LMFDB - Embedded newform 324.6.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,6,Mod(1,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("324.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	324.a (trivial)

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:	yes
Analytic conductor:	$51.9643576194$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 110x^{3} + 39x^{2} + 2214x - 1944 $ x^5 - 2x^4 - 110x^3 + 39x^2 + 2214x - 1944
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 36)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-7.49663$ of defining polynomial
Character	$\chi$	$=$	324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+110.399 q^{5} +101.745 q^{7} +O(q^{10})$	$q+110.399 q^{5} +101.745 q^{7} -150.312 q^{11} +635.424 q^{13} +1498.54 q^{17} +1437.69 q^{19} -1264.11 q^{23} +9063.00 q^{25} -2777.50 q^{29} -6968.68 q^{31} +11232.5 q^{35} -7950.71 q^{37} -2027.53 q^{41} -12523.3 q^{43} -6482.34 q^{47} -6455.01 q^{49} +9827.54 q^{53} -16594.3 q^{55} +47087.9 q^{59} +8337.84 q^{61} +70150.3 q^{65} -7260.89 q^{67} +3582.33 q^{71} +58077.5 q^{73} -15293.5 q^{77} -63742.8 q^{79} +82846.5 q^{83} +165438. q^{85} -3861.51 q^{89} +64651.0 q^{91} +158720. q^{95} +69277.2 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 21 q^{5} - 29 q^{7}+O(q^{10}) $ 5 * q + 21 * q^5 - 29 * q^7	$ 5 q + 21 q^{5} - 29 q^{7} - 177 q^{11} + 181 q^{13} + 1140 q^{17} - 416 q^{19} - 399 q^{23} + 4778 q^{25} + 6033 q^{29} - 2759 q^{31} + 18573 q^{35} - 7586 q^{37} + 18435 q^{41} - 1469 q^{43} + 25155 q^{47} + 4056 q^{49} + 58422 q^{53} + 7389 q^{55} + 90537 q^{59} - 1403 q^{61} + 148407 q^{65} - 13907 q^{67} + 114684 q^{71} + 7600 q^{73} + 211983 q^{77} - 29993 q^{79} + 228951 q^{83} + 49662 q^{85} + 299166 q^{89} + 62465 q^{91} + 394764 q^{95} - 40541 q^{97}+O(q^{100}) $ 5 * q + 21 * q^5 - 29 * q^7 - 177 * q^11 + 181 * q^13 + 1140 * q^17 - 416 * q^19 - 399 * q^23 + 4778 * q^25 + 6033 * q^29 - 2759 * q^31 + 18573 * q^35 - 7586 * q^37 + 18435 * q^41 - 1469 * q^43 + 25155 * q^47 + 4056 * q^49 + 58422 * q^53 + 7389 * q^55 + 90537 * q^59 - 1403 * q^61 + 148407 * q^65 - 13907 * q^67 + 114684 * q^71 + 7600 * q^73 + 211983 * q^77 - 29993 * q^79 + 228951 * q^83 + 49662 * q^85 + 299166 * q^89 + 62465 * q^91 + 394764 * q^95 - 40541 * q^97

Display 100 coefficients 10 coefficients

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
$5$	110.399	1.97488	0.987441	−	0.157988i	$-0.0505007\pi$
$5$	110.399	1.97488	0.987441	+	0.157988i	$0.0505007\pi$
$6$	0	0
$7$	101.745	0.784814	0.392407	−	0.919792i	$-0.371642\pi$
$7$	101.745	0.784814	0.392407	+	0.919792i	$0.371642\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−150.312	−0.374552	−0.187276	−	0.982307i	$-0.559966\pi$
$11$	−150.312	−0.374552	−0.187276	+	0.982307i	$0.559966\pi$
$12$	0	0
$13$	635.424	1.04281	0.521405	−	0.853309i	$-0.325408\pi$
$13$	635.424	1.04281	0.521405	+	0.853309i	$0.325408\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1498.54	1.25761	0.628806	−	0.777562i	$-0.283544\pi$
$17$	1498.54	1.25761	0.628806	+	0.777562i	$0.283544\pi$
$18$	0	0
$19$	1437.69	0.913651	0.456825	−	0.889556i	$-0.348986\pi$
$19$	1437.69	0.913651	0.456825	+	0.889556i	$0.348986\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1264.11	−0.498269	−0.249134	−	0.968469i	$-0.580146\pi$
$23$	−1264.11	−0.498269	−0.249134	+	0.968469i	$0.580146\pi$
$24$	0	0
$25$	9063.00	2.90016
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2777.50	−0.613280	−0.306640	−	0.951826i	$-0.599205\pi$
$29$	−2777.50	−0.613280	−0.306640	+	0.951826i	$0.599205\pi$
$30$	0	0
$31$	−6968.68	−1.30241	−0.651203	−	0.758904i	$-0.725735\pi$
$31$	−6968.68	−1.30241	−0.651203	+	0.758904i	$0.725735\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	11232.5	1.54992
$36$	0	0
$37$	−7950.71	−0.954776	−0.477388	−	0.878693i	$-0.658416\pi$
$37$	−7950.71	−0.954776	−0.477388	+	0.878693i	$0.658416\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2027.53	−0.188369	−0.0941843	−	0.995555i	$-0.530024\pi$
$41$	−2027.53	−0.188369	−0.0941843	+	0.995555i	$0.530024\pi$
$42$	0	0
$43$	−12523.3	−1.03288	−0.516438	−	0.856325i	$-0.672742\pi$
$43$	−12523.3	−1.03288	−0.516438	+	0.856325i	$0.672742\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6482.34	−0.428043	−0.214021	−	0.976829i	$-0.568656\pi$
$47$	−6482.34	−0.428043	−0.214021	+	0.976829i	$0.568656\pi$
$48$	0	0
$49$	−6455.01	−0.384067
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9827.54	0.480568	0.240284	−	0.970703i	$-0.422759\pi$
$53$	9827.54	0.480568	0.240284	+	0.970703i	$0.422759\pi$
$54$	0	0
$55$	−16594.3	−0.739696
$56$	0	0
$57$	0	0
$58$	0	0
$59$	47087.9	1.76108	0.880541	−	0.473970i	$-0.157179\pi$
$59$	47087.9	1.76108	0.880541	+	0.473970i	$0.157179\pi$
$60$	0	0
$61$	8337.84	0.286899	0.143450	−	0.989658i	$-0.454181\pi$
$61$	8337.84	0.286899	0.143450	+	0.989658i	$0.454181\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	70150.3	2.05943
$66$	0	0
$67$	−7260.89	−0.197607	−0.0988036	−	0.995107i	$-0.531502\pi$
$67$	−7260.89	−0.197607	−0.0988036	+	0.995107i	$0.531502\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3582.33	0.0843372	0.0421686	−	0.999111i	$-0.486573\pi$
$71$	3582.33	0.0843372	0.0421686	+	0.999111i	$0.486573\pi$
$72$	0	0
$73$	58077.5	1.27556	0.637780	−	0.770218i	$-0.279853\pi$
$73$	58077.5	1.27556	0.637780	+	0.770218i	$0.279853\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15293.5	−0.293954
$78$	0	0
$79$	−63742.8	−1.14912	−0.574558	−	0.818464i	$-0.694826\pi$
$79$	−63742.8	−1.14912	−0.574558	+	0.818464i	$0.694826\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	82846.5	1.32002	0.660008	−	0.751259i	$-0.270553\pi$
$83$	82846.5	1.32002	0.660008	+	0.751259i	$0.270553\pi$
$84$	0	0
$85$	165438.	2.48363
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3861.51	−0.0516752	−0.0258376	−	0.999666i	$-0.508225\pi$
$89$	−3861.51	−0.0516752	−0.0258376	+	0.999666i	$0.508225\pi$
$90$	0	0
$91$	64651.0	0.818412
$92$	0	0
$93$	0	0
$94$	0	0
$95$	158720.	1.80435
$96$	0	0
$97$	69277.2	0.747585	0.373793	−	0.927512i	$-0.378057\pi$
$97$	69277.2	0.747585	0.373793	+	0.927512i	$0.378057\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−25445.7	−0.248205	−0.124103	−	0.992269i	$-0.539605\pi$
$101$	−25445.7	−0.248205	−0.124103	+	0.992269i	$0.539605\pi$
$102$	0	0
$103$	−63387.8	−0.588725	−0.294363	−	0.955694i	$-0.595107\pi$
$103$	−63387.8	−0.588725	−0.294363	+	0.955694i	$0.595107\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−61158.9	−0.516417	−0.258208	−	0.966089i	$-0.583132\pi$
$107$	−61158.9	−0.516417	−0.258208	+	0.966089i	$0.583132\pi$
$108$	0	0
$109$	−124036.	−0.999957	−0.499979	−	0.866038i	$-0.666659\pi$
$109$	−124036.	−0.999957	−0.499979	+	0.866038i	$0.666659\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	117437.	0.865185	0.432592	−	0.901590i	$-0.357599\pi$
$113$	117437.	0.865185	0.432592	+	0.901590i	$0.357599\pi$
$114$	0	0
$115$	−139556.	−0.984022
$116$	0	0
$117$	0	0
$118$	0	0
$119$	152469.	0.986991
$120$	0	0
$121$	−138457.	−0.859711
$122$	0	0
$123$	0	0
$124$	0	0
$125$	655551.	3.75259
$126$	0	0
$127$	−29838.6	−0.164161	−0.0820803	−	0.996626i	$-0.526156\pi$
$127$	−29838.6	−0.164161	−0.0820803	+	0.996626i	$0.526156\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−172058.	−0.875983	−0.437992	−	0.898979i	$-0.644310\pi$
$131$	−172058.	−0.875983	−0.437992	+	0.898979i	$0.644310\pi$
$132$	0	0
$133$	146277.	0.717046
$134$	0	0
$135$	0	0
$136$	0	0
$137$	190361.	0.866518	0.433259	−	0.901270i	$-0.357364\pi$
$137$	190361.	0.866518	0.433259	+	0.901270i	$0.357364\pi$
$138$	0	0
$139$	−337063.	−1.47970	−0.739852	−	0.672770i	$-0.765104\pi$
$139$	−337063.	−1.47970	−0.739852	+	0.672770i	$0.765104\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−95511.9	−0.390586
$144$	0	0
$145$	−306634.	−1.21115
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−55098.7	−0.203318	−0.101659	−	0.994819i	$-0.532415\pi$
$149$	−55098.7	−0.203318	−0.101659	+	0.994819i	$0.532415\pi$
$150$	0	0
$151$	334064.	1.19230	0.596152	−	0.802872i	$-0.296696\pi$
$151$	334064.	1.19230	0.596152	+	0.802872i	$0.296696\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−769337.	−2.57210
$156$	0	0
$157$	−120107.	−0.388882	−0.194441	−	0.980914i	$-0.562289\pi$
$157$	−120107.	−0.388882	−0.194441	+	0.980914i	$0.562289\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−128616.	−0.391048
$162$	0	0
$163$	367083.	1.08217	0.541085	−	0.840968i	$-0.318014\pi$
$163$	367083.	1.08217	0.541085	+	0.840968i	$0.318014\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	589448.	1.63552	0.817758	−	0.575563i	$-0.195217\pi$
$167$	589448.	1.63552	0.817758	+	0.575563i	$0.195217\pi$
$168$	0	0
$169$	32470.6	0.0874527
$170$	0	0
$171$	0	0
$172$	0	0
$173$	436457.	1.10873	0.554366	−	0.832273i	$-0.312961\pi$
$173$	436457.	1.10873	0.554366	+	0.832273i	$0.312961\pi$
$174$	0	0
$175$	922112.	2.27609
$176$	0	0
$177$	0	0
$178$	0	0
$179$	241064.	0.562341	0.281171	−	0.959658i	$-0.409277\pi$
$179$	241064.	0.562341	0.281171	+	0.959658i	$0.409277\pi$
$180$	0	0
$181$	27128.8	0.0615510	0.0307755	−	0.999526i	$-0.490202\pi$
$181$	27128.8	0.0615510	0.0307755	+	0.999526i	$0.490202\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−877752.	−1.88557
$186$	0	0
$187$	−225249.	−0.471041
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−56072.5	−0.111216	−0.0556079	−	0.998453i	$-0.517710\pi$
$191$	−56072.5	−0.111216	−0.0556079	+	0.998453i	$0.517710\pi$
$192$	0	0
$193$	354483.	0.685019	0.342510	−	0.939514i	$-0.388723\pi$
$193$	354483.	0.685019	0.342510	+	0.939514i	$0.388723\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−816895.	−1.49969	−0.749844	−	0.661615i	$-0.769871\pi$
$197$	−816895.	−1.49969	−0.749844	+	0.661615i	$0.769871\pi$
$198$	0	0
$199$	860396.	1.54016	0.770079	−	0.637948i	$-0.220217\pi$
$199$	860396.	1.54016	0.770079	+	0.637948i	$0.220217\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−282596.	−0.481310
$204$	0	0
$205$	−223838.	−0.372006
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−216102.	−0.342210
$210$	0	0
$211$	−387171.	−0.598683	−0.299341	−	0.954146i	$-0.596767\pi$
$211$	−387171.	−0.598683	−0.299341	+	0.954146i	$0.596767\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.38256e6	−2.03981
$216$	0	0
$217$	−709026.	−1.02215
$218$	0	0
$219$	0	0
$220$	0	0
$221$	952209.	1.31145
$222$	0	0
$223$	−294994.	−0.397238	−0.198619	−	0.980077i	$-0.563646\pi$
$223$	−294994.	−0.397238	−0.198619	+	0.980077i	$0.563646\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	687980.	0.886158	0.443079	−	0.896483i	$-0.353886\pi$
$227$	687980.	0.886158	0.443079	+	0.896483i	$0.353886\pi$
$228$	0	0
$229$	−1.08647e6	−1.36908	−0.684538	−	0.728977i	$-0.739996\pi$
$229$	−1.08647e6	−1.36908	−0.684538	+	0.728977i	$0.739996\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	168058.	0.202801	0.101400	−	0.994846i	$-0.467668\pi$
$233$	168058.	0.202801	0.101400	+	0.994846i	$0.467668\pi$
$234$	0	0
$235$	−715646.	−0.845334
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.54628e6	−1.75102	−0.875512	−	0.483197i	$-0.839476\pi$
$239$	−1.54628e6	−1.75102	−0.875512	+	0.483197i	$0.839476\pi$
$240$	0	0
$241$	−1.15373e6	−1.27956	−0.639782	−	0.768557i	$-0.720975\pi$
$241$	−1.15373e6	−1.27956	−0.639782	+	0.768557i	$0.720975\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−712629.	−0.758487
$246$	0	0
$247$	913541.	0.952764
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−586711.	−0.587814	−0.293907	−	0.955834i	$-0.594956\pi$
$251$	−586711.	−0.587814	−0.293907	+	0.955834i	$0.594956\pi$
$252$	0	0
$253$	190010.	0.186628
$254$	0	0
$255$	0	0
$256$	0	0
$257$	108342.	0.102321	0.0511604	−	0.998690i	$-0.483708\pi$
$257$	108342.	0.102321	0.0511604	+	0.998690i	$0.483708\pi$
$258$	0	0
$259$	−808942.	−0.749321
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.73671e6	−1.54823	−0.774117	−	0.633042i	$-0.781806\pi$
$263$	−1.73671e6	−1.54823	−0.774117	+	0.633042i	$0.781806\pi$
$264$	0	0
$265$	1.08495e6	0.949065
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18297.3	0.0154172	0.00770860	−	0.999970i	$-0.497546\pi$
$269$	18297.3	0.0154172	0.00770860	+	0.999970i	$0.497546\pi$
$270$	0	0
$271$	7105.09	0.00587688	0.00293844	−	0.999996i	$-0.499065\pi$
$271$	7105.09	0.00587688	0.00293844	+	0.999996i	$0.499065\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.36228e6	−1.08626
$276$	0	0
$277$	710403.	0.556296	0.278148	−	0.960538i	$-0.410280\pi$
$277$	710403.	0.556296	0.278148	+	0.960538i	$0.410280\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.02845e6	−0.776991	−0.388496	−	0.921451i	$-0.627005\pi$
$281$	−1.02845e6	−0.776991	−0.388496	+	0.921451i	$0.627005\pi$
$282$	0	0
$283$	−1.84463e6	−1.36913	−0.684564	−	0.728953i	$-0.740007\pi$
$283$	−1.84463e6	−1.36913	−0.684564	+	0.728953i	$0.740007\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−206291.	−0.147834
$288$	0	0
$289$	825769.	0.581586
$290$	0	0
$291$	0	0
$292$	0	0
$293$	940972.	0.640336	0.320168	−	0.947361i	$-0.396261\pi$
$293$	940972.	0.640336	0.320168	+	0.947361i	$0.396261\pi$
$294$	0	0
$295$	5.19847e6	3.47793
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−803243.	−0.519600
$300$	0	0
$301$	−1.27418e6	−0.810615
$302$	0	0
$303$	0	0
$304$	0	0
$305$	920492.	0.566592
$306$	0	0
$307$	−2.93094e6	−1.77485	−0.887425	−	0.460952i	$-0.847508\pi$
$307$	−2.93094e6	−1.77485	−0.887425	+	0.460952i	$0.847508\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.28251e6	−1.33817	−0.669086	−	0.743185i	$-0.733314\pi$
$311$	−2.28251e6	−1.33817	−0.669086	+	0.743185i	$0.733314\pi$
$312$	0	0
$313$	802648.	0.463089	0.231544	−	0.972824i	$-0.425622\pi$
$313$	802648.	0.463089	0.231544	+	0.972824i	$0.425622\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.11196e6	0.621502	0.310751	−	0.950491i	$-0.399419\pi$
$317$	1.11196e6	0.621502	0.310751	+	0.950491i	$0.399419\pi$
$318$	0	0
$319$	417491.	0.229705
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.15443e6	1.14902
$324$	0	0
$325$	5.75885e6	3.02432
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−659544.	−0.335934
$330$	0	0
$331$	−1.22658e6	−0.615355	−0.307678	−	0.951491i	$-0.599552\pi$
$331$	−1.22658e6	−0.615355	−0.307678	+	0.951491i	$0.599552\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−801597.	−0.390251
$336$	0	0
$337$	2.30128e6	1.10381	0.551907	−	0.833906i	$-0.313900\pi$
$337$	2.30128e6	1.10381	0.551907	+	0.833906i	$0.313900\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.04748e6	0.487818
$342$	0	0
$343$	−2.36679e6	−1.08624
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.61930e6	0.721945	0.360973	−	0.932576i	$-0.382445\pi$
$347$	1.61930e6	0.721945	0.360973	+	0.932576i	$0.382445\pi$
$348$	0	0
$349$	−278614.	−0.122445	−0.0612223	−	0.998124i	$-0.519500\pi$
$349$	−278614.	−0.122445	−0.0612223	+	0.998124i	$0.519500\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.36993e6	1.43941	0.719704	−	0.694281i	$-0.244277\pi$
$353$	3.36993e6	1.43941	0.719704	+	0.694281i	$0.244277\pi$
$354$	0	0
$355$	395486.	0.166556
$356$	0	0
$357$	0	0
$358$	0	0
$359$	272571.	0.111621	0.0558103	−	0.998441i	$-0.482226\pi$
$359$	272571.	0.111621	0.0558103	+	0.998441i	$0.482226\pi$
$360$	0	0
$361$	−409156.	−0.165242
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.41172e6	2.51908
$366$	0	0
$367$	−3.95765e6	−1.53381	−0.766906	−	0.641759i	$-0.778205\pi$
$367$	−3.95765e6	−1.53381	−0.766906	+	0.641759i	$0.778205\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	999900.	0.377157
$372$	0	0
$373$	−1.92388e6	−0.715987	−0.357994	−	0.933724i	$-0.616539\pi$
$373$	−1.92388e6	−0.715987	−0.357994	+	0.933724i	$0.616539\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.76489e6	−0.639534
$378$	0	0
$379$	−3.11015e6	−1.11220	−0.556101	−	0.831115i	$-0.687703\pi$
$379$	−3.11015e6	−1.11220	−0.556101	+	0.831115i	$0.687703\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.83421e6	−0.987269	−0.493634	−	0.869670i	$-0.664332\pi$
$383$	−2.83421e6	−0.987269	−0.493634	+	0.869670i	$0.664332\pi$
$384$	0	0
$385$	−1.68839e6	−0.580524
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.05383e6	−0.688162	−0.344081	−	0.938940i	$-0.611810\pi$
$389$	−2.05383e6	−0.688162	−0.344081	+	0.938940i	$0.611810\pi$
$390$	0	0
$391$	−1.89431e6	−0.626628
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.03716e6	−2.26937
$396$	0	0
$397$	−3.82167e6	−1.21696	−0.608480	−	0.793569i	$-0.708220\pi$
$397$	−3.82167e6	−1.21696	−0.608480	+	0.793569i	$0.708220\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.31729e6	1.65131	0.825656	−	0.564173i	$-0.190805\pi$
$401$	5.31729e6	1.65131	0.825656	+	0.564173i	$0.190805\pi$
$402$	0	0
$403$	−4.42807e6	−1.35816
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.19509e6	0.357613
$408$	0	0
$409$	3.15843e6	0.933605	0.466803	−	0.884362i	$-0.345406\pi$
$409$	3.15843e6	0.933605	0.466803	+	0.884362i	$0.345406\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.79095e6	1.38212
$414$	0	0
$415$	9.14620e6	2.60688
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.61564e6	−1.00612	−0.503060	−	0.864251i	$-0.667793\pi$
$419$	−3.61564e6	−1.00612	−0.503060	+	0.864251i	$0.667793\pi$
$420$	0	0
$421$	1.02889e6	0.282920	0.141460	−	0.989944i	$-0.454820\pi$
$421$	1.02889e6	0.282920	0.141460	+	0.989944i	$0.454820\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.35813e7	3.64727
$426$	0	0
$427$	848331.	0.225162
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.21668e6	1.61200	0.806001	−	0.591914i	$-0.201628\pi$
$431$	6.21668e6	1.61200	0.806001	+	0.591914i	$0.201628\pi$
$432$	0	0
$433$	−598070.	−0.153297	−0.0766483	−	0.997058i	$-0.524422\pi$
$433$	−598070.	−0.153297	−0.0766483	+	0.997058i	$0.524422\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.81739e6	−0.455244
$438$	0	0
$439$	−493097.	−0.122115	−0.0610577	−	0.998134i	$-0.519447\pi$
$439$	−493097.	−0.122115	−0.0610577	+	0.998134i	$0.519447\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−195512.	−0.0473329	−0.0236665	−	0.999720i	$-0.507534\pi$
$443$	−195512.	−0.0473329	−0.0236665	+	0.999720i	$0.507534\pi$
$444$	0	0
$445$	−426308.	−0.102053
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.51225e6	−0.354004	−0.177002	−	0.984210i	$-0.556640\pi$
$449$	−1.51225e6	−0.354004	−0.177002	+	0.984210i	$0.556640\pi$
$450$	0	0
$451$	304763.	0.0705538
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.13743e6	1.61627
$456$	0	0
$457$	2.56472e6	0.574445	0.287223	−	0.957864i	$-0.407268\pi$
$457$	2.56472e6	0.574445	0.287223	+	0.957864i	$0.407268\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.36673e6	−1.17613	−0.588067	−	0.808812i	$-0.700111\pi$
$461$	−5.36673e6	−1.17613	−0.588067	+	0.808812i	$0.700111\pi$
$462$	0	0
$463$	5.50215e6	1.19283	0.596417	−	0.802675i	$-0.296591\pi$
$463$	5.50215e6	1.19283	0.596417	+	0.802675i	$0.296591\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	964114.	0.204567	0.102284	−	0.994755i	$-0.467385\pi$
$467$	964114.	0.204567	0.102284	+	0.994755i	$0.467385\pi$
$468$	0	0
$469$	−738757.	−0.155085
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.88240e6	0.386865
$474$	0	0
$475$	1.30298e7	2.64973
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−562534.	−0.112024	−0.0560118	−	0.998430i	$-0.517838\pi$
$479$	−562534.	−0.112024	−0.0560118	+	0.998430i	$0.517838\pi$
$480$	0	0
$481$	−5.05207e6	−0.995650
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.64815e6	1.47639
$486$	0	0
$487$	3.14185e6	0.600292	0.300146	−	0.953893i	$-0.402965\pi$
$487$	3.14185e6	0.600292	0.300146	+	0.953893i	$0.402965\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.73361e6	1.07331	0.536654	−	0.843802i	$-0.319688\pi$
$491$	5.73361e6	1.07331	0.536654	+	0.843802i	$0.319688\pi$
$492$	0	0
$493$	−4.16219e6	−0.771267
$494$	0	0
$495$	0	0
$496$	0	0
$497$	364483.	0.0661890
$498$	0	0
$499$	1.71562e6	0.308439	0.154220	−	0.988037i	$-0.450714\pi$
$499$	1.71562e6	0.308439	0.154220	+	0.988037i	$0.450714\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.15229e6	−0.907989	−0.453995	−	0.891004i	$-0.650001\pi$
$503$	−5.15229e6	−0.907989	−0.453995	+	0.891004i	$0.650001\pi$
$504$	0	0
$505$	−2.80919e6	−0.490176
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.04256e6	−0.178364	−0.0891819	−	0.996015i	$-0.528425\pi$
$509$	−1.04256e6	−0.178364	−0.0891819	+	0.996015i	$0.528425\pi$
$510$	0	0
$511$	5.90908e6	1.00108
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.99796e6	−1.16266
$516$	0	0
$517$	974374.	0.160324
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.52239e6	1.53692	0.768461	−	0.639897i	$-0.221023\pi$
$521$	9.52239e6	1.53692	0.768461	+	0.639897i	$0.221023\pi$
$522$	0	0
$523$	−2.97573e6	−0.475706	−0.237853	−	0.971301i	$-0.576444\pi$
$523$	−2.97573e6	−0.475706	−0.237853	+	0.971301i	$0.576444\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.04429e7	−1.63792
$528$	0	0
$529$	−4.83838e6	−0.751728
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.28834e6	−0.196433
$534$	0	0
$535$	−6.75190e6	−1.01986
$536$	0	0
$537$	0	0
$538$	0	0
$539$	970266.	0.143853
$540$	0	0
$541$	1.80579e6	0.265261	0.132631	−	0.991166i	$-0.457658\pi$
$541$	1.80579e6	0.265261	0.132631	+	0.991166i	$0.457658\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.36935e7	−1.97480
$546$	0	0
$547$	−9.36973e6	−1.33893	−0.669466	−	0.742843i	$-0.733477\pi$
$547$	−9.36973e6	−1.33893	−0.669466	+	0.742843i	$0.733477\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.99317e6	−0.560323
$552$	0	0
$553$	−6.48550e6	−0.901842
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.53733e6	0.483101	0.241550	−	0.970388i	$-0.422344\pi$
$557$	3.53733e6	0.483101	0.241550	+	0.970388i	$0.422344\pi$
$558$	0	0
$559$	−7.95761e6	−1.07709
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.78060e6	0.369716	0.184858	−	0.982765i	$-0.440818\pi$
$563$	2.78060e6	0.369716	0.184858	+	0.982765i	$0.440818\pi$
$564$	0	0
$565$	1.29650e7	1.70864
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.35293e6	0.952094	0.476047	−	0.879420i	$-0.342069\pi$
$569$	7.35293e6	0.952094	0.476047	+	0.879420i	$0.342069\pi$
$570$	0	0
$571$	−5.75171e6	−0.738255	−0.369128	−	0.929379i	$-0.620343\pi$
$571$	−5.75171e6	−0.738255	−0.369128	+	0.929379i	$0.620343\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.14566e7	−1.44506
$576$	0	0
$577$	−1.30488e7	−1.63167	−0.815835	−	0.578285i	$-0.803722\pi$
$577$	−1.30488e7	−1.63167	−0.815835	+	0.578285i	$0.803722\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.42920e6	1.03597
$582$	0	0
$583$	−1.47720e6	−0.179998
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.24724e6	0.149401	0.0747005	−	0.997206i	$-0.476200\pi$
$587$	1.24724e6	0.149401	0.0747005	+	0.997206i	$0.476200\pi$
$588$	0	0
$589$	−1.00188e7	−1.18994
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.32769e7	−1.55045	−0.775226	−	0.631684i	$-0.782364\pi$
$593$	−1.32769e7	−1.55045	−0.775226	+	0.631684i	$0.782364\pi$
$594$	0	0
$595$	1.68324e7	1.94919
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.04914e6	0.461101	0.230551	−	0.973060i	$-0.425947\pi$
$599$	4.04914e6	0.461101	0.230551	+	0.973060i	$0.425947\pi$
$600$	0	0
$601$	−6.83714e6	−0.772126	−0.386063	−	0.922472i	$-0.626165\pi$
$601$	−6.83714e6	−0.772126	−0.386063	+	0.922472i	$0.626165\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.52856e7	−1.69783
$606$	0	0
$607$	696200.	0.0766941	0.0383471	−	0.999264i	$-0.487791\pi$
$607$	696200.	0.0766941	0.0383471	+	0.999264i	$0.487791\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.11904e6	−0.446367
$612$	0	0
$613$	−1.42785e6	−0.153473	−0.0767363	−	0.997051i	$-0.524450\pi$
$613$	−1.42785e6	−0.153473	−0.0767363	+	0.997051i	$0.524450\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.50335e7	−1.58982	−0.794908	−	0.606730i	$-0.792481\pi$
$617$	−1.50335e7	−1.58982	−0.794908	+	0.606730i	$0.792481\pi$
$618$	0	0
$619$	−7.16303e6	−0.751398	−0.375699	−	0.926742i	$-0.622597\pi$
$619$	−7.16303e6	−0.751398	−0.375699	+	0.926742i	$0.622597\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−392889.	−0.0405555
$624$	0	0
$625$	4.40504e7	4.51076
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.19145e7	−1.20074
$630$	0	0
$631$	1.38021e7	1.37998	0.689989	−	0.723820i	$-0.257615\pi$
$631$	1.38021e7	1.37998	0.689989	+	0.723820i	$0.257615\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.29416e6	−0.324198
$636$	0	0
$637$	−4.10167e6	−0.400509
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4.04659e6	−0.388995	−0.194497	−	0.980903i	$-0.562308\pi$
$641$	−4.04659e6	−0.388995	−0.194497	+	0.980903i	$0.562308\pi$
$642$	0	0
$643$	1.08876e7	1.03850	0.519249	−	0.854623i	$-0.326212\pi$
$643$	1.08876e7	1.03850	0.519249	+	0.854623i	$0.326212\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.45797e6	−0.794339	−0.397169	−	0.917745i	$-0.630007\pi$
$647$	−8.45797e6	−0.794339	−0.397169	+	0.917745i	$0.630007\pi$
$648$	0	0
$649$	−7.07788e6	−0.659617
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.95548e6	0.454782	0.227391	−	0.973804i	$-0.426980\pi$
$653$	4.95548e6	0.454782	0.227391	+	0.973804i	$0.426980\pi$
$654$	0	0
$655$	−1.89950e7	−1.72996
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.67430e7	1.50183	0.750915	−	0.660398i	$-0.229613\pi$
$659$	1.67430e7	1.50183	0.750915	+	0.660398i	$0.229613\pi$
$660$	0	0
$661$	−1.50798e7	−1.34243	−0.671217	−	0.741260i	$-0.734228\pi$
$661$	−1.50798e7	−1.34243	−0.671217	+	0.741260i	$0.734228\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.61489e7	1.41608
$666$	0	0
$667$	3.51105e6	0.305578
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.25328e6	−0.107459
$672$	0	0
$673$	1.41419e7	1.20357	0.601784	−	0.798659i	$-0.294457\pi$
$673$	1.41419e7	1.20357	0.601784	+	0.798659i	$0.294457\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.45879e7	1.22327	0.611635	−	0.791140i	$-0.290512\pi$
$677$	1.45879e7	1.22327	0.611635	+	0.791140i	$0.290512\pi$
$678$	0	0
$679$	7.04859e6	0.586715
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.10096e7	−0.903065	−0.451532	−	0.892255i	$-0.649122\pi$
$683$	−1.10096e7	−0.903065	−0.451532	+	0.892255i	$0.649122\pi$
$684$	0	0
$685$	2.10158e7	1.71127
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.24465e6	0.501141
$690$	0	0
$691$	1.10359e7	0.879247	0.439623	−	0.898182i	$-0.355112\pi$
$691$	1.10359e7	0.879247	0.439623	+	0.898182i	$0.355112\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.72116e7	−2.92224
$696$	0	0
$697$	−3.03834e6	−0.236894
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.51294e7	−1.93146	−0.965732	−	0.259542i	$-0.916428\pi$
$701$	−2.51294e7	−1.93146	−0.965732	+	0.259542i	$0.916428\pi$
$702$	0	0
$703$	−1.14306e7	−0.872332
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.58896e6	−0.194795
$708$	0	0
$709$	1.25301e7	0.936137	0.468068	−	0.883692i	$-0.344950\pi$
$709$	1.25301e7	0.936137	0.468068	+	0.883692i	$0.344950\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.80915e6	0.648948
$714$	0	0
$715$	−1.05444e7	−0.771362
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.72246e6	−0.124258	−0.0621292	−	0.998068i	$-0.519789\pi$
$719$	−1.72246e6	−0.124258	−0.0621292	+	0.998068i	$0.519789\pi$
$720$	0	0
$721$	−6.44937e6	−0.462040
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.51724e7	−1.77861
$726$	0	0
$727$	6.98849e6	0.490397	0.245198	−	0.969473i	$-0.421147\pi$
$727$	6.98849e6	0.490397	0.245198	+	0.969473i	$0.421147\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.87667e7	−1.29896
$732$	0	0
$733$	−5.92609e6	−0.407388	−0.203694	−	0.979035i	$-0.565295\pi$
$733$	−5.92609e6	−0.407388	−0.203694	+	0.979035i	$0.565295\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.09140e6	0.0740142
$738$	0	0
$739$	−1.06632e7	−0.718254	−0.359127	−	0.933289i	$-0.616925\pi$
$739$	−1.06632e7	−0.718254	−0.359127	+	0.933289i	$0.616925\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.84986e7	−1.22933	−0.614664	−	0.788789i	$-0.710708\pi$
$743$	−1.84986e7	−1.22933	−0.614664	+	0.788789i	$0.710708\pi$
$744$	0	0
$745$	−6.08286e6	−0.401529
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6.22259e6	−0.405291
$750$	0	0
$751$	−2.61247e7	−1.69025	−0.845127	−	0.534566i	$-0.820475\pi$
$751$	−2.61247e7	−1.69025	−0.845127	+	0.534566i	$0.820475\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.68804e7	2.35466
$756$	0	0
$757$	2.49174e7	1.58039	0.790193	−	0.612858i	$-0.209980\pi$
$757$	2.49174e7	1.58039	0.790193	+	0.612858i	$0.209980\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.12439e7	0.703808	0.351904	−	0.936036i	$-0.385534\pi$
$761$	1.12439e7	0.703808	0.351904	+	0.936036i	$0.385534\pi$
$762$	0	0
$763$	−1.26200e7	−0.784781
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.99208e7	1.83647
$768$	0	0
$769$	8.67461e6	0.528974	0.264487	−	0.964389i	$-0.414797\pi$
$769$	8.67461e6	0.528974	0.264487	+	0.964389i	$0.414797\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.17066e7	1.30660	0.653300	−	0.757099i	$-0.273384\pi$
$773$	2.17066e7	1.30660	0.653300	+	0.757099i	$0.273384\pi$
$774$	0	0
$775$	−6.31571e7	−3.77718
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.91496e6	−0.172103
$780$	0	0
$781$	−538467.	−0.0315887
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.32597e7	−0.767997
$786$	0	0
$787$	7.86429e6	0.452609	0.226304	−	0.974057i	$-0.427336\pi$
$787$	7.86429e6	0.452609	0.226304	+	0.974057i	$0.427336\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.19486e7	0.679009
$792$	0	0
$793$	5.29807e6	0.299181
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.89618e7	1.05739	0.528693	−	0.848813i	$-0.322682\pi$
$797$	1.89618e7	1.05739	0.528693	+	0.848813i	$0.322682\pi$
$798$	0	0
$799$	−9.71406e6	−0.538312
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.72975e6	−0.477764
$804$	0	0
$805$	−1.41991e7	−0.772274
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.09408e7	−1.12492	−0.562459	−	0.826825i	$-0.690145\pi$
$809$	−2.09408e7	−1.12492	−0.562459	+	0.826825i	$0.690145\pi$
$810$	0	0
$811$	3.21466e7	1.71626	0.858130	−	0.513432i	$-0.171626\pi$
$811$	3.21466e7	1.71626	0.858130	+	0.513432i	$0.171626\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.05257e7	2.13716
$816$	0	0
$817$	−1.80046e7	−0.943687
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.97605e7	−1.54093	−0.770463	−	0.637485i	$-0.779975\pi$
$821$	−2.97605e7	−1.54093	−0.770463	+	0.637485i	$0.779975\pi$
$822$	0	0
$823$	−1.65220e7	−0.850284	−0.425142	−	0.905127i	$-0.639776\pi$
$823$	−1.65220e7	−0.850284	−0.425142	+	0.905127i	$0.639776\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.49588e7	1.26900	0.634498	−	0.772925i	$-0.281207\pi$
$827$	2.49588e7	1.26900	0.634498	+	0.772925i	$0.281207\pi$
$828$	0	0
$829$	−2.92909e7	−1.48029	−0.740145	−	0.672447i	$-0.765243\pi$
$829$	−2.92909e7	−1.48029	−0.740145	+	0.672447i	$0.765243\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.67311e6	−0.483007
$834$	0	0
$835$	6.50747e7	3.22995
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.74835e7	1.83838	0.919189	−	0.393817i	$-0.128845\pi$
$839$	3.74835e7	1.83838	0.919189	+	0.393817i	$0.128845\pi$
$840$	0	0
$841$	−1.27967e7	−0.623888
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.58473e6	0.172709
$846$	0	0
$847$	−1.40873e7	−0.674713
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.00505e7	0.475735
$852$	0	0
$853$	1.48271e7	0.697726	0.348863	−	0.937174i	$-0.386568\pi$
$853$	1.48271e7	0.697726	0.348863	+	0.937174i	$0.386568\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.32370e6	0.247606	0.123803	−	0.992307i	$-0.460491\pi$
$857$	5.32370e6	0.247606	0.123803	+	0.992307i	$0.460491\pi$
$858$	0	0
$859$	2.68868e7	1.24324	0.621621	−	0.783318i	$-0.286474\pi$
$859$	2.68868e7	1.24324	0.621621	+	0.783318i	$0.286474\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.21441e6	−0.101212	−0.0506060	−	0.998719i	$-0.516115\pi$
$863$	−2.21441e6	−0.101212	−0.0506060	+	0.998719i	$0.516115\pi$
$864$	0	0
$865$	4.81846e7	2.18961
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.58131e6	0.430403
$870$	0	0
$871$	−4.61374e6	−0.206067
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.66988e7	2.94509
$876$	0	0
$877$	3.24646e7	1.42531	0.712657	−	0.701513i	$-0.247492\pi$
$877$	3.24646e7	1.42531	0.712657	+	0.701513i	$0.247492\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.78418e6	0.337888	0.168944	−	0.985626i	$-0.445964\pi$
$881$	7.78418e6	0.337888	0.168944	+	0.985626i	$0.445964\pi$
$882$	0	0
$883$	1.71890e7	0.741906	0.370953	−	0.928652i	$-0.379031\pi$
$883$	1.71890e7	0.741906	0.370953	+	0.928652i	$0.379031\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.63433e7	−1.55101	−0.775506	−	0.631340i	$-0.782505\pi$
$887$	−3.63433e7	−1.55101	−0.775506	+	0.631340i	$0.782505\pi$
$888$	0	0
$889$	−3.03592e6	−0.128836
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.31958e6	−0.391082
$894$	0	0
$895$	2.66133e7	1.11056
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.93555e7	0.798739
$900$	0	0
$901$	1.47270e7	0.604368
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.99500e6	0.121556
$906$	0	0
$907$	−8.78418e6	−0.354555	−0.177277	−	0.984161i	$-0.556729\pi$
$907$	−8.78418e6	−0.354555	−0.177277	+	0.984161i	$0.556729\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.67006e6	−0.146513	−0.0732566	−	0.997313i	$-0.523339\pi$
$911$	−3.67006e6	−0.146513	−0.0732566	+	0.997313i	$0.523339\pi$
$912$	0	0
$913$	−1.24528e7	−0.494414
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.75060e7	−0.687484
$918$	0	0
$919$	3.38118e7	1.32063	0.660313	−	0.750990i	$-0.270424\pi$
$919$	3.38118e7	1.32063	0.660313	+	0.750990i	$0.270424\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.27630e6	0.0879477
$924$	0	0
$925$	−7.20572e7	−2.76900
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.17835e7	−0.447954	−0.223977	−	0.974594i	$-0.571904\pi$
$929$	−1.17835e7	−0.447954	−0.223977	+	0.974594i	$0.571904\pi$
$930$	0	0
$931$	−9.28029e6	−0.350903
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.48673e7	−0.930250
$936$	0	0
$937$	−3.78550e7	−1.40856	−0.704278	−	0.709924i	$-0.748729\pi$
$937$	−3.78550e7	−1.40856	−0.704278	+	0.709924i	$0.748729\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.56271e7	1.67977	0.839884	−	0.542766i	$-0.182623\pi$
$941$	4.56271e7	1.67977	0.839884	+	0.542766i	$0.182623\pi$
$942$	0	0
$943$	2.56302e6	0.0938582
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.15884e7	1.50694	0.753472	−	0.657480i	$-0.228378\pi$
$947$	4.15884e7	1.50694	0.753472	+	0.657480i	$0.228378\pi$
$948$	0	0
$949$	3.69039e7	1.33017
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.40042e7	−1.92617	−0.963087	−	0.269190i	$-0.913244\pi$
$953$	−5.40042e7	−1.92617	−0.963087	+	0.269190i	$0.913244\pi$
$954$	0	0
$955$	−6.19036e6	−0.219638
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.93683e7	0.680055
$960$	0	0
$961$	1.99334e7	0.696261
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.91347e7	1.35283
$966$	0	0
$967$	1.72935e7	0.594724	0.297362	−	0.954765i	$-0.403893\pi$
$967$	1.72935e7	0.594724	0.297362	+	0.954765i	$0.403893\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.55169e7	1.54926	0.774630	−	0.632415i	$-0.217936\pi$
$971$	4.55169e7	1.54926	0.774630	+	0.632415i	$0.217936\pi$
$972$	0	0
$973$	−3.42944e7	−1.16129
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.59652e6	0.221095	0.110547	−	0.993871i	$-0.464740\pi$
$977$	6.59652e6	0.221095	0.110547	+	0.993871i	$0.464740\pi$
$978$	0	0
$979$	580432.	0.0193551
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.69457e7	−1.21950	−0.609748	−	0.792595i	$-0.708729\pi$
$983$	−3.69457e7	−1.21950	−0.609748	+	0.792595i	$0.708729\pi$
$984$	0	0
$985$	−9.01846e7	−2.96171
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.58308e7	0.514649
$990$	0	0
$991$	−7.56015e6	−0.244538	−0.122269	−	0.992497i	$-0.539017\pi$
$991$	−7.56015e6	−0.244538	−0.122269	+	0.992497i	$0.539017\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.49870e7	3.04163
$996$	0	0
$997$	−1.56153e7	−0.497521	−0.248761	−	0.968565i	$-0.580023\pi$
$997$	−1.56153e7	−0.497521	−0.248761	+	0.968565i	$0.580023\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.6.a.e.1.5		5
3.2	odd	2		324.6.a.d.1.1		5
9.2	odd	6		108.6.e.a.37.5		10
9.4	even	3		36.6.e.a.25.5	yes	10
9.5	odd	6		108.6.e.a.73.5		10
9.7	even	3		36.6.e.a.13.5	✓	10
36.7	odd	6		144.6.i.d.49.1		10
36.11	even	6		432.6.i.d.145.5		10
36.23	even	6		432.6.i.d.289.5		10
36.31	odd	6		144.6.i.d.97.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.e.a.13.5	✓	10	9.7	even	3
36.6.e.a.25.5	yes	10	9.4	even	3
108.6.e.a.37.5		10	9.2	odd	6
108.6.e.a.73.5		10	9.5	odd	6
144.6.i.d.49.1		10	36.7	odd	6
144.6.i.d.97.1		10	36.31	odd	6
324.6.a.d.1.1		5	3.2	odd	2
324.6.a.e.1.5		5	1.1	even	1	trivial
432.6.i.d.145.5		10	36.11	even	6
432.6.i.d.289.5		10	36.23	even	6

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)

\(f(q)\)	\(=\)	\(q+110.399 q^{5} +101.745 q^{7} +O(q^{10})\)	\(q+110.399 q^{5} +101.745 q^{7} -150.312 q^{11} +635.424 q^{13} +1498.54 q^{17} +1437.69 q^{19} -1264.11 q^{23} +9063.00 q^{25} -2777.50 q^{29} -6968.68 q^{31} +11232.5 q^{35} -7950.71 q^{37} -2027.53 q^{41} -12523.3 q^{43} -6482.34 q^{47} -6455.01 q^{49} +9827.54 q^{53} -16594.3 q^{55} +47087.9 q^{59} +8337.84 q^{61} +70150.3 q^{65} -7260.89 q^{67} +3582.33 q^{71} +58077.5 q^{73} -15293.5 q^{77} -63742.8 q^{79} +82846.5 q^{83} +165438. q^{85} -3861.51 q^{89} +64651.0 q^{91} +158720. q^{95} +69277.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 21 q^{5} - 29 q^{7}+O(q^{10}) \) 5 * q + 21 * q^5 - 29 * q^7	\( 5 q + 21 q^{5} - 29 q^{7} - 177 q^{11} + 181 q^{13} + 1140 q^{17} - 416 q^{19} - 399 q^{23} + 4778 q^{25} + 6033 q^{29} - 2759 q^{31} + 18573 q^{35} - 7586 q^{37} + 18435 q^{41} - 1469 q^{43} + 25155 q^{47} + 4056 q^{49} + 58422 q^{53} + 7389 q^{55} + 90537 q^{59} - 1403 q^{61} + 148407 q^{65} - 13907 q^{67} + 114684 q^{71} + 7600 q^{73} + 211983 q^{77} - 29993 q^{79} + 228951 q^{83} + 49662 q^{85} + 299166 q^{89} + 62465 q^{91} + 394764 q^{95} - 40541 q^{97}+O(q^{100}) \) 5 * q + 21 * q^5 - 29 * q^7 - 177 * q^11 + 181 * q^13 + 1140 * q^17 - 416 * q^19 - 399 * q^23 + 4778 * q^25 + 6033 * q^29 - 2759 * q^31 + 18573 * q^35 - 7586 * q^37 + 18435 * q^41 - 1469 * q^43 + 25155 * q^47 + 4056 * q^49 + 58422 * q^53 + 7389 * q^55 + 90537 * q^59 - 1403 * q^61 + 148407 * q^65 - 13907 * q^67 + 114684 * q^71 + 7600 * q^73 + 211983 * q^77 - 29993 * q^79 + 228951 * q^83 + 49662 * q^85 + 299166 * q^89 + 62465 * q^91 + 394764 * q^95 - 40541 * q^97

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