LMFDB - Embedded newform 2178.4.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,4,Mod(1,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, 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("2178.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2178.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:	$128.506159993$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.978025.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 99x^{2} + 100x + 2420 $ x^4 - 2x^3 - 99x^2 + 100*x + 2420
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.92695$ of defining polynomial
Character	$\chi$	$=$	2178.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -12.7359 q^{5} +23.4611 q^{7} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} -12.7359 q^{5} +23.4611 q^{7} +8.00000 q^{8} -25.4718 q^{10} +11.4654 q^{13} +46.9222 q^{14} +16.0000 q^{16} +65.5022 q^{17} -7.25013 q^{19} -50.9436 q^{20} +104.072 q^{23} +37.2034 q^{25} +22.9309 q^{26} +93.8445 q^{28} +127.351 q^{29} -288.811 q^{31} +32.0000 q^{32} +131.004 q^{34} -298.799 q^{35} -85.4023 q^{37} -14.5003 q^{38} -101.887 q^{40} +135.444 q^{41} +353.691 q^{43} +208.145 q^{46} +134.604 q^{47} +207.424 q^{49} +74.4068 q^{50} +45.8618 q^{52} -501.431 q^{53} +187.689 q^{56} +254.702 q^{58} -651.658 q^{59} +365.787 q^{61} -577.623 q^{62} +64.0000 q^{64} -146.023 q^{65} -294.576 q^{67} +262.009 q^{68} -597.597 q^{70} -132.446 q^{71} +469.489 q^{73} -170.805 q^{74} -29.0005 q^{76} +408.033 q^{79} -203.775 q^{80} +270.887 q^{82} +1359.54 q^{83} -834.230 q^{85} +707.381 q^{86} +260.255 q^{89} +268.992 q^{91} +416.289 q^{92} +269.207 q^{94} +92.3370 q^{95} +1414.53 q^{97} +414.848 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} - 25 q^{5} - 3 q^{7} + 32 q^{8}+O(q^{10}) $ 4 * q + 8 * q^2 + 16 * q^4 - 25 * q^5 - 3 * q^7 + 32 * q^8	$ 4 q + 8 q^{2} + 16 q^{4} - 25 q^{5} - 3 q^{7} + 32 q^{8} - 50 q^{10} + 41 q^{13} - 6 q^{14} + 64 q^{16} + 52 q^{17} - 16 q^{19} - 100 q^{20} - 314 q^{23} - 21 q^{25} + 82 q^{26} - 12 q^{28} + 561 q^{29} + 199 q^{31} + 128 q^{32} + 104 q^{34} - 714 q^{35} + 357 q^{37} - 32 q^{38} - 200 q^{40} + 32 q^{41} + 721 q^{43} - 628 q^{46} - 403 q^{47} + 823 q^{49} - 42 q^{50} + 164 q^{52} + 133 q^{53} - 24 q^{56} + 1122 q^{58} - 1016 q^{59} + 919 q^{61} + 398 q^{62} + 256 q^{64} + 69 q^{65} + 289 q^{67} + 208 q^{68} - 1428 q^{70} + 1205 q^{71} + 1234 q^{73} + 714 q^{74} - 64 q^{76} + 603 q^{79} - 400 q^{80} + 64 q^{82} + 1514 q^{83} - 717 q^{85} + 1442 q^{86} + 1101 q^{89} - 2306 q^{91} - 1256 q^{92} - 806 q^{94} - 1766 q^{95} + 2116 q^{97} + 1646 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 - 25 * q^5 - 3 * q^7 + 32 * q^8 - 50 * q^10 + 41 * q^13 - 6 * q^14 + 64 * q^16 + 52 * q^17 - 16 * q^19 - 100 * q^20 - 314 * q^23 - 21 * q^25 + 82 * q^26 - 12 * q^28 + 561 * q^29 + 199 * q^31 + 128 * q^32 + 104 * q^34 - 714 * q^35 + 357 * q^37 - 32 * q^38 - 200 * q^40 + 32 * q^41 + 721 * q^43 - 628 * q^46 - 403 * q^47 + 823 * q^49 - 42 * q^50 + 164 * q^52 + 133 * q^53 - 24 * q^56 + 1122 * q^58 - 1016 * q^59 + 919 * q^61 + 398 * q^62 + 256 * q^64 + 69 * q^65 + 289 * q^67 + 208 * q^68 - 1428 * q^70 + 1205 * q^71 + 1234 * q^73 + 714 * q^74 - 64 * q^76 + 603 * q^79 - 400 * q^80 + 64 * q^82 + 1514 * q^83 - 717 * q^85 + 1442 * q^86 + 1101 * q^89 - 2306 * q^91 - 1256 * q^92 - 806 * q^94 - 1766 * q^95 + 2116 * q^97 + 1646 * q^98

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
$5$	−12.7359	−1.13913	−0.569567	−	0.821945i	$-0.692889\pi$
$5$	−12.7359	−1.13913	−0.569567	+	0.821945i	$0.692889\pi$
$6$	0	0
$7$	23.4611	1.26678	0.633391	−	0.773832i	$-0.281663\pi$
$7$	23.4611	1.26678	0.633391	+	0.773832i	$0.281663\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	−25.4718	−0.805490
$11$	0	0
$11$	0	0
$12$	0	0
$13$	11.4654	0.244611	0.122305	−	0.992493i	$-0.460971\pi$
$13$	11.4654	0.244611	0.122305	+	0.992493i	$0.460971\pi$
$14$	46.9222	0.895750
$15$	0	0
$16$	16.0000	0.250000
$17$	65.5022	0.934507	0.467253	−	0.884124i	$-0.345244\pi$
$17$	65.5022	0.934507	0.467253	+	0.884124i	$0.345244\pi$
$18$	0	0
$19$	−7.25013	−0.0875417	−0.0437709	−	0.999042i	$-0.513937\pi$
$19$	−7.25013	−0.0875417	−0.0437709	+	0.999042i	$0.513937\pi$
$20$	−50.9436	−0.569567
$21$	0	0
$22$	0	0
$23$	104.072	0.943504	0.471752	−	0.881731i	$-0.343622\pi$
$23$	104.072	0.943504	0.471752	+	0.881731i	$0.343622\pi$
$24$	0	0
$25$	37.2034	0.297627
$26$	22.9309	0.172966
$27$	0	0
$28$	93.8445	0.633391
$29$	127.351	0.815466	0.407733	−	0.913101i	$-0.366319\pi$
$29$	127.351	0.815466	0.407733	+	0.913101i	$0.366319\pi$
$30$	0	0
$31$	−288.811	−1.67329	−0.836646	−	0.547744i	$-0.815487\pi$
$31$	−288.811	−1.67329	−0.836646	+	0.547744i	$0.815487\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	131.004	0.660796
$35$	−298.799	−1.44303
$36$	0	0
$37$	−85.4023	−0.379461	−0.189731	−	0.981836i	$-0.560761\pi$
$37$	−85.4023	−0.379461	−0.189731	+	0.981836i	$0.560761\pi$
$38$	−14.5003	−0.0619014
$39$	0	0
$40$	−101.887	−0.402745
$41$	135.444	0.515921	0.257960	−	0.966156i	$-0.416950\pi$
$41$	135.444	0.515921	0.257960	+	0.966156i	$0.416950\pi$
$42$	0	0
$43$	353.691	1.25436	0.627178	−	0.778876i	$-0.284210\pi$
$43$	353.691	1.25436	0.627178	+	0.778876i	$0.284210\pi$
$44$	0	0
$45$	0	0
$46$	208.145	0.667158
$47$	134.604	0.417744	0.208872	−	0.977943i	$-0.433021\pi$
$47$	134.604	0.417744	0.208872	+	0.977943i	$0.433021\pi$
$48$	0	0
$49$	207.424	0.604735
$50$	74.4068	0.210454
$51$	0	0
$52$	45.8618	0.122305
$53$	−501.431	−1.29956	−0.649782	−	0.760121i	$-0.725140\pi$
$53$	−501.431	−1.29956	−0.649782	+	0.760121i	$0.725140\pi$
$54$	0	0
$55$	0	0
$56$	187.689	0.447875
$57$	0	0
$58$	254.702	0.576621
$59$	−651.658	−1.43794	−0.718971	−	0.695040i	$-0.755387\pi$
$59$	−651.658	−1.43794	−0.718971	+	0.695040i	$0.755387\pi$
$60$	0	0
$61$	365.787	0.767775	0.383887	−	0.923380i	$-0.374585\pi$
$61$	365.787	0.767775	0.383887	+	0.923380i	$0.374585\pi$
$62$	−577.623	−1.18320
$63$	0	0
$64$	64.0000	0.125000
$65$	−146.023	−0.278645
$66$	0	0
$67$	−294.576	−0.537136	−0.268568	−	0.963261i	$-0.586550\pi$
$67$	−294.576	−0.537136	−0.268568	+	0.963261i	$0.586550\pi$
$68$	262.009	0.467253
$69$	0	0
$70$	−597.597	−1.02038
$71$	−132.446	−0.221387	−0.110694	−	0.993855i	$-0.535307\pi$
$71$	−132.446	−0.221387	−0.110694	+	0.993855i	$0.535307\pi$
$72$	0	0
$73$	469.489	0.752733	0.376367	−	0.926471i	$-0.377173\pi$
$73$	469.489	0.752733	0.376367	+	0.926471i	$0.377173\pi$
$74$	−170.805	−0.268319
$75$	0	0
$76$	−29.0005	−0.0437709
$77$	0	0
$78$	0	0
$79$	408.033	0.581105	0.290552	−	0.956859i	$-0.406161\pi$
$79$	408.033	0.581105	0.290552	+	0.956859i	$0.406161\pi$
$80$	−203.775	−0.284784
$81$	0	0
$82$	270.887	0.364811
$83$	1359.54	1.79794	0.898971	−	0.438009i	$-0.144316\pi$
$83$	1359.54	1.79794	0.898971	+	0.438009i	$0.144316\pi$
$84$	0	0
$85$	−834.230	−1.06453
$86$	707.381	0.886964
$87$	0	0
$88$	0	0
$89$	260.255	0.309966	0.154983	−	0.987917i	$-0.450468\pi$
$89$	260.255	0.309966	0.154983	+	0.987917i	$0.450468\pi$
$90$	0	0
$91$	268.992	0.309869
$92$	416.289	0.471752
$93$	0	0
$94$	269.207	0.295390
$95$	92.3370	0.0997218
$96$	0	0
$97$	1414.53	1.48066	0.740329	−	0.672245i	$-0.234670\pi$
$97$	1414.53	1.48066	0.740329	+	0.672245i	$0.234670\pi$
$98$	414.848	0.427612
$99$	0	0
$100$	148.814	0.148814
$101$	186.617	0.183852	0.0919261	−	0.995766i	$-0.470698\pi$
$101$	186.617	0.183852	0.0919261	+	0.995766i	$0.470698\pi$
$102$	0	0
$103$	−1172.91	−1.12205	−0.561023	−	0.827800i	$-0.689592\pi$
$103$	−1172.91	−1.12205	−0.561023	+	0.827800i	$0.689592\pi$
$104$	91.7236	0.0864830
$105$	0	0
$106$	−1002.86	−0.918930
$107$	712.714	0.643931	0.321966	−	0.946751i	$-0.395656\pi$
$107$	712.714	0.643931	0.321966	+	0.946751i	$0.395656\pi$
$108$	0	0
$109$	1247.22	1.09598	0.547989	−	0.836486i	$-0.315394\pi$
$109$	1247.22	1.09598	0.547989	+	0.836486i	$0.315394\pi$
$110$	0	0
$111$	0	0
$112$	375.378	0.316695
$113$	982.002	0.817513	0.408756	−	0.912644i	$-0.365963\pi$
$113$	982.002	0.817513	0.408756	+	0.912644i	$0.365963\pi$
$114$	0	0
$115$	−1325.46	−1.07478
$116$	509.404	0.407733
$117$	0	0
$118$	−1303.32	−1.01678
$119$	1536.75	1.18382
$120$	0	0
$121$	0	0
$122$	731.574	0.542899
$123$	0	0
$124$	−1155.25	−0.836646
$125$	1118.17	0.800097
$126$	0	0
$127$	−1550.67	−1.08346	−0.541730	−	0.840552i	$-0.682231\pi$
$127$	−1550.67	−1.08346	−0.541730	+	0.840552i	$0.682231\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−292.046	−0.197032
$131$	−742.114	−0.494953	−0.247476	−	0.968894i	$-0.579601\pi$
$131$	−742.114	−0.494953	−0.247476	+	0.968894i	$0.579601\pi$
$132$	0	0
$133$	−170.096	−0.110896
$134$	−589.151	−0.379813
$135$	0	0
$136$	524.017	0.330398
$137$	−515.678	−0.321586	−0.160793	−	0.986988i	$-0.551405\pi$
$137$	−515.678	−0.321586	−0.160793	+	0.986988i	$0.551405\pi$
$138$	0	0
$139$	2463.17	1.50304	0.751522	−	0.659709i	$-0.229320\pi$
$139$	2463.17	1.50304	0.751522	+	0.659709i	$0.229320\pi$
$140$	−1195.19	−0.721517
$141$	0	0
$142$	−264.893	−0.156544
$143$	0	0
$144$	0	0
$145$	−1621.93	−0.928925
$146$	938.978	0.532263
$147$	0	0
$148$	−341.609	−0.189731
$149$	434.757	0.239038	0.119519	−	0.992832i	$-0.461865\pi$
$149$	434.757	0.239038	0.119519	+	0.992832i	$0.461865\pi$
$150$	0	0
$151$	−780.992	−0.420902	−0.210451	−	0.977604i	$-0.567493\pi$
$151$	−780.992	−0.420902	−0.210451	+	0.977604i	$0.567493\pi$
$152$	−58.0010	−0.0309507
$153$	0	0
$154$	0	0
$155$	3678.27	1.90610
$156$	0	0
$157$	522.382	0.265545	0.132773	−	0.991147i	$-0.457612\pi$
$157$	522.382	0.265545	0.132773	+	0.991147i	$0.457612\pi$
$158$	816.065	0.410903
$159$	0	0
$160$	−407.549	−0.201372
$161$	2441.65	1.19521
$162$	0	0
$163$	2543.29	1.22212	0.611060	−	0.791584i	$-0.290743\pi$
$163$	2543.29	1.22212	0.611060	+	0.791584i	$0.290743\pi$
$164$	541.775	0.257960
$165$	0	0
$166$	2719.08	1.27134
$167$	−1385.44	−0.641968	−0.320984	−	0.947085i	$-0.604014\pi$
$167$	−1385.44	−0.641968	−0.320984	+	0.947085i	$0.604014\pi$
$168$	0	0
$169$	−2065.54	−0.940165
$170$	−1668.46	−0.752735
$171$	0	0
$172$	1414.76	0.627178
$173$	443.511	0.194911	0.0974553	−	0.995240i	$-0.468930\pi$
$173$	443.511	0.194911	0.0974553	+	0.995240i	$0.468930\pi$
$174$	0	0
$175$	872.833	0.377029
$176$	0	0
$177$	0	0
$178$	520.510	0.219179
$179$	−376.639	−0.157270	−0.0786349	−	0.996903i	$-0.525056\pi$
$179$	−376.639	−0.157270	−0.0786349	+	0.996903i	$0.525056\pi$
$180$	0	0
$181$	3293.76	1.35261	0.676307	−	0.736620i	$-0.263579\pi$
$181$	3293.76	1.35261	0.676307	+	0.736620i	$0.263579\pi$
$182$	537.984	0.219110
$183$	0	0
$184$	832.579	0.333579
$185$	1087.68	0.432257
$186$	0	0
$187$	0	0
$188$	538.415	0.208872
$189$	0	0
$190$	184.674	0.0705140
$191$	−2290.26	−0.867631	−0.433816	−	0.901002i	$-0.642833\pi$
$191$	−2290.26	−0.867631	−0.433816	+	0.901002i	$0.642833\pi$
$192$	0	0
$193$	−2303.40	−0.859079	−0.429540	−	0.903048i	$-0.641324\pi$
$193$	−2303.40	−0.859079	−0.429540	+	0.903048i	$0.641324\pi$
$194$	2829.06	1.04698
$195$	0	0
$196$	829.696	0.302367
$197$	1041.86	0.376801	0.188400	−	0.982092i	$-0.439670\pi$
$197$	1041.86	0.376801	0.188400	+	0.982092i	$0.439670\pi$
$198$	0	0
$199$	3463.83	1.23389	0.616946	−	0.787005i	$-0.288370\pi$
$199$	3463.83	1.23389	0.616946	+	0.787005i	$0.288370\pi$
$200$	297.627	0.105227
$201$	0	0
$202$	373.234	0.130003
$203$	2987.80	1.03302
$204$	0	0
$205$	−1725.00	−0.587703
$206$	−2345.83	−0.793406
$207$	0	0
$208$	183.447	0.0611527
$209$	0	0
$210$	0	0
$211$	2091.98	0.682548	0.341274	−	0.939964i	$-0.389142\pi$
$211$	2091.98	0.682548	0.341274	+	0.939964i	$0.389142\pi$
$212$	−2005.72	−0.649782
$213$	0	0
$214$	1425.43	0.455328
$215$	−4504.57	−1.42888
$216$	0	0
$217$	−6775.84	−2.11969
$218$	2494.43	0.774973
$219$	0	0
$220$	0	0
$221$	751.012	0.228591
$222$	0	0
$223$	3703.53	1.11214	0.556068	−	0.831137i	$-0.312309\pi$
$223$	3703.53	1.11214	0.556068	+	0.831137i	$0.312309\pi$
$224$	750.756	0.223937
$225$	0	0
$226$	1964.00	0.578069
$227$	4853.50	1.41911	0.709555	−	0.704650i	$-0.248896\pi$
$227$	4853.50	1.41911	0.709555	+	0.704650i	$0.248896\pi$
$228$	0	0
$229$	−1336.85	−0.385770	−0.192885	−	0.981221i	$-0.561784\pi$
$229$	−1336.85	−0.385770	−0.192885	+	0.981221i	$0.561784\pi$
$230$	−2650.91	−0.759983
$231$	0	0
$232$	1018.81	0.288311
$233$	5903.91	1.65999	0.829995	−	0.557770i	$-0.188343\pi$
$233$	5903.91	1.65999	0.829995	+	0.557770i	$0.188343\pi$
$234$	0	0
$235$	−1714.30	−0.475866
$236$	−2606.63	−0.718971
$237$	0	0
$238$	3073.51	0.837084
$239$	3319.79	0.898490	0.449245	−	0.893409i	$-0.351693\pi$
$239$	3319.79	0.898490	0.449245	+	0.893409i	$0.351693\pi$
$240$	0	0
$241$	−5275.95	−1.41018	−0.705091	−	0.709116i	$-0.749094\pi$
$241$	−5275.95	−1.41018	−0.705091	+	0.709116i	$0.749094\pi$
$242$	0	0
$243$	0	0
$244$	1463.15	0.383887
$245$	−2641.73	−0.688874
$246$	0	0
$247$	−83.1259	−0.0214137
$248$	−2310.49	−0.591598
$249$	0	0
$250$	2236.34	0.565754
$251$	1961.85	0.493350	0.246675	−	0.969098i	$-0.420662\pi$
$251$	1961.85	0.493350	0.246675	+	0.969098i	$0.420662\pi$
$252$	0	0
$253$	0	0
$254$	−3101.34	−0.766123
$255$	0	0
$256$	256.000	0.0625000
$257$	−5941.97	−1.44222	−0.721108	−	0.692822i	$-0.756367\pi$
$257$	−5941.97	−1.44222	−0.721108	+	0.692822i	$0.756367\pi$
$258$	0	0
$259$	−2003.63	−0.480694
$260$	−584.092	−0.139322
$261$	0	0
$262$	−1484.23	−0.349985
$263$	1704.11	0.399544	0.199772	−	0.979842i	$-0.435980\pi$
$263$	1704.11	0.399544	0.199772	+	0.979842i	$0.435980\pi$
$264$	0	0
$265$	6386.18	1.48038
$266$	−340.192	−0.0784155
$267$	0	0
$268$	−1178.30	−0.268568
$269$	5845.33	1.32489	0.662446	−	0.749109i	$-0.269518\pi$
$269$	5845.33	1.32489	0.662446	+	0.749109i	$0.269518\pi$
$270$	0	0
$271$	5905.84	1.32382	0.661908	−	0.749585i	$-0.269747\pi$
$271$	5905.84	1.32382	0.661908	+	0.749585i	$0.269747\pi$
$272$	1048.03	0.233627
$273$	0	0
$274$	−1031.36	−0.227396
$275$	0	0
$276$	0	0
$277$	8882.38	1.92668	0.963341	−	0.268281i	$-0.0864556\pi$
$277$	8882.38	1.92668	0.963341	+	0.268281i	$0.0864556\pi$
$278$	4926.33	1.06281
$279$	0	0
$280$	−2390.39	−0.510190
$281$	8151.77	1.73058	0.865291	−	0.501269i	$-0.167133\pi$
$281$	8151.77	1.73058	0.865291	+	0.501269i	$0.167133\pi$
$282$	0	0
$283$	5863.96	1.23172	0.615859	−	0.787856i	$-0.288809\pi$
$283$	5863.96	1.23172	0.615859	+	0.787856i	$0.288809\pi$
$284$	−529.785	−0.110694
$285$	0	0
$286$	0	0
$287$	3177.66	0.653559
$288$	0	0
$289$	−622.465	−0.126698
$290$	−3243.86	−0.656849
$291$	0	0
$292$	1877.96	0.376367
$293$	−7616.57	−1.51865	−0.759326	−	0.650711i	$-0.774471\pi$
$293$	−7616.57	−1.51865	−0.759326	+	0.650711i	$0.774471\pi$
$294$	0	0
$295$	8299.45	1.63801
$296$	−683.219	−0.134160
$297$	0	0
$298$	869.515	0.169026
$299$	1193.24	0.230791
$300$	0	0
$301$	8297.98	1.58899
$302$	−1561.98	−0.297623
$303$	0	0
$304$	−116.002	−0.0218854
$305$	−4658.63	−0.874598
$306$	0	0
$307$	4100.68	0.762339	0.381170	−	0.924505i	$-0.375521\pi$
$307$	4100.68	0.762339	0.381170	+	0.924505i	$0.375521\pi$
$308$	0	0
$309$	0	0
$310$	7356.55	1.34782
$311$	−1456.13	−0.265497	−0.132749	−	0.991150i	$-0.542380\pi$
$311$	−1456.13	−0.265497	−0.132749	+	0.991150i	$0.542380\pi$
$312$	0	0
$313$	−3461.68	−0.625129	−0.312565	−	0.949896i	$-0.601188\pi$
$313$	−3461.68	−0.625129	−0.312565	+	0.949896i	$0.601188\pi$
$314$	1044.76	0.187769
$315$	0	0
$316$	1632.13	0.290552
$317$	−4992.31	−0.884530	−0.442265	−	0.896884i	$-0.645825\pi$
$317$	−4992.31	−0.884530	−0.442265	+	0.896884i	$0.645825\pi$
$318$	0	0
$319$	0	0
$320$	−815.098	−0.142392
$321$	0	0
$322$	4883.31	0.845143
$323$	−474.899	−0.0818083
$324$	0	0
$325$	426.554	0.0728029
$326$	5086.57	0.864169
$327$	0	0
$328$	1083.55	0.182406
$329$	3157.95	0.529190
$330$	0	0
$331$	10199.3	1.69366	0.846832	−	0.531861i	$-0.178507\pi$
$331$	10199.3	1.69366	0.846832	+	0.531861i	$0.178507\pi$
$332$	5438.17	0.898971
$333$	0	0
$334$	−2770.88	−0.453940
$335$	3751.69	0.611870
$336$	0	0
$337$	−1680.74	−0.271678	−0.135839	−	0.990731i	$-0.543373\pi$
$337$	−1680.74	−0.271678	−0.135839	+	0.990731i	$0.543373\pi$
$338$	−4131.09	−0.664797
$339$	0	0
$340$	−3336.92	−0.532264
$341$	0	0
$342$	0	0
$343$	−3180.76	−0.500715
$344$	2829.53	0.443482
$345$	0	0
$346$	887.022	0.137823
$347$	−10526.3	−1.62847	−0.814236	−	0.580534i	$-0.802844\pi$
$347$	−10526.3	−1.62847	−0.814236	+	0.580534i	$0.802844\pi$
$348$	0	0
$349$	−5300.36	−0.812956	−0.406478	−	0.913660i	$-0.633243\pi$
$349$	−5300.36	−0.812956	−0.406478	+	0.913660i	$0.633243\pi$
$350$	1745.67	0.266599
$351$	0	0
$352$	0	0
$353$	7438.40	1.12155	0.560773	−	0.827969i	$-0.310504\pi$
$353$	7438.40	1.12155	0.560773	+	0.827969i	$0.310504\pi$
$354$	0	0
$355$	1686.82	0.252190
$356$	1041.02	0.154983
$357$	0	0
$358$	−753.278	−0.111207
$359$	−10151.3	−1.49238	−0.746191	−	0.665731i	$-0.768120\pi$
$359$	−10151.3	−1.49238	−0.746191	+	0.665731i	$0.768120\pi$
$360$	0	0
$361$	−6806.44	−0.992336
$362$	6587.52	0.956443
$363$	0	0
$364$	1075.97	0.154934
$365$	−5979.37	−0.857464
$366$	0	0
$367$	476.803	0.0678172	0.0339086	−	0.999425i	$-0.489204\pi$
$367$	476.803	0.0678172	0.0339086	+	0.999425i	$0.489204\pi$
$368$	1665.16	0.235876
$369$	0	0
$370$	2175.35	0.305652
$371$	−11764.1	−1.64626
$372$	0	0
$373$	−12738.5	−1.76829	−0.884146	−	0.467210i	$-0.845259\pi$
$373$	−12738.5	−1.76829	−0.884146	+	0.467210i	$0.845259\pi$
$374$	0	0
$375$	0	0
$376$	1076.83	0.147695
$377$	1460.14	0.199472
$378$	0	0
$379$	−1298.69	−0.176013	−0.0880067	−	0.996120i	$-0.528050\pi$
$379$	−1298.69	−0.176013	−0.0880067	+	0.996120i	$0.528050\pi$
$380$	369.348	0.0498609
$381$	0	0
$382$	−4580.53	−0.613508
$383$	670.568	0.0894632	0.0447316	−	0.998999i	$-0.485757\pi$
$383$	670.568	0.0894632	0.0447316	+	0.998999i	$0.485757\pi$
$384$	0	0
$385$	0	0
$386$	−4606.80	−0.607461
$387$	0	0
$388$	5658.12	0.740329
$389$	−5235.90	−0.682444	−0.341222	−	0.939983i	$-0.610841\pi$
$389$	−5235.90	−0.682444	−0.341222	+	0.939983i	$0.610841\pi$
$390$	0	0
$391$	6816.97	0.881711
$392$	1659.39	0.213806
$393$	0	0
$394$	2083.73	0.266438
$395$	−5196.67	−0.661956
$396$	0	0
$397$	1751.64	0.221442	0.110721	−	0.993852i	$-0.464684\pi$
$397$	1751.64	0.221442	0.110721	+	0.993852i	$0.464684\pi$
$398$	6927.67	0.872494
$399$	0	0
$400$	595.254	0.0744068
$401$	−3612.78	−0.449909	−0.224954	−	0.974369i	$-0.572223\pi$
$401$	−3612.78	−0.449909	−0.224954	+	0.974369i	$0.572223\pi$
$402$	0	0
$403$	−3311.35	−0.409305
$404$	746.468	0.0919261
$405$	0	0
$406$	5975.60	0.730453
$407$	0	0
$408$	0	0
$409$	−11295.0	−1.36553	−0.682764	−	0.730639i	$-0.739222\pi$
$409$	−11295.0	−1.36553	−0.682764	+	0.730639i	$0.739222\pi$
$410$	−3450.00	−0.415569
$411$	0	0
$412$	−4691.66	−0.561023
$413$	−15288.6	−1.82156
$414$	0	0
$415$	−17315.0	−2.04810
$416$	366.894	0.0432415
$417$	0	0
$418$	0	0
$419$	3680.45	0.429121	0.214560	−	0.976711i	$-0.431168\pi$
$419$	3680.45	0.429121	0.214560	+	0.976711i	$0.431168\pi$
$420$	0	0
$421$	9256.17	1.07154	0.535770	−	0.844364i	$-0.320021\pi$
$421$	9256.17	1.07154	0.535770	+	0.844364i	$0.320021\pi$
$422$	4183.96	0.482635
$423$	0	0
$424$	−4011.45	−0.459465
$425$	2436.90	0.278135
$426$	0	0
$427$	8581.77	0.972602
$428$	2850.85	0.321966
$429$	0	0
$430$	−9009.14	−1.01037
$431$	−3584.04	−0.400550	−0.200275	−	0.979740i	$-0.564184\pi$
$431$	−3584.04	−0.400550	−0.200275	+	0.979740i	$0.564184\pi$
$432$	0	0
$433$	−8306.80	−0.921939	−0.460969	−	0.887416i	$-0.652498\pi$
$433$	−8306.80	−0.921939	−0.460969	+	0.887416i	$0.652498\pi$
$434$	−13551.7	−1.49885
$435$	0	0
$436$	4988.86	0.547989
$437$	−754.538	−0.0825960
$438$	0	0
$439$	11932.4	1.29727	0.648634	−	0.761101i	$-0.275341\pi$
$439$	11932.4	1.29727	0.648634	+	0.761101i	$0.275341\pi$
$440$	0	0
$441$	0	0
$442$	1502.02	0.161638
$443$	−1078.82	−0.115703	−0.0578514	−	0.998325i	$-0.518425\pi$
$443$	−1078.82	−0.115703	−0.0578514	+	0.998325i	$0.518425\pi$
$444$	0	0
$445$	−3314.58	−0.353093
$446$	7407.05	0.786399
$447$	0	0
$448$	1501.51	0.158348
$449$	−12765.9	−1.34179	−0.670893	−	0.741554i	$-0.734089\pi$
$449$	−12765.9	−1.34179	−0.670893	+	0.741554i	$0.734089\pi$
$450$	0	0
$451$	0	0
$452$	3928.01	0.408756
$453$	0	0
$454$	9707.00	1.00346
$455$	−3425.86	−0.352982
$456$	0	0
$457$	−11226.8	−1.14917	−0.574583	−	0.818446i	$-0.694836\pi$
$457$	−11226.8	−1.14917	−0.574583	+	0.818446i	$0.694836\pi$
$458$	−2673.69	−0.272780
$459$	0	0
$460$	−5301.83	−0.537389
$461$	2160.58	0.218283	0.109141	−	0.994026i	$-0.465190\pi$
$461$	2160.58	0.218283	0.109141	+	0.994026i	$0.465190\pi$
$462$	0	0
$463$	11469.5	1.15125	0.575627	−	0.817712i	$-0.304758\pi$
$463$	11469.5	1.15125	0.575627	+	0.817712i	$0.304758\pi$
$464$	2037.62	0.203866
$465$	0	0
$466$	11807.8	1.17379
$467$	−2018.39	−0.200000	−0.100000	−	0.994987i	$-0.531884\pi$
$467$	−2018.39	−0.200000	−0.100000	+	0.994987i	$0.531884\pi$
$468$	0	0
$469$	−6911.07	−0.680434
$470$	−3428.60	−0.336488
$471$	0	0
$472$	−5213.26	−0.508389
$473$	0	0
$474$	0	0
$475$	−269.729	−0.0260548
$476$	6147.02	0.591908
$477$	0	0
$478$	6639.57	0.635328
$479$	7327.00	0.698913	0.349457	−	0.936953i	$-0.386366\pi$
$479$	7327.00	0.698913	0.349457	+	0.936953i	$0.386366\pi$
$480$	0	0
$481$	−979.176	−0.0928203
$482$	−10551.9	−0.997150
$483$	0	0
$484$	0	0
$485$	−18015.3	−1.68667
$486$	0	0
$487$	−6189.80	−0.575948	−0.287974	−	0.957638i	$-0.592982\pi$
$487$	−6189.80	−0.575948	−0.287974	+	0.957638i	$0.592982\pi$
$488$	2926.30	0.271449
$489$	0	0
$490$	−5283.47	−0.487107
$491$	10120.9	0.930246	0.465123	−	0.885246i	$-0.346010\pi$
$491$	10120.9	0.930246	0.465123	+	0.885246i	$0.346010\pi$
$492$	0	0
$493$	8341.77	0.762058
$494$	−166.252	−0.0151418
$495$	0	0
$496$	−4620.98	−0.418323
$497$	−3107.34	−0.280449
$498$	0	0
$499$	2562.81	0.229914	0.114957	−	0.993370i	$-0.463327\pi$
$499$	2562.81	0.229914	0.114957	+	0.993370i	$0.463327\pi$
$500$	4472.68	0.400049
$501$	0	0
$502$	3923.70	0.348851
$503$	−8601.51	−0.762470	−0.381235	−	0.924478i	$-0.624501\pi$
$503$	−8601.51	−0.762470	−0.381235	+	0.924478i	$0.624501\pi$
$504$	0	0
$505$	−2376.74	−0.209432
$506$	0	0
$507$	0	0
$508$	−6202.67	−0.541730
$509$	13945.1	1.21435	0.607176	−	0.794567i	$-0.292302\pi$
$509$	13945.1	1.21435	0.607176	+	0.794567i	$0.292302\pi$
$510$	0	0
$511$	11014.7	0.953548
$512$	512.000	0.0441942
$513$	0	0
$514$	−11883.9	−1.01980
$515$	14938.1	1.27816
$516$	0	0
$517$	0	0
$518$	−4007.27	−0.339902
$519$	0	0
$520$	−1168.18	−0.0985158
$521$	12092.8	1.01688	0.508442	−	0.861096i	$-0.330222\pi$
$521$	12092.8	1.01688	0.508442	+	0.861096i	$0.330222\pi$
$522$	0	0
$523$	−16691.8	−1.39557	−0.697784	−	0.716308i	$-0.745831\pi$
$523$	−16691.8	−1.39557	−0.697784	+	0.716308i	$0.745831\pi$
$524$	−2968.46	−0.247476
$525$	0	0
$526$	3408.22	0.282520
$527$	−18917.8	−1.56370
$528$	0	0
$529$	−1335.94	−0.109800
$530$	12772.4	1.04678
$531$	0	0
$532$	−680.384	−0.0554481
$533$	1552.92	0.126200
$534$	0	0
$535$	−9077.06	−0.733524
$536$	−2356.60	−0.189906
$537$	0	0
$538$	11690.7	0.936840
$539$	0	0
$540$	0	0
$541$	20673.3	1.64291	0.821456	−	0.570272i	$-0.193162\pi$
$541$	20673.3	1.64291	0.821456	+	0.570272i	$0.193162\pi$
$542$	11811.7	0.936079
$543$	0	0
$544$	2096.07	0.165199
$545$	−15884.4	−1.24847
$546$	0	0
$547$	9169.80	0.716768	0.358384	−	0.933574i	$-0.383328\pi$
$547$	9169.80	0.716768	0.358384	+	0.933574i	$0.383328\pi$
$548$	−2062.71	−0.160793
$549$	0	0
$550$	0	0
$551$	−923.311	−0.0713873
$552$	0	0
$553$	9572.90	0.736132
$554$	17764.8	1.36237
$555$	0	0
$556$	9852.66	0.751522
$557$	−20310.8	−1.54506	−0.772528	−	0.634981i	$-0.781008\pi$
$557$	−20310.8	−1.54506	−0.772528	+	0.634981i	$0.781008\pi$
$558$	0	0
$559$	4055.22	0.306829
$560$	−4780.78	−0.360759
$561$	0	0
$562$	16303.5	1.22371
$563$	−5563.15	−0.416445	−0.208223	−	0.978081i	$-0.566768\pi$
$563$	−5563.15	−0.416445	−0.208223	+	0.978081i	$0.566768\pi$
$564$	0	0
$565$	−12506.7	−0.931257
$566$	11727.9	0.870956
$567$	0	0
$568$	−1059.57	−0.0782721
$569$	22785.5	1.67876	0.839381	−	0.543543i	$-0.182917\pi$
$569$	22785.5	1.67876	0.839381	+	0.543543i	$0.182917\pi$
$570$	0	0
$571$	−11157.6	−0.817743	−0.408871	−	0.912592i	$-0.634078\pi$
$571$	−11157.6	−0.817743	−0.408871	+	0.912592i	$0.634078\pi$
$572$	0	0
$573$	0	0
$574$	6355.32	0.462136
$575$	3871.85	0.280812
$576$	0	0
$577$	12429.1	0.896759	0.448380	−	0.893843i	$-0.352001\pi$
$577$	12429.1	0.896759	0.448380	+	0.893843i	$0.352001\pi$
$578$	−1244.93	−0.0895887
$579$	0	0
$580$	−6487.73	−0.464462
$581$	31896.4	2.27760
$582$	0	0
$583$	0	0
$584$	3755.91	0.266131
$585$	0	0
$586$	−15233.1	−1.07385
$587$	−6114.94	−0.429967	−0.214984	−	0.976618i	$-0.568970\pi$
$587$	−6114.94	−0.429967	−0.214984	+	0.976618i	$0.568970\pi$
$588$	0	0
$589$	2093.92	0.146483
$590$	16598.9	1.15825
$591$	0	0
$592$	−1366.44	−0.0948653
$593$	−7188.32	−0.497789	−0.248894	−	0.968531i	$-0.580067\pi$
$593$	−7188.32	−0.497789	−0.248894	+	0.968531i	$0.580067\pi$
$594$	0	0
$595$	−19572.0	−1.34852
$596$	1739.03	0.119519
$597$	0	0
$598$	2386.47	0.163194
$599$	1012.18	0.0690429	0.0345215	−	0.999404i	$-0.489009\pi$
$599$	1012.18	0.0690429	0.0345215	+	0.999404i	$0.489009\pi$
$600$	0	0
$601$	13380.9	0.908184	0.454092	−	0.890955i	$-0.349964\pi$
$601$	13380.9	0.908184	0.454092	+	0.890955i	$0.349964\pi$
$602$	16596.0	1.12359
$603$	0	0
$604$	−3123.97	−0.210451
$605$	0	0
$606$	0	0
$607$	546.951	0.0365734	0.0182867	−	0.999833i	$-0.494179\pi$
$607$	546.951	0.0365734	0.0182867	+	0.999833i	$0.494179\pi$
$608$	−232.004	−0.0154753
$609$	0	0
$610$	−9317.26	−0.618434
$611$	1543.29	0.102185
$612$	0	0
$613$	13745.6	0.905676	0.452838	−	0.891593i	$-0.350412\pi$
$613$	13745.6	0.905676	0.452838	+	0.891593i	$0.350412\pi$
$614$	8201.36	0.539055
$615$	0	0
$616$	0	0
$617$	3323.39	0.216847	0.108423	−	0.994105i	$-0.465420\pi$
$617$	3323.39	0.216847	0.108423	+	0.994105i	$0.465420\pi$
$618$	0	0
$619$	−21679.1	−1.40768	−0.703842	−	0.710357i	$-0.748533\pi$
$619$	−21679.1	−1.40768	−0.703842	+	0.710357i	$0.748533\pi$
$620$	14713.1	0.953052
$621$	0	0
$622$	−2912.26	−0.187735
$623$	6105.87	0.392659
$624$	0	0
$625$	−18891.3	−1.20905
$626$	−6923.35	−0.442033
$627$	0	0
$628$	2089.53	0.132773
$629$	−5594.04	−0.354609
$630$	0	0
$631$	−29888.8	−1.88566	−0.942831	−	0.333271i	$-0.891847\pi$
$631$	−29888.8	−1.88566	−0.942831	+	0.333271i	$0.891847\pi$
$632$	3264.26	0.205452
$633$	0	0
$634$	−9984.62	−0.625457
$635$	19749.2	1.23421
$636$	0	0
$637$	2378.21	0.147925
$638$	0	0
$639$	0	0
$640$	−1630.20	−0.100686
$641$	−27178.0	−1.67467	−0.837336	−	0.546689i	$-0.815888\pi$
$641$	−27178.0	−1.67467	−0.837336	+	0.546689i	$0.815888\pi$
$642$	0	0
$643$	16374.5	1.00427	0.502136	−	0.864789i	$-0.332548\pi$
$643$	16374.5	1.00427	0.502136	+	0.864789i	$0.332548\pi$
$644$	9766.62	0.597607
$645$	0	0
$646$	−949.798	−0.0578472
$647$	−2287.67	−0.139007	−0.0695034	−	0.997582i	$-0.522141\pi$
$647$	−2287.67	−0.139007	−0.0695034	+	0.997582i	$0.522141\pi$
$648$	0	0
$649$	0	0
$650$	853.107	0.0514794
$651$	0	0
$652$	10173.1	0.611060
$653$	1146.92	0.0687329	0.0343665	−	0.999409i	$-0.489059\pi$
$653$	1146.92	0.0687329	0.0343665	+	0.999409i	$0.489059\pi$
$654$	0	0
$655$	9451.50	0.563818
$656$	2167.10	0.128980
$657$	0	0
$658$	6315.90	0.374194
$659$	377.923	0.0223396	0.0111698	−	0.999938i	$-0.496444\pi$
$659$	377.923	0.0223396	0.0111698	+	0.999938i	$0.496444\pi$
$660$	0	0
$661$	−17500.4	−1.02978	−0.514892	−	0.857255i	$-0.672168\pi$
$661$	−17500.4	−1.02978	−0.514892	+	0.857255i	$0.672168\pi$
$662$	20398.5	1.19760
$663$	0	0
$664$	10876.3	0.635668
$665$	2166.33	0.126326
$666$	0	0
$667$	13253.7	0.769395
$668$	−5541.77	−0.320984
$669$	0	0
$670$	7503.38	0.432658
$671$	0	0
$672$	0	0
$673$	−14510.5	−0.831114	−0.415557	−	0.909567i	$-0.636413\pi$
$673$	−14510.5	−0.831114	−0.415557	+	0.909567i	$0.636413\pi$
$674$	−3361.47	−0.192106
$675$	0	0
$676$	−8262.17	−0.470083
$677$	−469.109	−0.0266312	−0.0133156	−	0.999911i	$-0.504239\pi$
$677$	−469.109	−0.0266312	−0.0133156	+	0.999911i	$0.504239\pi$
$678$	0	0
$679$	33186.5	1.87567
$680$	−6673.84	−0.376368
$681$	0	0
$682$	0	0
$683$	15892.3	0.890342	0.445171	−	0.895446i	$-0.353143\pi$
$683$	15892.3	0.890342	0.445171	+	0.895446i	$0.353143\pi$
$684$	0	0
$685$	6567.63	0.366330
$686$	−6361.53	−0.354059
$687$	0	0
$688$	5659.05	0.313589
$689$	−5749.13	−0.317887
$690$	0	0
$691$	−6205.64	−0.341641	−0.170820	−	0.985302i	$-0.554642\pi$
$691$	−6205.64	−0.341641	−0.170820	+	0.985302i	$0.554642\pi$
$692$	1774.04	0.0974553
$693$	0	0
$694$	−21052.5	−1.15150
$695$	−31370.7	−1.71217
$696$	0	0
$697$	8871.85	0.482131
$698$	−10600.7	−0.574847
$699$	0	0
$700$	3491.33	0.188514
$701$	−1872.54	−0.100891	−0.0504456	−	0.998727i	$-0.516064\pi$
$701$	−1872.54	−0.100891	−0.0504456	+	0.998727i	$0.516064\pi$
$702$	0	0
$703$	619.178	0.0332187
$704$	0	0
$705$	0	0
$706$	14876.8	0.793053
$707$	4378.24	0.232901
$708$	0	0
$709$	5236.01	0.277352	0.138676	−	0.990338i	$-0.455715\pi$
$709$	5236.01	0.277352	0.138676	+	0.990338i	$0.455715\pi$
$710$	3373.65	0.178325
$711$	0	0
$712$	2082.04	0.109589
$713$	−30057.3	−1.57876
$714$	0	0
$715$	0	0
$716$	−1506.56	−0.0786349
$717$	0	0
$718$	−20302.6	−1.05527
$719$	17274.8	0.896023	0.448011	−	0.894028i	$-0.352132\pi$
$719$	17274.8	0.896023	0.448011	+	0.894028i	$0.352132\pi$
$720$	0	0
$721$	−27517.9	−1.42139
$722$	−13612.9	−0.701688
$723$	0	0
$724$	13175.0	0.676307
$725$	4737.89	0.242705
$726$	0	0
$727$	17292.7	0.882190	0.441095	−	0.897461i	$-0.354590\pi$
$727$	17292.7	0.882190	0.441095	+	0.897461i	$0.354590\pi$
$728$	2151.94	0.109555
$729$	0	0
$730$	−11958.7	−0.606319
$731$	23167.5	1.17220
$732$	0	0
$733$	−13881.3	−0.699480	−0.349740	−	0.936847i	$-0.613730\pi$
$733$	−13881.3	−0.699480	−0.349740	+	0.936847i	$0.613730\pi$
$734$	953.606	0.0479540
$735$	0	0
$736$	3330.32	0.166790
$737$	0	0
$738$	0	0
$739$	6379.26	0.317544	0.158772	−	0.987315i	$-0.449247\pi$
$739$	6379.26	0.317544	0.158772	+	0.987315i	$0.449247\pi$
$740$	4350.71	0.216129
$741$	0	0
$742$	−23528.3	−1.16408
$743$	−23042.7	−1.13776	−0.568880	−	0.822421i	$-0.692623\pi$
$743$	−23042.7	−1.13776	−0.568880	+	0.822421i	$0.692623\pi$
$744$	0	0
$745$	−5537.03	−0.272297
$746$	−25477.0	−1.25037
$747$	0	0
$748$	0	0
$749$	16721.1	0.815720
$750$	0	0
$751$	−20806.1	−1.01095	−0.505476	−	0.862841i	$-0.668683\pi$
$751$	−20806.1	−1.01095	−0.505476	+	0.862841i	$0.668683\pi$
$752$	2153.66	0.104436
$753$	0	0
$754$	2920.27	0.141048
$755$	9946.65	0.479464
$756$	0	0
$757$	−21837.7	−1.04849	−0.524245	−	0.851568i	$-0.675652\pi$
$757$	−21837.7	−1.04849	−0.524245	+	0.851568i	$0.675652\pi$
$758$	−2597.38	−0.124460
$759$	0	0
$760$	738.696	0.0352570
$761$	7139.46	0.340086	0.170043	−	0.985437i	$-0.445609\pi$
$761$	7139.46	0.340086	0.170043	+	0.985437i	$0.445609\pi$
$762$	0	0
$763$	29261.1	1.38836
$764$	−9161.05	−0.433816
$765$	0	0
$766$	1341.14	0.0632600
$767$	−7471.55	−0.351737
$768$	0	0
$769$	15473.8	0.725618	0.362809	−	0.931864i	$-0.381818\pi$
$769$	15473.8	0.725618	0.362809	+	0.931864i	$0.381818\pi$
$770$	0	0
$771$	0	0
$772$	−9213.60	−0.429540
$773$	−15240.9	−0.709155	−0.354577	−	0.935027i	$-0.615375\pi$
$773$	−15240.9	−0.709155	−0.354577	+	0.935027i	$0.615375\pi$
$774$	0	0
$775$	−10744.8	−0.498017
$776$	11316.2	0.523492
$777$	0	0
$778$	−10471.8	−0.482561
$779$	−981.984	−0.0451646
$780$	0	0
$781$	0	0
$782$	13633.9	0.623464
$783$	0	0
$784$	3318.78	0.151184
$785$	−6653.01	−0.302492
$786$	0	0
$787$	36081.4	1.63426	0.817131	−	0.576452i	$-0.195563\pi$
$787$	36081.4	1.63426	0.817131	+	0.576452i	$0.195563\pi$
$788$	4167.46	0.188400
$789$	0	0
$790$	−10393.3	−0.468074
$791$	23038.9	1.03561
$792$	0	0
$793$	4193.91	0.187806
$794$	3503.29	0.156583
$795$	0	0
$796$	13855.3	0.616946
$797$	−3383.49	−0.150375	−0.0751877	−	0.997169i	$-0.523956\pi$
$797$	−3383.49	−0.150375	−0.0751877	+	0.997169i	$0.523956\pi$
$798$	0	0
$799$	8816.83	0.390384
$800$	1190.51	0.0526136
$801$	0	0
$802$	−7225.55	−0.318134
$803$	0	0
$804$	0	0
$805$	−31096.7	−1.36151
$806$	−6622.70	−0.289423
$807$	0	0
$808$	1492.94	0.0650016
$809$	−13480.3	−0.585835	−0.292917	−	0.956138i	$-0.594626\pi$
$809$	−13480.3	−0.585835	−0.292917	+	0.956138i	$0.594626\pi$
$810$	0	0
$811$	−21423.1	−0.927579	−0.463790	−	0.885945i	$-0.653511\pi$
$811$	−21423.1	−0.927579	−0.463790	+	0.885945i	$0.653511\pi$
$812$	11951.2	0.516508
$813$	0	0
$814$	0	0
$815$	−32391.1	−1.39216
$816$	0	0
$817$	−2564.30	−0.109809
$818$	−22590.0	−0.965574
$819$	0	0
$820$	−6899.99	−0.293852
$821$	−10664.8	−0.453355	−0.226677	−	0.973970i	$-0.572786\pi$
$821$	−10664.8	−0.453355	−0.226677	+	0.973970i	$0.572786\pi$
$822$	0	0
$823$	−11996.9	−0.508123	−0.254062	−	0.967188i	$-0.581767\pi$
$823$	−11996.9	−0.508123	−0.254062	+	0.967188i	$0.581767\pi$
$824$	−9383.31	−0.396703
$825$	0	0
$826$	−30577.2	−1.28804
$827$	−40880.7	−1.71894	−0.859469	−	0.511188i	$-0.829206\pi$
$827$	−40880.7	−1.71894	−0.859469	+	0.511188i	$0.829206\pi$
$828$	0	0
$829$	−33237.0	−1.39248	−0.696241	−	0.717808i	$-0.745145\pi$
$829$	−33237.0	−1.39248	−0.696241	+	0.717808i	$0.745145\pi$
$830$	−34630.0	−1.44822
$831$	0	0
$832$	733.789	0.0305764
$833$	13586.7	0.565128
$834$	0	0
$835$	17644.9	0.731288
$836$	0	0
$837$	0	0
$838$	7360.89	0.303434
$839$	6368.57	0.262059	0.131029	−	0.991378i	$-0.458172\pi$
$839$	6368.57	0.262059	0.131029	+	0.991378i	$0.458172\pi$
$840$	0	0
$841$	−8170.70	−0.335016
$842$	18512.3	0.757693
$843$	0	0
$844$	8367.91	0.341274
$845$	26306.6	1.07097
$846$	0	0
$847$	0	0
$848$	−8022.90	−0.324891
$849$	0	0
$850$	4873.81	0.196671
$851$	−8888.02	−0.358023
$852$	0	0
$853$	3036.41	0.121881	0.0609406	−	0.998141i	$-0.480590\pi$
$853$	3036.41	0.121881	0.0609406	+	0.998141i	$0.480590\pi$
$854$	17163.5	0.687734
$855$	0	0
$856$	5701.71	0.227664
$857$	10184.8	0.405959	0.202979	−	0.979183i	$-0.434938\pi$
$857$	10184.8	0.405959	0.202979	+	0.979183i	$0.434938\pi$
$858$	0	0
$859$	−34929.8	−1.38742	−0.693708	−	0.720256i	$-0.744024\pi$
$859$	−34929.8	−1.38742	−0.693708	+	0.720256i	$0.744024\pi$
$860$	−18018.3	−0.714440
$861$	0	0
$862$	−7168.08	−0.283232
$863$	12283.3	0.484504	0.242252	−	0.970213i	$-0.422114\pi$
$863$	12283.3	0.484504	0.242252	+	0.970213i	$0.422114\pi$
$864$	0	0
$865$	−5648.52	−0.222029
$866$	−16613.6	−0.651909
$867$	0	0
$868$	−27103.3	−1.05985
$869$	0	0
$870$	0	0
$871$	−3377.44	−0.131389
$872$	9977.72	0.387487
$873$	0	0
$874$	−1509.08	−0.0584042
$875$	26233.5	1.01355
$876$	0	0
$877$	19137.9	0.736877	0.368438	−	0.929652i	$-0.379893\pi$
$877$	19137.9	0.736877	0.368438	+	0.929652i	$0.379893\pi$
$878$	23864.7	0.917307
$879$	0	0
$880$	0	0
$881$	16737.0	0.640049	0.320024	−	0.947409i	$-0.396309\pi$
$881$	16737.0	0.640049	0.320024	+	0.947409i	$0.396309\pi$
$882$	0	0
$883$	−18604.2	−0.709039	−0.354520	−	0.935049i	$-0.615356\pi$
$883$	−18604.2	−0.709039	−0.354520	+	0.935049i	$0.615356\pi$
$884$	3004.05	0.114295
$885$	0	0
$886$	−2157.64	−0.0818142
$887$	−15912.8	−0.602367	−0.301184	−	0.953566i	$-0.597382\pi$
$887$	−15912.8	−0.602367	−0.301184	+	0.953566i	$0.597382\pi$
$888$	0	0
$889$	−36380.4	−1.37251
$890$	−6629.16	−0.249674
$891$	0	0
$892$	14814.1	0.556068
$893$	−975.894	−0.0365700
$894$	0	0
$895$	4796.84	0.179152
$896$	3003.02	0.111969
$897$	0	0
$898$	−25531.9	−0.948786
$899$	−36780.4	−1.36451
$900$	0	0
$901$	−32844.8	−1.21445
$902$	0	0
$903$	0	0
$904$	7856.01	0.289034
$905$	−41949.0	−1.54081
$906$	0	0
$907$	−22555.3	−0.825729	−0.412864	−	0.910793i	$-0.635472\pi$
$907$	−22555.3	−0.825729	−0.412864	+	0.910793i	$0.635472\pi$
$908$	19414.0	0.709555
$909$	0	0
$910$	−6851.72	−0.249596
$911$	−48359.6	−1.75876	−0.879378	−	0.476125i	$-0.842041\pi$
$911$	−48359.6	−1.75876	−0.879378	+	0.476125i	$0.842041\pi$
$912$	0	0
$913$	0	0
$914$	−22453.6	−0.812583
$915$	0	0
$916$	−5347.38	−0.192885
$917$	−17410.8	−0.626997
$918$	0	0
$919$	21014.3	0.754295	0.377148	−	0.926153i	$-0.376905\pi$
$919$	21014.3	0.754295	0.377148	+	0.926153i	$0.376905\pi$
$920$	−10603.7	−0.379991
$921$	0	0
$922$	4321.17	0.154349
$923$	−1518.56	−0.0541537
$924$	0	0
$925$	−3177.26	−0.112938
$926$	22938.9	0.814060
$927$	0	0
$928$	4075.23	0.144155
$929$	−4919.67	−0.173745	−0.0868725	−	0.996219i	$-0.527687\pi$
$929$	−4919.67	−0.173745	−0.0868725	+	0.996219i	$0.527687\pi$
$930$	0	0
$931$	−1503.85	−0.0529395
$932$	23615.6	0.829995
$933$	0	0
$934$	−4036.78	−0.141421
$935$	0	0
$936$	0	0
$937$	−17180.3	−0.598991	−0.299496	−	0.954098i	$-0.596818\pi$
$937$	−17180.3	−0.598991	−0.299496	+	0.954098i	$0.596818\pi$
$938$	−13822.1	−0.481140
$939$	0	0
$940$	−6857.20	−0.237933
$941$	51486.9	1.78366	0.891830	−	0.452370i	$-0.149421\pi$
$941$	51486.9	1.78366	0.891830	+	0.452370i	$0.149421\pi$
$942$	0	0
$943$	14095.9	0.486773
$944$	−10426.5	−0.359486
$945$	0	0
$946$	0	0
$947$	−45107.8	−1.54784	−0.773920	−	0.633283i	$-0.781707\pi$
$947$	−45107.8	−1.54784	−0.773920	+	0.633283i	$0.781707\pi$
$948$	0	0
$949$	5382.90	0.184127
$950$	−539.459	−0.0184235
$951$	0	0
$952$	12294.0	0.418542
$953$	−25385.8	−0.862881	−0.431440	−	0.902141i	$-0.641994\pi$
$953$	−25385.8	−0.862881	−0.431440	+	0.902141i	$0.641994\pi$
$954$	0	0
$955$	29168.6	0.988349
$956$	13279.1	0.449245
$957$	0	0
$958$	14654.0	0.494206
$959$	−12098.4	−0.407380
$960$	0	0
$961$	53621.0	1.79990
$962$	−1958.35	−0.0656339
$963$	0	0
$964$	−21103.8	−0.705091
$965$	29335.9	0.978607
$966$	0	0
$967$	26837.3	0.892482	0.446241	−	0.894913i	$-0.352762\pi$
$967$	26837.3	0.892482	0.446241	+	0.894913i	$0.352762\pi$
$968$	0	0
$969$	0	0
$970$	−36030.7	−1.19265
$971$	5451.84	0.180183	0.0900916	−	0.995933i	$-0.471284\pi$
$971$	5451.84	0.180183	0.0900916	+	0.995933i	$0.471284\pi$
$972$	0	0
$973$	57788.6	1.90403
$974$	−12379.6	−0.407257
$975$	0	0
$976$	5852.59	0.191944
$977$	3608.24	0.118155	0.0590776	−	0.998253i	$-0.481184\pi$
$977$	3608.24	0.118155	0.0590776	+	0.998253i	$0.481184\pi$
$978$	0	0
$979$	0	0
$980$	−10566.9	−0.344437
$981$	0	0
$982$	20241.9	0.657784
$983$	−58045.5	−1.88338	−0.941691	−	0.336478i	$-0.890764\pi$
$983$	−58045.5	−1.88338	−0.941691	+	0.336478i	$0.890764\pi$
$984$	0	0
$985$	−13269.1	−0.429227
$986$	16683.5	0.538856
$987$	0	0
$988$	−332.504	−0.0107068
$989$	36809.4	1.18349
$990$	0	0
$991$	−18977.5	−0.608315	−0.304157	−	0.952622i	$-0.598375\pi$
$991$	−18977.5	−0.608315	−0.304157	+	0.952622i	$0.598375\pi$
$992$	−9241.96	−0.295799
$993$	0	0
$994$	−6214.68	−0.198307
$995$	−44115.1	−1.40557
$996$	0	0
$997$	−24260.2	−0.770641	−0.385321	−	0.922783i	$-0.625909\pi$
$997$	−24260.2	−0.770641	−0.385321	+	0.922783i	$0.625909\pi$
$998$	5125.62	0.162574
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.by.1.2		4
3.2	odd	2		242.4.a.n.1.4		4
11.5	even	5		198.4.f.d.91.2		8
11.9	even	5		198.4.f.d.37.2		8
11.10	odd	2		2178.4.a.bt.1.2		4
12.11	even	2		1936.4.a.bn.1.1		4
33.2	even	10		242.4.c.q.81.2		8
33.5	odd	10		22.4.c.b.3.2	✓	8
33.8	even	10		242.4.c.n.9.1		8
33.14	odd	10		242.4.c.r.9.1		8
33.17	even	10		242.4.c.q.3.2		8
33.20	odd	10		22.4.c.b.15.2	yes	8
33.26	odd	10		242.4.c.r.27.1		8
33.29	even	10		242.4.c.n.27.1		8
33.32	even	2		242.4.a.o.1.4		4
132.71	even	10		176.4.m.b.113.1		8
132.119	even	10		176.4.m.b.81.1		8
132.131	odd	2		1936.4.a.bm.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.3.2	✓	8	33.5	odd	10
22.4.c.b.15.2	yes	8	33.20	odd	10
176.4.m.b.81.1		8	132.119	even	10
176.4.m.b.113.1		8	132.71	even	10
198.4.f.d.37.2		8	11.9	even	5
198.4.f.d.91.2		8	11.5	even	5
242.4.a.n.1.4		4	3.2	odd	2
242.4.a.o.1.4		4	33.32	even	2
242.4.c.n.9.1		8	33.8	even	10
242.4.c.n.27.1		8	33.29	even	10
242.4.c.q.3.2		8	33.17	even	10
242.4.c.q.81.2		8	33.2	even	10
242.4.c.r.9.1		8	33.14	odd	10
242.4.c.r.27.1		8	33.26	odd	10
1936.4.a.bm.1.1		4	132.131	odd	2
1936.4.a.bn.1.1		4	12.11	even	2
2178.4.a.bt.1.2		4	11.10	odd	2
2178.4.a.by.1.2		4	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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -12.7359 q^{5} +23.4611 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -12.7359 q^{5} +23.4611 q^{7} +8.00000 q^{8} -25.4718 q^{10} +11.4654 q^{13} +46.9222 q^{14} +16.0000 q^{16} +65.5022 q^{17} -7.25013 q^{19} -50.9436 q^{20} +104.072 q^{23} +37.2034 q^{25} +22.9309 q^{26} +93.8445 q^{28} +127.351 q^{29} -288.811 q^{31} +32.0000 q^{32} +131.004 q^{34} -298.799 q^{35} -85.4023 q^{37} -14.5003 q^{38} -101.887 q^{40} +135.444 q^{41} +353.691 q^{43} +208.145 q^{46} +134.604 q^{47} +207.424 q^{49} +74.4068 q^{50} +45.8618 q^{52} -501.431 q^{53} +187.689 q^{56} +254.702 q^{58} -651.658 q^{59} +365.787 q^{61} -577.623 q^{62} +64.0000 q^{64} -146.023 q^{65} -294.576 q^{67} +262.009 q^{68} -597.597 q^{70} -132.446 q^{71} +469.489 q^{73} -170.805 q^{74} -29.0005 q^{76} +408.033 q^{79} -203.775 q^{80} +270.887 q^{82} +1359.54 q^{83} -834.230 q^{85} +707.381 q^{86} +260.255 q^{89} +268.992 q^{91} +416.289 q^{92} +269.207 q^{94} +92.3370 q^{95} +1414.53 q^{97} +414.848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} - 25 q^{5} - 3 q^{7} + 32 q^{8}+O(q^{10}) \) 4 * q + 8 * q^2 + 16 * q^4 - 25 * q^5 - 3 * q^7 + 32 * q^8	\( 4 q + 8 q^{2} + 16 q^{4} - 25 q^{5} - 3 q^{7} + 32 q^{8} - 50 q^{10} + 41 q^{13} - 6 q^{14} + 64 q^{16} + 52 q^{17} - 16 q^{19} - 100 q^{20} - 314 q^{23} - 21 q^{25} + 82 q^{26} - 12 q^{28} + 561 q^{29} + 199 q^{31} + 128 q^{32} + 104 q^{34} - 714 q^{35} + 357 q^{37} - 32 q^{38} - 200 q^{40} + 32 q^{41} + 721 q^{43} - 628 q^{46} - 403 q^{47} + 823 q^{49} - 42 q^{50} + 164 q^{52} + 133 q^{53} - 24 q^{56} + 1122 q^{58} - 1016 q^{59} + 919 q^{61} + 398 q^{62} + 256 q^{64} + 69 q^{65} + 289 q^{67} + 208 q^{68} - 1428 q^{70} + 1205 q^{71} + 1234 q^{73} + 714 q^{74} - 64 q^{76} + 603 q^{79} - 400 q^{80} + 64 q^{82} + 1514 q^{83} - 717 q^{85} + 1442 q^{86} + 1101 q^{89} - 2306 q^{91} - 1256 q^{92} - 806 q^{94} - 1766 q^{95} + 2116 q^{97} + 1646 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 - 25 * q^5 - 3 * q^7 + 32 * q^8 - 50 * q^10 + 41 * q^13 - 6 * q^14 + 64 * q^16 + 52 * q^17 - 16 * q^19 - 100 * q^20 - 314 * q^23 - 21 * q^25 + 82 * q^26 - 12 * q^28 + 561 * q^29 + 199 * q^31 + 128 * q^32 + 104 * q^34 - 714 * q^35 + 357 * q^37 - 32 * q^38 - 200 * q^40 + 32 * q^41 + 721 * q^43 - 628 * q^46 - 403 * q^47 + 823 * q^49 - 42 * q^50 + 164 * q^52 + 133 * q^53 - 24 * q^56 + 1122 * q^58 - 1016 * q^59 + 919 * q^61 + 398 * q^62 + 256 * q^64 + 69 * q^65 + 289 * q^67 + 208 * q^68 - 1428 * q^70 + 1205 * q^71 + 1234 * q^73 + 714 * q^74 - 64 * q^76 + 603 * q^79 - 400 * q^80 + 64 * q^82 + 1514 * q^83 - 717 * q^85 + 1442 * q^86 + 1101 * q^89 - 2306 * q^91 - 1256 * q^92 - 806 * q^94 - 1766 * q^95 + 2116 * q^97 + 1646 * q^98

Introduction

L-functions

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