LMFDB - Embedded newform 882.6.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,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, 6, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$141.458529075$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{11}]$
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			1.2
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)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} +61.0711 q^{5} -64.0000 q^{8} +O(q^{10})$	$q-4.00000 q^{2} +16.0000 q^{4} +61.0711 q^{5} -64.0000 q^{8} -244.284 q^{10} +36.5442 q^{11} +34.5656 q^{13} +256.000 q^{16} +2061.04 q^{17} +452.238 q^{19} +977.137 q^{20} -146.177 q^{22} -1684.25 q^{23} +604.675 q^{25} -138.262 q^{26} +4765.22 q^{29} -5260.87 q^{31} -1024.00 q^{32} -8244.16 q^{34} -12822.0 q^{37} -1808.95 q^{38} -3908.55 q^{40} +7126.74 q^{41} +11141.7 q^{43} +584.706 q^{44} +6736.99 q^{46} +23443.2 q^{47} -2418.70 q^{50} +553.049 q^{52} +7030.30 q^{53} +2231.79 q^{55} -19060.9 q^{58} +44222.8 q^{59} +19380.5 q^{61} +21043.5 q^{62} +4096.00 q^{64} +2110.96 q^{65} +20944.1 q^{67} +32976.6 q^{68} -79843.4 q^{71} -37065.8 q^{73} +51287.8 q^{74} +7235.81 q^{76} +42072.0 q^{79} +15634.2 q^{80} -28506.9 q^{82} +6311.34 q^{83} +125870. q^{85} -44566.8 q^{86} -2338.83 q^{88} +51495.8 q^{89} -26948.0 q^{92} -93772.8 q^{94} +27618.7 q^{95} -127357. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} + 108 q^{5} - 128 q^{8}+O(q^{10}) $ 2 * q - 8 * q^2 + 32 * q^4 + 108 * q^5 - 128 * q^8	$ 2 q - 8 q^{2} + 32 q^{4} + 108 q^{5} - 128 q^{8} - 432 q^{10} + 124 q^{11} - 720 q^{13} + 512 q^{16} + 612 q^{17} - 2088 q^{19} + 1728 q^{20} - 496 q^{22} - 772 q^{23} - 318 q^{25} + 2880 q^{26} + 4592 q^{29} - 9792 q^{31} - 2048 q^{32} - 2448 q^{34} - 5992 q^{37} + 8352 q^{38} - 6912 q^{40} + 20196 q^{41} - 1136 q^{43} + 1984 q^{44} + 3088 q^{46} + 36936 q^{47} + 1272 q^{50} - 11520 q^{52} + 16708 q^{53} + 6336 q^{55} - 18368 q^{58} + 74592 q^{59} + 18648 q^{61} + 39168 q^{62} + 8192 q^{64} - 33300 q^{65} + 67344 q^{67} + 9792 q^{68} - 76548 q^{71} - 47304 q^{73} + 23968 q^{74} - 33408 q^{76} + 140656 q^{79} + 27648 q^{80} - 80784 q^{82} + 94104 q^{83} + 57868 q^{85} + 4544 q^{86} - 7936 q^{88} - 17604 q^{89} - 12352 q^{92} - 147744 q^{94} - 91592 q^{95} - 85176 q^{97}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 + 108 * q^5 - 128 * q^8 - 432 * q^10 + 124 * q^11 - 720 * q^13 + 512 * q^16 + 612 * q^17 - 2088 * q^19 + 1728 * q^20 - 496 * q^22 - 772 * q^23 - 318 * q^25 + 2880 * q^26 + 4592 * q^29 - 9792 * q^31 - 2048 * q^32 - 2448 * q^34 - 5992 * q^37 + 8352 * q^38 - 6912 * q^40 + 20196 * q^41 - 1136 * q^43 + 1984 * q^44 + 3088 * q^46 + 36936 * q^47 + 1272 * q^50 - 11520 * q^52 + 16708 * q^53 + 6336 * q^55 - 18368 * q^58 + 74592 * q^59 + 18648 * q^61 + 39168 * q^62 + 8192 * q^64 - 33300 * q^65 + 67344 * q^67 + 9792 * q^68 - 76548 * q^71 - 47304 * q^73 + 23968 * q^74 - 33408 * q^76 + 140656 * q^79 + 27648 * q^80 - 80784 * q^82 + 94104 * q^83 + 57868 * q^85 + 4544 * q^86 - 7936 * q^88 - 17604 * q^89 - 12352 * q^92 - 147744 * q^94 - 91592 * q^95 - 85176 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	61.0711	1.09247	0.546236	−	0.837631i	$-0.316060\pi$
$5$	61.0711	1.09247	0.546236	+	0.837631i	$0.316060\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−244.284	−0.772495
$11$	36.5442	0.0910618	0.0455309	−	0.998963i	$-0.485502\pi$
$11$	36.5442	0.0910618	0.0455309	+	0.998963i	$0.485502\pi$
$12$	0	0
$13$	34.5656	0.0567264	0.0283632	−	0.999598i	$-0.490970\pi$
$13$	34.5656	0.0567264	0.0283632	+	0.999598i	$0.490970\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	2061.04	1.72967	0.864836	−	0.502054i	$-0.167422\pi$
$17$	2061.04	1.72967	0.864836	+	0.502054i	$0.167422\pi$
$18$	0	0
$19$	452.238	0.287398	0.143699	−	0.989621i	$-0.454100\pi$
$19$	452.238	0.287398	0.143699	+	0.989621i	$0.454100\pi$
$20$	977.137	0.546236
$21$	0	0
$22$	−146.177	−0.0643904
$23$	−1684.25	−0.663875	−0.331938	−	0.943301i	$-0.607702\pi$
$23$	−1684.25	−0.663875	−0.331938	+	0.943301i	$0.607702\pi$
$24$	0	0
$25$	604.675	0.193496
$26$	−138.262	−0.0401116
$27$	0	0
$28$	0	0
$29$	4765.22	1.05217	0.526087	−	0.850431i	$-0.323659\pi$
$29$	4765.22	1.05217	0.526087	+	0.850431i	$0.323659\pi$
$30$	0	0
$31$	−5260.87	−0.983225	−0.491613	−	0.870814i	$-0.663592\pi$
$31$	−5260.87	−0.983225	−0.491613	+	0.870814i	$0.663592\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−8244.16	−1.22306
$35$	0	0
$36$	0	0
$37$	−12822.0	−1.53975	−0.769875	−	0.638195i	$-0.779681\pi$
$37$	−12822.0	−1.53975	−0.769875	+	0.638195i	$0.779681\pi$
$38$	−1808.95	−0.203221
$39$	0	0
$40$	−3908.55	−0.386247
$41$	7126.74	0.662111	0.331056	−	0.943611i	$-0.392595\pi$
$41$	7126.74	0.662111	0.331056	+	0.943611i	$0.392595\pi$
$42$	0	0
$43$	11141.7	0.918925	0.459462	−	0.888197i	$-0.348042\pi$
$43$	11141.7	0.918925	0.459462	+	0.888197i	$0.348042\pi$
$44$	584.706	0.0455309
$45$	0	0
$46$	6736.99	0.469431
$47$	23443.2	1.54800	0.774002	−	0.633183i	$-0.218252\pi$
$47$	23443.2	1.54800	0.774002	+	0.633183i	$0.218252\pi$
$48$	0	0
$49$	0	0
$50$	−2418.70	−0.136822
$51$	0	0
$52$	553.049	0.0283632
$53$	7030.30	0.343783	0.171891	−	0.985116i	$-0.445012\pi$
$53$	7030.30	0.343783	0.171891	+	0.985116i	$0.445012\pi$
$54$	0	0
$55$	2231.79	0.0994825
$56$	0	0
$57$	0	0
$58$	−19060.9	−0.744000
$59$	44222.8	1.65393	0.826964	−	0.562255i	$-0.190066\pi$
$59$	44222.8	1.65393	0.826964	+	0.562255i	$0.190066\pi$
$60$	0	0
$61$	19380.5	0.666868	0.333434	−	0.942773i	$-0.391793\pi$
$61$	19380.5	0.666868	0.333434	+	0.942773i	$0.391793\pi$
$62$	21043.5	0.695245
$63$	0	0
$64$	4096.00	0.125000
$65$	2110.96	0.0619721
$66$	0	0
$67$	20944.1	0.569999	0.285000	−	0.958528i	$-0.408007\pi$
$67$	20944.1	0.569999	0.285000	+	0.958528i	$0.408007\pi$
$68$	32976.6	0.864836
$69$	0	0
$70$	0	0
$71$	−79843.4	−1.87972	−0.939860	−	0.341560i	$-0.889045\pi$
$71$	−79843.4	−1.87972	−0.939860	+	0.341560i	$0.889045\pi$
$72$	0	0
$73$	−37065.8	−0.814079	−0.407039	−	0.913411i	$-0.633439\pi$
$73$	−37065.8	−0.814079	−0.407039	+	0.913411i	$0.633439\pi$
$74$	51287.8	1.08877
$75$	0	0
$76$	7235.81	0.143699
$77$	0	0
$78$	0	0
$79$	42072.0	0.758448	0.379224	−	0.925305i	$-0.376191\pi$
$79$	42072.0	0.758448	0.379224	+	0.925305i	$0.376191\pi$
$80$	15634.2	0.273118
$81$	0	0
$82$	−28506.9	−0.468184
$83$	6311.34	0.100560	0.0502801	−	0.998735i	$-0.483989\pi$
$83$	6311.34	0.100560	0.0502801	+	0.998735i	$0.483989\pi$
$84$	0	0
$85$	125870.	1.88962
$86$	−44566.8	−0.649778
$87$	0	0
$88$	−2338.83	−0.0321952
$89$	51495.8	0.689123	0.344562	−	0.938764i	$-0.388028\pi$
$89$	51495.8	0.689123	0.344562	+	0.938764i	$0.388028\pi$
$90$	0	0
$91$	0	0
$92$	−26948.0	−0.331938
$93$	0	0
$94$	−93772.8	−1.09460
$95$	27618.7	0.313974
$96$	0	0
$97$	−127357.	−1.37434	−0.687171	−	0.726496i	$-0.741148\pi$
$97$	−127357.	−1.37434	−0.687171	+	0.726496i	$0.741148\pi$
$98$	0	0
$99$	0	0
$100$	9674.81	0.0967481
$101$	−65757.9	−0.641423	−0.320712	−	0.947177i	$-0.603922\pi$
$101$	−65757.9	−0.641423	−0.320712	+	0.947177i	$0.603922\pi$
$102$	0	0
$103$	170010.	1.57900	0.789499	−	0.613751i	$-0.210340\pi$
$103$	170010.	1.57900	0.789499	+	0.613751i	$0.210340\pi$
$104$	−2212.20	−0.0200558
$105$	0	0
$106$	−28121.2	−0.243091
$107$	107933.	0.911369	0.455685	−	0.890141i	$-0.349395\pi$
$107$	107933.	0.911369	0.455685	+	0.890141i	$0.349395\pi$
$108$	0	0
$109$	−222698.	−1.79536	−0.897679	−	0.440651i	$-0.854748\pi$
$109$	−222698.	−1.79536	−0.897679	+	0.440651i	$0.854748\pi$
$110$	−8927.16	−0.0703448
$111$	0	0
$112$	0	0
$113$	−181055.	−1.33387	−0.666936	−	0.745115i	$-0.732395\pi$
$113$	−181055.	−1.33387	−0.666936	+	0.745115i	$0.732395\pi$
$114$	0	0
$115$	−102859.	−0.725265
$116$	76243.5	0.526087
$117$	0	0
$118$	−176891.	−1.16950
$119$	0	0
$120$	0	0
$121$	−159716.	−0.991708
$122$	−77521.9	−0.471547
$123$	0	0
$124$	−84173.9	−0.491613
$125$	−153919.	−0.881083
$126$	0	0
$127$	91474.7	0.503259	0.251630	−	0.967824i	$-0.419034\pi$
$127$	91474.7	0.503259	0.251630	+	0.967824i	$0.419034\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−8443.83	−0.0438209
$131$	19343.9	0.0984840	0.0492420	−	0.998787i	$-0.484319\pi$
$131$	19343.9	0.0984840	0.0492420	+	0.998787i	$0.484319\pi$
$132$	0	0
$133$	0	0
$134$	−83776.3	−0.403050
$135$	0	0
$136$	−131906.	−0.611532
$137$	3077.04	0.0140065	0.00700327	−	0.999975i	$-0.497771\pi$
$137$	3077.04	0.0140065	0.00700327	+	0.999975i	$0.497771\pi$
$138$	0	0
$139$	370001.	1.62430	0.812149	−	0.583450i	$-0.198298\pi$
$139$	370001.	1.62430	0.812149	+	0.583450i	$0.198298\pi$
$140$	0	0
$141$	0	0
$142$	319374.	1.32916
$143$	1263.17	0.00516561
$144$	0	0
$145$	291017.	1.14947
$146$	148263.	0.575641
$147$	0	0
$148$	−205151.	−0.769875
$149$	454120.	1.67573	0.837866	−	0.545875i	$-0.183803\pi$
$149$	454120.	1.67573	0.837866	+	0.545875i	$0.183803\pi$
$150$	0	0
$151$	177592.	0.633841	0.316921	−	0.948452i	$-0.397351\pi$
$151$	177592.	0.633841	0.316921	+	0.948452i	$0.397351\pi$
$152$	−28943.2	−0.101610
$153$	0	0
$154$	0	0
$155$	−321287.	−1.07415
$156$	0	0
$157$	−248981.	−0.806153	−0.403076	−	0.915166i	$-0.632059\pi$
$157$	−248981.	−0.806153	−0.403076	+	0.915166i	$0.632059\pi$
$158$	−168288.	−0.536303
$159$	0	0
$160$	−62536.8	−0.193124
$161$	0	0
$162$	0	0
$163$	202248.	0.596231	0.298116	−	0.954530i	$-0.403642\pi$
$163$	202248.	0.596231	0.298116	+	0.954530i	$0.403642\pi$
$164$	114028.	0.331056
$165$	0	0
$166$	−25245.3	−0.0711068
$167$	−475790.	−1.32015	−0.660076	−	0.751199i	$-0.729476\pi$
$167$	−475790.	−1.32015	−0.660076	+	0.751199i	$0.729476\pi$
$168$	0	0
$169$	−370098.	−0.996782
$170$	−503479.	−1.33616
$171$	0	0
$172$	178267.	0.459462
$173$	284703.	0.723231	0.361616	−	0.932327i	$-0.382225\pi$
$173$	284703.	0.723231	0.361616	+	0.932327i	$0.382225\pi$
$174$	0	0
$175$	0	0
$176$	9355.30	0.0227654
$177$	0	0
$178$	−205983.	−0.487284
$179$	−347958.	−0.811699	−0.405849	−	0.913940i	$-0.633024\pi$
$179$	−347958.	−0.811699	−0.405849	+	0.913940i	$0.633024\pi$
$180$	0	0
$181$	379706.	0.861492	0.430746	−	0.902473i	$-0.358250\pi$
$181$	379706.	0.861492	0.430746	+	0.902473i	$0.358250\pi$
$182$	0	0
$183$	0	0
$184$	107792.	0.234715
$185$	−783051.	−1.68213
$186$	0	0
$187$	75318.9	0.157507
$188$	375091.	0.774002
$189$	0	0
$190$	−110475.	−0.222013
$191$	−194492.	−0.385762	−0.192881	−	0.981222i	$-0.561783\pi$
$191$	−194492.	−0.385762	−0.192881	+	0.981222i	$0.561783\pi$
$192$	0	0
$193$	877401.	1.69553	0.847763	−	0.530375i	$-0.177949\pi$
$193$	877401.	1.69553	0.847763	+	0.530375i	$0.177949\pi$
$194$	509430.	0.971806
$195$	0	0
$196$	0	0
$197$	752368.	1.38123	0.690613	−	0.723225i	$-0.257341\pi$
$197$	752368.	1.38123	0.690613	+	0.723225i	$0.257341\pi$
$198$	0	0
$199$	−155453.	−0.278270	−0.139135	−	0.990273i	$-0.544432\pi$
$199$	−155453.	−0.278270	−0.139135	+	0.990273i	$0.544432\pi$
$200$	−38699.2	−0.0684112
$201$	0	0
$202$	263032.	0.453555
$203$	0	0
$204$	0	0
$205$	435237.	0.723339
$206$	−680041.	−1.11652
$207$	0	0
$208$	8848.79	0.0141816
$209$	16526.7	0.0261709
$210$	0	0
$211$	3898.23	0.00602783	0.00301391	−	0.999995i	$-0.499041\pi$
$211$	3898.23	0.00602783	0.00301391	+	0.999995i	$0.499041\pi$
$212$	112485.	0.171891
$213$	0	0
$214$	−431731.	−0.644435
$215$	680435.	1.00390
$216$	0	0
$217$	0	0
$218$	890794.	1.26951
$219$	0	0
$220$	35708.6	0.0497413
$221$	71241.0	0.0981182
$222$	0	0
$223$	782574.	1.05381	0.526906	−	0.849924i	$-0.323352\pi$
$223$	782574.	1.05381	0.526906	+	0.849924i	$0.323352\pi$
$224$	0	0
$225$	0	0
$226$	724220.	0.943190
$227$	−796543.	−1.02599	−0.512997	−	0.858391i	$-0.671465\pi$
$227$	−796543.	−1.02599	−0.512997	+	0.858391i	$0.671465\pi$
$228$	0	0
$229$	387045.	0.487722	0.243861	−	0.969810i	$-0.421586\pi$
$229$	387045.	0.487722	0.243861	+	0.969810i	$0.421586\pi$
$230$	411435.	0.512840
$231$	0	0
$232$	−304974.	−0.372000
$233$	383759.	0.463093	0.231547	−	0.972824i	$-0.425621\pi$
$233$	383759.	0.463093	0.231547	+	0.972824i	$0.425621\pi$
$234$	0	0
$235$	1.43170e6	1.69115
$236$	707565.	0.826964
$237$	0	0
$238$	0	0
$239$	465409.	0.527036	0.263518	−	0.964654i	$-0.415117\pi$
$239$	465409.	0.527036	0.263518	+	0.964654i	$0.415117\pi$
$240$	0	0
$241$	−348744.	−0.386781	−0.193390	−	0.981122i	$-0.561948\pi$
$241$	−348744.	−0.386781	−0.193390	+	0.981122i	$0.561948\pi$
$242$	638862.	0.701243
$243$	0	0
$244$	310088.	0.333434
$245$	0	0
$246$	0	0
$247$	15631.9	0.0163030
$248$	336695.	0.347623
$249$	0	0
$250$	615676.	0.623020
$251$	186543.	0.186894	0.0934468	−	0.995624i	$-0.470211\pi$
$251$	186543.	0.186894	0.0934468	+	0.995624i	$0.470211\pi$
$252$	0	0
$253$	−61549.4	−0.0604537
$254$	−365899.	−0.355858
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.55355e6	−1.46721	−0.733605	−	0.679577i	$-0.762164\pi$
$257$	−1.55355e6	−1.46721	−0.733605	+	0.679577i	$0.762164\pi$
$258$	0	0
$259$	0	0
$260$	33775.3	0.0309860
$261$	0	0
$262$	−77375.6	−0.0696387
$263$	1.25960e6	1.12290	0.561451	−	0.827510i	$-0.310243\pi$
$263$	1.25960e6	1.12290	0.561451	+	0.827510i	$0.310243\pi$
$264$	0	0
$265$	429348.	0.375573
$266$	0	0
$267$	0	0
$268$	335105.	0.285000
$269$	170461.	0.143630	0.0718149	−	0.997418i	$-0.477121\pi$
$269$	170461.	0.143630	0.0718149	+	0.997418i	$0.477121\pi$
$270$	0	0
$271$	2.22743e6	1.84239	0.921194	−	0.389104i	$-0.127215\pi$
$271$	2.22743e6	1.84239	0.921194	+	0.389104i	$0.127215\pi$
$272$	527626.	0.432418
$273$	0	0
$274$	−12308.1	−0.00990412
$275$	22097.3	0.0176201
$276$	0	0
$277$	1.98339e6	1.55314	0.776568	−	0.630034i	$-0.216959\pi$
$277$	1.98339e6	1.55314	0.776568	+	0.630034i	$0.216959\pi$
$278$	−1.48000e6	−1.14855
$279$	0	0
$280$	0	0
$281$	1.16687e6	0.881573	0.440786	−	0.897612i	$-0.354700\pi$
$281$	1.16687e6	0.881573	0.440786	+	0.897612i	$0.354700\pi$
$282$	0	0
$283$	−1.90090e6	−1.41089	−0.705445	−	0.708765i	$-0.749253\pi$
$283$	−1.90090e6	−1.41089	−0.705445	+	0.708765i	$0.749253\pi$
$284$	−1.27749e6	−0.939860
$285$	0	0
$286$	−5052.68	−0.00365264
$287$	0	0
$288$	0	0
$289$	2.82802e6	1.99177
$290$	−1.16407e6	−0.812799
$291$	0	0
$292$	−593053.	−0.407039
$293$	−1.65009e6	−1.12289	−0.561445	−	0.827514i	$-0.689755\pi$
$293$	−1.65009e6	−1.12289	−0.561445	+	0.827514i	$0.689755\pi$
$294$	0	0
$295$	2.70073e6	1.80687
$296$	820605.	0.544384
$297$	0	0
$298$	−1.81648e6	−1.18492
$299$	−58217.0	−0.0376593
$300$	0	0
$301$	0	0
$302$	−710367.	−0.448194
$303$	0	0
$304$	115773.	0.0718494
$305$	1.18359e6	0.728535
$306$	0	0
$307$	597936.	0.362084	0.181042	−	0.983475i	$-0.442053\pi$
$307$	597936.	0.362084	0.181042	+	0.983475i	$0.442053\pi$
$308$	0	0
$309$	0	0
$310$	1.28515e6	0.759536
$311$	−230480.	−0.135124	−0.0675620	−	0.997715i	$-0.521522\pi$
$311$	−230480.	−0.135124	−0.0675620	+	0.997715i	$0.521522\pi$
$312$	0	0
$313$	442577.	0.255346	0.127673	−	0.991816i	$-0.459249\pi$
$313$	442577.	0.255346	0.127673	+	0.991816i	$0.459249\pi$
$314$	995925.	0.570036
$315$	0	0
$316$	673152.	0.379224
$317$	1.26248e6	0.705627	0.352813	−	0.935694i	$-0.385225\pi$
$317$	1.26248e6	0.705627	0.352813	+	0.935694i	$0.385225\pi$
$318$	0	0
$319$	174141.	0.0958129
$320$	250147.	0.136559
$321$	0	0
$322$	0	0
$323$	932080.	0.497104
$324$	0	0
$325$	20901.0	0.0109763
$326$	−808991.	−0.421599
$327$	0	0
$328$	−456111.	−0.234092
$329$	0	0
$330$	0	0
$331$	−468936.	−0.235257	−0.117629	−	0.993058i	$-0.537529\pi$
$331$	−468936.	−0.235257	−0.117629	+	0.993058i	$0.537529\pi$
$332$	100981.	0.0502801
$333$	0	0
$334$	1.90316e6	0.933489
$335$	1.27908e6	0.622708
$336$	0	0
$337$	−2.46798e6	−1.18377	−0.591884	−	0.806023i	$-0.701616\pi$
$337$	−2.46798e6	−1.18377	−0.591884	+	0.806023i	$0.701616\pi$
$338$	1.48039e6	0.704831
$339$	0	0
$340$	2.01392e6	0.944810
$341$	−192254.	−0.0895343
$342$	0	0
$343$	0	0
$344$	−713068.	−0.324889
$345$	0	0
$346$	−1.13881e6	−0.511402
$347$	4.28250e6	1.90930	0.954648	−	0.297736i	$-0.0962315\pi$
$347$	4.28250e6	1.90930	0.954648	+	0.297736i	$0.0962315\pi$
$348$	0	0
$349$	−1.84999e6	−0.813028	−0.406514	−	0.913645i	$-0.633256\pi$
$349$	−1.84999e6	−0.813028	−0.406514	+	0.913645i	$0.633256\pi$
$350$	0	0
$351$	0	0
$352$	−37421.2	−0.0160976
$353$	2.27380e6	0.971213	0.485607	−	0.874177i	$-0.338599\pi$
$353$	2.27380e6	0.971213	0.485607	+	0.874177i	$0.338599\pi$
$354$	0	0
$355$	−4.87612e6	−2.05354
$356$	823933.	0.344562
$357$	0	0
$358$	1.39183e6	0.573958
$359$	4.39992e6	1.80181	0.900905	−	0.434017i	$-0.142904\pi$
$359$	4.39992e6	1.80181	0.900905	+	0.434017i	$0.142904\pi$
$360$	0	0
$361$	−2.27158e6	−0.917403
$362$	−1.51882e6	−0.609167
$363$	0	0
$364$	0	0
$365$	−2.26365e6	−0.889359
$366$	0	0
$367$	−1.89743e6	−0.735362	−0.367681	−	0.929952i	$-0.619848\pi$
$367$	−1.89743e6	−0.735362	−0.367681	+	0.929952i	$0.619848\pi$
$368$	−431168.	−0.165969
$369$	0	0
$370$	3.13220e6	1.18945
$371$	0	0
$372$	0	0
$373$	4.56260e6	1.69801	0.849005	−	0.528385i	$-0.177202\pi$
$373$	4.56260e6	1.69801	0.849005	+	0.528385i	$0.177202\pi$
$374$	−301276.	−0.111374
$375$	0	0
$376$	−1.50037e6	−0.547302
$377$	164713.	0.0596861
$378$	0	0
$379$	4.99400e6	1.78587	0.892936	−	0.450184i	$-0.148642\pi$
$379$	4.99400e6	1.78587	0.892936	+	0.450184i	$0.148642\pi$
$380$	441898.	0.156987
$381$	0	0
$382$	777970.	0.272775
$383$	3.67907e6	1.28157	0.640783	−	0.767722i	$-0.278610\pi$
$383$	3.67907e6	1.28157	0.640783	+	0.767722i	$0.278610\pi$
$384$	0	0
$385$	0	0
$386$	−3.50960e6	−1.19892
$387$	0	0
$388$	−2.03772e6	−0.687171
$389$	−257234.	−0.0861894	−0.0430947	−	0.999071i	$-0.513722\pi$
$389$	−257234.	−0.0861894	−0.0430947	+	0.999071i	$0.513722\pi$
$390$	0	0
$391$	−3.47130e6	−1.14829
$392$	0	0
$393$	0	0
$394$	−3.00947e6	−0.976674
$395$	2.56938e6	0.828583
$396$	0	0
$397$	433280.	0.137972	0.0689862	−	0.997618i	$-0.478024\pi$
$397$	433280.	0.137972	0.0689862	+	0.997618i	$0.478024\pi$
$398$	621811.	0.196766
$399$	0	0
$400$	154797.	0.0483740
$401$	1.13824e6	0.353488	0.176744	−	0.984257i	$-0.443444\pi$
$401$	1.13824e6	0.353488	0.176744	+	0.984257i	$0.443444\pi$
$402$	0	0
$403$	−181845.	−0.0557749
$404$	−1.05213e6	−0.320712
$405$	0	0
$406$	0	0
$407$	−468568.	−0.140212
$408$	0	0
$409$	−5.02438e6	−1.48516	−0.742581	−	0.669756i	$-0.766399\pi$
$409$	−5.02438e6	−1.48516	−0.742581	+	0.669756i	$0.766399\pi$
$410$	−1.74095e6	−0.511478
$411$	0	0
$412$	2.72016e6	0.789499
$413$	0	0
$414$	0	0
$415$	385440.	0.109859
$416$	−35395.2	−0.0100279
$417$	0	0
$418$	−66106.6	−0.0185056
$419$	2.57295e6	0.715974	0.357987	−	0.933727i	$-0.383463\pi$
$419$	2.57295e6	0.715974	0.357987	+	0.933727i	$0.383463\pi$
$420$	0	0
$421$	336425.	0.0925089	0.0462545	−	0.998930i	$-0.485271\pi$
$421$	336425.	0.0925089	0.0462545	+	0.998930i	$0.485271\pi$
$422$	−15592.9	−0.00426232
$423$	0	0
$424$	−449939.	−0.121546
$425$	1.24626e6	0.334685
$426$	0	0
$427$	0	0
$428$	1.72693e6	0.455685
$429$	0	0
$430$	−2.72174e6	−0.709864
$431$	−236410.	−0.0613018	−0.0306509	−	0.999530i	$-0.509758\pi$
$431$	−236410.	−0.0613018	−0.0306509	+	0.999530i	$0.509758\pi$
$432$	0	0
$433$	−2.33004e6	−0.597232	−0.298616	−	0.954373i	$-0.596525\pi$
$433$	−2.33004e6	−0.597232	−0.298616	+	0.954373i	$0.596525\pi$
$434$	0	0
$435$	0	0
$436$	−3.56317e6	−0.897679
$437$	−761681.	−0.190796
$438$	0	0
$439$	3.19962e6	0.792387	0.396193	−	0.918167i	$-0.370331\pi$
$439$	3.19962e6	0.792387	0.396193	+	0.918167i	$0.370331\pi$
$440$	−142835.	−0.0351724
$441$	0	0
$442$	−284964.	−0.0693800
$443$	−1.13240e6	−0.274152	−0.137076	−	0.990561i	$-0.543770\pi$
$443$	−1.13240e6	−0.274152	−0.137076	+	0.990561i	$0.543770\pi$
$444$	0	0
$445$	3.14490e6	0.752848
$446$	−3.13029e6	−0.745157
$447$	0	0
$448$	0	0
$449$	8.23687e6	1.92817	0.964087	−	0.265585i	$-0.0855651\pi$
$449$	8.23687e6	1.92817	0.964087	+	0.265585i	$0.0855651\pi$
$450$	0	0
$451$	260441.	0.0602931
$452$	−2.89688e6	−0.666936
$453$	0	0
$454$	3.18617e6	0.725487
$455$	0	0
$456$	0	0
$457$	−365132.	−0.0817824	−0.0408912	−	0.999164i	$-0.513020\pi$
$457$	−365132.	−0.0817824	−0.0408912	+	0.999164i	$0.513020\pi$
$458$	−1.54818e6	−0.344872
$459$	0	0
$460$	−1.64574e6	−0.362633
$461$	−332567.	−0.0728832	−0.0364416	−	0.999336i	$-0.511602\pi$
$461$	−332567.	−0.0728832	−0.0364416	+	0.999336i	$0.511602\pi$
$462$	0	0
$463$	1.69992e6	0.368533	0.184266	−	0.982876i	$-0.441009\pi$
$463$	1.69992e6	0.368533	0.184266	+	0.982876i	$0.441009\pi$
$464$	1.21990e6	0.263044
$465$	0	0
$466$	−1.53504e6	−0.327456
$467$	990465.	0.210159	0.105079	−	0.994464i	$-0.466490\pi$
$467$	990465.	0.210159	0.105079	+	0.994464i	$0.466490\pi$
$468$	0	0
$469$	0	0
$470$	−5.72681e6	−1.19583
$471$	0	0
$472$	−2.83026e6	−0.584752
$473$	407164.	0.0836789
$474$	0	0
$475$	273457.	0.0556103
$476$	0	0
$477$	0	0
$478$	−1.86164e6	−0.372671
$479$	6.08065e6	1.21091	0.605454	−	0.795880i	$-0.292991\pi$
$479$	6.08065e6	1.21091	0.605454	+	0.795880i	$0.292991\pi$
$480$	0	0
$481$	−443198.	−0.0873445
$482$	1.39498e6	0.273495
$483$	0	0
$484$	−2.55545e6	−0.495854
$485$	−7.77785e6	−1.50143
$486$	0	0
$487$	−6.71098e6	−1.28222	−0.641112	−	0.767448i	$-0.721526\pi$
$487$	−6.71098e6	−1.28222	−0.641112	+	0.767448i	$0.721526\pi$
$488$	−1.24035e6	−0.235773
$489$	0	0
$490$	0	0
$491$	914042.	0.171105	0.0855525	−	0.996334i	$-0.472734\pi$
$491$	914042.	0.171105	0.0855525	+	0.996334i	$0.472734\pi$
$492$	0	0
$493$	9.82130e6	1.81992
$494$	−62527.5	−0.0115280
$495$	0	0
$496$	−1.34678e6	−0.245806
$497$	0	0
$498$	0	0
$499$	−7.75492e6	−1.39420	−0.697101	−	0.716972i	$-0.745527\pi$
$499$	−7.75492e6	−1.39420	−0.697101	+	0.716972i	$0.745527\pi$
$500$	−2.46270e6	−0.440542
$501$	0	0
$502$	−746172.	−0.132154
$503$	3.79381e6	0.668584	0.334292	−	0.942470i	$-0.391503\pi$
$503$	3.79381e6	0.668584	0.334292	+	0.942470i	$0.391503\pi$
$504$	0	0
$505$	−4.01591e6	−0.700737
$506$	246198.	0.0427472
$507$	0	0
$508$	1.46360e6	0.251630
$509$	8.53160e6	1.45961	0.729804	−	0.683657i	$-0.239611\pi$
$509$	8.53160e6	1.45961	0.729804	+	0.683657i	$0.239611\pi$
$510$	0	0
$511$	0	0
$512$	−262144.	−0.0441942
$513$	0	0
$514$	6.21419e6	1.03747
$515$	1.03827e7	1.72501
$516$	0	0
$517$	856712.	0.140964
$518$	0	0
$519$	0	0
$520$	−135101.	−0.0219104
$521$	−3.19066e6	−0.514976	−0.257488	−	0.966282i	$-0.582895\pi$
$521$	−3.19066e6	−0.514976	−0.257488	+	0.966282i	$0.582895\pi$
$522$	0	0
$523$	9.60678e6	1.53576	0.767880	−	0.640593i	$-0.221311\pi$
$523$	9.60678e6	1.53576	0.767880	+	0.640593i	$0.221311\pi$
$524$	309502.	0.0492420
$525$	0	0
$526$	−5.03838e6	−0.794012
$527$	−1.08429e7	−1.70066
$528$	0	0
$529$	−3.59965e6	−0.559270
$530$	−1.71739e6	−0.265570
$531$	0	0
$532$	0	0
$533$	246340.	0.0375592
$534$	0	0
$535$	6.59158e6	0.995646
$536$	−1.34042e6	−0.201525
$537$	0	0
$538$	−681844.	−0.101562
$539$	0	0
$540$	0	0
$541$	1.02366e7	1.50371	0.751855	−	0.659328i	$-0.229159\pi$
$541$	1.02366e7	1.50371	0.751855	+	0.659328i	$0.229159\pi$
$542$	−8.90972e6	−1.30276
$543$	0	0
$544$	−2.11050e6	−0.305766
$545$	−1.36004e7	−1.96138
$546$	0	0
$547$	−9.27757e6	−1.32576	−0.662882	−	0.748724i	$-0.730667\pi$
$547$	−9.27757e6	−1.32576	−0.662882	+	0.748724i	$0.730667\pi$
$548$	49232.6	0.00700327
$549$	0	0
$550$	−88389.4	−0.0124593
$551$	2.15501e6	0.302392
$552$	0	0
$553$	0	0
$554$	−7.93357e6	−1.09823
$555$	0	0
$556$	5.92001e6	0.812149
$557$	−620098.	−0.0846881	−0.0423441	−	0.999103i	$-0.513483\pi$
$557$	−620098.	−0.0846881	−0.0423441	+	0.999103i	$0.513483\pi$
$558$	0	0
$559$	385119.	0.0521273
$560$	0	0
$561$	0	0
$562$	−4.66750e6	−0.623366
$563$	7.36664e6	0.979487	0.489744	−	0.871867i	$-0.337090\pi$
$563$	7.36664e6	0.979487	0.489744	+	0.871867i	$0.337090\pi$
$564$	0	0
$565$	−1.10572e7	−1.45722
$566$	7.60360e6	0.997650
$567$	0	0
$568$	5.10998e6	0.664581
$569$	−3.68222e6	−0.476792	−0.238396	−	0.971168i	$-0.576622\pi$
$569$	−3.68222e6	−0.476792	−0.238396	+	0.971168i	$0.576622\pi$
$570$	0	0
$571$	1.03320e7	1.32615	0.663077	−	0.748551i	$-0.269250\pi$
$571$	1.03320e7	1.32615	0.663077	+	0.748551i	$0.269250\pi$
$572$	20210.7	0.00258281
$573$	0	0
$574$	0	0
$575$	−1.01842e6	−0.128457
$576$	0	0
$577$	1.25123e7	1.56458	0.782292	−	0.622911i	$-0.214050\pi$
$577$	1.25123e7	1.56458	0.782292	+	0.622911i	$0.214050\pi$
$578$	−1.13121e7	−1.40839
$579$	0	0
$580$	4.65627e6	0.574736
$581$	0	0
$582$	0	0
$583$	256916.	0.0313055
$584$	2.37221e6	0.287820
$585$	0	0
$586$	6.60034e6	0.794004
$587$	−1.60865e7	−1.92694	−0.963468	−	0.267823i	$-0.913696\pi$
$587$	−1.60865e7	−1.92694	−0.963468	+	0.267823i	$0.913696\pi$
$588$	0	0
$589$	−2.37916e6	−0.282577
$590$	−1.08029e7	−1.27765
$591$	0	0
$592$	−3.28242e6	−0.384937
$593$	−9.46330e6	−1.10511	−0.552556	−	0.833476i	$-0.686347\pi$
$593$	−9.46330e6	−1.10511	−0.552556	+	0.833476i	$0.686347\pi$
$594$	0	0
$595$	0	0
$596$	7.26592e6	0.837866
$597$	0	0
$598$	232868.	0.0266291
$599$	−5.01498e6	−0.571087	−0.285544	−	0.958366i	$-0.592174\pi$
$599$	−5.01498e6	−0.571087	−0.285544	+	0.958366i	$0.592174\pi$
$600$	0	0
$601$	−1.26473e7	−1.42828	−0.714139	−	0.700004i	$-0.753182\pi$
$601$	−1.26473e7	−1.42828	−0.714139	+	0.700004i	$0.753182\pi$
$602$	0	0
$603$	0	0
$604$	2.84147e6	0.316921
$605$	−9.75400e6	−1.08341
$606$	0	0
$607$	2.87702e6	0.316936	0.158468	−	0.987364i	$-0.449345\pi$
$607$	2.87702e6	0.316936	0.158468	+	0.987364i	$0.449345\pi$
$608$	−463092.	−0.0508052
$609$	0	0
$610$	−4.73434e6	−0.515152
$611$	810328.	0.0878128
$612$	0	0
$613$	−3.72582e6	−0.400471	−0.200236	−	0.979748i	$-0.564171\pi$
$613$	−3.72582e6	−0.400471	−0.200236	+	0.979748i	$0.564171\pi$
$614$	−2.39174e6	−0.256032
$615$	0	0
$616$	0	0
$617$	1.15861e7	1.22525	0.612625	−	0.790374i	$-0.290114\pi$
$617$	1.15861e7	1.22525	0.612625	+	0.790374i	$0.290114\pi$
$618$	0	0
$619$	−1.85608e6	−0.194702	−0.0973508	−	0.995250i	$-0.531037\pi$
$619$	−1.85608e6	−0.194702	−0.0973508	+	0.995250i	$0.531037\pi$
$620$	−5.14059e6	−0.537073
$621$	0	0
$622$	921921.	0.0955471
$623$	0	0
$624$	0	0
$625$	−1.12896e7	−1.15606
$626$	−1.77031e6	−0.180557
$627$	0	0
$628$	−3.98370e6	−0.403076
$629$	−2.64266e7	−2.66326
$630$	0	0
$631$	8.92135e6	0.891984	0.445992	−	0.895037i	$-0.352851\pi$
$631$	8.92135e6	0.891984	0.445992	+	0.895037i	$0.352851\pi$
$632$	−2.69261e6	−0.268152
$633$	0	0
$634$	−5.04990e6	−0.498953
$635$	5.58646e6	0.549797
$636$	0	0
$637$	0	0
$638$	−696563.	−0.0677499
$639$	0	0
$640$	−1.00059e6	−0.0965618
$641$	−1.31866e7	−1.26762	−0.633810	−	0.773489i	$-0.718510\pi$
$641$	−1.31866e7	−1.26762	−0.633810	+	0.773489i	$0.718510\pi$
$642$	0	0
$643$	−9.67813e6	−0.923132	−0.461566	−	0.887106i	$-0.652712\pi$
$643$	−9.67813e6	−0.923132	−0.461566	+	0.887106i	$0.652712\pi$
$644$	0	0
$645$	0	0
$646$	−3.72832e6	−0.351505
$647$	1.72491e6	0.161996	0.0809982	−	0.996714i	$-0.474189\pi$
$647$	1.72491e6	0.161996	0.0809982	+	0.996714i	$0.474189\pi$
$648$	0	0
$649$	1.61609e6	0.150610
$650$	−83603.8	−0.00776145
$651$	0	0
$652$	3.23596e6	0.298116
$653$	−597510.	−0.0548356	−0.0274178	−	0.999624i	$-0.508728\pi$
$653$	−597510.	−0.0548356	−0.0274178	+	0.999624i	$0.508728\pi$
$654$	0	0
$655$	1.18135e6	0.107591
$656$	1.82444e6	0.165528
$657$	0	0
$658$	0	0
$659$	−1.47605e7	−1.32400	−0.662001	−	0.749503i	$-0.730293\pi$
$659$	−1.47605e7	−1.32400	−0.662001	+	0.749503i	$0.730293\pi$
$660$	0	0
$661$	1.69041e7	1.50484	0.752418	−	0.658686i	$-0.228888\pi$
$661$	1.69041e7	1.50484	0.752418	+	0.658686i	$0.228888\pi$
$662$	1.87574e6	0.166352
$663$	0	0
$664$	−403925.	−0.0355534
$665$	0	0
$666$	0	0
$667$	−8.02581e6	−0.698512
$668$	−7.61264e6	−0.660076
$669$	0	0
$670$	−5.11631e6	−0.440321
$671$	708243.	0.0607262
$672$	0	0
$673$	−1.58960e7	−1.35285	−0.676425	−	0.736512i	$-0.736472\pi$
$673$	−1.58960e7	−1.35285	−0.676425	+	0.736512i	$0.736472\pi$
$674$	9.87191e6	0.837050
$675$	0	0
$676$	−5.92157e6	−0.498391
$677$	1.95361e6	0.163820	0.0819099	−	0.996640i	$-0.473898\pi$
$677$	1.95361e6	0.163820	0.0819099	+	0.996640i	$0.473898\pi$
$678$	0	0
$679$	0	0
$680$	−8.05567e6	−0.668081
$681$	0	0
$682$	769016.	0.0633103
$683$	9.23970e6	0.757890	0.378945	−	0.925419i	$-0.376287\pi$
$683$	9.23970e6	0.757890	0.378945	+	0.925419i	$0.376287\pi$
$684$	0	0
$685$	187918.	0.0153018
$686$	0	0
$687$	0	0
$688$	2.85227e6	0.229731
$689$	243006.	0.0195016
$690$	0	0
$691$	−1.45698e7	−1.16080	−0.580401	−	0.814331i	$-0.697104\pi$
$691$	−1.45698e7	−1.16080	−0.580401	+	0.814331i	$0.697104\pi$
$692$	4.55525e6	0.361616
$693$	0	0
$694$	−1.71300e7	−1.35008
$695$	2.25964e7	1.77450
$696$	0	0
$697$	1.46885e7	1.14524
$698$	7.39995e6	0.574898
$699$	0	0
$700$	0	0
$701$	3.70190e6	0.284531	0.142266	−	0.989829i	$-0.454561\pi$
$701$	3.70190e6	0.284531	0.142266	+	0.989829i	$0.454561\pi$
$702$	0	0
$703$	−5.79858e6	−0.442520
$704$	149685.	0.0113827
$705$	0	0
$706$	−9.09518e6	−0.686751
$707$	0	0
$708$	0	0
$709$	−2.52937e7	−1.88971	−0.944857	−	0.327484i	$-0.893799\pi$
$709$	−2.52937e7	−1.88971	−0.944857	+	0.327484i	$0.893799\pi$
$710$	1.95045e7	1.45207
$711$	0	0
$712$	−3.29573e6	−0.243642
$713$	8.86061e6	0.652739
$714$	0	0
$715$	77143.1	0.00564329
$716$	−5.56734e6	−0.405849
$717$	0	0
$718$	−1.75997e7	−1.27407
$719$	−1.45185e7	−1.04737	−0.523683	−	0.851913i	$-0.675443\pi$
$719$	−1.45185e7	−1.04737	−0.523683	+	0.851913i	$0.675443\pi$
$720$	0	0
$721$	0	0
$722$	9.08632e6	0.648702
$723$	0	0
$724$	6.07530e6	0.430746
$725$	2.88141e6	0.203592
$726$	0	0
$727$	−1.42855e7	−1.00244	−0.501222	−	0.865319i	$-0.667116\pi$
$727$	−1.42855e7	−1.00244	−0.501222	+	0.865319i	$0.667116\pi$
$728$	0	0
$729$	0	0
$730$	9.05460e6	0.628872
$731$	2.29635e7	1.58944
$732$	0	0
$733$	−8.21556e6	−0.564777	−0.282389	−	0.959300i	$-0.591127\pi$
$733$	−8.21556e6	−0.564777	−0.282389	+	0.959300i	$0.591127\pi$
$734$	7.58973e6	0.519979
$735$	0	0
$736$	1.72467e6	0.117358
$737$	765384.	0.0519052
$738$	0	0
$739$	−3.43896e6	−0.231641	−0.115821	−	0.993270i	$-0.536950\pi$
$739$	−3.43896e6	−0.231641	−0.115821	+	0.993270i	$0.536950\pi$
$740$	−1.25288e7	−0.841067
$741$	0	0
$742$	0	0
$743$	−1.53588e6	−0.102067	−0.0510334	−	0.998697i	$-0.516252\pi$
$743$	−1.53588e6	−0.102067	−0.0510334	+	0.998697i	$0.516252\pi$
$744$	0	0
$745$	2.77336e7	1.83069
$746$	−1.82504e7	−1.20067
$747$	0	0
$748$	1.20510e6	0.0787535
$749$	0	0
$750$	0	0
$751$	−2.01146e7	−1.30140	−0.650700	−	0.759335i	$-0.725524\pi$
$751$	−2.01146e7	−1.30140	−0.650700	+	0.759335i	$0.725524\pi$
$752$	6.00146e6	0.387001
$753$	0	0
$754$	−658850.	−0.0422044
$755$	1.08457e7	0.692454
$756$	0	0
$757$	202045.	0.0128147	0.00640733	−	0.999979i	$-0.497960\pi$
$757$	202045.	0.0128147	0.00640733	+	0.999979i	$0.497960\pi$
$758$	−1.99760e7	−1.26280
$759$	0	0
$760$	−1.76759e6	−0.111007
$761$	1.08202e6	0.0677290	0.0338645	−	0.999426i	$-0.489219\pi$
$761$	1.08202e6	0.0677290	0.0338645	+	0.999426i	$0.489219\pi$
$762$	0	0
$763$	0	0
$764$	−3.11188e6	−0.192881
$765$	0	0
$766$	−1.47163e7	−0.906205
$767$	1.52859e6	0.0938214
$768$	0	0
$769$	1.19305e7	0.727517	0.363759	−	0.931493i	$-0.381493\pi$
$769$	1.19305e7	0.727517	0.363759	+	0.931493i	$0.381493\pi$
$770$	0	0
$771$	0	0
$772$	1.40384e7	0.847763
$773$	6.58657e6	0.396470	0.198235	−	0.980155i	$-0.436479\pi$
$773$	6.58657e6	0.396470	0.198235	+	0.980155i	$0.436479\pi$
$774$	0	0
$775$	−3.18112e6	−0.190250
$776$	8.15087e6	0.485903
$777$	0	0
$778$	1.02894e6	0.0609451
$779$	3.22298e6	0.190289
$780$	0	0
$781$	−2.91781e6	−0.171171
$782$	1.38852e7	0.811961
$783$	0	0
$784$	0	0
$785$	−1.52055e7	−0.880700
$786$	0	0
$787$	−1.92720e7	−1.10915	−0.554575	−	0.832134i	$-0.687119\pi$
$787$	−1.92720e7	−1.10915	−0.554575	+	0.832134i	$0.687119\pi$
$788$	1.20379e7	0.690613
$789$	0	0
$790$	−1.02775e7	−0.585897
$791$	0	0
$792$	0	0
$793$	669897.	0.0378290
$794$	−1.73312e6	−0.0975613
$795$	0	0
$796$	−2.48724e6	−0.139135
$797$	−1.54906e7	−0.863816	−0.431908	−	0.901918i	$-0.642159\pi$
$797$	−1.54906e7	−0.863816	−0.431908	+	0.901918i	$0.642159\pi$
$798$	0	0
$799$	4.83174e7	2.67754
$800$	−619188.	−0.0342056
$801$	0	0
$802$	−4.55298e6	−0.249954
$803$	−1.35454e6	−0.0741315
$804$	0	0
$805$	0	0
$806$	727380.	0.0394388
$807$	0	0
$808$	4.20851e6	0.226777
$809$	1.77071e6	0.0951210	0.0475605	−	0.998868i	$-0.484855\pi$
$809$	1.77071e6	0.0951210	0.0475605	+	0.998868i	$0.484855\pi$
$810$	0	0
$811$	6.80547e6	0.363334	0.181667	−	0.983360i	$-0.441851\pi$
$811$	6.80547e6	0.363334	0.181667	+	0.983360i	$0.441851\pi$
$812$	0	0
$813$	0	0
$814$	1.87427e6	0.0991451
$815$	1.23515e7	0.651366
$816$	0	0
$817$	5.03869e6	0.264097
$818$	2.00975e7	1.05017
$819$	0	0
$820$	6.96380e6	0.361669
$821$	−2.27127e7	−1.17601	−0.588006	−	0.808857i	$-0.700087\pi$
$821$	−2.27127e7	−1.17601	−0.588006	+	0.808857i	$0.700087\pi$
$822$	0	0
$823$	−3.03576e7	−1.56231	−0.781156	−	0.624336i	$-0.785370\pi$
$823$	−3.03576e7	−1.56231	−0.781156	+	0.624336i	$0.785370\pi$
$824$	−1.08807e7	−0.558260
$825$	0	0
$826$	0	0
$827$	5.80776e6	0.295287	0.147644	−	0.989041i	$-0.452831\pi$
$827$	5.80776e6	0.295287	0.147644	+	0.989041i	$0.452831\pi$
$828$	0	0
$829$	−9.04916e6	−0.457322	−0.228661	−	0.973506i	$-0.573435\pi$
$829$	−9.04916e6	−0.457322	−0.228661	+	0.973506i	$0.573435\pi$
$830$	−1.54176e6	−0.0776822
$831$	0	0
$832$	141581.	0.00709080
$833$	0	0
$834$	0	0
$835$	−2.90570e7	−1.44223
$836$	264426.	0.0130855
$837$	0	0
$838$	−1.02918e7	−0.506270
$839$	8.17101e6	0.400747	0.200374	−	0.979720i	$-0.435784\pi$
$839$	8.17101e6	0.400747	0.200374	+	0.979720i	$0.435784\pi$
$840$	0	0
$841$	2.19614e6	0.107071
$842$	−1.34570e6	−0.0654137
$843$	0	0
$844$	62371.6	0.00301391
$845$	−2.26023e7	−1.08896
$846$	0	0
$847$	0	0
$848$	1.79976e6	0.0859457
$849$	0	0
$850$	−4.98504e6	−0.236658
$851$	2.15954e7	1.02220
$852$	0	0
$853$	2.88437e7	1.35731	0.678654	−	0.734458i	$-0.262564\pi$
$853$	2.88437e7	1.35731	0.678654	+	0.734458i	$0.262564\pi$
$854$	0	0
$855$	0	0
$856$	−6.90770e6	−0.322218
$857$	2.69462e7	1.25327	0.626637	−	0.779311i	$-0.284431\pi$
$857$	2.69462e7	1.25327	0.626637	+	0.779311i	$0.284431\pi$
$858$	0	0
$859$	−7.49124e6	−0.346394	−0.173197	−	0.984887i	$-0.555410\pi$
$859$	−7.49124e6	−0.346394	−0.173197	+	0.984887i	$0.555410\pi$
$860$	1.08870e7	0.501950
$861$	0	0
$862$	945641.	0.0433469
$863$	1.41552e6	0.0646977	0.0323489	−	0.999477i	$-0.489701\pi$
$863$	1.41552e6	0.0646977	0.0323489	+	0.999477i	$0.489701\pi$
$864$	0	0
$865$	1.73871e7	0.790110
$866$	9.32015e6	0.422307
$867$	0	0
$868$	0	0
$869$	1.53749e6	0.0690656
$870$	0	0
$871$	723944.	0.0323340
$872$	1.42527e7	0.634755
$873$	0	0
$874$	3.04672e6	0.134913
$875$	0	0
$876$	0	0
$877$	2.55232e7	1.12056	0.560281	−	0.828303i	$-0.310693\pi$
$877$	2.55232e7	1.12056	0.560281	+	0.828303i	$0.310693\pi$
$878$	−1.27985e7	−0.560302
$879$	0	0
$880$	571338.	0.0248706
$881$	2.07516e6	0.0900764	0.0450382	−	0.998985i	$-0.485659\pi$
$881$	2.07516e6	0.0900764	0.0450382	+	0.998985i	$0.485659\pi$
$882$	0	0
$883$	1.80641e7	0.779677	0.389839	−	0.920883i	$-0.372531\pi$
$883$	1.80641e7	0.779677	0.389839	+	0.920883i	$0.372531\pi$
$884$	1.13986e6	0.0490591
$885$	0	0
$886$	4.52961e6	0.193855
$887$	9.28389e6	0.396206	0.198103	−	0.980181i	$-0.436522\pi$
$887$	9.28389e6	0.396206	0.198103	+	0.980181i	$0.436522\pi$
$888$	0	0
$889$	0	0
$890$	−1.25796e7	−0.532344
$891$	0	0
$892$	1.25212e7	0.526906
$893$	1.06019e7	0.444893
$894$	0	0
$895$	−2.12502e7	−0.886758
$896$	0	0
$897$	0	0
$898$	−3.29475e7	−1.36343
$899$	−2.50692e7	−1.03452
$900$	0	0
$901$	1.44897e7	0.594631
$902$	−1.04176e6	−0.0426336
$903$	0	0
$904$	1.15875e7	0.471595
$905$	2.31891e7	0.941156
$906$	0	0
$907$	−4.20818e7	−1.69854	−0.849270	−	0.527958i	$-0.822958\pi$
$907$	−4.20818e7	−1.69854	−0.849270	+	0.527958i	$0.822958\pi$
$908$	−1.27447e7	−0.512997
$909$	0	0
$910$	0	0
$911$	−2.33510e7	−0.932199	−0.466100	−	0.884732i	$-0.654341\pi$
$911$	−2.33510e7	−0.932199	−0.466100	+	0.884732i	$0.654341\pi$
$912$	0	0
$913$	230642.	0.00915719
$914$	1.46053e6	0.0578289
$915$	0	0
$916$	6.19272e6	0.243861
$917$	0	0
$918$	0	0
$919$	1.54594e7	0.603817	0.301908	−	0.953337i	$-0.402376\pi$
$919$	1.54594e7	0.603817	0.301908	+	0.953337i	$0.402376\pi$
$920$	6.58296e6	0.256420
$921$	0	0
$922$	1.33027e6	0.0515362
$923$	−2.75983e6	−0.106630
$924$	0	0
$925$	−7.75312e6	−0.297935
$926$	−6.79968e6	−0.260592
$927$	0	0
$928$	−4.87958e6	−0.186000
$929$	−1.49993e7	−0.570204	−0.285102	−	0.958497i	$-0.592028\pi$
$929$	−1.49993e7	−0.570204	−0.285102	+	0.958497i	$0.592028\pi$
$930$	0	0
$931$	0	0
$932$	6.14014e6	0.231547
$933$	0	0
$934$	−3.96186e6	−0.148605
$935$	4.59981e6	0.172072
$936$	0	0
$937$	1.52742e7	0.568343	0.284172	−	0.958773i	$-0.408281\pi$
$937$	1.52742e7	0.568343	0.284172	+	0.958773i	$0.408281\pi$
$938$	0	0
$939$	0	0
$940$	2.29072e7	0.845576
$941$	5.13600e6	0.189082	0.0945412	−	0.995521i	$-0.469862\pi$
$941$	5.13600e6	0.189082	0.0945412	+	0.995521i	$0.469862\pi$
$942$	0	0
$943$	−1.20032e7	−0.439559
$944$	1.13210e7	0.413482
$945$	0	0
$946$	−1.62865e6	−0.0591699
$947$	4.50054e7	1.63076	0.815379	−	0.578928i	$-0.196529\pi$
$947$	4.50054e7	1.63076	0.815379	+	0.578928i	$0.196529\pi$
$948$	0	0
$949$	−1.28120e6	−0.0461798
$950$	−1.09383e6	−0.0393224
$951$	0	0
$952$	0	0
$953$	4.49596e7	1.60358	0.801789	−	0.597607i	$-0.203882\pi$
$953$	4.49596e7	1.60358	0.801789	+	0.597607i	$0.203882\pi$
$954$	0	0
$955$	−1.18779e7	−0.421434
$956$	7.44655e6	0.263518
$957$	0	0
$958$	−2.43226e7	−0.856242
$959$	0	0
$960$	0	0
$961$	−952428.	−0.0332678
$962$	1.77279e6	0.0617619
$963$	0	0
$964$	−5.57991e6	−0.193390
$965$	5.35838e7	1.85232
$966$	0	0
$967$	1.04364e7	0.358907	0.179454	−	0.983766i	$-0.442567\pi$
$967$	1.04364e7	0.358907	0.179454	+	0.983766i	$0.442567\pi$
$968$	1.02218e7	0.350622
$969$	0	0
$970$	3.11114e7	1.06167
$971$	1.72207e7	0.586142	0.293071	−	0.956091i	$-0.405323\pi$
$971$	1.72207e7	0.586142	0.293071	+	0.956091i	$0.405323\pi$
$972$	0	0
$973$	0	0
$974$	2.68439e7	0.906669
$975$	0	0
$976$	4.96140e6	0.166717
$977$	−3.24376e7	−1.08721	−0.543603	−	0.839343i	$-0.682940\pi$
$977$	−3.24376e7	−1.08721	−0.543603	+	0.839343i	$0.682940\pi$
$978$	0	0
$979$	1.88187e6	0.0627528
$980$	0	0
$981$	0	0
$982$	−3.65617e6	−0.120989
$983$	−3.19804e7	−1.05560	−0.527801	−	0.849368i	$-0.676983\pi$
$983$	−3.19804e7	−1.05560	−0.527801	+	0.849368i	$0.676983\pi$
$984$	0	0
$985$	4.59479e7	1.50895
$986$	−3.92852e7	−1.28688
$987$	0	0
$988$	250110.	0.00815152
$989$	−1.87654e7	−0.610051
$990$	0	0
$991$	5.49614e7	1.77776	0.888880	−	0.458139i	$-0.151484\pi$
$991$	5.49614e7	1.77776	0.888880	+	0.458139i	$0.151484\pi$
$992$	5.38713e6	0.173811
$993$	0	0
$994$	0	0
$995$	−9.49367e6	−0.304002
$996$	0	0
$997$	7.51010e6	0.239281	0.119640	−	0.992817i	$-0.461826\pi$
$997$	7.51010e6	0.239281	0.119640	+	0.992817i	$0.461826\pi$
$998$	3.10197e7	0.985850
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.bj.1.2		2
3.2	odd	2		294.6.a.u.1.1	yes	2
7.6	odd	2		882.6.a.z.1.1		2
21.2	odd	6		294.6.e.u.67.2		4
21.5	even	6		294.6.e.v.67.1		4
21.11	odd	6		294.6.e.u.79.2		4
21.17	even	6		294.6.e.v.79.1		4
21.20	even	2		294.6.a.t.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.6.a.t.1.2	✓	2	21.20	even	2
294.6.a.u.1.1	yes	2	3.2	odd	2
294.6.e.u.67.2		4	21.2	odd	6
294.6.e.u.79.2		4	21.11	odd	6
294.6.e.v.67.1		4	21.5	even	6
294.6.e.v.79.1		4	21.17	even	6
882.6.a.z.1.1		2	7.6	odd	2
882.6.a.bj.1.2		2	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +61.0711 q^{5} -64.0000 q^{8} +O(q^{10})\)	\(q-4.00000 q^{2} +16.0000 q^{4} +61.0711 q^{5} -64.0000 q^{8} -244.284 q^{10} +36.5442 q^{11} +34.5656 q^{13} +256.000 q^{16} +2061.04 q^{17} +452.238 q^{19} +977.137 q^{20} -146.177 q^{22} -1684.25 q^{23} +604.675 q^{25} -138.262 q^{26} +4765.22 q^{29} -5260.87 q^{31} -1024.00 q^{32} -8244.16 q^{34} -12822.0 q^{37} -1808.95 q^{38} -3908.55 q^{40} +7126.74 q^{41} +11141.7 q^{43} +584.706 q^{44} +6736.99 q^{46} +23443.2 q^{47} -2418.70 q^{50} +553.049 q^{52} +7030.30 q^{53} +2231.79 q^{55} -19060.9 q^{58} +44222.8 q^{59} +19380.5 q^{61} +21043.5 q^{62} +4096.00 q^{64} +2110.96 q^{65} +20944.1 q^{67} +32976.6 q^{68} -79843.4 q^{71} -37065.8 q^{73} +51287.8 q^{74} +7235.81 q^{76} +42072.0 q^{79} +15634.2 q^{80} -28506.9 q^{82} +6311.34 q^{83} +125870. q^{85} -44566.8 q^{86} -2338.83 q^{88} +51495.8 q^{89} -26948.0 q^{92} -93772.8 q^{94} +27618.7 q^{95} -127357. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} + 108 q^{5} - 128 q^{8}+O(q^{10}) \) 2 * q - 8 * q^2 + 32 * q^4 + 108 * q^5 - 128 * q^8	\( 2 q - 8 q^{2} + 32 q^{4} + 108 q^{5} - 128 q^{8} - 432 q^{10} + 124 q^{11} - 720 q^{13} + 512 q^{16} + 612 q^{17} - 2088 q^{19} + 1728 q^{20} - 496 q^{22} - 772 q^{23} - 318 q^{25} + 2880 q^{26} + 4592 q^{29} - 9792 q^{31} - 2048 q^{32} - 2448 q^{34} - 5992 q^{37} + 8352 q^{38} - 6912 q^{40} + 20196 q^{41} - 1136 q^{43} + 1984 q^{44} + 3088 q^{46} + 36936 q^{47} + 1272 q^{50} - 11520 q^{52} + 16708 q^{53} + 6336 q^{55} - 18368 q^{58} + 74592 q^{59} + 18648 q^{61} + 39168 q^{62} + 8192 q^{64} - 33300 q^{65} + 67344 q^{67} + 9792 q^{68} - 76548 q^{71} - 47304 q^{73} + 23968 q^{74} - 33408 q^{76} + 140656 q^{79} + 27648 q^{80} - 80784 q^{82} + 94104 q^{83} + 57868 q^{85} + 4544 q^{86} - 7936 q^{88} - 17604 q^{89} - 12352 q^{92} - 147744 q^{94} - 91592 q^{95} - 85176 q^{97}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 + 108 * q^5 - 128 * q^8 - 432 * q^10 + 124 * q^11 - 720 * q^13 + 512 * q^16 + 612 * q^17 - 2088 * q^19 + 1728 * q^20 - 496 * q^22 - 772 * q^23 - 318 * q^25 + 2880 * q^26 + 4592 * q^29 - 9792 * q^31 - 2048 * q^32 - 2448 * q^34 - 5992 * q^37 + 8352 * q^38 - 6912 * q^40 + 20196 * q^41 - 1136 * q^43 + 1984 * q^44 + 3088 * q^46 + 36936 * q^47 + 1272 * q^50 - 11520 * q^52 + 16708 * q^53 + 6336 * q^55 - 18368 * q^58 + 74592 * q^59 + 18648 * q^61 + 39168 * q^62 + 8192 * q^64 - 33300 * q^65 + 67344 * q^67 + 9792 * q^68 - 76548 * q^71 - 47304 * q^73 + 23968 * q^74 - 33408 * q^76 + 140656 * q^79 + 27648 * q^80 - 80784 * q^82 + 94104 * q^83 + 57868 * q^85 + 4544 * q^86 - 7936 * q^88 - 17604 * q^89 - 12352 * q^92 - 147744 * q^94 - 91592 * q^95 - 85176 * q^97

Introduction

L-functions

Modular forms

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