LMFDB - Embedded newform 121.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,2,Mod(1,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 121 = 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	121.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:	$0.966189864457$
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	$\chi$	$=$	121.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{12} +1.00000 q^{13} -2.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} +2.00000 q^{23} -6.00000 q^{24} -4.00000 q^{25} +1.00000 q^{26} -4.00000 q^{27} +2.00000 q^{28} +9.00000 q^{29} +2.00000 q^{30} -2.00000 q^{31} +5.00000 q^{32} -5.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} -3.00000 q^{37} +6.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} -5.00000 q^{41} -4.00000 q^{42} +1.00000 q^{45} +2.00000 q^{46} +2.00000 q^{47} -2.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} -10.0000 q^{51} -1.00000 q^{52} +9.00000 q^{53} -4.00000 q^{54} +6.00000 q^{56} +12.0000 q^{57} +9.00000 q^{58} +8.00000 q^{59} -2.00000 q^{60} +6.00000 q^{61} -2.00000 q^{62} -2.00000 q^{63} +7.00000 q^{64} +1.00000 q^{65} +2.00000 q^{67} +5.00000 q^{68} +4.00000 q^{69} -2.00000 q^{70} +12.0000 q^{71} -3.00000 q^{72} -2.00000 q^{73} -3.00000 q^{74} -8.00000 q^{75} -6.00000 q^{76} +2.00000 q^{78} -10.0000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -5.00000 q^{82} +6.00000 q^{83} +4.00000 q^{84} -5.00000 q^{85} +18.0000 q^{87} -9.00000 q^{89} +1.00000 q^{90} -2.00000 q^{91} -2.00000 q^{92} -4.00000 q^{93} +2.00000 q^{94} +6.00000 q^{95} +10.0000 q^{96} -13.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	2.00000	0.816497
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−2.00000	−0.577350
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−2.00000	−0.534522
$15$	2.00000	0.516398
$16$	−1.00000	−0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	1.00000	0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	−0.223607
$21$	−4.00000	−0.872872
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−6.00000	−1.22474
$25$	−4.00000	−0.800000
$26$	1.00000	0.196116
$27$	−4.00000	−0.769800
$28$	2.00000	0.377964
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	2.00000	0.365148
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	−5.00000	−0.857493
$35$	−2.00000	−0.338062
$36$	−1.00000	−0.166667
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	6.00000	0.973329
$39$	2.00000	0.320256
$40$	−3.00000	−0.474342
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	−4.00000	−0.617213
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	2.00000	0.294884
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	−2.00000	−0.288675
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	−10.0000	−1.40028
$52$	−1.00000	−0.138675
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	6.00000	0.801784
$57$	12.0000	1.58944
$58$	9.00000	1.18176
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	−2.00000	−0.258199
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−2.00000	−0.254000
$63$	−2.00000	−0.251976
$64$	7.00000	0.875000
$65$	1.00000	0.124035
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	5.00000	0.606339
$69$	4.00000	0.481543
$70$	−2.00000	−0.239046
$71$	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
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−3.00000	−0.348743
$75$	−8.00000	−0.923760
$76$	−6.00000	−0.688247
$77$	0	0
$78$	2.00000	0.226455
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	−1.00000	−0.111803
$81$	−11.0000	−1.22222
$82$	−5.00000	−0.552158
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	4.00000	0.436436
$85$	−5.00000	−0.542326
$86$	0	0
$87$	18.0000	1.92980
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	1.00000	0.105409
$91$	−2.00000	−0.209657
$92$	−2.00000	−0.208514
$93$	−4.00000	−0.414781
$94$	2.00000	0.206284
$95$	6.00000	0.615587
$96$	10.0000	1.02062
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	4.00000	0.400000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−10.0000	−0.990148
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−3.00000	−0.294174
$105$	−4.00000	−0.390360
$106$	9.00000	0.874157
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	4.00000	0.384900
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	2.00000	0.188982
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	12.0000	1.12390
$115$	2.00000	0.186501
$116$	−9.00000	−0.835629
$117$	1.00000	0.0924500
$118$	8.00000	0.736460
$119$	10.0000	0.916698
$120$	−6.00000	−0.547723
$121$	0	0
$122$	6.00000	0.543214
$123$	−10.0000	−0.901670
$124$	2.00000	0.179605
$125$	−9.00000	−0.804984
$126$	−2.00000	−0.178174
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	1.00000	0.0877058
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	2.00000	0.172774
$135$	−4.00000	−0.344265
$136$	15.0000	1.28624
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	4.00000	0.340503
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	2.00000	0.169031
$141$	4.00000	0.336861
$142$	12.0000	1.00702
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	9.00000	0.747409
$146$	−2.00000	−0.165521
$147$	−6.00000	−0.494872
$148$	3.00000	0.246598
$149$	17.0000	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$149$	17.0000	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$150$	−8.00000	−0.653197
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−18.0000	−1.45999
$153$	−5.00000	−0.404226
$154$	0	0
$155$	−2.00000	−0.160644
$156$	−2.00000	−0.160128
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−10.0000	−0.795557
$159$	18.0000	1.42749
$160$	5.00000	0.395285
$161$	−4.00000	−0.315244
$162$	−11.0000	−0.864242
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	6.00000	0.465690
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	12.0000	0.925820
$169$	−12.0000	−0.923077
$170$	−5.00000	−0.383482
$171$	6.00000	0.458831
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	18.0000	1.36458
$175$	8.00000	0.604743
$176$	0	0
$177$	16.0000	1.20263
$178$	−9.00000	−0.674579
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	−1.00000	−0.0745356
$181$	1.00000	0.0743294	0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000	0.0743294	0.0371647	+	0.999309i	$0.488167\pi$
$182$	−2.00000	−0.148250
$183$	12.0000	0.887066
$184$	−6.00000	−0.442326
$185$	−3.00000	−0.220564
$186$	−4.00000	−0.293294
$187$	0	0
$188$	−2.00000	−0.145865
$189$	8.00000	0.581914
$190$	6.00000	0.435286
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	14.0000	1.01036
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	−13.0000	−0.933346
$195$	2.00000	0.143223
$196$	3.00000	0.214286
$197$	−11.0000	−0.783718	−0.391859	−	0.920025i	$-0.628168\pi$
$197$	−11.0000	−0.783718	−0.391859	+	0.920025i	$0.628168\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	12.0000	0.848528
$201$	4.00000	0.282138
$202$	−10.0000	−0.703598
$203$	−18.0000	−1.26335
$204$	10.0000	0.700140
$205$	−5.00000	−0.349215
$206$	8.00000	0.557386
$207$	2.00000	0.139010
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	−4.00000	−0.276026
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−9.00000	−0.618123
$213$	24.0000	1.64445
$214$	6.00000	0.410152
$215$	0	0
$216$	12.0000	0.816497
$217$	4.00000	0.271538
$218$	−11.0000	−0.745014
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−5.00000	−0.336336
$222$	−6.00000	−0.402694
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	−10.0000	−0.668153
$225$	−4.00000	−0.266667
$226$	−9.00000	−0.598671
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−12.0000	−0.794719
$229$	9.00000	0.594737	0.297368	−	0.954763i	$-0.403891\pi$
$229$	9.00000	0.594737	0.297368	+	0.954763i	$0.403891\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	−27.0000	−1.77264
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	1.00000	0.0653720
$235$	2.00000	0.130466
$236$	−8.00000	−0.520756
$237$	−20.0000	−1.29914
$238$	10.0000	0.648204
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	−2.00000	−0.129099
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	−6.00000	−0.384111
$245$	−3.00000	−0.191663
$246$	−10.0000	−0.637577
$247$	6.00000	0.381771
$248$	6.00000	0.381000
$249$	12.0000	0.760469
$250$	−9.00000	−0.569210
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−16.0000	−1.00393
$255$	−10.0000	−0.626224
$256$	−17.0000	−1.06250
$257$	19.0000	1.18519	0.592594	−	0.805502i	$-0.298104\pi$
$257$	19.0000	1.18519	0.592594	+	0.805502i	$0.298104\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	−1.00000	−0.0620174
$261$	9.00000	0.557086
$262$	0	0
$263$	−22.0000	−1.35658	−0.678289	−	0.734795i	$-0.737278\pi$
$263$	−22.0000	−1.35658	−0.678289	+	0.734795i	$0.737278\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	−12.0000	−0.735767
$267$	−18.0000	−1.10158
$268$	−2.00000	−0.122169
$269$	1.00000	0.0609711	0.0304855	−	0.999535i	$-0.490295\pi$
$269$	1.00000	0.0609711	0.0304855	+	0.999535i	$0.490295\pi$
$270$	−4.00000	−0.243432
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	5.00000	0.303170
$273$	−4.00000	−0.242091
$274$	−10.0000	−0.604122
$275$	0	0
$276$	−4.00000	−0.240772
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	−2.00000	−0.119952
$279$	−2.00000	−0.119737
$280$	6.00000	0.358569
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	4.00000	0.238197
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	−12.0000	−0.712069
$285$	12.0000	0.710819
$286$	0	0
$287$	10.0000	0.590281
$288$	5.00000	0.294628
$289$	8.00000	0.470588
$290$	9.00000	0.528498
$291$	−26.0000	−1.52415
$292$	2.00000	0.117041
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	−6.00000	−0.349927
$295$	8.00000	0.465778
$296$	9.00000	0.523114
$297$	0	0
$298$	17.0000	0.984784
$299$	2.00000	0.115663
$300$	8.00000	0.461880
$301$	0	0
$302$	−16.0000	−0.920697
$303$	−20.0000	−1.14897
$304$	−6.00000	−0.344124
$305$	6.00000	0.343559
$306$	−5.00000	−0.285831
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	−2.00000	−0.113592
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	−6.00000	−0.339683
$313$	23.0000	1.30004	0.650018	−	0.759918i	$-0.274761\pi$
$313$	23.0000	1.30004	0.650018	+	0.759918i	$0.274761\pi$
$314$	2.00000	0.112867
$315$	−2.00000	−0.112687
$316$	10.0000	0.562544
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	18.0000	1.00939
$319$	0	0
$320$	7.00000	0.391312
$321$	12.0000	0.669775
$322$	−4.00000	−0.222911
$323$	−30.0000	−1.66924
$324$	11.0000	0.611111
$325$	−4.00000	−0.221880
$326$	−2.00000	−0.110770
$327$	−22.0000	−1.21660
$328$	15.0000	0.828236
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−6.00000	−0.329293
$333$	−3.00000	−0.164399
$334$	12.0000	0.656611
$335$	2.00000	0.109272
$336$	4.00000	0.218218
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−12.0000	−0.652714
$339$	−18.0000	−0.977626
$340$	5.00000	0.271163
$341$	0	0
$342$	6.00000	0.324443
$343$	20.0000	1.07990
$344$	0	0
$345$	4.00000	0.215353
$346$	6.00000	0.322562
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	−18.0000	−0.964901
$349$	−27.0000	−1.44528	−0.722638	−	0.691226i	$-0.757071\pi$
$349$	−27.0000	−1.44528	−0.722638	+	0.691226i	$0.757071\pi$
$350$	8.00000	0.427618
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−9.00000	−0.479022	−0.239511	−	0.970894i	$-0.576987\pi$
$353$	−9.00000	−0.479022	−0.239511	+	0.970894i	$0.576987\pi$
$354$	16.0000	0.850390
$355$	12.0000	0.636894
$356$	9.00000	0.476999
$357$	20.0000	1.05851
$358$	24.0000	1.26844
$359$	−2.00000	−0.105556	−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000	−0.105556	−0.0527780	+	0.998606i	$0.516808\pi$
$360$	−3.00000	−0.158114
$361$	17.0000	0.894737
$362$	1.00000	0.0525588
$363$	0	0
$364$	2.00000	0.104828
$365$	−2.00000	−0.104685
$366$	12.0000	0.627250
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	−2.00000	−0.104257
$369$	−5.00000	−0.260290
$370$	−3.00000	−0.155963
$371$	−18.0000	−0.934513
$372$	4.00000	0.207390
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	−18.0000	−0.929516
$376$	−6.00000	−0.309426
$377$	9.00000	0.463524
$378$	8.00000	0.411476
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	−6.00000	−0.307794
$381$	−32.0000	−1.63941
$382$	8.00000	0.409316
$383$	20.0000	1.02195	0.510976	−	0.859595i	$-0.329284\pi$
$383$	20.0000	1.02195	0.510976	+	0.859595i	$0.329284\pi$
$384$	−6.00000	−0.306186
$385$	0	0
$386$	−5.00000	−0.254493
$387$	0	0
$388$	13.0000	0.659975
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	2.00000	0.101274
$391$	−10.0000	−0.505722
$392$	9.00000	0.454569
$393$	0	0
$394$	−11.0000	−0.554172
$395$	−10.0000	−0.503155
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	24.0000	1.20301
$399$	−24.0000	−1.20150
$400$	4.00000	0.200000
$401$	23.0000	1.14857	0.574283	−	0.818657i	$-0.305281\pi$
$401$	23.0000	1.14857	0.574283	+	0.818657i	$0.305281\pi$
$402$	4.00000	0.199502
$403$	−2.00000	−0.0996271
$404$	10.0000	0.497519
$405$	−11.0000	−0.546594
$406$	−18.0000	−0.893325
$407$	0	0
$408$	30.0000	1.48522
$409$	−21.0000	−1.03838	−0.519192	−	0.854658i	$-0.673767\pi$
$409$	−21.0000	−1.03838	−0.519192	+	0.854658i	$0.673767\pi$
$410$	−5.00000	−0.246932
$411$	−20.0000	−0.986527
$412$	−8.00000	−0.394132
$413$	−16.0000	−0.787309
$414$	2.00000	0.0982946
$415$	6.00000	0.294528
$416$	5.00000	0.245145
$417$	−4.00000	−0.195881
$418$	0	0
$419$	2.00000	0.0977064	0.0488532	−	0.998806i	$-0.484443\pi$
$419$	2.00000	0.0977064	0.0488532	+	0.998806i	$0.484443\pi$
$420$	4.00000	0.195180
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	12.0000	0.584151
$423$	2.00000	0.0972433
$424$	−27.0000	−1.31124
$425$	20.0000	0.970143
$426$	24.0000	1.16280
$427$	−12.0000	−0.580721
$428$	−6.00000	−0.290021
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	4.00000	0.192450
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	4.00000	0.192006
$435$	18.0000	0.863034
$436$	11.0000	0.526804
$437$	12.0000	0.574038
$438$	−4.00000	−0.191127
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−5.00000	−0.237826
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	6.00000	0.284747
$445$	−9.00000	−0.426641
$446$	−20.0000	−0.947027
$447$	34.0000	1.60814
$448$	−14.0000	−0.661438
$449$	−13.0000	−0.613508	−0.306754	−	0.951789i	$-0.599243\pi$
$449$	−13.0000	−0.613508	−0.306754	+	0.951789i	$0.599243\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	9.00000	0.423324
$453$	−32.0000	−1.50349
$454$	−24.0000	−1.12638
$455$	−2.00000	−0.0937614
$456$	−36.0000	−1.68585
$457$	39.0000	1.82434	0.912172	−	0.409809i	$-0.134405\pi$
$457$	39.0000	1.82434	0.912172	+	0.409809i	$0.134405\pi$
$458$	9.00000	0.420542
$459$	20.0000	0.933520
$460$	−2.00000	−0.0932505
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	−9.00000	−0.417815
$465$	−4.00000	−0.185496
$466$	−21.0000	−0.972806
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−1.00000	−0.0462250
$469$	−4.00000	−0.184703
$470$	2.00000	0.0922531
$471$	4.00000	0.184310
$472$	−24.0000	−1.10469
$473$	0	0
$474$	−20.0000	−0.918630
$475$	−24.0000	−1.10120
$476$	−10.0000	−0.458349
$477$	9.00000	0.412082
$478$	6.00000	0.274434
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	10.0000	0.456435
$481$	−3.00000	−0.136788
$482$	22.0000	1.00207
$483$	−8.00000	−0.364013
$484$	0	0
$485$	−13.0000	−0.590300
$486$	−10.0000	−0.453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−18.0000	−0.814822
$489$	−4.00000	−0.180886
$490$	−3.00000	−0.135526
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	10.0000	0.450835
$493$	−45.0000	−2.02670
$494$	6.00000	0.269953
$495$	0	0
$496$	2.00000	0.0898027
$497$	−24.0000	−1.07655
$498$	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	$0.442693\pi$
$500$	9.00000	0.402492
$501$	24.0000	1.07224
$502$	−2.00000	−0.0892644
$503$	−38.0000	−1.69434	−0.847168	−	0.531325i	$-0.821694\pi$
$503$	−38.0000	−1.69434	−0.847168	+	0.531325i	$0.821694\pi$
$504$	6.00000	0.267261
$505$	−10.0000	−0.444994
$506$	0	0
$507$	−24.0000	−1.06588
$508$	16.0000	0.709885
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	−10.0000	−0.442807
$511$	4.00000	0.176950
$512$	−11.0000	−0.486136
$513$	−24.0000	−1.05963
$514$	19.0000	0.838054
$515$	8.00000	0.352522
$516$	0	0
$517$	0	0
$518$	6.00000	0.263625
$519$	12.0000	0.526742
$520$	−3.00000	−0.131559
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	9.00000	0.393919
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	16.0000	0.698297
$526$	−22.0000	−0.959246
$527$	10.0000	0.435607
$528$	0	0
$529$	−19.0000	−0.826087
$530$	9.00000	0.390935
$531$	8.00000	0.347170
$532$	12.0000	0.520266
$533$	−5.00000	−0.216574
$534$	−18.0000	−0.778936
$535$	6.00000	0.259403
$536$	−6.00000	−0.259161
$537$	48.0000	2.07135
$538$	1.00000	0.0431131
$539$	0	0
$540$	4.00000	0.172133
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	20.0000	0.859074
$543$	2.00000	0.0858282
$544$	−25.0000	−1.07187
$545$	−11.0000	−0.471188
$546$	−4.00000	−0.171184
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	10.0000	0.427179
$549$	6.00000	0.256074
$550$	0	0
$551$	54.0000	2.30048
$552$	−12.0000	−0.510754
$553$	20.0000	0.850487
$554$	1.00000	0.0424859
$555$	−6.00000	−0.254686
$556$	2.00000	0.0848189
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	−2.00000	−0.0846668
$559$	0	0
$560$	2.00000	0.0845154
$561$	0	0
$562$	6.00000	0.253095
$563$	34.0000	1.43293	0.716465	−	0.697623i	$-0.245759\pi$
$563$	34.0000	1.43293	0.716465	+	0.697623i	$0.245759\pi$
$564$	−4.00000	−0.168430
$565$	−9.00000	−0.378633
$566$	28.0000	1.17693
$567$	22.0000	0.923913
$568$	−36.0000	−1.51053
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	12.0000	0.502625
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	10.0000	0.417392
$575$	−8.00000	−0.333623
$576$	7.00000	0.291667
$577$	−21.0000	−0.874241	−0.437121	−	0.899403i	$-0.644002\pi$
$577$	−21.0000	−0.874241	−0.437121	+	0.899403i	$0.644002\pi$
$578$	8.00000	0.332756
$579$	−10.0000	−0.415586
$580$	−9.00000	−0.373705
$581$	−12.0000	−0.497844
$582$	−26.0000	−1.07773
$583$	0	0
$584$	6.00000	0.248282
$585$	1.00000	0.0413449
$586$	9.00000	0.371787
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	6.00000	0.247436
$589$	−12.0000	−0.494451
$590$	8.00000	0.329355
$591$	−22.0000	−0.904959
$592$	3.00000	0.123299
$593$	11.0000	0.451716	0.225858	−	0.974160i	$-0.427481\pi$
$593$	11.0000	0.451716	0.225858	+	0.974160i	$0.427481\pi$
$594$	0	0
$595$	10.0000	0.409960
$596$	−17.0000	−0.696347
$597$	48.0000	1.96451
$598$	2.00000	0.0817861
$599$	34.0000	1.38920	0.694601	−	0.719395i	$-0.255581\pi$
$599$	34.0000	1.38920	0.694601	+	0.719395i	$0.255581\pi$
$600$	24.0000	0.979796
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	16.0000	0.651031
$605$	0	0
$606$	−20.0000	−0.812444
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	30.0000	1.21666
$609$	−36.0000	−1.45879
$610$	6.00000	0.242933
$611$	2.00000	0.0809113
$612$	5.00000	0.202113
$613$	17.0000	0.686624	0.343312	−	0.939222i	$-0.388451\pi$
$613$	17.0000	0.686624	0.343312	+	0.939222i	$0.388451\pi$
$614$	−22.0000	−0.887848
$615$	−10.0000	−0.403239
$616$	0	0
$617$	−9.00000	−0.362326	−0.181163	−	0.983453i	$-0.557986\pi$
$617$	−9.00000	−0.362326	−0.181163	+	0.983453i	$0.557986\pi$
$618$	16.0000	0.643614
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	2.00000	0.0803219
$621$	−8.00000	−0.321029
$622$	24.0000	0.962312
$623$	18.0000	0.721155
$624$	−2.00000	−0.0800641
$625$	11.0000	0.440000
$626$	23.0000	0.919265
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	15.0000	0.598089
$630$	−2.00000	−0.0796819
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	30.0000	1.19334
$633$	24.0000	0.953914
$634$	−2.00000	−0.0794301
$635$	−16.0000	−0.634941
$636$	−18.0000	−0.713746
$637$	−3.00000	−0.118864
$638$	0	0
$639$	12.0000	0.474713
$640$	−3.00000	−0.118585
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	12.0000	0.473602
$643$	−10.0000	−0.394362	−0.197181	−	0.980367i	$-0.563179\pi$
$643$	−10.0000	−0.394362	−0.197181	+	0.980367i	$0.563179\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	−30.0000	−1.18033
$647$	20.0000	0.786281	0.393141	−	0.919478i	$-0.371389\pi$
$647$	20.0000	0.786281	0.393141	+	0.919478i	$0.371389\pi$
$648$	33.0000	1.29636
$649$	0	0
$650$	−4.00000	−0.156893
$651$	8.00000	0.313545
$652$	2.00000	0.0783260
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	−22.0000	−0.860268
$655$	0	0
$656$	5.00000	0.195217
$657$	−2.00000	−0.0780274
$658$	−4.00000	−0.155936
$659$	22.0000	0.856998	0.428499	−	0.903542i	$-0.359042\pi$
$659$	22.0000	0.856998	0.428499	+	0.903542i	$0.359042\pi$
$660$	0	0
$661$	13.0000	0.505641	0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000	0.505641	0.252821	+	0.967513i	$0.418642\pi$
$662$	−20.0000	−0.777322
$663$	−10.0000	−0.388368
$664$	−18.0000	−0.698535
$665$	−12.0000	−0.465340
$666$	−3.00000	−0.116248
$667$	18.0000	0.696963
$668$	−12.0000	−0.464294
$669$	−40.0000	−1.54649
$670$	2.00000	0.0772667
$671$	0	0
$672$	−20.0000	−0.771517
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−13.0000	−0.500741
$675$	16.0000	0.615840
$676$	12.0000	0.461538
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	−18.0000	−0.691286
$679$	26.0000	0.997788
$680$	15.0000	0.575224
$681$	−48.0000	−1.83936
$682$	0	0
$683$	2.00000	0.0765279	0.0382639	−	0.999268i	$-0.487817\pi$
$683$	2.00000	0.0765279	0.0382639	+	0.999268i	$0.487817\pi$
$684$	−6.00000	−0.229416
$685$	−10.0000	−0.382080
$686$	20.0000	0.763604
$687$	18.0000	0.686743
$688$	0	0
$689$	9.00000	0.342873
$690$	4.00000	0.152277
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	28.0000	1.06287
$695$	−2.00000	−0.0758643
$696$	−54.0000	−2.04686
$697$	25.0000	0.946943
$698$	−27.0000	−1.02197
$699$	−42.0000	−1.58859
$700$	−8.00000	−0.302372
$701$	17.0000	0.642081	0.321041	−	0.947065i	$-0.395967\pi$
$701$	17.0000	0.642081	0.321041	+	0.947065i	$0.395967\pi$
$702$	−4.00000	−0.150970
$703$	−18.0000	−0.678883
$704$	0	0
$705$	4.00000	0.150649
$706$	−9.00000	−0.338719
$707$	20.0000	0.752177
$708$	−16.0000	−0.601317
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	12.0000	0.450352
$711$	−10.0000	−0.375029
$712$	27.0000	1.01187
$713$	−4.00000	−0.149801
$714$	20.0000	0.748481
$715$	0	0
$716$	−24.0000	−0.896922
$717$	12.0000	0.448148
$718$	−2.00000	−0.0746393
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	−1.00000	−0.0372678
$721$	−16.0000	−0.595871
$722$	17.0000	0.632674
$723$	44.0000	1.63638
$724$	−1.00000	−0.0371647
$725$	−36.0000	−1.33701
$726$	0	0
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	6.00000	0.222375
$729$	13.0000	0.481481
$730$	−2.00000	−0.0740233
$731$	0	0
$732$	−12.0000	−0.443533
$733$	9.00000	0.332423	0.166211	−	0.986090i	$-0.446847\pi$
$733$	9.00000	0.332423	0.166211	+	0.986090i	$0.446847\pi$
$734$	−14.0000	−0.516749
$735$	−6.00000	−0.221313
$736$	10.0000	0.368605
$737$	0	0
$738$	−5.00000	−0.184053
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	3.00000	0.110282
$741$	12.0000	0.440831
$742$	−18.0000	−0.660801
$743$	−38.0000	−1.39408	−0.697042	−	0.717030i	$-0.745501\pi$
$743$	−38.0000	−1.39408	−0.697042	+	0.717030i	$0.745501\pi$
$744$	12.0000	0.439941
$745$	17.0000	0.622832
$746$	22.0000	0.805477
$747$	6.00000	0.219529
$748$	0	0
$749$	−12.0000	−0.438470
$750$	−18.0000	−0.657267
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	−2.00000	−0.0729325
$753$	−4.00000	−0.145768
$754$	9.00000	0.327761
$755$	−16.0000	−0.582300
$756$	−8.00000	−0.290957
$757$	53.0000	1.92632	0.963159	−	0.268933i	$-0.0866710\pi$
$757$	53.0000	1.92632	0.963159	+	0.268933i	$0.0866710\pi$
$758$	−32.0000	−1.16229
$759$	0	0
$760$	−18.0000	−0.652929
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	−32.0000	−1.15924
$763$	22.0000	0.796453
$764$	−8.00000	−0.289430
$765$	−5.00000	−0.180775
$766$	20.0000	0.722629
$767$	8.00000	0.288863
$768$	−34.0000	−1.22687
$769$	11.0000	0.396670	0.198335	−	0.980134i	$-0.436447\pi$
$769$	11.0000	0.396670	0.198335	+	0.980134i	$0.436447\pi$
$770$	0	0
$771$	38.0000	1.36854
$772$	5.00000	0.179954
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	39.0000	1.40002
$777$	12.0000	0.430498
$778$	−3.00000	−0.107555
$779$	−30.0000	−1.07486
$780$	−2.00000	−0.0716115
$781$	0	0
$782$	−10.0000	−0.357599
$783$	−36.0000	−1.28654
$784$	3.00000	0.107143
$785$	2.00000	0.0713831
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	11.0000	0.391859
$789$	−44.0000	−1.56644
$790$	−10.0000	−0.355784
$791$	18.0000	0.640006
$792$	0	0
$793$	6.00000	0.213066
$794$	13.0000	0.461353
$795$	18.0000	0.638394
$796$	−24.0000	−0.850657
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	−24.0000	−0.849591
$799$	−10.0000	−0.353775
$800$	−20.0000	−0.707107
$801$	−9.00000	−0.317999
$802$	23.0000	0.812158
$803$	0	0
$804$	−4.00000	−0.141069
$805$	−4.00000	−0.140981
$806$	−2.00000	−0.0704470
$807$	2.00000	0.0704033
$808$	30.0000	1.05540
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	−11.0000	−0.386501
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	18.0000	0.631676
$813$	40.0000	1.40286
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	10.0000	0.350070
$817$	0	0
$818$	−21.0000	−0.734248
$819$	−2.00000	−0.0698857
$820$	5.00000	0.174608
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	−20.0000	−0.697580
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	−16.0000	−0.556711
$827$	−10.0000	−0.347734	−0.173867	−	0.984769i	$-0.555626\pi$
$827$	−10.0000	−0.347734	−0.173867	+	0.984769i	$0.555626\pi$
$828$	−2.00000	−0.0695048
$829$	−47.0000	−1.63238	−0.816189	−	0.577785i	$-0.803917\pi$
$829$	−47.0000	−1.63238	−0.816189	+	0.577785i	$0.803917\pi$
$830$	6.00000	0.208263
$831$	2.00000	0.0693792
$832$	7.00000	0.242681
$833$	15.0000	0.519719
$834$	−4.00000	−0.138509
$835$	12.0000	0.415277
$836$	0	0
$837$	8.00000	0.276520
$838$	2.00000	0.0690889
$839$	46.0000	1.58810	0.794048	−	0.607855i	$-0.207970\pi$
$839$	46.0000	1.58810	0.794048	+	0.607855i	$0.207970\pi$
$840$	12.0000	0.414039
$841$	52.0000	1.79310
$842$	13.0000	0.448010
$843$	12.0000	0.413302
$844$	−12.0000	−0.413057
$845$	−12.0000	−0.412813
$846$	2.00000	0.0687614
$847$	0	0
$848$	−9.00000	−0.309061
$849$	56.0000	1.92192
$850$	20.0000	0.685994
$851$	−6.00000	−0.205677
$852$	−24.0000	−0.822226
$853$	17.0000	0.582069	0.291034	−	0.956713i	$-0.406001\pi$
$853$	17.0000	0.582069	0.291034	+	0.956713i	$0.406001\pi$
$854$	−12.0000	−0.410632
$855$	6.00000	0.205196
$856$	−18.0000	−0.615227
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	0	0
$861$	20.0000	0.681598
$862$	12.0000	0.408722
$863$	−54.0000	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$863$	−54.0000	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$864$	−20.0000	−0.680414
$865$	6.00000	0.204006
$866$	19.0000	0.645646
$867$	16.0000	0.543388
$868$	−4.00000	−0.135769
$869$	0	0
$870$	18.0000	0.610257
$871$	2.00000	0.0677674
$872$	33.0000	1.11752
$873$	−13.0000	−0.439983
$874$	12.0000	0.405906
$875$	18.0000	0.608511
$876$	4.00000	0.135147
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	22.0000	0.742464
$879$	18.0000	0.607125
$880$	0	0
$881$	35.0000	1.17918	0.589590	−	0.807703i	$-0.299289\pi$
$881$	35.0000	1.17918	0.589590	+	0.807703i	$0.299289\pi$
$882$	−3.00000	−0.101015
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	5.00000	0.168168
$885$	16.0000	0.537834
$886$	−20.0000	−0.671913
$887$	−46.0000	−1.54453	−0.772264	−	0.635301i	$-0.780876\pi$
$887$	−46.0000	−1.54453	−0.772264	+	0.635301i	$0.780876\pi$
$888$	18.0000	0.604040
$889$	32.0000	1.07325
$890$	−9.00000	−0.301681
$891$	0	0
$892$	20.0000	0.669650
$893$	12.0000	0.401565
$894$	34.0000	1.13713
$895$	24.0000	0.802232
$896$	6.00000	0.200446
$897$	4.00000	0.133556
$898$	−13.0000	−0.433816
$899$	−18.0000	−0.600334
$900$	4.00000	0.133333
$901$	−45.0000	−1.49917
$902$	0	0
$903$	0	0
$904$	27.0000	0.898007
$905$	1.00000	0.0332411
$906$	−32.0000	−1.06313
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	24.0000	0.796468
$909$	−10.0000	−0.331679
$910$	−2.00000	−0.0662994
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	−12.0000	−0.397360
$913$	0	0
$914$	39.0000	1.29001
$915$	12.0000	0.396708
$916$	−9.00000	−0.297368
$917$	0	0
$918$	20.0000	0.660098
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	−6.00000	−0.197814
$921$	−44.0000	−1.44985
$922$	33.0000	1.08680
$923$	12.0000	0.394985
$924$	0	0
$925$	12.0000	0.394558
$926$	−20.0000	−0.657241
$927$	8.00000	0.262754
$928$	45.0000	1.47720
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	−4.00000	−0.131165
$931$	−18.0000	−0.589926
$932$	21.0000	0.687878
$933$	48.0000	1.57145
$934$	12.0000	0.392652
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	23.0000	0.751377	0.375689	−	0.926746i	$-0.377406\pi$
$937$	23.0000	0.751377	0.375689	+	0.926746i	$0.377406\pi$
$938$	−4.00000	−0.130605
$939$	46.0000	1.50115
$940$	−2.00000	−0.0652328
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	4.00000	0.130327
$943$	−10.0000	−0.325645
$944$	−8.00000	−0.260378
$945$	8.00000	0.260240
$946$	0	0
$947$	−42.0000	−1.36482	−0.682408	−	0.730971i	$-0.739067\pi$
$947$	−42.0000	−1.36482	−0.682408	+	0.730971i	$0.739067\pi$
$948$	20.0000	0.649570
$949$	−2.00000	−0.0649227
$950$	−24.0000	−0.778663
$951$	−4.00000	−0.129709
$952$	−30.0000	−0.972306
$953$	31.0000	1.00419	0.502094	−	0.864813i	$-0.332563\pi$
$953$	31.0000	1.00419	0.502094	+	0.864813i	$0.332563\pi$
$954$	9.00000	0.291386
$955$	8.00000	0.258874
$956$	−6.00000	−0.194054
$957$	0	0
$958$	−16.0000	−0.516937
$959$	20.0000	0.645834
$960$	14.0000	0.451848
$961$	−27.0000	−0.870968
$962$	−3.00000	−0.0967239
$963$	6.00000	0.193347
$964$	−22.0000	−0.708572
$965$	−5.00000	−0.160956
$966$	−8.00000	−0.257396
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	−60.0000	−1.92748
$970$	−13.0000	−0.417405
$971$	2.00000	0.0641831	0.0320915	−	0.999485i	$-0.489783\pi$
$971$	2.00000	0.0641831	0.0320915	+	0.999485i	$0.489783\pi$
$972$	10.0000	0.320750
$973$	4.00000	0.128234
$974$	2.00000	0.0640841
$975$	−8.00000	−0.256205
$976$	−6.00000	−0.192055
$977$	−57.0000	−1.82359	−0.911796	−	0.410644i	$-0.865304\pi$
$977$	−57.0000	−1.82359	−0.911796	+	0.410644i	$0.865304\pi$
$978$	−4.00000	−0.127906
$979$	0	0
$980$	3.00000	0.0958315
$981$	−11.0000	−0.351203
$982$	−2.00000	−0.0638226
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	30.0000	0.956365
$985$	−11.0000	−0.350489
$986$	−45.0000	−1.43309
$987$	−8.00000	−0.254643
$988$	−6.00000	−0.190885
$989$	0	0
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	−10.0000	−0.317500
$993$	−40.0000	−1.26936
$994$	−24.0000	−0.761234
$995$	24.0000	0.760851
$996$	−12.0000	−0.380235
$997$	53.0000	1.67853	0.839263	−	0.543725i	$-0.182987\pi$
$997$	53.0000	1.67853	0.839263	+	0.543725i	$0.182987\pi$
$998$	8.00000	0.253236
$999$	12.0000	0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.2.a.c.1.1	yes	1
3.2	odd	2		1089.2.a.c.1.1		1
4.3	odd	2		1936.2.a.b.1.1		1
5.4	even	2		3025.2.a.b.1.1		1
7.6	odd	2		5929.2.a.g.1.1		1
8.3	odd	2		7744.2.a.bf.1.1		1
8.5	even	2		7744.2.a.c.1.1		1
11.2	odd	10		121.2.c.d.81.1		4
11.3	even	5		121.2.c.b.9.1		4
11.4	even	5		121.2.c.b.27.1		4
11.5	even	5		121.2.c.b.3.1		4
11.6	odd	10		121.2.c.d.3.1		4
11.7	odd	10		121.2.c.d.27.1		4
11.8	odd	10		121.2.c.d.9.1		4
11.9	even	5		121.2.c.b.81.1		4
11.10	odd	2		121.2.a.a.1.1	✓	1
33.32	even	2		1089.2.a.i.1.1		1
44.43	even	2		1936.2.a.a.1.1		1
55.54	odd	2		3025.2.a.e.1.1		1
77.76	even	2		5929.2.a.a.1.1		1
88.21	odd	2		7744.2.a.f.1.1		1
88.43	even	2		7744.2.a.be.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.2.a.a.1.1	✓	1	11.10	odd	2
121.2.a.c.1.1	yes	1	1.1	even	1	trivial
121.2.c.b.3.1		4	11.5	even	5
121.2.c.b.9.1		4	11.3	even	5
121.2.c.b.27.1		4	11.4	even	5
121.2.c.b.81.1		4	11.9	even	5
121.2.c.d.3.1		4	11.6	odd	10
121.2.c.d.9.1		4	11.8	odd	10
121.2.c.d.27.1		4	11.7	odd	10
121.2.c.d.81.1		4	11.2	odd	10
1089.2.a.c.1.1		1	3.2	odd	2
1089.2.a.i.1.1		1	33.32	even	2
1936.2.a.a.1.1		1	44.43	even	2
1936.2.a.b.1.1		1	4.3	odd	2
3025.2.a.b.1.1		1	5.4	even	2
3025.2.a.e.1.1		1	55.54	odd	2
5929.2.a.a.1.1		1	77.76	even	2
5929.2.a.g.1.1		1	7.6	odd	2
7744.2.a.c.1.1		1	8.5	even	2
7744.2.a.f.1.1		1	88.21	odd	2
7744.2.a.be.1.1		1	88.43	even	2
7744.2.a.bf.1.1		1	8.3	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:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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