LMFDB - Embedded newform 350.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,2,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, 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("350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	350.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:	$2.79476407074$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{9} -4.89898 q^{11} -2.44949 q^{12} +4.44949 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} +1.55051 q^{19} +2.44949 q^{21} +4.89898 q^{22} +2.89898 q^{23} +2.44949 q^{24} -4.44949 q^{26} -1.00000 q^{28} +6.89898 q^{29} +8.89898 q^{31} -1.00000 q^{32} +12.0000 q^{33} -2.00000 q^{34} +3.00000 q^{36} +2.00000 q^{37} -1.55051 q^{38} -10.8990 q^{39} -1.10102 q^{41} -2.44949 q^{42} -0.898979 q^{43} -4.89898 q^{44} -2.89898 q^{46} +8.89898 q^{47} -2.44949 q^{48} +1.00000 q^{49} -4.89898 q^{51} +4.44949 q^{52} -10.8990 q^{53} +1.00000 q^{56} -3.79796 q^{57} -6.89898 q^{58} -1.55051 q^{59} +3.55051 q^{61} -8.89898 q^{62} -3.00000 q^{63} +1.00000 q^{64} -12.0000 q^{66} -8.00000 q^{67} +2.00000 q^{68} -7.10102 q^{69} -1.10102 q^{71} -3.00000 q^{72} +2.89898 q^{73} -2.00000 q^{74} +1.55051 q^{76} +4.89898 q^{77} +10.8990 q^{78} +6.89898 q^{79} -9.00000 q^{81} +1.10102 q^{82} -2.44949 q^{83} +2.44949 q^{84} +0.898979 q^{86} -16.8990 q^{87} +4.89898 q^{88} -10.0000 q^{89} -4.44949 q^{91} +2.89898 q^{92} -21.7980 q^{93} -8.89898 q^{94} +2.44949 q^{96} +15.7980 q^{97} -1.00000 q^{98} -14.6969 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 6 q^{9} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 8 q^{19} - 4 q^{23} - 4 q^{26} - 2 q^{28} + 4 q^{29} + 8 q^{31} - 2 q^{32} + 24 q^{33} - 4 q^{34}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 6 * q^9 + 4 * q^13 + 2 * q^14 + 2 * q^16 + 4 * q^17 - 6 * q^18 + 8 * q^19 - 4 * q^23 - 4 * q^26 - 2 * q^28 + 4 * q^29 + 8 * q^31 - 2 * q^32 + 24 * q^33 - 4 * q^34 + 6 * q^36 + 4 * q^37 - 8 * q^38 - 12 * q^39 - 12 * q^41 + 8 * q^43 + 4 * q^46 + 8 * q^47 + 2 * q^49 + 4 * q^52 - 12 * q^53 + 2 * q^56 + 12 * q^57 - 4 * q^58 - 8 * q^59 + 12 * q^61 - 8 * q^62 - 6 * q^63 + 2 * q^64 - 24 * q^66 - 16 * q^67 + 4 * q^68 - 24 * q^69 - 12 * q^71 - 6 * q^72 - 4 * q^73 - 4 * q^74 + 8 * q^76 + 12 * q^78 + 4 * q^79 - 18 * q^81 + 12 * q^82 - 8 * q^86 - 24 * q^87 - 20 * q^89 - 4 * q^91 - 4 * q^92 - 24 * q^93 - 8 * q^94 + 12 * q^97 - 2 * 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
$3$	−2.44949	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−2.44949	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.44949	1.00000
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	3.00000	1.00000
$10$	0	0
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	−2.44949	−0.707107
$13$	4.44949	1.23407	0.617033	−	0.786937i	$-0.288334\pi$
$13$	4.44949	1.23407	0.617033	+	0.786937i	$0.288334\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−3.00000	−0.707107
$19$	1.55051	0.355711	0.177856	−	0.984057i	$-0.443084\pi$
$19$	1.55051	0.355711	0.177856	+	0.984057i	$0.443084\pi$
$20$	0	0
$21$	2.44949	0.534522
$22$	4.89898	1.04447
$23$	2.89898	0.604479	0.302240	−	0.953232i	$-0.402266\pi$
$23$	2.89898	0.604479	0.302240	+	0.953232i	$0.402266\pi$
$24$	2.44949	0.500000
$25$	0	0
$26$	−4.44949	−0.872617
$27$	0	0
$28$	−1.00000	−0.188982
$29$	6.89898	1.28111	0.640554	−	0.767913i	$-0.278705\pi$
$29$	6.89898	1.28111	0.640554	+	0.767913i	$0.278705\pi$
$30$	0	0
$31$	8.89898	1.59830	0.799152	−	0.601129i	$-0.205282\pi$
$31$	8.89898	1.59830	0.799152	+	0.601129i	$0.205282\pi$
$32$	−1.00000	−0.176777
$33$	12.0000	2.08893
$34$	−2.00000	−0.342997
$35$	0	0
$36$	3.00000	0.500000
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−1.55051	−0.251526
$39$	−10.8990	−1.74523
$40$	0	0
$41$	−1.10102	−0.171951	−0.0859753	−	0.996297i	$-0.527401\pi$
$41$	−1.10102	−0.171951	−0.0859753	+	0.996297i	$0.527401\pi$
$42$	−2.44949	−0.377964
$43$	−0.898979	−0.137093	−0.0685465	−	0.997648i	$-0.521836\pi$
$43$	−0.898979	−0.137093	−0.0685465	+	0.997648i	$0.521836\pi$
$44$	−4.89898	−0.738549
$45$	0	0
$46$	−2.89898	−0.427431
$47$	8.89898	1.29805	0.649025	−	0.760767i	$-0.275177\pi$
$47$	8.89898	1.29805	0.649025	+	0.760767i	$0.275177\pi$
$48$	−2.44949	−0.353553
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.89898	−0.685994
$52$	4.44949	0.617033
$53$	−10.8990	−1.49709	−0.748545	−	0.663084i	$-0.769247\pi$
$53$	−10.8990	−1.49709	−0.748545	+	0.663084i	$0.769247\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	−3.79796	−0.503052
$58$	−6.89898	−0.905880
$59$	−1.55051	−0.201859	−0.100930	−	0.994894i	$-0.532182\pi$
$59$	−1.55051	−0.201859	−0.100930	+	0.994894i	$0.532182\pi$
$60$	0	0
$61$	3.55051	0.454596	0.227298	−	0.973825i	$-0.427011\pi$
$61$	3.55051	0.454596	0.227298	+	0.973825i	$0.427011\pi$
$62$	−8.89898	−1.13017
$63$	−3.00000	−0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	−12.0000	−1.47710
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	2.00000	0.242536
$69$	−7.10102	−0.854862
$70$	0	0
$71$	−1.10102	−0.130667	−0.0653335	−	0.997863i	$-0.520811\pi$
$71$	−1.10102	−0.130667	−0.0653335	+	0.997863i	$0.520811\pi$
$72$	−3.00000	−0.353553
$73$	2.89898	0.339300	0.169650	−	0.985504i	$-0.445736\pi$
$73$	2.89898	0.339300	0.169650	+	0.985504i	$0.445736\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	1.55051	0.177856
$77$	4.89898	0.558291
$78$	10.8990	1.23407
$79$	6.89898	0.776196	0.388098	−	0.921618i	$-0.373132\pi$
$79$	6.89898	0.776196	0.388098	+	0.921618i	$0.373132\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	1.10102	0.121587
$83$	−2.44949	−0.268866	−0.134433	−	0.990923i	$-0.542921\pi$
$83$	−2.44949	−0.268866	−0.134433	+	0.990923i	$0.542921\pi$
$84$	2.44949	0.267261
$85$	0	0
$86$	0.898979	0.0969395
$87$	−16.8990	−1.81176
$88$	4.89898	0.522233
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−4.44949	−0.466433
$92$	2.89898	0.302240
$93$	−21.7980	−2.26034
$94$	−8.89898	−0.917860
$95$	0	0
$96$	2.44949	0.250000
$97$	15.7980	1.60404	0.802020	−	0.597297i	$-0.203759\pi$
$97$	15.7980	1.60404	0.802020	+	0.597297i	$0.203759\pi$
$98$	−1.00000	−0.101015
$99$	−14.6969	−1.47710
$100$	0	0
$101$	3.55051	0.353289	0.176644	−	0.984275i	$-0.443476\pi$
$101$	3.55051	0.353289	0.176644	+	0.984275i	$0.443476\pi$
$102$	4.89898	0.485071
$103$	12.8990	1.27097	0.635487	−	0.772111i	$-0.280799\pi$
$103$	12.8990	1.27097	0.635487	+	0.772111i	$0.280799\pi$
$104$	−4.44949	−0.436308
$105$	0	0
$106$	10.8990	1.05860
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	6.89898	0.660802	0.330401	−	0.943841i	$-0.392816\pi$
$109$	6.89898	0.660802	0.330401	+	0.943841i	$0.392816\pi$
$110$	0	0
$111$	−4.89898	−0.464991
$112$	−1.00000	−0.0944911
$113$	19.7980	1.86244	0.931218	−	0.364464i	$-0.118748\pi$
$113$	19.7980	1.86244	0.931218	+	0.364464i	$0.118748\pi$
$114$	3.79796	0.355711
$115$	0	0
$116$	6.89898	0.640554
$117$	13.3485	1.23407
$118$	1.55051	0.142736
$119$	−2.00000	−0.183340
$120$	0	0
$121$	13.0000	1.18182
$122$	−3.55051	−0.321448
$123$	2.69694	0.243175
$124$	8.89898	0.799152
$125$	0	0
$126$	3.00000	0.267261
$127$	−14.8990	−1.32207	−0.661035	−	0.750355i	$-0.729883\pi$
$127$	−14.8990	−1.32207	−0.661035	+	0.750355i	$0.729883\pi$
$128$	−1.00000	−0.0883883
$129$	2.20204	0.193879
$130$	0	0
$131$	−6.44949	−0.563495	−0.281747	−	0.959489i	$-0.590914\pi$
$131$	−6.44949	−0.563495	−0.281747	+	0.959489i	$0.590914\pi$
$132$	12.0000	1.04447
$133$	−1.55051	−0.134446
$134$	8.00000	0.691095
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−1.79796	−0.153610	−0.0768050	−	0.997046i	$-0.524472\pi$
$137$	−1.79796	−0.153610	−0.0768050	+	0.997046i	$0.524472\pi$
$138$	7.10102	0.604479
$139$	1.55051	0.131513	0.0657563	−	0.997836i	$-0.479054\pi$
$139$	1.55051	0.131513	0.0657563	+	0.997836i	$0.479054\pi$
$140$	0	0
$141$	−21.7980	−1.83572
$142$	1.10102	0.0923956
$143$	−21.7980	−1.82284
$144$	3.00000	0.250000
$145$	0	0
$146$	−2.89898	−0.239921
$147$	−2.44949	−0.202031
$148$	2.00000	0.164399
$149$	−3.79796	−0.311141	−0.155570	−	0.987825i	$-0.549722\pi$
$149$	−3.79796	−0.311141	−0.155570	+	0.987825i	$0.549722\pi$
$150$	0	0
$151$	19.5959	1.59469	0.797347	−	0.603522i	$-0.206236\pi$
$151$	19.5959	1.59469	0.797347	+	0.603522i	$0.206236\pi$
$152$	−1.55051	−0.125763
$153$	6.00000	0.485071
$154$	−4.89898	−0.394771
$155$	0	0
$156$	−10.8990	−0.872617
$157$	3.55051	0.283362	0.141681	−	0.989912i	$-0.454749\pi$
$157$	3.55051	0.283362	0.141681	+	0.989912i	$0.454749\pi$
$158$	−6.89898	−0.548853
$159$	26.6969	2.11720
$160$	0	0
$161$	−2.89898	−0.228472
$162$	9.00000	0.707107
$163$	−7.10102	−0.556195	−0.278097	−	0.960553i	$-0.589704\pi$
$163$	−7.10102	−0.556195	−0.278097	+	0.960553i	$0.589704\pi$
$164$	−1.10102	−0.0859753
$165$	0	0
$166$	2.44949	0.190117
$167$	−4.89898	−0.379094	−0.189547	−	0.981872i	$-0.560702\pi$
$167$	−4.89898	−0.379094	−0.189547	+	0.981872i	$0.560702\pi$
$168$	−2.44949	−0.188982
$169$	6.79796	0.522920
$170$	0	0
$171$	4.65153	0.355711
$172$	−0.898979	−0.0685465
$173$	−6.24745	−0.474985	−0.237492	−	0.971389i	$-0.576325\pi$
$173$	−6.24745	−0.474985	−0.237492	+	0.971389i	$0.576325\pi$
$174$	16.8990	1.28111
$175$	0	0
$176$	−4.89898	−0.369274
$177$	3.79796	0.285472
$178$	10.0000	0.749532
$179$	13.7980	1.03131	0.515654	−	0.856797i	$-0.327549\pi$
$179$	13.7980	1.03131	0.515654	+	0.856797i	$0.327549\pi$
$180$	0	0
$181$	−10.2474	−0.761687	−0.380843	−	0.924640i	$-0.624366\pi$
$181$	−10.2474	−0.761687	−0.380843	+	0.924640i	$0.624366\pi$
$182$	4.44949	0.329818
$183$	−8.69694	−0.642896
$184$	−2.89898	−0.213716
$185$	0	0
$186$	21.7980	1.59830
$187$	−9.79796	−0.716498
$188$	8.89898	0.649025
$189$	0	0
$190$	0	0
$191$	12.6969	0.918718	0.459359	−	0.888251i	$-0.348079\pi$
$191$	12.6969	0.918718	0.459359	+	0.888251i	$0.348079\pi$
$192$	−2.44949	−0.176777
$193$	−21.5959	−1.55451	−0.777254	−	0.629187i	$-0.783388\pi$
$193$	−21.5959	−1.55451	−0.777254	+	0.629187i	$0.783388\pi$
$194$	−15.7980	−1.13423
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.8990	1.34650	0.673248	−	0.739417i	$-0.264899\pi$
$197$	18.8990	1.34650	0.673248	+	0.739417i	$0.264899\pi$
$198$	14.6969	1.04447
$199$	16.8990	1.19794	0.598968	−	0.800773i	$-0.295577\pi$
$199$	16.8990	1.19794	0.598968	+	0.800773i	$0.295577\pi$
$200$	0	0
$201$	19.5959	1.38219
$202$	−3.55051	−0.249813
$203$	−6.89898	−0.484213
$204$	−4.89898	−0.342997
$205$	0	0
$206$	−12.8990	−0.898714
$207$	8.69694	0.604479
$208$	4.44949	0.308517
$209$	−7.59592	−0.525421
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−10.8990	−0.748545
$213$	2.69694	0.184791
$214$	8.00000	0.546869
$215$	0	0
$216$	0	0
$217$	−8.89898	−0.604102
$218$	−6.89898	−0.467258
$219$	−7.10102	−0.479842
$220$	0	0
$221$	8.89898	0.598610
$222$	4.89898	0.328798
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−19.7980	−1.31694
$227$	7.34847	0.487735	0.243868	−	0.969809i	$-0.421584\pi$
$227$	7.34847	0.487735	0.243868	+	0.969809i	$0.421584\pi$
$228$	−3.79796	−0.251526
$229$	−19.1464	−1.26523	−0.632616	−	0.774466i	$-0.718019\pi$
$229$	−19.1464	−1.26523	−0.632616	+	0.774466i	$0.718019\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	−6.89898	−0.452940
$233$	29.7980	1.95213	0.976065	−	0.217481i	$-0.0697840\pi$
$233$	29.7980	1.95213	0.976065	+	0.217481i	$0.0697840\pi$
$234$	−13.3485	−0.872617
$235$	0	0
$236$	−1.55051	−0.100930
$237$	−16.8990	−1.09771
$238$	2.00000	0.129641
$239$	−6.20204	−0.401177	−0.200588	−	0.979676i	$-0.564285\pi$
$239$	−6.20204	−0.401177	−0.200588	+	0.979676i	$0.564285\pi$
$240$	0	0
$241$	−8.69694	−0.560219	−0.280110	−	0.959968i	$-0.590371\pi$
$241$	−8.69694	−0.560219	−0.280110	+	0.959968i	$0.590371\pi$
$242$	−13.0000	−0.835672
$243$	22.0454	1.41421
$244$	3.55051	0.227298
$245$	0	0
$246$	−2.69694	−0.171951
$247$	6.89898	0.438972
$248$	−8.89898	−0.565086
$249$	6.00000	0.380235
$250$	0	0
$251$	−6.44949	−0.407088	−0.203544	−	0.979066i	$-0.565246\pi$
$251$	−6.44949	−0.407088	−0.203544	+	0.979066i	$0.565246\pi$
$252$	−3.00000	−0.188982
$253$	−14.2020	−0.892875
$254$	14.8990	0.934845
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.69694	−0.542500	−0.271250	−	0.962509i	$-0.587437\pi$
$257$	−8.69694	−0.542500	−0.271250	+	0.962509i	$0.587437\pi$
$258$	−2.20204	−0.137093
$259$	−2.00000	−0.124274
$260$	0	0
$261$	20.6969	1.28111
$262$	6.44949	0.398451
$263$	9.79796	0.604168	0.302084	−	0.953281i	$-0.402318\pi$
$263$	9.79796	0.604168	0.302084	+	0.953281i	$0.402318\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	1.55051	0.0950679
$267$	24.4949	1.49906
$268$	−8.00000	−0.488678
$269$	19.1464	1.16738	0.583689	−	0.811977i	$-0.301609\pi$
$269$	19.1464	1.16738	0.583689	+	0.811977i	$0.301609\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	2.00000	0.121268
$273$	10.8990	0.659636
$274$	1.79796	0.108619
$275$	0	0
$276$	−7.10102	−0.427431
$277$	−14.8990	−0.895193	−0.447596	−	0.894236i	$-0.647720\pi$
$277$	−14.8990	−0.895193	−0.447596	+	0.894236i	$0.647720\pi$
$278$	−1.55051	−0.0929934
$279$	26.6969	1.59830
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	21.7980	1.29805
$283$	3.75255	0.223066	0.111533	−	0.993761i	$-0.464424\pi$
$283$	3.75255	0.223066	0.111533	+	0.993761i	$0.464424\pi$
$284$	−1.10102	−0.0653335
$285$	0	0
$286$	21.7980	1.28894
$287$	1.10102	0.0649912
$288$	−3.00000	−0.176777
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−38.6969	−2.26845
$292$	2.89898	0.169650
$293$	18.2474	1.06603	0.533014	−	0.846107i	$-0.321059\pi$
$293$	18.2474	1.06603	0.533014	+	0.846107i	$0.321059\pi$
$294$	2.44949	0.142857
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	3.79796	0.220010
$299$	12.8990	0.745967
$300$	0	0
$301$	0.898979	0.0518163
$302$	−19.5959	−1.12762
$303$	−8.69694	−0.499626
$304$	1.55051	0.0889279
$305$	0	0
$306$	−6.00000	−0.342997
$307$	−20.2474	−1.15558	−0.577791	−	0.816184i	$-0.696085\pi$
$307$	−20.2474	−1.15558	−0.577791	+	0.816184i	$0.696085\pi$
$308$	4.89898	0.279145
$309$	−31.5959	−1.79743
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	10.8990	0.617033
$313$	−21.5959	−1.22067	−0.610337	−	0.792142i	$-0.708966\pi$
$313$	−21.5959	−1.22067	−0.610337	+	0.792142i	$0.708966\pi$
$314$	−3.55051	−0.200367
$315$	0	0
$316$	6.89898	0.388098
$317$	−22.4949	−1.26344	−0.631720	−	0.775197i	$-0.717651\pi$
$317$	−22.4949	−1.26344	−0.631720	+	0.775197i	$0.717651\pi$
$318$	−26.6969	−1.49709
$319$	−33.7980	−1.89232
$320$	0	0
$321$	19.5959	1.09374
$322$	2.89898	0.161554
$323$	3.10102	0.172545
$324$	−9.00000	−0.500000
$325$	0	0
$326$	7.10102	0.393289
$327$	−16.8990	−0.934516
$328$	1.10102	0.0607937
$329$	−8.89898	−0.490617
$330$	0	0
$331$	−18.6969	−1.02768	−0.513838	−	0.857887i	$-0.671777\pi$
$331$	−18.6969	−1.02768	−0.513838	+	0.857887i	$0.671777\pi$
$332$	−2.44949	−0.134433
$333$	6.00000	0.328798
$334$	4.89898	0.268060
$335$	0	0
$336$	2.44949	0.133631
$337$	9.59592	0.522723	0.261361	−	0.965241i	$-0.415829\pi$
$337$	9.59592	0.522723	0.261361	+	0.965241i	$0.415829\pi$
$338$	−6.79796	−0.369760
$339$	−48.4949	−2.63388
$340$	0	0
$341$	−43.5959	−2.36085
$342$	−4.65153	−0.251526
$343$	−1.00000	−0.0539949
$344$	0.898979	0.0484697
$345$	0	0
$346$	6.24745	0.335865
$347$	28.8990	1.55138	0.775689	−	0.631115i	$-0.217402\pi$
$347$	28.8990	1.55138	0.775689	+	0.631115i	$0.217402\pi$
$348$	−16.8990	−0.905880
$349$	−8.44949	−0.452291	−0.226145	−	0.974094i	$-0.572612\pi$
$349$	−8.44949	−0.452291	−0.226145	+	0.974094i	$0.572612\pi$
$350$	0	0
$351$	0	0
$352$	4.89898	0.261116
$353$	22.8990	1.21879	0.609395	−	0.792867i	$-0.291412\pi$
$353$	22.8990	1.21879	0.609395	+	0.792867i	$0.291412\pi$
$354$	−3.79796	−0.201859
$355$	0	0
$356$	−10.0000	−0.529999
$357$	4.89898	0.259281
$358$	−13.7980	−0.729245
$359$	−27.5959	−1.45646	−0.728228	−	0.685334i	$-0.759656\pi$
$359$	−27.5959	−1.45646	−0.728228	+	0.685334i	$0.759656\pi$
$360$	0	0
$361$	−16.5959	−0.873469
$362$	10.2474	0.538594
$363$	−31.8434	−1.67134
$364$	−4.44949	−0.233217
$365$	0	0
$366$	8.69694	0.454596
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	2.89898	0.151120
$369$	−3.30306	−0.171951
$370$	0	0
$371$	10.8990	0.565847
$372$	−21.7980	−1.13017
$373$	−4.69694	−0.243198	−0.121599	−	0.992579i	$-0.538802\pi$
$373$	−4.69694	−0.243198	−0.121599	+	0.992579i	$0.538802\pi$
$374$	9.79796	0.506640
$375$	0	0
$376$	−8.89898	−0.458930
$377$	30.6969	1.58097
$378$	0	0
$379$	30.6969	1.57680	0.788398	−	0.615166i	$-0.210911\pi$
$379$	30.6969	1.57680	0.788398	+	0.615166i	$0.210911\pi$
$380$	0	0
$381$	36.4949	1.86969
$382$	−12.6969	−0.649632
$383$	−7.10102	−0.362845	−0.181423	−	0.983405i	$-0.558070\pi$
$383$	−7.10102	−0.362845	−0.181423	+	0.983405i	$0.558070\pi$
$384$	2.44949	0.125000
$385$	0	0
$386$	21.5959	1.09920
$387$	−2.69694	−0.137093
$388$	15.7980	0.802020
$389$	13.1010	0.664248	0.332124	−	0.943236i	$-0.392235\pi$
$389$	13.1010	0.664248	0.332124	+	0.943236i	$0.392235\pi$
$390$	0	0
$391$	5.79796	0.293215
$392$	−1.00000	−0.0505076
$393$	15.7980	0.796902
$394$	−18.8990	−0.952117
$395$	0	0
$396$	−14.6969	−0.738549
$397$	−2.65153	−0.133077	−0.0665383	−	0.997784i	$-0.521195\pi$
$397$	−2.65153	−0.133077	−0.0665383	+	0.997784i	$0.521195\pi$
$398$	−16.8990	−0.847069
$399$	3.79796	0.190136
$400$	0	0
$401$	−29.3939	−1.46786	−0.733930	−	0.679225i	$-0.762316\pi$
$401$	−29.3939	−1.46786	−0.733930	+	0.679225i	$0.762316\pi$
$402$	−19.5959	−0.977356
$403$	39.5959	1.97241
$404$	3.55051	0.176644
$405$	0	0
$406$	6.89898	0.342391
$407$	−9.79796	−0.485667
$408$	4.89898	0.242536
$409$	34.4949	1.70566	0.852831	−	0.522186i	$-0.174883\pi$
$409$	34.4949	1.70566	0.852831	+	0.522186i	$0.174883\pi$
$410$	0	0
$411$	4.40408	0.217237
$412$	12.8990	0.635487
$413$	1.55051	0.0762956
$414$	−8.69694	−0.427431
$415$	0	0
$416$	−4.44949	−0.218154
$417$	−3.79796	−0.185987
$418$	7.59592	0.371528
$419$	−1.55051	−0.0757474	−0.0378737	−	0.999283i	$-0.512058\pi$
$419$	−1.55051	−0.0757474	−0.0378737	+	0.999283i	$0.512058\pi$
$420$	0	0
$421$	−4.20204	−0.204795	−0.102397	−	0.994744i	$-0.532651\pi$
$421$	−4.20204	−0.204795	−0.102397	+	0.994744i	$0.532651\pi$
$422$	−12.0000	−0.584151
$423$	26.6969	1.29805
$424$	10.8990	0.529301
$425$	0	0
$426$	−2.69694	−0.130667
$427$	−3.55051	−0.171821
$428$	−8.00000	−0.386695
$429$	53.3939	2.57788
$430$	0	0
$431$	−1.79796	−0.0866046	−0.0433023	−	0.999062i	$-0.513788\pi$
$431$	−1.79796	−0.0866046	−0.0433023	+	0.999062i	$0.513788\pi$
$432$	0	0
$433$	−0.202041	−0.00970947	−0.00485474	−	0.999988i	$-0.501545\pi$
$433$	−0.202041	−0.00970947	−0.00485474	+	0.999988i	$0.501545\pi$
$434$	8.89898	0.427165
$435$	0	0
$436$	6.89898	0.330401
$437$	4.49490	0.215020
$438$	7.10102	0.339300
$439$	−21.3939	−1.02107	−0.510537	−	0.859856i	$-0.670553\pi$
$439$	−21.3939	−1.02107	−0.510537	+	0.859856i	$0.670553\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	−8.89898	−0.423281
$443$	9.79796	0.465515	0.232758	−	0.972535i	$-0.425225\pi$
$443$	9.79796	0.465515	0.232758	+	0.972535i	$0.425225\pi$
$444$	−4.89898	−0.232495
$445$	0	0
$446$	4.00000	0.189405
$447$	9.30306	0.440020
$448$	−1.00000	−0.0472456
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	5.39388	0.253988
$452$	19.7980	0.931218
$453$	−48.0000	−2.25524
$454$	−7.34847	−0.344881
$455$	0	0
$456$	3.79796	0.177856
$457$	29.5959	1.38444	0.692219	−	0.721687i	$-0.256633\pi$
$457$	29.5959	1.38444	0.692219	+	0.721687i	$0.256633\pi$
$458$	19.1464	0.894654
$459$	0	0
$460$	0	0
$461$	17.3485	0.807999	0.403999	−	0.914759i	$-0.367620\pi$
$461$	17.3485	0.807999	0.403999	+	0.914759i	$0.367620\pi$
$462$	12.0000	0.558291
$463$	3.59592	0.167116	0.0835582	−	0.996503i	$-0.473372\pi$
$463$	3.59592	0.167116	0.0835582	+	0.996503i	$0.473372\pi$
$464$	6.89898	0.320277
$465$	0	0
$466$	−29.7980	−1.38036
$467$	10.4495	0.483545	0.241772	−	0.970333i	$-0.422271\pi$
$467$	10.4495	0.483545	0.241772	+	0.970333i	$0.422271\pi$
$468$	13.3485	0.617033
$469$	8.00000	0.369406
$470$	0	0
$471$	−8.69694	−0.400734
$472$	1.55051	0.0713680
$473$	4.40408	0.202500
$474$	16.8990	0.776196
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	−32.6969	−1.49709
$478$	6.20204	0.283675
$479$	9.30306	0.425068	0.212534	−	0.977154i	$-0.431828\pi$
$479$	9.30306	0.425068	0.212534	+	0.977154i	$0.431828\pi$
$480$	0	0
$481$	8.89898	0.405759
$482$	8.69694	0.396135
$483$	7.10102	0.323108
$484$	13.0000	0.590909
$485$	0	0
$486$	−22.0454	−1.00000
$487$	−7.30306	−0.330933	−0.165467	−	0.986215i	$-0.552913\pi$
$487$	−7.30306	−0.330933	−0.165467	+	0.986215i	$0.552913\pi$
$488$	−3.55051	−0.160724
$489$	17.3939	0.786578
$490$	0	0
$491$	19.5959	0.884351	0.442176	−	0.896928i	$-0.354207\pi$
$491$	19.5959	0.884351	0.442176	+	0.896928i	$0.354207\pi$
$492$	2.69694	0.121587
$493$	13.7980	0.621429
$494$	−6.89898	−0.310400
$495$	0	0
$496$	8.89898	0.399576
$497$	1.10102	0.0493875
$498$	−6.00000	−0.268866
$499$	6.20204	0.277641	0.138821	−	0.990318i	$-0.455669\pi$
$499$	6.20204	0.277641	0.138821	+	0.990318i	$0.455669\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	6.44949	0.287855
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	14.2020	0.631358
$507$	−16.6515	−0.739520
$508$	−14.8990	−0.661035
$509$	−31.5505	−1.39845	−0.699226	−	0.714901i	$-0.746472\pi$
$509$	−31.5505	−1.39845	−0.699226	+	0.714901i	$0.746472\pi$
$510$	0	0
$511$	−2.89898	−0.128243
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	8.69694	0.383606
$515$	0	0
$516$	2.20204	0.0969395
$517$	−43.5959	−1.91735
$518$	2.00000	0.0878750
$519$	15.3031	0.671730
$520$	0	0
$521$	32.6969	1.43248	0.716239	−	0.697855i	$-0.245862\pi$
$521$	32.6969	1.43248	0.716239	+	0.697855i	$0.245862\pi$
$522$	−20.6969	−0.905880
$523$	−33.1464	−1.44939	−0.724696	−	0.689069i	$-0.758020\pi$
$523$	−33.1464	−1.44939	−0.724696	+	0.689069i	$0.758020\pi$
$524$	−6.44949	−0.281747
$525$	0	0
$526$	−9.79796	−0.427211
$527$	17.7980	0.775291
$528$	12.0000	0.522233
$529$	−14.5959	−0.634605
$530$	0	0
$531$	−4.65153	−0.201859
$532$	−1.55051	−0.0672231
$533$	−4.89898	−0.212198
$534$	−24.4949	−1.06000
$535$	0	0
$536$	8.00000	0.345547
$537$	−33.7980	−1.45849
$538$	−19.1464	−0.825461
$539$	−4.89898	−0.211014
$540$	0	0
$541$	9.59592	0.412561	0.206280	−	0.978493i	$-0.433864\pi$
$541$	9.59592	0.412561	0.206280	+	0.978493i	$0.433864\pi$
$542$	−12.0000	−0.515444
$543$	25.1010	1.07719
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	−10.8990	−0.466433
$547$	−18.6969	−0.799423	−0.399712	−	0.916641i	$-0.630890\pi$
$547$	−18.6969	−0.799423	−0.399712	+	0.916641i	$0.630890\pi$
$548$	−1.79796	−0.0768050
$549$	10.6515	0.454596
$550$	0	0
$551$	10.6969	0.455705
$552$	7.10102	0.302240
$553$	−6.89898	−0.293374
$554$	14.8990	0.632997
$555$	0	0
$556$	1.55051	0.0657563
$557$	12.6969	0.537987	0.268993	−	0.963142i	$-0.413309\pi$
$557$	12.6969	0.537987	0.268993	+	0.963142i	$0.413309\pi$
$558$	−26.6969	−1.13017
$559$	−4.00000	−0.169182
$560$	0	0
$561$	24.0000	1.01328
$562$	18.0000	0.759284
$563$	−30.0454	−1.26626	−0.633131	−	0.774044i	$-0.718231\pi$
$563$	−30.0454	−1.26626	−0.633131	+	0.774044i	$0.718231\pi$
$564$	−21.7980	−0.917860
$565$	0	0
$566$	−3.75255	−0.157731
$567$	9.00000	0.377964
$568$	1.10102	0.0461978
$569$	33.7980	1.41688	0.708442	−	0.705769i	$-0.249398\pi$
$569$	33.7980	1.41688	0.708442	+	0.705769i	$0.249398\pi$
$570$	0	0
$571$	−11.1010	−0.464563	−0.232282	−	0.972649i	$-0.574619\pi$
$571$	−11.1010	−0.464563	−0.232282	+	0.972649i	$0.574619\pi$
$572$	−21.7980	−0.911418
$573$	−31.1010	−1.29926
$574$	−1.10102	−0.0459557
$575$	0	0
$576$	3.00000	0.125000
$577$	−2.49490	−0.103864	−0.0519320	−	0.998651i	$-0.516538\pi$
$577$	−2.49490	−0.103864	−0.0519320	+	0.998651i	$0.516538\pi$
$578$	13.0000	0.540729
$579$	52.8990	2.19841
$580$	0	0
$581$	2.44949	0.101622
$582$	38.6969	1.60404
$583$	53.3939	2.21135
$584$	−2.89898	−0.119961
$585$	0	0
$586$	−18.2474	−0.753795
$587$	1.14643	0.0473182	0.0236591	−	0.999720i	$-0.492468\pi$
$587$	1.14643	0.0473182	0.0236591	+	0.999720i	$0.492468\pi$
$588$	−2.44949	−0.101015
$589$	13.7980	0.568535
$590$	0	0
$591$	−46.2929	−1.90423
$592$	2.00000	0.0821995
$593$	−10.8990	−0.447567	−0.223784	−	0.974639i	$-0.571841\pi$
$593$	−10.8990	−0.447567	−0.223784	+	0.974639i	$0.571841\pi$
$594$	0	0
$595$	0	0
$596$	−3.79796	−0.155570
$597$	−41.3939	−1.69414
$598$	−12.8990	−0.527478
$599$	−13.1010	−0.535293	−0.267647	−	0.963517i	$-0.586246\pi$
$599$	−13.1010	−0.535293	−0.267647	+	0.963517i	$0.586246\pi$
$600$	0	0
$601$	−39.3939	−1.60691	−0.803455	−	0.595366i	$-0.797007\pi$
$601$	−39.3939	−1.60691	−0.803455	+	0.595366i	$0.797007\pi$
$602$	−0.898979	−0.0366397
$603$	−24.0000	−0.977356
$604$	19.5959	0.797347
$605$	0	0
$606$	8.69694	0.353289
$607$	33.3939	1.35542	0.677708	−	0.735331i	$-0.262973\pi$
$607$	33.3939	1.35542	0.677708	+	0.735331i	$0.262973\pi$
$608$	−1.55051	−0.0628815
$609$	16.8990	0.684781
$610$	0	0
$611$	39.5959	1.60188
$612$	6.00000	0.242536
$613$	−27.7980	−1.12275	−0.561374	−	0.827562i	$-0.689727\pi$
$613$	−27.7980	−1.12275	−0.561374	+	0.827562i	$0.689727\pi$
$614$	20.2474	0.817121
$615$	0	0
$616$	−4.89898	−0.197386
$617$	29.5959	1.19149	0.595743	−	0.803175i	$-0.296858\pi$
$617$	29.5959	1.19149	0.595743	+	0.803175i	$0.296858\pi$
$618$	31.5959	1.27097
$619$	−41.5505	−1.67006	−0.835028	−	0.550207i	$-0.814549\pi$
$619$	−41.5505	−1.67006	−0.835028	+	0.550207i	$0.814549\pi$
$620$	0	0
$621$	0	0
$622$	−12.0000	−0.481156
$623$	10.0000	0.400642
$624$	−10.8990	−0.436308
$625$	0	0
$626$	21.5959	0.863146
$627$	18.6061	0.743057
$628$	3.55051	0.141681
$629$	4.00000	0.159490
$630$	0	0
$631$	−42.4949	−1.69170	−0.845848	−	0.533425i	$-0.820905\pi$
$631$	−42.4949	−1.69170	−0.845848	+	0.533425i	$0.820905\pi$
$632$	−6.89898	−0.274427
$633$	−29.3939	−1.16830
$634$	22.4949	0.893387
$635$	0	0
$636$	26.6969	1.05860
$637$	4.44949	0.176295
$638$	33.7980	1.33807
$639$	−3.30306	−0.130667
$640$	0	0
$641$	25.7980	1.01896	0.509479	−	0.860483i	$-0.329838\pi$
$641$	25.7980	1.01896	0.509479	+	0.860483i	$0.329838\pi$
$642$	−19.5959	−0.773389
$643$	25.1464	0.991678	0.495839	−	0.868414i	$-0.334861\pi$
$643$	25.1464	0.991678	0.495839	+	0.868414i	$0.334861\pi$
$644$	−2.89898	−0.114236
$645$	0	0
$646$	−3.10102	−0.122008
$647$	−46.2929	−1.81996	−0.909980	−	0.414652i	$-0.863903\pi$
$647$	−46.2929	−1.81996	−0.909980	+	0.414652i	$0.863903\pi$
$648$	9.00000	0.353553
$649$	7.59592	0.298166
$650$	0	0
$651$	21.7980	0.854329
$652$	−7.10102	−0.278097
$653$	−20.2020	−0.790567	−0.395283	−	0.918559i	$-0.629354\pi$
$653$	−20.2020	−0.790567	−0.395283	+	0.918559i	$0.629354\pi$
$654$	16.8990	0.660802
$655$	0	0
$656$	−1.10102	−0.0429876
$657$	8.69694	0.339300
$658$	8.89898	0.346918
$659$	16.8990	0.658291	0.329145	−	0.944279i	$-0.393239\pi$
$659$	16.8990	0.658291	0.329145	+	0.944279i	$0.393239\pi$
$660$	0	0
$661$	−40.9444	−1.59255	−0.796276	−	0.604933i	$-0.793200\pi$
$661$	−40.9444	−1.59255	−0.796276	+	0.604933i	$0.793200\pi$
$662$	18.6969	0.726677
$663$	−21.7980	−0.846563
$664$	2.44949	0.0950586
$665$	0	0
$666$	−6.00000	−0.232495
$667$	20.0000	0.774403
$668$	−4.89898	−0.189547
$669$	9.79796	0.378811
$670$	0	0
$671$	−17.3939	−0.671483
$672$	−2.44949	−0.0944911
$673$	−17.7980	−0.686061	−0.343030	−	0.939324i	$-0.611453\pi$
$673$	−17.7980	−0.686061	−0.343030	+	0.939324i	$0.611453\pi$
$674$	−9.59592	−0.369621
$675$	0	0
$676$	6.79796	0.261460
$677$	−36.4495	−1.40087	−0.700434	−	0.713717i	$-0.747010\pi$
$677$	−36.4495	−1.40087	−0.700434	+	0.713717i	$0.747010\pi$
$678$	48.4949	1.86244
$679$	−15.7980	−0.606270
$680$	0	0
$681$	−18.0000	−0.689761
$682$	43.5959	1.66937
$683$	3.59592	0.137594	0.0687970	−	0.997631i	$-0.478084\pi$
$683$	3.59592	0.137594	0.0687970	+	0.997631i	$0.478084\pi$
$684$	4.65153	0.177856
$685$	0	0
$686$	1.00000	0.0381802
$687$	46.8990	1.78931
$688$	−0.898979	−0.0342733
$689$	−48.4949	−1.84751
$690$	0	0
$691$	21.1464	0.804448	0.402224	−	0.915541i	$-0.368237\pi$
$691$	21.1464	0.804448	0.402224	+	0.915541i	$0.368237\pi$
$692$	−6.24745	−0.237492
$693$	14.6969	0.558291
$694$	−28.8990	−1.09699
$695$	0	0
$696$	16.8990	0.640554
$697$	−2.20204	−0.0834083
$698$	8.44949	0.319818
$699$	−72.9898	−2.76073
$700$	0	0
$701$	11.3031	0.426911	0.213455	−	0.976953i	$-0.431528\pi$
$701$	11.3031	0.426911	0.213455	+	0.976953i	$0.431528\pi$
$702$	0	0
$703$	3.10102	0.116957
$704$	−4.89898	−0.184637
$705$	0	0
$706$	−22.8990	−0.861814
$707$	−3.55051	−0.133531
$708$	3.79796	0.142736
$709$	−28.2929	−1.06256	−0.531280	−	0.847196i	$-0.678289\pi$
$709$	−28.2929	−1.06256	−0.531280	+	0.847196i	$0.678289\pi$
$710$	0	0
$711$	20.6969	0.776196
$712$	10.0000	0.374766
$713$	25.7980	0.966141
$714$	−4.89898	−0.183340
$715$	0	0
$716$	13.7980	0.515654
$717$	15.1918	0.567350
$718$	27.5959	1.02987
$719$	−4.49490	−0.167631	−0.0838157	−	0.996481i	$-0.526711\pi$
$719$	−4.49490	−0.167631	−0.0838157	+	0.996481i	$0.526711\pi$
$720$	0	0
$721$	−12.8990	−0.480383
$722$	16.5959	0.617636
$723$	21.3031	0.792269
$724$	−10.2474	−0.380843
$725$	0	0
$726$	31.8434	1.18182
$727$	22.6969	0.841783	0.420891	−	0.907111i	$-0.361717\pi$
$727$	22.6969	0.841783	0.420891	+	0.907111i	$0.361717\pi$
$728$	4.44949	0.164909
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−1.79796	−0.0664999
$732$	−8.69694	−0.321448
$733$	39.6413	1.46419	0.732093	−	0.681205i	$-0.238544\pi$
$733$	39.6413	1.46419	0.732093	+	0.681205i	$0.238544\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	−2.89898	−0.106858
$737$	39.1918	1.44365
$738$	3.30306	0.121587
$739$	4.49490	0.165347	0.0826737	−	0.996577i	$-0.473654\pi$
$739$	4.49490	0.165347	0.0826737	+	0.996577i	$0.473654\pi$
$740$	0	0
$741$	−16.8990	−0.620800
$742$	−10.8990	−0.400114
$743$	−44.6969	−1.63977	−0.819886	−	0.572527i	$-0.805963\pi$
$743$	−44.6969	−1.63977	−0.819886	+	0.572527i	$0.805963\pi$
$744$	21.7980	0.799152
$745$	0	0
$746$	4.69694	0.171967
$747$	−7.34847	−0.268866
$748$	−9.79796	−0.358249
$749$	8.00000	0.292314
$750$	0	0
$751$	−41.7980	−1.52523	−0.762615	−	0.646853i	$-0.776085\pi$
$751$	−41.7980	−1.52523	−0.762615	+	0.646853i	$0.776085\pi$
$752$	8.89898	0.324512
$753$	15.7980	0.575710
$754$	−30.6969	−1.11792
$755$	0	0
$756$	0	0
$757$	−51.7980	−1.88263	−0.941314	−	0.337531i	$-0.890408\pi$
$757$	−51.7980	−1.88263	−0.941314	+	0.337531i	$0.890408\pi$
$758$	−30.6969	−1.11496
$759$	34.7878	1.26272
$760$	0	0
$761$	−21.1010	−0.764911	−0.382456	−	0.923974i	$-0.624922\pi$
$761$	−21.1010	−0.764911	−0.382456	+	0.923974i	$0.624922\pi$
$762$	−36.4949	−1.32207
$763$	−6.89898	−0.249760
$764$	12.6969	0.459359
$765$	0	0
$766$	7.10102	0.256570
$767$	−6.89898	−0.249108
$768$	−2.44949	−0.0883883
$769$	−40.6969	−1.46757	−0.733785	−	0.679382i	$-0.762248\pi$
$769$	−40.6969	−1.46757	−0.733785	+	0.679382i	$0.762248\pi$
$770$	0	0
$771$	21.3031	0.767211
$772$	−21.5959	−0.777254
$773$	1.34847	0.0485011	0.0242505	−	0.999706i	$-0.492280\pi$
$773$	1.34847	0.0485011	0.0242505	+	0.999706i	$0.492280\pi$
$774$	2.69694	0.0969395
$775$	0	0
$776$	−15.7980	−0.567114
$777$	4.89898	0.175750
$778$	−13.1010	−0.469694
$779$	−1.70714	−0.0611648
$780$	0	0
$781$	5.39388	0.193008
$782$	−5.79796	−0.207335
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	−15.7980	−0.563495
$787$	50.4495	1.79833	0.899165	−	0.437610i	$-0.144175\pi$
$787$	50.4495	1.79833	0.899165	+	0.437610i	$0.144175\pi$
$788$	18.8990	0.673248
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−19.7980	−0.703934
$792$	14.6969	0.522233
$793$	15.7980	0.561002
$794$	2.65153	0.0940993
$795$	0	0
$796$	16.8990	0.598968
$797$	−0.944387	−0.0334519	−0.0167260	−	0.999860i	$-0.505324\pi$
$797$	−0.944387	−0.0334519	−0.0167260	+	0.999860i	$0.505324\pi$
$798$	−3.79796	−0.134446
$799$	17.7980	0.629647
$800$	0	0
$801$	−30.0000	−1.06000
$802$	29.3939	1.03793
$803$	−14.2020	−0.501179
$804$	19.5959	0.691095
$805$	0	0
$806$	−39.5959	−1.39471
$807$	−46.8990	−1.65092
$808$	−3.55051	−0.124907
$809$	47.5959	1.67338	0.836692	−	0.547674i	$-0.184487\pi$
$809$	47.5959	1.67338	0.836692	+	0.547674i	$0.184487\pi$
$810$	0	0
$811$	14.9444	0.524768	0.262384	−	0.964963i	$-0.415491\pi$
$811$	14.9444	0.524768	0.262384	+	0.964963i	$0.415491\pi$
$812$	−6.89898	−0.242107
$813$	−29.3939	−1.03089
$814$	9.79796	0.343418
$815$	0	0
$816$	−4.89898	−0.171499
$817$	−1.39388	−0.0487656
$818$	−34.4949	−1.20609
$819$	−13.3485	−0.466433
$820$	0	0
$821$	8.20204	0.286253	0.143127	−	0.989704i	$-0.454284\pi$
$821$	8.20204	0.286253	0.143127	+	0.989704i	$0.454284\pi$
$822$	−4.40408	−0.153610
$823$	−39.1918	−1.36614	−0.683071	−	0.730352i	$-0.739356\pi$
$823$	−39.1918	−1.36614	−0.683071	+	0.730352i	$0.739356\pi$
$824$	−12.8990	−0.449357
$825$	0	0
$826$	−1.55051	−0.0539492
$827$	−15.5959	−0.542323	−0.271162	−	0.962534i	$-0.587408\pi$
$827$	−15.5959	−0.542323	−0.271162	+	0.962534i	$0.587408\pi$
$828$	8.69694	0.302240
$829$	−43.6413	−1.51573	−0.757863	−	0.652414i	$-0.773756\pi$
$829$	−43.6413	−1.51573	−0.757863	+	0.652414i	$0.773756\pi$
$830$	0	0
$831$	36.4949	1.26599
$832$	4.44949	0.154258
$833$	2.00000	0.0692959
$834$	3.79796	0.131513
$835$	0	0
$836$	−7.59592	−0.262710
$837$	0	0
$838$	1.55051	0.0535615
$839$	36.8990	1.27389	0.636947	−	0.770907i	$-0.280197\pi$
$839$	36.8990	1.27389	0.636947	+	0.770907i	$0.280197\pi$
$840$	0	0
$841$	18.5959	0.641239
$842$	4.20204	0.144812
$843$	44.0908	1.51857
$844$	12.0000	0.413057
$845$	0	0
$846$	−26.6969	−0.917860
$847$	−13.0000	−0.446685
$848$	−10.8990	−0.374272
$849$	−9.19184	−0.315463
$850$	0	0
$851$	5.79796	0.198751
$852$	2.69694	0.0923956
$853$	−33.8434	−1.15877	−0.579387	−	0.815052i	$-0.696708\pi$
$853$	−33.8434	−1.15877	−0.579387	+	0.815052i	$0.696708\pi$
$854$	3.55051	0.121496
$855$	0	0
$856$	8.00000	0.273434
$857$	−53.1918	−1.81700	−0.908499	−	0.417886i	$-0.862771\pi$
$857$	−53.1918	−1.81700	−0.908499	+	0.417886i	$0.862771\pi$
$858$	−53.3939	−1.82284
$859$	53.6413	1.83022	0.915109	−	0.403206i	$-0.132104\pi$
$859$	53.6413	1.83022	0.915109	+	0.403206i	$0.132104\pi$
$860$	0	0
$861$	−2.69694	−0.0919114
$862$	1.79796	0.0612387
$863$	−45.3939	−1.54523	−0.772613	−	0.634878i	$-0.781051\pi$
$863$	−45.3939	−1.54523	−0.772613	+	0.634878i	$0.781051\pi$
$864$	0	0
$865$	0	0
$866$	0.202041	0.00686563
$867$	31.8434	1.08146
$868$	−8.89898	−0.302051
$869$	−33.7980	−1.14652
$870$	0	0
$871$	−35.5959	−1.20612
$872$	−6.89898	−0.233629
$873$	47.3939	1.60404
$874$	−4.49490	−0.152042
$875$	0	0
$876$	−7.10102	−0.239921
$877$	−39.3939	−1.33024	−0.665118	−	0.746738i	$-0.731619\pi$
$877$	−39.3939	−1.33024	−0.665118	+	0.746738i	$0.731619\pi$
$878$	21.3939	0.722008
$879$	−44.6969	−1.50759
$880$	0	0
$881$	8.20204	0.276334	0.138167	−	0.990409i	$-0.455879\pi$
$881$	8.20204	0.276334	0.138167	+	0.990409i	$0.455879\pi$
$882$	−3.00000	−0.101015
$883$	22.2020	0.747158	0.373579	−	0.927598i	$-0.378130\pi$
$883$	22.2020	0.747158	0.373579	+	0.927598i	$0.378130\pi$
$884$	8.89898	0.299305
$885$	0	0
$886$	−9.79796	−0.329169
$887$	2.69694	0.0905543	0.0452772	−	0.998974i	$-0.485583\pi$
$887$	2.69694	0.0905543	0.0452772	+	0.998974i	$0.485583\pi$
$888$	4.89898	0.164399
$889$	14.8990	0.499696
$890$	0	0
$891$	44.0908	1.47710
$892$	−4.00000	−0.133930
$893$	13.7980	0.461731
$894$	−9.30306	−0.311141
$895$	0	0
$896$	1.00000	0.0334077
$897$	−31.5959	−1.05496
$898$	10.0000	0.333704
$899$	61.3939	2.04760
$900$	0	0
$901$	−21.7980	−0.726195
$902$	−5.39388	−0.179596
$903$	−2.20204	−0.0732793
$904$	−19.7980	−0.658470
$905$	0	0
$906$	48.0000	1.59469
$907$	−41.7980	−1.38788	−0.693939	−	0.720034i	$-0.744126\pi$
$907$	−41.7980	−1.38788	−0.693939	+	0.720034i	$0.744126\pi$
$908$	7.34847	0.243868
$909$	10.6515	0.353289
$910$	0	0
$911$	−35.5959	−1.17935	−0.589673	−	0.807642i	$-0.700743\pi$
$911$	−35.5959	−1.17935	−0.589673	+	0.807642i	$0.700743\pi$
$912$	−3.79796	−0.125763
$913$	12.0000	0.397142
$914$	−29.5959	−0.978946
$915$	0	0
$916$	−19.1464	−0.632616
$917$	6.44949	0.212981
$918$	0	0
$919$	−26.8990	−0.887315	−0.443658	−	0.896196i	$-0.646319\pi$
$919$	−26.8990	−0.887315	−0.443658	+	0.896196i	$0.646319\pi$
$920$	0	0
$921$	49.5959	1.63424
$922$	−17.3485	−0.571341
$923$	−4.89898	−0.161252
$924$	−12.0000	−0.394771
$925$	0	0
$926$	−3.59592	−0.118169
$927$	38.6969	1.27097
$928$	−6.89898	−0.226470
$929$	−28.2929	−0.928259	−0.464129	−	0.885767i	$-0.653633\pi$
$929$	−28.2929	−0.928259	−0.464129	+	0.885767i	$0.653633\pi$
$930$	0	0
$931$	1.55051	0.0508159
$932$	29.7980	0.976065
$933$	−29.3939	−0.962312
$934$	−10.4495	−0.341918
$935$	0	0
$936$	−13.3485	−0.436308
$937$	−41.1010	−1.34271	−0.671356	−	0.741135i	$-0.734288\pi$
$937$	−41.1010	−1.34271	−0.671356	+	0.741135i	$0.734288\pi$
$938$	−8.00000	−0.261209
$939$	52.8990	1.72629
$940$	0	0
$941$	−19.5505	−0.637328	−0.318664	−	0.947868i	$-0.603234\pi$
$941$	−19.5505	−0.637328	−0.318664	+	0.947868i	$0.603234\pi$
$942$	8.69694	0.283362
$943$	−3.19184	−0.103940
$944$	−1.55051	−0.0504648
$945$	0	0
$946$	−4.40408	−0.143189
$947$	44.0908	1.43276	0.716379	−	0.697711i	$-0.245798\pi$
$947$	44.0908	1.43276	0.716379	+	0.697711i	$0.245798\pi$
$948$	−16.8990	−0.548853
$949$	12.8990	0.418719
$950$	0	0
$951$	55.1010	1.78677
$952$	2.00000	0.0648204
$953$	2.20204	0.0713311	0.0356656	−	0.999364i	$-0.488645\pi$
$953$	2.20204	0.0713311	0.0356656	+	0.999364i	$0.488645\pi$
$954$	32.6969	1.05860
$955$	0	0
$956$	−6.20204	−0.200588
$957$	82.7878	2.67615
$958$	−9.30306	−0.300568
$959$	1.79796	0.0580591
$960$	0	0
$961$	48.1918	1.55458
$962$	−8.89898	−0.286915
$963$	−24.0000	−0.773389
$964$	−8.69694	−0.280110
$965$	0	0
$966$	−7.10102	−0.228472
$967$	−36.2929	−1.16710	−0.583550	−	0.812077i	$-0.698337\pi$
$967$	−36.2929	−1.16710	−0.583550	+	0.812077i	$0.698337\pi$
$968$	−13.0000	−0.417836
$969$	−7.59592	−0.244016
$970$	0	0
$971$	−9.55051	−0.306490	−0.153245	−	0.988188i	$-0.548972\pi$
$971$	−9.55051	−0.306490	−0.153245	+	0.988188i	$0.548972\pi$
$972$	22.0454	0.707107
$973$	−1.55051	−0.0497071
$974$	7.30306	0.234005
$975$	0	0
$976$	3.55051	0.113649
$977$	−29.3939	−0.940393	−0.470197	−	0.882562i	$-0.655817\pi$
$977$	−29.3939	−0.940393	−0.470197	+	0.882562i	$0.655817\pi$
$978$	−17.3939	−0.556195
$979$	48.9898	1.56572
$980$	0	0
$981$	20.6969	0.660802
$982$	−19.5959	−0.625331
$983$	−13.3031	−0.424302	−0.212151	−	0.977237i	$-0.568047\pi$
$983$	−13.3031	−0.424302	−0.212151	+	0.977237i	$0.568047\pi$
$984$	−2.69694	−0.0859753
$985$	0	0
$986$	−13.7980	−0.439417
$987$	21.7980	0.693837
$988$	6.89898	0.219486
$989$	−2.60612	−0.0828699
$990$	0	0
$991$	31.3031	0.994375	0.497187	−	0.867643i	$-0.334366\pi$
$991$	31.3031	0.994375	0.497187	+	0.867643i	$0.334366\pi$
$992$	−8.89898	−0.282543
$993$	45.7980	1.45335
$994$	−1.10102	−0.0349223
$995$	0	0
$996$	6.00000	0.190117
$997$	57.3485	1.81624	0.908122	−	0.418705i	$-0.137516\pi$
$997$	57.3485	1.81624	0.908122	+	0.418705i	$0.137516\pi$
$998$	−6.20204	−0.196322
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.a.g.1.1		2
3.2	odd	2		3150.2.a.bt.1.2		2
4.3	odd	2		2800.2.a.bm.1.2		2
5.2	odd	4		70.2.c.a.29.2	✓	4
5.3	odd	4		70.2.c.a.29.3	yes	4
5.4	even	2		350.2.a.h.1.2		2
7.6	odd	2		2450.2.a.bl.1.2		2
15.2	even	4		630.2.g.g.379.3		4
15.8	even	4		630.2.g.g.379.1		4
15.14	odd	2		3150.2.a.bs.1.2		2
20.3	even	4		560.2.g.e.449.4		4
20.7	even	4		560.2.g.e.449.2		4
20.19	odd	2		2800.2.a.bl.1.1		2
35.2	odd	12		490.2.i.c.459.1		8
35.3	even	12		490.2.i.f.79.2		8
35.12	even	12		490.2.i.f.459.2		8
35.13	even	4		490.2.c.e.99.4		4
35.17	even	12		490.2.i.f.79.3		8
35.18	odd	12		490.2.i.c.79.1		8
35.23	odd	12		490.2.i.c.459.4		8
35.27	even	4		490.2.c.e.99.1		4
35.32	odd	12		490.2.i.c.79.4		8
35.33	even	12		490.2.i.f.459.3		8
35.34	odd	2		2450.2.a.bq.1.1		2
40.3	even	4		2240.2.g.i.449.1		4
40.13	odd	4		2240.2.g.j.449.3		4
40.27	even	4		2240.2.g.i.449.3		4
40.37	odd	4		2240.2.g.j.449.1		4
60.23	odd	4		5040.2.t.t.1009.1		4
60.47	odd	4		5040.2.t.t.1009.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.c.a.29.2	✓	4	5.2	odd	4
70.2.c.a.29.3	yes	4	5.3	odd	4
350.2.a.g.1.1		2	1.1	even	1	trivial
350.2.a.h.1.2		2	5.4	even	2
490.2.c.e.99.1		4	35.27	even	4
490.2.c.e.99.4		4	35.13	even	4
490.2.i.c.79.1		8	35.18	odd	12
490.2.i.c.79.4		8	35.32	odd	12
490.2.i.c.459.1		8	35.2	odd	12
490.2.i.c.459.4		8	35.23	odd	12
490.2.i.f.79.2		8	35.3	even	12
490.2.i.f.79.3		8	35.17	even	12
490.2.i.f.459.2		8	35.12	even	12
490.2.i.f.459.3		8	35.33	even	12
560.2.g.e.449.2		4	20.7	even	4
560.2.g.e.449.4		4	20.3	even	4
630.2.g.g.379.1		4	15.8	even	4
630.2.g.g.379.3		4	15.2	even	4
2240.2.g.i.449.1		4	40.3	even	4
2240.2.g.i.449.3		4	40.27	even	4
2240.2.g.j.449.1		4	40.37	odd	4
2240.2.g.j.449.3		4	40.13	odd	4
2450.2.a.bl.1.2		2	7.6	odd	2
2450.2.a.bq.1.1		2	35.34	odd	2
2800.2.a.bl.1.1		2	20.19	odd	2
2800.2.a.bm.1.2		2	4.3	odd	2
3150.2.a.bs.1.2		2	15.14	odd	2
3150.2.a.bt.1.2		2	3.2	odd	2
5040.2.t.t.1009.1		4	60.23	odd	4
5040.2.t.t.1009.2		4	60.47	odd	4

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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{9} -4.89898 q^{11} -2.44949 q^{12} +4.44949 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} +1.55051 q^{19} +2.44949 q^{21} +4.89898 q^{22} +2.89898 q^{23} +2.44949 q^{24} -4.44949 q^{26} -1.00000 q^{28} +6.89898 q^{29} +8.89898 q^{31} -1.00000 q^{32} +12.0000 q^{33} -2.00000 q^{34} +3.00000 q^{36} +2.00000 q^{37} -1.55051 q^{38} -10.8990 q^{39} -1.10102 q^{41} -2.44949 q^{42} -0.898979 q^{43} -4.89898 q^{44} -2.89898 q^{46} +8.89898 q^{47} -2.44949 q^{48} +1.00000 q^{49} -4.89898 q^{51} +4.44949 q^{52} -10.8990 q^{53} +1.00000 q^{56} -3.79796 q^{57} -6.89898 q^{58} -1.55051 q^{59} +3.55051 q^{61} -8.89898 q^{62} -3.00000 q^{63} +1.00000 q^{64} -12.0000 q^{66} -8.00000 q^{67} +2.00000 q^{68} -7.10102 q^{69} -1.10102 q^{71} -3.00000 q^{72} +2.89898 q^{73} -2.00000 q^{74} +1.55051 q^{76} +4.89898 q^{77} +10.8990 q^{78} +6.89898 q^{79} -9.00000 q^{81} +1.10102 q^{82} -2.44949 q^{83} +2.44949 q^{84} +0.898979 q^{86} -16.8990 q^{87} +4.89898 q^{88} -10.0000 q^{89} -4.44949 q^{91} +2.89898 q^{92} -21.7980 q^{93} -8.89898 q^{94} +2.44949 q^{96} +15.7980 q^{97} -1.00000 q^{98} -14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 6 q^{9} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 8 q^{19} - 4 q^{23} - 4 q^{26} - 2 q^{28} + 4 q^{29} + 8 q^{31} - 2 q^{32} + 24 q^{33} - 4 q^{34}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 6 * q^9 + 4 * q^13 + 2 * q^14 + 2 * q^16 + 4 * q^17 - 6 * q^18 + 8 * q^19 - 4 * q^23 - 4 * q^26 - 2 * q^28 + 4 * q^29 + 8 * q^31 - 2 * q^32 + 24 * q^33 - 4 * q^34 + 6 * q^36 + 4 * q^37 - 8 * q^38 - 12 * q^39 - 12 * q^41 + 8 * q^43 + 4 * q^46 + 8 * q^47 + 2 * q^49 + 4 * q^52 - 12 * q^53 + 2 * q^56 + 12 * q^57 - 4 * q^58 - 8 * q^59 + 12 * q^61 - 8 * q^62 - 6 * q^63 + 2 * q^64 - 24 * q^66 - 16 * q^67 + 4 * q^68 - 24 * q^69 - 12 * q^71 - 6 * q^72 - 4 * q^73 - 4 * q^74 + 8 * q^76 + 12 * q^78 + 4 * q^79 - 18 * q^81 + 12 * q^82 - 8 * q^86 - 24 * q^87 - 20 * q^89 - 4 * q^91 - 4 * q^92 - 24 * q^93 - 8 * q^94 + 12 * q^97 - 2 * q^98

Introduction

L-functions

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