LMFDB - Embedded newform 144.7.e.d.17.2

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 $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+173.948i q^{5} +484.000 q^{7} +O(q^{10})$	$q+173.948i q^{5} +484.000 q^{7} -1340.67i q^{11} +3368.00 q^{13} +12.7279i q^{17} -5744.00 q^{19} +3377.14i q^{23} -14633.0 q^{25} +29354.8i q^{29} +39796.0 q^{31} +84191.0i q^{35} +52526.0 q^{37} -37042.5i q^{41} -3800.00 q^{43} -76791.8i q^{47} +116607. q^{49} +238738. i q^{53} +233208. q^{55} +249841. i q^{59} +13250.0 q^{61} +585858. i q^{65} -168968. q^{67} +531467. i q^{71} +236144. q^{73} -648886. i q^{77} +35116.0 q^{79} +10980.0i q^{83} -2214.00 q^{85} -129328. i q^{89} +1.63011e6 q^{91} -999159. i q^{95} -321424. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 968 q^{7}+O(q^{10}) $ 2 * q + 968 * q^7	$ 2 q + 968 q^{7} + 6736 q^{13} - 11488 q^{19} - 29266 q^{25} + 79592 q^{31} + 105052 q^{37} - 7600 q^{43} + 233214 q^{49} + 466416 q^{55} + 26500 q^{61} - 337936 q^{67} + 472288 q^{73} + 70232 q^{79} - 4428 q^{85} + 3260224 q^{91} - 642848 q^{97}+O(q^{100}) $ 2 * q + 968 * q^7 + 6736 * q^13 - 11488 * q^19 - 29266 * q^25 + 79592 * q^31 + 105052 * q^37 - 7600 * q^43 + 233214 * q^49 + 466416 * q^55 + 26500 * q^61 - 337936 * q^67 + 472288 * q^73 + 70232 * q^79 - 4428 * q^85 + 3260224 * q^91 - 642848 * 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
$5$	173.948i	1.39159i	0.718242	+	0.695793i	$0.244947\pi$
$5$	173.948i	1.39159i	−0.718242	+	0.695793i	$0.755053\pi$
$6$	0	0
$7$	484.000	1.41108	0.705539	−	0.708671i	$-0.250705\pi$
$7$	484.000	1.41108	0.705539	+	0.708671i	$0.250705\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1340.67i	− 1.00727i	−0.863917	−	0.503634i	$-0.831996\pi$
$11$	− 1340.67i	− 1.00727i	0.863917	−	0.503634i	$-0.168004\pi$
$12$	0	0
$13$	3368.00	1.53300	0.766500	−	0.642245i	$-0.221997\pi$
$13$	3368.00	1.53300	0.766500	+	0.642245i	$0.221997\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	12.7279i	0.00259066i	0.999999	+	0.00129533i	$0.000412317\pi$
$17$	12.7279i	0.00259066i	−0.999999	+	0.00129533i	$0.999588\pi$
$18$	0	0
$19$	−5744.00	−0.837440	−0.418720	−	0.908115i	$-0.637521\pi$
$19$	−5744.00	−0.837440	−0.418720	+	0.908115i	$0.637521\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3377.14i	0.277566i	0.990323	+	0.138783i	$0.0443190\pi$
$23$	3377.14i	0.277566i	−0.990323	+	0.138783i	$0.955681\pi$
$24$	0	0
$25$	−14633.0	−0.936512
$26$	0	0
$27$	0	0
$28$	0	0
$29$	29354.8i	1.20361i	0.798643	+	0.601805i	$0.205551\pi$
$29$	29354.8i	1.20361i	−0.798643	+	0.601805i	$0.794449\pi$
$30$	0	0
$31$	39796.0	1.33584	0.667920	−	0.744233i	$-0.267185\pi$
$31$	39796.0	1.33584	0.667920	+	0.744233i	$0.267185\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	84191.0i	1.96364i
$36$	0	0
$37$	52526.0	1.03698	0.518489	−	0.855085i	$-0.326495\pi$
$37$	52526.0	1.03698	0.518489	+	0.855085i	$0.326495\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 37042.5i	− 0.537463i	−0.963215	−	0.268732i	$-0.913396\pi$
$41$	− 37042.5i	− 0.537463i	0.963215	−	0.268732i	$-0.0866045\pi$
$42$	0	0
$43$	−3800.00	−0.0477945	−0.0238973	−	0.999714i	$-0.507607\pi$
$43$	−3800.00	−0.0477945	−0.0238973	+	0.999714i	$0.507607\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 76791.8i	− 0.739641i	−0.929103	−	0.369821i	$-0.879419\pi$
$47$	− 76791.8i	− 0.739641i	0.929103	−	0.369821i	$-0.120581\pi$
$48$	0	0
$49$	116607.	0.991143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	238738.i	1.60359i	0.597599	+	0.801795i	$0.296121\pi$
$53$	238738.i	1.60359i	−0.597599	+	0.801795i	$0.703879\pi$
$54$	0	0
$55$	233208.	1.40170
$56$	0	0
$57$	0	0
$58$	0	0
$59$	249841.i	1.21649i	0.793751	+	0.608243i	$0.208125\pi$
$59$	249841.i	1.21649i	−0.793751	+	0.608243i	$0.791875\pi$
$60$	0	0
$61$	13250.0	0.0583749	0.0291875	−	0.999574i	$-0.490708\pi$
$61$	13250.0	0.0583749	0.0291875	+	0.999574i	$0.490708\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	585858.i	2.13330i
$66$	0	0
$67$	−168968.	−0.561798	−0.280899	−	0.959737i	$-0.590633\pi$
$67$	−168968.	−0.561798	−0.280899	+	0.959737i	$0.590633\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	531467.i	1.48491i	0.669894	+	0.742457i	$0.266340\pi$
$71$	531467.i	1.48491i	−0.669894	+	0.742457i	$0.733660\pi$
$72$	0	0
$73$	236144.	0.607027	0.303514	−	0.952827i	$-0.401840\pi$
$73$	236144.	0.607027	0.303514	+	0.952827i	$0.401840\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 648886.i	− 1.42134i
$78$	0	0
$79$	35116.0	0.0712236	0.0356118	−	0.999366i	$-0.488662\pi$
$79$	35116.0	0.0712236	0.0356118	+	0.999366i	$0.488662\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10980.0i	0.0192029i	0.999954	+	0.00960144i	$0.00305628\pi$
$83$	10980.0i	0.0192029i	−0.999954	+	0.00960144i	$0.996944\pi$
$84$	0	0
$85$	−2214.00	−0.00360513
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 129328.i	− 0.183453i	−0.995784	−	0.0917263i	$-0.970762\pi$
$89$	− 129328.i	− 0.183453i	0.995784	−	0.0917263i	$-0.0292385\pi$
$90$	0	0
$91$	1.63011e6	2.16318
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 999159.i	− 1.16537i
$96$	0	0
$97$	−321424.	−0.352179	−0.176089	−	0.984374i	$-0.556345\pi$
$97$	−321424.	−0.352179	−0.176089	+	0.984374i	$0.556345\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	668780.i	0.649111i	0.945867	+	0.324556i	$0.105215\pi$
$101$	668780.i	0.649111i	−0.945867	+	0.324556i	$0.894785\pi$
$102$	0	0
$103$	−1.99341e6	−1.82425	−0.912127	−	0.409907i	$-0.865561\pi$
$103$	−1.99341e6	−1.82425	−0.912127	+	0.409907i	$0.865561\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	260668.i	0.212783i	0.994324	+	0.106391i	$0.0339296\pi$
$107$	260668.i	0.212783i	−0.994324	+	0.106391i	$0.966070\pi$
$108$	0	0
$109$	194456.	0.150156	0.0750779	−	0.997178i	$-0.476079\pi$
$109$	194456.	0.150156	0.0750779	+	0.997178i	$0.476079\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 821897.i	− 0.569616i	−0.958585	−	0.284808i	$-0.908070\pi$
$113$	− 821897.i	− 0.569616i	0.958585	−	0.284808i	$-0.0919298\pi$
$114$	0	0
$115$	−587448.	−0.386257
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6160.31i	0.00365563i
$120$	0	0
$121$	−25847.0	−0.0145900
$122$	0	0
$123$	0	0
$124$	0	0
$125$	172557.i	0.0883490i
$126$	0	0
$127$	−3.05721e6	−1.49250	−0.746250	−	0.665666i	$-0.768148\pi$
$127$	−3.05721e6	−1.49250	−0.746250	+	0.665666i	$0.768148\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 3.07388e6i	− 1.36733i	−0.729797	−	0.683664i	$-0.760385\pi$
$131$	− 3.07388e6i	− 1.36733i	0.729797	−	0.683664i	$-0.239615\pi$
$132$	0	0
$133$	−2.78010e6	−1.18169
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.48412e6i	− 1.74388i	−0.489617	−	0.871938i	$-0.662863\pi$
$137$	− 4.48412e6i	− 1.74388i	0.489617	−	0.871938i	$-0.337137\pi$
$138$	0	0
$139$	1.09233e6	0.406732	0.203366	−	0.979103i	$-0.434812\pi$
$139$	1.09233e6	0.406732	0.203366	+	0.979103i	$0.434812\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 4.51539e6i	− 1.54414i
$144$	0	0
$145$	−5.10622e6	−1.67493
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2.22087e6i	− 0.671375i	−0.941973	−	0.335687i	$-0.891031\pi$
$149$	− 2.22087e6i	− 0.671375i	0.941973	−	0.335687i	$-0.108969\pi$
$150$	0	0
$151$	4.07871e6	1.18465	0.592327	−	0.805697i	$-0.298209\pi$
$151$	4.07871e6	1.18465	0.592327	+	0.805697i	$0.298209\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.92245e6i	1.85894i
$156$	0	0
$157$	6.15568e6	1.59066	0.795329	−	0.606178i	$-0.207298\pi$
$157$	6.15568e6	1.59066	0.795329	+	0.606178i	$0.207298\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.63454e6i	0.391667i
$162$	0	0
$163$	−800696.	−0.184886	−0.0924432	−	0.995718i	$-0.529468\pi$
$163$	−800696.	−0.184886	−0.0924432	+	0.995718i	$0.529468\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.80467e6i	− 1.03161i	−0.856707	−	0.515804i	$-0.827493\pi$
$167$	− 4.80467e6i	− 1.03161i	0.856707	−	0.515804i	$-0.172507\pi$
$168$	0	0
$169$	6.51661e6	1.35009
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.56992e6i	− 0.689478i	−0.938699	−	0.344739i	$-0.887967\pi$
$173$	− 3.56992e6i	− 0.689478i	0.938699	−	0.344739i	$-0.112033\pi$
$174$	0	0
$175$	−7.08237e6	−1.32149
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.43698e6i	1.29669i	0.761345	+	0.648347i	$0.224539\pi$
$179$	7.43698e6i	1.29669i	−0.761345	+	0.648347i	$0.775461\pi$
$180$	0	0
$181$	−1.03812e7	−1.75070	−0.875350	−	0.483491i	$-0.839369\pi$
$181$	−1.03812e7	−1.75070	−0.875350	+	0.483491i	$0.839369\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.13681e6i	1.44304i
$186$	0	0
$187$	17064.0	0.00260949
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.29941e7i	− 1.86485i	−0.361360	−	0.932426i	$-0.617687\pi$
$191$	− 1.29941e7i	− 1.86485i	0.361360	−	0.932426i	$-0.382313\pi$
$192$	0	0
$193$	−3.93195e6	−0.546936	−0.273468	−	0.961881i	$-0.588171\pi$
$193$	−3.93195e6	−0.546936	−0.273468	+	0.961881i	$0.588171\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.37967e6i	0.703651i	0.936066	+	0.351825i	$0.114439\pi$
$197$	5.37967e6i	0.703651i	−0.936066	+	0.351825i	$0.885561\pi$
$198$	0	0
$199$	565900.	0.0718093	0.0359046	−	0.999355i	$-0.488569\pi$
$199$	565900.	0.0718093	0.0359046	+	0.999355i	$0.488569\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.42077e7i	1.69839i
$204$	0	0
$205$	6.44348e6	0.747926
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.70083e6i	0.843527i
$210$	0	0
$211$	1.35165e7	1.43885	0.719427	−	0.694568i	$-0.244404\pi$
$211$	1.35165e7	1.43885	0.719427	+	0.694568i	$0.244404\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 661003.i	− 0.0665102i
$216$	0	0
$217$	1.92613e7	1.88497
$218$	0	0
$219$	0	0
$220$	0	0
$221$	42867.6i	0.00397148i
$222$	0	0
$223$	5.35484e6	0.482872	0.241436	−	0.970417i	$-0.422382\pi$
$223$	5.35484e6	0.482872	0.241436	+	0.970417i	$0.422382\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.36063e7i	1.16322i	0.813466	+	0.581612i	$0.197578\pi$
$227$	1.36063e7i	1.16322i	−0.813466	+	0.581612i	$0.802422\pi$
$228$	0	0
$229$	4.34641e6	0.361930	0.180965	−	0.983490i	$-0.442078\pi$
$229$	4.34641e6	0.361930	0.180965	+	0.983490i	$0.442078\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2.02333e7i	− 1.59956i	−0.600297	−	0.799778i	$-0.704951\pi$
$233$	− 2.02333e7i	− 1.59956i	0.600297	−	0.799778i	$-0.295049\pi$
$234$	0	0
$235$	1.33578e7	1.02927
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.03947e7i	1.49391i	0.664877	+	0.746953i	$0.268484\pi$
$239$	2.03947e7i	1.49391i	−0.664877	+	0.746953i	$0.731516\pi$
$240$	0	0
$241$	−3.12093e6	−0.222963	−0.111481	−	0.993767i	$-0.535560\pi$
$241$	−3.12093e6	−0.222963	−0.111481	+	0.993767i	$0.535560\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.02836e7i	1.37926i
$246$	0	0
$247$	−1.93458e7	−1.28379
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 5.09519e6i	− 0.322210i	−0.986937	−	0.161105i	$-0.948494\pi$
$251$	− 5.09519e6i	− 0.322210i	0.986937	−	0.161105i	$-0.0515058\pi$
$252$	0	0
$253$	4.52765e6	0.279583
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.44374e7i	0.850529i	0.905069	+	0.425264i	$0.139819\pi$
$257$	1.44374e7i	0.850529i	−0.905069	+	0.425264i	$0.860181\pi$
$258$	0	0
$259$	2.54226e7	1.46326
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 3.12567e7i	− 1.71821i	−0.511801	−	0.859104i	$-0.671022\pi$
$263$	− 3.12567e7i	− 1.71821i	0.511801	−	0.859104i	$-0.328978\pi$
$264$	0	0
$265$	−4.15280e7	−2.23153
$266$	0	0
$267$	0	0
$268$	0	0
$269$	251338.i	0.0129122i	0.999979	+	0.00645612i	$0.00205506\pi$
$269$	251338.i	0.0129122i	−0.999979	+	0.00645612i	$0.997945\pi$
$270$	0	0
$271$	−2.96399e7	−1.48925	−0.744627	−	0.667481i	$-0.767373\pi$
$271$	−2.96399e7	−1.48925	−0.744627	+	0.667481i	$0.767373\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.96181e7i	0.943319i
$276$	0	0
$277$	1.32213e7	0.622062	0.311031	−	0.950400i	$-0.399326\pi$
$277$	1.32213e7	0.622062	0.311031	+	0.950400i	$0.399326\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 6.12360e6i	− 0.275987i	−0.990433	−	0.137993i	$-0.955935\pi$
$281$	− 6.12360e6i	− 0.275987i	0.990433	−	0.137993i	$-0.0440652\pi$
$282$	0	0
$283$	6.74325e6	0.297516	0.148758	−	0.988874i	$-0.452473\pi$
$283$	6.74325e6	0.297516	0.148758	+	0.988874i	$0.452473\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.79286e7i	− 0.758403i
$288$	0	0
$289$	2.41374e7	0.999993
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.00239e7i	− 0.398505i	−0.979948	−	0.199253i	$-0.936149\pi$
$293$	− 1.00239e7i	− 0.398505i	0.979948	−	0.199253i	$-0.0638514\pi$
$294$	0	0
$295$	−4.34593e7	−1.69284
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.13742e7i	0.425508i
$300$	0	0
$301$	−1.83920e6	−0.0674418
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.30481e6i	0.0812337i
$306$	0	0
$307$	5.23060e6	0.180774	0.0903871	−	0.995907i	$-0.471190\pi$
$307$	5.23060e6	0.180774	0.0903871	+	0.995907i	$0.471190\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 3.12221e7i	− 1.03796i	−0.854786	−	0.518981i	$-0.826312\pi$
$311$	− 3.12221e7i	− 1.03796i	0.854786	−	0.518981i	$-0.173688\pi$
$312$	0	0
$313$	2.24778e7	0.733029	0.366515	−	0.930412i	$-0.380551\pi$
$313$	2.24778e7	0.733029	0.366515	+	0.930412i	$0.380551\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.76211e7i	− 0.867088i	−0.901132	−	0.433544i	$-0.857263\pi$
$317$	− 2.76211e7i	− 0.867088i	0.901132	−	0.433544i	$-0.142737\pi$
$318$	0	0
$319$	3.93553e7	1.21236
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 73109.2i	− 0.00216952i
$324$	0	0
$325$	−4.92839e7	−1.43567
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.71672e7i	− 1.04369i
$330$	0	0
$331$	5.76138e6	0.158870	0.0794352	−	0.996840i	$-0.474688\pi$
$331$	5.76138e6	0.158870	0.0794352	+	0.996840i	$0.474688\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 2.93917e7i	− 0.781790i
$336$	0	0
$337$	−4.01052e7	−1.04788	−0.523939	−	0.851756i	$-0.675538\pi$
$337$	−4.01052e7	−1.04788	−0.523939	+	0.851756i	$0.675538\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 5.33535e7i	− 1.34555i
$342$	0	0
$343$	−504328.	−0.0124977
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.78127e7i	− 1.62302i	−0.584341	−	0.811508i	$-0.698647\pi$
$347$	− 6.78127e7i	− 1.62302i	0.584341	−	0.811508i	$-0.301353\pi$
$348$	0	0
$349$	−4.20638e7	−0.989538	−0.494769	−	0.869024i	$-0.664747\pi$
$349$	−4.20638e7	−0.989538	−0.494769	+	0.869024i	$0.664747\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.75976e7i	− 0.400063i	−0.979789	−	0.200032i	$-0.935896\pi$
$353$	− 1.75976e7i	− 0.400063i	0.979789	−	0.200032i	$-0.0641045\pi$
$354$	0	0
$355$	−9.24478e7	−2.06639
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.39920e7i	0.302410i	0.988502	+	0.151205i	$0.0483154\pi$
$359$	1.39920e7i	0.302410i	−0.988502	+	0.151205i	$0.951685\pi$
$360$	0	0
$361$	−1.40523e7	−0.298694
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.10768e7i	0.844731i
$366$	0	0
$367$	2.65855e7	0.537832	0.268916	−	0.963164i	$-0.413335\pi$
$367$	2.65855e7	0.537832	0.268916	+	0.963164i	$0.413335\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.15549e8i	2.26279i
$372$	0	0
$373$	1.78829e7	0.344598	0.172299	−	0.985045i	$-0.444881\pi$
$373$	1.78829e7	0.344598	0.172299	+	0.985045i	$0.444881\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.88671e7i	1.84513i
$378$	0	0
$379$	−7.20978e7	−1.32435	−0.662177	−	0.749347i	$-0.730367\pi$
$379$	−7.20978e7	−1.32435	−0.662177	+	0.749347i	$0.730367\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 8.68648e6i	− 0.154614i	−0.997007	−	0.0773068i	$-0.975368\pi$
$383$	− 8.68648e6i	− 0.154614i	0.997007	−	0.0773068i	$-0.0246321\pi$
$384$	0	0
$385$	1.12873e8	1.97791
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 4.94411e7i	− 0.839923i	−0.907542	−	0.419962i	$-0.862044\pi$
$389$	− 4.94411e7i	− 0.839923i	0.907542	−	0.419962i	$-0.137956\pi$
$390$	0	0
$391$	−42984.0	−0.000719079 0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.10837e6i	0.0991137i
$396$	0	0
$397$	1.56911e7	0.250774	0.125387	−	0.992108i	$-0.459983\pi$
$397$	1.56911e7	0.250774	0.125387	+	0.992108i	$0.459983\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4.74514e7i	− 0.735895i	−0.929847	−	0.367947i	$-0.880061\pi$
$401$	− 4.74514e7i	− 0.735895i	0.929847	−	0.367947i	$-0.119939\pi$
$402$	0	0
$403$	1.34033e8	2.04784
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7.04203e7i	− 1.04451i
$408$	0	0
$409$	−1.15512e8	−1.68832	−0.844162	−	0.536088i	$-0.819901\pi$
$409$	−1.15512e8	−1.68832	−0.844162	+	0.536088i	$0.819901\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.20923e8i	1.71656i
$414$	0	0
$415$	−1.90994e6	−0.0267225
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.46693e8i	− 1.99420i	−0.0761306	−	0.997098i	$-0.524257\pi$
$419$	− 1.46693e8i	− 1.99420i	0.0761306	−	0.997098i	$-0.475743\pi$
$420$	0	0
$421$	1.39239e8	1.86601	0.933005	−	0.359863i	$-0.117176\pi$
$421$	1.39239e8	1.86601	0.933005	+	0.359863i	$0.117176\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 186248.i	− 0.00242619i
$426$	0	0
$427$	6.41300e6	0.0823716
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.00392e8i	1.25391i	0.779056	+	0.626954i	$0.215699\pi$
$431$	1.00392e8i	1.25391i	−0.779056	+	0.626954i	$0.784301\pi$
$432$	0	0
$433$	−4.00631e7	−0.493493	−0.246747	−	0.969080i	$-0.579362\pi$
$433$	−4.00631e7	−0.493493	−0.246747	+	0.969080i	$0.579362\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.93983e7i	− 0.232445i
$438$	0	0
$439$	1.38592e8	1.63811	0.819057	−	0.573712i	$-0.194497\pi$
$439$	1.38592e8	1.63811	0.819057	+	0.573712i	$0.194497\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.11443e8i	1.28186i	0.767600	+	0.640929i	$0.221451\pi$
$443$	1.11443e8i	1.28186i	−0.767600	+	0.640929i	$0.778549\pi$
$444$	0	0
$445$	2.24965e7	0.255290
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 6.11166e7i	− 0.675181i	−0.941293	−	0.337591i	$-0.890388\pi$
$449$	− 6.11166e7i	− 0.675181i	0.941293	−	0.337591i	$-0.109612\pi$
$450$	0	0
$451$	−4.96619e7	−0.541370
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.83555e8i	3.01026i
$456$	0	0
$457$	3.56665e7	0.373690	0.186845	−	0.982389i	$-0.440174\pi$
$457$	3.56665e7	0.373690	0.186845	+	0.982389i	$0.440174\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.51983e8i	1.55128i	0.631173	+	0.775642i	$0.282574\pi$
$461$	1.51983e8i	1.55128i	−0.631173	+	0.775642i	$0.717426\pi$
$462$	0	0
$463$	−1.14978e8	−1.15844	−0.579218	−	0.815173i	$-0.696642\pi$
$463$	−1.14978e8	−1.15844	−0.579218	+	0.815173i	$0.696642\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.81705e7i	− 0.865711i	−0.901463	−	0.432855i	$-0.857506\pi$
$467$	− 8.81705e7i	− 0.865711i	0.901463	−	0.432855i	$-0.142494\pi$
$468$	0	0
$469$	−8.17805e7	−0.792741
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.09456e6i	0.0481419i
$474$	0	0
$475$	8.40520e7	0.784272
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.94388e7i	0.813803i	0.913472	+	0.406902i	$0.133391\pi$
$479$	8.94388e7i	0.813803i	−0.913472	+	0.406902i	$0.866609\pi$
$480$	0	0
$481$	1.76908e8	1.58969
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 5.59111e7i	− 0.490087i
$486$	0	0
$487$	7.51688e7	0.650805	0.325403	−	0.945576i	$-0.394500\pi$
$487$	7.51688e7	0.650805	0.325403	+	0.945576i	$0.394500\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 4.50822e7i	− 0.380856i	−0.981701	−	0.190428i	$-0.939012\pi$
$491$	− 4.50822e7i	− 0.380856i	0.981701	−	0.190428i	$-0.0609876\pi$
$492$	0	0
$493$	−373626.	−0.00311815
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.57230e8i	2.09533i
$498$	0	0
$499$	−9.15458e7	−0.736778	−0.368389	−	0.929672i	$-0.620091\pi$
$499$	−9.15458e7	−0.736778	−0.368389	+	0.929672i	$0.620091\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.61043e8i	1.26543i	0.774386	+	0.632713i	$0.218059\pi$
$503$	1.61043e8i	1.26543i	−0.774386	+	0.632713i	$0.781941\pi$
$504$	0	0
$505$	−1.16333e8	−0.903295
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.39995e7i	0.181990i	0.995851	+	0.0909951i	$0.0290048\pi$
$509$	2.39995e7i	0.181990i	−0.995851	+	0.0909951i	$0.970995\pi$
$510$	0	0
$511$	1.14294e8	0.856564
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3.46751e8i	− 2.53861i
$516$	0	0
$517$	−1.02953e8	−0.745018
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 9.00897e7i	− 0.637033i	−0.947917	−	0.318517i	$-0.896815\pi$
$521$	− 9.00897e7i	− 0.637033i	0.947917	−	0.318517i	$-0.103185\pi$
$522$	0	0
$523$	3.77691e7	0.264016	0.132008	−	0.991249i	$-0.457857\pi$
$523$	3.77691e7	0.264016	0.132008	+	0.991249i	$0.457857\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	506520.i	0.00346071i
$528$	0	0
$529$	1.36631e8	0.922957
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.24759e8i	− 0.823931i
$534$	0	0
$535$	−4.53427e7	−0.296105
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.56332e8i	− 0.998347i
$540$	0	0
$541$	2.54800e7	0.160919	0.0804595	−	0.996758i	$-0.474361\pi$
$541$	2.54800e7	0.160919	0.0804595	+	0.996758i	$0.474361\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.38253e7i	0.208955i
$546$	0	0
$547$	−2.05216e8	−1.25386	−0.626930	−	0.779076i	$-0.715689\pi$
$547$	−2.05216e8	−1.25386	−0.626930	+	0.779076i	$0.715689\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1.68614e8i	− 1.00795i
$552$	0	0
$553$	1.69961e7	0.100502
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.41143e8i	− 1.39543i	−0.716375	−	0.697715i	$-0.754200\pi$
$557$	− 2.41143e8i	− 1.39543i	0.716375	−	0.697715i	$-0.245800\pi$
$558$	0	0
$559$	−1.27984e7	−0.0732690
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.68877e8i	0.946337i	0.880972	+	0.473168i	$0.156890\pi$
$563$	1.68877e8i	0.946337i	−0.880972	+	0.473168i	$0.843110\pi$
$564$	0	0
$565$	1.42968e8	0.792670
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 2.43995e8i	− 1.32448i	−0.749293	−	0.662238i	$-0.769607\pi$
$569$	− 2.43995e8i	− 1.32448i	0.749293	−	0.662238i	$-0.230393\pi$
$570$	0	0
$571$	−2.41502e8	−1.29722	−0.648608	−	0.761123i	$-0.724648\pi$
$571$	−2.41502e8	−1.29722	−0.648608	+	0.761123i	$0.724648\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 4.94177e7i	− 0.259944i
$576$	0	0
$577$	−4.93979e7	−0.257147	−0.128573	−	0.991700i	$-0.541040\pi$
$577$	−4.93979e7	−0.257147	−0.128573	+	0.991700i	$0.541040\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.31430e6i	0.0270968i
$582$	0	0
$583$	3.20069e8	1.61525
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.72052e8i	0.850639i	0.905043	+	0.425320i	$0.139838\pi$
$587$	1.72052e8i	0.850639i	−0.905043	+	0.425320i	$0.860162\pi$
$588$	0	0
$589$	−2.28588e8	−1.11869
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.70643e8i	1.29788i	0.760841	+	0.648938i	$0.224787\pi$
$593$	2.70643e8i	1.29788i	−0.760841	+	0.648938i	$0.775213\pi$
$594$	0	0
$595$	−1.07158e6	−0.00508712
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.73299e8i	0.806337i	0.915126	+	0.403169i	$0.132091\pi$
$599$	1.73299e8i	0.806337i	−0.915126	+	0.403169i	$0.867909\pi$
$600$	0	0
$601$	−4.31090e8	−1.98584	−0.992921	−	0.118775i	$-0.962103\pi$
$601$	−4.31090e8	−1.98584	−0.992921	+	0.118775i	$0.962103\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.49604e6i	− 0.0203032i
$606$	0	0
$607$	−1.66991e7	−0.0746665	−0.0373332	−	0.999303i	$-0.511886\pi$
$607$	−1.66991e7	−0.0746665	−0.0373332	+	0.999303i	$0.511886\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2.58635e8i	− 1.13387i
$612$	0	0
$613$	−1.92321e8	−0.834920	−0.417460	−	0.908695i	$-0.637080\pi$
$613$	−1.92321e8	−0.834920	−0.417460	+	0.908695i	$0.637080\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.87023e8i	− 0.796233i	−0.917335	−	0.398117i	$-0.869664\pi$
$617$	− 1.87023e8i	− 0.796233i	0.917335	−	0.398117i	$-0.130336\pi$
$618$	0	0
$619$	−2.54873e8	−1.07461	−0.537307	−	0.843387i	$-0.680558\pi$
$619$	−2.54873e8	−1.07461	−0.537307	+	0.843387i	$0.680558\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 6.25950e7i	− 0.258866i
$624$	0	0
$625$	−2.58657e8	−1.05946
$626$	0	0
$627$	0	0
$628$	0	0
$629$	668547.i	0.00268646i
$630$	0	0
$631$	−9.23602e7	−0.367618	−0.183809	−	0.982962i	$-0.558843\pi$
$631$	−9.23602e7	−0.367618	−0.183809	+	0.982962i	$0.558843\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 5.31797e8i	− 2.07694i
$636$	0	0
$637$	3.92732e8	1.51942
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 4.24666e8i	− 1.61240i	−0.591643	−	0.806200i	$-0.701520\pi$
$641$	− 4.24666e8i	− 1.61240i	0.591643	−	0.806200i	$-0.298480\pi$
$642$	0	0
$643$	−3.75946e8	−1.41414	−0.707071	−	0.707143i	$-0.749984\pi$
$643$	−3.75946e8	−1.41414	−0.707071	+	0.707143i	$0.749984\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.63747e7i	− 0.0973813i	−0.998814	−	0.0486906i	$-0.984495\pi$
$647$	− 2.63747e7i	− 0.0973813i	0.998814	−	0.0486906i	$-0.0155048\pi$
$648$	0	0
$649$	3.34955e8	1.22533
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.58756e8i	0.929291i	0.885497	+	0.464645i	$0.153818\pi$
$653$	2.58756e8i	0.929291i	−0.885497	+	0.464645i	$0.846182\pi$
$654$	0	0
$655$	5.34696e8	1.90275
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.39345e8i	0.486895i	0.969914	+	0.243447i	$0.0782783\pi$
$659$	1.39345e8i	0.486895i	−0.969914	+	0.243447i	$0.921722\pi$
$660$	0	0
$661$	−4.72545e8	−1.63621	−0.818104	−	0.575070i	$-0.804975\pi$
$661$	−4.72545e8	−1.63621	−0.818104	+	0.575070i	$0.804975\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 4.83593e8i	− 1.64443i
$666$	0	0
$667$	−9.91354e7	−0.334081
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1.77639e7i	− 0.0587992i
$672$	0	0
$673$	5.48833e8	1.80051	0.900254	−	0.435364i	$-0.143380\pi$
$673$	5.48833e8	1.80051	0.900254	+	0.435364i	$0.143380\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.00760e8i	− 0.324731i	−0.986731	−	0.162365i	$-0.948088\pi$
$677$	− 1.00760e8i	− 0.324731i	0.986731	−	0.162365i	$-0.0519123\pi$
$678$	0	0
$679$	−1.55569e8	−0.496952
$680$	0	0
$681$	0	0
$682$	0	0
$683$	313056.i	0 0.000982562i	1.00000		0.000491281i	$0.000156380\pi$
$683$	313056.i	0 0.000982562i	−1.00000		0.000491281i	$0.999844\pi$
$684$	0	0
$685$	7.80005e8	2.42675
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.04068e8i	2.45830i
$690$	0	0
$691$	3.72812e8	1.12994	0.564971	−	0.825111i	$-0.308887\pi$
$691$	3.72812e8	1.12994	0.564971	+	0.825111i	$0.308887\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.90009e8i	0.566003i
$696$	0	0
$697$	471474.	0.00139239
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.21170e8i	1.80325i	0.432517	+	0.901626i	$0.357626\pi$
$701$	6.21170e8i	1.80325i	−0.432517	+	0.901626i	$0.642374\pi$
$702$	0	0
$703$	−3.01709e8	−0.868406
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.23690e8i	0.915947i
$708$	0	0
$709$	−2.46510e8	−0.691666	−0.345833	−	0.938296i	$-0.612404\pi$
$709$	−2.46510e8	−0.691666	−0.345833	+	0.938296i	$0.612404\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.34397e8i	0.370783i
$714$	0	0
$715$	7.85445e8	2.14881
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.60389e7i	0.258381i	0.991620	+	0.129191i	$0.0412379\pi$
$719$	9.60389e7i	0.258381i	−0.991620	+	0.129191i	$0.958762\pi$
$720$	0	0
$721$	−9.64811e8	−2.57417
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 4.29549e8i	− 1.12719i
$726$	0	0
$727$	−3.91371e8	−1.01856	−0.509278	−	0.860602i	$-0.670088\pi$
$727$	−3.91371e8	−1.01856	−0.509278	+	0.860602i	$0.670088\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 48366.1i	0 0.000123819i
$732$	0	0
$733$	3.49078e7	0.0886361	0.0443181	−	0.999017i	$-0.485889\pi$
$733$	3.49078e7	0.0886361	0.0443181	+	0.999017i	$0.485889\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.26531e8i	0.565881i
$738$	0	0
$739$	3.02999e8	0.750773	0.375386	−	0.926868i	$-0.377510\pi$
$739$	3.02999e8	0.750773	0.375386	+	0.926868i	$0.377510\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 2.45628e8i	− 0.598842i	−0.954121	−	0.299421i	$-0.903207\pi$
$743$	− 2.45628e8i	− 0.598842i	0.954121	−	0.299421i	$-0.0967935\pi$
$744$	0	0
$745$	3.86317e8	0.934276
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.26163e8i	0.300253i
$750$	0	0
$751$	8.23270e7	0.194367	0.0971835	−	0.995266i	$-0.469017\pi$
$751$	8.23270e7	0.194367	0.0971835	+	0.995266i	$0.469017\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.09484e8i	1.64855i
$756$	0	0
$757$	−6.03579e8	−1.39138	−0.695691	−	0.718341i	$-0.744902\pi$
$757$	−6.03579e8	−1.39138	−0.695691	+	0.718341i	$0.744902\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 2.32982e8i	− 0.528651i	−0.964434	−	0.264325i	$-0.914851\pi$
$761$	− 2.32982e8i	− 0.528651i	0.964434	−	0.264325i	$-0.0851493\pi$
$762$	0	0
$763$	9.41167e7	0.211882
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.41463e8i	1.86487i
$768$	0	0
$769$	8.15796e8	1.79392	0.896958	−	0.442115i	$-0.145772\pi$
$769$	8.15796e8	1.79392	0.896958	+	0.442115i	$0.145772\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 3.66587e8i	− 0.793667i	−0.917891	−	0.396833i	$-0.870109\pi$
$773$	− 3.66587e8i	− 0.793667i	0.917891	−	0.396833i	$-0.129891\pi$
$774$	0	0
$775$	−5.82335e8	−1.25103
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.12772e8i	0.450093i
$780$	0	0
$781$	7.12524e8	1.49571
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.07077e9i	2.21354i
$786$	0	0
$787$	4.02462e8	0.825659	0.412830	−	0.910808i	$-0.364540\pi$
$787$	4.02462e8	0.825659	0.412830	+	0.910808i	$0.364540\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 3.97798e8i	− 0.803773i
$792$	0	0
$793$	4.46260e7	0.0894887
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 5.18940e8i	− 1.02504i	−0.858675	−	0.512521i	$-0.828712\pi$
$797$	− 5.18940e8i	− 1.02504i	0.858675	−	0.512521i	$-0.171288\pi$
$798$	0	0
$799$	977400.	0.00191616
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 3.16592e8i	− 0.611440i
$804$	0	0
$805$	−2.84325e8	−0.545038
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 3.04036e6i	− 0.00574221i	−0.999996	−	0.00287110i	$-0.999086\pi$
$809$	− 3.04036e6i	− 0.00574221i	0.999996	−	0.00287110i	$-0.000913902\pi$
$810$	0	0
$811$	−2.25521e8	−0.422790	−0.211395	−	0.977401i	$-0.567801\pi$
$811$	−2.25521e8	−0.422790	−0.211395	+	0.977401i	$0.567801\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1.39280e8i	− 0.257285i
$816$	0	0
$817$	2.18272e7	0.0400250
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.77035e8i	0.500617i	0.968166	+	0.250309i	$0.0805321\pi$
$821$	2.77035e8i	0.500617i	−0.968166	+	0.250309i	$0.919468\pi$
$822$	0	0
$823$	7.07336e8	1.26890	0.634448	−	0.772965i	$-0.281227\pi$
$823$	7.07336e8	1.26890	0.634448	+	0.772965i	$0.281227\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.66346e8i	0.470900i	0.971886	+	0.235450i	$0.0756564\pi$
$827$	2.66346e8i	0.470900i	−0.971886	+	0.235450i	$0.924344\pi$
$828$	0	0
$829$	5.03826e8	0.884336	0.442168	−	0.896932i	$-0.354209\pi$
$829$	5.03826e8	0.884336	0.442168	+	0.896932i	$0.354209\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.48416e6i	0.00256772i
$834$	0	0
$835$	8.35764e8	1.43557
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 7.63364e8i	− 1.29255i	−0.763106	−	0.646273i	$-0.776327\pi$
$839$	− 7.63364e8i	− 1.29255i	0.763106	−	0.646273i	$-0.223673\pi$
$840$	0	0
$841$	−2.66883e8	−0.448676
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.13355e9i	1.87876i
$846$	0	0
$847$	−1.25099e7	−0.0205876
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.77388e8i	0.287829i
$852$	0	0
$853$	−1.87985e7	−0.0302884	−0.0151442	−	0.999885i	$-0.504821\pi$
$853$	−1.87985e7	−0.0302884	−0.0151442	+	0.999885i	$0.504821\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 6.86427e8i	− 1.09057i	−0.838252	−	0.545283i	$-0.816422\pi$
$857$	− 6.86427e8i	− 1.09057i	0.838252	−	0.545283i	$-0.183578\pi$
$858$	0	0
$859$	−5.51932e8	−0.870775	−0.435387	−	0.900243i	$-0.643389\pi$
$859$	−5.51932e8	−0.870775	−0.435387	+	0.900243i	$0.643389\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 3.65665e8i	− 0.568920i	−0.958688	−	0.284460i	$-0.908186\pi$
$863$	− 3.65665e8i	− 0.568920i	0.958688	−	0.284460i	$-0.0918142\pi$
$864$	0	0
$865$	6.20982e8	0.959468
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 4.70791e7i	− 0.0717413i
$870$	0	0
$871$	−5.69084e8	−0.861236
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.35174e7i	0.124667i
$876$	0	0
$877$	−5.85387e8	−0.867849	−0.433925	−	0.900949i	$-0.642872\pi$
$877$	−5.85387e8	−0.867849	−0.433925	+	0.900949i	$0.642872\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 4.29761e8i	− 0.628491i	−0.949342	−	0.314246i	$-0.898248\pi$
$881$	− 4.29761e8i	− 0.628491i	0.949342	−	0.314246i	$-0.101752\pi$
$882$	0	0
$883$	−2.20085e8	−0.319675	−0.159837	−	0.987143i	$-0.551097\pi$
$883$	−2.20085e8	−0.319675	−0.159837	+	0.987143i	$0.551097\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.17196e9i	− 1.67936i	−0.543084	−	0.839678i	$-0.682744\pi$
$887$	− 1.17196e9i	− 1.67936i	0.543084	−	0.839678i	$-0.317256\pi$
$888$	0	0
$889$	−1.47969e9	−2.10604
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.41092e8i	0.619405i
$894$	0	0
$895$	−1.29365e9	−1.80446
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.16820e9i	1.60783i
$900$	0	0
$901$	−3.03863e6	−0.00415436
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.80579e9i	− 2.43625i
$906$	0	0
$907$	−7.31614e8	−0.980529	−0.490264	−	0.871574i	$-0.663100\pi$
$907$	−7.31614e8	−0.980529	−0.490264	+	0.871574i	$0.663100\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 9.18595e8i	− 1.21498i	−0.794327	−	0.607490i	$-0.792177\pi$
$911$	− 9.18595e8i	− 1.21498i	0.794327	−	0.607490i	$-0.207823\pi$
$912$	0	0
$913$	1.47205e7	0.0193425
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.48776e9i	− 1.92941i
$918$	0	0
$919$	2.15987e8	0.278279	0.139139	−	0.990273i	$-0.455566\pi$
$919$	2.15987e8	0.278279	0.139139	+	0.990273i	$0.455566\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.78998e9i	2.27637i
$924$	0	0
$925$	−7.68613e8	−0.971141
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.10124e8i	0.386802i	0.981120	+	0.193401i	$0.0619518\pi$
$929$	3.10124e8i	0.386802i	−0.981120	+	0.193401i	$0.938048\pi$
$930$	0	0
$931$	−6.69791e8	−0.830023
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.96825e6i	0.00363133i
$936$	0	0
$937$	−7.42448e8	−0.902501	−0.451250	−	0.892397i	$-0.649022\pi$
$937$	−7.42448e8	−0.902501	−0.451250	+	0.892397i	$0.649022\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.81766e8i	0.218144i	0.994034	+	0.109072i	$0.0347879\pi$
$941$	1.81766e8i	0.218144i	−0.994034	+	0.109072i	$0.965212\pi$
$942$	0	0
$943$	1.25098e8	0.149181
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.59189e8i	− 1.01167i	−0.862630	−	0.505835i	$-0.831184\pi$
$947$	− 8.59189e8i	− 1.01167i	0.862630	−	0.505835i	$-0.168816\pi$
$948$	0	0
$949$	7.95333e8	0.930573
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.86819e8i	0.793530i	0.917920	+	0.396765i	$0.129867\pi$
$953$	6.86819e8i	0.793530i	−0.917920	+	0.396765i	$0.870133\pi$
$954$	0	0
$955$	2.26029e9	2.59510
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2.17031e9i	− 2.46075i
$960$	0	0
$961$	6.96218e8	0.784468
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 6.83957e8i	− 0.761109i
$966$	0	0
$967$	1.09411e9	1.20999	0.604995	−	0.796230i	$-0.293175\pi$
$967$	1.09411e9	1.20999	0.604995	+	0.796230i	$0.293175\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 4.43115e8i	− 0.484014i	−0.970274	−	0.242007i	$-0.922194\pi$
$971$	− 4.43115e8i	− 0.484014i	0.970274	−	0.242007i	$-0.0778058\pi$
$972$	0	0
$973$	5.28687e8	0.573931
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.19004e9i	1.27608i	0.770001	+	0.638042i	$0.220256\pi$
$977$	1.19004e9i	1.27608i	−0.770001	+	0.638042i	$0.779744\pi$
$978$	0	0
$979$	−1.73387e8	−0.184786
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.18187e9i	1.24425i	0.782918	+	0.622125i	$0.213730\pi$
$983$	1.18187e9i	1.24425i	−0.782918	+	0.622125i	$0.786270\pi$
$984$	0	0
$985$	−9.35785e8	−0.979191
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.28331e7i	− 0.0132661i
$990$	0	0
$991$	−5.09602e8	−0.523613	−0.261806	−	0.965120i	$-0.584318\pi$
$991$	−5.09602e8	−0.523613	−0.261806	+	0.965120i	$0.584318\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.84373e7i	0.0999288i
$996$	0	0
$997$	−9.90780e8	−0.999751	−0.499875	−	0.866097i	$-0.666621\pi$
$997$	−9.90780e8	−0.999751	−0.499875	+	0.866097i	$0.666621\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.7.e.d.17.2		2
3.2	odd	2	inner	144.7.e.d.17.1		2
4.3	odd	2		18.7.b.a.17.2	yes	2
8.3	odd	2		576.7.e.b.449.1		2
8.5	even	2		576.7.e.k.449.1		2
12.11	even	2		18.7.b.a.17.1	✓	2
20.3	even	4		450.7.b.a.449.4		4
20.7	even	4		450.7.b.a.449.1		4
20.19	odd	2		450.7.d.a.251.1		2
24.5	odd	2		576.7.e.k.449.2		2
24.11	even	2		576.7.e.b.449.2		2
36.7	odd	6		162.7.d.d.53.2		4
36.11	even	6		162.7.d.d.53.1		4
36.23	even	6		162.7.d.d.107.2		4
36.31	odd	6		162.7.d.d.107.1		4
60.23	odd	4		450.7.b.a.449.2		4
60.47	odd	4		450.7.b.a.449.3		4
60.59	even	2		450.7.d.a.251.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.7.b.a.17.1	✓	2	12.11	even	2
18.7.b.a.17.2	yes	2	4.3	odd	2
144.7.e.d.17.1		2	3.2	odd	2	inner
144.7.e.d.17.2		2	1.1	even	1	trivial
162.7.d.d.53.1		4	36.11	even	6
162.7.d.d.53.2		4	36.7	odd	6
162.7.d.d.107.1		4	36.31	odd	6
162.7.d.d.107.2		4	36.23	even	6
450.7.b.a.449.1		4	20.7	even	4
450.7.b.a.449.2		4	60.23	odd	4
450.7.b.a.449.3		4	60.47	odd	4
450.7.b.a.449.4		4	20.3	even	4
450.7.d.a.251.1		2	20.19	odd	2
450.7.d.a.251.2		2	60.59	even	2
576.7.e.b.449.1		2	8.3	odd	2
576.7.e.b.449.2		2	24.11	even	2
576.7.e.k.449.1		2	8.5	even	2
576.7.e.k.449.2		2	24.5	odd	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)

\(f(q)\)	\(=\)	\(q+173.948i q^{5} +484.000 q^{7} +O(q^{10})\)	\(q+173.948i q^{5} +484.000 q^{7} -1340.67i q^{11} +3368.00 q^{13} +12.7279i q^{17} -5744.00 q^{19} +3377.14i q^{23} -14633.0 q^{25} +29354.8i q^{29} +39796.0 q^{31} +84191.0i q^{35} +52526.0 q^{37} -37042.5i q^{41} -3800.00 q^{43} -76791.8i q^{47} +116607. q^{49} +238738. i q^{53} +233208. q^{55} +249841. i q^{59} +13250.0 q^{61} +585858. i q^{65} -168968. q^{67} +531467. i q^{71} +236144. q^{73} -648886. i q^{77} +35116.0 q^{79} +10980.0i q^{83} -2214.00 q^{85} -129328. i q^{89} +1.63011e6 q^{91} -999159. i q^{95} -321424. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 968 q^{7}+O(q^{10}) \) 2 * q + 968 * q^7	\( 2 q + 968 q^{7} + 6736 q^{13} - 11488 q^{19} - 29266 q^{25} + 79592 q^{31} + 105052 q^{37} - 7600 q^{43} + 233214 q^{49} + 466416 q^{55} + 26500 q^{61} - 337936 q^{67} + 472288 q^{73} + 70232 q^{79} - 4428 q^{85} + 3260224 q^{91} - 642848 q^{97}+O(q^{100}) \) 2 * q + 968 * q^7 + 6736 * q^13 - 11488 * q^19 - 29266 * q^25 + 79592 * q^31 + 105052 * q^37 - 7600 * q^43 + 233214 * q^49 + 466416 * q^55 + 26500 * q^61 - 337936 * q^67 + 472288 * q^73 + 70232 * q^79 - 4428 * q^85 + 3260224 * q^91 - 642848 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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