LMFDB - Embedded newform 704.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,4,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, 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("704.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	704.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:	$41.5373446440$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-7.00000 q^{3} -9.00000 q^{5} +2.00000 q^{7} +22.0000 q^{9} +O(q^{10})$

$q-7.00000 q^{3} -9.00000 q^{5} +2.00000 q^{7} +22.0000 q^{9} +11.0000 q^{11} +63.0000 q^{15} -38.0000 q^{17} -44.0000 q^{19} -14.0000 q^{21} +175.000 q^{23} -44.0000 q^{25} +35.0000 q^{27} +264.000 q^{29} +159.000 q^{31} -77.0000 q^{33} -18.0000 q^{35} +173.000 q^{37} -220.000 q^{41} +542.000 q^{43} -198.000 q^{45} -264.000 q^{47} -339.000 q^{49} +266.000 q^{51} -682.000 q^{53} -99.0000 q^{55} +308.000 q^{57} -421.000 q^{59} -308.000 q^{61} +44.0000 q^{63} -177.000 q^{67} -1225.00 q^{69} +365.000 q^{71} -528.000 q^{73} +308.000 q^{75} +22.0000 q^{77} +686.000 q^{79} -839.000 q^{81} -698.000 q^{83} +342.000 q^{85} -1848.00 q^{87} +967.000 q^{89} -1113.00 q^{93} +396.000 q^{95} -1127.00 q^{97} +242.000 q^{99} +O(q^{100})$

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
$3$	−7.00000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	0	0
$5$	−9.00000	−0.804984	−0.402492	−	0.915423i	$-0.631856\pi$
$5$	−9.00000	−0.804984	−0.402492	+	0.915423i	$0.631856\pi$
$6$	0	0
$7$	2.00000	0.107990	0.0539949	−	0.998541i	$-0.482805\pi$
$7$	2.00000	0.107990	0.0539949	+	0.998541i	$0.482805\pi$
$8$	0	0
$9$	22.0000	0.814815
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	63.0000	1.08444
$16$	0	0
$17$	−38.0000	−0.542138	−0.271069	−	0.962560i	$-0.587377\pi$
$17$	−38.0000	−0.542138	−0.271069	+	0.962560i	$0.587377\pi$
$18$	0	0
$19$	−44.0000	−0.531279	−0.265639	−	0.964072i	$-0.585583\pi$
$19$	−44.0000	−0.531279	−0.265639	+	0.964072i	$0.585583\pi$
$20$	0	0
$21$	−14.0000	−0.145479
$22$	0	0
$23$	175.000	1.58652	0.793261	−	0.608881i	$-0.208381\pi$
$23$	175.000	1.58652	0.793261	+	0.608881i	$0.208381\pi$
$24$	0	0
$25$	−44.0000	−0.352000
$26$	0	0
$27$	35.0000	0.249472
$28$	0	0
$29$	264.000	1.69047	0.845234	−	0.534396i	$-0.179461\pi$
$29$	264.000	1.69047	0.845234	+	0.534396i	$0.179461\pi$
$30$	0	0
$31$	159.000	0.921201	0.460601	−	0.887607i	$-0.347634\pi$
$31$	159.000	0.921201	0.460601	+	0.887607i	$0.347634\pi$
$32$	0	0
$33$	−77.0000	−0.406181
$34$	0	0
$35$	−18.0000	−0.0869302
$36$	0	0
$37$	173.000	0.768676	0.384338	−	0.923192i	$-0.374430\pi$
$37$	173.000	0.768676	0.384338	+	0.923192i	$0.374430\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−220.000	−0.838006	−0.419003	−	0.907985i	$-0.637620\pi$
$41$	−220.000	−0.838006	−0.419003	+	0.907985i	$0.637620\pi$
$42$	0	0
$43$	542.000	1.92219	0.961096	−	0.276216i	$-0.0890805\pi$
$43$	542.000	1.92219	0.961096	+	0.276216i	$0.0890805\pi$
$44$	0	0
$45$	−198.000	−0.655913
$46$	0	0
$47$	−264.000	−0.819327	−0.409663	−	0.912237i	$-0.634354\pi$
$47$	−264.000	−0.819327	−0.409663	+	0.912237i	$0.634354\pi$
$48$	0	0
$49$	−339.000	−0.988338
$50$	0	0
$51$	266.000	0.730342
$52$	0	0
$53$	−682.000	−1.76755	−0.883773	−	0.467916i	$-0.845005\pi$
$53$	−682.000	−1.76755	−0.883773	+	0.467916i	$0.845005\pi$
$54$	0	0
$55$	−99.0000	−0.242712
$56$	0	0
$57$	308.000	0.715712
$58$	0	0
$59$	−421.000	−0.928975	−0.464488	−	0.885580i	$-0.653761\pi$
$59$	−421.000	−0.928975	−0.464488	+	0.885580i	$0.653761\pi$
$60$	0	0
$61$	−308.000	−0.646481	−0.323241	−	0.946317i	$-0.604772\pi$
$61$	−308.000	−0.646481	−0.323241	+	0.946317i	$0.604772\pi$
$62$	0	0
$63$	44.0000	0.0879917
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−177.000	−0.322746	−0.161373	−	0.986893i	$-0.551592\pi$
$67$	−177.000	−0.322746	−0.161373	+	0.986893i	$0.551592\pi$
$68$	0	0
$69$	−1225.00	−2.13729
$70$	0	0
$71$	365.000	0.610106	0.305053	−	0.952335i	$-0.401326\pi$
$71$	365.000	0.610106	0.305053	+	0.952335i	$0.401326\pi$
$72$	0	0
$73$	−528.000	−0.846544	−0.423272	−	0.906003i	$-0.639119\pi$
$73$	−528.000	−0.846544	−0.423272	+	0.906003i	$0.639119\pi$
$74$	0	0
$75$	308.000	0.474197
$76$	0	0
$77$	22.0000	0.0325602
$78$	0	0
$79$	686.000	0.976975	0.488488	−	0.872571i	$-0.337549\pi$
$79$	686.000	0.976975	0.488488	+	0.872571i	$0.337549\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	0	0
$83$	−698.000	−0.923078	−0.461539	−	0.887120i	$-0.652703\pi$
$83$	−698.000	−0.923078	−0.461539	+	0.887120i	$0.652703\pi$
$84$	0	0
$85$	342.000	0.436413
$86$	0	0
$87$	−1848.00	−2.27731
$88$	0	0
$89$	967.000	1.15171	0.575853	−	0.817553i	$-0.304670\pi$
$89$	967.000	1.15171	0.575853	+	0.817553i	$0.304670\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1113.00	−1.24100
$94$	0	0
$95$	396.000	0.427671
$96$	0	0
$97$	−1127.00	−1.17969	−0.589843	−	0.807518i	$-0.700810\pi$
$97$	−1127.00	−1.17969	−0.589843	+	0.807518i	$0.700810\pi$
$98$	0	0
$99$	242.000	0.245676
$100$	0	0
$101$	−510.000	−0.502445	−0.251222	−	0.967929i	$-0.580833\pi$
$101$	−510.000	−0.502445	−0.251222	+	0.967929i	$0.580833\pi$
$102$	0	0
$103$	−1056.00	−1.01020	−0.505101	−	0.863060i	$-0.668545\pi$
$103$	−1056.00	−1.01020	−0.505101	+	0.863060i	$0.668545\pi$
$104$	0	0
$105$	126.000	0.117108
$106$	0	0
$107$	−1046.00	−0.945053	−0.472526	−	0.881317i	$-0.656658\pi$
$107$	−1046.00	−0.945053	−0.472526	+	0.881317i	$0.656658\pi$
$108$	0	0
$109$	−250.000	−0.219685	−0.109842	−	0.993949i	$-0.535035\pi$
$109$	−250.000	−0.219685	−0.109842	+	0.993949i	$0.535035\pi$
$110$	0	0
$111$	−1211.00	−1.03552
$112$	0	0
$113$	1401.00	1.16633	0.583164	−	0.812355i	$-0.301815\pi$
$113$	1401.00	1.16633	0.583164	+	0.812355i	$0.301815\pi$
$114$	0	0
$115$	−1575.00	−1.27713
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−76.0000	−0.0585455
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1540.00	1.12892
$124$	0	0
$125$	1521.00	1.08834
$126$	0	0
$127$	−132.000	−0.0922292	−0.0461146	−	0.998936i	$-0.514684\pi$
$127$	−132.000	−0.0922292	−0.0461146	+	0.998936i	$0.514684\pi$
$128$	0	0
$129$	−3794.00	−2.58948
$130$	0	0
$131$	2054.00	1.36991	0.684957	−	0.728583i	$-0.259821\pi$
$131$	2054.00	1.36991	0.684957	+	0.728583i	$0.259821\pi$
$132$	0	0
$133$	−88.0000	−0.0573727
$134$	0	0
$135$	−315.000	−0.200821
$136$	0	0
$137$	889.000	0.554397	0.277199	−	0.960813i	$-0.410594\pi$
$137$	889.000	0.554397	0.277199	+	0.960813i	$0.410594\pi$
$138$	0	0
$139$	−2638.00	−1.60973	−0.804864	−	0.593459i	$-0.797762\pi$
$139$	−2638.00	−1.60973	−0.804864	+	0.593459i	$0.797762\pi$
$140$	0	0
$141$	1848.00	1.10376
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2376.00	−1.36080
$146$	0	0
$147$	2373.00	1.33144
$148$	0	0
$149$	−3338.00	−1.83530	−0.917650	−	0.397390i	$-0.869916\pi$
$149$	−3338.00	−1.83530	−0.917650	+	0.397390i	$0.869916\pi$
$150$	0	0
$151$	−430.000	−0.231741	−0.115871	−	0.993264i	$-0.536966\pi$
$151$	−430.000	−0.231741	−0.115871	+	0.993264i	$0.536966\pi$
$152$	0	0
$153$	−836.000	−0.441742
$154$	0	0
$155$	−1431.00	−0.741553
$156$	0	0
$157$	1159.00	0.589161	0.294580	−	0.955627i	$-0.404820\pi$
$157$	1159.00	0.589161	0.294580	+	0.955627i	$0.404820\pi$
$158$	0	0
$159$	4774.00	2.38115
$160$	0	0
$161$	350.000	0.171328
$162$	0	0
$163$	−1012.00	−0.486294	−0.243147	−	0.969989i	$-0.578180\pi$
$163$	−1012.00	−0.486294	−0.243147	+	0.969989i	$0.578180\pi$
$164$	0	0
$165$	693.000	0.326970
$166$	0	0
$167$	1584.00	0.733974	0.366987	−	0.930226i	$-0.380389\pi$
$167$	1584.00	0.733974	0.366987	+	0.930226i	$0.380389\pi$
$168$	0	0
$169$	−2197.00	−1.00000
$170$	0	0
$171$	−968.000	−0.432894
$172$	0	0
$173$	−474.000	−0.208310	−0.104155	−	0.994561i	$-0.533214\pi$
$173$	−474.000	−0.208310	−0.104155	+	0.994561i	$0.533214\pi$
$174$	0	0
$175$	−88.0000	−0.0380124
$176$	0	0
$177$	2947.00	1.25147
$178$	0	0
$179$	−1665.00	−0.695240	−0.347620	−	0.937636i	$-0.613010\pi$
$179$	−1665.00	−0.695240	−0.347620	+	0.937636i	$0.613010\pi$
$180$	0	0
$181$	−2543.00	−1.04431	−0.522154	−	0.852851i	$-0.674871\pi$
$181$	−2543.00	−1.04431	−0.522154	+	0.852851i	$0.674871\pi$
$182$	0	0
$183$	2156.00	0.870908
$184$	0	0
$185$	−1557.00	−0.618773
$186$	0	0
$187$	−418.000	−0.163461
$188$	0	0
$189$	70.0000	0.0269405
$190$	0	0
$191$	−1631.00	−0.617880	−0.308940	−	0.951082i	$-0.599974\pi$
$191$	−1631.00	−0.617880	−0.308940	+	0.951082i	$0.599974\pi$
$192$	0	0
$193$	484.000	0.180513	0.0902567	−	0.995919i	$-0.471231\pi$
$193$	484.000	0.180513	0.0902567	+	0.995919i	$0.471231\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1966.00	−0.711024	−0.355512	−	0.934672i	$-0.615693\pi$
$197$	−1966.00	−0.711024	−0.355512	+	0.934672i	$0.615693\pi$
$198$	0	0
$199$	968.000	0.344823	0.172411	−	0.985025i	$-0.444844\pi$
$199$	968.000	0.344823	0.172411	+	0.985025i	$0.444844\pi$
$200$	0	0
$201$	1239.00	0.434788
$202$	0	0
$203$	528.000	0.182553
$204$	0	0
$205$	1980.00	0.674581
$206$	0	0
$207$	3850.00	1.29272
$208$	0	0
$209$	−484.000	−0.160187
$210$	0	0
$211$	2948.00	0.961842	0.480921	−	0.876764i	$-0.340302\pi$
$211$	2948.00	0.961842	0.480921	+	0.876764i	$0.340302\pi$
$212$	0	0
$213$	−2555.00	−0.821905
$214$	0	0
$215$	−4878.00	−1.54733
$216$	0	0
$217$	318.000	0.0994804
$218$	0	0
$219$	3696.00	1.14042
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1995.00	0.599081	0.299541	−	0.954084i	$-0.403167\pi$
$223$	1995.00	0.599081	0.299541	+	0.954084i	$0.403167\pi$
$224$	0	0
$225$	−968.000	−0.286815
$226$	0	0
$227$	6026.00	1.76194	0.880968	−	0.473175i	$-0.156892\pi$
$227$	6026.00	1.76194	0.880968	+	0.473175i	$0.156892\pi$
$228$	0	0
$229$	4417.00	1.27460	0.637300	−	0.770615i	$-0.280051\pi$
$229$	4417.00	1.27460	0.637300	+	0.770615i	$0.280051\pi$
$230$	0	0
$231$	−154.000	−0.0438634
$232$	0	0
$233$	5808.00	1.63302	0.816512	−	0.577328i	$-0.195905\pi$
$233$	5808.00	1.63302	0.816512	+	0.577328i	$0.195905\pi$
$234$	0	0
$235$	2376.00	0.659545
$236$	0	0
$237$	−4802.00	−1.31613
$238$	0	0
$239$	626.000	0.169425	0.0847125	−	0.996405i	$-0.473003\pi$
$239$	626.000	0.169425	0.0847125	+	0.996405i	$0.473003\pi$
$240$	0	0
$241$	−3520.00	−0.940843	−0.470421	−	0.882442i	$-0.655898\pi$
$241$	−3520.00	−0.940843	−0.470421	+	0.882442i	$0.655898\pi$
$242$	0	0
$243$	4928.00	1.30095
$244$	0	0
$245$	3051.00	0.795597
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4886.00	1.24352
$250$	0	0
$251$	−3921.00	−0.986021	−0.493011	−	0.870023i	$-0.664104\pi$
$251$	−3921.00	−0.986021	−0.493011	+	0.870023i	$0.664104\pi$
$252$	0	0
$253$	1925.00	0.478355
$254$	0	0
$255$	−2394.00	−0.587914
$256$	0	0
$257$	4598.00	1.11601	0.558007	−	0.829837i	$-0.311566\pi$
$257$	4598.00	1.11601	0.558007	+	0.829837i	$0.311566\pi$
$258$	0	0
$259$	346.000	0.0830092
$260$	0	0
$261$	5808.00	1.37742
$262$	0	0
$263$	−5838.00	−1.36877	−0.684385	−	0.729121i	$-0.739929\pi$
$263$	−5838.00	−1.36877	−0.684385	+	0.729121i	$0.739929\pi$
$264$	0	0
$265$	6138.00	1.42285
$266$	0	0
$267$	−6769.00	−1.55152
$268$	0	0
$269$	5038.00	1.14190	0.570952	−	0.820983i	$-0.306574\pi$
$269$	5038.00	1.14190	0.570952	+	0.820983i	$0.306574\pi$
$270$	0	0
$271$	−8096.00	−1.81475	−0.907374	−	0.420323i	$-0.861917\pi$
$271$	−8096.00	−1.81475	−0.907374	+	0.420323i	$0.861917\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−484.000	−0.106132
$276$	0	0
$277$	−7014.00	−1.52141	−0.760705	−	0.649098i	$-0.775146\pi$
$277$	−7014.00	−1.52141	−0.760705	+	0.649098i	$0.775146\pi$
$278$	0	0
$279$	3498.00	0.750609
$280$	0	0
$281$	−4362.00	−0.926032	−0.463016	−	0.886350i	$-0.653233\pi$
$281$	−4362.00	−0.926032	−0.463016	+	0.886350i	$0.653233\pi$
$282$	0	0
$283$	−4620.00	−0.970426	−0.485213	−	0.874396i	$-0.661258\pi$
$283$	−4620.00	−0.970426	−0.485213	+	0.874396i	$0.661258\pi$
$284$	0	0
$285$	−2772.00	−0.576137
$286$	0	0
$287$	−440.000	−0.0904961
$288$	0	0
$289$	−3469.00	−0.706086
$290$	0	0
$291$	7889.00	1.58921
$292$	0	0
$293$	7172.00	1.43001	0.715005	−	0.699120i	$-0.246425\pi$
$293$	7172.00	1.43001	0.715005	+	0.699120i	$0.246425\pi$
$294$	0	0
$295$	3789.00	0.747811
$296$	0	0
$297$	385.000	0.0752187
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1084.00	0.207577
$302$	0	0
$303$	3570.00	0.676868
$304$	0	0
$305$	2772.00	0.520407
$306$	0	0
$307$	−7348.00	−1.36603	−0.683017	−	0.730402i	$-0.739333\pi$
$307$	−7348.00	−1.36603	−0.683017	+	0.730402i	$0.739333\pi$
$308$	0	0
$309$	7392.00	1.36089
$310$	0	0
$311$	−5508.00	−1.00428	−0.502138	−	0.864787i	$-0.667453\pi$
$311$	−5508.00	−1.00428	−0.502138	+	0.864787i	$0.667453\pi$
$312$	0	0
$313$	−7009.00	−1.26573	−0.632863	−	0.774264i	$-0.718120\pi$
$313$	−7009.00	−1.26573	−0.632863	+	0.774264i	$0.718120\pi$
$314$	0	0
$315$	−396.000	−0.0708320
$316$	0	0
$317$	−853.000	−0.151133	−0.0755666	−	0.997141i	$-0.524077\pi$
$317$	−853.000	−0.151133	−0.0755666	+	0.997141i	$0.524077\pi$
$318$	0	0
$319$	2904.00	0.509695
$320$	0	0
$321$	7322.00	1.27313
$322$	0	0
$323$	1672.00	0.288027
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1750.00	0.295949
$328$	0	0
$329$	−528.000	−0.0884790
$330$	0	0
$331$	−3631.00	−0.602954	−0.301477	−	0.953473i	$-0.597480\pi$
$331$	−3631.00	−0.602954	−0.301477	+	0.953473i	$0.597480\pi$
$332$	0	0
$333$	3806.00	0.626329
$334$	0	0
$335$	1593.00	0.259806
$336$	0	0
$337$	11574.0	1.87085	0.935424	−	0.353527i	$-0.115018\pi$
$337$	11574.0	1.87085	0.935424	+	0.353527i	$0.115018\pi$
$338$	0	0
$339$	−9807.00	−1.57122
$340$	0	0
$341$	1749.00	0.277753
$342$	0	0
$343$	−1364.00	−0.214720
$344$	0	0
$345$	11025.0	1.72048
$346$	0	0
$347$	1056.00	0.163369	0.0816845	−	0.996658i	$-0.473970\pi$
$347$	1056.00	0.163369	0.0816845	+	0.996658i	$0.473970\pi$
$348$	0	0
$349$	−6810.00	−1.04450	−0.522251	−	0.852792i	$-0.674907\pi$
$349$	−6810.00	−1.04450	−0.522251	+	0.852792i	$0.674907\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9923.00	1.49617	0.748085	−	0.663603i	$-0.230974\pi$
$353$	9923.00	1.49617	0.748085	+	0.663603i	$0.230974\pi$
$354$	0	0
$355$	−3285.00	−0.491126
$356$	0	0
$357$	532.000	0.0788695
$358$	0	0
$359$	−2200.00	−0.323431	−0.161715	−	0.986837i	$-0.551703\pi$
$359$	−2200.00	−0.323431	−0.161715	+	0.986837i	$0.551703\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	0	0
$363$	−847.000	−0.122468
$364$	0	0
$365$	4752.00	0.681455
$366$	0	0
$367$	8935.00	1.27085	0.635427	−	0.772161i	$-0.280824\pi$
$367$	8935.00	1.27085	0.635427	+	0.772161i	$0.280824\pi$
$368$	0	0
$369$	−4840.00	−0.682819
$370$	0	0
$371$	−1364.00	−0.190877
$372$	0	0
$373$	−9122.00	−1.26627	−0.633136	−	0.774041i	$-0.718233\pi$
$373$	−9122.00	−1.26627	−0.633136	+	0.774041i	$0.718233\pi$
$374$	0	0
$375$	−10647.0	−1.46616
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−8963.00	−1.21477	−0.607386	−	0.794407i	$-0.707782\pi$
$379$	−8963.00	−1.21477	−0.607386	+	0.794407i	$0.707782\pi$
$380$	0	0
$381$	924.000	0.124247
$382$	0	0
$383$	−1721.00	−0.229606	−0.114803	−	0.993388i	$-0.536624\pi$
$383$	−1721.00	−0.229606	−0.114803	+	0.993388i	$0.536624\pi$
$384$	0	0
$385$	−198.000	−0.0262104
$386$	0	0
$387$	11924.0	1.56623
$388$	0	0
$389$	7351.00	0.958125	0.479062	−	0.877781i	$-0.340977\pi$
$389$	7351.00	0.958125	0.479062	+	0.877781i	$0.340977\pi$
$390$	0	0
$391$	−6650.00	−0.860115
$392$	0	0
$393$	−14378.0	−1.84548
$394$	0	0
$395$	−6174.00	−0.786450
$396$	0	0
$397$	8338.00	1.05409	0.527043	−	0.849839i	$-0.323301\pi$
$397$	8338.00	1.05409	0.527043	+	0.849839i	$0.323301\pi$
$398$	0	0
$399$	616.000	0.0772897
$400$	0	0
$401$	−11814.0	−1.47123	−0.735615	−	0.677400i	$-0.763107\pi$
$401$	−11814.0	−1.47123	−0.735615	+	0.677400i	$0.763107\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	7551.00	0.926450
$406$	0	0
$407$	1903.00	0.231765
$408$	0	0
$409$	−7278.00	−0.879887	−0.439944	−	0.898025i	$-0.645002\pi$
$409$	−7278.00	−0.879887	−0.439944	+	0.898025i	$0.645002\pi$
$410$	0	0
$411$	−6223.00	−0.746856
$412$	0	0
$413$	−842.000	−0.100320
$414$	0	0
$415$	6282.00	0.743063
$416$	0	0
$417$	18466.0	2.16855
$418$	0	0
$419$	3828.00	0.446325	0.223162	−	0.974781i	$-0.428362\pi$
$419$	3828.00	0.446325	0.223162	+	0.974781i	$0.428362\pi$
$420$	0	0
$421$	15466.0	1.79042	0.895210	−	0.445645i	$-0.147026\pi$
$421$	15466.0	1.79042	0.895210	+	0.445645i	$0.147026\pi$
$422$	0	0
$423$	−5808.00	−0.667600
$424$	0	0
$425$	1672.00	0.190833
$426$	0	0
$427$	−616.000	−0.0698134
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2710.00	−0.302868	−0.151434	−	0.988467i	$-0.548389\pi$
$431$	−2710.00	−0.302868	−0.151434	+	0.988467i	$0.548389\pi$
$432$	0	0
$433$	−3347.00	−0.371470	−0.185735	−	0.982600i	$-0.559467\pi$
$433$	−3347.00	−0.371470	−0.185735	+	0.982600i	$0.559467\pi$
$434$	0	0
$435$	16632.0	1.83320
$436$	0	0
$437$	−7700.00	−0.842885
$438$	0	0
$439$	2464.00	0.267882	0.133941	−	0.990989i	$-0.457237\pi$
$439$	2464.00	0.267882	0.133941	+	0.990989i	$0.457237\pi$
$440$	0	0
$441$	−7458.00	−0.805313
$442$	0	0
$443$	5955.00	0.638669	0.319335	−	0.947642i	$-0.396541\pi$
$443$	5955.00	0.638669	0.319335	+	0.947642i	$0.396541\pi$
$444$	0	0
$445$	−8703.00	−0.927105
$446$	0	0
$447$	23366.0	2.47242
$448$	0	0
$449$	−13173.0	−1.38457	−0.692285	−	0.721624i	$-0.743396\pi$
$449$	−13173.0	−1.38457	−0.692285	+	0.721624i	$0.743396\pi$
$450$	0	0
$451$	−2420.00	−0.252668
$452$	0	0
$453$	3010.00	0.312190
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6292.00	−0.644042	−0.322021	−	0.946732i	$-0.604362\pi$
$457$	−6292.00	−0.644042	−0.322021	+	0.946732i	$0.604362\pi$
$458$	0	0
$459$	−1330.00	−0.135249
$460$	0	0
$461$	−10164.0	−1.02686	−0.513432	−	0.858130i	$-0.671626\pi$
$461$	−10164.0	−1.02686	−0.513432	+	0.858130i	$0.671626\pi$
$462$	0	0
$463$	5677.00	0.569833	0.284916	−	0.958552i	$-0.408034\pi$
$463$	5677.00	0.569833	0.284916	+	0.958552i	$0.408034\pi$
$464$	0	0
$465$	10017.0	0.998983
$466$	0	0
$467$	6675.00	0.661418	0.330709	−	0.943733i	$-0.392712\pi$
$467$	6675.00	0.661418	0.330709	+	0.943733i	$0.392712\pi$
$468$	0	0
$469$	−354.000	−0.0348533
$470$	0	0
$471$	−8113.00	−0.793689
$472$	0	0
$473$	5962.00	0.579562
$474$	0	0
$475$	1936.00	0.187010
$476$	0	0
$477$	−15004.0	−1.44022
$478$	0	0
$479$	6424.00	0.612777	0.306388	−	0.951907i	$-0.400879\pi$
$479$	6424.00	0.612777	0.306388	+	0.951907i	$0.400879\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−2450.00	−0.230805
$484$	0	0
$485$	10143.0	0.949629
$486$	0	0
$487$	−21369.0	−1.98834	−0.994170	−	0.107822i	$-0.965612\pi$
$487$	−21369.0	−1.98834	−0.994170	+	0.107822i	$0.965612\pi$
$488$	0	0
$489$	7084.00	0.655112
$490$	0	0
$491$	−1848.00	−0.169856	−0.0849278	−	0.996387i	$-0.527066\pi$
$491$	−1848.00	−0.169856	−0.0849278	+	0.996387i	$0.527066\pi$
$492$	0	0
$493$	−10032.0	−0.916468
$494$	0	0
$495$	−2178.00	−0.197765
$496$	0	0
$497$	730.000	0.0658853
$498$	0	0
$499$	19228.0	1.72498	0.862488	−	0.506077i	$-0.168905\pi$
$499$	19228.0	1.72498	0.862488	+	0.506077i	$0.168905\pi$
$500$	0	0
$501$	−11088.0	−0.988773
$502$	0	0
$503$	10878.0	0.964266	0.482133	−	0.876098i	$-0.339862\pi$
$503$	10878.0	0.964266	0.482133	+	0.876098i	$0.339862\pi$
$504$	0	0
$505$	4590.00	0.404460
$506$	0	0
$507$	15379.0	1.34715
$508$	0	0
$509$	−10047.0	−0.874903	−0.437451	−	0.899242i	$-0.644119\pi$
$509$	−10047.0	−0.874903	−0.437451	+	0.899242i	$0.644119\pi$
$510$	0	0
$511$	−1056.00	−0.0914182
$512$	0	0
$513$	−1540.00	−0.132539
$514$	0	0
$515$	9504.00	0.813197
$516$	0	0
$517$	−2904.00	−0.247036
$518$	0	0
$519$	3318.00	0.280624
$520$	0	0
$521$	−5827.00	−0.489991	−0.244996	−	0.969524i	$-0.578787\pi$
$521$	−5827.00	−0.489991	−0.244996	+	0.969524i	$0.578787\pi$
$522$	0	0
$523$	−836.000	−0.0698962	−0.0349481	−	0.999389i	$-0.511127\pi$
$523$	−836.000	−0.0698962	−0.0349481	+	0.999389i	$0.511127\pi$
$524$	0	0
$525$	616.000	0.0512085
$526$	0	0
$527$	−6042.00	−0.499419
$528$	0	0
$529$	18458.0	1.51705
$530$	0	0
$531$	−9262.00	−0.756943
$532$	0	0
$533$	0	0
$534$	0	0
$535$	9414.00	0.760753
$536$	0	0
$537$	11655.0	0.936593
$538$	0	0
$539$	−3729.00	−0.297995
$540$	0	0
$541$	7172.00	0.569960	0.284980	−	0.958533i	$-0.408013\pi$
$541$	7172.00	0.569960	0.284980	+	0.958533i	$0.408013\pi$
$542$	0	0
$543$	17801.0	1.40684
$544$	0	0
$545$	2250.00	0.176843
$546$	0	0
$547$	11264.0	0.880464	0.440232	−	0.897884i	$-0.354896\pi$
$547$	11264.0	0.880464	0.440232	+	0.897884i	$0.354896\pi$
$548$	0	0
$549$	−6776.00	−0.526763
$550$	0	0
$551$	−11616.0	−0.898109
$552$	0	0
$553$	1372.00	0.105503
$554$	0	0
$555$	10899.0	0.833580
$556$	0	0
$557$	−306.000	−0.0232776	−0.0116388	−	0.999932i	$-0.503705\pi$
$557$	−306.000	−0.0232776	−0.0116388	+	0.999932i	$0.503705\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2926.00	0.220206
$562$	0	0
$563$	−23804.0	−1.78192	−0.890958	−	0.454085i	$-0.849966\pi$
$563$	−23804.0	−1.78192	−0.890958	+	0.454085i	$0.849966\pi$
$564$	0	0
$565$	−12609.0	−0.938875
$566$	0	0
$567$	−1678.00	−0.124285
$568$	0	0
$569$	−4664.00	−0.343629	−0.171815	−	0.985129i	$-0.554963\pi$
$569$	−4664.00	−0.343629	−0.171815	+	0.985129i	$0.554963\pi$
$570$	0	0
$571$	−11572.0	−0.848114	−0.424057	−	0.905635i	$-0.639394\pi$
$571$	−11572.0	−0.848114	−0.424057	+	0.905635i	$0.639394\pi$
$572$	0	0
$573$	11417.0	0.832377
$574$	0	0
$575$	−7700.00	−0.558456
$576$	0	0
$577$	−24407.0	−1.76096	−0.880482	−	0.474079i	$-0.842781\pi$
$577$	−24407.0	−1.76096	−0.880482	+	0.474079i	$0.842781\pi$
$578$	0	0
$579$	−3388.00	−0.243179
$580$	0	0
$581$	−1396.00	−0.0996830
$582$	0	0
$583$	−7502.00	−0.532935
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18436.0	−1.29631	−0.648156	−	0.761508i	$-0.724460\pi$
$587$	−18436.0	−1.29631	−0.648156	+	0.761508i	$0.724460\pi$
$588$	0	0
$589$	−6996.00	−0.489415
$590$	0	0
$591$	13762.0	0.957856
$592$	0	0
$593$	−11044.0	−0.764794	−0.382397	−	0.923998i	$-0.624901\pi$
$593$	−11044.0	−0.764794	−0.382397	+	0.923998i	$0.624901\pi$
$594$	0	0
$595$	684.000	0.0471282
$596$	0	0
$597$	−6776.00	−0.464528
$598$	0	0
$599$	8800.00	0.600264	0.300132	−	0.953898i	$-0.402969\pi$
$599$	8800.00	0.600264	0.300132	+	0.953898i	$0.402969\pi$
$600$	0	0
$601$	−10826.0	−0.734778	−0.367389	−	0.930067i	$-0.619748\pi$
$601$	−10826.0	−0.734778	−0.367389	+	0.930067i	$0.619748\pi$
$602$	0	0
$603$	−3894.00	−0.262978
$604$	0	0
$605$	−1089.00	−0.0731804
$606$	0	0
$607$	4066.00	0.271884	0.135942	−	0.990717i	$-0.456594\pi$
$607$	4066.00	0.271884	0.135942	+	0.990717i	$0.456594\pi$
$608$	0	0
$609$	−3696.00	−0.245927
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−4752.00	−0.313102	−0.156551	−	0.987670i	$-0.550038\pi$
$613$	−4752.00	−0.313102	−0.156551	+	0.987670i	$0.550038\pi$
$614$	0	0
$615$	−13860.0	−0.908763
$616$	0	0
$617$	−3806.00	−0.248337	−0.124168	−	0.992261i	$-0.539626\pi$
$617$	−3806.00	−0.248337	−0.124168	+	0.992261i	$0.539626\pi$
$618$	0	0
$619$	5185.00	0.336676	0.168338	−	0.985729i	$-0.446160\pi$
$619$	5185.00	0.336676	0.168338	+	0.985729i	$0.446160\pi$
$620$	0	0
$621$	6125.00	0.395794
$622$	0	0
$623$	1934.00	0.124373
$624$	0	0
$625$	−8189.00	−0.524096
$626$	0	0
$627$	3388.00	0.215795
$628$	0	0
$629$	−6574.00	−0.416729
$630$	0	0
$631$	−13049.0	−0.823253	−0.411626	−	0.911353i	$-0.635039\pi$
$631$	−13049.0	−0.823253	−0.411626	+	0.911353i	$0.635039\pi$
$632$	0	0
$633$	−20636.0	−1.29575
$634$	0	0
$635$	1188.00	0.0742431
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8030.00	0.497123
$640$	0	0
$641$	9959.00	0.613661	0.306831	−	0.951764i	$-0.400731\pi$
$641$	9959.00	0.613661	0.306831	+	0.951764i	$0.400731\pi$
$642$	0	0
$643$	−12197.0	−0.748060	−0.374030	−	0.927417i	$-0.622024\pi$
$643$	−12197.0	−0.748060	−0.374030	+	0.927417i	$0.622024\pi$
$644$	0	0
$645$	34146.0	2.08449
$646$	0	0
$647$	−5759.00	−0.349938	−0.174969	−	0.984574i	$-0.555982\pi$
$647$	−5759.00	−0.349938	−0.174969	+	0.984574i	$0.555982\pi$
$648$	0	0
$649$	−4631.00	−0.280097
$650$	0	0
$651$	−2226.00	−0.134015
$652$	0	0
$653$	25209.0	1.51073	0.755363	−	0.655306i	$-0.227460\pi$
$653$	25209.0	1.51073	0.755363	+	0.655306i	$0.227460\pi$
$654$	0	0
$655$	−18486.0	−1.10276
$656$	0	0
$657$	−11616.0	−0.689777
$658$	0	0
$659$	4062.00	0.240111	0.120055	−	0.992767i	$-0.461693\pi$
$659$	4062.00	0.240111	0.120055	+	0.992767i	$0.461693\pi$
$660$	0	0
$661$	−31173.0	−1.83433	−0.917163	−	0.398513i	$-0.869526\pi$
$661$	−31173.0	−1.83433	−0.917163	+	0.398513i	$0.869526\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	792.000	0.0461841
$666$	0	0
$667$	46200.0	2.68197
$668$	0	0
$669$	−13965.0	−0.807052
$670$	0	0
$671$	−3388.00	−0.194921
$672$	0	0
$673$	−26690.0	−1.52871	−0.764357	−	0.644794i	$-0.776943\pi$
$673$	−26690.0	−1.52871	−0.764357	+	0.644794i	$0.776943\pi$
$674$	0	0
$675$	−1540.00	−0.0878143
$676$	0	0
$677$	7398.00	0.419983	0.209991	−	0.977703i	$-0.432656\pi$
$677$	7398.00	0.419983	0.209991	+	0.977703i	$0.432656\pi$
$678$	0	0
$679$	−2254.00	−0.127394
$680$	0	0
$681$	−42182.0	−2.37359
$682$	0	0
$683$	−27960.0	−1.56641	−0.783206	−	0.621762i	$-0.786417\pi$
$683$	−27960.0	−1.56641	−0.783206	+	0.621762i	$0.786417\pi$
$684$	0	0
$685$	−8001.00	−0.446281
$686$	0	0
$687$	−30919.0	−1.71708
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−19465.0	−1.07161	−0.535806	−	0.844341i	$-0.679992\pi$
$691$	−19465.0	−1.07161	−0.535806	+	0.844341i	$0.679992\pi$
$692$	0	0
$693$	484.000	0.0265305
$694$	0	0
$695$	23742.0	1.29581
$696$	0	0
$697$	8360.00	0.454315
$698$	0	0
$699$	−40656.0	−2.19993
$700$	0	0
$701$	−13518.0	−0.728342	−0.364171	−	0.931332i	$-0.618648\pi$
$701$	−13518.0	−0.728342	−0.364171	+	0.931332i	$0.618648\pi$
$702$	0	0
$703$	−7612.00	−0.408381
$704$	0	0
$705$	−16632.0	−0.888507
$706$	0	0
$707$	−1020.00	−0.0542589
$708$	0	0
$709$	−14727.0	−0.780090	−0.390045	−	0.920796i	$-0.627541\pi$
$709$	−14727.0	−0.780090	−0.390045	+	0.920796i	$0.627541\pi$
$710$	0	0
$711$	15092.0	0.796054
$712$	0	0
$713$	27825.0	1.46151
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4382.00	−0.228241
$718$	0	0
$719$	15711.0	0.814912	0.407456	−	0.913225i	$-0.366416\pi$
$719$	15711.0	0.814912	0.407456	+	0.913225i	$0.366416\pi$
$720$	0	0
$721$	−2112.00	−0.109092
$722$	0	0
$723$	24640.0	1.26746
$724$	0	0
$725$	−11616.0	−0.595045
$726$	0	0
$727$	12099.0	0.617231	0.308616	−	0.951187i	$-0.400134\pi$
$727$	12099.0	0.617231	0.308616	+	0.951187i	$0.400134\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	0	0
$731$	−20596.0	−1.04209
$732$	0	0
$733$	−9812.00	−0.494426	−0.247213	−	0.968961i	$-0.579515\pi$
$733$	−9812.00	−0.494426	−0.247213	+	0.968961i	$0.579515\pi$
$734$	0	0
$735$	−21357.0	−1.07179
$736$	0	0
$737$	−1947.00	−0.0973116
$738$	0	0
$739$	11854.0	0.590063	0.295031	−	0.955488i	$-0.404670\pi$
$739$	11854.0	0.590063	0.295031	+	0.955488i	$0.404670\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18788.0	0.927678	0.463839	−	0.885919i	$-0.346472\pi$
$743$	18788.0	0.927678	0.463839	+	0.885919i	$0.346472\pi$
$744$	0	0
$745$	30042.0	1.47739
$746$	0	0
$747$	−15356.0	−0.752137
$748$	0	0
$749$	−2092.00	−0.102056
$750$	0	0
$751$	−16559.0	−0.804589	−0.402295	−	0.915510i	$-0.631787\pi$
$751$	−16559.0	−0.804589	−0.402295	+	0.915510i	$0.631787\pi$
$752$	0	0
$753$	27447.0	1.32832
$754$	0	0
$755$	3870.00	0.186548
$756$	0	0
$757$	−26610.0	−1.27762	−0.638809	−	0.769365i	$-0.720573\pi$
$757$	−26610.0	−1.27762	−0.638809	+	0.769365i	$0.720573\pi$
$758$	0	0
$759$	−13475.0	−0.644416
$760$	0	0
$761$	7040.00	0.335348	0.167674	−	0.985843i	$-0.446374\pi$
$761$	7040.00	0.335348	0.167674	+	0.985843i	$0.446374\pi$
$762$	0	0
$763$	−500.000	−0.0237237
$764$	0	0
$765$	7524.00	0.355596
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−18128.0	−0.850081	−0.425041	−	0.905174i	$-0.639740\pi$
$769$	−18128.0	−0.850081	−0.425041	+	0.905174i	$0.639740\pi$
$770$	0	0
$771$	−32186.0	−1.50344
$772$	0	0
$773$	−9570.00	−0.445290	−0.222645	−	0.974900i	$-0.571469\pi$
$773$	−9570.00	−0.445290	−0.222645	+	0.974900i	$0.571469\pi$
$774$	0	0
$775$	−6996.00	−0.324263
$776$	0	0
$777$	−2422.00	−0.111826
$778$	0	0
$779$	9680.00	0.445214
$780$	0	0
$781$	4015.00	0.183954
$782$	0	0
$783$	9240.00	0.421725
$784$	0	0
$785$	−10431.0	−0.474265
$786$	0	0
$787$	16280.0	0.737382	0.368691	−	0.929552i	$-0.379806\pi$
$787$	16280.0	0.737382	0.368691	+	0.929552i	$0.379806\pi$
$788$	0	0
$789$	40866.0	1.84394
$790$	0	0
$791$	2802.00	0.125952
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−42966.0	−1.91679
$796$	0	0
$797$	603.000	0.0267997	0.0133998	−	0.999910i	$-0.495735\pi$
$797$	603.000	0.0267997	0.0133998	+	0.999910i	$0.495735\pi$
$798$	0	0
$799$	10032.0	0.444189
$800$	0	0
$801$	21274.0	0.938427
$802$	0	0
$803$	−5808.00	−0.255243
$804$	0	0
$805$	−3150.00	−0.137917
$806$	0	0
$807$	−35266.0	−1.53832
$808$	0	0
$809$	23276.0	1.01155	0.505773	−	0.862667i	$-0.331207\pi$
$809$	23276.0	1.01155	0.505773	+	0.862667i	$0.331207\pi$
$810$	0	0
$811$	45662.0	1.97708	0.988539	−	0.150968i	$-0.0482390\pi$
$811$	45662.0	1.97708	0.988539	+	0.150968i	$0.0482390\pi$
$812$	0	0
$813$	56672.0	2.44474
$814$	0	0
$815$	9108.00	0.391459
$816$	0	0
$817$	−23848.0	−1.02122
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−33222.0	−1.41225	−0.706124	−	0.708088i	$-0.749558\pi$
$821$	−33222.0	−1.41225	−0.706124	+	0.708088i	$0.749558\pi$
$822$	0	0
$823$	−3569.00	−0.151163	−0.0755817	−	0.997140i	$-0.524081\pi$
$823$	−3569.00	−0.151163	−0.0755817	+	0.997140i	$0.524081\pi$
$824$	0	0
$825$	3388.00	0.142976
$826$	0	0
$827$	41008.0	1.72429	0.862145	−	0.506662i	$-0.169121\pi$
$827$	41008.0	1.72429	0.862145	+	0.506662i	$0.169121\pi$
$828$	0	0
$829$	17839.0	0.747375	0.373688	−	0.927555i	$-0.378093\pi$
$829$	17839.0	0.747375	0.373688	+	0.927555i	$0.378093\pi$
$830$	0	0
$831$	49098.0	2.04957
$832$	0	0
$833$	12882.0	0.535816
$834$	0	0
$835$	−14256.0	−0.590837
$836$	0	0
$837$	5565.00	0.229814
$838$	0	0
$839$	20731.0	0.853056	0.426528	−	0.904474i	$-0.359737\pi$
$839$	20731.0	0.853056	0.426528	+	0.904474i	$0.359737\pi$
$840$	0	0
$841$	45307.0	1.85768
$842$	0	0
$843$	30534.0	1.24751
$844$	0	0
$845$	19773.0	0.804984
$846$	0	0
$847$	242.000	0.00981726
$848$	0	0
$849$	32340.0	1.30731
$850$	0	0
$851$	30275.0	1.21952
$852$	0	0
$853$	−9102.00	−0.365354	−0.182677	−	0.983173i	$-0.558476\pi$
$853$	−9102.00	−0.365354	−0.182677	+	0.983173i	$0.558476\pi$
$854$	0	0
$855$	8712.00	0.348473
$856$	0	0
$857$	12936.0	0.515619	0.257809	−	0.966196i	$-0.416999\pi$
$857$	12936.0	0.515619	0.257809	+	0.966196i	$0.416999\pi$
$858$	0	0
$859$	−15113.0	−0.600290	−0.300145	−	0.953894i	$-0.597035\pi$
$859$	−15113.0	−0.600290	−0.300145	+	0.953894i	$0.597035\pi$
$860$	0	0
$861$	3080.00	0.121912
$862$	0	0
$863$	29216.0	1.15240	0.576202	−	0.817308i	$-0.304534\pi$
$863$	29216.0	1.15240	0.576202	+	0.817308i	$0.304534\pi$
$864$	0	0
$865$	4266.00	0.167686
$866$	0	0
$867$	24283.0	0.951204
$868$	0	0
$869$	7546.00	0.294569
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−24794.0	−0.961225
$874$	0	0
$875$	3042.00	0.117530
$876$	0	0
$877$	−29304.0	−1.12831	−0.564154	−	0.825670i	$-0.690798\pi$
$877$	−29304.0	−1.12831	−0.564154	+	0.825670i	$0.690798\pi$
$878$	0	0
$879$	−50204.0	−1.92644
$880$	0	0
$881$	−23715.0	−0.906900	−0.453450	−	0.891282i	$-0.649807\pi$
$881$	−23715.0	−0.906900	−0.453450	+	0.891282i	$0.649807\pi$
$882$	0	0
$883$	−6028.00	−0.229738	−0.114869	−	0.993381i	$-0.536645\pi$
$883$	−6028.00	−0.229738	−0.114869	+	0.993381i	$0.536645\pi$
$884$	0	0
$885$	−26523.0	−1.00741
$886$	0	0
$887$	18786.0	0.711130	0.355565	−	0.934652i	$-0.384288\pi$
$887$	18786.0	0.711130	0.355565	+	0.934652i	$0.384288\pi$
$888$	0	0
$889$	−264.000	−0.00995982
$890$	0	0
$891$	−9229.00	−0.347007
$892$	0	0
$893$	11616.0	0.435291
$894$	0	0
$895$	14985.0	0.559657
$896$	0	0
$897$	0	0
$898$	0	0
$899$	41976.0	1.55726
$900$	0	0
$901$	25916.0	0.958254
$902$	0	0
$903$	−7588.00	−0.279638
$904$	0	0
$905$	22887.0	0.840652
$906$	0	0
$907$	2156.00	0.0789292	0.0394646	−	0.999221i	$-0.487435\pi$
$907$	2156.00	0.0789292	0.0394646	+	0.999221i	$0.487435\pi$
$908$	0	0
$909$	−11220.0	−0.409399
$910$	0	0
$911$	−51732.0	−1.88140	−0.940701	−	0.339236i	$-0.889831\pi$
$911$	−51732.0	−1.88140	−0.940701	+	0.339236i	$0.889831\pi$
$912$	0	0
$913$	−7678.00	−0.278318
$914$	0	0
$915$	−19404.0	−0.701067
$916$	0	0
$917$	4108.00	0.147937
$918$	0	0
$919$	−18254.0	−0.655216	−0.327608	−	0.944814i	$-0.606243\pi$
$919$	−18254.0	−0.655216	−0.327608	+	0.944814i	$0.606243\pi$
$920$	0	0
$921$	51436.0	1.84025
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−7612.00	−0.270574
$926$	0	0
$927$	−23232.0	−0.823127
$928$	0	0
$929$	5610.00	0.198125	0.0990625	−	0.995081i	$-0.468416\pi$
$929$	5610.00	0.198125	0.0990625	+	0.995081i	$0.468416\pi$
$930$	0	0
$931$	14916.0	0.525083
$932$	0	0
$933$	38556.0	1.35291
$934$	0	0
$935$	3762.00	0.131583
$936$	0	0
$937$	−37180.0	−1.29628	−0.648142	−	0.761520i	$-0.724454\pi$
$937$	−37180.0	−1.29628	−0.648142	+	0.761520i	$0.724454\pi$
$938$	0	0
$939$	49063.0	1.70512
$940$	0	0
$941$	31278.0	1.08356	0.541782	−	0.840519i	$-0.317750\pi$
$941$	31278.0	1.08356	0.541782	+	0.840519i	$0.317750\pi$
$942$	0	0
$943$	−38500.0	−1.32951
$944$	0	0
$945$	−630.000	−0.0216867
$946$	0	0
$947$	47475.0	1.62907	0.814535	−	0.580114i	$-0.196992\pi$
$947$	47475.0	1.62907	0.814535	+	0.580114i	$0.196992\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	5971.00	0.203599
$952$	0	0
$953$	20462.0	0.695519	0.347759	−	0.937584i	$-0.386943\pi$
$953$	20462.0	0.695519	0.347759	+	0.937584i	$0.386943\pi$
$954$	0	0
$955$	14679.0	0.497384
$956$	0	0
$957$	−20328.0	−0.686636
$958$	0	0
$959$	1778.00	0.0598693
$960$	0	0
$961$	−4510.00	−0.151388
$962$	0	0
$963$	−23012.0	−0.770043
$964$	0	0
$965$	−4356.00	−0.145310
$966$	0	0
$967$	52360.0	1.74125	0.870623	−	0.491952i	$-0.163716\pi$
$967$	52360.0	1.74125	0.870623	+	0.491952i	$0.163716\pi$
$968$	0	0
$969$	−11704.0	−0.388015
$970$	0	0
$971$	−56247.0	−1.85896	−0.929481	−	0.368870i	$-0.879745\pi$
$971$	−56247.0	−1.85896	−0.929481	+	0.368870i	$0.879745\pi$
$972$	0	0
$973$	−5276.00	−0.173834
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18155.0	−0.594503	−0.297252	−	0.954799i	$-0.596070\pi$
$977$	−18155.0	−0.594503	−0.297252	+	0.954799i	$0.596070\pi$
$978$	0	0
$979$	10637.0	0.347252
$980$	0	0
$981$	−5500.00	−0.179003
$982$	0	0
$983$	1191.00	0.0386439	0.0193220	−	0.999813i	$-0.493849\pi$
$983$	1191.00	0.0386439	0.0193220	+	0.999813i	$0.493849\pi$
$984$	0	0
$985$	17694.0	0.572363
$986$	0	0
$987$	3696.00	0.119195
$988$	0	0
$989$	94850.0	3.04960
$990$	0	0
$991$	28864.0	0.925222	0.462611	−	0.886561i	$-0.346913\pi$
$991$	28864.0	0.925222	0.462611	+	0.886561i	$0.346913\pi$
$992$	0	0
$993$	25417.0	0.812270
$994$	0	0
$995$	−8712.00	−0.277577
$996$	0	0
$997$	−51610.0	−1.63942	−0.819712	−	0.572776i	$-0.805866\pi$
$997$	−51610.0	−1.63942	−0.819712	+	0.572776i	$0.805866\pi$
$998$	0	0
$999$	6055.00	0.191763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.4.a.a.1.1		1
4.3	odd	2		704.4.a.k.1.1		1
8.3	odd	2		176.4.a.a.1.1		1
8.5	even	2		88.4.a.b.1.1	✓	1
24.5	odd	2		792.4.a.b.1.1		1
24.11	even	2		1584.4.a.g.1.1		1
40.29	even	2		2200.4.a.a.1.1		1
88.21	odd	2		968.4.a.e.1.1		1
88.43	even	2		1936.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.a.b.1.1	✓	1	8.5	even	2
176.4.a.a.1.1		1	8.3	odd	2
704.4.a.a.1.1		1	1.1	even	1	trivial
704.4.a.k.1.1		1	4.3	odd	2
792.4.a.b.1.1		1	24.5	odd	2
968.4.a.e.1.1		1	88.21	odd	2
1584.4.a.g.1.1		1	24.11	even	2
1936.4.a.b.1.1		1	88.43	even	2
2200.4.a.a.1.1		1	40.29	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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