LMFDB - Embedded newform 7623.2.a.cg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 2x + 1 $ x^4 - 2x^3 - 4x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.43091$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.301143 q^{2} -1.90931 q^{4} +0.698857 q^{5} -1.00000 q^{7} +1.17726 q^{8} +O(q^{10})$	$q-0.301143 q^{2} -1.90931 q^{4} +0.698857 q^{5} -1.00000 q^{7} +1.17726 q^{8} -0.210456 q^{10} +4.43091 q^{13} +0.301143 q^{14} +3.46410 q^{16} +6.80432 q^{17} +1.78954 q^{19} -1.33434 q^{20} +9.20204 q^{23} -4.51160 q^{25} -1.33434 q^{26} +1.90931 q^{28} -7.89932 q^{29} +10.4600 q^{31} -3.39771 q^{32} -2.04907 q^{34} -0.698857 q^{35} +6.16884 q^{37} -0.538909 q^{38} +0.822738 q^{40} -1.33434 q^{41} +6.33434 q^{43} -2.77113 q^{46} -8.20634 q^{47} +1.00000 q^{49} +1.35864 q^{50} -8.45999 q^{52} +2.35452 q^{53} -1.17726 q^{56} +2.37882 q^{58} +4.76524 q^{59} -1.69455 q^{61} -3.14995 q^{62} -5.90501 q^{64} +3.09657 q^{65} -12.5423 q^{67} -12.9916 q^{68} +0.210456 q^{70} -7.58387 q^{71} +6.96681 q^{73} -1.85770 q^{74} -3.41680 q^{76} +8.85751 q^{79} +2.42091 q^{80} +0.401826 q^{82} -5.71775 q^{83} +4.75525 q^{85} -1.90754 q^{86} +2.22935 q^{89} -4.43091 q^{91} -17.5696 q^{92} +2.47128 q^{94} +1.25064 q^{95} -0.171378 q^{97} -0.301143 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^7	$ 4 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{7} + 10 q^{10} + 10 q^{13} + 2 q^{14} - 6 q^{17} + 18 q^{19} + 2 q^{23} - 8 q^{25} - 4 q^{28} - 6 q^{29} - 12 q^{32} - 2 q^{34} - 2 q^{35} - 4 q^{37} + 8 q^{40} + 20 q^{43} + 16 q^{46} + 6 q^{47} + 4 q^{49} + 24 q^{50} + 8 q^{52} + 24 q^{58} + 6 q^{59} - 10 q^{61} - 16 q^{64} + 10 q^{65} + 4 q^{67} - 28 q^{68} - 10 q^{70} + 6 q^{71} + 34 q^{73} + 36 q^{74} + 36 q^{76} + 24 q^{79} - 12 q^{80} + 28 q^{82} - 6 q^{83} - 8 q^{85} - 38 q^{86} - 18 q^{89} - 10 q^{91} - 24 q^{92} - 6 q^{94} + 18 q^{95} - 10 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^7 + 10 * q^10 + 10 * q^13 + 2 * q^14 - 6 * q^17 + 18 * q^19 + 2 * q^23 - 8 * q^25 - 4 * q^28 - 6 * q^29 - 12 * q^32 - 2 * q^34 - 2 * q^35 - 4 * q^37 + 8 * q^40 + 20 * q^43 + 16 * q^46 + 6 * q^47 + 4 * q^49 + 24 * q^50 + 8 * q^52 + 24 * q^58 + 6 * q^59 - 10 * q^61 - 16 * q^64 + 10 * q^65 + 4 * q^67 - 28 * q^68 - 10 * q^70 + 6 * q^71 + 34 * q^73 + 36 * q^74 + 36 * q^76 + 24 * q^79 - 12 * q^80 + 28 * q^82 - 6 * q^83 - 8 * q^85 - 38 * q^86 - 18 * q^89 - 10 * q^91 - 24 * q^92 - 6 * q^94 + 18 * q^95 - 10 * q^97 - 2 * 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 $

$2$	−0.301143	−0.212940	−0.106470	−	0.994316i	$-0.533955\pi$
$2$	−0.301143	−0.212940	−0.106470	+	0.994316i	$0.533955\pi$
$3$	0	0
$3$	0	0
$4$	−1.90931	−0.954656
$5$	0.698857	0.312538	0.156269	−	0.987715i	$-0.450053\pi$
$5$	0.698857	0.312538	0.156269	+	0.987715i	$0.450053\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.17726	0.416225
$9$	0	0
$10$	−0.210456	−0.0665520
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.43091	1.22891	0.614456	−	0.788951i	$-0.289375\pi$
$13$	4.43091	1.22891	0.614456	+	0.788951i	$0.289375\pi$
$14$	0.301143	0.0804838
$15$	0	0
$16$	3.46410	0.866025
$17$	6.80432	1.65029	0.825145	−	0.564921i	$-0.191093\pi$
$17$	6.80432	1.65029	0.825145	+	0.564921i	$0.191093\pi$
$18$	0	0
$19$	1.78954	0.410550	0.205275	−	0.978704i	$-0.434191\pi$
$19$	1.78954	0.410550	0.205275	+	0.978704i	$0.434191\pi$
$20$	−1.33434	−0.298367
$21$	0	0
$22$	0	0
$23$	9.20204	1.91876	0.959379	−	0.282122i	$-0.0910382\pi$
$23$	9.20204	1.91876	0.959379	+	0.282122i	$0.0910382\pi$
$24$	0	0
$25$	−4.51160	−0.902320
$26$	−1.33434	−0.261685
$27$	0	0
$28$	1.90931	0.360826
$29$	−7.89932	−1.46687	−0.733433	−	0.679762i	$-0.762083\pi$
$29$	−7.89932	−1.46687	−0.733433	+	0.679762i	$0.762083\pi$
$30$	0	0
$31$	10.4600	1.87867	0.939335	−	0.343002i	$-0.111444\pi$
$31$	10.4600	1.87867	0.939335	+	0.343002i	$0.111444\pi$
$32$	−3.39771	−0.600637
$33$	0	0
$34$	−2.04907	−0.351413
$35$	−0.698857	−0.118128
$36$	0	0
$37$	6.16884	1.01415	0.507076	−	0.861901i	$-0.330726\pi$
$37$	6.16884	1.01415	0.507076	+	0.861901i	$0.330726\pi$
$38$	−0.538909	−0.0874225
$39$	0	0
$40$	0.822738	0.130086
$41$	−1.33434	−0.208388	−0.104194	−	0.994557i	$-0.533226\pi$
$41$	−1.33434	−0.208388	−0.104194	+	0.994557i	$0.533226\pi$
$42$	0	0
$43$	6.33434	0.965977	0.482989	−	0.875627i	$-0.339551\pi$
$43$	6.33434	0.965977	0.482989	+	0.875627i	$0.339551\pi$
$44$	0	0
$45$	0	0
$46$	−2.77113	−0.408581
$47$	−8.20634	−1.19702	−0.598509	−	0.801116i	$-0.704240\pi$
$47$	−8.20634	−1.19702	−0.598509	+	0.801116i	$0.704240\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.35864	0.192140
$51$	0	0
$52$	−8.45999	−1.17319
$53$	2.35452	0.323419	0.161709	−	0.986838i	$-0.448299\pi$
$53$	2.35452	0.323419	0.161709	+	0.986838i	$0.448299\pi$
$54$	0	0
$55$	0	0
$56$	−1.17726	−0.157318
$57$	0	0
$58$	2.37882	0.312355
$59$	4.76524	0.620382	0.310191	−	0.950674i	$-0.399607\pi$
$59$	4.76524	0.620382	0.310191	+	0.950674i	$0.399607\pi$
$60$	0	0
$61$	−1.69455	−0.216965	−0.108482	−	0.994098i	$-0.534599\pi$
$61$	−1.69455	−0.216965	−0.108482	+	0.994098i	$0.534599\pi$
$62$	−3.14995	−0.400044
$63$	0	0
$64$	−5.90501	−0.738126
$65$	3.09657	0.384082
$66$	0	0
$67$	−12.5423	−1.53228	−0.766140	−	0.642673i	$-0.777825\pi$
$67$	−12.5423	−1.53228	−0.766140	+	0.642673i	$0.777825\pi$
$68$	−12.9916	−1.57546
$69$	0	0
$70$	0.210456	0.0251543
$71$	−7.58387	−0.900040	−0.450020	−	0.893019i	$-0.648583\pi$
$71$	−7.58387	−0.900040	−0.450020	+	0.893019i	$0.648583\pi$
$72$	0	0
$73$	6.96681	0.815403	0.407701	−	0.913115i	$-0.366330\pi$
$73$	6.96681	0.815403	0.407701	+	0.913115i	$0.366330\pi$
$74$	−1.85770	−0.215954
$75$	0	0
$76$	−3.41680	−0.391934
$77$	0	0
$78$	0	0
$79$	8.85751	0.996548	0.498274	−	0.867020i	$-0.333967\pi$
$79$	8.85751	0.996548	0.498274	+	0.867020i	$0.333967\pi$
$80$	2.42091	0.270666
$81$	0	0
$82$	0.401826	0.0443743
$83$	−5.71775	−0.627604	−0.313802	−	0.949488i	$-0.601603\pi$
$83$	−5.71775	−0.627604	−0.313802	+	0.949488i	$0.601603\pi$
$84$	0	0
$85$	4.75525	0.515779
$86$	−1.90754	−0.205695
$87$	0	0
$88$	0	0
$89$	2.22935	0.236310	0.118155	−	0.992995i	$-0.462302\pi$
$89$	2.22935	0.236310	0.118155	+	0.992995i	$0.462302\pi$
$90$	0	0
$91$	−4.43091	−0.464485
$92$	−17.5696	−1.83175
$93$	0	0
$94$	2.47128	0.254893
$95$	1.25064	0.128312
$96$	0	0
$97$	−0.171378	−0.0174008	−0.00870041	−	0.999962i	$-0.502769\pi$
$97$	−0.171378	−0.0174008	−0.00870041	+	0.999962i	$0.502769\pi$
$98$	−0.301143	−0.0304200
$99$	0	0
$100$	8.61405	0.861405
$101$	−18.0723	−1.79826	−0.899129	−	0.437683i	$-0.855799\pi$
$101$	−18.0723	−1.79826	−0.899129	+	0.437683i	$0.855799\pi$
$102$	0	0
$103$	−1.73047	−0.170509	−0.0852543	−	0.996359i	$-0.527170\pi$
$103$	−1.73047	−0.170509	−0.0852543	+	0.996359i	$0.527170\pi$
$104$	5.21634	0.511504
$105$	0	0
$106$	−0.709048	−0.0688689
$107$	6.96982	0.673798	0.336899	−	0.941541i	$-0.390622\pi$
$107$	6.96982	0.673798	0.336899	+	0.941541i	$0.390622\pi$
$108$	0	0
$109$	17.6531	1.69086	0.845432	−	0.534084i	$-0.179343\pi$
$109$	17.6531	1.69086	0.845432	+	0.534084i	$0.179343\pi$
$110$	0	0
$111$	0	0
$112$	−3.46410	−0.327327
$113$	−11.7236	−1.10287	−0.551433	−	0.834219i	$-0.685919\pi$
$113$	−11.7236	−1.10287	−0.551433	+	0.834219i	$0.685919\pi$
$114$	0	0
$115$	6.43091	0.599685
$116$	15.0823	1.40035
$117$	0	0
$118$	−1.43502	−0.132104
$119$	−6.80432	−0.623751
$120$	0	0
$121$	0	0
$122$	0.510302	0.0462005
$123$	0	0
$124$	−19.9714	−1.79348
$125$	−6.64725	−0.594548
$126$	0	0
$127$	12.1530	1.07840	0.539201	−	0.842177i	$-0.318726\pi$
$127$	12.1530	1.07840	0.539201	+	0.842177i	$0.318726\pi$
$128$	8.57368	0.757813
$129$	0	0
$130$	−0.932511	−0.0817866
$131$	−18.9111	−1.65227	−0.826135	−	0.563473i	$-0.809465\pi$
$131$	−18.9111	−1.65227	−0.826135	+	0.563473i	$0.809465\pi$
$132$	0	0
$133$	−1.78954	−0.155173
$134$	3.77701	0.326284
$135$	0	0
$136$	8.01047	0.686892
$137$	8.37930	0.715892	0.357946	−	0.933742i	$-0.383477\pi$
$137$	8.37930	0.715892	0.357946	+	0.933742i	$0.383477\pi$
$138$	0	0
$139$	9.84039	0.834651	0.417325	−	0.908757i	$-0.362968\pi$
$139$	9.84039	0.834651	0.417325	+	0.908757i	$0.362968\pi$
$140$	1.33434	0.112772
$141$	0	0
$142$	2.28383	0.191655
$143$	0	0
$144$	0	0
$145$	−5.52049	−0.458452
$146$	−2.09800	−0.173632
$147$	0	0
$148$	−11.7783	−0.968166
$149$	−7.13435	−0.584469	−0.292234	−	0.956347i	$-0.594399\pi$
$149$	−7.13435	−0.584469	−0.292234	+	0.956347i	$0.594399\pi$
$150$	0	0
$151$	7.83881	0.637914	0.318957	−	0.947769i	$-0.396668\pi$
$151$	7.83881	0.637914	0.318957	+	0.947769i	$0.396668\pi$
$152$	2.10676	0.170881
$153$	0	0
$154$	0	0
$155$	7.31004	0.587156
$156$	0	0
$157$	−18.9959	−1.51604	−0.758018	−	0.652233i	$-0.773832\pi$
$157$	−18.9959	−1.51604	−0.758018	+	0.652233i	$0.773832\pi$
$158$	−2.66738	−0.212205
$159$	0	0
$160$	−2.37452	−0.187722
$161$	−9.20204	−0.725222
$162$	0	0
$163$	−12.8779	−1.00867	−0.504337	−	0.863507i	$-0.668263\pi$
$163$	−12.8779	−1.00867	−0.504337	+	0.863507i	$0.668263\pi$
$164$	2.54767	0.198939
$165$	0	0
$166$	1.72186	0.133642
$167$	−0.00588400	−0.000455318 0	−0.000227659	−	1.00000i	$-0.500072\pi$
$167$	−0.00588400	−0.000455318 0	−0.000227659		1.00000i	$0.500072\pi$
$168$	0	0
$169$	6.63294	0.510226
$170$	−1.43201	−0.109830
$171$	0	0
$172$	−12.0942	−0.922176
$173$	12.9193	0.982237	0.491118	−	0.871093i	$-0.336588\pi$
$173$	12.9193	0.982237	0.491118	+	0.871093i	$0.336588\pi$
$174$	0	0
$175$	4.51160	0.341045
$176$	0	0
$177$	0	0
$178$	−0.671352	−0.0503200
$179$	1.31133	0.0980137	0.0490069	−	0.998798i	$-0.484394\pi$
$179$	1.31133	0.0980137	0.0490069	+	0.998798i	$0.484394\pi$
$180$	0	0
$181$	4.43138	0.329382	0.164691	−	0.986345i	$-0.447337\pi$
$181$	4.43138	0.329382	0.164691	+	0.986345i	$0.447337\pi$
$182$	1.33434	0.0989076
$183$	0	0
$184$	10.8332	0.798635
$185$	4.31114	0.316961
$186$	0	0
$187$	0	0
$188$	15.6685	1.14274
$189$	0	0
$190$	−0.376620	−0.0273229
$191$	−5.34563	−0.386796	−0.193398	−	0.981120i	$-0.561951\pi$
$191$	−5.34563	−0.386796	−0.193398	+	0.981120i	$0.561951\pi$
$192$	0	0
$193$	0.803848	0.0578622	0.0289311	−	0.999581i	$-0.490790\pi$
$193$	0.803848	0.0578622	0.0289311	+	0.999581i	$0.490790\pi$
$194$	0.0516093	0.00370533
$195$	0	0
$196$	−1.90931	−0.136379
$197$	23.4082	1.66776	0.833882	−	0.551943i	$-0.186113\pi$
$197$	23.4082	1.66776	0.833882	+	0.551943i	$0.186113\pi$
$198$	0	0
$199$	23.3118	1.65253	0.826265	−	0.563281i	$-0.190461\pi$
$199$	23.3118	1.65253	0.826265	+	0.563281i	$0.190461\pi$
$200$	−5.31133	−0.375568
$201$	0	0
$202$	5.44234	0.382922
$203$	7.89932	0.554423
$204$	0	0
$205$	−0.932511	−0.0651294
$206$	0.521120	0.0363082
$207$	0	0
$208$	15.3491	1.06427
$209$	0	0
$210$	0	0
$211$	10.1455	0.698445	0.349223	−	0.937040i	$-0.386446\pi$
$211$	10.1455	0.698445	0.349223	+	0.937040i	$0.386446\pi$
$212$	−4.49552	−0.308754
$213$	0	0
$214$	−2.09891	−0.143479
$215$	4.42680	0.301905
$216$	0	0
$217$	−10.4600	−0.710070
$218$	−5.31612	−0.360053
$219$	0	0
$220$	0	0
$221$	30.1493	2.02806
$222$	0	0
$223$	−6.86052	−0.459414	−0.229707	−	0.973260i	$-0.573777\pi$
$223$	−6.86052	−0.459414	−0.229707	+	0.973260i	$0.573777\pi$
$224$	3.39771	0.227019
$225$	0	0
$226$	3.53049	0.234845
$227$	−9.65136	−0.640583	−0.320292	−	0.947319i	$-0.603781\pi$
$227$	−9.65136	−0.640583	−0.320292	+	0.947319i	$0.603781\pi$
$228$	0	0
$229$	2.37452	0.156912	0.0784562	−	0.996918i	$-0.475001\pi$
$229$	2.37452	0.156912	0.0784562	+	0.996918i	$0.475001\pi$
$230$	−1.93662	−0.127697
$231$	0	0
$232$	−9.29957	−0.610546
$233$	−16.2643	−1.06551	−0.532755	−	0.846269i	$-0.678843\pi$
$233$	−16.2643	−1.06551	−0.532755	+	0.846269i	$0.678843\pi$
$234$	0	0
$235$	−5.73506	−0.374114
$236$	−9.09834	−0.592252
$237$	0	0
$238$	2.04907	0.132822
$239$	17.2386	1.11507	0.557535	−	0.830153i	$-0.311747\pi$
$239$	17.2386	1.11507	0.557535	+	0.830153i	$0.311747\pi$
$240$	0	0
$241$	5.41818	0.349016	0.174508	−	0.984656i	$-0.444167\pi$
$241$	5.41818	0.349016	0.174508	+	0.984656i	$0.444167\pi$
$242$	0	0
$243$	0	0
$244$	3.23543	0.207127
$245$	0.698857	0.0446483
$246$	0	0
$247$	7.92931	0.504530
$248$	12.3141	0.781949
$249$	0	0
$250$	2.00177	0.126603
$251$	−15.1995	−0.959384	−0.479692	−	0.877437i	$-0.659252\pi$
$251$	−15.1995	−0.959384	−0.479692	+	0.877437i	$0.659252\pi$
$252$	0	0
$253$	0	0
$254$	−3.65978	−0.229635
$255$	0	0
$256$	9.22811	0.576757
$257$	26.6304	1.66116	0.830580	−	0.556900i	$-0.188009\pi$
$257$	26.6304	1.66116	0.830580	+	0.556900i	$0.188009\pi$
$258$	0	0
$259$	−6.16884	−0.383313
$260$	−5.91232	−0.366667
$261$	0	0
$262$	5.69494	0.351835
$263$	5.63979	0.347764	0.173882	−	0.984766i	$-0.444369\pi$
$263$	5.63979	0.347764	0.173882	+	0.984766i	$0.444369\pi$
$264$	0	0
$265$	1.64548	0.101081
$266$	0.538909	0.0330426
$267$	0	0
$268$	23.9471	1.46280
$269$	1.10958	0.0676521	0.0338261	−	0.999428i	$-0.489231\pi$
$269$	1.10958	0.0676521	0.0338261	+	0.999428i	$0.489231\pi$
$270$	0	0
$271$	−13.7713	−0.836548	−0.418274	−	0.908321i	$-0.637365\pi$
$271$	−13.7713	−0.836548	−0.418274	+	0.908321i	$0.637365\pi$
$272$	23.5709	1.42919
$273$	0	0
$274$	−2.52337	−0.152442
$275$	0	0
$276$	0	0
$277$	−7.90089	−0.474719	−0.237359	−	0.971422i	$-0.576282\pi$
$277$	−7.90089	−0.474719	−0.237359	+	0.971422i	$0.576282\pi$
$278$	−2.96336	−0.177731
$279$	0	0
$280$	−0.822738	−0.0491680
$281$	13.7596	0.820826	0.410413	−	0.911900i	$-0.365384\pi$
$281$	13.7596	0.820826	0.410413	+	0.911900i	$0.365384\pi$
$282$	0	0
$283$	17.8650	1.06197	0.530983	−	0.847383i	$-0.321823\pi$
$283$	17.8650	1.06197	0.530983	+	0.847383i	$0.321823\pi$
$284$	14.4800	0.859229
$285$	0	0
$286$	0	0
$287$	1.33434	0.0787634
$288$	0	0
$289$	29.2988	1.72346
$290$	1.66246	0.0976229
$291$	0	0
$292$	−13.3018	−0.778430
$293$	23.9721	1.40046	0.700231	−	0.713916i	$-0.253080\pi$
$293$	23.9721	1.40046	0.700231	+	0.713916i	$0.253080\pi$
$294$	0	0
$295$	3.33022	0.193893
$296$	7.26234	0.422115
$297$	0	0
$298$	2.14846	0.124457
$299$	40.7734	2.35799
$300$	0	0
$301$	−6.33434	−0.365105
$302$	−2.36060	−0.135837
$303$	0	0
$304$	6.19916	0.355546
$305$	−1.18425	−0.0678098
$306$	0	0
$307$	−24.2718	−1.38526	−0.692632	−	0.721292i	$-0.743549\pi$
$307$	−24.2718	−1.38526	−0.692632	+	0.721292i	$0.743549\pi$
$308$	0	0
$309$	0	0
$310$	−2.20137	−0.125029
$311$	1.97542	0.112016	0.0560079	−	0.998430i	$-0.482163\pi$
$311$	1.97542	0.112016	0.0560079	+	0.998430i	$0.482163\pi$
$312$	0	0
$313$	3.04189	0.171938	0.0859690	−	0.996298i	$-0.472601\pi$
$313$	3.04189	0.171938	0.0859690	+	0.996298i	$0.472601\pi$
$314$	5.72048	0.322825
$315$	0	0
$316$	−16.9118	−0.951361
$317$	−27.7329	−1.55763	−0.778816	−	0.627252i	$-0.784180\pi$
$317$	−27.7329	−1.55763	−0.778816	+	0.627252i	$0.784180\pi$
$318$	0	0
$319$	0	0
$320$	−4.12675	−0.230693
$321$	0	0
$322$	2.77113	0.154429
$323$	12.1766	0.677526
$324$	0	0
$325$	−19.9905	−1.10887
$326$	3.87809	0.214787
$327$	0	0
$328$	−1.57086	−0.0867365
$329$	8.20634	0.452430
$330$	0	0
$331$	−13.3502	−0.733794	−0.366897	−	0.930262i	$-0.619580\pi$
$331$	−13.3502	−0.733794	−0.366897	+	0.930262i	$0.619580\pi$
$332$	10.9170	0.599147
$333$	0	0
$334$	0.00177192	9.69554e−5 0
$335$	−8.76524	−0.478896
$336$	0	0
$337$	6.41345	0.349363	0.174681	−	0.984625i	$-0.444110\pi$
$337$	6.41345	0.349363	0.174681	+	0.984625i	$0.444110\pi$
$338$	−1.99746	−0.108648
$339$	0	0
$340$	−9.07926	−0.492392
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	7.45717	0.402064
$345$	0	0
$346$	−3.89056	−0.209158
$347$	9.97432	0.535450	0.267725	−	0.963495i	$-0.413728\pi$
$347$	9.97432	0.535450	0.267725	+	0.963495i	$0.413728\pi$
$348$	0	0
$349$	8.18267	0.438008	0.219004	−	0.975724i	$-0.429719\pi$
$349$	8.18267	0.438008	0.219004	+	0.975724i	$0.429719\pi$
$350$	−1.35864	−0.0726222
$351$	0	0
$352$	0	0
$353$	0.173150	0.00921585	0.00460792	−	0.999989i	$-0.498533\pi$
$353$	0.173150	0.00921585	0.00460792	+	0.999989i	$0.498533\pi$
$354$	0	0
$355$	−5.30004	−0.281297
$356$	−4.25652	−0.225595
$357$	0	0
$358$	−0.394899	−0.0208711
$359$	24.9257	1.31553	0.657763	−	0.753225i	$-0.271503\pi$
$359$	24.9257	1.31553	0.657763	+	0.753225i	$0.271503\pi$
$360$	0	0
$361$	−15.7975	−0.831449
$362$	−1.33448	−0.0701387
$363$	0	0
$364$	8.45999	0.443424
$365$	4.86880	0.254845
$366$	0	0
$367$	−34.1736	−1.78385	−0.891924	−	0.452185i	$-0.850645\pi$
$367$	−34.1736	−1.78385	−0.891924	+	0.452185i	$0.850645\pi$
$368$	31.8768	1.66169
$369$	0	0
$370$	−1.29827	−0.0674938
$371$	−2.35452	−0.122241
$372$	0	0
$373$	−1.51030	−0.0782005	−0.0391002	−	0.999235i	$-0.512449\pi$
$373$	−1.51030	−0.0782005	−0.0391002	+	0.999235i	$0.512449\pi$
$374$	0	0
$375$	0	0
$376$	−9.66102	−0.498229
$377$	−35.0011	−1.80265
$378$	0	0
$379$	−7.46013	−0.383201	−0.191601	−	0.981473i	$-0.561368\pi$
$379$	−7.46013	−0.383201	−0.191601	+	0.981473i	$0.561368\pi$
$380$	−2.38785	−0.122494
$381$	0	0
$382$	1.60980	0.0823645
$383$	−25.7482	−1.31567	−0.657836	−	0.753161i	$-0.728528\pi$
$383$	−25.7482	−1.31567	−0.657836	+	0.753161i	$0.728528\pi$
$384$	0	0
$385$	0	0
$386$	−0.242073	−0.0123212
$387$	0	0
$388$	0.327214	0.0166118
$389$	33.6464	1.70594	0.852971	−	0.521959i	$-0.174799\pi$
$389$	33.6464	1.70594	0.852971	+	0.521959i	$0.174799\pi$
$390$	0	0
$391$	62.6136	3.16651
$392$	1.17726	0.0594607
$393$	0	0
$394$	−7.04921	−0.355134
$395$	6.19013	0.311459
$396$	0	0
$397$	−19.6950	−0.988465	−0.494232	−	0.869330i	$-0.664551\pi$
$397$	−19.6950	−0.988465	−0.494232	+	0.869330i	$0.664551\pi$
$398$	−7.02019	−0.351890
$399$	0	0
$400$	−15.6286	−0.781432
$401$	−17.4368	−0.870752	−0.435376	−	0.900249i	$-0.643385\pi$
$401$	−17.4368	−0.870752	−0.435376	+	0.900249i	$0.643385\pi$
$402$	0	0
$403$	46.3472	2.30872
$404$	34.5056	1.71672
$405$	0	0
$406$	−2.37882	−0.118059
$407$	0	0
$408$	0	0
$409$	7.92547	0.391889	0.195945	−	0.980615i	$-0.437223\pi$
$409$	7.92547	0.391889	0.195945	+	0.980615i	$0.437223\pi$
$410$	0.280819	0.0138687
$411$	0	0
$412$	3.30402	0.162777
$413$	−4.76524	−0.234482
$414$	0	0
$415$	−3.99589	−0.196150
$416$	−15.0550	−0.738130
$417$	0	0
$418$	0	0
$419$	12.0302	0.587713	0.293856	−	0.955850i	$-0.405061\pi$
$419$	12.0302	0.587713	0.293856	+	0.955850i	$0.405061\pi$
$420$	0	0
$421$	−1.36314	−0.0664353	−0.0332177	−	0.999448i	$-0.510575\pi$
$421$	−1.36314	−0.0664353	−0.0332177	+	0.999448i	$0.510575\pi$
$422$	−3.05525	−0.148727
$423$	0	0
$424$	2.77189	0.134615
$425$	−30.6984	−1.48909
$426$	0	0
$427$	1.69455	0.0820050
$428$	−13.3076	−0.643245
$429$	0	0
$430$	−1.33310	−0.0642877
$431$	8.23954	0.396885	0.198442	−	0.980113i	$-0.436412\pi$
$431$	8.23954	0.396885	0.198442	+	0.980113i	$0.436412\pi$
$432$	0	0
$433$	13.7708	0.661785	0.330892	−	0.943668i	$-0.392650\pi$
$433$	13.7708	0.661785	0.330892	+	0.943668i	$0.392650\pi$
$434$	3.14995	0.151203
$435$	0	0
$436$	−33.7054	−1.61419
$437$	16.4675	0.787745
$438$	0	0
$439$	−18.7090	−0.892934	−0.446467	−	0.894800i	$-0.647318\pi$
$439$	−18.7090	−0.892934	−0.446467	+	0.894800i	$0.647318\pi$
$440$	0	0
$441$	0	0
$442$	−9.07926	−0.431856
$443$	−33.3148	−1.58283	−0.791417	−	0.611276i	$-0.790656\pi$
$443$	−33.3148	−1.58283	−0.791417	+	0.611276i	$0.790656\pi$
$444$	0	0
$445$	1.55799	0.0738560
$446$	2.06600	0.0978278
$447$	0	0
$448$	5.90501	0.278985
$449$	3.74233	0.176611	0.0883057	−	0.996093i	$-0.471855\pi$
$449$	3.74233	0.176611	0.0883057	+	0.996093i	$0.471855\pi$
$450$	0	0
$451$	0	0
$452$	22.3841	1.05286
$453$	0	0
$454$	2.90644	0.136406
$455$	−3.09657	−0.145169
$456$	0	0
$457$	19.9747	0.934379	0.467190	−	0.884157i	$-0.345267\pi$
$457$	19.9747	0.934379	0.467190	+	0.884157i	$0.345267\pi$
$458$	−0.715069	−0.0334130
$459$	0	0
$460$	−12.2786	−0.572493
$461$	0.904388	0.0421215	0.0210608	−	0.999778i	$-0.493296\pi$
$461$	0.904388	0.0421215	0.0210608	+	0.999778i	$0.493296\pi$
$462$	0	0
$463$	23.2784	1.08184	0.540920	−	0.841074i	$-0.318076\pi$
$463$	23.2784	1.08184	0.540920	+	0.841074i	$0.318076\pi$
$464$	−27.3640	−1.27034
$465$	0	0
$466$	4.89788	0.226890
$467$	−3.75018	−0.173538	−0.0867688	−	0.996228i	$-0.527654\pi$
$467$	−3.75018	−0.173538	−0.0867688	+	0.996228i	$0.527654\pi$
$468$	0	0
$469$	12.5423	0.579148
$470$	1.72707	0.0796640
$471$	0	0
$472$	5.60994	0.258219
$473$	0	0
$474$	0	0
$475$	−8.07371	−0.370447
$476$	12.9916	0.595468
$477$	0	0
$478$	−5.19128	−0.237443
$479$	29.0418	1.32695	0.663477	−	0.748197i	$-0.269080\pi$
$479$	29.0418	1.32695	0.663477	+	0.748197i	$0.269080\pi$
$480$	0	0
$481$	27.3336	1.24630
$482$	−1.63165	−0.0743195
$483$	0	0
$484$	0	0
$485$	−0.119769	−0.00543842
$486$	0	0
$487$	−12.4381	−0.563624	−0.281812	−	0.959470i	$-0.590935\pi$
$487$	−12.4381	−0.563624	−0.281812	+	0.959470i	$0.590935\pi$
$488$	−1.99493	−0.0903062
$489$	0	0
$490$	−0.210456	−0.00950743
$491$	−6.46343	−0.291691	−0.145845	−	0.989307i	$-0.546590\pi$
$491$	−6.46343	−0.291691	−0.145845	+	0.989307i	$0.546590\pi$
$492$	0	0
$493$	−53.7495	−2.42076
$494$	−2.38785	−0.107435
$495$	0	0
$496$	36.2345	1.62698
$497$	7.58387	0.340183
$498$	0	0
$499$	−13.2848	−0.594708	−0.297354	−	0.954767i	$-0.596104\pi$
$499$	−13.2848	−0.594708	−0.297354	+	0.954767i	$0.596104\pi$
$500$	12.6917	0.567589
$501$	0	0
$502$	4.57722	0.204291
$503$	−36.5805	−1.63104	−0.815522	−	0.578726i	$-0.803550\pi$
$503$	−36.5805	−1.63104	−0.815522	+	0.578726i	$0.803550\pi$
$504$	0	0
$505$	−12.6299	−0.562025
$506$	0	0
$507$	0	0
$508$	−23.2038	−1.02950
$509$	4.36993	0.193694	0.0968469	−	0.995299i	$-0.469124\pi$
$509$	4.36993	0.193694	0.0968469	+	0.995299i	$0.469124\pi$
$510$	0	0
$511$	−6.96681	−0.308193
$512$	−19.9263	−0.880628
$513$	0	0
$514$	−8.01956	−0.353728
$515$	−1.20935	−0.0532905
$516$	0	0
$517$	0	0
$518$	1.85770	0.0816228
$519$	0	0
$520$	3.64548	0.159865
$521$	36.1430	1.58345	0.791726	−	0.610876i	$-0.209183\pi$
$521$	36.1430	1.58345	0.791726	+	0.610876i	$0.209183\pi$
$522$	0	0
$523$	−28.5269	−1.24739	−0.623696	−	0.781667i	$-0.714370\pi$
$523$	−28.5269	−1.24739	−0.623696	+	0.781667i	$0.714370\pi$
$524$	36.1072	1.57735
$525$	0	0
$526$	−1.69838	−0.0740530
$527$	71.1731	3.10035
$528$	0	0
$529$	61.6775	2.68163
$530$	−0.495523	−0.0215242
$531$	0	0
$532$	3.41680	0.148137
$533$	−5.91232	−0.256091
$534$	0	0
$535$	4.87091	0.210588
$536$	−14.7655	−0.637774
$537$	0	0
$538$	−0.334141	−0.0144059
$539$	0	0
$540$	0	0
$541$	−26.3725	−1.13384	−0.566921	−	0.823772i	$-0.691866\pi$
$541$	−26.3725	−1.13384	−0.566921	+	0.823772i	$0.691866\pi$
$542$	4.14714	0.178135
$543$	0	0
$544$	−23.1191	−0.991225
$545$	12.3370	0.528460
$546$	0	0
$547$	32.2848	1.38040	0.690199	−	0.723620i	$-0.257523\pi$
$547$	32.2848	1.38040	0.690199	+	0.723620i	$0.257523\pi$
$548$	−15.9987	−0.683431
$549$	0	0
$550$	0	0
$551$	−14.1362	−0.602221
$552$	0	0
$553$	−8.85751	−0.376660
$554$	2.37930	0.101087
$555$	0	0
$556$	−18.7884	−0.796805
$557$	37.4768	1.58794	0.793972	−	0.607954i	$-0.208010\pi$
$557$	37.4768	1.58794	0.793972	+	0.607954i	$0.208010\pi$
$558$	0	0
$559$	28.0669	1.18710
$560$	−2.42091	−0.102302
$561$	0	0
$562$	−4.14359	−0.174787
$563$	14.3834	0.606188	0.303094	−	0.952961i	$-0.401980\pi$
$563$	14.3834	0.606188	0.303094	+	0.952961i	$0.401980\pi$
$564$	0	0
$565$	−8.19314	−0.344688
$566$	−5.37993	−0.226135
$567$	0	0
$568$	−8.92820	−0.374619
$569$	−29.2888	−1.22785	−0.613925	−	0.789364i	$-0.710410\pi$
$569$	−29.2888	−1.22785	−0.613925	+	0.789364i	$0.710410\pi$
$570$	0	0
$571$	18.5737	0.777284	0.388642	−	0.921389i	$-0.372944\pi$
$571$	18.5737	0.777284	0.388642	+	0.921389i	$0.372944\pi$
$572$	0	0
$573$	0	0
$574$	−0.401826	−0.0167719
$575$	−41.5159	−1.73133
$576$	0	0
$577$	20.3524	0.847282	0.423641	−	0.905830i	$-0.360752\pi$
$577$	20.3524	0.847282	0.423641	+	0.905830i	$0.360752\pi$
$578$	−8.82313	−0.366994
$579$	0	0
$580$	10.5403	0.437664
$581$	5.71775	0.237212
$582$	0	0
$583$	0	0
$584$	8.20176	0.339391
$585$	0	0
$586$	−7.21902	−0.298215
$587$	30.8643	1.27391	0.636954	−	0.770902i	$-0.280194\pi$
$587$	30.8643	1.27391	0.636954	+	0.770902i	$0.280194\pi$
$588$	0	0
$589$	18.7186	0.771287
$590$	−1.00287	−0.0412877
$591$	0	0
$592$	21.3695	0.878281
$593$	−4.76749	−0.195777	−0.0978887	−	0.995197i	$-0.531209\pi$
$593$	−4.76749	−0.195777	−0.0978887	+	0.995197i	$0.531209\pi$
$594$	0	0
$595$	−4.75525	−0.194946
$596$	13.6217	0.557967
$597$	0	0
$598$	−12.2786	−0.502110
$599$	−31.5423	−1.28878	−0.644391	−	0.764696i	$-0.722889\pi$
$599$	−31.5423	−1.28878	−0.644391	+	0.764696i	$0.722889\pi$
$600$	0	0
$601$	−2.59544	−0.105870	−0.0529352	−	0.998598i	$-0.516858\pi$
$601$	−2.59544	−0.105870	−0.0529352	+	0.998598i	$0.516858\pi$
$602$	1.90754	0.0777456
$603$	0	0
$604$	−14.9667	−0.608988
$605$	0	0
$606$	0	0
$607$	24.2991	0.986269	0.493135	−	0.869953i	$-0.335851\pi$
$607$	24.2991	0.986269	0.493135	+	0.869953i	$0.335851\pi$
$608$	−6.08036	−0.246591
$609$	0	0
$610$	0.356628	0.0144394
$611$	−36.3616	−1.47103
$612$	0	0
$613$	−4.32611	−0.174730	−0.0873650	−	0.996176i	$-0.527845\pi$
$613$	−4.32611	−0.174730	−0.0873650	+	0.996176i	$0.527845\pi$
$614$	7.30927	0.294978
$615$	0	0
$616$	0	0
$617$	−27.3809	−1.10231	−0.551156	−	0.834402i	$-0.685813\pi$
$617$	−27.3809	−1.10231	−0.551156	+	0.834402i	$0.685813\pi$
$618$	0	0
$619$	9.13728	0.367258	0.183629	−	0.982996i	$-0.441215\pi$
$619$	9.13728	0.367258	0.183629	+	0.982996i	$0.441215\pi$
$620$	−13.9571	−0.560533
$621$	0	0
$622$	−0.594884	−0.0238527
$623$	−2.22935	−0.0893169
$624$	0	0
$625$	17.9125	0.716501
$626$	−0.916045	−0.0366125
$627$	0	0
$628$	36.2691	1.44729
$629$	41.9748	1.67364
$630$	0	0
$631$	29.7629	1.18484	0.592421	−	0.805628i	$-0.298172\pi$
$631$	29.7629	1.18484	0.592421	+	0.805628i	$0.298172\pi$
$632$	10.4276	0.414788
$633$	0	0
$634$	8.35156	0.331683
$635$	8.49318	0.337042
$636$	0	0
$637$	4.43091	0.175559
$638$	0	0
$639$	0	0
$640$	5.99178	0.236846
$641$	17.6506	0.697159	0.348579	−	0.937279i	$-0.386664\pi$
$641$	17.6506	0.697159	0.348579	+	0.937279i	$0.386664\pi$
$642$	0	0
$643$	6.56451	0.258879	0.129439	−	0.991587i	$-0.458682\pi$
$643$	6.56451	0.258879	0.129439	+	0.991587i	$0.458682\pi$
$644$	17.5696	0.692338
$645$	0	0
$646$	−3.66691	−0.144273
$647$	−28.8178	−1.13294	−0.566472	−	0.824081i	$-0.691692\pi$
$647$	−28.8178	−1.13294	−0.566472	+	0.824081i	$0.691692\pi$
$648$	0	0
$649$	0	0
$650$	6.01999	0.236124
$651$	0	0
$652$	24.5879	0.962937
$653$	−32.3666	−1.26660	−0.633302	−	0.773905i	$-0.718301\pi$
$653$	−32.3666	−1.26660	−0.633302	+	0.773905i	$0.718301\pi$
$654$	0	0
$655$	−13.2161	−0.516397
$656$	−4.62228	−0.180470
$657$	0	0
$658$	−2.47128	−0.0963406
$659$	3.30435	0.128719	0.0643596	−	0.997927i	$-0.479500\pi$
$659$	3.30435	0.128719	0.0643596	+	0.997927i	$0.479500\pi$
$660$	0	0
$661$	−7.07404	−0.275148	−0.137574	−	0.990491i	$-0.543931\pi$
$661$	−7.07404	−0.275148	−0.137574	+	0.990491i	$0.543931\pi$
$662$	4.02032	0.156254
$663$	0	0
$664$	−6.73129	−0.261225
$665$	−1.25064	−0.0484976
$666$	0	0
$667$	−72.6898	−2.81456
$668$	0.0112344	0.000434672 0
$669$	0	0
$670$	2.63959	0.101976
$671$	0	0
$672$	0	0
$673$	21.2582	0.819445	0.409722	−	0.912210i	$-0.365626\pi$
$673$	21.2582	0.819445	0.409722	+	0.912210i	$0.365626\pi$
$674$	−1.93137	−0.0743934
$675$	0	0
$676$	−12.6644	−0.487091
$677$	43.2282	1.66140	0.830698	−	0.556723i	$-0.187942\pi$
$677$	43.2282	1.66140	0.830698	+	0.556723i	$0.187942\pi$
$678$	0	0
$679$	0.171378	0.00657689
$680$	5.59817	0.214680
$681$	0	0
$682$	0	0
$683$	−30.9708	−1.18507	−0.592533	−	0.805546i	$-0.701872\pi$
$683$	−30.9708	−1.18507	−0.592533	+	0.805546i	$0.701872\pi$
$684$	0	0
$685$	5.85593	0.223744
$686$	0.301143	0.0114977
$687$	0	0
$688$	21.9428	0.836561
$689$	10.4327	0.397453
$690$	0	0
$691$	−6.50729	−0.247549	−0.123775	−	0.992310i	$-0.539500\pi$
$691$	−6.50729	−0.247549	−0.123775	+	0.992310i	$0.539500\pi$
$692$	−24.6670	−0.937699
$693$	0	0
$694$	−3.00370	−0.114019
$695$	6.87703	0.260860
$696$	0	0
$697$	−9.07926	−0.343901
$698$	−2.46415	−0.0932696
$699$	0	0
$700$	−8.61405	−0.325581
$701$	13.3174	0.502992	0.251496	−	0.967858i	$-0.419078\pi$
$701$	13.3174	0.502992	0.251496	+	0.967858i	$0.419078\pi$
$702$	0	0
$703$	11.0394	0.416359
$704$	0	0
$705$	0	0
$706$	−0.0521429	−0.00196242
$707$	18.0723	0.679678
$708$	0	0
$709$	48.2055	1.81040	0.905198	−	0.424990i	$-0.139722\pi$
$709$	48.2055	1.81040	0.905198	+	0.424990i	$0.139722\pi$
$710$	1.59607	0.0598994
$711$	0	0
$712$	2.62452	0.0983582
$713$	96.2532	3.60471
$714$	0	0
$715$	0	0
$716$	−2.50375	−0.0935694
$717$	0	0
$718$	−7.50619	−0.280129
$719$	11.4733	0.427881	0.213940	−	0.976847i	$-0.431370\pi$
$719$	11.4733	0.427881	0.213940	+	0.976847i	$0.431370\pi$
$720$	0	0
$721$	1.73047	0.0644462
$722$	4.75732	0.177049
$723$	0	0
$724$	−8.46090	−0.314447
$725$	35.6385	1.32358
$726$	0	0
$727$	−29.5904	−1.09745	−0.548724	−	0.836004i	$-0.684886\pi$
$727$	−29.5904	−1.09745	−0.548724	+	0.836004i	$0.684886\pi$
$728$	−5.21634	−0.193330
$729$	0	0
$730$	−1.46621	−0.0542667
$731$	43.1009	1.59414
$732$	0	0
$733$	39.1482	1.44597	0.722986	−	0.690862i	$-0.242769\pi$
$733$	39.1482	1.44597	0.722986	+	0.690862i	$0.242769\pi$
$734$	10.2911	0.379853
$735$	0	0
$736$	−31.2659	−1.15248
$737$	0	0
$738$	0	0
$739$	15.6126	0.574317	0.287159	−	0.957883i	$-0.407289\pi$
$739$	15.6126	0.574317	0.287159	+	0.957883i	$0.407289\pi$
$740$	−8.23131	−0.302589
$741$	0	0
$742$	0.709048	0.0260300
$743$	−29.8447	−1.09490	−0.547448	−	0.836840i	$-0.684401\pi$
$743$	−29.8447	−1.09490	−0.547448	+	0.836840i	$0.684401\pi$
$744$	0	0
$745$	−4.98589	−0.182669
$746$	0.454817	0.0166520
$747$	0	0
$748$	0	0
$749$	−6.96982	−0.254672
$750$	0	0
$751$	1.52452	0.0556306	0.0278153	−	0.999613i	$-0.491145\pi$
$751$	1.52452	0.0556306	0.0278153	+	0.999613i	$0.491145\pi$
$752$	−28.4276	−1.03665
$753$	0	0
$754$	10.5403	0.383857
$755$	5.47821	0.199372
$756$	0	0
$757$	−13.3841	−0.486453	−0.243226	−	0.969970i	$-0.578206\pi$
$757$	−13.3841	−0.486453	−0.243226	+	0.969970i	$0.578206\pi$
$758$	2.24657	0.0815990
$759$	0	0
$760$	1.47233	0.0534069
$761$	17.1185	0.620544	0.310272	−	0.950648i	$-0.399580\pi$
$761$	17.1185	0.620544	0.310272	+	0.950648i	$0.399580\pi$
$762$	0	0
$763$	−17.6531	−0.639086
$764$	10.2065	0.369257
$765$	0	0
$766$	7.75389	0.280160
$767$	21.1144	0.762395
$768$	0	0
$769$	−15.9428	−0.574912	−0.287456	−	0.957794i	$-0.592809\pi$
$769$	−15.9428	−0.574912	−0.287456	+	0.957794i	$0.592809\pi$
$770$	0	0
$771$	0	0
$772$	−1.53480	−0.0552385
$773$	11.9350	0.429274	0.214637	−	0.976694i	$-0.431143\pi$
$773$	11.9350	0.429274	0.214637	+	0.976694i	$0.431143\pi$
$774$	0	0
$775$	−47.1913	−1.69516
$776$	−0.201757	−0.00724265
$777$	0	0
$778$	−10.1324	−0.363264
$779$	−2.38785	−0.0855538
$780$	0	0
$781$	0	0
$782$	−18.8557	−0.674277
$783$	0	0
$784$	3.46410	0.123718
$785$	−13.2754	−0.473820
$786$	0	0
$787$	−2.80275	−0.0999071	−0.0499535	−	0.998752i	$-0.515907\pi$
$787$	−2.80275	−0.0999071	−0.0499535	+	0.998752i	$0.515907\pi$
$788$	−44.6935	−1.59214
$789$	0	0
$790$	−1.86411	−0.0663222
$791$	11.7236	0.416844
$792$	0	0
$793$	−7.50839	−0.266631
$794$	5.93102	0.210484
$795$	0	0
$796$	−44.5095	−1.57760
$797$	13.6873	0.484828	0.242414	−	0.970173i	$-0.422061\pi$
$797$	13.6873	0.484828	0.242414	+	0.970173i	$0.422061\pi$
$798$	0	0
$799$	−55.8386	−1.97543
$800$	15.3291	0.541966
$801$	0	0
$802$	5.25097	0.185418
$803$	0	0
$804$	0	0
$805$	−6.43091	−0.226660
$806$	−13.9571	−0.491620
$807$	0	0
$808$	−21.2758	−0.748480
$809$	25.4980	0.896464	0.448232	−	0.893917i	$-0.352054\pi$
$809$	25.4980	0.896464	0.448232	+	0.893917i	$0.352054\pi$
$810$	0	0
$811$	−36.7404	−1.29013	−0.645065	−	0.764128i	$-0.723170\pi$
$811$	−36.7404	−1.29013	−0.645065	+	0.764128i	$0.723170\pi$
$812$	−15.0823	−0.529284
$813$	0	0
$814$	0	0
$815$	−8.99980	−0.315249
$816$	0	0
$817$	11.3356	0.396582
$818$	−2.38670	−0.0834490
$819$	0	0
$820$	1.78045	0.0621762
$821$	−13.0737	−0.456274	−0.228137	−	0.973629i	$-0.573263\pi$
$821$	−13.0737	−0.456274	−0.228137	+	0.973629i	$0.573263\pi$
$822$	0	0
$823$	48.7149	1.69809	0.849047	−	0.528318i	$-0.177177\pi$
$823$	48.7149	1.69809	0.849047	+	0.528318i	$0.177177\pi$
$824$	−2.03722	−0.0709700
$825$	0	0
$826$	1.43502	0.0499307
$827$	−53.8443	−1.87235	−0.936176	−	0.351532i	$-0.885661\pi$
$827$	−53.8443	−1.87235	−0.936176	+	0.351532i	$0.885661\pi$
$828$	0	0
$829$	11.1183	0.386154	0.193077	−	0.981184i	$-0.438153\pi$
$829$	11.1183	0.386154	0.193077	+	0.981184i	$0.438153\pi$
$830$	1.20333	0.0417683
$831$	0	0
$832$	−26.1645	−0.907092
$833$	6.80432	0.235756
$834$	0	0
$835$	−0.00411207	−0.000142304 0
$836$	0	0
$837$	0	0
$838$	−3.62281	−0.125148
$839$	−26.8372	−0.926524	−0.463262	−	0.886221i	$-0.653321\pi$
$839$	−26.8372	−0.926524	−0.463262	+	0.886221i	$0.653321\pi$
$840$	0	0
$841$	33.3992	1.15170
$842$	0.410500	0.0141468
$843$	0	0
$844$	−19.3709	−0.666775
$845$	4.63548	0.159465
$846$	0	0
$847$	0	0
$848$	8.15631	0.280089
$849$	0	0
$850$	9.24460	0.317087
$851$	56.7659	1.94591
$852$	0	0
$853$	−34.8276	−1.19247	−0.596237	−	0.802809i	$-0.703338\pi$
$853$	−34.8276	−1.19247	−0.596237	+	0.802809i	$0.703338\pi$
$854$	−0.510302	−0.0174622
$855$	0	0
$856$	8.20530	0.280451
$857$	−38.4741	−1.31425	−0.657126	−	0.753781i	$-0.728228\pi$
$857$	−38.4741	−1.31425	−0.657126	+	0.753781i	$0.728228\pi$
$858$	0	0
$859$	−7.58039	−0.258639	−0.129320	−	0.991603i	$-0.541279\pi$
$859$	−7.58039	−0.258639	−0.129320	+	0.991603i	$0.541279\pi$
$860$	−8.45214	−0.288216
$861$	0	0
$862$	−2.48128	−0.0845127
$863$	32.4054	1.10309	0.551546	−	0.834145i	$-0.314038\pi$
$863$	32.4054	1.10309	0.551546	+	0.834145i	$0.314038\pi$
$864$	0	0
$865$	9.02875	0.306987
$866$	−4.14699	−0.140921
$867$	0	0
$868$	19.9714	0.677873
$869$	0	0
$870$	0	0
$871$	−55.5736	−1.88304
$872$	20.7824	0.703780
$873$	0	0
$874$	−4.95906	−0.167743
$875$	6.64725	0.224718
$876$	0	0
$877$	11.2438	0.379676	0.189838	−	0.981815i	$-0.439204\pi$
$877$	11.2438	0.379676	0.189838	+	0.981815i	$0.439204\pi$
$878$	5.63410	0.190142
$879$	0	0
$880$	0	0
$881$	35.8854	1.20901	0.604505	−	0.796601i	$-0.293371\pi$
$881$	35.8854	1.20901	0.604505	+	0.796601i	$0.293371\pi$
$882$	0	0
$883$	−12.8564	−0.432653	−0.216326	−	0.976321i	$-0.569407\pi$
$883$	−12.8564	−0.432653	−0.216326	+	0.976321i	$0.569407\pi$
$884$	−57.5645	−1.93610
$885$	0	0
$886$	10.0325	0.337049
$887$	−44.5324	−1.49525	−0.747627	−	0.664119i	$-0.768807\pi$
$887$	−44.5324	−1.49525	−0.747627	+	0.664119i	$0.768807\pi$
$888$	0	0
$889$	−12.1530	−0.407597
$890$	−0.469179	−0.0157269
$891$	0	0
$892$	13.0989	0.438583
$893$	−14.6856	−0.491435
$894$	0	0
$895$	0.916435	0.0306331
$896$	−8.57368	−0.286427
$897$	0	0
$898$	−1.12698	−0.0376077
$899$	−82.6268	−2.75576
$900$	0	0
$901$	16.0209	0.533735
$902$	0	0
$903$	0	0
$904$	−13.8018	−0.459041
$905$	3.09690	0.102945
$906$	0	0
$907$	−0.321610	−0.0106789	−0.00533944	−	0.999986i	$-0.501700\pi$
$907$	−0.321610	−0.0106789	−0.00533944	+	0.999986i	$0.501700\pi$
$908$	18.4275	0.611537
$909$	0	0
$910$	0.932511	0.0309124
$911$	−42.8345	−1.41917	−0.709585	−	0.704620i	$-0.751118\pi$
$911$	−42.8345	−1.41917	−0.709585	+	0.704620i	$0.751118\pi$
$912$	0	0
$913$	0	0
$914$	−6.01525	−0.198967
$915$	0	0
$916$	−4.53369	−0.149797
$917$	18.9111	0.624499
$918$	0	0
$919$	41.3657	1.36453	0.682265	−	0.731105i	$-0.260995\pi$
$919$	41.3657	1.36453	0.682265	+	0.731105i	$0.260995\pi$
$920$	7.57086	0.249604
$921$	0	0
$922$	−0.272350	−0.00896937
$923$	−33.6034	−1.10607
$924$	0	0
$925$	−27.8313	−0.915089
$926$	−7.01013	−0.230367
$927$	0	0
$928$	26.8396	0.881054
$929$	−39.4522	−1.29439	−0.647193	−	0.762326i	$-0.724057\pi$
$929$	−39.4522	−1.29439	−0.647193	+	0.762326i	$0.724057\pi$
$930$	0	0
$931$	1.78954	0.0586499
$932$	31.0537	1.01720
$933$	0	0
$934$	1.12934	0.0369531
$935$	0	0
$936$	0	0
$937$	17.9805	0.587397	0.293698	−	0.955898i	$-0.405114\pi$
$937$	17.9805	0.587397	0.293698	+	0.955898i	$0.405114\pi$
$938$	−3.77701	−0.123324
$939$	0	0
$940$	10.9500	0.357150
$941$	30.5804	0.996892	0.498446	−	0.866921i	$-0.333904\pi$
$941$	30.5804	0.996892	0.498446	+	0.866921i	$0.333904\pi$
$942$	0	0
$943$	−12.2786	−0.399847
$944$	16.5073	0.537267
$945$	0	0
$946$	0	0
$947$	8.17056	0.265507	0.132754	−	0.991149i	$-0.457618\pi$
$947$	8.17056	0.265507	0.132754	+	0.991149i	$0.457618\pi$
$948$	0	0
$949$	30.8693	1.00206
$950$	2.43134	0.0788831
$951$	0	0
$952$	−8.01047	−0.259621
$953$	−8.38538	−0.271629	−0.135815	−	0.990734i	$-0.543365\pi$
$953$	−8.38538	−0.271629	−0.135815	+	0.990734i	$0.543365\pi$
$954$	0	0
$955$	−3.73583	−0.120889
$956$	−32.9138	−1.06451
$957$	0	0
$958$	−8.74573	−0.282562
$959$	−8.37930	−0.270582
$960$	0	0
$961$	78.4114	2.52940
$962$	−8.23131	−0.265388
$963$	0	0
$964$	−10.3450	−0.333190
$965$	0.561775	0.0180842
$966$	0	0
$967$	51.4778	1.65541	0.827707	−	0.561161i	$-0.189645\pi$
$967$	51.4778	1.65541	0.827707	+	0.561161i	$0.189645\pi$
$968$	0	0
$969$	0	0
$970$	0.0360675	0.00115806
$971$	13.5974	0.436360	0.218180	−	0.975909i	$-0.429988\pi$
$971$	13.5974	0.436360	0.218180	+	0.975909i	$0.429988\pi$
$972$	0	0
$973$	−9.84039	−0.315468
$974$	3.74564	0.120018
$975$	0	0
$976$	−5.87009	−0.187897
$977$	−19.6433	−0.628446	−0.314223	−	0.949349i	$-0.601744\pi$
$977$	−19.6433	−0.628446	−0.314223	+	0.949349i	$0.601744\pi$
$978$	0	0
$979$	0	0
$980$	−1.33434	−0.0426238
$981$	0	0
$982$	1.94642	0.0621127
$983$	57.0855	1.82074	0.910372	−	0.413791i	$-0.135796\pi$
$983$	57.0855	1.82074	0.910372	+	0.413791i	$0.135796\pi$
$984$	0	0
$985$	16.3590	0.521240
$986$	16.1863	0.515476
$987$	0	0
$988$	−15.1395	−0.481652
$989$	58.2888	1.85348
$990$	0	0
$991$	−32.7803	−1.04130	−0.520650	−	0.853770i	$-0.674310\pi$
$991$	−32.7803	−1.04130	−0.520650	+	0.853770i	$0.674310\pi$
$992$	−35.5401	−1.12840
$993$	0	0
$994$	−2.28383	−0.0724387
$995$	16.2916	0.516479
$996$	0	0
$997$	33.3145	1.05508	0.527540	−	0.849530i	$-0.323114\pi$
$997$	33.3145	1.05508	0.527540	+	0.849530i	$0.323114\pi$
$998$	4.00062	0.126637
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cg.1.3		4
3.2	odd	2		2541.2.a.bp.1.2	yes	4
11.10	odd	2		7623.2.a.cn.1.2		4
33.32	even	2		2541.2.a.bl.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.bl.1.3	✓	4	33.32	even	2
2541.2.a.bp.1.2	yes	4	3.2	odd	2
7623.2.a.cg.1.3		4	1.1	even	1	trivial
7623.2.a.cn.1.2		4	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-0.301143 q^{2} -1.90931 q^{4} +0.698857 q^{5} -1.00000 q^{7} +1.17726 q^{8} +O(q^{10})\)	\(q-0.301143 q^{2} -1.90931 q^{4} +0.698857 q^{5} -1.00000 q^{7} +1.17726 q^{8} -0.210456 q^{10} +4.43091 q^{13} +0.301143 q^{14} +3.46410 q^{16} +6.80432 q^{17} +1.78954 q^{19} -1.33434 q^{20} +9.20204 q^{23} -4.51160 q^{25} -1.33434 q^{26} +1.90931 q^{28} -7.89932 q^{29} +10.4600 q^{31} -3.39771 q^{32} -2.04907 q^{34} -0.698857 q^{35} +6.16884 q^{37} -0.538909 q^{38} +0.822738 q^{40} -1.33434 q^{41} +6.33434 q^{43} -2.77113 q^{46} -8.20634 q^{47} +1.00000 q^{49} +1.35864 q^{50} -8.45999 q^{52} +2.35452 q^{53} -1.17726 q^{56} +2.37882 q^{58} +4.76524 q^{59} -1.69455 q^{61} -3.14995 q^{62} -5.90501 q^{64} +3.09657 q^{65} -12.5423 q^{67} -12.9916 q^{68} +0.210456 q^{70} -7.58387 q^{71} +6.96681 q^{73} -1.85770 q^{74} -3.41680 q^{76} +8.85751 q^{79} +2.42091 q^{80} +0.401826 q^{82} -5.71775 q^{83} +4.75525 q^{85} -1.90754 q^{86} +2.22935 q^{89} -4.43091 q^{91} -17.5696 q^{92} +2.47128 q^{94} +1.25064 q^{95} -0.171378 q^{97} -0.301143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^7	\( 4 q - 2 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{7} + 10 q^{10} + 10 q^{13} + 2 q^{14} - 6 q^{17} + 18 q^{19} + 2 q^{23} - 8 q^{25} - 4 q^{28} - 6 q^{29} - 12 q^{32} - 2 q^{34} - 2 q^{35} - 4 q^{37} + 8 q^{40} + 20 q^{43} + 16 q^{46} + 6 q^{47} + 4 q^{49} + 24 q^{50} + 8 q^{52} + 24 q^{58} + 6 q^{59} - 10 q^{61} - 16 q^{64} + 10 q^{65} + 4 q^{67} - 28 q^{68} - 10 q^{70} + 6 q^{71} + 34 q^{73} + 36 q^{74} + 36 q^{76} + 24 q^{79} - 12 q^{80} + 28 q^{82} - 6 q^{83} - 8 q^{85} - 38 q^{86} - 18 q^{89} - 10 q^{91} - 24 q^{92} - 6 q^{94} + 18 q^{95} - 10 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^7 + 10 * q^10 + 10 * q^13 + 2 * q^14 - 6 * q^17 + 18 * q^19 + 2 * q^23 - 8 * q^25 - 4 * q^28 - 6 * q^29 - 12 * q^32 - 2 * q^34 - 2 * q^35 - 4 * q^37 + 8 * q^40 + 20 * q^43 + 16 * q^46 + 6 * q^47 + 4 * q^49 + 24 * q^50 + 8 * q^52 + 24 * q^58 + 6 * q^59 - 10 * q^61 - 16 * q^64 + 10 * q^65 + 4 * q^67 - 28 * q^68 - 10 * q^70 + 6 * q^71 + 34 * q^73 + 36 * q^74 + 36 * q^76 + 24 * q^79 - 12 * q^80 + 28 * q^82 - 6 * q^83 - 8 * q^85 - 38 * q^86 - 18 * q^89 - 10 * q^91 - 24 * q^92 - 6 * q^94 + 18 * q^95 - 10 * q^97 - 2 * q^98

Introduction

L-functions

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