LMFDB - Embedded newform 6032.2.a.be.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 159 x^{9} - 827 x^{8} - 362 x^{7} + 3029 x^{6} - 142 x^{5} + \cdots + 16 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 159x^9 - 827x^8 - 362x^7 + 3029x^6 - 142x^5 - 4316x^4 + 208x^3 + 2228x^2 + 376x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$3.23329$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23329 q^{3} +1.87064 q^{5} +2.94834 q^{7} +7.45419 q^{9} +O(q^{10})$	$q+3.23329 q^{3} +1.87064 q^{5} +2.94834 q^{7} +7.45419 q^{9} +3.98110 q^{11} -1.00000 q^{13} +6.04832 q^{15} -5.02033 q^{17} -0.0553702 q^{19} +9.53287 q^{21} -1.07065 q^{23} -1.50072 q^{25} +14.4017 q^{27} +1.00000 q^{29} -10.2306 q^{31} +12.8721 q^{33} +5.51528 q^{35} -9.39262 q^{37} -3.23329 q^{39} +7.68406 q^{41} +1.72051 q^{43} +13.9441 q^{45} +12.7366 q^{47} +1.69274 q^{49} -16.2322 q^{51} +7.71821 q^{53} +7.44719 q^{55} -0.179028 q^{57} -2.19031 q^{59} -9.35530 q^{61} +21.9775 q^{63} -1.87064 q^{65} +8.67155 q^{67} -3.46172 q^{69} +13.1909 q^{71} +4.26003 q^{73} -4.85226 q^{75} +11.7377 q^{77} -5.03787 q^{79} +24.2024 q^{81} -10.2861 q^{83} -9.39122 q^{85} +3.23329 q^{87} -2.62492 q^{89} -2.94834 q^{91} -33.0784 q^{93} -0.103577 q^{95} -5.93685 q^{97} +29.6759 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * 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
$3$	3.23329	1.86674	0.933372	−	0.358911i	$-0.116852\pi$
$3$	3.23329	1.86674	0.933372	+	0.358911i	$0.116852\pi$
$4$	0	0
$5$	1.87064	0.836574	0.418287	−	0.908315i	$-0.362631\pi$
$5$	1.87064	0.836574	0.418287	+	0.908315i	$0.362631\pi$
$6$	0	0
$7$	2.94834	1.11437	0.557185	−	0.830389i	$-0.311882\pi$
$7$	2.94834	1.11437	0.557185	+	0.830389i	$0.311882\pi$
$8$	0	0
$9$	7.45419	2.48473
$10$	0	0
$11$	3.98110	1.20035	0.600173	−	0.799870i	$-0.295098\pi$
$11$	3.98110	1.20035	0.600173	+	0.799870i	$0.295098\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	6.04832	1.56167
$16$	0	0
$17$	−5.02033	−1.21761	−0.608805	−	0.793320i	$-0.708351\pi$
$17$	−5.02033	−1.21761	−0.608805	+	0.793320i	$0.708351\pi$
$18$	0	0
$19$	−0.0553702	−0.0127028	−0.00635139	−	0.999980i	$-0.502022\pi$
$19$	−0.0553702	−0.0127028	−0.00635139	+	0.999980i	$0.502022\pi$
$20$	0	0
$21$	9.53287	2.08024
$22$	0	0
$23$	−1.07065	−0.223246	−0.111623	−	0.993751i	$-0.535605\pi$
$23$	−1.07065	−0.223246	−0.111623	+	0.993751i	$0.535605\pi$
$24$	0	0
$25$	−1.50072	−0.300143
$26$	0	0
$27$	14.4017	2.77161
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−10.2306	−1.83746	−0.918731	−	0.394883i	$-0.870785\pi$
$31$	−10.2306	−1.83746	−0.918731	+	0.394883i	$0.870785\pi$
$32$	0	0
$33$	12.8721	2.24074
$34$	0	0
$35$	5.51528	0.932253
$36$	0	0
$37$	−9.39262	−1.54414	−0.772068	−	0.635540i	$-0.780778\pi$
$37$	−9.39262	−1.54414	−0.772068	+	0.635540i	$0.780778\pi$
$38$	0	0
$39$	−3.23329	−0.517741
$40$	0	0
$41$	7.68406	1.20005	0.600025	−	0.799982i	$-0.295157\pi$
$41$	7.68406	1.20005	0.600025	+	0.799982i	$0.295157\pi$
$42$	0	0
$43$	1.72051	0.262375	0.131188	−	0.991358i	$-0.458121\pi$
$43$	1.72051	0.262375	0.131188	+	0.991358i	$0.458121\pi$
$44$	0	0
$45$	13.9441	2.07866
$46$	0	0
$47$	12.7366	1.85782	0.928912	−	0.370301i	$-0.120746\pi$
$47$	12.7366	1.85782	0.928912	+	0.370301i	$0.120746\pi$
$48$	0	0
$49$	1.69274	0.241819
$50$	0	0
$51$	−16.2322	−2.27296
$52$	0	0
$53$	7.71821	1.06018	0.530089	−	0.847942i	$-0.322158\pi$
$53$	7.71821	1.06018	0.530089	+	0.847942i	$0.322158\pi$
$54$	0	0
$55$	7.44719	1.00418
$56$	0	0
$57$	−0.179028	−0.0237128
$58$	0	0
$59$	−2.19031	−0.285154	−0.142577	−	0.989784i	$-0.545539\pi$
$59$	−2.19031	−0.285154	−0.142577	+	0.989784i	$0.545539\pi$
$60$	0	0
$61$	−9.35530	−1.19782	−0.598912	−	0.800815i	$-0.704400\pi$
$61$	−9.35530	−1.19782	−0.598912	+	0.800815i	$0.704400\pi$
$62$	0	0
$63$	21.9775	2.76891
$64$	0	0
$65$	−1.87064	−0.232024
$66$	0	0
$67$	8.67155	1.05940	0.529700	−	0.848185i	$-0.322305\pi$
$67$	8.67155	1.05940	0.529700	+	0.848185i	$0.322305\pi$
$68$	0	0
$69$	−3.46172	−0.416742
$70$	0	0
$71$	13.1909	1.56547	0.782733	−	0.622358i	$-0.213825\pi$
$71$	13.1909	1.56547	0.782733	+	0.622358i	$0.213825\pi$
$72$	0	0
$73$	4.26003	0.498598	0.249299	−	0.968427i	$-0.419800\pi$
$73$	4.26003	0.498598	0.249299	+	0.968427i	$0.419800\pi$
$74$	0	0
$75$	−4.85226	−0.560291
$76$	0	0
$77$	11.7377	1.33763
$78$	0	0
$79$	−5.03787	−0.566804	−0.283402	−	0.959001i	$-0.591463\pi$
$79$	−5.03787	−0.566804	−0.283402	+	0.959001i	$0.591463\pi$
$80$	0	0
$81$	24.2024	2.68916
$82$	0	0
$83$	−10.2861	−1.12905	−0.564525	−	0.825416i	$-0.690941\pi$
$83$	−10.2861	−1.12905	−0.564525	+	0.825416i	$0.690941\pi$
$84$	0	0
$85$	−9.39122	−1.01862
$86$	0	0
$87$	3.23329	0.346646
$88$	0	0
$89$	−2.62492	−0.278241	−0.139121	−	0.990275i	$-0.544428\pi$
$89$	−2.62492	−0.278241	−0.139121	+	0.990275i	$0.544428\pi$
$90$	0	0
$91$	−2.94834	−0.309070
$92$	0	0
$93$	−33.0784	−3.43007
$94$	0	0
$95$	−0.103577	−0.0106268
$96$	0	0
$97$	−5.93685	−0.602796	−0.301398	−	0.953498i	$-0.597453\pi$
$97$	−5.93685	−0.602796	−0.301398	+	0.953498i	$0.597453\pi$
$98$	0	0
$99$	29.6759	2.98254
$100$	0	0
$101$	−1.66876	−0.166048	−0.0830240	−	0.996548i	$-0.526458\pi$
$101$	−1.66876	−0.166048	−0.0830240	+	0.996548i	$0.526458\pi$
$102$	0	0
$103$	−18.0594	−1.77945	−0.889725	−	0.456497i	$-0.849104\pi$
$103$	−18.0594	−1.77945	−0.889725	+	0.456497i	$0.849104\pi$
$104$	0	0
$105$	17.8325	1.74028
$106$	0	0
$107$	0.421870	0.0407837	0.0203919	−	0.999792i	$-0.493509\pi$
$107$	0.421870	0.0407837	0.0203919	+	0.999792i	$0.493509\pi$
$108$	0	0
$109$	12.7056	1.21697	0.608486	−	0.793565i	$-0.291777\pi$
$109$	12.7056	1.21697	0.608486	+	0.793565i	$0.291777\pi$
$110$	0	0
$111$	−30.3691	−2.88251
$112$	0	0
$113$	10.4929	0.987085	0.493543	−	0.869722i	$-0.335702\pi$
$113$	10.4929	0.987085	0.493543	+	0.869722i	$0.335702\pi$
$114$	0	0
$115$	−2.00279	−0.186762
$116$	0	0
$117$	−7.45419	−0.689140
$118$	0	0
$119$	−14.8017	−1.35687
$120$	0	0
$121$	4.84916	0.440833
$122$	0	0
$123$	24.8448	2.24018
$124$	0	0
$125$	−12.1605	−1.08767
$126$	0	0
$127$	1.36193	0.120852	0.0604261	−	0.998173i	$-0.480754\pi$
$127$	1.36193	0.120852	0.0604261	+	0.998173i	$0.480754\pi$
$128$	0	0
$129$	5.56291	0.489787
$130$	0	0
$131$	−3.80828	−0.332731	−0.166365	−	0.986064i	$-0.553203\pi$
$131$	−3.80828	−0.332731	−0.166365	+	0.986064i	$0.553203\pi$
$132$	0	0
$133$	−0.163250	−0.0141556
$134$	0	0
$135$	26.9404	2.31866
$136$	0	0
$137$	0.694953	0.0593739	0.0296869	−	0.999559i	$-0.490549\pi$
$137$	0.694953	0.0593739	0.0296869	+	0.999559i	$0.490549\pi$
$138$	0	0
$139$	14.9705	1.26978	0.634890	−	0.772603i	$-0.281045\pi$
$139$	14.9705	1.26978	0.634890	+	0.772603i	$0.281045\pi$
$140$	0	0
$141$	41.1812	3.46808
$142$	0	0
$143$	−3.98110	−0.332916
$144$	0	0
$145$	1.87064	0.155348
$146$	0	0
$147$	5.47311	0.451415
$148$	0	0
$149$	5.48085	0.449009	0.224504	−	0.974473i	$-0.427924\pi$
$149$	5.48085	0.449009	0.224504	+	0.974473i	$0.427924\pi$
$150$	0	0
$151$	−19.8683	−1.61686	−0.808429	−	0.588594i	$-0.799682\pi$
$151$	−19.8683	−1.61686	−0.808429	+	0.588594i	$0.799682\pi$
$152$	0	0
$153$	−37.4225	−3.02543
$154$	0	0
$155$	−19.1377	−1.53717
$156$	0	0
$157$	−2.33811	−0.186601	−0.0933006	−	0.995638i	$-0.529742\pi$
$157$	−2.33811	−0.186601	−0.0933006	+	0.995638i	$0.529742\pi$
$158$	0	0
$159$	24.9553	1.97908
$160$	0	0
$161$	−3.15664	−0.248778
$162$	0	0
$163$	−18.4930	−1.44848	−0.724240	−	0.689548i	$-0.757809\pi$
$163$	−18.4930	−1.44848	−0.724240	+	0.689548i	$0.757809\pi$
$164$	0	0
$165$	24.0790	1.87455
$166$	0	0
$167$	−25.0232	−1.93635	−0.968175	−	0.250273i	$-0.919479\pi$
$167$	−25.0232	−1.93635	−0.968175	+	0.250273i	$0.919479\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.412740	−0.0315630
$172$	0	0
$173$	−14.1066	−1.07250	−0.536251	−	0.844059i	$-0.680160\pi$
$173$	−14.1066	−1.07250	−0.536251	+	0.844059i	$0.680160\pi$
$174$	0	0
$175$	−4.42463	−0.334471
$176$	0	0
$177$	−7.08191	−0.532309
$178$	0	0
$179$	−14.3061	−1.06928	−0.534642	−	0.845079i	$-0.679554\pi$
$179$	−14.3061	−1.06928	−0.534642	+	0.845079i	$0.679554\pi$
$180$	0	0
$181$	1.67706	0.124655	0.0623274	−	0.998056i	$-0.480148\pi$
$181$	1.67706	0.124655	0.0623274	+	0.998056i	$0.480148\pi$
$182$	0	0
$183$	−30.2484	−2.23603
$184$	0	0
$185$	−17.5702	−1.29179
$186$	0	0
$187$	−19.9864	−1.46155
$188$	0	0
$189$	42.4612	3.08860
$190$	0	0
$191$	−9.00243	−0.651393	−0.325697	−	0.945474i	$-0.605599\pi$
$191$	−9.00243	−0.651393	−0.325697	+	0.945474i	$0.605599\pi$
$192$	0	0
$193$	26.9660	1.94106	0.970528	−	0.240988i	$-0.0774715\pi$
$193$	26.9660	1.94106	0.970528	+	0.240988i	$0.0774715\pi$
$194$	0	0
$195$	−6.04832	−0.433129
$196$	0	0
$197$	−4.32782	−0.308345	−0.154172	−	0.988044i	$-0.549271\pi$
$197$	−4.32782	−0.308345	−0.154172	+	0.988044i	$0.549271\pi$
$198$	0	0
$199$	−11.0548	−0.783655	−0.391828	−	0.920039i	$-0.628157\pi$
$199$	−11.0548	−0.783655	−0.391828	+	0.920039i	$0.628157\pi$
$200$	0	0
$201$	28.0377	1.97763
$202$	0	0
$203$	2.94834	0.206933
$204$	0	0
$205$	14.3741	1.00393
$206$	0	0
$207$	−7.98082	−0.554705
$208$	0	0
$209$	−0.220434	−0.0152477
$210$	0	0
$211$	−4.90648	−0.337776	−0.168888	−	0.985635i	$-0.554018\pi$
$211$	−4.90648	−0.337776	−0.168888	+	0.985635i	$0.554018\pi$
$212$	0	0
$213$	42.6499	2.92232
$214$	0	0
$215$	3.21845	0.219496
$216$	0	0
$217$	−30.1632	−2.04761
$218$	0	0
$219$	13.7739	0.930755
$220$	0	0
$221$	5.02033	0.337704
$222$	0	0
$223$	−5.11524	−0.342542	−0.171271	−	0.985224i	$-0.554787\pi$
$223$	−5.11524	−0.342542	−0.171271	+	0.985224i	$0.554787\pi$
$224$	0	0
$225$	−11.1866	−0.745775
$226$	0	0
$227$	−15.2095	−1.00949	−0.504745	−	0.863269i	$-0.668413\pi$
$227$	−15.2095	−1.00949	−0.504745	+	0.863269i	$0.668413\pi$
$228$	0	0
$229$	−6.57274	−0.434339	−0.217169	−	0.976134i	$-0.569682\pi$
$229$	−6.57274	−0.434339	−0.217169	+	0.976134i	$0.569682\pi$
$230$	0	0
$231$	37.9513	2.49701
$232$	0	0
$233$	−18.5362	−1.21435	−0.607173	−	0.794570i	$-0.707696\pi$
$233$	−18.5362	−1.21435	−0.607173	+	0.794570i	$0.707696\pi$
$234$	0	0
$235$	23.8255	1.55421
$236$	0	0
$237$	−16.2889	−1.05808
$238$	0	0
$239$	−2.93194	−0.189651	−0.0948257	−	0.995494i	$-0.530229\pi$
$239$	−2.93194	−0.189651	−0.0948257	+	0.995494i	$0.530229\pi$
$240$	0	0
$241$	26.6260	1.71513	0.857567	−	0.514372i	$-0.171975\pi$
$241$	26.6260	1.71513	0.857567	+	0.514372i	$0.171975\pi$
$242$	0	0
$243$	35.0483	2.24835
$244$	0	0
$245$	3.16649	0.202300
$246$	0	0
$247$	0.0553702	0.00352312
$248$	0	0
$249$	−33.2581	−2.10765
$250$	0	0
$251$	−15.7359	−0.993239	−0.496620	−	0.867968i	$-0.665426\pi$
$251$	−15.7359	−0.993239	−0.496620	+	0.867968i	$0.665426\pi$
$252$	0	0
$253$	−4.26236	−0.267972
$254$	0	0
$255$	−30.3646	−1.90150
$256$	0	0
$257$	10.4171	0.649802	0.324901	−	0.945748i	$-0.394669\pi$
$257$	10.4171	0.649802	0.324901	+	0.945748i	$0.394669\pi$
$258$	0	0
$259$	−27.6927	−1.72074
$260$	0	0
$261$	7.45419	0.461403
$262$	0	0
$263$	26.9739	1.66328	0.831641	−	0.555314i	$-0.187402\pi$
$263$	26.9739	1.66328	0.831641	+	0.555314i	$0.187402\pi$
$264$	0	0
$265$	14.4380	0.886918
$266$	0	0
$267$	−8.48714	−0.519405
$268$	0	0
$269$	10.5962	0.646063	0.323031	−	0.946388i	$-0.395298\pi$
$269$	10.5962	0.646063	0.323031	+	0.946388i	$0.395298\pi$
$270$	0	0
$271$	21.0259	1.27723	0.638615	−	0.769526i	$-0.279508\pi$
$271$	21.0259	1.27723	0.638615	+	0.769526i	$0.279508\pi$
$272$	0	0
$273$	−9.53287	−0.576955
$274$	0	0
$275$	−5.97450	−0.360276
$276$	0	0
$277$	23.2439	1.39659	0.698296	−	0.715809i	$-0.253942\pi$
$277$	23.2439	1.39659	0.698296	+	0.715809i	$0.253942\pi$
$278$	0	0
$279$	−76.2605	−4.56560
$280$	0	0
$281$	17.4687	1.04209	0.521047	−	0.853528i	$-0.325542\pi$
$281$	17.4687	1.04209	0.521047	+	0.853528i	$0.325542\pi$
$282$	0	0
$283$	−0.916339	−0.0544707	−0.0272354	−	0.999629i	$-0.508670\pi$
$283$	−0.916339	−0.0544707	−0.0272354	+	0.999629i	$0.508670\pi$
$284$	0	0
$285$	−0.334896	−0.0198376
$286$	0	0
$287$	22.6553	1.33730
$288$	0	0
$289$	8.20373	0.482572
$290$	0	0
$291$	−19.1956	−1.12527
$292$	0	0
$293$	−18.1337	−1.05938	−0.529692	−	0.848190i	$-0.677692\pi$
$293$	−18.1337	−1.05938	−0.529692	+	0.848190i	$0.677692\pi$
$294$	0	0
$295$	−4.09727	−0.238552
$296$	0	0
$297$	57.3347	3.32689
$298$	0	0
$299$	1.07065	0.0619172
$300$	0	0
$301$	5.07265	0.292383
$302$	0	0
$303$	−5.39560	−0.309969
$304$	0	0
$305$	−17.5004	−1.00207
$306$	0	0
$307$	30.8869	1.76281	0.881405	−	0.472362i	$-0.156599\pi$
$307$	30.8869	1.76281	0.881405	+	0.472362i	$0.156599\pi$
$308$	0	0
$309$	−58.3915	−3.32178
$310$	0	0
$311$	5.42114	0.307405	0.153702	−	0.988117i	$-0.450880\pi$
$311$	5.42114	0.307405	0.153702	+	0.988117i	$0.450880\pi$
$312$	0	0
$313$	3.20494	0.181154	0.0905770	−	0.995889i	$-0.471129\pi$
$313$	3.20494	0.181154	0.0905770	+	0.995889i	$0.471129\pi$
$314$	0	0
$315$	41.1120	2.31640
$316$	0	0
$317$	−17.0265	−0.956305	−0.478152	−	0.878277i	$-0.658693\pi$
$317$	−17.0265	−0.956305	−0.478152	+	0.878277i	$0.658693\pi$
$318$	0	0
$319$	3.98110	0.222899
$320$	0	0
$321$	1.36403	0.0761327
$322$	0	0
$323$	0.277977	0.0154670
$324$	0	0
$325$	1.50072	0.0832448
$326$	0	0
$327$	41.0808	2.27177
$328$	0	0
$329$	37.5519	2.07030
$330$	0	0
$331$	29.2721	1.60894	0.804471	−	0.593992i	$-0.202449\pi$
$331$	29.2721	1.60894	0.804471	+	0.593992i	$0.202449\pi$
$332$	0	0
$333$	−70.0144	−3.83676
$334$	0	0
$335$	16.2213	0.886266
$336$	0	0
$337$	31.7222	1.72802	0.864010	−	0.503475i	$-0.167945\pi$
$337$	31.7222	1.72802	0.864010	+	0.503475i	$0.167945\pi$
$338$	0	0
$339$	33.9265	1.84263
$340$	0	0
$341$	−40.7289	−2.20559
$342$	0	0
$343$	−15.6476	−0.844893
$344$	0	0
$345$	−6.47563	−0.348636
$346$	0	0
$347$	2.05550	0.110345	0.0551726	−	0.998477i	$-0.482429\pi$
$347$	2.05550	0.110345	0.0551726	+	0.998477i	$0.482429\pi$
$348$	0	0
$349$	−19.8011	−1.05993	−0.529965	−	0.848020i	$-0.677795\pi$
$349$	−19.8011	−1.05993	−0.529965	+	0.848020i	$0.677795\pi$
$350$	0	0
$351$	−14.4017	−0.768707
$352$	0	0
$353$	14.7159	0.783250	0.391625	−	0.920125i	$-0.371913\pi$
$353$	14.7159	0.783250	0.391625	+	0.920125i	$0.371913\pi$
$354$	0	0
$355$	24.6753	1.30963
$356$	0	0
$357$	−47.8581	−2.53292
$358$	0	0
$359$	4.49276	0.237119	0.118559	−	0.992947i	$-0.462172\pi$
$359$	4.49276	0.237119	0.118559	+	0.992947i	$0.462172\pi$
$360$	0	0
$361$	−18.9969	−0.999839
$362$	0	0
$363$	15.6788	0.822921
$364$	0	0
$365$	7.96896	0.417114
$366$	0	0
$367$	−15.5338	−0.810856	−0.405428	−	0.914127i	$-0.632877\pi$
$367$	−15.5338	−0.810856	−0.405428	+	0.914127i	$0.632877\pi$
$368$	0	0
$369$	57.2785	2.98180
$370$	0	0
$371$	22.7560	1.18143
$372$	0	0
$373$	14.4988	0.750720	0.375360	−	0.926879i	$-0.377519\pi$
$373$	14.4988	0.750720	0.375360	+	0.926879i	$0.377519\pi$
$374$	0	0
$375$	−39.3184	−2.03039
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−13.2405	−0.680118	−0.340059	−	0.940404i	$-0.610447\pi$
$379$	−13.2405	−0.680118	−0.340059	+	0.940404i	$0.610447\pi$
$380$	0	0
$381$	4.40353	0.225600
$382$	0	0
$383$	−17.0392	−0.870663	−0.435332	−	0.900270i	$-0.643369\pi$
$383$	−17.0392	−0.870663	−0.435332	+	0.900270i	$0.643369\pi$
$384$	0	0
$385$	21.9569	1.11903
$386$	0	0
$387$	12.8250	0.651932
$388$	0	0
$389$	33.7583	1.71161	0.855806	−	0.517297i	$-0.173062\pi$
$389$	33.7583	1.71161	0.855806	+	0.517297i	$0.173062\pi$
$390$	0	0
$391$	5.37501	0.271826
$392$	0	0
$393$	−12.3133	−0.621123
$394$	0	0
$395$	−9.42402	−0.474174
$396$	0	0
$397$	−15.2027	−0.763003	−0.381502	−	0.924368i	$-0.624593\pi$
$397$	−15.2027	−0.763003	−0.381502	+	0.924368i	$0.624593\pi$
$398$	0	0
$399$	−0.527836	−0.0264249
$400$	0	0
$401$	13.7789	0.688088	0.344044	−	0.938954i	$-0.388203\pi$
$401$	13.7789	0.688088	0.344044	+	0.938954i	$0.388203\pi$
$402$	0	0
$403$	10.2306	0.509620
$404$	0	0
$405$	45.2739	2.24968
$406$	0	0
$407$	−37.3929	−1.85350
$408$	0	0
$409$	−12.8331	−0.634556	−0.317278	−	0.948333i	$-0.602769\pi$
$409$	−12.8331	−0.634556	−0.317278	+	0.948333i	$0.602769\pi$
$410$	0	0
$411$	2.24699	0.110836
$412$	0	0
$413$	−6.45778	−0.317767
$414$	0	0
$415$	−19.2416	−0.944535
$416$	0	0
$417$	48.4040	2.37035
$418$	0	0
$419$	−8.79496	−0.429662	−0.214831	−	0.976651i	$-0.568920\pi$
$419$	−8.79496	−0.429662	−0.214831	+	0.976651i	$0.568920\pi$
$420$	0	0
$421$	−28.7720	−1.40226	−0.701130	−	0.713033i	$-0.747321\pi$
$421$	−28.7720	−1.40226	−0.701130	+	0.713033i	$0.747321\pi$
$422$	0	0
$423$	94.9410	4.61619
$424$	0	0
$425$	7.53410	0.365457
$426$	0	0
$427$	−27.5827	−1.33482
$428$	0	0
$429$	−12.8721	−0.621469
$430$	0	0
$431$	−16.5694	−0.798121	−0.399061	−	0.916924i	$-0.630664\pi$
$431$	−16.5694	−0.798121	−0.399061	+	0.916924i	$0.630664\pi$
$432$	0	0
$433$	22.9741	1.10407	0.552033	−	0.833822i	$-0.313852\pi$
$433$	22.9741	1.10407	0.552033	+	0.833822i	$0.313852\pi$
$434$	0	0
$435$	6.04832	0.289995
$436$	0	0
$437$	0.0592820	0.00283584
$438$	0	0
$439$	34.4945	1.64633	0.823166	−	0.567801i	$-0.192206\pi$
$439$	34.4945	1.64633	0.823166	+	0.567801i	$0.192206\pi$
$440$	0	0
$441$	12.6180	0.600856
$442$	0	0
$443$	11.7525	0.558377	0.279188	−	0.960236i	$-0.409935\pi$
$443$	11.7525	0.558377	0.279188	+	0.960236i	$0.409935\pi$
$444$	0	0
$445$	−4.91028	−0.232769
$446$	0	0
$447$	17.7212	0.838184
$448$	0	0
$449$	16.3034	0.769403	0.384702	−	0.923041i	$-0.374304\pi$
$449$	16.3034	0.769403	0.384702	+	0.923041i	$0.374304\pi$
$450$	0	0
$451$	30.5910	1.44048
$452$	0	0
$453$	−64.2400	−3.01826
$454$	0	0
$455$	−5.51528	−0.258560
$456$	0	0
$457$	2.68684	0.125685	0.0628424	−	0.998023i	$-0.479983\pi$
$457$	2.68684	0.125685	0.0628424	+	0.998023i	$0.479983\pi$
$458$	0	0
$459$	−72.3014	−3.37474
$460$	0	0
$461$	11.6623	0.543166	0.271583	−	0.962415i	$-0.412453\pi$
$461$	11.6623	0.543166	0.271583	+	0.962415i	$0.412453\pi$
$462$	0	0
$463$	−17.4785	−0.812297	−0.406148	−	0.913807i	$-0.633128\pi$
$463$	−17.4785	−0.812297	−0.406148	+	0.913807i	$0.633128\pi$
$464$	0	0
$465$	−61.8777	−2.86951
$466$	0	0
$467$	5.11365	0.236632	0.118316	−	0.992976i	$-0.462250\pi$
$467$	5.11365	0.236632	0.118316	+	0.992976i	$0.462250\pi$
$468$	0	0
$469$	25.5667	1.18056
$470$	0	0
$471$	−7.55979	−0.348337
$472$	0	0
$473$	6.84952	0.314941
$474$	0	0
$475$	0.0830949	0.00381266
$476$	0	0
$477$	57.5330	2.63426
$478$	0	0
$479$	−8.39706	−0.383671	−0.191836	−	0.981427i	$-0.561444\pi$
$479$	−8.39706	−0.383671	−0.191836	+	0.981427i	$0.561444\pi$
$480$	0	0
$481$	9.39262	0.428266
$482$	0	0
$483$	−10.2063	−0.464405
$484$	0	0
$485$	−11.1057	−0.504284
$486$	0	0
$487$	−25.0750	−1.13626	−0.568128	−	0.822940i	$-0.692332\pi$
$487$	−25.0750	−1.13626	−0.568128	+	0.822940i	$0.692332\pi$
$488$	0	0
$489$	−59.7932	−2.70394
$490$	0	0
$491$	−40.4121	−1.82377	−0.911887	−	0.410441i	$-0.865375\pi$
$491$	−40.4121	−1.82377	−0.911887	+	0.410441i	$0.865375\pi$
$492$	0	0
$493$	−5.02033	−0.226104
$494$	0	0
$495$	55.5128	2.49512
$496$	0	0
$497$	38.8912	1.74451
$498$	0	0
$499$	−31.5582	−1.41274	−0.706371	−	0.707842i	$-0.749669\pi$
$499$	−31.5582	−1.41274	−0.706371	+	0.707842i	$0.749669\pi$
$500$	0	0
$501$	−80.9072	−3.61467
$502$	0	0
$503$	−1.28858	−0.0574550	−0.0287275	−	0.999587i	$-0.509146\pi$
$503$	−1.28858	−0.0574550	−0.0287275	+	0.999587i	$0.509146\pi$
$504$	0	0
$505$	−3.12165	−0.138911
$506$	0	0
$507$	3.23329	0.143596
$508$	0	0
$509$	−39.9662	−1.77147	−0.885734	−	0.464192i	$-0.846345\pi$
$509$	−39.9662	−1.77147	−0.885734	+	0.464192i	$0.846345\pi$
$510$	0	0
$511$	12.5600	0.555623
$512$	0	0
$513$	−0.797425	−0.0352072
$514$	0	0
$515$	−33.7827	−1.48864
$516$	0	0
$517$	50.7057	2.23003
$518$	0	0
$519$	−45.6106	−2.00209
$520$	0	0
$521$	−37.8830	−1.65968	−0.829841	−	0.557999i	$-0.811569\pi$
$521$	−37.8830	−1.65968	−0.829841	+	0.557999i	$0.811569\pi$
$522$	0	0
$523$	40.2996	1.76218	0.881090	−	0.472949i	$-0.156810\pi$
$523$	40.2996	1.76218	0.881090	+	0.472949i	$0.156810\pi$
$524$	0	0
$525$	−14.3061	−0.624371
$526$	0	0
$527$	51.3608	2.23731
$528$	0	0
$529$	−21.8537	−0.950161
$530$	0	0
$531$	−16.3270	−0.708531
$532$	0	0
$533$	−7.68406	−0.332834
$534$	0	0
$535$	0.789166	0.0341186
$536$	0	0
$537$	−46.2557	−1.99608
$538$	0	0
$539$	6.73895	0.290267
$540$	0	0
$541$	−17.1720	−0.738282	−0.369141	−	0.929373i	$-0.620348\pi$
$541$	−17.1720	−0.738282	−0.369141	+	0.929373i	$0.620348\pi$
$542$	0	0
$543$	5.42242	0.232698
$544$	0	0
$545$	23.7675	1.01809
$546$	0	0
$547$	2.66773	0.114064	0.0570320	−	0.998372i	$-0.481836\pi$
$547$	2.66773	0.114064	0.0570320	+	0.998372i	$0.481836\pi$
$548$	0	0
$549$	−69.7362	−2.97627
$550$	0	0
$551$	−0.0553702	−0.00235885
$552$	0	0
$553$	−14.8534	−0.631630
$554$	0	0
$555$	−56.8096	−2.41143
$556$	0	0
$557$	9.29998	0.394053	0.197026	−	0.980398i	$-0.436872\pi$
$557$	9.29998	0.394053	0.197026	+	0.980398i	$0.436872\pi$
$558$	0	0
$559$	−1.72051	−0.0727698
$560$	0	0
$561$	−64.6220	−2.72835
$562$	0	0
$563$	30.8407	1.29978	0.649889	−	0.760029i	$-0.274815\pi$
$563$	30.8407	1.29978	0.649889	+	0.760029i	$0.274815\pi$
$564$	0	0
$565$	19.6283	0.825770
$566$	0	0
$567$	71.3570	2.99671
$568$	0	0
$569$	8.09222	0.339244	0.169622	−	0.985509i	$-0.445745\pi$
$569$	8.09222	0.339244	0.169622	+	0.985509i	$0.445745\pi$
$570$	0	0
$571$	23.4744	0.982373	0.491187	−	0.871054i	$-0.336563\pi$
$571$	23.4744	0.982373	0.491187	+	0.871054i	$0.336563\pi$
$572$	0	0
$573$	−29.1075	−1.21598
$574$	0	0
$575$	1.60674	0.0670057
$576$	0	0
$577$	5.08314	0.211614	0.105807	−	0.994387i	$-0.466257\pi$
$577$	5.08314	0.211614	0.105807	+	0.994387i	$0.466257\pi$
$578$	0	0
$579$	87.1891	3.62345
$580$	0	0
$581$	−30.3271	−1.25818
$582$	0	0
$583$	30.7270	1.27258
$584$	0	0
$585$	−13.9441	−0.576517
$586$	0	0
$587$	−16.4058	−0.677141	−0.338571	−	0.940941i	$-0.609943\pi$
$587$	−16.4058	−0.677141	−0.338571	+	0.940941i	$0.609943\pi$
$588$	0	0
$589$	0.566468	0.0233409
$590$	0	0
$591$	−13.9931	−0.575600
$592$	0	0
$593$	−8.92633	−0.366560	−0.183280	−	0.983061i	$-0.558672\pi$
$593$	−8.92633	−0.366560	−0.183280	+	0.983061i	$0.558672\pi$
$594$	0	0
$595$	−27.6885	−1.13512
$596$	0	0
$597$	−35.7435	−1.46288
$598$	0	0
$599$	−8.07143	−0.329790	−0.164895	−	0.986311i	$-0.552729\pi$
$599$	−8.07143	−0.329790	−0.164895	+	0.986311i	$0.552729\pi$
$600$	0	0
$601$	−8.71252	−0.355391	−0.177696	−	0.984086i	$-0.556864\pi$
$601$	−8.71252	−0.355391	−0.177696	+	0.984086i	$0.556864\pi$
$602$	0	0
$603$	64.6394	2.63232
$604$	0	0
$605$	9.07102	0.368789
$606$	0	0
$607$	−40.0013	−1.62360	−0.811802	−	0.583933i	$-0.801513\pi$
$607$	−40.0013	−1.62360	−0.811802	+	0.583933i	$0.801513\pi$
$608$	0	0
$609$	9.53287	0.386291
$610$	0	0
$611$	−12.7366	−0.515268
$612$	0	0
$613$	18.7952	0.759133	0.379566	−	0.925164i	$-0.376073\pi$
$613$	18.7952	0.759133	0.379566	+	0.925164i	$0.376073\pi$
$614$	0	0
$615$	46.4757	1.87408
$616$	0	0
$617$	31.1493	1.25402	0.627012	−	0.779009i	$-0.284278\pi$
$617$	31.1493	1.25402	0.627012	+	0.779009i	$0.284278\pi$
$618$	0	0
$619$	−26.4475	−1.06301	−0.531507	−	0.847054i	$-0.678374\pi$
$619$	−26.4475	−1.06301	−0.531507	+	0.847054i	$0.678374\pi$
$620$	0	0
$621$	−15.4192	−0.618750
$622$	0	0
$623$	−7.73917	−0.310063
$624$	0	0
$625$	−15.2443	−0.609771
$626$	0	0
$627$	−0.712728	−0.0284636
$628$	0	0
$629$	47.1540	1.88016
$630$	0	0
$631$	49.0635	1.95319	0.976594	−	0.215093i	$-0.0690055\pi$
$631$	49.0635	1.95319	0.976594	+	0.215093i	$0.0690055\pi$
$632$	0	0
$633$	−15.8641	−0.630541
$634$	0	0
$635$	2.54768	0.101102
$636$	0	0
$637$	−1.69274	−0.0670686
$638$	0	0
$639$	98.3271	3.88976
$640$	0	0
$641$	−3.84553	−0.151889	−0.0759447	−	0.997112i	$-0.524197\pi$
$641$	−3.84553	−0.151889	−0.0759447	+	0.997112i	$0.524197\pi$
$642$	0	0
$643$	13.1571	0.518866	0.259433	−	0.965761i	$-0.416464\pi$
$643$	13.1571	0.518866	0.259433	+	0.965761i	$0.416464\pi$
$644$	0	0
$645$	10.4062	0.409743
$646$	0	0
$647$	32.8785	1.29259	0.646294	−	0.763088i	$-0.276318\pi$
$647$	32.8785	1.29259	0.646294	+	0.763088i	$0.276318\pi$
$648$	0	0
$649$	−8.71984	−0.342284
$650$	0	0
$651$	−97.5265	−3.82237
$652$	0	0
$653$	−20.5775	−0.805260	−0.402630	−	0.915363i	$-0.631904\pi$
$653$	−20.5775	−0.805260	−0.402630	+	0.915363i	$0.631904\pi$
$654$	0	0
$655$	−7.12391	−0.278354
$656$	0	0
$657$	31.7550	1.23888
$658$	0	0
$659$	46.0812	1.79507	0.897535	−	0.440943i	$-0.145356\pi$
$659$	46.0812	1.79507	0.897535	+	0.440943i	$0.145356\pi$
$660$	0	0
$661$	−27.6489	−1.07542	−0.537708	−	0.843131i	$-0.680710\pi$
$661$	−27.6489	−1.07542	−0.537708	+	0.843131i	$0.680710\pi$
$662$	0	0
$663$	16.2322	0.630407
$664$	0	0
$665$	−0.305382	−0.0118422
$666$	0	0
$667$	−1.07065	−0.0414557
$668$	0	0
$669$	−16.5391	−0.639437
$670$	0	0
$671$	−37.2444	−1.43780
$672$	0	0
$673$	−26.4071	−1.01792	−0.508959	−	0.860791i	$-0.669969\pi$
$673$	−26.4071	−1.01792	−0.508959	+	0.860791i	$0.669969\pi$
$674$	0	0
$675$	−21.6129	−0.831881
$676$	0	0
$677$	−17.2633	−0.663484	−0.331742	−	0.943370i	$-0.607636\pi$
$677$	−17.2633	−0.663484	−0.331742	+	0.943370i	$0.607636\pi$
$678$	0	0
$679$	−17.5039	−0.671737
$680$	0	0
$681$	−49.1768	−1.88446
$682$	0	0
$683$	−17.0030	−0.650604	−0.325302	−	0.945610i	$-0.605466\pi$
$683$	−17.0030	−0.650604	−0.325302	+	0.945610i	$0.605466\pi$
$684$	0	0
$685$	1.30001	0.0496707
$686$	0	0
$687$	−21.2516	−0.810799
$688$	0	0
$689$	−7.71821	−0.294040
$690$	0	0
$691$	10.8625	0.413231	0.206615	−	0.978422i	$-0.433755\pi$
$691$	10.8625	0.413231	0.206615	+	0.978422i	$0.433755\pi$
$692$	0	0
$693$	87.4947	3.32365
$694$	0	0
$695$	28.0043	1.06227
$696$	0	0
$697$	−38.5765	−1.46119
$698$	0	0
$699$	−59.9329	−2.26687
$700$	0	0
$701$	13.7479	0.519252	0.259626	−	0.965709i	$-0.416401\pi$
$701$	13.7479	0.519252	0.259626	+	0.965709i	$0.416401\pi$
$702$	0	0
$703$	0.520071	0.0196148
$704$	0	0
$705$	77.0350	2.90131
$706$	0	0
$707$	−4.92008	−0.185039
$708$	0	0
$709$	21.0269	0.789680	0.394840	−	0.918750i	$-0.370800\pi$
$709$	21.0269	0.789680	0.394840	+	0.918750i	$0.370800\pi$
$710$	0	0
$711$	−37.5532	−1.40836
$712$	0	0
$713$	10.9533	0.410206
$714$	0	0
$715$	−7.44719	−0.278509
$716$	0	0
$717$	−9.47982	−0.354031
$718$	0	0
$719$	−10.2127	−0.380868	−0.190434	−	0.981700i	$-0.560989\pi$
$719$	−10.2127	−0.380868	−0.190434	+	0.981700i	$0.560989\pi$
$720$	0	0
$721$	−53.2455	−1.98297
$722$	0	0
$723$	86.0899	3.20172
$724$	0	0
$725$	−1.50072	−0.0557352
$726$	0	0
$727$	16.2204	0.601581	0.300791	−	0.953690i	$-0.402749\pi$
$727$	16.2204	0.601581	0.300791	+	0.953690i	$0.402749\pi$
$728$	0	0
$729$	40.7144	1.50794
$730$	0	0
$731$	−8.63753	−0.319470
$732$	0	0
$733$	−42.6819	−1.57649	−0.788245	−	0.615361i	$-0.789010\pi$
$733$	−42.6819	−1.57649	−0.788245	+	0.615361i	$0.789010\pi$
$734$	0	0
$735$	10.2382	0.377642
$736$	0	0
$737$	34.5223	1.27165
$738$	0	0
$739$	6.50757	0.239385	0.119692	−	0.992811i	$-0.461809\pi$
$739$	6.50757	0.239385	0.119692	+	0.992811i	$0.461809\pi$
$740$	0	0
$741$	0.179028	0.00657676
$742$	0	0
$743$	−49.6589	−1.82181	−0.910904	−	0.412618i	$-0.864614\pi$
$743$	−49.6589	−1.82181	−0.910904	+	0.412618i	$0.864614\pi$
$744$	0	0
$745$	10.2527	0.375629
$746$	0	0
$747$	−76.6749	−2.80539
$748$	0	0
$749$	1.24382	0.0454481
$750$	0	0
$751$	6.81481	0.248676	0.124338	−	0.992240i	$-0.460319\pi$
$751$	6.81481	0.248676	0.124338	+	0.992240i	$0.460319\pi$
$752$	0	0
$753$	−50.8787	−1.85412
$754$	0	0
$755$	−37.1663	−1.35262
$756$	0	0
$757$	35.7042	1.29769	0.648845	−	0.760920i	$-0.275252\pi$
$757$	35.7042	1.29769	0.648845	+	0.760920i	$0.275252\pi$
$758$	0	0
$759$	−13.7815	−0.500235
$760$	0	0
$761$	34.5251	1.25153	0.625767	−	0.780010i	$-0.284786\pi$
$761$	34.5251	1.25153	0.625767	+	0.780010i	$0.284786\pi$
$762$	0	0
$763$	37.4604	1.35616
$764$	0	0
$765$	−70.0039	−2.53100
$766$	0	0
$767$	2.19031	0.0790875
$768$	0	0
$769$	−39.2415	−1.41508	−0.707542	−	0.706671i	$-0.750196\pi$
$769$	−39.2415	−1.41508	−0.707542	+	0.706671i	$0.750196\pi$
$770$	0	0
$771$	33.6816	1.21301
$772$	0	0
$773$	17.7974	0.640128	0.320064	−	0.947396i	$-0.396296\pi$
$773$	17.7974	0.640128	0.320064	+	0.947396i	$0.396296\pi$
$774$	0	0
$775$	15.3532	0.551502
$776$	0	0
$777$	−89.5386	−3.21218
$778$	0	0
$779$	−0.425468	−0.0152440
$780$	0	0
$781$	52.5141	1.87910
$782$	0	0
$783$	14.4017	0.514675
$784$	0	0
$785$	−4.37375	−0.156106
$786$	0	0
$787$	−12.6121	−0.449572	−0.224786	−	0.974408i	$-0.572168\pi$
$787$	−12.6121	−0.449572	−0.224786	+	0.974408i	$0.572168\pi$
$788$	0	0
$789$	87.2145	3.10492
$790$	0	0
$791$	30.9366	1.09998
$792$	0	0
$793$	9.35530	0.332217
$794$	0	0
$795$	46.6822	1.65565
$796$	0	0
$797$	4.09072	0.144901	0.0724504	−	0.997372i	$-0.476918\pi$
$797$	4.09072	0.144901	0.0724504	+	0.997372i	$0.476918\pi$
$798$	0	0
$799$	−63.9419	−2.26210
$800$	0	0
$801$	−19.5667	−0.691354
$802$	0	0
$803$	16.9596	0.598491
$804$	0	0
$805$	−5.90493	−0.208121
$806$	0	0
$807$	34.2607	1.20603
$808$	0	0
$809$	36.4945	1.28308	0.641540	−	0.767090i	$-0.278296\pi$
$809$	36.4945	1.28308	0.641540	+	0.767090i	$0.278296\pi$
$810$	0	0
$811$	40.4487	1.42035	0.710173	−	0.704027i	$-0.248617\pi$
$811$	40.4487	1.42035	0.710173	+	0.704027i	$0.248617\pi$
$812$	0	0
$813$	67.9828	2.38426
$814$	0	0
$815$	−34.5936	−1.21176
$816$	0	0
$817$	−0.0952649	−0.00333290
$818$	0	0
$819$	−21.9775	−0.767957
$820$	0	0
$821$	13.5135	0.471625	0.235813	−	0.971799i	$-0.424225\pi$
$821$	13.5135	0.471625	0.235813	+	0.971799i	$0.424225\pi$
$822$	0	0
$823$	19.0158	0.662850	0.331425	−	0.943482i	$-0.392471\pi$
$823$	19.0158	0.662850	0.331425	+	0.943482i	$0.392471\pi$
$824$	0	0
$825$	−19.3173	−0.672543
$826$	0	0
$827$	−4.31644	−0.150097	−0.0750487	−	0.997180i	$-0.523911\pi$
$827$	−4.31644	−0.150097	−0.0750487	+	0.997180i	$0.523911\pi$
$828$	0	0
$829$	28.6249	0.994184	0.497092	−	0.867698i	$-0.334401\pi$
$829$	28.6249	0.994184	0.497092	+	0.867698i	$0.334401\pi$
$830$	0	0
$831$	75.1544	2.60708
$832$	0	0
$833$	−8.49809	−0.294441
$834$	0	0
$835$	−46.8093	−1.61990
$836$	0	0
$837$	−147.338	−5.09273
$838$	0	0
$839$	5.19941	0.179504	0.0897518	−	0.995964i	$-0.471393\pi$
$839$	5.19941	0.179504	0.0897518	+	0.995964i	$0.471393\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	56.4814	1.94532
$844$	0	0
$845$	1.87064	0.0643519
$846$	0	0
$847$	14.2970	0.491250
$848$	0	0
$849$	−2.96279	−0.101683
$850$	0	0
$851$	10.0562	0.344722
$852$	0	0
$853$	8.63222	0.295562	0.147781	−	0.989020i	$-0.452787\pi$
$853$	8.63222	0.295562	0.147781	+	0.989020i	$0.452787\pi$
$854$	0	0
$855$	−0.772086	−0.0264048
$856$	0	0
$857$	11.8303	0.404117	0.202058	−	0.979373i	$-0.435237\pi$
$857$	11.8303	0.404117	0.202058	+	0.979373i	$0.435237\pi$
$858$	0	0
$859$	−20.1757	−0.688388	−0.344194	−	0.938899i	$-0.611848\pi$
$859$	−20.1757	−0.688388	−0.344194	+	0.938899i	$0.611848\pi$
$860$	0	0
$861$	73.2512	2.49639
$862$	0	0
$863$	2.81312	0.0957598	0.0478799	−	0.998853i	$-0.484754\pi$
$863$	2.81312	0.0957598	0.0478799	+	0.998853i	$0.484754\pi$
$864$	0	0
$865$	−26.3882	−0.897227
$866$	0	0
$867$	26.5251	0.900838
$868$	0	0
$869$	−20.0563	−0.680362
$870$	0	0
$871$	−8.67155	−0.293824
$872$	0	0
$873$	−44.2544	−1.49779
$874$	0	0
$875$	−35.8533	−1.21206
$876$	0	0
$877$	−12.5814	−0.424843	−0.212421	−	0.977178i	$-0.568135\pi$
$877$	−12.5814	−0.424843	−0.212421	+	0.977178i	$0.568135\pi$
$878$	0	0
$879$	−58.6317	−1.97760
$880$	0	0
$881$	−33.3609	−1.12396	−0.561979	−	0.827152i	$-0.689960\pi$
$881$	−33.3609	−1.12396	−0.561979	+	0.827152i	$0.689960\pi$
$882$	0	0
$883$	−28.8089	−0.969497	−0.484748	−	0.874654i	$-0.661089\pi$
$883$	−28.8089	−0.969497	−0.484748	+	0.874654i	$0.661089\pi$
$884$	0	0
$885$	−13.2477	−0.445316
$886$	0	0
$887$	7.28232	0.244516	0.122258	−	0.992498i	$-0.460986\pi$
$887$	7.28232	0.244516	0.122258	+	0.992498i	$0.460986\pi$
$888$	0	0
$889$	4.01545	0.134674
$890$	0	0
$891$	96.3522	3.22792
$892$	0	0
$893$	−0.705227	−0.0235995
$894$	0	0
$895$	−26.7614	−0.894536
$896$	0	0
$897$	3.46172	0.115584
$898$	0	0
$899$	−10.2306	−0.341208
$900$	0	0
$901$	−38.7480	−1.29088
$902$	0	0
$903$	16.4014	0.545804
$904$	0	0
$905$	3.13717	0.104283
$906$	0	0
$907$	47.8321	1.58824	0.794119	−	0.607762i	$-0.207933\pi$
$907$	47.8321	1.58824	0.794119	+	0.607762i	$0.207933\pi$
$908$	0	0
$909$	−12.4393	−0.412585
$910$	0	0
$911$	−42.1992	−1.39812	−0.699061	−	0.715062i	$-0.746398\pi$
$911$	−42.1992	−1.39812	−0.699061	+	0.715062i	$0.746398\pi$
$912$	0	0
$913$	−40.9502	−1.35525
$914$	0	0
$915$	−56.5839	−1.87061
$916$	0	0
$917$	−11.2281	−0.370785
$918$	0	0
$919$	13.7487	0.453528	0.226764	−	0.973950i	$-0.427185\pi$
$919$	13.7487	0.453528	0.226764	+	0.973950i	$0.427185\pi$
$920$	0	0
$921$	99.8665	3.29071
$922$	0	0
$923$	−13.1909	−0.434182
$924$	0	0
$925$	14.0957	0.463462
$926$	0	0
$927$	−134.619	−4.42145
$928$	0	0
$929$	26.1501	0.857957	0.428979	−	0.903315i	$-0.358874\pi$
$929$	26.1501	0.857957	0.428979	+	0.903315i	$0.358874\pi$
$930$	0	0
$931$	−0.0937270	−0.00307178
$932$	0	0
$933$	17.5282	0.573846
$934$	0	0
$935$	−37.3874	−1.22270
$936$	0	0
$937$	−3.23874	−0.105805	−0.0529024	−	0.998600i	$-0.516847\pi$
$937$	−3.23874	−0.105805	−0.0529024	+	0.998600i	$0.516847\pi$
$938$	0	0
$939$	10.3625	0.338168
$940$	0	0
$941$	13.9599	0.455080	0.227540	−	0.973769i	$-0.426932\pi$
$941$	13.9599	0.455080	0.227540	+	0.973769i	$0.426932\pi$
$942$	0	0
$943$	−8.22693	−0.267906
$944$	0	0
$945$	79.4295	2.58384
$946$	0	0
$947$	57.6838	1.87447	0.937236	−	0.348696i	$-0.113375\pi$
$947$	57.6838	1.87447	0.937236	+	0.348696i	$0.113375\pi$
$948$	0	0
$949$	−4.26003	−0.138286
$950$	0	0
$951$	−55.0517	−1.78518
$952$	0	0
$953$	−4.41087	−0.142882	−0.0714411	−	0.997445i	$-0.522760\pi$
$953$	−4.41087	−0.142882	−0.0714411	+	0.997445i	$0.522760\pi$
$954$	0	0
$955$	−16.8403	−0.544939
$956$	0	0
$957$	12.8721	0.416095
$958$	0	0
$959$	2.04896	0.0661644
$960$	0	0
$961$	73.6643	2.37627
$962$	0	0
$963$	3.14470	0.101337
$964$	0	0
$965$	50.4436	1.62384
$966$	0	0
$967$	38.3242	1.23242	0.616211	−	0.787581i	$-0.288667\pi$
$967$	38.3242	1.23242	0.616211	+	0.787581i	$0.288667\pi$
$968$	0	0
$969$	0.898780	0.0288730
$970$	0	0
$971$	−3.90856	−0.125432	−0.0627158	−	0.998031i	$-0.519976\pi$
$971$	−3.90856	−0.125432	−0.0627158	+	0.998031i	$0.519976\pi$
$972$	0	0
$973$	44.1381	1.41500
$974$	0	0
$975$	4.85226	0.155397
$976$	0	0
$977$	27.8747	0.891791	0.445895	−	0.895085i	$-0.352885\pi$
$977$	27.8747	0.891791	0.445895	+	0.895085i	$0.352885\pi$
$978$	0	0
$979$	−10.4501	−0.333986
$980$	0	0
$981$	94.7097	3.02385
$982$	0	0
$983$	4.60724	0.146948	0.0734741	−	0.997297i	$-0.476591\pi$
$983$	4.60724	0.146948	0.0734741	+	0.997297i	$0.476591\pi$
$984$	0	0
$985$	−8.09578	−0.257953
$986$	0	0
$987$	121.416	3.86472
$988$	0	0
$989$	−1.84206	−0.0585741
$990$	0	0
$991$	−41.8824	−1.33044	−0.665219	−	0.746648i	$-0.731662\pi$
$991$	−41.8824	−1.33044	−0.665219	+	0.746648i	$0.731662\pi$
$992$	0	0
$993$	94.6455	3.00348
$994$	0	0
$995$	−20.6796	−0.655586
$996$	0	0
$997$	−20.8831	−0.661376	−0.330688	−	0.943740i	$-0.607281\pi$
$997$	−20.8831	−0.661376	−0.330688	+	0.943740i	$0.607281\pi$
$998$	0	0
$999$	−135.270	−4.27975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.be.1.12		13
4.3	odd	2		3016.2.a.k.1.2	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.k.1.2	✓	13	4.3	odd	2
6032.2.a.be.1.12		13	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q+3.23329 q^{3} +1.87064 q^{5} +2.94834 q^{7} +7.45419 q^{9} +O(q^{10})\)	\(q+3.23329 q^{3} +1.87064 q^{5} +2.94834 q^{7} +7.45419 q^{9} +3.98110 q^{11} -1.00000 q^{13} +6.04832 q^{15} -5.02033 q^{17} -0.0553702 q^{19} +9.53287 q^{21} -1.07065 q^{23} -1.50072 q^{25} +14.4017 q^{27} +1.00000 q^{29} -10.2306 q^{31} +12.8721 q^{33} +5.51528 q^{35} -9.39262 q^{37} -3.23329 q^{39} +7.68406 q^{41} +1.72051 q^{43} +13.9441 q^{45} +12.7366 q^{47} +1.69274 q^{49} -16.2322 q^{51} +7.71821 q^{53} +7.44719 q^{55} -0.179028 q^{57} -2.19031 q^{59} -9.35530 q^{61} +21.9775 q^{63} -1.87064 q^{65} +8.67155 q^{67} -3.46172 q^{69} +13.1909 q^{71} +4.26003 q^{73} -4.85226 q^{75} +11.7377 q^{77} -5.03787 q^{79} +24.2024 q^{81} -10.2861 q^{83} -9.39122 q^{85} +3.23329 q^{87} -2.62492 q^{89} -2.94834 q^{91} -33.0784 q^{93} -0.103577 q^{95} -5.93685 q^{97} +29.6759 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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