LMFDB - Embedded newform 2070.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.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:	$16.5290332184$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 690)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -5.12311 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{10} -5.12311 q^{11} +2.00000 q^{13} +5.12311 q^{14} +1.00000 q^{16} -7.12311 q^{17} +4.00000 q^{19} -1.00000 q^{20} +5.12311 q^{22} -1.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} -5.12311 q^{28} -2.00000 q^{29} -1.00000 q^{32} +7.12311 q^{34} +5.12311 q^{35} -7.12311 q^{37} -4.00000 q^{38} +1.00000 q^{40} -2.00000 q^{41} -5.12311 q^{44} +1.00000 q^{46} +8.00000 q^{47} +19.2462 q^{49} -1.00000 q^{50} +2.00000 q^{52} +4.24621 q^{53} +5.12311 q^{55} +5.12311 q^{56} +2.00000 q^{58} +14.2462 q^{59} +0.876894 q^{61} +1.00000 q^{64} -2.00000 q^{65} -8.00000 q^{67} -7.12311 q^{68} -5.12311 q^{70} -6.24621 q^{71} +12.2462 q^{73} +7.12311 q^{74} +4.00000 q^{76} +26.2462 q^{77} -5.12311 q^{79} -1.00000 q^{80} +2.00000 q^{82} -11.3693 q^{83} +7.12311 q^{85} +5.12311 q^{88} -3.12311 q^{89} -10.2462 q^{91} -1.00000 q^{92} -8.00000 q^{94} -4.00000 q^{95} +0.246211 q^{97} -19.2462 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 6 q^{17} + 8 q^{19} - 2 q^{20} + 2 q^{22} - 2 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{28} - 4 q^{29} - 2 q^{32} + 6 q^{34} + 2 q^{35} - 6 q^{37} - 8 q^{38} + 2 q^{40} - 4 q^{41} - 2 q^{44} + 2 q^{46} + 16 q^{47} + 22 q^{49} - 2 q^{50} + 4 q^{52} - 8 q^{53} + 2 q^{55} + 2 q^{56} + 4 q^{58} + 12 q^{59} + 10 q^{61} + 2 q^{64} - 4 q^{65} - 16 q^{67} - 6 q^{68} - 2 q^{70} + 4 q^{71} + 8 q^{73} + 6 q^{74} + 8 q^{76} + 36 q^{77} - 2 q^{79} - 2 q^{80} + 4 q^{82} + 2 q^{83} + 6 q^{85} + 2 q^{88} + 2 q^{89} - 4 q^{91} - 2 q^{92} - 16 q^{94} - 8 q^{95} - 16 q^{97} - 22 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 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 6 * q^17 + 8 * q^19 - 2 * q^20 + 2 * q^22 - 2 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^28 - 4 * q^29 - 2 * q^32 + 6 * q^34 + 2 * q^35 - 6 * q^37 - 8 * q^38 + 2 * q^40 - 4 * q^41 - 2 * q^44 + 2 * q^46 + 16 * q^47 + 22 * q^49 - 2 * q^50 + 4 * q^52 - 8 * q^53 + 2 * q^55 + 2 * q^56 + 4 * q^58 + 12 * q^59 + 10 * q^61 + 2 * q^64 - 4 * q^65 - 16 * q^67 - 6 * q^68 - 2 * q^70 + 4 * q^71 + 8 * q^73 + 6 * q^74 + 8 * q^76 + 36 * q^77 - 2 * q^79 - 2 * q^80 + 4 * q^82 + 2 * q^83 + 6 * q^85 + 2 * q^88 + 2 * q^89 - 4 * q^91 - 2 * q^92 - 16 * q^94 - 8 * q^95 - 16 * q^97 - 22 * 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
$7$	−5.12311	−1.93635	−0.968176	−	0.250270i	$-0.919480\pi$
$7$	−5.12311	−1.93635	−0.968176	+	0.250270i	$0.919480\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−5.12311	−1.54467	−0.772337	−	0.635213i	$-0.780912\pi$
$11$	−5.12311	−1.54467	−0.772337	+	0.635213i	$0.780912\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	5.12311	1.36921
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.12311	−1.72761	−0.863803	−	0.503829i	$-0.831924\pi$
$17$	−7.12311	−1.72761	−0.863803	+	0.503829i	$0.831924\pi$
$18$	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
$22$	5.12311	1.09225
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−5.12311	−0.968176
$29$	−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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.12311	1.22160
$35$	5.12311	0.865963
$36$	0	0
$37$	−7.12311	−1.17103	−0.585516	−	0.810661i	$-0.699108\pi$
$37$	−7.12311	−1.17103	−0.585516	+	0.810661i	$0.699108\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	1.00000	0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−5.12311	−0.772337
$45$	0	0
$46$	1.00000	0.147442
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	19.2462	2.74946
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	4.24621	0.583262	0.291631	−	0.956531i	$-0.405802\pi$
$53$	4.24621	0.583262	0.291631	+	0.956531i	$0.405802\pi$
$54$	0	0
$55$	5.12311	0.690799
$56$	5.12311	0.684604
$57$	0	0
$58$	2.00000	0.262613
$59$	14.2462	1.85470	0.927349	−	0.374197i	$-0.122082\pi$
$59$	14.2462	1.85470	0.927349	+	0.374197i	$0.122082\pi$
$60$	0	0
$61$	0.876894	0.112275	0.0561374	−	0.998423i	$-0.482122\pi$
$61$	0.876894	0.112275	0.0561374	+	0.998423i	$0.482122\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−7.12311	−0.863803
$69$	0	0
$70$	−5.12311	−0.612328
$71$	−6.24621	−0.741289	−0.370644	−	0.928775i	$-0.620863\pi$
$71$	−6.24621	−0.741289	−0.370644	+	0.928775i	$0.620863\pi$
$72$	0	0
$73$	12.2462	1.43331	0.716655	−	0.697428i	$-0.245672\pi$
$73$	12.2462	1.43331	0.716655	+	0.697428i	$0.245672\pi$
$74$	7.12311	0.828044
$75$	0	0
$76$	4.00000	0.458831
$77$	26.2462	2.99103
$78$	0	0
$79$	−5.12311	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$79$	−5.12311	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	2.00000	0.220863
$83$	−11.3693	−1.24794	−0.623972	−	0.781446i	$-0.714482\pi$
$83$	−11.3693	−1.24794	−0.623972	+	0.781446i	$0.714482\pi$
$84$	0	0
$85$	7.12311	0.772609
$86$	0	0
$87$	0	0
$88$	5.12311	0.546125
$89$	−3.12311	−0.331049	−0.165524	−	0.986206i	$-0.552932\pi$
$89$	−3.12311	−0.331049	−0.165524	+	0.986206i	$0.552932\pi$
$90$	0	0
$91$	−10.2462	−1.07409
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−4.00000	−0.410391
$96$	0	0
$97$	0.246211	0.0249990	0.0124995	−	0.999922i	$-0.496021\pi$
$97$	0.246211	0.0249990	0.0124995	+	0.999922i	$0.496021\pi$
$98$	−19.2462	−1.94416
$99$	0	0
$100$	1.00000	0.100000
$101$	16.2462	1.61656	0.808279	−	0.588799i	$-0.200399\pi$
$101$	16.2462	1.61656	0.808279	+	0.588799i	$0.200399\pi$
$102$	0	0
$103$	15.3693	1.51438	0.757192	−	0.653192i	$-0.226571\pi$
$103$	15.3693	1.51438	0.757192	+	0.653192i	$0.226571\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−4.24621	−0.412428
$107$	−11.3693	−1.09911	−0.549557	−	0.835456i	$-0.685203\pi$
$107$	−11.3693	−1.09911	−0.549557	+	0.835456i	$0.685203\pi$
$108$	0	0
$109$	11.1231	1.06540	0.532700	−	0.846304i	$-0.321177\pi$
$109$	11.1231	1.06540	0.532700	+	0.846304i	$0.321177\pi$
$110$	−5.12311	−0.488469
$111$	0	0
$112$	−5.12311	−0.484088
$113$	0.876894	0.0824913	0.0412456	−	0.999149i	$-0.486867\pi$
$113$	0.876894	0.0824913	0.0412456	+	0.999149i	$0.486867\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−14.2462	−1.31147
$119$	36.4924	3.34525
$120$	0	0
$121$	15.2462	1.38602
$122$	−0.876894	−0.0793903
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−22.2462	−1.97403	−0.987016	−	0.160622i	$-0.948650\pi$
$127$	−22.2462	−1.97403	−0.987016	+	0.160622i	$0.948650\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.00000	0.175412
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	−20.4924	−1.77692
$134$	8.00000	0.691095
$135$	0	0
$136$	7.12311	0.610801
$137$	19.1231	1.63380	0.816899	−	0.576781i	$-0.195692\pi$
$137$	19.1231	1.63380	0.816899	+	0.576781i	$0.195692\pi$
$138$	0	0
$139$	−16.4924	−1.39887	−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924	−1.39887	−0.699435	+	0.714697i	$0.746565\pi$
$140$	5.12311	0.432981
$141$	0	0
$142$	6.24621	0.524170
$143$	−10.2462	−0.856831
$144$	0	0
$145$	2.00000	0.166091
$146$	−12.2462	−1.01350
$147$	0	0
$148$	−7.12311	−0.585516
$149$	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
$151$	−10.2462	−0.833825	−0.416912	−	0.908947i	$-0.636888\pi$
$151$	−10.2462	−0.833825	−0.416912	+	0.908947i	$0.636888\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	−26.2462	−2.11498
$155$	0	0
$156$	0	0
$157$	0.876894	0.0699838	0.0349919	−	0.999388i	$-0.488859\pi$
$157$	0.876894	0.0699838	0.0349919	+	0.999388i	$0.488859\pi$
$158$	5.12311	0.407572
$159$	0	0
$160$	1.00000	0.0790569
$161$	5.12311	0.403757
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	11.3693	0.882430
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−7.12311	−0.546317
$171$	0	0
$172$	0	0
$173$	−3.75379	−0.285395	−0.142698	−	0.989766i	$-0.545578\pi$
$173$	−3.75379	−0.285395	−0.142698	+	0.989766i	$0.545578\pi$
$174$	0	0
$175$	−5.12311	−0.387270
$176$	−5.12311	−0.386169
$177$	0	0
$178$	3.12311	0.234087
$179$	−24.4924	−1.83065	−0.915325	−	0.402716i	$-0.868066\pi$
$179$	−24.4924	−1.83065	−0.915325	+	0.402716i	$0.868066\pi$
$180$	0	0
$181$	19.1231	1.42141	0.710705	−	0.703491i	$-0.248376\pi$
$181$	19.1231	1.42141	0.710705	+	0.703491i	$0.248376\pi$
$182$	10.2462	0.759500
$183$	0	0
$184$	1.00000	0.0737210
$185$	7.12311	0.523701
$186$	0	0
$187$	36.4924	2.66859
$188$	8.00000	0.583460
$189$	0	0
$190$	4.00000	0.290191
$191$	20.4924	1.48278	0.741390	−	0.671075i	$-0.234167\pi$
$191$	20.4924	1.48278	0.741390	+	0.671075i	$0.234167\pi$
$192$	0	0
$193$	−8.24621	−0.593575	−0.296788	−	0.954944i	$-0.595915\pi$
$193$	−8.24621	−0.593575	−0.296788	+	0.954944i	$0.595915\pi$
$194$	−0.246211	−0.0176769
$195$	0	0
$196$	19.2462	1.37473
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	10.8769	0.771043	0.385521	−	0.922699i	$-0.374022\pi$
$199$	10.8769	0.771043	0.385521	+	0.922699i	$0.374022\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−16.2462	−1.14308
$203$	10.2462	0.719143
$204$	0	0
$205$	2.00000	0.139686
$206$	−15.3693	−1.07083
$207$	0	0
$208$	2.00000	0.138675
$209$	−20.4924	−1.41749
$210$	0	0
$211$	16.4924	1.13539	0.567693	−	0.823241i	$-0.307836\pi$
$211$	16.4924	1.13539	0.567693	+	0.823241i	$0.307836\pi$
$212$	4.24621	0.291631
$213$	0	0
$214$	11.3693	0.777191
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−11.1231	−0.753352
$219$	0	0
$220$	5.12311	0.345400
$221$	−14.2462	−0.958304
$222$	0	0
$223$	6.24621	0.418277	0.209139	−	0.977886i	$-0.432934\pi$
$223$	6.24621	0.418277	0.209139	+	0.977886i	$0.432934\pi$
$224$	5.12311	0.342302
$225$	0	0
$226$	−0.876894	−0.0583301
$227$	−19.3693	−1.28559	−0.642793	−	0.766040i	$-0.722225\pi$
$227$	−19.3693	−1.28559	−0.642793	+	0.766040i	$0.722225\pi$
$228$	0	0
$229$	−1.36932	−0.0904870	−0.0452435	−	0.998976i	$-0.514406\pi$
$229$	−1.36932	−0.0904870	−0.0452435	+	0.998976i	$0.514406\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	2.00000	0.131306
$233$	20.7386	1.35863	0.679317	−	0.733845i	$-0.262276\pi$
$233$	20.7386	1.35863	0.679317	+	0.733845i	$0.262276\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	14.2462	0.927349
$237$	0	0
$238$	−36.4924	−2.36545
$239$	1.75379	0.113443	0.0567216	−	0.998390i	$-0.481935\pi$
$239$	1.75379	0.113443	0.0567216	+	0.998390i	$0.481935\pi$
$240$	0	0
$241$	−2.49242	−0.160551	−0.0802755	−	0.996773i	$-0.525580\pi$
$241$	−2.49242	−0.160551	−0.0802755	+	0.996773i	$0.525580\pi$
$242$	−15.2462	−0.980064
$243$	0	0
$244$	0.876894	0.0561374
$245$	−19.2462	−1.22960
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	0	0
$250$	1.00000	0.0632456
$251$	7.36932	0.465147	0.232574	−	0.972579i	$-0.425285\pi$
$251$	7.36932	0.465147	0.232574	+	0.972579i	$0.425285\pi$
$252$	0	0
$253$	5.12311	0.322087
$254$	22.2462	1.39585
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.24621	0.514385	0.257192	−	0.966360i	$-0.417203\pi$
$257$	8.24621	0.514385	0.257192	+	0.966360i	$0.417203\pi$
$258$	0	0
$259$	36.4924	2.26753
$260$	−2.00000	−0.124035
$261$	0	0
$262$	−4.00000	−0.247121
$263$	−10.2462	−0.631808	−0.315904	−	0.948791i	$-0.602308\pi$
$263$	−10.2462	−0.631808	−0.315904	+	0.948791i	$0.602308\pi$
$264$	0	0
$265$	−4.24621	−0.260843
$266$	20.4924	1.25647
$267$	0	0
$268$	−8.00000	−0.488678
$269$	16.2462	0.990549	0.495274	−	0.868737i	$-0.335067\pi$
$269$	16.2462	0.990549	0.495274	+	0.868737i	$0.335067\pi$
$270$	0	0
$271$	10.2462	0.622413	0.311207	−	0.950342i	$-0.399267\pi$
$271$	10.2462	0.622413	0.311207	+	0.950342i	$0.399267\pi$
$272$	−7.12311	−0.431902
$273$	0	0
$274$	−19.1231	−1.15527
$275$	−5.12311	−0.308935
$276$	0	0
$277$	20.2462	1.21648	0.608238	−	0.793754i	$-0.291876\pi$
$277$	20.2462	1.21648	0.608238	+	0.793754i	$0.291876\pi$
$278$	16.4924	0.989150
$279$	0	0
$280$	−5.12311	−0.306164
$281$	12.8769	0.768171	0.384086	−	0.923298i	$-0.374517\pi$
$281$	12.8769	0.768171	0.384086	+	0.923298i	$0.374517\pi$
$282$	0	0
$283$	−2.24621	−0.133523	−0.0667617	−	0.997769i	$-0.521267\pi$
$283$	−2.24621	−0.133523	−0.0667617	+	0.997769i	$0.521267\pi$
$284$	−6.24621	−0.370644
$285$	0	0
$286$	10.2462	0.605871
$287$	10.2462	0.604815
$288$	0	0
$289$	33.7386	1.98463
$290$	−2.00000	−0.117444
$291$	0	0
$292$	12.2462	0.716655
$293$	−11.7538	−0.686664	−0.343332	−	0.939214i	$-0.611556\pi$
$293$	−11.7538	−0.686664	−0.343332	+	0.939214i	$0.611556\pi$
$294$	0	0
$295$	−14.2462	−0.829446
$296$	7.12311	0.414022
$297$	0	0
$298$	−10.0000	−0.579284
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	10.2462	0.589603
$303$	0	0
$304$	4.00000	0.229416
$305$	−0.876894	−0.0502108
$306$	0	0
$307$	0.492423	0.0281040	0.0140520	−	0.999901i	$-0.495527\pi$
$307$	0.492423	0.0281040	0.0140520	+	0.999901i	$0.495527\pi$
$308$	26.2462	1.49552
$309$	0	0
$310$	0	0
$311$	−26.7386	−1.51621	−0.758104	−	0.652133i	$-0.773874\pi$
$311$	−26.7386	−1.51621	−0.758104	+	0.652133i	$0.773874\pi$
$312$	0	0
$313$	10.4924	0.593067	0.296533	−	0.955022i	$-0.404169\pi$
$313$	10.4924	0.593067	0.296533	+	0.955022i	$0.404169\pi$
$314$	−0.876894	−0.0494860
$315$	0	0
$316$	−5.12311	−0.288197
$317$	−20.7386	−1.16480	−0.582399	−	0.812903i	$-0.697886\pi$
$317$	−20.7386	−1.16480	−0.582399	+	0.812903i	$0.697886\pi$
$318$	0	0
$319$	10.2462	0.573678
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−5.12311	−0.285500
$323$	−28.4924	−1.58536
$324$	0	0
$325$	2.00000	0.110940
$326$	12.0000	0.664619
$327$	0	0
$328$	2.00000	0.110432
$329$	−40.9848	−2.25957
$330$	0	0
$331$	−32.4924	−1.78595	−0.892973	−	0.450111i	$-0.851384\pi$
$331$	−32.4924	−1.78595	−0.892973	+	0.450111i	$0.851384\pi$
$332$	−11.3693	−0.623972
$333$	0	0
$334$	−8.00000	−0.437741
$335$	8.00000	0.437087
$336$	0	0
$337$	−6.49242	−0.353665	−0.176832	−	0.984241i	$-0.556585\pi$
$337$	−6.49242	−0.353665	−0.176832	+	0.984241i	$0.556585\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	7.12311	0.386305
$341$	0	0
$342$	0	0
$343$	−62.7386	−3.38757
$344$	0	0
$345$	0	0
$346$	3.75379	0.201805
$347$	34.7386	1.86487	0.932434	−	0.361341i	$-0.117681\pi$
$347$	34.7386	1.86487	0.932434	+	0.361341i	$0.117681\pi$
$348$	0	0
$349$	−24.7386	−1.32423	−0.662114	−	0.749403i	$-0.730341\pi$
$349$	−24.7386	−1.32423	−0.662114	+	0.749403i	$0.730341\pi$
$350$	5.12311	0.273842
$351$	0	0
$352$	5.12311	0.273062
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	6.24621	0.331514
$356$	−3.12311	−0.165524
$357$	0	0
$358$	24.4924	1.29446
$359$	12.4924	0.659325	0.329662	−	0.944099i	$-0.393065\pi$
$359$	12.4924	0.659325	0.329662	+	0.944099i	$0.393065\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−19.1231	−1.00509
$363$	0	0
$364$	−10.2462	−0.537047
$365$	−12.2462	−0.640996
$366$	0	0
$367$	13.1231	0.685021	0.342510	−	0.939514i	$-0.388723\pi$
$367$	13.1231	0.685021	0.342510	+	0.939514i	$0.388723\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−7.12311	−0.370313
$371$	−21.7538	−1.12940
$372$	0	0
$373$	13.3693	0.692237	0.346118	−	0.938191i	$-0.387500\pi$
$373$	13.3693	0.692237	0.346118	+	0.938191i	$0.387500\pi$
$374$	−36.4924	−1.88698
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−4.00000	−0.206010
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	−20.4924	−1.04848
$383$	−13.7538	−0.702786	−0.351393	−	0.936228i	$-0.614292\pi$
$383$	−13.7538	−0.702786	−0.351393	+	0.936228i	$0.614292\pi$
$384$	0	0
$385$	−26.2462	−1.33763
$386$	8.24621	0.419721
$387$	0	0
$388$	0.246211	0.0124995
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	7.12311	0.360231
$392$	−19.2462	−0.972080
$393$	0	0
$394$	−10.0000	−0.503793
$395$	5.12311	0.257771
$396$	0	0
$397$	−34.4924	−1.73113	−0.865563	−	0.500801i	$-0.833039\pi$
$397$	−34.4924	−1.73113	−0.865563	+	0.500801i	$0.833039\pi$
$398$	−10.8769	−0.545209
$399$	0	0
$400$	1.00000	0.0500000
$401$	17.3693	0.867382	0.433691	−	0.901062i	$-0.357211\pi$
$401$	17.3693	0.867382	0.433691	+	0.901062i	$0.357211\pi$
$402$	0	0
$403$	0	0
$404$	16.2462	0.808279
$405$	0	0
$406$	−10.2462	−0.508511
$407$	36.4924	1.80886
$408$	0	0
$409$	−16.2462	−0.803323	−0.401662	−	0.915788i	$-0.631567\pi$
$409$	−16.2462	−0.803323	−0.401662	+	0.915788i	$0.631567\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	15.3693	0.757192
$413$	−72.9848	−3.59135
$414$	0	0
$415$	11.3693	0.558098
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	20.4924	1.00232
$419$	1.61553	0.0789237	0.0394619	−	0.999221i	$-0.487436\pi$
$419$	1.61553	0.0789237	0.0394619	+	0.999221i	$0.487436\pi$
$420$	0	0
$421$	13.3693	0.651581	0.325790	−	0.945442i	$-0.394370\pi$
$421$	13.3693	0.651581	0.325790	+	0.945442i	$0.394370\pi$
$422$	−16.4924	−0.802839
$423$	0	0
$424$	−4.24621	−0.206214
$425$	−7.12311	−0.345521
$426$	0	0
$427$	−4.49242	−0.217404
$428$	−11.3693	−0.549557
$429$	0	0
$430$	0	0
$431$	−26.2462	−1.26424	−0.632118	−	0.774872i	$-0.717814\pi$
$431$	−26.2462	−1.26424	−0.632118	+	0.774872i	$0.717814\pi$
$432$	0	0
$433$	10.4924	0.504234	0.252117	−	0.967697i	$-0.418873\pi$
$433$	10.4924	0.504234	0.252117	+	0.967697i	$0.418873\pi$
$434$	0	0
$435$	0	0
$436$	11.1231	0.532700
$437$	−4.00000	−0.191346
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	−5.12311	−0.244234
$441$	0	0
$442$	14.2462	0.677623
$443$	24.4924	1.16367	0.581835	−	0.813307i	$-0.302335\pi$
$443$	24.4924	1.16367	0.581835	+	0.813307i	$0.302335\pi$
$444$	0	0
$445$	3.12311	0.148049
$446$	−6.24621	−0.295767
$447$	0	0
$448$	−5.12311	−0.242044
$449$	10.4924	0.495168	0.247584	−	0.968866i	$-0.420363\pi$
$449$	10.4924	0.495168	0.247584	+	0.968866i	$0.420363\pi$
$450$	0	0
$451$	10.2462	0.482475
$452$	0.876894	0.0412456
$453$	0	0
$454$	19.3693	0.909047
$455$	10.2462	0.480350
$456$	0	0
$457$	−32.7386	−1.53145	−0.765724	−	0.643169i	$-0.777619\pi$
$457$	−32.7386	−1.53145	−0.765724	+	0.643169i	$0.777619\pi$
$458$	1.36932	0.0639840
$459$	0	0
$460$	1.00000	0.0466252
$461$	−32.7386	−1.52479	−0.762395	−	0.647112i	$-0.775977\pi$
$461$	−32.7386	−1.52479	−0.762395	+	0.647112i	$0.775977\pi$
$462$	0	0
$463$	−36.0000	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$463$	−36.0000	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−20.7386	−0.960699
$467$	3.36932	0.155913	0.0779567	−	0.996957i	$-0.475160\pi$
$467$	3.36932	0.155913	0.0779567	+	0.996957i	$0.475160\pi$
$468$	0	0
$469$	40.9848	1.89250
$470$	8.00000	0.369012
$471$	0	0
$472$	−14.2462	−0.655735
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	36.4924	1.67263
$477$	0	0
$478$	−1.75379	−0.0802164
$479$	20.4924	0.936323	0.468161	−	0.883643i	$-0.344917\pi$
$479$	20.4924	0.936323	0.468161	+	0.883643i	$0.344917\pi$
$480$	0	0
$481$	−14.2462	−0.649571
$482$	2.49242	0.113527
$483$	0	0
$484$	15.2462	0.693010
$485$	−0.246211	−0.0111799
$486$	0	0
$487$	40.4924	1.83489	0.917443	−	0.397866i	$-0.130249\pi$
$487$	40.4924	1.83489	0.917443	+	0.397866i	$0.130249\pi$
$488$	−0.876894	−0.0396951
$489$	0	0
$490$	19.2462	0.869455
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	14.2462	0.641617
$494$	−8.00000	−0.359937
$495$	0	0
$496$	0	0
$497$	32.0000	1.43540
$498$	0	0
$499$	−28.9848	−1.29754	−0.648770	−	0.760985i	$-0.724716\pi$
$499$	−28.9848	−1.29754	−0.648770	+	0.760985i	$0.724716\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−7.36932	−0.328909
$503$	6.73863	0.300461	0.150230	−	0.988651i	$-0.451998\pi$
$503$	6.73863	0.300461	0.150230	+	0.988651i	$0.451998\pi$
$504$	0	0
$505$	−16.2462	−0.722947
$506$	−5.12311	−0.227750
$507$	0	0
$508$	−22.2462	−0.987016
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	−62.7386	−2.77539
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−8.24621	−0.363725
$515$	−15.3693	−0.677253
$516$	0	0
$517$	−40.9848	−1.80251
$518$	−36.4924	−1.60338
$519$	0	0
$520$	2.00000	0.0877058
$521$	−8.87689	−0.388904	−0.194452	−	0.980912i	$-0.562293\pi$
$521$	−8.87689	−0.388904	−0.194452	+	0.980912i	$0.562293\pi$
$522$	0	0
$523$	−28.4924	−1.24589	−0.622943	−	0.782267i	$-0.714063\pi$
$523$	−28.4924	−1.24589	−0.622943	+	0.782267i	$0.714063\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	10.2462	0.446756
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	4.24621	0.184444
$531$	0	0
$532$	−20.4924	−0.888459
$533$	−4.00000	−0.173259
$534$	0	0
$535$	11.3693	0.491538
$536$	8.00000	0.345547
$537$	0	0
$538$	−16.2462	−0.700424
$539$	−98.6004	−4.24702
$540$	0	0
$541$	−15.7538	−0.677308	−0.338654	−	0.940911i	$-0.609972\pi$
$541$	−15.7538	−0.677308	−0.338654	+	0.940911i	$0.609972\pi$
$542$	−10.2462	−0.440112
$543$	0	0
$544$	7.12311	0.305401
$545$	−11.1231	−0.476461
$546$	0	0
$547$	0.492423	0.0210545	0.0105272	−	0.999945i	$-0.496649\pi$
$547$	0.492423	0.0210545	0.0105272	+	0.999945i	$0.496649\pi$
$548$	19.1231	0.816899
$549$	0	0
$550$	5.12311	0.218450
$551$	−8.00000	−0.340811
$552$	0	0
$553$	26.2462	1.11610
$554$	−20.2462	−0.860179
$555$	0	0
$556$	−16.4924	−0.699435
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	0	0
$560$	5.12311	0.216491
$561$	0	0
$562$	−12.8769	−0.543179
$563$	−11.3693	−0.479160	−0.239580	−	0.970877i	$-0.577010\pi$
$563$	−11.3693	−0.479160	−0.239580	+	0.970877i	$0.577010\pi$
$564$	0	0
$565$	−0.876894	−0.0368912
$566$	2.24621	0.0944153
$567$	0	0
$568$	6.24621	0.262085
$569$	−27.1231	−1.13706	−0.568530	−	0.822663i	$-0.692488\pi$
$569$	−27.1231	−1.13706	−0.568530	+	0.822663i	$0.692488\pi$
$570$	0	0
$571$	34.7386	1.45377	0.726883	−	0.686761i	$-0.240968\pi$
$571$	34.7386	1.45377	0.726883	+	0.686761i	$0.240968\pi$
$572$	−10.2462	−0.428416
$573$	0	0
$574$	−10.2462	−0.427669
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	14.4924	0.603327	0.301664	−	0.953414i	$-0.402458\pi$
$577$	14.4924	0.603327	0.301664	+	0.953414i	$0.402458\pi$
$578$	−33.7386	−1.40334
$579$	0	0
$580$	2.00000	0.0830455
$581$	58.2462	2.41646
$582$	0	0
$583$	−21.7538	−0.900950
$584$	−12.2462	−0.506752
$585$	0	0
$586$	11.7538	0.485545
$587$	9.75379	0.402582	0.201291	−	0.979531i	$-0.435486\pi$
$587$	9.75379	0.402582	0.201291	+	0.979531i	$0.435486\pi$
$588$	0	0
$589$	0	0
$590$	14.2462	0.586507
$591$	0	0
$592$	−7.12311	−0.292758
$593$	−7.75379	−0.318410	−0.159205	−	0.987246i	$-0.550893\pi$
$593$	−7.75379	−0.318410	−0.159205	+	0.987246i	$0.550893\pi$
$594$	0	0
$595$	−36.4924	−1.49604
$596$	10.0000	0.409616
$597$	0	0
$598$	2.00000	0.0817861
$599$	6.24621	0.255213	0.127607	−	0.991825i	$-0.459270\pi$
$599$	6.24621	0.255213	0.127607	+	0.991825i	$0.459270\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	0	0
$604$	−10.2462	−0.416912
$605$	−15.2462	−0.619847
$606$	0	0
$607$	26.7386	1.08529	0.542644	−	0.839963i	$-0.317423\pi$
$607$	26.7386	1.08529	0.542644	+	0.839963i	$0.317423\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	0.876894	0.0355044
$611$	16.0000	0.647291
$612$	0	0
$613$	43.1231	1.74173	0.870863	−	0.491526i	$-0.163561\pi$
$613$	43.1231	1.74173	0.870863	+	0.491526i	$0.163561\pi$
$614$	−0.492423	−0.0198726
$615$	0	0
$616$	−26.2462	−1.05749
$617$	−40.1080	−1.61469	−0.807343	−	0.590083i	$-0.799095\pi$
$617$	−40.1080	−1.61469	−0.807343	+	0.590083i	$0.799095\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	0	0
$622$	26.7386	1.07212
$623$	16.0000	0.641026
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.4924	−0.419362
$627$	0	0
$628$	0.876894	0.0349919
$629$	50.7386	2.02308
$630$	0	0
$631$	−39.3693	−1.56727	−0.783634	−	0.621223i	$-0.786636\pi$
$631$	−39.3693	−1.56727	−0.783634	+	0.621223i	$0.786636\pi$
$632$	5.12311	0.203786
$633$	0	0
$634$	20.7386	0.823636
$635$	22.2462	0.882814
$636$	0	0
$637$	38.4924	1.52513
$638$	−10.2462	−0.405651
$639$	0	0
$640$	1.00000	0.0395285
$641$	9.36932	0.370066	0.185033	−	0.982732i	$-0.440761\pi$
$641$	9.36932	0.370066	0.185033	+	0.982732i	$0.440761\pi$
$642$	0	0
$643$	36.4924	1.43912	0.719560	−	0.694430i	$-0.244343\pi$
$643$	36.4924	1.43912	0.719560	+	0.694430i	$0.244343\pi$
$644$	5.12311	0.201879
$645$	0	0
$646$	28.4924	1.12102
$647$	−11.5076	−0.452410	−0.226205	−	0.974080i	$-0.572632\pi$
$647$	−11.5076	−0.452410	−0.226205	+	0.974080i	$0.572632\pi$
$648$	0	0
$649$	−72.9848	−2.86491
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−12.0000	−0.469956
$653$	−10.4924	−0.410600	−0.205300	−	0.978699i	$-0.565817\pi$
$653$	−10.4924	−0.410600	−0.205300	+	0.978699i	$0.565817\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	40.9848	1.59776
$659$	7.36932	0.287068	0.143534	−	0.989645i	$-0.454153\pi$
$659$	7.36932	0.287068	0.143534	+	0.989645i	$0.454153\pi$
$660$	0	0
$661$	33.8617	1.31707	0.658535	−	0.752551i	$-0.271177\pi$
$661$	33.8617	1.31707	0.658535	+	0.752551i	$0.271177\pi$
$662$	32.4924	1.26285
$663$	0	0
$664$	11.3693	0.441215
$665$	20.4924	0.794662
$666$	0	0
$667$	2.00000	0.0774403
$668$	8.00000	0.309529
$669$	0	0
$670$	−8.00000	−0.309067
$671$	−4.49242	−0.173428
$672$	0	0
$673$	−46.9848	−1.81113	−0.905566	−	0.424205i	$-0.860554\pi$
$673$	−46.9848	−1.81113	−0.905566	+	0.424205i	$0.860554\pi$
$674$	6.49242	0.250079
$675$	0	0
$676$	−9.00000	−0.346154
$677$	46.4924	1.78685	0.893424	−	0.449213i	$-0.148296\pi$
$677$	46.4924	1.78685	0.893424	+	0.449213i	$0.148296\pi$
$678$	0	0
$679$	−1.26137	−0.0484068
$680$	−7.12311	−0.273159
$681$	0	0
$682$	0	0
$683$	−17.7538	−0.679330	−0.339665	−	0.940547i	$-0.610314\pi$
$683$	−17.7538	−0.679330	−0.339665	+	0.940547i	$0.610314\pi$
$684$	0	0
$685$	−19.1231	−0.730656
$686$	62.7386	2.39537
$687$	0	0
$688$	0	0
$689$	8.49242	0.323536
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−3.75379	−0.142698
$693$	0	0
$694$	−34.7386	−1.31866
$695$	16.4924	0.625593
$696$	0	0
$697$	14.2462	0.539614
$698$	24.7386	0.936371
$699$	0	0
$700$	−5.12311	−0.193635
$701$	36.2462	1.36900	0.684500	−	0.729013i	$-0.260020\pi$
$701$	36.2462	1.36900	0.684500	+	0.729013i	$0.260020\pi$
$702$	0	0
$703$	−28.4924	−1.07461
$704$	−5.12311	−0.193084
$705$	0	0
$706$	−14.0000	−0.526897
$707$	−83.2311	−3.13023
$708$	0	0
$709$	13.3693	0.502095	0.251048	−	0.967975i	$-0.419225\pi$
$709$	13.3693	0.502095	0.251048	+	0.967975i	$0.419225\pi$
$710$	−6.24621	−0.234416
$711$	0	0
$712$	3.12311	0.117043
$713$	0	0
$714$	0	0
$715$	10.2462	0.383187
$716$	−24.4924	−0.915325
$717$	0	0
$718$	−12.4924	−0.466213
$719$	9.75379	0.363755	0.181877	−	0.983321i	$-0.441783\pi$
$719$	9.75379	0.363755	0.181877	+	0.983321i	$0.441783\pi$
$720$	0	0
$721$	−78.7386	−2.93238
$722$	3.00000	0.111648
$723$	0	0
$724$	19.1231	0.710705
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	27.8617	1.03333	0.516667	−	0.856186i	$-0.327172\pi$
$727$	27.8617	1.03333	0.516667	+	0.856186i	$0.327172\pi$
$728$	10.2462	0.379750
$729$	0	0
$730$	12.2462	0.453253
$731$	0	0
$732$	0	0
$733$	−13.8617	−0.511995	−0.255998	−	0.966677i	$-0.582404\pi$
$733$	−13.8617	−0.511995	−0.255998	+	0.966677i	$0.582404\pi$
$734$	−13.1231	−0.484383
$735$	0	0
$736$	1.00000	0.0368605
$737$	40.9848	1.50970
$738$	0	0
$739$	40.4924	1.48954	0.744769	−	0.667322i	$-0.232560\pi$
$739$	40.4924	1.48954	0.744769	+	0.667322i	$0.232560\pi$
$740$	7.12311	0.261851
$741$	0	0
$742$	21.7538	0.798607
$743$	14.7386	0.540708	0.270354	−	0.962761i	$-0.412859\pi$
$743$	14.7386	0.540708	0.270354	+	0.962761i	$0.412859\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	−13.3693	−0.489485
$747$	0	0
$748$	36.4924	1.33430
$749$	58.2462	2.12827
$750$	0	0
$751$	43.8617	1.60054	0.800269	−	0.599641i	$-0.204690\pi$
$751$	43.8617	1.60054	0.800269	+	0.599641i	$0.204690\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	4.00000	0.145671
$755$	10.2462	0.372898
$756$	0	0
$757$	9.86174	0.358431	0.179216	−	0.983810i	$-0.442644\pi$
$757$	9.86174	0.358431	0.179216	+	0.983810i	$0.442644\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	4.00000	0.145095
$761$	48.2462	1.74892	0.874462	−	0.485094i	$-0.161215\pi$
$761$	48.2462	1.74892	0.874462	+	0.485094i	$0.161215\pi$
$762$	0	0
$763$	−56.9848	−2.06299
$764$	20.4924	0.741390
$765$	0	0
$766$	13.7538	0.496945
$767$	28.4924	1.02880
$768$	0	0
$769$	−20.7386	−0.747854	−0.373927	−	0.927458i	$-0.621989\pi$
$769$	−20.7386	−0.747854	−0.373927	+	0.927458i	$0.621989\pi$
$770$	26.2462	0.945848
$771$	0	0
$772$	−8.24621	−0.296788
$773$	36.2462	1.30369	0.651843	−	0.758354i	$-0.273996\pi$
$773$	36.2462	1.30369	0.651843	+	0.758354i	$0.273996\pi$
$774$	0	0
$775$	0	0
$776$	−0.246211	−0.00883847
$777$	0	0
$778$	−18.0000	−0.645331
$779$	−8.00000	−0.286630
$780$	0	0
$781$	32.0000	1.14505
$782$	−7.12311	−0.254722
$783$	0	0
$784$	19.2462	0.687365
$785$	−0.876894	−0.0312977
$786$	0	0
$787$	48.0000	1.71102	0.855508	−	0.517790i	$-0.173245\pi$
$787$	48.0000	1.71102	0.855508	+	0.517790i	$0.173245\pi$
$788$	10.0000	0.356235
$789$	0	0
$790$	−5.12311	−0.182272
$791$	−4.49242	−0.159732
$792$	0	0
$793$	1.75379	0.0622789
$794$	34.4924	1.22409
$795$	0	0
$796$	10.8769	0.385521
$797$	22.4924	0.796722	0.398361	−	0.917229i	$-0.369579\pi$
$797$	22.4924	0.796722	0.398361	+	0.917229i	$0.369579\pi$
$798$	0	0
$799$	−56.9848	−2.01598
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−17.3693	−0.613332
$803$	−62.7386	−2.21400
$804$	0	0
$805$	−5.12311	−0.180566
$806$	0	0
$807$	0	0
$808$	−16.2462	−0.571540
$809$	−4.24621	−0.149289	−0.0746444	−	0.997210i	$-0.523782\pi$
$809$	−4.24621	−0.149289	−0.0746444	+	0.997210i	$0.523782\pi$
$810$	0	0
$811$	−36.0000	−1.26413	−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000	−1.26413	−0.632065	+	0.774915i	$0.717793\pi$
$812$	10.2462	0.359572
$813$	0	0
$814$	−36.4924	−1.27906
$815$	12.0000	0.420342
$816$	0	0
$817$	0	0
$818$	16.2462	0.568035
$819$	0	0
$820$	2.00000	0.0698430
$821$	19.7538	0.689412	0.344706	−	0.938711i	$-0.387979\pi$
$821$	19.7538	0.689412	0.344706	+	0.938711i	$0.387979\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−15.3693	−0.535416
$825$	0	0
$826$	72.9848	2.53947
$827$	−17.1231	−0.595429	−0.297714	−	0.954655i	$-0.596224\pi$
$827$	−17.1231	−0.595429	−0.297714	+	0.954655i	$0.596224\pi$
$828$	0	0
$829$	32.2462	1.11996	0.559979	−	0.828507i	$-0.310809\pi$
$829$	32.2462	1.11996	0.559979	+	0.828507i	$0.310809\pi$
$830$	−11.3693	−0.394635
$831$	0	0
$832$	2.00000	0.0693375
$833$	−137.093	−4.74998
$834$	0	0
$835$	−8.00000	−0.276851
$836$	−20.4924	−0.708745
$837$	0	0
$838$	−1.61553	−0.0558075
$839$	26.2462	0.906120	0.453060	−	0.891480i	$-0.350332\pi$
$839$	26.2462	0.906120	0.453060	+	0.891480i	$0.350332\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−13.3693	−0.460737
$843$	0	0
$844$	16.4924	0.567693
$845$	9.00000	0.309609
$846$	0	0
$847$	−78.1080	−2.68382
$848$	4.24621	0.145815
$849$	0	0
$850$	7.12311	0.244321
$851$	7.12311	0.244177
$852$	0	0
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	4.49242	0.153728
$855$	0	0
$856$	11.3693	0.388595
$857$	−24.7386	−0.845056	−0.422528	−	0.906350i	$-0.638857\pi$
$857$	−24.7386	−0.845056	−0.422528	+	0.906350i	$0.638857\pi$
$858$	0	0
$859$	48.4924	1.65454	0.827270	−	0.561804i	$-0.189893\pi$
$859$	48.4924	1.65454	0.827270	+	0.561804i	$0.189893\pi$
$860$	0	0
$861$	0	0
$862$	26.2462	0.893950
$863$	−48.9848	−1.66746	−0.833732	−	0.552170i	$-0.813800\pi$
$863$	−48.9848	−1.66746	−0.833732	+	0.552170i	$0.813800\pi$
$864$	0	0
$865$	3.75379	0.127633
$866$	−10.4924	−0.356547
$867$	0	0
$868$	0	0
$869$	26.2462	0.890342
$870$	0	0
$871$	−16.0000	−0.542139
$872$	−11.1231	−0.376676
$873$	0	0
$874$	4.00000	0.135302
$875$	5.12311	0.173193
$876$	0	0
$877$	3.26137	0.110129	0.0550643	−	0.998483i	$-0.482464\pi$
$877$	3.26137	0.110129	0.0550643	+	0.998483i	$0.482464\pi$
$878$	0	0
$879$	0	0
$880$	5.12311	0.172700
$881$	−31.6155	−1.06515	−0.532577	−	0.846381i	$-0.678776\pi$
$881$	−31.6155	−1.06515	−0.532577	+	0.846381i	$0.678776\pi$
$882$	0	0
$883$	−20.9848	−0.706196	−0.353098	−	0.935586i	$-0.614872\pi$
$883$	−20.9848	−0.706196	−0.353098	+	0.935586i	$0.614872\pi$
$884$	−14.2462	−0.479152
$885$	0	0
$886$	−24.4924	−0.822839
$887$	−52.4924	−1.76252	−0.881262	−	0.472629i	$-0.843305\pi$
$887$	−52.4924	−1.76252	−0.881262	+	0.472629i	$0.843305\pi$
$888$	0	0
$889$	113.970	3.82242
$890$	−3.12311	−0.104687
$891$	0	0
$892$	6.24621	0.209139
$893$	32.0000	1.07084
$894$	0	0
$895$	24.4924	0.818691
$896$	5.12311	0.171151
$897$	0	0
$898$	−10.4924	−0.350137
$899$	0	0
$900$	0	0
$901$	−30.2462	−1.00765
$902$	−10.2462	−0.341162
$903$	0	0
$904$	−0.876894	−0.0291651
$905$	−19.1231	−0.635674
$906$	0	0
$907$	37.7538	1.25359	0.626797	−	0.779183i	$-0.284366\pi$
$907$	37.7538	1.25359	0.626797	+	0.779183i	$0.284366\pi$
$908$	−19.3693	−0.642793
$909$	0	0
$910$	−10.2462	−0.339659
$911$	13.7538	0.455683	0.227842	−	0.973698i	$-0.426833\pi$
$911$	13.7538	0.455683	0.227842	+	0.973698i	$0.426833\pi$
$912$	0	0
$913$	58.2462	1.92767
$914$	32.7386	1.08290
$915$	0	0
$916$	−1.36932	−0.0452435
$917$	−20.4924	−0.676719
$918$	0	0
$919$	10.8769	0.358796	0.179398	−	0.983777i	$-0.442585\pi$
$919$	10.8769	0.358796	0.179398	+	0.983777i	$0.442585\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	32.7386	1.07819
$923$	−12.4924	−0.411193
$924$	0	0
$925$	−7.12311	−0.234206
$926$	36.0000	1.18303
$927$	0	0
$928$	2.00000	0.0656532
$929$	30.9848	1.01658	0.508290	−	0.861186i	$-0.330278\pi$
$929$	30.9848	1.01658	0.508290	+	0.861186i	$0.330278\pi$
$930$	0	0
$931$	76.9848	2.52308
$932$	20.7386	0.679317
$933$	0	0
$934$	−3.36932	−0.110247
$935$	−36.4924	−1.19343
$936$	0	0
$937$	−16.7386	−0.546827	−0.273414	−	0.961897i	$-0.588153\pi$
$937$	−16.7386	−0.546827	−0.273414	+	0.961897i	$0.588153\pi$
$938$	−40.9848	−1.33820
$939$	0	0
$940$	−8.00000	−0.260931
$941$	6.49242	0.211647	0.105823	−	0.994385i	$-0.466252\pi$
$941$	6.49242	0.211647	0.105823	+	0.994385i	$0.466252\pi$
$942$	0	0
$943$	2.00000	0.0651290
$944$	14.2462	0.463675
$945$	0	0
$946$	0	0
$947$	−22.2462	−0.722905	−0.361452	−	0.932391i	$-0.617719\pi$
$947$	−22.2462	−0.722905	−0.361452	+	0.932391i	$0.617719\pi$
$948$	0	0
$949$	24.4924	0.795058
$950$	−4.00000	−0.129777
$951$	0	0
$952$	−36.4924	−1.18273
$953$	33.8617	1.09689	0.548445	−	0.836187i	$-0.315220\pi$
$953$	33.8617	1.09689	0.548445	+	0.836187i	$0.315220\pi$
$954$	0	0
$955$	−20.4924	−0.663119
$956$	1.75379	0.0567216
$957$	0	0
$958$	−20.4924	−0.662080
$959$	−97.9697	−3.16361
$960$	0	0
$961$	−31.0000	−1.00000
$962$	14.2462	0.459316
$963$	0	0
$964$	−2.49242	−0.0802755
$965$	8.24621	0.265455
$966$	0	0
$967$	−4.98485	−0.160302	−0.0801509	−	0.996783i	$-0.525540\pi$
$967$	−4.98485	−0.160302	−0.0801509	+	0.996783i	$0.525540\pi$
$968$	−15.2462	−0.490032
$969$	0	0
$970$	0.246211	0.00790537
$971$	−34.8769	−1.11925	−0.559626	−	0.828745i	$-0.689055\pi$
$971$	−34.8769	−1.11925	−0.559626	+	0.828745i	$0.689055\pi$
$972$	0	0
$973$	84.4924	2.70870
$974$	−40.4924	−1.29746
$975$	0	0
$976$	0.876894	0.0280687
$977$	−21.8617	−0.699419	−0.349710	−	0.936858i	$-0.613720\pi$
$977$	−21.8617	−0.699419	−0.349710	+	0.936858i	$0.613720\pi$
$978$	0	0
$979$	16.0000	0.511362
$980$	−19.2462	−0.614798
$981$	0	0
$982$	−12.0000	−0.382935
$983$	26.2462	0.837124	0.418562	−	0.908188i	$-0.362534\pi$
$983$	26.2462	0.837124	0.418562	+	0.908188i	$0.362534\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	−14.2462	−0.453692
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	0	0
$991$	−37.4773	−1.19050	−0.595252	−	0.803539i	$-0.702948\pi$
$991$	−37.4773	−1.19050	−0.595252	+	0.803539i	$0.702948\pi$
$992$	0	0
$993$	0	0
$994$	−32.0000	−1.01498
$995$	−10.8769	−0.344821
$996$	0	0
$997$	20.2462	0.641204	0.320602	−	0.947214i	$-0.396115\pi$
$997$	20.2462	0.641204	0.320602	+	0.947214i	$0.396115\pi$
$998$	28.9848	0.917499
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.a.t.1.1		2
3.2	odd	2		690.2.a.l.1.1	✓	2
12.11	even	2		5520.2.a.bs.1.2		2
15.2	even	4		3450.2.d.v.2899.3		4
15.8	even	4		3450.2.d.v.2899.2		4
15.14	odd	2		3450.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.l.1.1	✓	2	3.2	odd	2
2070.2.a.t.1.1		2	1.1	even	1	trivial
3450.2.a.bi.1.2		2	15.14	odd	2
3450.2.d.v.2899.2		4	15.8	even	4
3450.2.d.v.2899.3		4	15.2	even	4
5520.2.a.bs.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -5.12311 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{10} -5.12311 q^{11} +2.00000 q^{13} +5.12311 q^{14} +1.00000 q^{16} -7.12311 q^{17} +4.00000 q^{19} -1.00000 q^{20} +5.12311 q^{22} -1.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} -5.12311 q^{28} -2.00000 q^{29} -1.00000 q^{32} +7.12311 q^{34} +5.12311 q^{35} -7.12311 q^{37} -4.00000 q^{38} +1.00000 q^{40} -2.00000 q^{41} -5.12311 q^{44} +1.00000 q^{46} +8.00000 q^{47} +19.2462 q^{49} -1.00000 q^{50} +2.00000 q^{52} +4.24621 q^{53} +5.12311 q^{55} +5.12311 q^{56} +2.00000 q^{58} +14.2462 q^{59} +0.876894 q^{61} +1.00000 q^{64} -2.00000 q^{65} -8.00000 q^{67} -7.12311 q^{68} -5.12311 q^{70} -6.24621 q^{71} +12.2462 q^{73} +7.12311 q^{74} +4.00000 q^{76} +26.2462 q^{77} -5.12311 q^{79} -1.00000 q^{80} +2.00000 q^{82} -11.3693 q^{83} +7.12311 q^{85} +5.12311 q^{88} -3.12311 q^{89} -10.2462 q^{91} -1.00000 q^{92} -8.00000 q^{94} -4.00000 q^{95} +0.246211 q^{97} -19.2462 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 2 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 6 q^{17} + 8 q^{19} - 2 q^{20} + 2 q^{22} - 2 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{28} - 4 q^{29} - 2 q^{32} + 6 q^{34} + 2 q^{35} - 6 q^{37} - 8 q^{38} + 2 q^{40} - 4 q^{41} - 2 q^{44} + 2 q^{46} + 16 q^{47} + 22 q^{49} - 2 q^{50} + 4 q^{52} - 8 q^{53} + 2 q^{55} + 2 q^{56} + 4 q^{58} + 12 q^{59} + 10 q^{61} + 2 q^{64} - 4 q^{65} - 16 q^{67} - 6 q^{68} - 2 q^{70} + 4 q^{71} + 8 q^{73} + 6 q^{74} + 8 q^{76} + 36 q^{77} - 2 q^{79} - 2 q^{80} + 4 q^{82} + 2 q^{83} + 6 q^{85} + 2 q^{88} + 2 q^{89} - 4 q^{91} - 2 q^{92} - 16 q^{94} - 8 q^{95} - 16 q^{97} - 22 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 2 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 6 * q^17 + 8 * q^19 - 2 * q^20 + 2 * q^22 - 2 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^28 - 4 * q^29 - 2 * q^32 + 6 * q^34 + 2 * q^35 - 6 * q^37 - 8 * q^38 + 2 * q^40 - 4 * q^41 - 2 * q^44 + 2 * q^46 + 16 * q^47 + 22 * q^49 - 2 * q^50 + 4 * q^52 - 8 * q^53 + 2 * q^55 + 2 * q^56 + 4 * q^58 + 12 * q^59 + 10 * q^61 + 2 * q^64 - 4 * q^65 - 16 * q^67 - 6 * q^68 - 2 * q^70 + 4 * q^71 + 8 * q^73 + 6 * q^74 + 8 * q^76 + 36 * q^77 - 2 * q^79 - 2 * q^80 + 4 * q^82 + 2 * q^83 + 6 * q^85 + 2 * q^88 + 2 * q^89 - 4 * q^91 - 2 * q^92 - 16 * q^94 - 8 * q^95 - 16 * q^97 - 22 * q^98

Introduction

L-functions

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