LMFDB - Embedded newform 8550.2.a.cm.1.2

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:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.476452$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.47645 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.47645 q^{7} +1.00000 q^{8} +2.29654 q^{11} -5.29654 q^{13} -1.47645 q^{14} +1.00000 q^{16} +1.47645 q^{17} -1.00000 q^{19} +2.29654 q^{22} -1.86719 q^{23} -5.29654 q^{26} -1.47645 q^{28} -7.16373 q^{29} +0.0470959 q^{31} +1.00000 q^{32} +1.47645 q^{34} +6.59308 q^{37} -1.00000 q^{38} -0.179911 q^{41} +7.98382 q^{43} +2.29654 q^{44} -1.86719 q^{46} +12.4132 q^{47} -4.82009 q^{49} -5.29654 q^{52} +11.9367 q^{53} -1.47645 q^{56} -7.16373 q^{58} -6.34364 q^{59} -9.93672 q^{61} +0.0470959 q^{62} +1.00000 q^{64} -14.8039 q^{67} +1.47645 q^{68} -14.0695 q^{71} -14.4603 q^{73} +6.59308 q^{74} -1.00000 q^{76} -3.39073 q^{77} -12.5460 q^{79} -0.179911 q^{82} +6.11663 q^{83} +7.98382 q^{86} +2.29654 q^{88} +6.46027 q^{89} +7.82009 q^{91} -1.86719 q^{92} +12.4132 q^{94} +4.42936 q^{97} -4.82009 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 5 q^{11} - 4 q^{13} - 2 q^{14} + 3 q^{16} + 2 q^{17} - 3 q^{19} - 5 q^{22} - q^{23} - 4 q^{26} - 2 q^{28} - 5 q^{29} + 5 q^{31} + 3 q^{32} + 2 q^{34} - 4 q^{37} - 3 q^{38} - 10 q^{41} - 2 q^{43} - 5 q^{44} - q^{46} + 4 q^{47} - 5 q^{49} - 4 q^{52} + 5 q^{53} - 2 q^{56} - 5 q^{58} - 12 q^{59} + q^{61} + 5 q^{62} + 3 q^{64} - 9 q^{67} + 2 q^{68} - 16 q^{71} - 15 q^{73} - 4 q^{74} - 3 q^{76} - 8 q^{77} - 9 q^{79} - 10 q^{82} - 3 q^{83} - 2 q^{86} - 5 q^{88} - 9 q^{89} + 14 q^{91} - q^{92} + 4 q^{94} + 6 q^{97} - 5 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 5 * q^11 - 4 * q^13 - 2 * q^14 + 3 * q^16 + 2 * q^17 - 3 * q^19 - 5 * q^22 - q^23 - 4 * q^26 - 2 * q^28 - 5 * q^29 + 5 * q^31 + 3 * q^32 + 2 * q^34 - 4 * q^37 - 3 * q^38 - 10 * q^41 - 2 * q^43 - 5 * q^44 - q^46 + 4 * q^47 - 5 * q^49 - 4 * q^52 + 5 * q^53 - 2 * q^56 - 5 * q^58 - 12 * q^59 + q^61 + 5 * q^62 + 3 * q^64 - 9 * q^67 + 2 * q^68 - 16 * q^71 - 15 * q^73 - 4 * q^74 - 3 * q^76 - 8 * q^77 - 9 * q^79 - 10 * q^82 - 3 * q^83 - 2 * q^86 - 5 * q^88 - 9 * q^89 + 14 * q^91 - q^92 + 4 * q^94 + 6 * q^97 - 5 * 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
$7$	−1.47645	−0.558046	−0.279023	−	0.960284i	$-0.590011\pi$
$7$	−1.47645	−0.558046	−0.279023	+	0.960284i	$0.590011\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	2.29654	0.692433	0.346217	−	0.938155i	$-0.387466\pi$
$11$	2.29654	0.692433	0.346217	+	0.938155i	$0.387466\pi$
$12$	0	0
$13$	−5.29654	−1.46900	−0.734498	−	0.678611i	$-0.762582\pi$
$13$	−5.29654	−1.46900	−0.734498	+	0.678611i	$0.762582\pi$
$14$	−1.47645	−0.394598
$15$	0	0
$16$	1.00000	0.250000
$17$	1.47645	0.358092	0.179046	−	0.983841i	$-0.442699\pi$
$17$	1.47645	0.358092	0.179046	+	0.983841i	$0.442699\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	2.29654	0.489624
$23$	−1.86719	−0.389335	−0.194668	−	0.980869i	$-0.562363\pi$
$23$	−1.86719	−0.389335	−0.194668	+	0.980869i	$0.562363\pi$
$24$	0	0
$25$	0	0
$26$	−5.29654	−1.03874
$27$	0	0
$28$	−1.47645	−0.279023
$29$	−7.16373	−1.33027	−0.665135	−	0.746723i	$-0.731626\pi$
$29$	−7.16373	−1.33027	−0.665135	+	0.746723i	$0.731626\pi$
$30$	0	0
$31$	0.0470959	0.00845868	0.00422934	−	0.999991i	$-0.498654\pi$
$31$	0.0470959	0.00845868	0.00422934	+	0.999991i	$0.498654\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.47645	0.253209
$35$	0	0
$36$	0	0
$37$	6.59308	1.08390	0.541948	−	0.840412i	$-0.317687\pi$
$37$	6.59308	1.08390	0.541948	+	0.840412i	$0.317687\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	−0.179911	−0.0280973	−0.0140487	−	0.999901i	$-0.504472\pi$
$41$	−0.179911	−0.0280973	−0.0140487	+	0.999901i	$0.504472\pi$
$42$	0	0
$43$	7.98382	1.21752	0.608760	−	0.793354i	$-0.291667\pi$
$43$	7.98382	1.21752	0.608760	+	0.793354i	$0.291667\pi$
$44$	2.29654	0.346217
$45$	0	0
$46$	−1.86719	−0.275301
$47$	12.4132	1.81065	0.905324	−	0.424722i	$-0.139628\pi$
$47$	12.4132	1.81065	0.905324	+	0.424722i	$0.139628\pi$
$48$	0	0
$49$	−4.82009	−0.688584
$50$	0	0
$51$	0	0
$52$	−5.29654	−0.734498
$53$	11.9367	1.63963	0.819817	−	0.572625i	$-0.194075\pi$
$53$	11.9367	1.63963	0.819817	+	0.572625i	$0.194075\pi$
$54$	0	0
$55$	0	0
$56$	−1.47645	−0.197299
$57$	0	0
$58$	−7.16373	−0.940643
$59$	−6.34364	−0.825871	−0.412936	−	0.910760i	$-0.635497\pi$
$59$	−6.34364	−0.825871	−0.412936	+	0.910760i	$0.635497\pi$
$60$	0	0
$61$	−9.93672	−1.27227	−0.636133	−	0.771579i	$-0.719467\pi$
$61$	−9.93672	−1.27227	−0.636133	+	0.771579i	$0.719467\pi$
$62$	0.0470959	0.00598119
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−14.8039	−1.80858	−0.904292	−	0.426914i	$-0.859601\pi$
$67$	−14.8039	−1.80858	−0.904292	+	0.426914i	$0.859601\pi$
$68$	1.47645	0.179046
$69$	0	0
$70$	0	0
$71$	−14.0695	−1.66975	−0.834873	−	0.550442i	$-0.814459\pi$
$71$	−14.0695	−1.66975	−0.834873	+	0.550442i	$0.814459\pi$
$72$	0	0
$73$	−14.4603	−1.69245	−0.846223	−	0.532829i	$-0.821129\pi$
$73$	−14.4603	−1.69245	−0.846223	+	0.532829i	$0.821129\pi$
$74$	6.59308	0.766430
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−3.39073	−0.386410
$78$	0	0
$79$	−12.5460	−1.41153	−0.705767	−	0.708444i	$-0.749397\pi$
$79$	−12.5460	−1.41153	−0.705767	+	0.708444i	$0.749397\pi$
$80$	0	0
$81$	0	0
$82$	−0.179911	−0.0198678
$83$	6.11663	0.671387	0.335694	−	0.941971i	$-0.391029\pi$
$83$	6.11663	0.671387	0.335694	+	0.941971i	$0.391029\pi$
$84$	0	0
$85$	0	0
$86$	7.98382	0.860917
$87$	0	0
$88$	2.29654	0.244812
$89$	6.46027	0.684787	0.342394	−	0.939557i	$-0.388762\pi$
$89$	6.46027	0.684787	0.342394	+	0.939557i	$0.388762\pi$
$90$	0	0
$91$	7.82009	0.819768
$92$	−1.86719	−0.194668
$93$	0	0
$94$	12.4132	1.28032
$95$	0	0
$96$	0	0
$97$	4.42936	0.449733	0.224866	−	0.974390i	$-0.427805\pi$
$97$	4.42936	0.449733	0.224866	+	0.974390i	$0.427805\pi$
$98$	−4.82009	−0.486903
$99$	0	0
$100$	0	0
$101$	−13.2965	−1.32306	−0.661528	−	0.749921i	$-0.730092\pi$
$101$	−13.2965	−1.32306	−0.661528	+	0.749921i	$0.730092\pi$
$102$	0	0
$103$	−0.227007	−0.0223676	−0.0111838	−	0.999937i	$-0.503560\pi$
$103$	−0.227007	−0.0223676	−0.0111838	+	0.999937i	$0.503560\pi$
$104$	−5.29654	−0.519369
$105$	0	0
$106$	11.9367	1.15940
$107$	−7.20235	−0.696277	−0.348139	−	0.937443i	$-0.613186\pi$
$107$	−7.20235	−0.696277	−0.348139	+	0.937443i	$0.613186\pi$
$108$	0	0
$109$	−7.04710	−0.674989	−0.337495	−	0.941327i	$-0.609580\pi$
$109$	−7.04710	−0.674989	−0.337495	+	0.941327i	$0.609580\pi$
$110$	0	0
$111$	0	0
$112$	−1.47645	−0.139512
$113$	−20.1004	−1.89089	−0.945445	−	0.325780i	$-0.894373\pi$
$113$	−20.1004	−1.89089	−0.945445	+	0.325780i	$0.894373\pi$
$114$	0	0
$115$	0	0
$116$	−7.16373	−0.665135
$117$	0	0
$118$	−6.34364	−0.583979
$119$	−2.17991	−0.199832
$120$	0	0
$121$	−5.72590	−0.520536
$122$	−9.93672	−0.899628
$123$	0	0
$124$	0.0470959	0.00422934
$125$	0	0
$126$	0	0
$127$	5.68727	0.504664	0.252332	−	0.967641i	$-0.418802\pi$
$127$	5.68727	0.504664	0.252332	+	0.967641i	$0.418802\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−12.2108	−1.06686	−0.533432	−	0.845843i	$-0.679098\pi$
$131$	−12.2108	−1.06686	−0.533432	+	0.845843i	$0.679098\pi$
$132$	0	0
$133$	1.47645	0.128025
$134$	−14.8039	−1.27886
$135$	0	0
$136$	1.47645	0.126605
$137$	−8.23326	−0.703415	−0.351708	−	0.936110i	$-0.614399\pi$
$137$	−8.23326	−0.703415	−0.351708	+	0.936110i	$0.614399\pi$
$138$	0	0
$139$	−13.2023	−1.11981	−0.559904	−	0.828557i	$-0.689162\pi$
$139$	−13.2023	−1.11981	−0.559904	+	0.828557i	$0.689162\pi$
$140$	0	0
$141$	0	0
$142$	−14.0695	−1.18069
$143$	−12.1637	−1.01718
$144$	0	0
$145$	0	0
$146$	−14.4603	−1.19674
$147$	0	0
$148$	6.59308	0.541948
$149$	−6.85871	−0.561888	−0.280944	−	0.959724i	$-0.590647\pi$
$149$	−6.85871	−0.561888	−0.280944	+	0.959724i	$0.590647\pi$
$150$	0	0
$151$	10.6788	0.869029	0.434514	−	0.900665i	$-0.356920\pi$
$151$	10.6788	0.869029	0.434514	+	0.900665i	$0.356920\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−3.39073	−0.273233
$155$	0	0
$156$	0	0
$157$	3.22701	0.257543	0.128772	−	0.991674i	$-0.458897\pi$
$157$	3.22701	0.257543	0.128772	+	0.991674i	$0.458897\pi$
$158$	−12.5460	−0.998105
$159$	0	0
$160$	0	0
$161$	2.75681	0.217267
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−0.179911	−0.0140487
$165$	0	0
$166$	6.11663	0.474743
$167$	20.9205	1.61888	0.809440	−	0.587203i	$-0.199771\pi$
$167$	20.9205	1.61888	0.809440	+	0.587203i	$0.199771\pi$
$168$	0	0
$169$	15.0534	1.15795
$170$	0	0
$171$	0	0
$172$	7.98382	0.608760
$173$	1.25792	0.0956378	0.0478189	−	0.998856i	$-0.484773\pi$
$173$	1.25792	0.0956378	0.0478189	+	0.998856i	$0.484773\pi$
$174$	0	0
$175$	0	0
$176$	2.29654	0.173108
$177$	0	0
$178$	6.46027	0.484218
$179$	6.49889	0.485750	0.242875	−	0.970058i	$-0.421910\pi$
$179$	6.49889	0.485750	0.242875	+	0.970058i	$0.421910\pi$
$180$	0	0
$181$	19.6240	1.45864	0.729320	−	0.684173i	$-0.239837\pi$
$181$	19.6240	1.45864	0.729320	+	0.684173i	$0.239837\pi$
$182$	7.82009	0.579664
$183$	0	0
$184$	−1.86719	−0.137651
$185$	0	0
$186$	0	0
$187$	3.39073	0.247955
$188$	12.4132	0.905324
$189$	0	0
$190$	0	0
$191$	−0.406917	−0.0294435	−0.0147217	−	0.999892i	$-0.504686\pi$
$191$	−0.406917	−0.0294435	−0.0147217	+	0.999892i	$0.504686\pi$
$192$	0	0
$193$	12.3970	0.892355	0.446177	−	0.894945i	$-0.352785\pi$
$193$	12.3970	0.892355	0.446177	+	0.894945i	$0.352785\pi$
$194$	4.42936	0.318009
$195$	0	0
$196$	−4.82009	−0.344292
$197$	−2.24945	−0.160266	−0.0801332	−	0.996784i	$-0.525535\pi$
$197$	−2.24945	−0.160266	−0.0801332	+	0.996784i	$0.525535\pi$
$198$	0	0
$199$	18.4665	1.30906	0.654529	−	0.756037i	$-0.272867\pi$
$199$	18.4665	1.30906	0.654529	+	0.756037i	$0.272867\pi$
$200$	0	0
$201$	0	0
$202$	−13.2965	−0.935541
$203$	10.5769	0.742353
$204$	0	0
$205$	0	0
$206$	−0.227007	−0.0158163
$207$	0	0
$208$	−5.29654	−0.367249
$209$	−2.29654	−0.158855
$210$	0	0
$211$	24.3584	1.67690	0.838450	−	0.544979i	$-0.183462\pi$
$211$	24.3584	1.67690	0.838450	+	0.544979i	$0.183462\pi$
$212$	11.9367	0.819817
$213$	0	0
$214$	−7.20235	−0.492342
$215$	0	0
$216$	0	0
$217$	−0.0695349	−0.00472033
$218$	−7.04710	−0.477290
$219$	0	0
$220$	0	0
$221$	−7.82009	−0.526036
$222$	0	0
$223$	−20.1946	−1.35233	−0.676167	−	0.736749i	$-0.736360\pi$
$223$	−20.1946	−1.35233	−0.676167	+	0.736749i	$0.736360\pi$
$224$	−1.47645	−0.0986496
$225$	0	0
$226$	−20.1004	−1.33706
$227$	6.35982	0.422116	0.211058	−	0.977474i	$-0.432309\pi$
$227$	6.35982	0.422116	0.211058	+	0.977474i	$0.432309\pi$
$228$	0	0
$229$	−11.8510	−0.783136	−0.391568	−	0.920149i	$-0.628067\pi$
$229$	−11.8510	−0.783136	−0.391568	+	0.920149i	$0.628067\pi$
$230$	0	0
$231$	0	0
$232$	−7.16373	−0.470322
$233$	−27.6155	−1.80915	−0.904576	−	0.426311i	$-0.859813\pi$
$233$	−27.6155	−1.80915	−0.904576	+	0.426311i	$0.859813\pi$
$234$	0	0
$235$	0	0
$236$	−6.34364	−0.412936
$237$	0	0
$238$	−2.17991	−0.141303
$239$	−28.2248	−1.82571	−0.912855	−	0.408284i	$-0.866127\pi$
$239$	−28.2248	−1.82571	−0.912855	+	0.408284i	$0.866127\pi$
$240$	0	0
$241$	7.38226	0.475533	0.237767	−	0.971322i	$-0.423585\pi$
$241$	7.38226	0.475533	0.237767	+	0.971322i	$0.423585\pi$
$242$	−5.72590	−0.368075
$243$	0	0
$244$	−9.93672	−0.636133
$245$	0	0
$246$	0	0
$247$	5.29654	0.337011
$248$	0.0470959	0.00299059
$249$	0	0
$250$	0	0
$251$	−15.6317	−0.986665	−0.493332	−	0.869841i	$-0.664221\pi$
$251$	−15.6317	−0.986665	−0.493332	+	0.869841i	$0.664221\pi$
$252$	0	0
$253$	−4.28807	−0.269589
$254$	5.68727	0.356851
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.13281	0.507311	0.253656	−	0.967295i	$-0.418367\pi$
$257$	8.13281	0.507311	0.253656	+	0.967295i	$0.418367\pi$
$258$	0	0
$259$	−9.73437	−0.604864
$260$	0	0
$261$	0	0
$262$	−12.2108	−0.754387
$263$	−10.2270	−0.630624	−0.315312	−	0.948988i	$-0.602109\pi$
$263$	−10.2270	−0.630624	−0.315312	+	0.948988i	$0.602109\pi$
$264$	0	0
$265$	0	0
$266$	1.47645	0.0905271
$267$	0	0
$268$	−14.8039	−0.904292
$269$	8.59308	0.523930	0.261965	−	0.965077i	$-0.415630\pi$
$269$	8.59308	0.523930	0.261965	+	0.965077i	$0.415630\pi$
$270$	0	0
$271$	8.78918	0.533905	0.266952	−	0.963710i	$-0.413983\pi$
$271$	8.78918	0.533905	0.266952	+	0.963710i	$0.413983\pi$
$272$	1.47645	0.0895231
$273$	0	0
$274$	−8.23326	−0.497390
$275$	0	0
$276$	0	0
$277$	24.7954	1.48981	0.744907	−	0.667169i	$-0.232494\pi$
$277$	24.7954	1.48981	0.744907	+	0.667169i	$0.232494\pi$
$278$	−13.2023	−0.791824
$279$	0	0
$280$	0	0
$281$	−23.5931	−1.40745	−0.703723	−	0.710475i	$-0.748480\pi$
$281$	−23.5931	−1.40745	−0.703723	+	0.710475i	$0.748480\pi$
$282$	0	0
$283$	−13.5460	−0.805225	−0.402613	−	0.915370i	$-0.631898\pi$
$283$	−13.5460	−0.805225	−0.402613	+	0.915370i	$0.631898\pi$
$284$	−14.0695	−0.834873
$285$	0	0
$286$	−12.1637	−0.719256
$287$	0.265629	0.0156796
$288$	0	0
$289$	−14.8201	−0.871770
$290$	0	0
$291$	0	0
$292$	−14.4603	−0.846223
$293$	−24.7954	−1.44856	−0.724282	−	0.689504i	$-0.757829\pi$
$293$	−24.7954	−1.44856	−0.724282	+	0.689504i	$0.757829\pi$
$294$	0	0
$295$	0	0
$296$	6.59308	0.383215
$297$	0	0
$298$	−6.85871	−0.397315
$299$	9.88962	0.571932
$300$	0	0
$301$	−11.7877	−0.679433
$302$	10.6788	0.614496
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	−11.6626	−0.665621	−0.332810	−	0.942994i	$-0.607997\pi$
$307$	−11.6626	−0.665621	−0.332810	+	0.942994i	$0.607997\pi$
$308$	−3.39073	−0.193205
$309$	0	0
$310$	0	0
$311$	3.72590	0.211276	0.105638	−	0.994405i	$-0.466311\pi$
$311$	3.72590	0.211276	0.105638	+	0.994405i	$0.466311\pi$
$312$	0	0
$313$	5.00847	0.283096	0.141548	−	0.989931i	$-0.454792\pi$
$313$	5.00847	0.283096	0.141548	+	0.989931i	$0.454792\pi$
$314$	3.22701	0.182111
$315$	0	0
$316$	−12.5460	−0.705767
$317$	27.1637	1.52567	0.762833	−	0.646595i	$-0.223808\pi$
$317$	27.1637	1.52567	0.762833	+	0.646595i	$0.223808\pi$
$318$	0	0
$319$	−16.4518	−0.921124
$320$	0	0
$321$	0	0
$322$	2.75681	0.153631
$323$	−1.47645	−0.0821520
$324$	0	0
$325$	0	0
$326$	−6.00000	−0.332309
$327$	0	0
$328$	−0.179911	−0.00993390
$329$	−18.3275	−1.01043
$330$	0	0
$331$	−2.80391	−0.154117	−0.0770583	−	0.997027i	$-0.524553\pi$
$331$	−2.80391	−0.154117	−0.0770583	+	0.997027i	$0.524553\pi$
$332$	6.11663	0.335694
$333$	0	0
$334$	20.9205	1.14472
$335$	0	0
$336$	0	0
$337$	6.85871	0.373618	0.186809	−	0.982396i	$-0.440185\pi$
$337$	6.85871	0.373618	0.186809	+	0.982396i	$0.440185\pi$
$338$	15.0534	0.818794
$339$	0	0
$340$	0	0
$341$	0.108158	0.00585707
$342$	0	0
$343$	17.4518	0.942308
$344$	7.98382	0.430459
$345$	0	0
$346$	1.25792	0.0676261
$347$	−19.7792	−1.06181	−0.530903	−	0.847433i	$-0.678147\pi$
$347$	−19.7792	−1.06181	−0.530903	+	0.847433i	$0.678147\pi$
$348$	0	0
$349$	−7.39699	−0.395952	−0.197976	−	0.980207i	$-0.563437\pi$
$349$	−7.39699	−0.395952	−0.197976	+	0.980207i	$0.563437\pi$
$350$	0	0
$351$	0	0
$352$	2.29654	0.122406
$353$	−36.0372	−1.91806	−0.959032	−	0.283296i	$-0.908572\pi$
$353$	−36.0372	−1.91806	−0.959032	+	0.283296i	$0.908572\pi$
$354$	0	0
$355$	0	0
$356$	6.46027	0.342394
$357$	0	0
$358$	6.49889	0.343477
$359$	−24.4132	−1.28848	−0.644239	−	0.764824i	$-0.722826\pi$
$359$	−24.4132	−1.28848	−0.644239	+	0.764824i	$0.722826\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	19.6240	1.03141
$363$	0	0
$364$	7.82009	0.409884
$365$	0	0
$366$	0	0
$367$	10.0695	0.525625	0.262813	−	0.964847i	$-0.415350\pi$
$367$	10.0695	0.525625	0.262813	+	0.964847i	$0.415350\pi$
$368$	−1.86719	−0.0973338
$369$	0	0
$370$	0	0
$371$	−17.6240	−0.914992
$372$	0	0
$373$	−24.5769	−1.27254	−0.636272	−	0.771465i	$-0.719524\pi$
$373$	−24.5769	−1.27254	−0.636272	+	0.771465i	$0.719524\pi$
$374$	3.39073	0.175331
$375$	0	0
$376$	12.4132	0.640160
$377$	37.9430	1.95416
$378$	0	0
$379$	−3.50111	−0.179840	−0.0899199	−	0.995949i	$-0.528661\pi$
$379$	−3.50111	−0.179840	−0.0899199	+	0.995949i	$0.528661\pi$
$380$	0	0
$381$	0	0
$382$	−0.406917	−0.0208197
$383$	18.9901	0.970347	0.485174	−	0.874418i	$-0.338756\pi$
$383$	18.9901	0.970347	0.485174	+	0.874418i	$0.338756\pi$
$384$	0	0
$385$	0	0
$386$	12.3970	0.630990
$387$	0	0
$388$	4.42936	0.224866
$389$	−14.2333	−0.721655	−0.360828	−	0.932633i	$-0.617506\pi$
$389$	−14.2333	−0.721655	−0.360828	+	0.932633i	$0.617506\pi$
$390$	0	0
$391$	−2.75681	−0.139418
$392$	−4.82009	−0.243451
$393$	0	0
$394$	−2.24945	−0.113325
$395$	0	0
$396$	0	0
$397$	29.3885	1.47497	0.737484	−	0.675365i	$-0.236014\pi$
$397$	29.3885	1.47497	0.737484	+	0.675365i	$0.236014\pi$
$398$	18.4665	0.925643
$399$	0	0
$400$	0	0
$401$	14.5460	0.726392	0.363196	−	0.931713i	$-0.381686\pi$
$401$	14.5460	0.726392	0.363196	+	0.931713i	$0.381686\pi$
$402$	0	0
$403$	−0.249445	−0.0124258
$404$	−13.2965	−0.661528
$405$	0	0
$406$	10.5769	0.524923
$407$	15.1413	0.750526
$408$	0	0
$409$	18.5931	0.919369	0.459684	−	0.888082i	$-0.347963\pi$
$409$	18.5931	0.919369	0.459684	+	0.888082i	$0.347963\pi$
$410$	0	0
$411$	0	0
$412$	−0.227007	−0.0111838
$413$	9.36608	0.460874
$414$	0	0
$415$	0	0
$416$	−5.29654	−0.259684
$417$	0	0
$418$	−2.29654	−0.112328
$419$	10.2333	0.499928	0.249964	−	0.968255i	$-0.419581\pi$
$419$	10.2333	0.499928	0.249964	+	0.968255i	$0.419581\pi$
$420$	0	0
$421$	9.82856	0.479015	0.239507	−	0.970895i	$-0.423014\pi$
$421$	9.82856	0.479015	0.239507	+	0.970895i	$0.423014\pi$
$422$	24.3584	1.18575
$423$	0	0
$424$	11.9367	0.579698
$425$	0	0
$426$	0	0
$427$	14.6711	0.709984
$428$	−7.20235	−0.348139
$429$	0	0
$430$	0	0
$431$	−36.0449	−1.73622	−0.868110	−	0.496371i	$-0.834665\pi$
$431$	−36.0449	−1.73622	−0.868110	+	0.496371i	$0.834665\pi$
$432$	0	0
$433$	19.2881	0.926925	0.463463	−	0.886116i	$-0.346607\pi$
$433$	19.2881	0.926925	0.463463	+	0.886116i	$0.346607\pi$
$434$	−0.0695349	−0.00333778
$435$	0	0
$436$	−7.04710	−0.337495
$437$	1.86719	0.0893196
$438$	0	0
$439$	24.4194	1.16548	0.582738	−	0.812660i	$-0.301981\pi$
$439$	24.4194	1.16548	0.582738	+	0.812660i	$0.301981\pi$
$440$	0	0
$441$	0	0
$442$	−7.82009	−0.371964
$443$	−13.8834	−0.659619	−0.329809	−	0.944048i	$-0.606984\pi$
$443$	−13.8834	−0.659619	−0.329809	+	0.944048i	$0.606984\pi$
$444$	0	0
$445$	0	0
$446$	−20.1946	−0.956244
$447$	0	0
$448$	−1.47645	−0.0697558
$449$	−13.6016	−0.641897	−0.320949	−	0.947097i	$-0.604002\pi$
$449$	−13.6016	−0.641897	−0.320949	+	0.947097i	$0.604002\pi$
$450$	0	0
$451$	−0.413172	−0.0194555
$452$	−20.1004	−0.945445
$453$	0	0
$454$	6.35982	0.298481
$455$	0	0
$456$	0	0
$457$	−25.5051	−1.19308	−0.596540	−	0.802583i	$-0.703458\pi$
$457$	−25.5051	−1.19308	−0.596540	+	0.802583i	$0.703458\pi$
$458$	−11.8510	−0.553761
$459$	0	0
$460$	0	0
$461$	−16.3598	−0.761953	−0.380976	−	0.924585i	$-0.624412\pi$
$461$	−16.3598	−0.761953	−0.380976	+	0.924585i	$0.624412\pi$
$462$	0	0
$463$	−14.7815	−0.686953	−0.343477	−	0.939161i	$-0.611605\pi$
$463$	−14.7815	−0.686953	−0.343477	+	0.939161i	$0.611605\pi$
$464$	−7.16373	−0.332568
$465$	0	0
$466$	−27.6155	−1.27926
$467$	−13.4055	−0.620331	−0.310165	−	0.950683i	$-0.600384\pi$
$467$	−13.4055	−0.620331	−0.310165	+	0.950683i	$0.600384\pi$
$468$	0	0
$469$	21.8573	1.00927
$470$	0	0
$471$	0	0
$472$	−6.34364	−0.291990
$473$	18.3352	0.843052
$474$	0	0
$475$	0	0
$476$	−2.17991	−0.0999160
$477$	0	0
$478$	−28.2248	−1.29097
$479$	37.0534	1.69301	0.846505	−	0.532380i	$-0.178702\pi$
$479$	37.0534	1.69301	0.846505	+	0.532380i	$0.178702\pi$
$480$	0	0
$481$	−34.9205	−1.59224
$482$	7.38226	0.336253
$483$	0	0
$484$	−5.72590	−0.260268
$485$	0	0
$486$	0	0
$487$	−13.0471	−0.591220	−0.295610	−	0.955309i	$-0.595523\pi$
$487$	−13.0471	−0.591220	−0.295610	+	0.955309i	$0.595523\pi$
$488$	−9.93672	−0.449814
$489$	0	0
$490$	0	0
$491$	30.7322	1.38692	0.693461	−	0.720494i	$-0.256085\pi$
$491$	30.7322	1.38692	0.693461	+	0.720494i	$0.256085\pi$
$492$	0	0
$493$	−10.5769	−0.476360
$494$	5.29654	0.238303
$495$	0	0
$496$	0.0470959	0.00211467
$497$	20.7730	0.931796
$498$	0	0
$499$	−17.3414	−0.776309	−0.388154	−	0.921594i	$-0.626887\pi$
$499$	−17.3414	−0.776309	−0.388154	+	0.921594i	$0.626887\pi$
$500$	0	0
$501$	0	0
$502$	−15.6317	−0.697677
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	0	0
$506$	−4.28807	−0.190628
$507$	0	0
$508$	5.68727	0.252332
$509$	25.3885	1.12533	0.562663	−	0.826686i	$-0.309777\pi$
$509$	25.3885	1.12533	0.562663	+	0.826686i	$0.309777\pi$
$510$	0	0
$511$	21.3499	0.944464
$512$	1.00000	0.0441942
$513$	0	0
$514$	8.13281	0.358723
$515$	0	0
$516$	0	0
$517$	28.5074	1.25375
$518$	−9.73437	−0.427704
$519$	0	0
$520$	0	0
$521$	−13.7645	−0.603035	−0.301517	−	0.953461i	$-0.597493\pi$
$521$	−13.7645	−0.603035	−0.301517	+	0.953461i	$0.597493\pi$
$522$	0	0
$523$	18.5931	0.813019	0.406509	−	0.913647i	$-0.366746\pi$
$523$	18.5931	0.813019	0.406509	+	0.913647i	$0.366746\pi$
$524$	−12.2108	−0.533432
$525$	0	0
$526$	−10.2270	−0.445919
$527$	0.0695349	0.00302899
$528$	0	0
$529$	−19.5136	−0.848418
$530$	0	0
$531$	0	0
$532$	1.47645	0.0640123
$533$	0.952904	0.0412749
$534$	0	0
$535$	0	0
$536$	−14.8039	−0.639431
$537$	0	0
$538$	8.59308	0.370474
$539$	−11.0695	−0.476799
$540$	0	0
$541$	34.2557	1.47277	0.736384	−	0.676564i	$-0.236532\pi$
$541$	34.2557	1.47277	0.736384	+	0.676564i	$0.236532\pi$
$542$	8.78918	0.377527
$543$	0	0
$544$	1.47645	0.0633024
$545$	0	0
$546$	0	0
$547$	−7.93672	−0.339350	−0.169675	−	0.985500i	$-0.554272\pi$
$547$	−7.93672	−0.339350	−0.169675	+	0.985500i	$0.554272\pi$
$548$	−8.23326	−0.351708
$549$	0	0
$550$	0	0
$551$	7.16373	0.305185
$552$	0	0
$553$	18.5235	0.787701
$554$	24.7954	1.05346
$555$	0	0
$556$	−13.2023	−0.559904
$557$	−1.46798	−0.0622003	−0.0311001	−	0.999516i	$-0.509901\pi$
$557$	−1.46798	−0.0622003	−0.0311001	+	0.999516i	$0.509901\pi$
$558$	0	0
$559$	−42.2866	−1.78853
$560$	0	0
$561$	0	0
$562$	−23.5931	−0.995214
$563$	−32.7160	−1.37881	−0.689407	−	0.724374i	$-0.742129\pi$
$563$	−32.7160	−1.37881	−0.689407	+	0.724374i	$0.742129\pi$
$564$	0	0
$565$	0	0
$566$	−13.5460	−0.569380
$567$	0	0
$568$	−14.0695	−0.590345
$569$	22.6873	0.951100	0.475550	−	0.879689i	$-0.342249\pi$
$569$	22.6873	0.951100	0.475550	+	0.879689i	$0.342249\pi$
$570$	0	0
$571$	8.34364	0.349170	0.174585	−	0.984642i	$-0.444142\pi$
$571$	8.34364	0.349170	0.174585	+	0.984642i	$0.444142\pi$
$572$	−12.1637	−0.508591
$573$	0	0
$574$	0.265629	0.0110872
$575$	0	0
$576$	0	0
$577$	−0.451795	−0.0188085	−0.00940424	−	0.999956i	$-0.502994\pi$
$577$	−0.451795	−0.0188085	−0.00940424	+	0.999956i	$0.502994\pi$
$578$	−14.8201	−0.616434
$579$	0	0
$580$	0	0
$581$	−9.03091	−0.374665
$582$	0	0
$583$	27.4132	1.13534
$584$	−14.4603	−0.598370
$585$	0	0
$586$	−24.7954	−1.02429
$587$	32.8812	1.35715	0.678575	−	0.734531i	$-0.262598\pi$
$587$	32.8812	1.35715	0.678575	+	0.734531i	$0.262598\pi$
$588$	0	0
$589$	−0.0470959	−0.00194055
$590$	0	0
$591$	0	0
$592$	6.59308	0.270974
$593$	36.8882	1.51482	0.757408	−	0.652942i	$-0.226466\pi$
$593$	36.8882	1.51482	0.757408	+	0.652942i	$0.226466\pi$
$594$	0	0
$595$	0	0
$596$	−6.85871	−0.280944
$597$	0	0
$598$	9.88962	0.404417
$599$	43.7715	1.78846	0.894228	−	0.447611i	$-0.147725\pi$
$599$	43.7715	1.78846	0.894228	+	0.447611i	$0.147725\pi$
$600$	0	0
$601$	15.9430	0.650328	0.325164	−	0.945658i	$-0.394581\pi$
$601$	15.9430	0.650328	0.325164	+	0.945658i	$0.394581\pi$
$602$	−11.7877	−0.480432
$603$	0	0
$604$	10.6788	0.434514
$605$	0	0
$606$	0	0
$607$	−19.9444	−0.809519	−0.404760	−	0.914423i	$-0.632645\pi$
$607$	−19.9444	−0.809519	−0.404760	+	0.914423i	$0.632645\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	−65.7469	−2.65983
$612$	0	0
$613$	36.9205	1.49121	0.745603	−	0.666390i	$-0.232161\pi$
$613$	36.9205	1.49121	0.745603	+	0.666390i	$0.232161\pi$
$614$	−11.6626	−0.470665
$615$	0	0
$616$	−3.39073	−0.136617
$617$	−23.1538	−0.932137	−0.466068	−	0.884749i	$-0.654330\pi$
$617$	−23.1538	−0.932137	−0.466068	+	0.884749i	$0.654330\pi$
$618$	0	0
$619$	−5.84475	−0.234920	−0.117460	−	0.993078i	$-0.537475\pi$
$619$	−5.84475	−0.234920	−0.117460	+	0.993078i	$0.537475\pi$
$620$	0	0
$621$	0	0
$622$	3.72590	0.149395
$623$	−9.53828	−0.382143
$624$	0	0
$625$	0	0
$626$	5.00847	0.200179
$627$	0	0
$628$	3.22701	0.128772
$629$	9.73437	0.388135
$630$	0	0
$631$	44.5607	1.77393	0.886967	−	0.461833i	$-0.152808\pi$
$631$	44.5607	1.77393	0.886967	+	0.461833i	$0.152808\pi$
$632$	−12.5460	−0.499053
$633$	0	0
$634$	27.1637	1.07881
$635$	0	0
$636$	0	0
$637$	25.5298	1.01153
$638$	−16.4518	−0.651333
$639$	0	0
$640$	0	0
$641$	4.50736	0.178030	0.0890151	−	0.996030i	$-0.471628\pi$
$641$	4.50736	0.178030	0.0890151	+	0.996030i	$0.471628\pi$
$642$	0	0
$643$	19.0596	0.751637	0.375819	−	0.926693i	$-0.377362\pi$
$643$	19.0596	0.751637	0.375819	+	0.926693i	$0.377362\pi$
$644$	2.75681	0.108634
$645$	0	0
$646$	−1.47645	−0.0580902
$647$	−21.0857	−0.828965	−0.414483	−	0.910057i	$-0.636037\pi$
$647$	−21.0857	−0.828965	−0.414483	+	0.910057i	$0.636037\pi$
$648$	0	0
$649$	−14.5684	−0.571861
$650$	0	0
$651$	0	0
$652$	−6.00000	−0.234978
$653$	−11.3745	−0.445121	−0.222560	−	0.974919i	$-0.571441\pi$
$653$	−11.3745	−0.445121	−0.222560	+	0.974919i	$0.571441\pi$
$654$	0	0
$655$	0	0
$656$	−0.179911	−0.00702433
$657$	0	0
$658$	−18.3275	−0.714479
$659$	−1.48493	−0.0578445	−0.0289222	−	0.999582i	$-0.509208\pi$
$659$	−1.48493	−0.0578445	−0.0289222	+	0.999582i	$0.509208\pi$
$660$	0	0
$661$	−20.4216	−0.794310	−0.397155	−	0.917752i	$-0.630002\pi$
$661$	−20.4216	−0.794310	−0.397155	+	0.917752i	$0.630002\pi$
$662$	−2.80391	−0.108977
$663$	0	0
$664$	6.11663	0.237371
$665$	0	0
$666$	0	0
$667$	13.3760	0.517921
$668$	20.9205	0.809440
$669$	0	0
$670$	0	0
$671$	−22.8201	−0.880960
$672$	0	0
$673$	23.0224	0.887450	0.443725	−	0.896163i	$-0.353657\pi$
$673$	23.0224	0.887450	0.443725	+	0.896163i	$0.353657\pi$
$674$	6.85871	0.264188
$675$	0	0
$676$	15.0534	0.578975
$677$	−25.9198	−0.996178	−0.498089	−	0.867126i	$-0.665965\pi$
$677$	−25.9198	−0.996178	−0.498089	+	0.867126i	$0.665965\pi$
$678$	0	0
$679$	−6.53973	−0.250972
$680$	0	0
$681$	0	0
$682$	0.108158	0.00414157
$683$	−3.56217	−0.136303	−0.0681513	−	0.997675i	$-0.521710\pi$
$683$	−3.56217	−0.136303	−0.0681513	+	0.997675i	$0.521710\pi$
$684$	0	0
$685$	0	0
$686$	17.4518	0.666313
$687$	0	0
$688$	7.98382	0.304380
$689$	−63.2233	−2.40862
$690$	0	0
$691$	−50.6218	−1.92574	−0.962872	−	0.269960i	$-0.912990\pi$
$691$	−50.6218	−1.92574	−0.962872	+	0.269960i	$0.912990\pi$
$692$	1.25792	0.0478189
$693$	0	0
$694$	−19.7792	−0.750810
$695$	0	0
$696$	0	0
$697$	−0.265629	−0.0100614
$698$	−7.39699	−0.279980
$699$	0	0
$700$	0	0
$701$	6.82634	0.257827	0.128914	−	0.991656i	$-0.458851\pi$
$701$	6.82634	0.257827	0.128914	+	0.991656i	$0.458851\pi$
$702$	0	0
$703$	−6.59308	−0.248663
$704$	2.29654	0.0865542
$705$	0	0
$706$	−36.0372	−1.35628
$707$	19.6317	0.738326
$708$	0	0
$709$	38.3499	1.44026	0.720130	−	0.693839i	$-0.244082\pi$
$709$	38.3499	1.44026	0.720130	+	0.693839i	$0.244082\pi$
$710$	0	0
$711$	0	0
$712$	6.46027	0.242109
$713$	−0.0879368	−0.00329326
$714$	0	0
$715$	0	0
$716$	6.49889	0.242875
$717$	0	0
$718$	−24.4132	−0.911091
$719$	−28.7020	−1.07040	−0.535202	−	0.844724i	$-0.679765\pi$
$719$	−28.7020	−1.07040	−0.535202	+	0.844724i	$0.679765\pi$
$720$	0	0
$721$	0.335164	0.0124822
$722$	1.00000	0.0372161
$723$	0	0
$724$	19.6240	0.729320
$725$	0	0
$726$	0	0
$727$	−1.51362	−0.0561370	−0.0280685	−	0.999606i	$-0.508936\pi$
$727$	−1.51362	−0.0561370	−0.0280685	+	0.999606i	$0.508936\pi$
$728$	7.82009	0.289832
$729$	0	0
$730$	0	0
$731$	11.7877	0.435985
$732$	0	0
$733$	−18.8488	−0.696196	−0.348098	−	0.937458i	$-0.613172\pi$
$733$	−18.8488	−0.696196	−0.348098	+	0.937458i	$0.613172\pi$
$734$	10.0695	0.371673
$735$	0	0
$736$	−1.86719	−0.0688254
$737$	−33.9978	−1.25232
$738$	0	0
$739$	−28.4989	−1.04835	−0.524174	−	0.851611i	$-0.675626\pi$
$739$	−28.4989	−1.04835	−0.524174	+	0.851611i	$0.675626\pi$
$740$	0	0
$741$	0	0
$742$	−17.6240	−0.646997
$743$	−11.1089	−0.407547	−0.203773	−	0.979018i	$-0.565321\pi$
$743$	−11.1089	−0.407547	−0.203773	+	0.979018i	$0.565321\pi$
$744$	0	0
$745$	0	0
$746$	−24.5769	−0.899824
$747$	0	0
$748$	3.39073	0.123977
$749$	10.6339	0.388555
$750$	0	0
$751$	−21.6487	−0.789971	−0.394985	−	0.918687i	$-0.629250\pi$
$751$	−21.6487	−0.789971	−0.394985	+	0.918687i	$0.629250\pi$
$752$	12.4132	0.452662
$753$	0	0
$754$	37.9430	1.38180
$755$	0	0
$756$	0	0
$757$	−5.99007	−0.217713	−0.108856	−	0.994057i	$-0.534719\pi$
$757$	−5.99007	−0.217713	−0.108856	+	0.994057i	$0.534719\pi$
$758$	−3.50111	−0.127166
$759$	0	0
$760$	0	0
$761$	−19.1615	−0.694604	−0.347302	−	0.937753i	$-0.612902\pi$
$761$	−19.1615	−0.694604	−0.347302	+	0.937753i	$0.612902\pi$
$762$	0	0
$763$	10.4047	0.376675
$764$	−0.406917	−0.0147217
$765$	0	0
$766$	18.9901	0.686139
$767$	33.5993	1.21320
$768$	0	0
$769$	−18.9761	−0.684296	−0.342148	−	0.939646i	$-0.611154\pi$
$769$	−18.9761	−0.684296	−0.342148	+	0.939646i	$0.611154\pi$
$770$	0	0
$771$	0	0
$772$	12.3970	0.446177
$773$	36.5438	1.31439	0.657194	−	0.753721i	$-0.271743\pi$
$773$	36.5438	1.31439	0.657194	+	0.753721i	$0.271743\pi$
$774$	0	0
$775$	0	0
$776$	4.42936	0.159005
$777$	0	0
$778$	−14.2333	−0.510287
$779$	0.179911	0.00644597
$780$	0	0
$781$	−32.3113	−1.15619
$782$	−2.75681	−0.0985833
$783$	0	0
$784$	−4.82009	−0.172146
$785$	0	0
$786$	0	0
$787$	32.5213	1.15926	0.579630	−	0.814880i	$-0.303197\pi$
$787$	32.5213	1.15926	0.579630	+	0.814880i	$0.303197\pi$
$788$	−2.24945	−0.0801332
$789$	0	0
$790$	0	0
$791$	29.6773	1.05520
$792$	0	0
$793$	52.6302	1.86895
$794$	29.3885	1.04296
$795$	0	0
$796$	18.4665	0.654529
$797$	41.4110	1.46685	0.733426	−	0.679770i	$-0.237920\pi$
$797$	41.4110	1.46685	0.733426	+	0.679770i	$0.237920\pi$
$798$	0	0
$799$	18.3275	0.648379
$800$	0	0
$801$	0	0
$802$	14.5460	0.513637
$803$	−33.2086	−1.17191
$804$	0	0
$805$	0	0
$806$	−0.249445	−0.00878634
$807$	0	0
$808$	−13.2965	−0.467771
$809$	8.72444	0.306735	0.153368	−	0.988169i	$-0.450988\pi$
$809$	8.72444	0.306735	0.153368	+	0.988169i	$0.450988\pi$
$810$	0	0
$811$	−11.8425	−0.415847	−0.207924	−	0.978145i	$-0.566671\pi$
$811$	−11.8425	−0.415847	−0.207924	+	0.978145i	$0.566671\pi$
$812$	10.5769	0.371176
$813$	0	0
$814$	15.1413	0.530702
$815$	0	0
$816$	0	0
$817$	−7.98382	−0.279318
$818$	18.5931	0.650092
$819$	0	0
$820$	0	0
$821$	−5.03091	−0.175580	−0.0877900	−	0.996139i	$-0.527980\pi$
$821$	−5.03091	−0.175580	−0.0877900	+	0.996139i	$0.527980\pi$
$822$	0	0
$823$	−7.81162	−0.272296	−0.136148	−	0.990689i	$-0.543472\pi$
$823$	−7.81162	−0.272296	−0.136148	+	0.990689i	$0.543472\pi$
$824$	−0.227007	−0.00790815
$825$	0	0
$826$	9.36608	0.325887
$827$	−2.78070	−0.0966946	−0.0483473	−	0.998831i	$-0.515395\pi$
$827$	−2.78070	−0.0966946	−0.0483473	+	0.998831i	$0.515395\pi$
$828$	0	0
$829$	47.9963	1.66698	0.833491	−	0.552534i	$-0.186339\pi$
$829$	47.9963	1.66698	0.833491	+	0.552534i	$0.186339\pi$
$830$	0	0
$831$	0	0
$832$	−5.29654	−0.183625
$833$	−7.11663	−0.246577
$834$	0	0
$835$	0	0
$836$	−2.29654	−0.0794275
$837$	0	0
$838$	10.2333	0.353502
$839$	0.290286	0.0100218	0.00501090	−	0.999987i	$-0.498405\pi$
$839$	0.290286	0.0100218	0.00501090	+	0.999987i	$0.498405\pi$
$840$	0	0
$841$	22.3190	0.769620
$842$	9.82856	0.338715
$843$	0	0
$844$	24.3584	0.838450
$845$	0	0
$846$	0	0
$847$	8.45401	0.290483
$848$	11.9367	0.409909
$849$	0	0
$850$	0	0
$851$	−12.3105	−0.421999
$852$	0	0
$853$	13.6725	0.468139	0.234070	−	0.972220i	$-0.424796\pi$
$853$	13.6725	0.468139	0.234070	+	0.972220i	$0.424796\pi$
$854$	14.6711	0.502034
$855$	0	0
$856$	−7.20235	−0.246171
$857$	48.9569	1.67234	0.836169	−	0.548472i	$-0.184790\pi$
$857$	48.9569	1.67234	0.836169	+	0.548472i	$0.184790\pi$
$858$	0	0
$859$	−0.985272	−0.0336170	−0.0168085	−	0.999859i	$-0.505351\pi$
$859$	−0.985272	−0.0336170	−0.0168085	+	0.999859i	$0.505351\pi$
$860$	0	0
$861$	0	0
$862$	−36.0449	−1.22769
$863$	−9.31273	−0.317009	−0.158504	−	0.987358i	$-0.550667\pi$
$863$	−9.31273	−0.317009	−0.158504	+	0.987358i	$0.550667\pi$
$864$	0	0
$865$	0	0
$866$	19.2881	0.655435
$867$	0	0
$868$	−0.0695349	−0.00236017
$869$	−28.8124	−0.977393
$870$	0	0
$871$	78.4095	2.65680
$872$	−7.04710	−0.238645
$873$	0	0
$874$	1.86719	0.0631585
$875$	0	0
$876$	0	0
$877$	−31.7469	−1.07202	−0.536008	−	0.844213i	$-0.680068\pi$
$877$	−31.7469	−1.07202	−0.536008	+	0.844213i	$0.680068\pi$
$878$	24.4194	0.824116
$879$	0	0
$880$	0	0
$881$	−8.78147	−0.295855	−0.147928	−	0.988998i	$-0.547260\pi$
$881$	−8.78147	−0.295855	−0.147928	+	0.988998i	$0.547260\pi$
$882$	0	0
$883$	−16.9044	−0.568877	−0.284438	−	0.958694i	$-0.591807\pi$
$883$	−16.9044	−0.568877	−0.284438	+	0.958694i	$0.591807\pi$
$884$	−7.82009	−0.263018
$885$	0	0
$886$	−13.8834	−0.466421
$887$	17.6851	0.593806	0.296903	−	0.954908i	$-0.404046\pi$
$887$	17.6851	0.593806	0.296903	+	0.954908i	$0.404046\pi$
$888$	0	0
$889$	−8.39699	−0.281626
$890$	0	0
$891$	0	0
$892$	−20.1946	−0.676167
$893$	−12.4132	−0.415391
$894$	0	0
$895$	0	0
$896$	−1.47645	−0.0493248
$897$	0	0
$898$	−13.6016	−0.453890
$899$	−0.337382	−0.0112523
$900$	0	0
$901$	17.6240	0.587140
$902$	−0.413172	−0.0137571
$903$	0	0
$904$	−20.1004	−0.668531
$905$	0	0
$906$	0	0
$907$	46.0210	1.52810	0.764051	−	0.645156i	$-0.223208\pi$
$907$	46.0210	1.52810	0.764051	+	0.645156i	$0.223208\pi$
$908$	6.35982	0.211058
$909$	0	0
$910$	0	0
$911$	−10.1560	−0.336484	−0.168242	−	0.985746i	$-0.553809\pi$
$911$	−10.1560	−0.336484	−0.168242	+	0.985746i	$0.553809\pi$
$912$	0	0
$913$	14.0471	0.464891
$914$	−25.5051	−0.843635
$915$	0	0
$916$	−11.8510	−0.391568
$917$	18.0287	0.595360
$918$	0	0
$919$	−40.0820	−1.32218	−0.661092	−	0.750305i	$-0.729907\pi$
$919$	−40.0820	−1.32218	−0.661092	+	0.750305i	$0.729907\pi$
$920$	0	0
$921$	0	0
$922$	−16.3598	−0.538782
$923$	74.5199	2.45285
$924$	0	0
$925$	0	0
$926$	−14.7815	−0.485749
$927$	0	0
$928$	−7.16373	−0.235161
$929$	−29.2634	−0.960101	−0.480051	−	0.877241i	$-0.659382\pi$
$929$	−29.2634	−0.960101	−0.480051	+	0.877241i	$0.659382\pi$
$930$	0	0
$931$	4.82009	0.157972
$932$	−27.6155	−0.904576
$933$	0	0
$934$	−13.4055	−0.438640
$935$	0	0
$936$	0	0
$937$	26.4989	0.865681	0.432841	−	0.901471i	$-0.357511\pi$
$937$	26.4989	0.865681	0.432841	+	0.901471i	$0.357511\pi$
$938$	21.8573	0.713665
$939$	0	0
$940$	0	0
$941$	19.8749	0.647903	0.323952	−	0.946074i	$-0.394989\pi$
$941$	19.8749	0.647903	0.323952	+	0.946074i	$0.394989\pi$
$942$	0	0
$943$	0.335926	0.0109393
$944$	−6.34364	−0.206468
$945$	0	0
$946$	18.3352	0.596128
$947$	−31.0766	−1.00985	−0.504926	−	0.863163i	$-0.668480\pi$
$947$	−31.0766	−1.00985	−0.504926	+	0.863163i	$0.668480\pi$
$948$	0	0
$949$	76.5894	2.48620
$950$	0	0
$951$	0	0
$952$	−2.17991	−0.0706513
$953$	−17.3275	−0.561291	−0.280646	−	0.959811i	$-0.590549\pi$
$953$	−17.3275	−0.561291	−0.280646	+	0.959811i	$0.590549\pi$
$954$	0	0
$955$	0	0
$956$	−28.2248	−0.912855
$957$	0	0
$958$	37.0534	1.19714
$959$	12.1560	0.392538
$960$	0	0
$961$	−30.9978	−0.999928
$962$	−34.9205	−1.12588
$963$	0	0
$964$	7.38226	0.237767
$965$	0	0
$966$	0	0
$967$	−12.4989	−0.401937	−0.200969	−	0.979598i	$-0.564409\pi$
$967$	−12.4989	−0.401937	−0.200969	+	0.979598i	$0.564409\pi$
$968$	−5.72590	−0.184037
$969$	0	0
$970$	0	0
$971$	50.3113	1.61457	0.807283	−	0.590165i	$-0.200937\pi$
$971$	50.3113	1.61457	0.807283	+	0.590165i	$0.200937\pi$
$972$	0	0
$973$	19.4926	0.624905
$974$	−13.0471	−0.418056
$975$	0	0
$976$	−9.93672	−0.318067
$977$	−30.9121	−0.988965	−0.494482	−	0.869188i	$-0.664642\pi$
$977$	−30.9121	−0.988965	−0.494482	+	0.869188i	$0.664642\pi$
$978$	0	0
$979$	14.8363	0.474169
$980$	0	0
$981$	0	0
$982$	30.7322	0.980702
$983$	40.9128	1.30492	0.652458	−	0.757825i	$-0.273738\pi$
$983$	40.9128	1.30492	0.652458	+	0.757825i	$0.273738\pi$
$984$	0	0
$985$	0	0
$986$	−10.5769	−0.336837
$987$	0	0
$988$	5.29654	0.168505
$989$	−14.9073	−0.474023
$990$	0	0
$991$	4.25937	0.135303	0.0676517	−	0.997709i	$-0.478449\pi$
$991$	4.25937	0.135303	0.0676517	+	0.997709i	$0.478449\pi$
$992$	0.0470959	0.00149530
$993$	0	0
$994$	20.7730	0.658879
$995$	0	0
$996$	0	0
$997$	50.6070	1.60274	0.801371	−	0.598168i	$-0.204104\pi$
$997$	50.6070	1.60274	0.801371	+	0.598168i	$0.204104\pi$
$998$	−17.3414	−0.548933
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cm.1.2	yes	3
3.2	odd	2		8550.2.a.cc.1.2	✓	3
5.4	even	2		8550.2.a.ch.1.2	yes	3
15.14	odd	2		8550.2.a.ct.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8550.2.a.cc.1.2	✓	3	3.2	odd	2
8550.2.a.ch.1.2	yes	3	5.4	even	2
8550.2.a.cm.1.2	yes	3	1.1	even	1	trivial
8550.2.a.ct.1.2	yes	3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.47645 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.47645 q^{7} +1.00000 q^{8} +2.29654 q^{11} -5.29654 q^{13} -1.47645 q^{14} +1.00000 q^{16} +1.47645 q^{17} -1.00000 q^{19} +2.29654 q^{22} -1.86719 q^{23} -5.29654 q^{26} -1.47645 q^{28} -7.16373 q^{29} +0.0470959 q^{31} +1.00000 q^{32} +1.47645 q^{34} +6.59308 q^{37} -1.00000 q^{38} -0.179911 q^{41} +7.98382 q^{43} +2.29654 q^{44} -1.86719 q^{46} +12.4132 q^{47} -4.82009 q^{49} -5.29654 q^{52} +11.9367 q^{53} -1.47645 q^{56} -7.16373 q^{58} -6.34364 q^{59} -9.93672 q^{61} +0.0470959 q^{62} +1.00000 q^{64} -14.8039 q^{67} +1.47645 q^{68} -14.0695 q^{71} -14.4603 q^{73} +6.59308 q^{74} -1.00000 q^{76} -3.39073 q^{77} -12.5460 q^{79} -0.179911 q^{82} +6.11663 q^{83} +7.98382 q^{86} +2.29654 q^{88} +6.46027 q^{89} +7.82009 q^{91} -1.86719 q^{92} +12.4132 q^{94} +4.42936 q^{97} -4.82009 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 5 q^{11} - 4 q^{13} - 2 q^{14} + 3 q^{16} + 2 q^{17} - 3 q^{19} - 5 q^{22} - q^{23} - 4 q^{26} - 2 q^{28} - 5 q^{29} + 5 q^{31} + 3 q^{32} + 2 q^{34} - 4 q^{37} - 3 q^{38} - 10 q^{41} - 2 q^{43} - 5 q^{44} - q^{46} + 4 q^{47} - 5 q^{49} - 4 q^{52} + 5 q^{53} - 2 q^{56} - 5 q^{58} - 12 q^{59} + q^{61} + 5 q^{62} + 3 q^{64} - 9 q^{67} + 2 q^{68} - 16 q^{71} - 15 q^{73} - 4 q^{74} - 3 q^{76} - 8 q^{77} - 9 q^{79} - 10 q^{82} - 3 q^{83} - 2 q^{86} - 5 q^{88} - 9 q^{89} + 14 q^{91} - q^{92} + 4 q^{94} + 6 q^{97} - 5 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 5 * q^11 - 4 * q^13 - 2 * q^14 + 3 * q^16 + 2 * q^17 - 3 * q^19 - 5 * q^22 - q^23 - 4 * q^26 - 2 * q^28 - 5 * q^29 + 5 * q^31 + 3 * q^32 + 2 * q^34 - 4 * q^37 - 3 * q^38 - 10 * q^41 - 2 * q^43 - 5 * q^44 - q^46 + 4 * q^47 - 5 * q^49 - 4 * q^52 + 5 * q^53 - 2 * q^56 - 5 * q^58 - 12 * q^59 + q^61 + 5 * q^62 + 3 * q^64 - 9 * q^67 + 2 * q^68 - 16 * q^71 - 15 * q^73 - 4 * q^74 - 3 * q^76 - 8 * q^77 - 9 * q^79 - 10 * q^82 - 3 * q^83 - 2 * q^86 - 5 * q^88 - 9 * q^89 + 14 * q^91 - q^92 + 4 * q^94 + 6 * q^97 - 5 * q^98

Introduction

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