LMFDB - Embedded newform 3450.2.a.bg.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 690)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.24264 q^{11} -1.00000 q^{12} +0.828427 q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.82843 q^{17} -1.00000 q^{18} +6.24264 q^{19} -2.00000 q^{21} +4.24264 q^{22} +1.00000 q^{23} +1.00000 q^{24} -0.828427 q^{26} -1.00000 q^{27} +2.00000 q^{28} +3.65685 q^{29} +6.00000 q^{31} -1.00000 q^{32} +4.24264 q^{33} +6.82843 q^{34} +1.00000 q^{36} -0.585786 q^{37} -6.24264 q^{38} -0.828427 q^{39} -6.82843 q^{41} +2.00000 q^{42} -10.2426 q^{43} -4.24264 q^{44} -1.00000 q^{46} +0.828427 q^{47} -1.00000 q^{48} -3.00000 q^{49} +6.82843 q^{51} +0.828427 q^{52} -10.5858 q^{53} +1.00000 q^{54} -2.00000 q^{56} -6.24264 q^{57} -3.65685 q^{58} +8.48528 q^{59} -0.585786 q^{61} -6.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -4.24264 q^{66} +3.41421 q^{67} -6.82843 q^{68} -1.00000 q^{69} -5.65685 q^{71} -1.00000 q^{72} -7.65685 q^{73} +0.585786 q^{74} +6.24264 q^{76} -8.48528 q^{77} +0.828427 q^{78} -3.65685 q^{79} +1.00000 q^{81} +6.82843 q^{82} -1.41421 q^{83} -2.00000 q^{84} +10.2426 q^{86} -3.65685 q^{87} +4.24264 q^{88} +9.17157 q^{89} +1.65685 q^{91} +1.00000 q^{92} -6.00000 q^{93} -0.828427 q^{94} +1.00000 q^{96} +6.00000 q^{97} +3.00000 q^{98} -4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 4 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 + 4 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{12} - 4 q^{13} - 4 q^{14} + 2 q^{16} - 8 q^{17} - 2 q^{18} + 4 q^{19} - 4 q^{21} + 2 q^{23} + 2 q^{24} + 4 q^{26} - 2 q^{27} + 4 q^{28} - 4 q^{29} + 12 q^{31} - 2 q^{32} + 8 q^{34} + 2 q^{36} - 4 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{41} + 4 q^{42} - 12 q^{43} - 2 q^{46} - 4 q^{47} - 2 q^{48} - 6 q^{49} + 8 q^{51} - 4 q^{52} - 24 q^{53} + 2 q^{54} - 4 q^{56} - 4 q^{57} + 4 q^{58} - 4 q^{61} - 12 q^{62} + 4 q^{63} + 2 q^{64} + 4 q^{67} - 8 q^{68} - 2 q^{69} - 2 q^{72} - 4 q^{73} + 4 q^{74} + 4 q^{76} - 4 q^{78} + 4 q^{79} + 2 q^{81} + 8 q^{82} - 4 q^{84} + 12 q^{86} + 4 q^{87} + 24 q^{89} - 8 q^{91} + 2 q^{92} - 12 q^{93} + 4 q^{94} + 2 q^{96} + 12 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 4 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^12 - 4 * q^13 - 4 * q^14 + 2 * q^16 - 8 * q^17 - 2 * q^18 + 4 * q^19 - 4 * q^21 + 2 * q^23 + 2 * q^24 + 4 * q^26 - 2 * q^27 + 4 * q^28 - 4 * q^29 + 12 * q^31 - 2 * q^32 + 8 * q^34 + 2 * q^36 - 4 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^41 + 4 * q^42 - 12 * q^43 - 2 * q^46 - 4 * q^47 - 2 * q^48 - 6 * q^49 + 8 * q^51 - 4 * q^52 - 24 * q^53 + 2 * q^54 - 4 * q^56 - 4 * q^57 + 4 * q^58 - 4 * q^61 - 12 * q^62 + 4 * q^63 + 2 * q^64 + 4 * q^67 - 8 * q^68 - 2 * q^69 - 2 * q^72 - 4 * q^73 + 4 * q^74 + 4 * q^76 - 4 * q^78 + 4 * q^79 + 2 * q^81 + 8 * q^82 - 4 * q^84 + 12 * q^86 + 4 * q^87 + 24 * q^89 - 8 * q^91 + 2 * q^92 - 12 * q^93 + 4 * q^94 + 2 * q^96 + 12 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−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
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	−1.00000	−0.288675
$13$	0.828427	0.229764	0.114882	−	0.993379i	$-0.463351\pi$
$13$	0.828427	0.229764	0.114882	+	0.993379i	$0.463351\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	−1.00000	−0.235702
$19$	6.24264	1.43216	0.716080	−	0.698018i	$-0.245935\pi$
$19$	6.24264	1.43216	0.716080	+	0.698018i	$0.245935\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	4.24264	0.904534
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	0	0
$26$	−0.828427	−0.162468
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	3.65685	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	4.24264	0.738549
$34$	6.82843	1.17107
$35$	0	0
$36$	1.00000	0.166667
$37$	−0.585786	−0.0963027	−0.0481513	−	0.998840i	$-0.515333\pi$
$37$	−0.585786	−0.0963027	−0.0481513	+	0.998840i	$0.515333\pi$
$38$	−6.24264	−1.01269
$39$	−0.828427	−0.132655
$40$	0	0
$41$	−6.82843	−1.06642	−0.533211	−	0.845983i	$-0.679015\pi$
$41$	−6.82843	−1.06642	−0.533211	+	0.845983i	$0.679015\pi$
$42$	2.00000	0.308607
$43$	−10.2426	−1.56199	−0.780994	−	0.624538i	$-0.785287\pi$
$43$	−10.2426	−1.56199	−0.780994	+	0.624538i	$0.785287\pi$
$44$	−4.24264	−0.639602
$45$	0	0
$46$	−1.00000	−0.147442
$47$	0.828427	0.120839	0.0604193	−	0.998173i	$-0.480756\pi$
$47$	0.828427	0.120839	0.0604193	+	0.998173i	$0.480756\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	6.82843	0.956171
$52$	0.828427	0.114882
$53$	−10.5858	−1.45407	−0.727035	−	0.686601i	$-0.759102\pi$
$53$	−10.5858	−1.45407	−0.727035	+	0.686601i	$0.759102\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	−6.24264	−0.826858
$58$	−3.65685	−0.480168
$59$	8.48528	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$60$	0	0
$61$	−0.585786	−0.0750023	−0.0375011	−	0.999297i	$-0.511940\pi$
$61$	−0.585786	−0.0750023	−0.0375011	+	0.999297i	$0.511940\pi$
$62$	−6.00000	−0.762001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.24264	−0.522233
$67$	3.41421	0.417113	0.208556	−	0.978010i	$-0.433124\pi$
$67$	3.41421	0.417113	0.208556	+	0.978010i	$0.433124\pi$
$68$	−6.82843	−0.828068
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	−1.00000	−0.117851
$73$	−7.65685	−0.896167	−0.448084	−	0.893992i	$-0.647893\pi$
$73$	−7.65685	−0.896167	−0.448084	+	0.893992i	$0.647893\pi$
$74$	0.585786	0.0680963
$75$	0	0
$76$	6.24264	0.716080
$77$	−8.48528	−0.966988
$78$	0.828427	0.0938009
$79$	−3.65685	−0.411428	−0.205714	−	0.978612i	$-0.565952\pi$
$79$	−3.65685	−0.411428	−0.205714	+	0.978612i	$0.565952\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.82843	0.754074
$83$	−1.41421	−0.155230	−0.0776151	−	0.996983i	$-0.524731\pi$
$83$	−1.41421	−0.155230	−0.0776151	+	0.996983i	$0.524731\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	10.2426	1.10449
$87$	−3.65685	−0.392056
$88$	4.24264	0.452267
$89$	9.17157	0.972185	0.486092	−	0.873907i	$-0.338422\pi$
$89$	9.17157	0.972185	0.486092	+	0.873907i	$0.338422\pi$
$90$	0	0
$91$	1.65685	0.173686
$92$	1.00000	0.104257
$93$	−6.00000	−0.622171
$94$	−0.828427	−0.0854457
$95$	0	0
$96$	1.00000	0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	3.00000	0.303046
$99$	−4.24264	−0.426401
$100$	0	0
$101$	13.3137	1.32476	0.662382	−	0.749166i	$-0.269546\pi$
$101$	13.3137	1.32476	0.662382	+	0.749166i	$0.269546\pi$
$102$	−6.82843	−0.676115
$103$	−16.1421	−1.59053	−0.795266	−	0.606261i	$-0.792669\pi$
$103$	−16.1421	−1.59053	−0.795266	+	0.606261i	$0.792669\pi$
$104$	−0.828427	−0.0812340
$105$	0	0
$106$	10.5858	1.02818
$107$	−0.928932	−0.0898033	−0.0449016	−	0.998991i	$-0.514297\pi$
$107$	−0.928932	−0.0898033	−0.0449016	+	0.998991i	$0.514297\pi$
$108$	−1.00000	−0.0962250
$109$	12.5858	1.20550	0.602750	−	0.797930i	$-0.294072\pi$
$109$	12.5858	1.20550	0.602750	+	0.797930i	$0.294072\pi$
$110$	0	0
$111$	0.585786	0.0556004
$112$	2.00000	0.188982
$113$	16.9706	1.59646	0.798228	−	0.602355i	$-0.205771\pi$
$113$	16.9706	1.59646	0.798228	+	0.602355i	$0.205771\pi$
$114$	6.24264	0.584677
$115$	0	0
$116$	3.65685	0.339530
$117$	0.828427	0.0765881
$118$	−8.48528	−0.781133
$119$	−13.6569	−1.25192
$120$	0	0
$121$	7.00000	0.636364
$122$	0.585786	0.0530346
$123$	6.82843	0.615699
$124$	6.00000	0.538816
$125$	0	0
$126$	−2.00000	−0.178174
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−1.00000	−0.0883883
$129$	10.2426	0.901814
$130$	0	0
$131$	−21.6569	−1.89217	−0.946084	−	0.323921i	$-0.894999\pi$
$131$	−21.6569	−1.89217	−0.946084	+	0.323921i	$0.894999\pi$
$132$	4.24264	0.369274
$133$	12.4853	1.08261
$134$	−3.41421	−0.294943
$135$	0	0
$136$	6.82843	0.585533
$137$	8.48528	0.724947	0.362473	−	0.931994i	$-0.381932\pi$
$137$	8.48528	0.724947	0.362473	+	0.931994i	$0.381932\pi$
$138$	1.00000	0.0851257
$139$	−16.9706	−1.43942	−0.719712	−	0.694273i	$-0.755726\pi$
$139$	−16.9706	−1.43942	−0.719712	+	0.694273i	$0.755726\pi$
$140$	0	0
$141$	−0.828427	−0.0697661
$142$	5.65685	0.474713
$143$	−3.51472	−0.293916
$144$	1.00000	0.0833333
$145$	0	0
$146$	7.65685	0.633686
$147$	3.00000	0.247436
$148$	−0.585786	−0.0481513
$149$	12.7279	1.04271	0.521356	−	0.853339i	$-0.325426\pi$
$149$	12.7279	1.04271	0.521356	+	0.853339i	$0.325426\pi$
$150$	0	0
$151$	−4.34315	−0.353440	−0.176720	−	0.984261i	$-0.556549\pi$
$151$	−4.34315	−0.353440	−0.176720	+	0.984261i	$0.556549\pi$
$152$	−6.24264	−0.506345
$153$	−6.82843	−0.552046
$154$	8.48528	0.683763
$155$	0	0
$156$	−0.828427	−0.0663273
$157$	−21.0711	−1.68165	−0.840827	−	0.541304i	$-0.817931\pi$
$157$	−21.0711	−1.68165	−0.840827	+	0.541304i	$0.817931\pi$
$158$	3.65685	0.290924
$159$	10.5858	0.839507
$160$	0	0
$161$	2.00000	0.157622
$162$	−1.00000	−0.0785674
$163$	−6.82843	−0.534844	−0.267422	−	0.963580i	$-0.586172\pi$
$163$	−6.82843	−0.534844	−0.267422	+	0.963580i	$0.586172\pi$
$164$	−6.82843	−0.533211
$165$	0	0
$166$	1.41421	0.109764
$167$	3.17157	0.245424	0.122712	−	0.992442i	$-0.460841\pi$
$167$	3.17157	0.245424	0.122712	+	0.992442i	$0.460841\pi$
$168$	2.00000	0.154303
$169$	−12.3137	−0.947208
$170$	0	0
$171$	6.24264	0.477387
$172$	−10.2426	−0.780994
$173$	−8.82843	−0.671213	−0.335606	−	0.942002i	$-0.608941\pi$
$173$	−8.82843	−0.671213	−0.335606	+	0.942002i	$0.608941\pi$
$174$	3.65685	0.277225
$175$	0	0
$176$	−4.24264	−0.319801
$177$	−8.48528	−0.637793
$178$	−9.17157	−0.687438
$179$	−21.6569	−1.61871	−0.809355	−	0.587320i	$-0.800183\pi$
$179$	−21.6569	−1.61871	−0.809355	+	0.587320i	$0.800183\pi$
$180$	0	0
$181$	−20.3848	−1.51519	−0.757594	−	0.652726i	$-0.773625\pi$
$181$	−20.3848	−1.51519	−0.757594	+	0.652726i	$0.773625\pi$
$182$	−1.65685	−0.122814
$183$	0.585786	0.0433026
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	6.00000	0.439941
$187$	28.9706	2.11854
$188$	0.828427	0.0604193
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−10.8284	−0.783517	−0.391759	−	0.920068i	$-0.628133\pi$
$191$	−10.8284	−0.783517	−0.391759	+	0.920068i	$0.628133\pi$
$192$	−1.00000	−0.0721688
$193$	20.9706	1.50949	0.754747	−	0.656016i	$-0.227760\pi$
$193$	20.9706	1.50949	0.754747	+	0.656016i	$0.227760\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−20.6274	−1.46964	−0.734821	−	0.678261i	$-0.762734\pi$
$197$	−20.6274	−1.46964	−0.734821	+	0.678261i	$0.762734\pi$
$198$	4.24264	0.301511
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	−3.41421	−0.240820
$202$	−13.3137	−0.936749
$203$	7.31371	0.513322
$204$	6.82843	0.478086
$205$	0	0
$206$	16.1421	1.12468
$207$	1.00000	0.0695048
$208$	0.828427	0.0574411
$209$	−26.4853	−1.83203
$210$	0	0
$211$	−3.51472	−0.241963	−0.120982	−	0.992655i	$-0.538604\pi$
$211$	−3.51472	−0.241963	−0.120982	+	0.992655i	$0.538604\pi$
$212$	−10.5858	−0.727035
$213$	5.65685	0.387601
$214$	0.928932	0.0635005
$215$	0	0
$216$	1.00000	0.0680414
$217$	12.0000	0.814613
$218$	−12.5858	−0.852417
$219$	7.65685	0.517402
$220$	0	0
$221$	−5.65685	−0.380521
$222$	−0.585786	−0.0393154
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−16.9706	−1.12887
$227$	22.3848	1.48573	0.742865	−	0.669441i	$-0.233466\pi$
$227$	22.3848	1.48573	0.742865	+	0.669441i	$0.233466\pi$
$228$	−6.24264	−0.413429
$229$	10.2426	0.676853	0.338426	−	0.940993i	$-0.390105\pi$
$229$	10.2426	0.676853	0.338426	+	0.940993i	$0.390105\pi$
$230$	0	0
$231$	8.48528	0.558291
$232$	−3.65685	−0.240084
$233$	−2.14214	−0.140336	−0.0701680	−	0.997535i	$-0.522354\pi$
$233$	−2.14214	−0.140336	−0.0701680	+	0.997535i	$0.522354\pi$
$234$	−0.828427	−0.0541560
$235$	0	0
$236$	8.48528	0.552345
$237$	3.65685	0.237538
$238$	13.6569	0.885242
$239$	−16.8284	−1.08854	−0.544270	−	0.838910i	$-0.683193\pi$
$239$	−16.8284	−1.08854	−0.544270	+	0.838910i	$0.683193\pi$
$240$	0	0
$241$	6.48528	0.417754	0.208877	−	0.977942i	$-0.433019\pi$
$241$	6.48528	0.417754	0.208877	+	0.977942i	$0.433019\pi$
$242$	−7.00000	−0.449977
$243$	−1.00000	−0.0641500
$244$	−0.585786	−0.0375011
$245$	0	0
$246$	−6.82843	−0.435365
$247$	5.17157	0.329059
$248$	−6.00000	−0.381000
$249$	1.41421	0.0896221
$250$	0	0
$251$	0.727922	0.0459460	0.0229730	−	0.999736i	$-0.492687\pi$
$251$	0.727922	0.0459460	0.0229730	+	0.999736i	$0.492687\pi$
$252$	2.00000	0.125988
$253$	−4.24264	−0.266733
$254$	−2.00000	−0.125491
$255$	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	$0.689739\pi$
$258$	−10.2426	−0.637679
$259$	−1.17157	−0.0727980
$260$	0	0
$261$	3.65685	0.226354
$262$	21.6569	1.33796
$263$	6.34315	0.391135	0.195568	−	0.980690i	$-0.437345\pi$
$263$	6.34315	0.391135	0.195568	+	0.980690i	$0.437345\pi$
$264$	−4.24264	−0.261116
$265$	0	0
$266$	−12.4853	−0.765522
$267$	−9.17157	−0.561291
$268$	3.41421	0.208556
$269$	−19.1716	−1.16891	−0.584456	−	0.811426i	$-0.698692\pi$
$269$	−19.1716	−1.16891	−0.584456	+	0.811426i	$0.698692\pi$
$270$	0	0
$271$	9.65685	0.586612	0.293306	−	0.956019i	$-0.405245\pi$
$271$	9.65685	0.586612	0.293306	+	0.956019i	$0.405245\pi$
$272$	−6.82843	−0.414034
$273$	−1.65685	−0.100277
$274$	−8.48528	−0.512615
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−13.5147	−0.812021	−0.406010	−	0.913868i	$-0.633080\pi$
$277$	−13.5147	−0.812021	−0.406010	+	0.913868i	$0.633080\pi$
$278$	16.9706	1.01783
$279$	6.00000	0.359211
$280$	0	0
$281$	−1.85786	−0.110831	−0.0554154	−	0.998463i	$-0.517648\pi$
$281$	−1.85786	−0.110831	−0.0554154	+	0.998463i	$0.517648\pi$
$282$	0.828427	0.0493321
$283$	−15.2132	−0.904331	−0.452166	−	0.891934i	$-0.649348\pi$
$283$	−15.2132	−0.904331	−0.452166	+	0.891934i	$0.649348\pi$
$284$	−5.65685	−0.335673
$285$	0	0
$286$	3.51472	0.207830
$287$	−13.6569	−0.806139
$288$	−1.00000	−0.0589256
$289$	29.6274	1.74279
$290$	0	0
$291$	−6.00000	−0.351726
$292$	−7.65685	−0.448084
$293$	−20.0416	−1.17084	−0.585422	−	0.810729i	$-0.699071\pi$
$293$	−20.0416	−1.17084	−0.585422	+	0.810729i	$0.699071\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	0.585786	0.0340481
$297$	4.24264	0.246183
$298$	−12.7279	−0.737309
$299$	0.828427	0.0479092
$300$	0	0
$301$	−20.4853	−1.18075
$302$	4.34315	0.249920
$303$	−13.3137	−0.764853
$304$	6.24264	0.358040
$305$	0	0
$306$	6.82843	0.390355
$307$	−25.4558	−1.45284	−0.726421	−	0.687250i	$-0.758818\pi$
$307$	−25.4558	−1.45284	−0.726421	+	0.687250i	$0.758818\pi$
$308$	−8.48528	−0.483494
$309$	16.1421	0.918294
$310$	0	0
$311$	−21.7990	−1.23611	−0.618054	−	0.786136i	$-0.712079\pi$
$311$	−21.7990	−1.23611	−0.618054	+	0.786136i	$0.712079\pi$
$312$	0.828427	0.0469005
$313$	−18.0000	−1.01742	−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000	−1.01742	−0.508710	+	0.860938i	$0.669877\pi$
$314$	21.0711	1.18911
$315$	0	0
$316$	−3.65685	−0.205714
$317$	14.4853	0.813574	0.406787	−	0.913523i	$-0.366649\pi$
$317$	14.4853	0.813574	0.406787	+	0.913523i	$0.366649\pi$
$318$	−10.5858	−0.593621
$319$	−15.5147	−0.868657
$320$	0	0
$321$	0.928932	0.0518479
$322$	−2.00000	−0.111456
$323$	−42.6274	−2.37185
$324$	1.00000	0.0555556
$325$	0	0
$326$	6.82843	0.378192
$327$	−12.5858	−0.695996
$328$	6.82843	0.377037
$329$	1.65685	0.0913453
$330$	0	0
$331$	−20.4853	−1.12597	−0.562986	−	0.826466i	$-0.690348\pi$
$331$	−20.4853	−1.12597	−0.562986	+	0.826466i	$0.690348\pi$
$332$	−1.41421	−0.0776151
$333$	−0.585786	−0.0321009
$334$	−3.17157	−0.173541
$335$	0	0
$336$	−2.00000	−0.109109
$337$	26.4853	1.44275	0.721373	−	0.692547i	$-0.243512\pi$
$337$	26.4853	1.44275	0.721373	+	0.692547i	$0.243512\pi$
$338$	12.3137	0.669777
$339$	−16.9706	−0.921714
$340$	0	0
$341$	−25.4558	−1.37851
$342$	−6.24264	−0.337563
$343$	−20.0000	−1.07990
$344$	10.2426	0.552246
$345$	0	0
$346$	8.82843	0.474619
$347$	31.7990	1.70706	0.853530	−	0.521044i	$-0.174457\pi$
$347$	31.7990	1.70706	0.853530	+	0.521044i	$0.174457\pi$
$348$	−3.65685	−0.196028
$349$	2.68629	0.143794	0.0718969	−	0.997412i	$-0.477095\pi$
$349$	2.68629	0.143794	0.0718969	+	0.997412i	$0.477095\pi$
$350$	0	0
$351$	−0.828427	−0.0442182
$352$	4.24264	0.226134
$353$	−2.14214	−0.114014	−0.0570072	−	0.998374i	$-0.518156\pi$
$353$	−2.14214	−0.114014	−0.0570072	+	0.998374i	$0.518156\pi$
$354$	8.48528	0.450988
$355$	0	0
$356$	9.17157	0.486092
$357$	13.6569	0.722797
$358$	21.6569	1.14460
$359$	19.7990	1.04495	0.522475	−	0.852654i	$-0.325009\pi$
$359$	19.7990	1.04495	0.522475	+	0.852654i	$0.325009\pi$
$360$	0	0
$361$	19.9706	1.05108
$362$	20.3848	1.07140
$363$	−7.00000	−0.367405
$364$	1.65685	0.0868428
$365$	0	0
$366$	−0.585786	−0.0306195
$367$	−1.02944	−0.0537362	−0.0268681	−	0.999639i	$-0.508553\pi$
$367$	−1.02944	−0.0537362	−0.0268681	+	0.999639i	$0.508553\pi$
$368$	1.00000	0.0521286
$369$	−6.82843	−0.355474
$370$	0	0
$371$	−21.1716	−1.09917
$372$	−6.00000	−0.311086
$373$	23.2132	1.20193	0.600967	−	0.799274i	$-0.294782\pi$
$373$	23.2132	1.20193	0.600967	+	0.799274i	$0.294782\pi$
$374$	−28.9706	−1.49803
$375$	0	0
$376$	−0.828427	−0.0427229
$377$	3.02944	0.156024
$378$	2.00000	0.102869
$379$	23.2132	1.19238	0.596191	−	0.802843i	$-0.296680\pi$
$379$	23.2132	1.19238	0.596191	+	0.802843i	$0.296680\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	10.8284	0.554031
$383$	−26.6274	−1.36060	−0.680299	−	0.732935i	$-0.738150\pi$
$383$	−26.6274	−1.36060	−0.680299	+	0.732935i	$0.738150\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−20.9706	−1.06737
$387$	−10.2426	−0.520663
$388$	6.00000	0.304604
$389$	−30.3848	−1.54057	−0.770285	−	0.637700i	$-0.779886\pi$
$389$	−30.3848	−1.54057	−0.770285	+	0.637700i	$0.779886\pi$
$390$	0	0
$391$	−6.82843	−0.345328
$392$	3.00000	0.151523
$393$	21.6569	1.09244
$394$	20.6274	1.03919
$395$	0	0
$396$	−4.24264	−0.213201
$397$	19.6569	0.986549	0.493275	−	0.869874i	$-0.335800\pi$
$397$	19.6569	0.986549	0.493275	+	0.869874i	$0.335800\pi$
$398$	14.0000	0.701757
$399$	−12.4853	−0.625046
$400$	0	0
$401$	4.97056	0.248218	0.124109	−	0.992269i	$-0.460393\pi$
$401$	4.97056	0.248218	0.124109	+	0.992269i	$0.460393\pi$
$402$	3.41421	0.170285
$403$	4.97056	0.247601
$404$	13.3137	0.662382
$405$	0	0
$406$	−7.31371	−0.362973
$407$	2.48528	0.123191
$408$	−6.82843	−0.338058
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	−8.48528	−0.418548
$412$	−16.1421	−0.795266
$413$	16.9706	0.835067
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−0.828427	−0.0406170
$417$	16.9706	0.831052
$418$	26.4853	1.29544
$419$	−39.0711	−1.90875	−0.954373	−	0.298616i	$-0.903475\pi$
$419$	−39.0711	−1.90875	−0.954373	+	0.298616i	$0.903475\pi$
$420$	0	0
$421$	1.75736	0.0856485	0.0428242	−	0.999083i	$-0.486364\pi$
$421$	1.75736	0.0856485	0.0428242	+	0.999083i	$0.486364\pi$
$422$	3.51472	0.171094
$423$	0.828427	0.0402795
$424$	10.5858	0.514091
$425$	0	0
$426$	−5.65685	−0.274075
$427$	−1.17157	−0.0566964
$428$	−0.928932	−0.0449016
$429$	3.51472	0.169692
$430$	0	0
$431$	−8.48528	−0.408722	−0.204361	−	0.978896i	$-0.565512\pi$
$431$	−8.48528	−0.408722	−0.204361	+	0.978896i	$0.565512\pi$
$432$	−1.00000	−0.0481125
$433$	6.97056	0.334984	0.167492	−	0.985873i	$-0.446433\pi$
$433$	6.97056	0.334984	0.167492	+	0.985873i	$0.446433\pi$
$434$	−12.0000	−0.576018
$435$	0	0
$436$	12.5858	0.602750
$437$	6.24264	0.298626
$438$	−7.65685	−0.365859
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	5.65685	0.269069
$443$	0.686292	0.0326067	0.0163033	−	0.999867i	$-0.494810\pi$
$443$	0.686292	0.0326067	0.0163033	+	0.999867i	$0.494810\pi$
$444$	0.585786	0.0278002
$445$	0	0
$446$	16.0000	0.757622
$447$	−12.7279	−0.602010
$448$	2.00000	0.0944911
$449$	−17.1716	−0.810377	−0.405188	−	0.914233i	$-0.632794\pi$
$449$	−17.1716	−0.810377	−0.405188	+	0.914233i	$0.632794\pi$
$450$	0	0
$451$	28.9706	1.36417
$452$	16.9706	0.798228
$453$	4.34315	0.204059
$454$	−22.3848	−1.05057
$455$	0	0
$456$	6.24264	0.292338
$457$	−34.9706	−1.63585	−0.817927	−	0.575322i	$-0.804877\pi$
$457$	−34.9706	−1.63585	−0.817927	+	0.575322i	$0.804877\pi$
$458$	−10.2426	−0.478607
$459$	6.82843	0.318724
$460$	0	0
$461$	−7.17157	−0.334013	−0.167007	−	0.985956i	$-0.553410\pi$
$461$	−7.17157	−0.334013	−0.167007	+	0.985956i	$0.553410\pi$
$462$	−8.48528	−0.394771
$463$	30.9706	1.43932	0.719662	−	0.694325i	$-0.244297\pi$
$463$	30.9706	1.43932	0.719662	+	0.694325i	$0.244297\pi$
$464$	3.65685	0.169765
$465$	0	0
$466$	2.14214	0.0992325
$467$	31.3553	1.45095	0.725476	−	0.688247i	$-0.241620\pi$
$467$	31.3553	1.45095	0.725476	+	0.688247i	$0.241620\pi$
$468$	0.828427	0.0382941
$469$	6.82843	0.315307
$470$	0	0
$471$	21.0711	0.970904
$472$	−8.48528	−0.390567
$473$	43.4558	1.99810
$474$	−3.65685	−0.167965
$475$	0	0
$476$	−13.6569	−0.625961
$477$	−10.5858	−0.484690
$478$	16.8284	0.769714
$479$	27.3137	1.24800	0.623998	−	0.781426i	$-0.285507\pi$
$479$	27.3137	1.24800	0.623998	+	0.781426i	$0.285507\pi$
$480$	0	0
$481$	−0.485281	−0.0221269
$482$	−6.48528	−0.295396
$483$	−2.00000	−0.0910032
$484$	7.00000	0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	−39.9411	−1.80991	−0.904953	−	0.425512i	$-0.860094\pi$
$487$	−39.9411	−1.80991	−0.904953	+	0.425512i	$0.860094\pi$
$488$	0.585786	0.0265173
$489$	6.82843	0.308792
$490$	0	0
$491$	−18.6274	−0.840644	−0.420322	−	0.907375i	$-0.638083\pi$
$491$	−18.6274	−0.840644	−0.420322	+	0.907375i	$0.638083\pi$
$492$	6.82843	0.307849
$493$	−24.9706	−1.12462
$494$	−5.17157	−0.232680
$495$	0	0
$496$	6.00000	0.269408
$497$	−11.3137	−0.507489
$498$	−1.41421	−0.0633724
$499$	−37.4558	−1.67675	−0.838377	−	0.545091i	$-0.816495\pi$
$499$	−37.4558	−1.67675	−0.838377	+	0.545091i	$0.816495\pi$
$500$	0	0
$501$	−3.17157	−0.141695
$502$	−0.727922	−0.0324888
$503$	6.14214	0.273864	0.136932	−	0.990580i	$-0.456276\pi$
$503$	6.14214	0.273864	0.136932	+	0.990580i	$0.456276\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	4.24264	0.188608
$507$	12.3137	0.546871
$508$	2.00000	0.0887357
$509$	−10.9706	−0.486262	−0.243131	−	0.969994i	$-0.578174\pi$
$509$	−10.9706	−0.486262	−0.243131	+	0.969994i	$0.578174\pi$
$510$	0	0
$511$	−15.3137	−0.677439
$512$	−1.00000	−0.0441942
$513$	−6.24264	−0.275619
$514$	18.0000	0.793946
$515$	0	0
$516$	10.2426	0.450907
$517$	−3.51472	−0.154577
$518$	1.17157	0.0514760
$519$	8.82843	0.387525
$520$	0	0
$521$	40.2843	1.76489	0.882443	−	0.470419i	$-0.155897\pi$
$521$	40.2843	1.76489	0.882443	+	0.470419i	$0.155897\pi$
$522$	−3.65685	−0.160056
$523$	17.0711	0.746466	0.373233	−	0.927738i	$-0.378249\pi$
$523$	17.0711	0.746466	0.373233	+	0.927738i	$0.378249\pi$
$524$	−21.6569	−0.946084
$525$	0	0
$526$	−6.34315	−0.276574
$527$	−40.9706	−1.78471
$528$	4.24264	0.184637
$529$	1.00000	0.0434783
$530$	0	0
$531$	8.48528	0.368230
$532$	12.4853	0.541306
$533$	−5.65685	−0.245026
$534$	9.17157	0.396893
$535$	0	0
$536$	−3.41421	−0.147472
$537$	21.6569	0.934562
$538$	19.1716	0.826545
$539$	12.7279	0.548230
$540$	0	0
$541$	−31.9411	−1.37326	−0.686628	−	0.727009i	$-0.740910\pi$
$541$	−31.9411	−1.37326	−0.686628	+	0.727009i	$0.740910\pi$
$542$	−9.65685	−0.414797
$543$	20.3848	0.874794
$544$	6.82843	0.292766
$545$	0	0
$546$	1.65685	0.0709068
$547$	−12.4853	−0.533832	−0.266916	−	0.963720i	$-0.586005\pi$
$547$	−12.4853	−0.533832	−0.266916	+	0.963720i	$0.586005\pi$
$548$	8.48528	0.362473
$549$	−0.585786	−0.0250008
$550$	0	0
$551$	22.8284	0.972524
$552$	1.00000	0.0425628
$553$	−7.31371	−0.311011
$554$	13.5147	0.574185
$555$	0	0
$556$	−16.9706	−0.719712
$557$	−38.8701	−1.64698	−0.823489	−	0.567333i	$-0.807975\pi$
$557$	−38.8701	−1.64698	−0.823489	+	0.567333i	$0.807975\pi$
$558$	−6.00000	−0.254000
$559$	−8.48528	−0.358889
$560$	0	0
$561$	−28.9706	−1.22314
$562$	1.85786	0.0783693
$563$	2.10051	0.0885257	0.0442629	−	0.999020i	$-0.485906\pi$
$563$	2.10051	0.0885257	0.0442629	+	0.999020i	$0.485906\pi$
$564$	−0.828427	−0.0348831
$565$	0	0
$566$	15.2132	0.639459
$567$	2.00000	0.0839921
$568$	5.65685	0.237356
$569$	14.3431	0.601296	0.300648	−	0.953735i	$-0.402797\pi$
$569$	14.3431	0.601296	0.300648	+	0.953735i	$0.402797\pi$
$570$	0	0
$571$	−17.2721	−0.722814	−0.361407	−	0.932408i	$-0.617703\pi$
$571$	−17.2721	−0.722814	−0.361407	+	0.932408i	$0.617703\pi$
$572$	−3.51472	−0.146958
$573$	10.8284	0.452364
$574$	13.6569	0.570026
$575$	0	0
$576$	1.00000	0.0416667
$577$	−24.6274	−1.02525	−0.512626	−	0.858612i	$-0.671327\pi$
$577$	−24.6274	−1.02525	−0.512626	+	0.858612i	$0.671327\pi$
$578$	−29.6274	−1.23234
$579$	−20.9706	−0.871507
$580$	0	0
$581$	−2.82843	−0.117343
$582$	6.00000	0.248708
$583$	44.9117	1.86005
$584$	7.65685	0.316843
$585$	0	0
$586$	20.0416	0.827912
$587$	−21.1716	−0.873844	−0.436922	−	0.899499i	$-0.643931\pi$
$587$	−21.1716	−0.873844	−0.436922	+	0.899499i	$0.643931\pi$
$588$	3.00000	0.123718
$589$	37.4558	1.54334
$590$	0	0
$591$	20.6274	0.848499
$592$	−0.585786	−0.0240757
$593$	23.6569	0.971471	0.485735	−	0.874106i	$-0.338552\pi$
$593$	23.6569	0.971471	0.485735	+	0.874106i	$0.338552\pi$
$594$	−4.24264	−0.174078
$595$	0	0
$596$	12.7279	0.521356
$597$	14.0000	0.572982
$598$	−0.828427	−0.0338769
$599$	−11.8579	−0.484499	−0.242250	−	0.970214i	$-0.577885\pi$
$599$	−11.8579	−0.484499	−0.242250	+	0.970214i	$0.577885\pi$
$600$	0	0
$601$	12.9706	0.529080	0.264540	−	0.964375i	$-0.414780\pi$
$601$	12.9706	0.529080	0.264540	+	0.964375i	$0.414780\pi$
$602$	20.4853	0.834918
$603$	3.41421	0.139038
$604$	−4.34315	−0.176720
$605$	0	0
$606$	13.3137	0.540832
$607$	−16.9706	−0.688814	−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706	−0.688814	−0.344407	+	0.938820i	$0.611920\pi$
$608$	−6.24264	−0.253173
$609$	−7.31371	−0.296366
$610$	0	0
$611$	0.686292	0.0277644
$612$	−6.82843	−0.276023
$613$	5.75736	0.232538	0.116269	−	0.993218i	$-0.462907\pi$
$613$	5.75736	0.232538	0.116269	+	0.993218i	$0.462907\pi$
$614$	25.4558	1.02731
$615$	0	0
$616$	8.48528	0.341882
$617$	8.68629	0.349697	0.174848	−	0.984595i	$-0.444056\pi$
$617$	8.68629	0.349697	0.174848	+	0.984595i	$0.444056\pi$
$618$	−16.1421	−0.649332
$619$	23.2132	0.933017	0.466509	−	0.884517i	$-0.345512\pi$
$619$	23.2132	0.933017	0.466509	+	0.884517i	$0.345512\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	21.7990	0.874060
$623$	18.3431	0.734903
$624$	−0.828427	−0.0331636
$625$	0	0
$626$	18.0000	0.719425
$627$	26.4853	1.05772
$628$	−21.0711	−0.840827
$629$	4.00000	0.159490
$630$	0	0
$631$	40.8284	1.62535	0.812677	−	0.582714i	$-0.198009\pi$
$631$	40.8284	1.62535	0.812677	+	0.582714i	$0.198009\pi$
$632$	3.65685	0.145462
$633$	3.51472	0.139698
$634$	−14.4853	−0.575284
$635$	0	0
$636$	10.5858	0.419754
$637$	−2.48528	−0.0984704
$638$	15.5147	0.614234
$639$	−5.65685	−0.223782
$640$	0	0
$641$	48.2843	1.90711	0.953557	−	0.301213i	$-0.0973914\pi$
$641$	48.2843	1.90711	0.953557	+	0.301213i	$0.0973914\pi$
$642$	−0.928932	−0.0366620
$643$	−4.38478	−0.172919	−0.0864593	−	0.996255i	$-0.527555\pi$
$643$	−4.38478	−0.172919	−0.0864593	+	0.996255i	$0.527555\pi$
$644$	2.00000	0.0788110
$645$	0	0
$646$	42.6274	1.67715
$647$	18.3431	0.721143	0.360572	−	0.932731i	$-0.382582\pi$
$647$	18.3431	0.721143	0.360572	+	0.932731i	$0.382582\pi$
$648$	−1.00000	−0.0392837
$649$	−36.0000	−1.41312
$650$	0	0
$651$	−12.0000	−0.470317
$652$	−6.82843	−0.267422
$653$	7.45584	0.291770	0.145885	−	0.989302i	$-0.453397\pi$
$653$	7.45584	0.291770	0.145885	+	0.989302i	$0.453397\pi$
$654$	12.5858	0.492143
$655$	0	0
$656$	−6.82843	−0.266605
$657$	−7.65685	−0.298722
$658$	−1.65685	−0.0645909
$659$	−45.8995	−1.78799	−0.893995	−	0.448076i	$-0.852109\pi$
$659$	−45.8995	−1.78799	−0.893995	+	0.448076i	$0.852109\pi$
$660$	0	0
$661$	34.7279	1.35076	0.675380	−	0.737470i	$-0.263980\pi$
$661$	34.7279	1.35076	0.675380	+	0.737470i	$0.263980\pi$
$662$	20.4853	0.796183
$663$	5.65685	0.219694
$664$	1.41421	0.0548821
$665$	0	0
$666$	0.585786	0.0226988
$667$	3.65685	0.141594
$668$	3.17157	0.122712
$669$	16.0000	0.618596
$670$	0	0
$671$	2.48528	0.0959432
$672$	2.00000	0.0771517
$673$	−27.9411	−1.07705	−0.538526	−	0.842609i	$-0.681018\pi$
$673$	−27.9411	−1.07705	−0.538526	+	0.842609i	$0.681018\pi$
$674$	−26.4853	−1.02017
$675$	0	0
$676$	−12.3137	−0.473604
$677$	40.5269	1.55758	0.778788	−	0.627287i	$-0.215835\pi$
$677$	40.5269	1.55758	0.778788	+	0.627287i	$0.215835\pi$
$678$	16.9706	0.651751
$679$	12.0000	0.460518
$680$	0	0
$681$	−22.3848	−0.857786
$682$	25.4558	0.974755
$683$	8.48528	0.324680	0.162340	−	0.986735i	$-0.448096\pi$
$683$	8.48528	0.324680	0.162340	+	0.986735i	$0.448096\pi$
$684$	6.24264	0.238693
$685$	0	0
$686$	20.0000	0.763604
$687$	−10.2426	−0.390781
$688$	−10.2426	−0.390497
$689$	−8.76955	−0.334093
$690$	0	0
$691$	−13.1716	−0.501070	−0.250535	−	0.968108i	$-0.580607\pi$
$691$	−13.1716	−0.501070	−0.250535	+	0.968108i	$0.580607\pi$
$692$	−8.82843	−0.335606
$693$	−8.48528	−0.322329
$694$	−31.7990	−1.20707
$695$	0	0
$696$	3.65685	0.138613
$697$	46.6274	1.76614
$698$	−2.68629	−0.101678
$699$	2.14214	0.0810230
$700$	0	0
$701$	−38.1838	−1.44218	−0.721090	−	0.692841i	$-0.756359\pi$
$701$	−38.1838	−1.44218	−0.721090	+	0.692841i	$0.756359\pi$
$702$	0.828427	0.0312670
$703$	−3.65685	−0.137921
$704$	−4.24264	−0.159901
$705$	0	0
$706$	2.14214	0.0806203
$707$	26.6274	1.00143
$708$	−8.48528	−0.318896
$709$	32.5858	1.22378	0.611892	−	0.790941i	$-0.290409\pi$
$709$	32.5858	1.22378	0.611892	+	0.790941i	$0.290409\pi$
$710$	0	0
$711$	−3.65685	−0.137143
$712$	−9.17157	−0.343719
$713$	6.00000	0.224702
$714$	−13.6569	−0.511095
$715$	0	0
$716$	−21.6569	−0.809355
$717$	16.8284	0.628469
$718$	−19.7990	−0.738892
$719$	−0.142136	−0.00530076	−0.00265038	−	0.999996i	$-0.500844\pi$
$719$	−0.142136	−0.00530076	−0.00265038	+	0.999996i	$0.500844\pi$
$720$	0	0
$721$	−32.2843	−1.20233
$722$	−19.9706	−0.743227
$723$	−6.48528	−0.241190
$724$	−20.3848	−0.757594
$725$	0	0
$726$	7.00000	0.259794
$727$	20.3431	0.754486	0.377243	−	0.926114i	$-0.376872\pi$
$727$	20.3431	0.754486	0.377243	+	0.926114i	$0.376872\pi$
$728$	−1.65685	−0.0614071
$729$	1.00000	0.0370370
$730$	0	0
$731$	69.9411	2.58687
$732$	0.585786	0.0216513
$733$	−11.4142	−0.421594	−0.210797	−	0.977530i	$-0.567606\pi$
$733$	−11.4142	−0.421594	−0.210797	+	0.977530i	$0.567606\pi$
$734$	1.02944	0.0379972
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−14.4853	−0.533572
$738$	6.82843	0.251358
$739$	−2.62742	−0.0966511	−0.0483255	−	0.998832i	$-0.515388\pi$
$739$	−2.62742	−0.0966511	−0.0483255	+	0.998832i	$0.515388\pi$
$740$	0	0
$741$	−5.17157	−0.189982
$742$	21.1716	0.777233
$743$	−13.6569	−0.501021	−0.250511	−	0.968114i	$-0.580599\pi$
$743$	−13.6569	−0.501021	−0.250511	+	0.968114i	$0.580599\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	−23.2132	−0.849896
$747$	−1.41421	−0.0517434
$748$	28.9706	1.05927
$749$	−1.85786	−0.0678849
$750$	0	0
$751$	−43.1716	−1.57535	−0.787677	−	0.616089i	$-0.788716\pi$
$751$	−43.1716	−1.57535	−0.787677	+	0.616089i	$0.788716\pi$
$752$	0.828427	0.0302096
$753$	−0.727922	−0.0265270
$754$	−3.02944	−0.110326
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	−33.7574	−1.22693	−0.613466	−	0.789721i	$-0.710225\pi$
$757$	−33.7574	−1.22693	−0.613466	+	0.789721i	$0.710225\pi$
$758$	−23.2132	−0.843142
$759$	4.24264	0.153998
$760$	0	0
$761$	−35.6569	−1.29256	−0.646280	−	0.763100i	$-0.723676\pi$
$761$	−35.6569	−1.29256	−0.646280	+	0.763100i	$0.723676\pi$
$762$	2.00000	0.0724524
$763$	25.1716	0.911272
$764$	−10.8284	−0.391759
$765$	0	0
$766$	26.6274	0.962088
$767$	7.02944	0.253818
$768$	−1.00000	−0.0360844
$769$	6.20101	0.223614	0.111807	−	0.993730i	$-0.464336\pi$
$769$	6.20101	0.223614	0.111807	+	0.993730i	$0.464336\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	20.9706	0.754747
$773$	31.0711	1.11755	0.558774	−	0.829320i	$-0.311272\pi$
$773$	31.0711	1.11755	0.558774	+	0.829320i	$0.311272\pi$
$774$	10.2426	0.368164
$775$	0	0
$776$	−6.00000	−0.215387
$777$	1.17157	0.0420299
$778$	30.3848	1.08935
$779$	−42.6274	−1.52729
$780$	0	0
$781$	24.0000	0.858788
$782$	6.82843	0.244184
$783$	−3.65685	−0.130685
$784$	−3.00000	−0.107143
$785$	0	0
$786$	−21.6569	−0.772474
$787$	22.7279	0.810163	0.405081	−	0.914281i	$-0.367243\pi$
$787$	22.7279	0.810163	0.405081	+	0.914281i	$0.367243\pi$
$788$	−20.6274	−0.734821
$789$	−6.34315	−0.225822
$790$	0	0
$791$	33.9411	1.20681
$792$	4.24264	0.150756
$793$	−0.485281	−0.0172328
$794$	−19.6569	−0.697596
$795$	0	0
$796$	−14.0000	−0.496217
$797$	18.5858	0.658342	0.329171	−	0.944270i	$-0.393231\pi$
$797$	18.5858	0.658342	0.329171	+	0.944270i	$0.393231\pi$
$798$	12.4853	0.441974
$799$	−5.65685	−0.200125
$800$	0	0
$801$	9.17157	0.324062
$802$	−4.97056	−0.175517
$803$	32.4853	1.14638
$804$	−3.41421	−0.120410
$805$	0	0
$806$	−4.97056	−0.175081
$807$	19.1716	0.674871
$808$	−13.3137	−0.468375
$809$	−9.31371	−0.327453	−0.163726	−	0.986506i	$-0.552351\pi$
$809$	−9.31371	−0.327453	−0.163726	+	0.986506i	$0.552351\pi$
$810$	0	0
$811$	1.17157	0.0411395	0.0205697	−	0.999788i	$-0.493452\pi$
$811$	1.17157	0.0411395	0.0205697	+	0.999788i	$0.493452\pi$
$812$	7.31371	0.256661
$813$	−9.65685	−0.338681
$814$	−2.48528	−0.0871091
$815$	0	0
$816$	6.82843	0.239043
$817$	−63.9411	−2.23702
$818$	−18.0000	−0.629355
$819$	1.65685	0.0578952
$820$	0	0
$821$	−29.3137	−1.02306	−0.511528	−	0.859267i	$-0.670920\pi$
$821$	−29.3137	−1.02306	−0.511528	+	0.859267i	$0.670920\pi$
$822$	8.48528	0.295958
$823$	24.2843	0.846496	0.423248	−	0.906014i	$-0.360890\pi$
$823$	24.2843	0.846496	0.423248	+	0.906014i	$0.360890\pi$
$824$	16.1421	0.562338
$825$	0	0
$826$	−16.9706	−0.590481
$827$	−37.8995	−1.31789	−0.658947	−	0.752189i	$-0.728998\pi$
$827$	−37.8995	−1.31789	−0.658947	+	0.752189i	$0.728998\pi$
$828$	1.00000	0.0347524
$829$	−6.28427	−0.218262	−0.109131	−	0.994027i	$-0.534807\pi$
$829$	−6.28427	−0.218262	−0.109131	+	0.994027i	$0.534807\pi$
$830$	0	0
$831$	13.5147	0.468820
$832$	0.828427	0.0287205
$833$	20.4853	0.709773
$834$	−16.9706	−0.587643
$835$	0	0
$836$	−26.4853	−0.916013
$837$	−6.00000	−0.207390
$838$	39.0711	1.34969
$839$	33.9411	1.17178	0.585889	−	0.810391i	$-0.300745\pi$
$839$	33.9411	1.17178	0.585889	+	0.810391i	$0.300745\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	−1.75736	−0.0605626
$843$	1.85786	0.0639882
$844$	−3.51472	−0.120982
$845$	0	0
$846$	−0.828427	−0.0284819
$847$	14.0000	0.481046
$848$	−10.5858	−0.363517
$849$	15.2132	0.522116
$850$	0	0
$851$	−0.585786	−0.0200805
$852$	5.65685	0.193801
$853$	21.3137	0.729767	0.364884	−	0.931053i	$-0.381109\pi$
$853$	21.3137	0.729767	0.364884	+	0.931053i	$0.381109\pi$
$854$	1.17157	0.0400904
$855$	0	0
$856$	0.928932	0.0317502
$857$	50.8284	1.73627	0.868133	−	0.496332i	$-0.165320\pi$
$857$	50.8284	1.73627	0.868133	+	0.496332i	$0.165320\pi$
$858$	−3.51472	−0.119991
$859$	24.2843	0.828569	0.414284	−	0.910148i	$-0.364032\pi$
$859$	24.2843	0.828569	0.414284	+	0.910148i	$0.364032\pi$
$860$	0	0
$861$	13.6569	0.465424
$862$	8.48528	0.289010
$863$	22.6274	0.770246	0.385123	−	0.922865i	$-0.374159\pi$
$863$	22.6274	0.770246	0.385123	+	0.922865i	$0.374159\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−6.97056	−0.236869
$867$	−29.6274	−1.00620
$868$	12.0000	0.407307
$869$	15.5147	0.526301
$870$	0	0
$871$	2.82843	0.0958376
$872$	−12.5858	−0.426209
$873$	6.00000	0.203069
$874$	−6.24264	−0.211160
$875$	0	0
$876$	7.65685	0.258701
$877$	16.8284	0.568256	0.284128	−	0.958786i	$-0.408296\pi$
$877$	16.8284	0.568256	0.284128	+	0.958786i	$0.408296\pi$
$878$	0	0
$879$	20.0416	0.675987
$880$	0	0
$881$	26.1421	0.880751	0.440375	−	0.897814i	$-0.354845\pi$
$881$	26.1421	0.880751	0.440375	+	0.897814i	$0.354845\pi$
$882$	3.00000	0.101015
$883$	−18.8284	−0.633627	−0.316814	−	0.948488i	$-0.602613\pi$
$883$	−18.8284	−0.633627	−0.316814	+	0.948488i	$0.602613\pi$
$884$	−5.65685	−0.190261
$885$	0	0
$886$	−0.686292	−0.0230564
$887$	−36.4264	−1.22308	−0.611540	−	0.791214i	$-0.709449\pi$
$887$	−36.4264	−1.22308	−0.611540	+	0.791214i	$0.709449\pi$
$888$	−0.585786	−0.0196577
$889$	4.00000	0.134156
$890$	0	0
$891$	−4.24264	−0.142134
$892$	−16.0000	−0.535720
$893$	5.17157	0.173060
$894$	12.7279	0.425685
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	−0.828427	−0.0276604
$898$	17.1716	0.573023
$899$	21.9411	0.731778
$900$	0	0
$901$	72.2843	2.40814
$902$	−28.9706	−0.964614
$903$	20.4853	0.681707
$904$	−16.9706	−0.564433
$905$	0	0
$906$	−4.34315	−0.144291
$907$	15.2132	0.505146	0.252573	−	0.967578i	$-0.418723\pi$
$907$	15.2132	0.505146	0.252573	+	0.967578i	$0.418723\pi$
$908$	22.3848	0.742865
$909$	13.3137	0.441588
$910$	0	0
$911$	−39.7990	−1.31860	−0.659300	−	0.751880i	$-0.729147\pi$
$911$	−39.7990	−1.31860	−0.659300	+	0.751880i	$0.729147\pi$
$912$	−6.24264	−0.206714
$913$	6.00000	0.198571
$914$	34.9706	1.15672
$915$	0	0
$916$	10.2426	0.338426
$917$	−43.3137	−1.43034
$918$	−6.82843	−0.225372
$919$	38.9706	1.28552	0.642760	−	0.766068i	$-0.277789\pi$
$919$	38.9706	1.28552	0.642760	+	0.766068i	$0.277789\pi$
$920$	0	0
$921$	25.4558	0.838799
$922$	7.17157	0.236183
$923$	−4.68629	−0.154251
$924$	8.48528	0.279145
$925$	0	0
$926$	−30.9706	−1.01776
$927$	−16.1421	−0.530177
$928$	−3.65685	−0.120042
$929$	13.8579	0.454662	0.227331	−	0.973818i	$-0.427000\pi$
$929$	13.8579	0.454662	0.227331	+	0.973818i	$0.427000\pi$
$930$	0	0
$931$	−18.7279	−0.613783
$932$	−2.14214	−0.0701680
$933$	21.7990	0.713667
$934$	−31.3553	−1.02598
$935$	0	0
$936$	−0.828427	−0.0270780
$937$	40.4264	1.32067	0.660337	−	0.750970i	$-0.270414\pi$
$937$	40.4264	1.32067	0.660337	+	0.750970i	$0.270414\pi$
$938$	−6.82843	−0.222956
$939$	18.0000	0.587408
$940$	0	0
$941$	−0.443651	−0.0144626	−0.00723130	−	0.999974i	$-0.502302\pi$
$941$	−0.443651	−0.0144626	−0.00723130	+	0.999974i	$0.502302\pi$
$942$	−21.0711	−0.686532
$943$	−6.82843	−0.222364
$944$	8.48528	0.276172
$945$	0	0
$946$	−43.4558	−1.41287
$947$	49.4558	1.60710	0.803549	−	0.595238i	$-0.202942\pi$
$947$	49.4558	1.60710	0.803549	+	0.595238i	$0.202942\pi$
$948$	3.65685	0.118769
$949$	−6.34315	−0.205907
$950$	0	0
$951$	−14.4853	−0.469717
$952$	13.6569	0.442621
$953$	56.0833	1.81672	0.908358	−	0.418194i	$-0.137337\pi$
$953$	56.0833	1.81672	0.908358	+	0.418194i	$0.137337\pi$
$954$	10.5858	0.342727
$955$	0	0
$956$	−16.8284	−0.544270
$957$	15.5147	0.501520
$958$	−27.3137	−0.882466
$959$	16.9706	0.548008
$960$	0	0
$961$	5.00000	0.161290
$962$	0.485281	0.0156461
$963$	−0.928932	−0.0299344
$964$	6.48528	0.208877
$965$	0	0
$966$	2.00000	0.0643489
$967$	28.9706	0.931630	0.465815	−	0.884882i	$-0.345761\pi$
$967$	28.9706	0.931630	0.465815	+	0.884882i	$0.345761\pi$
$968$	−7.00000	−0.224989
$969$	42.6274	1.36939
$970$	0	0
$971$	−54.3848	−1.74529	−0.872645	−	0.488355i	$-0.837597\pi$
$971$	−54.3848	−1.74529	−0.872645	+	0.488355i	$0.837597\pi$
$972$	−1.00000	−0.0320750
$973$	−33.9411	−1.08810
$974$	39.9411	1.27980
$975$	0	0
$976$	−0.585786	−0.0187506
$977$	19.7990	0.633426	0.316713	−	0.948521i	$-0.397421\pi$
$977$	19.7990	0.633426	0.316713	+	0.948521i	$0.397421\pi$
$978$	−6.82843	−0.218349
$979$	−38.9117	−1.24362
$980$	0	0
$981$	12.5858	0.401833
$982$	18.6274	0.594425
$983$	3.51472	0.112102	0.0560511	−	0.998428i	$-0.482149\pi$
$983$	3.51472	0.112102	0.0560511	+	0.998428i	$0.482149\pi$
$984$	−6.82843	−0.217682
$985$	0	0
$986$	24.9706	0.795225
$987$	−1.65685	−0.0527383
$988$	5.17157	0.164530
$989$	−10.2426	−0.325697
$990$	0	0
$991$	−15.9411	−0.506387	−0.253193	−	0.967416i	$-0.581481\pi$
$991$	−15.9411	−0.506387	−0.253193	+	0.967416i	$0.581481\pi$
$992$	−6.00000	−0.190500
$993$	20.4853	0.650081
$994$	11.3137	0.358849
$995$	0	0
$996$	1.41421	0.0448111
$997$	13.0294	0.412646	0.206323	−	0.978484i	$-0.433850\pi$
$997$	13.0294	0.412646	0.206323	+	0.978484i	$0.433850\pi$
$998$	37.4558	1.18564
$999$	0.585786	0.0185335

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.24264 q^{11} -1.00000 q^{12} +0.828427 q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.82843 q^{17} -1.00000 q^{18} +6.24264 q^{19} -2.00000 q^{21} +4.24264 q^{22} +1.00000 q^{23} +1.00000 q^{24} -0.828427 q^{26} -1.00000 q^{27} +2.00000 q^{28} +3.65685 q^{29} +6.00000 q^{31} -1.00000 q^{32} +4.24264 q^{33} +6.82843 q^{34} +1.00000 q^{36} -0.585786 q^{37} -6.24264 q^{38} -0.828427 q^{39} -6.82843 q^{41} +2.00000 q^{42} -10.2426 q^{43} -4.24264 q^{44} -1.00000 q^{46} +0.828427 q^{47} -1.00000 q^{48} -3.00000 q^{49} +6.82843 q^{51} +0.828427 q^{52} -10.5858 q^{53} +1.00000 q^{54} -2.00000 q^{56} -6.24264 q^{57} -3.65685 q^{58} +8.48528 q^{59} -0.585786 q^{61} -6.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -4.24264 q^{66} +3.41421 q^{67} -6.82843 q^{68} -1.00000 q^{69} -5.65685 q^{71} -1.00000 q^{72} -7.65685 q^{73} +0.585786 q^{74} +6.24264 q^{76} -8.48528 q^{77} +0.828427 q^{78} -3.65685 q^{79} +1.00000 q^{81} +6.82843 q^{82} -1.41421 q^{83} -2.00000 q^{84} +10.2426 q^{86} -3.65685 q^{87} +4.24264 q^{88} +9.17157 q^{89} +1.65685 q^{91} +1.00000 q^{92} -6.00000 q^{93} -0.828427 q^{94} +1.00000 q^{96} +6.00000 q^{97} +3.00000 q^{98} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 4 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{12} - 4 q^{13} - 4 q^{14} + 2 q^{16} - 8 q^{17} - 2 q^{18} + 4 q^{19} - 4 q^{21} + 2 q^{23} + 2 q^{24} + 4 q^{26} - 2 q^{27} + 4 q^{28} - 4 q^{29} + 12 q^{31} - 2 q^{32} + 8 q^{34} + 2 q^{36} - 4 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{41} + 4 q^{42} - 12 q^{43} - 2 q^{46} - 4 q^{47} - 2 q^{48} - 6 q^{49} + 8 q^{51} - 4 q^{52} - 24 q^{53} + 2 q^{54} - 4 q^{56} - 4 q^{57} + 4 q^{58} - 4 q^{61} - 12 q^{62} + 4 q^{63} + 2 q^{64} + 4 q^{67} - 8 q^{68} - 2 q^{69} - 2 q^{72} - 4 q^{73} + 4 q^{74} + 4 q^{76} - 4 q^{78} + 4 q^{79} + 2 q^{81} + 8 q^{82} - 4 q^{84} + 12 q^{86} + 4 q^{87} + 24 q^{89} - 8 q^{91} + 2 q^{92} - 12 q^{93} + 4 q^{94} + 2 q^{96} + 12 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 4 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^12 - 4 * q^13 - 4 * q^14 + 2 * q^16 - 8 * q^17 - 2 * q^18 + 4 * q^19 - 4 * q^21 + 2 * q^23 + 2 * q^24 + 4 * q^26 - 2 * q^27 + 4 * q^28 - 4 * q^29 + 12 * q^31 - 2 * q^32 + 8 * q^34 + 2 * q^36 - 4 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^41 + 4 * q^42 - 12 * q^43 - 2 * q^46 - 4 * q^47 - 2 * q^48 - 6 * q^49 + 8 * q^51 - 4 * q^52 - 24 * q^53 + 2 * q^54 - 4 * q^56 - 4 * q^57 + 4 * q^58 - 4 * q^61 - 12 * q^62 + 4 * q^63 + 2 * q^64 + 4 * q^67 - 8 * q^68 - 2 * q^69 - 2 * q^72 - 4 * q^73 + 4 * q^74 + 4 * q^76 - 4 * q^78 + 4 * q^79 + 2 * q^81 + 8 * q^82 - 4 * q^84 + 12 * q^86 + 4 * q^87 + 24 * q^89 - 8 * q^91 + 2 * q^92 - 12 * q^93 + 4 * q^94 + 2 * q^96 + 12 * q^97 + 6 * q^98

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