LMFDB - Embedded newform 2310.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2310,4,Mod(1,2310)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2310, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2310.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2310.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:	$136.294412113$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2310.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -11.0000 q^{11} +12.0000 q^{12} -58.0000 q^{13} -14.0000 q^{14} +15.0000 q^{15} +16.0000 q^{16} -6.00000 q^{17} -18.0000 q^{18} -124.000 q^{19} +20.0000 q^{20} +21.0000 q^{21} +22.0000 q^{22} +96.0000 q^{23} -24.0000 q^{24} +25.0000 q^{25} +116.000 q^{26} +27.0000 q^{27} +28.0000 q^{28} +78.0000 q^{29} -30.0000 q^{30} +224.000 q^{31} -32.0000 q^{32} -33.0000 q^{33} +12.0000 q^{34} +35.0000 q^{35} +36.0000 q^{36} -82.0000 q^{37} +248.000 q^{38} -174.000 q^{39} -40.0000 q^{40} +114.000 q^{41} -42.0000 q^{42} -52.0000 q^{43} -44.0000 q^{44} +45.0000 q^{45} -192.000 q^{46} +144.000 q^{47} +48.0000 q^{48} +49.0000 q^{49} -50.0000 q^{50} -18.0000 q^{51} -232.000 q^{52} +126.000 q^{53} -54.0000 q^{54} -55.0000 q^{55} -56.0000 q^{56} -372.000 q^{57} -156.000 q^{58} -828.000 q^{59} +60.0000 q^{60} +230.000 q^{61} -448.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} -290.000 q^{65} +66.0000 q^{66} +452.000 q^{67} -24.0000 q^{68} +288.000 q^{69} -70.0000 q^{70} -72.0000 q^{72} -1150.00 q^{73} +164.000 q^{74} +75.0000 q^{75} -496.000 q^{76} -77.0000 q^{77} +348.000 q^{78} -1168.00 q^{79} +80.0000 q^{80} +81.0000 q^{81} -228.000 q^{82} -108.000 q^{83} +84.0000 q^{84} -30.0000 q^{85} +104.000 q^{86} +234.000 q^{87} +88.0000 q^{88} -630.000 q^{89} -90.0000 q^{90} -406.000 q^{91} +384.000 q^{92} +672.000 q^{93} -288.000 q^{94} -620.000 q^{95} -96.0000 q^{96} -574.000 q^{97} -98.0000 q^{98} -99.0000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−6.00000	−0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	12.0000	0.288675
$13$	−58.0000	−1.23741	−0.618704	−	0.785624i	$-0.712342\pi$
$13$	−58.0000	−1.23741	−0.618704	+	0.785624i	$0.712342\pi$
$14$	−14.0000	−0.267261
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	−6.00000	−0.0856008	−0.0428004	−	0.999084i	$-0.513628\pi$
$17$	−6.00000	−0.0856008	−0.0428004	+	0.999084i	$0.513628\pi$
$18$	−18.0000	−0.235702
$19$	−124.000	−1.49724	−0.748620	−	0.663000i	$-0.769283\pi$
$19$	−124.000	−1.49724	−0.748620	+	0.663000i	$0.769283\pi$
$20$	20.0000	0.223607
$21$	21.0000	0.218218
$22$	22.0000	0.213201
$23$	96.0000	0.870321	0.435161	−	0.900353i	$-0.356692\pi$
$23$	96.0000	0.870321	0.435161	+	0.900353i	$0.356692\pi$
$24$	−24.0000	−0.204124
$25$	25.0000	0.200000
$26$	116.000	0.874980
$27$	27.0000	0.192450
$28$	28.0000	0.188982
$29$	78.0000	0.499456	0.249728	−	0.968316i	$-0.419659\pi$
$29$	78.0000	0.499456	0.249728	+	0.968316i	$0.419659\pi$
$30$	−30.0000	−0.182574
$31$	224.000	1.29779	0.648897	−	0.760877i	$-0.275231\pi$
$31$	224.000	1.29779	0.648897	+	0.760877i	$0.275231\pi$
$32$	−32.0000	−0.176777
$33$	−33.0000	−0.174078
$34$	12.0000	0.0605289
$35$	35.0000	0.169031
$36$	36.0000	0.166667
$37$	−82.0000	−0.364344	−0.182172	−	0.983267i	$-0.558313\pi$
$37$	−82.0000	−0.364344	−0.182172	+	0.983267i	$0.558313\pi$
$38$	248.000	1.05871
$39$	−174.000	−0.714418
$40$	−40.0000	−0.158114
$41$	114.000	0.434239	0.217120	−	0.976145i	$-0.430334\pi$
$41$	114.000	0.434239	0.217120	+	0.976145i	$0.430334\pi$
$42$	−42.0000	−0.154303
$43$	−52.0000	−0.184417	−0.0922084	−	0.995740i	$-0.529393\pi$
$43$	−52.0000	−0.184417	−0.0922084	+	0.995740i	$0.529393\pi$
$44$	−44.0000	−0.150756
$45$	45.0000	0.149071
$46$	−192.000	−0.615410
$47$	144.000	0.446906	0.223453	−	0.974715i	$-0.428267\pi$
$47$	144.000	0.446906	0.223453	+	0.974715i	$0.428267\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	−50.0000	−0.141421
$51$	−18.0000	−0.0494217
$52$	−232.000	−0.618704
$53$	126.000	0.326555	0.163278	−	0.986580i	$-0.447793\pi$
$53$	126.000	0.326555	0.163278	+	0.986580i	$0.447793\pi$
$54$	−54.0000	−0.136083
$55$	−55.0000	−0.134840
$56$	−56.0000	−0.133631
$57$	−372.000	−0.864432
$58$	−156.000	−0.353169
$59$	−828.000	−1.82706	−0.913529	−	0.406774i	$-0.866654\pi$
$59$	−828.000	−1.82706	−0.913529	+	0.406774i	$0.866654\pi$
$60$	60.0000	0.129099
$61$	230.000	0.482762	0.241381	−	0.970430i	$-0.422400\pi$
$61$	230.000	0.482762	0.241381	+	0.970430i	$0.422400\pi$
$62$	−448.000	−0.917678
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	−290.000	−0.553386
$66$	66.0000	0.123091
$67$	452.000	0.824188	0.412094	−	0.911141i	$-0.364798\pi$
$67$	452.000	0.824188	0.412094	+	0.911141i	$0.364798\pi$
$68$	−24.0000	−0.0428004
$69$	288.000	0.502480
$70$	−70.0000	−0.119523
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−72.0000	−0.117851
$73$	−1150.00	−1.84380	−0.921899	−	0.387429i	$-0.873363\pi$
$73$	−1150.00	−1.84380	−0.921899	+	0.387429i	$0.873363\pi$
$74$	164.000	0.257630
$75$	75.0000	0.115470
$76$	−496.000	−0.748620
$77$	−77.0000	−0.113961
$78$	348.000	0.505170
$79$	−1168.00	−1.66342	−0.831711	−	0.555209i	$-0.812638\pi$
$79$	−1168.00	−1.66342	−0.831711	+	0.555209i	$0.812638\pi$
$80$	80.0000	0.111803
$81$	81.0000	0.111111
$82$	−228.000	−0.307054
$83$	−108.000	−0.142826	−0.0714129	−	0.997447i	$-0.522751\pi$
$83$	−108.000	−0.142826	−0.0714129	+	0.997447i	$0.522751\pi$
$84$	84.0000	0.109109
$85$	−30.0000	−0.0382818
$86$	104.000	0.130402
$87$	234.000	0.288361
$88$	88.0000	0.106600
$89$	−630.000	−0.750336	−0.375168	−	0.926957i	$-0.622415\pi$
$89$	−630.000	−0.750336	−0.375168	+	0.926957i	$0.622415\pi$
$90$	−90.0000	−0.105409
$91$	−406.000	−0.467696
$92$	384.000	0.435161
$93$	672.000	0.749281
$94$	−288.000	−0.316010
$95$	−620.000	−0.669586
$96$	−96.0000	−0.102062
$97$	−574.000	−0.600834	−0.300417	−	0.953808i	$-0.597126\pi$
$97$	−574.000	−0.600834	−0.300417	+	0.953808i	$0.597126\pi$
$98$	−98.0000	−0.101015
$99$	−99.0000	−0.100504
$100$	100.000	0.100000
$101$	−450.000	−0.443333	−0.221667	−	0.975122i	$-0.571150\pi$
$101$	−450.000	−0.443333	−0.221667	+	0.975122i	$0.571150\pi$
$102$	36.0000	0.0349464
$103$	1208.00	1.15561	0.577805	−	0.816175i	$-0.303910\pi$
$103$	1208.00	1.15561	0.577805	+	0.816175i	$0.303910\pi$
$104$	464.000	0.437490
$105$	105.000	0.0975900
$106$	−252.000	−0.230909
$107$	−564.000	−0.509570	−0.254785	−	0.966998i	$-0.582005\pi$
$107$	−564.000	−0.509570	−0.254785	+	0.966998i	$0.582005\pi$
$108$	108.000	0.0962250
$109$	1646.00	1.44641	0.723203	−	0.690636i	$-0.242669\pi$
$109$	1646.00	1.44641	0.723203	+	0.690636i	$0.242669\pi$
$110$	110.000	0.0953463
$111$	−246.000	−0.210354
$112$	112.000	0.0944911
$113$	−1518.00	−1.26373	−0.631865	−	0.775079i	$-0.717710\pi$
$113$	−1518.00	−1.26373	−0.631865	+	0.775079i	$0.717710\pi$
$114$	744.000	0.611245
$115$	480.000	0.389219
$116$	312.000	0.249728
$117$	−522.000	−0.412469
$118$	1656.00	1.29193
$119$	−42.0000	−0.0323541
$120$	−120.000	−0.0912871
$121$	121.000	0.0909091
$122$	−460.000	−0.341364
$123$	342.000	0.250708
$124$	896.000	0.648897
$125$	125.000	0.0894427
$126$	−126.000	−0.0890871
$127$	−2512.00	−1.75515	−0.877575	−	0.479440i	$-0.840840\pi$
$127$	−2512.00	−1.75515	−0.877575	+	0.479440i	$0.840840\pi$
$128$	−128.000	−0.0883883
$129$	−156.000	−0.106473
$130$	580.000	0.391303
$131$	−1356.00	−0.904384	−0.452192	−	0.891921i	$-0.649358\pi$
$131$	−1356.00	−0.904384	−0.452192	+	0.891921i	$0.649358\pi$
$132$	−132.000	−0.0870388
$133$	−868.000	−0.565903
$134$	−904.000	−0.582789
$135$	135.000	0.0860663
$136$	48.0000	0.0302645
$137$	−678.000	−0.422813	−0.211407	−	0.977398i	$-0.567804\pi$
$137$	−678.000	−0.422813	−0.211407	+	0.977398i	$0.567804\pi$
$138$	−576.000	−0.355307
$139$	1580.00	0.964128	0.482064	−	0.876136i	$-0.339887\pi$
$139$	1580.00	0.964128	0.482064	+	0.876136i	$0.339887\pi$
$140$	140.000	0.0845154
$141$	432.000	0.258021
$142$	0	0
$143$	638.000	0.373093
$144$	144.000	0.0833333
$145$	390.000	0.223364
$146$	2300.00	1.30376
$147$	147.000	0.0824786
$148$	−328.000	−0.182172
$149$	−1818.00	−0.999573	−0.499786	−	0.866149i	$-0.666588\pi$
$149$	−1818.00	−0.999573	−0.499786	+	0.866149i	$0.666588\pi$
$150$	−150.000	−0.0816497
$151$	−1000.00	−0.538933	−0.269466	−	0.963010i	$-0.586847\pi$
$151$	−1000.00	−0.538933	−0.269466	+	0.963010i	$0.586847\pi$
$152$	992.000	0.529354
$153$	−54.0000	−0.0285336
$154$	154.000	0.0805823
$155$	1120.00	0.580391
$156$	−696.000	−0.357209
$157$	2342.00	1.19052	0.595261	−	0.803532i	$-0.297049\pi$
$157$	2342.00	1.19052	0.595261	+	0.803532i	$0.297049\pi$
$158$	2336.00	1.17622
$159$	378.000	0.188537
$160$	−160.000	−0.0790569
$161$	672.000	0.328950
$162$	−162.000	−0.0785674
$163$	−1276.00	−0.613154	−0.306577	−	0.951846i	$-0.599184\pi$
$163$	−1276.00	−0.613154	−0.306577	+	0.951846i	$0.599184\pi$
$164$	456.000	0.217120
$165$	−165.000	−0.0778499
$166$	216.000	0.100993
$167$	−2856.00	−1.32338	−0.661688	−	0.749779i	$-0.730160\pi$
$167$	−2856.00	−1.32338	−0.661688	+	0.749779i	$0.730160\pi$
$168$	−168.000	−0.0771517
$169$	1167.00	0.531179
$170$	60.0000	0.0270694
$171$	−1116.00	−0.499080
$172$	−208.000	−0.0922084
$173$	1542.00	0.677665	0.338833	−	0.940847i	$-0.389968\pi$
$173$	1542.00	0.677665	0.338833	+	0.940847i	$0.389968\pi$
$174$	−468.000	−0.203902
$175$	175.000	0.0755929
$176$	−176.000	−0.0753778
$177$	−2484.00	−1.05485
$178$	1260.00	0.530567
$179$	−3804.00	−1.58840	−0.794202	−	0.607654i	$-0.792111\pi$
$179$	−3804.00	−1.58840	−0.794202	+	0.607654i	$0.792111\pi$
$180$	180.000	0.0745356
$181$	−3634.00	−1.49234	−0.746169	−	0.665757i	$-0.768109\pi$
$181$	−3634.00	−1.49234	−0.746169	+	0.665757i	$0.768109\pi$
$182$	812.000	0.330711
$183$	690.000	0.278723
$184$	−768.000	−0.307705
$185$	−410.000	−0.162939
$186$	−1344.00	−0.529822
$187$	66.0000	0.0258096
$188$	576.000	0.223453
$189$	189.000	0.0727393
$190$	1240.00	0.473469
$191$	−3480.00	−1.31835	−0.659173	−	0.751992i	$-0.729093\pi$
$191$	−3480.00	−1.31835	−0.659173	+	0.751992i	$0.729093\pi$
$192$	192.000	0.0721688
$193$	−3790.00	−1.41352	−0.706762	−	0.707451i	$-0.749845\pi$
$193$	−3790.00	−1.41352	−0.706762	+	0.707451i	$0.749845\pi$
$194$	1148.00	0.424854
$195$	−870.000	−0.319497
$196$	196.000	0.0714286
$197$	2598.00	0.939593	0.469797	−	0.882775i	$-0.344327\pi$
$197$	2598.00	0.939593	0.469797	+	0.882775i	$0.344327\pi$
$198$	198.000	0.0710669
$199$	5144.00	1.83240	0.916202	−	0.400716i	$-0.131239\pi$
$199$	5144.00	1.83240	0.916202	+	0.400716i	$0.131239\pi$
$200$	−200.000	−0.0707107
$201$	1356.00	0.475845
$202$	900.000	0.313484
$203$	546.000	0.188777
$204$	−72.0000	−0.0247108
$205$	570.000	0.194198
$206$	−2416.00	−0.817139
$207$	864.000	0.290107
$208$	−928.000	−0.309352
$209$	1364.00	0.451435
$210$	−210.000	−0.0690066
$211$	−2380.00	−0.776521	−0.388261	−	0.921550i	$-0.626924\pi$
$211$	−2380.00	−0.776521	−0.388261	+	0.921550i	$0.626924\pi$
$212$	504.000	0.163278
$213$	0	0
$214$	1128.00	0.360320
$215$	−260.000	−0.0824737
$216$	−216.000	−0.0680414
$217$	1568.00	0.490520
$218$	−3292.00	−1.02276
$219$	−3450.00	−1.06452
$220$	−220.000	−0.0674200
$221$	348.000	0.105923
$222$	492.000	0.148743
$223$	4736.00	1.42218	0.711090	−	0.703101i	$-0.248202\pi$
$223$	4736.00	1.42218	0.711090	+	0.703101i	$0.248202\pi$
$224$	−224.000	−0.0668153
$225$	225.000	0.0666667
$226$	3036.00	0.893592
$227$	2148.00	0.628052	0.314026	−	0.949414i	$-0.398322\pi$
$227$	2148.00	0.628052	0.314026	+	0.949414i	$0.398322\pi$
$228$	−1488.00	−0.432216
$229$	542.000	0.156403	0.0782017	−	0.996938i	$-0.475082\pi$
$229$	542.000	0.156403	0.0782017	+	0.996938i	$0.475082\pi$
$230$	−960.000	−0.275220
$231$	−231.000	−0.0657952
$232$	−624.000	−0.176585
$233$	−7062.00	−1.98561	−0.992805	−	0.119745i	$-0.961792\pi$
$233$	−7062.00	−1.98561	−0.992805	+	0.119745i	$0.961792\pi$
$234$	1044.00	0.291660
$235$	720.000	0.199862
$236$	−3312.00	−0.913529
$237$	−3504.00	−0.960377
$238$	84.0000	0.0228778
$239$	2640.00	0.714508	0.357254	−	0.934007i	$-0.383713\pi$
$239$	2640.00	0.714508	0.357254	+	0.934007i	$0.383713\pi$
$240$	240.000	0.0645497
$241$	5066.00	1.35407	0.677033	−	0.735953i	$-0.263266\pi$
$241$	5066.00	1.35407	0.677033	+	0.735953i	$0.263266\pi$
$242$	−242.000	−0.0642824
$243$	243.000	0.0641500
$244$	920.000	0.241381
$245$	245.000	0.0638877
$246$	−684.000	−0.177277
$247$	7192.00	1.85270
$248$	−1792.00	−0.458839
$249$	−324.000	−0.0824605
$250$	−250.000	−0.0632456
$251$	−2460.00	−0.618621	−0.309310	−	0.950961i	$-0.600098\pi$
$251$	−2460.00	−0.618621	−0.309310	+	0.950961i	$0.600098\pi$
$252$	252.000	0.0629941
$253$	−1056.00	−0.262412
$254$	5024.00	1.24108
$255$	−90.0000	−0.0221020
$256$	256.000	0.0625000
$257$	−750.000	−0.182038	−0.0910189	−	0.995849i	$-0.529012\pi$
$257$	−750.000	−0.182038	−0.0910189	+	0.995849i	$0.529012\pi$
$258$	312.000	0.0752879
$259$	−574.000	−0.137709
$260$	−1160.00	−0.276693
$261$	702.000	0.166485
$262$	2712.00	0.639496
$263$	456.000	0.106913	0.0534566	−	0.998570i	$-0.482976\pi$
$263$	456.000	0.106913	0.0534566	+	0.998570i	$0.482976\pi$
$264$	264.000	0.0615457
$265$	630.000	0.146040
$266$	1736.00	0.400154
$267$	−1890.00	−0.433206
$268$	1808.00	0.412094
$269$	−42.0000	−0.00951965	−0.00475982	−	0.999989i	$-0.501515\pi$
$269$	−42.0000	−0.00951965	−0.00475982	+	0.999989i	$0.501515\pi$
$270$	−270.000	−0.0608581
$271$	−5248.00	−1.17636	−0.588180	−	0.808730i	$-0.700155\pi$
$271$	−5248.00	−1.17636	−0.588180	+	0.808730i	$0.700155\pi$
$272$	−96.0000	−0.0214002
$273$	−1218.00	−0.270025
$274$	1356.00	0.298974
$275$	−275.000	−0.0603023
$276$	1152.00	0.251240
$277$	950.000	0.206065	0.103032	−	0.994678i	$-0.467145\pi$
$277$	950.000	0.206065	0.103032	+	0.994678i	$0.467145\pi$
$278$	−3160.00	−0.681742
$279$	2016.00	0.432598
$280$	−280.000	−0.0597614
$281$	5370.00	1.14003	0.570013	−	0.821636i	$-0.306938\pi$
$281$	5370.00	1.14003	0.570013	+	0.821636i	$0.306938\pi$
$282$	−864.000	−0.182448
$283$	4748.00	0.997312	0.498656	−	0.866800i	$-0.333827\pi$
$283$	4748.00	0.997312	0.498656	+	0.866800i	$0.333827\pi$
$284$	0	0
$285$	−1860.00	−0.386586
$286$	−1276.00	−0.263816
$287$	798.000	0.164127
$288$	−288.000	−0.0589256
$289$	−4877.00	−0.992673
$290$	−780.000	−0.157942
$291$	−1722.00	−0.346892
$292$	−4600.00	−0.921899
$293$	702.000	0.139970	0.0699851	−	0.997548i	$-0.477705\pi$
$293$	702.000	0.139970	0.0699851	+	0.997548i	$0.477705\pi$
$294$	−294.000	−0.0583212
$295$	−4140.00	−0.817085
$296$	656.000	0.128815
$297$	−297.000	−0.0580259
$298$	3636.00	0.706805
$299$	−5568.00	−1.07694
$300$	300.000	0.0577350
$301$	−364.000	−0.0697030
$302$	2000.00	0.381083
$303$	−1350.00	−0.255959
$304$	−1984.00	−0.374310
$305$	1150.00	0.215898
$306$	108.000	0.0201763
$307$	−6796.00	−1.26341	−0.631707	−	0.775207i	$-0.717645\pi$
$307$	−6796.00	−1.26341	−0.631707	+	0.775207i	$0.717645\pi$
$308$	−308.000	−0.0569803
$309$	3624.00	0.667191
$310$	−2240.00	−0.410398
$311$	792.000	0.144406	0.0722029	−	0.997390i	$-0.476997\pi$
$311$	792.000	0.144406	0.0722029	+	0.997390i	$0.476997\pi$
$312$	1392.00	0.252585
$313$	218.000	0.0393677	0.0196838	−	0.999806i	$-0.493734\pi$
$313$	218.000	0.0393677	0.0196838	+	0.999806i	$0.493734\pi$
$314$	−4684.00	−0.841826
$315$	315.000	0.0563436
$316$	−4672.00	−0.831711
$317$	3078.00	0.545356	0.272678	−	0.962105i	$-0.412091\pi$
$317$	3078.00	0.545356	0.272678	+	0.962105i	$0.412091\pi$
$318$	−756.000	−0.133316
$319$	−858.000	−0.150592
$320$	320.000	0.0559017
$321$	−1692.00	−0.294200
$322$	−1344.00	−0.232603
$323$	744.000	0.128165
$324$	324.000	0.0555556
$325$	−1450.00	−0.247482
$326$	2552.00	0.433565
$327$	4938.00	0.835083
$328$	−912.000	−0.153527
$329$	1008.00	0.168914
$330$	330.000	0.0550482
$331$	−5236.00	−0.869476	−0.434738	−	0.900557i	$-0.643159\pi$
$331$	−5236.00	−0.869476	−0.434738	+	0.900557i	$0.643159\pi$
$332$	−432.000	−0.0714129
$333$	−738.000	−0.121448
$334$	5712.00	0.935769
$335$	2260.00	0.368588
$336$	336.000	0.0545545
$337$	−1438.00	−0.232442	−0.116221	−	0.993223i	$-0.537078\pi$
$337$	−1438.00	−0.232442	−0.116221	+	0.993223i	$0.537078\pi$
$338$	−2334.00	−0.375600
$339$	−4554.00	−0.729615
$340$	−120.000	−0.0191409
$341$	−2464.00	−0.391299
$342$	2232.00	0.352903
$343$	343.000	0.0539949
$344$	416.000	0.0652012
$345$	1440.00	0.224716
$346$	−3084.00	−0.479182
$347$	2604.00	0.402853	0.201427	−	0.979504i	$-0.435442\pi$
$347$	2604.00	0.402853	0.201427	+	0.979504i	$0.435442\pi$
$348$	936.000	0.144181
$349$	−9994.00	−1.53286	−0.766428	−	0.642331i	$-0.777968\pi$
$349$	−9994.00	−1.53286	−0.766428	+	0.642331i	$0.777968\pi$
$350$	−350.000	−0.0534522
$351$	−1566.00	−0.238139
$352$	352.000	0.0533002
$353$	−5646.00	−0.851293	−0.425646	−	0.904890i	$-0.639953\pi$
$353$	−5646.00	−0.851293	−0.425646	+	0.904890i	$0.639953\pi$
$354$	4968.00	0.745893
$355$	0	0
$356$	−2520.00	−0.375168
$357$	−126.000	−0.0186796
$358$	7608.00	1.12317
$359$	−1272.00	−0.187002	−0.0935008	−	0.995619i	$-0.529806\pi$
$359$	−1272.00	−0.187002	−0.0935008	+	0.995619i	$0.529806\pi$
$360$	−360.000	−0.0527046
$361$	8517.00	1.24173
$362$	7268.00	1.05524
$363$	363.000	0.0524864
$364$	−1624.00	−0.233848
$365$	−5750.00	−0.824572
$366$	−1380.00	−0.197087
$367$	3008.00	0.427837	0.213919	−	0.976851i	$-0.431377\pi$
$367$	3008.00	0.427837	0.213919	+	0.976851i	$0.431377\pi$
$368$	1536.00	0.217580
$369$	1026.00	0.144746
$370$	820.000	0.115216
$371$	882.000	0.123426
$372$	2688.00	0.374641
$373$	−4138.00	−0.574417	−0.287208	−	0.957868i	$-0.592727\pi$
$373$	−4138.00	−0.574417	−0.287208	+	0.957868i	$0.592727\pi$
$374$	−132.000	−0.0182502
$375$	375.000	0.0516398
$376$	−1152.00	−0.158005
$377$	−4524.00	−0.618031
$378$	−378.000	−0.0514344
$379$	4316.00	0.584955	0.292478	−	0.956272i	$-0.405520\pi$
$379$	4316.00	0.584955	0.292478	+	0.956272i	$0.405520\pi$
$380$	−2480.00	−0.334793
$381$	−7536.00	−1.01334
$382$	6960.00	0.932211
$383$	5232.00	0.698023	0.349011	−	0.937118i	$-0.386517\pi$
$383$	5232.00	0.698023	0.349011	+	0.937118i	$0.386517\pi$
$384$	−384.000	−0.0510310
$385$	−385.000	−0.0509647
$386$	7580.00	0.999513
$387$	−468.000	−0.0614723
$388$	−2296.00	−0.300417
$389$	−9522.00	−1.24109	−0.620546	−	0.784170i	$-0.713089\pi$
$389$	−9522.00	−1.24109	−0.620546	+	0.784170i	$0.713089\pi$
$390$	1740.00	0.225919
$391$	−576.000	−0.0745002
$392$	−392.000	−0.0505076
$393$	−4068.00	−0.522146
$394$	−5196.00	−0.664393
$395$	−5840.00	−0.743905
$396$	−396.000	−0.0502519
$397$	2966.00	0.374960	0.187480	−	0.982268i	$-0.439968\pi$
$397$	2966.00	0.374960	0.187480	+	0.982268i	$0.439968\pi$
$398$	−10288.0	−1.29571
$399$	−2604.00	−0.326724
$400$	400.000	0.0500000
$401$	−462.000	−0.0575341	−0.0287671	−	0.999586i	$-0.509158\pi$
$401$	−462.000	−0.0575341	−0.0287671	+	0.999586i	$0.509158\pi$
$402$	−2712.00	−0.336473
$403$	−12992.0	−1.60590
$404$	−1800.00	−0.221667
$405$	405.000	0.0496904
$406$	−1092.00	−0.133485
$407$	902.000	0.109854
$408$	144.000	0.0174732
$409$	10610.0	1.28272	0.641358	−	0.767242i	$-0.278371\pi$
$409$	10610.0	1.28272	0.641358	+	0.767242i	$0.278371\pi$
$410$	−1140.00	−0.137319
$411$	−2034.00	−0.244111
$412$	4832.00	0.577805
$413$	−5796.00	−0.690563
$414$	−1728.00	−0.205137
$415$	−540.000	−0.0638736
$416$	1856.00	0.218745
$417$	4740.00	0.556640
$418$	−2728.00	−0.319213
$419$	−16452.0	−1.91822	−0.959108	−	0.283039i	$-0.908657\pi$
$419$	−16452.0	−1.91822	−0.959108	+	0.283039i	$0.908657\pi$
$420$	420.000	0.0487950
$421$	−3058.00	−0.354009	−0.177005	−	0.984210i	$-0.556641\pi$
$421$	−3058.00	−0.354009	−0.177005	+	0.984210i	$0.556641\pi$
$422$	4760.00	0.549083
$423$	1296.00	0.148969
$424$	−1008.00	−0.115455
$425$	−150.000	−0.0171202
$426$	0	0
$427$	1610.00	0.182467
$428$	−2256.00	−0.254785
$429$	1914.00	0.215405
$430$	520.000	0.0583177
$431$	−192.000	−0.0214578	−0.0107289	−	0.999942i	$-0.503415\pi$
$431$	−192.000	−0.0214578	−0.0107289	+	0.999942i	$0.503415\pi$
$432$	432.000	0.0481125
$433$	−3502.00	−0.388673	−0.194336	−	0.980935i	$-0.562255\pi$
$433$	−3502.00	−0.388673	−0.194336	+	0.980935i	$0.562255\pi$
$434$	−3136.00	−0.346850
$435$	1170.00	0.128959
$436$	6584.00	0.723203
$437$	−11904.0	−1.30308
$438$	6900.00	0.752728
$439$	10136.0	1.10197	0.550985	−	0.834515i	$-0.314252\pi$
$439$	10136.0	1.10197	0.550985	+	0.834515i	$0.314252\pi$
$440$	440.000	0.0476731
$441$	441.000	0.0476190
$442$	−696.000	−0.0748990
$443$	6300.00	0.675671	0.337835	−	0.941205i	$-0.390305\pi$
$443$	6300.00	0.675671	0.337835	+	0.941205i	$0.390305\pi$
$444$	−984.000	−0.105177
$445$	−3150.00	−0.335560
$446$	−9472.00	−1.00563
$447$	−5454.00	−0.577104
$448$	448.000	0.0472456
$449$	−7038.00	−0.739741	−0.369871	−	0.929083i	$-0.620598\pi$
$449$	−7038.00	−0.739741	−0.369871	+	0.929083i	$0.620598\pi$
$450$	−450.000	−0.0471405
$451$	−1254.00	−0.130928
$452$	−6072.00	−0.631865
$453$	−3000.00	−0.311153
$454$	−4296.00	−0.444100
$455$	−2030.00	−0.209160
$456$	2976.00	0.305623
$457$	14474.0	1.48154	0.740772	−	0.671757i	$-0.234460\pi$
$457$	14474.0	1.48154	0.740772	+	0.671757i	$0.234460\pi$
$458$	−1084.00	−0.110594
$459$	−162.000	−0.0164739
$460$	1920.00	0.194610
$461$	14598.0	1.47483	0.737415	−	0.675440i	$-0.236046\pi$
$461$	14598.0	1.47483	0.737415	+	0.675440i	$0.236046\pi$
$462$	462.000	0.0465242
$463$	−6568.00	−0.659267	−0.329634	−	0.944109i	$-0.606925\pi$
$463$	−6568.00	−0.659267	−0.329634	+	0.944109i	$0.606925\pi$
$464$	1248.00	0.124864
$465$	3360.00	0.335089
$466$	14124.0	1.40404
$467$	−12516.0	−1.24020	−0.620098	−	0.784524i	$-0.712907\pi$
$467$	−12516.0	−1.24020	−0.620098	+	0.784524i	$0.712907\pi$
$468$	−2088.00	−0.206235
$469$	3164.00	0.311514
$470$	−1440.00	−0.141324
$471$	7026.00	0.687348
$472$	6624.00	0.645963
$473$	572.000	0.0556038
$474$	7008.00	0.679089
$475$	−3100.00	−0.299448
$476$	−168.000	−0.0161770
$477$	1134.00	0.108852
$478$	−5280.00	−0.505233
$479$	−9600.00	−0.915731	−0.457866	−	0.889021i	$-0.651386\pi$
$479$	−9600.00	−0.915731	−0.457866	+	0.889021i	$0.651386\pi$
$480$	−480.000	−0.0456435
$481$	4756.00	0.450842
$482$	−10132.0	−0.957469
$483$	2016.00	0.189920
$484$	484.000	0.0454545
$485$	−2870.00	−0.268701
$486$	−486.000	−0.0453609
$487$	−4768.00	−0.443652	−0.221826	−	0.975086i	$-0.571202\pi$
$487$	−4768.00	−0.443652	−0.221826	+	0.975086i	$0.571202\pi$
$488$	−1840.00	−0.170682
$489$	−3828.00	−0.354004
$490$	−490.000	−0.0451754
$491$	14364.0	1.32024	0.660120	−	0.751160i	$-0.270505\pi$
$491$	14364.0	1.32024	0.660120	+	0.751160i	$0.270505\pi$
$492$	1368.00	0.125354
$493$	−468.000	−0.0427539
$494$	−14384.0	−1.31005
$495$	−495.000	−0.0449467
$496$	3584.00	0.324448
$497$	0	0
$498$	648.000	0.0583084
$499$	−5788.00	−0.519251	−0.259626	−	0.965709i	$-0.583599\pi$
$499$	−5788.00	−0.519251	−0.259626	+	0.965709i	$0.583599\pi$
$500$	500.000	0.0447214
$501$	−8568.00	−0.764052
$502$	4920.00	0.437431
$503$	−10104.0	−0.895656	−0.447828	−	0.894120i	$-0.647802\pi$
$503$	−10104.0	−0.895656	−0.447828	+	0.894120i	$0.647802\pi$
$504$	−504.000	−0.0445435
$505$	−2250.00	−0.198265
$506$	2112.00	0.185553
$507$	3501.00	0.306676
$508$	−10048.0	−0.877575
$509$	3750.00	0.326554	0.163277	−	0.986580i	$-0.447794\pi$
$509$	3750.00	0.326554	0.163277	+	0.986580i	$0.447794\pi$
$510$	180.000	0.0156285
$511$	−8050.00	−0.696890
$512$	−512.000	−0.0441942
$513$	−3348.00	−0.288144
$514$	1500.00	0.128720
$515$	6040.00	0.516804
$516$	−624.000	−0.0532366
$517$	−1584.00	−0.134747
$518$	1148.00	0.0973750
$519$	4626.00	0.391250
$520$	2320.00	0.195651
$521$	−3894.00	−0.327446	−0.163723	−	0.986506i	$-0.552350\pi$
$521$	−3894.00	−0.327446	−0.163723	+	0.986506i	$0.552350\pi$
$522$	−1404.00	−0.117723
$523$	−1972.00	−0.164875	−0.0824374	−	0.996596i	$-0.526270\pi$
$523$	−1972.00	−0.164875	−0.0824374	+	0.996596i	$0.526270\pi$
$524$	−5424.00	−0.452192
$525$	525.000	0.0436436
$526$	−912.000	−0.0755990
$527$	−1344.00	−0.111092
$528$	−528.000	−0.0435194
$529$	−2951.00	−0.242541
$530$	−1260.00	−0.103266
$531$	−7452.00	−0.609019
$532$	−3472.00	−0.282952
$533$	−6612.00	−0.537331
$534$	3780.00	0.306323
$535$	−2820.00	−0.227886
$536$	−3616.00	−0.291394
$537$	−11412.0	−0.917065
$538$	84.0000	0.00673141
$539$	−539.000	−0.0430730
$540$	540.000	0.0430331
$541$	−5938.00	−0.471894	−0.235947	−	0.971766i	$-0.575819\pi$
$541$	−5938.00	−0.471894	−0.235947	+	0.971766i	$0.575819\pi$
$542$	10496.0	0.831811
$543$	−10902.0	−0.861601
$544$	192.000	0.0151322
$545$	8230.00	0.646852
$546$	2436.00	0.190936
$547$	−12412.0	−0.970199	−0.485099	−	0.874459i	$-0.661217\pi$
$547$	−12412.0	−0.970199	−0.485099	+	0.874459i	$0.661217\pi$
$548$	−2712.00	−0.211407
$549$	2070.00	0.160921
$550$	550.000	0.0426401
$551$	−9672.00	−0.747806
$552$	−2304.00	−0.177654
$553$	−8176.00	−0.628714
$554$	−1900.00	−0.145710
$555$	−1230.00	−0.0940731
$556$	6320.00	0.482064
$557$	−1314.00	−0.0999569	−0.0499784	−	0.998750i	$-0.515915\pi$
$557$	−1314.00	−0.0999569	−0.0499784	+	0.998750i	$0.515915\pi$
$558$	−4032.00	−0.305893
$559$	3016.00	0.228199
$560$	560.000	0.0422577
$561$	198.000	0.0149012
$562$	−10740.0	−0.806120
$563$	−4668.00	−0.349436	−0.174718	−	0.984618i	$-0.555901\pi$
$563$	−4668.00	−0.349436	−0.174718	+	0.984618i	$0.555901\pi$
$564$	1728.00	0.129011
$565$	−7590.00	−0.565157
$566$	−9496.00	−0.705206
$567$	567.000	0.0419961
$568$	0	0
$569$	−19254.0	−1.41858	−0.709288	−	0.704919i	$-0.750983\pi$
$569$	−19254.0	−1.41858	−0.709288	+	0.704919i	$0.750983\pi$
$570$	3720.00	0.273357
$571$	2828.00	0.207265	0.103632	−	0.994616i	$-0.466953\pi$
$571$	2828.00	0.207265	0.103632	+	0.994616i	$0.466953\pi$
$572$	2552.00	0.186546
$573$	−10440.0	−0.761147
$574$	−1596.00	−0.116055
$575$	2400.00	0.174064
$576$	576.000	0.0416667
$577$	7874.00	0.568109	0.284055	−	0.958808i	$-0.408320\pi$
$577$	7874.00	0.568109	0.284055	+	0.958808i	$0.408320\pi$
$578$	9754.00	0.701925
$579$	−11370.0	−0.816099
$580$	1560.00	0.111682
$581$	−756.000	−0.0539831
$582$	3444.00	0.245289
$583$	−1386.00	−0.0984601
$584$	9200.00	0.651881
$585$	−2610.00	−0.184462
$586$	−1404.00	−0.0989739
$587$	−22476.0	−1.58038	−0.790191	−	0.612861i	$-0.790018\pi$
$587$	−22476.0	−1.58038	−0.790191	+	0.612861i	$0.790018\pi$
$588$	588.000	0.0412393
$589$	−27776.0	−1.94311
$590$	8280.00	0.577766
$591$	7794.00	0.542474
$592$	−1312.00	−0.0910859
$593$	−1926.00	−0.133375	−0.0666875	−	0.997774i	$-0.521243\pi$
$593$	−1926.00	−0.133375	−0.0666875	+	0.997774i	$0.521243\pi$
$594$	594.000	0.0410305
$595$	−210.000	−0.0144692
$596$	−7272.00	−0.499786
$597$	15432.0	1.05794
$598$	11136.0	0.761513
$599$	−20112.0	−1.37188	−0.685938	−	0.727660i	$-0.740608\pi$
$599$	−20112.0	−1.37188	−0.685938	+	0.727660i	$0.740608\pi$
$600$	−600.000	−0.0408248
$601$	12722.0	0.863463	0.431731	−	0.902002i	$-0.357903\pi$
$601$	12722.0	0.863463	0.431731	+	0.902002i	$0.357903\pi$
$602$	728.000	0.0492875
$603$	4068.00	0.274729
$604$	−4000.00	−0.269466
$605$	605.000	0.0406558
$606$	2700.00	0.180990
$607$	10928.0	0.730731	0.365366	−	0.930864i	$-0.380944\pi$
$607$	10928.0	0.730731	0.365366	+	0.930864i	$0.380944\pi$
$608$	3968.00	0.264677
$609$	1638.00	0.108990
$610$	−2300.00	−0.152663
$611$	−8352.00	−0.553005
$612$	−216.000	−0.0142668
$613$	3782.00	0.249190	0.124595	−	0.992208i	$-0.460237\pi$
$613$	3782.00	0.249190	0.124595	+	0.992208i	$0.460237\pi$
$614$	13592.0	0.893369
$615$	1710.00	0.112120
$616$	616.000	0.0402911
$617$	18618.0	1.21480	0.607401	−	0.794396i	$-0.292212\pi$
$617$	18618.0	1.21480	0.607401	+	0.794396i	$0.292212\pi$
$618$	−7248.00	−0.471776
$619$	−4540.00	−0.294795	−0.147397	−	0.989077i	$-0.547090\pi$
$619$	−4540.00	−0.294795	−0.147397	+	0.989077i	$0.547090\pi$
$620$	4480.00	0.290195
$621$	2592.00	0.167493
$622$	−1584.00	−0.102110
$623$	−4410.00	−0.283600
$624$	−2784.00	−0.178604
$625$	625.000	0.0400000
$626$	−436.000	−0.0278372
$627$	4092.00	0.260636
$628$	9368.00	0.595261
$629$	492.000	0.0311881
$630$	−630.000	−0.0398410
$631$	29888.0	1.88561	0.942807	−	0.333339i	$-0.108175\pi$
$631$	29888.0	1.88561	0.942807	+	0.333339i	$0.108175\pi$
$632$	9344.00	0.588108
$633$	−7140.00	−0.448325
$634$	−6156.00	−0.385625
$635$	−12560.0	−0.784927
$636$	1512.00	0.0942684
$637$	−2842.00	−0.176773
$638$	1716.00	0.106484
$639$	0	0
$640$	−640.000	−0.0395285
$641$	−31422.0	−1.93618	−0.968092	−	0.250594i	$-0.919374\pi$
$641$	−31422.0	−1.93618	−0.968092	+	0.250594i	$0.919374\pi$
$642$	3384.00	0.208031
$643$	5372.00	0.329473	0.164736	−	0.986338i	$-0.447323\pi$
$643$	5372.00	0.329473	0.164736	+	0.986338i	$0.447323\pi$
$644$	2688.00	0.164475
$645$	−780.000	−0.0476162
$646$	−1488.00	−0.0906263
$647$	−2136.00	−0.129791	−0.0648955	−	0.997892i	$-0.520671\pi$
$647$	−2136.00	−0.129791	−0.0648955	+	0.997892i	$0.520671\pi$
$648$	−648.000	−0.0392837
$649$	9108.00	0.550879
$650$	2900.00	0.174996
$651$	4704.00	0.283202
$652$	−5104.00	−0.306577
$653$	11910.0	0.713743	0.356872	−	0.934153i	$-0.383843\pi$
$653$	11910.0	0.713743	0.356872	+	0.934153i	$0.383843\pi$
$654$	−9876.00	−0.590493
$655$	−6780.00	−0.404453
$656$	1824.00	0.108560
$657$	−10350.0	−0.614600
$658$	−2016.00	−0.119441
$659$	−8796.00	−0.519945	−0.259972	−	0.965616i	$-0.583713\pi$
$659$	−8796.00	−0.519945	−0.259972	+	0.965616i	$0.583713\pi$
$660$	−660.000	−0.0389249
$661$	−20050.0	−1.17981	−0.589905	−	0.807473i	$-0.700835\pi$
$661$	−20050.0	−1.17981	−0.589905	+	0.807473i	$0.700835\pi$
$662$	10472.0	0.614812
$663$	1044.00	0.0611548
$664$	864.000	0.0504965
$665$	−4340.00	−0.253080
$666$	1476.00	0.0858766
$667$	7488.00	0.434687
$668$	−11424.0	−0.661688
$669$	14208.0	0.821096
$670$	−4520.00	−0.260631
$671$	−2530.00	−0.145558
$672$	−672.000	−0.0385758
$673$	4322.00	0.247550	0.123775	−	0.992310i	$-0.460500\pi$
$673$	4322.00	0.247550	0.123775	+	0.992310i	$0.460500\pi$
$674$	2876.00	0.164361
$675$	675.000	0.0384900
$676$	4668.00	0.265589
$677$	−19986.0	−1.13460	−0.567300	−	0.823511i	$-0.692012\pi$
$677$	−19986.0	−1.13460	−0.567300	+	0.823511i	$0.692012\pi$
$678$	9108.00	0.515915
$679$	−4018.00	−0.227094
$680$	240.000	0.0135347
$681$	6444.00	0.362606
$682$	4928.00	0.276690
$683$	−7284.00	−0.408074	−0.204037	−	0.978963i	$-0.565406\pi$
$683$	−7284.00	−0.408074	−0.204037	+	0.978963i	$0.565406\pi$
$684$	−4464.00	−0.249540
$685$	−3390.00	−0.189088
$686$	−686.000	−0.0381802
$687$	1626.00	0.0902995
$688$	−832.000	−0.0461042
$689$	−7308.00	−0.404082
$690$	−2880.00	−0.158898
$691$	19724.0	1.08587	0.542935	−	0.839775i	$-0.317313\pi$
$691$	19724.0	1.08587	0.542935	+	0.839775i	$0.317313\pi$
$692$	6168.00	0.338833
$693$	−693.000	−0.0379869
$694$	−5208.00	−0.284860
$695$	7900.00	0.431171
$696$	−1872.00	−0.101951
$697$	−684.000	−0.0371712
$698$	19988.0	1.08389
$699$	−21186.0	−1.14639
$700$	700.000	0.0377964
$701$	−24162.0	−1.30183	−0.650917	−	0.759149i	$-0.725616\pi$
$701$	−24162.0	−1.30183	−0.650917	+	0.759149i	$0.725616\pi$
$702$	3132.00	0.168390
$703$	10168.0	0.545510
$704$	−704.000	−0.0376889
$705$	2160.00	0.115391
$706$	11292.0	0.601955
$707$	−3150.00	−0.167564
$708$	−9936.00	−0.527426
$709$	22814.0	1.20846	0.604230	−	0.796810i	$-0.293481\pi$
$709$	22814.0	1.20846	0.604230	+	0.796810i	$0.293481\pi$
$710$	0	0
$711$	−10512.0	−0.554474
$712$	5040.00	0.265284
$713$	21504.0	1.12950
$714$	252.000	0.0132085
$715$	3190.00	0.166852
$716$	−15216.0	−0.794202
$717$	7920.00	0.412521
$718$	2544.00	0.132230
$719$	12432.0	0.644834	0.322417	−	0.946598i	$-0.395505\pi$
$719$	12432.0	0.644834	0.322417	+	0.946598i	$0.395505\pi$
$720$	720.000	0.0372678
$721$	8456.00	0.436779
$722$	−17034.0	−0.878033
$723$	15198.0	0.781770
$724$	−14536.0	−0.746169
$725$	1950.00	0.0998913
$726$	−726.000	−0.0371135
$727$	29000.0	1.47944	0.739718	−	0.672917i	$-0.234959\pi$
$727$	29000.0	1.47944	0.739718	+	0.672917i	$0.234959\pi$
$728$	3248.00	0.165356
$729$	729.000	0.0370370
$730$	11500.0	0.583060
$731$	312.000	0.0157862
$732$	2760.00	0.139361
$733$	−5722.00	−0.288331	−0.144166	−	0.989554i	$-0.546050\pi$
$733$	−5722.00	−0.288331	−0.144166	+	0.989554i	$0.546050\pi$
$734$	−6016.00	−0.302527
$735$	735.000	0.0368856
$736$	−3072.00	−0.153852
$737$	−4972.00	−0.248502
$738$	−2052.00	−0.102351
$739$	−17212.0	−0.856771	−0.428385	−	0.903596i	$-0.640917\pi$
$739$	−17212.0	−0.856771	−0.428385	+	0.903596i	$0.640917\pi$
$740$	−1640.00	−0.0814697
$741$	21576.0	1.06965
$742$	−1764.00	−0.0872756
$743$	26040.0	1.28575	0.642877	−	0.765970i	$-0.277741\pi$
$743$	26040.0	1.28575	0.642877	+	0.765970i	$0.277741\pi$
$744$	−5376.00	−0.264911
$745$	−9090.00	−0.447023
$746$	8276.00	0.406174
$747$	−972.000	−0.0476086
$748$	264.000	0.0129048
$749$	−3948.00	−0.192599
$750$	−750.000	−0.0365148
$751$	24392.0	1.18519	0.592594	−	0.805501i	$-0.298104\pi$
$751$	24392.0	1.18519	0.592594	+	0.805501i	$0.298104\pi$
$752$	2304.00	0.111726
$753$	−7380.00	−0.357161
$754$	9048.00	0.437014
$755$	−5000.00	−0.241018
$756$	756.000	0.0363696
$757$	28046.0	1.34656	0.673282	−	0.739386i	$-0.264884\pi$
$757$	28046.0	1.34656	0.673282	+	0.739386i	$0.264884\pi$
$758$	−8632.00	−0.413626
$759$	−3168.00	−0.151503
$760$	4960.00	0.236734
$761$	24690.0	1.17610	0.588050	−	0.808825i	$-0.299896\pi$
$761$	24690.0	1.17610	0.588050	+	0.808825i	$0.299896\pi$
$762$	15072.0	0.716537
$763$	11522.0	0.546690
$764$	−13920.0	−0.659173
$765$	−270.000	−0.0127606
$766$	−10464.0	−0.493577
$767$	48024.0	2.26082
$768$	768.000	0.0360844
$769$	−17206.0	−0.806846	−0.403423	−	0.915014i	$-0.632180\pi$
$769$	−17206.0	−0.806846	−0.403423	+	0.915014i	$0.632180\pi$
$770$	770.000	0.0360375
$771$	−2250.00	−0.105100
$772$	−15160.0	−0.706762
$773$	−33474.0	−1.55754	−0.778768	−	0.627311i	$-0.784155\pi$
$773$	−33474.0	−1.55754	−0.778768	+	0.627311i	$0.784155\pi$
$774$	936.000	0.0434675
$775$	5600.00	0.259559
$776$	4592.00	0.212427
$777$	−1722.00	−0.0795063
$778$	19044.0	0.877584
$779$	−14136.0	−0.650160
$780$	−3480.00	−0.159749
$781$	0	0
$782$	1152.00	0.0526796
$783$	2106.00	0.0961204
$784$	784.000	0.0357143
$785$	11710.0	0.532418
$786$	8136.00	0.369213
$787$	38564.0	1.74671	0.873353	−	0.487087i	$-0.161941\pi$
$787$	38564.0	1.74671	0.873353	+	0.487087i	$0.161941\pi$
$788$	10392.0	0.469797
$789$	1368.00	0.0617263
$790$	11680.0	0.526020
$791$	−10626.0	−0.477645
$792$	792.000	0.0355335
$793$	−13340.0	−0.597374
$794$	−5932.00	−0.265137
$795$	1890.00	0.0843162
$796$	20576.0	0.916202
$797$	−15450.0	−0.686659	−0.343329	−	0.939215i	$-0.611555\pi$
$797$	−15450.0	−0.686659	−0.343329	+	0.939215i	$0.611555\pi$
$798$	5208.00	0.231029
$799$	−864.000	−0.0382555
$800$	−800.000	−0.0353553
$801$	−5670.00	−0.250112
$802$	924.000	0.0406828
$803$	12650.0	0.555926
$804$	5424.00	0.237923
$805$	3360.00	0.147111
$806$	25984.0	1.13554
$807$	−126.000	−0.00549617
$808$	3600.00	0.156742
$809$	1194.00	0.0518897	0.0259449	−	0.999663i	$-0.491741\pi$
$809$	1194.00	0.0518897	0.0259449	+	0.999663i	$0.491741\pi$
$810$	−810.000	−0.0351364
$811$	−20932.0	−0.906316	−0.453158	−	0.891430i	$-0.649703\pi$
$811$	−20932.0	−0.906316	−0.453158	+	0.891430i	$0.649703\pi$
$812$	2184.00	0.0943884
$813$	−15744.0	−0.679171
$814$	−1804.00	−0.0776783
$815$	−6380.00	−0.274211
$816$	−288.000	−0.0123554
$817$	6448.00	0.276116
$818$	−21220.0	−0.907017
$819$	−3654.00	−0.155899
$820$	2280.00	0.0970988
$821$	24390.0	1.03680	0.518402	−	0.855137i	$-0.326527\pi$
$821$	24390.0	1.03680	0.518402	+	0.855137i	$0.326527\pi$
$822$	4068.00	0.172613
$823$	−12496.0	−0.529263	−0.264631	−	0.964350i	$-0.585250\pi$
$823$	−12496.0	−0.529263	−0.264631	+	0.964350i	$0.585250\pi$
$824$	−9664.00	−0.408570
$825$	−825.000	−0.0348155
$826$	11592.0	0.488302
$827$	−22884.0	−0.962218	−0.481109	−	0.876661i	$-0.659766\pi$
$827$	−22884.0	−0.962218	−0.481109	+	0.876661i	$0.659766\pi$
$828$	3456.00	0.145054
$829$	−25306.0	−1.06021	−0.530105	−	0.847932i	$-0.677847\pi$
$829$	−25306.0	−1.06021	−0.530105	+	0.847932i	$0.677847\pi$
$830$	1080.00	0.0451655
$831$	2850.00	0.118972
$832$	−3712.00	−0.154676
$833$	−294.000	−0.0122287
$834$	−9480.00	−0.393604
$835$	−14280.0	−0.591832
$836$	5456.00	0.225717
$837$	6048.00	0.249760
$838$	32904.0	1.35638
$839$	−31560.0	−1.29866	−0.649328	−	0.760509i	$-0.724950\pi$
$839$	−31560.0	−1.29866	−0.649328	+	0.760509i	$0.724950\pi$
$840$	−840.000	−0.0345033
$841$	−18305.0	−0.750543
$842$	6116.00	0.250322
$843$	16110.0	0.658194
$844$	−9520.00	−0.388261
$845$	5835.00	0.237550
$846$	−2592.00	−0.105337
$847$	847.000	0.0343604
$848$	2016.00	0.0816388
$849$	14244.0	0.575798
$850$	300.000	0.0121058
$851$	−7872.00	−0.317096
$852$	0	0
$853$	10046.0	0.403246	0.201623	−	0.979463i	$-0.435378\pi$
$853$	10046.0	0.403246	0.201623	+	0.979463i	$0.435378\pi$
$854$	−3220.00	−0.129024
$855$	−5580.00	−0.223195
$856$	4512.00	0.180160
$857$	30786.0	1.22711	0.613553	−	0.789654i	$-0.289740\pi$
$857$	30786.0	1.22711	0.613553	+	0.789654i	$0.289740\pi$
$858$	−3828.00	−0.152314
$859$	33332.0	1.32395	0.661975	−	0.749526i	$-0.269718\pi$
$859$	33332.0	1.32395	0.661975	+	0.749526i	$0.269718\pi$
$860$	−1040.00	−0.0412369
$861$	2394.00	0.0947588
$862$	384.000	0.0151730
$863$	38424.0	1.51561	0.757803	−	0.652483i	$-0.226273\pi$
$863$	38424.0	1.51561	0.757803	+	0.652483i	$0.226273\pi$
$864$	−864.000	−0.0340207
$865$	7710.00	0.303061
$866$	7004.00	0.274833
$867$	−14631.0	−0.573120
$868$	6272.00	0.245260
$869$	12848.0	0.501540
$870$	−2340.00	−0.0911879
$871$	−26216.0	−1.01986
$872$	−13168.0	−0.511382
$873$	−5166.00	−0.200278
$874$	23808.0	0.921416
$875$	875.000	0.0338062
$876$	−13800.0	−0.532259
$877$	43502.0	1.67498	0.837490	−	0.546452i	$-0.184022\pi$
$877$	43502.0	1.67498	0.837490	+	0.546452i	$0.184022\pi$
$878$	−20272.0	−0.779211
$879$	2106.00	0.0808119
$880$	−880.000	−0.0337100
$881$	−36894.0	−1.41089	−0.705443	−	0.708767i	$-0.749252\pi$
$881$	−36894.0	−1.41089	−0.705443	+	0.708767i	$0.749252\pi$
$882$	−882.000	−0.0336718
$883$	45236.0	1.72402	0.862012	−	0.506888i	$-0.169204\pi$
$883$	45236.0	1.72402	0.862012	+	0.506888i	$0.169204\pi$
$884$	1392.00	0.0529616
$885$	−12420.0	−0.471744
$886$	−12600.0	−0.477771
$887$	27384.0	1.03660	0.518300	−	0.855199i	$-0.326565\pi$
$887$	27384.0	1.03660	0.518300	+	0.855199i	$0.326565\pi$
$888$	1968.00	0.0743713
$889$	−17584.0	−0.663384
$890$	6300.00	0.237277
$891$	−891.000	−0.0335013
$892$	18944.0	0.711090
$893$	−17856.0	−0.669125
$894$	10908.0	0.408074
$895$	−19020.0	−0.710356
$896$	−896.000	−0.0334077
$897$	−16704.0	−0.621773
$898$	14076.0	0.523076
$899$	17472.0	0.648191
$900$	900.000	0.0333333
$901$	−756.000	−0.0279534
$902$	2508.00	0.0925801
$903$	−1092.00	−0.0402431
$904$	12144.0	0.446796
$905$	−18170.0	−0.667394
$906$	6000.00	0.220018
$907$	6236.00	0.228294	0.114147	−	0.993464i	$-0.463586\pi$
$907$	6236.00	0.228294	0.114147	+	0.993464i	$0.463586\pi$
$908$	8592.00	0.314026
$909$	−4050.00	−0.147778
$910$	4060.00	0.147899
$911$	−6120.00	−0.222574	−0.111287	−	0.993788i	$-0.535497\pi$
$911$	−6120.00	−0.222574	−0.111287	+	0.993788i	$0.535497\pi$
$912$	−5952.00	−0.216108
$913$	1188.00	0.0430636
$914$	−28948.0	−1.04761
$915$	3450.00	0.124649
$916$	2168.00	0.0782017
$917$	−9492.00	−0.341825
$918$	324.000	0.0116488
$919$	−30568.0	−1.09722	−0.548610	−	0.836078i	$-0.684843\pi$
$919$	−30568.0	−1.09722	−0.548610	+	0.836078i	$0.684843\pi$
$920$	−3840.00	−0.137610
$921$	−20388.0	−0.729433
$922$	−29196.0	−1.04286
$923$	0	0
$924$	−924.000	−0.0328976
$925$	−2050.00	−0.0728687
$926$	13136.0	0.466173
$927$	10872.0	0.385203
$928$	−2496.00	−0.0882923
$929$	8082.00	0.285427	0.142714	−	0.989764i	$-0.454417\pi$
$929$	8082.00	0.285427	0.142714	+	0.989764i	$0.454417\pi$
$930$	−6720.00	−0.236944
$931$	−6076.00	−0.213891
$932$	−28248.0	−0.992805
$933$	2376.00	0.0833727
$934$	25032.0	0.876951
$935$	330.000	0.0115424
$936$	4176.00	0.145830
$937$	16994.0	0.592497	0.296249	−	0.955111i	$-0.404264\pi$
$937$	16994.0	0.592497	0.296249	+	0.955111i	$0.404264\pi$
$938$	−6328.00	−0.220273
$939$	654.000	0.0227289
$940$	2880.00	0.0999311
$941$	−37962.0	−1.31512	−0.657559	−	0.753403i	$-0.728411\pi$
$941$	−37962.0	−1.31512	−0.657559	+	0.753403i	$0.728411\pi$
$942$	−14052.0	−0.486029
$943$	10944.0	0.377928
$944$	−13248.0	−0.456764
$945$	945.000	0.0325300
$946$	−1144.00	−0.0393178
$947$	30612.0	1.05043	0.525215	−	0.850970i	$-0.323985\pi$
$947$	30612.0	1.05043	0.525215	+	0.850970i	$0.323985\pi$
$948$	−14016.0	−0.480188
$949$	66700.0	2.28153
$950$	6200.00	0.211742
$951$	9234.00	0.314861
$952$	336.000	0.0114389
$953$	13050.0	0.443579	0.221790	−	0.975095i	$-0.428810\pi$
$953$	13050.0	0.443579	0.221790	+	0.975095i	$0.428810\pi$
$954$	−2268.00	−0.0769698
$955$	−17400.0	−0.589582
$956$	10560.0	0.357254
$957$	−2574.00	−0.0869442
$958$	19200.0	0.647520
$959$	−4746.00	−0.159808
$960$	960.000	0.0322749
$961$	20385.0	0.684267
$962$	−9512.00	−0.318793
$963$	−5076.00	−0.169857
$964$	20264.0	0.677033
$965$	−18950.0	−0.632147
$966$	−4032.00	−0.134293
$967$	40904.0	1.36027	0.680136	−	0.733086i	$-0.261921\pi$
$967$	40904.0	1.36027	0.680136	+	0.733086i	$0.261921\pi$
$968$	−968.000	−0.0321412
$969$	2232.00	0.0739960
$970$	5740.00	0.190000
$971$	−8364.00	−0.276430	−0.138215	−	0.990402i	$-0.544136\pi$
$971$	−8364.00	−0.276430	−0.138215	+	0.990402i	$0.544136\pi$
$972$	972.000	0.0320750
$973$	11060.0	0.364406
$974$	9536.00	0.313710
$975$	−4350.00	−0.142884
$976$	3680.00	0.120691
$977$	38514.0	1.26118	0.630590	−	0.776116i	$-0.282813\pi$
$977$	38514.0	1.26118	0.630590	+	0.776116i	$0.282813\pi$
$978$	7656.00	0.250319
$979$	6930.00	0.226235
$980$	980.000	0.0319438
$981$	14814.0	0.482135
$982$	−28728.0	−0.933551
$983$	15768.0	0.511619	0.255809	−	0.966727i	$-0.417658\pi$
$983$	15768.0	0.511619	0.255809	+	0.966727i	$0.417658\pi$
$984$	−2736.00	−0.0886387
$985$	12990.0	0.420199
$986$	936.000	0.0302316
$987$	3024.00	0.0975228
$988$	28768.0	0.926348
$989$	−4992.00	−0.160502
$990$	990.000	0.0317821
$991$	45752.0	1.46656	0.733280	−	0.679927i	$-0.237989\pi$
$991$	45752.0	1.46656	0.733280	+	0.679927i	$0.237989\pi$
$992$	−7168.00	−0.229420
$993$	−15708.0	−0.501992
$994$	0	0
$995$	25720.0	0.819476
$996$	−1296.00	−0.0412303
$997$	7934.00	0.252028	0.126014	−	0.992028i	$-0.459782\pi$
$997$	7934.00	0.252028	0.126014	+	0.992028i	$0.459782\pi$
$998$	11576.0	0.367166
$999$	−2214.00	−0.0701180

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2310.4.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.4.a.e.1.1	✓	1	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:	\( N \)	\(=\)	\( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2310.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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