LMFDB - Embedded newform 3450.2.a.bn.1.1

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			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -2.47214 q^{13} +4.00000 q^{14} +1.00000 q^{16} +2.47214 q^{17} +1.00000 q^{18} +2.00000 q^{19} +4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} -2.47214 q^{26} +1.00000 q^{27} +4.00000 q^{28} -0.472136 q^{29} +1.00000 q^{32} +2.47214 q^{34} +1.00000 q^{36} +0.472136 q^{37} +2.00000 q^{38} -2.47214 q^{39} +10.9443 q^{41} +4.00000 q^{42} -1.00000 q^{46} +4.94427 q^{47} +1.00000 q^{48} +9.00000 q^{49} +2.47214 q^{51} -2.47214 q^{52} -8.94427 q^{53} +1.00000 q^{54} +4.00000 q^{56} +2.00000 q^{57} -0.472136 q^{58} -6.00000 q^{59} -0.472136 q^{61} +4.00000 q^{63} +1.00000 q^{64} -4.94427 q^{67} +2.47214 q^{68} -1.00000 q^{69} -7.52786 q^{71} +1.00000 q^{72} +4.94427 q^{73} +0.472136 q^{74} +2.00000 q^{76} -2.47214 q^{78} +12.4721 q^{79} +1.00000 q^{81} +10.9443 q^{82} +1.52786 q^{83} +4.00000 q^{84} -0.472136 q^{87} -16.4721 q^{89} -9.88854 q^{91} -1.00000 q^{92} +4.94427 q^{94} +1.00000 q^{96} -13.4164 q^{97} +9.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	−2.47214	−0.685647	−0.342824	−	0.939400i	$-0.611383\pi$
$13$	−2.47214	−0.685647	−0.342824	+	0.939400i	$0.611383\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	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
$26$	−2.47214	−0.484826
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	−0.472136	−0.0876734	−0.0438367	−	0.999039i	$-0.513958\pi$
$29$	−0.472136	−0.0876734	−0.0438367	+	0.999039i	$0.513958\pi$
$30$	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
$34$	2.47214	0.423968
$35$	0	0
$36$	1.00000	0.166667
$37$	0.472136	0.0776187	0.0388093	−	0.999247i	$-0.487644\pi$
$37$	0.472136	0.0776187	0.0388093	+	0.999247i	$0.487644\pi$
$38$	2.00000	0.324443
$39$	−2.47214	−0.395859
$40$	0	0
$41$	10.9443	1.70921	0.854604	−	0.519280i	$-0.173800\pi$
$41$	10.9443	1.70921	0.854604	+	0.519280i	$0.173800\pi$
$42$	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
$47$	4.94427	0.721196	0.360598	−	0.932721i	$-0.382573\pi$
$47$	4.94427	0.721196	0.360598	+	0.932721i	$0.382573\pi$
$48$	1.00000	0.144338
$49$	9.00000	1.28571
$50$	0	0
$51$	2.47214	0.346168
$52$	−2.47214	−0.342824
$53$	−8.94427	−1.22859	−0.614295	−	0.789076i	$-0.710560\pi$
$53$	−8.94427	−1.22859	−0.614295	+	0.789076i	$0.710560\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	4.00000	0.534522
$57$	2.00000	0.264906
$58$	−0.472136	−0.0619945
$59$	−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
$61$	−0.472136	−0.0604508	−0.0302254	−	0.999543i	$-0.509623\pi$
$61$	−0.472136	−0.0604508	−0.0302254	+	0.999543i	$0.509623\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.94427	−0.604039	−0.302019	−	0.953302i	$-0.597661\pi$
$67$	−4.94427	−0.604039	−0.302019	+	0.953302i	$0.597661\pi$
$68$	2.47214	0.299791
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−7.52786	−0.893393	−0.446697	−	0.894686i	$-0.647400\pi$
$71$	−7.52786	−0.893393	−0.446697	+	0.894686i	$0.647400\pi$
$72$	1.00000	0.117851
$73$	4.94427	0.578683	0.289342	−	0.957226i	$-0.406564\pi$
$73$	4.94427	0.578683	0.289342	+	0.957226i	$0.406564\pi$
$74$	0.472136	0.0548847
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	−2.47214	−0.279914
$79$	12.4721	1.40322	0.701612	−	0.712559i	$-0.252464\pi$
$79$	12.4721	1.40322	0.701612	+	0.712559i	$0.252464\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.9443	1.20859
$83$	1.52786	0.167705	0.0838524	−	0.996478i	$-0.473278\pi$
$83$	1.52786	0.167705	0.0838524	+	0.996478i	$0.473278\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	0	0
$87$	−0.472136	−0.0506183
$88$	0	0
$89$	−16.4721	−1.74604	−0.873021	−	0.487682i	$-0.837843\pi$
$89$	−16.4721	−1.74604	−0.873021	+	0.487682i	$0.837843\pi$
$90$	0	0
$91$	−9.88854	−1.03660
$92$	−1.00000	−0.104257
$93$	0	0
$94$	4.94427	0.509963
$95$	0	0
$96$	1.00000	0.102062
$97$	−13.4164	−1.36223	−0.681115	−	0.732177i	$-0.738505\pi$
$97$	−13.4164	−1.36223	−0.681115	+	0.732177i	$0.738505\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	0	0
$101$	0.472136	0.0469793	0.0234896	−	0.999724i	$-0.492522\pi$
$101$	0.472136	0.0469793	0.0234896	+	0.999724i	$0.492522\pi$
$102$	2.47214	0.244778
$103$	16.9443	1.66957	0.834784	−	0.550577i	$-0.185592\pi$
$103$	16.9443	1.66957	0.834784	+	0.550577i	$0.185592\pi$
$104$	−2.47214	−0.242413
$105$	0	0
$106$	−8.94427	−0.868744
$107$	−2.47214	−0.238990	−0.119495	−	0.992835i	$-0.538128\pi$
$107$	−2.47214	−0.238990	−0.119495	+	0.992835i	$0.538128\pi$
$108$	1.00000	0.0962250
$109$	0.472136	0.0452224	0.0226112	−	0.999744i	$-0.492802\pi$
$109$	0.472136	0.0452224	0.0226112	+	0.999744i	$0.492802\pi$
$110$	0	0
$111$	0.472136	0.0448132
$112$	4.00000	0.377964
$113$	19.4164	1.82654	0.913271	−	0.407353i	$-0.133548\pi$
$113$	19.4164	1.82654	0.913271	+	0.407353i	$0.133548\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	−0.472136	−0.0438367
$117$	−2.47214	−0.228549
$118$	−6.00000	−0.552345
$119$	9.88854	0.906481
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−0.472136	−0.0427452
$123$	10.9443	0.986812
$124$	0	0
$125$	0	0
$126$	4.00000	0.356348
$127$	11.4164	1.01304	0.506521	−	0.862228i	$-0.330931\pi$
$127$	11.4164	1.01304	0.506521	+	0.862228i	$0.330931\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−6.94427	−0.606724	−0.303362	−	0.952875i	$-0.598109\pi$
$131$	−6.94427	−0.606724	−0.303362	+	0.952875i	$0.598109\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	−4.94427	−0.427120
$135$	0	0
$136$	2.47214	0.211984
$137$	−19.4164	−1.65886	−0.829428	−	0.558614i	$-0.811333\pi$
$137$	−19.4164	−1.65886	−0.829428	+	0.558614i	$0.811333\pi$
$138$	−1.00000	−0.0851257
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	4.94427	0.416383
$142$	−7.52786	−0.631724
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.94427	0.409191
$147$	9.00000	0.742307
$148$	0.472136	0.0388093
$149$	−13.4164	−1.09911	−0.549557	−	0.835456i	$-0.685204\pi$
$149$	−13.4164	−1.09911	−0.549557	+	0.835456i	$0.685204\pi$
$150$	0	0
$151$	16.9443	1.37891	0.689453	−	0.724331i	$-0.257851\pi$
$151$	16.9443	1.37891	0.689453	+	0.724331i	$0.257851\pi$
$152$	2.00000	0.162221
$153$	2.47214	0.199860
$154$	0	0
$155$	0	0
$156$	−2.47214	−0.197929
$157$	−11.5279	−0.920024	−0.460012	−	0.887913i	$-0.652155\pi$
$157$	−11.5279	−0.920024	−0.460012	+	0.887913i	$0.652155\pi$
$158$	12.4721	0.992230
$159$	−8.94427	−0.709327
$160$	0	0
$161$	−4.00000	−0.315244
$162$	1.00000	0.0785674
$163$	−4.94427	−0.387265	−0.193633	−	0.981074i	$-0.562027\pi$
$163$	−4.94427	−0.387265	−0.193633	+	0.981074i	$0.562027\pi$
$164$	10.9443	0.854604
$165$	0	0
$166$	1.52786	0.118585
$167$	3.05573	0.236459	0.118230	−	0.992986i	$-0.462278\pi$
$167$	3.05573	0.236459	0.118230	+	0.992986i	$0.462278\pi$
$168$	4.00000	0.308607
$169$	−6.88854	−0.529888
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	−14.9443	−1.13619	−0.568096	−	0.822962i	$-0.692320\pi$
$173$	−14.9443	−1.13619	−0.568096	+	0.822962i	$0.692320\pi$
$174$	−0.472136	−0.0357925
$175$	0	0
$176$	0	0
$177$	−6.00000	−0.450988
$178$	−16.4721	−1.23464
$179$	−18.9443	−1.41596	−0.707981	−	0.706232i	$-0.750394\pi$
$179$	−18.9443	−1.41596	−0.707981	+	0.706232i	$0.750394\pi$
$180$	0	0
$181$	8.47214	0.629729	0.314864	−	0.949137i	$-0.398041\pi$
$181$	8.47214	0.629729	0.314864	+	0.949137i	$0.398041\pi$
$182$	−9.88854	−0.732988
$183$	−0.472136	−0.0349013
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	0	0
$187$	0	0
$188$	4.94427	0.360598
$189$	4.00000	0.290957
$190$	0	0
$191$	−25.8885	−1.87323	−0.936615	−	0.350361i	$-0.886059\pi$
$191$	−25.8885	−1.87323	−0.936615	+	0.350361i	$0.886059\pi$
$192$	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
$197$	18.9443	1.34972	0.674862	−	0.737944i	$-0.264203\pi$
$197$	18.9443	1.34972	0.674862	+	0.737944i	$0.264203\pi$
$198$	0	0
$199$	17.4164	1.23462	0.617308	−	0.786721i	$-0.288223\pi$
$199$	17.4164	1.23462	0.617308	+	0.786721i	$0.288223\pi$
$200$	0	0
$201$	−4.94427	−0.348742
$202$	0.472136	0.0332194
$203$	−1.88854	−0.132550
$204$	2.47214	0.173084
$205$	0	0
$206$	16.9443	1.18056
$207$	−1.00000	−0.0695048
$208$	−2.47214	−0.171412
$209$	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
$213$	−7.52786	−0.515801
$214$	−2.47214	−0.168992
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	0.472136	0.0319771
$219$	4.94427	0.334103
$220$	0	0
$221$	−6.11146	−0.411101
$222$	0.472136	0.0316877
$223$	11.4164	0.764499	0.382250	−	0.924059i	$-0.375149\pi$
$223$	11.4164	0.764499	0.382250	+	0.924059i	$0.375149\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	19.4164	1.29156
$227$	18.4721	1.22604	0.613019	−	0.790068i	$-0.289955\pi$
$227$	18.4721	1.22604	0.613019	+	0.790068i	$0.289955\pi$
$228$	2.00000	0.132453
$229$	25.4164	1.67956	0.839782	−	0.542924i	$-0.182683\pi$
$229$	25.4164	1.67956	0.839782	+	0.542924i	$0.182683\pi$
$230$	0	0
$231$	0	0
$232$	−0.472136	−0.0309972
$233$	14.9443	0.979032	0.489516	−	0.871994i	$-0.337174\pi$
$233$	14.9443	0.979032	0.489516	+	0.871994i	$0.337174\pi$
$234$	−2.47214	−0.161609
$235$	0	0
$236$	−6.00000	−0.390567
$237$	12.4721	0.810152
$238$	9.88854	0.640979
$239$	−12.4721	−0.806755	−0.403378	−	0.915034i	$-0.632164\pi$
$239$	−12.4721	−0.806755	−0.403378	+	0.915034i	$0.632164\pi$
$240$	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
$244$	−0.472136	−0.0302254
$245$	0	0
$246$	10.9443	0.697781
$247$	−4.94427	−0.314596
$248$	0	0
$249$	1.52786	0.0968244
$250$	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
$254$	11.4164	0.716329
$255$	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
$259$	1.88854	0.117348
$260$	0	0
$261$	−0.472136	−0.0292245
$262$	−6.94427	−0.429019
$263$	20.9443	1.29148	0.645740	−	0.763558i	$-0.276549\pi$
$263$	20.9443	1.29148	0.645740	+	0.763558i	$0.276549\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	−16.4721	−1.00808
$268$	−4.94427	−0.302019
$269$	0.472136	0.0287866	0.0143933	−	0.999896i	$-0.495418\pi$
$269$	0.472136	0.0287866	0.0143933	+	0.999896i	$0.495418\pi$
$270$	0	0
$271$	0.944272	0.0573604	0.0286802	−	0.999589i	$-0.490870\pi$
$271$	0.944272	0.0573604	0.0286802	+	0.999589i	$0.490870\pi$
$272$	2.47214	0.149895
$273$	−9.88854	−0.598482
$274$	−19.4164	−1.17299
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−27.4164	−1.64729	−0.823646	−	0.567104i	$-0.808064\pi$
$277$	−27.4164	−1.64729	−0.823646	+	0.567104i	$0.808064\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.52786	0.210455	0.105227	−	0.994448i	$-0.466443\pi$
$281$	3.52786	0.210455	0.105227	+	0.994448i	$0.466443\pi$
$282$	4.94427	0.294427
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	−7.52786	−0.446697
$285$	0	0
$286$	0	0
$287$	43.7771	2.58408
$288$	1.00000	0.0589256
$289$	−10.8885	−0.640503
$290$	0	0
$291$	−13.4164	−0.786484
$292$	4.94427	0.289342
$293$	−0.944272	−0.0551650	−0.0275825	−	0.999620i	$-0.508781\pi$
$293$	−0.944272	−0.0551650	−0.0275825	+	0.999620i	$0.508781\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	0.472136	0.0274423
$297$	0	0
$298$	−13.4164	−0.777192
$299$	2.47214	0.142967
$300$	0	0
$301$	0	0
$302$	16.9443	0.975033
$303$	0.472136	0.0271235
$304$	2.00000	0.114708
$305$	0	0
$306$	2.47214	0.141323
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	16.9443	0.963926
$310$	0	0
$311$	−24.4721	−1.38769	−0.693844	−	0.720126i	$-0.744084\pi$
$311$	−24.4721	−1.38769	−0.693844	+	0.720126i	$0.744084\pi$
$312$	−2.47214	−0.139957
$313$	−25.4164	−1.43662	−0.718310	−	0.695723i	$-0.755084\pi$
$313$	−25.4164	−1.43662	−0.718310	+	0.695723i	$0.755084\pi$
$314$	−11.5279	−0.650555
$315$	0	0
$316$	12.4721	0.701612
$317$	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
$321$	−2.47214	−0.137981
$322$	−4.00000	−0.222911
$323$	4.94427	0.275107
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.94427	−0.273838
$327$	0.472136	0.0261092
$328$	10.9443	0.604296
$329$	19.7771	1.09035
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	1.52786	0.0838524
$333$	0.472136	0.0258729
$334$	3.05573	0.167202
$335$	0	0
$336$	4.00000	0.218218
$337$	25.4164	1.38452	0.692260	−	0.721648i	$-0.256615\pi$
$337$	25.4164	1.38452	0.692260	+	0.721648i	$0.256615\pi$
$338$	−6.88854	−0.374687
$339$	19.4164	1.05455
$340$	0	0
$341$	0	0
$342$	2.00000	0.108148
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	−14.9443	−0.803409
$347$	8.94427	0.480154	0.240077	−	0.970754i	$-0.422827\pi$
$347$	8.94427	0.480154	0.240077	+	0.970754i	$0.422827\pi$
$348$	−0.472136	−0.0253091
$349$	−27.8885	−1.49284	−0.746420	−	0.665475i	$-0.768229\pi$
$349$	−27.8885	−1.49284	−0.746420	+	0.665475i	$0.768229\pi$
$350$	0	0
$351$	−2.47214	−0.131953
$352$	0	0
$353$	2.94427	0.156708	0.0783539	−	0.996926i	$-0.475034\pi$
$353$	2.94427	0.156708	0.0783539	+	0.996926i	$0.475034\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	−16.4721	−0.873021
$357$	9.88854	0.523357
$358$	−18.9443	−1.00124
$359$	−32.9443	−1.73873	−0.869366	−	0.494169i	$-0.835473\pi$
$359$	−32.9443	−1.73873	−0.869366	+	0.494169i	$0.835473\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	8.47214	0.445286
$363$	−11.0000	−0.577350
$364$	−9.88854	−0.518301
$365$	0	0
$366$	−0.472136	−0.0246789
$367$	−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
$369$	10.9443	0.569736
$370$	0	0
$371$	−35.7771	−1.85745
$372$	0	0
$373$	−21.4164	−1.10890	−0.554450	−	0.832217i	$-0.687071\pi$
$373$	−21.4164	−1.10890	−0.554450	+	0.832217i	$0.687071\pi$
$374$	0	0
$375$	0	0
$376$	4.94427	0.254981
$377$	1.16718	0.0601130
$378$	4.00000	0.205738
$379$	16.8328	0.864644	0.432322	−	0.901719i	$-0.357694\pi$
$379$	16.8328	0.864644	0.432322	+	0.901719i	$0.357694\pi$
$380$	0	0
$381$	11.4164	0.584880
$382$	−25.8885	−1.32457
$383$	−13.8885	−0.709671	−0.354836	−	0.934929i	$-0.615463\pi$
$383$	−13.8885	−0.709671	−0.354836	+	0.934929i	$0.615463\pi$
$384$	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	$-0.463834\pi$
$389$	4.47214	0.226746	0.113373	+	0.993552i	$0.463834\pi$
$390$	0	0
$391$	−2.47214	−0.125021
$392$	9.00000	0.454569
$393$	−6.94427	−0.350292
$394$	18.9443	0.954399
$395$	0	0
$396$	0	0
$397$	29.3050	1.47077	0.735387	−	0.677648i	$-0.237001\pi$
$397$	29.3050	1.47077	0.735387	+	0.677648i	$0.237001\pi$
$398$	17.4164	0.873006
$399$	8.00000	0.400501
$400$	0	0
$401$	25.4164	1.26923	0.634617	−	0.772826i	$-0.281158\pi$
$401$	25.4164	1.26923	0.634617	+	0.772826i	$0.281158\pi$
$402$	−4.94427	−0.246598
$403$	0	0
$404$	0.472136	0.0234896
$405$	0	0
$406$	−1.88854	−0.0937269
$407$	0	0
$408$	2.47214	0.122389
$409$	−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
$411$	−19.4164	−0.957741
$412$	16.9443	0.834784
$413$	−24.0000	−1.18096
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−2.47214	−0.121206
$417$	0	0
$418$	0	0
$419$	−37.8885	−1.85098	−0.925488	−	0.378776i	$-0.876345\pi$
$419$	−37.8885	−1.85098	−0.925488	+	0.378776i	$0.876345\pi$
$420$	0	0
$421$	−37.4164	−1.82356	−0.911782	−	0.410674i	$-0.865293\pi$
$421$	−37.4164	−1.82356	−0.911782	+	0.410674i	$0.865293\pi$
$422$	−12.0000	−0.584151
$423$	4.94427	0.240399
$424$	−8.94427	−0.434372
$425$	0	0
$426$	−7.52786	−0.364726
$427$	−1.88854	−0.0913930
$428$	−2.47214	−0.119495
$429$	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
$433$	−21.4164	−1.02921	−0.514603	−	0.857428i	$-0.672061\pi$
$433$	−21.4164	−1.02921	−0.514603	+	0.857428i	$0.672061\pi$
$434$	0	0
$435$	0	0
$436$	0.472136	0.0226112
$437$	−2.00000	−0.0956730
$438$	4.94427	0.236246
$439$	−26.8328	−1.28066	−0.640330	−	0.768100i	$-0.721202\pi$
$439$	−26.8328	−1.28066	−0.640330	+	0.768100i	$0.721202\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	−6.11146	−0.290692
$443$	−8.94427	−0.424955	−0.212478	−	0.977166i	$-0.568153\pi$
$443$	−8.94427	−0.424955	−0.212478	+	0.977166i	$0.568153\pi$
$444$	0.472136	0.0224066
$445$	0	0
$446$	11.4164	0.540583
$447$	−13.4164	−0.634574
$448$	4.00000	0.188982
$449$	−34.9443	−1.64912	−0.824561	−	0.565773i	$-0.808578\pi$
$449$	−34.9443	−1.64912	−0.824561	+	0.565773i	$0.808578\pi$
$450$	0	0
$451$	0	0
$452$	19.4164	0.913271
$453$	16.9443	0.796111
$454$	18.4721	0.866940
$455$	0	0
$456$	2.00000	0.0936586
$457$	25.4164	1.18893	0.594465	−	0.804122i	$-0.297364\pi$
$457$	25.4164	1.18893	0.594465	+	0.804122i	$0.297364\pi$
$458$	25.4164	1.18763
$459$	2.47214	0.115389
$460$	0	0
$461$	−14.3607	−0.668844	−0.334422	−	0.942424i	$-0.608541\pi$
$461$	−14.3607	−0.668844	−0.334422	+	0.942424i	$0.608541\pi$
$462$	0	0
$463$	−15.4164	−0.716461	−0.358231	−	0.933633i	$-0.616620\pi$
$463$	−15.4164	−0.716461	−0.358231	+	0.933633i	$0.616620\pi$
$464$	−0.472136	−0.0219184
$465$	0	0
$466$	14.9443	0.692280
$467$	26.4721	1.22498	0.612492	−	0.790477i	$-0.290167\pi$
$467$	26.4721	1.22498	0.612492	+	0.790477i	$0.290167\pi$
$468$	−2.47214	−0.114275
$469$	−19.7771	−0.913221
$470$	0	0
$471$	−11.5279	−0.531176
$472$	−6.00000	−0.276172
$473$	0	0
$474$	12.4721	0.572864
$475$	0	0
$476$	9.88854	0.453241
$477$	−8.94427	−0.409530
$478$	−12.4721	−0.570462
$479$	−16.9443	−0.774204	−0.387102	−	0.922037i	$-0.626524\pi$
$479$	−16.9443	−0.774204	−0.387102	+	0.922037i	$0.626524\pi$
$480$	0	0
$481$	−1.16718	−0.0532190
$482$	−2.00000	−0.0910975
$483$	−4.00000	−0.182006
$484$	−11.0000	−0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	−31.4164	−1.42361	−0.711807	−	0.702375i	$-0.752123\pi$
$487$	−31.4164	−1.42361	−0.711807	+	0.702375i	$0.752123\pi$
$488$	−0.472136	−0.0213726
$489$	−4.94427	−0.223588
$490$	0	0
$491$	27.8885	1.25859	0.629296	−	0.777166i	$-0.283343\pi$
$491$	27.8885	1.25859	0.629296	+	0.777166i	$0.283343\pi$
$492$	10.9443	0.493406
$493$	−1.16718	−0.0525673
$494$	−4.94427	−0.222453
$495$	0	0
$496$	0	0
$497$	−30.1115	−1.35068
$498$	1.52786	0.0684652
$499$	−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
$501$	3.05573	0.136520
$502$	−12.0000	−0.535586
$503$	13.8885	0.619260	0.309630	−	0.950857i	$-0.399795\pi$
$503$	13.8885	0.619260	0.309630	+	0.950857i	$0.399795\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	0	0
$507$	−6.88854	−0.305931
$508$	11.4164	0.506521
$509$	1.41641	0.0627812	0.0313906	−	0.999507i	$-0.490006\pi$
$509$	1.41641	0.0627812	0.0313906	+	0.999507i	$0.490006\pi$
$510$	0	0
$511$	19.7771	0.874887
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	−18.0000	−0.793946
$515$	0	0
$516$	0	0
$517$	0	0
$518$	1.88854	0.0829779
$519$	−14.9443	−0.655981
$520$	0	0
$521$	−19.5279	−0.855531	−0.427766	−	0.903890i	$-0.640699\pi$
$521$	−19.5279	−0.855531	−0.427766	+	0.903890i	$0.640699\pi$
$522$	−0.472136	−0.0206648
$523$	4.94427	0.216198	0.108099	−	0.994140i	$-0.465524\pi$
$523$	4.94427	0.216198	0.108099	+	0.994140i	$0.465524\pi$
$524$	−6.94427	−0.303362
$525$	0	0
$526$	20.9443	0.913214
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.00000	−0.260378
$532$	8.00000	0.346844
$533$	−27.0557	−1.17191
$534$	−16.4721	−0.712819
$535$	0	0
$536$	−4.94427	−0.213560
$537$	−18.9443	−0.817506
$538$	0.472136	0.0203552
$539$	0	0
$540$	0	0
$541$	40.8328	1.75554	0.877770	−	0.479082i	$-0.159030\pi$
$541$	40.8328	1.75554	0.877770	+	0.479082i	$0.159030\pi$
$542$	0.944272	0.0405600
$543$	8.47214	0.363574
$544$	2.47214	0.105992
$545$	0	0
$546$	−9.88854	−0.423191
$547$	−22.8328	−0.976261	−0.488130	−	0.872771i	$-0.662321\pi$
$547$	−22.8328	−0.976261	−0.488130	+	0.872771i	$0.662321\pi$
$548$	−19.4164	−0.829428
$549$	−0.472136	−0.0201503
$550$	0	0
$551$	−0.944272	−0.0402273
$552$	−1.00000	−0.0425628
$553$	49.8885	2.12148
$554$	−27.4164	−1.16481
$555$	0	0
$556$	0	0
$557$	−29.8885	−1.26642	−0.633209	−	0.773981i	$-0.718263\pi$
$557$	−29.8885	−1.26642	−0.633209	+	0.773981i	$0.718263\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	3.52786	0.148814
$563$	−37.3050	−1.57222	−0.786108	−	0.618089i	$-0.787907\pi$
$563$	−37.3050	−1.57222	−0.786108	+	0.618089i	$0.787907\pi$
$564$	4.94427	0.208191
$565$	0	0
$566$	24.0000	1.00880
$567$	4.00000	0.167984
$568$	−7.52786	−0.315862
$569$	39.3050	1.64775	0.823875	−	0.566772i	$-0.191808\pi$
$569$	39.3050	1.64775	0.823875	+	0.566772i	$0.191808\pi$
$570$	0	0
$571$	32.8328	1.37401	0.687005	−	0.726652i	$-0.258925\pi$
$571$	32.8328	1.37401	0.687005	+	0.726652i	$0.258925\pi$
$572$	0	0
$573$	−25.8885	−1.08151
$574$	43.7771	1.82722
$575$	0	0
$576$	1.00000	0.0416667
$577$	16.9443	0.705399	0.352700	−	0.935737i	$-0.385264\pi$
$577$	16.9443	0.705399	0.352700	+	0.935737i	$0.385264\pi$
$578$	−10.8885	−0.452904
$579$	20.0000	0.831172
$580$	0	0
$581$	6.11146	0.253546
$582$	−13.4164	−0.556128
$583$	0	0
$584$	4.94427	0.204595
$585$	0	0
$586$	−0.944272	−0.0390075
$587$	−29.8885	−1.23363	−0.616816	−	0.787107i	$-0.711578\pi$
$587$	−29.8885	−1.23363	−0.616816	+	0.787107i	$0.711578\pi$
$588$	9.00000	0.371154
$589$	0	0
$590$	0	0
$591$	18.9443	0.779263
$592$	0.472136	0.0194047
$593$	23.8885	0.980985	0.490492	−	0.871445i	$-0.336817\pi$
$593$	23.8885	0.980985	0.490492	+	0.871445i	$0.336817\pi$
$594$	0	0
$595$	0	0
$596$	−13.4164	−0.549557
$597$	17.4164	0.712806
$598$	2.47214	0.101093
$599$	26.3607	1.07707	0.538534	−	0.842604i	$-0.318978\pi$
$599$	26.3607	1.07707	0.538534	+	0.842604i	$0.318978\pi$
$600$	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
$603$	−4.94427	−0.201346
$604$	16.9443	0.689453
$605$	0	0
$606$	0.472136	0.0191792
$607$	7.41641	0.301023	0.150511	−	0.988608i	$-0.451908\pi$
$607$	7.41641	0.301023	0.150511	+	0.988608i	$0.451908\pi$
$608$	2.00000	0.0811107
$609$	−1.88854	−0.0765277
$610$	0	0
$611$	−12.2229	−0.494486
$612$	2.47214	0.0999302
$613$	9.41641	0.380325	0.190163	−	0.981753i	$-0.439098\pi$
$613$	9.41641	0.380325	0.190163	+	0.981753i	$0.439098\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.4721	0.582626	0.291313	−	0.956628i	$-0.405908\pi$
$617$	14.4721	0.582626	0.291313	+	0.956628i	$0.405908\pi$
$618$	16.9443	0.681599
$619$	40.8328	1.64121	0.820605	−	0.571496i	$-0.193637\pi$
$619$	40.8328	1.64121	0.820605	+	0.571496i	$0.193637\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	−24.4721	−0.981243
$623$	−65.8885	−2.63977
$624$	−2.47214	−0.0989646
$625$	0	0
$626$	−25.4164	−1.01584
$627$	0	0
$628$	−11.5279	−0.460012
$629$	1.16718	0.0465387
$630$	0	0
$631$	−2.36068	−0.0939772	−0.0469886	−	0.998895i	$-0.514962\pi$
$631$	−2.36068	−0.0939772	−0.0469886	+	0.998895i	$0.514962\pi$
$632$	12.4721	0.496115
$633$	−12.0000	−0.476957
$634$	18.0000	0.714871
$635$	0	0
$636$	−8.94427	−0.354663
$637$	−22.2492	−0.881546
$638$	0	0
$639$	−7.52786	−0.297798
$640$	0	0
$641$	−11.5279	−0.455323	−0.227662	−	0.973740i	$-0.573108\pi$
$641$	−11.5279	−0.455323	−0.227662	+	0.973740i	$0.573108\pi$
$642$	−2.47214	−0.0975674
$643$	12.9443	0.510472	0.255236	−	0.966879i	$-0.417847\pi$
$643$	12.9443	0.510472	0.255236	+	0.966879i	$0.417847\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	4.94427	0.194530
$647$	−41.8885	−1.64681	−0.823404	−	0.567455i	$-0.807928\pi$
$647$	−41.8885	−1.64681	−0.823404	+	0.567455i	$0.807928\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−4.94427	−0.193633
$653$	20.8328	0.815251	0.407626	−	0.913149i	$-0.366357\pi$
$653$	20.8328	0.815251	0.407626	+	0.913149i	$0.366357\pi$
$654$	0.472136	0.0184620
$655$	0	0
$656$	10.9443	0.427302
$657$	4.94427	0.192894
$658$	19.7771	0.770991
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	−36.2492	−1.40993	−0.704966	−	0.709241i	$-0.749038\pi$
$661$	−36.2492	−1.40993	−0.704966	+	0.709241i	$0.749038\pi$
$662$	0	0
$663$	−6.11146	−0.237349
$664$	1.52786	0.0592926
$665$	0	0
$666$	0.472136	0.0182949
$667$	0.472136	0.0182812
$668$	3.05573	0.118230
$669$	11.4164	0.441384
$670$	0	0
$671$	0	0
$672$	4.00000	0.154303
$673$	−38.8328	−1.49690	−0.748448	−	0.663194i	$-0.769200\pi$
$673$	−38.8328	−1.49690	−0.748448	+	0.663194i	$0.769200\pi$
$674$	25.4164	0.979003
$675$	0	0
$676$	−6.88854	−0.264944
$677$	−25.8885	−0.994978	−0.497489	−	0.867470i	$-0.665744\pi$
$677$	−25.8885	−0.994978	−0.497489	+	0.867470i	$0.665744\pi$
$678$	19.4164	0.745683
$679$	−53.6656	−2.05950
$680$	0	0
$681$	18.4721	0.707854
$682$	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
$687$	25.4164	0.969696
$688$	0	0
$689$	22.1115	0.842379
$690$	0	0
$691$	−39.7771	−1.51319	−0.756596	−	0.653883i	$-0.773139\pi$
$691$	−39.7771	−1.51319	−0.756596	+	0.653883i	$0.773139\pi$
$692$	−14.9443	−0.568096
$693$	0	0
$694$	8.94427	0.339520
$695$	0	0
$696$	−0.472136	−0.0178963
$697$	27.0557	1.02481
$698$	−27.8885	−1.05560
$699$	14.9443	0.565244
$700$	0	0
$701$	37.4164	1.41320	0.706599	−	0.707614i	$-0.250228\pi$
$701$	37.4164	1.41320	0.706599	+	0.707614i	$0.250228\pi$
$702$	−2.47214	−0.0933048
$703$	0.944272	0.0356139
$704$	0	0
$705$	0	0
$706$	2.94427	0.110809
$707$	1.88854	0.0710260
$708$	−6.00000	−0.225494
$709$	20.4721	0.768847	0.384424	−	0.923157i	$-0.374400\pi$
$709$	20.4721	0.768847	0.384424	+	0.923157i	$0.374400\pi$
$710$	0	0
$711$	12.4721	0.467742
$712$	−16.4721	−0.617319
$713$	0	0
$714$	9.88854	0.370069
$715$	0	0
$716$	−18.9443	−0.707981
$717$	−12.4721	−0.465780
$718$	−32.9443	−1.22947
$719$	−35.5279	−1.32497	−0.662483	−	0.749077i	$-0.730497\pi$
$719$	−35.5279	−1.32497	−0.662483	+	0.749077i	$0.730497\pi$
$720$	0	0
$721$	67.7771	2.52415
$722$	−15.0000	−0.558242
$723$	−2.00000	−0.0743808
$724$	8.47214	0.314864
$725$	0	0
$726$	−11.0000	−0.408248
$727$	0.944272	0.0350211	0.0175106	−	0.999847i	$-0.494426\pi$
$727$	0.944272	0.0350211	0.0175106	+	0.999847i	$0.494426\pi$
$728$	−9.88854	−0.366494
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−0.472136	−0.0174506
$733$	−0.472136	−0.0174387	−0.00871937	−	0.999962i	$-0.502775\pi$
$733$	−0.472136	−0.0174387	−0.00871937	+	0.999962i	$0.502775\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	10.9443	0.402864
$739$	1.88854	0.0694712	0.0347356	−	0.999397i	$-0.488941\pi$
$739$	1.88854	0.0694712	0.0347356	+	0.999397i	$0.488941\pi$
$740$	0	0
$741$	−4.94427	−0.181632
$742$	−35.7771	−1.31342
$743$	2.11146	0.0774618	0.0387309	−	0.999250i	$-0.487668\pi$
$743$	2.11146	0.0774618	0.0387309	+	0.999250i	$0.487668\pi$
$744$	0	0
$745$	0	0
$746$	−21.4164	−0.784110
$747$	1.52786	0.0559016
$748$	0	0
$749$	−9.88854	−0.361320
$750$	0	0
$751$	36.4721	1.33089	0.665444	−	0.746448i	$-0.268242\pi$
$751$	36.4721	1.33089	0.665444	+	0.746448i	$0.268242\pi$
$752$	4.94427	0.180299
$753$	−12.0000	−0.437304
$754$	1.16718	0.0425063
$755$	0	0
$756$	4.00000	0.145479
$757$	45.4164	1.65069	0.825344	−	0.564631i	$-0.190981\pi$
$757$	45.4164	1.65069	0.825344	+	0.564631i	$0.190981\pi$
$758$	16.8328	0.611395
$759$	0	0
$760$	0	0
$761$	−17.7771	−0.644419	−0.322209	−	0.946668i	$-0.604426\pi$
$761$	−17.7771	−0.644419	−0.322209	+	0.946668i	$0.604426\pi$
$762$	11.4164	0.413573
$763$	1.88854	0.0683699
$764$	−25.8885	−0.936615
$765$	0	0
$766$	−13.8885	−0.501813
$767$	14.8328	0.535582
$768$	1.00000	0.0360844
$769$	34.9443	1.26012	0.630061	−	0.776545i	$-0.283030\pi$
$769$	34.9443	1.26012	0.630061	+	0.776545i	$0.283030\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	20.0000	0.719816
$773$	−3.05573	−0.109907	−0.0549535	−	0.998489i	$-0.517501\pi$
$773$	−3.05573	−0.109907	−0.0549535	+	0.998489i	$0.517501\pi$
$774$	0	0
$775$	0	0
$776$	−13.4164	−0.481621
$777$	1.88854	0.0677511
$778$	4.47214	0.160334
$779$	21.8885	0.784238
$780$	0	0
$781$	0	0
$782$	−2.47214	−0.0884034
$783$	−0.472136	−0.0168728
$784$	9.00000	0.321429
$785$	0	0
$786$	−6.94427	−0.247694
$787$	−1.16718	−0.0416056	−0.0208028	−	0.999784i	$-0.506622\pi$
$787$	−1.16718	−0.0416056	−0.0208028	+	0.999784i	$0.506622\pi$
$788$	18.9443	0.674862
$789$	20.9443	0.745636
$790$	0	0
$791$	77.6656	2.76147
$792$	0	0
$793$	1.16718	0.0414479
$794$	29.3050	1.03999
$795$	0	0
$796$	17.4164	0.617308
$797$	−37.8885	−1.34208	−0.671041	−	0.741421i	$-0.734152\pi$
$797$	−37.8885	−1.34208	−0.671041	+	0.741421i	$0.734152\pi$
$798$	8.00000	0.283197
$799$	12.2229	0.432416
$800$	0	0
$801$	−16.4721	−0.582014
$802$	25.4164	0.897485
$803$	0	0
$804$	−4.94427	−0.174371
$805$	0	0
$806$	0	0
$807$	0.472136	0.0166200
$808$	0.472136	0.0166097
$809$	−3.88854	−0.136714	−0.0683570	−	0.997661i	$-0.521776\pi$
$809$	−3.88854	−0.136714	−0.0683570	+	0.997661i	$0.521776\pi$
$810$	0	0
$811$	−49.8885	−1.75182	−0.875912	−	0.482471i	$-0.839739\pi$
$811$	−49.8885	−1.75182	−0.875912	+	0.482471i	$0.839739\pi$
$812$	−1.88854	−0.0662749
$813$	0.944272	0.0331171
$814$	0	0
$815$	0	0
$816$	2.47214	0.0865421
$817$	0	0
$818$	−18.0000	−0.629355
$819$	−9.88854	−0.345534
$820$	0	0
$821$	−16.4721	−0.574882	−0.287441	−	0.957798i	$-0.592804\pi$
$821$	−16.4721	−0.574882	−0.287441	+	0.957798i	$0.592804\pi$
$822$	−19.4164	−0.677225
$823$	20.3607	0.709729	0.354864	−	0.934918i	$-0.384527\pi$
$823$	20.3607	0.709729	0.354864	+	0.934918i	$0.384527\pi$
$824$	16.9443	0.590282
$825$	0	0
$826$	−24.0000	−0.835067
$827$	−42.4721	−1.47690	−0.738450	−	0.674308i	$-0.764442\pi$
$827$	−42.4721	−1.47690	−0.738450	+	0.674308i	$0.764442\pi$
$828$	−1.00000	−0.0347524
$829$	30.9443	1.07474	0.537369	−	0.843347i	$-0.319418\pi$
$829$	30.9443	1.07474	0.537369	+	0.843347i	$0.319418\pi$
$830$	0	0
$831$	−27.4164	−0.951065
$832$	−2.47214	−0.0857059
$833$	22.2492	0.770890
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−37.8885	−1.30884
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	−37.4164	−1.28945
$843$	3.52786	0.121506
$844$	−12.0000	−0.413057
$845$	0	0
$846$	4.94427	0.169988
$847$	−44.0000	−1.51186
$848$	−8.94427	−0.307148
$849$	24.0000	0.823678
$850$	0	0
$851$	−0.472136	−0.0161846
$852$	−7.52786	−0.257900
$853$	9.52786	0.326228	0.163114	−	0.986607i	$-0.447846\pi$
$853$	9.52786	0.326228	0.163114	+	0.986607i	$0.447846\pi$
$854$	−1.88854	−0.0646246
$855$	0	0
$856$	−2.47214	−0.0844959
$857$	−23.8885	−0.816017	−0.408009	−	0.912978i	$-0.633777\pi$
$857$	−23.8885	−0.816017	−0.408009	+	0.912978i	$0.633777\pi$
$858$	0	0
$859$	−39.7771	−1.35718	−0.678588	−	0.734519i	$-0.737408\pi$
$859$	−39.7771	−1.35718	−0.678588	+	0.734519i	$0.737408\pi$
$860$	0	0
$861$	43.7771	1.49192
$862$	0	0
$863$	3.05573	0.104018	0.0520091	−	0.998647i	$-0.483438\pi$
$863$	3.05573	0.104018	0.0520091	+	0.998647i	$0.483438\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−21.4164	−0.727759
$867$	−10.8885	−0.369794
$868$	0	0
$869$	0	0
$870$	0	0
$871$	12.2229	0.414158
$872$	0.472136	0.0159885
$873$	−13.4164	−0.454077
$874$	−2.00000	−0.0676510
$875$	0	0
$876$	4.94427	0.167051
$877$	20.3607	0.687531	0.343766	−	0.939055i	$-0.388297\pi$
$877$	20.3607	0.687531	0.343766	+	0.939055i	$0.388297\pi$
$878$	−26.8328	−0.905564
$879$	−0.944272	−0.0318495
$880$	0	0
$881$	−4.47214	−0.150670	−0.0753350	−	0.997158i	$-0.524003\pi$
$881$	−4.47214	−0.150670	−0.0753350	+	0.997158i	$0.524003\pi$
$882$	9.00000	0.303046
$883$	−19.7771	−0.665552	−0.332776	−	0.943006i	$-0.607985\pi$
$883$	−19.7771	−0.665552	−0.332776	+	0.943006i	$0.607985\pi$
$884$	−6.11146	−0.205551
$885$	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	$0.367995\pi$
$888$	0.472136	0.0158438
$889$	45.6656	1.53158
$890$	0	0
$891$	0	0
$892$	11.4164	0.382250
$893$	9.88854	0.330908
$894$	−13.4164	−0.448712
$895$	0	0
$896$	4.00000	0.133631
$897$	2.47214	0.0825422
$898$	−34.9443	−1.16611
$899$	0	0
$900$	0	0
$901$	−22.1115	−0.736639
$902$	0	0
$903$	0	0
$904$	19.4164	0.645780
$905$	0	0
$906$	16.9443	0.562936
$907$	14.8328	0.492516	0.246258	−	0.969204i	$-0.420799\pi$
$907$	14.8328	0.492516	0.246258	+	0.969204i	$0.420799\pi$
$908$	18.4721	0.613019
$909$	0.472136	0.0156598
$910$	0	0
$911$	24.9443	0.826441	0.413220	−	0.910631i	$-0.364404\pi$
$911$	24.9443	0.826441	0.413220	+	0.910631i	$0.364404\pi$
$912$	2.00000	0.0662266
$913$	0	0
$914$	25.4164	0.840700
$915$	0	0
$916$	25.4164	0.839782
$917$	−27.7771	−0.917280
$918$	2.47214	0.0815926
$919$	−9.41641	−0.310619	−0.155309	−	0.987866i	$-0.549637\pi$
$919$	−9.41641	−0.310619	−0.155309	+	0.987866i	$0.549637\pi$
$920$	0	0
$921$	0	0
$922$	−14.3607	−0.472944
$923$	18.6099	0.612552
$924$	0	0
$925$	0	0
$926$	−15.4164	−0.506615
$927$	16.9443	0.556523
$928$	−0.472136	−0.0154986
$929$	−10.9443	−0.359070	−0.179535	−	0.983752i	$-0.557459\pi$
$929$	−10.9443	−0.359070	−0.179535	+	0.983752i	$0.557459\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	14.9443	0.489516
$933$	−24.4721	−0.801182
$934$	26.4721	0.866195
$935$	0	0
$936$	−2.47214	−0.0808043
$937$	6.58359	0.215077	0.107538	−	0.994201i	$-0.465703\pi$
$937$	6.58359	0.215077	0.107538	+	0.994201i	$0.465703\pi$
$938$	−19.7771	−0.645745
$939$	−25.4164	−0.829433
$940$	0	0
$941$	36.4721	1.18896	0.594479	−	0.804111i	$-0.297358\pi$
$941$	36.4721	1.18896	0.594479	+	0.804111i	$0.297358\pi$
$942$	−11.5279	−0.375598
$943$	−10.9443	−0.356395
$944$	−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	$0.562462\pi$
$948$	12.4721	0.405076
$949$	−12.2229	−0.396773
$950$	0	0
$951$	18.0000	0.583690
$952$	9.88854	0.320490
$953$	−12.3607	−0.400402	−0.200201	−	0.979755i	$-0.564159\pi$
$953$	−12.3607	−0.400402	−0.200201	+	0.979755i	$0.564159\pi$
$954$	−8.94427	−0.289581
$955$	0	0
$956$	−12.4721	−0.403378
$957$	0	0
$958$	−16.9443	−0.547445
$959$	−77.6656	−2.50795
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−1.16718	−0.0376315
$963$	−2.47214	−0.0796635
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	−4.00000	−0.128698
$967$	55.4164	1.78207	0.891036	−	0.453933i	$-0.149979\pi$
$967$	55.4164	1.78207	0.891036	+	0.453933i	$0.149979\pi$
$968$	−11.0000	−0.353553
$969$	4.94427	0.158833
$970$	0	0
$971$	−18.1115	−0.581224	−0.290612	−	0.956841i	$-0.593859\pi$
$971$	−18.1115	−0.581224	−0.290612	+	0.956841i	$0.593859\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−31.4164	−1.00665
$975$	0	0
$976$	−0.472136	−0.0151127
$977$	1.52786	0.0488807	0.0244404	−	0.999701i	$-0.492220\pi$
$977$	1.52786	0.0488807	0.0244404	+	0.999701i	$0.492220\pi$
$978$	−4.94427	−0.158100
$979$	0	0
$980$	0	0
$981$	0.472136	0.0150741
$982$	27.8885	0.889959
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	10.9443	0.348891
$985$	0	0
$986$	−1.16718	−0.0371707
$987$	19.7771	0.629512
$988$	−4.94427	−0.157298
$989$	0	0
$990$	0	0
$991$	53.6656	1.70474	0.852372	−	0.522935i	$-0.175163\pi$
$991$	53.6656	1.70474	0.852372	+	0.522935i	$0.175163\pi$
$992$	0	0
$993$	0	0
$994$	−30.1115	−0.955077
$995$	0	0
$996$	1.52786	0.0484122
$997$	−7.41641	−0.234880	−0.117440	−	0.993080i	$-0.537469\pi$
$997$	−7.41641	−0.234880	−0.117440	+	0.993080i	$0.537469\pi$
$998$	−24.0000	−0.759707
$999$	0.472136	0.0149377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.bn.1.1		2
5.2	odd	4		690.2.d.a.139.4	yes	4
5.3	odd	4		690.2.d.a.139.2	✓	4
5.4	even	2		3450.2.a.bc.1.2		2
15.2	even	4		2070.2.d.b.829.1		4
15.8	even	4		2070.2.d.b.829.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.d.a.139.2	✓	4	5.3	odd	4
690.2.d.a.139.4	yes	4	5.2	odd	4
2070.2.d.b.829.1		4	15.2	even	4
2070.2.d.b.829.3		4	15.8	even	4
3450.2.a.bc.1.2		2	5.4	even	2
3450.2.a.bn.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -2.47214 q^{13} +4.00000 q^{14} +1.00000 q^{16} +2.47214 q^{17} +1.00000 q^{18} +2.00000 q^{19} +4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} -2.47214 q^{26} +1.00000 q^{27} +4.00000 q^{28} -0.472136 q^{29} +1.00000 q^{32} +2.47214 q^{34} +1.00000 q^{36} +0.472136 q^{37} +2.00000 q^{38} -2.47214 q^{39} +10.9443 q^{41} +4.00000 q^{42} -1.00000 q^{46} +4.94427 q^{47} +1.00000 q^{48} +9.00000 q^{49} +2.47214 q^{51} -2.47214 q^{52} -8.94427 q^{53} +1.00000 q^{54} +4.00000 q^{56} +2.00000 q^{57} -0.472136 q^{58} -6.00000 q^{59} -0.472136 q^{61} +4.00000 q^{63} +1.00000 q^{64} -4.94427 q^{67} +2.47214 q^{68} -1.00000 q^{69} -7.52786 q^{71} +1.00000 q^{72} +4.94427 q^{73} +0.472136 q^{74} +2.00000 q^{76} -2.47214 q^{78} +12.4721 q^{79} +1.00000 q^{81} +10.9443 q^{82} +1.52786 q^{83} +4.00000 q^{84} -0.472136 q^{87} -16.4721 q^{89} -9.88854 q^{91} -1.00000 q^{92} +4.94427 q^{94} +1.00000 q^{96} -13.4164 q^{97} +9.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 8 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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

Introduction

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Database

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