LMFDB - Embedded newform 6032.2.a.be.1.7

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			1.7
Root			$-0.0772773$ 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)

$f(q)$	$=$	$q-0.0772773 q^{3} +0.908464 q^{5} -0.210501 q^{7} -2.99403 q^{9} +O(q^{10})$	$q-0.0772773 q^{3} +0.908464 q^{5} -0.210501 q^{7} -2.99403 q^{9} -2.13255 q^{11} -1.00000 q^{13} -0.0702036 q^{15} +0.611120 q^{17} -4.52114 q^{19} +0.0162669 q^{21} -3.27282 q^{23} -4.17469 q^{25} +0.463202 q^{27} +1.00000 q^{29} +7.05193 q^{31} +0.164797 q^{33} -0.191232 q^{35} +10.8205 q^{37} +0.0772773 q^{39} +3.27928 q^{41} -12.6360 q^{43} -2.71997 q^{45} +2.15776 q^{47} -6.95569 q^{49} -0.0472257 q^{51} +11.0650 q^{53} -1.93734 q^{55} +0.349381 q^{57} +10.1558 q^{59} +0.968865 q^{61} +0.630245 q^{63} -0.908464 q^{65} -2.04949 q^{67} +0.252915 q^{69} +5.77541 q^{71} +13.7477 q^{73} +0.322609 q^{75} +0.448903 q^{77} -7.24856 q^{79} +8.94629 q^{81} -1.00931 q^{83} +0.555181 q^{85} -0.0772773 q^{87} -1.77261 q^{89} +0.210501 q^{91} -0.544954 q^{93} -4.10729 q^{95} +9.66212 q^{97} +6.38490 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.0772773	−0.0446160	−0.0223080	−	0.999751i	$-0.507101\pi$
$3$	−0.0772773	−0.0446160	−0.0223080	+	0.999751i	$0.507101\pi$
$4$	0	0
$5$	0.908464	0.406278	0.203139	−	0.979150i	$-0.434886\pi$
$5$	0.908464	0.406278	0.203139	+	0.979150i	$0.434886\pi$
$6$	0	0
$7$	−0.210501	−0.0795618	−0.0397809	−	0.999208i	$-0.512666\pi$
$7$	−0.210501	−0.0795618	−0.0397809	+	0.999208i	$0.512666\pi$
$8$	0	0
$9$	−2.99403	−0.998009
$10$	0	0
$11$	−2.13255	−0.642987	−0.321493	−	0.946912i	$-0.604185\pi$
$11$	−2.13255	−0.642987	−0.321493	+	0.946912i	$0.604185\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.0702036	−0.0181265
$16$	0	0
$17$	0.611120	0.148218	0.0741092	−	0.997250i	$-0.476389\pi$
$17$	0.611120	0.148218	0.0741092	+	0.997250i	$0.476389\pi$
$18$	0	0
$19$	−4.52114	−1.03722	−0.518610	−	0.855011i	$-0.673551\pi$
$19$	−4.52114	−1.03722	−0.518610	+	0.855011i	$0.673551\pi$
$20$	0	0
$21$	0.0162669	0.00354973
$22$	0	0
$23$	−3.27282	−0.682430	−0.341215	−	0.939985i	$-0.610838\pi$
$23$	−3.27282	−0.682430	−0.341215	+	0.939985i	$0.610838\pi$
$24$	0	0
$25$	−4.17469	−0.834939
$26$	0	0
$27$	0.463202	0.0891433
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	7.05193	1.26656	0.633282	−	0.773921i	$-0.281707\pi$
$31$	7.05193	1.26656	0.633282	+	0.773921i	$0.281707\pi$
$32$	0	0
$33$	0.164797	0.0286875
$34$	0	0
$35$	−0.191232	−0.0323242
$36$	0	0
$37$	10.8205	1.77889	0.889443	−	0.457045i	$-0.151092\pi$
$37$	10.8205	1.77889	0.889443	+	0.457045i	$0.151092\pi$
$38$	0	0
$39$	0.0772773	0.0123743
$40$	0	0
$41$	3.27928	0.512137	0.256069	−	0.966659i	$-0.417573\pi$
$41$	3.27928	0.512137	0.256069	+	0.966659i	$0.417573\pi$
$42$	0	0
$43$	−12.6360	−1.92697	−0.963485	−	0.267761i	$-0.913716\pi$
$43$	−12.6360	−1.92697	−0.963485	+	0.267761i	$0.913716\pi$
$44$	0	0
$45$	−2.71997	−0.405469
$46$	0	0
$47$	2.15776	0.314742	0.157371	−	0.987540i	$-0.449698\pi$
$47$	2.15776	0.314742	0.157371	+	0.987540i	$0.449698\pi$
$48$	0	0
$49$	−6.95569	−0.993670
$50$	0	0
$51$	−0.0472257	−0.00661292
$52$	0	0
$53$	11.0650	1.51990	0.759948	−	0.649983i	$-0.225224\pi$
$53$	11.0650	1.51990	0.759948	+	0.649983i	$0.225224\pi$
$54$	0	0
$55$	−1.93734	−0.261231
$56$	0	0
$57$	0.349381	0.0462767
$58$	0	0
$59$	10.1558	1.32217	0.661084	−	0.750311i	$-0.270096\pi$
$59$	10.1558	1.32217	0.661084	+	0.750311i	$0.270096\pi$
$60$	0	0
$61$	0.968865	0.124050	0.0620252	−	0.998075i	$-0.480244\pi$
$61$	0.968865	0.124050	0.0620252	+	0.998075i	$0.480244\pi$
$62$	0	0
$63$	0.630245	0.0794034
$64$	0	0
$65$	−0.908464	−0.112681
$66$	0	0
$67$	−2.04949	−0.250385	−0.125193	−	0.992132i	$-0.539955\pi$
$67$	−2.04949	−0.250385	−0.125193	+	0.992132i	$0.539955\pi$
$68$	0	0
$69$	0.252915	0.0304473
$70$	0	0
$71$	5.77541	0.685415	0.342708	−	0.939442i	$-0.388656\pi$
$71$	5.77541	0.685415	0.342708	+	0.939442i	$0.388656\pi$
$72$	0	0
$73$	13.7477	1.60904	0.804522	−	0.593922i	$-0.202421\pi$
$73$	13.7477	1.60904	0.804522	+	0.593922i	$0.202421\pi$
$74$	0	0
$75$	0.322609	0.0372517
$76$	0	0
$77$	0.448903	0.0511572
$78$	0	0
$79$	−7.24856	−0.815527	−0.407763	−	0.913088i	$-0.633691\pi$
$79$	−7.24856	−0.815527	−0.407763	+	0.913088i	$0.633691\pi$
$80$	0	0
$81$	8.94629	0.994032
$82$	0	0
$83$	−1.00931	−0.110786	−0.0553929	−	0.998465i	$-0.517641\pi$
$83$	−1.00931	−0.110786	−0.0553929	+	0.998465i	$0.517641\pi$
$84$	0	0
$85$	0.555181	0.0602178
$86$	0	0
$87$	−0.0772773	−0.00828499
$88$	0	0
$89$	−1.77261	−0.187897	−0.0939484	−	0.995577i	$-0.529949\pi$
$89$	−1.77261	−0.187897	−0.0939484	+	0.995577i	$0.529949\pi$
$90$	0	0
$91$	0.210501	0.0220665
$92$	0	0
$93$	−0.544954	−0.0565091
$94$	0	0
$95$	−4.10729	−0.421400
$96$	0	0
$97$	9.66212	0.981040	0.490520	−	0.871430i	$-0.336807\pi$
$97$	9.66212	0.981040	0.490520	+	0.871430i	$0.336807\pi$
$98$	0	0
$99$	6.38490	0.641707
$100$	0	0
$101$	6.13137	0.610094	0.305047	−	0.952337i	$-0.401328\pi$
$101$	6.13137	0.610094	0.305047	+	0.952337i	$0.401328\pi$
$102$	0	0
$103$	13.6652	1.34647	0.673237	−	0.739426i	$-0.264903\pi$
$103$	13.6652	1.34647	0.673237	+	0.739426i	$0.264903\pi$
$104$	0	0
$105$	0.0147779	0.00144218
$106$	0	0
$107$	−3.51983	−0.340275	−0.170138	−	0.985420i	$-0.554421\pi$
$107$	−3.51983	−0.340275	−0.170138	+	0.985420i	$0.554421\pi$
$108$	0	0
$109$	18.4025	1.76264	0.881320	−	0.472520i	$-0.156656\pi$
$109$	18.4025	1.76264	0.881320	+	0.472520i	$0.156656\pi$
$110$	0	0
$111$	−0.836182	−0.0793669
$112$	0	0
$113$	17.8888	1.68284	0.841420	−	0.540382i	$-0.181720\pi$
$113$	17.8888	1.68284	0.841420	+	0.540382i	$0.181720\pi$
$114$	0	0
$115$	−2.97324	−0.277256
$116$	0	0
$117$	2.99403	0.276798
$118$	0	0
$119$	−0.128641	−0.0117925
$120$	0	0
$121$	−6.45224	−0.586568
$122$	0	0
$123$	−0.253414	−0.0228495
$124$	0	0
$125$	−8.33488	−0.745494
$126$	0	0
$127$	−18.7014	−1.65948	−0.829742	−	0.558147i	$-0.811512\pi$
$127$	−18.7014	−1.65948	−0.829742	+	0.558147i	$0.811512\pi$
$128$	0	0
$129$	0.976475	0.0859738
$130$	0	0
$131$	3.34195	0.291987	0.145994	−	0.989286i	$-0.453362\pi$
$131$	3.34195	0.291987	0.145994	+	0.989286i	$0.453362\pi$
$132$	0	0
$133$	0.951704	0.0825232
$134$	0	0
$135$	0.420803	0.0362169
$136$	0	0
$137$	−5.44458	−0.465162	−0.232581	−	0.972577i	$-0.574717\pi$
$137$	−5.44458	−0.465162	−0.232581	+	0.972577i	$0.574717\pi$
$138$	0	0
$139$	15.0695	1.27818	0.639090	−	0.769132i	$-0.279311\pi$
$139$	15.0695	1.27818	0.639090	+	0.769132i	$0.279311\pi$
$140$	0	0
$141$	−0.166746	−0.0140426
$142$	0	0
$143$	2.13255	0.178333
$144$	0	0
$145$	0.908464	0.0754438
$146$	0	0
$147$	0.537517	0.0443336
$148$	0	0
$149$	0.698881	0.0572546	0.0286273	−	0.999590i	$-0.490886\pi$
$149$	0.698881	0.0572546	0.0286273	+	0.999590i	$0.490886\pi$
$150$	0	0
$151$	5.26828	0.428727	0.214363	−	0.976754i	$-0.431232\pi$
$151$	5.26828	0.428727	0.214363	+	0.976754i	$0.431232\pi$
$152$	0	0
$153$	−1.82971	−0.147923
$154$	0	0
$155$	6.40642	0.514576
$156$	0	0
$157$	−19.1594	−1.52908	−0.764542	−	0.644574i	$-0.777035\pi$
$157$	−19.1594	−1.52908	−0.764542	+	0.644574i	$0.777035\pi$
$158$	0	0
$159$	−0.855074	−0.0678118
$160$	0	0
$161$	0.688931	0.0542954
$162$	0	0
$163$	9.00656	0.705448	0.352724	−	0.935727i	$-0.385255\pi$
$163$	9.00656	0.705448	0.352724	+	0.935727i	$0.385255\pi$
$164$	0	0
$165$	0.149713	0.0116551
$166$	0	0
$167$	6.42725	0.497355	0.248678	−	0.968586i	$-0.420004\pi$
$167$	6.42725	0.497355	0.248678	+	0.968586i	$0.420004\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	13.5364	1.03516
$172$	0	0
$173$	5.89563	0.448237	0.224118	−	0.974562i	$-0.428050\pi$
$173$	5.89563	0.448237	0.224118	+	0.974562i	$0.428050\pi$
$174$	0	0
$175$	0.878776	0.0664292
$176$	0	0
$177$	−0.784810	−0.0589900
$178$	0	0
$179$	−16.6745	−1.24631	−0.623155	−	0.782099i	$-0.714149\pi$
$179$	−16.6745	−1.24631	−0.623155	+	0.782099i	$0.714149\pi$
$180$	0	0
$181$	20.5016	1.52388	0.761938	−	0.647650i	$-0.224248\pi$
$181$	20.5016	1.52388	0.761938	+	0.647650i	$0.224248\pi$
$182$	0	0
$183$	−0.0748712	−0.00553464
$184$	0	0
$185$	9.83008	0.722722
$186$	0	0
$187$	−1.30324	−0.0953025
$188$	0	0
$189$	−0.0975044	−0.00709240
$190$	0	0
$191$	−21.9627	−1.58916	−0.794582	−	0.607156i	$-0.792310\pi$
$191$	−21.9627	−1.58916	−0.794582	+	0.607156i	$0.792310\pi$
$192$	0	0
$193$	−4.69972	−0.338293	−0.169147	−	0.985591i	$-0.554101\pi$
$193$	−4.69972	−0.338293	−0.169147	+	0.985591i	$0.554101\pi$
$194$	0	0
$195$	0.0702036	0.00502739
$196$	0	0
$197$	−19.0082	−1.35427	−0.677137	−	0.735857i	$-0.736780\pi$
$197$	−19.0082	−1.35427	−0.677137	+	0.735857i	$0.736780\pi$
$198$	0	0
$199$	−4.99025	−0.353750	−0.176875	−	0.984233i	$-0.556599\pi$
$199$	−4.99025	−0.353750	−0.176875	+	0.984233i	$0.556599\pi$
$200$	0	0
$201$	0.158379	0.0111712
$202$	0	0
$203$	−0.210501	−0.0147743
$204$	0	0
$205$	2.97911	0.208070
$206$	0	0
$207$	9.79892	0.681072
$208$	0	0
$209$	9.64154	0.666920
$210$	0	0
$211$	−2.98975	−0.205823	−0.102911	−	0.994691i	$-0.532816\pi$
$211$	−2.98975	−0.205823	−0.102911	+	0.994691i	$0.532816\pi$
$212$	0	0
$213$	−0.446308	−0.0305805
$214$	0	0
$215$	−11.4793	−0.782885
$216$	0	0
$217$	−1.48444	−0.100770
$218$	0	0
$219$	−1.06238	−0.0717892
$220$	0	0
$221$	−0.611120	−0.0411084
$222$	0	0
$223$	3.68321	0.246646	0.123323	−	0.992367i	$-0.460645\pi$
$223$	3.68321	0.246646	0.123323	+	0.992367i	$0.460645\pi$
$224$	0	0
$225$	12.4991	0.833277
$226$	0	0
$227$	−18.7789	−1.24640	−0.623199	−	0.782063i	$-0.714168\pi$
$227$	−18.7789	−1.24640	−0.623199	+	0.782063i	$0.714168\pi$
$228$	0	0
$229$	10.7692	0.711648	0.355824	−	0.934553i	$-0.384200\pi$
$229$	10.7692	0.711648	0.355824	+	0.934553i	$0.384200\pi$
$230$	0	0
$231$	−0.0346900	−0.00228243
$232$	0	0
$233$	14.2198	0.931570	0.465785	−	0.884898i	$-0.345772\pi$
$233$	14.2198	0.931570	0.465785	+	0.884898i	$0.345772\pi$
$234$	0	0
$235$	1.96025	0.127873
$236$	0	0
$237$	0.560149	0.0363856
$238$	0	0
$239$	4.26205	0.275689	0.137844	−	0.990454i	$-0.455983\pi$
$239$	4.26205	0.275689	0.137844	+	0.990454i	$0.455983\pi$
$240$	0	0
$241$	−3.96023	−0.255101	−0.127550	−	0.991832i	$-0.540711\pi$
$241$	−3.96023	−0.255101	−0.127550	+	0.991832i	$0.540711\pi$
$242$	0	0
$243$	−2.08095	−0.133493
$244$	0	0
$245$	−6.31899	−0.403706
$246$	0	0
$247$	4.52114	0.287673
$248$	0	0
$249$	0.0779965	0.00494283
$250$	0	0
$251$	−5.01221	−0.316368	−0.158184	−	0.987410i	$-0.550564\pi$
$251$	−5.01221	−0.316368	−0.158184	+	0.987410i	$0.550564\pi$
$252$	0	0
$253$	6.97944	0.438794
$254$	0	0
$255$	−0.0429029	−0.00268668
$256$	0	0
$257$	−4.80268	−0.299583	−0.149792	−	0.988718i	$-0.547860\pi$
$257$	−4.80268	−0.299583	−0.149792	+	0.988718i	$0.547860\pi$
$258$	0	0
$259$	−2.27773	−0.141531
$260$	0	0
$261$	−2.99403	−0.185326
$262$	0	0
$263$	24.3839	1.50358	0.751789	−	0.659404i	$-0.229191\pi$
$263$	24.3839	1.50358	0.751789	+	0.659404i	$0.229191\pi$
$264$	0	0
$265$	10.0522	0.617500
$266$	0	0
$267$	0.136983	0.00838321
$268$	0	0
$269$	4.00165	0.243985	0.121993	−	0.992531i	$-0.461072\pi$
$269$	4.00165	0.243985	0.121993	+	0.992531i	$0.461072\pi$
$270$	0	0
$271$	12.8186	0.778671	0.389336	−	0.921096i	$-0.372705\pi$
$271$	12.8186	0.778671	0.389336	+	0.921096i	$0.372705\pi$
$272$	0	0
$273$	−0.0162669	−0.000984519 0
$274$	0	0
$275$	8.90273	0.536855
$276$	0	0
$277$	23.4816	1.41087	0.705435	−	0.708775i	$-0.250752\pi$
$277$	23.4816	1.41087	0.705435	+	0.708775i	$0.250752\pi$
$278$	0	0
$279$	−21.1137	−1.26404
$280$	0	0
$281$	29.4553	1.75716	0.878578	−	0.477599i	$-0.158493\pi$
$281$	29.4553	1.75716	0.878578	+	0.477599i	$0.158493\pi$
$282$	0	0
$283$	28.6338	1.70210	0.851051	−	0.525083i	$-0.175966\pi$
$283$	28.6338	1.70210	0.851051	+	0.525083i	$0.175966\pi$
$284$	0	0
$285$	0.317401	0.0188012
$286$	0	0
$287$	−0.690291	−0.0407466
$288$	0	0
$289$	−16.6265	−0.978031
$290$	0	0
$291$	−0.746663	−0.0437701
$292$	0	0
$293$	31.2611	1.82629	0.913146	−	0.407634i	$-0.133646\pi$
$293$	31.2611	1.82629	0.913146	+	0.407634i	$0.133646\pi$
$294$	0	0
$295$	9.22616	0.537168
$296$	0	0
$297$	−0.987800	−0.0573180
$298$	0	0
$299$	3.27282	0.189272
$300$	0	0
$301$	2.65989	0.153313
$302$	0	0
$303$	−0.473815	−0.0272200
$304$	0	0
$305$	0.880179	0.0503989
$306$	0	0
$307$	−10.8216	−0.617619	−0.308810	−	0.951124i	$-0.599931\pi$
$307$	−10.8216	−0.617619	−0.308810	+	0.951124i	$0.599931\pi$
$308$	0	0
$309$	−1.05601	−0.0600744
$310$	0	0
$311$	16.5982	0.941195	0.470597	−	0.882348i	$-0.344038\pi$
$311$	16.5982	0.941195	0.470597	+	0.882348i	$0.344038\pi$
$312$	0	0
$313$	−16.8910	−0.954738	−0.477369	−	0.878703i	$-0.658409\pi$
$313$	−16.8910	−0.954738	−0.477369	+	0.878703i	$0.658409\pi$
$314$	0	0
$315$	0.572555	0.0322598
$316$	0	0
$317$	−28.0346	−1.57458	−0.787290	−	0.616583i	$-0.788516\pi$
$317$	−28.0346	−1.57458	−0.787290	+	0.616583i	$0.788516\pi$
$318$	0	0
$319$	−2.13255	−0.119400
$320$	0	0
$321$	0.272003	0.0151817
$322$	0	0
$323$	−2.76296	−0.153735
$324$	0	0
$325$	4.17469	0.231570
$326$	0	0
$327$	−1.42209	−0.0786420
$328$	0	0
$329$	−0.454211	−0.0250415
$330$	0	0
$331$	8.13448	0.447111	0.223556	−	0.974691i	$-0.428234\pi$
$331$	8.13448	0.447111	0.223556	+	0.974691i	$0.428234\pi$
$332$	0	0
$333$	−32.3970	−1.77535
$334$	0	0
$335$	−1.86189	−0.101726
$336$	0	0
$337$	−16.6228	−0.905501	−0.452750	−	0.891637i	$-0.649557\pi$
$337$	−16.6228	−0.905501	−0.452750	+	0.891637i	$0.649557\pi$
$338$	0	0
$339$	−1.38240	−0.0750817
$340$	0	0
$341$	−15.0386	−0.814384
$342$	0	0
$343$	2.93768	0.158620
$344$	0	0
$345$	0.229764	0.0123701
$346$	0	0
$347$	−27.7011	−1.48707	−0.743536	−	0.668696i	$-0.766853\pi$
$347$	−27.7011	−1.48707	−0.743536	+	0.668696i	$0.766853\pi$
$348$	0	0
$349$	27.9381	1.49549	0.747747	−	0.663983i	$-0.231135\pi$
$349$	27.9381	1.49549	0.747747	+	0.663983i	$0.231135\pi$
$350$	0	0
$351$	−0.463202	−0.0247239
$352$	0	0
$353$	−12.3417	−0.656882	−0.328441	−	0.944525i	$-0.606523\pi$
$353$	−12.3417	−0.656882	−0.328441	+	0.944525i	$0.606523\pi$
$354$	0	0
$355$	5.24675	0.278469
$356$	0	0
$357$	0.00994105	0.000526136 0
$358$	0	0
$359$	−19.8176	−1.04593	−0.522967	−	0.852353i	$-0.675175\pi$
$359$	−19.8176	−1.04593	−0.522967	+	0.852353i	$0.675175\pi$
$360$	0	0
$361$	1.44072	0.0758273
$362$	0	0
$363$	0.498612	0.0261703
$364$	0	0
$365$	12.4893	0.653719
$366$	0	0
$367$	−5.77769	−0.301593	−0.150796	−	0.988565i	$-0.548184\pi$
$367$	−5.77769	−0.301593	−0.150796	+	0.988565i	$0.548184\pi$
$368$	0	0
$369$	−9.81826	−0.511118
$370$	0	0
$371$	−2.32919	−0.120926
$372$	0	0
$373$	12.3590	0.639927	0.319963	−	0.947430i	$-0.396329\pi$
$373$	12.3590	0.639927	0.319963	+	0.947430i	$0.396329\pi$
$374$	0	0
$375$	0.644097	0.0332610
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−25.9637	−1.33367	−0.666833	−	0.745207i	$-0.732351\pi$
$379$	−25.9637	−1.33367	−0.666833	+	0.745207i	$0.732351\pi$
$380$	0	0
$381$	1.44520	0.0740396
$382$	0	0
$383$	−19.8250	−1.01301	−0.506506	−	0.862236i	$-0.669063\pi$
$383$	−19.8250	−1.01301	−0.506506	+	0.862236i	$0.669063\pi$
$384$	0	0
$385$	0.407812	0.0207840
$386$	0	0
$387$	37.8325	1.92314
$388$	0	0
$389$	17.2076	0.872461	0.436231	−	0.899835i	$-0.356313\pi$
$389$	17.2076	0.872461	0.436231	+	0.899835i	$0.356313\pi$
$390$	0	0
$391$	−2.00009	−0.101149
$392$	0	0
$393$	−0.258257	−0.0130273
$394$	0	0
$395$	−6.58506	−0.331330
$396$	0	0
$397$	−30.0950	−1.51042	−0.755212	−	0.655481i	$-0.772466\pi$
$397$	−30.0950	−1.51042	−0.755212	+	0.655481i	$0.772466\pi$
$398$	0	0
$399$	−0.0735451	−0.00368186
$400$	0	0
$401$	−12.2147	−0.609973	−0.304986	−	0.952357i	$-0.598652\pi$
$401$	−12.2147	−0.609973	−0.304986	+	0.952357i	$0.598652\pi$
$402$	0	0
$403$	−7.05193	−0.351282
$404$	0	0
$405$	8.12738	0.403853
$406$	0	0
$407$	−23.0753	−1.14380
$408$	0	0
$409$	30.3928	1.50283	0.751414	−	0.659831i	$-0.229372\pi$
$409$	30.3928	1.50283	0.751414	+	0.659831i	$0.229372\pi$
$410$	0	0
$411$	0.420742	0.0207537
$412$	0	0
$413$	−2.13780	−0.105194
$414$	0	0
$415$	−0.916920	−0.0450098
$416$	0	0
$417$	−1.16453	−0.0570273
$418$	0	0
$419$	39.6964	1.93930	0.969649	−	0.244503i	$-0.0786247\pi$
$419$	39.6964	1.93930	0.969649	+	0.244503i	$0.0786247\pi$
$420$	0	0
$421$	−33.3746	−1.62658	−0.813289	−	0.581860i	$-0.802325\pi$
$421$	−33.3746	−1.62658	−0.813289	+	0.581860i	$0.802325\pi$
$422$	0	0
$423$	−6.46041	−0.314116
$424$	0	0
$425$	−2.55124	−0.123753
$426$	0	0
$427$	−0.203947	−0.00986968
$428$	0	0
$429$	−0.164797	−0.00795649
$430$	0	0
$431$	19.6961	0.948725	0.474363	−	0.880329i	$-0.342679\pi$
$431$	19.6961	0.948725	0.474363	+	0.880329i	$0.342679\pi$
$432$	0	0
$433$	−4.79537	−0.230451	−0.115225	−	0.993339i	$-0.536759\pi$
$433$	−4.79537	−0.230451	−0.115225	+	0.993339i	$0.536759\pi$
$434$	0	0
$435$	−0.0702036	−0.00336601
$436$	0	0
$437$	14.7969	0.707831
$438$	0	0
$439$	−17.1651	−0.819248	−0.409624	−	0.912255i	$-0.634340\pi$
$439$	−17.1651	−0.819248	−0.409624	+	0.912255i	$0.634340\pi$
$440$	0	0
$441$	20.8255	0.991692
$442$	0	0
$443$	−2.15966	−0.102609	−0.0513043	−	0.998683i	$-0.516338\pi$
$443$	−2.15966	−0.102609	−0.0513043	+	0.998683i	$0.516338\pi$
$444$	0	0
$445$	−1.61036	−0.0763382
$446$	0	0
$447$	−0.0540076	−0.00255447
$448$	0	0
$449$	−19.0022	−0.896768	−0.448384	−	0.893841i	$-0.648000\pi$
$449$	−19.0022	−0.896768	−0.448384	+	0.893841i	$0.648000\pi$
$450$	0	0
$451$	−6.99322	−0.329298
$452$	0	0
$453$	−0.407119	−0.0191281
$454$	0	0
$455$	0.191232	0.00896511
$456$	0	0
$457$	0.731322	0.0342098	0.0171049	−	0.999854i	$-0.494555\pi$
$457$	0.731322	0.0342098	0.0171049	+	0.999854i	$0.494555\pi$
$458$	0	0
$459$	0.283072	0.0132127
$460$	0	0
$461$	−1.80036	−0.0838512	−0.0419256	−	0.999121i	$-0.513349\pi$
$461$	−1.80036	−0.0838512	−0.0419256	+	0.999121i	$0.513349\pi$
$462$	0	0
$463$	23.8652	1.10911	0.554554	−	0.832148i	$-0.312889\pi$
$463$	23.8652	1.10911	0.554554	+	0.832148i	$0.312889\pi$
$464$	0	0
$465$	−0.495071	−0.0229584
$466$	0	0
$467$	36.2789	1.67879	0.839393	−	0.543525i	$-0.182911\pi$
$467$	36.2789	1.67879	0.839393	+	0.543525i	$0.182911\pi$
$468$	0	0
$469$	0.431419	0.0199211
$470$	0	0
$471$	1.48058	0.0682216
$472$	0	0
$473$	26.9468	1.23902
$474$	0	0
$475$	18.8744	0.866016
$476$	0	0
$477$	−33.1290	−1.51687
$478$	0	0
$479$	−14.3094	−0.653812	−0.326906	−	0.945057i	$-0.606006\pi$
$479$	−14.3094	−0.653812	−0.326906	+	0.945057i	$0.606006\pi$
$480$	0	0
$481$	−10.8205	−0.493374
$482$	0	0
$483$	−0.0532387	−0.00242245
$484$	0	0
$485$	8.77769	0.398575
$486$	0	0
$487$	31.8738	1.44434	0.722169	−	0.691716i	$-0.243145\pi$
$487$	31.8738	1.44434	0.722169	+	0.691716i	$0.243145\pi$
$488$	0	0
$489$	−0.696002	−0.0314743
$490$	0	0
$491$	35.4365	1.59923	0.799613	−	0.600515i	$-0.205038\pi$
$491$	35.4365	1.59923	0.799613	+	0.600515i	$0.205038\pi$
$492$	0	0
$493$	0.611120	0.0275235
$494$	0	0
$495$	5.80046	0.260711
$496$	0	0
$497$	−1.21573	−0.0545329
$498$	0	0
$499$	13.7049	0.613517	0.306758	−	0.951787i	$-0.400756\pi$
$499$	13.7049	0.613517	0.306758	+	0.951787i	$0.400756\pi$
$500$	0	0
$501$	−0.496680	−0.0221900
$502$	0	0
$503$	−23.7834	−1.06045	−0.530224	−	0.847858i	$-0.677892\pi$
$503$	−23.7834	−1.06045	−0.530224	+	0.847858i	$0.677892\pi$
$504$	0	0
$505$	5.57013	0.247868
$506$	0	0
$507$	−0.0772773	−0.00343200
$508$	0	0
$509$	−12.1455	−0.538341	−0.269170	−	0.963093i	$-0.586750\pi$
$509$	−12.1455	−0.538341	−0.269170	+	0.963093i	$0.586750\pi$
$510$	0	0
$511$	−2.89390	−0.128019
$512$	0	0
$513$	−2.09420	−0.0924613
$514$	0	0
$515$	12.4144	0.547042
$516$	0	0
$517$	−4.60153	−0.202375
$518$	0	0
$519$	−0.455598	−0.0199985
$520$	0	0
$521$	−6.58701	−0.288582	−0.144291	−	0.989535i	$-0.546090\pi$
$521$	−6.58701	−0.288582	−0.144291	+	0.989535i	$0.546090\pi$
$522$	0	0
$523$	26.4397	1.15613	0.578063	−	0.815992i	$-0.303809\pi$
$523$	26.4397	1.15613	0.578063	+	0.815992i	$0.303809\pi$
$524$	0	0
$525$	−0.0679094	−0.00296381
$526$	0	0
$527$	4.30958	0.187728
$528$	0	0
$529$	−12.2886	−0.534289
$530$	0	0
$531$	−30.4067	−1.31954
$532$	0	0
$533$	−3.27928	−0.142041
$534$	0	0
$535$	−3.19764	−0.138246
$536$	0	0
$537$	1.28856	0.0556054
$538$	0	0
$539$	14.8333	0.638917
$540$	0	0
$541$	6.24394	0.268448	0.134224	−	0.990951i	$-0.457146\pi$
$541$	6.24394	0.268448	0.134224	+	0.990951i	$0.457146\pi$
$542$	0	0
$543$	−1.58431	−0.0679893
$544$	0	0
$545$	16.7180	0.716121
$546$	0	0
$547$	27.1883	1.16249	0.581245	−	0.813729i	$-0.302566\pi$
$547$	27.1883	1.16249	0.581245	+	0.813729i	$0.302566\pi$
$548$	0	0
$549$	−2.90081	−0.123803
$550$	0	0
$551$	−4.52114	−0.192607
$552$	0	0
$553$	1.52583	0.0648848
$554$	0	0
$555$	−0.759642	−0.0322450
$556$	0	0
$557$	23.6805	1.00337	0.501687	−	0.865049i	$-0.332713\pi$
$557$	23.6805	1.00337	0.501687	+	0.865049i	$0.332713\pi$
$558$	0	0
$559$	12.6360	0.534446
$560$	0	0
$561$	0.100711	0.00425202
$562$	0	0
$563$	43.0319	1.81358	0.906789	−	0.421584i	$-0.138526\pi$
$563$	43.0319	1.81358	0.906789	+	0.421584i	$0.138526\pi$
$564$	0	0
$565$	16.2514	0.683700
$566$	0	0
$567$	−1.88320	−0.0790870
$568$	0	0
$569$	30.1621	1.26446	0.632230	−	0.774781i	$-0.282140\pi$
$569$	30.1621	1.26446	0.632230	+	0.774781i	$0.282140\pi$
$570$	0	0
$571$	4.10516	0.171795	0.0858977	−	0.996304i	$-0.472624\pi$
$571$	4.10516	0.171795	0.0858977	+	0.996304i	$0.472624\pi$
$572$	0	0
$573$	1.69722	0.0709023
$574$	0	0
$575$	13.6630	0.569787
$576$	0	0
$577$	−20.8475	−0.867893	−0.433947	−	0.900939i	$-0.642879\pi$
$577$	−20.8475	−0.867893	−0.433947	+	0.900939i	$0.642879\pi$
$578$	0	0
$579$	0.363181	0.0150933
$580$	0	0
$581$	0.212460	0.00881433
$582$	0	0
$583$	−23.5967	−0.977274
$584$	0	0
$585$	2.71997	0.112457
$586$	0	0
$587$	28.3610	1.17059	0.585293	−	0.810822i	$-0.300980\pi$
$587$	28.3610	1.17059	0.585293	+	0.810822i	$0.300980\pi$
$588$	0	0
$589$	−31.8828	−1.31371
$590$	0	0
$591$	1.46890	0.0604224
$592$	0	0
$593$	47.9398	1.96865	0.984325	−	0.176364i	$-0.0564335\pi$
$593$	47.9398	1.96865	0.984325	+	0.176364i	$0.0564335\pi$
$594$	0	0
$595$	−0.116866	−0.00479104
$596$	0	0
$597$	0.385633	0.0157829
$598$	0	0
$599$	8.27768	0.338217	0.169108	−	0.985597i	$-0.445911\pi$
$599$	8.27768	0.338217	0.169108	+	0.985597i	$0.445911\pi$
$600$	0	0
$601$	21.3911	0.872561	0.436281	−	0.899811i	$-0.356295\pi$
$601$	21.3911	0.872561	0.436281	+	0.899811i	$0.356295\pi$
$602$	0	0
$603$	6.13623	0.249887
$604$	0	0
$605$	−5.86163	−0.238309
$606$	0	0
$607$	41.3871	1.67985	0.839924	−	0.542703i	$-0.182599\pi$
$607$	41.3871	1.67985	0.839924	+	0.542703i	$0.182599\pi$
$608$	0	0
$609$	0.0162669	0.000659169 0
$610$	0	0
$611$	−2.15776	−0.0872938
$612$	0	0
$613$	−30.3047	−1.22400	−0.611998	−	0.790860i	$-0.709634\pi$
$613$	−30.3047	−1.22400	−0.611998	+	0.790860i	$0.709634\pi$
$614$	0	0
$615$	−0.230217	−0.00928326
$616$	0	0
$617$	−20.4860	−0.824735	−0.412367	−	0.911018i	$-0.635298\pi$
$617$	−20.4860	−0.824735	−0.412367	+	0.911018i	$0.635298\pi$
$618$	0	0
$619$	15.6576	0.629332	0.314666	−	0.949202i	$-0.398107\pi$
$619$	15.6576	0.629332	0.314666	+	0.949202i	$0.398107\pi$
$620$	0	0
$621$	−1.51598	−0.0608341
$622$	0	0
$623$	0.373137	0.0149494
$624$	0	0
$625$	13.3015	0.532061
$626$	0	0
$627$	−0.745072	−0.0297553
$628$	0	0
$629$	6.61265	0.263664
$630$	0	0
$631$	−6.98036	−0.277884	−0.138942	−	0.990301i	$-0.544370\pi$
$631$	−6.98036	−0.277884	−0.138942	+	0.990301i	$0.544370\pi$
$632$	0	0
$633$	0.231040	0.00918301
$634$	0	0
$635$	−16.9896	−0.674211
$636$	0	0
$637$	6.95569	0.275594
$638$	0	0
$639$	−17.2917	−0.684051
$640$	0	0
$641$	−2.88127	−0.113804	−0.0569018	−	0.998380i	$-0.518122\pi$
$641$	−2.88127	−0.113804	−0.0569018	+	0.998380i	$0.518122\pi$
$642$	0	0
$643$	0.409700	0.0161570	0.00807850	−	0.999967i	$-0.497429\pi$
$643$	0.409700	0.0161570	0.00807850	+	0.999967i	$0.497429\pi$
$644$	0	0
$645$	0.887093	0.0349292
$646$	0	0
$647$	28.4889	1.12001	0.560006	−	0.828488i	$-0.310799\pi$
$647$	28.4889	1.12001	0.560006	+	0.828488i	$0.310799\pi$
$648$	0	0
$649$	−21.6577	−0.850137
$650$	0	0
$651$	0.114713	0.00449596
$652$	0	0
$653$	11.1679	0.437035	0.218517	−	0.975833i	$-0.429878\pi$
$653$	11.1679	0.437035	0.218517	+	0.975833i	$0.429878\pi$
$654$	0	0
$655$	3.03604	0.118628
$656$	0	0
$657$	−41.1610	−1.60584
$658$	0	0
$659$	−5.57957	−0.217349	−0.108675	−	0.994077i	$-0.534661\pi$
$659$	−5.57957	−0.217349	−0.108675	+	0.994077i	$0.534661\pi$
$660$	0	0
$661$	−29.9349	−1.16433	−0.582167	−	0.813069i	$-0.697795\pi$
$661$	−29.9349	−1.16433	−0.582167	+	0.813069i	$0.697795\pi$
$662$	0	0
$663$	0.0472257	0.00183409
$664$	0	0
$665$	0.864589	0.0335273
$666$	0	0
$667$	−3.27282	−0.126724
$668$	0	0
$669$	−0.284628	−0.0110044
$670$	0	0
$671$	−2.06615	−0.0797628
$672$	0	0
$673$	−30.8901	−1.19072	−0.595362	−	0.803457i	$-0.702991\pi$
$673$	−30.8901	−1.19072	−0.595362	+	0.803457i	$0.702991\pi$
$674$	0	0
$675$	−1.93373	−0.0744292
$676$	0	0
$677$	−47.6624	−1.83181	−0.915907	−	0.401390i	$-0.868527\pi$
$677$	−47.6624	−1.83181	−0.915907	+	0.401390i	$0.868527\pi$
$678$	0	0
$679$	−2.03388	−0.0780533
$680$	0	0
$681$	1.45118	0.0556094
$682$	0	0
$683$	−40.2599	−1.54050	−0.770251	−	0.637741i	$-0.779869\pi$
$683$	−40.2599	−1.54050	−0.770251	+	0.637741i	$0.779869\pi$
$684$	0	0
$685$	−4.94621	−0.188985
$686$	0	0
$687$	−0.832213	−0.0317509
$688$	0	0
$689$	−11.0650	−0.421543
$690$	0	0
$691$	29.1058	1.10724	0.553618	−	0.832771i	$-0.313247\pi$
$691$	29.1058	1.10724	0.553618	+	0.832771i	$0.313247\pi$
$692$	0	0
$693$	−1.34403	−0.0510554
$694$	0	0
$695$	13.6901	0.519296
$696$	0	0
$697$	2.00403	0.0759082
$698$	0	0
$699$	−1.09887	−0.0415630
$700$	0	0
$701$	−13.0402	−0.492522	−0.246261	−	0.969204i	$-0.579202\pi$
$701$	−13.0402	−0.492522	−0.246261	+	0.969204i	$0.579202\pi$
$702$	0	0
$703$	−48.9212	−1.84510
$704$	0	0
$705$	−0.151483	−0.00570518
$706$	0	0
$707$	−1.29066	−0.0485402
$708$	0	0
$709$	21.8669	0.821228	0.410614	−	0.911809i	$-0.365314\pi$
$709$	21.8669	0.821228	0.410614	+	0.911809i	$0.365314\pi$
$710$	0	0
$711$	21.7024	0.813903
$712$	0	0
$713$	−23.0797	−0.864341
$714$	0	0
$715$	1.93734	0.0724525
$716$	0	0
$717$	−0.329359	−0.0123002
$718$	0	0
$719$	−46.0521	−1.71745	−0.858726	−	0.512435i	$-0.828743\pi$
$719$	−46.0521	−1.71745	−0.858726	+	0.512435i	$0.828743\pi$
$720$	0	0
$721$	−2.87654	−0.107128
$722$	0	0
$723$	0.306036	0.0113816
$724$	0	0
$725$	−4.17469	−0.155044
$726$	0	0
$727$	−18.9855	−0.704133	−0.352066	−	0.935975i	$-0.614521\pi$
$727$	−18.9855	−0.704133	−0.352066	+	0.935975i	$0.614521\pi$
$728$	0	0
$729$	−26.6781	−0.988076
$730$	0	0
$731$	−7.72211	−0.285613
$732$	0	0
$733$	51.3965	1.89837	0.949186	−	0.314714i	$-0.101909\pi$
$733$	51.3965	1.89837	0.949186	+	0.314714i	$0.101909\pi$
$734$	0	0
$735$	0.488315	0.0180118
$736$	0	0
$737$	4.37063	0.160994
$738$	0	0
$739$	41.2258	1.51651	0.758257	−	0.651955i	$-0.226051\pi$
$739$	41.2258	1.51651	0.758257	+	0.651955i	$0.226051\pi$
$740$	0	0
$741$	−0.349381	−0.0128348
$742$	0	0
$743$	−19.2985	−0.707993	−0.353996	−	0.935247i	$-0.615178\pi$
$743$	−19.2985	−0.707993	−0.353996	+	0.935247i	$0.615178\pi$
$744$	0	0
$745$	0.634909	0.0232612
$746$	0	0
$747$	3.02190	0.110565
$748$	0	0
$749$	0.740928	0.0270729
$750$	0	0
$751$	21.3780	0.780096	0.390048	−	0.920795i	$-0.372458\pi$
$751$	21.3780	0.780096	0.390048	+	0.920795i	$0.372458\pi$
$752$	0	0
$753$	0.387330	0.0141151
$754$	0	0
$755$	4.78605	0.174182
$756$	0	0
$757$	47.7650	1.73605	0.868025	−	0.496521i	$-0.165389\pi$
$757$	47.7650	1.73605	0.868025	+	0.496521i	$0.165389\pi$
$758$	0	0
$759$	−0.539352	−0.0195772
$760$	0	0
$761$	32.4565	1.17655	0.588274	−	0.808662i	$-0.299808\pi$
$761$	32.4565	1.17655	0.588274	+	0.808662i	$0.299808\pi$
$762$	0	0
$763$	−3.87374	−0.140239
$764$	0	0
$765$	−1.66223	−0.0600979
$766$	0	0
$767$	−10.1558	−0.366704
$768$	0	0
$769$	9.65815	0.348282	0.174141	−	0.984721i	$-0.444285\pi$
$769$	9.65815	0.348282	0.174141	+	0.984721i	$0.444285\pi$
$770$	0	0
$771$	0.371138	0.0133662
$772$	0	0
$773$	4.98903	0.179443	0.0897215	−	0.995967i	$-0.471402\pi$
$773$	4.98903	0.179443	0.0897215	+	0.995967i	$0.471402\pi$
$774$	0	0
$775$	−29.4396	−1.05750
$776$	0	0
$777$	0.176017	0.00631457
$778$	0	0
$779$	−14.8261	−0.531200
$780$	0	0
$781$	−12.3163	−0.440713
$782$	0	0
$783$	0.463202	0.0165535
$784$	0	0
$785$	−17.4056	−0.621232
$786$	0	0
$787$	26.8899	0.958523	0.479261	−	0.877672i	$-0.340905\pi$
$787$	26.8899	0.958523	0.479261	+	0.877672i	$0.340905\pi$
$788$	0	0
$789$	−1.88432	−0.0670837
$790$	0	0
$791$	−3.76561	−0.133890
$792$	0	0
$793$	−0.968865	−0.0344054
$794$	0	0
$795$	−0.776804	−0.0275504
$796$	0	0
$797$	18.7965	0.665808	0.332904	−	0.942961i	$-0.391972\pi$
$797$	18.7965	0.665808	0.332904	+	0.942961i	$0.391972\pi$
$798$	0	0
$799$	1.31865	0.0466506
$800$	0	0
$801$	5.30726	0.187523
$802$	0	0
$803$	−29.3176	−1.03459
$804$	0	0
$805$	0.625869	0.0220590
$806$	0	0
$807$	−0.309237	−0.0108857
$808$	0	0
$809$	−42.7846	−1.50423	−0.752114	−	0.659033i	$-0.770966\pi$
$809$	−42.7846	−1.50423	−0.752114	+	0.659033i	$0.770966\pi$
$810$	0	0
$811$	43.3046	1.52063	0.760314	−	0.649555i	$-0.225045\pi$
$811$	43.3046	1.52063	0.760314	+	0.649555i	$0.225045\pi$
$812$	0	0
$813$	−0.990582	−0.0347412
$814$	0	0
$815$	8.18213	0.286608
$816$	0	0
$817$	57.1291	1.99869
$818$	0	0
$819$	−0.630245	−0.0220226
$820$	0	0
$821$	−13.7699	−0.480571	−0.240286	−	0.970702i	$-0.577241\pi$
$821$	−13.7699	−0.480571	−0.240286	+	0.970702i	$0.577241\pi$
$822$	0	0
$823$	−24.5860	−0.857013	−0.428507	−	0.903539i	$-0.640960\pi$
$823$	−24.5860	−0.857013	−0.428507	+	0.903539i	$0.640960\pi$
$824$	0	0
$825$	−0.687978	−0.0239523
$826$	0	0
$827$	39.3333	1.36775	0.683877	−	0.729598i	$-0.260293\pi$
$827$	39.3333	1.36775	0.683877	+	0.729598i	$0.260293\pi$
$828$	0	0
$829$	−14.0053	−0.486424	−0.243212	−	0.969973i	$-0.578201\pi$
$829$	−14.0053	−0.486424	−0.243212	+	0.969973i	$0.578201\pi$
$830$	0	0
$831$	−1.81459	−0.0629474
$832$	0	0
$833$	−4.25076	−0.147280
$834$	0	0
$835$	5.83892	0.202064
$836$	0	0
$837$	3.26647	0.112906
$838$	0	0
$839$	−38.4377	−1.32702	−0.663509	−	0.748168i	$-0.730933\pi$
$839$	−38.4377	−1.32702	−0.663509	+	0.748168i	$0.730933\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−2.27622	−0.0783973
$844$	0	0
$845$	0.908464	0.0312521
$846$	0	0
$847$	1.35820	0.0466684
$848$	0	0
$849$	−2.21274	−0.0759411
$850$	0	0
$851$	−35.4137	−1.21397
$852$	0	0
$853$	13.2300	0.452987	0.226493	−	0.974013i	$-0.427274\pi$
$853$	13.2300	0.452987	0.226493	+	0.974013i	$0.427274\pi$
$854$	0	0
$855$	12.2974	0.420561
$856$	0	0
$857$	−21.3629	−0.729742	−0.364871	−	0.931058i	$-0.618887\pi$
$857$	−21.3629	−0.729742	−0.364871	+	0.931058i	$0.618887\pi$
$858$	0	0
$859$	47.5708	1.62310	0.811548	−	0.584286i	$-0.198625\pi$
$859$	47.5708	1.62310	0.811548	+	0.584286i	$0.198625\pi$
$860$	0	0
$861$	0.0533438	0.00181795
$862$	0	0
$863$	−38.9503	−1.32588	−0.662942	−	0.748671i	$-0.730692\pi$
$863$	−38.9503	−1.32588	−0.662942	+	0.748671i	$0.730692\pi$
$864$	0	0
$865$	5.35597	0.182108
$866$	0	0
$867$	1.28485	0.0436359
$868$	0	0
$869$	15.4579	0.524373
$870$	0	0
$871$	2.04949	0.0694444
$872$	0	0
$873$	−28.9287	−0.979087
$874$	0	0
$875$	1.75450	0.0593129
$876$	0	0
$877$	−48.1248	−1.62506	−0.812530	−	0.582920i	$-0.801910\pi$
$877$	−48.1248	−1.62506	−0.812530	+	0.582920i	$0.801910\pi$
$878$	0	0
$879$	−2.41577	−0.0814819
$880$	0	0
$881$	−25.2603	−0.851042	−0.425521	−	0.904949i	$-0.639909\pi$
$881$	−25.2603	−0.851042	−0.425521	+	0.904949i	$0.639909\pi$
$882$	0	0
$883$	−7.64771	−0.257366	−0.128683	−	0.991686i	$-0.541075\pi$
$883$	−7.64771	−0.257366	−0.128683	+	0.991686i	$0.541075\pi$
$884$	0	0
$885$	−0.712972	−0.0239663
$886$	0	0
$887$	−56.0273	−1.88121	−0.940607	−	0.339497i	$-0.889743\pi$
$887$	−56.0273	−1.88121	−0.940607	+	0.339497i	$0.889743\pi$
$888$	0	0
$889$	3.93667	0.132032
$890$	0	0
$891$	−19.0784	−0.639150
$892$	0	0
$893$	−9.75556	−0.326457
$894$	0	0
$895$	−15.1482	−0.506347
$896$	0	0
$897$	−0.252915	−0.00844457
$898$	0	0
$899$	7.05193	0.235195
$900$	0	0
$901$	6.76205	0.225277
$902$	0	0
$903$	−0.205549	−0.00684023
$904$	0	0
$905$	18.6250	0.619116
$906$	0	0
$907$	−3.24468	−0.107738	−0.0538690	−	0.998548i	$-0.517155\pi$
$907$	−3.24468	−0.107738	−0.0538690	+	0.998548i	$0.517155\pi$
$908$	0	0
$909$	−18.3575	−0.608880
$910$	0	0
$911$	56.5861	1.87478	0.937390	−	0.348281i	$-0.113234\pi$
$911$	56.5861	1.87478	0.937390	+	0.348281i	$0.113234\pi$
$912$	0	0
$913$	2.15240	0.0712339
$914$	0	0
$915$	−0.0680178	−0.00224860
$916$	0	0
$917$	−0.703483	−0.0232310
$918$	0	0
$919$	33.9657	1.12043	0.560213	−	0.828348i	$-0.310719\pi$
$919$	33.9657	1.12043	0.560213	+	0.828348i	$0.310719\pi$
$920$	0	0
$921$	0.836261	0.0275557
$922$	0	0
$923$	−5.77541	−0.190100
$924$	0	0
$925$	−45.1725	−1.48526
$926$	0	0
$927$	−40.9141	−1.34379
$928$	0	0
$929$	16.9209	0.555156	0.277578	−	0.960703i	$-0.410468\pi$
$929$	16.9209	0.555156	0.277578	+	0.960703i	$0.410468\pi$
$930$	0	0
$931$	31.4477	1.03066
$932$	0	0
$933$	−1.28266	−0.0419924
$934$	0	0
$935$	−1.18395	−0.0387193
$936$	0	0
$937$	−43.0130	−1.40517	−0.702586	−	0.711598i	$-0.747972\pi$
$937$	−43.0130	−1.40517	−0.702586	+	0.711598i	$0.747972\pi$
$938$	0	0
$939$	1.30529	0.0425966
$940$	0	0
$941$	26.0198	0.848223	0.424111	−	0.905610i	$-0.360587\pi$
$941$	26.0198	0.848223	0.424111	+	0.905610i	$0.360587\pi$
$942$	0	0
$943$	−10.7325	−0.349498
$944$	0	0
$945$	−0.0885793	−0.00288148
$946$	0	0
$947$	8.81747	0.286529	0.143265	−	0.989684i	$-0.454240\pi$
$947$	8.81747	0.286529	0.143265	+	0.989684i	$0.454240\pi$
$948$	0	0
$949$	−13.7477	−0.446269
$950$	0	0
$951$	2.16644	0.0702515
$952$	0	0
$953$	−45.8495	−1.48521	−0.742605	−	0.669730i	$-0.766410\pi$
$953$	−45.8495	−1.48521	−0.742605	+	0.669730i	$0.766410\pi$
$954$	0	0
$955$	−19.9523	−0.645642
$956$	0	0
$957$	0.164797	0.00532714
$958$	0	0
$959$	1.14609	0.0370091
$960$	0	0
$961$	18.7297	0.604184
$962$	0	0
$963$	10.5385	0.339598
$964$	0	0
$965$	−4.26953	−0.137441
$966$	0	0
$967$	−14.6828	−0.472167	−0.236083	−	0.971733i	$-0.575864\pi$
$967$	−14.6828	−0.472167	−0.236083	+	0.971733i	$0.575864\pi$
$968$	0	0
$969$	0.213514	0.00685906
$970$	0	0
$971$	35.1531	1.12812	0.564059	−	0.825734i	$-0.309239\pi$
$971$	35.1531	1.12812	0.564059	+	0.825734i	$0.309239\pi$
$972$	0	0
$973$	−3.17214	−0.101694
$974$	0	0
$975$	−0.322609	−0.0103318
$976$	0	0
$977$	−30.7196	−0.982807	−0.491404	−	0.870932i	$-0.663516\pi$
$977$	−30.7196	−0.982807	−0.491404	+	0.870932i	$0.663516\pi$
$978$	0	0
$979$	3.78018	0.120815
$980$	0	0
$981$	−55.0976	−1.75913
$982$	0	0
$983$	−36.1013	−1.15145	−0.575727	−	0.817642i	$-0.695281\pi$
$983$	−36.1013	−1.15145	−0.575727	+	0.817642i	$0.695281\pi$
$984$	0	0
$985$	−17.2682	−0.550211
$986$	0	0
$987$	0.0351002	0.00111725
$988$	0	0
$989$	41.3553	1.31502
$990$	0	0
$991$	31.5459	1.00209	0.501044	−	0.865422i	$-0.332949\pi$
$991$	31.5459	1.00209	0.501044	+	0.865422i	$0.332949\pi$
$992$	0	0
$993$	−0.628610	−0.0199483
$994$	0	0
$995$	−4.53347	−0.143721
$996$	0	0
$997$	42.1984	1.33644	0.668218	−	0.743966i	$-0.267057\pi$
$997$	42.1984	1.33644	0.668218	+	0.743966i	$0.267057\pi$
$998$	0	0
$999$	5.01210	0.158576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.be.1.7		13
4.3	odd	2		3016.2.a.k.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.k.1.7	✓	13	4.3	odd	2
6032.2.a.be.1.7		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)

\(f(q)\)	\(=\)	\(q-0.0772773 q^{3} +0.908464 q^{5} -0.210501 q^{7} -2.99403 q^{9} +O(q^{10})\)	\(q-0.0772773 q^{3} +0.908464 q^{5} -0.210501 q^{7} -2.99403 q^{9} -2.13255 q^{11} -1.00000 q^{13} -0.0702036 q^{15} +0.611120 q^{17} -4.52114 q^{19} +0.0162669 q^{21} -3.27282 q^{23} -4.17469 q^{25} +0.463202 q^{27} +1.00000 q^{29} +7.05193 q^{31} +0.164797 q^{33} -0.191232 q^{35} +10.8205 q^{37} +0.0772773 q^{39} +3.27928 q^{41} -12.6360 q^{43} -2.71997 q^{45} +2.15776 q^{47} -6.95569 q^{49} -0.0472257 q^{51} +11.0650 q^{53} -1.93734 q^{55} +0.349381 q^{57} +10.1558 q^{59} +0.968865 q^{61} +0.630245 q^{63} -0.908464 q^{65} -2.04949 q^{67} +0.252915 q^{69} +5.77541 q^{71} +13.7477 q^{73} +0.322609 q^{75} +0.448903 q^{77} -7.24856 q^{79} +8.94629 q^{81} -1.00931 q^{83} +0.555181 q^{85} -0.0772773 q^{87} -1.77261 q^{89} +0.210501 q^{91} -0.544954 q^{93} -4.10729 q^{95} +9.66212 q^{97} +6.38490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) \) 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

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