LMFDB - Embedded newform 882.4.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.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:	$52.0396846251$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
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:	no (minimal twist has level 294)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +4.58579 q^{5} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} +4.58579 q^{5} -8.00000 q^{8} -9.17157 q^{10} +6.48528 q^{11} -45.2132 q^{13} +16.0000 q^{16} +81.5563 q^{17} +5.05382 q^{19} +18.3431 q^{20} -12.9706 q^{22} -106.250 q^{23} -103.971 q^{25} +90.4264 q^{26} +268.132 q^{29} -292.368 q^{31} -32.0000 q^{32} -163.113 q^{34} +114.558 q^{37} -10.1076 q^{38} -36.6863 q^{40} -161.605 q^{41} -471.294 q^{43} +25.9411 q^{44} +212.500 q^{46} +346.004 q^{47} +207.941 q^{50} -180.853 q^{52} -405.529 q^{53} +29.7401 q^{55} -536.264 q^{58} -253.436 q^{59} +751.217 q^{61} +584.735 q^{62} +64.0000 q^{64} -207.338 q^{65} +11.6468 q^{67} +326.225 q^{68} +681.661 q^{71} -685.457 q^{73} -229.117 q^{74} +20.2153 q^{76} +0.264069 q^{79} +73.3726 q^{80} +323.210 q^{82} -437.137 q^{83} +374.000 q^{85} +942.587 q^{86} -51.8823 q^{88} -58.5126 q^{89} -424.999 q^{92} -692.008 q^{94} +23.1758 q^{95} +1280.09 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} + 12 q^{5} - 16 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 + 8 * q^4 + 12 * q^5 - 16 * q^8	$ 2 q - 4 q^{2} + 8 q^{4} + 12 q^{5} - 16 q^{8} - 24 q^{10} - 4 q^{11} - 48 q^{13} + 32 q^{16} + 132 q^{17} - 120 q^{19} + 48 q^{20} + 8 q^{22} + 76 q^{23} - 174 q^{25} + 96 q^{26} + 112 q^{29} - 432 q^{31} - 64 q^{32} - 264 q^{34} - 280 q^{37} + 240 q^{38} - 96 q^{40} + 36 q^{41} - 128 q^{43} - 16 q^{44} - 152 q^{46} - 264 q^{47} + 348 q^{50} - 192 q^{52} - 268 q^{53} - 48 q^{55} - 224 q^{58} + 336 q^{59} + 504 q^{61} + 864 q^{62} + 128 q^{64} - 228 q^{65} - 384 q^{67} + 528 q^{68} + 396 q^{71} + 312 q^{73} + 560 q^{74} - 480 q^{76} - 848 q^{79} + 192 q^{80} - 72 q^{82} - 648 q^{83} + 748 q^{85} + 256 q^{86} + 32 q^{88} - 612 q^{89} + 304 q^{92} + 528 q^{94} - 904 q^{95} + 2184 q^{97}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 + 12 * q^5 - 16 * q^8 - 24 * q^10 - 4 * q^11 - 48 * q^13 + 32 * q^16 + 132 * q^17 - 120 * q^19 + 48 * q^20 + 8 * q^22 + 76 * q^23 - 174 * q^25 + 96 * q^26 + 112 * q^29 - 432 * q^31 - 64 * q^32 - 264 * q^34 - 280 * q^37 + 240 * q^38 - 96 * q^40 + 36 * q^41 - 128 * q^43 - 16 * q^44 - 152 * q^46 - 264 * q^47 + 348 * q^50 - 192 * q^52 - 268 * q^53 - 48 * q^55 - 224 * q^58 + 336 * q^59 + 504 * q^61 + 864 * q^62 + 128 * q^64 - 228 * q^65 - 384 * q^67 + 528 * q^68 + 396 * q^71 + 312 * q^73 + 560 * q^74 - 480 * q^76 - 848 * q^79 + 192 * q^80 - 72 * q^82 - 648 * q^83 + 748 * q^85 + 256 * q^86 + 32 * q^88 - 612 * q^89 + 304 * q^92 + 528 * q^94 - 904 * q^95 + 2184 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	4.58579	0.410165	0.205083	−	0.978745i	$-0.434254\pi$
$5$	4.58579	0.410165	0.205083	+	0.978745i	$0.434254\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−9.17157	−0.290031
$11$	6.48528	0.177762	0.0888812	−	0.996042i	$-0.471671\pi$
$11$	6.48528	0.177762	0.0888812	+	0.996042i	$0.471671\pi$
$12$	0	0
$13$	−45.2132	−0.964607	−0.482303	−	0.876004i	$-0.660200\pi$
$13$	−45.2132	−0.964607	−0.482303	+	0.876004i	$0.660200\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	81.5563	1.16355	0.581774	−	0.813350i	$-0.302359\pi$
$17$	81.5563	1.16355	0.581774	+	0.813350i	$0.302359\pi$
$18$	0	0
$19$	5.05382	0.0610225	0.0305112	−	0.999534i	$-0.490286\pi$
$19$	5.05382	0.0610225	0.0305112	+	0.999534i	$0.490286\pi$
$20$	18.3431	0.205083
$21$	0	0
$22$	−12.9706	−0.125697
$23$	−106.250	−0.963244	−0.481622	−	0.876379i	$-0.659952\pi$
$23$	−106.250	−0.963244	−0.481622	+	0.876379i	$0.659952\pi$
$24$	0	0
$25$	−103.971	−0.831765
$26$	90.4264	0.682080
$27$	0	0
$28$	0	0
$29$	268.132	1.71693	0.858463	−	0.512875i	$-0.171420\pi$
$29$	268.132	1.71693	0.858463	+	0.512875i	$0.171420\pi$
$30$	0	0
$31$	−292.368	−1.69390	−0.846948	−	0.531676i	$-0.821562\pi$
$31$	−292.368	−1.69390	−0.846948	+	0.531676i	$0.821562\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−163.113	−0.822753
$35$	0	0
$36$	0	0
$37$	114.558	0.509008	0.254504	−	0.967072i	$-0.418088\pi$
$37$	114.558	0.509008	0.254504	+	0.967072i	$0.418088\pi$
$38$	−10.1076	−0.0431494
$39$	0	0
$40$	−36.6863	−0.145015
$41$	−161.605	−0.615573	−0.307786	−	0.951456i	$-0.599588\pi$
$41$	−161.605	−0.615573	−0.307786	+	0.951456i	$0.599588\pi$
$42$	0	0
$43$	−471.294	−1.67143	−0.835716	−	0.549162i	$-0.814947\pi$
$43$	−471.294	−1.67143	−0.835716	+	0.549162i	$0.814947\pi$
$44$	25.9411	0.0888812
$45$	0	0
$46$	212.500	0.681116
$47$	346.004	1.07383	0.536914	−	0.843637i	$-0.319590\pi$
$47$	346.004	1.07383	0.536914	+	0.843637i	$0.319590\pi$
$48$	0	0
$49$	0	0
$50$	207.941	0.588146
$51$	0	0
$52$	−180.853	−0.482303
$53$	−405.529	−1.05101	−0.525507	−	0.850790i	$-0.676124\pi$
$53$	−405.529	−1.05101	−0.525507	+	0.850790i	$0.676124\pi$
$54$	0	0
$55$	29.7401	0.0729119
$56$	0	0
$57$	0	0
$58$	−536.264	−1.21405
$59$	−253.436	−0.559229	−0.279614	−	0.960112i	$-0.590207\pi$
$59$	−253.436	−0.559229	−0.279614	+	0.960112i	$0.590207\pi$
$60$	0	0
$61$	751.217	1.57678	0.788390	−	0.615176i	$-0.210915\pi$
$61$	751.217	1.57678	0.788390	+	0.615176i	$0.210915\pi$
$62$	584.735	1.19776
$63$	0	0
$64$	64.0000	0.125000
$65$	−207.338	−0.395648
$66$	0	0
$67$	11.6468	0.0212370	0.0106185	−	0.999944i	$-0.496620\pi$
$67$	11.6468	0.0212370	0.0106185	+	0.999944i	$0.496620\pi$
$68$	326.225	0.581774
$69$	0	0
$70$	0	0
$71$	681.661	1.13941	0.569706	−	0.821848i	$-0.307057\pi$
$71$	681.661	1.13941	0.569706	+	0.821848i	$0.307057\pi$
$72$	0	0
$73$	−685.457	−1.09900	−0.549498	−	0.835495i	$-0.685181\pi$
$73$	−685.457	−1.09900	−0.549498	+	0.835495i	$0.685181\pi$
$74$	−229.117	−0.359923
$75$	0	0
$76$	20.2153	0.0305112
$77$	0	0
$78$	0	0
$79$	0.264069	0.000376077 0	0.000188038	−	1.00000i	$-0.499940\pi$
$79$	0.264069	0.000376077 0	0.000188038		1.00000i	$0.499940\pi$
$80$	73.3726	0.102541
$81$	0	0
$82$	323.210	0.435276
$83$	−437.137	−0.578097	−0.289048	−	0.957314i	$-0.593339\pi$
$83$	−437.137	−0.578097	−0.289048	+	0.957314i	$0.593339\pi$
$84$	0	0
$85$	374.000	0.477247
$86$	942.587	1.18188
$87$	0	0
$88$	−51.8823	−0.0628485
$89$	−58.5126	−0.0696891	−0.0348445	−	0.999393i	$-0.511094\pi$
$89$	−58.5126	−0.0696891	−0.0348445	+	0.999393i	$0.511094\pi$
$90$	0	0
$91$	0	0
$92$	−424.999	−0.481622
$93$	0	0
$94$	−692.008	−0.759311
$95$	23.1758	0.0250293
$96$	0	0
$97$	1280.09	1.33993	0.669966	−	0.742391i	$-0.266308\pi$
$97$	1280.09	1.33993	0.669966	+	0.742391i	$0.266308\pi$
$98$	0	0
$99$	0	0
$100$	−415.882	−0.415882
$101$	−1306.39	−1.28704	−0.643518	−	0.765431i	$-0.722526\pi$
$101$	−1306.39	−1.28704	−0.643518	+	0.765431i	$0.722526\pi$
$102$	0	0
$103$	758.975	0.726058	0.363029	−	0.931778i	$-0.381743\pi$
$103$	758.975	0.726058	0.363029	+	0.931778i	$0.381743\pi$
$104$	361.706	0.341040
$105$	0	0
$106$	811.058	0.743178
$107$	−1262.51	−1.14067	−0.570336	−	0.821412i	$-0.693187\pi$
$107$	−1262.51	−1.14067	−0.570336	+	0.821412i	$0.693187\pi$
$108$	0	0
$109$	−2105.53	−1.85021	−0.925105	−	0.379711i	$-0.876023\pi$
$109$	−2105.53	−1.85021	−0.925105	+	0.379711i	$0.876023\pi$
$110$	−59.4802	−0.0515565
$111$	0	0
$112$	0	0
$113$	−1535.76	−1.27852	−0.639258	−	0.768992i	$-0.720759\pi$
$113$	−1535.76	−1.27852	−0.639258	+	0.768992i	$0.720759\pi$
$114$	0	0
$115$	−487.239	−0.395089
$116$	1072.53	0.858463
$117$	0	0
$118$	506.871	0.395435
$119$	0	0
$120$	0	0
$121$	−1288.94	−0.968401
$122$	−1502.43	−1.11495
$123$	0	0
$124$	−1169.47	−0.846948
$125$	−1050.01	−0.751326
$126$	0	0
$127$	24.1749	0.0168911	0.00844557	−	0.999964i	$-0.497312\pi$
$127$	24.1749	0.0168911	0.00844557	+	0.999964i	$0.497312\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	414.676	0.279765
$131$	1581.53	1.05480	0.527400	−	0.849617i	$-0.323167\pi$
$131$	1581.53	1.05480	0.527400	+	0.849617i	$0.323167\pi$
$132$	0	0
$133$	0	0
$134$	−23.2935	−0.0150168
$135$	0	0
$136$	−652.451	−0.411376
$137$	745.188	0.464713	0.232357	−	0.972631i	$-0.425356\pi$
$137$	745.188	0.464713	0.232357	+	0.972631i	$0.425356\pi$
$138$	0	0
$139$	1373.60	0.838179	0.419090	−	0.907945i	$-0.362349\pi$
$139$	1373.60	0.838179	0.419090	+	0.907945i	$0.362349\pi$
$140$	0	0
$141$	0	0
$142$	−1363.32	−0.805686
$143$	−293.220	−0.171471
$144$	0	0
$145$	1229.60	0.704224
$146$	1370.91	0.777107
$147$	0	0
$148$	458.234	0.254504
$149$	−620.530	−0.341180	−0.170590	−	0.985342i	$-0.554567\pi$
$149$	−620.530	−0.341180	−0.170590	+	0.985342i	$0.554567\pi$
$150$	0	0
$151$	−1939.26	−1.04513	−0.522567	−	0.852598i	$-0.675025\pi$
$151$	−1939.26	−1.04513	−0.522567	+	0.852598i	$0.675025\pi$
$152$	−40.4306	−0.0215747
$153$	0	0
$154$	0	0
$155$	−1340.74	−0.694777
$156$	0	0
$157$	412.843	0.209863	0.104931	−	0.994479i	$-0.466538\pi$
$157$	412.843	0.209863	0.104931	+	0.994479i	$0.466538\pi$
$158$	−0.528137	−0.000265926 0
$159$	0	0
$160$	−146.745	−0.0725077
$161$	0	0
$162$	0	0
$163$	−3907.44	−1.87763	−0.938817	−	0.344417i	$-0.888077\pi$
$163$	−3907.44	−1.87763	−0.938817	+	0.344417i	$0.888077\pi$
$164$	−646.420	−0.307786
$165$	0	0
$166$	874.274	0.408776
$167$	−1286.41	−0.596082	−0.298041	−	0.954553i	$-0.596333\pi$
$167$	−1286.41	−0.596082	−0.298041	+	0.954553i	$0.596333\pi$
$168$	0	0
$169$	−152.766	−0.0695340
$170$	−748.000	−0.337465
$171$	0	0
$172$	−1885.17	−0.835716
$173$	−1251.26	−0.549892	−0.274946	−	0.961460i	$-0.588660\pi$
$173$	−1251.26	−0.549892	−0.274946	+	0.961460i	$0.588660\pi$
$174$	0	0
$175$	0	0
$176$	103.765	0.0444406
$177$	0	0
$178$	117.025	0.0492776
$179$	−3623.51	−1.51304	−0.756520	−	0.653971i	$-0.773102\pi$
$179$	−3623.51	−1.51304	−0.756520	+	0.653971i	$0.773102\pi$
$180$	0	0
$181$	−181.727	−0.0746280	−0.0373140	−	0.999304i	$-0.511880\pi$
$181$	−181.727	−0.0746280	−0.0373140	+	0.999304i	$0.511880\pi$
$182$	0	0
$183$	0	0
$184$	849.998	0.340558
$185$	525.341	0.208777
$186$	0	0
$187$	528.916	0.206835
$188$	1384.02	0.536914
$189$	0	0
$190$	−46.3515	−0.0176984
$191$	−1481.28	−0.561160	−0.280580	−	0.959831i	$-0.590527\pi$
$191$	−1481.28	−0.561160	−0.280580	+	0.959831i	$0.590527\pi$
$192$	0	0
$193$	−356.708	−0.133038	−0.0665192	−	0.997785i	$-0.521189\pi$
$193$	−356.708	−0.133038	−0.0665192	+	0.997785i	$0.521189\pi$
$194$	−2560.18	−0.947476
$195$	0	0
$196$	0	0
$197$	−4890.53	−1.76871	−0.884355	−	0.466816i	$-0.845401\pi$
$197$	−4890.53	−1.76871	−0.884355	+	0.466816i	$0.845401\pi$
$198$	0	0
$199$	−3542.85	−1.26204	−0.631020	−	0.775766i	$-0.717364\pi$
$199$	−3542.85	−1.26204	−0.631020	+	0.775766i	$0.717364\pi$
$200$	831.765	0.294073
$201$	0	0
$202$	2612.78	0.910071
$203$	0	0
$204$	0	0
$205$	−741.087	−0.252486
$206$	−1517.95	−0.513401
$207$	0	0
$208$	−723.411	−0.241152
$209$	32.7755	0.0108475
$210$	0	0
$211$	−4289.50	−1.39953	−0.699765	−	0.714373i	$-0.746712\pi$
$211$	−4289.50	−1.39953	−0.699765	+	0.714373i	$0.746712\pi$
$212$	−1622.12	−0.525507
$213$	0	0
$214$	2525.03	0.806576
$215$	−2161.25	−0.685563
$216$	0	0
$217$	0	0
$218$	4211.05	1.30830
$219$	0	0
$220$	118.960	0.0364560
$221$	−3687.42	−1.12237
$222$	0	0
$223$	−5795.73	−1.74041	−0.870204	−	0.492692i	$-0.836013\pi$
$223$	−5795.73	−1.74041	−0.870204	+	0.492692i	$0.836013\pi$
$224$	0	0
$225$	0	0
$226$	3071.53	0.904048
$227$	4104.04	1.19998	0.599989	−	0.800008i	$-0.295172\pi$
$227$	4104.04	1.19998	0.599989	+	0.800008i	$0.295172\pi$
$228$	0	0
$229$	1296.83	0.374223	0.187111	−	0.982339i	$-0.440087\pi$
$229$	1296.83	0.374223	0.187111	+	0.982339i	$0.440087\pi$
$230$	974.478	0.279370
$231$	0	0
$232$	−2145.06	−0.607025
$233$	1478.33	0.415660	0.207830	−	0.978165i	$-0.433360\pi$
$233$	1478.33	0.415660	0.207830	+	0.978165i	$0.433360\pi$
$234$	0	0
$235$	1586.70	0.440447
$236$	−1013.74	−0.279614
$237$	0	0
$238$	0	0
$239$	3776.92	1.02221	0.511106	−	0.859518i	$-0.329236\pi$
$239$	3776.92	1.02221	0.511106	+	0.859518i	$0.329236\pi$
$240$	0	0
$241$	3996.38	1.06817	0.534086	−	0.845430i	$-0.320656\pi$
$241$	3996.38	1.06817	0.534086	+	0.845430i	$0.320656\pi$
$242$	2577.88	0.684763
$243$	0	0
$244$	3004.87	0.788390
$245$	0	0
$246$	0	0
$247$	−228.500	−0.0588627
$248$	2338.94	0.598882
$249$	0	0
$250$	2100.02	0.531268
$251$	5423.58	1.36388	0.681939	−	0.731409i	$-0.261137\pi$
$251$	5423.58	1.36388	0.681939	+	0.731409i	$0.261137\pi$
$252$	0	0
$253$	−689.060	−0.171229
$254$	−48.3498	−0.0119438
$255$	0	0
$256$	256.000	0.0625000
$257$	5964.22	1.44762	0.723809	−	0.690000i	$-0.242389\pi$
$257$	5964.22	1.44762	0.723809	+	0.690000i	$0.242389\pi$
$258$	0	0
$259$	0	0
$260$	−829.352	−0.197824
$261$	0	0
$262$	−3163.05	−0.745856
$263$	−5166.01	−1.21122	−0.605608	−	0.795763i	$-0.707070\pi$
$263$	−5166.01	−1.21122	−0.605608	+	0.795763i	$0.707070\pi$
$264$	0	0
$265$	−1859.67	−0.431089
$266$	0	0
$267$	0	0
$268$	46.5870	0.0106185
$269$	3883.29	0.880180	0.440090	−	0.897954i	$-0.354946\pi$
$269$	3883.29	0.880180	0.440090	+	0.897954i	$0.354946\pi$
$270$	0	0
$271$	−5527.65	−1.23904	−0.619522	−	0.784979i	$-0.712674\pi$
$271$	−5527.65	−1.23904	−0.619522	+	0.784979i	$0.712674\pi$
$272$	1304.90	0.290887
$273$	0	0
$274$	−1490.38	−0.328602
$275$	−674.278	−0.147856
$276$	0	0
$277$	2268.12	0.491979	0.245989	−	0.969273i	$-0.420887\pi$
$277$	2268.12	0.491979	0.245989	+	0.969273i	$0.420887\pi$
$278$	−2747.19	−0.592682
$279$	0	0
$280$	0	0
$281$	−725.656	−0.154053	−0.0770267	−	0.997029i	$-0.524543\pi$
$281$	−725.656	−0.154053	−0.0770267	+	0.997029i	$0.524543\pi$
$282$	0	0
$283$	−4237.00	−0.889976	−0.444988	−	0.895536i	$-0.646792\pi$
$283$	−4237.00	−0.889976	−0.444988	+	0.895536i	$0.646792\pi$
$284$	2726.64	0.569706
$285$	0	0
$286$	586.441	0.121248
$287$	0	0
$288$	0	0
$289$	1738.44	0.353845
$290$	−2459.19	−0.497961
$291$	0	0
$292$	−2741.83	−0.549498
$293$	−4373.78	−0.872079	−0.436039	−	0.899928i	$-0.643619\pi$
$293$	−4373.78	−0.872079	−0.436039	+	0.899928i	$0.643619\pi$
$294$	0	0
$295$	−1162.20	−0.229376
$296$	−916.468	−0.179961
$297$	0	0
$298$	1241.06	0.241251
$299$	4803.89	0.929152
$300$	0	0
$301$	0	0
$302$	3878.53	0.739021
$303$	0	0
$304$	80.8612	0.0152556
$305$	3444.92	0.646740
$306$	0	0
$307$	−4133.47	−0.768435	−0.384217	−	0.923243i	$-0.625529\pi$
$307$	−4133.47	−0.768435	−0.384217	+	0.923243i	$0.625529\pi$
$308$	0	0
$309$	0	0
$310$	2681.47	0.491282
$311$	5063.23	0.923182	0.461591	−	0.887093i	$-0.347279\pi$
$311$	5063.23	0.923182	0.461591	+	0.887093i	$0.347279\pi$
$312$	0	0
$313$	−7411.56	−1.33842	−0.669211	−	0.743073i	$-0.733368\pi$
$313$	−7411.56	−1.33842	−0.669211	+	0.743073i	$0.733368\pi$
$314$	−825.686	−0.148395
$315$	0	0
$316$	1.05627	0.000188038 0
$317$	6737.05	1.19366	0.596831	−	0.802367i	$-0.296426\pi$
$317$	6737.05	1.19366	0.596831	+	0.802367i	$0.296426\pi$
$318$	0	0
$319$	1738.91	0.305205
$320$	293.490	0.0512707
$321$	0	0
$322$	0	0
$323$	412.171	0.0710026
$324$	0	0
$325$	4700.84	0.802326
$326$	7814.88	1.32769
$327$	0	0
$328$	1292.84	0.217638
$329$	0	0
$330$	0	0
$331$	11175.9	1.85585	0.927923	−	0.372771i	$-0.121592\pi$
$331$	11175.9	1.85585	0.927923	+	0.372771i	$0.121592\pi$
$332$	−1748.55	−0.289048
$333$	0	0
$334$	2572.83	0.421494
$335$	53.4095	0.00871067
$336$	0	0
$337$	9379.78	1.51617	0.758085	−	0.652156i	$-0.226135\pi$
$337$	9379.78	1.51617	0.758085	+	0.652156i	$0.226135\pi$
$338$	305.532	0.0491680
$339$	0	0
$340$	1496.00	0.238624
$341$	−1896.09	−0.301111
$342$	0	0
$343$	0	0
$344$	3770.35	0.590941
$345$	0	0
$346$	2502.51	0.388832
$347$	−5681.46	−0.878953	−0.439476	−	0.898254i	$-0.644836\pi$
$347$	−5681.46	−0.878953	−0.439476	+	0.898254i	$0.644836\pi$
$348$	0	0
$349$	704.250	0.108016	0.0540080	−	0.998541i	$-0.482800\pi$
$349$	704.250	0.108016	0.0540080	+	0.998541i	$0.482800\pi$
$350$	0	0
$351$	0	0
$352$	−207.529	−0.0314242
$353$	−4284.96	−0.646078	−0.323039	−	0.946386i	$-0.604705\pi$
$353$	−4284.96	−0.646078	−0.323039	+	0.946386i	$0.604705\pi$
$354$	0	0
$355$	3125.95	0.467347
$356$	−234.051	−0.0348445
$357$	0	0
$358$	7247.03	1.06988
$359$	−4661.27	−0.685272	−0.342636	−	0.939468i	$-0.611320\pi$
$359$	−4661.27	−0.685272	−0.342636	+	0.939468i	$0.611320\pi$
$360$	0	0
$361$	−6833.46	−0.996276
$362$	363.454	0.0527700
$363$	0	0
$364$	0	0
$365$	−3143.36	−0.450770
$366$	0	0
$367$	6935.30	0.986430	0.493215	−	0.869907i	$-0.335822\pi$
$367$	6935.30	0.986430	0.493215	+	0.869907i	$0.335822\pi$
$368$	−1700.00	−0.240811
$369$	0	0
$370$	−1050.68	−0.147628
$371$	0	0
$372$	0	0
$373$	−3081.10	−0.427704	−0.213852	−	0.976866i	$-0.568601\pi$
$373$	−3081.10	−0.427704	−0.213852	+	0.976866i	$0.568601\pi$
$374$	−1057.83	−0.146254
$375$	0	0
$376$	−2768.03	−0.379655
$377$	−12123.1	−1.65616
$378$	0	0
$379$	941.827	0.127647	0.0638237	−	0.997961i	$-0.479670\pi$
$379$	941.827	0.127647	0.0638237	+	0.997961i	$0.479670\pi$
$380$	92.7030	0.0125146
$381$	0	0
$382$	2962.56	0.396800
$383$	−677.526	−0.0903915	−0.0451958	−	0.998978i	$-0.514391\pi$
$383$	−677.526	−0.0903915	−0.0451958	+	0.998978i	$0.514391\pi$
$384$	0	0
$385$	0	0
$386$	713.416	0.0940724
$387$	0	0
$388$	5120.36	0.669966
$389$	11865.4	1.54653	0.773266	−	0.634082i	$-0.218622\pi$
$389$	11865.4	1.54653	0.773266	+	0.634082i	$0.218622\pi$
$390$	0	0
$391$	−8665.34	−1.12078
$392$	0	0
$393$	0	0
$394$	9781.06	1.25067
$395$	1.21096	0.000154254 0
$396$	0	0
$397$	5140.76	0.649893	0.324947	−	0.945732i	$-0.394654\pi$
$397$	5140.76	0.649893	0.324947	+	0.945732i	$0.394654\pi$
$398$	7085.70	0.892397
$399$	0	0
$400$	−1663.53	−0.207941
$401$	12382.0	1.54196	0.770981	−	0.636858i	$-0.219766\pi$
$401$	12382.0	1.54196	0.770981	+	0.636858i	$0.219766\pi$
$402$	0	0
$403$	13218.9	1.63394
$404$	−5225.56	−0.643518
$405$	0	0
$406$	0	0
$407$	742.944	0.0904824
$408$	0	0
$409$	15875.6	1.91931	0.959657	−	0.281173i	$-0.0907235\pi$
$409$	15875.6	1.91931	0.959657	+	0.281173i	$0.0907235\pi$
$410$	1482.17	0.178535
$411$	0	0
$412$	3035.90	0.363029
$413$	0	0
$414$	0	0
$415$	−2004.62	−0.237115
$416$	1446.82	0.170520
$417$	0	0
$418$	−65.5509	−0.00767034
$419$	−16111.9	−1.87857	−0.939283	−	0.343145i	$-0.888508\pi$
$419$	−16111.9	−1.87857	−0.939283	+	0.343145i	$0.888508\pi$
$420$	0	0
$421$	8691.58	1.00618	0.503090	−	0.864234i	$-0.332197\pi$
$421$	8691.58	1.00618	0.503090	+	0.864234i	$0.332197\pi$
$422$	8578.99	0.989618
$423$	0	0
$424$	3244.23	0.371589
$425$	−8479.46	−0.967798
$426$	0	0
$427$	0	0
$428$	−5050.06	−0.570336
$429$	0	0
$430$	4322.50	0.484766
$431$	4195.10	0.468842	0.234421	−	0.972135i	$-0.424681\pi$
$431$	4195.10	0.468842	0.234421	+	0.972135i	$0.424681\pi$
$432$	0	0
$433$	5426.54	0.602270	0.301135	−	0.953582i	$-0.402635\pi$
$433$	5426.54	0.602270	0.301135	+	0.953582i	$0.402635\pi$
$434$	0	0
$435$	0	0
$436$	−8422.11	−0.925105
$437$	−536.968	−0.0587795
$438$	0	0
$439$	3771.24	0.410003	0.205002	−	0.978762i	$-0.434280\pi$
$439$	3771.24	0.410003	0.205002	+	0.978762i	$0.434280\pi$
$440$	−237.921	−0.0257783
$441$	0	0
$442$	7374.85	0.793633
$443$	5930.30	0.636020	0.318010	−	0.948087i	$-0.396985\pi$
$443$	5930.30	0.636020	0.318010	+	0.948087i	$0.396985\pi$
$444$	0	0
$445$	−268.326	−0.0285840
$446$	11591.5	1.23065
$447$	0	0
$448$	0	0
$449$	529.065	0.0556083	0.0278041	−	0.999613i	$-0.491149\pi$
$449$	529.065	0.0556083	0.0278041	+	0.999613i	$0.491149\pi$
$450$	0	0
$451$	−1048.05	−0.109426
$452$	−6143.05	−0.639258
$453$	0	0
$454$	−8208.09	−0.848512
$455$	0	0
$456$	0	0
$457$	10057.2	1.02944	0.514721	−	0.857358i	$-0.327896\pi$
$457$	10057.2	1.02944	0.514721	+	0.857358i	$0.327896\pi$
$458$	−2593.66	−0.264616
$459$	0	0
$460$	−1948.96	−0.197545
$461$	−5010.31	−0.506190	−0.253095	−	0.967441i	$-0.581448\pi$
$461$	−5010.31	−0.506190	−0.253095	+	0.967441i	$0.581448\pi$
$462$	0	0
$463$	−7124.38	−0.715114	−0.357557	−	0.933891i	$-0.616390\pi$
$463$	−7124.38	−0.715114	−0.357557	+	0.933891i	$0.616390\pi$
$464$	4290.11	0.429232
$465$	0	0
$466$	−2956.67	−0.293916
$467$	7501.44	0.743309	0.371654	−	0.928371i	$-0.378791\pi$
$467$	7501.44	0.743309	0.371654	+	0.928371i	$0.378791\pi$
$468$	0	0
$469$	0	0
$470$	−3173.40	−0.311443
$471$	0	0
$472$	2027.49	0.197717
$473$	−3056.47	−0.297118
$474$	0	0
$475$	−525.449	−0.0507563
$476$	0	0
$477$	0	0
$478$	−7553.84	−0.722813
$479$	−8173.80	−0.779688	−0.389844	−	0.920881i	$-0.627471\pi$
$479$	−8173.80	−0.779688	−0.389844	+	0.920881i	$0.627471\pi$
$480$	0	0
$481$	−5179.55	−0.490992
$482$	−7992.76	−0.755312
$483$	0	0
$484$	−5155.76	−0.484200
$485$	5870.22	0.549594
$486$	0	0
$487$	−11968.8	−1.11367	−0.556835	−	0.830623i	$-0.687985\pi$
$487$	−11968.8	−1.11367	−0.556835	+	0.830623i	$0.687985\pi$
$488$	−6009.74	−0.557476
$489$	0	0
$490$	0	0
$491$	−2079.96	−0.191176	−0.0955878	−	0.995421i	$-0.530473\pi$
$491$	−2079.96	−0.191176	−0.0955878	+	0.995421i	$0.530473\pi$
$492$	0	0
$493$	21867.9	1.99773
$494$	456.999	0.0416222
$495$	0	0
$496$	−4677.88	−0.423474
$497$	0	0
$498$	0	0
$499$	12834.4	1.15140	0.575699	−	0.817662i	$-0.304730\pi$
$499$	12834.4	1.15140	0.575699	+	0.817662i	$0.304730\pi$
$500$	−4200.04	−0.375663
$501$	0	0
$502$	−10847.2	−0.964408
$503$	16808.8	1.48999	0.744997	−	0.667068i	$-0.232451\pi$
$503$	16808.8	1.48999	0.744997	+	0.667068i	$0.232451\pi$
$504$	0	0
$505$	−5990.82	−0.527897
$506$	1378.12	0.121077
$507$	0	0
$508$	96.6996	0.00844557
$509$	−4270.26	−0.371859	−0.185929	−	0.982563i	$-0.559530\pi$
$509$	−4270.26	−0.371859	−0.185929	+	0.982563i	$0.559530\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−11928.4	−1.02362
$515$	3480.50	0.297804
$516$	0	0
$517$	2243.93	0.190886
$518$	0	0
$519$	0	0
$520$	1658.70	0.139883
$521$	15283.3	1.28517	0.642584	−	0.766215i	$-0.277862\pi$
$521$	15283.3	1.28517	0.642584	+	0.766215i	$0.277862\pi$
$522$	0	0
$523$	4499.33	0.376180	0.188090	−	0.982152i	$-0.439770\pi$
$523$	4499.33	0.376180	0.188090	+	0.982152i	$0.439770\pi$
$524$	6326.11	0.527400
$525$	0	0
$526$	10332.0	0.856459
$527$	−23844.4	−1.97093
$528$	0	0
$529$	−877.984	−0.0721611
$530$	3719.34	0.304826
$531$	0	0
$532$	0	0
$533$	7306.69	0.593785
$534$	0	0
$535$	−5789.62	−0.467864
$536$	−93.1740	−0.00750840
$537$	0	0
$538$	−7766.59	−0.622382
$539$	0	0
$540$	0	0
$541$	3970.82	0.315561	0.157781	−	0.987474i	$-0.449566\pi$
$541$	3970.82	0.315561	0.157781	+	0.987474i	$0.449566\pi$
$542$	11055.3	0.876137
$543$	0	0
$544$	−2609.80	−0.205688
$545$	−9655.50	−0.758892
$546$	0	0
$547$	−2703.90	−0.211353	−0.105677	−	0.994401i	$-0.533701\pi$
$547$	−2703.90	−0.211353	−0.105677	+	0.994401i	$0.533701\pi$
$548$	2980.75	0.232357
$549$	0	0
$550$	1348.56	0.104550
$551$	1355.09	0.104771
$552$	0	0
$553$	0	0
$554$	−4536.24	−0.347881
$555$	0	0
$556$	5494.38	0.419090
$557$	−790.824	−0.0601585	−0.0300793	−	0.999548i	$-0.509576\pi$
$557$	−790.824	−0.0601585	−0.0300793	+	0.999548i	$0.509576\pi$
$558$	0	0
$559$	21308.7	1.61227
$560$	0	0
$561$	0	0
$562$	1451.31	0.108932
$563$	7517.15	0.562718	0.281359	−	0.959603i	$-0.409215\pi$
$563$	7517.15	0.562718	0.281359	+	0.959603i	$0.409215\pi$
$564$	0	0
$565$	−7042.68	−0.524403
$566$	8473.99	0.629308
$567$	0	0
$568$	−5453.29	−0.402843
$569$	−13945.4	−1.02746	−0.513728	−	0.857953i	$-0.671736\pi$
$569$	−13945.4	−1.02746	−0.513728	+	0.857953i	$0.671736\pi$
$570$	0	0
$571$	−2118.49	−0.155265	−0.0776323	−	0.996982i	$-0.524736\pi$
$571$	−2118.49	−0.155265	−0.0776323	+	0.996982i	$0.524736\pi$
$572$	−1172.88	−0.0857354
$573$	0	0
$574$	0	0
$575$	11046.8	0.801192
$576$	0	0
$577$	−22857.7	−1.64918	−0.824592	−	0.565728i	$-0.808595\pi$
$577$	−22857.7	−1.64918	−0.824592	+	0.565728i	$0.808595\pi$
$578$	−3476.88	−0.250206
$579$	0	0
$580$	4918.38	0.352112
$581$	0	0
$582$	0	0
$583$	−2629.97	−0.186831
$584$	5483.66	0.388554
$585$	0	0
$586$	8747.56	0.616653
$587$	23955.3	1.68440	0.842199	−	0.539167i	$-0.181261\pi$
$587$	23955.3	1.68440	0.842199	+	0.539167i	$0.181261\pi$
$588$	0	0
$589$	−1477.57	−0.103366
$590$	2324.40	0.162194
$591$	0	0
$592$	1832.94	0.127252
$593$	−10778.0	−0.746373	−0.373186	−	0.927756i	$-0.621735\pi$
$593$	−10778.0	−0.746373	−0.373186	+	0.927756i	$0.621735\pi$
$594$	0	0
$595$	0	0
$596$	−2482.12	−0.170590
$597$	0	0
$598$	−9607.79	−0.657009
$599$	−7597.58	−0.518245	−0.259122	−	0.965845i	$-0.583433\pi$
$599$	−7597.58	−0.518245	−0.259122	+	0.965845i	$0.583433\pi$
$600$	0	0
$601$	−19956.1	−1.35445	−0.677225	−	0.735776i	$-0.736818\pi$
$601$	−19956.1	−1.35445	−0.677225	+	0.735776i	$0.736818\pi$
$602$	0	0
$603$	0	0
$604$	−7757.06	−0.522567
$605$	−5910.81	−0.397204
$606$	0	0
$607$	236.311	0.0158016	0.00790079	−	0.999969i	$-0.497485\pi$
$607$	236.311	0.0158016	0.00790079	+	0.999969i	$0.497485\pi$
$608$	−161.722	−0.0107873
$609$	0	0
$610$	−6889.85	−0.457314
$611$	−15644.0	−1.03582
$612$	0	0
$613$	26414.9	1.74044	0.870219	−	0.492664i	$-0.163977\pi$
$613$	26414.9	1.74044	0.870219	+	0.492664i	$0.163977\pi$
$614$	8266.94	0.543365
$615$	0	0
$616$	0	0
$617$	−18473.6	−1.20538	−0.602689	−	0.797976i	$-0.705904\pi$
$617$	−18473.6	−1.20538	−0.602689	+	0.797976i	$0.705904\pi$
$618$	0	0
$619$	−16047.9	−1.04204	−0.521018	−	0.853546i	$-0.674448\pi$
$619$	−16047.9	−1.04204	−0.521018	+	0.853546i	$0.674448\pi$
$620$	−5362.94	−0.347388
$621$	0	0
$622$	−10126.5	−0.652788
$623$	0	0
$624$	0	0
$625$	8181.20	0.523597
$626$	14823.1	0.946407
$627$	0	0
$628$	1651.37	0.104931
$629$	9342.97	0.592255
$630$	0	0
$631$	−15065.7	−0.950487	−0.475243	−	0.879854i	$-0.657640\pi$
$631$	−15065.7	−0.950487	−0.475243	+	0.879854i	$0.657640\pi$
$632$	−2.11255	−0.000132963 0
$633$	0	0
$634$	−13474.1	−0.844046
$635$	110.861	0.00692816
$636$	0	0
$637$	0	0
$638$	−3477.82	−0.215812
$639$	0	0
$640$	−586.981	−0.0362538
$641$	31198.4	1.92241	0.961203	−	0.275842i	$-0.0889568\pi$
$641$	31198.4	1.92241	0.961203	+	0.275842i	$0.0889568\pi$
$642$	0	0
$643$	12497.9	0.766517	0.383259	−	0.923641i	$-0.374802\pi$
$643$	12497.9	0.766517	0.383259	+	0.923641i	$0.374802\pi$
$644$	0	0
$645$	0	0
$646$	−824.343	−0.0502064
$647$	−9929.72	−0.603366	−0.301683	−	0.953408i	$-0.597548\pi$
$647$	−9929.72	−0.603366	−0.301683	+	0.953408i	$0.597548\pi$
$648$	0	0
$649$	−1643.60	−0.0994099
$650$	−9401.68	−0.567330
$651$	0	0
$652$	−15629.8	−0.938817
$653$	8145.58	0.488149	0.244075	−	0.969756i	$-0.421516\pi$
$653$	8145.58	0.488149	0.244075	+	0.969756i	$0.421516\pi$
$654$	0	0
$655$	7252.55	0.432642
$656$	−2585.68	−0.153893
$657$	0	0
$658$	0	0
$659$	16975.8	1.00347	0.501733	−	0.865022i	$-0.332696\pi$
$659$	16975.8	1.00347	0.501733	+	0.865022i	$0.332696\pi$
$660$	0	0
$661$	−20637.8	−1.21440	−0.607199	−	0.794550i	$-0.707707\pi$
$661$	−20637.8	−1.21440	−0.607199	+	0.794550i	$0.707707\pi$
$662$	−22351.9	−1.31228
$663$	0	0
$664$	3497.10	0.204388
$665$	0	0
$666$	0	0
$667$	−28489.0	−1.65382
$668$	−5145.66	−0.298041
$669$	0	0
$670$	−106.819	−0.00615937
$671$	4871.86	0.280292
$672$	0	0
$673$	−2150.29	−0.123161	−0.0615807	−	0.998102i	$-0.519614\pi$
$673$	−2150.29	−0.123161	−0.0615807	+	0.998102i	$0.519614\pi$
$674$	−18759.6	−1.07209
$675$	0	0
$676$	−611.065	−0.0347670
$677$	−27783.4	−1.57726	−0.788628	−	0.614871i	$-0.789208\pi$
$677$	−27783.4	−1.57726	−0.788628	+	0.614871i	$0.789208\pi$
$678$	0	0
$679$	0	0
$680$	−2992.00	−0.168732
$681$	0	0
$682$	3792.17	0.212918
$683$	−18181.8	−1.01860	−0.509302	−	0.860588i	$-0.670096\pi$
$683$	−18181.8	−1.01860	−0.509302	+	0.860588i	$0.670096\pi$
$684$	0	0
$685$	3417.27	0.190609
$686$	0	0
$687$	0	0
$688$	−7540.70	−0.417858
$689$	18335.3	1.01381
$690$	0	0
$691$	−23935.1	−1.31771	−0.658853	−	0.752272i	$-0.728958\pi$
$691$	−23935.1	−1.31771	−0.658853	+	0.752272i	$0.728958\pi$
$692$	−5005.02	−0.274946
$693$	0	0
$694$	11362.9	0.621513
$695$	6299.02	0.343792
$696$	0	0
$697$	−13179.9	−0.716249
$698$	−1408.50	−0.0763789
$699$	0	0
$700$	0	0
$701$	20627.2	1.11138	0.555691	−	0.831389i	$-0.312454\pi$
$701$	20627.2	1.11138	0.555691	+	0.831389i	$0.312454\pi$
$702$	0	0
$703$	578.958	0.0310609
$704$	415.058	0.0222203
$705$	0	0
$706$	8569.93	0.456846
$707$	0	0
$708$	0	0
$709$	−6584.64	−0.348789	−0.174394	−	0.984676i	$-0.555797\pi$
$709$	−6584.64	−0.348789	−0.174394	+	0.984676i	$0.555797\pi$
$710$	−6251.90	−0.330464
$711$	0	0
$712$	468.101	0.0246388
$713$	31064.0	1.63163
$714$	0	0
$715$	−1344.65	−0.0703313
$716$	−14494.1	−0.756520
$717$	0	0
$718$	9322.54	0.484560
$719$	170.886	0.00886366	0.00443183	−	0.999990i	$-0.498589\pi$
$719$	170.886	0.00886366	0.00443183	+	0.999990i	$0.498589\pi$
$720$	0	0
$721$	0	0
$722$	13666.9	0.704474
$723$	0	0
$724$	−726.908	−0.0373140
$725$	−27877.8	−1.42808
$726$	0	0
$727$	11127.5	0.567671	0.283836	−	0.958873i	$-0.408393\pi$
$727$	11127.5	0.567671	0.283836	+	0.958873i	$0.408393\pi$
$728$	0	0
$729$	0	0
$730$	6286.72	0.318742
$731$	−38437.0	−1.94479
$732$	0	0
$733$	22575.9	1.13760	0.568800	−	0.822476i	$-0.307408\pi$
$733$	22575.9	1.13760	0.568800	+	0.822476i	$0.307408\pi$
$734$	−13870.6	−0.697511
$735$	0	0
$736$	3399.99	0.170279
$737$	75.5325	0.00377513
$738$	0	0
$739$	−22936.4	−1.14172	−0.570860	−	0.821048i	$-0.693390\pi$
$739$	−22936.4	−1.14172	−0.570860	+	0.821048i	$0.693390\pi$
$740$	2101.36	0.104389
$741$	0	0
$742$	0	0
$743$	−16973.4	−0.838081	−0.419041	−	0.907967i	$-0.637634\pi$
$743$	−16973.4	−0.838081	−0.419041	+	0.907967i	$0.637634\pi$
$744$	0	0
$745$	−2845.62	−0.139940
$746$	6162.21	0.302432
$747$	0	0
$748$	2115.66	0.103418
$749$	0	0
$750$	0	0
$751$	−21197.9	−1.02999	−0.514994	−	0.857194i	$-0.672206\pi$
$751$	−21197.9	−1.02999	−0.514994	+	0.857194i	$0.672206\pi$
$752$	5536.07	0.268457
$753$	0	0
$754$	24246.2	1.17108
$755$	−8893.05	−0.428677
$756$	0	0
$757$	−7962.24	−0.382289	−0.191144	−	0.981562i	$-0.561220\pi$
$757$	−7962.24	−0.382289	−0.191144	+	0.981562i	$0.561220\pi$
$758$	−1883.65	−0.0902604
$759$	0	0
$760$	−185.406	−0.00884919
$761$	26856.2	1.27928	0.639642	−	0.768673i	$-0.279083\pi$
$761$	26856.2	1.27928	0.639642	+	0.768673i	$0.279083\pi$
$762$	0	0
$763$	0	0
$764$	−5925.12	−0.280580
$765$	0	0
$766$	1355.05	0.0639165
$767$	11458.6	0.539436
$768$	0	0
$769$	12183.6	0.571331	0.285666	−	0.958329i	$-0.407785\pi$
$769$	12183.6	0.571331	0.285666	+	0.958329i	$0.407785\pi$
$770$	0	0
$771$	0	0
$772$	−1426.83	−0.0665192
$773$	−17455.2	−0.812187	−0.406093	−	0.913832i	$-0.633109\pi$
$773$	−17455.2	−0.812187	−0.406093	+	0.913832i	$0.633109\pi$
$774$	0	0
$775$	30397.6	1.40892
$776$	−10240.7	−0.473738
$777$	0	0
$778$	−23730.8	−1.09356
$779$	−816.724	−0.0375638
$780$	0	0
$781$	4420.76	0.202545
$782$	17330.7	0.792512
$783$	0	0
$784$	0	0
$785$	1893.21	0.0860785
$786$	0	0
$787$	−30981.0	−1.40324	−0.701622	−	0.712549i	$-0.747540\pi$
$787$	−30981.0	−1.40324	−0.701622	+	0.712549i	$0.747540\pi$
$788$	−19562.1	−0.884355
$789$	0	0
$790$	−2.42193	−0.000109074 0
$791$	0	0
$792$	0	0
$793$	−33964.9	−1.52097
$794$	−10281.5	−0.459544
$795$	0	0
$796$	−14171.4	−0.631020
$797$	−10517.6	−0.467446	−0.233723	−	0.972303i	$-0.575091\pi$
$797$	−10517.6	−0.467446	−0.233723	+	0.972303i	$0.575091\pi$
$798$	0	0
$799$	28218.8	1.24945
$800$	3327.06	0.147037
$801$	0	0
$802$	−24764.0	−1.09033
$803$	−4445.38	−0.195360
$804$	0	0
$805$	0	0
$806$	−26437.7	−1.15537
$807$	0	0
$808$	10451.1	0.455036
$809$	41778.4	1.81564	0.907819	−	0.419361i	$-0.137746\pi$
$809$	41778.4	1.81564	0.907819	+	0.419361i	$0.137746\pi$
$810$	0	0
$811$	12935.4	0.560079	0.280039	−	0.959988i	$-0.409652\pi$
$811$	12935.4	0.560079	0.280039	+	0.959988i	$0.409652\pi$
$812$	0	0
$813$	0	0
$814$	−1485.89	−0.0639807
$815$	−17918.7	−0.770140
$816$	0	0
$817$	−2381.83	−0.101995
$818$	−31751.3	−1.35716
$819$	0	0
$820$	−2964.35	−0.126243
$821$	14499.2	0.616354	0.308177	−	0.951329i	$-0.400281\pi$
$821$	14499.2	0.616354	0.308177	+	0.951329i	$0.400281\pi$
$822$	0	0
$823$	−11977.1	−0.507287	−0.253643	−	0.967298i	$-0.581629\pi$
$823$	−11977.1	−0.507287	−0.253643	+	0.967298i	$0.581629\pi$
$824$	−6071.80	−0.256700
$825$	0	0
$826$	0	0
$827$	−27613.5	−1.16108	−0.580541	−	0.814231i	$-0.697159\pi$
$827$	−27613.5	−1.16108	−0.580541	+	0.814231i	$0.697159\pi$
$828$	0	0
$829$	677.836	0.0283983	0.0141992	−	0.999899i	$-0.495480\pi$
$829$	677.836	0.0283983	0.0141992	+	0.999899i	$0.495480\pi$
$830$	4009.23	0.167666
$831$	0	0
$832$	−2893.65	−0.120576
$833$	0	0
$834$	0	0
$835$	−5899.22	−0.244492
$836$	131.102	0.00542375
$837$	0	0
$838$	32223.8	1.32835
$839$	42209.6	1.73687	0.868436	−	0.495801i	$-0.165126\pi$
$839$	42209.6	1.73687	0.868436	+	0.495801i	$0.165126\pi$
$840$	0	0
$841$	47505.8	1.94784
$842$	−17383.2	−0.711476
$843$	0	0
$844$	−17158.0	−0.699765
$845$	−700.553	−0.0285204
$846$	0	0
$847$	0	0
$848$	−6488.46	−0.262753
$849$	0	0
$850$	16958.9	0.684337
$851$	−12171.8	−0.490299
$852$	0	0
$853$	2796.45	0.112249	0.0561247	−	0.998424i	$-0.482126\pi$
$853$	2796.45	0.112249	0.0561247	+	0.998424i	$0.482126\pi$
$854$	0	0
$855$	0	0
$856$	10100.1	0.403288
$857$	23181.1	0.923979	0.461989	−	0.886886i	$-0.347136\pi$
$857$	23181.1	0.923979	0.461989	+	0.886886i	$0.347136\pi$
$858$	0	0
$859$	11897.5	0.472569	0.236285	−	0.971684i	$-0.424070\pi$
$859$	11897.5	0.472569	0.236285	+	0.971684i	$0.424070\pi$
$860$	−8645.01	−0.342782
$861$	0	0
$862$	−8390.21	−0.331521
$863$	29815.9	1.17607	0.588033	−	0.808837i	$-0.299902\pi$
$863$	29815.9	1.17607	0.588033	+	0.808837i	$0.299902\pi$
$864$	0	0
$865$	−5737.99	−0.225546
$866$	−10853.1	−0.425869
$867$	0	0
$868$	0	0
$869$	1.71256	6.68523e−5 0
$870$	0	0
$871$	−526.587	−0.0204853
$872$	16844.2	0.654148
$873$	0	0
$874$	1073.94	0.0415634
$875$	0	0
$876$	0	0
$877$	−10338.8	−0.398082	−0.199041	−	0.979991i	$-0.563783\pi$
$877$	−10338.8	−0.398082	−0.199041	+	0.979991i	$0.563783\pi$
$878$	−7542.48	−0.289916
$879$	0	0
$880$	475.842	0.0182280
$881$	24140.2	0.923160	0.461580	−	0.887099i	$-0.347283\pi$
$881$	24140.2	0.923160	0.461580	+	0.887099i	$0.347283\pi$
$882$	0	0
$883$	12997.6	0.495361	0.247681	−	0.968842i	$-0.420332\pi$
$883$	12997.6	0.495361	0.247681	+	0.968842i	$0.420332\pi$
$884$	−14749.7	−0.561183
$885$	0	0
$886$	−11860.6	−0.449734
$887$	45266.6	1.71353	0.856766	−	0.515705i	$-0.172470\pi$
$887$	45266.6	1.71353	0.856766	+	0.515705i	$0.172470\pi$
$888$	0	0
$889$	0	0
$890$	536.653	0.0202120
$891$	0	0
$892$	−23182.9	−0.870204
$893$	1748.64	0.0655276
$894$	0	0
$895$	−16616.7	−0.620596
$896$	0	0
$897$	0	0
$898$	−1058.13	−0.0393210
$899$	−78393.1	−2.90829
$900$	0	0
$901$	−33073.5	−1.22290
$902$	2096.11	0.0773756
$903$	0	0
$904$	12286.1	0.452024
$905$	−833.361	−0.0306098
$906$	0	0
$907$	−27766.5	−1.01651	−0.508253	−	0.861208i	$-0.669708\pi$
$907$	−27766.5	−1.01651	−0.508253	+	0.861208i	$0.669708\pi$
$908$	16416.2	0.599989
$909$	0	0
$910$	0	0
$911$	−18531.2	−0.673948	−0.336974	−	0.941514i	$-0.609403\pi$
$911$	−18531.2	−0.673948	−0.336974	+	0.941514i	$0.609403\pi$
$912$	0	0
$913$	−2834.96	−0.102764
$914$	−20114.4	−0.727925
$915$	0	0
$916$	5187.33	0.187111
$917$	0	0
$918$	0	0
$919$	−18093.4	−0.649452	−0.324726	−	0.945808i	$-0.605272\pi$
$919$	−18093.4	−0.649452	−0.324726	+	0.945808i	$0.605272\pi$
$920$	3897.91	0.139685
$921$	0	0
$922$	10020.6	0.357930
$923$	−30820.1	−1.09908
$924$	0	0
$925$	−11910.7	−0.423375
$926$	14248.8	0.505662
$927$	0	0
$928$	−8580.23	−0.303513
$929$	10622.5	0.375149	0.187574	−	0.982250i	$-0.439937\pi$
$929$	10622.5	0.375149	0.187574	+	0.982250i	$0.439937\pi$
$930$	0	0
$931$	0	0
$932$	5913.34	0.207830
$933$	0	0
$934$	−15002.9	−0.525599
$935$	2425.50	0.0848366
$936$	0	0
$937$	−16057.6	−0.559851	−0.279925	−	0.960022i	$-0.590310\pi$
$937$	−16057.6	−0.559851	−0.279925	+	0.960022i	$0.590310\pi$
$938$	0	0
$939$	0	0
$940$	6346.81	0.220223
$941$	55153.3	1.91068	0.955338	−	0.295516i	$-0.0954915\pi$
$941$	55153.3	1.91068	0.955338	+	0.295516i	$0.0954915\pi$
$942$	0	0
$943$	17170.5	0.592947
$944$	−4054.97	−0.139807
$945$	0	0
$946$	6112.94	0.210094
$947$	−18785.7	−0.644619	−0.322309	−	0.946634i	$-0.604459\pi$
$947$	−18785.7	−0.644619	−0.322309	+	0.946634i	$0.604459\pi$
$948$	0	0
$949$	30991.7	1.06010
$950$	1050.90	0.0358901
$951$	0	0
$952$	0	0
$953$	36499.4	1.24064	0.620321	−	0.784348i	$-0.287002\pi$
$953$	36499.4	1.24064	0.620321	+	0.784348i	$0.287002\pi$
$954$	0	0
$955$	−6792.83	−0.230168
$956$	15107.7	0.511106
$957$	0	0
$958$	16347.6	0.551323
$959$	0	0
$960$	0	0
$961$	55687.8	1.86928
$962$	10359.1	0.347184
$963$	0	0
$964$	15985.5	0.534086
$965$	−1635.79	−0.0545677
$966$	0	0
$967$	26059.9	0.866627	0.433314	−	0.901243i	$-0.357344\pi$
$967$	26059.9	0.866627	0.433314	+	0.901243i	$0.357344\pi$
$968$	10311.5	0.342381
$969$	0	0
$970$	−11740.4	−0.388622
$971$	21689.0	0.716820	0.358410	−	0.933564i	$-0.383319\pi$
$971$	21689.0	0.716820	0.358410	+	0.933564i	$0.383319\pi$
$972$	0	0
$973$	0	0
$974$	23937.6	0.787484
$975$	0	0
$976$	12019.5	0.394195
$977$	−2119.19	−0.0693950	−0.0346975	−	0.999398i	$-0.511047\pi$
$977$	−2119.19	−0.0693950	−0.0346975	+	0.999398i	$0.511047\pi$
$978$	0	0
$979$	−379.471	−0.0123881
$980$	0	0
$981$	0	0
$982$	4159.92	0.135182
$983$	−48504.1	−1.57380	−0.786898	−	0.617083i	$-0.788314\pi$
$983$	−48504.1	−1.57380	−0.786898	+	0.617083i	$0.788314\pi$
$984$	0	0
$985$	−22426.9	−0.725463
$986$	−43735.7	−1.41261
$987$	0	0
$988$	−913.998	−0.0294313
$989$	50074.8	1.61000
$990$	0	0
$991$	2043.84	0.0655144	0.0327572	−	0.999463i	$-0.489571\pi$
$991$	2043.84	0.0655144	0.0327572	+	0.999463i	$0.489571\pi$
$992$	9355.76	0.299441
$993$	0	0
$994$	0	0
$995$	−16246.8	−0.517645
$996$	0	0
$997$	26738.9	0.849378	0.424689	−	0.905339i	$-0.360383\pi$
$997$	26738.9	0.849378	0.424689	+	0.905339i	$0.360383\pi$
$998$	−25668.8	−0.814161
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.bb.1.1		2
3.2	odd	2		294.4.a.l.1.2	✓	2
7.2	even	3		882.4.g.be.361.2		4
7.3	odd	6		882.4.g.bk.667.1		4
7.4	even	3		882.4.g.be.667.2		4
7.5	odd	6		882.4.g.bk.361.1		4
7.6	odd	2		882.4.a.t.1.2		2
12.11	even	2		2352.4.a.bw.1.2		2
21.2	odd	6		294.4.e.m.67.1		4
21.5	even	6		294.4.e.k.67.2		4
21.11	odd	6		294.4.e.m.79.1		4
21.17	even	6		294.4.e.k.79.2		4
21.20	even	2		294.4.a.o.1.1	yes	2
84.83	odd	2		2352.4.a.bu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.4.a.l.1.2	✓	2	3.2	odd	2
294.4.a.o.1.1	yes	2	21.20	even	2
294.4.e.k.67.2		4	21.5	even	6
294.4.e.k.79.2		4	21.17	even	6
294.4.e.m.67.1		4	21.2	odd	6
294.4.e.m.79.1		4	21.11	odd	6
882.4.a.t.1.2		2	7.6	odd	2
882.4.a.bb.1.1		2	1.1	even	1	trivial
882.4.g.be.361.2		4	7.2	even	3
882.4.g.be.667.2		4	7.4	even	3
882.4.g.bk.361.1		4	7.5	odd	6
882.4.g.bk.667.1		4	7.3	odd	6
2352.4.a.bu.1.1		2	84.83	odd	2
2352.4.a.bw.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +4.58579 q^{5} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +4.58579 q^{5} -8.00000 q^{8} -9.17157 q^{10} +6.48528 q^{11} -45.2132 q^{13} +16.0000 q^{16} +81.5563 q^{17} +5.05382 q^{19} +18.3431 q^{20} -12.9706 q^{22} -106.250 q^{23} -103.971 q^{25} +90.4264 q^{26} +268.132 q^{29} -292.368 q^{31} -32.0000 q^{32} -163.113 q^{34} +114.558 q^{37} -10.1076 q^{38} -36.6863 q^{40} -161.605 q^{41} -471.294 q^{43} +25.9411 q^{44} +212.500 q^{46} +346.004 q^{47} +207.941 q^{50} -180.853 q^{52} -405.529 q^{53} +29.7401 q^{55} -536.264 q^{58} -253.436 q^{59} +751.217 q^{61} +584.735 q^{62} +64.0000 q^{64} -207.338 q^{65} +11.6468 q^{67} +326.225 q^{68} +681.661 q^{71} -685.457 q^{73} -229.117 q^{74} +20.2153 q^{76} +0.264069 q^{79} +73.3726 q^{80} +323.210 q^{82} -437.137 q^{83} +374.000 q^{85} +942.587 q^{86} -51.8823 q^{88} -58.5126 q^{89} -424.999 q^{92} -692.008 q^{94} +23.1758 q^{95} +1280.09 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} + 12 q^{5} - 16 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 + 8 * q^4 + 12 * q^5 - 16 * q^8	\( 2 q - 4 q^{2} + 8 q^{4} + 12 q^{5} - 16 q^{8} - 24 q^{10} - 4 q^{11} - 48 q^{13} + 32 q^{16} + 132 q^{17} - 120 q^{19} + 48 q^{20} + 8 q^{22} + 76 q^{23} - 174 q^{25} + 96 q^{26} + 112 q^{29} - 432 q^{31} - 64 q^{32} - 264 q^{34} - 280 q^{37} + 240 q^{38} - 96 q^{40} + 36 q^{41} - 128 q^{43} - 16 q^{44} - 152 q^{46} - 264 q^{47} + 348 q^{50} - 192 q^{52} - 268 q^{53} - 48 q^{55} - 224 q^{58} + 336 q^{59} + 504 q^{61} + 864 q^{62} + 128 q^{64} - 228 q^{65} - 384 q^{67} + 528 q^{68} + 396 q^{71} + 312 q^{73} + 560 q^{74} - 480 q^{76} - 848 q^{79} + 192 q^{80} - 72 q^{82} - 648 q^{83} + 748 q^{85} + 256 q^{86} + 32 q^{88} - 612 q^{89} + 304 q^{92} + 528 q^{94} - 904 q^{95} + 2184 q^{97}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 + 12 * q^5 - 16 * q^8 - 24 * q^10 - 4 * q^11 - 48 * q^13 + 32 * q^16 + 132 * q^17 - 120 * q^19 + 48 * q^20 + 8 * q^22 + 76 * q^23 - 174 * q^25 + 96 * q^26 + 112 * q^29 - 432 * q^31 - 64 * q^32 - 264 * q^34 - 280 * q^37 + 240 * q^38 - 96 * q^40 + 36 * q^41 - 128 * q^43 - 16 * q^44 - 152 * q^46 - 264 * q^47 + 348 * q^50 - 192 * q^52 - 268 * q^53 - 48 * q^55 - 224 * q^58 + 336 * q^59 + 504 * q^61 + 864 * q^62 + 128 * q^64 - 228 * q^65 - 384 * q^67 + 528 * q^68 + 396 * q^71 + 312 * q^73 + 560 * q^74 - 480 * q^76 - 848 * q^79 + 192 * q^80 - 72 * q^82 - 648 * q^83 + 748 * q^85 + 256 * q^86 + 32 * q^88 - 612 * q^89 + 304 * q^92 + 528 * q^94 - 904 * q^95 + 2184 * q^97

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