LMFDB - Embedded newform 242.4.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [242,4,Mod(1,242)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(242, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("242.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	242.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:	$14.2784622214$
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:	$\Z[a_1, \ldots, a_{7}]$
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			1.4
Root			$-5.92695$ of defining polynomial
Character	$\chi$	$=$	242.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +8.54499 q^{3} +4.00000 q^{4} +12.7359 q^{5} -17.0900 q^{6} +23.4611 q^{7} -8.00000 q^{8} +46.0168 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +8.54499 q^{3} +4.00000 q^{4} +12.7359 q^{5} -17.0900 q^{6} +23.4611 q^{7} -8.00000 q^{8} +46.0168 q^{9} -25.4718 q^{10} +34.1799 q^{12} +11.4654 q^{13} -46.9222 q^{14} +108.828 q^{15} +16.0000 q^{16} -65.5022 q^{17} -92.0336 q^{18} -7.25013 q^{19} +50.9436 q^{20} +200.475 q^{21} -104.072 q^{23} -68.3599 q^{24} +37.2034 q^{25} -22.9309 q^{26} +162.498 q^{27} +93.8445 q^{28} -127.351 q^{29} -217.656 q^{30} -288.811 q^{31} -32.0000 q^{32} +131.004 q^{34} +298.799 q^{35} +184.067 q^{36} -85.4023 q^{37} +14.5003 q^{38} +97.9721 q^{39} -101.887 q^{40} -135.444 q^{41} -400.950 q^{42} +353.691 q^{43} +586.066 q^{45} +208.145 q^{46} -134.604 q^{47} +136.720 q^{48} +207.424 q^{49} -74.4068 q^{50} -559.715 q^{51} +45.8618 q^{52} +501.431 q^{53} -324.997 q^{54} -187.689 q^{56} -61.9522 q^{57} +254.702 q^{58} +651.658 q^{59} +435.313 q^{60} +365.787 q^{61} +577.623 q^{62} +1079.61 q^{63} +64.0000 q^{64} +146.023 q^{65} -294.576 q^{67} -262.009 q^{68} -889.297 q^{69} -597.597 q^{70} +132.446 q^{71} -368.134 q^{72} +469.489 q^{73} +170.805 q^{74} +317.903 q^{75} -29.0005 q^{76} -195.944 q^{78} +408.033 q^{79} +203.775 q^{80} +146.093 q^{81} +270.887 q^{82} -1359.54 q^{83} +801.900 q^{84} -834.230 q^{85} -707.381 q^{86} -1088.21 q^{87} -260.255 q^{89} -1172.13 q^{90} +268.992 q^{91} -416.289 q^{92} -2467.89 q^{93} +269.207 q^{94} -92.3370 q^{95} -273.440 q^{96} +1414.53 q^{97} -414.848 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} - 8 q^{6} - 3 q^{7} - 32 q^{8} + 102 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 - 8 * q^6 - 3 * q^7 - 32 * q^8 + 102 * q^9	\( 4 q - 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} - 8 q^{6} - 3 q^{7} - 32 q^{8} + 102 q^{9} - 50 q^{10} + 16 q^{12} + 41 q^{13} + 6 q^{14} + 68 q^{15} + 64 q^{16} - 52 q^{17} - 204 q^{18} - 16 q^{19} + 100 q^{20} - 25 q^{21} + 314 q^{23} - 32 q^{24} - 21 q^{25} - 82 q^{26} + 286 q^{27} - 12 q^{28} - 561 q^{29} - 136 q^{30} + 199 q^{31} - 128 q^{32} + 104 q^{34} + 714 q^{35} + 408 q^{36} + 357 q^{37} + 32 q^{38} + 1038 q^{39} - 200 q^{40} - 32 q^{41} + 50 q^{42} + 721 q^{43} + 1326 q^{45} - 628 q^{46} + 403 q^{47} + 64 q^{48} + 823 q^{49} + 42 q^{50} + 174 q^{51} + 164 q^{52} - 133 q^{53} - 572 q^{54} + 24 q^{56} + 1031 q^{57} + 1122 q^{58} + 1016 q^{59} + 272 q^{60} + 919 q^{61} - 398 q^{62} + 1367 q^{63} + 256 q^{64} - 69 q^{65} + 289 q^{67} - 208 q^{68} - 1620 q^{69} - 1428 q^{70} - 1205 q^{71} - 816 q^{72} + 1234 q^{73} - 714 q^{74} - 911 q^{75} - 64 q^{76} - 2076 q^{78} + 603 q^{79} + 400 q^{80} - 1400 q^{81} + 64 q^{82} - 1514 q^{83} - 100 q^{84} - 717 q^{85} - 1442 q^{86} + 1061 q^{87} - 1101 q^{89} - 2652 q^{90} - 2306 q^{91} + 1256 q^{92} - 2298 q^{93} - 806 q^{94} + 1766 q^{95} - 128 q^{96} + 2116 q^{97} - 1646 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 - 8 * q^6 - 3 * q^7 - 32 * q^8 + 102 * q^9 - 50 * q^10 + 16 * q^12 + 41 * q^13 + 6 * q^14 + 68 * q^15 + 64 * q^16 - 52 * q^17 - 204 * q^18 - 16 * q^19 + 100 * q^20 - 25 * q^21 + 314 * q^23 - 32 * q^24 - 21 * q^25 - 82 * q^26 + 286 * q^27 - 12 * q^28 - 561 * q^29 - 136 * q^30 + 199 * q^31 - 128 * q^32 + 104 * q^34 + 714 * q^35 + 408 * q^36 + 357 * q^37 + 32 * q^38 + 1038 * q^39 - 200 * q^40 - 32 * q^41 + 50 * q^42 + 721 * q^43 + 1326 * q^45 - 628 * q^46 + 403 * q^47 + 64 * q^48 + 823 * q^49 + 42 * q^50 + 174 * q^51 + 164 * q^52 - 133 * q^53 - 572 * q^54 + 24 * q^56 + 1031 * q^57 + 1122 * q^58 + 1016 * q^59 + 272 * q^60 + 919 * q^61 - 398 * q^62 + 1367 * q^63 + 256 * q^64 - 69 * q^65 + 289 * q^67 - 208 * q^68 - 1620 * q^69 - 1428 * q^70 - 1205 * q^71 - 816 * q^72 + 1234 * q^73 - 714 * q^74 - 911 * q^75 - 64 * q^76 - 2076 * q^78 + 603 * q^79 + 400 * q^80 - 1400 * q^81 + 64 * q^82 - 1514 * q^83 - 100 * q^84 - 717 * q^85 - 1442 * q^86 + 1061 * q^87 - 1101 * q^89 - 2652 * q^90 - 2306 * q^91 + 1256 * q^92 - 2298 * q^93 - 806 * q^94 + 1766 * q^95 - 128 * q^96 + 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
$3$	8.54499	1.64448	0.822242	−	0.569138i	$-0.192723\pi$
$3$	8.54499	1.64448	0.822242	+	0.569138i	$0.192723\pi$
$4$	4.00000	0.500000
$5$	12.7359	1.13913	0.569567	−	0.821945i	$-0.307111\pi$
$5$	12.7359	1.13913	0.569567	+	0.821945i	$0.307111\pi$
$6$	−17.0900	−1.16283
$7$	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
$9$	46.0168	1.70433
$10$	−25.4718	−0.805490
$11$	0	0
$11$	0	0
$12$	34.1799	0.822242
$13$	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
$15$	108.828	1.87329
$16$	16.0000	0.250000
$17$	−65.5022	−0.934507	−0.467253	−	0.884124i	$-0.654756\pi$
$17$	−65.5022	−0.934507	−0.467253	+	0.884124i	$0.654756\pi$
$18$	−92.0336	−1.20514
$19$	−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
$21$	200.475	2.08320
$22$	0	0
$23$	−104.072	−0.943504	−0.471752	−	0.881731i	$-0.656378\pi$
$23$	−104.072	−0.943504	−0.471752	+	0.881731i	$0.656378\pi$
$24$	−68.3599	−0.581413
$25$	37.2034	0.297627
$26$	−22.9309	−0.172966
$27$	162.498	1.15825
$28$	93.8445	0.633391
$29$	−127.351	−0.815466	−0.407733	−	0.913101i	$-0.633681\pi$
$29$	−127.351	−0.815466	−0.407733	+	0.913101i	$0.633681\pi$
$30$	−217.656	−1.32461
$31$	−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
$36$	184.067	0.852163
$37$	−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
$39$	97.9721	0.402259
$40$	−101.887	−0.402745
$41$	−135.444	−0.515921	−0.257960	−	0.966156i	$-0.583050\pi$
$41$	−135.444	−0.515921	−0.257960	+	0.966156i	$0.583050\pi$
$42$	−400.950	−1.47305
$43$	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
$45$	586.066	1.94146
$46$	208.145	0.667158
$47$	−134.604	−0.417744	−0.208872	−	0.977943i	$-0.566979\pi$
$47$	−134.604	−0.417744	−0.208872	+	0.977943i	$0.566979\pi$
$48$	136.720	0.411121
$49$	207.424	0.604735
$50$	−74.4068	−0.210454
$51$	−559.715	−1.53678
$52$	45.8618	0.122305
$53$	501.431	1.29956	0.649782	−	0.760121i	$-0.274860\pi$
$53$	501.431	1.29956	0.649782	+	0.760121i	$0.274860\pi$
$54$	−324.997	−0.819008
$55$	0	0
$56$	−187.689	−0.447875
$57$	−61.9522	−0.143961
$58$	254.702	0.576621
$59$	651.658	1.43794	0.718971	−	0.695040i	$-0.244613\pi$
$59$	651.658	1.43794	0.718971	+	0.695040i	$0.244613\pi$
$60$	435.313	0.936644
$61$	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
$63$	1079.61	2.15901
$64$	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
$69$	−889.297	−1.55158
$70$	−597.597	−1.02038
$71$	132.446	0.221387	0.110694	−	0.993855i	$-0.464693\pi$
$71$	132.446	0.221387	0.110694	+	0.993855i	$0.464693\pi$
$72$	−368.134	−0.602570
$73$	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
$75$	317.903	0.489443
$76$	−29.0005	−0.0437709
$77$	0	0
$78$	−195.944	−0.284440
$79$	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
$81$	146.093	0.200402
$82$	270.887	0.364811
$83$	−1359.54	−1.79794	−0.898971	−	0.438009i	$-0.855684\pi$
$83$	−1359.54	−1.79794	−0.898971	+	0.438009i	$0.855684\pi$
$84$	801.900	1.04160
$85$	−834.230	−1.06453
$86$	−707.381	−0.886964
$87$	−1088.21	−1.34102
$88$	0	0
$89$	−260.255	−0.309966	−0.154983	−	0.987917i	$-0.549532\pi$
$89$	−260.255	−0.309966	−0.154983	+	0.987917i	$0.549532\pi$
$90$	−1172.13	−1.37282
$91$	268.992	0.309869
$92$	−416.289	−0.471752
$93$	−2467.89	−2.75170
$94$	269.207	0.295390
$95$	−92.3370	−0.0997218
$96$	−273.440	−0.290706
$97$	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	$-0.529302\pi$
$101$	−186.617	−0.183852	−0.0919261	+	0.995766i	$0.529302\pi$
$102$	1119.43	1.08667
$103$	−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
$105$	2553.23	2.37305
$106$	−1002.86	−0.918930
$107$	−712.714	−0.643931	−0.321966	−	0.946751i	$-0.604344\pi$
$107$	−712.714	−0.643931	−0.321966	+	0.946751i	$0.604344\pi$
$108$	649.993	0.579126
$109$	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
$111$	−729.762	−0.624017
$112$	375.378	0.316695
$113$	−982.002	−0.817513	−0.408756	−	0.912644i	$-0.634037\pi$
$113$	−982.002	−0.817513	−0.408756	+	0.912644i	$0.634037\pi$
$114$	123.904	0.101796
$115$	−1325.46	−1.07478
$116$	−509.404	−0.407733
$117$	527.603	0.416897
$118$	−1303.32	−1.01678
$119$	−1536.75	−1.18382
$120$	−870.625	−0.662307
$121$	0	0
$122$	−731.574	−0.542899
$123$	−1157.36	−0.848423
$124$	−1155.25	−0.836646
$125$	−1118.17	−0.800097
$126$	−2159.21	−1.52665
$127$	−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
$129$	3022.28	2.06277
$130$	−292.046	−0.197032
$131$	742.114	0.494953	0.247476	−	0.968894i	$-0.420399\pi$
$131$	742.114	0.494953	0.247476	+	0.968894i	$0.420399\pi$
$132$	0	0
$133$	−170.096	−0.110896
$134$	589.151	0.379813
$135$	2069.56	1.31941
$136$	524.017	0.330398
$137$	515.678	0.321586	0.160793	−	0.986988i	$-0.448595\pi$
$137$	515.678	0.321586	0.160793	+	0.986988i	$0.448595\pi$
$138$	1778.59	1.09713
$139$	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
$141$	−1150.19	−0.686973
$142$	−264.893	−0.156544
$143$	0	0
$144$	736.269	0.426082
$145$	−1621.93	−0.928925
$146$	−938.978	−0.532263
$147$	1772.44	0.994476
$148$	−341.609	−0.189731
$149$	−434.757	−0.239038	−0.119519	−	0.992832i	$-0.538135\pi$
$149$	−434.757	−0.239038	−0.119519	+	0.992832i	$0.538135\pi$
$150$	−635.805	−0.346088
$151$	−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
$153$	−3014.20	−1.59270
$154$	0	0
$155$	−3678.27	−1.90610
$156$	391.888	0.201129
$157$	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
$159$	4284.72	2.13711
$160$	−407.549	−0.201372
$161$	−2441.65	−1.19521
$162$	−292.185	−0.141705
$163$	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	$-0.395986\pi$
$167$	1385.44	0.641968	0.320984	+	0.947085i	$0.395986\pi$
$168$	−1603.80	−0.736523
$169$	−2065.54	−0.940165
$170$	1668.46	0.752735
$171$	−333.628	−0.149200
$172$	1414.76	0.627178
$173$	−443.511	−0.194911	−0.0974553	−	0.995240i	$-0.531070\pi$
$173$	−443.511	−0.194911	−0.0974553	+	0.995240i	$0.531070\pi$
$174$	2176.43	0.948244
$175$	872.833	0.377029
$176$	0	0
$177$	5568.41	2.36467
$178$	520.510	0.219179
$179$	376.639	0.157270	0.0786349	−	0.996903i	$-0.474944\pi$
$179$	376.639	0.157270	0.0786349	+	0.996903i	$0.474944\pi$
$180$	2344.26	0.970728
$181$	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
$183$	3125.65	1.26259
$184$	832.579	0.333579
$185$	−1087.68	−0.432257
$186$	4935.78	1.94575
$187$	0	0
$188$	−538.415	−0.208872
$189$	3812.39	1.46725
$190$	184.674	0.0705140
$191$	2290.26	0.867631	0.433816	−	0.901002i	$-0.357167\pi$
$191$	2290.26	0.867631	0.433816	+	0.901002i	$0.357167\pi$
$192$	546.879	0.205560
$193$	−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
$195$	1247.76	0.458227
$196$	829.696	0.302367
$197$	−1041.86	−0.376801	−0.188400	−	0.982092i	$-0.560330\pi$
$197$	−1041.86	−0.376801	−0.188400	+	0.982092i	$0.560330\pi$
$198$	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
$201$	−2517.14	−0.883312
$202$	373.234	0.130003
$203$	−2987.80	−1.03302
$204$	−2238.86	−0.768390
$205$	−1725.00	−0.587703
$206$	2345.83	0.793406
$207$	−4789.08	−1.60804
$208$	183.447	0.0611527
$209$	0	0
$210$	−5106.46	−1.67800
$211$	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
$213$	1131.75	0.364067
$214$	1425.43	0.455328
$215$	4504.57	1.42888
$216$	−1299.99	−0.409504
$217$	−6775.84	−2.11969
$218$	−2494.43	−0.774973
$219$	4011.78	1.23786
$220$	0	0
$221$	−751.012	−0.228591
$222$	1459.52	0.441247
$223$	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
$225$	1711.98	0.507254
$226$	1964.00	0.578069
$227$	−4853.50	−1.41911	−0.709555	−	0.704650i	$-0.751104\pi$
$227$	−4853.50	−1.41911	−0.709555	+	0.704650i	$0.751104\pi$
$228$	−247.809	−0.0719805
$229$	−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	$-0.811657\pi$
$233$	−5903.91	−1.65999	−0.829995	+	0.557770i	$0.811657\pi$
$234$	−1055.21	−0.294791
$235$	−1714.30	−0.475866
$236$	2606.63	0.718971
$237$	3486.63	0.955617
$238$	3073.51	0.837084
$239$	−3319.79	−0.898490	−0.449245	−	0.893409i	$-0.648307\pi$
$239$	−3319.79	−0.898490	−0.449245	+	0.893409i	$0.648307\pi$
$240$	1741.25	0.468322
$241$	−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
$243$	−3139.10	−0.828696
$244$	1463.15	0.383887
$245$	2641.73	0.688874
$246$	2314.73	0.599926
$247$	−83.1259	−0.0214137
$248$	2310.49	0.591598
$249$	−11617.3	−2.95669
$250$	2236.34	0.565754
$251$	−1961.85	−0.493350	−0.246675	−	0.969098i	$-0.579338\pi$
$251$	−1961.85	−0.493350	−0.246675	+	0.969098i	$0.579338\pi$
$252$	4318.42	1.07950
$253$	0	0
$254$	3101.34	0.766123
$255$	−7128.48	−1.75060
$256$	256.000	0.0625000
$257$	5941.97	1.44222	0.721108	−	0.692822i	$-0.243633\pi$
$257$	5941.97	1.44222	0.721108	+	0.692822i	$0.243633\pi$
$258$	−6044.56	−1.45860
$259$	−2003.63	−0.480694
$260$	584.092	0.139322
$261$	−5860.29	−1.38982
$262$	−1484.23	−0.349985
$263$	−1704.11	−0.399544	−0.199772	−	0.979842i	$-0.564020\pi$
$263$	−1704.11	−0.399544	−0.199772	+	0.979842i	$0.564020\pi$
$264$	0	0
$265$	6386.18	1.48038
$266$	340.192	0.0784155
$267$	−2223.87	−0.509734
$268$	−1178.30	−0.268568
$269$	−5845.33	−1.32489	−0.662446	−	0.749109i	$-0.730482\pi$
$269$	−5845.33	−1.32489	−0.662446	+	0.749109i	$0.730482\pi$
$270$	−4139.13	−0.932961
$271$	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
$273$	2298.53	0.509574
$274$	−1031.36	−0.227396
$275$	0	0
$276$	−3557.19	−0.775788
$277$	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
$279$	−13290.2	−2.85183
$280$	−2390.39	−0.510190
$281$	−8151.77	−1.73058	−0.865291	−	0.501269i	$-0.832867\pi$
$281$	−8151.77	−1.73058	−0.865291	+	0.501269i	$0.832867\pi$
$282$	2300.37	0.485763
$283$	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
$285$	−789.018	−0.163991
$286$	0	0
$287$	−3177.66	−0.653559
$288$	−1472.54	−0.301285
$289$	−622.465	−0.126698
$290$	3243.86	0.656849
$291$	12087.1	2.43492
$292$	1877.96	0.376367
$293$	7616.57	1.51865	0.759326	−	0.650711i	$-0.225529\pi$
$293$	7616.57	1.51865	0.759326	+	0.650711i	$0.225529\pi$
$294$	−3544.87	−0.703201
$295$	8299.45	1.63801
$296$	683.219	0.134160
$297$	0	0
$298$	869.515	0.169026
$299$	−1193.24	−0.230791
$300$	1271.61	0.244722
$301$	8297.98	1.58899
$302$	1561.98	0.297623
$303$	−1594.64	−0.302342
$304$	−116.002	−0.0218854
$305$	4658.63	0.874598
$306$	6028.40	1.12621
$307$	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
$309$	−10022.5	−1.84519
$310$	7356.55	1.34782
$311$	1456.13	0.265497	0.132749	−	0.991150i	$-0.457620\pi$
$311$	1456.13	0.265497	0.132749	+	0.991150i	$0.457620\pi$
$312$	−783.777	−0.142220
$313$	−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
$315$	13749.8	2.45940
$316$	1632.13	0.290552
$317$	4992.31	0.884530	0.442265	−	0.896884i	$-0.354175\pi$
$317$	4992.31	0.884530	0.442265	+	0.896884i	$0.354175\pi$
$318$	−8569.44	−1.51117
$319$	0	0
$320$	815.098	0.142392
$321$	−6090.13	−1.05893
$322$	4883.31	0.845143
$323$	474.899	0.0818083
$324$	584.371	0.100201
$325$	426.554	0.0728029
$326$	−5086.57	−0.864169
$327$	10657.4	1.80232
$328$	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
$333$	−3929.94	−0.646725
$334$	−2770.88	−0.453940
$335$	−3751.69	−0.611870
$336$	3207.60	0.520800
$337$	−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
$339$	−8391.19	−1.34439
$340$	−3336.92	−0.532264
$341$	0	0
$342$	667.255	0.105500
$343$	−3180.76	−0.500715
$344$	−2829.53	−0.443482
$345$	−11326.0	−1.76745
$346$	887.022	0.137823
$347$	10526.3	1.62847	0.814236	−	0.580534i	$-0.197156\pi$
$347$	10526.3	1.62847	0.814236	+	0.580534i	$0.197156\pi$
$348$	−4352.85	−0.670510
$349$	−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
$351$	1863.12	0.283321
$352$	0	0
$353$	−7438.40	−1.12155	−0.560773	−	0.827969i	$-0.689496\pi$
$353$	−7438.40	−1.12155	−0.560773	+	0.827969i	$0.689496\pi$
$354$	−11136.8	−1.67208
$355$	1686.82	0.252190
$356$	−1041.02	−0.154983
$357$	−13131.5	−1.94676
$358$	−753.278	−0.111207
$359$	10151.3	1.49238	0.746191	−	0.665731i	$-0.231880\pi$
$359$	10151.3	1.49238	0.746191	+	0.665731i	$0.231880\pi$
$360$	−4688.53	−0.686409
$361$	−6806.44	−0.992336
$362$	−6587.52	−0.956443
$363$	0	0
$364$	1075.97	0.154934
$365$	5979.37	0.857464
$366$	−6251.29	−0.892788
$367$	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
$369$	−6232.69	−0.879297
$370$	2175.35	0.305652
$371$	11764.1	1.64626
$372$	−9871.56	−1.37585
$373$	−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
$375$	−9554.74	−1.31575
$376$	1076.83	0.147695
$377$	−1460.14	−0.199472
$378$	−7624.79	−1.03750
$379$	−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
$381$	−13250.4	−1.78173
$382$	−4580.53	−0.613508
$383$	−670.568	−0.0894632	−0.0447316	−	0.998999i	$-0.514243\pi$
$383$	−670.568	−0.0894632	−0.0447316	+	0.998999i	$0.514243\pi$
$384$	−1093.76	−0.145353
$385$	0	0
$386$	4606.80	0.607461
$387$	16275.7	2.13783
$388$	5658.12	0.740329
$389$	5235.90	0.682444	0.341222	−	0.939983i	$-0.389159\pi$
$389$	5235.90	0.682444	0.341222	+	0.939983i	$0.389159\pi$
$390$	−2495.53	−0.324015
$391$	6816.97	0.881711
$392$	−1659.39	−0.213806
$393$	6341.36	0.813942
$394$	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
$399$	−1453.47	−0.182367
$400$	595.254	0.0744068
$401$	3612.78	0.449909	0.224954	−	0.974369i	$-0.427777\pi$
$401$	3612.78	0.449909	0.224954	+	0.974369i	$0.427777\pi$
$402$	5034.29	0.624596
$403$	−3311.35	−0.409305
$404$	−746.468	−0.0919261
$405$	1860.62	0.228284
$406$	5975.60	0.730453
$407$	0	0
$408$	4477.72	0.543334
$409$	−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
$411$	4406.46	0.528844
$412$	−4691.66	−0.561023
$413$	15288.6	1.82156
$414$	9578.16	1.13705
$415$	−17315.0	−2.04810
$416$	−366.894	−0.0432415
$417$	21047.7	2.47173
$418$	0	0
$419$	−3680.45	−0.429121	−0.214560	−	0.976711i	$-0.568832\pi$
$419$	−3680.45	−0.429121	−0.214560	+	0.976711i	$0.568832\pi$
$420$	10212.9	1.18652
$421$	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
$423$	−6194.03	−0.711972
$424$	−4011.45	−0.459465
$425$	−2436.90	−0.278135
$426$	−2263.50	−0.257435
$427$	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	$-0.435816\pi$
$431$	3584.04	0.400550	0.200275	+	0.979740i	$0.435816\pi$
$432$	2599.97	0.289563
$433$	−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
$435$	−13859.4	−1.52760
$436$	4988.86	0.547989
$437$	754.538	0.0825960
$438$	−8023.55	−0.875297
$439$	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
$441$	9544.99	1.03067
$442$	1502.02	0.161638
$443$	1078.82	0.115703	0.0578514	−	0.998325i	$-0.481575\pi$
$443$	1078.82	0.115703	0.0578514	+	0.998325i	$0.481575\pi$
$444$	−2919.05	−0.312009
$445$	−3314.58	−0.353093
$446$	−7407.05	−0.786399
$447$	−3715.00	−0.393095
$448$	1501.51	0.158348
$449$	12765.9	1.34179	0.670893	−	0.741554i	$-0.265911\pi$
$449$	12765.9	1.34179	0.670893	+	0.741554i	$0.265911\pi$
$450$	−3423.96	−0.358683
$451$	0	0
$452$	−3928.01	−0.408756
$453$	−6673.57	−0.692167
$454$	9707.00	1.00346
$455$	3425.86	0.352982
$456$	495.618	0.0508979
$457$	−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
$459$	−10644.0	−1.08239
$460$	−5301.83	−0.537389
$461$	−2160.58	−0.218283	−0.109141	−	0.994026i	$-0.534810\pi$
$461$	−2160.58	−0.218283	−0.109141	+	0.994026i	$0.534810\pi$
$462$	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
$465$	−31430.8	−3.13456
$466$	11807.8	1.17379
$467$	2018.39	0.200000	0.100000	−	0.994987i	$-0.468116\pi$
$467$	2018.39	0.200000	0.100000	+	0.994987i	$0.468116\pi$
$468$	2110.41	0.208448
$469$	−6911.07	−0.680434
$470$	3428.60	0.336488
$471$	4463.75	0.436685
$472$	−5213.26	−0.508389
$473$	0	0
$474$	−6973.27	−0.675723
$475$	−269.729	−0.0260548
$476$	−6147.02	−0.591908
$477$	23074.3	2.21488
$478$	6639.57	0.635328
$479$	−7327.00	−0.698913	−0.349457	−	0.936953i	$-0.613634\pi$
$479$	−7327.00	−0.698913	−0.349457	+	0.936953i	$0.613634\pi$
$480$	−3482.50	−0.331154
$481$	−979.176	−0.0928203
$482$	10551.9	0.997150
$483$	−20863.9	−1.96551
$484$	0	0
$485$	18015.3	1.68667
$486$	6278.19	0.585976
$487$	−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
$489$	21732.3	2.00976
$490$	−5283.47	−0.487107
$491$	−10120.9	−0.930246	−0.465123	−	0.885246i	$-0.653990\pi$
$491$	−10120.9	−0.930246	−0.465123	+	0.885246i	$0.653990\pi$
$492$	−4629.46	−0.424212
$493$	8341.77	0.762058
$494$	166.252	0.0151418
$495$	0	0
$496$	−4620.98	−0.418323
$497$	3107.34	0.280449
$498$	23234.5	2.09069
$499$	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
$501$	11838.6	1.05571
$502$	3923.70	0.348851
$503$	8601.51	0.762470	0.381235	−	0.924478i	$-0.375499\pi$
$503$	8601.51	0.762470	0.381235	+	0.924478i	$0.375499\pi$
$504$	−8636.85	−0.763325
$505$	−2376.74	−0.209432
$506$	0	0
$507$	−17650.0	−1.54609
$508$	−6202.67	−0.541730
$509$	−13945.1	−1.21435	−0.607176	−	0.794567i	$-0.707698\pi$
$509$	−13945.1	−1.21435	−0.607176	+	0.794567i	$0.707698\pi$
$510$	14257.0	1.23786
$511$	11014.7	0.953548
$512$	−512.000	−0.0441942
$513$	−1178.13	−0.101395
$514$	−11883.9	−1.01980
$515$	−14938.1	−1.27816
$516$	12089.1	1.03138
$517$	0	0
$518$	4007.27	0.339902
$519$	−3789.80	−0.320527
$520$	−1168.18	−0.0985158
$521$	−12092.8	−1.01688	−0.508442	−	0.861096i	$-0.669778\pi$
$521$	−12092.8	−1.01688	−0.508442	+	0.861096i	$0.669778\pi$
$522$	11720.6	0.982751
$523$	−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
$525$	7458.35	0.620017
$526$	3408.22	0.282520
$527$	18917.8	1.56370
$528$	0	0
$529$	−1335.94	−0.109800
$530$	−12772.4	−1.04678
$531$	29987.2	2.45072
$532$	−680.384	−0.0554481
$533$	−1552.92	−0.126200
$534$	4447.75	0.360436
$535$	−9077.06	−0.733524
$536$	2356.60	0.189906
$537$	3218.37	0.258628
$538$	11690.7	0.936840
$539$	0	0
$540$	8278.26	0.659703
$541$	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
$543$	28145.1	2.22435
$544$	2096.07	0.165199
$545$	15884.4	1.24847
$546$	−4597.07	−0.360323
$547$	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
$549$	16832.4	1.30854
$550$	0	0
$551$	923.311	0.0713873
$552$	7114.38	0.548565
$553$	9572.90	0.736132
$554$	−17764.8	−1.36237
$555$	−9294.18	−0.710840
$556$	9852.66	0.751522
$557$	20310.8	1.54506	0.772528	−	0.634981i	$-0.218992\pi$
$557$	20310.8	1.54506	0.772528	+	0.634981i	$0.218992\pi$
$558$	26580.3	2.01655
$559$	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	$-0.433232\pi$
$563$	5563.15	0.416445	0.208223	+	0.978081i	$0.433232\pi$
$564$	−4600.75	−0.343486
$565$	−12506.7	−0.931257
$566$	−11727.9	−0.870956
$567$	3427.50	0.253865
$568$	−1059.57	−0.0782721
$569$	−22785.5	−1.67876	−0.839381	−	0.543543i	$-0.817083\pi$
$569$	−22785.5	−1.67876	−0.839381	+	0.543543i	$0.817083\pi$
$570$	1578.04	0.115959
$571$	−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
$573$	19570.3	1.42681
$574$	6355.32	0.462136
$575$	−3871.85	−0.280812
$576$	2945.08	0.213041
$577$	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
$579$	−19682.5	−1.41274
$580$	−6487.73	−0.464462
$581$	−31896.4	−2.27760
$582$	−24174.3	−1.72175
$583$	0	0
$584$	−3755.91	−0.266131
$585$	6719.51	0.474902
$586$	−15233.1	−1.07385
$587$	6114.94	0.429967	0.214984	−	0.976618i	$-0.431030\pi$
$587$	6114.94	0.429967	0.214984	+	0.976618i	$0.431030\pi$
$588$	7089.74	0.497238
$589$	2093.92	0.146483
$590$	−16598.9	−1.15825
$591$	−8902.72	−0.619643
$592$	−1366.44	−0.0948653
$593$	7188.32	0.497789	0.248894	−	0.968531i	$-0.419933\pi$
$593$	7188.32	0.497789	0.248894	+	0.968531i	$0.419933\pi$
$594$	0	0
$595$	−19572.0	−1.34852
$596$	−1739.03	−0.119519
$597$	29598.4	2.02912
$598$	2386.47	0.163194
$599$	−1012.18	−0.0690429	−0.0345215	−	0.999404i	$-0.510991\pi$
$599$	−1012.18	−0.0690429	−0.0345215	+	0.999404i	$0.510991\pi$
$600$	−2543.22	−0.173044
$601$	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
$603$	−13555.4	−0.915455
$604$	−3123.97	−0.210451
$605$	0	0
$606$	3189.28	0.213788
$607$	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
$609$	−25530.7	−1.69878
$610$	−9317.26	−0.618434
$611$	−1543.29	−0.102185
$612$	−12056.8	−0.796352
$613$	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
$615$	−14740.1	−0.966468
$616$	0	0
$617$	−3323.39	−0.216847	−0.108423	−	0.994105i	$-0.534580\pi$
$617$	−3323.39	−0.216847	−0.108423	+	0.994105i	$0.534580\pi$
$618$	20045.1	1.30474
$619$	−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
$621$	−16911.6	−1.09282
$622$	−2912.26	−0.187735
$623$	−6105.87	−0.392659
$624$	1567.55	0.100565
$625$	−18891.3	−1.20905
$626$	6923.35	0.442033
$627$	0	0
$628$	2089.53	0.132773
$629$	5594.04	0.354609
$630$	−27499.5	−1.73906
$631$	−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
$633$	17875.9	1.12244
$634$	−9984.62	−0.625457
$635$	−19749.2	−1.23421
$636$	17138.9	1.06856
$637$	2378.21	0.147925
$638$	0	0
$639$	6094.75	0.377316
$640$	−1630.20	−0.100686
$641$	27178.0	1.67467	0.837336	−	0.546689i	$-0.184112\pi$
$641$	27178.0	1.67467	0.837336	+	0.546689i	$0.184112\pi$
$642$	12180.3	0.748780
$643$	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
$645$	38491.5	2.34977
$646$	−949.798	−0.0578472
$647$	2287.67	0.139007	0.0695034	−	0.997582i	$-0.477859\pi$
$647$	2287.67	0.139007	0.0695034	+	0.997582i	$0.477859\pi$
$648$	−1168.74	−0.0708526
$649$	0	0
$650$	−853.107	−0.0514794
$651$	−57899.4	−3.48580
$652$	10173.1	0.611060
$653$	−1146.92	−0.0687329	−0.0343665	−	0.999409i	$-0.510941\pi$
$653$	−1146.92	−0.0687329	−0.0343665	+	0.999409i	$0.510941\pi$
$654$	−21314.9	−1.27443
$655$	9451.50	0.563818
$656$	−2167.10	−0.128980
$657$	21604.4	1.28290
$658$	6315.90	0.374194
$659$	−377.923	−0.0223396	−0.0111698	−	0.999938i	$-0.503556\pi$
$659$	−377.923	−0.0223396	−0.0111698	+	0.999938i	$0.503556\pi$
$660$	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
$663$	−6417.38	−0.375913
$664$	10876.3	0.635668
$665$	−2166.33	−0.126326
$666$	7859.89	0.457304
$667$	13253.7	0.769395
$668$	5541.77	0.320984
$669$	31646.6	1.82889
$670$	7503.38	0.432658
$671$	0	0
$672$	−6415.20	−0.368261
$673$	−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
$675$	6045.49	0.344728
$676$	−8262.17	−0.470083
$677$	469.109	0.0266312	0.0133156	−	0.999911i	$-0.495761\pi$
$677$	469.109	0.0266312	0.0133156	+	0.999911i	$0.495761\pi$
$678$	16782.4	0.950625
$679$	33186.5	1.87567
$680$	6673.84	0.376368
$681$	−41473.1	−2.33370
$682$	0	0
$683$	−15892.3	−0.890342	−0.445171	−	0.895446i	$-0.646857\pi$
$683$	−15892.3	−0.890342	−0.445171	+	0.895446i	$0.646857\pi$
$684$	−1334.51	−0.0745998
$685$	6567.63	0.366330
$686$	6361.53	0.354059
$687$	−11423.3	−0.634392
$688$	5659.05	0.313589
$689$	5749.13	0.317887
$690$	22652.0	1.24978
$691$	−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
$696$	8705.71	0.474122
$697$	8871.85	0.482131
$698$	10600.7	0.574847
$699$	−50448.8	−2.72983
$700$	3491.33	0.188514
$701$	1872.54	0.100891	0.0504456	−	0.998727i	$-0.483936\pi$
$701$	1872.54	0.100891	0.0504456	+	0.998727i	$0.483936\pi$
$702$	−3726.23	−0.200338
$703$	619.178	0.0332187
$704$	0	0
$705$	−14648.7	−0.782554
$706$	14876.8	0.793053
$707$	−4378.24	−0.232901
$708$	22273.6	1.18234
$709$	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
$711$	18776.4	0.990392
$712$	2082.04	0.109589
$713$	30057.3	1.57876
$714$	26263.1	1.37657
$715$	0	0
$716$	1506.56	0.0786349
$717$	−28367.5	−1.47755
$718$	−20302.6	−1.05527
$719$	−17274.8	−0.896023	−0.448011	−	0.894028i	$-0.647868\pi$
$719$	−17274.8	−0.896023	−0.448011	+	0.894028i	$0.647868\pi$
$720$	9377.05	0.485364
$721$	−27517.9	−1.42139
$722$	13612.9	0.701688
$723$	−45083.0	−2.31902
$724$	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
$729$	−30768.0	−1.56318
$730$	−11958.7	−0.606319
$731$	−23167.5	−1.17220
$732$	12502.6	0.631296
$733$	−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
$735$	22573.6	1.13284
$736$	3330.32	0.166790
$737$	0	0
$738$	12465.4	0.621757
$739$	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
$741$	−710.310	−0.0352144
$742$	−23528.3	−1.16408
$743$	23042.7	1.13776	0.568880	−	0.822421i	$-0.307377\pi$
$743$	23042.7	1.13776	0.568880	+	0.822421i	$0.307377\pi$
$744$	19743.1	0.972873
$745$	−5537.03	−0.272297
$746$	25477.0	1.25037
$747$	−62561.8	−3.06428
$748$	0	0
$749$	−16721.1	−0.815720
$750$	19109.5	0.930373
$751$	−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
$753$	−16764.0	−0.811306
$754$	2920.27	0.141048
$755$	−9946.65	−0.479464
$756$	15249.6	0.733626
$757$	−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	$-0.554391\pi$
$761$	−7139.46	−0.340086	−0.170043	+	0.985437i	$0.554391\pi$
$762$	26500.9	1.25988
$763$	29261.1	1.38836
$764$	9161.05	0.433816
$765$	−38388.6	−1.81430
$766$	1341.14	0.0632600
$767$	7471.55	0.351737
$768$	2187.52	0.102780
$769$	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
$771$	50774.0	2.37170
$772$	−9213.60	−0.429540
$773$	15240.9	0.709155	0.354577	−	0.935027i	$-0.384625\pi$
$773$	15240.9	0.709155	0.354577	+	0.935027i	$0.384625\pi$
$774$	−32551.4	−1.51168
$775$	−10744.8	−0.498017
$776$	−11316.2	−0.523492
$777$	−17121.0	−0.790494
$778$	−10471.8	−0.482561
$779$	981.984	0.0451646
$780$	4991.05	0.229113
$781$	0	0
$782$	−13633.9	−0.623464
$783$	−20694.3	−0.944515
$784$	3318.78	0.151184
$785$	6653.01	0.302492
$786$	−12682.7	−0.575544
$787$	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
$789$	−14561.6	−0.657043
$790$	−10393.3	−0.468074
$791$	−23038.9	−1.03561
$792$	0	0
$793$	4193.91	0.187806
$794$	−3503.29	−0.156583
$795$	54569.8	2.43446
$796$	13855.3	0.616946
$797$	3383.49	0.150375	0.0751877	−	0.997169i	$-0.476044\pi$
$797$	3383.49	0.150375	0.0751877	+	0.997169i	$0.476044\pi$
$798$	2906.94	0.128953
$799$	8816.83	0.390384
$800$	−1190.51	−0.0526136
$801$	−11976.1	−0.528283
$802$	−7225.55	−0.318134
$803$	0	0
$804$	−10068.6	−0.441656
$805$	−31096.7	−1.36151
$806$	6622.70	0.289423
$807$	−49948.3	−2.17876
$808$	1492.94	0.0650016
$809$	13480.3	0.585835	0.292917	−	0.956138i	$-0.405374\pi$
$809$	13480.3	0.585835	0.292917	+	0.956138i	$0.405374\pi$
$810$	−3721.25	−0.161421
$811$	−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
$813$	50465.3	2.17699
$814$	0	0
$815$	32391.1	1.39216
$816$	−8955.44	−0.384195
$817$	−2564.30	−0.109809
$818$	22590.0	0.965574
$819$	12378.2	0.528117
$820$	−6899.99	−0.293852
$821$	10664.8	0.453355	0.226677	−	0.973970i	$-0.427214\pi$
$821$	10664.8	0.453355	0.226677	+	0.973970i	$0.427214\pi$
$822$	−8812.92	−0.373949
$823$	−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	$-0.170794\pi$
$827$	40880.7	1.71894	0.859469	+	0.511188i	$0.170794\pi$
$828$	−19156.3	−0.804019
$829$	−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
$831$	75899.9	3.16840
$832$	733.789	0.0305764
$833$	−13586.7	−0.565128
$834$	−42095.4	−1.74778
$835$	17644.9	0.731288
$836$	0	0
$837$	−46931.4	−1.93809
$838$	7360.89	0.303434
$839$	−6368.57	−0.262059	−0.131029	−	0.991378i	$-0.541828\pi$
$839$	−6368.57	−0.262059	−0.131029	+	0.991378i	$0.541828\pi$
$840$	−20425.8	−0.838998
$841$	−8170.70	−0.335016
$842$	−18512.3	−0.757693
$843$	−69656.8	−2.84591
$844$	8367.91	0.341274
$845$	−26306.6	−1.07097
$846$	12388.1	0.503440
$847$	0	0
$848$	8022.90	0.324891
$849$	50107.5	2.02554
$850$	4873.81	0.196671
$851$	8888.02	0.358023
$852$	4527.01	0.182034
$853$	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
$855$	−4249.05	−0.169958
$856$	5701.71	0.227664
$857$	−10184.8	−0.405959	−0.202979	−	0.979183i	$-0.565062\pi$
$857$	−10184.8	−0.405959	−0.202979	+	0.979183i	$0.565062\pi$
$858$	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
$861$	−27153.1	−1.07477
$862$	−7168.08	−0.283232
$863$	−12283.3	−0.484504	−0.242252	−	0.970213i	$-0.577886\pi$
$863$	−12283.3	−0.484504	−0.242252	+	0.970213i	$0.577886\pi$
$864$	−5199.95	−0.204752
$865$	−5648.52	−0.222029
$866$	16613.6	0.651909
$867$	−5318.96	−0.208352
$868$	−27103.3	−1.05985
$869$	0	0
$870$	27718.8	1.08018
$871$	−3377.44	−0.131389
$872$	−9977.72	−0.387487
$873$	65092.2	2.52352
$874$	−1509.08	−0.0584042
$875$	−26233.5	−1.01355
$876$	16047.1	0.618929
$877$	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
$879$	65083.5	2.49740
$880$	0	0
$881$	−16737.0	−0.640049	−0.320024	−	0.947409i	$-0.603691\pi$
$881$	−16737.0	−0.640049	−0.320024	+	0.947409i	$0.603691\pi$
$882$	−19090.0	−0.728790
$883$	−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
$885$	70918.7	2.69368
$886$	−2157.64	−0.0818142
$887$	15912.8	0.602367	0.301184	−	0.953566i	$-0.402618\pi$
$887$	15912.8	0.602367	0.301184	+	0.953566i	$0.402618\pi$
$888$	5838.09	0.220623
$889$	−36380.4	−1.37251
$890$	6629.16	0.249674
$891$	0	0
$892$	14814.1	0.556068
$893$	975.894	0.0365700
$894$	7429.99	0.277960
$895$	4796.84	0.179152
$896$	−3003.02	−0.111969
$897$	−10196.2	−0.379533
$898$	−25531.9	−0.948786
$899$	36780.4	1.36451
$900$	6847.93	0.253627
$901$	−32844.8	−1.21445
$902$	0	0
$903$	70906.1	2.61308
$904$	7856.01	0.289034
$905$	41949.0	1.54081
$906$	13347.1	0.489436
$907$	−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
$909$	−8587.51	−0.313344
$910$	−6851.72	−0.249596
$911$	48359.6	1.75876	0.879378	−	0.476125i	$-0.157959\pi$
$911$	48359.6	1.75876	0.879378	+	0.476125i	$0.157959\pi$
$912$	−991.236	−0.0359902
$913$	0	0
$914$	22453.6	0.812583
$915$	39808.0	1.43826
$916$	−5347.38	−0.192885
$917$	17410.8	0.626997
$918$	21288.0	0.765369
$919$	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
$921$	35040.3	1.25365
$922$	4321.17	0.154349
$923$	1518.56	0.0541537
$924$	0	0
$925$	−3177.26	−0.112938
$926$	−22938.9	−0.814060
$927$	−53973.8	−1.91233
$928$	4075.23	0.144155
$929$	4919.67	0.173745	0.0868725	−	0.996219i	$-0.472313\pi$
$929$	4919.67	0.173745	0.0868725	+	0.996219i	$0.472313\pi$
$930$	62861.6	2.21647
$931$	−1503.85	−0.0529395
$932$	−23615.6	−0.829995
$933$	12442.6	0.436606
$934$	−4036.78	−0.141421
$935$	0	0
$936$	−4220.83	−0.147395
$937$	−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
$939$	−29580.0	−1.02802
$940$	−6857.20	−0.237933
$941$	−51486.9	−1.78366	−0.891830	−	0.452370i	$-0.850579\pi$
$941$	−51486.9	−1.78366	−0.891830	+	0.452370i	$0.850579\pi$
$942$	−8927.49	−0.308783
$943$	14095.9	0.486773
$944$	10426.5	0.359486
$945$	48554.3	1.67140
$946$	0	0
$947$	45107.8	1.54784	0.773920	−	0.633283i	$-0.218293\pi$
$947$	45107.8	1.54784	0.773920	+	0.633283i	$0.218293\pi$
$948$	13946.5	0.477809
$949$	5382.90	0.184127
$950$	539.459	0.0184235
$951$	42659.2	1.45460
$952$	12294.0	0.418542
$953$	25385.8	0.862881	0.431440	−	0.902141i	$-0.358006\pi$
$953$	25385.8	0.862881	0.431440	+	0.902141i	$0.358006\pi$
$954$	−46148.5	−1.56616
$955$	29168.6	0.988349
$956$	−13279.1	−0.449245
$957$	0	0
$958$	14654.0	0.494206
$959$	12098.4	0.407380
$960$	6965.00	0.234161
$961$	53621.0	1.79990
$962$	1958.35	0.0656339
$963$	−32796.8	−1.09747
$964$	−21103.8	−0.705091
$965$	−29335.9	−0.978607
$966$	41727.8	1.38982
$967$	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
$969$	4058.01	0.134532
$970$	−36030.7	−1.19265
$971$	−5451.84	−0.180183	−0.0900916	−	0.995933i	$-0.528716\pi$
$971$	−5451.84	−0.180183	−0.0900916	+	0.995933i	$0.528716\pi$
$972$	−12556.4	−0.414348
$973$	57788.6	1.90403
$974$	12379.6	0.407257
$975$	3644.89	0.119723
$976$	5852.59	0.191944
$977$	−3608.24	−0.118155	−0.0590776	−	0.998253i	$-0.518816\pi$
$977$	−3608.24	−0.118155	−0.0590776	+	0.998253i	$0.518816\pi$
$978$	−43464.7	−1.42111
$979$	0	0
$980$	10566.9	0.344437
$981$	57392.9	1.86790
$982$	20241.9	0.657784
$983$	58045.5	1.88338	0.941691	−	0.336478i	$-0.109236\pi$
$983$	58045.5	1.88338	0.941691	+	0.336478i	$0.109236\pi$
$984$	9258.92	0.299963
$985$	−13269.1	−0.429227
$986$	−16683.5	−0.538856
$987$	−26984.7	−0.870244
$988$	−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
$993$	87152.7	2.78520
$994$	−6214.68	−0.198307
$995$	44115.1	1.40557
$996$	−46469.1	−1.47834
$997$	−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
$999$	−13877.7	−0.439512

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.4.a.n.1.4		4
3.2	odd	2		2178.4.a.by.1.2		4
4.3	odd	2		1936.4.a.bn.1.1		4
11.2	odd	10		242.4.c.q.81.2		8
11.3	even	5		242.4.c.r.9.1		8
11.4	even	5		242.4.c.r.27.1		8
11.5	even	5		22.4.c.b.3.2	✓	8
11.6	odd	10		242.4.c.q.3.2		8
11.7	odd	10		242.4.c.n.27.1		8
11.8	odd	10		242.4.c.n.9.1		8
11.9	even	5		22.4.c.b.15.2	yes	8
11.10	odd	2		242.4.a.o.1.4		4
33.5	odd	10		198.4.f.d.91.2		8
33.20	odd	10		198.4.f.d.37.2		8
33.32	even	2		2178.4.a.bt.1.2		4
44.27	odd	10		176.4.m.b.113.1		8
44.31	odd	10		176.4.m.b.81.1		8
44.43	even	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	11.5	even	5
22.4.c.b.15.2	yes	8	11.9	even	5
176.4.m.b.81.1		8	44.31	odd	10
176.4.m.b.113.1		8	44.27	odd	10
198.4.f.d.37.2		8	33.20	odd	10
198.4.f.d.91.2		8	33.5	odd	10
242.4.a.n.1.4		4	1.1	even	1	trivial
242.4.a.o.1.4		4	11.10	odd	2
242.4.c.n.9.1		8	11.8	odd	10
242.4.c.n.27.1		8	11.7	odd	10
242.4.c.q.3.2		8	11.6	odd	10
242.4.c.q.81.2		8	11.2	odd	10
242.4.c.r.9.1		8	11.3	even	5
242.4.c.r.27.1		8	11.4	even	5
1936.4.a.bm.1.1		4	44.43	even	2
1936.4.a.bn.1.1		4	4.3	odd	2
2178.4.a.bt.1.2		4	33.32	even	2
2178.4.a.by.1.2		4	3.2	odd	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:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	242.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +8.54499 q^{3} +4.00000 q^{4} +12.7359 q^{5} -17.0900 q^{6} +23.4611 q^{7} -8.00000 q^{8} +46.0168 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +8.54499 q^{3} +4.00000 q^{4} +12.7359 q^{5} -17.0900 q^{6} +23.4611 q^{7} -8.00000 q^{8} +46.0168 q^{9} -25.4718 q^{10} +34.1799 q^{12} +11.4654 q^{13} -46.9222 q^{14} +108.828 q^{15} +16.0000 q^{16} -65.5022 q^{17} -92.0336 q^{18} -7.25013 q^{19} +50.9436 q^{20} +200.475 q^{21} -104.072 q^{23} -68.3599 q^{24} +37.2034 q^{25} -22.9309 q^{26} +162.498 q^{27} +93.8445 q^{28} -127.351 q^{29} -217.656 q^{30} -288.811 q^{31} -32.0000 q^{32} +131.004 q^{34} +298.799 q^{35} +184.067 q^{36} -85.4023 q^{37} +14.5003 q^{38} +97.9721 q^{39} -101.887 q^{40} -135.444 q^{41} -400.950 q^{42} +353.691 q^{43} +586.066 q^{45} +208.145 q^{46} -134.604 q^{47} +136.720 q^{48} +207.424 q^{49} -74.4068 q^{50} -559.715 q^{51} +45.8618 q^{52} +501.431 q^{53} -324.997 q^{54} -187.689 q^{56} -61.9522 q^{57} +254.702 q^{58} +651.658 q^{59} +435.313 q^{60} +365.787 q^{61} +577.623 q^{62} +1079.61 q^{63} +64.0000 q^{64} +146.023 q^{65} -294.576 q^{67} -262.009 q^{68} -889.297 q^{69} -597.597 q^{70} +132.446 q^{71} -368.134 q^{72} +469.489 q^{73} +170.805 q^{74} +317.903 q^{75} -29.0005 q^{76} -195.944 q^{78} +408.033 q^{79} +203.775 q^{80} +146.093 q^{81} +270.887 q^{82} -1359.54 q^{83} +801.900 q^{84} -834.230 q^{85} -707.381 q^{86} -1088.21 q^{87} -260.255 q^{89} -1172.13 q^{90} +268.992 q^{91} -416.289 q^{92} -2467.89 q^{93} +269.207 q^{94} -92.3370 q^{95} -273.440 q^{96} +1414.53 q^{97} -414.848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} - 8 q^{6} - 3 q^{7} - 32 q^{8} + 102 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 - 8 * q^6 - 3 * q^7 - 32 * q^8 + 102 * q^9	\( 4 q - 8 q^{2} + 4 q^{3} + 16 q^{4} + 25 q^{5} - 8 q^{6} - 3 q^{7} - 32 q^{8} + 102 q^{9} - 50 q^{10} + 16 q^{12} + 41 q^{13} + 6 q^{14} + 68 q^{15} + 64 q^{16} - 52 q^{17} - 204 q^{18} - 16 q^{19} + 100 q^{20} - 25 q^{21} + 314 q^{23} - 32 q^{24} - 21 q^{25} - 82 q^{26} + 286 q^{27} - 12 q^{28} - 561 q^{29} - 136 q^{30} + 199 q^{31} - 128 q^{32} + 104 q^{34} + 714 q^{35} + 408 q^{36} + 357 q^{37} + 32 q^{38} + 1038 q^{39} - 200 q^{40} - 32 q^{41} + 50 q^{42} + 721 q^{43} + 1326 q^{45} - 628 q^{46} + 403 q^{47} + 64 q^{48} + 823 q^{49} + 42 q^{50} + 174 q^{51} + 164 q^{52} - 133 q^{53} - 572 q^{54} + 24 q^{56} + 1031 q^{57} + 1122 q^{58} + 1016 q^{59} + 272 q^{60} + 919 q^{61} - 398 q^{62} + 1367 q^{63} + 256 q^{64} - 69 q^{65} + 289 q^{67} - 208 q^{68} - 1620 q^{69} - 1428 q^{70} - 1205 q^{71} - 816 q^{72} + 1234 q^{73} - 714 q^{74} - 911 q^{75} - 64 q^{76} - 2076 q^{78} + 603 q^{79} + 400 q^{80} - 1400 q^{81} + 64 q^{82} - 1514 q^{83} - 100 q^{84} - 717 q^{85} - 1442 q^{86} + 1061 q^{87} - 1101 q^{89} - 2652 q^{90} - 2306 q^{91} + 1256 q^{92} - 2298 q^{93} - 806 q^{94} + 1766 q^{95} - 128 q^{96} + 2116 q^{97} - 1646 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 4 * q^3 + 16 * q^4 + 25 * q^5 - 8 * q^6 - 3 * q^7 - 32 * q^8 + 102 * q^9 - 50 * q^10 + 16 * q^12 + 41 * q^13 + 6 * q^14 + 68 * q^15 + 64 * q^16 - 52 * q^17 - 204 * q^18 - 16 * q^19 + 100 * q^20 - 25 * q^21 + 314 * q^23 - 32 * q^24 - 21 * q^25 - 82 * q^26 + 286 * q^27 - 12 * q^28 - 561 * q^29 - 136 * q^30 + 199 * q^31 - 128 * q^32 + 104 * q^34 + 714 * q^35 + 408 * q^36 + 357 * q^37 + 32 * q^38 + 1038 * q^39 - 200 * q^40 - 32 * q^41 + 50 * q^42 + 721 * q^43 + 1326 * q^45 - 628 * q^46 + 403 * q^47 + 64 * q^48 + 823 * q^49 + 42 * q^50 + 174 * q^51 + 164 * q^52 - 133 * q^53 - 572 * q^54 + 24 * q^56 + 1031 * q^57 + 1122 * q^58 + 1016 * q^59 + 272 * q^60 + 919 * q^61 - 398 * q^62 + 1367 * q^63 + 256 * q^64 - 69 * q^65 + 289 * q^67 - 208 * q^68 - 1620 * q^69 - 1428 * q^70 - 1205 * q^71 - 816 * q^72 + 1234 * q^73 - 714 * q^74 - 911 * q^75 - 64 * q^76 - 2076 * q^78 + 603 * q^79 + 400 * q^80 - 1400 * q^81 + 64 * q^82 - 1514 * q^83 - 100 * q^84 - 717 * q^85 - 1442 * q^86 + 1061 * q^87 - 1101 * q^89 - 2652 * q^90 - 2306 * q^91 + 1256 * q^92 - 2298 * q^93 - 806 * q^94 + 1766 * q^95 - 128 * q^96 + 2116 * q^97 - 1646 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more