LMFDB - Embedded newform 3234.2.a.w.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)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.64575 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.64575 q^{10} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +2.64575 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -5.29150 q^{19} -2.64575 q^{20} -1.00000 q^{22} +2.64575 q^{23} +1.00000 q^{24} +2.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} -2.64575 q^{30} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -9.29150 q^{37} +5.29150 q^{38} +4.00000 q^{39} +2.64575 q^{40} +9.00000 q^{41} -1.29150 q^{43} +1.00000 q^{44} -2.64575 q^{45} -2.64575 q^{46} +11.9373 q^{47} -1.00000 q^{48} -2.00000 q^{50} +3.00000 q^{51} -4.00000 q^{52} +4.00000 q^{53} +1.00000 q^{54} -2.64575 q^{55} +5.29150 q^{57} -2.00000 q^{58} -14.5830 q^{59} +2.64575 q^{60} -3.93725 q^{61} +4.00000 q^{62} +1.00000 q^{64} +10.5830 q^{65} +1.00000 q^{66} -13.5830 q^{67} -3.00000 q^{68} -2.64575 q^{69} -13.2915 q^{71} -1.00000 q^{72} -4.70850 q^{73} +9.29150 q^{74} -2.00000 q^{75} -5.29150 q^{76} -4.00000 q^{78} +5.35425 q^{79} -2.64575 q^{80} +1.00000 q^{81} -9.00000 q^{82} -5.58301 q^{83} +7.93725 q^{85} +1.29150 q^{86} -2.00000 q^{87} -1.00000 q^{88} +13.2915 q^{89} +2.64575 q^{90} +2.64575 q^{92} +4.00000 q^{93} -11.9373 q^{94} +14.0000 q^{95} +1.00000 q^{96} +9.58301 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	2.64575	0.683130
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−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	$-0.411044\pi$
$23$	2.64575	0.551677	0.275839	+	0.961204i	$0.411044\pi$
$24$	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	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	−9.29150	−1.52751	−0.763757	−	0.645504i	$-0.776647\pi$
$37$	−9.29150	−1.52751	−0.763757	+	0.645504i	$0.776647\pi$
$38$	5.29150	0.858395
$39$	4.00000	0.640513
$40$	2.64575	0.418330
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−1.29150	−0.196952	−0.0984762	−	0.995139i	$-0.531397\pi$
$43$	−1.29150	−0.196952	−0.0984762	+	0.995139i	$0.531397\pi$
$44$	1.00000	0.150756
$45$	−2.64575	−0.394405
$46$	−2.64575	−0.390095
$47$	11.9373	1.74123	0.870614	−	0.491967i	$-0.163722\pi$
$47$	11.9373	1.74123	0.870614	+	0.491967i	$0.163722\pi$
$48$	−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
$59$	−14.5830	−1.89855	−0.949273	−	0.314454i	$-0.898179\pi$
$59$	−14.5830	−1.89855	−0.949273	+	0.314454i	$0.898179\pi$
$60$	2.64575	0.341565
$61$	−3.93725	−0.504114	−0.252057	−	0.967712i	$-0.581107\pi$
$61$	−3.93725	−0.504114	−0.252057	+	0.967712i	$0.581107\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	10.5830	1.31266
$66$	1.00000	0.123091
$67$	−13.5830	−1.65943	−0.829714	−	0.558189i	$-0.811497\pi$
$67$	−13.5830	−1.65943	−0.829714	+	0.558189i	$0.811497\pi$
$68$	−3.00000	−0.363803
$69$	−2.64575	−0.318511
$70$	0	0
$71$	−13.2915	−1.57741	−0.788706	−	0.614771i	$-0.789248\pi$
$71$	−13.2915	−1.57741	−0.788706	+	0.614771i	$0.789248\pi$
$72$	−1.00000	−0.117851
$73$	−4.70850	−0.551088	−0.275544	−	0.961288i	$-0.588858\pi$
$73$	−4.70850	−0.551088	−0.275544	+	0.961288i	$0.588858\pi$
$74$	9.29150	1.08012
$75$	−2.00000	−0.230940
$76$	−5.29150	−0.606977
$77$	0	0
$78$	−4.00000	−0.452911
$79$	5.35425	0.602400	0.301200	−	0.953561i	$-0.402613\pi$
$79$	5.35425	0.602400	0.301200	+	0.953561i	$0.402613\pi$
$80$	−2.64575	−0.295804
$81$	1.00000	0.111111
$82$	−9.00000	−0.993884
$83$	−5.58301	−0.612814	−0.306407	−	0.951901i	$-0.599127\pi$
$83$	−5.58301	−0.612814	−0.306407	+	0.951901i	$0.599127\pi$
$84$	0	0
$85$	7.93725	0.860916
$86$	1.29150	0.139266
$87$	−2.00000	−0.214423
$88$	−1.00000	−0.106600
$89$	13.2915	1.40890	0.704448	−	0.709755i	$-0.251195\pi$
$89$	13.2915	1.40890	0.704448	+	0.709755i	$0.251195\pi$
$90$	2.64575	0.278887
$91$	0	0
$92$	2.64575	0.275839
$93$	4.00000	0.414781
$94$	−11.9373	−1.23123
$95$	14.0000	1.43637
$96$	1.00000	0.102062
$97$	9.58301	0.973007	0.486503	−	0.873679i	$-0.338272\pi$
$97$	9.58301	0.973007	0.486503	+	0.873679i	$0.338272\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	2.00000	0.200000
$101$	−3.29150	−0.327517	−0.163758	−	0.986500i	$-0.552362\pi$
$101$	−3.29150	−0.327517	−0.163758	+	0.986500i	$0.552362\pi$
$102$	−3.00000	−0.297044
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	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
$109$	11.9373	1.14338	0.571691	−	0.820469i	$-0.306288\pi$
$109$	11.9373	1.14338	0.571691	+	0.820469i	$0.306288\pi$
$110$	2.64575	0.252262
$111$	9.29150	0.881910
$112$	0	0
$113$	−8.58301	−0.807421	−0.403711	−	0.914887i	$-0.632280\pi$
$113$	−8.58301	−0.807421	−0.403711	+	0.914887i	$0.632280\pi$
$114$	−5.29150	−0.495595
$115$	−7.00000	−0.652753
$116$	2.00000	0.185695
$117$	−4.00000	−0.369800
$118$	14.5830	1.34247
$119$	0	0
$120$	−2.64575	−0.241523
$121$	1.00000	0.0909091
$122$	3.93725	0.356462
$123$	−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	$-0.537452\pi$
$127$	−2.64575	−0.234772	−0.117386	+	0.993086i	$0.537452\pi$
$128$	−1.00000	−0.0883883
$129$	1.29150	0.113710
$130$	−10.5830	−0.928191
$131$	2.58301	0.225678	0.112839	−	0.993613i	$-0.464006\pi$
$131$	2.58301	0.225678	0.112839	+	0.993613i	$0.464006\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	13.5830	1.17339
$135$	2.64575	0.227710
$136$	3.00000	0.257248
$137$	−2.70850	−0.231403	−0.115701	−	0.993284i	$-0.536912\pi$
$137$	−2.70850	−0.231403	−0.115701	+	0.993284i	$0.536912\pi$
$138$	2.64575	0.225221
$139$	8.58301	0.728001	0.364001	−	0.931399i	$-0.381411\pi$
$139$	8.58301	0.728001	0.364001	+	0.931399i	$0.381411\pi$
$140$	0	0
$141$	−11.9373	−1.00530
$142$	13.2915	1.11540
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	−5.29150	−0.439435
$146$	4.70850	0.389678
$147$	0	0
$148$	−9.29150	−0.763757
$149$	−11.8745	−0.972798	−0.486399	−	0.873737i	$-0.661690\pi$
$149$	−11.8745	−0.972798	−0.486399	+	0.873737i	$0.661690\pi$
$150$	2.00000	0.163299
$151$	17.3542	1.41227	0.706134	−	0.708078i	$-0.250437\pi$
$151$	17.3542	1.41227	0.706134	+	0.708078i	$0.250437\pi$
$152$	5.29150	0.429198
$153$	−3.00000	−0.242536
$154$	0	0
$155$	10.5830	0.850047
$156$	4.00000	0.320256
$157$	−11.2915	−0.901160	−0.450580	−	0.892736i	$-0.648783\pi$
$157$	−11.2915	−0.901160	−0.450580	+	0.892736i	$0.648783\pi$
$158$	−5.35425	−0.425961
$159$	−4.00000	−0.317221
$160$	2.64575	0.209165
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	15.5830	1.22056	0.610278	−	0.792188i	$-0.291058\pi$
$163$	15.5830	1.22056	0.610278	+	0.792188i	$0.291058\pi$
$164$	9.00000	0.702782
$165$	2.64575	0.205971
$166$	5.58301	0.433325
$167$	19.2915	1.49282	0.746411	−	0.665486i	$-0.231776\pi$
$167$	19.2915	1.49282	0.746411	+	0.665486i	$0.231776\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−7.93725	−0.608760
$171$	−5.29150	−0.404651
$172$	−1.29150	−0.0984762
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	1.00000	0.0753778
$177$	14.5830	1.09613
$178$	−13.2915	−0.996240
$179$	15.2915	1.14294	0.571470	−	0.820623i	$-0.306373\pi$
$179$	15.2915	1.14294	0.571470	+	0.820623i	$0.306373\pi$
$180$	−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
$183$	3.93725	0.291050
$184$	−2.64575	−0.195047
$185$	24.5830	1.80738
$186$	−4.00000	−0.293294
$187$	−3.00000	−0.219382
$188$	11.9373	0.870614
$189$	0	0
$190$	−14.0000	−1.01567
$191$	17.2915	1.25117	0.625585	−	0.780156i	$-0.284861\pi$
$191$	17.2915	1.25117	0.625585	+	0.780156i	$0.284861\pi$
$192$	−1.00000	−0.0721688
$193$	15.8745	1.14267	0.571336	−	0.820716i	$-0.306425\pi$
$193$	15.8745	1.14267	0.571336	+	0.820716i	$0.306425\pi$
$194$	−9.58301	−0.688020
$195$	−10.5830	−0.757865
$196$	0	0
$197$	−9.87451	−0.703530	−0.351765	−	0.936088i	$-0.614418\pi$
$197$	−9.87451	−0.703530	−0.351765	+	0.936088i	$0.614418\pi$
$198$	−1.00000	−0.0710669
$199$	23.1660	1.64219	0.821097	−	0.570788i	$-0.193362\pi$
$199$	23.1660	1.64219	0.821097	+	0.570788i	$0.193362\pi$
$200$	−2.00000	−0.141421
$201$	13.5830	0.958071
$202$	3.29150	0.231589
$203$	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
$211$	10.7085	0.737203	0.368602	−	0.929587i	$-0.379837\pi$
$211$	10.7085	0.737203	0.368602	+	0.929587i	$0.379837\pi$
$212$	4.00000	0.274721
$213$	13.2915	0.910719
$214$	−9.00000	−0.615227
$215$	3.41699	0.233037
$216$	1.00000	0.0680414
$217$	0	0
$218$	−11.9373	−0.808493
$219$	4.70850	0.318171
$220$	−2.64575	−0.178377
$221$	12.0000	0.807207
$222$	−9.29150	−0.623605
$223$	−24.4575	−1.63780	−0.818898	−	0.573939i	$-0.805415\pi$
$223$	−24.4575	−1.63780	−0.818898	+	0.573939i	$0.805415\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	8.58301	0.570933
$227$	23.5830	1.56526	0.782630	−	0.622488i	$-0.213878\pi$
$227$	23.5830	1.56526	0.782630	+	0.622488i	$0.213878\pi$
$228$	5.29150	0.350438
$229$	17.2915	1.14265	0.571327	−	0.820722i	$-0.306429\pi$
$229$	17.2915	1.14265	0.571327	+	0.820722i	$0.306429\pi$
$230$	7.00000	0.461566
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−22.1660	−1.45214	−0.726072	−	0.687619i	$-0.758656\pi$
$233$	−22.1660	−1.45214	−0.726072	+	0.687619i	$0.758656\pi$
$234$	4.00000	0.261488
$235$	−31.5830	−2.06025
$236$	−14.5830	−0.949273
$237$	−5.35425	−0.347796
$238$	0	0
$239$	13.2915	0.859756	0.429878	−	0.902887i	$-0.358557\pi$
$239$	13.2915	0.859756	0.429878	+	0.902887i	$0.358557\pi$
$240$	2.64575	0.170783
$241$	15.1660	0.976929	0.488464	−	0.872584i	$-0.337557\pi$
$241$	15.1660	0.976929	0.488464	+	0.872584i	$0.337557\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−3.93725	−0.252057
$245$	0	0
$246$	9.00000	0.573819
$247$	21.1660	1.34676
$248$	4.00000	0.254000
$249$	5.58301	0.353809
$250$	−7.93725	−0.501996
$251$	12.7085	0.802153	0.401077	−	0.916045i	$-0.368636\pi$
$251$	12.7085	0.802153	0.401077	+	0.916045i	$0.368636\pi$
$252$	0	0
$253$	2.64575	0.166337
$254$	2.64575	0.166009
$255$	−7.93725	−0.497050
$256$	1.00000	0.0625000
$257$	−1.29150	−0.0805617	−0.0402809	−	0.999188i	$-0.512825\pi$
$257$	−1.29150	−0.0805617	−0.0402809	+	0.999188i	$0.512825\pi$
$258$	−1.29150	−0.0804054
$259$	0	0
$260$	10.5830	0.656330
$261$	2.00000	0.123797
$262$	−2.58301	−0.159579
$263$	−9.87451	−0.608888	−0.304444	−	0.952530i	$-0.598471\pi$
$263$	−9.87451	−0.608888	−0.304444	+	0.952530i	$0.598471\pi$
$264$	1.00000	0.0615457
$265$	−10.5830	−0.650109
$266$	0	0
$267$	−13.2915	−0.813427
$268$	−13.5830	−0.829714
$269$	−5.35425	−0.326454	−0.163227	−	0.986589i	$-0.552190\pi$
$269$	−5.35425	−0.326454	−0.163227	+	0.986589i	$0.552190\pi$
$270$	−2.64575	−0.161015
$271$	−1.29150	−0.0784532	−0.0392266	−	0.999230i	$-0.512489\pi$
$271$	−1.29150	−0.0784532	−0.0392266	+	0.999230i	$0.512489\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	2.70850	0.163626
$275$	2.00000	0.120605
$276$	−2.64575	−0.159256
$277$	25.1660	1.51208	0.756040	−	0.654526i	$-0.227132\pi$
$277$	25.1660	1.51208	0.756040	+	0.654526i	$0.227132\pi$
$278$	−8.58301	−0.514774
$279$	−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	$0.528521\pi$
$282$	11.9373	0.710853
$283$	−22.4575	−1.33496	−0.667480	−	0.744627i	$-0.732627\pi$
$283$	−22.4575	−1.33496	−0.667480	+	0.744627i	$0.732627\pi$
$284$	−13.2915	−0.788706
$285$	−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
$291$	−9.58301	−0.561766
$292$	−4.70850	−0.275544
$293$	6.58301	0.384583	0.192292	−	0.981338i	$-0.438408\pi$
$293$	6.58301	0.384583	0.192292	+	0.981338i	$0.438408\pi$
$294$	0	0
$295$	38.5830	2.24639
$296$	9.29150	0.540058
$297$	−1.00000	−0.0580259
$298$	11.8745	0.687872
$299$	−10.5830	−0.612031
$300$	−2.00000	−0.115470
$301$	0	0
$302$	−17.3542	−0.998625
$303$	3.29150	0.189092
$304$	−5.29150	−0.303488
$305$	10.4170	0.596475
$306$	3.00000	0.171499
$307$	−32.5830	−1.85961	−0.929805	−	0.368052i	$-0.880025\pi$
$307$	−32.5830	−1.85961	−0.929805	+	0.368052i	$0.880025\pi$
$308$	0	0
$309$	10.0000	0.568880
$310$	−10.5830	−0.601074
$311$	−14.5203	−0.823368	−0.411684	−	0.911327i	$-0.635059\pi$
$311$	−14.5203	−0.823368	−0.411684	+	0.911327i	$0.635059\pi$
$312$	−4.00000	−0.226455
$313$	0.583005	0.0329534	0.0164767	−	0.999864i	$-0.494755\pi$
$313$	0.583005	0.0329534	0.0164767	+	0.999864i	$0.494755\pi$
$314$	11.2915	0.637216
$315$	0	0
$316$	5.35425	0.301200
$317$	−17.2288	−0.967663	−0.483832	−	0.875161i	$-0.660755\pi$
$317$	−17.2288	−0.967663	−0.483832	+	0.875161i	$0.660755\pi$
$318$	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
$326$	−15.5830	−0.863063
$327$	−11.9373	−0.660132
$328$	−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	$0.383707\pi$
$332$	−5.58301	−0.306407
$333$	−9.29150	−0.509171
$334$	−19.2915	−1.05558
$335$	35.9373	1.96346
$336$	0	0
$337$	17.2915	0.941928	0.470964	−	0.882152i	$-0.343906\pi$
$337$	17.2915	0.941928	0.470964	+	0.882152i	$0.343906\pi$
$338$	−3.00000	−0.163178
$339$	8.58301	0.466165
$340$	7.93725	0.430458
$341$	−4.00000	−0.216612
$342$	5.29150	0.286132
$343$	0	0
$344$	1.29150	0.0696332
$345$	7.00000	0.376867
$346$	−4.00000	−0.215041
$347$	8.41699	0.451848	0.225924	−	0.974145i	$-0.427460\pi$
$347$	8.41699	0.451848	0.225924	+	0.974145i	$0.427460\pi$
$348$	−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
$353$	−12.7085	−0.676405	−0.338203	−	0.941073i	$-0.609819\pi$
$353$	−12.7085	−0.676405	−0.338203	+	0.941073i	$0.609819\pi$
$354$	−14.5830	−0.775078
$355$	35.1660	1.86642
$356$	13.2915	0.704448
$357$	0	0
$358$	−15.2915	−0.808181
$359$	−0.583005	−0.0307698	−0.0153849	−	0.999882i	$-0.504897\pi$
$359$	−0.583005	−0.0307698	−0.0153849	+	0.999882i	$0.504897\pi$
$360$	2.64575	0.139443
$361$	9.00000	0.473684
$362$	−5.29150	−0.278115
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	12.4575	0.652056
$366$	−3.93725	−0.205804
$367$	21.2915	1.11141	0.555704	−	0.831380i	$-0.312449\pi$
$367$	21.2915	1.11141	0.555704	+	0.831380i	$0.312449\pi$
$368$	2.64575	0.137919
$369$	9.00000	0.468521
$370$	−24.5830	−1.27801
$371$	0	0
$372$	4.00000	0.207390
$373$	31.8118	1.64715	0.823575	−	0.567207i	$-0.191976\pi$
$373$	31.8118	1.64715	0.823575	+	0.567207i	$0.191976\pi$
$374$	3.00000	0.155126
$375$	−7.93725	−0.409878
$376$	−11.9373	−0.615617
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−6.41699	−0.329619	−0.164809	−	0.986325i	$-0.552701\pi$
$379$	−6.41699	−0.329619	−0.164809	+	0.986325i	$0.552701\pi$
$380$	14.0000	0.718185
$381$	2.64575	0.135546
$382$	−17.2915	−0.884710
$383$	29.2915	1.49673	0.748363	−	0.663289i	$-0.230840\pi$
$383$	29.2915	1.49673	0.748363	+	0.663289i	$0.230840\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−15.8745	−0.807991
$387$	−1.29150	−0.0656508
$388$	9.58301	0.486503
$389$	16.0627	0.814414	0.407207	−	0.913336i	$-0.366503\pi$
$389$	16.0627	0.814414	0.407207	+	0.913336i	$0.366503\pi$
$390$	10.5830	0.535891
$391$	−7.93725	−0.401404
$392$	0	0
$393$	−2.58301	−0.130295
$394$	9.87451	0.497471
$395$	−14.1660	−0.712769
$396$	1.00000	0.0502519
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	−23.1660	−1.16121
$399$	0	0
$400$	2.00000	0.100000
$401$	−4.12549	−0.206017	−0.103009	−	0.994680i	$-0.532847\pi$
$401$	−4.12549	−0.206017	−0.103009	+	0.994680i	$0.532847\pi$
$402$	−13.5830	−0.677459
$403$	16.0000	0.797017
$404$	−3.29150	−0.163758
$405$	−2.64575	−0.131468
$406$	0	0
$407$	−9.29150	−0.460563
$408$	−3.00000	−0.148522
$409$	4.12549	0.203992	0.101996	−	0.994785i	$-0.467477\pi$
$409$	4.12549	0.203992	0.101996	+	0.994785i	$0.467477\pi$
$410$	23.8118	1.17598
$411$	2.70850	0.133600
$412$	−10.0000	−0.492665
$413$	0	0
$414$	−2.64575	−0.130032
$415$	14.7712	0.725092
$416$	4.00000	0.196116
$417$	−8.58301	−0.420312
$418$	5.29150	0.258816
$419$	21.8745	1.06864	0.534320	−	0.845282i	$-0.320568\pi$
$419$	21.8745	1.06864	0.534320	+	0.845282i	$0.320568\pi$
$420$	0	0
$421$	37.8745	1.84589	0.922945	−	0.384931i	$-0.125775\pi$
$421$	37.8745	1.84589	0.922945	+	0.384931i	$0.125775\pi$
$422$	−10.7085	−0.521281
$423$	11.9373	0.580409
$424$	−4.00000	−0.194257
$425$	−6.00000	−0.291043
$426$	−13.2915	−0.643975
$427$	0	0
$428$	9.00000	0.435031
$429$	4.00000	0.193122
$430$	−3.41699	−0.164782
$431$	−6.70850	−0.323137	−0.161568	−	0.986862i	$-0.551655\pi$
$431$	−6.70850	−0.323137	−0.161568	+	0.986862i	$0.551655\pi$
$432$	−1.00000	−0.0481125
$433$	−19.5830	−0.941099	−0.470550	−	0.882374i	$-0.655944\pi$
$433$	−19.5830	−0.941099	−0.470550	+	0.882374i	$0.655944\pi$
$434$	0	0
$435$	5.29150	0.253708
$436$	11.9373	0.571691
$437$	−14.0000	−0.669711
$438$	−4.70850	−0.224981
$439$	−14.6458	−0.699004	−0.349502	−	0.936936i	$-0.613649\pi$
$439$	−14.6458	−0.699004	−0.349502	+	0.936936i	$0.613649\pi$
$440$	2.64575	0.126131
$441$	0	0
$442$	−12.0000	−0.570782
$443$	11.1660	0.530513	0.265257	−	0.964178i	$-0.414543\pi$
$443$	11.1660	0.530513	0.265257	+	0.964178i	$0.414543\pi$
$444$	9.29150	0.440955
$445$	−35.1660	−1.66703
$446$	24.4575	1.15810
$447$	11.8745	0.561645
$448$	0	0
$449$	−22.4575	−1.05984	−0.529918	−	0.848049i	$-0.677777\pi$
$449$	−22.4575	−1.05984	−0.529918	+	0.848049i	$0.677777\pi$
$450$	−2.00000	−0.0942809
$451$	9.00000	0.423793
$452$	−8.58301	−0.403711
$453$	−17.3542	−0.815374
$454$	−23.5830	−1.10681
$455$	0	0
$456$	−5.29150	−0.247797
$457$	−19.4170	−0.908289	−0.454144	−	0.890928i	$-0.650055\pi$
$457$	−19.4170	−0.908289	−0.454144	+	0.890928i	$0.650055\pi$
$458$	−17.2915	−0.807979
$459$	3.00000	0.140028
$460$	−7.00000	−0.326377
$461$	−3.41699	−0.159145	−0.0795727	−	0.996829i	$-0.525356\pi$
$461$	−3.41699	−0.159145	−0.0795727	+	0.996829i	$0.525356\pi$
$462$	0	0
$463$	−35.0405	−1.62847	−0.814235	−	0.580535i	$-0.802844\pi$
$463$	−35.0405	−1.62847	−0.814235	+	0.580535i	$0.802844\pi$
$464$	2.00000	0.0928477
$465$	−10.5830	−0.490775
$466$	22.1660	1.02682
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	−4.00000	−0.184900
$469$	0	0
$470$	31.5830	1.45682
$471$	11.2915	0.520285
$472$	14.5830	0.671237
$473$	−1.29150	−0.0593834
$474$	5.35425	0.245929
$475$	−10.5830	−0.485582
$476$	0	0
$477$	4.00000	0.183147
$478$	−13.2915	−0.607939
$479$	−2.12549	−0.0971162	−0.0485581	−	0.998820i	$-0.515463\pi$
$479$	−2.12549	−0.0971162	−0.0485581	+	0.998820i	$0.515463\pi$
$480$	−2.64575	−0.120761
$481$	37.1660	1.69462
$482$	−15.1660	−0.690793
$483$	0	0
$484$	1.00000	0.0454545
$485$	−25.3542	−1.15128
$486$	1.00000	0.0453609
$487$	−9.87451	−0.447457	−0.223728	−	0.974652i	$-0.571823\pi$
$487$	−9.87451	−0.447457	−0.223728	+	0.974652i	$0.571823\pi$
$488$	3.93725	0.178231
$489$	−15.5830	−0.704688
$490$	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
$498$	−5.58301	−0.250180
$499$	2.58301	0.115631	0.0578156	−	0.998327i	$-0.481586\pi$
$499$	2.58301	0.115631	0.0578156	+	0.998327i	$0.481586\pi$
$500$	7.93725	0.354965
$501$	−19.2915	−0.861881
$502$	−12.7085	−0.567208
$503$	9.29150	0.414288	0.207144	−	0.978311i	$-0.433583\pi$
$503$	9.29150	0.414288	0.207144	+	0.978311i	$0.433583\pi$
$504$	0	0
$505$	8.70850	0.387523
$506$	−2.64575	−0.117618
$507$	−3.00000	−0.133235
$508$	−2.64575	−0.117386
$509$	−17.1660	−0.760870	−0.380435	−	0.924808i	$-0.624226\pi$
$509$	−17.1660	−0.760870	−0.380435	+	0.924808i	$0.624226\pi$
$510$	7.93725	0.351468
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	5.29150	0.233626
$514$	1.29150	0.0569657
$515$	26.4575	1.16586
$516$	1.29150	0.0568552
$517$	11.9373	0.525000
$518$	0	0
$519$	−4.00000	−0.175581
$520$	−10.5830	−0.464095
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	−2.00000	−0.0875376
$523$	13.2915	0.581197	0.290598	−	0.956845i	$-0.406146\pi$
$523$	13.2915	0.581197	0.290598	+	0.956845i	$0.406146\pi$
$524$	2.58301	0.112839
$525$	0	0
$526$	9.87451	0.430549
$527$	12.0000	0.522728
$528$	−1.00000	−0.0435194
$529$	−16.0000	−0.695652
$530$	10.5830	0.459696
$531$	−14.5830	−0.632849
$532$	0	0
$533$	−36.0000	−1.55933
$534$	13.2915	0.575180
$535$	−23.8118	−1.02947
$536$	13.5830	0.586696
$537$	−15.2915	−0.659877
$538$	5.35425	0.230838
$539$	0	0
$540$	2.64575	0.113855
$541$	−43.8118	−1.88361	−0.941807	−	0.336153i	$-0.890874\pi$
$541$	−43.8118	−1.88361	−0.941807	+	0.336153i	$0.890874\pi$
$542$	1.29150	0.0554748
$543$	−5.29150	−0.227080
$544$	3.00000	0.128624
$545$	−31.5830	−1.35287
$546$	0	0
$547$	25.2915	1.08139	0.540693	−	0.841220i	$-0.318162\pi$
$547$	25.2915	1.08139	0.540693	+	0.841220i	$0.318162\pi$
$548$	−2.70850	−0.115701
$549$	−3.93725	−0.168038
$550$	−2.00000	−0.0852803
$551$	−10.5830	−0.450851
$552$	2.64575	0.112611
$553$	0	0
$554$	−25.1660	−1.06920
$555$	−24.5830	−1.04349
$556$	8.58301	0.364001
$557$	−8.70850	−0.368991	−0.184495	−	0.982833i	$-0.559065\pi$
$557$	−8.70850	−0.368991	−0.184495	+	0.982833i	$0.559065\pi$
$558$	4.00000	0.169334
$559$	5.16601	0.218499
$560$	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	$0.5\pi$
$564$	−11.9373	−0.502649
$565$	22.7085	0.955354
$566$	22.4575	0.943960
$567$	0	0
$568$	13.2915	0.557699
$569$	27.1660	1.13886	0.569429	−	0.822040i	$-0.307164\pi$
$569$	27.1660	1.13886	0.569429	+	0.822040i	$0.307164\pi$
$570$	14.0000	0.586395
$571$	33.7490	1.41235	0.706176	−	0.708036i	$-0.250419\pi$
$571$	33.7490	1.41235	0.706176	+	0.708036i	$0.250419\pi$
$572$	−4.00000	−0.167248
$573$	−17.2915	−0.722363
$574$	0	0
$575$	5.29150	0.220671
$576$	1.00000	0.0416667
$577$	−40.7490	−1.69640	−0.848202	−	0.529673i	$-0.822315\pi$
$577$	−40.7490	−1.69640	−0.848202	+	0.529673i	$0.822315\pi$
$578$	8.00000	0.332756
$579$	−15.8745	−0.659722
$580$	−5.29150	−0.219718
$581$	0	0
$582$	9.58301	0.397228
$583$	4.00000	0.165663
$584$	4.70850	0.194839
$585$	10.5830	0.437553
$586$	−6.58301	−0.271941
$587$	−21.8745	−0.902858	−0.451429	−	0.892307i	$-0.649086\pi$
$587$	−21.8745	−0.902858	−0.451429	+	0.892307i	$0.649086\pi$
$588$	0	0
$589$	21.1660	0.872130
$590$	−38.5830	−1.58844
$591$	9.87451	0.406183
$592$	−9.29150	−0.381878
$593$	37.7490	1.55017	0.775083	−	0.631859i	$-0.217708\pi$
$593$	37.7490	1.55017	0.775083	+	0.631859i	$0.217708\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	−11.8745	−0.486399
$597$	−23.1660	−0.948122
$598$	10.5830	0.432771
$599$	34.6458	1.41559	0.707794	−	0.706419i	$-0.249691\pi$
$599$	34.6458	1.41559	0.707794	+	0.706419i	$0.249691\pi$
$600$	2.00000	0.0816497
$601$	−17.4170	−0.710454	−0.355227	−	0.934780i	$-0.615596\pi$
$601$	−17.4170	−0.710454	−0.355227	+	0.934780i	$0.615596\pi$
$602$	0	0
$603$	−13.5830	−0.553143
$604$	17.3542	0.706134
$605$	−2.64575	−0.107565
$606$	−3.29150	−0.133708
$607$	45.1033	1.83069	0.915343	−	0.402676i	$-0.131920\pi$
$607$	45.1033	1.83069	0.915343	+	0.402676i	$0.131920\pi$
$608$	5.29150	0.214599
$609$	0	0
$610$	−10.4170	−0.421772
$611$	−47.7490	−1.93172
$612$	−3.00000	−0.121268
$613$	−39.8118	−1.60798	−0.803991	−	0.594642i	$-0.797294\pi$
$613$	−39.8118	−1.60798	−0.803991	+	0.594642i	$0.797294\pi$
$614$	32.5830	1.31494
$615$	23.8118	0.960183
$616$	0	0
$617$	−25.2915	−1.01820	−0.509099	−	0.860708i	$-0.670021\pi$
$617$	−25.2915	−1.01820	−0.509099	+	0.860708i	$0.670021\pi$
$618$	−10.0000	−0.402259
$619$	−41.0000	−1.64793	−0.823965	−	0.566641i	$-0.808243\pi$
$619$	−41.0000	−1.64793	−0.823965	+	0.566641i	$0.808243\pi$
$620$	10.5830	0.425024
$621$	−2.64575	−0.106170
$622$	14.5203	0.582209
$623$	0	0
$624$	4.00000	0.160128
$625$	−31.0000	−1.24000
$626$	−0.583005	−0.0233016
$627$	5.29150	0.211322
$628$	−11.2915	−0.450580
$629$	27.8745	1.11143
$630$	0	0
$631$	10.1255	0.403089	0.201545	−	0.979479i	$-0.435404\pi$
$631$	10.1255	0.403089	0.201545	+	0.979479i	$0.435404\pi$
$632$	−5.35425	−0.212981
$633$	−10.7085	−0.425625
$634$	17.2288	0.684241
$635$	7.00000	0.277787
$636$	−4.00000	−0.158610
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	−13.2915	−0.525804
$640$	2.64575	0.104583
$641$	−5.41699	−0.213958	−0.106979	−	0.994261i	$-0.534118\pi$
$641$	−5.41699	−0.213958	−0.106979	+	0.994261i	$0.534118\pi$
$642$	9.00000	0.355202
$643$	41.1660	1.62343	0.811714	−	0.584054i	$-0.198535\pi$
$643$	41.1660	1.62343	0.811714	+	0.584054i	$0.198535\pi$
$644$	0	0
$645$	−3.41699	−0.134544
$646$	−15.8745	−0.624574
$647$	42.3948	1.66671	0.833355	−	0.552738i	$-0.186417\pi$
$647$	42.3948	1.66671	0.833355	+	0.552738i	$0.186417\pi$
$648$	−1.00000	−0.0392837
$649$	−14.5830	−0.572433
$650$	8.00000	0.313786
$651$	0	0
$652$	15.5830	0.610278
$653$	13.1033	0.512770	0.256385	−	0.966575i	$-0.417468\pi$
$653$	13.1033	0.512770	0.256385	+	0.966575i	$0.417468\pi$
$654$	11.9373	0.466784
$655$	−6.83399	−0.267026
$656$	9.00000	0.351391
$657$	−4.70850	−0.183696
$658$	0	0
$659$	18.1660	0.707647	0.353824	−	0.935312i	$-0.384881\pi$
$659$	18.1660	0.707647	0.353824	+	0.935312i	$0.384881\pi$
$660$	2.64575	0.102986
$661$	31.2915	1.21710	0.608549	−	0.793516i	$-0.291752\pi$
$661$	31.2915	1.21710	0.608549	+	0.793516i	$0.291752\pi$
$662$	−13.0000	−0.505259
$663$	−12.0000	−0.466041
$664$	5.58301	0.216663
$665$	0	0
$666$	9.29150	0.360038
$667$	5.29150	0.204888
$668$	19.2915	0.746411
$669$	24.4575	0.945582
$670$	−35.9373	−1.38838
$671$	−3.93725	−0.151996
$672$	0	0
$673$	−40.5830	−1.56436	−0.782180	−	0.623053i	$-0.785892\pi$
$673$	−40.5830	−1.56436	−0.782180	+	0.623053i	$0.785892\pi$
$674$	−17.2915	−0.666044
$675$	−2.00000	−0.0769800
$676$	3.00000	0.115385
$677$	−39.8745	−1.53250	−0.766251	−	0.642541i	$-0.777880\pi$
$677$	−39.8745	−1.53250	−0.766251	+	0.642541i	$0.777880\pi$
$678$	−8.58301	−0.329628
$679$	0	0
$680$	−7.93725	−0.304380
$681$	−23.5830	−0.903703
$682$	4.00000	0.153168
$683$	25.1660	0.962951	0.481475	−	0.876460i	$-0.340101\pi$
$683$	25.1660	0.962951	0.481475	+	0.876460i	$0.340101\pi$
$684$	−5.29150	−0.202326
$685$	7.16601	0.273799
$686$	0	0
$687$	−17.2915	−0.659712
$688$	−1.29150	−0.0492381
$689$	−16.0000	−0.609551
$690$	−7.00000	−0.266485
$691$	−28.1660	−1.07149	−0.535743	−	0.844381i	$-0.679968\pi$
$691$	−28.1660	−1.07149	−0.535743	+	0.844381i	$0.679968\pi$
$692$	4.00000	0.152057
$693$	0	0
$694$	−8.41699	−0.319505
$695$	−22.7085	−0.861382
$696$	2.00000	0.0758098
$697$	−27.0000	−1.02270
$698$	−13.2288	−0.500716
$699$	22.1660	0.838396
$700$	0	0
$701$	−10.4575	−0.394975	−0.197487	−	0.980305i	$-0.563278\pi$
$701$	−10.4575	−0.394975	−0.197487	+	0.980305i	$0.563278\pi$
$702$	−4.00000	−0.150970
$703$	49.1660	1.85433
$704$	1.00000	0.0376889
$705$	31.5830	1.18948
$706$	12.7085	0.478291
$707$	0	0
$708$	14.5830	0.548063
$709$	−47.7490	−1.79325	−0.896626	−	0.442789i	$-0.853989\pi$
$709$	−47.7490	−1.79325	−0.896626	+	0.442789i	$0.853989\pi$
$710$	−35.1660	−1.31976
$711$	5.35425	0.200800
$712$	−13.2915	−0.498120
$713$	−10.5830	−0.396337
$714$	0	0
$715$	10.5830	0.395782
$716$	15.2915	0.571470
$717$	−13.2915	−0.496380
$718$	0.583005	0.0217576
$719$	−43.9373	−1.63858	−0.819292	−	0.573377i	$-0.805633\pi$
$719$	−43.9373	−1.63858	−0.819292	+	0.573377i	$0.805633\pi$
$720$	−2.64575	−0.0986013
$721$	0	0
$722$	−9.00000	−0.334945
$723$	−15.1660	−0.564030
$724$	5.29150	0.196657
$725$	4.00000	0.148556
$726$	1.00000	0.0371135
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−12.4575	−0.461073
$731$	3.87451	0.143304
$732$	3.93725	0.145525
$733$	−3.93725	−0.145426	−0.0727129	−	0.997353i	$-0.523166\pi$
$733$	−3.93725	−0.145426	−0.0727129	+	0.997353i	$0.523166\pi$
$734$	−21.2915	−0.785884
$735$	0	0
$736$	−2.64575	−0.0975237
$737$	−13.5830	−0.500336
$738$	−9.00000	−0.331295
$739$	−3.41699	−0.125696	−0.0628481	−	0.998023i	$-0.520018\pi$
$739$	−3.41699	−0.125696	−0.0628481	+	0.998023i	$0.520018\pi$
$740$	24.5830	0.903689
$741$	−21.1660	−0.777553
$742$	0	0
$743$	−29.2915	−1.07460	−0.537301	−	0.843391i	$-0.680556\pi$
$743$	−29.2915	−1.07460	−0.537301	+	0.843391i	$0.680556\pi$
$744$	−4.00000	−0.146647
$745$	31.4170	1.15103
$746$	−31.8118	−1.16471
$747$	−5.58301	−0.204271
$748$	−3.00000	−0.109691
$749$	0	0
$750$	7.93725	0.289828
$751$	−18.7085	−0.682683	−0.341341	−	0.939939i	$-0.610881\pi$
$751$	−18.7085	−0.682683	−0.341341	+	0.939939i	$0.610881\pi$
$752$	11.9373	0.435307
$753$	−12.7085	−0.463123
$754$	8.00000	0.291343
$755$	−45.9150	−1.67102
$756$	0	0
$757$	−11.4170	−0.414958	−0.207479	−	0.978240i	$-0.566526\pi$
$757$	−11.4170	−0.414958	−0.207479	+	0.978240i	$0.566526\pi$
$758$	6.41699	0.233076
$759$	−2.64575	−0.0960347
$760$	−14.0000	−0.507833
$761$	26.4170	0.957615	0.478808	−	0.877920i	$-0.341069\pi$
$761$	26.4170	0.957615	0.478808	+	0.877920i	$0.341069\pi$
$762$	−2.64575	−0.0958455
$763$	0	0
$764$	17.2915	0.625585
$765$	7.93725	0.286972
$766$	−29.2915	−1.05835
$767$	58.3320	2.10625
$768$	−1.00000	−0.0360844
$769$	−41.2915	−1.48901	−0.744505	−	0.667617i	$-0.767314\pi$
$769$	−41.2915	−1.48901	−0.744505	+	0.667617i	$0.767314\pi$
$770$	0	0
$771$	1.29150	0.0465123
$772$	15.8745	0.571336
$773$	−23.9373	−0.860963	−0.430482	−	0.902599i	$-0.641656\pi$
$773$	−23.9373	−0.860963	−0.430482	+	0.902599i	$0.641656\pi$
$774$	1.29150	0.0464221
$775$	−8.00000	−0.287368
$776$	−9.58301	−0.344010
$777$	0	0
$778$	−16.0627	−0.575877
$779$	−47.6235	−1.70629
$780$	−10.5830	−0.378932
$781$	−13.2915	−0.475607
$782$	7.93725	0.283836
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	29.8745	1.06627
$786$	2.58301	0.0921327
$787$	−1.29150	−0.0460371	−0.0230185	−	0.999735i	$-0.507328\pi$
$787$	−1.29150	−0.0460371	−0.0230185	+	0.999735i	$0.507328\pi$
$788$	−9.87451	−0.351765
$789$	9.87451	0.351542
$790$	14.1660	0.504004
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	15.7490	0.559264
$794$	−6.00000	−0.212932
$795$	10.5830	0.375341
$796$	23.1660	0.821097
$797$	−13.1033	−0.464141	−0.232071	−	0.972699i	$-0.574550\pi$
$797$	−13.1033	−0.464141	−0.232071	+	0.972699i	$0.574550\pi$
$798$	0	0
$799$	−35.8118	−1.26693
$800$	−2.00000	−0.0707107
$801$	13.2915	0.469632
$802$	4.12549	0.145676
$803$	−4.70850	−0.166159
$804$	13.5830	0.479036
$805$	0	0
$806$	−16.0000	−0.563576
$807$	5.35425	0.188478
$808$	3.29150	0.115795
$809$	−39.3320	−1.38284	−0.691420	−	0.722453i	$-0.743015\pi$
$809$	−39.3320	−1.38284	−0.691420	+	0.722453i	$0.743015\pi$
$810$	2.64575	0.0929622
$811$	−4.58301	−0.160931	−0.0804655	−	0.996757i	$-0.525641\pi$
$811$	−4.58301	−0.160931	−0.0804655	+	0.996757i	$0.525641\pi$
$812$	0	0
$813$	1.29150	0.0452950
$814$	9.29150	0.325667
$815$	−41.2288	−1.44418
$816$	3.00000	0.105021
$817$	6.83399	0.239091
$818$	−4.12549	−0.144244
$819$	0	0
$820$	−23.8118	−0.831543
$821$	54.3320	1.89620	0.948100	−	0.317971i	$-0.103002\pi$
$821$	54.3320	1.89620	0.948100	+	0.317971i	$0.103002\pi$
$822$	−2.70850	−0.0944697
$823$	−43.8745	−1.52937	−0.764685	−	0.644405i	$-0.777105\pi$
$823$	−43.8745	−1.52937	−0.764685	+	0.644405i	$0.777105\pi$
$824$	10.0000	0.348367
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	−49.3320	−1.71544	−0.857721	−	0.514115i	$-0.828120\pi$
$827$	−49.3320	−1.71544	−0.857721	+	0.514115i	$0.828120\pi$
$828$	2.64575	0.0919462
$829$	−11.8745	−0.412419	−0.206209	−	0.978508i	$-0.566113\pi$
$829$	−11.8745	−0.412419	−0.206209	+	0.978508i	$0.566113\pi$
$830$	−14.7712	−0.512717
$831$	−25.1660	−0.872999
$832$	−4.00000	−0.138675
$833$	0	0
$834$	8.58301	0.297205
$835$	−51.0405	−1.76633
$836$	−5.29150	−0.183010
$837$	4.00000	0.138260
$838$	−21.8745	−0.755642
$839$	21.1033	0.728566	0.364283	−	0.931288i	$-0.381314\pi$
$839$	21.1033	0.728566	0.364283	+	0.931288i	$0.381314\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−37.8745	−1.30524
$843$	3.00000	0.103325
$844$	10.7085	0.368602
$845$	−7.93725	−0.273050
$846$	−11.9373	−0.410411
$847$	0	0
$848$	4.00000	0.137361
$849$	22.4575	0.770740
$850$	6.00000	0.205798
$851$	−24.5830	−0.842695
$852$	13.2915	0.455359
$853$	−22.5203	−0.771079	−0.385539	−	0.922691i	$-0.625985\pi$
$853$	−22.5203	−0.771079	−0.385539	+	0.922691i	$0.625985\pi$
$854$	0	0
$855$	14.0000	0.478790
$856$	−9.00000	−0.307614
$857$	52.1660	1.78196	0.890978	−	0.454046i	$-0.150020\pi$
$857$	52.1660	1.78196	0.890978	+	0.454046i	$0.150020\pi$
$858$	−4.00000	−0.136558
$859$	−14.7490	−0.503230	−0.251615	−	0.967827i	$-0.580962\pi$
$859$	−14.7490	−0.503230	−0.251615	+	0.967827i	$0.580962\pi$
$860$	3.41699	0.116519
$861$	0	0
$862$	6.70850	0.228492
$863$	25.1033	0.854525	0.427263	−	0.904128i	$-0.359478\pi$
$863$	25.1033	0.854525	0.427263	+	0.904128i	$0.359478\pi$
$864$	1.00000	0.0340207
$865$	−10.5830	−0.359833
$866$	19.5830	0.665458
$867$	8.00000	0.271694
$868$	0	0
$869$	5.35425	0.181630
$870$	−5.29150	−0.179399
$871$	54.3320	1.84097
$872$	−11.9373	−0.404246
$873$	9.58301	0.324336
$874$	14.0000	0.473557
$875$	0	0
$876$	4.70850	0.159085
$877$	37.2288	1.25713	0.628563	−	0.777759i	$-0.283643\pi$
$877$	37.2288	1.25713	0.628563	+	0.777759i	$0.283643\pi$
$878$	14.6458	0.494270
$879$	−6.58301	−0.222039
$880$	−2.64575	−0.0891883
$881$	−19.1660	−0.645719	−0.322860	−	0.946447i	$-0.604644\pi$
$881$	−19.1660	−0.645719	−0.322860	+	0.946447i	$0.604644\pi$
$882$	0	0
$883$	−55.5830	−1.87052	−0.935259	−	0.353965i	$-0.884833\pi$
$883$	−55.5830	−1.87052	−0.935259	+	0.353965i	$0.884833\pi$
$884$	12.0000	0.403604
$885$	−38.5830	−1.29695
$886$	−11.1660	−0.375129
$887$	43.0405	1.44516	0.722580	−	0.691288i	$-0.242956\pi$
$887$	43.0405	1.44516	0.722580	+	0.691288i	$0.242956\pi$
$888$	−9.29150	−0.311802
$889$	0	0
$890$	35.1660	1.17877
$891$	1.00000	0.0335013
$892$	−24.4575	−0.818898
$893$	−63.1660	−2.11377
$894$	−11.8745	−0.397143
$895$	−40.4575	−1.35235
$896$	0	0
$897$	10.5830	0.353356
$898$	22.4575	0.749417
$899$	−8.00000	−0.266815
$900$	2.00000	0.0666667
$901$	−12.0000	−0.399778
$902$	−9.00000	−0.299667
$903$	0	0
$904$	8.58301	0.285467
$905$	−14.0000	−0.465376
$906$	17.3542	0.576556
$907$	−58.1660	−1.93137	−0.965685	−	0.259715i	$-0.916371\pi$
$907$	−58.1660	−1.93137	−0.965685	+	0.259715i	$0.916371\pi$
$908$	23.5830	0.782630
$909$	−3.29150	−0.109172
$910$	0	0
$911$	50.3948	1.66965	0.834827	−	0.550513i	$-0.185568\pi$
$911$	50.3948	1.66965	0.834827	+	0.550513i	$0.185568\pi$
$912$	5.29150	0.175219
$913$	−5.58301	−0.184771
$914$	19.4170	0.642257
$915$	−10.4170	−0.344375
$916$	17.2915	0.571327
$917$	0	0
$918$	−3.00000	−0.0990148
$919$	14.6458	0.483119	0.241559	−	0.970386i	$-0.422341\pi$
$919$	14.6458	0.483119	0.241559	+	0.970386i	$0.422341\pi$
$920$	7.00000	0.230783
$921$	32.5830	1.07365
$922$	3.41699	0.112533
$923$	53.1660	1.74998
$924$	0	0
$925$	−18.5830	−0.611005
$926$	35.0405	1.15150
$927$	−10.0000	−0.328443
$928$	−2.00000	−0.0656532
$929$	18.4575	0.605571	0.302786	−	0.953059i	$-0.402083\pi$
$929$	18.4575	0.605571	0.302786	+	0.953059i	$0.402083\pi$
$930$	10.5830	0.347030
$931$	0	0
$932$	−22.1660	−0.726072
$933$	14.5203	0.475372
$934$	−8.00000	−0.261768
$935$	7.93725	0.259576
$936$	4.00000	0.130744
$937$	50.0000	1.63343	0.816714	−	0.577042i	$-0.195793\pi$
$937$	50.0000	1.63343	0.816714	+	0.577042i	$0.195793\pi$
$938$	0	0
$939$	−0.583005	−0.0190257
$940$	−31.5830	−1.03012
$941$	24.5830	0.801383	0.400692	−	0.916213i	$-0.368770\pi$
$941$	24.5830	0.801383	0.400692	+	0.916213i	$0.368770\pi$
$942$	−11.2915	−0.367897
$943$	23.8118	0.775418
$944$	−14.5830	−0.474636
$945$	0	0
$946$	1.29150	0.0419904
$947$	−36.5830	−1.18879	−0.594394	−	0.804174i	$-0.702608\pi$
$947$	−36.5830	−1.18879	−0.594394	+	0.804174i	$0.702608\pi$
$948$	−5.35425	−0.173898
$949$	18.8340	0.611377
$950$	10.5830	0.343358
$951$	17.2288	0.558681
$952$	0	0
$953$	−36.7490	−1.19042	−0.595209	−	0.803571i	$-0.702931\pi$
$953$	−36.7490	−1.19042	−0.595209	+	0.803571i	$0.702931\pi$
$954$	−4.00000	−0.129505
$955$	−45.7490	−1.48040
$956$	13.2915	0.429878
$957$	−2.00000	−0.0646508
$958$	2.12549	0.0686715
$959$	0	0
$960$	2.64575	0.0853913
$961$	−15.0000	−0.483871
$962$	−37.1660	−1.19828
$963$	9.00000	0.290021
$964$	15.1660	0.488464
$965$	−42.0000	−1.35203
$966$	0	0
$967$	55.9373	1.79882	0.899410	−	0.437105i	$-0.143996\pi$
$967$	55.9373	1.79882	0.899410	+	0.437105i	$0.143996\pi$
$968$	−1.00000	−0.0321412
$969$	−15.8745	−0.509963
$970$	25.3542	0.814076
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	9.87451	0.316400
$975$	8.00000	0.256205
$976$	−3.93725	−0.126028
$977$	−29.1660	−0.933103	−0.466552	−	0.884494i	$-0.654504\pi$
$977$	−29.1660	−0.933103	−0.466552	+	0.884494i	$0.654504\pi$
$978$	15.5830	0.498290
$979$	13.2915	0.424798
$980$	0	0
$981$	11.9373	0.381127
$982$	1.00000	0.0319113
$983$	5.47974	0.174777	0.0873883	−	0.996174i	$-0.472148\pi$
$983$	5.47974	0.174777	0.0873883	+	0.996174i	$0.472148\pi$
$984$	9.00000	0.286910
$985$	26.1255	0.832427
$986$	6.00000	0.191079
$987$	0	0
$988$	21.1660	0.673380
$989$	−3.41699	−0.108654
$990$	2.64575	0.0840875
$991$	−12.1255	−0.385179	−0.192589	−	0.981279i	$-0.561689\pi$
$991$	−12.1255	−0.385179	−0.192589	+	0.981279i	$0.561689\pi$
$992$	4.00000	0.127000
$993$	−13.0000	−0.412543
$994$	0	0
$995$	−61.2915	−1.94307
$996$	5.58301	0.176904
$997$	49.1660	1.55710	0.778552	−	0.627581i	$-0.215955\pi$
$997$	49.1660	1.55710	0.778552	+	0.627581i	$0.215955\pi$
$998$	−2.58301	−0.0817636
$999$	9.29150	0.293970

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.w.1.1		2
3.2	odd	2		9702.2.a.db.1.2		2
7.2	even	3		462.2.i.e.67.2	✓	4
7.4	even	3		462.2.i.e.331.2	yes	4
7.6	odd	2		3234.2.a.ba.1.2		2
21.2	odd	6		1386.2.k.r.991.1		4
21.11	odd	6		1386.2.k.r.793.1		4
21.20	even	2		9702.2.a.dm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.e.67.2	✓	4	7.2	even	3
462.2.i.e.331.2	yes	4	7.4	even	3
1386.2.k.r.793.1		4	21.11	odd	6
1386.2.k.r.991.1		4	21.2	odd	6
3234.2.a.w.1.1		2	1.1	even	1	trivial
3234.2.a.ba.1.2		2	7.6	odd	2
9702.2.a.db.1.2		2	3.2	odd	2
9702.2.a.dm.1.1		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.64575 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.64575 q^{10} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +2.64575 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -5.29150 q^{19} -2.64575 q^{20} -1.00000 q^{22} +2.64575 q^{23} +1.00000 q^{24} +2.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} -2.64575 q^{30} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -9.29150 q^{37} +5.29150 q^{38} +4.00000 q^{39} +2.64575 q^{40} +9.00000 q^{41} -1.29150 q^{43} +1.00000 q^{44} -2.64575 q^{45} -2.64575 q^{46} +11.9373 q^{47} -1.00000 q^{48} -2.00000 q^{50} +3.00000 q^{51} -4.00000 q^{52} +4.00000 q^{53} +1.00000 q^{54} -2.64575 q^{55} +5.29150 q^{57} -2.00000 q^{58} -14.5830 q^{59} +2.64575 q^{60} -3.93725 q^{61} +4.00000 q^{62} +1.00000 q^{64} +10.5830 q^{65} +1.00000 q^{66} -13.5830 q^{67} -3.00000 q^{68} -2.64575 q^{69} -13.2915 q^{71} -1.00000 q^{72} -4.70850 q^{73} +9.29150 q^{74} -2.00000 q^{75} -5.29150 q^{76} -4.00000 q^{78} +5.35425 q^{79} -2.64575 q^{80} +1.00000 q^{81} -9.00000 q^{82} -5.58301 q^{83} +7.93725 q^{85} +1.29150 q^{86} -2.00000 q^{87} -1.00000 q^{88} +13.2915 q^{89} +2.64575 q^{90} +2.64575 q^{92} +4.00000 q^{93} -11.9373 q^{94} +14.0000 q^{95} +1.00000 q^{96} +9.58301 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 8 q^{13} + 2 q^{16} - 6 q^{17} - 2 q^{18} - 2 q^{22} + 2 q^{24} + 4 q^{25} + 8 q^{26} - 2 q^{27} + 4 q^{29} - 8 q^{31}+ \cdots + 2 q^{99}+O(q^{100}) \) 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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