LMFDB - Embedded newform 40.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [40,4,Mod(1,40)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(40, 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("40.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 40 = 2^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	40.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:	$2.36007640023$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	40.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+10.0000 q^{3} -5.00000 q^{5} -18.0000 q^{7} +73.0000 q^{9} +O(q^{10})$

$q+10.0000 q^{3} -5.00000 q^{5} -18.0000 q^{7} +73.0000 q^{9} -16.0000 q^{11} -6.00000 q^{13} -50.0000 q^{15} -6.00000 q^{17} -124.000 q^{19} -180.000 q^{21} +42.0000 q^{23} +25.0000 q^{25} +460.000 q^{27} +142.000 q^{29} -188.000 q^{31} -160.000 q^{33} +90.0000 q^{35} +202.000 q^{37} -60.0000 q^{39} +54.0000 q^{41} +66.0000 q^{43} -365.000 q^{45} +38.0000 q^{47} -19.0000 q^{49} -60.0000 q^{51} +738.000 q^{53} +80.0000 q^{55} -1240.00 q^{57} +564.000 q^{59} -262.000 q^{61} -1314.00 q^{63} +30.0000 q^{65} -554.000 q^{67} +420.000 q^{69} +140.000 q^{71} +882.000 q^{73} +250.000 q^{75} +288.000 q^{77} -1160.00 q^{79} +2629.00 q^{81} +642.000 q^{83} +30.0000 q^{85} +1420.00 q^{87} -854.000 q^{89} +108.000 q^{91} -1880.00 q^{93} +620.000 q^{95} -478.000 q^{97} -1168.00 q^{99} +O(q^{100})$

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
$3$	10.0000	1.92450	0.962250	−	0.272166i	$-0.0877398\pi$
$3$	10.0000	1.92450	0.962250	+	0.272166i	$0.0877398\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−18.0000	−0.971909	−0.485954	−	0.873984i	$-0.661528\pi$
$7$	−18.0000	−0.971909	−0.485954	+	0.873984i	$0.661528\pi$
$8$	0	0
$9$	73.0000	2.70370
$10$	0	0
$11$	−16.0000	−0.438562	−0.219281	−	0.975662i	$-0.570371\pi$
$11$	−16.0000	−0.438562	−0.219281	+	0.975662i	$0.570371\pi$
$12$	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
$15$	−50.0000	−0.860663
$16$	0	0
$17$	−6.00000	−0.0856008	−0.0428004	−	0.999084i	$-0.513628\pi$
$17$	−6.00000	−0.0856008	−0.0428004	+	0.999084i	$0.513628\pi$
$18$	0	0
$19$	−124.000	−1.49724	−0.748620	−	0.663000i	$-0.769283\pi$
$19$	−124.000	−1.49724	−0.748620	+	0.663000i	$0.769283\pi$
$20$	0	0
$21$	−180.000	−1.87044
$22$	0	0
$23$	42.0000	0.380765	0.190383	−	0.981710i	$-0.439027\pi$
$23$	42.0000	0.380765	0.190383	+	0.981710i	$0.439027\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	460.000	3.27878
$28$	0	0
$29$	142.000	0.909267	0.454633	−	0.890679i	$-0.349770\pi$
$29$	142.000	0.909267	0.454633	+	0.890679i	$0.349770\pi$
$30$	0	0
$31$	−188.000	−1.08922	−0.544610	−	0.838690i	$-0.683322\pi$
$31$	−188.000	−1.08922	−0.544610	+	0.838690i	$0.683322\pi$
$32$	0	0
$33$	−160.000	−0.844013
$34$	0	0
$35$	90.0000	0.434651
$36$	0	0
$37$	202.000	0.897530	0.448765	−	0.893650i	$-0.351864\pi$
$37$	202.000	0.897530	0.448765	+	0.893650i	$0.351864\pi$
$38$	0	0
$39$	−60.0000	−0.246351
$40$	0	0
$41$	54.0000	0.205692	0.102846	−	0.994697i	$-0.467205\pi$
$41$	54.0000	0.205692	0.102846	+	0.994697i	$0.467205\pi$
$42$	0	0
$43$	66.0000	0.234068	0.117034	−	0.993128i	$-0.462661\pi$
$43$	66.0000	0.234068	0.117034	+	0.993128i	$0.462661\pi$
$44$	0	0
$45$	−365.000	−1.20913
$46$	0	0
$47$	38.0000	0.117933	0.0589667	−	0.998260i	$-0.481219\pi$
$47$	38.0000	0.117933	0.0589667	+	0.998260i	$0.481219\pi$
$48$	0	0
$49$	−19.0000	−0.0553936
$50$	0	0
$51$	−60.0000	−0.164739
$52$	0	0
$53$	738.000	1.91268	0.956341	−	0.292255i	$-0.0944055\pi$
$53$	738.000	1.91268	0.956341	+	0.292255i	$0.0944055\pi$
$54$	0	0
$55$	80.0000	0.196131
$56$	0	0
$57$	−1240.00	−2.88144
$58$	0	0
$59$	564.000	1.24452	0.622259	−	0.782812i	$-0.286215\pi$
$59$	564.000	1.24452	0.622259	+	0.782812i	$0.286215\pi$
$60$	0	0
$61$	−262.000	−0.549929	−0.274964	−	0.961454i	$-0.588666\pi$
$61$	−262.000	−0.549929	−0.274964	+	0.961454i	$0.588666\pi$
$62$	0	0
$63$	−1314.00	−2.62775
$64$	0	0
$65$	30.0000	0.0572468
$66$	0	0
$67$	−554.000	−1.01018	−0.505089	−	0.863067i	$-0.668540\pi$
$67$	−554.000	−1.01018	−0.505089	+	0.863067i	$0.668540\pi$
$68$	0	0
$69$	420.000	0.732783
$70$	0	0
$71$	140.000	0.234013	0.117007	−	0.993131i	$-0.462670\pi$
$71$	140.000	0.234013	0.117007	+	0.993131i	$0.462670\pi$
$72$	0	0
$73$	882.000	1.41411	0.707057	−	0.707157i	$-0.250023\pi$
$73$	882.000	1.41411	0.707057	+	0.707157i	$0.250023\pi$
$74$	0	0
$75$	250.000	0.384900
$76$	0	0
$77$	288.000	0.426242
$78$	0	0
$79$	−1160.00	−1.65203	−0.826014	−	0.563650i	$-0.809397\pi$
$79$	−1160.00	−1.65203	−0.826014	+	0.563650i	$0.809397\pi$
$80$	0	0
$81$	2629.00	3.60631
$82$	0	0
$83$	642.000	0.849020	0.424510	−	0.905423i	$-0.360446\pi$
$83$	642.000	0.849020	0.424510	+	0.905423i	$0.360446\pi$
$84$	0	0
$85$	30.0000	0.0382818
$86$	0	0
$87$	1420.00	1.74988
$88$	0	0
$89$	−854.000	−1.01712	−0.508561	−	0.861026i	$-0.669822\pi$
$89$	−854.000	−1.01712	−0.508561	+	0.861026i	$0.669822\pi$
$90$	0	0
$91$	108.000	0.124412
$92$	0	0
$93$	−1880.00	−2.09620
$94$	0	0
$95$	620.000	0.669586
$96$	0	0
$97$	−478.000	−0.500346	−0.250173	−	0.968201i	$-0.580487\pi$
$97$	−478.000	−0.500346	−0.250173	+	0.968201i	$0.580487\pi$
$98$	0	0
$99$	−1168.00	−1.18574
$100$	0	0
$101$	−1794.00	−1.76742	−0.883711	−	0.468033i	$-0.844963\pi$
$101$	−1794.00	−1.76742	−0.883711	+	0.468033i	$0.844963\pi$
$102$	0	0
$103$	642.000	0.614157	0.307078	−	0.951684i	$-0.400649\pi$
$103$	642.000	0.614157	0.307078	+	0.951684i	$0.400649\pi$
$104$	0	0
$105$	900.000	0.836486
$106$	0	0
$107$	−850.000	−0.767968	−0.383984	−	0.923340i	$-0.625448\pi$
$107$	−850.000	−0.767968	−0.383984	+	0.923340i	$0.625448\pi$
$108$	0	0
$109$	666.000	0.585241	0.292620	−	0.956229i	$-0.405473\pi$
$109$	666.000	0.585241	0.292620	+	0.956229i	$0.405473\pi$
$110$	0	0
$111$	2020.00	1.72730
$112$	0	0
$113$	−1446.00	−1.20379	−0.601895	−	0.798575i	$-0.705587\pi$
$113$	−1446.00	−1.20379	−0.601895	+	0.798575i	$0.705587\pi$
$114$	0	0
$115$	−210.000	−0.170283
$116$	0	0
$117$	−438.000	−0.346095
$118$	0	0
$119$	108.000	0.0831962
$120$	0	0
$121$	−1075.00	−0.807663
$122$	0	0
$123$	540.000	0.395855
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1154.00	−0.806307	−0.403153	−	0.915132i	$-0.632086\pi$
$127$	−1154.00	−0.806307	−0.403153	+	0.915132i	$0.632086\pi$
$128$	0	0
$129$	660.000	0.450463
$130$	0	0
$131$	−368.000	−0.245437	−0.122719	−	0.992441i	$-0.539161\pi$
$131$	−368.000	−0.245437	−0.122719	+	0.992441i	$0.539161\pi$
$132$	0	0
$133$	2232.00	1.45518
$134$	0	0
$135$	−2300.00	−1.46631
$136$	0	0
$137$	−670.000	−0.417825	−0.208912	−	0.977934i	$-0.566992\pi$
$137$	−670.000	−0.417825	−0.208912	+	0.977934i	$0.566992\pi$
$138$	0	0
$139$	−572.000	−0.349039	−0.174519	−	0.984654i	$-0.555837\pi$
$139$	−572.000	−0.349039	−0.174519	+	0.984654i	$0.555837\pi$
$140$	0	0
$141$	380.000	0.226963
$142$	0	0
$143$	96.0000	0.0561393
$144$	0	0
$145$	−710.000	−0.406636
$146$	0	0
$147$	−190.000	−0.106605
$148$	0	0
$149$	1730.00	0.951189	0.475594	−	0.879665i	$-0.342233\pi$
$149$	1730.00	0.951189	0.475594	+	0.879665i	$0.342233\pi$
$150$	0	0
$151$	1324.00	0.713547	0.356773	−	0.934191i	$-0.383877\pi$
$151$	1324.00	0.713547	0.356773	+	0.934191i	$0.383877\pi$
$152$	0	0
$153$	−438.000	−0.231439
$154$	0	0
$155$	940.000	0.487114
$156$	0	0
$157$	2946.00	1.49756	0.748778	−	0.662820i	$-0.230641\pi$
$157$	2946.00	1.49756	0.748778	+	0.662820i	$0.230641\pi$
$158$	0	0
$159$	7380.00	3.68096
$160$	0	0
$161$	−756.000	−0.370069
$162$	0	0
$163$	2098.00	1.00815	0.504074	−	0.863661i	$-0.331834\pi$
$163$	2098.00	1.00815	0.504074	+	0.863661i	$0.331834\pi$
$164$	0	0
$165$	800.000	0.377454
$166$	0	0
$167$	−866.000	−0.401276	−0.200638	−	0.979665i	$-0.564301\pi$
$167$	−866.000	−0.401276	−0.200638	+	0.979665i	$0.564301\pi$
$168$	0	0
$169$	−2161.00	−0.983614
$170$	0	0
$171$	−9052.00	−4.04809
$172$	0	0
$173$	−1678.00	−0.737433	−0.368717	−	0.929542i	$-0.620203\pi$
$173$	−1678.00	−0.737433	−0.368717	+	0.929542i	$0.620203\pi$
$174$	0	0
$175$	−450.000	−0.194382
$176$	0	0
$177$	5640.00	2.39508
$178$	0	0
$179$	−1620.00	−0.676450	−0.338225	−	0.941065i	$-0.609826\pi$
$179$	−1620.00	−0.676450	−0.338225	+	0.941065i	$0.609826\pi$
$180$	0	0
$181$	2510.00	1.03076	0.515378	−	0.856963i	$-0.327652\pi$
$181$	2510.00	1.03076	0.515378	+	0.856963i	$0.327652\pi$
$182$	0	0
$183$	−2620.00	−1.05834
$184$	0	0
$185$	−1010.00	−0.401387
$186$	0	0
$187$	96.0000	0.0375413
$188$	0	0
$189$	−8280.00	−3.18667
$190$	0	0
$191$	−372.000	−0.140927	−0.0704633	−	0.997514i	$-0.522448\pi$
$191$	−372.000	−0.140927	−0.0704633	+	0.997514i	$0.522448\pi$
$192$	0	0
$193$	2938.00	1.09576	0.547880	−	0.836557i	$-0.315435\pi$
$193$	2938.00	1.09576	0.547880	+	0.836557i	$0.315435\pi$
$194$	0	0
$195$	300.000	0.110172
$196$	0	0
$197$	2234.00	0.807949	0.403974	−	0.914770i	$-0.367628\pi$
$197$	2234.00	0.807949	0.403974	+	0.914770i	$0.367628\pi$
$198$	0	0
$199$	−3048.00	−1.08576	−0.542882	−	0.839809i	$-0.682667\pi$
$199$	−3048.00	−1.08576	−0.542882	+	0.839809i	$0.682667\pi$
$200$	0	0
$201$	−5540.00	−1.94409
$202$	0	0
$203$	−2556.00	−0.883724
$204$	0	0
$205$	−270.000	−0.0919884
$206$	0	0
$207$	3066.00	1.02948
$208$	0	0
$209$	1984.00	0.656632
$210$	0	0
$211$	4896.00	1.59741	0.798707	−	0.601720i	$-0.205518\pi$
$211$	4896.00	1.59741	0.798707	+	0.601720i	$0.205518\pi$
$212$	0	0
$213$	1400.00	0.450359
$214$	0	0
$215$	−330.000	−0.104678
$216$	0	0
$217$	3384.00	1.05862
$218$	0	0
$219$	8820.00	2.72146
$220$	0	0
$221$	36.0000	0.0109576
$222$	0	0
$223$	−5302.00	−1.59214	−0.796072	−	0.605202i	$-0.793092\pi$
$223$	−5302.00	−1.59214	−0.796072	+	0.605202i	$0.793092\pi$
$224$	0	0
$225$	1825.00	0.540741
$226$	0	0
$227$	−3778.00	−1.10465	−0.552323	−	0.833630i	$-0.686259\pi$
$227$	−3778.00	−1.10465	−0.552323	+	0.833630i	$0.686259\pi$
$228$	0	0
$229$	−3034.00	−0.875513	−0.437756	−	0.899094i	$-0.644227\pi$
$229$	−3034.00	−0.875513	−0.437756	+	0.899094i	$0.644227\pi$
$230$	0	0
$231$	2880.00	0.820303
$232$	0	0
$233$	−3478.00	−0.977903	−0.488951	−	0.872311i	$-0.662620\pi$
$233$	−3478.00	−0.977903	−0.488951	+	0.872311i	$0.662620\pi$
$234$	0	0
$235$	−190.000	−0.0527414
$236$	0	0
$237$	−11600.0	−3.17933
$238$	0	0
$239$	1560.00	0.422209	0.211105	−	0.977463i	$-0.432294\pi$
$239$	1560.00	0.422209	0.211105	+	0.977463i	$0.432294\pi$
$240$	0	0
$241$	−3218.00	−0.860123	−0.430061	−	0.902800i	$-0.641508\pi$
$241$	−3218.00	−0.860123	−0.430061	+	0.902800i	$0.641508\pi$
$242$	0	0
$243$	13870.0	3.66157
$244$	0	0
$245$	95.0000	0.0247728
$246$	0	0
$247$	744.000	0.191658
$248$	0	0
$249$	6420.00	1.63394
$250$	0	0
$251$	688.000	0.173013	0.0865063	−	0.996251i	$-0.472430\pi$
$251$	688.000	0.173013	0.0865063	+	0.996251i	$0.472430\pi$
$252$	0	0
$253$	−672.000	−0.166989
$254$	0	0
$255$	300.000	0.0736734
$256$	0	0
$257$	2170.00	0.526696	0.263348	−	0.964701i	$-0.415173\pi$
$257$	2170.00	0.526696	0.263348	+	0.964701i	$0.415173\pi$
$258$	0	0
$259$	−3636.00	−0.872317
$260$	0	0
$261$	10366.0	2.45839
$262$	0	0
$263$	2274.00	0.533159	0.266580	−	0.963813i	$-0.414106\pi$
$263$	2274.00	0.533159	0.266580	+	0.963813i	$0.414106\pi$
$264$	0	0
$265$	−3690.00	−0.855377
$266$	0	0
$267$	−8540.00	−1.95745
$268$	0	0
$269$	7146.00	1.61970	0.809850	−	0.586637i	$-0.199548\pi$
$269$	7146.00	1.61970	0.809850	+	0.586637i	$0.199548\pi$
$270$	0	0
$271$	2604.00	0.583696	0.291848	−	0.956465i	$-0.405730\pi$
$271$	2604.00	0.583696	0.291848	+	0.956465i	$0.405730\pi$
$272$	0	0
$273$	1080.00	0.239431
$274$	0	0
$275$	−400.000	−0.0877124
$276$	0	0
$277$	−5150.00	−1.11709	−0.558544	−	0.829475i	$-0.688640\pi$
$277$	−5150.00	−1.11709	−0.558544	+	0.829475i	$0.688640\pi$
$278$	0	0
$279$	−13724.0	−2.94493
$280$	0	0
$281$	5270.00	1.11880	0.559398	−	0.828899i	$-0.311032\pi$
$281$	5270.00	1.11880	0.559398	+	0.828899i	$0.311032\pi$
$282$	0	0
$283$	3434.00	0.721308	0.360654	−	0.932700i	$-0.382554\pi$
$283$	3434.00	0.721308	0.360654	+	0.932700i	$0.382554\pi$
$284$	0	0
$285$	6200.00	1.28862
$286$	0	0
$287$	−972.000	−0.199914
$288$	0	0
$289$	−4877.00	−0.992673
$290$	0	0
$291$	−4780.00	−0.962916
$292$	0	0
$293$	−9878.00	−1.96955	−0.984776	−	0.173826i	$-0.944387\pi$
$293$	−9878.00	−1.96955	−0.984776	+	0.173826i	$0.944387\pi$
$294$	0	0
$295$	−2820.00	−0.556565
$296$	0	0
$297$	−7360.00	−1.43795
$298$	0	0
$299$	−252.000	−0.0487409
$300$	0	0
$301$	−1188.00	−0.227492
$302$	0	0
$303$	−17940.0	−3.40141
$304$	0	0
$305$	1310.00	0.245936
$306$	0	0
$307$	8054.00	1.49728	0.748642	−	0.662975i	$-0.230706\pi$
$307$	8054.00	1.49728	0.748642	+	0.662975i	$0.230706\pi$
$308$	0	0
$309$	6420.00	1.18195
$310$	0	0
$311$	5492.00	1.00136	0.500680	−	0.865633i	$-0.333083\pi$
$311$	5492.00	1.00136	0.500680	+	0.865633i	$0.333083\pi$
$312$	0	0
$313$	−422.000	−0.0762072	−0.0381036	−	0.999274i	$-0.512132\pi$
$313$	−422.000	−0.0762072	−0.0381036	+	0.999274i	$0.512132\pi$
$314$	0	0
$315$	6570.00	1.17517
$316$	0	0
$317$	6194.00	1.09744	0.548722	−	0.836005i	$-0.315115\pi$
$317$	6194.00	1.09744	0.548722	+	0.836005i	$0.315115\pi$
$318$	0	0
$319$	−2272.00	−0.398770
$320$	0	0
$321$	−8500.00	−1.47796
$322$	0	0
$323$	744.000	0.128165
$324$	0	0
$325$	−150.000	−0.0256015
$326$	0	0
$327$	6660.00	1.12630
$328$	0	0
$329$	−684.000	−0.114620
$330$	0	0
$331$	7688.00	1.27665	0.638324	−	0.769768i	$-0.279628\pi$
$331$	7688.00	1.27665	0.638324	+	0.769768i	$0.279628\pi$
$332$	0	0
$333$	14746.0	2.42665
$334$	0	0
$335$	2770.00	0.451765
$336$	0	0
$337$	−1438.00	−0.232442	−0.116221	−	0.993223i	$-0.537078\pi$
$337$	−1438.00	−0.232442	−0.116221	+	0.993223i	$0.537078\pi$
$338$	0	0
$339$	−14460.0	−2.31669
$340$	0	0
$341$	3008.00	0.477690
$342$	0	0
$343$	6516.00	1.02575
$344$	0	0
$345$	−2100.00	−0.327711
$346$	0	0
$347$	8838.00	1.36729	0.683644	−	0.729816i	$-0.260394\pi$
$347$	8838.00	1.36729	0.683644	+	0.729816i	$0.260394\pi$
$348$	0	0
$349$	−7810.00	−1.19788	−0.598939	−	0.800794i	$-0.704411\pi$
$349$	−7810.00	−1.19788	−0.598939	+	0.800794i	$0.704411\pi$
$350$	0	0
$351$	−2760.00	−0.419709
$352$	0	0
$353$	5906.00	0.890495	0.445247	−	0.895408i	$-0.353116\pi$
$353$	5906.00	0.890495	0.445247	+	0.895408i	$0.353116\pi$
$354$	0	0
$355$	−700.000	−0.104654
$356$	0	0
$357$	1080.00	0.160111
$358$	0	0
$359$	8904.00	1.30901	0.654506	−	0.756057i	$-0.272877\pi$
$359$	8904.00	1.30901	0.654506	+	0.756057i	$0.272877\pi$
$360$	0	0
$361$	8517.00	1.24173
$362$	0	0
$363$	−10750.0	−1.55435
$364$	0	0
$365$	−4410.00	−0.632411
$366$	0	0
$367$	−7370.00	−1.04826	−0.524129	−	0.851639i	$-0.675609\pi$
$367$	−7370.00	−1.04826	−0.524129	+	0.851639i	$0.675609\pi$
$368$	0	0
$369$	3942.00	0.556131
$370$	0	0
$371$	−13284.0	−1.85895
$372$	0	0
$373$	−734.000	−0.101890	−0.0509451	−	0.998701i	$-0.516223\pi$
$373$	−734.000	−0.101890	−0.0509451	+	0.998701i	$0.516223\pi$
$374$	0	0
$375$	−1250.00	−0.172133
$376$	0	0
$377$	−852.000	−0.116393
$378$	0	0
$379$	10300.0	1.39598	0.697989	−	0.716109i	$-0.254079\pi$
$379$	10300.0	1.39598	0.697989	+	0.716109i	$0.254079\pi$
$380$	0	0
$381$	−11540.0	−1.55174
$382$	0	0
$383$	2682.00	0.357817	0.178908	−	0.983866i	$-0.442743\pi$
$383$	2682.00	0.357817	0.178908	+	0.983866i	$0.442743\pi$
$384$	0	0
$385$	−1440.00	−0.190621
$386$	0	0
$387$	4818.00	0.632849
$388$	0	0
$389$	6114.00	0.796895	0.398447	−	0.917191i	$-0.369549\pi$
$389$	6114.00	0.796895	0.398447	+	0.917191i	$0.369549\pi$
$390$	0	0
$391$	−252.000	−0.0325938
$392$	0	0
$393$	−3680.00	−0.472345
$394$	0	0
$395$	5800.00	0.738809
$396$	0	0
$397$	−7174.00	−0.906934	−0.453467	−	0.891273i	$-0.649813\pi$
$397$	−7174.00	−0.906934	−0.453467	+	0.891273i	$0.649813\pi$
$398$	0	0
$399$	22320.0	2.80050
$400$	0	0
$401$	10498.0	1.30734	0.653672	−	0.756778i	$-0.273228\pi$
$401$	10498.0	1.30734	0.653672	+	0.756778i	$0.273228\pi$
$402$	0	0
$403$	1128.00	0.139428
$404$	0	0
$405$	−13145.0	−1.61279
$406$	0	0
$407$	−3232.00	−0.393622
$408$	0	0
$409$	−1810.00	−0.218823	−0.109412	−	0.993997i	$-0.534897\pi$
$409$	−1810.00	−0.218823	−0.109412	+	0.993997i	$0.534897\pi$
$410$	0	0
$411$	−6700.00	−0.804104
$412$	0	0
$413$	−10152.0	−1.20956
$414$	0	0
$415$	−3210.00	−0.379693
$416$	0	0
$417$	−5720.00	−0.671726
$418$	0	0
$419$	−3396.00	−0.395956	−0.197978	−	0.980206i	$-0.563437\pi$
$419$	−3396.00	−0.395956	−0.197978	+	0.980206i	$0.563437\pi$
$420$	0	0
$421$	−14974.0	−1.73346	−0.866732	−	0.498775i	$-0.833784\pi$
$421$	−14974.0	−1.73346	−0.866732	+	0.498775i	$0.833784\pi$
$422$	0	0
$423$	2774.00	0.318857
$424$	0	0
$425$	−150.000	−0.0171202
$426$	0	0
$427$	4716.00	0.534481
$428$	0	0
$429$	960.000	0.108040
$430$	0	0
$431$	−13540.0	−1.51322	−0.756611	−	0.653865i	$-0.773146\pi$
$431$	−13540.0	−1.51322	−0.756611	+	0.653865i	$0.773146\pi$
$432$	0	0
$433$	15426.0	1.71207	0.856035	−	0.516918i	$-0.172921\pi$
$433$	15426.0	1.71207	0.856035	+	0.516918i	$0.172921\pi$
$434$	0	0
$435$	−7100.00	−0.782572
$436$	0	0
$437$	−5208.00	−0.570097
$438$	0	0
$439$	−10472.0	−1.13850	−0.569250	−	0.822165i	$-0.692766\pi$
$439$	−10472.0	−1.13850	−0.569250	+	0.822165i	$0.692766\pi$
$440$	0	0
$441$	−1387.00	−0.149768
$442$	0	0
$443$	722.000	0.0774340	0.0387170	−	0.999250i	$-0.487673\pi$
$443$	722.000	0.0774340	0.0387170	+	0.999250i	$0.487673\pi$
$444$	0	0
$445$	4270.00	0.454871
$446$	0	0
$447$	17300.0	1.83056
$448$	0	0
$449$	−11898.0	−1.25056	−0.625280	−	0.780401i	$-0.715015\pi$
$449$	−11898.0	−1.25056	−0.625280	+	0.780401i	$0.715015\pi$
$450$	0	0
$451$	−864.000	−0.0902088
$452$	0	0
$453$	13240.0	1.37322
$454$	0	0
$455$	−540.000	−0.0556387
$456$	0	0
$457$	−790.000	−0.0808635	−0.0404318	−	0.999182i	$-0.512873\pi$
$457$	−790.000	−0.0808635	−0.0404318	+	0.999182i	$0.512873\pi$
$458$	0	0
$459$	−2760.00	−0.280666
$460$	0	0
$461$	−3418.00	−0.345319	−0.172660	−	0.984982i	$-0.555236\pi$
$461$	−3418.00	−0.345319	−0.172660	+	0.984982i	$0.555236\pi$
$462$	0	0
$463$	−7534.00	−0.756230	−0.378115	−	0.925759i	$-0.623428\pi$
$463$	−7534.00	−0.756230	−0.378115	+	0.925759i	$0.623428\pi$
$464$	0	0
$465$	9400.00	0.937451
$466$	0	0
$467$	−14314.0	−1.41836	−0.709179	−	0.705029i	$-0.750934\pi$
$467$	−14314.0	−1.41836	−0.709179	+	0.705029i	$0.750934\pi$
$468$	0	0
$469$	9972.00	0.981800
$470$	0	0
$471$	29460.0	2.88205
$472$	0	0
$473$	−1056.00	−0.102653
$474$	0	0
$475$	−3100.00	−0.299448
$476$	0	0
$477$	53874.0	5.17132
$478$	0	0
$479$	−7016.00	−0.669247	−0.334623	−	0.942352i	$-0.608609\pi$
$479$	−7016.00	−0.669247	−0.334623	+	0.942352i	$0.608609\pi$
$480$	0	0
$481$	−1212.00	−0.114891
$482$	0	0
$483$	−7560.00	−0.712199
$484$	0	0
$485$	2390.00	0.223761
$486$	0	0
$487$	15190.0	1.41340	0.706699	−	0.707515i	$-0.250184\pi$
$487$	15190.0	1.41340	0.706699	+	0.707515i	$0.250184\pi$
$488$	0	0
$489$	20980.0	1.94018
$490$	0	0
$491$	−12624.0	−1.16031	−0.580156	−	0.814505i	$-0.697008\pi$
$491$	−12624.0	−1.16031	−0.580156	+	0.814505i	$0.697008\pi$
$492$	0	0
$493$	−852.000	−0.0778340
$494$	0	0
$495$	5840.00	0.530280
$496$	0	0
$497$	−2520.00	−0.227440
$498$	0	0
$499$	−2492.00	−0.223562	−0.111781	−	0.993733i	$-0.535655\pi$
$499$	−2492.00	−0.223562	−0.111781	+	0.993733i	$0.535655\pi$
$500$	0	0
$501$	−8660.00	−0.772256
$502$	0	0
$503$	11714.0	1.03837	0.519186	−	0.854661i	$-0.326235\pi$
$503$	11714.0	1.03837	0.519186	+	0.854661i	$0.326235\pi$
$504$	0	0
$505$	8970.00	0.790415
$506$	0	0
$507$	−21610.0	−1.89297
$508$	0	0
$509$	−5618.00	−0.489221	−0.244610	−	0.969621i	$-0.578660\pi$
$509$	−5618.00	−0.489221	−0.244610	+	0.969621i	$0.578660\pi$
$510$	0	0
$511$	−15876.0	−1.37439
$512$	0	0
$513$	−57040.0	−4.90912
$514$	0	0
$515$	−3210.00	−0.274659
$516$	0	0
$517$	−608.000	−0.0517211
$518$	0	0
$519$	−16780.0	−1.41919
$520$	0	0
$521$	13770.0	1.15792	0.578958	−	0.815357i	$-0.303459\pi$
$521$	13770.0	1.15792	0.578958	+	0.815357i	$0.303459\pi$
$522$	0	0
$523$	6986.00	0.584085	0.292042	−	0.956405i	$-0.405665\pi$
$523$	6986.00	0.584085	0.292042	+	0.956405i	$0.405665\pi$
$524$	0	0
$525$	−4500.00	−0.374088
$526$	0	0
$527$	1128.00	0.0932380
$528$	0	0
$529$	−10403.0	−0.855018
$530$	0	0
$531$	41172.0	3.36481
$532$	0	0
$533$	−324.000	−0.0263302
$534$	0	0
$535$	4250.00	0.343446
$536$	0	0
$537$	−16200.0	−1.30183
$538$	0	0
$539$	304.000	0.0242935
$540$	0	0
$541$	11958.0	0.950304	0.475152	−	0.879904i	$-0.342393\pi$
$541$	11958.0	0.950304	0.475152	+	0.879904i	$0.342393\pi$
$542$	0	0
$543$	25100.0	1.98369
$544$	0	0
$545$	−3330.00	−0.261728
$546$	0	0
$547$	−4194.00	−0.327829	−0.163915	−	0.986475i	$-0.552412\pi$
$547$	−4194.00	−0.327829	−0.163915	+	0.986475i	$0.552412\pi$
$548$	0	0
$549$	−19126.0	−1.48684
$550$	0	0
$551$	−17608.0	−1.36139
$552$	0	0
$553$	20880.0	1.60562
$554$	0	0
$555$	−10100.0	−0.772470
$556$	0	0
$557$	−5382.00	−0.409412	−0.204706	−	0.978823i	$-0.565624\pi$
$557$	−5382.00	−0.409412	−0.204706	+	0.978823i	$0.565624\pi$
$558$	0	0
$559$	−396.000	−0.0299625
$560$	0	0
$561$	960.000	0.0722482
$562$	0	0
$563$	15418.0	1.15416	0.577079	−	0.816688i	$-0.304192\pi$
$563$	15418.0	1.15416	0.577079	+	0.816688i	$0.304192\pi$
$564$	0	0
$565$	7230.00	0.538351
$566$	0	0
$567$	−47322.0	−3.50500
$568$	0	0
$569$	−5778.00	−0.425705	−0.212853	−	0.977084i	$-0.568275\pi$
$569$	−5778.00	−0.425705	−0.212853	+	0.977084i	$0.568275\pi$
$570$	0	0
$571$	6024.00	0.441500	0.220750	−	0.975330i	$-0.429149\pi$
$571$	6024.00	0.441500	0.220750	+	0.975330i	$0.429149\pi$
$572$	0	0
$573$	−3720.00	−0.271213
$574$	0	0
$575$	1050.00	0.0761531
$576$	0	0
$577$	554.000	0.0399711	0.0199855	−	0.999800i	$-0.493638\pi$
$577$	554.000	0.0399711	0.0199855	+	0.999800i	$0.493638\pi$
$578$	0	0
$579$	29380.0	2.10879
$580$	0	0
$581$	−11556.0	−0.825170
$582$	0	0
$583$	−11808.0	−0.838829
$584$	0	0
$585$	2190.00	0.154778
$586$	0	0
$587$	−2386.00	−0.167770	−0.0838848	−	0.996475i	$-0.526733\pi$
$587$	−2386.00	−0.167770	−0.0838848	+	0.996475i	$0.526733\pi$
$588$	0	0
$589$	23312.0	1.63082
$590$	0	0
$591$	22340.0	1.55490
$592$	0	0
$593$	−846.000	−0.0585853	−0.0292926	−	0.999571i	$-0.509325\pi$
$593$	−846.000	−0.0585853	−0.0292926	+	0.999571i	$0.509325\pi$
$594$	0	0
$595$	−540.000	−0.0372065
$596$	0	0
$597$	−30480.0	−2.08955
$598$	0	0
$599$	22304.0	1.52140	0.760698	−	0.649105i	$-0.224857\pi$
$599$	22304.0	1.52140	0.760698	+	0.649105i	$0.224857\pi$
$600$	0	0
$601$	5510.00	0.373973	0.186986	−	0.982363i	$-0.440128\pi$
$601$	5510.00	0.373973	0.186986	+	0.982363i	$0.440128\pi$
$602$	0	0
$603$	−40442.0	−2.73122
$604$	0	0
$605$	5375.00	0.361198
$606$	0	0
$607$	−8234.00	−0.550589	−0.275295	−	0.961360i	$-0.588775\pi$
$607$	−8234.00	−0.550589	−0.275295	+	0.961360i	$0.588775\pi$
$608$	0	0
$609$	−25560.0	−1.70073
$610$	0	0
$611$	−228.000	−0.0150964
$612$	0	0
$613$	−1046.00	−0.0689193	−0.0344597	−	0.999406i	$-0.510971\pi$
$613$	−1046.00	−0.0689193	−0.0344597	+	0.999406i	$0.510971\pi$
$614$	0	0
$615$	−2700.00	−0.177032
$616$	0	0
$617$	−3862.00	−0.251991	−0.125995	−	0.992031i	$-0.540212\pi$
$617$	−3862.00	−0.251991	−0.125995	+	0.992031i	$0.540212\pi$
$618$	0	0
$619$	13964.0	0.906721	0.453361	−	0.891327i	$-0.350225\pi$
$619$	13964.0	0.906721	0.453361	+	0.891327i	$0.350225\pi$
$620$	0	0
$621$	19320.0	1.24845
$622$	0	0
$623$	15372.0	0.988549
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	19840.0	1.26369
$628$	0	0
$629$	−1212.00	−0.0768293
$630$	0	0
$631$	−14884.0	−0.939022	−0.469511	−	0.882927i	$-0.655570\pi$
$631$	−14884.0	−0.939022	−0.469511	+	0.882927i	$0.655570\pi$
$632$	0	0
$633$	48960.0	3.07423
$634$	0	0
$635$	5770.00	0.360591
$636$	0	0
$637$	114.000	0.00709081
$638$	0	0
$639$	10220.0	0.632703
$640$	0	0
$641$	17838.0	1.09916	0.549578	−	0.835443i	$-0.314789\pi$
$641$	17838.0	1.09916	0.549578	+	0.835443i	$0.314789\pi$
$642$	0	0
$643$	−7814.00	−0.479244	−0.239622	−	0.970866i	$-0.577024\pi$
$643$	−7814.00	−0.479244	−0.239622	+	0.970866i	$0.577024\pi$
$644$	0	0
$645$	−3300.00	−0.201453
$646$	0	0
$647$	774.000	0.0470310	0.0235155	−	0.999723i	$-0.492514\pi$
$647$	774.000	0.0470310	0.0235155	+	0.999723i	$0.492514\pi$
$648$	0	0
$649$	−9024.00	−0.545798
$650$	0	0
$651$	33840.0	2.03732
$652$	0	0
$653$	−23422.0	−1.40364	−0.701818	−	0.712357i	$-0.747628\pi$
$653$	−23422.0	−1.40364	−0.701818	+	0.712357i	$0.747628\pi$
$654$	0	0
$655$	1840.00	0.109763
$656$	0	0
$657$	64386.0	3.82334
$658$	0	0
$659$	−13508.0	−0.798478	−0.399239	−	0.916847i	$-0.630726\pi$
$659$	−13508.0	−0.798478	−0.399239	+	0.916847i	$0.630726\pi$
$660$	0	0
$661$	−6222.00	−0.366124	−0.183062	−	0.983101i	$-0.558601\pi$
$661$	−6222.00	−0.366124	−0.183062	+	0.983101i	$0.558601\pi$
$662$	0	0
$663$	360.000	0.0210878
$664$	0	0
$665$	−11160.0	−0.650776
$666$	0	0
$667$	5964.00	0.346217
$668$	0	0
$669$	−53020.0	−3.06408
$670$	0	0
$671$	4192.00	0.241178
$672$	0	0
$673$	−15566.0	−0.891568	−0.445784	−	0.895141i	$-0.647075\pi$
$673$	−15566.0	−0.891568	−0.445784	+	0.895141i	$0.647075\pi$
$674$	0	0
$675$	11500.0	0.655756
$676$	0	0
$677$	2234.00	0.126824	0.0634118	−	0.997987i	$-0.479802\pi$
$677$	2234.00	0.126824	0.0634118	+	0.997987i	$0.479802\pi$
$678$	0	0
$679$	8604.00	0.486290
$680$	0	0
$681$	−37780.0	−2.12589
$682$	0	0
$683$	13282.0	0.744102	0.372051	−	0.928212i	$-0.378655\pi$
$683$	13282.0	0.744102	0.372051	+	0.928212i	$0.378655\pi$
$684$	0	0
$685$	3350.00	0.186857
$686$	0	0
$687$	−30340.0	−1.68492
$688$	0	0
$689$	−4428.00	−0.244838
$690$	0	0
$691$	−27416.0	−1.50934	−0.754670	−	0.656105i	$-0.772203\pi$
$691$	−27416.0	−1.50934	−0.754670	+	0.656105i	$0.772203\pi$
$692$	0	0
$693$	21024.0	1.15243
$694$	0	0
$695$	2860.00	0.156095
$696$	0	0
$697$	−324.000	−0.0176074
$698$	0	0
$699$	−34780.0	−1.88197
$700$	0	0
$701$	25626.0	1.38071	0.690357	−	0.723469i	$-0.257453\pi$
$701$	25626.0	1.38071	0.690357	+	0.723469i	$0.257453\pi$
$702$	0	0
$703$	−25048.0	−1.34382
$704$	0	0
$705$	−1900.00	−0.101501
$706$	0	0
$707$	32292.0	1.71777
$708$	0	0
$709$	11702.0	0.619856	0.309928	−	0.950760i	$-0.399695\pi$
$709$	11702.0	0.619856	0.309928	+	0.950760i	$0.399695\pi$
$710$	0	0
$711$	−84680.0	−4.46659
$712$	0	0
$713$	−7896.00	−0.414737
$714$	0	0
$715$	−480.000	−0.0251063
$716$	0	0
$717$	15600.0	0.812542
$718$	0	0
$719$	−28008.0	−1.45274	−0.726371	−	0.687302i	$-0.758795\pi$
$719$	−28008.0	−1.45274	−0.726371	+	0.687302i	$0.758795\pi$
$720$	0	0
$721$	−11556.0	−0.596904
$722$	0	0
$723$	−32180.0	−1.65531
$724$	0	0
$725$	3550.00	0.181853
$726$	0	0
$727$	−7682.00	−0.391898	−0.195949	−	0.980614i	$-0.562779\pi$
$727$	−7682.00	−0.391898	−0.195949	+	0.980614i	$0.562779\pi$
$728$	0	0
$729$	67717.0	3.44038
$730$	0	0
$731$	−396.000	−0.0200364
$732$	0	0
$733$	−14270.0	−0.719065	−0.359532	−	0.933133i	$-0.617064\pi$
$733$	−14270.0	−0.719065	−0.359532	+	0.933133i	$0.617064\pi$
$734$	0	0
$735$	950.000	0.0476752
$736$	0	0
$737$	8864.00	0.443025
$738$	0	0
$739$	29324.0	1.45968	0.729838	−	0.683620i	$-0.239595\pi$
$739$	29324.0	1.45968	0.729838	+	0.683620i	$0.239595\pi$
$740$	0	0
$741$	7440.00	0.368846
$742$	0	0
$743$	29258.0	1.44465	0.722323	−	0.691556i	$-0.243074\pi$
$743$	29258.0	1.44465	0.722323	+	0.691556i	$0.243074\pi$
$744$	0	0
$745$	−8650.00	−0.425385
$746$	0	0
$747$	46866.0	2.29550
$748$	0	0
$749$	15300.0	0.746395
$750$	0	0
$751$	19076.0	0.926888	0.463444	−	0.886126i	$-0.346613\pi$
$751$	19076.0	0.926888	0.463444	+	0.886126i	$0.346613\pi$
$752$	0	0
$753$	6880.00	0.332963
$754$	0	0
$755$	−6620.00	−0.319108
$756$	0	0
$757$	−22670.0	−1.08845	−0.544224	−	0.838940i	$-0.683176\pi$
$757$	−22670.0	−1.08845	−0.544224	+	0.838940i	$0.683176\pi$
$758$	0	0
$759$	−6720.00	−0.321371
$760$	0	0
$761$	−23206.0	−1.10541	−0.552705	−	0.833377i	$-0.686404\pi$
$761$	−23206.0	−1.10541	−0.552705	+	0.833377i	$0.686404\pi$
$762$	0	0
$763$	−11988.0	−0.568800
$764$	0	0
$765$	2190.00	0.103503
$766$	0	0
$767$	−3384.00	−0.159308
$768$	0	0
$769$	−1854.00	−0.0869401	−0.0434701	−	0.999055i	$-0.513841\pi$
$769$	−1854.00	−0.0869401	−0.0434701	+	0.999055i	$0.513841\pi$
$770$	0	0
$771$	21700.0	1.01363
$772$	0	0
$773$	6474.00	0.301234	0.150617	−	0.988592i	$-0.451874\pi$
$773$	6474.00	0.301234	0.150617	+	0.988592i	$0.451874\pi$
$774$	0	0
$775$	−4700.00	−0.217844
$776$	0	0
$777$	−36360.0	−1.67877
$778$	0	0
$779$	−6696.00	−0.307971
$780$	0	0
$781$	−2240.00	−0.102629
$782$	0	0
$783$	65320.0	2.98129
$784$	0	0
$785$	−14730.0	−0.669728
$786$	0	0
$787$	−20354.0	−0.921908	−0.460954	−	0.887424i	$-0.652493\pi$
$787$	−20354.0	−0.921908	−0.460954	+	0.887424i	$0.652493\pi$
$788$	0	0
$789$	22740.0	1.02607
$790$	0	0
$791$	26028.0	1.16997
$792$	0	0
$793$	1572.00	0.0703952
$794$	0	0
$795$	−36900.0	−1.64617
$796$	0	0
$797$	−1886.00	−0.0838213	−0.0419106	−	0.999121i	$-0.513344\pi$
$797$	−1886.00	−0.0838213	−0.0419106	+	0.999121i	$0.513344\pi$
$798$	0	0
$799$	−228.000	−0.0100952
$800$	0	0
$801$	−62342.0	−2.75000
$802$	0	0
$803$	−14112.0	−0.620176
$804$	0	0
$805$	3780.00	0.165500
$806$	0	0
$807$	71460.0	3.11711
$808$	0	0
$809$	−9462.00	−0.411207	−0.205603	−	0.978635i	$-0.565916\pi$
$809$	−9462.00	−0.411207	−0.205603	+	0.978635i	$0.565916\pi$
$810$	0	0
$811$	24512.0	1.06132	0.530661	−	0.847584i	$-0.321944\pi$
$811$	24512.0	1.06132	0.530661	+	0.847584i	$0.321944\pi$
$812$	0	0
$813$	26040.0	1.12332
$814$	0	0
$815$	−10490.0	−0.450857
$816$	0	0
$817$	−8184.00	−0.350455
$818$	0	0
$819$	7884.00	0.336373
$820$	0	0
$821$	36242.0	1.54063	0.770313	−	0.637666i	$-0.220100\pi$
$821$	36242.0	1.54063	0.770313	+	0.637666i	$0.220100\pi$
$822$	0	0
$823$	−17718.0	−0.750438	−0.375219	−	0.926936i	$-0.622433\pi$
$823$	−17718.0	−0.750438	−0.375219	+	0.926936i	$0.622433\pi$
$824$	0	0
$825$	−4000.00	−0.168803
$826$	0	0
$827$	6726.00	0.282812	0.141406	−	0.989952i	$-0.454838\pi$
$827$	6726.00	0.282812	0.141406	+	0.989952i	$0.454838\pi$
$828$	0	0
$829$	41722.0	1.74797	0.873984	−	0.485955i	$-0.161528\pi$
$829$	41722.0	1.74797	0.873984	+	0.485955i	$0.161528\pi$
$830$	0	0
$831$	−51500.0	−2.14984
$832$	0	0
$833$	114.000	0.00474174
$834$	0	0
$835$	4330.00	0.179456
$836$	0	0
$837$	−86480.0	−3.57131
$838$	0	0
$839$	16720.0	0.688008	0.344004	−	0.938968i	$-0.388217\pi$
$839$	16720.0	0.688008	0.344004	+	0.938968i	$0.388217\pi$
$840$	0	0
$841$	−4225.00	−0.173234
$842$	0	0
$843$	52700.0	2.15313
$844$	0	0
$845$	10805.0	0.439886
$846$	0	0
$847$	19350.0	0.784975
$848$	0	0
$849$	34340.0	1.38816
$850$	0	0
$851$	8484.00	0.341748
$852$	0	0
$853$	−33286.0	−1.33610	−0.668049	−	0.744118i	$-0.732870\pi$
$853$	−33286.0	−1.33610	−0.668049	+	0.744118i	$0.732870\pi$
$854$	0	0
$855$	45260.0	1.81036
$856$	0	0
$857$	38978.0	1.55363	0.776816	−	0.629727i	$-0.216833\pi$
$857$	38978.0	1.55363	0.776816	+	0.629727i	$0.216833\pi$
$858$	0	0
$859$	−1916.00	−0.0761037	−0.0380518	−	0.999276i	$-0.512115\pi$
$859$	−1916.00	−0.0761037	−0.0380518	+	0.999276i	$0.512115\pi$
$860$	0	0
$861$	−9720.00	−0.384735
$862$	0	0
$863$	−2374.00	−0.0936407	−0.0468203	−	0.998903i	$-0.514909\pi$
$863$	−2374.00	−0.0936407	−0.0468203	+	0.998903i	$0.514909\pi$
$864$	0	0
$865$	8390.00	0.329790
$866$	0	0
$867$	−48770.0	−1.91040
$868$	0	0
$869$	18560.0	0.724517
$870$	0	0
$871$	3324.00	0.129310
$872$	0	0
$873$	−34894.0	−1.35279
$874$	0	0
$875$	2250.00	0.0869302
$876$	0	0
$877$	32722.0	1.25991	0.629956	−	0.776631i	$-0.283073\pi$
$877$	32722.0	1.25991	0.629956	+	0.776631i	$0.283073\pi$
$878$	0	0
$879$	−98780.0	−3.79041
$880$	0	0
$881$	5390.00	0.206122	0.103061	−	0.994675i	$-0.467136\pi$
$881$	5390.00	0.206122	0.103061	+	0.994675i	$0.467136\pi$
$882$	0	0
$883$	−43238.0	−1.64788	−0.823938	−	0.566680i	$-0.808228\pi$
$883$	−43238.0	−1.64788	−0.823938	+	0.566680i	$0.808228\pi$
$884$	0	0
$885$	−28200.0	−1.07111
$886$	0	0
$887$	−11010.0	−0.416775	−0.208388	−	0.978046i	$-0.566822\pi$
$887$	−11010.0	−0.416775	−0.208388	+	0.978046i	$0.566822\pi$
$888$	0	0
$889$	20772.0	0.783656
$890$	0	0
$891$	−42064.0	−1.58159
$892$	0	0
$893$	−4712.00	−0.176575
$894$	0	0
$895$	8100.00	0.302517
$896$	0	0
$897$	−2520.00	−0.0938020
$898$	0	0
$899$	−26696.0	−0.990391
$900$	0	0
$901$	−4428.00	−0.163727
$902$	0	0
$903$	−11880.0	−0.437809
$904$	0	0
$905$	−12550.0	−0.460968
$906$	0	0
$907$	−74.0000	−0.00270907	−0.00135454	−	0.999999i	$-0.500431\pi$
$907$	−74.0000	−0.00270907	−0.00135454	+	0.999999i	$0.500431\pi$
$908$	0	0
$909$	−130962.	−4.77859
$910$	0	0
$911$	17460.0	0.634990	0.317495	−	0.948260i	$-0.397158\pi$
$911$	17460.0	0.634990	0.317495	+	0.948260i	$0.397158\pi$
$912$	0	0
$913$	−10272.0	−0.372348
$914$	0	0
$915$	13100.0	0.473303
$916$	0	0
$917$	6624.00	0.238543
$918$	0	0
$919$	17072.0	0.612789	0.306395	−	0.951905i	$-0.400877\pi$
$919$	17072.0	0.612789	0.306395	+	0.951905i	$0.400877\pi$
$920$	0	0
$921$	80540.0	2.88152
$922$	0	0
$923$	−840.000	−0.0299555
$924$	0	0
$925$	5050.00	0.179506
$926$	0	0
$927$	46866.0	1.66050
$928$	0	0
$929$	−14826.0	−0.523601	−0.261800	−	0.965122i	$-0.584316\pi$
$929$	−14826.0	−0.523601	−0.261800	+	0.965122i	$0.584316\pi$
$930$	0	0
$931$	2356.00	0.0829375
$932$	0	0
$933$	54920.0	1.92712
$934$	0	0
$935$	−480.000	−0.0167890
$936$	0	0
$937$	3354.00	0.116937	0.0584687	−	0.998289i	$-0.481378\pi$
$937$	3354.00	0.116937	0.0584687	+	0.998289i	$0.481378\pi$
$938$	0	0
$939$	−4220.00	−0.146661
$940$	0	0
$941$	−15434.0	−0.534680	−0.267340	−	0.963602i	$-0.586145\pi$
$941$	−15434.0	−0.534680	−0.267340	+	0.963602i	$0.586145\pi$
$942$	0	0
$943$	2268.00	0.0783205
$944$	0	0
$945$	41400.0	1.42512
$946$	0	0
$947$	−9306.00	−0.319329	−0.159664	−	0.987171i	$-0.551041\pi$
$947$	−9306.00	−0.319329	−0.159664	+	0.987171i	$0.551041\pi$
$948$	0	0
$949$	−5292.00	−0.181017
$950$	0	0
$951$	61940.0	2.11203
$952$	0	0
$953$	12202.0	0.414755	0.207378	−	0.978261i	$-0.433507\pi$
$953$	12202.0	0.414755	0.207378	+	0.978261i	$0.433507\pi$
$954$	0	0
$955$	1860.00	0.0630243
$956$	0	0
$957$	−22720.0	−0.767433
$958$	0	0
$959$	12060.0	0.406087
$960$	0	0
$961$	5553.00	0.186399
$962$	0	0
$963$	−62050.0	−2.07636
$964$	0	0
$965$	−14690.0	−0.490039
$966$	0	0
$967$	17478.0	0.581235	0.290618	−	0.956839i	$-0.406139\pi$
$967$	17478.0	0.581235	0.290618	+	0.956839i	$0.406139\pi$
$968$	0	0
$969$	7440.00	0.246653
$970$	0	0
$971$	10920.0	0.360906	0.180453	−	0.983584i	$-0.442244\pi$
$971$	10920.0	0.360906	0.180453	+	0.983584i	$0.442244\pi$
$972$	0	0
$973$	10296.0	0.339234
$974$	0	0
$975$	−1500.00	−0.0492702
$976$	0	0
$977$	10834.0	0.354770	0.177385	−	0.984142i	$-0.443236\pi$
$977$	10834.0	0.354770	0.177385	+	0.984142i	$0.443236\pi$
$978$	0	0
$979$	13664.0	0.446071
$980$	0	0
$981$	48618.0	1.58232
$982$	0	0
$983$	−36862.0	−1.19605	−0.598024	−	0.801478i	$-0.704047\pi$
$983$	−36862.0	−1.19605	−0.598024	+	0.801478i	$0.704047\pi$
$984$	0	0
$985$	−11170.0	−0.361326
$986$	0	0
$987$	−6840.00	−0.220587
$988$	0	0
$989$	2772.00	0.0891248
$990$	0	0
$991$	5380.00	0.172453	0.0862267	−	0.996276i	$-0.472519\pi$
$991$	5380.00	0.172453	0.0862267	+	0.996276i	$0.472519\pi$
$992$	0	0
$993$	76880.0	2.45691
$994$	0	0
$995$	15240.0	0.485568
$996$	0	0
$997$	31266.0	0.993184	0.496592	−	0.867984i	$-0.334585\pi$
$997$	31266.0	0.993184	0.496592	+	0.867984i	$0.334585\pi$
$998$	0	0
$999$	92920.0	2.94280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.4.a.c.1.1	✓	1
3.2	odd	2		360.4.a.i.1.1		1
4.3	odd	2		80.4.a.a.1.1		1
5.2	odd	4		200.4.c.a.49.1		2
5.3	odd	4		200.4.c.a.49.2		2
5.4	even	2		200.4.a.a.1.1		1
7.6	odd	2		1960.4.a.a.1.1		1
8.3	odd	2		320.4.a.n.1.1		1
8.5	even	2		320.4.a.a.1.1		1
12.11	even	2		720.4.a.ba.1.1		1
15.2	even	4		1800.4.f.n.649.1		2
15.8	even	4		1800.4.f.n.649.2		2
15.14	odd	2		1800.4.a.bd.1.1		1
16.3	odd	4		1280.4.d.b.641.1		2
16.5	even	4		1280.4.d.o.641.1		2
16.11	odd	4		1280.4.d.b.641.2		2
16.13	even	4		1280.4.d.o.641.2		2
20.3	even	4		400.4.c.a.49.1		2
20.7	even	4		400.4.c.a.49.2		2
20.19	odd	2		400.4.a.u.1.1		1
40.19	odd	2		1600.4.a.a.1.1		1
40.29	even	2		1600.4.a.ca.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.a.c.1.1	✓	1	1.1	even	1	trivial
80.4.a.a.1.1		1	4.3	odd	2
200.4.a.a.1.1		1	5.4	even	2
200.4.c.a.49.1		2	5.2	odd	4
200.4.c.a.49.2		2	5.3	odd	4
320.4.a.a.1.1		1	8.5	even	2
320.4.a.n.1.1		1	8.3	odd	2
360.4.a.i.1.1		1	3.2	odd	2
400.4.a.u.1.1		1	20.19	odd	2
400.4.c.a.49.1		2	20.3	even	4
400.4.c.a.49.2		2	20.7	even	4
720.4.a.ba.1.1		1	12.11	even	2
1280.4.d.b.641.1		2	16.3	odd	4
1280.4.d.b.641.2		2	16.11	odd	4
1280.4.d.o.641.1		2	16.5	even	4
1280.4.d.o.641.2		2	16.13	even	4
1600.4.a.a.1.1		1	40.19	odd	2
1600.4.a.ca.1.1		1	40.29	even	2
1800.4.a.bd.1.1		1	15.14	odd	2
1800.4.f.n.649.1		2	15.2	even	4
1800.4.f.n.649.2		2	15.8	even	4
1960.4.a.a.1.1		1	7.6	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:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	40.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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