LMFDB - Embedded newform 4598.2.a.bo.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, 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("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4598 = 2 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4598.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:	$36.7152148494$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.66908$ of defining polynomial
Character	$\chi$	$=$	4598.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.66908 q^{3} +1.00000 q^{4} -4.12398 q^{5} +2.66908 q^{6} +4.21417 q^{7} +1.00000 q^{8} +4.12398 q^{9} -4.12398 q^{10} +2.66908 q^{12} +2.21417 q^{13} +4.21417 q^{14} -11.0072 q^{15} +1.00000 q^{16} +3.45490 q^{17} +4.12398 q^{18} +1.00000 q^{19} -4.12398 q^{20} +11.2480 q^{21} -5.45490 q^{23} +2.66908 q^{24} +12.0072 q^{25} +2.21417 q^{26} +3.00000 q^{27} +4.21417 q^{28} -5.57889 q^{29} -11.0072 q^{30} +7.00724 q^{31} +1.00000 q^{32} +3.45490 q^{34} -17.3792 q^{35} +4.12398 q^{36} -2.90981 q^{37} +1.00000 q^{38} +5.90981 q^{39} -4.12398 q^{40} +11.9170 q^{41} +11.2480 q^{42} -1.46214 q^{43} -17.0072 q^{45} -5.45490 q^{46} +7.58612 q^{47} +2.66908 q^{48} +10.7593 q^{49} +12.0072 q^{50} +9.22141 q^{51} +2.21417 q^{52} -13.2214 q^{53} +3.00000 q^{54} +4.21417 q^{56} +2.66908 q^{57} -5.57889 q^{58} +4.79306 q^{59} -11.0072 q^{60} -8.90981 q^{61} +7.00724 q^{62} +17.3792 q^{63} +1.00000 q^{64} -9.13122 q^{65} +1.30437 q^{67} +3.45490 q^{68} -14.5596 q^{69} -17.3792 q^{70} -6.80030 q^{71} +4.12398 q^{72} +1.45490 q^{73} -2.90981 q^{74} +32.0483 q^{75} +1.00000 q^{76} +5.90981 q^{78} +9.15777 q^{79} -4.12398 q^{80} -4.36471 q^{81} +11.9170 q^{82} +13.2142 q^{83} +11.2480 q^{84} -14.2480 q^{85} -1.46214 q^{86} -14.8905 q^{87} +8.24797 q^{89} -17.0072 q^{90} +9.33092 q^{91} -5.45490 q^{92} +18.7029 q^{93} +7.58612 q^{94} -4.12398 q^{95} +2.66908 q^{96} +2.18038 q^{97} +10.7593 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 6 q^{14} - 9 q^{15} + 3 q^{16} + 9 q^{17} + 3 q^{18} + 3 q^{19} - 3 q^{20} + 15 q^{21} - 15 q^{23} + 12 q^{25} + 9 q^{27} + 6 q^{28}+ \cdots + 27 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + 6 * q^14 - 9 * q^15 + 3 * q^16 + 9 * q^17 + 3 * q^18 + 3 * q^19 - 3 * q^20 + 15 * q^21 - 15 * q^23 + 12 * q^25 + 9 * q^27 + 6 * q^28 - 6 * q^29 - 9 * q^30 - 3 * q^31 + 3 * q^32 + 9 * q^34 + 3 * q^36 - 6 * q^37 + 3 * q^38 + 15 * q^39 - 3 * q^40 + 9 * q^41 + 15 * q^42 + 21 * q^43 - 27 * q^45 - 15 * q^46 - 12 * q^47 + 27 * q^49 + 12 * q^50 - 3 * q^51 - 9 * q^53 + 9 * q^54 + 6 * q^56 - 6 * q^58 - 3 * q^59 - 9 * q^60 - 24 * q^61 - 3 * q^62 + 3 * q^64 + 6 * q^65 + 9 * q^68 + 3 * q^69 + 21 * q^71 + 3 * q^72 + 3 * q^73 - 6 * q^74 + 36 * q^75 + 3 * q^76 + 15 * q^78 + 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 + 33 * q^83 + 15 * q^84 - 24 * q^85 + 21 * q^86 - 6 * q^87 + 6 * q^89 - 27 * q^90 + 36 * q^91 - 15 * q^92 + 36 * q^93 - 12 * q^94 - 3 * q^95 + 12 * q^97 + 27 * q^98

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.66908	1.54099	0.770497	−	0.637444i	$-0.220008\pi$
$3$	2.66908	1.54099	0.770497	+	0.637444i	$0.220008\pi$
$4$	1.00000	0.500000
$5$	−4.12398	−1.84430	−0.922151	−	0.386831i	$-0.873570\pi$
$5$	−4.12398	−1.84430	−0.922151	+	0.386831i	$0.873570\pi$
$6$	2.66908	1.08965
$7$	4.21417	1.59281	0.796404	−	0.604765i	$-0.206733\pi$
$7$	4.21417	1.59281	0.796404	+	0.604765i	$0.206733\pi$
$8$	1.00000	0.353553
$9$	4.12398	1.37466
$10$	−4.12398	−1.30412
$11$	0	0
$11$	0	0
$12$	2.66908	0.770497
$13$	2.21417	0.614102	0.307051	−	0.951693i	$-0.400658\pi$
$13$	2.21417	0.614102	0.307051	+	0.951693i	$0.400658\pi$
$14$	4.21417	1.12629
$15$	−11.0072	−2.84206
$16$	1.00000	0.250000
$17$	3.45490	0.837937	0.418969	−	0.908001i	$-0.362392\pi$
$17$	3.45490	0.837937	0.418969	+	0.908001i	$0.362392\pi$
$18$	4.12398	0.972032
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−4.12398	−0.922151
$21$	11.2480	2.45451
$22$	0	0
$23$	−5.45490	−1.13743	−0.568713	−	0.822536i	$-0.692559\pi$
$23$	−5.45490	−1.13743	−0.568713	+	0.822536i	$0.692559\pi$
$24$	2.66908	0.544823
$25$	12.0072	2.40145
$26$	2.21417	0.434235
$27$	3.00000	0.577350
$28$	4.21417	0.796404
$29$	−5.57889	−1.03597	−0.517987	−	0.855389i	$-0.673318\pi$
$29$	−5.57889	−1.03597	−0.517987	+	0.855389i	$0.673318\pi$
$30$	−11.0072	−2.00964
$31$	7.00724	1.25854	0.629268	−	0.777188i	$-0.283355\pi$
$31$	7.00724	1.25854	0.629268	+	0.777188i	$0.283355\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.45490	0.592511
$35$	−17.3792	−2.93762
$36$	4.12398	0.687331
$37$	−2.90981	−0.478370	−0.239185	−	0.970974i	$-0.576880\pi$
$37$	−2.90981	−0.478370	−0.239185	+	0.970974i	$0.576880\pi$
$38$	1.00000	0.162221
$39$	5.90981	0.946327
$40$	−4.12398	−0.652059
$41$	11.9170	1.86113	0.930565	−	0.366127i	$-0.119316\pi$
$41$	11.9170	1.86113	0.930565	+	0.366127i	$0.119316\pi$
$42$	11.2480	1.73560
$43$	−1.46214	−0.222974	−0.111487	−	0.993766i	$-0.535561\pi$
$43$	−1.46214	−0.222974	−0.111487	+	0.993766i	$0.535561\pi$
$44$	0	0
$45$	−17.0072	−2.53529
$46$	−5.45490	−0.804282
$47$	7.58612	1.10655	0.553275	−	0.832999i	$-0.313378\pi$
$47$	7.58612	1.10655	0.553275	+	0.832999i	$0.313378\pi$
$48$	2.66908	0.385248
$49$	10.7593	1.53704
$50$	12.0072	1.69808
$51$	9.22141	1.29126
$52$	2.21417	0.307051
$53$	−13.2214	−1.81610	−0.908050	−	0.418861i	$-0.862429\pi$
$53$	−13.2214	−1.81610	−0.908050	+	0.418861i	$0.862429\pi$
$54$	3.00000	0.408248
$55$	0	0
$56$	4.21417	0.563143
$57$	2.66908	0.353528
$58$	−5.57889	−0.732544
$59$	4.79306	0.624004	0.312002	−	0.950082i	$-0.399001\pi$
$59$	4.79306	0.624004	0.312002	+	0.950082i	$0.399001\pi$
$60$	−11.0072	−1.42103
$61$	−8.90981	−1.14078	−0.570392	−	0.821373i	$-0.693209\pi$
$61$	−8.90981	−1.14078	−0.570392	+	0.821373i	$0.693209\pi$
$62$	7.00724	0.889920
$63$	17.3792	2.18957
$64$	1.00000	0.125000
$65$	−9.13122	−1.13259
$66$	0	0
$67$	1.30437	0.159354	0.0796769	−	0.996821i	$-0.474611\pi$
$67$	1.30437	0.159354	0.0796769	+	0.996821i	$0.474611\pi$
$68$	3.45490	0.418969
$69$	−14.5596	−1.75277
$70$	−17.3792	−2.07721
$71$	−6.80030	−0.807047	−0.403524	−	0.914969i	$-0.632215\pi$
$71$	−6.80030	−0.807047	−0.403524	+	0.914969i	$0.632215\pi$
$72$	4.12398	0.486016
$73$	1.45490	0.170284	0.0851418	−	0.996369i	$-0.472866\pi$
$73$	1.45490	0.170284	0.0851418	+	0.996369i	$0.472866\pi$
$74$	−2.90981	−0.338258
$75$	32.0483	3.70061
$76$	1.00000	0.114708
$77$	0	0
$78$	5.90981	0.669154
$79$	9.15777	1.03033	0.515165	−	0.857091i	$-0.327731\pi$
$79$	9.15777	1.03033	0.515165	+	0.857091i	$0.327731\pi$
$80$	−4.12398	−0.461075
$81$	−4.36471	−0.484968
$82$	11.9170	1.31602
$83$	13.2142	1.45044	0.725222	−	0.688515i	$-0.241737\pi$
$83$	13.2142	1.45044	0.725222	+	0.688515i	$0.241737\pi$
$84$	11.2480	1.22725
$85$	−14.2480	−1.54541
$86$	−1.46214	−0.157667
$87$	−14.8905	−1.59643
$88$	0	0
$89$	8.24797	0.874283	0.437141	−	0.899393i	$-0.355991\pi$
$89$	8.24797	0.874283	0.437141	+	0.899393i	$0.355991\pi$
$90$	−17.0072	−1.79272
$91$	9.33092	0.978146
$92$	−5.45490	−0.568713
$93$	18.7029	1.93940
$94$	7.58612	0.782449
$95$	−4.12398	−0.423112
$96$	2.66908	0.272412
$97$	2.18038	0.221384	0.110692	−	0.993855i	$-0.464693\pi$
$97$	2.18038	0.221384	0.110692	+	0.993855i	$0.464693\pi$
$98$	10.7593	1.08685
$99$	0	0
$100$	12.0072	1.20072
$101$	6.24797	0.621696	0.310848	−	0.950460i	$-0.399387\pi$
$101$	6.24797	0.621696	0.310848	+	0.950460i	$0.399387\pi$
$102$	9.22141	0.913056
$103$	0.605441	0.0596559	0.0298280	−	0.999555i	$-0.490504\pi$
$103$	0.605441	0.0596559	0.0298280	+	0.999555i	$0.490504\pi$
$104$	2.21417	0.217118
$105$	−46.3864	−4.52685
$106$	−13.2214	−1.28418
$107$	−3.88325	−0.375408	−0.187704	−	0.982226i	$-0.560105\pi$
$107$	−3.88325	−0.375408	−0.187704	+	0.982226i	$0.560105\pi$
$108$	3.00000	0.288675
$109$	4.97345	0.476370	0.238185	−	0.971220i	$-0.423448\pi$
$109$	4.97345	0.476370	0.238185	+	0.971220i	$0.423448\pi$
$110$	0	0
$111$	−7.76651	−0.737164
$112$	4.21417	0.398202
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	2.66908	0.249982
$115$	22.4959	2.09776
$116$	−5.57889	−0.517987
$117$	9.13122	0.844182
$118$	4.79306	0.441237
$119$	14.5596	1.33467
$120$	−11.0072	−1.00482
$121$	0	0
$122$	−8.90981	−0.806656
$123$	31.8075	2.86799
$124$	7.00724	0.629268
$125$	−28.8977	−2.58469
$126$	17.3792	1.54826
$127$	2.48146	0.220194	0.110097	−	0.993921i	$-0.464884\pi$
$127$	2.48146	0.220194	0.110097	+	0.993921i	$0.464884\pi$
$128$	1.00000	0.0883883
$129$	−3.90257	−0.343602
$130$	−9.13122	−0.800861
$131$	−11.0072	−0.961707	−0.480853	−	0.876801i	$-0.659673\pi$
$131$	−11.0072	−0.961707	−0.480853	+	0.876801i	$0.659673\pi$
$132$	0	0
$133$	4.21417	0.365415
$134$	1.30437	0.112680
$135$	−12.3719	−1.06481
$136$	3.45490	0.296256
$137$	6.94360	0.593232	0.296616	−	0.954997i	$-0.404142\pi$
$137$	6.94360	0.593232	0.296616	+	0.954997i	$0.404142\pi$
$138$	−14.5596	−1.23939
$139$	4.37195	0.370824	0.185412	−	0.982661i	$-0.440638\pi$
$139$	4.37195	0.370824	0.185412	+	0.982661i	$0.440638\pi$
$140$	−17.3792	−1.46881
$141$	20.2480	1.70519
$142$	−6.80030	−0.570668
$143$	0	0
$144$	4.12398	0.343665
$145$	23.0072	1.91065
$146$	1.45490	0.120409
$147$	28.7173	2.36857
$148$	−2.90981	−0.239185
$149$	−11.7665	−0.963950	−0.481975	−	0.876185i	$-0.660080\pi$
$149$	−11.7665	−0.963950	−0.481975	+	0.876185i	$0.660080\pi$
$150$	32.0483	2.61673
$151$	19.5861	1.59390	0.796948	−	0.604048i	$-0.206446\pi$
$151$	19.5861	1.59390	0.796948	+	0.604048i	$0.206446\pi$
$152$	1.00000	0.0811107
$153$	14.2480	1.15188
$154$	0	0
$155$	−28.8977	−2.32112
$156$	5.90981	0.473163
$157$	20.6199	1.64565	0.822824	−	0.568296i	$-0.192397\pi$
$157$	20.6199	1.64565	0.822824	+	0.568296i	$0.192397\pi$
$158$	9.15777	0.728553
$159$	−35.2890	−2.79860
$160$	−4.12398	−0.326029
$161$	−22.9879	−1.81170
$162$	−4.36471	−0.342924
$163$	−17.8341	−1.39687	−0.698437	−	0.715672i	$-0.746121\pi$
$163$	−17.8341	−1.39687	−0.698437	+	0.715672i	$0.746121\pi$
$164$	11.9170	0.930565
$165$	0	0
$166$	13.2142	1.02562
$167$	−4.18038	−0.323488	−0.161744	−	0.986833i	$-0.551712\pi$
$167$	−4.18038	−0.323488	−0.161744	+	0.986833i	$0.551712\pi$
$168$	11.2480	0.867799
$169$	−8.09743	−0.622879
$170$	−14.2480	−1.09277
$171$	4.12398	0.315369
$172$	−1.46214	−0.111487
$173$	−23.7101	−1.80265	−0.901323	−	0.433147i	$-0.857403\pi$
$173$	−23.7101	−1.80265	−0.901323	+	0.433147i	$0.857403\pi$
$174$	−14.8905	−1.12885
$175$	50.6006	3.82505
$176$	0	0
$177$	12.7931	0.961585
$178$	8.24797	0.618211
$179$	−10.8269	−0.809237	−0.404619	−	0.914486i	$-0.632596\pi$
$179$	−10.8269	−0.809237	−0.404619	+	0.914486i	$0.632596\pi$
$180$	−17.0072	−1.26764
$181$	−17.8341	−1.32560	−0.662799	−	0.748798i	$-0.730632\pi$
$181$	−17.8341	−1.32560	−0.662799	+	0.748798i	$0.730632\pi$
$182$	9.33092	0.691654
$183$	−23.7810	−1.75794
$184$	−5.45490	−0.402141
$185$	12.0000	0.882258
$186$	18.7029	1.37136
$187$	0	0
$188$	7.58612	0.553275
$189$	12.6425	0.919608
$190$	−4.12398	−0.299185
$191$	0.612679	0.0443319	0.0221659	−	0.999754i	$-0.492944\pi$
$191$	0.612679	0.0443319	0.0221659	+	0.999754i	$0.492944\pi$
$192$	2.66908	0.192624
$193$	−11.6425	−0.838047	−0.419024	−	0.907975i	$-0.637628\pi$
$193$	−11.6425	−0.838047	−0.419024	+	0.907975i	$0.637628\pi$
$194$	2.18038	0.156542
$195$	−24.3719	−1.74531
$196$	10.7593	0.768519
$197$	12.9098	0.919786	0.459893	−	0.887974i	$-0.347888\pi$
$197$	12.9098	0.919786	0.459893	+	0.887974i	$0.347888\pi$
$198$	0	0
$199$	−18.3647	−1.30184	−0.650920	−	0.759146i	$-0.725617\pi$
$199$	−18.3647	−1.30184	−0.650920	+	0.759146i	$0.725617\pi$
$200$	12.0072	0.849040
$201$	3.48146	0.245563
$202$	6.24797	0.439605
$203$	−23.5104	−1.65011
$204$	9.22141	0.645628
$205$	−49.1457	−3.43248
$206$	0.605441	0.0421831
$207$	−22.4959	−1.56358
$208$	2.21417	0.153525
$209$	0	0
$210$	−46.3864	−3.20097
$211$	−2.13122	−0.146719	−0.0733596	−	0.997306i	$-0.523372\pi$
$211$	−2.13122	−0.146719	−0.0733596	+	0.997306i	$0.523372\pi$
$212$	−13.2214	−0.908050
$213$	−18.1505	−1.24365
$214$	−3.88325	−0.265454
$215$	6.02985	0.411232
$216$	3.00000	0.204124
$217$	29.5297	2.00461
$218$	4.97345	0.336844
$219$	3.88325	0.262406
$220$	0	0
$221$	7.64976	0.514579
$222$	−7.76651	−0.521254
$223$	6.90981	0.462715	0.231357	−	0.972869i	$-0.425683\pi$
$223$	6.90981	0.462715	0.231357	+	0.972869i	$0.425683\pi$
$224$	4.21417	0.281571
$225$	49.5176	3.30118
$226$	−8.00000	−0.532152
$227$	2.72548	0.180896	0.0904482	−	0.995901i	$-0.471170\pi$
$227$	2.72548	0.180896	0.0904482	+	0.995901i	$0.471170\pi$
$228$	2.66908	0.176764
$229$	3.48870	0.230539	0.115270	−	0.993334i	$-0.463227\pi$
$229$	3.48870	0.230539	0.115270	+	0.993334i	$0.463227\pi$
$230$	22.4959	1.48334
$231$	0	0
$232$	−5.57889	−0.366272
$233$	17.0902	1.11962	0.559808	−	0.828622i	$-0.310875\pi$
$233$	17.0902	1.11962	0.559808	+	0.828622i	$0.310875\pi$
$234$	9.13122	0.596927
$235$	−31.2850	−2.04081
$236$	4.79306	0.312002
$237$	24.4428	1.58773
$238$	14.5596	0.943757
$239$	−6.06035	−0.392011	−0.196006	−	0.980603i	$-0.562797\pi$
$239$	−6.06035	−0.392011	−0.196006	+	0.980603i	$0.562797\pi$
$240$	−11.0072	−0.710514
$241$	−4.85341	−0.312635	−0.156318	−	0.987707i	$-0.549962\pi$
$241$	−4.85341	−0.312635	−0.156318	+	0.987707i	$0.549962\pi$
$242$	0	0
$243$	−20.6498	−1.32468
$244$	−8.90981	−0.570392
$245$	−44.3711	−2.83476
$246$	31.8075	2.02797
$247$	2.21417	0.140885
$248$	7.00724	0.444960
$249$	35.2697	2.23513
$250$	−28.8977	−1.82765
$251$	−1.58612	−0.100115	−0.0500576	−	0.998746i	$-0.515941\pi$
$251$	−1.58612	−0.100115	−0.0500576	+	0.998746i	$0.515941\pi$
$252$	17.3792	1.09479
$253$	0	0
$254$	2.48146	0.155701
$255$	−38.0289	−2.38147
$256$	1.00000	0.0625000
$257$	19.5185	1.21753	0.608767	−	0.793349i	$-0.291665\pi$
$257$	19.5185	1.21753	0.608767	+	0.793349i	$0.291665\pi$
$258$	−3.90257	−0.242963
$259$	−12.2624	−0.761951
$260$	−9.13122	−0.566294
$261$	−23.0072	−1.42411
$262$	−11.0072	−0.680029
$263$	13.0483	0.804591	0.402295	−	0.915510i	$-0.368213\pi$
$263$	13.0483	0.804591	0.402295	+	0.915510i	$0.368213\pi$
$264$	0	0
$265$	54.5249	3.34944
$266$	4.21417	0.258388
$267$	22.0145	1.34726
$268$	1.30437	0.0796769
$269$	23.6006	1.43895	0.719477	−	0.694516i	$-0.244382\pi$
$269$	23.6006	1.43895	0.719477	+	0.694516i	$0.244382\pi$
$270$	−12.3719	−0.752933
$271$	16.4090	0.996778	0.498389	−	0.866954i	$-0.333925\pi$
$271$	16.4090	0.996778	0.498389	+	0.866954i	$0.333925\pi$
$272$	3.45490	0.209484
$273$	24.9050	1.50732
$274$	6.94360	0.419478
$275$	0	0
$276$	−14.5596	−0.876383
$277$	11.5861	0.696143	0.348071	−	0.937468i	$-0.386837\pi$
$277$	11.5861	0.696143	0.348071	+	0.937468i	$0.386837\pi$
$278$	4.37195	0.262212
$279$	28.8977	1.73006
$280$	−17.3792	−1.03861
$281$	−7.03379	−0.419601	−0.209800	−	0.977744i	$-0.567281\pi$
$281$	−7.03379	−0.419601	−0.209800	+	0.977744i	$0.567281\pi$
$282$	20.2480	1.20575
$283$	−8.33092	−0.495222	−0.247611	−	0.968860i	$-0.579645\pi$
$283$	−8.33092	−0.495222	−0.247611	+	0.968860i	$0.579645\pi$
$284$	−6.80030	−0.403524
$285$	−11.0072	−0.652012
$286$	0	0
$287$	50.2205	2.96442
$288$	4.12398	0.243008
$289$	−5.06364	−0.297861
$290$	23.0072	1.35103
$291$	5.81962	0.341152
$292$	1.45490	0.0851418
$293$	−10.2142	−0.596718	−0.298359	−	0.954454i	$-0.596439\pi$
$293$	−10.2142	−0.596718	−0.298359	+	0.954454i	$0.596439\pi$
$294$	28.7173	1.67483
$295$	−19.7665	−1.15085
$296$	−2.90981	−0.169129
$297$	0	0
$298$	−11.7665	−0.681616
$299$	−12.0781	−0.698495
$300$	32.0483	1.85031
$301$	−6.16172	−0.355156
$302$	19.5861	1.12705
$303$	16.6763	0.958029
$304$	1.00000	0.0573539
$305$	36.7439	2.10395
$306$	14.2480	0.814502
$307$	1.93242	0.110289	0.0551444	−	0.998478i	$-0.482438\pi$
$307$	1.93242	0.110289	0.0551444	+	0.998478i	$0.482438\pi$
$308$	0	0
$309$	1.61597	0.0919294
$310$	−28.8977	−1.64128
$311$	−27.7173	−1.57171	−0.785853	−	0.618413i	$-0.787776\pi$
$311$	−27.7173	−1.57171	−0.785853	+	0.618413i	$0.787776\pi$
$312$	5.90981	0.334577
$313$	11.5258	0.651476	0.325738	−	0.945460i	$-0.394387\pi$
$313$	11.5258	0.651476	0.325738	+	0.945460i	$0.394387\pi$
$314$	20.6199	1.16365
$315$	−71.6715	−4.03823
$316$	9.15777	0.515165
$317$	−33.8977	−1.90389	−0.951943	−	0.306275i	$-0.900917\pi$
$317$	−33.8977	−1.90389	−0.951943	+	0.306275i	$0.900917\pi$
$318$	−35.2890	−1.97891
$319$	0	0
$320$	−4.12398	−0.230538
$321$	−10.3647	−0.578502
$322$	−22.9879	−1.28107
$323$	3.45490	0.192236
$324$	−4.36471	−0.242484
$325$	26.5861	1.47473
$326$	−17.8341	−0.987739
$327$	13.2745	0.734083
$328$	11.9170	0.658009
$329$	31.9693	1.76252
$330$	0	0
$331$	−16.2697	−0.894262	−0.447131	−	0.894468i	$-0.647554\pi$
$331$	−16.2697	−0.894262	−0.447131	+	0.894468i	$0.647554\pi$
$332$	13.2142	0.725222
$333$	−12.0000	−0.657596
$334$	−4.18038	−0.228740
$335$	−5.37919	−0.293896
$336$	11.2480	0.613627
$337$	6.82685	0.371882	0.185941	−	0.982561i	$-0.440467\pi$
$337$	6.82685	0.371882	0.185941	+	0.982561i	$0.440467\pi$
$338$	−8.09743	−0.440442
$339$	−21.3526	−1.15972
$340$	−14.2480	−0.772704
$341$	0	0
$342$	4.12398	0.222999
$343$	15.8422	0.855400
$344$	−1.46214	−0.0788334
$345$	60.0434	3.23263
$346$	−23.7101	−1.27466
$347$	5.09019	0.273256	0.136628	−	0.990622i	$-0.456374\pi$
$347$	5.09019	0.273256	0.136628	+	0.990622i	$0.456374\pi$
$348$	−14.8905	−0.798214
$349$	−30.9919	−1.65896	−0.829478	−	0.558539i	$-0.811362\pi$
$349$	−30.9919	−1.65896	−0.829478	+	0.558539i	$0.811362\pi$
$350$	50.6006	2.70472
$351$	6.64252	0.354552
$352$	0	0
$353$	−17.8301	−0.949003	−0.474501	−	0.880255i	$-0.657372\pi$
$353$	−17.8301	−0.949003	−0.474501	+	0.880255i	$0.657372\pi$
$354$	12.7931	0.679944
$355$	28.0443	1.48844
$356$	8.24797	0.437141
$357$	38.8606	2.05672
$358$	−10.8269	−0.572217
$359$	−24.4090	−1.28826	−0.644130	−	0.764916i	$-0.722780\pi$
$359$	−24.4090	−1.28826	−0.644130	+	0.764916i	$0.722780\pi$
$360$	−17.0072	−0.896360
$361$	1.00000	0.0526316
$362$	−17.8341	−0.937339
$363$	0	0
$364$	9.33092	0.489073
$365$	−6.00000	−0.314054
$366$	−23.7810	−1.24305
$367$	−26.6232	−1.38972	−0.694860	−	0.719145i	$-0.744534\pi$
$367$	−26.6232	−1.38972	−0.694860	+	0.719145i	$0.744534\pi$
$368$	−5.45490	−0.284357
$369$	49.1457	2.55842
$370$	12.0000	0.623850
$371$	−55.7173	−2.89270
$372$	18.7029	0.969699
$373$	12.6160	0.653230	0.326615	−	0.945157i	$-0.394092\pi$
$373$	12.6160	0.653230	0.326615	+	0.945157i	$0.394092\pi$
$374$	0	0
$375$	−77.1303	−3.98299
$376$	7.58612	0.391225
$377$	−12.3526	−0.636193
$378$	12.6425	0.650261
$379$	−22.8905	−1.17581	−0.587903	−	0.808932i	$-0.700046\pi$
$379$	−22.8905	−1.17581	−0.587903	+	0.808932i	$0.700046\pi$
$380$	−4.12398	−0.211556
$381$	6.62321	0.339317
$382$	0.612679	0.0313474
$383$	−0.0298464	−0.00152508	−0.000762539	−	1.00000i	$-0.500243\pi$
$383$	−0.0298464	−0.00152508	−0.000762539		1.00000i	$0.500243\pi$
$384$	2.66908	0.136206
$385$	0	0
$386$	−11.6425	−0.592589
$387$	−6.02985	−0.306514
$388$	2.18038	0.110692
$389$	9.22865	0.467911	0.233956	−	0.972247i	$-0.424833\pi$
$389$	9.22865	0.467911	0.233956	+	0.972247i	$0.424833\pi$
$390$	−24.3719	−1.23412
$391$	−18.8462	−0.953092
$392$	10.7593	0.543425
$393$	−29.3792	−1.48198
$394$	12.9098	0.650387
$395$	−37.7665	−1.90024
$396$	0	0
$397$	17.2697	0.866740	0.433370	−	0.901216i	$-0.357324\pi$
$397$	17.2697	0.866740	0.433370	+	0.901216i	$0.357324\pi$
$398$	−18.3647	−0.920540
$399$	11.2480	0.563103
$400$	12.0072	0.600362
$401$	−13.2850	−0.663424	−0.331712	−	0.943381i	$-0.607626\pi$
$401$	−13.2850	−0.663424	−0.331712	+	0.943381i	$0.607626\pi$
$402$	3.48146	0.173639
$403$	15.5152	0.772870
$404$	6.24797	0.310848
$405$	18.0000	0.894427
$406$	−23.5104	−1.16680
$407$	0	0
$408$	9.22141	0.456528
$409$	−1.94360	−0.0961048	−0.0480524	−	0.998845i	$-0.515301\pi$
$409$	−1.94360	−0.0961048	−0.0480524	+	0.998845i	$0.515301\pi$
$410$	−49.1457	−2.42713
$411$	18.5330	0.914166
$412$	0.605441	0.0298280
$413$	20.1988	0.993918
$414$	−22.4959	−1.10561
$415$	−54.4950	−2.67506
$416$	2.21417	0.108559
$417$	11.6691	0.571437
$418$	0	0
$419$	−2.49593	−0.121934	−0.0609671	−	0.998140i	$-0.519418\pi$
$419$	−2.49593	−0.121934	−0.0609671	+	0.998140i	$0.519418\pi$
$420$	−46.3864	−2.26343
$421$	15.6353	0.762017	0.381009	−	0.924571i	$-0.375577\pi$
$421$	15.6353	0.762017	0.381009	+	0.924571i	$0.375577\pi$
$422$	−2.13122	−0.103746
$423$	31.2850	1.52113
$424$	−13.2214	−0.642089
$425$	41.4839	2.01226
$426$	−18.1505	−0.879396
$427$	−37.5475	−1.81705
$428$	−3.88325	−0.187704
$429$	0	0
$430$	6.02985	0.290785
$431$	−0.262441	−0.0126413	−0.00632067	−	0.999980i	$-0.502012\pi$
$431$	−0.262441	−0.0126413	−0.00632067	+	0.999980i	$0.502012\pi$
$432$	3.00000	0.144338
$433$	−32.2769	−1.55113	−0.775565	−	0.631268i	$-0.782535\pi$
$433$	−32.2769	−1.55113	−0.775565	+	0.631268i	$0.782535\pi$
$434$	29.5297	1.41747
$435$	61.4081	2.94429
$436$	4.97345	0.238185
$437$	−5.45490	−0.260943
$438$	3.88325	0.185549
$439$	17.0902	0.815670	0.407835	−	0.913056i	$-0.366284\pi$
$439$	17.0902	0.815670	0.407835	+	0.913056i	$0.366284\pi$
$440$	0	0
$441$	44.3711	2.11291
$442$	7.64976	0.363862
$443$	16.2624	0.772652	0.386326	−	0.922362i	$-0.373744\pi$
$443$	16.2624	0.772652	0.386326	+	0.922362i	$0.373744\pi$
$444$	−7.76651	−0.368582
$445$	−34.0145	−1.61244
$446$	6.90981	0.327189
$447$	−31.4057	−1.48544
$448$	4.21417	0.199101
$449$	−27.3382	−1.29017	−0.645084	−	0.764112i	$-0.723178\pi$
$449$	−27.3382	−1.29017	−0.645084	+	0.764112i	$0.723178\pi$
$450$	49.5176	2.33428
$451$	0	0
$452$	−8.00000	−0.376288
$453$	52.2769	2.45618
$454$	2.72548	0.127913
$455$	−38.4806	−1.80400
$456$	2.66908	0.124991
$457$	−10.4920	−0.490794	−0.245397	−	0.969423i	$-0.578918\pi$
$457$	−10.4920	−0.490794	−0.245397	+	0.969423i	$0.578918\pi$
$458$	3.48870	0.163016
$459$	10.3647	0.483783
$460$	22.4959	1.04888
$461$	−6.18038	−0.287849	−0.143925	−	0.989589i	$-0.545972\pi$
$461$	−6.18038	−0.287849	−0.143925	+	0.989589i	$0.545972\pi$
$462$	0	0
$463$	−11.1433	−0.517873	−0.258937	−	0.965894i	$-0.583372\pi$
$463$	−11.1433	−0.517873	−0.258937	+	0.965894i	$0.583372\pi$
$464$	−5.57889	−0.258993
$465$	−77.1303	−3.57683
$466$	17.0902	0.791688
$467$	−34.5104	−1.59695	−0.798476	−	0.602027i	$-0.794360\pi$
$467$	−34.5104	−1.59695	−0.798476	+	0.602027i	$0.794360\pi$
$468$	9.13122	0.422091
$469$	5.49683	0.253820
$470$	−31.2850	−1.44307
$471$	55.0362	2.53593
$472$	4.79306	0.220619
$473$	0	0
$474$	24.4428	1.12270
$475$	12.0072	0.550930
$476$	14.5596	0.667337
$477$	−54.5249	−2.49652
$478$	−6.06035	−0.277194
$479$	−41.6980	−1.90523	−0.952616	−	0.304176i	$-0.901619\pi$
$479$	−41.6980	−1.90523	−0.952616	+	0.304176i	$0.901619\pi$
$480$	−11.0072	−0.502409
$481$	−6.44282	−0.293768
$482$	−4.85341	−0.221067
$483$	−61.3566	−2.79182
$484$	0	0
$485$	−8.99187	−0.408300
$486$	−20.6498	−0.936692
$487$	9.10137	0.412423	0.206211	−	0.978507i	$-0.433887\pi$
$487$	9.10137	0.412423	0.206211	+	0.978507i	$0.433887\pi$
$488$	−8.90981	−0.403328
$489$	−47.6006	−2.15257
$490$	−44.3711	−2.00448
$491$	30.8413	1.39185	0.695925	−	0.718115i	$-0.254995\pi$
$491$	30.8413	1.39185	0.695925	+	0.718115i	$0.254995\pi$
$492$	31.8075	1.43399
$493$	−19.2745	−0.868081
$494$	2.21417	0.0996204
$495$	0	0
$496$	7.00724	0.314634
$497$	−28.6577	−1.28547
$498$	35.2697	1.58047
$499$	21.7665	0.974403	0.487201	−	0.873290i	$-0.338018\pi$
$499$	21.7665	0.974403	0.487201	+	0.873290i	$0.338018\pi$
$500$	−28.8977	−1.29235
$501$	−11.1578	−0.498493
$502$	−1.58612	−0.0707922
$503$	−38.0893	−1.69832	−0.849159	−	0.528138i	$-0.822891\pi$
$503$	−38.0893	−1.69832	−0.849159	+	0.528138i	$0.822891\pi$
$504$	17.3792	0.774131
$505$	−25.7665	−1.14659
$506$	0	0
$507$	−21.6127	−0.959853
$508$	2.48146	0.110097
$509$	14.9388	0.662149	0.331074	−	0.943605i	$-0.392589\pi$
$509$	14.9388	0.662149	0.331074	+	0.943605i	$0.392589\pi$
$510$	−38.0289	−1.68395
$511$	6.13122	0.271229
$512$	1.00000	0.0441942
$513$	3.00000	0.132453
$514$	19.5185	0.860926
$515$	−2.49683	−0.110023
$516$	−3.90257	−0.171801
$517$	0	0
$518$	−12.2624	−0.538781
$519$	−63.2842	−2.77787
$520$	−9.13122	−0.400431
$521$	7.22536	0.316549	0.158274	−	0.987395i	$-0.449407\pi$
$521$	7.22536	0.316549	0.158274	+	0.987395i	$0.449407\pi$
$522$	−23.0072	−1.00700
$523$	−4.90586	−0.214518	−0.107259	−	0.994231i	$-0.534207\pi$
$523$	−4.90586	−0.214518	−0.107259	+	0.994231i	$0.534207\pi$
$524$	−11.0072	−0.480853
$525$	135.057	5.89437
$526$	13.0483	0.568931
$527$	24.2093	1.05458
$528$	0	0
$529$	6.75598	0.293738
$530$	54.5249	2.36841
$531$	19.7665	0.857793
$532$	4.21417	0.182708
$533$	26.3864	1.14292
$534$	22.0145	0.952659
$535$	16.0145	0.692366
$536$	1.30437	0.0563401
$537$	−28.8977	−1.24703
$538$	23.6006	1.01749
$539$	0	0
$540$	−12.3719	−0.532404
$541$	−30.8712	−1.32726	−0.663628	−	0.748063i	$-0.730984\pi$
$541$	−30.8712	−1.32726	−0.663628	+	0.748063i	$0.730984\pi$
$542$	16.4090	0.704828
$543$	−47.6006	−2.04274
$544$	3.45490	0.148128
$545$	−20.5104	−0.878569
$546$	24.9050	1.06583
$547$	−2.54115	−0.108652	−0.0543259	−	0.998523i	$-0.517301\pi$
$547$	−2.54115	−0.108652	−0.0543259	+	0.998523i	$0.517301\pi$
$548$	6.94360	0.296616
$549$	−36.7439	−1.56819
$550$	0	0
$551$	−5.57889	−0.237669
$552$	−14.5596	−0.619696
$553$	38.5925	1.64112
$554$	11.5861	0.492247
$555$	32.0289	1.35955
$556$	4.37195	0.185412
$557$	−23.9017	−1.01275	−0.506373	−	0.862314i	$-0.669014\pi$
$557$	−23.9017	−1.01275	−0.506373	+	0.862314i	$0.669014\pi$
$558$	28.8977	1.22334
$559$	−3.23744	−0.136929
$560$	−17.3792	−0.734405
$561$	0	0
$562$	−7.03379	−0.296703
$563$	10.6232	0.447715	0.223857	−	0.974622i	$-0.428135\pi$
$563$	10.6232	0.447715	0.223857	+	0.974622i	$0.428135\pi$
$564$	20.2480	0.852593
$565$	32.9919	1.38798
$566$	−8.33092	−0.350175
$567$	−18.3937	−0.772461
$568$	−6.80030	−0.285334
$569$	−29.3373	−1.22988	−0.614941	−	0.788573i	$-0.710820\pi$
$569$	−29.3373	−1.22988	−0.614941	+	0.788573i	$0.710820\pi$
$570$	−11.0072	−0.461042
$571$	−19.8639	−0.831280	−0.415640	−	0.909529i	$-0.636442\pi$
$571$	−19.8639	−0.831280	−0.415640	+	0.909529i	$0.636442\pi$
$572$	0	0
$573$	1.63529	0.0683151
$574$	50.2205	2.09616
$575$	−65.4983	−2.73147
$576$	4.12398	0.171833
$577$	−3.17709	−0.132264	−0.0661320	−	0.997811i	$-0.521066\pi$
$577$	−3.17709	−0.132264	−0.0661320	+	0.997811i	$0.521066\pi$
$578$	−5.06364	−0.210620
$579$	−31.0748	−1.29143
$580$	23.0072	0.955324
$581$	55.6868	2.31028
$582$	5.81962	0.241231
$583$	0	0
$584$	1.45490	0.0602044
$585$	−37.6570	−1.55693
$586$	−10.2142	−0.421944
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	28.7173	1.18428
$589$	7.00724	0.288728
$590$	−19.7665	−0.813774
$591$	34.4573	1.41738
$592$	−2.90981	−0.119592
$593$	3.45885	0.142038	0.0710190	−	0.997475i	$-0.477375\pi$
$593$	3.45885	0.142038	0.0710190	+	0.997475i	$0.477375\pi$
$594$	0	0
$595$	−60.0434	−2.46154
$596$	−11.7665	−0.481975
$597$	−49.0169	−2.00613
$598$	−12.0781	−0.493911
$599$	40.3864	1.65014	0.825072	−	0.565027i	$-0.191134\pi$
$599$	40.3864	1.65014	0.825072	+	0.565027i	$0.191134\pi$
$600$	32.0483	1.30836
$601$	−18.3309	−0.747734	−0.373867	−	0.927482i	$-0.621968\pi$
$601$	−18.3309	−0.747734	−0.373867	+	0.927482i	$0.621968\pi$
$602$	−6.16172	−0.251133
$603$	5.37919	0.219057
$604$	19.5861	0.796948
$605$	0	0
$606$	16.6763	0.677429
$607$	−12.4283	−0.504451	−0.252226	−	0.967668i	$-0.581162\pi$
$607$	−12.4283	−0.504451	−0.252226	+	0.967668i	$0.581162\pi$
$608$	1.00000	0.0405554
$609$	−62.7511	−2.54280
$610$	36.7439	1.48772
$611$	16.7970	0.679534
$612$	14.2480	0.575940
$613$	0.0531082	0.00214502	0.00107251	−	0.999999i	$-0.499659\pi$
$613$	0.0531082	0.00214502	0.00107251	+	0.999999i	$0.499659\pi$
$614$	1.93242	0.0779860
$615$	−131.174	−5.28944
$616$	0	0
$617$	−17.4887	−0.704068	−0.352034	−	0.935987i	$-0.614510\pi$
$617$	−17.4887	−0.704068	−0.352034	+	0.935987i	$0.614510\pi$
$618$	1.61597	0.0650039
$619$	25.2995	1.01687	0.508437	−	0.861099i	$-0.330224\pi$
$619$	25.2995	1.01687	0.508437	+	0.861099i	$0.330224\pi$
$620$	−28.8977	−1.16056
$621$	−16.3647	−0.656693
$622$	−27.7173	−1.11136
$623$	34.7584	1.39256
$624$	5.90981	0.236582
$625$	59.1376	2.36550
$626$	11.5258	0.460663
$627$	0	0
$628$	20.6199	0.822824
$629$	−10.0531	−0.400844
$630$	−71.6715	−2.85546
$631$	9.02261	0.359184	0.179592	−	0.983741i	$-0.442522\pi$
$631$	9.02261	0.359184	0.179592	+	0.983741i	$0.442522\pi$
$632$	9.15777	0.364277
$633$	−5.68840	−0.226093
$634$	−33.8977	−1.34625
$635$	−10.2335	−0.406104
$636$	−35.2890	−1.39930
$637$	23.8229	0.943898
$638$	0	0
$639$	−28.0443	−1.10942
$640$	−4.12398	−0.163015
$641$	30.4573	1.20299	0.601495	−	0.798876i	$-0.294572\pi$
$641$	30.4573	1.20299	0.601495	+	0.798876i	$0.294572\pi$
$642$	−10.3647	−0.409063
$643$	−36.7584	−1.44961	−0.724804	−	0.688955i	$-0.758070\pi$
$643$	−36.7584	−1.44961	−0.724804	+	0.688955i	$0.758070\pi$
$644$	−22.9879	−0.905851
$645$	16.0941	0.633706
$646$	3.45490	0.135931
$647$	−41.0700	−1.61463	−0.807314	−	0.590122i	$-0.799079\pi$
$647$	−41.0700	−1.61463	−0.807314	+	0.590122i	$0.799079\pi$
$648$	−4.36471	−0.171462
$649$	0	0
$650$	26.5861	1.04279
$651$	78.8172	3.08909
$652$	−17.8341	−0.698437
$653$	28.4806	1.11453	0.557265	−	0.830335i	$-0.311851\pi$
$653$	28.4806	1.11453	0.557265	+	0.830335i	$0.311851\pi$
$654$	13.2745	0.519075
$655$	45.3937	1.77368
$656$	11.9170	0.465282
$657$	6.00000	0.234082
$658$	31.9693	1.24629
$659$	10.5740	0.411906	0.205953	−	0.978562i	$-0.433971\pi$
$659$	10.5740	0.411906	0.205953	+	0.978562i	$0.433971\pi$
$660$	0	0
$661$	7.70287	0.299607	0.149803	−	0.988716i	$-0.452136\pi$
$661$	7.70287	0.299607	0.149803	+	0.988716i	$0.452136\pi$
$662$	−16.2697	−0.632339
$663$	20.4178	0.792962
$664$	13.2142	0.512809
$665$	−17.3792	−0.673936
$666$	−12.0000	−0.464991
$667$	30.4323	1.17834
$668$	−4.18038	−0.161744
$669$	18.4428	0.713041
$670$	−5.37919	−0.207816
$671$	0	0
$672$	11.2480	0.433900
$673$	31.5176	1.21492	0.607458	−	0.794352i	$-0.292189\pi$
$673$	31.5176	1.21492	0.607458	+	0.794352i	$0.292189\pi$
$674$	6.82685	0.262961
$675$	36.0217	1.38648
$676$	−8.09743	−0.311440
$677$	−11.9928	−0.460919	−0.230460	−	0.973082i	$-0.574023\pi$
$677$	−11.9928	−0.460919	−0.230460	+	0.973082i	$0.574023\pi$
$678$	−21.3526	−0.820043
$679$	9.18852	0.352623
$680$	−14.2480	−0.546385
$681$	7.27452	0.278760
$682$	0	0
$683$	39.1722	1.49888	0.749442	−	0.662070i	$-0.230322\pi$
$683$	39.1722	1.49888	0.749442	+	0.662070i	$0.230322\pi$
$684$	4.12398	0.157684
$685$	−28.6353	−1.09410
$686$	15.8422	0.604859
$687$	9.31160	0.355260
$688$	−1.46214	−0.0557436
$689$	−29.2745	−1.11527
$690$	60.0434	2.28581
$691$	−49.1722	−1.87060	−0.935300	−	0.353855i	$-0.884871\pi$
$691$	−49.1722	−1.87060	−0.935300	+	0.353855i	$0.884871\pi$
$692$	−23.7101	−0.901323
$693$	0	0
$694$	5.09019	0.193221
$695$	−18.0298	−0.683911
$696$	−14.8905	−0.564423
$697$	41.1722	1.55951
$698$	−30.9919	−1.17306
$699$	45.6151	1.72532
$700$	50.6006	1.91252
$701$	−37.8486	−1.42952	−0.714760	−	0.699370i	$-0.753464\pi$
$701$	−37.8486	−1.42952	−0.714760	+	0.699370i	$0.753464\pi$
$702$	6.64252	0.250706
$703$	−2.90981	−0.109745
$704$	0	0
$705$	−83.5023	−3.14488
$706$	−17.8301	−0.671046
$707$	26.3300	0.990242
$708$	12.7931	0.480793
$709$	27.7511	1.04222	0.521108	−	0.853491i	$-0.325519\pi$
$709$	27.7511	1.04222	0.521108	+	0.853491i	$0.325519\pi$
$710$	28.0443	1.05248
$711$	37.7665	1.41635
$712$	8.24797	0.309106
$713$	−38.2238	−1.43149
$714$	38.8606	1.45432
$715$	0	0
$716$	−10.8269	−0.404619
$717$	−16.1755	−0.604087
$718$	−24.4090	−0.910937
$719$	−17.0410	−0.635523	−0.317762	−	0.948171i	$-0.602931\pi$
$719$	−17.0410	−0.635523	−0.317762	+	0.948171i	$0.602931\pi$
$720$	−17.0072	−0.633822
$721$	2.55144	0.0950204
$722$	1.00000	0.0372161
$723$	−12.9541	−0.481769
$724$	−17.8341	−0.662799
$725$	−66.9870	−2.48784
$726$	0	0
$727$	13.5225	0.501521	0.250761	−	0.968049i	$-0.419319\pi$
$727$	13.5225	0.501521	0.250761	+	0.968049i	$0.419319\pi$
$728$	9.33092	0.345827
$729$	−42.0217	−1.55636
$730$	−6.00000	−0.222070
$731$	−5.05156	−0.186839
$732$	−23.7810	−0.878970
$733$	25.6682	0.948076	0.474038	−	0.880504i	$-0.342796\pi$
$733$	25.6682	0.948076	0.474038	+	0.880504i	$0.342796\pi$
$734$	−26.6232	−0.982681
$735$	−118.430	−4.36835
$736$	−5.45490	−0.201070
$737$	0	0
$738$	49.1457	1.80908
$739$	−21.3493	−0.785348	−0.392674	−	0.919678i	$-0.628450\pi$
$739$	−21.3493	−0.785348	−0.392674	+	0.919678i	$0.628450\pi$
$740$	12.0000	0.441129
$741$	5.90981	0.217102
$742$	−55.7173	−2.04545
$743$	7.55718	0.277246	0.138623	−	0.990345i	$-0.455732\pi$
$743$	7.55718	0.277246	0.138623	+	0.990345i	$0.455732\pi$
$744$	18.7029	0.685680
$745$	48.5249	1.77781
$746$	12.6160	0.461904
$747$	54.4950	1.99387
$748$	0	0
$749$	−16.3647	−0.597954
$750$	−77.1303	−2.81640
$751$	−53.5861	−1.95539	−0.977693	−	0.210040i	$-0.932640\pi$
$751$	−53.5861	−1.95539	−0.977693	+	0.210040i	$0.932640\pi$
$752$	7.58612	0.276637
$753$	−4.23349	−0.154277
$754$	−12.3526	−0.449856
$755$	−80.7728	−2.93962
$756$	12.6425	0.459804
$757$	−13.9436	−0.506789	−0.253394	−	0.967363i	$-0.581547\pi$
$757$	−13.9436	−0.506789	−0.253394	+	0.967363i	$0.581547\pi$
$758$	−22.8905	−0.831420
$759$	0	0
$760$	−4.12398	−0.149593
$761$	45.7994	1.66023	0.830114	−	0.557594i	$-0.188276\pi$
$761$	45.7994	1.66023	0.830114	+	0.557594i	$0.188276\pi$
$762$	6.62321	0.239934
$763$	20.9590	0.758766
$764$	0.612679	0.0221659
$765$	−58.7584	−2.12441
$766$	−0.0298464	−0.00107839
$767$	10.6127	0.383202
$768$	2.66908	0.0963121
$769$	40.8751	1.47399	0.736997	−	0.675896i	$-0.236243\pi$
$769$	40.8751	1.47399	0.736997	+	0.675896i	$0.236243\pi$
$770$	0	0
$771$	52.0965	1.87621
$772$	−11.6425	−0.419024
$773$	37.4983	1.34872	0.674361	−	0.738402i	$-0.264419\pi$
$773$	37.4983	1.34872	0.674361	+	0.738402i	$0.264419\pi$
$774$	−6.02985	−0.216738
$775$	84.1376	3.02231
$776$	2.18038	0.0782712
$777$	−32.7294	−1.17416
$778$	9.22865	0.330863
$779$	11.9170	0.426972
$780$	−24.3719	−0.872656
$781$	0	0
$782$	−18.8462	−0.673938
$783$	−16.7367	−0.598119
$784$	10.7593	0.384260
$785$	−85.0362	−3.03507
$786$	−29.3792	−1.04792
$787$	−8.80754	−0.313955	−0.156977	−	0.987602i	$-0.550175\pi$
$787$	−8.80754	−0.313955	−0.156977	+	0.987602i	$0.550175\pi$
$788$	12.9098	0.459893
$789$	34.8269	1.23987
$790$	−37.7665	−1.34367
$791$	−33.7134	−1.19871
$792$	0	0
$793$	−19.7279	−0.700557
$794$	17.2697	0.612878
$795$	145.531	5.16146
$796$	−18.3647	−0.650920
$797$	−26.1312	−0.925615	−0.462808	−	0.886459i	$-0.653158\pi$
$797$	−26.1312	−0.925615	−0.462808	+	0.886459i	$0.653158\pi$
$798$	11.2480	0.398174
$799$	26.2093	0.927220
$800$	12.0072	0.424520
$801$	34.0145	1.20184
$802$	−13.2850	−0.469111
$803$	0	0
$804$	3.48146	0.122782
$805$	94.8018	3.34132
$806$	15.5152	0.546501
$807$	62.9919	2.21742
$808$	6.24797	0.219803
$809$	6.01053	0.211319	0.105659	−	0.994402i	$-0.466305\pi$
$809$	6.01053	0.211319	0.105659	+	0.994402i	$0.466305\pi$
$810$	18.0000	0.632456
$811$	40.2809	1.41445	0.707226	−	0.706987i	$-0.249946\pi$
$811$	40.2809	1.41445	0.707226	+	0.706987i	$0.249946\pi$
$812$	−23.5104	−0.825054
$813$	43.7970	1.53603
$814$	0	0
$815$	73.5475	2.57626
$816$	9.22141	0.322814
$817$	−1.46214	−0.0511539
$818$	−1.94360	−0.0679564
$819$	38.4806	1.34462
$820$	−49.1457	−1.71624
$821$	−50.1496	−1.75023	−0.875117	−	0.483911i	$-0.839216\pi$
$821$	−50.1496	−1.75023	−0.875117	+	0.483911i	$0.839216\pi$
$822$	18.5330	0.646413
$823$	−42.7400	−1.48982	−0.744911	−	0.667164i	$-0.767508\pi$
$823$	−42.7400	−1.48982	−0.744911	+	0.667164i	$0.767508\pi$
$824$	0.605441	0.0210915
$825$	0	0
$826$	20.1988	0.702806
$827$	−49.0555	−1.70583	−0.852913	−	0.522052i	$-0.825167\pi$
$827$	−49.0555	−1.70583	−0.852913	+	0.522052i	$0.825167\pi$
$828$	−22.4959	−0.781788
$829$	49.9653	1.73537	0.867683	−	0.497117i	$-0.165608\pi$
$829$	49.9653	1.73537	0.867683	+	0.497117i	$0.165608\pi$
$830$	−54.4950	−1.89155
$831$	30.9243	1.07275
$832$	2.21417	0.0767627
$833$	37.1722	1.28794
$834$	11.6691	0.404067
$835$	17.2398	0.596609
$836$	0	0
$837$	21.0217	0.726617
$838$	−2.49593	−0.0862206
$839$	−52.0257	−1.79613	−0.898063	−	0.439868i	$-0.855025\pi$
$839$	−52.0257	−1.79613	−0.898063	+	0.439868i	$0.855025\pi$
$840$	−46.3864	−1.60048
$841$	2.12398	0.0732408
$842$	15.6353	0.538828
$843$	−18.7737	−0.646602
$844$	−2.13122	−0.0733596
$845$	33.3937	1.14878
$846$	31.2850	1.07560
$847$	0	0
$848$	−13.2214	−0.454025
$849$	−22.2359	−0.763134
$850$	41.4839	1.42288
$851$	15.8727	0.544110
$852$	−18.1505	−0.621827
$853$	−16.2335	−0.555824	−0.277912	−	0.960607i	$-0.589642\pi$
$853$	−16.2335	−0.555824	−0.277912	+	0.960607i	$0.589642\pi$
$854$	−37.5475	−1.28485
$855$	−17.0072	−0.581635
$856$	−3.88325	−0.132727
$857$	−9.60150	−0.327981	−0.163990	−	0.986462i	$-0.552437\pi$
$857$	−9.60150	−0.327981	−0.163990	+	0.986462i	$0.552437\pi$
$858$	0	0
$859$	19.4057	0.662115	0.331058	−	0.943611i	$-0.392595\pi$
$859$	19.4057	0.662115	0.331058	+	0.943611i	$0.392595\pi$
$860$	6.02985	0.205616
$861$	134.043	4.56816
$862$	−0.262441	−0.00893877
$863$	1.54839	0.0527077	0.0263539	−	0.999653i	$-0.491610\pi$
$863$	1.54839	0.0527077	0.0263539	+	0.999653i	$0.491610\pi$
$864$	3.00000	0.102062
$865$	97.7801	3.32462
$866$	−32.2769	−1.09681
$867$	−13.5152	−0.459002
$868$	29.5297	1.00230
$869$	0	0
$870$	61.4081	2.08193
$871$	2.88810	0.0978594
$872$	4.97345	0.168422
$873$	8.99187	0.304329
$874$	−5.45490	−0.184515
$875$	−121.780	−4.11692
$876$	3.88325	0.131203
$877$	12.1200	0.409265	0.204632	−	0.978839i	$-0.434400\pi$
$877$	12.1200	0.409265	0.204632	+	0.978839i	$0.434400\pi$
$878$	17.0902	0.576766
$879$	−27.2624	−0.919539
$880$	0	0
$881$	53.6836	1.80864	0.904322	−	0.426850i	$-0.140377\pi$
$881$	53.6836	1.80864	0.904322	+	0.426850i	$0.140377\pi$
$882$	44.3711	1.49405
$883$	34.4283	1.15861	0.579303	−	0.815112i	$-0.303325\pi$
$883$	34.4283	1.15861	0.579303	+	0.815112i	$0.303325\pi$
$884$	7.64976	0.257289
$885$	−52.7584	−1.77345
$886$	16.2624	0.546347
$887$	27.5330	0.924468	0.462234	−	0.886758i	$-0.347048\pi$
$887$	27.5330	0.924468	0.462234	+	0.886758i	$0.347048\pi$
$888$	−7.76651	−0.260627
$889$	10.4573	0.350727
$890$	−34.0145	−1.14017
$891$	0	0
$892$	6.90981	0.231357
$893$	7.58612	0.253860
$894$	−31.4057	−1.05037
$895$	44.6498	1.49248
$896$	4.21417	0.140786
$897$	−32.2374	−1.07638
$898$	−27.3382	−0.912286
$899$	−39.0926	−1.30381
$900$	49.5176	1.65059
$901$	−45.6787	−1.52178
$902$	0	0
$903$	−16.4461	−0.547292
$904$	−8.00000	−0.266076
$905$	73.5475	2.44480
$906$	52.2769	1.73678
$907$	15.5677	0.516917	0.258459	−	0.966022i	$-0.416785\pi$
$907$	15.5677	0.516917	0.258459	+	0.966022i	$0.416785\pi$
$908$	2.72548	0.0904482
$909$	25.7665	0.854621
$910$	−38.4806	−1.27562
$911$	−0.623208	−0.0206478	−0.0103239	−	0.999947i	$-0.503286\pi$
$911$	−0.623208	−0.0206478	−0.0103239	+	0.999947i	$0.503286\pi$
$912$	2.66908	0.0883820
$913$	0	0
$914$	−10.4920	−0.347044
$915$	98.0724	3.24217
$916$	3.48870	0.115270
$917$	−46.3864	−1.53181
$918$	10.3647	0.342086
$919$	44.4887	1.46755	0.733773	−	0.679394i	$-0.237757\pi$
$919$	44.4887	1.46755	0.733773	+	0.679394i	$0.237757\pi$
$920$	22.4959	0.741669
$921$	5.15777	0.169954
$922$	−6.18038	−0.203540
$923$	−15.0571	−0.495609
$924$	0	0
$925$	−34.9388	−1.14878
$926$	−11.1433	−0.366192
$927$	2.49683	0.0820067
$928$	−5.57889	−0.183136
$929$	44.1224	1.44761	0.723805	−	0.690005i	$-0.242391\pi$
$929$	44.1224	1.44761	0.723805	+	0.690005i	$0.242391\pi$
$930$	−77.1303	−2.52920
$931$	10.7593	0.352621
$932$	17.0902	0.559808
$933$	−73.9798	−2.42199
$934$	−34.5104	−1.12922
$935$	0	0
$936$	9.13122	0.298463
$937$	−24.8672	−0.812377	−0.406188	−	0.913789i	$-0.633142\pi$
$937$	−24.8672	−0.812377	−0.406188	+	0.913789i	$0.633142\pi$
$938$	5.49683	0.179478
$939$	30.7632	1.00392
$940$	−31.2850	−1.02041
$941$	−29.8301	−0.972435	−0.486217	−	0.873838i	$-0.661624\pi$
$941$	−29.8301	−0.972435	−0.486217	+	0.873838i	$0.661624\pi$
$942$	55.0362	1.79318
$943$	−65.0063	−2.11690
$944$	4.79306	0.156001
$945$	−52.1376	−1.69603
$946$	0	0
$947$	9.07572	0.294921	0.147461	−	0.989068i	$-0.452890\pi$
$947$	9.07572	0.294921	0.147461	+	0.989068i	$0.452890\pi$
$948$	24.4428	0.793866
$949$	3.22141	0.104571
$950$	12.0072	0.389566
$951$	−90.4757	−2.93388
$952$	14.5596	0.471878
$953$	−1.27058	−0.0411580	−0.0205790	−	0.999788i	$-0.506551\pi$
$953$	−1.27058	−0.0411580	−0.0205790	+	0.999788i	$0.506551\pi$
$954$	−54.5249	−1.76531
$955$	−2.52668	−0.0817613
$956$	−6.06035	−0.196006
$957$	0	0
$958$	−41.6980	−1.34720
$959$	29.2615	0.944905
$960$	−11.0072	−0.355257
$961$	18.1014	0.583915
$962$	−6.44282	−0.207725
$963$	−16.0145	−0.516059
$964$	−4.85341	−0.156318
$965$	48.0136	1.54561
$966$	−61.3566	−1.97412
$967$	−6.67632	−0.214696	−0.107348	−	0.994222i	$-0.534236\pi$
$967$	−6.67632	−0.214696	−0.107348	+	0.994222i	$0.534236\pi$
$968$	0	0
$969$	9.22141	0.296234
$970$	−8.99187	−0.288711
$971$	−0.191566	−0.00614764	−0.00307382	−	0.999995i	$-0.500978\pi$
$971$	−0.191566	−0.00614764	−0.00307382	+	0.999995i	$0.500978\pi$
$972$	−20.6498	−0.662341
$973$	18.4242	0.590651
$974$	9.10137	0.291627
$975$	70.9605	2.27255
$976$	−8.90981	−0.285196
$977$	16.4959	0.527752	0.263876	−	0.964557i	$-0.414999\pi$
$977$	16.4959	0.527752	0.263876	+	0.964557i	$0.414999\pi$
$978$	−47.6006	−1.52210
$979$	0	0
$980$	−44.3711	−1.41738
$981$	20.5104	0.654847
$982$	30.8413	0.984186
$983$	−35.0072	−1.11656	−0.558279	−	0.829653i	$-0.688538\pi$
$983$	−35.0072	−1.11656	−0.558279	+	0.829653i	$0.688538\pi$
$984$	31.8075	1.01399
$985$	−53.2398	−1.69636
$986$	−19.2745	−0.613826
$987$	85.3285	2.71604
$988$	2.21417	0.0704423
$989$	7.97584	0.253617
$990$	0	0
$991$	−11.3759	−0.361367	−0.180684	−	0.983541i	$-0.557831\pi$
$991$	−11.3759	−0.361367	−0.180684	+	0.983541i	$0.557831\pi$
$992$	7.00724	0.222480
$993$	−43.4251	−1.37805
$994$	−28.6577	−0.908966
$995$	75.7358	2.40099
$996$	35.2697	1.11756
$997$	−8.89533	−0.281718	−0.140859	−	0.990030i	$-0.544986\pi$
$997$	−8.89533	−0.281718	−0.140859	+	0.990030i	$0.544986\pi$
$998$	21.7665	0.689007
$999$	−8.72942	−0.276187

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.bo.1.3		3
11.10	odd	2		418.2.a.g.1.3	✓	3
33.32	even	2		3762.2.a.bg.1.3		3
44.43	even	2		3344.2.a.q.1.1		3
209.208	even	2		7942.2.a.bi.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.3	✓	3	11.10	odd	2
3344.2.a.q.1.1		3	44.43	even	2
3762.2.a.bg.1.3		3	33.32	even	2
4598.2.a.bo.1.3		3	1.1	even	1	trivial
7942.2.a.bi.1.1		3	209.208	even	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:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.66908 q^{3} +1.00000 q^{4} -4.12398 q^{5} +2.66908 q^{6} +4.21417 q^{7} +1.00000 q^{8} +4.12398 q^{9} -4.12398 q^{10} +2.66908 q^{12} +2.21417 q^{13} +4.21417 q^{14} -11.0072 q^{15} +1.00000 q^{16} +3.45490 q^{17} +4.12398 q^{18} +1.00000 q^{19} -4.12398 q^{20} +11.2480 q^{21} -5.45490 q^{23} +2.66908 q^{24} +12.0072 q^{25} +2.21417 q^{26} +3.00000 q^{27} +4.21417 q^{28} -5.57889 q^{29} -11.0072 q^{30} +7.00724 q^{31} +1.00000 q^{32} +3.45490 q^{34} -17.3792 q^{35} +4.12398 q^{36} -2.90981 q^{37} +1.00000 q^{38} +5.90981 q^{39} -4.12398 q^{40} +11.9170 q^{41} +11.2480 q^{42} -1.46214 q^{43} -17.0072 q^{45} -5.45490 q^{46} +7.58612 q^{47} +2.66908 q^{48} +10.7593 q^{49} +12.0072 q^{50} +9.22141 q^{51} +2.21417 q^{52} -13.2214 q^{53} +3.00000 q^{54} +4.21417 q^{56} +2.66908 q^{57} -5.57889 q^{58} +4.79306 q^{59} -11.0072 q^{60} -8.90981 q^{61} +7.00724 q^{62} +17.3792 q^{63} +1.00000 q^{64} -9.13122 q^{65} +1.30437 q^{67} +3.45490 q^{68} -14.5596 q^{69} -17.3792 q^{70} -6.80030 q^{71} +4.12398 q^{72} +1.45490 q^{73} -2.90981 q^{74} +32.0483 q^{75} +1.00000 q^{76} +5.90981 q^{78} +9.15777 q^{79} -4.12398 q^{80} -4.36471 q^{81} +11.9170 q^{82} +13.2142 q^{83} +11.2480 q^{84} -14.2480 q^{85} -1.46214 q^{86} -14.8905 q^{87} +8.24797 q^{89} -17.0072 q^{90} +9.33092 q^{91} -5.45490 q^{92} +18.7029 q^{93} +7.58612 q^{94} -4.12398 q^{95} +2.66908 q^{96} +2.18038 q^{97} +10.7593 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 6 q^{14} - 9 q^{15} + 3 q^{16} + 9 q^{17} + 3 q^{18} + 3 q^{19} - 3 q^{20} + 15 q^{21} - 15 q^{23} + 12 q^{25} + 9 q^{27} + 6 q^{28}+ \cdots + 27 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^10 + 6 * q^14 - 9 * q^15 + 3 * q^16 + 9 * q^17 + 3 * q^18 + 3 * q^19 - 3 * q^20 + 15 * q^21 - 15 * q^23 + 12 * q^25 + 9 * q^27 + 6 * q^28 - 6 * q^29 - 9 * q^30 - 3 * q^31 + 3 * q^32 + 9 * q^34 + 3 * q^36 - 6 * q^37 + 3 * q^38 + 15 * q^39 - 3 * q^40 + 9 * q^41 + 15 * q^42 + 21 * q^43 - 27 * q^45 - 15 * q^46 - 12 * q^47 + 27 * q^49 + 12 * q^50 - 3 * q^51 - 9 * q^53 + 9 * q^54 + 6 * q^56 - 6 * q^58 - 3 * q^59 - 9 * q^60 - 24 * q^61 - 3 * q^62 + 3 * q^64 + 6 * q^65 + 9 * q^68 + 3 * q^69 + 21 * q^71 + 3 * q^72 + 3 * q^73 - 6 * q^74 + 36 * q^75 + 3 * q^76 + 15 * q^78 + 6 * q^79 - 3 * q^80 - 9 * q^81 + 9 * q^82 + 33 * q^83 + 15 * q^84 - 24 * q^85 + 21 * q^86 - 6 * q^87 + 6 * q^89 - 27 * q^90 + 36 * q^91 - 15 * q^92 + 36 * q^93 - 12 * q^94 - 3 * q^95 + 12 * q^97 + 27 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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