LMFDB - Embedded newform 320.6.f.d.289.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(289,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("320.289");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.f (of order $2$, degree $1$, 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:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.19
Character	$\chi$	$=$	320.289
Dual form			320.6.f.d.289.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.76807 q^{3} +(-43.2814 - 35.3796i) q^{5} -82.4139i q^{7} -220.266 q^{9} +O(q^{10})$	$q+4.76807 q^{3} +(-43.2814 - 35.3796i) q^{5} -82.4139i q^{7} -220.266 q^{9} -109.227i q^{11} -621.825 q^{13} +(-206.369 - 168.693i) q^{15} +503.868i q^{17} +853.149i q^{19} -392.955i q^{21} -1424.25i q^{23} +(621.562 + 3062.56i) q^{25} -2208.88 q^{27} -275.098i q^{29} +6206.91 q^{31} -520.801i q^{33} +(-2915.77 + 3566.99i) q^{35} +3668.93 q^{37} -2964.90 q^{39} -1266.95 q^{41} +7494.09 q^{43} +(9533.40 + 7792.91i) q^{45} +15862.9i q^{47} +10014.9 q^{49} +2402.48i q^{51} +10023.0 q^{53} +(-3864.41 + 4727.49i) q^{55} +4067.88i q^{57} +27988.4i q^{59} -39901.8i q^{61} +18152.9i q^{63} +(26913.5 + 21999.9i) q^{65} -66510.4 q^{67} -6790.95i q^{69} +32714.8 q^{71} +71539.0i q^{73} +(2963.65 + 14602.5i) q^{75} -9001.81 q^{77} +20741.8 q^{79} +42992.4 q^{81} -108500. q^{83} +(17826.7 - 21808.1i) q^{85} -1311.69i q^{87} +84870.1 q^{89} +51247.0i q^{91} +29595.0 q^{93} +(30184.1 - 36925.5i) q^{95} +38393.1i q^{97} +24058.9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 2944 q^{9}+O(q^{10}) $ 32 * q + 2944 * q^9	$ 32 q + 2944 q^{9} - 46528 q^{25} - 36768 q^{41} - 113920 q^{49} + 25376 q^{65} + 633472 q^{81} + 968704 q^{89}+O(q^{100}) $ 32 * q + 2944 * q^9 - 46528 * q^25 - 36768 * q^41 - 113920 * q^49 + 25376 * q^65 + 633472 * q^81 + 968704 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	4.76807			0.305872			0.152936	−	0.988236i	$-0.451127\pi$
$3$	4.76807			0.305872			0.152936	+	0.988236i	$0.451127\pi$
$4$		0			0
$5$	−43.2814	−	35.3796i	−0.774242	−	0.632890i
$5$	−43.2814	−	35.3796i	−0.774242	−	0.632890i
$6$		0			0
$7$		−	82.4139i		−	0.635705i	−0.948140	−	0.317852i	$-0.897038\pi$
$7$		−	82.4139i		−	0.635705i	0.948140	−	0.317852i	$-0.102962\pi$
$8$		0			0
$9$	−220.266			−0.906442
$10$		0			0
$11$		−	109.227i		−	0.272175i	−0.990697	−	0.136087i	$-0.956547\pi$
$11$		−	109.227i		−	0.272175i	0.990697	−	0.136087i	$-0.0434528\pi$
$12$		0			0
$13$	−621.825			−1.02049			−0.510246	−	0.860029i	$-0.670446\pi$
$13$	−621.825			−1.02049			−0.510246	+	0.860029i	$0.670446\pi$
$14$		0			0
$15$	−206.369	−	168.693i	−0.236819	−	0.193583i
$16$		0			0
$17$			503.868i			0.422858i	0.977393	+	0.211429i	$0.0678118\pi$
$17$			503.868i			0.422858i	−0.977393	+	0.211429i	$0.932188\pi$
$18$		0			0
$19$			853.149i			0.542177i	0.962554	+	0.271089i	$0.0873836\pi$
$19$			853.149i			0.542177i	−0.962554	+	0.271089i	$0.912616\pi$
$20$		0			0
$21$		−	392.955i		−	0.194444i
$22$		0			0
$23$		−	1424.25i		−	0.561394i	−0.959796	−	0.280697i	$-0.909434\pi$
$23$		−	1424.25i		−	0.561394i	0.959796	−	0.280697i	$-0.0905657\pi$
$24$		0			0
$25$	621.562	+	3062.56i	0.198900	+	0.980020i
$26$		0			0
$27$	−2208.88			−0.583127
$28$		0			0
$29$		−	275.098i		−	0.0607424i	−0.999539	−	0.0303712i	$-0.990331\pi$
$29$		−	275.098i		−	0.0607424i	0.999539	−	0.0303712i	$-0.00966894\pi$
$30$		0			0
$31$	6206.91			1.16004			0.580018	−	0.814604i	$-0.303046\pi$
$31$	6206.91			1.16004			0.580018	+	0.814604i	$0.303046\pi$
$32$		0			0
$33$		−	520.801i		−	0.0832505i
$34$		0			0
$35$	−2915.77	+	3566.99i	−0.402331	+	0.492189i
$36$		0			0
$37$	3668.93			0.440590			0.220295	−	0.975433i	$-0.429298\pi$
$37$	3668.93			0.440590			0.220295	+	0.975433i	$0.429298\pi$
$38$		0			0
$39$	−2964.90			−0.312140
$40$		0			0
$41$	−1266.95			−0.117706			−0.0588530	−	0.998267i	$-0.518744\pi$
$41$	−1266.95			−0.117706			−0.0588530	+	0.998267i	$0.518744\pi$
$42$		0			0
$43$	7494.09			0.618084			0.309042	−	0.951048i	$-0.399992\pi$
$43$	7494.09			0.618084			0.309042	+	0.951048i	$0.399992\pi$
$44$		0			0
$45$	9533.40	+	7792.91i	0.701805	+	0.573679i
$46$		0			0
$47$			15862.9i			1.04746i	0.851884	+	0.523731i	$0.175460\pi$
$47$			15862.9i			1.04746i	−0.851884	+	0.523731i	$0.824540\pi$
$48$		0			0
$49$	10014.9			0.595880
$50$		0			0
$51$			2402.48i			0.129340i
$52$		0			0
$53$	10023.0			0.490127			0.245064	−	0.969507i	$-0.421191\pi$
$53$	10023.0			0.490127			0.245064	+	0.969507i	$0.421191\pi$
$54$		0			0
$55$	−3864.41	+	4727.49i	−0.172257	+	0.210729i
$56$		0			0
$57$			4067.88i			0.165837i
$58$		0			0
$59$			27988.4i			1.04676i	0.852098	+	0.523382i	$0.175330\pi$
$59$			27988.4i			1.04676i	−0.852098	+	0.523382i	$0.824670\pi$
$60$		0			0
$61$		−	39901.8i		−	1.37299i	−0.727134	−	0.686496i	$-0.759148\pi$
$61$		−	39901.8i		−	1.37299i	0.727134	−	0.686496i	$-0.240852\pi$
$62$		0			0
$63$			18152.9i			0.576230i
$64$		0			0
$65$	26913.5	+	21999.9i	0.790107	+	0.645859i
$66$		0			0
$67$	−66510.4			−1.81010			−0.905049	−	0.425306i	$-0.860166\pi$
$67$	−66510.4			−1.81010			−0.905049	+	0.425306i	$0.860166\pi$
$68$		0			0
$69$		−	6790.95i		−	0.171715i
$70$		0			0
$71$	32714.8			0.770191			0.385095	−	0.922877i	$-0.374169\pi$
$71$	32714.8			0.770191			0.385095	+	0.922877i	$0.374169\pi$
$72$		0			0
$73$			71539.0i			1.57121i	0.618726	+	0.785607i	$0.287649\pi$
$73$			71539.0i			1.57121i	−0.618726	+	0.785607i	$0.712351\pi$
$74$		0			0
$75$	2963.65	+	14602.5i	0.0608379	+	0.299760i
$76$		0			0
$77$	−9001.81			−0.173023
$78$		0			0
$79$	20741.8			0.373920			0.186960	−	0.982368i	$-0.440137\pi$
$79$	20741.8			0.373920			0.186960	+	0.982368i	$0.440137\pi$
$80$		0			0
$81$	42992.4			0.728080
$82$		0			0
$83$	−108500.			−1.72876			−0.864381	−	0.502838i	$-0.832289\pi$
$83$	−108500.			−1.72876			−0.864381	+	0.502838i	$0.832289\pi$
$84$		0			0
$85$	17826.7	−	21808.1i	0.267623	−	0.327394i
$86$		0			0
$87$		−	1311.69i		−	0.0185794i
$88$		0			0
$89$	84870.1			1.13574			0.567871	−	0.823117i	$-0.307767\pi$
$89$	84870.1			1.13574			0.567871	+	0.823117i	$0.307767\pi$
$90$		0			0
$91$			51247.0i			0.648731i
$92$		0			0
$93$	29595.0			0.354822
$94$		0			0
$95$	30184.1	−	36925.5i	0.343139	−	0.419776i
$96$		0			0
$97$			38393.1i			0.414309i	0.978308	+	0.207155i	$0.0664203\pi$
$97$			38393.1i			0.414309i	−0.978308	+	0.207155i	$0.933580\pi$
$98$		0			0
$99$			24058.9i			0.246711i
$100$		0			0
$101$			1331.92i			0.0129919i	0.999979	+	0.00649597i	$0.00206775\pi$
$101$			1331.92i			0.0129919i	−0.999979	+	0.00649597i	$0.997932\pi$
$102$		0			0
$103$			114505.i			1.06348i	0.846907	+	0.531741i	$0.178462\pi$
$103$			114505.i			1.06348i	−0.846907	+	0.531741i	$0.821538\pi$
$104$		0			0
$105$	−13902.6	+	17007.7i	−0.123062	+	0.150547i
$106$		0			0
$107$	−78186.5			−0.660196			−0.330098	−	0.943947i	$-0.607082\pi$
$107$	−78186.5			−0.660196			−0.330098	+	0.943947i	$0.607082\pi$
$108$		0			0
$109$		−	48709.9i		−	0.392691i	−0.980535	−	0.196346i	$-0.937093\pi$
$109$		−	48709.9i		−	0.392691i	0.980535	−	0.196346i	$-0.0629075\pi$
$110$		0			0
$111$	17493.7			0.134764
$112$		0			0
$113$			162096.i			1.19420i	0.802168	+	0.597098i	$0.203680\pi$
$113$			162096.i			1.19420i	−0.802168	+	0.597098i	$0.796320\pi$
$114$		0			0
$115$	−50389.6	+	61643.8i	−0.355301	+	0.434655i
$116$		0			0
$117$	136967.			0.925017
$118$		0			0
$119$	41525.8			0.268813
$120$		0			0
$121$	149121.			0.925921
$122$		0			0
$123$	−6040.88			−0.0360029
$124$		0			0
$125$	81450.2	−	154543.i	0.466248	−	0.884654i
$126$		0			0
$127$			227166.i			1.24978i	0.780712	+	0.624891i	$0.214857\pi$
$127$			227166.i			1.24978i	−0.780712	+	0.624891i	$0.785143\pi$
$128$		0			0
$129$	35732.3			0.189055
$130$		0			0
$131$			194060.i			0.987999i	0.869462	+	0.494000i	$0.164466\pi$
$131$			194060.i			0.987999i	−0.869462	+	0.494000i	$0.835534\pi$
$132$		0			0
$133$	70311.4			0.344664
$134$		0			0
$135$	95603.6	+	78149.5i	0.451481	+	0.369055i
$136$		0			0
$137$			153702.i			0.699647i	0.936816	+	0.349823i	$0.113758\pi$
$137$			153702.i			0.699647i	−0.936816	+	0.349823i	$0.886242\pi$
$138$		0			0
$139$			242909.i			1.06637i	0.846000	+	0.533183i	$0.179004\pi$
$139$			242909.i			1.06637i	−0.846000	+	0.533183i	$0.820996\pi$
$140$		0			0
$141$			75635.5i			0.320389i
$142$		0			0
$143$			67919.9i			0.277752i
$144$		0			0
$145$	−9732.86	+	11906.6i	−0.0384433	+	0.0470293i
$146$		0			0
$147$	47752.0			0.182263
$148$		0			0
$149$			97400.3i			0.359414i	0.983720	+	0.179707i	$0.0575149\pi$
$149$			97400.3i			0.359414i	−0.983720	+	0.179707i	$0.942485\pi$
$150$		0			0
$151$	−161098.			−0.574972			−0.287486	−	0.957785i	$-0.592819\pi$
$151$	−161098.			−0.574972			−0.287486	+	0.957785i	$0.592819\pi$
$152$		0			0
$153$		−	110985.i		−	0.383297i
$154$		0			0
$155$	−268644.	−	219598.i	−0.898147	−	0.734175i
$156$		0			0
$157$	168524.			0.545647			0.272824	−	0.962064i	$-0.412042\pi$
$157$	168524.			0.545647			0.272824	+	0.962064i	$0.412042\pi$
$158$		0			0
$159$	47790.5			0.149916
$160$		0			0
$161$	−117378.			−0.356881
$162$		0			0
$163$	231681.			0.683000			0.341500	−	0.939882i	$-0.389065\pi$
$163$	231681.			0.683000			0.341500	+	0.939882i	$0.389065\pi$
$164$		0			0
$165$	−18425.8	+	22541.0i	−0.0526885	+	0.0644560i
$166$		0			0
$167$		−	674343.i		−	1.87107i	−0.353238	−	0.935534i	$-0.614919\pi$
$167$		−	674343.i		−	1.87107i	0.353238	−	0.935534i	$-0.385081\pi$
$168$		0			0
$169$	15372.9			0.0414037
$170$		0			0
$171$		−	187919.i		−	0.491452i
$172$		0			0
$173$	−637772.			−1.62013			−0.810065	−	0.586340i	$-0.800568\pi$
$173$	−637772.			−1.62013			−0.810065	+	0.586340i	$0.800568\pi$
$174$		0			0
$175$	252398.	−	51225.4i	0.623003	−	0.126442i
$176$		0			0
$177$			133451.i			0.320176i
$178$		0			0
$179$		−	341699.i		−	0.797096i	−0.917147	−	0.398548i	$-0.869514\pi$
$179$		−	341699.i		−	0.797096i	0.917147	−	0.398548i	$-0.130486\pi$
$180$		0			0
$181$		−	123083.i		−	0.279256i	−0.990204	−	0.139628i	$-0.955409\pi$
$181$		−	123083.i		−	0.279256i	0.990204	−	0.139628i	$-0.0445907\pi$
$182$		0			0
$183$		−	190255.i		−	0.419960i
$184$		0			0
$185$	−158796.	−	129805.i	−0.341123	−	0.278845i
$186$		0			0
$187$	55035.9			0.115091
$188$		0			0
$189$			182043.i			0.370697i
$190$		0			0
$191$	354073.			0.702279			0.351139	−	0.936323i	$-0.385794\pi$
$191$	354073.			0.702279			0.351139	+	0.936323i	$0.385794\pi$
$192$		0			0
$193$		−	357838.i		−	0.691502i	−0.938326	−	0.345751i	$-0.887624\pi$
$193$		−	357838.i		−	0.691502i	0.938326	−	0.345751i	$-0.112376\pi$
$194$		0			0
$195$	128325.	+	104897.i	0.241672	+	0.197550i
$196$		0			0
$197$	−418271.			−0.767878			−0.383939	−	0.923359i	$-0.625433\pi$
$197$	−418271.			−0.767878			−0.383939	+	0.923359i	$0.625433\pi$
$198$		0			0
$199$	−969952.			−1.73627			−0.868136	−	0.496326i	$-0.834682\pi$
$199$	−969952.			−1.73627			−0.868136	+	0.496326i	$0.834682\pi$
$200$		0			0
$201$	−317126.			−0.553658
$202$		0			0
$203$	−22671.9			−0.0386142
$204$		0			0
$205$	54835.2	+	44824.1i	0.0911328	+	0.0744949i
$206$		0			0
$207$			313714.i			0.508872i
$208$		0			0
$209$	93186.8			0.147567
$210$		0			0
$211$		−	545362.i		−	0.843293i	−0.906760	−	0.421647i	$-0.861452\pi$
$211$		−	545362.i		−	0.843293i	0.906760	−	0.421647i	$-0.138548\pi$
$212$		0			0
$213$	155986.			0.235580
$214$		0			0
$215$	−324355.	−	265138.i	−0.478546	−	0.391179i
$216$		0			0
$217$		−	511536.i		−	0.737440i
$218$		0			0
$219$			341103.i			0.480590i
$220$		0			0
$221$		−	313318.i		−	0.431523i
$222$		0			0
$223$			627315.i			0.844741i	0.906423	+	0.422370i	$0.138802\pi$
$223$			627315.i			0.844741i	−0.906423	+	0.422370i	$0.861198\pi$
$224$		0			0
$225$	−136909.	−	674577.i	−0.180291	−	0.888332i
$226$		0			0
$227$	1.00091e6			1.28923			0.644615	−	0.764508i	$-0.277018\pi$
$227$	1.00091e6			1.28923			0.644615	+	0.764508i	$0.277018\pi$
$228$		0			0
$229$			920954.i			1.16051i	0.814435	+	0.580255i	$0.197047\pi$
$229$			920954.i			1.16051i	−0.814435	+	0.580255i	$0.802953\pi$
$230$		0			0
$231$	−42921.3			−0.0529228
$232$		0			0
$233$		−	185771.i		−	0.224176i	−0.993698	−	0.112088i	$-0.964246\pi$
$233$		−	185771.i		−	0.224176i	0.993698	−	0.112088i	$-0.0357538\pi$
$234$		0			0
$235$	561224.	−	686570.i	0.662929	−	0.810989i
$236$		0			0
$237$	98898.3			0.114371
$238$		0			0
$239$	−665553.			−0.753682			−0.376841	−	0.926278i	$-0.622990\pi$
$239$	−665553.			−0.753682			−0.376841	+	0.926278i	$0.622990\pi$
$240$		0			0
$241$	−761168.			−0.844186			−0.422093	−	0.906553i	$-0.638704\pi$
$241$	−761168.			−0.844186			−0.422093	+	0.906553i	$0.638704\pi$
$242$		0			0
$243$	741749.			0.805826
$244$		0			0
$245$	−433461.	−	354325.i	−0.461355	−	0.377126i
$246$		0			0
$247$		−	530509.i		−	0.553287i
$248$		0			0
$249$	−517336.			−0.528780
$250$		0			0
$251$		−	1.24971e6i		−	1.25206i	−0.779799	−	0.626030i	$-0.784679\pi$
$251$		−	1.24971e6i		−	1.25206i	0.779799	−	0.626030i	$-0.215321\pi$
$252$		0			0
$253$	−155567.			−0.152797
$254$		0			0
$255$	84998.9	−	103983.i	0.0818583	−	0.100141i
$256$		0			0
$257$		−	1.80658e6i		−	1.70617i	−0.521769	−	0.853087i	$-0.674728\pi$
$257$		−	1.80658e6i		−	1.70617i	0.521769	−	0.853087i	$-0.325272\pi$
$258$		0			0
$259$		−	302371.i		−	0.280085i
$260$		0			0
$261$			60594.5i			0.0550595i
$262$		0			0
$263$		−	1.14041e6i		−	1.01665i	−0.861166	−	0.508324i	$-0.830265\pi$
$263$		−	1.14041e6i		−	1.01665i	0.861166	−	0.508324i	$-0.169735\pi$
$264$		0			0
$265$	−433811.	−	354611.i	−0.379477	−	0.310197i
$266$		0			0
$267$	404667.			0.347392
$268$		0			0
$269$			1.86107e6i			1.56813i	0.620679	+	0.784065i	$0.286857\pi$
$269$			1.86107e6i			1.56813i	−0.620679	+	0.784065i	$0.713143\pi$
$270$		0			0
$271$	1.08713e6			0.899204			0.449602	−	0.893229i	$-0.351566\pi$
$271$	1.08713e6			0.899204			0.449602	+	0.893229i	$0.351566\pi$
$272$		0			0
$273$			244349.i			0.198429i
$274$		0			0
$275$	334514.	−	67891.3i	0.266736	−	0.0541355i
$276$		0			0
$277$	−628877.			−0.492455			−0.246227	−	0.969212i	$-0.579191\pi$
$277$	−628877.			−0.492455			−0.246227	+	0.969212i	$0.579191\pi$
$278$		0			0
$279$	−1.36717e6			−1.05150
$280$		0			0
$281$	−817929.			−0.617945			−0.308972	−	0.951071i	$-0.599985\pi$
$281$	−817929.			−0.617945			−0.308972	+	0.951071i	$0.599985\pi$
$282$		0			0
$283$	519909.			0.385888			0.192944	−	0.981210i	$-0.438196\pi$
$283$	519909.			0.385888			0.192944	+	0.981210i	$0.438196\pi$
$284$		0			0
$285$	143920.	−	176063.i	0.104956	−	0.128398i
$286$		0			0
$287$			104414.i			0.0748262i
$288$		0			0
$289$	1.16597e6			0.821191
$290$		0			0
$291$			183061.i			0.126725i
$292$		0			0
$293$	−61931.1			−0.0421444			−0.0210722	−	0.999778i	$-0.506708\pi$
$293$	−61931.1			−0.0421444			−0.0210722	+	0.999778i	$0.506708\pi$
$294$		0			0
$295$	990221.	−	1.21138e6i	0.662486	−	0.810448i
$296$		0			0
$297$			241269.i			0.158712i
$298$		0			0
$299$			885637.i			0.572898i
$300$		0			0
$301$		−	617617.i		−	0.392919i
$302$		0			0
$303$			6350.68i			0.00397387i
$304$		0			0
$305$	−1.41171e6	+	1.72701e6i	−0.868953	+	1.06303i
$306$		0			0
$307$	−2.02897e6			−1.22865			−0.614327	−	0.789051i	$-0.710573\pi$
$307$	−2.02897e6			−1.22865			−0.614327	+	0.789051i	$0.710573\pi$
$308$		0			0
$309$			545966.i			0.325289i
$310$		0			0
$311$	−1.44330e6			−0.846166			−0.423083	−	0.906091i	$-0.639052\pi$
$311$	−1.44330e6			−0.846166			−0.423083	+	0.906091i	$0.639052\pi$
$312$		0			0
$313$		−	210398.i		−	0.121389i	−0.998156	−	0.0606947i	$-0.980668\pi$
$313$		−	210398.i		−	0.121389i	0.998156	−	0.0606947i	$-0.0193316\pi$
$314$		0			0
$315$	642244.	−	785685.i	0.364690	−	0.446141i
$316$		0			0
$317$	−719655.			−0.402231			−0.201116	−	0.979567i	$-0.564457\pi$
$317$	−719655.			−0.402231			−0.201116	+	0.979567i	$0.564457\pi$
$318$		0			0
$319$	−30048.0			−0.0165325
$320$		0			0
$321$	−372799.			−0.201935
$322$		0			0
$323$	−429875.			−0.229264
$324$		0			0
$325$	−386503.	−	1.90438e6i	−0.202976	−	1.00010i
$326$		0			0
$327$		−	232252.i		−	0.120113i
$328$		0			0
$329$	1.30733e6			0.665877
$330$		0			0
$331$			1.34910e6i			0.676822i	0.940998	+	0.338411i	$0.109889\pi$
$331$			1.34910e6i			0.676822i	−0.940998	+	0.338411i	$0.890111\pi$
$332$		0			0
$333$	−808138.			−0.399370
$334$		0			0
$335$	2.87866e6	+	2.35311e6i	1.40145	+	1.14559i
$336$		0			0
$337$			1.81431e6i			0.870234i	0.900374	+	0.435117i	$0.143293\pi$
$337$			1.81431e6i			0.870234i	−0.900374	+	0.435117i	$0.856707\pi$
$338$		0			0
$339$			772884.i			0.365271i
$340$		0			0
$341$		−	677961.i		−	0.315732i
$342$		0			0
$343$		−	2.21050e6i		−	1.01451i
$344$		0			0
$345$	−240261.	+	293922.i	−0.108677	+	0.132949i
$346$		0			0
$347$	826969.			0.368693			0.184347	−	0.982861i	$-0.440983\pi$
$347$	826969.			0.368693			0.184347	+	0.982861i	$0.440983\pi$
$348$		0			0
$349$			1.90016e6i			0.835077i	0.908659	+	0.417538i	$0.137107\pi$
$349$			1.90016e6i			0.835077i	−0.908659	+	0.417538i	$0.862893\pi$
$350$		0			0
$351$	1.37354e6			0.595076
$352$		0			0
$353$		−	1.54658e6i		−	0.660595i	−0.943877	−	0.330298i	$-0.892851\pi$
$353$		−	1.54658e6i		−	0.660595i	0.943877	−	0.330298i	$-0.107149\pi$
$354$		0			0
$355$	−1.41594e6	−	1.15744e6i	−0.596314	−	0.487446i
$356$		0			0
$357$	197998.			0.0822223
$358$		0			0
$359$	−4.20737e6			−1.72296			−0.861479	−	0.507792i	$-0.830462\pi$
$359$	−4.20737e6			−1.72296			−0.861479	+	0.507792i	$0.830462\pi$
$360$		0			0
$361$	1.74824e6			0.706044
$362$		0			0
$363$	711017.			0.283213
$364$		0			0
$365$	2.53102e6	−	3.09631e6i	0.994406	−	1.21650i
$366$		0			0
$367$			3.08199e6i			1.19445i	0.802075	+	0.597223i	$0.203729\pi$
$367$			3.08199e6i			1.19445i	−0.802075	+	0.597223i	$0.796271\pi$
$368$		0			0
$369$	279064.			0.106694
$370$		0			0
$371$		−	826036.i		−	0.311576i
$372$		0			0
$373$	3.22578e6			1.20050			0.600252	−	0.799811i	$-0.295067\pi$
$373$	3.22578e6			1.20050			0.600252	+	0.799811i	$0.295067\pi$
$374$		0			0
$375$	388361.	−	736870.i	0.142612	−	0.270591i
$376$		0			0
$377$			171063.i			0.0619871i
$378$		0			0
$379$			3.54075e6i			1.26619i	0.774076	+	0.633093i	$0.218215\pi$
$379$			3.54075e6i			1.26619i	−0.774076	+	0.633093i	$0.781785\pi$
$380$		0			0
$381$			1.08314e6i			0.382273i
$382$		0			0
$383$			4.84417e6i			1.68742i	0.536803	+	0.843708i	$0.319632\pi$
$383$			4.84417e6i			1.68742i	−0.536803	+	0.843708i	$0.680368\pi$
$384$		0			0
$385$	389611.	+	318481.i	0.133961	+	0.109504i
$386$		0			0
$387$	−1.65069e6			−0.560258
$388$		0			0
$389$			2.70292e6i			0.905648i	0.891600	+	0.452824i	$0.149583\pi$
$389$			2.70292e6i			0.905648i	−0.891600	+	0.452824i	$0.850417\pi$
$390$		0			0
$391$	717637.			0.237390
$392$		0			0
$393$			925289.i			0.302201i
$394$		0			0
$395$	−897733.	−	733837.i	−0.289504	−	0.236650i
$396$		0			0
$397$	2.85969e6			0.910631			0.455316	−	0.890330i	$-0.349526\pi$
$397$	2.85969e6			0.910631			0.455316	+	0.890330i	$0.349526\pi$
$398$		0			0
$399$	335250.			0.105423
$400$		0			0
$401$	2.78230e6			0.864058			0.432029	−	0.901860i	$-0.357798\pi$
$401$	2.78230e6			0.864058			0.432029	+	0.901860i	$0.357798\pi$
$402$		0			0
$403$	−3.85961e6			−1.18381
$404$		0			0
$405$	−1.86077e6	−	1.52106e6i	−0.563710	−	0.460795i
$406$		0			0
$407$		−	400745.i		−	0.119917i
$408$		0			0
$409$	6.46827e6			1.91196			0.955982	−	0.293425i	$-0.0947950\pi$
$409$	6.46827e6			1.91196			0.955982	+	0.293425i	$0.0947950\pi$
$410$		0			0
$411$			732863.i			0.214002i
$412$		0			0
$413$	2.30664e6			0.665432
$414$		0			0
$415$	4.69604e6	+	3.83870e6i	1.33848	+	1.09412i
$416$		0			0
$417$			1.15821e6i			0.326171i
$418$		0			0
$419$			5.24272e6i			1.45889i	0.684041	+	0.729444i	$0.260221\pi$
$419$			5.24272e6i			1.45889i	−0.684041	+	0.729444i	$0.739779\pi$
$420$		0			0
$421$		−	4.34887e6i		−	1.19583i	−0.801558	−	0.597917i	$-0.795995\pi$
$421$		−	4.34887e6i		−	1.19583i	0.801558	−	0.597917i	$-0.204005\pi$
$422$		0			0
$423$		−	3.49405e6i		−	0.949464i
$424$		0			0
$425$	−1.54313e6	+	313186.i	−0.414409	+	0.0841065i
$426$		0			0
$427$	−3.28846e6			−0.872817
$428$		0			0
$429$			323847.i			0.0849565i
$430$		0			0
$431$	5.96989e6			1.54801			0.774003	−	0.633181i	$-0.218251\pi$
$431$	5.96989e6			1.54801			0.774003	+	0.633181i	$0.218251\pi$
$432$		0			0
$433$			6.35336e6i			1.62848i	0.580525	+	0.814242i	$0.302847\pi$
$433$			6.35336e6i			1.62848i	−0.580525	+	0.814242i	$0.697153\pi$
$434$		0			0
$435$	−46407.0	+	56771.6i	−0.0117587	+	0.0143849i
$436$		0			0
$437$	1.21510e6			0.304375
$438$		0			0
$439$	−6.83485e6			−1.69265			−0.846326	−	0.532666i	$-0.821190\pi$
$439$	−6.83485e6			−1.69265			−0.846326	+	0.532666i	$0.821190\pi$
$440$		0			0
$441$	−2.20595e6			−0.540131
$442$		0			0
$443$	−6.02270e6			−1.45808			−0.729041	−	0.684470i	$-0.760034\pi$
$443$	−6.02270e6			−1.45808			−0.729041	+	0.684470i	$0.760034\pi$
$444$		0			0
$445$	−3.67330e6	−	3.00267e6i	−0.879339	−	0.718800i
$446$		0			0
$447$			464412.i			0.109935i
$448$		0			0
$449$	4.11636e6			0.963602			0.481801	−	0.876281i	$-0.339983\pi$
$449$	4.11636e6			0.963602			0.481801	+	0.876281i	$0.339983\pi$
$450$		0			0
$451$			138384.i			0.0320366i
$452$		0			0
$453$	−768124.			−0.175868
$454$		0			0
$455$	1.81310e6	−	2.21804e6i	0.410576	−	0.502275i
$456$		0			0
$457$			7.61258e6i			1.70507i	0.522673	+	0.852533i	$0.324935\pi$
$457$			7.61258e6i			1.70507i	−0.522673	+	0.852533i	$0.675065\pi$
$458$		0			0
$459$		−	1.11299e6i		−	0.246580i
$460$		0			0
$461$		−	4.54787e6i		−	0.996680i	−0.866982	−	0.498340i	$-0.833943\pi$
$461$		−	4.54787e6i		−	0.996680i	0.866982	−	0.498340i	$-0.166057\pi$
$462$		0			0
$463$			3.15983e6i			0.685033i	0.939512	+	0.342516i	$0.111279\pi$
$463$			3.15983e6i			0.685033i	−0.939512	+	0.342516i	$0.888721\pi$
$464$		0			0
$465$	−1.28091e6	−	1.04706e6i	−0.274718	−	0.224563i
$466$		0			0
$467$	5.77790e6			1.22596			0.612982	−	0.790097i	$-0.289970\pi$
$467$	5.77790e6			1.22596			0.612982	+	0.790097i	$0.289970\pi$
$468$		0			0
$469$			5.48138e6i			1.15069i
$470$		0			0
$471$	803533.			0.166898
$472$		0			0
$473$		−	818555.i		−	0.168227i
$474$		0			0
$475$	−2.61282e6	+	530285.i	−0.531344	+	0.107839i
$476$		0			0
$477$	−2.20773e6			−0.444272
$478$		0			0
$479$	−4.83200e6			−0.962250			−0.481125	−	0.876652i	$-0.659772\pi$
$479$	−4.83200e6			−0.962250			−0.481125	+	0.876652i	$0.659772\pi$
$480$		0			0
$481$	−2.28143e6			−0.449619
$482$		0			0
$483$	−559668.			−0.109160
$484$		0			0
$485$	1.35834e6	−	1.66171e6i	0.262212	−	0.320775i
$486$		0			0
$487$		−	2.82344e6i		−	0.539455i	−0.962937	−	0.269728i	$-0.913066\pi$
$487$		−	2.82344e6i		−	0.539455i	0.962937	−	0.269728i	$-0.0869337\pi$
$488$		0			0
$489$	1.10467e6			0.208911
$490$		0			0
$491$		−	425391.i		−	0.0796314i	−0.999207	−	0.0398157i	$-0.987323\pi$
$491$		−	425391.i		−	0.0796314i	0.999207	−	0.0398157i	$-0.0126771\pi$
$492$		0			0
$493$	138613.			0.0256854
$494$		0			0
$495$	851195.	−	1.04130e6i	0.156141	−	0.191014i
$496$		0			0
$497$		−	2.69615e6i		−	0.489614i
$498$		0			0
$499$			1.09645e7i			1.97122i	0.169022	+	0.985612i	$0.445939\pi$
$499$			1.09645e7i			1.97122i	−0.169022	+	0.985612i	$0.554061\pi$
$500$		0			0
$501$		−	3.21531e6i		−	0.572307i
$502$		0			0
$503$		−	673131.i		−	0.118626i	−0.998239	−	0.0593130i	$-0.981109\pi$
$503$		−	673131.i		−	0.118626i	0.998239	−	0.0593130i	$-0.0188910\pi$
$504$		0			0
$505$	47122.8	−	57647.3i	0.00822247	−	0.0100589i
$506$		0			0
$507$	73299.1			0.0126642
$508$		0			0
$509$			1.03530e7i			1.77122i	0.464434	+	0.885608i	$0.346258\pi$
$509$			1.03530e7i			1.77122i	−0.464434	+	0.885608i	$0.653742\pi$
$510$		0			0
$511$	5.89580e6			0.998828
$512$		0			0
$513$		−	1.88451e6i		−	0.316158i
$514$		0			0
$515$	4.05113e6	−	4.95592e6i	0.673067	−	0.823392i
$516$		0			0
$517$	1.73266e6			0.285093
$518$		0			0
$519$	−3.04094e6			−0.495552
$520$		0			0
$521$	−5.28373e6			−0.852799			−0.426399	−	0.904535i	$-0.640218\pi$
$521$	−5.28373e6			−0.852799			−0.426399	+	0.904535i	$0.640218\pi$
$522$		0			0
$523$	−208807.			−0.0333803			−0.0166901	−	0.999861i	$-0.505313\pi$
$523$	−208807.			−0.0333803			−0.0166901	+	0.999861i	$0.505313\pi$
$524$		0			0
$525$	1.20345e6	−	244246.i	0.190559	−	0.0386749i
$526$		0			0
$527$			3.12747e6i			0.490530i
$528$		0			0
$529$	4.40784e6			0.684836
$530$		0			0
$531$		−	6.16489e6i		−	0.948831i
$532$		0			0
$533$	787818.			0.120118
$534$		0			0
$535$	3.38402e6	+	2.76621e6i	0.511151	+	0.417831i
$536$		0			0
$537$		−	1.62924e6i		−	0.243809i
$538$		0			0
$539$		−	1.09390e6i		−	0.162183i
$540$		0			0
$541$		−	7.19366e6i		−	1.05671i	−0.849023	−	0.528356i	$-0.822809\pi$
$541$		−	7.19366e6i		−	1.05671i	0.849023	−	0.528356i	$-0.177191\pi$
$542$		0			0
$543$		−	586870.i		−	0.0854165i
$544$		0			0
$545$	−1.72334e6	+	2.10824e6i	−0.248531	+	0.304038i
$546$		0			0
$547$	−1.32353e6			−0.189133			−0.0945664	−	0.995519i	$-0.530146\pi$
$547$	−1.32353e6			−0.189133			−0.0945664	+	0.995519i	$0.530146\pi$
$548$		0			0
$549$			8.78899e6i			1.24454i
$550$		0			0
$551$	234699.			0.0329331
$552$		0			0
$553$		−	1.70941e6i		−	0.237702i
$554$		0			0
$555$	−757152.	−	618921.i	−0.104340	−	0.0852909i
$556$		0			0
$557$	919685.			0.125603			0.0628017	−	0.998026i	$-0.479996\pi$
$557$	919685.			0.125603			0.0628017	+	0.998026i	$0.479996\pi$
$558$		0			0
$559$	−4.66001e6			−0.630750
$560$		0			0
$561$	262415.			0.0352032
$562$		0			0
$563$	9.78497e6			1.30103			0.650517	−	0.759492i	$-0.274552\pi$
$563$	9.78497e6			1.30103			0.650517	+	0.759492i	$0.274552\pi$
$564$		0			0
$565$	5.73489e6	−	7.01574e6i	0.755795	−	0.924596i
$566$		0			0
$567$		−	3.54317e6i		−	0.462844i
$568$		0			0
$569$	−289576.			−0.0374958			−0.0187479	−	0.999824i	$-0.505968\pi$
$569$	−289576.			−0.0374958			−0.0187479	+	0.999824i	$0.505968\pi$
$570$		0			0
$571$			1.52990e6i			0.196369i	0.995168	+	0.0981845i	$0.0313035\pi$
$571$			1.52990e6i			0.196369i	−0.995168	+	0.0981845i	$0.968696\pi$
$572$		0			0
$573$	1.68825e6			0.214807
$574$		0			0
$575$	4.36187e6	−	885263.i	0.550178	−	0.111661i
$576$		0			0
$577$		−	3.25795e6i		−	0.407385i	−0.979035	−	0.203692i	$-0.934706\pi$
$577$		−	3.25795e6i		−	0.407385i	0.979035	−	0.203692i	$-0.0652942\pi$
$578$		0			0
$579$		−	1.70620e6i		−	0.211511i
$580$		0			0
$581$			8.94192e6i			1.09898i
$582$		0			0
$583$		−	1.09478e6i		−	0.133400i
$584$		0			0
$585$	−5.92811e6	−	4.84583e6i	−0.716187	−	0.585434i
$586$		0			0
$587$	902249.			0.108076			0.0540382	−	0.998539i	$-0.482791\pi$
$587$	902249.			0.108076			0.0540382	+	0.998539i	$0.482791\pi$
$588$		0			0
$589$			5.29542e6i			0.628944i
$590$		0			0
$591$	−1.99434e6			−0.234872
$592$		0			0
$593$		−	1.13280e7i		−	1.32287i	−0.750004	−	0.661433i	$-0.769949\pi$
$593$		−	1.13280e7i		−	1.32287i	0.750004	−	0.661433i	$-0.230051\pi$
$594$		0			0
$595$	−1.79729e6	−	1.46917e6i	−0.208126	−	0.170129i
$596$		0			0
$597$	−4.62480e6			−0.531077
$598$		0			0
$599$	−1.65503e7			−1.88468			−0.942342	−	0.334651i	$-0.891382\pi$
$599$	−1.65503e7			−1.88468			−0.942342	+	0.334651i	$0.891382\pi$
$600$		0			0
$601$	204374.			0.0230802			0.0115401	−	0.999933i	$-0.496327\pi$
$601$	204374.			0.0230802			0.0115401	+	0.999933i	$0.496327\pi$
$602$		0			0
$603$	1.46499e7			1.64075
$604$		0			0
$605$	−6.45415e6	−	5.27583e6i	−0.716886	−	0.586006i
$606$		0			0
$607$			5.18844e6i			0.571565i	0.958295	+	0.285782i	$0.0922534\pi$
$607$			5.18844e6i			0.571565i	−0.958295	+	0.285782i	$0.907747\pi$
$608$		0			0
$609$	−108101.			−0.0118110
$610$		0			0
$611$		−	9.86396e6i		−	1.06893i
$612$		0			0
$613$	−1.06424e7			−1.14390			−0.571951	−	0.820288i	$-0.693813\pi$
$613$	−1.06424e7			−1.14390			−0.571951	+	0.820288i	$0.693813\pi$
$614$		0			0
$615$	261458.	+	213724.i	0.0278750	+	0.0227859i
$616$		0			0
$617$		−	6.94619e6i		−	0.734571i	−0.930108	−	0.367286i	$-0.880287\pi$
$617$		−	6.94619e6i		−	0.734571i	0.930108	−	0.367286i	$-0.119713\pi$
$618$		0			0
$619$		−	2.38878e6i		−	0.250582i	−0.992120	−	0.125291i	$-0.960014\pi$
$619$		−	2.38878e6i		−	0.250582i	0.992120	−	0.125291i	$-0.0399864\pi$
$620$		0			0
$621$			3.14601e6i			0.327364i
$622$		0			0
$623$		−	6.99448e6i		−	0.721997i
$624$		0			0
$625$	−8.99295e6	+	3.80715e6i	−0.920878	+	0.389852i
$626$		0			0
$627$	444321.			0.0451365
$628$		0			0
$629$			1.84866e6i			0.186307i
$630$		0			0
$631$	−1.38123e6			−0.138100			−0.0690499	−	0.997613i	$-0.521997\pi$
$631$	−1.38123e6			−0.138100			−0.0690499	+	0.997613i	$0.521997\pi$
$632$		0			0
$633$		−	2.60032e6i		−	0.257940i
$634$		0			0
$635$	8.03705e6	−	9.83207e6i	0.790975	−	0.967633i
$636$		0			0
$637$	−6.22754e6			−0.608090
$638$		0			0
$639$	−7.20594e6			−0.698133
$640$		0			0
$641$	−1.39783e7			−1.34372			−0.671861	−	0.740677i	$-0.734505\pi$
$641$	−1.39783e7			−1.34372			−0.671861	+	0.740677i	$0.734505\pi$
$642$		0			0
$643$	−1.28822e6			−0.122875			−0.0614373	−	0.998111i	$-0.519568\pi$
$643$	−1.28822e6			−0.122875			−0.0614373	+	0.998111i	$0.519568\pi$
$644$		0			0
$645$	−1.54655e6	−	1.26420e6i	−0.146374	−	0.119651i
$646$		0			0
$647$		−	1.51716e7i		−	1.42485i	−0.701746	−	0.712427i	$-0.747596\pi$
$647$		−	1.51716e7i		−	1.42485i	0.701746	−	0.712427i	$-0.252404\pi$
$648$		0			0
$649$	3.05709e6			0.284902
$650$		0			0
$651$		−	2.43904e6i		−	0.225562i
$652$		0			0
$653$	−1.46639e7			−1.34575			−0.672877	−	0.739754i	$-0.734942\pi$
$653$	−1.46639e7			−1.34575			−0.672877	+	0.739754i	$0.734942\pi$
$654$		0			0
$655$	6.86576e6	−	8.39917e6i	0.625295	−	0.764950i
$656$		0			0
$657$		−	1.57576e7i		−	1.42422i
$658$		0			0
$659$		−	91638.5i		−	0.00821986i	−0.999992	−	0.00410993i	$-0.998692\pi$
$659$		−	91638.5i		−	0.00821986i	0.999992	−	0.00410993i	$-0.00130824\pi$
$660$		0			0
$661$		−	3.06350e6i		−	0.272718i	−0.990659	−	0.136359i	$-0.956460\pi$
$661$		−	3.06350e6i		−	0.272718i	0.990659	−	0.136359i	$-0.0435400\pi$
$662$		0			0
$663$		−	1.49392e6i		−	0.131991i
$664$		0			0
$665$	−3.04318e6	−	2.48759e6i	−0.266854	−	0.218135i
$666$		0			0
$667$	−391809.			−0.0341004
$668$		0			0
$669$			2.99108e6i			0.258382i
$670$		0			0
$671$	−4.35835e6			−0.373694
$672$		0			0
$673$			1.55354e7i			1.32216i	0.750316	+	0.661079i	$0.229901\pi$
$673$			1.55354e7i			1.32216i	−0.750316	+	0.661079i	$0.770099\pi$
$674$		0			0
$675$	−1.37296e6	−	6.76484e6i	−0.115984	−	0.571476i
$676$		0			0
$677$	−5.41038e6			−0.453687			−0.226843	−	0.973931i	$-0.572841\pi$
$677$	−5.41038e6			−0.453687			−0.226843	+	0.973931i	$0.572841\pi$
$678$		0			0
$679$	3.16413e6			0.263378
$680$		0			0
$681$	4.77241e6			0.394339
$682$		0			0
$683$	−1.26434e6			−0.103708			−0.0518542	−	0.998655i	$-0.516513\pi$
$683$	−1.26434e6			−0.103708			−0.0518542	+	0.998655i	$0.516513\pi$
$684$		0			0
$685$	5.43793e6	−	6.65245e6i	0.442800	−	0.541696i
$686$		0			0
$687$			4.39117e6i			0.354968i
$688$		0			0
$689$	−6.23256e6			−0.500171
$690$		0			0
$691$		−	867712.i		−	0.0691323i	−0.999402	−	0.0345661i	$-0.988995\pi$
$691$		−	867712.i		−	0.0691323i	0.999402	−	0.0345661i	$-0.0110049\pi$
$692$		0			0
$693$	1.98279e6			0.156835
$694$		0			0
$695$	8.59402e6	−	1.05134e7i	0.674892	−	0.825624i
$696$		0			0
$697$		−	638374.i		−	0.0497729i
$698$		0			0
$699$		−	885770.i		−	0.0685690i
$700$		0			0
$701$			6.82392e6i			0.524492i	0.965001	+	0.262246i	$0.0844632\pi$
$701$			6.82392e6i			0.524492i	−0.965001	+	0.262246i	$0.915537\pi$
$702$		0			0
$703$			3.13014e6i			0.238878i
$704$		0			0
$705$	2.67596e6	−	3.27361e6i	0.202771	−	0.248059i
$706$		0			0
$707$	109769.			0.00825904
$708$		0			0
$709$			1.03780e7i			0.775350i	0.921796	+	0.387675i	$0.126722\pi$
$709$			1.03780e7i			0.775350i	−0.921796	+	0.387675i	$0.873278\pi$
$710$		0			0
$711$	−4.56870e6			−0.338937
$712$		0			0
$713$		−	8.84022e6i		−	0.651237i
$714$		0			0
$715$	2.40298e6	−	2.93967e6i	0.175786	−	0.215047i
$716$		0			0
$717$	−3.17341e6			−0.230530
$718$		0			0
$719$	−5.77556e6			−0.416650			−0.208325	−	0.978060i	$-0.566801\pi$
$719$	−5.77556e6			−0.416650			−0.208325	+	0.978060i	$0.566801\pi$
$720$		0			0
$721$	9.43677e6			0.676060
$722$		0			0
$723$	−3.62930e6			−0.258213
$724$		0			0
$725$	842504.	−	170990.i	0.0595288	−	0.0120817i
$726$		0			0
$727$			1.08256e7i			0.759653i	0.925058	+	0.379827i	$0.124016\pi$
$727$			1.08256e7i			0.759653i	−0.925058	+	0.379827i	$0.875984\pi$
$728$		0			0
$729$	−6.91044e6			−0.481601
$730$		0			0
$731$			3.77603e6i			0.261362i
$732$		0			0
$733$	2.58032e7			1.77383			0.886917	−	0.461929i	$-0.152842\pi$
$733$	2.58032e7			1.77383			0.886917	+	0.461929i	$0.152842\pi$
$734$		0			0
$735$	−2.06677e6	−	1.68945e6i	−0.141115	−	0.115352i
$736$		0			0
$737$			7.26471e6i			0.492663i
$738$		0			0
$739$			1.32347e7i			0.891459i	0.895168	+	0.445730i	$0.147056\pi$
$739$			1.32347e7i			0.891459i	−0.895168	+	0.445730i	$0.852944\pi$
$740$		0			0
$741$		−	2.52951e6i		−	0.169235i
$742$		0			0
$743$			1.33971e7i			0.890304i	0.895455	+	0.445152i	$0.146850\pi$
$743$			1.33971e7i			0.890304i	−0.895455	+	0.445152i	$0.853150\pi$
$744$		0			0
$745$	3.44599e6	−	4.21562e6i	0.227469	−	0.278273i
$746$		0			0
$747$	2.38988e7			1.56702
$748$		0			0
$749$			6.44366e6i			0.419689i
$750$		0			0
$751$	3.63485e6			0.235173			0.117586	−	0.993063i	$-0.462484\pi$
$751$	3.63485e6			0.235173			0.117586	+	0.993063i	$0.462484\pi$
$752$		0			0
$753$		−	5.95870e6i		−	0.382970i
$754$		0			0
$755$	6.97253e6	+	5.69957e6i	0.445167	+	0.363894i
$756$		0			0
$757$	2.53597e7			1.60844			0.804219	−	0.594332i	$-0.202584\pi$
$757$	2.53597e7			1.60844			0.804219	+	0.594332i	$0.202584\pi$
$758$		0			0
$759$	−741753.			−0.0467364
$760$		0			0
$761$	−9.42741e6			−0.590107			−0.295053	−	0.955481i	$-0.595337\pi$
$761$	−9.42741e6			−0.590107			−0.295053	+	0.955481i	$0.595337\pi$
$762$		0			0
$763$	−4.01438e6			−0.249636
$764$		0			0
$765$	−3.92660e6	+	4.80358e6i	−0.242585	+	0.296764i
$766$		0			0
$767$		−	1.74039e7i		−	1.06821i
$768$		0			0
$769$	1.77861e7			1.08459			0.542293	−	0.840189i	$-0.317556\pi$
$769$	1.77861e7			1.08459			0.542293	+	0.840189i	$0.317556\pi$
$770$		0			0
$771$		−	8.61388e6i		−	0.521871i
$772$		0			0
$773$	−2.23501e7			−1.34534			−0.672668	−	0.739944i	$-0.734852\pi$
$773$	−2.23501e7			−1.34534			−0.672668	+	0.739944i	$0.734852\pi$
$774$		0			0
$775$	3.85798e6	+	1.90090e7i	0.230731	+	1.13686i
$776$		0			0
$777$		−	1.44172e6i		−	0.0856702i
$778$		0			0
$779$		−	1.08089e6i		−	0.0638174i
$780$		0			0
$781$		−	3.57333e6i		−	0.209626i
$782$		0			0
$783$			607659.i			0.0354205i
$784$		0			0
$785$	−7.29395e6	−	5.96231e6i	−0.422463	−	0.345335i
$786$		0			0
$787$	−1.53456e7			−0.883177			−0.441589	−	0.897218i	$-0.645585\pi$
$787$	−1.53456e7			−0.883177			−0.441589	+	0.897218i	$0.645585\pi$
$788$		0			0
$789$		−	5.43754e6i		−	0.310964i
$790$		0			0
$791$	1.33589e7			0.759156
$792$		0			0
$793$			2.48119e7i			1.40113i
$794$		0			0
$795$	−2.06844e6	−	1.69081e6i	−0.116071	−	0.0948805i
$796$		0			0
$797$	−986256.			−0.0549976			−0.0274988	−	0.999622i	$-0.508754\pi$
$797$	−986256.			−0.0549976			−0.0274988	+	0.999622i	$0.508754\pi$
$798$		0			0
$799$	−7.99282e6			−0.442928
$800$		0			0
$801$	−1.86940e7			−1.02949
$802$		0			0
$803$	7.81397e6			0.427645
$804$		0			0
$805$	5.08030e6	+	4.15280e6i	0.276312	+	0.225866i
$806$		0			0
$807$			8.87371e6i			0.479647i
$808$		0			0
$809$	1.05342e6			0.0565890			0.0282945	−	0.999600i	$-0.490992\pi$
$809$	1.05342e6			0.0565890			0.0282945	+	0.999600i	$0.490992\pi$
$810$		0			0
$811$			2.33495e7i			1.24660i	0.781985	+	0.623298i	$0.214208\pi$
$811$			2.33495e7i			1.24660i	−0.781985	+	0.623298i	$0.785792\pi$
$812$		0			0
$813$	5.18351e6			0.275041
$814$		0			0
$815$	−1.00275e7	−	8.19678e6i	−0.528807	−	0.432264i
$816$		0			0
$817$			6.39358e6i			0.335111i
$818$		0			0
$819$		−	1.12879e7i		−	0.588038i
$820$		0			0
$821$		−	2.20305e7i		−	1.14069i	−0.821405	−	0.570345i	$-0.806810\pi$
$821$		−	2.20305e7i		−	1.14069i	0.821405	−	0.570345i	$-0.193190\pi$
$822$		0			0
$823$			1.96355e7i			1.01051i	0.862969	+	0.505257i	$0.168602\pi$
$823$			1.96355e7i			1.01051i	−0.862969	+	0.505257i	$0.831398\pi$
$824$		0			0
$825$	1.59499e6	−	323710.i	0.0815872	−	0.0165585i
$826$		0			0
$827$	−1.52997e7			−0.777891			−0.388945	−	0.921261i	$-0.627161\pi$
$827$	−1.52997e7			−0.777891			−0.388945	+	0.921261i	$0.627161\pi$
$828$		0			0
$829$		−	1.30033e7i		−	0.657153i	−0.944477	−	0.328577i	$-0.893431\pi$
$829$		−	1.30033e7i		−	0.657153i	0.944477	−	0.328577i	$-0.106569\pi$
$830$		0			0
$831$	−2.99853e6			−0.150628
$832$		0			0
$833$			5.04622e6i			0.251973i
$834$		0			0
$835$	−2.38580e7	+	2.91865e7i	−1.18418	+	1.44866i
$836$		0			0
$837$	−1.37103e7			−0.676448
$838$		0			0
$839$	1.78281e7			0.874381			0.437191	−	0.899369i	$-0.355973\pi$
$839$	1.78281e7			0.874381			0.437191	+	0.899369i	$0.355973\pi$
$840$		0			0
$841$	2.04355e7			0.996310
$842$		0			0
$843$	−3.89994e6			−0.189012
$844$		0			0
$845$	−665361.	−	543888.i	−0.0320565	−	0.0262040i
$846$		0			0
$847$		−	1.22896e7i		−	0.588612i
$848$		0			0
$849$	2.47896e6			0.118032
$850$		0			0
$851$		−	5.22549e6i		−	0.247345i
$852$		0			0
$853$	1.39396e7			0.655963			0.327981	−	0.944684i	$-0.393632\pi$
$853$	1.39396e7			0.655963			0.327981	+	0.944684i	$0.393632\pi$
$854$		0			0
$855$	−6.64852e6	+	8.13342e6i	−0.311035	+	0.380503i
$856$		0			0
$857$			4.09468e7i			1.90444i	0.305412	+	0.952220i	$0.401206\pi$
$857$			4.09468e7i			1.90444i	−0.305412	+	0.952220i	$0.598794\pi$
$858$		0			0
$859$			2.68350e7i			1.24085i	0.784267	+	0.620423i	$0.213039\pi$
$859$			2.68350e7i			1.24085i	−0.784267	+	0.620423i	$0.786961\pi$
$860$		0			0
$861$			497853.i			0.0228872i
$862$		0			0
$863$		−	2.28078e7i		−	1.04245i	−0.853419	−	0.521226i	$-0.825475\pi$
$863$		−	2.28078e7i		−	1.04245i	0.853419	−	0.521226i	$-0.174525\pi$
$864$		0			0
$865$	2.76037e7	+	2.25641e7i	1.25437	+	1.02536i
$866$		0			0
$867$	5.55944e6			0.251179
$868$		0			0
$869$		−	2.26556e6i		−	0.101771i
$870$		0			0
$871$	4.13578e7			1.84719
$872$		0			0
$873$		−	8.45669e6i		−	0.375547i
$874$		0			0
$875$	−1.27365e7	−	6.71263e6i	−0.562379	−	0.296396i
$876$		0			0
$877$	1.79568e7			0.788368			0.394184	−	0.919031i	$-0.371027\pi$
$877$	1.79568e7			0.788368			0.394184	+	0.919031i	$0.371027\pi$
$878$		0			0
$879$	−295292.			−0.0128908
$880$		0			0
$881$	3.16608e7			1.37430			0.687150	−	0.726515i	$-0.258861\pi$
$881$	3.16608e7			1.37430			0.687150	+	0.726515i	$0.258861\pi$
$882$		0			0
$883$	−2.60611e7			−1.12484			−0.562420	−	0.826852i	$-0.690129\pi$
$883$	−2.60611e7			−1.12484			−0.562420	+	0.826852i	$0.690129\pi$
$884$		0			0
$885$	4.72144e6	−	5.77594e6i	0.202636	−	0.247893i
$886$		0			0
$887$			2.57048e7i			1.09700i	0.836152	+	0.548498i	$0.184800\pi$
$887$			2.57048e7i			1.09700i	−0.836152	+	0.548498i	$0.815200\pi$
$888$		0			0
$889$	1.87216e7			0.794492
$890$		0			0
$891$		−	4.69592e6i		−	0.198165i
$892$		0			0
$893$	−1.35334e7			−0.567910
$894$		0			0
$895$	−1.20892e7	+	1.47892e7i	−0.504474	+	0.617145i
$896$		0			0
$897$			4.22278e6i			0.175233i
$898$		0			0
$899$		−	1.70751e6i		−	0.0704633i
$900$		0			0
$901$			5.05028e6i			0.207254i
$902$		0			0
$903$		−	2.94484e6i		−	0.120183i
$904$		0			0
$905$	−4.35464e6	+	5.32722e6i	−0.176738	+	0.216212i
$906$		0			0
$907$	−3.57667e7			−1.44364			−0.721822	−	0.692079i	$-0.756695\pi$
$907$	−3.57667e7			−1.44364			−0.721822	+	0.692079i	$0.756695\pi$
$908$		0			0
$909$		−	293376.i		−	0.0117764i
$910$		0			0
$911$	−2.63397e7			−1.05151			−0.525757	−	0.850635i	$-0.676218\pi$
$911$	−2.63397e7			−1.05151			−0.525757	+	0.850635i	$0.676218\pi$
$912$		0			0
$913$			1.18511e7i			0.470525i
$914$		0			0
$915$	−6.73114e6	+	8.23449e6i	−0.265788	+	0.325150i
$916$		0			0
$917$	1.59932e7			0.628076
$918$		0			0
$919$	4.57454e7			1.78673			0.893364	−	0.449333i	$-0.148338\pi$
$919$	4.57454e7			1.78673			0.893364	+	0.449333i	$0.148338\pi$
$920$		0			0
$921$	−9.67428e6			−0.375811
$922$		0			0
$923$	−2.03429e7			−0.785973
$924$		0			0
$925$	2.28047e6	+	1.12363e7i	0.0876334	+	0.431787i
$926$		0			0
$927$		−	2.52214e7i		−	0.963985i
$928$		0			0
$929$	−1.27869e7			−0.486100			−0.243050	−	0.970014i	$-0.578148\pi$
$929$	−1.27869e7			−0.486100			−0.243050	+	0.970014i	$0.578148\pi$
$930$		0			0
$931$			8.54425e6i			0.323072i
$932$		0			0
$933$	−6.88175e6			−0.258818
$934$		0			0
$935$	−2.38203e6	−	1.94715e6i	−0.0891084	−	0.0728401i
$936$		0			0
$937$			2.18690e7i			0.813728i	0.913489	+	0.406864i	$0.133378\pi$
$937$			2.18690e7i			0.813728i	−0.913489	+	0.406864i	$0.866622\pi$
$938$		0			0
$939$		−	1.00319e6i		−	0.0371296i
$940$		0			0
$941$		−	1.18344e7i		−	0.435683i	−0.975984	−	0.217842i	$-0.930098\pi$
$941$		−	1.18344e7i		−	0.435683i	0.975984	−	0.217842i	$-0.0699016\pi$
$942$		0			0
$943$			1.80445e6i			0.0660794i
$944$		0			0
$945$	6.44060e6	−	7.87906e6i	0.234610	−	0.287009i
$946$		0			0
$947$	1.94759e7			0.705704			0.352852	−	0.935679i	$-0.385212\pi$
$947$	1.94759e7			0.705704			0.352852	+	0.935679i	$0.385212\pi$
$948$		0			0
$949$		−	4.44847e7i		−	1.60341i
$950$		0			0
$951$	−3.43136e6			−0.123031
$952$		0			0
$953$		−	3.02336e7i		−	1.07834i	−0.842195	−	0.539172i	$-0.818737\pi$
$953$		−	3.02336e7i		−	1.07834i	0.842195	−	0.539172i	$-0.181263\pi$
$954$		0			0
$955$	−1.53248e7	−	1.25270e7i	−0.543733	−	0.444465i
$956$		0			0
$957$	−143271.			−0.00505684
$958$		0			0
$959$	1.26672e7			0.444769
$960$		0			0
$961$	9.89656e6			0.345681
$962$		0			0
$963$	1.72218e7			0.598429
$964$		0			0
$965$	−1.26602e7	+	1.54878e7i	−0.437645	+	0.535390i
$966$		0			0
$967$			4.84618e7i			1.66661i	0.552814	+	0.833304i	$0.313554\pi$
$967$			4.84618e7i			1.66661i	−0.552814	+	0.833304i	$0.686446\pi$
$968$		0			0
$969$	−2.04967e6			−0.0701254
$970$		0			0
$971$			1.73687e7i			0.591181i	0.955315	+	0.295590i	$0.0955163\pi$
$971$			1.73687e7i			0.591181i	−0.955315	+	0.295590i	$0.904484\pi$
$972$		0			0
$973$	2.00190e7			0.677893
$974$		0			0
$975$	−1.84287e6	−	9.08020e6i	−0.0620846	−	0.305903i
$976$		0			0
$977$		−	1.62432e7i		−	0.544422i	−0.962238	−	0.272211i	$-0.912245\pi$
$977$		−	1.62432e7i		−	0.544422i	0.962238	−	0.272211i	$-0.0877549\pi$
$978$		0			0
$979$		−	9.27009e6i		−	0.309120i
$980$		0			0
$981$			1.07291e7i			0.355952i
$982$		0			0
$983$			2.04107e7i			0.673713i	0.941556	+	0.336856i	$0.109364\pi$
$983$			2.04107e7i			0.673713i	−0.941556	+	0.336856i	$0.890636\pi$
$984$		0			0
$985$	1.81034e7	+	1.47983e7i	0.594523	+	0.485982i
$986$		0			0
$987$	6.23342e6			0.203673
$988$		0			0
$989$		−	1.06735e7i		−	0.346989i
$990$		0			0
$991$	−1.63999e7			−0.530464			−0.265232	−	0.964185i	$-0.585449\pi$
$991$	−1.63999e7			−0.530464			−0.265232	+	0.964185i	$0.585449\pi$
$992$		0			0
$993$			6.43261e6i			0.207021i
$994$		0			0
$995$	4.19809e7	+	3.43166e7i	1.34429	+	1.09887i
$996$		0			0
$997$	−1.61288e7			−0.513881			−0.256941	−	0.966427i	$-0.582715\pi$
$997$	−1.61288e7			−0.513881			−0.256941	+	0.966427i	$0.582715\pi$
$998$		0			0
$999$	−8.10423e6			−0.256920

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.f.d.289.19	yes	32
4.3	odd	2	inner	320.6.f.d.289.14	yes	32
5.4	even	2	inner	320.6.f.d.289.13	✓	32
8.3	odd	2	inner	320.6.f.d.289.17	yes	32
8.5	even	2	inner	320.6.f.d.289.16	yes	32
20.19	odd	2	inner	320.6.f.d.289.20	yes	32
40.19	odd	2	inner	320.6.f.d.289.15	yes	32
40.29	even	2	inner	320.6.f.d.289.18	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.f.d.289.13	✓	32	5.4	even	2	inner
320.6.f.d.289.14	yes	32	4.3	odd	2	inner
320.6.f.d.289.15	yes	32	40.19	odd	2	inner
320.6.f.d.289.16	yes	32	8.5	even	2	inner
320.6.f.d.289.17	yes	32	8.3	odd	2	inner
320.6.f.d.289.18	yes	32	40.29	even	2	inner
320.6.f.d.289.19	yes	32	1.1	even	1	trivial
320.6.f.d.289.20	yes	32	20.19	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.76807 q^{3} +(-43.2814 - 35.3796i) q^{5} -82.4139i q^{7} -220.266 q^{9} +O(q^{10})\)	\(q+4.76807 q^{3} +(-43.2814 - 35.3796i) q^{5} -82.4139i q^{7} -220.266 q^{9} -109.227i q^{11} -621.825 q^{13} +(-206.369 - 168.693i) q^{15} +503.868i q^{17} +853.149i q^{19} -392.955i q^{21} -1424.25i q^{23} +(621.562 + 3062.56i) q^{25} -2208.88 q^{27} -275.098i q^{29} +6206.91 q^{31} -520.801i q^{33} +(-2915.77 + 3566.99i) q^{35} +3668.93 q^{37} -2964.90 q^{39} -1266.95 q^{41} +7494.09 q^{43} +(9533.40 + 7792.91i) q^{45} +15862.9i q^{47} +10014.9 q^{49} +2402.48i q^{51} +10023.0 q^{53} +(-3864.41 + 4727.49i) q^{55} +4067.88i q^{57} +27988.4i q^{59} -39901.8i q^{61} +18152.9i q^{63} +(26913.5 + 21999.9i) q^{65} -66510.4 q^{67} -6790.95i q^{69} +32714.8 q^{71} +71539.0i q^{73} +(2963.65 + 14602.5i) q^{75} -9001.81 q^{77} +20741.8 q^{79} +42992.4 q^{81} -108500. q^{83} +(17826.7 - 21808.1i) q^{85} -1311.69i q^{87} +84870.1 q^{89} +51247.0i q^{91} +29595.0 q^{93} +(30184.1 - 36925.5i) q^{95} +38393.1i q^{97} +24058.9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2944 q^{9}+O(q^{10}) \) 32 * q + 2944 * q^9	\( 32 q + 2944 q^{9} - 46528 q^{25} - 36768 q^{41} - 113920 q^{49} + 25376 q^{65} + 633472 q^{81} + 968704 q^{89}+O(q^{100}) \) 32 * q + 2944 * q^9 - 46528 * q^25 - 36768 * q^41 - 113920 * q^49 + 25376 * q^65 + 633472 * q^81 + 968704 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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