LMFDB - Embedded newform 3640.2.a.r.1.2

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:	$4$
Coefficient field:	4.4.13676.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 7x + 1 $ x^4 - x^3 - 6x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.18363$ 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)

$f(q)$	$=$	$q-1.76823 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.126652 q^{9} +O(q^{10})$	$q-1.76823 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.126652 q^{9} -2.98782 q^{11} -1.00000 q^{13} +1.76823 q^{15} +5.39764 q^{17} +4.87335 q^{19} +1.76823 q^{21} -4.00000 q^{23} +1.00000 q^{25} +5.08075 q^{27} +9.74387 q^{29} -0.756053 q^{31} +5.28316 q^{33} +1.00000 q^{35} -11.4906 q^{37} +1.76823 q^{39} +0.114471 q^{41} +2.52429 q^{43} -0.126652 q^{45} +1.74670 q^{47} +1.00000 q^{49} -9.54428 q^{51} +12.0608 q^{53} +2.98782 q^{55} -8.61722 q^{57} -9.95410 q^{59} +11.2832 q^{61} -0.126652 q^{63} +1.00000 q^{65} -5.33688 q^{67} +7.07294 q^{69} -9.51211 q^{71} -6.46353 q^{73} -1.76823 q^{75} +2.98782 q^{77} +10.8490 q^{79} -9.36392 q^{81} -0.463532 q^{83} -5.39764 q^{85} -17.2294 q^{87} -16.5027 q^{89} +1.00000 q^{91} +1.33688 q^{93} -4.87335 q^{95} -13.2588 q^{97} -0.378414 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - 4 q^{5} - 4 q^{7} + 11 q^{9}+O(q^{10}) $ 4 * q - q^3 - 4 * q^5 - 4 * q^7 + 11 * q^9	$ 4 q - q^{3} - 4 q^{5} - 4 q^{7} + 11 q^{9} - 2 q^{11} - 4 q^{13} + q^{15} - 11 q^{17} + 9 q^{19} + q^{21} - 16 q^{23} + 4 q^{25} - 4 q^{27} + 13 q^{29} + 13 q^{31} - 12 q^{33} + 4 q^{35} + q^{37} + q^{39} + q^{41} - 12 q^{43} - 11 q^{45} - 14 q^{47} + 4 q^{49} - 26 q^{51} + 14 q^{53} + 2 q^{55} + 2 q^{57} - 5 q^{59} + 12 q^{61} - 11 q^{63} + 4 q^{65} - 23 q^{67} + 4 q^{69} - 6 q^{71} - 38 q^{73} - q^{75} + 2 q^{77} + 13 q^{79} + 20 q^{81} - 14 q^{83} + 11 q^{85} - q^{87} - 29 q^{89} + 4 q^{91} + 7 q^{93} - 9 q^{95} - 28 q^{99}+O(q^{100}) $ 4 * q - q^3 - 4 * q^5 - 4 * q^7 + 11 * q^9 - 2 * q^11 - 4 * q^13 + q^15 - 11 * q^17 + 9 * q^19 + q^21 - 16 * q^23 + 4 * q^25 - 4 * q^27 + 13 * q^29 + 13 * q^31 - 12 * q^33 + 4 * q^35 + q^37 + q^39 + q^41 - 12 * q^43 - 11 * q^45 - 14 * q^47 + 4 * q^49 - 26 * q^51 + 14 * q^53 + 2 * q^55 + 2 * q^57 - 5 * q^59 + 12 * q^61 - 11 * q^63 + 4 * q^65 - 23 * q^67 + 4 * q^69 - 6 * q^71 - 38 * q^73 - q^75 + 2 * q^77 + 13 * q^79 + 20 * q^81 - 14 * q^83 + 11 * q^85 - q^87 - 29 * q^89 + 4 * q^91 + 7 * q^93 - 9 * q^95 - 28 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.76823	−1.02089	−0.510445	−	0.859910i	$-0.670519\pi$
$3$	−1.76823	−1.02089	−0.510445	+	0.859910i	$0.670519\pi$
$4$	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
$9$	0.126652	0.0422174
$10$	0	0
$11$	−2.98782	−0.900861	−0.450431	−	0.892811i	$-0.648730\pi$
$11$	−2.98782	−0.900861	−0.450431	+	0.892811i	$0.648730\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.76823	0.456556
$16$	0	0
$17$	5.39764	1.30912	0.654559	−	0.756011i	$-0.272854\pi$
$17$	5.39764	1.30912	0.654559	+	0.756011i	$0.272854\pi$
$18$	0	0
$19$	4.87335	1.11802	0.559011	−	0.829160i	$-0.311181\pi$
$19$	4.87335	1.11802	0.559011	+	0.829160i	$0.311181\pi$
$20$	0	0
$21$	1.76823	0.385860
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.08075	0.977791
$28$	0	0
$29$	9.74387	1.80939	0.904696	−	0.426058i	$-0.140098\pi$
$29$	9.74387	1.80939	0.904696	+	0.426058i	$0.140098\pi$
$30$	0	0
$31$	−0.756053	−0.135791	−0.0678956	−	0.997692i	$-0.521628\pi$
$31$	−0.756053	−0.135791	−0.0678956	+	0.997692i	$0.521628\pi$
$32$	0	0
$33$	5.28316	0.919681
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−11.4906	−1.88904	−0.944519	−	0.328457i	$-0.893471\pi$
$37$	−11.4906	−1.88904	−0.944519	+	0.328457i	$0.893471\pi$
$38$	0	0
$39$	1.76823	0.283144
$40$	0	0
$41$	0.114471	0.0178774	0.00893871	−	0.999960i	$-0.497155\pi$
$41$	0.114471	0.0178774	0.00893871	+	0.999960i	$0.497155\pi$
$42$	0	0
$43$	2.52429	0.384950	0.192475	−	0.981302i	$-0.438349\pi$
$43$	2.52429	0.384950	0.192475	+	0.981302i	$0.438349\pi$
$44$	0	0
$45$	−0.126652	−0.0188802
$46$	0	0
$47$	1.74670	0.254782	0.127391	−	0.991853i	$-0.459340\pi$
$47$	1.74670	0.254782	0.127391	+	0.991853i	$0.459340\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.54428	−1.33647
$52$	0	0
$53$	12.0608	1.65667	0.828336	−	0.560231i	$-0.189288\pi$
$53$	12.0608	1.65667	0.828336	+	0.560231i	$0.189288\pi$
$54$	0	0
$55$	2.98782	0.402877
$56$	0	0
$57$	−8.61722	−1.14138
$58$	0	0
$59$	−9.95410	−1.29591	−0.647957	−	0.761677i	$-0.724376\pi$
$59$	−9.95410	−1.29591	−0.647957	+	0.761677i	$0.724376\pi$
$60$	0	0
$61$	11.2832	1.44466	0.722331	−	0.691548i	$-0.243071\pi$
$61$	11.2832	1.44466	0.722331	+	0.691548i	$0.243071\pi$
$62$	0	0
$63$	−0.126652	−0.0159567
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−5.33688	−0.652004	−0.326002	−	0.945369i	$-0.605702\pi$
$67$	−5.33688	−0.652004	−0.326002	+	0.945369i	$0.605702\pi$
$68$	0	0
$69$	7.07294	0.851482
$70$	0	0
$71$	−9.51211	−1.12888	−0.564440	−	0.825474i	$-0.690908\pi$
$71$	−9.51211	−1.12888	−0.564440	+	0.825474i	$0.690908\pi$
$72$	0	0
$73$	−6.46353	−0.756499	−0.378250	−	0.925704i	$-0.623474\pi$
$73$	−6.46353	−0.756499	−0.378250	+	0.925704i	$0.623474\pi$
$74$	0	0
$75$	−1.76823	−0.204178
$76$	0	0
$77$	2.98782	0.340494
$78$	0	0
$79$	10.8490	1.22061	0.610303	−	0.792168i	$-0.291048\pi$
$79$	10.8490	1.22061	0.610303	+	0.792168i	$0.291048\pi$
$80$	0	0
$81$	−9.36392	−1.04044
$82$	0	0
$83$	−0.463532	−0.0508792	−0.0254396	−	0.999676i	$-0.508099\pi$
$83$	−0.463532	−0.0508792	−0.0254396	+	0.999676i	$0.508099\pi$
$84$	0	0
$85$	−5.39764	−0.585456
$86$	0	0
$87$	−17.2294	−1.84719
$88$	0	0
$89$	−16.5027	−1.74929	−0.874644	−	0.484766i	$-0.838905\pi$
$89$	−16.5027	−1.74929	−0.874644	+	0.484766i	$0.838905\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	1.33688	0.138628
$94$	0	0
$95$	−4.87335	−0.499995
$96$	0	0
$97$	−13.2588	−1.34623	−0.673114	−	0.739539i	$-0.735044\pi$
$97$	−13.2588	−1.34623	−0.673114	+	0.739539i	$0.735044\pi$
$98$	0	0
$99$	−0.378414	−0.0380320
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−17.1443	−1.68928	−0.844641	−	0.535334i	$-0.820186\pi$
$103$	−17.1443	−1.68928	−0.844641	+	0.535334i	$0.820186\pi$
$104$	0	0
$105$	−1.76823	−0.172562
$106$	0	0
$107$	−4.54865	−0.439735	−0.219867	−	0.975530i	$-0.570562\pi$
$107$	−4.54865	−0.439735	−0.219867	+	0.975530i	$0.570562\pi$
$108$	0	0
$109$	16.5243	1.58274	0.791370	−	0.611338i	$-0.209368\pi$
$109$	16.5243	1.58274	0.791370	+	0.611338i	$0.209368\pi$
$110$	0	0
$111$	20.3180	1.92850
$112$	0	0
$113$	10.8196	1.01783	0.508913	−	0.860818i	$-0.330048\pi$
$113$	10.8196	1.01783	0.508913	+	0.860818i	$0.330048\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−0.126652	−0.0117090
$118$	0	0
$119$	−5.39764	−0.494800
$120$	0	0
$121$	−2.07294	−0.188449
$122$	0	0
$123$	−0.202412	−0.0182509
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.32624	−0.117685	−0.0588424	−	0.998267i	$-0.518741\pi$
$127$	−1.32624	−0.117685	−0.0588424	+	0.998267i	$0.518741\pi$
$128$	0	0
$129$	−4.46353	−0.392992
$130$	0	0
$131$	−6.02436	−0.526351	−0.263175	−	0.964748i	$-0.584770\pi$
$131$	−6.02436	−0.526351	−0.263175	+	0.964748i	$0.584770\pi$
$132$	0	0
$133$	−4.87335	−0.422573
$134$	0	0
$135$	−5.08075	−0.437281
$136$	0	0
$137$	9.46621	0.808753	0.404376	−	0.914593i	$-0.367489\pi$
$137$	9.46621	0.808753	0.404376	+	0.914593i	$0.367489\pi$
$138$	0	0
$139$	−9.28316	−0.787388	−0.393694	−	0.919242i	$-0.628803\pi$
$139$	−9.28316	−0.787388	−0.393694	+	0.919242i	$0.628803\pi$
$140$	0	0
$141$	−3.08857	−0.260104
$142$	0	0
$143$	2.98782	0.249854
$144$	0	0
$145$	−9.74387	−0.809185
$146$	0	0
$147$	−1.76823	−0.145841
$148$	0	0
$149$	0.524288	0.0429513	0.0214757	−	0.999769i	$-0.493164\pi$
$149$	0.524288	0.0429513	0.0214757	+	0.999769i	$0.493164\pi$
$150$	0	0
$151$	6.21023	0.505381	0.252691	−	0.967547i	$-0.418685\pi$
$151$	6.21023	0.505381	0.252691	+	0.967547i	$0.418685\pi$
$152$	0	0
$153$	0.683622	0.0552676
$154$	0	0
$155$	0.756053	0.0607277
$156$	0	0
$157$	3.99718	0.319009	0.159505	−	0.987197i	$-0.449010\pi$
$157$	3.99718	0.319009	0.159505	+	0.987197i	$0.449010\pi$
$158$	0	0
$159$	−21.3262	−1.69128
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	4.20740	0.329549	0.164775	−	0.986331i	$-0.447310\pi$
$163$	4.20740	0.329549	0.164775	+	0.986331i	$0.447310\pi$
$164$	0	0
$165$	−5.28316	−0.411294
$166$	0	0
$167$	−11.2588	−0.871232	−0.435616	−	0.900133i	$-0.643469\pi$
$167$	−11.2588	−0.871232	−0.435616	+	0.900133i	$0.643469\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.617220	0.0472000
$172$	0	0
$173$	−11.4906	−0.873612	−0.436806	−	0.899556i	$-0.643890\pi$
$173$	−11.4906	−0.873612	−0.436806	+	0.899556i	$0.643890\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	17.6012	1.32299
$178$	0	0
$179$	−15.9463	−1.19188	−0.595941	−	0.803028i	$-0.703221\pi$
$179$	−15.9463	−1.19188	−0.595941	+	0.803028i	$0.703221\pi$
$180$	0	0
$181$	6.41481	0.476809	0.238404	−	0.971166i	$-0.423376\pi$
$181$	6.41481	0.476809	0.238404	+	0.971166i	$0.423376\pi$
$182$	0	0
$183$	−19.9513	−1.47484
$184$	0	0
$185$	11.4906	0.844803
$186$	0	0
$187$	−16.1272	−1.17933
$188$	0	0
$189$	−5.08075	−0.369570
$190$	0	0
$191$	−7.51493	−0.543761	−0.271881	−	0.962331i	$-0.587646\pi$
$191$	−7.51493	−0.543761	−0.271881	+	0.962331i	$0.587646\pi$
$192$	0	0
$193$	22.1322	1.59311	0.796554	−	0.604568i	$-0.206654\pi$
$193$	22.1322	1.59311	0.796554	+	0.604568i	$0.206654\pi$
$194$	0	0
$195$	−1.76823	−0.126626
$196$	0	0
$197$	−3.53365	−0.251762	−0.125881	−	0.992045i	$-0.540176\pi$
$197$	−3.53365	−0.251762	−0.125881	+	0.992045i	$0.540176\pi$
$198$	0	0
$199$	12.1215	0.859271	0.429636	−	0.903002i	$-0.358642\pi$
$199$	12.1215	0.859271	0.429636	+	0.903002i	$0.358642\pi$
$200$	0	0
$201$	9.43685	0.665624
$202$	0	0
$203$	−9.74387	−0.683886
$204$	0	0
$205$	−0.114471	−0.00799503
$206$	0	0
$207$	−0.506609	−0.0352117
$208$	0	0
$209$	−14.5607	−1.00718
$210$	0	0
$211$	25.4153	1.74966	0.874831	−	0.484428i	$-0.160972\pi$
$211$	25.4153	1.74966	0.874831	+	0.484428i	$0.160972\pi$
$212$	0	0
$213$	16.8196	1.15246
$214$	0	0
$215$	−2.52429	−0.172155
$216$	0	0
$217$	0.756053	0.0513242
$218$	0	0
$219$	11.4290	0.772303
$220$	0	0
$221$	−5.39764	−0.363084
$222$	0	0
$223$	−13.9756	−0.935878	−0.467939	−	0.883761i	$-0.655003\pi$
$223$	−13.9756	−0.935878	−0.467939	+	0.883761i	$0.655003\pi$
$224$	0	0
$225$	0.126652	0.00844348
$226$	0	0
$227$	−6.43917	−0.427383	−0.213691	−	0.976901i	$-0.568549\pi$
$227$	−6.43917	−0.427383	−0.213691	+	0.976901i	$0.568549\pi$
$228$	0	0
$229$	−2.22395	−0.146963	−0.0734814	−	0.997297i	$-0.523411\pi$
$229$	−2.22395	−0.146963	−0.0734814	+	0.997297i	$0.523411\pi$
$230$	0	0
$231$	−5.28316	−0.347607
$232$	0	0
$233$	−26.0784	−1.70846	−0.854228	−	0.519899i	$-0.825969\pi$
$233$	−26.0784	−1.70846	−0.854228	+	0.519899i	$0.825969\pi$
$234$	0	0
$235$	−1.74670	−0.113942
$236$	0	0
$237$	−19.1835	−1.24611
$238$	0	0
$239$	−28.7278	−1.85825	−0.929124	−	0.369767i	$-0.879437\pi$
$239$	−28.7278	−1.85825	−0.929124	+	0.369767i	$0.879437\pi$
$240$	0	0
$241$	−23.6079	−1.52072	−0.760358	−	0.649504i	$-0.774977\pi$
$241$	−23.6079	−1.52072	−0.760358	+	0.649504i	$0.774977\pi$
$242$	0	0
$243$	1.31534	0.0843791
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−4.87335	−0.310084
$248$	0	0
$249$	0.819632	0.0519421
$250$	0	0
$251$	29.4690	1.86007	0.930034	−	0.367473i	$-0.119777\pi$
$251$	29.4690	1.86007	0.930034	+	0.367473i	$0.119777\pi$
$252$	0	0
$253$	11.9513	0.751370
$254$	0	0
$255$	9.54428	0.597686
$256$	0	0
$257$	−19.3626	−1.20781	−0.603904	−	0.797057i	$-0.706389\pi$
$257$	−19.3626	−1.20781	−0.603904	+	0.797057i	$0.706389\pi$
$258$	0	0
$259$	11.4906	0.713989
$260$	0	0
$261$	1.23408	0.0763878
$262$	0	0
$263$	−31.5662	−1.94645	−0.973227	−	0.229846i	$-0.926178\pi$
$263$	−31.5662	−1.94645	−0.973227	+	0.229846i	$0.926178\pi$
$264$	0	0
$265$	−12.0608	−0.740886
$266$	0	0
$267$	29.1807	1.78583
$268$	0	0
$269$	5.02986	0.306676	0.153338	−	0.988174i	$-0.450998\pi$
$269$	5.02986	0.306676	0.153338	+	0.988174i	$0.450998\pi$
$270$	0	0
$271$	2.71015	0.164630	0.0823150	−	0.996606i	$-0.473769\pi$
$271$	2.71015	0.164630	0.0823150	+	0.996606i	$0.473769\pi$
$272$	0	0
$273$	−1.76823	−0.107018
$274$	0	0
$275$	−2.98782	−0.180172
$276$	0	0
$277$	10.5486	0.633807	0.316903	−	0.948458i	$-0.397357\pi$
$277$	10.5486	0.633807	0.316903	+	0.948458i	$0.397357\pi$
$278$	0	0
$279$	−0.0957558	−0.00573275
$280$	0	0
$281$	27.4047	1.63483	0.817413	−	0.576052i	$-0.195407\pi$
$281$	27.4047	1.63483	0.817413	+	0.576052i	$0.195407\pi$
$282$	0	0
$283$	−25.1807	−1.49684	−0.748419	−	0.663226i	$-0.769187\pi$
$283$	−25.1807	−1.49684	−0.748419	+	0.663226i	$0.769187\pi$
$284$	0	0
$285$	8.61722	0.510440
$286$	0	0
$287$	−0.114471	−0.00675703
$288$	0	0
$289$	12.1345	0.713792
$290$	0	0
$291$	23.4447	1.37435
$292$	0	0
$293$	−25.4290	−1.48558	−0.742790	−	0.669524i	$-0.766498\pi$
$293$	−25.4290	−1.48558	−0.742790	+	0.669524i	$0.766498\pi$
$294$	0	0
$295$	9.95410	0.579550
$296$	0	0
$297$	−15.1804	−0.880854
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	−2.52429	−0.145498
$302$	0	0
$303$	3.53647	0.203165
$304$	0	0
$305$	−11.2832	−0.646072
$306$	0	0
$307$	−27.2532	−1.55542	−0.777710	−	0.628623i	$-0.783619\pi$
$307$	−27.2532	−1.55542	−0.777710	+	0.628623i	$0.783619\pi$
$308$	0	0
$309$	30.3152	1.72457
$310$	0	0
$311$	−20.3130	−1.15185	−0.575923	−	0.817504i	$-0.695357\pi$
$311$	−20.3130	−1.15185	−0.575923	+	0.817504i	$0.695357\pi$
$312$	0	0
$313$	15.8046	0.893330	0.446665	−	0.894701i	$-0.352612\pi$
$313$	15.8046	0.893330	0.446665	+	0.894701i	$0.352612\pi$
$314$	0	0
$315$	0.126652	0.00713604
$316$	0	0
$317$	−27.0486	−1.51920	−0.759600	−	0.650391i	$-0.774605\pi$
$317$	−27.0486	−1.51920	−0.759600	+	0.650391i	$0.774605\pi$
$318$	0	0
$319$	−29.1129	−1.63001
$320$	0	0
$321$	8.04308	0.448921
$322$	0	0
$323$	26.3046	1.46362
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−29.2188	−1.61580
$328$	0	0
$329$	−1.74670	−0.0962984
$330$	0	0
$331$	−27.0175	−1.48502	−0.742509	−	0.669836i	$-0.766364\pi$
$331$	−27.0175	−1.48502	−0.742509	+	0.669836i	$0.766364\pi$
$332$	0	0
$333$	−1.45531	−0.0797503
$334$	0	0
$335$	5.33688	0.291585
$336$	0	0
$337$	−30.5420	−1.66373	−0.831863	−	0.554980i	$-0.812726\pi$
$337$	−30.5420	−1.66373	−0.831863	+	0.554980i	$0.812726\pi$
$338$	0	0
$339$	−19.1316	−1.03909
$340$	0	0
$341$	2.25895	0.122329
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−7.07294	−0.380794
$346$	0	0
$347$	−14.9878	−0.804588	−0.402294	−	0.915510i	$-0.631787\pi$
$347$	−14.9878	−0.804588	−0.402294	+	0.915510i	$0.631787\pi$
$348$	0	0
$349$	20.6035	1.10288	0.551440	−	0.834215i	$-0.314079\pi$
$349$	20.6035	1.10288	0.551440	+	0.834215i	$0.314079\pi$
$350$	0	0
$351$	−5.08075	−0.271190
$352$	0	0
$353$	−3.09730	−0.164853	−0.0824263	−	0.996597i	$-0.526267\pi$
$353$	−3.09730	−0.164853	−0.0824263	+	0.996597i	$0.526267\pi$
$354$	0	0
$355$	9.51211	0.504850
$356$	0	0
$357$	9.54428	0.505137
$358$	0	0
$359$	1.51211	0.0798059	0.0399030	−	0.999204i	$-0.487295\pi$
$359$	1.51211	0.0798059	0.0399030	+	0.999204i	$0.487295\pi$
$360$	0	0
$361$	4.74952	0.249975
$362$	0	0
$363$	3.66544	0.192386
$364$	0	0
$365$	6.46353	0.338317
$366$	0	0
$367$	−10.7589	−0.561609	−0.280804	−	0.959765i	$-0.590601\pi$
$367$	−10.7589	−0.561609	−0.280804	+	0.959765i	$0.590601\pi$
$368$	0	0
$369$	0.0144980	0.000754738 0
$370$	0	0
$371$	−12.0608	−0.626163
$372$	0	0
$373$	0.216910	0.0112312	0.00561559	−	0.999984i	$-0.498212\pi$
$373$	0.216910	0.0112312	0.00561559	+	0.999984i	$0.498212\pi$
$374$	0	0
$375$	1.76823	0.0913112
$376$	0	0
$377$	−9.74387	−0.501835
$378$	0	0
$379$	5.59722	0.287510	0.143755	−	0.989613i	$-0.454082\pi$
$379$	5.59722	0.287510	0.143755	+	0.989613i	$0.454082\pi$
$380$	0	0
$381$	2.34511	0.120143
$382$	0	0
$383$	−8.64940	−0.441964	−0.220982	−	0.975278i	$-0.570926\pi$
$383$	−8.64940	−0.441964	−0.220982	+	0.975278i	$0.570926\pi$
$384$	0	0
$385$	−2.98782	−0.152273
$386$	0	0
$387$	0.319707	0.0162516
$388$	0	0
$389$	9.94628	0.504297	0.252148	−	0.967689i	$-0.418863\pi$
$389$	9.94628	0.504297	0.252148	+	0.967689i	$0.418863\pi$
$390$	0	0
$391$	−21.5905	−1.09188
$392$	0	0
$393$	10.6525	0.537347
$394$	0	0
$395$	−10.8490	−0.545872
$396$	0	0
$397$	−13.0299	−0.653950	−0.326975	−	0.945033i	$-0.606029\pi$
$397$	−13.0299	−0.653950	−0.326975	+	0.945033i	$0.606029\pi$
$398$	0	0
$399$	8.61722	0.431401
$400$	0	0
$401$	30.7278	1.53447	0.767237	−	0.641363i	$-0.221631\pi$
$401$	30.7278	1.53447	0.767237	+	0.641363i	$0.221631\pi$
$402$	0	0
$403$	0.756053	0.0376617
$404$	0	0
$405$	9.36392	0.465297
$406$	0	0
$407$	34.3317	1.70176
$408$	0	0
$409$	−12.6614	−0.626067	−0.313034	−	0.949742i	$-0.601345\pi$
$409$	−12.6614	−0.626067	−0.313034	+	0.949742i	$0.601345\pi$
$410$	0	0
$411$	−16.7385	−0.825648
$412$	0	0
$413$	9.95410	0.489809
$414$	0	0
$415$	0.463532	0.0227539
$416$	0	0
$417$	16.4148	0.803837
$418$	0	0
$419$	14.5663	0.711612	0.355806	−	0.934560i	$-0.384206\pi$
$419$	14.5663	0.711612	0.355806	+	0.934560i	$0.384206\pi$
$420$	0	0
$421$	36.8804	1.79744	0.898720	−	0.438523i	$-0.144498\pi$
$421$	36.8804	1.79744	0.898720	+	0.438523i	$0.144498\pi$
$422$	0	0
$423$	0.221223	0.0107562
$424$	0	0
$425$	5.39764	0.261824
$426$	0	0
$427$	−11.2832	−0.546031
$428$	0	0
$429$	−5.28316	−0.255074
$430$	0	0
$431$	14.2102	0.684483	0.342241	−	0.939612i	$-0.388814\pi$
$431$	14.2102	0.684483	0.342241	+	0.939612i	$0.388814\pi$
$432$	0	0
$433$	20.4706	0.983753	0.491876	−	0.870665i	$-0.336311\pi$
$433$	20.4706	0.983753	0.491876	+	0.870665i	$0.336311\pi$
$434$	0	0
$435$	17.2294	0.826089
$436$	0	0
$437$	−19.4934	−0.932495
$438$	0	0
$439$	−0.692474	−0.0330500	−0.0165250	−	0.999863i	$-0.505260\pi$
$439$	−0.692474	−0.0330500	−0.0165250	+	0.999863i	$0.505260\pi$
$440$	0	0
$441$	0.126652	0.00603106
$442$	0	0
$443$	21.7400	1.03290	0.516450	−	0.856318i	$-0.327253\pi$
$443$	21.7400	1.03290	0.516450	+	0.856318i	$0.327253\pi$
$444$	0	0
$445$	16.5027	0.782305
$446$	0	0
$447$	−0.927063	−0.0438486
$448$	0	0
$449$	−19.8682	−0.937639	−0.468819	−	0.883294i	$-0.655320\pi$
$449$	−19.8682	−0.937639	−0.468819	+	0.883294i	$0.655320\pi$
$450$	0	0
$451$	−0.342020	−0.0161051
$452$	0	0
$453$	−10.9811	−0.515939
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−38.2215	−1.78793	−0.893963	−	0.448141i	$-0.852086\pi$
$457$	−38.2215	−1.78793	−0.893963	+	0.448141i	$0.852086\pi$
$458$	0	0
$459$	27.4240	1.28004
$460$	0	0
$461$	42.2371	1.96718	0.983589	−	0.180422i	$-0.0577464\pi$
$461$	42.2371	1.96718	0.983589	+	0.180422i	$0.0577464\pi$
$462$	0	0
$463$	27.1564	1.26206	0.631032	−	0.775757i	$-0.282632\pi$
$463$	27.1564	1.26206	0.631032	+	0.775757i	$0.282632\pi$
$464$	0	0
$465$	−1.33688	−0.0619963
$466$	0	0
$467$	−42.2780	−1.95639	−0.978197	−	0.207680i	$-0.933409\pi$
$467$	−42.2780	−1.95639	−0.978197	+	0.207680i	$0.933409\pi$
$468$	0	0
$469$	5.33688	0.246434
$470$	0	0
$471$	−7.06794	−0.325674
$472$	0	0
$473$	−7.54211	−0.346787
$474$	0	0
$475$	4.87335	0.223605
$476$	0	0
$477$	1.52752	0.0699404
$478$	0	0
$479$	38.3035	1.75013	0.875066	−	0.484003i	$-0.160818\pi$
$479$	38.3035	1.75013	0.875066	+	0.484003i	$0.160818\pi$
$480$	0	0
$481$	11.4906	0.523925
$482$	0	0
$483$	−7.07294	−0.321830
$484$	0	0
$485$	13.2588	0.602051
$486$	0	0
$487$	25.6764	1.16351	0.581755	−	0.813364i	$-0.302366\pi$
$487$	25.6764	1.16351	0.581755	+	0.813364i	$0.302366\pi$
$488$	0	0
$489$	−7.43968	−0.336434
$490$	0	0
$491$	7.95128	0.358836	0.179418	−	0.983773i	$-0.442579\pi$
$491$	7.95128	0.358836	0.179418	+	0.983773i	$0.442579\pi$
$492$	0	0
$493$	52.5939	2.36871
$494$	0	0
$495$	0.378414	0.0170084
$496$	0	0
$497$	9.51211	0.426676
$498$	0	0
$499$	15.0663	0.674458	0.337229	−	0.941423i	$-0.390510\pi$
$499$	15.0663	0.674458	0.337229	+	0.941423i	$0.390510\pi$
$500$	0	0
$501$	19.9082	0.889433
$502$	0	0
$503$	8.73451	0.389453	0.194726	−	0.980858i	$-0.437618\pi$
$503$	8.73451	0.389453	0.194726	+	0.980858i	$0.437618\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	−1.76823	−0.0785300
$508$	0	0
$509$	−29.2482	−1.29640	−0.648201	−	0.761469i	$-0.724478\pi$
$509$	−29.2482	−1.29640	−0.648201	+	0.761469i	$0.724478\pi$
$510$	0	0
$511$	6.46353	0.285930
$512$	0	0
$513$	24.7603	1.09319
$514$	0	0
$515$	17.1443	0.755469
$516$	0	0
$517$	−5.21881	−0.229523
$518$	0	0
$519$	20.3180	0.891862
$520$	0	0
$521$	−9.70670	−0.425258	−0.212629	−	0.977133i	$-0.568203\pi$
$521$	−9.70670	−0.425258	−0.212629	+	0.977133i	$0.568203\pi$
$522$	0	0
$523$	17.0486	0.745482	0.372741	−	0.927935i	$-0.378418\pi$
$523$	17.0486	0.745482	0.372741	+	0.927935i	$0.378418\pi$
$524$	0	0
$525$	1.76823	0.0771721
$526$	0	0
$527$	−4.08090	−0.177767
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−1.26071	−0.0547101
$532$	0	0
$533$	−0.114471	−0.00495830
$534$	0	0
$535$	4.54865	0.196655
$536$	0	0
$537$	28.1968	1.21678
$538$	0	0
$539$	−2.98782	−0.128694
$540$	0	0
$541$	−0.402776	−0.0173167	−0.00865834	−	0.999963i	$-0.502756\pi$
$541$	−0.402776	−0.0173167	−0.00865834	+	0.999963i	$0.502756\pi$
$542$	0	0
$543$	−11.3429	−0.486770
$544$	0	0
$545$	−16.5243	−0.707823
$546$	0	0
$547$	−17.5111	−0.748719	−0.374360	−	0.927284i	$-0.622137\pi$
$547$	−17.5111	−0.748719	−0.374360	+	0.927284i	$0.622137\pi$
$548$	0	0
$549$	1.42904	0.0609898
$550$	0	0
$551$	47.4853	2.02294
$552$	0	0
$553$	−10.8490	−0.461346
$554$	0	0
$555$	−20.3180	−0.862452
$556$	0	0
$557$	−25.4340	−1.07767	−0.538837	−	0.842410i	$-0.681136\pi$
$557$	−25.4340	−1.07767	−0.538837	+	0.842410i	$0.681136\pi$
$558$	0	0
$559$	−2.52429	−0.106766
$560$	0	0
$561$	28.5166	1.20397
$562$	0	0
$563$	−0.458540	−0.0193251	−0.00966257	−	0.999953i	$-0.503076\pi$
$563$	−0.458540	−0.0193251	−0.00966257	+	0.999953i	$0.503076\pi$
$564$	0	0
$565$	−10.8196	−0.455185
$566$	0	0
$567$	9.36392	0.393248
$568$	0	0
$569$	−33.7439	−1.41462	−0.707308	−	0.706905i	$-0.750091\pi$
$569$	−33.7439	−1.41462	−0.707308	+	0.706905i	$0.750091\pi$
$570$	0	0
$571$	−14.4120	−0.603123	−0.301561	−	0.953447i	$-0.597508\pi$
$571$	−14.4120	−0.603123	−0.301561	+	0.953447i	$0.597508\pi$
$572$	0	0
$573$	13.2882	0.555121
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	0.981136	0.0408452	0.0204226	−	0.999791i	$-0.493499\pi$
$577$	0.981136	0.0408452	0.0204226	+	0.999791i	$0.493499\pi$
$578$	0	0
$579$	−39.1348	−1.62639
$580$	0	0
$581$	0.463532	0.0192305
$582$	0	0
$583$	−36.0354	−1.49243
$584$	0	0
$585$	0.126652	0.00523642
$586$	0	0
$587$	12.5850	0.519440	0.259720	−	0.965684i	$-0.416370\pi$
$587$	12.5850	0.519440	0.259720	+	0.965684i	$0.416370\pi$
$588$	0	0
$589$	−3.68451	−0.151818
$590$	0	0
$591$	6.24831	0.257021
$592$	0	0
$593$	12.4235	0.510174	0.255087	−	0.966918i	$-0.417896\pi$
$593$	12.4235	0.510174	0.255087	+	0.966918i	$0.417896\pi$
$594$	0	0
$595$	5.39764	0.221281
$596$	0	0
$597$	−21.4337	−0.877222
$598$	0	0
$599$	5.51775	0.225449	0.112725	−	0.993626i	$-0.464042\pi$
$599$	5.51775	0.225449	0.112725	+	0.993626i	$0.464042\pi$
$600$	0	0
$601$	−19.1857	−0.782602	−0.391301	−	0.920263i	$-0.627975\pi$
$601$	−19.1857	−0.782602	−0.391301	+	0.920263i	$0.627975\pi$
$602$	0	0
$603$	−0.675928	−0.0275259
$604$	0	0
$605$	2.07294	0.0842769
$606$	0	0
$607$	33.4839	1.35907	0.679534	−	0.733644i	$-0.262182\pi$
$607$	33.4839	1.35907	0.679534	+	0.733644i	$0.262182\pi$
$608$	0	0
$609$	17.2294	0.698172
$610$	0	0
$611$	−1.74670	−0.0706637
$612$	0	0
$613$	−14.9999	−0.605838	−0.302919	−	0.953016i	$-0.597961\pi$
$613$	−14.9999	−0.605838	−0.302919	+	0.953016i	$0.597961\pi$
$614$	0	0
$615$	0.202412	0.00816205
$616$	0	0
$617$	16.8059	0.676580	0.338290	−	0.941042i	$-0.390151\pi$
$617$	16.8059	0.676580	0.338290	+	0.941042i	$0.390151\pi$
$618$	0	0
$619$	0.899878	0.0361692	0.0180846	−	0.999836i	$-0.494243\pi$
$619$	0.899878	0.0361692	0.0180846	+	0.999836i	$0.494243\pi$
$620$	0	0
$621$	−20.3230	−0.815534
$622$	0	0
$623$	16.5027	0.661169
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	25.7467	1.02822
$628$	0	0
$629$	−62.0219	−2.47297
$630$	0	0
$631$	−7.02986	−0.279854	−0.139927	−	0.990162i	$-0.544687\pi$
$631$	−7.02986	−0.279854	−0.139927	+	0.990162i	$0.544687\pi$
$632$	0	0
$633$	−44.9402	−1.78621
$634$	0	0
$635$	1.32624	0.0526303
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−1.20473	−0.0476584
$640$	0	0
$641$	−15.0914	−0.596074	−0.298037	−	0.954554i	$-0.596332\pi$
$641$	−15.0914	−0.596074	−0.298037	+	0.954554i	$0.596332\pi$
$642$	0	0
$643$	19.4778	0.768128	0.384064	−	0.923306i	$-0.374524\pi$
$643$	19.4778	0.768128	0.384064	+	0.923306i	$0.374524\pi$
$644$	0	0
$645$	4.46353	0.175751
$646$	0	0
$647$	−15.9718	−0.627915	−0.313958	−	0.949437i	$-0.601655\pi$
$647$	−15.9718	−0.627915	−0.313958	+	0.949437i	$0.601655\pi$
$648$	0	0
$649$	29.7410	1.16744
$650$	0	0
$651$	−1.33688	−0.0523964
$652$	0	0
$653$	22.4138	0.877119	0.438559	−	0.898702i	$-0.355489\pi$
$653$	22.4138	0.877119	0.438559	+	0.898702i	$0.355489\pi$
$654$	0	0
$655$	6.02436	0.235391
$656$	0	0
$657$	−0.818621	−0.0319374
$658$	0	0
$659$	3.82477	0.148992	0.0744960	−	0.997221i	$-0.476265\pi$
$659$	3.82477	0.148992	0.0744960	+	0.997221i	$0.476265\pi$
$660$	0	0
$661$	20.5713	0.800132	0.400066	−	0.916486i	$-0.368987\pi$
$661$	20.5713	0.800132	0.400066	+	0.916486i	$0.368987\pi$
$662$	0	0
$663$	9.54428	0.370669
$664$	0	0
$665$	4.87335	0.188980
$666$	0	0
$667$	−38.9755	−1.50914
$668$	0	0
$669$	24.7122	0.955429
$670$	0	0
$671$	−33.7121	−1.30144
$672$	0	0
$673$	20.9811	0.808763	0.404382	−	0.914590i	$-0.367487\pi$
$673$	20.9811	0.808763	0.404382	+	0.914590i	$0.367487\pi$
$674$	0	0
$675$	5.08075	0.195558
$676$	0	0
$677$	−28.9868	−1.11405	−0.557026	−	0.830495i	$-0.688058\pi$
$677$	−28.9868	−1.11405	−0.557026	+	0.830495i	$0.688058\pi$
$678$	0	0
$679$	13.2588	0.508826
$680$	0	0
$681$	11.3860	0.436311
$682$	0	0
$683$	18.2780	0.699389	0.349695	−	0.936864i	$-0.386285\pi$
$683$	18.2780	0.699389	0.349695	+	0.936864i	$0.386285\pi$
$684$	0	0
$685$	−9.46621	−0.361685
$686$	0	0
$687$	3.93247	0.150033
$688$	0	0
$689$	−12.0608	−0.459478
$690$	0	0
$691$	14.9299	0.567960	0.283980	−	0.958830i	$-0.408345\pi$
$691$	14.9299	0.567960	0.283980	+	0.958830i	$0.408345\pi$
$692$	0	0
$693$	0.378414	0.0143748
$694$	0	0
$695$	9.28316	0.352130
$696$	0	0
$697$	0.617875	0.0234037
$698$	0	0
$699$	46.1128	1.74415
$700$	0	0
$701$	−5.77605	−0.218158	−0.109079	−	0.994033i	$-0.534790\pi$
$701$	−5.77605	−0.218158	−0.109079	+	0.994033i	$0.534790\pi$
$702$	0	0
$703$	−55.9975	−2.11199
$704$	0	0
$705$	3.08857	0.116322
$706$	0	0
$707$	2.00000	0.0752177
$708$	0	0
$709$	6.42714	0.241376	0.120688	−	0.992690i	$-0.461490\pi$
$709$	6.42714	0.241376	0.120688	+	0.992690i	$0.461490\pi$
$710$	0	0
$711$	1.37405	0.0515308
$712$	0	0
$713$	3.02421	0.113258
$714$	0	0
$715$	−2.98782	−0.111738
$716$	0	0
$717$	50.7975	1.89707
$718$	0	0
$719$	−8.82962	−0.329289	−0.164645	−	0.986353i	$-0.552648\pi$
$719$	−8.82962	−0.329289	−0.164645	+	0.986353i	$0.552648\pi$
$720$	0	0
$721$	17.1443	0.638488
$722$	0	0
$723$	41.7442	1.55248
$724$	0	0
$725$	9.74387	0.361878
$726$	0	0
$727$	27.7216	1.02814	0.514068	−	0.857750i	$-0.328138\pi$
$727$	27.7216	1.02814	0.514068	+	0.857750i	$0.328138\pi$
$728$	0	0
$729$	25.7659	0.954293
$730$	0	0
$731$	13.6252	0.503946
$732$	0	0
$733$	46.3561	1.71220	0.856101	−	0.516809i	$-0.172880\pi$
$733$	46.3561	1.71220	0.856101	+	0.516809i	$0.172880\pi$
$734$	0	0
$735$	1.76823	0.0652223
$736$	0	0
$737$	15.9456	0.587365
$738$	0	0
$739$	−27.6372	−1.01665	−0.508326	−	0.861165i	$-0.669735\pi$
$739$	−27.6372	−1.01665	−0.508326	+	0.861165i	$0.669735\pi$
$740$	0	0
$741$	8.61722	0.316561
$742$	0	0
$743$	−40.8543	−1.49880	−0.749400	−	0.662117i	$-0.769658\pi$
$743$	−40.8543	−1.49880	−0.749400	+	0.662117i	$0.769658\pi$
$744$	0	0
$745$	−0.524288	−0.0192084
$746$	0	0
$747$	−0.0587073	−0.00214799
$748$	0	0
$749$	4.54865	0.166204
$750$	0	0
$751$	38.0325	1.38783	0.693913	−	0.720058i	$-0.255885\pi$
$751$	38.0325	1.38783	0.693913	+	0.720058i	$0.255885\pi$
$752$	0	0
$753$	−52.1081	−1.89893
$754$	0	0
$755$	−6.21023	−0.226013
$756$	0	0
$757$	4.29534	0.156117	0.0780585	−	0.996949i	$-0.475128\pi$
$757$	4.29534	0.156117	0.0780585	+	0.996949i	$0.475128\pi$
$758$	0	0
$759$	−21.1327	−0.767067
$760$	0	0
$761$	−8.67298	−0.314395	−0.157198	−	0.987567i	$-0.550246\pi$
$761$	−8.67298	−0.314395	−0.157198	+	0.987567i	$0.550246\pi$
$762$	0	0
$763$	−16.5243	−0.598219
$764$	0	0
$765$	−0.683622	−0.0247164
$766$	0	0
$767$	9.95410	0.359422
$768$	0	0
$769$	7.19240	0.259365	0.129682	−	0.991556i	$-0.458604\pi$
$769$	7.19240	0.259365	0.129682	+	0.991556i	$0.458604\pi$
$770$	0	0
$771$	34.2377	1.23304
$772$	0	0
$773$	10.8196	0.389155	0.194578	−	0.980887i	$-0.437666\pi$
$773$	10.8196	0.389155	0.194578	+	0.980887i	$0.437666\pi$
$774$	0	0
$775$	−0.756053	−0.0271582
$776$	0	0
$777$	−20.3180	−0.728905
$778$	0	0
$779$	0.557859	0.0199874
$780$	0	0
$781$	28.4205	1.01696
$782$	0	0
$783$	49.5062	1.76921
$784$	0	0
$785$	−3.99718	−0.142665
$786$	0	0
$787$	40.0354	1.42711	0.713553	−	0.700601i	$-0.247085\pi$
$787$	40.0354	1.42711	0.713553	+	0.700601i	$0.247085\pi$
$788$	0	0
$789$	55.8164	1.98712
$790$	0	0
$791$	−10.8196	−0.384702
$792$	0	0
$793$	−11.2832	−0.400677
$794$	0	0
$795$	21.3262	0.756364
$796$	0	0
$797$	9.58772	0.339614	0.169807	−	0.985477i	$-0.445686\pi$
$797$	9.58772	0.339614	0.169807	+	0.985477i	$0.445686\pi$
$798$	0	0
$799$	9.42803	0.333540
$800$	0	0
$801$	−2.09011	−0.0738504
$802$	0	0
$803$	19.3119	0.681501
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	−8.89397	−0.313083
$808$	0	0
$809$	−6.36674	−0.223843	−0.111921	−	0.993717i	$-0.535700\pi$
$809$	−6.36674	−0.223843	−0.111921	+	0.993717i	$0.535700\pi$
$810$	0	0
$811$	3.40931	0.119717	0.0598585	−	0.998207i	$-0.480935\pi$
$811$	3.40931	0.119717	0.0598585	+	0.998207i	$0.480935\pi$
$812$	0	0
$813$	−4.79219	−0.168069
$814$	0	0
$815$	−4.20740	−0.147379
$816$	0	0
$817$	12.3017	0.430383
$818$	0	0
$819$	0.126652	0.00442559
$820$	0	0
$821$	−14.4843	−0.505505	−0.252753	−	0.967531i	$-0.581336\pi$
$821$	−14.4843	−0.505505	−0.252753	+	0.967531i	$0.581336\pi$
$822$	0	0
$823$	43.7951	1.52660	0.763301	−	0.646043i	$-0.223577\pi$
$823$	43.7951	1.52660	0.763301	+	0.646043i	$0.223577\pi$
$824$	0	0
$825$	5.28316	0.183936
$826$	0	0
$827$	13.5552	0.471360	0.235680	−	0.971831i	$-0.424268\pi$
$827$	13.5552	0.471360	0.235680	+	0.971831i	$0.424268\pi$
$828$	0	0
$829$	−41.2942	−1.43421	−0.717103	−	0.696967i	$-0.754532\pi$
$829$	−41.2942	−1.43421	−0.717103	+	0.696967i	$0.754532\pi$
$830$	0	0
$831$	−18.6525	−0.647047
$832$	0	0
$833$	5.39764	0.187017
$834$	0	0
$835$	11.2588	0.389627
$836$	0	0
$837$	−3.84132	−0.132775
$838$	0	0
$839$	13.5329	0.467207	0.233603	−	0.972332i	$-0.424948\pi$
$839$	13.5329	0.467207	0.233603	+	0.972332i	$0.424948\pi$
$840$	0	0
$841$	65.9431	2.27390
$842$	0	0
$843$	−48.4579	−1.66898
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	2.07294	0.0712270
$848$	0	0
$849$	44.5254	1.52811
$850$	0	0
$851$	45.9623	1.57557
$852$	0	0
$853$	−23.6492	−0.809735	−0.404867	−	0.914375i	$-0.632682\pi$
$853$	−23.6492	−0.809735	−0.404867	+	0.914375i	$0.632682\pi$
$854$	0	0
$855$	−0.617220	−0.0211085
$856$	0	0
$857$	−31.1709	−1.06478	−0.532388	−	0.846500i	$-0.678705\pi$
$857$	−31.1709	−1.06478	−0.532388	+	0.846500i	$0.678705\pi$
$858$	0	0
$859$	−46.1812	−1.57568	−0.787842	−	0.615878i	$-0.788801\pi$
$859$	−46.1812	−1.57568	−0.787842	+	0.615878i	$0.788801\pi$
$860$	0	0
$861$	0.202412	0.00689819
$862$	0	0
$863$	−41.6873	−1.41905	−0.709527	−	0.704679i	$-0.751091\pi$
$863$	−41.6873	−1.41905	−0.709527	+	0.704679i	$0.751091\pi$
$864$	0	0
$865$	11.4906	0.390691
$866$	0	0
$867$	−21.4566	−0.728704
$868$	0	0
$869$	−32.4148	−1.09960
$870$	0	0
$871$	5.33688	0.180833
$872$	0	0
$873$	−1.67926	−0.0568342
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−29.7471	−1.00449	−0.502243	−	0.864726i	$-0.667492\pi$
$877$	−29.7471	−1.00449	−0.502243	+	0.864726i	$0.667492\pi$
$878$	0	0
$879$	44.9645	1.51661
$880$	0	0
$881$	−48.8837	−1.64693	−0.823467	−	0.567365i	$-0.807963\pi$
$881$	−48.8837	−1.64693	−0.823467	+	0.567365i	$0.807963\pi$
$882$	0	0
$883$	−36.5784	−1.23096	−0.615480	−	0.788153i	$-0.711038\pi$
$883$	−36.5784	−1.23096	−0.615480	+	0.788153i	$0.711038\pi$
$884$	0	0
$885$	−17.6012	−0.591657
$886$	0	0
$887$	3.50825	0.117795	0.0588977	−	0.998264i	$-0.481241\pi$
$887$	3.50825	0.117795	0.0588977	+	0.998264i	$0.481241\pi$
$888$	0	0
$889$	1.32624	0.0444807
$890$	0	0
$891$	27.9777	0.937288
$892$	0	0
$893$	8.51226	0.284852
$894$	0	0
$895$	15.9463	0.533026
$896$	0	0
$897$	−7.07294	−0.236158
$898$	0	0
$899$	−7.36689	−0.245699
$900$	0	0
$901$	65.0996	2.16878
$902$	0	0
$903$	4.46353	0.148537
$904$	0	0
$905$	−6.41481	−0.213235
$906$	0	0
$907$	33.1493	1.10071	0.550353	−	0.834932i	$-0.314493\pi$
$907$	33.1493	1.10071	0.550353	+	0.834932i	$0.314493\pi$
$908$	0	0
$909$	−0.253304	−0.00840158
$910$	0	0
$911$	−16.9108	−0.560279	−0.280140	−	0.959959i	$-0.590381\pi$
$911$	−16.9108	−0.560279	−0.280140	+	0.959959i	$0.590381\pi$
$912$	0	0
$913$	1.38495	0.0458351
$914$	0	0
$915$	19.9513	0.659569
$916$	0	0
$917$	6.02436	0.198942
$918$	0	0
$919$	−3.39342	−0.111939	−0.0559693	−	0.998432i	$-0.517825\pi$
$919$	−3.39342	−0.111939	−0.0559693	+	0.998432i	$0.517825\pi$
$920$	0	0
$921$	48.1900	1.58791
$922$	0	0
$923$	9.51211	0.313095
$924$	0	0
$925$	−11.4906	−0.377808
$926$	0	0
$927$	−2.17137	−0.0713171
$928$	0	0
$929$	−55.1906	−1.81074	−0.905372	−	0.424619i	$-0.860408\pi$
$929$	−55.1906	−1.81074	−0.905372	+	0.424619i	$0.860408\pi$
$930$	0	0
$931$	4.87335	0.159718
$932$	0	0
$933$	35.9182	1.17591
$934$	0	0
$935$	16.1272	0.527414
$936$	0	0
$937$	−34.2173	−1.11783	−0.558915	−	0.829225i	$-0.688782\pi$
$937$	−34.2173	−1.11783	−0.558915	+	0.829225i	$0.688782\pi$
$938$	0	0
$939$	−27.9463	−0.911993
$940$	0	0
$941$	−29.9188	−0.975326	−0.487663	−	0.873032i	$-0.662151\pi$
$941$	−29.9188	−0.975326	−0.487663	+	0.873032i	$0.662151\pi$
$942$	0	0
$943$	−0.457885	−0.0149108
$944$	0	0
$945$	5.08075	0.165277
$946$	0	0
$947$	7.36124	0.239208	0.119604	−	0.992822i	$-0.461837\pi$
$947$	7.36124	0.239208	0.119604	+	0.992822i	$0.461837\pi$
$948$	0	0
$949$	6.46353	0.209815
$950$	0	0
$951$	47.8282	1.55094
$952$	0	0
$953$	−7.44775	−0.241256	−0.120628	−	0.992698i	$-0.538491\pi$
$953$	−7.44775	−0.241256	−0.120628	+	0.992698i	$0.538491\pi$
$954$	0	0
$955$	7.51493	0.243177
$956$	0	0
$957$	51.4785	1.66406
$958$	0	0
$959$	−9.46621	−0.305680
$960$	0	0
$961$	−30.4284	−0.981561
$962$	0	0
$963$	−0.576096	−0.0185644
$964$	0	0
$965$	−22.1322	−0.712459
$966$	0	0
$967$	34.9009	1.12234	0.561168	−	0.827702i	$-0.310352\pi$
$967$	34.9009	1.12234	0.561168	+	0.827702i	$0.310352\pi$
$968$	0	0
$969$	−46.5126	−1.49420
$970$	0	0
$971$	−26.1559	−0.839381	−0.419691	−	0.907667i	$-0.637861\pi$
$971$	−26.1559	−0.839381	−0.419691	+	0.907667i	$0.637861\pi$
$972$	0	0
$973$	9.28316	0.297605
$974$	0	0
$975$	1.76823	0.0566288
$976$	0	0
$977$	−50.9518	−1.63009	−0.815046	−	0.579396i	$-0.803288\pi$
$977$	−50.9518	−1.63009	−0.815046	+	0.579396i	$0.803288\pi$
$978$	0	0
$979$	49.3072	1.57587
$980$	0	0
$981$	2.09284	0.0668192
$982$	0	0
$983$	−1.14038	−0.0363723	−0.0181862	−	0.999835i	$-0.505789\pi$
$983$	−1.14038	−0.0363723	−0.0181862	+	0.999835i	$0.505789\pi$
$984$	0	0
$985$	3.53365	0.112591
$986$	0	0
$987$	3.08857	0.0983102
$988$	0	0
$989$	−10.0972	−0.321071
$990$	0	0
$991$	−49.8873	−1.58472	−0.792361	−	0.610053i	$-0.791148\pi$
$991$	−49.8873	−1.58472	−0.792361	+	0.610053i	$0.791148\pi$
$992$	0	0
$993$	47.7733	1.51604
$994$	0	0
$995$	−12.1215	−0.384278
$996$	0	0
$997$	39.0105	1.23547	0.617737	−	0.786385i	$-0.288050\pi$
$997$	39.0105	1.23547	0.617737	+	0.786385i	$0.288050\pi$
$998$	0	0
$999$	−58.3807	−1.84708

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.r.1.2	✓	4
4.3	odd	2		7280.2.a.bz.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.r.1.2	✓	4	1.1	even	1	trivial
7280.2.a.bz.1.3		4	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)

\(f(q)\)	\(=\)	\(q-1.76823 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.126652 q^{9} +O(q^{10})\)	\(q-1.76823 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.126652 q^{9} -2.98782 q^{11} -1.00000 q^{13} +1.76823 q^{15} +5.39764 q^{17} +4.87335 q^{19} +1.76823 q^{21} -4.00000 q^{23} +1.00000 q^{25} +5.08075 q^{27} +9.74387 q^{29} -0.756053 q^{31} +5.28316 q^{33} +1.00000 q^{35} -11.4906 q^{37} +1.76823 q^{39} +0.114471 q^{41} +2.52429 q^{43} -0.126652 q^{45} +1.74670 q^{47} +1.00000 q^{49} -9.54428 q^{51} +12.0608 q^{53} +2.98782 q^{55} -8.61722 q^{57} -9.95410 q^{59} +11.2832 q^{61} -0.126652 q^{63} +1.00000 q^{65} -5.33688 q^{67} +7.07294 q^{69} -9.51211 q^{71} -6.46353 q^{73} -1.76823 q^{75} +2.98782 q^{77} +10.8490 q^{79} -9.36392 q^{81} -0.463532 q^{83} -5.39764 q^{85} -17.2294 q^{87} -16.5027 q^{89} +1.00000 q^{91} +1.33688 q^{93} -4.87335 q^{95} -13.2588 q^{97} -0.378414 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - 4 q^{5} - 4 q^{7} + 11 q^{9}+O(q^{10}) \) 4 * q - q^3 - 4 * q^5 - 4 * q^7 + 11 * q^9	\( 4 q - q^{3} - 4 q^{5} - 4 q^{7} + 11 q^{9} - 2 q^{11} - 4 q^{13} + q^{15} - 11 q^{17} + 9 q^{19} + q^{21} - 16 q^{23} + 4 q^{25} - 4 q^{27} + 13 q^{29} + 13 q^{31} - 12 q^{33} + 4 q^{35} + q^{37} + q^{39} + q^{41} - 12 q^{43} - 11 q^{45} - 14 q^{47} + 4 q^{49} - 26 q^{51} + 14 q^{53} + 2 q^{55} + 2 q^{57} - 5 q^{59} + 12 q^{61} - 11 q^{63} + 4 q^{65} - 23 q^{67} + 4 q^{69} - 6 q^{71} - 38 q^{73} - q^{75} + 2 q^{77} + 13 q^{79} + 20 q^{81} - 14 q^{83} + 11 q^{85} - q^{87} - 29 q^{89} + 4 q^{91} + 7 q^{93} - 9 q^{95} - 28 q^{99}+O(q^{100}) \) 4 * q - q^3 - 4 * q^5 - 4 * q^7 + 11 * q^9 - 2 * q^11 - 4 * q^13 + q^15 - 11 * q^17 + 9 * q^19 + q^21 - 16 * q^23 + 4 * q^25 - 4 * q^27 + 13 * q^29 + 13 * q^31 - 12 * q^33 + 4 * q^35 + q^37 + q^39 + q^41 - 12 * q^43 - 11 * q^45 - 14 * q^47 + 4 * q^49 - 26 * q^51 + 14 * q^53 + 2 * q^55 + 2 * q^57 - 5 * q^59 + 12 * q^61 - 11 * q^63 + 4 * q^65 - 23 * q^67 + 4 * q^69 - 6 * q^71 - 38 * q^73 - q^75 + 2 * q^77 + 13 * q^79 + 20 * q^81 - 14 * q^83 + 11 * q^85 - q^87 - 29 * q^89 + 4 * q^91 + 7 * q^93 - 9 * q^95 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

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