LMFDB - Embedded newform 3234.2.a.ba.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:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
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			$-2.64575$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.64575 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.64575 q^{10} +1.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} -2.64575 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -5.29150 q^{19} -2.64575 q^{20} -1.00000 q^{22} -2.64575 q^{23} -1.00000 q^{24} +2.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{29} +2.64575 q^{30} +4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} +1.29150 q^{37} +5.29150 q^{38} +4.00000 q^{39} +2.64575 q^{40} -9.00000 q^{41} +9.29150 q^{43} +1.00000 q^{44} -2.64575 q^{45} +2.64575 q^{46} +3.93725 q^{47} +1.00000 q^{48} -2.00000 q^{50} +3.00000 q^{51} +4.00000 q^{52} +4.00000 q^{53} -1.00000 q^{54} -2.64575 q^{55} -5.29150 q^{57} -2.00000 q^{58} -6.58301 q^{59} -2.64575 q^{60} -11.9373 q^{61} -4.00000 q^{62} +1.00000 q^{64} -10.5830 q^{65} -1.00000 q^{66} +7.58301 q^{67} +3.00000 q^{68} -2.64575 q^{69} -2.70850 q^{71} -1.00000 q^{72} +15.2915 q^{73} -1.29150 q^{74} +2.00000 q^{75} -5.29150 q^{76} -4.00000 q^{78} +10.6458 q^{79} -2.64575 q^{80} +1.00000 q^{81} +9.00000 q^{82} -15.5830 q^{83} -7.93725 q^{85} -9.29150 q^{86} +2.00000 q^{87} -1.00000 q^{88} -2.70850 q^{89} +2.64575 q^{90} -2.64575 q^{92} +4.00000 q^{93} -3.93725 q^{94} +14.0000 q^{95} -1.00000 q^{96} +11.5830 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} + 2 q^{12} + 8 q^{13} + 2 q^{16} + 6 q^{17} - 2 q^{18} - 2 q^{22} - 2 q^{24} + 4 q^{25} - 8 q^{26} + 2 q^{27} + 4 q^{29} + 8 q^{31}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^11 + 2 * q^12 + 8 * q^13 + 2 * q^16 + 6 * q^17 - 2 * q^18 - 2 * q^22 - 2 * q^24 + 4 * q^25 - 8 * q^26 + 2 * q^27 + 4 * q^29 + 8 * q^31 - 2 * q^32 + 2 * q^33 - 6 * q^34 + 2 * q^36 - 8 * q^37 + 8 * q^39 - 18 * q^41 + 8 * q^43 + 2 * q^44 - 8 * q^47 + 2 * q^48 - 4 * q^50 + 6 * q^51 + 8 * q^52 + 8 * q^53 - 2 * q^54 - 4 * q^58 + 8 * q^59 - 8 * q^61 - 8 * q^62 + 2 * q^64 - 2 * q^66 - 6 * q^67 + 6 * q^68 - 16 * q^71 - 2 * q^72 + 20 * q^73 + 8 * q^74 + 4 * q^75 - 8 * q^78 + 16 * q^79 + 2 * q^81 + 18 * q^82 - 10 * q^83 - 8 * q^86 + 4 * q^87 - 2 * q^88 - 16 * q^89 + 8 * q^93 + 8 * q^94 + 28 * q^95 - 2 * q^96 + 2 * q^97 + 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
$5$	−2.64575	−1.18322	−0.591608	−	0.806226i	$-0.701507\pi$
$5$	−2.64575	−1.18322	−0.591608	+	0.806226i	$0.701507\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.64575	0.836660
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.00000	0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−2.64575	−0.683130
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−1.00000	−0.235702
$19$	−5.29150	−1.21395	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	−5.29150	−1.21395	−0.606977	+	0.794719i	$0.707618\pi$
$20$	−2.64575	−0.591608
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−2.64575	−0.551677	−0.275839	−	0.961204i	$-0.588956\pi$
$23$	−2.64575	−0.551677	−0.275839	+	0.961204i	$0.588956\pi$
$24$	−1.00000	−0.204124
$25$	2.00000	0.400000
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.64575	0.483046
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	1.29150	0.212322	0.106161	−	0.994349i	$-0.466144\pi$
$37$	1.29150	0.212322	0.106161	+	0.994349i	$0.466144\pi$
$38$	5.29150	0.858395
$39$	4.00000	0.640513
$40$	2.64575	0.418330
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	9.29150	1.41694	0.708470	−	0.705740i	$-0.249386\pi$
$43$	9.29150	1.41694	0.708470	+	0.705740i	$0.249386\pi$
$44$	1.00000	0.150756
$45$	−2.64575	−0.394405
$46$	2.64575	0.390095
$47$	3.93725	0.574308	0.287154	−	0.957885i	$-0.407291\pi$
$47$	3.93725	0.574308	0.287154	+	0.957885i	$0.407291\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−2.00000	−0.282843
$51$	3.00000	0.420084
$52$	4.00000	0.554700
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	−1.00000	−0.136083
$55$	−2.64575	−0.356753
$56$	0	0
$57$	−5.29150	−0.700877
$58$	−2.00000	−0.262613
$59$	−6.58301	−0.857034	−0.428517	−	0.903534i	$-0.640964\pi$
$59$	−6.58301	−0.857034	−0.428517	+	0.903534i	$0.640964\pi$
$60$	−2.64575	−0.341565
$61$	−11.9373	−1.52841	−0.764204	−	0.644974i	$-0.776868\pi$
$61$	−11.9373	−1.52841	−0.764204	+	0.644974i	$0.776868\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−10.5830	−1.31266
$66$	−1.00000	−0.123091
$67$	7.58301	0.926412	0.463206	−	0.886251i	$-0.346699\pi$
$67$	7.58301	0.926412	0.463206	+	0.886251i	$0.346699\pi$
$68$	3.00000	0.363803
$69$	−2.64575	−0.318511
$70$	0	0
$71$	−2.70850	−0.321440	−0.160720	−	0.987000i	$-0.551382\pi$
$71$	−2.70850	−0.321440	−0.160720	+	0.987000i	$0.551382\pi$
$72$	−1.00000	−0.117851
$73$	15.2915	1.78974	0.894868	−	0.446332i	$-0.147270\pi$
$73$	15.2915	1.78974	0.894868	+	0.446332i	$0.147270\pi$
$74$	−1.29150	−0.150134
$75$	2.00000	0.230940
$76$	−5.29150	−0.606977
$77$	0	0
$78$	−4.00000	−0.452911
$79$	10.6458	1.19774	0.598870	−	0.800846i	$-0.295616\pi$
$79$	10.6458	1.19774	0.598870	+	0.800846i	$0.295616\pi$
$80$	−2.64575	−0.295804
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	−15.5830	−1.71046	−0.855229	−	0.518251i	$-0.826583\pi$
$83$	−15.5830	−1.71046	−0.855229	+	0.518251i	$0.826583\pi$
$84$	0	0
$85$	−7.93725	−0.860916
$86$	−9.29150	−1.00193
$87$	2.00000	0.214423
$88$	−1.00000	−0.106600
$89$	−2.70850	−0.287100	−0.143550	−	0.989643i	$-0.545852\pi$
$89$	−2.70850	−0.287100	−0.143550	+	0.989643i	$0.545852\pi$
$90$	2.64575	0.278887
$91$	0	0
$92$	−2.64575	−0.275839
$93$	4.00000	0.414781
$94$	−3.93725	−0.406097
$95$	14.0000	1.43637
$96$	−1.00000	−0.102062
$97$	11.5830	1.17608	0.588038	−	0.808833i	$-0.299901\pi$
$97$	11.5830	1.17608	0.588038	+	0.808833i	$0.299901\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	2.00000	0.200000
$101$	−7.29150	−0.725532	−0.362766	−	0.931880i	$-0.618168\pi$
$101$	−7.29150	−0.725532	−0.362766	+	0.931880i	$0.618168\pi$
$102$	−3.00000	−0.297044
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−4.00000	−0.388514
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	1.00000	0.0962250
$109$	−3.93725	−0.377121	−0.188560	−	0.982062i	$-0.560382\pi$
$109$	−3.93725	−0.377121	−0.188560	+	0.982062i	$0.560382\pi$
$110$	2.64575	0.252262
$111$	1.29150	0.122584
$112$	0	0
$113$	12.5830	1.18371	0.591855	−	0.806045i	$-0.298396\pi$
$113$	12.5830	1.18371	0.591855	+	0.806045i	$0.298396\pi$
$114$	5.29150	0.495595
$115$	7.00000	0.652753
$116$	2.00000	0.185695
$117$	4.00000	0.369800
$118$	6.58301	0.606015
$119$	0	0
$120$	2.64575	0.241523
$121$	1.00000	0.0909091
$122$	11.9373	1.08075
$123$	−9.00000	−0.811503
$124$	4.00000	0.359211
$125$	7.93725	0.709930
$126$	0	0
$127$	2.64575	0.234772	0.117386	−	0.993086i	$-0.462548\pi$
$127$	2.64575	0.234772	0.117386	+	0.993086i	$0.462548\pi$
$128$	−1.00000	−0.0883883
$129$	9.29150	0.818071
$130$	10.5830	0.928191
$131$	18.5830	1.62360	0.811802	−	0.583932i	$-0.198487\pi$
$131$	18.5830	1.62360	0.811802	+	0.583932i	$0.198487\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−7.58301	−0.655072
$135$	−2.64575	−0.227710
$136$	−3.00000	−0.257248
$137$	−13.2915	−1.13557	−0.567785	−	0.823177i	$-0.692199\pi$
$137$	−13.2915	−1.13557	−0.567785	+	0.823177i	$0.692199\pi$
$138$	2.64575	0.225221
$139$	12.5830	1.06728	0.533638	−	0.845713i	$-0.320824\pi$
$139$	12.5830	1.06728	0.533638	+	0.845713i	$0.320824\pi$
$140$	0	0
$141$	3.93725	0.331577
$142$	2.70850	0.227292
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	−5.29150	−0.439435
$146$	−15.2915	−1.26553
$147$	0	0
$148$	1.29150	0.106161
$149$	19.8745	1.62818	0.814092	−	0.580737i	$-0.197235\pi$
$149$	19.8745	1.62818	0.814092	+	0.580737i	$0.197235\pi$
$150$	−2.00000	−0.163299
$151$	22.6458	1.84289	0.921443	−	0.388515i	$-0.127012\pi$
$151$	22.6458	1.84289	0.921443	+	0.388515i	$0.127012\pi$
$152$	5.29150	0.429198
$153$	3.00000	0.242536
$154$	0	0
$155$	−10.5830	−0.850047
$156$	4.00000	0.320256
$157$	0.708497	0.0565442	0.0282721	−	0.999600i	$-0.491000\pi$
$157$	0.708497	0.0565442	0.0282721	+	0.999600i	$0.491000\pi$
$158$	−10.6458	−0.846931
$159$	4.00000	0.317221
$160$	2.64575	0.209165
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−5.58301	−0.437295	−0.218647	−	0.975804i	$-0.570164\pi$
$163$	−5.58301	−0.437295	−0.218647	+	0.975804i	$0.570164\pi$
$164$	−9.00000	−0.702782
$165$	−2.64575	−0.205971
$166$	15.5830	1.20948
$167$	−8.70850	−0.673884	−0.336942	−	0.941525i	$-0.609393\pi$
$167$	−8.70850	−0.673884	−0.336942	+	0.941525i	$0.609393\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	7.93725	0.608760
$171$	−5.29150	−0.404651
$172$	9.29150	0.708470
$173$	−4.00000	−0.304114	−0.152057	−	0.988372i	$-0.548590\pi$
$173$	−4.00000	−0.304114	−0.152057	+	0.988372i	$0.548590\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	1.00000	0.0753778
$177$	−6.58301	−0.494809
$178$	2.70850	0.203010
$179$	4.70850	0.351930	0.175965	−	0.984396i	$-0.443696\pi$
$179$	4.70850	0.351930	0.175965	+	0.984396i	$0.443696\pi$
$180$	−2.64575	−0.197203
$181$	5.29150	0.393314	0.196657	−	0.980472i	$-0.436991\pi$
$181$	5.29150	0.393314	0.196657	+	0.980472i	$0.436991\pi$
$182$	0	0
$183$	−11.9373	−0.882427
$184$	2.64575	0.195047
$185$	−3.41699	−0.251222
$186$	−4.00000	−0.293294
$187$	3.00000	0.219382
$188$	3.93725	0.287154
$189$	0	0
$190$	−14.0000	−1.01567
$191$	6.70850	0.485410	0.242705	−	0.970100i	$-0.421965\pi$
$191$	6.70850	0.485410	0.242705	+	0.970100i	$0.421965\pi$
$192$	1.00000	0.0721688
$193$	−15.8745	−1.14267	−0.571336	−	0.820716i	$-0.693575\pi$
$193$	−15.8745	−1.14267	−0.571336	+	0.820716i	$0.693575\pi$
$194$	−11.5830	−0.831611
$195$	−10.5830	−0.757865
$196$	0	0
$197$	21.8745	1.55849	0.779247	−	0.626717i	$-0.215602\pi$
$197$	21.8745	1.55849	0.779247	+	0.626717i	$0.215602\pi$
$198$	−1.00000	−0.0710669
$199$	19.1660	1.35864	0.679321	−	0.733841i	$-0.262274\pi$
$199$	19.1660	1.35864	0.679321	+	0.733841i	$0.262274\pi$
$200$	−2.00000	−0.141421
$201$	7.58301	0.534864
$202$	7.29150	0.513028
$203$	0	0
$204$	3.00000	0.210042
$205$	23.8118	1.66309
$206$	−10.0000	−0.696733
$207$	−2.64575	−0.183892
$208$	4.00000	0.277350
$209$	−5.29150	−0.366021
$210$	0	0
$211$	21.2915	1.46577	0.732884	−	0.680354i	$-0.238174\pi$
$211$	21.2915	1.46577	0.732884	+	0.680354i	$0.238174\pi$
$212$	4.00000	0.274721
$213$	−2.70850	−0.185583
$214$	−9.00000	−0.615227
$215$	−24.5830	−1.67655
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	3.93725	0.266664
$219$	15.2915	1.03330
$220$	−2.64575	−0.178377
$221$	12.0000	0.807207
$222$	−1.29150	−0.0866800
$223$	−28.4575	−1.90566	−0.952828	−	0.303511i	$-0.901841\pi$
$223$	−28.4575	−1.90566	−0.952828	+	0.303511i	$0.901841\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	−12.5830	−0.837009
$227$	−2.41699	−0.160422	−0.0802108	−	0.996778i	$-0.525559\pi$
$227$	−2.41699	−0.160422	−0.0802108	+	0.996778i	$0.525559\pi$
$228$	−5.29150	−0.350438
$229$	−6.70850	−0.443310	−0.221655	−	0.975125i	$-0.571146\pi$
$229$	−6.70850	−0.443310	−0.221655	+	0.975125i	$0.571146\pi$
$230$	−7.00000	−0.461566
$231$	0	0
$232$	−2.00000	−0.131306
$233$	20.1660	1.32112	0.660560	−	0.750774i	$-0.270319\pi$
$233$	20.1660	1.32112	0.660560	+	0.750774i	$0.270319\pi$
$234$	−4.00000	−0.261488
$235$	−10.4170	−0.679530
$236$	−6.58301	−0.428517
$237$	10.6458	0.691516
$238$	0	0
$239$	2.70850	0.175198	0.0875991	−	0.996156i	$-0.472081\pi$
$239$	2.70850	0.175198	0.0875991	+	0.996156i	$0.472081\pi$
$240$	−2.64575	−0.170783
$241$	27.1660	1.74992	0.874958	−	0.484198i	$-0.160889\pi$
$241$	27.1660	1.74992	0.874958	+	0.484198i	$0.160889\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	−11.9373	−0.764204
$245$	0	0
$246$	9.00000	0.573819
$247$	−21.1660	−1.34676
$248$	−4.00000	−0.254000
$249$	−15.5830	−0.987533
$250$	−7.93725	−0.501996
$251$	−23.2915	−1.47015	−0.735073	−	0.677988i	$-0.762852\pi$
$251$	−23.2915	−1.47015	−0.735073	+	0.677988i	$0.762852\pi$
$252$	0	0
$253$	−2.64575	−0.166337
$254$	−2.64575	−0.166009
$255$	−7.93725	−0.497050
$256$	1.00000	0.0625000
$257$	−9.29150	−0.579588	−0.289794	−	0.957089i	$-0.593587\pi$
$257$	−9.29150	−0.579588	−0.289794	+	0.957089i	$0.593587\pi$
$258$	−9.29150	−0.578464
$259$	0	0
$260$	−10.5830	−0.656330
$261$	2.00000	0.123797
$262$	−18.5830	−1.14806
$263$	21.8745	1.34884	0.674420	−	0.738348i	$-0.264394\pi$
$263$	21.8745	1.34884	0.674420	+	0.738348i	$0.264394\pi$
$264$	−1.00000	−0.0615457
$265$	−10.5830	−0.650109
$266$	0	0
$267$	−2.70850	−0.165757
$268$	7.58301	0.463206
$269$	10.6458	0.649083	0.324541	−	0.945871i	$-0.394790\pi$
$269$	10.6458	0.649083	0.324541	+	0.945871i	$0.394790\pi$
$270$	2.64575	0.161015
$271$	−9.29150	−0.564419	−0.282209	−	0.959353i	$-0.591067\pi$
$271$	−9.29150	−0.564419	−0.282209	+	0.959353i	$0.591067\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	13.2915	0.802969
$275$	2.00000	0.120605
$276$	−2.64575	−0.159256
$277$	−17.1660	−1.03141	−0.515703	−	0.856768i	$-0.672469\pi$
$277$	−17.1660	−1.03141	−0.515703	+	0.856768i	$0.672469\pi$
$278$	−12.5830	−0.754679
$279$	4.00000	0.239474
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	−3.93725	−0.234460
$283$	−30.4575	−1.81051	−0.905256	−	0.424867i	$-0.860321\pi$
$283$	−30.4575	−1.81051	−0.905256	+	0.424867i	$0.860321\pi$
$284$	−2.70850	−0.160720
$285$	14.0000	0.829288
$286$	−4.00000	−0.236525
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−8.00000	−0.470588
$290$	5.29150	0.310728
$291$	11.5830	0.679008
$292$	15.2915	0.894868
$293$	14.5830	0.851948	0.425974	−	0.904735i	$-0.359931\pi$
$293$	14.5830	0.851948	0.425974	+	0.904735i	$0.359931\pi$
$294$	0	0
$295$	17.4170	1.01406
$296$	−1.29150	−0.0750671
$297$	1.00000	0.0580259
$298$	−19.8745	−1.15130
$299$	−10.5830	−0.612031
$300$	2.00000	0.115470
$301$	0	0
$302$	−22.6458	−1.30312
$303$	−7.29150	−0.418886
$304$	−5.29150	−0.303488
$305$	31.5830	1.80844
$306$	−3.00000	−0.171499
$307$	11.4170	0.651602	0.325801	−	0.945438i	$-0.394366\pi$
$307$	11.4170	0.651602	0.325801	+	0.945438i	$0.394366\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	10.5830	0.601074
$311$	−22.5203	−1.27701	−0.638503	−	0.769619i	$-0.720446\pi$
$311$	−22.5203	−1.27701	−0.638503	+	0.769619i	$0.720446\pi$
$312$	−4.00000	−0.226455
$313$	20.5830	1.16342	0.581710	−	0.813396i	$-0.302384\pi$
$313$	20.5830	1.16342	0.581710	+	0.813396i	$0.302384\pi$
$314$	−0.708497	−0.0399828
$315$	0	0
$316$	10.6458	0.598870
$317$	9.22876	0.518339	0.259169	−	0.965832i	$-0.416551\pi$
$317$	9.22876	0.518339	0.259169	+	0.965832i	$0.416551\pi$
$318$	−4.00000	−0.224309
$319$	2.00000	0.111979
$320$	−2.64575	−0.147902
$321$	9.00000	0.502331
$322$	0	0
$323$	−15.8745	−0.883281
$324$	1.00000	0.0555556
$325$	8.00000	0.443760
$326$	5.58301	0.309214
$327$	−3.93725	−0.217731
$328$	9.00000	0.496942
$329$	0	0
$330$	2.64575	0.145644
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	−15.5830	−0.855229
$333$	1.29150	0.0707739
$334$	8.70850	0.476508
$335$	−20.0627	−1.09614
$336$	0	0
$337$	6.70850	0.365435	0.182718	−	0.983165i	$-0.441511\pi$
$337$	6.70850	0.365435	0.182718	+	0.983165i	$0.441511\pi$
$338$	−3.00000	−0.163178
$339$	12.5830	0.683415
$340$	−7.93725	−0.430458
$341$	4.00000	0.216612
$342$	5.29150	0.286132
$343$	0	0
$344$	−9.29150	−0.500964
$345$	7.00000	0.376867
$346$	4.00000	0.215041
$347$	29.5830	1.58810	0.794049	−	0.607853i	$-0.207969\pi$
$347$	29.5830	1.58810	0.794049	+	0.607853i	$0.207969\pi$
$348$	2.00000	0.107211
$349$	13.2288	0.708119	0.354060	−	0.935223i	$-0.384801\pi$
$349$	13.2288	0.708119	0.354060	+	0.935223i	$0.384801\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	−1.00000	−0.0533002
$353$	23.2915	1.23968	0.619841	−	0.784728i	$-0.287197\pi$
$353$	23.2915	1.23968	0.619841	+	0.784728i	$0.287197\pi$
$354$	6.58301	0.349883
$355$	7.16601	0.380332
$356$	−2.70850	−0.143550
$357$	0	0
$358$	−4.70850	−0.248852
$359$	20.5830	1.08633	0.543165	−	0.839626i	$-0.317226\pi$
$359$	20.5830	1.08633	0.543165	+	0.839626i	$0.317226\pi$
$360$	2.64575	0.139443
$361$	9.00000	0.473684
$362$	−5.29150	−0.278115
$363$	1.00000	0.0524864
$364$	0	0
$365$	−40.4575	−2.11764
$366$	11.9373	0.623970
$367$	−10.7085	−0.558979	−0.279490	−	0.960149i	$-0.590165\pi$
$367$	−10.7085	−0.558979	−0.279490	+	0.960149i	$0.590165\pi$
$368$	−2.64575	−0.137919
$369$	−9.00000	−0.468521
$370$	3.41699	0.177641
$371$	0	0
$372$	4.00000	0.207390
$373$	−15.8118	−0.818702	−0.409351	−	0.912377i	$-0.634245\pi$
$373$	−15.8118	−0.818702	−0.409351	+	0.912377i	$0.634245\pi$
$374$	−3.00000	−0.155126
$375$	7.93725	0.409878
$376$	−3.93725	−0.203048
$377$	8.00000	0.412021
$378$	0	0
$379$	−27.5830	−1.41684	−0.708422	−	0.705789i	$-0.750593\pi$
$379$	−27.5830	−1.41684	−0.708422	+	0.705789i	$0.750593\pi$
$380$	14.0000	0.718185
$381$	2.64575	0.135546
$382$	−6.70850	−0.343237
$383$	−18.7085	−0.955960	−0.477980	−	0.878371i	$-0.658631\pi$
$383$	−18.7085	−0.955960	−0.477980	+	0.878371i	$0.658631\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	15.8745	0.807991
$387$	9.29150	0.472314
$388$	11.5830	0.588038
$389$	31.9373	1.61928	0.809642	−	0.586925i	$-0.199662\pi$
$389$	31.9373	1.61928	0.809642	+	0.586925i	$0.199662\pi$
$390$	10.5830	0.535891
$391$	−7.93725	−0.401404
$392$	0	0
$393$	18.5830	0.937389
$394$	−21.8745	−1.10202
$395$	−28.1660	−1.41719
$396$	1.00000	0.0502519
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	−19.1660	−0.960705
$399$	0	0
$400$	2.00000	0.100000
$401$	−35.8745	−1.79149	−0.895744	−	0.444571i	$-0.853356\pi$
$401$	−35.8745	−1.79149	−0.895744	+	0.444571i	$0.853356\pi$
$402$	−7.58301	−0.378206
$403$	16.0000	0.797017
$404$	−7.29150	−0.362766
$405$	−2.64575	−0.131468
$406$	0	0
$407$	1.29150	0.0640174
$408$	−3.00000	−0.148522
$409$	−35.8745	−1.77388	−0.886940	−	0.461884i	$-0.847173\pi$
$409$	−35.8745	−1.77388	−0.886940	+	0.461884i	$0.847173\pi$
$410$	−23.8118	−1.17598
$411$	−13.2915	−0.655621
$412$	10.0000	0.492665
$413$	0	0
$414$	2.64575	0.130032
$415$	41.2288	2.02384
$416$	−4.00000	−0.196116
$417$	12.5830	0.616192
$418$	5.29150	0.258816
$419$	9.87451	0.482401	0.241201	−	0.970475i	$-0.422459\pi$
$419$	9.87451	0.482401	0.241201	+	0.970475i	$0.422459\pi$
$420$	0	0
$421$	6.12549	0.298538	0.149269	−	0.988797i	$-0.452308\pi$
$421$	6.12549	0.298538	0.149269	+	0.988797i	$0.452308\pi$
$422$	−21.2915	−1.03645
$423$	3.93725	0.191436
$424$	−4.00000	−0.194257
$425$	6.00000	0.291043
$426$	2.70850	0.131227
$427$	0	0
$428$	9.00000	0.435031
$429$	4.00000	0.193122
$430$	24.5830	1.18550
$431$	−17.2915	−0.832902	−0.416451	−	0.909158i	$-0.636726\pi$
$431$	−17.2915	−0.832902	−0.416451	+	0.909158i	$0.636726\pi$
$432$	1.00000	0.0481125
$433$	−1.58301	−0.0760744	−0.0380372	−	0.999276i	$-0.512111\pi$
$433$	−1.58301	−0.0760744	−0.0380372	+	0.999276i	$0.512111\pi$
$434$	0	0
$435$	−5.29150	−0.253708
$436$	−3.93725	−0.188560
$437$	14.0000	0.669711
$438$	−15.2915	−0.730656
$439$	9.35425	0.446454	0.223227	−	0.974766i	$-0.428341\pi$
$439$	9.35425	0.446454	0.223227	+	0.974766i	$0.428341\pi$
$440$	2.64575	0.126131
$441$	0	0
$442$	−12.0000	−0.570782
$443$	−31.1660	−1.48074	−0.740371	−	0.672199i	$-0.765350\pi$
$443$	−31.1660	−1.48074	−0.740371	+	0.672199i	$0.765350\pi$
$444$	1.29150	0.0612920
$445$	7.16601	0.339701
$446$	28.4575	1.34750
$447$	19.8745	0.940032
$448$	0	0
$449$	30.4575	1.43738	0.718689	−	0.695331i	$-0.244742\pi$
$449$	30.4575	1.43738	0.718689	+	0.695331i	$0.244742\pi$
$450$	−2.00000	−0.0942809
$451$	−9.00000	−0.423793
$452$	12.5830	0.591855
$453$	22.6458	1.06399
$454$	2.41699	0.113435
$455$	0	0
$456$	5.29150	0.247797
$457$	−40.5830	−1.89839	−0.949196	−	0.314684i	$-0.898101\pi$
$457$	−40.5830	−1.89839	−0.949196	+	0.314684i	$0.898101\pi$
$458$	6.70850	0.313467
$459$	3.00000	0.140028
$460$	7.00000	0.326377
$461$	24.5830	1.14494	0.572472	−	0.819924i	$-0.305984\pi$
$461$	24.5830	1.14494	0.572472	+	0.819924i	$0.305984\pi$
$462$	0	0
$463$	39.0405	1.81437	0.907183	−	0.420735i	$-0.138228\pi$
$463$	39.0405	1.81437	0.907183	+	0.420735i	$0.138228\pi$
$464$	2.00000	0.0928477
$465$	−10.5830	−0.490775
$466$	−20.1660	−0.934172
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	10.4170	0.480500
$471$	0.708497	0.0326458
$472$	6.58301	0.303007
$473$	9.29150	0.427224
$474$	−10.6458	−0.488976
$475$	−10.5830	−0.485582
$476$	0	0
$477$	4.00000	0.183147
$478$	−2.70850	−0.123884
$479$	33.8745	1.54777	0.773883	−	0.633329i	$-0.218312\pi$
$479$	33.8745	1.54777	0.773883	+	0.633329i	$0.218312\pi$
$480$	2.64575	0.120761
$481$	5.16601	0.235550
$482$	−27.1660	−1.23738
$483$	0	0
$484$	1.00000	0.0454545
$485$	−30.6458	−1.39155
$486$	−1.00000	−0.0453609
$487$	21.8745	0.991229	0.495614	−	0.868543i	$-0.334943\pi$
$487$	21.8745	0.991229	0.495614	+	0.868543i	$0.334943\pi$
$488$	11.9373	0.540374
$489$	−5.58301	−0.252472
$490$	0	0
$491$	−1.00000	−0.0451294	−0.0225647	−	0.999745i	$-0.507183\pi$
$491$	−1.00000	−0.0451294	−0.0225647	+	0.999745i	$0.507183\pi$
$492$	−9.00000	−0.405751
$493$	6.00000	0.270226
$494$	21.1660	0.952304
$495$	−2.64575	−0.118918
$496$	4.00000	0.179605
$497$	0	0
$498$	15.5830	0.698291
$499$	−18.5830	−0.831890	−0.415945	−	0.909390i	$-0.636549\pi$
$499$	−18.5830	−0.831890	−0.415945	+	0.909390i	$0.636549\pi$
$500$	7.93725	0.354965
$501$	−8.70850	−0.389067
$502$	23.2915	1.03955
$503$	1.29150	0.0575853	0.0287926	−	0.999585i	$-0.490834\pi$
$503$	1.29150	0.0575853	0.0287926	+	0.999585i	$0.490834\pi$
$504$	0	0
$505$	19.2915	0.858461
$506$	2.64575	0.117618
$507$	3.00000	0.133235
$508$	2.64575	0.117386
$509$	−25.1660	−1.11546	−0.557732	−	0.830021i	$-0.688328\pi$
$509$	−25.1660	−1.11546	−0.557732	+	0.830021i	$0.688328\pi$
$510$	7.93725	0.351468
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−5.29150	−0.233626
$514$	9.29150	0.409831
$515$	−26.4575	−1.16586
$516$	9.29150	0.409036
$517$	3.93725	0.173160
$518$	0	0
$519$	−4.00000	−0.175581
$520$	10.5830	0.464095
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	−2.00000	−0.0875376
$523$	−2.70850	−0.118434	−0.0592172	−	0.998245i	$-0.518860\pi$
$523$	−2.70850	−0.118434	−0.0592172	+	0.998245i	$0.518860\pi$
$524$	18.5830	0.811802
$525$	0	0
$526$	−21.8745	−0.953774
$527$	12.0000	0.522728
$528$	1.00000	0.0435194
$529$	−16.0000	−0.695652
$530$	10.5830	0.459696
$531$	−6.58301	−0.285678
$532$	0	0
$533$	−36.0000	−1.55933
$534$	2.70850	0.117208
$535$	−23.8118	−1.02947
$536$	−7.58301	−0.327536
$537$	4.70850	0.203187
$538$	−10.6458	−0.458971
$539$	0	0
$540$	−2.64575	−0.113855
$541$	3.81176	0.163880	0.0819402	−	0.996637i	$-0.473888\pi$
$541$	3.81176	0.163880	0.0819402	+	0.996637i	$0.473888\pi$
$542$	9.29150	0.399104
$543$	5.29150	0.227080
$544$	−3.00000	−0.128624
$545$	10.4170	0.446215
$546$	0	0
$547$	14.7085	0.628890	0.314445	−	0.949276i	$-0.398182\pi$
$547$	14.7085	0.628890	0.314445	+	0.949276i	$0.398182\pi$
$548$	−13.2915	−0.567785
$549$	−11.9373	−0.509470
$550$	−2.00000	−0.0852803
$551$	−10.5830	−0.450851
$552$	2.64575	0.112611
$553$	0	0
$554$	17.1660	0.729314
$555$	−3.41699	−0.145043
$556$	12.5830	0.533638
$557$	−19.2915	−0.817407	−0.408704	−	0.912667i	$-0.634019\pi$
$557$	−19.2915	−0.817407	−0.408704	+	0.912667i	$0.634019\pi$
$558$	−4.00000	−0.169334
$559$	37.1660	1.57195
$560$	0	0
$561$	3.00000	0.126660
$562$	3.00000	0.126547
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	3.93725	0.165788
$565$	−33.2915	−1.40058
$566$	30.4575	1.28022
$567$	0	0
$568$	2.70850	0.113646
$569$	−15.1660	−0.635792	−0.317896	−	0.948126i	$-0.602976\pi$
$569$	−15.1660	−0.635792	−0.317896	+	0.948126i	$0.602976\pi$
$570$	−14.0000	−0.586395
$571$	−29.7490	−1.24496	−0.622479	−	0.782637i	$-0.713874\pi$
$571$	−29.7490	−1.24496	−0.622479	+	0.782637i	$0.713874\pi$
$572$	4.00000	0.167248
$573$	6.70850	0.280251
$574$	0	0
$575$	−5.29150	−0.220671
$576$	1.00000	0.0416667
$577$	−22.7490	−0.947054	−0.473527	−	0.880779i	$-0.657019\pi$
$577$	−22.7490	−0.947054	−0.473527	+	0.880779i	$0.657019\pi$
$578$	8.00000	0.332756
$579$	−15.8745	−0.659722
$580$	−5.29150	−0.219718
$581$	0	0
$582$	−11.5830	−0.480131
$583$	4.00000	0.165663
$584$	−15.2915	−0.632767
$585$	−10.5830	−0.437553
$586$	−14.5830	−0.602418
$587$	−9.87451	−0.407565	−0.203782	−	0.979016i	$-0.565323\pi$
$587$	−9.87451	−0.407565	−0.203782	+	0.979016i	$0.565323\pi$
$588$	0	0
$589$	−21.1660	−0.872130
$590$	−17.4170	−0.717046
$591$	21.8745	0.899797
$592$	1.29150	0.0530804
$593$	25.7490	1.05739	0.528693	−	0.848813i	$-0.322682\pi$
$593$	25.7490	1.05739	0.528693	+	0.848813i	$0.322682\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	19.8745	0.814092
$597$	19.1660	0.784413
$598$	10.5830	0.432771
$599$	29.3542	1.19938	0.599691	−	0.800232i	$-0.295290\pi$
$599$	29.3542	1.19938	0.599691	+	0.800232i	$0.295290\pi$
$600$	−2.00000	−0.0816497
$601$	38.5830	1.57383	0.786917	−	0.617059i	$-0.211676\pi$
$601$	38.5830	1.57383	0.786917	+	0.617059i	$0.211676\pi$
$602$	0	0
$603$	7.58301	0.308804
$604$	22.6458	0.921443
$605$	−2.64575	−0.107565
$606$	7.29150	0.296197
$607$	13.1033	0.531845	0.265923	−	0.963994i	$-0.414323\pi$
$607$	13.1033	0.531845	0.265923	+	0.963994i	$0.414323\pi$
$608$	5.29150	0.214599
$609$	0	0
$610$	−31.5830	−1.27876
$611$	15.7490	0.637137
$612$	3.00000	0.121268
$613$	7.81176	0.315514	0.157757	−	0.987478i	$-0.449574\pi$
$613$	7.81176	0.315514	0.157757	+	0.987478i	$0.449574\pi$
$614$	−11.4170	−0.460752
$615$	23.8118	0.960183
$616$	0	0
$617$	−14.7085	−0.592142	−0.296071	−	0.955166i	$-0.595676\pi$
$617$	−14.7085	−0.592142	−0.296071	+	0.955166i	$0.595676\pi$
$618$	−10.0000	−0.402259
$619$	41.0000	1.64793	0.823965	−	0.566641i	$-0.191757\pi$
$619$	41.0000	1.64793	0.823965	+	0.566641i	$0.191757\pi$
$620$	−10.5830	−0.425024
$621$	−2.64575	−0.106170
$622$	22.5203	0.902980
$623$	0	0
$624$	4.00000	0.160128
$625$	−31.0000	−1.24000
$626$	−20.5830	−0.822662
$627$	−5.29150	−0.211322
$628$	0.708497	0.0282721
$629$	3.87451	0.154487
$630$	0	0
$631$	41.8745	1.66700	0.833499	−	0.552521i	$-0.186334\pi$
$631$	41.8745	1.66700	0.833499	+	0.552521i	$0.186334\pi$
$632$	−10.6458	−0.423465
$633$	21.2915	0.846261
$634$	−9.22876	−0.366521
$635$	−7.00000	−0.277787
$636$	4.00000	0.158610
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	−2.70850	−0.107147
$640$	2.64575	0.104583
$641$	−26.5830	−1.04997	−0.524983	−	0.851113i	$-0.675928\pi$
$641$	−26.5830	−1.04997	−0.524983	+	0.851113i	$0.675928\pi$
$642$	−9.00000	−0.355202
$643$	1.16601	0.0459830	0.0229915	−	0.999736i	$-0.492681\pi$
$643$	1.16601	0.0459830	0.0229915	+	0.999736i	$0.492681\pi$
$644$	0	0
$645$	−24.5830	−0.967955
$646$	15.8745	0.624574
$647$	26.3948	1.03769	0.518843	−	0.854870i	$-0.326363\pi$
$647$	26.3948	1.03769	0.518843	+	0.854870i	$0.326363\pi$
$648$	−1.00000	−0.0392837
$649$	−6.58301	−0.258406
$650$	−8.00000	−0.313786
$651$	0	0
$652$	−5.58301	−0.218647
$653$	−45.1033	−1.76503	−0.882514	−	0.470287i	$-0.844150\pi$
$653$	−45.1033	−1.76503	−0.882514	+	0.470287i	$0.844150\pi$
$654$	3.93725	0.153959
$655$	−49.1660	−1.92107
$656$	−9.00000	−0.351391
$657$	15.2915	0.596578
$658$	0	0
$659$	−24.1660	−0.941374	−0.470687	−	0.882300i	$-0.655994\pi$
$659$	−24.1660	−0.941374	−0.470687	+	0.882300i	$0.655994\pi$
$660$	−2.64575	−0.102986
$661$	−20.7085	−0.805467	−0.402734	−	0.915317i	$-0.631940\pi$
$661$	−20.7085	−0.805467	−0.402734	+	0.915317i	$0.631940\pi$
$662$	−13.0000	−0.505259
$663$	12.0000	0.466041
$664$	15.5830	0.604738
$665$	0	0
$666$	−1.29150	−0.0500447
$667$	−5.29150	−0.204888
$668$	−8.70850	−0.336942
$669$	−28.4575	−1.10023
$670$	20.0627	0.775092
$671$	−11.9373	−0.460833
$672$	0	0
$673$	−19.4170	−0.748470	−0.374235	−	0.927334i	$-0.622095\pi$
$673$	−19.4170	−0.748470	−0.374235	+	0.927334i	$0.622095\pi$
$674$	−6.70850	−0.258402
$675$	2.00000	0.0769800
$676$	3.00000	0.115385
$677$	8.12549	0.312288	0.156144	−	0.987734i	$-0.450094\pi$
$677$	8.12549	0.312288	0.156144	+	0.987734i	$0.450094\pi$
$678$	−12.5830	−0.483247
$679$	0	0
$680$	7.93725	0.304380
$681$	−2.41699	−0.0926194
$682$	−4.00000	−0.153168
$683$	−17.1660	−0.656839	−0.328420	−	0.944532i	$-0.606516\pi$
$683$	−17.1660	−0.656839	−0.328420	+	0.944532i	$0.606516\pi$
$684$	−5.29150	−0.202326
$685$	35.1660	1.34362
$686$	0	0
$687$	−6.70850	−0.255945
$688$	9.29150	0.354235
$689$	16.0000	0.609551
$690$	−7.00000	−0.266485
$691$	−14.1660	−0.538900	−0.269450	−	0.963014i	$-0.586842\pi$
$691$	−14.1660	−0.538900	−0.269450	+	0.963014i	$0.586842\pi$
$692$	−4.00000	−0.152057
$693$	0	0
$694$	−29.5830	−1.12296
$695$	−33.2915	−1.26282
$696$	−2.00000	−0.0758098
$697$	−27.0000	−1.02270
$698$	−13.2288	−0.500716
$699$	20.1660	0.762749
$700$	0	0
$701$	42.4575	1.60360	0.801799	−	0.597594i	$-0.203876\pi$
$701$	42.4575	1.60360	0.801799	+	0.597594i	$0.203876\pi$
$702$	−4.00000	−0.150970
$703$	−6.83399	−0.257749
$704$	1.00000	0.0376889
$705$	−10.4170	−0.392327
$706$	−23.2915	−0.876587
$707$	0	0
$708$	−6.58301	−0.247404
$709$	15.7490	0.591467	0.295733	−	0.955271i	$-0.404436\pi$
$709$	15.7490	0.591467	0.295733	+	0.955271i	$0.404436\pi$
$710$	−7.16601	−0.268936
$711$	10.6458	0.399247
$712$	2.70850	0.101505
$713$	−10.5830	−0.396337
$714$	0	0
$715$	−10.5830	−0.395782
$716$	4.70850	0.175965
$717$	2.70850	0.101151
$718$	−20.5830	−0.768151
$719$	28.0627	1.04656	0.523282	−	0.852160i	$-0.324707\pi$
$719$	28.0627	1.04656	0.523282	+	0.852160i	$0.324707\pi$
$720$	−2.64575	−0.0986013
$721$	0	0
$722$	−9.00000	−0.334945
$723$	27.1660	1.01031
$724$	5.29150	0.196657
$725$	4.00000	0.148556
$726$	−1.00000	−0.0371135
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	40.4575	1.49740
$731$	27.8745	1.03098
$732$	−11.9373	−0.441214
$733$	−11.9373	−0.440913	−0.220456	−	0.975397i	$-0.570755\pi$
$733$	−11.9373	−0.440913	−0.220456	+	0.975397i	$0.570755\pi$
$734$	10.7085	0.395258
$735$	0	0
$736$	2.64575	0.0975237
$737$	7.58301	0.279324
$738$	9.00000	0.331295
$739$	−24.5830	−0.904300	−0.452150	−	0.891942i	$-0.649343\pi$
$739$	−24.5830	−0.904300	−0.452150	+	0.891942i	$0.649343\pi$
$740$	−3.41699	−0.125611
$741$	−21.1660	−0.777553
$742$	0	0
$743$	−18.7085	−0.686348	−0.343174	−	0.939272i	$-0.611502\pi$
$743$	−18.7085	−0.686348	−0.343174	+	0.939272i	$0.611502\pi$
$744$	−4.00000	−0.146647
$745$	−52.5830	−1.92649
$746$	15.8118	0.578910
$747$	−15.5830	−0.570152
$748$	3.00000	0.109691
$749$	0	0
$750$	−7.93725	−0.289828
$751$	−29.2915	−1.06886	−0.534431	−	0.845212i	$-0.679474\pi$
$751$	−29.2915	−1.06886	−0.534431	+	0.845212i	$0.679474\pi$
$752$	3.93725	0.143577
$753$	−23.2915	−0.848790
$754$	−8.00000	−0.291343
$755$	−59.9150	−2.18053
$756$	0	0
$757$	−32.5830	−1.18425	−0.592125	−	0.805846i	$-0.701711\pi$
$757$	−32.5830	−1.18425	−0.592125	+	0.805846i	$0.701711\pi$
$758$	27.5830	1.00186
$759$	−2.64575	−0.0960347
$760$	−14.0000	−0.507833
$761$	−47.5830	−1.72488	−0.862441	−	0.506157i	$-0.831066\pi$
$761$	−47.5830	−1.72488	−0.862441	+	0.506157i	$0.831066\pi$
$762$	−2.64575	−0.0958455
$763$	0	0
$764$	6.70850	0.242705
$765$	−7.93725	−0.286972
$766$	18.7085	0.675965
$767$	−26.3320	−0.950794
$768$	1.00000	0.0360844
$769$	30.7085	1.10738	0.553688	−	0.832724i	$-0.313220\pi$
$769$	30.7085	1.10738	0.553688	+	0.832724i	$0.313220\pi$
$770$	0	0
$771$	−9.29150	−0.334625
$772$	−15.8745	−0.571336
$773$	8.06275	0.289997	0.144998	−	0.989432i	$-0.453682\pi$
$773$	8.06275	0.289997	0.144998	+	0.989432i	$0.453682\pi$
$774$	−9.29150	−0.333976
$775$	8.00000	0.287368
$776$	−11.5830	−0.415806
$777$	0	0
$778$	−31.9373	−1.14501
$779$	47.6235	1.70629
$780$	−10.5830	−0.378932
$781$	−2.70850	−0.0969177
$782$	7.93725	0.283836
$783$	2.00000	0.0714742
$784$	0	0
$785$	−1.87451	−0.0669041
$786$	−18.5830	−0.662834
$787$	−9.29150	−0.331206	−0.165603	−	0.986192i	$-0.552957\pi$
$787$	−9.29150	−0.331206	−0.165603	+	0.986192i	$0.552957\pi$
$788$	21.8745	0.779247
$789$	21.8745	0.778753
$790$	28.1660	1.00210
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	−47.7490	−1.69562
$794$	6.00000	0.212932
$795$	−10.5830	−0.375341
$796$	19.1660	0.679321
$797$	−45.1033	−1.59764	−0.798820	−	0.601570i	$-0.794542\pi$
$797$	−45.1033	−1.59764	−0.798820	+	0.601570i	$0.794542\pi$
$798$	0	0
$799$	11.8118	0.417870
$800$	−2.00000	−0.0707107
$801$	−2.70850	−0.0957000
$802$	35.8745	1.26677
$803$	15.2915	0.539625
$804$	7.58301	0.267432
$805$	0	0
$806$	−16.0000	−0.563576
$807$	10.6458	0.374748
$808$	7.29150	0.256514
$809$	45.3320	1.59379	0.796894	−	0.604119i	$-0.206475\pi$
$809$	45.3320	1.59379	0.796894	+	0.604119i	$0.206475\pi$
$810$	2.64575	0.0929622
$811$	−16.5830	−0.582308	−0.291154	−	0.956676i	$-0.594039\pi$
$811$	−16.5830	−0.582308	−0.291154	+	0.956676i	$0.594039\pi$
$812$	0	0
$813$	−9.29150	−0.325867
$814$	−1.29150	−0.0452671
$815$	14.7712	0.517414
$816$	3.00000	0.105021
$817$	−49.1660	−1.72010
$818$	35.8745	1.25432
$819$	0	0
$820$	23.8118	0.831543
$821$	−30.3320	−1.05859	−0.529297	−	0.848436i	$-0.677544\pi$
$821$	−30.3320	−1.05859	−0.529297	+	0.848436i	$0.677544\pi$
$822$	13.2915	0.463594
$823$	−12.1255	−0.422668	−0.211334	−	0.977414i	$-0.567781\pi$
$823$	−12.1255	−0.422668	−0.211334	+	0.977414i	$0.567781\pi$
$824$	−10.0000	−0.348367
$825$	2.00000	0.0696311
$826$	0	0
$827$	35.3320	1.22861	0.614307	−	0.789067i	$-0.289435\pi$
$827$	35.3320	1.22861	0.614307	+	0.789067i	$0.289435\pi$
$828$	−2.64575	−0.0919462
$829$	−19.8745	−0.690270	−0.345135	−	0.938553i	$-0.612167\pi$
$829$	−19.8745	−0.690270	−0.345135	+	0.938553i	$0.612167\pi$
$830$	−41.2288	−1.43107
$831$	−17.1660	−0.595482
$832$	4.00000	0.138675
$833$	0	0
$834$	−12.5830	−0.435714
$835$	23.0405	0.797350
$836$	−5.29150	−0.183010
$837$	4.00000	0.138260
$838$	−9.87451	−0.341109
$839$	37.1033	1.28095	0.640473	−	0.767980i	$-0.278738\pi$
$839$	37.1033	1.28095	0.640473	+	0.767980i	$0.278738\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−6.12549	−0.211098
$843$	−3.00000	−0.103325
$844$	21.2915	0.732884
$845$	−7.93725	−0.273050
$846$	−3.93725	−0.135366
$847$	0	0
$848$	4.00000	0.137361
$849$	−30.4575	−1.04530
$850$	−6.00000	−0.205798
$851$	−3.41699	−0.117133
$852$	−2.70850	−0.0927916
$853$	−14.5203	−0.497164	−0.248582	−	0.968611i	$-0.579965\pi$
$853$	−14.5203	−0.497164	−0.248582	+	0.968611i	$0.579965\pi$
$854$	0	0
$855$	14.0000	0.478790
$856$	−9.00000	−0.307614
$857$	−9.83399	−0.335923	−0.167961	−	0.985794i	$-0.553718\pi$
$857$	−9.83399	−0.335923	−0.167961	+	0.985794i	$0.553718\pi$
$858$	−4.00000	−0.136558
$859$	−48.7490	−1.66329	−0.831647	−	0.555304i	$-0.812602\pi$
$859$	−48.7490	−1.66329	−0.831647	+	0.555304i	$0.812602\pi$
$860$	−24.5830	−0.838274
$861$	0	0
$862$	17.2915	0.588951
$863$	−33.1033	−1.12685	−0.563424	−	0.826168i	$-0.690516\pi$
$863$	−33.1033	−1.12685	−0.563424	+	0.826168i	$0.690516\pi$
$864$	−1.00000	−0.0340207
$865$	10.5830	0.359833
$866$	1.58301	0.0537927
$867$	−8.00000	−0.271694
$868$	0	0
$869$	10.6458	0.361132
$870$	5.29150	0.179399
$871$	30.3320	1.02776
$872$	3.93725	0.133332
$873$	11.5830	0.392025
$874$	−14.0000	−0.473557
$875$	0	0
$876$	15.2915	0.516652
$877$	10.7712	0.363719	0.181860	−	0.983325i	$-0.441788\pi$
$877$	10.7712	0.363719	0.181860	+	0.983325i	$0.441788\pi$
$878$	−9.35425	−0.315691
$879$	14.5830	0.491873
$880$	−2.64575	−0.0891883
$881$	−23.1660	−0.780483	−0.390241	−	0.920713i	$-0.627608\pi$
$881$	−23.1660	−0.780483	−0.390241	+	0.920713i	$0.627608\pi$
$882$	0	0
$883$	−34.4170	−1.15822	−0.579112	−	0.815248i	$-0.696601\pi$
$883$	−34.4170	−1.15822	−0.579112	+	0.815248i	$0.696601\pi$
$884$	12.0000	0.403604
$885$	17.4170	0.585466
$886$	31.1660	1.04704
$887$	31.0405	1.04224	0.521119	−	0.853484i	$-0.325515\pi$
$887$	31.0405	1.04224	0.521119	+	0.853484i	$0.325515\pi$
$888$	−1.29150	−0.0433400
$889$	0	0
$890$	−7.16601	−0.240205
$891$	1.00000	0.0335013
$892$	−28.4575	−0.952828
$893$	−20.8340	−0.697183
$894$	−19.8745	−0.664703
$895$	−12.4575	−0.416409
$896$	0	0
$897$	−10.5830	−0.353356
$898$	−30.4575	−1.01638
$899$	8.00000	0.266815
$900$	2.00000	0.0666667
$901$	12.0000	0.399778
$902$	9.00000	0.299667
$903$	0	0
$904$	−12.5830	−0.418505
$905$	−14.0000	−0.465376
$906$	−22.6458	−0.752355
$907$	−15.8340	−0.525759	−0.262879	−	0.964829i	$-0.584672\pi$
$907$	−15.8340	−0.525759	−0.262879	+	0.964829i	$0.584672\pi$
$908$	−2.41699	−0.0802108
$909$	−7.29150	−0.241844
$910$	0	0
$911$	−18.3948	−0.609446	−0.304723	−	0.952441i	$-0.598564\pi$
$911$	−18.3948	−0.609446	−0.304723	+	0.952441i	$0.598564\pi$
$912$	−5.29150	−0.175219
$913$	−15.5830	−0.515722
$914$	40.5830	1.34237
$915$	31.5830	1.04410
$916$	−6.70850	−0.221655
$917$	0	0
$918$	−3.00000	−0.0990148
$919$	9.35425	0.308568	0.154284	−	0.988027i	$-0.450693\pi$
$919$	9.35425	0.308568	0.154284	+	0.988027i	$0.450693\pi$
$920$	−7.00000	−0.230783
$921$	11.4170	0.376203
$922$	−24.5830	−0.809598
$923$	−10.8340	−0.356605
$924$	0	0
$925$	2.58301	0.0849287
$926$	−39.0405	−1.28295
$927$	10.0000	0.328443
$928$	−2.00000	−0.0656532
$929$	34.4575	1.13051	0.565257	−	0.824915i	$-0.308777\pi$
$929$	34.4575	1.13051	0.565257	+	0.824915i	$0.308777\pi$
$930$	10.5830	0.347030
$931$	0	0
$932$	20.1660	0.660560
$933$	−22.5203	−0.737280
$934$	8.00000	0.261768
$935$	−7.93725	−0.259576
$936$	−4.00000	−0.130744
$937$	−50.0000	−1.63343	−0.816714	−	0.577042i	$-0.804207\pi$
$937$	−50.0000	−1.63343	−0.816714	+	0.577042i	$0.804207\pi$
$938$	0	0
$939$	20.5830	0.671701
$940$	−10.4170	−0.339765
$941$	−3.41699	−0.111391	−0.0556954	−	0.998448i	$-0.517738\pi$
$941$	−3.41699	−0.111391	−0.0556954	+	0.998448i	$0.517738\pi$
$942$	−0.708497	−0.0230841
$943$	23.8118	0.775418
$944$	−6.58301	−0.214259
$945$	0	0
$946$	−9.29150	−0.302093
$947$	−15.4170	−0.500985	−0.250493	−	0.968119i	$-0.580593\pi$
$947$	−15.4170	−0.500985	−0.250493	+	0.968119i	$0.580593\pi$
$948$	10.6458	0.345758
$949$	61.1660	1.98553
$950$	10.5830	0.343358
$951$	9.22876	0.299263
$952$	0	0
$953$	26.7490	0.866486	0.433243	−	0.901277i	$-0.357369\pi$
$953$	26.7490	0.866486	0.433243	+	0.901277i	$0.357369\pi$
$954$	−4.00000	−0.129505
$955$	−17.7490	−0.574345
$956$	2.70850	0.0875991
$957$	2.00000	0.0646508
$958$	−33.8745	−1.09444
$959$	0	0
$960$	−2.64575	−0.0853913
$961$	−15.0000	−0.483871
$962$	−5.16601	−0.166559
$963$	9.00000	0.290021
$964$	27.1660	0.874958
$965$	42.0000	1.35203
$966$	0	0
$967$	40.0627	1.28833	0.644166	−	0.764886i	$-0.277205\pi$
$967$	40.0627	1.28833	0.644166	+	0.764886i	$0.277205\pi$
$968$	−1.00000	−0.0321412
$969$	−15.8745	−0.509963
$970$	30.6458	0.983976
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−21.8745	−0.700904
$975$	8.00000	0.256205
$976$	−11.9373	−0.382102
$977$	13.1660	0.421218	0.210609	−	0.977570i	$-0.432455\pi$
$977$	13.1660	0.421218	0.210609	+	0.977570i	$0.432455\pi$
$978$	5.58301	0.178525
$979$	−2.70850	−0.0865640
$980$	0	0
$981$	−3.93725	−0.125707
$982$	1.00000	0.0319113
$983$	−42.5203	−1.35619	−0.678093	−	0.734976i	$-0.737193\pi$
$983$	−42.5203	−1.35619	−0.678093	+	0.734976i	$0.737193\pi$
$984$	9.00000	0.286910
$985$	−57.8745	−1.84404
$986$	−6.00000	−0.191079
$987$	0	0
$988$	−21.1660	−0.673380
$989$	−24.5830	−0.781694
$990$	2.64575	0.0840875
$991$	−43.8745	−1.39372	−0.696860	−	0.717207i	$-0.745420\pi$
$991$	−43.8745	−1.39372	−0.696860	+	0.717207i	$0.745420\pi$
$992$	−4.00000	−0.127000
$993$	13.0000	0.412543
$994$	0	0
$995$	−50.7085	−1.60757
$996$	−15.5830	−0.493766
$997$	−6.83399	−0.216435	−0.108217	−	0.994127i	$-0.534514\pi$
$997$	−6.83399	−0.216435	−0.108217	+	0.994127i	$0.534514\pi$
$998$	18.5830	0.588235
$999$	1.29150	0.0408613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.ba.1.1		2
3.2	odd	2		9702.2.a.dm.1.2		2
7.3	odd	6		462.2.i.e.331.1	yes	4
7.5	odd	6		462.2.i.e.67.1	✓	4
7.6	odd	2		3234.2.a.w.1.2		2
21.5	even	6		1386.2.k.r.991.2		4
21.17	even	6		1386.2.k.r.793.2		4
21.20	even	2		9702.2.a.db.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.e.67.1	✓	4	7.5	odd	6
462.2.i.e.331.1	yes	4	7.3	odd	6
1386.2.k.r.793.2		4	21.17	even	6
1386.2.k.r.991.2		4	21.5	even	6
3234.2.a.w.1.2		2	7.6	odd	2
3234.2.a.ba.1.1		2	1.1	even	1	trivial
9702.2.a.db.1.1		2	21.20	even	2
9702.2.a.dm.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.64575 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.64575 q^{10} +1.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} -2.64575 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -5.29150 q^{19} -2.64575 q^{20} -1.00000 q^{22} -2.64575 q^{23} -1.00000 q^{24} +2.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{29} +2.64575 q^{30} +4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} +1.29150 q^{37} +5.29150 q^{38} +4.00000 q^{39} +2.64575 q^{40} -9.00000 q^{41} +9.29150 q^{43} +1.00000 q^{44} -2.64575 q^{45} +2.64575 q^{46} +3.93725 q^{47} +1.00000 q^{48} -2.00000 q^{50} +3.00000 q^{51} +4.00000 q^{52} +4.00000 q^{53} -1.00000 q^{54} -2.64575 q^{55} -5.29150 q^{57} -2.00000 q^{58} -6.58301 q^{59} -2.64575 q^{60} -11.9373 q^{61} -4.00000 q^{62} +1.00000 q^{64} -10.5830 q^{65} -1.00000 q^{66} +7.58301 q^{67} +3.00000 q^{68} -2.64575 q^{69} -2.70850 q^{71} -1.00000 q^{72} +15.2915 q^{73} -1.29150 q^{74} +2.00000 q^{75} -5.29150 q^{76} -4.00000 q^{78} +10.6458 q^{79} -2.64575 q^{80} +1.00000 q^{81} +9.00000 q^{82} -15.5830 q^{83} -7.93725 q^{85} -9.29150 q^{86} +2.00000 q^{87} -1.00000 q^{88} -2.70850 q^{89} +2.64575 q^{90} -2.64575 q^{92} +4.00000 q^{93} -3.93725 q^{94} +14.0000 q^{95} -1.00000 q^{96} +11.5830 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} + 2 q^{12} + 8 q^{13} + 2 q^{16} + 6 q^{17} - 2 q^{18} - 2 q^{22} - 2 q^{24} + 4 q^{25} - 8 q^{26} + 2 q^{27} + 4 q^{29} + 8 q^{31}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^11 + 2 * q^12 + 8 * q^13 + 2 * q^16 + 6 * q^17 - 2 * q^18 - 2 * q^22 - 2 * q^24 + 4 * q^25 - 8 * q^26 + 2 * q^27 + 4 * q^29 + 8 * q^31 - 2 * q^32 + 2 * q^33 - 6 * q^34 + 2 * q^36 - 8 * q^37 + 8 * q^39 - 18 * q^41 + 8 * q^43 + 2 * q^44 - 8 * q^47 + 2 * q^48 - 4 * q^50 + 6 * q^51 + 8 * q^52 + 8 * q^53 - 2 * q^54 - 4 * q^58 + 8 * q^59 - 8 * q^61 - 8 * q^62 + 2 * q^64 - 2 * q^66 - 6 * q^67 + 6 * q^68 - 16 * q^71 - 2 * q^72 + 20 * q^73 + 8 * q^74 + 4 * q^75 - 8 * q^78 + 16 * q^79 + 2 * q^81 + 18 * q^82 - 10 * q^83 - 8 * q^86 + 4 * q^87 - 2 * q^88 - 16 * q^89 + 8 * q^93 + 8 * q^94 + 28 * q^95 - 2 * q^96 + 2 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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