LMFDB - Embedded newform 1840.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} +5.00000 q^{5} +32.0000 q^{7} -26.0000 q^{9} +30.0000 q^{11} +19.0000 q^{13} +5.00000 q^{15} -60.0000 q^{17} +58.0000 q^{19} +32.0000 q^{21} -23.0000 q^{23} +25.0000 q^{25} -53.0000 q^{27} +85.0000 q^{29} +65.0000 q^{31} +30.0000 q^{33} +160.000 q^{35} -34.0000 q^{37} +19.0000 q^{39} +143.000 q^{41} +332.000 q^{43} -130.000 q^{45} +561.000 q^{47} +681.000 q^{49} -60.0000 q^{51} -422.000 q^{53} +150.000 q^{55} +58.0000 q^{57} -392.000 q^{59} -246.000 q^{61} -832.000 q^{63} +95.0000 q^{65} -894.000 q^{67} -23.0000 q^{69} +737.000 q^{71} +1041.00 q^{73} +25.0000 q^{75} +960.000 q^{77} -1114.00 q^{79} +649.000 q^{81} +936.000 q^{83} -300.000 q^{85} +85.0000 q^{87} +824.000 q^{89} +608.000 q^{91} +65.0000 q^{93} +290.000 q^{95} -868.000 q^{97} -780.000 q^{99} +O(q^{100})$

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
$3$	1.00000	0.192450	0.0962250	−	0.995360i	$-0.469323\pi$
$3$	1.00000	0.192450	0.0962250	+	0.995360i	$0.469323\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	32.0000	1.72784	0.863919	−	0.503631i	$-0.168003\pi$
$7$	32.0000	1.72784	0.863919	+	0.503631i	$0.168003\pi$
$8$	0	0
$9$	−26.0000	−0.962963
$10$	0	0
$11$	30.0000	0.822304	0.411152	−	0.911567i	$-0.365127\pi$
$11$	30.0000	0.822304	0.411152	+	0.911567i	$0.365127\pi$
$12$	0	0
$13$	19.0000	0.405358	0.202679	−	0.979245i	$-0.435035\pi$
$13$	19.0000	0.405358	0.202679	+	0.979245i	$0.435035\pi$
$14$	0	0
$15$	5.00000	0.0860663
$16$	0	0
$17$	−60.0000	−0.856008	−0.428004	−	0.903777i	$-0.640783\pi$
$17$	−60.0000	−0.856008	−0.428004	+	0.903777i	$0.640783\pi$
$18$	0	0
$19$	58.0000	0.700322	0.350161	−	0.936690i	$-0.386127\pi$
$19$	58.0000	0.700322	0.350161	+	0.936690i	$0.386127\pi$
$20$	0	0
$21$	32.0000	0.332522
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−53.0000	−0.377772
$28$	0	0
$29$	85.0000	0.544279	0.272140	−	0.962258i	$-0.412269\pi$
$29$	85.0000	0.544279	0.272140	+	0.962258i	$0.412269\pi$
$30$	0	0
$31$	65.0000	0.376592	0.188296	−	0.982112i	$-0.439704\pi$
$31$	65.0000	0.376592	0.188296	+	0.982112i	$0.439704\pi$
$32$	0	0
$33$	30.0000	0.158252
$34$	0	0
$35$	160.000	0.772712
$36$	0	0
$37$	−34.0000	−0.151069	−0.0755347	−	0.997143i	$-0.524066\pi$
$37$	−34.0000	−0.151069	−0.0755347	+	0.997143i	$0.524066\pi$
$38$	0	0
$39$	19.0000	0.0780112
$40$	0	0
$41$	143.000	0.544704	0.272352	−	0.962198i	$-0.412199\pi$
$41$	143.000	0.544704	0.272352	+	0.962198i	$0.412199\pi$
$42$	0	0
$43$	332.000	1.17743	0.588715	−	0.808340i	$-0.299634\pi$
$43$	332.000	1.17743	0.588715	+	0.808340i	$0.299634\pi$
$44$	0	0
$45$	−130.000	−0.430650
$46$	0	0
$47$	561.000	1.74107	0.870535	−	0.492107i	$-0.163773\pi$
$47$	561.000	1.74107	0.870535	+	0.492107i	$0.163773\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	−60.0000	−0.164739
$52$	0	0
$53$	−422.000	−1.09370	−0.546851	−	0.837230i	$-0.684173\pi$
$53$	−422.000	−1.09370	−0.546851	+	0.837230i	$0.684173\pi$
$54$	0	0
$55$	150.000	0.367745
$56$	0	0
$57$	58.0000	0.134777
$58$	0	0
$59$	−392.000	−0.864984	−0.432492	−	0.901638i	$-0.642366\pi$
$59$	−392.000	−0.864984	−0.432492	+	0.901638i	$0.642366\pi$
$60$	0	0
$61$	−246.000	−0.516345	−0.258173	−	0.966099i	$-0.583120\pi$
$61$	−246.000	−0.516345	−0.258173	+	0.966099i	$0.583120\pi$
$62$	0	0
$63$	−832.000	−1.66384
$64$	0	0
$65$	95.0000	0.181282
$66$	0	0
$67$	−894.000	−1.63014	−0.815071	−	0.579361i	$-0.803302\pi$
$67$	−894.000	−1.63014	−0.815071	+	0.579361i	$0.803302\pi$
$68$	0	0
$69$	−23.0000	−0.0401286
$70$	0	0
$71$	737.000	1.23191	0.615956	−	0.787780i	$-0.288770\pi$
$71$	737.000	1.23191	0.615956	+	0.787780i	$0.288770\pi$
$72$	0	0
$73$	1041.00	1.66904	0.834519	−	0.550979i	$-0.185745\pi$
$73$	1041.00	1.66904	0.834519	+	0.550979i	$0.185745\pi$
$74$	0	0
$75$	25.0000	0.0384900
$76$	0	0
$77$	960.000	1.42081
$78$	0	0
$79$	−1114.00	−1.58652	−0.793258	−	0.608885i	$-0.791617\pi$
$79$	−1114.00	−1.58652	−0.793258	+	0.608885i	$0.791617\pi$
$80$	0	0
$81$	649.000	0.890261
$82$	0	0
$83$	936.000	1.23782	0.618912	−	0.785461i	$-0.287574\pi$
$83$	936.000	1.23782	0.618912	+	0.785461i	$0.287574\pi$
$84$	0	0
$85$	−300.000	−0.382818
$86$	0	0
$87$	85.0000	0.104747
$88$	0	0
$89$	824.000	0.981391	0.490696	−	0.871331i	$-0.336743\pi$
$89$	824.000	0.981391	0.490696	+	0.871331i	$0.336743\pi$
$90$	0	0
$91$	608.000	0.700393
$92$	0	0
$93$	65.0000	0.0724751
$94$	0	0
$95$	290.000	0.313193
$96$	0	0
$97$	−868.000	−0.908578	−0.454289	−	0.890854i	$-0.650107\pi$
$97$	−868.000	−0.908578	−0.454289	+	0.890854i	$0.650107\pi$
$98$	0	0
$99$	−780.000	−0.791848
$100$	0	0
$101$	50.0000	0.0492593	0.0246296	−	0.999697i	$-0.492159\pi$
$101$	50.0000	0.0492593	0.0246296	+	0.999697i	$0.492159\pi$
$102$	0	0
$103$	−440.000	−0.420917	−0.210459	−	0.977603i	$-0.567496\pi$
$103$	−440.000	−0.420917	−0.210459	+	0.977603i	$0.567496\pi$
$104$	0	0
$105$	160.000	0.148709
$106$	0	0
$107$	318.000	0.287310	0.143655	−	0.989628i	$-0.454114\pi$
$107$	318.000	0.287310	0.143655	+	0.989628i	$0.454114\pi$
$108$	0	0
$109$	−828.000	−0.727596	−0.363798	−	0.931478i	$-0.618520\pi$
$109$	−828.000	−0.727596	−0.363798	+	0.931478i	$0.618520\pi$
$110$	0	0
$111$	−34.0000	−0.0290733
$112$	0	0
$113$	208.000	0.173159	0.0865796	−	0.996245i	$-0.472406\pi$
$113$	208.000	0.173159	0.0865796	+	0.996245i	$0.472406\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−494.000	−0.390345
$118$	0	0
$119$	−1920.00	−1.47904
$120$	0	0
$121$	−431.000	−0.323817
$122$	0	0
$123$	143.000	0.104828
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	495.000	0.345859	0.172930	−	0.984934i	$-0.444677\pi$
$127$	495.000	0.345859	0.172930	+	0.984934i	$0.444677\pi$
$128$	0	0
$129$	332.000	0.226597
$130$	0	0
$131$	−335.000	−0.223428	−0.111714	−	0.993740i	$-0.535634\pi$
$131$	−335.000	−0.223428	−0.111714	+	0.993740i	$0.535634\pi$
$132$	0	0
$133$	1856.00	1.21004
$134$	0	0
$135$	−265.000	−0.168945
$136$	0	0
$137$	−652.000	−0.406599	−0.203300	−	0.979117i	$-0.565167\pi$
$137$	−652.000	−0.406599	−0.203300	+	0.979117i	$0.565167\pi$
$138$	0	0
$139$	1999.00	1.21981	0.609903	−	0.792476i	$-0.291208\pi$
$139$	1999.00	1.21981	0.609903	+	0.792476i	$0.291208\pi$
$140$	0	0
$141$	561.000	0.335069
$142$	0	0
$143$	570.000	0.333327
$144$	0	0
$145$	425.000	0.243409
$146$	0	0
$147$	681.000	0.382095
$148$	0	0
$149$	1870.00	1.02816	0.514082	−	0.857741i	$-0.328133\pi$
$149$	1870.00	1.02816	0.514082	+	0.857741i	$0.328133\pi$
$150$	0	0
$151$	−1463.00	−0.788459	−0.394229	−	0.919012i	$-0.628988\pi$
$151$	−1463.00	−0.788459	−0.394229	+	0.919012i	$0.628988\pi$
$152$	0	0
$153$	1560.00	0.824304
$154$	0	0
$155$	325.000	0.168417
$156$	0	0
$157$	−3004.00	−1.52704	−0.763520	−	0.645784i	$-0.776531\pi$
$157$	−3004.00	−1.52704	−0.763520	+	0.645784i	$0.776531\pi$
$158$	0	0
$159$	−422.000	−0.210483
$160$	0	0
$161$	−736.000	−0.360279
$162$	0	0
$163$	4047.00	1.94470	0.972348	−	0.233536i	$-0.0750296\pi$
$163$	4047.00	1.94470	0.972348	+	0.233536i	$0.0750296\pi$
$164$	0	0
$165$	150.000	0.0707726
$166$	0	0
$167$	2112.00	0.978632	0.489316	−	0.872107i	$-0.337247\pi$
$167$	2112.00	0.978632	0.489316	+	0.872107i	$0.337247\pi$
$168$	0	0
$169$	−1836.00	−0.835685
$170$	0	0
$171$	−1508.00	−0.674384
$172$	0	0
$173$	342.000	0.150299	0.0751496	−	0.997172i	$-0.476057\pi$
$173$	342.000	0.150299	0.0751496	+	0.997172i	$0.476057\pi$
$174$	0	0
$175$	800.000	0.345568
$176$	0	0
$177$	−392.000	−0.166466
$178$	0	0
$179$	−1689.00	−0.705261	−0.352631	−	0.935763i	$-0.614713\pi$
$179$	−1689.00	−0.705261	−0.352631	+	0.935763i	$0.614713\pi$
$180$	0	0
$181$	−194.000	−0.0796680	−0.0398340	−	0.999206i	$-0.512683\pi$
$181$	−194.000	−0.0796680	−0.0398340	+	0.999206i	$0.512683\pi$
$182$	0	0
$183$	−246.000	−0.0993707
$184$	0	0
$185$	−170.000	−0.0675603
$186$	0	0
$187$	−1800.00	−0.703899
$188$	0	0
$189$	−1696.00	−0.652729
$190$	0	0
$191$	298.000	0.112893	0.0564464	−	0.998406i	$-0.482023\pi$
$191$	298.000	0.112893	0.0564464	+	0.998406i	$0.482023\pi$
$192$	0	0
$193$	2301.00	0.858184	0.429092	−	0.903261i	$-0.358834\pi$
$193$	2301.00	0.858184	0.429092	+	0.903261i	$0.358834\pi$
$194$	0	0
$195$	95.0000	0.0348876
$196$	0	0
$197$	−535.000	−0.193488	−0.0967441	−	0.995309i	$-0.530843\pi$
$197$	−535.000	−0.193488	−0.0967441	+	0.995309i	$0.530843\pi$
$198$	0	0
$199$	−278.000	−0.0990296	−0.0495148	−	0.998773i	$-0.515768\pi$
$199$	−278.000	−0.0990296	−0.0495148	+	0.998773i	$0.515768\pi$
$200$	0	0
$201$	−894.000	−0.313721
$202$	0	0
$203$	2720.00	0.940426
$204$	0	0
$205$	715.000	0.243599
$206$	0	0
$207$	598.000	0.200792
$208$	0	0
$209$	1740.00	0.575877
$210$	0	0
$211$	2444.00	0.797402	0.398701	−	0.917081i	$-0.369461\pi$
$211$	2444.00	0.797402	0.398701	+	0.917081i	$0.369461\pi$
$212$	0	0
$213$	737.000	0.237082
$214$	0	0
$215$	1660.00	0.526563
$216$	0	0
$217$	2080.00	0.650689
$218$	0	0
$219$	1041.00	0.321207
$220$	0	0
$221$	−1140.00	−0.346990
$222$	0	0
$223$	5368.00	1.61196	0.805982	−	0.591940i	$-0.201638\pi$
$223$	5368.00	1.61196	0.805982	+	0.591940i	$0.201638\pi$
$224$	0	0
$225$	−650.000	−0.192593
$226$	0	0
$227$	5780.00	1.69001	0.845005	−	0.534759i	$-0.179598\pi$
$227$	5780.00	1.69001	0.845005	+	0.534759i	$0.179598\pi$
$228$	0	0
$229$	−1064.00	−0.307035	−0.153518	−	0.988146i	$-0.549060\pi$
$229$	−1064.00	−0.307035	−0.153518	+	0.988146i	$0.549060\pi$
$230$	0	0
$231$	960.000	0.273434
$232$	0	0
$233$	−2833.00	−0.796549	−0.398275	−	0.917266i	$-0.630391\pi$
$233$	−2833.00	−0.796549	−0.398275	+	0.917266i	$0.630391\pi$
$234$	0	0
$235$	2805.00	0.778630
$236$	0	0
$237$	−1114.00	−0.305325
$238$	0	0
$239$	−6831.00	−1.84879	−0.924395	−	0.381437i	$-0.875429\pi$
$239$	−6831.00	−1.84879	−0.924395	+	0.381437i	$0.875429\pi$
$240$	0	0
$241$	−1794.00	−0.479509	−0.239755	−	0.970834i	$-0.577067\pi$
$241$	−1794.00	−0.479509	−0.239755	+	0.970834i	$0.577067\pi$
$242$	0	0
$243$	2080.00	0.549103
$244$	0	0
$245$	3405.00	0.887908
$246$	0	0
$247$	1102.00	0.283881
$248$	0	0
$249$	936.000	0.238219
$250$	0	0
$251$	6672.00	1.67782	0.838910	−	0.544270i	$-0.183193\pi$
$251$	6672.00	1.67782	0.838910	+	0.544270i	$0.183193\pi$
$252$	0	0
$253$	−690.000	−0.171462
$254$	0	0
$255$	−300.000	−0.0736734
$256$	0	0
$257$	291.000	0.0706307	0.0353153	−	0.999376i	$-0.488756\pi$
$257$	291.000	0.0706307	0.0353153	+	0.999376i	$0.488756\pi$
$258$	0	0
$259$	−1088.00	−0.261023
$260$	0	0
$261$	−2210.00	−0.524121
$262$	0	0
$263$	5306.00	1.24404	0.622019	−	0.783002i	$-0.286313\pi$
$263$	5306.00	1.24404	0.622019	+	0.783002i	$0.286313\pi$
$264$	0	0
$265$	−2110.00	−0.489118
$266$	0	0
$267$	824.000	0.188869
$268$	0	0
$269$	7785.00	1.76453	0.882267	−	0.470749i	$-0.156016\pi$
$269$	7785.00	1.76453	0.882267	+	0.470749i	$0.156016\pi$
$270$	0	0
$271$	−3440.00	−0.771089	−0.385544	−	0.922689i	$-0.625986\pi$
$271$	−3440.00	−0.771089	−0.385544	+	0.922689i	$0.625986\pi$
$272$	0	0
$273$	608.000	0.134791
$274$	0	0
$275$	750.000	0.164461
$276$	0	0
$277$	−7643.00	−1.65785	−0.828923	−	0.559363i	$-0.811046\pi$
$277$	−7643.00	−1.65785	−0.828923	+	0.559363i	$0.811046\pi$
$278$	0	0
$279$	−1690.00	−0.362644
$280$	0	0
$281$	5408.00	1.14809	0.574047	−	0.818823i	$-0.305373\pi$
$281$	5408.00	1.14809	0.574047	+	0.818823i	$0.305373\pi$
$282$	0	0
$283$	−4402.00	−0.924635	−0.462318	−	0.886714i	$-0.652982\pi$
$283$	−4402.00	−0.924635	−0.462318	+	0.886714i	$0.652982\pi$
$284$	0	0
$285$	290.000	0.0602741
$286$	0	0
$287$	4576.00	0.941159
$288$	0	0
$289$	−1313.00	−0.267250
$290$	0	0
$291$	−868.000	−0.174856
$292$	0	0
$293$	9808.00	1.95560	0.977798	−	0.209551i	$-0.0672002\pi$
$293$	9808.00	1.95560	0.977798	+	0.209551i	$0.0672002\pi$
$294$	0	0
$295$	−1960.00	−0.386833
$296$	0	0
$297$	−1590.00	−0.310644
$298$	0	0
$299$	−437.000	−0.0845230
$300$	0	0
$301$	10624.0	2.03441
$302$	0	0
$303$	50.0000	0.00947995
$304$	0	0
$305$	−1230.00	−0.230917
$306$	0	0
$307$	3288.00	0.611258	0.305629	−	0.952151i	$-0.401133\pi$
$307$	3288.00	0.611258	0.305629	+	0.952151i	$0.401133\pi$
$308$	0	0
$309$	−440.000	−0.0810056
$310$	0	0
$311$	−3919.00	−0.714553	−0.357277	−	0.933999i	$-0.616295\pi$
$311$	−3919.00	−0.714553	−0.357277	+	0.933999i	$0.616295\pi$
$312$	0	0
$313$	5112.00	0.923154	0.461577	−	0.887100i	$-0.347284\pi$
$313$	5112.00	0.923154	0.461577	+	0.887100i	$0.347284\pi$
$314$	0	0
$315$	−4160.00	−0.744093
$316$	0	0
$317$	8058.00	1.42770	0.713852	−	0.700296i	$-0.246949\pi$
$317$	8058.00	1.42770	0.713852	+	0.700296i	$0.246949\pi$
$318$	0	0
$319$	2550.00	0.447563
$320$	0	0
$321$	318.000	0.0552929
$322$	0	0
$323$	−3480.00	−0.599481
$324$	0	0
$325$	475.000	0.0810716
$326$	0	0
$327$	−828.000	−0.140026
$328$	0	0
$329$	17952.0	3.00829
$330$	0	0
$331$	2555.00	0.424276	0.212138	−	0.977240i	$-0.431957\pi$
$331$	2555.00	0.424276	0.212138	+	0.977240i	$0.431957\pi$
$332$	0	0
$333$	884.000	0.145474
$334$	0	0
$335$	−4470.00	−0.729021
$336$	0	0
$337$	−6272.00	−1.01382	−0.506910	−	0.861999i	$-0.669213\pi$
$337$	−6272.00	−1.01382	−0.506910	+	0.861999i	$0.669213\pi$
$338$	0	0
$339$	208.000	0.0333245
$340$	0	0
$341$	1950.00	0.309673
$342$	0	0
$343$	10816.0	1.70265
$344$	0	0
$345$	−115.000	−0.0179461
$346$	0	0
$347$	−6844.00	−1.05880	−0.529402	−	0.848371i	$-0.677584\pi$
$347$	−6844.00	−1.05880	−0.529402	+	0.848371i	$0.677584\pi$
$348$	0	0
$349$	4057.00	0.622253	0.311126	−	0.950369i	$-0.399294\pi$
$349$	4057.00	0.622253	0.311126	+	0.950369i	$0.399294\pi$
$350$	0	0
$351$	−1007.00	−0.153133
$352$	0	0
$353$	−7155.00	−1.07882	−0.539408	−	0.842044i	$-0.681352\pi$
$353$	−7155.00	−1.07882	−0.539408	+	0.842044i	$0.681352\pi$
$354$	0	0
$355$	3685.00	0.550928
$356$	0	0
$357$	−1920.00	−0.284642
$358$	0	0
$359$	5434.00	0.798873	0.399437	−	0.916761i	$-0.369206\pi$
$359$	5434.00	0.798873	0.399437	+	0.916761i	$0.369206\pi$
$360$	0	0
$361$	−3495.00	−0.509549
$362$	0	0
$363$	−431.000	−0.0623185
$364$	0	0
$365$	5205.00	0.746417
$366$	0	0
$367$	2944.00	0.418734	0.209367	−	0.977837i	$-0.432860\pi$
$367$	2944.00	0.418734	0.209367	+	0.977837i	$0.432860\pi$
$368$	0	0
$369$	−3718.00	−0.524529
$370$	0	0
$371$	−13504.0	−1.88974
$372$	0	0
$373$	−5962.00	−0.827616	−0.413808	−	0.910364i	$-0.635801\pi$
$373$	−5962.00	−0.827616	−0.413808	+	0.910364i	$0.635801\pi$
$374$	0	0
$375$	125.000	0.0172133
$376$	0	0
$377$	1615.00	0.220628
$378$	0	0
$379$	5856.00	0.793674	0.396837	−	0.917889i	$-0.370108\pi$
$379$	5856.00	0.793674	0.396837	+	0.917889i	$0.370108\pi$
$380$	0	0
$381$	495.000	0.0665607
$382$	0	0
$383$	−6456.00	−0.861322	−0.430661	−	0.902514i	$-0.641719\pi$
$383$	−6456.00	−0.861322	−0.430661	+	0.902514i	$0.641719\pi$
$384$	0	0
$385$	4800.00	0.635404
$386$	0	0
$387$	−8632.00	−1.13382
$388$	0	0
$389$	−152.000	−0.0198116	−0.00990579	−	0.999951i	$-0.503153\pi$
$389$	−152.000	−0.0198116	−0.00990579	+	0.999951i	$0.503153\pi$
$390$	0	0
$391$	1380.00	0.178490
$392$	0	0
$393$	−335.000	−0.0429988
$394$	0	0
$395$	−5570.00	−0.709512
$396$	0	0
$397$	11603.0	1.46685	0.733423	−	0.679773i	$-0.237921\pi$
$397$	11603.0	1.46685	0.733423	+	0.679773i	$0.237921\pi$
$398$	0	0
$399$	1856.00	0.232873
$400$	0	0
$401$	602.000	0.0749687	0.0374843	−	0.999297i	$-0.488066\pi$
$401$	602.000	0.0749687	0.0374843	+	0.999297i	$0.488066\pi$
$402$	0	0
$403$	1235.00	0.152654
$404$	0	0
$405$	3245.00	0.398137
$406$	0	0
$407$	−1020.00	−0.124225
$408$	0	0
$409$	3629.00	0.438735	0.219367	−	0.975642i	$-0.429601\pi$
$409$	3629.00	0.438735	0.219367	+	0.975642i	$0.429601\pi$
$410$	0	0
$411$	−652.000	−0.0782501
$412$	0	0
$413$	−12544.0	−1.49455
$414$	0	0
$415$	4680.00	0.553571
$416$	0	0
$417$	1999.00	0.234752
$418$	0	0
$419$	−92.0000	−0.0107267	−0.00536336	−	0.999986i	$-0.501707\pi$
$419$	−92.0000	−0.0107267	−0.00536336	+	0.999986i	$0.501707\pi$
$420$	0	0
$421$	−8280.00	−0.958533	−0.479267	−	0.877669i	$-0.659097\pi$
$421$	−8280.00	−0.958533	−0.479267	+	0.877669i	$0.659097\pi$
$422$	0	0
$423$	−14586.0	−1.67659
$424$	0	0
$425$	−1500.00	−0.171202
$426$	0	0
$427$	−7872.00	−0.892161
$428$	0	0
$429$	570.000	0.0641489
$430$	0	0
$431$	15372.0	1.71797	0.858983	−	0.512004i	$-0.171097\pi$
$431$	15372.0	1.71797	0.858983	+	0.512004i	$0.171097\pi$
$432$	0	0
$433$	−11342.0	−1.25880	−0.629402	−	0.777080i	$-0.716700\pi$
$433$	−11342.0	−1.25880	−0.629402	+	0.777080i	$0.716700\pi$
$434$	0	0
$435$	425.000	0.0468441
$436$	0	0
$437$	−1334.00	−0.146027
$438$	0	0
$439$	−4593.00	−0.499344	−0.249672	−	0.968330i	$-0.580323\pi$
$439$	−4593.00	−0.499344	−0.249672	+	0.968330i	$0.580323\pi$
$440$	0	0
$441$	−17706.0	−1.91189
$442$	0	0
$443$	−967.000	−0.103710	−0.0518550	−	0.998655i	$-0.516513\pi$
$443$	−967.000	−0.103710	−0.0518550	+	0.998655i	$0.516513\pi$
$444$	0	0
$445$	4120.00	0.438892
$446$	0	0
$447$	1870.00	0.197870
$448$	0	0
$449$	−11210.0	−1.17825	−0.589123	−	0.808043i	$-0.700527\pi$
$449$	−11210.0	−1.17825	−0.589123	+	0.808043i	$0.700527\pi$
$450$	0	0
$451$	4290.00	0.447912
$452$	0	0
$453$	−1463.00	−0.151739
$454$	0	0
$455$	3040.00	0.313225
$456$	0	0
$457$	−72.0000	−0.00736984	−0.00368492	−	0.999993i	$-0.501173\pi$
$457$	−72.0000	−0.00736984	−0.00368492	+	0.999993i	$0.501173\pi$
$458$	0	0
$459$	3180.00	0.323376
$460$	0	0
$461$	5137.00	0.518989	0.259495	−	0.965745i	$-0.416444\pi$
$461$	5137.00	0.518989	0.259495	+	0.965745i	$0.416444\pi$
$462$	0	0
$463$	19528.0	1.96014	0.980068	−	0.198661i	$-0.0636594\pi$
$463$	19528.0	1.96014	0.980068	+	0.198661i	$0.0636594\pi$
$464$	0	0
$465$	325.000	0.0324119
$466$	0	0
$467$	−14874.0	−1.47385	−0.736924	−	0.675976i	$-0.763722\pi$
$467$	−14874.0	−1.47385	−0.736924	+	0.675976i	$0.763722\pi$
$468$	0	0
$469$	−28608.0	−2.81662
$470$	0	0
$471$	−3004.00	−0.293879
$472$	0	0
$473$	9960.00	0.968206
$474$	0	0
$475$	1450.00	0.140064
$476$	0	0
$477$	10972.0	1.05319
$478$	0	0
$479$	−10568.0	−1.00807	−0.504034	−	0.863684i	$-0.668151\pi$
$479$	−10568.0	−1.00807	−0.504034	+	0.863684i	$0.668151\pi$
$480$	0	0
$481$	−646.000	−0.0612371
$482$	0	0
$483$	−736.000	−0.0693357
$484$	0	0
$485$	−4340.00	−0.406328
$486$	0	0
$487$	9311.00	0.866369	0.433184	−	0.901305i	$-0.357390\pi$
$487$	9311.00	0.866369	0.433184	+	0.901305i	$0.357390\pi$
$488$	0	0
$489$	4047.00	0.374257
$490$	0	0
$491$	16511.0	1.51758	0.758789	−	0.651336i	$-0.225791\pi$
$491$	16511.0	1.51758	0.758789	+	0.651336i	$0.225791\pi$
$492$	0	0
$493$	−5100.00	−0.465908
$494$	0	0
$495$	−3900.00	−0.354125
$496$	0	0
$497$	23584.0	2.12855
$498$	0	0
$499$	−2331.00	−0.209118	−0.104559	−	0.994519i	$-0.533343\pi$
$499$	−2331.00	−0.209118	−0.104559	+	0.994519i	$0.533343\pi$
$500$	0	0
$501$	2112.00	0.188338
$502$	0	0
$503$	−17062.0	−1.51244	−0.756220	−	0.654318i	$-0.772956\pi$
$503$	−17062.0	−1.51244	−0.756220	+	0.654318i	$0.772956\pi$
$504$	0	0
$505$	250.000	0.0220294
$506$	0	0
$507$	−1836.00	−0.160828
$508$	0	0
$509$	17169.0	1.49509	0.747547	−	0.664209i	$-0.231232\pi$
$509$	17169.0	1.49509	0.747547	+	0.664209i	$0.231232\pi$
$510$	0	0
$511$	33312.0	2.88383
$512$	0	0
$513$	−3074.00	−0.264562
$514$	0	0
$515$	−2200.00	−0.188240
$516$	0	0
$517$	16830.0	1.43169
$518$	0	0
$519$	342.000	0.0289251
$520$	0	0
$521$	−10060.0	−0.845944	−0.422972	−	0.906143i	$-0.639013\pi$
$521$	−10060.0	−0.845944	−0.422972	+	0.906143i	$0.639013\pi$
$522$	0	0
$523$	−15244.0	−1.27452	−0.637260	−	0.770649i	$-0.719932\pi$
$523$	−15244.0	−1.27452	−0.637260	+	0.770649i	$0.719932\pi$
$524$	0	0
$525$	800.000	0.0665045
$526$	0	0
$527$	−3900.00	−0.322366
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	10192.0	0.832948
$532$	0	0
$533$	2717.00	0.220800
$534$	0	0
$535$	1590.00	0.128489
$536$	0	0
$537$	−1689.00	−0.135728
$538$	0	0
$539$	20430.0	1.63262
$540$	0	0
$541$	16389.0	1.30244	0.651218	−	0.758891i	$-0.274258\pi$
$541$	16389.0	1.30244	0.651218	+	0.758891i	$0.274258\pi$
$542$	0	0
$543$	−194.000	−0.0153321
$544$	0	0
$545$	−4140.00	−0.325391
$546$	0	0
$547$	−17913.0	−1.40019	−0.700096	−	0.714049i	$-0.746859\pi$
$547$	−17913.0	−1.40019	−0.700096	+	0.714049i	$0.746859\pi$
$548$	0	0
$549$	6396.00	0.497222
$550$	0	0
$551$	4930.00	0.381171
$552$	0	0
$553$	−35648.0	−2.74124
$554$	0	0
$555$	−170.000	−0.0130020
$556$	0	0
$557$	−18224.0	−1.38631	−0.693156	−	0.720788i	$-0.743780\pi$
$557$	−18224.0	−1.38631	−0.693156	+	0.720788i	$0.743780\pi$
$558$	0	0
$559$	6308.00	0.477281
$560$	0	0
$561$	−1800.00	−0.135465
$562$	0	0
$563$	4860.00	0.363809	0.181905	−	0.983316i	$-0.441774\pi$
$563$	4860.00	0.363809	0.181905	+	0.983316i	$0.441774\pi$
$564$	0	0
$565$	1040.00	0.0774392
$566$	0	0
$567$	20768.0	1.53823
$568$	0	0
$569$	−12438.0	−0.916394	−0.458197	−	0.888851i	$-0.651505\pi$
$569$	−12438.0	−0.916394	−0.458197	+	0.888851i	$0.651505\pi$
$570$	0	0
$571$	−7948.00	−0.582510	−0.291255	−	0.956645i	$-0.594073\pi$
$571$	−7948.00	−0.582510	−0.291255	+	0.956645i	$0.594073\pi$
$572$	0	0
$573$	298.000	0.0217262
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	9061.00	0.653751	0.326876	−	0.945067i	$-0.394004\pi$
$577$	9061.00	0.653751	0.326876	+	0.945067i	$0.394004\pi$
$578$	0	0
$579$	2301.00	0.165158
$580$	0	0
$581$	29952.0	2.13876
$582$	0	0
$583$	−12660.0	−0.899354
$584$	0	0
$585$	−2470.00	−0.174567
$586$	0	0
$587$	−8559.00	−0.601819	−0.300909	−	0.953653i	$-0.597290\pi$
$587$	−8559.00	−0.601819	−0.300909	+	0.953653i	$0.597290\pi$
$588$	0	0
$589$	3770.00	0.263735
$590$	0	0
$591$	−535.000	−0.0372368
$592$	0	0
$593$	−8850.00	−0.612860	−0.306430	−	0.951893i	$-0.599135\pi$
$593$	−8850.00	−0.612860	−0.306430	+	0.951893i	$0.599135\pi$
$594$	0	0
$595$	−9600.00	−0.661448
$596$	0	0
$597$	−278.000	−0.0190583
$598$	0	0
$599$	7952.00	0.542421	0.271210	−	0.962520i	$-0.412576\pi$
$599$	7952.00	0.542421	0.271210	+	0.962520i	$0.412576\pi$
$600$	0	0
$601$	27085.0	1.83830	0.919152	−	0.393904i	$-0.128876\pi$
$601$	27085.0	1.83830	0.919152	+	0.393904i	$0.128876\pi$
$602$	0	0
$603$	23244.0	1.56977
$604$	0	0
$605$	−2155.00	−0.144815
$606$	0	0
$607$	5204.00	0.347980	0.173990	−	0.984747i	$-0.444334\pi$
$607$	5204.00	0.347980	0.173990	+	0.984747i	$0.444334\pi$
$608$	0	0
$609$	2720.00	0.180985
$610$	0	0
$611$	10659.0	0.705756
$612$	0	0
$613$	−4348.00	−0.286483	−0.143241	−	0.989688i	$-0.545753\pi$
$613$	−4348.00	−0.286483	−0.143241	+	0.989688i	$0.545753\pi$
$614$	0	0
$615$	715.000	0.0468806
$616$	0	0
$617$	−4066.00	−0.265301	−0.132651	−	0.991163i	$-0.542349\pi$
$617$	−4066.00	−0.265301	−0.132651	+	0.991163i	$0.542349\pi$
$618$	0	0
$619$	6080.00	0.394791	0.197396	−	0.980324i	$-0.436752\pi$
$619$	6080.00	0.394791	0.197396	+	0.980324i	$0.436752\pi$
$620$	0	0
$621$	1219.00	0.0787710
$622$	0	0
$623$	26368.0	1.69568
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	1740.00	0.110828
$628$	0	0
$629$	2040.00	0.129317
$630$	0	0
$631$	−21200.0	−1.33749	−0.668747	−	0.743490i	$-0.733169\pi$
$631$	−21200.0	−1.33749	−0.668747	+	0.743490i	$0.733169\pi$
$632$	0	0
$633$	2444.00	0.153460
$634$	0	0
$635$	2475.00	0.154673
$636$	0	0
$637$	12939.0	0.804807
$638$	0	0
$639$	−19162.0	−1.18629
$640$	0	0
$641$	14016.0	0.863649	0.431824	−	0.901958i	$-0.357870\pi$
$641$	14016.0	0.863649	0.431824	+	0.901958i	$0.357870\pi$
$642$	0	0
$643$	−23894.0	−1.46545	−0.732727	−	0.680522i	$-0.761753\pi$
$643$	−23894.0	−1.46545	−0.732727	+	0.680522i	$0.761753\pi$
$644$	0	0
$645$	1660.00	0.101337
$646$	0	0
$647$	−27111.0	−1.64736	−0.823681	−	0.567053i	$-0.808083\pi$
$647$	−27111.0	−1.64736	−0.823681	+	0.567053i	$0.808083\pi$
$648$	0	0
$649$	−11760.0	−0.711279
$650$	0	0
$651$	2080.00	0.125225
$652$	0	0
$653$	7819.00	0.468578	0.234289	−	0.972167i	$-0.424724\pi$
$653$	7819.00	0.468578	0.234289	+	0.972167i	$0.424724\pi$
$654$	0	0
$655$	−1675.00	−0.0999201
$656$	0	0
$657$	−27066.0	−1.60722
$658$	0	0
$659$	8712.00	0.514979	0.257490	−	0.966281i	$-0.417105\pi$
$659$	8712.00	0.514979	0.257490	+	0.966281i	$0.417105\pi$
$660$	0	0
$661$	−12746.0	−0.750018	−0.375009	−	0.927021i	$-0.622360\pi$
$661$	−12746.0	−0.750018	−0.375009	+	0.927021i	$0.622360\pi$
$662$	0	0
$663$	−1140.00	−0.0667782
$664$	0	0
$665$	9280.00	0.541147
$666$	0	0
$667$	−1955.00	−0.113490
$668$	0	0
$669$	5368.00	0.310223
$670$	0	0
$671$	−7380.00	−0.424593
$672$	0	0
$673$	−14937.0	−0.855541	−0.427770	−	0.903887i	$-0.640701\pi$
$673$	−14937.0	−0.855541	−0.427770	+	0.903887i	$0.640701\pi$
$674$	0	0
$675$	−1325.00	−0.0755545
$676$	0	0
$677$	8226.00	0.466988	0.233494	−	0.972358i	$-0.424984\pi$
$677$	8226.00	0.466988	0.233494	+	0.972358i	$0.424984\pi$
$678$	0	0
$679$	−27776.0	−1.56987
$680$	0	0
$681$	5780.00	0.325242
$682$	0	0
$683$	−5797.00	−0.324767	−0.162384	−	0.986728i	$-0.551918\pi$
$683$	−5797.00	−0.324767	−0.162384	+	0.986728i	$0.551918\pi$
$684$	0	0
$685$	−3260.00	−0.181837
$686$	0	0
$687$	−1064.00	−0.0590890
$688$	0	0
$689$	−8018.00	−0.443340
$690$	0	0
$691$	−16892.0	−0.929959	−0.464980	−	0.885321i	$-0.653938\pi$
$691$	−16892.0	−0.929959	−0.464980	+	0.885321i	$0.653938\pi$
$692$	0	0
$693$	−24960.0	−1.36818
$694$	0	0
$695$	9995.00	0.545514
$696$	0	0
$697$	−8580.00	−0.466271
$698$	0	0
$699$	−2833.00	−0.153296
$700$	0	0
$701$	−19008.0	−1.02414	−0.512070	−	0.858944i	$-0.671121\pi$
$701$	−19008.0	−1.02414	−0.512070	+	0.858944i	$0.671121\pi$
$702$	0	0
$703$	−1972.00	−0.105797
$704$	0	0
$705$	2805.00	0.149847
$706$	0	0
$707$	1600.00	0.0851120
$708$	0	0
$709$	−29484.0	−1.56177	−0.780885	−	0.624675i	$-0.785232\pi$
$709$	−29484.0	−1.56177	−0.780885	+	0.624675i	$0.785232\pi$
$710$	0	0
$711$	28964.0	1.52776
$712$	0	0
$713$	−1495.00	−0.0785248
$714$	0	0
$715$	2850.00	0.149068
$716$	0	0
$717$	−6831.00	−0.355800
$718$	0	0
$719$	−13760.0	−0.713715	−0.356858	−	0.934159i	$-0.616152\pi$
$719$	−13760.0	−0.713715	−0.356858	+	0.934159i	$0.616152\pi$
$720$	0	0
$721$	−14080.0	−0.727277
$722$	0	0
$723$	−1794.00	−0.0922816
$724$	0	0
$725$	2125.00	0.108856
$726$	0	0
$727$	30554.0	1.55871	0.779357	−	0.626580i	$-0.215546\pi$
$727$	30554.0	1.55871	0.779357	+	0.626580i	$0.215546\pi$
$728$	0	0
$729$	−15443.0	−0.784586
$730$	0	0
$731$	−19920.0	−1.00789
$732$	0	0
$733$	37404.0	1.88479	0.942393	−	0.334508i	$-0.108570\pi$
$733$	37404.0	1.88479	0.942393	+	0.334508i	$0.108570\pi$
$734$	0	0
$735$	3405.00	0.170878
$736$	0	0
$737$	−26820.0	−1.34047
$738$	0	0
$739$	−6873.00	−0.342121	−0.171060	−	0.985261i	$-0.554719\pi$
$739$	−6873.00	−0.342121	−0.171060	+	0.985261i	$0.554719\pi$
$740$	0	0
$741$	1102.00	0.0546329
$742$	0	0
$743$	−25216.0	−1.24507	−0.622534	−	0.782593i	$-0.713897\pi$
$743$	−25216.0	−1.24507	−0.622534	+	0.782593i	$0.713897\pi$
$744$	0	0
$745$	9350.00	0.459809
$746$	0	0
$747$	−24336.0	−1.19198
$748$	0	0
$749$	10176.0	0.496426
$750$	0	0
$751$	27556.0	1.33893	0.669463	−	0.742846i	$-0.266524\pi$
$751$	27556.0	1.33893	0.669463	+	0.742846i	$0.266524\pi$
$752$	0	0
$753$	6672.00	0.322897
$754$	0	0
$755$	−7315.00	−0.352609
$756$	0	0
$757$	−30550.0	−1.46679	−0.733394	−	0.679804i	$-0.762065\pi$
$757$	−30550.0	−1.46679	−0.733394	+	0.679804i	$0.762065\pi$
$758$	0	0
$759$	−690.000	−0.0329979
$760$	0	0
$761$	10125.0	0.482301	0.241150	−	0.970488i	$-0.422475\pi$
$761$	10125.0	0.482301	0.241150	+	0.970488i	$0.422475\pi$
$762$	0	0
$763$	−26496.0	−1.25717
$764$	0	0
$765$	7800.00	0.368640
$766$	0	0
$767$	−7448.00	−0.350628
$768$	0	0
$769$	−35210.0	−1.65111	−0.825556	−	0.564320i	$-0.809138\pi$
$769$	−35210.0	−1.65111	−0.825556	+	0.564320i	$0.809138\pi$
$770$	0	0
$771$	291.000	0.0135929
$772$	0	0
$773$	15696.0	0.730331	0.365166	−	0.930943i	$-0.381012\pi$
$773$	15696.0	0.730331	0.365166	+	0.930943i	$0.381012\pi$
$774$	0	0
$775$	1625.00	0.0753184
$776$	0	0
$777$	−1088.00	−0.0502340
$778$	0	0
$779$	8294.00	0.381468
$780$	0	0
$781$	22110.0	1.01301
$782$	0	0
$783$	−4505.00	−0.205614
$784$	0	0
$785$	−15020.0	−0.682913
$786$	0	0
$787$	−9356.00	−0.423768	−0.211884	−	0.977295i	$-0.567960\pi$
$787$	−9356.00	−0.423768	−0.211884	+	0.977295i	$0.567960\pi$
$788$	0	0
$789$	5306.00	0.239415
$790$	0	0
$791$	6656.00	0.299191
$792$	0	0
$793$	−4674.00	−0.209305
$794$	0	0
$795$	−2110.00	−0.0941308
$796$	0	0
$797$	−31110.0	−1.38265	−0.691325	−	0.722544i	$-0.742973\pi$
$797$	−31110.0	−1.38265	−0.691325	+	0.722544i	$0.742973\pi$
$798$	0	0
$799$	−33660.0	−1.49037
$800$	0	0
$801$	−21424.0	−0.945043
$802$	0	0
$803$	31230.0	1.37246
$804$	0	0
$805$	−3680.00	−0.161122
$806$	0	0
$807$	7785.00	0.339585
$808$	0	0
$809$	20890.0	0.907853	0.453927	−	0.891039i	$-0.350023\pi$
$809$	20890.0	0.907853	0.453927	+	0.891039i	$0.350023\pi$
$810$	0	0
$811$	−6291.00	−0.272388	−0.136194	−	0.990682i	$-0.543487\pi$
$811$	−6291.00	−0.272388	−0.136194	+	0.990682i	$0.543487\pi$
$812$	0	0
$813$	−3440.00	−0.148396
$814$	0	0
$815$	20235.0	0.869695
$816$	0	0
$817$	19256.0	0.824580
$818$	0	0
$819$	−15808.0	−0.674452
$820$	0	0
$821$	36210.0	1.53927	0.769633	−	0.638486i	$-0.220439\pi$
$821$	36210.0	1.53927	0.769633	+	0.638486i	$0.220439\pi$
$822$	0	0
$823$	−40867.0	−1.73090	−0.865452	−	0.500992i	$-0.832969\pi$
$823$	−40867.0	−1.73090	−0.865452	+	0.500992i	$0.832969\pi$
$824$	0	0
$825$	750.000	0.0316505
$826$	0	0
$827$	−2088.00	−0.0877955	−0.0438977	−	0.999036i	$-0.513978\pi$
$827$	−2088.00	−0.0877955	−0.0438977	+	0.999036i	$0.513978\pi$
$828$	0	0
$829$	−25942.0	−1.08686	−0.543428	−	0.839456i	$-0.682874\pi$
$829$	−25942.0	−1.08686	−0.543428	+	0.839456i	$0.682874\pi$
$830$	0	0
$831$	−7643.00	−0.319053
$832$	0	0
$833$	−40860.0	−1.69954
$834$	0	0
$835$	10560.0	0.437657
$836$	0	0
$837$	−3445.00	−0.142266
$838$	0	0
$839$	−19636.0	−0.807998	−0.403999	−	0.914760i	$-0.632380\pi$
$839$	−19636.0	−0.807998	−0.403999	+	0.914760i	$0.632380\pi$
$840$	0	0
$841$	−17164.0	−0.703760
$842$	0	0
$843$	5408.00	0.220951
$844$	0	0
$845$	−9180.00	−0.373730
$846$	0	0
$847$	−13792.0	−0.559503
$848$	0	0
$849$	−4402.00	−0.177946
$850$	0	0
$851$	782.000	0.0315001
$852$	0	0
$853$	−15954.0	−0.640392	−0.320196	−	0.947351i	$-0.603749\pi$
$853$	−15954.0	−0.640392	−0.320196	+	0.947351i	$0.603749\pi$
$854$	0	0
$855$	−7540.00	−0.301594
$856$	0	0
$857$	−36353.0	−1.44900	−0.724501	−	0.689274i	$-0.757930\pi$
$857$	−36353.0	−1.44900	−0.724501	+	0.689274i	$0.757930\pi$
$858$	0	0
$859$	34173.0	1.35735	0.678677	−	0.734437i	$-0.262553\pi$
$859$	34173.0	1.35735	0.678677	+	0.734437i	$0.262553\pi$
$860$	0	0
$861$	4576.00	0.181126
$862$	0	0
$863$	−9129.00	−0.360087	−0.180043	−	0.983659i	$-0.557624\pi$
$863$	−9129.00	−0.360087	−0.180043	+	0.983659i	$0.557624\pi$
$864$	0	0
$865$	1710.00	0.0672159
$866$	0	0
$867$	−1313.00	−0.0514323
$868$	0	0
$869$	−33420.0	−1.30460
$870$	0	0
$871$	−16986.0	−0.660791
$872$	0	0
$873$	22568.0	0.874927
$874$	0	0
$875$	4000.00	0.154542
$876$	0	0
$877$	−8106.00	−0.312110	−0.156055	−	0.987748i	$-0.549878\pi$
$877$	−8106.00	−0.312110	−0.156055	+	0.987748i	$0.549878\pi$
$878$	0	0
$879$	9808.00	0.376355
$880$	0	0
$881$	14124.0	0.540124	0.270062	−	0.962843i	$-0.412956\pi$
$881$	14124.0	0.540124	0.270062	+	0.962843i	$0.412956\pi$
$882$	0	0
$883$	1700.00	0.0647900	0.0323950	−	0.999475i	$-0.489687\pi$
$883$	1700.00	0.0647900	0.0323950	+	0.999475i	$0.489687\pi$
$884$	0	0
$885$	−1960.00	−0.0744460
$886$	0	0
$887$	−26711.0	−1.01112	−0.505562	−	0.862790i	$-0.668715\pi$
$887$	−26711.0	−1.01112	−0.505562	+	0.862790i	$0.668715\pi$
$888$	0	0
$889$	15840.0	0.597589
$890$	0	0
$891$	19470.0	0.732065
$892$	0	0
$893$	32538.0	1.21931
$894$	0	0
$895$	−8445.00	−0.315402
$896$	0	0
$897$	−437.000	−0.0162664
$898$	0	0
$899$	5525.00	0.204971
$900$	0	0
$901$	25320.0	0.936217
$902$	0	0
$903$	10624.0	0.391522
$904$	0	0
$905$	−970.000	−0.0356286
$906$	0	0
$907$	18758.0	0.686714	0.343357	−	0.939205i	$-0.388436\pi$
$907$	18758.0	0.686714	0.343357	+	0.939205i	$0.388436\pi$
$908$	0	0
$909$	−1300.00	−0.0474348
$910$	0	0
$911$	−20298.0	−0.738203	−0.369101	−	0.929389i	$-0.620334\pi$
$911$	−20298.0	−0.738203	−0.369101	+	0.929389i	$0.620334\pi$
$912$	0	0
$913$	28080.0	1.01787
$914$	0	0
$915$	−1230.00	−0.0444399
$916$	0	0
$917$	−10720.0	−0.386048
$918$	0	0
$919$	−39276.0	−1.40979	−0.704894	−	0.709312i	$-0.749006\pi$
$919$	−39276.0	−1.40979	−0.704894	+	0.709312i	$0.749006\pi$
$920$	0	0
$921$	3288.00	0.117637
$922$	0	0
$923$	14003.0	0.499366
$924$	0	0
$925$	−850.000	−0.0302139
$926$	0	0
$927$	11440.0	0.405328
$928$	0	0
$929$	−51265.0	−1.81050	−0.905248	−	0.424884i	$-0.860315\pi$
$929$	−51265.0	−1.81050	−0.905248	+	0.424884i	$0.860315\pi$
$930$	0	0
$931$	39498.0	1.39043
$932$	0	0
$933$	−3919.00	−0.137516
$934$	0	0
$935$	−9000.00	−0.314793
$936$	0	0
$937$	12182.0	0.424726	0.212363	−	0.977191i	$-0.431884\pi$
$937$	12182.0	0.424726	0.212363	+	0.977191i	$0.431884\pi$
$938$	0	0
$939$	5112.00	0.177661
$940$	0	0
$941$	33192.0	1.14987	0.574935	−	0.818199i	$-0.305027\pi$
$941$	33192.0	1.14987	0.574935	+	0.818199i	$0.305027\pi$
$942$	0	0
$943$	−3289.00	−0.113579
$944$	0	0
$945$	−8480.00	−0.291909
$946$	0	0
$947$	−18127.0	−0.622015	−0.311008	−	0.950407i	$-0.600666\pi$
$947$	−18127.0	−0.622015	−0.311008	+	0.950407i	$0.600666\pi$
$948$	0	0
$949$	19779.0	0.676558
$950$	0	0
$951$	8058.00	0.274762
$952$	0	0
$953$	−31722.0	−1.07825	−0.539127	−	0.842224i	$-0.681246\pi$
$953$	−31722.0	−1.07825	−0.539127	+	0.842224i	$0.681246\pi$
$954$	0	0
$955$	1490.00	0.0504872
$956$	0	0
$957$	2550.00	0.0861335
$958$	0	0
$959$	−20864.0	−0.702538
$960$	0	0
$961$	−25566.0	−0.858179
$962$	0	0
$963$	−8268.00	−0.276669
$964$	0	0
$965$	11505.0	0.383792
$966$	0	0
$967$	1459.00	0.0485194	0.0242597	−	0.999706i	$-0.492277\pi$
$967$	1459.00	0.0485194	0.0242597	+	0.999706i	$0.492277\pi$
$968$	0	0
$969$	−3480.00	−0.115370
$970$	0	0
$971$	29626.0	0.979139	0.489569	−	0.871964i	$-0.337154\pi$
$971$	29626.0	0.979139	0.489569	+	0.871964i	$0.337154\pi$
$972$	0	0
$973$	63968.0	2.10763
$974$	0	0
$975$	475.000	0.0156022
$976$	0	0
$977$	−40202.0	−1.31645	−0.658227	−	0.752819i	$-0.728693\pi$
$977$	−40202.0	−1.31645	−0.658227	+	0.752819i	$0.728693\pi$
$978$	0	0
$979$	24720.0	0.807002
$980$	0	0
$981$	21528.0	0.700648
$982$	0	0
$983$	9278.00	0.301040	0.150520	−	0.988607i	$-0.451905\pi$
$983$	9278.00	0.301040	0.150520	+	0.988607i	$0.451905\pi$
$984$	0	0
$985$	−2675.00	−0.0865305
$986$	0	0
$987$	17952.0	0.578945
$988$	0	0
$989$	−7636.00	−0.245511
$990$	0	0
$991$	−37264.0	−1.19448	−0.597240	−	0.802062i	$-0.703736\pi$
$991$	−37264.0	−1.19448	−0.597240	+	0.802062i	$0.703736\pi$
$992$	0	0
$993$	2555.00	0.0816520
$994$	0	0
$995$	−1390.00	−0.0442874
$996$	0	0
$997$	−41930.0	−1.33193	−0.665966	−	0.745982i	$-0.731980\pi$
$997$	−41930.0	−1.33193	−0.665966	+	0.745982i	$0.731980\pi$
$998$	0	0
$999$	1802.00	0.0570698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.e.1.1		1
4.3	odd	2		230.4.a.d.1.1	✓	1
12.11	even	2		2070.4.a.a.1.1		1
20.3	even	4		1150.4.b.d.599.1		2
20.7	even	4		1150.4.b.d.599.2		2
20.19	odd	2		1150.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.d.1.1	✓	1	4.3	odd	2
1150.4.a.c.1.1		1	20.19	odd	2
1150.4.b.d.599.1		2	20.3	even	4
1150.4.b.d.599.2		2	20.7	even	4
1840.4.a.e.1.1		1	1.1	even	1	trivial
2070.4.a.a.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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