LMFDB - Embedded newform 4650.2.d.bg.3349.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4650,2,Mod(3349,4650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4650.3349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,4,0,0,-4,0,10,0,0,2,0,4,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{65})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 33x^{2} + 256 $ x^4 + 33*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.4
Root			$3.53113i$ of defining polynomial
Character	$\chi$	$=$	4650.3349
Dual form			4650.2.d.bg.3349.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	3.53113i	1.33464i	0.744771	+	0.667321i	$0.232559\pi$
$7$	3.53113i	1.33464i	−0.744771	+	0.667321i	$0.767441\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−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	$0.574143\pi$
$12$	1.00000i	0.288675i
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	−3.53113	−0.943734
$15$	0	0
$16$	1.00000	0.250000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	− 1.00000i	− 0.235702i
$19$	3.53113	0.810097	0.405048	−	0.914295i	$-0.367255\pi$
$19$	3.53113	0.810097	0.405048	+	0.914295i	$0.367255\pi$
$20$	0	0
$21$	3.53113	0.770555
$22$	− 1.53113i	− 0.326438i
$23$	1.53113i	0.319262i	0.987177	+	0.159631i	$0.0510305\pi$
$23$	1.53113i	0.319262i	−0.987177	+	0.159631i	$0.948969\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	1.00000i	0.192450i
$28$	− 3.53113i	− 0.667321i
$29$	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
$32$	1.00000i	0.176777i
$33$	1.53113i	0.266535i
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	9.06226i	1.48983i	0.667162	+	0.744913i	$0.267509\pi$
$37$	9.06226i	1.48983i	−0.667162	+	0.744913i	$0.732491\pi$
$38$	3.53113i	0.572825i
$39$	−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	$0.750242\pi$
$42$	3.53113i	0.544865i
$43$	0.468871i	0.0715022i	0.999361	+	0.0357511i	$0.0113824\pi$
$43$	0.468871i	0.0715022i	−0.999361	+	0.0357511i	$0.988618\pi$
$44$	1.53113	0.230826
$45$	0	0
$46$	−1.53113	−0.225753
$47$	11.0623i	1.61360i	0.590827	+	0.806798i	$0.298801\pi$
$47$	11.0623i	1.61360i	−0.590827	+	0.806798i	$0.701199\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−5.46887	−0.781267
$50$	0	0
$51$	−4.00000	−0.560112
$52$	6.00000i	0.832050i
$53$	− 5.53113i	− 0.759759i	−0.925036	−	0.379879i	$-0.875965\pi$
$53$	− 5.53113i	− 0.759759i	0.925036	−	0.379879i	$-0.124035\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.53113	0.471867
$57$	− 3.53113i	− 0.467709i
$58$	0	0
$59$	−7.06226	−0.919428	−0.459714	−	0.888067i	$-0.652048\pi$
$59$	−7.06226	−0.919428	−0.459714	+	0.888067i	$0.652048\pi$
$60$	0	0
$61$	−11.0623	−1.41638	−0.708188	−	0.706023i	$-0.750487\pi$
$61$	−11.0623	−1.41638	−0.708188	+	0.706023i	$0.750487\pi$
$62$	1.00000i	0.127000i
$63$	− 3.53113i	− 0.444880i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.53113	−0.188469
$67$	− 11.0623i	− 1.35147i	−0.737145	−	0.675735i	$-0.763826\pi$
$67$	− 11.0623i	− 1.35147i	0.737145	−	0.675735i	$-0.236174\pi$
$68$	4.00000i	0.485071i
$69$	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	$0.585431\pi$
$72$	1.00000i	0.117851i
$73$	0.468871i	0.0548772i	0.999623	+	0.0274386i	$0.00873508\pi$
$73$	0.468871i	0.0548772i	−0.999623	+	0.0274386i	$0.991265\pi$
$74$	−9.06226	−1.05347
$75$	0	0
$76$	−3.53113	−0.405048
$77$	− 5.40661i	− 0.616141i
$78$	− 6.00000i	− 0.679366i
$79$	0.468871	0.0527521	0.0263761	−	0.999652i	$-0.491603\pi$
$79$	0.468871	0.0527521	0.0263761	+	0.999652i	$0.491603\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 9.06226i	− 1.00076i
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	−3.53113	−0.385278
$85$	0	0
$86$	−0.468871	−0.0505597
$87$	0	0
$88$	1.53113i	0.163219i
$89$	−1.53113	−0.162299	−0.0811497	−	0.996702i	$-0.525859\pi$
$89$	−1.53113	−0.162299	−0.0811497	+	0.996702i	$0.525859\pi$
$90$	0	0
$91$	21.1868	2.22098
$92$	− 1.53113i	− 0.159631i
$93$	− 1.00000i	− 0.103695i
$94$	−11.0623	−1.14098
$95$	0	0
$96$	1.00000	0.102062
$97$	− 16.1245i	− 1.63720i	−0.574367	−	0.818598i	$-0.694752\pi$
$97$	− 16.1245i	− 1.63720i	0.574367	−	0.818598i	$-0.305248\pi$
$98$	− 5.46887i	− 0.552439i
$99$	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	$0.837311\pi$
$102$	− 4.00000i	− 0.396059i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	5.53113	0.537231
$107$	− 14.5934i	− 1.41080i	−0.708811	−	0.705398i	$-0.750768\pi$
$107$	− 14.5934i	− 1.41080i	0.708811	−	0.705398i	$-0.249232\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	1.06226	0.101746	0.0508729	−	0.998705i	$-0.483800\pi$
$109$	1.06226	0.101746	0.0508729	+	0.998705i	$0.483800\pi$
$110$	0	0
$111$	9.06226	0.860151
$112$	3.53113i	0.333660i
$113$	− 16.5934i	− 1.56097i	−0.625172	−	0.780487i	$-0.714971\pi$
$113$	− 16.5934i	− 1.56097i	0.625172	−	0.780487i	$-0.285029\pi$
$114$	3.53113	0.330721
$115$	0	0
$116$	0	0
$117$	6.00000i	0.554700i
$118$	− 7.06226i	− 0.650134i
$119$	14.1245	1.29479
$120$	0	0
$121$	−8.65564	−0.786877
$122$	− 11.0623i	− 1.00153i
$123$	9.06226i	0.817116i
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	3.53113	0.314578
$127$	− 9.06226i	− 0.804145i	−0.915608	−	0.402073i	$-0.868290\pi$
$127$	− 9.06226i	− 0.804145i	0.915608	−	0.402073i	$-0.131710\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	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	$0.400168\pi$
$132$	− 1.53113i	− 0.133268i
$133$	12.4689i	1.08119i
$134$	11.0623	0.955634
$135$	0	0
$136$	−4.00000	−0.342997
$137$	0.937742i	0.0801167i	0.999197	+	0.0400584i	$0.0127544\pi$
$137$	0.937742i	0.0801167i	−0.999197	+	0.0400584i	$0.987246\pi$
$138$	1.53113i	0.130338i
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	11.0623	0.931610
$142$	− 4.46887i	− 0.375019i
$143$	9.18677i	0.768237i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−0.468871	−0.0388041
$147$	5.46887i	0.451065i
$148$	− 9.06226i	− 0.744913i
$149$	10.4689	0.857643	0.428822	−	0.903389i	$-0.358929\pi$
$149$	10.4689	0.857643	0.428822	+	0.903389i	$0.358929\pi$
$150$	0	0
$151$	−18.1245	−1.47495	−0.737476	−	0.675373i	$-0.763983\pi$
$151$	−18.1245	−1.47495	−0.737476	+	0.675373i	$0.763983\pi$
$152$	− 3.53113i	− 0.286412i
$153$	4.00000i	0.323381i
$154$	5.40661	0.435677
$155$	0	0
$156$	6.00000	0.480384
$157$	− 14.4689i	− 1.15474i	−0.816482	−	0.577371i	$-0.804079\pi$
$157$	− 14.4689i	− 1.15474i	0.816482	−	0.577371i	$-0.195921\pi$
$158$	0.468871i	0.0373014i
$159$	−5.53113	−0.438647
$160$	0	0
$161$	−5.40661	−0.426101
$162$	1.00000i	0.0785674i
$163$	11.0623i	0.866463i	0.901283	+	0.433231i	$0.142627\pi$
$163$	11.0623i	0.866463i	−0.901283	+	0.433231i	$0.857373\pi$
$164$	9.06226	0.707643
$165$	0	0
$166$	−8.00000	−0.620920
$167$	0.593387i	0.0459176i	0.999736	+	0.0229588i	$0.00730866\pi$
$167$	0.593387i	0.0459176i	−0.999736	+	0.0229588i	$0.992691\pi$
$168$	− 3.53113i	− 0.272433i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−3.53113	−0.270032
$172$	− 0.468871i	− 0.0357511i
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	0	0
$176$	−1.53113	−0.115413
$177$	7.06226i	0.530832i
$178$	− 1.53113i	− 0.114763i
$179$	−13.0623	−0.976319	−0.488159	−	0.872754i	$-0.662332\pi$
$179$	−13.0623	−0.976319	−0.488159	+	0.872754i	$0.662332\pi$
$180$	0	0
$181$	26.5934	1.97667	0.988335	−	0.152293i	$-0.0486657\pi$
$181$	26.5934	1.97667	0.988335	+	0.152293i	$0.0486657\pi$
$182$	21.1868i	1.57047i
$183$	11.0623i	0.817746i
$184$	1.53113	0.112876
$185$	0	0
$186$	1.00000	0.0733236
$187$	6.12452i	0.447869i
$188$	− 11.0623i	− 0.806798i
$189$	−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	$0.593466\pi$
$192$	1.00000i	0.0721688i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	16.1245	1.15767
$195$	0	0
$196$	5.46887	0.390634
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	1.53113i	0.108813i
$199$	0.468871	0.0332374	0.0166187	−	0.999862i	$-0.494710\pi$
$199$	0.468871	0.0332374	0.0166187	+	0.999862i	$0.494710\pi$
$200$	0	0
$201$	−11.0623	−0.780272
$202$	− 17.5311i	− 1.23349i
$203$	0	0
$204$	4.00000	0.280056
$205$	0	0
$206$	0	0
$207$	− 1.53113i	− 0.106421i
$208$	− 6.00000i	− 0.416025i
$209$	−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	$0.216388\pi$
$212$	5.53113i	0.379879i
$213$	4.46887i	0.306202i
$214$	14.5934	0.997583
$215$	0	0
$216$	1.00000	0.0680414
$217$	3.53113i	0.239709i
$218$	1.06226i	0.0719452i
$219$	0.468871	0.0316834
$220$	0	0
$221$	−24.0000	−1.61441
$222$	9.06226i	0.608219i
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	−3.53113	−0.235933
$225$	0	0
$226$	16.5934	1.10378
$227$	− 16.4689i	− 1.09308i	−0.837434	−	0.546539i	$-0.815945\pi$
$227$	− 16.4689i	− 1.09308i	0.837434	−	0.546539i	$-0.184055\pi$
$228$	3.53113i	0.233855i
$229$	−18.5934	−1.22869	−0.614343	−	0.789039i	$-0.710579\pi$
$229$	−18.5934	−1.22869	−0.614343	+	0.789039i	$0.710579\pi$
$230$	0	0
$231$	−5.40661	−0.355729
$232$	0	0
$233$	− 9.53113i	− 0.624405i	−0.950016	−	0.312203i	$-0.898933\pi$
$233$	− 9.53113i	− 0.624405i	0.950016	−	0.312203i	$-0.101067\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	7.06226	0.459714
$237$	− 0.468871i	− 0.0304565i
$238$	14.1245i	0.915556i
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	− 8.65564i	− 0.556406i
$243$	− 1.00000i	− 0.0641500i
$244$	11.0623	0.708188
$245$	0	0
$246$	−9.06226	−0.577788
$247$	− 21.1868i	− 1.34808i
$248$	− 1.00000i	− 0.0635001i
$249$	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	$0.635254\pi$
$252$	3.53113i	0.222440i
$253$	− 2.34436i	− 0.147388i
$254$	9.06226	0.568617
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 27.6556i	− 1.72511i	−0.505962	−	0.862556i	$-0.668862\pi$
$257$	− 27.6556i	− 1.72511i	0.505962	−	0.862556i	$-0.331138\pi$
$258$	0.468871i	0.0291906i
$259$	−32.0000	−1.98838
$260$	0	0
$261$	0	0
$262$	7.06226i	0.436308i
$263$	6.93774i	0.427800i	0.976856	+	0.213900i	$0.0686166\pi$
$263$	6.93774i	0.427800i	−0.976856	+	0.213900i	$0.931383\pi$
$264$	1.53113	0.0942345
$265$	0	0
$266$	−12.4689	−0.764516
$267$	1.53113i	0.0937036i
$268$	11.0623i	0.675735i
$269$	−29.1868	−1.77955	−0.889774	−	0.456400i	$-0.849138\pi$
$269$	−29.1868	−1.77955	−0.889774	+	0.456400i	$0.849138\pi$
$270$	0	0
$271$	−19.5311	−1.18643	−0.593216	−	0.805043i	$-0.702142\pi$
$271$	−19.5311	−1.18643	−0.593216	+	0.805043i	$0.702142\pi$
$272$	− 4.00000i	− 0.242536i
$273$	− 21.1868i	− 1.28228i
$274$	−0.937742	−0.0566511
$275$	0	0
$276$	−1.53113	−0.0921631
$277$	1.06226i	0.0638249i	0.999491	+	0.0319124i	$0.0101598\pi$
$277$	1.06226i	0.0638249i	−0.999491	+	0.0319124i	$0.989840\pi$
$278$	− 6.00000i	− 0.359856i
$279$	−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	$0.510087\pi$
$282$	11.0623i	0.658748i
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	4.46887	0.265179
$285$	0	0
$286$	−9.18677	−0.543225
$287$	− 32.0000i	− 1.88890i
$288$	− 1.00000i	− 0.0589256i
$289$	1.00000	0.0588235
$290$	0	0
$291$	−16.1245	−0.945236
$292$	− 0.468871i	− 0.0274386i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	−5.46887	−0.318951
$295$	0	0
$296$	9.06226	0.526733
$297$	− 1.53113i	− 0.0888451i
$298$	10.4689i	0.606445i
$299$	9.18677	0.531285
$300$	0	0
$301$	−1.65564	−0.0954298
$302$	− 18.1245i	− 1.04295i
$303$	17.5311i	1.00714i
$304$	3.53113	0.202524
$305$	0	0
$306$	−4.00000	−0.228665
$307$	2.12452i	0.121253i	0.998161	+	0.0606263i	$0.0193098\pi$
$307$	2.12452i	0.121253i	−0.998161	+	0.0606263i	$0.980690\pi$
$308$	5.40661i	0.308070i
$309$	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	$0.572833\pi$
$312$	6.00000i	0.339683i
$313$	− 27.0623i	− 1.52965i	−0.644239	−	0.764825i	$-0.722826\pi$
$313$	− 27.0623i	− 1.52965i	0.644239	−	0.764825i	$-0.277174\pi$
$314$	14.4689	0.816526
$315$	0	0
$316$	−0.468871	−0.0263761
$317$	29.0623i	1.63230i	0.577841	+	0.816150i	$0.303895\pi$
$317$	29.0623i	1.63230i	−0.577841	+	0.816150i	$0.696105\pi$
$318$	− 5.53113i	− 0.310170i
$319$	0	0
$320$	0	0
$321$	−14.5934	−0.814523
$322$	− 5.40661i	− 0.301299i
$323$	− 14.1245i	− 0.785909i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−11.0623	−0.612682
$327$	− 1.06226i	− 0.0587430i
$328$	9.06226i	0.500379i
$329$	−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	$0.205520\pi$
$332$	− 8.00000i	− 0.439057i
$333$	− 9.06226i	− 0.496609i
$334$	−0.593387	−0.0324687
$335$	0	0
$336$	3.53113	0.192639
$337$	29.1868i	1.58990i	0.606672	+	0.794952i	$0.292504\pi$
$337$	29.1868i	1.58990i	−0.606672	+	0.794952i	$0.707496\pi$
$338$	− 23.0000i	− 1.25104i
$339$	−16.5934	−0.901229
$340$	0	0
$341$	−1.53113	−0.0829153
$342$	− 3.53113i	− 0.190942i
$343$	5.40661i	0.291930i
$344$	0.468871	0.0252798
$345$	0	0
$346$	14.0000	0.752645
$347$	− 22.1245i	− 1.18771i	−0.804573	−	0.593853i	$-0.797606\pi$
$347$	− 22.1245i	− 1.18771i	0.804573	−	0.593853i	$-0.202394\pi$
$348$	0	0
$349$	−25.0623	−1.34155	−0.670776	−	0.741660i	$-0.734039\pi$
$349$	−25.0623	−1.34155	−0.670776	+	0.741660i	$0.734039\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	− 1.53113i	− 0.0816094i
$353$	23.0623i	1.22748i	0.789508	+	0.613740i	$0.210336\pi$
$353$	23.0623i	1.22748i	−0.789508	+	0.613740i	$0.789664\pi$
$354$	−7.06226	−0.375355
$355$	0	0
$356$	1.53113	0.0811497
$357$	− 14.1245i	− 0.747549i
$358$	− 13.0623i	− 0.690362i
$359$	−11.5311	−0.608590	−0.304295	−	0.952578i	$-0.598421\pi$
$359$	−11.5311	−0.608590	−0.304295	+	0.952578i	$0.598421\pi$
$360$	0	0
$361$	−6.53113	−0.343744
$362$	26.5934i	1.39772i
$363$	8.65564i	0.454304i
$364$	−21.1868	−1.11049
$365$	0	0
$366$	−11.0623	−0.578233
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	1.53113i	0.0798156i
$369$	9.06226	0.471762
$370$	0	0
$371$	19.5311	1.01401
$372$	1.00000i	0.0518476i
$373$	10.4689i	0.542058i	0.962571	+	0.271029i	$0.0873639\pi$
$373$	10.4689i	0.542058i	−0.962571	+	0.271029i	$0.912636\pi$
$374$	−6.12452	−0.316691
$375$	0	0
$376$	11.0623	0.570492
$377$	0	0
$378$	− 3.53113i	− 0.181622i
$379$	2.59339	0.133213	0.0666067	−	0.997779i	$-0.478783\pi$
$379$	2.59339	0.133213	0.0666067	+	0.997779i	$0.478783\pi$
$380$	0	0
$381$	−9.06226	−0.464274
$382$	− 8.00000i	− 0.409316i
$383$	− 13.0623i	− 0.667450i	−0.942670	−	0.333725i	$-0.891694\pi$
$383$	− 13.0623i	− 0.667450i	0.942670	−	0.333725i	$-0.108306\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	− 0.468871i	− 0.0238341i
$388$	16.1245i	0.818598i
$389$	−27.0623	−1.37211	−0.686055	−	0.727549i	$-0.740659\pi$
$389$	−27.0623	−1.37211	−0.686055	+	0.727549i	$0.740659\pi$
$390$	0	0
$391$	6.12452	0.309730
$392$	5.46887i	0.276220i
$393$	− 7.06226i	− 0.356244i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−1.53113	−0.0769421
$397$	− 18.7179i	− 0.939425i	−0.882820	−	0.469712i	$-0.844358\pi$
$397$	− 18.7179i	− 0.939425i	0.882820	−	0.469712i	$-0.155642\pi$
$398$	0.468871i	0.0235024i
$399$	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	$0.833868\pi$
$402$	− 11.0623i	− 0.551735i
$403$	− 6.00000i	− 0.298881i
$404$	17.5311	0.872206
$405$	0	0
$406$	0	0
$407$	− 13.8755i	− 0.687782i
$408$	4.00000i	0.198030i
$409$	9.06226	0.448100	0.224050	−	0.974578i	$-0.428072\pi$
$409$	9.06226	0.448100	0.224050	+	0.974578i	$0.428072\pi$
$410$	0	0
$411$	0.937742	0.0462554
$412$	0	0
$413$	− 24.9377i	− 1.22711i
$414$	1.53113	0.0752509
$415$	0	0
$416$	6.00000	0.294174
$417$	6.00000i	0.293821i
$418$	− 5.40661i	− 0.264446i
$419$	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	$0.117999\pi$
$422$	22.5934i	1.09983i
$423$	− 11.0623i	− 0.537865i
$424$	−5.53113	−0.268615
$425$	0	0
$426$	−4.46887	−0.216518
$427$	− 39.0623i	− 1.89036i
$428$	14.5934i	0.705398i
$429$	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	$0.696174\pi$
$432$	1.00000i	0.0481125i
$433$	1.40661i	0.0675975i	0.999429	+	0.0337988i	$0.0107605\pi$
$433$	1.40661i	0.0675975i	−0.999429	+	0.0337988i	$0.989239\pi$
$434$	−3.53113	−0.169500
$435$	0	0
$436$	−1.06226	−0.0508729
$437$	5.40661i	0.258633i
$438$	0.468871i	0.0224035i
$439$	8.93774	0.426575	0.213288	−	0.976989i	$-0.431583\pi$
$439$	8.93774	0.426575	0.213288	+	0.976989i	$0.431583\pi$
$440$	0	0
$441$	5.46887	0.260422
$442$	− 24.0000i	− 1.14156i
$443$	− 38.5934i	− 1.83363i	−0.399316	−	0.916814i	$-0.630752\pi$
$443$	− 38.5934i	− 1.83363i	0.399316	−	0.916814i	$-0.369248\pi$
$444$	−9.06226	−0.430076
$445$	0	0
$446$	2.00000	0.0947027
$447$	− 10.4689i	− 0.495161i
$448$	− 3.53113i	− 0.166830i
$449$	−28.1245	−1.32728	−0.663639	−	0.748053i	$-0.730989\pi$
$449$	−28.1245	−1.32728	−0.663639	+	0.748053i	$0.730989\pi$
$450$	0	0
$451$	13.8755	0.653371
$452$	16.5934i	0.780487i
$453$	18.1245i	0.851564i
$454$	16.4689	0.772922
$455$	0	0
$456$	−3.53113	−0.165360
$457$	31.0623i	1.45303i	0.687150	+	0.726516i	$0.258862\pi$
$457$	31.0623i	1.45303i	−0.687150	+	0.726516i	$0.741138\pi$
$458$	− 18.5934i	− 0.868812i
$459$	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	$0.311225\pi$
$462$	− 5.40661i	− 0.251538i
$463$	35.1868i	1.63527i	0.575738	+	0.817634i	$0.304715\pi$
$463$	35.1868i	1.63527i	−0.575738	+	0.817634i	$0.695285\pi$
$464$	0	0
$465$	0	0
$466$	9.53113	0.441521
$467$	10.1245i	0.468507i	0.972176	+	0.234253i	$0.0752645\pi$
$467$	10.1245i	0.468507i	−0.972176	+	0.234253i	$0.924735\pi$
$468$	− 6.00000i	− 0.277350i
$469$	39.0623	1.80373
$470$	0	0
$471$	−14.4689	−0.666690
$472$	7.06226i	0.325067i
$473$	− 0.717902i	− 0.0330092i
$474$	0.468871	0.0215360
$475$	0	0
$476$	−14.1245	−0.647396
$477$	5.53113i	0.253253i
$478$	8.00000i	0.365911i
$479$	−41.6556	−1.90329	−0.951647	−	0.307192i	$-0.900611\pi$
$479$	−41.6556	−1.90329	−0.951647	+	0.307192i	$0.900611\pi$
$480$	0	0
$481$	54.3735	2.47922
$482$	2.00000i	0.0910975i
$483$	5.40661i	0.246009i
$484$	8.65564	0.393438
$485$	0	0
$486$	1.00000	0.0453609
$487$	6.93774i	0.314379i	0.987568	+	0.157190i	$0.0502434\pi$
$487$	6.93774i	0.314379i	−0.987568	+	0.157190i	$0.949757\pi$
$488$	11.0623i	0.500765i
$489$	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	$0.377840\pi$
$492$	− 9.06226i	− 0.408558i
$493$	0	0
$494$	21.1868	0.953238
$495$	0	0
$496$	1.00000	0.0449013
$497$	− 15.7802i	− 0.707837i
$498$	8.00000i	0.358489i
$499$	−9.06226	−0.405682	−0.202841	−	0.979212i	$-0.565018\pi$
$499$	−9.06226	−0.405682	−0.202841	+	0.979212i	$0.565018\pi$
$500$	0	0
$501$	0.593387	0.0265106
$502$	− 13.0623i	− 0.582997i
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	−3.53113	−0.157289
$505$	0	0
$506$	2.34436	0.104219
$507$	23.0000i	1.02147i
$508$	9.06226i	0.402073i
$509$	−5.87548	−0.260426	−0.130213	−	0.991486i	$-0.541566\pi$
$509$	−5.87548	−0.260426	−0.130213	+	0.991486i	$0.541566\pi$
$510$	0	0
$511$	−1.65564	−0.0732414
$512$	1.00000i	0.0441942i
$513$	3.53113i	0.155903i
$514$	27.6556	1.21984
$515$	0	0
$516$	−0.468871	−0.0206409
$517$	− 16.9377i	− 0.744921i
$518$	− 32.0000i	− 1.40600i
$519$	−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
$523$	14.5934i	0.638124i	0.947734	+	0.319062i	$0.103368\pi$
$523$	14.5934i	0.638124i	−0.947734	+	0.319062i	$0.896632\pi$
$524$	−7.06226	−0.308516
$525$	0	0
$526$	−6.93774	−0.302500
$527$	− 4.00000i	− 0.174243i
$528$	1.53113i	0.0666338i
$529$	20.6556	0.898071
$530$	0	0
$531$	7.06226	0.306476
$532$	− 12.4689i	− 0.540594i
$533$	54.3735i	2.35518i
$534$	−1.53113	−0.0662584
$535$	0	0
$536$	−11.0623	−0.477817
$537$	13.0623i	0.563678i
$538$	− 29.1868i	− 1.25833i
$539$	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	$0.293340\pi$
$542$	− 19.5311i	− 0.838934i
$543$	− 26.5934i	− 1.14123i
$544$	4.00000	0.171499
$545$	0	0
$546$	21.1868	0.906710
$547$	41.1868i	1.76102i	0.474028	+	0.880510i	$0.342799\pi$
$547$	41.1868i	1.76102i	−0.474028	+	0.880510i	$0.657201\pi$
$548$	− 0.937742i	− 0.0400584i
$549$	11.0623	0.472126
$550$	0	0
$551$	0	0
$552$	− 1.53113i	− 0.0651692i
$553$	1.65564i	0.0704052i
$554$	−1.06226	−0.0451310
$555$	0	0
$556$	6.00000	0.254457
$557$	45.7802i	1.93977i	0.243568	+	0.969884i	$0.421682\pi$
$557$	45.7802i	1.93977i	−0.243568	+	0.969884i	$0.578318\pi$
$558$	− 1.00000i	− 0.0423334i
$559$	2.81323	0.118987
$560$	0	0
$561$	6.12452	0.258577
$562$	− 1.06226i	− 0.0448086i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	−11.0623	−0.465805
$565$	0	0
$566$	−12.0000	−0.504398
$567$	3.53113i	0.148293i
$568$	4.46887i	0.187510i
$569$	−17.5311	−0.734943	−0.367472	−	0.930035i	$-0.619776\pi$
$569$	−17.5311	−0.734943	−0.367472	+	0.930035i	$0.619776\pi$
$570$	0	0
$571$	3.87548	0.162184	0.0810920	−	0.996707i	$-0.474159\pi$
$571$	3.87548	0.162184	0.0810920	+	0.996707i	$0.474159\pi$
$572$	− 9.18677i	− 0.384118i
$573$	8.00000i	0.334205i
$574$	32.0000	1.33565
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 11.1868i	− 0.465711i	−0.972511	−	0.232856i	$-0.925193\pi$
$577$	− 11.1868i	− 0.465711i	0.972511	−	0.232856i	$-0.0748070\pi$
$578$	1.00000i	0.0415945i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−28.2490	−1.17197
$582$	− 16.1245i	− 0.668383i
$583$	8.46887i	0.350745i
$584$	0.468871	0.0194020
$585$	0	0
$586$	−14.0000	−0.578335
$587$	20.2490i	0.835767i	0.908501	+	0.417883i	$0.137228\pi$
$587$	20.2490i	0.835767i	−0.908501	+	0.417883i	$0.862772\pi$
$588$	− 5.46887i	− 0.225532i
$589$	3.53113	0.145498
$590$	0	0
$591$	6.00000	0.246807
$592$	9.06226i	0.372456i
$593$	14.0000i	0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$	14.0000i	0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$	1.53113	0.0628230
$595$	0	0
$596$	−10.4689	−0.428822
$597$	− 0.468871i	− 0.0191896i
$598$	9.18677i	0.375675i
$599$	−10.5934	−0.432834	−0.216417	−	0.976301i	$-0.569437\pi$
$599$	−10.5934	−0.432834	−0.216417	+	0.976301i	$0.569437\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	− 1.65564i	− 0.0674790i
$603$	11.0623i	0.450490i
$604$	18.1245	0.737476
$605$	0	0
$606$	−17.5311	−0.712153
$607$	− 38.5934i	− 1.56646i	−0.621734	−	0.783229i	$-0.713571\pi$
$607$	− 38.5934i	− 1.56646i	0.621734	−	0.783229i	$-0.286429\pi$
$608$	3.53113i	0.143206i
$609$	0	0
$610$	0	0
$611$	66.3735	2.68519
$612$	− 4.00000i	− 0.161690i
$613$	30.0000i	1.21169i	0.795583	+	0.605844i	$0.207165\pi$
$613$	30.0000i	1.21169i	−0.795583	+	0.605844i	$0.792835\pi$
$614$	−2.12452	−0.0857385
$615$	0	0
$616$	−5.40661	−0.217839
$617$	20.5934i	0.829059i	0.910036	+	0.414529i	$0.136054\pi$
$617$	20.5934i	0.829059i	−0.910036	+	0.414529i	$0.863946\pi$
$618$	0	0
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	0	0
$621$	−1.53113	−0.0614421
$622$	− 8.00000i	− 0.320771i
$623$	− 5.40661i	− 0.216611i
$624$	−6.00000	−0.240192
$625$	0	0
$626$	27.0623	1.08163
$627$	5.40661i	0.215919i
$628$	14.4689i	0.577371i
$629$	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	$0.561560\pi$
$632$	− 0.468871i	− 0.0186507i
$633$	− 22.5934i	− 0.898006i
$634$	−29.0623	−1.15421
$635$	0	0
$636$	5.53113	0.219324
$637$	32.8132i	1.30011i
$638$	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	$0.643066\pi$
$642$	− 14.5934i	− 0.575955i
$643$	− 6.59339i	− 0.260018i	−0.991513	−	0.130009i	$-0.958499\pi$
$643$	− 6.59339i	− 0.260018i	0.991513	−	0.130009i	$-0.0415006\pi$
$644$	5.40661	0.213050
$645$	0	0
$646$	14.1245	0.555722
$647$	1.53113i	0.0601949i	0.999547	+	0.0300974i	$0.00958176\pi$
$647$	1.53113i	0.0601949i	−0.999547	+	0.0300974i	$0.990418\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	10.8132	0.424456
$650$	0	0
$651$	3.53113	0.138396
$652$	− 11.0623i	− 0.433231i
$653$	26.2490i	1.02720i	0.858029	+	0.513602i	$0.171689\pi$
$653$	26.2490i	1.02720i	−0.858029	+	0.513602i	$0.828311\pi$
$654$	1.06226	0.0415376
$655$	0	0
$656$	−9.06226	−0.353822
$657$	− 0.468871i	− 0.0182924i
$658$	− 39.0623i	− 1.52281i
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	29.0623i	1.12954i
$663$	24.0000i	0.932083i
$664$	8.00000	0.310460
$665$	0	0
$666$	9.06226	0.351155
$667$	0	0
$668$	− 0.593387i	− 0.0229588i
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	16.9377	0.653874
$672$	3.53113i	0.136216i
$673$	− 7.06226i	− 0.272230i	−0.990693	−	0.136115i	$-0.956538\pi$
$673$	− 7.06226i	− 0.272230i	0.990693	−	0.136115i	$-0.0434617\pi$
$674$	−29.1868	−1.12423
$675$	0	0
$676$	23.0000	0.884615
$677$	3.40661i	0.130927i	0.997855	+	0.0654634i	$0.0208526\pi$
$677$	3.40661i	0.130927i	−0.997855	+	0.0654634i	$0.979147\pi$
$678$	− 16.5934i	− 0.637265i
$679$	56.9377	2.18507
$680$	0	0
$681$	−16.4689	−0.631089
$682$	− 1.53113i	− 0.0586300i
$683$	− 15.5311i	− 0.594282i	−0.954834	−	0.297141i	$-0.903967\pi$
$683$	− 15.5311i	− 0.594282i	0.954834	−	0.297141i	$-0.0960332\pi$
$684$	3.53113	0.135016
$685$	0	0
$686$	−5.40661	−0.206425
$687$	18.5934i	0.709382i
$688$	0.468871i	0.0178755i
$689$	−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	$0.545755\pi$
$692$	14.0000i	0.532200i
$693$	5.40661i	0.205380i
$694$	22.1245	0.839835
$695$	0	0
$696$	0	0
$697$	36.2490i	1.37303i
$698$	− 25.0623i	− 0.948620i
$699$	−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	$0.633289\pi$
$702$	6.00000i	0.226455i
$703$	32.0000i	1.20690i
$704$	1.53113	0.0577066
$705$	0	0
$706$	−23.0623	−0.867960
$707$	− 61.9047i	− 2.32816i
$708$	− 7.06226i	− 0.265416i
$709$	−25.4066	−0.954165	−0.477083	−	0.878858i	$-0.658306\pi$
$709$	−25.4066	−0.954165	−0.477083	+	0.878858i	$0.658306\pi$
$710$	0	0
$711$	−0.468871	−0.0175840
$712$	1.53113i	0.0573815i
$713$	1.53113i	0.0573412i
$714$	14.1245	0.528597
$715$	0	0
$716$	13.0623	0.488159
$717$	− 8.00000i	− 0.298765i
$718$	− 11.5311i	− 0.430338i
$719$	33.1868	1.23766	0.618829	−	0.785526i	$-0.287607\pi$
$719$	33.1868	1.23766	0.618829	+	0.785526i	$0.287607\pi$
$720$	0	0
$721$	0	0
$722$	− 6.53113i	− 0.243063i
$723$	− 2.00000i	− 0.0743808i
$724$	−26.5934	−0.988335
$725$	0	0
$726$	−8.65564	−0.321241
$727$	− 50.8424i	− 1.88564i	−0.333301	−	0.942820i	$-0.608163\pi$
$727$	− 50.8424i	− 1.88564i	0.333301	−	0.942820i	$-0.391837\pi$
$728$	− 21.1868i	− 0.785234i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	1.87548	0.0693673
$732$	− 11.0623i	− 0.408873i
$733$	− 4.12452i	− 0.152342i	−0.997095	−	0.0761712i	$-0.975730\pi$
$733$	− 4.12452i	− 0.152342i	0.997095	−	0.0761712i	$-0.0242696\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	−1.53113	−0.0564382
$737$	16.9377i	0.623910i
$738$	9.06226i	0.333586i
$739$	−27.1868	−1.00008	−0.500041	−	0.866002i	$-0.666682\pi$
$739$	−27.1868	−1.00008	−0.500041	+	0.866002i	$0.666682\pi$
$740$	0	0
$741$	−21.1868	−0.778316
$742$	19.5311i	0.717010i
$743$	− 11.6556i	− 0.427604i	−0.976877	−	0.213802i	$-0.931415\pi$
$743$	− 11.6556i	− 0.427604i	0.976877	−	0.213802i	$-0.0685848\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−10.4689	−0.383293
$747$	− 8.00000i	− 0.292705i
$748$	− 6.12452i	− 0.223934i
$749$	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	$0.171296\pi$
$752$	11.0623i	0.403399i
$753$	13.0623i	0.476015i
$754$	0	0
$755$	0	0
$756$	3.53113	0.128426
$757$	26.0000i	0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$	26.0000i	0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	2.59339i	0.0941960i
$759$	−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	$0.573302\pi$
$762$	− 9.06226i	− 0.328291i
$763$	3.75097i	0.135794i
$764$	8.00000	0.289430
$765$	0	0
$766$	13.0623	0.471959
$767$	42.3735i	1.53002i
$768$	− 1.00000i	− 0.0360844i
$769$	28.5934	1.03110	0.515552	−	0.856858i	$-0.327587\pi$
$769$	28.5934	1.03110	0.515552	+	0.856858i	$0.327587\pi$
$770$	0	0
$771$	−27.6556	−0.995994
$772$	14.0000i	0.503871i
$773$	− 14.4689i	− 0.520409i	−0.965554	−	0.260205i	$-0.916210\pi$
$773$	− 14.4689i	− 0.520409i	0.965554	−	0.260205i	$-0.0837900\pi$
$774$	0.468871	0.0168532
$775$	0	0
$776$	−16.1245	−0.578836
$777$	32.0000i	1.14799i
$778$	− 27.0623i	− 0.970229i
$779$	−32.0000	−1.14652
$780$	0	0
$781$	6.84242	0.244841
$782$	6.12452i	0.219012i
$783$	0	0
$784$	−5.46887	−0.195317
$785$	0	0
$786$	7.06226	0.251902
$787$	− 28.4689i	− 1.01481i	−0.861709	−	0.507403i	$-0.830606\pi$
$787$	− 28.4689i	− 1.01481i	0.861709	−	0.507403i	$-0.169394\pi$
$788$	− 6.00000i	− 0.213741i
$789$	6.93774	0.246990
$790$	0	0
$791$	58.5934	2.08334
$792$	− 1.53113i	− 0.0544063i
$793$	66.3735i	2.35699i
$794$	18.7179	0.664273
$795$	0	0
$796$	−0.468871	−0.0166187
$797$	− 20.1245i	− 0.712847i	−0.934325	−	0.356423i	$-0.883996\pi$
$797$	− 20.1245i	− 0.712847i	0.934325	−	0.356423i	$-0.116004\pi$
$798$	12.4689i	0.441393i
$799$	44.2490	1.56542
$800$	0	0
$801$	1.53113	0.0540998
$802$	− 34.7179i	− 1.22593i
$803$	− 0.717902i	− 0.0253342i
$804$	11.0623	0.390136
$805$	0	0
$806$	6.00000	0.211341
$807$	29.1868i	1.02742i
$808$	17.5311i	0.616743i
$809$	8.59339	0.302127	0.151064	−	0.988524i	$-0.451730\pi$
$809$	8.59339	0.302127	0.151064	+	0.988524i	$0.451730\pi$
$810$	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
$813$	19.5311i	0.684987i
$814$	13.8755	0.486335
$815$	0	0
$816$	−4.00000	−0.140028
$817$	1.65564i	0.0579237i
$818$	9.06226i	0.316854i
$819$	−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	$0.444871\pi$
$822$	0.937742i	0.0327075i
$823$	− 7.87548i	− 0.274522i	−0.990535	−	0.137261i	$-0.956170\pi$
$823$	− 7.87548i	− 0.274522i	0.990535	−	0.137261i	$-0.0438299\pi$
$824$	0	0
$825$	0	0
$826$	24.9377	0.867695
$827$	8.00000i	0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$	8.00000i	0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$	1.53113i	0.0532104i
$829$	9.65564	0.335354	0.167677	−	0.985842i	$-0.446373\pi$
$829$	9.65564	0.335354	0.167677	+	0.985842i	$0.446373\pi$
$830$	0	0
$831$	1.06226	0.0368493
$832$	6.00000i	0.208013i
$833$	21.8755i	0.757941i
$834$	−6.00000	−0.207763
$835$	0	0
$836$	5.40661	0.186992
$837$	1.00000i	0.0345651i
$838$	0	0
$839$	54.8424	1.89337	0.946685	−	0.322160i	$-0.104409\pi$
$839$	54.8424	1.89337	0.946685	+	0.322160i	$0.104409\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	38.2490i	1.31815i
$843$	1.06226i	0.0365861i
$844$	−22.5934	−0.777696
$845$	0	0
$846$	11.0623	0.380328
$847$	− 30.5642i	− 1.05020i
$848$	− 5.53113i	− 0.189940i
$849$	12.0000	0.411839
$850$	0	0
$851$	−13.8755	−0.475645
$852$	− 4.46887i	− 0.153101i
$853$	− 1.28210i	− 0.0438982i	−0.999759	−	0.0219491i	$-0.993013\pi$
$853$	− 1.28210i	− 0.0438982i	0.999759	−	0.0219491i	$-0.00698718\pi$
$854$	39.0623	1.33668
$855$	0	0
$856$	−14.5934	−0.498792
$857$	14.0000i	0.478231i	0.970991	+	0.239115i	$0.0768574\pi$
$857$	14.0000i	0.478231i	−0.970991	+	0.239115i	$0.923143\pi$
$858$	9.18677i	0.313631i
$859$	6.93774	0.236713	0.118356	−	0.992971i	$-0.462237\pi$
$859$	6.93774	0.236713	0.118356	+	0.992971i	$0.462237\pi$
$860$	0	0
$861$	−32.0000	−1.09056
$862$	− 24.0000i	− 0.817443i
$863$	− 41.5311i	− 1.41374i	−0.707345	−	0.706868i	$-0.750107\pi$
$863$	− 41.5311i	− 1.41374i	0.707345	−	0.706868i	$-0.249893\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−1.40661	−0.0477987
$867$	− 1.00000i	− 0.0339618i
$868$	− 3.53113i	− 0.119854i
$869$	−0.717902	−0.0243532
$870$	0	0
$871$	−66.3735	−2.24898
$872$	− 1.06226i	− 0.0359726i
$873$	16.1245i	0.545732i
$874$	−5.40661	−0.182881
$875$	0	0
$876$	−0.468871	−0.0158417
$877$	44.1245i	1.48998i	0.667076	+	0.744990i	$0.267546\pi$
$877$	44.1245i	1.48998i	−0.667076	+	0.744990i	$0.732454\pi$
$878$	8.93774i	0.301634i
$879$	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	$0.291758\pi$
$882$	5.46887i	0.184146i
$883$	53.6556i	1.80566i	0.430002	+	0.902828i	$0.358513\pi$
$883$	53.6556i	1.80566i	−0.430002	+	0.902828i	$0.641487\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	38.5934	1.29657
$887$	− 25.1868i	− 0.845689i	−0.906202	−	0.422845i	$-0.861032\pi$
$887$	− 25.1868i	− 0.845689i	0.906202	−	0.422845i	$-0.138968\pi$
$888$	− 9.06226i	− 0.304109i
$889$	32.0000	1.07325
$890$	0	0
$891$	−1.53113	−0.0512947
$892$	2.00000i	0.0669650i
$893$	39.0623i	1.30717i
$894$	10.4689	0.350131
$895$	0	0
$896$	3.53113	0.117967
$897$	− 9.18677i	− 0.306737i
$898$	− 28.1245i	− 0.938527i
$899$	0	0
$900$	0	0
$901$	−22.1245	−0.737074
$902$	13.8755i	0.462003i
$903$	1.65564i	0.0550964i
$904$	−16.5934	−0.551888
$905$	0	0
$906$	−18.1245	−0.602147
$907$	− 32.2490i	− 1.07081i	−0.844595	−	0.535406i	$-0.820159\pi$
$907$	− 32.2490i	− 1.07081i	0.844595	−	0.535406i	$-0.179841\pi$
$908$	16.4689i	0.546539i
$909$	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	$0.585389\pi$
$912$	− 3.53113i	− 0.116927i
$913$	− 12.2490i	− 0.405384i
$914$	−31.0623	−1.02745
$915$	0	0
$916$	18.5934	0.614343
$917$	24.9377i	0.823517i
$918$	4.00000i	0.132020i
$919$	8.93774	0.294829	0.147414	−	0.989075i	$-0.452905\pi$
$919$	8.93774	0.294829	0.147414	+	0.989075i	$0.452905\pi$
$920$	0	0
$921$	2.12452	0.0700052
$922$	24.0000i	0.790398i
$923$	26.8132i	0.882568i
$924$	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	$-0.706579\pi$
$929$	−36.8424	−1.20876	−0.604380	+	0.796696i	$0.706579\pi$
$930$	0	0
$931$	−19.3113	−0.632902
$932$	9.53113i	0.312203i
$933$	8.00000i	0.261908i
$934$	−10.1245	−0.331284
$935$	0	0
$936$	6.00000	0.196116
$937$	− 25.0623i	− 0.818748i	−0.912367	−	0.409374i	$-0.865747\pi$
$937$	− 25.0623i	− 0.818748i	0.912367	−	0.409374i	$-0.134253\pi$
$938$	39.0623i	1.27543i
$939$	−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	$0.616051\pi$
$942$	− 14.4689i	− 0.471421i
$943$	− 13.8755i	− 0.451848i
$944$	−7.06226	−0.229857
$945$	0	0
$946$	0.717902	0.0233410
$947$	35.0623i	1.13937i	0.821863	+	0.569685i	$0.192935\pi$
$947$	35.0623i	1.13937i	−0.821863	+	0.569685i	$0.807065\pi$
$948$	0.468871i	0.0152282i
$949$	2.81323	0.0913212
$950$	0	0
$951$	29.0623	0.942408
$952$	− 14.1245i	− 0.457778i
$953$	− 54.1245i	− 1.75327i	−0.481161	−	0.876633i	$-0.659785\pi$
$953$	− 54.1245i	− 1.75327i	0.481161	−	0.876633i	$-0.340215\pi$
$954$	−5.53113	−0.179077
$955$	0	0
$956$	−8.00000	−0.258738
$957$	0	0
$958$	− 41.6556i	− 1.34583i
$959$	−3.31129	−0.106927
$960$	0	0
$961$	1.00000	0.0322581
$962$	54.3735i	1.75307i
$963$	14.5934i	0.470265i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	−5.40661	−0.173955
$967$	− 17.0623i	− 0.548685i	−0.961632	−	0.274343i	$-0.911540\pi$
$967$	− 17.0623i	− 0.548685i	0.961632	−	0.274343i	$-0.0884602\pi$
$968$	8.65564i	0.278203i
$969$	−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	$0.678750\pi$
$972$	1.00000i	0.0320750i
$973$	− 21.1868i	− 0.679217i
$974$	−6.93774	−0.222300
$975$	0	0
$976$	−11.0623	−0.354094
$977$	− 38.0000i	− 1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$	− 38.0000i	− 1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$	11.0623i	0.353732i
$979$	2.34436	0.0749259
$980$	0	0
$981$	−1.06226	−0.0339153
$982$	16.5934i	0.529516i
$983$	− 17.0623i	− 0.544202i	−0.962269	−	0.272101i	$-0.912282\pi$
$983$	− 17.0623i	− 0.544202i	0.962269	−	0.272101i	$-0.0877184\pi$
$984$	9.06226	0.288894
$985$	0	0
$986$	0	0
$987$	39.0623i	1.24337i
$988$	21.1868i	0.674041i
$989$	−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	$0.627055\pi$
$992$	1.00000i	0.0317500i
$993$	− 29.0623i	− 0.922263i
$994$	15.7802	0.500516
$995$	0	0
$996$	−8.00000	−0.253490
$997$	62.2490i	1.97145i	0.168373	+	0.985723i	$0.446149\pi$
$997$	62.2490i	1.97145i	−0.168373	+	0.985723i	$0.553851\pi$
$998$	− 9.06226i	− 0.286861i
$999$	−9.06226	−0.286717

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.q.1.2	✓	2	5.3	odd	4
2790.2.a.bf.1.2		2	15.8	even	4
4650.2.a.bz.1.1		2	5.2	odd	4
4650.2.d.bg.3349.1		4	5.4	even	2	inner
4650.2.d.bg.3349.4		4	1.1	even	1	trivial
7440.2.a.bd.1.1		2	20.3	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.d (of order \(2\), degree \(1\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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