LMFDB - Embedded newform 18.7.b.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,7,Mod(17,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("18.17");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	18.b (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:	$4.14097350516$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	18.17
Dual form			18.7.b.a.17.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.65685i q^{2} -32.0000 q^{4} -173.948i q^{5} -484.000 q^{7} +181.019i q^{8} -984.000 q^{10} -1340.67i q^{11} +3368.00 q^{13} +2737.92i q^{14} +1024.00 q^{16} -12.7279i q^{17} +5744.00 q^{19} +5566.34i q^{20} -7584.00 q^{22} +3377.14i q^{23} -14633.0 q^{25} -19052.3i q^{26} +15488.0 q^{28} -29354.8i q^{29} -39796.0 q^{31} -5792.62i q^{32} -72.0000 q^{34} +84191.0i q^{35} +52526.0 q^{37} -32493.0i q^{38} +31488.0 q^{40} +37042.5i q^{41} +3800.00 q^{43} +42901.6i q^{44} +19104.0 q^{46} -76791.8i q^{47} +116607. q^{49} +82776.7i q^{50} -107776. q^{52} -238738. i q^{53} -233208. q^{55} -87613.4i q^{56} -166056. q^{58} +249841. i q^{59} +13250.0 q^{61} +225120. i q^{62} -32768.0 q^{64} -585858. i q^{65} +168968. q^{67} +407.294i q^{68} +476256. q^{70} +531467. i q^{71} +236144. q^{73} -297132. i q^{74} -183808. q^{76} +648886. i q^{77} -35116.0 q^{79} -178123. i q^{80} +209544. q^{82} +10980.0i q^{83} -2214.00 q^{85} -21496.0i q^{86} +242688. q^{88} +129328. i q^{89} -1.63011e6 q^{91} -108069. i q^{92} -434400. q^{94} -999159. i q^{95} -321424. q^{97} -659629. i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 64 q^{4} - 968 q^{7} - 1968 q^{10} + 6736 q^{13} + 2048 q^{16} + 11488 q^{19} - 15168 q^{22} - 29266 q^{25} + 30976 q^{28} - 79592 q^{31} - 144 q^{34} + 105052 q^{37} + 62976 q^{40} + 7600 q^{43}+ \cdots - 642848 q^{97}+O(q^{100}) $ 2 * q - 64 * q^4 - 968 * q^7 - 1968 * q^10 + 6736 * q^13 + 2048 * q^16 + 11488 * q^19 - 15168 * q^22 - 29266 * q^25 + 30976 * q^28 - 79592 * q^31 - 144 * q^34 + 105052 * q^37 + 62976 * q^40 + 7600 * q^43 + 38208 * q^46 + 233214 * q^49 - 215552 * q^52 - 466416 * q^55 - 332112 * q^58 + 26500 * q^61 - 65536 * q^64 + 337936 * q^67 + 952512 * q^70 + 472288 * q^73 - 367616 * q^76 - 70232 * q^79 + 419088 * q^82 - 4428 * q^85 + 485376 * q^88 - 3260224 * q^91 - 868800 * q^94 - 642848 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$.

$n$	$11$
$\chi(n)$	$-1$

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 $

$2$	− 5.65685i	− 0.707107i
$2$	− 5.65685i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−32.0000	−0.500000
$5$	− 173.948i	− 1.39159i	−0.718242	−	0.695793i	$-0.755053\pi$
$5$	− 173.948i	− 1.39159i	0.718242	−	0.695793i	$-0.244947\pi$
$6$	0	0
$7$	−484.000	−1.41108	−0.705539	−	0.708671i	$-0.749295\pi$
$7$	−484.000	−1.41108	−0.705539	+	0.708671i	$0.749295\pi$
$8$	181.019i	0.353553i
$9$	0	0
$10$	−984.000	−0.984000
$11$	− 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	$0.221997\pi$
$14$	2737.92i	0.997783i
$15$	0	0
$16$	1024.00	0.250000
$17$	− 12.7279i	− 0.00259066i	−0.999999	−	0.00129533i	$-0.999588\pi$
$17$	− 12.7279i	− 0.00259066i	0.999999	−	0.00129533i	$-0.000412317\pi$
$18$	0	0
$19$	5744.00	0.837440	0.418720	−	0.908115i	$-0.362479\pi$
$19$	5744.00	0.837440	0.418720	+	0.908115i	$0.362479\pi$
$20$	5566.34i	0.695793i
$21$	0	0
$22$	−7584.00	−0.712246
$23$	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
$26$	− 19052.3i	− 1.08399i
$27$	0	0
$28$	15488.0	0.705539
$29$	− 29354.8i	− 1.20361i	−0.798643	−	0.601805i	$-0.794449\pi$
$29$	− 29354.8i	− 1.20361i	0.798643	−	0.601805i	$-0.205551\pi$
$30$	0	0
$31$	−39796.0	−1.33584	−0.667920	−	0.744233i	$-0.732815\pi$
$31$	−39796.0	−1.33584	−0.667920	+	0.744233i	$0.732815\pi$
$32$	− 5792.62i	− 0.176777i
$33$	0	0
$34$	−72.0000	−0.00183187
$35$	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	$0.326495\pi$
$38$	− 32493.0i	− 0.592159i
$39$	0	0
$40$	31488.0	0.492000
$41$	37042.5i	0.537463i	0.963215	+	0.268732i	$0.0866045\pi$
$41$	37042.5i	0.537463i	−0.963215	+	0.268732i	$0.913396\pi$
$42$	0	0
$43$	3800.00	0.0477945	0.0238973	−	0.999714i	$-0.492393\pi$
$43$	3800.00	0.0477945	0.0238973	+	0.999714i	$0.492393\pi$
$44$	42901.6i	0.503634i
$45$	0	0
$46$	19104.0	0.196269
$47$	− 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
$50$	82776.7i	0.662214i
$51$	0	0
$52$	−107776.	−0.766500
$53$	− 238738.i	− 1.60359i	−0.597599	−	0.801795i	$-0.703879\pi$
$53$	− 238738.i	− 1.60359i	0.597599	−	0.801795i	$-0.296121\pi$
$54$	0	0
$55$	−233208.	−1.40170
$56$	− 87613.4i	− 0.498892i
$57$	0	0
$58$	−166056.	−0.851080
$59$	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	$0.490708\pi$
$62$	225120.i	0.944581i
$63$	0	0
$64$	−32768.0	−0.125000
$65$	− 585858.i	− 2.13330i
$66$	0	0
$67$	168968.	0.561798	0.280899	−	0.959737i	$-0.409367\pi$
$67$	168968.	0.561798	0.280899	+	0.959737i	$0.409367\pi$
$68$	407.294i	0.00129533i
$69$	0	0
$70$	476256.	1.38850
$71$	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	$0.401840\pi$
$74$	− 297132.i	− 0.733254i
$75$	0	0
$76$	−183808.	−0.418720
$77$	648886.i	1.42134i
$78$	0	0
$79$	−35116.0	−0.0712236	−0.0356118	−	0.999366i	$-0.511338\pi$
$79$	−35116.0	−0.0712236	−0.0356118	+	0.999366i	$0.511338\pi$
$80$	− 178123.i	− 0.347897i
$81$	0	0
$82$	209544.	0.380044
$83$	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
$86$	− 21496.0i	− 0.0337958i
$87$	0	0
$88$	242688.	0.356123
$89$	129328.i	0.183453i	0.995784	+	0.0917263i	$0.0292385\pi$
$89$	129328.i	0.183453i	−0.995784	+	0.0917263i	$0.970762\pi$
$90$	0	0
$91$	−1.63011e6	−2.16318
$92$	− 108069.i	− 0.138783i
$93$	0	0
$94$	−434400.	−0.523005
$95$	− 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	$0.556345\pi$
$98$	− 659629.i	− 0.700844i
$99$	0	0
$100$	468256.	0.468256
$101$	− 668780.i	− 0.649111i	−0.945867	−	0.324556i	$-0.894785\pi$
$101$	− 668780.i	− 0.649111i	0.945867	−	0.324556i	$-0.105215\pi$
$102$	0	0
$103$	1.99341e6	1.82425	0.912127	−	0.409907i	$-0.134439\pi$
$103$	1.99341e6	1.82425	0.912127	+	0.409907i	$0.134439\pi$
$104$	609673.i	0.541997i
$105$	0	0
$106$	−1.35050e6	−1.13391
$107$	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	$0.476079\pi$
$110$	1.31922e6i	0.991152i
$111$	0	0
$112$	−495616.	−0.352770
$113$	821897.i	0.569616i	0.958585	+	0.284808i	$0.0919298\pi$
$113$	821897.i	0.569616i	−0.958585	+	0.284808i	$0.908070\pi$
$114$	0	0
$115$	587448.	0.386257
$116$	939355.i	0.601805i
$117$	0	0
$118$	1.41331e6	0.860185
$119$	6160.31i	0.00365563i
$120$	0	0
$121$	−25847.0	−0.0145900
$122$	− 74953.3i	− 0.0412773i
$123$	0	0
$124$	1.27347e6	0.667920
$125$	− 172557.i	− 0.0883490i
$126$	0	0
$127$	3.05721e6	1.49250	0.746250	−	0.665666i	$-0.231852\pi$
$127$	3.05721e6	1.49250	0.746250	+	0.665666i	$0.231852\pi$
$128$	185364.i	0.0883883i
$129$	0	0
$130$	−3.31411e6	−1.50847
$131$	− 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
$134$	− 955827.i	− 0.397251i
$135$	0	0
$136$	2304.00	0.000915937 0
$137$	4.48412e6i	1.74388i	0.489617	+	0.871938i	$0.337137\pi$
$137$	4.48412e6i	1.74388i	−0.489617	+	0.871938i	$0.662863\pi$
$138$	0	0
$139$	−1.09233e6	−0.406732	−0.203366	−	0.979103i	$-0.565188\pi$
$139$	−1.09233e6	−0.406732	−0.203366	+	0.979103i	$0.565188\pi$
$140$	− 2.69411e6i	− 0.981819i
$141$	0	0
$142$	3.00643e6	1.04999
$143$	− 4.51539e6i	− 1.54414i
$144$	0	0
$145$	−5.10622e6	−1.67493
$146$	− 1.33583e6i	− 0.429233i
$147$	0	0
$148$	−1.68083e6	−0.518489
$149$	2.22087e6i	0.671375i	0.941973	+	0.335687i	$0.108969\pi$
$149$	2.22087e6i	0.671375i	−0.941973	+	0.335687i	$0.891031\pi$
$150$	0	0
$151$	−4.07871e6	−1.18465	−0.592327	−	0.805697i	$-0.701791\pi$
$151$	−4.07871e6	−1.18465	−0.592327	+	0.805697i	$0.701791\pi$
$152$	1.03978e6i	0.296080i
$153$	0	0
$154$	3.67066e6	1.00504
$155$	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	$0.207298\pi$
$158$	198646.i	0.0503627i
$159$	0	0
$160$	−1.00762e6	−0.246000
$161$	− 1.63454e6i	− 0.391667i
$162$	0	0
$163$	800696.	0.184886	0.0924432	−	0.995718i	$-0.470532\pi$
$163$	800696.	0.184886	0.0924432	+	0.995718i	$0.470532\pi$
$164$	− 1.18536e6i	− 0.268732i
$165$	0	0
$166$	62112.0	0.0135785
$167$	− 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
$170$	12524.3i	0.00254921i
$171$	0	0
$172$	−121600.	−0.0238973
$173$	3.56992e6i	0.689478i	0.938699	+	0.344739i	$0.112033\pi$
$173$	3.56992e6i	0.689478i	−0.938699	+	0.344739i	$0.887967\pi$
$174$	0	0
$175$	7.08237e6	1.32149
$176$	− 1.37285e6i	− 0.251817i
$177$	0	0
$178$	731592.	0.129721
$179$	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	$0.839369\pi$
$182$	9.22131e6i	1.52960i
$183$	0	0
$184$	−611328.	−0.0981343
$185$	− 9.13681e6i	− 1.44304i
$186$	0	0
$187$	−17064.0	−0.00260949
$188$	2.45734e6i	0.369821i
$189$	0	0
$190$	−5.65210e6	−0.824041
$191$	− 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	$0.588171\pi$
$194$	1.81825e6i	0.249028i
$195$	0	0
$196$	−3.73142e6	−0.495572
$197$	− 5.37967e6i	− 0.703651i	−0.936066	−	0.351825i	$-0.885561\pi$
$197$	− 5.37967e6i	− 0.703651i	0.936066	−	0.351825i	$-0.114439\pi$
$198$	0	0
$199$	−565900.	−0.0718093	−0.0359046	−	0.999355i	$-0.511431\pi$
$199$	−565900.	−0.0718093	−0.0359046	+	0.999355i	$0.511431\pi$
$200$	− 2.64886e6i	− 0.331107i
$201$	0	0
$202$	−3.78319e6	−0.458991
$203$	1.42077e7i	1.69839i
$204$	0	0
$205$	6.44348e6	0.747926
$206$	− 1.12764e7i	− 1.28994i
$207$	0	0
$208$	3.44883e6	0.383250
$209$	− 7.70083e6i	− 0.843527i
$210$	0	0
$211$	−1.35165e7	−1.43885	−0.719427	−	0.694568i	$-0.755596\pi$
$211$	−1.35165e7	−1.43885	−0.719427	+	0.694568i	$0.755596\pi$
$212$	7.63960e6i	0.801795i
$213$	0	0
$214$	1.47456e6	0.150460
$215$	− 661003.i	− 0.0665102i
$216$	0	0
$217$	1.92613e7	1.88497
$218$	− 1.10001e6i	− 0.106176i
$219$	0	0
$220$	7.46266e6	0.700850
$221$	− 42867.6i	− 0.00397148i
$222$	0	0
$223$	−5.35484e6	−0.482872	−0.241436	−	0.970417i	$-0.577618\pi$
$223$	−5.35484e6	−0.482872	−0.241436	+	0.970417i	$0.577618\pi$
$224$	2.80363e6i	0.249446i
$225$	0	0
$226$	4.64935e6	0.402779
$227$	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	$0.442078\pi$
$230$	− 3.32311e6i	− 0.273125i
$231$	0	0
$232$	5.31379e6	0.425540
$233$	2.02333e7i	1.59956i	0.600297	+	0.799778i	$0.295049\pi$
$233$	2.02333e7i	1.59956i	−0.600297	+	0.799778i	$0.704951\pi$
$234$	0	0
$235$	−1.33578e7	−1.02927
$236$	− 7.99490e6i	− 0.608243i
$237$	0	0
$238$	34848.0	0.00258492
$239$	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	$0.535560\pi$
$242$	146213.i	0.0103167i
$243$	0	0
$244$	−424000.	−0.0291875
$245$	− 2.02836e7i	− 1.37926i
$246$	0	0
$247$	1.93458e7	1.28379
$248$	− 7.20385e6i	− 0.472291i
$249$	0	0
$250$	−976128.	−0.0624722
$251$	− 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
$254$	− 1.72942e7i	− 1.05536i
$255$	0	0
$256$	1.04858e6	0.0625000
$257$	− 1.44374e7i	− 0.850529i	−0.905069	−	0.425264i	$-0.860181\pi$
$257$	− 1.44374e7i	− 0.850529i	0.905069	−	0.425264i	$-0.139819\pi$
$258$	0	0
$259$	−2.54226e7	−1.46326
$260$	1.87474e7i	1.06665i
$261$	0	0
$262$	−1.73885e7	−0.966847
$263$	− 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
$266$	1.57266e7i	0.835584i
$267$	0	0
$268$	−5.40698e6	−0.280899
$269$	− 251338.i	− 0.0129122i	−0.999979	−	0.00645612i	$-0.997945\pi$
$269$	− 251338.i	− 0.0129122i	0.999979	−	0.00645612i	$-0.00205506\pi$
$270$	0	0
$271$	2.96399e7	1.48925	0.744627	−	0.667481i	$-0.232627\pi$
$271$	2.96399e7	1.48925	0.744627	+	0.667481i	$0.232627\pi$
$272$	− 13033.4i	0 0.000647665i
$273$	0	0
$274$	2.53660e7	1.23311
$275$	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	$0.399326\pi$
$278$	6.17914e6i	0.287603i
$279$	0	0
$280$	−1.52402e7	−0.694251
$281$	6.12360e6i	0.275987i	0.990433	+	0.137993i	$0.0440652\pi$
$281$	6.12360e6i	0.275987i	−0.990433	+	0.137993i	$0.955935\pi$
$282$	0	0
$283$	−6.74325e6	−0.297516	−0.148758	−	0.988874i	$-0.547527\pi$
$283$	−6.74325e6	−0.297516	−0.148758	+	0.988874i	$0.547527\pi$
$284$	− 1.70069e7i	− 0.742457i
$285$	0	0
$286$	−2.55429e7	−1.09187
$287$	− 1.79286e7i	− 0.758403i
$288$	0	0
$289$	2.41374e7	0.999993
$290$	2.88852e7i	1.18435i
$291$	0	0
$292$	−7.55661e6	−0.303514
$293$	1.00239e7i	0.398505i	0.979948	+	0.199253i	$0.0638514\pi$
$293$	1.00239e7i	0.398505i	−0.979948	+	0.199253i	$0.936149\pi$
$294$	0	0
$295$	4.34593e7	1.69284
$296$	9.50822e6i	0.366627i
$297$	0	0
$298$	1.25632e7	0.474734
$299$	1.13742e7i	0.425508i
$300$	0	0
$301$	−1.83920e6	−0.0674418
$302$	2.30727e7i	0.837677i
$303$	0	0
$304$	5.88186e6	0.209360
$305$	− 2.30481e6i	− 0.0812337i
$306$	0	0
$307$	−5.23060e6	−0.180774	−0.0903871	−	0.995907i	$-0.528810\pi$
$307$	−5.23060e6	−0.180774	−0.0903871	+	0.995907i	$0.528810\pi$
$308$	− 2.07644e7i	− 0.710668i
$309$	0	0
$310$	3.91593e7	1.31447
$311$	− 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	$0.380551\pi$
$314$	− 3.48218e7i	− 1.12477i
$315$	0	0
$316$	1.12371e6	0.0356118
$317$	2.76211e7i	0.867088i	0.901132	+	0.433544i	$0.142737\pi$
$317$	2.76211e7i	0.867088i	−0.901132	+	0.433544i	$0.857263\pi$
$318$	0	0
$319$	−3.93553e7	−1.21236
$320$	5.69994e6i	0.173948i
$321$	0	0
$322$	−9.24634e6	−0.276950
$323$	− 73109.2i	− 0.00216952i
$324$	0	0
$325$	−4.92839e7	−1.43567
$326$	− 4.52942e6i	− 0.130734i
$327$	0	0
$328$	−6.70541e6	−0.190022
$329$	3.71672e7i	1.04369i
$330$	0	0
$331$	−5.76138e6	−0.158870	−0.0794352	−	0.996840i	$-0.525312\pi$
$331$	−5.76138e6	−0.158870	−0.0794352	+	0.996840i	$0.525312\pi$
$332$	− 351359.i	− 0.00960144i
$333$	0	0
$334$	−2.71793e7	−0.729456
$335$	− 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	$0.675538\pi$
$338$	− 3.68635e7i	− 0.954656i
$339$	0	0
$340$	70848.0	0.00180256
$341$	5.33535e7i	1.34555i
$342$	0	0
$343$	504328.	0.0124977
$344$	687873.i	0.0168979i
$345$	0	0
$346$	2.01945e7	0.487535
$347$	− 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	$0.664747\pi$
$350$	− 4.00639e7i	− 0.934436i
$351$	0	0
$352$	−7.76602e6	−0.178062
$353$	1.75976e7i	0.400063i	0.979789	+	0.200032i	$0.0641045\pi$
$353$	1.75976e7i	0.400063i	−0.979789	+	0.200032i	$0.935896\pi$
$354$	0	0
$355$	9.24478e7	2.06639
$356$	− 4.13851e6i	− 0.0917263i
$357$	0	0
$358$	4.20699e7	0.916901
$359$	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
$362$	5.87249e7i	1.23793i
$363$	0	0
$364$	5.21636e7	1.08159
$365$	− 4.10768e7i	− 0.844731i
$366$	0	0
$367$	−2.65855e7	−0.537832	−0.268916	−	0.963164i	$-0.586665\pi$
$367$	−2.65855e7	−0.537832	−0.268916	+	0.963164i	$0.586665\pi$
$368$	3.45819e6i	0.0693914i
$369$	0	0
$370$	−5.16856e7	−1.02039
$371$	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	$0.444881\pi$
$374$	96528.6i	0.00184519i
$375$	0	0
$376$	1.39008e7	0.261503
$377$	− 9.88671e7i	− 1.84513i
$378$	0	0
$379$	7.20978e7	1.32435	0.662177	−	0.749347i	$-0.269633\pi$
$379$	7.20978e7	1.32435	0.662177	+	0.749347i	$0.269633\pi$
$380$	3.19731e7i	0.582685i
$381$	0	0
$382$	−7.35055e7	−1.31865
$383$	− 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
$386$	2.22425e7i	0.386742i
$387$	0	0
$388$	1.02856e7	0.176089
$389$	4.94411e7i	0.839923i	0.907542	+	0.419962i	$0.137956\pi$
$389$	4.94411e7i	0.839923i	−0.907542	+	0.419962i	$0.862044\pi$
$390$	0	0
$391$	42984.0	0.000719079 0
$392$	2.11081e7i	0.350422i
$393$	0	0
$394$	−3.04320e7	−0.497556
$395$	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	$0.459983\pi$
$398$	3.20121e6i	0.0507768i
$399$	0	0
$400$	−1.49842e7	−0.234128
$401$	4.74514e7i	0.735895i	0.929847	+	0.367947i	$0.119939\pi$
$401$	4.74514e7i	0.735895i	−0.929847	+	0.367947i	$0.880061\pi$
$402$	0	0
$403$	−1.34033e8	−2.04784
$404$	2.14010e7i	0.324556i
$405$	0	0
$406$	8.03711e7	1.20094
$407$	− 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	$0.819901\pi$
$410$	− 3.64498e7i	− 0.528864i
$411$	0	0
$412$	−6.37892e7	−0.912127
$413$	− 1.20923e8i	− 1.71656i
$414$	0	0
$415$	1.90994e6	0.0267225
$416$	− 1.95095e7i	− 0.270999i
$417$	0	0
$418$	−4.35625e7	−0.596464
$419$	− 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	$0.117176\pi$
$422$	7.64609e7i	1.01742i
$423$	0	0
$424$	4.32161e7	0.566955
$425$	186248.i	0.00242619i
$426$	0	0
$427$	−6.41300e6	−0.0823716
$428$	− 8.34137e6i	− 0.106391i
$429$	0	0
$430$	−3.73920e6	−0.0470298
$431$	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	$0.579362\pi$
$434$	− 1.08958e8i	− 1.33288i
$435$	0	0
$436$	−6.22259e6	−0.0750779
$437$	1.93983e7i	0.232445i
$438$	0	0
$439$	−1.38592e8	−1.63811	−0.819057	−	0.573712i	$-0.805503\pi$
$439$	−1.38592e8	−1.63811	−0.819057	+	0.573712i	$0.805503\pi$
$440$	− 4.22152e7i	− 0.495576i
$441$	0	0
$442$	−242496.	−0.00280826
$443$	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
$446$	3.02915e7i	0.341442i
$447$	0	0
$448$	1.58597e7	0.176385
$449$	6.11166e7i	0.675181i	0.941293	+	0.337591i	$0.109612\pi$
$449$	6.11166e7i	0.675181i	−0.941293	+	0.337591i	$0.890388\pi$
$450$	0	0
$451$	4.96619e7	0.541370
$452$	− 2.63007e7i	− 0.284808i
$453$	0	0
$454$	7.69691e7	0.822524
$455$	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	$0.440174\pi$
$458$	− 2.45870e7i	− 0.255923i
$459$	0	0
$460$	−1.87983e7	−0.193128
$461$	− 1.51983e8i	− 1.55128i	−0.631173	−	0.775642i	$-0.717426\pi$
$461$	− 1.51983e8i	− 1.55128i	0.631173	−	0.775642i	$-0.282574\pi$
$462$	0	0
$463$	1.14978e8	1.15844	0.579218	−	0.815173i	$-0.303358\pi$
$463$	1.14978e8	1.15844	0.579218	+	0.815173i	$0.303358\pi$
$464$	− 3.00593e7i	− 0.300902i
$465$	0	0
$466$	1.14457e8	1.13106
$467$	− 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
$470$	7.55631e7i	0.727807i
$471$	0	0
$472$	−4.52260e7	−0.430093
$473$	− 5.09456e6i	− 0.0481419i
$474$	0	0
$475$	−8.40520e7	−0.784272
$476$	− 197130.i	− 0.00182781i
$477$	0	0
$478$	1.15370e8	1.05635
$479$	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
$482$	1.76546e7i	0.157659i
$483$	0	0
$484$	827104.	0.00729498
$485$	5.59111e7i	0.490087i
$486$	0	0
$487$	−7.51688e7	−0.650805	−0.325403	−	0.945576i	$-0.605500\pi$
$487$	−7.51688e7	−0.650805	−0.325403	+	0.945576i	$0.605500\pi$
$488$	2.39851e6i	0.0206387i
$489$	0	0
$490$	−1.14741e8	−0.975285
$491$	− 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
$494$	− 1.09436e8i	− 0.907780i
$495$	0	0
$496$	−4.07511e7	−0.333960
$497$	− 2.57230e8i	− 2.09533i
$498$	0	0
$499$	9.15458e7	0.736778	0.368389	−	0.929672i	$-0.379909\pi$
$499$	9.15458e7	0.736778	0.368389	+	0.929672i	$0.379909\pi$
$500$	5.52181e6i	0.0441745i
$501$	0	0
$502$	−2.88228e7	−0.227837
$503$	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
$506$	− 2.56122e7i	− 0.197695i
$507$	0	0
$508$	−9.78308e7	−0.746250
$509$	− 2.39995e7i	− 0.181990i	−0.995851	−	0.0909951i	$-0.970995\pi$
$509$	− 2.39995e7i	− 0.181990i	0.995851	−	0.0909951i	$-0.0290048\pi$
$510$	0	0
$511$	−1.14294e8	−0.856564
$512$	− 5.93164e6i	− 0.0441942i
$513$	0	0
$514$	−8.16702e7	−0.601415
$515$	− 3.46751e8i	− 2.53861i
$516$	0	0
$517$	−1.02953e8	−0.745018
$518$	1.43812e8i	1.03468i
$519$	0	0
$520$	1.06052e8	0.754236
$521$	9.00897e7i	0.637033i	0.947917	+	0.318517i	$0.103185\pi$
$521$	9.00897e7i	0.637033i	−0.947917	+	0.318517i	$0.896815\pi$
$522$	0	0
$523$	−3.77691e7	−0.264016	−0.132008	−	0.991249i	$-0.542143\pi$
$523$	−3.77691e7	−0.264016	−0.132008	+	0.991249i	$0.542143\pi$
$524$	9.83641e7i	0.683664i
$525$	0	0
$526$	−1.76815e8	−1.21496
$527$	506520.i	0.00346071i
$528$	0	0
$529$	1.36631e8	0.922957
$530$	2.34918e8i	1.57793i
$531$	0	0
$532$	8.89631e7	0.590847
$533$	1.24759e8i	0.823931i
$534$	0	0
$535$	4.53427e7	0.296105
$536$	3.05865e7i	0.198626i
$537$	0	0
$538$	−1.42178e6	−0.00913034
$539$	− 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	$0.474361\pi$
$542$	− 1.67669e8i	− 1.05306i
$543$	0	0
$544$	−73728.0	−0.000457969 0
$545$	− 3.38253e7i	− 0.208955i
$546$	0	0
$547$	2.05216e8	1.25386	0.626930	−	0.779076i	$-0.284311\pi$
$547$	2.05216e8	1.25386	0.626930	+	0.779076i	$0.284311\pi$
$548$	− 1.43492e8i	− 0.871938i
$549$	0	0
$550$	1.10977e8	0.667027
$551$	− 1.68614e8i	− 1.00795i
$552$	0	0
$553$	1.69961e7	0.100502
$554$	− 7.47908e7i	− 0.439865i
$555$	0	0
$556$	3.49545e7	0.203366
$557$	2.41143e8i	1.39543i	0.716375	+	0.697715i	$0.245800\pi$
$557$	2.41143e8i	1.39543i	−0.716375	+	0.697715i	$0.754200\pi$
$558$	0	0
$559$	1.27984e7	0.0732690
$560$	8.62115e7i	0.490909i
$561$	0	0
$562$	3.46403e7	0.195152
$563$	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
$566$	3.81456e7i	0.210375i
$567$	0	0
$568$	−9.62058e7	−0.524996
$569$	2.43995e8i	1.32448i	0.749293	+	0.662238i	$0.230393\pi$
$569$	2.43995e8i	1.32448i	−0.749293	+	0.662238i	$0.769607\pi$
$570$	0	0
$571$	2.41502e8	1.29722	0.648608	−	0.761123i	$-0.275352\pi$
$571$	2.41502e8	1.29722	0.648608	+	0.761123i	$0.275352\pi$
$572$	1.44493e8i	0.772071i
$573$	0	0
$574$	−1.01419e8	−0.536272
$575$	− 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	$0.541040\pi$
$578$	− 1.36542e8i	− 0.707102i
$579$	0	0
$580$	1.63399e8	0.837463
$581$	− 5.31430e6i	− 0.0270968i
$582$	0	0
$583$	−3.20069e8	−1.61525
$584$	4.27466e7i	0.214617i
$585$	0	0
$586$	5.67038e7	0.281786
$587$	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
$590$	− 2.45843e8i	− 1.19702i
$591$	0	0
$592$	5.37866e7	0.259244
$593$	− 2.70643e8i	− 1.29788i	−0.760841	−	0.648938i	$-0.775213\pi$
$593$	− 2.70643e8i	− 1.29788i	0.760841	−	0.648938i	$-0.224787\pi$
$594$	0	0
$595$	1.07158e6	0.00508712
$596$	− 7.10680e7i	− 0.335687i
$597$	0	0
$598$	6.43423e7	0.300880
$599$	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	$0.962103\pi$
$602$	1.04041e7i	0.0476886i
$603$	0	0
$604$	1.30519e8	0.592327
$605$	4.49604e6i	0.0203032i
$606$	0	0
$607$	1.66991e7	0.0746665	0.0373332	−	0.999303i	$-0.488114\pi$
$607$	1.66991e7	0.0746665	0.0373332	+	0.999303i	$0.488114\pi$
$608$	− 3.32728e7i	− 0.148040i
$609$	0	0
$610$	−1.30380e7	−0.0574409
$611$	− 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	$0.637080\pi$
$614$	2.95887e7i	0.127827i
$615$	0	0
$616$	−1.17461e8	−0.502518
$617$	1.87023e8i	0.796233i	0.917335	+	0.398117i	$0.130336\pi$
$617$	1.87023e8i	0.796233i	−0.917335	+	0.398117i	$0.869664\pi$
$618$	0	0
$619$	2.54873e8	1.07461	0.537307	−	0.843387i	$-0.319442\pi$
$619$	2.54873e8	1.07461	0.537307	+	0.843387i	$0.319442\pi$
$620$	− 2.21518e8i	− 0.929468i
$621$	0	0
$622$	−1.76619e8	−0.733950
$623$	− 6.25950e7i	− 0.258866i
$624$	0	0
$625$	−2.58657e8	−1.05946
$626$	− 1.27154e8i	− 0.518330i
$627$	0	0
$628$	−1.96982e8	−0.795329
$629$	− 668547.i	− 0.00268646i
$630$	0	0
$631$	9.23602e7	0.367618	0.183809	−	0.982962i	$-0.441157\pi$
$631$	9.23602e7	0.367618	0.183809	+	0.982962i	$0.441157\pi$
$632$	− 6.35668e6i	− 0.0251813i
$633$	0	0
$634$	1.56249e8	0.613124
$635$	− 5.31797e8i	− 2.07694i
$636$	0	0
$637$	3.92732e8	1.51942
$638$	2.22627e8i	0.857267i
$639$	0	0
$640$	3.22437e7	0.123000
$641$	4.24666e8i	1.61240i	0.591643	+	0.806200i	$0.298480\pi$
$641$	4.24666e8i	1.61240i	−0.591643	+	0.806200i	$0.701520\pi$
$642$	0	0
$643$	3.75946e8	1.41414	0.707071	−	0.707143i	$-0.250016\pi$
$643$	3.75946e8	1.41414	0.707071	+	0.707143i	$0.250016\pi$
$644$	5.23052e7i	0.195834i
$645$	0	0
$646$	−413568.	−0.00153408
$647$	− 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
$650$	2.78792e8i	1.01517i
$651$	0	0
$652$	−2.56223e7	−0.0924432
$653$	− 2.58756e8i	− 0.929291i	−0.885497	−	0.464645i	$-0.846182\pi$
$653$	− 2.58756e8i	− 0.929291i	0.885497	−	0.464645i	$-0.153818\pi$
$654$	0	0
$655$	−5.34696e8	−1.90275
$656$	3.79315e7i	0.134366i
$657$	0	0
$658$	2.10250e8	0.738002
$659$	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	$0.804975\pi$
$662$	3.25913e7i	0.112338i
$663$	0	0
$664$	−1.98758e6	−0.00678924
$665$	4.83593e8i	1.64443i
$666$	0	0
$667$	9.91354e7	0.334081
$668$	1.53749e8i	0.515804i
$669$	0	0
$670$	−1.66265e8	−0.552809
$671$	− 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	$0.143380\pi$
$674$	2.26869e8i	0.740961i
$675$	0	0
$676$	−2.08532e8	−0.675044
$677$	1.00760e8i	0.324731i	0.986731	+	0.162365i	$0.0519123\pi$
$677$	1.00760e8i	0.324731i	−0.986731	+	0.162365i	$0.948088\pi$
$678$	0	0
$679$	1.55569e8	0.496952
$680$	− 400777.i	− 0.00127461i
$681$	0	0
$682$	3.01813e8	0.951447
$683$	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
$686$	− 2.85291e6i	− 0.00883722i
$687$	0	0
$688$	3.89120e6	0.0119486
$689$	− 8.04068e8i	− 2.45830i
$690$	0	0
$691$	−3.72812e8	−1.12994	−0.564971	−	0.825111i	$-0.691113\pi$
$691$	−3.72812e8	−1.12994	−0.564971	+	0.825111i	$0.691113\pi$
$692$	− 1.14238e8i	− 0.344739i
$693$	0	0
$694$	−3.83607e8	−1.14765
$695$	1.90009e8i	0.566003i
$696$	0	0
$697$	471474.	0.00139239
$698$	2.37949e8i	0.699709i
$699$	0	0
$700$	−2.26636e8	−0.660746
$701$	− 6.21170e8i	− 1.80325i	−0.432517	−	0.901626i	$-0.642374\pi$
$701$	− 6.21170e8i	− 1.80325i	0.432517	−	0.901626i	$-0.357626\pi$
$702$	0	0
$703$	3.01709e8	0.868406
$704$	4.39312e7i	0.125909i
$705$	0	0
$706$	9.95469e7	0.282887
$707$	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	$0.612404\pi$
$710$	− 5.22964e8i	− 1.46116i
$711$	0	0
$712$	−2.34109e7	−0.0648603
$713$	− 1.34397e8i	− 0.370783i
$714$	0	0
$715$	−7.85445e8	−2.14881
$716$	− 2.37983e8i	− 0.648347i
$717$	0	0
$718$	7.91508e7	0.213836
$719$	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
$722$	7.94921e7i	0.211209i
$723$	0	0
$724$	3.32198e8	0.875350
$725$	4.29549e8i	1.12719i
$726$	0	0
$727$	3.91371e8	1.01856	0.509278	−	0.860602i	$-0.329912\pi$
$727$	3.91371e8	1.01856	0.509278	+	0.860602i	$0.329912\pi$
$728$	− 2.95082e8i	− 0.764801i
$729$	0	0
$730$	−2.32366e8	−0.597315
$731$	− 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	$0.485889\pi$
$734$	1.50390e8i	0.380304i
$735$	0	0
$736$	1.95625e7	0.0490671
$737$	− 2.26531e8i	− 0.565881i
$738$	0	0
$739$	−3.02999e8	−0.750773	−0.375386	−	0.926868i	$-0.622490\pi$
$739$	−3.02999e8	−0.750773	−0.375386	+	0.926868i	$0.622490\pi$
$740$	2.92378e8i	0.721521i
$741$	0	0
$742$	6.53644e8	1.60004
$743$	− 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
$746$	− 1.01161e8i	− 0.243667i
$747$	0	0
$748$	546048.	0.00130475
$749$	− 1.26163e8i	− 0.300253i
$750$	0	0
$751$	−8.23270e7	−0.194367	−0.0971835	−	0.995266i	$-0.530983\pi$
$751$	−8.23270e7	−0.194367	−0.0971835	+	0.995266i	$0.530983\pi$
$752$	− 7.86348e7i	− 0.184910i
$753$	0	0
$754$	−5.59277e8	−1.30471
$755$	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	$0.744902\pi$
$758$	− 4.07847e8i	− 0.936460i
$759$	0	0
$760$	1.80867e8	0.412020
$761$	2.32982e8i	0.528651i	0.964434	+	0.264325i	$0.0851493\pi$
$761$	2.32982e8i	0.528651i	−0.964434	+	0.264325i	$0.914851\pi$
$762$	0	0
$763$	−9.41167e7	−0.211882
$764$	4.15810e8i	0.932426i
$765$	0	0
$766$	−4.91382e7	−0.109328
$767$	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	$0.145772\pi$
$770$	− 6.38504e8i	− 1.39859i
$771$	0	0
$772$	1.25823e8	0.273468
$773$	3.66587e8i	0.793667i	0.917891	+	0.396833i	$0.129891\pi$
$773$	3.66587e8i	0.793667i	−0.917891	+	0.396833i	$0.870109\pi$
$774$	0	0
$775$	5.82335e8	1.25103
$776$	− 5.81840e7i	− 0.124514i
$777$	0	0
$778$	2.79681e8	0.593915
$779$	2.12772e8i	0.450093i
$780$	0	0
$781$	7.12524e8	1.49571
$782$	− 243154.i	0 0.000508466i
$783$	0	0
$784$	1.19406e8	0.247786
$785$	− 1.07077e9i	− 2.21354i
$786$	0	0
$787$	−4.02462e8	−0.825659	−0.412830	−	0.910808i	$-0.635460\pi$
$787$	−4.02462e8	−0.825659	−0.412830	+	0.910808i	$0.635460\pi$
$788$	1.72150e8i	0.351825i
$789$	0	0
$790$	3.45541e7	0.0700840
$791$	− 3.97798e8i	− 0.803773i
$792$	0	0
$793$	4.46260e7	0.0894887
$794$	− 8.87623e7i	− 0.177324i
$795$	0	0
$796$	1.81088e7	0.0359046
$797$	5.18940e8i	1.02504i	0.858675	+	0.512521i	$0.171288\pi$
$797$	5.18940e8i	1.02504i	−0.858675	+	0.512521i	$0.828712\pi$
$798$	0	0
$799$	−977400.	−0.00191616
$800$	8.47634e7i	0.165553i
$801$	0	0
$802$	2.68426e8	0.520356
$803$	− 3.16592e8i	− 0.611440i
$804$	0	0
$805$	−2.84325e8	−0.545038
$806$	7.58205e8i	1.44804i
$807$	0	0
$808$	1.21062e8	0.229496
$809$	3.04036e6i	0.00574221i	0.999996	+	0.00287110i	$0.000913902\pi$
$809$	3.04036e6i	0.00574221i	−0.999996	+	0.00287110i	$0.999086\pi$
$810$	0	0
$811$	2.25521e8	0.422790	0.211395	−	0.977401i	$-0.432199\pi$
$811$	2.25521e8	0.422790	0.211395	+	0.977401i	$0.432199\pi$
$812$	− 4.54648e8i	− 0.849194i
$813$	0	0
$814$	−3.98357e8	−0.738583
$815$	− 1.39280e8i	− 0.257285i
$816$	0	0
$817$	2.18272e7	0.0400250
$818$	6.53432e8i	1.19383i
$819$	0	0
$820$	−2.06191e8	−0.373963
$821$	− 2.77035e8i	− 0.500617i	−0.968166	−	0.250309i	$-0.919468\pi$
$821$	− 2.77035e8i	− 0.500617i	0.968166	−	0.250309i	$-0.0805321\pi$
$822$	0	0
$823$	−7.07336e8	−1.26890	−0.634448	−	0.772965i	$-0.718773\pi$
$823$	−7.07336e8	−1.26890	−0.634448	+	0.772965i	$0.718773\pi$
$824$	3.60846e8i	0.644971i
$825$	0	0
$826$	−6.84043e8	−1.21379
$827$	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	$0.354209\pi$
$830$	− 1.08043e7i	− 0.0188956i
$831$	0	0
$832$	−1.10363e8	−0.191625
$833$	− 1.48416e6i	− 0.00256772i
$834$	0	0
$835$	−8.35764e8	−1.43557
$836$	2.46427e8i	0.421763i
$837$	0	0
$838$	−8.29822e8	−1.41011
$839$	− 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
$842$	− 7.87654e8i	− 1.31947i
$843$	0	0
$844$	4.32528e8	0.719427
$845$	− 1.13355e9i	− 1.87876i
$846$	0	0
$847$	1.25099e7	0.0205876
$848$	− 2.44467e8i	− 0.400897i
$849$	0	0
$850$	1.05358e6	0.00171557
$851$	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	$0.504821\pi$
$854$	3.62774e7i	0.0582455i
$855$	0	0
$856$	−4.71859e7	−0.0752300
$857$	6.86427e8i	1.09057i	0.838252	+	0.545283i	$0.183578\pi$
$857$	6.86427e8i	1.09057i	−0.838252	+	0.545283i	$0.816422\pi$
$858$	0	0
$859$	5.51932e8	0.870775	0.435387	−	0.900243i	$-0.356611\pi$
$859$	5.51932e8	0.870775	0.435387	+	0.900243i	$0.356611\pi$
$860$	2.11521e7i	0.0332551i
$861$	0	0
$862$	5.67901e8	0.886647
$863$	− 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
$866$	2.26631e8i	0.348952i
$867$	0	0
$868$	−6.16360e8	−0.942487
$869$	4.70791e7i	0.0717413i
$870$	0	0
$871$	5.69084e8	0.861236
$872$	3.52003e7i	0.0530881i
$873$	0	0
$874$	1.09733e8	0.164363
$875$	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	$0.642872\pi$
$878$	7.83994e8i	1.15832i
$879$	0	0
$880$	−2.38805e8	−0.350425
$881$	4.29761e8i	0.628491i	0.949342	+	0.314246i	$0.101752\pi$
$881$	4.29761e8i	0.628491i	−0.949342	+	0.314246i	$0.898248\pi$
$882$	0	0
$883$	2.20085e8	0.319675	0.159837	−	0.987143i	$-0.448903\pi$
$883$	2.20085e8	0.319675	0.159837	+	0.987143i	$0.448903\pi$
$884$	1.37176e6i	0.00198574i
$885$	0	0
$886$	6.30415e8	0.906411
$887$	− 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
$890$	− 1.27259e8i	− 0.180517i
$891$	0	0
$892$	1.71355e8	0.241436
$893$	− 4.41092e8i	− 0.619405i
$894$	0	0
$895$	1.29365e9	1.80446
$896$	− 8.97161e7i	− 0.124723i
$897$	0	0
$898$	3.45728e8	0.477425
$899$	1.16820e9i	1.60783i
$900$	0	0
$901$	−3.03863e6	−0.00415436
$902$	− 2.80930e8i	− 0.382806i
$903$	0	0
$904$	−1.48779e8	−0.201390
$905$	1.80579e9i	2.43625i
$906$	0	0
$907$	7.31614e8	0.980529	0.490264	−	0.871574i	$-0.336900\pi$
$907$	7.31614e8	0.980529	0.490264	+	0.871574i	$0.336900\pi$
$908$	− 4.35403e8i	− 0.581612i
$909$	0	0
$910$	1.60403e9	2.12857
$911$	− 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
$914$	− 2.01760e8i	− 0.264239i
$915$	0	0
$916$	−1.39085e8	−0.180965
$917$	1.48776e9i	1.92941i
$918$	0	0
$919$	−2.15987e8	−0.278279	−0.139139	−	0.990273i	$-0.544434\pi$
$919$	−2.15987e8	−0.278279	−0.139139	+	0.990273i	$0.544434\pi$
$920$	1.06339e8i	0.136562i
$921$	0	0
$922$	−8.59744e8	−1.09692
$923$	1.78998e9i	2.27637i
$924$	0	0
$925$	−7.68613e8	−0.971141
$926$	− 6.50414e8i	− 0.819138i
$927$	0	0
$928$	−1.70041e8	−0.212770
$929$	− 3.10124e8i	− 0.386802i	−0.981120	−	0.193401i	$-0.938048\pi$
$929$	− 3.10124e8i	− 0.386802i	0.981120	−	0.193401i	$-0.0619518\pi$
$930$	0	0
$931$	6.69791e8	0.830023
$932$	− 6.47466e8i	− 0.799778i
$933$	0	0
$934$	−4.98768e8	−0.612150
$935$	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	$0.649022\pi$
$938$	4.62620e8i	0.560553i
$939$	0	0
$940$	4.27450e8	0.514637
$941$	− 1.81766e8i	− 0.218144i	−0.994034	−	0.109072i	$-0.965212\pi$
$941$	− 1.81766e8i	− 0.218144i	0.994034	−	0.109072i	$-0.0347879\pi$
$942$	0	0
$943$	−1.25098e8	−0.149181
$944$	2.55837e8i	0.304121i
$945$	0	0
$946$	−2.88192e7	−0.0340415
$947$	− 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
$950$	4.75470e8i	0.554564i
$951$	0	0
$952$	−1.11514e6	−0.00129246
$953$	− 6.86819e8i	− 0.793530i	−0.917920	−	0.396765i	$-0.870133\pi$
$953$	− 6.86819e8i	− 0.793530i	0.917920	−	0.396765i	$-0.129867\pi$
$954$	0	0
$955$	−2.26029e9	−2.59510
$956$	− 6.52630e8i	− 0.746953i
$957$	0	0
$958$	5.05942e8	0.575446
$959$	− 2.17031e9i	− 2.46075i
$960$	0	0
$961$	6.96218e8	0.784468
$962$	− 1.00074e9i	− 1.12408i
$963$	0	0
$964$	9.98697e7	0.111481
$965$	6.83957e8i	0.761109i
$966$	0	0
$967$	−1.09411e9	−1.20999	−0.604995	−	0.796230i	$-0.706825\pi$
$967$	−1.09411e9	−1.20999	−0.604995	+	0.796230i	$0.706825\pi$
$968$	− 4.67881e6i	− 0.00515833i
$969$	0	0
$970$	3.16281e8	0.346544
$971$	− 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
$974$	4.25219e8i	0.460189i
$975$	0	0
$976$	1.35680e7	0.0145937
$977$	− 1.19004e9i	− 1.27608i	−0.770001	−	0.638042i	$-0.779744\pi$
$977$	− 1.19004e9i	− 1.27608i	0.770001	−	0.638042i	$-0.220256\pi$
$978$	0	0
$979$	1.73387e8	0.184786
$980$	6.49075e8i	0.689631i
$981$	0	0
$982$	−2.55024e8	−0.269306
$983$	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
$986$	2.11355e6i	0.00220486i
$987$	0	0
$988$	−6.19065e8	−0.641897
$989$	1.28331e7i	0.0132661i
$990$	0	0
$991$	5.09602e8	0.523613	0.261806	−	0.965120i	$-0.415682\pi$
$991$	5.09602e8	0.523613	0.261806	+	0.965120i	$0.415682\pi$
$992$	2.30523e8i	0.236145i
$993$	0	0
$994$	−1.45511e9	−1.48162
$995$	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	$0.666621\pi$
$998$	− 5.17861e8i	− 0.520981i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.7.b.a.17.1	✓	2
3.2	odd	2	inner	18.7.b.a.17.2	yes	2
4.3	odd	2		144.7.e.d.17.1		2
5.2	odd	4		450.7.b.a.449.3		4
5.3	odd	4		450.7.b.a.449.2		4
5.4	even	2		450.7.d.a.251.2		2
8.3	odd	2		576.7.e.k.449.2		2
8.5	even	2		576.7.e.b.449.2		2
9.2	odd	6		162.7.d.d.53.2		4
9.4	even	3		162.7.d.d.107.2		4
9.5	odd	6		162.7.d.d.107.1		4
9.7	even	3		162.7.d.d.53.1		4
12.11	even	2		144.7.e.d.17.2		2
15.2	even	4		450.7.b.a.449.1		4
15.8	even	4		450.7.b.a.449.4		4
15.14	odd	2		450.7.d.a.251.1		2
24.5	odd	2		576.7.e.b.449.1		2
24.11	even	2		576.7.e.k.449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.7.b.a.17.1	✓	2	1.1	even	1	trivial
18.7.b.a.17.2	yes	2	3.2	odd	2	inner
144.7.e.d.17.1		2	4.3	odd	2
144.7.e.d.17.2		2	12.11	even	2
162.7.d.d.53.1		4	9.7	even	3
162.7.d.d.53.2		4	9.2	odd	6
162.7.d.d.107.1		4	9.5	odd	6
162.7.d.d.107.2		4	9.4	even	3
450.7.b.a.449.1		4	15.2	even	4
450.7.b.a.449.2		4	5.3	odd	4
450.7.b.a.449.3		4	5.2	odd	4
450.7.b.a.449.4		4	15.8	even	4
450.7.d.a.251.1		2	15.14	odd	2
450.7.d.a.251.2		2	5.4	even	2
576.7.e.b.449.1		2	24.5	odd	2
576.7.e.b.449.2		2	8.5	even	2
576.7.e.k.449.1		2	24.11	even	2
576.7.e.k.449.2		2	8.3	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:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	18.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.65685i q^{2} -32.0000 q^{4} -173.948i q^{5} -484.000 q^{7} +181.019i q^{8} -984.000 q^{10} -1340.67i q^{11} +3368.00 q^{13} +2737.92i q^{14} +1024.00 q^{16} -12.7279i q^{17} +5744.00 q^{19} +5566.34i q^{20} -7584.00 q^{22} +3377.14i q^{23} -14633.0 q^{25} -19052.3i q^{26} +15488.0 q^{28} -29354.8i q^{29} -39796.0 q^{31} -5792.62i q^{32} -72.0000 q^{34} +84191.0i q^{35} +52526.0 q^{37} -32493.0i q^{38} +31488.0 q^{40} +37042.5i q^{41} +3800.00 q^{43} +42901.6i q^{44} +19104.0 q^{46} -76791.8i q^{47} +116607. q^{49} +82776.7i q^{50} -107776. q^{52} -238738. i q^{53} -233208. q^{55} -87613.4i q^{56} -166056. q^{58} +249841. i q^{59} +13250.0 q^{61} +225120. i q^{62} -32768.0 q^{64} -585858. i q^{65} +168968. q^{67} +407.294i q^{68} +476256. q^{70} +531467. i q^{71} +236144. q^{73} -297132. i q^{74} -183808. q^{76} +648886. i q^{77} -35116.0 q^{79} -178123. i q^{80} +209544. q^{82} +10980.0i q^{83} -2214.00 q^{85} -21496.0i q^{86} +242688. q^{88} +129328. i q^{89} -1.63011e6 q^{91} -108069. i q^{92} -434400. q^{94} -999159. i q^{95} -321424. q^{97} -659629. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 64 q^{4} - 968 q^{7} - 1968 q^{10} + 6736 q^{13} + 2048 q^{16} + 11488 q^{19} - 15168 q^{22} - 29266 q^{25} + 30976 q^{28} - 79592 q^{31} - 144 q^{34} + 105052 q^{37} + 62976 q^{40} + 7600 q^{43}+ \cdots - 642848 q^{97}+O(q^{100}) \) 2 * q - 64 * q^4 - 968 * q^7 - 1968 * q^10 + 6736 * q^13 + 2048 * q^16 + 11488 * q^19 - 15168 * q^22 - 29266 * q^25 + 30976 * q^28 - 79592 * q^31 - 144 * q^34 + 105052 * q^37 + 62976 * q^40 + 7600 * q^43 + 38208 * q^46 + 233214 * q^49 - 215552 * q^52 - 466416 * q^55 - 332112 * q^58 + 26500 * q^61 - 65536 * q^64 + 337936 * q^67 + 952512 * q^70 + 472288 * q^73 - 367616 * q^76 - 70232 * q^79 + 419088 * q^82 - 4428 * q^85 + 485376 * q^88 - 3260224 * q^91 - 868800 * q^94 - 642848 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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