LMFDB - Embedded newform 3762.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3762,2,Mod(1,3762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3762, 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("3762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3762.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:	$30.0397212404$
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:	$\Z[a_1, \ldots, a_{7}]$
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	$\chi$	$=$	3762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.12398 q^{5} -4.21417 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +4.12398 q^{5} -4.21417 q^{7} +1.00000 q^{8} +4.12398 q^{10} +1.00000 q^{11} -2.21417 q^{13} -4.21417 q^{14} +1.00000 q^{16} +3.45490 q^{17} -1.00000 q^{19} +4.12398 q^{20} +1.00000 q^{22} +5.45490 q^{23} +12.0072 q^{25} -2.21417 q^{26} -4.21417 q^{28} -5.57889 q^{29} +7.00724 q^{31} +1.00000 q^{32} +3.45490 q^{34} -17.3792 q^{35} -2.90981 q^{37} -1.00000 q^{38} +4.12398 q^{40} +11.9170 q^{41} +1.46214 q^{43} +1.00000 q^{44} +5.45490 q^{46} -7.58612 q^{47} +10.7593 q^{49} +12.0072 q^{50} -2.21417 q^{52} +13.2214 q^{53} +4.12398 q^{55} -4.21417 q^{56} -5.57889 q^{58} -4.79306 q^{59} +8.90981 q^{61} +7.00724 q^{62} +1.00000 q^{64} -9.13122 q^{65} +1.30437 q^{67} +3.45490 q^{68} -17.3792 q^{70} +6.80030 q^{71} -1.45490 q^{73} -2.90981 q^{74} -1.00000 q^{76} -4.21417 q^{77} -9.15777 q^{79} +4.12398 q^{80} +11.9170 q^{82} +13.2142 q^{83} +14.2480 q^{85} +1.46214 q^{86} +1.00000 q^{88} -8.24797 q^{89} +9.33092 q^{91} +5.45490 q^{92} -7.58612 q^{94} -4.12398 q^{95} +2.18038 q^{97} +10.7593 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} - 6 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 15 q^{23} + 12 q^{25} - 6 q^{28} - 6 q^{29} - 3 q^{31} + 3 q^{32} + 9 q^{34} - 6 q^{37} - 3 q^{38} + 3 q^{40} + 9 q^{41} - 21 q^{43} + 3 q^{44} + 15 q^{46} + 12 q^{47} + 27 q^{49} + 12 q^{50} + 9 q^{53} + 3 q^{55} - 6 q^{56} - 6 q^{58} + 3 q^{59} + 24 q^{61} - 3 q^{62} + 3 q^{64} + 6 q^{65} + 9 q^{68} - 21 q^{71} - 3 q^{73} - 6 q^{74} - 3 q^{76} - 6 q^{77} - 6 q^{79} + 3 q^{80} + 9 q^{82} + 33 q^{83} + 24 q^{85} - 21 q^{86} + 3 q^{88} - 6 q^{89} + 36 q^{91} + 15 q^{92} + 12 q^{94} - 3 q^{95} + 12 q^{97} + 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^10 + 3 * q^11 - 6 * q^14 + 3 * q^16 + 9 * q^17 - 3 * q^19 + 3 * q^20 + 3 * q^22 + 15 * q^23 + 12 * q^25 - 6 * q^28 - 6 * q^29 - 3 * q^31 + 3 * q^32 + 9 * q^34 - 6 * q^37 - 3 * q^38 + 3 * q^40 + 9 * q^41 - 21 * q^43 + 3 * q^44 + 15 * q^46 + 12 * q^47 + 27 * q^49 + 12 * q^50 + 9 * q^53 + 3 * q^55 - 6 * q^56 - 6 * q^58 + 3 * q^59 + 24 * q^61 - 3 * q^62 + 3 * q^64 + 6 * q^65 + 9 * q^68 - 21 * q^71 - 3 * q^73 - 6 * q^74 - 3 * q^76 - 6 * q^77 - 6 * q^79 + 3 * q^80 + 9 * q^82 + 33 * q^83 + 24 * q^85 - 21 * q^86 + 3 * q^88 - 6 * q^89 + 36 * q^91 + 15 * q^92 + 12 * q^94 - 3 * q^95 + 12 * q^97 + 27 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	4.12398	1.84430	0.922151	−	0.386831i	$-0.126430\pi$
$5$	4.12398	1.84430	0.922151	+	0.386831i	$0.126430\pi$
$6$	0	0
$7$	−4.21417	−1.59281	−0.796404	−	0.604765i	$-0.793267\pi$
$7$	−4.21417	−1.59281	−0.796404	+	0.604765i	$0.793267\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	4.12398	1.30412
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.21417	−0.614102	−0.307051	−	0.951693i	$-0.599342\pi$
$13$	−2.21417	−0.614102	−0.307051	+	0.951693i	$0.599342\pi$
$14$	−4.21417	−1.12629
$15$	0	0
$16$	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	$0.362392\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	4.12398	0.922151
$21$	0	0
$22$	1.00000	0.213201
$23$	5.45490	1.13743	0.568713	−	0.822536i	$-0.307441\pi$
$23$	5.45490	1.13743	0.568713	+	0.822536i	$0.307441\pi$
$24$	0	0
$25$	12.0072	2.40145
$26$	−2.21417	−0.434235
$27$	0	0
$28$	−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	$0.673318\pi$
$30$	0	0
$31$	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
$36$	0	0
$37$	−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
$39$	0	0
$40$	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	$0.119316\pi$
$42$	0	0
$43$	1.46214	0.222974	0.111487	−	0.993766i	$-0.464439\pi$
$43$	1.46214	0.222974	0.111487	+	0.993766i	$0.464439\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	5.45490	0.804282
$47$	−7.58612	−1.10655	−0.553275	−	0.832999i	$-0.686622\pi$
$47$	−7.58612	−1.10655	−0.553275	+	0.832999i	$0.686622\pi$
$48$	0	0
$49$	10.7593	1.53704
$50$	12.0072	1.69808
$51$	0	0
$52$	−2.21417	−0.307051
$53$	13.2214	1.81610	0.908050	−	0.418861i	$-0.137571\pi$
$53$	13.2214	1.81610	0.908050	+	0.418861i	$0.137571\pi$
$54$	0	0
$55$	4.12398	0.556078
$56$	−4.21417	−0.563143
$57$	0	0
$58$	−5.57889	−0.732544
$59$	−4.79306	−0.624004	−0.312002	−	0.950082i	$-0.600999\pi$
$59$	−4.79306	−0.624004	−0.312002	+	0.950082i	$0.600999\pi$
$60$	0	0
$61$	8.90981	1.14078	0.570392	−	0.821373i	$-0.306791\pi$
$61$	8.90981	1.14078	0.570392	+	0.821373i	$0.306791\pi$
$62$	7.00724	0.889920
$63$	0	0
$64$	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
$69$	0	0
$70$	−17.3792	−2.07721
$71$	6.80030	0.807047	0.403524	−	0.914969i	$-0.367785\pi$
$71$	6.80030	0.807047	0.403524	+	0.914969i	$0.367785\pi$
$72$	0	0
$73$	−1.45490	−0.170284	−0.0851418	−	0.996369i	$-0.527134\pi$
$73$	−1.45490	−0.170284	−0.0851418	+	0.996369i	$0.527134\pi$
$74$	−2.90981	−0.338258
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−4.21417	−0.480250
$78$	0	0
$79$	−9.15777	−1.03033	−0.515165	−	0.857091i	$-0.672269\pi$
$79$	−9.15777	−1.03033	−0.515165	+	0.857091i	$0.672269\pi$
$80$	4.12398	0.461075
$81$	0	0
$82$	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	$0.241737\pi$
$84$	0	0
$85$	14.2480	1.54541
$86$	1.46214	0.157667
$87$	0	0
$88$	1.00000	0.106600
$89$	−8.24797	−0.874283	−0.437141	−	0.899393i	$-0.644009\pi$
$89$	−8.24797	−0.874283	−0.437141	+	0.899393i	$0.644009\pi$
$90$	0	0
$91$	9.33092	0.978146
$92$	5.45490	0.568713
$93$	0	0
$94$	−7.58612	−0.782449
$95$	−4.12398	−0.423112
$96$	0	0
$97$	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	$0.399387\pi$
$102$	0	0
$103$	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
$105$	0	0
$106$	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	$0.560105\pi$
$108$	0	0
$109$	−4.97345	−0.476370	−0.238185	−	0.971220i	$-0.576552\pi$
$109$	−4.97345	−0.476370	−0.238185	+	0.971220i	$0.576552\pi$
$110$	4.12398	0.393206
$111$	0	0
$112$	−4.21417	−0.398202
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	22.4959	2.09776
$116$	−5.57889	−0.517987
$117$	0	0
$118$	−4.79306	−0.441237
$119$	−14.5596	−1.33467
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.90981	0.806656
$123$	0	0
$124$	7.00724	0.629268
$125$	28.8977	2.58469
$126$	0	0
$127$	−2.48146	−0.220194	−0.110097	−	0.993921i	$-0.535116\pi$
$127$	−2.48146	−0.220194	−0.110097	+	0.993921i	$0.535116\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−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
$135$	0	0
$136$	3.45490	0.296256
$137$	−6.94360	−0.593232	−0.296616	−	0.954997i	$-0.595858\pi$
$137$	−6.94360	−0.593232	−0.296616	+	0.954997i	$0.595858\pi$
$138$	0	0
$139$	−4.37195	−0.370824	−0.185412	−	0.982661i	$-0.559362\pi$
$139$	−4.37195	−0.370824	−0.185412	+	0.982661i	$0.559362\pi$
$140$	−17.3792	−1.46881
$141$	0	0
$142$	6.80030	0.570668
$143$	−2.21417	−0.185159
$144$	0	0
$145$	−23.0072	−1.91065
$146$	−1.45490	−0.120409
$147$	0	0
$148$	−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	$0.660080\pi$
$150$	0	0
$151$	−19.5861	−1.59390	−0.796948	−	0.604048i	$-0.793554\pi$
$151$	−19.5861	−1.59390	−0.796948	+	0.604048i	$0.793554\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−4.21417	−0.339588
$155$	28.8977	2.32112
$156$	0	0
$157$	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
$159$	0	0
$160$	4.12398	0.326029
$161$	−22.9879	−1.81170
$162$	0	0
$163$	−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	$0.551712\pi$
$168$	0	0
$169$	−8.09743	−0.622879
$170$	14.2480	1.09277
$171$	0	0
$172$	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	$0.857403\pi$
$174$	0	0
$175$	−50.6006	−3.82505
$176$	1.00000	0.0753778
$177$	0	0
$178$	−8.24797	−0.618211
$179$	10.8269	0.809237	0.404619	−	0.914486i	$-0.367404\pi$
$179$	10.8269	0.809237	0.404619	+	0.914486i	$0.367404\pi$
$180$	0	0
$181$	−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
$183$	0	0
$184$	5.45490	0.402141
$185$	−12.0000	−0.882258
$186$	0	0
$187$	3.45490	0.252648
$188$	−7.58612	−0.553275
$189$	0	0
$190$	−4.12398	−0.299185
$191$	−0.612679	−0.0443319	−0.0221659	−	0.999754i	$-0.507056\pi$
$191$	−0.612679	−0.0443319	−0.0221659	+	0.999754i	$0.507056\pi$
$192$	0	0
$193$	11.6425	0.838047	0.419024	−	0.907975i	$-0.362372\pi$
$193$	11.6425	0.838047	0.419024	+	0.907975i	$0.362372\pi$
$194$	2.18038	0.156542
$195$	0	0
$196$	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
$201$	0	0
$202$	6.24797	0.439605
$203$	23.5104	1.65011
$204$	0	0
$205$	49.1457	3.43248
$206$	0.605441	0.0421831
$207$	0	0
$208$	−2.21417	−0.153525
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	2.13122	0.146719	0.0733596	−	0.997306i	$-0.476628\pi$
$211$	2.13122	0.146719	0.0733596	+	0.997306i	$0.476628\pi$
$212$	13.2214	0.908050
$213$	0	0
$214$	−3.88325	−0.265454
$215$	6.02985	0.411232
$216$	0	0
$217$	−29.5297	−2.00461
$218$	−4.97345	−0.336844
$219$	0	0
$220$	4.12398	0.278039
$221$	−7.64976	−0.514579
$222$	0	0
$223$	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
$225$	0	0
$226$	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	$0.471170\pi$
$228$	0	0
$229$	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	$0.310875\pi$
$234$	0	0
$235$	−31.2850	−2.04081
$236$	−4.79306	−0.312002
$237$	0	0
$238$	−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	$0.562797\pi$
$240$	0	0
$241$	4.85341	0.312635	0.156318	−	0.987707i	$-0.450038\pi$
$241$	4.85341	0.312635	0.156318	+	0.987707i	$0.450038\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	8.90981	0.570392
$245$	44.3711	2.83476
$246$	0	0
$247$	2.21417	0.140885
$248$	7.00724	0.444960
$249$	0	0
$250$	28.8977	1.82765
$251$	1.58612	0.100115	0.0500576	−	0.998746i	$-0.484059\pi$
$251$	1.58612	0.100115	0.0500576	+	0.998746i	$0.484059\pi$
$252$	0	0
$253$	5.45490	0.342947
$254$	−2.48146	−0.155701
$255$	0	0
$256$	1.00000	0.0625000
$257$	−19.5185	−1.21753	−0.608767	−	0.793349i	$-0.708335\pi$
$257$	−19.5185	−1.21753	−0.608767	+	0.793349i	$0.708335\pi$
$258$	0	0
$259$	12.2624	0.761951
$260$	−9.13122	−0.566294
$261$	0	0
$262$	−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
$267$	0	0
$268$	1.30437	0.0796769
$269$	−23.6006	−1.43895	−0.719477	−	0.694516i	$-0.755618\pi$
$269$	−23.6006	−1.43895	−0.719477	+	0.694516i	$0.755618\pi$
$270$	0	0
$271$	−16.4090	−0.996778	−0.498389	−	0.866954i	$-0.666075\pi$
$271$	−16.4090	−0.996778	−0.498389	+	0.866954i	$0.666075\pi$
$272$	3.45490	0.209484
$273$	0	0
$274$	−6.94360	−0.419478
$275$	12.0072	0.724064
$276$	0	0
$277$	−11.5861	−0.696143	−0.348071	−	0.937468i	$-0.613163\pi$
$277$	−11.5861	−0.696143	−0.348071	+	0.937468i	$0.613163\pi$
$278$	−4.37195	−0.262212
$279$	0	0
$280$	−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	$0.567281\pi$
$282$	0	0
$283$	8.33092	0.495222	0.247611	−	0.968860i	$-0.420355\pi$
$283$	8.33092	0.495222	0.247611	+	0.968860i	$0.420355\pi$
$284$	6.80030	0.403524
$285$	0	0
$286$	−2.21417	−0.130927
$287$	−50.2205	−2.96442
$288$	0	0
$289$	−5.06364	−0.297861
$290$	−23.0072	−1.35103
$291$	0	0
$292$	−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	$0.596439\pi$
$294$	0	0
$295$	−19.7665	−1.15085
$296$	−2.90981	−0.169129
$297$	0	0
$298$	−11.7665	−0.681616
$299$	−12.0781	−0.698495
$300$	0	0
$301$	−6.16172	−0.355156
$302$	−19.5861	−1.12705
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	36.7439	2.10395
$306$	0	0
$307$	−1.93242	−0.110289	−0.0551444	−	0.998478i	$-0.517562\pi$
$307$	−1.93242	−0.110289	−0.0551444	+	0.998478i	$0.517562\pi$
$308$	−4.21417	−0.240125
$309$	0	0
$310$	28.8977	1.64128
$311$	27.7173	1.57171	0.785853	−	0.618413i	$-0.212224\pi$
$311$	27.7173	1.57171	0.785853	+	0.618413i	$0.212224\pi$
$312$	0	0
$313$	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
$315$	0	0
$316$	−9.15777	−0.515165
$317$	33.8977	1.90389	0.951943	−	0.306275i	$-0.0990827\pi$
$317$	33.8977	1.90389	0.951943	+	0.306275i	$0.0990827\pi$
$318$	0	0
$319$	−5.57889	−0.312358
$320$	4.12398	0.230538
$321$	0	0
$322$	−22.9879	−1.28107
$323$	−3.45490	−0.192236
$324$	0	0
$325$	−26.5861	−1.47473
$326$	−17.8341	−0.987739
$327$	0	0
$328$	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
$333$	0	0
$334$	−4.18038	−0.228740
$335$	5.37919	0.293896
$336$	0	0
$337$	−6.82685	−0.371882	−0.185941	−	0.982561i	$-0.559533\pi$
$337$	−6.82685	−0.371882	−0.185941	+	0.982561i	$0.559533\pi$
$338$	−8.09743	−0.440442
$339$	0	0
$340$	14.2480	0.772704
$341$	7.00724	0.379463
$342$	0	0
$343$	−15.8422	−0.855400
$344$	1.46214	0.0788334
$345$	0	0
$346$	−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	$0.456374\pi$
$348$	0	0
$349$	30.9919	1.65896	0.829478	−	0.558539i	$-0.188638\pi$
$349$	30.9919	1.65896	0.829478	+	0.558539i	$0.188638\pi$
$350$	−50.6006	−2.70472
$351$	0	0
$352$	1.00000	0.0533002
$353$	17.8301	0.949003	0.474501	−	0.880255i	$-0.342628\pi$
$353$	17.8301	0.949003	0.474501	+	0.880255i	$0.342628\pi$
$354$	0	0
$355$	28.0443	1.48844
$356$	−8.24797	−0.437141
$357$	0	0
$358$	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	$0.722780\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−17.8341	−0.937339
$363$	0	0
$364$	9.33092	0.489073
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−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
$369$	0	0
$370$	−12.0000	−0.623850
$371$	−55.7173	−2.89270
$372$	0	0
$373$	−12.6160	−0.653230	−0.326615	−	0.945157i	$-0.605908\pi$
$373$	−12.6160	−0.653230	−0.326615	+	0.945157i	$0.605908\pi$
$374$	3.45490	0.178649
$375$	0	0
$376$	−7.58612	−0.391225
$377$	12.3526	0.636193
$378$	0	0
$379$	−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
$381$	0	0
$382$	−0.612679	−0.0313474
$383$	0.0298464	0.00152508	0.000762539	−	1.00000i	$-0.499757\pi$
$383$	0.0298464	0.00152508	0.000762539		1.00000i	$0.499757\pi$
$384$	0	0
$385$	−17.3792	−0.885725
$386$	11.6425	0.592589
$387$	0	0
$388$	2.18038	0.110692
$389$	−9.22865	−0.467911	−0.233956	−	0.972247i	$-0.575167\pi$
$389$	−9.22865	−0.467911	−0.233956	+	0.972247i	$0.575167\pi$
$390$	0	0
$391$	18.8462	0.953092
$392$	10.7593	0.543425
$393$	0	0
$394$	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
$399$	0	0
$400$	12.0072	0.600362
$401$	13.2850	0.663424	0.331712	−	0.943381i	$-0.392374\pi$
$401$	13.2850	0.663424	0.331712	+	0.943381i	$0.392374\pi$
$402$	0	0
$403$	−15.5152	−0.772870
$404$	6.24797	0.310848
$405$	0	0
$406$	23.5104	1.16680
$407$	−2.90981	−0.144234
$408$	0	0
$409$	1.94360	0.0961048	0.0480524	−	0.998845i	$-0.484699\pi$
$409$	1.94360	0.0961048	0.0480524	+	0.998845i	$0.484699\pi$
$410$	49.1457	2.42713
$411$	0	0
$412$	0.605441	0.0298280
$413$	20.1988	0.993918
$414$	0	0
$415$	54.4950	2.67506
$416$	−2.21417	−0.108559
$417$	0	0
$418$	−1.00000	−0.0489116
$419$	2.49593	0.121934	0.0609671	−	0.998140i	$-0.480582\pi$
$419$	2.49593	0.121934	0.0609671	+	0.998140i	$0.480582\pi$
$420$	0	0
$421$	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
$423$	0	0
$424$	13.2214	0.642089
$425$	41.4839	2.01226
$426$	0	0
$427$	−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	$0.502012\pi$
$432$	0	0
$433$	−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
$435$	0	0
$436$	−4.97345	−0.238185
$437$	−5.45490	−0.260943
$438$	0	0
$439$	−17.0902	−0.815670	−0.407835	−	0.913056i	$-0.633716\pi$
$439$	−17.0902	−0.815670	−0.407835	+	0.913056i	$0.633716\pi$
$440$	4.12398	0.196603
$441$	0	0
$442$	−7.64976	−0.363862
$443$	−16.2624	−0.772652	−0.386326	−	0.922362i	$-0.626256\pi$
$443$	−16.2624	−0.772652	−0.386326	+	0.922362i	$0.626256\pi$
$444$	0	0
$445$	−34.0145	−1.61244
$446$	6.90981	0.327189
$447$	0	0
$448$	−4.21417	−0.199101
$449$	27.3382	1.29017	0.645084	−	0.764112i	$-0.276822\pi$
$449$	27.3382	1.29017	0.645084	+	0.764112i	$0.276822\pi$
$450$	0	0
$451$	11.9170	0.561152
$452$	8.00000	0.376288
$453$	0	0
$454$	2.72548	0.127913
$455$	38.4806	1.80400
$456$	0	0
$457$	10.4920	0.490794	0.245397	−	0.969423i	$-0.421082\pi$
$457$	10.4920	0.490794	0.245397	+	0.969423i	$0.421082\pi$
$458$	3.48870	0.163016
$459$	0	0
$460$	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
$465$	0	0
$466$	17.0902	0.791688
$467$	34.5104	1.59695	0.798476	−	0.602027i	$-0.205640\pi$
$467$	34.5104	1.59695	0.798476	+	0.602027i	$0.205640\pi$
$468$	0	0
$469$	−5.49683	−0.253820
$470$	−31.2850	−1.44307
$471$	0	0
$472$	−4.79306	−0.220619
$473$	1.46214	0.0672293
$474$	0	0
$475$	−12.0072	−0.550930
$476$	−14.5596	−0.667337
$477$	0	0
$478$	−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	$0.901619\pi$
$480$	0	0
$481$	6.44282	0.293768
$482$	4.85341	0.221067
$483$	0	0
$484$	1.00000	0.0454545
$485$	8.99187	0.408300
$486$	0	0
$487$	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
$489$	0	0
$490$	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	$0.254995\pi$
$492$	0	0
$493$	−19.2745	−0.868081
$494$	2.21417	0.0996204
$495$	0	0
$496$	7.00724	0.314634
$497$	−28.6577	−1.28547
$498$	0	0
$499$	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
$501$	0	0
$502$	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	$0.822891\pi$
$504$	0	0
$505$	25.7665	1.14659
$506$	5.45490	0.242500
$507$	0	0
$508$	−2.48146	−0.110097
$509$	−14.9388	−0.662149	−0.331074	−	0.943605i	$-0.607411\pi$
$509$	−14.9388	−0.662149	−0.331074	+	0.943605i	$0.607411\pi$
$510$	0	0
$511$	6.13122	0.271229
$512$	1.00000	0.0441942
$513$	0	0
$514$	−19.5185	−0.860926
$515$	2.49683	0.110023
$516$	0	0
$517$	−7.58612	−0.333637
$518$	12.2624	0.538781
$519$	0	0
$520$	−9.13122	−0.400431
$521$	−7.22536	−0.316549	−0.158274	−	0.987395i	$-0.550593\pi$
$521$	−7.22536	−0.316549	−0.158274	+	0.987395i	$0.550593\pi$
$522$	0	0
$523$	4.90586	0.214518	0.107259	−	0.994231i	$-0.465793\pi$
$523$	4.90586	0.214518	0.107259	+	0.994231i	$0.465793\pi$
$524$	−11.0072	−0.480853
$525$	0	0
$526$	13.0483	0.568931
$527$	24.2093	1.05458
$528$	0	0
$529$	6.75598	0.293738
$530$	54.5249	2.36841
$531$	0	0
$532$	4.21417	0.182708
$533$	−26.3864	−1.14292
$534$	0	0
$535$	−16.0145	−0.692366
$536$	1.30437	0.0563401
$537$	0	0
$538$	−23.6006	−1.01749
$539$	10.7593	0.463435
$540$	0	0
$541$	30.8712	1.32726	0.663628	−	0.748063i	$-0.269016\pi$
$541$	30.8712	1.32726	0.663628	+	0.748063i	$0.269016\pi$
$542$	−16.4090	−0.704828
$543$	0	0
$544$	3.45490	0.148128
$545$	−20.5104	−0.878569
$546$	0	0
$547$	2.54115	0.108652	0.0543259	−	0.998523i	$-0.482699\pi$
$547$	2.54115	0.108652	0.0543259	+	0.998523i	$0.482699\pi$
$548$	−6.94360	−0.296616
$549$	0	0
$550$	12.0072	0.511990
$551$	5.57889	0.237669
$552$	0	0
$553$	38.5925	1.64112
$554$	−11.5861	−0.492247
$555$	0	0
$556$	−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	$0.669014\pi$
$558$	0	0
$559$	−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	$0.428135\pi$
$564$	0	0
$565$	32.9919	1.38798
$566$	8.33092	0.350175
$567$	0	0
$568$	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	$0.710820\pi$
$570$	0	0
$571$	19.8639	0.831280	0.415640	−	0.909529i	$-0.363558\pi$
$571$	19.8639	0.831280	0.415640	+	0.909529i	$0.363558\pi$
$572$	−2.21417	−0.0925793
$573$	0	0
$574$	−50.2205	−2.09616
$575$	65.4983	2.73147
$576$	0	0
$577$	−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
$579$	0	0
$580$	−23.0072	−0.955324
$581$	−55.6868	−2.31028
$582$	0	0
$583$	13.2214	0.547575
$584$	−1.45490	−0.0602044
$585$	0	0
$586$	−10.2142	−0.421944
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	−7.00724	−0.288728
$590$	−19.7665	−0.813774
$591$	0	0
$592$	−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
$597$	0	0
$598$	−12.0781	−0.493911
$599$	−40.3864	−1.65014	−0.825072	−	0.565027i	$-0.808866\pi$
$599$	−40.3864	−1.65014	−0.825072	+	0.565027i	$0.808866\pi$
$600$	0	0
$601$	18.3309	0.747734	0.373867	−	0.927482i	$-0.378032\pi$
$601$	18.3309	0.747734	0.373867	+	0.927482i	$0.378032\pi$
$602$	−6.16172	−0.251133
$603$	0	0
$604$	−19.5861	−0.796948
$605$	4.12398	0.167664
$606$	0	0
$607$	12.4283	0.504451	0.252226	−	0.967668i	$-0.418838\pi$
$607$	12.4283	0.504451	0.252226	+	0.967668i	$0.418838\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	36.7439	1.48772
$611$	16.7970	0.679534
$612$	0	0
$613$	−0.0531082	−0.00214502	−0.00107251	−	0.999999i	$-0.500341\pi$
$613$	−0.0531082	−0.00214502	−0.00107251	+	0.999999i	$0.500341\pi$
$614$	−1.93242	−0.0779860
$615$	0	0
$616$	−4.21417	−0.169794
$617$	17.4887	0.704068	0.352034	−	0.935987i	$-0.385490\pi$
$617$	17.4887	0.704068	0.352034	+	0.935987i	$0.385490\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	27.7173	1.11136
$623$	34.7584	1.39256
$624$	0	0
$625$	59.1376	2.36550
$626$	11.5258	0.460663
$627$	0	0
$628$	20.6199	0.822824
$629$	−10.0531	−0.400844
$630$	0	0
$631$	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
$633$	0	0
$634$	33.8977	1.34625
$635$	−10.2335	−0.406104
$636$	0	0
$637$	−23.8229	−0.943898
$638$	−5.57889	−0.220870
$639$	0	0
$640$	4.12398	0.163015
$641$	−30.4573	−1.20299	−0.601495	−	0.798876i	$-0.705428\pi$
$641$	−30.4573	−1.20299	−0.601495	+	0.798876i	$0.705428\pi$
$642$	0	0
$643$	−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
$645$	0	0
$646$	−3.45490	−0.135931
$647$	41.0700	1.61463	0.807314	−	0.590122i	$-0.200921\pi$
$647$	41.0700	1.61463	0.807314	+	0.590122i	$0.200921\pi$
$648$	0	0
$649$	−4.79306	−0.188144
$650$	−26.5861	−1.04279
$651$	0	0
$652$	−17.8341	−0.698437
$653$	−28.4806	−1.11453	−0.557265	−	0.830335i	$-0.688149\pi$
$653$	−28.4806	−1.11453	−0.557265	+	0.830335i	$0.688149\pi$
$654$	0	0
$655$	−45.3937	−1.77368
$656$	11.9170	0.465282
$657$	0	0
$658$	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
$663$	0	0
$664$	13.2142	0.512809
$665$	17.3792	0.673936
$666$	0	0
$667$	−30.4323	−1.17834
$668$	−4.18038	−0.161744
$669$	0	0
$670$	5.37919	0.207816
$671$	8.90981	0.343959
$672$	0	0
$673$	−31.5176	−1.21492	−0.607458	−	0.794352i	$-0.707811\pi$
$673$	−31.5176	−1.21492	−0.607458	+	0.794352i	$0.707811\pi$
$674$	−6.82685	−0.262961
$675$	0	0
$676$	−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	$0.574023\pi$
$678$	0	0
$679$	−9.18852	−0.352623
$680$	14.2480	0.546385
$681$	0	0
$682$	7.00724	0.268321
$683$	−39.1722	−1.49888	−0.749442	−	0.662070i	$-0.769678\pi$
$683$	−39.1722	−1.49888	−0.749442	+	0.662070i	$0.769678\pi$
$684$	0	0
$685$	−28.6353	−1.09410
$686$	−15.8422	−0.604859
$687$	0	0
$688$	1.46214	0.0557436
$689$	−29.2745	−1.11527
$690$	0	0
$691$	−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
$696$	0	0
$697$	41.1722	1.55951
$698$	30.9919	1.17306
$699$	0	0
$700$	−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	$0.753464\pi$
$702$	0	0
$703$	2.90981	0.109745
$704$	1.00000	0.0376889
$705$	0	0
$706$	17.8301	0.671046
$707$	−26.3300	−0.990242
$708$	0	0
$709$	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
$711$	0	0
$712$	−8.24797	−0.309106
$713$	38.2238	1.43149
$714$	0	0
$715$	−9.13122	−0.341488
$716$	10.8269	0.404619
$717$	0	0
$718$	−24.4090	−0.910937
$719$	17.0410	0.635523	0.317762	−	0.948171i	$-0.397069\pi$
$719$	17.0410	0.635523	0.317762	+	0.948171i	$0.397069\pi$
$720$	0	0
$721$	−2.55144	−0.0950204
$722$	1.00000	0.0372161
$723$	0	0
$724$	−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
$729$	0	0
$730$	−6.00000	−0.222070
$731$	5.05156	0.186839
$732$	0	0
$733$	−25.6682	−0.948076	−0.474038	−	0.880504i	$-0.657204\pi$
$733$	−25.6682	−0.948076	−0.474038	+	0.880504i	$0.657204\pi$
$734$	−26.6232	−0.982681
$735$	0	0
$736$	5.45490	0.201070
$737$	1.30437	0.0480470
$738$	0	0
$739$	21.3493	0.785348	0.392674	−	0.919678i	$-0.371550\pi$
$739$	21.3493	0.785348	0.392674	+	0.919678i	$0.371550\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−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	$0.455732\pi$
$744$	0	0
$745$	−48.5249	−1.77781
$746$	−12.6160	−0.461904
$747$	0	0
$748$	3.45490	0.126324
$749$	16.3647	0.597954
$750$	0	0
$751$	−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
$753$	0	0
$754$	12.3526	0.449856
$755$	−80.7728	−2.93962
$756$	0	0
$757$	−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	$0.188276\pi$
$762$	0	0
$763$	20.9590	0.758766
$764$	−0.612679	−0.0221659
$765$	0	0
$766$	0.0298464	0.00107839
$767$	10.6127	0.383202
$768$	0	0
$769$	−40.8751	−1.47399	−0.736997	−	0.675896i	$-0.763757\pi$
$769$	−40.8751	−1.47399	−0.736997	+	0.675896i	$0.763757\pi$
$770$	−17.3792	−0.626302
$771$	0	0
$772$	11.6425	0.419024
$773$	−37.4983	−1.34872	−0.674361	−	0.738402i	$-0.735581\pi$
$773$	−37.4983	−1.34872	−0.674361	+	0.738402i	$0.735581\pi$
$774$	0	0
$775$	84.1376	3.02231
$776$	2.18038	0.0782712
$777$	0	0
$778$	−9.22865	−0.330863
$779$	−11.9170	−0.426972
$780$	0	0
$781$	6.80030	0.243334
$782$	18.8462	0.673938
$783$	0	0
$784$	10.7593	0.384260
$785$	85.0362	3.03507
$786$	0	0
$787$	8.80754	0.313955	0.156977	−	0.987602i	$-0.449825\pi$
$787$	8.80754	0.313955	0.156977	+	0.987602i	$0.449825\pi$
$788$	12.9098	0.459893
$789$	0	0
$790$	−37.7665	−1.34367
$791$	−33.7134	−1.19871
$792$	0	0
$793$	−19.7279	−0.700557
$794$	17.2697	0.612878
$795$	0	0
$796$	−18.3647	−0.650920
$797$	26.1312	0.925615	0.462808	−	0.886459i	$-0.346842\pi$
$797$	26.1312	0.925615	0.462808	+	0.886459i	$0.346842\pi$
$798$	0	0
$799$	−26.2093	−0.927220
$800$	12.0072	0.424520
$801$	0	0
$802$	13.2850	0.469111
$803$	−1.45490	−0.0513425
$804$	0	0
$805$	−94.8018	−3.34132
$806$	−15.5152	−0.546501
$807$	0	0
$808$	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	$0.466305\pi$
$810$	0	0
$811$	−40.2809	−1.41445	−0.707226	−	0.706987i	$-0.750054\pi$
$811$	−40.2809	−1.41445	−0.707226	+	0.706987i	$0.750054\pi$
$812$	23.5104	0.825054
$813$	0	0
$814$	−2.90981	−0.101989
$815$	−73.5475	−2.57626
$816$	0	0
$817$	−1.46214	−0.0511539
$818$	1.94360	0.0679564
$819$	0	0
$820$	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	$0.839216\pi$
$822$	0	0
$823$	−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	$0.825167\pi$
$828$	0	0
$829$	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
$831$	0	0
$832$	−2.21417	−0.0767627
$833$	37.1722	1.28794
$834$	0	0
$835$	−17.2398	−0.596609
$836$	−1.00000	−0.0345857
$837$	0	0
$838$	2.49593	0.0862206
$839$	52.0257	1.79613	0.898063	−	0.439868i	$-0.144975\pi$
$839$	52.0257	1.79613	0.898063	+	0.439868i	$0.144975\pi$
$840$	0	0
$841$	2.12398	0.0732408
$842$	15.6353	0.538828
$843$	0	0
$844$	2.13122	0.0733596
$845$	−33.3937	−1.14878
$846$	0	0
$847$	−4.21417	−0.144801
$848$	13.2214	0.454025
$849$	0	0
$850$	41.4839	1.42288
$851$	−15.8727	−0.544110
$852$	0	0
$853$	16.2335	0.555824	0.277912	−	0.960607i	$-0.410358\pi$
$853$	16.2335	0.555824	0.277912	+	0.960607i	$0.410358\pi$
$854$	−37.5475	−1.28485
$855$	0	0
$856$	−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
$861$	0	0
$862$	−0.262441	−0.00893877
$863$	−1.54839	−0.0527077	−0.0263539	−	0.999653i	$-0.508390\pi$
$863$	−1.54839	−0.0527077	−0.0263539	+	0.999653i	$0.508390\pi$
$864$	0	0
$865$	−97.7801	−3.32462
$866$	−32.2769	−1.09681
$867$	0	0
$868$	−29.5297	−1.00230
$869$	−9.15777	−0.310656
$870$	0	0
$871$	−2.88810	−0.0978594
$872$	−4.97345	−0.168422
$873$	0	0
$874$	−5.45490	−0.184515
$875$	−121.780	−4.11692
$876$	0	0
$877$	−12.1200	−0.409265	−0.204632	−	0.978839i	$-0.565600\pi$
$877$	−12.1200	−0.409265	−0.204632	+	0.978839i	$0.565600\pi$
$878$	−17.0902	−0.576766
$879$	0	0
$880$	4.12398	0.139019
$881$	−53.6836	−1.80864	−0.904322	−	0.426850i	$-0.859623\pi$
$881$	−53.6836	−1.80864	−0.904322	+	0.426850i	$0.859623\pi$
$882$	0	0
$883$	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
$885$	0	0
$886$	−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	$0.347048\pi$
$888$	0	0
$889$	10.4573	0.350727
$890$	−34.0145	−1.14017
$891$	0	0
$892$	6.90981	0.231357
$893$	7.58612	0.253860
$894$	0	0
$895$	44.6498	1.49248
$896$	−4.21417	−0.140786
$897$	0	0
$898$	27.3382	0.912286
$899$	−39.0926	−1.30381
$900$	0	0
$901$	45.6787	1.52178
$902$	11.9170	0.396794
$903$	0	0
$904$	8.00000	0.266076
$905$	−73.5475	−2.44480
$906$	0	0
$907$	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
$909$	0	0
$910$	38.4806	1.27562
$911$	0.623208	0.0206478	0.0103239	−	0.999947i	$-0.496714\pi$
$911$	0.623208	0.0206478	0.0103239	+	0.999947i	$0.496714\pi$
$912$	0	0
$913$	13.2142	0.437325
$914$	10.4920	0.347044
$915$	0	0
$916$	3.48870	0.115270
$917$	46.3864	1.53181
$918$	0	0
$919$	−44.4887	−1.46755	−0.733773	−	0.679394i	$-0.762243\pi$
$919$	−44.4887	−1.46755	−0.733773	+	0.679394i	$0.762243\pi$
$920$	22.4959	0.741669
$921$	0	0
$922$	−6.18038	−0.203540
$923$	−15.0571	−0.495609
$924$	0	0
$925$	−34.9388	−1.14878
$926$	−11.1433	−0.366192
$927$	0	0
$928$	−5.57889	−0.183136
$929$	−44.1224	−1.44761	−0.723805	−	0.690005i	$-0.757609\pi$
$929$	−44.1224	−1.44761	−0.723805	+	0.690005i	$0.757609\pi$
$930$	0	0
$931$	−10.7593	−0.352621
$932$	17.0902	0.559808
$933$	0	0
$934$	34.5104	1.12922
$935$	14.2480	0.465958
$936$	0	0
$937$	24.8672	0.812377	0.406188	−	0.913789i	$-0.366858\pi$
$937$	24.8672	0.812377	0.406188	+	0.913789i	$0.366858\pi$
$938$	−5.49683	−0.179478
$939$	0	0
$940$	−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	$0.661624\pi$
$942$	0	0
$943$	65.0063	2.11690
$944$	−4.79306	−0.156001
$945$	0	0
$946$	1.46214	0.0475383
$947$	−9.07572	−0.294921	−0.147461	−	0.989068i	$-0.547110\pi$
$947$	−9.07572	−0.294921	−0.147461	+	0.989068i	$0.547110\pi$
$948$	0	0
$949$	3.22141	0.104571
$950$	−12.0072	−0.389566
$951$	0	0
$952$	−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	$0.506551\pi$
$954$	0	0
$955$	−2.52668	−0.0817613
$956$	−6.06035	−0.196006
$957$	0	0
$958$	−41.6980	−1.34720
$959$	29.2615	0.944905
$960$	0	0
$961$	18.1014	0.583915
$962$	6.44282	0.207725
$963$	0	0
$964$	4.85341	0.156318
$965$	48.0136	1.54561
$966$	0	0
$967$	6.67632	0.214696	0.107348	−	0.994222i	$-0.465764\pi$
$967$	6.67632	0.214696	0.107348	+	0.994222i	$0.465764\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	8.99187	0.288711
$971$	0.191566	0.00614764	0.00307382	−	0.999995i	$-0.499022\pi$
$971$	0.191566	0.00614764	0.00307382	+	0.999995i	$0.499022\pi$
$972$	0	0
$973$	18.4242	0.590651
$974$	9.10137	0.291627
$975$	0	0
$976$	8.90981	0.285196
$977$	−16.4959	−0.527752	−0.263876	−	0.964557i	$-0.585001\pi$
$977$	−16.4959	−0.527752	−0.263876	+	0.964557i	$0.585001\pi$
$978$	0	0
$979$	−8.24797	−0.263606
$980$	44.3711	1.41738
$981$	0	0
$982$	30.8413	0.984186
$983$	35.0072	1.11656	0.558279	−	0.829653i	$-0.311462\pi$
$983$	35.0072	1.11656	0.558279	+	0.829653i	$0.311462\pi$
$984$	0	0
$985$	53.2398	1.69636
$986$	−19.2745	−0.613826
$987$	0	0
$988$	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
$993$	0	0
$994$	−28.6577	−0.908966
$995$	−75.7358	−2.40099
$996$	0	0
$997$	8.89533	0.281718	0.140859	−	0.990030i	$-0.455014\pi$
$997$	8.89533	0.281718	0.140859	+	0.990030i	$0.455014\pi$
$998$	21.7665	0.689007
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3762.2.a.bg.1.3		3
3.2	odd	2		418.2.a.g.1.3	✓	3
12.11	even	2		3344.2.a.q.1.1		3
33.32	even	2		4598.2.a.bo.1.3		3
57.56	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	3.2	odd	2
3344.2.a.q.1.1		3	12.11	even	2
3762.2.a.bg.1.3		3	1.1	even	1	trivial
4598.2.a.bo.1.3		3	33.32	even	2
7942.2.a.bi.1.1		3	57.56	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:	\( N \)	\(=\)	\( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3762.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.12398 q^{5} -4.21417 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.12398 q^{5} -4.21417 q^{7} +1.00000 q^{8} +4.12398 q^{10} +1.00000 q^{11} -2.21417 q^{13} -4.21417 q^{14} +1.00000 q^{16} +3.45490 q^{17} -1.00000 q^{19} +4.12398 q^{20} +1.00000 q^{22} +5.45490 q^{23} +12.0072 q^{25} -2.21417 q^{26} -4.21417 q^{28} -5.57889 q^{29} +7.00724 q^{31} +1.00000 q^{32} +3.45490 q^{34} -17.3792 q^{35} -2.90981 q^{37} -1.00000 q^{38} +4.12398 q^{40} +11.9170 q^{41} +1.46214 q^{43} +1.00000 q^{44} +5.45490 q^{46} -7.58612 q^{47} +10.7593 q^{49} +12.0072 q^{50} -2.21417 q^{52} +13.2214 q^{53} +4.12398 q^{55} -4.21417 q^{56} -5.57889 q^{58} -4.79306 q^{59} +8.90981 q^{61} +7.00724 q^{62} +1.00000 q^{64} -9.13122 q^{65} +1.30437 q^{67} +3.45490 q^{68} -17.3792 q^{70} +6.80030 q^{71} -1.45490 q^{73} -2.90981 q^{74} -1.00000 q^{76} -4.21417 q^{77} -9.15777 q^{79} +4.12398 q^{80} +11.9170 q^{82} +13.2142 q^{83} +14.2480 q^{85} +1.46214 q^{86} +1.00000 q^{88} -8.24797 q^{89} +9.33092 q^{91} +5.45490 q^{92} -7.58612 q^{94} -4.12398 q^{95} +2.18038 q^{97} +10.7593 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} - 6 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 15 q^{23} + 12 q^{25} - 6 q^{28} - 6 q^{29} - 3 q^{31} + 3 q^{32} + 9 q^{34} - 6 q^{37} - 3 q^{38} + 3 q^{40} + 9 q^{41} - 21 q^{43} + 3 q^{44} + 15 q^{46} + 12 q^{47} + 27 q^{49} + 12 q^{50} + 9 q^{53} + 3 q^{55} - 6 q^{56} - 6 q^{58} + 3 q^{59} + 24 q^{61} - 3 q^{62} + 3 q^{64} + 6 q^{65} + 9 q^{68} - 21 q^{71} - 3 q^{73} - 6 q^{74} - 3 q^{76} - 6 q^{77} - 6 q^{79} + 3 q^{80} + 9 q^{82} + 33 q^{83} + 24 q^{85} - 21 q^{86} + 3 q^{88} - 6 q^{89} + 36 q^{91} + 15 q^{92} + 12 q^{94} - 3 q^{95} + 12 q^{97} + 27 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^7 + 3 * q^8 + 3 * q^10 + 3 * q^11 - 6 * q^14 + 3 * q^16 + 9 * q^17 - 3 * q^19 + 3 * q^20 + 3 * q^22 + 15 * q^23 + 12 * q^25 - 6 * q^28 - 6 * q^29 - 3 * q^31 + 3 * q^32 + 9 * q^34 - 6 * q^37 - 3 * q^38 + 3 * q^40 + 9 * q^41 - 21 * q^43 + 3 * q^44 + 15 * q^46 + 12 * q^47 + 27 * q^49 + 12 * q^50 + 9 * q^53 + 3 * q^55 - 6 * q^56 - 6 * q^58 + 3 * q^59 + 24 * q^61 - 3 * q^62 + 3 * q^64 + 6 * q^65 + 9 * q^68 - 21 * q^71 - 3 * q^73 - 6 * q^74 - 3 * q^76 - 6 * q^77 - 6 * q^79 + 3 * q^80 + 9 * q^82 + 33 * q^83 + 24 * q^85 - 21 * q^86 + 3 * q^88 - 6 * q^89 + 36 * q^91 + 15 * q^92 + 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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