LMFDB - Embedded newform 3450.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 690)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +6.47214 q^{13} +4.00000 q^{14} +1.00000 q^{16} -6.47214 q^{17} +1.00000 q^{18} +2.00000 q^{19} +4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} +6.47214 q^{26} +1.00000 q^{27} +4.00000 q^{28} +8.47214 q^{29} +1.00000 q^{32} -6.47214 q^{34} +1.00000 q^{36} -8.47214 q^{37} +2.00000 q^{38} +6.47214 q^{39} -6.94427 q^{41} +4.00000 q^{42} -1.00000 q^{46} -12.9443 q^{47} +1.00000 q^{48} +9.00000 q^{49} -6.47214 q^{51} +6.47214 q^{52} +8.94427 q^{53} +1.00000 q^{54} +4.00000 q^{56} +2.00000 q^{57} +8.47214 q^{58} -6.00000 q^{59} +8.47214 q^{61} +4.00000 q^{63} +1.00000 q^{64} +12.9443 q^{67} -6.47214 q^{68} -1.00000 q^{69} -16.4721 q^{71} +1.00000 q^{72} -12.9443 q^{73} -8.47214 q^{74} +2.00000 q^{76} +6.47214 q^{78} +3.52786 q^{79} +1.00000 q^{81} -6.94427 q^{82} +10.4721 q^{83} +4.00000 q^{84} +8.47214 q^{87} -7.52786 q^{89} +25.8885 q^{91} -1.00000 q^{92} -12.9443 q^{94} +1.00000 q^{96} +13.4164 q^{97} +9.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 8 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 + 8 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 8 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{12} + 4 q^{13} + 8 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 4 q^{19} + 8 q^{21} - 2 q^{23} + 2 q^{24} + 4 q^{26} + 2 q^{27} + 8 q^{28} + 8 q^{29} + 2 q^{32} - 4 q^{34} + 2 q^{36} - 8 q^{37} + 4 q^{38} + 4 q^{39} + 4 q^{41} + 8 q^{42} - 2 q^{46} - 8 q^{47} + 2 q^{48} + 18 q^{49} - 4 q^{51} + 4 q^{52} + 2 q^{54} + 8 q^{56} + 4 q^{57} + 8 q^{58} - 12 q^{59} + 8 q^{61} + 8 q^{63} + 2 q^{64} + 8 q^{67} - 4 q^{68} - 2 q^{69} - 24 q^{71} + 2 q^{72} - 8 q^{73} - 8 q^{74} + 4 q^{76} + 4 q^{78} + 16 q^{79} + 2 q^{81} + 4 q^{82} + 12 q^{83} + 8 q^{84} + 8 q^{87} - 24 q^{89} + 16 q^{91} - 2 q^{92} - 8 q^{94} + 2 q^{96} + 18 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 8 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^12 + 4 * q^13 + 8 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 4 * q^19 + 8 * q^21 - 2 * q^23 + 2 * q^24 + 4 * q^26 + 2 * q^27 + 8 * q^28 + 8 * q^29 + 2 * q^32 - 4 * q^34 + 2 * q^36 - 8 * q^37 + 4 * q^38 + 4 * q^39 + 4 * q^41 + 8 * q^42 - 2 * q^46 - 8 * q^47 + 2 * q^48 + 18 * q^49 - 4 * q^51 + 4 * q^52 + 2 * q^54 + 8 * q^56 + 4 * q^57 + 8 * q^58 - 12 * q^59 + 8 * q^61 + 8 * q^63 + 2 * q^64 + 8 * q^67 - 4 * q^68 - 2 * q^69 - 24 * q^71 + 2 * q^72 - 8 * q^73 - 8 * q^74 + 4 * q^76 + 4 * q^78 + 16 * q^79 + 2 * q^81 + 4 * q^82 + 12 * q^83 + 8 * q^84 + 8 * q^87 - 24 * q^89 + 16 * q^91 - 2 * q^92 - 8 * q^94 + 2 * q^96 + 18 * 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
$3$	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
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	6.47214	1.79505	0.897524	−	0.440966i	$-0.145364\pi$
$13$	6.47214	1.79505	0.897524	+	0.440966i	$0.145364\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.47214	−1.56972	−0.784862	−	0.619671i	$-0.787266\pi$
$17$	−6.47214	−1.56972	−0.784862	+	0.619671i	$0.787266\pi$
$18$	1.00000	0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	0	0
$26$	6.47214	1.26929
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.47214	−1.10996
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.47214	−1.39281	−0.696405	−	0.717649i	$-0.745218\pi$
$37$	−8.47214	−1.39281	−0.696405	+	0.717649i	$0.745218\pi$
$38$	2.00000	0.324443
$39$	6.47214	1.03637
$40$	0	0
$41$	−6.94427	−1.08451	−0.542257	−	0.840213i	$-0.682430\pi$
$41$	−6.94427	−1.08451	−0.542257	+	0.840213i	$0.682430\pi$
$42$	4.00000	0.617213
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−12.9443	−1.88812	−0.944058	−	0.329779i	$-0.893026\pi$
$47$	−12.9443	−1.88812	−0.944058	+	0.329779i	$0.893026\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	0	0
$51$	−6.47214	−0.906280
$52$	6.47214	0.897524
$53$	8.94427	1.22859	0.614295	−	0.789076i	$-0.289440\pi$
$53$	8.94427	1.22859	0.614295	+	0.789076i	$0.289440\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	4.00000	0.534522
$57$	2.00000	0.264906
$58$	8.47214	1.11245
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	8.47214	1.08475	0.542373	−	0.840138i	$-0.317526\pi$
$61$	8.47214	1.08475	0.542373	+	0.840138i	$0.317526\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	12.9443	1.58139	0.790697	−	0.612207i	$-0.209718\pi$
$67$	12.9443	1.58139	0.790697	+	0.612207i	$0.209718\pi$
$68$	−6.47214	−0.784862
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−16.4721	−1.95488	−0.977441	−	0.211207i	$-0.932261\pi$
$71$	−16.4721	−1.95488	−0.977441	+	0.211207i	$0.932261\pi$
$72$	1.00000	0.117851
$73$	−12.9443	−1.51501	−0.757506	−	0.652828i	$-0.773582\pi$
$73$	−12.9443	−1.51501	−0.757506	+	0.652828i	$0.773582\pi$
$74$	−8.47214	−0.984866
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	6.47214	0.732825
$79$	3.52786	0.396916	0.198458	−	0.980109i	$-0.436407\pi$
$79$	3.52786	0.396916	0.198458	+	0.980109i	$0.436407\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.94427	−0.766867
$83$	10.4721	1.14947	0.574733	−	0.818341i	$-0.305106\pi$
$83$	10.4721	1.14947	0.574733	+	0.818341i	$0.305106\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	0	0
$87$	8.47214	0.908308
$88$	0	0
$89$	−7.52786	−0.797952	−0.398976	−	0.916961i	$-0.630634\pi$
$89$	−7.52786	−0.797952	−0.398976	+	0.916961i	$0.630634\pi$
$90$	0	0
$91$	25.8885	2.71386
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−12.9443	−1.33510
$95$	0	0
$96$	1.00000	0.102062
$97$	13.4164	1.36223	0.681115	−	0.732177i	$-0.261495\pi$
$97$	13.4164	1.36223	0.681115	+	0.732177i	$0.261495\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	0	0
$101$	−8.47214	−0.843009	−0.421505	−	0.906826i	$-0.638498\pi$
$101$	−8.47214	−0.843009	−0.421505	+	0.906826i	$0.638498\pi$
$102$	−6.47214	−0.640837
$103$	−0.944272	−0.0930419	−0.0465209	−	0.998917i	$-0.514813\pi$
$103$	−0.944272	−0.0930419	−0.0465209	+	0.998917i	$0.514813\pi$
$104$	6.47214	0.634645
$105$	0	0
$106$	8.94427	0.868744
$107$	6.47214	0.625685	0.312842	−	0.949805i	$-0.398719\pi$
$107$	6.47214	0.625685	0.312842	+	0.949805i	$0.398719\pi$
$108$	1.00000	0.0962250
$109$	−8.47214	−0.811483	−0.405742	−	0.913988i	$-0.632987\pi$
$109$	−8.47214	−0.811483	−0.405742	+	0.913988i	$0.632987\pi$
$110$	0	0
$111$	−8.47214	−0.804140
$112$	4.00000	0.377964
$113$	−7.41641	−0.697677	−0.348838	−	0.937183i	$-0.613424\pi$
$113$	−7.41641	−0.697677	−0.348838	+	0.937183i	$0.613424\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	8.47214	0.786618
$117$	6.47214	0.598349
$118$	−6.00000	−0.552345
$119$	−25.8885	−2.37320
$120$	0	0
$121$	−11.0000	−1.00000
$122$	8.47214	0.767031
$123$	−6.94427	−0.626144
$124$	0	0
$125$	0	0
$126$	4.00000	0.356348
$127$	−15.4164	−1.36798	−0.683992	−	0.729489i	$-0.739758\pi$
$127$	−15.4164	−1.36798	−0.683992	+	0.729489i	$0.739758\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	10.9443	0.956205	0.478103	−	0.878304i	$-0.341325\pi$
$131$	10.9443	0.956205	0.478103	+	0.878304i	$0.341325\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	12.9443	1.11821
$135$	0	0
$136$	−6.47214	−0.554981
$137$	7.41641	0.633626	0.316813	−	0.948488i	$-0.397387\pi$
$137$	7.41641	0.633626	0.316813	+	0.948488i	$0.397387\pi$
$138$	−1.00000	−0.0851257
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−12.9443	−1.09010
$142$	−16.4721	−1.38231
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−12.9443	−1.07128
$147$	9.00000	0.742307
$148$	−8.47214	−0.696405
$149$	13.4164	1.09911	0.549557	−	0.835456i	$-0.314796\pi$
$149$	13.4164	1.09911	0.549557	+	0.835456i	$0.314796\pi$
$150$	0	0
$151$	−0.944272	−0.0768438	−0.0384219	−	0.999262i	$-0.512233\pi$
$151$	−0.944272	−0.0768438	−0.0384219	+	0.999262i	$0.512233\pi$
$152$	2.00000	0.162221
$153$	−6.47214	−0.523241
$154$	0	0
$155$	0	0
$156$	6.47214	0.518186
$157$	−20.4721	−1.63385	−0.816927	−	0.576741i	$-0.804324\pi$
$157$	−20.4721	−1.63385	−0.816927	+	0.576741i	$0.804324\pi$
$158$	3.52786	0.280662
$159$	8.94427	0.709327
$160$	0	0
$161$	−4.00000	−0.315244
$162$	1.00000	0.0785674
$163$	12.9443	1.01387	0.506937	−	0.861983i	$-0.330778\pi$
$163$	12.9443	1.01387	0.506937	+	0.861983i	$0.330778\pi$
$164$	−6.94427	−0.542257
$165$	0	0
$166$	10.4721	0.812795
$167$	20.9443	1.62072	0.810358	−	0.585935i	$-0.199273\pi$
$167$	20.9443	1.62072	0.810358	+	0.585935i	$0.199273\pi$
$168$	4.00000	0.308607
$169$	28.8885	2.22220
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	2.94427	0.223849	0.111924	−	0.993717i	$-0.464299\pi$
$173$	2.94427	0.223849	0.111924	+	0.993717i	$0.464299\pi$
$174$	8.47214	0.642271
$175$	0	0
$176$	0	0
$177$	−6.00000	−0.450988
$178$	−7.52786	−0.564237
$179$	−1.05573	−0.0789088	−0.0394544	−	0.999221i	$-0.512562\pi$
$179$	−1.05573	−0.0789088	−0.0394544	+	0.999221i	$0.512562\pi$
$180$	0	0
$181$	−0.472136	−0.0350936	−0.0175468	−	0.999846i	$-0.505586\pi$
$181$	−0.472136	−0.0350936	−0.0175468	+	0.999846i	$0.505586\pi$
$182$	25.8885	1.91899
$183$	8.47214	0.626278
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−12.9443	−0.944058
$189$	4.00000	0.290957
$190$	0	0
$191$	9.88854	0.715510	0.357755	−	0.933816i	$-0.383542\pi$
$191$	9.88854	0.715510	0.357755	+	0.933816i	$0.383542\pi$
$192$	1.00000	0.0721688
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	13.4164	0.963242
$195$	0	0
$196$	9.00000	0.642857
$197$	1.05573	0.0752175	0.0376088	−	0.999293i	$-0.488026\pi$
$197$	1.05573	0.0752175	0.0376088	+	0.999293i	$0.488026\pi$
$198$	0	0
$199$	−9.41641	−0.667511	−0.333756	−	0.942660i	$-0.608316\pi$
$199$	−9.41641	−0.667511	−0.333756	+	0.942660i	$0.608316\pi$
$200$	0	0
$201$	12.9443	0.913019
$202$	−8.47214	−0.596097
$203$	33.8885	2.37851
$204$	−6.47214	−0.453140
$205$	0	0
$206$	−0.944272	−0.0657905
$207$	−1.00000	−0.0695048
$208$	6.47214	0.448762
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	8.94427	0.614295
$213$	−16.4721	−1.12865
$214$	6.47214	0.442426
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	−8.47214	−0.573805
$219$	−12.9443	−0.874693
$220$	0	0
$221$	−41.8885	−2.81773
$222$	−8.47214	−0.568613
$223$	−15.4164	−1.03236	−0.516180	−	0.856480i	$-0.672646\pi$
$223$	−15.4164	−1.03236	−0.516180	+	0.856480i	$0.672646\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−7.41641	−0.493332
$227$	9.52786	0.632387	0.316193	−	0.948695i	$-0.397595\pi$
$227$	9.52786	0.632387	0.316193	+	0.948695i	$0.397595\pi$
$228$	2.00000	0.132453
$229$	−1.41641	−0.0935989	−0.0467994	−	0.998904i	$-0.514902\pi$
$229$	−1.41641	−0.0935989	−0.0467994	+	0.998904i	$0.514902\pi$
$230$	0	0
$231$	0	0
$232$	8.47214	0.556223
$233$	−2.94427	−0.192886	−0.0964428	−	0.995339i	$-0.530746\pi$
$233$	−2.94427	−0.192886	−0.0964428	+	0.995339i	$0.530746\pi$
$234$	6.47214	0.423097
$235$	0	0
$236$	−6.00000	−0.390567
$237$	3.52786	0.229159
$238$	−25.8885	−1.67811
$239$	−3.52786	−0.228199	−0.114099	−	0.993469i	$-0.536398\pi$
$239$	−3.52786	−0.228199	−0.114099	+	0.993469i	$0.536398\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−11.0000	−0.707107
$243$	1.00000	0.0641500
$244$	8.47214	0.542373
$245$	0	0
$246$	−6.94427	−0.442751
$247$	12.9443	0.823624
$248$	0	0
$249$	10.4721	0.663645
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	−15.4164	−0.967311
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−33.8885	−2.10573
$260$	0	0
$261$	8.47214	0.524412
$262$	10.9443	0.676139
$263$	3.05573	0.188424	0.0942121	−	0.995552i	$-0.469967\pi$
$263$	3.05573	0.188424	0.0942121	+	0.995552i	$0.469967\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	−7.52786	−0.460698
$268$	12.9443	0.790697
$269$	−8.47214	−0.516555	−0.258278	−	0.966071i	$-0.583155\pi$
$269$	−8.47214	−0.516555	−0.258278	+	0.966071i	$0.583155\pi$
$270$	0	0
$271$	−16.9443	−1.02929	−0.514646	−	0.857403i	$-0.672076\pi$
$271$	−16.9443	−1.02929	−0.514646	+	0.857403i	$0.672076\pi$
$272$	−6.47214	−0.392431
$273$	25.8885	1.56685
$274$	7.41641	0.448042
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−0.583592	−0.0350647	−0.0175323	−	0.999846i	$-0.505581\pi$
$277$	−0.583592	−0.0350647	−0.0175323	+	0.999846i	$0.505581\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.4721	0.744025	0.372013	−	0.928228i	$-0.378668\pi$
$281$	12.4721	0.744025	0.372013	+	0.928228i	$0.378668\pi$
$282$	−12.9443	−0.770820
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	−16.4721	−0.977441
$285$	0	0
$286$	0	0
$287$	−27.7771	−1.63963
$288$	1.00000	0.0589256
$289$	24.8885	1.46403
$290$	0	0
$291$	13.4164	0.786484
$292$	−12.9443	−0.757506
$293$	16.9443	0.989895	0.494947	−	0.868923i	$-0.335187\pi$
$293$	16.9443	0.989895	0.494947	+	0.868923i	$0.335187\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	−8.47214	−0.492433
$297$	0	0
$298$	13.4164	0.777192
$299$	−6.47214	−0.374293
$300$	0	0
$301$	0	0
$302$	−0.944272	−0.0543367
$303$	−8.47214	−0.486711
$304$	2.00000	0.114708
$305$	0	0
$306$	−6.47214	−0.369987
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−0.944272	−0.0537178
$310$	0	0
$311$	−15.5279	−0.880504	−0.440252	−	0.897874i	$-0.645111\pi$
$311$	−15.5279	−0.880504	−0.440252	+	0.897874i	$0.645111\pi$
$312$	6.47214	0.366413
$313$	1.41641	0.0800601	0.0400301	−	0.999198i	$-0.487255\pi$
$313$	1.41641	0.0800601	0.0400301	+	0.999198i	$0.487255\pi$
$314$	−20.4721	−1.15531
$315$	0	0
$316$	3.52786	0.198458
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	8.94427	0.501570
$319$	0	0
$320$	0	0
$321$	6.47214	0.361239
$322$	−4.00000	−0.222911
$323$	−12.9443	−0.720239
$324$	1.00000	0.0555556
$325$	0	0
$326$	12.9443	0.716917
$327$	−8.47214	−0.468510
$328$	−6.94427	−0.383433
$329$	−51.7771	−2.85456
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	10.4721	0.574733
$333$	−8.47214	−0.464270
$334$	20.9443	1.14602
$335$	0	0
$336$	4.00000	0.218218
$337$	−1.41641	−0.0771567	−0.0385783	−	0.999256i	$-0.512283\pi$
$337$	−1.41641	−0.0771567	−0.0385783	+	0.999256i	$0.512283\pi$
$338$	28.8885	1.57133
$339$	−7.41641	−0.402804
$340$	0	0
$341$	0	0
$342$	2.00000	0.108148
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	2.94427	0.158285
$347$	−8.94427	−0.480154	−0.240077	−	0.970754i	$-0.577173\pi$
$347$	−8.94427	−0.480154	−0.240077	+	0.970754i	$0.577173\pi$
$348$	8.47214	0.454154
$349$	7.88854	0.422264	0.211132	−	0.977458i	$-0.432285\pi$
$349$	7.88854	0.422264	0.211132	+	0.977458i	$0.432285\pi$
$350$	0	0
$351$	6.47214	0.345457
$352$	0	0
$353$	−14.9443	−0.795403	−0.397702	−	0.917515i	$-0.630192\pi$
$353$	−14.9443	−0.795403	−0.397702	+	0.917515i	$0.630192\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	−7.52786	−0.398976
$357$	−25.8885	−1.37017
$358$	−1.05573	−0.0557970
$359$	−15.0557	−0.794611	−0.397305	−	0.917686i	$-0.630055\pi$
$359$	−15.0557	−0.794611	−0.397305	+	0.917686i	$0.630055\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−0.472136	−0.0248149
$363$	−11.0000	−0.577350
$364$	25.8885	1.35693
$365$	0	0
$366$	8.47214	0.442846
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	−1.00000	−0.0521286
$369$	−6.94427	−0.361504
$370$	0	0
$371$	35.7771	1.85745
$372$	0	0
$373$	5.41641	0.280451	0.140225	−	0.990120i	$-0.455217\pi$
$373$	5.41641	0.280451	0.140225	+	0.990120i	$0.455217\pi$
$374$	0	0
$375$	0	0
$376$	−12.9443	−0.667550
$377$	54.8328	2.82403
$378$	4.00000	0.205738
$379$	−36.8328	−1.89197	−0.945987	−	0.324204i	$-0.894904\pi$
$379$	−36.8328	−1.89197	−0.945987	+	0.324204i	$0.894904\pi$
$380$	0	0
$381$	−15.4164	−0.789807
$382$	9.88854	0.505942
$383$	21.8885	1.11845	0.559226	−	0.829015i	$-0.311098\pi$
$383$	21.8885	1.11845	0.559226	+	0.829015i	$0.311098\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	20.0000	1.01797
$387$	0	0
$388$	13.4164	0.681115
$389$	−4.47214	−0.226746	−0.113373	−	0.993552i	$-0.536166\pi$
$389$	−4.47214	−0.226746	−0.113373	+	0.993552i	$0.536166\pi$
$390$	0	0
$391$	6.47214	0.327310
$392$	9.00000	0.454569
$393$	10.9443	0.552065
$394$	1.05573	0.0531868
$395$	0	0
$396$	0	0
$397$	−33.3050	−1.67153	−0.835764	−	0.549089i	$-0.814975\pi$
$397$	−33.3050	−1.67153	−0.835764	+	0.549089i	$0.814975\pi$
$398$	−9.41641	−0.472002
$399$	8.00000	0.400501
$400$	0	0
$401$	−1.41641	−0.0707320	−0.0353660	−	0.999374i	$-0.511260\pi$
$401$	−1.41641	−0.0707320	−0.0353660	+	0.999374i	$0.511260\pi$
$402$	12.9443	0.645602
$403$	0	0
$404$	−8.47214	−0.421505
$405$	0	0
$406$	33.8885	1.68186
$407$	0	0
$408$	−6.47214	−0.320418
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	7.41641	0.365824
$412$	−0.944272	−0.0465209
$413$	−24.0000	−1.18096
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	6.47214	0.317323
$417$	0	0
$418$	0	0
$419$	−2.11146	−0.103151	−0.0515757	−	0.998669i	$-0.516424\pi$
$419$	−2.11146	−0.103151	−0.0515757	+	0.998669i	$0.516424\pi$
$420$	0	0
$421$	−10.5836	−0.515813	−0.257906	−	0.966170i	$-0.583033\pi$
$421$	−10.5836	−0.515813	−0.257906	+	0.966170i	$0.583033\pi$
$422$	−12.0000	−0.584151
$423$	−12.9443	−0.629372
$424$	8.94427	0.434372
$425$	0	0
$426$	−16.4721	−0.798078
$427$	33.8885	1.63998
$428$	6.47214	0.312842
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	5.41641	0.260296	0.130148	−	0.991495i	$-0.458455\pi$
$433$	5.41641	0.260296	0.130148	+	0.991495i	$0.458455\pi$
$434$	0	0
$435$	0	0
$436$	−8.47214	−0.405742
$437$	−2.00000	−0.0956730
$438$	−12.9443	−0.618501
$439$	26.8328	1.28066	0.640330	−	0.768100i	$-0.278798\pi$
$439$	26.8328	1.28066	0.640330	+	0.768100i	$0.278798\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	−41.8885	−1.99243
$443$	8.94427	0.424955	0.212478	−	0.977166i	$-0.431847\pi$
$443$	8.94427	0.424955	0.212478	+	0.977166i	$0.431847\pi$
$444$	−8.47214	−0.402070
$445$	0	0
$446$	−15.4164	−0.729988
$447$	13.4164	0.634574
$448$	4.00000	0.188982
$449$	−17.0557	−0.804910	−0.402455	−	0.915440i	$-0.631843\pi$
$449$	−17.0557	−0.804910	−0.402455	+	0.915440i	$0.631843\pi$
$450$	0	0
$451$	0	0
$452$	−7.41641	−0.348838
$453$	−0.944272	−0.0443658
$454$	9.52786	0.447165
$455$	0	0
$456$	2.00000	0.0936586
$457$	−1.41641	−0.0662568	−0.0331284	−	0.999451i	$-0.510547\pi$
$457$	−1.41641	−0.0662568	−0.0331284	+	0.999451i	$0.510547\pi$
$458$	−1.41641	−0.0661844
$459$	−6.47214	−0.302093
$460$	0	0
$461$	30.3607	1.41404	0.707019	−	0.707195i	$-0.250040\pi$
$461$	30.3607	1.41404	0.707019	+	0.707195i	$0.250040\pi$
$462$	0	0
$463$	11.4164	0.530565	0.265283	−	0.964171i	$-0.414535\pi$
$463$	11.4164	0.530565	0.265283	+	0.964171i	$0.414535\pi$
$464$	8.47214	0.393309
$465$	0	0
$466$	−2.94427	−0.136391
$467$	17.5279	0.811093	0.405546	−	0.914074i	$-0.367081\pi$
$467$	17.5279	0.811093	0.405546	+	0.914074i	$0.367081\pi$
$468$	6.47214	0.299175
$469$	51.7771	2.39084
$470$	0	0
$471$	−20.4721	−0.943306
$472$	−6.00000	−0.276172
$473$	0	0
$474$	3.52786	0.162040
$475$	0	0
$476$	−25.8885	−1.18660
$477$	8.94427	0.409530
$478$	−3.52786	−0.161361
$479$	0.944272	0.0431449	0.0215724	−	0.999767i	$-0.493133\pi$
$479$	0.944272	0.0431449	0.0215724	+	0.999767i	$0.493133\pi$
$480$	0	0
$481$	−54.8328	−2.50016
$482$	−2.00000	−0.0910975
$483$	−4.00000	−0.182006
$484$	−11.0000	−0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	−4.58359	−0.207702	−0.103851	−	0.994593i	$-0.533117\pi$
$487$	−4.58359	−0.207702	−0.103851	+	0.994593i	$0.533117\pi$
$488$	8.47214	0.383516
$489$	12.9443	0.585360
$490$	0	0
$491$	−7.88854	−0.356005	−0.178002	−	0.984030i	$-0.556964\pi$
$491$	−7.88854	−0.356005	−0.178002	+	0.984030i	$0.556964\pi$
$492$	−6.94427	−0.313072
$493$	−54.8328	−2.46955
$494$	12.9443	0.582390
$495$	0	0
$496$	0	0
$497$	−65.8885	−2.95551
$498$	10.4721	0.469268
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	20.9443	0.935721
$502$	−12.0000	−0.535586
$503$	−21.8885	−0.975962	−0.487981	−	0.872854i	$-0.662266\pi$
$503$	−21.8885	−0.975962	−0.487981	+	0.872854i	$0.662266\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	0	0
$507$	28.8885	1.28299
$508$	−15.4164	−0.683992
$509$	−25.4164	−1.12656	−0.563281	−	0.826265i	$-0.690461\pi$
$509$	−25.4164	−1.12656	−0.563281	+	0.826265i	$0.690461\pi$
$510$	0	0
$511$	−51.7771	−2.29048
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	−18.0000	−0.793946
$515$	0	0
$516$	0	0
$517$	0	0
$518$	−33.8885	−1.48898
$519$	2.94427	0.129239
$520$	0	0
$521$	−28.4721	−1.24739	−0.623693	−	0.781669i	$-0.714369\pi$
$521$	−28.4721	−1.24739	−0.623693	+	0.781669i	$0.714369\pi$
$522$	8.47214	0.370815
$523$	−12.9443	−0.566013	−0.283007	−	0.959118i	$-0.591332\pi$
$523$	−12.9443	−0.566013	−0.283007	+	0.959118i	$0.591332\pi$
$524$	10.9443	0.478103
$525$	0	0
$526$	3.05573	0.133236
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.00000	−0.260378
$532$	8.00000	0.346844
$533$	−44.9443	−1.94675
$534$	−7.52786	−0.325763
$535$	0	0
$536$	12.9443	0.559107
$537$	−1.05573	−0.0455580
$538$	−8.47214	−0.365260
$539$	0	0
$540$	0	0
$541$	−12.8328	−0.551726	−0.275863	−	0.961197i	$-0.588964\pi$
$541$	−12.8328	−0.551726	−0.275863	+	0.961197i	$0.588964\pi$
$542$	−16.9443	−0.727819
$543$	−0.472136	−0.0202613
$544$	−6.47214	−0.277491
$545$	0	0
$546$	25.8885	1.10793
$547$	30.8328	1.31832	0.659158	−	0.752004i	$-0.270913\pi$
$547$	30.8328	1.31832	0.659158	+	0.752004i	$0.270913\pi$
$548$	7.41641	0.316813
$549$	8.47214	0.361582
$550$	0	0
$551$	16.9443	0.721850
$552$	−1.00000	−0.0425628
$553$	14.1115	0.600080
$554$	−0.583592	−0.0247945
$555$	0	0
$556$	0	0
$557$	5.88854	0.249506	0.124753	−	0.992188i	$-0.460186\pi$
$557$	5.88854	0.249506	0.124753	+	0.992188i	$0.460186\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	12.4721	0.526105
$563$	25.3050	1.06648	0.533238	−	0.845965i	$-0.320975\pi$
$563$	25.3050	1.06648	0.533238	+	0.845965i	$0.320975\pi$
$564$	−12.9443	−0.545052
$565$	0	0
$566$	24.0000	1.00880
$567$	4.00000	0.167984
$568$	−16.4721	−0.691155
$569$	−23.3050	−0.976994	−0.488497	−	0.872565i	$-0.662455\pi$
$569$	−23.3050	−0.976994	−0.488497	+	0.872565i	$0.662455\pi$
$570$	0	0
$571$	−20.8328	−0.871826	−0.435913	−	0.899989i	$-0.643575\pi$
$571$	−20.8328	−0.871826	−0.435913	+	0.899989i	$0.643575\pi$
$572$	0	0
$573$	9.88854	0.413100
$574$	−27.7771	−1.15939
$575$	0	0
$576$	1.00000	0.0416667
$577$	−0.944272	−0.0393106	−0.0196553	−	0.999807i	$-0.506257\pi$
$577$	−0.944272	−0.0393106	−0.0196553	+	0.999807i	$0.506257\pi$
$578$	24.8885	1.03523
$579$	20.0000	0.831172
$580$	0	0
$581$	41.8885	1.73783
$582$	13.4164	0.556128
$583$	0	0
$584$	−12.9443	−0.535638
$585$	0	0
$586$	16.9443	0.699961
$587$	5.88854	0.243046	0.121523	−	0.992589i	$-0.461222\pi$
$587$	5.88854	0.243046	0.121523	+	0.992589i	$0.461222\pi$
$588$	9.00000	0.371154
$589$	0	0
$590$	0	0
$591$	1.05573	0.0434269
$592$	−8.47214	−0.348203
$593$	−11.8885	−0.488204	−0.244102	−	0.969750i	$-0.578493\pi$
$593$	−11.8885	−0.488204	−0.244102	+	0.969750i	$0.578493\pi$
$594$	0	0
$595$	0	0
$596$	13.4164	0.549557
$597$	−9.41641	−0.385388
$598$	−6.47214	−0.264665
$599$	−18.3607	−0.750197	−0.375099	−	0.926985i	$-0.622391\pi$
$599$	−18.3607	−0.750197	−0.375099	+	0.926985i	$0.622391\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	12.9443	0.527132
$604$	−0.944272	−0.0384219
$605$	0	0
$606$	−8.47214	−0.344157
$607$	−19.4164	−0.788088	−0.394044	−	0.919092i	$-0.628924\pi$
$607$	−19.4164	−0.788088	−0.394044	+	0.919092i	$0.628924\pi$
$608$	2.00000	0.0811107
$609$	33.8885	1.37323
$610$	0	0
$611$	−83.7771	−3.38926
$612$	−6.47214	−0.261621
$613$	−17.4164	−0.703442	−0.351721	−	0.936105i	$-0.614403\pi$
$613$	−17.4164	−0.703442	−0.351721	+	0.936105i	$0.614403\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.52786	0.222543	0.111272	−	0.993790i	$-0.464508\pi$
$617$	5.52786	0.222543	0.111272	+	0.993790i	$0.464508\pi$
$618$	−0.944272	−0.0379842
$619$	−12.8328	−0.515794	−0.257897	−	0.966172i	$-0.583030\pi$
$619$	−12.8328	−0.515794	−0.257897	+	0.966172i	$0.583030\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	−15.5279	−0.622611
$623$	−30.1115	−1.20639
$624$	6.47214	0.259093
$625$	0	0
$626$	1.41641	0.0566110
$627$	0	0
$628$	−20.4721	−0.816927
$629$	54.8328	2.18633
$630$	0	0
$631$	42.3607	1.68635	0.843176	−	0.537638i	$-0.180683\pi$
$631$	42.3607	1.68635	0.843176	+	0.537638i	$0.180683\pi$
$632$	3.52786	0.140331
$633$	−12.0000	−0.476957
$634$	18.0000	0.714871
$635$	0	0
$636$	8.94427	0.354663
$637$	58.2492	2.30792
$638$	0	0
$639$	−16.4721	−0.651628
$640$	0	0
$641$	−20.4721	−0.808601	−0.404300	−	0.914626i	$-0.632485\pi$
$641$	−20.4721	−0.808601	−0.404300	+	0.914626i	$0.632485\pi$
$642$	6.47214	0.255435
$643$	−4.94427	−0.194983	−0.0974915	−	0.995236i	$-0.531082\pi$
$643$	−4.94427	−0.194983	−0.0974915	+	0.995236i	$0.531082\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	−12.9443	−0.509286
$647$	−6.11146	−0.240266	−0.120133	−	0.992758i	$-0.538332\pi$
$647$	−6.11146	−0.240266	−0.120133	+	0.992758i	$0.538332\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	0	0
$652$	12.9443	0.506937
$653$	−32.8328	−1.28485	−0.642424	−	0.766350i	$-0.722071\pi$
$653$	−32.8328	−1.28485	−0.642424	+	0.766350i	$0.722071\pi$
$654$	−8.47214	−0.331287
$655$	0	0
$656$	−6.94427	−0.271128
$657$	−12.9443	−0.505004
$658$	−51.7771	−2.01848
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	44.2492	1.72110	0.860548	−	0.509370i	$-0.170121\pi$
$661$	44.2492	1.72110	0.860548	+	0.509370i	$0.170121\pi$
$662$	0	0
$663$	−41.8885	−1.62682
$664$	10.4721	0.406398
$665$	0	0
$666$	−8.47214	−0.328289
$667$	−8.47214	−0.328042
$668$	20.9443	0.810358
$669$	−15.4164	−0.596033
$670$	0	0
$671$	0	0
$672$	4.00000	0.154303
$673$	14.8328	0.571763	0.285882	−	0.958265i	$-0.407714\pi$
$673$	14.8328	0.571763	0.285882	+	0.958265i	$0.407714\pi$
$674$	−1.41641	−0.0545580
$675$	0	0
$676$	28.8885	1.11110
$677$	9.88854	0.380048	0.190024	−	0.981779i	$-0.439143\pi$
$677$	9.88854	0.380048	0.190024	+	0.981779i	$0.439143\pi$
$678$	−7.41641	−0.284825
$679$	53.6656	2.05950
$680$	0	0
$681$	9.52786	0.365109
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	8.00000	0.305441
$687$	−1.41641	−0.0540393
$688$	0	0
$689$	57.8885	2.20538
$690$	0	0
$691$	31.7771	1.20886	0.604429	−	0.796659i	$-0.293401\pi$
$691$	31.7771	1.20886	0.604429	+	0.796659i	$0.293401\pi$
$692$	2.94427	0.111924
$693$	0	0
$694$	−8.94427	−0.339520
$695$	0	0
$696$	8.47214	0.321135
$697$	44.9443	1.70239
$698$	7.88854	0.298586
$699$	−2.94427	−0.111363
$700$	0	0
$701$	10.5836	0.399737	0.199868	−	0.979823i	$-0.435949\pi$
$701$	10.5836	0.399737	0.199868	+	0.979823i	$0.435949\pi$
$702$	6.47214	0.244275
$703$	−16.9443	−0.639065
$704$	0	0
$705$	0	0
$706$	−14.9443	−0.562435
$707$	−33.8885	−1.27451
$708$	−6.00000	−0.225494
$709$	11.5279	0.432938	0.216469	−	0.976289i	$-0.430546\pi$
$709$	11.5279	0.432938	0.216469	+	0.976289i	$0.430546\pi$
$710$	0	0
$711$	3.52786	0.132305
$712$	−7.52786	−0.282119
$713$	0	0
$714$	−25.8885	−0.968854
$715$	0	0
$716$	−1.05573	−0.0394544
$717$	−3.52786	−0.131750
$718$	−15.0557	−0.561875
$719$	−44.4721	−1.65853	−0.829265	−	0.558855i	$-0.811241\pi$
$719$	−44.4721	−1.65853	−0.829265	+	0.558855i	$0.811241\pi$
$720$	0	0
$721$	−3.77709	−0.140666
$722$	−15.0000	−0.558242
$723$	−2.00000	−0.0743808
$724$	−0.472136	−0.0175468
$725$	0	0
$726$	−11.0000	−0.408248
$727$	−16.9443	−0.628428	−0.314214	−	0.949352i	$-0.601741\pi$
$727$	−16.9443	−0.628428	−0.314214	+	0.949352i	$0.601741\pi$
$728$	25.8885	0.959493
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	8.47214	0.313139
$733$	8.47214	0.312925	0.156463	−	0.987684i	$-0.449991\pi$
$733$	8.47214	0.312925	0.156463	+	0.987684i	$0.449991\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	−6.94427	−0.255622
$739$	−33.8885	−1.24661	−0.623305	−	0.781979i	$-0.714211\pi$
$739$	−33.8885	−1.24661	−0.623305	+	0.781979i	$0.714211\pi$
$740$	0	0
$741$	12.9443	0.475520
$742$	35.7771	1.31342
$743$	37.8885	1.39000	0.694998	−	0.719012i	$-0.255405\pi$
$743$	37.8885	1.39000	0.694998	+	0.719012i	$0.255405\pi$
$744$	0	0
$745$	0	0
$746$	5.41641	0.198309
$747$	10.4721	0.383155
$748$	0	0
$749$	25.8885	0.945947
$750$	0	0
$751$	27.5279	1.00451	0.502253	−	0.864721i	$-0.332505\pi$
$751$	27.5279	1.00451	0.502253	+	0.864721i	$0.332505\pi$
$752$	−12.9443	−0.472029
$753$	−12.0000	−0.437304
$754$	54.8328	1.99689
$755$	0	0
$756$	4.00000	0.145479
$757$	18.5836	0.675432	0.337716	−	0.941248i	$-0.390346\pi$
$757$	18.5836	0.675432	0.337716	+	0.941248i	$0.390346\pi$
$758$	−36.8328	−1.33783
$759$	0	0
$760$	0	0
$761$	53.7771	1.94942	0.974709	−	0.223478i	$-0.0717411\pi$
$761$	53.7771	1.94942	0.974709	+	0.223478i	$0.0717411\pi$
$762$	−15.4164	−0.558478
$763$	−33.8885	−1.22685
$764$	9.88854	0.357755
$765$	0	0
$766$	21.8885	0.790865
$767$	−38.8328	−1.40217
$768$	1.00000	0.0360844
$769$	17.0557	0.615045	0.307523	−	0.951541i	$-0.400500\pi$
$769$	17.0557	0.615045	0.307523	+	0.951541i	$0.400500\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	20.0000	0.719816
$773$	−20.9443	−0.753313	−0.376657	−	0.926353i	$-0.622926\pi$
$773$	−20.9443	−0.753313	−0.376657	+	0.926353i	$0.622926\pi$
$774$	0	0
$775$	0	0
$776$	13.4164	0.481621
$777$	−33.8885	−1.21574
$778$	−4.47214	−0.160334
$779$	−13.8885	−0.497609
$780$	0	0
$781$	0	0
$782$	6.47214	0.231443
$783$	8.47214	0.302769
$784$	9.00000	0.321429
$785$	0	0
$786$	10.9443	0.390369
$787$	−54.8328	−1.95458	−0.977289	−	0.211909i	$-0.932032\pi$
$787$	−54.8328	−1.95458	−0.977289	+	0.211909i	$0.932032\pi$
$788$	1.05573	0.0376088
$789$	3.05573	0.108787
$790$	0	0
$791$	−29.6656	−1.05479
$792$	0	0
$793$	54.8328	1.94717
$794$	−33.3050	−1.18195
$795$	0	0
$796$	−9.41641	−0.333756
$797$	−2.11146	−0.0747916	−0.0373958	−	0.999301i	$-0.511906\pi$
$797$	−2.11146	−0.0747916	−0.0373958	+	0.999301i	$0.511906\pi$
$798$	8.00000	0.283197
$799$	83.7771	2.96382
$800$	0	0
$801$	−7.52786	−0.265984
$802$	−1.41641	−0.0500151
$803$	0	0
$804$	12.9443	0.456509
$805$	0	0
$806$	0	0
$807$	−8.47214	−0.298233
$808$	−8.47214	−0.298049
$809$	31.8885	1.12114	0.560571	−	0.828107i	$-0.310582\pi$
$809$	31.8885	1.12114	0.560571	+	0.828107i	$0.310582\pi$
$810$	0	0
$811$	−14.1115	−0.495520	−0.247760	−	0.968821i	$-0.579694\pi$
$811$	−14.1115	−0.495520	−0.247760	+	0.968821i	$0.579694\pi$
$812$	33.8885	1.18925
$813$	−16.9443	−0.594262
$814$	0	0
$815$	0	0
$816$	−6.47214	−0.226570
$817$	0	0
$818$	−18.0000	−0.629355
$819$	25.8885	0.904619
$820$	0	0
$821$	−7.52786	−0.262724	−0.131362	−	0.991334i	$-0.541935\pi$
$821$	−7.52786	−0.262724	−0.131362	+	0.991334i	$0.541935\pi$
$822$	7.41641	0.258677
$823$	−24.3607	−0.849160	−0.424580	−	0.905390i	$-0.639578\pi$
$823$	−24.3607	−0.849160	−0.424580	+	0.905390i	$0.639578\pi$
$824$	−0.944272	−0.0328953
$825$	0	0
$826$	−24.0000	−0.835067
$827$	−33.5279	−1.16588	−0.582939	−	0.812516i	$-0.698097\pi$
$827$	−33.5279	−1.16588	−0.582939	+	0.812516i	$0.698097\pi$
$828$	−1.00000	−0.0347524
$829$	13.0557	0.453444	0.226722	−	0.973959i	$-0.427199\pi$
$829$	13.0557	0.453444	0.226722	+	0.973959i	$0.427199\pi$
$830$	0	0
$831$	−0.583592	−0.0202446
$832$	6.47214	0.224381
$833$	−58.2492	−2.01822
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−2.11146	−0.0729390
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	−10.5836	−0.364735
$843$	12.4721	0.429563
$844$	−12.0000	−0.413057
$845$	0	0
$846$	−12.9443	−0.445033
$847$	−44.0000	−1.51186
$848$	8.94427	0.307148
$849$	24.0000	0.823678
$850$	0	0
$851$	8.47214	0.290421
$852$	−16.4721	−0.564326
$853$	18.4721	0.632474	0.316237	−	0.948680i	$-0.397581\pi$
$853$	18.4721	0.632474	0.316237	+	0.948680i	$0.397581\pi$
$854$	33.8885	1.15964
$855$	0	0
$856$	6.47214	0.221213
$857$	11.8885	0.406105	0.203052	−	0.979168i	$-0.434914\pi$
$857$	11.8885	0.406105	0.203052	+	0.979168i	$0.434914\pi$
$858$	0	0
$859$	31.7771	1.08422	0.542110	−	0.840307i	$-0.317626\pi$
$859$	31.7771	1.08422	0.542110	+	0.840307i	$0.317626\pi$
$860$	0	0
$861$	−27.7771	−0.946641
$862$	0	0
$863$	20.9443	0.712951	0.356476	−	0.934305i	$-0.383978\pi$
$863$	20.9443	0.712951	0.356476	+	0.934305i	$0.383978\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	5.41641	0.184057
$867$	24.8885	0.845259
$868$	0	0
$869$	0	0
$870$	0	0
$871$	83.7771	2.83868
$872$	−8.47214	−0.286903
$873$	13.4164	0.454077
$874$	−2.00000	−0.0676510
$875$	0	0
$876$	−12.9443	−0.437346
$877$	−24.3607	−0.822602	−0.411301	−	0.911500i	$-0.634925\pi$
$877$	−24.3607	−0.822602	−0.411301	+	0.911500i	$0.634925\pi$
$878$	26.8328	0.905564
$879$	16.9443	0.571516
$880$	0	0
$881$	4.47214	0.150670	0.0753350	−	0.997158i	$-0.475997\pi$
$881$	4.47214	0.150670	0.0753350	+	0.997158i	$0.475997\pi$
$882$	9.00000	0.303046
$883$	51.7771	1.74244	0.871219	−	0.490895i	$-0.163330\pi$
$883$	51.7771	1.74244	0.871219	+	0.490895i	$0.163330\pi$
$884$	−41.8885	−1.40886
$885$	0	0
$886$	8.94427	0.300489
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−8.47214	−0.284306
$889$	−61.6656	−2.06820
$890$	0	0
$891$	0	0
$892$	−15.4164	−0.516180
$893$	−25.8885	−0.866327
$894$	13.4164	0.448712
$895$	0	0
$896$	4.00000	0.133631
$897$	−6.47214	−0.216098
$898$	−17.0557	−0.569157
$899$	0	0
$900$	0	0
$901$	−57.8885	−1.92855
$902$	0	0
$903$	0	0
$904$	−7.41641	−0.246666
$905$	0	0
$906$	−0.944272	−0.0313713
$907$	−38.8328	−1.28942	−0.644711	−	0.764426i	$-0.723022\pi$
$907$	−38.8328	−1.28942	−0.644711	+	0.764426i	$0.723022\pi$
$908$	9.52786	0.316193
$909$	−8.47214	−0.281003
$910$	0	0
$911$	7.05573	0.233767	0.116883	−	0.993146i	$-0.462710\pi$
$911$	7.05573	0.233767	0.116883	+	0.993146i	$0.462710\pi$
$912$	2.00000	0.0662266
$913$	0	0
$914$	−1.41641	−0.0468506
$915$	0	0
$916$	−1.41641	−0.0467994
$917$	43.7771	1.44565
$918$	−6.47214	−0.213612
$919$	17.4164	0.574514	0.287257	−	0.957854i	$-0.407257\pi$
$919$	17.4164	0.574514	0.287257	+	0.957854i	$0.407257\pi$
$920$	0	0
$921$	0	0
$922$	30.3607	0.999876
$923$	−106.610	−3.50911
$924$	0	0
$925$	0	0
$926$	11.4164	0.375166
$927$	−0.944272	−0.0310140
$928$	8.47214	0.278111
$929$	6.94427	0.227834	0.113917	−	0.993490i	$-0.463660\pi$
$929$	6.94427	0.227834	0.113917	+	0.993490i	$0.463660\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	−2.94427	−0.0964428
$933$	−15.5279	−0.508359
$934$	17.5279	0.573529
$935$	0	0
$936$	6.47214	0.211548
$937$	33.4164	1.09167	0.545833	−	0.837894i	$-0.316213\pi$
$937$	33.4164	1.09167	0.545833	+	0.837894i	$0.316213\pi$
$938$	51.7771	1.69058
$939$	1.41641	0.0462227
$940$	0	0
$941$	27.5279	0.897383	0.448691	−	0.893687i	$-0.351890\pi$
$941$	27.5279	0.897383	0.448691	+	0.893687i	$0.351890\pi$
$942$	−20.4721	−0.667018
$943$	6.94427	0.226137
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	3.52786	0.114580
$949$	−83.7771	−2.71952
$950$	0	0
$951$	18.0000	0.583690
$952$	−25.8885	−0.839053
$953$	32.3607	1.04827	0.524133	−	0.851637i	$-0.324390\pi$
$953$	32.3607	1.04827	0.524133	+	0.851637i	$0.324390\pi$
$954$	8.94427	0.289581
$955$	0	0
$956$	−3.52786	−0.114099
$957$	0	0
$958$	0.944272	0.0305080
$959$	29.6656	0.957953
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−54.8328	−1.76788
$963$	6.47214	0.208562
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	−4.00000	−0.128698
$967$	28.5836	0.919186	0.459593	−	0.888130i	$-0.347995\pi$
$967$	28.5836	0.919186	0.459593	+	0.888130i	$0.347995\pi$
$968$	−11.0000	−0.353553
$969$	−12.9443	−0.415830
$970$	0	0
$971$	−53.8885	−1.72937	−0.864683	−	0.502318i	$-0.832481\pi$
$971$	−53.8885	−1.72937	−0.864683	+	0.502318i	$0.832481\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−4.58359	−0.146868
$975$	0	0
$976$	8.47214	0.271186
$977$	10.4721	0.335033	0.167517	−	0.985869i	$-0.446425\pi$
$977$	10.4721	0.335033	0.167517	+	0.985869i	$0.446425\pi$
$978$	12.9443	0.413912
$979$	0	0
$980$	0	0
$981$	−8.47214	−0.270494
$982$	−7.88854	−0.251734
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−6.94427	−0.221375
$985$	0	0
$986$	−54.8328	−1.74623
$987$	−51.7771	−1.64808
$988$	12.9443	0.411812
$989$	0	0
$990$	0	0
$991$	−53.6656	−1.70474	−0.852372	−	0.522935i	$-0.824837\pi$
$991$	−53.6656	−1.70474	−0.852372	+	0.522935i	$0.824837\pi$
$992$	0	0
$993$	0	0
$994$	−65.8885	−2.08986
$995$	0	0
$996$	10.4721	0.331822
$997$	19.4164	0.614924	0.307462	−	0.951560i	$-0.400520\pi$
$997$	19.4164	0.614924	0.307462	+	0.951560i	$0.400520\pi$
$998$	−24.0000	−0.759707
$999$	−8.47214	−0.268047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.bn.1.2		2
5.2	odd	4		690.2.d.a.139.3	yes	4
5.3	odd	4		690.2.d.a.139.1	✓	4
5.4	even	2		3450.2.a.bc.1.1		2
15.2	even	4		2070.2.d.b.829.2		4
15.8	even	4		2070.2.d.b.829.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.d.a.139.1	✓	4	5.3	odd	4
690.2.d.a.139.3	yes	4	5.2	odd	4
2070.2.d.b.829.2		4	15.2	even	4
2070.2.d.b.829.4		4	15.8	even	4
3450.2.a.bc.1.1		2	5.4	even	2
3450.2.a.bn.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +6.47214 q^{13} +4.00000 q^{14} +1.00000 q^{16} -6.47214 q^{17} +1.00000 q^{18} +2.00000 q^{19} +4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} +6.47214 q^{26} +1.00000 q^{27} +4.00000 q^{28} +8.47214 q^{29} +1.00000 q^{32} -6.47214 q^{34} +1.00000 q^{36} -8.47214 q^{37} +2.00000 q^{38} +6.47214 q^{39} -6.94427 q^{41} +4.00000 q^{42} -1.00000 q^{46} -12.9443 q^{47} +1.00000 q^{48} +9.00000 q^{49} -6.47214 q^{51} +6.47214 q^{52} +8.94427 q^{53} +1.00000 q^{54} +4.00000 q^{56} +2.00000 q^{57} +8.47214 q^{58} -6.00000 q^{59} +8.47214 q^{61} +4.00000 q^{63} +1.00000 q^{64} +12.9443 q^{67} -6.47214 q^{68} -1.00000 q^{69} -16.4721 q^{71} +1.00000 q^{72} -12.9443 q^{73} -8.47214 q^{74} +2.00000 q^{76} +6.47214 q^{78} +3.52786 q^{79} +1.00000 q^{81} -6.94427 q^{82} +10.4721 q^{83} +4.00000 q^{84} +8.47214 q^{87} -7.52786 q^{89} +25.8885 q^{91} -1.00000 q^{92} -12.9443 q^{94} +1.00000 q^{96} +13.4164 q^{97} +9.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 8 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 8 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 8 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{12} + 4 q^{13} + 8 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 4 q^{19} + 8 q^{21} - 2 q^{23} + 2 q^{24} + 4 q^{26} + 2 q^{27} + 8 q^{28} + 8 q^{29} + 2 q^{32} - 4 q^{34} + 2 q^{36} - 8 q^{37} + 4 q^{38} + 4 q^{39} + 4 q^{41} + 8 q^{42} - 2 q^{46} - 8 q^{47} + 2 q^{48} + 18 q^{49} - 4 q^{51} + 4 q^{52} + 2 q^{54} + 8 q^{56} + 4 q^{57} + 8 q^{58} - 12 q^{59} + 8 q^{61} + 8 q^{63} + 2 q^{64} + 8 q^{67} - 4 q^{68} - 2 q^{69} - 24 q^{71} + 2 q^{72} - 8 q^{73} - 8 q^{74} + 4 q^{76} + 4 q^{78} + 16 q^{79} + 2 q^{81} + 4 q^{82} + 12 q^{83} + 8 q^{84} + 8 q^{87} - 24 q^{89} + 16 q^{91} - 2 q^{92} - 8 q^{94} + 2 q^{96} + 18 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 8 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^12 + 4 * q^13 + 8 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^18 + 4 * q^19 + 8 * q^21 - 2 * q^23 + 2 * q^24 + 4 * q^26 + 2 * q^27 + 8 * q^28 + 8 * q^29 + 2 * q^32 - 4 * q^34 + 2 * q^36 - 8 * q^37 + 4 * q^38 + 4 * q^39 + 4 * q^41 + 8 * q^42 - 2 * q^46 - 8 * q^47 + 2 * q^48 + 18 * q^49 - 4 * q^51 + 4 * q^52 + 2 * q^54 + 8 * q^56 + 4 * q^57 + 8 * q^58 - 12 * q^59 + 8 * q^61 + 8 * q^63 + 2 * q^64 + 8 * q^67 - 4 * q^68 - 2 * q^69 - 24 * q^71 + 2 * q^72 - 8 * q^73 - 8 * q^74 + 4 * q^76 + 4 * q^78 + 16 * q^79 + 2 * q^81 + 4 * q^82 + 12 * q^83 + 8 * q^84 + 8 * q^87 - 24 * q^89 + 16 * q^91 - 2 * q^92 - 8 * q^94 + 2 * q^96 + 18 * q^98

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