LMFDB - Embedded newform 3234.2.a.bf.1.2

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			1.2
Root			$-1.61323$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.61323 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.61323 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.61323 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.61323 q^{15} +1.00000 q^{16} -7.34206 q^{17} +1.00000 q^{18} +1.17103 q^{19} -1.61323 q^{20} -1.00000 q^{22} +3.55780 q^{23} -1.00000 q^{24} -2.39749 q^{25} -1.00000 q^{27} +10.3975 q^{29} +1.61323 q^{30} +6.34206 q^{31} +1.00000 q^{32} +1.00000 q^{33} -7.34206 q^{34} +1.00000 q^{36} +2.82897 q^{37} +1.17103 q^{38} -1.61323 q^{40} +4.94457 q^{41} -11.5131 q^{43} -1.00000 q^{44} -1.61323 q^{45} +3.55780 q^{46} +6.78426 q^{47} -1.00000 q^{48} -2.39749 q^{50} +7.34206 q^{51} +8.00000 q^{53} -1.00000 q^{54} +1.61323 q^{55} -1.17103 q^{57} +10.3975 q^{58} +4.39749 q^{59} +1.61323 q^{60} -11.9553 q^{61} +6.34206 q^{62} +1.00000 q^{64} +1.00000 q^{66} +2.94457 q^{67} -7.34206 q^{68} -3.55780 q^{69} +15.5131 q^{71} +1.00000 q^{72} -5.11560 q^{73} +2.82897 q^{74} +2.39749 q^{75} +1.17103 q^{76} +5.61323 q^{79} -1.61323 q^{80} +1.00000 q^{81} +4.94457 q^{82} +5.05543 q^{83} +11.8444 q^{85} -11.5131 q^{86} -10.3975 q^{87} -1.00000 q^{88} -0.773540 q^{89} -1.61323 q^{90} +3.55780 q^{92} -6.34206 q^{93} +6.78426 q^{94} -1.88914 q^{95} -1.00000 q^{96} +7.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.61323	−0.721458	−0.360729	−	0.932671i	$-0.617472\pi$
$5$	−1.61323	−0.721458	−0.360729	+	0.932671i	$0.617472\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.61323	−0.510148
$11$	−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
$15$	1.61323	0.416534
$16$	1.00000	0.250000
$17$	−7.34206	−1.78071	−0.890356	−	0.455266i	$-0.849544\pi$
$17$	−7.34206	−1.78071	−0.890356	+	0.455266i	$0.849544\pi$
$18$	1.00000	0.235702
$19$	1.17103	0.268653	0.134326	−	0.990937i	$-0.457113\pi$
$19$	1.17103	0.268653	0.134326	+	0.990937i	$0.457113\pi$
$20$	−1.61323	−0.360729
$21$	0	0
$22$	−1.00000	−0.213201
$23$	3.55780	0.741853	0.370926	−	0.928662i	$-0.379040\pi$
$23$	3.55780	0.741853	0.370926	+	0.928662i	$0.379040\pi$
$24$	−1.00000	−0.204124
$25$	−2.39749	−0.479498
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	10.3975	1.93077	0.965383	−	0.260838i	$-0.0839988\pi$
$29$	10.3975	1.93077	0.965383	+	0.260838i	$0.0839988\pi$
$30$	1.61323	0.294534
$31$	6.34206	1.13907	0.569534	−	0.821968i	$-0.307124\pi$
$31$	6.34206	1.13907	0.569534	+	0.821968i	$0.307124\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	−7.34206	−1.25915
$35$	0	0
$36$	1.00000	0.166667
$37$	2.82897	0.465080	0.232540	−	0.972587i	$-0.425296\pi$
$37$	2.82897	0.465080	0.232540	+	0.972587i	$0.425296\pi$
$38$	1.17103	0.189966
$39$	0	0
$40$	−1.61323	−0.255074
$41$	4.94457	0.772212	0.386106	−	0.922454i	$-0.373820\pi$
$41$	4.94457	0.772212	0.386106	+	0.922454i	$0.373820\pi$
$42$	0	0
$43$	−11.5131	−1.75573	−0.877865	−	0.478908i	$-0.841033\pi$
$43$	−11.5131	−1.75573	−0.877865	+	0.478908i	$0.841033\pi$
$44$	−1.00000	−0.150756
$45$	−1.61323	−0.240486
$46$	3.55780	0.524569
$47$	6.78426	0.989586	0.494793	−	0.869011i	$-0.335244\pi$
$47$	6.78426	0.989586	0.494793	+	0.869011i	$0.335244\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−2.39749	−0.339056
$51$	7.34206	1.02809
$52$	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
$55$	1.61323	0.217528
$56$	0	0
$57$	−1.17103	−0.155107
$58$	10.3975	1.36526
$59$	4.39749	0.572504	0.286252	−	0.958154i	$-0.407590\pi$
$59$	4.39749	0.572504	0.286252	+	0.958154i	$0.407590\pi$
$60$	1.61323	0.208267
$61$	−11.9553	−1.53072	−0.765359	−	0.643604i	$-0.777439\pi$
$61$	−11.9553	−1.53072	−0.765359	+	0.643604i	$0.777439\pi$
$62$	6.34206	0.805442
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	2.94457	0.359736	0.179868	−	0.983691i	$-0.442433\pi$
$67$	2.94457	0.359736	0.179868	+	0.983691i	$0.442433\pi$
$68$	−7.34206	−0.890356
$69$	−3.55780	−0.428309
$70$	0	0
$71$	15.5131	1.84106	0.920532	−	0.390666i	$-0.127755\pi$
$71$	15.5131	1.84106	0.920532	+	0.390666i	$0.127755\pi$
$72$	1.00000	0.117851
$73$	−5.11560	−0.598736	−0.299368	−	0.954138i	$-0.596776\pi$
$73$	−5.11560	−0.598736	−0.299368	+	0.954138i	$0.596776\pi$
$74$	2.82897	0.328861
$75$	2.39749	0.276838
$76$	1.17103	0.134326
$77$	0	0
$78$	0	0
$79$	5.61323	0.631538	0.315769	−	0.948836i	$-0.397738\pi$
$79$	5.61323	0.631538	0.315769	+	0.948836i	$0.397738\pi$
$80$	−1.61323	−0.180365
$81$	1.00000	0.111111
$82$	4.94457	0.546036
$83$	5.05543	0.554906	0.277453	−	0.960739i	$-0.410510\pi$
$83$	5.05543	0.554906	0.277453	+	0.960739i	$0.410510\pi$
$84$	0	0
$85$	11.8444	1.28471
$86$	−11.5131	−1.24149
$87$	−10.3975	−1.11473
$88$	−1.00000	−0.106600
$89$	−0.773540	−0.0819951	−0.0409976	−	0.999159i	$-0.513054\pi$
$89$	−0.773540	−0.0819951	−0.0409976	+	0.999159i	$0.513054\pi$
$90$	−1.61323	−0.170049
$91$	0	0
$92$	3.55780	0.370926
$93$	−6.34206	−0.657641
$94$	6.78426	0.699743
$95$	−1.88914	−0.193822
$96$	−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
$100$	−2.39749	−0.239749
$101$	13.6239	1.35563	0.677817	−	0.735231i	$-0.262926\pi$
$101$	13.6239	1.35563	0.677817	+	0.735231i	$0.262926\pi$
$102$	7.34206	0.726972
$103$	−12.3421	−1.21610	−0.608050	−	0.793899i	$-0.708048\pi$
$103$	−12.3421	−1.21610	−0.608050	+	0.793899i	$0.708048\pi$
$104$	0	0
$105$	0	0
$106$	8.00000	0.777029
$107$	−15.7395	−1.52160	−0.760800	−	0.648987i	$-0.775193\pi$
$107$	−15.7395	−1.52160	−0.760800	+	0.648987i	$0.775193\pi$
$108$	−1.00000	−0.0962250
$109$	3.95529	0.378848	0.189424	−	0.981895i	$-0.439338\pi$
$109$	3.95529	0.378848	0.189424	+	0.981895i	$0.439338\pi$
$110$	1.61323	0.153815
$111$	−2.82897	−0.268514
$112$	0	0
$113$	−4.34206	−0.408467	−0.204233	−	0.978922i	$-0.565470\pi$
$113$	−4.34206	−0.408467	−0.204233	+	0.978922i	$0.565470\pi$
$114$	−1.17103	−0.109677
$115$	−5.73955	−0.535216
$116$	10.3975	0.965383
$117$	0	0
$118$	4.39749	0.404822
$119$	0	0
$120$	1.61323	0.147267
$121$	1.00000	0.0909091
$122$	−11.9553	−1.08238
$123$	−4.94457	−0.445837
$124$	6.34206	0.569534
$125$	11.9339	1.06740
$126$	0	0
$127$	−0.442200	−0.0392389	−0.0196195	−	0.999808i	$-0.506245\pi$
$127$	−0.442200	−0.0392389	−0.0196195	+	0.999808i	$0.506245\pi$
$128$	1.00000	0.0883883
$129$	11.5131	1.01367
$130$	0	0
$131$	−2.34206	−0.204627	−0.102313	−	0.994752i	$-0.532624\pi$
$131$	−2.34206	−0.204627	−0.102313	+	0.994752i	$0.532624\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	2.94457	0.254372
$135$	1.61323	0.138845
$136$	−7.34206	−0.629576
$137$	3.22646	0.275655	0.137828	−	0.990456i	$-0.455988\pi$
$137$	3.22646	0.275655	0.137828	+	0.990456i	$0.455988\pi$
$138$	−3.55780	−0.302860
$139$	18.3975	1.56045	0.780227	−	0.625496i	$-0.215103\pi$
$139$	18.3975	1.56045	0.780227	+	0.625496i	$0.215103\pi$
$140$	0	0
$141$	−6.78426	−0.571338
$142$	15.5131	1.30183
$143$	0	0
$144$	1.00000	0.0833333
$145$	−16.7735	−1.39297
$146$	−5.11560	−0.423370
$147$	0	0
$148$	2.82897	0.232540
$149$	19.6239	1.60766	0.803828	−	0.594861i	$-0.202793\pi$
$149$	19.6239	1.60766	0.803828	+	0.594861i	$0.202793\pi$
$150$	2.39749	0.195754
$151$	−0.331340	−0.0269641	−0.0134820	−	0.999909i	$-0.504292\pi$
$151$	−0.331340	−0.0269641	−0.0134820	+	0.999909i	$0.504292\pi$
$152$	1.17103	0.0949831
$153$	−7.34206	−0.593570
$154$	0	0
$155$	−10.2312	−0.821790
$156$	0	0
$157$	19.9660	1.59346	0.796730	−	0.604335i	$-0.206561\pi$
$157$	19.9660	1.59346	0.796730	+	0.604335i	$0.206561\pi$
$158$	5.61323	0.446565
$159$	−8.00000	−0.634441
$160$	−1.61323	−0.127537
$161$	0	0
$162$	1.00000	0.0785674
$163$	21.2866	1.66730	0.833649	−	0.552295i	$-0.186248\pi$
$163$	21.2866	1.66730	0.833649	+	0.552295i	$0.186248\pi$
$164$	4.94457	0.386106
$165$	−1.61323	−0.125590
$166$	5.05543	0.392377
$167$	22.0214	1.70407	0.852035	−	0.523485i	$-0.175368\pi$
$167$	22.0214	1.70407	0.852035	+	0.523485i	$0.175368\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	11.8444	0.908426
$171$	1.17103	0.0895509
$172$	−11.5131	−0.877865
$173$	1.65794	0.126051	0.0630254	−	0.998012i	$-0.479925\pi$
$173$	1.65794	0.126051	0.0630254	+	0.998012i	$0.479925\pi$
$174$	−10.3975	−0.788232
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−4.39749	−0.330536
$178$	−0.773540	−0.0579793
$179$	15.0602	1.12565	0.562825	−	0.826576i	$-0.309715\pi$
$179$	15.0602	1.12565	0.562825	+	0.826576i	$0.309715\pi$
$180$	−1.61323	−0.120243
$181$	15.1156	1.12353	0.561767	−	0.827296i	$-0.310122\pi$
$181$	15.1156	1.12353	0.561767	+	0.827296i	$0.310122\pi$
$182$	0	0
$183$	11.9553	0.883760
$184$	3.55780	0.262284
$185$	−4.56378	−0.335536
$186$	−6.34206	−0.465022
$187$	7.34206	0.536905
$188$	6.78426	0.494793
$189$	0	0
$190$	−1.88914	−0.137053
$191$	−4.77354	−0.345401	−0.172701	−	0.984974i	$-0.555249\pi$
$191$	−4.77354	−0.345401	−0.172701	+	0.984974i	$0.555249\pi$
$192$	−1.00000	−0.0721688
$193$	3.11560	0.224266	0.112133	−	0.993693i	$-0.464232\pi$
$193$	3.11560	0.224266	0.112133	+	0.993693i	$0.464232\pi$
$194$	7.00000	0.502571
$195$	0	0
$196$	0	0
$197$	−4.82897	−0.344050	−0.172025	−	0.985093i	$-0.555031\pi$
$197$	−4.82897	−0.344050	−0.172025	+	0.985093i	$0.555031\pi$
$198$	−1.00000	−0.0710669
$199$	5.88914	0.417470	0.208735	−	0.977972i	$-0.433065\pi$
$199$	5.88914	0.417470	0.208735	+	0.977972i	$0.433065\pi$
$200$	−2.39749	−0.169528
$201$	−2.94457	−0.207694
$202$	13.6239	0.958578
$203$	0	0
$204$	7.34206	0.514047
$205$	−7.97673	−0.557119
$206$	−12.3421	−0.859912
$207$	3.55780	0.247284
$208$	0	0
$209$	−1.17103	−0.0810018
$210$	0	0
$211$	−13.6794	−0.941727	−0.470864	−	0.882206i	$-0.656058\pi$
$211$	−13.6794	−0.941727	−0.470864	+	0.882206i	$0.656058\pi$
$212$	8.00000	0.549442
$213$	−15.5131	−1.06294
$214$	−15.7395	−1.07593
$215$	18.5733	1.26669
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	3.95529	0.267886
$219$	5.11560	0.345680
$220$	1.61323	0.108764
$221$	0	0
$222$	−2.82897	−0.189868
$223$	−5.11560	−0.342566	−0.171283	−	0.985222i	$-0.554791\pi$
$223$	−5.11560	−0.342566	−0.171283	+	0.985222i	$0.554791\pi$
$224$	0	0
$225$	−2.39749	−0.159833
$226$	−4.34206	−0.288830
$227$	−5.28663	−0.350886	−0.175443	−	0.984490i	$-0.556136\pi$
$227$	−5.28663	−0.350886	−0.175443	+	0.984490i	$0.556136\pi$
$228$	−1.17103	−0.0775533
$229$	−4.88440	−0.322770	−0.161385	−	0.986892i	$-0.551596\pi$
$229$	−4.88440	−0.322770	−0.161385	+	0.986892i	$0.551596\pi$
$230$	−5.73955	−0.378455
$231$	0	0
$232$	10.3975	0.682629
$233$	9.79498	0.641690	0.320845	−	0.947132i	$-0.396033\pi$
$233$	9.79498	0.641690	0.320845	+	0.947132i	$0.396033\pi$
$234$	0	0
$235$	−10.9446	−0.713945
$236$	4.39749	0.286252
$237$	−5.61323	−0.364618
$238$	0	0
$239$	−23.9106	−1.54665	−0.773323	−	0.634012i	$-0.781407\pi$
$239$	−23.9106	−1.54665	−0.773323	+	0.634012i	$0.781407\pi$
$240$	1.61323	0.104134
$241$	1.88914	0.121690	0.0608451	−	0.998147i	$-0.480620\pi$
$241$	1.88914	0.121690	0.0608451	+	0.998147i	$0.480620\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	−11.9553	−0.765359
$245$	0	0
$246$	−4.94457	−0.315254
$247$	0	0
$248$	6.34206	0.402721
$249$	−5.05543	−0.320375
$250$	11.9339	0.754763
$251$	0.939830	0.0593215	0.0296608	−	0.999560i	$-0.490557\pi$
$251$	0.939830	0.0593215	0.0296608	+	0.999560i	$0.490557\pi$
$252$	0	0
$253$	−3.55780	−0.223677
$254$	−0.442200	−0.0277461
$255$	−11.8444	−0.741727
$256$	1.00000	0.0625000
$257$	−11.2265	−0.700287	−0.350144	−	0.936696i	$-0.613867\pi$
$257$	−11.2265	−0.700287	−0.350144	+	0.936696i	$0.613867\pi$
$258$	11.5131	0.716774
$259$	0	0
$260$	0	0
$261$	10.3975	0.643588
$262$	−2.34206	−0.144693
$263$	−1.22646	−0.0756267	−0.0378134	−	0.999285i	$-0.512039\pi$
$263$	−1.22646	−0.0756267	−0.0378134	+	0.999285i	$0.512039\pi$
$264$	1.00000	0.0615457
$265$	−12.9058	−0.792799
$266$	0	0
$267$	0.773540	0.0473399
$268$	2.94457	0.179868
$269$	9.50237	0.579370	0.289685	−	0.957122i	$-0.406449\pi$
$269$	9.50237	0.579370	0.289685	+	0.957122i	$0.406449\pi$
$270$	1.61323	0.0981780
$271$	3.22646	0.195993	0.0979967	−	0.995187i	$-0.468757\pi$
$271$	3.22646	0.195993	0.0979967	+	0.995187i	$0.468757\pi$
$272$	−7.34206	−0.445178
$273$	0	0
$274$	3.22646	0.194918
$275$	2.39749	0.144574
$276$	−3.55780	−0.214154
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	18.3975	1.10341
$279$	6.34206	0.379689
$280$	0	0
$281$	−11.2312	−0.669997	−0.334999	−	0.942219i	$-0.608736\pi$
$281$	−11.2312	−0.669997	−0.334999	+	0.942219i	$0.608736\pi$
$282$	−6.78426	−0.403997
$283$	−25.4577	−1.51330	−0.756650	−	0.653820i	$-0.773165\pi$
$283$	−25.4577	−1.51330	−0.756650	+	0.653820i	$0.773165\pi$
$284$	15.5131	0.920532
$285$	1.88914	0.111903
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	36.9058	2.17093
$290$	−16.7735	−0.984976
$291$	−7.00000	−0.410347
$292$	−5.11560	−0.299368
$293$	17.9446	1.04833	0.524166	−	0.851616i	$-0.324377\pi$
$293$	17.9446	1.04833	0.524166	+	0.851616i	$0.324377\pi$
$294$	0	0
$295$	−7.09416	−0.413038
$296$	2.82897	0.164431
$297$	1.00000	0.0580259
$298$	19.6239	1.13678
$299$	0	0
$300$	2.39749	0.138419
$301$	0	0
$302$	−0.331340	−0.0190665
$303$	−13.6239	−0.782675
$304$	1.17103	0.0671632
$305$	19.2866	1.10435
$306$	−7.34206	−0.419718
$307$	−33.0262	−1.88490	−0.942452	−	0.334342i	$-0.891486\pi$
$307$	−33.0262	−1.88490	−0.942452	+	0.334342i	$0.891486\pi$
$308$	0	0
$309$	12.3421	0.702115
$310$	−10.2312	−0.581093
$311$	−0.331340	−0.0187886	−0.00939429	−	0.999956i	$-0.502990\pi$
$311$	−0.331340	−0.0187886	−0.00939429	+	0.999956i	$0.502990\pi$
$312$	0	0
$313$	−14.3975	−0.813794	−0.406897	−	0.913474i	$-0.633389\pi$
$313$	−14.3975	−0.813794	−0.406897	+	0.913474i	$0.633389\pi$
$314$	19.9660	1.12675
$315$	0	0
$316$	5.61323	0.315769
$317$	17.5238	0.984235	0.492118	−	0.870529i	$-0.336223\pi$
$317$	17.5238	0.984235	0.492118	+	0.870529i	$0.336223\pi$
$318$	−8.00000	−0.448618
$319$	−10.3975	−0.582148
$320$	−1.61323	−0.0901823
$321$	15.7395	0.878496
$322$	0	0
$323$	−8.59777	−0.478393
$324$	1.00000	0.0555556
$325$	0	0
$326$	21.2866	1.17896
$327$	−3.95529	−0.218728
$328$	4.94457	0.273018
$329$	0	0
$330$	−1.61323	−0.0888054
$331$	−8.94457	−0.491638	−0.245819	−	0.969316i	$-0.579057\pi$
$331$	−8.94457	−0.491638	−0.245819	+	0.969316i	$0.579057\pi$
$332$	5.05543	0.277453
$333$	2.82897	0.155027
$334$	22.0214	1.20496
$335$	−4.75027	−0.259535
$336$	0	0
$337$	−3.33732	−0.181795	−0.0908977	−	0.995860i	$-0.528974\pi$
$337$	−3.33732	−0.181795	−0.0908977	+	0.995860i	$0.528974\pi$
$338$	−13.0000	−0.707107
$339$	4.34206	0.235828
$340$	11.8444	0.642354
$341$	−6.34206	−0.343442
$342$	1.17103	0.0633220
$343$	0	0
$344$	−11.5131	−0.620744
$345$	5.73955	0.309007
$346$	1.65794	0.0891314
$347$	−30.4237	−1.63323	−0.816614	−	0.577184i	$-0.804151\pi$
$347$	−30.4237	−1.63323	−0.816614	+	0.577184i	$0.804151\pi$
$348$	−10.3975	−0.557364
$349$	17.5238	0.938028	0.469014	−	0.883191i	$-0.344609\pi$
$349$	17.5238	0.938028	0.469014	+	0.883191i	$0.344609\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	5.11560	0.272276	0.136138	−	0.990690i	$-0.456531\pi$
$353$	5.11560	0.272276	0.136138	+	0.990690i	$0.456531\pi$
$354$	−4.39749	−0.233724
$355$	−25.0262	−1.32825
$356$	−0.773540	−0.0409976
$357$	0	0
$358$	15.0602	0.795955
$359$	−30.6841	−1.61945	−0.809723	−	0.586812i	$-0.800383\pi$
$359$	−30.6841	−1.61945	−0.809723	+	0.586812i	$0.800383\pi$
$360$	−1.61323	−0.0850247
$361$	−17.6287	−0.927826
$362$	15.1156	0.794458
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	8.25264	0.431963
$366$	11.9553	0.624913
$367$	1.45766	0.0760892	0.0380446	−	0.999276i	$-0.487887\pi$
$367$	1.45766	0.0760892	0.0380446	+	0.999276i	$0.487887\pi$
$368$	3.55780	0.185463
$369$	4.94457	0.257404
$370$	−4.56378	−0.237260
$371$	0	0
$372$	−6.34206	−0.328820
$373$	−34.1865	−1.77011	−0.885055	−	0.465487i	$-0.845879\pi$
$373$	−34.1865	−1.77011	−0.885055	+	0.465487i	$0.845879\pi$
$374$	7.34206	0.379649
$375$	−11.9339	−0.616261
$376$	6.78426	0.349871
$377$	0	0
$378$	0	0
$379$	−9.28663	−0.477022	−0.238511	−	0.971140i	$-0.576659\pi$
$379$	−9.28663	−0.477022	−0.238511	+	0.971140i	$0.576659\pi$
$380$	−1.88914	−0.0969108
$381$	0.442200	0.0226546
$382$	−4.77354	−0.244236
$383$	−3.62395	−0.185175	−0.0925876	−	0.995705i	$-0.529514\pi$
$383$	−3.62395	−0.185175	−0.0925876	+	0.995705i	$0.529514\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	3.11560	0.158580
$387$	−11.5131	−0.585243
$388$	7.00000	0.355371
$389$	7.16031	0.363042	0.181521	−	0.983387i	$-0.441898\pi$
$389$	7.16031	0.363042	0.181521	+	0.983387i	$0.441898\pi$
$390$	0	0
$391$	−26.1216	−1.32103
$392$	0	0
$393$	2.34206	0.118141
$394$	−4.82897	−0.243280
$395$	−9.05543	−0.455628
$396$	−1.00000	−0.0502519
$397$	−30.3975	−1.52561	−0.762803	−	0.646631i	$-0.776177\pi$
$397$	−30.3975	−1.52561	−0.762803	+	0.646631i	$0.776177\pi$
$398$	5.88914	0.295196
$399$	0	0
$400$	−2.39749	−0.119874
$401$	−12.0214	−0.600322	−0.300161	−	0.953889i	$-0.597040\pi$
$401$	−12.0214	−0.600322	−0.300161	+	0.953889i	$0.597040\pi$
$402$	−2.94457	−0.146862
$403$	0	0
$404$	13.6239	0.677817
$405$	−1.61323	−0.0801620
$406$	0	0
$407$	−2.82897	−0.140227
$408$	7.34206	0.363486
$409$	17.5685	0.868707	0.434354	−	0.900742i	$-0.356977\pi$
$409$	17.5685	0.868707	0.434354	+	0.900742i	$0.356977\pi$
$410$	−7.97673	−0.393943
$411$	−3.22646	−0.159150
$412$	−12.3421	−0.608050
$413$	0	0
$414$	3.55780	0.174856
$415$	−8.15557	−0.400341
$416$	0	0
$417$	−18.3975	−0.900929
$418$	−1.17103	−0.0572769
$419$	−27.1710	−1.32739	−0.663696	−	0.748003i	$-0.731013\pi$
$419$	−27.1710	−1.32739	−0.663696	+	0.748003i	$0.731013\pi$
$420$	0	0
$421$	23.1710	1.12929	0.564643	−	0.825335i	$-0.309014\pi$
$421$	23.1710	1.12929	0.564643	+	0.825335i	$0.309014\pi$
$422$	−13.6794	−0.665902
$423$	6.78426	0.329862
$424$	8.00000	0.388514
$425$	17.6025	0.853847
$426$	−15.5131	−0.751612
$427$	0	0
$428$	−15.7395	−0.760800
$429$	0	0
$430$	18.5733	0.895682
$431$	−16.7735	−0.807953	−0.403977	−	0.914769i	$-0.632372\pi$
$431$	−16.7735	−0.807953	−0.403977	+	0.914769i	$0.632372\pi$
$432$	−1.00000	−0.0481125
$433$	17.3421	0.833406	0.416703	−	0.909043i	$-0.363185\pi$
$433$	17.3421	0.833406	0.416703	+	0.909043i	$0.363185\pi$
$434$	0	0
$435$	16.7735	0.804230
$436$	3.95529	0.189424
$437$	4.16629	0.199301
$438$	5.11560	0.244433
$439$	33.1478	1.58206	0.791028	−	0.611780i	$-0.209546\pi$
$439$	33.1478	1.58206	0.791028	+	0.611780i	$0.209546\pi$
$440$	1.61323	0.0769077
$441$	0	0
$442$	0	0
$443$	−12.0554	−0.572771	−0.286385	−	0.958115i	$-0.592454\pi$
$443$	−12.0554	−0.572771	−0.286385	+	0.958115i	$0.592454\pi$
$444$	−2.82897	−0.134257
$445$	1.24790	0.0591561
$446$	−5.11560	−0.242231
$447$	−19.6239	−0.928181
$448$	0	0
$449$	−36.5947	−1.72701	−0.863505	−	0.504340i	$-0.831736\pi$
$449$	−36.5947	−1.72701	−0.863505	+	0.504340i	$0.831736\pi$
$450$	−2.39749	−0.113019
$451$	−4.94457	−0.232831
$452$	−4.34206	−0.204233
$453$	0.331340	0.0155677
$454$	−5.28663	−0.248114
$455$	0	0
$456$	−1.17103	−0.0548385
$457$	37.8212	1.76920	0.884600	−	0.466351i	$-0.154432\pi$
$457$	37.8212	1.76920	0.884600	+	0.466351i	$0.154432\pi$
$458$	−4.88440	−0.228233
$459$	7.34206	0.342698
$460$	−5.73955	−0.267608
$461$	34.9707	1.62875	0.814375	−	0.580339i	$-0.197080\pi$
$461$	34.9707	1.62875	0.814375	+	0.580339i	$0.197080\pi$
$462$	0	0
$463$	7.56852	0.351739	0.175869	−	0.984413i	$-0.443726\pi$
$463$	7.56852	0.351739	0.175869	+	0.984413i	$0.443726\pi$
$464$	10.3975	0.482691
$465$	10.2312	0.474461
$466$	9.79498	0.453744
$467$	−14.6287	−0.676935	−0.338468	−	0.940978i	$-0.609909\pi$
$467$	−14.6287	−0.676935	−0.338468	+	0.940978i	$0.609909\pi$
$468$	0	0
$469$	0	0
$470$	−10.9446	−0.504835
$471$	−19.9660	−0.919985
$472$	4.39749	0.202411
$473$	11.5131	0.529372
$474$	−5.61323	−0.257824
$475$	−2.80753	−0.128818
$476$	0	0
$477$	8.00000	0.366295
$478$	−23.9106	−1.09364
$479$	−1.22646	−0.0560384	−0.0280192	−	0.999607i	$-0.508920\pi$
$479$	−1.22646	−0.0560384	−0.0280192	+	0.999607i	$0.508920\pi$
$480$	1.61323	0.0736335
$481$	0	0
$482$	1.88914	0.0860480
$483$	0	0
$484$	1.00000	0.0454545
$485$	−11.2926	−0.512771
$486$	−1.00000	−0.0453609
$487$	30.7056	1.39140	0.695701	−	0.718332i	$-0.255094\pi$
$487$	30.7056	1.39140	0.695701	+	0.718332i	$0.255094\pi$
$488$	−11.9553	−0.541191
$489$	−21.2866	−0.962615
$490$	0	0
$491$	30.9446	1.39651	0.698254	−	0.715850i	$-0.253960\pi$
$491$	30.9446	1.39651	0.698254	+	0.715850i	$0.253960\pi$
$492$	−4.94457	−0.222918
$493$	−76.3390	−3.43814
$494$	0	0
$495$	1.61323	0.0725093
$496$	6.34206	0.284767
$497$	0	0
$498$	−5.05543	−0.226539
$499$	−10.3421	−0.462974	−0.231487	−	0.972838i	$-0.574359\pi$
$499$	−10.3421	−0.462974	−0.231487	+	0.972838i	$0.574359\pi$
$500$	11.9339	0.533698
$501$	−22.0214	−0.983845
$502$	0.939830	0.0419467
$503$	4.66268	0.207899	0.103949	−	0.994583i	$-0.466852\pi$
$503$	4.66268	0.207899	0.103949	+	0.994583i	$0.466852\pi$
$504$	0	0
$505$	−21.9786	−0.978033
$506$	−3.55780	−0.158163
$507$	13.0000	0.577350
$508$	−0.442200	−0.0196195
$509$	25.5900	1.13425	0.567127	−	0.823630i	$-0.308055\pi$
$509$	25.5900	1.13425	0.567127	+	0.823630i	$0.308055\pi$
$510$	−11.8444	−0.524480
$511$	0	0
$512$	1.00000	0.0441942
$513$	−1.17103	−0.0517022
$514$	−11.2265	−0.495178
$515$	19.9106	0.877365
$516$	11.5131	0.506835
$517$	−6.78426	−0.298371
$518$	0	0
$519$	−1.65794	−0.0727755
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	10.3975	0.455086
$523$	30.3635	1.32770	0.663852	−	0.747864i	$-0.268921\pi$
$523$	30.3635	1.32770	0.663852	+	0.747864i	$0.268921\pi$
$524$	−2.34206	−0.102313
$525$	0	0
$526$	−1.22646	−0.0534762
$527$	−46.5638	−2.02835
$528$	1.00000	0.0435194
$529$	−10.3421	−0.449655
$530$	−12.9058	−0.560594
$531$	4.39749	0.190835
$532$	0	0
$533$	0	0
$534$	0.773540	0.0334744
$535$	25.3915	1.09777
$536$	2.94457	0.127186
$537$	−15.0602	−0.649894
$538$	9.50237	0.409676
$539$	0	0
$540$	1.61323	0.0694224
$541$	1.61323	0.0693582	0.0346791	−	0.999398i	$-0.488959\pi$
$541$	1.61323	0.0693582	0.0346791	+	0.999398i	$0.488959\pi$
$542$	3.22646	0.138588
$543$	−15.1156	−0.648672
$544$	−7.34206	−0.314788
$545$	−6.38079	−0.273323
$546$	0	0
$547$	15.7348	0.672772	0.336386	−	0.941724i	$-0.390795\pi$
$547$	15.7348	0.672772	0.336386	+	0.941724i	$0.390795\pi$
$548$	3.22646	0.137828
$549$	−11.9553	−0.510239
$550$	2.39749	0.102229
$551$	12.1758	0.518705
$552$	−3.55780	−0.151430
$553$	0	0
$554$	−16.0000	−0.679775
$555$	4.56378	0.193722
$556$	18.3975	0.780227
$557$	11.2819	0.478029	0.239015	−	0.971016i	$-0.423176\pi$
$557$	11.2819	0.478029	0.239015	+	0.971016i	$0.423176\pi$
$558$	6.34206	0.268481
$559$	0	0
$560$	0	0
$561$	−7.34206	−0.309982
$562$	−11.2312	−0.473760
$563$	20.9058	0.881076	0.440538	−	0.897734i	$-0.354788\pi$
$563$	20.9058	0.881076	0.440538	+	0.897734i	$0.354788\pi$
$564$	−6.78426	−0.285669
$565$	7.00474	0.294692
$566$	−25.4577	−1.07007
$567$	0	0
$568$	15.5131	0.650915
$569$	−7.83371	−0.328406	−0.164203	−	0.986427i	$-0.552505\pi$
$569$	−7.83371	−0.328406	−0.164203	+	0.986427i	$0.552505\pi$
$570$	1.88914	0.0791274
$571$	2.39749	0.100332	0.0501659	−	0.998741i	$-0.484025\pi$
$571$	2.39749	0.100332	0.0501659	+	0.998741i	$0.484025\pi$
$572$	0	0
$573$	4.77354	0.199418
$574$	0	0
$575$	−8.52979	−0.355717
$576$	1.00000	0.0416667
$577$	25.6287	1.06694	0.533468	−	0.845820i	$-0.320888\pi$
$577$	25.6287	1.06694	0.533468	+	0.845820i	$0.320888\pi$
$578$	36.9058	1.53508
$579$	−3.11560	−0.129480
$580$	−16.7735	−0.696483
$581$	0	0
$582$	−7.00000	−0.290159
$583$	−8.00000	−0.331326
$584$	−5.11560	−0.211685
$585$	0	0
$586$	17.9446	0.741283
$587$	5.79972	0.239380	0.119690	−	0.992811i	$-0.461810\pi$
$587$	5.79972	0.239380	0.119690	+	0.992811i	$0.461810\pi$
$588$	0	0
$589$	7.42674	0.306014
$590$	−7.09416	−0.292062
$591$	4.82897	0.198637
$592$	2.82897	0.116270
$593$	31.1925	1.28092	0.640461	−	0.767991i	$-0.278743\pi$
$593$	31.1925	1.28092	0.640461	+	0.767991i	$0.278743\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	19.6239	0.803828
$597$	−5.88914	−0.241026
$598$	0	0
$599$	−31.2926	−1.27858	−0.639291	−	0.768965i	$-0.720772\pi$
$599$	−31.2926	−1.27858	−0.639291	+	0.768965i	$0.720772\pi$
$600$	2.39749	0.0978771
$601$	−22.1203	−0.902307	−0.451154	−	0.892446i	$-0.648987\pi$
$601$	−22.1203	−0.902307	−0.451154	+	0.892446i	$0.648987\pi$
$602$	0	0
$603$	2.94457	0.119912
$604$	−0.331340	−0.0134820
$605$	−1.61323	−0.0655871
$606$	−13.6239	−0.553435
$607$	−20.6180	−0.836858	−0.418429	−	0.908250i	$-0.637419\pi$
$607$	−20.6180	−0.836858	−0.418429	+	0.908250i	$0.637419\pi$
$608$	1.17103	0.0474915
$609$	0	0
$610$	19.2866	0.780893
$611$	0	0
$612$	−7.34206	−0.296785
$613$	−15.7550	−0.636339	−0.318169	−	0.948034i	$-0.603068\pi$
$613$	−15.7550	−0.636339	−0.318169	+	0.948034i	$0.603068\pi$
$614$	−33.0262	−1.33283
$615$	7.97673	0.321653
$616$	0	0
$617$	34.2526	1.37896	0.689480	−	0.724305i	$-0.257839\pi$
$617$	34.2526	1.37896	0.689480	+	0.724305i	$0.257839\pi$
$618$	12.3421	0.496470
$619$	−21.9707	−0.883079	−0.441539	−	0.897242i	$-0.645567\pi$
$619$	−21.9707	−0.883079	−0.441539	+	0.897242i	$0.645567\pi$
$620$	−10.2312	−0.410895
$621$	−3.55780	−0.142770
$622$	−0.331340	−0.0132855
$623$	0	0
$624$	0	0
$625$	−7.26460	−0.290584
$626$	−14.3975	−0.575439
$627$	1.17103	0.0467664
$628$	19.9660	0.796730
$629$	−20.7705	−0.828173
$630$	0	0
$631$	−27.5685	−1.09749	−0.548743	−	0.835991i	$-0.684893\pi$
$631$	−27.5685	−1.09749	−0.548743	+	0.835991i	$0.684893\pi$
$632$	5.61323	0.223282
$633$	13.6794	0.543707
$634$	17.5238	0.695959
$635$	0.713370	0.0283092
$636$	−8.00000	−0.317221
$637$	0	0
$638$	−10.3975	−0.411641
$639$	15.5131	0.613688
$640$	−1.61323	−0.0637685
$641$	5.65794	0.223475	0.111738	−	0.993738i	$-0.464358\pi$
$641$	5.65794	0.223475	0.111738	+	0.993738i	$0.464358\pi$
$642$	15.7395	0.621190
$643$	17.4791	0.689308	0.344654	−	0.938730i	$-0.387996\pi$
$643$	17.4791	0.689308	0.344654	+	0.938730i	$0.387996\pi$
$644$	0	0
$645$	−18.5733	−0.731321
$646$	−8.59777	−0.338275
$647$	−8.83969	−0.347524	−0.173762	−	0.984788i	$-0.555592\pi$
$647$	−8.83969	−0.347524	−0.173762	+	0.984788i	$0.555592\pi$
$648$	1.00000	0.0392837
$649$	−4.39749	−0.172617
$650$	0	0
$651$	0	0
$652$	21.2866	0.833649
$653$	−24.7503	−0.968553	−0.484276	−	0.874915i	$-0.660917\pi$
$653$	−24.7503	−0.968553	−0.484276	+	0.874915i	$0.660917\pi$
$654$	−3.95529	−0.154664
$655$	3.77828	0.147630
$656$	4.94457	0.193053
$657$	−5.11560	−0.199579
$658$	0	0
$659$	5.62869	0.219263	0.109631	−	0.993972i	$-0.465033\pi$
$659$	5.62869	0.219263	0.109631	+	0.993972i	$0.465033\pi$
$660$	−1.61323	−0.0627949
$661$	−13.4022	−0.521286	−0.260643	−	0.965435i	$-0.583935\pi$
$661$	−13.4022	−0.521286	−0.260643	+	0.965435i	$0.583935\pi$
$662$	−8.94457	−0.347641
$663$	0	0
$664$	5.05543	0.196189
$665$	0	0
$666$	2.82897	0.109620
$667$	36.9922	1.43234
$668$	22.0214	0.852035
$669$	5.11560	0.197781
$670$	−4.75027	−0.183519
$671$	11.9553	0.461529
$672$	0	0
$673$	−29.0262	−1.11888	−0.559438	−	0.828872i	$-0.688983\pi$
$673$	−29.0262	−1.11888	−0.559438	+	0.828872i	$0.688983\pi$
$674$	−3.33732	−0.128549
$675$	2.39749	0.0922794
$676$	−13.0000	−0.500000
$677$	−20.3081	−0.780502	−0.390251	−	0.920708i	$-0.627612\pi$
$677$	−20.3081	−0.780502	−0.390251	+	0.920708i	$0.627612\pi$
$678$	4.34206	0.166756
$679$	0	0
$680$	11.8444	0.454213
$681$	5.28663	0.202584
$682$	−6.34206	−0.242850
$683$	−10.7395	−0.410937	−0.205469	−	0.978664i	$-0.565872\pi$
$683$	−10.7395	−0.410937	−0.205469	+	0.978664i	$0.565872\pi$
$684$	1.17103	0.0447754
$685$	−5.20502	−0.198874
$686$	0	0
$687$	4.88440	0.186351
$688$	−11.5131	−0.438932
$689$	0	0
$690$	5.73955	0.218501
$691$	5.97075	0.227138	0.113569	−	0.993530i	$-0.463772\pi$
$691$	5.97075	0.227138	0.113569	+	0.993530i	$0.463772\pi$
$692$	1.65794	0.0630254
$693$	0	0
$694$	−30.4237	−1.15487
$695$	−29.6794	−1.12580
$696$	−10.3975	−0.394116
$697$	−36.3033	−1.37509
$698$	17.5238	0.663286
$699$	−9.79498	−0.370480
$700$	0	0
$701$	−5.28189	−0.199494	−0.0997471	−	0.995013i	$-0.531803\pi$
$701$	−5.28189	−0.199494	−0.0997471	+	0.995013i	$0.531803\pi$
$702$	0	0
$703$	3.31281	0.124945
$704$	−1.00000	−0.0376889
$705$	10.9446	0.412196
$706$	5.11560	0.192528
$707$	0	0
$708$	−4.39749	−0.165268
$709$	39.3128	1.47642	0.738212	−	0.674569i	$-0.235671\pi$
$709$	39.3128	1.47642	0.738212	+	0.674569i	$0.235671\pi$
$710$	−25.0262	−0.939216
$711$	5.61323	0.210513
$712$	−0.773540	−0.0289896
$713$	22.5638	0.845020
$714$	0	0
$715$	0	0
$716$	15.0602	0.562825
$717$	23.9106	0.892957
$718$	−30.6841	−1.14512
$719$	−35.3575	−1.31861	−0.659306	−	0.751874i	$-0.729150\pi$
$719$	−35.3575	−1.31861	−0.659306	+	0.751874i	$0.729150\pi$
$720$	−1.61323	−0.0601215
$721$	0	0
$722$	−17.6287	−0.656072
$723$	−1.88914	−0.0702579
$724$	15.1156	0.561767
$725$	−24.9279	−0.925798
$726$	−1.00000	−0.0371135
$727$	−40.2312	−1.49209	−0.746046	−	0.665894i	$-0.768050\pi$
$727$	−40.2312	−1.49209	−0.746046	+	0.665894i	$0.768050\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	8.25264	0.305444
$731$	84.5298	3.12645
$732$	11.9553	0.441880
$733$	28.7288	1.06112	0.530562	−	0.847646i	$-0.321981\pi$
$733$	28.7288	1.06112	0.530562	+	0.847646i	$0.321981\pi$
$734$	1.45766	0.0538032
$735$	0	0
$736$	3.55780	0.131142
$737$	−2.94457	−0.108465
$738$	4.94457	0.182012
$739$	−16.3421	−0.601152	−0.300576	−	0.953758i	$-0.597179\pi$
$739$	−16.3421	−0.601152	−0.300576	+	0.953758i	$0.597179\pi$
$740$	−4.56378	−0.167768
$741$	0	0
$742$	0	0
$743$	45.3897	1.66519	0.832593	−	0.553885i	$-0.186855\pi$
$743$	45.3897	1.66519	0.832593	+	0.553885i	$0.186855\pi$
$744$	−6.34206	−0.232511
$745$	−31.6579	−1.15986
$746$	−34.1865	−1.25166
$747$	5.05543	0.184969
$748$	7.34206	0.268452
$749$	0	0
$750$	−11.9339	−0.435763
$751$	−27.7997	−1.01443	−0.507213	−	0.861821i	$-0.669324\pi$
$751$	−27.7997	−1.01443	−0.507213	+	0.861821i	$0.669324\pi$
$752$	6.78426	0.247396
$753$	−0.939830	−0.0342493
$754$	0	0
$755$	0.534528	0.0194535
$756$	0	0
$757$	19.0816	0.693533	0.346766	−	0.937952i	$-0.387280\pi$
$757$	19.0816	0.693533	0.346766	+	0.937952i	$0.387280\pi$
$758$	−9.28663	−0.337306
$759$	3.55780	0.129140
$760$	−1.88914	−0.0685263
$761$	47.2186	1.71167	0.855837	−	0.517245i	$-0.173042\pi$
$761$	47.2186	1.71167	0.855837	+	0.517245i	$0.173042\pi$
$762$	0.442200	0.0160192
$763$	0	0
$764$	−4.77354	−0.172701
$765$	11.8444	0.428236
$766$	−3.62395	−0.130939
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−9.56852	−0.345050	−0.172525	−	0.985005i	$-0.555192\pi$
$769$	−9.56852	−0.345050	−0.172525	+	0.985005i	$0.555192\pi$
$770$	0	0
$771$	11.2265	0.404311
$772$	3.11560	0.112133
$773$	15.9553	0.573872	0.286936	−	0.957950i	$-0.407363\pi$
$773$	15.9553	0.573872	0.286936	+	0.957950i	$0.407363\pi$
$774$	−11.5131	−0.413829
$775$	−15.2050	−0.546180
$776$	7.00000	0.251285
$777$	0	0
$778$	7.16031	0.256710
$779$	5.79024	0.207457
$780$	0	0
$781$	−15.5131	−0.555102
$782$	−26.1216	−0.934106
$783$	−10.3975	−0.371576
$784$	0	0
$785$	−32.2098	−1.14962
$786$	2.34206	0.0835385
$787$	49.8551	1.77714	0.888572	−	0.458737i	$-0.151698\pi$
$787$	49.8551	1.77714	0.888572	+	0.458737i	$0.151698\pi$
$788$	−4.82897	−0.172025
$789$	1.22646	0.0436631
$790$	−9.05543	−0.322178
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	0	0
$794$	−30.3975	−1.07877
$795$	12.9058	0.457723
$796$	5.88914	0.208735
$797$	36.4177	1.28998	0.644990	−	0.764191i	$-0.276861\pi$
$797$	36.4177	1.28998	0.644990	+	0.764191i	$0.276861\pi$
$798$	0	0
$799$	−49.8104	−1.76217
$800$	−2.39749	−0.0847641
$801$	−0.773540	−0.0273317
$802$	−12.0214	−0.424492
$803$	5.11560	0.180526
$804$	−2.94457	−0.103847
$805$	0	0
$806$	0	0
$807$	−9.50237	−0.334499
$808$	13.6239	0.479289
$809$	48.4237	1.70249	0.851243	−	0.524772i	$-0.175849\pi$
$809$	48.4237	1.70249	0.851243	+	0.524772i	$0.175849\pi$
$810$	−1.61323	−0.0566831
$811$	−41.0262	−1.44062	−0.720312	−	0.693650i	$-0.756001\pi$
$811$	−41.0262	−1.44062	−0.720312	+	0.693650i	$0.756001\pi$
$812$	0	0
$813$	−3.22646	−0.113157
$814$	−2.82897	−0.0991554
$815$	−34.3402	−1.20289
$816$	7.34206	0.257023
$817$	−13.4822	−0.471681
$818$	17.5685	0.614269
$819$	0	0
$820$	−7.97673	−0.278559
$821$	39.9320	1.39364	0.696819	−	0.717247i	$-0.254598\pi$
$821$	39.9320	1.39364	0.696819	+	0.717247i	$0.254598\pi$
$822$	−3.22646	−0.112536
$823$	−15.7997	−0.550744	−0.275372	−	0.961338i	$-0.588801\pi$
$823$	−15.7997	−0.550744	−0.275372	+	0.961338i	$0.588801\pi$
$824$	−12.3421	−0.429956
$825$	−2.39749	−0.0834699
$826$	0	0
$827$	−26.8766	−0.934591	−0.467295	−	0.884101i	$-0.654771\pi$
$827$	−26.8766	−0.934591	−0.467295	+	0.884101i	$0.654771\pi$
$828$	3.55780	0.123642
$829$	−8.30807	−0.288551	−0.144276	−	0.989538i	$-0.546085\pi$
$829$	−8.30807	−0.288551	−0.144276	+	0.989538i	$0.546085\pi$
$830$	−8.15557	−0.283084
$831$	16.0000	0.555034
$832$	0	0
$833$	0	0
$834$	−18.3975	−0.637053
$835$	−35.5256	−1.22942
$836$	−1.17103	−0.0405009
$837$	−6.34206	−0.219214
$838$	−27.1710	−0.938608
$839$	−4.28787	−0.148034	−0.0740168	−	0.997257i	$-0.523582\pi$
$839$	−4.28787	−0.148034	−0.0740168	+	0.997257i	$0.523582\pi$
$840$	0	0
$841$	79.1078	2.72785
$842$	23.1710	0.798526
$843$	11.2312	0.386823
$844$	−13.6794	−0.470864
$845$	20.9720	0.721458
$846$	6.78426	0.233248
$847$	0	0
$848$	8.00000	0.274721
$849$	25.4577	0.873705
$850$	17.6025	0.603761
$851$	10.0649	0.345021
$852$	−15.5131	−0.531470
$853$	−46.2973	−1.58519	−0.792596	−	0.609748i	$-0.791271\pi$
$853$	−46.2973	−1.58519	−0.792596	+	0.609748i	$0.791271\pi$
$854$	0	0
$855$	−1.88914	−0.0646072
$856$	−15.7395	−0.537967
$857$	−27.4624	−0.938098	−0.469049	−	0.883172i	$-0.655403\pi$
$857$	−27.4624	−0.938098	−0.469049	+	0.883172i	$0.655403\pi$
$858$	0	0
$859$	−12.1925	−0.416002	−0.208001	−	0.978129i	$-0.566696\pi$
$859$	−12.1925	−0.416002	−0.208001	+	0.978129i	$0.566696\pi$
$860$	18.5733	0.633343
$861$	0	0
$862$	−16.7735	−0.571309
$863$	0.839690	0.0285834	0.0142917	−	0.999898i	$-0.495451\pi$
$863$	0.839690	0.0285834	0.0142917	+	0.999898i	$0.495451\pi$
$864$	−1.00000	−0.0340207
$865$	−2.67464	−0.0909405
$866$	17.3421	0.589307
$867$	−36.9058	−1.25339
$868$	0	0
$869$	−5.61323	−0.190416
$870$	16.7735	0.568676
$871$	0	0
$872$	3.95529	0.133943
$873$	7.00000	0.236914
$874$	4.16629	0.140927
$875$	0	0
$876$	5.11560	0.172840
$877$	−20.5285	−0.693200	−0.346600	−	0.938013i	$-0.612664\pi$
$877$	−20.5285	−0.693200	−0.346600	+	0.938013i	$0.612664\pi$
$878$	33.1478	1.11868
$879$	−17.9446	−0.605255
$880$	1.61323	0.0543820
$881$	1.88914	0.0636468	0.0318234	−	0.999494i	$-0.489869\pi$
$881$	1.88914	0.0636468	0.0318234	+	0.999494i	$0.489869\pi$
$882$	0	0
$883$	26.5345	0.892958	0.446479	−	0.894794i	$-0.352678\pi$
$883$	26.5345	0.892958	0.446479	+	0.894794i	$0.352678\pi$
$884$	0	0
$885$	7.09416	0.238468
$886$	−12.0554	−0.405010
$887$	17.1156	0.574686	0.287343	−	0.957828i	$-0.407228\pi$
$887$	17.1156	0.574686	0.287343	+	0.957828i	$0.407228\pi$
$888$	−2.82897	−0.0949340
$889$	0	0
$890$	1.24790	0.0418296
$891$	−1.00000	−0.0335013
$892$	−5.11560	−0.171283
$893$	7.94457	0.265855
$894$	−19.6239	−0.656323
$895$	−24.2955	−0.812110
$896$	0	0
$897$	0	0
$898$	−36.5947	−1.22118
$899$	65.9415	2.19927
$900$	−2.39749	−0.0799163
$901$	−58.7365	−1.95680
$902$	−4.94457	−0.164636
$903$	0	0
$904$	−4.34206	−0.144415
$905$	−24.3849	−0.810583
$906$	0.331340	0.0110080
$907$	3.28663	0.109131	0.0545654	−	0.998510i	$-0.482623\pi$
$907$	3.28663	0.109131	0.0545654	+	0.998510i	$0.482623\pi$
$908$	−5.28663	−0.175443
$909$	13.6239	0.451878
$910$	0	0
$911$	14.0107	0.464196	0.232098	−	0.972692i	$-0.425441\pi$
$911$	14.0107	0.464196	0.232098	+	0.972692i	$0.425441\pi$
$912$	−1.17103	−0.0387767
$913$	−5.05543	−0.167310
$914$	37.8212	1.25101
$915$	−19.2866	−0.637596
$916$	−4.88440	−0.161385
$917$	0	0
$918$	7.34206	0.242324
$919$	−46.5160	−1.53442	−0.767211	−	0.641395i	$-0.778356\pi$
$919$	−46.5160	−1.53442	−0.767211	+	0.641395i	$0.778356\pi$
$920$	−5.73955	−0.189227
$921$	33.0262	1.08825
$922$	34.9707	1.15170
$923$	0	0
$924$	0	0
$925$	−6.78243	−0.223005
$926$	7.56852	0.248717
$927$	−12.3421	−0.405366
$928$	10.3975	0.341314
$929$	−24.8844	−0.816431	−0.408215	−	0.912886i	$-0.633849\pi$
$929$	−24.8844	−0.816431	−0.408215	+	0.912886i	$0.633849\pi$
$930$	10.2312	0.335494
$931$	0	0
$932$	9.79498	0.320845
$933$	0.331340	0.0108476
$934$	−14.6287	−0.478665
$935$	−11.8444	−0.387354
$936$	0	0
$937$	−46.7950	−1.52873	−0.764363	−	0.644787i	$-0.776946\pi$
$937$	−46.7950	−1.52873	−0.764363	+	0.644787i	$0.776946\pi$
$938$	0	0
$939$	14.3975	0.469844
$940$	−10.9446	−0.356973
$941$	17.4917	0.570212	0.285106	−	0.958496i	$-0.407971\pi$
$941$	17.4917	0.570212	0.285106	+	0.958496i	$0.407971\pi$
$942$	−19.9660	−0.650528
$943$	17.5918	0.572868
$944$	4.39749	0.143126
$945$	0	0
$946$	11.5131	0.374323
$947$	14.6192	0.475060	0.237530	−	0.971380i	$-0.423662\pi$
$947$	14.6192	0.475060	0.237530	+	0.971380i	$0.423662\pi$
$948$	−5.61323	−0.182309
$949$	0	0
$950$	−2.80753	−0.0910884
$951$	−17.5238	−0.568248
$952$	0	0
$953$	−20.0816	−0.650507	−0.325254	−	0.945627i	$-0.605450\pi$
$953$	−20.0816	−0.650507	−0.325254	+	0.945627i	$0.605450\pi$
$954$	8.00000	0.259010
$955$	7.70082	0.249193
$956$	−23.9106	−0.773323
$957$	10.3975	0.336103
$958$	−1.22646	−0.0396251
$959$	0	0
$960$	1.61323	0.0520668
$961$	9.22172	0.297475
$962$	0	0
$963$	−15.7395	−0.507200
$964$	1.88914	0.0608451
$965$	−5.02618	−0.161798
$966$	0	0
$967$	−60.2634	−1.93794	−0.968969	−	0.247180i	$-0.920496\pi$
$967$	−60.2634	−1.93794	−0.968969	+	0.247180i	$0.920496\pi$
$968$	1.00000	0.0321412
$969$	8.59777	0.276200
$970$	−11.2926	−0.362584
$971$	−52.8473	−1.69595	−0.847976	−	0.530035i	$-0.822179\pi$
$971$	−52.8473	−1.69595	−0.847976	+	0.530035i	$0.822179\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	30.7056	0.983870
$975$	0	0
$976$	−11.9553	−0.382679
$977$	24.9058	0.796808	0.398404	−	0.917210i	$-0.369564\pi$
$977$	24.9058	0.796808	0.398404	+	0.917210i	$0.369564\pi$
$978$	−21.2866	−0.680671
$979$	0.773540	0.0247225
$980$	0	0
$981$	3.95529	0.126283
$982$	30.9446	0.987481
$983$	−25.1263	−0.801405	−0.400703	−	0.916208i	$-0.631234\pi$
$983$	−25.1263	−0.801405	−0.400703	+	0.916208i	$0.631234\pi$
$984$	−4.94457	−0.157627
$985$	7.79024	0.248218
$986$	−76.3390	−2.43113
$987$	0	0
$988$	0	0
$989$	−40.9613	−1.30249
$990$	1.61323	0.0512718
$991$	57.9201	1.83989	0.919946	−	0.392046i	$-0.128233\pi$
$991$	57.9201	1.83989	0.919946	+	0.392046i	$0.128233\pi$
$992$	6.34206	0.201361
$993$	8.94457	0.283847
$994$	0	0
$995$	−9.50054	−0.301187
$996$	−5.05543	−0.160187
$997$	17.3682	0.550058	0.275029	−	0.961436i	$-0.411313\pi$
$997$	17.3682	0.550058	0.275029	+	0.961436i	$0.411313\pi$
$998$	−10.3421	−0.327372
$999$	−2.82897	−0.0895047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.bf.1.2		3
3.2	odd	2		9702.2.a.dv.1.2		3
7.2	even	3		462.2.i.g.67.2	✓	6
7.4	even	3		462.2.i.g.331.2	yes	6
7.6	odd	2		3234.2.a.bh.1.2		3
21.2	odd	6		1386.2.k.v.991.2		6
21.11	odd	6		1386.2.k.v.793.2		6
21.20	even	2		9702.2.a.dw.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.g.67.2	✓	6	7.2	even	3
462.2.i.g.331.2	yes	6	7.4	even	3
1386.2.k.v.793.2		6	21.11	odd	6
1386.2.k.v.991.2		6	21.2	odd	6
3234.2.a.bf.1.2		3	1.1	even	1	trivial
3234.2.a.bh.1.2		3	7.6	odd	2
9702.2.a.dv.1.2		3	3.2	odd	2
9702.2.a.dw.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.61323 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.61323 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.61323 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.61323 q^{15} +1.00000 q^{16} -7.34206 q^{17} +1.00000 q^{18} +1.17103 q^{19} -1.61323 q^{20} -1.00000 q^{22} +3.55780 q^{23} -1.00000 q^{24} -2.39749 q^{25} -1.00000 q^{27} +10.3975 q^{29} +1.61323 q^{30} +6.34206 q^{31} +1.00000 q^{32} +1.00000 q^{33} -7.34206 q^{34} +1.00000 q^{36} +2.82897 q^{37} +1.17103 q^{38} -1.61323 q^{40} +4.94457 q^{41} -11.5131 q^{43} -1.00000 q^{44} -1.61323 q^{45} +3.55780 q^{46} +6.78426 q^{47} -1.00000 q^{48} -2.39749 q^{50} +7.34206 q^{51} +8.00000 q^{53} -1.00000 q^{54} +1.61323 q^{55} -1.17103 q^{57} +10.3975 q^{58} +4.39749 q^{59} +1.61323 q^{60} -11.9553 q^{61} +6.34206 q^{62} +1.00000 q^{64} +1.00000 q^{66} +2.94457 q^{67} -7.34206 q^{68} -3.55780 q^{69} +15.5131 q^{71} +1.00000 q^{72} -5.11560 q^{73} +2.82897 q^{74} +2.39749 q^{75} +1.17103 q^{76} +5.61323 q^{79} -1.61323 q^{80} +1.00000 q^{81} +4.94457 q^{82} +5.05543 q^{83} +11.8444 q^{85} -11.5131 q^{86} -10.3975 q^{87} -1.00000 q^{88} -0.773540 q^{89} -1.61323 q^{90} +3.55780 q^{92} -6.34206 q^{93} +6.78426 q^{94} -1.88914 q^{95} -1.00000 q^{96} +7.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 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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