LMFDB - Embedded newform 825.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.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:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+5.00000 q^{2} -3.00000 q^{3} +17.0000 q^{4} -15.0000 q^{6} +32.0000 q^{7} +45.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+5.00000 q^{2} -3.00000 q^{3} +17.0000 q^{4} -15.0000 q^{6} +32.0000 q^{7} +45.0000 q^{8} +9.00000 q^{9} -11.0000 q^{11} -51.0000 q^{12} +38.0000 q^{13} +160.000 q^{14} +89.0000 q^{16} +2.00000 q^{17} +45.0000 q^{18} +72.0000 q^{19} -96.0000 q^{21} -55.0000 q^{22} -68.0000 q^{23} -135.000 q^{24} +190.000 q^{26} -27.0000 q^{27} +544.000 q^{28} -54.0000 q^{29} -152.000 q^{31} +85.0000 q^{32} +33.0000 q^{33} +10.0000 q^{34} +153.000 q^{36} -174.000 q^{37} +360.000 q^{38} -114.000 q^{39} +94.0000 q^{41} -480.000 q^{42} +528.000 q^{43} -187.000 q^{44} -340.000 q^{46} +340.000 q^{47} -267.000 q^{48} +681.000 q^{49} -6.00000 q^{51} +646.000 q^{52} +438.000 q^{53} -135.000 q^{54} +1440.00 q^{56} -216.000 q^{57} -270.000 q^{58} +20.0000 q^{59} +570.000 q^{61} -760.000 q^{62} +288.000 q^{63} -287.000 q^{64} +165.000 q^{66} +460.000 q^{67} +34.0000 q^{68} +204.000 q^{69} -1092.00 q^{71} +405.000 q^{72} -562.000 q^{73} -870.000 q^{74} +1224.00 q^{76} -352.000 q^{77} -570.000 q^{78} -16.0000 q^{79} +81.0000 q^{81} +470.000 q^{82} -372.000 q^{83} -1632.00 q^{84} +2640.00 q^{86} +162.000 q^{87} -495.000 q^{88} -966.000 q^{89} +1216.00 q^{91} -1156.00 q^{92} +456.000 q^{93} +1700.00 q^{94} -255.000 q^{96} +526.000 q^{97} +3405.00 q^{98} -99.0000 q^{99} +O(q^{100})\)

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 $

$2$	5.00000	1.76777	0.883883	−	0.467707i	$-0.154920\pi$
$2$	5.00000	1.76777	0.883883	+	0.467707i	$0.154920\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	17.0000	2.12500
$5$	0	0
$5$	0	0
$6$	−15.0000	−1.02062
$7$	32.0000	1.72784	0.863919	−	0.503631i	$-0.168003\pi$
$7$	32.0000	1.72784	0.863919	+	0.503631i	$0.168003\pi$
$8$	45.0000	1.98874
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−51.0000	−1.22687
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	160.000	3.05441
$15$	0	0
$16$	89.0000	1.39062
$17$	2.00000	0.0285336	0.0142668	−	0.999898i	$-0.495459\pi$
$17$	2.00000	0.0285336	0.0142668	+	0.999898i	$0.495459\pi$
$18$	45.0000	0.589256
$19$	72.0000	0.869365	0.434682	−	0.900584i	$-0.356861\pi$
$19$	72.0000	0.869365	0.434682	+	0.900584i	$0.356861\pi$
$20$	0	0
$21$	−96.0000	−0.997567
$22$	−55.0000	−0.533002
$23$	−68.0000	−0.616477	−0.308239	−	0.951309i	$-0.599740\pi$
$23$	−68.0000	−0.616477	−0.308239	+	0.951309i	$0.599740\pi$
$24$	−135.000	−1.14820
$25$	0	0
$26$	190.000	1.43316
$27$	−27.0000	−0.192450
$28$	544.000	3.67165
$29$	−54.0000	−0.345778	−0.172889	−	0.984941i	$-0.555310\pi$
$29$	−54.0000	−0.345778	−0.172889	+	0.984941i	$0.555310\pi$
$30$	0	0
$31$	−152.000	−0.880645	−0.440323	−	0.897840i	$-0.645136\pi$
$31$	−152.000	−0.880645	−0.440323	+	0.897840i	$0.645136\pi$
$32$	85.0000	0.469563
$33$	33.0000	0.174078
$34$	10.0000	0.0504408
$35$	0	0
$36$	153.000	0.708333
$37$	−174.000	−0.773120	−0.386560	−	0.922264i	$-0.626337\pi$
$37$	−174.000	−0.773120	−0.386560	+	0.922264i	$0.626337\pi$
$38$	360.000	1.53683
$39$	−114.000	−0.468067
$40$	0	0
$41$	94.0000	0.358057	0.179028	−	0.983844i	$-0.442705\pi$
$41$	94.0000	0.358057	0.179028	+	0.983844i	$0.442705\pi$
$42$	−480.000	−1.76347
$43$	528.000	1.87254	0.936270	−	0.351280i	$-0.114254\pi$
$43$	528.000	1.87254	0.936270	+	0.351280i	$0.114254\pi$
$44$	−187.000	−0.640712
$45$	0	0
$46$	−340.000	−1.08979
$47$	340.000	1.05519	0.527597	−	0.849495i	$-0.323093\pi$
$47$	340.000	1.05519	0.527597	+	0.849495i	$0.323093\pi$
$48$	−267.000	−0.802878
$49$	681.000	1.98542
$50$	0	0
$51$	−6.00000	−0.0164739
$52$	646.000	1.72277
$53$	438.000	1.13517	0.567584	−	0.823315i	$-0.307878\pi$
$53$	438.000	1.13517	0.567584	+	0.823315i	$0.307878\pi$
$54$	−135.000	−0.340207
$55$	0	0
$56$	1440.00	3.43622
$57$	−216.000	−0.501928
$58$	−270.000	−0.611254
$59$	20.0000	0.0441318	0.0220659	−	0.999757i	$-0.492976\pi$
$59$	20.0000	0.0441318	0.0220659	+	0.999757i	$0.492976\pi$
$60$	0	0
$61$	570.000	1.19641	0.598205	−	0.801343i	$-0.295881\pi$
$61$	570.000	1.19641	0.598205	+	0.801343i	$0.295881\pi$
$62$	−760.000	−1.55678
$63$	288.000	0.575946
$64$	−287.000	−0.560547
$65$	0	0
$66$	165.000	0.307729
$67$	460.000	0.838775	0.419388	−	0.907807i	$-0.362245\pi$
$67$	460.000	0.838775	0.419388	+	0.907807i	$0.362245\pi$
$68$	34.0000	0.0606339
$69$	204.000	0.355923
$70$	0	0
$71$	−1092.00	−1.82530	−0.912652	−	0.408738i	$-0.865969\pi$
$71$	−1092.00	−1.82530	−0.912652	+	0.408738i	$0.865969\pi$
$72$	405.000	0.662913
$73$	−562.000	−0.901057	−0.450528	−	0.892762i	$-0.648764\pi$
$73$	−562.000	−0.901057	−0.450528	+	0.892762i	$0.648764\pi$
$74$	−870.000	−1.36670
$75$	0	0
$76$	1224.00	1.84740
$77$	−352.000	−0.520963
$78$	−570.000	−0.827433
$79$	−16.0000	−0.0227866	−0.0113933	−	0.999935i	$-0.503627\pi$
$79$	−16.0000	−0.0227866	−0.0113933	+	0.999935i	$0.503627\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	470.000	0.632961
$83$	−372.000	−0.491955	−0.245978	−	0.969275i	$-0.579109\pi$
$83$	−372.000	−0.491955	−0.245978	+	0.969275i	$0.579109\pi$
$84$	−1632.00	−2.11983
$85$	0	0
$86$	2640.00	3.31022
$87$	162.000	0.199635
$88$	−495.000	−0.599627
$89$	−966.000	−1.15051	−0.575257	−	0.817973i	$-0.695098\pi$
$89$	−966.000	−1.15051	−0.575257	+	0.817973i	$0.695098\pi$
$90$	0	0
$91$	1216.00	1.40079
$92$	−1156.00	−1.31001
$93$	456.000	0.508441
$94$	1700.00	1.86534
$95$	0	0
$96$	−255.000	−0.271102
$97$	526.000	0.550590	0.275295	−	0.961360i	$-0.411225\pi$
$97$	526.000	0.550590	0.275295	+	0.961360i	$0.411225\pi$
$98$	3405.00	3.50976
$99$	−99.0000	−0.100504
$100$	0	0
$101$	50.0000	0.0492593	0.0246296	−	0.999697i	$-0.492159\pi$
$101$	50.0000	0.0492593	0.0246296	+	0.999697i	$0.492159\pi$
$102$	−30.0000	−0.0291220
$103$	−944.000	−0.903059	−0.451530	−	0.892256i	$-0.649121\pi$
$103$	−944.000	−0.903059	−0.451530	+	0.892256i	$0.649121\pi$
$104$	1710.00	1.61230
$105$	0	0
$106$	2190.00	2.00671
$107$	−468.000	−0.422834	−0.211417	−	0.977396i	$-0.567808\pi$
$107$	−468.000	−0.422834	−0.211417	+	0.977396i	$0.567808\pi$
$108$	−459.000	−0.408956
$109$	154.000	0.135326	0.0676630	−	0.997708i	$-0.478446\pi$
$109$	154.000	0.135326	0.0676630	+	0.997708i	$0.478446\pi$
$110$	0	0
$111$	522.000	0.446361
$112$	2848.00	2.40277
$113$	54.0000	0.0449548	0.0224774	−	0.999747i	$-0.492845\pi$
$113$	54.0000	0.0449548	0.0224774	+	0.999747i	$0.492845\pi$
$114$	−1080.00	−0.887292
$115$	0	0
$116$	−918.000	−0.734777
$117$	342.000	0.270239
$118$	100.000	0.0780148
$119$	64.0000	0.0493014
$120$	0	0
$121$	121.000	0.0909091
$122$	2850.00	2.11497
$123$	−282.000	−0.206724
$124$	−2584.00	−1.87137
$125$	0	0
$126$	1440.00	1.01814
$127$	2224.00	1.55392	0.776961	−	0.629549i	$-0.216760\pi$
$127$	2224.00	1.55392	0.776961	+	0.629549i	$0.216760\pi$
$128$	−2115.00	−1.46048
$129$	−1584.00	−1.08111
$130$	0	0
$131$	−2772.00	−1.84878	−0.924392	−	0.381443i	$-0.875427\pi$
$131$	−2772.00	−1.84878	−0.924392	+	0.381443i	$0.875427\pi$
$132$	561.000	0.369915
$133$	2304.00	1.50212
$134$	2300.00	1.48276
$135$	0	0
$136$	90.0000	0.0567459
$137$	−1130.00	−0.704689	−0.352345	−	0.935870i	$-0.614615\pi$
$137$	−1130.00	−0.704689	−0.352345	+	0.935870i	$0.614615\pi$
$138$	1020.00	0.629190
$139$	−1616.00	−0.986096	−0.493048	−	0.870002i	$-0.664117\pi$
$139$	−1616.00	−0.986096	−0.493048	+	0.870002i	$0.664117\pi$
$140$	0	0
$141$	−1020.00	−0.609216
$142$	−5460.00	−3.22671
$143$	−418.000	−0.244440
$144$	801.000	0.463542
$145$	0	0
$146$	−2810.00	−1.59286
$147$	−2043.00	−1.14628
$148$	−2958.00	−1.64288
$149$	2066.00	1.13593	0.567964	−	0.823053i	$-0.307731\pi$
$149$	2066.00	1.13593	0.567964	+	0.823053i	$0.307731\pi$
$150$	0	0
$151$	248.000	0.133655	0.0668277	−	0.997765i	$-0.478712\pi$
$151$	248.000	0.133655	0.0668277	+	0.997765i	$0.478712\pi$
$152$	3240.00	1.72894
$153$	18.0000	0.00951120
$154$	−1760.00	−0.920941
$155$	0	0
$156$	−1938.00	−0.994642
$157$	−2366.00	−1.20272	−0.601361	−	0.798977i	$-0.705375\pi$
$157$	−2366.00	−1.20272	−0.601361	+	0.798977i	$0.705375\pi$
$158$	−80.0000	−0.0402814
$159$	−1314.00	−0.655390
$160$	0	0
$161$	−2176.00	−1.06517
$162$	405.000	0.196419
$163$	284.000	0.136470	0.0682350	−	0.997669i	$-0.478263\pi$
$163$	284.000	0.136470	0.0682350	+	0.997669i	$0.478263\pi$
$164$	1598.00	0.760871
$165$	0	0
$166$	−1860.00	−0.869663
$167$	−600.000	−0.278020	−0.139010	−	0.990291i	$-0.544392\pi$
$167$	−600.000	−0.278020	−0.139010	+	0.990291i	$0.544392\pi$
$168$	−4320.00	−1.98390
$169$	−753.000	−0.342740
$170$	0	0
$171$	648.000	0.289788
$172$	8976.00	3.97915
$173$	−138.000	−0.0606471	−0.0303235	−	0.999540i	$-0.509654\pi$
$173$	−138.000	−0.0606471	−0.0303235	+	0.999540i	$0.509654\pi$
$174$	810.000	0.352908
$175$	0	0
$176$	−979.000	−0.419289
$177$	−60.0000	−0.0254795
$178$	−4830.00	−2.03384
$179$	3972.00	1.65855	0.829277	−	0.558838i	$-0.188752\pi$
$179$	3972.00	1.65855	0.829277	+	0.558838i	$0.188752\pi$
$180$	0	0
$181$	2230.00	0.915771	0.457886	−	0.889011i	$-0.348607\pi$
$181$	2230.00	0.915771	0.457886	+	0.889011i	$0.348607\pi$
$182$	6080.00	2.47626
$183$	−1710.00	−0.690748
$184$	−3060.00	−1.22601
$185$	0	0
$186$	2280.00	0.898805
$187$	−22.0000	−0.00860320
$188$	5780.00	2.24229
$189$	−864.000	−0.332522
$190$	0	0
$191$	−772.000	−0.292461	−0.146230	−	0.989251i	$-0.546714\pi$
$191$	−772.000	−0.292461	−0.146230	+	0.989251i	$0.546714\pi$
$192$	861.000	0.323632
$193$	−394.000	−0.146947	−0.0734734	−	0.997297i	$-0.523408\pi$
$193$	−394.000	−0.146947	−0.0734734	+	0.997297i	$0.523408\pi$
$194$	2630.00	0.973314
$195$	0	0
$196$	11577.0	4.21902
$197$	−3058.00	−1.10596	−0.552978	−	0.833196i	$-0.686509\pi$
$197$	−3058.00	−1.10596	−0.552978	+	0.833196i	$0.686509\pi$
$198$	−495.000	−0.177667
$199$	2664.00	0.948975	0.474487	−	0.880262i	$-0.342633\pi$
$199$	2664.00	0.948975	0.474487	+	0.880262i	$0.342633\pi$
$200$	0	0
$201$	−1380.00	−0.484267
$202$	250.000	0.0870789
$203$	−1728.00	−0.597447
$204$	−102.000	−0.0350070
$205$	0	0
$206$	−4720.00	−1.59640
$207$	−612.000	−0.205492
$208$	3382.00	1.12740
$209$	−792.000	−0.262123
$210$	0	0
$211$	−6000.00	−1.95762	−0.978808	−	0.204779i	$-0.934352\pi$
$211$	−6000.00	−1.95762	−0.978808	+	0.204779i	$0.934352\pi$
$212$	7446.00	2.41223
$213$	3276.00	1.05384
$214$	−2340.00	−0.747472
$215$	0	0
$216$	−1215.00	−0.382733
$217$	−4864.00	−1.52161
$218$	770.000	0.239225
$219$	1686.00	0.520225
$220$	0	0
$221$	76.0000	0.0231326
$222$	2610.00	0.789062
$223$	560.000	0.168163	0.0840816	−	0.996459i	$-0.473204\pi$
$223$	560.000	0.168163	0.0840816	+	0.996459i	$0.473204\pi$
$224$	2720.00	0.811329
$225$	0	0
$226$	270.000	0.0794696
$227$	−5292.00	−1.54732	−0.773662	−	0.633599i	$-0.781577\pi$
$227$	−5292.00	−1.54732	−0.773662	+	0.633599i	$0.781577\pi$
$228$	−3672.00	−1.06660
$229$	−5322.00	−1.53575	−0.767877	−	0.640597i	$-0.778687\pi$
$229$	−5322.00	−1.53575	−0.767877	+	0.640597i	$0.778687\pi$
$230$	0	0
$231$	1056.00	0.300778
$232$	−2430.00	−0.687661
$233$	3954.00	1.11174	0.555869	−	0.831270i	$-0.312385\pi$
$233$	3954.00	1.11174	0.555869	+	0.831270i	$0.312385\pi$
$234$	1710.00	0.477719
$235$	0	0
$236$	340.000	0.0937801
$237$	48.0000	0.0131558
$238$	320.000	0.0871534
$239$	−3360.00	−0.909374	−0.454687	−	0.890651i	$-0.650249\pi$
$239$	−3360.00	−0.909374	−0.454687	+	0.890651i	$0.650249\pi$
$240$	0	0
$241$	−3278.00	−0.876160	−0.438080	−	0.898936i	$-0.644341\pi$
$241$	−3278.00	−0.876160	−0.438080	+	0.898936i	$0.644341\pi$
$242$	605.000	0.160706
$243$	−243.000	−0.0641500
$244$	9690.00	2.54237
$245$	0	0
$246$	−1410.00	−0.365440
$247$	2736.00	0.704808
$248$	−6840.00	−1.75137
$249$	1116.00	0.284031
$250$	0	0
$251$	2092.00	0.526079	0.263040	−	0.964785i	$-0.415275\pi$
$251$	2092.00	0.526079	0.263040	+	0.964785i	$0.415275\pi$
$252$	4896.00	1.22388
$253$	748.000	0.185875
$254$	11120.0	2.74697
$255$	0	0
$256$	−8279.00	−2.02124
$257$	−658.000	−0.159708	−0.0798539	−	0.996807i	$-0.525445\pi$
$257$	−658.000	−0.159708	−0.0798539	+	0.996807i	$0.525445\pi$
$258$	−7920.00	−1.91115
$259$	−5568.00	−1.33583
$260$	0	0
$261$	−486.000	−0.115259
$262$	−13860.0	−3.26822
$263$	5104.00	1.19668	0.598339	−	0.801243i	$-0.295828\pi$
$263$	5104.00	1.19668	0.598339	+	0.801243i	$0.295828\pi$
$264$	1485.00	0.346195
$265$	0	0
$266$	11520.0	2.65540
$267$	2898.00	0.664250
$268$	7820.00	1.78240
$269$	−4238.00	−0.960578	−0.480289	−	0.877110i	$-0.659468\pi$
$269$	−4238.00	−0.960578	−0.480289	+	0.877110i	$0.659468\pi$
$270$	0	0
$271$	−3376.00	−0.756743	−0.378372	−	0.925654i	$-0.623516\pi$
$271$	−3376.00	−0.756743	−0.378372	+	0.925654i	$0.623516\pi$
$272$	178.000	0.0396795
$273$	−3648.00	−0.808744
$274$	−5650.00	−1.24573
$275$	0	0
$276$	3468.00	0.756337
$277$	−2074.00	−0.449872	−0.224936	−	0.974374i	$-0.572217\pi$
$277$	−2074.00	−0.449872	−0.224936	+	0.974374i	$0.572217\pi$
$278$	−8080.00	−1.74319
$279$	−1368.00	−0.293548
$280$	0	0
$281$	702.000	0.149031	0.0745157	−	0.997220i	$-0.476259\pi$
$281$	702.000	0.149031	0.0745157	+	0.997220i	$0.476259\pi$
$282$	−5100.00	−1.07695
$283$	−4912.00	−1.03176	−0.515880	−	0.856661i	$-0.672535\pi$
$283$	−4912.00	−1.03176	−0.515880	+	0.856661i	$0.672535\pi$
$284$	−18564.0	−3.87877
$285$	0	0
$286$	−2090.00	−0.432113
$287$	3008.00	0.618664
$288$	765.000	0.156521
$289$	−4909.00	−0.999186
$290$	0	0
$291$	−1578.00	−0.317883
$292$	−9554.00	−1.91475
$293$	3486.00	0.695066	0.347533	−	0.937668i	$-0.387019\pi$
$293$	3486.00	0.695066	0.347533	+	0.937668i	$0.387019\pi$
$294$	−10215.0	−2.02636
$295$	0	0
$296$	−7830.00	−1.53753
$297$	297.000	0.0580259
$298$	10330.0	2.00806
$299$	−2584.00	−0.499788
$300$	0	0
$301$	16896.0	3.23545
$302$	1240.00	0.236271
$303$	−150.000	−0.0284399
$304$	6408.00	1.20896
$305$	0	0
$306$	90.0000	0.0168136
$307$	−8360.00	−1.55417	−0.777085	−	0.629395i	$-0.783303\pi$
$307$	−8360.00	−1.55417	−0.777085	+	0.629395i	$0.783303\pi$
$308$	−5984.00	−1.10705
$309$	2832.00	0.521381
$310$	0	0
$311$	−5532.00	−1.00865	−0.504326	−	0.863513i	$-0.668259\pi$
$311$	−5532.00	−1.00865	−0.504326	+	0.863513i	$0.668259\pi$
$312$	−5130.00	−0.930862
$313$	−4826.00	−0.871507	−0.435753	−	0.900066i	$-0.643518\pi$
$313$	−4826.00	−0.871507	−0.435753	+	0.900066i	$0.643518\pi$
$314$	−11830.0	−2.12613
$315$	0	0
$316$	−272.000	−0.0484215
$317$	−7570.00	−1.34124	−0.670621	−	0.741800i	$-0.733972\pi$
$317$	−7570.00	−1.34124	−0.670621	+	0.741800i	$0.733972\pi$
$318$	−6570.00	−1.15858
$319$	594.000	0.104256
$320$	0	0
$321$	1404.00	0.244123
$322$	−10880.0	−1.88298
$323$	144.000	0.0248061
$324$	1377.00	0.236111
$325$	0	0
$326$	1420.00	0.241247
$327$	−462.000	−0.0781305
$328$	4230.00	0.712081
$329$	10880.0	1.82320
$330$	0	0
$331$	3676.00	0.610427	0.305213	−	0.952284i	$-0.401272\pi$
$331$	3676.00	0.610427	0.305213	+	0.952284i	$0.401272\pi$
$332$	−6324.00	−1.04541
$333$	−1566.00	−0.257707
$334$	−3000.00	−0.491475
$335$	0	0
$336$	−8544.00	−1.38724
$337$	5686.00	0.919098	0.459549	−	0.888152i	$-0.348011\pi$
$337$	5686.00	0.919098	0.459549	+	0.888152i	$0.348011\pi$
$338$	−3765.00	−0.605885
$339$	−162.000	−0.0259547
$340$	0	0
$341$	1672.00	0.265525
$342$	3240.00	0.512278
$343$	10816.0	1.70265
$344$	23760.0	3.72399
$345$	0	0
$346$	−690.000	−0.107210
$347$	1652.00	0.255574	0.127787	−	0.991802i	$-0.459213\pi$
$347$	1652.00	0.255574	0.127787	+	0.991802i	$0.459213\pi$
$348$	2754.00	0.424224
$349$	−6990.00	−1.07211	−0.536055	−	0.844183i	$-0.680086\pi$
$349$	−6990.00	−1.07211	−0.536055	+	0.844183i	$0.680086\pi$
$350$	0	0
$351$	−1026.00	−0.156022
$352$	−935.000	−0.141579
$353$	8094.00	1.22040	0.610199	−	0.792249i	$-0.291090\pi$
$353$	8094.00	1.22040	0.610199	+	0.792249i	$0.291090\pi$
$354$	−300.000	−0.0450419
$355$	0	0
$356$	−16422.0	−2.44484
$357$	−192.000	−0.0284642
$358$	19860.0	2.93194
$359$	1024.00	0.150542	0.0752711	−	0.997163i	$-0.476018\pi$
$359$	1024.00	0.150542	0.0752711	+	0.997163i	$0.476018\pi$
$360$	0	0
$361$	−1675.00	−0.244205
$362$	11150.0	1.61887
$363$	−363.000	−0.0524864
$364$	20672.0	2.97667
$365$	0	0
$366$	−8550.00	−1.22108
$367$	13664.0	1.94347	0.971737	−	0.236066i	$-0.0758581\pi$
$367$	13664.0	1.94347	0.971737	+	0.236066i	$0.0758581\pi$
$368$	−6052.00	−0.857289
$369$	846.000	0.119352
$370$	0	0
$371$	14016.0	1.96139
$372$	7752.00	1.08044
$373$	1958.00	0.271800	0.135900	−	0.990723i	$-0.456607\pi$
$373$	1958.00	0.271800	0.135900	+	0.990723i	$0.456607\pi$
$374$	−110.000	−0.0152085
$375$	0	0
$376$	15300.0	2.09850
$377$	−2052.00	−0.280327
$378$	−4320.00	−0.587822
$379$	6124.00	0.829997	0.414998	−	0.909822i	$-0.363782\pi$
$379$	6124.00	0.829997	0.414998	+	0.909822i	$0.363782\pi$
$380$	0	0
$381$	−6672.00	−0.897157
$382$	−3860.00	−0.517002
$383$	−5612.00	−0.748720	−0.374360	−	0.927283i	$-0.622138\pi$
$383$	−5612.00	−0.748720	−0.374360	+	0.927283i	$0.622138\pi$
$384$	6345.00	0.843208
$385$	0	0
$386$	−1970.00	−0.259768
$387$	4752.00	0.624180
$388$	8942.00	1.17000
$389$	12450.0	1.62273	0.811363	−	0.584543i	$-0.198726\pi$
$389$	12450.0	1.62273	0.811363	+	0.584543i	$0.198726\pi$
$390$	0	0
$391$	−136.000	−0.0175903
$392$	30645.0	3.94849
$393$	8316.00	1.06740
$394$	−15290.0	−1.95507
$395$	0	0
$396$	−1683.00	−0.213571
$397$	−14830.0	−1.87480	−0.937401	−	0.348252i	$-0.886775\pi$
$397$	−14830.0	−1.87480	−0.937401	+	0.348252i	$0.886775\pi$
$398$	13320.0	1.67757
$399$	−6912.00	−0.867250
$400$	0	0
$401$	−3358.00	−0.418181	−0.209090	−	0.977896i	$-0.567050\pi$
$401$	−3358.00	−0.418181	−0.209090	+	0.977896i	$0.567050\pi$
$402$	−6900.00	−0.856071
$403$	−5776.00	−0.713953
$404$	850.000	0.104676
$405$	0	0
$406$	−8640.00	−1.05615
$407$	1914.00	0.233104
$408$	−270.000	−0.0327622
$409$	10698.0	1.29335	0.646677	−	0.762764i	$-0.276158\pi$
$409$	10698.0	1.29335	0.646677	+	0.762764i	$0.276158\pi$
$410$	0	0
$411$	3390.00	0.406852
$412$	−16048.0	−1.91900
$413$	640.000	0.0762526
$414$	−3060.00	−0.363263
$415$	0	0
$416$	3230.00	0.380682
$417$	4848.00	0.569323
$418$	−3960.00	−0.463373
$419$	−2044.00	−0.238320	−0.119160	−	0.992875i	$-0.538020\pi$
$419$	−2044.00	−0.238320	−0.119160	+	0.992875i	$0.538020\pi$
$420$	0	0
$421$	3070.00	0.355398	0.177699	−	0.984085i	$-0.443135\pi$
$421$	3070.00	0.355398	0.177699	+	0.984085i	$0.443135\pi$
$422$	−30000.0	−3.46061
$423$	3060.00	0.351731
$424$	19710.0	2.25755
$425$	0	0
$426$	16380.0	1.86294
$427$	18240.0	2.06720
$428$	−7956.00	−0.898523
$429$	1254.00	0.141127
$430$	0	0
$431$	−12600.0	−1.40817	−0.704084	−	0.710116i	$-0.748642\pi$
$431$	−12600.0	−1.40817	−0.704084	+	0.710116i	$0.748642\pi$
$432$	−2403.00	−0.267626
$433$	9902.00	1.09898	0.549492	−	0.835499i	$-0.314821\pi$
$433$	9902.00	1.09898	0.549492	+	0.835499i	$0.314821\pi$
$434$	−24320.0	−2.68986
$435$	0	0
$436$	2618.00	0.287568
$437$	−4896.00	−0.535944
$438$	8430.00	0.919637
$439$	11440.0	1.24374	0.621869	−	0.783121i	$-0.286373\pi$
$439$	11440.0	1.24374	0.621869	+	0.783121i	$0.286373\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	380.000	0.0408931
$443$	5180.00	0.555551	0.277776	−	0.960646i	$-0.410403\pi$
$443$	5180.00	0.555551	0.277776	+	0.960646i	$0.410403\pi$
$444$	8874.00	0.948517
$445$	0	0
$446$	2800.00	0.297273
$447$	−6198.00	−0.655829
$448$	−9184.00	−0.968534
$449$	10826.0	1.13789	0.568943	−	0.822377i	$-0.307353\pi$
$449$	10826.0	1.13789	0.568943	+	0.822377i	$0.307353\pi$
$450$	0	0
$451$	−1034.00	−0.107958
$452$	918.000	0.0955290
$453$	−744.000	−0.0771659
$454$	−26460.0	−2.73531
$455$	0	0
$456$	−9720.00	−0.998203
$457$	15798.0	1.61707	0.808533	−	0.588451i	$-0.200262\pi$
$457$	15798.0	1.61707	0.808533	+	0.588451i	$0.200262\pi$
$458$	−26610.0	−2.71486
$459$	−54.0000	−0.00549129
$460$	0	0
$461$	−3894.00	−0.393409	−0.196705	−	0.980463i	$-0.563024\pi$
$461$	−3894.00	−0.393409	−0.196705	+	0.980463i	$0.563024\pi$
$462$	5280.00	0.531705
$463$	15992.0	1.60521	0.802604	−	0.596512i	$-0.203447\pi$
$463$	15992.0	1.60521	0.802604	+	0.596512i	$0.203447\pi$
$464$	−4806.00	−0.480847
$465$	0	0
$466$	19770.0	1.96530
$467$	−11844.0	−1.17361	−0.586804	−	0.809729i	$-0.699614\pi$
$467$	−11844.0	−1.17361	−0.586804	+	0.809729i	$0.699614\pi$
$468$	5814.00	0.574257
$469$	14720.0	1.44927
$470$	0	0
$471$	7098.00	0.694392
$472$	900.000	0.0877666
$473$	−5808.00	−0.564592
$474$	240.000	0.0232565
$475$	0	0
$476$	1088.00	0.104766
$477$	3942.00	0.378389
$478$	−16800.0	−1.60756
$479$	14936.0	1.42472	0.712362	−	0.701812i	$-0.247625\pi$
$479$	14936.0	1.42472	0.712362	+	0.701812i	$0.247625\pi$
$480$	0	0
$481$	−6612.00	−0.626780
$482$	−16390.0	−1.54885
$483$	6528.00	0.614978
$484$	2057.00	0.193182
$485$	0	0
$486$	−1215.00	−0.113402
$487$	2056.00	0.191306	0.0956532	−	0.995415i	$-0.469506\pi$
$487$	2056.00	0.191306	0.0956532	+	0.995415i	$0.469506\pi$
$488$	25650.0	2.37935
$489$	−852.000	−0.0787909
$490$	0	0
$491$	−17852.0	−1.64083	−0.820417	−	0.571766i	$-0.806259\pi$
$491$	−17852.0	−1.64083	−0.820417	+	0.571766i	$0.806259\pi$
$492$	−4794.00	−0.439289
$493$	−108.000	−0.00986628
$494$	13680.0	1.24594
$495$	0	0
$496$	−13528.0	−1.22465
$497$	−34944.0	−3.15383
$498$	5580.00	0.502100
$499$	4508.00	0.404420	0.202210	−	0.979342i	$-0.435188\pi$
$499$	4508.00	0.404420	0.202210	+	0.979342i	$0.435188\pi$
$500$	0	0
$501$	1800.00	0.160515
$502$	10460.0	0.929985
$503$	5912.00	0.524062	0.262031	−	0.965059i	$-0.415608\pi$
$503$	5912.00	0.524062	0.262031	+	0.965059i	$0.415608\pi$
$504$	12960.0	1.14541
$505$	0	0
$506$	3740.00	0.328584
$507$	2259.00	0.197881
$508$	37808.0	3.30208
$509$	−11406.0	−0.993246	−0.496623	−	0.867966i	$-0.665427\pi$
$509$	−11406.0	−0.993246	−0.496623	+	0.867966i	$0.665427\pi$
$510$	0	0
$511$	−17984.0	−1.55688
$512$	−24475.0	−2.11260
$513$	−1944.00	−0.167309
$514$	−3290.00	−0.282326
$515$	0	0
$516$	−26928.0	−2.29736
$517$	−3740.00	−0.318153
$518$	−27840.0	−2.36143
$519$	414.000	0.0350146
$520$	0	0
$521$	−1542.00	−0.129667	−0.0648333	−	0.997896i	$-0.520652\pi$
$521$	−1542.00	−0.129667	−0.0648333	+	0.997896i	$0.520652\pi$
$522$	−2430.00	−0.203751
$523$	7504.00	0.627394	0.313697	−	0.949523i	$-0.398432\pi$
$523$	7504.00	0.627394	0.313697	+	0.949523i	$0.398432\pi$
$524$	−47124.0	−3.92867
$525$	0	0
$526$	25520.0	2.11545
$527$	−304.000	−0.0251280
$528$	2937.00	0.242077
$529$	−7543.00	−0.619956
$530$	0	0
$531$	180.000	0.0147106
$532$	39168.0	3.19201
$533$	3572.00	0.290282
$534$	14490.0	1.17424
$535$	0	0
$536$	20700.0	1.66810
$537$	−11916.0	−0.957567
$538$	−21190.0	−1.69808
$539$	−7491.00	−0.598627
$540$	0	0
$541$	1018.00	0.0809006	0.0404503	−	0.999182i	$-0.487121\pi$
$541$	1018.00	0.0809006	0.0404503	+	0.999182i	$0.487121\pi$
$542$	−16880.0	−1.33775
$543$	−6690.00	−0.528721
$544$	170.000	0.0133983
$545$	0	0
$546$	−18240.0	−1.42967
$547$	−7904.00	−0.617826	−0.308913	−	0.951090i	$-0.599965\pi$
$547$	−7904.00	−0.617826	−0.308913	+	0.951090i	$0.599965\pi$
$548$	−19210.0	−1.49746
$549$	5130.00	0.398803
$550$	0	0
$551$	−3888.00	−0.300607
$552$	9180.00	0.707838
$553$	−512.000	−0.0393715
$554$	−10370.0	−0.795269
$555$	0	0
$556$	−27472.0	−2.09545
$557$	22934.0	1.74460	0.872302	−	0.488967i	$-0.162626\pi$
$557$	22934.0	1.74460	0.872302	+	0.488967i	$0.162626\pi$
$558$	−6840.00	−0.518925
$559$	20064.0	1.51810
$560$	0	0
$561$	66.0000	0.00496706
$562$	3510.00	0.263453
$563$	−14020.0	−1.04951	−0.524754	−	0.851254i	$-0.675843\pi$
$563$	−14020.0	−1.04951	−0.524754	+	0.851254i	$0.675843\pi$
$564$	−17340.0	−1.29458
$565$	0	0
$566$	−24560.0	−1.82391
$567$	2592.00	0.191982
$568$	−49140.0	−3.63005
$569$	4230.00	0.311653	0.155827	−	0.987784i	$-0.450196\pi$
$569$	4230.00	0.311653	0.155827	+	0.987784i	$0.450196\pi$
$570$	0	0
$571$	−8536.00	−0.625605	−0.312803	−	0.949818i	$-0.601268\pi$
$571$	−8536.00	−0.625605	−0.312803	+	0.949818i	$0.601268\pi$
$572$	−7106.00	−0.519435
$573$	2316.00	0.168852
$574$	15040.0	1.09365
$575$	0	0
$576$	−2583.00	−0.186849
$577$	11982.0	0.864501	0.432251	−	0.901754i	$-0.357720\pi$
$577$	11982.0	0.864501	0.432251	+	0.901754i	$0.357720\pi$
$578$	−24545.0	−1.76633
$579$	1182.00	0.0848398
$580$	0	0
$581$	−11904.0	−0.850019
$582$	−7890.00	−0.561943
$583$	−4818.00	−0.342266
$584$	−25290.0	−1.79197
$585$	0	0
$586$	17430.0	1.22871
$587$	20396.0	1.43413	0.717064	−	0.697007i	$-0.245486\pi$
$587$	20396.0	1.43413	0.717064	+	0.697007i	$0.245486\pi$
$588$	−34731.0	−2.43585
$589$	−10944.0	−0.765602
$590$	0	0
$591$	9174.00	0.638524
$592$	−15486.0	−1.07512
$593$	−12518.0	−0.866868	−0.433434	−	0.901185i	$-0.642698\pi$
$593$	−12518.0	−0.866868	−0.433434	+	0.901185i	$0.642698\pi$
$594$	1485.00	0.102576
$595$	0	0
$596$	35122.0	2.41385
$597$	−7992.00	−0.547891
$598$	−12920.0	−0.883509
$599$	−25292.0	−1.72521	−0.862607	−	0.505875i	$-0.831170\pi$
$599$	−25292.0	−1.72521	−0.862607	+	0.505875i	$0.831170\pi$
$600$	0	0
$601$	15962.0	1.08337	0.541683	−	0.840583i	$-0.317787\pi$
$601$	15962.0	1.08337	0.541683	+	0.840583i	$0.317787\pi$
$602$	84480.0	5.71951
$603$	4140.00	0.279592
$604$	4216.00	0.284018
$605$	0	0
$606$	−750.000	−0.0502750
$607$	1600.00	0.106988	0.0534942	−	0.998568i	$-0.482964\pi$
$607$	1600.00	0.106988	0.0534942	+	0.998568i	$0.482964\pi$
$608$	6120.00	0.408222
$609$	5184.00	0.344936
$610$	0	0
$611$	12920.0	0.855462
$612$	306.000	0.0202113
$613$	−2162.00	−0.142451	−0.0712254	−	0.997460i	$-0.522691\pi$
$613$	−2162.00	−0.142451	−0.0712254	+	0.997460i	$0.522691\pi$
$614$	−41800.0	−2.74741
$615$	0	0
$616$	−15840.0	−1.03606
$617$	18126.0	1.18270	0.591350	−	0.806415i	$-0.298595\pi$
$617$	18126.0	1.18270	0.591350	+	0.806415i	$0.298595\pi$
$618$	14160.0	0.921681
$619$	17348.0	1.12645	0.563227	−	0.826302i	$-0.309560\pi$
$619$	17348.0	1.12645	0.563227	+	0.826302i	$0.309560\pi$
$620$	0	0
$621$	1836.00	0.118641
$622$	−27660.0	−1.78306
$623$	−30912.0	−1.98790
$624$	−10146.0	−0.650906
$625$	0	0
$626$	−24130.0	−1.54062
$627$	2376.00	0.151337
$628$	−40222.0	−2.55578
$629$	−348.000	−0.0220599
$630$	0	0
$631$	10096.0	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	10096.0	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−720.000	−0.0453166
$633$	18000.0	1.13023
$634$	−37850.0	−2.37100
$635$	0	0
$636$	−22338.0	−1.39270
$637$	25878.0	1.60961
$638$	2970.00	0.184300
$639$	−9828.00	−0.608435
$640$	0	0
$641$	8922.00	0.549763	0.274881	−	0.961478i	$-0.411361\pi$
$641$	8922.00	0.549763	0.274881	+	0.961478i	$0.411361\pi$
$642$	7020.00	0.431553
$643$	14644.0	0.898138	0.449069	−	0.893497i	$-0.351756\pi$
$643$	14644.0	0.898138	0.449069	+	0.893497i	$0.351756\pi$
$644$	−36992.0	−2.26349
$645$	0	0
$646$	720.000	0.0438514
$647$	−6932.00	−0.421213	−0.210607	−	0.977571i	$-0.567544\pi$
$647$	−6932.00	−0.421213	−0.210607	+	0.977571i	$0.567544\pi$
$648$	3645.00	0.220971
$649$	−220.000	−0.0133062
$650$	0	0
$651$	14592.0	0.878503
$652$	4828.00	0.289999
$653$	5942.00	0.356093	0.178046	−	0.984022i	$-0.443022\pi$
$653$	5942.00	0.356093	0.178046	+	0.984022i	$0.443022\pi$
$654$	−2310.00	−0.138116
$655$	0	0
$656$	8366.00	0.497923
$657$	−5058.00	−0.300352
$658$	54400.0	3.22300
$659$	484.000	0.0286100	0.0143050	−	0.999898i	$-0.495446\pi$
$659$	484.000	0.0286100	0.0143050	+	0.999898i	$0.495446\pi$
$660$	0	0
$661$	−17114.0	−1.00705	−0.503523	−	0.863982i	$-0.667963\pi$
$661$	−17114.0	−1.00705	−0.503523	+	0.863982i	$0.667963\pi$
$662$	18380.0	1.07909
$663$	−228.000	−0.0133556
$664$	−16740.0	−0.978370
$665$	0	0
$666$	−7830.00	−0.455565
$667$	3672.00	0.213164
$668$	−10200.0	−0.590793
$669$	−1680.00	−0.0970890
$670$	0	0
$671$	−6270.00	−0.360731
$672$	−8160.00	−0.468421
$673$	−16154.0	−0.925247	−0.462623	−	0.886555i	$-0.653092\pi$
$673$	−16154.0	−0.925247	−0.462623	+	0.886555i	$0.653092\pi$
$674$	28430.0	1.62475
$675$	0	0
$676$	−12801.0	−0.728323
$677$	3390.00	0.192449	0.0962247	−	0.995360i	$-0.469323\pi$
$677$	3390.00	0.192449	0.0962247	+	0.995360i	$0.469323\pi$
$678$	−810.000	−0.0458818
$679$	16832.0	0.951330
$680$	0	0
$681$	15876.0	0.893347
$682$	8360.00	0.469386
$683$	25540.0	1.43084	0.715418	−	0.698697i	$-0.246236\pi$
$683$	25540.0	1.43084	0.715418	+	0.698697i	$0.246236\pi$
$684$	11016.0	0.615800
$685$	0	0
$686$	54080.0	3.00989
$687$	15966.0	0.886668
$688$	46992.0	2.60400
$689$	16644.0	0.920299
$690$	0	0
$691$	12476.0	0.686844	0.343422	−	0.939181i	$-0.388414\pi$
$691$	12476.0	0.686844	0.343422	+	0.939181i	$0.388414\pi$
$692$	−2346.00	−0.128875
$693$	−3168.00	−0.173654
$694$	8260.00	0.451794
$695$	0	0
$696$	7290.00	0.397021
$697$	188.000	0.0102167
$698$	−34950.0	−1.89524
$699$	−11862.0	−0.641863
$700$	0	0
$701$	−20806.0	−1.12102	−0.560508	−	0.828149i	$-0.689394\pi$
$701$	−20806.0	−1.12102	−0.560508	+	0.828149i	$0.689394\pi$
$702$	−5130.00	−0.275811
$703$	−12528.0	−0.672123
$704$	3157.00	0.169011
$705$	0	0
$706$	40470.0	2.15738
$707$	1600.00	0.0851120
$708$	−1020.00	−0.0541440
$709$	14198.0	0.752069	0.376035	−	0.926606i	$-0.377287\pi$
$709$	14198.0	0.752069	0.376035	+	0.926606i	$0.377287\pi$
$710$	0	0
$711$	−144.000	−0.00759553
$712$	−43470.0	−2.28807
$713$	10336.0	0.542898
$714$	−960.000	−0.0503181
$715$	0	0
$716$	67524.0	3.52443
$717$	10080.0	0.525027
$718$	5120.00	0.266124
$719$	4596.00	0.238389	0.119195	−	0.992871i	$-0.461969\pi$
$719$	4596.00	0.238389	0.119195	+	0.992871i	$0.461969\pi$
$720$	0	0
$721$	−30208.0	−1.56034
$722$	−8375.00	−0.431697
$723$	9834.00	0.505851
$724$	37910.0	1.94601
$725$	0	0
$726$	−1815.00	−0.0927837
$727$	−19560.0	−0.997855	−0.498927	−	0.866644i	$-0.666273\pi$
$727$	−19560.0	−0.997855	−0.498927	+	0.866644i	$0.666273\pi$
$728$	54720.0	2.78579
$729$	729.000	0.0370370
$730$	0	0
$731$	1056.00	0.0534303
$732$	−29070.0	−1.46784
$733$	1638.00	0.0825388	0.0412694	−	0.999148i	$-0.486860\pi$
$733$	1638.00	0.0825388	0.0412694	+	0.999148i	$0.486860\pi$
$734$	68320.0	3.43561
$735$	0	0
$736$	−5780.00	−0.289475
$737$	−5060.00	−0.252900
$738$	4230.00	0.210987
$739$	−15592.0	−0.776131	−0.388066	−	0.921632i	$-0.626857\pi$
$739$	−15592.0	−0.776131	−0.388066	+	0.921632i	$0.626857\pi$
$740$	0	0
$741$	−8208.00	−0.406921
$742$	70080.0	3.46727
$743$	−592.000	−0.0292307	−0.0146153	−	0.999893i	$-0.504652\pi$
$743$	−592.000	−0.0292307	−0.0146153	+	0.999893i	$0.504652\pi$
$744$	20520.0	1.01116
$745$	0	0
$746$	9790.00	0.480479
$747$	−3348.00	−0.163985
$748$	−374.000	−0.0182818
$749$	−14976.0	−0.730589
$750$	0	0
$751$	39832.0	1.93541	0.967703	−	0.252092i	$-0.0811186\pi$
$751$	39832.0	1.93541	0.967703	+	0.252092i	$0.0811186\pi$
$752$	30260.0	1.46738
$753$	−6276.00	−0.303732
$754$	−10260.0	−0.495553
$755$	0	0
$756$	−14688.0	−0.706610
$757$	−10958.0	−0.526123	−0.263062	−	0.964779i	$-0.584732\pi$
$757$	−10958.0	−0.526123	−0.263062	+	0.964779i	$0.584732\pi$
$758$	30620.0	1.46724
$759$	−2244.00	−0.107315
$760$	0	0
$761$	−8970.00	−0.427283	−0.213641	−	0.976912i	$-0.568532\pi$
$761$	−8970.00	−0.427283	−0.213641	+	0.976912i	$0.568532\pi$
$762$	−33360.0	−1.58596
$763$	4928.00	0.233821
$764$	−13124.0	−0.621479
$765$	0	0
$766$	−28060.0	−1.32356
$767$	760.000	0.0357784
$768$	24837.0	1.16696
$769$	−10054.0	−0.471465	−0.235732	−	0.971818i	$-0.575749\pi$
$769$	−10054.0	−0.471465	−0.235732	+	0.971818i	$0.575749\pi$
$770$	0	0
$771$	1974.00	0.0922074
$772$	−6698.00	−0.312262
$773$	−26346.0	−1.22587	−0.612936	−	0.790132i	$-0.710012\pi$
$773$	−26346.0	−1.22587	−0.612936	+	0.790132i	$0.710012\pi$
$774$	23760.0	1.10341
$775$	0	0
$776$	23670.0	1.09498
$777$	16704.0	0.771239
$778$	62250.0	2.86860
$779$	6768.00	0.311282
$780$	0	0
$781$	12012.0	0.550350
$782$	−680.000	−0.0310956
$783$	1458.00	0.0665449
$784$	60609.0	2.76098
$785$	0	0
$786$	41580.0	1.88691
$787$	16040.0	0.726511	0.363256	−	0.931690i	$-0.381665\pi$
$787$	16040.0	0.726511	0.363256	+	0.931690i	$0.381665\pi$
$788$	−51986.0	−2.35016
$789$	−15312.0	−0.690902
$790$	0	0
$791$	1728.00	0.0776746
$792$	−4455.00	−0.199876
$793$	21660.0	0.969948
$794$	−74150.0	−3.31421
$795$	0	0
$796$	45288.0	2.01657
$797$	−32810.0	−1.45821	−0.729103	−	0.684404i	$-0.760062\pi$
$797$	−32810.0	−1.45821	−0.729103	+	0.684404i	$0.760062\pi$
$798$	−34560.0	−1.53310
$799$	680.000	0.0301085
$800$	0	0
$801$	−8694.00	−0.383505
$802$	−16790.0	−0.739246
$803$	6182.00	0.271679
$804$	−23460.0	−1.02907
$805$	0	0
$806$	−28880.0	−1.26210
$807$	12714.0	0.554590
$808$	2250.00	0.0979638
$809$	18918.0	0.822153	0.411076	−	0.911601i	$-0.365153\pi$
$809$	18918.0	0.822153	0.411076	+	0.911601i	$0.365153\pi$
$810$	0	0
$811$	−8552.00	−0.370285	−0.185143	−	0.982712i	$-0.559275\pi$
$811$	−8552.00	−0.370285	−0.185143	+	0.982712i	$0.559275\pi$
$812$	−29376.0	−1.26958
$813$	10128.0	0.436906
$814$	9570.00	0.412074
$815$	0	0
$816$	−534.000	−0.0229090
$817$	38016.0	1.62792
$818$	53490.0	2.28635
$819$	10944.0	0.466928
$820$	0	0
$821$	−46430.0	−1.97371	−0.986856	−	0.161600i	$-0.948335\pi$
$821$	−46430.0	−1.97371	−0.986856	+	0.161600i	$0.948335\pi$
$822$	16950.0	0.719220
$823$	−16392.0	−0.694276	−0.347138	−	0.937814i	$-0.612846\pi$
$823$	−16392.0	−0.694276	−0.347138	+	0.937814i	$0.612846\pi$
$824$	−42480.0	−1.79595
$825$	0	0
$826$	3200.00	0.134797
$827$	13876.0	0.583453	0.291727	−	0.956502i	$-0.405770\pi$
$827$	13876.0	0.583453	0.291727	+	0.956502i	$0.405770\pi$
$828$	−10404.0	−0.436671
$829$	−24554.0	−1.02870	−0.514352	−	0.857579i	$-0.671968\pi$
$829$	−24554.0	−1.02870	−0.514352	+	0.857579i	$0.671968\pi$
$830$	0	0
$831$	6222.00	0.259734
$832$	−10906.0	−0.454444
$833$	1362.00	0.0566513
$834$	24240.0	1.00643
$835$	0	0
$836$	−13464.0	−0.557012
$837$	4104.00	0.169480
$838$	−10220.0	−0.421294
$839$	19900.0	0.818861	0.409430	−	0.912341i	$-0.365727\pi$
$839$	19900.0	0.818861	0.409430	+	0.912341i	$0.365727\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	15350.0	0.628261
$843$	−2106.00	−0.0860433
$844$	−102000.	−4.15993
$845$	0	0
$846$	15300.0	0.621779
$847$	3872.00	0.157076
$848$	38982.0	1.57859
$849$	14736.0	0.595687
$850$	0	0
$851$	11832.0	0.476611
$852$	55692.0	2.23941
$853$	−41138.0	−1.65128	−0.825638	−	0.564200i	$-0.809185\pi$
$853$	−41138.0	−1.65128	−0.825638	+	0.564200i	$0.809185\pi$
$854$	91200.0	3.65433
$855$	0	0
$856$	−21060.0	−0.840907
$857$	−19910.0	−0.793597	−0.396799	−	0.917906i	$-0.629879\pi$
$857$	−19910.0	−0.793597	−0.396799	+	0.917906i	$0.629879\pi$
$858$	6270.00	0.249481
$859$	42924.0	1.70495	0.852473	−	0.522772i	$-0.175102\pi$
$859$	42924.0	1.70495	0.852473	+	0.522772i	$0.175102\pi$
$860$	0	0
$861$	−9024.00	−0.357186
$862$	−63000.0	−2.48931
$863$	46236.0	1.82374	0.911872	−	0.410474i	$-0.134637\pi$
$863$	46236.0	1.82374	0.911872	+	0.410474i	$0.134637\pi$
$864$	−2295.00	−0.0903675
$865$	0	0
$866$	49510.0	1.94275
$867$	14727.0	0.576880
$868$	−82688.0	−3.23343
$869$	176.000	0.00687042
$870$	0	0
$871$	17480.0	0.680008
$872$	6930.00	0.269128
$873$	4734.00	0.183530
$874$	−24480.0	−0.947424
$875$	0	0
$876$	28662.0	1.10548
$877$	−25746.0	−0.991312	−0.495656	−	0.868519i	$-0.665072\pi$
$877$	−25746.0	−0.991312	−0.495656	+	0.868519i	$0.665072\pi$
$878$	57200.0	2.19864
$879$	−10458.0	−0.401296
$880$	0	0
$881$	−24550.0	−0.938831	−0.469416	−	0.882977i	$-0.655535\pi$
$881$	−24550.0	−0.938831	−0.469416	+	0.882977i	$0.655535\pi$
$882$	30645.0	1.16992
$883$	19436.0	0.740740	0.370370	−	0.928884i	$-0.379231\pi$
$883$	19436.0	0.740740	0.370370	+	0.928884i	$0.379231\pi$
$884$	1292.00	0.0491569
$885$	0	0
$886$	25900.0	0.982085
$887$	22912.0	0.867316	0.433658	−	0.901077i	$-0.357223\pi$
$887$	22912.0	0.867316	0.433658	+	0.901077i	$0.357223\pi$
$888$	23490.0	0.887695
$889$	71168.0	2.68492
$890$	0	0
$891$	−891.000	−0.0335013
$892$	9520.00	0.357347
$893$	24480.0	0.917348
$894$	−30990.0	−1.15935
$895$	0	0
$896$	−67680.0	−2.52347
$897$	7752.00	0.288553
$898$	54130.0	2.01152
$899$	8208.00	0.304507
$900$	0	0
$901$	876.000	0.0323904
$902$	−5170.00	−0.190845
$903$	−50688.0	−1.86799
$904$	2430.00	0.0894033
$905$	0	0
$906$	−3720.00	−0.136411
$907$	39900.0	1.46070	0.730352	−	0.683071i	$-0.239356\pi$
$907$	39900.0	1.46070	0.730352	+	0.683071i	$0.239356\pi$
$908$	−89964.0	−3.28806
$909$	450.000	0.0164198
$910$	0	0
$911$	29460.0	1.07141	0.535704	−	0.844406i	$-0.320046\pi$
$911$	29460.0	1.07141	0.535704	+	0.844406i	$0.320046\pi$
$912$	−19224.0	−0.697994
$913$	4092.00	0.148330
$914$	78990.0	2.85860
$915$	0	0
$916$	−90474.0	−3.26348
$917$	−88704.0	−3.19440
$918$	−270.000	−0.00970733
$919$	29368.0	1.05415	0.527073	−	0.849820i	$-0.323289\pi$
$919$	29368.0	1.05415	0.527073	+	0.849820i	$0.323289\pi$
$920$	0	0
$921$	25080.0	0.897301
$922$	−19470.0	−0.695456
$923$	−41496.0	−1.47980
$924$	17952.0	0.639153
$925$	0	0
$926$	79960.0	2.83763
$927$	−8496.00	−0.301020
$928$	−4590.00	−0.162364
$929$	33954.0	1.19913	0.599567	−	0.800325i	$-0.295340\pi$
$929$	33954.0	1.19913	0.599567	+	0.800325i	$0.295340\pi$
$930$	0	0
$931$	49032.0	1.72606
$932$	67218.0	2.36245
$933$	16596.0	0.582346
$934$	−59220.0	−2.07467
$935$	0	0
$936$	15390.0	0.537434
$937$	2854.00	0.0995049	0.0497525	−	0.998762i	$-0.484157\pi$
$937$	2854.00	0.0995049	0.0497525	+	0.998762i	$0.484157\pi$
$938$	73600.0	2.56197
$939$	14478.0	0.503165
$940$	0	0
$941$	−6294.00	−0.218043	−0.109022	−	0.994039i	$-0.534772\pi$
$941$	−6294.00	−0.218043	−0.109022	+	0.994039i	$0.534772\pi$
$942$	35490.0	1.22752
$943$	−6392.00	−0.220734
$944$	1780.00	0.0613708
$945$	0	0
$946$	−29040.0	−0.998067
$947$	−2268.00	−0.0778248	−0.0389124	−	0.999243i	$-0.512389\pi$
$947$	−2268.00	−0.0778248	−0.0389124	+	0.999243i	$0.512389\pi$
$948$	816.000	0.0279562
$949$	−21356.0	−0.730501
$950$	0	0
$951$	22710.0	0.774366
$952$	2880.00	0.0980476
$953$	−26566.0	−0.902998	−0.451499	−	0.892272i	$-0.649111\pi$
$953$	−26566.0	−0.902998	−0.451499	+	0.892272i	$0.649111\pi$
$954$	19710.0	0.668904
$955$	0	0
$956$	−57120.0	−1.93242
$957$	−1782.00	−0.0601921
$958$	74680.0	2.51858
$959$	−36160.0	−1.21759
$960$	0	0
$961$	−6687.00	−0.224464
$962$	−33060.0	−1.10800
$963$	−4212.00	−0.140945
$964$	−55726.0	−1.86184
$965$	0	0
$966$	32640.0	1.08714
$967$	−11176.0	−0.371661	−0.185830	−	0.982582i	$-0.559497\pi$
$967$	−11176.0	−0.371661	−0.185830	+	0.982582i	$0.559497\pi$
$968$	5445.00	0.180794
$969$	−432.000	−0.0143218
$970$	0	0
$971$	−42316.0	−1.39854	−0.699271	−	0.714856i	$-0.746492\pi$
$971$	−42316.0	−1.39854	−0.699271	+	0.714856i	$0.746492\pi$
$972$	−4131.00	−0.136319
$973$	−51712.0	−1.70381
$974$	10280.0	0.338185
$975$	0	0
$976$	50730.0	1.66376
$977$	45054.0	1.47534	0.737669	−	0.675163i	$-0.235927\pi$
$977$	45054.0	1.47534	0.737669	+	0.675163i	$0.235927\pi$
$978$	−4260.00	−0.139284
$979$	10626.0	0.346893
$980$	0	0
$981$	1386.00	0.0451086
$982$	−89260.0	−2.90061
$983$	12300.0	0.399094	0.199547	−	0.979888i	$-0.436053\pi$
$983$	12300.0	0.399094	0.199547	+	0.979888i	$0.436053\pi$
$984$	−12690.0	−0.411120
$985$	0	0
$986$	−540.000	−0.0174413
$987$	−32640.0	−1.05263
$988$	46512.0	1.49772
$989$	−35904.0	−1.15438
$990$	0	0
$991$	36280.0	1.16294	0.581469	−	0.813568i	$-0.302478\pi$
$991$	36280.0	1.16294	0.581469	+	0.813568i	$0.302478\pi$
$992$	−12920.0	−0.413519
$993$	−11028.0	−0.352430
$994$	−174720.	−5.57523
$995$	0	0
$996$	18972.0	0.603565
$997$	−3290.00	−0.104509	−0.0522544	−	0.998634i	$-0.516641\pi$
$997$	−3290.00	−0.104509	−0.0522544	+	0.998634i	$0.516641\pi$
$998$	22540.0	0.714921
$999$	4698.00	0.148787

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.i.1.1		1
3.2	odd	2		2475.4.a.b.1.1		1
5.2	odd	4		825.4.c.a.199.2		2
5.3	odd	4		825.4.c.a.199.1		2
5.4	even	2		33.4.a.a.1.1	✓	1
15.14	odd	2		99.4.a.b.1.1		1
20.19	odd	2		528.4.a.a.1.1		1
35.34	odd	2		1617.4.a.a.1.1		1
40.19	odd	2		2112.4.a.y.1.1		1
40.29	even	2		2112.4.a.l.1.1		1
55.54	odd	2		363.4.a.h.1.1		1
60.59	even	2		1584.4.a.t.1.1		1
165.164	even	2		1089.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.a.1.1	✓	1	5.4	even	2
99.4.a.b.1.1		1	15.14	odd	2
363.4.a.h.1.1		1	55.54	odd	2
528.4.a.a.1.1		1	20.19	odd	2
825.4.a.i.1.1		1	1.1	even	1	trivial
825.4.c.a.199.1		2	5.3	odd	4
825.4.c.a.199.2		2	5.2	odd	4
1089.4.a.a.1.1		1	165.164	even	2
1584.4.a.t.1.1		1	60.59	even	2
1617.4.a.a.1.1		1	35.34	odd	2
2112.4.a.l.1.1		1	40.29	even	2
2112.4.a.y.1.1		1	40.19	odd	2
2475.4.a.b.1.1		1	3.2	odd	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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