LMFDB - Embedded newform 6930.2.a.bt.1.1

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			1.1
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +1.00000 q^{20} +1.00000 q^{22} -5.46410 q^{23} +1.00000 q^{25} +3.46410 q^{26} +1.00000 q^{28} -2.00000 q^{29} -6.92820 q^{31} -1.00000 q^{32} +3.46410 q^{34} +1.00000 q^{35} +0.535898 q^{37} -1.00000 q^{40} -4.92820 q^{41} +10.9282 q^{43} -1.00000 q^{44} +5.46410 q^{46} +6.92820 q^{47} +1.00000 q^{49} -1.00000 q^{50} -3.46410 q^{52} +0.928203 q^{53} -1.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} -10.9282 q^{59} +8.92820 q^{61} +6.92820 q^{62} +1.00000 q^{64} -3.46410 q^{65} +2.53590 q^{67} -3.46410 q^{68} -1.00000 q^{70} +4.00000 q^{71} +12.9282 q^{73} -0.535898 q^{74} -1.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} +4.92820 q^{82} +6.92820 q^{83} -3.46410 q^{85} -10.9282 q^{86} +1.00000 q^{88} +3.46410 q^{89} -3.46410 q^{91} -5.46410 q^{92} -6.92820 q^{94} +15.8564 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	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
$23$	−5.46410	−1.13934	−0.569672	−	0.821872i	$-0.692930\pi$
$23$	−5.46410	−1.13934	−0.569672	+	0.821872i	$0.692930\pi$
$24$	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	$-0.713749\pi$
$31$	−6.92820	−1.24434	−0.622171	+	0.782881i	$0.713749\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.46410	0.594089
$35$	1.00000	0.169031
$36$	0	0
$37$	0.535898	0.0881012	0.0440506	−	0.999029i	$-0.485974\pi$
$37$	0.535898	0.0881012	0.0440506	+	0.999029i	$0.485974\pi$
$38$	0	0
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−4.92820	−0.769656	−0.384828	−	0.922988i	$-0.625739\pi$
$41$	−4.92820	−0.769656	−0.384828	+	0.922988i	$0.625739\pi$
$42$	0	0
$43$	10.9282	1.66654	0.833268	−	0.552870i	$-0.186467\pi$
$43$	10.9282	1.66654	0.833268	+	0.552870i	$0.186467\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	5.46410	0.805638
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.46410	−0.480384
$53$	0.928203	0.127499	0.0637493	−	0.997966i	$-0.479694\pi$
$53$	0.928203	0.127499	0.0637493	+	0.997966i	$0.479694\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.00000	0.262613
$59$	−10.9282	−1.42273	−0.711365	−	0.702822i	$-0.751923\pi$
$59$	−10.9282	−1.42273	−0.711365	+	0.702822i	$0.751923\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	6.92820	0.879883
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.46410	−0.429669
$66$	0	0
$67$	2.53590	0.309809	0.154905	−	0.987929i	$-0.450493\pi$
$67$	2.53590	0.309809	0.154905	+	0.987929i	$0.450493\pi$
$68$	−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
$73$	12.9282	1.51313	0.756566	−	0.653917i	$-0.226876\pi$
$73$	12.9282	1.51313	0.756566	+	0.653917i	$0.226876\pi$
$74$	−0.535898	−0.0622969
$75$	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
$82$	4.92820	0.544229
$83$	6.92820	0.760469	0.380235	−	0.924890i	$-0.375843\pi$
$83$	6.92820	0.760469	0.380235	+	0.924890i	$0.375843\pi$
$84$	0	0
$85$	−3.46410	−0.375735
$86$	−10.9282	−1.17842
$87$	0	0
$88$	1.00000	0.106600
$89$	3.46410	0.367194	0.183597	−	0.983002i	$-0.441226\pi$
$89$	3.46410	0.367194	0.183597	+	0.983002i	$0.441226\pi$
$90$	0	0
$91$	−3.46410	−0.363137
$92$	−5.46410	−0.569672
$93$	0	0
$94$	−6.92820	−0.714590
$95$	0	0
$96$	0	0
$97$	15.8564	1.60997	0.804987	−	0.593292i	$-0.202172\pi$
$97$	15.8564	1.60997	0.804987	+	0.593292i	$0.202172\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	15.8564	1.57777	0.788886	−	0.614540i	$-0.210658\pi$
$101$	15.8564	1.57777	0.788886	+	0.614540i	$0.210658\pi$
$102$	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
$106$	−0.928203	−0.0901551
$107$	−6.92820	−0.669775	−0.334887	−	0.942258i	$-0.608698\pi$
$107$	−6.92820	−0.669775	−0.334887	+	0.942258i	$0.608698\pi$
$108$	0	0
$109$	−6.39230	−0.612272	−0.306136	−	0.951988i	$-0.599036\pi$
$109$	−6.39230	−0.612272	−0.306136	+	0.951988i	$0.599036\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	1.00000	0.0944911
$113$	−4.53590	−0.426701	−0.213351	−	0.976976i	$-0.568438\pi$
$113$	−4.53590	−0.426701	−0.213351	+	0.976976i	$0.568438\pi$
$114$	0	0
$115$	−5.46410	−0.509530
$116$	−2.00000	−0.185695
$117$	0	0
$118$	10.9282	1.00602
$119$	−3.46410	−0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	−8.92820	−0.808322
$123$	0	0
$124$	−6.92820	−0.622171
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.92820	−0.259836	−0.129918	−	0.991525i	$-0.541471\pi$
$127$	−2.92820	−0.259836	−0.129918	+	0.991525i	$0.541471\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	3.46410	0.303822
$131$	14.9282	1.30428	0.652142	−	0.758097i	$-0.273871\pi$
$131$	14.9282	1.30428	0.652142	+	0.758097i	$0.273871\pi$
$132$	0	0
$133$	0	0
$134$	−2.53590	−0.219068
$135$	0	0
$136$	3.46410	0.297044
$137$	11.4641	0.979444	0.489722	−	0.871879i	$-0.337098\pi$
$137$	11.4641	0.979444	0.489722	+	0.871879i	$0.337098\pi$
$138$	0	0
$139$	10.9282	0.926918	0.463459	−	0.886118i	$-0.346608\pi$
$139$	10.9282	0.926918	0.463459	+	0.886118i	$0.346608\pi$
$140$	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
$146$	−12.9282	−1.06995
$147$	0	0
$148$	0.535898	0.0440506
$149$	0.928203	0.0760414	0.0380207	−	0.999277i	$-0.487895\pi$
$149$	0.928203	0.0760414	0.0380207	+	0.999277i	$0.487895\pi$
$150$	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
$157$	22.7846	1.81841	0.909205	−	0.416349i	$-0.136691\pi$
$157$	22.7846	1.81841	0.909205	+	0.416349i	$0.136691\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−5.46410	−0.430632
$162$	0	0
$163$	2.53590	0.198627	0.0993134	−	0.995056i	$-0.468335\pi$
$163$	2.53590	0.198627	0.0993134	+	0.995056i	$0.468335\pi$
$164$	−4.92820	−0.384828
$165$	0	0
$166$	−6.92820	−0.537733
$167$	2.53590	0.196234	0.0981169	−	0.995175i	$-0.468718\pi$
$167$	2.53590	0.196234	0.0981169	+	0.995175i	$0.468718\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	3.46410	0.265684
$171$	0	0
$172$	10.9282	0.833268
$173$	−4.92820	−0.374684	−0.187342	−	0.982295i	$-0.559987\pi$
$173$	−4.92820	−0.374684	−0.187342	+	0.982295i	$0.559987\pi$
$174$	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	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	0	0
$181$	−23.4641	−1.74407	−0.872036	−	0.489441i	$-0.837201\pi$
$181$	−23.4641	−1.74407	−0.872036	+	0.489441i	$0.837201\pi$
$182$	3.46410	0.256776
$183$	0	0
$184$	5.46410	0.402819
$185$	0.535898	0.0394000
$186$	0	0
$187$	3.46410	0.253320
$188$	6.92820	0.505291
$189$	0	0
$190$	0	0
$191$	−1.07180	−0.0775525	−0.0387762	−	0.999248i	$-0.512346\pi$
$191$	−1.07180	−0.0775525	−0.0387762	+	0.999248i	$0.512346\pi$
$192$	0	0
$193$	24.9282	1.79437	0.897186	−	0.441654i	$-0.145608\pi$
$193$	24.9282	1.79437	0.897186	+	0.441654i	$0.145608\pi$
$194$	−15.8564	−1.13842
$195$	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	$-0.421027\pi$
$199$	6.92820	0.491127	0.245564	+	0.969380i	$0.421027\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−15.8564	−1.11565
$203$	−2.00000	−0.140372
$204$	0	0
$205$	−4.92820	−0.344201
$206$	−16.0000	−1.11477
$207$	0	0
$208$	−3.46410	−0.240192
$209$	0	0
$210$	0	0
$211$	11.3205	0.779336	0.389668	−	0.920955i	$-0.372590\pi$
$211$	11.3205	0.779336	0.389668	+	0.920955i	$0.372590\pi$
$212$	0.928203	0.0637493
$213$	0	0
$214$	6.92820	0.473602
$215$	10.9282	0.745297
$216$	0	0
$217$	−6.92820	−0.470317
$218$	6.39230	0.432942
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	12.0000	0.807207
$222$	0	0
$223$	−16.7846	−1.12398	−0.561990	−	0.827144i	$-0.689964\pi$
$223$	−16.7846	−1.12398	−0.561990	+	0.827144i	$0.689964\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	4.53590	0.301723
$227$	6.92820	0.459841	0.229920	−	0.973209i	$-0.426153\pi$
$227$	6.92820	0.459841	0.229920	+	0.973209i	$0.426153\pi$
$228$	0	0
$229$	−18.3923	−1.21540	−0.607699	−	0.794168i	$-0.707907\pi$
$229$	−18.3923	−1.21540	−0.607699	+	0.794168i	$0.707907\pi$
$230$	5.46410	0.360292
$231$	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
$236$	−10.9282	−0.711365
$237$	0	0
$238$	3.46410	0.224544
$239$	16.3923	1.06033	0.530165	−	0.847894i	$-0.322130\pi$
$239$	16.3923	1.06033	0.530165	+	0.847894i	$0.322130\pi$
$240$	0	0
$241$	−20.9282	−1.34810	−0.674052	−	0.738684i	$-0.735448\pi$
$241$	−20.9282	−1.34810	−0.674052	+	0.738684i	$0.735448\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	8.92820	0.571570
$245$	1.00000	0.0638877
$246$	0	0
$247$	0	0
$248$	6.92820	0.439941
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	18.9282	1.19474	0.597369	−	0.801967i	$-0.296213\pi$
$251$	18.9282	1.19474	0.597369	+	0.801967i	$0.296213\pi$
$252$	0	0
$253$	5.46410	0.343525
$254$	2.92820	0.183732
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.07180	0.191613	0.0958067	−	0.995400i	$-0.469457\pi$
$257$	3.07180	0.191613	0.0958067	+	0.995400i	$0.469457\pi$
$258$	0	0
$259$	0.535898	0.0332991
$260$	−3.46410	−0.214834
$261$	0	0
$262$	−14.9282	−0.922267
$263$	−24.7846	−1.52828	−0.764142	−	0.645048i	$-0.776837\pi$
$263$	−24.7846	−1.52828	−0.764142	+	0.645048i	$0.776837\pi$
$264$	0	0
$265$	0.928203	0.0570191
$266$	0	0
$267$	0	0
$268$	2.53590	0.154905
$269$	3.07180	0.187291	0.0936454	−	0.995606i	$-0.470148\pi$
$269$	3.07180	0.187291	0.0936454	+	0.995606i	$0.470148\pi$
$270$	0	0
$271$	21.8564	1.32768	0.663841	−	0.747874i	$-0.268925\pi$
$271$	21.8564	1.32768	0.663841	+	0.747874i	$0.268925\pi$
$272$	−3.46410	−0.210042
$273$	0	0
$274$	−11.4641	−0.692572
$275$	−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	$0.597127\pi$
$278$	−10.9282	−0.655430
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−6.39230	−0.381333	−0.190666	−	0.981655i	$-0.561065\pi$
$281$	−6.39230	−0.381333	−0.190666	+	0.981655i	$0.561065\pi$
$282$	0	0
$283$	−5.46410	−0.324807	−0.162404	−	0.986724i	$-0.551925\pi$
$283$	−5.46410	−0.324807	−0.162404	+	0.986724i	$0.551925\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	−3.46410	−0.204837
$287$	−4.92820	−0.290903
$288$	0	0
$289$	−5.00000	−0.294118
$290$	2.00000	0.117444
$291$	0	0
$292$	12.9282	0.756566
$293$	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
$295$	−10.9282	−0.636265
$296$	−0.535898	−0.0311485
$297$	0	0
$298$	−0.928203	−0.0537694
$299$	18.9282	1.09465
$300$	0	0
$301$	10.9282	0.629891
$302$	−8.00000	−0.460348
$303$	0	0
$304$	0	0
$305$	8.92820	0.511227
$306$	0	0
$307$	−27.3205	−1.55926	−0.779632	−	0.626238i	$-0.784594\pi$
$307$	−27.3205	−1.55926	−0.779632	+	0.626238i	$0.784594\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	6.92820	0.393496
$311$	−13.4641	−0.763479	−0.381740	−	0.924270i	$-0.624675\pi$
$311$	−13.4641	−0.763479	−0.381740	+	0.924270i	$0.624675\pi$
$312$	0	0
$313$	−33.7128	−1.90556	−0.952780	−	0.303660i	$-0.901791\pi$
$313$	−33.7128	−1.90556	−0.952780	+	0.303660i	$0.901791\pi$
$314$	−22.7846	−1.28581
$315$	0	0
$316$	8.00000	0.450035
$317$	16.9282	0.950783	0.475391	−	0.879774i	$-0.342306\pi$
$317$	16.9282	0.950783	0.475391	+	0.879774i	$0.342306\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	1.00000	0.0559017
$321$	0	0
$322$	5.46410	0.304502
$323$	0	0
$324$	0	0
$325$	−3.46410	−0.192154
$326$	−2.53590	−0.140450
$327$	0	0
$328$	4.92820	0.272115
$329$	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
$334$	−2.53590	−0.138758
$335$	2.53590	0.138551
$336$	0	0
$337$	−18.7846	−1.02326	−0.511631	−	0.859205i	$-0.670959\pi$
$337$	−18.7846	−1.02326	−0.511631	+	0.859205i	$0.670959\pi$
$338$	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
$344$	−10.9282	−0.589209
$345$	0	0
$346$	4.92820	0.264942
$347$	14.9282	0.801388	0.400694	−	0.916212i	$-0.368769\pi$
$347$	14.9282	0.801388	0.400694	+	0.916212i	$0.368769\pi$
$348$	0	0
$349$	−15.0718	−0.806775	−0.403387	−	0.915029i	$-0.632167\pi$
$349$	−15.0718	−0.806775	−0.403387	+	0.915029i	$0.632167\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	1.00000	0.0533002
$353$	−7.07180	−0.376394	−0.188197	−	0.982131i	$-0.560264\pi$
$353$	−7.07180	−0.376394	−0.188197	+	0.982131i	$0.560264\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	3.46410	0.183597
$357$	0	0
$358$	20.7846	1.09850
$359$	−2.53590	−0.133840	−0.0669198	−	0.997758i	$-0.521317\pi$
$359$	−2.53590	−0.133840	−0.0669198	+	0.997758i	$0.521317\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	23.4641	1.23325
$363$	0	0
$364$	−3.46410	−0.181568
$365$	12.9282	0.676693
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	−5.46410	−0.284836
$369$	0	0
$370$	−0.535898	−0.0278600
$371$	0.928203	0.0481899
$372$	0	0
$373$	8.92820	0.462285	0.231142	−	0.972920i	$-0.425754\pi$
$373$	8.92820	0.462285	0.231142	+	0.972920i	$0.425754\pi$
$374$	−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
$382$	1.07180	0.0548379
$383$	−28.7846	−1.47082	−0.735412	−	0.677620i	$-0.763012\pi$
$383$	−28.7846	−1.47082	−0.735412	+	0.677620i	$0.763012\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−24.9282	−1.26881
$387$	0	0
$388$	15.8564	0.804987
$389$	33.7128	1.70931	0.854654	−	0.519198i	$-0.173769\pi$
$389$	33.7128	1.70931	0.854654	+	0.519198i	$0.173769\pi$
$390$	0	0
$391$	18.9282	0.957240
$392$	−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
$401$	−29.7128	−1.48379	−0.741894	−	0.670518i	$-0.766072\pi$
$401$	−29.7128	−1.48379	−0.741894	+	0.670518i	$0.766072\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	15.8564	0.788886
$405$	0	0
$406$	2.00000	0.0992583
$407$	−0.535898	−0.0265635
$408$	0	0
$409$	0.928203	0.0458967	0.0229483	−	0.999737i	$-0.492695\pi$
$409$	0.928203	0.0458967	0.0229483	+	0.999737i	$0.492695\pi$
$410$	4.92820	0.243387
$411$	0	0
$412$	16.0000	0.788263
$413$	−10.9282	−0.537742
$414$	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	$-0.390092\pi$
$419$	13.8564	0.676930	0.338465	+	0.940979i	$0.390092\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−11.3205	−0.551074
$423$	0	0
$424$	−0.928203	−0.0450775
$425$	−3.46410	−0.168034
$426$	0	0
$427$	8.92820	0.432066
$428$	−6.92820	−0.334887
$429$	0	0
$430$	−10.9282	−0.527005
$431$	−11.3205	−0.545290	−0.272645	−	0.962115i	$-0.587898\pi$
$431$	−11.3205	−0.545290	−0.272645	+	0.962115i	$0.587898\pi$
$432$	0	0
$433$	37.7128	1.81236	0.906181	−	0.422890i	$-0.138984\pi$
$433$	37.7128	1.81236	0.906181	+	0.422890i	$0.138984\pi$
$434$	6.92820	0.332564
$435$	0	0
$436$	−6.39230	−0.306136
$437$	0	0
$438$	0	0
$439$	24.7846	1.18290	0.591452	−	0.806340i	$-0.298555\pi$
$439$	24.7846	1.18290	0.591452	+	0.806340i	$0.298555\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	−12.0000	−0.570782
$443$	−13.0718	−0.621060	−0.310530	−	0.950564i	$-0.600506\pi$
$443$	−13.0718	−0.621060	−0.310530	+	0.950564i	$0.600506\pi$
$444$	0	0
$445$	3.46410	0.164214
$446$	16.7846	0.794774
$447$	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
$451$	4.92820	0.232060
$452$	−4.53590	−0.213351
$453$	0	0
$454$	−6.92820	−0.325157
$455$	−3.46410	−0.162400
$456$	0	0
$457$	32.9282	1.54032	0.770158	−	0.637853i	$-0.220177\pi$
$457$	32.9282	1.54032	0.770158	+	0.637853i	$0.220177\pi$
$458$	18.3923	0.859416
$459$	0	0
$460$	−5.46410	−0.254765
$461$	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	$-0.660442\pi$
$463$	−20.7846	−0.965943	−0.482971	+	0.875636i	$0.660442\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	2.00000	0.0926482
$467$	17.0718	0.789989	0.394994	−	0.918684i	$-0.370747\pi$
$467$	17.0718	0.789989	0.394994	+	0.918684i	$0.370747\pi$
$468$	0	0
$469$	2.53590	0.117097
$470$	−6.92820	−0.319574
$471$	0	0
$472$	10.9282	0.503011
$473$	−10.9282	−0.502479
$474$	0	0
$475$	0	0
$476$	−3.46410	−0.158777
$477$	0	0
$478$	−16.3923	−0.749767
$479$	13.8564	0.633115	0.316558	−	0.948573i	$-0.397473\pi$
$479$	13.8564	0.633115	0.316558	+	0.948573i	$0.397473\pi$
$480$	0	0
$481$	−1.85641	−0.0846448
$482$	20.9282	0.953254
$483$	0	0
$484$	1.00000	0.0454545
$485$	15.8564	0.720002
$486$	0	0
$487$	14.9282	0.676461	0.338231	−	0.941063i	$-0.390172\pi$
$487$	14.9282	0.676461	0.338231	+	0.941063i	$0.390172\pi$
$488$	−8.92820	−0.404161
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	25.8564	1.16688	0.583442	−	0.812155i	$-0.301706\pi$
$491$	25.8564	1.16688	0.583442	+	0.812155i	$0.301706\pi$
$492$	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
$499$	30.9282	1.38454	0.692268	−	0.721640i	$-0.256611\pi$
$499$	30.9282	1.38454	0.692268	+	0.721640i	$0.256611\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−18.9282	−0.844807
$503$	8.39230	0.374194	0.187097	−	0.982341i	$-0.440092\pi$
$503$	8.39230	0.374194	0.187097	+	0.982341i	$0.440092\pi$
$504$	0	0
$505$	15.8564	0.705601
$506$	−5.46410	−0.242909
$507$	0	0
$508$	−2.92820	−0.129918
$509$	19.0718	0.845343	0.422671	−	0.906283i	$-0.361092\pi$
$509$	19.0718	0.845343	0.422671	+	0.906283i	$0.361092\pi$
$510$	0	0
$511$	12.9282	0.571910
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−3.07180	−0.135491
$515$	16.0000	0.705044
$516$	0	0
$517$	−6.92820	−0.304702
$518$	−0.535898	−0.0235460
$519$	0	0
$520$	3.46410	0.151911
$521$	−20.5359	−0.899694	−0.449847	−	0.893106i	$-0.648521\pi$
$521$	−20.5359	−0.899694	−0.449847	+	0.893106i	$0.648521\pi$
$522$	0	0
$523$	23.6077	1.03229	0.516146	−	0.856500i	$-0.327366\pi$
$523$	23.6077	1.03229	0.516146	+	0.856500i	$0.327366\pi$
$524$	14.9282	0.652142
$525$	0	0
$526$	24.7846	1.08066
$527$	24.0000	1.04546
$528$	0	0
$529$	6.85641	0.298105
$530$	−0.928203	−0.0403186
$531$	0	0
$532$	0	0
$533$	17.0718	0.739462
$534$	0	0
$535$	−6.92820	−0.299532
$536$	−2.53590	−0.109534
$537$	0	0
$538$	−3.07180	−0.132435
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−25.3205	−1.08861	−0.544307	−	0.838886i	$-0.683207\pi$
$541$	−25.3205	−1.08861	−0.544307	+	0.838886i	$0.683207\pi$
$542$	−21.8564	−0.938813
$543$	0	0
$544$	3.46410	0.148522
$545$	−6.39230	−0.273816
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	11.4641	0.489722
$549$	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
$556$	10.9282	0.463459
$557$	4.14359	0.175570	0.0877848	−	0.996139i	$-0.472021\pi$
$557$	4.14359	0.175570	0.0877848	+	0.996139i	$0.472021\pi$
$558$	0	0
$559$	−37.8564	−1.60116
$560$	1.00000	0.0422577
$561$	0	0
$562$	6.39230	0.269643
$563$	−23.7128	−0.999376	−0.499688	−	0.866205i	$-0.666552\pi$
$563$	−23.7128	−0.999376	−0.499688	+	0.866205i	$0.666552\pi$
$564$	0	0
$565$	−4.53590	−0.190827
$566$	5.46410	0.229673
$567$	0	0
$568$	−4.00000	−0.167836
$569$	−23.1769	−0.971627	−0.485813	−	0.874063i	$-0.661477\pi$
$569$	−23.1769	−0.971627	−0.485813	+	0.874063i	$0.661477\pi$
$570$	0	0
$571$	−11.3205	−0.473749	−0.236874	−	0.971540i	$-0.576123\pi$
$571$	−11.3205	−0.473749	−0.236874	+	0.971540i	$0.576123\pi$
$572$	3.46410	0.144841
$573$	0	0
$574$	4.92820	0.205699
$575$	−5.46410	−0.227869
$576$	0	0
$577$	−9.71281	−0.404350	−0.202175	−	0.979349i	$-0.564801\pi$
$577$	−9.71281	−0.404350	−0.202175	+	0.979349i	$0.564801\pi$
$578$	5.00000	0.207973
$579$	0	0
$580$	−2.00000	−0.0830455
$581$	6.92820	0.287430
$582$	0	0
$583$	−0.928203	−0.0384422
$584$	−12.9282	−0.534973
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−20.7846	−0.857873	−0.428936	−	0.903335i	$-0.641112\pi$
$587$	−20.7846	−0.857873	−0.428936	+	0.903335i	$0.641112\pi$
$588$	0	0
$589$	0	0
$590$	10.9282	0.449907
$591$	0	0
$592$	0.535898	0.0220253
$593$	35.1769	1.44454	0.722271	−	0.691610i	$-0.243098\pi$
$593$	35.1769	1.44454	0.722271	+	0.691610i	$0.243098\pi$
$594$	0	0
$595$	−3.46410	−0.142014
$596$	0.928203	0.0380207
$597$	0	0
$598$	−18.9282	−0.774032
$599$	44.7846	1.82985	0.914925	−	0.403624i	$-0.132250\pi$
$599$	44.7846	1.82985	0.914925	+	0.403624i	$0.132250\pi$
$600$	0	0
$601$	−36.9282	−1.50633	−0.753166	−	0.657830i	$-0.771475\pi$
$601$	−36.9282	−1.50633	−0.753166	+	0.657830i	$0.771475\pi$
$602$	−10.9282	−0.445400
$603$	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
$610$	−8.92820	−0.361492
$611$	−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	$0.512860\pi$
$614$	27.3205	1.10257
$615$	0	0
$616$	1.00000	0.0402911
$617$	−18.3923	−0.740446	−0.370223	−	0.928943i	$-0.620719\pi$
$617$	−18.3923	−0.740446	−0.370223	+	0.928943i	$0.620719\pi$
$618$	0	0
$619$	−19.3205	−0.776557	−0.388278	−	0.921542i	$-0.626930\pi$
$619$	−19.3205	−0.776557	−0.388278	+	0.921542i	$0.626930\pi$
$620$	−6.92820	−0.278243
$621$	0	0
$622$	13.4641	0.539861
$623$	3.46410	0.138786
$624$	0	0
$625$	1.00000	0.0400000
$626$	33.7128	1.34743
$627$	0	0
$628$	22.7846	0.909205
$629$	−1.85641	−0.0740198
$630$	0	0
$631$	−24.7846	−0.986660	−0.493330	−	0.869842i	$-0.664220\pi$
$631$	−24.7846	−0.986660	−0.493330	+	0.869842i	$0.664220\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	−16.9282	−0.672305
$635$	−2.92820	−0.116202
$636$	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
$643$	31.7128	1.25063	0.625316	−	0.780372i	$-0.284970\pi$
$643$	31.7128	1.25063	0.625316	+	0.780372i	$0.284970\pi$
$644$	−5.46410	−0.215316
$645$	0	0
$646$	0	0
$647$	9.07180	0.356649	0.178325	−	0.983972i	$-0.442932\pi$
$647$	9.07180	0.356649	0.178325	+	0.983972i	$0.442932\pi$
$648$	0	0
$649$	10.9282	0.428969
$650$	3.46410	0.135873
$651$	0	0
$652$	2.53590	0.0993134
$653$	−4.14359	−0.162151	−0.0810757	−	0.996708i	$-0.525836\pi$
$653$	−4.14359	−0.162151	−0.0810757	+	0.996708i	$0.525836\pi$
$654$	0	0
$655$	14.9282	0.583293
$656$	−4.92820	−0.192414
$657$	0	0
$658$	−6.92820	−0.270089
$659$	−15.7128	−0.612084	−0.306042	−	0.952018i	$-0.599005\pi$
$659$	−15.7128	−0.612084	−0.306042	+	0.952018i	$0.599005\pi$
$660$	0	0
$661$	22.3923	0.870960	0.435480	−	0.900198i	$-0.356579\pi$
$661$	22.3923	0.870960	0.435480	+	0.900198i	$0.356579\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−6.92820	−0.268866
$665$	0	0
$666$	0	0
$667$	10.9282	0.423142
$668$	2.53590	0.0981169
$669$	0	0
$670$	−2.53590	−0.0979703
$671$	−8.92820	−0.344669
$672$	0	0
$673$	−28.9282	−1.11510	−0.557550	−	0.830143i	$-0.688259\pi$
$673$	−28.9282	−1.11510	−0.557550	+	0.830143i	$0.688259\pi$
$674$	18.7846	0.723556
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	27.0718	1.04045	0.520227	−	0.854028i	$-0.325847\pi$
$677$	27.0718	1.04045	0.520227	+	0.854028i	$0.325847\pi$
$678$	0	0
$679$	15.8564	0.608513
$680$	3.46410	0.132842
$681$	0	0
$682$	−6.92820	−0.265295
$683$	−13.0718	−0.500178	−0.250089	−	0.968223i	$-0.580460\pi$
$683$	−13.0718	−0.500178	−0.250089	+	0.968223i	$0.580460\pi$
$684$	0	0
$685$	11.4641	0.438021
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	10.9282	0.416634
$689$	−3.21539	−0.122497
$690$	0	0
$691$	24.3923	0.927927	0.463964	−	0.885854i	$-0.346427\pi$
$691$	24.3923	0.927927	0.463964	+	0.885854i	$0.346427\pi$
$692$	−4.92820	−0.187342
$693$	0	0
$694$	−14.9282	−0.566667
$695$	10.9282	0.414530
$696$	0	0
$697$	17.0718	0.646640
$698$	15.0718	0.570476
$699$	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
$706$	7.07180	0.266151
$707$	15.8564	0.596342
$708$	0	0
$709$	19.8564	0.745723	0.372861	−	0.927887i	$-0.378377\pi$
$709$	19.8564	0.745723	0.372861	+	0.927887i	$0.378377\pi$
$710$	−4.00000	−0.150117
$711$	0	0
$712$	−3.46410	−0.129823
$713$	37.8564	1.41773
$714$	0	0
$715$	3.46410	0.129550
$716$	−20.7846	−0.776757
$717$	0	0
$718$	2.53590	0.0946389
$719$	−16.3923	−0.611330	−0.305665	−	0.952139i	$-0.598879\pi$
$719$	−16.3923	−0.611330	−0.305665	+	0.952139i	$0.598879\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	19.0000	0.707107
$723$	0	0
$724$	−23.4641	−0.872036
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	26.9282	0.998712	0.499356	−	0.866397i	$-0.333570\pi$
$727$	26.9282	0.998712	0.499356	+	0.866397i	$0.333570\pi$
$728$	3.46410	0.128388
$729$	0	0
$730$	−12.9282	−0.478494
$731$	−37.8564	−1.40017
$732$	0	0
$733$	24.2487	0.895647	0.447823	−	0.894122i	$-0.352199\pi$
$733$	24.2487	0.895647	0.447823	+	0.894122i	$0.352199\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	5.46410	0.201409
$737$	−2.53590	−0.0934110
$738$	0	0
$739$	−19.3205	−0.710716	−0.355358	−	0.934730i	$-0.615641\pi$
$739$	−19.3205	−0.710716	−0.355358	+	0.934730i	$0.615641\pi$
$740$	0.535898	0.0197000
$741$	0	0
$742$	−0.928203	−0.0340754
$743$	13.8564	0.508342	0.254171	−	0.967159i	$-0.418197\pi$
$743$	13.8564	0.508342	0.254171	+	0.967159i	$0.418197\pi$
$744$	0	0
$745$	0.928203	0.0340067
$746$	−8.92820	−0.326885
$747$	0	0
$748$	3.46410	0.126660
$749$	−6.92820	−0.253151
$750$	0	0
$751$	40.7846	1.48825	0.744126	−	0.668040i	$-0.232866\pi$
$751$	40.7846	1.48825	0.744126	+	0.668040i	$0.232866\pi$
$752$	6.92820	0.252646
$753$	0	0
$754$	−6.92820	−0.252310
$755$	8.00000	0.291150
$756$	0	0
$757$	26.1051	0.948807	0.474403	−	0.880308i	$-0.342664\pi$
$757$	26.1051	0.948807	0.474403	+	0.880308i	$0.342664\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	0	0
$761$	−2.78461	−0.100942	−0.0504710	−	0.998726i	$-0.516072\pi$
$761$	−2.78461	−0.100942	−0.0504710	+	0.998726i	$0.516072\pi$
$762$	0	0
$763$	−6.39230	−0.231417
$764$	−1.07180	−0.0387762
$765$	0	0
$766$	28.7846	1.04003
$767$	37.8564	1.36692
$768$	0	0
$769$	−2.78461	−0.100416	−0.0502078	−	0.998739i	$-0.515988\pi$
$769$	−2.78461	−0.100416	−0.0502078	+	0.998739i	$0.515988\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	24.9282	0.897186
$773$	−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
$776$	−15.8564	−0.569212
$777$	0	0
$778$	−33.7128	−1.20866
$779$	0	0
$780$	0	0
$781$	−4.00000	−0.143131
$782$	−18.9282	−0.676871
$783$	0	0
$784$	1.00000	0.0357143
$785$	22.7846	0.813218
$786$	0	0
$787$	34.5359	1.23107	0.615536	−	0.788109i	$-0.288940\pi$
$787$	34.5359	1.23107	0.615536	+	0.788109i	$0.288940\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	−8.00000	−0.284627
$791$	−4.53590	−0.161278
$792$	0	0
$793$	−30.9282	−1.09829
$794$	10.0000	0.354887
$795$	0	0
$796$	6.92820	0.245564
$797$	−4.14359	−0.146774	−0.0733868	−	0.997304i	$-0.523381\pi$
$797$	−4.14359	−0.146774	−0.0733868	+	0.997304i	$0.523381\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	29.7128	1.04920
$803$	−12.9282	−0.456226
$804$	0	0
$805$	−5.46410	−0.192584
$806$	−24.0000	−0.845364
$807$	0	0
$808$	−15.8564	−0.557826
$809$	4.53590	0.159474	0.0797368	−	0.996816i	$-0.474592\pi$
$809$	4.53590	0.159474	0.0797368	+	0.996816i	$0.474592\pi$
$810$	0	0
$811$	−13.8564	−0.486564	−0.243282	−	0.969956i	$-0.578224\pi$
$811$	−13.8564	−0.486564	−0.243282	+	0.969956i	$0.578224\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	0.535898	0.0187832
$815$	2.53590	0.0888286
$816$	0	0
$817$	0	0
$818$	−0.928203	−0.0324539
$819$	0	0
$820$	−4.92820	−0.172100
$821$	−39.8564	−1.39100	−0.695499	−	0.718527i	$-0.744817\pi$
$821$	−39.8564	−1.39100	−0.695499	+	0.718527i	$0.744817\pi$
$822$	0	0
$823$	22.1436	0.771877	0.385939	−	0.922524i	$-0.373878\pi$
$823$	22.1436	0.771877	0.385939	+	0.922524i	$0.373878\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	10.9282	0.380241
$827$	20.7846	0.722752	0.361376	−	0.932420i	$-0.382307\pi$
$827$	20.7846	0.722752	0.361376	+	0.932420i	$0.382307\pi$
$828$	0	0
$829$	−11.1769	−0.388190	−0.194095	−	0.980983i	$-0.562177\pi$
$829$	−11.1769	−0.388190	−0.194095	+	0.980983i	$0.562177\pi$
$830$	−6.92820	−0.240481
$831$	0	0
$832$	−3.46410	−0.120096
$833$	−3.46410	−0.120024
$834$	0	0
$835$	2.53590	0.0877584
$836$	0	0
$837$	0	0
$838$	−13.8564	−0.478662
$839$	−37.4641	−1.29340	−0.646702	−	0.762743i	$-0.723852\pi$
$839$	−37.4641	−1.29340	−0.646702	+	0.762743i	$0.723852\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	26.0000	0.896019
$843$	0	0
$844$	11.3205	0.389668
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	1.00000	0.0343604
$848$	0.928203	0.0318746
$849$	0	0
$850$	3.46410	0.118818
$851$	−2.92820	−0.100378
$852$	0	0
$853$	−47.1769	−1.61531	−0.807653	−	0.589658i	$-0.799263\pi$
$853$	−47.1769	−1.61531	−0.807653	+	0.589658i	$0.799263\pi$
$854$	−8.92820	−0.305517
$855$	0	0
$856$	6.92820	0.236801
$857$	7.46410	0.254969	0.127484	−	0.991841i	$-0.459310\pi$
$857$	7.46410	0.254969	0.127484	+	0.991841i	$0.459310\pi$
$858$	0	0
$859$	−0.392305	−0.0133853	−0.00669263	−	0.999978i	$-0.502130\pi$
$859$	−0.392305	−0.0133853	−0.00669263	+	0.999978i	$0.502130\pi$
$860$	10.9282	0.372649
$861$	0	0
$862$	11.3205	0.385578
$863$	6.24871	0.212709	0.106354	−	0.994328i	$-0.466082\pi$
$863$	6.24871	0.212709	0.106354	+	0.994328i	$0.466082\pi$
$864$	0	0
$865$	−4.92820	−0.167564
$866$	−37.7128	−1.28153
$867$	0	0
$868$	−6.92820	−0.235159
$869$	−8.00000	−0.271381
$870$	0	0
$871$	−8.78461	−0.297655
$872$	6.39230	0.216471
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	19.8564	0.670503	0.335252	−	0.942129i	$-0.391179\pi$
$877$	19.8564	0.670503	0.335252	+	0.942129i	$0.391179\pi$
$878$	−24.7846	−0.836440
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	17.3205	0.583543	0.291771	−	0.956488i	$-0.405755\pi$
$881$	17.3205	0.583543	0.291771	+	0.956488i	$0.405755\pi$
$882$	0	0
$883$	28.1051	0.945813	0.472906	−	0.881113i	$-0.343205\pi$
$883$	28.1051	0.945813	0.472906	+	0.881113i	$0.343205\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	13.0718	0.439156
$887$	51.3205	1.72317	0.861587	−	0.507610i	$-0.169471\pi$
$887$	51.3205	1.72317	0.861587	+	0.507610i	$0.169471\pi$
$888$	0	0
$889$	−2.92820	−0.0982088
$890$	−3.46410	−0.116117
$891$	0	0
$892$	−16.7846	−0.561990
$893$	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
$901$	−3.21539	−0.107120
$902$	−4.92820	−0.164091
$903$	0	0
$904$	4.53590	0.150862
$905$	−23.4641	−0.779973
$906$	0	0
$907$	41.1769	1.36726	0.683629	−	0.729830i	$-0.260401\pi$
$907$	41.1769	1.36726	0.683629	+	0.729830i	$0.260401\pi$
$908$	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
$914$	−32.9282	−1.08917
$915$	0	0
$916$	−18.3923	−0.607699
$917$	14.9282	0.492973
$918$	0	0
$919$	58.9282	1.94386	0.971931	−	0.235266i	$-0.0755961\pi$
$919$	58.9282	1.94386	0.971931	+	0.235266i	$0.0755961\pi$
$920$	5.46410	0.180146
$921$	0	0
$922$	−18.0000	−0.592798
$923$	−13.8564	−0.456089
$924$	0	0
$925$	0.535898	0.0176202
$926$	20.7846	0.683025
$927$	0	0
$928$	2.00000	0.0656532
$929$	46.3923	1.52208	0.761041	−	0.648704i	$-0.224689\pi$
$929$	46.3923	1.52208	0.761041	+	0.648704i	$0.224689\pi$
$930$	0	0
$931$	0	0
$932$	−2.00000	−0.0655122
$933$	0	0
$934$	−17.0718	−0.558606
$935$	3.46410	0.113288
$936$	0	0
$937$	23.8564	0.779355	0.389677	−	0.920951i	$-0.372587\pi$
$937$	23.8564	0.779355	0.389677	+	0.920951i	$0.372587\pi$
$938$	−2.53590	−0.0828000
$939$	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
$943$	26.9282	0.876903
$944$	−10.9282	−0.355683
$945$	0	0
$946$	10.9282	0.355307
$947$	37.8564	1.23017	0.615084	−	0.788462i	$-0.289122\pi$
$947$	37.8564	1.23017	0.615084	+	0.788462i	$0.289122\pi$
$948$	0	0
$949$	−44.7846	−1.45377
$950$	0	0
$951$	0	0
$952$	3.46410	0.112272
$953$	−40.6410	−1.31649	−0.658246	−	0.752803i	$-0.728701\pi$
$953$	−40.6410	−1.31649	−0.658246	+	0.752803i	$0.728701\pi$
$954$	0	0
$955$	−1.07180	−0.0346825
$956$	16.3923	0.530165
$957$	0	0
$958$	−13.8564	−0.447680
$959$	11.4641	0.370195
$960$	0	0
$961$	17.0000	0.548387
$962$	1.85641	0.0598529
$963$	0	0
$964$	−20.9282	−0.674052
$965$	24.9282	0.802467
$966$	0	0
$967$	−13.0718	−0.420361	−0.210180	−	0.977663i	$-0.567405\pi$
$967$	−13.0718	−0.420361	−0.210180	+	0.977663i	$0.567405\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−15.8564	−0.509119
$971$	19.7128	0.632614	0.316307	−	0.948657i	$-0.397557\pi$
$971$	19.7128	0.632614	0.316307	+	0.948657i	$0.397557\pi$
$972$	0	0
$973$	10.9282	0.350342
$974$	−14.9282	−0.478330
$975$	0	0
$976$	8.92820	0.285785
$977$	2.67949	0.0857245	0.0428623	−	0.999081i	$-0.486352\pi$
$977$	2.67949	0.0857245	0.0428623	+	0.999081i	$0.486352\pi$
$978$	0	0
$979$	−3.46410	−0.110713
$980$	1.00000	0.0319438
$981$	0	0
$982$	−25.8564	−0.825111
$983$	−44.7846	−1.42841	−0.714204	−	0.699938i	$-0.753211\pi$
$983$	−44.7846	−1.42841	−0.714204	+	0.699938i	$0.753211\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	−6.92820	−0.220639
$987$	0	0
$988$	0	0
$989$	−59.7128	−1.89876
$990$	0	0
$991$	0.784610	0.0249239	0.0124620	−	0.999922i	$-0.496033\pi$
$991$	0.784610	0.0249239	0.0124620	+	0.999922i	$0.496033\pi$
$992$	6.92820	0.219971
$993$	0	0
$994$	−4.00000	−0.126872
$995$	6.92820	0.219639
$996$	0	0
$997$	53.3205	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$997$	53.3205	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$998$	−30.9282	−0.979015
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} +1.00000 q^{20} +1.00000 q^{22} -5.46410 q^{23} +1.00000 q^{25} +3.46410 q^{26} +1.00000 q^{28} -2.00000 q^{29} -6.92820 q^{31} -1.00000 q^{32} +3.46410 q^{34} +1.00000 q^{35} +0.535898 q^{37} -1.00000 q^{40} -4.92820 q^{41} +10.9282 q^{43} -1.00000 q^{44} +5.46410 q^{46} +6.92820 q^{47} +1.00000 q^{49} -1.00000 q^{50} -3.46410 q^{52} +0.928203 q^{53} -1.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} -10.9282 q^{59} +8.92820 q^{61} +6.92820 q^{62} +1.00000 q^{64} -3.46410 q^{65} +2.53590 q^{67} -3.46410 q^{68} -1.00000 q^{70} +4.00000 q^{71} +12.9282 q^{73} -0.535898 q^{74} -1.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} +4.92820 q^{82} +6.92820 q^{83} -3.46410 q^{85} -10.9282 q^{86} +1.00000 q^{88} +3.46410 q^{89} -3.46410 q^{91} -5.46410 q^{92} -6.92820 q^{94} +15.8564 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 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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