LMFDB - Embedded newform 3234.2.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, 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("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.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:	$25.8236200137$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2700.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 15x - 20 $ x^3 - 15*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.80560$ of defining polynomial
Character	$\chi$	$=$	3234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.80560 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.80560 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.80560 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.80560 q^{15} +1.00000 q^{16} +7.96523 q^{17} +1.00000 q^{18} -6.48261 q^{19} -2.80560 q^{20} -1.00000 q^{22} -5.28822 q^{23} -1.00000 q^{24} +2.87141 q^{25} -1.00000 q^{27} +5.12859 q^{29} +2.80560 q^{30} -8.96523 q^{31} +1.00000 q^{32} +1.00000 q^{33} +7.96523 q^{34} +1.00000 q^{36} +10.4826 q^{37} -6.48261 q^{38} -2.80560 q^{40} -5.09382 q^{41} +11.4478 q^{43} -1.00000 q^{44} -2.80560 q^{45} -5.28822 q^{46} +0.322990 q^{47} -1.00000 q^{48} +2.87141 q^{50} -7.96523 q^{51} +8.00000 q^{53} -1.00000 q^{54} +2.80560 q^{55} +6.48261 q^{57} +5.12859 q^{58} -0.871407 q^{59} +2.80560 q^{60} +2.15962 q^{61} -8.96523 q^{62} +1.00000 q^{64} +1.00000 q^{66} -7.09382 q^{67} +7.96523 q^{68} +5.28822 q^{69} -7.44784 q^{71} +1.00000 q^{72} +12.5764 q^{73} +10.4826 q^{74} -2.87141 q^{75} -6.48261 q^{76} +6.80560 q^{79} -2.80560 q^{80} +1.00000 q^{81} -5.09382 q^{82} +15.0938 q^{83} -22.3473 q^{85} +11.4478 q^{86} -5.12859 q^{87} -1.00000 q^{88} +1.61121 q^{89} -2.80560 q^{90} -5.28822 q^{92} +8.96523 q^{93} +0.322990 q^{94} +18.1876 q^{95} -1.00000 q^{96} +7.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 3 q^{11} - 3 q^{12} + 3 q^{16} + 3 q^{17} + 3 q^{18} - 9 q^{19} - 3 q^{22} + 3 q^{23} - 3 q^{24} + 15 q^{25} - 3 q^{27} + 9 q^{29} - 6 q^{31} + 3 q^{32} + 3 q^{33} + 3 q^{34} + 3 q^{36} + 21 q^{37} - 9 q^{38} + 12 q^{41} + 3 q^{43} - 3 q^{44} + 3 q^{46} + 3 q^{47} - 3 q^{48} + 15 q^{50} - 3 q^{51} + 24 q^{53} - 3 q^{54} + 9 q^{57} + 9 q^{58} - 9 q^{59} - 6 q^{61} - 6 q^{62} + 3 q^{64} + 3 q^{66} + 6 q^{67} + 3 q^{68} - 3 q^{69} + 9 q^{71} + 3 q^{72} + 21 q^{74} - 15 q^{75} - 9 q^{76} + 12 q^{79} + 3 q^{81} + 12 q^{82} + 18 q^{83} + 3 q^{86} - 9 q^{87} - 3 q^{88} - 12 q^{89} + 3 q^{92} + 6 q^{93} + 3 q^{94} - 3 q^{96} + 21 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 - 3 * q^11 - 3 * q^12 + 3 * q^16 + 3 * q^17 + 3 * q^18 - 9 * q^19 - 3 * q^22 + 3 * q^23 - 3 * q^24 + 15 * q^25 - 3 * q^27 + 9 * q^29 - 6 * q^31 + 3 * q^32 + 3 * q^33 + 3 * q^34 + 3 * q^36 + 21 * q^37 - 9 * q^38 + 12 * q^41 + 3 * q^43 - 3 * q^44 + 3 * q^46 + 3 * q^47 - 3 * q^48 + 15 * q^50 - 3 * q^51 + 24 * q^53 - 3 * q^54 + 9 * q^57 + 9 * q^58 - 9 * q^59 - 6 * q^61 - 6 * q^62 + 3 * q^64 + 3 * q^66 + 6 * q^67 + 3 * q^68 - 3 * q^69 + 9 * q^71 + 3 * q^72 + 21 * q^74 - 15 * q^75 - 9 * q^76 + 12 * q^79 + 3 * q^81 + 12 * q^82 + 18 * q^83 + 3 * q^86 - 9 * q^87 - 3 * q^88 - 12 * q^89 + 3 * q^92 + 6 * q^93 + 3 * q^94 - 3 * q^96 + 21 * q^97 - 3 * 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
$5$	−2.80560	−1.25470	−0.627352	−	0.778736i	$-0.715861\pi$
$5$	−2.80560	−1.25470	−0.627352	+	0.778736i	$0.715861\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.80560	−0.887210
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	2.80560	0.724404
$16$	1.00000	0.250000
$17$	7.96523	1.93185	0.965926	−	0.258820i	$-0.0833337\pi$
$17$	7.96523	1.93185	0.965926	+	0.258820i	$0.0833337\pi$
$18$	1.00000	0.235702
$19$	−6.48261	−1.48721	−0.743607	−	0.668617i	$-0.766886\pi$
$19$	−6.48261	−1.48721	−0.743607	+	0.668617i	$0.766886\pi$
$20$	−2.80560	−0.627352
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−5.28822	−1.10267	−0.551335	−	0.834284i	$-0.685881\pi$
$23$	−5.28822	−1.10267	−0.551335	+	0.834284i	$0.685881\pi$
$24$	−1.00000	−0.204124
$25$	2.87141	0.574281
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.12859	0.952356	0.476178	−	0.879349i	$-0.342022\pi$
$29$	5.12859	0.952356	0.476178	+	0.879349i	$0.342022\pi$
$30$	2.80560	0.512231
$31$	−8.96523	−1.61020	−0.805101	−	0.593138i	$-0.797889\pi$
$31$	−8.96523	−1.61020	−0.805101	+	0.593138i	$0.797889\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	7.96523	1.36602
$35$	0	0
$36$	1.00000	0.166667
$37$	10.4826	1.72333	0.861665	−	0.507477i	$-0.169422\pi$
$37$	10.4826	1.72333	0.861665	+	0.507477i	$0.169422\pi$
$38$	−6.48261	−1.05162
$39$	0	0
$40$	−2.80560	−0.443605
$41$	−5.09382	−0.795521	−0.397760	−	0.917489i	$-0.630212\pi$
$41$	−5.09382	−0.795521	−0.397760	+	0.917489i	$0.630212\pi$
$42$	0	0
$43$	11.4478	1.74578	0.872890	−	0.487918i	$-0.162243\pi$
$43$	11.4478	1.74578	0.872890	+	0.487918i	$0.162243\pi$
$44$	−1.00000	−0.150756
$45$	−2.80560	−0.418235
$46$	−5.28822	−0.779705
$47$	0.322990	0.0471129	0.0235565	−	0.999723i	$-0.492501\pi$
$47$	0.322990	0.0471129	0.0235565	+	0.999723i	$0.492501\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	2.87141	0.406078
$51$	−7.96523	−1.11535
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	−1.00000	−0.136083
$55$	2.80560	0.378307
$56$	0	0
$57$	6.48261	0.858643
$58$	5.12859	0.673417
$59$	−0.871407	−0.113448	−0.0567238	−	0.998390i	$-0.518065\pi$
$59$	−0.871407	−0.113448	−0.0567238	+	0.998390i	$0.518065\pi$
$60$	2.80560	0.362202
$61$	2.15962	0.276511	0.138256	−	0.990397i	$-0.455850\pi$
$61$	2.15962	0.276511	0.138256	+	0.990397i	$0.455850\pi$
$62$	−8.96523	−1.13858
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	−7.09382	−0.866648	−0.433324	−	0.901238i	$-0.642659\pi$
$67$	−7.09382	−0.866648	−0.433324	+	0.901238i	$0.642659\pi$
$68$	7.96523	0.965926
$69$	5.28822	0.636626
$70$	0	0
$71$	−7.44784	−0.883896	−0.441948	−	0.897041i	$-0.645712\pi$
$71$	−7.44784	−0.883896	−0.441948	+	0.897041i	$0.645712\pi$
$72$	1.00000	0.117851
$73$	12.5764	1.47196	0.735980	−	0.677003i	$-0.236722\pi$
$73$	12.5764	1.47196	0.735980	+	0.677003i	$0.236722\pi$
$74$	10.4826	1.21858
$75$	−2.87141	−0.331562
$76$	−6.48261	−0.743607
$77$	0	0
$78$	0	0
$79$	6.80560	0.765690	0.382845	−	0.923813i	$-0.374944\pi$
$79$	6.80560	0.765690	0.382845	+	0.923813i	$0.374944\pi$
$80$	−2.80560	−0.313676
$81$	1.00000	0.111111
$82$	−5.09382	−0.562518
$83$	15.0938	1.65676	0.828381	−	0.560165i	$-0.189262\pi$
$83$	15.0938	1.65676	0.828381	+	0.560165i	$0.189262\pi$
$84$	0	0
$85$	−22.3473	−2.42390
$86$	11.4478	1.23445
$87$	−5.12859	−0.549843
$88$	−1.00000	−0.106600
$89$	1.61121	0.170787	0.0853937	−	0.996347i	$-0.472785\pi$
$89$	1.61121	0.170787	0.0853937	+	0.996347i	$0.472785\pi$
$90$	−2.80560	−0.295737
$91$	0	0
$92$	−5.28822	−0.551335
$93$	8.96523	0.929651
$94$	0.322990	0.0333139
$95$	18.1876	1.86601
$96$	−1.00000	−0.102062
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	2.87141	0.287141
$101$	10.7398	1.06865	0.534325	−	0.845279i	$-0.320566\pi$
$101$	10.7398	1.06865	0.534325	+	0.845279i	$0.320566\pi$
$102$	−7.96523	−0.788675
$103$	2.96523	0.292172	0.146086	−	0.989272i	$-0.453332\pi$
$103$	2.96523	0.292172	0.146086	+	0.989272i	$0.453332\pi$
$104$	0	0
$105$	0	0
$106$	8.00000	0.777029
$107$	4.83663	0.467575	0.233787	−	0.972288i	$-0.424888\pi$
$107$	4.83663	0.467575	0.233787	+	0.972288i	$0.424888\pi$
$108$	−1.00000	−0.0962250
$109$	−10.1596	−0.973115	−0.486558	−	0.873648i	$-0.661748\pi$
$109$	−10.1596	−0.973115	−0.486558	+	0.873648i	$0.661748\pi$
$110$	2.80560	0.267504
$111$	−10.4826	−0.994966
$112$	0	0
$113$	10.9652	1.03152	0.515761	−	0.856733i	$-0.327509\pi$
$113$	10.9652	1.03152	0.515761	+	0.856733i	$0.327509\pi$
$114$	6.48261	0.607152
$115$	14.8366	1.38352
$116$	5.12859	0.476178
$117$	0	0
$118$	−0.871407	−0.0802195
$119$	0	0
$120$	2.80560	0.256115
$121$	1.00000	0.0909091
$122$	2.15962	0.195523
$123$	5.09382	0.459294
$124$	−8.96523	−0.805101
$125$	5.97199	0.534151
$126$	0	0
$127$	−9.28822	−0.824196	−0.412098	−	0.911140i	$-0.635204\pi$
$127$	−9.28822	−0.824196	−0.412098	+	0.911140i	$0.635204\pi$
$128$	1.00000	0.0883883
$129$	−11.4478	−1.00793
$130$	0	0
$131$	12.9652	1.13278	0.566389	−	0.824138i	$-0.308340\pi$
$131$	12.9652	1.13278	0.566389	+	0.824138i	$0.308340\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−7.09382	−0.612813
$135$	2.80560	0.241468
$136$	7.96523	0.683012
$137$	5.61121	0.479398	0.239699	−	0.970847i	$-0.422951\pi$
$137$	5.61121	0.479398	0.239699	+	0.970847i	$0.422951\pi$
$138$	5.28822	0.450163
$139$	13.1286	1.11355	0.556776	−	0.830662i	$-0.312038\pi$
$139$	13.1286	1.11355	0.556776	+	0.830662i	$0.312038\pi$
$140$	0	0
$141$	−0.322990	−0.0272007
$142$	−7.44784	−0.625009
$143$	0	0
$144$	1.00000	0.0833333
$145$	−14.3888	−1.19492
$146$	12.5764	1.04083
$147$	0	0
$148$	10.4826	0.861665
$149$	16.7398	1.37138	0.685689	−	0.727895i	$-0.259501\pi$
$149$	16.7398	1.37138	0.685689	+	0.727895i	$0.259501\pi$
$150$	−2.87141	−0.234449
$151$	10.8994	0.886982	0.443491	−	0.896279i	$-0.353740\pi$
$151$	10.8994	0.886982	0.443491	+	0.896279i	$0.353740\pi$
$152$	−6.48261	−0.525809
$153$	7.96523	0.643950
$154$	0	0
$155$	25.1529	2.02033
$156$	0	0
$157$	1.77457	0.141626	0.0708132	−	0.997490i	$-0.477441\pi$
$157$	1.77457	0.141626	0.0708132	+	0.997490i	$0.477441\pi$
$158$	6.80560	0.541425
$159$	−8.00000	−0.634441
$160$	−2.80560	−0.221802
$161$	0	0
$162$	1.00000	0.0785674
$163$	−4.05904	−0.317929	−0.158964	−	0.987284i	$-0.550816\pi$
$163$	−4.05904	−0.317929	−0.158964	+	0.987284i	$0.550816\pi$
$164$	−5.09382	−0.397760
$165$	−2.80560	−0.218416
$166$	15.0938	1.17151
$167$	13.8684	1.07317	0.536584	−	0.843847i	$-0.319714\pi$
$167$	13.8684	1.07317	0.536584	+	0.843847i	$0.319714\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−22.3473	−1.71396
$171$	−6.48261	−0.495738
$172$	11.4478	0.872890
$173$	16.9652	1.28984	0.644921	−	0.764249i	$-0.276890\pi$
$173$	16.9652	1.28984	0.644921	+	0.764249i	$0.276890\pi$
$174$	−5.12859	−0.388798
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0.871407	0.0654990
$178$	1.61121	0.120765
$179$	−12.6703	−0.947019	−0.473509	−	0.880789i	$-0.657013\pi$
$179$	−12.6703	−0.947019	−0.473509	+	0.880789i	$0.657013\pi$
$180$	−2.80560	−0.209117
$181$	−2.57643	−0.191505	−0.0957523	−	0.995405i	$-0.530526\pi$
$181$	−2.57643	−0.191505	−0.0957523	+	0.995405i	$0.530526\pi$
$182$	0	0
$183$	−2.15962	−0.159644
$184$	−5.28822	−0.389852
$185$	−29.4100	−2.16227
$186$	8.96523	0.657362
$187$	−7.96523	−0.582475
$188$	0.322990	0.0235565
$189$	0	0
$190$	18.1876	1.31947
$191$	−2.38879	−0.172847	−0.0864235	−	0.996258i	$-0.527544\pi$
$191$	−2.38879	−0.172847	−0.0864235	+	0.996258i	$0.527544\pi$
$192$	−1.00000	−0.0721688
$193$	−14.5764	−1.04923	−0.524617	−	0.851338i	$-0.675792\pi$
$193$	−14.5764	−1.04923	−0.524617	+	0.851338i	$0.675792\pi$
$194$	7.00000	0.502571
$195$	0	0
$196$	0	0
$197$	−12.4826	−0.889349	−0.444675	−	0.895692i	$-0.646681\pi$
$197$	−12.4826	−0.889349	−0.444675	+	0.895692i	$0.646681\pi$
$198$	−1.00000	−0.0710669
$199$	−14.1876	−1.00573	−0.502867	−	0.864364i	$-0.667722\pi$
$199$	−14.1876	−1.00573	−0.502867	+	0.864364i	$0.667722\pi$
$200$	2.87141	0.203039
$201$	7.09382	0.500359
$202$	10.7398	0.755650
$203$	0	0
$204$	−7.96523	−0.557677
$205$	14.2912	0.998143
$206$	2.96523	0.206597
$207$	−5.28822	−0.367556
$208$	0	0
$209$	6.48261	0.448412
$210$	0	0
$211$	−20.8336	−1.43425	−0.717123	−	0.696947i	$-0.754541\pi$
$211$	−20.8336	−1.43425	−0.717123	+	0.696947i	$0.754541\pi$
$212$	8.00000	0.549442
$213$	7.44784	0.510318
$214$	4.83663	0.330625
$215$	−32.1181	−2.19044
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−10.1596	−0.688096
$219$	−12.5764	−0.849836
$220$	2.80560	0.189154
$221$	0	0
$222$	−10.4826	−0.703547
$223$	12.5764	0.842180	0.421090	−	0.907019i	$-0.361648\pi$
$223$	12.5764	0.842180	0.421090	+	0.907019i	$0.361648\pi$
$224$	0	0
$225$	2.87141	0.191427
$226$	10.9652	0.729396
$227$	20.0590	1.33137	0.665683	−	0.746235i	$-0.268140\pi$
$227$	20.0590	1.33137	0.665683	+	0.746235i	$0.268140\pi$
$228$	6.48261	0.429322
$229$	−22.5764	−1.49189	−0.745946	−	0.666006i	$-0.768002\pi$
$229$	−22.5764	−1.49189	−0.745946	+	0.666006i	$0.768002\pi$
$230$	14.8366	0.978299
$231$	0	0
$232$	5.12859	0.336709
$233$	−0.742815	−0.0486634	−0.0243317	−	0.999704i	$-0.507746\pi$
$233$	−0.742815	−0.0486634	−0.0243317	+	0.999704i	$0.507746\pi$
$234$	0	0
$235$	−0.906181	−0.0591128
$236$	−0.871407	−0.0567238
$237$	−6.80560	−0.442071
$238$	0	0
$239$	4.31925	0.279389	0.139694	−	0.990195i	$-0.455388\pi$
$239$	4.31925	0.279389	0.139694	+	0.990195i	$0.455388\pi$
$240$	2.80560	0.181101
$241$	−18.1876	−1.17157	−0.585784	−	0.810467i	$-0.699213\pi$
$241$	−18.1876	−1.17157	−0.585784	+	0.810467i	$0.699213\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	2.15962	0.138256
$245$	0	0
$246$	5.09382	0.324770
$247$	0	0
$248$	−8.96523	−0.569292
$249$	−15.0938	−0.956532
$250$	5.97199	0.377702
$251$	28.6703	1.80965	0.904825	−	0.425784i	$-0.140001\pi$
$251$	28.6703	1.80965	0.904825	+	0.425784i	$0.140001\pi$
$252$	0	0
$253$	5.28822	0.332467
$254$	−9.28822	−0.582794
$255$	22.3473	1.39944
$256$	1.00000	0.0625000
$257$	−13.6112	−0.849044	−0.424522	−	0.905418i	$-0.639558\pi$
$257$	−13.6112	−0.849044	−0.424522	+	0.905418i	$0.639558\pi$
$258$	−11.4478	−0.712711
$259$	0	0
$260$	0	0
$261$	5.12859	0.317452
$262$	12.9652	0.800994
$263$	−3.61121	−0.222676	−0.111338	−	0.993783i	$-0.535514\pi$
$263$	−3.61121	−0.222676	−0.111338	+	0.993783i	$0.535514\pi$
$264$	1.00000	0.0615457
$265$	−22.4448	−1.37877
$266$	0	0
$267$	−1.61121	−0.0986042
$268$	−7.09382	−0.433324
$269$	−9.38203	−0.572033	−0.286016	−	0.958225i	$-0.592331\pi$
$269$	−9.38203	−0.572033	−0.286016	+	0.958225i	$0.592331\pi$
$270$	2.80560	0.170744
$271$	5.61121	0.340856	0.170428	−	0.985370i	$-0.445485\pi$
$271$	5.61121	0.340856	0.170428	+	0.985370i	$0.445485\pi$
$272$	7.96523	0.482963
$273$	0	0
$274$	5.61121	0.338985
$275$	−2.87141	−0.173152
$276$	5.28822	0.318313
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	13.1286	0.787401
$279$	−8.96523	−0.536734
$280$	0	0
$281$	24.1529	1.44084	0.720420	−	0.693539i	$-0.243949\pi$
$281$	24.1529	1.44084	0.720420	+	0.693539i	$0.243949\pi$
$282$	−0.322990	−0.0192338
$283$	7.54166	0.448305	0.224152	−	0.974554i	$-0.428039\pi$
$283$	7.54166	0.448305	0.224152	+	0.974554i	$0.428039\pi$
$284$	−7.44784	−0.441948
$285$	−18.1876	−1.07734
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	46.4448	2.73205
$290$	−14.3888	−0.844939
$291$	−7.00000	−0.410347
$292$	12.5764	0.735980
$293$	7.90618	0.461884	0.230942	−	0.972968i	$-0.425819\pi$
$293$	7.90618	0.461884	0.230942	+	0.972968i	$0.425819\pi$
$294$	0	0
$295$	2.44482	0.142343
$296$	10.4826	0.609289
$297$	1.00000	0.0580259
$298$	16.7398	0.969710
$299$	0	0
$300$	−2.87141	−0.165781
$301$	0	0
$302$	10.8994	0.627191
$303$	−10.7398	−0.616985
$304$	−6.48261	−0.371803
$305$	−6.05904	−0.346940
$306$	7.96523	0.455342
$307$	12.8957	0.735995	0.367998	−	0.929827i	$-0.380043\pi$
$307$	12.8957	0.735995	0.367998	+	0.929827i	$0.380043\pi$
$308$	0	0
$309$	−2.96523	−0.168686
$310$	25.1529	1.42859
$311$	10.8994	0.618049	0.309025	−	0.951054i	$-0.399997\pi$
$311$	10.8994	0.618049	0.309025	+	0.951054i	$0.399997\pi$
$312$	0	0
$313$	−9.12859	−0.515979	−0.257989	−	0.966148i	$-0.583060\pi$
$313$	−9.12859	−0.515979	−0.257989	+	0.966148i	$0.583060\pi$
$314$	1.77457	0.100145
$315$	0	0
$316$	6.80560	0.382845
$317$	−9.51364	−0.534339	−0.267170	−	0.963649i	$-0.586088\pi$
$317$	−9.51364	−0.534339	−0.267170	+	0.963649i	$0.586088\pi$
$318$	−8.00000	−0.448618
$319$	−5.12859	−0.287146
$320$	−2.80560	−0.156838
$321$	−4.83663	−0.269955
$322$	0	0
$323$	−51.6355	−2.87307
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.05904	−0.224810
$327$	10.1596	0.561828
$328$	−5.09382	−0.281259
$329$	0	0
$330$	−2.80560	−0.154443
$331$	1.09382	0.0601217	0.0300609	−	0.999548i	$-0.490430\pi$
$331$	1.09382	0.0601217	0.0300609	+	0.999548i	$0.490430\pi$
$332$	15.0938	0.828381
$333$	10.4826	0.574444
$334$	13.8684	0.758845
$335$	19.9024	1.08739
$336$	0	0
$337$	−25.7988	−1.40535	−0.702676	−	0.711510i	$-0.748012\pi$
$337$	−25.7988	−1.40535	−0.702676	+	0.711510i	$0.748012\pi$
$338$	−13.0000	−0.707107
$339$	−10.9652	−0.595549
$340$	−22.3473	−1.21195
$341$	8.96523	0.485494
$342$	−6.48261	−0.350540
$343$	0	0
$344$	11.4478	0.617226
$345$	−14.8366	−0.798777
$346$	16.9652	0.912056
$347$	20.7671	1.11484	0.557418	−	0.830232i	$-0.311792\pi$
$347$	20.7671	1.11484	0.557418	+	0.830232i	$0.311792\pi$
$348$	−5.12859	−0.274921
$349$	−9.51364	−0.509254	−0.254627	−	0.967039i	$-0.581953\pi$
$349$	−9.51364	−0.509254	−0.254627	+	0.967039i	$0.581953\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−12.5764	−0.669376	−0.334688	−	0.942329i	$-0.608631\pi$
$353$	−12.5764	−0.669376	−0.334688	+	0.942329i	$0.608631\pi$
$354$	0.871407	0.0463148
$355$	20.8957	1.10903
$356$	1.61121	0.0853937
$357$	0	0
$358$	−12.6703	−0.669644
$359$	−0.0695483	−0.00367062	−0.00183531	−	0.999998i	$-0.500584\pi$
$359$	−0.0695483	−0.00367062	−0.00183531	+	0.999998i	$0.500584\pi$
$360$	−2.80560	−0.147868
$361$	23.0243	1.21180
$362$	−2.57643	−0.135414
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−35.2845	−1.84687
$366$	−2.15962	−0.112885
$367$	−31.5417	−1.64646	−0.823231	−	0.567707i	$-0.807831\pi$
$367$	−31.5417	−1.64646	−0.823231	+	0.567707i	$0.807831\pi$
$368$	−5.28822	−0.275667
$369$	−5.09382	−0.265174
$370$	−29.4100	−1.52896
$371$	0	0
$372$	8.96523	0.464825
$373$	15.3125	0.792850	0.396425	−	0.918067i	$-0.370251\pi$
$373$	15.3125	0.792850	0.396425	+	0.918067i	$0.370251\pi$
$374$	−7.96523	−0.411872
$375$	−5.97199	−0.308392
$376$	0.322990	0.0166569
$377$	0	0
$378$	0	0
$379$	16.0590	0.824898	0.412449	−	0.910981i	$-0.364674\pi$
$379$	16.0590	0.824898	0.412449	+	0.910981i	$0.364674\pi$
$380$	18.1876	0.933006
$381$	9.28822	0.475850
$382$	−2.38879	−0.122221
$383$	−0.739798	−0.0378019	−0.0189010	−	0.999821i	$-0.506017\pi$
$383$	−0.739798	−0.0378019	−0.0189010	+	0.999821i	$0.506017\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.5764	−0.741921
$387$	11.4478	0.581926
$388$	7.00000	0.355371
$389$	3.58319	0.181675	0.0908375	−	0.995866i	$-0.471046\pi$
$389$	3.58319	0.181675	0.0908375	+	0.995866i	$0.471046\pi$
$390$	0	0
$391$	−42.1218	−2.13019
$392$	0	0
$393$	−12.9652	−0.654009
$394$	−12.4826	−0.628865
$395$	−19.0938	−0.960714
$396$	−1.00000	−0.0502519
$397$	−25.1286	−1.26117	−0.630584	−	0.776121i	$-0.717185\pi$
$397$	−25.1286	−1.26117	−0.630584	+	0.776121i	$0.717185\pi$
$398$	−14.1876	−0.711162
$399$	0	0
$400$	2.87141	0.143570
$401$	−3.86839	−0.193178	−0.0965891	−	0.995324i	$-0.530793\pi$
$401$	−3.86839	−0.193178	−0.0965891	+	0.995324i	$0.530793\pi$
$402$	7.09382	0.353808
$403$	0	0
$404$	10.7398	0.534325
$405$	−2.80560	−0.139412
$406$	0	0
$407$	−10.4826	−0.519604
$408$	−7.96523	−0.394337
$409$	4.64598	0.229729	0.114864	−	0.993381i	$-0.463357\pi$
$409$	4.64598	0.229729	0.114864	+	0.993381i	$0.463357\pi$
$410$	14.2912	0.705794
$411$	−5.61121	−0.276780
$412$	2.96523	0.146086
$413$	0	0
$414$	−5.28822	−0.259902
$415$	−42.3473	−2.07875
$416$	0	0
$417$	−13.1286	−0.642910
$418$	6.48261	0.317075
$419$	−19.5174	−0.953487	−0.476743	−	0.879043i	$-0.658183\pi$
$419$	−19.5174	−0.953487	−0.476743	+	0.879043i	$0.658183\pi$
$420$	0	0
$421$	15.5174	0.756271	0.378136	−	0.925750i	$-0.376565\pi$
$421$	15.5174	0.756271	0.378136	+	0.925750i	$0.376565\pi$
$422$	−20.8336	−1.01416
$423$	0.322990	0.0157043
$424$	8.00000	0.388514
$425$	22.8714	1.10943
$426$	7.44784	0.360849
$427$	0	0
$428$	4.83663	0.233787
$429$	0	0
$430$	−32.1181	−1.54887
$431$	−14.3888	−0.693084	−0.346542	−	0.938034i	$-0.612644\pi$
$431$	−14.3888	−0.693084	−0.346542	+	0.938034i	$0.612644\pi$
$432$	−1.00000	−0.0481125
$433$	2.03477	0.0977850	0.0488925	−	0.998804i	$-0.484431\pi$
$433$	2.03477	0.0977850	0.0488925	+	0.998804i	$0.484431\pi$
$434$	0	0
$435$	14.3888	0.689890
$436$	−10.1596	−0.486558
$437$	34.2815	1.63990
$438$	−12.5764	−0.600925
$439$	3.22616	0.153976	0.0769880	−	0.997032i	$-0.475470\pi$
$439$	3.22616	0.153976	0.0769880	+	0.997032i	$0.475470\pi$
$440$	2.80560	0.133752
$441$	0	0
$442$	0	0
$443$	−22.0938	−1.04971	−0.524854	−	0.851192i	$-0.675880\pi$
$443$	−22.0938	−1.04971	−0.524854	+	0.851192i	$0.675880\pi$
$444$	−10.4826	−0.497483
$445$	−4.52040	−0.214288
$446$	12.5764	0.595511
$447$	−16.7398	−0.791765
$448$	0	0
$449$	22.2497	1.05003	0.525014	−	0.851094i	$-0.324060\pi$
$449$	22.2497	1.05003	0.525014	+	0.851094i	$0.324060\pi$
$450$	2.87141	0.135359
$451$	5.09382	0.239859
$452$	10.9652	0.515761
$453$	−10.8994	−0.512099
$454$	20.0590	0.941418
$455$	0	0
$456$	6.48261	0.303576
$457$	−18.6385	−0.871872	−0.435936	−	0.899978i	$-0.643583\pi$
$457$	−18.6385	−0.871872	−0.435936	+	0.899978i	$0.643583\pi$
$458$	−22.5764	−1.05493
$459$	−7.96523	−0.371785
$460$	14.8366	0.691762
$461$	−20.9895	−0.977578	−0.488789	−	0.872402i	$-0.662561\pi$
$461$	−20.9895	−0.977578	−0.488789	+	0.872402i	$0.662561\pi$
$462$	0	0
$463$	−5.35402	−0.248822	−0.124411	−	0.992231i	$-0.539704\pi$
$463$	−5.35402	−0.248822	−0.124411	+	0.992231i	$0.539704\pi$
$464$	5.12859	0.238089
$465$	−25.1529	−1.16644
$466$	−0.742815	−0.0344102
$467$	26.0243	1.20426	0.602130	−	0.798398i	$-0.294319\pi$
$467$	26.0243	1.20426	0.602130	+	0.798398i	$0.294319\pi$
$468$	0	0
$469$	0	0
$470$	−0.906181	−0.0417990
$471$	−1.77457	−0.0817680
$472$	−0.871407	−0.0401098
$473$	−11.4478	−0.526372
$474$	−6.80560	−0.312592
$475$	−18.6142	−0.854079
$476$	0	0
$477$	8.00000	0.366295
$478$	4.31925	0.197558
$479$	−3.61121	−0.165000	−0.0825001	−	0.996591i	$-0.526290\pi$
$479$	−3.61121	−0.165000	−0.0825001	+	0.996591i	$0.526290\pi$
$480$	2.80560	0.128058
$481$	0	0
$482$	−18.1876	−0.828424
$483$	0	0
$484$	1.00000	0.0454545
$485$	−19.6392	−0.891771
$486$	−1.00000	−0.0453609
$487$	−8.06206	−0.365327	−0.182663	−	0.983176i	$-0.558472\pi$
$487$	−8.06206	−0.365327	−0.182663	+	0.983176i	$0.558472\pi$
$488$	2.15962	0.0977615
$489$	4.05904	0.183556
$490$	0	0
$491$	20.9062	0.943483	0.471741	−	0.881737i	$-0.343626\pi$
$491$	20.9062	0.943483	0.471741	+	0.881737i	$0.343626\pi$
$492$	5.09382	0.229647
$493$	40.8504	1.83981
$494$	0	0
$495$	2.80560	0.126102
$496$	−8.96523	−0.402551
$497$	0	0
$498$	−15.0938	−0.676370
$499$	4.96523	0.222274	0.111137	−	0.993805i	$-0.464551\pi$
$499$	4.96523	0.222274	0.111137	+	0.993805i	$0.464551\pi$
$500$	5.97199	0.267075
$501$	−13.8684	−0.619594
$502$	28.6703	1.27962
$503$	−17.7988	−0.793611	−0.396806	−	0.917903i	$-0.629881\pi$
$503$	−17.7988	−0.793611	−0.396806	+	0.917903i	$0.629881\pi$
$504$	0	0
$505$	−30.1316	−1.34084
$506$	5.28822	0.235090
$507$	13.0000	0.577350
$508$	−9.28822	−0.412098
$509$	4.51437	0.200096	0.100048	−	0.994983i	$-0.468100\pi$
$509$	4.51437	0.200096	0.100048	+	0.994983i	$0.468100\pi$
$510$	22.3473	0.989553
$511$	0	0
$512$	1.00000	0.0441942
$513$	6.48261	0.286214
$514$	−13.6112	−0.600365
$515$	−8.31925	−0.366590
$516$	−11.4478	−0.503963
$517$	−0.322990	−0.0142051
$518$	0	0
$519$	−16.9652	−0.744691
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	5.12859	0.224472
$523$	6.90317	0.301854	0.150927	−	0.988545i	$-0.451774\pi$
$523$	6.90317	0.301854	0.150927	+	0.988545i	$0.451774\pi$
$524$	12.9652	0.566389
$525$	0	0
$526$	−3.61121	−0.157456
$527$	−71.4100	−3.11067
$528$	1.00000	0.0435194
$529$	4.96523	0.215879
$530$	−22.4448	−0.974941
$531$	−0.871407	−0.0378159
$532$	0	0
$533$	0	0
$534$	−1.61121	−0.0697237
$535$	−13.5697	−0.586668
$536$	−7.09382	−0.306406
$537$	12.6703	0.546762
$538$	−9.38203	−0.404488
$539$	0	0
$540$	2.80560	0.120734
$541$	2.80560	0.120622	0.0603111	−	0.998180i	$-0.480791\pi$
$541$	2.80560	0.120622	0.0603111	+	0.998180i	$0.480791\pi$
$542$	5.61121	0.241022
$543$	2.57643	0.110565
$544$	7.96523	0.341506
$545$	28.5039	1.22097
$546$	0	0
$547$	32.9274	1.40788	0.703938	−	0.710262i	$-0.251423\pi$
$547$	32.9274	1.40788	0.703938	+	0.710262i	$0.251423\pi$
$548$	5.61121	0.239699
$549$	2.15962	0.0921705
$550$	−2.87141	−0.122437
$551$	−33.2467	−1.41636
$552$	5.28822	0.225081
$553$	0	0
$554$	−16.0000	−0.679775
$555$	29.4100	1.24839
$556$	13.1286	0.556776
$557$	23.7050	1.00441	0.502207	−	0.864747i	$-0.332522\pi$
$557$	23.7050	1.00441	0.502207	+	0.864747i	$0.332522\pi$
$558$	−8.96523	−0.379528
$559$	0	0
$560$	0	0
$561$	7.96523	0.336292
$562$	24.1529	1.01883
$563$	30.4448	1.28310	0.641548	−	0.767083i	$-0.278292\pi$
$563$	30.4448	1.28310	0.641548	+	0.767083i	$0.278292\pi$
$564$	−0.322990	−0.0136003
$565$	−30.7641	−1.29425
$566$	7.54166	0.317000
$567$	0	0
$568$	−7.44784	−0.312504
$569$	22.2815	0.934087	0.467044	−	0.884234i	$-0.345319\pi$
$569$	22.2815	0.934087	0.467044	+	0.884234i	$0.345319\pi$
$570$	−18.1876	−0.761796
$571$	−2.87141	−0.120165	−0.0600823	−	0.998193i	$-0.519136\pi$
$571$	−2.87141	−0.120165	−0.0600823	+	0.998193i	$0.519136\pi$
$572$	0	0
$573$	2.38879	0.0997933
$574$	0	0
$575$	−15.1846	−0.633242
$576$	1.00000	0.0416667
$577$	−15.0243	−0.625469	−0.312734	−	0.949841i	$-0.601245\pi$
$577$	−15.0243	−0.625469	−0.312734	+	0.949841i	$0.601245\pi$
$578$	46.4448	1.93185
$579$	14.5764	0.605776
$580$	−14.3888	−0.597462
$581$	0	0
$582$	−7.00000	−0.290159
$583$	−8.00000	−0.331326
$584$	12.5764	0.520416
$585$	0	0
$586$	7.90618	0.326601
$587$	−42.5069	−1.75445	−0.877223	−	0.480082i	$-0.840607\pi$
$587$	−42.5069	−1.75445	−0.877223	+	0.480082i	$0.840607\pi$
$588$	0	0
$589$	58.1181	2.39471
$590$	2.44482	0.100652
$591$	12.4826	0.513466
$592$	10.4826	0.430833
$593$	15.3858	0.631818	0.315909	−	0.948789i	$-0.397691\pi$
$593$	15.3858	0.631818	0.315909	+	0.948789i	$0.397691\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	16.7398	0.685689
$597$	14.1876	0.580661
$598$	0	0
$599$	−39.6392	−1.61961	−0.809807	−	0.586696i	$-0.800428\pi$
$599$	−39.6392	−1.61961	−0.809807	+	0.586696i	$0.800428\pi$
$600$	−2.87141	−0.117225
$601$	33.3405	1.35999	0.679994	−	0.733218i	$-0.261983\pi$
$601$	33.3405	1.35999	0.679994	+	0.733218i	$0.261983\pi$
$602$	0	0
$603$	−7.09382	−0.288883
$604$	10.8994	0.443491
$605$	−2.80560	−0.114064
$606$	−10.7398	−0.436274
$607$	15.9585	0.647734	0.323867	−	0.946103i	$-0.395017\pi$
$607$	15.9585	0.647734	0.323867	+	0.946103i	$0.395017\pi$
$608$	−6.48261	−0.262905
$609$	0	0
$610$	−6.05904	−0.245324
$611$	0	0
$612$	7.96523	0.321975
$613$	46.6665	1.88484	0.942421	−	0.334428i	$-0.108543\pi$
$613$	46.6665	1.88484	0.942421	+	0.334428i	$0.108543\pi$
$614$	12.8957	0.520427
$615$	−14.2912	−0.576278
$616$	0	0
$617$	−9.28447	−0.373779	−0.186889	−	0.982381i	$-0.559841\pi$
$617$	−9.28447	−0.373779	−0.186889	+	0.982381i	$0.559841\pi$
$618$	−2.96523	−0.119279
$619$	33.9895	1.36615	0.683077	−	0.730347i	$-0.260642\pi$
$619$	33.9895	1.36615	0.683077	+	0.730347i	$0.260642\pi$
$620$	25.1529	1.01016
$621$	5.28822	0.212209
$622$	10.8994	0.437027
$623$	0	0
$624$	0	0
$625$	−31.1121	−1.24448
$626$	−9.12859	−0.364852
$627$	−6.48261	−0.258891
$628$	1.77457	0.0708132
$629$	83.4964	3.32922
$630$	0	0
$631$	−14.6460	−0.583047	−0.291524	−	0.956564i	$-0.594162\pi$
$631$	−14.6460	−0.583047	−0.291524	+	0.956564i	$0.594162\pi$
$632$	6.80560	0.270712
$633$	20.8336	0.828062
$634$	−9.51364	−0.377835
$635$	26.0590	1.03412
$636$	−8.00000	−0.317221
$637$	0	0
$638$	−5.12859	−0.203043
$639$	−7.44784	−0.294632
$640$	−2.80560	−0.110901
$641$	20.9652	0.828077	0.414038	−	0.910259i	$-0.364118\pi$
$641$	20.9652	0.828077	0.414038	+	0.910259i	$0.364118\pi$
$642$	−4.83663	−0.190887
$643$	−23.6733	−0.933582	−0.466791	−	0.884368i	$-0.654590\pi$
$643$	−23.6733	−0.933582	−0.466791	+	0.884368i	$0.654590\pi$
$644$	0	0
$645$	32.1181	1.26465
$646$	−51.6355	−2.03157
$647$	−12.4168	−0.488155	−0.244078	−	0.969756i	$-0.578485\pi$
$647$	−12.4168	−0.488155	−0.244078	+	0.969756i	$0.578485\pi$
$648$	1.00000	0.0392837
$649$	0.871407	0.0342057
$650$	0	0
$651$	0	0
$652$	−4.05904	−0.158964
$653$	−0.0975625	−0.00381792	−0.00190896	−	0.999998i	$-0.500608\pi$
$653$	−0.0975625	−0.00381792	−0.00190896	+	0.999998i	$0.500608\pi$
$654$	10.1596	0.397273
$655$	−36.3753	−1.42130
$656$	−5.09382	−0.198880
$657$	12.5764	0.490653
$658$	0	0
$659$	−35.0243	−1.36435	−0.682176	−	0.731188i	$-0.738966\pi$
$659$	−35.0243	−1.36435	−0.682176	+	0.731188i	$0.738966\pi$
$660$	−2.80560	−0.109208
$661$	29.6355	1.15269	0.576343	−	0.817208i	$-0.304479\pi$
$661$	29.6355	1.15269	0.576343	+	0.817208i	$0.304479\pi$
$662$	1.09382	0.0425125
$663$	0	0
$664$	15.0938	0.585754
$665$	0	0
$666$	10.4826	0.406193
$667$	−27.1211	−1.05013
$668$	13.8684	0.536584
$669$	−12.5764	−0.486233
$670$	19.9024	0.768898
$671$	−2.15962	−0.0833713
$672$	0	0
$673$	16.8957	0.651281	0.325640	−	0.945494i	$-0.394420\pi$
$673$	16.8957	0.651281	0.325640	+	0.945494i	$0.394420\pi$
$674$	−25.7988	−0.993734
$675$	−2.87141	−0.110521
$676$	−13.0000	−0.500000
$677$	13.1907	0.506958	0.253479	−	0.967341i	$-0.418425\pi$
$677$	13.1907	0.506958	0.253479	+	0.967341i	$0.418425\pi$
$678$	−10.9652	−0.421117
$679$	0	0
$680$	−22.3473	−0.856978
$681$	−20.0590	−0.768664
$682$	8.96523	0.343296
$683$	9.83663	0.376388	0.188194	−	0.982132i	$-0.439737\pi$
$683$	9.83663	0.376388	0.188194	+	0.982132i	$0.439737\pi$
$684$	−6.48261	−0.247869
$685$	−15.7428	−0.601502
$686$	0	0
$687$	22.5764	0.861345
$688$	11.4478	0.436445
$689$	0	0
$690$	−14.8366	−0.564821
$691$	−49.9895	−1.90169	−0.950845	−	0.309667i	$-0.899782\pi$
$691$	−49.9895	−1.90169	−0.950845	+	0.309667i	$0.899782\pi$
$692$	16.9652	0.644921
$693$	0	0
$694$	20.7671	0.788308
$695$	−36.8336	−1.39718
$696$	−5.12859	−0.194399
$697$	−40.5734	−1.53683
$698$	−9.51364	−0.360097
$699$	0.742815	0.0280958
$700$	0	0
$701$	−17.7050	−0.668710	−0.334355	−	0.942447i	$-0.608518\pi$
$701$	−17.7050	−0.668710	−0.334355	+	0.942447i	$0.608518\pi$
$702$	0	0
$703$	−67.9547	−2.56296
$704$	−1.00000	−0.0376889
$705$	0.906181	0.0341288
$706$	−12.5764	−0.473320
$707$	0	0
$708$	0.871407	0.0327495
$709$	−31.9547	−1.20008	−0.600042	−	0.799968i	$-0.704850\pi$
$709$	−31.9547	−1.20008	−0.600042	+	0.799968i	$0.704850\pi$
$710$	20.8957	0.784201
$711$	6.80560	0.255230
$712$	1.61121	0.0603825
$713$	47.4100	1.77552
$714$	0	0
$715$	0	0
$716$	−12.6703	−0.473509
$717$	−4.31925	−0.161305
$718$	−0.0695483	−0.00259552
$719$	21.7951	0.812820	0.406410	−	0.913691i	$-0.366780\pi$
$719$	21.7951	0.812820	0.406410	+	0.913691i	$0.366780\pi$
$720$	−2.80560	−0.104559
$721$	0	0
$722$	23.0243	0.856875
$723$	18.1876	0.676406
$724$	−2.57643	−0.0957523
$725$	14.7263	0.546920
$726$	−1.00000	−0.0371135
$727$	−4.84714	−0.179770	−0.0898852	−	0.995952i	$-0.528650\pi$
$727$	−4.84714	−0.179770	−0.0898852	+	0.995952i	$0.528650\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−35.2845	−1.30594
$731$	91.1846	3.37259
$732$	−2.15962	−0.0798220
$733$	12.2292	0.451695	0.225847	−	0.974163i	$-0.427485\pi$
$733$	12.2292	0.451695	0.225847	+	0.974163i	$0.427485\pi$
$734$	−31.5417	−1.16422
$735$	0	0
$736$	−5.28822	−0.194926
$737$	7.09382	0.261304
$738$	−5.09382	−0.187506
$739$	−1.03477	−0.0380648	−0.0190324	−	0.999819i	$-0.506059\pi$
$739$	−1.03477	−0.0380648	−0.0190324	+	0.999819i	$0.506059\pi$
$740$	−29.4100	−1.08113
$741$	0	0
$742$	0	0
$743$	−23.9925	−0.880200	−0.440100	−	0.897949i	$-0.645057\pi$
$743$	−23.9925	−0.880200	−0.440100	+	0.897949i	$0.645057\pi$
$744$	8.96523	0.328681
$745$	−46.9652	−1.72067
$746$	15.3125	0.560630
$747$	15.0938	0.552254
$748$	−7.96523	−0.291238
$749$	0	0
$750$	−5.97199	−0.218066
$751$	20.5069	0.748307	0.374153	−	0.927367i	$-0.377933\pi$
$751$	20.5069	0.748307	0.374153	+	0.927367i	$0.377933\pi$
$752$	0.322990	0.0117782
$753$	−28.6703	−1.04480
$754$	0	0
$755$	−30.5794	−1.11290
$756$	0	0
$757$	−16.8019	−0.610674	−0.305337	−	0.952244i	$-0.598769\pi$
$757$	−16.8019	−0.610674	−0.305337	+	0.952244i	$0.598769\pi$
$758$	16.0590	0.583291
$759$	−5.28822	−0.191950
$760$	18.1876	0.659735
$761$	−14.5099	−0.525983	−0.262992	−	0.964798i	$-0.584709\pi$
$761$	−14.5099	−0.525983	−0.262992	+	0.964798i	$0.584709\pi$
$762$	9.28822	0.336477
$763$	0	0
$764$	−2.38879	−0.0864235
$765$	−22.3473	−0.807967
$766$	−0.739798	−0.0267300
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	3.35402	0.120949	0.0604745	−	0.998170i	$-0.480739\pi$
$769$	3.35402	0.120949	0.0604745	+	0.998170i	$0.480739\pi$
$770$	0	0
$771$	13.6112	0.490196
$772$	−14.5764	−0.524617
$773$	1.84038	0.0661938	0.0330969	−	0.999452i	$-0.489463\pi$
$773$	1.84038	0.0661938	0.0330969	+	0.999452i	$0.489463\pi$
$774$	11.4478	0.411484
$775$	−25.7428	−0.924709
$776$	7.00000	0.251285
$777$	0	0
$778$	3.58319	0.128464
$779$	33.0213	1.18311
$780$	0	0
$781$	7.44784	0.266505
$782$	−42.1218	−1.50627
$783$	−5.12859	−0.183281
$784$	0	0
$785$	−4.97875	−0.177699
$786$	−12.9652	−0.462454
$787$	11.5869	0.413030	0.206515	−	0.978443i	$-0.433788\pi$
$787$	11.5869	0.413030	0.206515	+	0.978443i	$0.433788\pi$
$788$	−12.4826	−0.444675
$789$	3.61121	0.128562
$790$	−19.0938	−0.679328
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	0	0
$794$	−25.1286	−0.891780
$795$	22.4448	0.796036
$796$	−14.1876	−0.502867
$797$	−48.4653	−1.71673	−0.858365	−	0.513039i	$-0.828520\pi$
$797$	−48.4653	−1.71673	−0.858365	+	0.513039i	$0.828520\pi$
$798$	0	0
$799$	2.57269	0.0910151
$800$	2.87141	0.101520
$801$	1.61121	0.0569292
$802$	−3.86839	−0.136598
$803$	−12.5764	−0.443813
$804$	7.09382	0.250180
$805$	0	0
$806$	0	0
$807$	9.38203	0.330263
$808$	10.7398	0.377825
$809$	−2.76708	−0.0972855	−0.0486428	−	0.998816i	$-0.515490\pi$
$809$	−2.76708	−0.0972855	−0.0486428	+	0.998816i	$0.515490\pi$
$810$	−2.80560	−0.0985788
$811$	4.89568	0.171910	0.0859552	−	0.996299i	$-0.472606\pi$
$811$	4.89568	0.171910	0.0859552	+	0.996299i	$0.472606\pi$
$812$	0	0
$813$	−5.61121	−0.196794
$814$	−10.4826	−0.367415
$815$	11.3881	0.398907
$816$	−7.96523	−0.278839
$817$	−74.2119	−2.59635
$818$	4.64598	0.162443
$819$	0	0
$820$	14.2912	0.499071
$821$	3.54915	0.123866	0.0619330	−	0.998080i	$-0.480273\pi$
$821$	3.54915	0.123866	0.0619330	+	0.998080i	$0.480273\pi$
$822$	−5.61121	−0.195713
$823$	32.5069	1.13312	0.566559	−	0.824021i	$-0.308274\pi$
$823$	32.5069	1.13312	0.566559	+	0.824021i	$0.308274\pi$
$824$	2.96523	0.103299
$825$	2.87141	0.0999696
$826$	0	0
$827$	19.5447	0.679635	0.339817	−	0.940491i	$-0.389635\pi$
$827$	19.5447	0.679635	0.339817	+	0.940491i	$0.389635\pi$
$828$	−5.28822	−0.183778
$829$	25.1907	0.874908	0.437454	−	0.899241i	$-0.355880\pi$
$829$	25.1907	0.874908	0.437454	+	0.899241i	$0.355880\pi$
$830$	−42.3473	−1.46989
$831$	16.0000	0.555034
$832$	0	0
$833$	0	0
$834$	−13.1286	−0.454606
$835$	−38.9092	−1.34651
$836$	6.48261	0.224206
$837$	8.96523	0.309884
$838$	−19.5174	−0.674217
$839$	−50.4033	−1.74011	−0.870057	−	0.492950i	$-0.835918\pi$
$839$	−50.4033	−1.74011	−0.870057	+	0.492950i	$0.835918\pi$
$840$	0	0
$841$	−2.69754	−0.0930185
$842$	15.5174	0.534764
$843$	−24.1529	−0.831869
$844$	−20.8336	−0.717123
$845$	36.4728	1.25470
$846$	0.322990	0.0111046
$847$	0	0
$848$	8.00000	0.274721
$849$	−7.54166	−0.258829
$850$	22.8714	0.784483
$851$	−55.4343	−1.90026
$852$	7.44784	0.255159
$853$	−16.8752	−0.577794	−0.288897	−	0.957360i	$-0.593289\pi$
$853$	−16.8752	−0.577794	−0.288897	+	0.957360i	$0.593289\pi$
$854$	0	0
$855$	18.1876	0.622004
$856$	4.83663	0.165313
$857$	43.3057	1.47930	0.739648	−	0.672994i	$-0.234992\pi$
$857$	43.3057	1.47930	0.739648	+	0.672994i	$0.234992\pi$
$858$	0	0
$859$	3.61422	0.123316	0.0616578	−	0.998097i	$-0.480361\pi$
$859$	3.61422	0.123316	0.0616578	+	0.998097i	$0.480361\pi$
$860$	−32.1181	−1.09522
$861$	0	0
$862$	−14.3888	−0.490084
$863$	4.41681	0.150350	0.0751750	−	0.997170i	$-0.476048\pi$
$863$	4.41681	0.150350	0.0751750	+	0.997170i	$0.476048\pi$
$864$	−1.00000	−0.0340207
$865$	−47.5977	−1.61837
$866$	2.03477	0.0691444
$867$	−46.4448	−1.57735
$868$	0	0
$869$	−6.80560	−0.230864
$870$	14.3888	0.487826
$871$	0	0
$872$	−10.1596	−0.344048
$873$	7.00000	0.236914
$874$	34.2815	1.15959
$875$	0	0
$876$	−12.5764	−0.424918
$877$	44.2777	1.49515	0.747576	−	0.664176i	$-0.231218\pi$
$877$	44.2777	1.49515	0.747576	+	0.664176i	$0.231218\pi$
$878$	3.22616	0.108877
$879$	−7.90618	−0.266669
$880$	2.80560	0.0945769
$881$	−18.1876	−0.612757	−0.306379	−	0.951910i	$-0.599117\pi$
$881$	−18.1876	−0.612757	−0.306379	+	0.951910i	$0.599117\pi$
$882$	0	0
$883$	−4.57945	−0.154111	−0.0770553	−	0.997027i	$-0.524552\pi$
$883$	−4.57945	−0.154111	−0.0770553	+	0.997027i	$0.524552\pi$
$884$	0	0
$885$	−2.44482	−0.0821818
$886$	−22.0938	−0.742256
$887$	−0.576432	−0.0193547	−0.00967734	−	0.999953i	$-0.503080\pi$
$887$	−0.576432	−0.0193547	−0.00967734	+	0.999953i	$0.503080\pi$
$888$	−10.4826	−0.351773
$889$	0	0
$890$	−4.52040	−0.151524
$891$	−1.00000	−0.0335013
$892$	12.5764	0.421090
$893$	−2.09382	−0.0700670
$894$	−16.7398	−0.559863
$895$	35.5477	1.18823
$896$	0	0
$897$	0	0
$898$	22.2497	0.742482
$899$	−45.9790	−1.53349
$900$	2.87141	0.0957136
$901$	63.7218	2.12288
$902$	5.09382	0.169606
$903$	0	0
$904$	10.9652	0.364698
$905$	7.22844	0.240282
$906$	−10.8994	−0.362109
$907$	−22.0590	−0.732459	−0.366229	−	0.930525i	$-0.619351\pi$
$907$	−22.0590	−0.732459	−0.366229	+	0.930525i	$0.619351\pi$
$908$	20.0590	0.665683
$909$	10.7398	0.356217
$910$	0	0
$911$	9.93420	0.329135	0.164567	−	0.986366i	$-0.447377\pi$
$911$	9.93420	0.329135	0.164567	+	0.986366i	$0.447377\pi$
$912$	6.48261	0.214661
$913$	−15.0938	−0.499532
$914$	−18.6385	−0.616507
$915$	6.05904	0.200306
$916$	−22.5764	−0.745946
$917$	0	0
$918$	−7.96523	−0.262892
$919$	44.6347	1.47236	0.736182	−	0.676783i	$-0.236627\pi$
$919$	44.6347	1.47236	0.736182	+	0.676783i	$0.236627\pi$
$920$	14.8366	0.489149
$921$	−12.8957	−0.424927
$922$	−20.9895	−0.691252
$923$	0	0
$924$	0	0
$925$	30.0999	0.989677
$926$	−5.35402	−0.175944
$927$	2.96523	0.0973908
$928$	5.12859	0.168354
$929$	−42.5764	−1.39689	−0.698444	−	0.715665i	$-0.746124\pi$
$929$	−42.5764	−1.39689	−0.698444	+	0.715665i	$0.746124\pi$
$930$	−25.1529	−0.824795
$931$	0	0
$932$	−0.742815	−0.0243317
$933$	−10.8994	−0.356831
$934$	26.0243	0.851540
$935$	22.3473	0.730834
$936$	0	0
$937$	−36.2572	−1.18447	−0.592235	−	0.805765i	$-0.701754\pi$
$937$	−36.2572	−1.18447	−0.592235	+	0.805765i	$0.701754\pi$
$938$	0	0
$939$	9.12859	0.297900
$940$	−0.906181	−0.0295564
$941$	2.68377	0.0874884	0.0437442	−	0.999043i	$-0.486071\pi$
$941$	2.68377	0.0874884	0.0437442	+	0.999043i	$0.486071\pi$
$942$	−1.77457	−0.0578187
$943$	26.9372	0.877196
$944$	−0.871407	−0.0283619
$945$	0	0
$946$	−11.4478	−0.372201
$947$	49.5039	1.60866	0.804330	−	0.594183i	$-0.202525\pi$
$947$	49.5039	1.60866	0.804330	+	0.594183i	$0.202525\pi$
$948$	−6.80560	−0.221036
$949$	0	0
$950$	−18.6142	−0.603925
$951$	9.51364	0.308501
$952$	0	0
$953$	15.8019	0.511872	0.255936	−	0.966694i	$-0.417616\pi$
$953$	15.8019	0.511872	0.255936	+	0.966694i	$0.417616\pi$
$954$	8.00000	0.259010
$955$	6.70201	0.216872
$956$	4.31925	0.139694
$957$	5.12859	0.165784
$958$	−3.61121	−0.116673
$959$	0	0
$960$	2.80560	0.0905504
$961$	49.3753	1.59275
$962$	0	0
$963$	4.83663	0.155858
$964$	−18.1876	−0.585784
$965$	40.8957	1.31648
$966$	0	0
$967$	−12.6497	−0.406788	−0.203394	−	0.979097i	$-0.565197\pi$
$967$	−12.6497	−0.406788	−0.203394	+	0.979097i	$0.565197\pi$
$968$	1.00000	0.0321412
$969$	51.6355	1.65877
$970$	−19.6392	−0.630577
$971$	49.5342	1.58963	0.794814	−	0.606854i	$-0.207569\pi$
$971$	49.5342	1.58963	0.794814	+	0.606854i	$0.207569\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−8.06206	−0.258325
$975$	0	0
$976$	2.15962	0.0691278
$977$	34.4448	1.10199	0.550994	−	0.834509i	$-0.314249\pi$
$977$	34.4448	1.10199	0.550994	+	0.834509i	$0.314249\pi$
$978$	4.05904	0.129794
$979$	−1.61121	−0.0514944
$980$	0	0
$981$	−10.1596	−0.324372
$982$	20.9062	0.667143
$983$	−3.35776	−0.107096	−0.0535480	−	0.998565i	$-0.517053\pi$
$983$	−3.35776	−0.107096	−0.0535480	+	0.998565i	$0.517053\pi$
$984$	5.09382	0.162385
$985$	35.0213	1.11587
$986$	40.8504	1.30094
$987$	0	0
$988$	0	0
$989$	−60.5386	−1.92502
$990$	2.80560	0.0891679
$991$	−45.8474	−1.45639	−0.728195	−	0.685370i	$-0.759641\pi$
$991$	−45.8474	−1.45639	−0.728195	+	0.685370i	$0.759641\pi$
$992$	−8.96523	−0.284646
$993$	−1.09382	−0.0347113
$994$	0	0
$995$	39.8049	1.26190
$996$	−15.0938	−0.478266
$997$	−43.8609	−1.38909	−0.694544	−	0.719450i	$-0.744394\pi$
$997$	−43.8609	−1.38909	−0.694544	+	0.719450i	$0.744394\pi$
$998$	4.96523	0.157171
$999$	−10.4826	−0.331655

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.bf.1.1		3
3.2	odd	2		9702.2.a.dv.1.3		3
7.2	even	3		462.2.i.g.67.3	✓	6
7.4	even	3		462.2.i.g.331.3	yes	6
7.6	odd	2		3234.2.a.bh.1.3		3
21.2	odd	6		1386.2.k.v.991.1		6
21.11	odd	6		1386.2.k.v.793.1		6
21.20	even	2		9702.2.a.dw.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.g.67.3	✓	6	7.2	even	3
462.2.i.g.331.3	yes	6	7.4	even	3
1386.2.k.v.793.1		6	21.11	odd	6
1386.2.k.v.991.1		6	21.2	odd	6
3234.2.a.bf.1.1		3	1.1	even	1	trivial
3234.2.a.bh.1.3		3	7.6	odd	2
9702.2.a.dv.1.3		3	3.2	odd	2
9702.2.a.dw.1.1		3	21.20	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:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.80560 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.80560 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.80560 q^{10} -1.00000 q^{11} -1.00000 q^{12} +2.80560 q^{15} +1.00000 q^{16} +7.96523 q^{17} +1.00000 q^{18} -6.48261 q^{19} -2.80560 q^{20} -1.00000 q^{22} -5.28822 q^{23} -1.00000 q^{24} +2.87141 q^{25} -1.00000 q^{27} +5.12859 q^{29} +2.80560 q^{30} -8.96523 q^{31} +1.00000 q^{32} +1.00000 q^{33} +7.96523 q^{34} +1.00000 q^{36} +10.4826 q^{37} -6.48261 q^{38} -2.80560 q^{40} -5.09382 q^{41} +11.4478 q^{43} -1.00000 q^{44} -2.80560 q^{45} -5.28822 q^{46} +0.322990 q^{47} -1.00000 q^{48} +2.87141 q^{50} -7.96523 q^{51} +8.00000 q^{53} -1.00000 q^{54} +2.80560 q^{55} +6.48261 q^{57} +5.12859 q^{58} -0.871407 q^{59} +2.80560 q^{60} +2.15962 q^{61} -8.96523 q^{62} +1.00000 q^{64} +1.00000 q^{66} -7.09382 q^{67} +7.96523 q^{68} +5.28822 q^{69} -7.44784 q^{71} +1.00000 q^{72} +12.5764 q^{73} +10.4826 q^{74} -2.87141 q^{75} -6.48261 q^{76} +6.80560 q^{79} -2.80560 q^{80} +1.00000 q^{81} -5.09382 q^{82} +15.0938 q^{83} -22.3473 q^{85} +11.4478 q^{86} -5.12859 q^{87} -1.00000 q^{88} +1.61121 q^{89} -2.80560 q^{90} -5.28822 q^{92} +8.96523 q^{93} +0.322990 q^{94} +18.1876 q^{95} -1.00000 q^{96} +7.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 3 q^{11} - 3 q^{12} + 3 q^{16} + 3 q^{17} + 3 q^{18} - 9 q^{19} - 3 q^{22} + 3 q^{23} - 3 q^{24} + 15 q^{25} - 3 q^{27} + 9 q^{29} - 6 q^{31} + 3 q^{32} + 3 q^{33} + 3 q^{34} + 3 q^{36} + 21 q^{37} - 9 q^{38} + 12 q^{41} + 3 q^{43} - 3 q^{44} + 3 q^{46} + 3 q^{47} - 3 q^{48} + 15 q^{50} - 3 q^{51} + 24 q^{53} - 3 q^{54} + 9 q^{57} + 9 q^{58} - 9 q^{59} - 6 q^{61} - 6 q^{62} + 3 q^{64} + 3 q^{66} + 6 q^{67} + 3 q^{68} - 3 q^{69} + 9 q^{71} + 3 q^{72} + 21 q^{74} - 15 q^{75} - 9 q^{76} + 12 q^{79} + 3 q^{81} + 12 q^{82} + 18 q^{83} + 3 q^{86} - 9 q^{87} - 3 q^{88} - 12 q^{89} + 3 q^{92} + 6 q^{93} + 3 q^{94} - 3 q^{96} + 21 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 - 3 * q^11 - 3 * q^12 + 3 * q^16 + 3 * q^17 + 3 * q^18 - 9 * q^19 - 3 * q^22 + 3 * q^23 - 3 * q^24 + 15 * q^25 - 3 * q^27 + 9 * q^29 - 6 * q^31 + 3 * q^32 + 3 * q^33 + 3 * q^34 + 3 * q^36 + 21 * q^37 - 9 * q^38 + 12 * q^41 + 3 * q^43 - 3 * q^44 + 3 * q^46 + 3 * q^47 - 3 * q^48 + 15 * q^50 - 3 * q^51 + 24 * q^53 - 3 * q^54 + 9 * q^57 + 9 * q^58 - 9 * q^59 - 6 * q^61 - 6 * q^62 + 3 * q^64 + 3 * q^66 + 6 * q^67 + 3 * q^68 - 3 * q^69 + 9 * q^71 + 3 * q^72 + 21 * q^74 - 15 * q^75 - 9 * q^76 + 12 * q^79 + 3 * q^81 + 12 * q^82 + 18 * q^83 + 3 * q^86 - 9 * q^87 - 3 * q^88 - 12 * q^89 + 3 * q^92 + 6 * q^93 + 3 * q^94 - 3 * q^96 + 21 * q^97 - 3 * q^99

Introduction

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