LMFDB - Embedded newform 441.4.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.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:	$26.0198423125$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.57516.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 24x + 6 $ x^3 - x^2 - 24*x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.55637$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.55637 q^{2} +12.7605 q^{4} -17.8732 q^{5} +21.6905 q^{8} -81.4369 q^{10} +11.3942 q^{11} -13.0987 q^{13} -3.25412 q^{16} -53.2674 q^{17} -42.4223 q^{19} -228.071 q^{20} +51.9159 q^{22} -152.085 q^{23} +194.451 q^{25} -59.6823 q^{26} -186.493 q^{29} -157.874 q^{31} -188.351 q^{32} -242.706 q^{34} +3.74588 q^{37} -193.291 q^{38} -387.678 q^{40} +39.3230 q^{41} +429.439 q^{43} +145.395 q^{44} -692.957 q^{46} -21.1869 q^{47} +885.992 q^{50} -167.145 q^{52} -365.904 q^{53} -203.650 q^{55} -849.732 q^{58} +226.578 q^{59} +651.973 q^{61} -719.331 q^{62} -832.161 q^{64} +234.115 q^{65} +145.433 q^{67} -679.717 q^{68} +368.962 q^{71} +608.906 q^{73} +17.0676 q^{74} -541.328 q^{76} +910.237 q^{79} +58.1615 q^{80} +179.170 q^{82} +327.929 q^{83} +952.058 q^{85} +1956.68 q^{86} +247.144 q^{88} +37.6118 q^{89} -1940.68 q^{92} -96.5352 q^{94} +758.222 q^{95} +722.013 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 25 q^{4} - 11 q^{5} - 39 q^{8} - 55 q^{10} - 35 q^{11} + 62 q^{13} + 241 q^{16} - 48 q^{17} - 202 q^{19} - 439 q^{20} - 7 q^{22} - 216 q^{23} + 130 q^{25} - 274 q^{26} - 53 q^{29} - 95 q^{31}+ \cdots + 19 q^{97}+O(q^{100}) $ 3 * q - q^2 + 25 * q^4 - 11 * q^5 - 39 * q^8 - 55 * q^10 - 35 * q^11 + 62 * q^13 + 241 * q^16 - 48 * q^17 - 202 * q^19 - 439 * q^20 - 7 * q^22 - 216 * q^23 + 130 * q^25 - 274 * q^26 - 53 * q^29 - 95 * q^31 - 683 * q^32 - 24 * q^34 + 262 * q^37 + 398 * q^38 + 21 * q^40 - 244 * q^41 + 360 * q^43 + 905 * q^44 - 1056 * q^46 + 210 * q^47 + 1378 * q^50 + 324 * q^52 - 393 * q^53 - 1031 * q^55 - 1249 * q^58 - 1143 * q^59 - 70 * q^61 - 1059 * q^62 - 399 * q^64 + 472 * q^65 - 628 * q^67 - 1944 * q^68 - 318 * q^71 + 988 * q^73 - 1002 * q^74 - 2340 * q^76 + 861 * q^79 - 175 * q^80 + 124 * q^82 - 519 * q^83 + 1800 * q^85 + 3208 * q^86 - 891 * q^88 - 1766 * q^89 + 672 * q^92 - 3294 * q^94 + 736 * q^95 + 19 * q^97

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 $

$2$	4.55637	1.61092	0.805459	−	0.592651i	$-0.201919\pi$
$2$	4.55637	1.61092	0.805459	+	0.592651i	$0.201919\pi$
$3$	0	0
$3$	0	0
$4$	12.7605	1.59506
$5$	−17.8732	−1.59863	−0.799314	−	0.600914i	$-0.794804\pi$
$5$	−17.8732	−1.59863	−0.799314	+	0.600914i	$0.794804\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	21.6905	0.958592
$9$	0	0
$10$	−81.4369	−2.57526
$11$	11.3942	0.312315	0.156158	−	0.987732i	$-0.450089\pi$
$11$	11.3942	0.312315	0.156158	+	0.987732i	$0.450089\pi$
$12$	0	0
$13$	−13.0987	−0.279455	−0.139728	−	0.990190i	$-0.544623\pi$
$13$	−13.0987	−0.279455	−0.139728	+	0.990190i	$0.544623\pi$
$14$	0	0
$15$	0	0
$16$	−3.25412	−0.0508456
$17$	−53.2674	−0.759955	−0.379977	−	0.924996i	$-0.624068\pi$
$17$	−53.2674	−0.759955	−0.379977	+	0.924996i	$0.624068\pi$
$18$	0	0
$19$	−42.4223	−0.512228	−0.256114	−	0.966647i	$-0.582442\pi$
$19$	−42.4223	−0.512228	−0.256114	+	0.966647i	$0.582442\pi$
$20$	−228.071	−2.54991
$21$	0	0
$22$	51.9159	0.503114
$23$	−152.085	−1.37878	−0.689391	−	0.724389i	$-0.742122\pi$
$23$	−152.085	−1.37878	−0.689391	+	0.724389i	$0.742122\pi$
$24$	0	0
$25$	194.451	1.55561
$26$	−59.6823	−0.450180
$27$	0	0
$28$	0	0
$29$	−186.493	−1.19417	−0.597085	−	0.802178i	$-0.703675\pi$
$29$	−186.493	−1.19417	−0.597085	+	0.802178i	$0.703675\pi$
$30$	0	0
$31$	−157.874	−0.914676	−0.457338	−	0.889293i	$-0.651197\pi$
$31$	−157.874	−0.914676	−0.457338	+	0.889293i	$0.651197\pi$
$32$	−188.351	−1.04050
$33$	0	0
$34$	−242.706	−1.22423
$35$	0	0
$36$	0	0
$37$	3.74588	0.0166438	0.00832188	−	0.999965i	$-0.497351\pi$
$37$	3.74588	0.0166438	0.00832188	+	0.999965i	$0.497351\pi$
$38$	−193.291	−0.825158
$39$	0	0
$40$	−387.678	−1.53243
$41$	39.3230	0.149786	0.0748930	−	0.997192i	$-0.476138\pi$
$41$	39.3230	0.149786	0.0748930	+	0.997192i	$0.476138\pi$
$42$	0	0
$43$	429.439	1.52300	0.761498	−	0.648168i	$-0.224464\pi$
$43$	429.439	1.52300	0.761498	+	0.648168i	$0.224464\pi$
$44$	145.395	0.498161
$45$	0	0
$46$	−692.957	−2.22111
$47$	−21.1869	−0.0657537	−0.0328768	−	0.999459i	$-0.510467\pi$
$47$	−21.1869	−0.0657537	−0.0328768	+	0.999459i	$0.510467\pi$
$48$	0	0
$49$	0	0
$50$	885.992	2.50596
$51$	0	0
$52$	−167.145	−0.445748
$53$	−365.904	−0.948317	−0.474158	−	0.880440i	$-0.657248\pi$
$53$	−365.904	−0.948317	−0.474158	+	0.880440i	$0.657248\pi$
$54$	0	0
$55$	−203.650	−0.499276
$56$	0	0
$57$	0	0
$58$	−849.732	−1.92371
$59$	226.578	0.499964	0.249982	−	0.968250i	$-0.419575\pi$
$59$	226.578	0.499964	0.249982	+	0.968250i	$0.419575\pi$
$60$	0	0
$61$	651.973	1.36847	0.684235	−	0.729262i	$-0.260136\pi$
$61$	651.973	1.36847	0.684235	+	0.729262i	$0.260136\pi$
$62$	−719.331	−1.47347
$63$	0	0
$64$	−832.161	−1.62532
$65$	234.115	0.446745
$66$	0	0
$67$	145.433	0.265186	0.132593	−	0.991171i	$-0.457670\pi$
$67$	145.433	0.265186	0.132593	+	0.991171i	$0.457670\pi$
$68$	−679.717	−1.21217
$69$	0	0
$70$	0	0
$71$	368.962	0.616728	0.308364	−	0.951268i	$-0.400218\pi$
$71$	368.962	0.616728	0.308364	+	0.951268i	$0.400218\pi$
$72$	0	0
$73$	608.906	0.976261	0.488130	−	0.872771i	$-0.337679\pi$
$73$	608.906	0.976261	0.488130	+	0.872771i	$0.337679\pi$
$74$	17.0676	0.0268117
$75$	0	0
$76$	−541.328	−0.817034
$77$	0	0
$78$	0	0
$79$	910.237	1.29633	0.648163	−	0.761502i	$-0.275538\pi$
$79$	910.237	1.29633	0.648163	+	0.761502i	$0.275538\pi$
$80$	58.1615	0.0812832
$81$	0	0
$82$	179.170	0.241293
$83$	327.929	0.433674	0.216837	−	0.976208i	$-0.430426\pi$
$83$	327.929	0.433674	0.216837	+	0.976208i	$0.430426\pi$
$84$	0	0
$85$	952.058	1.21489
$86$	1956.68	2.45342
$87$	0	0
$88$	247.144	0.299383
$89$	37.6118	0.0447960	0.0223980	−	0.999749i	$-0.492870\pi$
$89$	37.6118	0.0447960	0.0223980	+	0.999749i	$0.492870\pi$
$90$	0	0
$91$	0	0
$92$	−1940.68	−2.19924
$93$	0	0
$94$	−96.5352	−0.105924
$95$	758.222	0.818863
$96$	0	0
$97$	722.013	0.755766	0.377883	−	0.925853i	$-0.376652\pi$
$97$	722.013	0.755766	0.377883	+	0.925853i	$0.376652\pi$
$98$	0	0
$99$	0	0
$100$	2481.29	2.48129
$101$	−1518.67	−1.49617	−0.748087	−	0.663601i	$-0.769027\pi$
$101$	−1518.67	−1.49617	−0.748087	+	0.663601i	$0.769027\pi$
$102$	0	0
$103$	−1051.88	−1.00626	−0.503132	−	0.864210i	$-0.667819\pi$
$103$	−1051.88	−1.00626	−0.503132	+	0.864210i	$0.667819\pi$
$104$	−284.116	−0.267883
$105$	0	0
$106$	−1667.19	−1.52766
$107$	−766.520	−0.692545	−0.346273	−	0.938134i	$-0.612553\pi$
$107$	−766.520	−0.692545	−0.346273	+	0.938134i	$0.612553\pi$
$108$	0	0
$109$	−1427.05	−1.25400	−0.627002	−	0.779018i	$-0.715718\pi$
$109$	−1427.05	−1.25400	−0.627002	+	0.779018i	$0.715718\pi$
$110$	−927.904	−0.804292
$111$	0	0
$112$	0	0
$113$	−362.564	−0.301833	−0.150917	−	0.988546i	$-0.548222\pi$
$113$	−362.564	−0.301833	−0.150917	+	0.988546i	$0.548222\pi$
$114$	0	0
$115$	2718.25	2.20416
$116$	−2379.74	−1.90477
$117$	0	0
$118$	1032.37	0.805402
$119$	0	0
$120$	0	0
$121$	−1201.17	−0.902459
$122$	2970.63	2.20449
$123$	0	0
$124$	−2014.54	−1.45896
$125$	−1241.32	−0.888216
$126$	0	0
$127$	974.777	0.681082	0.340541	−	0.940230i	$-0.389390\pi$
$127$	974.777	0.681082	0.340541	+	0.940230i	$0.389390\pi$
$128$	−2284.83	−1.57775
$129$	0	0
$130$	1066.71	0.719670
$131$	1792.70	1.19564	0.597821	−	0.801629i	$-0.296033\pi$
$131$	1792.70	1.19564	0.597821	+	0.801629i	$0.296033\pi$
$132$	0	0
$133$	0	0
$134$	662.647	0.427194
$135$	0	0
$136$	−1155.39	−0.728486
$137$	−1684.42	−1.05043	−0.525217	−	0.850969i	$-0.676016\pi$
$137$	−1684.42	−1.05043	−0.525217	+	0.850969i	$0.676016\pi$
$138$	0	0
$139$	315.089	0.192270	0.0961350	−	0.995368i	$-0.469352\pi$
$139$	315.089	0.192270	0.0961350	+	0.995368i	$0.469352\pi$
$140$	0	0
$141$	0	0
$142$	1681.13	0.993499
$143$	−149.248	−0.0872781
$144$	0	0
$145$	3333.23	1.90903
$146$	2774.40	1.57268
$147$	0	0
$148$	47.7992	0.0265478
$149$	−1893.77	−1.04123	−0.520617	−	0.853790i	$-0.674298\pi$
$149$	−1893.77	−1.04123	−0.520617	+	0.853790i	$0.674298\pi$
$150$	0	0
$151$	−2011.84	−1.08425	−0.542124	−	0.840299i	$-0.682380\pi$
$151$	−2011.84	−1.08425	−0.542124	+	0.840299i	$0.682380\pi$
$152$	−920.159	−0.491018
$153$	0	0
$154$	0	0
$155$	2821.71	1.46223
$156$	0	0
$157$	−3828.50	−1.94616	−0.973082	−	0.230460i	$-0.925977\pi$
$157$	−3828.50	−1.94616	−0.973082	+	0.230460i	$0.925977\pi$
$158$	4147.38	2.08827
$159$	0	0
$160$	3366.43	1.66337
$161$	0	0
$162$	0	0
$163$	−3509.26	−1.68630	−0.843148	−	0.537682i	$-0.819300\pi$
$163$	−3509.26	−1.68630	−0.843148	+	0.537682i	$0.819300\pi$
$164$	501.780	0.238917
$165$	0	0
$166$	1494.17	0.698613
$167$	343.008	0.158939	0.0794694	−	0.996837i	$-0.474677\pi$
$167$	343.008	0.158939	0.0794694	+	0.996837i	$0.474677\pi$
$168$	0	0
$169$	−2025.42	−0.921905
$170$	4337.93	1.95708
$171$	0	0
$172$	5479.84	2.42927
$173$	4187.21	1.84016	0.920081	−	0.391729i	$-0.128123\pi$
$173$	4187.21	1.84016	0.920081	+	0.391729i	$0.128123\pi$
$174$	0	0
$175$	0	0
$176$	−37.0779	−0.0158798
$177$	0	0
$178$	171.373	0.0721627
$179$	1970.29	0.822716	0.411358	−	0.911474i	$-0.365055\pi$
$179$	1970.29	0.822716	0.411358	+	0.911474i	$0.365055\pi$
$180$	0	0
$181$	−3613.10	−1.48376	−0.741878	−	0.670535i	$-0.766065\pi$
$181$	−3613.10	−1.48376	−0.741878	+	0.670535i	$0.766065\pi$
$182$	0	0
$183$	0	0
$184$	−3298.80	−1.32169
$185$	−66.9509	−0.0266072
$186$	0	0
$187$	−606.936	−0.237345
$188$	−270.355	−0.104881
$189$	0	0
$190$	3454.74	1.31912
$191$	−1907.77	−0.722729	−0.361365	−	0.932425i	$-0.617689\pi$
$191$	−1907.77	−0.722729	−0.361365	+	0.932425i	$0.617689\pi$
$192$	0	0
$193$	2399.93	0.895080	0.447540	−	0.894264i	$-0.352300\pi$
$193$	2399.93	0.895080	0.447540	+	0.894264i	$0.352300\pi$
$194$	3289.75	1.21748
$195$	0	0
$196$	0	0
$197$	−1514.32	−0.547668	−0.273834	−	0.961777i	$-0.588292\pi$
$197$	−1514.32	−0.547668	−0.273834	+	0.961777i	$0.588292\pi$
$198$	0	0
$199$	1367.78	0.487232	0.243616	−	0.969872i	$-0.421666\pi$
$199$	1367.78	0.487232	0.243616	+	0.969872i	$0.421666\pi$
$200$	4217.74	1.49120
$201$	0	0
$202$	−6919.63	−2.41021
$203$	0	0
$204$	0	0
$205$	−702.828	−0.239452
$206$	−4792.76	−1.62101
$207$	0	0
$208$	42.6246	0.0142091
$209$	−483.366	−0.159977
$210$	0	0
$211$	4302.52	1.40378	0.701891	−	0.712285i	$-0.252339\pi$
$211$	4302.52	1.40378	0.701891	+	0.712285i	$0.252339\pi$
$212$	−4669.11	−1.51262
$213$	0	0
$214$	−3492.55	−1.11563
$215$	−7675.45	−2.43470
$216$	0	0
$217$	0	0
$218$	−6502.16	−2.02010
$219$	0	0
$220$	−2598.67	−0.796374
$221$	697.731	0.212373
$222$	0	0
$223$	−1497.19	−0.449592	−0.224796	−	0.974406i	$-0.572172\pi$
$223$	−1497.19	−0.449592	−0.224796	+	0.974406i	$0.572172\pi$
$224$	0	0
$225$	0	0
$226$	−1651.97	−0.486229
$227$	−1603.32	−0.468795	−0.234397	−	0.972141i	$-0.575312\pi$
$227$	−1603.32	−0.468795	−0.234397	+	0.972141i	$0.575312\pi$
$228$	0	0
$229$	1010.52	0.291603	0.145802	−	0.989314i	$-0.453424\pi$
$229$	1010.52	0.291603	0.145802	+	0.989314i	$0.453424\pi$
$230$	12385.4	3.55072
$231$	0	0
$232$	−4045.13	−1.14472
$233$	−198.217	−0.0557323	−0.0278661	−	0.999612i	$-0.508871\pi$
$233$	−198.217	−0.0557323	−0.0278661	+	0.999612i	$0.508871\pi$
$234$	0	0
$235$	378.677	0.105116
$236$	2891.24	0.797472
$237$	0	0
$238$	0	0
$239$	1201.19	0.325098	0.162549	−	0.986700i	$-0.448028\pi$
$239$	1201.19	0.325098	0.162549	+	0.986700i	$0.448028\pi$
$240$	0	0
$241$	−2732.69	−0.730407	−0.365204	−	0.930928i	$-0.619001\pi$
$241$	−2732.69	−0.730407	−0.365204	+	0.930928i	$0.619001\pi$
$242$	−5472.99	−1.45379
$243$	0	0
$244$	8319.49	2.18279
$245$	0	0
$246$	0	0
$247$	555.675	0.143145
$248$	−3424.35	−0.876801
$249$	0	0
$250$	−5655.91	−1.43084
$251$	−7565.82	−1.90259	−0.951295	−	0.308281i	$-0.900246\pi$
$251$	−7565.82	−1.90259	−0.951295	+	0.308281i	$0.900246\pi$
$252$	0	0
$253$	−1732.88	−0.430615
$254$	4441.44	1.09717
$255$	0	0
$256$	−3753.22	−0.916313
$257$	−5008.68	−1.21569	−0.607846	−	0.794055i	$-0.707966\pi$
$257$	−5008.68	−1.21569	−0.607846	+	0.794055i	$0.707966\pi$
$258$	0	0
$259$	0	0
$260$	2987.42	0.712584
$261$	0	0
$262$	8168.21	1.92608
$263$	6248.81	1.46509	0.732544	−	0.680720i	$-0.238333\pi$
$263$	6248.81	1.46509	0.732544	+	0.680720i	$0.238333\pi$
$264$	0	0
$265$	6539.88	1.51601
$266$	0	0
$267$	0	0
$268$	1855.80	0.422988
$269$	3588.45	0.813351	0.406676	−	0.913573i	$-0.366688\pi$
$269$	3588.45	0.813351	0.406676	+	0.913573i	$0.366688\pi$
$270$	0	0
$271$	−1983.14	−0.444529	−0.222264	−	0.974986i	$-0.571345\pi$
$271$	−1983.14	−0.444529	−0.222264	+	0.974986i	$0.571345\pi$
$272$	173.338	0.0386404
$273$	0	0
$274$	−7674.81	−1.69216
$275$	2215.61	0.485841
$276$	0	0
$277$	7363.91	1.59731	0.798654	−	0.601790i	$-0.205546\pi$
$277$	7363.91	1.59731	0.798654	+	0.601790i	$0.205546\pi$
$278$	1435.66	0.309731
$279$	0	0
$280$	0	0
$281$	5312.05	1.12772	0.563861	−	0.825869i	$-0.309315\pi$
$281$	5312.05	1.12772	0.563861	+	0.825869i	$0.309315\pi$
$282$	0	0
$283$	−1091.76	−0.229324	−0.114662	−	0.993405i	$-0.536578\pi$
$283$	−1091.76	−0.229324	−0.114662	+	0.993405i	$0.536578\pi$
$284$	4708.13	0.983718
$285$	0	0
$286$	−680.030	−0.140598
$287$	0	0
$288$	0	0
$289$	−2075.59	−0.422469
$290$	15187.4	3.07530
$291$	0	0
$292$	7769.92	1.55719
$293$	7191.86	1.43397	0.716985	−	0.697089i	$-0.245522\pi$
$293$	7191.86	1.43397	0.716985	+	0.697089i	$0.245522\pi$
$294$	0	0
$295$	−4049.67	−0.799257
$296$	81.2499	0.0159546
$297$	0	0
$298$	−8628.73	−1.67734
$299$	1992.12	0.385308
$300$	0	0
$301$	0	0
$302$	−9166.69	−1.74663
$303$	0	0
$304$	138.047	0.0260446
$305$	−11652.9	−2.18767
$306$	0	0
$307$	541.355	0.100641	0.0503204	−	0.998733i	$-0.483976\pi$
$307$	541.355	0.100641	0.0503204	+	0.998733i	$0.483976\pi$
$308$	0	0
$309$	0	0
$310$	12856.7	2.35553
$311$	−54.0168	−0.00984892	−0.00492446	−	0.999988i	$-0.501568\pi$
$311$	−54.0168	−0.00984892	−0.00492446	+	0.999988i	$0.501568\pi$
$312$	0	0
$313$	3772.94	0.681340	0.340670	−	0.940183i	$-0.389346\pi$
$313$	3772.94	0.681340	0.340670	+	0.940183i	$0.389346\pi$
$314$	−17444.1	−3.13511
$315$	0	0
$316$	11615.1	2.06772
$317$	−1719.24	−0.304612	−0.152306	−	0.988333i	$-0.548670\pi$
$317$	−1719.24	−0.304612	−0.152306	+	0.988333i	$0.548670\pi$
$318$	0	0
$319$	−2124.93	−0.372957
$320$	14873.4	2.59827
$321$	0	0
$322$	0	0
$323$	2259.72	0.389270
$324$	0	0
$325$	−2547.06	−0.434724
$326$	−15989.5	−2.71649
$327$	0	0
$328$	852.934	0.143584
$329$	0	0
$330$	0	0
$331$	8408.21	1.39625	0.698123	−	0.715978i	$-0.254019\pi$
$331$	8408.21	1.39625	0.698123	+	0.715978i	$0.254019\pi$
$332$	4184.53	0.691735
$333$	0	0
$334$	1562.87	0.256037
$335$	−2599.36	−0.423935
$336$	0	0
$337$	2789.46	0.450894	0.225447	−	0.974255i	$-0.427616\pi$
$337$	2789.46	0.450894	0.225447	+	0.974255i	$0.427616\pi$
$338$	−9228.58	−1.48511
$339$	0	0
$340$	12148.7	1.93781
$341$	−1798.84	−0.285667
$342$	0	0
$343$	0	0
$344$	9314.72	1.45993
$345$	0	0
$346$	19078.5	2.96435
$347$	3471.96	0.537132	0.268566	−	0.963261i	$-0.413450\pi$
$347$	3471.96	0.537132	0.268566	+	0.963261i	$0.413450\pi$
$348$	0	0
$349$	−6626.12	−1.01630	−0.508149	−	0.861269i	$-0.669670\pi$
$349$	−6626.12	−1.01630	−0.508149	+	0.861269i	$0.669670\pi$
$350$	0	0
$351$	0	0
$352$	−2146.10	−0.324964
$353$	−9468.40	−1.42763	−0.713813	−	0.700337i	$-0.753033\pi$
$353$	−9468.40	−1.42763	−0.713813	+	0.700337i	$0.753033\pi$
$354$	0	0
$355$	−6594.53	−0.985919
$356$	479.944	0.0714522
$357$	0	0
$358$	8977.35	1.32533
$359$	6279.55	0.923182	0.461591	−	0.887093i	$-0.347279\pi$
$359$	6279.55	0.923182	0.461591	+	0.887093i	$0.347279\pi$
$360$	0	0
$361$	−5059.35	−0.737622
$362$	−16462.6	−2.39021
$363$	0	0
$364$	0	0
$365$	−10883.1	−1.56068
$366$	0	0
$367$	−10827.8	−1.54008	−0.770038	−	0.637998i	$-0.779763\pi$
$367$	−10827.8	−1.54008	−0.770038	+	0.637998i	$0.779763\pi$
$368$	494.904	0.0701050
$369$	0	0
$370$	−305.053	−0.0428620
$371$	0	0
$372$	0	0
$373$	5239.23	0.727284	0.363642	−	0.931539i	$-0.381533\pi$
$373$	5239.23	0.727284	0.363642	+	0.931539i	$0.381533\pi$
$374$	−2765.42	−0.382344
$375$	0	0
$376$	−459.553	−0.0630310
$377$	2442.81	0.333717
$378$	0	0
$379$	−11050.4	−1.49768	−0.748839	−	0.662751i	$-0.769389\pi$
$379$	−11050.4	−1.49768	−0.748839	+	0.662751i	$0.769389\pi$
$380$	9675.27	1.30613
$381$	0	0
$382$	−8692.49	−1.16426
$383$	10468.0	1.39658	0.698292	−	0.715813i	$-0.253944\pi$
$383$	10468.0	1.39658	0.698292	+	0.715813i	$0.253944\pi$
$384$	0	0
$385$	0	0
$386$	10934.9	1.44190
$387$	0	0
$388$	9213.22	1.20549
$389$	11614.0	1.51377	0.756884	−	0.653550i	$-0.226721\pi$
$389$	11614.0	1.51377	0.756884	+	0.653550i	$0.226721\pi$
$390$	0	0
$391$	8101.19	1.04781
$392$	0	0
$393$	0	0
$394$	−6899.78	−0.882248
$395$	−16268.9	−2.07234
$396$	0	0
$397$	−6707.30	−0.847934	−0.423967	−	0.905678i	$-0.639363\pi$
$397$	−6707.30	−0.847934	−0.423967	+	0.905678i	$0.639363\pi$
$398$	6232.10	0.784892
$399$	0	0
$400$	−632.768	−0.0790960
$401$	−5526.38	−0.688215	−0.344107	−	0.938930i	$-0.611818\pi$
$401$	−5526.38	−0.688215	−0.344107	+	0.938930i	$0.611818\pi$
$402$	0	0
$403$	2067.94	0.255611
$404$	−19379.0	−2.38649
$405$	0	0
$406$	0	0
$407$	42.6811	0.00519810
$408$	0	0
$409$	−1318.91	−0.159452	−0.0797258	−	0.996817i	$-0.525404\pi$
$409$	−1318.91	−0.159452	−0.0797258	+	0.996817i	$0.525404\pi$
$410$	−3202.34	−0.385738
$411$	0	0
$412$	−13422.5	−1.60505
$413$	0	0
$414$	0	0
$415$	−5861.15	−0.693283
$416$	2467.14	0.290773
$417$	0	0
$418$	−2202.39	−0.257709
$419$	−3656.13	−0.426286	−0.213143	−	0.977021i	$-0.568370\pi$
$419$	−3656.13	−0.426286	−0.213143	+	0.977021i	$0.568370\pi$
$420$	0	0
$421$	−135.389	−0.0156733	−0.00783663	−	0.999969i	$-0.502495\pi$
$421$	−135.389	−0.0156733	−0.00783663	+	0.999969i	$0.502495\pi$
$422$	19603.9	2.26138
$423$	0	0
$424$	−7936.63	−0.909049
$425$	−10357.9	−1.18219
$426$	0	0
$427$	0	0
$428$	−9781.16	−1.10465
$429$	0	0
$430$	−34972.1	−3.92211
$431$	−8389.16	−0.937568	−0.468784	−	0.883313i	$-0.655308\pi$
$431$	−8389.16	−0.937568	−0.468784	+	0.883313i	$0.655308\pi$
$432$	0	0
$433$	−8243.02	−0.914859	−0.457430	−	0.889246i	$-0.651230\pi$
$433$	−8243.02	−0.914859	−0.457430	+	0.889246i	$0.651230\pi$
$434$	0	0
$435$	0	0
$436$	−18209.8	−2.00021
$437$	6451.81	0.706252
$438$	0	0
$439$	−18283.2	−1.98772	−0.993859	−	0.110649i	$-0.964707\pi$
$439$	−18283.2	−1.98772	−0.993859	+	0.110649i	$0.964707\pi$
$440$	−4417.26	−0.478602
$441$	0	0
$442$	3179.12	0.342116
$443$	−1210.44	−0.129818	−0.0649092	−	0.997891i	$-0.520676\pi$
$443$	−1210.44	−0.129818	−0.0649092	+	0.997891i	$0.520676\pi$
$444$	0	0
$445$	−672.243	−0.0716121
$446$	−6821.72	−0.724256
$447$	0	0
$448$	0	0
$449$	8301.16	0.872508	0.436254	−	0.899824i	$-0.356305\pi$
$449$	8301.16	0.872508	0.436254	+	0.899824i	$0.356305\pi$
$450$	0	0
$451$	448.052	0.0467804
$452$	−4626.49	−0.481442
$453$	0	0
$454$	−7305.34	−0.755190
$455$	0	0
$456$	0	0
$457$	−12293.8	−1.25838	−0.629188	−	0.777253i	$-0.716612\pi$
$457$	−12293.8	−1.25838	−0.629188	+	0.777253i	$0.716612\pi$
$458$	4604.31	0.469749
$459$	0	0
$460$	34686.2	3.51577
$461$	−19434.2	−1.96343	−0.981717	−	0.190346i	$-0.939039\pi$
$461$	−19434.2	−1.96343	−0.981717	+	0.190346i	$0.939039\pi$
$462$	0	0
$463$	−12491.1	−1.25380	−0.626902	−	0.779098i	$-0.715678\pi$
$463$	−12491.1	−1.25380	−0.626902	+	0.779098i	$0.715678\pi$
$464$	606.871	0.0607183
$465$	0	0
$466$	−903.149	−0.0897802
$467$	−3385.17	−0.335433	−0.167716	−	0.985835i	$-0.553639\pi$
$467$	−3385.17	−0.335433	−0.167716	+	0.985835i	$0.553639\pi$
$468$	0	0
$469$	0	0
$470$	1725.39	0.169333
$471$	0	0
$472$	4914.57	0.479262
$473$	4893.09	0.475654
$474$	0	0
$475$	−8249.07	−0.796828
$476$	0	0
$477$	0	0
$478$	5473.05	0.523706
$479$	−5979.41	−0.570368	−0.285184	−	0.958473i	$-0.592055\pi$
$479$	−5979.41	−0.570368	−0.285184	+	0.958473i	$0.592055\pi$
$480$	0	0
$481$	−49.0661	−0.00465119
$482$	−12451.1	−1.17663
$483$	0	0
$484$	−15327.5	−1.43948
$485$	−12904.7	−1.20819
$486$	0	0
$487$	−1114.96	−0.103745	−0.0518725	−	0.998654i	$-0.516519\pi$
$487$	−1114.96	−0.103745	−0.0518725	+	0.998654i	$0.516519\pi$
$488$	14141.6	1.31180
$489$	0	0
$490$	0	0
$491$	−1086.23	−0.0998387	−0.0499194	−	0.998753i	$-0.515896\pi$
$491$	−1086.23	−0.0998387	−0.0499194	+	0.998753i	$0.515896\pi$
$492$	0	0
$493$	9934.01	0.907516
$494$	2531.86	0.230595
$495$	0	0
$496$	513.740	0.0465073
$497$	0	0
$498$	0	0
$499$	−2213.50	−0.198577	−0.0992884	−	0.995059i	$-0.531657\pi$
$499$	−2213.50	−0.198577	−0.0992884	+	0.995059i	$0.531657\pi$
$500$	−15839.8	−1.41676
$501$	0	0
$502$	−34472.6	−3.06492
$503$	2643.32	0.234314	0.117157	−	0.993113i	$-0.462622\pi$
$503$	2643.32	0.234314	0.117157	+	0.993113i	$0.462622\pi$
$504$	0	0
$505$	27143.5	2.39183
$506$	−7895.66	−0.693685
$507$	0	0
$508$	12438.6	1.08637
$509$	665.169	0.0579236	0.0289618	−	0.999581i	$-0.490780\pi$
$509$	665.169	0.0579236	0.0289618	+	0.999581i	$0.490780\pi$
$510$	0	0
$511$	0	0
$512$	1177.58	0.101645
$513$	0	0
$514$	−22821.4	−1.95838
$515$	18800.5	1.60864
$516$	0	0
$517$	−241.406	−0.0205359
$518$	0	0
$519$	0	0
$520$	5078.07	0.428246
$521$	11762.0	0.989063	0.494531	−	0.869160i	$-0.335340\pi$
$521$	11762.0	0.989063	0.494531	+	0.869160i	$0.335340\pi$
$522$	0	0
$523$	10122.6	0.846330	0.423165	−	0.906053i	$-0.360919\pi$
$523$	10122.6	0.846330	0.423165	+	0.906053i	$0.360919\pi$
$524$	22875.7	1.90712
$525$	0	0
$526$	28471.9	2.36014
$527$	8409.52	0.695112
$528$	0	0
$529$	10963.0	0.901042
$530$	29798.1	2.44216
$531$	0	0
$532$	0	0
$533$	−515.079	−0.0418585
$534$	0	0
$535$	13700.2	1.10712
$536$	3154.51	0.254206
$537$	0	0
$538$	16350.3	1.31024
$539$	0	0
$540$	0	0
$541$	16118.0	1.28090	0.640449	−	0.768001i	$-0.278748\pi$
$541$	16118.0	1.28090	0.640449	+	0.768001i	$0.278748\pi$
$542$	−9035.92	−0.716100
$543$	0	0
$544$	10032.9	0.790733
$545$	25505.9	2.00469
$546$	0	0
$547$	−626.100	−0.0489399	−0.0244699	−	0.999701i	$-0.507790\pi$
$547$	−626.100	−0.0489399	−0.0244699	+	0.999701i	$0.507790\pi$
$548$	−21493.9	−1.67550
$549$	0	0
$550$	10095.1	0.782650
$551$	7911.47	0.611688
$552$	0	0
$553$	0	0
$554$	33552.7	2.57313
$555$	0	0
$556$	4020.69	0.306682
$557$	−20771.2	−1.58008	−0.790039	−	0.613057i	$-0.789940\pi$
$557$	−20771.2	−1.58008	−0.790039	+	0.613057i	$0.789940\pi$
$558$	0	0
$559$	−5625.08	−0.425609
$560$	0	0
$561$	0	0
$562$	24203.6	1.81667
$563$	5521.72	0.413344	0.206672	−	0.978410i	$-0.433737\pi$
$563$	5521.72	0.413344	0.206672	+	0.978410i	$0.433737\pi$
$564$	0	0
$565$	6480.18	0.482519
$566$	−4974.48	−0.369422
$567$	0	0
$568$	8002.95	0.591191
$569$	−7574.81	−0.558089	−0.279044	−	0.960278i	$-0.590018\pi$
$569$	−7574.81	−0.558089	−0.279044	+	0.960278i	$0.590018\pi$
$570$	0	0
$571$	−331.248	−0.0242772	−0.0121386	−	0.999926i	$-0.503864\pi$
$571$	−331.248	−0.0242772	−0.0121386	+	0.999926i	$0.503864\pi$
$572$	−1904.48	−0.139214
$573$	0	0
$574$	0	0
$575$	−29573.2	−2.14485
$576$	0	0
$577$	2038.11	0.147050	0.0735248	−	0.997293i	$-0.476575\pi$
$577$	2038.11	0.147050	0.0735248	+	0.997293i	$0.476575\pi$
$578$	−9457.14	−0.680563
$579$	0	0
$580$	42533.6	3.04502
$581$	0	0
$582$	0	0
$583$	−4169.17	−0.296174
$584$	13207.4	0.935835
$585$	0	0
$586$	32768.8	2.31001
$587$	−5232.90	−0.367947	−0.183973	−	0.982931i	$-0.558896\pi$
$587$	−5232.90	−0.367947	−0.183973	+	0.982931i	$0.558896\pi$
$588$	0	0
$589$	6697.36	0.468523
$590$	−18451.8	−1.28754
$591$	0	0
$592$	−12.1895	−0.000846262 0
$593$	5720.24	0.396125	0.198062	−	0.980189i	$-0.436535\pi$
$593$	5720.24	0.396125	0.198062	+	0.980189i	$0.436535\pi$
$594$	0	0
$595$	0	0
$596$	−24165.5	−1.66083
$597$	0	0
$598$	9076.81	0.620700
$599$	−18088.4	−1.23384	−0.616922	−	0.787024i	$-0.711621\pi$
$599$	−18088.4	−1.23384	−0.616922	+	0.787024i	$0.711621\pi$
$600$	0	0
$601$	−1821.43	−0.123623	−0.0618117	−	0.998088i	$-0.519688\pi$
$601$	−1821.43	−0.123623	−0.0618117	+	0.998088i	$0.519688\pi$
$602$	0	0
$603$	0	0
$604$	−25672.0	−1.72944
$605$	21468.8	1.44270
$606$	0	0
$607$	2372.20	0.158624	0.0793120	−	0.996850i	$-0.474728\pi$
$607$	2372.20	0.158624	0.0793120	+	0.996850i	$0.474728\pi$
$608$	7990.26	0.532974
$609$	0	0
$610$	−53094.7	−3.52416
$611$	277.520	0.0183752
$612$	0	0
$613$	9725.08	0.640770	0.320385	−	0.947287i	$-0.396188\pi$
$613$	9725.08	0.640770	0.320385	+	0.947287i	$0.396188\pi$
$614$	2466.61	0.162124
$615$	0	0
$616$	0	0
$617$	5329.51	0.347744	0.173872	−	0.984768i	$-0.444372\pi$
$617$	5329.51	0.347744	0.173872	+	0.984768i	$0.444372\pi$
$618$	0	0
$619$	−15976.6	−1.03740	−0.518702	−	0.854955i	$-0.673585\pi$
$619$	−15976.6	−1.03740	−0.518702	+	0.854955i	$0.673585\pi$
$620$	36006.3	2.33234
$621$	0	0
$622$	−246.120	−0.0158658
$623$	0	0
$624$	0	0
$625$	−2120.06	−0.135684
$626$	17190.9	1.09758
$627$	0	0
$628$	−48853.5	−3.10425
$629$	−199.533	−0.0126485
$630$	0	0
$631$	−4199.98	−0.264974	−0.132487	−	0.991185i	$-0.542296\pi$
$631$	−4199.98	−0.264974	−0.132487	+	0.991185i	$0.542296\pi$
$632$	19743.5	1.24265
$633$	0	0
$634$	−7833.47	−0.490705
$635$	−17422.4	−1.08880
$636$	0	0
$637$	0	0
$638$	−9681.97	−0.600804
$639$	0	0
$640$	40837.2	2.52224
$641$	−2648.51	−0.163198	−0.0815988	−	0.996665i	$-0.526003\pi$
$641$	−2648.51	−0.163198	−0.0815988	+	0.996665i	$0.526003\pi$
$642$	0	0
$643$	13.4305	0.000823715 0	0.000411857	−	1.00000i	$-0.499869\pi$
$643$	13.4305	0.000823715 0	0.000411857		1.00000i	$0.499869\pi$
$644$	0	0
$645$	0	0
$646$	10296.1	0.627083
$647$	−11624.1	−0.706324	−0.353162	−	0.935562i	$-0.614894\pi$
$647$	−11624.1	−0.706324	−0.353162	+	0.935562i	$0.614894\pi$
$648$	0	0
$649$	2581.66	0.156146
$650$	−11605.3	−0.700305
$651$	0	0
$652$	−44779.8	−2.68974
$653$	28516.6	1.70894	0.854471	−	0.519499i	$-0.173881\pi$
$653$	28516.6	1.70894	0.854471	+	0.519499i	$0.173881\pi$
$654$	0	0
$655$	−32041.3	−1.91139
$656$	−127.962	−0.00761595
$657$	0	0
$658$	0	0
$659$	−18048.6	−1.06688	−0.533440	−	0.845838i	$-0.679101\pi$
$659$	−18048.6	−1.06688	−0.533440	+	0.845838i	$0.679101\pi$
$660$	0	0
$661$	17841.4	1.04985	0.524926	−	0.851148i	$-0.324093\pi$
$661$	17841.4	1.04985	0.524926	+	0.851148i	$0.324093\pi$
$662$	38310.9	2.24924
$663$	0	0
$664$	7112.94	0.415716
$665$	0	0
$666$	0	0
$667$	28362.9	1.64650
$668$	4376.95	0.253517
$669$	0	0
$670$	−11843.6	−0.682924
$671$	7428.68	0.427394
$672$	0	0
$673$	−6826.13	−0.390978	−0.195489	−	0.980706i	$-0.562629\pi$
$673$	−6826.13	−0.390978	−0.195489	+	0.980706i	$0.562629\pi$
$674$	12709.8	0.726354
$675$	0	0
$676$	−25845.4	−1.47049
$677$	21286.9	1.20845	0.604225	−	0.796814i	$-0.293483\pi$
$677$	21286.9	1.20845	0.604225	+	0.796814i	$0.293483\pi$
$678$	0	0
$679$	0	0
$680$	20650.6	1.16458
$681$	0	0
$682$	−8196.16	−0.460187
$683$	20696.8	1.15951	0.579753	−	0.814793i	$-0.303149\pi$
$683$	20696.8	1.15951	0.579753	+	0.814793i	$0.303149\pi$
$684$	0	0
$685$	30105.9	1.67925
$686$	0	0
$687$	0	0
$688$	−1397.44	−0.0774376
$689$	4792.86	0.265012
$690$	0	0
$691$	31341.9	1.72548	0.862738	−	0.505652i	$-0.168748\pi$
$691$	31341.9	1.72548	0.862738	+	0.505652i	$0.168748\pi$
$692$	53430.8	2.93517
$693$	0	0
$694$	15819.5	0.865276
$695$	−5631.66	−0.307368
$696$	0	0
$697$	−2094.63	−0.113831
$698$	−30191.0	−1.63717
$699$	0	0
$700$	0	0
$701$	9213.32	0.496408	0.248204	−	0.968708i	$-0.420160\pi$
$701$	9213.32	0.496408	0.248204	+	0.968708i	$0.420160\pi$
$702$	0	0
$703$	−158.909	−0.00852541
$704$	−9481.77	−0.507610
$705$	0	0
$706$	−43141.5	−2.29979
$707$	0	0
$708$	0	0
$709$	14516.5	0.768942	0.384471	−	0.923137i	$-0.374384\pi$
$709$	14516.5	0.768942	0.384471	+	0.923137i	$0.374384\pi$
$710$	−30047.1	−1.58824
$711$	0	0
$712$	815.817	0.0429411
$713$	24010.3	1.26114
$714$	0	0
$715$	2667.54	0.139525
$716$	25141.8	1.31228
$717$	0	0
$718$	28611.9	1.48717
$719$	−25882.4	−1.34249	−0.671246	−	0.741235i	$-0.734240\pi$
$719$	−25882.4	−1.34249	−0.671246	+	0.741235i	$0.734240\pi$
$720$	0	0
$721$	0	0
$722$	−23052.3	−1.18825
$723$	0	0
$724$	−46104.9	−2.36668
$725$	−36263.9	−1.85767
$726$	0	0
$727$	32181.2	1.64172	0.820862	−	0.571127i	$-0.193494\pi$
$727$	32181.2	1.64172	0.820862	+	0.571127i	$0.193494\pi$
$728$	0	0
$729$	0	0
$730$	−49587.4	−2.51412
$731$	−22875.1	−1.15741
$732$	0	0
$733$	20836.1	1.04993	0.524966	−	0.851123i	$-0.324078\pi$
$733$	20836.1	1.04993	0.524966	+	0.851123i	$0.324078\pi$
$734$	−49335.5	−2.48094
$735$	0	0
$736$	28645.4	1.43462
$737$	1657.09	0.0828217
$738$	0	0
$739$	26434.9	1.31586	0.657931	−	0.753078i	$-0.271432\pi$
$739$	26434.9	1.31586	0.657931	+	0.753078i	$0.271432\pi$
$740$	−854.325	−0.0424400
$741$	0	0
$742$	0	0
$743$	−9954.69	−0.491524	−0.245762	−	0.969330i	$-0.579038\pi$
$743$	−9954.69	−0.491524	−0.245762	+	0.969330i	$0.579038\pi$
$744$	0	0
$745$	33847.8	1.66455
$746$	23871.8	1.17160
$747$	0	0
$748$	−7744.79	−0.378580
$749$	0	0
$750$	0	0
$751$	−33204.5	−1.61338	−0.806692	−	0.590973i	$-0.798744\pi$
$751$	−33204.5	−1.61338	−0.806692	+	0.590973i	$0.798744\pi$
$752$	68.9446	0.00334329
$753$	0	0
$754$	11130.4	0.537591
$755$	35958.1	1.73331
$756$	0	0
$757$	1964.06	0.0942998	0.0471499	−	0.998888i	$-0.484986\pi$
$757$	1964.06	0.0942998	0.0471499	+	0.998888i	$0.484986\pi$
$758$	−50349.6	−2.41264
$759$	0	0
$760$	16446.2	0.784955
$761$	38553.8	1.83650	0.918248	−	0.396005i	$-0.129604\pi$
$761$	38553.8	1.83650	0.918248	+	0.396005i	$0.129604\pi$
$762$	0	0
$763$	0	0
$764$	−24344.0	−1.15280
$765$	0	0
$766$	47696.2	2.24978
$767$	−2967.87	−0.139718
$768$	0	0
$769$	−19715.0	−0.924501	−0.462251	−	0.886749i	$-0.652958\pi$
$769$	−19715.0	−0.924501	−0.462251	+	0.886749i	$0.652958\pi$
$770$	0	0
$771$	0	0
$772$	30624.2	1.42771
$773$	14700.7	0.684019	0.342010	−	0.939696i	$-0.388892\pi$
$773$	14700.7	0.684019	0.342010	+	0.939696i	$0.388892\pi$
$774$	0	0
$775$	−30698.8	−1.42288
$776$	15660.8	0.724471
$777$	0	0
$778$	52917.8	2.43856
$779$	−1668.17	−0.0767246
$780$	0	0
$781$	4204.01	0.192614
$782$	36912.0	1.68794
$783$	0	0
$784$	0	0
$785$	68427.6	3.11119
$786$	0	0
$787$	−23918.6	−1.08336	−0.541681	−	0.840584i	$-0.682212\pi$
$787$	−23918.6	−1.08336	−0.541681	+	0.840584i	$0.682212\pi$
$788$	−19323.4	−0.873562
$789$	0	0
$790$	−74126.9	−3.33837
$791$	0	0
$792$	0	0
$793$	−8539.98	−0.382426
$794$	−30560.9	−1.36595
$795$	0	0
$796$	17453.5	0.777164
$797$	−38252.7	−1.70010	−0.850051	−	0.526700i	$-0.823429\pi$
$797$	−38252.7	−1.70010	−0.850051	+	0.526700i	$0.823429\pi$
$798$	0	0
$799$	1128.57	0.0499698
$800$	−36625.1	−1.61861
$801$	0	0
$802$	−25180.2	−1.10866
$803$	6937.96	0.304901
$804$	0	0
$805$	0	0
$806$	9422.27	0.411769
$807$	0	0
$808$	−32940.7	−1.43422
$809$	31435.6	1.36615	0.683075	−	0.730348i	$-0.260642\pi$
$809$	31435.6	1.36615	0.683075	+	0.730348i	$0.260642\pi$
$810$	0	0
$811$	11467.0	0.496501	0.248250	−	0.968696i	$-0.420144\pi$
$811$	11467.0	0.496501	0.248250	+	0.968696i	$0.420144\pi$
$812$	0	0
$813$	0	0
$814$	194.471	0.00837371
$815$	62721.7	2.69576
$816$	0	0
$817$	−18217.8	−0.780121
$818$	−6009.42	−0.256864
$819$	0	0
$820$	−8968.42	−0.381940
$821$	5030.57	0.213847	0.106923	−	0.994267i	$-0.465900\pi$
$821$	5030.57	0.213847	0.106923	+	0.994267i	$0.465900\pi$
$822$	0	0
$823$	−13985.2	−0.592336	−0.296168	−	0.955136i	$-0.595709\pi$
$823$	−13985.2	−0.592336	−0.296168	+	0.955136i	$0.595709\pi$
$824$	−22815.8	−0.964596
$825$	0	0
$826$	0	0
$827$	13939.5	0.586125	0.293063	−	0.956093i	$-0.405326\pi$
$827$	13939.5	0.586125	0.293063	+	0.956093i	$0.405326\pi$
$828$	0	0
$829$	−20104.4	−0.842286	−0.421143	−	0.906994i	$-0.638371\pi$
$829$	−20104.4	−0.842286	−0.421143	+	0.906994i	$0.638371\pi$
$830$	−26705.5	−1.11682
$831$	0	0
$832$	10900.2	0.454203
$833$	0	0
$834$	0	0
$835$	−6130.66	−0.254084
$836$	−6167.98	−0.255172
$837$	0	0
$838$	−16658.7	−0.686711
$839$	−15949.5	−0.656302	−0.328151	−	0.944625i	$-0.606425\pi$
$839$	−15949.5	−0.656302	−0.328151	+	0.944625i	$0.606425\pi$
$840$	0	0
$841$	10390.8	0.426043
$842$	−616.881	−0.0252484
$843$	0	0
$844$	54902.2	2.23911
$845$	36200.8	1.47378
$846$	0	0
$847$	0	0
$848$	1190.70	0.0482177
$849$	0	0
$850$	−47194.5	−1.90442
$851$	−569.694	−0.0229481
$852$	0	0
$853$	11802.0	0.473730	0.236865	−	0.971543i	$-0.423880\pi$
$853$	11802.0	0.473730	0.236865	+	0.971543i	$0.423880\pi$
$854$	0	0
$855$	0	0
$856$	−16626.2	−0.663868
$857$	−9595.28	−0.382460	−0.191230	−	0.981545i	$-0.561248\pi$
$857$	−9595.28	−0.382460	−0.191230	+	0.981545i	$0.561248\pi$
$858$	0	0
$859$	21840.9	0.867521	0.433760	−	0.901028i	$-0.357186\pi$
$859$	21840.9	0.867521	0.433760	+	0.901028i	$0.357186\pi$
$860$	−97942.3	−3.88350
$861$	0	0
$862$	−38224.1	−1.51035
$863$	26531.6	1.04652	0.523260	−	0.852173i	$-0.324716\pi$
$863$	26531.6	1.04652	0.523260	+	0.852173i	$0.324716\pi$
$864$	0	0
$865$	−74838.9	−2.94173
$866$	−37558.2	−1.47376
$867$	0	0
$868$	0	0
$869$	10371.4	0.404862
$870$	0	0
$871$	−1904.98	−0.0741077
$872$	−30953.3	−1.20208
$873$	0	0
$874$	29396.8	1.13771
$875$	0	0
$876$	0	0
$877$	−6832.46	−0.263074	−0.131537	−	0.991311i	$-0.541991\pi$
$877$	−6832.46	−0.263074	−0.131537	+	0.991311i	$0.541991\pi$
$878$	−83304.9	−3.20205
$879$	0	0
$880$	662.701	0.0253860
$881$	3994.77	0.152766	0.0763832	−	0.997079i	$-0.475663\pi$
$881$	3994.77	0.152766	0.0763832	+	0.997079i	$0.475663\pi$
$882$	0	0
$883$	13727.0	0.523161	0.261580	−	0.965182i	$-0.415756\pi$
$883$	13727.0	0.523161	0.261580	+	0.965182i	$0.415756\pi$
$884$	8903.38	0.338748
$885$	0	0
$886$	−5515.19	−0.209127
$887$	−44119.4	−1.67011	−0.835054	−	0.550169i	$-0.814563\pi$
$887$	−44119.4	−1.67011	−0.835054	+	0.550169i	$0.814563\pi$
$888$	0	0
$889$	0	0
$890$	−3062.99	−0.115361
$891$	0	0
$892$	−19104.8	−0.717125
$893$	898.796	0.0336809
$894$	0	0
$895$	−35215.3	−1.31522
$896$	0	0
$897$	0	0
$898$	37823.1	1.40554
$899$	29442.4	1.09228
$900$	0	0
$901$	19490.7	0.720678
$902$	2041.49	0.0753594
$903$	0	0
$904$	−7864.18	−0.289335
$905$	64577.7	2.37197
$906$	0	0
$907$	36905.8	1.35109	0.675545	−	0.737319i	$-0.263908\pi$
$907$	36905.8	1.35109	0.675545	+	0.737319i	$0.263908\pi$
$908$	−20459.2	−0.747755
$909$	0	0
$910$	0	0
$911$	3169.56	0.115271	0.0576356	−	0.998338i	$-0.481644\pi$
$911$	3169.56	0.115271	0.0576356	+	0.998338i	$0.481644\pi$
$912$	0	0
$913$	3736.48	0.135443
$914$	−56014.9	−2.02714
$915$	0	0
$916$	12894.7	0.465124
$917$	0	0
$918$	0	0
$919$	8727.00	0.313250	0.156625	−	0.987658i	$-0.449939\pi$
$919$	8727.00	0.313250	0.156625	+	0.987658i	$0.449939\pi$
$920$	58960.2	2.11289
$921$	0	0
$922$	−88549.5	−3.16293
$923$	−4832.91	−0.172348
$924$	0	0
$925$	728.392	0.0258912
$926$	−56914.1	−2.01978
$927$	0	0
$928$	35126.1	1.24253
$929$	−19405.1	−0.685317	−0.342659	−	0.939460i	$-0.611327\pi$
$929$	−19405.1	−0.685317	−0.342659	+	0.939460i	$0.611327\pi$
$930$	0	0
$931$	0	0
$932$	−2529.34	−0.0888963
$933$	0	0
$934$	−15424.1	−0.540355
$935$	10847.9	0.379427
$936$	0	0
$937$	−615.692	−0.0214662	−0.0107331	−	0.999942i	$-0.503417\pi$
$937$	−615.692	−0.0214662	−0.0107331	+	0.999942i	$0.503417\pi$
$938$	0	0
$939$	0	0
$940$	4832.10	0.167666
$941$	29602.0	1.02550	0.512751	−	0.858537i	$-0.328626\pi$
$941$	29602.0	1.02550	0.512751	+	0.858537i	$0.328626\pi$
$942$	0	0
$943$	−5980.46	−0.206522
$944$	−737.310	−0.0254210
$945$	0	0
$946$	22294.7	0.766240
$947$	−13537.4	−0.464527	−0.232264	−	0.972653i	$-0.574613\pi$
$947$	−13537.4	−0.464527	−0.232264	+	0.972653i	$0.574613\pi$
$948$	0	0
$949$	−7975.85	−0.272821
$950$	−37585.8	−1.28363
$951$	0	0
$952$	0	0
$953$	−33468.5	−1.13762	−0.568810	−	0.822469i	$-0.692596\pi$
$953$	−33468.5	−1.13762	−0.568810	+	0.822469i	$0.692596\pi$
$954$	0	0
$955$	34097.9	1.15538
$956$	15327.7	0.518550
$957$	0	0
$958$	−27244.4	−0.918817
$959$	0	0
$960$	0	0
$961$	−4866.88	−0.163368
$962$	−223.563	−0.00749268
$963$	0	0
$964$	−34870.4	−1.16504
$965$	−42894.4	−1.43090
$966$	0	0
$967$	−55733.5	−1.85343	−0.926715	−	0.375764i	$-0.877380\pi$
$967$	−55733.5	−1.85343	−0.926715	+	0.375764i	$0.877380\pi$
$968$	−26054.0	−0.865090
$969$	0	0
$970$	−58798.4	−1.94629
$971$	19491.3	0.644187	0.322094	−	0.946708i	$-0.395613\pi$
$971$	19491.3	0.644187	0.322094	+	0.946708i	$0.395613\pi$
$972$	0	0
$973$	0	0
$974$	−5080.18	−0.167125
$975$	0	0
$976$	−2121.60	−0.0695807
$977$	6241.56	0.204386	0.102193	−	0.994765i	$-0.467414\pi$
$977$	6241.56	0.204386	0.102193	+	0.994765i	$0.467414\pi$
$978$	0	0
$979$	428.554	0.0139905
$980$	0	0
$981$	0	0
$982$	−4949.25	−0.160832
$983$	−59694.6	−1.93689	−0.968444	−	0.249231i	$-0.919822\pi$
$983$	−59694.6	−1.93689	−0.968444	+	0.249231i	$0.919822\pi$
$984$	0	0
$985$	27065.7	0.875517
$986$	45263.0	1.46193
$987$	0	0
$988$	7090.68	0.228324
$989$	−65311.4	−2.09988
$990$	0	0
$991$	15561.6	0.498821	0.249411	−	0.968398i	$-0.419763\pi$
$991$	15561.6	0.498821	0.249411	+	0.968398i	$0.419763\pi$
$992$	29735.6	0.951720
$993$	0	0
$994$	0	0
$995$	−24446.6	−0.778903
$996$	0	0
$997$	19884.9	0.631657	0.315829	−	0.948816i	$-0.397718\pi$
$997$	19884.9	0.631657	0.315829	+	0.948816i	$0.397718\pi$
$998$	−10085.5	−0.319891
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.s.1.3		3
3.2	odd	2		147.4.a.l.1.1		3
7.2	even	3		63.4.e.c.46.1		6
7.3	odd	6		441.4.e.w.226.1		6
7.4	even	3		63.4.e.c.37.1		6
7.5	odd	6		441.4.e.w.361.1		6
7.6	odd	2		441.4.a.t.1.3		3
12.11	even	2		2352.4.a.ci.1.3		3
21.2	odd	6		21.4.e.b.4.3	✓	6
21.5	even	6		147.4.e.n.67.3		6
21.11	odd	6		21.4.e.b.16.3	yes	6
21.17	even	6		147.4.e.n.79.3		6
21.20	even	2		147.4.a.m.1.1		3
84.11	even	6		336.4.q.k.289.1		6
84.23	even	6		336.4.q.k.193.1		6
84.83	odd	2		2352.4.a.cg.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.e.b.4.3	✓	6	21.2	odd	6
21.4.e.b.16.3	yes	6	21.11	odd	6
63.4.e.c.37.1		6	7.4	even	3
63.4.e.c.46.1		6	7.2	even	3
147.4.a.l.1.1		3	3.2	odd	2
147.4.a.m.1.1		3	21.20	even	2
147.4.e.n.67.3		6	21.5	even	6
147.4.e.n.79.3		6	21.17	even	6
336.4.q.k.193.1		6	84.23	even	6
336.4.q.k.289.1		6	84.11	even	6
441.4.a.s.1.3		3	1.1	even	1	trivial
441.4.a.t.1.3		3	7.6	odd	2
441.4.e.w.226.1		6	7.3	odd	6
441.4.e.w.361.1		6	7.5	odd	6
2352.4.a.cg.1.1		3	84.83	odd	2
2352.4.a.ci.1.3		3	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+4.55637 q^{2} +12.7605 q^{4} -17.8732 q^{5} +21.6905 q^{8} -81.4369 q^{10} +11.3942 q^{11} -13.0987 q^{13} -3.25412 q^{16} -53.2674 q^{17} -42.4223 q^{19} -228.071 q^{20} +51.9159 q^{22} -152.085 q^{23} +194.451 q^{25} -59.6823 q^{26} -186.493 q^{29} -157.874 q^{31} -188.351 q^{32} -242.706 q^{34} +3.74588 q^{37} -193.291 q^{38} -387.678 q^{40} +39.3230 q^{41} +429.439 q^{43} +145.395 q^{44} -692.957 q^{46} -21.1869 q^{47} +885.992 q^{50} -167.145 q^{52} -365.904 q^{53} -203.650 q^{55} -849.732 q^{58} +226.578 q^{59} +651.973 q^{61} -719.331 q^{62} -832.161 q^{64} +234.115 q^{65} +145.433 q^{67} -679.717 q^{68} +368.962 q^{71} +608.906 q^{73} +17.0676 q^{74} -541.328 q^{76} +910.237 q^{79} +58.1615 q^{80} +179.170 q^{82} +327.929 q^{83} +952.058 q^{85} +1956.68 q^{86} +247.144 q^{88} +37.6118 q^{89} -1940.68 q^{92} -96.5352 q^{94} +758.222 q^{95} +722.013 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 25 q^{4} - 11 q^{5} - 39 q^{8} - 55 q^{10} - 35 q^{11} + 62 q^{13} + 241 q^{16} - 48 q^{17} - 202 q^{19} - 439 q^{20} - 7 q^{22} - 216 q^{23} + 130 q^{25} - 274 q^{26} - 53 q^{29} - 95 q^{31}+ \cdots + 19 q^{97}+O(q^{100}) \) 3 * q - q^2 + 25 * q^4 - 11 * q^5 - 39 * q^8 - 55 * q^10 - 35 * q^11 + 62 * q^13 + 241 * q^16 - 48 * q^17 - 202 * q^19 - 439 * q^20 - 7 * q^22 - 216 * q^23 + 130 * q^25 - 274 * q^26 - 53 * q^29 - 95 * q^31 - 683 * q^32 - 24 * q^34 + 262 * q^37 + 398 * q^38 + 21 * q^40 - 244 * q^41 + 360 * q^43 + 905 * q^44 - 1056 * q^46 + 210 * q^47 + 1378 * q^50 + 324 * q^52 - 393 * q^53 - 1031 * q^55 - 1249 * q^58 - 1143 * q^59 - 70 * q^61 - 1059 * q^62 - 399 * q^64 + 472 * q^65 - 628 * q^67 - 1944 * q^68 - 318 * q^71 + 988 * q^73 - 1002 * q^74 - 2340 * q^76 + 861 * q^79 - 175 * q^80 + 124 * q^82 - 519 * q^83 + 1800 * q^85 + 3208 * q^86 - 891 * q^88 - 1766 * q^89 + 672 * q^92 - 3294 * q^94 + 736 * q^95 + 19 * q^97

Introduction

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