LMFDB - Embedded newform 6930.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6930,2,Mod(1,6930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6930.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6930.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:	$55.3363286007$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2310)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	6930.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} +1.00000 q^{20} +1.00000 q^{22} +1.46410 q^{23} +1.00000 q^{25} -3.46410 q^{26} +1.00000 q^{28} -2.00000 q^{29} +6.92820 q^{31} -1.00000 q^{32} -3.46410 q^{34} +1.00000 q^{35} +7.46410 q^{37} -1.00000 q^{40} +8.92820 q^{41} -2.92820 q^{43} -1.00000 q^{44} -1.46410 q^{46} -6.92820 q^{47} +1.00000 q^{49} -1.00000 q^{50} +3.46410 q^{52} -12.9282 q^{53} -1.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} +2.92820 q^{59} -4.92820 q^{61} -6.92820 q^{62} +1.00000 q^{64} +3.46410 q^{65} +9.46410 q^{67} +3.46410 q^{68} -1.00000 q^{70} +4.00000 q^{71} -0.928203 q^{73} -7.46410 q^{74} -1.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} -8.92820 q^{82} -6.92820 q^{83} +3.46410 q^{85} +2.92820 q^{86} +1.00000 q^{88} -3.46410 q^{89} +3.46410 q^{91} +1.46410 q^{92} +6.92820 q^{94} -11.8564 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{25} + 2 q^{28} - 4 q^{29} - 2 q^{32} + 2 q^{35} + 8 q^{37} - 2 q^{40} + 4 q^{41} + 8 q^{43} - 2 q^{44} + 4 q^{46} + 2 q^{49} - 2 q^{50} - 12 q^{53} - 2 q^{55} - 2 q^{56} + 4 q^{58} - 8 q^{59} + 4 q^{61} + 2 q^{64} + 12 q^{67} - 2 q^{70} + 8 q^{71} + 12 q^{73} - 8 q^{74} - 2 q^{77} + 16 q^{79} + 2 q^{80} - 4 q^{82} - 8 q^{86} + 2 q^{88} - 4 q^{92} + 4 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 - 2 * q^8 - 2 * q^10 - 2 * q^11 - 2 * q^14 + 2 * q^16 + 2 * q^20 + 2 * q^22 - 4 * q^23 + 2 * q^25 + 2 * q^28 - 4 * q^29 - 2 * q^32 + 2 * q^35 + 8 * q^37 - 2 * q^40 + 4 * q^41 + 8 * q^43 - 2 * q^44 + 4 * q^46 + 2 * q^49 - 2 * q^50 - 12 * q^53 - 2 * q^55 - 2 * q^56 + 4 * q^58 - 8 * q^59 + 4 * q^61 + 2 * q^64 + 12 * q^67 - 2 * q^70 + 8 * q^71 + 12 * q^73 - 8 * q^74 - 2 * q^77 + 16 * q^79 + 2 * q^80 - 4 * q^82 - 8 * q^86 + 2 * q^88 - 4 * q^92 + 4 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	1.46410	0.305286	0.152643	−	0.988281i	$-0.451221\pi$
$23$	1.46410	0.305286	0.152643	+	0.988281i	$0.451221\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.46410	−0.679366
$27$	0	0
$28$	1.00000	0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	6.92820	1.24434	0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820	1.24434	0.622171	+	0.782881i	$0.286251\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.46410	−0.594089
$35$	1.00000	0.169031
$36$	0	0
$37$	7.46410	1.22709	0.613545	−	0.789659i	$-0.289743\pi$
$37$	7.46410	1.22709	0.613545	+	0.789659i	$0.289743\pi$
$38$	0	0
$39$	0	0
$40$	−1.00000	−0.158114
$41$	8.92820	1.39435	0.697176	−	0.716900i	$-0.254440\pi$
$41$	8.92820	1.39435	0.697176	+	0.716900i	$0.254440\pi$
$42$	0	0
$43$	−2.92820	−0.446547	−0.223273	−	0.974756i	$-0.571674\pi$
$43$	−2.92820	−0.446547	−0.223273	+	0.974756i	$0.571674\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−1.46410	−0.215870
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	3.46410	0.480384
$53$	−12.9282	−1.77583	−0.887913	−	0.460012i	$-0.847845\pi$
$53$	−12.9282	−1.77583	−0.887913	+	0.460012i	$0.847845\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.00000	0.262613
$59$	2.92820	0.381220	0.190610	−	0.981666i	$-0.438953\pi$
$59$	2.92820	0.381220	0.190610	+	0.981666i	$0.438953\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	−6.92820	−0.879883
$63$	0	0
$64$	1.00000	0.125000
$65$	3.46410	0.429669
$66$	0	0
$67$	9.46410	1.15622	0.578112	−	0.815957i	$-0.303790\pi$
$67$	9.46410	1.15622	0.578112	+	0.815957i	$0.303790\pi$
$68$	3.46410	0.420084
$69$	0	0
$70$	−1.00000	−0.119523
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−0.928203	−0.108638	−0.0543190	−	0.998524i	$-0.517299\pi$
$73$	−0.928203	−0.108638	−0.0543190	+	0.998524i	$0.517299\pi$
$74$	−7.46410	−0.867684
$75$	0	0
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−8.92820	−0.985955
$83$	−6.92820	−0.760469	−0.380235	−	0.924890i	$-0.624157\pi$
$83$	−6.92820	−0.760469	−0.380235	+	0.924890i	$0.624157\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	2.92820	0.315756
$87$	0	0
$88$	1.00000	0.106600
$89$	−3.46410	−0.367194	−0.183597	−	0.983002i	$-0.558774\pi$
$89$	−3.46410	−0.367194	−0.183597	+	0.983002i	$0.558774\pi$
$90$	0	0
$91$	3.46410	0.363137
$92$	1.46410	0.152643
$93$	0	0
$94$	6.92820	0.714590
$95$	0	0
$96$	0	0
$97$	−11.8564	−1.20384	−0.601918	−	0.798558i	$-0.705597\pi$
$97$	−11.8564	−1.20384	−0.601918	+	0.798558i	$0.705597\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−11.8564	−1.17976	−0.589878	−	0.807492i	$-0.700824\pi$
$101$	−11.8564	−1.17976	−0.589878	+	0.807492i	$0.700824\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−3.46410	−0.339683
$105$	0	0
$106$	12.9282	1.25570
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	0	0
$109$	14.3923	1.37853	0.689266	−	0.724508i	$-0.257933\pi$
$109$	14.3923	1.37853	0.689266	+	0.724508i	$0.257933\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	1.00000	0.0944911
$113$	−11.4641	−1.07845	−0.539226	−	0.842161i	$-0.681283\pi$
$113$	−11.4641	−1.07845	−0.539226	+	0.842161i	$0.681283\pi$
$114$	0	0
$115$	1.46410	0.136528
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−2.92820	−0.269563
$119$	3.46410	0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	4.92820	0.446179
$123$	0	0
$124$	6.92820	0.622171
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.9282	0.969721	0.484861	−	0.874591i	$-0.338870\pi$
$127$	10.9282	0.969721	0.484861	+	0.874591i	$0.338870\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.46410	−0.303822
$131$	1.07180	0.0936433	0.0468217	−	0.998903i	$-0.485091\pi$
$131$	1.07180	0.0936433	0.0468217	+	0.998903i	$0.485091\pi$
$132$	0	0
$133$	0	0
$134$	−9.46410	−0.817574
$135$	0	0
$136$	−3.46410	−0.297044
$137$	4.53590	0.387528	0.193764	−	0.981048i	$-0.437930\pi$
$137$	4.53590	0.387528	0.193764	+	0.981048i	$0.437930\pi$
$138$	0	0
$139$	−2.92820	−0.248367	−0.124183	−	0.992259i	$-0.539631\pi$
$139$	−2.92820	−0.248367	−0.124183	+	0.992259i	$0.539631\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−4.00000	−0.335673
$143$	−3.46410	−0.289683
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0.928203	0.0768186
$147$	0	0
$148$	7.46410	0.613545
$149$	−12.9282	−1.05912	−0.529560	−	0.848273i	$-0.677643\pi$
$149$	−12.9282	−1.05912	−0.529560	+	0.848273i	$0.677643\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	0	0
$154$	1.00000	0.0805823
$155$	6.92820	0.556487
$156$	0	0
$157$	−18.7846	−1.49918	−0.749588	−	0.661905i	$-0.769748\pi$
$157$	−18.7846	−1.49918	−0.749588	+	0.661905i	$0.769748\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	1.46410	0.115387
$162$	0	0
$163$	9.46410	0.741286	0.370643	−	0.928775i	$-0.379137\pi$
$163$	9.46410	0.741286	0.370643	+	0.928775i	$0.379137\pi$
$164$	8.92820	0.697176
$165$	0	0
$166$	6.92820	0.537733
$167$	9.46410	0.732354	0.366177	−	0.930545i	$-0.380666\pi$
$167$	9.46410	0.732354	0.366177	+	0.930545i	$0.380666\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	−3.46410	−0.265684
$171$	0	0
$172$	−2.92820	−0.223273
$173$	8.92820	0.678799	0.339399	−	0.940642i	$-0.389776\pi$
$173$	8.92820	0.678799	0.339399	+	0.940642i	$0.389776\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	3.46410	0.259645
$179$	20.7846	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	0	0
$181$	−16.5359	−1.22910	−0.614552	−	0.788876i	$-0.710663\pi$
$181$	−16.5359	−1.22910	−0.614552	+	0.788876i	$0.710663\pi$
$182$	−3.46410	−0.256776
$183$	0	0
$184$	−1.46410	−0.107935
$185$	7.46410	0.548772
$186$	0	0
$187$	−3.46410	−0.253320
$188$	−6.92820	−0.505291
$189$	0	0
$190$	0	0
$191$	−14.9282	−1.08017	−0.540083	−	0.841611i	$-0.681607\pi$
$191$	−14.9282	−1.08017	−0.540083	+	0.841611i	$0.681607\pi$
$192$	0	0
$193$	11.0718	0.796965	0.398483	−	0.917176i	$-0.369537\pi$
$193$	11.0718	0.796965	0.398483	+	0.917176i	$0.369537\pi$
$194$	11.8564	0.851240
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−6.92820	−0.491127	−0.245564	−	0.969380i	$-0.578973\pi$
$199$	−6.92820	−0.491127	−0.245564	+	0.969380i	$0.578973\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	11.8564	0.834214
$203$	−2.00000	−0.140372
$204$	0	0
$205$	8.92820	0.623573
$206$	−16.0000	−1.11477
$207$	0	0
$208$	3.46410	0.240192
$209$	0	0
$210$	0	0
$211$	−23.3205	−1.60545	−0.802725	−	0.596349i	$-0.796617\pi$
$211$	−23.3205	−1.60545	−0.802725	+	0.596349i	$0.796617\pi$
$212$	−12.9282	−0.887913
$213$	0	0
$214$	−6.92820	−0.473602
$215$	−2.92820	−0.199702
$216$	0	0
$217$	6.92820	0.470317
$218$	−14.3923	−0.974770
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	12.0000	0.807207
$222$	0	0
$223$	24.7846	1.65970	0.829850	−	0.557986i	$-0.188426\pi$
$223$	24.7846	1.65970	0.829850	+	0.557986i	$0.188426\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	11.4641	0.762581
$227$	−6.92820	−0.459841	−0.229920	−	0.973209i	$-0.573847\pi$
$227$	−6.92820	−0.459841	−0.229920	+	0.973209i	$0.573847\pi$
$228$	0	0
$229$	2.39230	0.158088	0.0790440	−	0.996871i	$-0.474813\pi$
$229$	2.39230	0.158088	0.0790440	+	0.996871i	$0.474813\pi$
$230$	−1.46410	−0.0965400
$231$	0	0
$232$	2.00000	0.131306
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	−6.92820	−0.451946
$236$	2.92820	0.190610
$237$	0	0
$238$	−3.46410	−0.224544
$239$	−4.39230	−0.284115	−0.142057	−	0.989858i	$-0.545372\pi$
$239$	−4.39230	−0.284115	−0.142057	+	0.989858i	$0.545372\pi$
$240$	0	0
$241$	−7.07180	−0.455534	−0.227767	−	0.973716i	$-0.573143\pi$
$241$	−7.07180	−0.455534	−0.227767	+	0.973716i	$0.573143\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−4.92820	−0.315496
$245$	1.00000	0.0638877
$246$	0	0
$247$	0	0
$248$	−6.92820	−0.439941
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	5.07180	0.320129	0.160064	−	0.987107i	$-0.448830\pi$
$251$	5.07180	0.320129	0.160064	+	0.987107i	$0.448830\pi$
$252$	0	0
$253$	−1.46410	−0.0920473
$254$	−10.9282	−0.685696
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.9282	1.05595	0.527976	−	0.849259i	$-0.322951\pi$
$257$	16.9282	1.05595	0.527976	+	0.849259i	$0.322951\pi$
$258$	0	0
$259$	7.46410	0.463797
$260$	3.46410	0.214834
$261$	0	0
$262$	−1.07180	−0.0662158
$263$	16.7846	1.03498	0.517492	−	0.855688i	$-0.326866\pi$
$263$	16.7846	1.03498	0.517492	+	0.855688i	$0.326866\pi$
$264$	0	0
$265$	−12.9282	−0.794173
$266$	0	0
$267$	0	0
$268$	9.46410	0.578112
$269$	16.9282	1.03213	0.516065	−	0.856549i	$-0.327396\pi$
$269$	16.9282	1.03213	0.516065	+	0.856549i	$0.327396\pi$
$270$	0	0
$271$	−5.85641	−0.355751	−0.177876	−	0.984053i	$-0.556922\pi$
$271$	−5.85641	−0.355751	−0.177876	+	0.984053i	$0.556922\pi$
$272$	3.46410	0.210042
$273$	0	0
$274$	−4.53590	−0.274024
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	2.92820	0.175622
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	14.3923	0.858573	0.429286	−	0.903168i	$-0.358765\pi$
$281$	14.3923	0.858573	0.429286	+	0.903168i	$0.358765\pi$
$282$	0	0
$283$	1.46410	0.0870318	0.0435159	−	0.999053i	$-0.486144\pi$
$283$	1.46410	0.0870318	0.0435159	+	0.999053i	$0.486144\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	3.46410	0.204837
$287$	8.92820	0.527015
$288$	0	0
$289$	−5.00000	−0.294118
$290$	2.00000	0.117444
$291$	0	0
$292$	−0.928203	−0.0543190
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	2.92820	0.170487
$296$	−7.46410	−0.433842
$297$	0	0
$298$	12.9282	0.748911
$299$	5.07180	0.293310
$300$	0	0
$301$	−2.92820	−0.168779
$302$	−8.00000	−0.460348
$303$	0	0
$304$	0	0
$305$	−4.92820	−0.282188
$306$	0	0
$307$	7.32051	0.417803	0.208902	−	0.977937i	$-0.433011\pi$
$307$	7.32051	0.417803	0.208902	+	0.977937i	$0.433011\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−6.92820	−0.393496
$311$	−6.53590	−0.370617	−0.185308	−	0.982680i	$-0.559328\pi$
$311$	−6.53590	−0.370617	−0.185308	+	0.982680i	$0.559328\pi$
$312$	0	0
$313$	21.7128	1.22728	0.613640	−	0.789586i	$-0.289704\pi$
$313$	21.7128	1.22728	0.613640	+	0.789586i	$0.289704\pi$
$314$	18.7846	1.06008
$315$	0	0
$316$	8.00000	0.450035
$317$	3.07180	0.172529	0.0862646	−	0.996272i	$-0.472507\pi$
$317$	3.07180	0.172529	0.0862646	+	0.996272i	$0.472507\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	1.00000	0.0559017
$321$	0	0
$322$	−1.46410	−0.0815912
$323$	0	0
$324$	0	0
$325$	3.46410	0.192154
$326$	−9.46410	−0.524168
$327$	0	0
$328$	−8.92820	−0.492978
$329$	−6.92820	−0.381964
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−6.92820	−0.380235
$333$	0	0
$334$	−9.46410	−0.517853
$335$	9.46410	0.517079
$336$	0	0
$337$	22.7846	1.24116	0.620578	−	0.784144i	$-0.286898\pi$
$337$	22.7846	1.24116	0.620578	+	0.784144i	$0.286898\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	3.46410	0.187867
$341$	−6.92820	−0.375183
$342$	0	0
$343$	1.00000	0.0539949
$344$	2.92820	0.157878
$345$	0	0
$346$	−8.92820	−0.479983
$347$	1.07180	0.0575371	0.0287685	−	0.999586i	$-0.490841\pi$
$347$	1.07180	0.0575371	0.0287685	+	0.999586i	$0.490841\pi$
$348$	0	0
$349$	−28.9282	−1.54849	−0.774246	−	0.632885i	$-0.781870\pi$
$349$	−28.9282	−1.54849	−0.774246	+	0.632885i	$0.781870\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	1.00000	0.0533002
$353$	−20.9282	−1.11390	−0.556948	−	0.830547i	$-0.688028\pi$
$353$	−20.9282	−1.11390	−0.556948	+	0.830547i	$0.688028\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	−3.46410	−0.183597
$357$	0	0
$358$	−20.7846	−1.09850
$359$	−9.46410	−0.499496	−0.249748	−	0.968311i	$-0.580348\pi$
$359$	−9.46410	−0.499496	−0.249748	+	0.968311i	$0.580348\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	16.5359	0.869108
$363$	0	0
$364$	3.46410	0.181568
$365$	−0.928203	−0.0485844
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	1.46410	0.0763216
$369$	0	0
$370$	−7.46410	−0.388040
$371$	−12.9282	−0.671199
$372$	0	0
$373$	−4.92820	−0.255173	−0.127586	−	0.991827i	$-0.540723\pi$
$373$	−4.92820	−0.255173	−0.127586	+	0.991827i	$0.540723\pi$
$374$	3.46410	0.179124
$375$	0	0
$376$	6.92820	0.357295
$377$	−6.92820	−0.356821
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	14.9282	0.763793
$383$	12.7846	0.653263	0.326632	−	0.945152i	$-0.394086\pi$
$383$	12.7846	0.653263	0.326632	+	0.945152i	$0.394086\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−11.0718	−0.563540
$387$	0	0
$388$	−11.8564	−0.601918
$389$	−21.7128	−1.10088	−0.550442	−	0.834874i	$-0.685541\pi$
$389$	−21.7128	−1.10088	−0.550442	+	0.834874i	$0.685541\pi$
$390$	0	0
$391$	5.07180	0.256492
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−18.0000	−0.906827
$395$	8.00000	0.402524
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	6.92820	0.347279
$399$	0	0
$400$	1.00000	0.0500000
$401$	25.7128	1.28404	0.642018	−	0.766689i	$-0.278097\pi$
$401$	25.7128	1.28404	0.642018	+	0.766689i	$0.278097\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	−11.8564	−0.589878
$405$	0	0
$406$	2.00000	0.0992583
$407$	−7.46410	−0.369982
$408$	0	0
$409$	−12.9282	−0.639259	−0.319629	−	0.947543i	$-0.603558\pi$
$409$	−12.9282	−0.639259	−0.319629	+	0.947543i	$0.603558\pi$
$410$	−8.92820	−0.440933
$411$	0	0
$412$	16.0000	0.788263
$413$	2.92820	0.144087
$414$	0	0
$415$	−6.92820	−0.340092
$416$	−3.46410	−0.169842
$417$	0	0
$418$	0	0
$419$	−13.8564	−0.676930	−0.338465	−	0.940979i	$-0.609908\pi$
$419$	−13.8564	−0.676930	−0.338465	+	0.940979i	$0.609908\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	23.3205	1.13522
$423$	0	0
$424$	12.9282	0.627849
$425$	3.46410	0.168034
$426$	0	0
$427$	−4.92820	−0.238492
$428$	6.92820	0.334887
$429$	0	0
$430$	2.92820	0.141210
$431$	23.3205	1.12331	0.561655	−	0.827372i	$-0.310165\pi$
$431$	23.3205	1.12331	0.561655	+	0.827372i	$0.310165\pi$
$432$	0	0
$433$	−17.7128	−0.851223	−0.425612	−	0.904906i	$-0.639941\pi$
$433$	−17.7128	−0.851223	−0.425612	+	0.904906i	$0.639941\pi$
$434$	−6.92820	−0.332564
$435$	0	0
$436$	14.3923	0.689266
$437$	0	0
$438$	0	0
$439$	−16.7846	−0.801086	−0.400543	−	0.916278i	$-0.631178\pi$
$439$	−16.7846	−0.801086	−0.400543	+	0.916278i	$0.631178\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	−12.0000	−0.570782
$443$	−26.9282	−1.27940	−0.639699	−	0.768626i	$-0.720941\pi$
$443$	−26.9282	−1.27940	−0.639699	+	0.768626i	$0.720941\pi$
$444$	0	0
$445$	−3.46410	−0.164214
$446$	−24.7846	−1.17359
$447$	0	0
$448$	1.00000	0.0472456
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	−8.92820	−0.420413
$452$	−11.4641	−0.539226
$453$	0	0
$454$	6.92820	0.325157
$455$	3.46410	0.162400
$456$	0	0
$457$	19.0718	0.892141	0.446071	−	0.894998i	$-0.352823\pi$
$457$	19.0718	0.892141	0.446071	+	0.894998i	$0.352823\pi$
$458$	−2.39230	−0.111785
$459$	0	0
$460$	1.46410	0.0682641
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	20.7846	0.965943	0.482971	−	0.875636i	$-0.339558\pi$
$463$	20.7846	0.965943	0.482971	+	0.875636i	$0.339558\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	2.00000	0.0926482
$467$	30.9282	1.43119	0.715593	−	0.698517i	$-0.246156\pi$
$467$	30.9282	1.43119	0.715593	+	0.698517i	$0.246156\pi$
$468$	0	0
$469$	9.46410	0.437012
$470$	6.92820	0.319574
$471$	0	0
$472$	−2.92820	−0.134781
$473$	2.92820	0.134639
$474$	0	0
$475$	0	0
$476$	3.46410	0.158777
$477$	0	0
$478$	4.39230	0.200899
$479$	−13.8564	−0.633115	−0.316558	−	0.948573i	$-0.602527\pi$
$479$	−13.8564	−0.633115	−0.316558	+	0.948573i	$0.602527\pi$
$480$	0	0
$481$	25.8564	1.17895
$482$	7.07180	0.322112
$483$	0	0
$484$	1.00000	0.0454545
$485$	−11.8564	−0.538372
$486$	0	0
$487$	1.07180	0.0485677	0.0242839	−	0.999705i	$-0.492269\pi$
$487$	1.07180	0.0485677	0.0242839	+	0.999705i	$0.492269\pi$
$488$	4.92820	0.223089
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	−1.85641	−0.0837785	−0.0418892	−	0.999122i	$-0.513338\pi$
$491$	−1.85641	−0.0837785	−0.0418892	+	0.999122i	$0.513338\pi$
$492$	0	0
$493$	−6.92820	−0.312031
$494$	0	0
$495$	0	0
$496$	6.92820	0.311086
$497$	4.00000	0.179425
$498$	0	0
$499$	17.0718	0.764239	0.382119	−	0.924113i	$-0.375194\pi$
$499$	17.0718	0.764239	0.382119	+	0.924113i	$0.375194\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−5.07180	−0.226365
$503$	−12.3923	−0.552546	−0.276273	−	0.961079i	$-0.589099\pi$
$503$	−12.3923	−0.552546	−0.276273	+	0.961079i	$0.589099\pi$
$504$	0	0
$505$	−11.8564	−0.527603
$506$	1.46410	0.0650873
$507$	0	0
$508$	10.9282	0.484861
$509$	32.9282	1.45952	0.729758	−	0.683705i	$-0.239633\pi$
$509$	32.9282	1.45952	0.729758	+	0.683705i	$0.239633\pi$
$510$	0	0
$511$	−0.928203	−0.0410613
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−16.9282	−0.746671
$515$	16.0000	0.705044
$516$	0	0
$517$	6.92820	0.304702
$518$	−7.46410	−0.327954
$519$	0	0
$520$	−3.46410	−0.151911
$521$	−27.4641	−1.20322	−0.601612	−	0.798788i	$-0.705475\pi$
$521$	−27.4641	−1.20322	−0.601612	+	0.798788i	$0.705475\pi$
$522$	0	0
$523$	44.3923	1.94114	0.970570	−	0.240819i	$-0.0774161\pi$
$523$	44.3923	1.94114	0.970570	+	0.240819i	$0.0774161\pi$
$524$	1.07180	0.0468217
$525$	0	0
$526$	−16.7846	−0.731844
$527$	24.0000	1.04546
$528$	0	0
$529$	−20.8564	−0.906800
$530$	12.9282	0.561565
$531$	0	0
$532$	0	0
$533$	30.9282	1.33965
$534$	0	0
$535$	6.92820	0.299532
$536$	−9.46410	−0.408787
$537$	0	0
$538$	−16.9282	−0.729827
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	9.32051	0.400720	0.200360	−	0.979722i	$-0.435789\pi$
$541$	9.32051	0.400720	0.200360	+	0.979722i	$0.435789\pi$
$542$	5.85641	0.251554
$543$	0	0
$544$	−3.46410	−0.148522
$545$	14.3923	0.616499
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	4.53590	0.193764
$549$	0	0
$550$	1.00000	0.0426401
$551$	0	0
$552$	0	0
$553$	8.00000	0.340195
$554$	10.0000	0.424859
$555$	0	0
$556$	−2.92820	−0.124183
$557$	31.8564	1.34980	0.674900	−	0.737910i	$-0.264187\pi$
$557$	31.8564	1.34980	0.674900	+	0.737910i	$0.264187\pi$
$558$	0	0
$559$	−10.1436	−0.429028
$560$	1.00000	0.0422577
$561$	0	0
$562$	−14.3923	−0.607103
$563$	31.7128	1.33654	0.668268	−	0.743921i	$-0.267036\pi$
$563$	31.7128	1.33654	0.668268	+	0.743921i	$0.267036\pi$
$564$	0	0
$565$	−11.4641	−0.482298
$566$	−1.46410	−0.0615408
$567$	0	0
$568$	−4.00000	−0.167836
$569$	39.1769	1.64238	0.821191	−	0.570654i	$-0.193310\pi$
$569$	39.1769	1.64238	0.821191	+	0.570654i	$0.193310\pi$
$570$	0	0
$571$	23.3205	0.975933	0.487966	−	0.872862i	$-0.337739\pi$
$571$	23.3205	0.975933	0.487966	+	0.872862i	$0.337739\pi$
$572$	−3.46410	−0.144841
$573$	0	0
$574$	−8.92820	−0.372656
$575$	1.46410	0.0610573
$576$	0	0
$577$	45.7128	1.90305	0.951525	−	0.307572i	$-0.0995167\pi$
$577$	45.7128	1.90305	0.951525	+	0.307572i	$0.0995167\pi$
$578$	5.00000	0.207973
$579$	0	0
$580$	−2.00000	−0.0830455
$581$	−6.92820	−0.287430
$582$	0	0
$583$	12.9282	0.535431
$584$	0.928203	0.0384093
$585$	0	0
$586$	−14.0000	−0.578335
$587$	20.7846	0.857873	0.428936	−	0.903335i	$-0.358888\pi$
$587$	20.7846	0.857873	0.428936	+	0.903335i	$0.358888\pi$
$588$	0	0
$589$	0	0
$590$	−2.92820	−0.120552
$591$	0	0
$592$	7.46410	0.306773
$593$	−27.1769	−1.11602	−0.558011	−	0.829834i	$-0.688435\pi$
$593$	−27.1769	−1.11602	−0.558011	+	0.829834i	$0.688435\pi$
$594$	0	0
$595$	3.46410	0.142014
$596$	−12.9282	−0.529560
$597$	0	0
$598$	−5.07180	−0.207401
$599$	3.21539	0.131377	0.0656886	−	0.997840i	$-0.479076\pi$
$599$	3.21539	0.131377	0.0656886	+	0.997840i	$0.479076\pi$
$600$	0	0
$601$	−23.0718	−0.941118	−0.470559	−	0.882368i	$-0.655948\pi$
$601$	−23.0718	−0.941118	−0.470559	+	0.882368i	$0.655948\pi$
$602$	2.92820	0.119345
$603$	0	0
$604$	8.00000	0.325515
$605$	1.00000	0.0406558
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	0	0
$610$	4.92820	0.199537
$611$	−24.0000	−0.970936
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−7.32051	−0.295432
$615$	0	0
$616$	1.00000	0.0402911
$617$	2.39230	0.0963106	0.0481553	−	0.998840i	$-0.484666\pi$
$617$	2.39230	0.0963106	0.0481553	+	0.998840i	$0.484666\pi$
$618$	0	0
$619$	15.3205	0.615783	0.307892	−	0.951421i	$-0.400377\pi$
$619$	15.3205	0.615783	0.307892	+	0.951421i	$0.400377\pi$
$620$	6.92820	0.278243
$621$	0	0
$622$	6.53590	0.262066
$623$	−3.46410	−0.138786
$624$	0	0
$625$	1.00000	0.0400000
$626$	−21.7128	−0.867819
$627$	0	0
$628$	−18.7846	−0.749588
$629$	25.8564	1.03096
$630$	0	0
$631$	16.7846	0.668185	0.334092	−	0.942540i	$-0.391570\pi$
$631$	16.7846	0.668185	0.334092	+	0.942540i	$0.391570\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	−3.07180	−0.121997
$635$	10.9282	0.433673
$636$	0	0
$637$	3.46410	0.137253
$638$	−2.00000	−0.0791808
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	−23.7128	−0.935142	−0.467571	−	0.883956i	$-0.654871\pi$
$643$	−23.7128	−0.935142	−0.467571	+	0.883956i	$0.654871\pi$
$644$	1.46410	0.0576937
$645$	0	0
$646$	0	0
$647$	22.9282	0.901401	0.450700	−	0.892675i	$-0.351174\pi$
$647$	22.9282	0.901401	0.450700	+	0.892675i	$0.351174\pi$
$648$	0	0
$649$	−2.92820	−0.114942
$650$	−3.46410	−0.135873
$651$	0	0
$652$	9.46410	0.370643
$653$	−31.8564	−1.24664	−0.623319	−	0.781968i	$-0.714216\pi$
$653$	−31.8564	−1.24664	−0.623319	+	0.781968i	$0.714216\pi$
$654$	0	0
$655$	1.07180	0.0418786
$656$	8.92820	0.348588
$657$	0	0
$658$	6.92820	0.270089
$659$	39.7128	1.54699	0.773496	−	0.633801i	$-0.218506\pi$
$659$	39.7128	1.54699	0.773496	+	0.633801i	$0.218506\pi$
$660$	0	0
$661$	1.60770	0.0625321	0.0312660	−	0.999511i	$-0.490046\pi$
$661$	1.60770	0.0625321	0.0312660	+	0.999511i	$0.490046\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	6.92820	0.268866
$665$	0	0
$666$	0	0
$667$	−2.92820	−0.113380
$668$	9.46410	0.366177
$669$	0	0
$670$	−9.46410	−0.365630
$671$	4.92820	0.190251
$672$	0	0
$673$	−15.0718	−0.580975	−0.290488	−	0.956879i	$-0.593817\pi$
$673$	−15.0718	−0.580975	−0.290488	+	0.956879i	$0.593817\pi$
$674$	−22.7846	−0.877630
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	40.9282	1.57300	0.786499	−	0.617591i	$-0.211891\pi$
$677$	40.9282	1.57300	0.786499	+	0.617591i	$0.211891\pi$
$678$	0	0
$679$	−11.8564	−0.455007
$680$	−3.46410	−0.132842
$681$	0	0
$682$	6.92820	0.265295
$683$	−26.9282	−1.03038	−0.515190	−	0.857076i	$-0.672278\pi$
$683$	−26.9282	−1.03038	−0.515190	+	0.857076i	$0.672278\pi$
$684$	0	0
$685$	4.53590	0.173308
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−2.92820	−0.111637
$689$	−44.7846	−1.70616
$690$	0	0
$691$	3.60770	0.137243	0.0686216	−	0.997643i	$-0.478140\pi$
$691$	3.60770	0.137243	0.0686216	+	0.997643i	$0.478140\pi$
$692$	8.92820	0.339399
$693$	0	0
$694$	−1.07180	−0.0406848
$695$	−2.92820	−0.111073
$696$	0	0
$697$	30.9282	1.17149
$698$	28.9282	1.09495
$699$	0	0
$700$	1.00000	0.0377964
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	20.9282	0.787643
$707$	−11.8564	−0.445906
$708$	0	0
$709$	−7.85641	−0.295054	−0.147527	−	0.989058i	$-0.547131\pi$
$709$	−7.85641	−0.295054	−0.147527	+	0.989058i	$0.547131\pi$
$710$	−4.00000	−0.150117
$711$	0	0
$712$	3.46410	0.129823
$713$	10.1436	0.379881
$714$	0	0
$715$	−3.46410	−0.129550
$716$	20.7846	0.776757
$717$	0	0
$718$	9.46410	0.353197
$719$	4.39230	0.163805	0.0819027	−	0.996640i	$-0.473900\pi$
$719$	4.39230	0.163805	0.0819027	+	0.996640i	$0.473900\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	19.0000	0.707107
$723$	0	0
$724$	−16.5359	−0.614552
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	13.0718	0.484806	0.242403	−	0.970176i	$-0.422064\pi$
$727$	13.0718	0.484806	0.242403	+	0.970176i	$0.422064\pi$
$728$	−3.46410	−0.128388
$729$	0	0
$730$	0.928203	0.0343543
$731$	−10.1436	−0.375174
$732$	0	0
$733$	−24.2487	−0.895647	−0.447823	−	0.894122i	$-0.647801\pi$
$733$	−24.2487	−0.895647	−0.447823	+	0.894122i	$0.647801\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−1.46410	−0.0539675
$737$	−9.46410	−0.348615
$738$	0	0
$739$	15.3205	0.563574	0.281787	−	0.959477i	$-0.409073\pi$
$739$	15.3205	0.563574	0.281787	+	0.959477i	$0.409073\pi$
$740$	7.46410	0.274386
$741$	0	0
$742$	12.9282	0.474609
$743$	−13.8564	−0.508342	−0.254171	−	0.967159i	$-0.581803\pi$
$743$	−13.8564	−0.508342	−0.254171	+	0.967159i	$0.581803\pi$
$744$	0	0
$745$	−12.9282	−0.473653
$746$	4.92820	0.180434
$747$	0	0
$748$	−3.46410	−0.126660
$749$	6.92820	0.253151
$750$	0	0
$751$	−0.784610	−0.0286308	−0.0143154	−	0.999898i	$-0.504557\pi$
$751$	−0.784610	−0.0286308	−0.0143154	+	0.999898i	$0.504557\pi$
$752$	−6.92820	−0.252646
$753$	0	0
$754$	6.92820	0.252310
$755$	8.00000	0.291150
$756$	0	0
$757$	−50.1051	−1.82110	−0.910551	−	0.413397i	$-0.864342\pi$
$757$	−50.1051	−1.82110	−0.910551	+	0.413397i	$0.864342\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	0	0
$761$	38.7846	1.40594	0.702971	−	0.711219i	$-0.251857\pi$
$761$	38.7846	1.40594	0.702971	+	0.711219i	$0.251857\pi$
$762$	0	0
$763$	14.3923	0.521036
$764$	−14.9282	−0.540083
$765$	0	0
$766$	−12.7846	−0.461927
$767$	10.1436	0.366264
$768$	0	0
$769$	38.7846	1.39861	0.699304	−	0.714824i	$-0.253493\pi$
$769$	38.7846	1.39861	0.699304	+	0.714824i	$0.253493\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	11.0718	0.398483
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	6.92820	0.248868
$776$	11.8564	0.425620
$777$	0	0
$778$	21.7128	0.778442
$779$	0	0
$780$	0	0
$781$	−4.00000	−0.143131
$782$	−5.07180	−0.181367
$783$	0	0
$784$	1.00000	0.0357143
$785$	−18.7846	−0.670451
$786$	0	0
$787$	41.4641	1.47804	0.739018	−	0.673686i	$-0.235290\pi$
$787$	41.4641	1.47804	0.739018	+	0.673686i	$0.235290\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	−8.00000	−0.284627
$791$	−11.4641	−0.407617
$792$	0	0
$793$	−17.0718	−0.606237
$794$	10.0000	0.354887
$795$	0	0
$796$	−6.92820	−0.245564
$797$	−31.8564	−1.12841	−0.564206	−	0.825634i	$-0.690818\pi$
$797$	−31.8564	−1.12841	−0.564206	+	0.825634i	$0.690818\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−25.7128	−0.907951
$803$	0.928203	0.0327556
$804$	0	0
$805$	1.46410	0.0516028
$806$	−24.0000	−0.845364
$807$	0	0
$808$	11.8564	0.417107
$809$	11.4641	0.403056	0.201528	−	0.979483i	$-0.435409\pi$
$809$	11.4641	0.403056	0.201528	+	0.979483i	$0.435409\pi$
$810$	0	0
$811$	13.8564	0.486564	0.243282	−	0.969956i	$-0.421776\pi$
$811$	13.8564	0.486564	0.243282	+	0.969956i	$0.421776\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	7.46410	0.261617
$815$	9.46410	0.331513
$816$	0	0
$817$	0	0
$818$	12.9282	0.452024
$819$	0	0
$820$	8.92820	0.311786
$821$	−12.1436	−0.423814	−0.211907	−	0.977290i	$-0.567967\pi$
$821$	−12.1436	−0.423814	−0.211907	+	0.977290i	$0.567967\pi$
$822$	0	0
$823$	49.8564	1.73789	0.868943	−	0.494913i	$-0.164800\pi$
$823$	49.8564	1.73789	0.868943	+	0.494913i	$0.164800\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	−2.92820	−0.101885
$827$	−20.7846	−0.722752	−0.361376	−	0.932420i	$-0.617693\pi$
$827$	−20.7846	−0.722752	−0.361376	+	0.932420i	$0.617693\pi$
$828$	0	0
$829$	51.1769	1.77745	0.888724	−	0.458443i	$-0.151593\pi$
$829$	51.1769	1.77745	0.888724	+	0.458443i	$0.151593\pi$
$830$	6.92820	0.240481
$831$	0	0
$832$	3.46410	0.120096
$833$	3.46410	0.120024
$834$	0	0
$835$	9.46410	0.327519
$836$	0	0
$837$	0	0
$838$	13.8564	0.478662
$839$	−30.5359	−1.05422	−0.527108	−	0.849798i	$-0.676724\pi$
$839$	−30.5359	−1.05422	−0.527108	+	0.849798i	$0.676724\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	26.0000	0.896019
$843$	0	0
$844$	−23.3205	−0.802725
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	1.00000	0.0343604
$848$	−12.9282	−0.443956
$849$	0	0
$850$	−3.46410	−0.118818
$851$	10.9282	0.374614
$852$	0	0
$853$	15.1769	0.519648	0.259824	−	0.965656i	$-0.416336\pi$
$853$	15.1769	0.519648	0.259824	+	0.965656i	$0.416336\pi$
$854$	4.92820	0.168640
$855$	0	0
$856$	−6.92820	−0.236801
$857$	0.535898	0.0183059	0.00915297	−	0.999958i	$-0.497086\pi$
$857$	0.535898	0.0183059	0.00915297	+	0.999958i	$0.497086\pi$
$858$	0	0
$859$	20.3923	0.695776	0.347888	−	0.937536i	$-0.386899\pi$
$859$	20.3923	0.695776	0.347888	+	0.937536i	$0.386899\pi$
$860$	−2.92820	−0.0998509
$861$	0	0
$862$	−23.3205	−0.794300
$863$	−42.2487	−1.43816	−0.719081	−	0.694926i	$-0.755437\pi$
$863$	−42.2487	−1.43816	−0.719081	+	0.694926i	$0.755437\pi$
$864$	0	0
$865$	8.92820	0.303568
$866$	17.7128	0.601906
$867$	0	0
$868$	6.92820	0.235159
$869$	−8.00000	−0.271381
$870$	0	0
$871$	32.7846	1.11086
$872$	−14.3923	−0.487385
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−7.85641	−0.265292	−0.132646	−	0.991163i	$-0.542347\pi$
$877$	−7.85641	−0.265292	−0.132646	+	0.991163i	$0.542347\pi$
$878$	16.7846	0.566453
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	−17.3205	−0.583543	−0.291771	−	0.956488i	$-0.594245\pi$
$881$	−17.3205	−0.583543	−0.291771	+	0.956488i	$0.594245\pi$
$882$	0	0
$883$	−48.1051	−1.61887	−0.809433	−	0.587212i	$-0.800225\pi$
$883$	−48.1051	−1.61887	−0.809433	+	0.587212i	$0.800225\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	26.9282	0.904671
$887$	16.6795	0.560043	0.280021	−	0.959994i	$-0.409658\pi$
$887$	16.6795	0.560043	0.280021	+	0.959994i	$0.409658\pi$
$888$	0	0
$889$	10.9282	0.366520
$890$	3.46410	0.116117
$891$	0	0
$892$	24.7846	0.829850
$893$	0	0
$894$	0	0
$895$	20.7846	0.694753
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−22.0000	−0.734150
$899$	−13.8564	−0.462137
$900$	0	0
$901$	−44.7846	−1.49199
$902$	8.92820	0.297277
$903$	0	0
$904$	11.4641	0.381290
$905$	−16.5359	−0.549672
$906$	0	0
$907$	−21.1769	−0.703168	−0.351584	−	0.936156i	$-0.614357\pi$
$907$	−21.1769	−0.703168	−0.351584	+	0.936156i	$0.614357\pi$
$908$	−6.92820	−0.229920
$909$	0	0
$910$	−3.46410	−0.114834
$911$	4.00000	0.132526	0.0662630	−	0.997802i	$-0.478892\pi$
$911$	4.00000	0.132526	0.0662630	+	0.997802i	$0.478892\pi$
$912$	0	0
$913$	6.92820	0.229290
$914$	−19.0718	−0.630839
$915$	0	0
$916$	2.39230	0.0790440
$917$	1.07180	0.0353938
$918$	0	0
$919$	45.0718	1.48678	0.743391	−	0.668857i	$-0.233216\pi$
$919$	45.0718	1.48678	0.743391	+	0.668857i	$0.233216\pi$
$920$	−1.46410	−0.0482700
$921$	0	0
$922$	−18.0000	−0.592798
$923$	13.8564	0.456089
$924$	0	0
$925$	7.46410	0.245418
$926$	−20.7846	−0.683025
$927$	0	0
$928$	2.00000	0.0656532
$929$	25.6077	0.840161	0.420081	−	0.907487i	$-0.362002\pi$
$929$	25.6077	0.840161	0.420081	+	0.907487i	$0.362002\pi$
$930$	0	0
$931$	0	0
$932$	−2.00000	−0.0655122
$933$	0	0
$934$	−30.9282	−1.01200
$935$	−3.46410	−0.113288
$936$	0	0
$937$	−3.85641	−0.125983	−0.0629917	−	0.998014i	$-0.520064\pi$
$937$	−3.85641	−0.125983	−0.0629917	+	0.998014i	$0.520064\pi$
$938$	−9.46410	−0.309014
$939$	0	0
$940$	−6.92820	−0.225973
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	13.0718	0.425676
$944$	2.92820	0.0953049
$945$	0	0
$946$	−2.92820	−0.0952041
$947$	10.1436	0.329622	0.164811	−	0.986325i	$-0.447299\pi$
$947$	10.1436	0.329622	0.164811	+	0.986325i	$0.447299\pi$
$948$	0	0
$949$	−3.21539	−0.104376
$950$	0	0
$951$	0	0
$952$	−3.46410	−0.112272
$953$	28.6410	0.927774	0.463887	−	0.885895i	$-0.346454\pi$
$953$	28.6410	0.927774	0.463887	+	0.885895i	$0.346454\pi$
$954$	0	0
$955$	−14.9282	−0.483065
$956$	−4.39230	−0.142057
$957$	0	0
$958$	13.8564	0.447680
$959$	4.53590	0.146472
$960$	0	0
$961$	17.0000	0.548387
$962$	−25.8564	−0.833644
$963$	0	0
$964$	−7.07180	−0.227767
$965$	11.0718	0.356414
$966$	0	0
$967$	−26.9282	−0.865953	−0.432976	−	0.901405i	$-0.642537\pi$
$967$	−26.9282	−0.865953	−0.432976	+	0.901405i	$0.642537\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	11.8564	0.380686
$971$	−35.7128	−1.14608	−0.573039	−	0.819528i	$-0.694236\pi$
$971$	−35.7128	−1.14608	−0.573039	+	0.819528i	$0.694236\pi$
$972$	0	0
$973$	−2.92820	−0.0938739
$974$	−1.07180	−0.0343426
$975$	0	0
$976$	−4.92820	−0.157748
$977$	37.3205	1.19399	0.596994	−	0.802245i	$-0.296361\pi$
$977$	37.3205	1.19399	0.596994	+	0.802245i	$0.296361\pi$
$978$	0	0
$979$	3.46410	0.110713
$980$	1.00000	0.0319438
$981$	0	0
$982$	1.85641	0.0592403
$983$	−3.21539	−0.102555	−0.0512775	−	0.998684i	$-0.516329\pi$
$983$	−3.21539	−0.102555	−0.0512775	+	0.998684i	$0.516329\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	6.92820	0.220639
$987$	0	0
$988$	0	0
$989$	−4.28719	−0.136325
$990$	0	0
$991$	−40.7846	−1.29557	−0.647783	−	0.761825i	$-0.724304\pi$
$991$	−40.7846	−1.29557	−0.647783	+	0.761825i	$0.724304\pi$
$992$	−6.92820	−0.219971
$993$	0	0
$994$	−4.00000	−0.126872
$995$	−6.92820	−0.219639
$996$	0	0
$997$	18.6795	0.591585	0.295793	−	0.955252i	$-0.404416\pi$
$997$	18.6795	0.591585	0.295793	+	0.955252i	$0.404416\pi$
$998$	−17.0718	−0.540398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.bt.1.2		2
3.2	odd	2		2310.2.a.bb.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.2.a.bb.1.2	✓	2	3.2	odd	2
6930.2.a.bt.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6930.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} +1.00000 q^{20} +1.00000 q^{22} +1.46410 q^{23} +1.00000 q^{25} -3.46410 q^{26} +1.00000 q^{28} -2.00000 q^{29} +6.92820 q^{31} -1.00000 q^{32} -3.46410 q^{34} +1.00000 q^{35} +7.46410 q^{37} -1.00000 q^{40} +8.92820 q^{41} -2.92820 q^{43} -1.00000 q^{44} -1.46410 q^{46} -6.92820 q^{47} +1.00000 q^{49} -1.00000 q^{50} +3.46410 q^{52} -12.9282 q^{53} -1.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} +2.92820 q^{59} -4.92820 q^{61} -6.92820 q^{62} +1.00000 q^{64} +3.46410 q^{65} +9.46410 q^{67} +3.46410 q^{68} -1.00000 q^{70} +4.00000 q^{71} -0.928203 q^{73} -7.46410 q^{74} -1.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} -8.92820 q^{82} -6.92820 q^{83} +3.46410 q^{85} +2.92820 q^{86} +1.00000 q^{88} -3.46410 q^{89} +3.46410 q^{91} +1.46410 q^{92} +6.92820 q^{94} -11.8564 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{20} + 2 q^{22} - 4 q^{23} + 2 q^{25} + 2 q^{28} - 4 q^{29} - 2 q^{32} + 2 q^{35} + 8 q^{37} - 2 q^{40} + 4 q^{41} + 8 q^{43} - 2 q^{44} + 4 q^{46} + 2 q^{49} - 2 q^{50} - 12 q^{53} - 2 q^{55} - 2 q^{56} + 4 q^{58} - 8 q^{59} + 4 q^{61} + 2 q^{64} + 12 q^{67} - 2 q^{70} + 8 q^{71} + 12 q^{73} - 8 q^{74} - 2 q^{77} + 16 q^{79} + 2 q^{80} - 4 q^{82} - 8 q^{86} + 2 q^{88} - 4 q^{92} + 4 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^7 - 2 * q^8 - 2 * q^10 - 2 * q^11 - 2 * q^14 + 2 * q^16 + 2 * q^20 + 2 * q^22 - 4 * q^23 + 2 * q^25 + 2 * q^28 - 4 * q^29 - 2 * q^32 + 2 * q^35 + 8 * q^37 - 2 * q^40 + 4 * q^41 + 8 * q^43 - 2 * q^44 + 4 * q^46 + 2 * q^49 - 2 * q^50 - 12 * q^53 - 2 * q^55 - 2 * q^56 + 4 * q^58 - 8 * q^59 + 4 * q^61 + 2 * q^64 + 12 * q^67 - 2 * q^70 + 8 * q^71 + 12 * q^73 - 8 * q^74 - 2 * q^77 + 16 * q^79 + 2 * q^80 - 4 * q^82 - 8 * q^86 + 2 * q^88 - 4 * q^92 + 4 * q^97 - 2 * q^98

Introduction

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