LMFDB - Embedded newform 8550.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2850)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.732051 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -0.732051 q^{7} -1.00000 q^{8} -3.73205 q^{11} +2.73205 q^{13} +0.732051 q^{14} +1.00000 q^{16} -6.19615 q^{17} -1.00000 q^{19} +3.73205 q^{22} +5.92820 q^{23} -2.73205 q^{26} -0.732051 q^{28} +1.73205 q^{29} -2.46410 q^{31} -1.00000 q^{32} +6.19615 q^{34} +2.00000 q^{37} +1.00000 q^{38} +2.92820 q^{41} +8.19615 q^{43} -3.73205 q^{44} -5.92820 q^{46} +3.46410 q^{47} -6.46410 q^{49} +2.73205 q^{52} -1.73205 q^{53} +0.732051 q^{56} -1.73205 q^{58} +8.19615 q^{59} +10.6603 q^{61} +2.46410 q^{62} +1.00000 q^{64} +0.267949 q^{67} -6.19615 q^{68} -12.1962 q^{71} +2.46410 q^{73} -2.00000 q^{74} -1.00000 q^{76} +2.73205 q^{77} -5.53590 q^{79} -2.92820 q^{82} +3.73205 q^{83} -8.19615 q^{86} +3.73205 q^{88} -10.8564 q^{89} -2.00000 q^{91} +5.92820 q^{92} -3.46410 q^{94} +15.1244 q^{97} +6.46410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{19} + 4 q^{22} - 2 q^{23} - 2 q^{26} + 2 q^{28} + 2 q^{31} - 2 q^{32} + 2 q^{34} + 4 q^{37} + 2 q^{38} - 8 q^{41} + 6 q^{43} - 4 q^{44} + 2 q^{46} - 6 q^{49} + 2 q^{52} - 2 q^{56} + 6 q^{59} + 4 q^{61} - 2 q^{62} + 2 q^{64} + 4 q^{67} - 2 q^{68} - 14 q^{71} - 2 q^{73} - 4 q^{74} - 2 q^{76} + 2 q^{77} - 18 q^{79} + 8 q^{82} + 4 q^{83} - 6 q^{86} + 4 q^{88} + 6 q^{89} - 4 q^{91} - 2 q^{92} + 6 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 4 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - 2 * q^19 + 4 * q^22 - 2 * q^23 - 2 * q^26 + 2 * q^28 + 2 * q^31 - 2 * q^32 + 2 * q^34 + 4 * q^37 + 2 * q^38 - 8 * q^41 + 6 * q^43 - 4 * q^44 + 2 * q^46 - 6 * q^49 + 2 * q^52 - 2 * q^56 + 6 * q^59 + 4 * q^61 - 2 * q^62 + 2 * q^64 + 4 * q^67 - 2 * q^68 - 14 * q^71 - 2 * q^73 - 4 * q^74 - 2 * q^76 + 2 * q^77 - 18 * q^79 + 8 * q^82 + 4 * q^83 - 6 * q^86 + 4 * q^88 + 6 * q^89 - 4 * q^91 - 2 * q^92 + 6 * q^97 + 6 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−3.73205	−1.12526	−0.562628	−	0.826710i	$-0.690210\pi$
$11$	−3.73205	−1.12526	−0.562628	+	0.826710i	$0.690210\pi$
$12$	0	0
$13$	2.73205	0.757735	0.378867	−	0.925451i	$-0.376314\pi$
$13$	2.73205	0.757735	0.378867	+	0.925451i	$0.376314\pi$
$14$	0.732051	0.195649
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.19615	−1.50279	−0.751394	−	0.659854i	$-0.770618\pi$
$17$	−6.19615	−1.50279	−0.751394	+	0.659854i	$0.770618\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	3.73205	0.795676
$23$	5.92820	1.23612	0.618058	−	0.786133i	$-0.287920\pi$
$23$	5.92820	1.23612	0.618058	+	0.786133i	$0.287920\pi$
$24$	0	0
$25$	0	0
$26$	−2.73205	−0.535799
$27$	0	0
$28$	−0.732051	−0.138345
$29$	1.73205	0.321634	0.160817	−	0.986984i	$-0.448587\pi$
$29$	1.73205	0.321634	0.160817	+	0.986984i	$0.448587\pi$
$30$	0	0
$31$	−2.46410	−0.442566	−0.221283	−	0.975210i	$-0.571024\pi$
$31$	−2.46410	−0.442566	−0.221283	+	0.975210i	$0.571024\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.19615	1.06263
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	2.92820	0.457309	0.228654	−	0.973508i	$-0.426567\pi$
$41$	2.92820	0.457309	0.228654	+	0.973508i	$0.426567\pi$
$42$	0	0
$43$	8.19615	1.24990	0.624951	−	0.780664i	$-0.285119\pi$
$43$	8.19615	1.24990	0.624951	+	0.780664i	$0.285119\pi$
$44$	−3.73205	−0.562628
$45$	0	0
$46$	−5.92820	−0.874066
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0	0
$52$	2.73205	0.378867
$53$	−1.73205	−0.237915	−0.118958	−	0.992899i	$-0.537955\pi$
$53$	−1.73205	−0.237915	−0.118958	+	0.992899i	$0.537955\pi$
$54$	0	0
$55$	0	0
$56$	0.732051	0.0978244
$57$	0	0
$58$	−1.73205	−0.227429
$59$	8.19615	1.06705	0.533524	−	0.845785i	$-0.320867\pi$
$59$	8.19615	1.06705	0.533524	+	0.845785i	$0.320867\pi$
$60$	0	0
$61$	10.6603	1.36491	0.682453	−	0.730930i	$-0.260913\pi$
$61$	10.6603	1.36491	0.682453	+	0.730930i	$0.260913\pi$
$62$	2.46410	0.312941
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	0.267949	0.0327352	0.0163676	−	0.999866i	$-0.494790\pi$
$67$	0.267949	0.0327352	0.0163676	+	0.999866i	$0.494790\pi$
$68$	−6.19615	−0.751394
$69$	0	0
$70$	0	0
$71$	−12.1962	−1.44742	−0.723708	−	0.690106i	$-0.757564\pi$
$71$	−12.1962	−1.44742	−0.723708	+	0.690106i	$0.757564\pi$
$72$	0	0
$73$	2.46410	0.288401	0.144201	−	0.989548i	$-0.453939\pi$
$73$	2.46410	0.288401	0.144201	+	0.989548i	$0.453939\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−1.00000	−0.114708
$77$	2.73205	0.311346
$78$	0	0
$79$	−5.53590	−0.622837	−0.311419	−	0.950273i	$-0.600804\pi$
$79$	−5.53590	−0.622837	−0.311419	+	0.950273i	$0.600804\pi$
$80$	0	0
$81$	0	0
$82$	−2.92820	−0.323366
$83$	3.73205	0.409646	0.204823	−	0.978799i	$-0.434338\pi$
$83$	3.73205	0.409646	0.204823	+	0.978799i	$0.434338\pi$
$84$	0	0
$85$	0	0
$86$	−8.19615	−0.883814
$87$	0	0
$88$	3.73205	0.397838
$89$	−10.8564	−1.15078	−0.575388	−	0.817880i	$-0.695149\pi$
$89$	−10.8564	−1.15078	−0.575388	+	0.817880i	$0.695149\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	5.92820	0.618058
$93$	0	0
$94$	−3.46410	−0.357295
$95$	0	0
$96$	0	0
$97$	15.1244	1.53565	0.767823	−	0.640662i	$-0.221340\pi$
$97$	15.1244	1.53565	0.767823	+	0.640662i	$0.221340\pi$
$98$	6.46410	0.652973
$99$	0	0
$100$	0	0
$101$	−9.26795	−0.922195	−0.461098	−	0.887349i	$-0.652544\pi$
$101$	−9.26795	−0.922195	−0.461098	+	0.887349i	$0.652544\pi$
$102$	0	0
$103$	3.53590	0.348402	0.174201	−	0.984710i	$-0.444266\pi$
$103$	3.53590	0.348402	0.174201	+	0.984710i	$0.444266\pi$
$104$	−2.73205	−0.267900
$105$	0	0
$106$	1.73205	0.168232
$107$	2.19615	0.212310	0.106155	−	0.994350i	$-0.466146\pi$
$107$	2.19615	0.212310	0.106155	+	0.994350i	$0.466146\pi$
$108$	0	0
$109$	2.39230	0.229141	0.114571	−	0.993415i	$-0.463451\pi$
$109$	2.39230	0.229141	0.114571	+	0.993415i	$0.463451\pi$
$110$	0	0
$111$	0	0
$112$	−0.732051	−0.0691723
$113$	−13.3923	−1.25984	−0.629921	−	0.776659i	$-0.716913\pi$
$113$	−13.3923	−1.25984	−0.629921	+	0.776659i	$0.716913\pi$
$114$	0	0
$115$	0	0
$116$	1.73205	0.160817
$117$	0	0
$118$	−8.19615	−0.754517
$119$	4.53590	0.415805
$120$	0	0
$121$	2.92820	0.266200
$122$	−10.6603	−0.965134
$123$	0	0
$124$	−2.46410	−0.221283
$125$	0	0
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−10.2679	−0.897115	−0.448557	−	0.893754i	$-0.648062\pi$
$131$	−10.2679	−0.897115	−0.448557	+	0.893754i	$0.648062\pi$
$132$	0	0
$133$	0.732051	0.0634769
$134$	−0.267949	−0.0231473
$135$	0	0
$136$	6.19615	0.531316
$137$	−12.3923	−1.05875	−0.529373	−	0.848389i	$-0.677573\pi$
$137$	−12.3923	−1.05875	−0.529373	+	0.848389i	$0.677573\pi$
$138$	0	0
$139$	−21.1244	−1.79174	−0.895872	−	0.444312i	$-0.853448\pi$
$139$	−21.1244	−1.79174	−0.895872	+	0.444312i	$0.853448\pi$
$140$	0	0
$141$	0	0
$142$	12.1962	1.02348
$143$	−10.1962	−0.852645
$144$	0	0
$145$	0	0
$146$	−2.46410	−0.203931
$147$	0	0
$148$	2.00000	0.164399
$149$	21.8564	1.79055	0.895273	−	0.445517i	$-0.146980\pi$
$149$	21.8564	1.79055	0.895273	+	0.445517i	$0.146980\pi$
$150$	0	0
$151$	−11.4641	−0.932935	−0.466468	−	0.884538i	$-0.654474\pi$
$151$	−11.4641	−0.932935	−0.466468	+	0.884538i	$0.654474\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−2.73205	−0.220155
$155$	0	0
$156$	0	0
$157$	−13.8564	−1.10586	−0.552931	−	0.833227i	$-0.686491\pi$
$157$	−13.8564	−1.10586	−0.552931	+	0.833227i	$0.686491\pi$
$158$	5.53590	0.440412
$159$	0	0
$160$	0	0
$161$	−4.33975	−0.342020
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	2.92820	0.228654
$165$	0	0
$166$	−3.73205	−0.289663
$167$	−8.92820	−0.690885	−0.345443	−	0.938440i	$-0.612271\pi$
$167$	−8.92820	−0.690885	−0.345443	+	0.938440i	$0.612271\pi$
$168$	0	0
$169$	−5.53590	−0.425838
$170$	0	0
$171$	0	0
$172$	8.19615	0.624951
$173$	13.5885	1.03311	0.516556	−	0.856254i	$-0.327214\pi$
$173$	13.5885	1.03311	0.516556	+	0.856254i	$0.327214\pi$
$174$	0	0
$175$	0	0
$176$	−3.73205	−0.281314
$177$	0	0
$178$	10.8564	0.813722
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−16.5885	−1.23301	−0.616505	−	0.787351i	$-0.711452\pi$
$181$	−16.5885	−1.23301	−0.616505	+	0.787351i	$0.711452\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	−5.92820	−0.437033
$185$	0	0
$186$	0	0
$187$	23.1244	1.69102
$188$	3.46410	0.252646
$189$	0	0
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	−22.0526	−1.58738	−0.793689	−	0.608324i	$-0.791842\pi$
$193$	−22.0526	−1.58738	−0.793689	+	0.608324i	$0.791842\pi$
$194$	−15.1244	−1.08587
$195$	0	0
$196$	−6.46410	−0.461722
$197$	24.5885	1.75186	0.875928	−	0.482443i	$-0.160250\pi$
$197$	24.5885	1.75186	0.875928	+	0.482443i	$0.160250\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	0	0
$202$	9.26795	0.652091
$203$	−1.26795	−0.0889926
$204$	0	0
$205$	0	0
$206$	−3.53590	−0.246358
$207$	0	0
$208$	2.73205	0.189434
$209$	3.73205	0.258151
$210$	0	0
$211$	−10.2679	−0.706875	−0.353437	−	0.935458i	$-0.614987\pi$
$211$	−10.2679	−0.706875	−0.353437	+	0.935458i	$0.614987\pi$
$212$	−1.73205	−0.118958
$213$	0	0
$214$	−2.19615	−0.150126
$215$	0	0
$216$	0	0
$217$	1.80385	0.122453
$218$	−2.39230	−0.162027
$219$	0	0
$220$	0	0
$221$	−16.9282	−1.13871
$222$	0	0
$223$	3.53590	0.236781	0.118391	−	0.992967i	$-0.462226\pi$
$223$	3.53590	0.236781	0.118391	+	0.992967i	$0.462226\pi$
$224$	0.732051	0.0489122
$225$	0	0
$226$	13.3923	0.890843
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	−6.26795	−0.414198	−0.207099	−	0.978320i	$-0.566402\pi$
$229$	−6.26795	−0.414198	−0.207099	+	0.978320i	$0.566402\pi$
$230$	0	0
$231$	0	0
$232$	−1.73205	−0.113715
$233$	−21.1244	−1.38390	−0.691951	−	0.721944i	$-0.743249\pi$
$233$	−21.1244	−1.38390	−0.691951	+	0.721944i	$0.743249\pi$
$234$	0	0
$235$	0	0
$236$	8.19615	0.533524
$237$	0	0
$238$	−4.53590	−0.294019
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	0	0
$241$	−30.1962	−1.94511	−0.972553	−	0.232683i	$-0.925249\pi$
$241$	−30.1962	−1.94511	−0.972553	+	0.232683i	$0.925249\pi$
$242$	−2.92820	−0.188232
$243$	0	0
$244$	10.6603	0.682453
$245$	0	0
$246$	0	0
$247$	−2.73205	−0.173836
$248$	2.46410	0.156471
$249$	0	0
$250$	0	0
$251$	−11.4641	−0.723608	−0.361804	−	0.932254i	$-0.617839\pi$
$251$	−11.4641	−0.723608	−0.361804	+	0.932254i	$0.617839\pi$
$252$	0	0
$253$	−22.1244	−1.39095
$254$	−5.00000	−0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.00000	0.561405	0.280702	−	0.959795i	$-0.409433\pi$
$257$	9.00000	0.561405	0.280702	+	0.959795i	$0.409433\pi$
$258$	0	0
$259$	−1.46410	−0.0909748
$260$	0	0
$261$	0	0
$262$	10.2679	0.634356
$263$	15.0000	0.924940	0.462470	−	0.886635i	$-0.346963\pi$
$263$	15.0000	0.924940	0.462470	+	0.886635i	$0.346963\pi$
$264$	0	0
$265$	0	0
$266$	−0.732051	−0.0448849
$267$	0	0
$268$	0.267949	0.0163676
$269$	20.5359	1.25210	0.626048	−	0.779785i	$-0.284671\pi$
$269$	20.5359	1.25210	0.626048	+	0.779785i	$0.284671\pi$
$270$	0	0
$271$	−15.8038	−0.960015	−0.480008	−	0.877264i	$-0.659366\pi$
$271$	−15.8038	−0.960015	−0.480008	+	0.877264i	$0.659366\pi$
$272$	−6.19615	−0.375697
$273$	0	0
$274$	12.3923	0.748647
$275$	0	0
$276$	0	0
$277$	−11.1962	−0.672712	−0.336356	−	0.941735i	$-0.609194\pi$
$277$	−11.1962	−0.672712	−0.336356	+	0.941735i	$0.609194\pi$
$278$	21.1244	1.26695
$279$	0	0
$280$	0	0
$281$	−17.7846	−1.06094	−0.530470	−	0.847703i	$-0.677985\pi$
$281$	−17.7846	−1.06094	−0.530470	+	0.847703i	$0.677985\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.1962	−0.723708
$285$	0	0
$286$	10.1962	0.602911
$287$	−2.14359	−0.126532
$288$	0	0
$289$	21.3923	1.25837
$290$	0	0
$291$	0	0
$292$	2.46410	0.144201
$293$	−27.7321	−1.62012	−0.810062	−	0.586344i	$-0.800567\pi$
$293$	−27.7321	−1.62012	−0.810062	+	0.586344i	$0.800567\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−21.8564	−1.26611
$299$	16.1962	0.936648
$300$	0	0
$301$	−6.00000	−0.345834
$302$	11.4641	0.659685
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	20.2679	1.15675	0.578376	−	0.815770i	$-0.303686\pi$
$307$	20.2679	1.15675	0.578376	+	0.815770i	$0.303686\pi$
$308$	2.73205	0.155673
$309$	0	0
$310$	0	0
$311$	29.3205	1.66261	0.831307	−	0.555814i	$-0.187593\pi$
$311$	29.3205	1.66261	0.831307	+	0.555814i	$0.187593\pi$
$312$	0	0
$313$	3.92820	0.222035	0.111018	−	0.993818i	$-0.464589\pi$
$313$	3.92820	0.222035	0.111018	+	0.993818i	$0.464589\pi$
$314$	13.8564	0.781962
$315$	0	0
$316$	−5.53590	−0.311419
$317$	−1.73205	−0.0972817	−0.0486408	−	0.998816i	$-0.515489\pi$
$317$	−1.73205	−0.0972817	−0.0486408	+	0.998816i	$0.515489\pi$
$318$	0	0
$319$	−6.46410	−0.361920
$320$	0	0
$321$	0	0
$322$	4.33975	0.241845
$323$	6.19615	0.344763
$324$	0	0
$325$	0	0
$326$	−2.00000	−0.110770
$327$	0	0
$328$	−2.92820	−0.161683
$329$	−2.53590	−0.139809
$330$	0	0
$331$	−22.2679	−1.22396	−0.611979	−	0.790874i	$-0.709626\pi$
$331$	−22.2679	−1.22396	−0.611979	+	0.790874i	$0.709626\pi$
$332$	3.73205	0.204823
$333$	0	0
$334$	8.92820	0.488530
$335$	0	0
$336$	0	0
$337$	−10.5359	−0.573927	−0.286963	−	0.957942i	$-0.592646\pi$
$337$	−10.5359	−0.573927	−0.286963	+	0.957942i	$0.592646\pi$
$338$	5.53590	0.301113
$339$	0	0
$340$	0	0
$341$	9.19615	0.498000
$342$	0	0
$343$	9.85641	0.532196
$344$	−8.19615	−0.441907
$345$	0	0
$346$	−13.5885	−0.730520
$347$	−20.7846	−1.11578	−0.557888	−	0.829916i	$-0.688388\pi$
$347$	−20.7846	−1.11578	−0.557888	+	0.829916i	$0.688388\pi$
$348$	0	0
$349$	−17.0526	−0.912803	−0.456401	−	0.889774i	$-0.650862\pi$
$349$	−17.0526	−0.912803	−0.456401	+	0.889774i	$0.650862\pi$
$350$	0	0
$351$	0	0
$352$	3.73205	0.198919
$353$	16.0526	0.854391	0.427196	−	0.904159i	$-0.359502\pi$
$353$	16.0526	0.854391	0.427196	+	0.904159i	$0.359502\pi$
$354$	0	0
$355$	0	0
$356$	−10.8564	−0.575388
$357$	0	0
$358$	−12.0000	−0.634220
$359$	16.9282	0.893436	0.446718	−	0.894675i	$-0.352593\pi$
$359$	16.9282	0.893436	0.446718	+	0.894675i	$0.352593\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	16.5885	0.871870
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	13.2679	0.692581	0.346291	−	0.938127i	$-0.387441\pi$
$367$	13.2679	0.692581	0.346291	+	0.938127i	$0.387441\pi$
$368$	5.92820	0.309029
$369$	0	0
$370$	0	0
$371$	1.26795	0.0658286
$372$	0	0
$373$	31.5167	1.63187	0.815935	−	0.578143i	$-0.196222\pi$
$373$	31.5167	1.63187	0.815935	+	0.578143i	$0.196222\pi$
$374$	−23.1244	−1.19573
$375$	0	0
$376$	−3.46410	−0.178647
$377$	4.73205	0.243713
$378$	0	0
$379$	−0.392305	−0.0201513	−0.0100757	−	0.999949i	$-0.503207\pi$
$379$	−0.392305	−0.0201513	−0.0100757	+	0.999949i	$0.503207\pi$
$380$	0	0
$381$	0	0
$382$	3.00000	0.153493
$383$	−19.5167	−0.997255	−0.498627	−	0.866816i	$-0.666162\pi$
$383$	−19.5167	−0.997255	−0.498627	+	0.866816i	$0.666162\pi$
$384$	0	0
$385$	0	0
$386$	22.0526	1.12245
$387$	0	0
$388$	15.1244	0.767823
$389$	−21.7128	−1.10088	−0.550442	−	0.834874i	$-0.685541\pi$
$389$	−21.7128	−1.10088	−0.550442	+	0.834874i	$0.685541\pi$
$390$	0	0
$391$	−36.7321	−1.85762
$392$	6.46410	0.326486
$393$	0	0
$394$	−24.5885	−1.23875
$395$	0	0
$396$	0	0
$397$	−11.7321	−0.588815	−0.294407	−	0.955680i	$-0.595122\pi$
$397$	−11.7321	−0.588815	−0.294407	+	0.955680i	$0.595122\pi$
$398$	14.0000	0.701757
$399$	0	0
$400$	0	0
$401$	−16.8564	−0.841769	−0.420884	−	0.907114i	$-0.638280\pi$
$401$	−16.8564	−0.841769	−0.420884	+	0.907114i	$0.638280\pi$
$402$	0	0
$403$	−6.73205	−0.335347
$404$	−9.26795	−0.461098
$405$	0	0
$406$	1.26795	0.0629273
$407$	−7.46410	−0.369982
$408$	0	0
$409$	1.60770	0.0794954	0.0397477	−	0.999210i	$-0.487345\pi$
$409$	1.60770	0.0794954	0.0397477	+	0.999210i	$0.487345\pi$
$410$	0	0
$411$	0	0
$412$	3.53590	0.174201
$413$	−6.00000	−0.295241
$414$	0	0
$415$	0	0
$416$	−2.73205	−0.133950
$417$	0	0
$418$	−3.73205	−0.182541
$419$	−13.8564	−0.676930	−0.338465	−	0.940979i	$-0.609908\pi$
$419$	−13.8564	−0.676930	−0.338465	+	0.940979i	$0.609908\pi$
$420$	0	0
$421$	6.78461	0.330662	0.165331	−	0.986238i	$-0.447131\pi$
$421$	6.78461	0.330662	0.165331	+	0.986238i	$0.447131\pi$
$422$	10.2679	0.499836
$423$	0	0
$424$	1.73205	0.0841158
$425$	0	0
$426$	0	0
$427$	−7.80385	−0.377655
$428$	2.19615	0.106155
$429$	0	0
$430$	0	0
$431$	29.3205	1.41232	0.706160	−	0.708053i	$-0.250426\pi$
$431$	29.3205	1.41232	0.706160	+	0.708053i	$0.250426\pi$
$432$	0	0
$433$	−24.1962	−1.16279	−0.581396	−	0.813620i	$-0.697493\pi$
$433$	−24.1962	−1.16279	−0.581396	+	0.813620i	$0.697493\pi$
$434$	−1.80385	−0.0865875
$435$	0	0
$436$	2.39230	0.114571
$437$	−5.92820	−0.283584
$438$	0	0
$439$	25.7846	1.23063	0.615316	−	0.788280i	$-0.289028\pi$
$439$	25.7846	1.23063	0.615316	+	0.788280i	$0.289028\pi$
$440$	0	0
$441$	0	0
$442$	16.9282	0.805193
$443$	3.33975	0.158676	0.0793381	−	0.996848i	$-0.474719\pi$
$443$	3.33975	0.158676	0.0793381	+	0.996848i	$0.474719\pi$
$444$	0	0
$445$	0	0
$446$	−3.53590	−0.167430
$447$	0	0
$448$	−0.732051	−0.0345861
$449$	−24.4641	−1.15453	−0.577266	−	0.816556i	$-0.695880\pi$
$449$	−24.4641	−1.15453	−0.577266	+	0.816556i	$0.695880\pi$
$450$	0	0
$451$	−10.9282	−0.514589
$452$	−13.3923	−0.629921
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	−5.07180	−0.237249	−0.118624	−	0.992939i	$-0.537848\pi$
$457$	−5.07180	−0.237249	−0.118624	+	0.992939i	$0.537848\pi$
$458$	6.26795	0.292882
$459$	0	0
$460$	0	0
$461$	17.8564	0.831656	0.415828	−	0.909443i	$-0.363492\pi$
$461$	17.8564	0.831656	0.415828	+	0.909443i	$0.363492\pi$
$462$	0	0
$463$	−35.3205	−1.64148	−0.820742	−	0.571300i	$-0.806439\pi$
$463$	−35.3205	−1.64148	−0.820742	+	0.571300i	$0.806439\pi$
$464$	1.73205	0.0804084
$465$	0	0
$466$	21.1244	0.978567
$467$	−7.19615	−0.332998	−0.166499	−	0.986042i	$-0.553246\pi$
$467$	−7.19615	−0.332998	−0.166499	+	0.986042i	$0.553246\pi$
$468$	0	0
$469$	−0.196152	−0.00905748
$470$	0	0
$471$	0	0
$472$	−8.19615	−0.377258
$473$	−30.5885	−1.40646
$474$	0	0
$475$	0	0
$476$	4.53590	0.207903
$477$	0	0
$478$	−14.0000	−0.640345
$479$	−3.14359	−0.143634	−0.0718172	−	0.997418i	$-0.522880\pi$
$479$	−3.14359	−0.143634	−0.0718172	+	0.997418i	$0.522880\pi$
$480$	0	0
$481$	5.46410	0.249142
$482$	30.1962	1.37540
$483$	0	0
$484$	2.92820	0.133100
$485$	0	0
$486$	0	0
$487$	2.14359	0.0971355	0.0485677	−	0.998820i	$-0.484534\pi$
$487$	2.14359	0.0971355	0.0485677	+	0.998820i	$0.484534\pi$
$488$	−10.6603	−0.482567
$489$	0	0
$490$	0	0
$491$	2.92820	0.132148	0.0660740	−	0.997815i	$-0.478953\pi$
$491$	2.92820	0.132148	0.0660740	+	0.997815i	$0.478953\pi$
$492$	0	0
$493$	−10.7321	−0.483347
$494$	2.73205	0.122921
$495$	0	0
$496$	−2.46410	−0.110641
$497$	8.92820	0.400485
$498$	0	0
$499$	41.3731	1.85211	0.926056	−	0.377385i	$-0.123177\pi$
$499$	41.3731	1.85211	0.926056	+	0.377385i	$0.123177\pi$
$500$	0	0
$501$	0	0
$502$	11.4641	0.511668
$503$	17.8564	0.796178	0.398089	−	0.917347i	$-0.369674\pi$
$503$	17.8564	0.796178	0.398089	+	0.917347i	$0.369674\pi$
$504$	0	0
$505$	0	0
$506$	22.1244	0.983548
$507$	0	0
$508$	5.00000	0.221839
$509$	−13.8756	−0.615027	−0.307514	−	0.951544i	$-0.599497\pi$
$509$	−13.8756	−0.615027	−0.307514	+	0.951544i	$0.599497\pi$
$510$	0	0
$511$	−1.80385	−0.0797975
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−9.00000	−0.396973
$515$	0	0
$516$	0	0
$517$	−12.9282	−0.568582
$518$	1.46410	0.0643289
$519$	0	0
$520$	0	0
$521$	−24.1769	−1.05921	−0.529605	−	0.848244i	$-0.677660\pi$
$521$	−24.1769	−1.05921	−0.529605	+	0.848244i	$0.677660\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−10.2679	−0.448557
$525$	0	0
$526$	−15.0000	−0.654031
$527$	15.2679	0.665082
$528$	0	0
$529$	12.1436	0.527982
$530$	0	0
$531$	0	0
$532$	0.732051	0.0317384
$533$	8.00000	0.346518
$534$	0	0
$535$	0	0
$536$	−0.267949	−0.0115736
$537$	0	0
$538$	−20.5359	−0.885365
$539$	24.1244	1.03911
$540$	0	0
$541$	17.1962	0.739320	0.369660	−	0.929167i	$-0.379474\pi$
$541$	17.1962	0.739320	0.369660	+	0.929167i	$0.379474\pi$
$542$	15.8038	0.678833
$543$	0	0
$544$	6.19615	0.265658
$545$	0	0
$546$	0	0
$547$	−43.4449	−1.85757	−0.928784	−	0.370621i	$-0.879145\pi$
$547$	−43.4449	−1.85757	−0.928784	+	0.370621i	$0.879145\pi$
$548$	−12.3923	−0.529373
$549$	0	0
$550$	0	0
$551$	−1.73205	−0.0737878
$552$	0	0
$553$	4.05256	0.172332
$554$	11.1962	0.475679
$555$	0	0
$556$	−21.1244	−0.895872
$557$	3.80385	0.161174	0.0805871	−	0.996748i	$-0.474320\pi$
$557$	3.80385	0.161174	0.0805871	+	0.996748i	$0.474320\pi$
$558$	0	0
$559$	22.3923	0.947094
$560$	0	0
$561$	0	0
$562$	17.7846	0.750198
$563$	7.66025	0.322841	0.161421	−	0.986886i	$-0.448392\pi$
$563$	7.66025	0.322841	0.161421	+	0.986886i	$0.448392\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	12.1962	0.511739
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	−6.19615	−0.259301	−0.129650	−	0.991560i	$-0.541386\pi$
$571$	−6.19615	−0.259301	−0.129650	+	0.991560i	$0.541386\pi$
$572$	−10.1962	−0.426323
$573$	0	0
$574$	2.14359	0.0894719
$575$	0	0
$576$	0	0
$577$	17.2487	0.718073	0.359037	−	0.933323i	$-0.383105\pi$
$577$	17.2487	0.718073	0.359037	+	0.933323i	$0.383105\pi$
$578$	−21.3923	−0.889803
$579$	0	0
$580$	0	0
$581$	−2.73205	−0.113345
$582$	0	0
$583$	6.46410	0.267716
$584$	−2.46410	−0.101965
$585$	0	0
$586$	27.7321	1.14560
$587$	−12.2679	−0.506352	−0.253176	−	0.967420i	$-0.581475\pi$
$587$	−12.2679	−0.506352	−0.253176	+	0.967420i	$0.581475\pi$
$588$	0	0
$589$	2.46410	0.101532
$590$	0	0
$591$	0	0
$592$	2.00000	0.0821995
$593$	−27.6077	−1.13371	−0.566856	−	0.823817i	$-0.691840\pi$
$593$	−27.6077	−1.13371	−0.566856	+	0.823817i	$0.691840\pi$
$594$	0	0
$595$	0	0
$596$	21.8564	0.895273
$597$	0	0
$598$	−16.1962	−0.662310
$599$	3.66025	0.149554	0.0747770	−	0.997200i	$-0.476176\pi$
$599$	3.66025	0.149554	0.0747770	+	0.997200i	$0.476176\pi$
$600$	0	0
$601$	−10.0526	−0.410052	−0.205026	−	0.978756i	$-0.565728\pi$
$601$	−10.0526	−0.410052	−0.205026	+	0.978756i	$0.565728\pi$
$602$	6.00000	0.244542
$603$	0	0
$604$	−11.4641	−0.466468
$605$	0	0
$606$	0	0
$607$	11.7846	0.478323	0.239161	−	0.970980i	$-0.423128\pi$
$607$	11.7846	0.478323	0.239161	+	0.970980i	$0.423128\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	9.46410	0.382877
$612$	0	0
$613$	−38.7846	−1.56650	−0.783248	−	0.621710i	$-0.786438\pi$
$613$	−38.7846	−1.56650	−0.783248	+	0.621710i	$0.786438\pi$
$614$	−20.2679	−0.817948
$615$	0	0
$616$	−2.73205	−0.110077
$617$	46.3923	1.86768	0.933842	−	0.357686i	$-0.116434\pi$
$617$	46.3923	1.86768	0.933842	+	0.357686i	$0.116434\pi$
$618$	0	0
$619$	−0.196152	−0.00788403	−0.00394202	−	0.999992i	$-0.501255\pi$
$619$	−0.196152	−0.00788403	−0.00394202	+	0.999992i	$0.501255\pi$
$620$	0	0
$621$	0	0
$622$	−29.3205	−1.17565
$623$	7.94744	0.318408
$624$	0	0
$625$	0	0
$626$	−3.92820	−0.157003
$627$	0	0
$628$	−13.8564	−0.552931
$629$	−12.3923	−0.494114
$630$	0	0
$631$	2.92820	0.116570	0.0582850	−	0.998300i	$-0.481437\pi$
$631$	2.92820	0.116570	0.0582850	+	0.998300i	$0.481437\pi$
$632$	5.53590	0.220206
$633$	0	0
$634$	1.73205	0.0687885
$635$	0	0
$636$	0	0
$637$	−17.6603	−0.699725
$638$	6.46410	0.255916
$639$	0	0
$640$	0	0
$641$	−10.5359	−0.416143	−0.208071	−	0.978114i	$-0.566719\pi$
$641$	−10.5359	−0.416143	−0.208071	+	0.978114i	$0.566719\pi$
$642$	0	0
$643$	−27.0718	−1.06761	−0.533804	−	0.845608i	$-0.679238\pi$
$643$	−27.0718	−1.06761	−0.533804	+	0.845608i	$0.679238\pi$
$644$	−4.33975	−0.171010
$645$	0	0
$646$	−6.19615	−0.243784
$647$	9.39230	0.369250	0.184625	−	0.982809i	$-0.440893\pi$
$647$	9.39230	0.369250	0.184625	+	0.982809i	$0.440893\pi$
$648$	0	0
$649$	−30.5885	−1.20070
$650$	0	0
$651$	0	0
$652$	2.00000	0.0783260
$653$	−17.4641	−0.683423	−0.341712	−	0.939805i	$-0.611007\pi$
$653$	−17.4641	−0.683423	−0.341712	+	0.939805i	$0.611007\pi$
$654$	0	0
$655$	0	0
$656$	2.92820	0.114327
$657$	0	0
$658$	2.53590	0.0988596
$659$	−7.41154	−0.288713	−0.144356	−	0.989526i	$-0.546111\pi$
$659$	−7.41154	−0.288713	−0.144356	+	0.989526i	$0.546111\pi$
$660$	0	0
$661$	−22.6410	−0.880633	−0.440317	−	0.897843i	$-0.645134\pi$
$661$	−22.6410	−0.880633	−0.440317	+	0.897843i	$0.645134\pi$
$662$	22.2679	0.865468
$663$	0	0
$664$	−3.73205	−0.144832
$665$	0	0
$666$	0	0
$667$	10.2679	0.397577
$668$	−8.92820	−0.345443
$669$	0	0
$670$	0	0
$671$	−39.7846	−1.53587
$672$	0	0
$673$	−6.98076	−0.269089	−0.134544	−	0.990908i	$-0.542957\pi$
$673$	−6.98076	−0.269089	−0.134544	+	0.990908i	$0.542957\pi$
$674$	10.5359	0.405828
$675$	0	0
$676$	−5.53590	−0.212919
$677$	−8.66025	−0.332841	−0.166420	−	0.986055i	$-0.553221\pi$
$677$	−8.66025	−0.332841	−0.166420	+	0.986055i	$0.553221\pi$
$678$	0	0
$679$	−11.0718	−0.424897
$680$	0	0
$681$	0	0
$682$	−9.19615	−0.352139
$683$	46.9808	1.79767	0.898834	−	0.438288i	$-0.144415\pi$
$683$	46.9808	1.79767	0.898834	+	0.438288i	$0.144415\pi$
$684$	0	0
$685$	0	0
$686$	−9.85641	−0.376319
$687$	0	0
$688$	8.19615	0.312475
$689$	−4.73205	−0.180277
$690$	0	0
$691$	−22.1962	−0.844381	−0.422191	−	0.906507i	$-0.638739\pi$
$691$	−22.1962	−0.844381	−0.422191	+	0.906507i	$0.638739\pi$
$692$	13.5885	0.516556
$693$	0	0
$694$	20.7846	0.788973
$695$	0	0
$696$	0	0
$697$	−18.1436	−0.687238
$698$	17.0526	0.645449
$699$	0	0
$700$	0	0
$701$	−3.60770	−0.136261	−0.0681304	−	0.997676i	$-0.521703\pi$
$701$	−3.60770	−0.136261	−0.0681304	+	0.997676i	$0.521703\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	−3.73205	−0.140657
$705$	0	0
$706$	−16.0526	−0.604146
$707$	6.78461	0.255162
$708$	0	0
$709$	47.3013	1.77644	0.888218	−	0.459422i	$-0.151943\pi$
$709$	47.3013	1.77644	0.888218	+	0.459422i	$0.151943\pi$
$710$	0	0
$711$	0	0
$712$	10.8564	0.406861
$713$	−14.6077	−0.547062
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	−16.9282	−0.631755
$719$	−12.1769	−0.454122	−0.227061	−	0.973881i	$-0.572912\pi$
$719$	−12.1769	−0.454122	−0.227061	+	0.973881i	$0.572912\pi$
$720$	0	0
$721$	−2.58846	−0.0963992
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−16.5885	−0.616505
$725$	0	0
$726$	0	0
$727$	38.1051	1.41324	0.706620	−	0.707593i	$-0.250219\pi$
$727$	38.1051	1.41324	0.706620	+	0.707593i	$0.250219\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	0	0
$731$	−50.7846	−1.87834
$732$	0	0
$733$	20.5167	0.757800	0.378900	−	0.925438i	$-0.376302\pi$
$733$	20.5167	0.757800	0.378900	+	0.925438i	$0.376302\pi$
$734$	−13.2679	−0.489729
$735$	0	0
$736$	−5.92820	−0.218516
$737$	−1.00000	−0.0368355
$738$	0	0
$739$	−53.1769	−1.95614	−0.978072	−	0.208266i	$-0.933218\pi$
$739$	−53.1769	−1.95614	−0.978072	+	0.208266i	$0.933218\pi$
$740$	0	0
$741$	0	0
$742$	−1.26795	−0.0465479
$743$	−5.07180	−0.186066	−0.0930331	−	0.995663i	$-0.529656\pi$
$743$	−5.07180	−0.186066	−0.0930331	+	0.995663i	$0.529656\pi$
$744$	0	0
$745$	0	0
$746$	−31.5167	−1.15391
$747$	0	0
$748$	23.1244	0.845510
$749$	−1.60770	−0.0587439
$750$	0	0
$751$	7.46410	0.272369	0.136184	−	0.990683i	$-0.456516\pi$
$751$	7.46410	0.272369	0.136184	+	0.990683i	$0.456516\pi$
$752$	3.46410	0.126323
$753$	0	0
$754$	−4.73205	−0.172331
$755$	0	0
$756$	0	0
$757$	13.5885	0.493881	0.246940	−	0.969031i	$-0.420575\pi$
$757$	13.5885	0.493881	0.246940	+	0.969031i	$0.420575\pi$
$758$	0.392305	0.0142492
$759$	0	0
$760$	0	0
$761$	−1.26795	−0.0459631	−0.0229816	−	0.999736i	$-0.507316\pi$
$761$	−1.26795	−0.0459631	−0.0229816	+	0.999736i	$0.507316\pi$
$762$	0	0
$763$	−1.75129	−0.0634009
$764$	−3.00000	−0.108536
$765$	0	0
$766$	19.5167	0.705166
$767$	22.3923	0.808539
$768$	0	0
$769$	−34.7128	−1.25178	−0.625888	−	0.779913i	$-0.715263\pi$
$769$	−34.7128	−1.25178	−0.625888	+	0.779913i	$0.715263\pi$
$770$	0	0
$771$	0	0
$772$	−22.0526	−0.793689
$773$	−45.3205	−1.63007	−0.815033	−	0.579415i	$-0.803281\pi$
$773$	−45.3205	−1.63007	−0.815033	+	0.579415i	$0.803281\pi$
$774$	0	0
$775$	0	0
$776$	−15.1244	−0.542933
$777$	0	0
$778$	21.7128	0.778442
$779$	−2.92820	−0.104914
$780$	0	0
$781$	45.5167	1.62871
$782$	36.7321	1.31354
$783$	0	0
$784$	−6.46410	−0.230861
$785$	0	0
$786$	0	0
$787$	−2.12436	−0.0757251	−0.0378626	−	0.999283i	$-0.512055\pi$
$787$	−2.12436	−0.0757251	−0.0378626	+	0.999283i	$0.512055\pi$
$788$	24.5885	0.875928
$789$	0	0
$790$	0	0
$791$	9.80385	0.348585
$792$	0	0
$793$	29.1244	1.03424
$794$	11.7321	0.416355
$795$	0	0
$796$	−14.0000	−0.496217
$797$	20.0000	0.708436	0.354218	−	0.935163i	$-0.384747\pi$
$797$	20.0000	0.708436	0.354218	+	0.935163i	$0.384747\pi$
$798$	0	0
$799$	−21.4641	−0.759345
$800$	0	0
$801$	0	0
$802$	16.8564	0.595220
$803$	−9.19615	−0.324525
$804$	0	0
$805$	0	0
$806$	6.73205	0.237126
$807$	0	0
$808$	9.26795	0.326045
$809$	−37.5167	−1.31902	−0.659508	−	0.751698i	$-0.729235\pi$
$809$	−37.5167	−1.31902	−0.659508	+	0.751698i	$0.729235\pi$
$810$	0	0
$811$	18.3731	0.645166	0.322583	−	0.946541i	$-0.395449\pi$
$811$	18.3731	0.645166	0.322583	+	0.946541i	$0.395449\pi$
$812$	−1.26795	−0.0444963
$813$	0	0
$814$	7.46410	0.261617
$815$	0	0
$816$	0	0
$817$	−8.19615	−0.286747
$818$	−1.60770	−0.0562117
$819$	0	0
$820$	0	0
$821$	−2.48334	−0.0866691	−0.0433346	−	0.999061i	$-0.513798\pi$
$821$	−2.48334	−0.0866691	−0.0433346	+	0.999061i	$0.513798\pi$
$822$	0	0
$823$	6.78461	0.236497	0.118248	−	0.992984i	$-0.462272\pi$
$823$	6.78461	0.236497	0.118248	+	0.992984i	$0.462272\pi$
$824$	−3.53590	−0.123179
$825$	0	0
$826$	6.00000	0.208767
$827$	6.73205	0.234096	0.117048	−	0.993126i	$-0.462657\pi$
$827$	6.73205	0.234096	0.117048	+	0.993126i	$0.462657\pi$
$828$	0	0
$829$	6.19615	0.215201	0.107601	−	0.994194i	$-0.465683\pi$
$829$	6.19615	0.215201	0.107601	+	0.994194i	$0.465683\pi$
$830$	0	0
$831$	0	0
$832$	2.73205	0.0947168
$833$	40.0526	1.38774
$834$	0	0
$835$	0	0
$836$	3.73205	0.129076
$837$	0	0
$838$	13.8564	0.478662
$839$	46.5885	1.60841	0.804206	−	0.594351i	$-0.202591\pi$
$839$	46.5885	1.60841	0.804206	+	0.594351i	$0.202591\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	−6.78461	−0.233813
$843$	0	0
$844$	−10.2679	−0.353437
$845$	0	0
$846$	0	0
$847$	−2.14359	−0.0736547
$848$	−1.73205	−0.0594789
$849$	0	0
$850$	0	0
$851$	11.8564	0.406432
$852$	0	0
$853$	4.53590	0.155306	0.0776531	−	0.996980i	$-0.475257\pi$
$853$	4.53590	0.155306	0.0776531	+	0.996980i	$0.475257\pi$
$854$	7.80385	0.267042
$855$	0	0
$856$	−2.19615	−0.0750629
$857$	−42.9282	−1.46640	−0.733200	−	0.680013i	$-0.761974\pi$
$857$	−42.9282	−1.46640	−0.733200	+	0.680013i	$0.761974\pi$
$858$	0	0
$859$	56.9282	1.94237	0.971183	−	0.238337i	$-0.0766021\pi$
$859$	56.9282	1.94237	0.971183	+	0.238337i	$0.0766021\pi$
$860$	0	0
$861$	0	0
$862$	−29.3205	−0.998660
$863$	−55.3205	−1.88313	−0.941566	−	0.336829i	$-0.890646\pi$
$863$	−55.3205	−1.88313	−0.941566	+	0.336829i	$0.890646\pi$
$864$	0	0
$865$	0	0
$866$	24.1962	0.822219
$867$	0	0
$868$	1.80385	0.0612266
$869$	20.6603	0.700851
$870$	0	0
$871$	0.732051	0.0248046
$872$	−2.39230	−0.0810137
$873$	0	0
$874$	5.92820	0.200524
$875$	0	0
$876$	0	0
$877$	−31.1769	−1.05277	−0.526385	−	0.850246i	$-0.676453\pi$
$877$	−31.1769	−1.05277	−0.526385	+	0.850246i	$0.676453\pi$
$878$	−25.7846	−0.870188
$879$	0	0
$880$	0	0
$881$	26.5359	0.894017	0.447009	−	0.894530i	$-0.352489\pi$
$881$	26.5359	0.894017	0.447009	+	0.894530i	$0.352489\pi$
$882$	0	0
$883$	−27.2679	−0.917640	−0.458820	−	0.888529i	$-0.651728\pi$
$883$	−27.2679	−0.917640	−0.458820	+	0.888529i	$0.651728\pi$
$884$	−16.9282	−0.569357
$885$	0	0
$886$	−3.33975	−0.112201
$887$	−7.60770	−0.255441	−0.127721	−	0.991810i	$-0.540766\pi$
$887$	−7.60770	−0.255441	−0.127721	+	0.991810i	$0.540766\pi$
$888$	0	0
$889$	−3.66025	−0.122761
$890$	0	0
$891$	0	0
$892$	3.53590	0.118391
$893$	−3.46410	−0.115922
$894$	0	0
$895$	0	0
$896$	0.732051	0.0244561
$897$	0	0
$898$	24.4641	0.816378
$899$	−4.26795	−0.142344
$900$	0	0
$901$	10.7321	0.357536
$902$	10.9282	0.363869
$903$	0	0
$904$	13.3923	0.445421
$905$	0	0
$906$	0	0
$907$	−49.3205	−1.63766	−0.818830	−	0.574036i	$-0.805377\pi$
$907$	−49.3205	−1.63766	−0.818830	+	0.574036i	$0.805377\pi$
$908$	−6.00000	−0.199117
$909$	0	0
$910$	0	0
$911$	−34.1051	−1.12995	−0.564976	−	0.825107i	$-0.691115\pi$
$911$	−34.1051	−1.12995	−0.564976	+	0.825107i	$0.691115\pi$
$912$	0	0
$913$	−13.9282	−0.460956
$914$	5.07180	0.167760
$915$	0	0
$916$	−6.26795	−0.207099
$917$	7.51666	0.248222
$918$	0	0
$919$	−4.48334	−0.147892	−0.0739459	−	0.997262i	$-0.523559\pi$
$919$	−4.48334	−0.147892	−0.0739459	+	0.997262i	$0.523559\pi$
$920$	0	0
$921$	0	0
$922$	−17.8564	−0.588069
$923$	−33.3205	−1.09676
$924$	0	0
$925$	0	0
$926$	35.3205	1.16070
$927$	0	0
$928$	−1.73205	−0.0568574
$929$	−42.9282	−1.40843	−0.704214	−	0.709987i	$-0.748701\pi$
$929$	−42.9282	−1.40843	−0.704214	+	0.709987i	$0.748701\pi$
$930$	0	0
$931$	6.46410	0.211852
$932$	−21.1244	−0.691951
$933$	0	0
$934$	7.19615	0.235465
$935$	0	0
$936$	0	0
$937$	13.2154	0.431728	0.215864	−	0.976423i	$-0.430743\pi$
$937$	13.2154	0.431728	0.215864	+	0.976423i	$0.430743\pi$
$938$	0.196152	0.00640460
$939$	0	0
$940$	0	0
$941$	12.2679	0.399924	0.199962	−	0.979804i	$-0.435918\pi$
$941$	12.2679	0.399924	0.199962	+	0.979804i	$0.435918\pi$
$942$	0	0
$943$	17.3590	0.565286
$944$	8.19615	0.266762
$945$	0	0
$946$	30.5885	0.994517
$947$	5.85641	0.190308	0.0951538	−	0.995463i	$-0.469666\pi$
$947$	5.85641	0.190308	0.0951538	+	0.995463i	$0.469666\pi$
$948$	0	0
$949$	6.73205	0.218532
$950$	0	0
$951$	0	0
$952$	−4.53590	−0.147009
$953$	5.24871	0.170022	0.0850112	−	0.996380i	$-0.472907\pi$
$953$	5.24871	0.170022	0.0850112	+	0.996380i	$0.472907\pi$
$954$	0	0
$955$	0	0
$956$	14.0000	0.452792
$957$	0	0
$958$	3.14359	0.101565
$959$	9.07180	0.292944
$960$	0	0
$961$	−24.9282	−0.804136
$962$	−5.46410	−0.176170
$963$	0	0
$964$	−30.1962	−0.972553
$965$	0	0
$966$	0	0
$967$	18.1051	0.582221	0.291111	−	0.956689i	$-0.405975\pi$
$967$	18.1051	0.582221	0.291111	+	0.956689i	$0.405975\pi$
$968$	−2.92820	−0.0941160
$969$	0	0
$970$	0	0
$971$	37.7654	1.21195	0.605974	−	0.795484i	$-0.292783\pi$
$971$	37.7654	1.21195	0.605974	+	0.795484i	$0.292783\pi$
$972$	0	0
$973$	15.4641	0.495756
$974$	−2.14359	−0.0686852
$975$	0	0
$976$	10.6603	0.341226
$977$	1.46410	0.0468408	0.0234204	−	0.999726i	$-0.492544\pi$
$977$	1.46410	0.0468408	0.0234204	+	0.999726i	$0.492544\pi$
$978$	0	0
$979$	40.5167	1.29492
$980$	0	0
$981$	0	0
$982$	−2.92820	−0.0934427
$983$	30.3397	0.967688	0.483844	−	0.875154i	$-0.339240\pi$
$983$	30.3397	0.967688	0.483844	+	0.875154i	$0.339240\pi$
$984$	0	0
$985$	0	0
$986$	10.7321	0.341778
$987$	0	0
$988$	−2.73205	−0.0869181
$989$	48.5885	1.54502
$990$	0	0
$991$	−40.3205	−1.28082	−0.640412	−	0.768032i	$-0.721236\pi$
$991$	−40.3205	−1.28082	−0.640412	+	0.768032i	$0.721236\pi$
$992$	2.46410	0.0782353
$993$	0	0
$994$	−8.92820	−0.283185
$995$	0	0
$996$	0	0
$997$	−63.0526	−1.99689	−0.998447	−	0.0557047i	$-0.982259\pi$
$997$	−63.0526	−1.99689	−0.998447	+	0.0557047i	$0.982259\pi$
$998$	−41.3731	−1.30964
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.bs.1.1		2
3.2	odd	2		2850.2.a.bi.1.1	yes	2
5.4	even	2		8550.2.a.by.1.2		2
15.2	even	4		2850.2.d.x.799.3		4
15.8	even	4		2850.2.d.x.799.2		4
15.14	odd	2		2850.2.a.bd.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.bd.1.2	✓	2	15.14	odd	2
2850.2.a.bi.1.1	yes	2	3.2	odd	2
2850.2.d.x.799.2		4	15.8	even	4
2850.2.d.x.799.3		4	15.2	even	4
8550.2.a.bs.1.1		2	1.1	even	1	trivial
8550.2.a.by.1.2		2	5.4	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.732051 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.732051 q^{7} -1.00000 q^{8} -3.73205 q^{11} +2.73205 q^{13} +0.732051 q^{14} +1.00000 q^{16} -6.19615 q^{17} -1.00000 q^{19} +3.73205 q^{22} +5.92820 q^{23} -2.73205 q^{26} -0.732051 q^{28} +1.73205 q^{29} -2.46410 q^{31} -1.00000 q^{32} +6.19615 q^{34} +2.00000 q^{37} +1.00000 q^{38} +2.92820 q^{41} +8.19615 q^{43} -3.73205 q^{44} -5.92820 q^{46} +3.46410 q^{47} -6.46410 q^{49} +2.73205 q^{52} -1.73205 q^{53} +0.732051 q^{56} -1.73205 q^{58} +8.19615 q^{59} +10.6603 q^{61} +2.46410 q^{62} +1.00000 q^{64} +0.267949 q^{67} -6.19615 q^{68} -12.1962 q^{71} +2.46410 q^{73} -2.00000 q^{74} -1.00000 q^{76} +2.73205 q^{77} -5.53590 q^{79} -2.92820 q^{82} +3.73205 q^{83} -8.19615 q^{86} +3.73205 q^{88} -10.8564 q^{89} -2.00000 q^{91} +5.92820 q^{92} -3.46410 q^{94} +15.1244 q^{97} +6.46410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 4 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{19} + 4 q^{22} - 2 q^{23} - 2 q^{26} + 2 q^{28} + 2 q^{31} - 2 q^{32} + 2 q^{34} + 4 q^{37} + 2 q^{38} - 8 q^{41} + 6 q^{43} - 4 q^{44} + 2 q^{46} - 6 q^{49} + 2 q^{52} - 2 q^{56} + 6 q^{59} + 4 q^{61} - 2 q^{62} + 2 q^{64} + 4 q^{67} - 2 q^{68} - 14 q^{71} - 2 q^{73} - 4 q^{74} - 2 q^{76} + 2 q^{77} - 18 q^{79} + 8 q^{82} + 4 q^{83} - 6 q^{86} + 4 q^{88} + 6 q^{89} - 4 q^{91} - 2 q^{92} + 6 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 4 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - 2 * q^19 + 4 * q^22 - 2 * q^23 - 2 * q^26 + 2 * q^28 + 2 * q^31 - 2 * q^32 + 2 * q^34 + 4 * q^37 + 2 * q^38 - 8 * q^41 + 6 * q^43 - 4 * q^44 + 2 * q^46 - 6 * q^49 + 2 * q^52 - 2 * q^56 + 6 * q^59 + 4 * q^61 - 2 * q^62 + 2 * q^64 + 4 * q^67 - 2 * q^68 - 14 * q^71 - 2 * q^73 - 4 * q^74 - 2 * q^76 + 2 * q^77 - 18 * q^79 + 8 * q^82 + 4 * q^83 - 6 * q^86 + 4 * q^88 + 6 * q^89 - 4 * q^91 - 2 * q^92 + 6 * q^97 + 6 * q^98

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