LMFDB - Embedded newform 4650.2.a.bz.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.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:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.53113$ of defining polynomial
Character	$\chi$	$=$	4650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.53113 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.53113 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.53113 q^{11} -1.00000 q^{12} -6.00000 q^{13} +3.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -3.53113 q^{19} +3.53113 q^{21} +1.53113 q^{22} +1.53113 q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} -3.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} +1.53113 q^{33} -4.00000 q^{34} +1.00000 q^{36} -9.06226 q^{37} +3.53113 q^{38} +6.00000 q^{39} -9.06226 q^{41} -3.53113 q^{42} +0.468871 q^{43} -1.53113 q^{44} -1.53113 q^{46} -11.0623 q^{47} -1.00000 q^{48} +5.46887 q^{49} -4.00000 q^{51} -6.00000 q^{52} -5.53113 q^{53} +1.00000 q^{54} +3.53113 q^{56} +3.53113 q^{57} +7.06226 q^{59} -11.0623 q^{61} -1.00000 q^{62} -3.53113 q^{63} +1.00000 q^{64} -1.53113 q^{66} +11.0623 q^{67} +4.00000 q^{68} -1.53113 q^{69} -4.46887 q^{71} -1.00000 q^{72} +0.468871 q^{73} +9.06226 q^{74} -3.53113 q^{76} +5.40661 q^{77} -6.00000 q^{78} -0.468871 q^{79} +1.00000 q^{81} +9.06226 q^{82} +8.00000 q^{83} +3.53113 q^{84} -0.468871 q^{86} +1.53113 q^{88} +1.53113 q^{89} +21.1868 q^{91} +1.53113 q^{92} -1.00000 q^{93} +11.0623 q^{94} +1.00000 q^{96} +16.1245 q^{97} -5.46887 q^{98} -1.53113 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} + 5 q^{11} - 2 q^{12} - 12 q^{13} - q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{18} + q^{19} - q^{21} - 5 q^{22} - 5 q^{23} + 2 q^{24} + 12 q^{26} - 2 q^{27} + q^{28} + 2 q^{31} - 2 q^{32} - 5 q^{33} - 8 q^{34} + 2 q^{36} - 2 q^{37} - q^{38} + 12 q^{39} - 2 q^{41} + q^{42} + 9 q^{43} + 5 q^{44} + 5 q^{46} - 6 q^{47} - 2 q^{48} + 19 q^{49} - 8 q^{51} - 12 q^{52} - 3 q^{53} + 2 q^{54} - q^{56} - q^{57} - 2 q^{59} - 6 q^{61} - 2 q^{62} + q^{63} + 2 q^{64} + 5 q^{66} + 6 q^{67} + 8 q^{68} + 5 q^{69} - 17 q^{71} - 2 q^{72} + 9 q^{73} + 2 q^{74} + q^{76} + 35 q^{77} - 12 q^{78} - 9 q^{79} + 2 q^{81} + 2 q^{82} + 16 q^{83} - q^{84} - 9 q^{86} - 5 q^{88} - 5 q^{89} - 6 q^{91} - 5 q^{92} - 2 q^{93} + 6 q^{94} + 2 q^{96} - 19 q^{98} + 5 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9 + 5 * q^11 - 2 * q^12 - 12 * q^13 - q^14 + 2 * q^16 + 8 * q^17 - 2 * q^18 + q^19 - q^21 - 5 * q^22 - 5 * q^23 + 2 * q^24 + 12 * q^26 - 2 * q^27 + q^28 + 2 * q^31 - 2 * q^32 - 5 * q^33 - 8 * q^34 + 2 * q^36 - 2 * q^37 - q^38 + 12 * q^39 - 2 * q^41 + q^42 + 9 * q^43 + 5 * q^44 + 5 * q^46 - 6 * q^47 - 2 * q^48 + 19 * q^49 - 8 * q^51 - 12 * q^52 - 3 * q^53 + 2 * q^54 - q^56 - q^57 - 2 * q^59 - 6 * q^61 - 2 * q^62 + q^63 + 2 * q^64 + 5 * q^66 + 6 * q^67 + 8 * q^68 + 5 * q^69 - 17 * q^71 - 2 * q^72 + 9 * q^73 + 2 * q^74 + q^76 + 35 * q^77 - 12 * q^78 - 9 * q^79 + 2 * q^81 + 2 * q^82 + 16 * q^83 - q^84 - 9 * q^86 - 5 * q^88 - 5 * q^89 - 6 * q^91 - 5 * q^92 - 2 * q^93 + 6 * q^94 + 2 * q^96 - 19 * q^98 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−3.53113	−1.33464	−0.667321	−	0.744771i	$-0.732559\pi$
$7$	−3.53113	−1.33464	−0.667321	+	0.744771i	$0.732559\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.53113	−0.461653	−0.230826	−	0.972995i	$-0.574143\pi$
$11$	−1.53113	−0.461653	−0.230826	+	0.972995i	$0.574143\pi$
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	3.53113	0.943734
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	−3.53113	−0.810097	−0.405048	−	0.914295i	$-0.632745\pi$
$19$	−3.53113	−0.810097	−0.405048	+	0.914295i	$0.632745\pi$
$20$	0	0
$21$	3.53113	0.770555
$22$	1.53113	0.326438
$23$	1.53113	0.319262	0.159631	−	0.987177i	$-0.448969\pi$
$23$	1.53113	0.319262	0.159631	+	0.987177i	$0.448969\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	−1.00000	−0.192450
$28$	−3.53113	−0.667321
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	1.53113	0.266535
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	−9.06226	−1.48983	−0.744913	−	0.667162i	$-0.767509\pi$
$37$	−9.06226	−1.48983	−0.744913	+	0.667162i	$0.767509\pi$
$38$	3.53113	0.572825
$39$	6.00000	0.960769
$40$	0	0
$41$	−9.06226	−1.41529	−0.707643	−	0.706570i	$-0.750242\pi$
$41$	−9.06226	−1.41529	−0.707643	+	0.706570i	$0.750242\pi$
$42$	−3.53113	−0.544865
$43$	0.468871	0.0715022	0.0357511	−	0.999361i	$-0.488618\pi$
$43$	0.468871	0.0715022	0.0357511	+	0.999361i	$0.488618\pi$
$44$	−1.53113	−0.230826
$45$	0	0
$46$	−1.53113	−0.225753
$47$	−11.0623	−1.61360	−0.806798	−	0.590827i	$-0.798801\pi$
$47$	−11.0623	−1.61360	−0.806798	+	0.590827i	$0.798801\pi$
$48$	−1.00000	−0.144338
$49$	5.46887	0.781267
$50$	0	0
$51$	−4.00000	−0.560112
$52$	−6.00000	−0.832050
$53$	−5.53113	−0.759759	−0.379879	−	0.925036i	$-0.624035\pi$
$53$	−5.53113	−0.759759	−0.379879	+	0.925036i	$0.624035\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	3.53113	0.471867
$57$	3.53113	0.467709
$58$	0	0
$59$	7.06226	0.919428	0.459714	−	0.888067i	$-0.347952\pi$
$59$	7.06226	0.919428	0.459714	+	0.888067i	$0.347952\pi$
$60$	0	0
$61$	−11.0623	−1.41638	−0.708188	−	0.706023i	$-0.750487\pi$
$61$	−11.0623	−1.41638	−0.708188	+	0.706023i	$0.750487\pi$
$62$	−1.00000	−0.127000
$63$	−3.53113	−0.444880
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.53113	−0.188469
$67$	11.0623	1.35147	0.675735	−	0.737145i	$-0.263826\pi$
$67$	11.0623	1.35147	0.675735	+	0.737145i	$0.263826\pi$
$68$	4.00000	0.485071
$69$	−1.53113	−0.184326
$70$	0	0
$71$	−4.46887	−0.530357	−0.265179	−	0.964199i	$-0.585431\pi$
$71$	−4.46887	−0.530357	−0.265179	+	0.964199i	$0.585431\pi$
$72$	−1.00000	−0.117851
$73$	0.468871	0.0548772	0.0274386	−	0.999623i	$-0.491265\pi$
$73$	0.468871	0.0548772	0.0274386	+	0.999623i	$0.491265\pi$
$74$	9.06226	1.05347
$75$	0	0
$76$	−3.53113	−0.405048
$77$	5.40661	0.616141
$78$	−6.00000	−0.679366
$79$	−0.468871	−0.0527521	−0.0263761	−	0.999652i	$-0.508397\pi$
$79$	−0.468871	−0.0527521	−0.0263761	+	0.999652i	$0.508397\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	9.06226	1.00076
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	3.53113	0.385278
$85$	0	0
$86$	−0.468871	−0.0505597
$87$	0	0
$88$	1.53113	0.163219
$89$	1.53113	0.162299	0.0811497	−	0.996702i	$-0.474141\pi$
$89$	1.53113	0.162299	0.0811497	+	0.996702i	$0.474141\pi$
$90$	0	0
$91$	21.1868	2.22098
$92$	1.53113	0.159631
$93$	−1.00000	−0.103695
$94$	11.0623	1.14098
$95$	0	0
$96$	1.00000	0.102062
$97$	16.1245	1.63720	0.818598	−	0.574367i	$-0.194752\pi$
$97$	16.1245	1.63720	0.818598	+	0.574367i	$0.194752\pi$
$98$	−5.46887	−0.552439
$99$	−1.53113	−0.153884
$100$	0	0
$101$	−17.5311	−1.74441	−0.872206	−	0.489138i	$-0.837311\pi$
$101$	−17.5311	−1.74441	−0.872206	+	0.489138i	$0.837311\pi$
$102$	4.00000	0.396059
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	5.53113	0.537231
$107$	14.5934	1.41080	0.705398	−	0.708811i	$-0.250768\pi$
$107$	14.5934	1.41080	0.705398	+	0.708811i	$0.250768\pi$
$108$	−1.00000	−0.0962250
$109$	−1.06226	−0.101746	−0.0508729	−	0.998705i	$-0.516200\pi$
$109$	−1.06226	−0.101746	−0.0508729	+	0.998705i	$0.516200\pi$
$110$	0	0
$111$	9.06226	0.860151
$112$	−3.53113	−0.333660
$113$	−16.5934	−1.56097	−0.780487	−	0.625172i	$-0.785029\pi$
$113$	−16.5934	−1.56097	−0.780487	+	0.625172i	$0.785029\pi$
$114$	−3.53113	−0.330721
$115$	0	0
$116$	0	0
$117$	−6.00000	−0.554700
$118$	−7.06226	−0.650134
$119$	−14.1245	−1.29479
$120$	0	0
$121$	−8.65564	−0.786877
$122$	11.0623	1.00153
$123$	9.06226	0.817116
$124$	1.00000	0.0898027
$125$	0	0
$126$	3.53113	0.314578
$127$	9.06226	0.804145	0.402073	−	0.915608i	$-0.368290\pi$
$127$	9.06226	0.804145	0.402073	+	0.915608i	$0.368290\pi$
$128$	−1.00000	−0.0883883
$129$	−0.468871	−0.0412818
$130$	0	0
$131$	7.06226	0.617032	0.308516	−	0.951219i	$-0.400168\pi$
$131$	7.06226	0.617032	0.308516	+	0.951219i	$0.400168\pi$
$132$	1.53113	0.133268
$133$	12.4689	1.08119
$134$	−11.0623	−0.955634
$135$	0	0
$136$	−4.00000	−0.342997
$137$	−0.937742	−0.0801167	−0.0400584	−	0.999197i	$-0.512754\pi$
$137$	−0.937742	−0.0801167	−0.0400584	+	0.999197i	$0.512754\pi$
$138$	1.53113	0.130338
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	0	0
$141$	11.0623	0.931610
$142$	4.46887	0.375019
$143$	9.18677	0.768237
$144$	1.00000	0.0833333
$145$	0	0
$146$	−0.468871	−0.0388041
$147$	−5.46887	−0.451065
$148$	−9.06226	−0.744913
$149$	−10.4689	−0.857643	−0.428822	−	0.903389i	$-0.641071\pi$
$149$	−10.4689	−0.857643	−0.428822	+	0.903389i	$0.641071\pi$
$150$	0	0
$151$	−18.1245	−1.47495	−0.737476	−	0.675373i	$-0.763983\pi$
$151$	−18.1245	−1.47495	−0.737476	+	0.675373i	$0.763983\pi$
$152$	3.53113	0.286412
$153$	4.00000	0.323381
$154$	−5.40661	−0.435677
$155$	0	0
$156$	6.00000	0.480384
$157$	14.4689	1.15474	0.577371	−	0.816482i	$-0.304079\pi$
$157$	14.4689	1.15474	0.577371	+	0.816482i	$0.304079\pi$
$158$	0.468871	0.0373014
$159$	5.53113	0.438647
$160$	0	0
$161$	−5.40661	−0.426101
$162$	−1.00000	−0.0785674
$163$	11.0623	0.866463	0.433231	−	0.901283i	$-0.357373\pi$
$163$	11.0623	0.866463	0.433231	+	0.901283i	$0.357373\pi$
$164$	−9.06226	−0.707643
$165$	0	0
$166$	−8.00000	−0.620920
$167$	−0.593387	−0.0459176	−0.0229588	−	0.999736i	$-0.507309\pi$
$167$	−0.593387	−0.0459176	−0.0229588	+	0.999736i	$0.507309\pi$
$168$	−3.53113	−0.272433
$169$	23.0000	1.76923
$170$	0	0
$171$	−3.53113	−0.270032
$172$	0.468871	0.0357511
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	0	0
$176$	−1.53113	−0.115413
$177$	−7.06226	−0.530832
$178$	−1.53113	−0.114763
$179$	13.0623	0.976319	0.488159	−	0.872754i	$-0.337668\pi$
$179$	13.0623	0.976319	0.488159	+	0.872754i	$0.337668\pi$
$180$	0	0
$181$	26.5934	1.97667	0.988335	−	0.152293i	$-0.0486657\pi$
$181$	26.5934	1.97667	0.988335	+	0.152293i	$0.0486657\pi$
$182$	−21.1868	−1.57047
$183$	11.0623	0.817746
$184$	−1.53113	−0.112876
$185$	0	0
$186$	1.00000	0.0733236
$187$	−6.12452	−0.447869
$188$	−11.0623	−0.806798
$189$	3.53113	0.256852
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−16.1245	−1.15767
$195$	0	0
$196$	5.46887	0.390634
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	1.53113	0.108813
$199$	−0.468871	−0.0332374	−0.0166187	−	0.999862i	$-0.505290\pi$
$199$	−0.468871	−0.0332374	−0.0166187	+	0.999862i	$0.505290\pi$
$200$	0	0
$201$	−11.0623	−0.780272
$202$	17.5311	1.23349
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	0	0
$207$	1.53113	0.106421
$208$	−6.00000	−0.416025
$209$	5.40661	0.373983
$210$	0	0
$211$	22.5934	1.55539	0.777696	−	0.628640i	$-0.216388\pi$
$211$	22.5934	1.55539	0.777696	+	0.628640i	$0.216388\pi$
$212$	−5.53113	−0.379879
$213$	4.46887	0.306202
$214$	−14.5934	−0.997583
$215$	0	0
$216$	1.00000	0.0680414
$217$	−3.53113	−0.239709
$218$	1.06226	0.0719452
$219$	−0.468871	−0.0316834
$220$	0	0
$221$	−24.0000	−1.61441
$222$	−9.06226	−0.608219
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	3.53113	0.235933
$225$	0	0
$226$	16.5934	1.10378
$227$	16.4689	1.09308	0.546539	−	0.837434i	$-0.315945\pi$
$227$	16.4689	1.09308	0.546539	+	0.837434i	$0.315945\pi$
$228$	3.53113	0.233855
$229$	18.5934	1.22869	0.614343	−	0.789039i	$-0.289421\pi$
$229$	18.5934	1.22869	0.614343	+	0.789039i	$0.289421\pi$
$230$	0	0
$231$	−5.40661	−0.355729
$232$	0	0
$233$	−9.53113	−0.624405	−0.312203	−	0.950016i	$-0.601067\pi$
$233$	−9.53113	−0.624405	−0.312203	+	0.950016i	$0.601067\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	7.06226	0.459714
$237$	0.468871	0.0304565
$238$	14.1245	0.915556
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	8.65564	0.556406
$243$	−1.00000	−0.0641500
$244$	−11.0623	−0.708188
$245$	0	0
$246$	−9.06226	−0.577788
$247$	21.1868	1.34808
$248$	−1.00000	−0.0635001
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−13.0623	−0.824482	−0.412241	−	0.911075i	$-0.635254\pi$
$251$	−13.0623	−0.824482	−0.412241	+	0.911075i	$0.635254\pi$
$252$	−3.53113	−0.222440
$253$	−2.34436	−0.147388
$254$	−9.06226	−0.568617
$255$	0	0
$256$	1.00000	0.0625000
$257$	27.6556	1.72511	0.862556	−	0.505962i	$-0.168862\pi$
$257$	27.6556	1.72511	0.862556	+	0.505962i	$0.168862\pi$
$258$	0.468871	0.0291906
$259$	32.0000	1.98838
$260$	0	0
$261$	0	0
$262$	−7.06226	−0.436308
$263$	6.93774	0.427800	0.213900	−	0.976856i	$-0.431383\pi$
$263$	6.93774	0.427800	0.213900	+	0.976856i	$0.431383\pi$
$264$	−1.53113	−0.0942345
$265$	0	0
$266$	−12.4689	−0.764516
$267$	−1.53113	−0.0937036
$268$	11.0623	0.675735
$269$	29.1868	1.77955	0.889774	−	0.456400i	$-0.150862\pi$
$269$	29.1868	1.77955	0.889774	+	0.456400i	$0.150862\pi$
$270$	0	0
$271$	−19.5311	−1.18643	−0.593216	−	0.805043i	$-0.702142\pi$
$271$	−19.5311	−1.18643	−0.593216	+	0.805043i	$0.702142\pi$
$272$	4.00000	0.242536
$273$	−21.1868	−1.28228
$274$	0.937742	0.0566511
$275$	0	0
$276$	−1.53113	−0.0921631
$277$	−1.06226	−0.0638249	−0.0319124	−	0.999491i	$-0.510160\pi$
$277$	−1.06226	−0.0638249	−0.0319124	+	0.999491i	$0.510160\pi$
$278$	−6.00000	−0.359856
$279$	1.00000	0.0598684
$280$	0	0
$281$	−1.06226	−0.0633690	−0.0316845	−	0.999498i	$-0.510087\pi$
$281$	−1.06226	−0.0633690	−0.0316845	+	0.999498i	$0.510087\pi$
$282$	−11.0623	−0.658748
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−4.46887	−0.265179
$285$	0	0
$286$	−9.18677	−0.543225
$287$	32.0000	1.88890
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−16.1245	−0.945236
$292$	0.468871	0.0274386
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	5.46887	0.318951
$295$	0	0
$296$	9.06226	0.526733
$297$	1.53113	0.0888451
$298$	10.4689	0.606445
$299$	−9.18677	−0.531285
$300$	0	0
$301$	−1.65564	−0.0954298
$302$	18.1245	1.04295
$303$	17.5311	1.00714
$304$	−3.53113	−0.202524
$305$	0	0
$306$	−4.00000	−0.228665
$307$	−2.12452	−0.121253	−0.0606263	−	0.998161i	$-0.519310\pi$
$307$	−2.12452	−0.121253	−0.0606263	+	0.998161i	$0.519310\pi$
$308$	5.40661	0.308070
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	−6.00000	−0.339683
$313$	−27.0623	−1.52965	−0.764825	−	0.644239i	$-0.777174\pi$
$313$	−27.0623	−1.52965	−0.764825	+	0.644239i	$0.777174\pi$
$314$	−14.4689	−0.816526
$315$	0	0
$316$	−0.468871	−0.0263761
$317$	−29.0623	−1.63230	−0.816150	−	0.577841i	$-0.803895\pi$
$317$	−29.0623	−1.63230	−0.816150	+	0.577841i	$0.803895\pi$
$318$	−5.53113	−0.310170
$319$	0	0
$320$	0	0
$321$	−14.5934	−0.814523
$322$	5.40661	0.301299
$323$	−14.1245	−0.785909
$324$	1.00000	0.0555556
$325$	0	0
$326$	−11.0623	−0.612682
$327$	1.06226	0.0587430
$328$	9.06226	0.500379
$329$	39.0623	2.15357
$330$	0	0
$331$	29.0623	1.59741	0.798703	−	0.601725i	$-0.205520\pi$
$331$	29.0623	1.59741	0.798703	+	0.601725i	$0.205520\pi$
$332$	8.00000	0.439057
$333$	−9.06226	−0.496609
$334$	0.593387	0.0324687
$335$	0	0
$336$	3.53113	0.192639
$337$	−29.1868	−1.58990	−0.794952	−	0.606672i	$-0.792504\pi$
$337$	−29.1868	−1.58990	−0.794952	+	0.606672i	$0.792504\pi$
$338$	−23.0000	−1.25104
$339$	16.5934	0.901229
$340$	0	0
$341$	−1.53113	−0.0829153
$342$	3.53113	0.190942
$343$	5.40661	0.291930
$344$	−0.468871	−0.0252798
$345$	0	0
$346$	14.0000	0.752645
$347$	22.1245	1.18771	0.593853	−	0.804573i	$-0.297606\pi$
$347$	22.1245	1.18771	0.593853	+	0.804573i	$0.297606\pi$
$348$	0	0
$349$	25.0623	1.34155	0.670776	−	0.741660i	$-0.265961\pi$
$349$	25.0623	1.34155	0.670776	+	0.741660i	$0.265961\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	1.53113	0.0816094
$353$	23.0623	1.22748	0.613740	−	0.789508i	$-0.289664\pi$
$353$	23.0623	1.22748	0.613740	+	0.789508i	$0.289664\pi$
$354$	7.06226	0.375355
$355$	0	0
$356$	1.53113	0.0811497
$357$	14.1245	0.747549
$358$	−13.0623	−0.690362
$359$	11.5311	0.608590	0.304295	−	0.952578i	$-0.401579\pi$
$359$	11.5311	0.608590	0.304295	+	0.952578i	$0.401579\pi$
$360$	0	0
$361$	−6.53113	−0.343744
$362$	−26.5934	−1.39772
$363$	8.65564	0.454304
$364$	21.1868	1.11049
$365$	0	0
$366$	−11.0623	−0.578233
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	1.53113	0.0798156
$369$	−9.06226	−0.471762
$370$	0	0
$371$	19.5311	1.01401
$372$	−1.00000	−0.0518476
$373$	10.4689	0.542058	0.271029	−	0.962571i	$-0.412636\pi$
$373$	10.4689	0.542058	0.271029	+	0.962571i	$0.412636\pi$
$374$	6.12452	0.316691
$375$	0	0
$376$	11.0623	0.570492
$377$	0	0
$378$	−3.53113	−0.181622
$379$	−2.59339	−0.133213	−0.0666067	−	0.997779i	$-0.521217\pi$
$379$	−2.59339	−0.133213	−0.0666067	+	0.997779i	$0.521217\pi$
$380$	0	0
$381$	−9.06226	−0.464274
$382$	8.00000	0.409316
$383$	−13.0623	−0.667450	−0.333725	−	0.942670i	$-0.608306\pi$
$383$	−13.0623	−0.667450	−0.333725	+	0.942670i	$0.608306\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	0.468871	0.0238341
$388$	16.1245	0.818598
$389$	27.0623	1.37211	0.686055	−	0.727549i	$-0.259341\pi$
$389$	27.0623	1.37211	0.686055	+	0.727549i	$0.259341\pi$
$390$	0	0
$391$	6.12452	0.309730
$392$	−5.46887	−0.276220
$393$	−7.06226	−0.356244
$394$	6.00000	0.302276
$395$	0	0
$396$	−1.53113	−0.0769421
$397$	18.7179	0.939425	0.469712	−	0.882820i	$-0.344358\pi$
$397$	18.7179	0.939425	0.469712	+	0.882820i	$0.344358\pi$
$398$	0.468871	0.0235024
$399$	−12.4689	−0.624224
$400$	0	0
$401$	−34.7179	−1.73373	−0.866865	−	0.498544i	$-0.833868\pi$
$401$	−34.7179	−1.73373	−0.866865	+	0.498544i	$0.833868\pi$
$402$	11.0623	0.551735
$403$	−6.00000	−0.298881
$404$	−17.5311	−0.872206
$405$	0	0
$406$	0	0
$407$	13.8755	0.687782
$408$	4.00000	0.198030
$409$	−9.06226	−0.448100	−0.224050	−	0.974578i	$-0.571928\pi$
$409$	−9.06226	−0.448100	−0.224050	+	0.974578i	$0.571928\pi$
$410$	0	0
$411$	0.937742	0.0462554
$412$	0	0
$413$	−24.9377	−1.22711
$414$	−1.53113	−0.0752509
$415$	0	0
$416$	6.00000	0.294174
$417$	−6.00000	−0.293821
$418$	−5.40661	−0.264446
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	38.2490	1.86414	0.932072	−	0.362273i	$-0.117999\pi$
$421$	38.2490	1.86414	0.932072	+	0.362273i	$0.117999\pi$
$422$	−22.5934	−1.09983
$423$	−11.0623	−0.537865
$424$	5.53113	0.268615
$425$	0	0
$426$	−4.46887	−0.216518
$427$	39.0623	1.89036
$428$	14.5934	0.705398
$429$	−9.18677	−0.443542
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−1.00000	−0.0481125
$433$	1.40661	0.0675975	0.0337988	−	0.999429i	$-0.489239\pi$
$433$	1.40661	0.0675975	0.0337988	+	0.999429i	$0.489239\pi$
$434$	3.53113	0.169500
$435$	0	0
$436$	−1.06226	−0.0508729
$437$	−5.40661	−0.258633
$438$	0.468871	0.0224035
$439$	−8.93774	−0.426575	−0.213288	−	0.976989i	$-0.568417\pi$
$439$	−8.93774	−0.426575	−0.213288	+	0.976989i	$0.568417\pi$
$440$	0	0
$441$	5.46887	0.260422
$442$	24.0000	1.14156
$443$	−38.5934	−1.83363	−0.916814	−	0.399316i	$-0.869248\pi$
$443$	−38.5934	−1.83363	−0.916814	+	0.399316i	$0.869248\pi$
$444$	9.06226	0.430076
$445$	0	0
$446$	2.00000	0.0947027
$447$	10.4689	0.495161
$448$	−3.53113	−0.166830
$449$	28.1245	1.32728	0.663639	−	0.748053i	$-0.269011\pi$
$449$	28.1245	1.32728	0.663639	+	0.748053i	$0.269011\pi$
$450$	0	0
$451$	13.8755	0.653371
$452$	−16.5934	−0.780487
$453$	18.1245	0.851564
$454$	−16.4689	−0.772922
$455$	0	0
$456$	−3.53113	−0.165360
$457$	−31.0623	−1.45303	−0.726516	−	0.687150i	$-0.758862\pi$
$457$	−31.0623	−1.45303	−0.726516	+	0.687150i	$0.758862\pi$
$458$	−18.5934	−0.868812
$459$	−4.00000	−0.186704
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	5.40661	0.251538
$463$	35.1868	1.63527	0.817634	−	0.575738i	$-0.195285\pi$
$463$	35.1868	1.63527	0.817634	+	0.575738i	$0.195285\pi$
$464$	0	0
$465$	0	0
$466$	9.53113	0.441521
$467$	−10.1245	−0.468507	−0.234253	−	0.972176i	$-0.575265\pi$
$467$	−10.1245	−0.468507	−0.234253	+	0.972176i	$0.575265\pi$
$468$	−6.00000	−0.277350
$469$	−39.0623	−1.80373
$470$	0	0
$471$	−14.4689	−0.666690
$472$	−7.06226	−0.325067
$473$	−0.717902	−0.0330092
$474$	−0.468871	−0.0215360
$475$	0	0
$476$	−14.1245	−0.647396
$477$	−5.53113	−0.253253
$478$	8.00000	0.365911
$479$	41.6556	1.90329	0.951647	−	0.307192i	$-0.0993895\pi$
$479$	41.6556	1.90329	0.951647	+	0.307192i	$0.0993895\pi$
$480$	0	0
$481$	54.3735	2.47922
$482$	−2.00000	−0.0910975
$483$	5.40661	0.246009
$484$	−8.65564	−0.393438
$485$	0	0
$486$	1.00000	0.0453609
$487$	−6.93774	−0.314379	−0.157190	−	0.987568i	$-0.550243\pi$
$487$	−6.93774	−0.314379	−0.157190	+	0.987568i	$0.550243\pi$
$488$	11.0623	0.500765
$489$	−11.0623	−0.500253
$490$	0	0
$491$	16.5934	0.748849	0.374425	−	0.927257i	$-0.377840\pi$
$491$	16.5934	0.748849	0.374425	+	0.927257i	$0.377840\pi$
$492$	9.06226	0.408558
$493$	0	0
$494$	−21.1868	−0.953238
$495$	0	0
$496$	1.00000	0.0449013
$497$	15.7802	0.707837
$498$	8.00000	0.358489
$499$	9.06226	0.405682	0.202841	−	0.979212i	$-0.434982\pi$
$499$	9.06226	0.405682	0.202841	+	0.979212i	$0.434982\pi$
$500$	0	0
$501$	0.593387	0.0265106
$502$	13.0623	0.582997
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	3.53113	0.157289
$505$	0	0
$506$	2.34436	0.104219
$507$	−23.0000	−1.02147
$508$	9.06226	0.402073
$509$	5.87548	0.260426	0.130213	−	0.991486i	$-0.458434\pi$
$509$	5.87548	0.260426	0.130213	+	0.991486i	$0.458434\pi$
$510$	0	0
$511$	−1.65564	−0.0732414
$512$	−1.00000	−0.0441942
$513$	3.53113	0.155903
$514$	−27.6556	−1.21984
$515$	0	0
$516$	−0.468871	−0.0206409
$517$	16.9377	0.744921
$518$	−32.0000	−1.40600
$519$	14.0000	0.614532
$520$	0	0
$521$	20.1245	0.881671	0.440836	−	0.897588i	$-0.354682\pi$
$521$	20.1245	0.881671	0.440836	+	0.897588i	$0.354682\pi$
$522$	0	0
$523$	14.5934	0.638124	0.319062	−	0.947734i	$-0.396632\pi$
$523$	14.5934	0.638124	0.319062	+	0.947734i	$0.396632\pi$
$524$	7.06226	0.308516
$525$	0	0
$526$	−6.93774	−0.302500
$527$	4.00000	0.174243
$528$	1.53113	0.0666338
$529$	−20.6556	−0.898071
$530$	0	0
$531$	7.06226	0.306476
$532$	12.4689	0.540594
$533$	54.3735	2.35518
$534$	1.53113	0.0662584
$535$	0	0
$536$	−11.0623	−0.477817
$537$	−13.0623	−0.563678
$538$	−29.1868	−1.25833
$539$	−8.37355	−0.360674
$540$	0	0
$541$	28.1245	1.20917	0.604584	−	0.796542i	$-0.293340\pi$
$541$	28.1245	1.20917	0.604584	+	0.796542i	$0.293340\pi$
$542$	19.5311	0.838934
$543$	−26.5934	−1.14123
$544$	−4.00000	−0.171499
$545$	0	0
$546$	21.1868	0.906710
$547$	−41.1868	−1.76102	−0.880510	−	0.474028i	$-0.842799\pi$
$547$	−41.1868	−1.76102	−0.880510	+	0.474028i	$0.842799\pi$
$548$	−0.937742	−0.0400584
$549$	−11.0623	−0.472126
$550$	0	0
$551$	0	0
$552$	1.53113	0.0651692
$553$	1.65564	0.0704052
$554$	1.06226	0.0451310
$555$	0	0
$556$	6.00000	0.254457
$557$	−45.7802	−1.93977	−0.969884	−	0.243568i	$-0.921682\pi$
$557$	−45.7802	−1.93977	−0.969884	+	0.243568i	$0.921682\pi$
$558$	−1.00000	−0.0423334
$559$	−2.81323	−0.118987
$560$	0	0
$561$	6.12452	0.258577
$562$	1.06226	0.0448086
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	11.0623	0.465805
$565$	0	0
$566$	−12.0000	−0.504398
$567$	−3.53113	−0.148293
$568$	4.46887	0.187510
$569$	17.5311	0.734943	0.367472	−	0.930035i	$-0.380224\pi$
$569$	17.5311	0.734943	0.367472	+	0.930035i	$0.380224\pi$
$570$	0	0
$571$	3.87548	0.162184	0.0810920	−	0.996707i	$-0.474159\pi$
$571$	3.87548	0.162184	0.0810920	+	0.996707i	$0.474159\pi$
$572$	9.18677	0.384118
$573$	8.00000	0.334205
$574$	−32.0000	−1.33565
$575$	0	0
$576$	1.00000	0.0416667
$577$	11.1868	0.465711	0.232856	−	0.972511i	$-0.425193\pi$
$577$	11.1868	0.465711	0.232856	+	0.972511i	$0.425193\pi$
$578$	1.00000	0.0415945
$579$	14.0000	0.581820
$580$	0	0
$581$	−28.2490	−1.17197
$582$	16.1245	0.668383
$583$	8.46887	0.350745
$584$	−0.468871	−0.0194020
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−20.2490	−0.835767	−0.417883	−	0.908501i	$-0.637228\pi$
$587$	−20.2490	−0.835767	−0.417883	+	0.908501i	$0.637228\pi$
$588$	−5.46887	−0.225532
$589$	−3.53113	−0.145498
$590$	0	0
$591$	6.00000	0.246807
$592$	−9.06226	−0.372456
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	−1.53113	−0.0628230
$595$	0	0
$596$	−10.4689	−0.428822
$597$	0.468871	0.0191896
$598$	9.18677	0.375675
$599$	10.5934	0.432834	0.216417	−	0.976301i	$-0.430563\pi$
$599$	10.5934	0.432834	0.216417	+	0.976301i	$0.430563\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	1.65564	0.0674790
$603$	11.0623	0.450490
$604$	−18.1245	−0.737476
$605$	0	0
$606$	−17.5311	−0.712153
$607$	38.5934	1.56646	0.783229	−	0.621734i	$-0.213571\pi$
$607$	38.5934	1.56646	0.783229	+	0.621734i	$0.213571\pi$
$608$	3.53113	0.143206
$609$	0	0
$610$	0	0
$611$	66.3735	2.68519
$612$	4.00000	0.161690
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	2.12452	0.0857385
$615$	0	0
$616$	−5.40661	−0.217839
$617$	−20.5934	−0.829059	−0.414529	−	0.910036i	$-0.636054\pi$
$617$	−20.5934	−0.829059	−0.414529	+	0.910036i	$0.636054\pi$
$618$	0	0
$619$	−34.0000	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$619$	−34.0000	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$620$	0	0
$621$	−1.53113	−0.0614421
$622$	8.00000	0.320771
$623$	−5.40661	−0.216611
$624$	6.00000	0.240192
$625$	0	0
$626$	27.0623	1.08163
$627$	−5.40661	−0.215919
$628$	14.4689	0.577371
$629$	−36.2490	−1.44534
$630$	0	0
$631$	−9.65564	−0.384385	−0.192193	−	0.981357i	$-0.561560\pi$
$631$	−9.65564	−0.384385	−0.192193	+	0.981357i	$0.561560\pi$
$632$	0.468871	0.0186507
$633$	−22.5934	−0.898006
$634$	29.0623	1.15421
$635$	0	0
$636$	5.53113	0.219324
$637$	−32.8132	−1.30011
$638$	0	0
$639$	−4.46887	−0.176786
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	14.5934	0.575955
$643$	−6.59339	−0.260018	−0.130009	−	0.991513i	$-0.541501\pi$
$643$	−6.59339	−0.260018	−0.130009	+	0.991513i	$0.541501\pi$
$644$	−5.40661	−0.213050
$645$	0	0
$646$	14.1245	0.555722
$647$	−1.53113	−0.0601949	−0.0300974	−	0.999547i	$-0.509582\pi$
$647$	−1.53113	−0.0601949	−0.0300974	+	0.999547i	$0.509582\pi$
$648$	−1.00000	−0.0392837
$649$	−10.8132	−0.424456
$650$	0	0
$651$	3.53113	0.138396
$652$	11.0623	0.433231
$653$	26.2490	1.02720	0.513602	−	0.858029i	$-0.328311\pi$
$653$	26.2490	1.02720	0.513602	+	0.858029i	$0.328311\pi$
$654$	−1.06226	−0.0415376
$655$	0	0
$656$	−9.06226	−0.353822
$657$	0.468871	0.0182924
$658$	−39.0623	−1.52281
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−29.0623	−1.12954
$663$	24.0000	0.932083
$664$	−8.00000	−0.310460
$665$	0	0
$666$	9.06226	0.351155
$667$	0	0
$668$	−0.593387	−0.0229588
$669$	2.00000	0.0773245
$670$	0	0
$671$	16.9377	0.653874
$672$	−3.53113	−0.136216
$673$	−7.06226	−0.272230	−0.136115	−	0.990693i	$-0.543462\pi$
$673$	−7.06226	−0.272230	−0.136115	+	0.990693i	$0.543462\pi$
$674$	29.1868	1.12423
$675$	0	0
$676$	23.0000	0.884615
$677$	−3.40661	−0.130927	−0.0654634	−	0.997855i	$-0.520853\pi$
$677$	−3.40661	−0.130927	−0.0654634	+	0.997855i	$0.520853\pi$
$678$	−16.5934	−0.637265
$679$	−56.9377	−2.18507
$680$	0	0
$681$	−16.4689	−0.631089
$682$	1.53113	0.0586300
$683$	−15.5311	−0.594282	−0.297141	−	0.954834i	$-0.596033\pi$
$683$	−15.5311	−0.594282	−0.297141	+	0.954834i	$0.596033\pi$
$684$	−3.53113	−0.135016
$685$	0	0
$686$	−5.40661	−0.206425
$687$	−18.5934	−0.709382
$688$	0.468871	0.0178755
$689$	33.1868	1.26432
$690$	0	0
$691$	−7.53113	−0.286498	−0.143249	−	0.989687i	$-0.545755\pi$
$691$	−7.53113	−0.286498	−0.143249	+	0.989687i	$0.545755\pi$
$692$	−14.0000	−0.532200
$693$	5.40661	0.205380
$694$	−22.1245	−0.839835
$695$	0	0
$696$	0	0
$697$	−36.2490	−1.37303
$698$	−25.0623	−0.948620
$699$	9.53113	0.360500
$700$	0	0
$701$	−21.5311	−0.813220	−0.406610	−	0.913602i	$-0.633289\pi$
$701$	−21.5311	−0.813220	−0.406610	+	0.913602i	$0.633289\pi$
$702$	−6.00000	−0.226455
$703$	32.0000	1.20690
$704$	−1.53113	−0.0577066
$705$	0	0
$706$	−23.0623	−0.867960
$707$	61.9047	2.32816
$708$	−7.06226	−0.265416
$709$	25.4066	0.954165	0.477083	−	0.878858i	$-0.341694\pi$
$709$	25.4066	0.954165	0.477083	+	0.878858i	$0.341694\pi$
$710$	0	0
$711$	−0.468871	−0.0175840
$712$	−1.53113	−0.0573815
$713$	1.53113	0.0573412
$714$	−14.1245	−0.528597
$715$	0	0
$716$	13.0623	0.488159
$717$	8.00000	0.298765
$718$	−11.5311	−0.430338
$719$	−33.1868	−1.23766	−0.618829	−	0.785526i	$-0.712393\pi$
$719$	−33.1868	−1.23766	−0.618829	+	0.785526i	$0.712393\pi$
$720$	0	0
$721$	0	0
$722$	6.53113	0.243063
$723$	−2.00000	−0.0743808
$724$	26.5934	0.988335
$725$	0	0
$726$	−8.65564	−0.321241
$727$	50.8424	1.88564	0.942820	−	0.333301i	$-0.108163\pi$
$727$	50.8424	1.88564	0.942820	+	0.333301i	$0.108163\pi$
$728$	−21.1868	−0.785234
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.87548	0.0693673
$732$	11.0623	0.408873
$733$	−4.12452	−0.152342	−0.0761712	−	0.997095i	$-0.524270\pi$
$733$	−4.12452	−0.152342	−0.0761712	+	0.997095i	$0.524270\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	−1.53113	−0.0564382
$737$	−16.9377	−0.623910
$738$	9.06226	0.333586
$739$	27.1868	1.00008	0.500041	−	0.866002i	$-0.333318\pi$
$739$	27.1868	1.00008	0.500041	+	0.866002i	$0.333318\pi$
$740$	0	0
$741$	−21.1868	−0.778316
$742$	−19.5311	−0.717010
$743$	−11.6556	−0.427604	−0.213802	−	0.976877i	$-0.568585\pi$
$743$	−11.6556	−0.427604	−0.213802	+	0.976877i	$0.568585\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	−10.4689	−0.383293
$747$	8.00000	0.292705
$748$	−6.12452	−0.223934
$749$	−51.5311	−1.88291
$750$	0	0
$751$	47.0623	1.71733	0.858663	−	0.512540i	$-0.171296\pi$
$751$	47.0623	1.71733	0.858663	+	0.512540i	$0.171296\pi$
$752$	−11.0623	−0.403399
$753$	13.0623	0.476015
$754$	0	0
$755$	0	0
$756$	3.53113	0.128426
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	2.59339	0.0941960
$759$	2.34436	0.0850947
$760$	0	0
$761$	−12.5934	−0.456510	−0.228255	−	0.973601i	$-0.573302\pi$
$761$	−12.5934	−0.456510	−0.228255	+	0.973601i	$0.573302\pi$
$762$	9.06226	0.328291
$763$	3.75097	0.135794
$764$	−8.00000	−0.289430
$765$	0	0
$766$	13.0623	0.471959
$767$	−42.3735	−1.53002
$768$	−1.00000	−0.0360844
$769$	−28.5934	−1.03110	−0.515552	−	0.856858i	$-0.672413\pi$
$769$	−28.5934	−1.03110	−0.515552	+	0.856858i	$0.672413\pi$
$770$	0	0
$771$	−27.6556	−0.995994
$772$	−14.0000	−0.503871
$773$	−14.4689	−0.520409	−0.260205	−	0.965554i	$-0.583790\pi$
$773$	−14.4689	−0.520409	−0.260205	+	0.965554i	$0.583790\pi$
$774$	−0.468871	−0.0168532
$775$	0	0
$776$	−16.1245	−0.578836
$777$	−32.0000	−1.14799
$778$	−27.0623	−0.970229
$779$	32.0000	1.14652
$780$	0	0
$781$	6.84242	0.244841
$782$	−6.12452	−0.219012
$783$	0	0
$784$	5.46887	0.195317
$785$	0	0
$786$	7.06226	0.251902
$787$	28.4689	1.01481	0.507403	−	0.861709i	$-0.330606\pi$
$787$	28.4689	1.01481	0.507403	+	0.861709i	$0.330606\pi$
$788$	−6.00000	−0.213741
$789$	−6.93774	−0.246990
$790$	0	0
$791$	58.5934	2.08334
$792$	1.53113	0.0544063
$793$	66.3735	2.35699
$794$	−18.7179	−0.664273
$795$	0	0
$796$	−0.468871	−0.0166187
$797$	20.1245	0.712847	0.356423	−	0.934325i	$-0.383996\pi$
$797$	20.1245	0.712847	0.356423	+	0.934325i	$0.383996\pi$
$798$	12.4689	0.441393
$799$	−44.2490	−1.56542
$800$	0	0
$801$	1.53113	0.0540998
$802$	34.7179	1.22593
$803$	−0.717902	−0.0253342
$804$	−11.0623	−0.390136
$805$	0	0
$806$	6.00000	0.211341
$807$	−29.1868	−1.02742
$808$	17.5311	0.616743
$809$	−8.59339	−0.302127	−0.151064	−	0.988524i	$-0.548270\pi$
$809$	−8.59339	−0.302127	−0.151064	+	0.988524i	$0.548270\pi$
$810$	0	0
$811$	−39.7802	−1.39687	−0.698435	−	0.715673i	$-0.746120\pi$
$811$	−39.7802	−1.39687	−0.698435	+	0.715673i	$0.746120\pi$
$812$	0	0
$813$	19.5311	0.684987
$814$	−13.8755	−0.486335
$815$	0	0
$816$	−4.00000	−0.140028
$817$	−1.65564	−0.0579237
$818$	9.06226	0.316854
$819$	21.1868	0.740326
$820$	0	0
$821$	9.87548	0.344657	0.172328	−	0.985040i	$-0.444871\pi$
$821$	9.87548	0.344657	0.172328	+	0.985040i	$0.444871\pi$
$822$	−0.937742	−0.0327075
$823$	−7.87548	−0.274522	−0.137261	−	0.990535i	$-0.543830\pi$
$823$	−7.87548	−0.274522	−0.137261	+	0.990535i	$0.543830\pi$
$824$	0	0
$825$	0	0
$826$	24.9377	0.867695
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	1.53113	0.0532104
$829$	−9.65564	−0.335354	−0.167677	−	0.985842i	$-0.553627\pi$
$829$	−9.65564	−0.335354	−0.167677	+	0.985842i	$0.553627\pi$
$830$	0	0
$831$	1.06226	0.0368493
$832$	−6.00000	−0.208013
$833$	21.8755	0.757941
$834$	6.00000	0.207763
$835$	0	0
$836$	5.40661	0.186992
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−54.8424	−1.89337	−0.946685	−	0.322160i	$-0.895591\pi$
$839$	−54.8424	−1.89337	−0.946685	+	0.322160i	$0.895591\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−38.2490	−1.31815
$843$	1.06226	0.0365861
$844$	22.5934	0.777696
$845$	0	0
$846$	11.0623	0.380328
$847$	30.5642	1.05020
$848$	−5.53113	−0.189940
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−13.8755	−0.475645
$852$	4.46887	0.153101
$853$	−1.28210	−0.0438982	−0.0219491	−	0.999759i	$-0.506987\pi$
$853$	−1.28210	−0.0438982	−0.0219491	+	0.999759i	$0.506987\pi$
$854$	−39.0623	−1.33668
$855$	0	0
$856$	−14.5934	−0.498792
$857$	−14.0000	−0.478231	−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000	−0.478231	−0.239115	+	0.970991i	$0.576857\pi$
$858$	9.18677	0.313631
$859$	−6.93774	−0.236713	−0.118356	−	0.992971i	$-0.537763\pi$
$859$	−6.93774	−0.236713	−0.118356	+	0.992971i	$0.537763\pi$
$860$	0	0
$861$	−32.0000	−1.09056
$862$	24.0000	0.817443
$863$	−41.5311	−1.41374	−0.706868	−	0.707345i	$-0.749893\pi$
$863$	−41.5311	−1.41374	−0.706868	+	0.707345i	$0.749893\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−1.40661	−0.0477987
$867$	1.00000	0.0339618
$868$	−3.53113	−0.119854
$869$	0.717902	0.0243532
$870$	0	0
$871$	−66.3735	−2.24898
$872$	1.06226	0.0359726
$873$	16.1245	0.545732
$874$	5.40661	0.182881
$875$	0	0
$876$	−0.468871	−0.0158417
$877$	−44.1245	−1.48998	−0.744990	−	0.667076i	$-0.767546\pi$
$877$	−44.1245	−1.48998	−0.744990	+	0.667076i	$0.767546\pi$
$878$	8.93774	0.301634
$879$	−14.0000	−0.472208
$880$	0	0
$881$	36.1245	1.21707	0.608533	−	0.793529i	$-0.291758\pi$
$881$	36.1245	1.21707	0.608533	+	0.793529i	$0.291758\pi$
$882$	−5.46887	−0.184146
$883$	53.6556	1.80566	0.902828	−	0.430002i	$-0.141487\pi$
$883$	53.6556	1.80566	0.902828	+	0.430002i	$0.141487\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	38.5934	1.29657
$887$	25.1868	0.845689	0.422845	−	0.906202i	$-0.361032\pi$
$887$	25.1868	0.845689	0.422845	+	0.906202i	$0.361032\pi$
$888$	−9.06226	−0.304109
$889$	−32.0000	−1.07325
$890$	0	0
$891$	−1.53113	−0.0512947
$892$	−2.00000	−0.0669650
$893$	39.0623	1.30717
$894$	−10.4689	−0.350131
$895$	0	0
$896$	3.53113	0.117967
$897$	9.18677	0.306737
$898$	−28.1245	−0.938527
$899$	0	0
$900$	0	0
$901$	−22.1245	−0.737074
$902$	−13.8755	−0.462003
$903$	1.65564	0.0550964
$904$	16.5934	0.551888
$905$	0	0
$906$	−18.1245	−0.602147
$907$	32.2490	1.07081	0.535406	−	0.844595i	$-0.320159\pi$
$907$	32.2490	1.07081	0.535406	+	0.844595i	$0.320159\pi$
$908$	16.4689	0.546539
$909$	−17.5311	−0.581471
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	3.53113	0.116927
$913$	−12.2490	−0.405384
$914$	31.0623	1.02745
$915$	0	0
$916$	18.5934	0.614343
$917$	−24.9377	−0.823517
$918$	4.00000	0.132020
$919$	−8.93774	−0.294829	−0.147414	−	0.989075i	$-0.547095\pi$
$919$	−8.93774	−0.294829	−0.147414	+	0.989075i	$0.547095\pi$
$920$	0	0
$921$	2.12452	0.0700052
$922$	−24.0000	−0.790398
$923$	26.8132	0.882568
$924$	−5.40661	−0.177865
$925$	0	0
$926$	−35.1868	−1.15631
$927$	0	0
$928$	0	0
$929$	36.8424	1.20876	0.604380	−	0.796696i	$-0.293421\pi$
$929$	36.8424	1.20876	0.604380	+	0.796696i	$0.293421\pi$
$930$	0	0
$931$	−19.3113	−0.632902
$932$	−9.53113	−0.312203
$933$	8.00000	0.261908
$934$	10.1245	0.331284
$935$	0	0
$936$	6.00000	0.196116
$937$	25.0623	0.818748	0.409374	−	0.912367i	$-0.365747\pi$
$937$	25.0623	0.818748	0.409374	+	0.912367i	$0.365747\pi$
$938$	39.0623	1.27543
$939$	27.0623	0.883143
$940$	0	0
$941$	−21.8755	−0.713120	−0.356560	−	0.934272i	$-0.616051\pi$
$941$	−21.8755	−0.713120	−0.356560	+	0.934272i	$0.616051\pi$
$942$	14.4689	0.471421
$943$	−13.8755	−0.451848
$944$	7.06226	0.229857
$945$	0	0
$946$	0.717902	0.0233410
$947$	−35.0623	−1.13937	−0.569685	−	0.821863i	$-0.692935\pi$
$947$	−35.0623	−1.13937	−0.569685	+	0.821863i	$0.692935\pi$
$948$	0.468871	0.0152282
$949$	−2.81323	−0.0913212
$950$	0	0
$951$	29.0623	0.942408
$952$	14.1245	0.457778
$953$	−54.1245	−1.75327	−0.876633	−	0.481161i	$-0.840215\pi$
$953$	−54.1245	−1.75327	−0.876633	+	0.481161i	$0.840215\pi$
$954$	5.53113	0.179077
$955$	0	0
$956$	−8.00000	−0.258738
$957$	0	0
$958$	−41.6556	−1.34583
$959$	3.31129	0.106927
$960$	0	0
$961$	1.00000	0.0322581
$962$	−54.3735	−1.75307
$963$	14.5934	0.470265
$964$	2.00000	0.0644157
$965$	0	0
$966$	−5.40661	−0.173955
$967$	17.0623	0.548685	0.274343	−	0.961632i	$-0.411540\pi$
$967$	17.0623	0.548685	0.274343	+	0.961632i	$0.411540\pi$
$968$	8.65564	0.278203
$969$	14.1245	0.453745
$970$	0	0
$971$	−33.1868	−1.06501	−0.532507	−	0.846426i	$-0.678750\pi$
$971$	−33.1868	−1.06501	−0.532507	+	0.846426i	$0.678750\pi$
$972$	−1.00000	−0.0320750
$973$	−21.1868	−0.679217
$974$	6.93774	0.222300
$975$	0	0
$976$	−11.0623	−0.354094
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	11.0623	0.353732
$979$	−2.34436	−0.0749259
$980$	0	0
$981$	−1.06226	−0.0339153
$982$	−16.5934	−0.529516
$983$	−17.0623	−0.544202	−0.272101	−	0.962269i	$-0.587718\pi$
$983$	−17.0623	−0.544202	−0.272101	+	0.962269i	$0.587718\pi$
$984$	−9.06226	−0.288894
$985$	0	0
$986$	0	0
$987$	−39.0623	−1.24337
$988$	21.1868	0.674041
$989$	0.717902	0.0228280
$990$	0	0
$991$	−24.4689	−0.777279	−0.388640	−	0.921390i	$-0.627055\pi$
$991$	−24.4689	−0.777279	−0.388640	+	0.921390i	$0.627055\pi$
$992$	−1.00000	−0.0317500
$993$	−29.0623	−0.922263
$994$	−15.7802	−0.500516
$995$	0	0
$996$	−8.00000	−0.253490
$997$	−62.2490	−1.97145	−0.985723	−	0.168373i	$-0.946149\pi$
$997$	−62.2490	−1.97145	−0.985723	+	0.168373i	$0.946149\pi$
$998$	−9.06226	−0.286861
$999$	9.06226	0.286717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.bz.1.1		2
5.2	odd	4		4650.2.d.bg.3349.1		4
5.3	odd	4		4650.2.d.bg.3349.4		4
5.4	even	2		930.2.a.q.1.2	✓	2
15.14	odd	2		2790.2.a.bf.1.2		2
20.19	odd	2		7440.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.q.1.2	✓	2	5.4	even	2
2790.2.a.bf.1.2		2	15.14	odd	2
4650.2.a.bz.1.1		2	1.1	even	1	trivial
4650.2.d.bg.3349.1		4	5.2	odd	4
4650.2.d.bg.3349.4		4	5.3	odd	4
7440.2.a.bd.1.1		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.53113 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.53113 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.53113 q^{11} -1.00000 q^{12} -6.00000 q^{13} +3.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -3.53113 q^{19} +3.53113 q^{21} +1.53113 q^{22} +1.53113 q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} -3.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} +1.53113 q^{33} -4.00000 q^{34} +1.00000 q^{36} -9.06226 q^{37} +3.53113 q^{38} +6.00000 q^{39} -9.06226 q^{41} -3.53113 q^{42} +0.468871 q^{43} -1.53113 q^{44} -1.53113 q^{46} -11.0623 q^{47} -1.00000 q^{48} +5.46887 q^{49} -4.00000 q^{51} -6.00000 q^{52} -5.53113 q^{53} +1.00000 q^{54} +3.53113 q^{56} +3.53113 q^{57} +7.06226 q^{59} -11.0623 q^{61} -1.00000 q^{62} -3.53113 q^{63} +1.00000 q^{64} -1.53113 q^{66} +11.0623 q^{67} +4.00000 q^{68} -1.53113 q^{69} -4.46887 q^{71} -1.00000 q^{72} +0.468871 q^{73} +9.06226 q^{74} -3.53113 q^{76} +5.40661 q^{77} -6.00000 q^{78} -0.468871 q^{79} +1.00000 q^{81} +9.06226 q^{82} +8.00000 q^{83} +3.53113 q^{84} -0.468871 q^{86} +1.53113 q^{88} +1.53113 q^{89} +21.1868 q^{91} +1.53113 q^{92} -1.00000 q^{93} +11.0623 q^{94} +1.00000 q^{96} +16.1245 q^{97} -5.46887 q^{98} -1.53113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} + 5 q^{11} - 2 q^{12} - 12 q^{13} - q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{18} + q^{19} - q^{21} - 5 q^{22} - 5 q^{23} + 2 q^{24} + 12 q^{26} - 2 q^{27} + q^{28} + 2 q^{31} - 2 q^{32} - 5 q^{33} - 8 q^{34} + 2 q^{36} - 2 q^{37} - q^{38} + 12 q^{39} - 2 q^{41} + q^{42} + 9 q^{43} + 5 q^{44} + 5 q^{46} - 6 q^{47} - 2 q^{48} + 19 q^{49} - 8 q^{51} - 12 q^{52} - 3 q^{53} + 2 q^{54} - q^{56} - q^{57} - 2 q^{59} - 6 q^{61} - 2 q^{62} + q^{63} + 2 q^{64} + 5 q^{66} + 6 q^{67} + 8 q^{68} + 5 q^{69} - 17 q^{71} - 2 q^{72} + 9 q^{73} + 2 q^{74} + q^{76} + 35 q^{77} - 12 q^{78} - 9 q^{79} + 2 q^{81} + 2 q^{82} + 16 q^{83} - q^{84} - 9 q^{86} - 5 q^{88} - 5 q^{89} - 6 q^{91} - 5 q^{92} - 2 q^{93} + 6 q^{94} + 2 q^{96} - 19 q^{98} + 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9 + 5 * q^11 - 2 * q^12 - 12 * q^13 - q^14 + 2 * q^16 + 8 * q^17 - 2 * q^18 + q^19 - q^21 - 5 * q^22 - 5 * q^23 + 2 * q^24 + 12 * q^26 - 2 * q^27 + q^28 + 2 * q^31 - 2 * q^32 - 5 * q^33 - 8 * q^34 + 2 * q^36 - 2 * q^37 - q^38 + 12 * q^39 - 2 * q^41 + q^42 + 9 * q^43 + 5 * q^44 + 5 * q^46 - 6 * q^47 - 2 * q^48 + 19 * q^49 - 8 * q^51 - 12 * q^52 - 3 * q^53 + 2 * q^54 - q^56 - q^57 - 2 * q^59 - 6 * q^61 - 2 * q^62 + q^63 + 2 * q^64 + 5 * q^66 + 6 * q^67 + 8 * q^68 + 5 * q^69 - 17 * q^71 - 2 * q^72 + 9 * q^73 + 2 * q^74 + q^76 + 35 * q^77 - 12 * q^78 - 9 * q^79 + 2 * q^81 + 2 * q^82 + 16 * q^83 - q^84 - 9 * q^86 - 5 * q^88 - 5 * q^89 - 6 * q^91 - 5 * q^92 - 2 * q^93 + 6 * q^94 + 2 * q^96 - 19 * q^98 + 5 * q^99

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