$	0	0
$45$	−3.11491	−0.464343
$46$	0	0
$47$	8.15604	1.18968	0.594840	−	0.803844i	$-0.297215\pi$
$47$	8.15604	1.18968	0.594840	+	0.803844i	$0.297215\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	19.0474	2.66717
$52$	0	0
$53$	5.17548	0.710907	0.355454	−	0.934694i	$-0.384326\pi$
$53$	5.17548	0.710907	0.355454	+	0.934694i	$0.384326\pi$
$54$	0	0
$55$	−3.58774	−0.483771
$56$	0	0
$57$	3.64207	0.482404
$58$	0	0
$59$	−1.45339	−0.189215	−0.0946074	−	0.995515i	$-0.530160\pi$
$59$	−1.45339	−0.189215	−0.0946074	+	0.995515i	$0.530160\pi$
$60$	0	0
$61$	−9.58774	−1.22758	−0.613792	−	0.789468i	$-0.710357\pi$
$61$	−9.58774	−1.22758	−0.613792	+	0.789468i	$0.710357\pi$
$62$	0	0
$63$	−3.11491	−0.392441
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	6.11491	0.747055	0.373527	−	0.927619i	$-0.378148\pi$
$67$	6.11491	0.747055	0.373527	+	0.927619i	$0.378148\pi$
$68$	0	0
$69$	7.10170	0.854945
$70$	0	0
$71$	12.1212	1.43852	0.719258	−	0.694743i	$-0.244482\pi$
$71$	12.1212	1.43852	0.719258	+	0.694743i	$0.244482\pi$
$72$	0	0
$73$	13.3315	1.56034	0.780168	−	0.625570i	$-0.215133\pi$
$73$	13.3315	1.56034	0.780168	+	0.625570i	$0.215133\pi$
$74$	0	0
$75$	−2.47283	−0.285538
$76$	0	0
$77$	−3.58774	−0.408861
$78$	0	0
$79$	9.06058	1.01939	0.509697	−	0.860354i	$-0.329757\pi$
$79$	9.06058	1.01939	0.509697	+	0.860354i	$0.329757\pi$
$80$	0	0
$81$	−8.64207	−0.960230
$82$	0	0
$83$	−14.4596	−1.58715	−0.793575	−	0.608472i	$-0.791783\pi$
$83$	−14.4596	−1.58715	−0.793575	+	0.608472i	$0.791783\pi$
$84$	0	0
$85$	7.70265	0.835470
$86$	0	0
$87$	−20.5008	−2.19791
$88$	0	0
$89$	−1.60095	−0.169700	−0.0848499	−	0.996394i	$-0.527041\pi$
$89$	−1.60095	−0.169700	−0.0848499	+	0.996394i	$0.527041\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−16.0062	−1.65977
$94$	0	0
$95$	1.47283	0.151110
$96$	0	0
$97$	−9.24926	−0.939120	−0.469560	−	0.882900i	$-0.655587\pi$
$97$	−9.24926	−0.939120	−0.469560	+	0.882900i	$0.655587\pi$
$98$	0	0
$99$	11.1755	1.12318
$100$	0	0
$101$	13.3315	1.32654	0.663268	−	0.748382i	$-0.269169\pi$
$101$	13.3315	1.32654	0.663268	+	0.748382i	$0.269169\pi$
$102$	0	0
$103$	−1.09546	−0.107939	−0.0539695	−	0.998543i	$-0.517187\pi$
$103$	−1.09546	−0.107939	−0.0539695	+	0.998543i	$0.517187\pi$
$104$	0	0
$105$	−2.47283	−0.241324
$106$	0	0
$107$	−9.66152	−0.934014	−0.467007	−	0.884254i	$-0.654668\pi$
$107$	−9.66152	−0.934014	−0.467007	+	0.884254i	$0.654668\pi$
$108$	0	0
$109$	−13.0279	−1.24785	−0.623924	−	0.781485i	$-0.714463\pi$
$109$	−13.0279	−1.24785	−0.623924	+	0.781485i	$0.714463\pi$
$110$	0	0
$111$	19.1817	1.82065
$112$	0	0
$113$	−4.49452	−0.422808	−0.211404	−	0.977399i	$-0.567804\pi$
$113$	−4.49452	−0.422808	−0.211404	+	0.977399i	$0.567804\pi$
$114$	0	0
$115$	2.87189	0.267805
$116$	0	0
$117$	3.11491	0.287973
$118$	0	0
$119$	7.70265	0.706101
$120$	0	0
$121$	1.87189	0.170172
$122$	0	0
$123$	10.7438	0.968734
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.9387	1.76927	0.884637	−	0.466280i	$-0.154406\pi$
$127$	19.9387	1.76927	0.884637	+	0.466280i	$0.154406\pi$
$128$	0	0
$129$	−2.33848	−0.205892
$130$	0	0
$131$	−15.2841	−1.33538	−0.667691	−	0.744438i	$-0.732717\pi$
$131$	−15.2841	−1.33538	−0.667691	+	0.744438i	$0.732717\pi$
$132$	0	0
$133$	1.47283	0.127711
$134$	0	0
$135$	0.284147	0.0244555
$136$	0	0
$137$	−5.72210	−0.488872	−0.244436	−	0.969665i	$-0.578603\pi$
$137$	−5.72210	−0.488872	−0.244436	+	0.969665i	$0.578603\pi$
$138$	0	0
$139$	3.62263	0.307267	0.153634	−	0.988128i	$-0.450902\pi$
$139$	3.62263	0.307267	0.153634	+	0.988128i	$0.450902\pi$
$140$	0	0
$141$	−20.1685	−1.69850
$142$	0	0
$143$	3.58774	0.300022
$144$	0	0
$145$	−8.29039	−0.688479
$146$	0	0
$147$	−2.47283	−0.203956
$148$	0	0
$149$	−21.8998	−1.79410	−0.897051	−	0.441926i	$-0.854295\pi$
$149$	−21.8998	−1.79410	−0.897051	+	0.441926i	$0.854295\pi$
$150$	0	0
$151$	−7.97359	−0.648882	−0.324441	−	0.945906i	$-0.605176\pi$
$151$	−7.97359	−0.648882	−0.324441	+	0.945906i	$0.605176\pi$
$152$	0	0
$153$	−23.9930	−1.93972
$154$	0	0
$155$	−6.47283	−0.519911
$156$	0	0
$157$	−11.9045	−0.950086	−0.475043	−	0.879963i	$-0.657567\pi$
$157$	−11.9045	−0.950086	−0.475043	+	0.879963i	$0.657567\pi$
$158$	0	0
$159$	−12.7981	−1.01496
$160$	0	0
$161$	2.87189	0.226337
$162$	0	0
$163$	13.9130	1.08975	0.544876	−	0.838517i	$-0.316577\pi$
$163$	13.9130	1.08975	0.544876	+	0.838517i	$0.316577\pi$
$164$	0	0
$165$	8.87189	0.690676
$166$	0	0
$167$	−17.5529	−1.35828	−0.679140	−	0.734008i	$-0.737647\pi$
$167$	−17.5529	−1.35828	−0.679140	+	0.734008i	$0.737647\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.58774	−0.350833
$172$	0	0
$173$	−9.26247	−0.704212	−0.352106	−	0.935960i	$-0.614534\pi$
$173$	−9.26247	−0.704212	−0.352106	+	0.935960i	$0.614534\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	3.59398	0.270140
$178$	0	0
$179$	5.24302	0.391882	0.195941	−	0.980616i	$-0.437224\pi$
$179$	5.24302	0.391882	0.195941	+	0.980616i	$0.437224\pi$
$180$	0	0
$181$	−23.4527	−1.74322	−0.871612	−	0.490197i	$-0.836925\pi$
$181$	−23.4527	−1.74322	−0.871612	+	0.490197i	$0.836925\pi$
$182$	0	0
$183$	23.7089	1.75261
$184$	0	0
$185$	7.75698	0.570305
$186$	0	0
$187$	−27.6351	−2.02088
$188$	0	0
$189$	0.284147	0.0206687
$190$	0	0
$191$	−15.3447	−1.11031	−0.555153	−	0.831749i	$-0.687340\pi$
$191$	−15.3447	−1.11031	−0.555153	+	0.831749i	$0.687340\pi$
$192$	0	0
$193$	6.27094	0.451392	0.225696	−	0.974198i	$-0.427534\pi$
$193$	6.27094	0.451392	0.225696	+	0.974198i	$0.427534\pi$
$194$	0	0
$195$	2.47283	0.177083
$196$	0	0
$197$	−17.9868	−1.28151	−0.640753	−	0.767747i	$-0.721378\pi$
$197$	−17.9868	−1.28151	−0.640753	+	0.767747i	$0.721378\pi$
$198$	0	0
$199$	−4.48604	−0.318007	−0.159003	−	0.987278i	$-0.550828\pi$
$199$	−4.48604	−0.318007	−0.159003	+	0.987278i	$0.550828\pi$
$200$	0	0
$201$	−15.1212	−1.06656
$202$	0	0
$203$	−8.29039	−0.581871
$204$	0	0
$205$	4.34472	0.303449
$206$	0	0
$207$	−8.94567	−0.621767
$208$	0	0
$209$	−5.28415	−0.365512
$210$	0	0
$211$	3.12187	0.214918	0.107459	−	0.994210i	$-0.465729\pi$
$211$	3.12187	0.214918	0.107459	+	0.994210i	$0.465729\pi$
$212$	0	0
$213$	−29.9736	−2.05376
$214$	0	0
$215$	−0.945668	−0.0644940
$216$	0	0
$217$	−6.47283	−0.439405
$218$	0	0
$219$	−32.9666	−2.22768
$220$	0	0
$221$	−7.70265	−0.518136
$222$	0	0
$223$	−3.81756	−0.255642	−0.127821	−	0.991797i	$-0.540798\pi$
$223$	−3.81756	−0.255642	−0.127821	+	0.991797i	$0.540798\pi$
$224$	0	0
$225$	3.11491	0.207661
$226$	0	0
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	−22.6483	−1.49664	−0.748321	−	0.663336i	$-0.769140\pi$
$229$	−22.6483	−1.49664	−0.748321	+	0.663336i	$0.769140\pi$
$230$	0	0
$231$	8.87189	0.583727
$232$	0	0
$233$	19.8998	1.30368	0.651840	−	0.758356i	$-0.273997\pi$
$233$	19.8998	1.30368	0.651840	+	0.758356i	$0.273997\pi$
$234$	0	0
$235$	−8.15604	−0.532041
$236$	0	0
$237$	−22.4053	−1.45538
$238$	0	0
$239$	−20.3510	−1.31639	−0.658197	−	0.752846i	$-0.728681\pi$
$239$	−20.3510	−1.31639	−0.658197	+	0.752846i	$0.728681\pi$
$240$	0	0
$241$	19.1079	1.23085	0.615426	−	0.788195i	$-0.288984\pi$
$241$	19.1079	1.23085	0.615426	+	0.788195i	$0.288984\pi$
$242$	0	0
$243$	22.2229	1.42560
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−1.47283	−0.0937142
$248$	0	0
$249$	35.7563	2.26596
$250$	0	0
$251$	−9.96511	−0.628992	−0.314496	−	0.949259i	$-0.601836\pi$
$251$	−9.96511	−0.628992	−0.314496	+	0.949259i	$0.601836\pi$
$252$	0	0
$253$	−10.3036	−0.647781
$254$	0	0
$255$	−19.0474	−1.19279
$256$	0	0
$257$	−19.7438	−1.23158	−0.615791	−	0.787909i	$-0.711164\pi$
$257$	−19.7438	−1.23158	−0.615791	+	0.787909i	$0.711164\pi$
$258$	0	0
$259$	7.75698	0.481995
$260$	0	0
$261$	25.8238	1.59845
$262$	0	0
$263$	−1.39281	−0.0858844	−0.0429422	−	0.999078i	$-0.513673\pi$
$263$	−1.39281	−0.0858844	−0.0429422	+	0.999078i	$0.513673\pi$
$264$	0	0
$265$	−5.17548	−0.317927
$266$	0	0
$267$	3.95887	0.242279
$268$	0	0
$269$	28.9666	1.76613	0.883063	−	0.469254i	$-0.155477\pi$
$269$	28.9666	1.76613	0.883063	+	0.469254i	$0.155477\pi$
$270$	0	0
$271$	2.52092	0.153135	0.0765676	−	0.997064i	$-0.475604\pi$
$271$	2.52092	0.153135	0.0765676	+	0.997064i	$0.475604\pi$
$272$	0	0
$273$	2.47283	0.149663
$274$	0	0
$275$	3.58774	0.216349
$276$	0	0
$277$	−29.8913	−1.79600	−0.897998	−	0.439999i	$-0.854979\pi$
$277$	−29.8913	−1.79600	−0.897998	+	0.439999i	$0.854979\pi$
$278$	0	0
$279$	20.1623	1.20708
$280$	0	0
$281$	0.486038	0.0289946	0.0144973	−	0.999895i	$-0.495385\pi$
$281$	0.486038	0.0289946	0.0144973	+	0.999895i	$0.495385\pi$
$282$	0	0
$283$	15.3293	0.911231	0.455616	−	0.890177i	$-0.349419\pi$
$283$	15.3293	0.911231	0.455616	+	0.890177i	$0.349419\pi$
$284$	0	0
$285$	−3.64207	−0.215738
$286$	0	0
$287$	4.34472	0.256461
$288$	0	0
$289$	42.3308	2.49005
$290$	0	0
$291$	22.8719	1.34077
$292$	0	0
$293$	24.5808	1.43602	0.718012	−	0.696030i	$-0.245052\pi$
$293$	24.5808	1.43602	0.718012	+	0.696030i	$0.245052\pi$
$294$	0	0
$295$	1.45339	0.0846195
$296$	0	0
$297$	−1.01945	−0.0591543
$298$	0	0
$299$	−2.87189	−0.166086
$300$	0	0
$301$	−0.945668	−0.0545074
$302$	0	0
$303$	−32.9666	−1.89388
$304$	0	0
$305$	9.58774	0.548992
$306$	0	0
$307$	−6.60719	−0.377092	−0.188546	−	0.982064i	$-0.560377\pi$
$307$	−6.60719	−0.377092	−0.188546	+	0.982064i	$0.560377\pi$
$308$	0	0
$309$	2.70889	0.154103
$310$	0	0
$311$	13.0543	0.740243	0.370122	−	0.928983i	$-0.379316\pi$
$311$	13.0543	0.740243	0.370122	+	0.928983i	$0.379316\pi$
$312$	0	0
$313$	−11.9519	−0.675562	−0.337781	−	0.941225i	$-0.609676\pi$
$313$	−11.9519	−0.675562	−0.337781	+	0.941225i	$0.609676\pi$
$314$	0	0
$315$	3.11491	0.175505
$316$	0	0
$317$	−8.45115	−0.474664	−0.237332	−	0.971429i	$-0.576273\pi$
$317$	−8.45115	−0.474664	−0.237332	+	0.971429i	$0.576273\pi$
$318$	0	0
$319$	29.7438	1.66533
$320$	0	0
$321$	23.8913	1.33348
$322$	0	0
$323$	11.3447	0.631237
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	32.2159	1.78154
$328$	0	0
$329$	−8.15604	−0.449657
$330$	0	0
$331$	−26.0863	−1.43383	−0.716915	−	0.697160i	$-0.754447\pi$
$331$	−26.0863	−1.43383	−0.716915	+	0.697160i	$0.754447\pi$
$332$	0	0
$333$	−24.1623	−1.32408
$334$	0	0
$335$	−6.11491	−0.334093
$336$	0	0
$337$	24.3036	1.32390	0.661951	−	0.749547i	$-0.269729\pi$
$337$	24.3036	1.32390	0.661951	+	0.749547i	$0.269729\pi$
$338$	0	0
$339$	11.1142	0.603640
$340$	0	0
$341$	23.2229	1.25759
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−7.10170	−0.382343
$346$	0	0
$347$	−20.4985	−1.10042	−0.550209	−	0.835027i	$-0.685452\pi$
$347$	−20.4985	−1.10042	−0.550209	+	0.835027i	$0.685452\pi$
$348$	0	0
$349$	−0.628870	−0.0336626	−0.0168313	−	0.999858i	$-0.505358\pi$
$349$	−0.628870	−0.0336626	−0.0168313	+	0.999858i	$0.505358\pi$
$350$	0	0
$351$	−0.284147	−0.0151667
$352$	0	0
$353$	−12.7632	−0.679318	−0.339659	−	0.940549i	$-0.610312\pi$
$353$	−12.7632	−0.679318	−0.339659	+	0.940549i	$0.610312\pi$
$354$	0	0
$355$	−12.1212	−0.643324
$356$	0	0
$357$	−19.0474	−1.00809
$358$	0	0
$359$	−22.8759	−1.20734	−0.603672	−	0.797233i	$-0.706296\pi$
$359$	−22.8759	−1.20734	−0.603672	+	0.797233i	$0.706296\pi$
$360$	0	0
$361$	−16.8308	−0.885829
$362$	0	0
$363$	−4.62887	−0.242953
$364$	0	0
$365$	−13.3315	−0.697804
$366$	0	0
$367$	−30.3510	−1.58431	−0.792154	−	0.610322i	$-0.791040\pi$
$367$	−30.3510	−1.58431	−0.792154	+	0.610322i	$0.791040\pi$
$368$	0	0
$369$	−13.5334	−0.704521
$370$	0	0
$371$	−5.17548	−0.268698
$372$	0	0
$373$	−19.2577	−0.997128	−0.498564	−	0.866853i	$-0.666139\pi$
$373$	−19.2577	−0.997128	−0.498564	+	0.866853i	$0.666139\pi$
$374$	0	0
$375$	2.47283	0.127697
$376$	0	0
$377$	8.29039	0.426977
$378$	0	0
$379$	−2.31207	−0.118763	−0.0593816	−	0.998235i	$-0.518913\pi$
$379$	−2.31207	−0.118763	−0.0593816	+	0.998235i	$0.518913\pi$
$380$	0	0
$381$	−49.3051	−2.52598
$382$	0	0
$383$	34.5334	1.76457	0.882287	−	0.470711i	$-0.156003\pi$
$383$	34.5334	1.76457	0.882287	+	0.470711i	$0.156003\pi$
$384$	0	0
$385$	3.58774	0.182848
$386$	0	0
$387$	2.94567	0.149737
$388$	0	0
$389$	26.8906	1.36341	0.681704	−	0.731628i	$-0.261239\pi$
$389$	26.8906	1.36341	0.681704	+	0.731628i	$0.261239\pi$
$390$	0	0
$391$	22.1212	1.11871
$392$	0	0
$393$	37.7952	1.90651
$394$	0	0
$395$	−9.06058	−0.455887
$396$	0	0
$397$	−31.2966	−1.57073	−0.785367	−	0.619031i	$-0.787525\pi$
$397$	−31.2966	−1.57073	−0.785367	+	0.619031i	$0.787525\pi$
$398$	0	0
$399$	−3.64207	−0.182332
$400$	0	0
$401$	−26.9317	−1.34491	−0.672454	−	0.740139i	$-0.734760\pi$
$401$	−26.9317	−1.34491	−0.672454	+	0.740139i	$0.734760\pi$
$402$	0	0
$403$	6.47283	0.322435
$404$	0	0
$405$	8.64207	0.429428
$406$	0	0
$407$	−27.8300	−1.37948
$408$	0	0
$409$	−36.5155	−1.80557	−0.902787	−	0.430088i	$-0.858483\pi$
$409$	−36.5155	−1.80557	−0.902787	+	0.430088i	$0.858483\pi$
$410$	0	0
$411$	14.1498	0.697958
$412$	0	0
$413$	1.45339	0.0715165
$414$	0	0
$415$	14.4596	0.709795
$416$	0	0
$417$	−8.95815	−0.438683
$418$	0	0
$419$	−18.6810	−0.912625	−0.456312	−	0.889820i	$-0.650830\pi$
$419$	−18.6810	−0.912625	−0.456312	+	0.889820i	$0.650830\pi$
$420$	0	0
$421$	7.33152	0.357316	0.178658	−	0.983911i	$-0.442824\pi$
$421$	7.33152	0.357316	0.178658	+	0.983911i	$0.442824\pi$
$422$	0	0
$423$	25.4053	1.23525
$424$	0	0
$425$	−7.70265	−0.373633
$426$	0	0
$427$	9.58774	0.463983
$428$	0	0
$429$	−8.87189	−0.428339
$430$	0	0
$431$	31.6476	1.52441	0.762206	−	0.647335i	$-0.224117\pi$
$431$	31.6476	1.52441	0.762206	+	0.647335i	$0.224117\pi$
$432$	0	0
$433$	23.7849	1.14303	0.571515	−	0.820592i	$-0.306356\pi$
$433$	23.7849	1.14303	0.571515	+	0.820592i	$0.306356\pi$
$434$	0	0
$435$	20.5008	0.982936
$436$	0	0
$437$	4.22982	0.202339
$438$	0	0
$439$	24.3121	1.16035	0.580176	−	0.814491i	$-0.302984\pi$
$439$	24.3121	1.16035	0.580176	+	0.814491i	$0.302984\pi$
$440$	0	0
$441$	3.11491	0.148329
$442$	0	0
$443$	3.13659	0.149024	0.0745119	−	0.997220i	$-0.476260\pi$
$443$	3.13659	0.149024	0.0745119	+	0.997220i	$0.476260\pi$
$444$	0	0
$445$	1.60095	0.0758921
$446$	0	0
$447$	54.1546	2.56142
$448$	0	0
$449$	7.86493	0.371169	0.185584	−	0.982628i	$-0.440582\pi$
$449$	7.86493	0.371169	0.185584	+	0.982628i	$0.440582\pi$
$450$	0	0
$451$	−15.5877	−0.733998
$452$	0	0
$453$	19.7174	0.926403
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	12.4115	0.580587	0.290294	−	0.956938i	$-0.406247\pi$
$457$	12.4115	0.580587	0.290294	+	0.956938i	$0.406247\pi$
$458$	0	0
$459$	2.18869	0.102159
$460$	0	0
$461$	16.9409	0.789018	0.394509	−	0.918892i	$-0.370915\pi$
$461$	16.9409	0.789018	0.394509	+	0.918892i	$0.370915\pi$
$462$	0	0
$463$	12.2709	0.570279	0.285140	−	0.958486i	$-0.407960\pi$
$463$	12.2709	0.570279	0.285140	+	0.958486i	$0.407960\pi$
$464$	0	0
$465$	16.0062	0.742272
$466$	0	0
$467$	−7.21661	−0.333945	−0.166972	−	0.985962i	$-0.553399\pi$
$467$	−7.21661	−0.333945	−0.166972	+	0.985962i	$0.553399\pi$
$468$	0	0
$469$	−6.11491	−0.282360
$470$	0	0
$471$	29.4379	1.35643
$472$	0	0
$473$	3.39281	0.156002
$474$	0	0
$475$	−1.47283	−0.0675783
$476$	0	0
$477$	16.1212	0.738137
$478$	0	0
$479$	−0.655277	−0.0299404	−0.0149702	−	0.999888i	$-0.504765\pi$
$479$	−0.655277	−0.0299404	−0.0149702	+	0.999888i	$0.504765\pi$
$480$	0	0
$481$	−7.75698	−0.353688
$482$	0	0
$483$	−7.10170	−0.323139
$484$	0	0
$485$	9.24926	0.419987
$486$	0	0
$487$	−21.2794	−0.964263	−0.482131	−	0.876099i	$-0.660137\pi$
$487$	−21.2794	−0.964263	−0.482131	+	0.876099i	$0.660137\pi$
$488$	0	0
$489$	−34.4046	−1.55583
$490$	0	0
$491$	6.10866	0.275680	0.137840	−	0.990454i	$-0.455984\pi$
$491$	6.10866	0.275680	0.137840	+	0.990454i	$0.455984\pi$
$492$	0	0
$493$	−63.8580	−2.87602
$494$	0	0
$495$	−11.1755	−0.502301
$496$	0	0
$497$	−12.1212	−0.543708
$498$	0	0
$499$	−5.93871	−0.265853	−0.132927	−	0.991126i	$-0.542437\pi$
$499$	−5.93871	−0.265853	−0.132927	+	0.991126i	$0.542437\pi$
$500$	0	0
$501$	43.4053	1.93921
$502$	0	0
$503$	−41.8385	−1.86549	−0.932744	−	0.360540i	$-0.882592\pi$
$503$	−41.8385	−1.86549	−0.932744	+	0.360540i	$0.882592\pi$
$504$	0	0
$505$	−13.3315	−0.593245
$506$	0	0
$507$	−2.47283	−0.109822
$508$	0	0
$509$	2.87813	0.127571	0.0637855	−	0.997964i	$-0.479683\pi$
$509$	2.87813	0.127571	0.0637855	+	0.997964i	$0.479683\pi$
$510$	0	0
$511$	−13.3315	−0.589752
$512$	0	0
$513$	0.418502	0.0184773
$514$	0	0
$515$	1.09546	0.0482718
$516$	0	0
$517$	29.2617	1.28693
$518$	0	0
$519$	22.9045	1.00540
$520$	0	0
$521$	21.8216	0.956020	0.478010	−	0.878354i	$-0.341358\pi$
$521$	21.8216	0.956020	0.478010	+	0.878354i	$0.341358\pi$
$522$	0	0
$523$	16.6546	0.728253	0.364127	−	0.931349i	$-0.381368\pi$
$523$	16.6546	0.728253	0.364127	+	0.931349i	$0.381368\pi$
$524$	0	0
$525$	2.47283	0.107923
$526$	0	0
$527$	−49.8580	−2.17185
$528$	0	0
$529$	−14.7523	−0.641402
$530$	0	0
$531$	−4.52717	−0.196462
$532$	0	0
$533$	−4.34472	−0.188191
$534$	0	0
$535$	9.66152	0.417704
$536$	0	0
$537$	−12.9651	−0.559486
$538$	0	0
$539$	3.58774	0.154535
$540$	0	0
$541$	7.97359	0.342812	0.171406	−	0.985201i	$-0.445169\pi$
$541$	7.97359	0.342812	0.171406	+	0.985201i	$0.445169\pi$
$542$	0	0
$543$	57.9946	2.48878
$544$	0	0
$545$	13.0279	0.558055
$546$	0	0
$547$	−9.17548	−0.392315	−0.196158	−	0.980572i	$-0.562846\pi$
$547$	−9.17548	−0.392315	−0.196158	+	0.980572i	$0.562846\pi$
$548$	0	0
$549$	−29.8649	−1.27460
$550$	0	0
$551$	−12.2104	−0.520179
$552$	0	0
$553$	−9.06058	−0.385295
$554$	0	0
$555$	−19.1817	−0.814219
$556$	0	0
$557$	−39.8323	−1.68775	−0.843874	−	0.536542i	$-0.819730\pi$
$557$	−39.8323	−1.68775	−0.843874	+	0.536542i	$0.819730\pi$
$558$	0	0
$559$	0.945668	0.0399975
$560$	0	0
$561$	68.3370	2.88519
$562$	0	0
$563$	−19.3991	−0.817573	−0.408786	−	0.912630i	$-0.634048\pi$
$563$	−19.3991	−0.817573	−0.408786	+	0.912630i	$0.634048\pi$
$564$	0	0
$565$	4.49452	0.189086
$566$	0	0
$567$	8.64207	0.362933
$568$	0	0
$569$	−34.7585	−1.45715	−0.728576	−	0.684965i	$-0.759817\pi$
$569$	−34.7585	−1.45715	−0.728576	+	0.684965i	$0.759817\pi$
$570$	0	0
$571$	−1.35721	−0.0567974	−0.0283987	−	0.999597i	$-0.509041\pi$
$571$	−1.35721	−0.0567974	−0.0283987	+	0.999597i	$0.509041\pi$
$572$	0	0
$573$	37.9450	1.58517
$574$	0	0
$575$	−2.87189	−0.119766
$576$	0	0
$577$	17.0668	0.710501	0.355250	−	0.934771i	$-0.384396\pi$
$577$	17.0668	0.710501	0.355250	+	0.934771i	$0.384396\pi$
$578$	0	0
$579$	−15.5070	−0.644449
$580$	0	0
$581$	14.4596	0.599887
$582$	0	0
$583$	18.5683	0.769020
$584$	0	0
$585$	−3.11491	−0.128786
$586$	0	0
$587$	6.58078	0.271618	0.135809	−	0.990735i	$-0.456637\pi$
$587$	6.58078	0.271618	0.135809	+	0.990735i	$0.456637\pi$
$588$	0	0
$589$	−9.53341	−0.392817
$590$	0	0
$591$	44.4784	1.82960
$592$	0	0
$593$	−1.28415	−0.0527336	−0.0263668	−	0.999652i	$-0.508394\pi$
$593$	−1.28415	−0.0527336	−0.0263668	+	0.999652i	$0.508394\pi$
$594$	0	0
$595$	−7.70265	−0.315778
$596$	0	0
$597$	11.0932	0.454016
$598$	0	0
$599$	−43.5349	−1.77879	−0.889394	−	0.457141i	$-0.848874\pi$
$599$	−43.5349	−1.77879	−0.889394	+	0.457141i	$0.848874\pi$
$600$	0	0
$601$	24.3774	0.994374	0.497187	−	0.867643i	$-0.334366\pi$
$601$	24.3774	0.994374	0.497187	+	0.867643i	$0.334366\pi$
$602$	0	0
$603$	19.0474	0.775669
$604$	0	0
$605$	−1.87189	−0.0761031
$606$	0	0
$607$	21.0621	0.854884	0.427442	−	0.904043i	$-0.359415\pi$
$607$	21.0621	0.854884	0.427442	+	0.904043i	$0.359415\pi$
$608$	0	0
$609$	20.5008	0.830733
$610$	0	0
$611$	8.15604	0.329958
$612$	0	0
$613$	17.5877	0.710362	0.355181	−	0.934798i	$-0.384419\pi$
$613$	17.5877	0.710362	0.355181	+	0.934798i	$0.384419\pi$
$614$	0	0
$615$	−10.7438	−0.433231
$616$	0	0
$617$	34.9604	1.40745	0.703726	−	0.710471i	$-0.251518\pi$
$617$	34.9604	1.40745	0.703726	+	0.710471i	$0.251518\pi$
$618$	0	0
$619$	22.0342	0.885628	0.442814	−	0.896614i	$-0.353980\pi$
$619$	22.0342	0.885628	0.442814	+	0.896614i	$0.353980\pi$
$620$	0	0
$621$	0.816039	0.0327465
$622$	0	0
$623$	1.60095	0.0641405
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	13.0668	0.521838
$628$	0	0
$629$	59.7493	2.38236
$630$	0	0
$631$	34.8759	1.38839	0.694194	−	0.719788i	$-0.255761\pi$
$631$	34.8759	1.38839	0.694194	+	0.719788i	$0.255761\pi$
$632$	0	0
$633$	−7.71986	−0.306837
$634$	0	0
$635$	−19.9387	−0.791243
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	37.7563	1.49361
$640$	0	0
$641$	−33.5521	−1.32523	−0.662615	−	0.748960i	$-0.730553\pi$
$641$	−33.5521	−1.32523	−0.662615	+	0.748960i	$0.730553\pi$
$642$	0	0
$643$	−36.8106	−1.45167	−0.725834	−	0.687870i	$-0.758546\pi$
$643$	−36.8106	−1.45167	−0.725834	+	0.687870i	$0.758546\pi$
$644$	0	0
$645$	2.33848	0.0920775
$646$	0	0
$647$	−41.0885	−1.61536	−0.807678	−	0.589624i	$-0.799276\pi$
$647$	−41.0885	−1.61536	−0.807678	+	0.589624i	$0.799276\pi$
$648$	0	0
$649$	−5.21438	−0.204682
$650$	0	0
$651$	16.0062	0.627334
$652$	0	0
$653$	−21.6226	−0.846159	−0.423079	−	0.906093i	$-0.639051\pi$
$653$	−21.6226	−0.846159	−0.423079	+	0.906093i	$0.639051\pi$
$654$	0	0
$655$	15.2841	0.597201
$656$	0	0
$657$	41.5264	1.62010
$658$	0	0
$659$	8.35320	0.325394	0.162697	−	0.986676i	$-0.447981\pi$
$659$	8.35320	0.325394	0.162697	+	0.986676i	$0.447981\pi$
$660$	0	0
$661$	17.0691	0.663909	0.331955	−	0.943295i	$-0.392292\pi$
$661$	17.0691	0.663909	0.331955	+	0.943295i	$0.392292\pi$
$662$	0	0
$663$	19.0474	0.739739
$664$	0	0
$665$	−1.47283	−0.0571140
$666$	0	0
$667$	−23.8091	−0.921891
$668$	0	0
$669$	9.44018	0.364979
$670$	0	0
$671$	−34.3983	−1.32793
$672$	0	0
$673$	30.8410	1.18883	0.594417	−	0.804157i	$-0.297383\pi$
$673$	30.8410	1.18883	0.594417	+	0.804157i	$0.297383\pi$
$674$	0	0
$675$	−0.284147	−0.0109368
$676$	0	0
$677$	−16.1560	−0.620927	−0.310463	−	0.950585i	$-0.600484\pi$
$677$	−16.1560	−0.620927	−0.310463	+	0.950585i	$0.600484\pi$
$678$	0	0
$679$	9.24926	0.354954
$680$	0	0
$681$	14.8370	0.568555
$682$	0	0
$683$	8.30136	0.317643	0.158821	−	0.987307i	$-0.449231\pi$
$683$	8.30136	0.317643	0.158821	+	0.987307i	$0.449231\pi$
$684$	0	0
$685$	5.72210	0.218630
$686$	0	0
$687$	56.0055	2.13674
$688$	0	0
$689$	5.17548	0.197170
$690$	0	0
$691$	−34.9534	−1.32969	−0.664846	−	0.746981i	$-0.731503\pi$
$691$	−34.9534	−1.32969	−0.664846	+	0.746981i	$0.731503\pi$
$692$	0	0
$693$	−11.1755	−0.424521
$694$	0	0
$695$	−3.62263	−0.137414
$696$	0	0
$697$	33.4659	1.26761
$698$	0	0
$699$	−49.2089	−1.86125
$700$	0	0
$701$	−6.04113	−0.228170	−0.114085	−	0.993471i	$-0.536394\pi$
$701$	−6.04113	−0.228170	−0.114085	+	0.993471i	$0.536394\pi$
$702$	0	0
$703$	11.4247	0.430892
$704$	0	0
$705$	20.1685	0.759590
$706$	0	0
$707$	−13.3315	−0.501383
$708$	0	0
$709$	10.6157	0.398680	0.199340	−	0.979930i	$-0.436120\pi$
$709$	10.6157	0.398680	0.199340	+	0.979930i	$0.436120\pi$
$710$	0	0
$711$	28.2229	1.05844
$712$	0	0
$713$	−18.5893	−0.696173
$714$	0	0
$715$	−3.58774	−0.134174
$716$	0	0
$717$	50.3246	1.87941
$718$	0	0
$719$	2.43322	0.0907439	0.0453719	−	0.998970i	$-0.485553\pi$
$719$	2.43322	0.0907439	0.0453719	+	0.998970i	$0.485553\pi$
$720$	0	0
$721$	1.09546	0.0407971
$722$	0	0
$723$	−47.2508	−1.75728
$724$	0	0
$725$	8.29039	0.307897
$726$	0	0
$727$	−3.44892	−0.127913	−0.0639566	−	0.997953i	$-0.520372\pi$
$727$	−3.44892	−0.127913	−0.0639566	+	0.997953i	$0.520372\pi$
$728$	0	0
$729$	−29.0272	−1.07508
$730$	0	0
$731$	−7.28415	−0.269414
$732$	0	0
$733$	13.1755	0.486648	0.243324	−	0.969945i	$-0.421762\pi$
$733$	13.1755	0.486648	0.243324	+	0.969945i	$0.421762\pi$
$734$	0	0
$735$	2.47283	0.0912119
$736$	0	0
$737$	21.9387	0.808123
$738$	0	0
$739$	37.9736	1.39688	0.698441	−	0.715668i	$-0.253878\pi$
$739$	37.9736	1.39688	0.698441	+	0.715668i	$0.253878\pi$
$740$	0	0
$741$	3.64207	0.133795
$742$	0	0
$743$	−8.96039	−0.328725	−0.164362	−	0.986400i	$-0.552557\pi$
$743$	−8.96039	−0.328725	−0.164362	+	0.986400i	$0.552557\pi$
$744$	0	0
$745$	21.8998	0.802347
$746$	0	0
$747$	−45.0404	−1.64794
$748$	0	0
$749$	9.66152	0.353024
$750$	0	0
$751$	−13.5272	−0.493613	−0.246807	−	0.969065i	$-0.579381\pi$
$751$	−13.5272	−0.493613	−0.246807	+	0.969065i	$0.579381\pi$
$752$	0	0
$753$	24.6421	0.898007
$754$	0	0
$755$	7.97359	0.290189
$756$	0	0
$757$	29.1755	1.06040	0.530200	−	0.847872i	$-0.322117\pi$
$757$	29.1755	1.06040	0.530200	+	0.847872i	$0.322117\pi$
$758$	0	0
$759$	25.4791	0.924832
$760$	0	0
$761$	0.919978	0.0333492	0.0166746	−	0.999861i	$-0.494692\pi$
$761$	0.919978	0.0333492	0.0166746	+	0.999861i	$0.494692\pi$
$762$	0	0
$763$	13.0279	0.471643
$764$	0	0
$765$	23.9930	0.867470
$766$	0	0
$767$	−1.45339	−0.0524788
$768$	0	0
$769$	−9.07530	−0.327264	−0.163632	−	0.986521i	$-0.552321\pi$
$769$	−9.07530	−0.327264	−0.163632	+	0.986521i	$0.552321\pi$
$770$	0	0
$771$	48.8231	1.75832
$772$	0	0
$773$	−43.8913	−1.57866	−0.789331	−	0.613968i	$-0.789572\pi$
$773$	−43.8913	−1.57866	−0.789331	+	0.613968i	$0.789572\pi$
$774$	0	0
$775$	6.47283	0.232511
$776$	0	0
$777$	−19.1817	−0.688141
$778$	0	0
$779$	6.39905	0.229270
$780$	0	0
$781$	43.4876	1.55611
$782$	0	0
$783$	−2.35569	−0.0841856
$784$	0	0
$785$	11.9045	0.424891
$786$	0	0
$787$	33.9083	1.20870	0.604350	−	0.796719i	$-0.293433\pi$
$787$	33.9083	1.20870	0.604350	+	0.796719i	$0.293433\pi$
$788$	0	0
$789$	3.44419	0.122616
$790$	0	0
$791$	4.49452	0.159807
$792$	0	0
$793$	−9.58774	−0.340471
$794$	0	0
$795$	12.7981	0.453902
$796$	0	0
$797$	−17.1887	−0.608855	−0.304427	−	0.952536i	$-0.598465\pi$
$797$	−17.1887	−0.608855	−0.304427	+	0.952536i	$0.598465\pi$
$798$	0	0
$799$	−62.8231	−2.22252
$800$	0	0
$801$	−4.98680	−0.176200
$802$	0	0
$803$	47.8300	1.68789
$804$	0	0
$805$	−2.87189	−0.101221
$806$	0	0
$807$	−71.6297	−2.52148
$808$	0	0
$809$	18.5132	0.650891	0.325446	−	0.945561i	$-0.394486\pi$
$809$	18.5132	0.650891	0.325446	+	0.945561i	$0.394486\pi$
$810$	0	0
$811$	14.5683	0.511562	0.255781	−	0.966735i	$-0.417667\pi$
$811$	14.5683	0.511562	0.255781	+	0.966735i	$0.417667\pi$
$812$	0	0
$813$	−6.23382	−0.218630
$814$	0	0
$815$	−13.9130	−0.487352
$816$	0	0
$817$	−1.39281	−0.0487283
$818$	0	0
$819$	−3.11491	−0.108844
$820$	0	0
$821$	−45.8121	−1.59885	−0.799427	−	0.600763i	$-0.794863\pi$
$821$	−45.8121	−1.59885	−0.799427	+	0.600763i	$0.794863\pi$
$822$	0	0
$823$	10.0085	0.348874	0.174437	−	0.984668i	$-0.444190\pi$
$823$	10.0085	0.348874	0.174437	+	0.984668i	$0.444190\pi$
$824$	0	0
$825$	−8.87189	−0.308880
$826$	0	0
$827$	43.0529	1.49710	0.748548	−	0.663081i	$-0.230751\pi$
$827$	43.0529	1.49710	0.748548	+	0.663081i	$0.230751\pi$
$828$	0	0
$829$	27.8774	0.968223	0.484111	−	0.875006i	$-0.339143\pi$
$829$	27.8774	0.968223	0.484111	+	0.875006i	$0.339143\pi$
$830$	0	0
$831$	73.9163	2.56413
$832$	0	0
$833$	−7.70265	−0.266881
$834$	0	0
$835$	17.5529	0.607442
$836$	0	0
$837$	−1.83924	−0.0635734
$838$	0	0
$839$	−33.8859	−1.16987	−0.584935	−	0.811080i	$-0.698880\pi$
$839$	−33.8859	−1.16987	−0.584935	+	0.811080i	$0.698880\pi$
$840$	0	0
$841$	39.7306	1.37002
$842$	0	0
$843$	−1.20189	−0.0413953
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−1.87189	−0.0643189
$848$	0	0
$849$	−37.9068	−1.30096
$850$	0	0
$851$	22.2772	0.763652
$852$	0	0
$853$	21.6740	0.742104	0.371052	−	0.928612i	$-0.378997\pi$
$853$	21.6740	0.742104	0.371052	+	0.928612i	$0.378997\pi$
$854$	0	0
$855$	4.58774	0.156897
$856$	0	0
$857$	1.62735	0.0555893	0.0277947	−	0.999614i	$-0.491152\pi$
$857$	1.62735	0.0555893	0.0277947	+	0.999614i	$0.491152\pi$
$858$	0	0
$859$	−0.260231	−0.00887897	−0.00443949	−	0.999990i	$-0.501413\pi$
$859$	−0.260231	−0.00887897	−0.00443949	+	0.999990i	$0.501413\pi$
$860$	0	0
$861$	−10.7438	−0.366147
$862$	0	0
$863$	37.8796	1.28944	0.644719	−	0.764420i	$-0.276974\pi$
$863$	37.8796	1.28944	0.644719	+	0.764420i	$0.276974\pi$
$864$	0	0
$865$	9.26247	0.314933
$866$	0	0
$867$	−104.677	−3.55502
$868$	0	0
$869$	32.5070	1.10272
$870$	0	0
$871$	6.11491	0.207196
$872$	0	0
$873$	−28.8106	−0.975091
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−7.19565	−0.242980	−0.121490	−	0.992593i	$-0.538767\pi$
$877$	−7.19565	−0.242980	−0.121490	+	0.992593i	$0.538767\pi$
$878$	0	0
$879$	−60.7842	−2.05020
$880$	0	0
$881$	35.9347	1.21067	0.605335	−	0.795971i	$-0.293039\pi$
$881$	35.9347	1.21067	0.605335	+	0.795971i	$0.293039\pi$
$882$	0	0
$883$	12.3649	0.416112	0.208056	−	0.978117i	$-0.433286\pi$
$883$	12.3649	0.416112	0.208056	+	0.978117i	$0.433286\pi$
$884$	0	0
$885$	−3.59398	−0.120810
$886$	0	0
$887$	−13.3836	−0.449378	−0.224689	−	0.974431i	$-0.572137\pi$
$887$	−13.3836	−0.449378	−0.224689	+	0.974431i	$0.572137\pi$
$888$	0	0
$889$	−19.9387	−0.668723
$890$	0	0
$891$	−31.0055	−1.03872
$892$	0	0
$893$	−12.0125	−0.401983
$894$	0	0
$895$	−5.24302	−0.175255
$896$	0	0
$897$	7.10170	0.237119
$898$	0	0
$899$	53.6623	1.78974
$900$	0	0
$901$	−39.8649	−1.32809
$902$	0	0
$903$	2.33848	0.0778197
$904$	0	0
$905$	23.4527	0.779593
$906$	0	0
$907$	35.3230	1.17288	0.586441	−	0.809992i	$-0.300529\pi$
$907$	35.3230	1.17288	0.586441	+	0.809992i	$0.300529\pi$
$908$	0	0
$909$	41.5264	1.37735
$910$	0	0
$911$	18.3796	0.608944	0.304472	−	0.952521i	$-0.401520\pi$
$911$	18.3796	0.608944	0.304472	+	0.952521i	$0.401520\pi$
$912$	0	0
$913$	−51.8774	−1.71689
$914$	0	0
$915$	−23.7089	−0.783792
$916$	0	0
$917$	15.2841	0.504727
$918$	0	0
$919$	6.95887	0.229552	0.114776	−	0.993391i	$-0.463385\pi$
$919$	6.95887	0.229552	0.114776	+	0.993391i	$0.463385\pi$
$920$	0	0
$921$	16.3385	0.538371
$922$	0	0
$923$	12.1212	0.398973
$924$	0	0
$925$	−7.75698	−0.255048
$926$	0	0
$927$	−3.41226	−0.112073
$928$	0	0
$929$	14.9758	0.491341	0.245670	−	0.969353i	$-0.420992\pi$
$929$	14.9758	0.491341	0.245670	+	0.969353i	$0.420992\pi$
$930$	0	0
$931$	−1.47283	−0.0482702
$932$	0	0
$933$	−32.2812	−1.05684
$934$	0	0
$935$	27.6351	0.903765
$936$	0	0
$937$	−9.90606	−0.323617	−0.161808	−	0.986822i	$-0.551733\pi$
$937$	−9.90606	−0.323617	−0.161808	+	0.986822i	$0.551733\pi$
$938$	0	0
$939$	29.5551	0.964494
$940$	0	0
$941$	−36.3223	−1.18407	−0.592037	−	0.805911i	$-0.701676\pi$
$941$	−36.3223	−1.18407	−0.592037	+	0.805911i	$0.701676\pi$
$942$	0	0
$943$	12.4776	0.406325
$944$	0	0
$945$	−0.284147	−0.00924331
$946$	0	0
$947$	−17.4603	−0.567385	−0.283693	−	0.958915i	$-0.591560\pi$
$947$	−17.4603	−0.567385	−0.283693	+	0.958915i	$0.591560\pi$
$948$	0	0
$949$	13.3315	0.432759
$950$	0	0
$951$	20.8983	0.677674
$952$	0	0
$953$	−21.9861	−0.712199	−0.356099	−	0.934448i	$-0.615894\pi$
$953$	−21.9861	−0.712199	−0.356099	+	0.934448i	$0.615894\pi$
$954$	0	0
$955$	15.3447	0.496544
$956$	0	0
$957$	−73.5514	−2.37758
$958$	0	0
$959$	5.72210	0.184776
$960$	0	0
$961$	10.8976	0.351535
$962$	0	0
$963$	−30.0947	−0.969790
$964$	0	0
$965$	−6.27094	−0.201869
$966$	0	0
$967$	−15.8044	−0.508234	−0.254117	−	0.967174i	$-0.581785\pi$
$967$	−15.8044	−0.508234	−0.254117	+	0.967174i	$0.581785\pi$
$968$	0	0
$969$	−28.0536	−0.901212
$970$	0	0
$971$	−5.69641	−0.182806	−0.0914032	−	0.995814i	$-0.529135\pi$
$971$	−5.69641	−0.182806	−0.0914032	+	0.995814i	$0.529135\pi$
$972$	0	0
$973$	−3.62263	−0.116136
$974$	0	0
$975$	−2.47283	−0.0791941
$976$	0	0
$977$	−34.1359	−1.09210	−0.546052	−	0.837752i	$-0.683870\pi$
$977$	−34.1359	−1.09210	−0.546052	+	0.837752i	$0.683870\pi$
$978$	0	0
$979$	−5.74378	−0.183572
$980$	0	0
$981$	−40.5808	−1.29564
$982$	0	0
$983$	−11.2841	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$983$	−11.2841	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$984$	0	0
$985$	17.9868	0.573107
$986$	0	0
$987$	20.1685	0.641971
$988$	0	0
$989$	−2.71585	−0.0863591
$990$	0	0
$991$	−11.8587	−0.376704	−0.188352	−	0.982102i	$-0.560315\pi$
$991$	−11.8587	−0.376704	−0.188352	+	0.982102i	$0.560315\pi$
$992$	0	0
$993$	64.5070	2.04707
$994$	0	0
$995$	4.48604	0.142217
$996$	0	0
$997$	37.5591	1.18951	0.594754	−	0.803908i	$-0.297249\pi$
$997$	37.5591	1.18951	0.594754	+	0.803908i	$0.297249\pi$
$998$	0	0
$999$	2.20412	0.0697354

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.o.1.1	✓	3
4.3	odd	2		7280.2.a.bp.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.o.1.1	✓	3	1.1	even	1	trivial
7280.2.a.bp.1.3		3	4.3	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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47283 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.11491 q^{9} +O(q^{10})\)	\(q-2.47283 q^{3} -1.00000 q^{5} -1.00000 q^{7} +3.11491 q^{9} +3.58774 q^{11} +1.00000 q^{13} +2.47283 q^{15} -7.70265 q^{17} -1.47283 q^{19} +2.47283 q^{21} -2.87189 q^{23} +1.00000 q^{25} -0.284147 q^{27} +8.29039 q^{29} +6.47283 q^{31} -8.87189 q^{33} +1.00000 q^{35} -7.75698 q^{37} -2.47283 q^{39} -4.34472 q^{41} +0.945668 q^{43} -3.11491 q^{45} +8.15604 q^{47} +1.00000 q^{49} +19.0474 q^{51} +5.17548 q^{53} -3.58774 q^{55} +3.64207 q^{57} -1.45339 q^{59} -9.58774 q^{61} -3.11491 q^{63} -1.00000 q^{65} +6.11491 q^{67} +7.10170 q^{69} +12.1212 q^{71} +13.3315 q^{73} -2.47283 q^{75} -3.58774 q^{77} +9.06058 q^{79} -8.64207 q^{81} -14.4596 q^{83} +7.70265 q^{85} -20.5008 q^{87} -1.60095 q^{89} -1.00000 q^{91} -16.0062 q^{93} +1.47283 q^{95} -9.24926 q^{97} +11.1755 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q - 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - q^{11} + 3 q^{13} + 2 q^{15} - 5 q^{17} + q^{19} + 2 q^{21} + 5 q^{23} + 3 q^{25} + q^{27} - 5 q^{29} + 14 q^{31} - 13 q^{33} + 3 q^{35} - 16 q^{37} - 2 q^{39} + 6 q^{41} - 8 q^{43} - 3 q^{45} + 9 q^{47} + 3 q^{49} + 20 q^{51} - 8 q^{53} + q^{55} + 10 q^{57} - 7 q^{59} - 17 q^{61} - 3 q^{63} - 3 q^{65} + 12 q^{67} - 5 q^{69} + 2 q^{71} + q^{73} - 2 q^{75} + q^{77} + 10 q^{79} - 25 q^{81} - 18 q^{83} + 5 q^{85} - 27 q^{87} - 13 q^{89} - 3 q^{91} - 20 q^{93} - q^{95} - 7 q^{97} + 10 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 - q^11 + 3 * q^13 + 2 * q^15 - 5 * q^17 + q^19 + 2 * q^21 + 5 * q^23 + 3 * q^25 + q^27 - 5 * q^29 + 14 * q^31 - 13 * q^33 + 3 * q^35 - 16 * q^37 - 2 * q^39 + 6 * q^41 - 8 * q^43 - 3 * q^45 + 9 * q^47 + 3 * q^49 + 20 * q^51 - 8 * q^53 + q^55 + 10 * q^57 - 7 * q^59 - 17 * q^61 - 3 * q^63 - 3 * q^65 + 12 * q^67 - 5 * q^69 + 2 * q^71 + q^73 - 2 * q^75 + q^77 + 10 * q^79 - 25 * q^81 - 18 * q^83 + 5 * q^85 - 27 * q^87 - 13 * q^89 - 3 * q^91 - 20 * q^93 - q^95 - 7 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more