LMFDB - Embedded newform 704.5.h.i.65.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,5,Mod(65,704)]

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, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("704.65");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	704.h (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$72.7724540110$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{553})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 271x^{2} + 272x + 19602 $ x^4 - 2x^3 - 271x^2 + 272*x + 19602
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 22)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.1
Root			$12.2580 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	704.65
Dual form			704.5.h.i.65.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.2580 q^{3} -4.25798 q^{5} -33.9411i q^{7} +69.2580 q^{9} +O(q^{10})$	$q-12.2580 q^{3} -4.25798 q^{5} -33.9411i q^{7} +69.2580 q^{9} +(92.0319 - 78.5565i) q^{11} -326.819i q^{13} +52.1942 q^{15} -280.285i q^{17} +12.5926i q^{19} +416.050i q^{21} +184.641 q^{23} -606.870 q^{25} +143.933 q^{27} +1039.58i q^{29} +1492.51 q^{31} +(-1128.12 + 962.944i) q^{33} +144.520i q^{35} +1400.64 q^{37} +4006.14i q^{39} -2563.10i q^{41} -870.960i q^{43} -294.899 q^{45} +3021.08 q^{47} +1249.00 q^{49} +3435.73i q^{51} +260.724 q^{53} +(-391.870 + 334.492i) q^{55} -154.359i q^{57} -293.024 q^{59} -3593.92i q^{61} -2350.69i q^{63} +1391.59i q^{65} +2359.87 q^{67} -2263.32 q^{69} -2494.04 q^{71} -4481.31i q^{73} +7438.99 q^{75} +(-2666.30 - 3123.67i) q^{77} +9693.98i q^{79} -7374.23 q^{81} +6237.99i q^{83} +1193.45i q^{85} -12743.2i q^{87} -9516.72 q^{89} -11092.6 q^{91} -18295.2 q^{93} -53.6188i q^{95} +6401.66 q^{97} +(6373.94 - 5440.67i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 30 q^{5} + 230 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 30 * q^5 + 230 * q^9	$ 4 q - 2 q^{3} + 30 q^{5} + 230 q^{9} + 180 q^{11} + 538 q^{15} - 1566 q^{23} - 1722 q^{25} - 506 q^{27} + 4418 q^{31} - 2302 q^{33} + 382 q^{37} + 1172 q^{45} + 5688 q^{47} + 4996 q^{49} + 8568 q^{53} - 862 q^{55} + 3390 q^{59} + 8734 q^{67} - 26314 q^{69} + 3522 q^{71} + 9156 q^{75} + 2880 q^{77} - 31096 q^{81} - 8766 q^{89} - 17280 q^{91} - 20458 q^{93} + 17282 q^{97} + 12562 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 30 * q^5 + 230 * q^9 + 180 * q^11 + 538 * q^15 - 1566 * q^23 - 1722 * q^25 - 506 * q^27 + 4418 * q^31 - 2302 * q^33 + 382 * q^37 + 1172 * q^45 + 5688 * q^47 + 4996 * q^49 + 8568 * q^53 - 862 * q^55 + 3390 * q^59 + 8734 * q^67 - 26314 * q^69 + 3522 * q^71 + 9156 * q^75 + 2880 * q^77 - 31096 * q^81 - 8766 * q^89 - 17280 * q^91 - 20458 * q^93 + 17282 * q^97 + 12562 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/704\mathbb{Z}\right)^\times$.

$n$	$133$	$321$	$639$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	−12.2580			−1.36200			−0.680999	−	0.732285i	$-0.738454\pi$
$3$	−12.2580			−1.36200			−0.680999	+	0.732285i	$0.738454\pi$
$4$		0			0
$5$	−4.25798			−0.170319			−0.0851595	−	0.996367i	$-0.527140\pi$
$5$	−4.25798			−0.170319			−0.0851595	+	0.996367i	$0.527140\pi$
$6$		0			0
$7$		−	33.9411i		−	0.692676i	−0.938110	−	0.346338i	$-0.887425\pi$
$7$		−	33.9411i		−	0.692676i	0.938110	−	0.346338i	$-0.112575\pi$
$8$		0			0
$9$	69.2580			0.855037
$10$		0			0
$11$	92.0319	−	78.5565i	0.760594	−	0.649228i
$11$	92.0319	−	78.5565i	0.760594	−	0.649228i
$12$		0			0
$13$		−	326.819i		−	1.93384i	−0.255083	−	0.966919i	$-0.582103\pi$
$13$		−	326.819i		−	1.93384i	0.255083	−	0.966919i	$-0.417897\pi$
$14$		0			0
$15$	52.1942			0.231974
$16$		0			0
$17$		−	280.285i		−	0.969844i	−0.874557	−	0.484922i	$-0.838848\pi$
$17$		−	280.285i		−	0.969844i	0.874557	−	0.484922i	$-0.161152\pi$
$18$		0			0
$19$			12.5926i			0.0348825i	0.999848	+	0.0174412i	$0.00555200\pi$
$19$			12.5926i			0.0348825i	−0.999848	+	0.0174412i	$0.994448\pi$
$20$		0			0
$21$			416.050i			0.943423i
$22$		0			0
$23$	184.641			0.349037			0.174519	−	0.984654i	$-0.444163\pi$
$23$	184.641			0.349037			0.174519	+	0.984654i	$0.444163\pi$
$24$		0			0
$25$	−606.870			−0.970991
$26$		0			0
$27$	143.933			0.197440
$28$		0			0
$29$			1039.58i			1.23613i	0.786128	+	0.618063i	$0.212082\pi$
$29$			1039.58i			1.23613i	−0.786128	+	0.618063i	$0.787918\pi$
$30$		0			0
$31$	1492.51			1.55308			0.776542	−	0.630066i	$-0.216972\pi$
$31$	1492.51			1.55308			0.776542	+	0.630066i	$0.216972\pi$
$32$		0			0
$33$	−1128.12	+	962.944i	−1.03593	+	0.884246i
$34$		0			0
$35$			144.520i			0.117976i
$36$		0			0
$37$	1400.64			1.02311			0.511554	−	0.859251i	$-0.329070\pi$
$37$	1400.64			1.02311			0.511554	+	0.859251i	$0.329070\pi$
$38$		0			0
$39$			4006.14i			2.63388i
$40$		0			0
$41$		−	2563.10i		−	1.52475i	−0.647138	−	0.762373i	$-0.724034\pi$
$41$		−	2563.10i		−	1.52475i	0.647138	−	0.762373i	$-0.275966\pi$
$42$		0			0
$43$		−	870.960i		−	0.471044i	−0.971869	−	0.235522i	$-0.924320\pi$
$43$		−	870.960i		−	0.471044i	0.971869	−	0.235522i	$-0.0756799\pi$
$44$		0			0
$45$	−294.899			−0.145629
$46$		0			0
$47$	3021.08			1.36763			0.683813	−	0.729658i	$-0.260321\pi$
$47$	3021.08			1.36763			0.683813	+	0.729658i	$0.260321\pi$
$48$		0			0
$49$	1249.00			0.520200
$50$		0			0
$51$			3435.73i			1.32093i
$52$		0			0
$53$	260.724			0.0928173			0.0464087	−	0.998923i	$-0.485222\pi$
$53$	260.724			0.0928173			0.0464087	+	0.998923i	$0.485222\pi$
$54$		0			0
$55$	−391.870	+	334.492i	−0.129544	+	0.110576i
$56$		0			0
$57$		−	154.359i		−	0.0475098i
$58$		0			0
$59$	−293.024			−0.0841780			−0.0420890	−	0.999114i	$-0.513401\pi$
$59$	−293.024			−0.0841780			−0.0420890	+	0.999114i	$0.513401\pi$
$60$		0			0
$61$		−	3593.92i		−	0.965849i	−0.875662	−	0.482924i	$-0.839575\pi$
$61$		−	3593.92i		−	0.965849i	0.875662	−	0.482924i	$-0.160425\pi$
$62$		0			0
$63$		−	2350.69i		−	0.592263i
$64$		0			0
$65$			1391.59i			0.329370i
$66$		0			0
$67$	2359.87			0.525701			0.262850	−	0.964837i	$-0.415337\pi$
$67$	2359.87			0.525701			0.262850	+	0.964837i	$0.415337\pi$
$68$		0			0
$69$	−2263.32			−0.475388
$70$		0			0
$71$	−2494.04			−0.494751			−0.247375	−	0.968920i	$-0.579568\pi$
$71$	−2494.04			−0.494751			−0.247375	+	0.968920i	$0.579568\pi$
$72$		0			0
$73$		−	4481.31i		−	0.840929i	−0.907309	−	0.420465i	$-0.861867\pi$
$73$		−	4481.31i		−	0.840929i	0.907309	−	0.420465i	$-0.138133\pi$
$74$		0			0
$75$	7438.99			1.32249
$76$		0			0
$77$	−2666.30	−	3123.67i	−0.449704	−	0.526845i
$78$		0			0
$79$			9693.98i			1.55327i	0.629949	+	0.776637i	$0.283076\pi$
$79$			9693.98i			1.55327i	−0.629949	+	0.776637i	$0.716924\pi$
$80$		0			0
$81$	−7374.23			−1.12395
$82$		0			0
$83$			6237.99i			0.905500i	0.891638	+	0.452750i	$0.149557\pi$
$83$			6237.99i			0.905500i	−0.891638	+	0.452750i	$0.850443\pi$
$84$		0			0
$85$			1193.45i			0.165183i
$86$		0			0
$87$		−	12743.2i		−	1.68360i
$88$		0			0
$89$	−9516.72			−1.20145			−0.600727	−	0.799454i	$-0.705122\pi$
$89$	−9516.72			−1.20145			−0.600727	+	0.799454i	$0.705122\pi$
$90$		0			0
$91$	−11092.6			−1.33952
$92$		0			0
$93$	−18295.2			−2.11530
$94$		0			0
$95$		−	53.6188i		−	0.00594115i
$96$		0			0
$97$	6401.66			0.680376			0.340188	−	0.940357i	$-0.389509\pi$
$97$	6401.66			0.680376			0.340188	+	0.940357i	$0.389509\pi$
$98$		0			0
$99$	6373.94	−	5440.67i	0.650336	−	0.555113i
$100$		0			0
$101$			15.9343i			0.00156203i	1.00000		0.000781015i	$0.000248605\pi$
$101$			15.9343i			0.00156203i	−1.00000		0.000781015i	$0.999751\pi$
$102$		0			0
$103$	12891.5			1.21514			0.607572	−	0.794265i	$-0.292144\pi$
$103$	12891.5			1.21514			0.607572	+	0.794265i	$0.292144\pi$
$104$		0			0
$105$		−	1771.53i		−	0.160683i
$106$		0			0
$107$		−	10805.9i		−	0.943827i	−0.881645	−	0.471913i	$-0.843564\pi$
$107$		−	10805.9i		−	0.943827i	0.881645	−	0.471913i	$-0.156436\pi$
$108$		0			0
$109$		−	8470.43i		−	0.712939i	−0.934307	−	0.356470i	$-0.883980\pi$
$109$		−	8470.43i		−	0.712939i	0.934307	−	0.356470i	$-0.116020\pi$
$110$		0			0
$111$	−17169.0			−1.39347
$112$		0			0
$113$	−1009.87			−0.0790874			−0.0395437	−	0.999218i	$-0.512590\pi$
$113$	−1009.87			−0.0790874			−0.0395437	+	0.999218i	$0.512590\pi$
$114$		0			0
$115$	−786.196			−0.0594477
$116$		0			0
$117$		−	22634.8i		−	1.65350i
$118$		0			0
$119$	−9513.19			−0.671788
$120$		0			0
$121$	2298.74	−	14459.4i	0.157007	−	0.987597i
$122$		0			0
$123$			31418.4i			2.07670i
$124$		0			0
$125$	5245.27			0.335697
$126$		0			0
$127$		−	7859.09i		−	0.487264i	−0.969868	−	0.243632i	$-0.921661\pi$
$127$		−	7859.09i		−	0.487264i	0.969868	−	0.243632i	$-0.0783390\pi$
$128$		0			0
$129$			10676.2i			0.641560i
$130$		0			0
$131$		−	24418.3i		−	1.42290i	−0.702738	−	0.711448i	$-0.748040\pi$
$131$		−	24418.3i		−	1.42290i	0.702738	−	0.711448i	$-0.251960\pi$
$132$		0			0
$133$	427.406			0.0241622
$134$		0			0
$135$	−612.865			−0.0336277
$136$		0			0
$137$	−2895.91			−0.154292			−0.0771462	−	0.997020i	$-0.524581\pi$
$137$	−2895.91			−0.154292			−0.0771462	+	0.997020i	$0.524581\pi$
$138$		0			0
$139$		−	14602.9i		−	0.755807i	−0.925845	−	0.377903i	$-0.876645\pi$
$139$		−	14602.9i		−	0.755807i	0.925845	−	0.377903i	$-0.123355\pi$
$140$		0			0
$141$	−37032.4			−1.86270
$142$		0			0
$143$	−25673.7	−	30077.7i	−1.25550	−	1.47087i
$144$		0			0
$145$		−	4426.52i		−	0.210536i
$146$		0			0
$147$	−15310.2			−0.708511
$148$		0			0
$149$			9469.08i			0.426516i	0.976996	+	0.213258i	$0.0684075\pi$
$149$			9469.08i			0.426516i	−0.976996	+	0.213258i	$0.931593\pi$
$150$		0			0
$151$		−	15335.9i		−	0.672597i	−0.941755	−	0.336298i	$-0.890825\pi$
$151$		−	15335.9i		−	0.672597i	0.941755	−	0.336298i	$-0.109175\pi$
$152$		0			0
$153$		−	19412.0i		−	0.829253i
$154$		0			0
$155$	−6355.09			−0.264520
$156$		0			0
$157$	4810.17			0.195147			0.0975733	−	0.995228i	$-0.468892\pi$
$157$	4810.17			0.195147			0.0975733	+	0.995228i	$0.468892\pi$
$158$		0			0
$159$	−3195.95			−0.126417
$160$		0			0
$161$		−	6266.92i		−	0.241770i
$162$		0			0
$163$	9875.98			0.371711			0.185855	−	0.982577i	$-0.440494\pi$
$163$	9875.98			0.371711			0.185855	+	0.982577i	$0.440494\pi$
$164$		0			0
$165$	4803.53	−	4100.19i	0.176438	−	0.150604i
$166$		0			0
$167$			13437.3i			0.481812i	0.970548	+	0.240906i	$0.0774445\pi$
$167$			13437.3i			0.481812i	−0.970548	+	0.240906i	$0.922555\pi$
$168$		0			0
$169$	−78249.5			−2.73973
$170$		0			0
$171$			872.136i			0.0298258i
$172$		0			0
$173$		−	37664.1i		−	1.25845i	−0.777223	−	0.629225i	$-0.783373\pi$
$173$		−	37664.1i		−	1.25845i	0.777223	−	0.629225i	$-0.216627\pi$
$174$		0			0
$175$			20597.8i			0.672582i
$176$		0			0
$177$	3591.88			0.114650
$178$		0			0
$179$	−53941.3			−1.68351			−0.841755	−	0.539860i	$-0.818477\pi$
$179$	−53941.3			−1.68351			−0.841755	+	0.539860i	$0.818477\pi$
$180$		0			0
$181$	−23379.6			−0.713642			−0.356821	−	0.934173i	$-0.616139\pi$
$181$	−23379.6			−0.713642			−0.356821	+	0.934173i	$0.616139\pi$
$182$		0			0
$183$			44054.2i			1.31548i
$184$		0			0
$185$	−5963.87			−0.174255
$186$		0			0
$187$	−22018.2	−	25795.2i	−0.629650	−	0.737658i
$188$		0			0
$189$		−	4885.26i		−	0.136762i
$190$		0			0
$191$	46611.4			1.27769			0.638844	−	0.769336i	$-0.279413\pi$
$191$	46611.4			1.27769			0.638844	+	0.769336i	$0.279413\pi$
$192$		0			0
$193$		−	57585.6i		−	1.54596i	−0.634428	−	0.772982i	$-0.718764\pi$
$193$		−	57585.6i		−	1.54596i	0.634428	−	0.772982i	$-0.281236\pi$
$194$		0			0
$195$		−	17058.0i		−	0.448600i
$196$		0			0
$197$			39775.2i			1.02490i	0.858718	+	0.512448i	$0.171261\pi$
$197$			39775.2i			1.02490i	−0.858718	+	0.512448i	$0.828739\pi$
$198$		0			0
$199$	18905.2			0.477393			0.238697	−	0.971094i	$-0.423280\pi$
$199$	18905.2			0.477393			0.238697	+	0.971094i	$0.423280\pi$
$200$		0			0
$201$	−28927.2			−0.716003
$202$		0			0
$203$	35284.6			0.856235
$204$		0			0
$205$			10913.6i			0.259693i
$206$		0			0
$207$	12787.8			0.298440
$208$		0			0
$209$	989.228	+	1158.92i	0.0226466	+	0.0265314i
$210$		0			0
$211$		−	30974.3i		−	0.695723i	−0.937546	−	0.347862i	$-0.886908\pi$
$211$		−	30974.3i		−	0.695723i	0.937546	−	0.347862i	$-0.113092\pi$
$212$		0			0
$213$	30571.9			0.673849
$214$		0			0
$215$			3708.52i			0.0802277i
$216$		0			0
$217$		−	50657.6i		−	1.07578i
$218$		0			0
$219$			54931.8i			1.14534i
$220$		0			0
$221$	−91602.4			−1.87552
$222$		0			0
$223$	−19890.6			−0.399980			−0.199990	−	0.979798i	$-0.564091\pi$
$223$	−19890.6			−0.399980			−0.199990	+	0.979798i	$0.564091\pi$
$224$		0			0
$225$	−42030.6			−0.830233
$226$		0			0
$227$			76528.9i			1.48516i	0.669757	+	0.742581i	$0.266398\pi$
$227$			76528.9i			1.48516i	−0.669757	+	0.742581i	$0.733602\pi$
$228$		0			0
$229$	56079.9			1.06939			0.534695	−	0.845045i	$-0.320426\pi$
$229$	56079.9			1.06939			0.534695	+	0.845045i	$0.320426\pi$
$230$		0			0
$231$	32683.4	+	38289.8i	0.612496	+	0.717562i
$232$		0			0
$233$		−	3353.39i		−	0.0617693i	−0.999523	−	0.0308847i	$-0.990168\pi$
$233$		−	3353.39i		−	0.0617693i	0.999523	−	0.0308847i	$-0.00983246\pi$
$234$		0			0
$235$	−12863.7			−0.232933
$236$		0			0
$237$		−	118829.i		−	2.11555i
$238$		0			0
$239$		−	8629.71i		−	0.151078i	−0.997143	−	0.0755388i	$-0.975932\pi$
$239$		−	8629.71i		−	0.151078i	0.997143	−	0.0755388i	$-0.0240677\pi$
$240$		0			0
$241$			101396.i			1.74577i	0.487924	+	0.872886i	$0.337754\pi$
$241$			101396.i			1.74577i	−0.487924	+	0.872886i	$0.662246\pi$
$242$		0			0
$243$	78734.5			1.33338
$244$		0			0
$245$	−5318.21			−0.0886000
$246$		0			0
$247$	4115.49			0.0674570
$248$		0			0
$249$		−	76465.1i		−	1.23329i
$250$		0			0
$251$	81244.8			1.28958			0.644790	−	0.764360i	$-0.276945\pi$
$251$	81244.8			1.28958			0.644790	+	0.764360i	$0.276945\pi$
$252$		0			0
$253$	16992.8	−	14504.7i	0.265476	−	0.226605i
$254$		0			0
$255$		−	14629.2i		−	0.224979i
$256$		0			0
$257$	56962.9			0.862434			0.431217	−	0.902248i	$-0.358084\pi$
$257$	56962.9			0.862434			0.431217	+	0.902248i	$0.358084\pi$
$258$		0			0
$259$		−	47539.1i		−	0.708683i
$260$		0			0
$261$			71999.4i			1.05693i
$262$		0			0
$263$			6000.71i			0.0867543i	0.999059	+	0.0433772i	$0.0138117\pi$
$263$			6000.71i			0.0867543i	−0.999059	+	0.0433772i	$0.986188\pi$
$264$		0			0
$265$	−1110.16			−0.0158086
$266$		0			0
$267$	116656.			1.63638
$268$		0			0
$269$	−24832.0			−0.343169			−0.171584	−	0.985169i	$-0.554889\pi$
$269$	−24832.0			−0.343169			−0.171584	+	0.985169i	$0.554889\pi$
$270$		0			0
$271$		−	65976.9i		−	0.898367i	−0.893440	−	0.449183i	$-0.851715\pi$
$271$		−	65976.9i		−	0.898367i	0.893440	−	0.449183i	$-0.148285\pi$
$272$		0			0
$273$	135973.			1.82443
$274$		0			0
$275$	−55851.4	+	47673.6i	−0.738530	+	0.630394i
$276$		0			0
$277$			31675.1i			0.412818i	0.978466	+	0.206409i	$0.0661778\pi$
$277$			31675.1i			0.412818i	−0.978466	+	0.206409i	$0.933822\pi$
$278$		0			0
$279$	103368.			1.32794
$280$		0			0
$281$			130542.i			1.65325i	0.562753	+	0.826625i	$0.309742\pi$
$281$			130542.i			1.65325i	−0.562753	+	0.826625i	$0.690258\pi$
$282$		0			0
$283$			77156.4i			0.963384i	0.876341	+	0.481692i	$0.159978\pi$
$283$			77156.4i			0.963384i	−0.876341	+	0.481692i	$0.840022\pi$
$284$		0			0
$285$			657.258i			0.00809182i
$286$		0			0
$287$	−86994.4			−1.05615
$288$		0			0
$289$	4961.32			0.0594020
$290$		0			0
$291$	−78471.4			−0.926671
$292$		0			0
$293$		−	73563.5i		−	0.856894i	−0.903567	−	0.428447i	$-0.859061\pi$
$293$		−	73563.5i		−	0.856894i	0.903567	−	0.428447i	$-0.140939\pi$
$294$		0			0
$295$	1247.69			0.0143371
$296$		0			0
$297$	13246.5	−	11306.9i	0.150171	−	0.128183i
$298$		0			0
$299$		−	60344.1i		−	0.674982i
$300$		0			0
$301$	−29561.3			−0.326281
$302$		0			0
$303$		−	195.322i		−	0.00212748i
$304$		0			0
$305$			15302.8i			0.164502i
$306$		0			0
$307$			14140.6i			0.150034i	0.997182	+	0.0750171i	$0.0239011\pi$
$307$			14140.6i			0.150034i	−0.997182	+	0.0750171i	$0.976099\pi$
$308$		0			0
$309$	−158023.			−1.65502
$310$		0			0
$311$	−115950.			−1.19881			−0.599406	−	0.800445i	$-0.704596\pi$
$311$	−115950.			−1.19881			−0.599406	+	0.800445i	$0.704596\pi$
$312$		0			0
$313$	−167682.			−1.71159			−0.855793	−	0.517318i	$-0.826930\pi$
$313$	−167682.			−1.71159			−0.855793	+	0.517318i	$0.826930\pi$
$314$		0			0
$315$			10009.2i			0.100874i
$316$		0			0
$317$	43569.8			0.433577			0.216789	−	0.976219i	$-0.430442\pi$
$317$	43569.8			0.433577			0.216789	+	0.976219i	$0.430442\pi$
$318$		0			0
$319$	81666.0	+	95674.7i	0.802527	+	0.940191i
$320$		0			0
$321$			132458.i			1.28549i
$322$		0			0
$323$	3529.51			0.0338305
$324$		0			0
$325$			198336.i			1.87774i
$326$		0			0
$327$			103830.i			0.971021i
$328$		0			0
$329$		−	102539.i		−	0.947321i
$330$		0			0
$331$	13167.8			0.120187			0.0600934	−	0.998193i	$-0.480860\pi$
$331$	13167.8			0.120187			0.0600934	+	0.998193i	$0.480860\pi$
$332$		0			0
$333$	97005.2			0.874795
$334$		0			0
$335$	−10048.3			−0.0895368
$336$		0			0
$337$			41741.2i			0.367540i	0.982969	+	0.183770i	$0.0588302\pi$
$337$			41741.2i			0.367540i	−0.982969	+	0.183770i	$0.941170\pi$
$338$		0			0
$339$	12378.9			0.107717
$340$		0			0
$341$	137359.	−	117247.i	1.18127	−	1.00830i
$342$		0			0
$343$		−	123885.i		−	1.05301i
$344$		0			0
$345$	9637.17			0.0809676
$346$		0			0
$347$		−	8110.63i		−	0.0673590i	−0.999433	−	0.0336795i	$-0.989277\pi$
$347$		−	8110.63i		−	0.0673590i	0.999433	−	0.0336795i	$-0.0107225\pi$
$348$		0			0
$349$			16786.2i			0.137817i	0.997623	+	0.0689083i	$0.0219516\pi$
$349$			16786.2i			0.137817i	−0.997623	+	0.0689083i	$0.978048\pi$
$350$		0			0
$351$		−	47040.1i		−	0.381816i
$352$		0			0
$353$	116711.			0.936618			0.468309	−	0.883565i	$-0.344863\pi$
$353$	116711.			0.936618			0.468309	+	0.883565i	$0.344863\pi$
$354$		0			0
$355$	10619.6			0.0842655
$356$		0			0
$357$	116612.			0.914973
$358$		0			0
$359$		−	35393.9i		−	0.274625i	−0.990528	−	0.137312i	$-0.956154\pi$
$359$		−	35393.9i		−	0.274625i	0.990528	−	0.137312i	$-0.0438464\pi$
$360$		0			0
$361$	130162.			0.998783
$362$		0			0
$363$	−28177.9	+	177243.i	−0.213843	+	1.34511i
$364$		0			0
$365$			19081.3i			0.143226i
$366$		0			0
$367$	−91892.5			−0.682257			−0.341128	−	0.940017i	$-0.610809\pi$
$367$	−91892.5			−0.682257			−0.341128	+	0.940017i	$0.610809\pi$
$368$		0			0
$369$		−	177515.i		−	1.30371i
$370$		0			0
$371$		−	8849.26i		−	0.0642923i
$372$		0			0
$373$		−	35452.4i		−	0.254817i	−0.991850	−	0.127408i	$-0.959334\pi$
$373$		−	35452.4i		−	0.254817i	0.991850	−	0.127408i	$-0.0406659\pi$
$374$		0			0
$375$	−64296.4			−0.457219
$376$		0			0
$377$	339755.			2.39047
$378$		0			0
$379$	−217027.			−1.51090			−0.755449	−	0.655207i	$-0.772581\pi$
$379$	−217027.			−1.51090			−0.755449	+	0.655207i	$0.772581\pi$
$380$		0			0
$381$			96336.5i			0.663653i
$382$		0			0
$383$	−199025.			−1.35678			−0.678390	−	0.734702i	$-0.737322\pi$
$383$	−199025.			−1.35678			−0.678390	+	0.734702i	$0.737322\pi$
$384$		0			0
$385$	11353.0	+	13300.5i	0.0765932	+	0.0897318i
$386$		0			0
$387$		−	60320.9i		−	0.402760i
$388$		0			0
$389$	141391.			0.934380			0.467190	−	0.884157i	$-0.345266\pi$
$389$	141391.			0.934380			0.467190	+	0.884157i	$0.345266\pi$
$390$		0			0
$391$		−	51752.1i		−	0.338512i
$392$		0			0
$393$			299319.i			1.93798i
$394$		0			0
$395$		−	41276.7i		−	0.264552i
$396$		0			0
$397$	−109000.			−0.691585			−0.345793	−	0.938311i	$-0.612390\pi$
$397$	−109000.			−0.691585			−0.345793	+	0.938311i	$0.612390\pi$
$398$		0			0
$399$	−5239.13			−0.0329089
$400$		0			0
$401$	−262604.			−1.63310			−0.816549	−	0.577277i	$-0.804115\pi$
$401$	−262604.			−1.63310			−0.816549	+	0.577277i	$0.804115\pi$
$402$		0			0
$403$		−	487781.i		−	3.00341i
$404$		0			0
$405$	31399.3			0.191430
$406$		0			0
$407$	128903.	−	110029.i	0.778170	−	0.664230i
$408$		0			0
$409$		−	132695.i		−	0.793244i	−0.917982	−	0.396622i	$-0.870182\pi$
$409$		−	132695.i		−	0.793244i	0.917982	−	0.396622i	$-0.129818\pi$
$410$		0			0
$411$	35498.0			0.210146
$412$		0			0
$413$			9945.55i			0.0583081i
$414$		0			0
$415$		−	26561.2i		−	0.154224i
$416$		0			0
$417$			179003.i			1.02941i
$418$		0			0
$419$	−54888.5			−0.312646			−0.156323	−	0.987706i	$-0.549964\pi$
$419$	−54888.5			−0.312646			−0.156323	+	0.987706i	$0.549964\pi$
$420$		0			0
$421$	−20412.0			−0.115165			−0.0575826	−	0.998341i	$-0.518339\pi$
$421$	−20412.0			−0.115165			−0.0575826	+	0.998341i	$0.518339\pi$
$422$		0			0
$423$	209234.			1.16937
$424$		0			0
$425$			170096.i			0.941710i
$426$		0			0
$427$	−121982.			−0.669020
$428$		0			0
$429$	314708.	+	368692.i	1.70999	+	2.00332i
$430$		0			0
$431$		−	157298.i		−	0.846776i	−0.905949	−	0.423388i	$-0.860841\pi$
$431$		−	157298.i		−	0.846776i	0.905949	−	0.423388i	$-0.139159\pi$
$432$		0			0
$433$	−134223.			−0.715900			−0.357950	−	0.933741i	$-0.616524\pi$
$433$	−134223.			−0.715900			−0.357950	+	0.933741i	$0.616524\pi$
$434$		0			0
$435$			54260.1i			0.286749i
$436$		0			0
$437$			2325.10i			0.0121753i
$438$		0			0
$439$			304612.i			1.58059i	0.612729	+	0.790293i	$0.290072\pi$
$439$			304612.i			1.58059i	−0.612729	+	0.790293i	$0.709928\pi$
$440$		0			0
$441$	86503.2			0.444790
$442$		0			0
$443$	116115.			0.591671			0.295835	−	0.955239i	$-0.404402\pi$
$443$	116115.			0.591671			0.295835	+	0.955239i	$0.404402\pi$
$444$		0			0
$445$	40522.0			0.204631
$446$		0			0
$447$		−	116072.i		−	0.580913i
$448$		0			0
$449$	156051.			0.774059			0.387030	−	0.922067i	$-0.373501\pi$
$449$	156051.			0.774059			0.387030	+	0.922067i	$0.373501\pi$
$450$		0			0
$451$	−201348.	−	235887.i	−0.989906	−	1.15971i
$452$		0			0
$453$			187987.i			0.916075i
$454$		0			0
$455$	47232.0			0.228146
$456$		0			0
$457$			154609.i			0.740290i	0.928974	+	0.370145i	$0.120692\pi$
$457$			154609.i			0.740290i	−0.928974	+	0.370145i	$0.879308\pi$
$458$		0			0
$459$		−	40342.4i		−	0.191486i
$460$		0			0
$461$			324927.i			1.52892i	0.644674	+	0.764458i	$0.276993\pi$
$461$			324927.i			1.52892i	−0.644674	+	0.764458i	$0.723007\pi$
$462$		0			0
$463$	−135116.			−0.630295			−0.315147	−	0.949043i	$-0.602054\pi$
$463$	−135116.			−0.630295			−0.315147	+	0.949043i	$0.602054\pi$
$464$		0			0
$465$	77900.5			0.360275
$466$		0			0
$467$	−217618.			−0.997840			−0.498920	−	0.866648i	$-0.666270\pi$
$467$	−217618.			−0.997840			−0.498920	+	0.866648i	$0.666270\pi$
$468$		0			0
$469$		−	80096.6i		−	0.364140i
$470$		0			0
$471$	−58962.9			−0.265789
$472$		0			0
$473$	−68419.6	−	80156.1i	−0.305814	−	0.358273i
$474$		0			0
$475$		−	7642.05i		−	0.0338706i
$476$		0			0
$477$	18057.2			0.0793622
$478$		0			0
$479$			2332.66i			0.0101667i	0.999987	+	0.00508335i	$0.00161809\pi$
$479$			2332.66i			0.0101667i	−0.999987	+	0.00508335i	$0.998382\pi$
$480$		0			0
$481$		−	457754.i		−	1.97853i
$482$		0			0
$483$			76819.7i			0.329290i
$484$		0			0
$485$	−27258.1			−0.115881
$486$		0			0
$487$	307695.			1.29737			0.648683	−	0.761059i	$-0.275320\pi$
$487$	307695.			1.29737			0.648683	+	0.761059i	$0.275320\pi$
$488$		0			0
$489$	−121060.			−0.506269
$490$		0			0
$491$			165203.i			0.685258i	0.939471	+	0.342629i	$0.111317\pi$
$491$			165203.i			0.685258i	−0.939471	+	0.342629i	$0.888683\pi$
$492$		0			0
$493$	291379.			1.19885
$494$		0			0
$495$	−27140.1	+	23166.2i	−0.110765	+	0.0945464i
$496$		0			0
$497$			84650.5i			0.342702i
$498$		0			0
$499$	−271209.			−1.08919			−0.544594	−	0.838700i	$-0.683316\pi$
$499$	−271209.			−1.08919			−0.544594	+	0.838700i	$0.683316\pi$
$500$		0			0
$501$		−	164713.i		−	0.656226i
$502$		0			0
$503$			242593.i			0.958831i	0.877588	+	0.479415i	$0.159151\pi$
$503$			242593.i			0.958831i	−0.877588	+	0.479415i	$0.840849\pi$
$504$		0			0
$505$		−	67.8477i		−	0.000266043i
$506$		0			0
$507$	959180.			3.73151
$508$		0			0
$509$	231272.			0.892663			0.446331	−	0.894868i	$-0.352730\pi$
$509$	231272.			0.892663			0.446331	+	0.894868i	$0.352730\pi$
$510$		0			0
$511$	−152101.			−0.582491
$512$		0			0
$513$			1812.49i			0.00688718i
$514$		0			0
$515$	−54891.6			−0.206962
$516$		0			0
$517$	278036.	−	237326.i	1.04021	−	0.887900i
$518$		0			0
$519$			461686.i			1.71400i
$520$		0			0
$521$	−19525.5			−0.0719327			−0.0359664	−	0.999353i	$-0.511451\pi$
$521$	−19525.5			−0.0719327			−0.0359664	+	0.999353i	$0.511451\pi$
$522$		0			0
$523$			493543.i			1.80435i	0.431366	+	0.902177i	$0.358032\pi$
$523$			493543.i			1.80435i	−0.431366	+	0.902177i	$0.641968\pi$
$524$		0			0
$525$		−	252488.i		−	0.916056i
$526$		0			0
$527$		−	418329.i		−	1.50625i
$528$		0			0
$529$	−245749.			−0.878173
$530$		0			0
$531$	−20294.2			−0.0719753
$532$		0			0
$533$	−837668.			−2.94861
$534$		0			0
$535$			46011.1i			0.160752i
$536$		0			0
$537$	661211.			2.29293
$538$		0			0
$539$	114948.	−	98117.1i	0.395661	−	0.337728i
$540$		0			0
$541$		−	211511.i		−	0.722669i	−0.932436	−	0.361334i	$-0.882321\pi$
$541$		−	211511.i		−	0.722669i	0.932436	−	0.361334i	$-0.117679\pi$
$542$		0			0
$543$	286587.			0.971978
$544$		0			0
$545$			36066.9i			0.121427i
$546$		0			0
$547$			271731.i			0.908163i	0.890960	+	0.454082i	$0.150033\pi$
$547$			271731.i			0.908163i	−0.890960	+	0.454082i	$0.849967\pi$
$548$		0			0
$549$		−	248908.i		−	0.825836i
$550$		0			0
$551$	−13091.0			−0.0431191
$552$		0			0
$553$	329025.			1.07592
$554$		0			0
$555$	73105.0			0.237335
$556$		0			0
$557$			420797.i			1.35632i	0.734914	+	0.678160i	$0.237223\pi$
$557$			420797.i			1.35632i	−0.734914	+	0.678160i	$0.762777\pi$
$558$		0			0
$559$	−284646.			−0.910922
$560$		0			0
$561$	269899.	+	316196.i	0.857581	+	1.00469i
$562$		0			0
$563$			199781.i			0.630285i	0.949044	+	0.315142i	$0.102052\pi$
$563$			199781.i			0.630285i	−0.949044	+	0.315142i	$0.897948\pi$
$564$		0			0
$565$	4299.99			0.0134701
$566$		0			0
$567$			250290.i			0.778532i
$568$		0			0
$569$			393771.i			1.21624i	0.793845	+	0.608120i	$0.208076\pi$
$569$			393771.i			1.21624i	−0.793845	+	0.608120i	$0.791924\pi$
$570$		0			0
$571$		−	152076.i		−	0.466431i	−0.972425	−	0.233215i	$-0.925075\pi$
$571$		−	152076.i		−	0.466431i	0.972425	−	0.233215i	$-0.0749247\pi$
$572$		0			0
$573$	−571361.			−1.74021
$574$		0			0
$575$	−112053.			−0.338912
$576$		0			0
$577$	−260379.			−0.782086			−0.391043	−	0.920372i	$-0.627886\pi$
$577$	−260379.			−0.782086			−0.391043	+	0.920372i	$0.627886\pi$
$578$		0			0
$579$			705883.i			2.10560i
$580$		0			0
$581$	211724.			0.627218
$582$		0			0
$583$	23994.9	−	20481.6i	0.0705963	−	0.0602596i
$584$		0			0
$585$			96378.4i			0.281623i
$586$		0			0
$587$	197623.			0.573538			0.286769	−	0.958000i	$-0.407419\pi$
$587$	197623.			0.573538			0.286769	+	0.958000i	$0.407419\pi$
$588$		0			0
$589$			18794.6i			0.0541754i
$590$		0			0
$591$		−	487563.i		−	1.39590i
$592$		0			0
$593$		−	103667.i		−	0.294803i	−0.989077	−	0.147401i	$-0.952909\pi$
$593$		−	103667.i		−	0.294803i	0.989077	−	0.147401i	$-0.0470909\pi$
$594$		0			0
$595$	40506.9			0.114418
$596$		0			0
$597$	−231740.			−0.650208
$598$		0			0
$599$	204660.			0.570399			0.285199	−	0.958468i	$-0.407940\pi$
$599$	204660.			0.570399			0.285199	+	0.958468i	$0.407940\pi$
$600$		0			0
$601$		−	140115.i		−	0.387913i	−0.981010	−	0.193957i	$-0.937868\pi$
$601$		−	140115.i		−	0.387913i	0.981010	−	0.193957i	$-0.0621321\pi$
$602$		0			0
$603$	163440.			0.449493
$604$		0			0
$605$	−9787.99	+	61567.8i	−0.0267413	+	0.168207i
$606$		0			0
$607$			69648.7i			0.189032i	0.995523	+	0.0945160i	$0.0301304\pi$
$607$			69648.7i			0.189032i	−0.995523	+	0.0945160i	$0.969870\pi$
$608$		0			0
$609$	−432518.			−1.16619
$610$		0			0
$611$		−	987347.i		−	2.64477i
$612$		0			0
$613$			619709.i			1.64917i	0.565735	+	0.824587i	$0.308593\pi$
$613$			619709.i			1.64917i	−0.565735	+	0.824587i	$0.691407\pi$
$614$		0			0
$615$		−	133779.i		−	0.353701i
$616$		0			0
$617$	−284417.			−0.747112			−0.373556	−	0.927608i	$-0.621862\pi$
$617$	−284417.			−0.747112			−0.373556	+	0.927608i	$0.621862\pi$
$618$		0			0
$619$	231315.			0.603702			0.301851	−	0.953355i	$-0.402395\pi$
$619$	231315.			0.603702			0.301851	+	0.953355i	$0.402395\pi$
$620$		0			0
$621$	26576.0			0.0689138
$622$		0			0
$623$			323008.i			0.832219i
$624$		0			0
$625$	356959.			0.913816
$626$		0			0
$627$	−12125.9	−	14206.0i	−0.0308447	−	0.0361357i
$628$		0			0
$629$		−	392577.i		−	0.992256i
$630$		0			0
$631$	311488.			0.782316			0.391158	−	0.920324i	$-0.372075\pi$
$631$	311488.			0.782316			0.391158	+	0.920324i	$0.372075\pi$
$632$		0			0
$633$			379682.i			0.947573i
$634$		0			0
$635$			33463.8i			0.0829904i
$636$		0			0
$637$		−	408197.i		−	1.00598i
$638$		0			0
$639$	−172732.			−0.423030
$640$		0			0
$641$	−297618.			−0.724341			−0.362170	−	0.932112i	$-0.617964\pi$
$641$	−297618.			−0.724341			−0.362170	+	0.932112i	$0.617964\pi$
$642$		0			0
$643$	121820.			0.294645			0.147322	−	0.989089i	$-0.452935\pi$
$643$	121820.			0.294645			0.147322	+	0.989089i	$0.452935\pi$
$644$		0			0
$645$		−	45459.0i		−	0.109270i
$646$		0			0
$647$	−377550.			−0.901917			−0.450958	−	0.892545i	$-0.648918\pi$
$647$	−377550.			−0.901917			−0.450958	+	0.892545i	$0.648918\pi$
$648$		0			0
$649$	−26967.5	+	23018.9i	−0.0640253	+	0.0546507i
$650$		0			0
$651$			620959.i			1.46521i
$652$		0			0
$653$	558170.			1.30900			0.654501	−	0.756061i	$-0.272879\pi$
$653$	558170.			1.30900			0.654501	+	0.756061i	$0.272879\pi$
$654$		0			0
$655$			103973.i			0.242346i
$656$		0			0
$657$		−	310367.i		−	0.719025i
$658$		0			0
$659$		−	357653.i		−	0.823551i	−0.911285	−	0.411776i	$-0.864909\pi$
$659$		−	357653.i		−	0.823551i	0.911285	−	0.411776i	$-0.135091\pi$
$660$		0			0
$661$	−290959.			−0.665931			−0.332965	−	0.942939i	$-0.608049\pi$
$661$	−290959.			−0.665931			−0.332965	+	0.942939i	$0.608049\pi$
$662$		0			0
$663$	1.12286e6			2.55446
$664$		0			0
$665$	−1819.88			−0.00411529
$666$		0			0
$667$			191949.i			0.431454i
$668$		0			0
$669$	243818.			0.544771
$670$		0			0
$671$	−282326.	−	330756.i	−0.627056	−	0.734619i
$672$		0			0
$673$		−	465570.i		−	1.02791i	−0.857817	−	0.513955i	$-0.828180\pi$
$673$		−	465570.i		−	1.02791i	0.857817	−	0.513955i	$-0.171820\pi$
$674$		0			0
$675$	−87348.8			−0.191712
$676$		0			0
$677$			361627.i			0.789012i	0.918893	+	0.394506i	$0.129084\pi$
$677$			361627.i			0.789012i	−0.918893	+	0.394506i	$0.870916\pi$
$678$		0			0
$679$		−	217280.i		−	0.471280i
$680$		0			0
$681$		−	938089.i		−	2.02279i
$682$		0			0
$683$	−673707.			−1.44421			−0.722103	−	0.691785i	$-0.756825\pi$
$683$	−673707.			−1.44421			−0.722103	+	0.691785i	$0.756825\pi$
$684$		0			0
$685$	12330.7			0.0262789
$686$		0			0
$687$	−687426.			−1.45651
$688$		0			0
$689$		−	85209.4i		−	0.179494i
$690$		0			0
$691$	−66171.2			−0.138584			−0.0692920	−	0.997596i	$-0.522074\pi$
$691$	−66171.2			−0.138584			−0.0692920	+	0.997596i	$0.522074\pi$
$692$		0			0
$693$	−184662.	−	216339.i	−0.384514	−	0.450472i
$694$		0			0
$695$			62179.0i			0.128728i
$696$		0			0
$697$	−718397.			−1.47877
$698$		0			0
$699$			41105.8i			0.0841297i
$700$		0			0
$701$		−	439863.i		−	0.895121i	−0.894254	−	0.447560i	$-0.852293\pi$
$701$		−	439863.i		−	0.895121i	0.894254	−	0.447560i	$-0.147707\pi$
$702$		0			0
$703$			17637.6i			0.0356885i
$704$		0			0
$705$	157683.			0.317254
$706$		0			0
$707$	540.827			0.00108198
$708$		0			0
$709$	−511529.			−1.01760			−0.508801	−	0.860884i	$-0.669911\pi$
$709$	−511529.			−1.01760			−0.508801	+	0.860884i	$0.669911\pi$
$710$		0			0
$711$			671386.i			1.32811i
$712$		0			0
$713$	275579.			0.542084
$714$		0			0
$715$	109318.	+	128070.i	0.213836	+	0.250517i
$716$		0			0
$717$			105783.i			0.205767i
$718$		0			0
$719$	683729.			1.32259			0.661297	−	0.750124i	$-0.270006\pi$
$719$	683729.			1.32259			0.661297	+	0.750124i	$0.270006\pi$
$720$		0			0
$721$		−	437551.i		−	0.841701i
$722$		0			0
$723$		−	1.24291e6i		−	2.37774i
$724$		0			0
$725$		−	630891.i		−	1.20027i
$726$		0			0
$727$	258952.			0.489949			0.244974	−	0.969530i	$-0.421220\pi$
$727$	258952.			0.489949			0.244974	+	0.969530i	$0.421220\pi$
$728$		0			0
$729$	−367813.			−0.692105
$730$		0			0
$731$	−244117.			−0.456839
$732$		0			0
$733$		−	474044.i		−	0.882288i	−0.897436	−	0.441144i	$-0.854573\pi$
$733$		−	474044.i		−	0.882288i	0.897436	−	0.441144i	$-0.145427\pi$
$734$		0			0
$735$	65190.5			0.120673
$736$		0			0
$737$	217183.	−	185383.i	0.399845	−	0.341299i
$738$		0			0
$739$		−	758640.i		−	1.38914i	−0.719424	−	0.694571i	$-0.755594\pi$
$739$		−	758640.i		−	1.38914i	0.719424	−	0.694571i	$-0.244406\pi$
$740$		0			0
$741$	−50447.5			−0.0918763
$742$		0			0
$743$		−	579823.i		−	1.05031i	−0.851006	−	0.525156i	$-0.824007\pi$
$743$		−	579823.i		−	1.05031i	0.851006	−	0.525156i	$-0.175993\pi$
$744$		0			0
$745$		−	40319.1i		−	0.0726438i
$746$		0			0
$747$			432030.i			0.774236i
$748$		0			0
$749$	−366763.			−0.653766
$750$		0			0
$751$	−844322.			−1.49702			−0.748511	−	0.663122i	$-0.769231\pi$
$751$	−844322.			−1.49702			−0.748511	+	0.663122i	$0.769231\pi$
$752$		0			0
$753$	−995897.			−1.75640
$754$		0			0
$755$			65299.8i			0.114556i
$756$		0			0
$757$	−801567.			−1.39877			−0.699387	−	0.714743i	$-0.746544\pi$
$757$	−801567.			−1.39877			−0.699387	+	0.714743i	$0.746544\pi$
$758$		0			0
$759$	−208298.	+	177799.i	−0.361577	+	0.308635i
$760$		0			0
$761$			6823.03i			0.0117817i	0.999983	+	0.00589085i	$0.00187513\pi$
$761$			6823.03i			0.0117817i	−0.999983	+	0.00589085i	$0.998125\pi$
$762$		0			0
$763$	−287496.			−0.493836
$764$		0			0
$765$			82655.7i			0.141237i
$766$		0			0
$767$			95765.6i			0.162787i
$768$		0			0
$769$			375862.i			0.635588i	0.948160	+	0.317794i	$0.102942\pi$
$769$			375862.i			0.635588i	−0.948160	+	0.317794i	$0.897058\pi$
$770$		0			0
$771$	−698250.			−1.17463
$772$		0			0
$773$	857421.			1.43494			0.717472	−	0.696587i	$-0.245299\pi$
$773$	857421.			1.43494			0.717472	+	0.696587i	$0.245299\pi$
$774$		0			0
$775$	−905761.			−1.50803
$776$		0			0
$777$			582734.i			0.965224i
$778$		0			0
$779$	32276.0			0.0531868
$780$		0			0
$781$	−229531.	+	195923.i	−0.376305	+	0.321206i
$782$		0			0
$783$			149631.i			0.244060i
$784$		0			0
$785$	−20481.6			−0.0332372
$786$		0			0
$787$			1.02294e6i			1.65158i	0.563976	+	0.825791i	$0.309271\pi$
$787$			1.02294e6i			1.65158i	−0.563976	+	0.825791i	$0.690729\pi$
$788$		0			0
$789$		−	73556.6i		−	0.118159i
$790$		0			0
$791$			34276.0i			0.0547820i
$792$		0			0
$793$	−1.17456e6			−1.86780
$794$		0			0
$795$	13608.3			0.0215312
$796$		0			0
$797$	−865636.			−1.36276			−0.681379	−	0.731931i	$-0.738619\pi$
$797$	−865636.			−1.36276			−0.681379	+	0.731931i	$0.738619\pi$
$798$		0			0
$799$		−	846765.i		−	1.32638i
$800$		0			0
$801$	−659109.			−1.02729
$802$		0			0
$803$	−352036.	−	412424.i	−0.545954	−	0.639606i
$804$		0			0
$805$			26684.4i			0.0411780i
$806$		0			0
$807$	304390.			0.467395
$808$		0			0
$809$		−	886947.i		−	1.35519i	−0.735435	−	0.677596i	$-0.763022\pi$
$809$		−	886947.i		−	1.35519i	0.735435	−	0.677596i	$-0.236978\pi$
$810$		0			0
$811$		−	608105.i		−	0.924563i	−0.886733	−	0.462282i	$-0.847031\pi$
$811$		−	608105.i		−	0.924563i	0.886733	−	0.462282i	$-0.152969\pi$
$812$		0			0
$813$			808744.i			1.22357i
$814$		0			0
$815$	−42051.7			−0.0633094
$816$		0			0
$817$	10967.6			0.0164312
$818$		0			0
$819$	−768251.			−1.14534
$820$		0			0
$821$			102607.i			0.152227i	0.997099	+	0.0761133i	$0.0242511\pi$
$821$			102607.i			0.152227i	−0.997099	+	0.0761133i	$0.975749\pi$
$822$		0			0
$823$	338801.			0.500202			0.250101	−	0.968220i	$-0.419536\pi$
$823$	338801.			0.500202			0.250101	+	0.968220i	$0.419536\pi$
$824$		0			0
$825$	684625.	−	584382.i	1.00588	−	0.858595i
$826$		0			0
$827$			303788.i			0.444181i	0.975026	+	0.222090i	$0.0712880\pi$
$827$			303788.i			0.444181i	−0.975026	+	0.222090i	$0.928712\pi$
$828$		0			0
$829$	−923889.			−1.34435			−0.672173	−	0.740394i	$-0.734639\pi$
$829$	−923889.			−1.34435			−0.672173	+	0.740394i	$0.734639\pi$
$830$		0			0
$831$		−	388273.i		−	0.562258i
$832$		0			0
$833$		−	350076.i		−	0.504513i
$834$		0			0
$835$		−	57215.5i		−	0.0820617i
$836$		0			0
$837$	214823.			0.306640
$838$		0			0
$839$	−44067.7			−0.0626032			−0.0313016	−	0.999510i	$-0.509965\pi$
$839$	−44067.7			−0.0626032			−0.0313016	+	0.999510i	$0.509965\pi$
$840$		0			0
$841$	−373450.			−0.528009
$842$		0			0
$843$		−	1.60018e6i		−	2.25172i
$844$		0			0
$845$	333184.			0.466628
$846$		0			0
$847$	−490769.	−	78021.9i	−0.684085	−	0.108755i
$848$		0			0
$849$		−	945782.i		−	1.31213i
$850$		0			0
$851$	258614.			0.357103
$852$		0			0
$853$		−	1.02952e6i		−	1.41494i	−0.706746	−	0.707468i	$-0.749837\pi$
$853$		−	1.02952e6i		−	1.41494i	0.706746	−	0.707468i	$-0.250163\pi$
$854$		0			0
$855$		−	3713.53i		−	0.00507990i
$856$		0			0
$857$			571557.i			0.778212i	0.921193	+	0.389106i	$0.127216\pi$
$857$			571557.i			0.778212i	−0.921193	+	0.389106i	$0.872784\pi$
$858$		0			0
$859$	479749.			0.650172			0.325086	−	0.945685i	$-0.394607\pi$
$859$	479749.			0.650172			0.325086	+	0.945685i	$0.394607\pi$
$860$		0			0
$861$	1.06637e6			1.43848
$862$		0			0
$863$	−220319.			−0.295822			−0.147911	−	0.989001i	$-0.547255\pi$
$863$	−220319.			−0.295822			−0.147911	+	0.989001i	$0.547255\pi$
$864$		0			0
$865$			160373.i			0.214338i
$866$		0			0
$867$	−60815.7			−0.0809054
$868$		0			0
$869$	761526.	+	892156.i	1.00843	+	1.18141i
$870$		0			0
$871$		−	771250.i		−	1.01662i
$872$		0			0
$873$	443366.			0.581747
$874$		0			0
$875$		−	178030.i		−	0.232530i
$876$		0			0
$877$		−	1.23303e6i		−	1.60315i	−0.597893	−	0.801576i	$-0.703995\pi$
$877$		−	1.23303e6i		−	1.60315i	0.597893	−	0.801576i	$-0.296005\pi$
$878$		0			0
$879$			901740.i			1.16709i
$880$		0			0
$881$	227150.			0.292659			0.146329	−	0.989236i	$-0.453254\pi$
$881$	227150.			0.292659			0.146329	+	0.989236i	$0.453254\pi$
$882$		0			0
$883$	136322.			0.174842			0.0874210	−	0.996171i	$-0.472137\pi$
$883$	136322.			0.174842			0.0874210	+	0.996171i	$0.472137\pi$
$884$		0			0
$885$	−15294.1			−0.0195271
$886$		0			0
$887$			852106.i			1.08304i	0.840686	+	0.541522i	$0.182152\pi$
$887$			852106.i			1.08304i	−0.840686	+	0.541522i	$0.817848\pi$
$888$		0			0
$889$	−266746.			−0.337516
$890$		0			0
$891$	−678664.	+	579294.i	−0.854869	+	0.729699i
$892$		0			0
$893$			38043.2i			0.0477061i
$894$		0			0
$895$	229681.			0.286734
$896$		0			0
$897$			739696.i			0.919324i
$898$		0			0
$899$			1.55159e6i			1.91981i
$900$		0			0
$901$		−	73077.0i		−	0.0900183i
$902$		0			0
$903$	362362.			0.444393
$904$		0			0
$905$	99549.8			0.121547
$906$		0			0
$907$	945937.			1.14987			0.574934	−	0.818200i	$-0.305028\pi$
$907$	945937.			1.14987			0.574934	+	0.818200i	$0.305028\pi$
$908$		0			0
$909$			1103.57i			0.00133559i
$910$		0			0
$911$	−13013.5			−0.0156803			−0.00784017	−	0.999969i	$-0.502496\pi$
$911$	−13013.5			−0.0156803			−0.00784017	+	0.999969i	$0.502496\pi$
$912$		0			0
$913$	490035.	+	574094.i	0.587875	+	0.688718i
$914$		0			0
$915$		−	187582.i		−	0.224052i
$916$		0			0
$917$	−828786.			−0.985607
$918$		0			0
$919$			809716.i			0.958742i	0.877612	+	0.479371i	$0.159135\pi$
$919$			809716.i			0.958742i	−0.877612	+	0.479371i	$0.840865\pi$
$920$		0			0
$921$		−	173335.i		−	0.204346i
$922$		0			0
$923$			815099.i			0.956768i
$924$		0			0
$925$	−850003.			−0.993429
$926$		0			0
$927$	892837.			1.03899
$928$		0			0
$929$	731171.			0.847203			0.423601	−	0.905849i	$-0.360766\pi$
$929$	731171.			0.847203			0.423601	+	0.905849i	$0.360766\pi$
$930$		0			0
$931$			15728.1i			0.0181458i
$932$		0			0
$933$	1.42131e6			1.63278
$934$		0			0
$935$	93753.0	+	109835.i	0.107241	+	0.125637i
$936$		0			0
$937$			637069.i			0.725617i	0.931864	+	0.362809i	$0.118182\pi$
$937$			637069.i			0.725617i	−0.931864	+	0.362809i	$0.881818\pi$
$938$		0			0
$939$	2.05545e6			2.33118
$940$		0			0
$941$		−	77429.7i		−	0.0874437i	−0.999044	−	0.0437218i	$-0.986078\pi$
$941$		−	77429.7i		−	0.0874437i	0.999044	−	0.0437218i	$-0.0139215\pi$
$942$		0			0
$943$		−	473252.i		−	0.532193i
$944$		0			0
$945$			20801.3i			0.0232931i
$946$		0			0
$947$	−362138.			−0.403807			−0.201904	−	0.979405i	$-0.564713\pi$
$947$	−362138.			−0.403807			−0.201904	+	0.979405i	$0.564713\pi$
$948$		0			0
$949$	−1.46458e6			−1.62622
$950$		0			0
$951$	−534077.			−0.590531
$952$		0			0
$953$		−	1.11862e6i		−	1.23168i	−0.787873	−	0.615838i	$-0.788818\pi$
$953$		−	1.11862e6i		−	1.23168i	0.787873	−	0.615838i	$-0.211182\pi$
$954$		0			0
$955$	−198470.			−0.217615
$956$		0			0
$957$	−1.00106e6	−	1.17278e6i	−1.09304	−	1.28054i
$958$		0			0
$959$			98290.6i			0.106875i
$960$		0			0
$961$	1.30407e6			1.41207
$962$		0			0
$963$		−	748393.i		−	0.807006i
$964$		0			0
$965$			245198.i			0.263307i
$966$		0			0
$967$		−	104836.i		−	0.112113i	−0.998428	−	0.0560565i	$-0.982147\pi$
$967$		−	104836.i		−	0.112113i	0.998428	−	0.0560565i	$-0.0178527\pi$
$968$		0			0
$969$	−43264.6			−0.0460771
$970$		0			0
$971$	1.63144e6			1.73034			0.865170	−	0.501478i	$-0.167210\pi$
$971$	1.63144e6			1.73034			0.865170	+	0.501478i	$0.167210\pi$
$972$		0			0
$973$	−495640.			−0.523529
$974$		0			0
$975$		−	2.43120e6i		−	2.55748i
$976$		0			0
$977$	4386.95			0.00459593			0.00229796	−	0.999997i	$-0.499269\pi$
$977$	4386.95			0.00459593			0.00229796	+	0.999997i	$0.499269\pi$
$978$		0			0
$979$	−875842.	+	747600.i	−0.913819	+	0.780017i
$980$		0			0
$981$		−	586645.i		−	0.609589i
$982$		0			0
$983$	147531.			0.152678			0.0763388	−	0.997082i	$-0.475677\pi$
$983$	147531.			0.152678			0.0763388	+	0.997082i	$0.475677\pi$
$984$		0			0
$985$		−	169362.i		−	0.174559i
$986$		0			0
$987$			1.25692e6i			1.29025i
$988$		0			0
$989$		−	160815.i		−	0.164412i
$990$		0			0
$991$	373627.			0.380444			0.190222	−	0.981741i	$-0.439079\pi$
$991$	373627.			0.380444			0.190222	+	0.981741i	$0.439079\pi$
$992$		0			0
$993$	−161410.			−0.163694
$994$		0			0
$995$	−80498.1			−0.0813092
$996$		0			0
$997$			71451.7i			0.0718824i	0.999354	+	0.0359412i	$0.0114429\pi$
$997$			71451.7i			0.0718824i	−0.999354	+	0.0359412i	$0.988557\pi$
$998$		0			0
$999$	201598.			0.202002

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.5.h.i.65.1		4
4.3	odd	2		704.5.h.j.65.4		4
8.3	odd	2		176.5.h.e.65.2		4
8.5	even	2		22.5.b.a.21.2	✓	4
11.10	odd	2	inner	704.5.h.i.65.2		4
24.5	odd	2		198.5.d.a.109.3		4
40.13	odd	4		550.5.c.a.549.4		8
40.29	even	2		550.5.d.a.351.3		4
40.37	odd	4		550.5.c.a.549.5		8
44.43	even	2		704.5.h.j.65.3		4
88.21	odd	2		22.5.b.a.21.4	yes	4
88.43	even	2		176.5.h.e.65.1		4
264.197	even	2		198.5.d.a.109.1		4
440.109	odd	2		550.5.d.a.351.1		4
440.197	even	4		550.5.c.a.549.1		8
440.373	even	4		550.5.c.a.549.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.5.b.a.21.2	✓	4	8.5	even	2
22.5.b.a.21.4	yes	4	88.21	odd	2
176.5.h.e.65.1		4	88.43	even	2
176.5.h.e.65.2		4	8.3	odd	2
198.5.d.a.109.1		4	264.197	even	2
198.5.d.a.109.3		4	24.5	odd	2
550.5.c.a.549.1		8	440.197	even	4
550.5.c.a.549.4		8	40.13	odd	4
550.5.c.a.549.5		8	40.37	odd	4
550.5.c.a.549.8		8	440.373	even	4
550.5.d.a.351.1		4	440.109	odd	2
550.5.d.a.351.3		4	40.29	even	2
704.5.h.i.65.1		4	1.1	even	1	trivial
704.5.h.i.65.2		4	11.10	odd	2	inner
704.5.h.j.65.3		4	44.43	even	2
704.5.h.j.65.4		4	4.3	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:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	704.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-12.2580 q^{3} -4.25798 q^{5} -33.9411i q^{7} +69.2580 q^{9} +O(q^{10})\)	\(q-12.2580 q^{3} -4.25798 q^{5} -33.9411i q^{7} +69.2580 q^{9} +(92.0319 - 78.5565i) q^{11} -326.819i q^{13} +52.1942 q^{15} -280.285i q^{17} +12.5926i q^{19} +416.050i q^{21} +184.641 q^{23} -606.870 q^{25} +143.933 q^{27} +1039.58i q^{29} +1492.51 q^{31} +(-1128.12 + 962.944i) q^{33} +144.520i q^{35} +1400.64 q^{37} +4006.14i q^{39} -2563.10i q^{41} -870.960i q^{43} -294.899 q^{45} +3021.08 q^{47} +1249.00 q^{49} +3435.73i q^{51} +260.724 q^{53} +(-391.870 + 334.492i) q^{55} -154.359i q^{57} -293.024 q^{59} -3593.92i q^{61} -2350.69i q^{63} +1391.59i q^{65} +2359.87 q^{67} -2263.32 q^{69} -2494.04 q^{71} -4481.31i q^{73} +7438.99 q^{75} +(-2666.30 - 3123.67i) q^{77} +9693.98i q^{79} -7374.23 q^{81} +6237.99i q^{83} +1193.45i q^{85} -12743.2i q^{87} -9516.72 q^{89} -11092.6 q^{91} -18295.2 q^{93} -53.6188i q^{95} +6401.66 q^{97} +(6373.94 - 5440.67i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 30 q^{5} + 230 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 30 * q^5 + 230 * q^9	\( 4 q - 2 q^{3} + 30 q^{5} + 230 q^{9} + 180 q^{11} + 538 q^{15} - 1566 q^{23} - 1722 q^{25} - 506 q^{27} + 4418 q^{31} - 2302 q^{33} + 382 q^{37} + 1172 q^{45} + 5688 q^{47} + 4996 q^{49} + 8568 q^{53} - 862 q^{55} + 3390 q^{59} + 8734 q^{67} - 26314 q^{69} + 3522 q^{71} + 9156 q^{75} + 2880 q^{77} - 31096 q^{81} - 8766 q^{89} - 17280 q^{91} - 20458 q^{93} + 17282 q^{97} + 12562 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 30 * q^5 + 230 * q^9 + 180 * q^11 + 538 * q^15 - 1566 * q^23 - 1722 * q^25 - 506 * q^27 + 4418 * q^31 - 2302 * q^33 + 382 * q^37 + 1172 * q^45 + 5688 * q^47 + 4996 * q^49 + 8568 * q^53 - 862 * q^55 + 3390 * q^59 + 8734 * q^67 - 26314 * q^69 + 3522 * q^71 + 9156 * q^75 + 2880 * q^77 - 31096 * q^81 - 8766 * q^89 - 17280 * q^91 - 20458 * q^93 + 17282 * q^97 + 12562 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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