LMFDB - Embedded newform 560.5.f.b.321.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,5,Mod(321,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("560.321");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	560.f (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:	$57.8871793270$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 109 x^{10} + 570 x^{9} + 5814 x^{8} - 22512 x^{7} - 151120 x^{6} + 300288 x^{5} + \cdots + 205833600 $ x^12 - 6x^11 - 109x^10 + 570x^9 + 5814x^8 - 22512x^7 - 151120x^6 + 300288x^5 + 2661184x^4 - 1881600x^3 - 12922560x^2 + 25948800*x + 205833600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{5} $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.11
Root			$5.81769 - 2.23607i$ of defining polynomial
Character	$\chi$	$=$	560.321
Dual form			560.5.f.b.321.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.5704i q^{3} +11.1803i q^{5} +(-44.8989 + 19.6236i) q^{7} -131.296 q^{9} +O(q^{10})$	$q+14.5704i q^{3} +11.1803i q^{5} +(-44.8989 + 19.6236i) q^{7} -131.296 q^{9} -201.717 q^{11} +258.711i q^{13} -162.902 q^{15} +242.930i q^{17} +253.614i q^{19} +(-285.923 - 654.195i) q^{21} +323.663 q^{23} -125.000 q^{25} -732.838i q^{27} -331.107 q^{29} +278.747i q^{31} -2939.09i q^{33} +(-219.398 - 501.985i) q^{35} -322.518 q^{37} -3769.52 q^{39} -578.646i q^{41} +1524.82 q^{43} -1467.94i q^{45} -389.432i q^{47} +(1630.83 - 1762.16i) q^{49} -3539.59 q^{51} +1071.73 q^{53} -2255.26i q^{55} -3695.26 q^{57} +1030.67i q^{59} +6681.90i q^{61} +(5895.07 - 2576.51i) q^{63} -2892.47 q^{65} +4053.08 q^{67} +4715.90i q^{69} -3210.95 q^{71} -2238.66i q^{73} -1821.30i q^{75} +(9056.86 - 3958.40i) q^{77} +5234.87 q^{79} +42.7274 q^{81} +3799.88i q^{83} -2716.04 q^{85} -4824.37i q^{87} -13693.9i q^{89} +(-5076.83 - 11615.8i) q^{91} -4061.45 q^{93} -2835.49 q^{95} +18557.4i q^{97} +26484.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 50 q^{7} - 434 q^{9}+O(q^{10}) $ 12 * q + 50 * q^7 - 434 * q^9	$ 12 q + 50 q^{7} - 434 q^{9} - 126 q^{11} - 50 q^{15} - 642 q^{21} + 756 q^{23} - 1500 q^{25} - 2190 q^{29} + 150 q^{35} + 5564 q^{37} - 8634 q^{39} - 3944 q^{43} - 8796 q^{49} - 7206 q^{51} + 11760 q^{53} - 12900 q^{57} + 4310 q^{63} - 750 q^{65} + 24096 q^{67} + 5664 q^{71} + 26904 q^{77} + 1590 q^{79} - 11912 q^{81} + 1050 q^{85} + 7182 q^{91} - 70980 q^{93} + 3000 q^{95} - 23084 q^{99}+O(q^{100}) $ 12 * q + 50 * q^7 - 434 * q^9 - 126 * q^11 - 50 * q^15 - 642 * q^21 + 756 * q^23 - 1500 * q^25 - 2190 * q^29 + 150 * q^35 + 5564 * q^37 - 8634 * q^39 - 3944 * q^43 - 8796 * q^49 - 7206 * q^51 + 11760 * q^53 - 12900 * q^57 + 4310 * q^63 - 750 * q^65 + 24096 * q^67 + 5664 * q^71 + 26904 * q^77 + 1590 * q^79 - 11912 * q^81 + 1050 * q^85 + 7182 * q^91 - 70980 * q^93 + 3000 * q^95 - 23084 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$			14.5704i			1.61893i	0.587166	+	0.809466i	$0.300243\pi$
$3$			14.5704i			1.61893i	−0.587166	+	0.809466i	$0.699757\pi$
$4$		0			0
$5$			11.1803i			0.447214i
$5$			11.1803i			0.447214i
$6$		0			0
$7$	−44.8989	+	19.6236i	−0.916305	+	0.400481i
$7$	−44.8989	+	19.6236i	−0.916305	+	0.400481i
$8$		0			0
$9$	−131.296			−1.62094
$10$		0			0
$11$	−201.717			−1.66708			−0.833539	−	0.552460i	$-0.813689\pi$
$11$	−201.717			−1.66708			−0.833539	+	0.552460i	$0.813689\pi$
$12$		0			0
$13$			258.711i			1.53083i	0.643535	+	0.765416i	$0.277467\pi$
$13$			258.711i			1.53083i	−0.643535	+	0.765416i	$0.722533\pi$
$14$		0			0
$15$	−162.902			−0.724009
$16$		0			0
$17$			242.930i			0.840588i	0.907388	+	0.420294i	$0.138073\pi$
$17$			242.930i			0.840588i	−0.907388	+	0.420294i	$0.861927\pi$
$18$		0			0
$19$			253.614i			0.702533i	0.936276	+	0.351266i	$0.114249\pi$
$19$			253.614i			0.702533i	−0.936276	+	0.351266i	$0.885751\pi$
$20$		0			0
$21$	−285.923	−	654.195i	−0.648352	−	1.48344i
$22$		0			0
$23$	323.663			0.611840			0.305920	−	0.952057i	$-0.401036\pi$
$23$	323.663			0.611840			0.305920	+	0.952057i	$0.401036\pi$
$24$		0			0
$25$	−125.000			−0.200000
$26$		0			0
$27$		−	732.838i		−	1.00526i
$28$		0			0
$29$	−331.107			−0.393707			−0.196853	−	0.980433i	$-0.563072\pi$
$29$	−331.107			−0.393707			−0.196853	+	0.980433i	$0.563072\pi$
$30$		0			0
$31$			278.747i			0.290059i	0.989427	+	0.145030i	$0.0463277\pi$
$31$			278.747i			0.290059i	−0.989427	+	0.145030i	$0.953672\pi$
$32$		0			0
$33$		−	2939.09i		−	2.69889i
$34$		0			0
$35$	−219.398	−	501.985i	−0.179101	−	0.409784i
$36$		0			0
$37$	−322.518			−0.235587			−0.117793	−	0.993038i	$-0.537582\pi$
$37$	−322.518			−0.235587			−0.117793	+	0.993038i	$0.537582\pi$
$38$		0			0
$39$	−3769.52			−2.47831
$40$		0			0
$41$		−	578.646i		−	0.344227i	−0.985077	−	0.172114i	$-0.944940\pi$
$41$		−	578.646i		−	0.344227i	0.985077	−	0.172114i	$-0.0550596\pi$
$42$		0			0
$43$	1524.82			0.824674			0.412337	−	0.911031i	$-0.364713\pi$
$43$	1524.82			0.824674			0.412337	+	0.911031i	$0.364713\pi$
$44$		0			0
$45$		−	1467.94i		−	0.724908i
$46$		0			0
$47$		−	389.432i		−	0.176293i	−0.996108	−	0.0881467i	$-0.971906\pi$
$47$		−	389.432i		−	0.176293i	0.996108	−	0.0881467i	$-0.0280944\pi$
$48$		0			0
$49$	1630.83	−	1762.16i	0.679229	−	0.733926i
$50$		0			0
$51$	−3539.59			−1.36086
$52$		0			0
$53$	1071.73			0.381535			0.190768	−	0.981635i	$-0.438902\pi$
$53$	1071.73			0.381535			0.190768	+	0.981635i	$0.438902\pi$
$54$		0			0
$55$		−	2255.26i		−	0.745540i
$56$		0			0
$57$	−3695.26			−1.13735
$58$		0			0
$59$			1030.67i			0.296085i	0.988981	+	0.148042i	$0.0472972\pi$
$59$			1030.67i			0.296085i	−0.988981	+	0.148042i	$0.952703\pi$
$60$		0			0
$61$			6681.90i			1.79573i	0.440273	+	0.897864i	$0.354882\pi$
$61$			6681.90i			1.79573i	−0.440273	+	0.897864i	$0.645118\pi$
$62$		0			0
$63$	5895.07	−	2576.51i	1.48528	−	0.649157i
$64$		0			0
$65$	−2892.47			−0.684609
$66$		0			0
$67$	4053.08			0.902892			0.451446	−	0.892299i	$-0.350908\pi$
$67$	4053.08			0.902892			0.451446	+	0.892299i	$0.350908\pi$
$68$		0			0
$69$			4715.90i			0.990527i
$70$		0			0
$71$	−3210.95			−0.636967			−0.318483	−	0.947928i	$-0.603173\pi$
$71$	−3210.95			−0.636967			−0.318483	+	0.947928i	$0.603173\pi$
$72$		0			0
$73$		−	2238.66i		−	0.420091i	−0.977692	−	0.210045i	$-0.932639\pi$
$73$		−	2238.66i		−	0.420091i	0.977692	−	0.210045i	$-0.0673612\pi$
$74$		0			0
$75$		−	1821.30i		−	0.323787i
$76$		0			0
$77$	9056.86	−	3958.40i	1.52755	−	0.667634i
$78$		0			0
$79$	5234.87			0.838787			0.419393	−	0.907805i	$-0.362243\pi$
$79$	5234.87			0.838787			0.419393	+	0.907805i	$0.362243\pi$
$80$		0			0
$81$	42.7274			0.00651233
$82$		0			0
$83$			3799.88i			0.551587i	0.961217	+	0.275793i	$0.0889405\pi$
$83$			3799.88i			0.551587i	−0.961217	+	0.275793i	$0.911059\pi$
$84$		0			0
$85$	−2716.04			−0.375923
$86$		0			0
$87$		−	4824.37i		−	0.637385i
$88$		0			0
$89$		−	13693.9i		−	1.72881i	−0.502799	−	0.864403i	$-0.667696\pi$
$89$		−	13693.9i		−	1.72881i	0.502799	−	0.864403i	$-0.332304\pi$
$90$		0			0
$91$	−5076.83	−	11615.8i	−0.613070	−	1.40271i
$92$		0			0
$93$	−4061.45			−0.469586
$94$		0			0
$95$	−2835.49			−0.314182
$96$		0			0
$97$			18557.4i			1.97230i	0.165847	+	0.986152i	$0.446964\pi$
$97$			18557.4i			1.97230i	−0.165847	+	0.986152i	$0.553036\pi$
$98$		0			0
$99$	26484.6			2.70224
$100$		0			0
$101$		−	3882.89i		−	0.380639i	−0.981722	−	0.190319i	$-0.939048\pi$
$101$		−	3882.89i		−	0.380639i	0.981722	−	0.190319i	$-0.0609523\pi$
$102$		0			0
$103$		−	5551.06i		−	0.523241i	−0.965171	−	0.261620i	$-0.915743\pi$
$103$		−	5551.06i		−	0.523241i	0.965171	−	0.261620i	$-0.0842569\pi$
$104$		0			0
$105$	7314.12	−	3196.72i	0.663413	−	0.289952i
$106$		0			0
$107$	1207.91			0.105504			0.0527518	−	0.998608i	$-0.483201\pi$
$107$	1207.91			0.105504			0.0527518	+	0.998608i	$0.483201\pi$
$108$		0			0
$109$	4780.21			0.402341			0.201170	−	0.979556i	$-0.435526\pi$
$109$	4780.21			0.402341			0.201170	+	0.979556i	$0.435526\pi$
$110$		0			0
$111$		−	4699.22i		−	0.381399i
$112$		0			0
$113$	17585.2			1.37718			0.688590	−	0.725151i	$-0.258230\pi$
$113$	17585.2			1.37718			0.688590	+	0.725151i	$0.258230\pi$
$114$		0			0
$115$			3618.66i			0.273623i
$116$		0			0
$117$		−	33967.8i		−	2.48139i
$118$		0			0
$119$	−4767.16	−	10907.3i	−0.336640	−	0.770235i
$120$		0			0
$121$	26048.6			1.77915
$122$		0			0
$123$	8431.10			0.557281
$124$		0			0
$125$		−	1397.54i		−	0.0894427i
$126$		0			0
$127$	−12120.0			−0.751439			−0.375720	−	0.926733i	$-0.622604\pi$
$127$	−12120.0			−0.751439			−0.375720	+	0.926733i	$0.622604\pi$
$128$		0			0
$129$			22217.2i			1.33509i
$130$		0			0
$131$			30779.1i			1.79355i	0.442486	+	0.896776i	$0.354097\pi$
$131$			30779.1i			1.79355i	−0.442486	+	0.896776i	$0.645903\pi$
$132$		0			0
$133$	−4976.82	−	11387.0i	−0.281351	−	0.643734i
$134$		0			0
$135$	8193.37			0.449568
$136$		0			0
$137$	−25010.4			−1.33254			−0.666268	−	0.745712i	$-0.732109\pi$
$137$	−25010.4			−1.33254			−0.666268	+	0.745712i	$0.732109\pi$
$138$		0			0
$139$			5006.55i			0.259125i	0.991571	+	0.129562i	$0.0413573\pi$
$139$			5006.55i			0.259125i	−0.991571	+	0.129562i	$0.958643\pi$
$140$		0			0
$141$	5674.18			0.285407
$142$		0			0
$143$		−	52186.2i		−	2.55202i
$144$		0			0
$145$		−	3701.89i		−	0.176071i
$146$		0			0
$147$	25675.3	+	23761.8i	1.18818	+	1.09963i
$148$		0			0
$149$	26261.1			1.18288			0.591439	−	0.806350i	$-0.298560\pi$
$149$	26261.1			1.18288			0.591439	+	0.806350i	$0.298560\pi$
$150$		0			0
$151$	−14133.8			−0.619876			−0.309938	−	0.950757i	$-0.600308\pi$
$151$	−14133.8			−0.619876			−0.309938	+	0.950757i	$0.600308\pi$
$152$		0			0
$153$		−	31895.8i		−	1.36255i
$154$		0			0
$155$	−3116.48			−0.129718
$156$		0			0
$157$		−	17409.6i		−	0.706300i	−0.935567	−	0.353150i	$-0.885111\pi$
$157$		−	17409.6i		−	0.706300i	0.935567	−	0.353150i	$-0.114889\pi$
$158$		0			0
$159$			15615.6i			0.617680i
$160$		0			0
$161$	−14532.1	+	6351.43i	−0.560632	+	0.245030i
$162$		0			0
$163$	41569.3			1.56458			0.782290	−	0.622915i	$-0.214052\pi$
$163$	41569.3			1.56458			0.782290	+	0.622915i	$0.214052\pi$
$164$		0			0
$165$	32860.0			1.20698
$166$		0			0
$167$			16314.1i			0.584967i	0.956271	+	0.292483i	$0.0944817\pi$
$167$			16314.1i			0.584967i	−0.956271	+	0.292483i	$0.905518\pi$
$168$		0			0
$169$	−38370.2			−1.34345
$170$		0			0
$171$		−	33298.6i		−	1.13877i
$172$		0			0
$173$		−	23077.3i		−	0.771069i	−0.922693	−	0.385535i	$-0.874017\pi$
$173$		−	23077.3i		−	0.771069i	0.922693	−	0.385535i	$-0.125983\pi$
$174$		0			0
$175$	5612.37	−	2452.95i	0.183261	−	0.0800963i
$176$		0			0
$177$	−15017.3			−0.479341
$178$		0			0
$179$	−4029.02			−0.125746			−0.0628729	−	0.998022i	$-0.520026\pi$
$179$	−4029.02			−0.125746			−0.0628729	+	0.998022i	$0.520026\pi$
$180$		0			0
$181$			4726.94i			0.144286i	0.997394	+	0.0721428i	$0.0229837\pi$
$181$			4726.94i			0.144286i	−0.997394	+	0.0721428i	$0.977016\pi$
$182$		0			0
$183$	−97358.0			−2.90716
$184$		0			0
$185$		−	3605.87i		−	0.105358i
$186$		0			0
$187$		−	49003.0i		−	1.40133i
$188$		0			0
$189$	14380.9	+	32903.6i	0.402590	+	0.921128i
$190$		0			0
$191$	−99.8028			−0.00273575			−0.00136787	−	0.999999i	$-0.500435\pi$
$191$	−99.8028			−0.00273575			−0.00136787	+	0.999999i	$0.500435\pi$
$192$		0			0
$193$	−27603.3			−0.741047			−0.370524	−	0.928823i	$-0.620822\pi$
$193$	−27603.3			−0.741047			−0.370524	+	0.928823i	$0.620822\pi$
$194$		0			0
$195$		−	42144.5i		−	1.10834i
$196$		0			0
$197$	−17085.7			−0.440251			−0.220125	−	0.975472i	$-0.570647\pi$
$197$	−17085.7			−0.440251			−0.220125	+	0.975472i	$0.570647\pi$
$198$		0			0
$199$		−	12618.2i		−	0.318634i	−0.987227	−	0.159317i	$-0.949071\pi$
$199$		−	12618.2i		−	0.318634i	0.987227	−	0.159317i	$-0.0509292\pi$
$200$		0			0
$201$			59055.0i			1.46172i
$202$		0			0
$203$	14866.4	−	6497.52i	0.360756	−	0.157672i
$204$		0			0
$205$	6469.46			0.153943
$206$		0			0
$207$	−42495.8			−0.991757
$208$		0			0
$209$		−	51158.2i		−	1.17118i
$210$		0			0
$211$	−27000.1			−0.606458			−0.303229	−	0.952918i	$-0.598065\pi$
$211$	−27000.1			−0.606458			−0.303229	+	0.952918i	$0.598065\pi$
$212$		0			0
$213$		−	46784.8i		−	1.03121i
$214$		0			0
$215$			17048.0i			0.368805i
$216$		0			0
$217$	−5470.01	−	12515.4i	−0.116163	−	0.265783i
$218$		0			0
$219$	32618.2			0.680099
$220$		0			0
$221$	−62848.6			−1.28680
$222$		0			0
$223$		−	63568.7i		−	1.27830i	−0.769081	−	0.639152i	$-0.779286\pi$
$223$		−	63568.7i		−	1.27830i	0.769081	−	0.639152i	$-0.220714\pi$
$224$		0			0
$225$	16412.0			0.324189
$226$		0			0
$227$			74171.9i			1.43942i	0.694275	+	0.719710i	$0.255725\pi$
$227$			74171.9i			1.43942i	−0.694275	+	0.719710i	$0.744275\pi$
$228$		0			0
$229$		−	79719.2i		−	1.52017i	−0.649824	−	0.760084i	$-0.725158\pi$
$229$		−	79719.2i		−	1.52017i	0.649824	−	0.760084i	$-0.274842\pi$
$230$		0			0
$231$	57675.5	+	131962.i	1.08085	+	2.47300i
$232$		0			0
$233$	34612.6			0.637562			0.318781	−	0.947828i	$-0.396727\pi$
$233$	34612.6			0.637562			0.318781	+	0.947828i	$0.396727\pi$
$234$		0			0
$235$	4353.98			0.0788408
$236$		0			0
$237$			76274.1i			1.35794i
$238$		0			0
$239$	−76161.8			−1.33334			−0.666671	−	0.745352i	$-0.732281\pi$
$239$	−76161.8			−1.33334			−0.666671	+	0.745352i	$0.732281\pi$
$240$		0			0
$241$			57912.8i			0.997103i	0.866860	+	0.498552i	$0.166135\pi$
$241$			57912.8i			0.997103i	−0.866860	+	0.498552i	$0.833865\pi$
$242$		0			0
$243$		−	58737.3i		−	0.994721i
$244$		0			0
$245$	19701.5	+	18233.2i	0.328222	+	0.303761i
$246$		0			0
$247$	−65612.7			−1.07546
$248$		0			0
$249$	−55365.8			−0.892982
$250$		0			0
$251$			36981.7i			0.587002i	0.955959	+	0.293501i	$0.0948203\pi$
$251$			36981.7i			0.587002i	−0.955959	+	0.293501i	$0.905180\pi$
$252$		0			0
$253$	−65288.2			−1.01998
$254$		0			0
$255$		−	39573.8i		−	0.608593i
$256$		0			0
$257$			75229.5i			1.13900i	0.821993	+	0.569498i	$0.192862\pi$
$257$			75229.5i			1.13900i	−0.821993	+	0.569498i	$0.807138\pi$
$258$		0			0
$259$	14480.7	−	6328.97i	0.215869	−	0.0943482i
$260$		0			0
$261$	43473.2			0.638176
$262$		0			0
$263$	31641.9			0.457458			0.228729	−	0.973490i	$-0.426543\pi$
$263$	31641.9			0.457458			0.228729	+	0.973490i	$0.426543\pi$
$264$		0			0
$265$			11982.3i			0.170628i
$266$		0			0
$267$	199525.			2.79882
$268$		0			0
$269$			79641.8i			1.10062i	0.834961	+	0.550309i	$0.185490\pi$
$269$			79641.8i			1.10062i	−0.834961	+	0.550309i	$0.814510\pi$
$270$		0			0
$271$			40912.1i			0.557075i	0.960425	+	0.278537i	$0.0898496\pi$
$271$			40912.1i			0.557075i	−0.960425	+	0.278537i	$0.910150\pi$
$272$		0			0
$273$	169247.	−	73971.5i	2.27089	−	0.992519i
$274$		0			0
$275$	25214.6			0.333416
$276$		0			0
$277$	−120604.			−1.57182			−0.785911	−	0.618340i	$-0.787806\pi$
$277$	−120604.			−1.57182			−0.785911	+	0.618340i	$0.787806\pi$
$278$		0			0
$279$		−	36598.4i		−	0.470169i
$280$		0			0
$281$	−48206.6			−0.610512			−0.305256	−	0.952270i	$-0.598742\pi$
$281$	−48206.6			−0.610512			−0.305256	+	0.952270i	$0.598742\pi$
$282$		0			0
$283$		−	45809.5i		−	0.571982i	−0.958232	−	0.285991i	$-0.907677\pi$
$283$		−	45809.5i		−	0.571982i	0.958232	−	0.285991i	$-0.0923228\pi$
$284$		0			0
$285$		−	41314.3i		−	0.508640i
$286$		0			0
$287$	11355.1	+	25980.6i	0.137857	+	0.315417i
$288$		0			0
$289$	24506.0			0.293411
$290$		0			0
$291$	−270389.			−3.19303
$292$		0			0
$293$		−	73600.8i		−	0.857328i	−0.903464	−	0.428664i	$-0.858984\pi$
$293$		−	73600.8i		−	0.857328i	0.903464	−	0.428664i	$-0.141016\pi$
$294$		0			0
$295$	−11523.2			−0.132413
$296$		0			0
$297$			147825.i			1.67585i
$298$		0			0
$299$			83735.1i			0.936624i
$300$		0			0
$301$	−68462.9	+	29922.5i	−0.755652	+	0.330266i
$302$		0			0
$303$	56575.3			0.616228
$304$		0			0
$305$	−74706.0			−0.803074
$306$		0			0
$307$		−	4729.94i		−	0.0501856i	−0.999685	−	0.0250928i	$-0.992012\pi$
$307$		−	4729.94i		−	0.0501856i	0.999685	−	0.0250928i	$-0.00798812\pi$
$308$		0			0
$309$	80881.2			0.847092
$310$		0			0
$311$		−	90455.7i		−	0.935223i	−0.883934	−	0.467611i	$-0.845115\pi$
$311$		−	90455.7i		−	0.935223i	0.883934	−	0.467611i	$-0.154885\pi$
$312$		0			0
$313$			108219.i			1.10462i	0.833638	+	0.552312i	$0.186254\pi$
$313$			108219.i			1.10462i	−0.833638	+	0.552312i	$0.813746\pi$
$314$		0			0
$315$	28806.2	+	65908.9i	0.290312	+	0.664236i
$316$		0			0
$317$	117203.			1.16633			0.583163	−	0.812356i	$-0.301815\pi$
$317$	117203.			1.16633			0.583163	+	0.812356i	$0.301815\pi$
$318$		0			0
$319$	66789.9			0.656340
$320$		0			0
$321$			17599.7i			0.170803i
$322$		0			0
$323$	−61610.5			−0.590541
$324$		0			0
$325$		−	32338.8i		−	0.306167i
$326$		0			0
$327$			69649.5i			0.651362i
$328$		0			0
$329$	7642.06	+	17485.1i	0.0706022	+	0.161538i
$330$		0			0
$331$	59340.4			0.541619			0.270810	−	0.962633i	$-0.412709\pi$
$331$	59340.4			0.541619			0.270810	+	0.962633i	$0.412709\pi$
$332$		0			0
$333$	42345.5			0.381873
$334$		0			0
$335$			45314.8i			0.403785i
$336$		0			0
$337$	157384.			1.38580			0.692902	−	0.721032i	$-0.256332\pi$
$337$	157384.			1.38580			0.692902	+	0.721032i	$0.256332\pi$
$338$		0			0
$339$			256223.i			2.22956i
$340$		0			0
$341$		−	56227.8i		−	0.483551i
$342$		0			0
$343$	−38642.7	+	111122.i	−0.328457	+	0.944519i
$344$		0			0
$345$	−52725.4			−0.442977
$346$		0			0
$347$	−165575.			−1.37510			−0.687551	−	0.726136i	$-0.741314\pi$
$347$	−165575.			−1.37510			−0.687551	+	0.726136i	$0.741314\pi$
$348$		0			0
$349$			187470.i			1.53915i	0.638558	+	0.769574i	$0.279531\pi$
$349$			187470.i			1.53915i	−0.638558	+	0.769574i	$0.720469\pi$
$350$		0			0
$351$	189593.			1.53889
$352$		0			0
$353$			132192.i			1.06085i	0.847731	+	0.530426i	$0.177968\pi$
$353$			132192.i			1.06085i	−0.847731	+	0.530426i	$0.822032\pi$
$354$		0			0
$355$		−	35899.5i		−	0.284860i
$356$		0			0
$357$	158924.	−	69459.4i	1.24696	−	0.544998i
$358$		0			0
$359$	216123.			1.67692			0.838459	−	0.544965i	$-0.183457\pi$
$359$	216123.			1.67692			0.838459	+	0.544965i	$0.183457\pi$
$360$		0			0
$361$	66000.8			0.506448
$362$		0			0
$363$			379538.i			2.88033i
$364$		0			0
$365$	25029.0			0.187870
$366$		0			0
$367$			36676.9i			0.272308i	0.990688	+	0.136154i	$0.0434742\pi$
$367$			36676.9i			0.272308i	−0.990688	+	0.136154i	$0.956526\pi$
$368$		0			0
$369$			75974.1i			0.557973i
$370$		0			0
$371$	−48119.6	+	21031.2i	−0.349602	+	0.152798i
$372$		0			0
$373$	−250019.			−1.79703			−0.898517	−	0.438940i	$-0.855354\pi$
$373$	−250019.			−1.79703			−0.898517	+	0.438940i	$0.855354\pi$
$374$		0			0
$375$	20362.7			0.144802
$376$		0			0
$377$		−	85661.1i		−	0.602699i
$378$		0			0
$379$	−159502.			−1.11042			−0.555211	−	0.831710i	$-0.687362\pi$
$379$	−159502.			−1.11042			−0.555211	+	0.831710i	$0.687362\pi$
$380$		0			0
$381$		−	176593.i		−	1.21653i
$382$		0			0
$383$		−	237902.i		−	1.62181i	−0.585178	−	0.810905i	$-0.698975\pi$
$383$		−	237902.i		−	1.62181i	0.585178	−	0.810905i	$-0.301025\pi$
$384$		0			0
$385$	44256.3	+	101259.i	0.298575	+	0.683142i
$386$		0			0
$387$	−200203.			−1.33675
$388$		0			0
$389$	−487.890			−0.00322420			−0.00161210	−	0.999999i	$-0.500513\pi$
$389$	−487.890			−0.00322420			−0.00161210	+	0.999999i	$0.500513\pi$
$390$		0			0
$391$			78627.5i			0.514305i
$392$		0			0
$393$	−448464.			−2.90364
$394$		0			0
$395$			58527.6i			0.375117i
$396$		0			0
$397$		−	213503.i		−	1.35464i	−0.735689	−	0.677319i	$-0.763142\pi$
$397$		−	213503.i		−	1.35464i	0.735689	−	0.677319i	$-0.236858\pi$
$398$		0			0
$399$	165913.	−	72514.3i	1.04216	−	0.455489i
$400$		0			0
$401$	−195733.			−1.21724			−0.608619	−	0.793463i	$-0.708276\pi$
$401$	−195733.			−1.21724			−0.608619	+	0.793463i	$0.708276\pi$
$402$		0			0
$403$	−72114.8			−0.444032
$404$		0			0
$405$			477.707i			0.00291240i
$406$		0			0
$407$	65057.3			0.392742
$408$		0			0
$409$			61109.4i			0.365310i	0.983177	+	0.182655i	$0.0584692\pi$
$409$			61109.4i			0.365310i	−0.983177	+	0.182655i	$0.941531\pi$
$410$		0			0
$411$		−	364411.i		−	2.15729i
$412$		0			0
$413$	−20225.5	−	46276.0i	−0.118576	−	0.271304i
$414$		0			0
$415$	−42484.0			−0.246677
$416$		0			0
$417$	−72947.4			−0.419506
$418$		0			0
$419$		−	131493.i		−	0.748988i	−0.927229	−	0.374494i	$-0.877816\pi$
$419$		−	131493.i		−	0.748988i	0.927229	−	0.374494i	$-0.122184\pi$
$420$		0			0
$421$	−35202.6			−0.198615			−0.0993073	−	0.995057i	$-0.531663\pi$
$421$	−35202.6			−0.198615			−0.0993073	+	0.995057i	$0.531663\pi$
$422$		0			0
$423$			51131.0i			0.285761i
$424$		0			0
$425$		−	30366.3i		−	0.168118i
$426$		0			0
$427$	−131123.	−	300010.i	−0.719156	−	1.64543i
$428$		0			0
$429$	760374.			4.13155
$430$		0			0
$431$	−285348.			−1.53610			−0.768052	−	0.640388i	$-0.778774\pi$
$431$	−285348.			−1.53610			−0.768052	+	0.640388i	$0.778774\pi$
$432$		0			0
$433$			255078.i			1.36050i	0.732982	+	0.680248i	$0.238128\pi$
$433$			255078.i			1.36050i	−0.732982	+	0.680248i	$0.761872\pi$
$434$		0			0
$435$	53938.1			0.285047
$436$		0			0
$437$			82085.6i			0.429837i
$438$		0			0
$439$			190309.i			0.987482i	0.869609	+	0.493741i	$0.164371\pi$
$439$			190309.i			0.987482i	−0.869609	+	0.493741i	$0.835629\pi$
$440$		0			0
$441$	−214122.	+	231365.i	−1.10099	+	1.18965i
$442$		0			0
$443$	−119088.			−0.606822			−0.303411	−	0.952860i	$-0.598125\pi$
$443$	−119088.			−0.606822			−0.303411	+	0.952860i	$0.598125\pi$
$444$		0			0
$445$	153102.			0.773146
$446$		0			0
$447$			382634.i			1.91500i
$448$		0			0
$449$	170376.			0.845114			0.422557	−	0.906336i	$-0.361133\pi$
$449$	170376.			0.845114			0.422557	+	0.906336i	$0.361133\pi$
$450$		0			0
$451$			116722.i			0.573854i
$452$		0			0
$453$		−	205935.i		−	1.00354i
$454$		0			0
$455$	129869.	−	56760.7i	0.627311	−	0.274173i
$456$		0			0
$457$	−341751.			−1.63635			−0.818176	−	0.574967i	$-0.805015\pi$
$457$	−341751.			−1.63635			−0.818176	+	0.574967i	$0.805015\pi$
$458$		0			0
$459$	178028.			0.845013
$460$		0			0
$461$		−	220317.i		−	1.03668i	−0.855174	−	0.518342i	$-0.826550\pi$
$461$		−	220317.i		−	1.03668i	0.855174	−	0.518342i	$-0.173450\pi$
$462$		0			0
$463$	−334907.			−1.56229			−0.781145	−	0.624349i	$-0.785364\pi$
$463$	−334907.			−1.56229			−0.781145	+	0.624349i	$0.785364\pi$
$464$		0			0
$465$		−	45408.4i		−	0.210005i
$466$		0			0
$467$		−	331727.i		−	1.52106i	−0.649302	−	0.760531i	$-0.724939\pi$
$467$		−	331727.i		−	1.52106i	0.649302	−	0.760531i	$-0.275061\pi$
$468$		0			0
$469$	−181979.	+	79536.0i	−0.827324	+	0.361591i
$470$		0			0
$471$	253665.			1.14345
$472$		0			0
$473$	−307582.			−1.37480
$474$		0			0
$475$		−	31701.8i		−	0.140507i
$476$		0			0
$477$	−140715.			−0.618447
$478$		0			0
$479$		−	339790.i		−	1.48095i	−0.672085	−	0.740474i	$-0.734601\pi$
$479$		−	339790.i		−	1.48095i	0.672085	−	0.740474i	$-0.265399\pi$
$480$		0			0
$481$		−	83439.0i		−	0.360644i
$482$		0			0
$483$	−92542.9	−	211739.i	−0.396688	−	0.907625i
$484$		0			0
$485$	−207478.			−0.882041
$486$		0			0
$487$	−74140.1			−0.312605			−0.156302	−	0.987709i	$-0.549957\pi$
$487$	−74140.1			−0.312605			−0.156302	+	0.987709i	$0.549957\pi$
$488$		0			0
$489$			605681.i			2.53295i
$490$		0			0
$491$	1259.76			0.00522547			0.00261273	−	0.999997i	$-0.499168\pi$
$491$	1259.76			0.00522547			0.00261273	+	0.999997i	$0.499168\pi$
$492$		0			0
$493$		−	80436.0i		−	0.330945i
$494$		0			0
$495$			296107.i			1.20848i
$496$		0			0
$497$	144168.	−	63010.4i	0.583656	−	0.255093i
$498$		0			0
$499$	77913.9			0.312906			0.156453	−	0.987685i	$-0.449994\pi$
$499$	77913.9			0.312906			0.156453	+	0.987685i	$0.449994\pi$
$500$		0			0
$501$	−237703.			−0.947022
$502$		0			0
$503$		−	221665.i		−	0.876116i	−0.898947	−	0.438058i	$-0.855666\pi$
$503$		−	221665.i		−	0.876116i	0.898947	−	0.438058i	$-0.144334\pi$
$504$		0			0
$505$	43412.1			0.170227
$506$		0			0
$507$		−	559069.i		−	2.17495i
$508$		0			0
$509$		−	339130.i		−	1.30897i	−0.756074	−	0.654486i	$-0.772885\pi$
$509$		−	339130.i		−	1.30897i	0.756074	−	0.654486i	$-0.227115\pi$
$510$		0			0
$511$	43930.6	+	100514.i	0.168239	+	0.384931i
$512$		0			0
$513$	185858.			0.706231
$514$		0			0
$515$	62062.8			0.234000
$516$		0			0
$517$			78554.9i			0.293895i
$518$		0			0
$519$	336246.			1.24831
$520$		0			0
$521$		−	342794.i		−	1.26287i	−0.775430	−	0.631433i	$-0.782467\pi$
$521$		−	342794.i		−	1.26287i	0.775430	−	0.631433i	$-0.217533\pi$
$522$		0			0
$523$			174970.i			0.639676i	0.947472	+	0.319838i	$0.103628\pi$
$523$			174970.i			0.639676i	−0.947472	+	0.319838i	$0.896372\pi$
$524$		0			0
$525$	35740.4	+	81774.4i	0.129670	+	0.296687i
$526$		0			0
$527$	−67716.0			−0.243820
$528$		0			0
$529$	−175083.			−0.625652
$530$		0			0
$531$		−	135323.i		−	0.479936i
$532$		0			0
$533$	149702.			0.526954
$534$		0			0
$535$			13504.8i			0.0471826i
$536$		0			0
$537$		−	58704.4i		−	0.203574i
$538$		0			0
$539$	−328965.	+	355456.i	−1.13233	+	1.22351i
$540$		0			0
$541$	316083.			1.07996			0.539978	−	0.841679i	$-0.318433\pi$
$541$	316083.			1.07996			0.539978	+	0.841679i	$0.318433\pi$
$542$		0			0
$543$	−68873.4			−0.233589
$544$		0			0
$545$			53444.4i			0.179932i
$546$		0			0
$547$	−94365.5			−0.315383			−0.157692	−	0.987488i	$-0.550405\pi$
$547$	−94365.5			−0.315383			−0.157692	+	0.987488i	$0.550405\pi$
$548$		0			0
$549$		−	877310.i		−	2.91077i
$550$		0			0
$551$		−	83973.6i		−	0.276592i
$552$		0			0
$553$	−235040.	+	102727.i	−0.768584	+	0.335918i
$554$		0			0
$555$	52538.9			0.170567
$556$		0			0
$557$	−135243.			−0.435916			−0.217958	−	0.975958i	$-0.569940\pi$
$557$	−135243.			−0.435916			−0.217958	+	0.975958i	$0.569940\pi$
$558$		0			0
$559$			394488.i			1.26244i
$560$		0			0
$561$	713993.			2.26865
$562$		0			0
$563$			243317.i			0.767637i	0.923408	+	0.383819i	$0.125391\pi$
$563$			243317.i			0.767637i	−0.923408	+	0.383819i	$0.874609\pi$
$564$		0			0
$565$			196609.i			0.615894i
$566$		0			0
$567$	−1918.41	+	838.465i	−0.00596728	+	0.00260807i
$568$		0			0
$569$	139613.			0.431222			0.215611	−	0.976479i	$-0.430826\pi$
$569$	139613.			0.431222			0.215611	+	0.976479i	$0.430826\pi$
$570$		0			0
$571$	548346.			1.68183			0.840915	−	0.541167i	$-0.182017\pi$
$571$	548346.			1.68183			0.840915	+	0.541167i	$0.182017\pi$
$572$		0			0
$573$		−	1454.17i		−	0.00442899i
$574$		0			0
$575$	−40457.9			−0.122368
$576$		0			0
$577$		−	445087.i		−	1.33688i	−0.743764	−	0.668442i	$-0.766962\pi$
$577$		−	445087.i		−	1.33688i	0.743764	−	0.668442i	$-0.233038\pi$
$578$		0			0
$579$		−	402190.i		−	1.19971i
$580$		0			0
$581$	−74567.3	−	170611.i	−0.220900	−	0.505422i
$582$		0			0
$583$	−216186.			−0.636049
$584$		0			0
$585$	379771.			1.10971
$586$		0			0
$587$		−	143426.i		−	0.416248i	−0.978102	−	0.208124i	$-0.933264\pi$
$587$		−	143426.i		−	0.416248i	0.978102	−	0.208124i	$-0.0667358\pi$
$588$		0			0
$589$	−70694.2			−0.203776
$590$		0			0
$591$		−	248945.i		−	0.712736i
$592$		0			0
$593$			299727.i			0.852347i	0.904642	+	0.426173i	$0.140139\pi$
$593$			299727.i			0.852347i	−0.904642	+	0.426173i	$0.859861\pi$
$594$		0			0
$595$	121947.	−	53298.5i	0.344460	−	0.150550i
$596$		0			0
$597$	183852.			0.515847
$598$		0			0
$599$	439671.			1.22539			0.612694	−	0.790320i	$-0.290086\pi$
$599$	439671.			1.22539			0.612694	+	0.790320i	$0.290086\pi$
$600$		0			0
$601$		−	188484.i		−	0.521827i	−0.965362	−	0.260913i	$-0.915976\pi$
$601$		−	188484.i		−	0.521827i	0.965362	−	0.260913i	$-0.0840237\pi$
$602$		0			0
$603$	−532155.			−1.46354
$604$		0			0
$605$			291232.i			0.795661i
$606$		0			0
$607$			10230.2i			0.0277656i	0.999904	+	0.0138828i	$0.00441917\pi$
$607$			10230.2i			0.0277656i	−0.999904	+	0.0138828i	$0.995581\pi$
$608$		0			0
$609$	94671.4	+	216609.i	0.255261	+	0.584039i
$610$		0			0
$611$	100750.			0.269876
$612$		0			0
$613$	436819.			1.16247			0.581233	−	0.813737i	$-0.302571\pi$
$613$	436819.			1.16247			0.581233	+	0.813737i	$0.302571\pi$
$614$		0			0
$615$			94262.5i			0.249223i
$616$		0			0
$617$	26556.4			0.0697588			0.0348794	−	0.999392i	$-0.488895\pi$
$617$	26556.4			0.0697588			0.0348794	+	0.999392i	$0.488895\pi$
$618$		0			0
$619$			510737.i			1.33296i	0.745525	+	0.666478i	$0.232199\pi$
$619$			510737.i			1.33296i	−0.745525	+	0.666478i	$0.767801\pi$
$620$		0			0
$621$		−	237193.i		−	0.615061i
$622$		0			0
$623$	268723.	+	614841.i	0.692355	+	1.58411i
$624$		0			0
$625$	15625.0			0.0400000
$626$		0			0
$627$	745395.			1.89606
$628$		0			0
$629$		−	78349.4i		−	0.198032i
$630$		0			0
$631$	3641.38			0.00914550			0.00457275	−	0.999990i	$-0.498544\pi$
$631$	3641.38			0.00914550			0.00457275	+	0.999990i	$0.498544\pi$
$632$		0			0
$633$		−	393403.i		−	0.981815i
$634$		0			0
$635$		−	135505.i		−	0.336054i
$636$		0			0
$637$	455889.	+	421913.i	1.12352	+	1.03979i
$638$		0			0
$639$	421586.			1.03249
$640$		0			0
$641$	330651.			0.804736			0.402368	−	0.915478i	$-0.368187\pi$
$641$	330651.			0.804736			0.402368	+	0.915478i	$0.368187\pi$
$642$		0			0
$643$			75104.8i			0.181654i	0.995867	+	0.0908272i	$0.0289511\pi$
$643$			75104.8i			0.181654i	−0.995867	+	0.0908272i	$0.971049\pi$
$644$		0			0
$645$	−248396.			−0.597071
$646$		0			0
$647$			381648.i			0.911706i	0.890055	+	0.455853i	$0.150666\pi$
$647$			381648.i			0.911706i	−0.890055	+	0.455853i	$0.849334\pi$
$648$		0			0
$649$		−	207903.i		−	0.493596i
$650$		0			0
$651$	182355.	−	79700.3i	0.430284	−	0.188061i
$652$		0			0
$653$	462477.			1.08458			0.542292	−	0.840190i	$-0.317557\pi$
$653$	462477.			1.08458			0.542292	+	0.840190i	$0.317557\pi$
$654$		0			0
$655$	−344121.			−0.802100
$656$		0			0
$657$			293929.i			0.680943i
$658$		0			0
$659$	187187.			0.431028			0.215514	−	0.976501i	$-0.430857\pi$
$659$	187187.			0.431028			0.215514	+	0.976501i	$0.430857\pi$
$660$		0			0
$661$			95871.7i			0.219426i	0.993963	+	0.109713i	$0.0349931\pi$
$661$			95871.7i			0.219426i	−0.993963	+	0.109713i	$0.965007\pi$
$662$		0			0
$663$		−	915729.i		−	2.08324i
$664$		0			0
$665$	127311.	−	55642.6i	0.287887	−	0.125824i
$666$		0			0
$667$	−107167.			−0.240885
$668$		0			0
$669$	926221.			2.06949
$670$		0			0
$671$		−	1.34785e6i		−	2.99362i
$672$		0			0
$673$	−118770.			−0.262226			−0.131113	−	0.991367i	$-0.541855\pi$
$673$	−118770.			−0.262226			−0.131113	+	0.991367i	$0.541855\pi$
$674$		0			0
$675$			91604.7i			0.201053i
$676$		0			0
$677$		−	340700.i		−	0.743352i	−0.928363	−	0.371676i	$-0.878783\pi$
$677$		−	340700.i		−	0.743352i	0.928363	−	0.371676i	$-0.121217\pi$
$678$		0			0
$679$	−364163.	−	833208.i	−0.789871	−	1.80723i
$680$		0			0
$681$	−1.08071e6			−2.33032
$682$		0			0
$683$	148626.			0.318605			0.159303	−	0.987230i	$-0.449075\pi$
$683$	148626.			0.318605			0.159303	+	0.987230i	$0.449075\pi$
$684$		0			0
$685$		−	279624.i		−	0.595928i
$686$		0			0
$687$	1.16154e6			2.46105
$688$		0			0
$689$			277269.i			0.584066i
$690$		0			0
$691$			68328.5i			0.143102i	0.997437	+	0.0715510i	$0.0227949\pi$
$691$			68328.5i			0.143102i	−0.997437	+	0.0715510i	$0.977205\pi$
$692$		0			0
$693$	−1.18913e6	+	519724.i	−2.47607	+	1.08220i
$694$		0			0
$695$	−55974.9			−0.115884
$696$		0			0
$697$	140570.			0.289353
$698$		0			0
$699$			504319.i			1.03217i
$700$		0			0
$701$	−8208.96			−0.0167052			−0.00835261	−	0.999965i	$-0.502659\pi$
$701$	−8208.96			−0.0167052			−0.00835261	+	0.999965i	$0.502659\pi$
$702$		0			0
$703$		−	81795.3i		−	0.165507i
$704$		0			0
$705$			63439.2i			0.127638i
$706$		0			0
$707$	76196.3	+	174338.i	0.152439	+	0.348781i
$708$		0			0
$709$	−678427.			−1.34962			−0.674808	−	0.737993i	$-0.735774\pi$
$709$	−678427.			−1.34962			−0.674808	+	0.737993i	$0.735774\pi$
$710$		0			0
$711$	−687319.			−1.35962
$712$		0			0
$713$			90220.1i			0.177470i
$714$		0			0
$715$	583460.			1.14130
$716$		0			0
$717$		−	1.10971e6i		−	2.15859i
$718$		0			0
$719$		−	589061.i		−	1.13947i	−0.821829	−	0.569735i	$-0.807046\pi$
$719$		−	589061.i		−	1.13947i	0.821829	−	0.569735i	$-0.192954\pi$
$720$		0			0
$721$	108932.	+	249237.i	0.209548	+	0.479448i
$722$		0			0
$723$	−843812.			−1.61424
$724$		0			0
$725$	41388.4			0.0787414
$726$		0			0
$727$			705964.i			1.33571i	0.744289	+	0.667857i	$0.232788\pi$
$727$			705964.i			1.33571i	−0.744289	+	0.667857i	$0.767212\pi$
$728$		0			0
$729$	859286.			1.61690
$730$		0			0
$731$			370425.i			0.693211i
$732$		0			0
$733$		−	318582.i		−	0.592944i	−0.955042	−	0.296472i	$-0.904190\pi$
$733$		−	318582.i		−	0.592944i	0.955042	−	0.296472i	$-0.0958101\pi$
$734$		0			0
$735$	−265665.	+	287059.i	−0.491768	+	0.531369i
$736$		0			0
$737$	−817573.			−1.50519
$738$		0			0
$739$	547274.			1.00211			0.501055	−	0.865415i	$-0.332945\pi$
$739$	547274.			1.00211			0.501055	+	0.865415i	$0.332945\pi$
$740$		0			0
$741$		−	956003.i		−	1.74110i
$742$		0			0
$743$	−233435.			−0.422851			−0.211426	−	0.977394i	$-0.567811\pi$
$743$	−233435.			−0.422851			−0.211426	+	0.977394i	$0.567811\pi$
$744$		0			0
$745$			293608.i			0.528999i
$746$		0			0
$747$		−	498910.i		−	0.894090i
$748$		0			0
$749$	−54233.9	+	23703.5i	−0.0966734	+	0.0422522i
$750$		0			0
$751$	−495140.			−0.877906			−0.438953	−	0.898510i	$-0.644651\pi$
$751$	−495140.			−0.877906			−0.438953	+	0.898510i	$0.644651\pi$
$752$		0			0
$753$	−538838.			−0.950316
$754$		0			0
$755$		−	158021.i		−	0.277217i
$756$		0			0
$757$	124849.			0.217868			0.108934	−	0.994049i	$-0.465256\pi$
$757$	124849.			0.217868			0.108934	+	0.994049i	$0.465256\pi$
$758$		0			0
$759$		−	951275.i		−	1.65129i
$760$		0			0
$761$		−	892015.i		−	1.54029i	−0.637868	−	0.770146i	$-0.720183\pi$
$761$		−	892015.i		−	1.54029i	0.637868	−	0.770146i	$-0.279817\pi$
$762$		0			0
$763$	−214626.	+	93804.9i	−0.368667	+	0.161130i
$764$		0			0
$765$	356606.			0.609349
$766$		0			0
$767$	−266645.			−0.453256
$768$		0			0
$769$			105869.i			0.179026i	0.995986	+	0.0895132i	$0.0285311\pi$
$769$			105869.i			0.179026i	−0.995986	+	0.0895132i	$0.971469\pi$
$770$		0			0
$771$	−1.09612e6			−1.84396
$772$		0			0
$773$			255665.i			0.427870i	0.976848	+	0.213935i	$0.0686281\pi$
$773$			255665.i			0.427870i	−0.976848	+	0.213935i	$0.931372\pi$
$774$		0			0
$775$		−	34843.4i		−	0.0580118i
$776$		0			0
$777$	92215.6	+	210990.i	0.152743	+	0.349478i
$778$		0			0
$779$	146753.			0.241831
$780$		0			0
$781$	647702.			1.06187
$782$		0			0
$783$			242648.i			0.395779i
$784$		0			0
$785$	194645.			0.315867
$786$		0			0
$787$		−	754940.i		−	1.21889i	−0.792830	−	0.609443i	$-0.791393\pi$
$787$		−	754940.i		−	1.21889i	0.792830	−	0.609443i	$-0.208607\pi$
$788$		0			0
$789$			461035.i			0.740594i
$790$		0			0
$791$	−789557.	+	345085.i	−1.26192	+	0.551535i
$792$		0			0
$793$	−1.72868e6			−2.74896
$794$		0			0
$795$	−174587.			−0.276235
$796$		0			0
$797$			360915.i			0.568183i	0.958797	+	0.284091i	$0.0916919\pi$
$797$			360915.i			0.568183i	−0.958797	+	0.284091i	$0.908308\pi$
$798$		0			0
$799$	94604.8			0.148190
$800$		0			0
$801$			1.79796e6i			2.80230i
$802$		0			0
$803$			451576.i			0.700325i
$804$		0			0
$805$	−71011.2	−	162474.i	−0.109581	−	0.250722i
$806$		0			0
$807$	−1.16041e6			−1.78183
$808$		0			0
$809$	52604.1			0.0803754			0.0401877	−	0.999192i	$-0.487204\pi$
$809$	52604.1			0.0803754			0.0401877	+	0.999192i	$0.487204\pi$
$810$		0			0
$811$			284790.i			0.432996i	0.976283	+	0.216498i	$0.0694634\pi$
$811$			284790.i			0.432996i	−0.976283	+	0.216498i	$0.930537\pi$
$812$		0			0
$813$	−596106.			−0.901867
$814$		0			0
$815$			464759.i			0.699701i
$816$		0			0
$817$			386716.i			0.579360i
$818$		0			0
$819$	666570.	+	1.52512e6i	0.993751	+	2.27371i
$820$		0			0
$821$	−459269.			−0.681366			−0.340683	−	0.940178i	$-0.610658\pi$
$821$	−459269.			−0.681366			−0.340683	+	0.940178i	$0.610658\pi$
$822$		0			0
$823$	750048.			1.10736			0.553680	−	0.832729i	$-0.313223\pi$
$823$	750048.			1.10736			0.553680	+	0.832729i	$0.313223\pi$
$824$		0			0
$825$			367386.i			0.539778i
$826$		0			0
$827$	892076.			1.30434			0.652170	−	0.758072i	$-0.273859\pi$
$827$	892076.			1.30434			0.652170	+	0.758072i	$0.273859\pi$
$828$		0			0
$829$			965908.i			1.40549i	0.711443	+	0.702743i	$0.248042\pi$
$829$			965908.i			1.40549i	−0.711443	+	0.702743i	$0.751958\pi$
$830$		0			0
$831$		−	1.75725e6i		−	2.54467i
$832$		0			0
$833$	428081.	+	396177.i	0.616930	+	0.570952i
$834$		0			0
$835$	−182398.			−0.261605
$836$		0			0
$837$	204276.			0.291586
$838$		0			0
$839$			1.19914e6i			1.70351i	0.523938	+	0.851757i	$0.324463\pi$
$839$			1.19914e6i			1.70351i	−0.523938	+	0.851757i	$0.675537\pi$
$840$		0			0
$841$	−597649.			−0.844995
$842$		0			0
$843$		−	702389.i		−	0.988377i
$844$		0			0
$845$		−	428992.i		−	0.600809i
$846$		0			0
$847$	−1.16955e6	+	511166.i	−1.63025	+	0.712517i
$848$		0			0
$849$	667462.			0.926000
$850$		0			0
$851$	−104387.			−0.144141
$852$		0			0
$853$		−	517748.i		−	0.711575i	−0.934567	−	0.355787i	$-0.884213\pi$
$853$		−	517748.i		−	0.711575i	0.934567	−	0.355787i	$-0.115787\pi$
$854$		0			0
$855$	372290.			0.509271
$856$		0			0
$857$			1446.87i			0.00197001i	1.00000		0.000985004i	$0.000313536\pi$
$857$			1446.87i			0.00197001i	−1.00000		0.000985004i	$0.999686\pi$
$858$		0			0
$859$		−	217289.i		−	0.294477i	−0.989101	−	0.147239i	$-0.952961\pi$
$859$		−	217289.i		−	0.294477i	0.989101	−	0.147239i	$-0.0470385\pi$
$860$		0			0
$861$	−378547.	+	165448.i	−0.510639	+	0.223181i
$862$		0			0
$863$	−608808.			−0.817446			−0.408723	−	0.912658i	$-0.634026\pi$
$863$	−608808.			−0.817446			−0.408723	+	0.912658i	$0.634026\pi$
$864$		0			0
$865$	258012.			0.344833
$866$		0			0
$867$			357062.i			0.475013i
$868$		0			0
$869$	−1.05596e6			−1.39832
$870$		0			0
$871$			1.04858e6i			1.38218i
$872$		0			0
$873$		−	2.43652e6i		−	3.19699i
$874$		0			0
$875$	27424.8	+	62748.2i	0.0358202	+	0.0819568i
$876$		0			0
$877$	461213.			0.599657			0.299828	−	0.953993i	$-0.403071\pi$
$877$	461213.			0.599657			0.299828	+	0.953993i	$0.403071\pi$
$878$		0			0
$879$	1.07239e6			1.38796
$880$		0			0
$881$			468049.i			0.603031i	0.953461	+	0.301515i	$0.0974925\pi$
$881$			468049.i			0.603031i	−0.953461	+	0.301515i	$0.902508\pi$
$882$		0			0
$883$	1.35057e6			1.73218			0.866092	−	0.499884i	$-0.166624\pi$
$883$	1.35057e6			1.73218			0.866092	+	0.499884i	$0.166624\pi$
$884$		0			0
$885$		−	167898.i		−	0.214368i
$886$		0			0
$887$			417192.i			0.530259i	0.964213	+	0.265130i	$0.0854148\pi$
$887$			417192.i			0.530259i	−0.964213	+	0.265130i	$0.914585\pi$
$888$		0			0
$889$	544174.	−	237837.i	0.688547	−	0.300937i
$890$		0			0
$891$	−8618.82			−0.0108566
$892$		0			0
$893$	98765.5			0.123852
$894$		0			0
$895$		−	45045.8i		−	0.0562352i
$896$		0			0
$897$	−1.22005e6			−1.51633
$898$		0			0
$899$		−	92295.2i		−	0.114198i
$900$		0			0
$901$			260356.i			0.320714i
$902$		0			0
$903$	−435982.	−	997531.i	−0.534679	−	1.22335i
$904$		0			0
$905$	−52848.8			−0.0645265
$906$		0			0
$907$	207532.			0.252273			0.126137	−	0.992013i	$-0.459742\pi$
$907$	207532.			0.252273			0.126137	+	0.992013i	$0.459742\pi$
$908$		0			0
$909$			509810.i			0.616993i
$910$		0			0
$911$	−1.34881e6			−1.62522			−0.812612	−	0.582805i	$-0.801955\pi$
$911$	−1.34881e6			−1.62522			−0.812612	+	0.582805i	$0.801955\pi$
$912$		0			0
$913$		−	766499.i		−	0.919538i
$914$		0			0
$915$		−	1.08850e6i		−	1.30012i
$916$		0			0
$917$	−603997.	−	1.38195e6i	−0.718284	−	1.64344i
$918$		0			0
$919$	−352887.			−0.417835			−0.208918	−	0.977933i	$-0.566994\pi$
$919$	−352887.			−0.417835			−0.208918	+	0.977933i	$0.566994\pi$
$920$		0			0
$921$	68917.1			0.0812471
$922$		0			0
$923$		−	830707.i		−	0.975089i
$924$		0			0
$925$	40314.8			0.0471174
$926$		0			0
$927$			728834.i			0.848143i
$928$		0			0
$929$			382601.i			0.443317i	0.975124	+	0.221658i	$0.0711470\pi$
$929$			382601.i			0.443317i	−0.975124	+	0.221658i	$0.928853\pi$
$930$		0			0
$931$	446908.	+	413602.i	0.515607	+	0.477181i
$932$		0			0
$933$	1.31797e6			1.51406
$934$		0			0
$935$	547870.			0.626693
$936$		0			0
$937$			1.43975e6i			1.63987i	0.572460	+	0.819933i	$0.305989\pi$
$937$			1.43975e6i			1.63987i	−0.572460	+	0.819933i	$0.694011\pi$
$938$		0			0
$939$	−1.57679e6			−1.78831
$940$		0			0
$941$			829098.i			0.936326i	0.883642	+	0.468163i	$0.155084\pi$
$941$			829098.i			0.936326i	−0.883642	+	0.468163i	$0.844916\pi$
$942$		0			0
$943$		−	187286.i		−	0.210612i
$944$		0			0
$945$	−367874.	+	160783.i	−0.411941	+	0.180044i
$946$		0			0
$947$	1.43840e6			1.60391			0.801957	−	0.597382i	$-0.203792\pi$
$947$	1.43840e6			1.60391			0.801957	+	0.597382i	$0.203792\pi$
$948$		0			0
$949$	579167.			0.643089
$950$		0			0
$951$			1.70769e6i			1.88820i
$952$		0			0
$953$	1.38772e6			1.52797			0.763985	−	0.645235i	$-0.223240\pi$
$953$	1.38772e6			1.52797			0.763985	+	0.645235i	$0.223240\pi$
$954$		0			0
$955$		−	1115.83i		−	0.00122346i
$956$		0			0
$957$			973154.i			1.06257i
$958$		0			0
$959$	1.12294e6	−	490793.i	1.22101	−	0.533656i
$960$		0			0
$961$	845821.			0.915866
$962$		0			0
$963$	−158594.			−0.171015
$964$		0			0
$965$		−	308614.i		−	0.331406i
$966$		0			0
$967$	−1.67888e6			−1.79542			−0.897711	−	0.440586i	$-0.854771\pi$
$967$	−1.67888e6			−1.79542			−0.897711	+	0.440586i	$0.854771\pi$
$968$		0			0
$969$		−	897690.i		−	0.956046i
$970$		0			0
$971$		−	9746.54i		−	0.0103374i	−0.999987	−	0.00516871i	$-0.998355\pi$
$971$		−	9746.54i		−	0.0103374i	0.999987	−	0.00516871i	$-0.00164526\pi$
$972$		0			0
$973$	−98246.5	−	224789.i	−0.103775	−	0.237437i
$974$		0			0
$975$	471190.			0.495663
$976$		0			0
$977$	1.26088e6			1.32095			0.660473	−	0.750850i	$-0.270356\pi$
$977$	1.26088e6			1.32095			0.660473	+	0.750850i	$0.270356\pi$
$978$		0			0
$979$			2.76228e6i			2.88206i
$980$		0			0
$981$	−627624.			−0.652171
$982$		0			0
$983$		−	1.04613e6i		−	1.08263i	−0.840819	−	0.541316i	$-0.817926\pi$
$983$		−	1.04613e6i		−	1.08263i	0.840819	−	0.541316i	$-0.182074\pi$
$984$		0			0
$985$		−	191024.i		−	0.196886i
$986$		0			0
$987$	−254765.	+	111348.i	−0.261520	+	0.114300i
$988$		0			0
$989$	493529.			0.504568
$990$		0			0
$991$	587932.			0.598659			0.299330	−	0.954150i	$-0.403237\pi$
$991$	587932.			0.598659			0.299330	+	0.954150i	$0.403237\pi$
$992$		0			0
$993$			864612.i			0.876845i
$994$		0			0
$995$	141076.			0.142497
$996$		0			0
$997$			480094.i			0.482988i	0.970402	+	0.241494i	$0.0776374\pi$
$997$			480094.i			0.482988i	−0.970402	+	0.241494i	$0.922363\pi$
$998$		0			0
$999$			236354.i			0.236827i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.5.f.b.321.11		12
4.3	odd	2		35.5.d.a.6.9	✓	12
7.6	odd	2	inner	560.5.f.b.321.2		12
12.11	even	2		315.5.h.a.181.3		12
20.3	even	4		175.5.c.d.174.1		24
20.7	even	4		175.5.c.d.174.24		24
20.19	odd	2		175.5.d.i.76.4		12
28.27	even	2		35.5.d.a.6.10	yes	12
84.83	odd	2		315.5.h.a.181.4		12
140.27	odd	4		175.5.c.d.174.2		24
140.83	odd	4		175.5.c.d.174.23		24
140.139	even	2		175.5.d.i.76.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.d.a.6.9	✓	12	4.3	odd	2
35.5.d.a.6.10	yes	12	28.27	even	2
175.5.c.d.174.1		24	20.3	even	4
175.5.c.d.174.2		24	140.27	odd	4
175.5.c.d.174.23		24	140.83	odd	4
175.5.c.d.174.24		24	20.7	even	4
175.5.d.i.76.3		12	140.139	even	2
175.5.d.i.76.4		12	20.19	odd	2
315.5.h.a.181.3		12	12.11	even	2
315.5.h.a.181.4		12	84.83	odd	2
560.5.f.b.321.2		12	7.6	odd	2	inner
560.5.f.b.321.11		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	560.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+14.5704i q^{3} +11.1803i q^{5} +(-44.8989 + 19.6236i) q^{7} -131.296 q^{9} +O(q^{10})\)	\(q+14.5704i q^{3} +11.1803i q^{5} +(-44.8989 + 19.6236i) q^{7} -131.296 q^{9} -201.717 q^{11} +258.711i q^{13} -162.902 q^{15} +242.930i q^{17} +253.614i q^{19} +(-285.923 - 654.195i) q^{21} +323.663 q^{23} -125.000 q^{25} -732.838i q^{27} -331.107 q^{29} +278.747i q^{31} -2939.09i q^{33} +(-219.398 - 501.985i) q^{35} -322.518 q^{37} -3769.52 q^{39} -578.646i q^{41} +1524.82 q^{43} -1467.94i q^{45} -389.432i q^{47} +(1630.83 - 1762.16i) q^{49} -3539.59 q^{51} +1071.73 q^{53} -2255.26i q^{55} -3695.26 q^{57} +1030.67i q^{59} +6681.90i q^{61} +(5895.07 - 2576.51i) q^{63} -2892.47 q^{65} +4053.08 q^{67} +4715.90i q^{69} -3210.95 q^{71} -2238.66i q^{73} -1821.30i q^{75} +(9056.86 - 3958.40i) q^{77} +5234.87 q^{79} +42.7274 q^{81} +3799.88i q^{83} -2716.04 q^{85} -4824.37i q^{87} -13693.9i q^{89} +(-5076.83 - 11615.8i) q^{91} -4061.45 q^{93} -2835.49 q^{95} +18557.4i q^{97} +26484.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 50 q^{7} - 434 q^{9}+O(q^{10}) \) 12 * q + 50 * q^7 - 434 * q^9	\( 12 q + 50 q^{7} - 434 q^{9} - 126 q^{11} - 50 q^{15} - 642 q^{21} + 756 q^{23} - 1500 q^{25} - 2190 q^{29} + 150 q^{35} + 5564 q^{37} - 8634 q^{39} - 3944 q^{43} - 8796 q^{49} - 7206 q^{51} + 11760 q^{53} - 12900 q^{57} + 4310 q^{63} - 750 q^{65} + 24096 q^{67} + 5664 q^{71} + 26904 q^{77} + 1590 q^{79} - 11912 q^{81} + 1050 q^{85} + 7182 q^{91} - 70980 q^{93} + 3000 q^{95} - 23084 q^{99}+O(q^{100}) \) 12 * q + 50 * q^7 - 434 * q^9 - 126 * q^11 - 50 * q^15 - 642 * q^21 + 756 * q^23 - 1500 * q^25 - 2190 * q^29 + 150 * q^35 + 5564 * q^37 - 8634 * q^39 - 3944 * q^43 - 8796 * q^49 - 7206 * q^51 + 11760 * q^53 - 12900 * q^57 + 4310 * q^63 - 750 * q^65 + 24096 * q^67 + 5664 * q^71 + 26904 * q^77 + 1590 * q^79 - 11912 * q^81 + 1050 * q^85 + 7182 * q^91 - 70980 * q^93 + 3000 * q^95 - 23084 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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