LMFDB - Embedded newform 3234.2.a.bb.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.23607 q^{10} -1.00000 q^{11} -1.00000 q^{12} -1.23607 q^{15} +1.00000 q^{16} -5.23607 q^{17} +1.00000 q^{18} -7.70820 q^{19} +1.23607 q^{20} -1.00000 q^{22} -2.47214 q^{23} -1.00000 q^{24} -3.47214 q^{25} -1.00000 q^{27} +4.47214 q^{29} -1.23607 q^{30} -2.76393 q^{31} +1.00000 q^{32} +1.00000 q^{33} -5.23607 q^{34} +1.00000 q^{36} -10.9443 q^{37} -7.70820 q^{38} +1.23607 q^{40} -5.23607 q^{41} +6.47214 q^{43} -1.00000 q^{44} +1.23607 q^{45} -2.47214 q^{46} +3.70820 q^{47} -1.00000 q^{48} -3.47214 q^{50} +5.23607 q^{51} -6.00000 q^{53} -1.00000 q^{54} -1.23607 q^{55} +7.70820 q^{57} +4.47214 q^{58} -1.52786 q^{59} -1.23607 q^{60} -2.76393 q^{62} +1.00000 q^{64} +1.00000 q^{66} +15.4164 q^{67} -5.23607 q^{68} +2.47214 q^{69} -2.47214 q^{71} +1.00000 q^{72} -15.7082 q^{73} -10.9443 q^{74} +3.47214 q^{75} -7.70820 q^{76} +11.4164 q^{79} +1.23607 q^{80} +1.00000 q^{81} -5.23607 q^{82} +1.23607 q^{83} -6.47214 q^{85} +6.47214 q^{86} -4.47214 q^{87} -1.00000 q^{88} -6.47214 q^{89} +1.23607 q^{90} -2.47214 q^{92} +2.76393 q^{93} +3.70820 q^{94} -9.52786 q^{95} -1.00000 q^{96} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 2 q^{12} + 2 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 2 * q^12 + 2 * q^15 + 2 * q^16 - 6 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 2 * q^22 + 4 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^27 + 2 * q^30 - 10 * q^31 + 2 * q^32 + 2 * q^33 - 6 * q^34 + 2 * q^36 - 4 * q^37 - 2 * q^38 - 2 * q^40 - 6 * q^41 + 4 * q^43 - 2 * q^44 - 2 * q^45 + 4 * q^46 - 6 * q^47 - 2 * q^48 + 2 * q^50 + 6 * q^51 - 12 * q^53 - 2 * q^54 + 2 * q^55 + 2 * q^57 - 12 * q^59 + 2 * q^60 - 10 * q^62 + 2 * q^64 + 2 * q^66 + 4 * q^67 - 6 * q^68 - 4 * q^69 + 4 * q^71 + 2 * q^72 - 18 * q^73 - 4 * q^74 - 2 * q^75 - 2 * q^76 - 4 * q^79 - 2 * q^80 + 2 * q^81 - 6 * q^82 - 2 * q^83 - 4 * q^85 + 4 * q^86 - 2 * q^88 - 4 * q^89 - 2 * q^90 + 4 * q^92 + 10 * q^93 - 6 * q^94 - 28 * q^95 - 2 * q^96 - 2 * q^99

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
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.23607	0.390879
$11$	−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
$15$	−1.23607	−0.319151
$16$	1.00000	0.250000
$17$	−5.23607	−1.26993	−0.634967	−	0.772540i	$-0.718986\pi$
$17$	−5.23607	−1.26993	−0.634967	+	0.772540i	$0.718986\pi$
$18$	1.00000	0.235702
$19$	−7.70820	−1.76838	−0.884192	−	0.467124i	$-0.845290\pi$
$19$	−7.70820	−1.76838	−0.884192	+	0.467124i	$0.845290\pi$
$20$	1.23607	0.276393
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−2.47214	−0.515476	−0.257738	−	0.966215i	$-0.582977\pi$
$23$	−2.47214	−0.515476	−0.257738	+	0.966215i	$0.582977\pi$
$24$	−1.00000	−0.204124
$25$	−3.47214	−0.694427
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	−1.23607	−0.225674
$31$	−2.76393	−0.496417	−0.248208	−	0.968707i	$-0.579842\pi$
$31$	−2.76393	−0.496417	−0.248208	+	0.968707i	$0.579842\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	−5.23607	−0.897978
$35$	0	0
$36$	1.00000	0.166667
$37$	−10.9443	−1.79923	−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443	−1.79923	−0.899614	+	0.436687i	$0.856152\pi$
$38$	−7.70820	−1.25044
$39$	0	0
$40$	1.23607	0.195440
$41$	−5.23607	−0.817736	−0.408868	−	0.912593i	$-0.634076\pi$
$41$	−5.23607	−0.817736	−0.408868	+	0.912593i	$0.634076\pi$
$42$	0	0
$43$	6.47214	0.986991	0.493496	−	0.869748i	$-0.335719\pi$
$43$	6.47214	0.986991	0.493496	+	0.869748i	$0.335719\pi$
$44$	−1.00000	−0.150756
$45$	1.23607	0.184262
$46$	−2.47214	−0.364497
$47$	3.70820	0.540897	0.270449	−	0.962734i	$-0.412828\pi$
$47$	3.70820	0.540897	0.270449	+	0.962734i	$0.412828\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−3.47214	−0.491034
$51$	5.23607	0.733196
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	−1.23607	−0.166671
$56$	0	0
$57$	7.70820	1.02098
$58$	4.47214	0.587220
$59$	−1.52786	−0.198911	−0.0994555	−	0.995042i	$-0.531710\pi$
$59$	−1.52786	−0.198911	−0.0994555	+	0.995042i	$0.531710\pi$
$60$	−1.23607	−0.159576
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−2.76393	−0.351020
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	15.4164	1.88341	0.941707	−	0.336434i	$-0.109221\pi$
$67$	15.4164	1.88341	0.941707	+	0.336434i	$0.109221\pi$
$68$	−5.23607	−0.634967
$69$	2.47214	0.297610
$70$	0	0
$71$	−2.47214	−0.293389	−0.146694	−	0.989182i	$-0.546863\pi$
$71$	−2.47214	−0.293389	−0.146694	+	0.989182i	$0.546863\pi$
$72$	1.00000	0.117851
$73$	−15.7082	−1.83851	−0.919253	−	0.393667i	$-0.871206\pi$
$73$	−15.7082	−1.83851	−0.919253	+	0.393667i	$0.871206\pi$
$74$	−10.9443	−1.27225
$75$	3.47214	0.400928
$76$	−7.70820	−0.884192
$77$	0	0
$78$	0	0
$79$	11.4164	1.28445	0.642223	−	0.766518i	$-0.278012\pi$
$79$	11.4164	1.28445	0.642223	+	0.766518i	$0.278012\pi$
$80$	1.23607	0.138197
$81$	1.00000	0.111111
$82$	−5.23607	−0.578227
$83$	1.23607	0.135676	0.0678380	−	0.997696i	$-0.478390\pi$
$83$	1.23607	0.135676	0.0678380	+	0.997696i	$0.478390\pi$
$84$	0	0
$85$	−6.47214	−0.702002
$86$	6.47214	0.697908
$87$	−4.47214	−0.479463
$88$	−1.00000	−0.106600
$89$	−6.47214	−0.686045	−0.343023	−	0.939327i	$-0.611451\pi$
$89$	−6.47214	−0.686045	−0.343023	+	0.939327i	$0.611451\pi$
$90$	1.23607	0.130293
$91$	0	0
$92$	−2.47214	−0.257738
$93$	2.76393	0.286606
$94$	3.70820	0.382472
$95$	−9.52786	−0.977538
$96$	−1.00000	−0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	−3.47214	−0.347214
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	5.23607	0.518448
$103$	10.7639	1.06060	0.530301	−	0.847810i	$-0.322079\pi$
$103$	10.7639	1.06060	0.530301	+	0.847810i	$0.322079\pi$
$104$	0	0
$105$	0	0
$106$	−6.00000	−0.582772
$107$	0.944272	0.0912862	0.0456431	−	0.998958i	$-0.485466\pi$
$107$	0.944272	0.0912862	0.0456431	+	0.998958i	$0.485466\pi$
$108$	−1.00000	−0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−1.23607	−0.117854
$111$	10.9443	1.03878
$112$	0	0
$113$	13.4164	1.26211	0.631055	−	0.775738i	$-0.282622\pi$
$113$	13.4164	1.26211	0.631055	+	0.775738i	$0.282622\pi$
$114$	7.70820	0.721939
$115$	−3.05573	−0.284948
$116$	4.47214	0.415227
$117$	0	0
$118$	−1.52786	−0.140651
$119$	0	0
$120$	−1.23607	−0.112837
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.23607	0.472120
$124$	−2.76393	−0.248208
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−6.47214	−0.574309	−0.287155	−	0.957884i	$-0.592709\pi$
$127$	−6.47214	−0.574309	−0.287155	+	0.957884i	$0.592709\pi$
$128$	1.00000	0.0883883
$129$	−6.47214	−0.569840
$130$	0	0
$131$	6.76393	0.590967	0.295484	−	0.955348i	$-0.404519\pi$
$131$	6.76393	0.590967	0.295484	+	0.955348i	$0.404519\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	15.4164	1.33177
$135$	−1.23607	−0.106384
$136$	−5.23607	−0.448989
$137$	−16.4721	−1.40731	−0.703655	−	0.710542i	$-0.748450\pi$
$137$	−16.4721	−1.40731	−0.703655	+	0.710542i	$0.748450\pi$
$138$	2.47214	0.210442
$139$	−10.1803	−0.863485	−0.431743	−	0.901997i	$-0.642101\pi$
$139$	−10.1803	−0.863485	−0.431743	+	0.901997i	$0.642101\pi$
$140$	0	0
$141$	−3.70820	−0.312287
$142$	−2.47214	−0.207457
$143$	0	0
$144$	1.00000	0.0833333
$145$	5.52786	0.459064
$146$	−15.7082	−1.30002
$147$	0	0
$148$	−10.9443	−0.899614
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	3.47214	0.283499
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−7.70820	−0.625218
$153$	−5.23607	−0.423311
$154$	0	0
$155$	−3.41641	−0.274412
$156$	0	0
$157$	−14.7639	−1.17829	−0.589145	−	0.808027i	$-0.700535\pi$
$157$	−14.7639	−1.17829	−0.589145	+	0.808027i	$0.700535\pi$
$158$	11.4164	0.908241
$159$	6.00000	0.475831
$160$	1.23607	0.0977198
$161$	0	0
$162$	1.00000	0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−5.23607	−0.408868
$165$	1.23607	0.0962278
$166$	1.23607	0.0959375
$167$	18.4721	1.42942	0.714708	−	0.699423i	$-0.246559\pi$
$167$	18.4721	1.42942	0.714708	+	0.699423i	$0.246559\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−6.47214	−0.496390
$171$	−7.70820	−0.589461
$172$	6.47214	0.493496
$173$	19.4164	1.47620	0.738101	−	0.674690i	$-0.235723\pi$
$173$	19.4164	1.47620	0.738101	+	0.674690i	$0.235723\pi$
$174$	−4.47214	−0.339032
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	1.52786	0.114841
$178$	−6.47214	−0.485107
$179$	−5.52786	−0.413172	−0.206586	−	0.978428i	$-0.566235\pi$
$179$	−5.52786	−0.413172	−0.206586	+	0.978428i	$0.566235\pi$
$180$	1.23607	0.0921311
$181$	11.7082	0.870264	0.435132	−	0.900367i	$-0.356702\pi$
$181$	11.7082	0.870264	0.435132	+	0.900367i	$0.356702\pi$
$182$	0	0
$183$	0	0
$184$	−2.47214	−0.182248
$185$	−13.5279	−0.994588
$186$	2.76393	0.202661
$187$	5.23607	0.382899
$188$	3.70820	0.270449
$189$	0	0
$190$	−9.52786	−0.691224
$191$	−10.4721	−0.757737	−0.378869	−	0.925450i	$-0.623687\pi$
$191$	−10.4721	−0.757737	−0.378869	+	0.925450i	$0.623687\pi$
$192$	−1.00000	−0.0721688
$193$	5.05573	0.363919	0.181960	−	0.983306i	$-0.441756\pi$
$193$	5.05573	0.363919	0.181960	+	0.983306i	$0.441756\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−22.3607	−1.59313	−0.796566	−	0.604551i	$-0.793352\pi$
$197$	−22.3607	−1.59313	−0.796566	+	0.604551i	$0.793352\pi$
$198$	−1.00000	−0.0710669
$199$	−23.1246	−1.63926	−0.819630	−	0.572893i	$-0.805821\pi$
$199$	−23.1246	−1.63926	−0.819630	+	0.572893i	$0.805821\pi$
$200$	−3.47214	−0.245517
$201$	−15.4164	−1.08739
$202$	−12.0000	−0.844317
$203$	0	0
$204$	5.23607	0.366598
$205$	−6.47214	−0.452034
$206$	10.7639	0.749959
$207$	−2.47214	−0.171825
$208$	0	0
$209$	7.70820	0.533188
$210$	0	0
$211$	3.41641	0.235195	0.117598	−	0.993061i	$-0.462481\pi$
$211$	3.41641	0.235195	0.117598	+	0.993061i	$0.462481\pi$
$212$	−6.00000	−0.412082
$213$	2.47214	0.169388
$214$	0.944272	0.0645491
$215$	8.00000	0.545595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	6.00000	0.406371
$219$	15.7082	1.06146
$220$	−1.23607	−0.0833357
$221$	0	0
$222$	10.9443	0.734531
$223$	−15.7082	−1.05190	−0.525950	−	0.850516i	$-0.676290\pi$
$223$	−15.7082	−1.05190	−0.525950	+	0.850516i	$0.676290\pi$
$224$	0	0
$225$	−3.47214	−0.231476
$226$	13.4164	0.892446
$227$	8.65248	0.574285	0.287142	−	0.957888i	$-0.407295\pi$
$227$	8.65248	0.574285	0.287142	+	0.957888i	$0.407295\pi$
$228$	7.70820	0.510488
$229$	19.7082	1.30235	0.651177	−	0.758926i	$-0.274275\pi$
$229$	19.7082	1.30235	0.651177	+	0.758926i	$0.274275\pi$
$230$	−3.05573	−0.201489
$231$	0	0
$232$	4.47214	0.293610
$233$	−9.05573	−0.593260	−0.296630	−	0.954992i	$-0.595863\pi$
$233$	−9.05573	−0.593260	−0.296630	+	0.954992i	$0.595863\pi$
$234$	0	0
$235$	4.58359	0.299001
$236$	−1.52786	−0.0994555
$237$	−11.4164	−0.741575
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−1.23607	−0.0797878
$241$	18.1803	1.17110	0.585549	−	0.810637i	$-0.300879\pi$
$241$	18.1803	1.17110	0.585549	+	0.810637i	$0.300879\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	5.23607	0.333840
$247$	0	0
$248$	−2.76393	−0.175510
$249$	−1.23607	−0.0783326
$250$	−10.4721	−0.662316
$251$	−6.47214	−0.408518	−0.204259	−	0.978917i	$-0.565478\pi$
$251$	−6.47214	−0.408518	−0.204259	+	0.978917i	$0.565478\pi$
$252$	0	0
$253$	2.47214	0.155422
$254$	−6.47214	−0.406098
$255$	6.47214	0.405301
$256$	1.00000	0.0625000
$257$	22.4721	1.40177	0.700887	−	0.713273i	$-0.252788\pi$
$257$	22.4721	1.40177	0.700887	+	0.713273i	$0.252788\pi$
$258$	−6.47214	−0.402938
$259$	0	0
$260$	0	0
$261$	4.47214	0.276818
$262$	6.76393	0.417877
$263$	−9.52786	−0.587513	−0.293757	−	0.955880i	$-0.594906\pi$
$263$	−9.52786	−0.587513	−0.293757	+	0.955880i	$0.594906\pi$
$264$	1.00000	0.0615457
$265$	−7.41641	−0.455586
$266$	0	0
$267$	6.47214	0.396088
$268$	15.4164	0.941707
$269$	−27.7082	−1.68940	−0.844700	−	0.535241i	$-0.820221\pi$
$269$	−27.7082	−1.68940	−0.844700	+	0.535241i	$0.820221\pi$
$270$	−1.23607	−0.0752247
$271$	−2.47214	−0.150172	−0.0750858	−	0.997177i	$-0.523923\pi$
$271$	−2.47214	−0.150172	−0.0750858	+	0.997177i	$0.523923\pi$
$272$	−5.23607	−0.317483
$273$	0	0
$274$	−16.4721	−0.995118
$275$	3.47214	0.209378
$276$	2.47214	0.148805
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−10.1803	−0.610576
$279$	−2.76393	−0.165472
$280$	0	0
$281$	19.8885	1.18645	0.593226	−	0.805036i	$-0.297854\pi$
$281$	19.8885	1.18645	0.593226	+	0.805036i	$0.297854\pi$
$282$	−3.70820	−0.220820
$283$	23.7082	1.40931	0.704653	−	0.709552i	$-0.251103\pi$
$283$	23.7082	1.40931	0.704653	+	0.709552i	$0.251103\pi$
$284$	−2.47214	−0.146694
$285$	9.52786	0.564382
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	10.4164	0.612730
$290$	5.52786	0.324607
$291$	0	0
$292$	−15.7082	−0.919253
$293$	−4.58359	−0.267776	−0.133888	−	0.990996i	$-0.542746\pi$
$293$	−4.58359	−0.267776	−0.133888	+	0.990996i	$0.542746\pi$
$294$	0	0
$295$	−1.88854	−0.109955
$296$	−10.9443	−0.636123
$297$	1.00000	0.0580259
$298$	−6.00000	−0.347571
$299$	0	0
$300$	3.47214	0.200464
$301$	0	0
$302$	0	0
$303$	12.0000	0.689382
$304$	−7.70820	−0.442096
$305$	0	0
$306$	−5.23607	−0.299326
$307$	25.5967	1.46088	0.730442	−	0.682975i	$-0.239314\pi$
$307$	25.5967	1.46088	0.730442	+	0.682975i	$0.239314\pi$
$308$	0	0
$309$	−10.7639	−0.612339
$310$	−3.41641	−0.194039
$311$	−8.65248	−0.490637	−0.245318	−	0.969443i	$-0.578893\pi$
$311$	−8.65248	−0.490637	−0.245318	+	0.969443i	$0.578893\pi$
$312$	0	0
$313$	5.52786	0.312453	0.156227	−	0.987721i	$-0.450067\pi$
$313$	5.52786	0.312453	0.156227	+	0.987721i	$0.450067\pi$
$314$	−14.7639	−0.833177
$315$	0	0
$316$	11.4164	0.642223
$317$	−31.8885	−1.79104	−0.895520	−	0.445022i	$-0.853196\pi$
$317$	−31.8885	−1.79104	−0.895520	+	0.445022i	$0.853196\pi$
$318$	6.00000	0.336463
$319$	−4.47214	−0.250392
$320$	1.23607	0.0690983
$321$	−0.944272	−0.0527041
$322$	0	0
$323$	40.3607	2.24573
$324$	1.00000	0.0555556
$325$	0	0
$326$	−12.0000	−0.664619
$327$	−6.00000	−0.331801
$328$	−5.23607	−0.289113
$329$	0	0
$330$	1.23607	0.0680433
$331$	−7.41641	−0.407643	−0.203821	−	0.979008i	$-0.565336\pi$
$331$	−7.41641	−0.407643	−0.203821	+	0.979008i	$0.565336\pi$
$332$	1.23607	0.0678380
$333$	−10.9443	−0.599742
$334$	18.4721	1.01075
$335$	19.0557	1.04113
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−13.0000	−0.707107
$339$	−13.4164	−0.728679
$340$	−6.47214	−0.351001
$341$	2.76393	0.149675
$342$	−7.70820	−0.416812
$343$	0	0
$344$	6.47214	0.348954
$345$	3.05573	0.164515
$346$	19.4164	1.04383
$347$	−26.8328	−1.44046	−0.720231	−	0.693735i	$-0.755964\pi$
$347$	−26.8328	−1.44046	−0.720231	+	0.693735i	$0.755964\pi$
$348$	−4.47214	−0.239732
$349$	−0.583592	−0.0312390	−0.0156195	−	0.999878i	$-0.504972\pi$
$349$	−0.583592	−0.0312390	−0.0156195	+	0.999878i	$0.504972\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	24.3607	1.29659	0.648294	−	0.761390i	$-0.275483\pi$
$353$	24.3607	1.29659	0.648294	+	0.761390i	$0.275483\pi$
$354$	1.52786	0.0812051
$355$	−3.05573	−0.162181
$356$	−6.47214	−0.343023
$357$	0	0
$358$	−5.52786	−0.292157
$359$	1.52786	0.0806376	0.0403188	−	0.999187i	$-0.487163\pi$
$359$	1.52786	0.0806376	0.0403188	+	0.999187i	$0.487163\pi$
$360$	1.23607	0.0651465
$361$	40.4164	2.12718
$362$	11.7082	0.615370
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−19.4164	−1.01630
$366$	0	0
$367$	8.29180	0.432828	0.216414	−	0.976302i	$-0.430564\pi$
$367$	8.29180	0.432828	0.216414	+	0.976302i	$0.430564\pi$
$368$	−2.47214	−0.128869
$369$	−5.23607	−0.272579
$370$	−13.5279	−0.703280
$371$	0	0
$372$	2.76393	0.143303
$373$	−1.41641	−0.0733388	−0.0366694	−	0.999327i	$-0.511675\pi$
$373$	−1.41641	−0.0733388	−0.0366694	+	0.999327i	$0.511675\pi$
$374$	5.23607	0.270751
$375$	10.4721	0.540779
$376$	3.70820	0.191236
$377$	0	0
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	−9.52786	−0.488769
$381$	6.47214	0.331578
$382$	−10.4721	−0.535801
$383$	−0.652476	−0.0333400	−0.0166700	−	0.999861i	$-0.505306\pi$
$383$	−0.652476	−0.0333400	−0.0166700	+	0.999861i	$0.505306\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	5.05573	0.257330
$387$	6.47214	0.328997
$388$	0	0
$389$	−6.94427	−0.352089	−0.176044	−	0.984382i	$-0.556330\pi$
$389$	−6.94427	−0.352089	−0.176044	+	0.984382i	$0.556330\pi$
$390$	0	0
$391$	12.9443	0.654620
$392$	0	0
$393$	−6.76393	−0.341195
$394$	−22.3607	−1.12651
$395$	14.1115	0.710024
$396$	−1.00000	−0.0502519
$397$	−1.81966	−0.0913261	−0.0456631	−	0.998957i	$-0.514540\pi$
$397$	−1.81966	−0.0913261	−0.0456631	+	0.998957i	$0.514540\pi$
$398$	−23.1246	−1.15913
$399$	0	0
$400$	−3.47214	−0.173607
$401$	−8.47214	−0.423078	−0.211539	−	0.977370i	$-0.567848\pi$
$401$	−8.47214	−0.423078	−0.211539	+	0.977370i	$0.567848\pi$
$402$	−15.4164	−0.768901
$403$	0	0
$404$	−12.0000	−0.597022
$405$	1.23607	0.0614207
$406$	0	0
$407$	10.9443	0.542487
$408$	5.23607	0.259224
$409$	2.76393	0.136668	0.0683338	−	0.997663i	$-0.478232\pi$
$409$	2.76393	0.136668	0.0683338	+	0.997663i	$0.478232\pi$
$410$	−6.47214	−0.319636
$411$	16.4721	0.812511
$412$	10.7639	0.530301
$413$	0	0
$414$	−2.47214	−0.121499
$415$	1.52786	0.0749999
$416$	0	0
$417$	10.1803	0.498533
$418$	7.70820	0.377021
$419$	15.0557	0.735520	0.367760	−	0.929921i	$-0.380125\pi$
$419$	15.0557	0.735520	0.367760	+	0.929921i	$0.380125\pi$
$420$	0	0
$421$	−5.05573	−0.246401	−0.123201	−	0.992382i	$-0.539316\pi$
$421$	−5.05573	−0.246401	−0.123201	+	0.992382i	$0.539316\pi$
$422$	3.41641	0.166308
$423$	3.70820	0.180299
$424$	−6.00000	−0.291386
$425$	18.1803	0.881876
$426$	2.47214	0.119775
$427$	0	0
$428$	0.944272	0.0456431
$429$	0	0
$430$	8.00000	0.385794
$431$	−22.4721	−1.08244	−0.541222	−	0.840880i	$-0.682038\pi$
$431$	−22.4721	−1.08244	−0.541222	+	0.840880i	$0.682038\pi$
$432$	−1.00000	−0.0481125
$433$	9.88854	0.475213	0.237607	−	0.971361i	$-0.423637\pi$
$433$	9.88854	0.475213	0.237607	+	0.971361i	$0.423637\pi$
$434$	0	0
$435$	−5.52786	−0.265041
$436$	6.00000	0.287348
$437$	19.0557	0.911559
$438$	15.7082	0.750567
$439$	31.4164	1.49942	0.749712	−	0.661765i	$-0.230192\pi$
$439$	31.4164	1.49942	0.749712	+	0.661765i	$0.230192\pi$
$440$	−1.23607	−0.0589272
$441$	0	0
$442$	0	0
$443$	7.41641	0.352364	0.176182	−	0.984358i	$-0.443625\pi$
$443$	7.41641	0.352364	0.176182	+	0.984358i	$0.443625\pi$
$444$	10.9443	0.519392
$445$	−8.00000	−0.379236
$446$	−15.7082	−0.743805
$447$	6.00000	0.283790
$448$	0	0
$449$	19.5279	0.921577	0.460788	−	0.887510i	$-0.347567\pi$
$449$	19.5279	0.921577	0.460788	+	0.887510i	$0.347567\pi$
$450$	−3.47214	−0.163678
$451$	5.23607	0.246557
$452$	13.4164	0.631055
$453$	0	0
$454$	8.65248	0.406081
$455$	0	0
$456$	7.70820	0.360970
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	19.7082	0.920904
$459$	5.23607	0.244399
$460$	−3.05573	−0.142474
$461$	−26.8328	−1.24973	−0.624864	−	0.780733i	$-0.714846\pi$
$461$	−26.8328	−1.24973	−0.624864	+	0.780733i	$0.714846\pi$
$462$	0	0
$463$	−1.88854	−0.0877681	−0.0438840	−	0.999037i	$-0.513973\pi$
$463$	−1.88854	−0.0877681	−0.0438840	+	0.999037i	$0.513973\pi$
$464$	4.47214	0.207614
$465$	3.41641	0.158432
$466$	−9.05573	−0.419499
$467$	−29.8885	−1.38308	−0.691538	−	0.722340i	$-0.743067\pi$
$467$	−29.8885	−1.38308	−0.691538	+	0.722340i	$0.743067\pi$
$468$	0	0
$469$	0	0
$470$	4.58359	0.211425
$471$	14.7639	0.680286
$472$	−1.52786	−0.0703256
$473$	−6.47214	−0.297589
$474$	−11.4164	−0.524373
$475$	26.7639	1.22801
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	35.7771	1.63470	0.817348	−	0.576144i	$-0.195443\pi$
$479$	35.7771	1.63470	0.817348	+	0.576144i	$0.195443\pi$
$480$	−1.23607	−0.0564185
$481$	0	0
$482$	18.1803	0.828092
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	36.9443	1.67410	0.837052	−	0.547123i	$-0.184277\pi$
$487$	36.9443	1.67410	0.837052	+	0.547123i	$0.184277\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	−29.8885	−1.34885	−0.674426	−	0.738343i	$-0.735609\pi$
$491$	−29.8885	−1.34885	−0.674426	+	0.738343i	$0.735609\pi$
$492$	5.23607	0.236060
$493$	−23.4164	−1.05462
$494$	0	0
$495$	−1.23607	−0.0555571
$496$	−2.76393	−0.124104
$497$	0	0
$498$	−1.23607	−0.0553895
$499$	7.41641	0.332004	0.166002	−	0.986125i	$-0.446914\pi$
$499$	7.41641	0.332004	0.166002	+	0.986125i	$0.446914\pi$
$500$	−10.4721	−0.468328
$501$	−18.4721	−0.825274
$502$	−6.47214	−0.288866
$503$	−41.3050	−1.84170	−0.920848	−	0.389921i	$-0.872502\pi$
$503$	−41.3050	−1.84170	−0.920848	+	0.389921i	$0.872502\pi$
$504$	0	0
$505$	−14.8328	−0.660052
$506$	2.47214	0.109900
$507$	13.0000	0.577350
$508$	−6.47214	−0.287155
$509$	−6.76393	−0.299806	−0.149903	−	0.988701i	$-0.547896\pi$
$509$	−6.76393	−0.299806	−0.149903	+	0.988701i	$0.547896\pi$
$510$	6.47214	0.286591
$511$	0	0
$512$	1.00000	0.0441942
$513$	7.70820	0.340326
$514$	22.4721	0.991203
$515$	13.3050	0.586286
$516$	−6.47214	−0.284920
$517$	−3.70820	−0.163087
$518$	0	0
$519$	−19.4164	−0.852286
$520$	0	0
$521$	−21.3050	−0.933387	−0.466693	−	0.884419i	$-0.654555\pi$
$521$	−21.3050	−0.933387	−0.466693	+	0.884419i	$0.654555\pi$
$522$	4.47214	0.195740
$523$	−24.2918	−1.06221	−0.531103	−	0.847307i	$-0.678222\pi$
$523$	−24.2918	−1.06221	−0.531103	+	0.847307i	$0.678222\pi$
$524$	6.76393	0.295484
$525$	0	0
$526$	−9.52786	−0.415435
$527$	14.4721	0.630416
$528$	1.00000	0.0435194
$529$	−16.8885	−0.734285
$530$	−7.41641	−0.322148
$531$	−1.52786	−0.0663037
$532$	0	0
$533$	0	0
$534$	6.47214	0.280077
$535$	1.16718	0.0504618
$536$	15.4164	0.665887
$537$	5.52786	0.238545
$538$	−27.7082	−1.19459
$539$	0	0
$540$	−1.23607	−0.0531919
$541$	21.4164	0.920763	0.460382	−	0.887721i	$-0.347713\pi$
$541$	21.4164	0.920763	0.460382	+	0.887721i	$0.347713\pi$
$542$	−2.47214	−0.106187
$543$	−11.7082	−0.502447
$544$	−5.23607	−0.224495
$545$	7.41641	0.317684
$546$	0	0
$547$	38.4721	1.64495	0.822475	−	0.568801i	$-0.192593\pi$
$547$	38.4721	1.64495	0.822475	+	0.568801i	$0.192593\pi$
$548$	−16.4721	−0.703655
$549$	0	0
$550$	3.47214	0.148052
$551$	−34.4721	−1.46856
$552$	2.47214	0.105221
$553$	0	0
$554$	26.0000	1.10463
$555$	13.5279	0.574226
$556$	−10.1803	−0.431743
$557$	−10.5836	−0.448441	−0.224221	−	0.974538i	$-0.571984\pi$
$557$	−10.5836	−0.448441	−0.224221	+	0.974538i	$0.571984\pi$
$558$	−2.76393	−0.117007
$559$	0	0
$560$	0	0
$561$	−5.23607	−0.221067
$562$	19.8885	0.838948
$563$	6.76393	0.285066	0.142533	−	0.989790i	$-0.454475\pi$
$563$	6.76393	0.285066	0.142533	+	0.989790i	$0.454475\pi$
$564$	−3.70820	−0.156144
$565$	16.5836	0.697677
$566$	23.7082	0.996530
$567$	0	0
$568$	−2.47214	−0.103729
$569$	−34.9443	−1.46494	−0.732470	−	0.680799i	$-0.761633\pi$
$569$	−34.9443	−1.46494	−0.732470	+	0.680799i	$0.761633\pi$
$570$	9.52786	0.399078
$571$	−17.5279	−0.733518	−0.366759	−	0.930316i	$-0.619533\pi$
$571$	−17.5279	−0.733518	−0.366759	+	0.930316i	$0.619533\pi$
$572$	0	0
$573$	10.4721	0.437480
$574$	0	0
$575$	8.58359	0.357961
$576$	1.00000	0.0416667
$577$	−39.4164	−1.64093	−0.820463	−	0.571699i	$-0.806284\pi$
$577$	−39.4164	−1.64093	−0.820463	+	0.571699i	$0.806284\pi$
$578$	10.4164	0.433265
$579$	−5.05573	−0.210109
$580$	5.52786	0.229532
$581$	0	0
$582$	0	0
$583$	6.00000	0.248495
$584$	−15.7082	−0.650010
$585$	0	0
$586$	−4.58359	−0.189346
$587$	17.5279	0.723452	0.361726	−	0.932284i	$-0.382188\pi$
$587$	17.5279	0.723452	0.361726	+	0.932284i	$0.382188\pi$
$588$	0	0
$589$	21.3050	0.877855
$590$	−1.88854	−0.0777501
$591$	22.3607	0.919795
$592$	−10.9443	−0.449807
$593$	18.7639	0.770542	0.385271	−	0.922803i	$-0.374108\pi$
$593$	18.7639	0.770542	0.385271	+	0.922803i	$0.374108\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	−6.00000	−0.245770
$597$	23.1246	0.946427
$598$	0	0
$599$	25.3050	1.03393	0.516966	−	0.856006i	$-0.327061\pi$
$599$	25.3050	1.03393	0.516966	+	0.856006i	$0.327061\pi$
$600$	3.47214	0.141749
$601$	−0.291796	−0.0119026	−0.00595130	−	0.999982i	$-0.501894\pi$
$601$	−0.291796	−0.0119026	−0.00595130	+	0.999982i	$0.501894\pi$
$602$	0	0
$603$	15.4164	0.627805
$604$	0	0
$605$	1.23607	0.0502533
$606$	12.0000	0.487467
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−7.70820	−0.312609
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−5.23607	−0.211656
$613$	12.4721	0.503745	0.251872	−	0.967760i	$-0.418954\pi$
$613$	12.4721	0.503745	0.251872	+	0.967760i	$0.418954\pi$
$614$	25.5967	1.03300
$615$	6.47214	0.260982
$616$	0	0
$617$	23.8885	0.961717	0.480858	−	0.876798i	$-0.340325\pi$
$617$	23.8885	0.961717	0.480858	+	0.876798i	$0.340325\pi$
$618$	−10.7639	−0.432989
$619$	18.8328	0.756955	0.378477	−	0.925611i	$-0.376448\pi$
$619$	18.8328	0.756955	0.378477	+	0.925611i	$0.376448\pi$
$620$	−3.41641	−0.137206
$621$	2.47214	0.0992034
$622$	−8.65248	−0.346933
$623$	0	0
$624$	0	0
$625$	4.41641	0.176656
$626$	5.52786	0.220938
$627$	−7.70820	−0.307836
$628$	−14.7639	−0.589145
$629$	57.3050	2.28490
$630$	0	0
$631$	9.88854	0.393657	0.196828	−	0.980438i	$-0.436936\pi$
$631$	9.88854	0.393657	0.196828	+	0.980438i	$0.436936\pi$
$632$	11.4164	0.454120
$633$	−3.41641	−0.135790
$634$	−31.8885	−1.26646
$635$	−8.00000	−0.317470
$636$	6.00000	0.237915
$637$	0	0
$638$	−4.47214	−0.177054
$639$	−2.47214	−0.0977962
$640$	1.23607	0.0488599
$641$	−35.8885	−1.41751	−0.708756	−	0.705454i	$-0.750743\pi$
$641$	−35.8885	−1.41751	−0.708756	+	0.705454i	$0.750743\pi$
$642$	−0.944272	−0.0372674
$643$	43.4164	1.71218	0.856088	−	0.516830i	$-0.172888\pi$
$643$	43.4164	1.71218	0.856088	+	0.516830i	$0.172888\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	40.3607	1.58797
$647$	45.0132	1.76965	0.884825	−	0.465924i	$-0.154278\pi$
$647$	45.0132	1.76965	0.884825	+	0.465924i	$0.154278\pi$
$648$	1.00000	0.0392837
$649$	1.52786	0.0599739
$650$	0	0
$651$	0	0
$652$	−12.0000	−0.469956
$653$	−14.9443	−0.584815	−0.292407	−	0.956294i	$-0.594456\pi$
$653$	−14.9443	−0.584815	−0.292407	+	0.956294i	$0.594456\pi$
$654$	−6.00000	−0.234619
$655$	8.36068	0.326679
$656$	−5.23607	−0.204434
$657$	−15.7082	−0.612835
$658$	0	0
$659$	13.8885	0.541021	0.270510	−	0.962717i	$-0.412808\pi$
$659$	13.8885	0.541021	0.270510	+	0.962717i	$0.412808\pi$
$660$	1.23607	0.0481139
$661$	5.59675	0.217688	0.108844	−	0.994059i	$-0.465285\pi$
$661$	5.59675	0.217688	0.108844	+	0.994059i	$0.465285\pi$
$662$	−7.41641	−0.288247
$663$	0	0
$664$	1.23607	0.0479687
$665$	0	0
$666$	−10.9443	−0.424082
$667$	−11.0557	−0.428080
$668$	18.4721	0.714708
$669$	15.7082	0.607314
$670$	19.0557	0.736187
$671$	0	0
$672$	0	0
$673$	34.9443	1.34700	0.673501	−	0.739186i	$-0.264790\pi$
$673$	34.9443	1.34700	0.673501	+	0.739186i	$0.264790\pi$
$674$	−18.0000	−0.693334
$675$	3.47214	0.133643
$676$	−13.0000	−0.500000
$677$	−46.4721	−1.78607	−0.893035	−	0.449988i	$-0.851428\pi$
$677$	−46.4721	−1.78607	−0.893035	+	0.449988i	$0.851428\pi$
$678$	−13.4164	−0.515254
$679$	0	0
$680$	−6.47214	−0.248195
$681$	−8.65248	−0.331564
$682$	2.76393	0.105836
$683$	−4.36068	−0.166857	−0.0834284	−	0.996514i	$-0.526587\pi$
$683$	−4.36068	−0.166857	−0.0834284	+	0.996514i	$0.526587\pi$
$684$	−7.70820	−0.294731
$685$	−20.3607	−0.777942
$686$	0	0
$687$	−19.7082	−0.751915
$688$	6.47214	0.246748
$689$	0	0
$690$	3.05573	0.116330
$691$	−17.5279	−0.666791	−0.333396	−	0.942787i	$-0.608194\pi$
$691$	−17.5279	−0.666791	−0.333396	+	0.942787i	$0.608194\pi$
$692$	19.4164	0.738101
$693$	0	0
$694$	−26.8328	−1.01856
$695$	−12.5836	−0.477323
$696$	−4.47214	−0.169516
$697$	27.4164	1.03847
$698$	−0.583592	−0.0220893
$699$	9.05573	0.342519
$700$	0	0
$701$	−8.11146	−0.306365	−0.153183	−	0.988198i	$-0.548952\pi$
$701$	−8.11146	−0.306365	−0.153183	+	0.988198i	$0.548952\pi$
$702$	0	0
$703$	84.3607	3.18172
$704$	−1.00000	−0.0376889
$705$	−4.58359	−0.172628
$706$	24.3607	0.916826
$707$	0	0
$708$	1.52786	0.0574206
$709$	5.05573	0.189872	0.0949359	−	0.995483i	$-0.469735\pi$
$709$	5.05573	0.189872	0.0949359	+	0.995483i	$0.469735\pi$
$710$	−3.05573	−0.114679
$711$	11.4164	0.428149
$712$	−6.47214	−0.242554
$713$	6.83282	0.255891
$714$	0	0
$715$	0	0
$716$	−5.52786	−0.206586
$717$	0	0
$718$	1.52786	0.0570194
$719$	20.2918	0.756756	0.378378	−	0.925651i	$-0.376482\pi$
$719$	20.2918	0.756756	0.378378	+	0.925651i	$0.376482\pi$
$720$	1.23607	0.0460655
$721$	0	0
$722$	40.4164	1.50414
$723$	−18.1803	−0.676134
$724$	11.7082	0.435132
$725$	−15.5279	−0.576690
$726$	−1.00000	−0.0371135
$727$	45.2361	1.67771	0.838856	−	0.544353i	$-0.183225\pi$
$727$	45.2361	1.67771	0.838856	+	0.544353i	$0.183225\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−19.4164	−0.718633
$731$	−33.8885	−1.25341
$732$	0	0
$733$	−22.8328	−0.843349	−0.421675	−	0.906747i	$-0.638558\pi$
$733$	−22.8328	−0.843349	−0.421675	+	0.906747i	$0.638558\pi$
$734$	8.29180	0.306056
$735$	0	0
$736$	−2.47214	−0.0911241
$737$	−15.4164	−0.567871
$738$	−5.23607	−0.192742
$739$	43.4164	1.59710	0.798549	−	0.601930i	$-0.205601\pi$
$739$	43.4164	1.59710	0.798549	+	0.601930i	$0.205601\pi$
$740$	−13.5279	−0.497294
$741$	0	0
$742$	0	0
$743$	−25.8885	−0.949759	−0.474879	−	0.880051i	$-0.657508\pi$
$743$	−25.8885	−0.949759	−0.474879	+	0.880051i	$0.657508\pi$
$744$	2.76393	0.101331
$745$	−7.41641	−0.271716
$746$	−1.41641	−0.0518584
$747$	1.23607	0.0452254
$748$	5.23607	0.191450
$749$	0	0
$750$	10.4721	0.382388
$751$	−14.8328	−0.541257	−0.270629	−	0.962684i	$-0.587232\pi$
$751$	−14.8328	−0.541257	−0.270629	+	0.962684i	$0.587232\pi$
$752$	3.70820	0.135224
$753$	6.47214	0.235858
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.9443	0.833924	0.416962	−	0.908924i	$-0.363095\pi$
$757$	22.9443	0.833924	0.416962	+	0.908924i	$0.363095\pi$
$758$	−4.00000	−0.145287
$759$	−2.47214	−0.0897329
$760$	−9.52786	−0.345612
$761$	23.1246	0.838267	0.419133	−	0.907925i	$-0.362334\pi$
$761$	23.1246	0.838267	0.419133	+	0.907925i	$0.362334\pi$
$762$	6.47214	0.234461
$763$	0	0
$764$	−10.4721	−0.378869
$765$	−6.47214	−0.234001
$766$	−0.652476	−0.0235749
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−12.0689	−0.435215	−0.217608	−	0.976036i	$-0.569825\pi$
$769$	−12.0689	−0.435215	−0.217608	+	0.976036i	$0.569825\pi$
$770$	0	0
$771$	−22.4721	−0.809314
$772$	5.05573	0.181960
$773$	−49.9574	−1.79684	−0.898422	−	0.439133i	$-0.855286\pi$
$773$	−49.9574	−1.79684	−0.898422	+	0.439133i	$0.855286\pi$
$774$	6.47214	0.232636
$775$	9.59675	0.344725
$776$	0	0
$777$	0	0
$778$	−6.94427	−0.248964
$779$	40.3607	1.44607
$780$	0	0
$781$	2.47214	0.0884600
$782$	12.9443	0.462886
$783$	−4.47214	−0.159821
$784$	0	0
$785$	−18.2492	−0.651343
$786$	−6.76393	−0.241261
$787$	−22.5410	−0.803501	−0.401750	−	0.915749i	$-0.631598\pi$
$787$	−22.5410	−0.803501	−0.401750	+	0.915749i	$0.631598\pi$
$788$	−22.3607	−0.796566
$789$	9.52786	0.339201
$790$	14.1115	0.502063
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	0	0
$794$	−1.81966	−0.0645773
$795$	7.41641	0.263033
$796$	−23.1246	−0.819630
$797$	12.8754	0.456070	0.228035	−	0.973653i	$-0.426770\pi$
$797$	12.8754	0.456070	0.228035	+	0.973653i	$0.426770\pi$
$798$	0	0
$799$	−19.4164	−0.686903
$800$	−3.47214	−0.122759
$801$	−6.47214	−0.228682
$802$	−8.47214	−0.299162
$803$	15.7082	0.554330
$804$	−15.4164	−0.543695
$805$	0	0
$806$	0	0
$807$	27.7082	0.975375
$808$	−12.0000	−0.422159
$809$	44.8328	1.57624	0.788119	−	0.615523i	$-0.211055\pi$
$809$	44.8328	1.57624	0.788119	+	0.615523i	$0.211055\pi$
$810$	1.23607	0.0434310
$811$	17.5967	0.617905	0.308953	−	0.951077i	$-0.400022\pi$
$811$	17.5967	0.617905	0.308953	+	0.951077i	$0.400022\pi$
$812$	0	0
$813$	2.47214	0.0867016
$814$	10.9443	0.383597
$815$	−14.8328	−0.519571
$816$	5.23607	0.183299
$817$	−49.8885	−1.74538
$818$	2.76393	0.0966386
$819$	0	0
$820$	−6.47214	−0.226017
$821$	29.7771	1.03923	0.519614	−	0.854401i	$-0.326076\pi$
$821$	29.7771	1.03923	0.519614	+	0.854401i	$0.326076\pi$
$822$	16.4721	0.574532
$823$	−30.8328	−1.07476	−0.537382	−	0.843339i	$-0.680587\pi$
$823$	−30.8328	−1.07476	−0.537382	+	0.843339i	$0.680587\pi$
$824$	10.7639	0.374979
$825$	−3.47214	−0.120884
$826$	0	0
$827$	36.7214	1.27693	0.638463	−	0.769652i	$-0.279570\pi$
$827$	36.7214	1.27693	0.638463	+	0.769652i	$0.279570\pi$
$828$	−2.47214	−0.0859127
$829$	47.4853	1.64923	0.824616	−	0.565693i	$-0.191391\pi$
$829$	47.4853	1.64923	0.824616	+	0.565693i	$0.191391\pi$
$830$	1.52786	0.0530329
$831$	−26.0000	−0.901930
$832$	0	0
$833$	0	0
$834$	10.1803	0.352516
$835$	22.8328	0.790162
$836$	7.70820	0.266594
$837$	2.76393	0.0955355
$838$	15.0557	0.520091
$839$	25.2361	0.871246	0.435623	−	0.900129i	$-0.356528\pi$
$839$	25.2361	0.871246	0.435623	+	0.900129i	$0.356528\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	−5.05573	−0.174232
$843$	−19.8885	−0.684998
$844$	3.41641	0.117598
$845$	−16.0689	−0.552786
$846$	3.70820	0.127491
$847$	0	0
$848$	−6.00000	−0.206041
$849$	−23.7082	−0.813663
$850$	18.1803	0.623581
$851$	27.0557	0.927458
$852$	2.47214	0.0846940
$853$	−33.8885	−1.16032	−0.580161	−	0.814502i	$-0.697010\pi$
$853$	−33.8885	−1.16032	−0.580161	+	0.814502i	$0.697010\pi$
$854$	0	0
$855$	−9.52786	−0.325846
$856$	0.944272	0.0322745
$857$	47.1246	1.60975	0.804873	−	0.593447i	$-0.202233\pi$
$857$	47.1246	1.60975	0.804873	+	0.593447i	$0.202233\pi$
$858$	0	0
$859$	−43.4164	−1.48135	−0.740674	−	0.671864i	$-0.765494\pi$
$859$	−43.4164	−1.48135	−0.740674	+	0.671864i	$0.765494\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−22.4721	−0.765404
$863$	−44.3607	−1.51006	−0.755028	−	0.655693i	$-0.772377\pi$
$863$	−44.3607	−1.51006	−0.755028	+	0.655693i	$0.772377\pi$
$864$	−1.00000	−0.0340207
$865$	24.0000	0.816024
$866$	9.88854	0.336026
$867$	−10.4164	−0.353760
$868$	0	0
$869$	−11.4164	−0.387275
$870$	−5.52786	−0.187412
$871$	0	0
$872$	6.00000	0.203186
$873$	0	0
$874$	19.0557	0.644570
$875$	0	0
$876$	15.7082	0.530731
$877$	30.0000	1.01303	0.506514	−	0.862232i	$-0.330934\pi$
$877$	30.0000	1.01303	0.506514	+	0.862232i	$0.330934\pi$
$878$	31.4164	1.06025
$879$	4.58359	0.154601
$880$	−1.23607	−0.0416678
$881$	26.8328	0.904021	0.452010	−	0.892013i	$-0.350707\pi$
$881$	26.8328	0.904021	0.452010	+	0.892013i	$0.350707\pi$
$882$	0	0
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	0	0
$885$	1.88854	0.0634827
$886$	7.41641	0.249159
$887$	−36.9443	−1.24047	−0.620234	−	0.784417i	$-0.712962\pi$
$887$	−36.9443	−1.24047	−0.620234	+	0.784417i	$0.712962\pi$
$888$	10.9443	0.367266
$889$	0	0
$890$	−8.00000	−0.268161
$891$	−1.00000	−0.0335013
$892$	−15.7082	−0.525950
$893$	−28.5836	−0.956513
$894$	6.00000	0.200670
$895$	−6.83282	−0.228396
$896$	0	0
$897$	0	0
$898$	19.5279	0.651653
$899$	−12.3607	−0.412252
$900$	−3.47214	−0.115738
$901$	31.4164	1.04663
$902$	5.23607	0.174342
$903$	0	0
$904$	13.4164	0.446223
$905$	14.4721	0.481070
$906$	0	0
$907$	−33.3050	−1.10587	−0.552936	−	0.833223i	$-0.686493\pi$
$907$	−33.3050	−1.10587	−0.552936	+	0.833223i	$0.686493\pi$
$908$	8.65248	0.287142
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−31.4164	−1.04087	−0.520436	−	0.853901i	$-0.674231\pi$
$911$	−31.4164	−1.04087	−0.520436	+	0.853901i	$0.674231\pi$
$912$	7.70820	0.255244
$913$	−1.23607	−0.0409079
$914$	18.0000	0.595387
$915$	0	0
$916$	19.7082	0.651177
$917$	0	0
$918$	5.23607	0.172816
$919$	−53.6656	−1.77027	−0.885133	−	0.465338i	$-0.845933\pi$
$919$	−53.6656	−1.77027	−0.885133	+	0.465338i	$0.845933\pi$
$920$	−3.05573	−0.100744
$921$	−25.5967	−0.843442
$922$	−26.8328	−0.883692
$923$	0	0
$924$	0	0
$925$	38.0000	1.24943
$926$	−1.88854	−0.0620614
$927$	10.7639	0.353534
$928$	4.47214	0.146805
$929$	8.36068	0.274305	0.137153	−	0.990550i	$-0.456205\pi$
$929$	8.36068	0.274305	0.137153	+	0.990550i	$0.456205\pi$
$930$	3.41641	0.112028
$931$	0	0
$932$	−9.05573	−0.296630
$933$	8.65248	0.283269
$934$	−29.8885	−0.977983
$935$	6.47214	0.211661
$936$	0	0
$937$	−1.59675	−0.0521635	−0.0260817	−	0.999660i	$-0.508303\pi$
$937$	−1.59675	−0.0521635	−0.0260817	+	0.999660i	$0.508303\pi$
$938$	0	0
$939$	−5.52786	−0.180395
$940$	4.58359	0.149500
$941$	−7.63932	−0.249035	−0.124517	−	0.992217i	$-0.539738\pi$
$941$	−7.63932	−0.249035	−0.124517	+	0.992217i	$0.539738\pi$
$942$	14.7639	0.481035
$943$	12.9443	0.421523
$944$	−1.52786	−0.0497277
$945$	0	0
$946$	−6.47214	−0.210427
$947$	−37.8885	−1.23121	−0.615606	−	0.788054i	$-0.711089\pi$
$947$	−37.8885	−1.23121	−0.615606	+	0.788054i	$0.711089\pi$
$948$	−11.4164	−0.370788
$949$	0	0
$950$	26.7639	0.868337
$951$	31.8885	1.03406
$952$	0	0
$953$	−30.9443	−1.00238	−0.501192	−	0.865336i	$-0.667105\pi$
$953$	−30.9443	−1.00238	−0.501192	+	0.865336i	$0.667105\pi$
$954$	−6.00000	−0.194257
$955$	−12.9443	−0.418867
$956$	0	0
$957$	4.47214	0.144564
$958$	35.7771	1.15591
$959$	0	0
$960$	−1.23607	−0.0398939
$961$	−23.3607	−0.753570
$962$	0	0
$963$	0.944272	0.0304287
$964$	18.1803	0.585549
$965$	6.24922	0.201170
$966$	0	0
$967$	54.8328	1.76330	0.881652	−	0.471900i	$-0.156432\pi$
$967$	54.8328	1.76330	0.881652	+	0.471900i	$0.156432\pi$
$968$	1.00000	0.0321412
$969$	−40.3607	−1.29657
$970$	0	0
$971$	−39.0557	−1.25336	−0.626679	−	0.779278i	$-0.715586\pi$
$971$	−39.0557	−1.25336	−0.626679	+	0.779278i	$0.715586\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	36.9443	1.18377
$975$	0	0
$976$	0	0
$977$	−11.8885	−0.380348	−0.190174	−	0.981750i	$-0.560905\pi$
$977$	−11.8885	−0.380348	−0.190174	+	0.981750i	$0.560905\pi$
$978$	12.0000	0.383718
$979$	6.47214	0.206850
$980$	0	0
$981$	6.00000	0.191565
$982$	−29.8885	−0.953782
$983$	51.7082	1.64924	0.824618	−	0.565690i	$-0.191390\pi$
$983$	51.7082	1.64924	0.824618	+	0.565690i	$0.191390\pi$
$984$	5.23607	0.166920
$985$	−27.6393	−0.880662
$986$	−23.4164	−0.745730
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	−1.23607	−0.0392848
$991$	−30.8328	−0.979437	−0.489718	−	0.871881i	$-0.662900\pi$
$991$	−30.8328	−0.979437	−0.489718	+	0.871881i	$0.662900\pi$
$992$	−2.76393	−0.0877549
$993$	7.41641	0.235353
$994$	0	0
$995$	−28.5836	−0.906161
$996$	−1.23607	−0.0391663
$997$	54.2492	1.71809	0.859045	−	0.511900i	$-0.171058\pi$
$997$	54.2492	1.71809	0.859045	+	0.511900i	$0.171058\pi$
$998$	7.41641	0.234762
$999$	10.9443	0.346261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.bb.1.2	✓	2
3.2	odd	2		9702.2.a.cw.1.1		2
7.6	odd	2		3234.2.a.be.1.1	yes	2
21.20	even	2		9702.2.a.ck.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.bb.1.2	✓	2	1.1	even	1	trivial
3234.2.a.be.1.1	yes	2	7.6	odd	2
9702.2.a.ck.1.2		2	21.20	even	2
9702.2.a.cw.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.23607 q^{10} -1.00000 q^{11} -1.00000 q^{12} -1.23607 q^{15} +1.00000 q^{16} -5.23607 q^{17} +1.00000 q^{18} -7.70820 q^{19} +1.23607 q^{20} -1.00000 q^{22} -2.47214 q^{23} -1.00000 q^{24} -3.47214 q^{25} -1.00000 q^{27} +4.47214 q^{29} -1.23607 q^{30} -2.76393 q^{31} +1.00000 q^{32} +1.00000 q^{33} -5.23607 q^{34} +1.00000 q^{36} -10.9443 q^{37} -7.70820 q^{38} +1.23607 q^{40} -5.23607 q^{41} +6.47214 q^{43} -1.00000 q^{44} +1.23607 q^{45} -2.47214 q^{46} +3.70820 q^{47} -1.00000 q^{48} -3.47214 q^{50} +5.23607 q^{51} -6.00000 q^{53} -1.00000 q^{54} -1.23607 q^{55} +7.70820 q^{57} +4.47214 q^{58} -1.52786 q^{59} -1.23607 q^{60} -2.76393 q^{62} +1.00000 q^{64} +1.00000 q^{66} +15.4164 q^{67} -5.23607 q^{68} +2.47214 q^{69} -2.47214 q^{71} +1.00000 q^{72} -15.7082 q^{73} -10.9443 q^{74} +3.47214 q^{75} -7.70820 q^{76} +11.4164 q^{79} +1.23607 q^{80} +1.00000 q^{81} -5.23607 q^{82} +1.23607 q^{83} -6.47214 q^{85} +6.47214 q^{86} -4.47214 q^{87} -1.00000 q^{88} -6.47214 q^{89} +1.23607 q^{90} -2.47214 q^{92} +2.76393 q^{93} +3.70820 q^{94} -9.52786 q^{95} -1.00000 q^{96} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 2 q^{12} + 2 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{24}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 2 * q^12 + 2 * q^15 + 2 * q^16 - 6 * q^17 + 2 * q^18 - 2 * q^19 - 2 * q^20 - 2 * q^22 + 4 * q^23 - 2 * q^24 + 2 * q^25 - 2 * q^27 + 2 * q^30 - 10 * q^31 + 2 * q^32 + 2 * q^33 - 6 * q^34 + 2 * q^36 - 4 * q^37 - 2 * q^38 - 2 * q^40 - 6 * q^41 + 4 * q^43 - 2 * q^44 - 2 * q^45 + 4 * q^46 - 6 * q^47 - 2 * q^48 + 2 * q^50 + 6 * q^51 - 12 * q^53 - 2 * q^54 + 2 * q^55 + 2 * q^57 - 12 * q^59 + 2 * q^60 - 10 * q^62 + 2 * q^64 + 2 * q^66 + 4 * q^67 - 6 * q^68 - 4 * q^69 + 4 * q^71 + 2 * q^72 - 18 * q^73 - 4 * q^74 - 2 * q^75 - 2 * q^76 - 4 * q^79 - 2 * q^80 + 2 * q^81 - 6 * q^82 - 2 * q^83 - 4 * q^85 + 4 * q^86 - 2 * q^88 - 4 * q^89 - 2 * q^90 + 4 * q^92 + 10 * q^93 - 6 * q^94 - 28 * q^95 - 2 * q^96 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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