LMFDB - Embedded newform 6930.2.a.bn.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(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
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			$2.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -5.12311 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.12311 q^{17} +4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} -3.12311 q^{23} +1.00000 q^{25} +5.12311 q^{26} -1.00000 q^{28} +8.24621 q^{29} +8.00000 q^{31} -1.00000 q^{32} +5.12311 q^{34} +1.00000 q^{35} +1.12311 q^{37} -4.00000 q^{38} +1.00000 q^{40} -8.24621 q^{41} -10.2462 q^{43} -1.00000 q^{44} +3.12311 q^{46} -8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -5.12311 q^{52} -12.2462 q^{53} +1.00000 q^{55} +1.00000 q^{56} -8.24621 q^{58} +2.24621 q^{59} +6.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +5.12311 q^{65} -7.12311 q^{67} -5.12311 q^{68} -1.00000 q^{70} -14.2462 q^{71} +8.24621 q^{73} -1.12311 q^{74} +4.00000 q^{76} +1.00000 q^{77} -1.00000 q^{80} +8.24621 q^{82} +2.24621 q^{83} +5.12311 q^{85} +10.2462 q^{86} +1.00000 q^{88} -11.3693 q^{89} +5.12311 q^{91} -3.12311 q^{92} +8.00000 q^{94} -4.00000 q^{95} -4.24621 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 8 q^{19} - 2 q^{20} + 2 q^{22} + 2 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} + 16 q^{31} - 2 q^{32} + 2 q^{34} + 2 q^{35} - 6 q^{37} - 8 q^{38} + 2 q^{40} - 4 q^{43} - 2 q^{44} - 2 q^{46} - 16 q^{47} + 2 q^{49} - 2 q^{50} - 2 q^{52} - 8 q^{53} + 2 q^{55} + 2 q^{56} - 12 q^{59} + 12 q^{61} - 16 q^{62} + 2 q^{64} + 2 q^{65} - 6 q^{67} - 2 q^{68} - 2 q^{70} - 12 q^{71} + 6 q^{74} + 8 q^{76} + 2 q^{77} - 2 q^{80} - 12 q^{83} + 2 q^{85} + 4 q^{86} + 2 q^{88} + 2 q^{89} + 2 q^{91} + 2 q^{92} + 16 q^{94} - 8 q^{95} + 8 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^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 + 8 * q^19 - 2 * q^20 + 2 * q^22 + 2 * q^23 + 2 * q^25 + 2 * q^26 - 2 * q^28 + 16 * q^31 - 2 * q^32 + 2 * q^34 + 2 * q^35 - 6 * q^37 - 8 * q^38 + 2 * q^40 - 4 * q^43 - 2 * q^44 - 2 * q^46 - 16 * q^47 + 2 * q^49 - 2 * q^50 - 2 * q^52 - 8 * q^53 + 2 * q^55 + 2 * q^56 - 12 * q^59 + 12 * q^61 - 16 * q^62 + 2 * q^64 + 2 * q^65 - 6 * q^67 - 2 * q^68 - 2 * q^70 - 12 * q^71 + 6 * q^74 + 8 * q^76 + 2 * q^77 - 2 * q^80 - 12 * q^83 + 2 * q^85 + 4 * q^86 + 2 * q^88 + 2 * q^89 + 2 * q^91 + 2 * q^92 + 16 * q^94 - 8 * q^95 + 8 * 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
$13$	−5.12311	−1.42089	−0.710447	−	0.703751i	$-0.751507\pi$
$13$	−5.12311	−1.42089	−0.710447	+	0.703751i	$0.751507\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.12311	−1.24254	−0.621268	−	0.783598i	$-0.713382\pi$
$17$	−5.12311	−1.24254	−0.621268	+	0.783598i	$0.713382\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	−3.12311	−0.651213	−0.325606	−	0.945505i	$-0.605568\pi$
$23$	−3.12311	−0.651213	−0.325606	+	0.945505i	$0.605568\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	5.12311	1.00472
$27$	0	0
$28$	−1.00000	−0.188982
$29$	8.24621	1.53128	0.765641	−	0.643268i	$-0.222422\pi$
$29$	8.24621	1.53128	0.765641	+	0.643268i	$0.222422\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.12311	0.878605
$35$	1.00000	0.169031
$36$	0	0
$37$	1.12311	0.184637	0.0923187	−	0.995730i	$-0.470572\pi$
$37$	1.12311	0.184637	0.0923187	+	0.995730i	$0.470572\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	1.00000	0.158114
$41$	−8.24621	−1.28784	−0.643921	−	0.765092i	$-0.722693\pi$
$41$	−8.24621	−1.28784	−0.643921	+	0.765092i	$0.722693\pi$
$42$	0	0
$43$	−10.2462	−1.56253	−0.781266	−	0.624198i	$-0.785426\pi$
$43$	−10.2462	−1.56253	−0.781266	+	0.624198i	$0.785426\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	3.12311	0.460477
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−5.12311	−0.710447
$53$	−12.2462	−1.68215	−0.841073	−	0.540921i	$-0.818076\pi$
$53$	−12.2462	−1.68215	−0.841073	+	0.540921i	$0.818076\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	−8.24621	−1.08278
$59$	2.24621	0.292432	0.146216	−	0.989253i	$-0.453291\pi$
$59$	2.24621	0.292432	0.146216	+	0.989253i	$0.453291\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	5.12311	0.635443
$66$	0	0
$67$	−7.12311	−0.870226	−0.435113	−	0.900376i	$-0.643292\pi$
$67$	−7.12311	−0.870226	−0.435113	+	0.900376i	$0.643292\pi$
$68$	−5.12311	−0.621268
$69$	0	0
$70$	−1.00000	−0.119523
$71$	−14.2462	−1.69071	−0.845357	−	0.534202i	$-0.820612\pi$
$71$	−14.2462	−1.69071	−0.845357	+	0.534202i	$0.820612\pi$
$72$	0	0
$73$	8.24621	0.965146	0.482573	−	0.875856i	$-0.339702\pi$
$73$	8.24621	0.965146	0.482573	+	0.875856i	$0.339702\pi$
$74$	−1.12311	−0.130558
$75$	0	0
$76$	4.00000	0.458831
$77$	1.00000	0.113961
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	8.24621	0.910642
$83$	2.24621	0.246554	0.123277	−	0.992372i	$-0.460660\pi$
$83$	2.24621	0.246554	0.123277	+	0.992372i	$0.460660\pi$
$84$	0	0
$85$	5.12311	0.555679
$86$	10.2462	1.10488
$87$	0	0
$88$	1.00000	0.106600
$89$	−11.3693	−1.20515	−0.602573	−	0.798064i	$-0.705858\pi$
$89$	−11.3693	−1.20515	−0.602573	+	0.798064i	$0.705858\pi$
$90$	0	0
$91$	5.12311	0.537047
$92$	−3.12311	−0.325606
$93$	0	0
$94$	8.00000	0.825137
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−4.24621	−0.431137	−0.215569	−	0.976489i	$-0.569161\pi$
$97$	−4.24621	−0.431137	−0.215569	+	0.976489i	$0.569161\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	14.4924	1.44205	0.721025	−	0.692909i	$-0.243671\pi$
$101$	14.4924	1.44205	0.721025	+	0.692909i	$0.243671\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	5.12311	0.502362
$105$	0	0
$106$	12.2462	1.18946
$107$	16.4924	1.59438	0.797191	−	0.603727i	$-0.206318\pi$
$107$	16.4924	1.59438	0.797191	+	0.603727i	$0.206318\pi$
$108$	0	0
$109$	−6.87689	−0.658687	−0.329344	−	0.944210i	$-0.606827\pi$
$109$	−6.87689	−0.658687	−0.329344	+	0.944210i	$0.606827\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	9.12311	0.858230	0.429115	−	0.903250i	$-0.358826\pi$
$113$	9.12311	0.858230	0.429115	+	0.903250i	$0.358826\pi$
$114$	0	0
$115$	3.12311	0.291231
$116$	8.24621	0.765641
$117$	0	0
$118$	−2.24621	−0.206781
$119$	5.12311	0.469634
$120$	0	0
$121$	1.00000	0.0909091
$122$	−6.00000	−0.543214
$123$	0	0
$124$	8.00000	0.718421
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−5.12311	−0.449326
$131$	−16.4924	−1.44095	−0.720475	−	0.693481i	$-0.756076\pi$
$131$	−16.4924	−1.44095	−0.720475	+	0.693481i	$0.756076\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	7.12311	0.615343
$135$	0	0
$136$	5.12311	0.439303
$137$	17.1231	1.46293	0.731463	−	0.681881i	$-0.238838\pi$
$137$	17.1231	1.46293	0.731463	+	0.681881i	$0.238838\pi$
$138$	0	0
$139$	18.2462	1.54762	0.773812	−	0.633416i	$-0.218348\pi$
$139$	18.2462	1.54762	0.773812	+	0.633416i	$0.218348\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	14.2462	1.19552
$143$	5.12311	0.428416
$144$	0	0
$145$	−8.24621	−0.684811
$146$	−8.24621	−0.682461
$147$	0	0
$148$	1.12311	0.0923187
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	−8.00000	−0.642575
$156$	0	0
$157$	7.75379	0.618820	0.309410	−	0.950929i	$-0.399869\pi$
$157$	7.75379	0.618820	0.309410	+	0.950929i	$0.399869\pi$
$158$	0	0
$159$	0	0
$160$	1.00000	0.0790569
$161$	3.12311	0.246135
$162$	0	0
$163$	5.36932	0.420557	0.210279	−	0.977641i	$-0.432563\pi$
$163$	5.36932	0.420557	0.210279	+	0.977641i	$0.432563\pi$
$164$	−8.24621	−0.643921
$165$	0	0
$166$	−2.24621	−0.174340
$167$	−9.36932	−0.725020	−0.362510	−	0.931980i	$-0.618080\pi$
$167$	−9.36932	−0.725020	−0.362510	+	0.931980i	$0.618080\pi$
$168$	0	0
$169$	13.2462	1.01894
$170$	−5.12311	−0.392924
$171$	0	0
$172$	−10.2462	−0.781266
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	11.3693	0.852166
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−11.3693	−0.845075	−0.422537	−	0.906346i	$-0.638860\pi$
$181$	−11.3693	−0.845075	−0.422537	+	0.906346i	$0.638860\pi$
$182$	−5.12311	−0.379750
$183$	0	0
$184$	3.12311	0.230238
$185$	−1.12311	−0.0825724
$186$	0	0
$187$	5.12311	0.374639
$188$	−8.00000	−0.583460
$189$	0	0
$190$	4.00000	0.290191
$191$	−12.4924	−0.903920	−0.451960	−	0.892038i	$-0.649275\pi$
$191$	−12.4924	−0.903920	−0.451960	+	0.892038i	$0.649275\pi$
$192$	0	0
$193$	8.24621	0.593575	0.296788	−	0.954944i	$-0.404085\pi$
$193$	8.24621	0.593575	0.296788	+	0.954944i	$0.404085\pi$
$194$	4.24621	0.304860
$195$	0	0
$196$	1.00000	0.0714286
$197$	0.246211	0.0175418	0.00877091	−	0.999962i	$-0.497208\pi$
$197$	0.246211	0.0175418	0.00877091	+	0.999962i	$0.497208\pi$
$198$	0	0
$199$	−12.4924	−0.885564	−0.442782	−	0.896629i	$-0.646009\pi$
$199$	−12.4924	−0.885564	−0.442782	+	0.896629i	$0.646009\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−14.4924	−1.01968
$203$	−8.24621	−0.578771
$204$	0	0
$205$	8.24621	0.575940
$206$	0	0
$207$	0	0
$208$	−5.12311	−0.355223
$209$	−4.00000	−0.276686
$210$	0	0
$211$	15.1231	1.04112	0.520559	−	0.853826i	$-0.325724\pi$
$211$	15.1231	1.04112	0.520559	+	0.853826i	$0.325724\pi$
$212$	−12.2462	−0.841073
$213$	0	0
$214$	−16.4924	−1.12740
$215$	10.2462	0.698786
$216$	0	0
$217$	−8.00000	−0.543075
$218$	6.87689	0.465762
$219$	0	0
$220$	1.00000	0.0674200
$221$	26.2462	1.76551
$222$	0	0
$223$	4.49242	0.300835	0.150417	−	0.988623i	$-0.451938\pi$
$223$	4.49242	0.300835	0.150417	+	0.988623i	$0.451938\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−9.12311	−0.606860
$227$	−22.7386	−1.50922	−0.754608	−	0.656176i	$-0.772173\pi$
$227$	−22.7386	−1.50922	−0.754608	+	0.656176i	$0.772173\pi$
$228$	0	0
$229$	−5.12311	−0.338544	−0.169272	−	0.985569i	$-0.554142\pi$
$229$	−5.12311	−0.338544	−0.169272	+	0.985569i	$0.554142\pi$
$230$	−3.12311	−0.205931
$231$	0	0
$232$	−8.24621	−0.541390
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	2.24621	0.146216
$237$	0	0
$238$	−5.12311	−0.332082
$239$	−1.36932	−0.0885737	−0.0442869	−	0.999019i	$-0.514102\pi$
$239$	−1.36932	−0.0885737	−0.0442869	+	0.999019i	$0.514102\pi$
$240$	0	0
$241$	−18.4924	−1.19120	−0.595601	−	0.803281i	$-0.703086\pi$
$241$	−18.4924	−1.19120	−0.595601	+	0.803281i	$0.703086\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	6.00000	0.384111
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−20.4924	−1.30390
$248$	−8.00000	−0.508001
$249$	0	0
$250$	1.00000	0.0632456
$251$	18.2462	1.15169	0.575845	−	0.817559i	$-0.304673\pi$
$251$	18.2462	1.15169	0.575845	+	0.817559i	$0.304673\pi$
$252$	0	0
$253$	3.12311	0.196348
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.7538	−0.733181	−0.366591	−	0.930382i	$-0.619475\pi$
$257$	−11.7538	−0.733181	−0.366591	+	0.930382i	$0.619475\pi$
$258$	0	0
$259$	−1.12311	−0.0697864
$260$	5.12311	0.317722
$261$	0	0
$262$	16.4924	1.01891
$263$	30.2462	1.86506	0.932531	−	0.361091i	$-0.117596\pi$
$263$	30.2462	1.86506	0.932531	+	0.361091i	$0.117596\pi$
$264$	0	0
$265$	12.2462	0.752279
$266$	4.00000	0.245256
$267$	0	0
$268$	−7.12311	−0.435113
$269$	11.7538	0.716641	0.358321	−	0.933599i	$-0.383349\pi$
$269$	11.7538	0.716641	0.358321	+	0.933599i	$0.383349\pi$
$270$	0	0
$271$	−12.4924	−0.758861	−0.379430	−	0.925220i	$-0.623880\pi$
$271$	−12.4924	−0.758861	−0.379430	+	0.925220i	$0.623880\pi$
$272$	−5.12311	−0.310634
$273$	0	0
$274$	−17.1231	−1.03444
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−3.75379	−0.225543	−0.112772	−	0.993621i	$-0.535973\pi$
$277$	−3.75379	−0.225543	−0.112772	+	0.993621i	$0.535973\pi$
$278$	−18.2462	−1.09434
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	17.1231	1.02148	0.510739	−	0.859736i	$-0.329372\pi$
$281$	17.1231	1.02148	0.510739	+	0.859736i	$0.329372\pi$
$282$	0	0
$283$	−15.1231	−0.898975	−0.449488	−	0.893287i	$-0.648393\pi$
$283$	−15.1231	−0.898975	−0.449488	+	0.893287i	$0.648393\pi$
$284$	−14.2462	−0.845357
$285$	0	0
$286$	−5.12311	−0.302936
$287$	8.24621	0.486758
$288$	0	0
$289$	9.24621	0.543895
$290$	8.24621	0.484234
$291$	0	0
$292$	8.24621	0.482573
$293$	24.2462	1.41648	0.708239	−	0.705972i	$-0.249490\pi$
$293$	24.2462	1.41648	0.708239	+	0.705972i	$0.249490\pi$
$294$	0	0
$295$	−2.24621	−0.130779
$296$	−1.12311	−0.0652792
$297$	0	0
$298$	−10.0000	−0.579284
$299$	16.0000	0.925304
$300$	0	0
$301$	10.2462	0.590582
$302$	−8.00000	−0.460348
$303$	0	0
$304$	4.00000	0.229416
$305$	−6.00000	−0.343559
$306$	0	0
$307$	21.3693	1.21961	0.609806	−	0.792551i	$-0.291247\pi$
$307$	21.3693	1.21961	0.609806	+	0.792551i	$0.291247\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	8.00000	0.454369
$311$	1.36932	0.0776468	0.0388234	−	0.999246i	$-0.487639\pi$
$311$	1.36932	0.0776468	0.0388234	+	0.999246i	$0.487639\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−7.75379	−0.437572
$315$	0	0
$316$	0	0
$317$	24.2462	1.36180	0.680901	−	0.732375i	$-0.261588\pi$
$317$	24.2462	1.36180	0.680901	+	0.732375i	$0.261588\pi$
$318$	0	0
$319$	−8.24621	−0.461699
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−3.12311	−0.174044
$323$	−20.4924	−1.14023
$324$	0	0
$325$	−5.12311	−0.284179
$326$	−5.36932	−0.297379
$327$	0	0
$328$	8.24621	0.455321
$329$	8.00000	0.441054
$330$	0	0
$331$	24.4924	1.34623	0.673113	−	0.739540i	$-0.264957\pi$
$331$	24.4924	1.34623	0.673113	+	0.739540i	$0.264957\pi$
$332$	2.24621	0.123277
$333$	0	0
$334$	9.36932	0.512666
$335$	7.12311	0.389177
$336$	0	0
$337$	30.4924	1.66103	0.830514	−	0.556998i	$-0.188047\pi$
$337$	30.4924	1.66103	0.830514	+	0.556998i	$0.188047\pi$
$338$	−13.2462	−0.720499
$339$	0	0
$340$	5.12311	0.277839
$341$	−8.00000	−0.433224
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	10.2462	0.552439
$345$	0	0
$346$	−10.0000	−0.537603
$347$	16.4924	0.885360	0.442680	−	0.896680i	$-0.354028\pi$
$347$	16.4924	0.885360	0.442680	+	0.896680i	$0.354028\pi$
$348$	0	0
$349$	−6.49242	−0.347531	−0.173766	−	0.984787i	$-0.555594\pi$
$349$	−6.49242	−0.347531	−0.173766	+	0.984787i	$0.555594\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	1.00000	0.0533002
$353$	20.2462	1.07760	0.538799	−	0.842435i	$-0.318878\pi$
$353$	20.2462	1.07760	0.538799	+	0.842435i	$0.318878\pi$
$354$	0	0
$355$	14.2462	0.756110
$356$	−11.3693	−0.602573
$357$	0	0
$358$	20.0000	1.05703
$359$	25.3693	1.33894	0.669471	−	0.742838i	$-0.266521\pi$
$359$	25.3693	1.33894	0.669471	+	0.742838i	$0.266521\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	11.3693	0.597558
$363$	0	0
$364$	5.12311	0.268524
$365$	−8.24621	−0.431626
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−3.12311	−0.162803
$369$	0	0
$370$	1.12311	0.0583875
$371$	12.2462	0.635792
$372$	0	0
$373$	−38.4924	−1.99306	−0.996531	−	0.0832219i	$-0.973479\pi$
$373$	−38.4924	−1.99306	−0.996531	+	0.0832219i	$0.973479\pi$
$374$	−5.12311	−0.264909
$375$	0	0
$376$	8.00000	0.412568
$377$	−42.2462	−2.17579
$378$	0	0
$379$	−0.492423	−0.0252940	−0.0126470	−	0.999920i	$-0.504026\pi$
$379$	−0.492423	−0.0252940	−0.0126470	+	0.999920i	$0.504026\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	12.4924	0.639168
$383$	36.4924	1.86468	0.932338	−	0.361588i	$-0.117765\pi$
$383$	36.4924	1.86468	0.932338	+	0.361588i	$0.117765\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−8.24621	−0.419721
$387$	0	0
$388$	−4.24621	−0.215569
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−0.246211	−0.0124039
$395$	0	0
$396$	0	0
$397$	−5.50758	−0.276417	−0.138209	−	0.990403i	$-0.544134\pi$
$397$	−5.50758	−0.276417	−0.138209	+	0.990403i	$0.544134\pi$
$398$	12.4924	0.626189
$399$	0	0
$400$	1.00000	0.0500000
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	−40.9848	−2.04160
$404$	14.4924	0.721025
$405$	0	0
$406$	8.24621	0.409253
$407$	−1.12311	−0.0556703
$408$	0	0
$409$	20.7386	1.02546	0.512730	−	0.858550i	$-0.328634\pi$
$409$	20.7386	1.02546	0.512730	+	0.858550i	$0.328634\pi$
$410$	−8.24621	−0.407251
$411$	0	0
$412$	0	0
$413$	−2.24621	−0.110529
$414$	0	0
$415$	−2.24621	−0.110262
$416$	5.12311	0.251181
$417$	0	0
$418$	4.00000	0.195646
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	−15.1231	−0.736181
$423$	0	0
$424$	12.2462	0.594729
$425$	−5.12311	−0.248507
$426$	0	0
$427$	−6.00000	−0.290360
$428$	16.4924	0.797191
$429$	0	0
$430$	−10.2462	−0.494116
$431$	14.6307	0.704735	0.352368	−	0.935862i	$-0.385377\pi$
$431$	14.6307	0.704735	0.352368	+	0.935862i	$0.385377\pi$
$432$	0	0
$433$	−32.7386	−1.57332	−0.786659	−	0.617388i	$-0.788191\pi$
$433$	−32.7386	−1.57332	−0.786659	+	0.617388i	$0.788191\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	−6.87689	−0.329344
$437$	−12.4924	−0.597594
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	−26.2462	−1.24840
$443$	13.7538	0.653462	0.326731	−	0.945117i	$-0.394053\pi$
$443$	13.7538	0.653462	0.326731	+	0.945117i	$0.394053\pi$
$444$	0	0
$445$	11.3693	0.538957
$446$	−4.49242	−0.212722
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	8.24621	0.388299
$452$	9.12311	0.429115
$453$	0	0
$454$	22.7386	1.06718
$455$	−5.12311	−0.240175
$456$	0	0
$457$	−15.7538	−0.736931	−0.368466	−	0.929641i	$-0.620117\pi$
$457$	−15.7538	−0.736931	−0.368466	+	0.929641i	$0.620117\pi$
$458$	5.12311	0.239387
$459$	0	0
$460$	3.12311	0.145616
$461$	19.7538	0.920026	0.460013	−	0.887912i	$-0.347845\pi$
$461$	19.7538	0.920026	0.460013	+	0.887912i	$0.347845\pi$
$462$	0	0
$463$	12.4924	0.580572	0.290286	−	0.956940i	$-0.406250\pi$
$463$	12.4924	0.580572	0.290286	+	0.956940i	$0.406250\pi$
$464$	8.24621	0.382821
$465$	0	0
$466$	−22.0000	−1.01913
$467$	−8.49242	−0.392982	−0.196491	−	0.980506i	$-0.562955\pi$
$467$	−8.49242	−0.392982	−0.196491	+	0.980506i	$0.562955\pi$
$468$	0	0
$469$	7.12311	0.328914
$470$	−8.00000	−0.369012
$471$	0	0
$472$	−2.24621	−0.103390
$473$	10.2462	0.471121
$474$	0	0
$475$	4.00000	0.183533
$476$	5.12311	0.234817
$477$	0	0
$478$	1.36932	0.0626311
$479$	−40.9848	−1.87265	−0.936323	−	0.351141i	$-0.885794\pi$
$479$	−40.9848	−1.87265	−0.936323	+	0.351141i	$0.885794\pi$
$480$	0	0
$481$	−5.75379	−0.262350
$482$	18.4924	0.842307
$483$	0	0
$484$	1.00000	0.0454545
$485$	4.24621	0.192811
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	1.00000	0.0451754
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	0	0
$493$	−42.2462	−1.90267
$494$	20.4924	0.921998
$495$	0	0
$496$	8.00000	0.359211
$497$	14.2462	0.639030
$498$	0	0
$499$	13.7538	0.615704	0.307852	−	0.951434i	$-0.400390\pi$
$499$	13.7538	0.615704	0.307852	+	0.951434i	$0.400390\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−18.2462	−0.814368
$503$	35.1231	1.56606	0.783031	−	0.621983i	$-0.213673\pi$
$503$	35.1231	1.56606	0.783031	+	0.621983i	$0.213673\pi$
$504$	0	0
$505$	−14.4924	−0.644904
$506$	−3.12311	−0.138839
$507$	0	0
$508$	16.0000	0.709885
$509$	−23.7538	−1.05287	−0.526434	−	0.850216i	$-0.676471\pi$
$509$	−23.7538	−1.05287	−0.526434	+	0.850216i	$0.676471\pi$
$510$	0	0
$511$	−8.24621	−0.364791
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	11.7538	0.518437
$515$	0	0
$516$	0	0
$517$	8.00000	0.351840
$518$	1.12311	0.0493464
$519$	0	0
$520$	−5.12311	−0.224663
$521$	−11.3693	−0.498099	−0.249049	−	0.968491i	$-0.580118\pi$
$521$	−11.3693	−0.498099	−0.249049	+	0.968491i	$0.580118\pi$
$522$	0	0
$523$	23.1231	1.01110	0.505551	−	0.862796i	$-0.331289\pi$
$523$	23.1231	1.01110	0.505551	+	0.862796i	$0.331289\pi$
$524$	−16.4924	−0.720475
$525$	0	0
$526$	−30.2462	−1.31880
$527$	−40.9848	−1.78533
$528$	0	0
$529$	−13.2462	−0.575922
$530$	−12.2462	−0.531941
$531$	0	0
$532$	−4.00000	−0.173422
$533$	42.2462	1.82989
$534$	0	0
$535$	−16.4924	−0.713030
$536$	7.12311	0.307671
$537$	0	0
$538$	−11.7538	−0.506742
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−41.6155	−1.78919	−0.894596	−	0.446877i	$-0.852536\pi$
$541$	−41.6155	−1.78919	−0.894596	+	0.446877i	$0.852536\pi$
$542$	12.4924	0.536595
$543$	0	0
$544$	5.12311	0.219651
$545$	6.87689	0.294574
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	17.1231	0.731463
$549$	0	0
$550$	1.00000	0.0426401
$551$	32.9848	1.40520
$552$	0	0
$553$	0	0
$554$	3.75379	0.159483
$555$	0	0
$556$	18.2462	0.773812
$557$	−20.2462	−0.857860	−0.428930	−	0.903338i	$-0.641109\pi$
$557$	−20.2462	−0.857860	−0.428930	+	0.903338i	$0.641109\pi$
$558$	0	0
$559$	52.4924	2.22019
$560$	1.00000	0.0422577
$561$	0	0
$562$	−17.1231	−0.722295
$563$	−0.492423	−0.0207531	−0.0103766	−	0.999946i	$-0.503303\pi$
$563$	−0.492423	−0.0207531	−0.0103766	+	0.999946i	$0.503303\pi$
$564$	0	0
$565$	−9.12311	−0.383812
$566$	15.1231	0.635672
$567$	0	0
$568$	14.2462	0.597758
$569$	−5.12311	−0.214772	−0.107386	−	0.994217i	$-0.534248\pi$
$569$	−5.12311	−0.214772	−0.107386	+	0.994217i	$0.534248\pi$
$570$	0	0
$571$	13.3693	0.559488	0.279744	−	0.960075i	$-0.409750\pi$
$571$	13.3693	0.559488	0.279744	+	0.960075i	$0.409750\pi$
$572$	5.12311	0.214208
$573$	0	0
$574$	−8.24621	−0.344190
$575$	−3.12311	−0.130243
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−9.24621	−0.384592
$579$	0	0
$580$	−8.24621	−0.342405
$581$	−2.24621	−0.0931885
$582$	0	0
$583$	12.2462	0.507186
$584$	−8.24621	−0.341231
$585$	0	0
$586$	−24.2462	−1.00160
$587$	−7.50758	−0.309871	−0.154935	−	0.987925i	$-0.549517\pi$
$587$	−7.50758	−0.309871	−0.154935	+	0.987925i	$0.549517\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	2.24621	0.0924751
$591$	0	0
$592$	1.12311	0.0461594
$593$	−43.3693	−1.78096	−0.890482	−	0.455018i	$-0.849633\pi$
$593$	−43.3693	−1.78096	−0.890482	+	0.455018i	$0.849633\pi$
$594$	0	0
$595$	−5.12311	−0.210027
$596$	10.0000	0.409616
$597$	0	0
$598$	−16.0000	−0.654289
$599$	11.5076	0.470187	0.235093	−	0.971973i	$-0.424460\pi$
$599$	11.5076	0.470187	0.235093	+	0.971973i	$0.424460\pi$
$600$	0	0
$601$	11.7538	0.479447	0.239724	−	0.970841i	$-0.422943\pi$
$601$	11.7538	0.479447	0.239724	+	0.970841i	$0.422943\pi$
$602$	−10.2462	−0.417604
$603$	0	0
$604$	8.00000	0.325515
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	6.00000	0.242933
$611$	40.9848	1.65807
$612$	0	0
$613$	40.7386	1.64542	0.822709	−	0.568463i	$-0.192462\pi$
$613$	40.7386	1.64542	0.822709	+	0.568463i	$0.192462\pi$
$614$	−21.3693	−0.862395
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	29.6155	1.19228	0.596138	−	0.802882i	$-0.296701\pi$
$617$	29.6155	1.19228	0.596138	+	0.802882i	$0.296701\pi$
$618$	0	0
$619$	−7.12311	−0.286302	−0.143151	−	0.989701i	$-0.545723\pi$
$619$	−7.12311	−0.286302	−0.143151	+	0.989701i	$0.545723\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	−1.36932	−0.0549046
$623$	11.3693	0.455502
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	0	0
$628$	7.75379	0.309410
$629$	−5.75379	−0.229419
$630$	0	0
$631$	11.5076	0.458109	0.229055	−	0.973414i	$-0.426437\pi$
$631$	11.5076	0.458109	0.229055	+	0.973414i	$0.426437\pi$
$632$	0	0
$633$	0	0
$634$	−24.2462	−0.962940
$635$	−16.0000	−0.634941
$636$	0	0
$637$	−5.12311	−0.202985
$638$	8.24621	0.326471
$639$	0	0
$640$	1.00000	0.0395285
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	36.9848	1.45854	0.729270	−	0.684226i	$-0.239860\pi$
$643$	36.9848	1.45854	0.729270	+	0.684226i	$0.239860\pi$
$644$	3.12311	0.123068
$645$	0	0
$646$	20.4924	0.806264
$647$	40.9848	1.61128	0.805640	−	0.592405i	$-0.201821\pi$
$647$	40.9848	1.61128	0.805640	+	0.592405i	$0.201821\pi$
$648$	0	0
$649$	−2.24621	−0.0881715
$650$	5.12311	0.200945
$651$	0	0
$652$	5.36932	0.210279
$653$	42.9848	1.68213	0.841063	−	0.540936i	$-0.181930\pi$
$653$	42.9848	1.68213	0.841063	+	0.540936i	$0.181930\pi$
$654$	0	0
$655$	16.4924	0.644412
$656$	−8.24621	−0.321960
$657$	0	0
$658$	−8.00000	−0.311872
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	29.6155	1.15191	0.575955	−	0.817481i	$-0.304630\pi$
$661$	29.6155	1.15191	0.575955	+	0.817481i	$0.304630\pi$
$662$	−24.4924	−0.951925
$663$	0	0
$664$	−2.24621	−0.0871699
$665$	4.00000	0.155113
$666$	0	0
$667$	−25.7538	−0.997191
$668$	−9.36932	−0.362510
$669$	0	0
$670$	−7.12311	−0.275190
$671$	−6.00000	−0.231627
$672$	0	0
$673$	46.4924	1.79215	0.896076	−	0.443901i	$-0.146406\pi$
$673$	46.4924	1.79215	0.896076	+	0.443901i	$0.146406\pi$
$674$	−30.4924	−1.17452
$675$	0	0
$676$	13.2462	0.509470
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	0	0
$679$	4.24621	0.162955
$680$	−5.12311	−0.196462
$681$	0	0
$682$	8.00000	0.306336
$683$	−46.7386	−1.78840	−0.894202	−	0.447664i	$-0.852256\pi$
$683$	−46.7386	−1.78840	−0.894202	+	0.447664i	$0.852256\pi$
$684$	0	0
$685$	−17.1231	−0.654240
$686$	1.00000	0.0381802
$687$	0	0
$688$	−10.2462	−0.390633
$689$	62.7386	2.39015
$690$	0	0
$691$	41.8617	1.59250	0.796248	−	0.604971i	$-0.206815\pi$
$691$	41.8617	1.59250	0.796248	+	0.604971i	$0.206815\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	−16.4924	−0.626044
$695$	−18.2462	−0.692118
$696$	0	0
$697$	42.2462	1.60019
$698$	6.49242	0.245742
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−16.7386	−0.632209	−0.316105	−	0.948724i	$-0.602375\pi$
$701$	−16.7386	−0.632209	−0.316105	+	0.948724i	$0.602375\pi$
$702$	0	0
$703$	4.49242	0.169435
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−20.2462	−0.761976
$707$	−14.4924	−0.545044
$708$	0	0
$709$	18.4924	0.694498	0.347249	−	0.937773i	$-0.387116\pi$
$709$	18.4924	0.694498	0.347249	+	0.937773i	$0.387116\pi$
$710$	−14.2462	−0.534651
$711$	0	0
$712$	11.3693	0.426083
$713$	−24.9848	−0.935690
$714$	0	0
$715$	−5.12311	−0.191593
$716$	−20.0000	−0.747435
$717$	0	0
$718$	−25.3693	−0.946774
$719$	−41.3693	−1.54282	−0.771408	−	0.636341i	$-0.780447\pi$
$719$	−41.3693	−1.54282	−0.771408	+	0.636341i	$0.780447\pi$
$720$	0	0
$721$	0	0
$722$	3.00000	0.111648
$723$	0	0
$724$	−11.3693	−0.422537
$725$	8.24621	0.306257
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−5.12311	−0.189875
$729$	0	0
$730$	8.24621	0.305206
$731$	52.4924	1.94150
$732$	0	0
$733$	26.8769	0.992721	0.496360	−	0.868117i	$-0.334669\pi$
$733$	26.8769	0.992721	0.496360	+	0.868117i	$0.334669\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	3.12311	0.115119
$737$	7.12311	0.262383
$738$	0	0
$739$	−35.6155	−1.31014	−0.655069	−	0.755569i	$-0.727361\pi$
$739$	−35.6155	−1.31014	−0.655069	+	0.755569i	$0.727361\pi$
$740$	−1.12311	−0.0412862
$741$	0	0
$742$	−12.2462	−0.449573
$743$	20.4924	0.751794	0.375897	−	0.926661i	$-0.377335\pi$
$743$	20.4924	0.751794	0.375897	+	0.926661i	$0.377335\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	38.4924	1.40931
$747$	0	0
$748$	5.12311	0.187319
$749$	−16.4924	−0.602620
$750$	0	0
$751$	−24.9848	−0.911710	−0.455855	−	0.890054i	$-0.650666\pi$
$751$	−24.9848	−0.911710	−0.455855	+	0.890054i	$0.650666\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	42.2462	1.53852
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−39.8617	−1.44880	−0.724400	−	0.689380i	$-0.757883\pi$
$757$	−39.8617	−1.44880	−0.724400	+	0.689380i	$0.757883\pi$
$758$	0.492423	0.0178856
$759$	0	0
$760$	4.00000	0.145095
$761$	4.24621	0.153925	0.0769625	−	0.997034i	$-0.475478\pi$
$761$	4.24621	0.153925	0.0769625	+	0.997034i	$0.475478\pi$
$762$	0	0
$763$	6.87689	0.248960
$764$	−12.4924	−0.451960
$765$	0	0
$766$	−36.4924	−1.31852
$767$	−11.5076	−0.415515
$768$	0	0
$769$	3.75379	0.135365	0.0676825	−	0.997707i	$-0.478439\pi$
$769$	3.75379	0.135365	0.0676825	+	0.997707i	$0.478439\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	8.24621	0.296788
$773$	−24.7386	−0.889787	−0.444893	−	0.895584i	$-0.646758\pi$
$773$	−24.7386	−0.889787	−0.444893	+	0.895584i	$0.646758\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	4.24621	0.152430
$777$	0	0
$778$	6.00000	0.215110
$779$	−32.9848	−1.18180
$780$	0	0
$781$	14.2462	0.509770
$782$	−16.0000	−0.572159
$783$	0	0
$784$	1.00000	0.0357143
$785$	−7.75379	−0.276745
$786$	0	0
$787$	−19.6155	−0.699218	−0.349609	−	0.936896i	$-0.613686\pi$
$787$	−19.6155	−0.699218	−0.349609	+	0.936896i	$0.613686\pi$
$788$	0.246211	0.00877091
$789$	0	0
$790$	0	0
$791$	−9.12311	−0.324380
$792$	0	0
$793$	−30.7386	−1.09156
$794$	5.50758	0.195457
$795$	0	0
$796$	−12.4924	−0.442782
$797$	11.7538	0.416341	0.208170	−	0.978093i	$-0.433249\pi$
$797$	11.7538	0.416341	0.208170	+	0.978093i	$0.433249\pi$
$798$	0	0
$799$	40.9848	1.44994
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−14.0000	−0.494357
$803$	−8.24621	−0.291002
$804$	0	0
$805$	−3.12311	−0.110075
$806$	40.9848	1.44363
$807$	0	0
$808$	−14.4924	−0.509842
$809$	19.8617	0.698302	0.349151	−	0.937067i	$-0.386470\pi$
$809$	19.8617	0.698302	0.349151	+	0.937067i	$0.386470\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	−8.24621	−0.289385
$813$	0	0
$814$	1.12311	0.0393648
$815$	−5.36932	−0.188079
$816$	0	0
$817$	−40.9848	−1.43388
$818$	−20.7386	−0.725109
$819$	0	0
$820$	8.24621	0.287970
$821$	−37.2311	−1.29937	−0.649686	−	0.760202i	$-0.725100\pi$
$821$	−37.2311	−1.29937	−0.649686	+	0.760202i	$0.725100\pi$
$822$	0	0
$823$	−10.7386	−0.374325	−0.187163	−	0.982329i	$-0.559929\pi$
$823$	−10.7386	−0.374325	−0.187163	+	0.982329i	$0.559929\pi$
$824$	0	0
$825$	0	0
$826$	2.24621	0.0781557
$827$	16.4924	0.573498	0.286749	−	0.958006i	$-0.407425\pi$
$827$	16.4924	0.573498	0.286749	+	0.958006i	$0.407425\pi$
$828$	0	0
$829$	18.1080	0.628915	0.314458	−	0.949272i	$-0.398177\pi$
$829$	18.1080	0.628915	0.314458	+	0.949272i	$0.398177\pi$
$830$	2.24621	0.0779671
$831$	0	0
$832$	−5.12311	−0.177612
$833$	−5.12311	−0.177505
$834$	0	0
$835$	9.36932	0.324239
$836$	−4.00000	−0.138343
$837$	0	0
$838$	−4.00000	−0.138178
$839$	13.8617	0.478560	0.239280	−	0.970951i	$-0.423089\pi$
$839$	13.8617	0.478560	0.239280	+	0.970951i	$0.423089\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	−6.00000	−0.206774
$843$	0	0
$844$	15.1231	0.520559
$845$	−13.2462	−0.455684
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−12.2462	−0.420537
$849$	0	0
$850$	5.12311	0.175721
$851$	−3.50758	−0.120238
$852$	0	0
$853$	34.8769	1.19416	0.597081	−	0.802181i	$-0.296327\pi$
$853$	34.8769	1.19416	0.597081	+	0.802181i	$0.296327\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	−16.4924	−0.563699
$857$	−6.87689	−0.234910	−0.117455	−	0.993078i	$-0.537474\pi$
$857$	−6.87689	−0.234910	−0.117455	+	0.993078i	$0.537474\pi$
$858$	0	0
$859$	−13.3693	−0.456155	−0.228078	−	0.973643i	$-0.573244\pi$
$859$	−13.3693	−0.456155	−0.228078	+	0.973643i	$0.573244\pi$
$860$	10.2462	0.349393
$861$	0	0
$862$	−14.6307	−0.498323
$863$	42.3542	1.44175	0.720876	−	0.693064i	$-0.243740\pi$
$863$	42.3542	1.44175	0.720876	+	0.693064i	$0.243740\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	32.7386	1.11250
$867$	0	0
$868$	−8.00000	−0.271538
$869$	0	0
$870$	0	0
$871$	36.4924	1.23650
$872$	6.87689	0.232881
$873$	0	0
$874$	12.4924	0.422562
$875$	1.00000	0.0338062
$876$	0	0
$877$	−52.7386	−1.78086	−0.890429	−	0.455123i	$-0.849595\pi$
$877$	−52.7386	−1.78086	−0.890429	+	0.455123i	$0.849595\pi$
$878$	24.0000	0.809961
$879$	0	0
$880$	1.00000	0.0337100
$881$	−6.87689	−0.231688	−0.115844	−	0.993267i	$-0.536957\pi$
$881$	−6.87689	−0.231688	−0.115844	+	0.993267i	$0.536957\pi$
$882$	0	0
$883$	−32.1080	−1.08052	−0.540259	−	0.841499i	$-0.681674\pi$
$883$	−32.1080	−1.08052	−0.540259	+	0.841499i	$0.681674\pi$
$884$	26.2462	0.882756
$885$	0	0
$886$	−13.7538	−0.462068
$887$	37.8617	1.27127	0.635636	−	0.771989i	$-0.280738\pi$
$887$	37.8617	1.27127	0.635636	+	0.771989i	$0.280738\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	−11.3693	−0.381100
$891$	0	0
$892$	4.49242	0.150417
$893$	−32.0000	−1.07084
$894$	0	0
$895$	20.0000	0.668526
$896$	1.00000	0.0334077
$897$	0	0
$898$	2.00000	0.0667409
$899$	65.9697	2.20021
$900$	0	0
$901$	62.7386	2.09013
$902$	−8.24621	−0.274569
$903$	0	0
$904$	−9.12311	−0.303430
$905$	11.3693	0.377929
$906$	0	0
$907$	−8.87689	−0.294752	−0.147376	−	0.989081i	$-0.547083\pi$
$907$	−8.87689	−0.294752	−0.147376	+	0.989081i	$0.547083\pi$
$908$	−22.7386	−0.754608
$909$	0	0
$910$	5.12311	0.169829
$911$	9.75379	0.323157	0.161579	−	0.986860i	$-0.448341\pi$
$911$	9.75379	0.323157	0.161579	+	0.986860i	$0.448341\pi$
$912$	0	0
$913$	−2.24621	−0.0743387
$914$	15.7538	0.521089
$915$	0	0
$916$	−5.12311	−0.169272
$917$	16.4924	0.544628
$918$	0	0
$919$	−10.7386	−0.354235	−0.177117	−	0.984190i	$-0.556677\pi$
$919$	−10.7386	−0.354235	−0.177117	+	0.984190i	$0.556677\pi$
$920$	−3.12311	−0.102966
$921$	0	0
$922$	−19.7538	−0.650556
$923$	72.9848	2.40233
$924$	0	0
$925$	1.12311	0.0369275
$926$	−12.4924	−0.410526
$927$	0	0
$928$	−8.24621	−0.270695
$929$	−45.1231	−1.48044	−0.740221	−	0.672364i	$-0.765279\pi$
$929$	−45.1231	−1.48044	−0.740221	+	0.672364i	$0.765279\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	22.0000	0.720634
$933$	0	0
$934$	8.49242	0.277881
$935$	−5.12311	−0.167543
$936$	0	0
$937$	−58.4924	−1.91086	−0.955432	−	0.295211i	$-0.904610\pi$
$937$	−58.4924	−1.91086	−0.955432	+	0.295211i	$0.904610\pi$
$938$	−7.12311	−0.232578
$939$	0	0
$940$	8.00000	0.260931
$941$	−34.4924	−1.12442	−0.562210	−	0.826994i	$-0.690049\pi$
$941$	−34.4924	−1.12442	−0.562210	+	0.826994i	$0.690049\pi$
$942$	0	0
$943$	25.7538	0.838659
$944$	2.24621	0.0731079
$945$	0	0
$946$	−10.2462	−0.333133
$947$	−57.4773	−1.86776	−0.933880	−	0.357586i	$-0.883600\pi$
$947$	−57.4773	−1.86776	−0.933880	+	0.357586i	$0.883600\pi$
$948$	0	0
$949$	−42.2462	−1.37137
$950$	−4.00000	−0.129777
$951$	0	0
$952$	−5.12311	−0.166041
$953$	−12.7386	−0.412645	−0.206322	−	0.978484i	$-0.566150\pi$
$953$	−12.7386	−0.412645	−0.206322	+	0.978484i	$0.566150\pi$
$954$	0	0
$955$	12.4924	0.404245
$956$	−1.36932	−0.0442869
$957$	0	0
$958$	40.9848	1.32416
$959$	−17.1231	−0.552934
$960$	0	0
$961$	33.0000	1.06452
$962$	5.75379	0.185510
$963$	0	0
$964$	−18.4924	−0.595601
$965$	−8.24621	−0.265455
$966$	0	0
$967$	52.4924	1.68804	0.844021	−	0.536310i	$-0.180182\pi$
$967$	52.4924	1.68804	0.844021	+	0.536310i	$0.180182\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−4.24621	−0.136338
$971$	−32.4924	−1.04273	−0.521366	−	0.853333i	$-0.674577\pi$
$971$	−32.4924	−1.04273	−0.521366	+	0.853333i	$0.674577\pi$
$972$	0	0
$973$	−18.2462	−0.584947
$974$	8.00000	0.256337
$975$	0	0
$976$	6.00000	0.192055
$977$	−50.6004	−1.61885	−0.809425	−	0.587224i	$-0.800221\pi$
$977$	−50.6004	−1.61885	−0.809425	+	0.587224i	$0.800221\pi$
$978$	0	0
$979$	11.3693	0.363365
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−4.00000	−0.127645
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	0	0
$985$	−0.246211	−0.00784494
$986$	42.2462	1.34539
$987$	0	0
$988$	−20.4924	−0.651951
$989$	32.0000	1.01754
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−8.00000	−0.254000
$993$	0	0
$994$	−14.2462	−0.451862
$995$	12.4924	0.396036
$996$	0	0
$997$	−35.3693	−1.12016	−0.560079	−	0.828439i	$-0.689229\pi$
$997$	−35.3693	−1.12016	−0.560079	+	0.828439i	$0.689229\pi$
$998$	−13.7538	−0.435369
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -5.12311 q^{13} +1.00000 q^{14} +1.00000 q^{16} -5.12311 q^{17} +4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} -3.12311 q^{23} +1.00000 q^{25} +5.12311 q^{26} -1.00000 q^{28} +8.24621 q^{29} +8.00000 q^{31} -1.00000 q^{32} +5.12311 q^{34} +1.00000 q^{35} +1.12311 q^{37} -4.00000 q^{38} +1.00000 q^{40} -8.24621 q^{41} -10.2462 q^{43} -1.00000 q^{44} +3.12311 q^{46} -8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -5.12311 q^{52} -12.2462 q^{53} +1.00000 q^{55} +1.00000 q^{56} -8.24621 q^{58} +2.24621 q^{59} +6.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +5.12311 q^{65} -7.12311 q^{67} -5.12311 q^{68} -1.00000 q^{70} -14.2462 q^{71} +8.24621 q^{73} -1.12311 q^{74} +4.00000 q^{76} +1.00000 q^{77} -1.00000 q^{80} +8.24621 q^{82} +2.24621 q^{83} +5.12311 q^{85} +10.2462 q^{86} +1.00000 q^{88} -11.3693 q^{89} +5.12311 q^{91} -3.12311 q^{92} +8.00000 q^{94} -4.00000 q^{95} -4.24621 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 2 q^{8} + 2 q^{10} - 2 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 8 q^{19} - 2 q^{20} + 2 q^{22} + 2 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} + 16 q^{31} - 2 q^{32} + 2 q^{34} + 2 q^{35} - 6 q^{37} - 8 q^{38} + 2 q^{40} - 4 q^{43} - 2 q^{44} - 2 q^{46} - 16 q^{47} + 2 q^{49} - 2 q^{50} - 2 q^{52} - 8 q^{53} + 2 q^{55} + 2 q^{56} - 12 q^{59} + 12 q^{61} - 16 q^{62} + 2 q^{64} + 2 q^{65} - 6 q^{67} - 2 q^{68} - 2 q^{70} - 12 q^{71} + 6 q^{74} + 8 q^{76} + 2 q^{77} - 2 q^{80} - 12 q^{83} + 2 q^{85} + 4 q^{86} + 2 q^{88} + 2 q^{89} + 2 q^{91} + 2 q^{92} + 16 q^{94} - 8 q^{95} + 8 q^{97} - 2 q^{98}+O(q^{100}) \) 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^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 + 8 * q^19 - 2 * q^20 + 2 * q^22 + 2 * q^23 + 2 * q^25 + 2 * q^26 - 2 * q^28 + 16 * q^31 - 2 * q^32 + 2 * q^34 + 2 * q^35 - 6 * q^37 - 8 * q^38 + 2 * q^40 - 4 * q^43 - 2 * q^44 - 2 * q^46 - 16 * q^47 + 2 * q^49 - 2 * q^50 - 2 * q^52 - 8 * q^53 + 2 * q^55 + 2 * q^56 - 12 * q^59 + 12 * q^61 - 16 * q^62 + 2 * q^64 + 2 * q^65 - 6 * q^67 - 2 * q^68 - 2 * q^70 - 12 * q^71 + 6 * q^74 + 8 * q^76 + 2 * q^77 - 2 * q^80 - 12 * q^83 + 2 * q^85 + 4 * q^86 + 2 * q^88 + 2 * q^89 + 2 * q^91 + 2 * q^92 + 16 * q^94 - 8 * q^95 + 8 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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