LMFDB - Embedded newform 144.7.e.c.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,7,Mod(17,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("144.17");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	144.e (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:	$33.1277880413$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	144.17
Dual form			144.7.e.c.17.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-63.6396i q^{5} +244.000 q^{7} +O(q^{10})$	$q-63.6396i q^{5} +244.000 q^{7} +2392.85i q^{11} -2728.00 q^{13} -5893.03i q^{17} +5392.00 q^{19} +2494.67i q^{23} +11575.0 q^{25} +42014.9i q^{29} -10172.0 q^{31} -15528.1i q^{35} +65006.0 q^{37} +69736.3i q^{41} +49480.0 q^{43} +164801. i q^{47} -58113.0 q^{49} -24883.1i q^{53} +152280. q^{55} +359131. i q^{59} +100610. q^{61} +173609. i q^{65} +435736. q^{67} -60126.7i q^{71} +619568. q^{73} +583855. i q^{77} -514340. q^{79} -243918. i q^{83} -375030. q^{85} +428256. i q^{89} -665632. q^{91} -343145. i q^{95} +42704.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 488 q^{7}+O(q^{10}) $ 2 * q + 488 * q^7	$ 2 q + 488 q^{7} - 5456 q^{13} + 10784 q^{19} + 23150 q^{25} - 20344 q^{31} + 130012 q^{37} + 98960 q^{43} - 116226 q^{49} + 304560 q^{55} + 201220 q^{61} + 871472 q^{67} + 1239136 q^{73} - 1028680 q^{79} - 750060 q^{85} - 1331264 q^{91} + 85408 q^{97}+O(q^{100}) $ 2 * q + 488 * q^7 - 5456 * q^13 + 10784 * q^19 + 23150 * q^25 - 20344 * q^31 + 130012 * q^37 + 98960 * q^43 - 116226 * q^49 + 304560 * q^55 + 201220 * q^61 + 871472 * q^67 + 1239136 * q^73 - 1028680 * q^79 - 750060 * q^85 - 1331264 * q^91 + 85408 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 63.6396i	− 0.509117i	−0.967057	−	0.254558i	$-0.918070\pi$
$5$	− 63.6396i	− 0.509117i	0.967057	−	0.254558i	$-0.0819301\pi$
$6$	0	0
$7$	244.000	0.711370	0.355685	−	0.934606i	$-0.384248\pi$
$7$	244.000	0.711370	0.355685	+	0.934606i	$0.384248\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2392.85i	1.79778i	0.438171	+	0.898892i	$0.355626\pi$
$11$	2392.85i	1.79778i	−0.438171	+	0.898892i	$0.644374\pi$
$12$	0	0
$13$	−2728.00	−1.24169	−0.620847	−	0.783932i	$-0.713211\pi$
$13$	−2728.00	−1.24169	−0.620847	+	0.783932i	$0.713211\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5893.03i	− 1.19948i	−0.800196	−	0.599738i	$-0.795271\pi$
$17$	− 5893.03i	− 1.19948i	0.800196	−	0.599738i	$-0.204729\pi$
$18$	0	0
$19$	5392.00	0.786120	0.393060	−	0.919513i	$-0.371416\pi$
$19$	5392.00	0.786120	0.393060	+	0.919513i	$0.371416\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2494.67i	0.205036i	0.994731	+	0.102518i	$0.0326899\pi$
$23$	2494.67i	0.205036i	−0.994731	+	0.102518i	$0.967310\pi$
$24$	0	0
$25$	11575.0	0.740800
$26$	0	0
$27$	0	0
$28$	0	0
$29$	42014.9i	1.72270i	0.508014	+	0.861349i	$0.330380\pi$
$29$	42014.9i	1.72270i	−0.508014	+	0.861349i	$0.669620\pi$
$30$	0	0
$31$	−10172.0	−0.341445	−0.170723	−	0.985319i	$-0.554610\pi$
$31$	−10172.0	−0.341445	−0.170723	+	0.985319i	$0.554610\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 15528.1i	− 0.362171i
$36$	0	0
$37$	65006.0	1.28336	0.641680	−	0.766973i	$-0.278238\pi$
$37$	65006.0	1.28336	0.641680	+	0.766973i	$0.278238\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	69736.3i	1.01183i	0.862584	+	0.505915i	$0.168845\pi$
$41$	69736.3i	1.01183i	−0.862584	+	0.505915i	$0.831155\pi$
$42$	0	0
$43$	49480.0	0.622335	0.311168	−	0.950355i	$-0.399280\pi$
$43$	49480.0	0.622335	0.311168	+	0.950355i	$0.399280\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	164801.i	1.58733i	0.608356	+	0.793664i	$0.291829\pi$
$47$	164801.i	1.58733i	−0.608356	+	0.793664i	$0.708171\pi$
$48$	0	0
$49$	−58113.0	−0.493952
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 24883.1i	− 0.167139i	−0.996502	−	0.0835693i	$-0.973368\pi$
$53$	− 24883.1i	− 0.167139i	0.996502	−	0.0835693i	$-0.0266320\pi$
$54$	0	0
$55$	152280.	0.915282
$56$	0	0
$57$	0	0
$58$	0	0
$59$	359131.i	1.74863i	0.485363	+	0.874313i	$0.338688\pi$
$59$	359131.i	1.74863i	−0.485363	+	0.874313i	$0.661312\pi$
$60$	0	0
$61$	100610.	0.443253	0.221626	−	0.975132i	$-0.428863\pi$
$61$	100610.	0.443253	0.221626	+	0.975132i	$0.428863\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	173609.i	0.632167i
$66$	0	0
$67$	435736.	1.44877	0.724384	−	0.689396i	$-0.242124\pi$
$67$	435736.	1.44877	0.724384	+	0.689396i	$0.242124\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 60126.7i	− 0.167993i	−0.996466	−	0.0839967i	$-0.973231\pi$
$71$	− 60126.7i	− 0.167993i	0.996466	−	0.0839967i	$-0.0267685\pi$
$72$	0	0
$73$	619568.	1.59265	0.796325	−	0.604869i	$-0.206774\pi$
$73$	619568.	1.59265	0.796325	+	0.604869i	$0.206774\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	583855.i	1.27889i
$78$	0	0
$79$	−514340.	−1.04320	−0.521602	−	0.853189i	$-0.674665\pi$
$79$	−514340.	−1.04320	−0.521602	+	0.853189i	$0.674665\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 243918.i	− 0.426589i	−0.976988	−	0.213294i	$-0.931581\pi$
$83$	− 243918.i	− 0.426589i	0.976988	−	0.213294i	$-0.0684193\pi$
$84$	0	0
$85$	−375030.	−0.610674
$86$	0	0
$87$	0	0
$88$	0	0
$89$	428256.i	0.607483i	0.952755	+	0.303741i	$0.0982359\pi$
$89$	428256.i	0.607483i	−0.952755	+	0.303741i	$0.901764\pi$
$90$	0	0
$91$	−665632.	−0.883304
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 343145.i	− 0.400227i
$96$	0	0
$97$	42704.0	0.0467900	0.0233950	−	0.999726i	$-0.492552\pi$
$97$	42704.0	0.0467900	0.0233950	+	0.999726i	$0.492552\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 911688.i	− 0.884876i	−0.896799	−	0.442438i	$-0.854114\pi$
$101$	− 911688.i	− 0.884876i	0.896799	−	0.442438i	$-0.145886\pi$
$102$	0	0
$103$	1.42059e6	1.30004	0.650020	−	0.759917i	$-0.274761\pi$
$103$	1.42059e6	1.30004	0.650020	+	0.759917i	$0.274761\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.31637e6i	1.07455i	0.843406	+	0.537276i	$0.180547\pi$
$107$	1.31637e6i	1.07455i	−0.843406	+	0.537276i	$0.819453\pi$
$108$	0	0
$109$	140456.	0.108458	0.0542289	−	0.998529i	$-0.482730\pi$
$109$	140456.	0.108458	0.0542289	+	0.998529i	$0.482730\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.89892e6i	− 1.31604i	−0.752998	−	0.658022i	$-0.771393\pi$
$113$	− 1.89892e6i	− 1.31604i	0.752998	−	0.658022i	$-0.228607\pi$
$114$	0	0
$115$	158760.	0.104387
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1.43790e6i	− 0.853272i
$120$	0	0
$121$	−3.95417e6	−2.23202
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.73100e6i	− 0.886271i
$126$	0	0
$127$	−3.29438e6	−1.60828	−0.804142	−	0.594438i	$-0.797375\pi$
$127$	−3.29438e6	−1.60828	−0.804142	+	0.594438i	$0.797375\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 641385.i	− 0.285302i	−0.989773	−	0.142651i	$-0.954437\pi$
$131$	− 641385.i	− 0.285302i	0.989773	−	0.142651i	$-0.0455627\pi$
$132$	0	0
$133$	1.31565e6	0.559223
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 762746.i	− 0.296632i	−0.988940	−	0.148316i	$-0.952615\pi$
$137$	− 762746.i	− 0.296632i	0.988940	−	0.148316i	$-0.0473853\pi$
$138$	0	0
$139$	1.63079e6	0.607231	0.303616	−	0.952795i	$-0.401806\pi$
$139$	1.63079e6	0.607231	0.303616	+	0.952795i	$0.401806\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 6.52769e6i	− 2.23230i
$144$	0	0
$145$	2.67381e6	0.877054
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.60023e6i	0.786054i	0.919527	+	0.393027i	$0.128572\pi$
$149$	2.60023e6i	0.786054i	−0.919527	+	0.393027i	$0.871428\pi$
$150$	0	0
$151$	−5.62886e6	−1.63489	−0.817447	−	0.576004i	$-0.804611\pi$
$151$	−5.62886e6	−1.63489	−0.817447	+	0.576004i	$0.804611\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	647342.i	0.173836i
$156$	0	0
$157$	−5.12528e6	−1.32440	−0.662199	−	0.749328i	$-0.730377\pi$
$157$	−5.12528e6	−1.32440	−0.662199	+	0.749328i	$0.730377\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	608700.i	0.145856i
$162$	0	0
$163$	−1.59682e6	−0.368718	−0.184359	−	0.982859i	$-0.559021\pi$
$163$	−1.59682e6	−0.368718	−0.184359	+	0.982859i	$0.559021\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 250180.i	− 0.0537159i	−0.999639	−	0.0268580i	$-0.991450\pi$
$167$	− 250180.i	− 0.0537159i	0.999639	−	0.0268580i	$-0.00855019\pi$
$168$	0	0
$169$	2.61518e6	0.541802
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.40512e6i	− 0.271379i	−0.990751	−	0.135690i	$-0.956675\pi$
$173$	− 1.40512e6i	− 0.271379i	0.990751	−	0.135690i	$-0.0433250\pi$
$174$	0	0
$175$	2.82430e6	0.526983
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.81300e6i	1.18790i	0.804502	+	0.593949i	$0.202432\pi$
$179$	6.81300e6i	1.18790i	−0.804502	+	0.593949i	$0.797568\pi$
$180$	0	0
$181$	−503992.	−0.0849939	−0.0424970	−	0.999097i	$-0.513531\pi$
$181$	−503992.	−0.0849939	−0.0424970	+	0.999097i	$0.513531\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 4.13696e6i	− 0.653380i
$186$	0	0
$187$	1.41011e7	2.15640
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 118013.i	− 0.0169368i	−0.999964	−	0.00846839i	$-0.997304\pi$
$191$	− 118013.i	− 0.0169368i	0.999964	−	0.00846839i	$-0.00269560\pi$
$192$	0	0
$193$	−1.42021e6	−0.197552	−0.0987758	−	0.995110i	$-0.531493\pi$
$193$	−1.42021e6	−0.197552	−0.0987758	+	0.995110i	$0.531493\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.82315e6i	− 0.892454i	−0.894920	−	0.446227i	$-0.852767\pi$
$197$	− 6.82315e6i	− 0.892454i	0.894920	−	0.446227i	$-0.147233\pi$
$198$	0	0
$199$	−5.53620e6	−0.702510	−0.351255	−	0.936280i	$-0.614245\pi$
$199$	−5.53620e6	−0.702510	−0.351255	+	0.936280i	$0.614245\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.02516e7i	1.22548i
$204$	0	0
$205$	4.43799e6	0.515139
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.29022e7i	1.41327i
$210$	0	0
$211$	4.13567e6	0.440249	0.220125	−	0.975472i	$-0.429354\pi$
$211$	4.13567e6	0.440249	0.220125	+	0.975472i	$0.429354\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 3.14889e6i	− 0.316841i
$216$	0	0
$217$	−2.48197e6	−0.242894
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.60762e7i	1.48938i
$222$	0	0
$223$	−5.80272e6	−0.523259	−0.261630	−	0.965168i	$-0.584260\pi$
$223$	−5.80272e6	−0.523259	−0.261630	+	0.965168i	$0.584260\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.75079e7i	1.49677i	0.663263	+	0.748386i	$0.269171\pi$
$227$	1.75079e7i	1.49677i	−0.663263	+	0.748386i	$0.730829\pi$
$228$	0	0
$229$	1.82843e7	1.52255	0.761276	−	0.648428i	$-0.224573\pi$
$229$	1.82843e7	1.52255	0.761276	+	0.648428i	$0.224573\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.00126e7i	1.58211i	0.611748	+	0.791053i	$0.290467\pi$
$233$	2.00126e7i	1.58211i	−0.611748	+	0.791053i	$0.709533\pi$
$234$	0	0
$235$	1.04879e7	0.808135
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 2.33685e7i	− 1.71173i	−0.517195	−	0.855867i	$-0.673024\pi$
$239$	− 2.33685e7i	− 1.71173i	0.517195	−	0.855867i	$-0.326976\pi$
$240$	0	0
$241$	1.23056e7	0.879127	0.439564	−	0.898211i	$-0.355133\pi$
$241$	1.23056e7	0.879127	0.439564	+	0.898211i	$0.355133\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.69829e6i	0.251479i
$246$	0	0
$247$	−1.47094e7	−0.976120
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.42559e7i	0.901518i	0.892646	+	0.450759i	$0.148847\pi$
$251$	1.42559e7i	0.901518i	−0.892646	+	0.450759i	$0.851153\pi$
$252$	0	0
$253$	−5.96938e6	−0.368610
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.51969e7i	− 1.48439i	−0.670185	−	0.742194i	$-0.733785\pi$
$257$	− 2.51969e7i	− 1.48439i	0.670185	−	0.742194i	$-0.266215\pi$
$258$	0	0
$259$	1.58615e7	0.912944
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 1.63580e7i	− 0.899215i	−0.893226	−	0.449608i	$-0.851564\pi$
$263$	− 1.63580e7i	− 0.899215i	0.893226	−	0.449608i	$-0.148436\pi$
$264$	0	0
$265$	−1.58355e6	−0.0850931
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.98259e6i	0.461471i	0.973016	+	0.230736i	$0.0741133\pi$
$269$	8.98259e6i	0.461471i	−0.973016	+	0.230736i	$0.925887\pi$
$270$	0	0
$271$	1.32419e7	0.665339	0.332669	−	0.943044i	$-0.392051\pi$
$271$	1.32419e7	0.665339	0.332669	+	0.943044i	$0.392051\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.76972e7i	1.33180i
$276$	0	0
$277$	−2.45331e7	−1.15428	−0.577142	−	0.816644i	$-0.695832\pi$
$277$	−2.45331e7	−1.15428	−0.577142	+	0.816644i	$0.695832\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 2.48398e7i	− 1.11951i	−0.828657	−	0.559757i	$-0.810894\pi$
$281$	− 2.48398e7i	− 1.11951i	0.828657	−	0.559757i	$-0.189106\pi$
$282$	0	0
$283$	−2.60467e7	−1.14919	−0.574597	−	0.818436i	$-0.694841\pi$
$283$	−2.60467e7	−1.14919	−0.574597	+	0.818436i	$0.694841\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.70157e7i	0.719785i
$288$	0	0
$289$	−1.05902e7	−0.438744
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.38405e7i	− 0.550235i	−0.961411	−	0.275118i	$-0.911283\pi$
$293$	− 1.38405e7i	− 0.550235i	0.961411	−	0.275118i	$-0.0887168\pi$
$294$	0	0
$295$	2.28550e7	0.890255
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 6.80547e6i	− 0.254592i
$300$	0	0
$301$	1.20731e7	0.442711
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 6.40278e6i	− 0.225668i
$306$	0	0
$307$	3.32320e7	1.14853	0.574263	−	0.818671i	$-0.305289\pi$
$307$	3.32320e7	1.14853	0.574263	+	0.818671i	$0.305289\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 4.68279e7i	− 1.55677i	−0.627790	−	0.778383i	$-0.716040\pi$
$311$	− 4.68279e7i	− 1.55677i	0.627790	−	0.778383i	$-0.283960\pi$
$312$	0	0
$313$	−762238.	−0.0248575	−0.0124288	−	0.999923i	$-0.503956\pi$
$313$	−762238.	−0.0248575	−0.0124288	+	0.999923i	$0.503956\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.26499e7i	− 0.711030i	−0.934671	−	0.355515i	$-0.884306\pi$
$317$	− 2.26499e7i	− 0.711030i	0.934671	−	0.355515i	$-0.115694\pi$
$318$	0	0
$319$	−1.00535e8	−3.09704
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3.17752e7i	− 0.942933i
$324$	0	0
$325$	−3.15766e7	−0.919846
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.02115e7i	1.12918i
$330$	0	0
$331$	−9.16460e6	−0.252714	−0.126357	−	0.991985i	$-0.540329\pi$
$331$	−9.16460e6	−0.252714	−0.126357	+	0.991985i	$0.540329\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 2.77301e7i	− 0.737593i
$336$	0	0
$337$	−1.76811e7	−0.461976	−0.230988	−	0.972957i	$-0.574196\pi$
$337$	−1.76811e7	−0.461976	−0.230988	+	0.972957i	$0.574196\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.43401e7i	− 0.613845i
$342$	0	0
$343$	−4.28859e7	−1.06275
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.85407e6i	0.116176i	0.998311	+	0.0580882i	$0.0185005\pi$
$347$	4.85407e6i	0.116176i	−0.998311	+	0.0580882i	$0.981500\pi$
$348$	0	0
$349$	4.79164e7	1.12722	0.563609	−	0.826042i	$-0.309413\pi$
$349$	4.79164e7	1.12722	0.563609	+	0.826042i	$0.309413\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.58647e7i	0.815349i	0.913127	+	0.407674i	$0.133660\pi$
$353$	3.58647e7i	0.815349i	−0.913127	+	0.407674i	$0.866340\pi$
$354$	0	0
$355$	−3.82644e6	−0.0855283
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 6.54618e7i	− 1.41483i	−0.706798	−	0.707415i	$-0.749861\pi$
$359$	− 6.54618e7i	− 1.41483i	0.706798	−	0.707415i	$-0.250139\pi$
$360$	0	0
$361$	−1.79722e7	−0.382015
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.94291e7i	− 0.810845i
$366$	0	0
$367$	8.10299e7	1.63926	0.819629	−	0.572895i	$-0.194180\pi$
$367$	8.10299e7	1.63926	0.819629	+	0.572895i	$0.194180\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 6.07147e6i	− 0.118897i
$372$	0	0
$373$	−1.25001e7	−0.240873	−0.120436	−	0.992721i	$-0.538429\pi$
$373$	−1.25001e7	−0.240873	−0.120436	+	0.992721i	$0.538429\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.14617e8i	− 2.13906i
$378$	0	0
$379$	−5.83107e6	−0.107110	−0.0535551	−	0.998565i	$-0.517055\pi$
$379$	−5.83107e6	−0.107110	−0.0535551	+	0.998565i	$0.517055\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.54454e7i	1.34288i	0.741060	+	0.671439i	$0.234323\pi$
$383$	7.54454e7i	1.34288i	−0.741060	+	0.671439i	$0.765677\pi$
$384$	0	0
$385$	3.71563e7	0.651104
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.10847e7i	0.528078i	0.964512	+	0.264039i	$0.0850547\pi$
$389$	3.10847e7i	0.528078i	−0.964512	+	0.264039i	$0.914945\pi$
$390$	0	0
$391$	1.47012e7	0.245936
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.27324e7i	0.531113i
$396$	0	0
$397$	−8.23020e7	−1.31534	−0.657672	−	0.753305i	$-0.728459\pi$
$397$	−8.23020e7	−1.31534	−0.657672	+	0.753305i	$0.728459\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.97466e7i	0.616405i	0.951321	+	0.308203i	$0.0997275\pi$
$401$	3.97466e7i	0.616405i	−0.951321	+	0.308203i	$0.900272\pi$
$402$	0	0
$403$	2.77492e7	0.423970
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.55550e8i	2.30720i
$408$	0	0
$409$	−3.66381e7	−0.535505	−0.267752	−	0.963488i	$-0.586281\pi$
$409$	−3.66381e7	−0.535505	−0.267752	+	0.963488i	$0.586281\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.76280e7i	1.24392i
$414$	0	0
$415$	−1.55228e7	−0.217184
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 3.87942e7i	− 0.527382i	−0.964607	−	0.263691i	$-0.915060\pi$
$419$	− 3.87942e7i	− 0.527382i	0.964607	−	0.263691i	$-0.0849399\pi$
$420$	0	0
$421$	4.55875e7	0.610942	0.305471	−	0.952201i	$-0.401186\pi$
$421$	4.55875e7	0.610942	0.305471	+	0.952201i	$0.401186\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 6.82118e7i	− 0.888572i
$426$	0	0
$427$	2.45488e7	0.315317
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 1.92433e7i	− 0.240353i	−0.992753	−	0.120176i	$-0.961654\pi$
$431$	− 1.92433e7i	− 0.240353i	0.992753	−	0.120176i	$-0.0383460\pi$
$432$	0	0
$433$	7.49525e7	0.923257	0.461629	−	0.887073i	$-0.347265\pi$
$433$	7.49525e7	0.923257	0.461629	+	0.887073i	$0.347265\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.34513e7i	0.161183i
$438$	0	0
$439$	5.71583e7	0.675593	0.337797	−	0.941219i	$-0.390318\pi$
$439$	5.71583e7	0.675593	0.337797	+	0.941219i	$0.390318\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 1.13525e8i	− 1.30581i	−0.757439	−	0.652906i	$-0.773550\pi$
$443$	− 1.13525e8i	− 1.30581i	0.757439	−	0.652906i	$-0.226450\pi$
$444$	0	0
$445$	2.72541e7	0.309280
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.42209e8i	1.57105i	0.618832	+	0.785523i	$0.287606\pi$
$449$	1.42209e8i	1.57105i	−0.618832	+	0.785523i	$0.712394\pi$
$450$	0	0
$451$	−1.66868e8	−1.81905
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.23606e7i	0.449705i
$456$	0	0
$457$	1.19006e8	1.24687	0.623436	−	0.781874i	$-0.285736\pi$
$457$	1.19006e8	1.24687	0.623436	+	0.781874i	$0.285736\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.01298e8i	1.03394i	0.856002	+	0.516972i	$0.172941\pi$
$461$	1.01298e8i	1.03394i	−0.856002	+	0.516972i	$0.827059\pi$
$462$	0	0
$463$	1.32408e8	1.33404	0.667022	−	0.745038i	$-0.267569\pi$
$463$	1.32408e8	1.33404	0.667022	+	0.745038i	$0.267569\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.05835e7i	0.889402i	0.895679	+	0.444701i	$0.146690\pi$
$467$	9.05835e7i	0.889402i	−0.895679	+	0.444701i	$0.853310\pi$
$468$	0	0
$469$	1.06320e8	1.03061
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.18398e8i	1.11882i
$474$	0	0
$475$	6.24124e7	0.582358
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 1.76007e7i	− 0.160149i	−0.996789	−	0.0800745i	$-0.974484\pi$
$479$	− 1.76007e7i	− 0.160149i	0.996789	−	0.0800745i	$-0.0255158\pi$
$480$	0	0
$481$	−1.77336e8	−1.59354
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 2.71767e6i	− 0.0238216i
$486$	0	0
$487$	1.25577e7	0.108723	0.0543617	−	0.998521i	$-0.482688\pi$
$487$	1.25577e7	0.108723	0.0543617	+	0.998521i	$0.482688\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.73001e7i	0.230632i	0.993329	+	0.115316i	$0.0367880\pi$
$491$	2.73001e7i	0.230632i	−0.993329	+	0.115316i	$0.963212\pi$
$492$	0	0
$493$	2.47595e8	2.06634
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.46709e7i	− 0.119506i
$498$	0	0
$499$	−8.84919e7	−0.712200	−0.356100	−	0.934448i	$-0.615894\pi$
$499$	−8.84919e7	−0.712200	−0.356100	+	0.934448i	$0.615894\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.81896e8i	1.42929i	0.699488	+	0.714644i	$0.253411\pi$
$503$	1.81896e8i	1.42929i	−0.699488	+	0.714644i	$0.746589\pi$
$504$	0	0
$505$	−5.80195e7	−0.450505
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1.32765e8i	− 1.00677i	−0.864063	−	0.503384i	$-0.832088\pi$
$509$	− 1.32765e8i	− 1.00677i	0.864063	−	0.503384i	$-0.167912\pi$
$510$	0	0
$511$	1.51175e8	1.13296
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 9.04057e7i	− 0.661872i
$516$	0	0
$517$	−3.94344e8	−2.85367
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.26305e8i	− 0.893117i	−0.894754	−	0.446558i	$-0.852649\pi$
$521$	− 1.26305e8i	− 0.893117i	0.894754	−	0.446558i	$-0.147351\pi$
$522$	0	0
$523$	−2.49848e7	−0.174651	−0.0873254	−	0.996180i	$-0.527832\pi$
$523$	−2.49848e7	−0.174651	−0.0873254	+	0.996180i	$0.527832\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.99439e7i	0.409556i
$528$	0	0
$529$	1.41812e8	0.957960
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.90241e8i	− 1.25638i
$534$	0	0
$535$	8.37734e7	0.547073
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.39056e8i	− 0.888019i
$540$	0	0
$541$	2.04308e6	0.0129031	0.00645154	−	0.999979i	$-0.497946\pi$
$541$	2.04308e6	0.0129031	0.00645154	+	0.999979i	$0.497946\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 8.93857e6i	− 0.0552177i
$546$	0	0
$547$	−4.53311e7	−0.276971	−0.138485	−	0.990364i	$-0.544223\pi$
$547$	−4.53311e7	−0.276971	−0.138485	+	0.990364i	$0.544223\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.26544e8i	1.35425i
$552$	0	0
$553$	−1.25499e8	−0.742104
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.33395e8i	0.771922i	0.922515	+	0.385961i	$0.126130\pi$
$557$	1.33395e8i	0.771922i	−0.922515	+	0.385961i	$0.873870\pi$
$558$	0	0
$559$	−1.34981e8	−0.772749
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2.52746e8i	− 1.41631i	−0.706056	−	0.708156i	$-0.749527\pi$
$563$	− 2.52746e8i	− 1.41631i	0.706056	−	0.708156i	$-0.250473\pi$
$564$	0	0
$565$	−1.20846e8	−0.670021
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.42745e7i	0.0774863i	0.999249	+	0.0387432i	$0.0123354\pi$
$569$	1.42745e7i	0.0774863i	−0.999249	+	0.0387432i	$0.987665\pi$
$570$	0	0
$571$	−2.59680e8	−1.39486	−0.697430	−	0.716653i	$-0.745673\pi$
$571$	−2.59680e8	−1.39486	−0.697430	+	0.716653i	$0.745673\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.88758e7i	0.151891i
$576$	0	0
$577$	1.75130e8	0.911658	0.455829	−	0.890067i	$-0.349343\pi$
$577$	1.75130e8	0.911658	0.455829	+	0.890067i	$0.349343\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 5.95160e7i	− 0.303463i
$582$	0	0
$583$	5.95415e7	0.300479
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.81499e8i	− 1.39175i	−0.718161	−	0.695877i	$-0.755016\pi$
$587$	− 2.81499e8i	− 1.39175i	0.718161	−	0.695877i	$-0.244984\pi$
$588$	0	0
$589$	−5.48474e7	−0.268417
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 3.14702e7i	− 0.150916i	−0.997149	−	0.0754581i	$-0.975958\pi$
$593$	− 3.14702e7i	− 0.150916i	0.997149	−	0.0754581i	$-0.0240419\pi$
$594$	0	0
$595$	−9.15073e7	−0.434415
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 2.44137e8i	− 1.13593i	−0.823052	−	0.567966i	$-0.807730\pi$
$599$	− 2.44137e8i	− 1.13593i	0.823052	−	0.567966i	$-0.192270\pi$
$600$	0	0
$601$	6.33608e7	0.291875	0.145938	−	0.989294i	$-0.453380\pi$
$601$	6.33608e7	0.291875	0.145938	+	0.989294i	$0.453380\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.51642e8i	1.13636i
$606$	0	0
$607$	−1.28655e8	−0.575254	−0.287627	−	0.957743i	$-0.592866\pi$
$607$	−1.28655e8	−0.575254	−0.287627	+	0.957743i	$0.592866\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 4.49577e8i	− 1.97097i
$612$	0	0
$613$	7.55398e7	0.327940	0.163970	−	0.986465i	$-0.447570\pi$
$613$	7.55398e7	0.327940	0.163970	+	0.986465i	$0.447570\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.27329e8i	− 1.39357i	−0.717279	−	0.696786i	$-0.754613\pi$
$617$	− 3.27329e8i	− 1.39357i	0.717279	−	0.696786i	$-0.245387\pi$
$618$	0	0
$619$	5.42356e7	0.228672	0.114336	−	0.993442i	$-0.463526\pi$
$619$	5.42356e7	0.228672	0.114336	+	0.993442i	$0.463526\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.04495e8i	0.432145i
$624$	0	0
$625$	7.06994e7	0.289585
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3.83082e8i	− 1.53936i
$630$	0	0
$631$	3.92478e7	0.156217	0.0781083	−	0.996945i	$-0.475112\pi$
$631$	3.92478e7	0.156217	0.0781083	+	0.996945i	$0.475112\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.09653e8i	0.818804i
$636$	0	0
$637$	1.58532e8	0.613337
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.07803e8i	0.409316i	0.978834	+	0.204658i	$0.0656082\pi$
$641$	1.07803e8i	0.409316i	−0.978834	+	0.204658i	$0.934392\pi$
$642$	0	0
$643$	−6.83678e7	−0.257169	−0.128585	−	0.991699i	$-0.541043\pi$
$643$	−6.83678e7	−0.257169	−0.128585	+	0.991699i	$0.541043\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.15029e8i	− 0.424712i	−0.977192	−	0.212356i	$-0.931886\pi$
$647$	− 1.15029e8i	− 0.424712i	0.977192	−	0.212356i	$-0.0681136\pi$
$648$	0	0
$649$	−8.59346e8	−3.14365
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.11435e8i	0.759343i	0.925121	+	0.379671i	$0.123963\pi$
$653$	2.11435e8i	0.759343i	−0.925121	+	0.379671i	$0.876037\pi$
$654$	0	0
$655$	−4.08175e7	−0.145252
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.71625e8i	1.29852i	0.760566	+	0.649260i	$0.224921\pi$
$659$	3.71625e8i	1.29852i	−0.760566	+	0.649260i	$0.775079\pi$
$660$	0	0
$661$	−2.02502e7	−0.0701174	−0.0350587	−	0.999385i	$-0.511162\pi$
$661$	−2.02502e7	−0.0701174	−0.0350587	+	0.999385i	$0.511162\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 8.37273e7i	− 0.284710i
$666$	0	0
$667$	−1.04813e8	−0.353215
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.40745e8i	0.796873i
$672$	0	0
$673$	−9.67161e7	−0.317288	−0.158644	−	0.987336i	$-0.550712\pi$
$673$	−9.67161e7	−0.317288	−0.158644	+	0.987336i	$0.550712\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 3.96716e8i	− 1.27854i	−0.768983	−	0.639270i	$-0.779237\pi$
$677$	− 3.96716e8i	− 1.27854i	0.768983	−	0.639270i	$-0.220763\pi$
$678$	0	0
$679$	1.04198e7	0.0332850
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.73415e8i	0.858143i	0.903271	+	0.429071i	$0.141159\pi$
$683$	2.73415e8i	0.858143i	−0.903271	+	0.429071i	$0.858841\pi$
$684$	0	0
$685$	−4.85409e7	−0.151020
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.78811e7i	0.207535i
$690$	0	0
$691$	6.10061e8	1.84901	0.924505	−	0.381170i	$-0.124479\pi$
$691$	6.10061e8	1.84901	0.924505	+	0.381170i	$0.124479\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.03783e8i	− 0.309152i
$696$	0	0
$697$	4.10958e8	1.21367
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.58578e7i	0.104095i	0.998645	+	0.0520474i	$0.0165747\pi$
$701$	3.58578e7i	0.104095i	−0.998645	+	0.0520474i	$0.983425\pi$
$702$	0	0
$703$	3.50512e8	1.00887
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.22452e8i	− 0.629474i
$708$	0	0
$709$	2.53957e8	0.712561	0.356281	−	0.934379i	$-0.384045\pi$
$709$	2.53957e8	0.712561	0.356281	+	0.934379i	$0.384045\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.53758e7i	− 0.0700086i
$714$	0	0
$715$	−4.15420e8	−1.13650
$716$	0	0
$717$	0	0
$718$	0	0
$719$	5.46106e8i	1.46923i	0.678483	+	0.734616i	$0.262638\pi$
$719$	5.46106e8i	1.46923i	−0.678483	+	0.734616i	$0.737362\pi$
$720$	0	0
$721$	3.46623e8	0.924809
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.86322e8i	1.27617i
$726$	0	0
$727$	2.43487e8	0.633684	0.316842	−	0.948478i	$-0.397378\pi$
$727$	2.43487e8	0.633684	0.316842	+	0.948478i	$0.397378\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2.91587e8i	− 0.746476i
$732$	0	0
$733$	5.17021e7	0.131279	0.0656397	−	0.997843i	$-0.479091\pi$
$733$	5.17021e7	0.131279	0.0656397	+	0.997843i	$0.479091\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.04265e9i	2.60457i
$738$	0	0
$739$	1.91749e8	0.475116	0.237558	−	0.971373i	$-0.423653\pi$
$739$	1.91749e8	0.475116	0.237558	+	0.971373i	$0.423653\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.33634e7i	0.154480i	0.997013	+	0.0772399i	$0.0246107\pi$
$743$	6.33634e7i	0.154480i	−0.997013	+	0.0772399i	$0.975389\pi$
$744$	0	0
$745$	1.65477e8	0.400193
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.21195e8i	0.764404i
$750$	0	0
$751$	−6.04218e8	−1.42651	−0.713253	−	0.700906i	$-0.752779\pi$
$751$	−6.04218e8	−1.42651	−0.713253	+	0.700906i	$0.752779\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.58218e8i	0.832352i
$756$	0	0
$757$	−3.57599e8	−0.824345	−0.412173	−	0.911106i	$-0.635230\pi$
$757$	−3.57599e8	−0.824345	−0.412173	+	0.911106i	$0.635230\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 6.04342e8i	− 1.37129i	−0.727937	−	0.685644i	$-0.759520\pi$
$761$	− 6.04342e8i	− 1.37129i	0.727937	−	0.685644i	$-0.240480\pi$
$762$	0	0
$763$	3.42713e7	0.0771537
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 9.79710e8i	− 2.17126i
$768$	0	0
$769$	7.64239e8	1.68055	0.840273	−	0.542164i	$-0.182395\pi$
$769$	7.64239e8	1.68055	0.840273	+	0.542164i	$0.182395\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1.35000e7i	− 0.0292277i	−0.999893	−	0.0146139i	$-0.995348\pi$
$773$	− 1.35000e7i	− 0.0292277i	0.999893	−	0.0146139i	$-0.00465190\pi$
$774$	0	0
$775$	−1.17741e8	−0.252943
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.76018e8i	0.795420i
$780$	0	0
$781$	1.43874e8	0.302016
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.26171e8i	0.674274i
$786$	0	0
$787$	−2.77809e8	−0.569931	−0.284966	−	0.958538i	$-0.591982\pi$
$787$	−2.77809e8	−0.569931	−0.284966	+	0.958538i	$0.591982\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 4.63336e8i	− 0.936195i
$792$	0	0
$793$	−2.74464e8	−0.550384
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.77395e8i	1.33803i	0.743247	+	0.669017i	$0.233285\pi$
$797$	6.77395e8i	1.33803i	−0.743247	+	0.669017i	$0.766715\pi$
$798$	0	0
$799$	9.71178e8	1.90396
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.48253e9i	2.86324i
$804$	0	0
$805$	3.87374e7	0.0742580
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 2.72493e8i	− 0.514646i	−0.966325	−	0.257323i	$-0.917159\pi$
$809$	− 2.72493e8i	− 0.514646i	0.966325	−	0.257323i	$-0.0828405\pi$
$810$	0	0
$811$	9.35831e8	1.75443	0.877213	−	0.480101i	$-0.159400\pi$
$811$	9.35831e8	1.75443	0.877213	+	0.480101i	$0.159400\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.01621e8i	0.187721i
$816$	0	0
$817$	2.66796e8	0.489230
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 8.06443e8i	− 1.45728i	−0.684895	−	0.728642i	$-0.740152\pi$
$821$	− 8.06443e8i	− 1.45728i	0.684895	−	0.728642i	$-0.259848\pi$
$822$	0	0
$823$	−3.70180e8	−0.664070	−0.332035	−	0.943267i	$-0.607735\pi$
$823$	−3.70180e8	−0.664070	−0.332035	+	0.943267i	$0.607735\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.68813e8i	0.298462i	0.988802	+	0.149231i	$0.0476799\pi$
$827$	1.68813e8i	0.298462i	−0.988802	+	0.149231i	$0.952320\pi$
$828$	0	0
$829$	4.66016e7	0.0817969	0.0408985	−	0.999163i	$-0.486978\pi$
$829$	4.66016e7	0.0817969	0.0408985	+	0.999163i	$0.486978\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.42462e8i	0.592484i
$834$	0	0
$835$	−1.59214e7	−0.0273477
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.51890e8i	0.934473i	0.884132	+	0.467237i	$0.154750\pi$
$839$	5.51890e8i	0.934473i	−0.884132	+	0.467237i	$0.845250\pi$
$840$	0	0
$841$	−1.17043e9	−1.96769
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.66429e8i	− 0.275841i
$846$	0	0
$847$	−9.64817e8	−1.58780
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.62169e8i	0.263135i
$852$	0	0
$853$	−5.59302e8	−0.901155	−0.450577	−	0.892737i	$-0.648782\pi$
$853$	−5.59302e8	−0.901155	−0.450577	+	0.892737i	$0.648782\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 5.49346e7i	− 0.0872778i	−0.999047	−	0.0436389i	$-0.986105\pi$
$857$	− 5.49346e7i	− 0.0872778i	0.999047	−	0.0436389i	$-0.0138951\pi$
$858$	0	0
$859$	1.07732e9	1.69967	0.849834	−	0.527050i	$-0.176702\pi$
$859$	1.07732e9	1.69967	0.849834	+	0.527050i	$0.176702\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 4.78807e8i	− 0.744952i	−0.928042	−	0.372476i	$-0.878509\pi$
$863$	− 4.78807e8i	− 0.744952i	0.928042	−	0.372476i	$-0.121491\pi$
$864$	0	0
$865$	−8.94216e7	−0.138164
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1.23074e9i	− 1.87545i
$870$	0	0
$871$	−1.18869e9	−1.79893
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 4.22363e8i	− 0.630467i
$876$	0	0
$877$	1.02515e9	1.51981	0.759904	−	0.650035i	$-0.225246\pi$
$877$	1.02515e9	1.51981	0.759904	+	0.650035i	$0.225246\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 5.58043e8i	− 0.816093i	−0.912961	−	0.408047i	$-0.866210\pi$
$881$	− 5.58043e8i	− 0.816093i	0.912961	−	0.408047i	$-0.133790\pi$
$882$	0	0
$883$	5.20043e8	0.755366	0.377683	−	0.925935i	$-0.376721\pi$
$883$	5.20043e8	0.755366	0.377683	+	0.925935i	$0.376721\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 7.26982e8i	− 1.04172i	−0.853641	−	0.520862i	$-0.825611\pi$
$887$	− 7.26982e8i	− 1.04172i	0.853641	−	0.520862i	$-0.174389\pi$
$888$	0	0
$889$	−8.03829e8	−1.14408
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.88608e8i	1.24783i
$894$	0	0
$895$	4.33577e8	0.604779
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 4.27375e8i	− 0.588207i
$900$	0	0
$901$	−1.46637e8	−0.200479
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.20739e7i	0.0432718i
$906$	0	0
$907$	−8.85539e8	−1.18682	−0.593411	−	0.804899i	$-0.702219\pi$
$907$	−8.85539e8	−1.18682	−0.593411	+	0.804899i	$0.702219\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1.16978e9i	− 1.54721i	−0.633670	−	0.773603i	$-0.718452\pi$
$911$	− 1.16978e9i	− 1.54721i	0.633670	−	0.773603i	$-0.281548\pi$
$912$	0	0
$913$	5.83659e8	0.766914
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.56498e8i	− 0.202956i
$918$	0	0
$919$	−8.29003e8	−1.06809	−0.534047	−	0.845455i	$-0.679329\pi$
$919$	−8.29003e8	−1.06809	−0.534047	+	0.845455i	$0.679329\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.64026e8i	0.208596i
$924$	0	0
$925$	7.52444e8	0.950713
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 2.69724e8i	− 0.336413i	−0.985752	−	0.168207i	$-0.946202\pi$
$929$	− 2.69724e8i	− 0.336413i	0.985752	−	0.168207i	$-0.0537975\pi$
$930$	0	0
$931$	−3.13345e8	−0.388306
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 8.97390e8i	− 1.09786i
$936$	0	0
$937$	7.62796e8	0.927234	0.463617	−	0.886036i	$-0.346551\pi$
$937$	7.62796e8	0.927234	0.463617	+	0.886036i	$0.346551\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8.04800e8i	0.965871i	0.875656	+	0.482935i	$0.160429\pi$
$941$	8.04800e8i	0.965871i	−0.875656	+	0.482935i	$0.839571\pi$
$942$	0	0
$943$	−1.73969e8	−0.207461
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 5.60784e8i	− 0.660307i	−0.943927	−	0.330153i	$-0.892900\pi$
$947$	− 5.60784e8i	− 0.660307i	0.943927	−	0.330153i	$-0.107100\pi$
$948$	0	0
$949$	−1.69018e9	−1.97758
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 9.97457e8i	− 1.15243i	−0.817297	−	0.576216i	$-0.804529\pi$
$953$	− 9.97457e8i	− 1.15243i	0.817297	−	0.576216i	$-0.195471\pi$
$954$	0	0
$955$	−7.51032e6	−0.00862280
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.86110e8i	− 0.211015i
$960$	0	0
$961$	−7.84034e8	−0.883415
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.03816e7i	0.100577i
$966$	0	0
$967$	1.06450e8	0.117725	0.0588623	−	0.998266i	$-0.481253\pi$
$967$	1.06450e8	0.117725	0.0588623	+	0.998266i	$0.481253\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.55459e8i	0.715959i	0.933729	+	0.357979i	$0.116534\pi$
$971$	6.55459e8i	0.715959i	−0.933729	+	0.357979i	$0.883466\pi$
$972$	0	0
$973$	3.97913e8	0.431966
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.75342e8i	0.938629i	0.883031	+	0.469314i	$0.155499\pi$
$977$	8.75342e8i	0.938629i	−0.883031	+	0.469314i	$0.844501\pi$
$978$	0	0
$979$	−1.02475e9	−1.09212
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.24589e8i	0.447000i	0.974704	+	0.223500i	$0.0717483\pi$
$983$	4.24589e8i	0.447000i	−0.974704	+	0.223500i	$0.928252\pi$
$984$	0	0
$985$	−4.34222e8	−0.454364
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.23436e8i	0.127601i
$990$	0	0
$991$	−1.44957e9	−1.48942	−0.744711	−	0.667387i	$-0.767413\pi$
$991$	−1.44957e9	−1.48942	−0.744711	+	0.667387i	$0.767413\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.52321e8i	0.357659i
$996$	0	0
$997$	−1.54558e9	−1.55957	−0.779785	−	0.626048i	$-0.784672\pi$
$997$	−1.54558e9	−1.55957	−0.779785	+	0.626048i	$0.784672\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.7.e.c.17.1		2
3.2	odd	2	inner	144.7.e.c.17.2		2
4.3	odd	2		36.7.c.a.17.1	✓	2
8.3	odd	2		576.7.e.c.449.2		2
8.5	even	2		576.7.e.j.449.2		2
12.11	even	2		36.7.c.a.17.2	yes	2
24.5	odd	2		576.7.e.j.449.1		2
24.11	even	2		576.7.e.c.449.1		2
36.7	odd	6		324.7.g.e.53.2		4
36.11	even	6		324.7.g.e.53.1		4
36.23	even	6		324.7.g.e.269.2		4
36.31	odd	6		324.7.g.e.269.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.7.c.a.17.1	✓	2	4.3	odd	2
36.7.c.a.17.2	yes	2	12.11	even	2
144.7.e.c.17.1		2	1.1	even	1	trivial
144.7.e.c.17.2		2	3.2	odd	2	inner
324.7.g.e.53.1		4	36.11	even	6
324.7.g.e.53.2		4	36.7	odd	6
324.7.g.e.269.1		4	36.31	odd	6
324.7.g.e.269.2		4	36.23	even	6
576.7.e.c.449.1		2	24.11	even	2
576.7.e.c.449.2		2	8.3	odd	2
576.7.e.j.449.1		2	24.5	odd	2
576.7.e.j.449.2		2	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	144.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-63.6396i q^{5} +244.000 q^{7} +O(q^{10})\)	\(q-63.6396i q^{5} +244.000 q^{7} +2392.85i q^{11} -2728.00 q^{13} -5893.03i q^{17} +5392.00 q^{19} +2494.67i q^{23} +11575.0 q^{25} +42014.9i q^{29} -10172.0 q^{31} -15528.1i q^{35} +65006.0 q^{37} +69736.3i q^{41} +49480.0 q^{43} +164801. i q^{47} -58113.0 q^{49} -24883.1i q^{53} +152280. q^{55} +359131. i q^{59} +100610. q^{61} +173609. i q^{65} +435736. q^{67} -60126.7i q^{71} +619568. q^{73} +583855. i q^{77} -514340. q^{79} -243918. i q^{83} -375030. q^{85} +428256. i q^{89} -665632. q^{91} -343145. i q^{95} +42704.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 488 q^{7}+O(q^{10}) \) 2 * q + 488 * q^7	\( 2 q + 488 q^{7} - 5456 q^{13} + 10784 q^{19} + 23150 q^{25} - 20344 q^{31} + 130012 q^{37} + 98960 q^{43} - 116226 q^{49} + 304560 q^{55} + 201220 q^{61} + 871472 q^{67} + 1239136 q^{73} - 1028680 q^{79} - 750060 q^{85} - 1331264 q^{91} + 85408 q^{97}+O(q^{100}) \) 2 * q + 488 * q^7 - 5456 * q^13 + 10784 * q^19 + 23150 * q^25 - 20344 * q^31 + 130012 * q^37 + 98960 * q^43 - 116226 * q^49 + 304560 * q^55 + 201220 * q^61 + 871472 * q^67 + 1239136 * q^73 - 1028680 * q^79 - 750060 * q^85 - 1331264 * q^91 + 85408 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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