LMFDB - Embedded newform 240.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,4,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, 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("240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	240.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:	$14.1604584014$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -5.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -5.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} -72.0000 q^{11} -6.00000 q^{13} -15.0000 q^{15} +38.0000 q^{17} -52.0000 q^{19} -12.0000 q^{21} -152.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -78.0000 q^{29} -120.000 q^{31} -216.000 q^{33} +20.0000 q^{35} -150.000 q^{37} -18.0000 q^{39} +362.000 q^{41} +484.000 q^{43} -45.0000 q^{45} -280.000 q^{47} -327.000 q^{49} +114.000 q^{51} -670.000 q^{53} +360.000 q^{55} -156.000 q^{57} -696.000 q^{59} +222.000 q^{61} -36.0000 q^{63} +30.0000 q^{65} +4.00000 q^{67} -456.000 q^{69} -96.0000 q^{71} +178.000 q^{73} +75.0000 q^{75} +288.000 q^{77} +632.000 q^{79} +81.0000 q^{81} +612.000 q^{83} -190.000 q^{85} -234.000 q^{87} +994.000 q^{89} +24.0000 q^{91} -360.000 q^{93} +260.000 q^{95} +1634.00 q^{97} -648.000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$7$	−4.00000	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−72.0000	−1.97353	−0.986764	−	0.162160i	$-0.948154\pi$
$11$	−72.0000	−1.97353	−0.986764	+	0.162160i	$0.948154\pi$
$12$	0	0
$13$	−6.00000	−0.128008	−0.0640039	−	0.997950i	$-0.520387\pi$
$13$	−6.00000	−0.128008	−0.0640039	+	0.997950i	$0.520387\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	38.0000	0.542138	0.271069	−	0.962560i	$-0.412623\pi$
$17$	38.0000	0.542138	0.271069	+	0.962560i	$0.412623\pi$
$18$	0	0
$19$	−52.0000	−0.627875	−0.313937	−	0.949444i	$-0.601648\pi$
$19$	−52.0000	−0.627875	−0.313937	+	0.949444i	$0.601648\pi$
$20$	0	0
$21$	−12.0000	−0.124696
$22$	0	0
$23$	−152.000	−1.37801	−0.689004	−	0.724757i	$-0.741952\pi$
$23$	−152.000	−1.37801	−0.689004	+	0.724757i	$0.741952\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−78.0000	−0.499456	−0.249728	−	0.968316i	$-0.580341\pi$
$29$	−78.0000	−0.499456	−0.249728	+	0.968316i	$0.580341\pi$
$30$	0	0
$31$	−120.000	−0.695246	−0.347623	−	0.937634i	$-0.613011\pi$
$31$	−120.000	−0.695246	−0.347623	+	0.937634i	$0.613011\pi$
$32$	0	0
$33$	−216.000	−1.13942
$34$	0	0
$35$	20.0000	0.0965891
$36$	0	0
$37$	−150.000	−0.666482	−0.333241	−	0.942842i	$-0.608142\pi$
$37$	−150.000	−0.666482	−0.333241	+	0.942842i	$0.608142\pi$
$38$	0	0
$39$	−18.0000	−0.0739053
$40$	0	0
$41$	362.000	1.37890	0.689450	−	0.724333i	$-0.257852\pi$
$41$	362.000	1.37890	0.689450	+	0.724333i	$0.257852\pi$
$42$	0	0
$43$	484.000	1.71650	0.858248	−	0.513236i	$-0.171553\pi$
$43$	484.000	1.71650	0.858248	+	0.513236i	$0.171553\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−280.000	−0.868983	−0.434491	−	0.900676i	$-0.643072\pi$
$47$	−280.000	−0.868983	−0.434491	+	0.900676i	$0.643072\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	114.000	0.313004
$52$	0	0
$53$	−670.000	−1.73644	−0.868222	−	0.496175i	$-0.834737\pi$
$53$	−670.000	−1.73644	−0.868222	+	0.496175i	$0.834737\pi$
$54$	0	0
$55$	360.000	0.882589
$56$	0	0
$57$	−156.000	−0.362504
$58$	0	0
$59$	−696.000	−1.53579	−0.767894	−	0.640577i	$-0.778695\pi$
$59$	−696.000	−1.53579	−0.767894	+	0.640577i	$0.778695\pi$
$60$	0	0
$61$	222.000	0.465970	0.232985	−	0.972480i	$-0.425151\pi$
$61$	222.000	0.465970	0.232985	+	0.972480i	$0.425151\pi$
$62$	0	0
$63$	−36.0000	−0.0719932
$64$	0	0
$65$	30.0000	0.0572468
$66$	0	0
$67$	4.00000	0.00729370	0.00364685	−	0.999993i	$-0.498839\pi$
$67$	4.00000	0.00729370	0.00364685	+	0.999993i	$0.498839\pi$
$68$	0	0
$69$	−456.000	−0.795593
$70$	0	0
$71$	−96.0000	−0.160466	−0.0802331	−	0.996776i	$-0.525566\pi$
$71$	−96.0000	−0.160466	−0.0802331	+	0.996776i	$0.525566\pi$
$72$	0	0
$73$	178.000	0.285388	0.142694	−	0.989767i	$-0.454424\pi$
$73$	178.000	0.285388	0.142694	+	0.989767i	$0.454424\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	288.000	0.426242
$78$	0	0
$79$	632.000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	632.000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	612.000	0.809346	0.404673	−	0.914461i	$-0.367385\pi$
$83$	612.000	0.809346	0.404673	+	0.914461i	$0.367385\pi$
$84$	0	0
$85$	−190.000	−0.242452
$86$	0	0
$87$	−234.000	−0.288361
$88$	0	0
$89$	994.000	1.18386	0.591931	−	0.805988i	$-0.298366\pi$
$89$	994.000	1.18386	0.591931	+	0.805988i	$0.298366\pi$
$90$	0	0
$91$	24.0000	0.0276471
$92$	0	0
$93$	−360.000	−0.401401
$94$	0	0
$95$	260.000	0.280794
$96$	0	0
$97$	1634.00	1.71039	0.855194	−	0.518309i	$-0.173438\pi$
$97$	1634.00	1.71039	0.855194	+	0.518309i	$0.173438\pi$
$98$	0	0
$99$	−648.000	−0.657843
$100$	0	0
$101$	890.000	0.876815	0.438407	−	0.898776i	$-0.355543\pi$
$101$	890.000	0.876815	0.438407	+	0.898776i	$0.355543\pi$
$102$	0	0
$103$	524.000	0.501274	0.250637	−	0.968081i	$-0.419360\pi$
$103$	524.000	0.501274	0.250637	+	0.968081i	$0.419360\pi$
$104$	0	0
$105$	60.0000	0.0557657
$106$	0	0
$107$	−932.000	−0.842055	−0.421027	−	0.907048i	$-0.638330\pi$
$107$	−932.000	−0.842055	−0.421027	+	0.907048i	$0.638330\pi$
$108$	0	0
$109$	446.000	0.391918	0.195959	−	0.980612i	$-0.437218\pi$
$109$	446.000	0.391918	0.195959	+	0.980612i	$0.437218\pi$
$110$	0	0
$111$	−450.000	−0.384794
$112$	0	0
$113$	−786.000	−0.654342	−0.327171	−	0.944965i	$-0.606095\pi$
$113$	−786.000	−0.654342	−0.327171	+	0.944965i	$0.606095\pi$
$114$	0	0
$115$	760.000	0.616264
$116$	0	0
$117$	−54.0000	−0.0426692
$118$	0	0
$119$	−152.000	−0.117091
$120$	0	0
$121$	3853.00	2.89482
$122$	0	0
$123$	1086.00	0.796108
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−716.000	−0.500273	−0.250137	−	0.968211i	$-0.580476\pi$
$127$	−716.000	−0.500273	−0.250137	+	0.968211i	$0.580476\pi$
$128$	0	0
$129$	1452.00	0.991019
$130$	0	0
$131$	808.000	0.538895	0.269448	−	0.963015i	$-0.413159\pi$
$131$	808.000	0.538895	0.269448	+	0.963015i	$0.413159\pi$
$132$	0	0
$133$	208.000	0.135608
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	−1770.00	−1.10381	−0.551903	−	0.833909i	$-0.686098\pi$
$137$	−1770.00	−1.10381	−0.551903	+	0.833909i	$0.686098\pi$
$138$	0	0
$139$	924.000	0.563832	0.281916	−	0.959439i	$-0.409030\pi$
$139$	924.000	0.563832	0.281916	+	0.959439i	$0.409030\pi$
$140$	0	0
$141$	−840.000	−0.501708
$142$	0	0
$143$	432.000	0.252627
$144$	0	0
$145$	390.000	0.223364
$146$	0	0
$147$	−981.000	−0.550418
$148$	0	0
$149$	−3198.00	−1.75832	−0.879162	−	0.476522i	$-0.841897\pi$
$149$	−3198.00	−1.75832	−0.879162	+	0.476522i	$0.841897\pi$
$150$	0	0
$151$	3384.00	1.82375	0.911874	−	0.410470i	$-0.134635\pi$
$151$	3384.00	1.82375	0.911874	+	0.410470i	$0.134635\pi$
$152$	0	0
$153$	342.000	0.180713
$154$	0	0
$155$	600.000	0.310924
$156$	0	0
$157$	−3302.00	−1.67852	−0.839262	−	0.543727i	$-0.817013\pi$
$157$	−3302.00	−1.67852	−0.839262	+	0.543727i	$0.817013\pi$
$158$	0	0
$159$	−2010.00	−1.00254
$160$	0	0
$161$	608.000	0.297622
$162$	0	0
$163$	−2252.00	−1.08215	−0.541074	−	0.840975i	$-0.681982\pi$
$163$	−2252.00	−1.08215	−0.541074	+	0.840975i	$0.681982\pi$
$164$	0	0
$165$	1080.00	0.509563
$166$	0	0
$167$	184.000	0.0852596	0.0426298	−	0.999091i	$-0.486426\pi$
$167$	184.000	0.0852596	0.0426298	+	0.999091i	$0.486426\pi$
$168$	0	0
$169$	−2161.00	−0.983614
$170$	0	0
$171$	−468.000	−0.209292
$172$	0	0
$173$	−2646.00	−1.16284	−0.581421	−	0.813603i	$-0.697503\pi$
$173$	−2646.00	−1.16284	−0.581421	+	0.813603i	$0.697503\pi$
$174$	0	0
$175$	−100.000	−0.0431959
$176$	0	0
$177$	−2088.00	−0.886688
$178$	0	0
$179$	608.000	0.253877	0.126939	−	0.991911i	$-0.459485\pi$
$179$	608.000	0.253877	0.126939	+	0.991911i	$0.459485\pi$
$180$	0	0
$181$	2246.00	0.922342	0.461171	−	0.887311i	$-0.347430\pi$
$181$	2246.00	0.922342	0.461171	+	0.887311i	$0.347430\pi$
$182$	0	0
$183$	666.000	0.269028
$184$	0	0
$185$	750.000	0.298060
$186$	0	0
$187$	−2736.00	−1.06993
$188$	0	0
$189$	−108.000	−0.0415653
$190$	0	0
$191$	3848.00	1.45776	0.728878	−	0.684643i	$-0.240042\pi$
$191$	3848.00	1.45776	0.728878	+	0.684643i	$0.240042\pi$
$192$	0	0
$193$	2058.00	0.767555	0.383777	−	0.923426i	$-0.374623\pi$
$193$	2058.00	0.767555	0.383777	+	0.923426i	$0.374623\pi$
$194$	0	0
$195$	90.0000	0.0330515
$196$	0	0
$197$	−3838.00	−1.38805	−0.694026	−	0.719950i	$-0.744165\pi$
$197$	−3838.00	−1.38805	−0.694026	+	0.719950i	$0.744165\pi$
$198$	0	0
$199$	1992.00	0.709594	0.354797	−	0.934943i	$-0.384550\pi$
$199$	1992.00	0.709594	0.354797	+	0.934943i	$0.384550\pi$
$200$	0	0
$201$	12.0000	0.00421102
$202$	0	0
$203$	312.000	0.107872
$204$	0	0
$205$	−1810.00	−0.616663
$206$	0	0
$207$	−1368.00	−0.459336
$208$	0	0
$209$	3744.00	1.23913
$210$	0	0
$211$	−4764.00	−1.55435	−0.777174	−	0.629286i	$-0.783347\pi$
$211$	−4764.00	−1.55435	−0.777174	+	0.629286i	$0.783347\pi$
$212$	0	0
$213$	−288.000	−0.0926452
$214$	0	0
$215$	−2420.00	−0.767640
$216$	0	0
$217$	480.000	0.150159
$218$	0	0
$219$	534.000	0.164769
$220$	0	0
$221$	−228.000	−0.0693979
$222$	0	0
$223$	−4092.00	−1.22879	−0.614396	−	0.788998i	$-0.710600\pi$
$223$	−4092.00	−1.22879	−0.614396	+	0.788998i	$0.710600\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−468.000	−0.136838	−0.0684191	−	0.997657i	$-0.521795\pi$
$227$	−468.000	−0.136838	−0.0684191	+	0.997657i	$0.521795\pi$
$228$	0	0
$229$	−5586.00	−1.61194	−0.805968	−	0.591959i	$-0.798355\pi$
$229$	−5586.00	−1.61194	−0.805968	+	0.591959i	$0.798355\pi$
$230$	0	0
$231$	864.000	0.246091
$232$	0	0
$233$	−1058.00	−0.297476	−0.148738	−	0.988877i	$-0.547521\pi$
$233$	−1058.00	−0.297476	−0.148738	+	0.988877i	$0.547521\pi$
$234$	0	0
$235$	1400.00	0.388621
$236$	0	0
$237$	1896.00	0.519656
$238$	0	0
$239$	−6840.00	−1.85123	−0.925613	−	0.378472i	$-0.876450\pi$
$239$	−6840.00	−1.85123	−0.925613	+	0.378472i	$0.876450\pi$
$240$	0	0
$241$	−6430.00	−1.71864	−0.859321	−	0.511437i	$-0.829113\pi$
$241$	−6430.00	−1.71864	−0.859321	+	0.511437i	$0.829113\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1635.00	0.426352
$246$	0	0
$247$	312.000	0.0803728
$248$	0	0
$249$	1836.00	0.467276
$250$	0	0
$251$	6352.00	1.59735	0.798675	−	0.601763i	$-0.205535\pi$
$251$	6352.00	1.59735	0.798675	+	0.601763i	$0.205535\pi$
$252$	0	0
$253$	10944.0	2.71954
$254$	0	0
$255$	−570.000	−0.139980
$256$	0	0
$257$	1422.00	0.345144	0.172572	−	0.984997i	$-0.444792\pi$
$257$	1422.00	0.345144	0.172572	+	0.984997i	$0.444792\pi$
$258$	0	0
$259$	600.000	0.143947
$260$	0	0
$261$	−702.000	−0.166485
$262$	0	0
$263$	−7224.00	−1.69373	−0.846865	−	0.531808i	$-0.821513\pi$
$263$	−7224.00	−1.69373	−0.846865	+	0.531808i	$0.821513\pi$
$264$	0	0
$265$	3350.00	0.776562
$266$	0	0
$267$	2982.00	0.683504
$268$	0	0
$269$	3186.00	0.722133	0.361067	−	0.932540i	$-0.382413\pi$
$269$	3186.00	0.722133	0.361067	+	0.932540i	$0.382413\pi$
$270$	0	0
$271$	256.000	0.0573834	0.0286917	−	0.999588i	$-0.490866\pi$
$271$	256.000	0.0573834	0.0286917	+	0.999588i	$0.490866\pi$
$272$	0	0
$273$	72.0000	0.0159620
$274$	0	0
$275$	−1800.00	−0.394706
$276$	0	0
$277$	−5942.00	−1.28888	−0.644441	−	0.764654i	$-0.722910\pi$
$277$	−5942.00	−1.28888	−0.644441	+	0.764654i	$0.722910\pi$
$278$	0	0
$279$	−1080.00	−0.231749
$280$	0	0
$281$	3202.00	0.679770	0.339885	−	0.940467i	$-0.389612\pi$
$281$	3202.00	0.679770	0.339885	+	0.940467i	$0.389612\pi$
$282$	0	0
$283$	−3940.00	−0.827593	−0.413796	−	0.910370i	$-0.635797\pi$
$283$	−3940.00	−0.827593	−0.413796	+	0.910370i	$0.635797\pi$
$284$	0	0
$285$	780.000	0.162117
$286$	0	0
$287$	−1448.00	−0.297814
$288$	0	0
$289$	−3469.00	−0.706086
$290$	0	0
$291$	4902.00	0.987493
$292$	0	0
$293$	1826.00	0.364082	0.182041	−	0.983291i	$-0.441730\pi$
$293$	1826.00	0.364082	0.182041	+	0.983291i	$0.441730\pi$
$294$	0	0
$295$	3480.00	0.686825
$296$	0	0
$297$	−1944.00	−0.379806
$298$	0	0
$299$	912.000	0.176396
$300$	0	0
$301$	−1936.00	−0.370728
$302$	0	0
$303$	2670.00	0.506229
$304$	0	0
$305$	−1110.00	−0.208388
$306$	0	0
$307$	−6580.00	−1.22326	−0.611629	−	0.791144i	$-0.709486\pi$
$307$	−6580.00	−1.22326	−0.611629	+	0.791144i	$0.709486\pi$
$308$	0	0
$309$	1572.00	0.289411
$310$	0	0
$311$	5728.00	1.04439	0.522195	−	0.852826i	$-0.325113\pi$
$311$	5728.00	1.04439	0.522195	+	0.852826i	$0.325113\pi$
$312$	0	0
$313$	−1742.00	−0.314580	−0.157290	−	0.987552i	$-0.550276\pi$
$313$	−1742.00	−0.314580	−0.157290	+	0.987552i	$0.550276\pi$
$314$	0	0
$315$	180.000	0.0321964
$316$	0	0
$317$	8746.00	1.54960	0.774802	−	0.632204i	$-0.217850\pi$
$317$	8746.00	1.54960	0.774802	+	0.632204i	$0.217850\pi$
$318$	0	0
$319$	5616.00	0.985692
$320$	0	0
$321$	−2796.00	−0.486160
$322$	0	0
$323$	−1976.00	−0.340395
$324$	0	0
$325$	−150.000	−0.0256015
$326$	0	0
$327$	1338.00	0.226274
$328$	0	0
$329$	1120.00	0.187683
$330$	0	0
$331$	2564.00	0.425771	0.212885	−	0.977077i	$-0.431714\pi$
$331$	2564.00	0.425771	0.212885	+	0.977077i	$0.431714\pi$
$332$	0	0
$333$	−1350.00	−0.222161
$334$	0	0
$335$	−20.0000	−0.00326184
$336$	0	0
$337$	−4166.00	−0.673402	−0.336701	−	0.941612i	$-0.609311\pi$
$337$	−4166.00	−0.673402	−0.336701	+	0.941612i	$0.609311\pi$
$338$	0	0
$339$	−2358.00	−0.377785
$340$	0	0
$341$	8640.00	1.37209
$342$	0	0
$343$	2680.00	0.421885
$344$	0	0
$345$	2280.00	0.355800
$346$	0	0
$347$	9444.00	1.46104	0.730519	−	0.682892i	$-0.239278\pi$
$347$	9444.00	1.46104	0.730519	+	0.682892i	$0.239278\pi$
$348$	0	0
$349$	−9218.00	−1.41383	−0.706917	−	0.707296i	$-0.749915\pi$
$349$	−9218.00	−1.41383	−0.706917	+	0.707296i	$0.749915\pi$
$350$	0	0
$351$	−162.000	−0.0246351
$352$	0	0
$353$	−4698.00	−0.708355	−0.354177	−	0.935178i	$-0.615239\pi$
$353$	−4698.00	−0.708355	−0.354177	+	0.935178i	$0.615239\pi$
$354$	0	0
$355$	480.000	0.0717627
$356$	0	0
$357$	−456.000	−0.0676025
$358$	0	0
$359$	6056.00	0.890316	0.445158	−	0.895452i	$-0.353148\pi$
$359$	6056.00	0.890316	0.445158	+	0.895452i	$0.353148\pi$
$360$	0	0
$361$	−4155.00	−0.605773
$362$	0	0
$363$	11559.0	1.67132
$364$	0	0
$365$	−890.000	−0.127629
$366$	0	0
$367$	8228.00	1.17029	0.585147	−	0.810927i	$-0.301037\pi$
$367$	8228.00	1.17029	0.585147	+	0.810927i	$0.301037\pi$
$368$	0	0
$369$	3258.00	0.459633
$370$	0	0
$371$	2680.00	0.375037
$372$	0	0
$373$	5954.00	0.826505	0.413253	−	0.910616i	$-0.364393\pi$
$373$	5954.00	0.826505	0.413253	+	0.910616i	$0.364393\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	468.000	0.0639343
$378$	0	0
$379$	−5284.00	−0.716150	−0.358075	−	0.933693i	$-0.616567\pi$
$379$	−5284.00	−0.716150	−0.358075	+	0.933693i	$0.616567\pi$
$380$	0	0
$381$	−2148.00	−0.288833
$382$	0	0
$383$	−9832.00	−1.31173	−0.655864	−	0.754879i	$-0.727695\pi$
$383$	−9832.00	−1.31173	−0.655864	+	0.754879i	$0.727695\pi$
$384$	0	0
$385$	−1440.00	−0.190621
$386$	0	0
$387$	4356.00	0.572165
$388$	0	0
$389$	−222.000	−0.0289353	−0.0144677	−	0.999895i	$-0.504605\pi$
$389$	−222.000	−0.0289353	−0.0144677	+	0.999895i	$0.504605\pi$
$390$	0	0
$391$	−5776.00	−0.747071
$392$	0	0
$393$	2424.00	0.311131
$394$	0	0
$395$	−3160.00	−0.402524
$396$	0	0
$397$	12098.0	1.52942	0.764712	−	0.644372i	$-0.222881\pi$
$397$	12098.0	1.52942	0.764712	+	0.644372i	$0.222881\pi$
$398$	0	0
$399$	624.000	0.0782934
$400$	0	0
$401$	−5958.00	−0.741966	−0.370983	−	0.928640i	$-0.620979\pi$
$401$	−5958.00	−0.741966	−0.370983	+	0.928640i	$0.620979\pi$
$402$	0	0
$403$	720.000	0.0889969
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	10800.0	1.31532
$408$	0	0
$409$	1930.00	0.233331	0.116665	−	0.993171i	$-0.462779\pi$
$409$	1930.00	0.233331	0.116665	+	0.993171i	$0.462779\pi$
$410$	0	0
$411$	−5310.00	−0.637282
$412$	0	0
$413$	2784.00	0.331699
$414$	0	0
$415$	−3060.00	−0.361951
$416$	0	0
$417$	2772.00	0.325529
$418$	0	0
$419$	−4744.00	−0.553125	−0.276563	−	0.960996i	$-0.589195\pi$
$419$	−4744.00	−0.553125	−0.276563	+	0.960996i	$0.589195\pi$
$420$	0	0
$421$	1614.00	0.186845	0.0934223	−	0.995627i	$-0.470219\pi$
$421$	1614.00	0.186845	0.0934223	+	0.995627i	$0.470219\pi$
$422$	0	0
$423$	−2520.00	−0.289661
$424$	0	0
$425$	950.000	0.108428
$426$	0	0
$427$	−888.000	−0.100640
$428$	0	0
$429$	1296.00	0.145854
$430$	0	0
$431$	−9296.00	−1.03892	−0.519458	−	0.854496i	$-0.673866\pi$
$431$	−9296.00	−1.03892	−0.519458	+	0.854496i	$0.673866\pi$
$432$	0	0
$433$	−3494.00	−0.387785	−0.193893	−	0.981023i	$-0.562111\pi$
$433$	−3494.00	−0.387785	−0.193893	+	0.981023i	$0.562111\pi$
$434$	0	0
$435$	1170.00	0.128959
$436$	0	0
$437$	7904.00	0.865216
$438$	0	0
$439$	12584.0	1.36811	0.684056	−	0.729429i	$-0.260214\pi$
$439$	12584.0	1.36811	0.684056	+	0.729429i	$0.260214\pi$
$440$	0	0
$441$	−2943.00	−0.317784
$442$	0	0
$443$	12852.0	1.37837	0.689184	−	0.724586i	$-0.257969\pi$
$443$	12852.0	1.37837	0.689184	+	0.724586i	$0.257969\pi$
$444$	0	0
$445$	−4970.00	−0.529440
$446$	0	0
$447$	−9594.00	−1.01517
$448$	0	0
$449$	14458.0	1.51963	0.759816	−	0.650138i	$-0.225289\pi$
$449$	14458.0	1.51963	0.759816	+	0.650138i	$0.225289\pi$
$450$	0	0
$451$	−26064.0	−2.72130
$452$	0	0
$453$	10152.0	1.05294
$454$	0	0
$455$	−120.000	−0.0123641
$456$	0	0
$457$	−4310.00	−0.441167	−0.220583	−	0.975368i	$-0.570796\pi$
$457$	−4310.00	−0.441167	−0.220583	+	0.975368i	$0.570796\pi$
$458$	0	0
$459$	1026.00	0.104335
$460$	0	0
$461$	5338.00	0.539296	0.269648	−	0.962959i	$-0.413093\pi$
$461$	5338.00	0.539296	0.269648	+	0.962959i	$0.413093\pi$
$462$	0	0
$463$	−1156.00	−0.116034	−0.0580171	−	0.998316i	$-0.518478\pi$
$463$	−1156.00	−0.116034	−0.0580171	+	0.998316i	$0.518478\pi$
$464$	0	0
$465$	1800.00	0.179512
$466$	0	0
$467$	−5948.00	−0.589380	−0.294690	−	0.955593i	$-0.595216\pi$
$467$	−5948.00	−0.589380	−0.294690	+	0.955593i	$0.595216\pi$
$468$	0	0
$469$	−16.0000	−0.00157529
$470$	0	0
$471$	−9906.00	−0.969096
$472$	0	0
$473$	−34848.0	−3.38755
$474$	0	0
$475$	−1300.00	−0.125575
$476$	0	0
$477$	−6030.00	−0.578815
$478$	0	0
$479$	−6888.00	−0.657037	−0.328519	−	0.944498i	$-0.606549\pi$
$479$	−6888.00	−0.657037	−0.328519	+	0.944498i	$0.606549\pi$
$480$	0	0
$481$	900.000	0.0853149
$482$	0	0
$483$	1824.00	0.171832
$484$	0	0
$485$	−8170.00	−0.764908
$486$	0	0
$487$	−2892.00	−0.269095	−0.134547	−	0.990907i	$-0.542958\pi$
$487$	−2892.00	−0.269095	−0.134547	+	0.990907i	$0.542958\pi$
$488$	0	0
$489$	−6756.00	−0.624779
$490$	0	0
$491$	−4096.00	−0.376476	−0.188238	−	0.982123i	$-0.560278\pi$
$491$	−4096.00	−0.376476	−0.188238	+	0.982123i	$0.560278\pi$
$492$	0	0
$493$	−2964.00	−0.270775
$494$	0	0
$495$	3240.00	0.294196
$496$	0	0
$497$	384.000	0.0346575
$498$	0	0
$499$	−11060.0	−0.992212	−0.496106	−	0.868262i	$-0.665237\pi$
$499$	−11060.0	−0.992212	−0.496106	+	0.868262i	$0.665237\pi$
$500$	0	0
$501$	552.000	0.0492246
$502$	0	0
$503$	9648.00	0.855235	0.427617	−	0.903960i	$-0.359353\pi$
$503$	9648.00	0.855235	0.427617	+	0.903960i	$0.359353\pi$
$504$	0	0
$505$	−4450.00	−0.392124
$506$	0	0
$507$	−6483.00	−0.567890
$508$	0	0
$509$	−10062.0	−0.876209	−0.438104	−	0.898924i	$-0.644350\pi$
$509$	−10062.0	−0.876209	−0.438104	+	0.898924i	$0.644350\pi$
$510$	0	0
$511$	−712.000	−0.0616380
$512$	0	0
$513$	−1404.00	−0.120835
$514$	0	0
$515$	−2620.00	−0.224177
$516$	0	0
$517$	20160.0	1.71496
$518$	0	0
$519$	−7938.00	−0.671367
$520$	0	0
$521$	−7966.00	−0.669859	−0.334930	−	0.942243i	$-0.608713\pi$
$521$	−7966.00	−0.669859	−0.334930	+	0.942243i	$0.608713\pi$
$522$	0	0
$523$	−7668.00	−0.641106	−0.320553	−	0.947231i	$-0.603869\pi$
$523$	−7668.00	−0.641106	−0.320553	+	0.947231i	$0.603869\pi$
$524$	0	0
$525$	−300.000	−0.0249392
$526$	0	0
$527$	−4560.00	−0.376920
$528$	0	0
$529$	10937.0	0.898907
$530$	0	0
$531$	−6264.00	−0.511929
$532$	0	0
$533$	−2172.00	−0.176510
$534$	0	0
$535$	4660.00	0.376578
$536$	0	0
$537$	1824.00	0.146576
$538$	0	0
$539$	23544.0	1.88147
$540$	0	0
$541$	6590.00	0.523708	0.261854	−	0.965107i	$-0.415666\pi$
$541$	6590.00	0.523708	0.261854	+	0.965107i	$0.415666\pi$
$542$	0	0
$543$	6738.00	0.532514
$544$	0	0
$545$	−2230.00	−0.175271
$546$	0	0
$547$	4700.00	0.367381	0.183691	−	0.982984i	$-0.441196\pi$
$547$	4700.00	0.367381	0.183691	+	0.982984i	$0.441196\pi$
$548$	0	0
$549$	1998.00	0.155323
$550$	0	0
$551$	4056.00	0.313596
$552$	0	0
$553$	−2528.00	−0.194397
$554$	0	0
$555$	2250.00	0.172085
$556$	0	0
$557$	−15766.0	−1.19933	−0.599665	−	0.800251i	$-0.704700\pi$
$557$	−15766.0	−1.19933	−0.599665	+	0.800251i	$0.704700\pi$
$558$	0	0
$559$	−2904.00	−0.219725
$560$	0	0
$561$	−8208.00	−0.617722
$562$	0	0
$563$	−22788.0	−1.70586	−0.852930	−	0.522025i	$-0.825177\pi$
$563$	−22788.0	−1.70586	−0.852930	+	0.522025i	$0.825177\pi$
$564$	0	0
$565$	3930.00	0.292631
$566$	0	0
$567$	−324.000	−0.0239977
$568$	0	0
$569$	−3358.00	−0.247407	−0.123704	−	0.992319i	$-0.539477\pi$
$569$	−3358.00	−0.247407	−0.123704	+	0.992319i	$0.539477\pi$
$570$	0	0
$571$	11444.0	0.838733	0.419366	−	0.907817i	$-0.362252\pi$
$571$	11444.0	0.838733	0.419366	+	0.907817i	$0.362252\pi$
$572$	0	0
$573$	11544.0	0.841636
$574$	0	0
$575$	−3800.00	−0.275602
$576$	0	0
$577$	−10622.0	−0.766377	−0.383189	−	0.923670i	$-0.625174\pi$
$577$	−10622.0	−0.766377	−0.383189	+	0.923670i	$0.625174\pi$
$578$	0	0
$579$	6174.00	0.443148
$580$	0	0
$581$	−2448.00	−0.174802
$582$	0	0
$583$	48240.0	3.42692
$584$	0	0
$585$	270.000	0.0190823
$586$	0	0
$587$	6588.00	0.463230	0.231615	−	0.972808i	$-0.425599\pi$
$587$	6588.00	0.463230	0.231615	+	0.972808i	$0.425599\pi$
$588$	0	0
$589$	6240.00	0.436528
$590$	0	0
$591$	−11514.0	−0.801392
$592$	0	0
$593$	−11362.0	−0.786815	−0.393408	−	0.919364i	$-0.628704\pi$
$593$	−11362.0	−0.786815	−0.393408	+	0.919364i	$0.628704\pi$
$594$	0	0
$595$	760.000	0.0523646
$596$	0	0
$597$	5976.00	0.409684
$598$	0	0
$599$	−1624.00	−0.110776	−0.0553880	−	0.998465i	$-0.517640\pi$
$599$	−1624.00	−0.110776	−0.0553880	+	0.998465i	$0.517640\pi$
$600$	0	0
$601$	−14950.0	−1.01468	−0.507340	−	0.861746i	$-0.669371\pi$
$601$	−14950.0	−1.01468	−0.507340	+	0.861746i	$0.669371\pi$
$602$	0	0
$603$	36.0000	0.00243123
$604$	0	0
$605$	−19265.0	−1.29460
$606$	0	0
$607$	−8244.00	−0.551258	−0.275629	−	0.961264i	$-0.588886\pi$
$607$	−8244.00	−0.551258	−0.275629	+	0.961264i	$0.588886\pi$
$608$	0	0
$609$	936.000	0.0622802
$610$	0	0
$611$	1680.00	0.111237
$612$	0	0
$613$	6698.00	0.441321	0.220660	−	0.975351i	$-0.429179\pi$
$613$	6698.00	0.441321	0.220660	+	0.975351i	$0.429179\pi$
$614$	0	0
$615$	−5430.00	−0.356030
$616$	0	0
$617$	22670.0	1.47919	0.739595	−	0.673053i	$-0.235017\pi$
$617$	22670.0	1.47919	0.739595	+	0.673053i	$0.235017\pi$
$618$	0	0
$619$	10060.0	0.653224	0.326612	−	0.945159i	$-0.394093\pi$
$619$	10060.0	0.653224	0.326612	+	0.945159i	$0.394093\pi$
$620$	0	0
$621$	−4104.00	−0.265198
$622$	0	0
$623$	−3976.00	−0.255690
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	11232.0	0.715411
$628$	0	0
$629$	−5700.00	−0.361326
$630$	0	0
$631$	−10240.0	−0.646035	−0.323017	−	0.946393i	$-0.604697\pi$
$631$	−10240.0	−0.646035	−0.323017	+	0.946393i	$0.604697\pi$
$632$	0	0
$633$	−14292.0	−0.897403
$634$	0	0
$635$	3580.00	0.223729
$636$	0	0
$637$	1962.00	0.122037
$638$	0	0
$639$	−864.000	−0.0534888
$640$	0	0
$641$	13218.0	0.814477	0.407238	−	0.913322i	$-0.366492\pi$
$641$	13218.0	0.814477	0.407238	+	0.913322i	$0.366492\pi$
$642$	0	0
$643$	23412.0	1.43589	0.717946	−	0.696098i	$-0.245082\pi$
$643$	23412.0	1.43589	0.717946	+	0.696098i	$0.245082\pi$
$644$	0	0
$645$	−7260.00	−0.443197
$646$	0	0
$647$	15264.0	0.927496	0.463748	−	0.885967i	$-0.346504\pi$
$647$	15264.0	0.927496	0.463748	+	0.885967i	$0.346504\pi$
$648$	0	0
$649$	50112.0	3.03092
$650$	0	0
$651$	1440.00	0.0866944
$652$	0	0
$653$	1482.00	0.0888134	0.0444067	−	0.999014i	$-0.485860\pi$
$653$	1482.00	0.0888134	0.0444067	+	0.999014i	$0.485860\pi$
$654$	0	0
$655$	−4040.00	−0.241001
$656$	0	0
$657$	1602.00	0.0951293
$658$	0	0
$659$	18920.0	1.11839	0.559195	−	0.829036i	$-0.311110\pi$
$659$	18920.0	1.11839	0.559195	+	0.829036i	$0.311110\pi$
$660$	0	0
$661$	−24218.0	−1.42507	−0.712535	−	0.701637i	$-0.752453\pi$
$661$	−24218.0	−1.42507	−0.712535	+	0.701637i	$0.752453\pi$
$662$	0	0
$663$	−684.000	−0.0400669
$664$	0	0
$665$	−1040.00	−0.0606458
$666$	0	0
$667$	11856.0	0.688255
$668$	0	0
$669$	−12276.0	−0.709443
$670$	0	0
$671$	−15984.0	−0.919606
$672$	0	0
$673$	890.000	0.0509762	0.0254881	−	0.999675i	$-0.491886\pi$
$673$	890.000	0.0509762	0.0254881	+	0.999675i	$0.491886\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	29250.0	1.66052	0.830258	−	0.557380i	$-0.188193\pi$
$677$	29250.0	1.66052	0.830258	+	0.557380i	$0.188193\pi$
$678$	0	0
$679$	−6536.00	−0.369409
$680$	0	0
$681$	−1404.00	−0.0790035
$682$	0	0
$683$	−14580.0	−0.816820	−0.408410	−	0.912799i	$-0.633917\pi$
$683$	−14580.0	−0.816820	−0.408410	+	0.912799i	$0.633917\pi$
$684$	0	0
$685$	8850.00	0.493637
$686$	0	0
$687$	−16758.0	−0.930651
$688$	0	0
$689$	4020.00	0.222278
$690$	0	0
$691$	−23668.0	−1.30300	−0.651500	−	0.758649i	$-0.725860\pi$
$691$	−23668.0	−1.30300	−0.651500	+	0.758649i	$0.725860\pi$
$692$	0	0
$693$	2592.00	0.142081
$694$	0	0
$695$	−4620.00	−0.252153
$696$	0	0
$697$	13756.0	0.747555
$698$	0	0
$699$	−3174.00	−0.171748
$700$	0	0
$701$	32402.0	1.74580	0.872901	−	0.487898i	$-0.162236\pi$
$701$	32402.0	1.74580	0.872901	+	0.487898i	$0.162236\pi$
$702$	0	0
$703$	7800.00	0.418467
$704$	0	0
$705$	4200.00	0.224370
$706$	0	0
$707$	−3560.00	−0.189374
$708$	0	0
$709$	−30626.0	−1.62226	−0.811131	−	0.584865i	$-0.801148\pi$
$709$	−30626.0	−1.62226	−0.811131	+	0.584865i	$0.801148\pi$
$710$	0	0
$711$	5688.00	0.300023
$712$	0	0
$713$	18240.0	0.958055
$714$	0	0
$715$	−2160.00	−0.112978
$716$	0	0
$717$	−20520.0	−1.06881
$718$	0	0
$719$	−13440.0	−0.697117	−0.348559	−	0.937287i	$-0.613329\pi$
$719$	−13440.0	−0.697117	−0.348559	+	0.937287i	$0.613329\pi$
$720$	0	0
$721$	−2096.00	−0.108265
$722$	0	0
$723$	−19290.0	−0.992258
$724$	0	0
$725$	−1950.00	−0.0998913
$726$	0	0
$727$	24820.0	1.26619	0.633097	−	0.774073i	$-0.281783\pi$
$727$	24820.0	1.26619	0.633097	+	0.774073i	$0.281783\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	18392.0	0.930578
$732$	0	0
$733$	21986.0	1.10787	0.553937	−	0.832559i	$-0.313125\pi$
$733$	21986.0	1.10787	0.553937	+	0.832559i	$0.313125\pi$
$734$	0	0
$735$	4905.00	0.246155
$736$	0	0
$737$	−288.000	−0.0143943
$738$	0	0
$739$	−4420.00	−0.220017	−0.110008	−	0.993931i	$-0.535088\pi$
$739$	−4420.00	−0.220017	−0.110008	+	0.993931i	$0.535088\pi$
$740$	0	0
$741$	936.000	0.0464033
$742$	0	0
$743$	−34560.0	−1.70644	−0.853219	−	0.521553i	$-0.825353\pi$
$743$	−34560.0	−1.70644	−0.853219	+	0.521553i	$0.825353\pi$
$744$	0	0
$745$	15990.0	0.786347
$746$	0	0
$747$	5508.00	0.269782
$748$	0	0
$749$	3728.00	0.181867
$750$	0	0
$751$	24792.0	1.20462	0.602312	−	0.798261i	$-0.294246\pi$
$751$	24792.0	1.20462	0.602312	+	0.798261i	$0.294246\pi$
$752$	0	0
$753$	19056.0	0.922230
$754$	0	0
$755$	−16920.0	−0.815605
$756$	0	0
$757$	−2166.00	−0.103996	−0.0519978	−	0.998647i	$-0.516559\pi$
$757$	−2166.00	−0.103996	−0.0519978	+	0.998647i	$0.516559\pi$
$758$	0	0
$759$	32832.0	1.57013
$760$	0	0
$761$	−10622.0	−0.505975	−0.252988	−	0.967470i	$-0.581413\pi$
$761$	−10622.0	−0.505975	−0.252988	+	0.967470i	$0.581413\pi$
$762$	0	0
$763$	−1784.00	−0.0846463
$764$	0	0
$765$	−1710.00	−0.0808172
$766$	0	0
$767$	4176.00	0.196593
$768$	0	0
$769$	29826.0	1.39864	0.699319	−	0.714809i	$-0.253487\pi$
$769$	29826.0	1.39864	0.699319	+	0.714809i	$0.253487\pi$
$770$	0	0
$771$	4266.00	0.199269
$772$	0	0
$773$	6386.00	0.297139	0.148570	−	0.988902i	$-0.452533\pi$
$773$	6386.00	0.297139	0.148570	+	0.988902i	$0.452533\pi$
$774$	0	0
$775$	−3000.00	−0.139049
$776$	0	0
$777$	1800.00	0.0831076
$778$	0	0
$779$	−18824.0	−0.865776
$780$	0	0
$781$	6912.00	0.316685
$782$	0	0
$783$	−2106.00	−0.0961204
$784$	0	0
$785$	16510.0	0.750659
$786$	0	0
$787$	−3516.00	−0.159253	−0.0796263	−	0.996825i	$-0.525373\pi$
$787$	−3516.00	−0.159253	−0.0796263	+	0.996825i	$0.525373\pi$
$788$	0	0
$789$	−21672.0	−0.977875
$790$	0	0
$791$	3144.00	0.141325
$792$	0	0
$793$	−1332.00	−0.0596478
$794$	0	0
$795$	10050.0	0.448348
$796$	0	0
$797$	−25030.0	−1.11243	−0.556216	−	0.831038i	$-0.687747\pi$
$797$	−25030.0	−1.11243	−0.556216	+	0.831038i	$0.687747\pi$
$798$	0	0
$799$	−10640.0	−0.471109
$800$	0	0
$801$	8946.00	0.394621
$802$	0	0
$803$	−12816.0	−0.563221
$804$	0	0
$805$	−3040.00	−0.133101
$806$	0	0
$807$	9558.00	0.416924
$808$	0	0
$809$	7962.00	0.346019	0.173009	−	0.984920i	$-0.444651\pi$
$809$	7962.00	0.346019	0.173009	+	0.984920i	$0.444651\pi$
$810$	0	0
$811$	34668.0	1.50106	0.750529	−	0.660837i	$-0.229799\pi$
$811$	34668.0	1.50106	0.750529	+	0.660837i	$0.229799\pi$
$812$	0	0
$813$	768.000	0.0331303
$814$	0	0
$815$	11260.0	0.483952
$816$	0	0
$817$	−25168.0	−1.07774
$818$	0	0
$819$	216.000	0.00921569
$820$	0	0
$821$	250.000	0.0106274	0.00531368	−	0.999986i	$-0.498309\pi$
$821$	250.000	0.0106274	0.00531368	+	0.999986i	$0.498309\pi$
$822$	0	0
$823$	6388.00	0.270561	0.135280	−	0.990807i	$-0.456806\pi$
$823$	6388.00	0.270561	0.135280	+	0.990807i	$0.456806\pi$
$824$	0	0
$825$	−5400.00	−0.227883
$826$	0	0
$827$	−3932.00	−0.165331	−0.0826657	−	0.996577i	$-0.526343\pi$
$827$	−3932.00	−0.165331	−0.0826657	+	0.996577i	$0.526343\pi$
$828$	0	0
$829$	−25906.0	−1.08535	−0.542673	−	0.839944i	$-0.682588\pi$
$829$	−25906.0	−1.08535	−0.542673	+	0.839944i	$0.682588\pi$
$830$	0	0
$831$	−17826.0	−0.744136
$832$	0	0
$833$	−12426.0	−0.516849
$834$	0	0
$835$	−920.000	−0.0381292
$836$	0	0
$837$	−3240.00	−0.133800
$838$	0	0
$839$	9944.00	0.409184	0.204592	−	0.978847i	$-0.434413\pi$
$839$	9944.00	0.409184	0.204592	+	0.978847i	$0.434413\pi$
$840$	0	0
$841$	−18305.0	−0.750543
$842$	0	0
$843$	9606.00	0.392465
$844$	0	0
$845$	10805.0	0.439886
$846$	0	0
$847$	−15412.0	−0.625221
$848$	0	0
$849$	−11820.0	−0.477811
$850$	0	0
$851$	22800.0	0.918418
$852$	0	0
$853$	−14630.0	−0.587247	−0.293623	−	0.955921i	$-0.594861\pi$
$853$	−14630.0	−0.587247	−0.293623	+	0.955921i	$0.594861\pi$
$854$	0	0
$855$	2340.00	0.0935980
$856$	0	0
$857$	478.000	0.0190527	0.00952635	−	0.999955i	$-0.496968\pi$
$857$	478.000	0.0190527	0.00952635	+	0.999955i	$0.496968\pi$
$858$	0	0
$859$	−24132.0	−0.958525	−0.479263	−	0.877672i	$-0.659096\pi$
$859$	−24132.0	−0.958525	−0.479263	+	0.877672i	$0.659096\pi$
$860$	0	0
$861$	−4344.00	−0.171943
$862$	0	0
$863$	−15776.0	−0.622273	−0.311136	−	0.950365i	$-0.600710\pi$
$863$	−15776.0	−0.622273	−0.311136	+	0.950365i	$0.600710\pi$
$864$	0	0
$865$	13230.0	0.520039
$866$	0	0
$867$	−10407.0	−0.407659
$868$	0	0
$869$	−45504.0	−1.77631
$870$	0	0
$871$	−24.0000	−0.000933650 0
$872$	0	0
$873$	14706.0	0.570129
$874$	0	0
$875$	500.000	0.0193178
$876$	0	0
$877$	−33542.0	−1.29149	−0.645743	−	0.763555i	$-0.723452\pi$
$877$	−33542.0	−1.29149	−0.645743	+	0.763555i	$0.723452\pi$
$878$	0	0
$879$	5478.00	0.210203
$880$	0	0
$881$	22858.0	0.874127	0.437063	−	0.899431i	$-0.356019\pi$
$881$	22858.0	0.874127	0.437063	+	0.899431i	$0.356019\pi$
$882$	0	0
$883$	−2764.00	−0.105341	−0.0526704	−	0.998612i	$-0.516773\pi$
$883$	−2764.00	−0.105341	−0.0526704	+	0.998612i	$0.516773\pi$
$884$	0	0
$885$	10440.0	0.396539
$886$	0	0
$887$	−6216.00	−0.235302	−0.117651	−	0.993055i	$-0.537536\pi$
$887$	−6216.00	−0.235302	−0.117651	+	0.993055i	$0.537536\pi$
$888$	0	0
$889$	2864.00	0.108049
$890$	0	0
$891$	−5832.00	−0.219281
$892$	0	0
$893$	14560.0	0.545612
$894$	0	0
$895$	−3040.00	−0.113537
$896$	0	0
$897$	2736.00	0.101842
$898$	0	0
$899$	9360.00	0.347245
$900$	0	0
$901$	−25460.0	−0.941394
$902$	0	0
$903$	−5808.00	−0.214040
$904$	0	0
$905$	−11230.0	−0.412484
$906$	0	0
$907$	18884.0	0.691326	0.345663	−	0.938359i	$-0.387654\pi$
$907$	18884.0	0.691326	0.345663	+	0.938359i	$0.387654\pi$
$908$	0	0
$909$	8010.00	0.292272
$910$	0	0
$911$	15232.0	0.553961	0.276981	−	0.960876i	$-0.410666\pi$
$911$	15232.0	0.553961	0.276981	+	0.960876i	$0.410666\pi$
$912$	0	0
$913$	−44064.0	−1.59727
$914$	0	0
$915$	−3330.00	−0.120313
$916$	0	0
$917$	−3232.00	−0.116390
$918$	0	0
$919$	−7744.00	−0.277966	−0.138983	−	0.990295i	$-0.544383\pi$
$919$	−7744.00	−0.277966	−0.138983	+	0.990295i	$0.544383\pi$
$920$	0	0
$921$	−19740.0	−0.706249
$922$	0	0
$923$	576.000	0.0205409
$924$	0	0
$925$	−3750.00	−0.133296
$926$	0	0
$927$	4716.00	0.167091
$928$	0	0
$929$	22266.0	0.786355	0.393177	−	0.919463i	$-0.371376\pi$
$929$	22266.0	0.786355	0.393177	+	0.919463i	$0.371376\pi$
$930$	0	0
$931$	17004.0	0.598586
$932$	0	0
$933$	17184.0	0.602978
$934$	0	0
$935$	13680.0	0.478485
$936$	0	0
$937$	16202.0	0.564884	0.282442	−	0.959284i	$-0.408856\pi$
$937$	16202.0	0.564884	0.282442	+	0.959284i	$0.408856\pi$
$938$	0	0
$939$	−5226.00	−0.181623
$940$	0	0
$941$	−53494.0	−1.85319	−0.926596	−	0.376057i	$-0.877280\pi$
$941$	−53494.0	−1.85319	−0.926596	+	0.376057i	$0.877280\pi$
$942$	0	0
$943$	−55024.0	−1.90014
$944$	0	0
$945$	540.000	0.0185886
$946$	0	0
$947$	2332.00	0.0800209	0.0400105	−	0.999199i	$-0.487261\pi$
$947$	2332.00	0.0800209	0.0400105	+	0.999199i	$0.487261\pi$
$948$	0	0
$949$	−1068.00	−0.0365319
$950$	0	0
$951$	26238.0	0.894664
$952$	0	0
$953$	15414.0	0.523933	0.261967	−	0.965077i	$-0.415629\pi$
$953$	15414.0	0.523933	0.261967	+	0.965077i	$0.415629\pi$
$954$	0	0
$955$	−19240.0	−0.651929
$956$	0	0
$957$	16848.0	0.569089
$958$	0	0
$959$	7080.00	0.238400
$960$	0	0
$961$	−15391.0	−0.516633
$962$	0	0
$963$	−8388.00	−0.280685
$964$	0	0
$965$	−10290.0	−0.343261
$966$	0	0
$967$	−35012.0	−1.16433	−0.582167	−	0.813070i	$-0.697795\pi$
$967$	−35012.0	−1.16433	−0.582167	+	0.813070i	$0.697795\pi$
$968$	0	0
$969$	−5928.00	−0.196527
$970$	0	0
$971$	−11360.0	−0.375448	−0.187724	−	0.982222i	$-0.560111\pi$
$971$	−11360.0	−0.375448	−0.187724	+	0.982222i	$0.560111\pi$
$972$	0	0
$973$	−3696.00	−0.121776
$974$	0	0
$975$	−450.000	−0.0147811
$976$	0	0
$977$	−24586.0	−0.805093	−0.402546	−	0.915400i	$-0.631875\pi$
$977$	−24586.0	−0.805093	−0.402546	+	0.915400i	$0.631875\pi$
$978$	0	0
$979$	−71568.0	−2.33639
$980$	0	0
$981$	4014.00	0.130639
$982$	0	0
$983$	−8832.00	−0.286569	−0.143284	−	0.989682i	$-0.545766\pi$
$983$	−8832.00	−0.286569	−0.143284	+	0.989682i	$0.545766\pi$
$984$	0	0
$985$	19190.0	0.620756
$986$	0	0
$987$	3360.00	0.108359
$988$	0	0
$989$	−73568.0	−2.36535
$990$	0	0
$991$	22912.0	0.734434	0.367217	−	0.930135i	$-0.380311\pi$
$991$	22912.0	0.734434	0.367217	+	0.930135i	$0.380311\pi$
$992$	0	0
$993$	7692.00	0.245819
$994$	0	0
$995$	−9960.00	−0.317340
$996$	0	0
$997$	−10974.0	−0.348596	−0.174298	−	0.984693i	$-0.555766\pi$
$997$	−10974.0	−0.348596	−0.174298	+	0.984693i	$0.555766\pi$
$998$	0	0
$999$	−4050.00	−0.128265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.4.a.h.1.1		1
3.2	odd	2		720.4.a.v.1.1		1
4.3	odd	2		120.4.a.a.1.1	✓	1
5.2	odd	4		1200.4.f.a.49.1		2
5.3	odd	4		1200.4.f.a.49.2		2
5.4	even	2		1200.4.a.k.1.1		1
8.3	odd	2		960.4.a.bf.1.1		1
8.5	even	2		960.4.a.o.1.1		1
12.11	even	2		360.4.a.l.1.1		1
20.3	even	4		600.4.f.i.49.1		2
20.7	even	4		600.4.f.i.49.2		2
20.19	odd	2		600.4.a.l.1.1		1
60.23	odd	4		1800.4.f.a.649.1		2
60.47	odd	4		1800.4.f.a.649.2		2
60.59	even	2		1800.4.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.a.1.1	✓	1	4.3	odd	2
240.4.a.h.1.1		1	1.1	even	1	trivial
360.4.a.l.1.1		1	12.11	even	2
600.4.a.l.1.1		1	20.19	odd	2
600.4.f.i.49.1		2	20.3	even	4
600.4.f.i.49.2		2	20.7	even	4
720.4.a.v.1.1		1	3.2	odd	2
960.4.a.o.1.1		1	8.5	even	2
960.4.a.bf.1.1		1	8.3	odd	2
1200.4.a.k.1.1		1	5.4	even	2
1200.4.f.a.49.1		2	5.2	odd	4
1200.4.f.a.49.2		2	5.3	odd	4
1800.4.a.n.1.1		1	60.59	even	2
1800.4.f.a.649.1		2	60.23	odd	4
1800.4.f.a.649.2		2	60.47	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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more