LMFDB - Embedded newform 3640.2.a.ba.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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 33x^{3} - 41x^{2} - 31x - 4 $ x^7 - 3x^6 - 11x^5 + 24x^4 + 33x^3 - 41x^2 - 31x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.70423$ 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)

$f(q)$	$=$	$q-0.704230 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.50406 q^{9} +O(q^{10})$	$q-0.704230 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.50406 q^{9} +1.86704 q^{11} +1.00000 q^{13} -0.704230 q^{15} +0.491240 q^{17} -1.86431 q^{19} -0.704230 q^{21} +3.76871 q^{23} +1.00000 q^{25} +3.87613 q^{27} +8.13179 q^{29} -5.24723 q^{31} -1.31483 q^{33} +1.00000 q^{35} -3.87416 q^{37} -0.704230 q^{39} -5.85586 q^{41} +6.02992 q^{43} -2.50406 q^{45} -3.69025 q^{47} +1.00000 q^{49} -0.345946 q^{51} -6.27709 q^{53} +1.86704 q^{55} +1.31290 q^{57} -1.51837 q^{59} +11.2899 q^{61} -2.50406 q^{63} +1.00000 q^{65} +11.2919 q^{67} -2.65404 q^{69} +8.55424 q^{71} +9.94676 q^{73} -0.704230 q^{75} +1.86704 q^{77} +8.08316 q^{79} +4.78249 q^{81} +5.46765 q^{83} +0.491240 q^{85} -5.72666 q^{87} -12.6748 q^{89} +1.00000 q^{91} +3.69526 q^{93} -1.86431 q^{95} -0.919953 q^{97} -4.67519 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9}+O(q^{10}) $ 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9	$ 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9} + 7 q^{11} + 7 q^{13} + 4 q^{15} + 3 q^{17} + 13 q^{19} + 4 q^{21} + 11 q^{23} + 7 q^{25} + 13 q^{27} + 3 q^{29} + 12 q^{31} - 11 q^{33} + 7 q^{35} - 4 q^{37} + 4 q^{39} + 6 q^{41} + 10 q^{43} + 11 q^{45} + 15 q^{47} + 7 q^{49} + 8 q^{53} + 7 q^{55} - 18 q^{57} + 13 q^{59} - q^{61} + 11 q^{63} + 7 q^{65} + 8 q^{67} - 9 q^{69} + 20 q^{71} - 17 q^{73} + 4 q^{75} + 7 q^{77} + 6 q^{79} + 39 q^{81} + 30 q^{83} + 3 q^{85} + 21 q^{87} + 5 q^{89} + 7 q^{91} - 24 q^{93} + 13 q^{95} - 9 q^{97} + 12 q^{99}+O(q^{100}) $ 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9 + 7 * q^11 + 7 * q^13 + 4 * q^15 + 3 * q^17 + 13 * q^19 + 4 * q^21 + 11 * q^23 + 7 * q^25 + 13 * q^27 + 3 * q^29 + 12 * q^31 - 11 * q^33 + 7 * q^35 - 4 * q^37 + 4 * q^39 + 6 * q^41 + 10 * q^43 + 11 * q^45 + 15 * q^47 + 7 * q^49 + 8 * q^53 + 7 * q^55 - 18 * q^57 + 13 * q^59 - q^61 + 11 * q^63 + 7 * q^65 + 8 * q^67 - 9 * q^69 + 20 * q^71 - 17 * q^73 + 4 * q^75 + 7 * q^77 + 6 * q^79 + 39 * q^81 + 30 * q^83 + 3 * q^85 + 21 * q^87 + 5 * q^89 + 7 * q^91 - 24 * q^93 + 13 * q^95 - 9 * 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
$3$	−0.704230	−0.406588	−0.203294	−	0.979118i	$-0.565165\pi$
$3$	−0.704230	−0.406588	−0.203294	+	0.979118i	$0.565165\pi$
$4$	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
$9$	−2.50406	−0.834687
$10$	0	0
$11$	1.86704	0.562935	0.281467	−	0.959571i	$-0.409179\pi$
$11$	1.86704	0.562935	0.281467	+	0.959571i	$0.409179\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.704230	−0.181831
$16$	0	0
$17$	0.491240	0.119143	0.0595716	−	0.998224i	$-0.481027\pi$
$17$	0.491240	0.119143	0.0595716	+	0.998224i	$0.481027\pi$
$18$	0	0
$19$	−1.86431	−0.427702	−0.213851	−	0.976866i	$-0.568601\pi$
$19$	−1.86431	−0.427702	−0.213851	+	0.976866i	$0.568601\pi$
$20$	0	0
$21$	−0.704230	−0.153676
$22$	0	0
$23$	3.76871	0.785831	0.392915	−	0.919575i	$-0.371467\pi$
$23$	3.76871	0.785831	0.392915	+	0.919575i	$0.371467\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.87613	0.745961
$28$	0	0
$29$	8.13179	1.51004	0.755018	−	0.655704i	$-0.227628\pi$
$29$	8.13179	1.51004	0.755018	+	0.655704i	$0.227628\pi$
$30$	0	0
$31$	−5.24723	−0.942430	−0.471215	−	0.882018i	$-0.656184\pi$
$31$	−5.24723	−0.942430	−0.471215	+	0.882018i	$0.656184\pi$
$32$	0	0
$33$	−1.31483	−0.228882
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−3.87416	−0.636908	−0.318454	−	0.947938i	$-0.603164\pi$
$37$	−3.87416	−0.636908	−0.318454	+	0.947938i	$0.603164\pi$
$38$	0	0
$39$	−0.704230	−0.112767
$40$	0	0
$41$	−5.85586	−0.914532	−0.457266	−	0.889330i	$-0.651171\pi$
$41$	−5.85586	−0.914532	−0.457266	+	0.889330i	$0.651171\pi$
$42$	0	0
$43$	6.02992	0.919555	0.459777	−	0.888034i	$-0.347929\pi$
$43$	6.02992	0.919555	0.459777	+	0.888034i	$0.347929\pi$
$44$	0	0
$45$	−2.50406	−0.373283
$46$	0	0
$47$	−3.69025	−0.538278	−0.269139	−	0.963101i	$-0.586739\pi$
$47$	−3.69025	−0.538278	−0.269139	+	0.963101i	$0.586739\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.345946	−0.0484422
$52$	0	0
$53$	−6.27709	−0.862224	−0.431112	−	0.902298i	$-0.641879\pi$
$53$	−6.27709	−0.862224	−0.431112	+	0.902298i	$0.641879\pi$
$54$	0	0
$55$	1.86704	0.251752
$56$	0	0
$57$	1.31290	0.173898
$58$	0	0
$59$	−1.51837	−0.197674	−0.0988372	−	0.995104i	$-0.531512\pi$
$59$	−1.51837	−0.197674	−0.0988372	+	0.995104i	$0.531512\pi$
$60$	0	0
$61$	11.2899	1.44552	0.722762	−	0.691097i	$-0.242872\pi$
$61$	11.2899	1.44552	0.722762	+	0.691097i	$0.242872\pi$
$62$	0	0
$63$	−2.50406	−0.315482
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	11.2919	1.37952	0.689761	−	0.724037i	$-0.257716\pi$
$67$	11.2919	1.37952	0.689761	+	0.724037i	$0.257716\pi$
$68$	0	0
$69$	−2.65404	−0.319509
$70$	0	0
$71$	8.55424	1.01520	0.507601	−	0.861593i	$-0.330533\pi$
$71$	8.55424	1.01520	0.507601	+	0.861593i	$0.330533\pi$
$72$	0	0
$73$	9.94676	1.16418	0.582090	−	0.813124i	$-0.302235\pi$
$73$	9.94676	1.16418	0.582090	+	0.813124i	$0.302235\pi$
$74$	0	0
$75$	−0.704230	−0.0813175
$76$	0	0
$77$	1.86704	0.212769
$78$	0	0
$79$	8.08316	0.909426	0.454713	−	0.890638i	$-0.349742\pi$
$79$	8.08316	0.909426	0.454713	+	0.890638i	$0.349742\pi$
$80$	0	0
$81$	4.78249	0.531388
$82$	0	0
$83$	5.46765	0.600153	0.300076	−	0.953915i	$-0.402988\pi$
$83$	5.46765	0.600153	0.300076	+	0.953915i	$0.402988\pi$
$84$	0	0
$85$	0.491240	0.0532825
$86$	0	0
$87$	−5.72666	−0.613962
$88$	0	0
$89$	−12.6748	−1.34353	−0.671763	−	0.740766i	$-0.734463\pi$
$89$	−12.6748	−1.34353	−0.671763	+	0.740766i	$0.734463\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	3.69526	0.383180
$94$	0	0
$95$	−1.86431	−0.191274
$96$	0	0
$97$	−0.919953	−0.0934071	−0.0467035	−	0.998909i	$-0.514872\pi$
$97$	−0.919953	−0.0934071	−0.0467035	+	0.998909i	$0.514872\pi$
$98$	0	0
$99$	−4.67519	−0.469874
$100$	0	0
$101$	17.0141	1.69296	0.846482	−	0.532418i	$-0.178716\pi$
$101$	17.0141	1.69296	0.846482	+	0.532418i	$0.178716\pi$
$102$	0	0
$103$	17.1495	1.68979	0.844896	−	0.534930i	$-0.179662\pi$
$103$	17.1495	1.68979	0.844896	+	0.534930i	$0.179662\pi$
$104$	0	0
$105$	−0.704230	−0.0687258
$106$	0	0
$107$	−17.8364	−1.72431	−0.862155	−	0.506645i	$-0.830886\pi$
$107$	−17.8364	−1.72431	−0.862155	+	0.506645i	$0.830886\pi$
$108$	0	0
$109$	−20.1061	−1.92581	−0.962906	−	0.269838i	$-0.913030\pi$
$109$	−20.1061	−1.92581	−0.962906	+	0.269838i	$0.913030\pi$
$110$	0	0
$111$	2.72830	0.258959
$112$	0	0
$113$	−5.88145	−0.553280	−0.276640	−	0.960974i	$-0.589221\pi$
$113$	−5.88145	−0.553280	−0.276640	+	0.960974i	$0.589221\pi$
$114$	0	0
$115$	3.76871	0.351434
$116$	0	0
$117$	−2.50406	−0.231500
$118$	0	0
$119$	0.491240	0.0450319
$120$	0	0
$121$	−7.51415	−0.683104
$122$	0	0
$123$	4.12387	0.371837
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.52844	0.845512	0.422756	−	0.906243i	$-0.361063\pi$
$127$	9.52844	0.845512	0.422756	+	0.906243i	$0.361063\pi$
$128$	0	0
$129$	−4.24646	−0.373880
$130$	0	0
$131$	6.64133	0.580256	0.290128	−	0.956988i	$-0.406302\pi$
$131$	6.64133	0.580256	0.290128	+	0.956988i	$0.406302\pi$
$132$	0	0
$133$	−1.86431	−0.161656
$134$	0	0
$135$	3.87613	0.333604
$136$	0	0
$137$	−17.5309	−1.49776	−0.748881	−	0.662705i	$-0.769409\pi$
$137$	−17.5309	−1.49776	−0.748881	+	0.662705i	$0.769409\pi$
$138$	0	0
$139$	3.94081	0.334255	0.167127	−	0.985935i	$-0.446551\pi$
$139$	3.94081	0.334255	0.167127	+	0.985935i	$0.446551\pi$
$140$	0	0
$141$	2.59878	0.218857
$142$	0	0
$143$	1.86704	0.156130
$144$	0	0
$145$	8.13179	0.675309
$146$	0	0
$147$	−0.704230	−0.0580839
$148$	0	0
$149$	15.8329	1.29708	0.648541	−	0.761180i	$-0.275379\pi$
$149$	15.8329	1.29708	0.648541	+	0.761180i	$0.275379\pi$
$150$	0	0
$151$	14.2533	1.15992	0.579958	−	0.814646i	$-0.303069\pi$
$151$	14.2533	1.15992	0.579958	+	0.814646i	$0.303069\pi$
$152$	0	0
$153$	−1.23009	−0.0994472
$154$	0	0
$155$	−5.24723	−0.421468
$156$	0	0
$157$	−19.7628	−1.57725	−0.788624	−	0.614876i	$-0.789206\pi$
$157$	−19.7628	−1.57725	−0.788624	+	0.614876i	$0.789206\pi$
$158$	0	0
$159$	4.42051	0.350570
$160$	0	0
$161$	3.76871	0.297016
$162$	0	0
$163$	−4.31287	−0.337810	−0.168905	−	0.985632i	$-0.554023\pi$
$163$	−4.31287	−0.337810	−0.168905	+	0.985632i	$0.554023\pi$
$164$	0	0
$165$	−1.31483	−0.102359
$166$	0	0
$167$	13.7740	1.06586	0.532930	−	0.846159i	$-0.321091\pi$
$167$	13.7740	1.06586	0.532930	+	0.846159i	$0.321091\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.66835	0.356997
$172$	0	0
$173$	12.2327	0.930033	0.465016	−	0.885302i	$-0.346048\pi$
$173$	12.2327	0.930033	0.465016	+	0.885302i	$0.346048\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	1.06928	0.0803719
$178$	0	0
$179$	9.01009	0.673446	0.336723	−	0.941604i	$-0.390681\pi$
$179$	9.01009	0.673446	0.336723	+	0.941604i	$0.390681\pi$
$180$	0	0
$181$	−16.7676	−1.24633	−0.623164	−	0.782092i	$-0.714153\pi$
$181$	−16.7676	−1.24633	−0.623164	+	0.782092i	$0.714153\pi$
$182$	0	0
$183$	−7.95069	−0.587732
$184$	0	0
$185$	−3.87416	−0.284834
$186$	0	0
$187$	0.917167	0.0670699
$188$	0	0
$189$	3.87613	0.281947
$190$	0	0
$191$	13.8303	1.00073	0.500363	−	0.865816i	$-0.333200\pi$
$191$	13.8303	1.00073	0.500363	+	0.865816i	$0.333200\pi$
$192$	0	0
$193$	26.3145	1.89416	0.947079	−	0.321000i	$-0.104019\pi$
$193$	26.3145	1.89416	0.947079	+	0.321000i	$0.104019\pi$
$194$	0	0
$195$	−0.704230	−0.0504310
$196$	0	0
$197$	−3.19315	−0.227503	−0.113751	−	0.993509i	$-0.536287\pi$
$197$	−3.19315	−0.227503	−0.113751	+	0.993509i	$0.536287\pi$
$198$	0	0
$199$	15.8718	1.12512	0.562561	−	0.826756i	$-0.309816\pi$
$199$	15.8718	1.12512	0.562561	+	0.826756i	$0.309816\pi$
$200$	0	0
$201$	−7.95208	−0.560896
$202$	0	0
$203$	8.13179	0.570740
$204$	0	0
$205$	−5.85586	−0.408991
$206$	0	0
$207$	−9.43708	−0.655922
$208$	0	0
$209$	−3.48075	−0.240769
$210$	0	0
$211$	24.7218	1.70192	0.850960	−	0.525231i	$-0.176021\pi$
$211$	24.7218	1.70192	0.850960	+	0.525231i	$0.176021\pi$
$212$	0	0
$213$	−6.02415	−0.412768
$214$	0	0
$215$	6.02992	0.411237
$216$	0	0
$217$	−5.24723	−0.356205
$218$	0	0
$219$	−7.00481	−0.473341
$220$	0	0
$221$	0.491240	0.0330444
$222$	0	0
$223$	−19.9444	−1.33558	−0.667789	−	0.744351i	$-0.732759\pi$
$223$	−19.9444	−1.33558	−0.667789	+	0.744351i	$0.732759\pi$
$224$	0	0
$225$	−2.50406	−0.166937
$226$	0	0
$227$	14.1788	0.941077	0.470539	−	0.882379i	$-0.344060\pi$
$227$	14.1788	0.941077	0.470539	+	0.882379i	$0.344060\pi$
$228$	0	0
$229$	13.6264	0.900458	0.450229	−	0.892913i	$-0.351342\pi$
$229$	13.6264	0.900458	0.450229	+	0.892913i	$0.351342\pi$
$230$	0	0
$231$	−1.31483	−0.0865094
$232$	0	0
$233$	−0.986646	−0.0646373	−0.0323187	−	0.999478i	$-0.510289\pi$
$233$	−0.986646	−0.0646373	−0.0323187	+	0.999478i	$0.510289\pi$
$234$	0	0
$235$	−3.69025	−0.240725
$236$	0	0
$237$	−5.69240	−0.369761
$238$	0	0
$239$	15.8911	1.02791	0.513954	−	0.857818i	$-0.328180\pi$
$239$	15.8911	1.02791	0.513954	+	0.857818i	$0.328180\pi$
$240$	0	0
$241$	9.14260	0.588927	0.294463	−	0.955663i	$-0.404859\pi$
$241$	9.14260	0.588927	0.294463	+	0.955663i	$0.404859\pi$
$242$	0	0
$243$	−14.9964	−0.962017
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−1.86431	−0.118623
$248$	0	0
$249$	−3.85049	−0.244015
$250$	0	0
$251$	17.6567	1.11448	0.557240	−	0.830351i	$-0.311860\pi$
$251$	17.6567	1.11448	0.557240	+	0.830351i	$0.311860\pi$
$252$	0	0
$253$	7.03635	0.442372
$254$	0	0
$255$	−0.345946	−0.0216640
$256$	0	0
$257$	26.0461	1.62471	0.812356	−	0.583162i	$-0.198185\pi$
$257$	26.0461	1.62471	0.812356	+	0.583162i	$0.198185\pi$
$258$	0	0
$259$	−3.87416	−0.240728
$260$	0	0
$261$	−20.3625	−1.26041
$262$	0	0
$263$	25.5810	1.57739	0.788697	−	0.614782i	$-0.210756\pi$
$263$	25.5810	1.57739	0.788697	+	0.614782i	$0.210756\pi$
$264$	0	0
$265$	−6.27709	−0.385598
$266$	0	0
$267$	8.92597	0.546261
$268$	0	0
$269$	−2.09757	−0.127891	−0.0639454	−	0.997953i	$-0.520368\pi$
$269$	−2.09757	−0.127891	−0.0639454	+	0.997953i	$0.520368\pi$
$270$	0	0
$271$	−23.9998	−1.45788	−0.728942	−	0.684575i	$-0.759988\pi$
$271$	−23.9998	−1.45788	−0.728942	+	0.684575i	$0.759988\pi$
$272$	0	0
$273$	−0.704230	−0.0426220
$274$	0	0
$275$	1.86704	0.112587
$276$	0	0
$277$	2.43716	0.146434	0.0732172	−	0.997316i	$-0.476673\pi$
$277$	2.43716	0.146434	0.0732172	+	0.997316i	$0.476673\pi$
$278$	0	0
$279$	13.1394	0.786634
$280$	0	0
$281$	19.0338	1.13546	0.567729	−	0.823215i	$-0.307822\pi$
$281$	19.0338	1.13546	0.567729	+	0.823215i	$0.307822\pi$
$282$	0	0
$283$	−31.5934	−1.87803	−0.939016	−	0.343873i	$-0.888261\pi$
$283$	−31.5934	−1.87803	−0.939016	+	0.343873i	$0.888261\pi$
$284$	0	0
$285$	1.31290	0.0777698
$286$	0	0
$287$	−5.85586	−0.345661
$288$	0	0
$289$	−16.7587	−0.985805
$290$	0	0
$291$	0.647859	0.0379782
$292$	0	0
$293$	7.88817	0.460832	0.230416	−	0.973092i	$-0.425991\pi$
$293$	7.88817	0.460832	0.230416	+	0.973092i	$0.425991\pi$
$294$	0	0
$295$	−1.51837	−0.0884026
$296$	0	0
$297$	7.23690	0.419927
$298$	0	0
$299$	3.76871	0.217950
$300$	0	0
$301$	6.02992	0.347559
$302$	0	0
$303$	−11.9818	−0.688338
$304$	0	0
$305$	11.2899	0.646458
$306$	0	0
$307$	−1.45825	−0.0832268	−0.0416134	−	0.999134i	$-0.513250\pi$
$307$	−1.45825	−0.0832268	−0.0416134	+	0.999134i	$0.513250\pi$
$308$	0	0
$309$	−12.0772	−0.687048
$310$	0	0
$311$	−14.0137	−0.794645	−0.397323	−	0.917679i	$-0.630061\pi$
$311$	−14.0137	−0.794645	−0.397323	+	0.917679i	$0.630061\pi$
$312$	0	0
$313$	−10.6963	−0.604591	−0.302295	−	0.953214i	$-0.597753\pi$
$313$	−10.6963	−0.604591	−0.302295	+	0.953214i	$0.597753\pi$
$314$	0	0
$315$	−2.50406	−0.141088
$316$	0	0
$317$	−33.1095	−1.85961	−0.929807	−	0.368048i	$-0.880026\pi$
$317$	−33.1095	−1.85961	−0.929807	+	0.368048i	$0.880026\pi$
$318$	0	0
$319$	15.1824	0.850052
$320$	0	0
$321$	12.5609	0.701083
$322$	0	0
$323$	−0.915824	−0.0509578
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	14.1593	0.783011
$328$	0	0
$329$	−3.69025	−0.203450
$330$	0	0
$331$	4.35526	0.239387	0.119693	−	0.992811i	$-0.461809\pi$
$331$	4.35526	0.239387	0.119693	+	0.992811i	$0.461809\pi$
$332$	0	0
$333$	9.70112	0.531618
$334$	0	0
$335$	11.2919	0.616941
$336$	0	0
$337$	−14.1237	−0.769369	−0.384684	−	0.923048i	$-0.625690\pi$
$337$	−14.1237	−0.769369	−0.384684	+	0.923048i	$0.625690\pi$
$338$	0	0
$339$	4.14189	0.224957
$340$	0	0
$341$	−9.79681	−0.530527
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.65404	−0.142889
$346$	0	0
$347$	13.8860	0.745437	0.372719	−	0.927944i	$-0.378426\pi$
$347$	13.8860	0.745437	0.372719	+	0.927944i	$0.378426\pi$
$348$	0	0
$349$	13.8505	0.741400	0.370700	−	0.928753i	$-0.379118\pi$
$349$	13.8505	0.741400	0.370700	+	0.928753i	$0.379118\pi$
$350$	0	0
$351$	3.87613	0.206892
$352$	0	0
$353$	−15.9307	−0.847906	−0.423953	−	0.905684i	$-0.639358\pi$
$353$	−15.9307	−0.847906	−0.423953	+	0.905684i	$0.639358\pi$
$354$	0	0
$355$	8.55424	0.454012
$356$	0	0
$357$	−0.345946	−0.0183094
$358$	0	0
$359$	−6.66619	−0.351828	−0.175914	−	0.984406i	$-0.556288\pi$
$359$	−6.66619	−0.351828	−0.175914	+	0.984406i	$0.556288\pi$
$360$	0	0
$361$	−15.5243	−0.817071
$362$	0	0
$363$	5.29169	0.277742
$364$	0	0
$365$	9.94676	0.520637
$366$	0	0
$367$	−3.57229	−0.186472	−0.0932360	−	0.995644i	$-0.529721\pi$
$367$	−3.57229	−0.186472	−0.0932360	+	0.995644i	$0.529721\pi$
$368$	0	0
$369$	14.6634	0.763347
$370$	0	0
$371$	−6.27709	−0.325890
$372$	0	0
$373$	18.5313	0.959513	0.479756	−	0.877402i	$-0.340725\pi$
$373$	18.5313	0.959513	0.479756	+	0.877402i	$0.340725\pi$
$374$	0	0
$375$	−0.704230	−0.0363663
$376$	0	0
$377$	8.13179	0.418809
$378$	0	0
$379$	36.6029	1.88016	0.940082	−	0.340948i	$-0.110748\pi$
$379$	36.6029	1.88016	0.940082	+	0.340948i	$0.110748\pi$
$380$	0	0
$381$	−6.71022	−0.343775
$382$	0	0
$383$	17.3646	0.887291	0.443645	−	0.896202i	$-0.353685\pi$
$383$	17.3646	0.887291	0.443645	+	0.896202i	$0.353685\pi$
$384$	0	0
$385$	1.86704	0.0951534
$386$	0	0
$387$	−15.0993	−0.767540
$388$	0	0
$389$	23.3154	1.18214	0.591068	−	0.806622i	$-0.298707\pi$
$389$	23.3154	1.18214	0.591068	+	0.806622i	$0.298707\pi$
$390$	0	0
$391$	1.85134	0.0936264
$392$	0	0
$393$	−4.67703	−0.235925
$394$	0	0
$395$	8.08316	0.406708
$396$	0	0
$397$	−33.7093	−1.69182	−0.845910	−	0.533326i	$-0.820942\pi$
$397$	−33.7093	−1.69182	−0.845910	+	0.533326i	$0.820942\pi$
$398$	0	0
$399$	1.31290	0.0657274
$400$	0	0
$401$	32.7144	1.63368	0.816839	−	0.576866i	$-0.195724\pi$
$401$	32.7144	1.63368	0.816839	+	0.576866i	$0.195724\pi$
$402$	0	0
$403$	−5.24723	−0.261383
$404$	0	0
$405$	4.78249	0.237644
$406$	0	0
$407$	−7.23322	−0.358538
$408$	0	0
$409$	−1.96685	−0.0972544	−0.0486272	−	0.998817i	$-0.515485\pi$
$409$	−1.96685	−0.0972544	−0.0486272	+	0.998817i	$0.515485\pi$
$410$	0	0
$411$	12.3458	0.608971
$412$	0	0
$413$	−1.51837	−0.0747139
$414$	0	0
$415$	5.46765	0.268397
$416$	0	0
$417$	−2.77524	−0.135904
$418$	0	0
$419$	−21.2123	−1.03629	−0.518144	−	0.855293i	$-0.673377\pi$
$419$	−21.2123	−1.03629	−0.518144	+	0.855293i	$0.673377\pi$
$420$	0	0
$421$	26.5621	1.29456	0.647280	−	0.762252i	$-0.275906\pi$
$421$	26.5621	1.29456	0.647280	+	0.762252i	$0.275906\pi$
$422$	0	0
$423$	9.24060	0.449293
$424$	0	0
$425$	0.491240	0.0238286
$426$	0	0
$427$	11.2899	0.546357
$428$	0	0
$429$	−1.31483	−0.0634805
$430$	0	0
$431$	11.5351	0.555624	0.277812	−	0.960635i	$-0.410391\pi$
$431$	11.5351	0.555624	0.277812	+	0.960635i	$0.410391\pi$
$432$	0	0
$433$	−2.70808	−0.130142	−0.0650709	−	0.997881i	$-0.520727\pi$
$433$	−2.70808	−0.130142	−0.0650709	+	0.997881i	$0.520727\pi$
$434$	0	0
$435$	−5.72666	−0.274572
$436$	0	0
$437$	−7.02605	−0.336102
$438$	0	0
$439$	−38.3831	−1.83193	−0.915964	−	0.401261i	$-0.868572\pi$
$439$	−38.3831	−1.83193	−0.915964	+	0.401261i	$0.868572\pi$
$440$	0	0
$441$	−2.50406	−0.119241
$442$	0	0
$443$	10.2131	0.485241	0.242620	−	0.970121i	$-0.421993\pi$
$443$	10.2131	0.485241	0.242620	+	0.970121i	$0.421993\pi$
$444$	0	0
$445$	−12.6748	−0.600843
$446$	0	0
$447$	−11.1500	−0.527377
$448$	0	0
$449$	20.1196	0.949501	0.474750	−	0.880120i	$-0.342538\pi$
$449$	20.1196	0.949501	0.474750	+	0.880120i	$0.342538\pi$
$450$	0	0
$451$	−10.9331	−0.514822
$452$	0	0
$453$	−10.0376	−0.471607
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	−13.4796	−0.630548	−0.315274	−	0.949001i	$-0.602096\pi$
$457$	−13.4796	−0.630548	−0.315274	+	0.949001i	$0.602096\pi$
$458$	0	0
$459$	1.90411	0.0888762
$460$	0	0
$461$	1.79297	0.0835069	0.0417535	−	0.999128i	$-0.486706\pi$
$461$	1.79297	0.0835069	0.0417535	+	0.999128i	$0.486706\pi$
$462$	0	0
$463$	1.85444	0.0861831	0.0430916	−	0.999071i	$-0.486279\pi$
$463$	1.85444	0.0861831	0.0430916	+	0.999071i	$0.486279\pi$
$464$	0	0
$465$	3.69526	0.171363
$466$	0	0
$467$	−37.7424	−1.74651	−0.873255	−	0.487263i	$-0.837995\pi$
$467$	−37.7424	−1.74651	−0.873255	+	0.487263i	$0.837995\pi$
$468$	0	0
$469$	11.2919	0.521410
$470$	0	0
$471$	13.9176	0.641289
$472$	0	0
$473$	11.2581	0.517650
$474$	0	0
$475$	−1.86431	−0.0855405
$476$	0	0
$477$	15.7182	0.719687
$478$	0	0
$479$	−30.0900	−1.37485	−0.687425	−	0.726256i	$-0.741259\pi$
$479$	−30.0900	−1.37485	−0.687425	+	0.726256i	$0.741259\pi$
$480$	0	0
$481$	−3.87416	−0.176646
$482$	0	0
$483$	−2.65404	−0.120763
$484$	0	0
$485$	−0.919953	−0.0417729
$486$	0	0
$487$	18.9586	0.859098	0.429549	−	0.903044i	$-0.358673\pi$
$487$	18.9586	0.859098	0.429549	+	0.903044i	$0.358673\pi$
$488$	0	0
$489$	3.03726	0.137349
$490$	0	0
$491$	−9.89035	−0.446345	−0.223173	−	0.974779i	$-0.571641\pi$
$491$	−9.89035	−0.446345	−0.223173	+	0.974779i	$0.571641\pi$
$492$	0	0
$493$	3.99466	0.179911
$494$	0	0
$495$	−4.67519	−0.210134
$496$	0	0
$497$	8.55424	0.383710
$498$	0	0
$499$	−22.7789	−1.01972	−0.509861	−	0.860257i	$-0.670303\pi$
$499$	−22.7789	−1.01972	−0.509861	+	0.860257i	$0.670303\pi$
$500$	0	0
$501$	−9.70004	−0.433366
$502$	0	0
$503$	−9.33202	−0.416094	−0.208047	−	0.978119i	$-0.566711\pi$
$503$	−9.33202	−0.416094	−0.208047	+	0.978119i	$0.566711\pi$
$504$	0	0
$505$	17.0141	0.757116
$506$	0	0
$507$	−0.704230	−0.0312760
$508$	0	0
$509$	−15.2586	−0.676326	−0.338163	−	0.941088i	$-0.609806\pi$
$509$	−15.2586	−0.676326	−0.338163	+	0.941088i	$0.609806\pi$
$510$	0	0
$511$	9.94676	0.440019
$512$	0	0
$513$	−7.22630	−0.319049
$514$	0	0
$515$	17.1495	0.755698
$516$	0	0
$517$	−6.88986	−0.303015
$518$	0	0
$519$	−8.61462	−0.378140
$520$	0	0
$521$	16.6150	0.727918	0.363959	−	0.931415i	$-0.381425\pi$
$521$	16.6150	0.727918	0.363959	+	0.931415i	$0.381425\pi$
$522$	0	0
$523$	−26.5396	−1.16050	−0.580249	−	0.814439i	$-0.697045\pi$
$523$	−26.5396	−1.16050	−0.580249	+	0.814439i	$0.697045\pi$
$524$	0	0
$525$	−0.704230	−0.0307351
$526$	0	0
$527$	−2.57765	−0.112284
$528$	0	0
$529$	−8.79681	−0.382470
$530$	0	0
$531$	3.80208	0.164996
$532$	0	0
$533$	−5.85586	−0.253645
$534$	0	0
$535$	−17.8364	−0.771135
$536$	0	0
$537$	−6.34518	−0.273815
$538$	0	0
$539$	1.86704	0.0804193
$540$	0	0
$541$	−18.7173	−0.804719	−0.402360	−	0.915482i	$-0.631810\pi$
$541$	−18.7173	−0.804719	−0.402360	+	0.915482i	$0.631810\pi$
$542$	0	0
$543$	11.8083	0.506741
$544$	0	0
$545$	−20.1061	−0.861249
$546$	0	0
$547$	−17.5634	−0.750957	−0.375479	−	0.926831i	$-0.622522\pi$
$547$	−17.5634	−0.750957	−0.375479	+	0.926831i	$0.622522\pi$
$548$	0	0
$549$	−28.2706	−1.20656
$550$	0	0
$551$	−15.1602	−0.645846
$552$	0	0
$553$	8.08316	0.343731
$554$	0	0
$555$	2.72830	0.115810
$556$	0	0
$557$	−26.9162	−1.14048	−0.570238	−	0.821479i	$-0.693149\pi$
$557$	−26.9162	−1.14048	−0.570238	+	0.821479i	$0.693149\pi$
$558$	0	0
$559$	6.02992	0.255039
$560$	0	0
$561$	−0.645897	−0.0272698
$562$	0	0
$563$	−19.1387	−0.806600	−0.403300	−	0.915068i	$-0.632137\pi$
$563$	−19.1387	−0.806600	−0.403300	+	0.915068i	$0.632137\pi$
$564$	0	0
$565$	−5.88145	−0.247434
$566$	0	0
$567$	4.78249	0.200846
$568$	0	0
$569$	−4.39555	−0.184271	−0.0921355	−	0.995746i	$-0.529369\pi$
$569$	−4.39555	−0.184271	−0.0921355	+	0.995746i	$0.529369\pi$
$570$	0	0
$571$	36.8400	1.54171	0.770854	−	0.637012i	$-0.219830\pi$
$571$	36.8400	1.54171	0.770854	+	0.637012i	$0.219830\pi$
$572$	0	0
$573$	−9.73973	−0.406883
$574$	0	0
$575$	3.76871	0.157166
$576$	0	0
$577$	−11.9308	−0.496685	−0.248342	−	0.968672i	$-0.579886\pi$
$577$	−11.9308	−0.496685	−0.248342	+	0.968672i	$0.579886\pi$
$578$	0	0
$579$	−18.5315	−0.770141
$580$	0	0
$581$	5.46765	0.226836
$582$	0	0
$583$	−11.7196	−0.485376
$584$	0	0
$585$	−2.50406	−0.103530
$586$	0	0
$587$	−28.8581	−1.19110	−0.595550	−	0.803318i	$-0.703066\pi$
$587$	−28.8581	−1.19110	−0.595550	+	0.803318i	$0.703066\pi$
$588$	0	0
$589$	9.78247	0.403080
$590$	0	0
$591$	2.24872	0.0924998
$592$	0	0
$593$	−31.5668	−1.29629	−0.648146	−	0.761516i	$-0.724455\pi$
$593$	−31.5668	−1.29629	−0.648146	+	0.761516i	$0.724455\pi$
$594$	0	0
$595$	0.491240	0.0201389
$596$	0	0
$597$	−11.1774	−0.457460
$598$	0	0
$599$	−33.0568	−1.35067	−0.675333	−	0.737513i	$-0.736000\pi$
$599$	−33.0568	−1.35067	−0.675333	+	0.737513i	$0.736000\pi$
$600$	0	0
$601$	42.7372	1.74329	0.871644	−	0.490139i	$-0.163054\pi$
$601$	42.7372	1.74329	0.871644	+	0.490139i	$0.163054\pi$
$602$	0	0
$603$	−28.2755	−1.15147
$604$	0	0
$605$	−7.51415	−0.305493
$606$	0	0
$607$	25.9643	1.05386	0.526929	−	0.849909i	$-0.323343\pi$
$607$	25.9643	1.05386	0.526929	+	0.849909i	$0.323343\pi$
$608$	0	0
$609$	−5.72666	−0.232056
$610$	0	0
$611$	−3.69025	−0.149291
$612$	0	0
$613$	−13.2090	−0.533507	−0.266753	−	0.963765i	$-0.585951\pi$
$613$	−13.2090	−0.533507	−0.266753	+	0.963765i	$0.585951\pi$
$614$	0	0
$615$	4.12387	0.166291
$616$	0	0
$617$	−27.5743	−1.11010	−0.555050	−	0.831817i	$-0.687301\pi$
$617$	−27.5743	−1.11010	−0.555050	+	0.831817i	$0.687301\pi$
$618$	0	0
$619$	11.5687	0.464985	0.232492	−	0.972598i	$-0.425312\pi$
$619$	11.5687	0.464985	0.232492	+	0.972598i	$0.425312\pi$
$620$	0	0
$621$	14.6080	0.586199
$622$	0	0
$623$	−12.6748	−0.507805
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.45125	0.0978935
$628$	0	0
$629$	−1.90314	−0.0758832
$630$	0	0
$631$	0.397114	0.0158089	0.00790444	−	0.999969i	$-0.497484\pi$
$631$	0.397114	0.0158089	0.00790444	+	0.999969i	$0.497484\pi$
$632$	0	0
$633$	−17.4098	−0.691979
$634$	0	0
$635$	9.52844	0.378125
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−21.4203	−0.847375
$640$	0	0
$641$	33.8975	1.33887	0.669435	−	0.742871i	$-0.266536\pi$
$641$	33.8975	1.33887	0.669435	+	0.742871i	$0.266536\pi$
$642$	0	0
$643$	14.1702	0.558818	0.279409	−	0.960172i	$-0.409862\pi$
$643$	14.1702	0.558818	0.279409	+	0.960172i	$0.409862\pi$
$644$	0	0
$645$	−4.24646	−0.167204
$646$	0	0
$647$	20.4370	0.803462	0.401731	−	0.915758i	$-0.368409\pi$
$647$	20.4370	0.803462	0.401731	+	0.915758i	$0.368409\pi$
$648$	0	0
$649$	−2.83485	−0.111278
$650$	0	0
$651$	3.69526	0.144829
$652$	0	0
$653$	5.83688	0.228415	0.114207	−	0.993457i	$-0.463567\pi$
$653$	5.83688	0.228415	0.114207	+	0.993457i	$0.463567\pi$
$654$	0	0
$655$	6.64133	0.259498
$656$	0	0
$657$	−24.9073	−0.971726
$658$	0	0
$659$	19.3389	0.753335	0.376667	−	0.926349i	$-0.377070\pi$
$659$	19.3389	0.753335	0.376667	+	0.926349i	$0.377070\pi$
$660$	0	0
$661$	−39.0270	−1.51797	−0.758987	−	0.651106i	$-0.774305\pi$
$661$	−39.0270	−1.51797	−0.758987	+	0.651106i	$0.774305\pi$
$662$	0	0
$663$	−0.345946	−0.0134354
$664$	0	0
$665$	−1.86431	−0.0722949
$666$	0	0
$667$	30.6464	1.18663
$668$	0	0
$669$	14.0455	0.543029
$670$	0	0
$671$	21.0788	0.813736
$672$	0	0
$673$	11.2702	0.434435	0.217217	−	0.976123i	$-0.430302\pi$
$673$	11.2702	0.434435	0.217217	+	0.976123i	$0.430302\pi$
$674$	0	0
$675$	3.87613	0.149192
$676$	0	0
$677$	−11.2408	−0.432018	−0.216009	−	0.976391i	$-0.569304\pi$
$677$	−11.2408	−0.432018	−0.216009	+	0.976391i	$0.569304\pi$
$678$	0	0
$679$	−0.919953	−0.0353046
$680$	0	0
$681$	−9.98511	−0.382630
$682$	0	0
$683$	−15.6549	−0.599016	−0.299508	−	0.954094i	$-0.596823\pi$
$683$	−15.6549	−0.599016	−0.299508	+	0.954094i	$0.596823\pi$
$684$	0	0
$685$	−17.5309	−0.669819
$686$	0	0
$687$	−9.59613	−0.366115
$688$	0	0
$689$	−6.27709	−0.239138
$690$	0	0
$691$	−27.7982	−1.05749	−0.528746	−	0.848780i	$-0.677338\pi$
$691$	−27.7982	−1.05749	−0.528746	+	0.848780i	$0.677338\pi$
$692$	0	0
$693$	−4.67519	−0.177596
$694$	0	0
$695$	3.94081	0.149483
$696$	0	0
$697$	−2.87663	−0.108960
$698$	0	0
$699$	0.694826	0.0262807
$700$	0	0
$701$	−34.5663	−1.30555	−0.652775	−	0.757551i	$-0.726395\pi$
$701$	−34.5663	−1.30555	−0.652775	+	0.757551i	$0.726395\pi$
$702$	0	0
$703$	7.22264	0.272407
$704$	0	0
$705$	2.59878	0.0978759
$706$	0	0
$707$	17.0141	0.639880
$708$	0	0
$709$	−51.9891	−1.95249	−0.976247	−	0.216662i	$-0.930483\pi$
$709$	−51.9891	−1.95249	−0.976247	+	0.216662i	$0.930483\pi$
$710$	0	0
$711$	−20.2407	−0.759086
$712$	0	0
$713$	−19.7753	−0.740591
$714$	0	0
$715$	1.86704	0.0698235
$716$	0	0
$717$	−11.1910	−0.417934
$718$	0	0
$719$	25.4436	0.948887	0.474443	−	0.880286i	$-0.342649\pi$
$719$	25.4436	0.948887	0.474443	+	0.880286i	$0.342649\pi$
$720$	0	0
$721$	17.1495	0.638681
$722$	0	0
$723$	−6.43850	−0.239450
$724$	0	0
$725$	8.13179	0.302007
$726$	0	0
$727$	19.2732	0.714802	0.357401	−	0.933951i	$-0.383663\pi$
$727$	19.2732	0.714802	0.357401	+	0.933951i	$0.383663\pi$
$728$	0	0
$729$	−3.78660	−0.140244
$730$	0	0
$731$	2.96214	0.109559
$732$	0	0
$733$	−31.4155	−1.16036	−0.580180	−	0.814489i	$-0.697018\pi$
$733$	−31.4155	−1.16036	−0.580180	+	0.814489i	$0.697018\pi$
$734$	0	0
$735$	−0.704230	−0.0259759
$736$	0	0
$737$	21.0824	0.776581
$738$	0	0
$739$	−44.8251	−1.64892	−0.824458	−	0.565923i	$-0.808520\pi$
$739$	−44.8251	−1.64892	−0.824458	+	0.565923i	$0.808520\pi$
$740$	0	0
$741$	1.31290	0.0482308
$742$	0	0
$743$	26.6184	0.976533	0.488266	−	0.872695i	$-0.337629\pi$
$743$	26.6184	0.976533	0.488266	+	0.872695i	$0.337629\pi$
$744$	0	0
$745$	15.8329	0.580073
$746$	0	0
$747$	−13.6913	−0.500940
$748$	0	0
$749$	−17.8364	−0.651728
$750$	0	0
$751$	−47.0282	−1.71608	−0.858042	−	0.513579i	$-0.828319\pi$
$751$	−47.0282	−1.71608	−0.858042	+	0.513579i	$0.828319\pi$
$752$	0	0
$753$	−12.4344	−0.453134
$754$	0	0
$755$	14.2533	0.518730
$756$	0	0
$757$	−37.5999	−1.36659	−0.683295	−	0.730142i	$-0.739454\pi$
$757$	−37.5999	−1.36659	−0.683295	+	0.730142i	$0.739454\pi$
$758$	0	0
$759$	−4.95521	−0.179863
$760$	0	0
$761$	15.9248	0.577275	0.288637	−	0.957439i	$-0.406798\pi$
$761$	15.9248	0.577275	0.288637	+	0.957439i	$0.406798\pi$
$762$	0	0
$763$	−20.1061	−0.727888
$764$	0	0
$765$	−1.23009	−0.0444742
$766$	0	0
$767$	−1.51837	−0.0548250
$768$	0	0
$769$	−12.1757	−0.439068	−0.219534	−	0.975605i	$-0.570454\pi$
$769$	−12.1757	−0.439068	−0.219534	+	0.975605i	$0.570454\pi$
$770$	0	0
$771$	−18.3425	−0.660587
$772$	0	0
$773$	5.78031	0.207903	0.103952	−	0.994582i	$-0.466851\pi$
$773$	5.78031	0.207903	0.103952	+	0.994582i	$0.466851\pi$
$774$	0	0
$775$	−5.24723	−0.188486
$776$	0	0
$777$	2.72830	0.0978772
$778$	0	0
$779$	10.9171	0.391147
$780$	0	0
$781$	15.9711	0.571492
$782$	0	0
$783$	31.5199	1.12643
$784$	0	0
$785$	−19.7628	−0.705366
$786$	0	0
$787$	−36.4098	−1.29787	−0.648935	−	0.760844i	$-0.724785\pi$
$787$	−36.4098	−1.29787	−0.648935	+	0.760844i	$0.724785\pi$
$788$	0	0
$789$	−18.0149	−0.641349
$790$	0	0
$791$	−5.88145	−0.209120
$792$	0	0
$793$	11.2899	0.400916
$794$	0	0
$795$	4.42051	0.156780
$796$	0	0
$797$	−33.7225	−1.19451	−0.597255	−	0.802051i	$-0.703742\pi$
$797$	−33.7225	−1.19451	−0.597255	+	0.802051i	$0.703742\pi$
$798$	0	0
$799$	−1.81280	−0.0641322
$800$	0	0
$801$	31.7384	1.12142
$802$	0	0
$803$	18.5710	0.655358
$804$	0	0
$805$	3.76871	0.132830
$806$	0	0
$807$	1.47717	0.0519988
$808$	0	0
$809$	44.9634	1.58083	0.790415	−	0.612571i	$-0.209865\pi$
$809$	44.9634	1.58083	0.790415	+	0.612571i	$0.209865\pi$
$810$	0	0
$811$	47.9310	1.68308	0.841542	−	0.540191i	$-0.181648\pi$
$811$	47.9310	1.68308	0.841542	+	0.540191i	$0.181648\pi$
$812$	0	0
$813$	16.9014	0.592758
$814$	0	0
$815$	−4.31287	−0.151073
$816$	0	0
$817$	−11.2417	−0.393296
$818$	0	0
$819$	−2.50406	−0.0874989
$820$	0	0
$821$	40.9142	1.42792	0.713958	−	0.700188i	$-0.246901\pi$
$821$	40.9142	1.42792	0.713958	+	0.700188i	$0.246901\pi$
$822$	0	0
$823$	31.6411	1.10294	0.551470	−	0.834195i	$-0.314067\pi$
$823$	31.6411	1.10294	0.551470	+	0.834195i	$0.314067\pi$
$824$	0	0
$825$	−1.31483	−0.0457765
$826$	0	0
$827$	1.10585	0.0384540	0.0192270	−	0.999815i	$-0.493879\pi$
$827$	1.10585	0.0384540	0.0192270	+	0.999815i	$0.493879\pi$
$828$	0	0
$829$	−5.78517	−0.200927	−0.100464	−	0.994941i	$-0.532033\pi$
$829$	−5.78517	−0.200927	−0.100464	+	0.994941i	$0.532033\pi$
$830$	0	0
$831$	−1.71632	−0.0595384
$832$	0	0
$833$	0.491240	0.0170205
$834$	0	0
$835$	13.7740	0.476667
$836$	0	0
$837$	−20.3389	−0.703016
$838$	0	0
$839$	25.1574	0.868530	0.434265	−	0.900785i	$-0.357008\pi$
$839$	25.1574	0.868530	0.434265	+	0.900785i	$0.357008\pi$
$840$	0	0
$841$	37.1261	1.28021
$842$	0	0
$843$	−13.4041	−0.461663
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−7.51415	−0.258189
$848$	0	0
$849$	22.2490	0.763585
$850$	0	0
$851$	−14.6006	−0.500502
$852$	0	0
$853$	−36.8030	−1.26011	−0.630056	−	0.776550i	$-0.716968\pi$
$853$	−36.8030	−1.26011	−0.630056	+	0.776550i	$0.716968\pi$
$854$	0	0
$855$	4.66835	0.159654
$856$	0	0
$857$	−49.1567	−1.67916	−0.839580	−	0.543236i	$-0.817199\pi$
$857$	−49.1567	−1.67916	−0.839580	+	0.543236i	$0.817199\pi$
$858$	0	0
$859$	−43.7397	−1.49238	−0.746190	−	0.665732i	$-0.768119\pi$
$859$	−43.7397	−1.49238	−0.746190	+	0.665732i	$0.768119\pi$
$860$	0	0
$861$	4.12387	0.140541
$862$	0	0
$863$	−22.2037	−0.755823	−0.377912	−	0.925842i	$-0.623358\pi$
$863$	−22.2037	−0.755823	−0.377912	+	0.925842i	$0.623358\pi$
$864$	0	0
$865$	12.2327	0.415923
$866$	0	0
$867$	11.8020	0.400816
$868$	0	0
$869$	15.0916	0.511948
$870$	0	0
$871$	11.2919	0.382611
$872$	0	0
$873$	2.30362	0.0779657
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	2.32808	0.0786137	0.0393069	−	0.999227i	$-0.487485\pi$
$877$	2.32808	0.0786137	0.0393069	+	0.999227i	$0.487485\pi$
$878$	0	0
$879$	−5.55509	−0.187368
$880$	0	0
$881$	−25.4933	−0.858890	−0.429445	−	0.903093i	$-0.641291\pi$
$881$	−25.4933	−0.858890	−0.429445	+	0.903093i	$0.641291\pi$
$882$	0	0
$883$	−52.3506	−1.76174	−0.880868	−	0.473362i	$-0.843040\pi$
$883$	−52.3506	−1.76174	−0.880868	+	0.473362i	$0.843040\pi$
$884$	0	0
$885$	1.06928	0.0359434
$886$	0	0
$887$	−6.51445	−0.218734	−0.109367	−	0.994001i	$-0.534882\pi$
$887$	−6.51445	−0.218734	−0.109367	+	0.994001i	$0.534882\pi$
$888$	0	0
$889$	9.52844	0.319574
$890$	0	0
$891$	8.92913	0.299137
$892$	0	0
$893$	6.87977	0.230223
$894$	0	0
$895$	9.01009	0.301174
$896$	0	0
$897$	−2.65404	−0.0886159
$898$	0	0
$899$	−42.6694	−1.42310
$900$	0	0
$901$	−3.08356	−0.102728
$902$	0	0
$903$	−4.24646	−0.141313
$904$	0	0
$905$	−16.7676	−0.557374
$906$	0	0
$907$	−15.7360	−0.522507	−0.261253	−	0.965270i	$-0.584136\pi$
$907$	−15.7360	−0.522507	−0.261253	+	0.965270i	$0.584136\pi$
$908$	0	0
$909$	−42.6042	−1.41309
$910$	0	0
$911$	23.9454	0.793346	0.396673	−	0.917960i	$-0.370165\pi$
$911$	23.9454	0.793346	0.396673	+	0.917960i	$0.370165\pi$
$912$	0	0
$913$	10.2083	0.337847
$914$	0	0
$915$	−7.95069	−0.262842
$916$	0	0
$917$	6.64133	0.219316
$918$	0	0
$919$	12.0583	0.397768	0.198884	−	0.980023i	$-0.436268\pi$
$919$	12.0583	0.397768	0.198884	+	0.980023i	$0.436268\pi$
$920$	0	0
$921$	1.02695	0.0338390
$922$	0	0
$923$	8.55424	0.281566
$924$	0	0
$925$	−3.87416	−0.127382
$926$	0	0
$927$	−42.9434	−1.41045
$928$	0	0
$929$	16.0664	0.527121	0.263561	−	0.964643i	$-0.415103\pi$
$929$	16.0664	0.527121	0.263561	+	0.964643i	$0.415103\pi$
$930$	0	0
$931$	−1.86431	−0.0611003
$932$	0	0
$933$	9.86889	0.323093
$934$	0	0
$935$	0.917167	0.0299946
$936$	0	0
$937$	40.1210	1.31070	0.655348	−	0.755327i	$-0.272522\pi$
$937$	40.1210	1.31070	0.655348	+	0.755327i	$0.272522\pi$
$938$	0	0
$939$	7.53266	0.245819
$940$	0	0
$941$	−14.2765	−0.465401	−0.232700	−	0.972548i	$-0.574756\pi$
$941$	−14.2765	−0.465401	−0.232700	+	0.972548i	$0.574756\pi$
$942$	0	0
$943$	−22.0691	−0.718667
$944$	0	0
$945$	3.87613	0.126090
$946$	0	0
$947$	12.7324	0.413746	0.206873	−	0.978368i	$-0.433671\pi$
$947$	12.7324	0.413746	0.206873	+	0.978368i	$0.433671\pi$
$948$	0	0
$949$	9.94676	0.322885
$950$	0	0
$951$	23.3167	0.756096
$952$	0	0
$953$	55.3830	1.79403	0.897016	−	0.441998i	$-0.145730\pi$
$953$	55.3830	1.79403	0.897016	+	0.441998i	$0.145730\pi$
$954$	0	0
$955$	13.8303	0.447539
$956$	0	0
$957$	−10.6919	−0.345621
$958$	0	0
$959$	−17.5309	−0.566101
$960$	0	0
$961$	−3.46658	−0.111825
$962$	0	0
$963$	44.6634	1.43926
$964$	0	0
$965$	26.3145	0.847093
$966$	0	0
$967$	−11.6473	−0.374551	−0.187275	−	0.982307i	$-0.559966\pi$
$967$	−11.6473	−0.374551	−0.187275	+	0.982307i	$0.559966\pi$
$968$	0	0
$969$	0.644951	0.0207188
$970$	0	0
$971$	24.7138	0.793102	0.396551	−	0.918013i	$-0.370207\pi$
$971$	24.7138	0.793102	0.396551	+	0.918013i	$0.370207\pi$
$972$	0	0
$973$	3.94081	0.126337
$974$	0	0
$975$	−0.704230	−0.0225534
$976$	0	0
$977$	4.24299	0.135745	0.0678727	−	0.997694i	$-0.478379\pi$
$977$	4.24299	0.135745	0.0678727	+	0.997694i	$0.478379\pi$
$978$	0	0
$979$	−23.6644	−0.756317
$980$	0	0
$981$	50.3468	1.60745
$982$	0	0
$983$	−40.4280	−1.28945	−0.644726	−	0.764413i	$-0.723029\pi$
$983$	−40.4280	−1.28945	−0.644726	+	0.764413i	$0.723029\pi$
$984$	0	0
$985$	−3.19315	−0.101742
$986$	0	0
$987$	2.59878	0.0827202
$988$	0	0
$989$	22.7250	0.722614
$990$	0	0
$991$	−1.72892	−0.0549209	−0.0274604	−	0.999623i	$-0.508742\pi$
$991$	−1.72892	−0.0549209	−0.0274604	+	0.999623i	$0.508742\pi$
$992$	0	0
$993$	−3.06711	−0.0973316
$994$	0	0
$995$	15.8718	0.503170
$996$	0	0
$997$	−38.7260	−1.22646	−0.613232	−	0.789903i	$-0.710131\pi$
$997$	−38.7260	−1.22646	−0.613232	+	0.789903i	$0.710131\pi$
$998$	0	0
$999$	−15.0167	−0.475108

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.ba.1.3	✓	7
4.3	odd	2		7280.2.a.cf.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.ba.1.3	✓	7	1.1	even	1	trivial
7280.2.a.cf.1.5		7	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)

\(f(q)\)	\(=\)	\(q-0.704230 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.50406 q^{9} +O(q^{10})\)	\(q-0.704230 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.50406 q^{9} +1.86704 q^{11} +1.00000 q^{13} -0.704230 q^{15} +0.491240 q^{17} -1.86431 q^{19} -0.704230 q^{21} +3.76871 q^{23} +1.00000 q^{25} +3.87613 q^{27} +8.13179 q^{29} -5.24723 q^{31} -1.31483 q^{33} +1.00000 q^{35} -3.87416 q^{37} -0.704230 q^{39} -5.85586 q^{41} +6.02992 q^{43} -2.50406 q^{45} -3.69025 q^{47} +1.00000 q^{49} -0.345946 q^{51} -6.27709 q^{53} +1.86704 q^{55} +1.31290 q^{57} -1.51837 q^{59} +11.2899 q^{61} -2.50406 q^{63} +1.00000 q^{65} +11.2919 q^{67} -2.65404 q^{69} +8.55424 q^{71} +9.94676 q^{73} -0.704230 q^{75} +1.86704 q^{77} +8.08316 q^{79} +4.78249 q^{81} +5.46765 q^{83} +0.491240 q^{85} -5.72666 q^{87} -12.6748 q^{89} +1.00000 q^{91} +3.69526 q^{93} -1.86431 q^{95} -0.919953 q^{97} -4.67519 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9}+O(q^{10}) \) 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9	\( 7 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 11 q^{9} + 7 q^{11} + 7 q^{13} + 4 q^{15} + 3 q^{17} + 13 q^{19} + 4 q^{21} + 11 q^{23} + 7 q^{25} + 13 q^{27} + 3 q^{29} + 12 q^{31} - 11 q^{33} + 7 q^{35} - 4 q^{37} + 4 q^{39} + 6 q^{41} + 10 q^{43} + 11 q^{45} + 15 q^{47} + 7 q^{49} + 8 q^{53} + 7 q^{55} - 18 q^{57} + 13 q^{59} - q^{61} + 11 q^{63} + 7 q^{65} + 8 q^{67} - 9 q^{69} + 20 q^{71} - 17 q^{73} + 4 q^{75} + 7 q^{77} + 6 q^{79} + 39 q^{81} + 30 q^{83} + 3 q^{85} + 21 q^{87} + 5 q^{89} + 7 q^{91} - 24 q^{93} + 13 q^{95} - 9 q^{97} + 12 q^{99}+O(q^{100}) \) 7 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 11 * q^9 + 7 * q^11 + 7 * q^13 + 4 * q^15 + 3 * q^17 + 13 * q^19 + 4 * q^21 + 11 * q^23 + 7 * q^25 + 13 * q^27 + 3 * q^29 + 12 * q^31 - 11 * q^33 + 7 * q^35 - 4 * q^37 + 4 * q^39 + 6 * q^41 + 10 * q^43 + 11 * q^45 + 15 * q^47 + 7 * q^49 + 8 * q^53 + 7 * q^55 - 18 * q^57 + 13 * q^59 - q^61 + 11 * q^63 + 7 * q^65 + 8 * q^67 - 9 * q^69 + 20 * q^71 - 17 * q^73 + 4 * q^75 + 7 * q^77 + 6 * q^79 + 39 * q^81 + 30 * q^83 + 3 * q^85 + 21 * q^87 + 5 * q^89 + 7 * q^91 - 24 * q^93 + 13 * q^95 - 9 * q^97 + 12 * q^99

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