LMFDB - Embedded newform 8280.2.a.bv.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 18x^{5} + 46x^{4} + 60x^{3} - 76x^{2} - 51x - 7 $ x^7 - 3x^6 - 18x^5 + 46x^4 + 60x^3 - 76x^2 - 51x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.330805$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.34984 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +2.34984 q^{7} +0.333733 q^{11} +3.37929 q^{13} +7.81531 q^{17} +1.20992 q^{19} -1.00000 q^{23} +1.00000 q^{25} +1.35450 q^{29} -10.2907 q^{31} -2.34984 q^{35} +2.83910 q^{37} +7.94371 q^{41} -7.92175 q^{43} +11.7314 q^{47} -1.47824 q^{49} +7.92062 q^{53} -0.333733 q^{55} +13.2728 q^{59} +2.88487 q^{61} -3.37929 q^{65} +7.70597 q^{67} -15.1945 q^{71} -7.02239 q^{73} +0.784221 q^{77} +0.325047 q^{79} +10.4828 q^{83} -7.81531 q^{85} +8.05475 q^{89} +7.94081 q^{91} -1.20992 q^{95} -1.00919 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{5} - 6 q^{7}+O(q^{10}) $ 7 * q - 7 * q^5 - 6 * q^7	$ 7 q - 7 q^{5} - 6 q^{7} + 2 q^{11} - 6 q^{13} + 4 q^{17} - 8 q^{19} - 7 q^{23} + 7 q^{25} - 2 q^{29} - 8 q^{31} + 6 q^{35} - 16 q^{37} - 2 q^{41} - 10 q^{43} + 8 q^{47} + 19 q^{49} + 10 q^{53} - 2 q^{55} + 24 q^{59} + 8 q^{61} + 6 q^{65} - 20 q^{67} + 8 q^{71} + 2 q^{73} + 12 q^{77} - 2 q^{79} + 22 q^{83} - 4 q^{85} + 16 q^{89} - 20 q^{91} + 8 q^{95} + 4 q^{97}+O(q^{100}) $ 7 * q - 7 * q^5 - 6 * q^7 + 2 * q^11 - 6 * q^13 + 4 * q^17 - 8 * q^19 - 7 * q^23 + 7 * q^25 - 2 * q^29 - 8 * q^31 + 6 * q^35 - 16 * q^37 - 2 * q^41 - 10 * q^43 + 8 * q^47 + 19 * q^49 + 10 * q^53 - 2 * q^55 + 24 * q^59 + 8 * q^61 + 6 * q^65 - 20 * q^67 + 8 * q^71 + 2 * q^73 + 12 * q^77 - 2 * q^79 + 22 * q^83 - 4 * q^85 + 16 * q^89 - 20 * q^91 + 8 * q^95 + 4 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.34984	0.888157	0.444078	−	0.895988i	$-0.353531\pi$
$7$	2.34984	0.888157	0.444078	+	0.895988i	$0.353531\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.333733	0.100624	0.0503122	−	0.998734i	$-0.483978\pi$
$11$	0.333733	0.100624	0.0503122	+	0.998734i	$0.483978\pi$
$12$	0	0
$13$	3.37929	0.937247	0.468624	−	0.883398i	$-0.344750\pi$
$13$	3.37929	0.937247	0.468624	+	0.883398i	$0.344750\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.81531	1.89549	0.947746	−	0.319026i	$-0.103356\pi$
$17$	7.81531	1.89549	0.947746	+	0.319026i	$0.103356\pi$
$18$	0	0
$19$	1.20992	0.277575	0.138788	−	0.990322i	$-0.455679\pi$
$19$	1.20992	0.277575	0.138788	+	0.990322i	$0.455679\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.35450	0.251524	0.125762	−	0.992060i	$-0.459862\pi$
$29$	1.35450	0.251524	0.125762	+	0.992060i	$0.459862\pi$
$30$	0	0
$31$	−10.2907	−1.84827	−0.924134	−	0.382070i	$-0.875211\pi$
$31$	−10.2907	−1.84827	−0.924134	+	0.382070i	$0.875211\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.34984	−0.397196
$36$	0	0
$37$	2.83910	0.466745	0.233372	−	0.972387i	$-0.425024\pi$
$37$	2.83910	0.466745	0.233372	+	0.972387i	$0.425024\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.94371	1.24060	0.620300	−	0.784365i	$-0.287011\pi$
$41$	7.94371	1.24060	0.620300	+	0.784365i	$0.287011\pi$
$42$	0	0
$43$	−7.92175	−1.20806	−0.604028	−	0.796963i	$-0.706438\pi$
$43$	−7.92175	−1.20806	−0.604028	+	0.796963i	$0.706438\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.7314	1.71120	0.855600	−	0.517638i	$-0.173189\pi$
$47$	11.7314	1.71120	0.855600	+	0.517638i	$0.173189\pi$
$48$	0	0
$49$	−1.47824	−0.211178
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.92062	1.08798	0.543990	−	0.839092i	$-0.316913\pi$
$53$	7.92062	1.08798	0.543990	+	0.839092i	$0.316913\pi$
$54$	0	0
$55$	−0.333733	−0.0450006
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.2728	1.72797	0.863985	−	0.503517i	$-0.167961\pi$
$59$	13.2728	1.72797	0.863985	+	0.503517i	$0.167961\pi$
$60$	0	0
$61$	2.88487	0.369370	0.184685	−	0.982798i	$-0.440873\pi$
$61$	2.88487	0.369370	0.184685	+	0.982798i	$0.440873\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.37929	−0.419150
$66$	0	0
$67$	7.70597	0.941434	0.470717	−	0.882284i	$-0.343995\pi$
$67$	7.70597	0.941434	0.470717	+	0.882284i	$0.343995\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.1945	−1.80326	−0.901630	−	0.432508i	$-0.857629\pi$
$71$	−15.1945	−1.80326	−0.901630	+	0.432508i	$0.857629\pi$
$72$	0	0
$73$	−7.02239	−0.821909	−0.410955	−	0.911656i	$-0.634805\pi$
$73$	−7.02239	−0.821909	−0.410955	+	0.911656i	$0.634805\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.784221	0.0893703
$78$	0	0
$79$	0.325047	0.0365707	0.0182853	−	0.999833i	$-0.494179\pi$
$79$	0.325047	0.0365707	0.0182853	+	0.999833i	$0.494179\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.4828	1.15063	0.575317	−	0.817930i	$-0.304879\pi$
$83$	10.4828	1.15063	0.575317	+	0.817930i	$0.304879\pi$
$84$	0	0
$85$	−7.81531	−0.847690
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.05475	0.853802	0.426901	−	0.904298i	$-0.359605\pi$
$89$	8.05475	0.853802	0.426901	+	0.904298i	$0.359605\pi$
$90$	0	0
$91$	7.94081	0.832423
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.20992	−0.124135
$96$	0	0
$97$	−1.00919	−0.102468	−0.0512341	−	0.998687i	$-0.516315\pi$
$97$	−1.00919	−0.102468	−0.0512341	+	0.998687i	$0.516315\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.6145	−1.15569	−0.577844	−	0.816147i	$-0.696106\pi$
$101$	−11.6145	−1.15569	−0.577844	+	0.816147i	$0.696106\pi$
$102$	0	0
$103$	12.8961	1.27069	0.635346	−	0.772228i	$-0.280858\pi$
$103$	12.8961	1.27069	0.635346	+	0.772228i	$0.280858\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.2078	−1.47019	−0.735097	−	0.677962i	$-0.762863\pi$
$107$	−15.2078	−1.47019	−0.735097	+	0.677962i	$0.762863\pi$
$108$	0	0
$109$	−12.0680	−1.15590	−0.577950	−	0.816072i	$-0.696147\pi$
$109$	−12.0680	−1.15590	−0.577950	+	0.816072i	$0.696147\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.15370	−0.108531	−0.0542657	−	0.998527i	$-0.517282\pi$
$113$	−1.15370	−0.108531	−0.0542657	+	0.998527i	$0.517282\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	18.3648	1.68349
$120$	0	0
$121$	−10.8886	−0.989875
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.77157	−0.689615	−0.344808	−	0.938673i	$-0.612056\pi$
$127$	−7.77157	−0.689615	−0.344808	+	0.938673i	$0.612056\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.59841	−0.139654	−0.0698268	−	0.997559i	$-0.522245\pi$
$131$	−1.59841	−0.139654	−0.0698268	+	0.997559i	$0.522245\pi$
$132$	0	0
$133$	2.84312	0.246530
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.98666	0.426039	0.213019	−	0.977048i	$-0.431670\pi$
$137$	4.98666	0.426039	0.213019	+	0.977048i	$0.431670\pi$
$138$	0	0
$139$	−11.3677	−0.964192	−0.482096	−	0.876118i	$-0.660124\pi$
$139$	−11.3677	−0.964192	−0.482096	+	0.876118i	$0.660124\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.12778	0.0943100
$144$	0	0
$145$	−1.35450	−0.112485
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.4841	−1.51428	−0.757139	−	0.653254i	$-0.773404\pi$
$149$	−18.4841	−1.51428	−0.757139	+	0.653254i	$0.773404\pi$
$150$	0	0
$151$	−14.6892	−1.19539	−0.597694	−	0.801724i	$-0.703916\pi$
$151$	−14.6892	−1.19539	−0.597694	+	0.801724i	$0.703916\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.2907	0.826570
$156$	0	0
$157$	7.70994	0.615320	0.307660	−	0.951496i	$-0.400454\pi$
$157$	7.70994	0.615320	0.307660	+	0.951496i	$0.400454\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.34984	−0.185193
$162$	0	0
$163$	−16.2704	−1.27440	−0.637198	−	0.770700i	$-0.719907\pi$
$163$	−16.2704	−1.27440	−0.637198	+	0.770700i	$0.719907\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.6828	1.21357	0.606786	−	0.794865i	$-0.292458\pi$
$167$	15.6828	1.21357	0.606786	+	0.794865i	$0.292458\pi$
$168$	0	0
$169$	−1.58037	−0.121567
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.7815	−1.12382	−0.561908	−	0.827200i	$-0.689932\pi$
$173$	−14.7815	−1.12382	−0.561908	+	0.827200i	$0.689932\pi$
$174$	0	0
$175$	2.34984	0.177631
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.66161	0.647399	0.323700	−	0.946160i	$-0.395073\pi$
$179$	8.66161	0.647399	0.323700	+	0.946160i	$0.395073\pi$
$180$	0	0
$181$	5.53481	0.411400	0.205700	−	0.978615i	$-0.434053\pi$
$181$	5.53481	0.411400	0.205700	+	0.978615i	$0.434053\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.83910	−0.208734
$186$	0	0
$187$	2.60823	0.190733
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.81157	0.131081	0.0655403	−	0.997850i	$-0.479123\pi$
$191$	1.81157	0.131081	0.0655403	+	0.997850i	$0.479123\pi$
$192$	0	0
$193$	26.9261	1.93819	0.969093	−	0.246695i	$-0.0793447\pi$
$193$	26.9261	1.93819	0.969093	+	0.246695i	$0.0793447\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.60772	0.542028	0.271014	−	0.962575i	$-0.412641\pi$
$197$	7.60772	0.542028	0.271014	+	0.962575i	$0.412641\pi$
$198$	0	0
$199$	−7.99167	−0.566515	−0.283257	−	0.959044i	$-0.591415\pi$
$199$	−7.99167	−0.566515	−0.283257	+	0.959044i	$0.591415\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.18286	0.223393
$204$	0	0
$205$	−7.94371	−0.554813
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.403791	0.0279308
$210$	0	0
$211$	4.70950	0.324215	0.162108	−	0.986773i	$-0.448171\pi$
$211$	4.70950	0.324215	0.162108	+	0.986773i	$0.448171\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.92175	0.540259
$216$	0	0
$217$	−24.1816	−1.64155
$218$	0	0
$219$	0	0
$220$	0	0
$221$	26.4102	1.77654
$222$	0	0
$223$	25.3766	1.69934	0.849670	−	0.527315i	$-0.176801\pi$
$223$	25.3766	1.69934	0.849670	+	0.527315i	$0.176801\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.4820	−0.695714	−0.347857	−	0.937548i	$-0.613091\pi$
$227$	−10.4820	−0.695714	−0.347857	+	0.937548i	$0.613091\pi$
$228$	0	0
$229$	29.3540	1.93977	0.969884	−	0.243566i	$-0.0783173\pi$
$229$	29.3540	1.93977	0.969884	+	0.243566i	$0.0783173\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.9472	1.37229	0.686147	−	0.727463i	$-0.259300\pi$
$233$	20.9472	1.37229	0.686147	+	0.727463i	$0.259300\pi$
$234$	0	0
$235$	−11.7314	−0.765272
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−0.00679442	−0.000439494 0	−0.000219747	−	1.00000i	$-0.500070\pi$
$239$	−0.00679442	−0.000439494 0	−0.000219747		1.00000i	$0.500070\pi$
$240$	0	0
$241$	28.2482	1.81963	0.909814	−	0.415017i	$-0.136224\pi$
$241$	28.2482	1.81963	0.909814	+	0.415017i	$0.136224\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.47824	0.0944415
$246$	0	0
$247$	4.08868	0.260156
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.85470	0.622023	0.311011	−	0.950406i	$-0.399332\pi$
$251$	9.85470	0.622023	0.311011	+	0.950406i	$0.399332\pi$
$252$	0	0
$253$	−0.333733	−0.0209816
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.06873	0.440935	0.220467	−	0.975394i	$-0.429242\pi$
$257$	7.06873	0.440935	0.220467	+	0.975394i	$0.429242\pi$
$258$	0	0
$259$	6.67143	0.414542
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.9113	0.734482	0.367241	−	0.930126i	$-0.380302\pi$
$263$	11.9113	0.734482	0.367241	+	0.930126i	$0.380302\pi$
$264$	0	0
$265$	−7.92062	−0.486560
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−30.5465	−1.86245	−0.931227	−	0.364440i	$-0.881261\pi$
$269$	−30.5465	−1.86245	−0.931227	+	0.364440i	$0.881261\pi$
$270$	0	0
$271$	12.1356	0.737183	0.368591	−	0.929592i	$-0.379840\pi$
$271$	12.1356	0.737183	0.368591	+	0.929592i	$0.379840\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.333733	0.0201249
$276$	0	0
$277$	13.7470	0.825974	0.412987	−	0.910737i	$-0.364485\pi$
$277$	13.7470	0.825974	0.412987	+	0.910737i	$0.364485\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.3351	1.27275	0.636374	−	0.771381i	$-0.280434\pi$
$281$	21.3351	1.27275	0.636374	+	0.771381i	$0.280434\pi$
$282$	0	0
$283$	16.1882	0.962287	0.481144	−	0.876642i	$-0.340222\pi$
$283$	16.1882	0.962287	0.481144	+	0.876642i	$0.340222\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	18.6665	1.10185
$288$	0	0
$289$	44.0791	2.59289
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−25.4006	−1.48392	−0.741959	−	0.670445i	$-0.766103\pi$
$293$	−25.4006	−1.48392	−0.741959	+	0.670445i	$0.766103\pi$
$294$	0	0
$295$	−13.2728	−0.772772
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.37929	−0.195430
$300$	0	0
$301$	−18.6149	−1.07294
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.88487	−0.165187
$306$	0	0
$307$	−17.8342	−1.01785	−0.508927	−	0.860810i	$-0.669958\pi$
$307$	−17.8342	−1.01785	−0.508927	+	0.860810i	$0.669958\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.21196	−0.465658	−0.232829	−	0.972518i	$-0.574798\pi$
$311$	−8.21196	−0.465658	−0.232829	+	0.972518i	$0.574798\pi$
$312$	0	0
$313$	26.6902	1.50862	0.754308	−	0.656520i	$-0.227972\pi$
$313$	26.6902	1.50862	0.754308	+	0.656520i	$0.227972\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.2587	1.02551	0.512756	−	0.858534i	$-0.328625\pi$
$317$	18.2587	1.02551	0.512756	+	0.858534i	$0.328625\pi$
$318$	0	0
$319$	0.452042	0.0253095
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.45592	0.526141
$324$	0	0
$325$	3.37929	0.187449
$326$	0	0
$327$	0	0
$328$	0	0
$329$	27.5669	1.51981
$330$	0	0
$331$	−17.2539	−0.948359	−0.474179	−	0.880428i	$-0.657255\pi$
$331$	−17.2539	−0.948359	−0.474179	+	0.880428i	$0.657255\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.70597	−0.421022
$336$	0	0
$337$	−34.1028	−1.85770	−0.928848	−	0.370460i	$-0.879200\pi$
$337$	−34.1028	−1.85770	−0.928848	+	0.370460i	$0.879200\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.43436	−0.185981
$342$	0	0
$343$	−19.9225	−1.07572
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.63528	−0.409883	−0.204942	−	0.978774i	$-0.565700\pi$
$347$	−7.63528	−0.409883	−0.204942	+	0.978774i	$0.565700\pi$
$348$	0	0
$349$	30.9799	1.65832	0.829158	−	0.559015i	$-0.188820\pi$
$349$	30.9799	1.65832	0.829158	+	0.559015i	$0.188820\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.93004	0.368849	0.184424	−	0.982847i	$-0.440958\pi$
$353$	6.93004	0.368849	0.184424	+	0.982847i	$0.440958\pi$
$354$	0	0
$355$	15.1945	0.806442
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.20979	−0.486074	−0.243037	−	0.970017i	$-0.578144\pi$
$359$	−9.20979	−0.486074	−0.243037	+	0.970017i	$0.578144\pi$
$360$	0	0
$361$	−17.5361	−0.922952
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.02239	0.367569
$366$	0	0
$367$	17.8092	0.929631	0.464816	−	0.885408i	$-0.346121\pi$
$367$	17.8092	0.929631	0.464816	+	0.885408i	$0.346121\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18.6122	0.966297
$372$	0	0
$373$	−27.0055	−1.39829	−0.699145	−	0.714980i	$-0.746436\pi$
$373$	−27.0055	−1.39829	−0.699145	+	0.714980i	$0.746436\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.57725	0.235740
$378$	0	0
$379$	−29.9380	−1.53781	−0.768907	−	0.639361i	$-0.779199\pi$
$379$	−29.9380	−1.53781	−0.768907	+	0.639361i	$0.779199\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.61303	−0.0824221	−0.0412110	−	0.999150i	$-0.513122\pi$
$383$	−1.61303	−0.0824221	−0.0412110	+	0.999150i	$0.513122\pi$
$384$	0	0
$385$	−0.784221	−0.0399676
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.1507	0.768172	0.384086	−	0.923297i	$-0.374517\pi$
$389$	15.1507	0.768172	0.384086	+	0.923297i	$0.374517\pi$
$390$	0	0
$391$	−7.81531	−0.395237
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.325047	−0.0163549
$396$	0	0
$397$	−5.69395	−0.285771	−0.142885	−	0.989739i	$-0.545638\pi$
$397$	−5.69395	−0.285771	−0.142885	+	0.989739i	$0.545638\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.72660	−0.435786	−0.217893	−	0.975973i	$-0.569918\pi$
$401$	−8.72660	−0.435786	−0.217893	+	0.975973i	$0.569918\pi$
$402$	0	0
$403$	−34.7754	−1.73228
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.947501	0.0469659
$408$	0	0
$409$	20.0768	0.992733	0.496367	−	0.868113i	$-0.334667\pi$
$409$	20.0768	0.992733	0.496367	+	0.868113i	$0.334667\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	31.1890	1.53471
$414$	0	0
$415$	−10.4828	−0.514579
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.8662	0.970527	0.485263	−	0.874368i	$-0.338724\pi$
$419$	19.8662	0.970527	0.485263	+	0.874368i	$0.338724\pi$
$420$	0	0
$421$	−20.3892	−0.993708	−0.496854	−	0.867834i	$-0.665512\pi$
$421$	−20.3892	−0.993708	−0.496854	+	0.867834i	$0.665512\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.81531	0.379098
$426$	0	0
$427$	6.77900	0.328059
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.44768	−0.117901	−0.0589503	−	0.998261i	$-0.518775\pi$
$431$	−2.44768	−0.117901	−0.0589503	+	0.998261i	$0.518775\pi$
$432$	0	0
$433$	26.6576	1.28108	0.640542	−	0.767923i	$-0.278710\pi$
$433$	26.6576	1.28108	0.640542	+	0.767923i	$0.278710\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.20992	−0.0578784
$438$	0	0
$439$	14.2544	0.680324	0.340162	−	0.940367i	$-0.389518\pi$
$439$	14.2544	0.680324	0.340162	+	0.940367i	$0.389518\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.350414	0.0166487	0.00832433	−	0.999965i	$-0.497350\pi$
$443$	0.350414	0.0166487	0.00832433	+	0.999965i	$0.497350\pi$
$444$	0	0
$445$	−8.05475	−0.381832
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.04679	−0.190980	−0.0954900	−	0.995430i	$-0.530442\pi$
$449$	−4.04679	−0.190980	−0.0954900	+	0.995430i	$0.530442\pi$
$450$	0	0
$451$	2.65108	0.124835
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.94081	−0.372271
$456$	0	0
$457$	−5.60780	−0.262322	−0.131161	−	0.991361i	$-0.541870\pi$
$457$	−5.60780	−0.262322	−0.131161	+	0.991361i	$0.541870\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.0088	−0.605880	−0.302940	−	0.953010i	$-0.597968\pi$
$461$	−13.0088	−0.605880	−0.302940	+	0.953010i	$0.597968\pi$
$462$	0	0
$463$	18.6108	0.864918	0.432459	−	0.901654i	$-0.357646\pi$
$463$	18.6108	0.864918	0.432459	+	0.901654i	$0.357646\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.4676	1.03968	0.519838	−	0.854265i	$-0.325992\pi$
$467$	22.4676	1.03968	0.519838	+	0.854265i	$0.325992\pi$
$468$	0	0
$469$	18.1078	0.836141
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.64375	−0.121560
$474$	0	0
$475$	1.20992	0.0555150
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.60035	0.301578	0.150789	−	0.988566i	$-0.451819\pi$
$479$	6.60035	0.301578	0.150789	+	0.988566i	$0.451819\pi$
$480$	0	0
$481$	9.59414	0.437455
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.00919	0.0458252
$486$	0	0
$487$	0.00974374	0.000441531 0	0.000220765	−	1.00000i	$-0.499930\pi$
$487$	0.00974374	0.000441531 0	0.000220765		1.00000i	$0.499930\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	38.1282	1.72070	0.860351	−	0.509702i	$-0.170244\pi$
$491$	38.1282	1.72070	0.860351	+	0.509702i	$0.170244\pi$
$492$	0	0
$493$	10.5858	0.476762
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−35.7048	−1.60158
$498$	0	0
$499$	17.1894	0.769505	0.384753	−	0.923020i	$-0.374287\pi$
$499$	17.1894	0.769505	0.384753	+	0.923020i	$0.374287\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.7526	0.524022	0.262011	−	0.965065i	$-0.415614\pi$
$503$	11.7526	0.524022	0.262011	+	0.965065i	$0.415614\pi$
$504$	0	0
$505$	11.6145	0.516839
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17.2543	0.764783	0.382391	−	0.924000i	$-0.375101\pi$
$509$	17.2543	0.764783	0.382391	+	0.924000i	$0.375101\pi$
$510$	0	0
$511$	−16.5015	−0.729984
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.8961	−0.568270
$516$	0	0
$517$	3.91516	0.172188
$518$	0	0
$519$	0	0
$520$	0	0
$521$	36.6710	1.60658	0.803292	−	0.595585i	$-0.203080\pi$
$521$	36.6710	1.60658	0.803292	+	0.595585i	$0.203080\pi$
$522$	0	0
$523$	−25.3419	−1.10813	−0.554063	−	0.832475i	$-0.686923\pi$
$523$	−25.3419	−1.10813	−0.554063	+	0.832475i	$0.686923\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−80.4252	−3.50338
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	26.8441	1.16275
$534$	0	0
$535$	15.2078	0.657491
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.493339	−0.0212496
$540$	0	0
$541$	31.2913	1.34532	0.672658	−	0.739953i	$-0.265152\pi$
$541$	31.2913	1.34532	0.672658	+	0.739953i	$0.265152\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.0680	0.516934
$546$	0	0
$547$	−39.8281	−1.70293	−0.851464	−	0.524412i	$-0.824285\pi$
$547$	−39.8281	−1.70293	−0.851464	+	0.524412i	$0.824285\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.63884	0.0698168
$552$	0	0
$553$	0.763810	0.0324805
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.5055	1.20782	0.603908	−	0.797054i	$-0.293610\pi$
$557$	28.5055	1.20782	0.603908	+	0.797054i	$0.293610\pi$
$558$	0	0
$559$	−26.7699	−1.13225
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.7082	−0.451296	−0.225648	−	0.974209i	$-0.572450\pi$
$563$	−10.7082	−0.451296	−0.225648	+	0.974209i	$0.572450\pi$
$564$	0	0
$565$	1.15370	0.0485367
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−39.8053	−1.66873	−0.834363	−	0.551216i	$-0.814164\pi$
$569$	−39.8053	−1.66873	−0.834363	+	0.551216i	$0.814164\pi$
$570$	0	0
$571$	−29.6536	−1.24097	−0.620483	−	0.784220i	$-0.713063\pi$
$571$	−29.6536	−1.24097	−0.620483	+	0.784220i	$0.713063\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	11.6699	0.485824	0.242912	−	0.970048i	$-0.421897\pi$
$577$	11.6699	0.485824	0.242912	+	0.970048i	$0.421897\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	24.6329	1.02194
$582$	0	0
$583$	2.64338	0.109477
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.01134	−0.330663	−0.165332	−	0.986238i	$-0.552869\pi$
$587$	−8.01134	−0.330663	−0.165332	+	0.986238i	$0.552869\pi$
$588$	0	0
$589$	−12.4510	−0.513033
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.83280	−0.280590	−0.140295	−	0.990110i	$-0.544805\pi$
$593$	−6.83280	−0.280590	−0.140295	+	0.990110i	$0.544805\pi$
$594$	0	0
$595$	−18.3648	−0.752881
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.5228	1.24713	0.623563	−	0.781773i	$-0.285684\pi$
$599$	30.5228	1.24713	0.623563	+	0.781773i	$0.285684\pi$
$600$	0	0
$601$	−22.2373	−0.907080	−0.453540	−	0.891236i	$-0.649839\pi$
$601$	−22.2373	−0.907080	−0.453540	+	0.891236i	$0.649839\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.8886	0.442685
$606$	0	0
$607$	−45.5255	−1.84782	−0.923912	−	0.382605i	$-0.875027\pi$
$607$	−45.5255	−1.84782	−0.923912	+	0.382605i	$0.875027\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	39.6438	1.60382
$612$	0	0
$613$	0.297344	0.0120096	0.00600481	−	0.999982i	$-0.498089\pi$
$613$	0.297344	0.0120096	0.00600481	+	0.999982i	$0.498089\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.4980	1.18755	0.593773	−	0.804633i	$-0.297638\pi$
$617$	29.4980	1.18755	0.593773	+	0.804633i	$0.297638\pi$
$618$	0	0
$619$	0.774004	0.0311098	0.0155549	−	0.999879i	$-0.495049\pi$
$619$	0.774004	0.0311098	0.0155549	+	0.999879i	$0.495049\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	18.9274	0.758310
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	22.1884	0.884710
$630$	0	0
$631$	26.4887	1.05450	0.527249	−	0.849711i	$-0.323224\pi$
$631$	26.4887	1.05450	0.527249	+	0.849711i	$0.323224\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.77157	0.308405
$636$	0	0
$637$	−4.99542	−0.197926
$638$	0	0
$639$	0	0
$640$	0	0
$641$	35.5092	1.40253	0.701264	−	0.712901i	$-0.252619\pi$
$641$	35.5092	1.40253	0.701264	+	0.712901i	$0.252619\pi$
$642$	0	0
$643$	−3.28511	−0.129552	−0.0647761	−	0.997900i	$-0.520633\pi$
$643$	−3.28511	−0.129552	−0.0647761	+	0.997900i	$0.520633\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.1929	−0.479352	−0.239676	−	0.970853i	$-0.577041\pi$
$647$	−12.1929	−0.479352	−0.239676	+	0.970853i	$0.577041\pi$
$648$	0	0
$649$	4.42957	0.173876
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.9398	−0.741171	−0.370585	−	0.928798i	$-0.620843\pi$
$653$	−18.9398	−0.741171	−0.370585	+	0.928798i	$0.620843\pi$
$654$	0	0
$655$	1.59841	0.0624550
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.9980	0.623192	0.311596	−	0.950215i	$-0.399136\pi$
$659$	15.9980	0.623192	0.311596	+	0.950215i	$0.399136\pi$
$660$	0	0
$661$	37.2112	1.44735	0.723675	−	0.690141i	$-0.242452\pi$
$661$	37.2112	1.44735	0.723675	+	0.690141i	$0.242452\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.84312	−0.110252
$666$	0	0
$667$	−1.35450	−0.0524464
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.962779	0.0371677
$672$	0	0
$673$	31.8952	1.22947	0.614736	−	0.788733i	$-0.289263\pi$
$673$	31.8952	1.22947	0.614736	+	0.788733i	$0.289263\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.4331	0.823743	0.411871	−	0.911242i	$-0.364875\pi$
$677$	21.4331	0.823743	0.411871	+	0.911242i	$0.364875\pi$
$678$	0	0
$679$	−2.37145	−0.0910078
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−30.5938	−1.17064	−0.585320	−	0.810802i	$-0.699031\pi$
$683$	−30.5938	−1.17064	−0.585320	+	0.810802i	$0.699031\pi$
$684$	0	0
$685$	−4.98666	−0.190530
$686$	0	0
$687$	0	0
$688$	0	0
$689$	26.7661	1.01971
$690$	0	0
$691$	−7.45708	−0.283681	−0.141840	−	0.989890i	$-0.545302\pi$
$691$	−7.45708	−0.283681	−0.141840	+	0.989890i	$0.545302\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.3677	0.431200
$696$	0	0
$697$	62.0826	2.35155
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−44.1300	−1.66677	−0.833383	−	0.552696i	$-0.813599\pi$
$701$	−44.1300	−1.66677	−0.833383	+	0.552696i	$0.813599\pi$
$702$	0	0
$703$	3.43508	0.129557
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−27.2923	−1.02643
$708$	0	0
$709$	19.5296	0.733451	0.366726	−	0.930329i	$-0.380479\pi$
$709$	19.5296	0.733451	0.366726	+	0.930329i	$0.380479\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.2907	0.385390
$714$	0	0
$715$	−1.12778	−0.0421767
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21.4663	−0.800557	−0.400278	−	0.916394i	$-0.631087\pi$
$719$	−21.4663	−0.800557	−0.400278	+	0.916394i	$0.631087\pi$
$720$	0	0
$721$	30.3038	1.12857
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.35450	0.0503048
$726$	0	0
$727$	5.33502	0.197865	0.0989325	−	0.995094i	$-0.468457\pi$
$727$	5.33502	0.197865	0.0989325	+	0.995094i	$0.468457\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−61.9109	−2.28986
$732$	0	0
$733$	−16.2267	−0.599349	−0.299674	−	0.954042i	$-0.596878\pi$
$733$	−16.2267	−0.599349	−0.299674	+	0.954042i	$0.596878\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.57174	0.0947313
$738$	0	0
$739$	47.0855	1.73207	0.866034	−	0.499985i	$-0.166661\pi$
$739$	47.0855	1.73207	0.866034	+	0.499985i	$0.166661\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.2973	1.62511	0.812555	−	0.582885i	$-0.198076\pi$
$743$	44.2973	1.62511	0.812555	+	0.582885i	$0.198076\pi$
$744$	0	0
$745$	18.4841	0.677206
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−35.7359	−1.30576
$750$	0	0
$751$	−36.6518	−1.33744	−0.668721	−	0.743513i	$-0.733158\pi$
$751$	−36.6518	−1.33744	−0.668721	+	0.743513i	$0.733158\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.6892	0.534594
$756$	0	0
$757$	3.88129	0.141068	0.0705339	−	0.997509i	$-0.477530\pi$
$757$	3.88129	0.141068	0.0705339	+	0.997509i	$0.477530\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−0.935040	−0.0338952	−0.0169476	−	0.999856i	$-0.505395\pi$
$761$	−0.935040	−0.0338952	−0.0169476	+	0.999856i	$0.505395\pi$
$762$	0	0
$763$	−28.3578	−1.02662
$764$	0	0
$765$	0	0
$766$	0	0
$767$	44.8527	1.61954
$768$	0	0
$769$	−4.23198	−0.152609	−0.0763046	−	0.997085i	$-0.524312\pi$
$769$	−4.23198	−0.152609	−0.0763046	+	0.997085i	$0.524312\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.17222	0.0781293	0.0390646	−	0.999237i	$-0.487562\pi$
$773$	2.17222	0.0781293	0.0390646	+	0.999237i	$0.487562\pi$
$774$	0	0
$775$	−10.2907	−0.369653
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.61127	0.344360
$780$	0	0
$781$	−5.07093	−0.181452
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.70994	−0.275180
$786$	0	0
$787$	−39.4188	−1.40513	−0.702564	−	0.711621i	$-0.747961\pi$
$787$	−39.4188	−1.40513	−0.702564	+	0.711621i	$0.747961\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.71102	−0.0963928
$792$	0	0
$793$	9.74884	0.346191
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.30785	0.188014	0.0940068	−	0.995572i	$-0.470032\pi$
$797$	5.30785	0.188014	0.0940068	+	0.995572i	$0.470032\pi$
$798$	0	0
$799$	91.6845	3.24356
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.34361	−0.0827041
$804$	0	0
$805$	2.34984	0.0828210
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−52.9515	−1.86168	−0.930838	−	0.365431i	$-0.880921\pi$
$809$	−52.9515	−1.86168	−0.930838	+	0.365431i	$0.880921\pi$
$810$	0	0
$811$	−41.8333	−1.46897	−0.734483	−	0.678627i	$-0.762575\pi$
$811$	−41.8333	−1.46897	−0.734483	+	0.678627i	$0.762575\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.2704	0.569927
$816$	0	0
$817$	−9.58469	−0.335326
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.5303	1.69372	0.846860	−	0.531816i	$-0.178490\pi$
$821$	48.5303	1.69372	0.846860	+	0.531816i	$0.178490\pi$
$822$	0	0
$823$	−32.4844	−1.13233	−0.566167	−	0.824291i	$-0.691574\pi$
$823$	−32.4844	−1.13233	−0.566167	+	0.824291i	$0.691574\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.9416	−1.56277	−0.781385	−	0.624049i	$-0.785487\pi$
$827$	−44.9416	−1.56277	−0.781385	+	0.624049i	$0.785487\pi$
$828$	0	0
$829$	0.782855	0.0271897	0.0135948	−	0.999908i	$-0.495672\pi$
$829$	0.782855	0.0271897	0.0135948	+	0.999908i	$0.495672\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−11.5529	−0.400285
$834$	0	0
$835$	−15.6828	−0.542726
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.55402	0.226270	0.113135	−	0.993580i	$-0.463911\pi$
$839$	6.55402	0.226270	0.113135	+	0.993580i	$0.463911\pi$
$840$	0	0
$841$	−27.1653	−0.936736
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.58037	0.0543665
$846$	0	0
$847$	−25.5865	−0.879164
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.83910	−0.0973230
$852$	0	0
$853$	−33.8079	−1.15756	−0.578780	−	0.815484i	$-0.696471\pi$
$853$	−33.8079	−1.15756	−0.578780	+	0.815484i	$0.696471\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−48.8990	−1.67036	−0.835180	−	0.549977i	$-0.814636\pi$
$857$	−48.8990	−1.67036	−0.835180	+	0.549977i	$0.814636\pi$
$858$	0	0
$859$	−8.46428	−0.288797	−0.144399	−	0.989520i	$-0.546125\pi$
$859$	−8.46428	−0.288797	−0.144399	+	0.989520i	$0.546125\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.62307	−0.123331	−0.0616654	−	0.998097i	$-0.519641\pi$
$863$	−3.62307	−0.123331	−0.0616654	+	0.998097i	$0.519641\pi$
$864$	0	0
$865$	14.7815	0.502586
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.108479	0.00367990
$870$	0	0
$871$	26.0407	0.882357
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.34984	−0.0794392
$876$	0	0
$877$	−10.9308	−0.369108	−0.184554	−	0.982822i	$-0.559084\pi$
$877$	−10.9308	−0.369108	−0.184554	+	0.982822i	$0.559084\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−16.0740	−0.541545	−0.270773	−	0.962643i	$-0.587279\pi$
$881$	−16.0740	−0.541545	−0.270773	+	0.962643i	$0.587279\pi$
$882$	0	0
$883$	24.7421	0.832639	0.416319	−	0.909218i	$-0.363320\pi$
$883$	24.7421	0.832639	0.416319	+	0.909218i	$0.363320\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.8153	1.26971	0.634856	−	0.772630i	$-0.281059\pi$
$887$	37.8153	1.26971	0.634856	+	0.772630i	$0.281059\pi$
$888$	0	0
$889$	−18.2620	−0.612487
$890$	0	0
$891$	0	0
$892$	0	0
$893$	14.1941	0.474986
$894$	0	0
$895$	−8.66161	−0.289526
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.9388	−0.464884
$900$	0	0
$901$	61.9021	2.06226
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.53481	−0.183983
$906$	0	0
$907$	−53.9550	−1.79155	−0.895774	−	0.444510i	$-0.853378\pi$
$907$	−53.9550	−1.79155	−0.895774	+	0.444510i	$0.853378\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.8570	1.05547	0.527735	−	0.849409i	$-0.323041\pi$
$911$	31.8570	1.05547	0.527735	+	0.849409i	$0.323041\pi$
$912$	0	0
$913$	3.49845	0.115782
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.75601	−0.124034
$918$	0	0
$919$	−26.8032	−0.884157	−0.442078	−	0.896976i	$-0.645759\pi$
$919$	−26.8032	−0.884157	−0.442078	+	0.896976i	$0.645759\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−51.3468	−1.69010
$924$	0	0
$925$	2.83910	0.0933489
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.25414	−0.139574	−0.0697870	−	0.997562i	$-0.522232\pi$
$929$	−4.25414	−0.139574	−0.0697870	+	0.997562i	$0.522232\pi$
$930$	0	0
$931$	−1.78856	−0.0586176
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.60823	−0.0852983
$936$	0	0
$937$	33.6296	1.09863	0.549316	−	0.835615i	$-0.314888\pi$
$937$	33.6296	1.09863	0.549316	+	0.835615i	$0.314888\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−37.3525	−1.21766	−0.608829	−	0.793302i	$-0.708360\pi$
$941$	−37.3525	−1.21766	−0.608829	+	0.793302i	$0.708360\pi$
$942$	0	0
$943$	−7.94371	−0.258683
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.4779	1.70530	0.852652	−	0.522479i	$-0.174993\pi$
$947$	52.4779	1.70530	0.852652	+	0.522479i	$0.174993\pi$
$948$	0	0
$949$	−23.7307	−0.770332
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.95962	−0.128265	−0.0641323	−	0.997941i	$-0.520428\pi$
$953$	−3.95962	−0.128265	−0.0641323	+	0.997941i	$0.520428\pi$
$954$	0	0
$955$	−1.81157	−0.0586210
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.7179	0.378389
$960$	0	0
$961$	74.8988	2.41609
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−26.9261	−0.866783
$966$	0	0
$967$	−24.7487	−0.795864	−0.397932	−	0.917415i	$-0.630272\pi$
$967$	−24.7487	−0.795864	−0.397932	+	0.917415i	$0.630272\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.3608	0.653408	0.326704	−	0.945127i	$-0.394062\pi$
$971$	20.3608	0.653408	0.326704	+	0.945127i	$0.394062\pi$
$972$	0	0
$973$	−26.7122	−0.856354
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.2849	1.57676	0.788381	−	0.615187i	$-0.210919\pi$
$977$	49.2849	1.57676	0.788381	+	0.615187i	$0.210919\pi$
$978$	0	0
$979$	2.68814	0.0859134
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15.9279	0.508022	0.254011	−	0.967201i	$-0.418250\pi$
$983$	15.9279	0.508022	0.254011	+	0.967201i	$0.418250\pi$
$984$	0	0
$985$	−7.60772	−0.242402
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.92175	0.251897
$990$	0	0
$991$	13.8100	0.438688	0.219344	−	0.975648i	$-0.429608\pi$
$991$	13.8100	0.438688	0.219344	+	0.975648i	$0.429608\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.99167	0.253353
$996$	0	0
$997$	−44.7356	−1.41679	−0.708396	−	0.705815i	$-0.750581\pi$
$997$	−44.7356	−1.41679	−0.708396	+	0.705815i	$0.750581\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bv.1.6	✓	7
3.2	odd	2		8280.2.a.bw.1.6	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.a.bv.1.6	✓	7	1.1	even	1	trivial
8280.2.a.bw.1.6	yes	7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.34984 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +2.34984 q^{7} +0.333733 q^{11} +3.37929 q^{13} +7.81531 q^{17} +1.20992 q^{19} -1.00000 q^{23} +1.00000 q^{25} +1.35450 q^{29} -10.2907 q^{31} -2.34984 q^{35} +2.83910 q^{37} +7.94371 q^{41} -7.92175 q^{43} +11.7314 q^{47} -1.47824 q^{49} +7.92062 q^{53} -0.333733 q^{55} +13.2728 q^{59} +2.88487 q^{61} -3.37929 q^{65} +7.70597 q^{67} -15.1945 q^{71} -7.02239 q^{73} +0.784221 q^{77} +0.325047 q^{79} +10.4828 q^{83} -7.81531 q^{85} +8.05475 q^{89} +7.94081 q^{91} -1.20992 q^{95} -1.00919 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{5} - 6 q^{7}+O(q^{10}) \) 7 * q - 7 * q^5 - 6 * q^7	\( 7 q - 7 q^{5} - 6 q^{7} + 2 q^{11} - 6 q^{13} + 4 q^{17} - 8 q^{19} - 7 q^{23} + 7 q^{25} - 2 q^{29} - 8 q^{31} + 6 q^{35} - 16 q^{37} - 2 q^{41} - 10 q^{43} + 8 q^{47} + 19 q^{49} + 10 q^{53} - 2 q^{55} + 24 q^{59} + 8 q^{61} + 6 q^{65} - 20 q^{67} + 8 q^{71} + 2 q^{73} + 12 q^{77} - 2 q^{79} + 22 q^{83} - 4 q^{85} + 16 q^{89} - 20 q^{91} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) 7 * q - 7 * q^5 - 6 * q^7 + 2 * q^11 - 6 * q^13 + 4 * q^17 - 8 * q^19 - 7 * q^23 + 7 * q^25 - 2 * q^29 - 8 * q^31 + 6 * q^35 - 16 * q^37 - 2 * q^41 - 10 * q^43 + 8 * q^47 + 19 * q^49 + 10 * q^53 - 2 * q^55 + 24 * q^59 + 8 * q^61 + 6 * q^65 - 20 * q^67 + 8 * q^71 + 2 * q^73 + 12 * q^77 - 2 * q^79 + 22 * q^83 - 4 * q^85 + 16 * q^89 - 20 * q^91 + 8 * q^95 + 4 * q^97

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