LMFDB - Embedded newform 7728.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1932)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.38197 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.38197 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -6.09017 q^{13} +1.38197 q^{15} -5.00000 q^{17} -5.47214 q^{19} +1.00000 q^{21} -1.00000 q^{23} -3.09017 q^{25} -1.00000 q^{27} -1.76393 q^{29} -4.70820 q^{31} -1.00000 q^{33} +1.38197 q^{35} -7.47214 q^{37} +6.09017 q^{39} -10.2361 q^{41} +9.32624 q^{43} -1.38197 q^{45} +9.70820 q^{47} +1.00000 q^{49} +5.00000 q^{51} +2.61803 q^{53} -1.38197 q^{55} +5.47214 q^{57} +1.38197 q^{59} +12.5623 q^{61} -1.00000 q^{63} +8.41641 q^{65} -8.32624 q^{67} +1.00000 q^{69} -11.3262 q^{71} -9.47214 q^{73} +3.09017 q^{75} -1.00000 q^{77} -2.70820 q^{79} +1.00000 q^{81} +13.1803 q^{83} +6.90983 q^{85} +1.76393 q^{87} -14.3820 q^{89} +6.09017 q^{91} +4.70820 q^{93} +7.56231 q^{95} -7.76393 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - q^{13} + 5 q^{15} - 10 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 5 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 2 q^{33} + 5 q^{35} - 6 q^{37} + q^{39} - 16 q^{41} + 3 q^{43} - 5 q^{45} + 6 q^{47} + 2 q^{49} + 10 q^{51} + 3 q^{53} - 5 q^{55} + 2 q^{57} + 5 q^{59} + 5 q^{61} - 2 q^{63} - 10 q^{65} - q^{67} + 2 q^{69} - 7 q^{71} - 10 q^{73} - 5 q^{75} - 2 q^{77} + 8 q^{79} + 2 q^{81} + 4 q^{83} + 25 q^{85} + 8 q^{87} - 31 q^{89} + q^{91} - 4 q^{93} - 5 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - q^13 + 5 * q^15 - 10 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 5 * q^25 - 2 * q^27 - 8 * q^29 + 4 * q^31 - 2 * q^33 + 5 * q^35 - 6 * q^37 + q^39 - 16 * q^41 + 3 * q^43 - 5 * q^45 + 6 * q^47 + 2 * q^49 + 10 * q^51 + 3 * q^53 - 5 * q^55 + 2 * q^57 + 5 * q^59 + 5 * q^61 - 2 * q^63 - 10 * q^65 - q^67 + 2 * q^69 - 7 * q^71 - 10 * q^73 - 5 * q^75 - 2 * q^77 + 8 * q^79 + 2 * q^81 + 4 * q^83 + 25 * q^85 + 8 * q^87 - 31 * q^89 + q^91 - 4 * q^93 - 5 * q^95 - 20 * q^97 + 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.38197	−0.618034	−0.309017	−	0.951057i	$-0.600000\pi$
$5$	−1.38197	−0.618034	−0.309017	+	0.951057i	$0.600000\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−6.09017	−1.68911	−0.844555	−	0.535469i	$-0.820135\pi$
$13$	−6.09017	−1.68911	−0.844555	+	0.535469i	$0.820135\pi$
$14$	0	0
$15$	1.38197	0.356822
$16$	0	0
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	−5.47214	−1.25539	−0.627697	−	0.778458i	$-0.716002\pi$
$19$	−5.47214	−1.25539	−0.627697	+	0.778458i	$0.716002\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.09017	−0.618034
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.76393	−0.327554	−0.163777	−	0.986497i	$-0.552368\pi$
$29$	−1.76393	−0.327554	−0.163777	+	0.986497i	$0.552368\pi$
$30$	0	0
$31$	−4.70820	−0.845618	−0.422809	−	0.906219i	$-0.638956\pi$
$31$	−4.70820	−0.845618	−0.422809	+	0.906219i	$0.638956\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	1.38197	0.233595
$36$	0	0
$37$	−7.47214	−1.22841	−0.614206	−	0.789146i	$-0.710524\pi$
$37$	−7.47214	−1.22841	−0.614206	+	0.789146i	$0.710524\pi$
$38$	0	0
$39$	6.09017	0.975208
$40$	0	0
$41$	−10.2361	−1.59861	−0.799303	−	0.600929i	$-0.794798\pi$
$41$	−10.2361	−1.59861	−0.799303	+	0.600929i	$0.794798\pi$
$42$	0	0
$43$	9.32624	1.42224	0.711119	−	0.703072i	$-0.248189\pi$
$43$	9.32624	1.42224	0.711119	+	0.703072i	$0.248189\pi$
$44$	0	0
$45$	−1.38197	−0.206011
$46$	0	0
$47$	9.70820	1.41609	0.708044	−	0.706169i	$-0.249578\pi$
$47$	9.70820	1.41609	0.708044	+	0.706169i	$0.249578\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.00000	0.700140
$52$	0	0
$53$	2.61803	0.359615	0.179807	−	0.983702i	$-0.442453\pi$
$53$	2.61803	0.359615	0.179807	+	0.983702i	$0.442453\pi$
$54$	0	0
$55$	−1.38197	−0.186344
$56$	0	0
$57$	5.47214	0.724802
$58$	0	0
$59$	1.38197	0.179917	0.0899583	−	0.995946i	$-0.471327\pi$
$59$	1.38197	0.179917	0.0899583	+	0.995946i	$0.471327\pi$
$60$	0	0
$61$	12.5623	1.60844	0.804219	−	0.594333i	$-0.202584\pi$
$61$	12.5623	1.60844	0.804219	+	0.594333i	$0.202584\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	8.41641	1.04393
$66$	0	0
$67$	−8.32624	−1.01721	−0.508606	−	0.860999i	$-0.669839\pi$
$67$	−8.32624	−1.01721	−0.508606	+	0.860999i	$0.669839\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−11.3262	−1.34418	−0.672089	−	0.740471i	$-0.734603\pi$
$71$	−11.3262	−1.34418	−0.672089	+	0.740471i	$0.734603\pi$
$72$	0	0
$73$	−9.47214	−1.10863	−0.554315	−	0.832307i	$-0.687020\pi$
$73$	−9.47214	−1.10863	−0.554315	+	0.832307i	$0.687020\pi$
$74$	0	0
$75$	3.09017	0.356822
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−2.70820	−0.304697	−0.152348	−	0.988327i	$-0.548684\pi$
$79$	−2.70820	−0.304697	−0.152348	+	0.988327i	$0.548684\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.1803	1.44673	0.723365	−	0.690466i	$-0.242594\pi$
$83$	13.1803	1.44673	0.723365	+	0.690466i	$0.242594\pi$
$84$	0	0
$85$	6.90983	0.749476
$86$	0	0
$87$	1.76393	0.189113
$88$	0	0
$89$	−14.3820	−1.52449	−0.762243	−	0.647291i	$-0.775902\pi$
$89$	−14.3820	−1.52449	−0.762243	+	0.647291i	$0.775902\pi$
$90$	0	0
$91$	6.09017	0.638423
$92$	0	0
$93$	4.70820	0.488218
$94$	0	0
$95$	7.56231	0.775876
$96$	0	0
$97$	−7.76393	−0.788308	−0.394154	−	0.919044i	$-0.628962\pi$
$97$	−7.76393	−0.788308	−0.394154	+	0.919044i	$0.628962\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−3.38197	−0.336518	−0.168259	−	0.985743i	$-0.553815\pi$
$101$	−3.38197	−0.336518	−0.168259	+	0.985743i	$0.553815\pi$
$102$	0	0
$103$	6.47214	0.637719	0.318859	−	0.947802i	$-0.396700\pi$
$103$	6.47214	0.637719	0.318859	+	0.947802i	$0.396700\pi$
$104$	0	0
$105$	−1.38197	−0.134866
$106$	0	0
$107$	12.3820	1.19701	0.598505	−	0.801119i	$-0.295762\pi$
$107$	12.3820	1.19701	0.598505	+	0.801119i	$0.295762\pi$
$108$	0	0
$109$	20.5066	1.96417	0.982087	−	0.188428i	$-0.0603393\pi$
$109$	20.5066	1.96417	0.982087	+	0.188428i	$0.0603393\pi$
$110$	0	0
$111$	7.47214	0.709224
$112$	0	0
$113$	9.85410	0.926996	0.463498	−	0.886098i	$-0.346594\pi$
$113$	9.85410	0.926996	0.463498	+	0.886098i	$0.346594\pi$
$114$	0	0
$115$	1.38197	0.128869
$116$	0	0
$117$	−6.09017	−0.563036
$118$	0	0
$119$	5.00000	0.458349
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	10.2361	0.922955
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	−2.38197	−0.211365	−0.105683	−	0.994400i	$-0.533703\pi$
$127$	−2.38197	−0.211365	−0.105683	+	0.994400i	$0.533703\pi$
$128$	0	0
$129$	−9.32624	−0.821129
$130$	0	0
$131$	−13.1803	−1.15157	−0.575786	−	0.817601i	$-0.695304\pi$
$131$	−13.1803	−1.15157	−0.575786	+	0.817601i	$0.695304\pi$
$132$	0	0
$133$	5.47214	0.474494
$134$	0	0
$135$	1.38197	0.118941
$136$	0	0
$137$	−13.4721	−1.15100	−0.575501	−	0.817801i	$-0.695193\pi$
$137$	−13.4721	−1.15100	−0.575501	+	0.817801i	$0.695193\pi$
$138$	0	0
$139$	−16.3262	−1.38477	−0.692387	−	0.721527i	$-0.743441\pi$
$139$	−16.3262	−1.38477	−0.692387	+	0.721527i	$0.743441\pi$
$140$	0	0
$141$	−9.70820	−0.817578
$142$	0	0
$143$	−6.09017	−0.509286
$144$	0	0
$145$	2.43769	0.202439
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−21.1246	−1.73060	−0.865298	−	0.501258i	$-0.832871\pi$
$149$	−21.1246	−1.73060	−0.865298	+	0.501258i	$0.832871\pi$
$150$	0	0
$151$	−0.763932	−0.0621679	−0.0310840	−	0.999517i	$-0.509896\pi$
$151$	−0.763932	−0.0621679	−0.0310840	+	0.999517i	$0.509896\pi$
$152$	0	0
$153$	−5.00000	−0.404226
$154$	0	0
$155$	6.50658	0.522621
$156$	0	0
$157$	−16.6525	−1.32901	−0.664506	−	0.747283i	$-0.731358\pi$
$157$	−16.6525	−1.32901	−0.664506	+	0.747283i	$0.731358\pi$
$158$	0	0
$159$	−2.61803	−0.207624
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	2.90983	0.227915	0.113958	−	0.993486i	$-0.463647\pi$
$163$	2.90983	0.227915	0.113958	+	0.993486i	$0.463647\pi$
$164$	0	0
$165$	1.38197	0.107586
$166$	0	0
$167$	−19.1803	−1.48422	−0.742110	−	0.670279i	$-0.766175\pi$
$167$	−19.1803	−1.48422	−0.742110	+	0.670279i	$0.766175\pi$
$168$	0	0
$169$	24.0902	1.85309
$170$	0	0
$171$	−5.47214	−0.418465
$172$	0	0
$173$	−13.1803	−1.00208	−0.501041	−	0.865423i	$-0.667050\pi$
$173$	−13.1803	−1.00208	−0.501041	+	0.865423i	$0.667050\pi$
$174$	0	0
$175$	3.09017	0.233595
$176$	0	0
$177$	−1.38197	−0.103875
$178$	0	0
$179$	23.2705	1.73932	0.869660	−	0.493652i	$-0.164338\pi$
$179$	23.2705	1.73932	0.869660	+	0.493652i	$0.164338\pi$
$180$	0	0
$181$	4.23607	0.314864	0.157432	−	0.987530i	$-0.449678\pi$
$181$	4.23607	0.314864	0.157432	+	0.987530i	$0.449678\pi$
$182$	0	0
$183$	−12.5623	−0.928632
$184$	0	0
$185$	10.3262	0.759200
$186$	0	0
$187$	−5.00000	−0.365636
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	14.6525	1.06022	0.530108	−	0.847930i	$-0.322151\pi$
$191$	14.6525	1.06022	0.530108	+	0.847930i	$0.322151\pi$
$192$	0	0
$193$	8.29180	0.596857	0.298428	−	0.954432i	$-0.403538\pi$
$193$	8.29180	0.596857	0.298428	+	0.954432i	$0.403538\pi$
$194$	0	0
$195$	−8.41641	−0.602711
$196$	0	0
$197$	20.3820	1.45215	0.726077	−	0.687613i	$-0.241341\pi$
$197$	20.3820	1.45215	0.726077	+	0.687613i	$0.241341\pi$
$198$	0	0
$199$	20.7984	1.47436	0.737179	−	0.675698i	$-0.236158\pi$
$199$	20.7984	1.47436	0.737179	+	0.675698i	$0.236158\pi$
$200$	0	0
$201$	8.32624	0.587288
$202$	0	0
$203$	1.76393	0.123804
$204$	0	0
$205$	14.1459	0.987992
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−5.47214	−0.378516
$210$	0	0
$211$	−13.7639	−0.947548	−0.473774	−	0.880646i	$-0.657109\pi$
$211$	−13.7639	−0.947548	−0.473774	+	0.880646i	$0.657109\pi$
$212$	0	0
$213$	11.3262	0.776061
$214$	0	0
$215$	−12.8885	−0.878991
$216$	0	0
$217$	4.70820	0.319614
$218$	0	0
$219$	9.47214	0.640068
$220$	0	0
$221$	30.4508	2.04835
$222$	0	0
$223$	9.56231	0.640339	0.320170	−	0.947360i	$-0.396260\pi$
$223$	9.56231	0.640339	0.320170	+	0.947360i	$0.396260\pi$
$224$	0	0
$225$	−3.09017	−0.206011
$226$	0	0
$227$	10.1459	0.673407	0.336703	−	0.941611i	$-0.390688\pi$
$227$	10.1459	0.673407	0.336703	+	0.941611i	$0.390688\pi$
$228$	0	0
$229$	16.7984	1.11007	0.555034	−	0.831828i	$-0.312705\pi$
$229$	16.7984	1.11007	0.555034	+	0.831828i	$0.312705\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−1.56231	−0.102350	−0.0511750	−	0.998690i	$-0.516297\pi$
$233$	−1.56231	−0.102350	−0.0511750	+	0.998690i	$0.516297\pi$
$234$	0	0
$235$	−13.4164	−0.875190
$236$	0	0
$237$	2.70820	0.175917
$238$	0	0
$239$	−28.3262	−1.83227	−0.916136	−	0.400868i	$-0.868709\pi$
$239$	−28.3262	−1.83227	−0.916136	+	0.400868i	$0.868709\pi$
$240$	0	0
$241$	5.65248	0.364108	0.182054	−	0.983289i	$-0.441725\pi$
$241$	5.65248	0.364108	0.182054	+	0.983289i	$0.441725\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.38197	−0.0882906
$246$	0	0
$247$	33.3262	2.12050
$248$	0	0
$249$	−13.1803	−0.835270
$250$	0	0
$251$	−26.1803	−1.65249	−0.826244	−	0.563312i	$-0.809527\pi$
$251$	−26.1803	−1.65249	−0.826244	+	0.563312i	$0.809527\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	−6.90983	−0.432710
$256$	0	0
$257$	−13.7082	−0.855094	−0.427547	−	0.903993i	$-0.640622\pi$
$257$	−13.7082	−0.855094	−0.427547	+	0.903993i	$0.640622\pi$
$258$	0	0
$259$	7.47214	0.464296
$260$	0	0
$261$	−1.76393	−0.109185
$262$	0	0
$263$	4.81966	0.297193	0.148596	−	0.988898i	$-0.452524\pi$
$263$	4.81966	0.297193	0.148596	+	0.988898i	$0.452524\pi$
$264$	0	0
$265$	−3.61803	−0.222254
$266$	0	0
$267$	14.3820	0.880162
$268$	0	0
$269$	13.1459	0.801520	0.400760	−	0.916183i	$-0.368746\pi$
$269$	13.1459	0.801520	0.400760	+	0.916183i	$0.368746\pi$
$270$	0	0
$271$	15.4721	0.939865	0.469933	−	0.882702i	$-0.344278\pi$
$271$	15.4721	0.939865	0.469933	+	0.882702i	$0.344278\pi$
$272$	0	0
$273$	−6.09017	−0.368594
$274$	0	0
$275$	−3.09017	−0.186344
$276$	0	0
$277$	19.9098	1.19627	0.598133	−	0.801397i	$-0.295909\pi$
$277$	19.9098	1.19627	0.598133	+	0.801397i	$0.295909\pi$
$278$	0	0
$279$	−4.70820	−0.281873
$280$	0	0
$281$	−14.1803	−0.845928	−0.422964	−	0.906146i	$-0.639010\pi$
$281$	−14.1803	−0.845928	−0.422964	+	0.906146i	$0.639010\pi$
$282$	0	0
$283$	−32.0344	−1.90425	−0.952125	−	0.305709i	$-0.901106\pi$
$283$	−32.0344	−1.90425	−0.952125	+	0.305709i	$0.901106\pi$
$284$	0	0
$285$	−7.56231	−0.447952
$286$	0	0
$287$	10.2361	0.604216
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	7.76393	0.455130
$292$	0	0
$293$	−6.47214	−0.378106	−0.189053	−	0.981967i	$-0.560542\pi$
$293$	−6.47214	−0.378106	−0.189053	+	0.981967i	$0.560542\pi$
$294$	0	0
$295$	−1.90983	−0.111195
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	6.09017	0.352204
$300$	0	0
$301$	−9.32624	−0.537555
$302$	0	0
$303$	3.38197	0.194289
$304$	0	0
$305$	−17.3607	−0.994070
$306$	0	0
$307$	29.0689	1.65905	0.829524	−	0.558470i	$-0.188612\pi$
$307$	29.0689	1.65905	0.829524	+	0.558470i	$0.188612\pi$
$308$	0	0
$309$	−6.47214	−0.368187
$310$	0	0
$311$	3.90983	0.221706	0.110853	−	0.993837i	$-0.464642\pi$
$311$	3.90983	0.221706	0.110853	+	0.993837i	$0.464642\pi$
$312$	0	0
$313$	−27.4164	−1.54967	−0.774833	−	0.632165i	$-0.782166\pi$
$313$	−27.4164	−1.54967	−0.774833	+	0.632165i	$0.782166\pi$
$314$	0	0
$315$	1.38197	0.0778650
$316$	0	0
$317$	28.6180	1.60735	0.803674	−	0.595069i	$-0.202875\pi$
$317$	28.6180	1.60735	0.803674	+	0.595069i	$0.202875\pi$
$318$	0	0
$319$	−1.76393	−0.0987612
$320$	0	0
$321$	−12.3820	−0.691094
$322$	0	0
$323$	27.3607	1.52239
$324$	0	0
$325$	18.8197	1.04393
$326$	0	0
$327$	−20.5066	−1.13402
$328$	0	0
$329$	−9.70820	−0.535231
$330$	0	0
$331$	6.58359	0.361867	0.180933	−	0.983495i	$-0.442088\pi$
$331$	6.58359	0.361867	0.180933	+	0.983495i	$0.442088\pi$
$332$	0	0
$333$	−7.47214	−0.409471
$334$	0	0
$335$	11.5066	0.628672
$336$	0	0
$337$	33.9787	1.85094	0.925469	−	0.378823i	$-0.123671\pi$
$337$	33.9787	1.85094	0.925469	+	0.378823i	$0.123671\pi$
$338$	0	0
$339$	−9.85410	−0.535201
$340$	0	0
$341$	−4.70820	−0.254964
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.38197	−0.0744025
$346$	0	0
$347$	−28.3050	−1.51949	−0.759745	−	0.650221i	$-0.774676\pi$
$347$	−28.3050	−1.51949	−0.759745	+	0.650221i	$0.774676\pi$
$348$	0	0
$349$	32.0344	1.71476	0.857382	−	0.514680i	$-0.172089\pi$
$349$	32.0344	1.71476	0.857382	+	0.514680i	$0.172089\pi$
$350$	0	0
$351$	6.09017	0.325069
$352$	0	0
$353$	15.6525	0.833097	0.416549	−	0.909113i	$-0.363240\pi$
$353$	15.6525	0.833097	0.416549	+	0.909113i	$0.363240\pi$
$354$	0	0
$355$	15.6525	0.830747
$356$	0	0
$357$	−5.00000	−0.264628
$358$	0	0
$359$	−3.56231	−0.188011	−0.0940057	−	0.995572i	$-0.529967\pi$
$359$	−3.56231	−0.188011	−0.0940057	+	0.995572i	$0.529967\pi$
$360$	0	0
$361$	10.9443	0.576014
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	13.0902	0.685171
$366$	0	0
$367$	−6.27051	−0.327318	−0.163659	−	0.986517i	$-0.552330\pi$
$367$	−6.27051	−0.327318	−0.163659	+	0.986517i	$0.552330\pi$
$368$	0	0
$369$	−10.2361	−0.532868
$370$	0	0
$371$	−2.61803	−0.135922
$372$	0	0
$373$	−2.88854	−0.149563	−0.0747816	−	0.997200i	$-0.523826\pi$
$373$	−2.88854	−0.149563	−0.0747816	+	0.997200i	$0.523826\pi$
$374$	0	0
$375$	−11.1803	−0.577350
$376$	0	0
$377$	10.7426	0.553274
$378$	0	0
$379$	−17.8885	−0.918873	−0.459436	−	0.888211i	$-0.651949\pi$
$379$	−17.8885	−0.918873	−0.459436	+	0.888211i	$0.651949\pi$
$380$	0	0
$381$	2.38197	0.122032
$382$	0	0
$383$	11.7639	0.601109	0.300554	−	0.953765i	$-0.402828\pi$
$383$	11.7639	0.601109	0.300554	+	0.953765i	$0.402828\pi$
$384$	0	0
$385$	1.38197	0.0704315
$386$	0	0
$387$	9.32624	0.474079
$388$	0	0
$389$	−20.1246	−1.02036	−0.510179	−	0.860068i	$-0.670421\pi$
$389$	−20.1246	−1.02036	−0.510179	+	0.860068i	$0.670421\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	13.1803	0.664860
$394$	0	0
$395$	3.74265	0.188313
$396$	0	0
$397$	20.1803	1.01282	0.506411	−	0.862292i	$-0.330972\pi$
$397$	20.1803	1.01282	0.506411	+	0.862292i	$0.330972\pi$
$398$	0	0
$399$	−5.47214	−0.273949
$400$	0	0
$401$	12.7082	0.634617	0.317309	−	0.948322i	$-0.397221\pi$
$401$	12.7082	0.634617	0.317309	+	0.948322i	$0.397221\pi$
$402$	0	0
$403$	28.6738	1.42834
$404$	0	0
$405$	−1.38197	−0.0686704
$406$	0	0
$407$	−7.47214	−0.370380
$408$	0	0
$409$	2.81966	0.139423	0.0697116	−	0.997567i	$-0.477792\pi$
$409$	2.81966	0.139423	0.0697116	+	0.997567i	$0.477792\pi$
$410$	0	0
$411$	13.4721	0.664531
$412$	0	0
$413$	−1.38197	−0.0680021
$414$	0	0
$415$	−18.2148	−0.894128
$416$	0	0
$417$	16.3262	0.799499
$418$	0	0
$419$	9.85410	0.481404	0.240702	−	0.970599i	$-0.422622\pi$
$419$	9.85410	0.481404	0.240702	+	0.970599i	$0.422622\pi$
$420$	0	0
$421$	8.96556	0.436955	0.218477	−	0.975842i	$-0.429891\pi$
$421$	8.96556	0.436955	0.218477	+	0.975842i	$0.429891\pi$
$422$	0	0
$423$	9.70820	0.472029
$424$	0	0
$425$	15.4508	0.749476
$426$	0	0
$427$	−12.5623	−0.607933
$428$	0	0
$429$	6.09017	0.294036
$430$	0	0
$431$	11.0344	0.531510	0.265755	−	0.964041i	$-0.414379\pi$
$431$	11.0344	0.531510	0.265755	+	0.964041i	$0.414379\pi$
$432$	0	0
$433$	7.70820	0.370433	0.185216	−	0.982698i	$-0.440701\pi$
$433$	7.70820	0.370433	0.185216	+	0.982698i	$0.440701\pi$
$434$	0	0
$435$	−2.43769	−0.116878
$436$	0	0
$437$	5.47214	0.261768
$438$	0	0
$439$	15.2918	0.729838	0.364919	−	0.931039i	$-0.381097\pi$
$439$	15.2918	0.729838	0.364919	+	0.931039i	$0.381097\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−8.94427	−0.424955	−0.212478	−	0.977166i	$-0.568153\pi$
$443$	−8.94427	−0.424955	−0.212478	+	0.977166i	$0.568153\pi$
$444$	0	0
$445$	19.8754	0.942184
$446$	0	0
$447$	21.1246	0.999160
$448$	0	0
$449$	−25.8541	−1.22013	−0.610065	−	0.792351i	$-0.708857\pi$
$449$	−25.8541	−1.22013	−0.610065	+	0.792351i	$0.708857\pi$
$450$	0	0
$451$	−10.2361	−0.481998
$452$	0	0
$453$	0.763932	0.0358927
$454$	0	0
$455$	−8.41641	−0.394567
$456$	0	0
$457$	27.2705	1.27566	0.637830	−	0.770177i	$-0.279832\pi$
$457$	27.2705	1.27566	0.637830	+	0.770177i	$0.279832\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	0	0
$461$	−37.2705	−1.73586	−0.867930	−	0.496686i	$-0.834550\pi$
$461$	−37.2705	−1.73586	−0.867930	+	0.496686i	$0.834550\pi$
$462$	0	0
$463$	−31.4721	−1.46263	−0.731317	−	0.682038i	$-0.761094\pi$
$463$	−31.4721	−1.46263	−0.731317	+	0.682038i	$0.761094\pi$
$464$	0	0
$465$	−6.50658	−0.301735
$466$	0	0
$467$	14.5967	0.675457	0.337728	−	0.941244i	$-0.390341\pi$
$467$	14.5967	0.675457	0.337728	+	0.941244i	$0.390341\pi$
$468$	0	0
$469$	8.32624	0.384470
$470$	0	0
$471$	16.6525	0.767306
$472$	0	0
$473$	9.32624	0.428821
$474$	0	0
$475$	16.9098	0.775876
$476$	0	0
$477$	2.61803	0.119872
$478$	0	0
$479$	−10.8885	−0.497510	−0.248755	−	0.968566i	$-0.580021\pi$
$479$	−10.8885	−0.497510	−0.248755	+	0.968566i	$0.580021\pi$
$480$	0	0
$481$	45.5066	2.07492
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	10.7295	0.487201
$486$	0	0
$487$	28.7771	1.30401	0.652007	−	0.758213i	$-0.273927\pi$
$487$	28.7771	1.30401	0.652007	+	0.758213i	$0.273927\pi$
$488$	0	0
$489$	−2.90983	−0.131587
$490$	0	0
$491$	6.09017	0.274846	0.137423	−	0.990512i	$-0.456118\pi$
$491$	6.09017	0.274846	0.137423	+	0.990512i	$0.456118\pi$
$492$	0	0
$493$	8.81966	0.397218
$494$	0	0
$495$	−1.38197	−0.0621148
$496$	0	0
$497$	11.3262	0.508051
$498$	0	0
$499$	−8.03444	−0.359671	−0.179836	−	0.983697i	$-0.557557\pi$
$499$	−8.03444	−0.359671	−0.179836	+	0.983697i	$0.557557\pi$
$500$	0	0
$501$	19.1803	0.856914
$502$	0	0
$503$	3.67376	0.163805	0.0819025	−	0.996640i	$-0.473900\pi$
$503$	3.67376	0.163805	0.0819025	+	0.996640i	$0.473900\pi$
$504$	0	0
$505$	4.67376	0.207980
$506$	0	0
$507$	−24.0902	−1.06988
$508$	0	0
$509$	11.2361	0.498030	0.249015	−	0.968500i	$-0.419893\pi$
$509$	11.2361	0.498030	0.249015	+	0.968500i	$0.419893\pi$
$510$	0	0
$511$	9.47214	0.419023
$512$	0	0
$513$	5.47214	0.241601
$514$	0	0
$515$	−8.94427	−0.394132
$516$	0	0
$517$	9.70820	0.426966
$518$	0	0
$519$	13.1803	0.578553
$520$	0	0
$521$	−31.3050	−1.37149	−0.685747	−	0.727840i	$-0.740525\pi$
$521$	−31.3050	−1.37149	−0.685747	+	0.727840i	$0.740525\pi$
$522$	0	0
$523$	−0.472136	−0.0206451	−0.0103225	−	0.999947i	$-0.503286\pi$
$523$	−0.472136	−0.0206451	−0.0103225	+	0.999947i	$0.503286\pi$
$524$	0	0
$525$	−3.09017	−0.134866
$526$	0	0
$527$	23.5410	1.02546
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.38197	0.0599722
$532$	0	0
$533$	62.3394	2.70022
$534$	0	0
$535$	−17.1115	−0.739793
$536$	0	0
$537$	−23.2705	−1.00420
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−0.763932	−0.0328440	−0.0164220	−	0.999865i	$-0.505228\pi$
$541$	−0.763932	−0.0328440	−0.0164220	+	0.999865i	$0.505228\pi$
$542$	0	0
$543$	−4.23607	−0.181787
$544$	0	0
$545$	−28.3394	−1.21393
$546$	0	0
$547$	−23.4508	−1.00269	−0.501343	−	0.865249i	$-0.667160\pi$
$547$	−23.4508	−1.00269	−0.501343	+	0.865249i	$0.667160\pi$
$548$	0	0
$549$	12.5623	0.536146
$550$	0	0
$551$	9.65248	0.411209
$552$	0	0
$553$	2.70820	0.115165
$554$	0	0
$555$	−10.3262	−0.438324
$556$	0	0
$557$	−1.81966	−0.0771015	−0.0385507	−	0.999257i	$-0.512274\pi$
$557$	−1.81966	−0.0771015	−0.0385507	+	0.999257i	$0.512274\pi$
$558$	0	0
$559$	−56.7984	−2.40232
$560$	0	0
$561$	5.00000	0.211100
$562$	0	0
$563$	−17.4377	−0.734911	−0.367456	−	0.930041i	$-0.619771\pi$
$563$	−17.4377	−0.734911	−0.367456	+	0.930041i	$0.619771\pi$
$564$	0	0
$565$	−13.6180	−0.572915
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−6.11146	−0.256206	−0.128103	−	0.991761i	$-0.540889\pi$
$569$	−6.11146	−0.256206	−0.128103	+	0.991761i	$0.540889\pi$
$570$	0	0
$571$	1.18034	0.0493957	0.0246978	−	0.999695i	$-0.492138\pi$
$571$	1.18034	0.0493957	0.0246978	+	0.999695i	$0.492138\pi$
$572$	0	0
$573$	−14.6525	−0.612116
$574$	0	0
$575$	3.09017	0.128869
$576$	0	0
$577$	21.4164	0.891577	0.445788	−	0.895138i	$-0.352923\pi$
$577$	21.4164	0.891577	0.445788	+	0.895138i	$0.352923\pi$
$578$	0	0
$579$	−8.29180	−0.344595
$580$	0	0
$581$	−13.1803	−0.546813
$582$	0	0
$583$	2.61803	0.108428
$584$	0	0
$585$	8.41641	0.347976
$586$	0	0
$587$	40.3820	1.66674	0.833371	−	0.552714i	$-0.186408\pi$
$587$	40.3820	1.66674	0.833371	+	0.552714i	$0.186408\pi$
$588$	0	0
$589$	25.7639	1.06158
$590$	0	0
$591$	−20.3820	−0.838402
$592$	0	0
$593$	41.0132	1.68421	0.842104	−	0.539315i	$-0.181317\pi$
$593$	41.0132	1.68421	0.842104	+	0.539315i	$0.181317\pi$
$594$	0	0
$595$	−6.90983	−0.283275
$596$	0	0
$597$	−20.7984	−0.851221
$598$	0	0
$599$	−19.7984	−0.808940	−0.404470	−	0.914551i	$-0.632544\pi$
$599$	−19.7984	−0.808940	−0.404470	+	0.914551i	$0.632544\pi$
$600$	0	0
$601$	−41.0902	−1.67610	−0.838051	−	0.545591i	$-0.816305\pi$
$601$	−41.0902	−1.67610	−0.838051	+	0.545591i	$0.816305\pi$
$602$	0	0
$603$	−8.32624	−0.339071
$604$	0	0
$605$	13.8197	0.561849
$606$	0	0
$607$	46.3394	1.88086	0.940429	−	0.339990i	$-0.110424\pi$
$607$	46.3394	1.88086	0.940429	+	0.339990i	$0.110424\pi$
$608$	0	0
$609$	−1.76393	−0.0714781
$610$	0	0
$611$	−59.1246	−2.39193
$612$	0	0
$613$	−21.0689	−0.850964	−0.425482	−	0.904967i	$-0.639895\pi$
$613$	−21.0689	−0.850964	−0.425482	+	0.904967i	$0.639895\pi$
$614$	0	0
$615$	−14.1459	−0.570418
$616$	0	0
$617$	−0.965558	−0.0388719	−0.0194360	−	0.999811i	$-0.506187\pi$
$617$	−0.965558	−0.0388719	−0.0194360	+	0.999811i	$0.506187\pi$
$618$	0	0
$619$	44.9230	1.80561	0.902804	−	0.430053i	$-0.141505\pi$
$619$	44.9230	1.80561	0.902804	+	0.430053i	$0.141505\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	14.3820	0.576201
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.47214	0.218536
$628$	0	0
$629$	37.3607	1.48967
$630$	0	0
$631$	−9.47214	−0.377080	−0.188540	−	0.982066i	$-0.560375\pi$
$631$	−9.47214	−0.377080	−0.188540	+	0.982066i	$0.560375\pi$
$632$	0	0
$633$	13.7639	0.547067
$634$	0	0
$635$	3.29180	0.130631
$636$	0	0
$637$	−6.09017	−0.241301
$638$	0	0
$639$	−11.3262	−0.448059
$640$	0	0
$641$	−14.3820	−0.568054	−0.284027	−	0.958816i	$-0.591670\pi$
$641$	−14.3820	−0.568054	−0.284027	+	0.958816i	$0.591670\pi$
$642$	0	0
$643$	22.0902	0.871151	0.435576	−	0.900152i	$-0.356545\pi$
$643$	22.0902	0.871151	0.435576	+	0.900152i	$0.356545\pi$
$644$	0	0
$645$	12.8885	0.507486
$646$	0	0
$647$	−27.0902	−1.06502	−0.532512	−	0.846422i	$-0.678752\pi$
$647$	−27.0902	−1.06502	−0.532512	+	0.846422i	$0.678752\pi$
$648$	0	0
$649$	1.38197	0.0542469
$650$	0	0
$651$	−4.70820	−0.184529
$652$	0	0
$653$	14.4377	0.564991	0.282495	−	0.959269i	$-0.408838\pi$
$653$	14.4377	0.564991	0.282495	+	0.959269i	$0.408838\pi$
$654$	0	0
$655$	18.2148	0.711710
$656$	0	0
$657$	−9.47214	−0.369543
$658$	0	0
$659$	1.00000	0.0389545	0.0194772	−	0.999810i	$-0.493800\pi$
$659$	1.00000	0.0389545	0.0194772	+	0.999810i	$0.493800\pi$
$660$	0	0
$661$	−25.2918	−0.983737	−0.491868	−	0.870670i	$-0.663686\pi$
$661$	−25.2918	−0.983737	−0.491868	+	0.870670i	$0.663686\pi$
$662$	0	0
$663$	−30.4508	−1.18261
$664$	0	0
$665$	−7.56231	−0.293254
$666$	0	0
$667$	1.76393	0.0682997
$668$	0	0
$669$	−9.56231	−0.369700
$670$	0	0
$671$	12.5623	0.484962
$672$	0	0
$673$	12.0557	0.464714	0.232357	−	0.972631i	$-0.425356\pi$
$673$	12.0557	0.464714	0.232357	+	0.972631i	$0.425356\pi$
$674$	0	0
$675$	3.09017	0.118941
$676$	0	0
$677$	−20.5623	−0.790274	−0.395137	−	0.918622i	$-0.629303\pi$
$677$	−20.5623	−0.790274	−0.395137	+	0.918622i	$0.629303\pi$
$678$	0	0
$679$	7.76393	0.297952
$680$	0	0
$681$	−10.1459	−0.388792
$682$	0	0
$683$	−1.94427	−0.0743955	−0.0371977	−	0.999308i	$-0.511843\pi$
$683$	−1.94427	−0.0743955	−0.0371977	+	0.999308i	$0.511843\pi$
$684$	0	0
$685$	18.6180	0.711359
$686$	0	0
$687$	−16.7984	−0.640898
$688$	0	0
$689$	−15.9443	−0.607428
$690$	0	0
$691$	37.2148	1.41572	0.707859	−	0.706354i	$-0.249661\pi$
$691$	37.2148	1.41572	0.707859	+	0.706354i	$0.249661\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	22.5623	0.855837
$696$	0	0
$697$	51.1803	1.93859
$698$	0	0
$699$	1.56231	0.0590918
$700$	0	0
$701$	−37.0902	−1.40088	−0.700438	−	0.713713i	$-0.747012\pi$
$701$	−37.0902	−1.40088	−0.700438	+	0.713713i	$0.747012\pi$
$702$	0	0
$703$	40.8885	1.54214
$704$	0	0
$705$	13.4164	0.505291
$706$	0	0
$707$	3.38197	0.127192
$708$	0	0
$709$	−42.1591	−1.58332	−0.791658	−	0.610964i	$-0.790782\pi$
$709$	−42.1591	−1.58332	−0.791658	+	0.610964i	$0.790782\pi$
$710$	0	0
$711$	−2.70820	−0.101566
$712$	0	0
$713$	4.70820	0.176324
$714$	0	0
$715$	8.41641	0.314756
$716$	0	0
$717$	28.3262	1.05786
$718$	0	0
$719$	−50.7771	−1.89367	−0.946833	−	0.321726i	$-0.895737\pi$
$719$	−50.7771	−1.89367	−0.946833	+	0.321726i	$0.895737\pi$
$720$	0	0
$721$	−6.47214	−0.241035
$722$	0	0
$723$	−5.65248	−0.210218
$724$	0	0
$725$	5.45085	0.202439
$726$	0	0
$727$	−43.2492	−1.60402	−0.802012	−	0.597307i	$-0.796237\pi$
$727$	−43.2492	−1.60402	−0.802012	+	0.597307i	$0.796237\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−46.6312	−1.72472
$732$	0	0
$733$	15.7082	0.580196	0.290098	−	0.956997i	$-0.406312\pi$
$733$	15.7082	0.580196	0.290098	+	0.956997i	$0.406312\pi$
$734$	0	0
$735$	1.38197	0.0509746
$736$	0	0
$737$	−8.32624	−0.306701
$738$	0	0
$739$	15.7639	0.579886	0.289943	−	0.957044i	$-0.406364\pi$
$739$	15.7639	0.579886	0.289943	+	0.957044i	$0.406364\pi$
$740$	0	0
$741$	−33.3262	−1.22427
$742$	0	0
$743$	−0.145898	−0.00535248	−0.00267624	−	0.999996i	$-0.500852\pi$
$743$	−0.145898	−0.00535248	−0.00267624	+	0.999996i	$0.500852\pi$
$744$	0	0
$745$	29.1935	1.06957
$746$	0	0
$747$	13.1803	0.482243
$748$	0	0
$749$	−12.3820	−0.452427
$750$	0	0
$751$	38.3262	1.39854	0.699272	−	0.714856i	$-0.253508\pi$
$751$	38.3262	1.39854	0.699272	+	0.714856i	$0.253508\pi$
$752$	0	0
$753$	26.1803	0.954065
$754$	0	0
$755$	1.05573	0.0384219
$756$	0	0
$757$	−10.4164	−0.378591	−0.189295	−	0.981920i	$-0.560620\pi$
$757$	−10.4164	−0.378591	−0.189295	+	0.981920i	$0.560620\pi$
$758$	0	0
$759$	1.00000	0.0362977
$760$	0	0
$761$	−24.9443	−0.904229	−0.452115	−	0.891960i	$-0.649330\pi$
$761$	−24.9443	−0.904229	−0.452115	+	0.891960i	$0.649330\pi$
$762$	0	0
$763$	−20.5066	−0.742388
$764$	0	0
$765$	6.90983	0.249825
$766$	0	0
$767$	−8.41641	−0.303899
$768$	0	0
$769$	−44.6525	−1.61021	−0.805105	−	0.593133i	$-0.797891\pi$
$769$	−44.6525	−1.61021	−0.805105	+	0.593133i	$0.797891\pi$
$770$	0	0
$771$	13.7082	0.493689
$772$	0	0
$773$	−14.7082	−0.529017	−0.264509	−	0.964383i	$-0.585210\pi$
$773$	−14.7082	−0.529017	−0.264509	+	0.964383i	$0.585210\pi$
$774$	0	0
$775$	14.5492	0.522621
$776$	0	0
$777$	−7.47214	−0.268061
$778$	0	0
$779$	56.0132	2.00688
$780$	0	0
$781$	−11.3262	−0.405285
$782$	0	0
$783$	1.76393	0.0630378
$784$	0	0
$785$	23.0132	0.821375
$786$	0	0
$787$	17.7426	0.632457	0.316229	−	0.948683i	$-0.397583\pi$
$787$	17.7426	0.632457	0.316229	+	0.948683i	$0.397583\pi$
$788$	0	0
$789$	−4.81966	−0.171584
$790$	0	0
$791$	−9.85410	−0.350372
$792$	0	0
$793$	−76.5066	−2.71683
$794$	0	0
$795$	3.61803	0.128318
$796$	0	0
$797$	−18.0689	−0.640033	−0.320016	−	0.947412i	$-0.603688\pi$
$797$	−18.0689	−0.640033	−0.320016	+	0.947412i	$0.603688\pi$
$798$	0	0
$799$	−48.5410	−1.71726
$800$	0	0
$801$	−14.3820	−0.508162
$802$	0	0
$803$	−9.47214	−0.334264
$804$	0	0
$805$	−1.38197	−0.0487079
$806$	0	0
$807$	−13.1459	−0.462758
$808$	0	0
$809$	49.6180	1.74448	0.872239	−	0.489081i	$-0.162668\pi$
$809$	49.6180	1.74448	0.872239	+	0.489081i	$0.162668\pi$
$810$	0	0
$811$	2.63932	0.0926791	0.0463395	−	0.998926i	$-0.485244\pi$
$811$	2.63932	0.0926791	0.0463395	+	0.998926i	$0.485244\pi$
$812$	0	0
$813$	−15.4721	−0.542631
$814$	0	0
$815$	−4.02129	−0.140860
$816$	0	0
$817$	−51.0344	−1.78547
$818$	0	0
$819$	6.09017	0.212808
$820$	0	0
$821$	38.6525	1.34898	0.674490	−	0.738284i	$-0.264363\pi$
$821$	38.6525	1.34898	0.674490	+	0.738284i	$0.264363\pi$
$822$	0	0
$823$	−10.7984	−0.376408	−0.188204	−	0.982130i	$-0.560267\pi$
$823$	−10.7984	−0.376408	−0.188204	+	0.982130i	$0.560267\pi$
$824$	0	0
$825$	3.09017	0.107586
$826$	0	0
$827$	−29.2016	−1.01544	−0.507720	−	0.861522i	$-0.669512\pi$
$827$	−29.2016	−1.01544	−0.507720	+	0.861522i	$0.669512\pi$
$828$	0	0
$829$	0.236068	0.00819898	0.00409949	−	0.999992i	$-0.498695\pi$
$829$	0.236068	0.00819898	0.00409949	+	0.999992i	$0.498695\pi$
$830$	0	0
$831$	−19.9098	−0.690664
$832$	0	0
$833$	−5.00000	−0.173240
$834$	0	0
$835$	26.5066	0.917298
$836$	0	0
$837$	4.70820	0.162739
$838$	0	0
$839$	−26.2705	−0.906959	−0.453479	−	0.891267i	$-0.649817\pi$
$839$	−26.2705	−0.906959	−0.453479	+	0.891267i	$0.649817\pi$
$840$	0	0
$841$	−25.8885	−0.892708
$842$	0	0
$843$	14.1803	0.488397
$844$	0	0
$845$	−33.2918	−1.14527
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	32.0344	1.09942
$850$	0	0
$851$	7.47214	0.256142
$852$	0	0
$853$	22.5410	0.771790	0.385895	−	0.922543i	$-0.373893\pi$
$853$	22.5410	0.771790	0.385895	+	0.922543i	$0.373893\pi$
$854$	0	0
$855$	7.56231	0.258625
$856$	0	0
$857$	−8.65248	−0.295563	−0.147781	−	0.989020i	$-0.547213\pi$
$857$	−8.65248	−0.295563	−0.147781	+	0.989020i	$0.547213\pi$
$858$	0	0
$859$	11.8885	0.405632	0.202816	−	0.979217i	$-0.434991\pi$
$859$	11.8885	0.405632	0.202816	+	0.979217i	$0.434991\pi$
$860$	0	0
$861$	−10.2361	−0.348844
$862$	0	0
$863$	−31.4853	−1.07177	−0.535886	−	0.844290i	$-0.680022\pi$
$863$	−31.4853	−1.07177	−0.535886	+	0.844290i	$0.680022\pi$
$864$	0	0
$865$	18.2148	0.619321
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−2.70820	−0.0918695
$870$	0	0
$871$	50.7082	1.71818
$872$	0	0
$873$	−7.76393	−0.262769
$874$	0	0
$875$	−11.1803	−0.377964
$876$	0	0
$877$	5.30495	0.179135	0.0895677	−	0.995981i	$-0.471451\pi$
$877$	5.30495	0.179135	0.0895677	+	0.995981i	$0.471451\pi$
$878$	0	0
$879$	6.47214	0.218300
$880$	0	0
$881$	−51.1246	−1.72243	−0.861216	−	0.508239i	$-0.830297\pi$
$881$	−51.1246	−1.72243	−0.861216	+	0.508239i	$0.830297\pi$
$882$	0	0
$883$	−10.6738	−0.359201	−0.179600	−	0.983740i	$-0.557480\pi$
$883$	−10.6738	−0.359201	−0.179600	+	0.983740i	$0.557480\pi$
$884$	0	0
$885$	1.90983	0.0641982
$886$	0	0
$887$	48.2705	1.62077	0.810383	−	0.585901i	$-0.199259\pi$
$887$	48.2705	1.62077	0.810383	+	0.585901i	$0.199259\pi$
$888$	0	0
$889$	2.38197	0.0798886
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−53.1246	−1.77775
$894$	0	0
$895$	−32.1591	−1.07496
$896$	0	0
$897$	−6.09017	−0.203345
$898$	0	0
$899$	8.30495	0.276986
$900$	0	0
$901$	−13.0902	−0.436097
$902$	0	0
$903$	9.32624	0.310358
$904$	0	0
$905$	−5.85410	−0.194597
$906$	0	0
$907$	−15.9230	−0.528714	−0.264357	−	0.964425i	$-0.585160\pi$
$907$	−15.9230	−0.528714	−0.264357	+	0.964425i	$0.585160\pi$
$908$	0	0
$909$	−3.38197	−0.112173
$910$	0	0
$911$	24.5410	0.813080	0.406540	−	0.913633i	$-0.366735\pi$
$911$	24.5410	0.813080	0.406540	+	0.913633i	$0.366735\pi$
$912$	0	0
$913$	13.1803	0.436206
$914$	0	0
$915$	17.3607	0.573926
$916$	0	0
$917$	13.1803	0.435253
$918$	0	0
$919$	−19.0000	−0.626752	−0.313376	−	0.949629i	$-0.601460\pi$
$919$	−19.0000	−0.626752	−0.313376	+	0.949629i	$0.601460\pi$
$920$	0	0
$921$	−29.0689	−0.957852
$922$	0	0
$923$	68.9787	2.27046
$924$	0	0
$925$	23.0902	0.759200
$926$	0	0
$927$	6.47214	0.212573
$928$	0	0
$929$	9.96556	0.326959	0.163480	−	0.986547i	$-0.447728\pi$
$929$	9.96556	0.326959	0.163480	+	0.986547i	$0.447728\pi$
$930$	0	0
$931$	−5.47214	−0.179342
$932$	0	0
$933$	−3.90983	−0.128002
$934$	0	0
$935$	6.90983	0.225976
$936$	0	0
$937$	−13.0000	−0.424691	−0.212346	−	0.977195i	$-0.568110\pi$
$937$	−13.0000	−0.424691	−0.212346	+	0.977195i	$0.568110\pi$
$938$	0	0
$939$	27.4164	0.894701
$940$	0	0
$941$	42.8328	1.39631	0.698155	−	0.715947i	$-0.254005\pi$
$941$	42.8328	1.39631	0.698155	+	0.715947i	$0.254005\pi$
$942$	0	0
$943$	10.2361	0.333332
$944$	0	0
$945$	−1.38197	−0.0449554
$946$	0	0
$947$	42.3607	1.37654	0.688269	−	0.725456i	$-0.258371\pi$
$947$	42.3607	1.37654	0.688269	+	0.725456i	$0.258371\pi$
$948$	0	0
$949$	57.6869	1.87260
$950$	0	0
$951$	−28.6180	−0.928003
$952$	0	0
$953$	−54.6312	−1.76968	−0.884839	−	0.465897i	$-0.845732\pi$
$953$	−54.6312	−1.76968	−0.884839	+	0.465897i	$0.845732\pi$
$954$	0	0
$955$	−20.2492	−0.655249
$956$	0	0
$957$	1.76393	0.0570198
$958$	0	0
$959$	13.4721	0.435038
$960$	0	0
$961$	−8.83282	−0.284930
$962$	0	0
$963$	12.3820	0.399003
$964$	0	0
$965$	−11.4590	−0.368878
$966$	0	0
$967$	29.0557	0.934369	0.467185	−	0.884160i	$-0.345268\pi$
$967$	29.0557	0.934369	0.467185	+	0.884160i	$0.345268\pi$
$968$	0	0
$969$	−27.3607	−0.878952
$970$	0	0
$971$	26.7426	0.858212	0.429106	−	0.903254i	$-0.358829\pi$
$971$	26.7426	0.858212	0.429106	+	0.903254i	$0.358829\pi$
$972$	0	0
$973$	16.3262	0.523395
$974$	0	0
$975$	−18.8197	−0.602711
$976$	0	0
$977$	17.2574	0.552112	0.276056	−	0.961142i	$-0.410973\pi$
$977$	17.2574	0.552112	0.276056	+	0.961142i	$0.410973\pi$
$978$	0	0
$979$	−14.3820	−0.459650
$980$	0	0
$981$	20.5066	0.654725
$982$	0	0
$983$	4.70820	0.150168	0.0750842	−	0.997177i	$-0.476077\pi$
$983$	4.70820	0.150168	0.0750842	+	0.997177i	$0.476077\pi$
$984$	0	0
$985$	−28.1672	−0.897481
$986$	0	0
$987$	9.70820	0.309016
$988$	0	0
$989$	−9.32624	−0.296557
$990$	0	0
$991$	44.2705	1.40630	0.703150	−	0.711042i	$-0.251776\pi$
$991$	44.2705	1.40630	0.703150	+	0.711042i	$0.251776\pi$
$992$	0	0
$993$	−6.58359	−0.208924
$994$	0	0
$995$	−28.7426	−0.911203
$996$	0	0
$997$	−22.2361	−0.704223	−0.352112	−	0.935958i	$-0.614536\pi$
$997$	−22.2361	−0.704223	−0.352112	+	0.935958i	$0.614536\pi$
$998$	0	0
$999$	7.47214	0.236408

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.w.1.2		2
4.3	odd	2		1932.2.a.f.1.2	✓	2
12.11	even	2		5796.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.f.1.2	✓	2	4.3	odd	2
5796.2.a.n.1.1		2	12.11	even	2
7728.2.a.w.1.2		2	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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.38197 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.38197 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -6.09017 q^{13} +1.38197 q^{15} -5.00000 q^{17} -5.47214 q^{19} +1.00000 q^{21} -1.00000 q^{23} -3.09017 q^{25} -1.00000 q^{27} -1.76393 q^{29} -4.70820 q^{31} -1.00000 q^{33} +1.38197 q^{35} -7.47214 q^{37} +6.09017 q^{39} -10.2361 q^{41} +9.32624 q^{43} -1.38197 q^{45} +9.70820 q^{47} +1.00000 q^{49} +5.00000 q^{51} +2.61803 q^{53} -1.38197 q^{55} +5.47214 q^{57} +1.38197 q^{59} +12.5623 q^{61} -1.00000 q^{63} +8.41641 q^{65} -8.32624 q^{67} +1.00000 q^{69} -11.3262 q^{71} -9.47214 q^{73} +3.09017 q^{75} -1.00000 q^{77} -2.70820 q^{79} +1.00000 q^{81} +13.1803 q^{83} +6.90983 q^{85} +1.76393 q^{87} -14.3820 q^{89} +6.09017 q^{91} +4.70820 q^{93} +7.56231 q^{95} -7.76393 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - q^{13} + 5 q^{15} - 10 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 5 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 2 q^{33} + 5 q^{35} - 6 q^{37} + q^{39} - 16 q^{41} + 3 q^{43} - 5 q^{45} + 6 q^{47} + 2 q^{49} + 10 q^{51} + 3 q^{53} - 5 q^{55} + 2 q^{57} + 5 q^{59} + 5 q^{61} - 2 q^{63} - 10 q^{65} - q^{67} + 2 q^{69} - 7 q^{71} - 10 q^{73} - 5 q^{75} - 2 q^{77} + 8 q^{79} + 2 q^{81} + 4 q^{83} + 25 q^{85} + 8 q^{87} - 31 q^{89} + q^{91} - 4 q^{93} - 5 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - q^13 + 5 * q^15 - 10 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 5 * q^25 - 2 * q^27 - 8 * q^29 + 4 * q^31 - 2 * q^33 + 5 * q^35 - 6 * q^37 + q^39 - 16 * q^41 + 3 * q^43 - 5 * q^45 + 6 * q^47 + 2 * q^49 + 10 * q^51 + 3 * q^53 - 5 * q^55 + 2 * q^57 + 5 * q^59 + 5 * q^61 - 2 * q^63 - 10 * q^65 - q^67 + 2 * q^69 - 7 * q^71 - 10 * q^73 - 5 * q^75 - 2 * q^77 + 8 * q^79 + 2 * q^81 + 4 * q^83 + 25 * q^85 + 8 * q^87 - 31 * q^89 + q^91 - 4 * q^93 - 5 * q^95 - 20 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more