LMFDB - Embedded newform 4761.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.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:	$38.0167764023$
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:	no (minimal twist has level 23)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} -3.23607 q^{5} +1.23607 q^{7} -2.23607 q^{8} +O(q^{10})$	$q+1.61803 q^{2} +0.618034 q^{4} -3.23607 q^{5} +1.23607 q^{7} -2.23607 q^{8} -5.23607 q^{10} -0.763932 q^{11} +3.00000 q^{13} +2.00000 q^{14} -4.85410 q^{16} +5.23607 q^{17} +2.00000 q^{19} -2.00000 q^{20} -1.23607 q^{22} +5.47214 q^{25} +4.85410 q^{26} +0.763932 q^{28} +3.00000 q^{29} -6.70820 q^{31} -3.38197 q^{32} +8.47214 q^{34} -4.00000 q^{35} -3.23607 q^{37} +3.23607 q^{38} +7.23607 q^{40} -5.47214 q^{41} -0.472136 q^{44} -2.23607 q^{47} -5.47214 q^{49} +8.85410 q^{50} +1.85410 q^{52} -8.47214 q^{53} +2.47214 q^{55} -2.76393 q^{56} +4.85410 q^{58} +2.47214 q^{59} -10.9443 q^{61} -10.8541 q^{62} +4.23607 q^{64} -9.70820 q^{65} +7.23607 q^{67} +3.23607 q^{68} -6.47214 q^{70} -7.76393 q^{71} +15.4721 q^{73} -5.23607 q^{74} +1.23607 q^{76} -0.944272 q^{77} -6.94427 q^{79} +15.7082 q^{80} -8.85410 q^{82} -13.2361 q^{83} -16.9443 q^{85} +1.70820 q^{88} -1.52786 q^{89} +3.70820 q^{91} -3.61803 q^{94} -6.47214 q^{95} -4.29180 q^{97} -8.85410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7	$ 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{10} - 6 q^{11} + 6 q^{13} + 4 q^{14} - 3 q^{16} + 6 q^{17} + 4 q^{19} - 4 q^{20} + 2 q^{22} + 2 q^{25} + 3 q^{26} + 6 q^{28} + 6 q^{29} - 9 q^{32} + 8 q^{34} - 8 q^{35} - 2 q^{37} + 2 q^{38} + 10 q^{40} - 2 q^{41} + 8 q^{44} - 2 q^{49} + 11 q^{50} - 3 q^{52} - 8 q^{53} - 4 q^{55} - 10 q^{56} + 3 q^{58} - 4 q^{59} - 4 q^{61} - 15 q^{62} + 4 q^{64} - 6 q^{65} + 10 q^{67} + 2 q^{68} - 4 q^{70} - 20 q^{71} + 22 q^{73} - 6 q^{74} - 2 q^{76} + 16 q^{77} + 4 q^{79} + 18 q^{80} - 11 q^{82} - 22 q^{83} - 16 q^{85} - 10 q^{88} - 12 q^{89} - 6 q^{91} - 5 q^{94} - 4 q^{95} - 22 q^{97} - 11 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7 - 6 * q^10 - 6 * q^11 + 6 * q^13 + 4 * q^14 - 3 * q^16 + 6 * q^17 + 4 * q^19 - 4 * q^20 + 2 * q^22 + 2 * q^25 + 3 * q^26 + 6 * q^28 + 6 * q^29 - 9 * q^32 + 8 * q^34 - 8 * q^35 - 2 * q^37 + 2 * q^38 + 10 * q^40 - 2 * q^41 + 8 * q^44 - 2 * q^49 + 11 * q^50 - 3 * q^52 - 8 * q^53 - 4 * q^55 - 10 * q^56 + 3 * q^58 - 4 * q^59 - 4 * q^61 - 15 * q^62 + 4 * q^64 - 6 * q^65 + 10 * q^67 + 2 * q^68 - 4 * q^70 - 20 * q^71 + 22 * q^73 - 6 * q^74 - 2 * q^76 + 16 * q^77 + 4 * q^79 + 18 * q^80 - 11 * q^82 - 22 * q^83 - 16 * q^85 - 10 * q^88 - 12 * q^89 - 6 * q^91 - 5 * q^94 - 4 * q^95 - 22 * q^97 - 11 * 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	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	−5.23607	−1.65579
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.85410	−1.21353
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−1.23607	−0.263531
$23$	0	0
$23$	0	0
$24$	0	0
$25$	5.47214	1.09443
$26$	4.85410	0.951968
$27$	0	0
$28$	0.763932	0.144370
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−6.70820	−1.20483	−0.602414	−	0.798183i	$-0.705795\pi$
$31$	−6.70820	−1.20483	−0.602414	+	0.798183i	$0.705795\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	8.47214	1.45296
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−3.23607	−0.532006	−0.266003	−	0.963972i	$-0.585703\pi$
$37$	−3.23607	−0.532006	−0.266003	+	0.963972i	$0.585703\pi$
$38$	3.23607	0.524960
$39$	0	0
$40$	7.23607	1.14412
$41$	−5.47214	−0.854604	−0.427302	−	0.904109i	$-0.640536\pi$
$41$	−5.47214	−0.854604	−0.427302	+	0.904109i	$0.640536\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−0.472136	−0.0711772
$45$	0	0
$46$	0	0
$47$	−2.23607	−0.326164	−0.163082	−	0.986613i	$-0.552144\pi$
$47$	−2.23607	−0.326164	−0.163082	+	0.986613i	$0.552144\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	8.85410	1.25216
$51$	0	0
$52$	1.85410	0.257118
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0	0
$55$	2.47214	0.333343
$56$	−2.76393	−0.369346
$57$	0	0
$58$	4.85410	0.637375
$59$	2.47214	0.321845	0.160922	−	0.986967i	$-0.448553\pi$
$59$	2.47214	0.321845	0.160922	+	0.986967i	$0.448553\pi$
$60$	0	0
$61$	−10.9443	−1.40127	−0.700635	−	0.713520i	$-0.747100\pi$
$61$	−10.9443	−1.40127	−0.700635	+	0.713520i	$0.747100\pi$
$62$	−10.8541	−1.37847
$63$	0	0
$64$	4.23607	0.529508
$65$	−9.70820	−1.20415
$66$	0	0
$67$	7.23607	0.884026	0.442013	−	0.897009i	$-0.354264\pi$
$67$	7.23607	0.884026	0.442013	+	0.897009i	$0.354264\pi$
$68$	3.23607	0.392431
$69$	0	0
$70$	−6.47214	−0.773568
$71$	−7.76393	−0.921409	−0.460705	−	0.887554i	$-0.652403\pi$
$71$	−7.76393	−0.921409	−0.460705	+	0.887554i	$0.652403\pi$
$72$	0	0
$73$	15.4721	1.81088	0.905438	−	0.424478i	$-0.139542\pi$
$73$	15.4721	1.81088	0.905438	+	0.424478i	$0.139542\pi$
$74$	−5.23607	−0.608681
$75$	0	0
$76$	1.23607	0.141787
$77$	−0.944272	−0.107610
$78$	0	0
$79$	−6.94427	−0.781292	−0.390646	−	0.920541i	$-0.627748\pi$
$79$	−6.94427	−0.781292	−0.390646	+	0.920541i	$0.627748\pi$
$80$	15.7082	1.75623
$81$	0	0
$82$	−8.85410	−0.977772
$83$	−13.2361	−1.45285	−0.726424	−	0.687247i	$-0.758819\pi$
$83$	−13.2361	−1.45285	−0.726424	+	0.687247i	$0.758819\pi$
$84$	0	0
$85$	−16.9443	−1.83786
$86$	0	0
$87$	0	0
$88$	1.70820	0.182095
$89$	−1.52786	−0.161953	−0.0809766	−	0.996716i	$-0.525804\pi$
$89$	−1.52786	−0.161953	−0.0809766	+	0.996716i	$0.525804\pi$
$90$	0	0
$91$	3.70820	0.388725
$92$	0	0
$93$	0	0
$94$	−3.61803	−0.373172
$95$	−6.47214	−0.664027
$96$	0	0
$97$	−4.29180	−0.435766	−0.217883	−	0.975975i	$-0.569915\pi$
$97$	−4.29180	−0.435766	−0.217883	+	0.975975i	$0.569915\pi$
$98$	−8.85410	−0.894399
$99$	0	0
$100$	3.38197	0.338197
$101$	4.47214	0.444994	0.222497	−	0.974933i	$-0.428579\pi$
$101$	4.47214	0.444994	0.222497	+	0.974933i	$0.428579\pi$
$102$	0	0
$103$	−18.1803	−1.79136	−0.895681	−	0.444697i	$-0.853311\pi$
$103$	−18.1803	−1.79136	−0.895681	+	0.444697i	$0.853311\pi$
$104$	−6.70820	−0.657794
$105$	0	0
$106$	−13.7082	−1.33146
$107$	−13.4164	−1.29701	−0.648507	−	0.761209i	$-0.724606\pi$
$107$	−13.4164	−1.29701	−0.648507	+	0.761209i	$0.724606\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	−6.00000	−0.566947
$113$	13.2361	1.24514	0.622572	−	0.782562i	$-0.286088\pi$
$113$	13.2361	1.24514	0.622572	+	0.782562i	$0.286088\pi$
$114$	0	0
$115$	0	0
$116$	1.85410	0.172149
$117$	0	0
$118$	4.00000	0.368230
$119$	6.47214	0.593300
$120$	0	0
$121$	−10.4164	−0.946946
$122$	−17.7082	−1.60323
$123$	0	0
$124$	−4.14590	−0.372313
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−20.7082	−1.83756	−0.918778	−	0.394775i	$-0.870823\pi$
$127$	−20.7082	−1.83756	−0.918778	+	0.394775i	$0.870823\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	−15.7082	−1.37770
$131$	−5.29180	−0.462346	−0.231173	−	0.972913i	$-0.574256\pi$
$131$	−5.29180	−0.462346	−0.231173	+	0.972913i	$0.574256\pi$
$132$	0	0
$133$	2.47214	0.214361
$134$	11.7082	1.01143
$135$	0	0
$136$	−11.7082	−1.00397
$137$	13.8885	1.18658	0.593289	−	0.804989i	$-0.297829\pi$
$137$	13.8885	1.18658	0.593289	+	0.804989i	$0.297829\pi$
$138$	0	0
$139$	2.70820	0.229707	0.114853	−	0.993382i	$-0.463360\pi$
$139$	2.70820	0.229707	0.114853	+	0.993382i	$0.463360\pi$
$140$	−2.47214	−0.208934
$141$	0	0
$142$	−12.5623	−1.05421
$143$	−2.29180	−0.191650
$144$	0	0
$145$	−9.70820	−0.806222
$146$	25.0344	2.07187
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−11.8885	−0.973947	−0.486974	−	0.873417i	$-0.661899\pi$
$149$	−11.8885	−0.973947	−0.486974	+	0.873417i	$0.661899\pi$
$150$	0	0
$151$	−0.236068	−0.0192109	−0.00960547	−	0.999954i	$-0.503058\pi$
$151$	−0.236068	−0.0192109	−0.00960547	+	0.999954i	$0.503058\pi$
$152$	−4.47214	−0.362738
$153$	0	0
$154$	−1.52786	−0.123119
$155$	21.7082	1.74364
$156$	0	0
$157$	−15.4164	−1.23036	−0.615182	−	0.788385i	$-0.710917\pi$
$157$	−15.4164	−1.23036	−0.615182	+	0.788385i	$0.710917\pi$
$158$	−11.2361	−0.893894
$159$	0	0
$160$	10.9443	0.865221
$161$	0	0
$162$	0	0
$163$	−10.2361	−0.801751	−0.400875	−	0.916133i	$-0.631294\pi$
$163$	−10.2361	−0.801751	−0.400875	+	0.916133i	$0.631294\pi$
$164$	−3.38197	−0.264087
$165$	0	0
$166$	−21.4164	−1.66224
$167$	−10.4721	−0.810358	−0.405179	−	0.914237i	$-0.632791\pi$
$167$	−10.4721	−0.810358	−0.405179	+	0.914237i	$0.632791\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	−27.4164	−2.10274
$171$	0	0
$172$	0	0
$173$	−5.05573	−0.384380	−0.192190	−	0.981358i	$-0.561559\pi$
$173$	−5.05573	−0.384380	−0.192190	+	0.981358i	$0.561559\pi$
$174$	0	0
$175$	6.76393	0.511305
$176$	3.70820	0.279516
$177$	0	0
$178$	−2.47214	−0.185294
$179$	12.7082	0.949856	0.474928	−	0.880025i	$-0.342474\pi$
$179$	12.7082	0.949856	0.474928	+	0.880025i	$0.342474\pi$
$180$	0	0
$181$	14.6525	1.08911	0.544555	−	0.838725i	$-0.316699\pi$
$181$	14.6525	1.08911	0.544555	+	0.838725i	$0.316699\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	0	0
$185$	10.4721	0.769927
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−1.38197	−0.100790
$189$	0	0
$190$	−10.4721	−0.759729
$191$	−3.81966	−0.276381	−0.138190	−	0.990406i	$-0.544129\pi$
$191$	−3.81966	−0.276381	−0.138190	+	0.990406i	$0.544129\pi$
$192$	0	0
$193$	−7.94427	−0.571841	−0.285921	−	0.958253i	$-0.592299\pi$
$193$	−7.94427	−0.571841	−0.285921	+	0.958253i	$0.592299\pi$
$194$	−6.94427	−0.498570
$195$	0	0
$196$	−3.38197	−0.241569
$197$	−7.47214	−0.532368	−0.266184	−	0.963922i	$-0.585763\pi$
$197$	−7.47214	−0.532368	−0.266184	+	0.963922i	$0.585763\pi$
$198$	0	0
$199$	25.7082	1.82241	0.911203	−	0.411957i	$-0.135155\pi$
$199$	25.7082	1.82241	0.911203	+	0.411957i	$0.135155\pi$
$200$	−12.2361	−0.865221
$201$	0	0
$202$	7.23607	0.509128
$203$	3.70820	0.260265
$204$	0	0
$205$	17.7082	1.23679
$206$	−29.4164	−2.04954
$207$	0	0
$208$	−14.5623	−1.00971
$209$	−1.52786	−0.105685
$210$	0	0
$211$	3.41641	0.235195	0.117598	−	0.993061i	$-0.462481\pi$
$211$	3.41641	0.235195	0.117598	+	0.993061i	$0.462481\pi$
$212$	−5.23607	−0.359615
$213$	0	0
$214$	−21.7082	−1.48394
$215$	0	0
$216$	0	0
$217$	−8.29180	−0.562884
$218$	0	0
$219$	0	0
$220$	1.52786	0.103009
$221$	15.7082	1.05665
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−4.18034	−0.279311
$225$	0	0
$226$	21.4164	1.42460
$227$	10.1803	0.675693	0.337846	−	0.941201i	$-0.390302\pi$
$227$	10.1803	0.675693	0.337846	+	0.941201i	$0.390302\pi$
$228$	0	0
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	0	0
$231$	0	0
$232$	−6.70820	−0.440415
$233$	15.4721	1.01361	0.506807	−	0.862060i	$-0.330826\pi$
$233$	15.4721	1.01361	0.506807	+	0.862060i	$0.330826\pi$
$234$	0	0
$235$	7.23607	0.472029
$236$	1.52786	0.0994555
$237$	0	0
$238$	10.4721	0.678808
$239$	−18.2361	−1.17959	−0.589797	−	0.807552i	$-0.700792\pi$
$239$	−18.2361	−1.17959	−0.589797	+	0.807552i	$0.700792\pi$
$240$	0	0
$241$	−17.1246	−1.10309	−0.551547	−	0.834144i	$-0.685962\pi$
$241$	−17.1246	−1.10309	−0.551547	+	0.834144i	$0.685962\pi$
$242$	−16.8541	−1.08342
$243$	0	0
$244$	−6.76393	−0.433016
$245$	17.7082	1.13134
$246$	0	0
$247$	6.00000	0.381771
$248$	15.0000	0.952501
$249$	0	0
$250$	−2.47214	−0.156352
$251$	15.7082	0.991493	0.495747	−	0.868467i	$-0.334895\pi$
$251$	15.7082	0.991493	0.495747	+	0.868467i	$0.334895\pi$
$252$	0	0
$253$	0	0
$254$	−33.5066	−2.10239
$255$	0	0
$256$	13.5623	0.847644
$257$	−1.47214	−0.0918293	−0.0459147	−	0.998945i	$-0.514620\pi$
$257$	−1.47214	−0.0918293	−0.0459147	+	0.998945i	$0.514620\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	−6.00000	−0.372104
$261$	0	0
$262$	−8.56231	−0.528981
$263$	−14.9443	−0.921503	−0.460752	−	0.887529i	$-0.652420\pi$
$263$	−14.9443	−0.921503	−0.460752	+	0.887529i	$0.652420\pi$
$264$	0	0
$265$	27.4164	1.68418
$266$	4.00000	0.245256
$267$	0	0
$268$	4.47214	0.273179
$269$	−9.94427	−0.606313	−0.303156	−	0.952941i	$-0.598040\pi$
$269$	−9.94427	−0.606313	−0.303156	+	0.952941i	$0.598040\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−25.4164	−1.54110
$273$	0	0
$274$	22.4721	1.35759
$275$	−4.18034	−0.252084
$276$	0	0
$277$	6.52786	0.392221	0.196111	−	0.980582i	$-0.437169\pi$
$277$	6.52786	0.392221	0.196111	+	0.980582i	$0.437169\pi$
$278$	4.38197	0.262813
$279$	0	0
$280$	8.94427	0.534522
$281$	−13.2361	−0.789598	−0.394799	−	0.918768i	$-0.629186\pi$
$281$	−13.2361	−0.789598	−0.394799	+	0.918768i	$0.629186\pi$
$282$	0	0
$283$	−14.2918	−0.849559	−0.424780	−	0.905297i	$-0.639648\pi$
$283$	−14.2918	−0.849559	−0.424780	+	0.905297i	$0.639648\pi$
$284$	−4.79837	−0.284731
$285$	0	0
$286$	−3.70820	−0.219271
$287$	−6.76393	−0.399262
$288$	0	0
$289$	10.4164	0.612730
$290$	−15.7082	−0.922417
$291$	0	0
$292$	9.56231	0.559592
$293$	−10.4721	−0.611789	−0.305894	−	0.952065i	$-0.598955\pi$
$293$	−10.4721	−0.611789	−0.305894	+	0.952065i	$0.598955\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	7.23607	0.420588
$297$	0	0
$298$	−19.2361	−1.11432
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−0.381966	−0.0219797
$303$	0	0
$304$	−9.70820	−0.556804
$305$	35.4164	2.02794
$306$	0	0
$307$	18.4721	1.05426	0.527130	−	0.849785i	$-0.323268\pi$
$307$	18.4721	1.05426	0.527130	+	0.849785i	$0.323268\pi$
$308$	−0.583592	−0.0332532
$309$	0	0
$310$	35.1246	1.99494
$311$	9.18034	0.520569	0.260285	−	0.965532i	$-0.416184\pi$
$311$	9.18034	0.520569	0.260285	+	0.965532i	$0.416184\pi$
$312$	0	0
$313$	20.3607	1.15085	0.575427	−	0.817853i	$-0.304836\pi$
$313$	20.3607	1.15085	0.575427	+	0.817853i	$0.304836\pi$
$314$	−24.9443	−1.40769
$315$	0	0
$316$	−4.29180	−0.241432
$317$	1.41641	0.0795534	0.0397767	−	0.999209i	$-0.487335\pi$
$317$	1.41641	0.0795534	0.0397767	+	0.999209i	$0.487335\pi$
$318$	0	0
$319$	−2.29180	−0.128316
$320$	−13.7082	−0.766312
$321$	0	0
$322$	0	0
$323$	10.4721	0.582685
$324$	0	0
$325$	16.4164	0.910618
$326$	−16.5623	−0.917301
$327$	0	0
$328$	12.2361	0.675624
$329$	−2.76393	−0.152381
$330$	0	0
$331$	11.6525	0.640478	0.320239	−	0.947337i	$-0.396237\pi$
$331$	11.6525	0.640478	0.320239	+	0.947337i	$0.396237\pi$
$332$	−8.18034	−0.448954
$333$	0	0
$334$	−16.9443	−0.927149
$335$	−23.4164	−1.27938
$336$	0	0
$337$	3.41641	0.186104	0.0930518	−	0.995661i	$-0.470338\pi$
$337$	3.41641	0.186104	0.0930518	+	0.995661i	$0.470338\pi$
$338$	−6.47214	−0.352038
$339$	0	0
$340$	−10.4721	−0.567931
$341$	5.12461	0.277513
$342$	0	0
$343$	−15.4164	−0.832408
$344$	0	0
$345$	0	0
$346$	−8.18034	−0.439778
$347$	−25.8885	−1.38977	−0.694885	−	0.719121i	$-0.744545\pi$
$347$	−25.8885	−1.38977	−0.694885	+	0.719121i	$0.744545\pi$
$348$	0	0
$349$	−2.41641	−0.129347	−0.0646737	−	0.997906i	$-0.520601\pi$
$349$	−2.41641	−0.129347	−0.0646737	+	0.997906i	$0.520601\pi$
$350$	10.9443	0.584996
$351$	0	0
$352$	2.58359	0.137706
$353$	35.3607	1.88206	0.941030	−	0.338324i	$-0.109860\pi$
$353$	35.3607	1.88206	0.941030	+	0.338324i	$0.109860\pi$
$354$	0	0
$355$	25.1246	1.33348
$356$	−0.944272	−0.0500463
$357$	0	0
$358$	20.5623	1.08675
$359$	15.8885	0.838565	0.419283	−	0.907856i	$-0.362282\pi$
$359$	15.8885	0.838565	0.419283	+	0.907856i	$0.362282\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	23.7082	1.24608
$363$	0	0
$364$	2.29180	0.120123
$365$	−50.0689	−2.62073
$366$	0	0
$367$	−18.1803	−0.949006	−0.474503	−	0.880254i	$-0.657372\pi$
$367$	−18.1803	−0.949006	−0.474503	+	0.880254i	$0.657372\pi$
$368$	0	0
$369$	0	0
$370$	16.9443	0.880891
$371$	−10.4721	−0.543686
$372$	0	0
$373$	5.70820	0.295560	0.147780	−	0.989020i	$-0.452787\pi$
$373$	5.70820	0.295560	0.147780	+	0.989020i	$0.452787\pi$
$374$	−6.47214	−0.334666
$375$	0	0
$376$	5.00000	0.257855
$377$	9.00000	0.463524
$378$	0	0
$379$	20.3607	1.04586	0.522929	−	0.852376i	$-0.324839\pi$
$379$	20.3607	1.04586	0.522929	+	0.852376i	$0.324839\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	−6.18034	−0.316214
$383$	24.9443	1.27459	0.637296	−	0.770619i	$-0.280053\pi$
$383$	24.9443	1.27459	0.637296	+	0.770619i	$0.280053\pi$
$384$	0	0
$385$	3.05573	0.155734
$386$	−12.8541	−0.654257
$387$	0	0
$388$	−2.65248	−0.134659
$389$	34.4721	1.74781	0.873903	−	0.486100i	$-0.161581\pi$
$389$	34.4721	1.74781	0.873903	+	0.486100i	$0.161581\pi$
$390$	0	0
$391$	0	0
$392$	12.2361	0.618015
$393$	0	0
$394$	−12.0902	−0.609094
$395$	22.4721	1.13070
$396$	0	0
$397$	2.41641	0.121276	0.0606380	−	0.998160i	$-0.480686\pi$
$397$	2.41641	0.121276	0.0606380	+	0.998160i	$0.480686\pi$
$398$	41.5967	2.08506
$399$	0	0
$400$	−26.5623	−1.32812
$401$	8.18034	0.408507	0.204253	−	0.978918i	$-0.434523\pi$
$401$	8.18034	0.408507	0.204253	+	0.978918i	$0.434523\pi$
$402$	0	0
$403$	−20.1246	−1.00248
$404$	2.76393	0.137511
$405$	0	0
$406$	6.00000	0.297775
$407$	2.47214	0.122539
$408$	0	0
$409$	−23.3607	−1.15511	−0.577556	−	0.816351i	$-0.695993\pi$
$409$	−23.3607	−1.15511	−0.577556	+	0.816351i	$0.695993\pi$
$410$	28.6525	1.41504
$411$	0	0
$412$	−11.2361	−0.553561
$413$	3.05573	0.150363
$414$	0	0
$415$	42.8328	2.10258
$416$	−10.1459	−0.497444
$417$	0	0
$418$	−2.47214	−0.120916
$419$	−31.4164	−1.53479	−0.767396	−	0.641173i	$-0.778448\pi$
$419$	−31.4164	−1.53479	−0.767396	+	0.641173i	$0.778448\pi$
$420$	0	0
$421$	23.7082	1.15547	0.577734	−	0.816225i	$-0.303937\pi$
$421$	23.7082	1.15547	0.577734	+	0.816225i	$0.303937\pi$
$422$	5.52786	0.269092
$423$	0	0
$424$	18.9443	0.920015
$425$	28.6525	1.38985
$426$	0	0
$427$	−13.5279	−0.654659
$428$	−8.29180	−0.400799
$429$	0	0
$430$	0	0
$431$	−26.4721	−1.27512	−0.637559	−	0.770402i	$-0.720056\pi$
$431$	−26.4721	−1.27512	−0.637559	+	0.770402i	$0.720056\pi$
$432$	0	0
$433$	−40.1803	−1.93094	−0.965472	−	0.260507i	$-0.916110\pi$
$433$	−40.1803	−1.93094	−0.965472	+	0.260507i	$0.916110\pi$
$434$	−13.4164	−0.644008
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−5.29180	−0.252564	−0.126282	−	0.991994i	$-0.540304\pi$
$439$	−5.29180	−0.252564	−0.126282	+	0.991994i	$0.540304\pi$
$440$	−5.52786	−0.263531
$441$	0	0
$442$	25.4164	1.20894
$443$	2.12461	0.100943	0.0504717	−	0.998725i	$-0.483928\pi$
$443$	2.12461	0.100943	0.0504717	+	0.998725i	$0.483928\pi$
$444$	0	0
$445$	4.94427	0.234381
$446$	6.47214	0.306465
$447$	0	0
$448$	5.23607	0.247381
$449$	−2.94427	−0.138949	−0.0694744	−	0.997584i	$-0.522132\pi$
$449$	−2.94427	−0.138949	−0.0694744	+	0.997584i	$0.522132\pi$
$450$	0	0
$451$	4.18034	0.196845
$452$	8.18034	0.384771
$453$	0	0
$454$	16.4721	0.773076
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−35.1246	−1.64306	−0.821530	−	0.570165i	$-0.806879\pi$
$457$	−35.1246	−1.64306	−0.821530	+	0.570165i	$0.806879\pi$
$458$	19.4164	0.907269
$459$	0	0
$460$	0	0
$461$	−7.47214	−0.348012	−0.174006	−	0.984745i	$-0.555671\pi$
$461$	−7.47214	−0.348012	−0.174006	+	0.984745i	$0.555671\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	−14.5623	−0.676038
$465$	0	0
$466$	25.0344	1.15970
$467$	−30.9443	−1.43193	−0.715965	−	0.698136i	$-0.754013\pi$
$467$	−30.9443	−1.43193	−0.715965	+	0.698136i	$0.754013\pi$
$468$	0	0
$469$	8.94427	0.413008
$470$	11.7082	0.540059
$471$	0	0
$472$	−5.52786	−0.254441
$473$	0	0
$474$	0	0
$475$	10.9443	0.502158
$476$	4.00000	0.183340
$477$	0	0
$478$	−29.5066	−1.34960
$479$	−17.5967	−0.804016	−0.402008	−	0.915636i	$-0.631688\pi$
$479$	−17.5967	−0.804016	−0.402008	+	0.915636i	$0.631688\pi$
$480$	0	0
$481$	−9.70820	−0.442656
$482$	−27.7082	−1.26207
$483$	0	0
$484$	−6.43769	−0.292622
$485$	13.8885	0.630646
$486$	0	0
$487$	−1.29180	−0.0585369	−0.0292684	−	0.999572i	$-0.509318\pi$
$487$	−1.29180	−0.0585369	−0.0292684	+	0.999572i	$0.509318\pi$
$488$	24.4721	1.10780
$489$	0	0
$490$	28.6525	1.29439
$491$	−39.6525	−1.78949	−0.894746	−	0.446576i	$-0.852643\pi$
$491$	−39.6525	−1.78949	−0.894746	+	0.446576i	$0.852643\pi$
$492$	0	0
$493$	15.7082	0.707462
$494$	9.70820	0.436793
$495$	0	0
$496$	32.5623	1.46209
$497$	−9.59675	−0.430473
$498$	0	0
$499$	32.7082	1.46422	0.732110	−	0.681186i	$-0.238536\pi$
$499$	32.7082	1.46422	0.732110	+	0.681186i	$0.238536\pi$
$500$	−0.944272	−0.0422291
$501$	0	0
$502$	25.4164	1.13439
$503$	−9.05573	−0.403775	−0.201887	−	0.979409i	$-0.564708\pi$
$503$	−9.05573	−0.403775	−0.201887	+	0.979409i	$0.564708\pi$
$504$	0	0
$505$	−14.4721	−0.644002
$506$	0	0
$507$	0	0
$508$	−12.7984	−0.567836
$509$	−34.3050	−1.52054	−0.760270	−	0.649607i	$-0.774933\pi$
$509$	−34.3050	−1.52054	−0.760270	+	0.649607i	$0.774933\pi$
$510$	0	0
$511$	19.1246	0.846023
$512$	−5.29180	−0.233867
$513$	0	0
$514$	−2.38197	−0.105064
$515$	58.8328	2.59248
$516$	0	0
$517$	1.70820	0.0751267
$518$	−6.47214	−0.284369
$519$	0	0
$520$	21.7082	0.951968
$521$	4.58359	0.200811	0.100405	−	0.994947i	$-0.467986\pi$
$521$	4.58359	0.200811	0.100405	+	0.994947i	$0.467986\pi$
$522$	0	0
$523$	−0.875388	−0.0382781	−0.0191390	−	0.999817i	$-0.506093\pi$
$523$	−0.875388	−0.0382781	−0.0191390	+	0.999817i	$0.506093\pi$
$524$	−3.27051	−0.142873
$525$	0	0
$526$	−24.1803	−1.05431
$527$	−35.1246	−1.53005
$528$	0	0
$529$	0	0
$530$	44.3607	1.92690
$531$	0	0
$532$	1.52786	0.0662413
$533$	−16.4164	−0.711074
$534$	0	0
$535$	43.4164	1.87705
$536$	−16.1803	−0.698884
$537$	0	0
$538$	−16.0902	−0.693696
$539$	4.18034	0.180060
$540$	0	0
$541$	−7.58359	−0.326044	−0.163022	−	0.986622i	$-0.552124\pi$
$541$	−7.58359	−0.326044	−0.163022	+	0.986622i	$0.552124\pi$
$542$	12.9443	0.556004
$543$	0	0
$544$	−17.7082	−0.759233
$545$	0	0
$546$	0	0
$547$	37.5410	1.60514	0.802569	−	0.596559i	$-0.203466\pi$
$547$	37.5410	1.60514	0.802569	+	0.596559i	$0.203466\pi$
$548$	8.58359	0.366673
$549$	0	0
$550$	−6.76393	−0.288415
$551$	6.00000	0.255609
$552$	0	0
$553$	−8.58359	−0.365011
$554$	10.5623	0.448749
$555$	0	0
$556$	1.67376	0.0709833
$557$	19.4164	0.822700	0.411350	−	0.911478i	$-0.365057\pi$
$557$	19.4164	0.822700	0.411350	+	0.911478i	$0.365057\pi$
$558$	0	0
$559$	0	0
$560$	19.4164	0.820493
$561$	0	0
$562$	−21.4164	−0.903397
$563$	−15.0557	−0.634523	−0.317262	−	0.948338i	$-0.602763\pi$
$563$	−15.0557	−0.634523	−0.317262	+	0.948338i	$0.602763\pi$
$564$	0	0
$565$	−42.8328	−1.80199
$566$	−23.1246	−0.972000
$567$	0	0
$568$	17.3607	0.728438
$569$	0.180340	0.00756024	0.00378012	−	0.999993i	$-0.498797\pi$
$569$	0.180340	0.00756024	0.00378012	+	0.999993i	$0.498797\pi$
$570$	0	0
$571$	27.7082	1.15955	0.579776	−	0.814776i	$-0.303140\pi$
$571$	27.7082	1.15955	0.579776	+	0.814776i	$0.303140\pi$
$572$	−1.41641	−0.0592230
$573$	0	0
$574$	−10.9443	−0.456805
$575$	0	0
$576$	0	0
$577$	−12.8885	−0.536557	−0.268279	−	0.963341i	$-0.586455\pi$
$577$	−12.8885	−0.536557	−0.268279	+	0.963341i	$0.586455\pi$
$578$	16.8541	0.701038
$579$	0	0
$580$	−6.00000	−0.249136
$581$	−16.3607	−0.678755
$582$	0	0
$583$	6.47214	0.268048
$584$	−34.5967	−1.43162
$585$	0	0
$586$	−16.9443	−0.699961
$587$	11.2918	0.466062	0.233031	−	0.972469i	$-0.425136\pi$
$587$	11.2918	0.466062	0.233031	+	0.972469i	$0.425136\pi$
$588$	0	0
$589$	−13.4164	−0.552813
$590$	−12.9443	−0.532907
$591$	0	0
$592$	15.7082	0.645603
$593$	−14.9443	−0.613688	−0.306844	−	0.951760i	$-0.599273\pi$
$593$	−14.9443	−0.613688	−0.306844	+	0.951760i	$0.599273\pi$
$594$	0	0
$595$	−20.9443	−0.858631
$596$	−7.34752	−0.300966
$597$	0	0
$598$	0	0
$599$	1.88854	0.0771638	0.0385819	−	0.999255i	$-0.487716\pi$
$599$	1.88854	0.0771638	0.0385819	+	0.999255i	$0.487716\pi$
$600$	0	0
$601$	11.1115	0.453246	0.226623	−	0.973983i	$-0.427232\pi$
$601$	11.1115	0.453246	0.226623	+	0.973983i	$0.427232\pi$
$602$	0	0
$603$	0	0
$604$	−0.145898	−0.00593651
$605$	33.7082	1.37043
$606$	0	0
$607$	17.5279	0.711434	0.355717	−	0.934594i	$-0.384237\pi$
$607$	17.5279	0.711434	0.355717	+	0.934594i	$0.384237\pi$
$608$	−6.76393	−0.274314
$609$	0	0
$610$	57.3050	2.32021
$611$	−6.70820	−0.271385
$612$	0	0
$613$	7.70820	0.311331	0.155666	−	0.987810i	$-0.450248\pi$
$613$	7.70820	0.311331	0.155666	+	0.987810i	$0.450248\pi$
$614$	29.8885	1.20620
$615$	0	0
$616$	2.11146	0.0850730
$617$	−16.4721	−0.663143	−0.331572	−	0.943430i	$-0.607579\pi$
$617$	−16.4721	−0.663143	−0.331572	+	0.943430i	$0.607579\pi$
$618$	0	0
$619$	7.41641	0.298091	0.149045	−	0.988830i	$-0.452380\pi$
$619$	7.41641	0.298091	0.149045	+	0.988830i	$0.452380\pi$
$620$	13.4164	0.538816
$621$	0	0
$622$	14.8541	0.595595
$623$	−1.88854	−0.0756629
$624$	0	0
$625$	−22.4164	−0.896656
$626$	32.9443	1.31672
$627$	0	0
$628$	−9.52786	−0.380203
$629$	−16.9443	−0.675612
$630$	0	0
$631$	32.3607	1.28826	0.644129	−	0.764917i	$-0.277220\pi$
$631$	32.3607	1.28826	0.644129	+	0.764917i	$0.277220\pi$
$632$	15.5279	0.617665
$633$	0	0
$634$	2.29180	0.0910188
$635$	67.0132	2.65934
$636$	0	0
$637$	−16.4164	−0.650442
$638$	−3.70820	−0.146809
$639$	0	0
$640$	−44.0689	−1.74198
$641$	45.3050	1.78944	0.894719	−	0.446629i	$-0.147376\pi$
$641$	45.3050	1.78944	0.894719	+	0.446629i	$0.147376\pi$
$642$	0	0
$643$	−19.5967	−0.772820	−0.386410	−	0.922327i	$-0.626285\pi$
$643$	−19.5967	−0.772820	−0.386410	+	0.922327i	$0.626285\pi$
$644$	0	0
$645$	0	0
$646$	16.9443	0.666663
$647$	6.70820	0.263727	0.131863	−	0.991268i	$-0.457904\pi$
$647$	6.70820	0.263727	0.131863	+	0.991268i	$0.457904\pi$
$648$	0	0
$649$	−1.88854	−0.0741318
$650$	26.5623	1.04186
$651$	0	0
$652$	−6.32624	−0.247755
$653$	−24.3050	−0.951126	−0.475563	−	0.879682i	$-0.657756\pi$
$653$	−24.3050	−0.951126	−0.475563	+	0.879682i	$0.657756\pi$
$654$	0	0
$655$	17.1246	0.669114
$656$	26.5623	1.03708
$657$	0	0
$658$	−4.47214	−0.174342
$659$	20.6525	0.804506	0.402253	−	0.915528i	$-0.368227\pi$
$659$	20.6525	0.804506	0.402253	+	0.915528i	$0.368227\pi$
$660$	0	0
$661$	5.05573	0.196645	0.0983225	−	0.995155i	$-0.468652\pi$
$661$	5.05573	0.196645	0.0983225	+	0.995155i	$0.468652\pi$
$662$	18.8541	0.732785
$663$	0	0
$664$	29.5967	1.14858
$665$	−8.00000	−0.310227
$666$	0	0
$667$	0	0
$668$	−6.47214	−0.250414
$669$	0	0
$670$	−37.8885	−1.46376
$671$	8.36068	0.322760
$672$	0	0
$673$	3.00000	0.115642	0.0578208	−	0.998327i	$-0.481585\pi$
$673$	3.00000	0.115642	0.0578208	+	0.998327i	$0.481585\pi$
$674$	5.52786	0.212925
$675$	0	0
$676$	−2.47214	−0.0950822
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	−5.30495	−0.203585
$680$	37.8885	1.45296
$681$	0	0
$682$	8.29180	0.317509
$683$	22.5967	0.864641	0.432320	−	0.901720i	$-0.357695\pi$
$683$	22.5967	0.864641	0.432320	+	0.901720i	$0.357695\pi$
$684$	0	0
$685$	−44.9443	−1.71723
$686$	−24.9443	−0.952377
$687$	0	0
$688$	0	0
$689$	−25.4164	−0.968288
$690$	0	0
$691$	24.9443	0.948925	0.474462	−	0.880276i	$-0.342642\pi$
$691$	24.9443	0.948925	0.474462	+	0.880276i	$0.342642\pi$
$692$	−3.12461	−0.118780
$693$	0	0
$694$	−41.8885	−1.59007
$695$	−8.76393	−0.332435
$696$	0	0
$697$	−28.6525	−1.08529
$698$	−3.90983	−0.147989
$699$	0	0
$700$	4.18034	0.158002
$701$	−26.1803	−0.988818	−0.494409	−	0.869229i	$-0.664615\pi$
$701$	−26.1803	−0.988818	−0.494409	+	0.869229i	$0.664615\pi$
$702$	0	0
$703$	−6.47214	−0.244101
$704$	−3.23607	−0.121964
$705$	0	0
$706$	57.2148	2.15331
$707$	5.52786	0.207897
$708$	0	0
$709$	−16.0689	−0.603480	−0.301740	−	0.953390i	$-0.597567\pi$
$709$	−16.0689	−0.603480	−0.301740	+	0.953390i	$0.597567\pi$
$710$	40.6525	1.52566
$711$	0	0
$712$	3.41641	0.128035
$713$	0	0
$714$	0	0
$715$	7.41641	0.277358
$716$	7.85410	0.293522
$717$	0	0
$718$	25.7082	0.959422
$719$	20.9443	0.781090	0.390545	−	0.920584i	$-0.372287\pi$
$719$	20.9443	0.781090	0.390545	+	0.920584i	$0.372287\pi$
$720$	0	0
$721$	−22.4721	−0.836906
$722$	−24.2705	−0.903255
$723$	0	0
$724$	9.05573	0.336553
$725$	16.4164	0.609690
$726$	0	0
$727$	14.2918	0.530053	0.265027	−	0.964241i	$-0.414619\pi$
$727$	14.2918	0.530053	0.265027	+	0.964241i	$0.414619\pi$
$728$	−8.29180	−0.307314
$729$	0	0
$730$	−81.0132	−2.99843
$731$	0	0
$732$	0	0
$733$	26.7639	0.988548	0.494274	−	0.869306i	$-0.335434\pi$
$733$	26.7639	0.988548	0.494274	+	0.869306i	$0.335434\pi$
$734$	−29.4164	−1.08578
$735$	0	0
$736$	0	0
$737$	−5.52786	−0.203621
$738$	0	0
$739$	49.1803	1.80913	0.904564	−	0.426338i	$-0.140197\pi$
$739$	49.1803	1.80913	0.904564	+	0.426338i	$0.140197\pi$
$740$	6.47214	0.237920
$741$	0	0
$742$	−16.9443	−0.622044
$743$	0.875388	0.0321149	0.0160574	−	0.999871i	$-0.494889\pi$
$743$	0.875388	0.0321149	0.0160574	+	0.999871i	$0.494889\pi$
$744$	0	0
$745$	38.4721	1.40951
$746$	9.23607	0.338156
$747$	0	0
$748$	−2.47214	−0.0903902
$749$	−16.5836	−0.605951
$750$	0	0
$751$	44.3607	1.61874	0.809372	−	0.587296i	$-0.199808\pi$
$751$	44.3607	1.61874	0.809372	+	0.587296i	$0.199808\pi$
$752$	10.8541	0.395808
$753$	0	0
$754$	14.5623	0.530328
$755$	0.763932	0.0278023
$756$	0	0
$757$	47.5967	1.72993	0.864967	−	0.501829i	$-0.167339\pi$
$757$	47.5967	1.72993	0.864967	+	0.501829i	$0.167339\pi$
$758$	32.9443	1.19659
$759$	0	0
$760$	14.4721	0.524960
$761$	16.3050	0.591054	0.295527	−	0.955334i	$-0.404505\pi$
$761$	16.3050	0.591054	0.295527	+	0.955334i	$0.404505\pi$
$762$	0	0
$763$	0	0
$764$	−2.36068	−0.0854064
$765$	0	0
$766$	40.3607	1.45829
$767$	7.41641	0.267791
$768$	0	0
$769$	−17.1246	−0.617529	−0.308765	−	0.951138i	$-0.599916\pi$
$769$	−17.1246	−0.617529	−0.308765	+	0.951138i	$0.599916\pi$
$770$	4.94427	0.178179
$771$	0	0
$772$	−4.90983	−0.176709
$773$	−14.4721	−0.520527	−0.260263	−	0.965538i	$-0.583809\pi$
$773$	−14.4721	−0.520527	−0.260263	+	0.965538i	$0.583809\pi$
$774$	0	0
$775$	−36.7082	−1.31860
$776$	9.59675	0.344503
$777$	0	0
$778$	55.7771	1.99971
$779$	−10.9443	−0.392119
$780$	0	0
$781$	5.93112	0.212232
$782$	0	0
$783$	0	0
$784$	26.5623	0.948654
$785$	49.8885	1.78060
$786$	0	0
$787$	−51.4164	−1.83280	−0.916399	−	0.400267i	$-0.868917\pi$
$787$	−51.4164	−1.83280	−0.916399	+	0.400267i	$0.868917\pi$
$788$	−4.61803	−0.164511
$789$	0	0
$790$	36.3607	1.29365
$791$	16.3607	0.581719
$792$	0	0
$793$	−32.8328	−1.16593
$794$	3.90983	0.138755
$795$	0	0
$796$	15.8885	0.563155
$797$	10.3607	0.366994	0.183497	−	0.983020i	$-0.441258\pi$
$797$	10.3607	0.366994	0.183497	+	0.983020i	$0.441258\pi$
$798$	0	0
$799$	−11.7082	−0.414206
$800$	−18.5066	−0.654306
$801$	0	0
$802$	13.2361	0.467382
$803$	−11.8197	−0.417107
$804$	0	0
$805$	0	0
$806$	−32.5623	−1.14696
$807$	0	0
$808$	−10.0000	−0.351799
$809$	−47.8885	−1.68367	−0.841836	−	0.539734i	$-0.818525\pi$
$809$	−47.8885	−1.68367	−0.841836	+	0.539734i	$0.818525\pi$
$810$	0	0
$811$	−55.6525	−1.95422	−0.977111	−	0.212728i	$-0.931765\pi$
$811$	−55.6525	−1.95422	−0.977111	+	0.212728i	$0.931765\pi$
$812$	2.29180	0.0804263
$813$	0	0
$814$	4.00000	0.140200
$815$	33.1246	1.16030
$816$	0	0
$817$	0	0
$818$	−37.7984	−1.32159
$819$	0	0
$820$	10.9443	0.382191
$821$	21.0557	0.734850	0.367425	−	0.930053i	$-0.380239\pi$
$821$	21.0557	0.734850	0.367425	+	0.930053i	$0.380239\pi$
$822$	0	0
$823$	27.5410	0.960020	0.480010	−	0.877263i	$-0.340633\pi$
$823$	27.5410	0.960020	0.480010	+	0.877263i	$0.340633\pi$
$824$	40.6525	1.41620
$825$	0	0
$826$	4.94427	0.172033
$827$	10.4721	0.364152	0.182076	−	0.983284i	$-0.441718\pi$
$827$	10.4721	0.364152	0.182076	+	0.983284i	$0.441718\pi$
$828$	0	0
$829$	−40.2492	−1.39791	−0.698957	−	0.715164i	$-0.746352\pi$
$829$	−40.2492	−1.39791	−0.698957	+	0.715164i	$0.746352\pi$
$830$	69.3050	2.40561
$831$	0	0
$832$	12.7082	0.440578
$833$	−28.6525	−0.992749
$834$	0	0
$835$	33.8885	1.17276
$836$	−0.944272	−0.0326583
$837$	0	0
$838$	−50.8328	−1.75599
$839$	−0.875388	−0.0302218	−0.0151109	−	0.999886i	$-0.504810\pi$
$839$	−0.875388	−0.0302218	−0.0151109	+	0.999886i	$0.504810\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	38.3607	1.32200
$843$	0	0
$844$	2.11146	0.0726793
$845$	12.9443	0.445296
$846$	0	0
$847$	−12.8754	−0.442404
$848$	41.1246	1.41222
$849$	0	0
$850$	46.3607	1.59016
$851$	0	0
$852$	0	0
$853$	−37.4164	−1.28111	−0.640557	−	0.767911i	$-0.721296\pi$
$853$	−37.4164	−1.28111	−0.640557	+	0.767911i	$0.721296\pi$
$854$	−21.8885	−0.749011
$855$	0	0
$856$	30.0000	1.02538
$857$	7.47214	0.255243	0.127622	−	0.991823i	$-0.459266\pi$
$857$	7.47214	0.255243	0.127622	+	0.991823i	$0.459266\pi$
$858$	0	0
$859$	−3.29180	−0.112315	−0.0561573	−	0.998422i	$-0.517885\pi$
$859$	−3.29180	−0.112315	−0.0561573	+	0.998422i	$0.517885\pi$
$860$	0	0
$861$	0	0
$862$	−42.8328	−1.45889
$863$	−45.5410	−1.55023	−0.775117	−	0.631818i	$-0.782309\pi$
$863$	−45.5410	−1.55023	−0.775117	+	0.631818i	$0.782309\pi$
$864$	0	0
$865$	16.3607	0.556280
$866$	−65.0132	−2.20924
$867$	0	0
$868$	−5.12461	−0.173941
$869$	5.30495	0.179958
$870$	0	0
$871$	21.7082	0.735554
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.88854	−0.0638444
$876$	0	0
$877$	−27.5279	−0.929550	−0.464775	−	0.885429i	$-0.653865\pi$
$877$	−27.5279	−0.929550	−0.464775	+	0.885429i	$0.653865\pi$
$878$	−8.56231	−0.288964
$879$	0	0
$880$	−12.0000	−0.404520
$881$	21.8197	0.735123	0.367562	−	0.929999i	$-0.380193\pi$
$881$	21.8197	0.735123	0.367562	+	0.929999i	$0.380193\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	9.70820	0.326522
$885$	0	0
$886$	3.43769	0.115492
$887$	35.0689	1.17750	0.588749	−	0.808316i	$-0.299621\pi$
$887$	35.0689	1.17750	0.588749	+	0.808316i	$0.299621\pi$
$888$	0	0
$889$	−25.5967	−0.858487
$890$	8.00000	0.268161
$891$	0	0
$892$	2.47214	0.0827732
$893$	−4.47214	−0.149654
$894$	0	0
$895$	−41.1246	−1.37464
$896$	16.8328	0.562345
$897$	0	0
$898$	−4.76393	−0.158974
$899$	−20.1246	−0.671193
$900$	0	0
$901$	−44.3607	−1.47787
$902$	6.76393	0.225214
$903$	0	0
$904$	−29.5967	−0.984373
$905$	−47.4164	−1.57617
$906$	0	0
$907$	−40.2492	−1.33645	−0.668227	−	0.743958i	$-0.732946\pi$
$907$	−40.2492	−1.33645	−0.668227	+	0.743958i	$0.732946\pi$
$908$	6.29180	0.208801
$909$	0	0
$910$	−19.4164	−0.643648
$911$	−31.3050	−1.03718	−0.518590	−	0.855023i	$-0.673543\pi$
$911$	−31.3050	−1.03718	−0.518590	+	0.855023i	$0.673543\pi$
$912$	0	0
$913$	10.1115	0.334640
$914$	−56.8328	−1.87986
$915$	0	0
$916$	7.41641	0.245045
$917$	−6.54102	−0.216003
$918$	0	0
$919$	−0.875388	−0.0288764	−0.0144382	−	0.999896i	$-0.504596\pi$
$919$	−0.875388	−0.0288764	−0.0144382	+	0.999896i	$0.504596\pi$
$920$	0	0
$921$	0	0
$922$	−12.0902	−0.398169
$923$	−23.2918	−0.766659
$924$	0	0
$925$	−17.7082	−0.582242
$926$	−32.3607	−1.06344
$927$	0	0
$928$	−10.1459	−0.333055
$929$	41.9443	1.37615	0.688073	−	0.725641i	$-0.258457\pi$
$929$	41.9443	1.37615	0.688073	+	0.725641i	$0.258457\pi$
$930$	0	0
$931$	−10.9443	−0.358684
$932$	9.56231	0.313224
$933$	0	0
$934$	−50.0689	−1.63830
$935$	12.9443	0.423323
$936$	0	0
$937$	−11.8197	−0.386131	−0.193066	−	0.981186i	$-0.561843\pi$
$937$	−11.8197	−0.386131	−0.193066	+	0.981186i	$0.561843\pi$
$938$	14.4721	0.472532
$939$	0	0
$940$	4.47214	0.145865
$941$	−24.6525	−0.803648	−0.401824	−	0.915717i	$-0.631624\pi$
$941$	−24.6525	−0.803648	−0.401824	+	0.915717i	$0.631624\pi$
$942$	0	0
$943$	0	0
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	33.1803	1.07822	0.539108	−	0.842237i	$-0.318761\pi$
$947$	33.1803	1.07822	0.539108	+	0.842237i	$0.318761\pi$
$948$	0	0
$949$	46.4164	1.50674
$950$	17.7082	0.574530
$951$	0	0
$952$	−14.4721	−0.469045
$953$	11.5279	0.373424	0.186712	−	0.982415i	$-0.440217\pi$
$953$	11.5279	0.373424	0.186712	+	0.982415i	$0.440217\pi$
$954$	0	0
$955$	12.3607	0.399982
$956$	−11.2705	−0.364514
$957$	0	0
$958$	−28.4721	−0.919893
$959$	17.1672	0.554357
$960$	0	0
$961$	14.0000	0.451613
$962$	−15.7082	−0.506453
$963$	0	0
$964$	−10.5836	−0.340875
$965$	25.7082	0.827576
$966$	0	0
$967$	−39.5410	−1.27155	−0.635777	−	0.771873i	$-0.719320\pi$
$967$	−39.5410	−1.27155	−0.635777	+	0.771873i	$0.719320\pi$
$968$	23.2918	0.748627
$969$	0	0
$970$	22.4721	0.721537
$971$	7.52786	0.241581	0.120790	−	0.992678i	$-0.461457\pi$
$971$	7.52786	0.241581	0.120790	+	0.992678i	$0.461457\pi$
$972$	0	0
$973$	3.34752	0.107317
$974$	−2.09017	−0.0669734
$975$	0	0
$976$	53.1246	1.70048
$977$	−54.6525	−1.74849	−0.874244	−	0.485487i	$-0.838642\pi$
$977$	−54.6525	−1.74849	−0.874244	+	0.485487i	$0.838642\pi$
$978$	0	0
$979$	1.16718	0.0373034
$980$	10.9443	0.349602
$981$	0	0
$982$	−64.1591	−2.04740
$983$	−31.5279	−1.00558	−0.502791	−	0.864408i	$-0.667694\pi$
$983$	−31.5279	−1.00558	−0.502791	+	0.864408i	$0.667694\pi$
$984$	0	0
$985$	24.1803	0.770450
$986$	25.4164	0.809423
$987$	0	0
$988$	3.70820	0.117974
$989$	0	0
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	22.6869	0.720310
$993$	0	0
$994$	−15.5279	−0.492514
$995$	−83.1935	−2.63741
$996$	0	0
$997$	−36.8328	−1.16651	−0.583253	−	0.812290i	$-0.698221\pi$
$997$	−36.8328	−1.16651	−0.583253	+	0.812290i	$0.698221\pi$
$998$	52.9230	1.67525
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.w.1.2		2
3.2	odd	2		529.2.a.a.1.1		2
12.11	even	2		8464.2.a.bb.1.1		2
23.22	odd	2		207.2.a.d.1.2		2
69.2	odd	22		529.2.c.n.487.1		20
69.5	even	22		529.2.c.o.255.1		20
69.8	odd	22		529.2.c.n.501.2		20
69.11	even	22		529.2.c.o.466.1		20
69.14	even	22		529.2.c.o.334.1		20
69.17	even	22		529.2.c.o.266.2		20
69.20	even	22		529.2.c.o.170.2		20
69.26	odd	22		529.2.c.n.170.2		20
69.29	odd	22		529.2.c.n.266.2		20
69.32	odd	22		529.2.c.n.334.1		20
69.35	odd	22		529.2.c.n.466.1		20
69.38	even	22		529.2.c.o.501.2		20
69.41	odd	22		529.2.c.n.255.1		20
69.44	even	22		529.2.c.o.487.1		20
69.50	odd	22		529.2.c.n.177.2		20
69.53	even	22		529.2.c.o.118.2		20
69.56	even	22		529.2.c.o.399.2		20
69.59	odd	22		529.2.c.n.399.2		20
69.62	odd	22		529.2.c.n.118.2		20
69.65	even	22		529.2.c.o.177.2		20
69.68	even	2		23.2.a.a.1.1	✓	2
92.91	even	2		3312.2.a.ba.1.2		2
115.114	odd	2		5175.2.a.be.1.1		2
276.275	odd	2		368.2.a.h.1.1		2
345.68	odd	4		575.2.b.d.24.4		4
345.137	odd	4		575.2.b.d.24.1		4
345.344	even	2		575.2.a.f.1.2		2
483.482	odd	2		1127.2.a.c.1.1		2
552.275	odd	2		1472.2.a.s.1.2		2
552.413	even	2		1472.2.a.t.1.1		2
759.758	odd	2		2783.2.a.c.1.2		2
897.896	even	2		3887.2.a.i.1.2		2
1173.1172	even	2		6647.2.a.b.1.1		2
1311.1310	odd	2		8303.2.a.e.1.2		2
1380.1379	odd	2		9200.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.2.a.a.1.1	✓	2	69.68	even	2
207.2.a.d.1.2		2	23.22	odd	2
368.2.a.h.1.1		2	276.275	odd	2
529.2.a.a.1.1		2	3.2	odd	2
529.2.c.n.118.2		20	69.62	odd	22
529.2.c.n.170.2		20	69.26	odd	22
529.2.c.n.177.2		20	69.50	odd	22
529.2.c.n.255.1		20	69.41	odd	22
529.2.c.n.266.2		20	69.29	odd	22
529.2.c.n.334.1		20	69.32	odd	22
529.2.c.n.399.2		20	69.59	odd	22
529.2.c.n.466.1		20	69.35	odd	22
529.2.c.n.487.1		20	69.2	odd	22
529.2.c.n.501.2		20	69.8	odd	22
529.2.c.o.118.2		20	69.53	even	22
529.2.c.o.170.2		20	69.20	even	22
529.2.c.o.177.2		20	69.65	even	22
529.2.c.o.255.1		20	69.5	even	22
529.2.c.o.266.2		20	69.17	even	22
529.2.c.o.334.1		20	69.14	even	22
529.2.c.o.399.2		20	69.56	even	22
529.2.c.o.466.1		20	69.11	even	22
529.2.c.o.487.1		20	69.44	even	22
529.2.c.o.501.2		20	69.38	even	22
575.2.a.f.1.2		2	345.344	even	2
575.2.b.d.24.1		4	345.137	odd	4
575.2.b.d.24.4		4	345.68	odd	4
1127.2.a.c.1.1		2	483.482	odd	2
1472.2.a.s.1.2		2	552.275	odd	2
1472.2.a.t.1.1		2	552.413	even	2
2783.2.a.c.1.2		2	759.758	odd	2
3312.2.a.ba.1.2		2	92.91	even	2
3887.2.a.i.1.2		2	897.896	even	2
4761.2.a.w.1.2		2	1.1	even	1	trivial
5175.2.a.be.1.1		2	115.114	odd	2
6647.2.a.b.1.1		2	1173.1172	even	2
8303.2.a.e.1.2		2	1311.1310	odd	2
8464.2.a.bb.1.1		2	12.11	even	2
9200.2.a.bt.1.2		2	1380.1379	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:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} -3.23607 q^{5} +1.23607 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+1.61803 q^{2} +0.618034 q^{4} -3.23607 q^{5} +1.23607 q^{7} -2.23607 q^{8} -5.23607 q^{10} -0.763932 q^{11} +3.00000 q^{13} +2.00000 q^{14} -4.85410 q^{16} +5.23607 q^{17} +2.00000 q^{19} -2.00000 q^{20} -1.23607 q^{22} +5.47214 q^{25} +4.85410 q^{26} +0.763932 q^{28} +3.00000 q^{29} -6.70820 q^{31} -3.38197 q^{32} +8.47214 q^{34} -4.00000 q^{35} -3.23607 q^{37} +3.23607 q^{38} +7.23607 q^{40} -5.47214 q^{41} -0.472136 q^{44} -2.23607 q^{47} -5.47214 q^{49} +8.85410 q^{50} +1.85410 q^{52} -8.47214 q^{53} +2.47214 q^{55} -2.76393 q^{56} +4.85410 q^{58} +2.47214 q^{59} -10.9443 q^{61} -10.8541 q^{62} +4.23607 q^{64} -9.70820 q^{65} +7.23607 q^{67} +3.23607 q^{68} -6.47214 q^{70} -7.76393 q^{71} +15.4721 q^{73} -5.23607 q^{74} +1.23607 q^{76} -0.944272 q^{77} -6.94427 q^{79} +15.7082 q^{80} -8.85410 q^{82} -13.2361 q^{83} -16.9443 q^{85} +1.70820 q^{88} -1.52786 q^{89} +3.70820 q^{91} -3.61803 q^{94} -6.47214 q^{95} -4.29180 q^{97} -8.85410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7	\( 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{10} - 6 q^{11} + 6 q^{13} + 4 q^{14} - 3 q^{16} + 6 q^{17} + 4 q^{19} - 4 q^{20} + 2 q^{22} + 2 q^{25} + 3 q^{26} + 6 q^{28} + 6 q^{29} - 9 q^{32} + 8 q^{34} - 8 q^{35} - 2 q^{37} + 2 q^{38} + 10 q^{40} - 2 q^{41} + 8 q^{44} - 2 q^{49} + 11 q^{50} - 3 q^{52} - 8 q^{53} - 4 q^{55} - 10 q^{56} + 3 q^{58} - 4 q^{59} - 4 q^{61} - 15 q^{62} + 4 q^{64} - 6 q^{65} + 10 q^{67} + 2 q^{68} - 4 q^{70} - 20 q^{71} + 22 q^{73} - 6 q^{74} - 2 q^{76} + 16 q^{77} + 4 q^{79} + 18 q^{80} - 11 q^{82} - 22 q^{83} - 16 q^{85} - 10 q^{88} - 12 q^{89} - 6 q^{91} - 5 q^{94} - 4 q^{95} - 22 q^{97} - 11 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7 - 6 * q^10 - 6 * q^11 + 6 * q^13 + 4 * q^14 - 3 * q^16 + 6 * q^17 + 4 * q^19 - 4 * q^20 + 2 * q^22 + 2 * q^25 + 3 * q^26 + 6 * q^28 + 6 * q^29 - 9 * q^32 + 8 * q^34 - 8 * q^35 - 2 * q^37 + 2 * q^38 + 10 * q^40 - 2 * q^41 + 8 * q^44 - 2 * q^49 + 11 * q^50 - 3 * q^52 - 8 * q^53 - 4 * q^55 - 10 * q^56 + 3 * q^58 - 4 * q^59 - 4 * q^61 - 15 * q^62 + 4 * q^64 - 6 * q^65 + 10 * q^67 + 2 * q^68 - 4 * q^70 - 20 * q^71 + 22 * q^73 - 6 * q^74 - 2 * q^76 + 16 * q^77 + 4 * q^79 + 18 * q^80 - 11 * q^82 - 22 * q^83 - 16 * q^85 - 10 * q^88 - 12 * q^89 - 6 * q^91 - 5 * q^94 - 4 * q^95 - 22 * q^97 - 11 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more