LMFDB - Embedded newform 5520.2.a.bu.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -0.732051 q^{11} -4.19615 q^{13} +1.00000 q^{15} -5.73205 q^{17} +6.73205 q^{19} +3.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.267949 q^{29} -3.92820 q^{31} -0.732051 q^{33} +3.00000 q^{35} +11.9282 q^{37} -4.19615 q^{39} -0.267949 q^{41} +4.53590 q^{43} +1.00000 q^{45} +9.66025 q^{47} +2.00000 q^{49} -5.73205 q^{51} -2.26795 q^{53} -0.732051 q^{55} +6.73205 q^{57} +3.19615 q^{59} +13.1244 q^{61} +3.00000 q^{63} -4.19615 q^{65} -0.464102 q^{67} -1.00000 q^{69} +0.267949 q^{71} +9.66025 q^{73} +1.00000 q^{75} -2.19615 q^{77} +6.92820 q^{79} +1.00000 q^{81} +10.2679 q^{83} -5.73205 q^{85} -0.267949 q^{87} +9.46410 q^{89} -12.5885 q^{91} -3.92820 q^{93} +6.73205 q^{95} -4.00000 q^{97} -0.732051 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} - 8 q^{17} + 10 q^{19} + 6 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} + 2 q^{33} + 6 q^{35} + 10 q^{37} + 2 q^{39} - 4 q^{41} + 16 q^{43} + 2 q^{45} + 2 q^{47} + 4 q^{49} - 8 q^{51} - 8 q^{53} + 2 q^{55} + 10 q^{57} - 4 q^{59} + 2 q^{61} + 6 q^{63} + 2 q^{65} + 6 q^{67} - 2 q^{69} + 4 q^{71} + 2 q^{73} + 2 q^{75} + 6 q^{77} + 2 q^{81} + 24 q^{83} - 8 q^{85} - 4 q^{87} + 12 q^{89} + 6 q^{91} + 6 q^{93} + 10 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 2 * q^15 - 8 * q^17 + 10 * q^19 + 6 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 + 6 * q^31 + 2 * q^33 + 6 * q^35 + 10 * q^37 + 2 * q^39 - 4 * q^41 + 16 * q^43 + 2 * q^45 + 2 * q^47 + 4 * q^49 - 8 * q^51 - 8 * q^53 + 2 * q^55 + 10 * q^57 - 4 * q^59 + 2 * q^61 + 6 * q^63 + 2 * q^65 + 6 * q^67 - 2 * q^69 + 4 * q^71 + 2 * q^73 + 2 * q^75 + 6 * q^77 + 2 * q^81 + 24 * q^83 - 8 * q^85 - 4 * q^87 + 12 * q^89 + 6 * q^91 + 6 * q^93 + 10 * q^95 - 8 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.732051	−0.220722	−0.110361	−	0.993892i	$-0.535201\pi$
$11$	−0.732051	−0.220722	−0.110361	+	0.993892i	$0.535201\pi$
$12$	0	0
$13$	−4.19615	−1.16380	−0.581902	−	0.813259i	$-0.697691\pi$
$13$	−4.19615	−1.16380	−0.581902	+	0.813259i	$0.697691\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−5.73205	−1.39023	−0.695113	−	0.718900i	$-0.744646\pi$
$17$	−5.73205	−1.39023	−0.695113	+	0.718900i	$0.744646\pi$
$18$	0	0
$19$	6.73205	1.54444	0.772219	−	0.635356i	$-0.219147\pi$
$19$	6.73205	1.54444	0.772219	+	0.635356i	$0.219147\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	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
$29$	−0.267949	−0.0497569	−0.0248785	−	0.999690i	$-0.507920\pi$
$29$	−0.267949	−0.0497569	−0.0248785	+	0.999690i	$0.507920\pi$
$30$	0	0
$31$	−3.92820	−0.705526	−0.352763	−	0.935713i	$-0.614758\pi$
$31$	−3.92820	−0.705526	−0.352763	+	0.935713i	$0.614758\pi$
$32$	0	0
$33$	−0.732051	−0.127434
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	11.9282	1.96098	0.980492	−	0.196558i	$-0.0629763\pi$
$37$	11.9282	1.96098	0.980492	+	0.196558i	$0.0629763\pi$
$38$	0	0
$39$	−4.19615	−0.671922
$40$	0	0
$41$	−0.267949	−0.0418466	−0.0209233	−	0.999781i	$-0.506661\pi$
$41$	−0.267949	−0.0418466	−0.0209233	+	0.999781i	$0.506661\pi$
$42$	0	0
$43$	4.53590	0.691718	0.345859	−	0.938286i	$-0.387588\pi$
$43$	4.53590	0.691718	0.345859	+	0.938286i	$0.387588\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	9.66025	1.40909	0.704546	−	0.709658i	$-0.251150\pi$
$47$	9.66025	1.40909	0.704546	+	0.709658i	$0.251150\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−5.73205	−0.802648
$52$	0	0
$53$	−2.26795	−0.311527	−0.155763	−	0.987794i	$-0.549784\pi$
$53$	−2.26795	−0.311527	−0.155763	+	0.987794i	$0.549784\pi$
$54$	0	0
$55$	−0.732051	−0.0987097
$56$	0	0
$57$	6.73205	0.891682
$58$	0	0
$59$	3.19615	0.416104	0.208052	−	0.978118i	$-0.433288\pi$
$59$	3.19615	0.416104	0.208052	+	0.978118i	$0.433288\pi$
$60$	0	0
$61$	13.1244	1.68040	0.840201	−	0.542275i	$-0.182437\pi$
$61$	13.1244	1.68040	0.840201	+	0.542275i	$0.182437\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	−4.19615	−0.520469
$66$	0	0
$67$	−0.464102	−0.0566990	−0.0283495	−	0.999598i	$-0.509025\pi$
$67$	−0.464102	−0.0566990	−0.0283495	+	0.999598i	$0.509025\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0.267949	0.0317997	0.0158999	−	0.999874i	$-0.494939\pi$
$71$	0.267949	0.0317997	0.0158999	+	0.999874i	$0.494939\pi$
$72$	0	0
$73$	9.66025	1.13065	0.565324	−	0.824869i	$-0.308751\pi$
$73$	9.66025	1.13065	0.565324	+	0.824869i	$0.308751\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.19615	−0.250275
$78$	0	0
$79$	6.92820	0.779484	0.389742	−	0.920924i	$-0.372564\pi$
$79$	6.92820	0.779484	0.389742	+	0.920924i	$0.372564\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.2679	1.12705	0.563527	−	0.826098i	$-0.309444\pi$
$83$	10.2679	1.12705	0.563527	+	0.826098i	$0.309444\pi$
$84$	0	0
$85$	−5.73205	−0.621728
$86$	0	0
$87$	−0.267949	−0.0287272
$88$	0	0
$89$	9.46410	1.00319	0.501596	−	0.865102i	$-0.332746\pi$
$89$	9.46410	1.00319	0.501596	+	0.865102i	$0.332746\pi$
$90$	0	0
$91$	−12.5885	−1.31963
$92$	0	0
$93$	−3.92820	−0.407336
$94$	0	0
$95$	6.73205	0.690694
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	−0.732051	−0.0735739
$100$	0	0
$101$	−5.19615	−0.517036	−0.258518	−	0.966006i	$-0.583234\pi$
$101$	−5.19615	−0.517036	−0.258518	+	0.966006i	$0.583234\pi$
$102$	0	0
$103$	11.8564	1.16825	0.584123	−	0.811665i	$-0.301438\pi$
$103$	11.8564	1.16825	0.584123	+	0.811665i	$0.301438\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	18.1244	1.75215	0.876074	−	0.482177i	$-0.160154\pi$
$107$	18.1244	1.75215	0.876074	+	0.482177i	$0.160154\pi$
$108$	0	0
$109$	6.73205	0.644814	0.322407	−	0.946601i	$-0.395508\pi$
$109$	6.73205	0.644814	0.322407	+	0.946601i	$0.395508\pi$
$110$	0	0
$111$	11.9282	1.13217
$112$	0	0
$113$	−9.19615	−0.865101	−0.432551	−	0.901610i	$-0.642386\pi$
$113$	−9.19615	−0.865101	−0.432551	+	0.901610i	$0.642386\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−4.19615	−0.387934
$118$	0	0
$119$	−17.1962	−1.57637
$120$	0	0
$121$	−10.4641	−0.951282
$122$	0	0
$123$	−0.267949	−0.0241602
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.80385	−0.160066	−0.0800328	−	0.996792i	$-0.525503\pi$
$127$	−1.80385	−0.160066	−0.0800328	+	0.996792i	$0.525503\pi$
$128$	0	0
$129$	4.53590	0.399364
$130$	0	0
$131$	−5.07180	−0.443125	−0.221562	−	0.975146i	$-0.571116\pi$
$131$	−5.07180	−0.443125	−0.221562	+	0.975146i	$0.571116\pi$
$132$	0	0
$133$	20.1962	1.75123
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	20.7846	1.77575	0.887875	−	0.460086i	$-0.152181\pi$
$137$	20.7846	1.77575	0.887875	+	0.460086i	$0.152181\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	9.66025	0.813540
$142$	0	0
$143$	3.07180	0.256877
$144$	0	0
$145$	−0.267949	−0.0222520
$146$	0	0
$147$	2.00000	0.164957
$148$	0	0
$149$	−16.1962	−1.32684	−0.663420	−	0.748247i	$-0.730896\pi$
$149$	−16.1962	−1.32684	−0.663420	+	0.748247i	$0.730896\pi$
$150$	0	0
$151$	18.3923	1.49674	0.748372	−	0.663279i	$-0.230836\pi$
$151$	18.3923	1.49674	0.748372	+	0.663279i	$0.230836\pi$
$152$	0	0
$153$	−5.73205	−0.463409
$154$	0	0
$155$	−3.92820	−0.315521
$156$	0	0
$157$	−13.9282	−1.11159	−0.555796	−	0.831319i	$-0.687586\pi$
$157$	−13.9282	−1.11159	−0.555796	+	0.831319i	$0.687586\pi$
$158$	0	0
$159$	−2.26795	−0.179860
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	0	0
$165$	−0.732051	−0.0569901
$166$	0	0
$167$	6.33975	0.490584	0.245292	−	0.969449i	$-0.421116\pi$
$167$	6.33975	0.490584	0.245292	+	0.969449i	$0.421116\pi$
$168$	0	0
$169$	4.60770	0.354438
$170$	0	0
$171$	6.73205	0.514813
$172$	0	0
$173$	−25.8564	−1.96583	−0.982913	−	0.184070i	$-0.941073\pi$
$173$	−25.8564	−1.96583	−0.982913	+	0.184070i	$0.941073\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	0	0
$177$	3.19615	0.240238
$178$	0	0
$179$	−11.4641	−0.856867	−0.428434	−	0.903573i	$-0.640934\pi$
$179$	−11.4641	−0.856867	−0.428434	+	0.903573i	$0.640934\pi$
$180$	0	0
$181$	−8.39230	−0.623795	−0.311898	−	0.950116i	$-0.600965\pi$
$181$	−8.39230	−0.623795	−0.311898	+	0.950116i	$0.600965\pi$
$182$	0	0
$183$	13.1244	0.970180
$184$	0	0
$185$	11.9282	0.876979
$186$	0	0
$187$	4.19615	0.306853
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	13.6603	0.988421	0.494211	−	0.869342i	$-0.335457\pi$
$191$	13.6603	0.988421	0.494211	+	0.869342i	$0.335457\pi$
$192$	0	0
$193$	17.4641	1.25709	0.628547	−	0.777772i	$-0.283650\pi$
$193$	17.4641	1.25709	0.628547	+	0.777772i	$0.283650\pi$
$194$	0	0
$195$	−4.19615	−0.300493
$196$	0	0
$197$	−22.3923	−1.59539	−0.797693	−	0.603064i	$-0.793946\pi$
$197$	−22.3923	−1.59539	−0.797693	+	0.603064i	$0.793946\pi$
$198$	0	0
$199$	4.53590	0.321541	0.160771	−	0.986992i	$-0.448602\pi$
$199$	4.53590	0.321541	0.160771	+	0.986992i	$0.448602\pi$
$200$	0	0
$201$	−0.464102	−0.0327352
$202$	0	0
$203$	−0.803848	−0.0564190
$204$	0	0
$205$	−0.267949	−0.0187144
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−4.92820	−0.340891
$210$	0	0
$211$	−19.7846	−1.36203	−0.681014	−	0.732270i	$-0.738461\pi$
$211$	−19.7846	−1.36203	−0.681014	+	0.732270i	$0.738461\pi$
$212$	0	0
$213$	0.267949	0.0183596
$214$	0	0
$215$	4.53590	0.309346
$216$	0	0
$217$	−11.7846	−0.799991
$218$	0	0
$219$	9.66025	0.652779
$220$	0	0
$221$	24.0526	1.61795
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−9.07180	−0.602116	−0.301058	−	0.953606i	$-0.597340\pi$
$227$	−9.07180	−0.602116	−0.301058	+	0.953606i	$0.597340\pi$
$228$	0	0
$229$	−11.3205	−0.748080	−0.374040	−	0.927413i	$-0.622028\pi$
$229$	−11.3205	−0.748080	−0.374040	+	0.927413i	$0.622028\pi$
$230$	0	0
$231$	−2.19615	−0.144496
$232$	0	0
$233$	25.8564	1.69391	0.846955	−	0.531665i	$-0.178433\pi$
$233$	25.8564	1.69391	0.846955	+	0.531665i	$0.178433\pi$
$234$	0	0
$235$	9.66025	0.630165
$236$	0	0
$237$	6.92820	0.450035
$238$	0	0
$239$	−21.0526	−1.36178	−0.680888	−	0.732387i	$-0.738406\pi$
$239$	−21.0526	−1.36178	−0.680888	+	0.732387i	$0.738406\pi$
$240$	0	0
$241$	−30.7321	−1.97963	−0.989813	−	0.142376i	$-0.954526\pi$
$241$	−30.7321	−1.97963	−0.989813	+	0.142376i	$0.954526\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	−28.2487	−1.79742
$248$	0	0
$249$	10.2679	0.650705
$250$	0	0
$251$	−19.8564	−1.25333	−0.626663	−	0.779291i	$-0.715580\pi$
$251$	−19.8564	−1.25333	−0.626663	+	0.779291i	$0.715580\pi$
$252$	0	0
$253$	0.732051	0.0460236
$254$	0	0
$255$	−5.73205	−0.358955
$256$	0	0
$257$	−5.66025	−0.353077	−0.176538	−	0.984294i	$-0.556490\pi$
$257$	−5.66025	−0.353077	−0.176538	+	0.984294i	$0.556490\pi$
$258$	0	0
$259$	35.7846	2.22355
$260$	0	0
$261$	−0.267949	−0.0165856
$262$	0	0
$263$	−8.26795	−0.509824	−0.254912	−	0.966964i	$-0.582046\pi$
$263$	−8.26795	−0.509824	−0.254912	+	0.966964i	$0.582046\pi$
$264$	0	0
$265$	−2.26795	−0.139319
$266$	0	0
$267$	9.46410	0.579194
$268$	0	0
$269$	28.6603	1.74745	0.873723	−	0.486423i	$-0.161699\pi$
$269$	28.6603	1.74745	0.873723	+	0.486423i	$0.161699\pi$
$270$	0	0
$271$	−5.39230	−0.327559	−0.163780	−	0.986497i	$-0.552369\pi$
$271$	−5.39230	−0.327559	−0.163780	+	0.986497i	$0.552369\pi$
$272$	0	0
$273$	−12.5885	−0.761888
$274$	0	0
$275$	−0.732051	−0.0441443
$276$	0	0
$277$	14.7846	0.888321	0.444161	−	0.895947i	$-0.353502\pi$
$277$	14.7846	0.888321	0.444161	+	0.895947i	$0.353502\pi$
$278$	0	0
$279$	−3.92820	−0.235175
$280$	0	0
$281$	−16.0526	−0.957615	−0.478808	−	0.877920i	$-0.658931\pi$
$281$	−16.0526	−0.957615	−0.478808	+	0.877920i	$0.658931\pi$
$282$	0	0
$283$	6.46410	0.384251	0.192125	−	0.981370i	$-0.438462\pi$
$283$	6.46410	0.384251	0.192125	+	0.981370i	$0.438462\pi$
$284$	0	0
$285$	6.73205	0.398772
$286$	0	0
$287$	−0.803848	−0.0474496
$288$	0	0
$289$	15.8564	0.932730
$290$	0	0
$291$	−4.00000	−0.234484
$292$	0	0
$293$	−7.73205	−0.451711	−0.225856	−	0.974161i	$-0.572518\pi$
$293$	−7.73205	−0.451711	−0.225856	+	0.974161i	$0.572518\pi$
$294$	0	0
$295$	3.19615	0.186087
$296$	0	0
$297$	−0.732051	−0.0424779
$298$	0	0
$299$	4.19615	0.242670
$300$	0	0
$301$	13.6077	0.784335
$302$	0	0
$303$	−5.19615	−0.298511
$304$	0	0
$305$	13.1244	0.751498
$306$	0	0
$307$	−24.1962	−1.38095	−0.690474	−	0.723358i	$-0.742598\pi$
$307$	−24.1962	−1.38095	−0.690474	+	0.723358i	$0.742598\pi$
$308$	0	0
$309$	11.8564	0.674487
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	5.92820	0.335082	0.167541	−	0.985865i	$-0.446417\pi$
$313$	5.92820	0.335082	0.167541	+	0.985865i	$0.446417\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	−27.5167	−1.54549	−0.772745	−	0.634717i	$-0.781117\pi$
$317$	−27.5167	−1.54549	−0.772745	+	0.634717i	$0.781117\pi$
$318$	0	0
$319$	0.196152	0.0109824
$320$	0	0
$321$	18.1244	1.01160
$322$	0	0
$323$	−38.5885	−2.14712
$324$	0	0
$325$	−4.19615	−0.232761
$326$	0	0
$327$	6.73205	0.372283
$328$	0	0
$329$	28.9808	1.59776
$330$	0	0
$331$	14.3205	0.787126	0.393563	−	0.919298i	$-0.371242\pi$
$331$	14.3205	0.787126	0.393563	+	0.919298i	$0.371242\pi$
$332$	0	0
$333$	11.9282	0.653662
$334$	0	0
$335$	−0.464102	−0.0253566
$336$	0	0
$337$	−22.7846	−1.24116	−0.620578	−	0.784144i	$-0.713102\pi$
$337$	−22.7846	−1.24116	−0.620578	+	0.784144i	$0.713102\pi$
$338$	0	0
$339$	−9.19615	−0.499466
$340$	0	0
$341$	2.87564	0.155725
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−7.32051	−0.392985	−0.196493	−	0.980505i	$-0.562955\pi$
$347$	−7.32051	−0.392985	−0.196493	+	0.980505i	$0.562955\pi$
$348$	0	0
$349$	30.3205	1.62302	0.811510	−	0.584339i	$-0.198646\pi$
$349$	30.3205	1.62302	0.811510	+	0.584339i	$0.198646\pi$
$350$	0	0
$351$	−4.19615	−0.223974
$352$	0	0
$353$	1.26795	0.0674861	0.0337431	−	0.999431i	$-0.489257\pi$
$353$	1.26795	0.0674861	0.0337431	+	0.999431i	$0.489257\pi$
$354$	0	0
$355$	0.267949	0.0142213
$356$	0	0
$357$	−17.1962	−0.910117
$358$	0	0
$359$	27.1244	1.43157	0.715784	−	0.698321i	$-0.246069\pi$
$359$	27.1244	1.43157	0.715784	+	0.698321i	$0.246069\pi$
$360$	0	0
$361$	26.3205	1.38529
$362$	0	0
$363$	−10.4641	−0.549223
$364$	0	0
$365$	9.66025	0.505641
$366$	0	0
$367$	15.0000	0.782994	0.391497	−	0.920179i	$-0.371957\pi$
$367$	15.0000	0.782994	0.391497	+	0.920179i	$0.371957\pi$
$368$	0	0
$369$	−0.267949	−0.0139489
$370$	0	0
$371$	−6.80385	−0.353238
$372$	0	0
$373$	32.3923	1.67721	0.838605	−	0.544740i	$-0.183372\pi$
$373$	32.3923	1.67721	0.838605	+	0.544740i	$0.183372\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	1.12436	0.0579073
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	−1.80385	−0.0924139
$382$	0	0
$383$	2.41154	0.123224	0.0616120	−	0.998100i	$-0.480376\pi$
$383$	2.41154	0.123224	0.0616120	+	0.998100i	$0.480376\pi$
$384$	0	0
$385$	−2.19615	−0.111926
$386$	0	0
$387$	4.53590	0.230573
$388$	0	0
$389$	3.46410	0.175637	0.0878185	−	0.996136i	$-0.472010\pi$
$389$	3.46410	0.175637	0.0878185	+	0.996136i	$0.472010\pi$
$390$	0	0
$391$	5.73205	0.289882
$392$	0	0
$393$	−5.07180	−0.255838
$394$	0	0
$395$	6.92820	0.348596
$396$	0	0
$397$	24.7846	1.24390	0.621952	−	0.783055i	$-0.286340\pi$
$397$	24.7846	1.24390	0.621952	+	0.783055i	$0.286340\pi$
$398$	0	0
$399$	20.1962	1.01107
$400$	0	0
$401$	5.32051	0.265693	0.132847	−	0.991137i	$-0.457588\pi$
$401$	5.32051	0.265693	0.132847	+	0.991137i	$0.457588\pi$
$402$	0	0
$403$	16.4833	0.821094
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−8.73205	−0.432832
$408$	0	0
$409$	−11.3923	−0.563313	−0.281657	−	0.959515i	$-0.590884\pi$
$409$	−11.3923	−0.563313	−0.281657	+	0.959515i	$0.590884\pi$
$410$	0	0
$411$	20.7846	1.02523
$412$	0	0
$413$	9.58846	0.471817
$414$	0	0
$415$	10.2679	0.504034
$416$	0	0
$417$	−5.00000	−0.244851
$418$	0	0
$419$	−11.2679	−0.550475	−0.275238	−	0.961376i	$-0.588757\pi$
$419$	−11.2679	−0.550475	−0.275238	+	0.961376i	$0.588757\pi$
$420$	0	0
$421$	21.6603	1.05566	0.527828	−	0.849351i	$-0.323007\pi$
$421$	21.6603	1.05566	0.527828	+	0.849351i	$0.323007\pi$
$422$	0	0
$423$	9.66025	0.469698
$424$	0	0
$425$	−5.73205	−0.278045
$426$	0	0
$427$	39.3731	1.90540
$428$	0	0
$429$	3.07180	0.148308
$430$	0	0
$431$	16.3923	0.789590	0.394795	−	0.918769i	$-0.370816\pi$
$431$	16.3923	0.789590	0.394795	+	0.918769i	$0.370816\pi$
$432$	0	0
$433$	24.0718	1.15682	0.578408	−	0.815747i	$-0.303674\pi$
$433$	24.0718	1.15682	0.578408	+	0.815747i	$0.303674\pi$
$434$	0	0
$435$	−0.267949	−0.0128472
$436$	0	0
$437$	−6.73205	−0.322038
$438$	0	0
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−19.5167	−0.927265	−0.463632	−	0.886028i	$-0.653454\pi$
$443$	−19.5167	−0.927265	−0.463632	+	0.886028i	$0.653454\pi$
$444$	0	0
$445$	9.46410	0.448641
$446$	0	0
$447$	−16.1962	−0.766052
$448$	0	0
$449$	−29.0526	−1.37108	−0.685538	−	0.728037i	$-0.740433\pi$
$449$	−29.0526	−1.37108	−0.685538	+	0.728037i	$0.740433\pi$
$450$	0	0
$451$	0.196152	0.00923646
$452$	0	0
$453$	18.3923	0.864146
$454$	0	0
$455$	−12.5885	−0.590156
$456$	0	0
$457$	−23.9282	−1.11931	−0.559657	−	0.828724i	$-0.689067\pi$
$457$	−23.9282	−1.11931	−0.559657	+	0.828724i	$0.689067\pi$
$458$	0	0
$459$	−5.73205	−0.267549
$460$	0	0
$461$	−27.4641	−1.27913	−0.639565	−	0.768737i	$-0.720886\pi$
$461$	−27.4641	−1.27913	−0.639565	+	0.768737i	$0.720886\pi$
$462$	0	0
$463$	−17.6603	−0.820742	−0.410371	−	0.911919i	$-0.634601\pi$
$463$	−17.6603	−0.820742	−0.410371	+	0.911919i	$0.634601\pi$
$464$	0	0
$465$	−3.92820	−0.182166
$466$	0	0
$467$	12.2679	0.567693	0.283846	−	0.958870i	$-0.408389\pi$
$467$	12.2679	0.567693	0.283846	+	0.958870i	$0.408389\pi$
$468$	0	0
$469$	−1.39230	−0.0642907
$470$	0	0
$471$	−13.9282	−0.641778
$472$	0	0
$473$	−3.32051	−0.152677
$474$	0	0
$475$	6.73205	0.308888
$476$	0	0
$477$	−2.26795	−0.103842
$478$	0	0
$479$	38.4449	1.75659	0.878295	−	0.478119i	$-0.158681\pi$
$479$	38.4449	1.75659	0.878295	+	0.478119i	$0.158681\pi$
$480$	0	0
$481$	−50.0526	−2.28220
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	−4.00000	−0.181631
$486$	0	0
$487$	−7.66025	−0.347119	−0.173560	−	0.984823i	$-0.555527\pi$
$487$	−7.66025	−0.347119	−0.173560	+	0.984823i	$0.555527\pi$
$488$	0	0
$489$	−14.0000	−0.633102
$490$	0	0
$491$	25.4449	1.14831	0.574155	−	0.818746i	$-0.305331\pi$
$491$	25.4449	1.14831	0.574155	+	0.818746i	$0.305331\pi$
$492$	0	0
$493$	1.53590	0.0691734
$494$	0	0
$495$	−0.732051	−0.0329032
$496$	0	0
$497$	0.803848	0.0360575
$498$	0	0
$499$	−14.6077	−0.653930	−0.326965	−	0.945036i	$-0.606026\pi$
$499$	−14.6077	−0.653930	−0.326965	+	0.945036i	$0.606026\pi$
$500$	0	0
$501$	6.33975	0.283239
$502$	0	0
$503$	5.33975	0.238088	0.119044	−	0.992889i	$-0.462017\pi$
$503$	5.33975	0.238088	0.119044	+	0.992889i	$0.462017\pi$
$504$	0	0
$505$	−5.19615	−0.231226
$506$	0	0
$507$	4.60770	0.204635
$508$	0	0
$509$	−34.9282	−1.54817	−0.774083	−	0.633084i	$-0.781789\pi$
$509$	−34.9282	−1.54817	−0.774083	+	0.633084i	$0.781789\pi$
$510$	0	0
$511$	28.9808	1.28203
$512$	0	0
$513$	6.73205	0.297227
$514$	0	0
$515$	11.8564	0.522456
$516$	0	0
$517$	−7.07180	−0.311017
$518$	0	0
$519$	−25.8564	−1.13497
$520$	0	0
$521$	−12.7321	−0.557801	−0.278901	−	0.960320i	$-0.589970\pi$
$521$	−12.7321	−0.557801	−0.278901	+	0.960320i	$0.589970\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	3.00000	0.130931
$526$	0	0
$527$	22.5167	0.980841
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.19615	0.138701
$532$	0	0
$533$	1.12436	0.0487012
$534$	0	0
$535$	18.1244	0.783584
$536$	0	0
$537$	−11.4641	−0.494713
$538$	0	0
$539$	−1.46410	−0.0630633
$540$	0	0
$541$	−1.85641	−0.0798131	−0.0399066	−	0.999203i	$-0.512706\pi$
$541$	−1.85641	−0.0798131	−0.0399066	+	0.999203i	$0.512706\pi$
$542$	0	0
$543$	−8.39230	−0.360148
$544$	0	0
$545$	6.73205	0.288369
$546$	0	0
$547$	−34.9282	−1.49342	−0.746711	−	0.665149i	$-0.768368\pi$
$547$	−34.9282	−1.49342	−0.746711	+	0.665149i	$0.768368\pi$
$548$	0	0
$549$	13.1244	0.560134
$550$	0	0
$551$	−1.80385	−0.0768465
$552$	0	0
$553$	20.7846	0.883852
$554$	0	0
$555$	11.9282	0.506324
$556$	0	0
$557$	3.58846	0.152048	0.0760239	−	0.997106i	$-0.475777\pi$
$557$	3.58846	0.152048	0.0760239	+	0.997106i	$0.475777\pi$
$558$	0	0
$559$	−19.0333	−0.805024
$560$	0	0
$561$	4.19615	0.177162
$562$	0	0
$563$	4.51666	0.190355	0.0951773	−	0.995460i	$-0.469658\pi$
$563$	4.51666	0.190355	0.0951773	+	0.995460i	$0.469658\pi$
$564$	0	0
$565$	−9.19615	−0.386885
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	−17.3205	−0.726113	−0.363057	−	0.931767i	$-0.618267\pi$
$569$	−17.3205	−0.726113	−0.363057	+	0.931767i	$0.618267\pi$
$570$	0	0
$571$	−14.5885	−0.610508	−0.305254	−	0.952271i	$-0.598741\pi$
$571$	−14.5885	−0.610508	−0.305254	+	0.952271i	$0.598741\pi$
$572$	0	0
$573$	13.6603	0.570665
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−10.9282	−0.454947	−0.227474	−	0.973784i	$-0.573047\pi$
$577$	−10.9282	−0.454947	−0.227474	+	0.973784i	$0.573047\pi$
$578$	0	0
$579$	17.4641	0.725783
$580$	0	0
$581$	30.8038	1.27796
$582$	0	0
$583$	1.66025	0.0687607
$584$	0	0
$585$	−4.19615	−0.173490
$586$	0	0
$587$	−19.8564	−0.819562	−0.409781	−	0.912184i	$-0.634395\pi$
$587$	−19.8564	−0.819562	−0.409781	+	0.912184i	$0.634395\pi$
$588$	0	0
$589$	−26.4449	−1.08964
$590$	0	0
$591$	−22.3923	−0.921096
$592$	0	0
$593$	4.58846	0.188425	0.0942127	−	0.995552i	$-0.469967\pi$
$593$	4.58846	0.188425	0.0942127	+	0.995552i	$0.469967\pi$
$594$	0	0
$595$	−17.1962	−0.704974
$596$	0	0
$597$	4.53590	0.185642
$598$	0	0
$599$	−27.3205	−1.11629	−0.558143	−	0.829745i	$-0.688486\pi$
$599$	−27.3205	−1.11629	−0.558143	+	0.829745i	$0.688486\pi$
$600$	0	0
$601$	2.46410	0.100513	0.0502564	−	0.998736i	$-0.483996\pi$
$601$	2.46410	0.100513	0.0502564	+	0.998736i	$0.483996\pi$
$602$	0	0
$603$	−0.464102	−0.0188997
$604$	0	0
$605$	−10.4641	−0.425426
$606$	0	0
$607$	−37.1244	−1.50683	−0.753416	−	0.657545i	$-0.771595\pi$
$607$	−37.1244	−1.50683	−0.753416	+	0.657545i	$0.771595\pi$
$608$	0	0
$609$	−0.803848	−0.0325735
$610$	0	0
$611$	−40.5359	−1.63991
$612$	0	0
$613$	20.1436	0.813592	0.406796	−	0.913519i	$-0.366646\pi$
$613$	20.1436	0.813592	0.406796	+	0.913519i	$0.366646\pi$
$614$	0	0
$615$	−0.267949	−0.0108048
$616$	0	0
$617$	−40.5167	−1.63114	−0.815570	−	0.578659i	$-0.803576\pi$
$617$	−40.5167	−1.63114	−0.815570	+	0.578659i	$0.803576\pi$
$618$	0	0
$619$	−40.7846	−1.63927	−0.819636	−	0.572885i	$-0.805824\pi$
$619$	−40.7846	−1.63927	−0.819636	+	0.572885i	$0.805824\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	28.3923	1.13751
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.92820	−0.196813
$628$	0	0
$629$	−68.3731	−2.72621
$630$	0	0
$631$	17.1244	0.681710	0.340855	−	0.940116i	$-0.389284\pi$
$631$	17.1244	0.681710	0.340855	+	0.940116i	$0.389284\pi$
$632$	0	0
$633$	−19.7846	−0.786368
$634$	0	0
$635$	−1.80385	−0.0715835
$636$	0	0
$637$	−8.39230	−0.332515
$638$	0	0
$639$	0.267949	0.0105999
$640$	0	0
$641$	−25.5167	−1.00785	−0.503924	−	0.863748i	$-0.668111\pi$
$641$	−25.5167	−1.00785	−0.503924	+	0.863748i	$0.668111\pi$
$642$	0	0
$643$	43.6410	1.72103	0.860517	−	0.509422i	$-0.170141\pi$
$643$	43.6410	1.72103	0.860517	+	0.509422i	$0.170141\pi$
$644$	0	0
$645$	4.53590	0.178601
$646$	0	0
$647$	4.33975	0.170613	0.0853065	−	0.996355i	$-0.472813\pi$
$647$	4.33975	0.170613	0.0853065	+	0.996355i	$0.472813\pi$
$648$	0	0
$649$	−2.33975	−0.0918431
$650$	0	0
$651$	−11.7846	−0.461875
$652$	0	0
$653$	9.41154	0.368302	0.184151	−	0.982898i	$-0.441046\pi$
$653$	9.41154	0.368302	0.184151	+	0.982898i	$0.441046\pi$
$654$	0	0
$655$	−5.07180	−0.198171
$656$	0	0
$657$	9.66025	0.376882
$658$	0	0
$659$	−5.80385	−0.226086	−0.113043	−	0.993590i	$-0.536060\pi$
$659$	−5.80385	−0.226086	−0.113043	+	0.993590i	$0.536060\pi$
$660$	0	0
$661$	9.32051	0.362526	0.181263	−	0.983435i	$-0.441982\pi$
$661$	9.32051	0.362526	0.181263	+	0.983435i	$0.441982\pi$
$662$	0	0
$663$	24.0526	0.934124
$664$	0	0
$665$	20.1962	0.783173
$666$	0	0
$667$	0.267949	0.0103750
$668$	0	0
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	−9.60770	−0.370901
$672$	0	0
$673$	23.6603	0.912036	0.456018	−	0.889971i	$-0.349275\pi$
$673$	23.6603	0.912036	0.456018	+	0.889971i	$0.349275\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	39.8372	1.53107	0.765533	−	0.643396i	$-0.222475\pi$
$677$	39.8372	1.53107	0.765533	+	0.643396i	$0.222475\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	0	0
$681$	−9.07180	−0.347632
$682$	0	0
$683$	−10.0526	−0.384650	−0.192325	−	0.981331i	$-0.561603\pi$
$683$	−10.0526	−0.384650	−0.192325	+	0.981331i	$0.561603\pi$
$684$	0	0
$685$	20.7846	0.794139
$686$	0	0
$687$	−11.3205	−0.431904
$688$	0	0
$689$	9.51666	0.362556
$690$	0	0
$691$	2.67949	0.101933	0.0509663	−	0.998700i	$-0.483770\pi$
$691$	2.67949	0.101933	0.0509663	+	0.998700i	$0.483770\pi$
$692$	0	0
$693$	−2.19615	−0.0834249
$694$	0	0
$695$	−5.00000	−0.189661
$696$	0	0
$697$	1.53590	0.0581763
$698$	0	0
$699$	25.8564	0.977979
$700$	0	0
$701$	−15.5167	−0.586056	−0.293028	−	0.956104i	$-0.594663\pi$
$701$	−15.5167	−0.586056	−0.293028	+	0.956104i	$0.594663\pi$
$702$	0	0
$703$	80.3013	3.02862
$704$	0	0
$705$	9.66025	0.363826
$706$	0	0
$707$	−15.5885	−0.586264
$708$	0	0
$709$	−7.66025	−0.287687	−0.143843	−	0.989600i	$-0.545946\pi$
$709$	−7.66025	−0.287687	−0.143843	+	0.989600i	$0.545946\pi$
$710$	0	0
$711$	6.92820	0.259828
$712$	0	0
$713$	3.92820	0.147112
$714$	0	0
$715$	3.07180	0.114879
$716$	0	0
$717$	−21.0526	−0.786222
$718$	0	0
$719$	−5.19615	−0.193784	−0.0968919	−	0.995295i	$-0.530890\pi$
$719$	−5.19615	−0.193784	−0.0968919	+	0.995295i	$0.530890\pi$
$720$	0	0
$721$	35.5692	1.32467
$722$	0	0
$723$	−30.7321	−1.14294
$724$	0	0
$725$	−0.267949	−0.00995138
$726$	0	0
$727$	50.1769	1.86096	0.930479	−	0.366344i	$-0.119391\pi$
$727$	50.1769	1.86096	0.930479	+	0.366344i	$0.119391\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−26.0000	−0.961645
$732$	0	0
$733$	−27.1051	−1.00115	−0.500575	−	0.865693i	$-0.666878\pi$
$733$	−27.1051	−1.00115	−0.500575	+	0.865693i	$0.666878\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	0.339746	0.0125147
$738$	0	0
$739$	−48.3205	−1.77750	−0.888749	−	0.458394i	$-0.848425\pi$
$739$	−48.3205	−1.77750	−0.888749	+	0.458394i	$0.848425\pi$
$740$	0	0
$741$	−28.2487	−1.03774
$742$	0	0
$743$	28.0000	1.02722	0.513610	−	0.858024i	$-0.328308\pi$
$743$	28.0000	1.02722	0.513610	+	0.858024i	$0.328308\pi$
$744$	0	0
$745$	−16.1962	−0.593381
$746$	0	0
$747$	10.2679	0.375685
$748$	0	0
$749$	54.3731	1.98675
$750$	0	0
$751$	−5.26795	−0.192230	−0.0961151	−	0.995370i	$-0.530642\pi$
$751$	−5.26795	−0.192230	−0.0961151	+	0.995370i	$0.530642\pi$
$752$	0	0
$753$	−19.8564	−0.723608
$754$	0	0
$755$	18.3923	0.669365
$756$	0	0
$757$	−18.6077	−0.676308	−0.338154	−	0.941091i	$-0.609803\pi$
$757$	−18.6077	−0.676308	−0.338154	+	0.941091i	$0.609803\pi$
$758$	0	0
$759$	0.732051	0.0265718
$760$	0	0
$761$	−3.33975	−0.121066	−0.0605328	−	0.998166i	$-0.519280\pi$
$761$	−3.33975	−0.121066	−0.0605328	+	0.998166i	$0.519280\pi$
$762$	0	0
$763$	20.1962	0.731150
$764$	0	0
$765$	−5.73205	−0.207243
$766$	0	0
$767$	−13.4115	−0.484263
$768$	0	0
$769$	−1.80385	−0.0650484	−0.0325242	−	0.999471i	$-0.510355\pi$
$769$	−1.80385	−0.0650484	−0.0325242	+	0.999471i	$0.510355\pi$
$770$	0	0
$771$	−5.66025	−0.203849
$772$	0	0
$773$	15.4641	0.556205	0.278103	−	0.960551i	$-0.410294\pi$
$773$	15.4641	0.556205	0.278103	+	0.960551i	$0.410294\pi$
$774$	0	0
$775$	−3.92820	−0.141105
$776$	0	0
$777$	35.7846	1.28377
$778$	0	0
$779$	−1.80385	−0.0646295
$780$	0	0
$781$	−0.196152	−0.00701889
$782$	0	0
$783$	−0.267949	−0.00957572
$784$	0	0
$785$	−13.9282	−0.497119
$786$	0	0
$787$	17.0000	0.605985	0.302992	−	0.952993i	$-0.402014\pi$
$787$	17.0000	0.605985	0.302992	+	0.952993i	$0.402014\pi$
$788$	0	0
$789$	−8.26795	−0.294347
$790$	0	0
$791$	−27.5885	−0.980933
$792$	0	0
$793$	−55.0718	−1.95566
$794$	0	0
$795$	−2.26795	−0.0804359
$796$	0	0
$797$	29.9808	1.06197	0.530987	−	0.847380i	$-0.321821\pi$
$797$	29.9808	1.06197	0.530987	+	0.847380i	$0.321821\pi$
$798$	0	0
$799$	−55.3731	−1.95896
$800$	0	0
$801$	9.46410	0.334398
$802$	0	0
$803$	−7.07180	−0.249558
$804$	0	0
$805$	−3.00000	−0.105736
$806$	0	0
$807$	28.6603	1.00889
$808$	0	0
$809$	10.1244	0.355953	0.177977	−	0.984035i	$-0.443045\pi$
$809$	10.1244	0.355953	0.177977	+	0.984035i	$0.443045\pi$
$810$	0	0
$811$	−3.00000	−0.105344	−0.0526721	−	0.998612i	$-0.516774\pi$
$811$	−3.00000	−0.105344	−0.0526721	+	0.998612i	$0.516774\pi$
$812$	0	0
$813$	−5.39230	−0.189116
$814$	0	0
$815$	−14.0000	−0.490399
$816$	0	0
$817$	30.5359	1.06832
$818$	0	0
$819$	−12.5885	−0.439876
$820$	0	0
$821$	−39.7128	−1.38599	−0.692993	−	0.720944i	$-0.743708\pi$
$821$	−39.7128	−1.38599	−0.692993	+	0.720944i	$0.743708\pi$
$822$	0	0
$823$	46.9282	1.63581	0.817907	−	0.575350i	$-0.195134\pi$
$823$	46.9282	1.63581	0.817907	+	0.575350i	$0.195134\pi$
$824$	0	0
$825$	−0.732051	−0.0254867
$826$	0	0
$827$	−32.3731	−1.12572	−0.562861	−	0.826552i	$-0.690299\pi$
$827$	−32.3731	−1.12572	−0.562861	+	0.826552i	$0.690299\pi$
$828$	0	0
$829$	−27.9282	−0.969987	−0.484993	−	0.874518i	$-0.661178\pi$
$829$	−27.9282	−0.969987	−0.484993	+	0.874518i	$0.661178\pi$
$830$	0	0
$831$	14.7846	0.512872
$832$	0	0
$833$	−11.4641	−0.397208
$834$	0	0
$835$	6.33975	0.219396
$836$	0	0
$837$	−3.92820	−0.135779
$838$	0	0
$839$	−11.8564	−0.409329	−0.204664	−	0.978832i	$-0.565610\pi$
$839$	−11.8564	−0.409329	−0.204664	+	0.978832i	$0.565610\pi$
$840$	0	0
$841$	−28.9282	−0.997524
$842$	0	0
$843$	−16.0526	−0.552879
$844$	0	0
$845$	4.60770	0.158510
$846$	0	0
$847$	−31.3923	−1.07865
$848$	0	0
$849$	6.46410	0.221847
$850$	0	0
$851$	−11.9282	−0.408894
$852$	0	0
$853$	15.4641	0.529481	0.264740	−	0.964320i	$-0.414714\pi$
$853$	15.4641	0.529481	0.264740	+	0.964320i	$0.414714\pi$
$854$	0	0
$855$	6.73205	0.230231
$856$	0	0
$857$	−4.92820	−0.168344	−0.0841721	−	0.996451i	$-0.526825\pi$
$857$	−4.92820	−0.168344	−0.0841721	+	0.996451i	$0.526825\pi$
$858$	0	0
$859$	−36.0718	−1.23075	−0.615377	−	0.788233i	$-0.710996\pi$
$859$	−36.0718	−1.23075	−0.615377	+	0.788233i	$0.710996\pi$
$860$	0	0
$861$	−0.803848	−0.0273951
$862$	0	0
$863$	6.39230	0.217597	0.108798	−	0.994064i	$-0.465300\pi$
$863$	6.39230	0.217597	0.108798	+	0.994064i	$0.465300\pi$
$864$	0	0
$865$	−25.8564	−0.879144
$866$	0	0
$867$	15.8564	0.538512
$868$	0	0
$869$	−5.07180	−0.172049
$870$	0	0
$871$	1.94744	0.0659865
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	20.5359	0.693448	0.346724	−	0.937967i	$-0.387294\pi$
$877$	20.5359	0.693448	0.346724	+	0.937967i	$0.387294\pi$
$878$	0	0
$879$	−7.73205	−0.260796
$880$	0	0
$881$	−24.1962	−0.815189	−0.407595	−	0.913163i	$-0.633632\pi$
$881$	−24.1962	−0.815189	−0.407595	+	0.913163i	$0.633632\pi$
$882$	0	0
$883$	44.0526	1.48249	0.741243	−	0.671236i	$-0.234236\pi$
$883$	44.0526	1.48249	0.741243	+	0.671236i	$0.234236\pi$
$884$	0	0
$885$	3.19615	0.107437
$886$	0	0
$887$	3.46410	0.116313	0.0581566	−	0.998307i	$-0.481478\pi$
$887$	3.46410	0.116313	0.0581566	+	0.998307i	$0.481478\pi$
$888$	0	0
$889$	−5.41154	−0.181497
$890$	0	0
$891$	−0.732051	−0.0245246
$892$	0	0
$893$	65.0333	2.17626
$894$	0	0
$895$	−11.4641	−0.383203
$896$	0	0
$897$	4.19615	0.140105
$898$	0	0
$899$	1.05256	0.0351048
$900$	0	0
$901$	13.0000	0.433093
$902$	0	0
$903$	13.6077	0.452836
$904$	0	0
$905$	−8.39230	−0.278970
$906$	0	0
$907$	−40.1769	−1.33405	−0.667026	−	0.745034i	$-0.732433\pi$
$907$	−40.1769	−1.33405	−0.667026	+	0.745034i	$0.732433\pi$
$908$	0	0
$909$	−5.19615	−0.172345
$910$	0	0
$911$	34.9282	1.15722	0.578612	−	0.815603i	$-0.303595\pi$
$911$	34.9282	1.15722	0.578612	+	0.815603i	$0.303595\pi$
$912$	0	0
$913$	−7.51666	−0.248765
$914$	0	0
$915$	13.1244	0.433878
$916$	0	0
$917$	−15.2154	−0.502456
$918$	0	0
$919$	33.8564	1.11682	0.558410	−	0.829565i	$-0.311412\pi$
$919$	33.8564	1.11682	0.558410	+	0.829565i	$0.311412\pi$
$920$	0	0
$921$	−24.1962	−0.797290
$922$	0	0
$923$	−1.12436	−0.0370086
$924$	0	0
$925$	11.9282	0.392197
$926$	0	0
$927$	11.8564	0.389415
$928$	0	0
$929$	29.9808	0.983637	0.491818	−	0.870698i	$-0.336332\pi$
$929$	29.9808	0.983637	0.491818	+	0.870698i	$0.336332\pi$
$930$	0	0
$931$	13.4641	0.441268
$932$	0	0
$933$	12.0000	0.392862
$934$	0	0
$935$	4.19615	0.137229
$936$	0	0
$937$	36.4974	1.19232	0.596159	−	0.802866i	$-0.296693\pi$
$937$	36.4974	1.19232	0.596159	+	0.802866i	$0.296693\pi$
$938$	0	0
$939$	5.92820	0.193460
$940$	0	0
$941$	48.6936	1.58737	0.793683	−	0.608332i	$-0.208161\pi$
$941$	48.6936	1.58737	0.793683	+	0.608332i	$0.208161\pi$
$942$	0	0
$943$	0.267949	0.00872563
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	−31.6077	−1.02711	−0.513556	−	0.858056i	$-0.671672\pi$
$947$	−31.6077	−1.02711	−0.513556	+	0.858056i	$0.671672\pi$
$948$	0	0
$949$	−40.5359	−1.31585
$950$	0	0
$951$	−27.5167	−0.892289
$952$	0	0
$953$	12.1436	0.393370	0.196685	−	0.980467i	$-0.436982\pi$
$953$	12.1436	0.393370	0.196685	+	0.980467i	$0.436982\pi$
$954$	0	0
$955$	13.6603	0.442035
$956$	0	0
$957$	0.196152	0.00634071
$958$	0	0
$959$	62.3538	2.01351
$960$	0	0
$961$	−15.5692	−0.502233
$962$	0	0
$963$	18.1244	0.584049
$964$	0	0
$965$	17.4641	0.562189
$966$	0	0
$967$	−5.80385	−0.186639	−0.0933196	−	0.995636i	$-0.529748\pi$
$967$	−5.80385	−0.186639	−0.0933196	+	0.995636i	$0.529748\pi$
$968$	0	0
$969$	−38.5885	−1.23964
$970$	0	0
$971$	−2.53590	−0.0813809	−0.0406904	−	0.999172i	$-0.512956\pi$
$971$	−2.53590	−0.0813809	−0.0406904	+	0.999172i	$0.512956\pi$
$972$	0	0
$973$	−15.0000	−0.480878
$974$	0	0
$975$	−4.19615	−0.134384
$976$	0	0
$977$	−16.1244	−0.515864	−0.257932	−	0.966163i	$-0.583041\pi$
$977$	−16.1244	−0.515864	−0.257932	+	0.966163i	$0.583041\pi$
$978$	0	0
$979$	−6.92820	−0.221426
$980$	0	0
$981$	6.73205	0.214938
$982$	0	0
$983$	13.9808	0.445917	0.222959	−	0.974828i	$-0.428429\pi$
$983$	13.9808	0.445917	0.222959	+	0.974828i	$0.428429\pi$
$984$	0	0
$985$	−22.3923	−0.713478
$986$	0	0
$987$	28.9808	0.922468
$988$	0	0
$989$	−4.53590	−0.144233
$990$	0	0
$991$	25.0000	0.794151	0.397076	−	0.917786i	$-0.370025\pi$
$991$	25.0000	0.794151	0.397076	+	0.917786i	$0.370025\pi$
$992$	0	0
$993$	14.3205	0.454448
$994$	0	0
$995$	4.53590	0.143798
$996$	0	0
$997$	−57.4641	−1.81991	−0.909953	−	0.414711i	$-0.863883\pi$
$997$	−57.4641	−1.81991	−0.909953	+	0.414711i	$0.863883\pi$
$998$	0	0
$999$	11.9282	0.377392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bu.1.1		2
4.3	odd	2		345.2.a.g.1.2	✓	2
12.11	even	2		1035.2.a.l.1.1		2
20.3	even	4		1725.2.b.p.1174.2		4
20.7	even	4		1725.2.b.p.1174.3		4
20.19	odd	2		1725.2.a.bd.1.1		2
60.59	even	2		5175.2.a.bd.1.2		2
92.91	even	2		7935.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.g.1.2	✓	2	4.3	odd	2
1035.2.a.l.1.1		2	12.11	even	2
1725.2.a.bd.1.1		2	20.19	odd	2
1725.2.b.p.1174.2		4	20.3	even	4
1725.2.b.p.1174.3		4	20.7	even	4
5175.2.a.bd.1.2		2	60.59	even	2
5520.2.a.bu.1.1		2	1.1	even	1	trivial
7935.2.a.n.1.2		2	92.91	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -0.732051 q^{11} -4.19615 q^{13} +1.00000 q^{15} -5.73205 q^{17} +6.73205 q^{19} +3.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.267949 q^{29} -3.92820 q^{31} -0.732051 q^{33} +3.00000 q^{35} +11.9282 q^{37} -4.19615 q^{39} -0.267949 q^{41} +4.53590 q^{43} +1.00000 q^{45} +9.66025 q^{47} +2.00000 q^{49} -5.73205 q^{51} -2.26795 q^{53} -0.732051 q^{55} +6.73205 q^{57} +3.19615 q^{59} +13.1244 q^{61} +3.00000 q^{63} -4.19615 q^{65} -0.464102 q^{67} -1.00000 q^{69} +0.267949 q^{71} +9.66025 q^{73} +1.00000 q^{75} -2.19615 q^{77} +6.92820 q^{79} +1.00000 q^{81} +10.2679 q^{83} -5.73205 q^{85} -0.267949 q^{87} +9.46410 q^{89} -12.5885 q^{91} -3.92820 q^{93} +6.73205 q^{95} -4.00000 q^{97} -0.732051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 2 q^{15} - 8 q^{17} + 10 q^{19} + 6 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} + 2 q^{33} + 6 q^{35} + 10 q^{37} + 2 q^{39} - 4 q^{41} + 16 q^{43} + 2 q^{45} + 2 q^{47} + 4 q^{49} - 8 q^{51} - 8 q^{53} + 2 q^{55} + 10 q^{57} - 4 q^{59} + 2 q^{61} + 6 q^{63} + 2 q^{65} + 6 q^{67} - 2 q^{69} + 4 q^{71} + 2 q^{73} + 2 q^{75} + 6 q^{77} + 2 q^{81} + 24 q^{83} - 8 q^{85} - 4 q^{87} + 12 q^{89} + 6 q^{91} + 6 q^{93} + 10 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 2 * q^15 - 8 * q^17 + 10 * q^19 + 6 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 + 6 * q^31 + 2 * q^33 + 6 * q^35 + 10 * q^37 + 2 * q^39 - 4 * q^41 + 16 * q^43 + 2 * q^45 + 2 * q^47 + 4 * q^49 - 8 * q^51 - 8 * q^53 + 2 * q^55 + 10 * q^57 - 4 * q^59 + 2 * q^61 + 6 * q^63 + 2 * q^65 + 6 * q^67 - 2 * q^69 + 4 * q^71 + 2 * q^73 + 2 * q^75 + 6 * q^77 + 2 * q^81 + 24 * q^83 - 8 * q^85 - 4 * q^87 + 12 * q^89 + 6 * q^91 + 6 * q^93 + 10 * q^95 - 8 * q^97 + 2 * q^99

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