LMFDB - Embedded newform 5445.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} +2.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} +2.00000 q^{7} +2.23607 q^{8} -1.61803 q^{10} -6.47214 q^{13} -3.23607 q^{14} -4.85410 q^{16} +5.47214 q^{17} +0.618034 q^{20} -8.23607 q^{23} +1.00000 q^{25} +10.4721 q^{26} +1.23607 q^{28} +0.472136 q^{29} +6.70820 q^{31} +3.38197 q^{32} -8.85410 q^{34} +2.00000 q^{35} +8.47214 q^{37} +2.23607 q^{40} -6.00000 q^{41} +6.00000 q^{43} +13.3262 q^{46} -3.76393 q^{47} -3.00000 q^{49} -1.61803 q^{50} -4.00000 q^{52} -11.9443 q^{53} +4.47214 q^{56} -0.763932 q^{58} -10.9443 q^{59} +5.47214 q^{61} -10.8541 q^{62} +4.23607 q^{64} -6.47214 q^{65} -15.4164 q^{67} +3.38197 q^{68} -3.23607 q^{70} -4.47214 q^{71} +12.9443 q^{73} -13.7082 q^{74} -10.7082 q^{79} -4.85410 q^{80} +9.70820 q^{82} +5.47214 q^{85} -9.70820 q^{86} +0.472136 q^{89} -12.9443 q^{91} -5.09017 q^{92} +6.09017 q^{94} -6.47214 q^{97} +4.85410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^5 + 4 * q^7	$ 2 q - q^{2} - q^{4} + 2 q^{5} + 4 q^{7} - q^{10} - 4 q^{13} - 2 q^{14} - 3 q^{16} + 2 q^{17} - q^{20} - 12 q^{23} + 2 q^{25} + 12 q^{26} - 2 q^{28} - 8 q^{29} + 9 q^{32} - 11 q^{34} + 4 q^{35} + 8 q^{37} - 12 q^{41} + 12 q^{43} + 11 q^{46} - 12 q^{47} - 6 q^{49} - q^{50} - 8 q^{52} - 6 q^{53} - 6 q^{58} - 4 q^{59} + 2 q^{61} - 15 q^{62} + 4 q^{64} - 4 q^{65} - 4 q^{67} + 9 q^{68} - 2 q^{70} + 8 q^{73} - 14 q^{74} - 8 q^{79} - 3 q^{80} + 6 q^{82} + 2 q^{85} - 6 q^{86} - 8 q^{89} - 8 q^{91} + q^{92} + q^{94} - 4 q^{97} + 3 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 + 4 * q^7 - q^10 - 4 * q^13 - 2 * q^14 - 3 * q^16 + 2 * q^17 - q^20 - 12 * q^23 + 2 * q^25 + 12 * q^26 - 2 * q^28 - 8 * q^29 + 9 * q^32 - 11 * q^34 + 4 * q^35 + 8 * q^37 - 12 * q^41 + 12 * q^43 + 11 * q^46 - 12 * q^47 - 6 * q^49 - q^50 - 8 * q^52 - 6 * q^53 - 6 * q^58 - 4 * q^59 + 2 * q^61 - 15 * q^62 + 4 * q^64 - 4 * q^65 - 4 * q^67 + 9 * q^68 - 2 * q^70 + 8 * q^73 - 14 * q^74 - 8 * q^79 - 3 * q^80 + 6 * q^82 + 2 * q^85 - 6 * q^86 - 8 * q^89 - 8 * q^91 + q^92 + q^94 - 4 * q^97 + 3 * q^98

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 $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	−1.61803	−0.511667
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	−3.23607	−0.864876
$15$	0	0
$16$	−4.85410	−1.21353
$17$	5.47214	1.32719	0.663594	−	0.748093i	$-0.269030\pi$
$17$	5.47214	1.32719	0.663594	+	0.748093i	$0.269030\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0.618034	0.138197
$21$	0	0
$22$	0	0
$23$	−8.23607	−1.71734	−0.858669	−	0.512530i	$-0.828708\pi$
$23$	−8.23607	−1.71734	−0.858669	+	0.512530i	$0.828708\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	10.4721	2.05375
$27$	0	0
$28$	1.23607	0.233595
$29$	0.472136	0.0876734	0.0438367	−	0.999039i	$-0.486042\pi$
$29$	0.472136	0.0876734	0.0438367	+	0.999039i	$0.486042\pi$
$30$	0	0
$31$	6.70820	1.20483	0.602414	−	0.798183i	$-0.294205\pi$
$31$	6.70820	1.20483	0.602414	+	0.798183i	$0.294205\pi$
$32$	3.38197	0.597853
$33$	0	0
$34$	−8.85410	−1.51847
$35$	2.00000	0.338062
$36$	0	0
$37$	8.47214	1.39281	0.696405	−	0.717649i	$-0.254782\pi$
$37$	8.47214	1.39281	0.696405	+	0.717649i	$0.254782\pi$
$38$	0	0
$39$	0	0
$40$	2.23607	0.353553
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	0	0
$46$	13.3262	1.96485
$47$	−3.76393	−0.549026	−0.274513	−	0.961583i	$-0.588517\pi$
$47$	−3.76393	−0.549026	−0.274513	+	0.961583i	$0.588517\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−1.61803	−0.228825
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−11.9443	−1.64067	−0.820336	−	0.571882i	$-0.806214\pi$
$53$	−11.9443	−1.64067	−0.820336	+	0.571882i	$0.806214\pi$
$54$	0	0
$55$	0	0
$56$	4.47214	0.597614
$57$	0	0
$58$	−0.763932	−0.100309
$59$	−10.9443	−1.42482	−0.712411	−	0.701762i	$-0.752397\pi$
$59$	−10.9443	−1.42482	−0.712411	+	0.701762i	$0.752397\pi$
$60$	0	0
$61$	5.47214	0.700635	0.350318	−	0.936631i	$-0.386074\pi$
$61$	5.47214	0.700635	0.350318	+	0.936631i	$0.386074\pi$
$62$	−10.8541	−1.37847
$63$	0	0
$64$	4.23607	0.529508
$65$	−6.47214	−0.802770
$66$	0	0
$67$	−15.4164	−1.88341	−0.941707	−	0.336434i	$-0.890779\pi$
$67$	−15.4164	−1.88341	−0.941707	+	0.336434i	$0.890779\pi$
$68$	3.38197	0.410124
$69$	0	0
$70$	−3.23607	−0.386784
$71$	−4.47214	−0.530745	−0.265372	−	0.964146i	$-0.585495\pi$
$71$	−4.47214	−0.530745	−0.265372	+	0.964146i	$0.585495\pi$
$72$	0	0
$73$	12.9443	1.51501	0.757506	−	0.652828i	$-0.226418\pi$
$73$	12.9443	1.51501	0.757506	+	0.652828i	$0.226418\pi$
$74$	−13.7082	−1.59355
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.7082	−1.20477	−0.602384	−	0.798207i	$-0.705782\pi$
$79$	−10.7082	−1.20477	−0.602384	+	0.798207i	$0.705782\pi$
$80$	−4.85410	−0.542705
$81$	0	0
$82$	9.70820	1.07209
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	5.47214	0.593536
$86$	−9.70820	−1.04686
$87$	0	0
$88$	0	0
$89$	0.472136	0.0500463	0.0250232	−	0.999687i	$-0.492034\pi$
$89$	0.472136	0.0500463	0.0250232	+	0.999687i	$0.492034\pi$
$90$	0	0
$91$	−12.9443	−1.35693
$92$	−5.09017	−0.530687
$93$	0	0
$94$	6.09017	0.628153
$95$	0	0
$96$	0	0
$97$	−6.47214	−0.657146	−0.328573	−	0.944479i	$-0.606568\pi$
$97$	−6.47214	−0.657146	−0.328573	+	0.944479i	$0.606568\pi$
$98$	4.85410	0.490338
$99$	0	0
$100$	0.618034	0.0618034
$101$	1.05573	0.105049	0.0525244	−	0.998620i	$-0.483273\pi$
$101$	1.05573	0.105049	0.0525244	+	0.998620i	$0.483273\pi$
$102$	0	0
$103$	−3.52786	−0.347611	−0.173805	−	0.984780i	$-0.555606\pi$
$103$	−3.52786	−0.347611	−0.173805	+	0.984780i	$0.555606\pi$
$104$	−14.4721	−1.41911
$105$	0	0
$106$	19.3262	1.87713
$107$	−1.76393	−0.170526	−0.0852629	−	0.996358i	$-0.527173\pi$
$107$	−1.76393	−0.170526	−0.0852629	+	0.996358i	$0.527173\pi$
$108$	0	0
$109$	−1.05573	−0.101120	−0.0505602	−	0.998721i	$-0.516101\pi$
$109$	−1.05573	−0.101120	−0.0505602	+	0.998721i	$0.516101\pi$
$110$	0	0
$111$	0	0
$112$	−9.70820	−0.917339
$113$	−6.52786	−0.614090	−0.307045	−	0.951695i	$-0.599340\pi$
$113$	−6.52786	−0.614090	−0.307045	+	0.951695i	$0.599340\pi$
$114$	0	0
$115$	−8.23607	−0.768017
$116$	0.291796	0.0270926
$117$	0	0
$118$	17.7082	1.63017
$119$	10.9443	1.00326
$120$	0	0
$121$	0	0
$122$	−8.85410	−0.801613
$123$	0	0
$124$	4.14590	0.372313
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.47214	−0.219367	−0.109683	−	0.993967i	$-0.534984\pi$
$127$	−2.47214	−0.219367	−0.109683	+	0.993967i	$0.534984\pi$
$128$	−13.6180	−1.20368
$129$	0	0
$130$	10.4721	0.918467
$131$	−4.47214	−0.390732	−0.195366	−	0.980730i	$-0.562590\pi$
$131$	−4.47214	−0.390732	−0.195366	+	0.980730i	$0.562590\pi$
$132$	0	0
$133$	0	0
$134$	24.9443	2.15486
$135$	0	0
$136$	12.2361	1.04923
$137$	10.4164	0.889934	0.444967	−	0.895547i	$-0.353215\pi$
$137$	10.4164	0.889934	0.444967	+	0.895547i	$0.353215\pi$
$138$	0	0
$139$	4.70820	0.399345	0.199672	−	0.979863i	$-0.436012\pi$
$139$	4.70820	0.399345	0.199672	+	0.979863i	$0.436012\pi$
$140$	1.23607	0.104467
$141$	0	0
$142$	7.23607	0.607237
$143$	0	0
$144$	0	0
$145$	0.472136	0.0392088
$146$	−20.9443	−1.73336
$147$	0	0
$148$	5.23607	0.430402
$149$	10.4721	0.857911	0.428955	−	0.903326i	$-0.358882\pi$
$149$	10.4721	0.857911	0.428955	+	0.903326i	$0.358882\pi$
$150$	0	0
$151$	6.70820	0.545906	0.272953	−	0.962027i	$-0.412000\pi$
$151$	6.70820	0.545906	0.272953	+	0.962027i	$0.412000\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.70820	0.538816
$156$	0	0
$157$	−15.4164	−1.23036	−0.615182	−	0.788385i	$-0.710917\pi$
$157$	−15.4164	−1.23036	−0.615182	+	0.788385i	$0.710917\pi$
$158$	17.3262	1.37840
$159$	0	0
$160$	3.38197	0.267368
$161$	−16.4721	−1.29819
$162$	0	0
$163$	15.4164	1.20751	0.603753	−	0.797171i	$-0.293671\pi$
$163$	15.4164	1.20751	0.603753	+	0.797171i	$0.293671\pi$
$164$	−3.70820	−0.289562
$165$	0	0
$166$	0	0
$167$	24.7082	1.91198	0.955989	−	0.293402i	$-0.0947875\pi$
$167$	24.7082	1.91198	0.955989	+	0.293402i	$0.0947875\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	−8.85410	−0.679079
$171$	0	0
$172$	3.70820	0.282748
$173$	−22.9443	−1.74442	−0.872210	−	0.489131i	$-0.837314\pi$
$173$	−22.9443	−1.74442	−0.872210	+	0.489131i	$0.837314\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	0	0
$178$	−0.763932	−0.0572591
$179$	−17.5279	−1.31009	−0.655047	−	0.755588i	$-0.727351\pi$
$179$	−17.5279	−1.31009	−0.655047	+	0.755588i	$0.727351\pi$
$180$	0	0
$181$	1.05573	0.0784717	0.0392358	−	0.999230i	$-0.487508\pi$
$181$	1.05573	0.0784717	0.0392358	+	0.999230i	$0.487508\pi$
$182$	20.9443	1.55249
$183$	0	0
$184$	−18.4164	−1.35768
$185$	8.47214	0.622884
$186$	0	0
$187$	0	0
$188$	−2.32624	−0.169658
$189$	0	0
$190$	0	0
$191$	−1.05573	−0.0763898	−0.0381949	−	0.999270i	$-0.512161\pi$
$191$	−1.05573	−0.0763898	−0.0381949	+	0.999270i	$0.512161\pi$
$192$	0	0
$193$	−6.47214	−0.465875	−0.232937	−	0.972492i	$-0.574834\pi$
$193$	−6.47214	−0.465875	−0.232937	+	0.972492i	$0.574834\pi$
$194$	10.4721	0.751856
$195$	0	0
$196$	−1.85410	−0.132436
$197$	−5.05573	−0.360206	−0.180103	−	0.983648i	$-0.557643\pi$
$197$	−5.05573	−0.360206	−0.180103	+	0.983648i	$0.557643\pi$
$198$	0	0
$199$	3.18034	0.225448	0.112724	−	0.993626i	$-0.464042\pi$
$199$	3.18034	0.225448	0.112724	+	0.993626i	$0.464042\pi$
$200$	2.23607	0.158114
$201$	0	0
$202$	−1.70820	−0.120189
$203$	0.944272	0.0662749
$204$	0	0
$205$	−6.00000	−0.419058
$206$	5.70820	0.397709
$207$	0	0
$208$	31.4164	2.17834
$209$	0	0
$210$	0	0
$211$	3.29180	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$211$	3.29180	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$212$	−7.38197	−0.506996
$213$	0	0
$214$	2.85410	0.195102
$215$	6.00000	0.409197
$216$	0	0
$217$	13.4164	0.910765
$218$	1.70820	0.115694
$219$	0	0
$220$	0	0
$221$	−35.4164	−2.38237
$222$	0	0
$223$	−24.4721	−1.63878	−0.819388	−	0.573240i	$-0.805686\pi$
$223$	−24.4721	−1.63878	−0.819388	+	0.573240i	$0.805686\pi$
$224$	6.76393	0.451934
$225$	0	0
$226$	10.5623	0.702594
$227$	7.18034	0.476576	0.238288	−	0.971195i	$-0.423414\pi$
$227$	7.18034	0.476576	0.238288	+	0.971195i	$0.423414\pi$
$228$	0	0
$229$	9.47214	0.625936	0.312968	−	0.949764i	$-0.398677\pi$
$229$	9.47214	0.625936	0.312968	+	0.949764i	$0.398677\pi$
$230$	13.3262	0.878706
$231$	0	0
$232$	1.05573	0.0693119
$233$	−17.3607	−1.13734	−0.568668	−	0.822567i	$-0.692541\pi$
$233$	−17.3607	−1.13734	−0.568668	+	0.822567i	$0.692541\pi$
$234$	0	0
$235$	−3.76393	−0.245532
$236$	−6.76393	−0.440294
$237$	0	0
$238$	−17.7082	−1.14785
$239$	28.3607	1.83450	0.917250	−	0.398312i	$-0.130404\pi$
$239$	28.3607	1.83450	0.917250	+	0.398312i	$0.130404\pi$
$240$	0	0
$241$	−15.0000	−0.966235	−0.483117	−	0.875556i	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	0	0
$243$	0	0
$244$	3.38197	0.216508
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0	0
$248$	15.0000	0.952501
$249$	0	0
$250$	−1.61803	−0.102333
$251$	7.41641	0.468120	0.234060	−	0.972222i	$-0.424799\pi$
$251$	7.41641	0.468120	0.234060	+	0.972222i	$0.424799\pi$
$252$	0	0
$253$	0	0
$254$	4.00000	0.250982
$255$	0	0
$256$	13.5623	0.847644
$257$	17.4721	1.08988	0.544941	−	0.838474i	$-0.316552\pi$
$257$	17.4721	1.08988	0.544941	+	0.838474i	$0.316552\pi$
$258$	0	0
$259$	16.9443	1.05287
$260$	−4.00000	−0.248069
$261$	0	0
$262$	7.23607	0.447046
$263$	12.7082	0.783621	0.391811	−	0.920046i	$-0.371849\pi$
$263$	12.7082	0.783621	0.391811	+	0.920046i	$0.371849\pi$
$264$	0	0
$265$	−11.9443	−0.733731
$266$	0	0
$267$	0	0
$268$	−9.52786	−0.582007
$269$	−17.8885	−1.09068	−0.545342	−	0.838214i	$-0.683600\pi$
$269$	−17.8885	−1.09068	−0.545342	+	0.838214i	$0.683600\pi$
$270$	0	0
$271$	6.23607	0.378814	0.189407	−	0.981899i	$-0.439343\pi$
$271$	6.23607	0.378814	0.189407	+	0.981899i	$0.439343\pi$
$272$	−26.5623	−1.61058
$273$	0	0
$274$	−16.8541	−1.01819
$275$	0	0
$276$	0	0
$277$	10.9443	0.657578	0.328789	−	0.944403i	$-0.393360\pi$
$277$	10.9443	0.657578	0.328789	+	0.944403i	$0.393360\pi$
$278$	−7.61803	−0.456899
$279$	0	0
$280$	4.47214	0.267261
$281$	−28.3607	−1.69186	−0.845928	−	0.533297i	$-0.820953\pi$
$281$	−28.3607	−1.69186	−0.845928	+	0.533297i	$0.820953\pi$
$282$	0	0
$283$	−14.3607	−0.853654	−0.426827	−	0.904333i	$-0.640369\pi$
$283$	−14.3607	−0.853654	−0.426827	+	0.904333i	$0.640369\pi$
$284$	−2.76393	−0.164009
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	12.9443	0.761428
$290$	−0.763932	−0.0448596
$291$	0	0
$292$	8.00000	0.468165
$293$	−11.9443	−0.697792	−0.348896	−	0.937161i	$-0.613443\pi$
$293$	−11.9443	−0.697792	−0.348896	+	0.937161i	$0.613443\pi$
$294$	0	0
$295$	−10.9443	−0.637200
$296$	18.9443	1.10111
$297$	0	0
$298$	−16.9443	−0.981555
$299$	53.3050	3.08270
$300$	0	0
$301$	12.0000	0.691669
$302$	−10.8541	−0.624583
$303$	0	0
$304$	0	0
$305$	5.47214	0.313334
$306$	0	0
$307$	−30.3607	−1.73278	−0.866388	−	0.499372i	$-0.833564\pi$
$307$	−30.3607	−1.73278	−0.866388	+	0.499372i	$0.833564\pi$
$308$	0	0
$309$	0	0
$310$	−10.8541	−0.616472
$311$	−17.5279	−0.993914	−0.496957	−	0.867775i	$-0.665549\pi$
$311$	−17.5279	−0.993914	−0.496957	+	0.867775i	$0.665549\pi$
$312$	0	0
$313$	−12.4721	−0.704967	−0.352483	−	0.935818i	$-0.614663\pi$
$313$	−12.4721	−0.704967	−0.352483	+	0.935818i	$0.614663\pi$
$314$	24.9443	1.40769
$315$	0	0
$316$	−6.61803	−0.372293
$317$	−27.9443	−1.56951	−0.784753	−	0.619808i	$-0.787210\pi$
$317$	−27.9443	−1.56951	−0.784753	+	0.619808i	$0.787210\pi$
$318$	0	0
$319$	0	0
$320$	4.23607	0.236803
$321$	0	0
$322$	26.6525	1.48528
$323$	0	0
$324$	0	0
$325$	−6.47214	−0.359010
$326$	−24.9443	−1.38154
$327$	0	0
$328$	−13.4164	−0.740797
$329$	−7.52786	−0.415025
$330$	0	0
$331$	24.2361	1.33213	0.666067	−	0.745892i	$-0.267976\pi$
$331$	24.2361	1.33213	0.666067	+	0.745892i	$0.267976\pi$
$332$	0	0
$333$	0	0
$334$	−39.9787	−2.18754
$335$	−15.4164	−0.842288
$336$	0	0
$337$	24.0000	1.30736	0.653682	−	0.756770i	$-0.273224\pi$
$337$	24.0000	1.30736	0.653682	+	0.756770i	$0.273224\pi$
$338$	−46.7426	−2.54246
$339$	0	0
$340$	3.38197	0.183413
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	13.4164	0.723364
$345$	0	0
$346$	37.1246	1.99583
$347$	−32.1246	−1.72454	−0.862270	−	0.506449i	$-0.830958\pi$
$347$	−32.1246	−1.72454	−0.862270	+	0.506449i	$0.830958\pi$
$348$	0	0
$349$	30.4164	1.62815	0.814076	−	0.580758i	$-0.197244\pi$
$349$	30.4164	1.62815	0.814076	+	0.580758i	$0.197244\pi$
$350$	−3.23607	−0.172975
$351$	0	0
$352$	0	0
$353$	−29.4721	−1.56864	−0.784322	−	0.620354i	$-0.786989\pi$
$353$	−29.4721	−1.56864	−0.784322	+	0.620354i	$0.786989\pi$
$354$	0	0
$355$	−4.47214	−0.237356
$356$	0.291796	0.0154652
$357$	0	0
$358$	28.3607	1.49891
$359$	−28.4721	−1.50270	−0.751351	−	0.659903i	$-0.770597\pi$
$359$	−28.4721	−1.50270	−0.751351	+	0.659903i	$0.770597\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−1.70820	−0.0897812
$363$	0	0
$364$	−8.00000	−0.419314
$365$	12.9443	0.677534
$366$	0	0
$367$	4.58359	0.239262	0.119631	−	0.992818i	$-0.461829\pi$
$367$	4.58359	0.239262	0.119631	+	0.992818i	$0.461829\pi$
$368$	39.9787	2.08403
$369$	0	0
$370$	−13.7082	−0.712656
$371$	−23.8885	−1.24023
$372$	0	0
$373$	2.11146	0.109327	0.0546635	−	0.998505i	$-0.482591\pi$
$373$	2.11146	0.109327	0.0546635	+	0.998505i	$0.482591\pi$
$374$	0	0
$375$	0	0
$376$	−8.41641	−0.434043
$377$	−3.05573	−0.157378
$378$	0	0
$379$	−26.5967	−1.36618	−0.683092	−	0.730333i	$-0.739365\pi$
$379$	−26.5967	−1.36618	−0.683092	+	0.730333i	$0.739365\pi$
$380$	0	0
$381$	0	0
$382$	1.70820	0.0873993
$383$	−24.9443	−1.27459	−0.637296	−	0.770619i	$-0.719947\pi$
$383$	−24.9443	−1.27459	−0.637296	+	0.770619i	$0.719947\pi$
$384$	0	0
$385$	0	0
$386$	10.4721	0.533018
$387$	0	0
$388$	−4.00000	−0.203069
$389$	0.472136	0.0239382	0.0119691	−	0.999928i	$-0.496190\pi$
$389$	0.472136	0.0239382	0.0119691	+	0.999928i	$0.496190\pi$
$390$	0	0
$391$	−45.0689	−2.27923
$392$	−6.70820	−0.338815
$393$	0	0
$394$	8.18034	0.412120
$395$	−10.7082	−0.538788
$396$	0	0
$397$	16.9443	0.850409	0.425204	−	0.905097i	$-0.360202\pi$
$397$	16.9443	0.850409	0.425204	+	0.905097i	$0.360202\pi$
$398$	−5.14590	−0.257941
$399$	0	0
$400$	−4.85410	−0.242705
$401$	17.8885	0.893311	0.446656	−	0.894706i	$-0.352615\pi$
$401$	17.8885	0.893311	0.446656	+	0.894706i	$0.352615\pi$
$402$	0	0
$403$	−43.4164	−2.16273
$404$	0.652476	0.0324619
$405$	0	0
$406$	−1.52786	−0.0758266
$407$	0	0
$408$	0	0
$409$	3.00000	0.148340	0.0741702	−	0.997246i	$-0.476369\pi$
$409$	3.00000	0.148340	0.0741702	+	0.997246i	$0.476369\pi$
$410$	9.70820	0.479454
$411$	0	0
$412$	−2.18034	−0.107418
$413$	−21.8885	−1.07706
$414$	0	0
$415$	0	0
$416$	−21.8885	−1.07317
$417$	0	0
$418$	0	0
$419$	31.7771	1.55241	0.776206	−	0.630479i	$-0.217142\pi$
$419$	31.7771	1.55241	0.776206	+	0.630479i	$0.217142\pi$
$420$	0	0
$421$	34.4164	1.67735	0.838677	−	0.544630i	$-0.183330\pi$
$421$	34.4164	1.67735	0.838677	+	0.544630i	$0.183330\pi$
$422$	−5.32624	−0.259277
$423$	0	0
$424$	−26.7082	−1.29707
$425$	5.47214	0.265438
$426$	0	0
$427$	10.9443	0.529630
$428$	−1.09017	−0.0526954
$429$	0	0
$430$	−9.70820	−0.468171
$431$	−23.8885	−1.15067	−0.575335	−	0.817918i	$-0.695128\pi$
$431$	−23.8885	−1.15067	−0.575335	+	0.817918i	$0.695128\pi$
$432$	0	0
$433$	−27.5279	−1.32290	−0.661452	−	0.749987i	$-0.730060\pi$
$433$	−27.5279	−1.32290	−0.661452	+	0.749987i	$0.730060\pi$
$434$	−21.7082	−1.04203
$435$	0	0
$436$	−0.652476	−0.0312479
$437$	0	0
$438$	0	0
$439$	10.7082	0.511075	0.255537	−	0.966799i	$-0.417748\pi$
$439$	10.7082	0.511075	0.255537	+	0.966799i	$0.417748\pi$
$440$	0	0
$441$	0	0
$442$	57.3050	2.72572
$443$	−20.9443	−0.995092	−0.497546	−	0.867437i	$-0.665765\pi$
$443$	−20.9443	−0.995092	−0.497546	+	0.867437i	$0.665765\pi$
$444$	0	0
$445$	0.472136	0.0223814
$446$	39.5967	1.87496
$447$	0	0
$448$	8.47214	0.400271
$449$	17.8885	0.844213	0.422106	−	0.906546i	$-0.361291\pi$
$449$	17.8885	0.844213	0.422106	+	0.906546i	$0.361291\pi$
$450$	0	0
$451$	0	0
$452$	−4.03444	−0.189764
$453$	0	0
$454$	−11.6180	−0.545261
$455$	−12.9443	−0.606837
$456$	0	0
$457$	−36.8328	−1.72297	−0.861483	−	0.507786i	$-0.830464\pi$
$457$	−36.8328	−1.72297	−0.861483	+	0.507786i	$0.830464\pi$
$458$	−15.3262	−0.716148
$459$	0	0
$460$	−5.09017	−0.237330
$461$	−14.4721	−0.674035	−0.337017	−	0.941498i	$-0.609418\pi$
$461$	−14.4721	−0.674035	−0.337017	+	0.941498i	$0.609418\pi$
$462$	0	0
$463$	27.3050	1.26897	0.634484	−	0.772936i	$-0.281212\pi$
$463$	27.3050	1.26897	0.634484	+	0.772936i	$0.281212\pi$
$464$	−2.29180	−0.106394
$465$	0	0
$466$	28.0902	1.30125
$467$	−3.65248	−0.169016	−0.0845082	−	0.996423i	$-0.526932\pi$
$467$	−3.65248	−0.169016	−0.0845082	+	0.996423i	$0.526932\pi$
$468$	0	0
$469$	−30.8328	−1.42373
$470$	6.09017	0.280919
$471$	0	0
$472$	−24.4721	−1.12642
$473$	0	0
$474$	0	0
$475$	0	0
$476$	6.76393	0.310024
$477$	0	0
$478$	−45.8885	−2.09889
$479$	7.41641	0.338864	0.169432	−	0.985542i	$-0.445807\pi$
$479$	7.41641	0.338864	0.169432	+	0.985542i	$0.445807\pi$
$480$	0	0
$481$	−54.8328	−2.50016
$482$	24.2705	1.10549
$483$	0	0
$484$	0	0
$485$	−6.47214	−0.293885
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	12.2361	0.553901
$489$	0	0
$490$	4.85410	0.219286
$491$	16.4721	0.743377	0.371689	−	0.928357i	$-0.378779\pi$
$491$	16.4721	0.743377	0.371689	+	0.928357i	$0.378779\pi$
$492$	0	0
$493$	2.58359	0.116359
$494$	0	0
$495$	0	0
$496$	−32.5623	−1.46209
$497$	−8.94427	−0.401205
$498$	0	0
$499$	12.9443	0.579465	0.289733	−	0.957108i	$-0.406434\pi$
$499$	12.9443	0.579465	0.289733	+	0.957108i	$0.406434\pi$
$500$	0.618034	0.0276393
$501$	0	0
$502$	−12.0000	−0.535586
$503$	32.2361	1.43734	0.718668	−	0.695354i	$-0.244752\pi$
$503$	32.2361	1.43734	0.718668	+	0.695354i	$0.244752\pi$
$504$	0	0
$505$	1.05573	0.0469793
$506$	0	0
$507$	0	0
$508$	−1.52786	−0.0677880
$509$	−14.3607	−0.636526	−0.318263	−	0.948002i	$-0.603099\pi$
$509$	−14.3607	−0.636526	−0.318263	+	0.948002i	$0.603099\pi$
$510$	0	0
$511$	25.8885	1.14524
$512$	5.29180	0.233867
$513$	0	0
$514$	−28.2705	−1.24696
$515$	−3.52786	−0.155456
$516$	0	0
$517$	0	0
$518$	−27.4164	−1.20461
$519$	0	0
$520$	−14.4721	−0.634645
$521$	17.5279	0.767910	0.383955	−	0.923352i	$-0.374562\pi$
$521$	17.5279	0.767910	0.383955	+	0.923352i	$0.374562\pi$
$522$	0	0
$523$	−2.47214	−0.108099	−0.0540495	−	0.998538i	$-0.517213\pi$
$523$	−2.47214	−0.108099	−0.0540495	+	0.998538i	$0.517213\pi$
$524$	−2.76393	−0.120743
$525$	0	0
$526$	−20.5623	−0.896559
$527$	36.7082	1.59903
$528$	0	0
$529$	44.8328	1.94925
$530$	19.3262	0.839478
$531$	0	0
$532$	0	0
$533$	38.8328	1.68204
$534$	0	0
$535$	−1.76393	−0.0762614
$536$	−34.4721	−1.48897
$537$	0	0
$538$	28.9443	1.24788
$539$	0	0
$540$	0	0
$541$	−20.8328	−0.895673	−0.447836	−	0.894116i	$-0.647805\pi$
$541$	−20.8328	−0.895673	−0.447836	+	0.894116i	$0.647805\pi$
$542$	−10.0902	−0.433410
$543$	0	0
$544$	18.5066	0.793463
$545$	−1.05573	−0.0452224
$546$	0	0
$547$	−12.5836	−0.538036	−0.269018	−	0.963135i	$-0.586699\pi$
$547$	−12.5836	−0.538036	−0.269018	+	0.963135i	$0.586699\pi$
$548$	6.43769	0.275005
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−21.4164	−0.910718
$554$	−17.7082	−0.752349
$555$	0	0
$556$	2.90983	0.123404
$557$	−17.9443	−0.760323	−0.380162	−	0.924920i	$-0.624132\pi$
$557$	−17.9443	−0.760323	−0.380162	+	0.924920i	$0.624132\pi$
$558$	0	0
$559$	−38.8328	−1.64245
$560$	−9.70820	−0.410246
$561$	0	0
$562$	45.8885	1.93569
$563$	−35.7771	−1.50782	−0.753912	−	0.656975i	$-0.771836\pi$
$563$	−35.7771	−1.50782	−0.753912	+	0.656975i	$0.771836\pi$
$564$	0	0
$565$	−6.52786	−0.274629
$566$	23.2361	0.976685
$567$	0	0
$568$	−10.0000	−0.419591
$569$	42.3607	1.77585	0.887926	−	0.459986i	$-0.152146\pi$
$569$	42.3607	1.77585	0.887926	+	0.459986i	$0.152146\pi$
$570$	0	0
$571$	−23.2918	−0.974731	−0.487366	−	0.873198i	$-0.662042\pi$
$571$	−23.2918	−0.974731	−0.487366	+	0.873198i	$0.662042\pi$
$572$	0	0
$573$	0	0
$574$	19.4164	0.810425
$575$	−8.23607	−0.343468
$576$	0	0
$577$	−16.0000	−0.666089	−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000	−0.666089	−0.333044	+	0.942911i	$0.608076\pi$
$578$	−20.9443	−0.871167
$579$	0	0
$580$	0.291796	0.0121162
$581$	0	0
$582$	0	0
$583$	0	0
$584$	28.9443	1.19772
$585$	0	0
$586$	19.3262	0.798360
$587$	−8.34752	−0.344539	−0.172270	−	0.985050i	$-0.555110\pi$
$587$	−8.34752	−0.344539	−0.172270	+	0.985050i	$0.555110\pi$
$588$	0	0
$589$	0	0
$590$	17.7082	0.729035
$591$	0	0
$592$	−41.1246	−1.69021
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	10.9443	0.448671
$596$	6.47214	0.265109
$597$	0	0
$598$	−86.2492	−3.52699
$599$	−35.8885	−1.46637	−0.733183	−	0.680031i	$-0.761966\pi$
$599$	−35.8885	−1.46637	−0.733183	+	0.680031i	$0.761966\pi$
$600$	0	0
$601$	−16.8328	−0.686625	−0.343312	−	0.939221i	$-0.611549\pi$
$601$	−16.8328	−0.686625	−0.343312	+	0.939221i	$0.611549\pi$
$602$	−19.4164	−0.791354
$603$	0	0
$604$	4.14590	0.168694
$605$	0	0
$606$	0	0
$607$	−2.47214	−0.100341	−0.0501705	−	0.998741i	$-0.515976\pi$
$607$	−2.47214	−0.100341	−0.0501705	+	0.998741i	$0.515976\pi$
$608$	0	0
$609$	0	0
$610$	−8.85410	−0.358492
$611$	24.3607	0.985528
$612$	0	0
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	49.1246	1.98251
$615$	0	0
$616$	0	0
$617$	−9.05573	−0.364570	−0.182285	−	0.983246i	$-0.558349\pi$
$617$	−9.05573	−0.364570	−0.182285	+	0.983246i	$0.558349\pi$
$618$	0	0
$619$	12.9443	0.520274	0.260137	−	0.965572i	$-0.416232\pi$
$619$	12.9443	0.520274	0.260137	+	0.965572i	$0.416232\pi$
$620$	4.14590	0.166503
$621$	0	0
$622$	28.3607	1.13716
$623$	0.944272	0.0378315
$624$	0	0
$625$	1.00000	0.0400000
$626$	20.1803	0.806569
$627$	0	0
$628$	−9.52786	−0.380203
$629$	46.3607	1.84852
$630$	0	0
$631$	29.2918	1.16609	0.583044	−	0.812441i	$-0.301862\pi$
$631$	29.2918	1.16609	0.583044	+	0.812441i	$0.301862\pi$
$632$	−23.9443	−0.952452
$633$	0	0
$634$	45.2148	1.79571
$635$	−2.47214	−0.0981037
$636$	0	0
$637$	19.4164	0.769306
$638$	0	0
$639$	0	0
$640$	−13.6180	−0.538300
$641$	31.4164	1.24087	0.620437	−	0.784256i	$-0.286955\pi$
$641$	31.4164	1.24087	0.620437	+	0.784256i	$0.286955\pi$
$642$	0	0
$643$	−10.5836	−0.417376	−0.208688	−	0.977982i	$-0.566919\pi$
$643$	−10.5836	−0.417376	−0.208688	+	0.977982i	$0.566919\pi$
$644$	−10.1803	−0.401162
$645$	0	0
$646$	0	0
$647$	−5.65248	−0.222222	−0.111111	−	0.993808i	$-0.535441\pi$
$647$	−5.65248	−0.222222	−0.111111	+	0.993808i	$0.535441\pi$
$648$	0	0
$649$	0	0
$650$	10.4721	0.410751
$651$	0	0
$652$	9.52786	0.373140
$653$	11.8885	0.465235	0.232617	−	0.972568i	$-0.425271\pi$
$653$	11.8885	0.465235	0.232617	+	0.972568i	$0.425271\pi$
$654$	0	0
$655$	−4.47214	−0.174741
$656$	29.1246	1.13713
$657$	0	0
$658$	12.1803	0.474839
$659$	−20.8328	−0.811531	−0.405766	−	0.913977i	$-0.632995\pi$
$659$	−20.8328	−0.811531	−0.405766	+	0.913977i	$0.632995\pi$
$660$	0	0
$661$	18.9443	0.736847	0.368423	−	0.929658i	$-0.379898\pi$
$661$	18.9443	0.736847	0.368423	+	0.929658i	$0.379898\pi$
$662$	−39.2148	−1.52413
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.88854	−0.150565
$668$	15.2705	0.590834
$669$	0	0
$670$	24.9443	0.963681
$671$	0	0
$672$	0	0
$673$	−18.5836	−0.716345	−0.358172	−	0.933655i	$-0.616600\pi$
$673$	−18.5836	−0.716345	−0.358172	+	0.933655i	$0.616600\pi$
$674$	−38.8328	−1.49578
$675$	0	0
$676$	17.8541	0.686696
$677$	19.8885	0.764379	0.382189	−	0.924084i	$-0.375170\pi$
$677$	19.8885	0.764379	0.382189	+	0.924084i	$0.375170\pi$
$678$	0	0
$679$	−12.9443	−0.496756
$680$	12.2361	0.469232
$681$	0	0
$682$	0	0
$683$	−35.7771	−1.36897	−0.684486	−	0.729026i	$-0.739973\pi$
$683$	−35.7771	−1.36897	−0.684486	+	0.729026i	$0.739973\pi$
$684$	0	0
$685$	10.4164	0.397990
$686$	32.3607	1.23554
$687$	0	0
$688$	−29.1246	−1.11037
$689$	77.3050	2.94508
$690$	0	0
$691$	−34.5967	−1.31612	−0.658061	−	0.752964i	$-0.728623\pi$
$691$	−34.5967	−1.31612	−0.658061	+	0.752964i	$0.728623\pi$
$692$	−14.1803	−0.539056
$693$	0	0
$694$	51.9787	1.97308
$695$	4.70820	0.178592
$696$	0	0
$697$	−32.8328	−1.24363
$698$	−49.2148	−1.86281
$699$	0	0
$700$	1.23607	0.0467190
$701$	−0.360680	−0.0136227	−0.00681134	−	0.999977i	$-0.502168\pi$
$701$	−0.360680	−0.0136227	−0.00681134	+	0.999977i	$0.502168\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	47.6869	1.79472
$707$	2.11146	0.0794095
$708$	0	0
$709$	24.5279	0.921163	0.460582	−	0.887617i	$-0.347641\pi$
$709$	24.5279	0.921163	0.460582	+	0.887617i	$0.347641\pi$
$710$	7.23607	0.271565
$711$	0	0
$712$	1.05573	0.0395651
$713$	−55.2492	−2.06910
$714$	0	0
$715$	0	0
$716$	−10.8328	−0.404841
$717$	0	0
$718$	46.0689	1.71928
$719$	−43.3050	−1.61500	−0.807501	−	0.589866i	$-0.799181\pi$
$719$	−43.3050	−1.61500	−0.807501	+	0.589866i	$0.799181\pi$
$720$	0	0
$721$	−7.05573	−0.262769
$722$	30.7426	1.14412
$723$	0	0
$724$	0.652476	0.0242491
$725$	0.472136	0.0175347
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	−28.9443	−1.07275
$729$	0	0
$730$	−20.9443	−0.775182
$731$	32.8328	1.21437
$732$	0	0
$733$	−44.0000	−1.62518	−0.812589	−	0.582838i	$-0.801942\pi$
$733$	−44.0000	−1.62518	−0.812589	+	0.582838i	$0.801942\pi$
$734$	−7.41641	−0.273745
$735$	0	0
$736$	−27.8541	−1.02672
$737$	0	0
$738$	0	0
$739$	28.7082	1.05605	0.528024	−	0.849229i	$-0.322933\pi$
$739$	28.7082	1.05605	0.528024	+	0.849229i	$0.322933\pi$
$740$	5.23607	0.192482
$741$	0	0
$742$	38.6525	1.41898
$743$	2.59675	0.0952654	0.0476327	−	0.998865i	$-0.484832\pi$
$743$	2.59675	0.0952654	0.0476327	+	0.998865i	$0.484832\pi$
$744$	0	0
$745$	10.4721	0.383669
$746$	−3.41641	−0.125084
$747$	0	0
$748$	0	0
$749$	−3.52786	−0.128905
$750$	0	0
$751$	−25.5410	−0.932005	−0.466003	−	0.884783i	$-0.654306\pi$
$751$	−25.5410	−0.932005	−0.466003	+	0.884783i	$0.654306\pi$
$752$	18.2705	0.666257
$753$	0	0
$754$	4.94427	0.180060
$755$	6.70820	0.244137
$756$	0	0
$757$	−6.94427	−0.252394	−0.126197	−	0.992005i	$-0.540277\pi$
$757$	−6.94427	−0.252394	−0.126197	+	0.992005i	$0.540277\pi$
$758$	43.0344	1.56308
$759$	0	0
$760$	0	0
$761$	31.4164	1.13884	0.569422	−	0.822045i	$-0.307167\pi$
$761$	31.4164	1.13884	0.569422	+	0.822045i	$0.307167\pi$
$762$	0	0
$763$	−2.11146	−0.0764398
$764$	−0.652476	−0.0236057
$765$	0	0
$766$	40.3607	1.45829
$767$	70.8328	2.55762
$768$	0	0
$769$	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
$771$	0	0
$772$	−4.00000	−0.143963
$773$	37.9443	1.36476	0.682380	−	0.730997i	$-0.260945\pi$
$773$	37.9443	1.36476	0.682380	+	0.730997i	$0.260945\pi$
$774$	0	0
$775$	6.70820	0.240966
$776$	−14.4721	−0.519519
$777$	0	0
$778$	−0.763932	−0.0273883
$779$	0	0
$780$	0	0
$781$	0	0
$782$	72.9230	2.60772
$783$	0	0
$784$	14.5623	0.520082
$785$	−15.4164	−0.550235
$786$	0	0
$787$	−33.4164	−1.19117	−0.595583	−	0.803294i	$-0.703079\pi$
$787$	−33.4164	−1.19117	−0.595583	+	0.803294i	$0.703079\pi$
$788$	−3.12461	−0.111310
$789$	0	0
$790$	17.3262	0.616440
$791$	−13.0557	−0.464208
$792$	0	0
$793$	−35.4164	−1.25767
$794$	−27.4164	−0.972972
$795$	0	0
$796$	1.96556	0.0696674
$797$	−18.9443	−0.671041	−0.335520	−	0.942033i	$-0.608912\pi$
$797$	−18.9443	−0.671041	−0.335520	+	0.942033i	$0.608912\pi$
$798$	0	0
$799$	−20.5967	−0.728661
$800$	3.38197	0.119571
$801$	0	0
$802$	−28.9443	−1.02206
$803$	0	0
$804$	0	0
$805$	−16.4721	−0.580567
$806$	70.2492	2.47442
$807$	0	0
$808$	2.36068	0.0830484
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−5.87539	−0.206313	−0.103156	−	0.994665i	$-0.532894\pi$
$811$	−5.87539	−0.206313	−0.103156	+	0.994665i	$0.532894\pi$
$812$	0.583592	0.0204801
$813$	0	0
$814$	0	0
$815$	15.4164	0.540013
$816$	0	0
$817$	0	0
$818$	−4.85410	−0.169720
$819$	0	0
$820$	−3.70820	−0.129496
$821$	−11.8885	−0.414913	−0.207457	−	0.978244i	$-0.566519\pi$
$821$	−11.8885	−0.414913	−0.207457	+	0.978244i	$0.566519\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	−7.88854	−0.274810
$825$	0	0
$826$	35.4164	1.23229
$827$	−20.9443	−0.728304	−0.364152	−	0.931340i	$-0.618641\pi$
$827$	−20.9443	−0.728304	−0.364152	+	0.931340i	$0.618641\pi$
$828$	0	0
$829$	1.58359	0.0550004	0.0275002	−	0.999622i	$-0.491245\pi$
$829$	1.58359	0.0550004	0.0275002	+	0.999622i	$0.491245\pi$
$830$	0	0
$831$	0	0
$832$	−27.4164	−0.950493
$833$	−16.4164	−0.568795
$834$	0	0
$835$	24.7082	0.855063
$836$	0	0
$837$	0	0
$838$	−51.4164	−1.77615
$839$	38.3607	1.32436	0.662179	−	0.749346i	$-0.269632\pi$
$839$	38.3607	1.32436	0.662179	+	0.749346i	$0.269632\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	−55.6869	−1.91910
$843$	0	0
$844$	2.03444	0.0700284
$845$	28.8885	0.993796
$846$	0	0
$847$	0	0
$848$	57.9787	1.99100
$849$	0	0
$850$	−8.85410	−0.303693
$851$	−69.7771	−2.39193
$852$	0	0
$853$	−20.8328	−0.713302	−0.356651	−	0.934238i	$-0.616081\pi$
$853$	−20.8328	−0.713302	−0.356651	+	0.934238i	$0.616081\pi$
$854$	−17.7082	−0.605962
$855$	0	0
$856$	−3.94427	−0.134812
$857$	−22.5279	−0.769537	−0.384769	−	0.923013i	$-0.625719\pi$
$857$	−22.5279	−0.769537	−0.384769	+	0.923013i	$0.625719\pi$
$858$	0	0
$859$	−39.7771	−1.35718	−0.678588	−	0.734519i	$-0.737408\pi$
$859$	−39.7771	−1.35718	−0.678588	+	0.734519i	$0.737408\pi$
$860$	3.70820	0.126449
$861$	0	0
$862$	38.6525	1.31651
$863$	44.9443	1.52992	0.764960	−	0.644077i	$-0.222759\pi$
$863$	44.9443	1.52992	0.764960	+	0.644077i	$0.222759\pi$
$864$	0	0
$865$	−22.9443	−0.780129
$866$	44.5410	1.51357
$867$	0	0
$868$	8.29180	0.281442
$869$	0	0
$870$	0	0
$871$	99.7771	3.38082
$872$	−2.36068	−0.0799427
$873$	0	0
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−26.8328	−0.906080	−0.453040	−	0.891490i	$-0.649660\pi$
$877$	−26.8328	−0.906080	−0.453040	+	0.891490i	$0.649660\pi$
$878$	−17.3262	−0.584732
$879$	0	0
$880$	0	0
$881$	−14.9443	−0.503485	−0.251743	−	0.967794i	$-0.581004\pi$
$881$	−14.9443	−0.503485	−0.251743	+	0.967794i	$0.581004\pi$
$882$	0	0
$883$	8.36068	0.281359	0.140680	−	0.990055i	$-0.455071\pi$
$883$	8.36068	0.281359	0.140680	+	0.990055i	$0.455071\pi$
$884$	−21.8885	−0.736191
$885$	0	0
$886$	33.8885	1.13851
$887$	−31.7771	−1.06697	−0.533485	−	0.845809i	$-0.679118\pi$
$887$	−31.7771	−1.06697	−0.533485	+	0.845809i	$0.679118\pi$
$888$	0	0
$889$	−4.94427	−0.165826
$890$	−0.763932	−0.0256071
$891$	0	0
$892$	−15.1246	−0.506409
$893$	0	0
$894$	0	0
$895$	−17.5279	−0.585892
$896$	−27.2361	−0.909893
$897$	0	0
$898$	−28.9443	−0.965883
$899$	3.16718	0.105632
$900$	0	0
$901$	−65.3607	−2.17748
$902$	0	0
$903$	0	0
$904$	−14.5967	−0.485481
$905$	1.05573	0.0350936
$906$	0	0
$907$	−38.8328	−1.28942	−0.644711	−	0.764426i	$-0.723022\pi$
$907$	−38.8328	−1.28942	−0.644711	+	0.764426i	$0.723022\pi$
$908$	4.43769	0.147270
$909$	0	0
$910$	20.9443	0.694296
$911$	−32.8328	−1.08780	−0.543900	−	0.839150i	$-0.683053\pi$
$911$	−32.8328	−1.08780	−0.543900	+	0.839150i	$0.683053\pi$
$912$	0	0
$913$	0	0
$914$	59.5967	1.97129
$915$	0	0
$916$	5.85410	0.193425
$917$	−8.94427	−0.295366
$918$	0	0
$919$	−18.8328	−0.621237	−0.310619	−	0.950535i	$-0.600536\pi$
$919$	−18.8328	−0.621237	−0.310619	+	0.950535i	$0.600536\pi$
$920$	−18.4164	−0.607171
$921$	0	0
$922$	23.4164	0.771178
$923$	28.9443	0.952712
$924$	0	0
$925$	8.47214	0.278562
$926$	−44.1803	−1.45186
$927$	0	0
$928$	1.59675	0.0524158
$929$	−38.7214	−1.27041	−0.635203	−	0.772345i	$-0.719084\pi$
$929$	−38.7214	−1.27041	−0.635203	+	0.772345i	$0.719084\pi$
$930$	0	0
$931$	0	0
$932$	−10.7295	−0.351456
$933$	0	0
$934$	5.90983	0.193376
$935$	0	0
$936$	0	0
$937$	−22.2492	−0.726850	−0.363425	−	0.931623i	$-0.618393\pi$
$937$	−22.2492	−0.726850	−0.363425	+	0.931623i	$0.618393\pi$
$938$	49.8885	1.62892
$939$	0	0
$940$	−2.32624	−0.0758735
$941$	−49.5279	−1.61456	−0.807281	−	0.590167i	$-0.799062\pi$
$941$	−49.5279	−1.61456	−0.807281	+	0.590167i	$0.799062\pi$
$942$	0	0
$943$	49.4164	1.60922
$944$	53.1246	1.72906
$945$	0	0
$946$	0	0
$947$	−47.6525	−1.54850	−0.774249	−	0.632881i	$-0.781872\pi$
$947$	−47.6525	−1.54850	−0.774249	+	0.632881i	$0.781872\pi$
$948$	0	0
$949$	−83.7771	−2.71952
$950$	0	0
$951$	0	0
$952$	24.4721	0.793146
$953$	−15.8885	−0.514680	−0.257340	−	0.966321i	$-0.582846\pi$
$953$	−15.8885	−0.514680	−0.257340	+	0.966321i	$0.582846\pi$
$954$	0	0
$955$	−1.05573	−0.0341626
$956$	17.5279	0.566892
$957$	0	0
$958$	−12.0000	−0.387702
$959$	20.8328	0.672727
$960$	0	0
$961$	14.0000	0.451613
$962$	88.7214	2.86049
$963$	0	0
$964$	−9.27051	−0.298583
$965$	−6.47214	−0.208345
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	0	0
$970$	10.4721	0.336240
$971$	−53.3050	−1.71064	−0.855319	−	0.518102i	$-0.826639\pi$
$971$	−53.3050	−1.71064	−0.855319	+	0.518102i	$0.826639\pi$
$972$	0	0
$973$	9.41641	0.301876
$974$	−3.23607	−0.103690
$975$	0	0
$976$	−26.5623	−0.850239
$977$	−21.4721	−0.686954	−0.343477	−	0.939161i	$-0.611605\pi$
$977$	−21.4721	−0.686954	−0.343477	+	0.939161i	$0.611605\pi$
$978$	0	0
$979$	0	0
$980$	−1.85410	−0.0592271
$981$	0	0
$982$	−26.6525	−0.850515
$983$	45.4296	1.44898	0.724489	−	0.689286i	$-0.242076\pi$
$983$	45.4296	1.44898	0.724489	+	0.689286i	$0.242076\pi$
$984$	0	0
$985$	−5.05573	−0.161089
$986$	−4.18034	−0.133129
$987$	0	0
$988$	0	0
$989$	−49.4164	−1.57135
$990$	0	0
$991$	18.7082	0.594286	0.297143	−	0.954833i	$-0.403966\pi$
$991$	18.7082	0.594286	0.297143	+	0.954833i	$0.403966\pi$
$992$	22.6869	0.720310
$993$	0	0
$994$	14.4721	0.459028
$995$	3.18034	0.100824
$996$	0	0
$997$	41.4164	1.31167	0.655835	−	0.754904i	$-0.272317\pi$
$997$	41.4164	1.31167	0.655835	+	0.754904i	$0.272317\pi$
$998$	−20.9443	−0.662979
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.p.1.1	yes	2
3.2	odd	2		5445.2.a.v.1.2	yes	2
11.10	odd	2		5445.2.a.w.1.2	yes	2
33.32	even	2		5445.2.a.n.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5445.2.a.n.1.1	✓	2	33.32	even	2
5445.2.a.p.1.1	yes	2	1.1	even	1	trivial
5445.2.a.v.1.2	yes	2	3.2	odd	2
5445.2.a.w.1.2	yes	2	11.10	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} +2.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-1.61803 q^{2} +0.618034 q^{4} +1.00000 q^{5} +2.00000 q^{7} +2.23607 q^{8} -1.61803 q^{10} -6.47214 q^{13} -3.23607 q^{14} -4.85410 q^{16} +5.47214 q^{17} +0.618034 q^{20} -8.23607 q^{23} +1.00000 q^{25} +10.4721 q^{26} +1.23607 q^{28} +0.472136 q^{29} +6.70820 q^{31} +3.38197 q^{32} -8.85410 q^{34} +2.00000 q^{35} +8.47214 q^{37} +2.23607 q^{40} -6.00000 q^{41} +6.00000 q^{43} +13.3262 q^{46} -3.76393 q^{47} -3.00000 q^{49} -1.61803 q^{50} -4.00000 q^{52} -11.9443 q^{53} +4.47214 q^{56} -0.763932 q^{58} -10.9443 q^{59} +5.47214 q^{61} -10.8541 q^{62} +4.23607 q^{64} -6.47214 q^{65} -15.4164 q^{67} +3.38197 q^{68} -3.23607 q^{70} -4.47214 q^{71} +12.9443 q^{73} -13.7082 q^{74} -10.7082 q^{79} -4.85410 q^{80} +9.70820 q^{82} +5.47214 q^{85} -9.70820 q^{86} +0.472136 q^{89} -12.9443 q^{91} -5.09017 q^{92} +6.09017 q^{94} -6.47214 q^{97} +4.85410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^5 + 4 * q^7	\( 2 q - q^{2} - q^{4} + 2 q^{5} + 4 q^{7} - q^{10} - 4 q^{13} - 2 q^{14} - 3 q^{16} + 2 q^{17} - q^{20} - 12 q^{23} + 2 q^{25} + 12 q^{26} - 2 q^{28} - 8 q^{29} + 9 q^{32} - 11 q^{34} + 4 q^{35} + 8 q^{37} - 12 q^{41} + 12 q^{43} + 11 q^{46} - 12 q^{47} - 6 q^{49} - q^{50} - 8 q^{52} - 6 q^{53} - 6 q^{58} - 4 q^{59} + 2 q^{61} - 15 q^{62} + 4 q^{64} - 4 q^{65} - 4 q^{67} + 9 q^{68} - 2 q^{70} + 8 q^{73} - 14 q^{74} - 8 q^{79} - 3 q^{80} + 6 q^{82} + 2 q^{85} - 6 q^{86} - 8 q^{89} - 8 q^{91} + q^{92} + q^{94} - 4 q^{97} + 3 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 + 4 * q^7 - q^10 - 4 * q^13 - 2 * q^14 - 3 * q^16 + 2 * q^17 - q^20 - 12 * q^23 + 2 * q^25 + 12 * q^26 - 2 * q^28 - 8 * q^29 + 9 * q^32 - 11 * q^34 + 4 * q^35 + 8 * q^37 - 12 * q^41 + 12 * q^43 + 11 * q^46 - 12 * q^47 - 6 * q^49 - q^50 - 8 * q^52 - 6 * q^53 - 6 * q^58 - 4 * q^59 + 2 * q^61 - 15 * q^62 + 4 * q^64 - 4 * q^65 - 4 * q^67 + 9 * q^68 - 2 * q^70 + 8 * q^73 - 14 * q^74 - 8 * q^79 - 3 * q^80 + 6 * q^82 + 2 * q^85 - 6 * q^86 - 8 * q^89 - 8 * q^91 + q^92 + q^94 - 4 * q^97 + 3 * q^98

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