LMFDB - Embedded newform 1050.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,4,Mod(1,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1050.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1050.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:	$61.9520055060$
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	$\chi$	$=$	1050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -9.00000 q^{11} +12.0000 q^{12} +32.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -114.000 q^{17} -18.0000 q^{18} -16.0000 q^{19} +21.0000 q^{21} +18.0000 q^{22} +21.0000 q^{23} -24.0000 q^{24} -64.0000 q^{26} +27.0000 q^{27} +28.0000 q^{28} -213.000 q^{29} +50.0000 q^{31} -32.0000 q^{32} -27.0000 q^{33} +228.000 q^{34} +36.0000 q^{36} -115.000 q^{37} +32.0000 q^{38} +96.0000 q^{39} -336.000 q^{41} -42.0000 q^{42} -103.000 q^{43} -36.0000 q^{44} -42.0000 q^{46} +240.000 q^{47} +48.0000 q^{48} +49.0000 q^{49} -342.000 q^{51} +128.000 q^{52} +342.000 q^{53} -54.0000 q^{54} -56.0000 q^{56} -48.0000 q^{57} +426.000 q^{58} +336.000 q^{59} -844.000 q^{61} -100.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} +54.0000 q^{66} +167.000 q^{67} -456.000 q^{68} +63.0000 q^{69} -1017.00 q^{71} -72.0000 q^{72} -130.000 q^{73} +230.000 q^{74} -64.0000 q^{76} -63.0000 q^{77} -192.000 q^{78} +155.000 q^{79} +81.0000 q^{81} +672.000 q^{82} -858.000 q^{83} +84.0000 q^{84} +206.000 q^{86} -639.000 q^{87} +72.0000 q^{88} -84.0000 q^{89} +224.000 q^{91} +84.0000 q^{92} +150.000 q^{93} -480.000 q^{94} -96.0000 q^{96} +938.000 q^{97} -98.0000 q^{98} -81.0000 q^{99} +O(q^{100})\)

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
$5$	0	0
$5$	0	0
$6$	−6.00000	−0.408248
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	−9.00000	−0.246691	−0.123346	−	0.992364i	$-0.539362\pi$
$11$	−9.00000	−0.246691	−0.123346	+	0.992364i	$0.539362\pi$
$12$	12.0000	0.288675
$13$	32.0000	0.682708	0.341354	−	0.939935i	$-0.389115\pi$
$13$	32.0000	0.682708	0.341354	+	0.939935i	$0.389115\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−114.000	−1.62642	−0.813208	−	0.581974i	$-0.802281\pi$
$17$	−114.000	−1.62642	−0.813208	+	0.581974i	$0.802281\pi$
$18$	−18.0000	−0.235702
$19$	−16.0000	−0.193192	−0.0965961	−	0.995324i	$-0.530796\pi$
$19$	−16.0000	−0.193192	−0.0965961	+	0.995324i	$0.530796\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	18.0000	0.174437
$23$	21.0000	0.190383	0.0951914	−	0.995459i	$-0.469654\pi$
$23$	21.0000	0.190383	0.0951914	+	0.995459i	$0.469654\pi$
$24$	−24.0000	−0.204124
$25$	0	0
$26$	−64.0000	−0.482747
$27$	27.0000	0.192450
$28$	28.0000	0.188982
$29$	−213.000	−1.36390	−0.681950	−	0.731399i	$-0.738868\pi$
$29$	−213.000	−1.36390	−0.681950	+	0.731399i	$0.738868\pi$
$30$	0	0
$31$	50.0000	0.289686	0.144843	−	0.989455i	$-0.453732\pi$
$31$	50.0000	0.289686	0.144843	+	0.989455i	$0.453732\pi$
$32$	−32.0000	−0.176777
$33$	−27.0000	−0.142427
$34$	228.000	1.15005
$35$	0	0
$36$	36.0000	0.166667
$37$	−115.000	−0.510970	−0.255485	−	0.966813i	$-0.582235\pi$
$37$	−115.000	−0.510970	−0.255485	+	0.966813i	$0.582235\pi$
$38$	32.0000	0.136608
$39$	96.0000	0.394162
$40$	0	0
$41$	−336.000	−1.27986	−0.639932	−	0.768432i	$-0.721037\pi$
$41$	−336.000	−1.27986	−0.639932	+	0.768432i	$0.721037\pi$
$42$	−42.0000	−0.154303
$43$	−103.000	−0.365287	−0.182644	−	0.983179i	$-0.558465\pi$
$43$	−103.000	−0.365287	−0.182644	+	0.983179i	$0.558465\pi$
$44$	−36.0000	−0.123346
$45$	0	0
$46$	−42.0000	−0.134621
$47$	240.000	0.744843	0.372421	−	0.928064i	$-0.378528\pi$
$47$	240.000	0.744843	0.372421	+	0.928064i	$0.378528\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	0	0
$51$	−342.000	−0.939011
$52$	128.000	0.341354
$53$	342.000	0.886364	0.443182	−	0.896432i	$-0.353849\pi$
$53$	342.000	0.886364	0.443182	+	0.896432i	$0.353849\pi$
$54$	−54.0000	−0.136083
$55$	0	0
$56$	−56.0000	−0.133631
$57$	−48.0000	−0.111540
$58$	426.000	0.964423
$59$	336.000	0.741415	0.370707	−	0.928750i	$-0.379115\pi$
$59$	336.000	0.741415	0.370707	+	0.928750i	$0.379115\pi$
$60$	0	0
$61$	−844.000	−1.77153	−0.885763	−	0.464137i	$-0.846364\pi$
$61$	−844.000	−1.77153	−0.885763	+	0.464137i	$0.846364\pi$
$62$	−100.000	−0.204839
$63$	63.0000	0.125988
$64$	64.0000	0.125000
$65$	0	0
$66$	54.0000	0.100711
$67$	167.000	0.304512	0.152256	−	0.988341i	$-0.451346\pi$
$67$	167.000	0.304512	0.152256	+	0.988341i	$0.451346\pi$
$68$	−456.000	−0.813208
$69$	63.0000	0.109918
$70$	0	0
$71$	−1017.00	−1.69994	−0.849970	−	0.526832i	$-0.823380\pi$
$71$	−1017.00	−1.69994	−0.849970	+	0.526832i	$0.823380\pi$
$72$	−72.0000	−0.117851
$73$	−130.000	−0.208429	−0.104215	−	0.994555i	$-0.533233\pi$
$73$	−130.000	−0.208429	−0.104215	+	0.994555i	$0.533233\pi$
$74$	230.000	0.361310
$75$	0	0
$76$	−64.0000	−0.0965961
$77$	−63.0000	−0.0932405
$78$	−192.000	−0.278714
$79$	155.000	0.220745	0.110373	−	0.993890i	$-0.464796\pi$
$79$	155.000	0.220745	0.110373	+	0.993890i	$0.464796\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	672.000	0.905000
$83$	−858.000	−1.13467	−0.567336	−	0.823487i	$-0.692026\pi$
$83$	−858.000	−1.13467	−0.567336	+	0.823487i	$0.692026\pi$
$84$	84.0000	0.109109
$85$	0	0
$86$	206.000	0.258297
$87$	−639.000	−0.787448
$88$	72.0000	0.0872185
$89$	−84.0000	−0.100045	−0.0500224	−	0.998748i	$-0.515929\pi$
$89$	−84.0000	−0.100045	−0.0500224	+	0.998748i	$0.515929\pi$
$90$	0	0
$91$	224.000	0.258039
$92$	84.0000	0.0951914
$93$	150.000	0.167250
$94$	−480.000	−0.526683
$95$	0	0
$96$	−96.0000	−0.102062
$97$	938.000	0.981850	0.490925	−	0.871202i	$-0.336659\pi$
$97$	938.000	0.981850	0.490925	+	0.871202i	$0.336659\pi$
$98$	−98.0000	−0.101015
$99$	−81.0000	−0.0822304
$100$	0	0
$101$	−702.000	−0.691600	−0.345800	−	0.938308i	$-0.612392\pi$
$101$	−702.000	−0.691600	−0.345800	+	0.938308i	$0.612392\pi$
$102$	684.000	0.663981
$103$	56.0000	0.0535713	0.0267857	−	0.999641i	$-0.491473\pi$
$103$	56.0000	0.0535713	0.0267857	+	0.999641i	$0.491473\pi$
$104$	−256.000	−0.241374
$105$	0	0
$106$	−684.000	−0.626754
$107$	−84.0000	−0.0758933	−0.0379467	−	0.999280i	$-0.512082\pi$
$107$	−84.0000	−0.0758933	−0.0379467	+	0.999280i	$0.512082\pi$
$108$	108.000	0.0962250
$109$	−403.000	−0.354132	−0.177066	−	0.984199i	$-0.556661\pi$
$109$	−403.000	−0.354132	−0.177066	+	0.984199i	$0.556661\pi$
$110$	0	0
$111$	−345.000	−0.295009
$112$	112.000	0.0944911
$113$	1677.00	1.39610	0.698048	−	0.716051i	$-0.254052\pi$
$113$	1677.00	1.39610	0.698048	+	0.716051i	$0.254052\pi$
$114$	96.0000	0.0788704
$115$	0	0
$116$	−852.000	−0.681950
$117$	288.000	0.227569
$118$	−672.000	−0.524259
$119$	−798.000	−0.614727
$120$	0	0
$121$	−1250.00	−0.939144
$122$	1688.00	1.25266
$123$	−1008.00	−0.738929
$124$	200.000	0.144843
$125$	0	0
$126$	−126.000	−0.0890871
$127$	1955.00	1.36597	0.682985	−	0.730432i	$-0.260681\pi$
$127$	1955.00	1.36597	0.682985	+	0.730432i	$0.260681\pi$
$128$	−128.000	−0.0883883
$129$	−309.000	−0.210899
$130$	0	0
$131$	−30.0000	−0.0200085	−0.0100042	−	0.999950i	$-0.503185\pi$
$131$	−30.0000	−0.0200085	−0.0100042	+	0.999950i	$0.503185\pi$
$132$	−108.000	−0.0712136
$133$	−112.000	−0.0730198
$134$	−334.000	−0.215322
$135$	0	0
$136$	912.000	0.575025
$137$	270.000	0.168377	0.0841885	−	0.996450i	$-0.473170\pi$
$137$	270.000	0.168377	0.0841885	+	0.996450i	$0.473170\pi$
$138$	−126.000	−0.0777234
$139$	1766.00	1.07763	0.538814	−	0.842425i	$-0.318873\pi$
$139$	1766.00	1.07763	0.538814	+	0.842425i	$0.318873\pi$
$140$	0	0
$141$	720.000	0.430035
$142$	2034.00	1.20204
$143$	−288.000	−0.168418
$144$	144.000	0.0833333
$145$	0	0
$146$	260.000	0.147382
$147$	147.000	0.0824786
$148$	−460.000	−0.255485
$149$	−831.000	−0.456900	−0.228450	−	0.973556i	$-0.573366\pi$
$149$	−831.000	−0.456900	−0.228450	+	0.973556i	$0.573366\pi$
$150$	0	0
$151$	−3103.00	−1.67231	−0.836154	−	0.548494i	$-0.815201\pi$
$151$	−3103.00	−1.67231	−0.836154	+	0.548494i	$0.815201\pi$
$152$	128.000	0.0683038
$153$	−1026.00	−0.542138
$154$	126.000	0.0659310
$155$	0	0
$156$	384.000	0.197081
$157$	−100.000	−0.0508336	−0.0254168	−	0.999677i	$-0.508091\pi$
$157$	−100.000	−0.0508336	−0.0254168	+	0.999677i	$0.508091\pi$
$158$	−310.000	−0.156090
$159$	1026.00	0.511743
$160$	0	0
$161$	147.000	0.0719579
$162$	−162.000	−0.0785674
$163$	−100.000	−0.0480528	−0.0240264	−	0.999711i	$-0.507649\pi$
$163$	−100.000	−0.0480528	−0.0240264	+	0.999711i	$0.507649\pi$
$164$	−1344.00	−0.639932
$165$	0	0
$166$	1716.00	0.802334
$167$	−2604.00	−1.20661	−0.603304	−	0.797511i	$-0.706149\pi$
$167$	−2604.00	−1.20661	−0.603304	+	0.797511i	$0.706149\pi$
$168$	−168.000	−0.0771517
$169$	−1173.00	−0.533910
$170$	0	0
$171$	−144.000	−0.0643974
$172$	−412.000	−0.182644
$173$	−2676.00	−1.17603	−0.588013	−	0.808851i	$-0.700090\pi$
$173$	−2676.00	−1.17603	−0.588013	+	0.808851i	$0.700090\pi$
$174$	1278.00	0.556810
$175$	0	0
$176$	−144.000	−0.0616728
$177$	1008.00	0.428056
$178$	168.000	0.0707423
$179$	−3828.00	−1.59843	−0.799213	−	0.601048i	$-0.794750\pi$
$179$	−3828.00	−1.59843	−0.799213	+	0.601048i	$0.794750\pi$
$180$	0	0
$181$	−2158.00	−0.886204	−0.443102	−	0.896471i	$-0.646122\pi$
$181$	−2158.00	−0.886204	−0.443102	+	0.896471i	$0.646122\pi$
$182$	−448.000	−0.182461
$183$	−2532.00	−1.02279
$184$	−168.000	−0.0673105
$185$	0	0
$186$	−300.000	−0.118264
$187$	1026.00	0.401222
$188$	960.000	0.372421
$189$	189.000	0.0727393
$190$	0	0
$191$	−2700.00	−1.02285	−0.511427	−	0.859327i	$-0.670883\pi$
$191$	−2700.00	−1.02285	−0.511427	+	0.859327i	$0.670883\pi$
$192$	192.000	0.0721688
$193$	−1321.00	−0.492682	−0.246341	−	0.969183i	$-0.579228\pi$
$193$	−1321.00	−0.492682	−0.246341	+	0.969183i	$0.579228\pi$
$194$	−1876.00	−0.694273
$195$	0	0
$196$	196.000	0.0714286
$197$	−435.000	−0.157322	−0.0786611	−	0.996901i	$-0.525064\pi$
$197$	−435.000	−0.157322	−0.0786611	+	0.996901i	$0.525064\pi$
$198$	162.000	0.0581456
$199$	−1648.00	−0.587053	−0.293527	−	0.955951i	$-0.594829\pi$
$199$	−1648.00	−0.587053	−0.293527	+	0.955951i	$0.594829\pi$
$200$	0	0
$201$	501.000	0.175810
$202$	1404.00	0.489035
$203$	−1491.00	−0.515506
$204$	−1368.00	−0.469506
$205$	0	0
$206$	−112.000	−0.0378806
$207$	189.000	0.0634609
$208$	512.000	0.170677
$209$	144.000	0.0476588
$210$	0	0
$211$	4220.00	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	4220.00	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	1368.00	0.443182
$213$	−3051.00	−0.981460
$214$	168.000	0.0536647
$215$	0	0
$216$	−216.000	−0.0680414
$217$	350.000	0.109491
$218$	806.000	0.250409
$219$	−390.000	−0.120337
$220$	0	0
$221$	−3648.00	−1.11037
$222$	690.000	0.208603
$223$	3698.00	1.11048	0.555239	−	0.831691i	$-0.312627\pi$
$223$	3698.00	1.11048	0.555239	+	0.831691i	$0.312627\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−3354.00	−0.987189
$227$	−3084.00	−0.901728	−0.450864	−	0.892593i	$-0.648884\pi$
$227$	−3084.00	−0.901728	−0.450864	+	0.892593i	$0.648884\pi$
$228$	−192.000	−0.0557698
$229$	3782.00	1.09136	0.545680	−	0.837993i	$-0.316271\pi$
$229$	3782.00	1.09136	0.545680	+	0.837993i	$0.316271\pi$
$230$	0	0
$231$	−189.000	−0.0538324
$232$	1704.00	0.482212
$233$	−789.000	−0.221842	−0.110921	−	0.993829i	$-0.535380\pi$
$233$	−789.000	−0.221842	−0.110921	+	0.993829i	$0.535380\pi$
$234$	−576.000	−0.160916
$235$	0	0
$236$	1344.00	0.370707
$237$	465.000	0.127447
$238$	1596.00	0.434678
$239$	2040.00	0.552120	0.276060	−	0.961140i	$-0.410971\pi$
$239$	2040.00	0.552120	0.276060	+	0.961140i	$0.410971\pi$
$240$	0	0
$241$	−4348.00	−1.16215	−0.581077	−	0.813848i	$-0.697369\pi$
$241$	−4348.00	−1.16215	−0.581077	+	0.813848i	$0.697369\pi$
$242$	2500.00	0.664075
$243$	243.000	0.0641500
$244$	−3376.00	−0.885763
$245$	0	0
$246$	2016.00	0.522502
$247$	−512.000	−0.131894
$248$	−400.000	−0.102419
$249$	−2574.00	−0.655103
$250$	0	0
$251$	2118.00	0.532617	0.266309	−	0.963888i	$-0.414196\pi$
$251$	2118.00	0.532617	0.266309	+	0.963888i	$0.414196\pi$
$252$	252.000	0.0629941
$253$	−189.000	−0.0469657
$254$	−3910.00	−0.965887
$255$	0	0
$256$	256.000	0.0625000
$257$	−1950.00	−0.473298	−0.236649	−	0.971595i	$-0.576049\pi$
$257$	−1950.00	−0.473298	−0.236649	+	0.971595i	$0.576049\pi$
$258$	618.000	0.149128
$259$	−805.000	−0.193128
$260$	0	0
$261$	−1917.00	−0.454633
$262$	60.0000	0.0141481
$263$	−4413.00	−1.03467	−0.517333	−	0.855784i	$-0.673075\pi$
$263$	−4413.00	−1.03467	−0.517333	+	0.855784i	$0.673075\pi$
$264$	216.000	0.0503556
$265$	0	0
$266$	224.000	0.0516328
$267$	−252.000	−0.0577609
$268$	668.000	0.152256
$269$	−3582.00	−0.811890	−0.405945	−	0.913898i	$-0.633058\pi$
$269$	−3582.00	−0.811890	−0.405945	+	0.913898i	$0.633058\pi$
$270$	0	0
$271$	−6220.00	−1.39424	−0.697118	−	0.716956i	$-0.745535\pi$
$271$	−6220.00	−1.39424	−0.697118	+	0.716956i	$0.745535\pi$
$272$	−1824.00	−0.406604
$273$	672.000	0.148979
$274$	−540.000	−0.119061
$275$	0	0
$276$	252.000	0.0549588
$277$	−1870.00	−0.405622	−0.202811	−	0.979218i	$-0.565008\pi$
$277$	−1870.00	−0.405622	−0.202811	+	0.979218i	$0.565008\pi$
$278$	−3532.00	−0.761997
$279$	450.000	0.0965620
$280$	0	0
$281$	−669.000	−0.142026	−0.0710128	−	0.997475i	$-0.522623\pi$
$281$	−669.000	−0.142026	−0.0710128	+	0.997475i	$0.522623\pi$
$282$	−1440.00	−0.304081
$283$	38.0000	0.00798186	0.00399093	−	0.999992i	$-0.498730\pi$
$283$	38.0000	0.00798186	0.00399093	+	0.999992i	$0.498730\pi$
$284$	−4068.00	−0.849970
$285$	0	0
$286$	576.000	0.119089
$287$	−2352.00	−0.483743
$288$	−288.000	−0.0589256
$289$	8083.00	1.64523
$290$	0	0
$291$	2814.00	0.566871
$292$	−520.000	−0.104215
$293$	−4380.00	−0.873319	−0.436659	−	0.899627i	$-0.643838\pi$
$293$	−4380.00	−0.873319	−0.436659	+	0.899627i	$0.643838\pi$
$294$	−294.000	−0.0583212
$295$	0	0
$296$	920.000	0.180655
$297$	−243.000	−0.0474757
$298$	1662.00	0.323077
$299$	672.000	0.129976
$300$	0	0
$301$	−721.000	−0.138066
$302$	6206.00	1.18250
$303$	−2106.00	−0.399296
$304$	−256.000	−0.0482980
$305$	0	0
$306$	2052.00	0.383350
$307$	−5566.00	−1.03475	−0.517375	−	0.855759i	$-0.673091\pi$
$307$	−5566.00	−1.03475	−0.517375	+	0.855759i	$0.673091\pi$
$308$	−252.000	−0.0466202
$309$	168.000	0.0309294
$310$	0	0
$311$	7602.00	1.38608	0.693038	−	0.720901i	$-0.256272\pi$
$311$	7602.00	1.38608	0.693038	+	0.720901i	$0.256272\pi$
$312$	−768.000	−0.139357
$313$	−10960.0	−1.97922	−0.989610	−	0.143778i	$-0.954075\pi$
$313$	−10960.0	−1.97922	−0.989610	+	0.143778i	$0.954075\pi$
$314$	200.000	0.0359448
$315$	0	0
$316$	620.000	0.110373
$317$	−699.000	−0.123848	−0.0619239	−	0.998081i	$-0.519724\pi$
$317$	−699.000	−0.123848	−0.0619239	+	0.998081i	$0.519724\pi$
$318$	−2052.00	−0.361857
$319$	1917.00	0.336462
$320$	0	0
$321$	−252.000	−0.0438170
$322$	−294.000	−0.0508819
$323$	1824.00	0.314211
$324$	324.000	0.0555556
$325$	0	0
$326$	200.000	0.0339785
$327$	−1209.00	−0.204458
$328$	2688.00	0.452500
$329$	1680.00	0.281524
$330$	0	0
$331$	1031.00	0.171205	0.0856025	−	0.996329i	$-0.472718\pi$
$331$	1031.00	0.171205	0.0856025	+	0.996329i	$0.472718\pi$
$332$	−3432.00	−0.567336
$333$	−1035.00	−0.170323
$334$	5208.00	0.853201
$335$	0	0
$336$	336.000	0.0545545
$337$	794.000	0.128344	0.0641720	−	0.997939i	$-0.479559\pi$
$337$	794.000	0.128344	0.0641720	+	0.997939i	$0.479559\pi$
$338$	2346.00	0.377531
$339$	5031.00	0.806037
$340$	0	0
$341$	−450.000	−0.0714630
$342$	288.000	0.0455358
$343$	343.000	0.0539949
$344$	824.000	0.129149
$345$	0	0
$346$	5352.00	0.831576
$347$	−8349.00	−1.29164	−0.645818	−	0.763491i	$-0.723484\pi$
$347$	−8349.00	−1.29164	−0.645818	+	0.763491i	$0.723484\pi$
$348$	−2556.00	−0.393724
$349$	6128.00	0.939898	0.469949	−	0.882694i	$-0.344272\pi$
$349$	6128.00	0.939898	0.469949	+	0.882694i	$0.344272\pi$
$350$	0	0
$351$	864.000	0.131387
$352$	288.000	0.0436092
$353$	9810.00	1.47913	0.739566	−	0.673084i	$-0.235031\pi$
$353$	9810.00	1.47913	0.739566	+	0.673084i	$0.235031\pi$
$354$	−2016.00	−0.302681
$355$	0	0
$356$	−336.000	−0.0500224
$357$	−2394.00	−0.354913
$358$	7656.00	1.13026
$359$	−5697.00	−0.837538	−0.418769	−	0.908093i	$-0.637538\pi$
$359$	−5697.00	−0.837538	−0.418769	+	0.908093i	$0.637538\pi$
$360$	0	0
$361$	−6603.00	−0.962677
$362$	4316.00	0.626641
$363$	−3750.00	−0.542215
$364$	896.000	0.129020
$365$	0	0
$366$	5064.00	0.723223
$367$	2756.00	0.391995	0.195997	−	0.980604i	$-0.437206\pi$
$367$	2756.00	0.391995	0.195997	+	0.980604i	$0.437206\pi$
$368$	336.000	0.0475957
$369$	−3024.00	−0.426621
$370$	0	0
$371$	2394.00	0.335014
$372$	600.000	0.0836251
$373$	13079.0	1.81556	0.907781	−	0.419444i	$-0.137775\pi$
$373$	13079.0	1.81556	0.907781	+	0.419444i	$0.137775\pi$
$374$	−2052.00	−0.283707
$375$	0	0
$376$	−1920.00	−0.263342
$377$	−6816.00	−0.931146
$378$	−378.000	−0.0514344
$379$	13775.0	1.86695	0.933475	−	0.358642i	$-0.116760\pi$
$379$	13775.0	1.86695	0.933475	+	0.358642i	$0.116760\pi$
$380$	0	0
$381$	5865.00	0.788643
$382$	5400.00	0.723267
$383$	−552.000	−0.0736446	−0.0368223	−	0.999322i	$-0.511724\pi$
$383$	−552.000	−0.0736446	−0.0368223	+	0.999322i	$0.511724\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	2642.00	0.348379
$387$	−927.000	−0.121762
$388$	3752.00	0.490925
$389$	831.000	0.108312	0.0541560	−	0.998532i	$-0.482753\pi$
$389$	831.000	0.108312	0.0541560	+	0.998532i	$0.482753\pi$
$390$	0	0
$391$	−2394.00	−0.309641
$392$	−392.000	−0.0505076
$393$	−90.0000	−0.0115519
$394$	870.000	0.111244
$395$	0	0
$396$	−324.000	−0.0411152
$397$	3218.00	0.406818	0.203409	−	0.979094i	$-0.434798\pi$
$397$	3218.00	0.406818	0.203409	+	0.979094i	$0.434798\pi$
$398$	3296.00	0.415109
$399$	−336.000	−0.0421580
$400$	0	0
$401$	−2697.00	−0.335865	−0.167932	−	0.985799i	$-0.553709\pi$
$401$	−2697.00	−0.335865	−0.167932	+	0.985799i	$0.553709\pi$
$402$	−1002.00	−0.124316
$403$	1600.00	0.197771
$404$	−2808.00	−0.345800
$405$	0	0
$406$	2982.00	0.364518
$407$	1035.00	0.126052
$408$	2736.00	0.331991
$409$	8852.00	1.07018	0.535090	−	0.844795i	$-0.320278\pi$
$409$	8852.00	1.07018	0.535090	+	0.844795i	$0.320278\pi$
$410$	0	0
$411$	810.000	0.0972125
$412$	224.000	0.0267857
$413$	2352.00	0.280228
$414$	−378.000	−0.0448736
$415$	0	0
$416$	−1024.00	−0.120687
$417$	5298.00	0.622168
$418$	−288.000	−0.0336999
$419$	13386.0	1.56074	0.780369	−	0.625320i	$-0.215031\pi$
$419$	13386.0	1.56074	0.780369	+	0.625320i	$0.215031\pi$
$420$	0	0
$421$	2291.00	0.265217	0.132609	−	0.991168i	$-0.457665\pi$
$421$	2291.00	0.265217	0.132609	+	0.991168i	$0.457665\pi$
$422$	−8440.00	−0.973585
$423$	2160.00	0.248281
$424$	−2736.00	−0.313377
$425$	0	0
$426$	6102.00	0.693997
$427$	−5908.00	−0.669574
$428$	−336.000	−0.0379467
$429$	−864.000	−0.0972362
$430$	0	0
$431$	9840.00	1.09971	0.549856	−	0.835259i	$-0.314682\pi$
$431$	9840.00	1.09971	0.549856	+	0.835259i	$0.314682\pi$
$432$	432.000	0.0481125
$433$	13064.0	1.44992	0.724960	−	0.688790i	$-0.241858\pi$
$433$	13064.0	1.44992	0.724960	+	0.688790i	$0.241858\pi$
$434$	−700.000	−0.0774218
$435$	0	0
$436$	−1612.00	−0.177066
$437$	−336.000	−0.0367805
$438$	780.000	0.0850910
$439$	5984.00	0.650571	0.325286	−	0.945616i	$-0.394540\pi$
$439$	5984.00	0.650571	0.325286	+	0.945616i	$0.394540\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	7296.00	0.785148
$443$	−108.000	−0.0115829	−0.00579146	−	0.999983i	$-0.501843\pi$
$443$	−108.000	−0.0115829	−0.00579146	+	0.999983i	$0.501843\pi$
$444$	−1380.00	−0.147504
$445$	0	0
$446$	−7396.00	−0.785226
$447$	−2493.00	−0.263792
$448$	448.000	0.0472456
$449$	−5739.00	−0.603207	−0.301604	−	0.953433i	$-0.597522\pi$
$449$	−5739.00	−0.603207	−0.301604	+	0.953433i	$0.597522\pi$
$450$	0	0
$451$	3024.00	0.315731
$452$	6708.00	0.698048
$453$	−9309.00	−0.965508
$454$	6168.00	0.637618
$455$	0	0
$456$	384.000	0.0394352
$457$	−4885.00	−0.500023	−0.250012	−	0.968243i	$-0.580434\pi$
$457$	−4885.00	−0.500023	−0.250012	+	0.968243i	$0.580434\pi$
$458$	−7564.00	−0.771709
$459$	−3078.00	−0.313004
$460$	0	0
$461$	5412.00	0.546772	0.273386	−	0.961904i	$-0.411856\pi$
$461$	5412.00	0.546772	0.273386	+	0.961904i	$0.411856\pi$
$462$	378.000	0.0380653
$463$	7196.00	0.722303	0.361152	−	0.932507i	$-0.382384\pi$
$463$	7196.00	0.722303	0.361152	+	0.932507i	$0.382384\pi$
$464$	−3408.00	−0.340975
$465$	0	0
$466$	1578.00	0.156866
$467$	−3066.00	−0.303806	−0.151903	−	0.988395i	$-0.548540\pi$
$467$	−3066.00	−0.303806	−0.151903	+	0.988395i	$0.548540\pi$
$468$	1152.00	0.113785
$469$	1169.00	0.115095
$470$	0	0
$471$	−300.000	−0.0293488
$472$	−2688.00	−0.262130
$473$	927.000	0.0901131
$474$	−930.000	−0.0901188
$475$	0	0
$476$	−3192.00	−0.307364
$477$	3078.00	0.295455
$478$	−4080.00	−0.390408
$479$	−6744.00	−0.643301	−0.321651	−	0.946858i	$-0.604238\pi$
$479$	−6744.00	−0.643301	−0.321651	+	0.946858i	$0.604238\pi$
$480$	0	0
$481$	−3680.00	−0.348843
$482$	8696.00	0.821768
$483$	441.000	0.0415449
$484$	−5000.00	−0.469572
$485$	0	0
$486$	−486.000	−0.0453609
$487$	2729.00	0.253928	0.126964	−	0.991907i	$-0.459477\pi$
$487$	2729.00	0.253928	0.126964	+	0.991907i	$0.459477\pi$
$488$	6752.00	0.626329
$489$	−300.000	−0.0277433
$490$	0	0
$491$	20667.0	1.89957	0.949785	−	0.312904i	$-0.101302\pi$
$491$	20667.0	1.89957	0.949785	+	0.312904i	$0.101302\pi$
$492$	−4032.00	−0.369465
$493$	24282.0	2.21827
$494$	1024.00	0.0932630
$495$	0	0
$496$	800.000	0.0724215
$497$	−7119.00	−0.642517
$498$	5148.00	0.463228
$499$	4844.00	0.434564	0.217282	−	0.976109i	$-0.430281\pi$
$499$	4844.00	0.434564	0.217282	+	0.976109i	$0.430281\pi$
$500$	0	0
$501$	−7812.00	−0.696636
$502$	−4236.00	−0.376617
$503$	5118.00	0.453679	0.226839	−	0.973932i	$-0.427161\pi$
$503$	5118.00	0.453679	0.226839	+	0.973932i	$0.427161\pi$
$504$	−504.000	−0.0445435
$505$	0	0
$506$	378.000	0.0332098
$507$	−3519.00	−0.308253
$508$	7820.00	0.682985
$509$	15144.0	1.31875	0.659377	−	0.751812i	$-0.270820\pi$
$509$	15144.0	1.31875	0.659377	+	0.751812i	$0.270820\pi$
$510$	0	0
$511$	−910.000	−0.0787789
$512$	−512.000	−0.0441942
$513$	−432.000	−0.0371799
$514$	3900.00	0.334672
$515$	0	0
$516$	−1236.00	−0.105449
$517$	−2160.00	−0.183746
$518$	1610.00	0.136562
$519$	−8028.00	−0.678979
$520$	0	0
$521$	20334.0	1.70988	0.854941	−	0.518725i	$-0.173593\pi$
$521$	20334.0	1.70988	0.854941	+	0.518725i	$0.173593\pi$
$522$	3834.00	0.321474
$523$	15320.0	1.28087	0.640437	−	0.768011i	$-0.278753\pi$
$523$	15320.0	1.28087	0.640437	+	0.768011i	$0.278753\pi$
$524$	−120.000	−0.0100042
$525$	0	0
$526$	8826.00	0.731620
$527$	−5700.00	−0.471150
$528$	−432.000	−0.0356068
$529$	−11726.0	−0.963754
$530$	0	0
$531$	3024.00	0.247138
$532$	−448.000	−0.0365099
$533$	−10752.0	−0.873773
$534$	504.000	0.0408431
$535$	0	0
$536$	−1336.00	−0.107661
$537$	−11484.0	−0.922851
$538$	7164.00	0.574093
$539$	−441.000	−0.0352416
$540$	0	0
$541$	−20005.0	−1.58980	−0.794900	−	0.606740i	$-0.792477\pi$
$541$	−20005.0	−1.58980	−0.794900	+	0.606740i	$0.792477\pi$
$542$	12440.0	0.985874
$543$	−6474.00	−0.511650
$544$	3648.00	0.287512
$545$	0	0
$546$	−1344.00	−0.105344
$547$	12419.0	0.970746	0.485373	−	0.874307i	$-0.338684\pi$
$547$	12419.0	0.970746	0.485373	+	0.874307i	$0.338684\pi$
$548$	1080.00	0.0841885
$549$	−7596.00	−0.590509
$550$	0	0
$551$	3408.00	0.263495
$552$	−504.000	−0.0388617
$553$	1085.00	0.0834338
$554$	3740.00	0.286818
$555$	0	0
$556$	7064.00	0.538814
$557$	9549.00	0.726399	0.363199	−	0.931711i	$-0.381684\pi$
$557$	9549.00	0.726399	0.363199	+	0.931711i	$0.381684\pi$
$558$	−900.000	−0.0682796
$559$	−3296.00	−0.249385
$560$	0	0
$561$	3078.00	0.231646
$562$	1338.00	0.100427
$563$	2358.00	0.176515	0.0882574	−	0.996098i	$-0.471870\pi$
$563$	2358.00	0.176515	0.0882574	+	0.996098i	$0.471870\pi$
$564$	2880.00	0.215018
$565$	0	0
$566$	−76.0000	−0.00564403
$567$	567.000	0.0419961
$568$	8136.00	0.601019
$569$	19251.0	1.41835	0.709177	−	0.705030i	$-0.249067\pi$
$569$	19251.0	1.41835	0.709177	+	0.705030i	$0.249067\pi$
$570$	0	0
$571$	−18313.0	−1.34216	−0.671082	−	0.741383i	$-0.734170\pi$
$571$	−18313.0	−1.34216	−0.671082	+	0.741383i	$0.734170\pi$
$572$	−1152.00	−0.0842090
$573$	−8100.00	−0.590545
$574$	4704.00	0.342058
$575$	0	0
$576$	576.000	0.0416667
$577$	9092.00	0.655988	0.327994	−	0.944680i	$-0.393627\pi$
$577$	9092.00	0.655988	0.327994	+	0.944680i	$0.393627\pi$
$578$	−16166.0	−1.16335
$579$	−3963.00	−0.284450
$580$	0	0
$581$	−6006.00	−0.428866
$582$	−5628.00	−0.400839
$583$	−3078.00	−0.218658
$584$	1040.00	0.0736909
$585$	0	0
$586$	8760.00	0.617529
$587$	−6174.00	−0.434120	−0.217060	−	0.976158i	$-0.569647\pi$
$587$	−6174.00	−0.434120	−0.217060	+	0.976158i	$0.569647\pi$
$588$	588.000	0.0412393
$589$	−800.000	−0.0559651
$590$	0	0
$591$	−1305.00	−0.0908300
$592$	−1840.00	−0.127742
$593$	6396.00	0.442921	0.221461	−	0.975169i	$-0.428918\pi$
$593$	6396.00	0.442921	0.221461	+	0.975169i	$0.428918\pi$
$594$	486.000	0.0335704
$595$	0	0
$596$	−3324.00	−0.228450
$597$	−4944.00	−0.338935
$598$	−1344.00	−0.0919068
$599$	−16959.0	−1.15680	−0.578402	−	0.815752i	$-0.696324\pi$
$599$	−16959.0	−1.15680	−0.578402	+	0.815752i	$0.696324\pi$
$600$	0	0
$601$	23636.0	1.60421	0.802107	−	0.597180i	$-0.203712\pi$
$601$	23636.0	1.60421	0.802107	+	0.597180i	$0.203712\pi$
$602$	1442.00	0.0976271
$603$	1503.00	0.101504
$604$	−12412.0	−0.836154
$605$	0	0
$606$	4212.00	0.282345
$607$	−10876.0	−0.727254	−0.363627	−	0.931545i	$-0.618462\pi$
$607$	−10876.0	−0.727254	−0.363627	+	0.931545i	$0.618462\pi$
$608$	512.000	0.0341519
$609$	−4473.00	−0.297627
$610$	0	0
$611$	7680.00	0.508510
$612$	−4104.00	−0.271069
$613$	13841.0	0.911962	0.455981	−	0.889990i	$-0.349288\pi$
$613$	13841.0	0.911962	0.455981	+	0.889990i	$0.349288\pi$
$614$	11132.0	0.731679
$615$	0	0
$616$	504.000	0.0329655
$617$	−7881.00	−0.514225	−0.257113	−	0.966381i	$-0.582771\pi$
$617$	−7881.00	−0.514225	−0.257113	+	0.966381i	$0.582771\pi$
$618$	−336.000	−0.0218704
$619$	−9556.00	−0.620498	−0.310249	−	0.950655i	$-0.600412\pi$
$619$	−9556.00	−0.620498	−0.310249	+	0.950655i	$0.600412\pi$
$620$	0	0
$621$	567.000	0.0366392
$622$	−15204.0	−0.980104
$623$	−588.000	−0.0378134
$624$	1536.00	0.0985404
$625$	0	0
$626$	21920.0	1.39952
$627$	432.000	0.0275158
$628$	−400.000	−0.0254168
$629$	13110.0	0.831049
$630$	0	0
$631$	14903.0	0.940220	0.470110	−	0.882608i	$-0.344214\pi$
$631$	14903.0	0.940220	0.470110	+	0.882608i	$0.344214\pi$
$632$	−1240.00	−0.0780452
$633$	12660.0	0.794929
$634$	1398.00	0.0875736
$635$	0	0
$636$	4104.00	0.255871
$637$	1568.00	0.0975297
$638$	−3834.00	−0.237915
$639$	−9153.00	−0.566646
$640$	0	0
$641$	−7113.00	−0.438294	−0.219147	−	0.975692i	$-0.570327\pi$
$641$	−7113.00	−0.438294	−0.219147	+	0.975692i	$0.570327\pi$
$642$	504.000	0.0309833
$643$	−16828.0	−1.03209	−0.516043	−	0.856563i	$-0.672596\pi$
$643$	−16828.0	−1.03209	−0.516043	+	0.856563i	$0.672596\pi$
$644$	588.000	0.0359790
$645$	0	0
$646$	−3648.00	−0.222181
$647$	−8598.00	−0.522446	−0.261223	−	0.965279i	$-0.584126\pi$
$647$	−8598.00	−0.522446	−0.261223	+	0.965279i	$0.584126\pi$
$648$	−648.000	−0.0392837
$649$	−3024.00	−0.182900
$650$	0	0
$651$	1050.00	0.0632147
$652$	−400.000	−0.0240264
$653$	−7206.00	−0.431842	−0.215921	−	0.976411i	$-0.569275\pi$
$653$	−7206.00	−0.431842	−0.215921	+	0.976411i	$0.569275\pi$
$654$	2418.00	0.144574
$655$	0	0
$656$	−5376.00	−0.319966
$657$	−1170.00	−0.0694765
$658$	−3360.00	−0.199068
$659$	−30492.0	−1.80243	−0.901214	−	0.433375i	$-0.857323\pi$
$659$	−30492.0	−1.80243	−0.901214	+	0.433375i	$0.857323\pi$
$660$	0	0
$661$	−32332.0	−1.90253	−0.951263	−	0.308382i	$-0.900212\pi$
$661$	−32332.0	−1.90253	−0.951263	+	0.308382i	$0.900212\pi$
$662$	−2062.00	−0.121060
$663$	−10944.0	−0.641070
$664$	6864.00	0.401167
$665$	0	0
$666$	2070.00	0.120437
$667$	−4473.00	−0.259663
$668$	−10416.0	−0.603304
$669$	11094.0	0.641134
$670$	0	0
$671$	7596.00	0.437020
$672$	−672.000	−0.0385758
$673$	24314.0	1.39262	0.696312	−	0.717739i	$-0.254823\pi$
$673$	24314.0	1.39262	0.696312	+	0.717739i	$0.254823\pi$
$674$	−1588.00	−0.0907529
$675$	0	0
$676$	−4692.00	−0.266955
$677$	−12774.0	−0.725177	−0.362588	−	0.931949i	$-0.618107\pi$
$677$	−12774.0	−0.725177	−0.362588	+	0.931949i	$0.618107\pi$
$678$	−10062.0	−0.569954
$679$	6566.00	0.371104
$680$	0	0
$681$	−9252.00	−0.520613
$682$	900.000	0.0505319
$683$	−23901.0	−1.33901	−0.669507	−	0.742806i	$-0.733495\pi$
$683$	−23901.0	−1.33901	−0.669507	+	0.742806i	$0.733495\pi$
$684$	−576.000	−0.0321987
$685$	0	0
$686$	−686.000	−0.0381802
$687$	11346.0	0.630097
$688$	−1648.00	−0.0913218
$689$	10944.0	0.605128
$690$	0	0
$691$	−9484.00	−0.522125	−0.261062	−	0.965322i	$-0.584073\pi$
$691$	−9484.00	−0.522125	−0.261062	+	0.965322i	$0.584073\pi$
$692$	−10704.0	−0.588013
$693$	−567.000	−0.0310802
$694$	16698.0	0.913325
$695$	0	0
$696$	5112.00	0.278405
$697$	38304.0	2.08159
$698$	−12256.0	−0.664608
$699$	−2367.00	−0.128080
$700$	0	0
$701$	28698.0	1.54623	0.773116	−	0.634265i	$-0.218697\pi$
$701$	28698.0	1.54623	0.773116	+	0.634265i	$0.218697\pi$
$702$	−1728.00	−0.0929048
$703$	1840.00	0.0987154
$704$	−576.000	−0.0308364
$705$	0	0
$706$	−19620.0	−1.04590
$707$	−4914.00	−0.261400
$708$	4032.00	0.214028
$709$	16310.0	0.863942	0.431971	−	0.901887i	$-0.357818\pi$
$709$	16310.0	0.863942	0.431971	+	0.901887i	$0.357818\pi$
$710$	0	0
$711$	1395.00	0.0735817
$712$	672.000	0.0353712
$713$	1050.00	0.0551512
$714$	4788.00	0.250961
$715$	0	0
$716$	−15312.0	−0.799213
$717$	6120.00	0.318767
$718$	11394.0	0.592229
$719$	22680.0	1.17639	0.588193	−	0.808721i	$-0.299840\pi$
$719$	22680.0	1.17639	0.588193	+	0.808721i	$0.299840\pi$
$720$	0	0
$721$	392.000	0.0202480
$722$	13206.0	0.680715
$723$	−13044.0	−0.670970
$724$	−8632.00	−0.443102
$725$	0	0
$726$	7500.00	0.383404
$727$	−34594.0	−1.76481	−0.882407	−	0.470486i	$-0.844079\pi$
$727$	−34594.0	−1.76481	−0.882407	+	0.470486i	$0.844079\pi$
$728$	−1792.00	−0.0912307
$729$	729.000	0.0370370
$730$	0	0
$731$	11742.0	0.594109
$732$	−10128.0	−0.511396
$733$	−20068.0	−1.01123	−0.505613	−	0.862760i	$-0.668734\pi$
$733$	−20068.0	−1.01123	−0.505613	+	0.862760i	$0.668734\pi$
$734$	−5512.00	−0.277182
$735$	0	0
$736$	−672.000	−0.0336552
$737$	−1503.00	−0.0751204
$738$	6048.00	0.301667
$739$	−19663.0	−0.978776	−0.489388	−	0.872066i	$-0.662780\pi$
$739$	−19663.0	−0.978776	−0.489388	+	0.872066i	$0.662780\pi$
$740$	0	0
$741$	−1536.00	−0.0761489
$742$	−4788.00	−0.236891
$743$	18768.0	0.926691	0.463345	−	0.886178i	$-0.346649\pi$
$743$	18768.0	0.926691	0.463345	+	0.886178i	$0.346649\pi$
$744$	−1200.00	−0.0591319
$745$	0	0
$746$	−26158.0	−1.28380
$747$	−7722.00	−0.378224
$748$	4104.00	0.200611
$749$	−588.000	−0.0286850
$750$	0	0
$751$	7808.00	0.379385	0.189692	−	0.981844i	$-0.439251\pi$
$751$	7808.00	0.379385	0.189692	+	0.981844i	$0.439251\pi$
$752$	3840.00	0.186211
$753$	6354.00	0.307507
$754$	13632.0	0.658419
$755$	0	0
$756$	756.000	0.0363696
$757$	30413.0	1.46021	0.730105	−	0.683335i	$-0.239471\pi$
$757$	30413.0	1.46021	0.730105	+	0.683335i	$0.239471\pi$
$758$	−27550.0	−1.32013
$759$	−567.000	−0.0271157
$760$	0	0
$761$	−37212.0	−1.77258	−0.886290	−	0.463130i	$-0.846726\pi$
$761$	−37212.0	−1.77258	−0.886290	+	0.463130i	$0.846726\pi$
$762$	−11730.0	−0.557655
$763$	−2821.00	−0.133849
$764$	−10800.0	−0.511427
$765$	0	0
$766$	1104.00	0.0520746
$767$	10752.0	0.506170
$768$	768.000	0.0360844
$769$	−21256.0	−0.996763	−0.498382	−	0.866958i	$-0.666072\pi$
$769$	−21256.0	−0.996763	−0.498382	+	0.866958i	$0.666072\pi$
$770$	0	0
$771$	−5850.00	−0.273259
$772$	−5284.00	−0.246341
$773$	−114.000	−0.00530439	−0.00265220	−	0.999996i	$-0.500844\pi$
$773$	−114.000	−0.00530439	−0.00265220	+	0.999996i	$0.500844\pi$
$774$	1854.00	0.0860990
$775$	0	0
$776$	−7504.00	−0.347136
$777$	−2415.00	−0.111503
$778$	−1662.00	−0.0765882
$779$	5376.00	0.247260
$780$	0	0
$781$	9153.00	0.419360
$782$	4788.00	0.218950
$783$	−5751.00	−0.262483
$784$	784.000	0.0357143
$785$	0	0
$786$	180.000	0.00816843
$787$	11162.0	0.505568	0.252784	−	0.967523i	$-0.418654\pi$
$787$	11162.0	0.505568	0.252784	+	0.967523i	$0.418654\pi$
$788$	−1740.00	−0.0786611
$789$	−13239.0	−0.597365
$790$	0	0
$791$	11739.0	0.527675
$792$	648.000	0.0290728
$793$	−27008.0	−1.20944
$794$	−6436.00	−0.287664
$795$	0	0
$796$	−6592.00	−0.293527
$797$	198.000	0.00879990	0.00439995	−	0.999990i	$-0.498599\pi$
$797$	198.000	0.00879990	0.00439995	+	0.999990i	$0.498599\pi$
$798$	672.000	0.0298102
$799$	−27360.0	−1.21142
$800$	0	0
$801$	−756.000	−0.0333482
$802$	5394.00	0.237492
$803$	1170.00	0.0514177
$804$	2004.00	0.0879050
$805$	0	0
$806$	−3200.00	−0.139845
$807$	−10746.0	−0.468745
$808$	5616.00	0.244518
$809$	1665.00	0.0723588	0.0361794	−	0.999345i	$-0.488481\pi$
$809$	1665.00	0.0723588	0.0361794	+	0.999345i	$0.488481\pi$
$810$	0	0
$811$	−17794.0	−0.770446	−0.385223	−	0.922823i	$-0.625876\pi$
$811$	−17794.0	−0.770446	−0.385223	+	0.922823i	$0.625876\pi$
$812$	−5964.00	−0.257753
$813$	−18660.0	−0.804963
$814$	−2070.00	−0.0891320
$815$	0	0
$816$	−5472.00	−0.234753
$817$	1648.00	0.0705707
$818$	−17704.0	−0.756731
$819$	2016.00	0.0860131
$820$	0	0
$821$	24498.0	1.04140	0.520698	−	0.853741i	$-0.325672\pi$
$821$	24498.0	1.04140	0.520698	+	0.853741i	$0.325672\pi$
$822$	−1620.00	−0.0687396
$823$	−4177.00	−0.176915	−0.0884575	−	0.996080i	$-0.528194\pi$
$823$	−4177.00	−0.176915	−0.0884575	+	0.996080i	$0.528194\pi$
$824$	−448.000	−0.0189403
$825$	0	0
$826$	−4704.00	−0.198151
$827$	38247.0	1.60820	0.804098	−	0.594496i	$-0.202649\pi$
$827$	38247.0	1.60820	0.804098	+	0.594496i	$0.202649\pi$
$828$	756.000	0.0317305
$829$	−18814.0	−0.788223	−0.394112	−	0.919063i	$-0.628948\pi$
$829$	−18814.0	−0.788223	−0.394112	+	0.919063i	$0.628948\pi$
$830$	0	0
$831$	−5610.00	−0.234186
$832$	2048.00	0.0853385
$833$	−5586.00	−0.232345
$834$	−10596.0	−0.439939
$835$	0	0
$836$	576.000	0.0238294
$837$	1350.00	0.0557501
$838$	−26772.0	−1.10361
$839$	19974.0	0.821906	0.410953	−	0.911657i	$-0.365196\pi$
$839$	19974.0	0.821906	0.410953	+	0.911657i	$0.365196\pi$
$840$	0	0
$841$	20980.0	0.860224
$842$	−4582.00	−0.187537
$843$	−2007.00	−0.0819985
$844$	16880.0	0.688428
$845$	0	0
$846$	−4320.00	−0.175561
$847$	−8750.00	−0.354963
$848$	5472.00	0.221591
$849$	114.000	0.00460833
$850$	0	0
$851$	−2415.00	−0.0972798
$852$	−12204.0	−0.490730
$853$	11828.0	0.474775	0.237387	−	0.971415i	$-0.423709\pi$
$853$	11828.0	0.474775	0.237387	+	0.971415i	$0.423709\pi$
$854$	11816.0	0.473460
$855$	0	0
$856$	672.000	0.0268323
$857$	34476.0	1.37419	0.687093	−	0.726569i	$-0.258886\pi$
$857$	34476.0	1.37419	0.687093	+	0.726569i	$0.258886\pi$
$858$	1728.00	0.0687563
$859$	−15310.0	−0.608115	−0.304057	−	0.952654i	$-0.598341\pi$
$859$	−15310.0	−0.608115	−0.304057	+	0.952654i	$0.598341\pi$
$860$	0	0
$861$	−7056.00	−0.279289
$862$	−19680.0	−0.777614
$863$	−35589.0	−1.40378	−0.701891	−	0.712284i	$-0.747661\pi$
$863$	−35589.0	−1.40378	−0.701891	+	0.712284i	$0.747661\pi$
$864$	−864.000	−0.0340207
$865$	0	0
$866$	−26128.0	−1.02525
$867$	24249.0	0.949872
$868$	1400.00	0.0547455
$869$	−1395.00	−0.0544559
$870$	0	0
$871$	5344.00	0.207893
$872$	3224.00	0.125205
$873$	8442.00	0.327283
$874$	672.000	0.0260077
$875$	0	0
$876$	−1560.00	−0.0601684
$877$	42734.0	1.64541	0.822705	−	0.568468i	$-0.192464\pi$
$877$	42734.0	1.64541	0.822705	+	0.568468i	$0.192464\pi$
$878$	−11968.0	−0.460023
$879$	−13140.0	−0.504211
$880$	0	0
$881$	−5502.00	−0.210405	−0.105203	−	0.994451i	$-0.533549\pi$
$881$	−5502.00	−0.210405	−0.105203	+	0.994451i	$0.533549\pi$
$882$	−882.000	−0.0336718
$883$	−37465.0	−1.42786	−0.713928	−	0.700219i	$-0.753086\pi$
$883$	−37465.0	−1.42786	−0.713928	+	0.700219i	$0.753086\pi$
$884$	−14592.0	−0.555183
$885$	0	0
$886$	216.000	0.00819036
$887$	12348.0	0.467424	0.233712	−	0.972306i	$-0.424913\pi$
$887$	12348.0	0.467424	0.233712	+	0.972306i	$0.424913\pi$
$888$	2760.00	0.104301
$889$	13685.0	0.516288
$890$	0	0
$891$	−729.000	−0.0274101
$892$	14792.0	0.555239
$893$	−3840.00	−0.143898
$894$	4986.00	0.186529
$895$	0	0
$896$	−896.000	−0.0334077
$897$	2016.00	0.0750416
$898$	11478.0	0.426532
$899$	−10650.0	−0.395103
$900$	0	0
$901$	−38988.0	−1.44160
$902$	−6048.00	−0.223255
$903$	−2163.00	−0.0797122
$904$	−13416.0	−0.493595
$905$	0	0
$906$	18618.0	0.682717
$907$	−7504.00	−0.274715	−0.137357	−	0.990522i	$-0.543861\pi$
$907$	−7504.00	−0.274715	−0.137357	+	0.990522i	$0.543861\pi$
$908$	−12336.0	−0.450864
$909$	−6318.00	−0.230533
$910$	0	0
$911$	−3075.00	−0.111832	−0.0559162	−	0.998435i	$-0.517808\pi$
$911$	−3075.00	−0.111832	−0.0559162	+	0.998435i	$0.517808\pi$
$912$	−768.000	−0.0278849
$913$	7722.00	0.279913
$914$	9770.00	0.353570
$915$	0	0
$916$	15128.0	0.545680
$917$	−210.000	−0.00756250
$918$	6156.00	0.221327
$919$	16295.0	0.584899	0.292450	−	0.956281i	$-0.405530\pi$
$919$	16295.0	0.584899	0.292450	+	0.956281i	$0.405530\pi$
$920$	0	0
$921$	−16698.0	−0.597413
$922$	−10824.0	−0.386626
$923$	−32544.0	−1.16056
$924$	−756.000	−0.0269162
$925$	0	0
$926$	−14392.0	−0.510746
$927$	504.000	0.0178571
$928$	6816.00	0.241106
$929$	−3798.00	−0.134132	−0.0670658	−	0.997749i	$-0.521364\pi$
$929$	−3798.00	−0.134132	−0.0670658	+	0.997749i	$0.521364\pi$
$930$	0	0
$931$	−784.000	−0.0275989
$932$	−3156.00	−0.110921
$933$	22806.0	0.800252
$934$	6132.00	0.214824
$935$	0	0
$936$	−2304.00	−0.0804579
$937$	−18556.0	−0.646956	−0.323478	−	0.946236i	$-0.604852\pi$
$937$	−18556.0	−0.646956	−0.323478	+	0.946236i	$0.604852\pi$
$938$	−2338.00	−0.0813842
$939$	−32880.0	−1.14270
$940$	0	0
$941$	−7266.00	−0.251716	−0.125858	−	0.992048i	$-0.540168\pi$
$941$	−7266.00	−0.251716	−0.125858	+	0.992048i	$0.540168\pi$
$942$	600.000	0.0207527
$943$	−7056.00	−0.243664
$944$	5376.00	0.185354
$945$	0	0
$946$	−1854.00	−0.0637196
$947$	−18348.0	−0.629599	−0.314799	−	0.949158i	$-0.601937\pi$
$947$	−18348.0	−0.629599	−0.314799	+	0.949158i	$0.601937\pi$
$948$	1860.00	0.0637236
$949$	−4160.00	−0.142296
$950$	0	0
$951$	−2097.00	−0.0715036
$952$	6384.00	0.217339
$953$	22437.0	0.762650	0.381325	−	0.924441i	$-0.375468\pi$
$953$	22437.0	0.762650	0.381325	+	0.924441i	$0.375468\pi$
$954$	−6156.00	−0.208918
$955$	0	0
$956$	8160.00	0.276060
$957$	5751.00	0.194256
$958$	13488.0	0.454883
$959$	1890.00	0.0636405
$960$	0	0
$961$	−27291.0	−0.916082
$962$	7360.00	0.246669
$963$	−756.000	−0.0252978
$964$	−17392.0	−0.581077
$965$	0	0
$966$	−882.000	−0.0293767
$967$	44624.0	1.48398	0.741991	−	0.670410i	$-0.233882\pi$
$967$	44624.0	1.48398	0.741991	+	0.670410i	$0.233882\pi$
$968$	10000.0	0.332037
$969$	5472.00	0.181410
$970$	0	0
$971$	−28692.0	−0.948270	−0.474135	−	0.880452i	$-0.657239\pi$
$971$	−28692.0	−0.948270	−0.474135	+	0.880452i	$0.657239\pi$
$972$	972.000	0.0320750
$973$	12362.0	0.407305
$974$	−5458.00	−0.179554
$975$	0	0
$976$	−13504.0	−0.442882
$977$	10239.0	0.335286	0.167643	−	0.985848i	$-0.446384\pi$
$977$	10239.0	0.335286	0.167643	+	0.985848i	$0.446384\pi$
$978$	600.000	0.0196175
$979$	756.000	0.0246801
$980$	0	0
$981$	−3627.00	−0.118044
$982$	−41334.0	−1.34320
$983$	36162.0	1.17334	0.586668	−	0.809828i	$-0.300439\pi$
$983$	36162.0	1.17334	0.586668	+	0.809828i	$0.300439\pi$
$984$	8064.00	0.261251
$985$	0	0
$986$	−48564.0	−1.56855
$987$	5040.00	0.162538
$988$	−2048.00	−0.0659469
$989$	−2163.00	−0.0695444
$990$	0	0
$991$	25319.0	0.811589	0.405794	−	0.913964i	$-0.366995\pi$
$991$	25319.0	0.811589	0.405794	+	0.913964i	$0.366995\pi$
$992$	−1600.00	−0.0512097
$993$	3093.00	0.0988453
$994$	14238.0	0.454328
$995$	0	0
$996$	−10296.0	−0.327551
$997$	2792.00	0.0886896	0.0443448	−	0.999016i	$-0.485880\pi$
$997$	2792.00	0.0886896	0.0443448	+	0.999016i	$0.485880\pi$
$998$	−9688.00	−0.307283
$999$	−3105.00	−0.0983362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.4.a.j.1.1	✓	1
5.2	odd	4		1050.4.g.b.799.1		2
5.3	odd	4		1050.4.g.b.799.2		2
5.4	even	2		1050.4.a.m.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.4.a.j.1.1	✓	1	1.1	even	1	trivial
1050.4.a.m.1.1	yes	1	5.4	even	2
1050.4.g.b.799.1		2	5.2	odd	4
1050.4.g.b.799.2		2	5.3	odd	4

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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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