LMFDB - Embedded newform 720.6.o.d.719.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,6,Mod(719,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("720.719");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	720.o (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:	$115.476350265$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			719.16
Character	$\chi$	$=$	720.719
Dual form			720.6.o.d.719.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-25.8892 + 49.5454i) q^{5} +154.886 q^{7} +O(q^{10})$	$q+(-25.8892 + 49.5454i) q^{5} +154.886 q^{7} -104.125 q^{11} -727.772i q^{13} -484.801 q^{17} -877.159i q^{19} +1026.82i q^{23} +(-1784.50 - 2565.38i) q^{25} +6378.27i q^{29} +7639.78i q^{31} +(-4009.88 + 7673.91i) q^{35} -5549.52i q^{37} -4746.20i q^{41} -4329.04 q^{43} +16288.7i q^{47} +7182.77 q^{49} +3180.41 q^{53} +(2695.71 - 5158.92i) q^{55} -13461.7 q^{59} -10129.2 q^{61} +(36057.8 + 18841.4i) q^{65} +40768.3 q^{67} -29907.5 q^{71} +42516.5i q^{73} -16127.5 q^{77} -27059.7i q^{79} -97816.4i q^{83} +(12551.1 - 24019.7i) q^{85} -54193.0i q^{89} -112722. i q^{91} +(43459.2 + 22708.9i) q^{95} +124356. i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q+O(q^{10}) $ 40 * q	$ 40 q + 10648 q^{25} + 36248 q^{49} - 133136 q^{61} - 336816 q^{85}+O(q^{100}) $ 40 * q + 10648 * q^25 + 36248 * q^49 - 133136 * q^61 - 336816 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−25.8892	+	49.5454i	−0.463119	+	0.886296i
$5$	−25.8892	+	49.5454i	−0.463119	+	0.886296i
$6$		0			0
$7$	154.886			1.19472			0.597362	−	0.801971i	$-0.296215\pi$
$7$	154.886			1.19472			0.597362	+	0.801971i	$0.296215\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−104.125			−0.259462			−0.129731	−	0.991549i	$-0.541411\pi$
$11$	−104.125			−0.259462			−0.129731	+	0.991549i	$0.541411\pi$
$12$		0			0
$13$		−	727.772i		−	1.19436i	−0.802106	−	0.597182i	$-0.796287\pi$
$13$		−	727.772i		−	1.19436i	0.802106	−	0.597182i	$-0.203713\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−484.801			−0.406856			−0.203428	−	0.979090i	$-0.565208\pi$
$17$	−484.801			−0.406856			−0.203428	+	0.979090i	$0.565208\pi$
$18$		0			0
$19$		−	877.159i		−	0.557435i	−0.960373	−	0.278718i	$-0.910091\pi$
$19$		−	877.159i		−	0.557435i	0.960373	−	0.278718i	$-0.0899094\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1026.82i			0.404740i	0.979309	+	0.202370i	$0.0648644\pi$
$23$			1026.82i			0.404740i	−0.979309	+	0.202370i	$0.935136\pi$
$24$		0			0
$25$	−1784.50	−	2565.38i	−0.571041	−	0.820922i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6378.27i			1.40834i	0.710031	+	0.704170i	$0.248681\pi$
$29$			6378.27i			1.40834i	−0.710031	+	0.704170i	$0.751319\pi$
$30$		0			0
$31$			7639.78i			1.42783i	0.700232	+	0.713915i	$0.253080\pi$
$31$			7639.78i			1.42783i	−0.700232	+	0.713915i	$0.746920\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4009.88	+	7673.91i	−0.553300	+	1.05888i
$36$		0			0
$37$		−	5549.52i		−	0.666425i	−0.942852	−	0.333213i	$-0.891867\pi$
$37$		−	5549.52i		−	0.666425i	0.942852	−	0.333213i	$-0.108133\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4746.20i		−	0.440947i	−0.975393	−	0.220473i	$-0.929240\pi$
$41$		−	4746.20i		−	0.440947i	0.975393	−	0.220473i	$-0.0707602\pi$
$42$		0			0
$43$	−4329.04			−0.357043			−0.178521	−	0.983936i	$-0.557131\pi$
$43$	−4329.04			−0.357043			−0.178521	+	0.983936i	$0.557131\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			16288.7i			1.07557i	0.843081	+	0.537787i	$0.180740\pi$
$47$			16288.7i			1.07557i	−0.843081	+	0.537787i	$0.819260\pi$
$48$		0			0
$49$	7182.77			0.427368
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3180.41			0.155523			0.0777614	−	0.996972i	$-0.475223\pi$
$53$	3180.41			0.155523			0.0777614	+	0.996972i	$0.475223\pi$
$54$		0			0
$55$	2695.71	−	5158.92i	0.120162	−	0.229960i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−13461.7			−0.503467			−0.251733	−	0.967797i	$-0.581001\pi$
$59$	−13461.7			−0.503467			−0.251733	+	0.967797i	$0.581001\pi$
$60$		0			0
$61$	−10129.2			−0.348539			−0.174270	−	0.984698i	$-0.555756\pi$
$61$	−10129.2			−0.348539			−0.174270	+	0.984698i	$0.555756\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	36057.8	+	18841.4i	1.05856	+	0.553133i
$66$		0			0
$67$	40768.3			1.10952			0.554761	−	0.832010i	$-0.312810\pi$
$67$	40768.3			1.10952			0.554761	+	0.832010i	$0.312810\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−29907.5			−0.704099			−0.352049	−	0.935981i	$-0.614515\pi$
$71$	−29907.5			−0.704099			−0.352049	+	0.935981i	$0.614515\pi$
$72$		0			0
$73$			42516.5i			0.933793i	0.884312	+	0.466896i	$0.154628\pi$
$73$			42516.5i			0.933793i	−0.884312	+	0.466896i	$0.845372\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−16127.5			−0.309985
$78$		0			0
$79$		−	27059.7i		−	0.487815i	−0.969799	−	0.243907i	$-0.921571\pi$
$79$		−	27059.7i		−	0.487815i	0.969799	−	0.243907i	$-0.0784293\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	97816.4i		−	1.55853i	−0.626692	−	0.779267i	$-0.715591\pi$
$83$		−	97816.4i		−	1.55853i	0.626692	−	0.779267i	$-0.284409\pi$
$84$		0			0
$85$	12551.1	−	24019.7i	0.188423	−	0.360595i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	54193.0i		−	0.725218i	−0.931941	−	0.362609i	$-0.881886\pi$
$89$		−	54193.0i		−	0.725218i	0.931941	−	0.362609i	$-0.118114\pi$
$90$		0			0
$91$		−	112722.i		−	1.42694i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	43459.2	+	22708.9i	0.494053	+	0.258159i
$96$		0			0
$97$			124356.i			1.34195i	0.741479	+	0.670976i	$0.234125\pi$
$97$			124356.i			1.34195i	−0.741479	+	0.670976i	$0.765875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			61508.4i			0.599972i	0.953944	+	0.299986i	$0.0969820\pi$
$101$			61508.4i			0.599972i	−0.953944	+	0.299986i	$0.903018\pi$
$102$		0			0
$103$	−207669.			−1.92877			−0.964383	−	0.264510i	$-0.914790\pi$
$103$	−207669.			−1.92877			−0.964383	+	0.264510i	$0.914790\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			83795.7i			0.707558i	0.935329	+	0.353779i	$0.115104\pi$
$107$			83795.7i			0.707558i	−0.935329	+	0.353779i	$0.884896\pi$
$108$		0			0
$109$	−72230.8			−0.582312			−0.291156	−	0.956676i	$-0.594040\pi$
$109$	−72230.8			−0.582312			−0.291156	+	0.956676i	$0.594040\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−43446.5			−0.320080			−0.160040	−	0.987111i	$-0.551162\pi$
$113$	−43446.5			−0.320080			−0.160040	+	0.987111i	$0.551162\pi$
$114$		0			0
$115$	−50874.5	−	26583.6i	−0.358720	−	0.187443i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−75089.0			−0.486081
$120$		0			0
$121$	−150209.			−0.932680
$122$		0			0
$123$		0			0
$124$		0			0
$125$	173302.	−	21998.4i	0.992040	−	0.125926i
$126$		0			0
$127$	−128198.			−0.705295			−0.352647	−	0.935756i	$-0.614719\pi$
$127$	−128198.			−0.705295			−0.352647	+	0.935756i	$0.614719\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11161.1			−0.0568234			−0.0284117	−	0.999596i	$-0.509045\pi$
$131$	−11161.1			−0.0568234			−0.0284117	+	0.999596i	$0.509045\pi$
$132$		0			0
$133$		−	135860.i		−	0.665982i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−409309.			−1.86316			−0.931580	−	0.363537i	$-0.881569\pi$
$137$	−409309.			−1.86316			−0.931580	+	0.363537i	$0.881569\pi$
$138$		0			0
$139$		−	338018.i		−	1.48389i	−0.670459	−	0.741947i	$-0.733903\pi$
$139$		−	338018.i		−	1.48389i	0.670459	−	0.741947i	$-0.266097\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			75779.2i			0.309892i
$144$		0			0
$145$	−316014.	−	165128.i	−1.24821	−	0.652230i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			351713.i			1.29785i	0.760854	+	0.648923i	$0.224780\pi$
$149$			351713.i			1.29785i	−0.760854	+	0.648923i	$0.775220\pi$
$150$		0			0
$151$		−	160296.i		−	0.572109i	−0.958213	−	0.286055i	$-0.907656\pi$
$151$		−	160296.i		−	0.572109i	0.958213	−	0.286055i	$-0.0923439\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−378516.	−	197788.i	−1.26548	−	0.661256i
$156$		0			0
$157$			291321.i			0.943240i	0.881802	+	0.471620i	$0.156331\pi$
$157$			291321.i			0.943240i	−0.881802	+	0.471620i	$0.843669\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			159041.i			0.483554i
$162$		0			0
$163$	41756.4			0.123099			0.0615495	−	0.998104i	$-0.480396\pi$
$163$	41756.4			0.123099			0.0615495	+	0.998104i	$0.480396\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			661493.i			1.83541i	0.397259	+	0.917707i	$0.369962\pi$
$167$			661493.i			1.83541i	−0.397259	+	0.917707i	$0.630038\pi$
$168$		0			0
$169$	−158358.			−0.426505
$170$		0			0
$171$		0			0
$172$		0			0
$173$	635477.			1.61430			0.807150	−	0.590346i	$-0.201009\pi$
$173$	635477.			1.61430			0.807150	+	0.590346i	$0.201009\pi$
$174$		0			0
$175$	−276395.	−	397342.i	−0.682237	−	0.980776i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−564674.			−1.31724			−0.658621	−	0.752475i	$-0.728860\pi$
$179$	−564674.			−1.31724			−0.658621	+	0.752475i	$0.728860\pi$
$180$		0			0
$181$	43204.7			0.0980245			0.0490122	−	0.998798i	$-0.484393\pi$
$181$	43204.7			0.0980245			0.0490122	+	0.998798i	$0.484393\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	274954.	+	143673.i	0.590650	+	0.308634i
$186$		0			0
$187$	50479.9			0.105564
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−689667.			−1.36790			−0.683952	−	0.729527i	$-0.739740\pi$
$191$	−689667.			−1.36790			−0.683952	+	0.729527i	$0.739740\pi$
$192$		0			0
$193$			190713.i			0.368542i	0.982875	+	0.184271i	$0.0589924\pi$
$193$			190713.i			0.368542i	−0.982875	+	0.184271i	$0.941008\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	301619.			0.553724			0.276862	−	0.960910i	$-0.410706\pi$
$197$	301619.			0.553724			0.276862	+	0.960910i	$0.410706\pi$
$198$		0			0
$199$		−	422972.i		−	0.757145i	−0.925572	−	0.378573i	$-0.876415\pi$
$199$		−	422972.i		−	0.757145i	0.925572	−	0.378573i	$-0.123585\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			987906.i			1.68258i
$204$		0			0
$205$	235153.	+	122875.i	0.390809	+	0.204211i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			91334.2i			0.144633i
$210$		0			0
$211$		−	923292.i		−	1.42769i	−0.700306	−	0.713843i	$-0.746953\pi$
$211$		−	923292.i		−	1.42769i	0.700306	−	0.713843i	$-0.253047\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	112075.	−	214484.i	0.165353	−	0.316445i
$216$		0			0
$217$			1.18330e6i			1.70586i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			352824.i			0.485934i
$222$		0			0
$223$	72349.8			0.0974261			0.0487131	−	0.998813i	$-0.484488\pi$
$223$	72349.8			0.0974261			0.0487131	+	0.998813i	$0.484488\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			66534.1i			0.0856997i	0.999082	+	0.0428499i	$0.0136437\pi$
$227$			66534.1i			0.0856997i	−0.999082	+	0.0428499i	$0.986356\pi$
$228$		0			0
$229$	−719131.			−0.906191			−0.453095	−	0.891462i	$-0.649680\pi$
$229$	−719131.			−0.906191			−0.453095	+	0.891462i	$0.649680\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.01104e6			1.22005			0.610023	−	0.792383i	$-0.291160\pi$
$233$	1.01104e6			1.22005			0.610023	+	0.792383i	$0.291160\pi$
$234$		0			0
$235$	−807029.	−	421700.i	−0.953277	−	0.498120i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.05941e6			−1.19969			−0.599844	−	0.800117i	$-0.704771\pi$
$239$	−1.05941e6			−1.19969			−0.599844	+	0.800117i	$0.704771\pi$
$240$		0			0
$241$	−1.34839e6			−1.49546			−0.747729	−	0.664005i	$-0.768856\pi$
$241$	−1.34839e6			−1.49546			−0.747729	+	0.664005i	$0.768856\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−185956.	+	355873.i	−0.197922	+	0.378774i
$246$		0			0
$247$	−638372.			−0.665781
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−945587.			−0.947364			−0.473682	−	0.880696i	$-0.657075\pi$
$251$	−945587.			−0.947364			−0.473682	+	0.880696i	$0.657075\pi$
$252$		0			0
$253$		−	106918.i		−	0.105015i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.35263e6			−1.27746			−0.638728	−	0.769433i	$-0.720539\pi$
$257$	−1.35263e6			−1.27746			−0.638728	+	0.769433i	$0.720539\pi$
$258$		0			0
$259$		−	859545.i		−	0.796195i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	239536.i		−	0.213541i	−0.994284	−	0.106771i	$-0.965949\pi$
$263$		−	239536.i		−	0.213541i	0.994284	−	0.106771i	$-0.0340511\pi$
$264$		0			0
$265$	−82338.3	+	157575.i	−0.0720256	+	0.137839i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			614017.i			0.517368i	0.965962	+	0.258684i	$0.0832889\pi$
$269$			614017.i			0.517368i	−0.965962	+	0.258684i	$0.916711\pi$
$270$		0			0
$271$		−	2.18206e6i		−	1.80486i	−0.430839	−	0.902429i	$-0.641782\pi$
$271$		−	2.18206e6i		−	1.80486i	0.430839	−	0.902429i	$-0.358218\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	185811.	+	267120.i	0.148163	+	0.212998i
$276$		0			0
$277$		−	1.52191e6i		−	1.19176i	−0.803073	−	0.595881i	$-0.796803\pi$
$277$		−	1.52191e6i		−	1.19176i	0.803073	−	0.595881i	$-0.203197\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			29215.0i			0.0220719i	0.999939	+	0.0110359i	$0.00351292\pi$
$281$			29215.0i			0.0220719i	−0.999939	+	0.0110359i	$0.996487\pi$
$282$		0			0
$283$	−1.74371e6			−1.29422			−0.647111	−	0.762395i	$-0.724023\pi$
$283$	−1.74371e6			−1.29422			−0.647111	+	0.762395i	$0.724023\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	735121.i		−	0.526810i
$288$		0			0
$289$	−1.18483e6			−0.834468
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.46650e6			−1.67847			−0.839233	−	0.543772i	$-0.816996\pi$
$293$	−2.46650e6			−1.67847			−0.839233	+	0.543772i	$0.816996\pi$
$294$		0			0
$295$	348513.	−	666968.i	0.233165	−	0.446221i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	747294.			0.483407
$300$		0			0
$301$	−670508.			−0.426568
$302$		0			0
$303$		0			0
$304$		0			0
$305$	262237.	−	501857.i	0.161415	−	0.308909i
$306$		0			0
$307$	−2.00904e6			−1.21658			−0.608292	−	0.793713i	$-0.708145\pi$
$307$	−2.00904e6			−1.21658			−0.608292	+	0.793713i	$0.708145\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	810096.			0.474937			0.237468	−	0.971395i	$-0.423682\pi$
$311$	810096.			0.474937			0.237468	+	0.971395i	$0.423682\pi$
$312$		0			0
$313$			893672.i			0.515605i	0.966198	+	0.257803i	$0.0829985\pi$
$313$			893672.i			0.515605i	−0.966198	+	0.257803i	$0.917002\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.79723e6			1.56344			0.781719	−	0.623631i	$-0.214343\pi$
$317$	2.79723e6			1.56344			0.781719	+	0.623631i	$0.214343\pi$
$318$		0			0
$319$		−	664137.i		−	0.365410i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			425247.i			0.226796i
$324$		0			0
$325$	−1.86701e6	+	1.29871e6i	−0.980479	+	0.682030i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			2.52289e6i			1.28502i
$330$		0			0
$331$			1.87721e6i			0.941766i	0.882196	+	0.470883i	$0.156065\pi$
$331$			1.87721e6i			0.941766i	−0.882196	+	0.470883i	$0.843935\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.05546e6	+	2.01989e6i	−0.513841	+	0.983365i
$336$		0			0
$337$			120244.i			0.0576750i	0.999584	+	0.0288375i	$0.00918054\pi$
$337$			120244.i			0.0576750i	−0.999584	+	0.0288375i	$0.990819\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	795492.i		−	0.370467i
$342$		0			0
$343$	−1.49066e6			−0.684138
$344$		0			0
$345$		0			0
$346$		0			0
$347$			108098.i			0.0481943i	0.999710	+	0.0240971i	$0.00767110\pi$
$347$			108098.i			0.0481943i	−0.999710	+	0.0240971i	$0.992329\pi$
$348$		0			0
$349$	2.91905e6			1.28286			0.641429	−	0.767183i	$-0.278342\pi$
$349$	2.91905e6			1.28286			0.641429	+	0.767183i	$0.278342\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	376244.			0.160706			0.0803531	−	0.996766i	$-0.474395\pi$
$353$	376244.			0.160706			0.0803531	+	0.996766i	$0.474395\pi$
$354$		0			0
$355$	774279.	−	1.48178e6i	0.326082	−	0.624040i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.92622e6			−1.19832			−0.599158	−	0.800631i	$-0.704498\pi$
$359$	−2.92622e6			−1.19832			−0.599158	+	0.800631i	$0.704498\pi$
$360$		0			0
$361$	1.70669e6			0.689266
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.10650e6	−	1.10072e6i	−0.827617	−	0.432458i
$366$		0			0
$367$	3.23268e6			1.25284			0.626422	−	0.779484i	$-0.284519\pi$
$367$	3.23268e6			1.25284			0.626422	+	0.779484i	$0.284519\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	492603.			0.185807
$372$		0			0
$373$			2.17558e6i			0.809662i	0.914391	+	0.404831i	$0.132670\pi$
$373$			2.17558e6i			0.809662i	−0.914391	+	0.404831i	$0.867330\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.64192e6			1.68207
$378$		0			0
$379$			1.04384e6i			0.373282i	0.982428	+	0.186641i	$0.0597602\pi$
$379$			1.04384e6i			0.373282i	−0.982428	+	0.186641i	$0.940240\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	3.50276e6i		−	1.22015i	−0.792343	−	0.610076i	$-0.791139\pi$
$383$		−	3.50276e6i		−	1.22015i	0.792343	−	0.610076i	$-0.208861\pi$
$384$		0			0
$385$	417528.	−	799046.i	0.143560	−	0.274739i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	3.67848e6i		−	1.23252i	−0.787542	−	0.616261i	$-0.788647\pi$
$389$		−	3.67848e6i		−	1.23252i	0.787542	−	0.616261i	$-0.211353\pi$
$390$		0			0
$391$		−	497805.i		−	0.164671i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.34068e6	+	700552.i	0.432348	+	0.225916i
$396$		0			0
$397$		−	2.33149e6i		−	0.742434i	−0.928546	−	0.371217i	$-0.878941\pi$
$397$		−	2.33149e6i		−	0.742434i	0.928546	−	0.371217i	$-0.121059\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			2.20669e6i			0.685300i	0.939463	+	0.342650i	$0.111324\pi$
$401$			2.20669e6i			0.685300i	−0.939463	+	0.342650i	$0.888676\pi$
$402$		0			0
$403$	5.56001e6			1.70535
$404$		0			0
$405$		0			0
$406$		0			0
$407$			577844.i			0.172912i
$408$		0			0
$409$	−1.32837e6			−0.392655			−0.196328	−	0.980538i	$-0.562902\pi$
$409$	−1.32837e6			−0.392655			−0.196328	+	0.980538i	$0.562902\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.08504e6			−0.601505
$414$		0			0
$415$	4.84636e6	+	2.53239e6i	1.38132	+	0.721788i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.51751e6			−0.978813			−0.489406	−	0.872056i	$-0.662787\pi$
$419$	−3.51751e6			−0.978813			−0.489406	+	0.872056i	$0.662787\pi$
$420$		0			0
$421$	1.95415e6			0.537344			0.268672	−	0.963232i	$-0.413415\pi$
$421$	1.95415e6			0.537344			0.268672	+	0.963232i	$0.413415\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	865128.	+	1.24370e6i	0.232331	+	0.333997i
$426$		0			0
$427$	−1.56888e6			−0.416408
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.35646e6			0.351733			0.175866	−	0.984414i	$-0.443727\pi$
$431$	1.35646e6			0.351733			0.175866	+	0.984414i	$0.443727\pi$
$432$		0			0
$433$		−	958941.i		−	0.245795i	−0.992419	−	0.122897i	$-0.960781\pi$
$433$		−	958941.i		−	0.245795i	0.992419	−	0.122897i	$-0.0392186\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	900689.			0.225617
$438$		0			0
$439$			7.29425e6i			1.80642i	0.429196	+	0.903211i	$0.358797\pi$
$439$			7.29425e6i			1.80642i	−0.429196	+	0.903211i	$0.641203\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			3.38040e6i			0.818387i	0.912448	+	0.409193i	$0.134190\pi$
$443$			3.38040e6i			0.818387i	−0.912448	+	0.409193i	$0.865810\pi$
$444$		0			0
$445$	2.68502e6	+	1.40301e6i	0.642758	+	0.335863i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			1.83219e6i			0.428898i	0.976735	+	0.214449i	$0.0687957\pi$
$449$			1.83219e6i			0.428898i	−0.976735	+	0.214449i	$0.931204\pi$
$450$		0			0
$451$			494198.i			0.114409i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.58485e6	+	2.91827e6i	1.26469	+	0.660842i
$456$		0			0
$457$		−	4.56773e6i		−	1.02308i	−0.859259	−	0.511540i	$-0.829075\pi$
$457$		−	4.56773e6i		−	1.02308i	0.859259	−	0.511540i	$-0.170925\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	3.30868e6i		−	0.725107i	−0.931963	−	0.362554i	$-0.881905\pi$
$461$		−	3.30868e6i		−	0.725107i	0.931963	−	0.362554i	$-0.118095\pi$
$462$		0			0
$463$	−2.15559e6			−0.467320			−0.233660	−	0.972318i	$-0.575070\pi$
$463$	−2.15559e6			−0.467320			−0.233660	+	0.972318i	$0.575070\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	4.77902e6i		−	1.01402i	−0.861940	−	0.507010i	$-0.830751\pi$
$467$		−	4.77902e6i		−	1.01402i	0.861940	−	0.507010i	$-0.169249\pi$
$468$		0			0
$469$	6.31446e6			1.32557
$470$		0			0
$471$		0			0
$472$		0			0
$473$	450761.			0.0926389
$474$		0			0
$475$	−2.25025e6	+	1.56529e6i	−0.457611	+	0.318318i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.43060e6			1.08146			0.540728	−	0.841198i	$-0.318149\pi$
$479$	5.43060e6			1.08146			0.540728	+	0.841198i	$0.318149\pi$
$480$		0			0
$481$	−4.03879e6			−0.795954
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.16127e6	−	3.21947e6i	−1.18937	−	0.621484i
$486$		0			0
$487$	−4.90831e6			−0.937799			−0.468899	−	0.883252i	$-0.655349\pi$
$487$	−4.90831e6			−0.937799			−0.468899	+	0.883252i	$0.655349\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.75489e6			1.07729			0.538645	−	0.842533i	$-0.318936\pi$
$491$	5.75489e6			1.07729			0.538645	+	0.842533i	$0.318936\pi$
$492$		0			0
$493$		−	3.09219e6i		−	0.572992i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.63225e6			−0.841204
$498$		0			0
$499$			7.74644e6i			1.39268i	0.717713	+	0.696339i	$0.245189\pi$
$499$			7.74644e6i			1.39268i	−0.717713	+	0.696339i	$0.754811\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	1.08354e6i		−	0.190952i	−0.995432	−	0.0954760i	$-0.969563\pi$
$503$		−	1.08354e6i		−	0.190952i	0.995432	−	0.0954760i	$-0.0304373\pi$
$504$		0			0
$505$	−3.04746e6	−	1.59240e6i	−0.531753	−	0.277859i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	1.29709e6i		−	0.221909i	−0.993825	−	0.110954i	$-0.964609\pi$
$509$		−	1.29709e6i		−	0.221909i	0.993825	−	0.110954i	$-0.0353908\pi$
$510$		0			0
$511$			6.58523e6i			1.11563i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.37639e6	−	1.02891e7i	0.893249	−	1.70946i
$516$		0			0
$517$		−	1.69606e6i		−	0.279070i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	2.87108e6i		−	0.463394i	−0.972788	−	0.231697i	$-0.925572\pi$
$521$		−	2.87108e6i		−	0.463394i	0.972788	−	0.231697i	$-0.0744278\pi$
$522$		0			0
$523$	7.74611e6			1.23831			0.619155	−	0.785269i	$-0.287475\pi$
$523$	7.74611e6			1.23831			0.619155	+	0.785269i	$0.287475\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	3.70377e6i		−	0.580922i
$528$		0			0
$529$	5.38197e6			0.836185
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.45415e6			−0.526651
$534$		0			0
$535$	−4.15170e6	−	2.16940e6i	−0.627106	−	0.327684i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−747906.			−0.110886
$540$		0			0
$541$	4.15367e6			0.610154			0.305077	−	0.952328i	$-0.401318\pi$
$541$	4.15367e6			0.610154			0.305077	+	0.952328i	$0.401318\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.86999e6	−	3.57870e6i	0.269680	−	0.516101i
$546$		0			0
$547$	9.82428e6			1.40389			0.701944	−	0.712232i	$-0.252316\pi$
$547$	9.82428e6			1.40389			0.701944	+	0.712232i	$0.252316\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.59476e6			0.785059
$552$		0			0
$553$		−	4.19117e6i		−	0.582804i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.24335e6			−1.12581			−0.562906	−	0.826521i	$-0.690317\pi$
$557$	−8.24335e6			−1.12581			−0.562906	+	0.826521i	$0.690317\pi$
$558$		0			0
$559$			3.15055e6i			0.426439i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			544090.i			0.0723436i	0.999346	+	0.0361718i	$0.0115164\pi$
$563$			544090.i			0.0723436i	−0.999346	+	0.0361718i	$0.988484\pi$
$564$		0			0
$565$	1.12479e6	−	2.15258e6i	0.148235	−	0.283686i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			7.86937e6i			1.01896i	0.860481	+	0.509482i	$0.170163\pi$
$569$			7.86937e6i			1.01896i	−0.860481	+	0.509482i	$0.829837\pi$
$570$		0			0
$571$			4.15081e6i			0.532774i	0.963866	+	0.266387i	$0.0858299\pi$
$571$			4.15081e6i			0.532774i	−0.963866	+	0.266387i	$0.914170\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.63420e6	−	1.83237e6i	0.332260	−	0.231123i
$576$		0			0
$577$			1.35134e7i			1.68977i	0.534951	+	0.844883i	$0.320330\pi$
$577$			1.35134e7i			1.68977i	−0.534951	+	0.844883i	$0.679670\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	1.51504e7i		−	1.86202i
$582$		0			0
$583$	−331161.			−0.0403522
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	7.07517e6i		−	0.847504i	−0.905778	−	0.423752i	$-0.860713\pi$
$587$		−	7.07517e6i		−	0.847504i	0.905778	−	0.423752i	$-0.139287\pi$
$588$		0			0
$589$	6.70130e6			0.795923
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.43880e6			0.635135			0.317567	−	0.948236i	$-0.397134\pi$
$593$	5.43880e6			0.635135			0.317567	+	0.948236i	$0.397134\pi$
$594$		0			0
$595$	1.94399e6	−	3.72032e6i	0.225114	−	0.430812i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.50637e6			−0.171539			−0.0857697	−	0.996315i	$-0.527335\pi$
$599$	−1.50637e6			−0.171539			−0.0857697	+	0.996315i	$0.527335\pi$
$600$		0			0
$601$	1.39751e6			0.157823			0.0789113	−	0.996882i	$-0.474856\pi$
$601$	1.39751e6			0.157823			0.0789113	+	0.996882i	$0.474856\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.88879e6	−	7.44217e6i	0.431942	−	0.826630i
$606$		0			0
$607$	1.32808e7			1.46302			0.731512	−	0.681828i	$-0.238815\pi$
$607$	1.32808e7			1.46302			0.731512	+	0.681828i	$0.238815\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.18544e7			1.28463
$612$		0			0
$613$		−	6.61279e6i		−	0.710778i	−0.934719	−	0.355389i	$-0.884348\pi$
$613$		−	6.61279e6i		−	0.710778i	0.934719	−	0.355389i	$-0.115652\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.99579e6			0.211058			0.105529	−	0.994416i	$-0.466346\pi$
$617$	1.99579e6			0.211058			0.105529	+	0.994416i	$0.466346\pi$
$618$		0			0
$619$			1.25288e7i			1.31427i	0.753774	+	0.657134i	$0.228232\pi$
$619$			1.25288e7i			1.31427i	−0.753774	+	0.657134i	$0.771768\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	8.39376e6i		−	0.866436i
$624$		0			0
$625$	−3.39673e6	+	9.15585e6i	−0.347825	+	0.937559i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			2.69041e6i			0.271139i
$630$		0			0
$631$			8.38620e6i			0.838479i	0.907876	+	0.419239i	$0.137703\pi$
$631$			8.38620e6i			0.838479i	−0.907876	+	0.419239i	$0.862297\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.31893e6	−	6.35161e6i	0.326636	−	0.625100i
$636$		0			0
$637$		−	5.22741e6i		−	0.510433i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1.36686e6i			0.131395i	0.997840	+	0.0656974i	$0.0209272\pi$
$641$			1.36686e6i			0.131395i	−0.997840	+	0.0656974i	$0.979073\pi$
$642$		0			0
$643$	7.02175e6			0.669758			0.334879	−	0.942261i	$-0.391305\pi$
$643$	7.02175e6			0.669758			0.334879	+	0.942261i	$0.391305\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.47870e7i			1.38873i	0.719621	+	0.694367i	$0.244316\pi$
$647$			1.47870e7i			1.38873i	−0.719621	+	0.694367i	$0.755684\pi$
$648$		0			0
$649$	1.40170e6			0.130630
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.24519e7			1.14275			0.571377	−	0.820688i	$-0.306409\pi$
$653$	1.24519e7			1.14275			0.571377	+	0.820688i	$0.306409\pi$
$654$		0			0
$655$	288950.	−	552979.i	0.0263160	−	0.0503623i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.95740e7			−1.75577			−0.877884	−	0.478874i	$-0.841045\pi$
$659$	−1.95740e7			−1.75577			−0.877884	+	0.478874i	$0.841045\pi$
$660$		0			0
$661$	2.10926e7			1.87770			0.938851	−	0.344324i	$-0.111892\pi$
$661$	2.10926e7			1.87770			0.938851	+	0.344324i	$0.111892\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.73124e6	+	3.51730e6i	0.590257	+	0.308429i
$666$		0			0
$667$	−6.54936e6			−0.570012
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.05471e6			0.0904325
$672$		0			0
$673$			1.25319e7i			1.06655i	0.845943	+	0.533273i	$0.179038\pi$
$673$			1.25319e7i			1.06655i	−0.845943	+	0.533273i	$0.820962\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.78634e7			−1.49793			−0.748967	−	0.662607i	$-0.769450\pi$
$677$	−1.78634e7			−1.49793			−0.748967	+	0.662607i	$0.769450\pi$
$678$		0			0
$679$			1.92610e7i			1.60326i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.58838e7i			1.30287i	0.758703	+	0.651436i	$0.225833\pi$
$683$			1.58838e7i			1.30287i	−0.758703	+	0.651436i	$0.774167\pi$
$684$		0			0
$685$	1.05967e7	−	2.02794e7i	0.862866	−	1.65131i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	2.31461e6i		−	0.185751i
$690$		0			0
$691$		−	1.27317e7i		−	1.01436i	−0.861841	−	0.507179i	$-0.830688\pi$
$691$		−	1.27317e7i		−	1.01436i	0.861841	−	0.507179i	$-0.169312\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.67472e7	+	8.75100e6i	1.31517	+	0.687220i
$696$		0			0
$697$			2.30096e6i			0.179402i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1.26599e7i			0.973049i	0.873667	+	0.486525i	$0.161736\pi$
$701$			1.26599e7i			0.973049i	−0.873667	+	0.486525i	$0.838264\pi$
$702$		0			0
$703$	−4.86782e6			−0.371489
$704$		0			0
$705$		0			0
$706$		0			0
$707$			9.52681e6i			0.716802i
$708$		0			0
$709$	6.55798e6			0.489953			0.244977	−	0.969529i	$-0.421220\pi$
$709$	6.55798e6			0.489953			0.244977	+	0.969529i	$0.421220\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.84471e6			−0.577901
$714$		0			0
$715$	−3.75451e6	−	1.96186e6i	−0.274656	−	0.143517i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.18837e7			−1.57869			−0.789347	−	0.613947i	$-0.789581\pi$
$719$	−2.18837e7			−1.57869			−0.789347	+	0.613947i	$0.789581\pi$
$720$		0			0
$721$	−3.21651e7			−2.30434
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.63627e7	−	1.13820e7i	1.15614	−	0.804220i
$726$		0			0
$727$	9.39022e6			0.658931			0.329465	−	0.944168i	$-0.393131\pi$
$727$	9.39022e6			0.658931			0.329465	+	0.944168i	$0.393131\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.09872e6			0.145265
$732$		0			0
$733$		−	2.13158e7i		−	1.46535i	−0.680577	−	0.732677i	$-0.738271\pi$
$733$		−	2.13158e7i		−	1.46535i	0.680577	−	0.732677i	$-0.261729\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.24500e6			−0.287879
$738$		0			0
$739$			1.72669e7i			1.16307i	0.813523	+	0.581533i	$0.197547\pi$
$739$			1.72669e7i			1.16307i	−0.813523	+	0.581533i	$0.802453\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	2.69724e7i		−	1.79245i	−0.443598	−	0.896226i	$-0.646298\pi$
$743$		−	2.69724e7i		−	1.79245i	0.443598	−	0.896226i	$-0.353702\pi$
$744$		0			0
$745$	−1.74258e7	−	9.10557e6i	−1.15028	−	0.601058i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1.29788e7i			0.845338i
$750$		0			0
$751$		−	1.88255e7i		−	1.21800i	−0.793171	−	0.608999i	$-0.791572\pi$
$751$		−	1.88255e7i		−	1.21800i	0.793171	−	0.608999i	$-0.208428\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.94191e6	+	4.14992e6i	0.507058	+	0.264955i
$756$		0			0
$757$			6.35520e6i			0.403079i	0.979480	+	0.201539i	$0.0645944\pi$
$757$			6.35520e6i			0.403079i	−0.979480	+	0.201539i	$0.935406\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			9.60819e6i			0.601423i	0.953715	+	0.300711i	$0.0972241\pi$
$761$			9.60819e6i			0.601423i	−0.953715	+	0.300711i	$0.902776\pi$
$762$		0			0
$763$	−1.11876e7			−0.695703
$764$		0			0
$765$		0			0
$766$		0			0
$767$			9.79707e6i			0.601323i
$768$		0			0
$769$	−1.33196e7			−0.812225			−0.406113	−	0.913823i	$-0.633116\pi$
$769$	−1.33196e7			−0.812225			−0.406113	+	0.913823i	$0.633116\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.03053e7			−0.620317			−0.310159	−	0.950685i	$-0.600382\pi$
$773$	−1.03053e7			−0.620317			−0.310159	+	0.950685i	$0.600382\pi$
$774$		0			0
$775$	1.95989e7	−	1.36332e7i	1.17214	−	0.815349i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.16317e6			−0.245799
$780$		0			0
$781$	3.11411e6			0.182687
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.44336e7	−	7.54205e6i	−0.835990	−	0.436833i
$786$		0			0
$787$	−1.64506e6			−0.0946772			−0.0473386	−	0.998879i	$-0.515074\pi$
$787$	−1.64506e6			−0.0946772			−0.0473386	+	0.998879i	$0.515074\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.72927e6			−0.382408
$792$		0			0
$793$			7.37176e6i			0.416282i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.23277e6			0.124508			0.0622542	−	0.998060i	$-0.480171\pi$
$797$	2.23277e6			0.124508			0.0622542	+	0.998060i	$0.480171\pi$
$798$		0			0
$799$		−	7.89675e6i		−	0.437604i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	4.42703e6i		−	0.242283i
$804$		0			0
$805$	−7.87976e6	−	4.11744e6i	−0.428571	−	0.223943i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			3.52749e7i			1.89494i	0.319851	+	0.947468i	$0.396367\pi$
$809$			3.52749e7i			1.89494i	−0.319851	+	0.947468i	$0.603633\pi$
$810$		0			0
$811$		−	5.36982e6i		−	0.286687i	−0.989673	−	0.143343i	$-0.954215\pi$
$811$		−	5.36982e6i		−	0.286687i	0.989673	−	0.143343i	$-0.0457853\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.08104e6	+	2.06884e6i	−0.0570095	+	0.109102i
$816$		0			0
$817$			3.79725e6i			0.199028i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	3.41519e7i		−	1.76830i	−0.467200	−	0.884152i	$-0.654737\pi$
$821$		−	3.41519e7i		−	1.76830i	0.467200	−	0.884152i	$-0.345263\pi$
$822$		0			0
$823$	−2.29782e7			−1.18254			−0.591270	−	0.806474i	$-0.701373\pi$
$823$	−2.29782e7			−1.18254			−0.591270	+	0.806474i	$0.701373\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			5.00130e6i			0.254284i	0.991885	+	0.127142i	$0.0405804\pi$
$827$			5.00130e6i			0.254284i	−0.991885	+	0.127142i	$0.959420\pi$
$828$		0			0
$829$	3.83727e7			1.93926			0.969629	−	0.244581i	$-0.0786504\pi$
$829$	3.83727e7			1.93926			0.969629	+	0.244581i	$0.0786504\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.48221e6			−0.173877
$834$		0			0
$835$	−3.27740e7	−	1.71255e7i	−1.62672	−	0.850016i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3.37076e7			−1.65319			−0.826596	−	0.562795i	$-0.809726\pi$
$839$	−3.37076e7			−1.65319			−0.826596	+	0.562795i	$0.809726\pi$
$840$		0			0
$841$	−2.01711e7			−0.983422
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.09977e6	−	7.84594e6i	0.197523	−	0.378010i
$846$		0			0
$847$	−2.32653e7			−1.11430
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.69839e6			0.269729
$852$		0			0
$853$			3.84619e7i			1.80992i	0.425500	+	0.904959i	$0.360098\pi$
$853$			3.84619e7i			1.80992i	−0.425500	+	0.904959i	$0.639902\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.81377e7			1.30869			0.654345	−	0.756197i	$-0.272945\pi$
$857$	2.81377e7			1.30869			0.654345	+	0.756197i	$0.272945\pi$
$858$		0			0
$859$		−	1.58144e7i		−	0.731256i	−0.930761	−	0.365628i	$-0.880854\pi$
$859$		−	1.58144e7i		−	0.731256i	0.930761	−	0.365628i	$-0.119146\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	5.22733e6i		−	0.238920i	−0.992839	−	0.119460i	$-0.961884\pi$
$863$		−	5.22733e6i		−	0.238920i	0.992839	−	0.119460i	$-0.0381164\pi$
$864$		0			0
$865$	−1.64520e7	+	3.14850e7i	−0.747614	+	1.43075i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			2.81759e6i			0.126569i
$870$		0			0
$871$		−	2.96700e7i		−	1.32517i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.68421e7	−	3.40725e6i	1.18521	−	0.150447i
$876$		0			0
$877$			7.13556e6i			0.313278i	0.987656	+	0.156639i	$0.0500658\pi$
$877$			7.13556e6i			0.313278i	−0.987656	+	0.156639i	$0.949934\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	2.90570e6i		−	0.126128i	−0.998009	−	0.0630638i	$-0.979913\pi$
$881$		−	2.90570e6i		−	0.126128i	0.998009	−	0.0630638i	$-0.0200872\pi$
$882$		0			0
$883$	−886730.			−0.0382727			−0.0191364	−	0.999817i	$-0.506092\pi$
$883$	−886730.			−0.0382727			−0.0191364	+	0.999817i	$0.506092\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.43504e7i		−	0.612426i	−0.951963	−	0.306213i	$-0.900938\pi$
$887$		−	1.43504e7i		−	0.612426i	0.951963	−	0.306213i	$-0.0990619\pi$
$888$		0			0
$889$	−1.98561e7			−0.842633
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.42877e7			0.599563
$894$		0			0
$895$	1.46189e7	−	2.79770e7i	0.610040	−	1.16747i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.87285e7			−2.01087
$900$		0			0
$901$	−1.54187e6			−0.0632754
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.11853e6	+	2.14060e6i	−0.0453970	+	0.0868787i
$906$		0			0
$907$	−2.28409e7			−0.921924			−0.460962	−	0.887420i	$-0.652495\pi$
$907$	−2.28409e7			−0.921924			−0.460962	+	0.887420i	$0.652495\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.77827e7			1.50833			0.754167	−	0.656683i	$-0.228041\pi$
$911$	3.77827e7			1.50833			0.754167	+	0.656683i	$0.228041\pi$
$912$		0			0
$913$			1.01851e7i			0.404380i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.72869e6			−0.0678883
$918$		0			0
$919$		−	3.23668e6i		−	0.126419i	−0.998000	−	0.0632093i	$-0.979866\pi$
$919$		−	3.23668e6i		−	0.126419i	0.998000	−	0.0632093i	$-0.0201336\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.17658e7i			0.840950i
$924$		0			0
$925$	−1.42366e7	+	9.90314e6i	−0.547083	+	0.380556i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			4.52545e7i			1.72037i	0.509980	+	0.860186i	$0.329653\pi$
$929$			4.52545e7i			1.72037i	−0.509980	+	0.860186i	$0.670347\pi$
$930$		0			0
$931$		−	6.30043e6i		−	0.238230i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.30688e6	+	2.50105e6i	−0.0488886	+	0.0935606i
$936$		0			0
$937$			7.85407e6i			0.292244i	0.989267	+	0.146122i	$0.0466792\pi$
$937$			7.85407e6i			0.292244i	−0.989267	+	0.146122i	$0.953321\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			721027.i			0.0265447i	0.999912	+	0.0132723i	$0.00422484\pi$
$941$			721027.i			0.0265447i	−0.999912	+	0.0132723i	$0.995775\pi$
$942$		0			0
$943$	4.87351e6			0.178469
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1.39438e7i		−	0.505250i	−0.967564	−	0.252625i	$-0.918706\pi$
$947$		−	1.39438e7i		−	0.505250i	0.967564	−	0.252625i	$-0.0812938\pi$
$948$		0			0
$949$	3.09423e7			1.11529
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.37454e7			0.490260			0.245130	−	0.969490i	$-0.421169\pi$
$953$	1.37454e7			0.490260			0.245130	+	0.969490i	$0.421169\pi$
$954$		0			0
$955$	1.78549e7	−	3.41698e7i	0.633503	−	1.21237i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.33964e7			−2.22596
$960$		0			0
$961$	−2.97371e7			−1.03870
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.44895e6	−	4.93740e6i	−0.326637	−	0.170679i
$966$		0			0
$967$	1.22441e7			0.421077			0.210539	−	0.977586i	$-0.432478\pi$
$967$	1.22441e7			0.421077			0.210539	+	0.977586i	$0.432478\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5.33008e7			−1.81420			−0.907102	−	0.420912i	$-0.861710\pi$
$971$	−5.33008e7			−1.81420			−0.907102	+	0.420912i	$0.861710\pi$
$972$		0			0
$973$		−	5.23543e7i		−	1.77284i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.05399e7			1.02360			0.511802	−	0.859104i	$-0.328978\pi$
$977$	3.05399e7			1.02360			0.511802	+	0.859104i	$0.328978\pi$
$978$		0			0
$979$			5.64285e6i			0.188166i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			4.17124e7i			1.37683i	0.725315	+	0.688417i	$0.241694\pi$
$983$			4.17124e7i			1.37683i	−0.725315	+	0.688417i	$0.758306\pi$
$984$		0			0
$985$	−7.80867e6	+	1.49439e7i	−0.256440	+	0.490763i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	4.44516e6i		−	0.144510i
$990$		0			0
$991$		−	2.89827e7i		−	0.937464i	−0.883341	−	0.468732i	$-0.844711\pi$
$991$		−	2.89827e7i		−	0.937464i	0.883341	−	0.468732i	$-0.155289\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.09564e7	+	1.09504e7i	0.671055	+	0.350649i
$996$		0			0
$997$		−	4.17250e7i		−	1.32941i	−0.747106	−	0.664705i	$-0.768557\pi$
$997$		−	4.17250e7i		−	1.32941i	0.747106	−	0.664705i	$-0.231443\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.o.d.719.16	yes	40
3.2	odd	2	inner	720.6.o.d.719.26	yes	40
4.3	odd	2	inner	720.6.o.d.719.15	yes	40
5.4	even	2	inner	720.6.o.d.719.27	yes	40
12.11	even	2	inner	720.6.o.d.719.25	yes	40
15.14	odd	2	inner	720.6.o.d.719.13	✓	40
20.19	odd	2	inner	720.6.o.d.719.28	yes	40
60.59	even	2	inner	720.6.o.d.719.14	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.6.o.d.719.13	✓	40	15.14	odd	2	inner
720.6.o.d.719.14	yes	40	60.59	even	2	inner
720.6.o.d.719.15	yes	40	4.3	odd	2	inner
720.6.o.d.719.16	yes	40	1.1	even	1	trivial
720.6.o.d.719.25	yes	40	12.11	even	2	inner
720.6.o.d.719.26	yes	40	3.2	odd	2	inner
720.6.o.d.719.27	yes	40	5.4	even	2	inner
720.6.o.d.719.28	yes	40	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-25.8892 + 49.5454i) q^{5} +154.886 q^{7} +O(q^{10})\)	\(q+(-25.8892 + 49.5454i) q^{5} +154.886 q^{7} -104.125 q^{11} -727.772i q^{13} -484.801 q^{17} -877.159i q^{19} +1026.82i q^{23} +(-1784.50 - 2565.38i) q^{25} +6378.27i q^{29} +7639.78i q^{31} +(-4009.88 + 7673.91i) q^{35} -5549.52i q^{37} -4746.20i q^{41} -4329.04 q^{43} +16288.7i q^{47} +7182.77 q^{49} +3180.41 q^{53} +(2695.71 - 5158.92i) q^{55} -13461.7 q^{59} -10129.2 q^{61} +(36057.8 + 18841.4i) q^{65} +40768.3 q^{67} -29907.5 q^{71} +42516.5i q^{73} -16127.5 q^{77} -27059.7i q^{79} -97816.4i q^{83} +(12551.1 - 24019.7i) q^{85} -54193.0i q^{89} -112722. i q^{91} +(43459.2 + 22708.9i) q^{95} +124356. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q+O(q^{10}) \) 40 * q	\( 40 q + 10648 q^{25} + 36248 q^{49} - 133136 q^{61} - 336816 q^{85}+O(q^{100}) \) 40 * q + 10648 * q^25 + 36248 * q^49 - 133136 * q^61 - 336816 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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