LMFDB - Embedded newform 5550.2.a.bx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5550,2,Mod(1,5550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5550, 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("5550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5550.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:	$44.3169731218$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{113}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 28 $ x^2 - x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.81507$ of defining polynomial
Character	$\chi$	$=$	5550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} -6.81507 q^{13} -3.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -2.81507 q^{19} +3.00000 q^{21} -1.00000 q^{22} +4.81507 q^{23} -1.00000 q^{24} -6.81507 q^{26} -1.00000 q^{27} -3.00000 q^{28} +3.81507 q^{29} -3.81507 q^{31} +1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -2.81507 q^{38} +6.81507 q^{39} +5.81507 q^{41} +3.00000 q^{42} -5.81507 q^{43} -1.00000 q^{44} +4.81507 q^{46} +8.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{51} -6.81507 q^{52} +10.6301 q^{53} -1.00000 q^{54} -3.00000 q^{56} +2.81507 q^{57} +3.81507 q^{58} +2.00000 q^{59} -3.81507 q^{61} -3.81507 q^{62} -3.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} +13.6301 q^{67} -1.00000 q^{68} -4.81507 q^{69} +9.63015 q^{71} +1.00000 q^{72} -4.81507 q^{73} -1.00000 q^{74} -2.81507 q^{76} +3.00000 q^{77} +6.81507 q^{78} +12.0000 q^{79} +1.00000 q^{81} +5.81507 q^{82} -6.81507 q^{83} +3.00000 q^{84} -5.81507 q^{86} -3.81507 q^{87} -1.00000 q^{88} +1.18493 q^{89} +20.4452 q^{91} +4.81507 q^{92} +3.81507 q^{93} +8.00000 q^{94} -1.00000 q^{96} -5.81507 q^{97} +2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{11} - 2 q^{12} - 3 q^{13} - 6 q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{18} + 5 q^{19} + 6 q^{21} - 2 q^{22} - q^{23} - 2 q^{24} - 3 q^{26} - 2 q^{27} - 6 q^{28} - 3 q^{29} + 3 q^{31} + 2 q^{32} + 2 q^{33} - 2 q^{34} + 2 q^{36} - 2 q^{37} + 5 q^{38} + 3 q^{39} + q^{41} + 6 q^{42} - q^{43} - 2 q^{44} - q^{46} + 16 q^{47} - 2 q^{48} + 4 q^{49} + 2 q^{51} - 3 q^{52} - 2 q^{54} - 6 q^{56} - 5 q^{57} - 3 q^{58} + 4 q^{59} + 3 q^{61} + 3 q^{62} - 6 q^{63} + 2 q^{64} + 2 q^{66} + 6 q^{67} - 2 q^{68} + q^{69} - 2 q^{71} + 2 q^{72} + q^{73} - 2 q^{74} + 5 q^{76} + 6 q^{77} + 3 q^{78} + 24 q^{79} + 2 q^{81} + q^{82} - 3 q^{83} + 6 q^{84} - q^{86} + 3 q^{87} - 2 q^{88} + 13 q^{89} + 9 q^{91} - q^{92} - 3 q^{93} + 16 q^{94} - 2 q^{96} - q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^11 - 2 * q^12 - 3 * q^13 - 6 * q^14 + 2 * q^16 - 2 * q^17 + 2 * q^18 + 5 * q^19 + 6 * q^21 - 2 * q^22 - q^23 - 2 * q^24 - 3 * q^26 - 2 * q^27 - 6 * q^28 - 3 * q^29 + 3 * q^31 + 2 * q^32 + 2 * q^33 - 2 * q^34 + 2 * q^36 - 2 * q^37 + 5 * q^38 + 3 * q^39 + q^41 + 6 * q^42 - q^43 - 2 * q^44 - q^46 + 16 * q^47 - 2 * q^48 + 4 * q^49 + 2 * q^51 - 3 * q^52 - 2 * q^54 - 6 * q^56 - 5 * q^57 - 3 * q^58 + 4 * q^59 + 3 * q^61 + 3 * q^62 - 6 * q^63 + 2 * q^64 + 2 * q^66 + 6 * q^67 - 2 * q^68 + q^69 - 2 * q^71 + 2 * q^72 + q^73 - 2 * q^74 + 5 * q^76 + 6 * q^77 + 3 * q^78 + 24 * q^79 + 2 * q^81 + q^82 - 3 * q^83 + 6 * q^84 - q^86 + 3 * q^87 - 2 * q^88 + 13 * q^89 + 9 * q^91 - q^92 - 3 * q^93 + 16 * q^94 - 2 * q^96 - q^97 + 4 * q^98 - 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−1.00000	−0.288675
$13$	−6.81507	−1.89016	−0.945081	−	0.326837i	$-0.894017\pi$
$13$	−6.81507	−1.89016	−0.945081	+	0.326837i	$0.894017\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	1.00000	0.235702
$19$	−2.81507	−0.645822	−0.322911	−	0.946429i	$-0.604661\pi$
$19$	−2.81507	−0.645822	−0.322911	+	0.946429i	$0.604661\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	−1.00000	−0.213201
$23$	4.81507	1.00401	0.502006	−	0.864864i	$-0.332596\pi$
$23$	4.81507	1.00401	0.502006	+	0.864864i	$0.332596\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−6.81507	−1.33655
$27$	−1.00000	−0.192450
$28$	−3.00000	−0.566947
$29$	3.81507	0.708441	0.354221	−	0.935162i	$-0.384746\pi$
$29$	3.81507	0.708441	0.354221	+	0.935162i	$0.384746\pi$
$30$	0	0
$31$	−3.81507	−0.685207	−0.342604	−	0.939480i	$-0.611309\pi$
$31$	−3.81507	−0.685207	−0.342604	+	0.939480i	$0.611309\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−2.81507	−0.456665
$39$	6.81507	1.09129
$40$	0	0
$41$	5.81507	0.908162	0.454081	−	0.890960i	$-0.349968\pi$
$41$	5.81507	0.908162	0.454081	+	0.890960i	$0.349968\pi$
$42$	3.00000	0.462910
$43$	−5.81507	−0.886790	−0.443395	−	0.896326i	$-0.646226\pi$
$43$	−5.81507	−0.886790	−0.443395	+	0.896326i	$0.646226\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	4.81507	0.709944
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	0	0
$51$	1.00000	0.140028
$52$	−6.81507	−0.945081
$53$	10.6301	1.46016	0.730081	−	0.683360i	$-0.239482\pi$
$53$	10.6301	1.46016	0.730081	+	0.683360i	$0.239482\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−3.00000	−0.400892
$57$	2.81507	0.372866
$58$	3.81507	0.500944
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	−3.81507	−0.488470	−0.244235	−	0.969716i	$-0.578537\pi$
$61$	−3.81507	−0.488470	−0.244235	+	0.969716i	$0.578537\pi$
$62$	−3.81507	−0.484515
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	13.6301	1.66519	0.832594	−	0.553884i	$-0.186855\pi$
$67$	13.6301	1.66519	0.832594	+	0.553884i	$0.186855\pi$
$68$	−1.00000	−0.121268
$69$	−4.81507	−0.579667
$70$	0	0
$71$	9.63015	1.14289	0.571444	−	0.820641i	$-0.306383\pi$
$71$	9.63015	1.14289	0.571444	+	0.820641i	$0.306383\pi$
$72$	1.00000	0.117851
$73$	−4.81507	−0.563562	−0.281781	−	0.959479i	$-0.590925\pi$
$73$	−4.81507	−0.563562	−0.281781	+	0.959479i	$0.590925\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−2.81507	−0.322911
$77$	3.00000	0.341882
$78$	6.81507	0.771655
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.81507	0.642167
$83$	−6.81507	−0.748051	−0.374026	−	0.927418i	$-0.622023\pi$
$83$	−6.81507	−0.748051	−0.374026	+	0.927418i	$0.622023\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	−5.81507	−0.627055
$87$	−3.81507	−0.409019
$88$	−1.00000	−0.106600
$89$	1.18493	0.125602	0.0628010	−	0.998026i	$-0.479997\pi$
$89$	1.18493	0.125602	0.0628010	+	0.998026i	$0.479997\pi$
$90$	0	0
$91$	20.4452	2.14324
$92$	4.81507	0.502006
$93$	3.81507	0.395605
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−5.81507	−0.590431	−0.295216	−	0.955431i	$-0.595391\pi$
$97$	−5.81507	−0.590431	−0.295216	+	0.955431i	$0.595391\pi$
$98$	2.00000	0.202031
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−9.63015	−0.958235	−0.479118	−	0.877751i	$-0.659043\pi$
$101$	−9.63015	−0.958235	−0.479118	+	0.877751i	$0.659043\pi$
$102$	1.00000	0.0990148
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−6.81507	−0.668273
$105$	0	0
$106$	10.6301	1.03249
$107$	18.8151	1.81892	0.909461	−	0.415790i	$-0.136495\pi$
$107$	18.8151	1.81892	0.909461	+	0.415790i	$0.136495\pi$
$108$	−1.00000	−0.0962250
$109$	13.0000	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	−3.00000	−0.283473
$113$	−3.81507	−0.358892	−0.179446	−	0.983768i	$-0.557430\pi$
$113$	−3.81507	−0.358892	−0.179446	+	0.983768i	$0.557430\pi$
$114$	2.81507	0.263656
$115$	0	0
$116$	3.81507	0.354221
$117$	−6.81507	−0.630054
$118$	2.00000	0.184115
$119$	3.00000	0.275010
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−3.81507	−0.345400
$123$	−5.81507	−0.524327
$124$	−3.81507	−0.342604
$125$	0	0
$126$	−3.00000	−0.267261
$127$	−10.4452	−0.926863	−0.463432	−	0.886133i	$-0.653382\pi$
$127$	−10.4452	−0.926863	−0.463432	+	0.886133i	$0.653382\pi$
$128$	1.00000	0.0883883
$129$	5.81507	0.511989
$130$	0	0
$131$	11.6301	1.01613	0.508065	−	0.861319i	$-0.330361\pi$
$131$	11.6301	1.01613	0.508065	+	0.861319i	$0.330361\pi$
$132$	1.00000	0.0870388
$133$	8.44522	0.732293
$134$	13.6301	1.17747
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−4.81507	−0.409886
$139$	12.1849	1.03351	0.516756	−	0.856133i	$-0.327139\pi$
$139$	12.1849	1.03351	0.516756	+	0.856133i	$0.327139\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	9.63015	0.808144
$143$	6.81507	0.569905
$144$	1.00000	0.0833333
$145$	0	0
$146$	−4.81507	−0.398498
$147$	−2.00000	−0.164957
$148$	−1.00000	−0.0821995
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−6.81507	−0.554603	−0.277301	−	0.960783i	$-0.589440\pi$
$151$	−6.81507	−0.554603	−0.277301	+	0.960783i	$0.589440\pi$
$152$	−2.81507	−0.228333
$153$	−1.00000	−0.0808452
$154$	3.00000	0.241747
$155$	0	0
$156$	6.81507	0.545643
$157$	2.18493	0.174376	0.0871881	−	0.996192i	$-0.472212\pi$
$157$	2.18493	0.174376	0.0871881	+	0.996192i	$0.472212\pi$
$158$	12.0000	0.954669
$159$	−10.6301	−0.843025
$160$	0	0
$161$	−14.4452	−1.13844
$162$	1.00000	0.0785674
$163$	4.63015	0.362661	0.181331	−	0.983422i	$-0.441960\pi$
$163$	4.63015	0.362661	0.181331	+	0.983422i	$0.441960\pi$
$164$	5.81507	0.454081
$165$	0	0
$166$	−6.81507	−0.528952
$167$	11.1849	0.865516	0.432758	−	0.901510i	$-0.357541\pi$
$167$	11.1849	0.865516	0.432758	+	0.901510i	$0.357541\pi$
$168$	3.00000	0.231455
$169$	33.4452	2.57271
$170$	0	0
$171$	−2.81507	−0.215274
$172$	−5.81507	−0.443395
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	−3.81507	−0.289220
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−2.00000	−0.150329
$178$	1.18493	0.0888140
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	20.4452	1.51550
$183$	3.81507	0.282018
$184$	4.81507	0.354972
$185$	0	0
$186$	3.81507	0.279735
$187$	1.00000	0.0731272
$188$	8.00000	0.583460
$189$	3.00000	0.218218
$190$	0	0
$191$	−16.6301	−1.20332	−0.601658	−	0.798754i	$-0.705493\pi$
$191$	−16.6301	−1.20332	−0.601658	+	0.798754i	$0.705493\pi$
$192$	−1.00000	−0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−5.81507	−0.417498
$195$	0	0
$196$	2.00000	0.142857
$197$	14.8151	1.05553	0.527765	−	0.849390i	$-0.323030\pi$
$197$	14.8151	1.05553	0.527765	+	0.849390i	$0.323030\pi$
$198$	−1.00000	−0.0710669
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−13.6301	−0.961396
$202$	−9.63015	−0.677575
$203$	−11.4452	−0.803297
$204$	1.00000	0.0700140
$205$	0	0
$206$	0	0
$207$	4.81507	0.334671
$208$	−6.81507	−0.472540
$209$	2.81507	0.194723
$210$	0	0
$211$	−23.4452	−1.61404	−0.807018	−	0.590527i	$-0.798920\pi$
$211$	−23.4452	−1.61404	−0.807018	+	0.590527i	$0.798920\pi$
$212$	10.6301	0.730081
$213$	−9.63015	−0.659847
$214$	18.8151	1.28617
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	11.4452	0.776952
$218$	13.0000	0.880471
$219$	4.81507	0.325372
$220$	0	0
$221$	6.81507	0.458431
$222$	1.00000	0.0671156
$223$	6.18493	0.414173	0.207087	−	0.978323i	$-0.433602\pi$
$223$	6.18493	0.414173	0.207087	+	0.978323i	$0.433602\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	−3.81507	−0.253775
$227$	14.1849	0.941487	0.470743	−	0.882270i	$-0.343986\pi$
$227$	14.1849	0.941487	0.470743	+	0.882270i	$0.343986\pi$
$228$	2.81507	0.186433
$229$	21.6301	1.42936	0.714680	−	0.699451i	$-0.246572\pi$
$229$	21.6301	1.42936	0.714680	+	0.699451i	$0.246572\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	3.81507	0.250472
$233$	−13.6301	−0.892941	−0.446470	−	0.894798i	$-0.647319\pi$
$233$	−13.6301	−0.892941	−0.446470	+	0.894798i	$0.647319\pi$
$234$	−6.81507	−0.445515
$235$	0	0
$236$	2.00000	0.130189
$237$	−12.0000	−0.779484
$238$	3.00000	0.194461
$239$	−25.4452	−1.64591	−0.822957	−	0.568103i	$-0.807677\pi$
$239$	−25.4452	−1.64591	−0.822957	+	0.568103i	$0.807677\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−10.0000	−0.642824
$243$	−1.00000	−0.0641500
$244$	−3.81507	−0.244235
$245$	0	0
$246$	−5.81507	−0.370756
$247$	19.1849	1.22071
$248$	−3.81507	−0.242257
$249$	6.81507	0.431888
$250$	0	0
$251$	9.63015	0.607849	0.303925	−	0.952696i	$-0.401703\pi$
$251$	9.63015	0.607849	0.303925	+	0.952696i	$0.401703\pi$
$252$	−3.00000	−0.188982
$253$	−4.81507	−0.302721
$254$	−10.4452	−0.655391
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.81507	−0.425113	−0.212556	−	0.977149i	$-0.568179\pi$
$257$	−6.81507	−0.425113	−0.212556	+	0.977149i	$0.568179\pi$
$258$	5.81507	0.362031
$259$	3.00000	0.186411
$260$	0	0
$261$	3.81507	0.236147
$262$	11.6301	0.718513
$263$	−1.44522	−0.0891160	−0.0445580	−	0.999007i	$-0.514188\pi$
$263$	−1.44522	−0.0891160	−0.0445580	+	0.999007i	$0.514188\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	8.44522	0.517810
$267$	−1.18493	−0.0725164
$268$	13.6301	0.832594
$269$	−0.815073	−0.0496959	−0.0248479	−	0.999691i	$-0.507910\pi$
$269$	−0.815073	−0.0496959	−0.0248479	+	0.999691i	$0.507910\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−1.00000	−0.0606339
$273$	−20.4452	−1.23740
$274$	−2.00000	−0.120824
$275$	0	0
$276$	−4.81507	−0.289833
$277$	10.4452	0.627592	0.313796	−	0.949490i	$-0.398399\pi$
$277$	10.4452	0.627592	0.313796	+	0.949490i	$0.398399\pi$
$278$	12.1849	0.730803
$279$	−3.81507	−0.228402
$280$	0	0
$281$	12.4452	0.742420	0.371210	−	0.928549i	$-0.378943\pi$
$281$	12.4452	0.742420	0.371210	+	0.928549i	$0.378943\pi$
$282$	−8.00000	−0.476393
$283$	11.1849	0.664875	0.332437	−	0.943125i	$-0.392129\pi$
$283$	11.1849	0.664875	0.332437	+	0.943125i	$0.392129\pi$
$284$	9.63015	0.571444
$285$	0	0
$286$	6.81507	0.402984
$287$	−17.4452	−1.02976
$288$	1.00000	0.0589256
$289$	−16.0000	−0.941176
$290$	0	0
$291$	5.81507	0.340886
$292$	−4.81507	−0.281781
$293$	14.2603	0.833095	0.416548	−	0.909114i	$-0.363240\pi$
$293$	14.2603	0.833095	0.416548	+	0.909114i	$0.363240\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	−1.00000	−0.0581238
$297$	1.00000	0.0580259
$298$	2.00000	0.115857
$299$	−32.8151	−1.89774
$300$	0	0
$301$	17.4452	1.00553
$302$	−6.81507	−0.392163
$303$	9.63015	0.553237
$304$	−2.81507	−0.161456
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	18.0000	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	0	0
$311$	−21.4452	−1.21605	−0.608023	−	0.793919i	$-0.708037\pi$
$311$	−21.4452	−1.21605	−0.608023	+	0.793919i	$0.708037\pi$
$312$	6.81507	0.385828
$313$	30.8904	1.74603	0.873015	−	0.487693i	$-0.162161\pi$
$313$	30.8904	1.74603	0.873015	+	0.487693i	$0.162161\pi$
$314$	2.18493	0.123303
$315$	0	0
$316$	12.0000	0.675053
$317$	−7.81507	−0.438938	−0.219469	−	0.975619i	$-0.570433\pi$
$317$	−7.81507	−0.438938	−0.219469	+	0.975619i	$0.570433\pi$
$318$	−10.6301	−0.596109
$319$	−3.81507	−0.213603
$320$	0	0
$321$	−18.8151	−1.05015
$322$	−14.4452	−0.805001
$323$	2.81507	0.156635
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.63015	0.256440
$327$	−13.0000	−0.718902
$328$	5.81507	0.321084
$329$	−24.0000	−1.32316
$330$	0	0
$331$	19.2603	1.05864	0.529321	−	0.848422i	$-0.322447\pi$
$331$	19.2603	1.05864	0.529321	+	0.848422i	$0.322447\pi$
$332$	−6.81507	−0.374026
$333$	−1.00000	−0.0547997
$334$	11.1849	0.612012
$335$	0	0
$336$	3.00000	0.163663
$337$	14.8151	0.807028	0.403514	−	0.914973i	$-0.367789\pi$
$337$	14.8151	0.807028	0.403514	+	0.914973i	$0.367789\pi$
$338$	33.4452	1.81918
$339$	3.81507	0.207206
$340$	0	0
$341$	3.81507	0.206598
$342$	−2.81507	−0.152222
$343$	15.0000	0.809924
$344$	−5.81507	−0.313528
$345$	0	0
$346$	1.00000	0.0537603
$347$	31.2603	1.67814	0.839070	−	0.544023i	$-0.183100\pi$
$347$	31.2603	1.67814	0.839070	+	0.544023i	$0.183100\pi$
$348$	−3.81507	−0.204509
$349$	−35.2603	−1.88744	−0.943720	−	0.330745i	$-0.892700\pi$
$349$	−35.2603	−1.88744	−0.943720	+	0.330745i	$0.892700\pi$
$350$	0	0
$351$	6.81507	0.363762
$352$	−1.00000	−0.0533002
$353$	23.8151	1.26755	0.633774	−	0.773518i	$-0.281505\pi$
$353$	23.8151	1.26755	0.633774	+	0.773518i	$0.281505\pi$
$354$	−2.00000	−0.106299
$355$	0	0
$356$	1.18493	0.0628010
$357$	−3.00000	−0.158777
$358$	16.0000	0.845626
$359$	27.6301	1.45826	0.729132	−	0.684373i	$-0.239924\pi$
$359$	27.6301	1.45826	0.729132	+	0.684373i	$0.239924\pi$
$360$	0	0
$361$	−11.0754	−0.582914
$362$	−10.0000	−0.525588
$363$	10.0000	0.524864
$364$	20.4452	1.07162
$365$	0	0
$366$	3.81507	0.199417
$367$	21.0000	1.09619	0.548096	−	0.836416i	$-0.315353\pi$
$367$	21.0000	1.09619	0.548096	+	0.836416i	$0.315353\pi$
$368$	4.81507	0.251003
$369$	5.81507	0.302721
$370$	0	0
$371$	−31.8904	−1.65567
$372$	3.81507	0.197802
$373$	−25.2603	−1.30793	−0.653964	−	0.756526i	$-0.726895\pi$
$373$	−25.2603	−1.30793	−0.653964	+	0.756526i	$0.726895\pi$
$374$	1.00000	0.0517088
$375$	0	0
$376$	8.00000	0.412568
$377$	−26.0000	−1.33907
$378$	3.00000	0.154303
$379$	1.63015	0.0837350	0.0418675	−	0.999123i	$-0.486669\pi$
$379$	1.63015	0.0837350	0.0418675	+	0.999123i	$0.486669\pi$
$380$	0	0
$381$	10.4452	0.535125
$382$	−16.6301	−0.850872
$383$	−1.55478	−0.0794456	−0.0397228	−	0.999211i	$-0.512647\pi$
$383$	−1.55478	−0.0794456	−0.0397228	+	0.999211i	$0.512647\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	−5.81507	−0.295597
$388$	−5.81507	−0.295216
$389$	38.7055	1.96245	0.981224	−	0.192873i	$-0.0617807\pi$
$389$	38.7055	1.96245	0.981224	+	0.192873i	$0.0617807\pi$
$390$	0	0
$391$	−4.81507	−0.243509
$392$	2.00000	0.101015
$393$	−11.6301	−0.586663
$394$	14.8151	0.746373
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	−15.6301	−0.784455	−0.392227	−	0.919868i	$-0.628295\pi$
$397$	−15.6301	−0.784455	−0.392227	+	0.919868i	$0.628295\pi$
$398$	−4.00000	−0.200502
$399$	−8.44522	−0.422790
$400$	0	0
$401$	32.4452	1.62024	0.810118	−	0.586266i	$-0.199403\pi$
$401$	32.4452	1.62024	0.810118	+	0.586266i	$0.199403\pi$
$402$	−13.6301	−0.679810
$403$	26.0000	1.29515
$404$	−9.63015	−0.479118
$405$	0	0
$406$	−11.4452	−0.568017
$407$	1.00000	0.0495682
$408$	1.00000	0.0495074
$409$	27.6301	1.36622	0.683111	−	0.730314i	$-0.260626\pi$
$409$	27.6301	1.36622	0.683111	+	0.730314i	$0.260626\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	−6.00000	−0.295241
$414$	4.81507	0.236648
$415$	0	0
$416$	−6.81507	−0.334136
$417$	−12.1849	−0.596698
$418$	2.81507	0.137690
$419$	6.44522	0.314870	0.157435	−	0.987529i	$-0.449678\pi$
$419$	6.44522	0.314870	0.157435	+	0.987529i	$0.449678\pi$
$420$	0	0
$421$	13.2603	0.646267	0.323134	−	0.946353i	$-0.395264\pi$
$421$	13.2603	0.646267	0.323134	+	0.946353i	$0.395264\pi$
$422$	−23.4452	−1.14130
$423$	8.00000	0.388973
$424$	10.6301	0.516246
$425$	0	0
$426$	−9.63015	−0.466582
$427$	11.4452	0.553873
$428$	18.8151	0.909461
$429$	−6.81507	−0.329035
$430$	0	0
$431$	21.0000	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$431$	21.0000	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$432$	−1.00000	−0.0481125
$433$	8.81507	0.423625	0.211813	−	0.977310i	$-0.432063\pi$
$433$	8.81507	0.423625	0.211813	+	0.977310i	$0.432063\pi$
$434$	11.4452	0.549388
$435$	0	0
$436$	13.0000	0.622587
$437$	−13.5548	−0.648413
$438$	4.81507	0.230073
$439$	13.4452	0.641705	0.320853	−	0.947129i	$-0.396031\pi$
$439$	13.4452	0.641705	0.320853	+	0.947129i	$0.396031\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	6.81507	0.324160
$443$	7.63015	0.362519	0.181260	−	0.983435i	$-0.441983\pi$
$443$	7.63015	0.362519	0.181260	+	0.983435i	$0.441983\pi$
$444$	1.00000	0.0474579
$445$	0	0
$446$	6.18493	0.292865
$447$	−2.00000	−0.0945968
$448$	−3.00000	−0.141737
$449$	−32.8904	−1.55220	−0.776098	−	0.630612i	$-0.782804\pi$
$449$	−32.8904	−1.55220	−0.776098	+	0.630612i	$0.782804\pi$
$450$	0	0
$451$	−5.81507	−0.273821
$452$	−3.81507	−0.179446
$453$	6.81507	0.320200
$454$	14.1849	0.665732
$455$	0	0
$456$	2.81507	0.131828
$457$	−25.0754	−1.17298	−0.586488	−	0.809958i	$-0.699490\pi$
$457$	−25.0754	−1.17298	−0.586488	+	0.809958i	$0.699490\pi$
$458$	21.6301	1.01071
$459$	1.00000	0.0466760
$460$	0	0
$461$	24.1849	1.12640	0.563202	−	0.826319i	$-0.309569\pi$
$461$	24.1849	1.12640	0.563202	+	0.826319i	$0.309569\pi$
$462$	−3.00000	−0.139573
$463$	−3.63015	−0.168707	−0.0843536	−	0.996436i	$-0.526883\pi$
$463$	−3.63015	−0.168707	−0.0843536	+	0.996436i	$0.526883\pi$
$464$	3.81507	0.177110
$465$	0	0
$466$	−13.6301	−0.631404
$467$	−18.1849	−0.841498	−0.420749	−	0.907177i	$-0.638233\pi$
$467$	−18.1849	−0.841498	−0.420749	+	0.907177i	$0.638233\pi$
$468$	−6.81507	−0.315027
$469$	−40.8904	−1.88814
$470$	0	0
$471$	−2.18493	−0.100676
$472$	2.00000	0.0920575
$473$	5.81507	0.267377
$474$	−12.0000	−0.551178
$475$	0	0
$476$	3.00000	0.137505
$477$	10.6301	0.486721
$478$	−25.4452	−1.16384
$479$	−0.815073	−0.0372416	−0.0186208	−	0.999827i	$-0.505928\pi$
$479$	−0.815073	−0.0372416	−0.0186208	+	0.999827i	$0.505928\pi$
$480$	0	0
$481$	6.81507	0.310741
$482$	−26.0000	−1.18427
$483$	14.4452	0.657280
$484$	−10.0000	−0.454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−3.81507	−0.172700
$489$	−4.63015	−0.209382
$490$	0	0
$491$	12.8151	0.578336	0.289168	−	0.957278i	$-0.406621\pi$
$491$	12.8151	0.578336	0.289168	+	0.957278i	$0.406621\pi$
$492$	−5.81507	−0.262164
$493$	−3.81507	−0.171822
$494$	19.1849	0.863171
$495$	0	0
$496$	−3.81507	−0.171302
$497$	−28.8904	−1.29591
$498$	6.81507	0.305391
$499$	14.4452	0.646657	0.323328	−	0.946287i	$-0.395198\pi$
$499$	14.4452	0.646657	0.323328	+	0.946287i	$0.395198\pi$
$500$	0	0
$501$	−11.1849	−0.499706
$502$	9.63015	0.429814
$503$	−43.2603	−1.92888	−0.964441	−	0.264300i	$-0.914859\pi$
$503$	−43.2603	−1.92888	−0.964441	+	0.264300i	$0.914859\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−4.81507	−0.214056
$507$	−33.4452	−1.48535
$508$	−10.4452	−0.463432
$509$	−14.4452	−0.640273	−0.320137	−	0.947371i	$-0.603729\pi$
$509$	−14.4452	−0.640273	−0.320137	+	0.947371i	$0.603729\pi$
$510$	0	0
$511$	14.4452	0.639019
$512$	1.00000	0.0441942
$513$	2.81507	0.124289
$514$	−6.81507	−0.300600
$515$	0	0
$516$	5.81507	0.255994
$517$	−8.00000	−0.351840
$518$	3.00000	0.131812
$519$	−1.00000	−0.0438951
$520$	0	0
$521$	−9.81507	−0.430006	−0.215003	−	0.976613i	$-0.568976\pi$
$521$	−9.81507	−0.430006	−0.215003	+	0.976613i	$0.568976\pi$
$522$	3.81507	0.166981
$523$	43.2603	1.89164	0.945820	−	0.324691i	$-0.105260\pi$
$523$	43.2603	1.89164	0.945820	+	0.324691i	$0.105260\pi$
$524$	11.6301	0.508065
$525$	0	0
$526$	−1.44522	−0.0630145
$527$	3.81507	0.166187
$528$	1.00000	0.0435194
$529$	0.184927	0.00804031
$530$	0	0
$531$	2.00000	0.0867926
$532$	8.44522	0.366147
$533$	−39.6301	−1.71657
$534$	−1.18493	−0.0512768
$535$	0	0
$536$	13.6301	0.588733
$537$	−16.0000	−0.690451
$538$	−0.815073	−0.0351403
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	17.1849	0.738838	0.369419	−	0.929263i	$-0.379557\pi$
$541$	17.1849	0.738838	0.369419	+	0.929263i	$0.379557\pi$
$542$	−20.0000	−0.859074
$543$	10.0000	0.429141
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−20.4452	−0.874975
$547$	39.8904	1.70559	0.852796	−	0.522244i	$-0.174905\pi$
$547$	39.8904	1.70559	0.852796	+	0.522244i	$0.174905\pi$
$548$	−2.00000	−0.0854358
$549$	−3.81507	−0.162823
$550$	0	0
$551$	−10.7397	−0.457527
$552$	−4.81507	−0.204943
$553$	−36.0000	−1.53088
$554$	10.4452	0.443775
$555$	0	0
$556$	12.1849	0.516756
$557$	0.369854	0.0156712	0.00783561	−	0.999969i	$-0.497506\pi$
$557$	0.369854	0.0156712	0.00783561	+	0.999969i	$0.497506\pi$
$558$	−3.81507	−0.161505
$559$	39.6301	1.67618
$560$	0	0
$561$	−1.00000	−0.0422200
$562$	12.4452	0.524970
$563$	−23.4452	−0.988098	−0.494049	−	0.869434i	$-0.664484\pi$
$563$	−23.4452	−0.988098	−0.494049	+	0.869434i	$0.664484\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	11.1849	0.470138
$567$	−3.00000	−0.125988
$568$	9.63015	0.404072
$569$	−25.1849	−1.05581	−0.527904	−	0.849304i	$-0.677022\pi$
$569$	−25.1849	−1.05581	−0.527904	+	0.849304i	$0.677022\pi$
$570$	0	0
$571$	−4.18493	−0.175134	−0.0875669	−	0.996159i	$-0.527909\pi$
$571$	−4.18493	−0.175134	−0.0875669	+	0.996159i	$0.527909\pi$
$572$	6.81507	0.284953
$573$	16.6301	0.694734
$574$	−17.4452	−0.728149
$575$	0	0
$576$	1.00000	0.0416667
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	−16.0000	−0.665512
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	20.4452	0.848211
$582$	5.81507	0.241043
$583$	−10.6301	−0.440256
$584$	−4.81507	−0.199249
$585$	0	0
$586$	14.2603	0.589087
$587$	−10.1849	−0.420377	−0.210188	−	0.977661i	$-0.567408\pi$
$587$	−10.1849	−0.420377	−0.210188	+	0.977661i	$0.567408\pi$
$588$	−2.00000	−0.0824786
$589$	10.7397	0.442522
$590$	0	0
$591$	−14.8151	−0.609411
$592$	−1.00000	−0.0410997
$593$	−7.63015	−0.313333	−0.156666	−	0.987652i	$-0.550075\pi$
$593$	−7.63015	−0.313333	−0.156666	+	0.987652i	$0.550075\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	2.00000	0.0819232
$597$	4.00000	0.163709
$598$	−32.8151	−1.34191
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−40.6301	−1.65734	−0.828669	−	0.559739i	$-0.810901\pi$
$601$	−40.6301	−1.65734	−0.828669	+	0.559739i	$0.810901\pi$
$602$	17.4452	0.711014
$603$	13.6301	0.555062
$604$	−6.81507	−0.277301
$605$	0	0
$606$	9.63015	0.391198
$607$	−5.26029	−0.213509	−0.106754	−	0.994285i	$-0.534046\pi$
$607$	−5.26029	−0.213509	−0.106754	+	0.994285i	$0.534046\pi$
$608$	−2.81507	−0.114166
$609$	11.4452	0.463784
$610$	0	0
$611$	−54.5206	−2.20567
$612$	−1.00000	−0.0404226
$613$	−47.0754	−1.90136	−0.950678	−	0.310179i	$-0.899611\pi$
$613$	−47.0754	−1.90136	−0.950678	+	0.310179i	$0.899611\pi$
$614$	18.0000	0.726421
$615$	0	0
$616$	3.00000	0.120873
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	−45.4452	−1.82660	−0.913299	−	0.407290i	$-0.866474\pi$
$619$	−45.4452	−1.82660	−0.913299	+	0.407290i	$0.866474\pi$
$620$	0	0
$621$	−4.81507	−0.193222
$622$	−21.4452	−0.859875
$623$	−3.55478	−0.142419
$624$	6.81507	0.272821
$625$	0	0
$626$	30.8904	1.23463
$627$	−2.81507	−0.112423
$628$	2.18493	0.0871881
$629$	1.00000	0.0398726
$630$	0	0
$631$	−40.7055	−1.62046	−0.810230	−	0.586112i	$-0.800658\pi$
$631$	−40.7055	−1.62046	−0.810230	+	0.586112i	$0.800658\pi$
$632$	12.0000	0.477334
$633$	23.4452	0.931864
$634$	−7.81507	−0.310376
$635$	0	0
$636$	−10.6301	−0.421513
$637$	−13.6301	−0.540046
$638$	−3.81507	−0.151040
$639$	9.63015	0.380963
$640$	0	0
$641$	−45.4452	−1.79498	−0.897489	−	0.441037i	$-0.854611\pi$
$641$	−45.4452	−1.79498	−0.897489	+	0.441037i	$0.854611\pi$
$642$	−18.8151	−0.742572
$643$	−17.0000	−0.670415	−0.335207	−	0.942144i	$-0.608806\pi$
$643$	−17.0000	−0.670415	−0.335207	+	0.942144i	$0.608806\pi$
$644$	−14.4452	−0.569221
$645$	0	0
$646$	2.81507	0.110758
$647$	28.0754	1.10376	0.551878	−	0.833925i	$-0.313911\pi$
$647$	28.0754	1.10376	0.551878	+	0.833925i	$0.313911\pi$
$648$	1.00000	0.0392837
$649$	−2.00000	−0.0785069
$650$	0	0
$651$	−11.4452	−0.448573
$652$	4.63015	0.181331
$653$	−4.00000	−0.156532	−0.0782660	−	0.996933i	$-0.524938\pi$
$653$	−4.00000	−0.156532	−0.0782660	+	0.996933i	$0.524938\pi$
$654$	−13.0000	−0.508340
$655$	0	0
$656$	5.81507	0.227040
$657$	−4.81507	−0.187854
$658$	−24.0000	−0.935617
$659$	11.2603	0.438639	0.219319	−	0.975653i	$-0.429616\pi$
$659$	11.2603	0.438639	0.219319	+	0.975653i	$0.429616\pi$
$660$	0	0
$661$	46.2603	1.79932	0.899658	−	0.436595i	$-0.143816\pi$
$661$	46.2603	1.79932	0.899658	+	0.436595i	$0.143816\pi$
$662$	19.2603	0.748572
$663$	−6.81507	−0.264675
$664$	−6.81507	−0.264476
$665$	0	0
$666$	−1.00000	−0.0387492
$667$	18.3699	0.711284
$668$	11.1849	0.432758
$669$	−6.18493	−0.239123
$670$	0	0
$671$	3.81507	0.147279
$672$	3.00000	0.115728
$673$	−16.8151	−0.648173	−0.324087	−	0.946027i	$-0.605057\pi$
$673$	−16.8151	−0.648173	−0.324087	+	0.946027i	$0.605057\pi$
$674$	14.8151	0.570655
$675$	0	0
$676$	33.4452	1.28635
$677$	6.07536	0.233495	0.116748	−	0.993162i	$-0.462753\pi$
$677$	6.07536	0.233495	0.116748	+	0.993162i	$0.462753\pi$
$678$	3.81507	0.146517
$679$	17.4452	0.669486
$680$	0	0
$681$	−14.1849	−0.543568
$682$	3.81507	0.146087
$683$	−48.3357	−1.84951	−0.924756	−	0.380560i	$-0.875731\pi$
$683$	−48.3357	−1.84951	−0.924756	+	0.380560i	$0.875731\pi$
$684$	−2.81507	−0.107637
$685$	0	0
$686$	15.0000	0.572703
$687$	−21.6301	−0.825242
$688$	−5.81507	−0.221698
$689$	−72.4452	−2.75994
$690$	0	0
$691$	19.0754	0.725661	0.362831	−	0.931855i	$-0.381810\pi$
$691$	19.0754	0.725661	0.362831	+	0.931855i	$0.381810\pi$
$692$	1.00000	0.0380143
$693$	3.00000	0.113961
$694$	31.2603	1.18662
$695$	0	0
$696$	−3.81507	−0.144610
$697$	−5.81507	−0.220262
$698$	−35.2603	−1.33462
$699$	13.6301	0.515539
$700$	0	0
$701$	12.3699	0.467203	0.233601	−	0.972332i	$-0.424949\pi$
$701$	12.3699	0.467203	0.233601	+	0.972332i	$0.424949\pi$
$702$	6.81507	0.257218
$703$	2.81507	0.106172
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	23.8151	0.896292
$707$	28.8904	1.08654
$708$	−2.00000	−0.0751646
$709$	−0.630146	−0.0236656	−0.0118328	−	0.999930i	$-0.503767\pi$
$709$	−0.630146	−0.0236656	−0.0118328	+	0.999930i	$0.503767\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	1.18493	0.0444070
$713$	−18.3699	−0.687956
$714$	−3.00000	−0.112272
$715$	0	0
$716$	16.0000	0.597948
$717$	25.4452	0.950269
$718$	27.6301	1.03115
$719$	34.0000	1.26799	0.633993	−	0.773339i	$-0.281415\pi$
$719$	34.0000	1.26799	0.633993	+	0.773339i	$0.281415\pi$
$720$	0	0
$721$	0	0
$722$	−11.0754	−0.412182
$723$	26.0000	0.966950
$724$	−10.0000	−0.371647
$725$	0	0
$726$	10.0000	0.371135
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	20.4452	0.757750
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.81507	0.215078
$732$	3.81507	0.141009
$733$	−8.55478	−0.315978	−0.157989	−	0.987441i	$-0.550501\pi$
$733$	−8.55478	−0.315978	−0.157989	+	0.987441i	$0.550501\pi$
$734$	21.0000	0.775124
$735$	0	0
$736$	4.81507	0.177486
$737$	−13.6301	−0.502073
$738$	5.81507	0.214056
$739$	−19.0754	−0.701699	−0.350849	−	0.936432i	$-0.614107\pi$
$739$	−19.0754	−0.701699	−0.350849	+	0.936432i	$0.614107\pi$
$740$	0	0
$741$	−19.1849	−0.704776
$742$	−31.8904	−1.17073
$743$	27.4452	1.00687	0.503434	−	0.864034i	$-0.332070\pi$
$743$	27.4452	1.00687	0.503434	+	0.864034i	$0.332070\pi$
$744$	3.81507	0.139867
$745$	0	0
$746$	−25.2603	−0.924845
$747$	−6.81507	−0.249350
$748$	1.00000	0.0365636
$749$	−56.4452	−2.06246
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	8.00000	0.291730
$753$	−9.63015	−0.350942
$754$	−26.0000	−0.946864
$755$	0	0
$756$	3.00000	0.109109
$757$	−13.1849	−0.479214	−0.239607	−	0.970870i	$-0.577019\pi$
$757$	−13.1849	−0.479214	−0.239607	+	0.970870i	$0.577019\pi$
$758$	1.63015	0.0592096
$759$	4.81507	0.174776
$760$	0	0
$761$	−19.8151	−0.718296	−0.359148	−	0.933281i	$-0.616933\pi$
$761$	−19.8151	−0.718296	−0.359148	+	0.933281i	$0.616933\pi$
$762$	10.4452	0.378390
$763$	−39.0000	−1.41189
$764$	−16.6301	−0.601658
$765$	0	0
$766$	−1.55478	−0.0561765
$767$	−13.6301	−0.492156
$768$	−1.00000	−0.0360844
$769$	6.73971	0.243040	0.121520	−	0.992589i	$-0.461223\pi$
$769$	6.73971	0.243040	0.121520	+	0.992589i	$0.461223\pi$
$770$	0	0
$771$	6.81507	0.245439
$772$	2.00000	0.0719816
$773$	28.6301	1.02975	0.514877	−	0.857264i	$-0.327837\pi$
$773$	28.6301	1.02975	0.514877	+	0.857264i	$0.327837\pi$
$774$	−5.81507	−0.209018
$775$	0	0
$776$	−5.81507	−0.208749
$777$	−3.00000	−0.107624
$778$	38.7055	1.38766
$779$	−16.3699	−0.586511
$780$	0	0
$781$	−9.63015	−0.344594
$782$	−4.81507	−0.172187
$783$	−3.81507	−0.136340
$784$	2.00000	0.0714286
$785$	0	0
$786$	−11.6301	−0.414834
$787$	18.3699	0.654815	0.327407	−	0.944883i	$-0.393825\pi$
$787$	18.3699	0.654815	0.327407	+	0.944883i	$0.393825\pi$
$788$	14.8151	0.527765
$789$	1.44522	0.0514511
$790$	0	0
$791$	11.4452	0.406945
$792$	−1.00000	−0.0355335
$793$	26.0000	0.923287
$794$	−15.6301	−0.554693
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−28.8904	−1.02335	−0.511676	−	0.859179i	$-0.670975\pi$
$797$	−28.8904	−1.02335	−0.511676	+	0.859179i	$0.670975\pi$
$798$	−8.44522	−0.298958
$799$	−8.00000	−0.283020
$800$	0	0
$801$	1.18493	0.0418673
$802$	32.4452	1.14568
$803$	4.81507	0.169920
$804$	−13.6301	−0.480698
$805$	0	0
$806$	26.0000	0.915811
$807$	0.815073	0.0286919
$808$	−9.63015	−0.338787
$809$	40.0754	1.40897	0.704487	−	0.709717i	$-0.251177\pi$
$809$	40.0754	1.40897	0.704487	+	0.709717i	$0.251177\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−11.4452	−0.401648
$813$	20.0000	0.701431
$814$	1.00000	0.0350500
$815$	0	0
$816$	1.00000	0.0350070
$817$	16.3699	0.572709
$818$	27.6301	0.966065
$819$	20.4452	0.714414
$820$	0	0
$821$	−42.0754	−1.46844	−0.734220	−	0.678911i	$-0.762452\pi$
$821$	−42.0754	−1.46844	−0.734220	+	0.678911i	$0.762452\pi$
$822$	2.00000	0.0697580
$823$	12.0754	0.420921	0.210460	−	0.977602i	$-0.432504\pi$
$823$	12.0754	0.420921	0.210460	+	0.977602i	$0.432504\pi$
$824$	0	0
$825$	0	0
$826$	−6.00000	−0.208767
$827$	−25.0754	−0.871956	−0.435978	−	0.899957i	$-0.643597\pi$
$827$	−25.0754	−0.871956	−0.435978	+	0.899957i	$0.643597\pi$
$828$	4.81507	0.167335
$829$	31.5206	1.09476	0.547378	−	0.836886i	$-0.315626\pi$
$829$	31.5206	1.09476	0.547378	+	0.836886i	$0.315626\pi$
$830$	0	0
$831$	−10.4452	−0.362341
$832$	−6.81507	−0.236270
$833$	−2.00000	−0.0692959
$834$	−12.1849	−0.421930
$835$	0	0
$836$	2.81507	0.0973613
$837$	3.81507	0.131868
$838$	6.44522	0.222646
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	−14.4452	−0.498111
$842$	13.2603	0.456980
$843$	−12.4452	−0.428636
$844$	−23.4452	−0.807018
$845$	0	0
$846$	8.00000	0.275046
$847$	30.0000	1.03081
$848$	10.6301	0.365041
$849$	−11.1849	−0.383866
$850$	0	0
$851$	−4.81507	−0.165059
$852$	−9.63015	−0.329923
$853$	−42.0754	−1.44063	−0.720317	−	0.693646i	$-0.756003\pi$
$853$	−42.0754	−1.44063	−0.720317	+	0.693646i	$0.756003\pi$
$854$	11.4452	0.391647
$855$	0	0
$856$	18.8151	0.643086
$857$	12.2603	0.418804	0.209402	−	0.977830i	$-0.432848\pi$
$857$	12.2603	0.418804	0.209402	+	0.977830i	$0.432848\pi$
$858$	−6.81507	−0.232663
$859$	−0.445219	−0.0151907	−0.00759533	−	0.999971i	$-0.502418\pi$
$859$	−0.445219	−0.0151907	−0.00759533	+	0.999971i	$0.502418\pi$
$860$	0	0
$861$	17.4452	0.594531
$862$	21.0000	0.715263
$863$	33.4452	1.13849	0.569244	−	0.822168i	$-0.307236\pi$
$863$	33.4452	1.13849	0.569244	+	0.822168i	$0.307236\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	8.81507	0.299548
$867$	16.0000	0.543388
$868$	11.4452	0.388476
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−92.8904	−3.14747
$872$	13.0000	0.440236
$873$	−5.81507	−0.196810
$874$	−13.5548	−0.458497
$875$	0	0
$876$	4.81507	0.162686
$877$	24.7055	0.834246	0.417123	−	0.908850i	$-0.363038\pi$
$877$	24.7055	0.834246	0.417123	+	0.908850i	$0.363038\pi$
$878$	13.4452	0.453754
$879$	−14.2603	−0.480988
$880$	0	0
$881$	53.8151	1.81308	0.906538	−	0.422124i	$-0.138715\pi$
$881$	53.8151	1.81308	0.906538	+	0.422124i	$0.138715\pi$
$882$	2.00000	0.0673435
$883$	−27.0000	−0.908622	−0.454311	−	0.890843i	$-0.650115\pi$
$883$	−27.0000	−0.908622	−0.454311	+	0.890843i	$0.650115\pi$
$884$	6.81507	0.229216
$885$	0	0
$886$	7.63015	0.256340
$887$	−39.0754	−1.31202	−0.656011	−	0.754751i	$-0.727758\pi$
$887$	−39.0754	−1.31202	−0.656011	+	0.754751i	$0.727758\pi$
$888$	1.00000	0.0335578
$889$	31.3357	1.05096
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	6.18493	0.207087
$893$	−22.5206	−0.753623
$894$	−2.00000	−0.0668900
$895$	0	0
$896$	−3.00000	−0.100223
$897$	32.8151	1.09566
$898$	−32.8904	−1.09757
$899$	−14.5548	−0.485429
$900$	0	0
$901$	−10.6301	−0.354142
$902$	−5.81507	−0.193621
$903$	−17.4452	−0.580541
$904$	−3.81507	−0.126887
$905$	0	0
$906$	6.81507	0.226416
$907$	35.1849	1.16830	0.584148	−	0.811647i	$-0.301429\pi$
$907$	35.1849	1.16830	0.584148	+	0.811647i	$0.301429\pi$
$908$	14.1849	0.470743
$909$	−9.63015	−0.319412
$910$	0	0
$911$	30.5206	1.01119	0.505596	−	0.862770i	$-0.331273\pi$
$911$	30.5206	1.01119	0.505596	+	0.862770i	$0.331273\pi$
$912$	2.81507	0.0932164
$913$	6.81507	0.225546
$914$	−25.0754	−0.829419
$915$	0	0
$916$	21.6301	0.714680
$917$	−34.8904	−1.15218
$918$	1.00000	0.0330049
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	−18.0000	−0.593120
$922$	24.1849	0.796488
$923$	−65.6301	−2.16024
$924$	−3.00000	−0.0986928
$925$	0	0
$926$	−3.63015	−0.119294
$927$	0	0
$928$	3.81507	0.125236
$929$	−7.44522	−0.244270	−0.122135	−	0.992514i	$-0.538974\pi$
$929$	−7.44522	−0.244270	−0.122135	+	0.992514i	$0.538974\pi$
$930$	0	0
$931$	−5.63015	−0.184521
$932$	−13.6301	−0.446470
$933$	21.4452	0.702085
$934$	−18.1849	−0.595029
$935$	0	0
$936$	−6.81507	−0.222758
$937$	−9.63015	−0.314603	−0.157302	−	0.987551i	$-0.550279\pi$
$937$	−9.63015	−0.314603	−0.157302	+	0.987551i	$0.550279\pi$
$938$	−40.8904	−1.33512
$939$	−30.8904	−1.00807
$940$	0	0
$941$	41.2603	1.34505	0.672524	−	0.740076i	$-0.265210\pi$
$941$	41.2603	1.34505	0.672524	+	0.740076i	$0.265210\pi$
$942$	−2.18493	−0.0711888
$943$	28.0000	0.911805
$944$	2.00000	0.0650945
$945$	0	0
$946$	5.81507	0.189064
$947$	57.4452	1.86672	0.933359	−	0.358943i	$-0.116863\pi$
$947$	57.4452	1.86672	0.933359	+	0.358943i	$0.116863\pi$
$948$	−12.0000	−0.389742
$949$	32.8151	1.06522
$950$	0	0
$951$	7.81507	0.253421
$952$	3.00000	0.0972306
$953$	48.8904	1.58372	0.791858	−	0.610705i	$-0.209114\pi$
$953$	48.8904	1.58372	0.791858	+	0.610705i	$0.209114\pi$
$954$	10.6301	0.344164
$955$	0	0
$956$	−25.4452	−0.822957
$957$	3.81507	0.123324
$958$	−0.815073	−0.0263338
$959$	6.00000	0.193750
$960$	0	0
$961$	−16.4452	−0.530491
$962$	6.81507	0.219727
$963$	18.8151	0.606307
$964$	−26.0000	−0.837404
$965$	0	0
$966$	14.4452	0.464767
$967$	13.6301	0.438316	0.219158	−	0.975689i	$-0.429669\pi$
$967$	13.6301	0.438316	0.219158	+	0.975689i	$0.429669\pi$
$968$	−10.0000	−0.321412
$969$	−2.81507	−0.0904332
$970$	0	0
$971$	5.07536	0.162876	0.0814381	−	0.996678i	$-0.474049\pi$
$971$	5.07536	0.162876	0.0814381	+	0.996678i	$0.474049\pi$
$972$	−1.00000	−0.0320750
$973$	−36.5548	−1.17189
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−3.81507	−0.122118
$977$	−21.0000	−0.671850	−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000	−0.671850	−0.335925	+	0.941889i	$0.609049\pi$
$978$	−4.63015	−0.148056
$979$	−1.18493	−0.0378704
$980$	0	0
$981$	13.0000	0.415058
$982$	12.8151	0.408945
$983$	38.3357	1.22272	0.611359	−	0.791354i	$-0.290623\pi$
$983$	38.3357	1.22272	0.611359	+	0.791354i	$0.290623\pi$
$984$	−5.81507	−0.185378
$985$	0	0
$986$	−3.81507	−0.121497
$987$	24.0000	0.763928
$988$	19.1849	0.610354
$989$	−28.0000	−0.890348
$990$	0	0
$991$	−6.55478	−0.208219	−0.104110	−	0.994566i	$-0.533199\pi$
$991$	−6.55478	−0.208219	−0.104110	+	0.994566i	$0.533199\pi$
$992$	−3.81507	−0.121129
$993$	−19.2603	−0.611207
$994$	−28.8904	−0.916349
$995$	0	0
$996$	6.81507	0.215944
$997$	−1.92464	−0.0609538	−0.0304769	−	0.999535i	$-0.509703\pi$
$997$	−1.92464	−0.0609538	−0.0304769	+	0.999535i	$0.509703\pi$
$998$	14.4452	0.457255
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5550.2.a.bx.1.1		2
5.4	even	2		1110.2.a.q.1.2	✓	2
15.14	odd	2		3330.2.a.bf.1.2		2
20.19	odd	2		8880.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.q.1.2	✓	2	5.4	even	2
3330.2.a.bf.1.2		2	15.14	odd	2
5550.2.a.bx.1.1		2	1.1	even	1	trivial
8880.2.a.bh.1.2		2	20.19	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:	\( N \)	\(=\)	\( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} -6.81507 q^{13} -3.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -2.81507 q^{19} +3.00000 q^{21} -1.00000 q^{22} +4.81507 q^{23} -1.00000 q^{24} -6.81507 q^{26} -1.00000 q^{27} -3.00000 q^{28} +3.81507 q^{29} -3.81507 q^{31} +1.00000 q^{32} +1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -2.81507 q^{38} +6.81507 q^{39} +5.81507 q^{41} +3.00000 q^{42} -5.81507 q^{43} -1.00000 q^{44} +4.81507 q^{46} +8.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} +1.00000 q^{51} -6.81507 q^{52} +10.6301 q^{53} -1.00000 q^{54} -3.00000 q^{56} +2.81507 q^{57} +3.81507 q^{58} +2.00000 q^{59} -3.81507 q^{61} -3.81507 q^{62} -3.00000 q^{63} +1.00000 q^{64} +1.00000 q^{66} +13.6301 q^{67} -1.00000 q^{68} -4.81507 q^{69} +9.63015 q^{71} +1.00000 q^{72} -4.81507 q^{73} -1.00000 q^{74} -2.81507 q^{76} +3.00000 q^{77} +6.81507 q^{78} +12.0000 q^{79} +1.00000 q^{81} +5.81507 q^{82} -6.81507 q^{83} +3.00000 q^{84} -5.81507 q^{86} -3.81507 q^{87} -1.00000 q^{88} +1.18493 q^{89} +20.4452 q^{91} +4.81507 q^{92} +3.81507 q^{93} +8.00000 q^{94} -1.00000 q^{96} -5.81507 q^{97} +2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} - 6 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{11} - 2 q^{12} - 3 q^{13} - 6 q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{18} + 5 q^{19} + 6 q^{21} - 2 q^{22} - q^{23} - 2 q^{24} - 3 q^{26} - 2 q^{27} - 6 q^{28} - 3 q^{29} + 3 q^{31} + 2 q^{32} + 2 q^{33} - 2 q^{34} + 2 q^{36} - 2 q^{37} + 5 q^{38} + 3 q^{39} + q^{41} + 6 q^{42} - q^{43} - 2 q^{44} - q^{46} + 16 q^{47} - 2 q^{48} + 4 q^{49} + 2 q^{51} - 3 q^{52} - 2 q^{54} - 6 q^{56} - 5 q^{57} - 3 q^{58} + 4 q^{59} + 3 q^{61} + 3 q^{62} - 6 q^{63} + 2 q^{64} + 2 q^{66} + 6 q^{67} - 2 q^{68} + q^{69} - 2 q^{71} + 2 q^{72} + q^{73} - 2 q^{74} + 5 q^{76} + 6 q^{77} + 3 q^{78} + 24 q^{79} + 2 q^{81} + q^{82} - 3 q^{83} + 6 q^{84} - q^{86} + 3 q^{87} - 2 q^{88} + 13 q^{89} + 9 q^{91} - q^{92} - 3 q^{93} + 16 q^{94} - 2 q^{96} - q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 - 6 * q^7 + 2 * q^8 + 2 * q^9 - 2 * q^11 - 2 * q^12 - 3 * q^13 - 6 * q^14 + 2 * q^16 - 2 * q^17 + 2 * q^18 + 5 * q^19 + 6 * q^21 - 2 * q^22 - q^23 - 2 * q^24 - 3 * q^26 - 2 * q^27 - 6 * q^28 - 3 * q^29 + 3 * q^31 + 2 * q^32 + 2 * q^33 - 2 * q^34 + 2 * q^36 - 2 * q^37 + 5 * q^38 + 3 * q^39 + q^41 + 6 * q^42 - q^43 - 2 * q^44 - q^46 + 16 * q^47 - 2 * q^48 + 4 * q^49 + 2 * q^51 - 3 * q^52 - 2 * q^54 - 6 * q^56 - 5 * q^57 - 3 * q^58 + 4 * q^59 + 3 * q^61 + 3 * q^62 - 6 * q^63 + 2 * q^64 + 2 * q^66 + 6 * q^67 - 2 * q^68 + q^69 - 2 * q^71 + 2 * q^72 + q^73 - 2 * q^74 + 5 * q^76 + 6 * q^77 + 3 * q^78 + 24 * q^79 + 2 * q^81 + q^82 - 3 * q^83 + 6 * q^84 - q^86 + 3 * q^87 - 2 * q^88 + 13 * q^89 + 9 * q^91 - q^92 - 3 * q^93 + 16 * q^94 - 2 * q^96 - q^97 + 4 * q^98 - 2 * q^99

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