LMFDB - Embedded newform 3311.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3311,2,Mod(1,3311)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3311, 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("3311.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3311 = 7 \cdot 11 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3311.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:	$26.4384681092$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3311.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} +1.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} -8.00000 q^{15} -1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} +4.00000 q^{20} -2.00000 q^{21} +1.00000 q^{22} -6.00000 q^{24} +11.0000 q^{25} -4.00000 q^{26} -4.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -8.00000 q^{30} +4.00000 q^{31} +5.00000 q^{32} +2.00000 q^{33} -4.00000 q^{34} +4.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} +8.00000 q^{38} -8.00000 q^{39} +12.0000 q^{40} +12.0000 q^{41} -2.00000 q^{42} -1.00000 q^{43} -1.00000 q^{44} -4.00000 q^{45} +8.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +11.0000 q^{50} -8.00000 q^{51} +4.00000 q^{52} -10.0000 q^{53} -4.00000 q^{54} -4.00000 q^{55} +3.00000 q^{56} +16.0000 q^{57} +6.00000 q^{58} +4.00000 q^{59} +8.00000 q^{60} -6.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} +16.0000 q^{65} +2.00000 q^{66} -12.0000 q^{67} +4.00000 q^{68} +4.00000 q^{70} +12.0000 q^{71} -3.00000 q^{72} +14.0000 q^{73} +2.00000 q^{74} +22.0000 q^{75} -8.00000 q^{76} -1.00000 q^{77} -8.00000 q^{78} -4.00000 q^{79} +4.00000 q^{80} -11.0000 q^{81} +12.0000 q^{82} -6.00000 q^{83} +2.00000 q^{84} +16.0000 q^{85} -1.00000 q^{86} +12.0000 q^{87} -3.00000 q^{88} -4.00000 q^{90} +4.00000 q^{91} +8.00000 q^{93} +8.00000 q^{94} -32.0000 q^{95} +10.0000 q^{96} -2.00000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.00000	−0.500000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.00000	−0.577350
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−1.00000	−0.267261
$15$	−8.00000	−2.06559
$16$	−1.00000	−0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.00000	0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	4.00000	0.894427
$21$	−2.00000	−0.436436
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−6.00000	−1.22474
$25$	11.0000	2.20000
$26$	−4.00000	−0.784465
$27$	−4.00000	−0.769800
$28$	1.00000	0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−8.00000	−1.46059
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.00000	0.883883
$33$	2.00000	0.348155
$34$	−4.00000	−0.685994
$35$	4.00000	0.676123
$36$	−1.00000	−0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	8.00000	1.29777
$39$	−8.00000	−1.28103
$40$	12.0000	1.89737
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	−2.00000	−0.308607
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−1.00000	−0.150756
$45$	−4.00000	−0.596285
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−2.00000	−0.288675
$49$	1.00000	0.142857
$50$	11.0000	1.55563
$51$	−8.00000	−1.12022
$52$	4.00000	0.554700
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	−4.00000	−0.544331
$55$	−4.00000	−0.539360
$56$	3.00000	0.400892
$57$	16.0000	2.11925
$58$	6.00000	0.787839
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	8.00000	1.03280
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	4.00000	0.508001
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	16.0000	1.98456
$66$	2.00000	0.246183
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	4.00000	0.478091
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−3.00000	−0.353553
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	2.00000	0.232495
$75$	22.0000	2.54034
$76$	−8.00000	−0.917663
$77$	−1.00000	−0.113961
$78$	−8.00000	−0.905822
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	4.00000	0.447214
$81$	−11.0000	−1.22222
$82$	12.0000	1.32518
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	2.00000	0.218218
$85$	16.0000	1.73544
$86$	−1.00000	−0.107833
$87$	12.0000	1.28654
$88$	−3.00000	−0.319801
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−4.00000	−0.421637
$91$	4.00000	0.419314
$92$	0	0
$93$	8.00000	0.829561
$94$	8.00000	0.825137
$95$	−32.0000	−3.28313
$96$	10.0000	1.02062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	1.00000	0.101015
$99$	1.00000	0.100504
$100$	−11.0000	−1.10000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	−8.00000	−0.792118
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	12.0000	1.17670
$105$	8.00000	0.780720
$106$	−10.0000	−0.971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	4.00000	0.384900
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−4.00000	−0.381385
$111$	4.00000	0.379663
$112$	1.00000	0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	16.0000	1.49854
$115$	0	0
$116$	−6.00000	−0.557086
$117$	−4.00000	−0.369800
$118$	4.00000	0.368230
$119$	4.00000	0.366679
$120$	24.0000	2.19089
$121$	1.00000	0.0909091
$122$	−6.00000	−0.543214
$123$	24.0000	2.16401
$124$	−4.00000	−0.359211
$125$	−24.0000	−2.14663
$126$	−1.00000	−0.0890871
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−3.00000	−0.265165
$129$	−2.00000	−0.176090
$130$	16.0000	1.40329
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−2.00000	−0.174078
$133$	−8.00000	−0.693688
$134$	−12.0000	−1.03664
$135$	16.0000	1.37706
$136$	12.0000	1.02899
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	−4.00000	−0.338062
$141$	16.0000	1.34744
$142$	12.0000	1.00702
$143$	−4.00000	−0.334497
$144$	−1.00000	−0.0833333
$145$	−24.0000	−1.99309
$146$	14.0000	1.15865
$147$	2.00000	0.164957
$148$	−2.00000	−0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	22.0000	1.79629
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−24.0000	−1.94666
$153$	−4.00000	−0.323381
$154$	−1.00000	−0.0805823
$155$	−16.0000	−1.28515
$156$	8.00000	0.640513
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	−4.00000	−0.318223
$159$	−20.0000	−1.58610
$160$	−20.0000	−1.58114
$161$	0	0
$162$	−11.0000	−0.864242
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−12.0000	−0.937043
$165$	−8.00000	−0.622799
$166$	−6.00000	−0.465690
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	6.00000	0.462910
$169$	3.00000	0.230769
$170$	16.0000	1.22714
$171$	8.00000	0.611775
$172$	1.00000	0.0762493
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	12.0000	0.909718
$175$	−11.0000	−0.831522
$176$	−1.00000	−0.0753778
$177$	8.00000	0.601317
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	4.00000	0.298142
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	4.00000	0.296500
$183$	−12.0000	−0.887066
$184$	0	0
$185$	−8.00000	−0.588172
$186$	8.00000	0.586588
$187$	−4.00000	−0.292509
$188$	−8.00000	−0.583460
$189$	4.00000	0.290957
$190$	−32.0000	−2.32152
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	14.0000	1.01036
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−2.00000	−0.143592
$195$	32.0000	2.29157
$196$	−1.00000	−0.0714286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	1.00000	0.0710669
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	−33.0000	−2.33345
$201$	−24.0000	−1.69283
$202$	−12.0000	−0.844317
$203$	−6.00000	−0.421117
$204$	8.00000	0.560112
$205$	−48.0000	−3.35247
$206$	4.00000	0.278693
$207$	0	0
$208$	4.00000	0.277350
$209$	8.00000	0.553372
$210$	8.00000	0.552052
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	10.0000	0.686803
$213$	24.0000	1.64445
$214$	−12.0000	−0.820303
$215$	4.00000	0.272798
$216$	12.0000	0.816497
$217$	−4.00000	−0.271538
$218$	10.0000	0.677285
$219$	28.0000	1.89206
$220$	4.00000	0.269680
$221$	16.0000	1.07628
$222$	4.00000	0.268462
$223$	18.0000	1.20537	0.602685	−	0.797980i	$-0.294098\pi$
$223$	18.0000	1.20537	0.602685	+	0.797980i	$0.294098\pi$
$224$	−5.00000	−0.334077
$225$	11.0000	0.733333
$226$	−14.0000	−0.931266
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−16.0000	−1.05963
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	−18.0000	−1.18176
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	−4.00000	−0.261488
$235$	−32.0000	−2.08745
$236$	−4.00000	−0.260378
$237$	−8.00000	−0.519656
$238$	4.00000	0.259281
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	8.00000	0.516398
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	1.00000	0.0642824
$243$	−10.0000	−0.641500
$244$	6.00000	0.384111
$245$	−4.00000	−0.255551
$246$	24.0000	1.53018
$247$	−32.0000	−2.03611
$248$	−12.0000	−0.762001
$249$	−12.0000	−0.760469
$250$	−24.0000	−1.51789
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	0	0
$255$	32.0000	2.00392
$256$	−17.0000	−1.06250
$257$	4.00000	0.249513	0.124757	−	0.992187i	$-0.460185\pi$
$257$	4.00000	0.249513	0.124757	+	0.992187i	$0.460185\pi$
$258$	−2.00000	−0.124515
$259$	−2.00000	−0.124274
$260$	−16.0000	−0.992278
$261$	6.00000	0.371391
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	−6.00000	−0.369274
$265$	40.0000	2.45718
$266$	−8.00000	−0.490511
$267$	0	0
$268$	12.0000	0.733017
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	16.0000	0.973729
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	4.00000	0.242536
$273$	8.00000	0.484182
$274$	6.00000	0.362473
$275$	11.0000	0.663325
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	14.0000	0.839664
$279$	4.00000	0.239474
$280$	−12.0000	−0.717137
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	16.0000	0.952786
$283$	30.0000	1.78331	0.891657	−	0.452711i	$-0.149543\pi$
$283$	30.0000	1.78331	0.891657	+	0.452711i	$0.149543\pi$
$284$	−12.0000	−0.712069
$285$	−64.0000	−3.79103
$286$	−4.00000	−0.236525
$287$	−12.0000	−0.708338
$288$	5.00000	0.294628
$289$	−1.00000	−0.0588235
$290$	−24.0000	−1.40933
$291$	−4.00000	−0.234484
$292$	−14.0000	−0.819288
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	2.00000	0.116642
$295$	−16.0000	−0.931556
$296$	−6.00000	−0.348743
$297$	−4.00000	−0.232104
$298$	−10.0000	−0.579284
$299$	0	0
$300$	−22.0000	−1.27017
$301$	1.00000	0.0576390
$302$	−8.00000	−0.460348
$303$	−24.0000	−1.37876
$304$	−8.00000	−0.458831
$305$	24.0000	1.37424
$306$	−4.00000	−0.228665
$307$	34.0000	1.94048	0.970241	−	0.242140i	$-0.0778494\pi$
$307$	34.0000	1.94048	0.970241	+	0.242140i	$0.0778494\pi$
$308$	1.00000	0.0569803
$309$	8.00000	0.455104
$310$	−16.0000	−0.908739
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	24.0000	1.35873
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	12.0000	0.677199
$315$	4.00000	0.225374
$316$	4.00000	0.225018
$317$	−34.0000	−1.90963	−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−34.0000	−1.90963	−0.954815	+	0.297200i	$0.903947\pi$
$318$	−20.0000	−1.12154
$319$	6.00000	0.335936
$320$	−28.0000	−1.56525
$321$	−24.0000	−1.33955
$322$	0	0
$323$	−32.0000	−1.78053
$324$	11.0000	0.611111
$325$	−44.0000	−2.44068
$326$	20.0000	1.10770
$327$	20.0000	1.10600
$328$	−36.0000	−1.98777
$329$	−8.00000	−0.441054
$330$	−8.00000	−0.440386
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	6.00000	0.329293
$333$	2.00000	0.109599
$334$	18.0000	0.984916
$335$	48.0000	2.62252
$336$	2.00000	0.109109
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	3.00000	0.163178
$339$	−28.0000	−1.52075
$340$	−16.0000	−0.867722
$341$	4.00000	0.216612
$342$	8.00000	0.432590
$343$	−1.00000	−0.0539949
$344$	3.00000	0.161749
$345$	0	0
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−12.0000	−0.643268
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	−11.0000	−0.587975
$351$	16.0000	0.854017
$352$	5.00000	0.266501
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	8.00000	0.425195
$355$	−48.0000	−2.54758
$356$	0	0
$357$	8.00000	0.423405
$358$	4.00000	0.211407
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	12.0000	0.632456
$361$	45.0000	2.36842
$362$	−14.0000	−0.735824
$363$	2.00000	0.104973
$364$	−4.00000	−0.209657
$365$	−56.0000	−2.93117
$366$	−12.0000	−0.627250
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	−8.00000	−0.415900
$371$	10.0000	0.519174
$372$	−8.00000	−0.414781
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−4.00000	−0.206835
$375$	−48.0000	−2.47871
$376$	−24.0000	−1.23771
$377$	−24.0000	−1.23606
$378$	4.00000	0.205738
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	32.0000	1.64157
$381$	0	0
$382$	12.0000	0.613973
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	−6.00000	−0.306186
$385$	4.00000	0.203859
$386$	10.0000	0.508987
$387$	−1.00000	−0.0508329
$388$	2.00000	0.101535
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	32.0000	1.62038
$391$	0	0
$392$	−3.00000	−0.151523
$393$	0	0
$394$	−2.00000	−0.100759
$395$	16.0000	0.805047
$396$	−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	$0.548110\pi$
$398$	18.0000	0.902258
$399$	−16.0000	−0.801002
$400$	−11.0000	−0.550000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−24.0000	−1.19701
$403$	−16.0000	−0.797017
$404$	12.0000	0.597022
$405$	44.0000	2.18638
$406$	−6.00000	−0.297775
$407$	2.00000	0.0991363
$408$	24.0000	1.18818
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	−48.0000	−2.37055
$411$	12.0000	0.591916
$412$	−4.00000	−0.197066
$413$	−4.00000	−0.196827
$414$	0	0
$415$	24.0000	1.17811
$416$	−20.0000	−0.980581
$417$	28.0000	1.37117
$418$	8.00000	0.391293
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	−8.00000	−0.390360
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	28.0000	1.36302
$423$	8.00000	0.388973
$424$	30.0000	1.45693
$425$	−44.0000	−2.13431
$426$	24.0000	1.16280
$427$	6.00000	0.290360
$428$	12.0000	0.580042
$429$	−8.00000	−0.386244
$430$	4.00000	0.192897
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	4.00000	0.192450
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	−4.00000	−0.192006
$435$	−48.0000	−2.30142
$436$	−10.0000	−0.478913
$437$	0	0
$438$	28.0000	1.33789
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	12.0000	0.572078
$441$	1.00000	0.0476190
$442$	16.0000	0.761042
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	18.0000	0.852325
$447$	−20.0000	−0.945968
$448$	−7.00000	−0.330719
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	11.0000	0.518545
$451$	12.0000	0.565058
$452$	14.0000	0.658505
$453$	−16.0000	−0.751746
$454$	−24.0000	−1.12638
$455$	−16.0000	−0.750092
$456$	−48.0000	−2.24781
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	2.00000	0.0934539
$459$	16.0000	0.746816
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	−2.00000	−0.0930484
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−6.00000	−0.278543
$465$	−32.0000	−1.48396
$466$	22.0000	1.01913
$467$	−30.0000	−1.38823	−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000	−1.38823	−0.694117	+	0.719862i	$0.744205\pi$
$468$	4.00000	0.184900
$469$	12.0000	0.554109
$470$	−32.0000	−1.47605
$471$	24.0000	1.10586
$472$	−12.0000	−0.552345
$473$	−1.00000	−0.0459800
$474$	−8.00000	−0.367452
$475$	88.0000	4.03772
$476$	−4.00000	−0.183340
$477$	−10.0000	−0.457869
$478$	12.0000	0.548867
$479$	38.0000	1.73626	0.868132	−	0.496333i	$-0.165321\pi$
$479$	38.0000	1.73626	0.868132	+	0.496333i	$0.165321\pi$
$480$	−40.0000	−1.82574
$481$	−8.00000	−0.364769
$482$	18.0000	0.819878
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	8.00000	0.363261
$486$	−10.0000	−0.453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	18.0000	0.814822
$489$	40.0000	1.80886
$490$	−4.00000	−0.180702
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	−24.0000	−1.08200
$493$	−24.0000	−1.08091
$494$	−32.0000	−1.43975
$495$	−4.00000	−0.179787
$496$	−4.00000	−0.179605
$497$	−12.0000	−0.538274
$498$	−12.0000	−0.537733
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	24.0000	1.07331
$501$	36.0000	1.60836
$502$	28.0000	1.24970
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	3.00000	0.133631
$505$	48.0000	2.13597
$506$	0	0
$507$	6.00000	0.266469
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	32.0000	1.41698
$511$	−14.0000	−0.619324
$512$	−11.0000	−0.486136
$513$	−32.0000	−1.41283
$514$	4.00000	0.176432
$515$	−16.0000	−0.705044
$516$	2.00000	0.0880451
$517$	8.00000	0.351840
$518$	−2.00000	−0.0878750
$519$	0	0
$520$	−48.0000	−2.10494
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	6.00000	0.262613
$523$	40.0000	1.74908	0.874539	−	0.484955i	$-0.161164\pi$
$523$	40.0000	1.74908	0.874539	+	0.484955i	$0.161164\pi$
$524$	0	0
$525$	−22.0000	−0.960159
$526$	24.0000	1.04645
$527$	−16.0000	−0.696971
$528$	−2.00000	−0.0870388
$529$	−23.0000	−1.00000
$530$	40.0000	1.73749
$531$	4.00000	0.173585
$532$	8.00000	0.346844
$533$	−48.0000	−2.07911
$534$	0	0
$535$	48.0000	2.07522
$536$	36.0000	1.55496
$537$	8.00000	0.345225
$538$	30.0000	1.29339
$539$	1.00000	0.0430730
$540$	−16.0000	−0.688530
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−2.00000	−0.0859074
$543$	−28.0000	−1.20160
$544$	−20.0000	−0.857493
$545$	−40.0000	−1.71341
$546$	8.00000	0.342368
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	−6.00000	−0.256307
$549$	−6.00000	−0.256074
$550$	11.0000	0.469042
$551$	48.0000	2.04487
$552$	0	0
$553$	4.00000	0.170097
$554$	−22.0000	−0.934690
$555$	−16.0000	−0.679162
$556$	−14.0000	−0.593732
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	4.00000	0.169334
$559$	4.00000	0.169182
$560$	−4.00000	−0.169031
$561$	−8.00000	−0.337760
$562$	−2.00000	−0.0843649
$563$	14.0000	0.590030	0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000	0.590030	0.295015	+	0.955493i	$0.404675\pi$
$564$	−16.0000	−0.673722
$565$	56.0000	2.35594
$566$	30.0000	1.26099
$567$	11.0000	0.461957
$568$	−36.0000	−1.51053
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	−64.0000	−2.68067
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	4.00000	0.167248
$573$	24.0000	1.00261
$574$	−12.0000	−0.500870
$575$	0	0
$576$	7.00000	0.291667
$577$	12.0000	0.499567	0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000	0.499567	0.249783	+	0.968302i	$0.419641\pi$
$578$	−1.00000	−0.0415945
$579$	20.0000	0.831172
$580$	24.0000	0.996546
$581$	6.00000	0.248922
$582$	−4.00000	−0.165805
$583$	−10.0000	−0.414158
$584$	−42.0000	−1.73797
$585$	16.0000	0.661519
$586$	−24.0000	−0.991431
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	−2.00000	−0.0824786
$589$	32.0000	1.31854
$590$	−16.0000	−0.658710
$591$	−4.00000	−0.164538
$592$	−2.00000	−0.0821995
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	−4.00000	−0.164122
$595$	−16.0000	−0.655936
$596$	10.0000	0.409616
$597$	36.0000	1.47338
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	−66.0000	−2.69444
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	1.00000	0.0407570
$603$	−12.0000	−0.488678
$604$	8.00000	0.325515
$605$	−4.00000	−0.162623
$606$	−24.0000	−0.974933
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	40.0000	1.62221
$609$	−12.0000	−0.486265
$610$	24.0000	0.971732
$611$	−32.0000	−1.29458
$612$	4.00000	0.161690
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	34.0000	1.37213
$615$	−96.0000	−3.87109
$616$	3.00000	0.120873
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	8.00000	0.321807
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	16.0000	0.642575
$621$	0	0
$622$	−12.0000	−0.481156
$623$	0	0
$624$	8.00000	0.320256
$625$	41.0000	1.64000
$626$	−4.00000	−0.159872
$627$	16.0000	0.638978
$628$	−12.0000	−0.478852
$629$	−8.00000	−0.318981
$630$	4.00000	0.159364
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	12.0000	0.477334
$633$	56.0000	2.22580
$634$	−34.0000	−1.35031
$635$	0	0
$636$	20.0000	0.793052
$637$	−4.00000	−0.158486
$638$	6.00000	0.237542
$639$	12.0000	0.474713
$640$	12.0000	0.474342
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	−24.0000	−0.947204
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	−32.0000	−1.25902
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	33.0000	1.29636
$649$	4.00000	0.157014
$650$	−44.0000	−1.72582
$651$	−8.00000	−0.313545
$652$	−20.0000	−0.783260
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	20.0000	0.782062
$655$	0	0
$656$	−12.0000	−0.468521
$657$	14.0000	0.546192
$658$	−8.00000	−0.311872
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	8.00000	0.311400
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	32.0000	1.24278
$664$	18.0000	0.698535
$665$	32.0000	1.24091
$666$	2.00000	0.0774984
$667$	0	0
$668$	−18.0000	−0.696441
$669$	36.0000	1.39184
$670$	48.0000	1.85440
$671$	−6.00000	−0.231627
$672$	−10.0000	−0.385758
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	30.0000	1.15556
$675$	−44.0000	−1.69356
$676$	−3.00000	−0.115385
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	−28.0000	−1.07533
$679$	2.00000	0.0767530
$680$	−48.0000	−1.84072
$681$	−48.0000	−1.83936
$682$	4.00000	0.153168
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−8.00000	−0.305888
$685$	−24.0000	−0.916993
$686$	−1.00000	−0.0381802
$687$	4.00000	0.152610
$688$	1.00000	0.0381246
$689$	40.0000	1.52388
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	−12.0000	−0.455514
$695$	−56.0000	−2.12420
$696$	−36.0000	−1.36458
$697$	−48.0000	−1.81813
$698$	−22.0000	−0.832712
$699$	44.0000	1.66423
$700$	11.0000	0.415761
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	16.0000	0.603881
$703$	16.0000	0.603451
$704$	7.00000	0.263822
$705$	−64.0000	−2.41038
$706$	−10.0000	−0.376355
$707$	12.0000	0.451306
$708$	−8.00000	−0.300658
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−48.0000	−1.80141
$711$	−4.00000	−0.150012
$712$	0	0
$713$	0	0
$714$	8.00000	0.299392
$715$	16.0000	0.598366
$716$	−4.00000	−0.149487
$717$	24.0000	0.896296
$718$	−12.0000	−0.447836
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	4.00000	0.149071
$721$	−4.00000	−0.148968
$722$	45.0000	1.67473
$723$	36.0000	1.33885
$724$	14.0000	0.520306
$725$	66.0000	2.45118
$726$	2.00000	0.0742270
$727$	−38.0000	−1.40934	−0.704671	−	0.709534i	$-0.748905\pi$
$727$	−38.0000	−1.40934	−0.704671	+	0.709534i	$0.748905\pi$
$728$	−12.0000	−0.444750
$729$	13.0000	0.481481
$730$	−56.0000	−2.07265
$731$	4.00000	0.147945
$732$	12.0000	0.443533
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	−28.0000	−1.03350
$735$	−8.00000	−0.295084
$736$	0	0
$737$	−12.0000	−0.442026
$738$	12.0000	0.441726
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	8.00000	0.294086
$741$	−64.0000	−2.35110
$742$	10.0000	0.367112
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	−24.0000	−0.879883
$745$	40.0000	1.46549
$746$	−10.0000	−0.366126
$747$	−6.00000	−0.219529
$748$	4.00000	0.146254
$749$	12.0000	0.438470
$750$	−48.0000	−1.75271
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	−8.00000	−0.291730
$753$	56.0000	2.04075
$754$	−24.0000	−0.874028
$755$	32.0000	1.16460
$756$	−4.00000	−0.145479
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	96.0000	3.48229
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	−12.0000	−0.434145
$765$	16.0000	0.578481
$766$	−14.0000	−0.505841
$767$	−16.0000	−0.577727
$768$	−34.0000	−1.22687
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	4.00000	0.144150
$771$	8.00000	0.288113
$772$	−10.0000	−0.359908
$773$	−28.0000	−1.00709	−0.503545	−	0.863969i	$-0.667971\pi$
$773$	−28.0000	−1.00709	−0.503545	+	0.863969i	$0.667971\pi$
$774$	−1.00000	−0.0359443
$775$	44.0000	1.58053
$776$	6.00000	0.215387
$777$	−4.00000	−0.143499
$778$	−30.0000	−1.07555
$779$	96.0000	3.43956
$780$	−32.0000	−1.14578
$781$	12.0000	0.429394
$782$	0	0
$783$	−24.0000	−0.857690
$784$	−1.00000	−0.0357143
$785$	−48.0000	−1.71319
$786$	0	0
$787$	−6.00000	−0.213877	−0.106938	−	0.994266i	$-0.534105\pi$
$787$	−6.00000	−0.213877	−0.106938	+	0.994266i	$0.534105\pi$
$788$	2.00000	0.0712470
$789$	48.0000	1.70885
$790$	16.0000	0.569254
$791$	14.0000	0.497783
$792$	−3.00000	−0.106600
$793$	24.0000	0.852265
$794$	−6.00000	−0.212932
$795$	80.0000	2.83731
$796$	−18.0000	−0.637993
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	−16.0000	−0.566394
$799$	−32.0000	−1.13208
$800$	55.0000	1.94454
$801$	0	0
$802$	18.0000	0.635602
$803$	14.0000	0.494049
$804$	24.0000	0.846415
$805$	0	0
$806$	−16.0000	−0.563576
$807$	60.0000	2.11210
$808$	36.0000	1.26648
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	44.0000	1.54600
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	6.00000	0.210559
$813$	−4.00000	−0.140286
$814$	2.00000	0.0701000
$815$	−80.0000	−2.80228
$816$	8.00000	0.280056
$817$	−8.00000	−0.279885
$818$	34.0000	1.18878
$819$	4.00000	0.139771
$820$	48.0000	1.67623
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	12.0000	0.418548
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	−12.0000	−0.418040
$825$	22.0000	0.765942
$826$	−4.00000	−0.139178
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	24.0000	0.833052
$831$	−44.0000	−1.52634
$832$	−28.0000	−0.970725
$833$	−4.00000	−0.138592
$834$	28.0000	0.969561
$835$	−72.0000	−2.49166
$836$	−8.00000	−0.276686
$837$	−16.0000	−0.553041
$838$	30.0000	1.03633
$839$	22.0000	0.759524	0.379762	−	0.925084i	$-0.376006\pi$
$839$	22.0000	0.759524	0.379762	+	0.925084i	$0.376006\pi$
$840$	−24.0000	−0.828079
$841$	7.00000	0.241379
$842$	22.0000	0.758170
$843$	−4.00000	−0.137767
$844$	−28.0000	−0.963800
$845$	−12.0000	−0.412813
$846$	8.00000	0.275046
$847$	−1.00000	−0.0343604
$848$	10.0000	0.343401
$849$	60.0000	2.05919
$850$	−44.0000	−1.50919
$851$	0	0
$852$	−24.0000	−0.822226
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	6.00000	0.205316
$855$	−32.0000	−1.09438
$856$	36.0000	1.23045
$857$	20.0000	0.683187	0.341593	−	0.939848i	$-0.389033\pi$
$857$	20.0000	0.683187	0.341593	+	0.939848i	$0.389033\pi$
$858$	−8.00000	−0.273115
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	−4.00000	−0.136399
$861$	−24.0000	−0.817918
$862$	−36.0000	−1.22616
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	−20.0000	−0.680414
$865$	0	0
$866$	−28.0000	−0.951479
$867$	−2.00000	−0.0679236
$868$	4.00000	0.135769
$869$	−4.00000	−0.135691
$870$	−48.0000	−1.62735
$871$	48.0000	1.62642
$872$	−30.0000	−1.01593
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	24.0000	0.811348
$876$	−28.0000	−0.946032
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	−14.0000	−0.472477
$879$	−48.0000	−1.61900
$880$	4.00000	0.134840
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	1.00000	0.0336718
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	−16.0000	−0.538138
$885$	−32.0000	−1.07567
$886$	−4.00000	−0.134383
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	−12.0000	−0.402694
$889$	0	0
$890$	0	0
$891$	−11.0000	−0.368514
$892$	−18.0000	−0.602685
$893$	64.0000	2.14168
$894$	−20.0000	−0.668900
$895$	−16.0000	−0.534821
$896$	3.00000	0.100223
$897$	0	0
$898$	10.0000	0.333704
$899$	24.0000	0.800445
$900$	−11.0000	−0.366667
$901$	40.0000	1.33259
$902$	12.0000	0.399556
$903$	2.00000	0.0665558
$904$	42.0000	1.39690
$905$	56.0000	1.86150
$906$	−16.0000	−0.531564
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	24.0000	0.796468
$909$	−12.0000	−0.398015
$910$	−16.0000	−0.530395
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	−16.0000	−0.529813
$913$	−6.00000	−0.198571
$914$	−22.0000	−0.727695
$915$	48.0000	1.58683
$916$	−2.00000	−0.0660819
$917$	0	0
$918$	16.0000	0.528079
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	68.0000	2.24068
$922$	−12.0000	−0.395199
$923$	−48.0000	−1.57994
$924$	2.00000	0.0657952
$925$	22.0000	0.723356
$926$	0	0
$927$	4.00000	0.131377
$928$	30.0000	0.984798
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	−32.0000	−1.04932
$931$	8.00000	0.262189
$932$	−22.0000	−0.720634
$933$	−24.0000	−0.785725
$934$	−30.0000	−0.981630
$935$	16.0000	0.523256
$936$	12.0000	0.392232
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	12.0000	0.391814
$939$	−8.00000	−0.261070
$940$	32.0000	1.04372
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	24.0000	0.781962
$943$	0	0
$944$	−4.00000	−0.130189
$945$	−16.0000	−0.520480
$946$	−1.00000	−0.0325128
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	8.00000	0.259828
$949$	−56.0000	−1.81784
$950$	88.0000	2.85510
$951$	−68.0000	−2.20505
$952$	−12.0000	−0.388922
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	−10.0000	−0.323762
$955$	−48.0000	−1.55324
$956$	−12.0000	−0.388108
$957$	12.0000	0.387905
$958$	38.0000	1.22772
$959$	−6.00000	−0.193750
$960$	−56.0000	−1.80739
$961$	−15.0000	−0.483871
$962$	−8.00000	−0.257930
$963$	−12.0000	−0.386695
$964$	−18.0000	−0.579741
$965$	−40.0000	−1.28765
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	−3.00000	−0.0964237
$969$	−64.0000	−2.05598
$970$	8.00000	0.256865
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	10.0000	0.320750
$973$	−14.0000	−0.448819
$974$	−8.00000	−0.256337
$975$	−88.0000	−2.81826
$976$	6.00000	0.192055
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	40.0000	1.27906
$979$	0	0
$980$	4.00000	0.127775
$981$	10.0000	0.319275
$982$	36.0000	1.14881
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	−72.0000	−2.29528
$985$	8.00000	0.254901
$986$	−24.0000	−0.764316
$987$	−16.0000	−0.509286
$988$	32.0000	1.01806
$989$	0	0
$990$	−4.00000	−0.127128
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	20.0000	0.635001
$993$	0	0
$994$	−12.0000	−0.380617
$995$	−72.0000	−2.28255
$996$	12.0000	0.380235
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	8.00000	0.253236
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3311.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3311.2.a.a.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3311 = 7 \cdot 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3311.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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