LMFDB - Embedded newform 560.5.f.b.321.5

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			321.5
Root			$7.50299 - 2.23607i$ of defining polynomial
Character	$\chi$	$=$	560.321
Dual form			560.5.f.b.321.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.52807i q^{3} +11.1803i q^{5} +(19.9894 + 44.7373i) q^{7} +50.4404 q^{9} +O(q^{10})$	$q-5.52807i q^{3} +11.1803i q^{5} +(19.9894 + 44.7373i) q^{7} +50.4404 q^{9} +105.424 q^{11} -234.089i q^{13} +61.8057 q^{15} +375.251i q^{17} +45.1673i q^{19} +(247.311 - 110.503i) q^{21} -46.3674 q^{23} -125.000 q^{25} -726.612i q^{27} -257.421 q^{29} -194.899i q^{31} -582.791i q^{33} +(-500.178 + 223.489i) q^{35} +2523.02 q^{37} -1294.06 q^{39} +2504.06i q^{41} -2288.24 q^{43} +563.941i q^{45} +2624.74i q^{47} +(-1601.84 + 1788.55i) q^{49} +2074.42 q^{51} -358.663 q^{53} +1178.68i q^{55} +249.688 q^{57} +854.723i q^{59} +2085.25i q^{61} +(1008.28 + 2256.57i) q^{63} +2617.20 q^{65} +7558.23 q^{67} +256.322i q^{69} +282.660 q^{71} -9190.85i q^{73} +691.009i q^{75} +(2107.37 + 4716.38i) q^{77} +7184.60 q^{79} +68.9152 q^{81} -5064.74i q^{83} -4195.44 q^{85} +1423.04i q^{87} -10243.7i q^{89} +(10472.5 - 4679.31i) q^{91} -1077.42 q^{93} -504.986 q^{95} +608.860i q^{97} +5317.63 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	5.52807i		−	0.614230i	−0.951672	−	0.307115i	$-0.900636\pi$
$3$		−	5.52807i		−	0.614230i	0.951672	−	0.307115i	$-0.0993636\pi$
$4$		0			0
$5$			11.1803i			0.447214i
$5$			11.1803i			0.447214i
$6$		0			0
$7$	19.9894	+	44.7373i	0.407948	+	0.913005i
$7$	19.9894	+	44.7373i	0.407948	+	0.913005i
$8$		0			0
$9$	50.4404			0.622722
$10$		0			0
$11$	105.424			0.871273			0.435636	−	0.900123i	$-0.356523\pi$
$11$	105.424			0.871273			0.435636	+	0.900123i	$0.356523\pi$
$12$		0			0
$13$		−	234.089i		−	1.38514i	−0.721349	−	0.692571i	$-0.756478\pi$
$13$		−	234.089i		−	1.38514i	0.721349	−	0.692571i	$-0.243522\pi$
$14$		0			0
$15$	61.8057			0.274692
$16$		0			0
$17$			375.251i			1.29845i	0.760597	+	0.649224i	$0.224906\pi$
$17$			375.251i			1.29845i	−0.760597	+	0.649224i	$0.775094\pi$
$18$		0			0
$19$			45.1673i			0.125117i	0.998041	+	0.0625586i	$0.0199260\pi$
$19$			45.1673i			0.125117i	−0.998041	+	0.0625586i	$0.980074\pi$
$20$		0			0
$21$	247.311	−	110.503i	0.560795	−	0.250574i
$22$		0			0
$23$	−46.3674			−0.0876510			−0.0438255	−	0.999039i	$-0.513955\pi$
$23$	−46.3674			−0.0876510			−0.0438255	+	0.999039i	$0.513955\pi$
$24$		0			0
$25$	−125.000			−0.200000
$26$		0			0
$27$		−	726.612i		−	0.996724i
$28$		0			0
$29$	−257.421			−0.306089			−0.153045	−	0.988219i	$-0.548908\pi$
$29$	−257.421			−0.306089			−0.153045	+	0.988219i	$0.548908\pi$
$30$		0			0
$31$		−	194.899i		−	0.202809i	−0.994845	−	0.101404i	$-0.967666\pi$
$31$		−	194.899i		−	0.202809i	0.994845	−	0.101404i	$-0.0323336\pi$
$32$		0			0
$33$		−	582.791i		−	0.535162i
$34$		0			0
$35$	−500.178	+	223.489i	−0.408308	+	0.182440i
$36$		0			0
$37$	2523.02			1.84296			0.921482	−	0.388421i	$-0.126979\pi$
$37$	2523.02			1.84296			0.921482	+	0.388421i	$0.126979\pi$
$38$		0			0
$39$	−1294.06			−0.850796
$40$		0			0
$41$			2504.06i			1.48963i	0.667274	+	0.744813i	$0.267461\pi$
$41$			2504.06i			1.48963i	−0.667274	+	0.744813i	$0.732539\pi$
$42$		0			0
$43$	−2288.24			−1.23756			−0.618779	−	0.785565i	$-0.712372\pi$
$43$	−2288.24			−1.23756			−0.618779	+	0.785565i	$0.712372\pi$
$44$		0			0
$45$			563.941i			0.278490i
$46$		0			0
$47$			2624.74i			1.18820i	0.804391	+	0.594101i	$0.202492\pi$
$47$			2624.74i			1.18820i	−0.804391	+	0.594101i	$0.797508\pi$
$48$		0			0
$49$	−1601.84	+	1788.55i	−0.667157	+	0.744917i
$50$		0			0
$51$	2074.42			0.797545
$52$		0			0
$53$	−358.663			−0.127683			−0.0638417	−	0.997960i	$-0.520335\pi$
$53$	−358.663			−0.127683			−0.0638417	+	0.997960i	$0.520335\pi$
$54$		0			0
$55$			1178.68i			0.389645i
$56$		0			0
$57$	249.688			0.0768508
$58$		0			0
$59$			854.723i			0.245540i	0.992435	+	0.122770i	$0.0391777\pi$
$59$			854.723i			0.245540i	−0.992435	+	0.122770i	$0.960822\pi$
$60$		0			0
$61$			2085.25i			0.560400i	0.959942	+	0.280200i	$0.0904007\pi$
$61$			2085.25i			0.560400i	−0.959942	+	0.280200i	$0.909599\pi$
$62$		0			0
$63$	1008.28	+	2256.57i	0.254038	+	0.568548i
$64$		0			0
$65$	2617.20			0.619455
$66$		0			0
$67$	7558.23			1.68372			0.841862	−	0.539693i	$-0.181460\pi$
$67$	7558.23			1.68372			0.841862	+	0.539693i	$0.181460\pi$
$68$		0			0
$69$			256.322i			0.0538379i
$70$		0			0
$71$	282.660			0.0560722			0.0280361	−	0.999607i	$-0.491075\pi$
$71$	282.660			0.0560722			0.0280361	+	0.999607i	$0.491075\pi$
$72$		0			0
$73$		−	9190.85i		−	1.72469i	−0.506325	−	0.862343i	$-0.668996\pi$
$73$		−	9190.85i		−	1.72469i	0.506325	−	0.862343i	$-0.331004\pi$
$74$		0			0
$75$			691.009i			0.122846i
$76$		0			0
$77$	2107.37	+	4716.38i	0.355434	+	0.795477i
$78$		0			0
$79$	7184.60			1.15119			0.575597	−	0.817734i	$-0.304770\pi$
$79$	7184.60			1.15119			0.575597	+	0.817734i	$0.304770\pi$
$80$		0			0
$81$	68.9152			0.0105038
$82$		0			0
$83$		−	5064.74i		−	0.735192i	−0.929986	−	0.367596i	$-0.880181\pi$
$83$		−	5064.74i		−	0.735192i	0.929986	−	0.367596i	$-0.119819\pi$
$84$		0			0
$85$	−4195.44			−0.580683
$86$		0			0
$87$			1423.04i			0.188009i
$88$		0			0
$89$		−	10243.7i		−	1.29323i	−0.762817	−	0.646614i	$-0.776184\pi$
$89$		−	10243.7i		−	1.29323i	0.762817	−	0.646614i	$-0.223816\pi$
$90$		0			0
$91$	10472.5	−	4679.31i	1.26464	−	0.565066i
$92$		0			0
$93$	−1077.42			−0.124571
$94$		0			0
$95$	−504.986			−0.0559541
$96$		0			0
$97$			608.860i			0.0647104i	0.999476	+	0.0323552i	$0.0103008\pi$
$97$			608.860i			0.0647104i	−0.999476	+	0.0323552i	$0.989699\pi$
$98$		0			0
$99$	5317.63			0.542560
$100$		0			0
$101$			14966.3i			1.46714i	0.679616	+	0.733568i	$0.262146\pi$
$101$			14966.3i			1.46714i	−0.679616	+	0.733568i	$0.737854\pi$
$102$		0			0
$103$			18840.1i			1.77586i	0.459979	+	0.887930i	$0.347857\pi$
$103$			18840.1i			1.77586i	−0.459979	+	0.887930i	$0.652143\pi$
$104$		0			0
$105$	1235.46	+	2765.02i	0.112060	+	0.250795i
$106$		0			0
$107$	−3903.78			−0.340971			−0.170485	−	0.985360i	$-0.554534\pi$
$107$	−3903.78			−0.340971			−0.170485	+	0.985360i	$0.554534\pi$
$108$		0			0
$109$	16429.7			1.38285			0.691427	−	0.722447i	$-0.256982\pi$
$109$	16429.7			1.38285			0.691427	+	0.722447i	$0.256982\pi$
$110$		0			0
$111$		−	13947.4i		−	1.13200i
$112$		0			0
$113$	19430.8			1.52172			0.760858	−	0.648918i	$-0.224778\pi$
$113$	19430.8			1.52172			0.760858	+	0.648918i	$0.224778\pi$
$114$		0			0
$115$		−	518.403i		−	0.0391987i
$116$		0			0
$117$		−	11807.6i		−	0.862558i
$118$		0			0
$119$	−16787.7	+	7501.06i	−1.18549	+	0.529699i
$120$		0			0
$121$	−3526.78			−0.240884
$122$		0			0
$123$	13842.6			0.914972
$124$		0			0
$125$		−	1397.54i		−	0.0894427i
$126$		0			0
$127$	3473.63			0.215365			0.107683	−	0.994185i	$-0.465657\pi$
$127$	3473.63			0.215365			0.107683	+	0.994185i	$0.465657\pi$
$128$		0			0
$129$			12649.6i			0.760145i
$130$		0			0
$131$			5214.48i			0.303856i	0.988392	+	0.151928i	$0.0485482\pi$
$131$			5214.48i			0.303856i	−0.988392	+	0.151928i	$0.951452\pi$
$132$		0			0
$133$	−2020.66	+	902.870i	−0.114233	+	0.0510413i
$134$		0			0
$135$	8123.77			0.445749
$136$		0			0
$137$	25017.6			1.33292			0.666461	−	0.745540i	$-0.267808\pi$
$137$	25017.6			1.33292			0.666461	+	0.745540i	$0.267808\pi$
$138$		0			0
$139$		−	9723.25i		−	0.503248i	−0.967825	−	0.251624i	$-0.919035\pi$
$139$		−	9723.25i		−	0.503248i	0.967825	−	0.251624i	$-0.0809646\pi$
$140$		0			0
$141$	14509.7			0.729829
$142$		0			0
$143$		−	24678.6i		−	1.20684i
$144$		0			0
$145$		−	2878.06i		−	0.136887i
$146$		0			0
$147$	9887.20	+	8855.11i	0.457550	+	0.409788i
$148$		0			0
$149$	4969.64			0.223847			0.111924	−	0.993717i	$-0.464299\pi$
$149$	4969.64			0.223847			0.111924	+	0.993717i	$0.464299\pi$
$150$		0			0
$151$	15572.0			0.682954			0.341477	−	0.939890i	$-0.389073\pi$
$151$	15572.0			0.682954			0.341477	+	0.939890i	$0.389073\pi$
$152$		0			0
$153$			18927.8i			0.808571i
$154$		0			0
$155$	2179.04			0.0906988
$156$		0			0
$157$			2245.97i			0.0911180i	0.998962	+	0.0455590i	$0.0145069\pi$
$157$			2245.97i			0.0911180i	−0.998962	+	0.0455590i	$0.985493\pi$
$158$		0			0
$159$			1982.71i			0.0784270i
$160$		0			0
$161$	−926.858	−	2074.35i	−0.0357570	−	0.0800259i
$162$		0			0
$163$	−27048.4			−1.01804			−0.509021	−	0.860754i	$-0.669993\pi$
$163$	−27048.4			−1.01804			−0.509021	+	0.860754i	$0.669993\pi$
$164$		0			0
$165$	6515.80			0.239332
$166$		0			0
$167$			36693.5i			1.31570i	0.753149	+	0.657849i	$0.228534\pi$
$167$			36693.5i			1.31570i	−0.753149	+	0.657849i	$0.771466\pi$
$168$		0			0
$169$	−26236.7			−0.918620
$170$		0			0
$171$			2278.26i			0.0779132i
$172$		0			0
$173$		−	3132.59i		−	0.104667i	−0.998630	−	0.0523337i	$-0.983334\pi$
$173$		−	3132.59i		−	0.104667i	0.998630	−	0.0523337i	$-0.0166660\pi$
$174$		0			0
$175$	−2498.68	−	5592.16i	−0.0815895	−	0.182601i
$176$		0			0
$177$	4724.97			0.150818
$178$		0			0
$179$	25816.5			0.805732			0.402866	−	0.915259i	$-0.368014\pi$
$179$	25816.5			0.805732			0.402866	+	0.915259i	$0.368014\pi$
$180$		0			0
$181$			31614.4i			0.965002i	0.875895	+	0.482501i	$0.160272\pi$
$181$			31614.4i			0.965002i	−0.875895	+	0.482501i	$0.839728\pi$
$182$		0			0
$183$	11527.4			0.344214
$184$		0			0
$185$			28208.2i			0.824199i
$186$		0			0
$187$			39560.5i			1.13130i
$188$		0			0
$189$	32506.6	−	14524.6i	0.910014	−	0.406611i
$190$		0			0
$191$	4727.49			0.129588			0.0647938	−	0.997899i	$-0.479361\pi$
$191$	4727.49			0.129588			0.0647938	+	0.997899i	$0.479361\pi$
$192$		0			0
$193$	−36953.1			−0.992057			−0.496028	−	0.868306i	$-0.665209\pi$
$193$	−36953.1			−0.992057			−0.496028	+	0.868306i	$0.665209\pi$
$194$		0			0
$195$		−	14468.0i		−	0.380488i
$196$		0			0
$197$	−3222.40			−0.0830322			−0.0415161	−	0.999138i	$-0.513219\pi$
$197$	−3222.40			−0.0830322			−0.0415161	+	0.999138i	$0.513219\pi$
$198$		0			0
$199$			51984.5i			1.31271i	0.754453	+	0.656354i	$0.227902\pi$
$199$			51984.5i			1.31271i	−0.754453	+	0.656354i	$0.772098\pi$
$200$		0			0
$201$		−	41782.4i		−	1.03419i
$202$		0			0
$203$	−5145.70	−	11516.3i	−0.124868	−	0.279461i
$204$		0			0
$205$	−27996.2			−0.666181
$206$		0			0
$207$	−2338.79			−0.0545822
$208$		0			0
$209$			4761.72i			0.109011i
$210$		0			0
$211$	15750.2			0.353770			0.176885	−	0.984232i	$-0.443398\pi$
$211$	15750.2			0.353770			0.176885	+	0.984232i	$0.443398\pi$
$212$		0			0
$213$		−	1562.56i		−	0.0344412i
$214$		0			0
$215$		−	25583.3i		−	0.553453i
$216$		0			0
$217$	8719.26	−	3895.93i	0.185165	−	0.0827354i
$218$		0			0
$219$	−50807.7			−1.05935
$220$		0			0
$221$	87842.3			1.79854
$222$		0			0
$223$			11942.3i			0.240148i	0.992765	+	0.120074i	$0.0383133\pi$
$223$			11942.3i			0.240148i	−0.992765	+	0.120074i	$0.961687\pi$
$224$		0			0
$225$	−6305.06			−0.124544
$226$		0			0
$227$		−	79946.4i		−	1.55148i	−0.631051	−	0.775742i	$-0.717376\pi$
$227$		−	79946.4i		−	1.55148i	0.631051	−	0.775742i	$-0.282624\pi$
$228$		0			0
$229$			40487.1i			0.772050i	0.922488	+	0.386025i	$0.126152\pi$
$229$			40487.1i			0.772050i	−0.922488	+	0.386025i	$0.873848\pi$
$230$		0			0
$231$	26072.5	−	11649.7i	0.488606	−	0.218318i
$232$		0			0
$233$	−40236.2			−0.741148			−0.370574	−	0.928803i	$-0.620839\pi$
$233$	−40236.2			−0.741148			−0.370574	+	0.928803i	$0.620839\pi$
$234$		0			0
$235$	−29345.5			−0.531380
$236$		0			0
$237$		−	39717.0i		−	0.707097i
$238$		0			0
$239$	12346.0			0.216137			0.108069	−	0.994143i	$-0.465533\pi$
$239$	12346.0			0.216137			0.108069	+	0.994143i	$0.465533\pi$
$240$		0			0
$241$			20969.2i			0.361034i	0.983572	+	0.180517i	$0.0577771\pi$
$241$			20969.2i			0.361034i	−0.983572	+	0.180517i	$0.942223\pi$
$242$		0			0
$243$		−	59236.5i		−	1.00318i
$244$		0			0
$245$	−19996.5	−	17909.2i	−0.333137	−	0.298362i
$246$		0			0
$247$	10573.2			0.173305
$248$		0			0
$249$	−27998.2			−0.451577
$250$		0			0
$251$			82331.2i			1.30682i	0.757003	+	0.653411i	$0.226663\pi$
$251$			82331.2i			1.30682i	−0.757003	+	0.653411i	$0.773337\pi$
$252$		0			0
$253$	−4888.24			−0.0763680
$254$		0			0
$255$			23192.7i			0.356673i
$256$		0			0
$257$		−	6751.23i		−	0.102216i	−0.998693	−	0.0511078i	$-0.983725\pi$
$257$		−	6751.23i		−	0.102216i	0.998693	−	0.0511078i	$-0.0162752\pi$
$258$		0			0
$259$	50433.7	+	112873.i	0.751833	+	1.68264i
$260$		0			0
$261$	−12984.4			−0.190608
$262$		0			0
$263$	−23786.1			−0.343884			−0.171942	−	0.985107i	$-0.555004\pi$
$263$	−23786.1			−0.343884			−0.171942	+	0.985107i	$0.555004\pi$
$264$		0			0
$265$		−	4009.97i		−	0.0571018i
$266$		0			0
$267$	−56627.7			−0.794340
$268$		0			0
$269$			74052.1i			1.02337i	0.859173	+	0.511685i	$0.170979\pi$
$269$			74052.1i			1.02337i	−0.859173	+	0.511685i	$0.829021\pi$
$270$		0			0
$271$		−	61472.9i		−	0.837037i	−0.908208	−	0.418519i	$-0.862549\pi$
$271$		−	61472.9i		−	0.837037i	0.908208	−	0.418519i	$-0.137451\pi$
$272$		0			0
$273$	−25867.5	−	57892.7i	−0.347080	−	0.776781i
$274$		0			0
$275$	−13178.0			−0.174255
$276$		0			0
$277$	−73318.6			−0.955552			−0.477776	−	0.878482i	$-0.658557\pi$
$277$	−73318.6			−0.955552			−0.477776	+	0.878482i	$0.658557\pi$
$278$		0			0
$279$		−	9830.80i		−	0.126293i
$280$		0			0
$281$	−63569.9			−0.805080			−0.402540	−	0.915403i	$-0.631872\pi$
$281$	−63569.9			−0.805080			−0.402540	+	0.915403i	$0.631872\pi$
$282$		0			0
$283$		−	99674.3i		−	1.24454i	−0.782801	−	0.622272i	$-0.786210\pi$
$283$		−	99674.3i		−	1.24454i	0.782801	−	0.622272i	$-0.213790\pi$
$284$		0			0
$285$			2791.60i			0.0343687i
$286$		0			0
$287$	−112025.	+	50054.7i	−1.36004	+	0.607689i
$288$		0			0
$289$	−57292.6			−0.685967
$290$		0			0
$291$	3365.82			0.0397471
$292$		0			0
$293$			33390.9i			0.388949i	0.980908	+	0.194475i	$0.0623002\pi$
$293$			33390.9i			0.388949i	−0.980908	+	0.194475i	$0.937700\pi$
$294$		0			0
$295$	−9556.10			−0.109809
$296$		0			0
$297$		−	76602.3i		−	0.868419i
$298$		0			0
$299$			10854.1i			0.121409i
$300$		0			0
$301$	−45740.7	−	102370.i	−0.504859	−	1.12990i
$302$		0			0
$303$	82734.5			0.901159
$304$		0			0
$305$	−23313.8			−0.250618
$306$		0			0
$307$		−	81081.7i		−	0.860292i	−0.902759	−	0.430146i	$-0.858462\pi$
$307$		−	81081.7i		−	0.860292i	0.902759	−	0.430146i	$-0.141538\pi$
$308$		0			0
$309$	104149.			1.09079
$310$		0			0
$311$		−	58811.2i		−	0.608050i	−0.952664	−	0.304025i	$-0.901669\pi$
$311$		−	58811.2i		−	0.608050i	0.952664	−	0.304025i	$-0.0983306\pi$
$312$		0			0
$313$		−	146677.i		−	1.49718i	−0.663033	−	0.748590i	$-0.730731\pi$
$313$		−	146677.i		−	1.49718i	0.663033	−	0.748590i	$-0.269269\pi$
$314$		0			0
$315$	−25229.2	+	11272.9i	−0.254262	+	0.113609i
$316$		0			0
$317$	106159.			1.05642			0.528210	−	0.849114i	$-0.322863\pi$
$317$	106159.			1.05642			0.528210	+	0.849114i	$0.322863\pi$
$318$		0			0
$319$	−27138.4			−0.266687
$320$		0			0
$321$			21580.3i			0.209435i
$322$		0			0
$323$	−16949.1			−0.162458
$324$		0			0
$325$			29261.1i			0.277029i
$326$		0			0
$327$		−	90824.4i		−	0.849390i
$328$		0			0
$329$	−117424.	+	52467.0i	−1.08483	+	0.484724i
$330$		0			0
$331$	24297.3			0.221769			0.110885	−	0.993833i	$-0.464632\pi$
$331$	24297.3			0.221769			0.110885	+	0.993833i	$0.464632\pi$
$332$		0			0
$333$	127262.			1.14765
$334$		0			0
$335$			84503.6i			0.752984i
$336$		0			0
$337$	−170981.			−1.50553			−0.752763	−	0.658292i	$-0.771279\pi$
$337$	−170981.			−1.50553			−0.752763	+	0.658292i	$0.771279\pi$
$338$		0			0
$339$		−	107415.i		−	0.934684i
$340$		0			0
$341$		−	20547.1i		−	0.176702i
$342$		0			0
$343$	−112035.	−	35910.1i	−0.952278	−	0.305231i
$344$		0			0
$345$	−2865.77			−0.0240770
$346$		0			0
$347$	202470.			1.68152			0.840761	−	0.541406i	$-0.182108\pi$
$347$	202470.			1.68152			0.840761	+	0.541406i	$0.182108\pi$
$348$		0			0
$349$		−	90133.9i		−	0.740009i	−0.929030	−	0.370005i	$-0.879356\pi$
$349$		−	90133.9i		−	0.740009i	0.929030	−	0.370005i	$-0.120644\pi$
$350$		0			0
$351$	−170092.			−1.38061
$352$		0			0
$353$		−	118178.i		−	0.948388i	−0.880420	−	0.474194i	$-0.842739\pi$
$353$		−	118178.i		−	0.948388i	0.880420	−	0.474194i	$-0.157261\pi$
$354$		0			0
$355$			3160.24i			0.0250763i
$356$		0			0
$357$	41466.4	+	92803.7i	0.325357	+	0.728163i
$358$		0			0
$359$	−194552.			−1.50955			−0.754774	−	0.655985i	$-0.772253\pi$
$359$	−194552.			−1.50955			−0.754774	+	0.655985i	$0.772253\pi$
$360$		0			0
$361$	128281.			0.984346
$362$		0			0
$363$			19496.3i			0.147958i
$364$		0			0
$365$	102757.			0.771303
$366$		0			0
$367$		−	105692.i		−	0.784710i	−0.919814	−	0.392355i	$-0.871661\pi$
$367$		−	105692.i		−	0.784710i	0.919814	−	0.392355i	$-0.128339\pi$
$368$		0			0
$369$			126306.i			0.927622i
$370$		0			0
$371$	−7169.46	−	16045.6i	−0.0520881	−	0.116576i
$372$		0			0
$373$	66392.5			0.477201			0.238600	−	0.971118i	$-0.423311\pi$
$373$	66392.5			0.477201			0.238600	+	0.971118i	$0.423311\pi$
$374$		0			0
$375$	−7725.71			−0.0549384
$376$		0			0
$377$			60259.5i			0.423978i
$378$		0			0
$379$	−175353.			−1.22077			−0.610387	−	0.792103i	$-0.708986\pi$
$379$	−175353.			−1.22077			−0.610387	+	0.792103i	$0.708986\pi$
$380$		0			0
$381$		−	19202.5i		−	0.132284i
$382$		0			0
$383$			4098.21i			0.0279381i	0.999902	+	0.0139690i	$0.00444662\pi$
$383$			4098.21i			0.0279381i	−0.999902	+	0.0139690i	$0.995553\pi$
$384$		0			0
$385$	−52730.7	+	23561.1i	−0.355748	+	0.158955i
$386$		0			0
$387$	−115420.			−0.770654
$388$		0			0
$389$	−207427.			−1.37077			−0.685387	−	0.728179i	$-0.740367\pi$
$389$	−207427.			−1.37077			−0.685387	+	0.728179i	$0.740367\pi$
$390$		0			0
$391$		−	17399.4i		−	0.113810i
$392$		0			0
$393$	28826.0			0.186638
$394$		0			0
$395$			80326.3i			0.514829i
$396$		0			0
$397$			115680.i			0.733968i	0.930227	+	0.366984i	$0.119610\pi$
$397$			115680.i			0.733968i	−0.930227	+	0.366984i	$0.880390\pi$
$398$		0			0
$399$	4991.13	+	11170.4i	0.0313511	+	0.0701652i
$400$		0			0
$401$	−198641.			−1.23532			−0.617661	−	0.786444i	$-0.711920\pi$
$401$	−198641.			−1.23532			−0.617661	+	0.786444i	$0.711920\pi$
$402$		0			0
$403$	−45623.8			−0.280919
$404$		0			0
$405$			770.495i			0.00469742i
$406$		0			0
$407$	265987.			1.60572
$408$		0			0
$409$		−	39642.8i		−	0.236983i	−0.992955	−	0.118492i	$-0.962194\pi$
$409$		−	39642.8i		−	0.236983i	0.992955	−	0.118492i	$-0.0378059\pi$
$410$		0			0
$411$		−	138299.i		−	0.818720i
$412$		0			0
$413$	−38238.0	+	17085.4i	−0.224179	+	0.100167i
$414$		0			0
$415$	56625.5			0.328788
$416$		0			0
$417$	−53750.8			−0.309110
$418$		0			0
$419$			157716.i			0.898354i	0.893443	+	0.449177i	$0.148283\pi$
$419$			157716.i			0.898354i	−0.893443	+	0.449177i	$0.851717\pi$
$420$		0			0
$421$	9811.94			0.0553593			0.0276797	−	0.999617i	$-0.491188\pi$
$421$	9811.94			0.0553593			0.0276797	+	0.999617i	$0.491188\pi$
$422$		0			0
$423$			132393.i			0.739919i
$424$		0			0
$425$		−	46906.4i		−	0.259690i
$426$		0			0
$427$	−93288.2	+	41682.9i	−0.511648	+	0.228614i
$428$		0			0
$429$	−136425.			−0.741276
$430$		0			0
$431$	−223189.			−1.20149			−0.600743	−	0.799442i	$-0.705128\pi$
$431$	−223189.			−1.20149			−0.600743	+	0.799442i	$0.705128\pi$
$432$		0			0
$433$		−	112634.i		−	0.600748i	−0.953821	−	0.300374i	$-0.902888\pi$
$433$		−	112634.i		−	0.600748i	0.953821	−	0.300374i	$-0.0971115\pi$
$434$		0			0
$435$	−15910.1			−0.0840803
$436$		0			0
$437$		−	2094.29i		−	0.0109667i
$438$		0			0
$439$		−	73105.8i		−	0.379335i	−0.981848	−	0.189667i	$-0.939259\pi$
$439$		−	73105.8i		−	0.379335i	0.981848	−	0.189667i	$-0.0607410\pi$
$440$		0			0
$441$	−80797.8	+	90215.0i	−0.415453	+	0.463876i
$442$		0			0
$443$	−50089.8			−0.255236			−0.127618	−	0.991823i	$-0.540733\pi$
$443$	−50089.8			−0.255236			−0.127618	+	0.991823i	$0.540733\pi$
$444$		0			0
$445$	114528.			0.578349
$446$		0			0
$447$		−	27472.5i		−	0.137494i
$448$		0			0
$449$	−5042.02			−0.0250099			−0.0125049	−	0.999922i	$-0.503981\pi$
$449$	−5042.02			−0.0250099			−0.0125049	+	0.999922i	$0.503981\pi$
$450$		0			0
$451$			263988.i			1.29787i
$452$		0			0
$453$		−	86083.3i		−	0.419491i
$454$		0			0
$455$	52316.3	+	117086.i	0.252705	+	0.565565i
$456$		0			0
$457$	−38281.4			−0.183297			−0.0916486	−	0.995791i	$-0.529214\pi$
$457$	−38281.4			−0.183297			−0.0916486	+	0.995791i	$0.529214\pi$
$458$		0			0
$459$	272662.			1.29419
$460$		0			0
$461$			37749.3i			0.177626i	0.996048	+	0.0888132i	$0.0283074\pi$
$461$			37749.3i			0.177626i	−0.996048	+	0.0888132i	$0.971693\pi$
$462$		0			0
$463$	343927.			1.60437			0.802185	−	0.597075i	$-0.203671\pi$
$463$	343927.			1.60437			0.802185	+	0.597075i	$0.203671\pi$
$464$		0			0
$465$		−	12045.9i		−	0.0557099i
$466$		0			0
$467$			96841.6i			0.444046i	0.975041	+	0.222023i	$0.0712661\pi$
$467$			96841.6i			0.444046i	−0.975041	+	0.222023i	$0.928734\pi$
$468$		0			0
$469$	151085.	+	338135.i	0.686871	+	1.53725i
$470$		0			0
$471$	12415.9			0.0559674
$472$		0			0
$473$	−241236.			−1.07825
$474$		0			0
$475$		−	5645.92i		−	0.0250235i
$476$		0			0
$477$	−18091.1			−0.0795112
$478$		0			0
$479$		−	41421.1i		−	0.180530i	−0.995918	−	0.0902652i	$-0.971229\pi$
$479$		−	41421.1i		−	0.180530i	0.995918	−	0.0902652i	$-0.0287715\pi$
$480$		0			0
$481$		−	590611.i		−	2.55277i
$482$		0			0
$483$	−11467.2	+	5123.74i	−0.0491543	+	0.0219630i
$484$		0			0
$485$	−6807.26			−0.0289394
$486$		0			0
$487$	29470.2			0.124258			0.0621290	−	0.998068i	$-0.480211\pi$
$487$	29470.2			0.124258			0.0621290	+	0.998068i	$0.480211\pi$
$488$		0			0
$489$			149525.i			0.625312i
$490$		0			0
$491$	198756.			0.824437			0.412218	−	0.911085i	$-0.364754\pi$
$491$	198756.			0.824437			0.412218	+	0.911085i	$0.364754\pi$
$492$		0			0
$493$		−	96597.7i		−	0.397441i
$494$		0			0
$495$			59453.0i			0.242640i
$496$		0			0
$497$	5650.22	+	12645.4i	0.0228745	+	0.0511942i
$498$		0			0
$499$	−62009.6			−0.249034			−0.124517	−	0.992218i	$-0.539738\pi$
$499$	−62009.6			−0.249034			−0.124517	+	0.992218i	$0.539738\pi$
$500$		0			0
$501$	202844.			0.808142
$502$		0			0
$503$		−	335191.i		−	1.32482i	−0.749143	−	0.662409i	$-0.769534\pi$
$503$		−	335191.i		−	1.32482i	0.749143	−	0.662409i	$-0.230466\pi$
$504$		0			0
$505$	−167328.			−0.656123
$506$		0			0
$507$			145038.i			0.564244i
$508$		0			0
$509$		−	27607.7i		−	0.106560i	−0.998580	−	0.0532800i	$-0.983032\pi$
$509$		−	27607.7i		−	0.106560i	0.998580	−	0.0532800i	$-0.0169676\pi$
$510$		0			0
$511$	411173.	−	183720.i	1.57465	−	0.703581i
$512$		0			0
$513$	32819.1			0.124707
$514$		0			0
$515$	−210639.			−0.794188
$516$		0			0
$517$			276710.i			1.03525i
$518$		0			0
$519$	−17317.2			−0.0642899
$520$		0			0
$521$		−	73225.7i		−	0.269767i	−0.990861	−	0.134883i	$-0.956934\pi$
$521$		−	73225.7i		−	0.269767i	0.990861	−	0.134883i	$-0.0430660\pi$
$522$		0			0
$523$		−	208338.i		−	0.761668i	−0.924643	−	0.380834i	$-0.875637\pi$
$523$		−	208338.i		−	0.761668i	0.924643	−	0.380834i	$-0.124363\pi$
$524$		0			0
$525$	−30913.8	+	13812.9i	−0.112159	+	0.0501147i
$526$		0			0
$527$	73136.2			0.263337
$528$		0			0
$529$	−277691.			−0.992317
$530$		0			0
$531$			43112.6i			0.152903i
$532$		0			0
$533$	586173.			2.06334
$534$		0			0
$535$		−	43645.5i		−	0.152487i
$536$		0			0
$537$		−	142715.i		−	0.494905i
$538$		0			0
$539$	−168873.	+	188556.i	−0.581276	+	0.649026i
$540$		0			0
$541$	71741.5			0.245118			0.122559	−	0.992461i	$-0.460890\pi$
$541$	71741.5			0.245118			0.122559	+	0.992461i	$0.460890\pi$
$542$		0			0
$543$	174767.			0.592733
$544$		0			0
$545$			183689.i			0.618431i
$546$		0			0
$547$	−184265.			−0.615839			−0.307919	−	0.951412i	$-0.599633\pi$
$547$	−184265.			−0.615839			−0.307919	+	0.951412i	$0.599633\pi$
$548$		0			0
$549$			105181.i			0.348973i
$550$		0			0
$551$		−	11627.0i		−	0.0382971i
$552$		0			0
$553$	143616.	+	321419.i	0.469627	+	1.05105i
$554$		0			0
$555$	155937.			0.506247
$556$		0			0
$557$	386198.			1.24480			0.622399	−	0.782700i	$-0.286158\pi$
$557$	386198.			1.24480			0.622399	+	0.782700i	$0.286158\pi$
$558$		0			0
$559$			535653.i			1.71419i
$560$		0			0
$561$	218693.			0.694880
$562$		0			0
$563$		−	352409.i		−	1.11181i	−0.831246	−	0.555904i	$-0.812372\pi$
$563$		−	352409.i		−	1.11181i	0.831246	−	0.555904i	$-0.187628\pi$
$564$		0			0
$565$			217243.i			0.680532i
$566$		0			0
$567$	1377.58	+	3083.08i	0.00428498	+	0.00958999i
$568$		0			0
$569$	265786.			0.820932			0.410466	−	0.911876i	$-0.365366\pi$
$569$	265786.			0.820932			0.410466	+	0.911876i	$0.365366\pi$
$570$		0			0
$571$	−455939.			−1.39841			−0.699205	−	0.714921i	$-0.746463\pi$
$571$	−455939.			−1.39841			−0.699205	+	0.714921i	$0.746463\pi$
$572$		0			0
$573$		−	26133.9i		−	0.0795966i
$574$		0			0
$575$	5795.92			0.0175302
$576$		0			0
$577$		−	211395.i		−	0.634956i	−0.948266	−	0.317478i	$-0.897164\pi$
$577$		−	211395.i		−	0.634956i	0.948266	−	0.317478i	$-0.102836\pi$
$578$		0			0
$579$			204279.i			0.609351i
$580$		0			0
$581$	226583.	−	101241.i	0.671234	−	0.299920i
$582$		0			0
$583$	−37811.7			−0.111247
$584$		0			0
$585$	132013.			0.385748
$586$		0			0
$587$		−	158002.i		−	0.458550i	−0.973362	−	0.229275i	$-0.926365\pi$
$587$		−	158002.i		−	0.458550i	0.973362	−	0.229275i	$-0.0736355\pi$
$588$		0			0
$589$	8803.08			0.0253749
$590$		0			0
$591$			17813.6i			0.0510009i
$592$		0			0
$593$		−	239670.i		−	0.681561i	−0.940143	−	0.340780i	$-0.889309\pi$
$593$		−	239670.i		−	0.681561i	0.940143	−	0.340780i	$-0.110691\pi$
$594$		0			0
$595$	−83864.4	−	187692.i	−0.236888	−	0.530167i
$596$		0			0
$597$	287374.			0.806304
$598$		0			0
$599$	−183572.			−0.511626			−0.255813	−	0.966726i	$-0.582343\pi$
$599$	−183572.			−0.511626			−0.255813	+	0.966726i	$0.582343\pi$
$600$		0			0
$601$			14783.5i			0.0409287i	0.999791	+	0.0204643i	$0.00651446\pi$
$601$			14783.5i			0.0409287i	−0.999791	+	0.0204643i	$0.993486\pi$
$602$		0			0
$603$	381241.			1.04849
$604$		0			0
$605$		−	39430.6i		−	0.107726i
$606$		0			0
$607$		−	591265.i		−	1.60474i	−0.596827	−	0.802370i	$-0.703572\pi$
$607$		−	591265.i		−	1.60474i	0.596827	−	0.802370i	$-0.296428\pi$
$608$		0			0
$609$	−63663.0	+	28445.8i	−0.171653	+	0.0766979i
$610$		0			0
$611$	614422.			1.64583
$612$		0			0
$613$	−90639.8			−0.241212			−0.120606	−	0.992700i	$-0.538484\pi$
$613$	−90639.8			−0.241212			−0.120606	+	0.992700i	$0.538484\pi$
$614$		0			0
$615$			154765.i			0.409188i
$616$		0			0
$617$	−504471.			−1.32515			−0.662576	−	0.748995i	$-0.730537\pi$
$617$	−504471.			−1.32515			−0.662576	+	0.748995i	$0.730537\pi$
$618$		0			0
$619$			732305.i			1.91122i	0.294634	+	0.955610i	$0.404802\pi$
$619$			732305.i			1.91122i	−0.294634	+	0.955610i	$0.595198\pi$
$620$		0			0
$621$			33691.1i			0.0873639i
$622$		0			0
$623$	458273.	−	204765.i	1.18072	−	0.527569i
$624$		0			0
$625$	15625.0			0.0400000
$626$		0			0
$627$	26323.1			0.0669580
$628$		0			0
$629$			946766.i			2.39299i
$630$		0			0
$631$	−537983.			−1.35117			−0.675585	−	0.737282i	$-0.736109\pi$
$631$	−537983.			−1.35117			−0.675585	+	0.737282i	$0.736109\pi$
$632$		0			0
$633$		−	87068.1i		−	0.217296i
$634$		0			0
$635$			38836.4i			0.0963144i
$636$		0			0
$637$	418679.	+	374974.i	1.03182	+	0.924108i
$638$		0			0
$639$	14257.5			0.0349174
$640$		0			0
$641$	226234.			0.550608			0.275304	−	0.961357i	$-0.411221\pi$
$641$	226234.			0.550608			0.275304	+	0.961357i	$0.411221\pi$
$642$		0			0
$643$			186548.i			0.451199i	0.974220	+	0.225599i	$0.0724340\pi$
$643$			186548.i			0.451199i	−0.974220	+	0.225599i	$0.927566\pi$
$644$		0			0
$645$	−141426.			−0.339947
$646$		0			0
$647$			580904.i			1.38770i	0.720120	+	0.693850i	$0.244087\pi$
$647$			580904.i			1.38770i	−0.720120	+	0.693850i	$0.755913\pi$
$648$		0			0
$649$			90108.4i			0.213932i
$650$		0			0
$651$	−21536.9	−	48200.7i	−0.0508185	−	0.113734i
$652$		0			0
$653$	242502.			0.568707			0.284354	−	0.958720i	$-0.408221\pi$
$653$	242502.			0.568707			0.284354	+	0.958720i	$0.408221\pi$
$654$		0			0
$655$	−58299.6			−0.135889
$656$		0			0
$657$		−	463591.i		−	1.07400i
$658$		0			0
$659$	377939.			0.870265			0.435132	−	0.900367i	$-0.356702\pi$
$659$	377939.			0.870265			0.435132	+	0.900367i	$0.356702\pi$
$660$		0			0
$661$		−	377336.i		−	0.863625i	−0.901963	−	0.431812i	$-0.857874\pi$
$661$		−	377336.i		−	0.863625i	0.901963	−	0.431812i	$-0.142126\pi$
$662$		0			0
$663$		−	485598.i		−	1.10471i
$664$		0			0
$665$	−10094.4	−	22591.7i	−0.0228264	−	0.0510864i
$666$		0			0
$667$	11936.0			0.0268291
$668$		0			0
$669$	66018.1			0.147506
$670$		0			0
$671$			219835.i			0.488261i
$672$		0			0
$673$	334342.			0.738177			0.369088	−	0.929394i	$-0.379670\pi$
$673$	334342.			0.738177			0.369088	+	0.929394i	$0.379670\pi$
$674$		0			0
$675$			90826.5i			0.199345i
$676$		0			0
$677$		−	823510.i		−	1.79677i	−0.439212	−	0.898383i	$-0.644742\pi$
$677$		−	823510.i		−	1.79677i	0.439212	−	0.898383i	$-0.355258\pi$
$678$		0			0
$679$	−27238.7	+	12170.8i	−0.0590809	+	0.0263984i
$680$		0			0
$681$	−441949.			−0.952967
$682$		0			0
$683$	821787.			1.76164			0.880822	−	0.473448i	$-0.156991\pi$
$683$	821787.			1.76164			0.880822	+	0.473448i	$0.156991\pi$
$684$		0			0
$685$			279705.i			0.596101i
$686$		0			0
$687$	223815.			0.474216
$688$		0			0
$689$			83959.0i			0.176860i
$690$		0			0
$691$		−	744548.i		−	1.55933i	−0.626200	−	0.779663i	$-0.715391\pi$
$691$		−	744548.i		−	1.55933i	0.626200	−	0.779663i	$-0.284609\pi$
$692$		0			0
$693$	106297.	+	237896.i	0.221336	+	0.495361i
$694$		0			0
$695$	108709.			0.225059
$696$		0			0
$697$	−939652.			−1.93420
$698$		0			0
$699$			222428.i			0.455235i
$700$		0			0
$701$	318730.			0.648616			0.324308	−	0.945952i	$-0.394869\pi$
$701$	318730.			0.648616			0.324308	+	0.945952i	$0.394869\pi$
$702$		0			0
$703$			113958.i			0.230587i
$704$		0			0
$705$			162224.i			0.326389i
$706$		0			0
$707$	−669549.	+	299167.i	−1.33950	+	0.598515i
$708$		0			0
$709$	150036.			0.298471			0.149235	−	0.988802i	$-0.452319\pi$
$709$	150036.			0.298471			0.149235	+	0.988802i	$0.452319\pi$
$710$		0			0
$711$	362394.			0.716873
$712$		0			0
$713$			9036.97i			0.0177764i
$714$		0			0
$715$	275915.			0.539714
$716$		0			0
$717$		−	68249.4i		−	0.132758i
$718$		0			0
$719$		−	124427.i		−	0.240690i	−0.992732	−	0.120345i	$-0.961600\pi$
$719$		−	124427.i		−	0.240690i	0.992732	−	0.120345i	$-0.0384001\pi$
$720$		0			0
$721$	−842854.	+	376603.i	−1.62137	+	0.724458i
$722$		0			0
$723$	115919.			0.221758
$724$		0			0
$725$	32177.6			0.0612179
$726$		0			0
$727$			534357.i			1.01103i	0.862819	+	0.505513i	$0.168697\pi$
$727$			534357.i			1.01103i	−0.862819	+	0.505513i	$0.831303\pi$
$728$		0			0
$729$	−321882.			−0.605677
$730$		0			0
$731$		−	858667.i		−	1.60690i
$732$		0			0
$733$		−	638734.i		−	1.18881i	−0.804166	−	0.594404i	$-0.797388\pi$
$733$		−	638734.i		−	1.18881i	0.804166	−	0.594404i	$-0.202612\pi$
$734$		0			0
$735$	−99003.1	+	110542.i	−0.183263	+	0.204623i
$736$		0			0
$737$	796819.			1.46698
$738$		0			0
$739$	−198812.			−0.364044			−0.182022	−	0.983295i	$-0.558264\pi$
$739$	−198812.			−0.364044			−0.182022	+	0.983295i	$0.558264\pi$
$740$		0			0
$741$		−	58449.3i		−	0.106449i
$742$		0			0
$743$	−865398.			−1.56761			−0.783806	−	0.621006i	$-0.786724\pi$
$743$	−865398.			−1.56761			−0.783806	+	0.621006i	$0.786724\pi$
$744$		0			0
$745$			55562.2i			0.100108i
$746$		0			0
$747$		−	255468.i		−	0.457820i
$748$		0			0
$749$	−78034.3	−	174644.i	−0.139098	−	0.311308i
$750$		0			0
$751$	18813.0			0.0333564			0.0166782	−	0.999861i	$-0.494691\pi$
$751$	18813.0			0.0333564			0.0166782	+	0.999861i	$0.494691\pi$
$752$		0			0
$753$	455132.			0.802690
$754$		0			0
$755$			174101.i			0.305426i
$756$		0			0
$757$	−247909.			−0.432614			−0.216307	−	0.976325i	$-0.569401\pi$
$757$	−247909.			−0.432614			−0.216307	+	0.976325i	$0.569401\pi$
$758$		0			0
$759$			27022.5i			0.0469075i
$760$		0			0
$761$			882187.i			1.52332i	0.647976	+	0.761661i	$0.275616\pi$
$761$			882187.i			1.52332i	−0.647976	+	0.761661i	$0.724384\pi$
$762$		0			0
$763$	328420.	+	735019.i	0.564132	+	1.26255i
$764$		0			0
$765$	−211620.			−0.361604
$766$		0			0
$767$	200081.			0.340107
$768$		0			0
$769$			10006.5i			0.0169212i	0.999964	+	0.00846060i	$0.00269312\pi$
$769$			10006.5i			0.0169212i	−0.999964	+	0.00846060i	$0.997307\pi$
$770$		0			0
$771$	−37321.3			−0.0627838
$772$		0			0
$773$		−	148529.i		−	0.248573i	−0.992246	−	0.124286i	$-0.960336\pi$
$773$		−	148529.i		−	0.248573i	0.992246	−	0.124286i	$-0.0396641\pi$
$774$		0			0
$775$			24362.4i			0.0405618i
$776$		0			0
$777$	623969.	−	278801.i	1.03353	−	0.461798i
$778$		0			0
$779$	−113102.			−0.186378
$780$		0			0
$781$	29799.2			0.0488542
$782$		0			0
$783$			187045.i			0.305087i
$784$		0			0
$785$	−25110.7			−0.0407492
$786$		0			0
$787$		−	954103.i		−	1.54044i	−0.637776	−	0.770222i	$-0.720145\pi$
$787$		−	954103.i		−	1.54044i	0.637776	−	0.770222i	$-0.279855\pi$
$788$		0			0
$789$			131491.i			0.211224i
$790$		0			0
$791$	388411.	+	869281.i	0.620781	+	1.38934i
$792$		0			0
$793$	488134.			0.776233
$794$		0			0
$795$	−22167.4			−0.0350736
$796$		0			0
$797$			105180.i			0.165583i	0.996567	+	0.0827914i	$0.0263835\pi$
$797$			105180.i			0.165583i	−0.996567	+	0.0827914i	$0.973616\pi$
$798$		0			0
$799$	−984936.			−1.54282
$800$		0			0
$801$		−	516695.i		−	0.805321i
$802$		0			0
$803$		−	968936.i		−	1.50267i
$804$		0			0
$805$	23191.9	−	10362.6i	0.0357887	−	0.0159910i
$806$		0			0
$807$	409365.			0.628585
$808$		0			0
$809$	13264.8			0.0202677			0.0101338	−	0.999949i	$-0.496774\pi$
$809$	13264.8			0.0202677			0.0101338	+	0.999949i	$0.496774\pi$
$810$		0			0
$811$			68424.4i			0.104033i	0.998646	+	0.0520163i	$0.0165648\pi$
$811$			68424.4i			0.104033i	−0.998646	+	0.0520163i	$0.983435\pi$
$812$		0			0
$813$	−339826.			−0.514133
$814$		0			0
$815$		−	302410.i		−	0.455283i
$816$		0			0
$817$		−	103354.i		−	0.154840i
$818$		0			0
$819$	528238.	−	236026.i	0.787520	−	0.351879i
$820$		0			0
$821$	23666.0			0.0351107			0.0175553	−	0.999846i	$-0.494412\pi$
$821$	23666.0			0.0351107			0.0175553	+	0.999846i	$0.494412\pi$
$822$		0			0
$823$	−541429.			−0.799358			−0.399679	−	0.916655i	$-0.630878\pi$
$823$	−541429.			−0.799358			−0.399679	+	0.916655i	$0.630878\pi$
$824$		0			0
$825$			72848.9i			0.107032i
$826$		0			0
$827$	486140.			0.710805			0.355402	−	0.934713i	$-0.384344\pi$
$827$	486140.			0.710805			0.355402	+	0.934713i	$0.384344\pi$
$828$		0			0
$829$			199179.i			0.289824i	0.989445	+	0.144912i	$0.0462900\pi$
$829$			199179.i			0.289824i	−0.989445	+	0.144912i	$0.953710\pi$
$830$		0			0
$831$			405310.i			0.586929i
$832$		0			0
$833$	−671154.	−	601095.i	−0.967236	−	0.866269i
$834$		0			0
$835$	−410246.			−0.588398
$836$		0			0
$837$	−141616.			−0.202144
$838$		0			0
$839$			1.05803e6i			1.50306i	0.659701	+	0.751528i	$0.270683\pi$
$839$			1.05803e6i			1.50306i	−0.659701	+	0.751528i	$0.729317\pi$
$840$		0			0
$841$	−641015.			−0.906309
$842$		0			0
$843$			351419.i			0.494504i
$844$		0			0
$845$		−	293335.i		−	0.410820i
$846$		0			0
$847$	−70498.3	−	157778.i	−0.0982679	−	0.219928i
$848$		0			0
$849$	−551007.			−0.764436
$850$		0			0
$851$	−116986.			−0.161538
$852$		0			0
$853$		−	659921.i		−	0.906972i	−0.891263	−	0.453486i	$-0.850180\pi$
$853$		−	659921.i		−	0.906972i	0.891263	−	0.453486i	$-0.149820\pi$
$854$		0			0
$855$	−25471.7			−0.0348439
$856$		0			0
$857$			776880.i			1.05777i	0.848693	+	0.528886i	$0.177390\pi$
$857$			776880.i			1.05777i	−0.848693	+	0.528886i	$0.822610\pi$
$858$		0			0
$859$			31227.0i			0.0423198i	0.999776	+	0.0211599i	$0.00673591\pi$
$859$			31227.0i			0.0423198i	−0.999776	+	0.0211599i	$0.993264\pi$
$860$		0			0
$861$	276706.	+	619281.i	0.373261	+	0.835375i
$862$		0			0
$863$	−197197.			−0.264776			−0.132388	−	0.991198i	$-0.542264\pi$
$863$	−197197.			−0.264776			−0.132388	+	0.991198i	$0.542264\pi$
$864$		0			0
$865$	35023.4			0.0468087
$866$		0			0
$867$			316718.i			0.421341i
$868$		0			0
$869$	757429.			1.00300
$870$		0			0
$871$		−	1.76930e6i		−	2.33220i
$872$		0			0
$873$			30711.2i			0.0402965i
$874$		0			0
$875$	62522.2	−	27936.1i	0.0816617	−	0.0364879i
$876$		0			0
$877$	−1.22846e6			−1.59720			−0.798602	−	0.601860i	$-0.794427\pi$
$877$	−1.22846e6			−1.59720			−0.798602	+	0.601860i	$0.794427\pi$
$878$		0			0
$879$	184587.			0.238904
$880$		0			0
$881$			1.23726e6i			1.59408i	0.603927	+	0.797040i	$0.293602\pi$
$881$			1.23726e6i			1.59408i	−0.603927	+	0.797040i	$0.706398\pi$
$882$		0			0
$883$	−708608.			−0.908835			−0.454417	−	0.890789i	$-0.650152\pi$
$883$	−708608.			−0.908835			−0.454417	+	0.890789i	$0.650152\pi$
$884$		0			0
$885$			52826.8i			0.0674478i
$886$		0			0
$887$		−	48621.1i		−	0.0617984i	−0.999523	−	0.0308992i	$-0.990163\pi$
$887$		−	48621.1i		−	0.0617984i	0.999523	−	0.0308992i	$-0.00983709\pi$
$888$		0			0
$889$	69435.9	+	155401.i	0.0878578	+	0.196630i
$890$		0			0
$891$	7265.31			0.00915164
$892$		0			0
$893$	−118552.			−0.148665
$894$		0			0
$895$			288637.i			0.360334i
$896$		0			0
$897$	60002.2			0.0745732
$898$		0			0
$899$			50171.2i			0.0620776i
$900$		0			0
$901$		−	134589.i		−	0.165790i
$902$		0			0
$903$	−565907.	+	252858.i	−0.694016	+	0.310099i
$904$		0			0
$905$	−353460.			−0.431562
$906$		0			0
$907$	−1.42767e6			−1.73546			−0.867730	−	0.497036i	$-0.834422\pi$
$907$	−1.42767e6			−1.73546			−0.867730	+	0.497036i	$0.834422\pi$
$908$		0			0
$909$			754905.i			0.913617i
$910$		0			0
$911$	583391.			0.702947			0.351474	−	0.936198i	$-0.385681\pi$
$911$	583391.			0.702947			0.351474	+	0.936198i	$0.385681\pi$
$912$		0			0
$913$		−	533945.i		−	0.640553i
$914$		0			0
$915$			128880.i			0.153937i
$916$		0			0
$917$	−233281.	+	104234.i	−0.277422	+	0.123957i
$918$		0			0
$919$	−911636.			−1.07942			−0.539710	−	0.841851i	$-0.681466\pi$
$919$	−911636.			−1.07942			−0.539710	+	0.841851i	$0.681466\pi$
$920$		0			0
$921$	−448225.			−0.528417
$922$		0			0
$923$		−	66167.7i		−	0.0776681i
$924$		0			0
$925$	−315377.			−0.368593
$926$		0			0
$927$			950303.i			1.10587i
$928$		0			0
$929$		−	1.18209e6i		−	1.36968i	−0.728696	−	0.684838i	$-0.759873\pi$
$929$		−	1.18209e6i		−	1.36968i	0.728696	−	0.684838i	$-0.240127\pi$
$930$		0			0
$931$	−80783.8	−	72351.1i	−0.0932020	−	0.0834729i
$932$		0			0
$933$	−325112.			−0.373482
$934$		0			0
$935$	−442300.			−0.505934
$936$		0			0
$937$		−	648753.i		−	0.738925i	−0.929246	−	0.369463i	$-0.879542\pi$
$937$		−	648753.i		−	0.738925i	0.929246	−	0.369463i	$-0.120458\pi$
$938$		0			0
$939$	−810842.			−0.919613
$940$		0			0
$941$		−	539679.i		−	0.609475i	−0.952436	−	0.304738i	$-0.901431\pi$
$941$		−	539679.i		−	0.609475i	0.952436	−	0.304738i	$-0.0985688\pi$
$942$		0			0
$943$		−	116107.i		−	0.130567i
$944$		0			0
$945$	162390.	+	363435.i	0.181842	+	0.406971i
$946$		0			0
$947$	−380310.			−0.424071			−0.212035	−	0.977262i	$-0.568009\pi$
$947$	−380310.			−0.424071			−0.212035	+	0.977262i	$0.568009\pi$
$948$		0			0
$949$	−2.15148e6			−2.38894
$950$		0			0
$951$		−	586852.i		−	0.648884i
$952$		0			0
$953$	−1.51323e6			−1.66617			−0.833084	−	0.553147i	$-0.813427\pi$
$953$	−1.51323e6			−1.66617			−0.833084	+	0.553147i	$0.813427\pi$
$954$		0			0
$955$			52854.9i			0.0579534i
$956$		0			0
$957$			150023.i			0.163807i
$958$		0			0
$959$	500088.	+	1.11922e6i	0.543762	+	1.21696i
$960$		0			0
$961$	885535.			0.958869
$962$		0			0
$963$	−196908.			−0.212330
$964$		0			0
$965$		−	413148.i		−	0.443661i
$966$		0			0
$967$	393218.			0.420514			0.210257	−	0.977646i	$-0.432570\pi$
$967$	393218.			0.420514			0.210257	+	0.977646i	$0.432570\pi$
$968$		0			0
$969$			93695.8i			0.0997867i
$970$		0			0
$971$		−	957332.i		−	1.01537i	−0.861543	−	0.507685i	$-0.830502\pi$
$971$		−	957332.i		−	1.01537i	0.861543	−	0.507685i	$-0.169498\pi$
$972$		0			0
$973$	434992.	−	194362.i	0.459468	−	0.205299i
$974$		0			0
$975$	161758.			0.170159
$976$		0			0
$977$	1.33928e6			1.40308			0.701539	−	0.712631i	$-0.252497\pi$
$977$	1.33928e6			1.40308			0.701539	+	0.712631i	$0.252497\pi$
$978$		0			0
$979$		−	1.07993e6i		−	1.12675i
$980$		0			0
$981$	828721.			0.861133
$982$		0			0
$983$		−	1.72782e6i		−	1.78810i	−0.447968	−	0.894050i	$-0.647852\pi$
$983$		−	1.72782e6i		−	1.78810i	0.447968	−	0.894050i	$-0.352148\pi$
$984$		0			0
$985$		−	36027.5i		−	0.0371331i
$986$		0			0
$987$	290041.	+	649126.i	0.297732	+	0.666338i
$988$		0			0
$989$	106100.			0.108473
$990$		0			0
$991$	−946703.			−0.963977			−0.481988	−	0.876178i	$-0.660085\pi$
$991$	−946703.			−0.963977			−0.481988	+	0.876178i	$0.660085\pi$
$992$		0			0
$993$		−	134317.i		−	0.136217i
$994$		0			0
$995$	−581205.			−0.587061
$996$		0			0
$997$			552328.i			0.555657i	0.960631	+	0.277829i	$0.0896148\pi$
$997$			552328.i			0.555657i	−0.960631	+	0.277829i	$0.910385\pi$
$998$		0			0
$999$		−	1.83325e6i		−	1.83693i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.5.f.b.321.5		12
4.3	odd	2		35.5.d.a.6.12	yes	12
7.6	odd	2	inner	560.5.f.b.321.8		12
12.11	even	2		315.5.h.a.181.1		12
20.3	even	4		175.5.c.d.174.17		24
20.7	even	4		175.5.c.d.174.8		24
20.19	odd	2		175.5.d.i.76.1		12
28.27	even	2		35.5.d.a.6.11	✓	12
84.83	odd	2		315.5.h.a.181.2		12
140.27	odd	4		175.5.c.d.174.18		24
140.83	odd	4		175.5.c.d.174.7		24
140.139	even	2		175.5.d.i.76.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.d.a.6.11	✓	12	28.27	even	2
35.5.d.a.6.12	yes	12	4.3	odd	2
175.5.c.d.174.7		24	140.83	odd	4
175.5.c.d.174.8		24	20.7	even	4
175.5.c.d.174.17		24	20.3	even	4
175.5.c.d.174.18		24	140.27	odd	4
175.5.d.i.76.1		12	20.19	odd	2
175.5.d.i.76.2		12	140.139	even	2
315.5.h.a.181.1		12	12.11	even	2
315.5.h.a.181.2		12	84.83	odd	2
560.5.f.b.321.5		12	1.1	even	1	trivial
560.5.f.b.321.8		12	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-5.52807i q^{3} +11.1803i q^{5} +(19.9894 + 44.7373i) q^{7} +50.4404 q^{9} +O(q^{10})\)	\(q-5.52807i q^{3} +11.1803i q^{5} +(19.9894 + 44.7373i) q^{7} +50.4404 q^{9} +105.424 q^{11} -234.089i q^{13} +61.8057 q^{15} +375.251i q^{17} +45.1673i q^{19} +(247.311 - 110.503i) q^{21} -46.3674 q^{23} -125.000 q^{25} -726.612i q^{27} -257.421 q^{29} -194.899i q^{31} -582.791i q^{33} +(-500.178 + 223.489i) q^{35} +2523.02 q^{37} -1294.06 q^{39} +2504.06i q^{41} -2288.24 q^{43} +563.941i q^{45} +2624.74i q^{47} +(-1601.84 + 1788.55i) q^{49} +2074.42 q^{51} -358.663 q^{53} +1178.68i q^{55} +249.688 q^{57} +854.723i q^{59} +2085.25i q^{61} +(1008.28 + 2256.57i) q^{63} +2617.20 q^{65} +7558.23 q^{67} +256.322i q^{69} +282.660 q^{71} -9190.85i q^{73} +691.009i q^{75} +(2107.37 + 4716.38i) q^{77} +7184.60 q^{79} +68.9152 q^{81} -5064.74i q^{83} -4195.44 q^{85} +1423.04i q^{87} -10243.7i q^{89} +(10472.5 - 4679.31i) q^{91} -1077.42 q^{93} -504.986 q^{95} +608.860i q^{97} +5317.63 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 50 q^{7} - 434 q^{9}+O(q^{10}) \) 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

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Database

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