LMFDB - Embedded newform 9016.2.a.bl.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9016,2,Mod(1,9016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9016, 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("9016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9016 = 2^{3} \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9016.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:	$71.9931224624$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4x^{10} - 14x^{9} + 63x^{8} + 51x^{7} - 305x^{6} + 16x^{5} + 429x^{4} - 234x^{3} - 42x^{2} + 39x - 3 $ x^11 - 4x^10 - 14x^9 + 63x^8 + 51x^7 - 305x^6 + 16x^5 + 429x^4 - 234x^3 - 42x^2 + 39x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1288)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.656305$ of defining polynomial
Character	$\chi$	$=$	9016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.656305 q^{3} +2.09142 q^{5} -2.56926 q^{9} +O(q^{10})$	$q-0.656305 q^{3} +2.09142 q^{5} -2.56926 q^{9} -1.84217 q^{11} -2.10767 q^{13} -1.37261 q^{15} +4.60636 q^{17} +8.45409 q^{19} +1.00000 q^{23} -0.625980 q^{25} +3.65513 q^{27} -2.24733 q^{29} -10.1139 q^{31} +1.20903 q^{33} -6.27743 q^{37} +1.38328 q^{39} +0.725868 q^{41} -10.8164 q^{43} -5.37340 q^{45} -2.14894 q^{47} -3.02317 q^{51} +12.8348 q^{53} -3.85275 q^{55} -5.54846 q^{57} +1.37532 q^{59} -1.80134 q^{61} -4.40802 q^{65} -5.83508 q^{67} -0.656305 q^{69} +4.43192 q^{71} -4.08864 q^{73} +0.410834 q^{75} +6.92763 q^{79} +5.30891 q^{81} +1.78309 q^{83} +9.63381 q^{85} +1.47494 q^{87} +3.69528 q^{89} +6.63778 q^{93} +17.6810 q^{95} -3.34404 q^{97} +4.73303 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 4 q^{3} - q^{5} + 11 q^{9}+O(q^{10}) $ 11 * q - 4 * q^3 - q^5 + 11 * q^9	$ 11 q - 4 q^{3} - q^{5} + 11 q^{9} + 3 q^{13} - 8 q^{15} - 5 q^{17} - 12 q^{19} + 11 q^{23} + 22 q^{25} - 19 q^{27} - 15 q^{29} - 16 q^{31} - 4 q^{33} + 3 q^{37} - q^{39} - 28 q^{41} - 9 q^{43} + 19 q^{45} - 31 q^{47} - 15 q^{51} + 13 q^{53} - 35 q^{55} - 21 q^{57} - 11 q^{59} + 19 q^{61} - 7 q^{65} + 19 q^{67} - 4 q^{69} - 5 q^{71} + 5 q^{73} - 28 q^{75} - 13 q^{79} + 35 q^{81} - 17 q^{83} - 39 q^{85} - 4 q^{87} - 10 q^{89} - 6 q^{93} + 33 q^{95} - 35 q^{97} + 13 q^{99}+O(q^{100}) $ 11 * q - 4 * q^3 - q^5 + 11 * q^9 + 3 * q^13 - 8 * q^15 - 5 * q^17 - 12 * q^19 + 11 * q^23 + 22 * q^25 - 19 * q^27 - 15 * q^29 - 16 * q^31 - 4 * q^33 + 3 * q^37 - q^39 - 28 * q^41 - 9 * q^43 + 19 * q^45 - 31 * q^47 - 15 * q^51 + 13 * q^53 - 35 * q^55 - 21 * q^57 - 11 * q^59 + 19 * q^61 - 7 * q^65 + 19 * q^67 - 4 * q^69 - 5 * q^71 + 5 * q^73 - 28 * q^75 - 13 * q^79 + 35 * q^81 - 17 * q^83 - 39 * q^85 - 4 * q^87 - 10 * q^89 - 6 * q^93 + 33 * q^95 - 35 * q^97 + 13 * 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
$3$	−0.656305	−0.378918	−0.189459	−	0.981889i	$-0.560673\pi$
$3$	−0.656305	−0.378918	−0.189459	+	0.981889i	$0.560673\pi$
$4$	0	0
$5$	2.09142	0.935310	0.467655	−	0.883911i	$-0.345099\pi$
$5$	2.09142	0.935310	0.467655	+	0.883911i	$0.345099\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.56926	−0.856421
$10$	0	0
$11$	−1.84217	−0.555436	−0.277718	−	0.960663i	$-0.589578\pi$
$11$	−1.84217	−0.555436	−0.277718	+	0.960663i	$0.589578\pi$
$12$	0	0
$13$	−2.10767	−0.584564	−0.292282	−	0.956332i	$-0.594415\pi$
$13$	−2.10767	−0.584564	−0.292282	+	0.956332i	$0.594415\pi$
$14$	0	0
$15$	−1.37261	−0.354405
$16$	0	0
$17$	4.60636	1.11721	0.558603	−	0.829435i	$-0.311338\pi$
$17$	4.60636	1.11721	0.558603	+	0.829435i	$0.311338\pi$
$18$	0	0
$19$	8.45409	1.93950	0.969751	−	0.244096i	$-0.0784912\pi$
$19$	8.45409	1.93950	0.969751	+	0.244096i	$0.0784912\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−0.625980	−0.125196
$26$	0	0
$27$	3.65513	0.703431
$28$	0	0
$29$	−2.24733	−0.417320	−0.208660	−	0.977988i	$-0.566910\pi$
$29$	−2.24733	−0.417320	−0.208660	+	0.977988i	$0.566910\pi$
$30$	0	0
$31$	−10.1139	−1.81650	−0.908252	−	0.418424i	$-0.862583\pi$
$31$	−10.1139	−1.81650	−0.908252	+	0.418424i	$0.862583\pi$
$32$	0	0
$33$	1.20903	0.210465
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.27743	−1.03200	−0.516002	−	0.856588i	$-0.672580\pi$
$37$	−6.27743	−1.03200	−0.516002	+	0.856588i	$0.672580\pi$
$38$	0	0
$39$	1.38328	0.221502
$40$	0	0
$41$	0.725868	0.113361	0.0566807	−	0.998392i	$-0.481948\pi$
$41$	0.725868	0.113361	0.0566807	+	0.998392i	$0.481948\pi$
$42$	0	0
$43$	−10.8164	−1.64949	−0.824744	−	0.565506i	$-0.808681\pi$
$43$	−10.8164	−1.64949	−0.824744	+	0.565506i	$0.808681\pi$
$44$	0	0
$45$	−5.37340	−0.801019
$46$	0	0
$47$	−2.14894	−0.313456	−0.156728	−	0.987642i	$-0.550095\pi$
$47$	−2.14894	−0.313456	−0.156728	+	0.987642i	$0.550095\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.02317	−0.423329
$52$	0	0
$53$	12.8348	1.76299	0.881496	−	0.472192i	$-0.156537\pi$
$53$	12.8348	1.76299	0.881496	+	0.472192i	$0.156537\pi$
$54$	0	0
$55$	−3.85275	−0.519505
$56$	0	0
$57$	−5.54846	−0.734912
$58$	0	0
$59$	1.37532	0.179052	0.0895258	−	0.995985i	$-0.471465\pi$
$59$	1.37532	0.179052	0.0895258	+	0.995985i	$0.471465\pi$
$60$	0	0
$61$	−1.80134	−0.230638	−0.115319	−	0.993329i	$-0.536789\pi$
$61$	−1.80134	−0.230638	−0.115319	+	0.993329i	$0.536789\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.40802	−0.546748
$66$	0	0
$67$	−5.83508	−0.712869	−0.356434	−	0.934320i	$-0.616008\pi$
$67$	−5.83508	−0.712869	−0.356434	+	0.934320i	$0.616008\pi$
$68$	0	0
$69$	−0.656305	−0.0790098
$70$	0	0
$71$	4.43192	0.525972	0.262986	−	0.964800i	$-0.415293\pi$
$71$	4.43192	0.525972	0.262986	+	0.964800i	$0.415293\pi$
$72$	0	0
$73$	−4.08864	−0.478539	−0.239269	−	0.970953i	$-0.576908\pi$
$73$	−4.08864	−0.478539	−0.239269	+	0.970953i	$0.576908\pi$
$74$	0	0
$75$	0.410834	0.0474390
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.92763	0.779419	0.389709	−	0.920938i	$-0.372575\pi$
$79$	6.92763	0.779419	0.389709	+	0.920938i	$0.372575\pi$
$80$	0	0
$81$	5.30891	0.589879
$82$	0	0
$83$	1.78309	0.195720	0.0978599	−	0.995200i	$-0.468800\pi$
$83$	1.78309	0.195720	0.0978599	+	0.995200i	$0.468800\pi$
$84$	0	0
$85$	9.63381	1.04493
$86$	0	0
$87$	1.47494	0.158130
$88$	0	0
$89$	3.69528	0.391699	0.195850	−	0.980634i	$-0.437254\pi$
$89$	3.69528	0.391699	0.195850	+	0.980634i	$0.437254\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.63778	0.688306
$94$	0	0
$95$	17.6810	1.81403
$96$	0	0
$97$	−3.34404	−0.339536	−0.169768	−	0.985484i	$-0.554302\pi$
$97$	−3.34404	−0.339536	−0.169768	+	0.985484i	$0.554302\pi$
$98$	0	0
$99$	4.73303	0.475687
$100$	0	0
$101$	2.32678	0.231523	0.115762	−	0.993277i	$-0.463069\pi$
$101$	2.32678	0.231523	0.115762	+	0.993277i	$0.463069\pi$
$102$	0	0
$103$	−3.18954	−0.314274	−0.157137	−	0.987577i	$-0.550226\pi$
$103$	−3.18954	−0.314274	−0.157137	+	0.987577i	$0.550226\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.3705	−1.00255	−0.501275	−	0.865288i	$-0.667135\pi$
$107$	−10.3705	−1.00255	−0.501275	+	0.865288i	$0.667135\pi$
$108$	0	0
$109$	−6.00681	−0.575348	−0.287674	−	0.957728i	$-0.592882\pi$
$109$	−6.00681	−0.575348	−0.287674	+	0.957728i	$0.592882\pi$
$110$	0	0
$111$	4.11991	0.391044
$112$	0	0
$113$	3.85151	0.362320	0.181160	−	0.983454i	$-0.442015\pi$
$113$	3.85151	0.362320	0.181160	+	0.983454i	$0.442015\pi$
$114$	0	0
$115$	2.09142	0.195026
$116$	0	0
$117$	5.41517	0.500633
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.60640	−0.691491
$122$	0	0
$123$	−0.476390	−0.0429547
$124$	0	0
$125$	−11.7663	−1.05241
$126$	0	0
$127$	−9.28567	−0.823970	−0.411985	−	0.911191i	$-0.635164\pi$
$127$	−9.28567	−0.823970	−0.411985	+	0.911191i	$0.635164\pi$
$128$	0	0
$129$	7.09887	0.625020
$130$	0	0
$131$	−9.47233	−0.827601	−0.413801	−	0.910368i	$-0.635799\pi$
$131$	−9.47233	−0.827601	−0.413801	+	0.910368i	$0.635799\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	7.64440	0.657926
$136$	0	0
$137$	10.4365	0.891650	0.445825	−	0.895120i	$-0.352910\pi$
$137$	10.4365	0.891650	0.445825	+	0.895120i	$0.352910\pi$
$138$	0	0
$139$	−22.3518	−1.89585	−0.947925	−	0.318492i	$-0.896823\pi$
$139$	−22.3518	−1.89585	−0.947925	+	0.318492i	$0.896823\pi$
$140$	0	0
$141$	1.41036	0.118774
$142$	0	0
$143$	3.88270	0.324688
$144$	0	0
$145$	−4.70011	−0.390323
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.9214	1.14048	0.570242	−	0.821477i	$-0.306850\pi$
$149$	13.9214	1.14048	0.570242	+	0.821477i	$0.306850\pi$
$150$	0	0
$151$	15.2367	1.23994	0.619971	−	0.784625i	$-0.287144\pi$
$151$	15.2367	1.23994	0.619971	+	0.784625i	$0.287144\pi$
$152$	0	0
$153$	−11.8350	−0.956799
$154$	0	0
$155$	−21.1523	−1.69899
$156$	0	0
$157$	8.70661	0.694863	0.347432	−	0.937705i	$-0.387054\pi$
$157$	8.70661	0.694863	0.347432	+	0.937705i	$0.387054\pi$
$158$	0	0
$159$	−8.42352	−0.668029
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3.40669	0.266833	0.133416	−	0.991060i	$-0.457405\pi$
$163$	3.40669	0.266833	0.133416	+	0.991060i	$0.457405\pi$
$164$	0	0
$165$	2.52858	0.196850
$166$	0	0
$167$	−10.4072	−0.805331	−0.402666	−	0.915347i	$-0.631916\pi$
$167$	−10.4072	−0.805331	−0.402666	+	0.915347i	$0.631916\pi$
$168$	0	0
$169$	−8.55771	−0.658285
$170$	0	0
$171$	−21.7208	−1.66103
$172$	0	0
$173$	8.24459	0.626825	0.313412	−	0.949617i	$-0.398528\pi$
$173$	8.24459	0.626825	0.313412	+	0.949617i	$0.398528\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.902630	−0.0678458
$178$	0	0
$179$	−19.1821	−1.43373	−0.716867	−	0.697209i	$-0.754425\pi$
$179$	−19.1821	−1.43373	−0.716867	+	0.697209i	$0.754425\pi$
$180$	0	0
$181$	−10.1160	−0.751916	−0.375958	−	0.926637i	$-0.622686\pi$
$181$	−10.1160	−0.751916	−0.375958	+	0.926637i	$0.622686\pi$
$182$	0	0
$183$	1.18223	0.0873929
$184$	0	0
$185$	−13.1287	−0.965242
$186$	0	0
$187$	−8.48571	−0.620537
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.9124	−1.22374	−0.611870	−	0.790958i	$-0.709582\pi$
$191$	−16.9124	−1.22374	−0.611870	+	0.790958i	$0.709582\pi$
$192$	0	0
$193$	7.29414	0.525044	0.262522	−	0.964926i	$-0.415446\pi$
$193$	7.29414	0.525044	0.262522	+	0.964926i	$0.415446\pi$
$194$	0	0
$195$	2.89301	0.207173
$196$	0	0
$197$	4.20119	0.299322	0.149661	−	0.988737i	$-0.452182\pi$
$197$	4.20119	0.299322	0.149661	+	0.988737i	$0.452182\pi$
$198$	0	0
$199$	−3.31070	−0.234689	−0.117345	−	0.993091i	$-0.537438\pi$
$199$	−3.31070	−0.234689	−0.117345	+	0.993091i	$0.537438\pi$
$200$	0	0
$201$	3.82959	0.270119
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.51809	0.106028
$206$	0	0
$207$	−2.56926	−0.178576
$208$	0	0
$209$	−15.5739	−1.07727
$210$	0	0
$211$	−3.70533	−0.255085	−0.127543	−	0.991833i	$-0.540709\pi$
$211$	−3.70533	−0.255085	−0.127543	+	0.991833i	$0.540709\pi$
$212$	0	0
$213$	−2.90869	−0.199300
$214$	0	0
$215$	−22.6216	−1.54278
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.68339	0.181327
$220$	0	0
$221$	−9.70871	−0.653078
$222$	0	0
$223$	−0.499565	−0.0334534	−0.0167267	−	0.999860i	$-0.505325\pi$
$223$	−0.499565	−0.0334534	−0.0167267	+	0.999860i	$0.505325\pi$
$224$	0	0
$225$	1.60831	0.107221
$226$	0	0
$227$	28.8372	1.91399	0.956997	−	0.290097i	$-0.0936877\pi$
$227$	28.8372	1.91399	0.956997	+	0.290097i	$0.0936877\pi$
$228$	0	0
$229$	19.8349	1.31073	0.655363	−	0.755314i	$-0.272516\pi$
$229$	19.8349	1.31073	0.655363	+	0.755314i	$0.272516\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.64925	0.632143	0.316072	−	0.948735i	$-0.397636\pi$
$233$	9.64925	0.632143	0.316072	+	0.948735i	$0.397636\pi$
$234$	0	0
$235$	−4.49433	−0.293178
$236$	0	0
$237$	−4.54663	−0.295336
$238$	0	0
$239$	−4.66004	−0.301433	−0.150716	−	0.988577i	$-0.548158\pi$
$239$	−4.66004	−0.301433	−0.150716	+	0.988577i	$0.548158\pi$
$240$	0	0
$241$	9.87452	0.636074	0.318037	−	0.948078i	$-0.396976\pi$
$241$	9.87452	0.636074	0.318037	+	0.948078i	$0.396976\pi$
$242$	0	0
$243$	−14.4497	−0.926946
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−17.8185	−1.13376
$248$	0	0
$249$	−1.17025	−0.0741617
$250$	0	0
$251$	−28.2589	−1.78368	−0.891842	−	0.452348i	$-0.850587\pi$
$251$	−28.2589	−1.78368	−0.891842	+	0.452348i	$0.850587\pi$
$252$	0	0
$253$	−1.84217	−0.115816
$254$	0	0
$255$	−6.32272	−0.395944
$256$	0	0
$257$	−22.9547	−1.43188	−0.715939	−	0.698163i	$-0.754001\pi$
$257$	−22.9547	−1.43188	−0.715939	+	0.698163i	$0.754001\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.77400	0.357401
$262$	0	0
$263$	−18.8159	−1.16024	−0.580120	−	0.814531i	$-0.696994\pi$
$263$	−18.8159	−1.16024	−0.580120	+	0.814531i	$0.696994\pi$
$264$	0	0
$265$	26.8428	1.64894
$266$	0	0
$267$	−2.42523	−0.148422
$268$	0	0
$269$	−16.6246	−1.01362	−0.506810	−	0.862058i	$-0.669175\pi$
$269$	−16.6246	−1.01362	−0.506810	+	0.862058i	$0.669175\pi$
$270$	0	0
$271$	−5.64711	−0.343038	−0.171519	−	0.985181i	$-0.554867\pi$
$271$	−5.64711	−0.343038	−0.171519	+	0.985181i	$0.554867\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.15316	0.0695384
$276$	0	0
$277$	−25.6748	−1.54265	−0.771324	−	0.636442i	$-0.780405\pi$
$277$	−25.6748	−1.54265	−0.771324	+	0.636442i	$0.780405\pi$
$278$	0	0
$279$	25.9852	1.55569
$280$	0	0
$281$	23.7618	1.41751	0.708754	−	0.705456i	$-0.249258\pi$
$281$	23.7618	1.41751	0.708754	+	0.705456i	$0.249258\pi$
$282$	0	0
$283$	7.23914	0.430322	0.215161	−	0.976579i	$-0.430972\pi$
$283$	7.23914	0.430322	0.215161	+	0.976579i	$0.430972\pi$
$284$	0	0
$285$	−11.6041	−0.687370
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4.21854	0.248149
$290$	0	0
$291$	2.19471	0.128656
$292$	0	0
$293$	−28.7605	−1.68021	−0.840103	−	0.542427i	$-0.817506\pi$
$293$	−28.7605	−1.68021	−0.840103	+	0.542427i	$0.817506\pi$
$294$	0	0
$295$	2.87637	0.167469
$296$	0	0
$297$	−6.73339	−0.390711
$298$	0	0
$299$	−2.10767	−0.121890
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.52708	−0.0877283
$304$	0	0
$305$	−3.76735	−0.215718
$306$	0	0
$307$	−20.7093	−1.18194	−0.590970	−	0.806693i	$-0.701255\pi$
$307$	−20.7093	−1.18194	−0.590970	+	0.806693i	$0.701255\pi$
$308$	0	0
$309$	2.09331	0.119084
$310$	0	0
$311$	−23.6473	−1.34092	−0.670458	−	0.741948i	$-0.733902\pi$
$311$	−23.6473	−1.34092	−0.670458	+	0.741948i	$0.733902\pi$
$312$	0	0
$313$	−34.8435	−1.96947	−0.984736	−	0.174057i	$-0.944312\pi$
$313$	−34.8435	−1.96947	−0.984736	+	0.174057i	$0.944312\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.0952	0.960163	0.480082	−	0.877224i	$-0.340607\pi$
$317$	17.0952	0.960163	0.480082	+	0.877224i	$0.340607\pi$
$318$	0	0
$319$	4.13998	0.231794
$320$	0	0
$321$	6.80618	0.379884
$322$	0	0
$323$	38.9426	2.16682
$324$	0	0
$325$	1.31936	0.0731851
$326$	0	0
$327$	3.94229	0.218009
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.4867	−1.23598	−0.617989	−	0.786186i	$-0.712053\pi$
$331$	−22.4867	−1.23598	−0.617989	+	0.786186i	$0.712053\pi$
$332$	0	0
$333$	16.1284	0.883830
$334$	0	0
$335$	−12.2036	−0.666753
$336$	0	0
$337$	−33.3104	−1.81453	−0.907265	−	0.420559i	$-0.861834\pi$
$337$	−33.3104	−1.81453	−0.907265	+	0.420559i	$0.861834\pi$
$338$	0	0
$339$	−2.52777	−0.137289
$340$	0	0
$341$	18.6315	1.00895
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.37261	−0.0738986
$346$	0	0
$347$	9.15658	0.491551	0.245775	−	0.969327i	$-0.420957\pi$
$347$	9.15658	0.491551	0.245775	+	0.969327i	$0.420957\pi$
$348$	0	0
$349$	14.4050	0.771080	0.385540	−	0.922691i	$-0.374015\pi$
$349$	14.4050	0.771080	0.385540	+	0.922691i	$0.374015\pi$
$350$	0	0
$351$	−7.70383	−0.411200
$352$	0	0
$353$	32.5934	1.73477	0.867387	−	0.497635i	$-0.165798\pi$
$353$	32.5934	1.73477	0.867387	+	0.497635i	$0.165798\pi$
$354$	0	0
$355$	9.26898	0.491947
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.2441	−0.804555	−0.402277	−	0.915518i	$-0.631781\pi$
$359$	−15.2441	−0.804555	−0.402277	+	0.915518i	$0.631781\pi$
$360$	0	0
$361$	52.4717	2.76167
$362$	0	0
$363$	4.99211	0.262018
$364$	0	0
$365$	−8.55104	−0.447582
$366$	0	0
$367$	17.1810	0.896840	0.448420	−	0.893823i	$-0.351987\pi$
$367$	17.1810	0.896840	0.448420	+	0.893823i	$0.351987\pi$
$368$	0	0
$369$	−1.86495	−0.0970852
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.5476	0.856800	0.428400	−	0.903589i	$-0.359077\pi$
$373$	16.5476	0.856800	0.428400	+	0.903589i	$0.359077\pi$
$374$	0	0
$375$	7.72225	0.398775
$376$	0	0
$377$	4.73665	0.243950
$378$	0	0
$379$	−7.38165	−0.379170	−0.189585	−	0.981864i	$-0.560714\pi$
$379$	−7.38165	−0.379170	−0.189585	+	0.981864i	$0.560714\pi$
$380$	0	0
$381$	6.09423	0.312217
$382$	0	0
$383$	7.38662	0.377439	0.188719	−	0.982031i	$-0.439566\pi$
$383$	7.38662	0.377439	0.188719	+	0.982031i	$0.439566\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	27.7902	1.41266
$388$	0	0
$389$	17.6130	0.893014	0.446507	−	0.894780i	$-0.352668\pi$
$389$	17.6130	0.893014	0.446507	+	0.894780i	$0.352668\pi$
$390$	0	0
$391$	4.60636	0.232954
$392$	0	0
$393$	6.21673	0.313593
$394$	0	0
$395$	14.4885	0.728998
$396$	0	0
$397$	−0.0589560	−0.00295892	−0.00147946	−	0.999999i	$-0.500471\pi$
$397$	−0.0589560	−0.00295892	−0.00147946	+	0.999999i	$0.500471\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.8210	−1.23950	−0.619750	−	0.784799i	$-0.712766\pi$
$401$	−24.8210	−1.23950	−0.619750	+	0.784799i	$0.712766\pi$
$402$	0	0
$403$	21.3167	1.06186
$404$	0	0
$405$	11.1031	0.551719
$406$	0	0
$407$	11.5641	0.573212
$408$	0	0
$409$	−9.25096	−0.457431	−0.228715	−	0.973493i	$-0.573453\pi$
$409$	−9.25096	−0.457431	−0.228715	+	0.973493i	$0.573453\pi$
$410$	0	0
$411$	−6.84952	−0.337862
$412$	0	0
$413$	0	0
$414$	0	0
$415$	3.72919	0.183059
$416$	0	0
$417$	14.6696	0.718371
$418$	0	0
$419$	−18.7225	−0.914652	−0.457326	−	0.889299i	$-0.651193\pi$
$419$	−18.7225	−0.914652	−0.457326	+	0.889299i	$0.651193\pi$
$420$	0	0
$421$	−9.89925	−0.482460	−0.241230	−	0.970468i	$-0.577551\pi$
$421$	−9.89925	−0.482460	−0.241230	+	0.970468i	$0.577551\pi$
$422$	0	0
$423$	5.52120	0.268450
$424$	0	0
$425$	−2.88349	−0.139870
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.54824	−0.123030
$430$	0	0
$431$	−6.23370	−0.300267	−0.150133	−	0.988666i	$-0.547970\pi$
$431$	−6.23370	−0.300267	−0.150133	+	0.988666i	$0.547970\pi$
$432$	0	0
$433$	−12.6098	−0.605988	−0.302994	−	0.952993i	$-0.597986\pi$
$433$	−12.6098	−0.605988	−0.302994	+	0.952993i	$0.597986\pi$
$434$	0	0
$435$	3.08470	0.147900
$436$	0	0
$437$	8.45409	0.404414
$438$	0	0
$439$	−8.16544	−0.389715	−0.194858	−	0.980832i	$-0.562425\pi$
$439$	−8.16544	−0.389715	−0.194858	+	0.980832i	$0.562425\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.46926	0.0698068	0.0349034	−	0.999391i	$-0.488888\pi$
$443$	1.46926	0.0698068	0.0349034	+	0.999391i	$0.488888\pi$
$444$	0	0
$445$	7.72838	0.366360
$446$	0	0
$447$	−9.13666	−0.432149
$448$	0	0
$449$	13.9254	0.657179	0.328589	−	0.944473i	$-0.393427\pi$
$449$	13.9254	0.657179	0.328589	+	0.944473i	$0.393427\pi$
$450$	0	0
$451$	−1.33717	−0.0629650
$452$	0	0
$453$	−9.99990	−0.469836
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.3337	−1.51251	−0.756253	−	0.654279i	$-0.772972\pi$
$457$	−32.3337	−1.51251	−0.756253	+	0.654279i	$0.772972\pi$
$458$	0	0
$459$	16.8369	0.785877
$460$	0	0
$461$	12.6308	0.588274	0.294137	−	0.955763i	$-0.404968\pi$
$461$	12.6308	0.588274	0.294137	+	0.955763i	$0.404968\pi$
$462$	0	0
$463$	35.5062	1.65011	0.825057	−	0.565050i	$-0.191143\pi$
$463$	35.5062	1.65011	0.825057	+	0.565050i	$0.191143\pi$
$464$	0	0
$465$	13.8824	0.643779
$466$	0	0
$467$	−42.2179	−1.95361	−0.976805	−	0.214131i	$-0.931308\pi$
$467$	−42.2179	−1.95361	−0.976805	+	0.214131i	$0.931308\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5.71419	−0.263296
$472$	0	0
$473$	19.9257	0.916185
$474$	0	0
$475$	−5.29210	−0.242818
$476$	0	0
$477$	−32.9759	−1.50986
$478$	0	0
$479$	−37.2828	−1.70350	−0.851748	−	0.523951i	$-0.824457\pi$
$479$	−37.2828	−1.70350	−0.851748	+	0.523951i	$0.824457\pi$
$480$	0	0
$481$	13.2308	0.603272
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.99378	−0.317571
$486$	0	0
$487$	32.3495	1.46590	0.732948	−	0.680285i	$-0.238144\pi$
$487$	32.3495	1.46590	0.732948	+	0.680285i	$0.238144\pi$
$488$	0	0
$489$	−2.23583	−0.101108
$490$	0	0
$491$	27.3117	1.23256	0.616281	−	0.787526i	$-0.288639\pi$
$491$	27.3117	1.23256	0.616281	+	0.787526i	$0.288639\pi$
$492$	0	0
$493$	−10.3520	−0.466232
$494$	0	0
$495$	9.89873	0.444915
$496$	0	0
$497$	0	0
$498$	0	0
$499$	28.5162	1.27656	0.638281	−	0.769804i	$-0.279646\pi$
$499$	28.5162	1.27656	0.638281	+	0.769804i	$0.279646\pi$
$500$	0	0
$501$	6.83027	0.305154
$502$	0	0
$503$	−30.0184	−1.33846	−0.669228	−	0.743057i	$-0.733375\pi$
$503$	−30.0184	−1.33846	−0.669228	+	0.743057i	$0.733375\pi$
$504$	0	0
$505$	4.86627	0.216546
$506$	0	0
$507$	5.61646	0.249436
$508$	0	0
$509$	−25.4238	−1.12689	−0.563445	−	0.826154i	$-0.690524\pi$
$509$	−25.4238	−1.12689	−0.563445	+	0.826154i	$0.690524\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	30.9008	1.36431
$514$	0	0
$515$	−6.67065	−0.293944
$516$	0	0
$517$	3.95873	0.174105
$518$	0	0
$519$	−5.41096	−0.237515
$520$	0	0
$521$	19.8447	0.869411	0.434705	−	0.900573i	$-0.356853\pi$
$521$	19.8447	0.869411	0.434705	+	0.900573i	$0.356853\pi$
$522$	0	0
$523$	−17.8154	−0.779014	−0.389507	−	0.921023i	$-0.627355\pi$
$523$	−17.8154	−0.779014	−0.389507	+	0.921023i	$0.627355\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−46.5881	−2.02941
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.53356	−0.153344
$532$	0	0
$533$	−1.52989	−0.0662670
$534$	0	0
$535$	−21.6889	−0.937695
$536$	0	0
$537$	12.5893	0.543268
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.05298	−0.260238	−0.130119	−	0.991498i	$-0.541536\pi$
$541$	−6.05298	−0.260238	−0.130119	+	0.991498i	$0.541536\pi$
$542$	0	0
$543$	6.63917	0.284914
$544$	0	0
$545$	−12.5627	−0.538128
$546$	0	0
$547$	−30.0674	−1.28559	−0.642794	−	0.766039i	$-0.722225\pi$
$547$	−30.0674	−1.28559	−0.642794	+	0.766039i	$0.722225\pi$
$548$	0	0
$549$	4.62812	0.197523
$550$	0	0
$551$	−18.9992	−0.809392
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.61644	0.365747
$556$	0	0
$557$	−8.85430	−0.375169	−0.187584	−	0.982248i	$-0.560066\pi$
$557$	−8.85430	−0.375169	−0.187584	+	0.982248i	$0.560066\pi$
$558$	0	0
$559$	22.7975	0.964231
$560$	0	0
$561$	5.56921	0.235132
$562$	0	0
$563$	−6.27094	−0.264288	−0.132144	−	0.991231i	$-0.542186\pi$
$563$	−6.27094	−0.264288	−0.132144	+	0.991231i	$0.542186\pi$
$564$	0	0
$565$	8.05512	0.338881
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.9798	0.544143	0.272071	−	0.962277i	$-0.412291\pi$
$569$	12.9798	0.544143	0.272071	+	0.962277i	$0.412291\pi$
$570$	0	0
$571$	−15.0547	−0.630020	−0.315010	−	0.949088i	$-0.602008\pi$
$571$	−15.0547	−0.630020	−0.315010	+	0.949088i	$0.602008\pi$
$572$	0	0
$573$	11.0997	0.463697
$574$	0	0
$575$	−0.625980	−0.0261052
$576$	0	0
$577$	23.9638	0.997627	0.498813	−	0.866709i	$-0.333769\pi$
$577$	23.9638	0.997627	0.498813	+	0.866709i	$0.333769\pi$
$578$	0	0
$579$	−4.78718	−0.198948
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−23.6439	−0.979229
$584$	0	0
$585$	11.3254	0.468247
$586$	0	0
$587$	−12.1473	−0.501374	−0.250687	−	0.968068i	$-0.580657\pi$
$587$	−12.1473	−0.501374	−0.250687	+	0.968068i	$0.580657\pi$
$588$	0	0
$589$	−85.5036	−3.52311
$590$	0	0
$591$	−2.75726	−0.113419
$592$	0	0
$593$	29.3662	1.20593	0.602963	−	0.797769i	$-0.293987\pi$
$593$	29.3662	1.20593	0.602963	+	0.797769i	$0.293987\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.17283	0.0889279
$598$	0	0
$599$	−6.22666	−0.254414	−0.127207	−	0.991876i	$-0.540601\pi$
$599$	−6.22666	−0.254414	−0.127207	+	0.991876i	$0.540601\pi$
$600$	0	0
$601$	25.7823	1.05168	0.525842	−	0.850582i	$-0.323750\pi$
$601$	25.7823	1.05168	0.525842	+	0.850582i	$0.323750\pi$
$602$	0	0
$603$	14.9919	0.610516
$604$	0	0
$605$	−15.9081	−0.646758
$606$	0	0
$607$	−24.0980	−0.978106	−0.489053	−	0.872254i	$-0.662658\pi$
$607$	−24.0980	−0.978106	−0.489053	+	0.872254i	$0.662658\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.52927	0.183235
$612$	0	0
$613$	−4.42526	−0.178734	−0.0893672	−	0.995999i	$-0.528484\pi$
$613$	−4.42526	−0.178734	−0.0893672	+	0.995999i	$0.528484\pi$
$614$	0	0
$615$	−0.996330	−0.0401759
$616$	0	0
$617$	−38.6078	−1.55429	−0.777145	−	0.629321i	$-0.783333\pi$
$617$	−38.6078	−1.55429	−0.777145	+	0.629321i	$0.783333\pi$
$618$	0	0
$619$	39.0428	1.56926	0.784632	−	0.619962i	$-0.212852\pi$
$619$	39.0428	1.56926	0.784632	+	0.619962i	$0.212852\pi$
$620$	0	0
$621$	3.65513	0.146675
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−21.4782	−0.859130
$626$	0	0
$627$	10.2212	0.408196
$628$	0	0
$629$	−28.9161	−1.15296
$630$	0	0
$631$	39.0520	1.55463	0.777317	−	0.629109i	$-0.216580\pi$
$631$	39.0520	1.55463	0.777317	+	0.629109i	$0.216580\pi$
$632$	0	0
$633$	2.43182	0.0966563
$634$	0	0
$635$	−19.4202	−0.770667
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−11.3868	−0.450454
$640$	0	0
$641$	32.2264	1.27287	0.636434	−	0.771331i	$-0.280409\pi$
$641$	32.2264	1.27287	0.636434	+	0.771331i	$0.280409\pi$
$642$	0	0
$643$	−5.83718	−0.230196	−0.115098	−	0.993354i	$-0.536718\pi$
$643$	−5.83718	−0.230196	−0.115098	+	0.993354i	$0.536718\pi$
$644$	0	0
$645$	14.8467	0.584587
$646$	0	0
$647$	−33.2734	−1.30811	−0.654056	−	0.756446i	$-0.726934\pi$
$647$	−33.2734	−1.30811	−0.654056	+	0.756446i	$0.726934\pi$
$648$	0	0
$649$	−2.53358	−0.0994517
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17.8525	−0.698624	−0.349312	−	0.937006i	$-0.613585\pi$
$653$	−17.8525	−0.698624	−0.349312	+	0.937006i	$0.613585\pi$
$654$	0	0
$655$	−19.8106	−0.774063
$656$	0	0
$657$	10.5048	0.409831
$658$	0	0
$659$	39.3673	1.53353	0.766765	−	0.641927i	$-0.221865\pi$
$659$	39.3673	1.53353	0.766765	+	0.641927i	$0.221865\pi$
$660$	0	0
$661$	−6.01467	−0.233944	−0.116972	−	0.993135i	$-0.537319\pi$
$661$	−6.01467	−0.233944	−0.116972	+	0.993135i	$0.537319\pi$
$662$	0	0
$663$	6.37187	0.247463
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.24733	−0.0870171
$668$	0	0
$669$	0.327867	0.0126761
$670$	0	0
$671$	3.31838	0.128105
$672$	0	0
$673$	−51.3147	−1.97804	−0.989018	−	0.147792i	$-0.952783\pi$
$673$	−51.3147	−1.97804	−0.989018	+	0.147792i	$0.952783\pi$
$674$	0	0
$675$	−2.28804	−0.0880668
$676$	0	0
$677$	45.3850	1.74429	0.872143	−	0.489251i	$-0.162730\pi$
$677$	45.3850	1.74429	0.872143	+	0.489251i	$0.162730\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−18.9260	−0.725246
$682$	0	0
$683$	−24.8901	−0.952394	−0.476197	−	0.879339i	$-0.657985\pi$
$683$	−24.8901	−0.952394	−0.476197	+	0.879339i	$0.657985\pi$
$684$	0	0
$685$	21.8271	0.833969
$686$	0	0
$687$	−13.0177	−0.496657
$688$	0	0
$689$	−27.0515	−1.03058
$690$	0	0
$691$	28.8890	1.09899	0.549495	−	0.835497i	$-0.314820\pi$
$691$	28.8890	1.09899	0.549495	+	0.835497i	$0.314820\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−46.7468	−1.77321
$696$	0	0
$697$	3.34361	0.126648
$698$	0	0
$699$	−6.33285	−0.239530
$700$	0	0
$701$	5.88167	0.222148	0.111074	−	0.993812i	$-0.464571\pi$
$701$	5.88167	0.222148	0.111074	+	0.993812i	$0.464571\pi$
$702$	0	0
$703$	−53.0700	−2.00157
$704$	0	0
$705$	2.94965	0.111090
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−40.5374	−1.52242	−0.761208	−	0.648508i	$-0.775393\pi$
$709$	−40.5374	−1.52242	−0.761208	+	0.648508i	$0.775393\pi$
$710$	0	0
$711$	−17.7989	−0.667511
$712$	0	0
$713$	−10.1139	−0.378767
$714$	0	0
$715$	8.12034	0.303684
$716$	0	0
$717$	3.05840	0.114218
$718$	0	0
$719$	46.1308	1.72039	0.860194	−	0.509966i	$-0.170342\pi$
$719$	46.1308	1.72039	0.860194	+	0.509966i	$0.170342\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−6.48069	−0.241020
$724$	0	0
$725$	1.40679	0.0522468
$726$	0	0
$727$	−38.2028	−1.41686	−0.708431	−	0.705780i	$-0.750597\pi$
$727$	−38.2028	−1.41686	−0.708431	+	0.705780i	$0.750597\pi$
$728$	0	0
$729$	−6.44335	−0.238643
$730$	0	0
$731$	−49.8243	−1.84282
$732$	0	0
$733$	−15.1384	−0.559150	−0.279575	−	0.960124i	$-0.590194\pi$
$733$	−15.1384	−0.559150	−0.279575	+	0.960124i	$0.590194\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.7492	0.395953
$738$	0	0
$739$	16.5448	0.608610	0.304305	−	0.952575i	$-0.401576\pi$
$739$	16.5448	0.608610	0.304305	+	0.952575i	$0.401576\pi$
$740$	0	0
$741$	11.6944	0.429603
$742$	0	0
$743$	−40.6663	−1.49190	−0.745951	−	0.666000i	$-0.768005\pi$
$743$	−40.6663	−1.49190	−0.745951	+	0.666000i	$0.768005\pi$
$744$	0	0
$745$	29.1154	1.06671
$746$	0	0
$747$	−4.58123	−0.167619
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−34.9837	−1.27657	−0.638287	−	0.769798i	$-0.720357\pi$
$751$	−34.9837	−1.27657	−0.638287	+	0.769798i	$0.720357\pi$
$752$	0	0
$753$	18.5464	0.675869
$754$	0	0
$755$	31.8662	1.15973
$756$	0	0
$757$	−4.88371	−0.177502	−0.0887508	−	0.996054i	$-0.528287\pi$
$757$	−4.88371	−0.177502	−0.0887508	+	0.996054i	$0.528287\pi$
$758$	0	0
$759$	1.20903	0.0438849
$760$	0	0
$761$	10.4743	0.379692	0.189846	−	0.981814i	$-0.439201\pi$
$761$	10.4743	0.379692	0.189846	+	0.981814i	$0.439201\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−24.7518	−0.894903
$766$	0	0
$767$	−2.89873	−0.104667
$768$	0	0
$769$	−20.0464	−0.722891	−0.361446	−	0.932393i	$-0.617717\pi$
$769$	−20.0464	−0.722891	−0.361446	+	0.932393i	$0.617717\pi$
$770$	0	0
$771$	15.0653	0.542564
$772$	0	0
$773$	−41.2519	−1.48373	−0.741864	−	0.670550i	$-0.766058\pi$
$773$	−41.2519	−1.48373	−0.741864	+	0.670550i	$0.766058\pi$
$774$	0	0
$775$	6.33108	0.227419
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.13655	0.219865
$780$	0	0
$781$	−8.16436	−0.292144
$782$	0	0
$783$	−8.21431	−0.293555
$784$	0	0
$785$	18.2091	0.649912
$786$	0	0
$787$	−17.3939	−0.620025	−0.310012	−	0.950733i	$-0.600333\pi$
$787$	−17.3939	−0.620025	−0.310012	+	0.950733i	$0.600333\pi$
$788$	0	0
$789$	12.3490	0.439635
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.79664	0.134823
$794$	0	0
$795$	−17.6171	−0.624814
$796$	0	0
$797$	2.80365	0.0993105	0.0496553	−	0.998766i	$-0.484188\pi$
$797$	2.80365	0.0993105	0.0496553	+	0.998766i	$0.484188\pi$
$798$	0	0
$799$	−9.89880	−0.350194
$800$	0	0
$801$	−9.49416	−0.335460
$802$	0	0
$803$	7.53198	0.265798
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.9108	0.384078
$808$	0	0
$809$	−35.3761	−1.24376	−0.621879	−	0.783114i	$-0.713630\pi$
$809$	−35.3761	−1.24376	−0.621879	+	0.783114i	$0.713630\pi$
$810$	0	0
$811$	5.70149	0.200207	0.100103	−	0.994977i	$-0.468083\pi$
$811$	5.70149	0.200207	0.100103	+	0.994977i	$0.468083\pi$
$812$	0	0
$813$	3.70623	0.129983
$814$	0	0
$815$	7.12480	0.249571
$816$	0	0
$817$	−91.4430	−3.19919
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44.8387	1.56488	0.782440	−	0.622726i	$-0.213975\pi$
$821$	44.8387	1.56488	0.782440	+	0.622726i	$0.213975\pi$
$822$	0	0
$823$	−0.510832	−0.0178065	−0.00890325	−	0.999960i	$-0.502834\pi$
$823$	−0.510832	−0.0178065	−0.00890325	+	0.999960i	$0.502834\pi$
$824$	0	0
$825$	−0.756827	−0.0263493
$826$	0	0
$827$	24.9250	0.866726	0.433363	−	0.901219i	$-0.357327\pi$
$827$	24.9250	0.866726	0.433363	+	0.901219i	$0.357327\pi$
$828$	0	0
$829$	−10.9851	−0.381527	−0.190763	−	0.981636i	$-0.561096\pi$
$829$	−10.9851	−0.381527	−0.190763	+	0.981636i	$0.561096\pi$
$830$	0	0
$831$	16.8505	0.584537
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−21.7657	−0.753234
$836$	0	0
$837$	−36.9675	−1.27779
$838$	0	0
$839$	7.42486	0.256335	0.128167	−	0.991753i	$-0.459091\pi$
$839$	7.42486	0.256335	0.128167	+	0.991753i	$0.459091\pi$
$840$	0	0
$841$	−23.9495	−0.825844
$842$	0	0
$843$	−15.5950	−0.537119
$844$	0	0
$845$	−17.8977	−0.615700
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.75108	−0.163057
$850$	0	0
$851$	−6.27743	−0.215188
$852$	0	0
$853$	18.4615	0.632111	0.316055	−	0.948741i	$-0.397641\pi$
$853$	18.4615	0.632111	0.316055	+	0.948741i	$0.397641\pi$
$854$	0	0
$855$	−45.4272	−1.55358
$856$	0	0
$857$	4.88736	0.166949	0.0834745	−	0.996510i	$-0.473398\pi$
$857$	4.88736	0.166949	0.0834745	+	0.996510i	$0.473398\pi$
$858$	0	0
$859$	−14.0678	−0.479988	−0.239994	−	0.970774i	$-0.577146\pi$
$859$	−14.0678	−0.479988	−0.239994	+	0.970774i	$0.577146\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.5800	0.802674	0.401337	−	0.915931i	$-0.368546\pi$
$863$	23.5800	0.802674	0.401337	+	0.915931i	$0.368546\pi$
$864$	0	0
$865$	17.2429	0.586275
$866$	0	0
$867$	−2.76865	−0.0940282
$868$	0	0
$869$	−12.7619	−0.432917
$870$	0	0
$871$	12.2985	0.416717
$872$	0	0
$873$	8.59172	0.290786
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.4140	0.452959	0.226479	−	0.974016i	$-0.427278\pi$
$877$	13.4140	0.452959	0.226479	+	0.974016i	$0.427278\pi$
$878$	0	0
$879$	18.8756	0.636660
$880$	0	0
$881$	−21.8130	−0.734898	−0.367449	−	0.930044i	$-0.619769\pi$
$881$	−21.8130	−0.734898	−0.367449	+	0.930044i	$0.619769\pi$
$882$	0	0
$883$	16.6473	0.560226	0.280113	−	0.959967i	$-0.409628\pi$
$883$	16.6473	0.560226	0.280113	+	0.959967i	$0.409628\pi$
$884$	0	0
$885$	−1.88777	−0.0634568
$886$	0	0
$887$	−27.0819	−0.909320	−0.454660	−	0.890665i	$-0.650239\pi$
$887$	−27.0819	−0.909320	−0.454660	+	0.890665i	$0.650239\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−9.77993	−0.327640
$892$	0	0
$893$	−18.1674	−0.607948
$894$	0	0
$895$	−40.1177	−1.34099
$896$	0	0
$897$	1.38328	0.0461863
$898$	0	0
$899$	22.7292	0.758063
$900$	0	0
$901$	59.1216	1.96962
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.1567	−0.703274
$906$	0	0
$907$	−1.72010	−0.0571149	−0.0285575	−	0.999592i	$-0.509091\pi$
$907$	−1.72010	−0.0571149	−0.0285575	+	0.999592i	$0.509091\pi$
$908$	0	0
$909$	−5.97811	−0.198282
$910$	0	0
$911$	−21.1481	−0.700669	−0.350335	−	0.936625i	$-0.613932\pi$
$911$	−21.1481	−0.700669	−0.350335	+	0.936625i	$0.613932\pi$
$912$	0	0
$913$	−3.28476	−0.108710
$914$	0	0
$915$	2.47253	0.0817394
$916$	0	0
$917$	0	0
$918$	0	0
$919$	38.7328	1.27768	0.638838	−	0.769341i	$-0.279415\pi$
$919$	38.7328	1.27768	0.638838	+	0.769341i	$0.279415\pi$
$920$	0	0
$921$	13.5916	0.447858
$922$	0	0
$923$	−9.34104	−0.307464
$924$	0	0
$925$	3.92955	0.129203
$926$	0	0
$927$	8.19476	0.269151
$928$	0	0
$929$	−3.93995	−0.129266	−0.0646328	−	0.997909i	$-0.520588\pi$
$929$	−3.93995	−0.129266	−0.0646328	+	0.997909i	$0.520588\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	15.5198	0.508097
$934$	0	0
$935$	−17.7471	−0.580394
$936$	0	0
$937$	−35.6176	−1.16358	−0.581788	−	0.813340i	$-0.697647\pi$
$937$	−35.6176	−1.16358	−0.581788	+	0.813340i	$0.697647\pi$
$938$	0	0
$939$	22.8680	0.746267
$940$	0	0
$941$	36.4565	1.18845	0.594223	−	0.804300i	$-0.297459\pi$
$941$	36.4565	1.18845	0.594223	+	0.804300i	$0.297459\pi$
$942$	0	0
$943$	0.725868	0.0236375
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.0725	0.944729	0.472365	−	0.881403i	$-0.343401\pi$
$947$	29.0725	0.944729	0.472365	+	0.881403i	$0.343401\pi$
$948$	0	0
$949$	8.61752	0.279737
$950$	0	0
$951$	−11.2197	−0.363823
$952$	0	0
$953$	56.7993	1.83991	0.919955	−	0.392025i	$-0.128225\pi$
$953$	56.7993	1.83991	0.919955	+	0.392025i	$0.128225\pi$
$954$	0	0
$955$	−35.3709	−1.14458
$956$	0	0
$957$	−2.71709	−0.0878310
$958$	0	0
$959$	0	0
$960$	0	0
$961$	71.2903	2.29969
$962$	0	0
$963$	26.6444	0.858605
$964$	0	0
$965$	15.2551	0.491078
$966$	0	0
$967$	24.5018	0.787924	0.393962	−	0.919127i	$-0.371104\pi$
$967$	24.5018	0.787924	0.393962	+	0.919127i	$0.371104\pi$
$968$	0	0
$969$	−25.5582	−0.821048
$970$	0	0
$971$	9.12898	0.292963	0.146482	−	0.989213i	$-0.453205\pi$
$971$	9.12898	0.292963	0.146482	+	0.989213i	$0.453205\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−0.865904	−0.0277311
$976$	0	0
$977$	−22.2173	−0.710795	−0.355397	−	0.934715i	$-0.615654\pi$
$977$	−22.2173	−0.710795	−0.355397	+	0.934715i	$0.615654\pi$
$978$	0	0
$979$	−6.80735	−0.217564
$980$	0	0
$981$	15.4331	0.492740
$982$	0	0
$983$	−2.07208	−0.0660889	−0.0330445	−	0.999454i	$-0.510520\pi$
$983$	−2.07208	−0.0660889	−0.0330445	+	0.999454i	$0.510520\pi$
$984$	0	0
$985$	8.78643	0.279959
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.8164	−0.343942
$990$	0	0
$991$	−6.06800	−0.192756	−0.0963782	−	0.995345i	$-0.530726\pi$
$991$	−6.06800	−0.192756	−0.0963782	+	0.995345i	$0.530726\pi$
$992$	0	0
$993$	14.7581	0.468334
$994$	0	0
$995$	−6.92405	−0.219507
$996$	0	0
$997$	46.4885	1.47231	0.736153	−	0.676816i	$-0.236641\pi$
$997$	46.4885	1.47231	0.736153	+	0.676816i	$0.236641\pi$
$998$	0	0
$999$	−22.9448	−0.725943

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.bl.1.5		11
7.3	odd	6		1288.2.q.a.737.5	✓	22
7.5	odd	6		1288.2.q.a.921.5	yes	22
7.6	odd	2		9016.2.a.bq.1.7		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.q.a.737.5	✓	22	7.3	odd	6
1288.2.q.a.921.5	yes	22	7.5	odd	6
9016.2.a.bl.1.5		11	1.1	even	1	trivial
9016.2.a.bq.1.7		11	7.6	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:	\( N \)	\(=\)	\( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9016.a (trivial)

\(f(q)\)	\(=\)	\(q-0.656305 q^{3} +2.09142 q^{5} -2.56926 q^{9} +O(q^{10})\)	\(q-0.656305 q^{3} +2.09142 q^{5} -2.56926 q^{9} -1.84217 q^{11} -2.10767 q^{13} -1.37261 q^{15} +4.60636 q^{17} +8.45409 q^{19} +1.00000 q^{23} -0.625980 q^{25} +3.65513 q^{27} -2.24733 q^{29} -10.1139 q^{31} +1.20903 q^{33} -6.27743 q^{37} +1.38328 q^{39} +0.725868 q^{41} -10.8164 q^{43} -5.37340 q^{45} -2.14894 q^{47} -3.02317 q^{51} +12.8348 q^{53} -3.85275 q^{55} -5.54846 q^{57} +1.37532 q^{59} -1.80134 q^{61} -4.40802 q^{65} -5.83508 q^{67} -0.656305 q^{69} +4.43192 q^{71} -4.08864 q^{73} +0.410834 q^{75} +6.92763 q^{79} +5.30891 q^{81} +1.78309 q^{83} +9.63381 q^{85} +1.47494 q^{87} +3.69528 q^{89} +6.63778 q^{93} +17.6810 q^{95} -3.34404 q^{97} +4.73303 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 4 q^{3} - q^{5} + 11 q^{9}+O(q^{10}) \) 11 * q - 4 * q^3 - q^5 + 11 * q^9	\( 11 q - 4 q^{3} - q^{5} + 11 q^{9} + 3 q^{13} - 8 q^{15} - 5 q^{17} - 12 q^{19} + 11 q^{23} + 22 q^{25} - 19 q^{27} - 15 q^{29} - 16 q^{31} - 4 q^{33} + 3 q^{37} - q^{39} - 28 q^{41} - 9 q^{43} + 19 q^{45} - 31 q^{47} - 15 q^{51} + 13 q^{53} - 35 q^{55} - 21 q^{57} - 11 q^{59} + 19 q^{61} - 7 q^{65} + 19 q^{67} - 4 q^{69} - 5 q^{71} + 5 q^{73} - 28 q^{75} - 13 q^{79} + 35 q^{81} - 17 q^{83} - 39 q^{85} - 4 q^{87} - 10 q^{89} - 6 q^{93} + 33 q^{95} - 35 q^{97} + 13 q^{99}+O(q^{100}) \) 11 * q - 4 * q^3 - q^5 + 11 * q^9 + 3 * q^13 - 8 * q^15 - 5 * q^17 - 12 * q^19 + 11 * q^23 + 22 * q^25 - 19 * q^27 - 15 * q^29 - 16 * q^31 - 4 * q^33 + 3 * q^37 - q^39 - 28 * q^41 - 9 * q^43 + 19 * q^45 - 31 * q^47 - 15 * q^51 + 13 * q^53 - 35 * q^55 - 21 * q^57 - 11 * q^59 + 19 * q^61 - 7 * q^65 + 19 * q^67 - 4 * q^69 - 5 * q^71 + 5 * q^73 - 28 * q^75 - 13 * q^79 + 35 * q^81 - 17 * q^83 - 39 * q^85 - 4 * q^87 - 10 * q^89 - 6 * q^93 + 33 * q^95 - 35 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more