LMFDB - Embedded newform 5520.2.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, 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("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -3.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -3.37228 q^{7} +1.00000 q^{9} -4.00000 q^{11} +4.74456 q^{13} +1.00000 q^{15} +5.37228 q^{17} -4.00000 q^{19} -3.37228 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -5.37228 q^{29} -3.37228 q^{31} -4.00000 q^{33} -3.37228 q^{35} -5.37228 q^{37} +4.74456 q^{39} -8.11684 q^{41} +4.00000 q^{43} +1.00000 q^{45} +6.74456 q^{47} +4.37228 q^{49} +5.37228 q^{51} +1.37228 q^{53} -4.00000 q^{55} -4.00000 q^{57} -8.62772 q^{59} -2.00000 q^{61} -3.37228 q^{63} +4.74456 q^{65} -15.3723 q^{67} -1.00000 q^{69} -3.37228 q^{71} -12.7446 q^{73} +1.00000 q^{75} +13.4891 q^{77} +13.4891 q^{79} +1.00000 q^{81} +12.8614 q^{83} +5.37228 q^{85} -5.37228 q^{87} -4.74456 q^{89} -16.0000 q^{91} -3.37228 q^{93} -4.00000 q^{95} +15.4891 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - 8 q^{11} - 2 q^{13} + 2 q^{15} + 5 q^{17} - 8 q^{19} - q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 5 q^{29} - q^{31} - 8 q^{33} - q^{35} - 5 q^{37} - 2 q^{39} + q^{41} + 8 q^{43} + 2 q^{45} + 2 q^{47} + 3 q^{49} + 5 q^{51} - 3 q^{53} - 8 q^{55} - 8 q^{57} - 23 q^{59} - 4 q^{61} - q^{63} - 2 q^{65} - 25 q^{67} - 2 q^{69} - q^{71} - 14 q^{73} + 2 q^{75} + 4 q^{77} + 4 q^{79} + 2 q^{81} - 3 q^{83} + 5 q^{85} - 5 q^{87} + 2 q^{89} - 32 q^{91} - q^{93} - 8 q^{95} + 8 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - 8 * q^11 - 2 * q^13 + 2 * q^15 + 5 * q^17 - 8 * q^19 - q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 - 5 * q^29 - q^31 - 8 * q^33 - q^35 - 5 * q^37 - 2 * q^39 + q^41 + 8 * q^43 + 2 * q^45 + 2 * q^47 + 3 * q^49 + 5 * q^51 - 3 * q^53 - 8 * q^55 - 8 * q^57 - 23 * q^59 - 4 * q^61 - q^63 - 2 * q^65 - 25 * q^67 - 2 * q^69 - q^71 - 14 * q^73 + 2 * q^75 + 4 * q^77 + 4 * q^79 + 2 * q^81 - 3 * q^83 + 5 * q^85 - 5 * q^87 + 2 * q^89 - 32 * q^91 - q^93 - 8 * q^95 + 8 * q^97 - 8 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.37228	−1.27460	−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.74456	1.31590	0.657952	−	0.753059i	$-0.271423\pi$
$13$	4.74456	1.31590	0.657952	+	0.753059i	$0.271423\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	5.37228	1.30297	0.651485	−	0.758662i	$-0.274146\pi$
$17$	5.37228	1.30297	0.651485	+	0.758662i	$0.274146\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−3.37228	−0.735892
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.37228	−0.997608	−0.498804	−	0.866715i	$-0.666227\pi$
$29$	−5.37228	−0.997608	−0.498804	+	0.866715i	$0.666227\pi$
$30$	0	0
$31$	−3.37228	−0.605680	−0.302840	−	0.953041i	$-0.597935\pi$
$31$	−3.37228	−0.605680	−0.302840	+	0.953041i	$0.597935\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	−3.37228	−0.570020
$36$	0	0
$37$	−5.37228	−0.883198	−0.441599	−	0.897213i	$-0.645589\pi$
$37$	−5.37228	−0.883198	−0.441599	+	0.897213i	$0.645589\pi$
$38$	0	0
$39$	4.74456	0.759738
$40$	0	0
$41$	−8.11684	−1.26764	−0.633819	−	0.773481i	$-0.718514\pi$
$41$	−8.11684	−1.26764	−0.633819	+	0.773481i	$0.718514\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	6.74456	0.983796	0.491898	−	0.870653i	$-0.336303\pi$
$47$	6.74456	0.983796	0.491898	+	0.870653i	$0.336303\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	5.37228	0.752270
$52$	0	0
$53$	1.37228	0.188497	0.0942487	−	0.995549i	$-0.469955\pi$
$53$	1.37228	0.188497	0.0942487	+	0.995549i	$0.469955\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−8.62772	−1.12323	−0.561617	−	0.827398i	$-0.689820\pi$
$59$	−8.62772	−1.12323	−0.561617	+	0.827398i	$0.689820\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−3.37228	−0.424868
$64$	0	0
$65$	4.74456	0.588491
$66$	0	0
$67$	−15.3723	−1.87802	−0.939012	−	0.343886i	$-0.888257\pi$
$67$	−15.3723	−1.87802	−0.939012	+	0.343886i	$0.888257\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−3.37228	−0.400216	−0.200108	−	0.979774i	$-0.564129\pi$
$71$	−3.37228	−0.400216	−0.200108	+	0.979774i	$0.564129\pi$
$72$	0	0
$73$	−12.7446	−1.49164	−0.745819	−	0.666149i	$-0.767942\pi$
$73$	−12.7446	−1.49164	−0.745819	+	0.666149i	$0.767942\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	13.4891	1.53723
$78$	0	0
$79$	13.4891	1.51765	0.758823	−	0.651297i	$-0.225775\pi$
$79$	13.4891	1.51765	0.758823	+	0.651297i	$0.225775\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.8614	1.41172	0.705861	−	0.708350i	$-0.250560\pi$
$83$	12.8614	1.41172	0.705861	+	0.708350i	$0.250560\pi$
$84$	0	0
$85$	5.37228	0.582706
$86$	0	0
$87$	−5.37228	−0.575969
$88$	0	0
$89$	−4.74456	−0.502923	−0.251461	−	0.967867i	$-0.580911\pi$
$89$	−4.74456	−0.502923	−0.251461	+	0.967867i	$0.580911\pi$
$90$	0	0
$91$	−16.0000	−1.67726
$92$	0	0
$93$	−3.37228	−0.349689
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	15.4891	1.57268	0.786341	−	0.617792i	$-0.211973\pi$
$97$	15.4891	1.57268	0.786341	+	0.617792i	$0.211973\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−13.3723	−1.33059	−0.665296	−	0.746580i	$-0.731695\pi$
$101$	−13.3723	−1.33059	−0.665296	+	0.746580i	$0.731695\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−3.37228	−0.329101
$106$	0	0
$107$	7.37228	0.712705	0.356353	−	0.934352i	$-0.384020\pi$
$107$	7.37228	0.712705	0.356353	+	0.934352i	$0.384020\pi$
$108$	0	0
$109$	4.74456	0.454447	0.227223	−	0.973843i	$-0.427035\pi$
$109$	4.74456	0.454447	0.227223	+	0.973843i	$0.427035\pi$
$110$	0	0
$111$	−5.37228	−0.509914
$112$	0	0
$113$	−8.11684	−0.763568	−0.381784	−	0.924251i	$-0.624690\pi$
$113$	−8.11684	−0.763568	−0.381784	+	0.924251i	$0.624690\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	4.74456	0.438635
$118$	0	0
$119$	−18.1168	−1.66077
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−8.11684	−0.731871
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.25544	−0.111402	−0.0557010	−	0.998447i	$-0.517739\pi$
$127$	−1.25544	−0.111402	−0.0557010	+	0.998447i	$0.517739\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	13.4891	1.16966
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−11.4891	−0.981582	−0.490791	−	0.871277i	$-0.663292\pi$
$137$	−11.4891	−0.981582	−0.490791	+	0.871277i	$0.663292\pi$
$138$	0	0
$139$	−8.62772	−0.731794	−0.365897	−	0.930655i	$-0.619238\pi$
$139$	−8.62772	−0.731794	−0.365897	+	0.930655i	$0.619238\pi$
$140$	0	0
$141$	6.74456	0.567995
$142$	0	0
$143$	−18.9783	−1.58704
$144$	0	0
$145$	−5.37228	−0.446144
$146$	0	0
$147$	4.37228	0.360620
$148$	0	0
$149$	3.48913	0.285840	0.142920	−	0.989734i	$-0.454351\pi$
$149$	3.48913	0.285840	0.142920	+	0.989734i	$0.454351\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	5.37228	0.434323
$154$	0	0
$155$	−3.37228	−0.270868
$156$	0	0
$157$	−17.6060	−1.40511	−0.702555	−	0.711630i	$-0.747957\pi$
$157$	−17.6060	−1.40511	−0.702555	+	0.711630i	$0.747957\pi$
$158$	0	0
$159$	1.37228	0.108829
$160$	0	0
$161$	3.37228	0.265773
$162$	0	0
$163$	−5.25544	−0.411638	−0.205819	−	0.978590i	$-0.565986\pi$
$163$	−5.25544	−0.411638	−0.205819	+	0.978590i	$0.565986\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	−1.25544	−0.0971487	−0.0485743	−	0.998820i	$-0.515468\pi$
$167$	−1.25544	−0.0971487	−0.0485743	+	0.998820i	$0.515468\pi$
$168$	0	0
$169$	9.51087	0.731606
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−14.2337	−1.08217	−0.541084	−	0.840969i	$-0.681986\pi$
$173$	−14.2337	−1.08217	−0.541084	+	0.840969i	$0.681986\pi$
$174$	0	0
$175$	−3.37228	−0.254921
$176$	0	0
$177$	−8.62772	−0.648499
$178$	0	0
$179$	−9.48913	−0.709251	−0.354625	−	0.935009i	$-0.615392\pi$
$179$	−9.48913	−0.709251	−0.354625	+	0.935009i	$0.615392\pi$
$180$	0	0
$181$	−14.2337	−1.05798	−0.528991	−	0.848628i	$-0.677429\pi$
$181$	−14.2337	−1.05798	−0.528991	+	0.848628i	$0.677429\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−5.37228	−0.394978
$186$	0	0
$187$	−21.4891	−1.57144
$188$	0	0
$189$	−3.37228	−0.245297
$190$	0	0
$191$	−20.2337	−1.46406	−0.732029	−	0.681273i	$-0.761427\pi$
$191$	−20.2337	−1.46406	−0.732029	+	0.681273i	$0.761427\pi$
$192$	0	0
$193$	−20.7446	−1.49323	−0.746613	−	0.665258i	$-0.768321\pi$
$193$	−20.7446	−1.49323	−0.746613	+	0.665258i	$0.768321\pi$
$194$	0	0
$195$	4.74456	0.339765
$196$	0	0
$197$	11.4891	0.818566	0.409283	−	0.912407i	$-0.365779\pi$
$197$	11.4891	0.818566	0.409283	+	0.912407i	$0.365779\pi$
$198$	0	0
$199$	−1.25544	−0.0889956	−0.0444978	−	0.999009i	$-0.514169\pi$
$199$	−1.25544	−0.0889956	−0.0444978	+	0.999009i	$0.514169\pi$
$200$	0	0
$201$	−15.3723	−1.08428
$202$	0	0
$203$	18.1168	1.27155
$204$	0	0
$205$	−8.11684	−0.566905
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	−20.8614	−1.43616	−0.718079	−	0.695961i	$-0.754978\pi$
$211$	−20.8614	−1.43616	−0.718079	+	0.695961i	$0.754978\pi$
$212$	0	0
$213$	−3.37228	−0.231065
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	11.3723	0.772001
$218$	0	0
$219$	−12.7446	−0.861198
$220$	0	0
$221$	25.4891	1.71458
$222$	0	0
$223$	−17.2554	−1.15551	−0.577755	−	0.816210i	$-0.696071\pi$
$223$	−17.2554	−1.15551	−0.577755	+	0.816210i	$0.696071\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	28.7446	1.89949	0.949747	−	0.313018i	$-0.101340\pi$
$229$	28.7446	1.89949	0.949747	+	0.313018i	$0.101340\pi$
$230$	0	0
$231$	13.4891	0.887519
$232$	0	0
$233$	−26.2337	−1.71863	−0.859313	−	0.511450i	$-0.829109\pi$
$233$	−26.2337	−1.71863	−0.859313	+	0.511450i	$0.829109\pi$
$234$	0	0
$235$	6.74456	0.439967
$236$	0	0
$237$	13.4891	0.876213
$238$	0	0
$239$	11.3723	0.735612	0.367806	−	0.929903i	$-0.380109\pi$
$239$	11.3723	0.735612	0.367806	+	0.929903i	$0.380109\pi$
$240$	0	0
$241$	−7.25544	−0.467364	−0.233682	−	0.972313i	$-0.575077\pi$
$241$	−7.25544	−0.467364	−0.233682	+	0.972313i	$0.575077\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	4.37228	0.279335
$246$	0	0
$247$	−18.9783	−1.20756
$248$	0	0
$249$	12.8614	0.815059
$250$	0	0
$251$	18.7446	1.18315	0.591573	−	0.806251i	$-0.298507\pi$
$251$	18.7446	1.18315	0.591573	+	0.806251i	$0.298507\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	5.37228	0.336425
$256$	0	0
$257$	−11.4891	−0.716672	−0.358336	−	0.933593i	$-0.616656\pi$
$257$	−11.4891	−0.716672	−0.358336	+	0.933593i	$0.616656\pi$
$258$	0	0
$259$	18.1168	1.12573
$260$	0	0
$261$	−5.37228	−0.332536
$262$	0	0
$263$	−0.861407	−0.0531166	−0.0265583	−	0.999647i	$-0.508455\pi$
$263$	−0.861407	−0.0531166	−0.0265583	+	0.999647i	$0.508455\pi$
$264$	0	0
$265$	1.37228	0.0842986
$266$	0	0
$267$	−4.74456	−0.290363
$268$	0	0
$269$	14.8614	0.906116	0.453058	−	0.891481i	$-0.350333\pi$
$269$	14.8614	0.906116	0.453058	+	0.891481i	$0.350333\pi$
$270$	0	0
$271$	5.88316	0.357376	0.178688	−	0.983906i	$-0.442815\pi$
$271$	5.88316	0.357376	0.178688	+	0.983906i	$0.442815\pi$
$272$	0	0
$273$	−16.0000	−0.968364
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	−3.37228	−0.201893
$280$	0	0
$281$	22.2337	1.32635	0.663175	−	0.748464i	$-0.269208\pi$
$281$	22.2337	1.32635	0.663175	+	0.748464i	$0.269208\pi$
$282$	0	0
$283$	−0.627719	−0.0373140	−0.0186570	−	0.999826i	$-0.505939\pi$
$283$	−0.627719	−0.0373140	−0.0186570	+	0.999826i	$0.505939\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	27.3723	1.61573
$288$	0	0
$289$	11.8614	0.697730
$290$	0	0
$291$	15.4891	0.907989
$292$	0	0
$293$	−18.8614	−1.10190	−0.550948	−	0.834540i	$-0.685734\pi$
$293$	−18.8614	−1.10190	−0.550948	+	0.834540i	$0.685734\pi$
$294$	0	0
$295$	−8.62772	−0.502325
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	−4.74456	−0.274385
$300$	0	0
$301$	−13.4891	−0.777500
$302$	0	0
$303$	−13.3723	−0.768217
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	2.86141	0.161736	0.0808681	−	0.996725i	$-0.474231\pi$
$313$	2.86141	0.161736	0.0808681	+	0.996725i	$0.474231\pi$
$314$	0	0
$315$	−3.37228	−0.190007
$316$	0	0
$317$	8.51087	0.478018	0.239009	−	0.971017i	$-0.423177\pi$
$317$	8.51087	0.478018	0.239009	+	0.971017i	$0.423177\pi$
$318$	0	0
$319$	21.4891	1.20316
$320$	0	0
$321$	7.37228	0.411481
$322$	0	0
$323$	−21.4891	−1.19569
$324$	0	0
$325$	4.74456	0.263181
$326$	0	0
$327$	4.74456	0.262375
$328$	0	0
$329$	−22.7446	−1.25395
$330$	0	0
$331$	−1.88316	−0.103508	−0.0517538	−	0.998660i	$-0.516481\pi$
$331$	−1.88316	−0.103508	−0.0517538	+	0.998660i	$0.516481\pi$
$332$	0	0
$333$	−5.37228	−0.294399
$334$	0	0
$335$	−15.3723	−0.839877
$336$	0	0
$337$	−27.4891	−1.49743	−0.748714	−	0.662893i	$-0.769328\pi$
$337$	−27.4891	−1.49743	−0.748714	+	0.662893i	$0.769328\pi$
$338$	0	0
$339$	−8.11684	−0.440846
$340$	0	0
$341$	13.4891	0.730477
$342$	0	0
$343$	8.86141	0.478471
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	−4.11684	−0.220370	−0.110185	−	0.993911i	$-0.535144\pi$
$349$	−4.11684	−0.220370	−0.110185	+	0.993911i	$0.535144\pi$
$350$	0	0
$351$	4.74456	0.253246
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	−3.37228	−0.178982
$356$	0	0
$357$	−18.1168	−0.958845
$358$	0	0
$359$	25.7228	1.35760	0.678799	−	0.734324i	$-0.262501\pi$
$359$	25.7228	1.35760	0.678799	+	0.734324i	$0.262501\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−12.7446	−0.667081
$366$	0	0
$367$	24.8614	1.29775	0.648877	−	0.760893i	$-0.275239\pi$
$367$	24.8614	1.29775	0.648877	+	0.760893i	$0.275239\pi$
$368$	0	0
$369$	−8.11684	−0.422546
$370$	0	0
$371$	−4.62772	−0.240259
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−25.4891	−1.31276
$378$	0	0
$379$	−9.48913	−0.487424	−0.243712	−	0.969848i	$-0.578365\pi$
$379$	−9.48913	−0.487424	−0.243712	+	0.969848i	$0.578365\pi$
$380$	0	0
$381$	−1.25544	−0.0643180
$382$	0	0
$383$	18.1168	0.925727	0.462864	−	0.886429i	$-0.346822\pi$
$383$	18.1168	0.925727	0.462864	+	0.886429i	$0.346822\pi$
$384$	0	0
$385$	13.4891	0.687469
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	3.48913	0.176906	0.0884528	−	0.996080i	$-0.471808\pi$
$389$	3.48913	0.176906	0.0884528	+	0.996080i	$0.471808\pi$
$390$	0	0
$391$	−5.37228	−0.271688
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	13.4891	0.678712
$396$	0	0
$397$	34.2337	1.71814	0.859070	−	0.511859i	$-0.171043\pi$
$397$	34.2337	1.71814	0.859070	+	0.511859i	$0.171043\pi$
$398$	0	0
$399$	13.4891	0.675301
$400$	0	0
$401$	23.4891	1.17299	0.586495	−	0.809953i	$-0.300507\pi$
$401$	23.4891	1.17299	0.586495	+	0.809953i	$0.300507\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	21.4891	1.06518
$408$	0	0
$409$	14.6277	0.723294	0.361647	−	0.932315i	$-0.382215\pi$
$409$	14.6277	0.723294	0.361647	+	0.932315i	$0.382215\pi$
$410$	0	0
$411$	−11.4891	−0.566717
$412$	0	0
$413$	29.0951	1.43168
$414$	0	0
$415$	12.8614	0.631342
$416$	0	0
$417$	−8.62772	−0.422501
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−3.25544	−0.158660	−0.0793302	−	0.996848i	$-0.525278\pi$
$421$	−3.25544	−0.158660	−0.0793302	+	0.996848i	$0.525278\pi$
$422$	0	0
$423$	6.74456	0.327932
$424$	0	0
$425$	5.37228	0.260594
$426$	0	0
$427$	6.74456	0.326392
$428$	0	0
$429$	−18.9783	−0.916279
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	31.0951	1.49433	0.747167	−	0.664636i	$-0.231413\pi$
$433$	31.0951	1.49433	0.747167	+	0.664636i	$0.231413\pi$
$434$	0	0
$435$	−5.37228	−0.257581
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	−26.9783	−1.28760	−0.643801	−	0.765193i	$-0.722643\pi$
$439$	−26.9783	−1.28760	−0.643801	+	0.765193i	$0.722643\pi$
$440$	0	0
$441$	4.37228	0.208204
$442$	0	0
$443$	21.2554	1.00988	0.504938	−	0.863156i	$-0.331515\pi$
$443$	21.2554	1.00988	0.504938	+	0.863156i	$0.331515\pi$
$444$	0	0
$445$	−4.74456	−0.224914
$446$	0	0
$447$	3.48913	0.165030
$448$	0	0
$449$	−6.86141	−0.323810	−0.161905	−	0.986806i	$-0.551764\pi$
$449$	−6.86141	−0.323810	−0.161905	+	0.986806i	$0.551764\pi$
$450$	0	0
$451$	32.4674	1.52883
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	−16.0000	−0.750092
$456$	0	0
$457$	−37.6060	−1.75913	−0.879567	−	0.475776i	$-0.842167\pi$
$457$	−37.6060	−1.75913	−0.879567	+	0.475776i	$0.842167\pi$
$458$	0	0
$459$	5.37228	0.250757
$460$	0	0
$461$	−34.4674	−1.60531	−0.802653	−	0.596446i	$-0.796579\pi$
$461$	−34.4674	−1.60531	−0.802653	+	0.596446i	$0.796579\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	−3.37228	−0.156386
$466$	0	0
$467$	42.3505	1.95975	0.979874	−	0.199615i	$-0.0639691\pi$
$467$	42.3505	1.95975	0.979874	+	0.199615i	$0.0639691\pi$
$468$	0	0
$469$	51.8397	2.39373
$470$	0	0
$471$	−17.6060	−0.811240
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	1.37228	0.0628324
$478$	0	0
$479$	33.7228	1.54083	0.770417	−	0.637540i	$-0.220048\pi$
$479$	33.7228	1.54083	0.770417	+	0.637540i	$0.220048\pi$
$480$	0	0
$481$	−25.4891	−1.16220
$482$	0	0
$483$	3.37228	0.153444
$484$	0	0
$485$	15.4891	0.703325
$486$	0	0
$487$	−33.7228	−1.52813	−0.764063	−	0.645141i	$-0.776798\pi$
$487$	−33.7228	−1.52813	−0.764063	+	0.645141i	$0.776798\pi$
$488$	0	0
$489$	−5.25544	−0.237659
$490$	0	0
$491$	16.6277	0.750398	0.375199	−	0.926944i	$-0.377574\pi$
$491$	16.6277	0.750398	0.375199	+	0.926944i	$0.377574\pi$
$492$	0	0
$493$	−28.8614	−1.29985
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	11.3723	0.510117
$498$	0	0
$499$	−43.6060	−1.95207	−0.976036	−	0.217611i	$-0.930174\pi$
$499$	−43.6060	−1.95207	−0.976036	+	0.217611i	$0.930174\pi$
$500$	0	0
$501$	−1.25544	−0.0560888
$502$	0	0
$503$	28.6277	1.27645	0.638223	−	0.769851i	$-0.279670\pi$
$503$	28.6277	1.27645	0.638223	+	0.769851i	$0.279670\pi$
$504$	0	0
$505$	−13.3723	−0.595059
$506$	0	0
$507$	9.51087	0.422393
$508$	0	0
$509$	−20.9783	−0.929845	−0.464922	−	0.885351i	$-0.653918\pi$
$509$	−20.9783	−0.929845	−0.464922	+	0.885351i	$0.653918\pi$
$510$	0	0
$511$	42.9783	1.90125
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	−26.9783	−1.18650
$518$	0	0
$519$	−14.2337	−0.624790
$520$	0	0
$521$	22.2337	0.974076	0.487038	−	0.873381i	$-0.338077\pi$
$521$	22.2337	0.974076	0.487038	+	0.873381i	$0.338077\pi$
$522$	0	0
$523$	−38.9783	−1.70440	−0.852200	−	0.523216i	$-0.824732\pi$
$523$	−38.9783	−1.70440	−0.852200	+	0.523216i	$0.824732\pi$
$524$	0	0
$525$	−3.37228	−0.147178
$526$	0	0
$527$	−18.1168	−0.789182
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−8.62772	−0.374411
$532$	0	0
$533$	−38.5109	−1.66809
$534$	0	0
$535$	7.37228	0.318732
$536$	0	0
$537$	−9.48913	−0.409486
$538$	0	0
$539$	−17.4891	−0.753310
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	0	0
$543$	−14.2337	−0.610826
$544$	0	0
$545$	4.74456	0.203235
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	21.4891	0.915468
$552$	0	0
$553$	−45.4891	−1.93439
$554$	0	0
$555$	−5.37228	−0.228041
$556$	0	0
$557$	36.3505	1.54022	0.770111	−	0.637910i	$-0.220201\pi$
$557$	36.3505	1.54022	0.770111	+	0.637910i	$0.220201\pi$
$558$	0	0
$559$	18.9783	0.802694
$560$	0	0
$561$	−21.4891	−0.907272
$562$	0	0
$563$	−7.37228	−0.310705	−0.155352	−	0.987859i	$-0.549651\pi$
$563$	−7.37228	−0.310705	−0.155352	+	0.987859i	$0.549651\pi$
$564$	0	0
$565$	−8.11684	−0.341478
$566$	0	0
$567$	−3.37228	−0.141623
$568$	0	0
$569$	−0.510875	−0.0214170	−0.0107085	−	0.999943i	$-0.503409\pi$
$569$	−0.510875	−0.0214170	−0.0107085	+	0.999943i	$0.503409\pi$
$570$	0	0
$571$	−0.233688	−0.00977954	−0.00488977	−	0.999988i	$-0.501556\pi$
$571$	−0.233688	−0.00977954	−0.00488977	+	0.999988i	$0.501556\pi$
$572$	0	0
$573$	−20.2337	−0.845274
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−7.25544	−0.302048	−0.151024	−	0.988530i	$-0.548257\pi$
$577$	−7.25544	−0.302048	−0.151024	+	0.988530i	$0.548257\pi$
$578$	0	0
$579$	−20.7446	−0.862115
$580$	0	0
$581$	−43.3723	−1.79939
$582$	0	0
$583$	−5.48913	−0.227336
$584$	0	0
$585$	4.74456	0.196164
$586$	0	0
$587$	−24.2337	−1.00023	−0.500116	−	0.865959i	$-0.666709\pi$
$587$	−24.2337	−1.00023	−0.500116	+	0.865959i	$0.666709\pi$
$588$	0	0
$589$	13.4891	0.555810
$590$	0	0
$591$	11.4891	0.472599
$592$	0	0
$593$	12.9783	0.532953	0.266476	−	0.963841i	$-0.414141\pi$
$593$	12.9783	0.532953	0.266476	+	0.963841i	$0.414141\pi$
$594$	0	0
$595$	−18.1168	−0.742718
$596$	0	0
$597$	−1.25544	−0.0513816
$598$	0	0
$599$	10.5109	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$599$	10.5109	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$600$	0	0
$601$	18.8614	0.769373	0.384686	−	0.923047i	$-0.374310\pi$
$601$	18.8614	0.769373	0.384686	+	0.923047i	$0.374310\pi$
$602$	0	0
$603$	−15.3723	−0.626008
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	33.2554	1.34980	0.674898	−	0.737911i	$-0.264187\pi$
$607$	33.2554	1.34980	0.674898	+	0.737911i	$0.264187\pi$
$608$	0	0
$609$	18.1168	0.734132
$610$	0	0
$611$	32.0000	1.29458
$612$	0	0
$613$	−39.4891	−1.59495	−0.797475	−	0.603351i	$-0.793832\pi$
$613$	−39.4891	−1.59495	−0.797475	+	0.603351i	$0.793832\pi$
$614$	0	0
$615$	−8.11684	−0.327303
$616$	0	0
$617$	−20.3505	−0.819282	−0.409641	−	0.912247i	$-0.634346\pi$
$617$	−20.3505	−0.819282	−0.409641	+	0.912247i	$0.634346\pi$
$618$	0	0
$619$	−38.9783	−1.56667	−0.783334	−	0.621601i	$-0.786483\pi$
$619$	−38.9783	−1.56667	−0.783334	+	0.621601i	$0.786483\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	16.0000	0.641026
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	16.0000	0.638978
$628$	0	0
$629$	−28.8614	−1.15078
$630$	0	0
$631$	17.2554	0.686928	0.343464	−	0.939166i	$-0.388400\pi$
$631$	17.2554	0.686928	0.343464	+	0.939166i	$0.388400\pi$
$632$	0	0
$633$	−20.8614	−0.829166
$634$	0	0
$635$	−1.25544	−0.0498205
$636$	0	0
$637$	20.7446	0.821929
$638$	0	0
$639$	−3.37228	−0.133405
$640$	0	0
$641$	43.7228	1.72695	0.863474	−	0.504394i	$-0.168284\pi$
$641$	43.7228	1.72695	0.863474	+	0.504394i	$0.168284\pi$
$642$	0	0
$643$	−47.3723	−1.86818	−0.934090	−	0.357037i	$-0.883787\pi$
$643$	−47.3723	−1.86818	−0.934090	+	0.357037i	$0.883787\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	21.4891	0.844825	0.422412	−	0.906404i	$-0.361183\pi$
$647$	21.4891	0.844825	0.422412	+	0.906404i	$0.361183\pi$
$648$	0	0
$649$	34.5109	1.35467
$650$	0	0
$651$	11.3723	0.445715
$652$	0	0
$653$	10.2337	0.400475	0.200238	−	0.979747i	$-0.435829\pi$
$653$	10.2337	0.400475	0.200238	+	0.979747i	$0.435829\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	−12.7446	−0.497213
$658$	0	0
$659$	−22.9783	−0.895106	−0.447553	−	0.894258i	$-0.647704\pi$
$659$	−22.9783	−0.895106	−0.447553	+	0.894258i	$0.647704\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	25.4891	0.989916
$664$	0	0
$665$	13.4891	0.523086
$666$	0	0
$667$	5.37228	0.208016
$668$	0	0
$669$	−17.2554	−0.667134
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	−16.5109	−0.636447	−0.318224	−	0.948016i	$-0.603086\pi$
$673$	−16.5109	−0.636447	−0.318224	+	0.948016i	$0.603086\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	40.1168	1.54182	0.770908	−	0.636947i	$-0.219803\pi$
$677$	40.1168	1.54182	0.770908	+	0.636947i	$0.219803\pi$
$678$	0	0
$679$	−52.2337	−2.00454
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	32.2337	1.23339	0.616694	−	0.787203i	$-0.288472\pi$
$683$	32.2337	1.23339	0.616694	+	0.787203i	$0.288472\pi$
$684$	0	0
$685$	−11.4891	−0.438977
$686$	0	0
$687$	28.7446	1.09667
$688$	0	0
$689$	6.51087	0.248045
$690$	0	0
$691$	1.48913	0.0566490	0.0283245	−	0.999599i	$-0.490983\pi$
$691$	1.48913	0.0566490	0.0283245	+	0.999599i	$0.490983\pi$
$692$	0	0
$693$	13.4891	0.512409
$694$	0	0
$695$	−8.62772	−0.327268
$696$	0	0
$697$	−43.6060	−1.65169
$698$	0	0
$699$	−26.2337	−0.992249
$700$	0	0
$701$	−39.9565	−1.50914	−0.754568	−	0.656222i	$-0.772154\pi$
$701$	−39.9565	−1.50914	−0.754568	+	0.656222i	$0.772154\pi$
$702$	0	0
$703$	21.4891	0.810478
$704$	0	0
$705$	6.74456	0.254015
$706$	0	0
$707$	45.0951	1.69598
$708$	0	0
$709$	−20.9783	−0.787855	−0.393927	−	0.919142i	$-0.628884\pi$
$709$	−20.9783	−0.787855	−0.393927	+	0.919142i	$0.628884\pi$
$710$	0	0
$711$	13.4891	0.505882
$712$	0	0
$713$	3.37228	0.126293
$714$	0	0
$715$	−18.9783	−0.709746
$716$	0	0
$717$	11.3723	0.424706
$718$	0	0
$719$	−29.0951	−1.08506	−0.542532	−	0.840035i	$-0.682534\pi$
$719$	−29.0951	−1.08506	−0.542532	+	0.840035i	$0.682534\pi$
$720$	0	0
$721$	−26.9783	−1.00472
$722$	0	0
$723$	−7.25544	−0.269833
$724$	0	0
$725$	−5.37228	−0.199522
$726$	0	0
$727$	3.37228	0.125071	0.0625355	−	0.998043i	$-0.480081\pi$
$727$	3.37228	0.125071	0.0625355	+	0.998043i	$0.480081\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	21.4891	0.794804
$732$	0	0
$733$	0.116844	0.00431573	0.00215787	−	0.999998i	$-0.499313\pi$
$733$	0.116844	0.00431573	0.00215787	+	0.999998i	$0.499313\pi$
$734$	0	0
$735$	4.37228	0.161274
$736$	0	0
$737$	61.4891	2.26498
$738$	0	0
$739$	−39.3723	−1.44833	−0.724166	−	0.689625i	$-0.757775\pi$
$739$	−39.3723	−1.44833	−0.724166	+	0.689625i	$0.757775\pi$
$740$	0	0
$741$	−18.9783	−0.697183
$742$	0	0
$743$	−34.9783	−1.28323	−0.641614	−	0.767028i	$-0.721735\pi$
$743$	−34.9783	−1.28323	−0.641614	+	0.767028i	$0.721735\pi$
$744$	0	0
$745$	3.48913	0.127832
$746$	0	0
$747$	12.8614	0.470574
$748$	0	0
$749$	−24.8614	−0.908416
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	18.7446	0.683090
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−25.6060	−0.930665	−0.465332	−	0.885136i	$-0.654065\pi$
$757$	−25.6060	−0.930665	−0.465332	+	0.885136i	$0.654065\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	45.8397	1.66169	0.830843	−	0.556507i	$-0.187859\pi$
$761$	45.8397	1.66169	0.830843	+	0.556507i	$0.187859\pi$
$762$	0	0
$763$	−16.0000	−0.579239
$764$	0	0
$765$	5.37228	0.194235
$766$	0	0
$767$	−40.9348	−1.47807
$768$	0	0
$769$	20.5109	0.739641	0.369821	−	0.929103i	$-0.379419\pi$
$769$	20.5109	0.739641	0.369821	+	0.929103i	$0.379419\pi$
$770$	0	0
$771$	−11.4891	−0.413771
$772$	0	0
$773$	−4.97825	−0.179055	−0.0895276	−	0.995984i	$-0.528536\pi$
$773$	−4.97825	−0.179055	−0.0895276	+	0.995984i	$0.528536\pi$
$774$	0	0
$775$	−3.37228	−0.121136
$776$	0	0
$777$	18.1168	0.649938
$778$	0	0
$779$	32.4674	1.16326
$780$	0	0
$781$	13.4891	0.482679
$782$	0	0
$783$	−5.37228	−0.191990
$784$	0	0
$785$	−17.6060	−0.628384
$786$	0	0
$787$	0.627719	0.0223758	0.0111879	−	0.999937i	$-0.496439\pi$
$787$	0.627719	0.0223758	0.0111879	+	0.999937i	$0.496439\pi$
$788$	0	0
$789$	−0.861407	−0.0306669
$790$	0	0
$791$	27.3723	0.973246
$792$	0	0
$793$	−9.48913	−0.336969
$794$	0	0
$795$	1.37228	0.0486698
$796$	0	0
$797$	16.1168	0.570888	0.285444	−	0.958395i	$-0.407859\pi$
$797$	16.1168	0.570888	0.285444	+	0.958395i	$0.407859\pi$
$798$	0	0
$799$	36.2337	1.28186
$800$	0	0
$801$	−4.74456	−0.167641
$802$	0	0
$803$	50.9783	1.79898
$804$	0	0
$805$	3.37228	0.118857
$806$	0	0
$807$	14.8614	0.523146
$808$	0	0
$809$	30.6277	1.07681	0.538407	−	0.842685i	$-0.319026\pi$
$809$	30.6277	1.07681	0.538407	+	0.842685i	$0.319026\pi$
$810$	0	0
$811$	−17.8832	−0.627963	−0.313981	−	0.949429i	$-0.601663\pi$
$811$	−17.8832	−0.627963	−0.313981	+	0.949429i	$0.601663\pi$
$812$	0	0
$813$	5.88316	0.206331
$814$	0	0
$815$	−5.25544	−0.184090
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	−16.0000	−0.559085
$820$	0	0
$821$	8.97825	0.313343	0.156672	−	0.987651i	$-0.449924\pi$
$821$	8.97825	0.313343	0.156672	+	0.987651i	$0.449924\pi$
$822$	0	0
$823$	10.9783	0.382678	0.191339	−	0.981524i	$-0.438717\pi$
$823$	10.9783	0.382678	0.191339	+	0.981524i	$0.438717\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	43.6060	1.51633	0.758164	−	0.652064i	$-0.226097\pi$
$827$	43.6060	1.51633	0.758164	+	0.652064i	$0.226097\pi$
$828$	0	0
$829$	43.0951	1.49675	0.748377	−	0.663273i	$-0.230833\pi$
$829$	43.0951	1.49675	0.748377	+	0.663273i	$0.230833\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	0	0
$833$	23.4891	0.813850
$834$	0	0
$835$	−1.25544	−0.0434462
$836$	0	0
$837$	−3.37228	−0.116563
$838$	0	0
$839$	−14.7446	−0.509039	−0.254519	−	0.967068i	$-0.581917\pi$
$839$	−14.7446	−0.509039	−0.254519	+	0.967068i	$0.581917\pi$
$840$	0	0
$841$	−0.138593	−0.00477908
$842$	0	0
$843$	22.2337	0.765769
$844$	0	0
$845$	9.51087	0.327184
$846$	0	0
$847$	−16.8614	−0.579365
$848$	0	0
$849$	−0.627719	−0.0215432
$850$	0	0
$851$	5.37228	0.184159
$852$	0	0
$853$	32.9783	1.12915	0.564577	−	0.825380i	$-0.309039\pi$
$853$	32.9783	1.12915	0.564577	+	0.825380i	$0.309039\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	50.4674	1.72393	0.861966	−	0.506965i	$-0.169233\pi$
$857$	50.4674	1.72393	0.861966	+	0.506965i	$0.169233\pi$
$858$	0	0
$859$	27.6060	0.941904	0.470952	−	0.882159i	$-0.343911\pi$
$859$	27.6060	0.941904	0.470952	+	0.882159i	$0.343911\pi$
$860$	0	0
$861$	27.3723	0.932845
$862$	0	0
$863$	6.74456	0.229588	0.114794	−	0.993389i	$-0.463379\pi$
$863$	6.74456	0.229588	0.114794	+	0.993389i	$0.463379\pi$
$864$	0	0
$865$	−14.2337	−0.483960
$866$	0	0
$867$	11.8614	0.402834
$868$	0	0
$869$	−53.9565	−1.83035
$870$	0	0
$871$	−72.9348	−2.47130
$872$	0	0
$873$	15.4891	0.524227
$874$	0	0
$875$	−3.37228	−0.114004
$876$	0	0
$877$	−20.5109	−0.692603	−0.346302	−	0.938123i	$-0.612563\pi$
$877$	−20.5109	−0.692603	−0.346302	+	0.938123i	$0.612563\pi$
$878$	0	0
$879$	−18.8614	−0.636179
$880$	0	0
$881$	−22.0000	−0.741199	−0.370599	−	0.928793i	$-0.620848\pi$
$881$	−22.0000	−0.741199	−0.370599	+	0.928793i	$0.620848\pi$
$882$	0	0
$883$	51.2119	1.72342	0.861709	−	0.507402i	$-0.169394\pi$
$883$	51.2119	1.72342	0.861709	+	0.507402i	$0.169394\pi$
$884$	0	0
$885$	−8.62772	−0.290018
$886$	0	0
$887$	25.7228	0.863688	0.431844	−	0.901948i	$-0.357863\pi$
$887$	25.7228	0.863688	0.431844	+	0.901948i	$0.357863\pi$
$888$	0	0
$889$	4.23369	0.141993
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	−26.9783	−0.902793
$894$	0	0
$895$	−9.48913	−0.317186
$896$	0	0
$897$	−4.74456	−0.158416
$898$	0	0
$899$	18.1168	0.604231
$900$	0	0
$901$	7.37228	0.245606
$902$	0	0
$903$	−13.4891	−0.448890
$904$	0	0
$905$	−14.2337	−0.473144
$906$	0	0
$907$	8.62772	0.286479	0.143239	−	0.989688i	$-0.454248\pi$
$907$	8.62772	0.286479	0.143239	+	0.989688i	$0.454248\pi$
$908$	0	0
$909$	−13.3723	−0.443531
$910$	0	0
$911$	56.4674	1.87085	0.935424	−	0.353528i	$-0.115018\pi$
$911$	56.4674	1.87085	0.935424	+	0.353528i	$0.115018\pi$
$912$	0	0
$913$	−51.4456	−1.70260
$914$	0	0
$915$	−2.00000	−0.0661180
$916$	0	0
$917$	−13.4891	−0.445450
$918$	0	0
$919$	−37.4891	−1.23665	−0.618326	−	0.785922i	$-0.712189\pi$
$919$	−37.4891	−1.23665	−0.618326	+	0.785922i	$0.712189\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	−16.0000	−0.526646
$924$	0	0
$925$	−5.37228	−0.176640
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	33.6060	1.10258	0.551288	−	0.834315i	$-0.314137\pi$
$929$	33.6060	1.10258	0.551288	+	0.834315i	$0.314137\pi$
$930$	0	0
$931$	−17.4891	−0.573183
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	−21.4891	−0.702770
$936$	0	0
$937$	34.4674	1.12600	0.563000	−	0.826457i	$-0.309647\pi$
$937$	34.4674	1.12600	0.563000	+	0.826457i	$0.309647\pi$
$938$	0	0
$939$	2.86141	0.0933785
$940$	0	0
$941$	−12.9783	−0.423079	−0.211539	−	0.977369i	$-0.567848\pi$
$941$	−12.9783	−0.423079	−0.211539	+	0.977369i	$0.567848\pi$
$942$	0	0
$943$	8.11684	0.264321
$944$	0	0
$945$	−3.37228	−0.109700
$946$	0	0
$947$	−54.9783	−1.78655	−0.893277	−	0.449508i	$-0.851599\pi$
$947$	−54.9783	−1.78655	−0.893277	+	0.449508i	$0.851599\pi$
$948$	0	0
$949$	−60.4674	−1.96285
$950$	0	0
$951$	8.51087	0.275984
$952$	0	0
$953$	15.4891	0.501742	0.250871	−	0.968021i	$-0.419283\pi$
$953$	15.4891	0.501742	0.250871	+	0.968021i	$0.419283\pi$
$954$	0	0
$955$	−20.2337	−0.654747
$956$	0	0
$957$	21.4891	0.694645
$958$	0	0
$959$	38.7446	1.25113
$960$	0	0
$961$	−19.6277	−0.633152
$962$	0	0
$963$	7.37228	0.237568
$964$	0	0
$965$	−20.7446	−0.667791
$966$	0	0
$967$	−11.7663	−0.378379	−0.189190	−	0.981941i	$-0.560586\pi$
$967$	−11.7663	−0.378379	−0.189190	+	0.981941i	$0.560586\pi$
$968$	0	0
$969$	−21.4891	−0.690330
$970$	0	0
$971$	−1.48913	−0.0477883	−0.0238942	−	0.999714i	$-0.507606\pi$
$971$	−1.48913	−0.0477883	−0.0238942	+	0.999714i	$0.507606\pi$
$972$	0	0
$973$	29.0951	0.932746
$974$	0	0
$975$	4.74456	0.151948
$976$	0	0
$977$	30.6277	0.979868	0.489934	−	0.871760i	$-0.337021\pi$
$977$	30.6277	0.979868	0.489934	+	0.871760i	$0.337021\pi$
$978$	0	0
$979$	18.9783	0.606548
$980$	0	0
$981$	4.74456	0.151482
$982$	0	0
$983$	48.8614	1.55844	0.779218	−	0.626752i	$-0.215616\pi$
$983$	48.8614	1.55844	0.779218	+	0.626752i	$0.215616\pi$
$984$	0	0
$985$	11.4891	0.366074
$986$	0	0
$987$	−22.7446	−0.723967
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	42.1168	1.33789	0.668943	−	0.743314i	$-0.266747\pi$
$991$	42.1168	1.33789	0.668943	+	0.743314i	$0.266747\pi$
$992$	0	0
$993$	−1.88316	−0.0597602
$994$	0	0
$995$	−1.25544	−0.0398000
$996$	0	0
$997$	−46.2337	−1.46424	−0.732118	−	0.681178i	$-0.761468\pi$
$997$	−46.2337	−1.46424	−0.732118	+	0.681178i	$0.761468\pi$
$998$	0	0
$999$	−5.37228	−0.169971

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bq.1.1		2
4.3	odd	2		2760.2.a.o.1.2	✓	2
12.11	even	2		8280.2.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.o.1.2	✓	2	4.3	odd	2
5520.2.a.bq.1.1		2	1.1	even	1	trivial
8280.2.a.z.1.2		2	12.11	even	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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -3.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -3.37228 q^{7} +1.00000 q^{9} -4.00000 q^{11} +4.74456 q^{13} +1.00000 q^{15} +5.37228 q^{17} -4.00000 q^{19} -3.37228 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -5.37228 q^{29} -3.37228 q^{31} -4.00000 q^{33} -3.37228 q^{35} -5.37228 q^{37} +4.74456 q^{39} -8.11684 q^{41} +4.00000 q^{43} +1.00000 q^{45} +6.74456 q^{47} +4.37228 q^{49} +5.37228 q^{51} +1.37228 q^{53} -4.00000 q^{55} -4.00000 q^{57} -8.62772 q^{59} -2.00000 q^{61} -3.37228 q^{63} +4.74456 q^{65} -15.3723 q^{67} -1.00000 q^{69} -3.37228 q^{71} -12.7446 q^{73} +1.00000 q^{75} +13.4891 q^{77} +13.4891 q^{79} +1.00000 q^{81} +12.8614 q^{83} +5.37228 q^{85} -5.37228 q^{87} -4.74456 q^{89} -16.0000 q^{91} -3.37228 q^{93} -4.00000 q^{95} +15.4891 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - 8 q^{11} - 2 q^{13} + 2 q^{15} + 5 q^{17} - 8 q^{19} - q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 5 q^{29} - q^{31} - 8 q^{33} - q^{35} - 5 q^{37} - 2 q^{39} + q^{41} + 8 q^{43} + 2 q^{45} + 2 q^{47} + 3 q^{49} + 5 q^{51} - 3 q^{53} - 8 q^{55} - 8 q^{57} - 23 q^{59} - 4 q^{61} - q^{63} - 2 q^{65} - 25 q^{67} - 2 q^{69} - q^{71} - 14 q^{73} + 2 q^{75} + 4 q^{77} + 4 q^{79} + 2 q^{81} - 3 q^{83} + 5 q^{85} - 5 q^{87} + 2 q^{89} - 32 q^{91} - q^{93} - 8 q^{95} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - 8 * q^11 - 2 * q^13 + 2 * q^15 + 5 * q^17 - 8 * q^19 - q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 - 5 * q^29 - q^31 - 8 * q^33 - q^35 - 5 * q^37 - 2 * q^39 + q^41 + 8 * q^43 + 2 * q^45 + 2 * q^47 + 3 * q^49 + 5 * q^51 - 3 * q^53 - 8 * q^55 - 8 * q^57 - 23 * q^59 - 4 * q^61 - q^63 - 2 * q^65 - 25 * q^67 - 2 * q^69 - q^71 - 14 * q^73 + 2 * q^75 + 4 * q^77 + 4 * q^79 + 2 * q^81 - 3 * q^83 + 5 * q^85 - 5 * q^87 + 2 * q^89 - 32 * q^91 - q^93 - 8 * q^95 + 8 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

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