LMFDB - Embedded newform 3640.2.a.y.1.3

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:	$5$
Coefficient field:	5.5.1194649.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 3x^{2} + 10x + 4 $ x^5 - 8x^3 - 3x^2 + 10*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.402509$ 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)

$f(q)$	$=$	$q+0.402509 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.83799 q^{9} +O(q^{10})$	$q+0.402509 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.83799 q^{9} +1.78532 q^{11} -1.00000 q^{13} -0.402509 q^{15} +7.18532 q^{17} -5.62331 q^{19} +0.402509 q^{21} +1.08564 q^{23} +1.00000 q^{25} -2.34984 q^{27} -8.01153 q^{29} -7.90219 q^{31} +0.718607 q^{33} -1.00000 q^{35} -9.53516 q^{37} -0.402509 q^{39} +1.73265 q^{41} +10.9564 q^{43} +2.83799 q^{45} +0.542313 q^{47} +1.00000 q^{49} +2.89215 q^{51} +6.30470 q^{53} -1.78532 q^{55} -2.26343 q^{57} -1.12112 q^{59} -2.71861 q^{61} -2.83799 q^{63} +1.00000 q^{65} -10.8567 q^{67} +0.436978 q^{69} -4.26169 q^{71} +14.1130 q^{73} +0.402509 q^{75} +1.78532 q^{77} -0.819299 q^{79} +7.56813 q^{81} -0.910352 q^{83} -7.18532 q^{85} -3.22471 q^{87} -9.09818 q^{89} -1.00000 q^{91} -3.18070 q^{93} +5.62331 q^{95} -10.2183 q^{97} -5.06671 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 5 q^{7} + q^{9}+O(q^{10}) $ 5 * q - 5 * q^5 + 5 * q^7 + q^9	$ 5 q - 5 q^{5} + 5 q^{7} + q^{9} - 7 q^{11} - 5 q^{13} + q^{17} + 3 q^{19} - 5 q^{23} + 5 q^{25} - 9 q^{27} - 9 q^{29} + 6 q^{31} + q^{33} - 5 q^{35} - 10 q^{37} - 8 q^{41} + 6 q^{43} - q^{45} - 13 q^{47} + 5 q^{49} - 4 q^{51} - 16 q^{53} + 7 q^{55} - 10 q^{57} - q^{59} - 11 q^{61} + q^{63} + 5 q^{65} - 30 q^{67} - 15 q^{69} - 12 q^{71} + 23 q^{73} - 7 q^{77} + 2 q^{79} - 11 q^{81} - 2 q^{83} - q^{85} - 5 q^{87} - 25 q^{89} - 5 q^{91} - 22 q^{93} - 3 q^{95} - 5 q^{97} - 12 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 + 5 * q^7 + q^9 - 7 * q^11 - 5 * q^13 + q^17 + 3 * q^19 - 5 * q^23 + 5 * q^25 - 9 * q^27 - 9 * q^29 + 6 * q^31 + q^33 - 5 * q^35 - 10 * q^37 - 8 * q^41 + 6 * q^43 - q^45 - 13 * q^47 + 5 * q^49 - 4 * q^51 - 16 * q^53 + 7 * q^55 - 10 * q^57 - q^59 - 11 * q^61 + q^63 + 5 * q^65 - 30 * q^67 - 15 * q^69 - 12 * q^71 + 23 * q^73 - 7 * q^77 + 2 * q^79 - 11 * q^81 - 2 * q^83 - q^85 - 5 * q^87 - 25 * q^89 - 5 * q^91 - 22 * q^93 - 3 * q^95 - 5 * q^97 - 12 * 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
$3$	0.402509	0.232389	0.116194	−	0.993226i	$-0.462930\pi$
$3$	0.402509	0.232389	0.116194	+	0.993226i	$0.462930\pi$
$4$	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
$9$	−2.83799	−0.945996
$10$	0	0
$11$	1.78532	0.538294	0.269147	−	0.963099i	$-0.413258\pi$
$11$	1.78532	0.538294	0.269147	+	0.963099i	$0.413258\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.402509	−0.103927
$16$	0	0
$17$	7.18532	1.74270	0.871348	−	0.490666i	$-0.163247\pi$
$17$	7.18532	1.74270	0.871348	+	0.490666i	$0.163247\pi$
$18$	0	0
$19$	−5.62331	−1.29007	−0.645037	−	0.764151i	$-0.723158\pi$
$19$	−5.62331	−1.29007	−0.645037	+	0.764151i	$0.723158\pi$
$20$	0	0
$21$	0.402509	0.0878347
$22$	0	0
$23$	1.08564	0.226371	0.113185	−	0.993574i	$-0.463895\pi$
$23$	1.08564	0.226371	0.113185	+	0.993574i	$0.463895\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.34984	−0.452227
$28$	0	0
$29$	−8.01153	−1.48770	−0.743852	−	0.668344i	$-0.767003\pi$
$29$	−8.01153	−1.48770	−0.743852	+	0.668344i	$0.767003\pi$
$30$	0	0
$31$	−7.90219	−1.41927	−0.709637	−	0.704567i	$-0.751141\pi$
$31$	−7.90219	−1.41927	−0.709637	+	0.704567i	$0.751141\pi$
$32$	0	0
$33$	0.718607	0.125093
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−9.53516	−1.56757	−0.783785	−	0.621032i	$-0.786714\pi$
$37$	−9.53516	−1.56757	−0.783785	+	0.621032i	$0.786714\pi$
$38$	0	0
$39$	−0.402509	−0.0644530
$40$	0	0
$41$	1.73265	0.270595	0.135297	−	0.990805i	$-0.456801\pi$
$41$	1.73265	0.270595	0.135297	+	0.990805i	$0.456801\pi$
$42$	0	0
$43$	10.9564	1.67083	0.835414	−	0.549621i	$-0.185228\pi$
$43$	10.9564	1.67083	0.835414	+	0.549621i	$0.185228\pi$
$44$	0	0
$45$	2.83799	0.423062
$46$	0	0
$47$	0.542313	0.0791044	0.0395522	−	0.999218i	$-0.487407\pi$
$47$	0.542313	0.0791044	0.0395522	+	0.999218i	$0.487407\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.89215	0.404983
$52$	0	0
$53$	6.30470	0.866017	0.433008	−	0.901390i	$-0.357452\pi$
$53$	6.30470	0.866017	0.433008	+	0.901390i	$0.357452\pi$
$54$	0	0
$55$	−1.78532	−0.240732
$56$	0	0
$57$	−2.26343	−0.299799
$58$	0	0
$59$	−1.12112	−0.145957	−0.0729784	−	0.997334i	$-0.523250\pi$
$59$	−1.12112	−0.145957	−0.0729784	+	0.997334i	$0.523250\pi$
$60$	0	0
$61$	−2.71861	−0.348082	−0.174041	−	0.984738i	$-0.555683\pi$
$61$	−2.71861	−0.348082	−0.174041	+	0.984738i	$0.555683\pi$
$62$	0	0
$63$	−2.83799	−0.357553
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−10.8567	−1.32635	−0.663177	−	0.748463i	$-0.730792\pi$
$67$	−10.8567	−1.32635	−0.663177	+	0.748463i	$0.730792\pi$
$68$	0	0
$69$	0.436978	0.0526060
$70$	0	0
$71$	−4.26169	−0.505770	−0.252885	−	0.967496i	$-0.581379\pi$
$71$	−4.26169	−0.505770	−0.252885	+	0.967496i	$0.581379\pi$
$72$	0	0
$73$	14.1130	1.65180	0.825898	−	0.563820i	$-0.190669\pi$
$73$	14.1130	1.65180	0.825898	+	0.563820i	$0.190669\pi$
$74$	0	0
$75$	0.402509	0.0464777
$76$	0	0
$77$	1.78532	0.203456
$78$	0	0
$79$	−0.819299	−0.0921783	−0.0460892	−	0.998937i	$-0.514676\pi$
$79$	−0.819299	−0.0921783	−0.0460892	+	0.998937i	$0.514676\pi$
$80$	0	0
$81$	7.56813	0.840903
$82$	0	0
$83$	−0.910352	−0.0999241	−0.0499621	−	0.998751i	$-0.515910\pi$
$83$	−0.910352	−0.0999241	−0.0499621	+	0.998751i	$0.515910\pi$
$84$	0	0
$85$	−7.18532	−0.779357
$86$	0	0
$87$	−3.22471	−0.345726
$88$	0	0
$89$	−9.09818	−0.964405	−0.482203	−	0.876060i	$-0.660163\pi$
$89$	−9.09818	−0.964405	−0.482203	+	0.876060i	$0.660163\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−3.18070	−0.329823
$94$	0	0
$95$	5.62331	0.576939
$96$	0	0
$97$	−10.2183	−1.03751	−0.518755	−	0.854923i	$-0.673604\pi$
$97$	−10.2183	−1.03751	−0.518755	+	0.854923i	$0.673604\pi$
$98$	0	0
$99$	−5.06671	−0.509224
$100$	0	0
$101$	3.19960	0.318372	0.159186	−	0.987249i	$-0.449113\pi$
$101$	3.19960	0.318372	0.159186	+	0.987249i	$0.449113\pi$
$102$	0	0
$103$	−10.8808	−1.07211	−0.536056	−	0.844182i	$-0.680086\pi$
$103$	−10.8808	−1.07211	−0.536056	+	0.844182i	$0.680086\pi$
$104$	0	0
$105$	−0.402509	−0.0392808
$106$	0	0
$107$	−19.4373	−1.87908	−0.939540	−	0.342440i	$-0.888747\pi$
$107$	−19.4373	−1.87908	−0.939540	+	0.342440i	$0.888747\pi$
$108$	0	0
$109$	−13.3519	−1.27888	−0.639442	−	0.768839i	$-0.720835\pi$
$109$	−13.3519	−1.27888	−0.639442	+	0.768839i	$0.720835\pi$
$110$	0	0
$111$	−3.83799	−0.364286
$112$	0	0
$113$	−1.91359	−0.180015	−0.0900077	−	0.995941i	$-0.528689\pi$
$113$	−1.91359	−0.180015	−0.0900077	+	0.995941i	$0.528689\pi$
$114$	0	0
$115$	−1.08564	−0.101236
$116$	0	0
$117$	2.83799	0.262372
$118$	0	0
$119$	7.18532	0.658677
$120$	0	0
$121$	−7.81263	−0.710240
$122$	0	0
$123$	0.697408	0.0628832
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.82910	0.428513	0.214257	−	0.976777i	$-0.431267\pi$
$127$	4.82910	0.428513	0.214257	+	0.976777i	$0.431267\pi$
$128$	0	0
$129$	4.41003	0.388282
$130$	0	0
$131$	−17.1363	−1.49720	−0.748601	−	0.663021i	$-0.769274\pi$
$131$	−17.1363	−1.49720	−0.748601	+	0.663021i	$0.769274\pi$
$132$	0	0
$133$	−5.62331	−0.487602
$134$	0	0
$135$	2.34984	0.202242
$136$	0	0
$137$	−17.0194	−1.45406	−0.727032	−	0.686603i	$-0.759101\pi$
$137$	−17.0194	−1.45406	−0.727032	+	0.686603i	$0.759101\pi$
$138$	0	0
$139$	−9.15211	−0.776272	−0.388136	−	0.921602i	$-0.626881\pi$
$139$	−9.15211	−0.776272	−0.388136	+	0.921602i	$0.626881\pi$
$140$	0	0
$141$	0.218286	0.0183830
$142$	0	0
$143$	−1.78532	−0.149296
$144$	0	0
$145$	8.01153	0.665322
$146$	0	0
$147$	0.402509	0.0331984
$148$	0	0
$149$	−11.4527	−0.938239	−0.469119	−	0.883135i	$-0.655429\pi$
$149$	−11.4527	−0.938239	−0.469119	+	0.883135i	$0.655429\pi$
$150$	0	0
$151$	20.6983	1.68440	0.842200	−	0.539164i	$-0.181260\pi$
$151$	20.6983	1.68440	0.842200	+	0.539164i	$0.181260\pi$
$152$	0	0
$153$	−20.3918	−1.64858
$154$	0	0
$155$	7.90219	0.634719
$156$	0	0
$157$	−12.4485	−0.993499	−0.496750	−	0.867894i	$-0.665473\pi$
$157$	−12.4485	−0.993499	−0.496750	+	0.867894i	$0.665473\pi$
$158$	0	0
$159$	2.53770	0.201252
$160$	0	0
$161$	1.08564	0.0855601
$162$	0	0
$163$	16.3822	1.28315	0.641575	−	0.767060i	$-0.278281\pi$
$163$	16.3822	1.28315	0.641575	+	0.767060i	$0.278281\pi$
$164$	0	0
$165$	−0.718607	−0.0559435
$166$	0	0
$167$	17.7650	1.37470	0.687348	−	0.726329i	$-0.258775\pi$
$167$	17.7650	1.37470	0.687348	+	0.726329i	$0.258775\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	15.9589	1.22041
$172$	0	0
$173$	−7.27748	−0.553296	−0.276648	−	0.960971i	$-0.589224\pi$
$173$	−7.27748	−0.553296	−0.276648	+	0.960971i	$0.589224\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−0.451259	−0.0339187
$178$	0	0
$179$	17.9366	1.34065	0.670323	−	0.742069i	$-0.266155\pi$
$179$	17.9366	1.34065	0.670323	+	0.742069i	$0.266155\pi$
$180$	0	0
$181$	4.35171	0.323460	0.161730	−	0.986835i	$-0.448293\pi$
$181$	4.35171	0.323460	0.161730	+	0.986835i	$0.448293\pi$
$182$	0	0
$183$	−1.09426	−0.0808903
$184$	0	0
$185$	9.53516	0.701039
$186$	0	0
$187$	12.8281	0.938083
$188$	0	0
$189$	−2.34984	−0.170926
$190$	0	0
$191$	4.50683	0.326103	0.163051	−	0.986618i	$-0.447866\pi$
$191$	4.50683	0.326103	0.163051	+	0.986618i	$0.447866\pi$
$192$	0	0
$193$	−16.7954	−1.20896	−0.604478	−	0.796622i	$-0.706618\pi$
$193$	−16.7954	−1.20896	−0.604478	+	0.796622i	$0.706618\pi$
$194$	0	0
$195$	0.402509	0.0288243
$196$	0	0
$197$	5.10580	0.363773	0.181887	−	0.983320i	$-0.441780\pi$
$197$	5.10580	0.363773	0.181887	+	0.983320i	$0.441780\pi$
$198$	0	0
$199$	8.28163	0.587069	0.293535	−	0.955948i	$-0.405168\pi$
$199$	8.28163	0.587069	0.293535	+	0.955948i	$0.405168\pi$
$200$	0	0
$201$	−4.36991	−0.308230
$202$	0	0
$203$	−8.01153	−0.562299
$204$	0	0
$205$	−1.73265	−0.121014
$206$	0	0
$207$	−3.08102	−0.214146
$208$	0	0
$209$	−10.0394	−0.694440
$210$	0	0
$211$	−10.2699	−0.707012	−0.353506	−	0.935432i	$-0.615011\pi$
$211$	−10.2699	−0.707012	−0.353506	+	0.935432i	$0.615011\pi$
$212$	0	0
$213$	−1.71537	−0.117535
$214$	0	0
$215$	−10.9564	−0.747217
$216$	0	0
$217$	−7.90219	−0.536435
$218$	0	0
$219$	5.68059	0.383859
$220$	0	0
$221$	−7.18532	−0.483337
$222$	0	0
$223$	−11.3079	−0.757235	−0.378618	−	0.925553i	$-0.623600\pi$
$223$	−11.3079	−0.757235	−0.378618	+	0.925553i	$0.623600\pi$
$224$	0	0
$225$	−2.83799	−0.189199
$226$	0	0
$227$	3.04378	0.202023	0.101011	−	0.994885i	$-0.467792\pi$
$227$	3.04378	0.202023	0.101011	+	0.994885i	$0.467792\pi$
$228$	0	0
$229$	−2.42472	−0.160230	−0.0801150	−	0.996786i	$-0.525529\pi$
$229$	−2.42472	−0.160230	−0.0801150	+	0.996786i	$0.525529\pi$
$230$	0	0
$231$	0.718607	0.0472809
$232$	0	0
$233$	−26.1990	−1.71635	−0.858175	−	0.513357i	$-0.828402\pi$
$233$	−26.1990	−1.71635	−0.858175	+	0.513357i	$0.828402\pi$
$234$	0	0
$235$	−0.542313	−0.0353766
$236$	0	0
$237$	−0.329775	−0.0214212
$238$	0	0
$239$	−24.5929	−1.59078	−0.795392	−	0.606095i	$-0.792735\pi$
$239$	−24.5929	−1.59078	−0.795392	+	0.606095i	$0.792735\pi$
$240$	0	0
$241$	−1.36461	−0.0879024	−0.0439512	−	0.999034i	$-0.513995\pi$
$241$	−1.36461	−0.0879024	−0.0439512	+	0.999034i	$0.513995\pi$
$242$	0	0
$243$	10.0958	0.647644
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	5.62331	0.357802
$248$	0	0
$249$	−0.366425	−0.0232212
$250$	0	0
$251$	5.74232	0.362452	0.181226	−	0.983441i	$-0.441993\pi$
$251$	5.74232	0.362452	0.181226	+	0.983441i	$0.441993\pi$
$252$	0	0
$253$	1.93821	0.121854
$254$	0	0
$255$	−2.89215	−0.181114
$256$	0	0
$257$	15.7041	0.979593	0.489796	−	0.871837i	$-0.337071\pi$
$257$	15.7041	0.979593	0.489796	+	0.871837i	$0.337071\pi$
$258$	0	0
$259$	−9.53516	−0.592486
$260$	0	0
$261$	22.7366	1.40736
$262$	0	0
$263$	−12.3843	−0.763648	−0.381824	−	0.924235i	$-0.624704\pi$
$263$	−12.3843	−0.763648	−0.381824	+	0.924235i	$0.624704\pi$
$264$	0	0
$265$	−6.30470	−0.387294
$266$	0	0
$267$	−3.66210	−0.224117
$268$	0	0
$269$	−15.1996	−0.926736	−0.463368	−	0.886166i	$-0.653359\pi$
$269$	−15.1996	−0.926736	−0.463368	+	0.886166i	$0.653359\pi$
$270$	0	0
$271$	6.56525	0.398810	0.199405	−	0.979917i	$-0.436099\pi$
$271$	6.56525	0.398810	0.199405	+	0.979917i	$0.436099\pi$
$272$	0	0
$273$	−0.402509	−0.0243609
$274$	0	0
$275$	1.78532	0.107659
$276$	0	0
$277$	−26.9565	−1.61966	−0.809829	−	0.586666i	$-0.800440\pi$
$277$	−26.9565	−1.61966	−0.809829	+	0.586666i	$0.800440\pi$
$278$	0	0
$279$	22.4263	1.34263
$280$	0	0
$281$	−13.1506	−0.784497	−0.392248	−	0.919859i	$-0.628303\pi$
$281$	−13.1506	−0.784497	−0.392248	+	0.919859i	$0.628303\pi$
$282$	0	0
$283$	32.5613	1.93557	0.967785	−	0.251776i	$-0.0810148\pi$
$283$	32.5613	1.93557	0.967785	+	0.251776i	$0.0810148\pi$
$284$	0	0
$285$	2.26343	0.134074
$286$	0	0
$287$	1.73265	0.102275
$288$	0	0
$289$	34.6288	2.03699
$290$	0	0
$291$	−4.11295	−0.241105
$292$	0	0
$293$	−27.8094	−1.62464	−0.812321	−	0.583210i	$-0.801796\pi$
$293$	−27.8094	−1.62464	−0.812321	+	0.583210i	$0.801796\pi$
$294$	0	0
$295$	1.12112	0.0652739
$296$	0	0
$297$	−4.19522	−0.243431
$298$	0	0
$299$	−1.08564	−0.0627839
$300$	0	0
$301$	10.9564	0.631514
$302$	0	0
$303$	1.28787	0.0739860
$304$	0	0
$305$	2.71861	0.155667
$306$	0	0
$307$	0.737531	0.0420931	0.0210465	−	0.999778i	$-0.493300\pi$
$307$	0.737531	0.0420931	0.0210465	+	0.999778i	$0.493300\pi$
$308$	0	0
$309$	−4.37960	−0.249147
$310$	0	0
$311$	13.3628	0.757734	0.378867	−	0.925451i	$-0.376314\pi$
$311$	13.3628	0.757734	0.378867	+	0.925451i	$0.376314\pi$
$312$	0	0
$313$	−8.64373	−0.488573	−0.244286	−	0.969703i	$-0.578554\pi$
$313$	−8.64373	−0.488573	−0.244286	+	0.969703i	$0.578554\pi$
$314$	0	0
$315$	2.83799	0.159902
$316$	0	0
$317$	20.7274	1.16416	0.582082	−	0.813130i	$-0.302238\pi$
$317$	20.7274	1.16416	0.582082	+	0.813130i	$0.302238\pi$
$318$	0	0
$319$	−14.3031	−0.800822
$320$	0	0
$321$	−7.82371	−0.436677
$322$	0	0
$323$	−40.4052	−2.24821
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−5.37428	−0.297198
$328$	0	0
$329$	0.542313	0.0298987
$330$	0	0
$331$	30.6513	1.68475	0.842373	−	0.538896i	$-0.181158\pi$
$331$	30.6513	1.68475	0.842373	+	0.538896i	$0.181158\pi$
$332$	0	0
$333$	27.0607	1.48291
$334$	0	0
$335$	10.8567	0.593163
$336$	0	0
$337$	−6.79538	−0.370168	−0.185084	−	0.982723i	$-0.559256\pi$
$337$	−6.79538	−0.370168	−0.185084	+	0.982723i	$0.559256\pi$
$338$	0	0
$339$	−0.770237	−0.0418335
$340$	0	0
$341$	−14.1079	−0.763987
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.436978	−0.0235261
$346$	0	0
$347$	6.14786	0.330035	0.165017	−	0.986291i	$-0.447232\pi$
$347$	6.14786	0.330035	0.165017	+	0.986291i	$0.447232\pi$
$348$	0	0
$349$	−11.1456	−0.596610	−0.298305	−	0.954471i	$-0.596421\pi$
$349$	−11.1456	−0.596610	−0.298305	+	0.954471i	$0.596421\pi$
$350$	0	0
$351$	2.34984	0.125425
$352$	0	0
$353$	9.72801	0.517770	0.258885	−	0.965908i	$-0.416645\pi$
$353$	9.72801	0.517770	0.258885	+	0.965908i	$0.416645\pi$
$354$	0	0
$355$	4.26169	0.226187
$356$	0	0
$357$	2.89215	0.153069
$358$	0	0
$359$	−30.5614	−1.61297	−0.806484	−	0.591256i	$-0.798632\pi$
$359$	−30.5614	−1.61297	−0.806484	+	0.591256i	$0.798632\pi$
$360$	0	0
$361$	12.6216	0.664293
$362$	0	0
$363$	−3.14466	−0.165052
$364$	0	0
$365$	−14.1130	−0.738706
$366$	0	0
$367$	−8.10392	−0.423021	−0.211511	−	0.977376i	$-0.567838\pi$
$367$	−8.10392	−0.423021	−0.211511	+	0.977376i	$0.567838\pi$
$368$	0	0
$369$	−4.91724	−0.255981
$370$	0	0
$371$	6.30470	0.327324
$372$	0	0
$373$	27.1169	1.40406	0.702031	−	0.712147i	$-0.252277\pi$
$373$	27.1169	1.40406	0.702031	+	0.712147i	$0.252277\pi$
$374$	0	0
$375$	−0.402509	−0.0207855
$376$	0	0
$377$	8.01153	0.412615
$378$	0	0
$379$	16.2424	0.834314	0.417157	−	0.908834i	$-0.363026\pi$
$379$	16.2424	0.834314	0.417157	+	0.908834i	$0.363026\pi$
$380$	0	0
$381$	1.94375	0.0995816
$382$	0	0
$383$	−3.44985	−0.176279	−0.0881396	−	0.996108i	$-0.528092\pi$
$383$	−3.44985	−0.176279	−0.0881396	+	0.996108i	$0.528092\pi$
$384$	0	0
$385$	−1.78532	−0.0909883
$386$	0	0
$387$	−31.0940	−1.58060
$388$	0	0
$389$	−22.8899	−1.16056	−0.580281	−	0.814416i	$-0.697057\pi$
$389$	−22.8899	−1.16056	−0.580281	+	0.814416i	$0.697057\pi$
$390$	0	0
$391$	7.80064	0.394495
$392$	0	0
$393$	−6.89750	−0.347933
$394$	0	0
$395$	0.819299	0.0412234
$396$	0	0
$397$	36.3778	1.82575	0.912876	−	0.408237i	$-0.133856\pi$
$397$	36.3778	1.82575	0.912876	+	0.408237i	$0.133856\pi$
$398$	0	0
$399$	−2.26343	−0.113313
$400$	0	0
$401$	−16.5435	−0.826141	−0.413070	−	0.910699i	$-0.635544\pi$
$401$	−16.5435	−0.826141	−0.413070	+	0.910699i	$0.635544\pi$
$402$	0	0
$403$	7.90219	0.393636
$404$	0	0
$405$	−7.56813	−0.376063
$406$	0	0
$407$	−17.0233	−0.843814
$408$	0	0
$409$	0.742553	0.0367169	0.0183584	−	0.999831i	$-0.494156\pi$
$409$	0.742553	0.0367169	0.0183584	+	0.999831i	$0.494156\pi$
$410$	0	0
$411$	−6.85046	−0.337908
$412$	0	0
$413$	−1.12112	−0.0551665
$414$	0	0
$415$	0.910352	0.0446874
$416$	0	0
$417$	−3.68381	−0.180397
$418$	0	0
$419$	29.9567	1.46348	0.731741	−	0.681583i	$-0.238708\pi$
$419$	29.9567	1.46348	0.731741	+	0.681583i	$0.238708\pi$
$420$	0	0
$421$	−9.15673	−0.446272	−0.223136	−	0.974787i	$-0.571629\pi$
$421$	−9.15673	−0.446272	−0.223136	+	0.974787i	$0.571629\pi$
$422$	0	0
$423$	−1.53908	−0.0748324
$424$	0	0
$425$	7.18532	0.348539
$426$	0	0
$427$	−2.71861	−0.131563
$428$	0	0
$429$	−0.718607	−0.0346947
$430$	0	0
$431$	−24.7499	−1.19216	−0.596080	−	0.802925i	$-0.703276\pi$
$431$	−24.7499	−1.19216	−0.596080	+	0.802925i	$0.703276\pi$
$432$	0	0
$433$	−9.77667	−0.469837	−0.234918	−	0.972015i	$-0.575482\pi$
$433$	−9.77667	−0.469837	−0.234918	+	0.972015i	$0.575482\pi$
$434$	0	0
$435$	3.22471	0.154613
$436$	0	0
$437$	−6.10486	−0.292035
$438$	0	0
$439$	26.3992	1.25996	0.629982	−	0.776609i	$-0.283062\pi$
$439$	26.3992	1.25996	0.629982	+	0.776609i	$0.283062\pi$
$440$	0	0
$441$	−2.83799	−0.135142
$442$	0	0
$443$	6.15683	0.292520	0.146260	−	0.989246i	$-0.453276\pi$
$443$	6.15683	0.292520	0.146260	+	0.989246i	$0.453276\pi$
$444$	0	0
$445$	9.09818	0.431295
$446$	0	0
$447$	−4.60980	−0.218036
$448$	0	0
$449$	35.9909	1.69851	0.849257	−	0.527979i	$-0.177050\pi$
$449$	35.9909	1.69851	0.849257	+	0.527979i	$0.177050\pi$
$450$	0	0
$451$	3.09334	0.145660
$452$	0	0
$453$	8.33124	0.391436
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	33.7593	1.57919	0.789596	−	0.613627i	$-0.210290\pi$
$457$	33.7593	1.57919	0.789596	+	0.613627i	$0.210290\pi$
$458$	0	0
$459$	−16.8844	−0.788094
$460$	0	0
$461$	−6.99585	−0.325829	−0.162915	−	0.986640i	$-0.552089\pi$
$461$	−6.99585	−0.325829	−0.162915	+	0.986640i	$0.552089\pi$
$462$	0	0
$463$	−0.388133	−0.0180381	−0.00901903	−	0.999959i	$-0.502871\pi$
$463$	−0.388133	−0.0180381	−0.00901903	+	0.999959i	$0.502871\pi$
$464$	0	0
$465$	3.18070	0.147501
$466$	0	0
$467$	−18.4707	−0.854723	−0.427362	−	0.904081i	$-0.640557\pi$
$467$	−18.4707	−0.854723	−0.427362	+	0.904081i	$0.640557\pi$
$468$	0	0
$469$	−10.8567	−0.501315
$470$	0	0
$471$	−5.01064	−0.230878
$472$	0	0
$473$	19.5606	0.899397
$474$	0	0
$475$	−5.62331	−0.258015
$476$	0	0
$477$	−17.8926	−0.819248
$478$	0	0
$479$	9.99709	0.456779	0.228389	−	0.973570i	$-0.426654\pi$
$479$	9.99709	0.456779	0.228389	+	0.973570i	$0.426654\pi$
$480$	0	0
$481$	9.53516	0.434766
$482$	0	0
$483$	0.436978	0.0198832
$484$	0	0
$485$	10.2183	0.463988
$486$	0	0
$487$	8.39221	0.380287	0.190144	−	0.981756i	$-0.439105\pi$
$487$	8.39221	0.380287	0.190144	+	0.981756i	$0.439105\pi$
$488$	0	0
$489$	6.59397	0.298190
$490$	0	0
$491$	28.6266	1.29190	0.645951	−	0.763379i	$-0.276461\pi$
$491$	28.6266	1.29190	0.645951	+	0.763379i	$0.276461\pi$
$492$	0	0
$493$	−57.5654	−2.59262
$494$	0	0
$495$	5.06671	0.227732
$496$	0	0
$497$	−4.26169	−0.191163
$498$	0	0
$499$	6.52740	0.292207	0.146103	−	0.989269i	$-0.453327\pi$
$499$	6.52740	0.292207	0.146103	+	0.989269i	$0.453327\pi$
$500$	0	0
$501$	7.15056	0.319464
$502$	0	0
$503$	23.6826	1.05595	0.527977	−	0.849259i	$-0.322951\pi$
$503$	23.6826	1.05595	0.527977	+	0.849259i	$0.322951\pi$
$504$	0	0
$505$	−3.19960	−0.142380
$506$	0	0
$507$	0.402509	0.0178760
$508$	0	0
$509$	−38.4801	−1.70560	−0.852800	−	0.522238i	$-0.825097\pi$
$509$	−38.4801	−1.70560	−0.852800	+	0.522238i	$0.825097\pi$
$510$	0	0
$511$	14.1130	0.624320
$512$	0	0
$513$	13.2139	0.583407
$514$	0	0
$515$	10.8808	0.479463
$516$	0	0
$517$	0.968201	0.0425814
$518$	0	0
$519$	−2.92925	−0.128580
$520$	0	0
$521$	15.8804	0.695731	0.347866	−	0.937544i	$-0.386907\pi$
$521$	15.8804	0.695731	0.347866	+	0.937544i	$0.386907\pi$
$522$	0	0
$523$	38.6226	1.68885	0.844425	−	0.535674i	$-0.179942\pi$
$523$	38.6226	1.68885	0.844425	+	0.535674i	$0.179942\pi$
$524$	0	0
$525$	0.402509	0.0175669
$526$	0	0
$527$	−56.7797	−2.47336
$528$	0	0
$529$	−21.8214	−0.948756
$530$	0	0
$531$	3.18171	0.138075
$532$	0	0
$533$	−1.73265	−0.0750495
$534$	0	0
$535$	19.4373	0.840350
$536$	0	0
$537$	7.21965	0.311551
$538$	0	0
$539$	1.78532	0.0768991
$540$	0	0
$541$	1.02873	0.0442287	0.0221143	−	0.999755i	$-0.492960\pi$
$541$	1.02873	0.0442287	0.0221143	+	0.999755i	$0.492960\pi$
$542$	0	0
$543$	1.75160	0.0751684
$544$	0	0
$545$	13.3519	0.571935
$546$	0	0
$547$	−8.85149	−0.378462	−0.189231	−	0.981933i	$-0.560600\pi$
$547$	−8.85149	−0.378462	−0.189231	+	0.981933i	$0.560600\pi$
$548$	0	0
$549$	7.71537	0.329284
$550$	0	0
$551$	45.0513	1.91925
$552$	0	0
$553$	−0.819299	−0.0348401
$554$	0	0
$555$	3.83799	0.162913
$556$	0	0
$557$	−32.5612	−1.37966	−0.689831	−	0.723970i	$-0.742315\pi$
$557$	−32.5612	−1.37966	−0.689831	+	0.723970i	$0.742315\pi$
$558$	0	0
$559$	−10.9564	−0.463404
$560$	0	0
$561$	5.16342	0.218000
$562$	0	0
$563$	5.22873	0.220365	0.110182	−	0.993911i	$-0.464857\pi$
$563$	5.22873	0.220365	0.110182	+	0.993911i	$0.464857\pi$
$564$	0	0
$565$	1.91359	0.0805053
$566$	0	0
$567$	7.56813	0.317831
$568$	0	0
$569$	−24.1510	−1.01246	−0.506232	−	0.862397i	$-0.668962\pi$
$569$	−24.1510	−1.01246	−0.506232	+	0.862397i	$0.668962\pi$
$570$	0	0
$571$	−1.17214	−0.0490526	−0.0245263	−	0.999699i	$-0.507808\pi$
$571$	−1.17214	−0.0490526	−0.0245263	+	0.999699i	$0.507808\pi$
$572$	0	0
$573$	1.81404	0.0757826
$574$	0	0
$575$	1.08564	0.0452741
$576$	0	0
$577$	32.9742	1.37273	0.686367	−	0.727255i	$-0.259204\pi$
$577$	32.9742	1.37273	0.686367	+	0.727255i	$0.259204\pi$
$578$	0	0
$579$	−6.76028	−0.280948
$580$	0	0
$581$	−0.910352	−0.0377678
$582$	0	0
$583$	11.2559	0.466172
$584$	0	0
$585$	−2.83799	−0.117336
$586$	0	0
$587$	−29.5099	−1.21800	−0.609002	−	0.793169i	$-0.708430\pi$
$587$	−29.5099	−1.21800	−0.609002	+	0.793169i	$0.708430\pi$
$588$	0	0
$589$	44.4364	1.83097
$590$	0	0
$591$	2.05513	0.0845367
$592$	0	0
$593$	35.3469	1.45152	0.725762	−	0.687946i	$-0.241487\pi$
$593$	35.3469	1.45152	0.725762	+	0.687946i	$0.241487\pi$
$594$	0	0
$595$	−7.18532	−0.294569
$596$	0	0
$597$	3.33343	0.136428
$598$	0	0
$599$	26.1581	1.06879	0.534396	−	0.845234i	$-0.320539\pi$
$599$	26.1581	1.06879	0.534396	+	0.845234i	$0.320539\pi$
$600$	0	0
$601$	35.4317	1.44529	0.722644	−	0.691221i	$-0.242927\pi$
$601$	35.4317	1.44529	0.722644	+	0.691221i	$0.242927\pi$
$602$	0	0
$603$	30.8111	1.25472
$604$	0	0
$605$	7.81263	0.317629
$606$	0	0
$607$	0.130519	0.00529759	0.00264879	−	0.999996i	$-0.499157\pi$
$607$	0.130519	0.00529759	0.00264879	+	0.999996i	$0.499157\pi$
$608$	0	0
$609$	−3.22471	−0.130672
$610$	0	0
$611$	−0.542313	−0.0219396
$612$	0	0
$613$	−35.1001	−1.41768	−0.708839	−	0.705370i	$-0.750781\pi$
$613$	−35.1001	−1.41768	−0.708839	+	0.705370i	$0.750781\pi$
$614$	0	0
$615$	−0.697408	−0.0281222
$616$	0	0
$617$	12.4806	0.502450	0.251225	−	0.967929i	$-0.419167\pi$
$617$	12.4806	0.502450	0.251225	+	0.967929i	$0.419167\pi$
$618$	0	0
$619$	28.6208	1.15037	0.575183	−	0.818024i	$-0.304931\pi$
$619$	28.6208	1.15037	0.575183	+	0.818024i	$0.304931\pi$
$620$	0	0
$621$	−2.55107	−0.102371
$622$	0	0
$623$	−9.09818	−0.364511
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.04095	−0.161380
$628$	0	0
$629$	−68.5131	−2.73180
$630$	0	0
$631$	42.0266	1.67305	0.836527	−	0.547926i	$-0.184583\pi$
$631$	42.0266	1.67305	0.836527	+	0.547926i	$0.184583\pi$
$632$	0	0
$633$	−4.13374	−0.164302
$634$	0	0
$635$	−4.82910	−0.191637
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	12.0946	0.478456
$640$	0	0
$641$	1.37819	0.0544354	0.0272177	−	0.999630i	$-0.491335\pi$
$641$	1.37819	0.0544354	0.0272177	+	0.999630i	$0.491335\pi$
$642$	0	0
$643$	−19.1298	−0.754405	−0.377202	−	0.926131i	$-0.623114\pi$
$643$	−19.1298	−0.754405	−0.377202	+	0.926131i	$0.623114\pi$
$644$	0	0
$645$	−4.41003	−0.173645
$646$	0	0
$647$	−23.3613	−0.918428	−0.459214	−	0.888326i	$-0.651869\pi$
$647$	−23.3613	−0.918428	−0.459214	+	0.888326i	$0.651869\pi$
$648$	0	0
$649$	−2.00155	−0.0785677
$650$	0	0
$651$	−3.18070	−0.124662
$652$	0	0
$653$	−10.1913	−0.398818	−0.199409	−	0.979916i	$-0.563902\pi$
$653$	−10.1913	−0.398818	−0.199409	+	0.979916i	$0.563902\pi$
$654$	0	0
$655$	17.1363	0.669569
$656$	0	0
$657$	−40.0524	−1.56259
$658$	0	0
$659$	12.0793	0.470544	0.235272	−	0.971930i	$-0.424402\pi$
$659$	12.0793	0.470544	0.235272	+	0.971930i	$0.424402\pi$
$660$	0	0
$661$	−1.05391	−0.0409925	−0.0204962	−	0.999790i	$-0.506525\pi$
$661$	−1.05391	−0.0409925	−0.0204962	+	0.999790i	$0.506525\pi$
$662$	0	0
$663$	−2.89215	−0.112322
$664$	0	0
$665$	5.62331	0.218062
$666$	0	0
$667$	−8.69761	−0.336773
$668$	0	0
$669$	−4.55154	−0.175973
$670$	0	0
$671$	−4.85358	−0.187370
$672$	0	0
$673$	−9.23697	−0.356059	−0.178030	−	0.984025i	$-0.556972\pi$
$673$	−9.23697	−0.356059	−0.178030	+	0.984025i	$0.556972\pi$
$674$	0	0
$675$	−2.34984	−0.0904455
$676$	0	0
$677$	40.7253	1.56520	0.782601	−	0.622524i	$-0.213893\pi$
$677$	40.7253	1.56520	0.782601	+	0.622524i	$0.213893\pi$
$678$	0	0
$679$	−10.2183	−0.392142
$680$	0	0
$681$	1.22515	0.0469478
$682$	0	0
$683$	−33.8560	−1.29547	−0.647733	−	0.761868i	$-0.724283\pi$
$683$	−33.8560	−1.29547	−0.647733	+	0.761868i	$0.724283\pi$
$684$	0	0
$685$	17.0194	0.650277
$686$	0	0
$687$	−0.975971	−0.0372356
$688$	0	0
$689$	−6.30470	−0.240190
$690$	0	0
$691$	−38.6654	−1.47090	−0.735450	−	0.677579i	$-0.763029\pi$
$691$	−38.6654	−1.47090	−0.735450	+	0.677579i	$0.763029\pi$
$692$	0	0
$693$	−5.06671	−0.192468
$694$	0	0
$695$	9.15211	0.347159
$696$	0	0
$697$	12.4497	0.471564
$698$	0	0
$699$	−10.5453	−0.398860
$700$	0	0
$701$	−37.6500	−1.42202	−0.711010	−	0.703182i	$-0.751762\pi$
$701$	−37.6500	−1.42202	−0.711010	+	0.703182i	$0.751762\pi$
$702$	0	0
$703$	53.6191	2.02228
$704$	0	0
$705$	−0.218286	−0.00822111
$706$	0	0
$707$	3.19960	0.120333
$708$	0	0
$709$	6.98893	0.262475	0.131237	−	0.991351i	$-0.458105\pi$
$709$	6.98893	0.262475	0.131237	+	0.991351i	$0.458105\pi$
$710$	0	0
$711$	2.32516	0.0872003
$712$	0	0
$713$	−8.57890	−0.321282
$714$	0	0
$715$	1.78532	0.0667672
$716$	0	0
$717$	−9.89887	−0.369680
$718$	0	0
$719$	46.2532	1.72495	0.862476	−	0.506098i	$-0.168912\pi$
$719$	46.2532	1.72495	0.862476	+	0.506098i	$0.168912\pi$
$720$	0	0
$721$	−10.8808	−0.405220
$722$	0	0
$723$	−0.549269	−0.0204275
$724$	0	0
$725$	−8.01153	−0.297541
$726$	0	0
$727$	35.2080	1.30579	0.652897	−	0.757447i	$-0.273553\pi$
$727$	35.2080	1.30579	0.652897	+	0.757447i	$0.273553\pi$
$728$	0	0
$729$	−18.6407	−0.690398
$730$	0	0
$731$	78.7249	2.91175
$732$	0	0
$733$	−15.5908	−0.575858	−0.287929	−	0.957652i	$-0.592967\pi$
$733$	−15.5908	−0.575858	−0.287929	+	0.957652i	$0.592967\pi$
$734$	0	0
$735$	−0.402509	−0.0148468
$736$	0	0
$737$	−19.3826	−0.713968
$738$	0	0
$739$	8.46889	0.311533	0.155767	−	0.987794i	$-0.450215\pi$
$739$	8.46889	0.311533	0.155767	+	0.987794i	$0.450215\pi$
$740$	0	0
$741$	2.26343	0.0831492
$742$	0	0
$743$	−10.4907	−0.384865	−0.192432	−	0.981310i	$-0.561638\pi$
$743$	−10.4907	−0.384865	−0.192432	+	0.981310i	$0.561638\pi$
$744$	0	0
$745$	11.4527	0.419593
$746$	0	0
$747$	2.58357	0.0945278
$748$	0	0
$749$	−19.4373	−0.710225
$750$	0	0
$751$	−15.4549	−0.563958	−0.281979	−	0.959421i	$-0.590991\pi$
$751$	−15.4549	−0.563958	−0.281979	+	0.959421i	$0.590991\pi$
$752$	0	0
$753$	2.31133	0.0842297
$754$	0	0
$755$	−20.6983	−0.753287
$756$	0	0
$757$	10.4366	0.379324	0.189662	−	0.981849i	$-0.439261\pi$
$757$	10.4366	0.379324	0.189662	+	0.981849i	$0.439261\pi$
$758$	0	0
$759$	0.780146	0.0283175
$760$	0	0
$761$	29.4893	1.06899	0.534493	−	0.845173i	$-0.320503\pi$
$761$	29.4893	1.06899	0.534493	+	0.845173i	$0.320503\pi$
$762$	0	0
$763$	−13.3519	−0.483373
$764$	0	0
$765$	20.3918	0.737268
$766$	0	0
$767$	1.12112	0.0404812
$768$	0	0
$769$	18.9834	0.684560	0.342280	−	0.939598i	$-0.388801\pi$
$769$	18.9834	0.684560	0.342280	+	0.939598i	$0.388801\pi$
$770$	0	0
$771$	6.32103	0.227646
$772$	0	0
$773$	15.2933	0.550061	0.275030	−	0.961436i	$-0.411312\pi$
$773$	15.2933	0.550061	0.275030	+	0.961436i	$0.411312\pi$
$774$	0	0
$775$	−7.90219	−0.283855
$776$	0	0
$777$	−3.83799	−0.137687
$778$	0	0
$779$	−9.74323	−0.349088
$780$	0	0
$781$	−7.60849	−0.272253
$782$	0	0
$783$	18.8258	0.672781
$784$	0	0
$785$	12.4485	0.444306
$786$	0	0
$787$	6.48399	0.231129	0.115565	−	0.993300i	$-0.463132\pi$
$787$	6.48399	0.231129	0.115565	+	0.993300i	$0.463132\pi$
$788$	0	0
$789$	−4.98479	−0.177463
$790$	0	0
$791$	−1.91359	−0.0680394
$792$	0	0
$793$	2.71861	0.0965406
$794$	0	0
$795$	−2.53770	−0.0900028
$796$	0	0
$797$	1.55102	0.0549398	0.0274699	−	0.999623i	$-0.491255\pi$
$797$	1.55102	0.0549398	0.0274699	+	0.999623i	$0.491255\pi$
$798$	0	0
$799$	3.89669	0.137855
$800$	0	0
$801$	25.8205	0.912323
$802$	0	0
$803$	25.1961	0.889152
$804$	0	0
$805$	−1.08564	−0.0382636
$806$	0	0
$807$	−6.11797	−0.215363
$808$	0	0
$809$	26.4126	0.928618	0.464309	−	0.885673i	$-0.346303\pi$
$809$	26.4126	0.928618	0.464309	+	0.885673i	$0.346303\pi$
$810$	0	0
$811$	21.5881	0.758060	0.379030	−	0.925384i	$-0.376258\pi$
$811$	21.5881	0.758060	0.379030	+	0.925384i	$0.376258\pi$
$812$	0	0
$813$	2.64257	0.0926790
$814$	0	0
$815$	−16.3822	−0.573842
$816$	0	0
$817$	−61.6109	−2.15549
$818$	0	0
$819$	2.83799	0.0991673
$820$	0	0
$821$	−1.77000	−0.0617735	−0.0308867	−	0.999523i	$-0.509833\pi$
$821$	−1.77000	−0.0617735	−0.0308867	+	0.999523i	$0.509833\pi$
$822$	0	0
$823$	−46.6186	−1.62502	−0.812511	−	0.582946i	$-0.801900\pi$
$823$	−46.6186	−1.62502	−0.812511	+	0.582946i	$0.801900\pi$
$824$	0	0
$825$	0.718607	0.0250187
$826$	0	0
$827$	−52.9620	−1.84167	−0.920834	−	0.389954i	$-0.872491\pi$
$827$	−52.9620	−1.84167	−0.920834	+	0.389954i	$0.872491\pi$
$828$	0	0
$829$	23.9551	0.831995	0.415998	−	0.909366i	$-0.363432\pi$
$829$	23.9551	0.831995	0.415998	+	0.909366i	$0.363432\pi$
$830$	0	0
$831$	−10.8502	−0.376390
$832$	0	0
$833$	7.18532	0.248956
$834$	0	0
$835$	−17.7650	−0.614782
$836$	0	0
$837$	18.5689	0.641835
$838$	0	0
$839$	−26.0800	−0.900381	−0.450190	−	0.892933i	$-0.648644\pi$
$839$	−26.0800	−0.900381	−0.450190	+	0.892933i	$0.648644\pi$
$840$	0	0
$841$	35.1847	1.21326
$842$	0	0
$843$	−5.29322	−0.182308
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−7.81263	−0.268445
$848$	0	0
$849$	13.1062	0.449805
$850$	0	0
$851$	−10.3517	−0.354852
$852$	0	0
$853$	−17.7449	−0.607574	−0.303787	−	0.952740i	$-0.598251\pi$
$853$	−17.7449	−0.607574	−0.303787	+	0.952740i	$0.598251\pi$
$854$	0	0
$855$	−15.9589	−0.545782
$856$	0	0
$857$	40.6660	1.38912	0.694562	−	0.719433i	$-0.255598\pi$
$857$	40.6660	1.38912	0.694562	+	0.719433i	$0.255598\pi$
$858$	0	0
$859$	−35.6986	−1.21802	−0.609011	−	0.793162i	$-0.708433\pi$
$859$	−35.6986	−1.21802	−0.609011	+	0.793162i	$0.708433\pi$
$860$	0	0
$861$	0.697408	0.0237676
$862$	0	0
$863$	−28.5658	−0.972392	−0.486196	−	0.873850i	$-0.661616\pi$
$863$	−28.5658	−0.972392	−0.486196	+	0.873850i	$0.661616\pi$
$864$	0	0
$865$	7.27748	0.247442
$866$	0	0
$867$	13.9384	0.473373
$868$	0	0
$869$	−1.46271	−0.0496190
$870$	0	0
$871$	10.8567	0.367864
$872$	0	0
$873$	28.9994	0.981480
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	35.9590	1.21425	0.607125	−	0.794606i	$-0.292323\pi$
$877$	35.9590	1.21425	0.607125	+	0.794606i	$0.292323\pi$
$878$	0	0
$879$	−11.1935	−0.377548
$880$	0	0
$881$	31.0997	1.04777	0.523887	−	0.851788i	$-0.324481\pi$
$881$	31.0997	1.04777	0.523887	+	0.851788i	$0.324481\pi$
$882$	0	0
$883$	−25.1700	−0.847037	−0.423518	−	0.905887i	$-0.639205\pi$
$883$	−25.1700	−0.847037	−0.423518	+	0.905887i	$0.639205\pi$
$884$	0	0
$885$	0.451259	0.0151689
$886$	0	0
$887$	0.168366	0.00565317	0.00282658	−	0.999996i	$-0.499100\pi$
$887$	0.168366	0.00565317	0.00282658	+	0.999996i	$0.499100\pi$
$888$	0	0
$889$	4.82910	0.161963
$890$	0	0
$891$	13.5115	0.452653
$892$	0	0
$893$	−3.04959	−0.102051
$894$	0	0
$895$	−17.9366	−0.599555
$896$	0	0
$897$	−0.436978	−0.0145903
$898$	0	0
$899$	63.3086	2.11146
$900$	0	0
$901$	45.3012	1.50920
$902$	0	0
$903$	4.41003	0.146757
$904$	0	0
$905$	−4.35171	−0.144656
$906$	0	0
$907$	−7.51178	−0.249425	−0.124712	−	0.992193i	$-0.539801\pi$
$907$	−7.51178	−0.249425	−0.124712	+	0.992193i	$0.539801\pi$
$908$	0	0
$909$	−9.08042	−0.301178
$910$	0	0
$911$	32.8041	1.08685	0.543424	−	0.839458i	$-0.317127\pi$
$911$	32.8041	1.08685	0.543424	+	0.839458i	$0.317127\pi$
$912$	0	0
$913$	−1.62527	−0.0537886
$914$	0	0
$915$	1.09426	0.0361752
$916$	0	0
$917$	−17.1363	−0.565889
$918$	0	0
$919$	32.9300	1.08626	0.543130	−	0.839649i	$-0.317239\pi$
$919$	32.9300	1.08626	0.543130	+	0.839649i	$0.317239\pi$
$920$	0	0
$921$	0.296863	0.00978196
$922$	0	0
$923$	4.26169	0.140275
$924$	0	0
$925$	−9.53516	−0.313514
$926$	0	0
$927$	30.8794	1.01421
$928$	0	0
$929$	1.81337	0.0594947	0.0297473	−	0.999557i	$-0.490530\pi$
$929$	1.81337	0.0594947	0.0297473	+	0.999557i	$0.490530\pi$
$930$	0	0
$931$	−5.62331	−0.184296
$932$	0	0
$933$	5.37864	0.176089
$934$	0	0
$935$	−12.8281	−0.419523
$936$	0	0
$937$	−52.7281	−1.72255	−0.861276	−	0.508137i	$-0.830334\pi$
$937$	−52.7281	−1.72255	−0.861276	+	0.508137i	$0.830334\pi$
$938$	0	0
$939$	−3.47918	−0.113539
$940$	0	0
$941$	55.3573	1.80460	0.902299	−	0.431112i	$-0.141878\pi$
$941$	55.3573	1.80460	0.902299	+	0.431112i	$0.141878\pi$
$942$	0	0
$943$	1.88103	0.0612547
$944$	0	0
$945$	2.34984	0.0764404
$946$	0	0
$947$	39.1840	1.27331	0.636655	−	0.771149i	$-0.280318\pi$
$947$	39.1840	1.27331	0.636655	+	0.771149i	$0.280318\pi$
$948$	0	0
$949$	−14.1130	−0.458126
$950$	0	0
$951$	8.34295	0.270539
$952$	0	0
$953$	29.8788	0.967870	0.483935	−	0.875104i	$-0.339207\pi$
$953$	29.8788	0.967870	0.483935	+	0.875104i	$0.339207\pi$
$954$	0	0
$955$	−4.50683	−0.145838
$956$	0	0
$957$	−5.75714	−0.186102
$958$	0	0
$959$	−17.0194	−0.549585
$960$	0	0
$961$	31.4446	1.01434
$962$	0	0
$963$	55.1629	1.77760
$964$	0	0
$965$	16.7954	0.540662
$966$	0	0
$967$	−21.9235	−0.705011	−0.352505	−	0.935810i	$-0.614670\pi$
$967$	−21.9235	−0.705011	−0.352505	+	0.935810i	$0.614670\pi$
$968$	0	0
$969$	−16.2635	−0.522458
$970$	0	0
$971$	−31.5933	−1.01388	−0.506939	−	0.861982i	$-0.669223\pi$
$971$	−31.5933	−1.01388	−0.506939	+	0.861982i	$0.669223\pi$
$972$	0	0
$973$	−9.15211	−0.293403
$974$	0	0
$975$	−0.402509	−0.0128906
$976$	0	0
$977$	−17.9568	−0.574487	−0.287244	−	0.957858i	$-0.592739\pi$
$977$	−17.9568	−0.574487	−0.287244	+	0.957858i	$0.592739\pi$
$978$	0	0
$979$	−16.2432	−0.519134
$980$	0	0
$981$	37.8926	1.20982
$982$	0	0
$983$	−6.36482	−0.203006	−0.101503	−	0.994835i	$-0.532365\pi$
$983$	−6.36482	−0.203006	−0.101503	+	0.994835i	$0.532365\pi$
$984$	0	0
$985$	−5.10580	−0.162684
$986$	0	0
$987$	0.218286	0.00694811
$988$	0	0
$989$	11.8946	0.378227
$990$	0	0
$991$	−29.4746	−0.936290	−0.468145	−	0.883652i	$-0.655078\pi$
$991$	−29.4746	−0.936290	−0.468145	+	0.883652i	$0.655078\pi$
$992$	0	0
$993$	12.3374	0.391516
$994$	0	0
$995$	−8.28163	−0.262545
$996$	0	0
$997$	12.9334	0.409606	0.204803	−	0.978803i	$-0.434345\pi$
$997$	12.9334	0.409606	0.204803	+	0.978803i	$0.434345\pi$
$998$	0	0
$999$	22.4061	0.708898

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.y.1.3	✓	5
4.3	odd	2		7280.2.a.cc.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.y.1.3	✓	5	1.1	even	1	trivial
7280.2.a.cc.1.3		5	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)

\(f(q)\)	\(=\)	\(q+0.402509 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.83799 q^{9} +O(q^{10})\)	\(q+0.402509 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.83799 q^{9} +1.78532 q^{11} -1.00000 q^{13} -0.402509 q^{15} +7.18532 q^{17} -5.62331 q^{19} +0.402509 q^{21} +1.08564 q^{23} +1.00000 q^{25} -2.34984 q^{27} -8.01153 q^{29} -7.90219 q^{31} +0.718607 q^{33} -1.00000 q^{35} -9.53516 q^{37} -0.402509 q^{39} +1.73265 q^{41} +10.9564 q^{43} +2.83799 q^{45} +0.542313 q^{47} +1.00000 q^{49} +2.89215 q^{51} +6.30470 q^{53} -1.78532 q^{55} -2.26343 q^{57} -1.12112 q^{59} -2.71861 q^{61} -2.83799 q^{63} +1.00000 q^{65} -10.8567 q^{67} +0.436978 q^{69} -4.26169 q^{71} +14.1130 q^{73} +0.402509 q^{75} +1.78532 q^{77} -0.819299 q^{79} +7.56813 q^{81} -0.910352 q^{83} -7.18532 q^{85} -3.22471 q^{87} -9.09818 q^{89} -1.00000 q^{91} -3.18070 q^{93} +5.62331 q^{95} -10.2183 q^{97} -5.06671 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 5 q^{7} + q^{9}+O(q^{10}) \) 5 * q - 5 * q^5 + 5 * q^7 + q^9	\( 5 q - 5 q^{5} + 5 q^{7} + q^{9} - 7 q^{11} - 5 q^{13} + q^{17} + 3 q^{19} - 5 q^{23} + 5 q^{25} - 9 q^{27} - 9 q^{29} + 6 q^{31} + q^{33} - 5 q^{35} - 10 q^{37} - 8 q^{41} + 6 q^{43} - q^{45} - 13 q^{47} + 5 q^{49} - 4 q^{51} - 16 q^{53} + 7 q^{55} - 10 q^{57} - q^{59} - 11 q^{61} + q^{63} + 5 q^{65} - 30 q^{67} - 15 q^{69} - 12 q^{71} + 23 q^{73} - 7 q^{77} + 2 q^{79} - 11 q^{81} - 2 q^{83} - q^{85} - 5 q^{87} - 25 q^{89} - 5 q^{91} - 22 q^{93} - 3 q^{95} - 5 q^{97} - 12 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 + 5 * q^7 + q^9 - 7 * q^11 - 5 * q^13 + q^17 + 3 * q^19 - 5 * q^23 + 5 * q^25 - 9 * q^27 - 9 * q^29 + 6 * q^31 + q^33 - 5 * q^35 - 10 * q^37 - 8 * q^41 + 6 * q^43 - q^45 - 13 * q^47 + 5 * q^49 - 4 * q^51 - 16 * q^53 + 7 * q^55 - 10 * q^57 - q^59 - 11 * q^61 + q^63 + 5 * q^65 - 30 * q^67 - 15 * q^69 - 12 * q^71 + 23 * q^73 - 7 * q^77 + 2 * q^79 - 11 * q^81 - 2 * q^83 - q^85 - 5 * q^87 - 25 * q^89 - 5 * q^91 - 22 * q^93 - 3 * q^95 - 5 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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