LMFDB - Embedded newform 6032.2.a.z.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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 17x^{8} + 47x^{7} + 104x^{6} - 235x^{5} - 283x^{4} + 364x^{3} + 330x^{2} + 12x - 8 $ x^10 - 3x^9 - 17x^8 + 47x^7 + 104x^6 - 235x^5 - 283x^4 + 364x^3 + 330x^2 + 12*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.696475$ 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)

$f(q)$	$=$	$q+0.696475 q^{3} -1.84500 q^{5} -0.816882 q^{7} -2.51492 q^{9} +O(q^{10})$	$q+0.696475 q^{3} -1.84500 q^{5} -0.816882 q^{7} -2.51492 q^{9} +0.991003 q^{11} -1.00000 q^{13} -1.28500 q^{15} +6.04150 q^{17} +1.01814 q^{19} -0.568938 q^{21} +2.81464 q^{23} -1.59597 q^{25} -3.84101 q^{27} -1.00000 q^{29} +5.49493 q^{31} +0.690209 q^{33} +1.50715 q^{35} +4.58285 q^{37} -0.696475 q^{39} -10.1252 q^{41} -0.776950 q^{43} +4.64004 q^{45} -8.28531 q^{47} -6.33270 q^{49} +4.20775 q^{51} +8.01468 q^{53} -1.82840 q^{55} +0.709110 q^{57} +5.45055 q^{59} +1.74531 q^{61} +2.05440 q^{63} +1.84500 q^{65} -11.3022 q^{67} +1.96033 q^{69} +1.41561 q^{71} +9.49931 q^{73} -1.11155 q^{75} -0.809533 q^{77} +2.74751 q^{79} +4.86960 q^{81} -14.1967 q^{83} -11.1466 q^{85} -0.696475 q^{87} +13.8083 q^{89} +0.816882 q^{91} +3.82708 q^{93} -1.87847 q^{95} +3.12782 q^{97} -2.49230 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9}+O(q^{10}) $ 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9	$ 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9} - 14 q^{11} - 10 q^{13} - 7 q^{15} + 5 q^{17} - 11 q^{19} - 7 q^{23} + 10 q^{25} - 21 q^{27} - 10 q^{29} - 5 q^{31} + 5 q^{33} - 11 q^{35} + 8 q^{37} + 3 q^{39} + 14 q^{41} - 35 q^{43} + 7 q^{45} - 7 q^{49} - 20 q^{51} - 11 q^{53} - 8 q^{55} + 4 q^{57} - 23 q^{59} - 8 q^{61} - 43 q^{63} - 4 q^{65} - 27 q^{67} + 10 q^{69} - 3 q^{71} + 7 q^{73} - 23 q^{75} + 2 q^{77} - 9 q^{79} - 6 q^{81} - 48 q^{83} - 6 q^{85} + 3 q^{87} + 20 q^{89} - 3 q^{91} - 11 q^{93} - 11 q^{95} + q^{97} - 54 q^{99}+O(q^{100}) $ 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9 - 14 * q^11 - 10 * q^13 - 7 * q^15 + 5 * q^17 - 11 * q^19 - 7 * q^23 + 10 * q^25 - 21 * q^27 - 10 * q^29 - 5 * q^31 + 5 * q^33 - 11 * q^35 + 8 * q^37 + 3 * q^39 + 14 * q^41 - 35 * q^43 + 7 * q^45 - 7 * q^49 - 20 * q^51 - 11 * q^53 - 8 * q^55 + 4 * q^57 - 23 * q^59 - 8 * q^61 - 43 * q^63 - 4 * q^65 - 27 * q^67 + 10 * q^69 - 3 * q^71 + 7 * q^73 - 23 * q^75 + 2 * q^77 - 9 * q^79 - 6 * q^81 - 48 * q^83 - 6 * q^85 + 3 * q^87 + 20 * q^89 - 3 * q^91 - 11 * q^93 - 11 * q^95 + q^97 - 54 * 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
$3$	0.696475	0.402110	0.201055	−	0.979580i	$-0.435563\pi$
$3$	0.696475	0.402110	0.201055	+	0.979580i	$0.435563\pi$
$4$	0	0
$5$	−1.84500	−0.825110	−0.412555	−	0.910933i	$-0.635364\pi$
$5$	−1.84500	−0.825110	−0.412555	+	0.910933i	$0.635364\pi$
$6$	0	0
$7$	−0.816882	−0.308752	−0.154376	−	0.988012i	$-0.549337\pi$
$7$	−0.816882	−0.308752	−0.154376	+	0.988012i	$0.549337\pi$
$8$	0	0
$9$	−2.51492	−0.838307
$10$	0	0
$11$	0.991003	0.298799	0.149399	−	0.988777i	$-0.452266\pi$
$11$	0.991003	0.298799	0.149399	+	0.988777i	$0.452266\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.28500	−0.331785
$16$	0	0
$17$	6.04150	1.46528	0.732639	−	0.680617i	$-0.238288\pi$
$17$	6.04150	1.46528	0.732639	+	0.680617i	$0.238288\pi$
$18$	0	0
$19$	1.01814	0.233578	0.116789	−	0.993157i	$-0.462740\pi$
$19$	1.01814	0.233578	0.116789	+	0.993157i	$0.462740\pi$
$20$	0	0
$21$	−0.568938	−0.124153
$22$	0	0
$23$	2.81464	0.586893	0.293446	−	0.955976i	$-0.405198\pi$
$23$	2.81464	0.586893	0.293446	+	0.955976i	$0.405198\pi$
$24$	0	0
$25$	−1.59597	−0.319193
$26$	0	0
$27$	−3.84101	−0.739202
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.49493	0.986918	0.493459	−	0.869769i	$-0.335732\pi$
$31$	5.49493	0.986918	0.493459	+	0.869769i	$0.335732\pi$
$32$	0	0
$33$	0.690209	0.120150
$34$	0	0
$35$	1.50715	0.254755
$36$	0	0
$37$	4.58285	0.753416	0.376708	−	0.926332i	$-0.377056\pi$
$37$	4.58285	0.753416	0.376708	+	0.926332i	$0.377056\pi$
$38$	0	0
$39$	−0.696475	−0.111525
$40$	0	0
$41$	−10.1252	−1.58129	−0.790643	−	0.612278i	$-0.790253\pi$
$41$	−10.1252	−1.58129	−0.790643	+	0.612278i	$0.790253\pi$
$42$	0	0
$43$	−0.776950	−0.118484	−0.0592419	−	0.998244i	$-0.518868\pi$
$43$	−0.776950	−0.118484	−0.0592419	+	0.998244i	$0.518868\pi$
$44$	0	0
$45$	4.64004	0.691696
$46$	0	0
$47$	−8.28531	−1.20854	−0.604269	−	0.796781i	$-0.706535\pi$
$47$	−8.28531	−1.20854	−0.604269	+	0.796781i	$0.706535\pi$
$48$	0	0
$49$	−6.33270	−0.904672
$50$	0	0
$51$	4.20775	0.589203
$52$	0	0
$53$	8.01468	1.10090	0.550450	−	0.834868i	$-0.314456\pi$
$53$	8.01468	1.10090	0.550450	+	0.834868i	$0.314456\pi$
$54$	0	0
$55$	−1.82840	−0.246542
$56$	0	0
$57$	0.709110	0.0939239
$58$	0	0
$59$	5.45055	0.709601	0.354801	−	0.934942i	$-0.384549\pi$
$59$	5.45055	0.709601	0.354801	+	0.934942i	$0.384549\pi$
$60$	0	0
$61$	1.74531	0.223464	0.111732	−	0.993738i	$-0.464360\pi$
$61$	1.74531	0.223464	0.111732	+	0.993738i	$0.464360\pi$
$62$	0	0
$63$	2.05440	0.258829
$64$	0	0
$65$	1.84500	0.228844
$66$	0	0
$67$	−11.3022	−1.38078	−0.690391	−	0.723437i	$-0.742561\pi$
$67$	−11.3022	−1.38078	−0.690391	+	0.723437i	$0.742561\pi$
$68$	0	0
$69$	1.96033	0.235995
$70$	0	0
$71$	1.41561	0.168003	0.0840013	−	0.996466i	$-0.473230\pi$
$71$	1.41561	0.168003	0.0840013	+	0.996466i	$0.473230\pi$
$72$	0	0
$73$	9.49931	1.11181	0.555905	−	0.831246i	$-0.312372\pi$
$73$	9.49931	1.11181	0.555905	+	0.831246i	$0.312372\pi$
$74$	0	0
$75$	−1.11155	−0.128351
$76$	0	0
$77$	−0.809533	−0.0922549
$78$	0	0
$79$	2.74751	0.309119	0.154559	−	0.987983i	$-0.450604\pi$
$79$	2.74751	0.309119	0.154559	+	0.987983i	$0.450604\pi$
$80$	0	0
$81$	4.86960	0.541067
$82$	0	0
$83$	−14.1967	−1.55830	−0.779148	−	0.626841i	$-0.784348\pi$
$83$	−14.1967	−1.55830	−0.779148	+	0.626841i	$0.784348\pi$
$84$	0	0
$85$	−11.1466	−1.20902
$86$	0	0
$87$	−0.696475	−0.0746700
$88$	0	0
$89$	13.8083	1.46368	0.731840	−	0.681477i	$-0.238662\pi$
$89$	13.8083	1.46368	0.731840	+	0.681477i	$0.238662\pi$
$90$	0	0
$91$	0.816882	0.0856325
$92$	0	0
$93$	3.82708	0.396850
$94$	0	0
$95$	−1.87847	−0.192727
$96$	0	0
$97$	3.12782	0.317582	0.158791	−	0.987312i	$-0.449240\pi$
$97$	3.12782	0.317582	0.158791	+	0.987312i	$0.449240\pi$
$98$	0	0
$99$	−2.49230	−0.250485
$100$	0	0
$101$	−12.2100	−1.21494	−0.607471	−	0.794342i	$-0.707816\pi$
$101$	−12.2100	−1.21494	−0.607471	+	0.794342i	$0.707816\pi$
$102$	0	0
$103$	−4.78793	−0.471768	−0.235884	−	0.971781i	$-0.575799\pi$
$103$	−4.78793	−0.471768	−0.235884	+	0.971781i	$0.575799\pi$
$104$	0	0
$105$	1.04969	0.102439
$106$	0	0
$107$	−18.2151	−1.76092	−0.880460	−	0.474121i	$-0.842766\pi$
$107$	−18.2151	−1.76092	−0.880460	+	0.474121i	$0.842766\pi$
$108$	0	0
$109$	9.09387	0.871035	0.435517	−	0.900180i	$-0.356565\pi$
$109$	9.09387	0.871035	0.435517	+	0.900180i	$0.356565\pi$
$110$	0	0
$111$	3.19184	0.302956
$112$	0	0
$113$	−7.53984	−0.709288	−0.354644	−	0.935001i	$-0.615398\pi$
$113$	−7.53984	−0.709288	−0.354644	+	0.935001i	$0.615398\pi$
$114$	0	0
$115$	−5.19301	−0.484251
$116$	0	0
$117$	2.51492	0.232505
$118$	0	0
$119$	−4.93519	−0.452408
$120$	0	0
$121$	−10.0179	−0.910719
$122$	0	0
$123$	−7.05193	−0.635851
$124$	0	0
$125$	12.1696	1.08848
$126$	0	0
$127$	2.65593	0.235675	0.117838	−	0.993033i	$-0.462404\pi$
$127$	2.65593	0.235675	0.117838	+	0.993033i	$0.462404\pi$
$128$	0	0
$129$	−0.541126	−0.0476435
$130$	0	0
$131$	−4.54062	−0.396715	−0.198358	−	0.980130i	$-0.563561\pi$
$131$	−4.54062	−0.396715	−0.198358	+	0.980130i	$0.563561\pi$
$132$	0	0
$133$	−0.831701	−0.0721177
$134$	0	0
$135$	7.08667	0.609923
$136$	0	0
$137$	−10.8553	−0.927434	−0.463717	−	0.885983i	$-0.653485\pi$
$137$	−10.8553	−0.927434	−0.463717	+	0.885983i	$0.653485\pi$
$138$	0	0
$139$	−20.6145	−1.74850	−0.874250	−	0.485477i	$-0.838646\pi$
$139$	−20.6145	−1.74850	−0.874250	+	0.485477i	$0.838646\pi$
$140$	0	0
$141$	−5.77052	−0.485965
$142$	0	0
$143$	−0.991003	−0.0828719
$144$	0	0
$145$	1.84500	0.153219
$146$	0	0
$147$	−4.41057	−0.363778
$148$	0	0
$149$	−0.222787	−0.0182514	−0.00912570	−	0.999958i	$-0.502905\pi$
$149$	−0.222787	−0.0182514	−0.00912570	+	0.999958i	$0.502905\pi$
$150$	0	0
$151$	−1.43515	−0.116791	−0.0583954	−	0.998294i	$-0.518598\pi$
$151$	−1.43515	−0.116791	−0.0583954	+	0.998294i	$0.518598\pi$
$152$	0	0
$153$	−15.1939	−1.22835
$154$	0	0
$155$	−10.1382	−0.814316
$156$	0	0
$157$	−5.17492	−0.413003	−0.206502	−	0.978446i	$-0.566208\pi$
$157$	−5.17492	−0.413003	−0.206502	+	0.978446i	$0.566208\pi$
$158$	0	0
$159$	5.58203	0.442683
$160$	0	0
$161$	−2.29923	−0.181205
$162$	0	0
$163$	−7.29769	−0.571599	−0.285800	−	0.958289i	$-0.592259\pi$
$163$	−7.29769	−0.571599	−0.285800	+	0.958289i	$0.592259\pi$
$164$	0	0
$165$	−1.27344	−0.0991370
$166$	0	0
$167$	2.43290	0.188263	0.0941316	−	0.995560i	$-0.469993\pi$
$167$	2.43290	0.188263	0.0941316	+	0.995560i	$0.469993\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.56055	−0.195810
$172$	0	0
$173$	9.75034	0.741305	0.370652	−	0.928772i	$-0.379134\pi$
$173$	9.75034	0.741305	0.370652	+	0.928772i	$0.379134\pi$
$174$	0	0
$175$	1.30372	0.0985517
$176$	0	0
$177$	3.79617	0.285338
$178$	0	0
$179$	−13.9515	−1.04278	−0.521392	−	0.853317i	$-0.674587\pi$
$179$	−13.9515	−1.04278	−0.521392	+	0.853317i	$0.674587\pi$
$180$	0	0
$181$	−16.2019	−1.20428	−0.602139	−	0.798392i	$-0.705685\pi$
$181$	−16.2019	−1.20428	−0.602139	+	0.798392i	$0.705685\pi$
$182$	0	0
$183$	1.21556	0.0898570
$184$	0	0
$185$	−8.45537	−0.621651
$186$	0	0
$187$	5.98714	0.437823
$188$	0	0
$189$	3.13765	0.228230
$190$	0	0
$191$	−5.87280	−0.424941	−0.212470	−	0.977167i	$-0.568151\pi$
$191$	−5.87280	−0.424941	−0.212470	+	0.977167i	$0.568151\pi$
$192$	0	0
$193$	−2.95666	−0.212825	−0.106413	−	0.994322i	$-0.533936\pi$
$193$	−2.95666	−0.212825	−0.106413	+	0.994322i	$0.533936\pi$
$194$	0	0
$195$	1.28500	0.0920207
$196$	0	0
$197$	9.34671	0.665926	0.332963	−	0.942940i	$-0.391952\pi$
$197$	9.34671	0.665926	0.332963	+	0.942940i	$0.391952\pi$
$198$	0	0
$199$	−9.40953	−0.667024	−0.333512	−	0.942746i	$-0.608234\pi$
$199$	−9.40953	−0.667024	−0.333512	+	0.942746i	$0.608234\pi$
$200$	0	0
$201$	−7.87169	−0.555226
$202$	0	0
$203$	0.816882	0.0573339
$204$	0	0
$205$	18.6810	1.30473
$206$	0	0
$207$	−7.07860	−0.491996
$208$	0	0
$209$	1.00898	0.0697927
$210$	0	0
$211$	−1.07601	−0.0740756	−0.0370378	−	0.999314i	$-0.511792\pi$
$211$	−1.07601	−0.0740756	−0.0370378	+	0.999314i	$0.511792\pi$
$212$	0	0
$213$	0.985940	0.0675555
$214$	0	0
$215$	1.43347	0.0977621
$216$	0	0
$217$	−4.48871	−0.304713
$218$	0	0
$219$	6.61604	0.447070
$220$	0	0
$221$	−6.04150	−0.406395
$222$	0	0
$223$	−17.6731	−1.18348	−0.591740	−	0.806129i	$-0.701559\pi$
$223$	−17.6731	−1.18348	−0.591740	+	0.806129i	$0.701559\pi$
$224$	0	0
$225$	4.01373	0.267582
$226$	0	0
$227$	−22.9054	−1.52029	−0.760144	−	0.649755i	$-0.774872\pi$
$227$	−22.9054	−1.52029	−0.760144	+	0.649755i	$0.774872\pi$
$228$	0	0
$229$	−16.6950	−1.10324	−0.551618	−	0.834097i	$-0.685989\pi$
$229$	−16.6950	−1.10324	−0.551618	+	0.834097i	$0.685989\pi$
$230$	0	0
$231$	−0.563820	−0.0370966
$232$	0	0
$233$	−3.26189	−0.213693	−0.106847	−	0.994276i	$-0.534075\pi$
$233$	−3.26189	−0.213693	−0.106847	+	0.994276i	$0.534075\pi$
$234$	0	0
$235$	15.2864	0.997176
$236$	0	0
$237$	1.91357	0.124300
$238$	0	0
$239$	−8.41118	−0.544074	−0.272037	−	0.962287i	$-0.587697\pi$
$239$	−8.41118	−0.544074	−0.272037	+	0.962287i	$0.587697\pi$
$240$	0	0
$241$	−1.36400	−0.0878631	−0.0439316	−	0.999035i	$-0.513988\pi$
$241$	−1.36400	−0.0878631	−0.0439316	+	0.999035i	$0.513988\pi$
$242$	0	0
$243$	14.9146	0.956771
$244$	0	0
$245$	11.6839	0.746454
$246$	0	0
$247$	−1.01814	−0.0647828
$248$	0	0
$249$	−9.88768	−0.626606
$250$	0	0
$251$	−18.3526	−1.15841	−0.579204	−	0.815182i	$-0.696637\pi$
$251$	−18.3526	−1.15841	−0.579204	+	0.815182i	$0.696637\pi$
$252$	0	0
$253$	2.78932	0.175363
$254$	0	0
$255$	−7.76331	−0.486158
$256$	0	0
$257$	4.80340	0.299628	0.149814	−	0.988714i	$-0.452133\pi$
$257$	4.80340	0.299628	0.149814	+	0.988714i	$0.452133\pi$
$258$	0	0
$259$	−3.74365	−0.232619
$260$	0	0
$261$	2.51492	0.155670
$262$	0	0
$263$	−15.6792	−0.966819	−0.483410	−	0.875394i	$-0.660602\pi$
$263$	−15.6792	−0.966819	−0.483410	+	0.875394i	$0.660602\pi$
$264$	0	0
$265$	−14.7871	−0.908364
$266$	0	0
$267$	9.61716	0.588561
$268$	0	0
$269$	29.5469	1.80151	0.900753	−	0.434331i	$-0.143015\pi$
$269$	29.5469	1.80151	0.900753	+	0.434331i	$0.143015\pi$
$270$	0	0
$271$	−10.4629	−0.635576	−0.317788	−	0.948162i	$-0.602940\pi$
$271$	−10.4629	−0.635576	−0.317788	+	0.948162i	$0.602940\pi$
$272$	0	0
$273$	0.568938	0.0344337
$274$	0	0
$275$	−1.58161	−0.0953746
$276$	0	0
$277$	−22.0577	−1.32532	−0.662661	−	0.748920i	$-0.730573\pi$
$277$	−22.0577	−1.32532	−0.662661	+	0.748920i	$0.730573\pi$
$278$	0	0
$279$	−13.8193	−0.827341
$280$	0	0
$281$	−5.74763	−0.342875	−0.171437	−	0.985195i	$-0.554841\pi$
$281$	−5.74763	−0.342875	−0.171437	+	0.985195i	$0.554841\pi$
$282$	0	0
$283$	24.1781	1.43724	0.718619	−	0.695404i	$-0.244775\pi$
$283$	24.1781	1.43724	0.718619	+	0.695404i	$0.244775\pi$
$284$	0	0
$285$	−1.30831	−0.0774976
$286$	0	0
$287$	8.27107	0.488226
$288$	0	0
$289$	19.4997	1.14704
$290$	0	0
$291$	2.17845	0.127703
$292$	0	0
$293$	−5.50852	−0.321811	−0.160906	−	0.986970i	$-0.551441\pi$
$293$	−5.50852	−0.321811	−0.160906	+	0.986970i	$0.551441\pi$
$294$	0	0
$295$	−10.0563	−0.585499
$296$	0	0
$297$	−3.80645	−0.220873
$298$	0	0
$299$	−2.81464	−0.162775
$300$	0	0
$301$	0.634677	0.0365821
$302$	0	0
$303$	−8.50397	−0.488541
$304$	0	0
$305$	−3.22010	−0.184382
$306$	0	0
$307$	0.471442	0.0269066	0.0134533	−	0.999910i	$-0.495718\pi$
$307$	0.471442	0.0269066	0.0134533	+	0.999910i	$0.495718\pi$
$308$	0	0
$309$	−3.33467	−0.189703
$310$	0	0
$311$	−19.2880	−1.09372	−0.546861	−	0.837224i	$-0.684177\pi$
$311$	−19.2880	−1.09372	−0.546861	+	0.837224i	$0.684177\pi$
$312$	0	0
$313$	2.33811	0.132158	0.0660788	−	0.997814i	$-0.478951\pi$
$313$	2.33811	0.132158	0.0660788	+	0.997814i	$0.478951\pi$
$314$	0	0
$315$	−3.79036	−0.213563
$316$	0	0
$317$	22.4368	1.26018	0.630088	−	0.776524i	$-0.283019\pi$
$317$	22.4368	1.26018	0.630088	+	0.776524i	$0.283019\pi$
$318$	0	0
$319$	−0.991003	−0.0554855
$320$	0	0
$321$	−12.6864	−0.708084
$322$	0	0
$323$	6.15110	0.342256
$324$	0	0
$325$	1.59597	0.0885283
$326$	0	0
$327$	6.33365	0.350252
$328$	0	0
$329$	6.76813	0.373139
$330$	0	0
$331$	−6.58740	−0.362076	−0.181038	−	0.983476i	$-0.557946\pi$
$331$	−6.58740	−0.362076	−0.181038	+	0.983476i	$0.557946\pi$
$332$	0	0
$333$	−11.5255	−0.631595
$334$	0	0
$335$	20.8526	1.13930
$336$	0	0
$337$	−2.48025	−0.135108	−0.0675539	−	0.997716i	$-0.521519\pi$
$337$	−2.48025	−0.135108	−0.0675539	+	0.997716i	$0.521519\pi$
$338$	0	0
$339$	−5.25131	−0.285212
$340$	0	0
$341$	5.44549	0.294890
$342$	0	0
$343$	10.8912	0.588072
$344$	0	0
$345$	−3.61681	−0.194722
$346$	0	0
$347$	17.4585	0.937224	0.468612	−	0.883404i	$-0.344754\pi$
$347$	17.4585	0.937224	0.468612	+	0.883404i	$0.344754\pi$
$348$	0	0
$349$	−10.4138	−0.557437	−0.278719	−	0.960373i	$-0.589910\pi$
$349$	−10.4138	−0.557437	−0.278719	+	0.960373i	$0.589910\pi$
$350$	0	0
$351$	3.84101	0.205018
$352$	0	0
$353$	−8.27263	−0.440308	−0.220154	−	0.975465i	$-0.570656\pi$
$353$	−8.27263	−0.440308	−0.220154	+	0.975465i	$0.570656\pi$
$354$	0	0
$355$	−2.61181	−0.138621
$356$	0	0
$357$	−3.43724	−0.181918
$358$	0	0
$359$	−19.3685	−1.02223	−0.511115	−	0.859513i	$-0.670767\pi$
$359$	−19.3685	−1.02223	−0.511115	+	0.859513i	$0.670767\pi$
$360$	0	0
$361$	−17.9634	−0.945441
$362$	0	0
$363$	−6.97723	−0.366210
$364$	0	0
$365$	−17.5263	−0.917366
$366$	0	0
$367$	22.4542	1.17210	0.586051	−	0.810274i	$-0.300682\pi$
$367$	22.4542	1.17210	0.586051	+	0.810274i	$0.300682\pi$
$368$	0	0
$369$	25.4640	1.32560
$370$	0	0
$371$	−6.54705	−0.339906
$372$	0	0
$373$	1.95724	0.101342	0.0506709	−	0.998715i	$-0.483864\pi$
$373$	1.95724	0.101342	0.0506709	+	0.998715i	$0.483864\pi$
$374$	0	0
$375$	8.47581	0.437689
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	23.1780	1.19057	0.595286	−	0.803514i	$-0.297039\pi$
$379$	23.1780	1.19057	0.595286	+	0.803514i	$0.297039\pi$
$380$	0	0
$381$	1.84979	0.0947675
$382$	0	0
$383$	30.2566	1.54604	0.773019	−	0.634383i	$-0.218746\pi$
$383$	30.2566	1.54604	0.773019	+	0.634383i	$0.218746\pi$
$384$	0	0
$385$	1.49359	0.0761204
$386$	0	0
$387$	1.95397	0.0993258
$388$	0	0
$389$	−14.6142	−0.740968	−0.370484	−	0.928839i	$-0.620808\pi$
$389$	−14.6142	−0.740968	−0.370484	+	0.928839i	$0.620808\pi$
$390$	0	0
$391$	17.0046	0.859961
$392$	0	0
$393$	−3.16243	−0.159523
$394$	0	0
$395$	−5.06916	−0.255057
$396$	0	0
$397$	17.9106	0.898909	0.449454	−	0.893303i	$-0.351618\pi$
$397$	17.9106	0.898909	0.449454	+	0.893303i	$0.351618\pi$
$398$	0	0
$399$	−0.579260	−0.0289992
$400$	0	0
$401$	32.8283	1.63937	0.819683	−	0.572818i	$-0.194150\pi$
$401$	32.8283	1.63937	0.819683	+	0.572818i	$0.194150\pi$
$402$	0	0
$403$	−5.49493	−0.273722
$404$	0	0
$405$	−8.98442	−0.446440
$406$	0	0
$407$	4.54162	0.225120
$408$	0	0
$409$	−35.9815	−1.77917	−0.889586	−	0.456768i	$-0.849007\pi$
$409$	−35.9815	−1.77917	−0.889586	+	0.456768i	$0.849007\pi$
$410$	0	0
$411$	−7.56047	−0.372931
$412$	0	0
$413$	−4.45246	−0.219091
$414$	0	0
$415$	26.1930	1.28576
$416$	0	0
$417$	−14.3575	−0.703089
$418$	0	0
$419$	−11.8447	−0.578650	−0.289325	−	0.957231i	$-0.593431\pi$
$419$	−11.8447	−0.578650	−0.289325	+	0.957231i	$0.593431\pi$
$420$	0	0
$421$	−16.4266	−0.800581	−0.400291	−	0.916388i	$-0.631091\pi$
$421$	−16.4266	−0.800581	−0.400291	+	0.916388i	$0.631091\pi$
$422$	0	0
$423$	20.8369	1.01313
$424$	0	0
$425$	−9.64203	−0.467707
$426$	0	0
$427$	−1.42571	−0.0689950
$428$	0	0
$429$	−0.690209	−0.0333236
$430$	0	0
$431$	27.7718	1.33772	0.668860	−	0.743389i	$-0.266783\pi$
$431$	27.7718	1.33772	0.668860	+	0.743389i	$0.266783\pi$
$432$	0	0
$433$	16.1562	0.776416	0.388208	−	0.921572i	$-0.373094\pi$
$433$	16.1562	0.776416	0.388208	+	0.921572i	$0.373094\pi$
$434$	0	0
$435$	1.28500	0.0616110
$436$	0	0
$437$	2.86570	0.137085
$438$	0	0
$439$	33.8000	1.61319	0.806594	−	0.591106i	$-0.201308\pi$
$439$	33.8000	1.61319	0.806594	+	0.591106i	$0.201308\pi$
$440$	0	0
$441$	15.9263	0.758393
$442$	0	0
$443$	−26.5513	−1.26149	−0.630746	−	0.775990i	$-0.717251\pi$
$443$	−26.5513	−1.26149	−0.630746	+	0.775990i	$0.717251\pi$
$444$	0	0
$445$	−25.4764	−1.20770
$446$	0	0
$447$	−0.155165	−0.00733908
$448$	0	0
$449$	37.6348	1.77610	0.888048	−	0.459751i	$-0.152061\pi$
$449$	37.6348	1.77610	0.888048	+	0.459751i	$0.152061\pi$
$450$	0	0
$451$	−10.0341	−0.472486
$452$	0	0
$453$	−0.999547	−0.0469628
$454$	0	0
$455$	−1.50715	−0.0706563
$456$	0	0
$457$	3.10831	0.145401	0.0727003	−	0.997354i	$-0.476838\pi$
$457$	3.10831	0.145401	0.0727003	+	0.997354i	$0.476838\pi$
$458$	0	0
$459$	−23.2054	−1.08314
$460$	0	0
$461$	−29.1336	−1.35688	−0.678442	−	0.734654i	$-0.737345\pi$
$461$	−29.1336	−1.35688	−0.678442	+	0.734654i	$0.737345\pi$
$462$	0	0
$463$	−26.2826	−1.22146	−0.610728	−	0.791840i	$-0.709123\pi$
$463$	−26.2826	−1.22146	−0.610728	+	0.791840i	$0.709123\pi$
$464$	0	0
$465$	−7.06097	−0.327445
$466$	0	0
$467$	33.7735	1.56285	0.781427	−	0.623997i	$-0.214492\pi$
$467$	33.7735	1.56285	0.781427	+	0.623997i	$0.214492\pi$
$468$	0	0
$469$	9.23255	0.426320
$470$	0	0
$471$	−3.60420	−0.166073
$472$	0	0
$473$	−0.769960	−0.0354028
$474$	0	0
$475$	−1.62492	−0.0745564
$476$	0	0
$477$	−20.1563	−0.922893
$478$	0	0
$479$	24.2473	1.10789	0.553944	−	0.832554i	$-0.313122\pi$
$479$	24.2473	1.10789	0.553944	+	0.832554i	$0.313122\pi$
$480$	0	0
$481$	−4.58285	−0.208960
$482$	0	0
$483$	−1.60136	−0.0728642
$484$	0	0
$485$	−5.77083	−0.262040
$486$	0	0
$487$	11.9997	0.543759	0.271880	−	0.962331i	$-0.412355\pi$
$487$	11.9997	0.543759	0.271880	+	0.962331i	$0.412355\pi$
$488$	0	0
$489$	−5.08266	−0.229846
$490$	0	0
$491$	−20.6230	−0.930702	−0.465351	−	0.885126i	$-0.654072\pi$
$491$	−20.6230	−0.930702	−0.465351	+	0.885126i	$0.654072\pi$
$492$	0	0
$493$	−6.04150	−0.272095
$494$	0	0
$495$	4.59829	0.206678
$496$	0	0
$497$	−1.15639	−0.0518712
$498$	0	0
$499$	−32.6101	−1.45983	−0.729914	−	0.683538i	$-0.760440\pi$
$499$	−32.6101	−1.45983	−0.729914	+	0.683538i	$0.760440\pi$
$500$	0	0
$501$	1.69445	0.0757026
$502$	0	0
$503$	−35.4598	−1.58107	−0.790536	−	0.612415i	$-0.790198\pi$
$503$	−35.4598	−1.58107	−0.790536	+	0.612415i	$0.790198\pi$
$504$	0	0
$505$	22.5275	1.00246
$506$	0	0
$507$	0.696475	0.0309316
$508$	0	0
$509$	−30.8198	−1.36607	−0.683033	−	0.730388i	$-0.739339\pi$
$509$	−30.8198	−1.36607	−0.683033	+	0.730388i	$0.739339\pi$
$510$	0	0
$511$	−7.75982	−0.343274
$512$	0	0
$513$	−3.91069	−0.172661
$514$	0	0
$515$	8.83374	0.389261
$516$	0	0
$517$	−8.21077	−0.361109
$518$	0	0
$519$	6.79087	0.298086
$520$	0	0
$521$	−3.86496	−0.169327	−0.0846635	−	0.996410i	$-0.526982\pi$
$521$	−3.86496	−0.169327	−0.0846635	+	0.996410i	$0.526982\pi$
$522$	0	0
$523$	−22.1711	−0.969475	−0.484738	−	0.874660i	$-0.661085\pi$
$523$	−22.1711	−0.969475	−0.484738	+	0.874660i	$0.661085\pi$
$524$	0	0
$525$	0.908007	0.0396287
$526$	0	0
$527$	33.1976	1.44611
$528$	0	0
$529$	−15.0778	−0.655557
$530$	0	0
$531$	−13.7077	−0.594864
$532$	0	0
$533$	10.1252	0.438570
$534$	0	0
$535$	33.6069	1.45295
$536$	0	0
$537$	−9.71688	−0.419314
$538$	0	0
$539$	−6.27573	−0.270315
$540$	0	0
$541$	−25.7011	−1.10498	−0.552489	−	0.833520i	$-0.686322\pi$
$541$	−25.7011	−1.10498	−0.552489	+	0.833520i	$0.686322\pi$
$542$	0	0
$543$	−11.2842	−0.484252
$544$	0	0
$545$	−16.7782	−0.718699
$546$	0	0
$547$	10.6484	0.455291	0.227646	−	0.973744i	$-0.426897\pi$
$547$	10.6484	0.455291	0.227646	+	0.973744i	$0.426897\pi$
$548$	0	0
$549$	−4.38931	−0.187331
$550$	0	0
$551$	−1.01814	−0.0433743
$552$	0	0
$553$	−2.24439	−0.0954412
$554$	0	0
$555$	−5.88896	−0.249972
$556$	0	0
$557$	23.1260	0.979879	0.489940	−	0.871756i	$-0.337019\pi$
$557$	23.1260	0.979879	0.489940	+	0.871756i	$0.337019\pi$
$558$	0	0
$559$	0.776950	0.0328615
$560$	0	0
$561$	4.16990	0.176053
$562$	0	0
$563$	−25.9648	−1.09429	−0.547143	−	0.837039i	$-0.684285\pi$
$563$	−25.9648	−1.09429	−0.547143	+	0.837039i	$0.684285\pi$
$564$	0	0
$565$	13.9110	0.585241
$566$	0	0
$567$	−3.97789	−0.167056
$568$	0	0
$569$	19.3442	0.810951	0.405475	−	0.914106i	$-0.367106\pi$
$569$	19.3442	0.810951	0.405475	+	0.914106i	$0.367106\pi$
$570$	0	0
$571$	26.4651	1.10753	0.553765	−	0.832673i	$-0.313190\pi$
$571$	26.4651	1.10753	0.553765	+	0.832673i	$0.313190\pi$
$572$	0	0
$573$	−4.09026	−0.170873
$574$	0	0
$575$	−4.49207	−0.187332
$576$	0	0
$577$	16.6179	0.691811	0.345905	−	0.938269i	$-0.387572\pi$
$577$	16.6179	0.691811	0.345905	+	0.938269i	$0.387572\pi$
$578$	0	0
$579$	−2.05924	−0.0855792
$580$	0	0
$581$	11.5971	0.481127
$582$	0	0
$583$	7.94257	0.328948
$584$	0	0
$585$	−4.64004	−0.191842
$586$	0	0
$587$	−5.15443	−0.212746	−0.106373	−	0.994326i	$-0.533924\pi$
$587$	−5.15443	−0.212746	−0.106373	+	0.994326i	$0.533924\pi$
$588$	0	0
$589$	5.59461	0.230522
$590$	0	0
$591$	6.50975	0.267776
$592$	0	0
$593$	27.2931	1.12079	0.560396	−	0.828225i	$-0.310649\pi$
$593$	27.2931	1.12079	0.560396	+	0.828225i	$0.310649\pi$
$594$	0	0
$595$	9.10544	0.373287
$596$	0	0
$597$	−6.55351	−0.268217
$598$	0	0
$599$	−2.80788	−0.114727	−0.0573634	−	0.998353i	$-0.518269\pi$
$599$	−2.80788	−0.114727	−0.0573634	+	0.998353i	$0.518269\pi$
$600$	0	0
$601$	−32.0868	−1.30885	−0.654424	−	0.756128i	$-0.727089\pi$
$601$	−32.0868	−1.30885	−0.654424	+	0.756128i	$0.727089\pi$
$602$	0	0
$603$	28.4241	1.15752
$604$	0	0
$605$	18.4831	0.751444
$606$	0	0
$607$	−33.4361	−1.35713	−0.678564	−	0.734541i	$-0.737397\pi$
$607$	−33.4361	−1.35713	−0.678564	+	0.734541i	$0.737397\pi$
$608$	0	0
$609$	0.568938	0.0230545
$610$	0	0
$611$	8.28531	0.335188
$612$	0	0
$613$	−13.7402	−0.554961	−0.277480	−	0.960731i	$-0.589499\pi$
$613$	−13.7402	−0.554961	−0.277480	+	0.960731i	$0.589499\pi$
$614$	0	0
$615$	13.0108	0.524647
$616$	0	0
$617$	5.61346	0.225989	0.112995	−	0.993596i	$-0.463956\pi$
$617$	5.61346	0.225989	0.112995	+	0.993596i	$0.463956\pi$
$618$	0	0
$619$	−21.0570	−0.846351	−0.423175	−	0.906048i	$-0.639085\pi$
$619$	−21.0570	−0.846351	−0.423175	+	0.906048i	$0.639085\pi$
$620$	0	0
$621$	−10.8110	−0.433832
$622$	0	0
$623$	−11.2798	−0.451915
$624$	0	0
$625$	−14.4731	−0.578922
$626$	0	0
$627$	0.702731	0.0280644
$628$	0	0
$629$	27.6873	1.10396
$630$	0	0
$631$	9.12929	0.363431	0.181716	−	0.983351i	$-0.441835\pi$
$631$	9.12929	0.363431	0.181716	+	0.983351i	$0.441835\pi$
$632$	0	0
$633$	−0.749414	−0.0297865
$634$	0	0
$635$	−4.90019	−0.194458
$636$	0	0
$637$	6.33270	0.250911
$638$	0	0
$639$	−3.56016	−0.140838
$640$	0	0
$641$	14.9526	0.590593	0.295296	−	0.955406i	$-0.404582\pi$
$641$	14.9526	0.590593	0.295296	+	0.955406i	$0.404582\pi$
$642$	0	0
$643$	−0.597526	−0.0235641	−0.0117821	−	0.999931i	$-0.503750\pi$
$643$	−0.597526	−0.0235641	−0.0117821	+	0.999931i	$0.503750\pi$
$644$	0	0
$645$	0.998379	0.0393112
$646$	0	0
$647$	50.5328	1.98665	0.993325	−	0.115347i	$-0.0367979\pi$
$647$	50.5328	1.98665	0.993325	+	0.115347i	$0.0367979\pi$
$648$	0	0
$649$	5.40151	0.212028
$650$	0	0
$651$	−3.12627	−0.122528
$652$	0	0
$653$	−15.3873	−0.602153	−0.301076	−	0.953600i	$-0.597346\pi$
$653$	−15.3873	−0.602153	−0.301076	+	0.953600i	$0.597346\pi$
$654$	0	0
$655$	8.37745	0.327334
$656$	0	0
$657$	−23.8900	−0.932039
$658$	0	0
$659$	44.5625	1.73591	0.867955	−	0.496643i	$-0.165434\pi$
$659$	44.5625	1.73591	0.867955	+	0.496643i	$0.165434\pi$
$660$	0	0
$661$	32.9362	1.28107	0.640534	−	0.767929i	$-0.278713\pi$
$661$	32.9362	1.28107	0.640534	+	0.767929i	$0.278713\pi$
$662$	0	0
$663$	−4.20775	−0.163416
$664$	0	0
$665$	1.53449	0.0595050
$666$	0	0
$667$	−2.81464	−0.108983
$668$	0	0
$669$	−12.3089	−0.475889
$670$	0	0
$671$	1.72961	0.0667707
$672$	0	0
$673$	−8.38943	−0.323389	−0.161694	−	0.986841i	$-0.551696\pi$
$673$	−8.38943	−0.323389	−0.161694	+	0.986841i	$0.551696\pi$
$674$	0	0
$675$	6.13012	0.235948
$676$	0	0
$677$	13.8719	0.533139	0.266570	−	0.963816i	$-0.414110\pi$
$677$	13.8719	0.533139	0.266570	+	0.963816i	$0.414110\pi$
$678$	0	0
$679$	−2.55506	−0.0980542
$680$	0	0
$681$	−15.9531	−0.611323
$682$	0	0
$683$	−15.2353	−0.582964	−0.291482	−	0.956576i	$-0.594148\pi$
$683$	−15.2353	−0.582964	−0.291482	+	0.956576i	$0.594148\pi$
$684$	0	0
$685$	20.0281	0.765235
$686$	0	0
$687$	−11.6276	−0.443622
$688$	0	0
$689$	−8.01468	−0.305335
$690$	0	0
$691$	4.91868	0.187115	0.0935577	−	0.995614i	$-0.470176\pi$
$691$	4.91868	0.187115	0.0935577	+	0.995614i	$0.470176\pi$
$692$	0	0
$693$	2.03591	0.0773379
$694$	0	0
$695$	38.0338	1.44270
$696$	0	0
$697$	−61.1712	−2.31702
$698$	0	0
$699$	−2.27182	−0.0859283
$700$	0	0
$701$	24.5897	0.928742	0.464371	−	0.885641i	$-0.346280\pi$
$701$	24.5897	0.928742	0.464371	+	0.885641i	$0.346280\pi$
$702$	0	0
$703$	4.66599	0.175981
$704$	0	0
$705$	10.6466	0.400975
$706$	0	0
$707$	9.97414	0.375116
$708$	0	0
$709$	−8.52728	−0.320249	−0.160124	−	0.987097i	$-0.551189\pi$
$709$	−8.52728	−0.320249	−0.160124	+	0.987097i	$0.551189\pi$
$710$	0	0
$711$	−6.90977	−0.259137
$712$	0	0
$713$	15.4662	0.579215
$714$	0	0
$715$	1.82840	0.0683784
$716$	0	0
$717$	−5.85818	−0.218778
$718$	0	0
$719$	50.7335	1.89204	0.946021	−	0.324106i	$-0.105064\pi$
$719$	50.7335	1.89204	0.946021	+	0.324106i	$0.105064\pi$
$720$	0	0
$721$	3.91117	0.145660
$722$	0	0
$723$	−0.949994	−0.0353306
$724$	0	0
$725$	1.59597	0.0592727
$726$	0	0
$727$	10.8557	0.402617	0.201309	−	0.979528i	$-0.435481\pi$
$727$	10.8557	0.402617	0.201309	+	0.979528i	$0.435481\pi$
$728$	0	0
$729$	−4.22117	−0.156339
$730$	0	0
$731$	−4.69394	−0.173612
$732$	0	0
$733$	−1.10215	−0.0407090	−0.0203545	−	0.999793i	$-0.506479\pi$
$733$	−1.10215	−0.0407090	−0.0203545	+	0.999793i	$0.506479\pi$
$734$	0	0
$735$	8.13751	0.300157
$736$	0	0
$737$	−11.2005	−0.412576
$738$	0	0
$739$	−9.48120	−0.348772	−0.174386	−	0.984677i	$-0.555794\pi$
$739$	−9.48120	−0.348772	−0.174386	+	0.984677i	$0.555794\pi$
$740$	0	0
$741$	−0.709110	−0.0260498
$742$	0	0
$743$	46.7686	1.71577	0.857887	−	0.513838i	$-0.171777\pi$
$743$	46.7686	1.71577	0.857887	+	0.513838i	$0.171777\pi$
$744$	0	0
$745$	0.411042	0.0150594
$746$	0	0
$747$	35.7037	1.30633
$748$	0	0
$749$	14.8796	0.543688
$750$	0	0
$751$	12.2108	0.445579	0.222789	−	0.974867i	$-0.428484\pi$
$751$	12.2108	0.445579	0.222789	+	0.974867i	$0.428484\pi$
$752$	0	0
$753$	−12.7822	−0.465808
$754$	0	0
$755$	2.64786	0.0963653
$756$	0	0
$757$	−34.7869	−1.26435	−0.632176	−	0.774825i	$-0.717838\pi$
$757$	−34.7869	−1.26435	−0.632176	+	0.774825i	$0.717838\pi$
$758$	0	0
$759$	1.94269	0.0705152
$760$	0	0
$761$	29.6670	1.07543	0.537714	−	0.843127i	$-0.319288\pi$
$761$	29.6670	1.07543	0.537714	+	0.843127i	$0.319288\pi$
$762$	0	0
$763$	−7.42862	−0.268934
$764$	0	0
$765$	28.0328	1.01353
$766$	0	0
$767$	−5.45055	−0.196808
$768$	0	0
$769$	41.1766	1.48486	0.742432	−	0.669921i	$-0.233672\pi$
$769$	41.1766	1.48486	0.742432	+	0.669921i	$0.233672\pi$
$770$	0	0
$771$	3.34545	0.120483
$772$	0	0
$773$	14.5631	0.523799	0.261900	−	0.965095i	$-0.415651\pi$
$773$	14.5631	0.523799	0.261900	+	0.965095i	$0.415651\pi$
$774$	0	0
$775$	−8.76972	−0.315018
$776$	0	0
$777$	−2.60736	−0.0935385
$778$	0	0
$779$	−10.3088	−0.369353
$780$	0	0
$781$	1.40288	0.0501989
$782$	0	0
$783$	3.84101	0.137266
$784$	0	0
$785$	9.54773	0.340773
$786$	0	0
$787$	−16.4343	−0.585820	−0.292910	−	0.956140i	$-0.594624\pi$
$787$	−16.4343	−0.585820	−0.292910	+	0.956140i	$0.594624\pi$
$788$	0	0
$789$	−10.9202	−0.388768
$790$	0	0
$791$	6.15916	0.218994
$792$	0	0
$793$	−1.74531	−0.0619777
$794$	0	0
$795$	−10.2989	−0.365263
$796$	0	0
$797$	0.842835	0.0298548	0.0149274	−	0.999889i	$-0.495248\pi$
$797$	0.842835	0.0298548	0.0149274	+	0.999889i	$0.495248\pi$
$798$	0	0
$799$	−50.0557	−1.77084
$800$	0	0
$801$	−34.7269	−1.22701
$802$	0	0
$803$	9.41385	0.332208
$804$	0	0
$805$	4.24208	0.149514
$806$	0	0
$807$	20.5787	0.724404
$808$	0	0
$809$	−45.3490	−1.59438	−0.797192	−	0.603725i	$-0.793682\pi$
$809$	−45.3490	−1.59438	−0.797192	+	0.603725i	$0.793682\pi$
$810$	0	0
$811$	−32.0855	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$811$	−32.0855	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$812$	0	0
$813$	−7.28715	−0.255572
$814$	0	0
$815$	13.4643	0.471632
$816$	0	0
$817$	−0.791045	−0.0276752
$818$	0	0
$819$	−2.05440	−0.0717864
$820$	0	0
$821$	24.9935	0.872277	0.436139	−	0.899879i	$-0.356346\pi$
$821$	24.9935	0.872277	0.436139	+	0.899879i	$0.356346\pi$
$822$	0	0
$823$	−41.7049	−1.45374	−0.726871	−	0.686774i	$-0.759026\pi$
$823$	−41.7049	−1.45374	−0.726871	+	0.686774i	$0.759026\pi$
$824$	0	0
$825$	−1.10155	−0.0383511
$826$	0	0
$827$	−8.23780	−0.286456	−0.143228	−	0.989690i	$-0.545748\pi$
$827$	−8.23780	−0.286456	−0.143228	+	0.989690i	$0.545748\pi$
$828$	0	0
$829$	−35.0797	−1.21837	−0.609183	−	0.793029i	$-0.708503\pi$
$829$	−35.0797	−1.21837	−0.609183	+	0.793029i	$0.708503\pi$
$830$	0	0
$831$	−15.3627	−0.532925
$832$	0	0
$833$	−38.2590	−1.32560
$834$	0	0
$835$	−4.48870	−0.155338
$836$	0	0
$837$	−21.1061	−0.729532
$838$	0	0
$839$	43.3031	1.49499	0.747495	−	0.664268i	$-0.231257\pi$
$839$	43.3031	1.49499	0.747495	+	0.664268i	$0.231257\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−4.00308	−0.137873
$844$	0	0
$845$	−1.84500	−0.0634700
$846$	0	0
$847$	8.18345	0.281187
$848$	0	0
$849$	16.8394	0.577928
$850$	0	0
$851$	12.8991	0.442174
$852$	0	0
$853$	−0.112994	−0.00386884	−0.00193442	−	0.999998i	$-0.500616\pi$
$853$	−0.112994	−0.00386884	−0.00193442	+	0.999998i	$0.500616\pi$
$854$	0	0
$855$	4.72421	0.161565
$856$	0	0
$857$	47.4178	1.61976	0.809881	−	0.586595i	$-0.199532\pi$
$857$	47.4178	1.61976	0.809881	+	0.586595i	$0.199532\pi$
$858$	0	0
$859$	−18.6771	−0.637253	−0.318626	−	0.947880i	$-0.603222\pi$
$859$	−18.6771	−0.637253	−0.318626	+	0.947880i	$0.603222\pi$
$860$	0	0
$861$	5.76059	0.196321
$862$	0	0
$863$	−1.97754	−0.0673161	−0.0336580	−	0.999433i	$-0.510716\pi$
$863$	−1.97754	−0.0673161	−0.0336580	+	0.999433i	$0.510716\pi$
$864$	0	0
$865$	−17.9894	−0.611658
$866$	0	0
$867$	13.5810	0.461237
$868$	0	0
$869$	2.72279	0.0923643
$870$	0	0
$871$	11.3022	0.382960
$872$	0	0
$873$	−7.86622	−0.266231
$874$	0	0
$875$	−9.94111	−0.336071
$876$	0	0
$877$	17.4744	0.590070	0.295035	−	0.955486i	$-0.404669\pi$
$877$	17.4744	0.590070	0.295035	+	0.955486i	$0.404669\pi$
$878$	0	0
$879$	−3.83655	−0.129403
$880$	0	0
$881$	−45.0465	−1.51766	−0.758828	−	0.651292i	$-0.774228\pi$
$881$	−45.0465	−1.51766	−0.758828	+	0.651292i	$0.774228\pi$
$882$	0	0
$883$	−16.1535	−0.543607	−0.271803	−	0.962353i	$-0.587620\pi$
$883$	−16.1535	−0.543607	−0.271803	+	0.962353i	$0.587620\pi$
$884$	0	0
$885$	−7.00395	−0.235435
$886$	0	0
$887$	−21.6340	−0.726398	−0.363199	−	0.931712i	$-0.618315\pi$
$887$	−21.6340	−0.726398	−0.363199	+	0.931712i	$0.618315\pi$
$888$	0	0
$889$	−2.16958	−0.0727653
$890$	0	0
$891$	4.82579	0.161670
$892$	0	0
$893$	−8.43562	−0.282287
$894$	0	0
$895$	25.7406	0.860412
$896$	0	0
$897$	−1.96033	−0.0654534
$898$	0	0
$899$	−5.49493	−0.183266
$900$	0	0
$901$	48.4207	1.61313
$902$	0	0
$903$	0.442037	0.0147101
$904$	0	0
$905$	29.8925	0.993661
$906$	0	0
$907$	−0.319386	−0.0106051	−0.00530253	−	0.999986i	$-0.501688\pi$
$907$	−0.319386	−0.0106051	−0.00530253	+	0.999986i	$0.501688\pi$
$908$	0	0
$909$	30.7072	1.01849
$910$	0	0
$911$	2.60096	0.0861736	0.0430868	−	0.999071i	$-0.486281\pi$
$911$	2.60096	0.0861736	0.0430868	+	0.999071i	$0.486281\pi$
$912$	0	0
$913$	−14.0690	−0.465617
$914$	0	0
$915$	−2.24272	−0.0741419
$916$	0	0
$917$	3.70915	0.122487
$918$	0	0
$919$	−12.6238	−0.416421	−0.208211	−	0.978084i	$-0.566764\pi$
$919$	−12.6238	−0.416421	−0.208211	+	0.978084i	$0.566764\pi$
$920$	0	0
$921$	0.328348	0.0108194
$922$	0	0
$923$	−1.41561	−0.0465955
$924$	0	0
$925$	−7.31408	−0.240485
$926$	0	0
$927$	12.0413	0.395487
$928$	0	0
$929$	39.3067	1.28961	0.644806	−	0.764346i	$-0.276938\pi$
$929$	39.3067	1.28961	0.644806	+	0.764346i	$0.276938\pi$
$930$	0	0
$931$	−6.44759	−0.211311
$932$	0	0
$933$	−13.4336	−0.439796
$934$	0	0
$935$	−11.0463	−0.361252
$936$	0	0
$937$	−48.5071	−1.58466	−0.792330	−	0.610093i	$-0.791132\pi$
$937$	−48.5071	−1.58466	−0.792330	+	0.610093i	$0.791132\pi$
$938$	0	0
$939$	1.62843	0.0531419
$940$	0	0
$941$	−44.5758	−1.45313	−0.726565	−	0.687098i	$-0.758884\pi$
$941$	−44.5758	−1.45313	−0.726565	+	0.687098i	$0.758884\pi$
$942$	0	0
$943$	−28.4987	−0.928044
$944$	0	0
$945$	−5.78897	−0.188315
$946$	0	0
$947$	−43.5071	−1.41379	−0.706895	−	0.707319i	$-0.749905\pi$
$947$	−43.5071	−1.41379	−0.706895	+	0.707319i	$0.749905\pi$
$948$	0	0
$949$	−9.49931	−0.308361
$950$	0	0
$951$	15.6267	0.506730
$952$	0	0
$953$	3.45716	0.111989	0.0559943	−	0.998431i	$-0.482167\pi$
$953$	3.45716	0.111989	0.0559943	+	0.998431i	$0.482167\pi$
$954$	0	0
$955$	10.8353	0.350623
$956$	0	0
$957$	−0.690209	−0.0223113
$958$	0	0
$959$	8.86753	0.286347
$960$	0	0
$961$	−0.805783	−0.0259930
$962$	0	0
$963$	45.8095	1.47619
$964$	0	0
$965$	5.45505	0.175604
$966$	0	0
$967$	−51.1168	−1.64381	−0.821903	−	0.569628i	$-0.807087\pi$
$967$	−51.1168	−1.64381	−0.821903	+	0.569628i	$0.807087\pi$
$968$	0	0
$969$	4.28409	0.137625
$970$	0	0
$971$	8.27101	0.265429	0.132715	−	0.991154i	$-0.457631\pi$
$971$	8.27101	0.265429	0.132715	+	0.991154i	$0.457631\pi$
$972$	0	0
$973$	16.8396	0.539853
$974$	0	0
$975$	1.11155	0.0355981
$976$	0	0
$977$	29.7178	0.950756	0.475378	−	0.879782i	$-0.342311\pi$
$977$	29.7178	0.950756	0.475378	+	0.879782i	$0.342311\pi$
$978$	0	0
$979$	13.6841	0.437346
$980$	0	0
$981$	−22.8704	−0.730195
$982$	0	0
$983$	5.82570	0.185811	0.0929055	−	0.995675i	$-0.470385\pi$
$983$	5.82570	0.185811	0.0929055	+	0.995675i	$0.470385\pi$
$984$	0	0
$985$	−17.2447	−0.549462
$986$	0	0
$987$	4.71383	0.150043
$988$	0	0
$989$	−2.18683	−0.0695372
$990$	0	0
$991$	59.7496	1.89801	0.949004	−	0.315265i	$-0.102093\pi$
$991$	59.7496	1.89801	0.949004	+	0.315265i	$0.102093\pi$
$992$	0	0
$993$	−4.58796	−0.145595
$994$	0	0
$995$	17.3606	0.550368
$996$	0	0
$997$	−61.9927	−1.96333	−0.981665	−	0.190615i	$-0.938952\pi$
$997$	−61.9927	−1.96333	−0.981665	+	0.190615i	$0.938952\pi$
$998$	0	0
$999$	−17.6028	−0.556927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.z.1.7		10
4.3	odd	2		3016.2.a.h.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.h.1.4	✓	10	4.3	odd	2
6032.2.a.z.1.7		10	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)

\(f(q)\)	\(=\)	\(q+0.696475 q^{3} -1.84500 q^{5} -0.816882 q^{7} -2.51492 q^{9} +O(q^{10})\)	\(q+0.696475 q^{3} -1.84500 q^{5} -0.816882 q^{7} -2.51492 q^{9} +0.991003 q^{11} -1.00000 q^{13} -1.28500 q^{15} +6.04150 q^{17} +1.01814 q^{19} -0.568938 q^{21} +2.81464 q^{23} -1.59597 q^{25} -3.84101 q^{27} -1.00000 q^{29} +5.49493 q^{31} +0.690209 q^{33} +1.50715 q^{35} +4.58285 q^{37} -0.696475 q^{39} -10.1252 q^{41} -0.776950 q^{43} +4.64004 q^{45} -8.28531 q^{47} -6.33270 q^{49} +4.20775 q^{51} +8.01468 q^{53} -1.82840 q^{55} +0.709110 q^{57} +5.45055 q^{59} +1.74531 q^{61} +2.05440 q^{63} +1.84500 q^{65} -11.3022 q^{67} +1.96033 q^{69} +1.41561 q^{71} +9.49931 q^{73} -1.11155 q^{75} -0.809533 q^{77} +2.74751 q^{79} +4.86960 q^{81} -14.1967 q^{83} -11.1466 q^{85} -0.696475 q^{87} +13.8083 q^{89} +0.816882 q^{91} +3.82708 q^{93} -1.87847 q^{95} +3.12782 q^{97} -2.49230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9}+O(q^{10}) \) 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9	\( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9} - 14 q^{11} - 10 q^{13} - 7 q^{15} + 5 q^{17} - 11 q^{19} - 7 q^{23} + 10 q^{25} - 21 q^{27} - 10 q^{29} - 5 q^{31} + 5 q^{33} - 11 q^{35} + 8 q^{37} + 3 q^{39} + 14 q^{41} - 35 q^{43} + 7 q^{45} - 7 q^{49} - 20 q^{51} - 11 q^{53} - 8 q^{55} + 4 q^{57} - 23 q^{59} - 8 q^{61} - 43 q^{63} - 4 q^{65} - 27 q^{67} + 10 q^{69} - 3 q^{71} + 7 q^{73} - 23 q^{75} + 2 q^{77} - 9 q^{79} - 6 q^{81} - 48 q^{83} - 6 q^{85} + 3 q^{87} + 20 q^{89} - 3 q^{91} - 11 q^{93} - 11 q^{95} + q^{97} - 54 q^{99}+O(q^{100}) \) 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9 - 14 * q^11 - 10 * q^13 - 7 * q^15 + 5 * q^17 - 11 * q^19 - 7 * q^23 + 10 * q^25 - 21 * q^27 - 10 * q^29 - 5 * q^31 + 5 * q^33 - 11 * q^35 + 8 * q^37 + 3 * q^39 + 14 * q^41 - 35 * q^43 + 7 * q^45 - 7 * q^49 - 20 * q^51 - 11 * q^53 - 8 * q^55 + 4 * q^57 - 23 * q^59 - 8 * q^61 - 43 * q^63 - 4 * q^65 - 27 * q^67 + 10 * q^69 - 3 * q^71 + 7 * q^73 - 23 * q^75 + 2 * q^77 - 9 * q^79 - 6 * q^81 - 48 * q^83 - 6 * q^85 + 3 * q^87 + 20 * q^89 - 3 * q^91 - 11 * q^93 - 11 * q^95 + q^97 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more