LMFDB - Embedded newform 2394.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, 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("2394.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.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:	$19.1161862439$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 798)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	2394.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.46410 q^{10} -1.46410 q^{11} +3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -0.535898 q^{17} +1.00000 q^{19} -3.46410 q^{20} -1.46410 q^{22} -1.46410 q^{23} +7.00000 q^{25} +3.46410 q^{26} -1.00000 q^{28} +6.00000 q^{29} -6.92820 q^{31} +1.00000 q^{32} -0.535898 q^{34} +3.46410 q^{35} +10.0000 q^{37} +1.00000 q^{38} -3.46410 q^{40} +2.00000 q^{41} +6.92820 q^{43} -1.46410 q^{44} -1.46410 q^{46} +2.92820 q^{47} +1.00000 q^{49} +7.00000 q^{50} +3.46410 q^{52} -2.00000 q^{53} +5.07180 q^{55} -1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} +8.92820 q^{61} -6.92820 q^{62} +1.00000 q^{64} -12.0000 q^{65} +2.53590 q^{67} -0.535898 q^{68} +3.46410 q^{70} +10.9282 q^{71} +10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{76} +1.46410 q^{77} +8.39230 q^{79} -3.46410 q^{80} +2.00000 q^{82} -8.00000 q^{83} +1.85641 q^{85} +6.92820 q^{86} -1.46410 q^{88} +2.00000 q^{89} -3.46410 q^{91} -1.46410 q^{92} +2.92820 q^{94} -3.46410 q^{95} +11.4641 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 4 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} + 14 q^{25} - 2 q^{28} + 12 q^{29} + 2 q^{32} - 8 q^{34} + 20 q^{37} + 2 q^{38} + 4 q^{41} + 4 q^{44} + 4 q^{46} - 8 q^{47} + 2 q^{49} + 14 q^{50} - 4 q^{53} + 24 q^{55} - 2 q^{56} + 12 q^{58} - 8 q^{59} + 4 q^{61} + 2 q^{64} - 24 q^{65} + 12 q^{67} - 8 q^{68} + 8 q^{71} + 20 q^{73} + 20 q^{74} + 2 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{82} - 16 q^{83} - 24 q^{85} + 4 q^{88} + 4 q^{89} + 4 q^{92} - 8 q^{94} + 16 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 4 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^17 + 2 * q^19 + 4 * q^22 + 4 * q^23 + 14 * q^25 - 2 * q^28 + 12 * q^29 + 2 * q^32 - 8 * q^34 + 20 * q^37 + 2 * q^38 + 4 * q^41 + 4 * q^44 + 4 * q^46 - 8 * q^47 + 2 * q^49 + 14 * q^50 - 4 * q^53 + 24 * q^55 - 2 * q^56 + 12 * q^58 - 8 * q^59 + 4 * q^61 + 2 * q^64 - 24 * q^65 + 12 * q^67 - 8 * q^68 + 8 * q^71 + 20 * q^73 + 20 * q^74 + 2 * q^76 - 4 * q^77 - 4 * q^79 + 4 * q^82 - 16 * q^83 - 24 * q^85 + 4 * q^88 + 4 * q^89 + 4 * q^92 - 8 * q^94 + 16 * 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
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.46410	−1.09545
$11$	−1.46410	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$11$	−1.46410	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.535898	−0.129974	−0.0649872	−	0.997886i	$-0.520701\pi$
$17$	−0.535898	−0.129974	−0.0649872	+	0.997886i	$0.520701\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.46410	−0.774597
$21$	0	0
$22$	−1.46410	−0.312148
$23$	−1.46410	−0.305286	−0.152643	−	0.988281i	$-0.548779\pi$
$23$	−1.46410	−0.305286	−0.152643	+	0.988281i	$0.548779\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	3.46410	0.679366
$27$	0	0
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	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
$34$	−0.535898	−0.0919058
$35$	3.46410	0.585540
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−3.46410	−0.547723
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	6.92820	1.05654	0.528271	−	0.849076i	$-0.322841\pi$
$43$	6.92820	1.05654	0.528271	+	0.849076i	$0.322841\pi$
$44$	−1.46410	−0.220722
$45$	0	0
$46$	−1.46410	−0.215870
$47$	2.92820	0.427122	0.213561	−	0.976930i	$-0.431494\pi$
$47$	2.92820	0.427122	0.213561	+	0.976930i	$0.431494\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.00000	0.989949
$51$	0	0
$52$	3.46410	0.480384
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	5.07180	0.683881
$56$	−1.00000	−0.133631
$57$	0	0
$58$	6.00000	0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	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
$65$	−12.0000	−1.48842
$66$	0	0
$67$	2.53590	0.309809	0.154905	−	0.987929i	$-0.450493\pi$
$67$	2.53590	0.309809	0.154905	+	0.987929i	$0.450493\pi$
$68$	−0.535898	−0.0649872
$69$	0	0
$70$	3.46410	0.414039
$71$	10.9282	1.29694	0.648470	−	0.761241i	$-0.275409\pi$
$71$	10.9282	1.29694	0.648470	+	0.761241i	$0.275409\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	1.00000	0.114708
$77$	1.46410	0.166850
$78$	0	0
$79$	8.39230	0.944208	0.472104	−	0.881543i	$-0.343495\pi$
$79$	8.39230	0.944208	0.472104	+	0.881543i	$0.343495\pi$
$80$	−3.46410	−0.387298
$81$	0	0
$82$	2.00000	0.220863
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	1.85641	0.201356
$86$	6.92820	0.747087
$87$	0	0
$88$	−1.46410	−0.156074
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−3.46410	−0.363137
$92$	−1.46410	−0.152643
$93$	0	0
$94$	2.92820	0.302021
$95$	−3.46410	−0.355409
$96$	0	0
$97$	11.4641	1.16400	0.582002	−	0.813188i	$-0.302270\pi$
$97$	11.4641	1.16400	0.582002	+	0.813188i	$0.302270\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	7.00000	0.700000
$101$	−11.4641	−1.14072	−0.570360	−	0.821395i	$-0.693196\pi$
$101$	−11.4641	−1.14072	−0.570360	+	0.821395i	$0.693196\pi$
$102$	0	0
$103$	6.92820	0.682656	0.341328	−	0.939944i	$-0.389123\pi$
$103$	6.92820	0.682656	0.341328	+	0.939944i	$0.389123\pi$
$104$	3.46410	0.339683
$105$	0	0
$106$	−2.00000	−0.194257
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	5.07180	0.483577
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	15.8564	1.49165	0.745823	−	0.666145i	$-0.232057\pi$
$113$	15.8564	1.49165	0.745823	+	0.666145i	$0.232057\pi$
$114$	0	0
$115$	5.07180	0.472947
$116$	6.00000	0.557086
$117$	0	0
$118$	−4.00000	−0.368230
$119$	0.535898	0.0491257
$120$	0	0
$121$	−8.85641	−0.805128
$122$	8.92820	0.808322
$123$	0	0
$124$	−6.92820	−0.622171
$125$	−6.92820	−0.619677
$126$	0	0
$127$	−10.5359	−0.934910	−0.467455	−	0.884017i	$-0.654829\pi$
$127$	−10.5359	−0.934910	−0.467455	+	0.884017i	$0.654829\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−12.0000	−1.05247
$131$	18.9282	1.65376	0.826882	−	0.562375i	$-0.190112\pi$
$131$	18.9282	1.65376	0.826882	+	0.562375i	$0.190112\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	2.53590	0.219068
$135$	0	0
$136$	−0.535898	−0.0459529
$137$	−12.9282	−1.10453	−0.552265	−	0.833668i	$-0.686237\pi$
$137$	−12.9282	−1.10453	−0.552265	+	0.833668i	$0.686237\pi$
$138$	0	0
$139$	−17.8564	−1.51456	−0.757280	−	0.653090i	$-0.773472\pi$
$139$	−17.8564	−1.51456	−0.757280	+	0.653090i	$0.773472\pi$
$140$	3.46410	0.292770
$141$	0	0
$142$	10.9282	0.917074
$143$	−5.07180	−0.424125
$144$	0	0
$145$	−20.7846	−1.72607
$146$	10.0000	0.827606
$147$	0	0
$148$	10.0000	0.821995
$149$	16.9282	1.38681	0.693406	−	0.720547i	$-0.256109\pi$
$149$	16.9282	1.38681	0.693406	+	0.720547i	$0.256109\pi$
$150$	0	0
$151$	−5.46410	−0.444662	−0.222331	−	0.974971i	$-0.571367\pi$
$151$	−5.46410	−0.444662	−0.222331	+	0.974971i	$0.571367\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	1.46410	0.117981
$155$	24.0000	1.92773
$156$	0	0
$157$	−7.85641	−0.627009	−0.313505	−	0.949587i	$-0.601503\pi$
$157$	−7.85641	−0.627009	−0.313505	+	0.949587i	$0.601503\pi$
$158$	8.39230	0.667656
$159$	0	0
$160$	−3.46410	−0.273861
$161$	1.46410	0.115387
$162$	0	0
$163$	14.9282	1.16927	0.584634	−	0.811297i	$-0.301238\pi$
$163$	14.9282	1.16927	0.584634	+	0.811297i	$0.301238\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−8.00000	−0.620920
$167$	5.07180	0.392467	0.196234	−	0.980557i	$-0.437129\pi$
$167$	5.07180	0.392467	0.196234	+	0.980557i	$0.437129\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	1.85641	0.142380
$171$	0	0
$172$	6.92820	0.528271
$173$	−12.9282	−0.982913	−0.491457	−	0.870902i	$-0.663535\pi$
$173$	−12.9282	−0.982913	−0.491457	+	0.870902i	$0.663535\pi$
$174$	0	0
$175$	−7.00000	−0.529150
$176$	−1.46410	−0.110361
$177$	0	0
$178$	2.00000	0.149906
$179$	−17.8564	−1.33465	−0.667325	−	0.744766i	$-0.732561\pi$
$179$	−17.8564	−1.33465	−0.667325	+	0.744766i	$0.732561\pi$
$180$	0	0
$181$	−15.4641	−1.14944	−0.574719	−	0.818351i	$-0.694889\pi$
$181$	−15.4641	−1.14944	−0.574719	+	0.818351i	$0.694889\pi$
$182$	−3.46410	−0.256776
$183$	0	0
$184$	−1.46410	−0.107935
$185$	−34.6410	−2.54686
$186$	0	0
$187$	0.784610	0.0573763
$188$	2.92820	0.213561
$189$	0	0
$190$	−3.46410	−0.251312
$191$	−1.46410	−0.105939	−0.0529693	−	0.998596i	$-0.516869\pi$
$191$	−1.46410	−0.105939	−0.0529693	+	0.998596i	$0.516869\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	11.4641	0.823075
$195$	0	0
$196$	1.00000	0.0714286
$197$	8.92820	0.636108	0.318054	−	0.948073i	$-0.396971\pi$
$197$	8.92820	0.636108	0.318054	+	0.948073i	$0.396971\pi$
$198$	0	0
$199$	16.7846	1.18983	0.594915	−	0.803789i	$-0.297186\pi$
$199$	16.7846	1.18983	0.594915	+	0.803789i	$0.297186\pi$
$200$	7.00000	0.494975
$201$	0	0
$202$	−11.4641	−0.806611
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−6.92820	−0.483887
$206$	6.92820	0.482711
$207$	0	0
$208$	3.46410	0.240192
$209$	−1.46410	−0.101274
$210$	0	0
$211$	−5.46410	−0.376164	−0.188082	−	0.982153i	$-0.560227\pi$
$211$	−5.46410	−0.376164	−0.188082	+	0.982153i	$0.560227\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	4.00000	0.273434
$215$	−24.0000	−1.63679
$216$	0	0
$217$	6.92820	0.470317
$218$	2.00000	0.135457
$219$	0	0
$220$	5.07180	0.341940
$221$	−1.85641	−0.124875
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	15.8564	1.05475
$227$	−1.07180	−0.0711377	−0.0355688	−	0.999367i	$-0.511324\pi$
$227$	−1.07180	−0.0711377	−0.0355688	+	0.999367i	$0.511324\pi$
$228$	0	0
$229$	16.9282	1.11865	0.559324	−	0.828949i	$-0.311061\pi$
$229$	16.9282	1.11865	0.559324	+	0.828949i	$0.311061\pi$
$230$	5.07180	0.334424
$231$	0	0
$232$	6.00000	0.393919
$233$	14.7846	0.968572	0.484286	−	0.874910i	$-0.339079\pi$
$233$	14.7846	0.968572	0.484286	+	0.874910i	$0.339079\pi$
$234$	0	0
$235$	−10.1436	−0.661695
$236$	−4.00000	−0.260378
$237$	0	0
$238$	0.535898	0.0347371
$239$	18.2487	1.18041	0.590206	−	0.807253i	$-0.299047\pi$
$239$	18.2487	1.18041	0.590206	+	0.807253i	$0.299047\pi$
$240$	0	0
$241$	−13.3205	−0.858049	−0.429025	−	0.903293i	$-0.641143\pi$
$241$	−13.3205	−0.858049	−0.429025	+	0.903293i	$0.641143\pi$
$242$	−8.85641	−0.569311
$243$	0	0
$244$	8.92820	0.571570
$245$	−3.46410	−0.221313
$246$	0	0
$247$	3.46410	0.220416
$248$	−6.92820	−0.439941
$249$	0	0
$250$	−6.92820	−0.438178
$251$	−18.9282	−1.19474	−0.597369	−	0.801967i	$-0.703787\pi$
$251$	−18.9282	−1.19474	−0.597369	+	0.801967i	$0.703787\pi$
$252$	0	0
$253$	2.14359	0.134767
$254$	−10.5359	−0.661081
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.0718	−0.690640	−0.345320	−	0.938485i	$-0.612230\pi$
$257$	−11.0718	−0.690640	−0.345320	+	0.938485i	$0.612230\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	−12.0000	−0.744208
$261$	0	0
$262$	18.9282	1.16939
$263$	23.3205	1.43800	0.719002	−	0.695008i	$-0.244599\pi$
$263$	23.3205	1.43800	0.719002	+	0.695008i	$0.244599\pi$
$264$	0	0
$265$	6.92820	0.425596
$266$	−1.00000	−0.0613139
$267$	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
$271$	−2.14359	−0.130214	−0.0651070	−	0.997878i	$-0.520739\pi$
$271$	−2.14359	−0.130214	−0.0651070	+	0.997878i	$0.520739\pi$
$272$	−0.535898	−0.0324936
$273$	0	0
$274$	−12.9282	−0.781021
$275$	−10.2487	−0.618021
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−17.8564	−1.07096
$279$	0	0
$280$	3.46410	0.207020
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	10.9282	0.648470
$285$	0	0
$286$	−5.07180	−0.299902
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−16.7128	−0.983107
$290$	−20.7846	−1.22051
$291$	0	0
$292$	10.0000	0.585206
$293$	−20.9282	−1.22264	−0.611319	−	0.791384i	$-0.709361\pi$
$293$	−20.9282	−1.22264	−0.611319	+	0.791384i	$0.709361\pi$
$294$	0	0
$295$	13.8564	0.806751
$296$	10.0000	0.581238
$297$	0	0
$298$	16.9282	0.980624
$299$	−5.07180	−0.293310
$300$	0	0
$301$	−6.92820	−0.399335
$302$	−5.46410	−0.314424
$303$	0	0
$304$	1.00000	0.0573539
$305$	−30.9282	−1.77094
$306$	0	0
$307$	−20.7846	−1.18624	−0.593120	−	0.805114i	$-0.702104\pi$
$307$	−20.7846	−1.18624	−0.593120	+	0.805114i	$0.702104\pi$
$308$	1.46410	0.0834249
$309$	0	0
$310$	24.0000	1.36311
$311$	−29.8564	−1.69300	−0.846501	−	0.532388i	$-0.821295\pi$
$311$	−29.8564	−1.69300	−0.846501	+	0.532388i	$0.821295\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−7.85641	−0.443363
$315$	0	0
$316$	8.39230	0.472104
$317$	19.8564	1.11525	0.557623	−	0.830094i	$-0.311713\pi$
$317$	19.8564	1.11525	0.557623	+	0.830094i	$0.311713\pi$
$318$	0	0
$319$	−8.78461	−0.491844
$320$	−3.46410	−0.193649
$321$	0	0
$322$	1.46410	0.0815912
$323$	−0.535898	−0.0298182
$324$	0	0
$325$	24.2487	1.34508
$326$	14.9282	0.826797
$327$	0	0
$328$	2.00000	0.110432
$329$	−2.92820	−0.161437
$330$	0	0
$331$	21.4641	1.17977	0.589887	−	0.807486i	$-0.299172\pi$
$331$	21.4641	1.17977	0.589887	+	0.807486i	$0.299172\pi$
$332$	−8.00000	−0.439057
$333$	0	0
$334$	5.07180	0.277516
$335$	−8.78461	−0.479954
$336$	0	0
$337$	−30.7846	−1.67694	−0.838472	−	0.544944i	$-0.816551\pi$
$337$	−30.7846	−1.67694	−0.838472	+	0.544944i	$0.816551\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	1.85641	0.100678
$341$	10.1436	0.549306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	6.92820	0.373544
$345$	0	0
$346$	−12.9282	−0.695025
$347$	7.32051	0.392985	0.196493	−	0.980505i	$-0.437045\pi$
$347$	7.32051	0.392985	0.196493	+	0.980505i	$0.437045\pi$
$348$	0	0
$349$	8.14359	0.435917	0.217958	−	0.975958i	$-0.430060\pi$
$349$	8.14359	0.435917	0.217958	+	0.975958i	$0.430060\pi$
$350$	−7.00000	−0.374166
$351$	0	0
$352$	−1.46410	−0.0780369
$353$	29.3205	1.56057	0.780287	−	0.625422i	$-0.215073\pi$
$353$	29.3205	1.56057	0.780287	+	0.625422i	$0.215073\pi$
$354$	0	0
$355$	−37.8564	−2.00921
$356$	2.00000	0.106000
$357$	0	0
$358$	−17.8564	−0.943740
$359$	−0.679492	−0.0358622	−0.0179311	−	0.999839i	$-0.505708\pi$
$359$	−0.679492	−0.0358622	−0.0179311	+	0.999839i	$0.505708\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−15.4641	−0.812775
$363$	0	0
$364$	−3.46410	−0.181568
$365$	−34.6410	−1.81319
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−1.46410	−0.0763216
$369$	0	0
$370$	−34.6410	−1.80090
$371$	2.00000	0.103835
$372$	0	0
$373$	−30.7846	−1.59397	−0.796983	−	0.604001i	$-0.793572\pi$
$373$	−30.7846	−1.59397	−0.796983	+	0.604001i	$0.793572\pi$
$374$	0.784610	0.0405712
$375$	0	0
$376$	2.92820	0.151011
$377$	20.7846	1.07046
$378$	0	0
$379$	5.46410	0.280672	0.140336	−	0.990104i	$-0.455182\pi$
$379$	5.46410	0.280672	0.140336	+	0.990104i	$0.455182\pi$
$380$	−3.46410	−0.177705
$381$	0	0
$382$	−1.46410	−0.0749100
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	−5.07180	−0.258483
$386$	−22.0000	−1.11977
$387$	0	0
$388$	11.4641	0.582002
$389$	25.7128	1.30369	0.651846	−	0.758352i	$-0.273995\pi$
$389$	25.7128	1.30369	0.651846	+	0.758352i	$0.273995\pi$
$390$	0	0
$391$	0.784610	0.0396794
$392$	1.00000	0.0505076
$393$	0	0
$394$	8.92820	0.449796
$395$	−29.0718	−1.46276
$396$	0	0
$397$	−7.85641	−0.394302	−0.197151	−	0.980373i	$-0.563169\pi$
$397$	−7.85641	−0.394302	−0.197151	+	0.980373i	$0.563169\pi$
$398$	16.7846	0.841336
$399$	0	0
$400$	7.00000	0.350000
$401$	7.85641	0.392330	0.196165	−	0.980571i	$-0.437151\pi$
$401$	7.85641	0.392330	0.196165	+	0.980571i	$0.437151\pi$
$402$	0	0
$403$	−24.0000	−1.19553
$404$	−11.4641	−0.570360
$405$	0	0
$406$	−6.00000	−0.297775
$407$	−14.6410	−0.725728
$408$	0	0
$409$	31.1769	1.54160	0.770800	−	0.637078i	$-0.219857\pi$
$409$	31.1769	1.54160	0.770800	+	0.637078i	$0.219857\pi$
$410$	−6.92820	−0.342160
$411$	0	0
$412$	6.92820	0.341328
$413$	4.00000	0.196827
$414$	0	0
$415$	27.7128	1.36037
$416$	3.46410	0.169842
$417$	0	0
$418$	−1.46410	−0.0716116
$419$	−5.07180	−0.247773	−0.123887	−	0.992296i	$-0.539536\pi$
$419$	−5.07180	−0.247773	−0.123887	+	0.992296i	$0.539536\pi$
$420$	0	0
$421$	−22.7846	−1.11045	−0.555227	−	0.831699i	$-0.687369\pi$
$421$	−22.7846	−1.11045	−0.555227	+	0.831699i	$0.687369\pi$
$422$	−5.46410	−0.265988
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	−3.75129	−0.181964
$426$	0	0
$427$	−8.92820	−0.432066
$428$	4.00000	0.193347
$429$	0	0
$430$	−24.0000	−1.15738
$431$	21.0718	1.01499	0.507496	−	0.861654i	$-0.330571\pi$
$431$	21.0718	1.01499	0.507496	+	0.861654i	$0.330571\pi$
$432$	0	0
$433$	41.3205	1.98574	0.992868	−	0.119215i	$-0.0380378\pi$
$433$	41.3205	1.98574	0.992868	+	0.119215i	$0.0380378\pi$
$434$	6.92820	0.332564
$435$	0	0
$436$	2.00000	0.0957826
$437$	−1.46410	−0.0700375
$438$	0	0
$439$	36.7846	1.75563	0.877817	−	0.478996i	$-0.158999\pi$
$439$	36.7846	1.75563	0.877817	+	0.478996i	$0.158999\pi$
$440$	5.07180	0.241788
$441$	0	0
$442$	−1.85641	−0.0883003
$443$	−23.3205	−1.10799	−0.553995	−	0.832520i	$-0.686897\pi$
$443$	−23.3205	−1.10799	−0.553995	+	0.832520i	$0.686897\pi$
$444$	0	0
$445$	−6.92820	−0.328428
$446$	−12.0000	−0.568216
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−19.8564	−0.937082	−0.468541	−	0.883442i	$-0.655220\pi$
$449$	−19.8564	−0.937082	−0.468541	+	0.883442i	$0.655220\pi$
$450$	0	0
$451$	−2.92820	−0.137884
$452$	15.8564	0.745823
$453$	0	0
$454$	−1.07180	−0.0503019
$455$	12.0000	0.562569
$456$	0	0
$457$	39.8564	1.86440	0.932202	−	0.361938i	$-0.117885\pi$
$457$	39.8564	1.86440	0.932202	+	0.361938i	$0.117885\pi$
$458$	16.9282	0.791003
$459$	0	0
$460$	5.07180	0.236474
$461$	38.1051	1.77473	0.887366	−	0.461065i	$-0.152533\pi$
$461$	38.1051	1.77473	0.887366	+	0.461065i	$0.152533\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	14.7846	0.684884
$467$	19.7128	0.912200	0.456100	−	0.889928i	$-0.349246\pi$
$467$	19.7128	0.912200	0.456100	+	0.889928i	$0.349246\pi$
$468$	0	0
$469$	−2.53590	−0.117097
$470$	−10.1436	−0.467889
$471$	0	0
$472$	−4.00000	−0.184115
$473$	−10.1436	−0.466403
$474$	0	0
$475$	7.00000	0.321182
$476$	0.535898	0.0245629
$477$	0	0
$478$	18.2487	0.834677
$479$	27.7128	1.26623	0.633115	−	0.774057i	$-0.281776\pi$
$479$	27.7128	1.26623	0.633115	+	0.774057i	$0.281776\pi$
$480$	0	0
$481$	34.6410	1.57949
$482$	−13.3205	−0.606733
$483$	0	0
$484$	−8.85641	−0.402564
$485$	−39.7128	−1.80327
$486$	0	0
$487$	−4.67949	−0.212048	−0.106024	−	0.994364i	$-0.533812\pi$
$487$	−4.67949	−0.212048	−0.106024	+	0.994364i	$0.533812\pi$
$488$	8.92820	0.404161
$489$	0	0
$490$	−3.46410	−0.156492
$491$	19.6077	0.884883	0.442441	−	0.896797i	$-0.354112\pi$
$491$	19.6077	0.884883	0.442441	+	0.896797i	$0.354112\pi$
$492$	0	0
$493$	−3.21539	−0.144814
$494$	3.46410	0.155857
$495$	0	0
$496$	−6.92820	−0.311086
$497$	−10.9282	−0.490197
$498$	0	0
$499$	−9.07180	−0.406109	−0.203055	−	0.979167i	$-0.565087\pi$
$499$	−9.07180	−0.406109	−0.203055	+	0.979167i	$0.565087\pi$
$500$	−6.92820	−0.309839
$501$	0	0
$502$	−18.9282	−0.844807
$503$	−5.85641	−0.261124	−0.130562	−	0.991440i	$-0.541678\pi$
$503$	−5.85641	−0.261124	−0.130562	+	0.991440i	$0.541678\pi$
$504$	0	0
$505$	39.7128	1.76720
$506$	2.14359	0.0952944
$507$	0	0
$508$	−10.5359	−0.467455
$509$	−24.6410	−1.09219	−0.546097	−	0.837722i	$-0.683887\pi$
$509$	−24.6410	−1.09219	−0.546097	+	0.837722i	$0.683887\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	1.00000	0.0441942
$513$	0	0
$514$	−11.0718	−0.488356
$515$	−24.0000	−1.05757
$516$	0	0
$517$	−4.28719	−0.188550
$518$	−10.0000	−0.439375
$519$	0	0
$520$	−12.0000	−0.526235
$521$	29.7128	1.30174	0.650871	−	0.759188i	$-0.274404\pi$
$521$	29.7128	1.30174	0.650871	+	0.759188i	$0.274404\pi$
$522$	0	0
$523$	−42.6410	−1.86456	−0.932281	−	0.361736i	$-0.882184\pi$
$523$	−42.6410	−1.86456	−0.932281	+	0.361736i	$0.882184\pi$
$524$	18.9282	0.826882
$525$	0	0
$526$	23.3205	1.01682
$527$	3.71281	0.161733
$528$	0	0
$529$	−20.8564	−0.906800
$530$	6.92820	0.300942
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	6.92820	0.300094
$534$	0	0
$535$	−13.8564	−0.599065
$536$	2.53590	0.109534
$537$	0	0
$538$	3.07180	0.132435
$539$	−1.46410	−0.0630633
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	−2.14359	−0.0920752
$543$	0	0
$544$	−0.535898	−0.0229765
$545$	−6.92820	−0.296772
$546$	0	0
$547$	38.2487	1.63540	0.817698	−	0.575647i	$-0.195250\pi$
$547$	38.2487	1.63540	0.817698	+	0.575647i	$0.195250\pi$
$548$	−12.9282	−0.552265
$549$	0	0
$550$	−10.2487	−0.437007
$551$	6.00000	0.255609
$552$	0	0
$553$	−8.39230	−0.356877
$554$	22.0000	0.934690
$555$	0	0
$556$	−17.8564	−0.757280
$557$	8.92820	0.378300	0.189150	−	0.981948i	$-0.439427\pi$
$557$	8.92820	0.378300	0.189150	+	0.981948i	$0.439427\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	3.46410	0.146385
$561$	0	0
$562$	26.0000	1.09674
$563$	1.85641	0.0782382	0.0391191	−	0.999235i	$-0.487545\pi$
$563$	1.85641	0.0782382	0.0391191	+	0.999235i	$0.487545\pi$
$564$	0	0
$565$	−54.9282	−2.31085
$566$	−20.0000	−0.840663
$567$	0	0
$568$	10.9282	0.458537
$569$	−9.71281	−0.407182	−0.203591	−	0.979056i	$-0.565261\pi$
$569$	−9.71281	−0.407182	−0.203591	+	0.979056i	$0.565261\pi$
$570$	0	0
$571$	−45.5692	−1.90701	−0.953506	−	0.301373i	$-0.902555\pi$
$571$	−45.5692	−1.90701	−0.953506	+	0.301373i	$0.902555\pi$
$572$	−5.07180	−0.212062
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	−10.2487	−0.427401
$576$	0	0
$577$	23.8564	0.993155	0.496578	−	0.867992i	$-0.334590\pi$
$577$	23.8564	0.993155	0.496578	+	0.867992i	$0.334590\pi$
$578$	−16.7128	−0.695161
$579$	0	0
$580$	−20.7846	−0.863034
$581$	8.00000	0.331896
$582$	0	0
$583$	2.92820	0.121274
$584$	10.0000	0.413803
$585$	0	0
$586$	−20.9282	−0.864536
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	−6.92820	−0.285472
$590$	13.8564	0.570459
$591$	0	0
$592$	10.0000	0.410997
$593$	−19.4641	−0.799295	−0.399647	−	0.916669i	$-0.630867\pi$
$593$	−19.4641	−0.799295	−0.399647	+	0.916669i	$0.630867\pi$
$594$	0	0
$595$	−1.85641	−0.0761052
$596$	16.9282	0.693406
$597$	0	0
$598$	−5.07180	−0.207401
$599$	−26.9282	−1.10026	−0.550128	−	0.835080i	$-0.685421\pi$
$599$	−26.9282	−1.10026	−0.550128	+	0.835080i	$0.685421\pi$
$600$	0	0
$601$	41.3205	1.68550	0.842749	−	0.538306i	$-0.180936\pi$
$601$	41.3205	1.68550	0.842749	+	0.538306i	$0.180936\pi$
$602$	−6.92820	−0.282372
$603$	0	0
$604$	−5.46410	−0.222331
$605$	30.6795	1.24730
$606$	0	0
$607$	9.85641	0.400059	0.200030	−	0.979790i	$-0.435896\pi$
$607$	9.85641	0.400059	0.200030	+	0.979790i	$0.435896\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−30.9282	−1.25225
$611$	10.1436	0.410366
$612$	0	0
$613$	3.85641	0.155759	0.0778794	−	0.996963i	$-0.475185\pi$
$613$	3.85641	0.155759	0.0778794	+	0.996963i	$0.475185\pi$
$614$	−20.7846	−0.838799
$615$	0	0
$616$	1.46410	0.0589903
$617$	0.928203	0.0373681	0.0186840	−	0.999825i	$-0.494052\pi$
$617$	0.928203	0.0373681	0.0186840	+	0.999825i	$0.494052\pi$
$618$	0	0
$619$	−42.6410	−1.71389	−0.856944	−	0.515410i	$-0.827640\pi$
$619$	−42.6410	−1.71389	−0.856944	+	0.515410i	$0.827640\pi$
$620$	24.0000	0.963863
$621$	0	0
$622$	−29.8564	−1.19713
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−6.00000	−0.239808
$627$	0	0
$628$	−7.85641	−0.313505
$629$	−5.35898	−0.213677
$630$	0	0
$631$	−24.7846	−0.986660	−0.493330	−	0.869842i	$-0.664220\pi$
$631$	−24.7846	−0.986660	−0.493330	+	0.869842i	$0.664220\pi$
$632$	8.39230	0.333828
$633$	0	0
$634$	19.8564	0.788599
$635$	36.4974	1.44836
$636$	0	0
$637$	3.46410	0.137253
$638$	−8.78461	−0.347786
$639$	0	0
$640$	−3.46410	−0.136931
$641$	−11.8564	−0.468300	−0.234150	−	0.972200i	$-0.575231\pi$
$641$	−11.8564	−0.468300	−0.234150	+	0.972200i	$0.575231\pi$
$642$	0	0
$643$	−1.07180	−0.0422675	−0.0211338	−	0.999777i	$-0.506728\pi$
$643$	−1.07180	−0.0422675	−0.0211338	+	0.999777i	$0.506728\pi$
$644$	1.46410	0.0576937
$645$	0	0
$646$	−0.535898	−0.0210846
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	5.85641	0.229884
$650$	24.2487	0.951113
$651$	0	0
$652$	14.9282	0.584634
$653$	−21.7128	−0.849688	−0.424844	−	0.905267i	$-0.639671\pi$
$653$	−21.7128	−0.849688	−0.424844	+	0.905267i	$0.639671\pi$
$654$	0	0
$655$	−65.5692	−2.56200
$656$	2.00000	0.0780869
$657$	0	0
$658$	−2.92820	−0.114153
$659$	−41.8564	−1.63049	−0.815247	−	0.579113i	$-0.803399\pi$
$659$	−41.8564	−1.63049	−0.815247	+	0.579113i	$0.803399\pi$
$660$	0	0
$661$	10.6795	0.415384	0.207692	−	0.978194i	$-0.433405\pi$
$661$	10.6795	0.415384	0.207692	+	0.978194i	$0.433405\pi$
$662$	21.4641	0.834226
$663$	0	0
$664$	−8.00000	−0.310460
$665$	3.46410	0.134332
$666$	0	0
$667$	−8.78461	−0.340141
$668$	5.07180	0.196234
$669$	0	0
$670$	−8.78461	−0.339379
$671$	−13.0718	−0.504631
$672$	0	0
$673$	−3.07180	−0.118409	−0.0592045	−	0.998246i	$-0.518856\pi$
$673$	−3.07180	−0.118409	−0.0592045	+	0.998246i	$0.518856\pi$
$674$	−30.7846	−1.18578
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	−7.85641	−0.301946	−0.150973	−	0.988538i	$-0.548241\pi$
$677$	−7.85641	−0.301946	−0.150973	+	0.988538i	$0.548241\pi$
$678$	0	0
$679$	−11.4641	−0.439952
$680$	1.85641	0.0711899
$681$	0	0
$682$	10.1436	0.388418
$683$	−17.0718	−0.653234	−0.326617	−	0.945157i	$-0.605909\pi$
$683$	−17.0718	−0.653234	−0.326617	+	0.945157i	$0.605909\pi$
$684$	0	0
$685$	44.7846	1.71113
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	6.92820	0.264135
$689$	−6.92820	−0.263944
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−12.9282	−0.491457
$693$	0	0
$694$	7.32051	0.277883
$695$	61.8564	2.34635
$696$	0	0
$697$	−1.07180	−0.0405972
$698$	8.14359	0.308240
$699$	0	0
$700$	−7.00000	−0.264575
$701$	41.7128	1.57547	0.787736	−	0.616013i	$-0.211253\pi$
$701$	41.7128	1.57547	0.787736	+	0.616013i	$0.211253\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	−1.46410	−0.0551804
$705$	0	0
$706$	29.3205	1.10349
$707$	11.4641	0.431152
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	−37.8564	−1.42073
$711$	0	0
$712$	2.00000	0.0749532
$713$	10.1436	0.379881
$714$	0	0
$715$	17.5692	0.657052
$716$	−17.8564	−0.667325
$717$	0	0
$718$	−0.679492	−0.0253584
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	−6.92820	−0.258020
$722$	1.00000	0.0372161
$723$	0	0
$724$	−15.4641	−0.574719
$725$	42.0000	1.55984
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−3.46410	−0.128388
$729$	0	0
$730$	−34.6410	−1.28212
$731$	−3.71281	−0.137323
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−1.46410	−0.0539675
$737$	−3.71281	−0.136763
$738$	0	0
$739$	17.8564	0.656859	0.328429	−	0.944529i	$-0.393481\pi$
$739$	17.8564	0.656859	0.328429	+	0.944529i	$0.393481\pi$
$740$	−34.6410	−1.27343
$741$	0	0
$742$	2.00000	0.0734223
$743$	27.7128	1.01668	0.508342	−	0.861155i	$-0.330258\pi$
$743$	27.7128	1.01668	0.508342	+	0.861155i	$0.330258\pi$
$744$	0	0
$745$	−58.6410	−2.14844
$746$	−30.7846	−1.12710
$747$	0	0
$748$	0.784610	0.0286882
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−0.392305	−0.0143154	−0.00715770	−	0.999974i	$-0.502278\pi$
$751$	−0.392305	−0.0143154	−0.00715770	+	0.999974i	$0.502278\pi$
$752$	2.92820	0.106781
$753$	0	0
$754$	20.7846	0.756931
$755$	18.9282	0.688868
$756$	0	0
$757$	−23.8564	−0.867076	−0.433538	−	0.901135i	$-0.642735\pi$
$757$	−23.8564	−0.867076	−0.433538	+	0.901135i	$0.642735\pi$
$758$	5.46410	0.198465
$759$	0	0
$760$	−3.46410	−0.125656
$761$	−11.4641	−0.415573	−0.207787	−	0.978174i	$-0.566626\pi$
$761$	−11.4641	−0.415573	−0.207787	+	0.978174i	$0.566626\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	−1.46410	−0.0529693
$765$	0	0
$766$	−8.00000	−0.289052
$767$	−13.8564	−0.500326
$768$	0	0
$769$	7.85641	0.283309	0.141655	−	0.989916i	$-0.454758\pi$
$769$	7.85641	0.283309	0.141655	+	0.989916i	$0.454758\pi$
$770$	−5.07180	−0.182775
$771$	0	0
$772$	−22.0000	−0.791797
$773$	38.7846	1.39499	0.697493	−	0.716592i	$-0.254299\pi$
$773$	38.7846	1.39499	0.697493	+	0.716592i	$0.254299\pi$
$774$	0	0
$775$	−48.4974	−1.74208
$776$	11.4641	0.411537
$777$	0	0
$778$	25.7128	0.921849
$779$	2.00000	0.0716574
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0.784610	0.0280576
$783$	0	0
$784$	1.00000	0.0357143
$785$	27.2154	0.971359
$786$	0	0
$787$	−52.7846	−1.88157	−0.940784	−	0.339006i	$-0.889909\pi$
$787$	−52.7846	−1.88157	−0.940784	+	0.339006i	$0.889909\pi$
$788$	8.92820	0.318054
$789$	0	0
$790$	−29.0718	−1.03433
$791$	−15.8564	−0.563789
$792$	0	0
$793$	30.9282	1.09829
$794$	−7.85641	−0.278813
$795$	0	0
$796$	16.7846	0.594915
$797$	−24.6410	−0.872830	−0.436415	−	0.899746i	$-0.643752\pi$
$797$	−24.6410	−0.872830	−0.436415	+	0.899746i	$0.643752\pi$
$798$	0	0
$799$	−1.56922	−0.0555150
$800$	7.00000	0.247487
$801$	0	0
$802$	7.85641	0.277419
$803$	−14.6410	−0.516670
$804$	0	0
$805$	−5.07180	−0.178757
$806$	−24.0000	−0.845364
$807$	0	0
$808$	−11.4641	−0.403306
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	46.9282	1.64787	0.823936	−	0.566683i	$-0.191773\pi$
$811$	46.9282	1.64787	0.823936	+	0.566683i	$0.191773\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	−14.6410	−0.513167
$815$	−51.7128	−1.81142
$816$	0	0
$817$	6.92820	0.242387
$818$	31.1769	1.09008
$819$	0	0
$820$	−6.92820	−0.241943
$821$	35.0718	1.22401	0.612007	−	0.790852i	$-0.290362\pi$
$821$	35.0718	1.22401	0.612007	+	0.790852i	$0.290362\pi$
$822$	0	0
$823$	5.85641	0.204141	0.102071	−	0.994777i	$-0.467453\pi$
$823$	5.85641	0.204141	0.102071	+	0.994777i	$0.467453\pi$
$824$	6.92820	0.241355
$825$	0	0
$826$	4.00000	0.139178
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−26.3923	−0.916643	−0.458321	−	0.888787i	$-0.651549\pi$
$829$	−26.3923	−0.916643	−0.458321	+	0.888787i	$0.651549\pi$
$830$	27.7128	0.961926
$831$	0	0
$832$	3.46410	0.120096
$833$	−0.535898	−0.0185678
$834$	0	0
$835$	−17.5692	−0.608008
$836$	−1.46410	−0.0506370
$837$	0	0
$838$	−5.07180	−0.175202
$839$	18.9282	0.653474	0.326737	−	0.945115i	$-0.394051\pi$
$839$	18.9282	0.653474	0.326737	+	0.945115i	$0.394051\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−22.7846	−0.785210
$843$	0	0
$844$	−5.46410	−0.188082
$845$	3.46410	0.119169
$846$	0	0
$847$	8.85641	0.304310
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	−3.75129	−0.128668
$851$	−14.6410	−0.501888
$852$	0	0
$853$	48.9282	1.67527	0.837635	−	0.546231i	$-0.183938\pi$
$853$	48.9282	1.67527	0.837635	+	0.546231i	$0.183938\pi$
$854$	−8.92820	−0.305517
$855$	0	0
$856$	4.00000	0.136717
$857$	35.5692	1.21502	0.607511	−	0.794311i	$-0.292168\pi$
$857$	35.5692	1.21502	0.607511	+	0.794311i	$0.292168\pi$
$858$	0	0
$859$	12.7846	0.436205	0.218103	−	0.975926i	$-0.430013\pi$
$859$	12.7846	0.436205	0.218103	+	0.975926i	$0.430013\pi$
$860$	−24.0000	−0.818393
$861$	0	0
$862$	21.0718	0.717708
$863$	−45.8564	−1.56097	−0.780485	−	0.625174i	$-0.785028\pi$
$863$	−45.8564	−1.56097	−0.780485	+	0.625174i	$0.785028\pi$
$864$	0	0
$865$	44.7846	1.52272
$866$	41.3205	1.40413
$867$	0	0
$868$	6.92820	0.235159
$869$	−12.2872	−0.416814
$870$	0	0
$871$	8.78461	0.297655
$872$	2.00000	0.0677285
$873$	0	0
$874$	−1.46410	−0.0495240
$875$	6.92820	0.234216
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	36.7846	1.24142
$879$	0	0
$880$	5.07180	0.170970
$881$	19.1769	0.646087	0.323043	−	0.946384i	$-0.395294\pi$
$881$	19.1769	0.646087	0.323043	+	0.946384i	$0.395294\pi$
$882$	0	0
$883$	−48.4974	−1.63207	−0.816034	−	0.578004i	$-0.803832\pi$
$883$	−48.4974	−1.63207	−0.816034	+	0.578004i	$0.803832\pi$
$884$	−1.85641	−0.0624377
$885$	0	0
$886$	−23.3205	−0.783468
$887$	−27.7128	−0.930505	−0.465253	−	0.885178i	$-0.654037\pi$
$887$	−27.7128	−0.930505	−0.465253	+	0.885178i	$0.654037\pi$
$888$	0	0
$889$	10.5359	0.353363
$890$	−6.92820	−0.232234
$891$	0	0
$892$	−12.0000	−0.401790
$893$	2.92820	0.0979886
$894$	0	0
$895$	61.8564	2.06763
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−19.8564	−0.662617
$899$	−41.5692	−1.38641
$900$	0	0
$901$	1.07180	0.0357067
$902$	−2.92820	−0.0974985
$903$	0	0
$904$	15.8564	0.527376
$905$	53.5692	1.78070
$906$	0	0
$907$	−34.5359	−1.14675	−0.573373	−	0.819295i	$-0.694365\pi$
$907$	−34.5359	−1.14675	−0.573373	+	0.819295i	$0.694365\pi$
$908$	−1.07180	−0.0355688
$909$	0	0
$910$	12.0000	0.397796
$911$	46.6410	1.54529	0.772643	−	0.634841i	$-0.218934\pi$
$911$	46.6410	1.54529	0.772643	+	0.634841i	$0.218934\pi$
$912$	0	0
$913$	11.7128	0.387638
$914$	39.8564	1.31833
$915$	0	0
$916$	16.9282	0.559324
$917$	−18.9282	−0.625064
$918$	0	0
$919$	33.5692	1.10735	0.553673	−	0.832734i	$-0.313226\pi$
$919$	33.5692	1.10735	0.553673	+	0.832734i	$0.313226\pi$
$920$	5.07180	0.167212
$921$	0	0
$922$	38.1051	1.25493
$923$	37.8564	1.24606
$924$	0	0
$925$	70.0000	2.30159
$926$	24.0000	0.788689
$927$	0	0
$928$	6.00000	0.196960
$929$	−46.3923	−1.52208	−0.761041	−	0.648704i	$-0.775311\pi$
$929$	−46.3923	−1.52208	−0.761041	+	0.648704i	$0.775311\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	14.7846	0.484286
$933$	0	0
$934$	19.7128	0.645023
$935$	−2.71797	−0.0888870
$936$	0	0
$937$	7.07180	0.231026	0.115513	−	0.993306i	$-0.463149\pi$
$937$	7.07180	0.231026	0.115513	+	0.993306i	$0.463149\pi$
$938$	−2.53590	−0.0828000
$939$	0	0
$940$	−10.1436	−0.330848
$941$	46.7846	1.52513	0.762567	−	0.646909i	$-0.223939\pi$
$941$	46.7846	1.52513	0.762567	+	0.646909i	$0.223939\pi$
$942$	0	0
$943$	−2.92820	−0.0953554
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−10.1436	−0.329797
$947$	−48.1051	−1.56321	−0.781603	−	0.623776i	$-0.785598\pi$
$947$	−48.1051	−1.56321	−0.781603	+	0.623776i	$0.785598\pi$
$948$	0	0
$949$	34.6410	1.12449
$950$	7.00000	0.227110
$951$	0	0
$952$	0.535898	0.0173686
$953$	4.14359	0.134224	0.0671121	−	0.997745i	$-0.478621\pi$
$953$	4.14359	0.134224	0.0671121	+	0.997745i	$0.478621\pi$
$954$	0	0
$955$	5.07180	0.164119
$956$	18.2487	0.590206
$957$	0	0
$958$	27.7128	0.895360
$959$	12.9282	0.417473
$960$	0	0
$961$	17.0000	0.548387
$962$	34.6410	1.11687
$963$	0	0
$964$	−13.3205	−0.429025
$965$	76.2102	2.45329
$966$	0	0
$967$	−5.07180	−0.163098	−0.0815490	−	0.996669i	$-0.525987\pi$
$967$	−5.07180	−0.163098	−0.0815490	+	0.996669i	$0.525987\pi$
$968$	−8.85641	−0.284656
$969$	0	0
$970$	−39.7128	−1.27510
$971$	−58.6410	−1.88188	−0.940940	−	0.338574i	$-0.890056\pi$
$971$	−58.6410	−1.88188	−0.940940	+	0.338574i	$0.890056\pi$
$972$	0	0
$973$	17.8564	0.572450
$974$	−4.67949	−0.149941
$975$	0	0
$976$	8.92820	0.285785
$977$	−19.8564	−0.635263	−0.317631	−	0.948214i	$-0.602887\pi$
$977$	−19.8564	−0.635263	−0.317631	+	0.948214i	$0.602887\pi$
$978$	0	0
$979$	−2.92820	−0.0935858
$980$	−3.46410	−0.110657
$981$	0	0
$982$	19.6077	0.625707
$983$	−8.78461	−0.280186	−0.140093	−	0.990138i	$-0.544740\pi$
$983$	−8.78461	−0.280186	−0.140093	+	0.990138i	$0.544740\pi$
$984$	0	0
$985$	−30.9282	−0.985454
$986$	−3.21539	−0.102399
$987$	0	0
$988$	3.46410	0.110208
$989$	−10.1436	−0.322548
$990$	0	0
$991$	56.3923	1.79136	0.895680	−	0.444699i	$-0.146689\pi$
$991$	56.3923	1.79136	0.895680	+	0.444699i	$0.146689\pi$
$992$	−6.92820	−0.219971
$993$	0	0
$994$	−10.9282	−0.346622
$995$	−58.1436	−1.84328
$996$	0	0
$997$	−20.1436	−0.637954	−0.318977	−	0.947762i	$-0.603339\pi$
$997$	−20.1436	−0.637954	−0.318977	+	0.947762i	$0.603339\pi$
$998$	−9.07180	−0.287163
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.x.1.1		2
3.2	odd	2		798.2.a.k.1.2	✓	2
12.11	even	2		6384.2.a.br.1.2		2
21.20	even	2		5586.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.a.k.1.2	✓	2	3.2	odd	2
2394.2.a.x.1.1		2	1.1	even	1	trivial
5586.2.a.bd.1.1		2	21.20	even	2
6384.2.a.br.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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.46410 q^{10} -1.46410 q^{11} +3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -0.535898 q^{17} +1.00000 q^{19} -3.46410 q^{20} -1.46410 q^{22} -1.46410 q^{23} +7.00000 q^{25} +3.46410 q^{26} -1.00000 q^{28} +6.00000 q^{29} -6.92820 q^{31} +1.00000 q^{32} -0.535898 q^{34} +3.46410 q^{35} +10.0000 q^{37} +1.00000 q^{38} -3.46410 q^{40} +2.00000 q^{41} +6.92820 q^{43} -1.46410 q^{44} -1.46410 q^{46} +2.92820 q^{47} +1.00000 q^{49} +7.00000 q^{50} +3.46410 q^{52} -2.00000 q^{53} +5.07180 q^{55} -1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} +8.92820 q^{61} -6.92820 q^{62} +1.00000 q^{64} -12.0000 q^{65} +2.53590 q^{67} -0.535898 q^{68} +3.46410 q^{70} +10.9282 q^{71} +10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{76} +1.46410 q^{77} +8.39230 q^{79} -3.46410 q^{80} +2.00000 q^{82} -8.00000 q^{83} +1.85641 q^{85} +6.92820 q^{86} -1.46410 q^{88} +2.00000 q^{89} -3.46410 q^{91} -1.46410 q^{92} +2.92820 q^{94} -3.46410 q^{95} +11.4641 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 4 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} + 14 q^{25} - 2 q^{28} + 12 q^{29} + 2 q^{32} - 8 q^{34} + 20 q^{37} + 2 q^{38} + 4 q^{41} + 4 q^{44} + 4 q^{46} - 8 q^{47} + 2 q^{49} + 14 q^{50} - 4 q^{53} + 24 q^{55} - 2 q^{56} + 12 q^{58} - 8 q^{59} + 4 q^{61} + 2 q^{64} - 24 q^{65} + 12 q^{67} - 8 q^{68} + 8 q^{71} + 20 q^{73} + 20 q^{74} + 2 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{82} - 16 q^{83} - 24 q^{85} + 4 q^{88} + 4 q^{89} + 4 q^{92} - 8 q^{94} + 16 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 4 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^17 + 2 * q^19 + 4 * q^22 + 4 * q^23 + 14 * q^25 - 2 * q^28 + 12 * q^29 + 2 * q^32 - 8 * q^34 + 20 * q^37 + 2 * q^38 + 4 * q^41 + 4 * q^44 + 4 * q^46 - 8 * q^47 + 2 * q^49 + 14 * q^50 - 4 * q^53 + 24 * q^55 - 2 * q^56 + 12 * q^58 - 8 * q^59 + 4 * q^61 + 2 * q^64 - 24 * q^65 + 12 * q^67 - 8 * q^68 + 8 * q^71 + 20 * q^73 + 20 * q^74 + 2 * q^76 - 4 * q^77 - 4 * q^79 + 4 * q^82 - 16 * q^83 - 24 * q^85 + 4 * q^88 + 4 * q^89 + 4 * q^92 - 8 * q^94 + 16 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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