LMFDB - Embedded newform 8034.2.a.u.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4x^{10} - 18x^{9} + 64x^{8} + 85x^{7} - 249x^{6} - 109x^{5} + 230x^{4} + 97x^{3} - 53x^{2} - 32x - 4 $ x^11 - 4x^10 - 18x^9 + 64x^8 + 85x^7 - 249x^6 - 109x^5 + 230x^4 + 97x^3 - 53x^2 - 32x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.444701$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.48470 q^{5} -1.00000 q^{6} -3.17363 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.48470 q^{5} -1.00000 q^{6} -3.17363 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.48470 q^{10} +0.113354 q^{11} -1.00000 q^{12} -1.00000 q^{13} -3.17363 q^{14} -1.48470 q^{15} +1.00000 q^{16} -3.88650 q^{17} +1.00000 q^{18} -0.323777 q^{19} +1.48470 q^{20} +3.17363 q^{21} +0.113354 q^{22} +3.28852 q^{23} -1.00000 q^{24} -2.79567 q^{25} -1.00000 q^{26} -1.00000 q^{27} -3.17363 q^{28} -2.92223 q^{29} -1.48470 q^{30} +5.22103 q^{31} +1.00000 q^{32} -0.113354 q^{33} -3.88650 q^{34} -4.71187 q^{35} +1.00000 q^{36} +9.23070 q^{37} -0.323777 q^{38} +1.00000 q^{39} +1.48470 q^{40} +10.1399 q^{41} +3.17363 q^{42} +2.13959 q^{43} +0.113354 q^{44} +1.48470 q^{45} +3.28852 q^{46} +4.85813 q^{47} -1.00000 q^{48} +3.07190 q^{49} -2.79567 q^{50} +3.88650 q^{51} -1.00000 q^{52} -10.9886 q^{53} -1.00000 q^{54} +0.168296 q^{55} -3.17363 q^{56} +0.323777 q^{57} -2.92223 q^{58} -3.11573 q^{59} -1.48470 q^{60} -12.7465 q^{61} +5.22103 q^{62} -3.17363 q^{63} +1.00000 q^{64} -1.48470 q^{65} -0.113354 q^{66} -12.7525 q^{67} -3.88650 q^{68} -3.28852 q^{69} -4.71187 q^{70} +3.57724 q^{71} +1.00000 q^{72} -8.63410 q^{73} +9.23070 q^{74} +2.79567 q^{75} -0.323777 q^{76} -0.359743 q^{77} +1.00000 q^{78} -1.98371 q^{79} +1.48470 q^{80} +1.00000 q^{81} +10.1399 q^{82} -9.53492 q^{83} +3.17363 q^{84} -5.77027 q^{85} +2.13959 q^{86} +2.92223 q^{87} +0.113354 q^{88} +8.30325 q^{89} +1.48470 q^{90} +3.17363 q^{91} +3.28852 q^{92} -5.22103 q^{93} +4.85813 q^{94} -0.480711 q^{95} -1.00000 q^{96} +2.69171 q^{97} +3.07190 q^{98} +0.113354 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} - 2 q^{5} - 11 q^{6} - 2 q^{7} + 11 q^{8} + 11 q^{9}+O(q^{10}) $ 11 * q + 11 * q^2 - 11 * q^3 + 11 * q^4 - 2 * q^5 - 11 * q^6 - 2 * q^7 + 11 * q^8 + 11 * q^9	\( 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} - 2 q^{5} - 11 q^{6} - 2 q^{7} + 11 q^{8} + 11 q^{9} - 2 q^{10} - 10 q^{11} - 11 q^{12} - 11 q^{13} - 2 q^{14} + 2 q^{15} + 11 q^{16} + 6 q^{17} + 11 q^{18} - 3 q^{19} - 2 q^{20} + 2 q^{21} - 10 q^{22} + 3 q^{23} - 11 q^{24} - q^{25} - 11 q^{26} - 11 q^{27} - 2 q^{28} - 22 q^{29} + 2 q^{30} - 5 q^{31} + 11 q^{32} + 10 q^{33} + 6 q^{34} - 20 q^{35} + 11 q^{36} - 26 q^{37} - 3 q^{38} + 11 q^{39} - 2 q^{40} - 6 q^{41} + 2 q^{42} - 8 q^{43} - 10 q^{44} - 2 q^{45} + 3 q^{46} + 6 q^{47} - 11 q^{48} - 5 q^{49} - q^{50} - 6 q^{51} - 11 q^{52} - 25 q^{53} - 11 q^{54} - 2 q^{56} + 3 q^{57} - 22 q^{58} + 7 q^{59} + 2 q^{60} - 36 q^{61} - 5 q^{62} - 2 q^{63} + 11 q^{64} + 2 q^{65} + 10 q^{66} - 12 q^{67} + 6 q^{68} - 3 q^{69} - 20 q^{70} - 15 q^{71} + 11 q^{72} - 12 q^{73} - 26 q^{74} + q^{75} - 3 q^{76} - q^{77} + 11 q^{78} - 15 q^{79} - 2 q^{80} + 11 q^{81} - 6 q^{82} - 16 q^{83} + 2 q^{84} - 25 q^{85} - 8 q^{86} + 22 q^{87} - 10 q^{88} - 2 q^{89} - 2 q^{90} + 2 q^{91} + 3 q^{92} + 5 q^{93} + 6 q^{94} + 16 q^{95} - 11 q^{96} - 10 q^{97} - 5 q^{98} - 10 q^{99}+O(q^{100}) \) 11 * q + 11 * q^2 - 11 * q^3 + 11 * q^4 - 2 * q^5 - 11 * q^6 - 2 * q^7 + 11 * q^8 + 11 * q^9 - 2 * q^10 - 10 * q^11 - 11 * q^12 - 11 * q^13 - 2 * q^14 + 2 * q^15 + 11 * q^16 + 6 * q^17 + 11 * q^18 - 3 * q^19 - 2 * q^20 + 2 * q^21 - 10 * q^22 + 3 * q^23 - 11 * q^24 - q^25 - 11 * q^26 - 11 * q^27 - 2 * q^28 - 22 * q^29 + 2 * q^30 - 5 * q^31 + 11 * q^32 + 10 * q^33 + 6 * q^34 - 20 * q^35 + 11 * q^36 - 26 * q^37 - 3 * q^38 + 11 * q^39 - 2 * q^40 - 6 * q^41 + 2 * q^42 - 8 * q^43 - 10 * q^44 - 2 * q^45 + 3 * q^46 + 6 * q^47 - 11 * q^48 - 5 * q^49 - q^50 - 6 * q^51 - 11 * q^52 - 25 * q^53 - 11 * q^54 - 2 * q^56 + 3 * q^57 - 22 * q^58 + 7 * q^59 + 2 * q^60 - 36 * q^61 - 5 * q^62 - 2 * q^63 + 11 * q^64 + 2 * q^65 + 10 * q^66 - 12 * q^67 + 6 * q^68 - 3 * q^69 - 20 * q^70 - 15 * q^71 + 11 * q^72 - 12 * q^73 - 26 * q^74 + q^75 - 3 * q^76 - q^77 + 11 * q^78 - 15 * q^79 - 2 * q^80 + 11 * q^81 - 6 * q^82 - 16 * q^83 + 2 * q^84 - 25 * q^85 - 8 * q^86 + 22 * q^87 - 10 * q^88 - 2 * q^89 - 2 * q^90 + 2 * q^91 + 3 * q^92 + 5 * q^93 + 6 * q^94 + 16 * q^95 - 11 * q^96 - 10 * q^97 - 5 * q^98 - 10 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.48470	0.663977	0.331988	−	0.943283i	$-0.392281\pi$
$5$	1.48470	0.663977	0.331988	+	0.943283i	$0.392281\pi$
$6$	−1.00000	−0.408248
$7$	−3.17363	−1.19952	−0.599759	−	0.800181i	$-0.704737\pi$
$7$	−3.17363	−1.19952	−0.599759	+	0.800181i	$0.704737\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.48470	0.469503
$11$	0.113354	0.0341775	0.0170888	−	0.999854i	$-0.494560\pi$
$11$	0.113354	0.0341775	0.0170888	+	0.999854i	$0.494560\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.17363	−0.848187
$15$	−1.48470	−0.383347
$16$	1.00000	0.250000
$17$	−3.88650	−0.942614	−0.471307	−	0.881969i	$-0.656218\pi$
$17$	−3.88650	−0.942614	−0.471307	+	0.881969i	$0.656218\pi$
$18$	1.00000	0.235702
$19$	−0.323777	−0.0742796	−0.0371398	−	0.999310i	$-0.511825\pi$
$19$	−0.323777	−0.0742796	−0.0371398	+	0.999310i	$0.511825\pi$
$20$	1.48470	0.331988
$21$	3.17363	0.692542
$22$	0.113354	0.0241672
$23$	3.28852	0.685705	0.342852	−	0.939389i	$-0.388607\pi$
$23$	3.28852	0.685705	0.342852	+	0.939389i	$0.388607\pi$
$24$	−1.00000	−0.204124
$25$	−2.79567	−0.559135
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	−3.17363	−0.599759
$29$	−2.92223	−0.542644	−0.271322	−	0.962489i	$-0.587461\pi$
$29$	−2.92223	−0.542644	−0.271322	+	0.962489i	$0.587461\pi$
$30$	−1.48470	−0.271067
$31$	5.22103	0.937725	0.468863	−	0.883271i	$-0.344664\pi$
$31$	5.22103	0.937725	0.468863	+	0.883271i	$0.344664\pi$
$32$	1.00000	0.176777
$33$	−0.113354	−0.0197324
$34$	−3.88650	−0.666529
$35$	−4.71187	−0.796452
$36$	1.00000	0.166667
$37$	9.23070	1.51752	0.758759	−	0.651371i	$-0.225806\pi$
$37$	9.23070	1.51752	0.758759	+	0.651371i	$0.225806\pi$
$38$	−0.323777	−0.0525236
$39$	1.00000	0.160128
$40$	1.48470	0.234751
$41$	10.1399	1.58359	0.791797	−	0.610785i	$-0.209146\pi$
$41$	10.1399	1.58359	0.791797	+	0.610785i	$0.209146\pi$
$42$	3.17363	0.489701
$43$	2.13959	0.326284	0.163142	−	0.986603i	$-0.447837\pi$
$43$	2.13959	0.326284	0.163142	+	0.986603i	$0.447837\pi$
$44$	0.113354	0.0170888
$45$	1.48470	0.221326
$46$	3.28852	0.484866
$47$	4.85813	0.708632	0.354316	−	0.935126i	$-0.384714\pi$
$47$	4.85813	0.708632	0.354316	+	0.935126i	$0.384714\pi$
$48$	−1.00000	−0.144338
$49$	3.07190	0.438843
$50$	−2.79567	−0.395368
$51$	3.88650	0.544218
$52$	−1.00000	−0.138675
$53$	−10.9886	−1.50939	−0.754697	−	0.656073i	$-0.772216\pi$
$53$	−10.9886	−1.50939	−0.754697	+	0.656073i	$0.772216\pi$
$54$	−1.00000	−0.136083
$55$	0.168296	0.0226931
$56$	−3.17363	−0.424094
$57$	0.323777	0.0428853
$58$	−2.92223	−0.383707
$59$	−3.11573	−0.405633	−0.202817	−	0.979217i	$-0.565010\pi$
$59$	−3.11573	−0.405633	−0.202817	+	0.979217i	$0.565010\pi$
$60$	−1.48470	−0.191674
$61$	−12.7465	−1.63203	−0.816013	−	0.578033i	$-0.803820\pi$
$61$	−12.7465	−1.63203	−0.816013	+	0.578033i	$0.803820\pi$
$62$	5.22103	0.663072
$63$	−3.17363	−0.399839
$64$	1.00000	0.125000
$65$	−1.48470	−0.184154
$66$	−0.113354	−0.0139529
$67$	−12.7525	−1.55796	−0.778982	−	0.627047i	$-0.784264\pi$
$67$	−12.7525	−1.55796	−0.778982	+	0.627047i	$0.784264\pi$
$68$	−3.88650	−0.471307
$69$	−3.28852	−0.395892
$70$	−4.71187	−0.563177
$71$	3.57724	0.424540	0.212270	−	0.977211i	$-0.431914\pi$
$71$	3.57724	0.424540	0.212270	+	0.977211i	$0.431914\pi$
$72$	1.00000	0.117851
$73$	−8.63410	−1.01054	−0.505272	−	0.862960i	$-0.668608\pi$
$73$	−8.63410	−1.01054	−0.505272	+	0.862960i	$0.668608\pi$
$74$	9.23070	1.07305
$75$	2.79567	0.322817
$76$	−0.323777	−0.0371398
$77$	−0.359743	−0.0409966
$78$	1.00000	0.113228
$79$	−1.98371	−0.223185	−0.111592	−	0.993754i	$-0.535595\pi$
$79$	−1.98371	−0.223185	−0.111592	+	0.993754i	$0.535595\pi$
$80$	1.48470	0.165994
$81$	1.00000	0.111111
$82$	10.1399	1.11977
$83$	−9.53492	−1.04659	−0.523297	−	0.852151i	$-0.675298\pi$
$83$	−9.53492	−1.04659	−0.523297	+	0.852151i	$0.675298\pi$
$84$	3.17363	0.346271
$85$	−5.77027	−0.625874
$86$	2.13959	0.230717
$87$	2.92223	0.313296
$88$	0.113354	0.0120836
$89$	8.30325	0.880143	0.440072	−	0.897963i	$-0.354953\pi$
$89$	8.30325	0.880143	0.440072	+	0.897963i	$0.354953\pi$
$90$	1.48470	0.156501
$91$	3.17363	0.332686
$92$	3.28852	0.342852
$93$	−5.22103	−0.541396
$94$	4.85813	0.501078
$95$	−0.480711	−0.0493199
$96$	−1.00000	−0.102062
$97$	2.69171	0.273301	0.136651	−	0.990619i	$-0.456366\pi$
$97$	2.69171	0.273301	0.136651	+	0.990619i	$0.456366\pi$
$98$	3.07190	0.310309
$99$	0.113354	0.0113925
$100$	−2.79567	−0.279567
$101$	−17.5503	−1.74632	−0.873161	−	0.487431i	$-0.837934\pi$
$101$	−17.5503	−1.74632	−0.873161	+	0.487431i	$0.837934\pi$
$102$	3.88650	0.384820
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−1.00000	−0.0980581
$105$	4.71187	0.459832
$106$	−10.9886	−1.06730
$107$	−15.4959	−1.49805	−0.749023	−	0.662544i	$-0.769477\pi$
$107$	−15.4959	−1.49805	−0.749023	+	0.662544i	$0.769477\pi$
$108$	−1.00000	−0.0962250
$109$	−6.28842	−0.602322	−0.301161	−	0.953573i	$-0.597374\pi$
$109$	−6.28842	−0.602322	−0.301161	+	0.953573i	$0.597374\pi$
$110$	0.168296	0.0160464
$111$	−9.23070	−0.876140
$112$	−3.17363	−0.299880
$113$	−3.33461	−0.313694	−0.156847	−	0.987623i	$-0.550133\pi$
$113$	−3.33461	−0.313694	−0.156847	+	0.987623i	$0.550133\pi$
$114$	0.323777	0.0303245
$115$	4.88246	0.455292
$116$	−2.92223	−0.271322
$117$	−1.00000	−0.0924500
$118$	−3.11573	−0.286826
$119$	12.3343	1.13068
$120$	−1.48470	−0.135534
$121$	−10.9872	−0.998832
$122$	−12.7465	−1.15402
$123$	−10.1399	−0.914288
$124$	5.22103	0.468863
$125$	−11.5742	−1.03523
$126$	−3.17363	−0.282729
$127$	3.22861	0.286493	0.143247	−	0.989687i	$-0.454246\pi$
$127$	3.22861	0.286493	0.143247	+	0.989687i	$0.454246\pi$
$128$	1.00000	0.0883883
$129$	−2.13959	−0.188380
$130$	−1.48470	−0.130217
$131$	−7.29169	−0.637078	−0.318539	−	0.947910i	$-0.603192\pi$
$131$	−7.29169	−0.637078	−0.318539	+	0.947910i	$0.603192\pi$
$132$	−0.113354	−0.00986620
$133$	1.02755	0.0890997
$134$	−12.7525	−1.10165
$135$	−1.48470	−0.127782
$136$	−3.88650	−0.333264
$137$	10.0244	0.856441	0.428221	−	0.903674i	$-0.359141\pi$
$137$	10.0244	0.856441	0.428221	+	0.903674i	$0.359141\pi$
$138$	−3.28852	−0.279938
$139$	−6.96503	−0.590766	−0.295383	−	0.955379i	$-0.595447\pi$
$139$	−6.96503	−0.590766	−0.295383	+	0.955379i	$0.595447\pi$
$140$	−4.71187	−0.398226
$141$	−4.85813	−0.409129
$142$	3.57724	0.300195
$143$	−0.113354	−0.00947914
$144$	1.00000	0.0833333
$145$	−4.33862	−0.360303
$146$	−8.63410	−0.714563
$147$	−3.07190	−0.253366
$148$	9.23070	0.758759
$149$	−12.2744	−1.00556	−0.502779	−	0.864415i	$-0.667689\pi$
$149$	−12.2744	−1.00556	−0.502779	+	0.864415i	$0.667689\pi$
$150$	2.79567	0.228266
$151$	−2.61066	−0.212452	−0.106226	−	0.994342i	$-0.533877\pi$
$151$	−2.61066	−0.212452	−0.106226	+	0.994342i	$0.533877\pi$
$152$	−0.323777	−0.0262618
$153$	−3.88650	−0.314205
$154$	−0.359743	−0.0289889
$155$	7.75165	0.622628
$156$	1.00000	0.0800641
$157$	9.09881	0.726164	0.363082	−	0.931757i	$-0.381724\pi$
$157$	9.09881	0.726164	0.363082	+	0.931757i	$0.381724\pi$
$158$	−1.98371	−0.157815
$159$	10.9886	0.871449
$160$	1.48470	0.117376
$161$	−10.4365	−0.822515
$162$	1.00000	0.0785674
$163$	9.51850	0.745547	0.372773	−	0.927922i	$-0.378407\pi$
$163$	9.51850	0.745547	0.372773	+	0.927922i	$0.378407\pi$
$164$	10.1399	0.791797
$165$	−0.168296	−0.0131019
$166$	−9.53492	−0.740053
$167$	10.7498	0.831842	0.415921	−	0.909401i	$-0.363459\pi$
$167$	10.7498	0.831842	0.415921	+	0.909401i	$0.363459\pi$
$168$	3.17363	0.244851
$169$	1.00000	0.0769231
$170$	−5.77027	−0.442560
$171$	−0.323777	−0.0247599
$172$	2.13959	0.163142
$173$	−12.4141	−0.943829	−0.471914	−	0.881644i	$-0.656437\pi$
$173$	−12.4141	−0.943829	−0.471914	+	0.881644i	$0.656437\pi$
$174$	2.92223	0.221533
$175$	8.87242	0.670692
$176$	0.113354	0.00854438
$177$	3.11573	0.234193
$178$	8.30325	0.622355
$179$	−8.17919	−0.611342	−0.305671	−	0.952137i	$-0.598881\pi$
$179$	−8.17919	−0.611342	−0.305671	+	0.952137i	$0.598881\pi$
$180$	1.48470	0.110663
$181$	−8.74842	−0.650265	−0.325132	−	0.945669i	$-0.605409\pi$
$181$	−8.74842	−0.650265	−0.325132	+	0.945669i	$0.605409\pi$
$182$	3.17363	0.235245
$183$	12.7465	0.942251
$184$	3.28852	0.242433
$185$	13.7048	1.00760
$186$	−5.22103	−0.382825
$187$	−0.440550	−0.0322162
$188$	4.85813	0.354316
$189$	3.17363	0.230847
$190$	−0.480711	−0.0348745
$191$	−24.0969	−1.74359	−0.871796	−	0.489869i	$-0.837044\pi$
$191$	−24.0969	−1.74359	−0.871796	+	0.489869i	$0.837044\pi$
$192$	−1.00000	−0.0721688
$193$	25.3188	1.82249	0.911244	−	0.411866i	$-0.135123\pi$
$193$	25.3188	1.82249	0.911244	+	0.411866i	$0.135123\pi$
$194$	2.69171	0.193253
$195$	1.48470	0.106321
$196$	3.07190	0.219422
$197$	−7.34948	−0.523629	−0.261814	−	0.965118i	$-0.584321\pi$
$197$	−7.34948	−0.523629	−0.261814	+	0.965118i	$0.584321\pi$
$198$	0.113354	0.00805572
$199$	16.4690	1.16745	0.583727	−	0.811950i	$-0.301594\pi$
$199$	16.4690	1.16745	0.583727	+	0.811950i	$0.301594\pi$
$200$	−2.79567	−0.197684
$201$	12.7525	0.899491
$202$	−17.5503	−1.23484
$203$	9.27406	0.650911
$204$	3.88650	0.272109
$205$	15.0548	1.05147
$206$	1.00000	0.0696733
$207$	3.28852	0.228568
$208$	−1.00000	−0.0693375
$209$	−0.0367015	−0.00253869
$210$	4.71187	0.325150
$211$	−6.81556	−0.469203	−0.234601	−	0.972092i	$-0.575378\pi$
$211$	−6.81556	−0.469203	−0.234601	+	0.972092i	$0.575378\pi$
$212$	−10.9886	−0.754697
$213$	−3.57724	−0.245108
$214$	−15.4959	−1.05928
$215$	3.17664	0.216645
$216$	−1.00000	−0.0680414
$217$	−16.5696	−1.12482
$218$	−6.28842	−0.425906
$219$	8.63410	0.583438
$220$	0.168296	0.0113465
$221$	3.88650	0.261434
$222$	−9.23070	−0.619524
$223$	8.92654	0.597765	0.298883	−	0.954290i	$-0.403386\pi$
$223$	8.92654	0.597765	0.298883	+	0.954290i	$0.403386\pi$
$224$	−3.17363	−0.212047
$225$	−2.79567	−0.186378
$226$	−3.33461	−0.221815
$227$	22.1880	1.47267	0.736336	−	0.676616i	$-0.236554\pi$
$227$	22.1880	1.47267	0.736336	+	0.676616i	$0.236554\pi$
$228$	0.323777	0.0214427
$229$	−14.2480	−0.941533	−0.470766	−	0.882258i	$-0.656023\pi$
$229$	−14.2480	−0.941533	−0.470766	+	0.882258i	$0.656023\pi$
$230$	4.88246	0.321940
$231$	0.359743	0.0236694
$232$	−2.92223	−0.191854
$233$	1.71660	0.112459	0.0562293	−	0.998418i	$-0.482092\pi$
$233$	1.71660	0.112459	0.0562293	+	0.998418i	$0.482092\pi$
$234$	−1.00000	−0.0653720
$235$	7.21286	0.470515
$236$	−3.11573	−0.202817
$237$	1.98371	0.128856
$238$	12.3343	0.799513
$239$	−12.2896	−0.794948	−0.397474	−	0.917613i	$-0.630113\pi$
$239$	−12.2896	−0.794948	−0.397474	+	0.917613i	$0.630113\pi$
$240$	−1.48470	−0.0958368
$241$	−11.1522	−0.718380	−0.359190	−	0.933265i	$-0.616947\pi$
$241$	−11.1522	−0.718380	−0.359190	+	0.933265i	$0.616947\pi$
$242$	−10.9872	−0.706281
$243$	−1.00000	−0.0641500
$244$	−12.7465	−0.816013
$245$	4.56085	0.291382
$246$	−10.1399	−0.646499
$247$	0.323777	0.0206015
$248$	5.22103	0.331536
$249$	9.53492	0.604251
$250$	−11.5742	−0.732018
$251$	−21.4365	−1.35306	−0.676531	−	0.736414i	$-0.736517\pi$
$251$	−21.4365	−1.35306	−0.676531	+	0.736414i	$0.736517\pi$
$252$	−3.17363	−0.199920
$253$	0.372767	0.0234357
$254$	3.22861	0.202581
$255$	5.77027	0.361348
$256$	1.00000	0.0625000
$257$	−2.50352	−0.156165	−0.0780826	−	0.996947i	$-0.524880\pi$
$257$	−2.50352	−0.156165	−0.0780826	+	0.996947i	$0.524880\pi$
$258$	−2.13959	−0.133205
$259$	−29.2948	−1.82029
$260$	−1.48470	−0.0920770
$261$	−2.92223	−0.180881
$262$	−7.29169	−0.450482
$263$	10.3482	0.638095	0.319047	−	0.947739i	$-0.396637\pi$
$263$	10.3482	0.638095	0.319047	+	0.947739i	$0.396637\pi$
$264$	−0.113354	−0.00697646
$265$	−16.3147	−1.00220
$266$	1.02755	0.0630030
$267$	−8.30325	−0.508151
$268$	−12.7525	−0.778982
$269$	−12.0134	−0.732467	−0.366234	−	0.930523i	$-0.619353\pi$
$269$	−12.0134	−0.732467	−0.366234	+	0.930523i	$0.619353\pi$
$270$	−1.48470	−0.0903558
$271$	−17.2845	−1.04996	−0.524980	−	0.851115i	$-0.675927\pi$
$271$	−17.2845	−1.04996	−0.524980	+	0.851115i	$0.675927\pi$
$272$	−3.88650	−0.235653
$273$	−3.17363	−0.192077
$274$	10.0244	0.605595
$275$	−0.316901	−0.0191098
$276$	−3.28852	−0.197946
$277$	10.5495	0.633858	0.316929	−	0.948449i	$-0.397348\pi$
$277$	10.5495	0.633858	0.316929	+	0.948449i	$0.397348\pi$
$278$	−6.96503	−0.417735
$279$	5.22103	0.312575
$280$	−4.71187	−0.281588
$281$	−15.1321	−0.902707	−0.451354	−	0.892345i	$-0.649059\pi$
$281$	−15.1321	−0.902707	−0.451354	+	0.892345i	$0.649059\pi$
$282$	−4.85813	−0.289298
$283$	14.4417	0.858472	0.429236	−	0.903192i	$-0.358783\pi$
$283$	14.4417	0.858472	0.429236	+	0.903192i	$0.358783\pi$
$284$	3.57724	0.212270
$285$	0.480711	0.0284749
$286$	−0.113354	−0.00670276
$287$	−32.1804	−1.89955
$288$	1.00000	0.0589256
$289$	−1.89515	−0.111479
$290$	−4.33862	−0.254773
$291$	−2.69171	−0.157791
$292$	−8.63410	−0.505272
$293$	10.2510	0.598868	0.299434	−	0.954117i	$-0.403202\pi$
$293$	10.2510	0.598868	0.299434	+	0.954117i	$0.403202\pi$
$294$	−3.07190	−0.179157
$295$	−4.62591	−0.269331
$296$	9.23070	0.536524
$297$	−0.113354	−0.00657747
$298$	−12.2744	−0.711036
$299$	−3.28852	−0.190180
$300$	2.79567	0.161408
$301$	−6.79024	−0.391383
$302$	−2.61066	−0.150226
$303$	17.5503	1.00824
$304$	−0.323777	−0.0185699
$305$	−18.9247	−1.08363
$306$	−3.88650	−0.222176
$307$	5.42553	0.309652	0.154826	−	0.987942i	$-0.450518\pi$
$307$	5.42553	0.309652	0.154826	+	0.987942i	$0.450518\pi$
$308$	−0.359743	−0.0204983
$309$	−1.00000	−0.0568880
$310$	7.75165	0.440264
$311$	−5.69268	−0.322802	−0.161401	−	0.986889i	$-0.551601\pi$
$311$	−5.69268	−0.322802	−0.161401	+	0.986889i	$0.551601\pi$
$312$	1.00000	0.0566139
$313$	−14.9124	−0.842899	−0.421449	−	0.906852i	$-0.638478\pi$
$313$	−14.9124	−0.842899	−0.421449	+	0.906852i	$0.638478\pi$
$314$	9.09881	0.513476
$315$	−4.71187	−0.265484
$316$	−1.98371	−0.111592
$317$	−2.52576	−0.141861	−0.0709304	−	0.997481i	$-0.522597\pi$
$317$	−2.52576	−0.141861	−0.0709304	+	0.997481i	$0.522597\pi$
$318$	10.9886	0.616208
$319$	−0.331246	−0.0185462
$320$	1.48470	0.0829971
$321$	15.4959	0.864897
$322$	−10.4365	−0.581606
$323$	1.25836	0.0700170
$324$	1.00000	0.0555556
$325$	2.79567	0.155076
$326$	9.51850	0.527181
$327$	6.28842	0.347751
$328$	10.1399	0.559885
$329$	−15.4179	−0.850017
$330$	−0.168296	−0.00926441
$331$	19.0758	1.04850	0.524250	−	0.851565i	$-0.324346\pi$
$331$	19.0758	1.04850	0.524250	+	0.851565i	$0.324346\pi$
$332$	−9.53492	−0.523297
$333$	9.23070	0.505839
$334$	10.7498	0.588201
$335$	−18.9336	−1.03445
$336$	3.17363	0.173136
$337$	2.83472	0.154417	0.0772085	−	0.997015i	$-0.475399\pi$
$337$	2.83472	0.154417	0.0772085	+	0.997015i	$0.475399\pi$
$338$	1.00000	0.0543928
$339$	3.33461	0.181111
$340$	−5.77027	−0.312937
$341$	0.591825	0.0320491
$342$	−0.323777	−0.0175079
$343$	12.4663	0.673117
$344$	2.13959	0.115359
$345$	−4.88246	−0.262863
$346$	−12.4141	−0.667388
$347$	15.5470	0.834605	0.417302	−	0.908768i	$-0.362976\pi$
$347$	15.5470	0.834605	0.417302	+	0.908768i	$0.362976\pi$
$348$	2.92223	0.156648
$349$	−11.2061	−0.599851	−0.299925	−	0.953963i	$-0.596962\pi$
$349$	−11.2061	−0.599851	−0.299925	+	0.953963i	$0.596962\pi$
$350$	8.87242	0.474251
$351$	1.00000	0.0533761
$352$	0.113354	0.00604179
$353$	6.15868	0.327794	0.163897	−	0.986477i	$-0.447594\pi$
$353$	6.15868	0.327794	0.163897	+	0.986477i	$0.447594\pi$
$354$	3.11573	0.165599
$355$	5.31111	0.281885
$356$	8.30325	0.440072
$357$	−12.3343	−0.652800
$358$	−8.17919	−0.432284
$359$	−17.8129	−0.940127	−0.470064	−	0.882633i	$-0.655769\pi$
$359$	−17.8129	−0.940127	−0.470064	+	0.882633i	$0.655769\pi$
$360$	1.48470	0.0782504
$361$	−18.8952	−0.994483
$362$	−8.74842	−0.459807
$363$	10.9872	0.576676
$364$	3.17363	0.166343
$365$	−12.8190	−0.670978
$366$	12.7465	0.666272
$367$	−22.0977	−1.15349	−0.576746	−	0.816924i	$-0.695678\pi$
$367$	−22.0977	−1.15349	−0.576746	+	0.816924i	$0.695678\pi$
$368$	3.28852	0.171426
$369$	10.1399	0.527865
$370$	13.7048	0.712479
$371$	34.8736	1.81055
$372$	−5.22103	−0.270698
$373$	−18.0937	−0.936854	−0.468427	−	0.883502i	$-0.655179\pi$
$373$	−18.0937	−0.936854	−0.468427	+	0.883502i	$0.655179\pi$
$374$	−0.440550	−0.0227803
$375$	11.5742	0.597690
$376$	4.85813	0.250539
$377$	2.92223	0.150502
$378$	3.17363	0.163234
$379$	31.2440	1.60490	0.802449	−	0.596721i	$-0.203530\pi$
$379$	31.2440	1.60490	0.802449	+	0.596721i	$0.203530\pi$
$380$	−0.480711	−0.0246600
$381$	−3.22861	−0.165407
$382$	−24.0969	−1.23291
$383$	36.2653	1.85307	0.926536	−	0.376207i	$-0.122772\pi$
$383$	36.2653	1.85307	0.926536	+	0.376207i	$0.122772\pi$
$384$	−1.00000	−0.0510310
$385$	−0.534110	−0.0272208
$386$	25.3188	1.28869
$387$	2.13959	0.108761
$388$	2.69171	0.136651
$389$	9.58000	0.485725	0.242863	−	0.970061i	$-0.421914\pi$
$389$	9.58000	0.485725	0.242863	+	0.970061i	$0.421914\pi$
$390$	1.48470	0.0751806
$391$	−12.7808	−0.646355
$392$	3.07190	0.155155
$393$	7.29169	0.367817
$394$	−7.34948	−0.370261
$395$	−2.94521	−0.148189
$396$	0.113354	0.00569625
$397$	−2.56093	−0.128530	−0.0642648	−	0.997933i	$-0.520470\pi$
$397$	−2.56093	−0.128530	−0.0642648	+	0.997933i	$0.520470\pi$
$398$	16.4690	0.825514
$399$	−1.02755	−0.0514417
$400$	−2.79567	−0.139784
$401$	−9.41791	−0.470308	−0.235154	−	0.971958i	$-0.575559\pi$
$401$	−9.41791	−0.470308	−0.235154	+	0.971958i	$0.575559\pi$
$402$	12.7525	0.636036
$403$	−5.22103	−0.260078
$404$	−17.5503	−0.873161
$405$	1.48470	0.0737752
$406$	9.27406	0.460264
$407$	1.04634	0.0518650
$408$	3.88650	0.192410
$409$	−9.09171	−0.449556	−0.224778	−	0.974410i	$-0.572166\pi$
$409$	−9.09171	−0.449556	−0.224778	+	0.974410i	$0.572166\pi$
$410$	15.0548	0.743501
$411$	−10.0244	−0.494467
$412$	1.00000	0.0492665
$413$	9.88816	0.486565
$414$	3.28852	0.161622
$415$	−14.1565	−0.694914
$416$	−1.00000	−0.0490290
$417$	6.96503	0.341079
$418$	−0.0367015	−0.00179513
$419$	20.2381	0.988695	0.494347	−	0.869264i	$-0.335407\pi$
$419$	20.2381	0.988695	0.494347	+	0.869264i	$0.335407\pi$
$420$	4.71187	0.229916
$421$	17.8977	0.872282	0.436141	−	0.899878i	$-0.356345\pi$
$421$	17.8977	0.872282	0.436141	+	0.899878i	$0.356345\pi$
$422$	−6.81556	−0.331776
$423$	4.85813	0.236211
$424$	−10.9886	−0.533651
$425$	10.8654	0.527048
$426$	−3.57724	−0.173318
$427$	40.4527	1.95765
$428$	−15.4959	−0.749023
$429$	0.113354	0.00547278
$430$	3.17664	0.153191
$431$	−23.3925	−1.12678	−0.563388	−	0.826193i	$-0.690502\pi$
$431$	−23.3925	−1.12678	−0.563388	+	0.826193i	$0.690502\pi$
$432$	−1.00000	−0.0481125
$433$	1.86560	0.0896551	0.0448276	−	0.998995i	$-0.485726\pi$
$433$	1.86560	0.0896551	0.0448276	+	0.998995i	$0.485726\pi$
$434$	−16.5696	−0.795367
$435$	4.33862	0.208021
$436$	−6.28842	−0.301161
$437$	−1.06475	−0.0509339
$438$	8.63410	0.412553
$439$	−29.4105	−1.40369	−0.701844	−	0.712331i	$-0.747639\pi$
$439$	−29.4105	−1.40369	−0.701844	+	0.712331i	$0.747639\pi$
$440$	0.168296	0.00802322
$441$	3.07190	0.146281
$442$	3.88650	0.184862
$443$	−21.4595	−1.01957	−0.509786	−	0.860301i	$-0.670275\pi$
$443$	−21.4595	−1.01957	−0.509786	+	0.860301i	$0.670275\pi$
$444$	−9.23070	−0.438070
$445$	12.3278	0.584395
$446$	8.92654	0.422684
$447$	12.2744	0.580559
$448$	−3.17363	−0.149940
$449$	−38.2227	−1.80384	−0.901921	−	0.431900i	$-0.857843\pi$
$449$	−38.2227	−1.80384	−0.901921	+	0.431900i	$0.857843\pi$
$450$	−2.79567	−0.131789
$451$	1.14940	0.0541233
$452$	−3.33461	−0.156847
$453$	2.61066	0.122659
$454$	22.1880	1.04134
$455$	4.71187	0.220896
$456$	0.323777	0.0151623
$457$	−20.7933	−0.972671	−0.486335	−	0.873772i	$-0.661667\pi$
$457$	−20.7933	−0.972671	−0.486335	+	0.873772i	$0.661667\pi$
$458$	−14.2480	−0.665764
$459$	3.88650	0.181406
$460$	4.88246	0.227646
$461$	−33.1573	−1.54429	−0.772145	−	0.635446i	$-0.780816\pi$
$461$	−33.1573	−1.54429	−0.772145	+	0.635446i	$0.780816\pi$
$462$	0.359743	0.0167368
$463$	4.25650	0.197817	0.0989083	−	0.995097i	$-0.468465\pi$
$463$	4.25650	0.197817	0.0989083	+	0.995097i	$0.468465\pi$
$464$	−2.92223	−0.135661
$465$	−7.75165	−0.359474
$466$	1.71660	0.0795202
$467$	−39.2985	−1.81852	−0.909259	−	0.416231i	$-0.863351\pi$
$467$	−39.2985	−1.81852	−0.909259	+	0.416231i	$0.863351\pi$
$468$	−1.00000	−0.0462250
$469$	40.4716	1.86881
$470$	7.21286	0.332704
$471$	−9.09881	−0.419251
$472$	−3.11573	−0.143413
$473$	0.242531	0.0111516
$474$	1.98371	0.0911147
$475$	0.905176	0.0415323
$476$	12.3343	0.565341
$477$	−10.9886	−0.503131
$478$	−12.2896	−0.562113
$479$	11.3113	0.516826	0.258413	−	0.966035i	$-0.416801\pi$
$479$	11.3113	0.516826	0.258413	+	0.966035i	$0.416801\pi$
$480$	−1.48470	−0.0677669
$481$	−9.23070	−0.420884
$482$	−11.1522	−0.507971
$483$	10.4365	0.474879
$484$	−10.9872	−0.499416
$485$	3.99637	0.181466
$486$	−1.00000	−0.0453609
$487$	−6.07314	−0.275200	−0.137600	−	0.990488i	$-0.543939\pi$
$487$	−6.07314	−0.275200	−0.137600	+	0.990488i	$0.543939\pi$
$488$	−12.7465	−0.577009
$489$	−9.51850	−0.430442
$490$	4.56085	0.206038
$491$	0.00695527	0.000313887 0	0.000156944	−	1.00000i	$-0.499950\pi$
$491$	0.00695527	0.000313887 0	0.000156944		1.00000i	$0.499950\pi$
$492$	−10.1399	−0.457144
$493$	11.3572	0.511504
$494$	0.323777	0.0145674
$495$	0.168296	0.00756436
$496$	5.22103	0.234431
$497$	−11.3528	−0.509243
$498$	9.53492	0.427270
$499$	12.9523	0.579826	0.289913	−	0.957053i	$-0.406374\pi$
$499$	12.9523	0.579826	0.289913	+	0.957053i	$0.406374\pi$
$500$	−11.5742	−0.517615
$501$	−10.7498	−0.480264
$502$	−21.4365	−0.956759
$503$	9.27561	0.413579	0.206790	−	0.978385i	$-0.433698\pi$
$503$	9.27561	0.413579	0.206790	+	0.978385i	$0.433698\pi$
$504$	−3.17363	−0.141365
$505$	−26.0569	−1.15952
$506$	0.372767	0.0165715
$507$	−1.00000	−0.0444116
$508$	3.22861	0.143247
$509$	3.46025	0.153373	0.0766864	−	0.997055i	$-0.475566\pi$
$509$	3.46025	0.153373	0.0766864	+	0.997055i	$0.475566\pi$
$510$	5.77027	0.255512
$511$	27.4014	1.21217
$512$	1.00000	0.0441942
$513$	0.323777	0.0142951
$514$	−2.50352	−0.110425
$515$	1.48470	0.0654236
$516$	−2.13959	−0.0941900
$517$	0.550689	0.0242193
$518$	−29.2948	−1.28714
$519$	12.4141	0.544920
$520$	−1.48470	−0.0651083
$521$	35.2695	1.54518	0.772592	−	0.634902i	$-0.218960\pi$
$521$	35.2695	1.54518	0.772592	+	0.634902i	$0.218960\pi$
$522$	−2.92223	−0.127902
$523$	17.0169	0.744097	0.372049	−	0.928213i	$-0.378655\pi$
$523$	17.0169	0.744097	0.372049	+	0.928213i	$0.378655\pi$
$524$	−7.29169	−0.318539
$525$	−8.87242	−0.387224
$526$	10.3482	0.451201
$527$	−20.2915	−0.883913
$528$	−0.113354	−0.00493310
$529$	−12.1856	−0.529809
$530$	−16.3147	−0.708664
$531$	−3.11573	−0.135211
$532$	1.02755	0.0445499
$533$	−10.1399	−0.439210
$534$	−8.30325	−0.359317
$535$	−23.0067	−0.994667
$536$	−12.7525	−0.550823
$537$	8.17919	0.352958
$538$	−12.0134	−0.517933
$539$	0.348213	0.0149986
$540$	−1.48470	−0.0638912
$541$	−42.1025	−1.81013	−0.905064	−	0.425274i	$-0.860178\pi$
$541$	−42.1025	−1.81013	−0.905064	+	0.425274i	$0.860178\pi$
$542$	−17.2845	−0.742433
$543$	8.74842	0.375430
$544$	−3.88650	−0.166632
$545$	−9.33640	−0.399928
$546$	−3.17363	−0.135819
$547$	−1.29684	−0.0554488	−0.0277244	−	0.999616i	$-0.508826\pi$
$547$	−1.29684	−0.0554488	−0.0277244	+	0.999616i	$0.508826\pi$
$548$	10.0244	0.428221
$549$	−12.7465	−0.544009
$550$	−0.316901	−0.0135127
$551$	0.946151	0.0403074
$552$	−3.28852	−0.139969
$553$	6.29555	0.267714
$554$	10.5495	0.448205
$555$	−13.7048	−0.581736
$556$	−6.96503	−0.295383
$557$	39.5327	1.67505	0.837527	−	0.546395i	$-0.184000\pi$
$557$	39.5327	1.67505	0.837527	+	0.546395i	$0.184000\pi$
$558$	5.22103	0.221024
$559$	−2.13959	−0.0904948
$560$	−4.71187	−0.199113
$561$	0.440550	0.0186000
$562$	−15.1321	−0.638311
$563$	28.7103	1.20999	0.604997	−	0.796228i	$-0.293174\pi$
$563$	28.7103	1.20999	0.604997	+	0.796228i	$0.293174\pi$
$564$	−4.85813	−0.204564
$565$	−4.95089	−0.208285
$566$	14.4417	0.607031
$567$	−3.17363	−0.133280
$568$	3.57724	0.150097
$569$	−5.86136	−0.245721	−0.122861	−	0.992424i	$-0.539207\pi$
$569$	−5.86136	−0.245721	−0.122861	+	0.992424i	$0.539207\pi$
$570$	0.480711	0.0201348
$571$	17.8186	0.745687	0.372843	−	0.927894i	$-0.378383\pi$
$571$	17.8186	0.745687	0.372843	+	0.927894i	$0.378383\pi$
$572$	−0.113354	−0.00473957
$573$	24.0969	1.00666
$574$	−32.1804	−1.34318
$575$	−9.19364	−0.383401
$576$	1.00000	0.0416667
$577$	−27.6035	−1.14915	−0.574574	−	0.818452i	$-0.694832\pi$
$577$	−27.6035	−1.14915	−0.574574	+	0.818452i	$0.694832\pi$
$578$	−1.89515	−0.0788279
$579$	−25.3188	−1.05221
$580$	−4.33862	−0.180151
$581$	30.2603	1.25541
$582$	−2.69171	−0.111575
$583$	−1.24560	−0.0515874
$584$	−8.63410	−0.357281
$585$	−1.48470	−0.0613847
$586$	10.2510	0.423463
$587$	−23.9578	−0.988846	−0.494423	−	0.869221i	$-0.664621\pi$
$587$	−23.9578	−0.988846	−0.494423	+	0.869221i	$0.664621\pi$
$588$	−3.07190	−0.126683
$589$	−1.69045	−0.0696538
$590$	−4.62591	−0.190446
$591$	7.34948	0.302317
$592$	9.23070	0.379380
$593$	24.1332	0.991032	0.495516	−	0.868599i	$-0.334979\pi$
$593$	24.1332	0.991032	0.495516	+	0.868599i	$0.334979\pi$
$594$	−0.113354	−0.00465097
$595$	18.3127	0.750747
$596$	−12.2744	−0.502779
$597$	−16.4690	−0.674030
$598$	−3.28852	−0.134478
$599$	35.8315	1.46403	0.732017	−	0.681286i	$-0.238579\pi$
$599$	35.8315	1.46403	0.732017	+	0.681286i	$0.238579\pi$
$600$	2.79567	0.114133
$601$	45.2491	1.84575	0.922874	−	0.385101i	$-0.125833\pi$
$601$	45.2491	1.84575	0.922874	+	0.385101i	$0.125833\pi$
$602$	−6.79024	−0.276750
$603$	−12.7525	−0.519321
$604$	−2.61066	−0.106226
$605$	−16.3126	−0.663201
$606$	17.5503	0.712933
$607$	5.20732	0.211359	0.105679	−	0.994400i	$-0.466298\pi$
$607$	5.20732	0.211359	0.105679	+	0.994400i	$0.466298\pi$
$608$	−0.323777	−0.0131309
$609$	−9.27406	−0.375804
$610$	−18.9247	−0.766241
$611$	−4.85813	−0.196539
$612$	−3.88650	−0.157102
$613$	−15.3307	−0.619201	−0.309600	−	0.950867i	$-0.600195\pi$
$613$	−15.3307	−0.619201	−0.309600	+	0.950867i	$0.600195\pi$
$614$	5.42553	0.218957
$615$	−15.0548	−0.607066
$616$	−0.359743	−0.0144945
$617$	7.13143	0.287101	0.143550	−	0.989643i	$-0.454148\pi$
$617$	7.13143	0.287101	0.143550	+	0.989643i	$0.454148\pi$
$618$	−1.00000	−0.0402259
$619$	15.0757	0.605944	0.302972	−	0.952999i	$-0.402021\pi$
$619$	15.0757	0.605944	0.302972	+	0.952999i	$0.402021\pi$
$620$	7.75165	0.311314
$621$	−3.28852	−0.131964
$622$	−5.69268	−0.228256
$623$	−26.3514	−1.05575
$624$	1.00000	0.0400320
$625$	−3.20584	−0.128234
$626$	−14.9124	−0.596019
$627$	0.0367015	0.00146572
$628$	9.09881	0.363082
$629$	−35.8751	−1.43043
$630$	−4.71187	−0.187726
$631$	6.68593	0.266163	0.133081	−	0.991105i	$-0.457513\pi$
$631$	6.68593	0.266163	0.133081	+	0.991105i	$0.457513\pi$
$632$	−1.98371	−0.0789077
$633$	6.81556	0.270894
$634$	−2.52576	−0.100311
$635$	4.79352	0.190225
$636$	10.9886	0.435725
$637$	−3.07190	−0.121713
$638$	−0.331246	−0.0131142
$639$	3.57724	0.141513
$640$	1.48470	0.0586878
$641$	14.9993	0.592439	0.296219	−	0.955120i	$-0.404274\pi$
$641$	14.9993	0.592439	0.296219	+	0.955120i	$0.404274\pi$
$642$	15.4959	0.611574
$643$	−10.1578	−0.400585	−0.200293	−	0.979736i	$-0.564189\pi$
$643$	−10.1578	−0.400585	−0.200293	+	0.979736i	$0.564189\pi$
$644$	−10.4365	−0.411258
$645$	−3.17664	−0.125080
$646$	1.25836	0.0495095
$647$	29.7550	1.16979	0.584895	−	0.811109i	$-0.301136\pi$
$647$	29.7550	1.16979	0.584895	+	0.811109i	$0.301136\pi$
$648$	1.00000	0.0392837
$649$	−0.353180	−0.0138635
$650$	2.79567	0.109655
$651$	16.5696	0.649414
$652$	9.51850	0.372773
$653$	−33.3661	−1.30572	−0.652859	−	0.757479i	$-0.726431\pi$
$653$	−33.3661	−1.30572	−0.652859	+	0.757479i	$0.726431\pi$
$654$	6.28842	0.245897
$655$	−10.8260	−0.423005
$656$	10.1399	0.395898
$657$	−8.63410	−0.336848
$658$	−15.4179	−0.601053
$659$	25.1238	0.978685	0.489343	−	0.872092i	$-0.337237\pi$
$659$	25.1238	0.978685	0.489343	+	0.872092i	$0.337237\pi$
$660$	−0.168296	−0.00655093
$661$	15.1693	0.590019	0.295010	−	0.955494i	$-0.404677\pi$
$661$	15.1693	0.590019	0.295010	+	0.955494i	$0.404677\pi$
$662$	19.0758	0.741401
$663$	−3.88650	−0.150939
$664$	−9.53492	−0.370027
$665$	1.52560	0.0591601
$666$	9.23070	0.357682
$667$	−9.60981	−0.372093
$668$	10.7498	0.415921
$669$	−8.92654	−0.345120
$670$	−18.9336	−0.731468
$671$	−1.44487	−0.0557786
$672$	3.17363	0.122425
$673$	−21.5866	−0.832103	−0.416051	−	0.909341i	$-0.636586\pi$
$673$	−21.5866	−0.832103	−0.416051	+	0.909341i	$0.636586\pi$
$674$	2.83472	0.109189
$675$	2.79567	0.107606
$676$	1.00000	0.0384615
$677$	35.9497	1.38166	0.690829	−	0.723018i	$-0.257246\pi$
$677$	35.9497	1.38166	0.690829	+	0.723018i	$0.257246\pi$
$678$	3.33461	0.128065
$679$	−8.54247	−0.327830
$680$	−5.77027	−0.221280
$681$	−22.1880	−0.850247
$682$	0.591825	0.0226622
$683$	15.5058	0.593312	0.296656	−	0.954984i	$-0.404129\pi$
$683$	15.5058	0.593312	0.296656	+	0.954984i	$0.404129\pi$
$684$	−0.323777	−0.0123799
$685$	14.8832	0.568657
$686$	12.4663	0.475966
$687$	14.2480	0.543594
$688$	2.13959	0.0815709
$689$	10.9886	0.418631
$690$	−4.88246	−0.185872
$691$	39.6153	1.50704	0.753519	−	0.657426i	$-0.228355\pi$
$691$	39.6153	1.50704	0.753519	+	0.657426i	$0.228355\pi$
$692$	−12.4141	−0.471914
$693$	−0.359743	−0.0136655
$694$	15.5470	0.590155
$695$	−10.3410	−0.392255
$696$	2.92223	0.110767
$697$	−39.4089	−1.49272
$698$	−11.2061	−0.424159
$699$	−1.71660	−0.0649280
$700$	8.87242	0.335346
$701$	−23.6571	−0.893518	−0.446759	−	0.894654i	$-0.647422\pi$
$701$	−23.6571	−0.893518	−0.446759	+	0.894654i	$0.647422\pi$
$702$	1.00000	0.0377426
$703$	−2.98869	−0.112721
$704$	0.113354	0.00427219
$705$	−7.21286	−0.271652
$706$	6.15868	0.231785
$707$	55.6982	2.09475
$708$	3.11573	0.117096
$709$	−22.1187	−0.830686	−0.415343	−	0.909665i	$-0.636338\pi$
$709$	−22.1187	−0.830686	−0.415343	+	0.909665i	$0.636338\pi$
$710$	5.31111	0.199323
$711$	−1.98371	−0.0743949
$712$	8.30325	0.311178
$713$	17.1695	0.643002
$714$	−12.3343	−0.461599
$715$	−0.168296	−0.00629393
$716$	−8.17919	−0.305671
$717$	12.2896	0.458963
$718$	−17.8129	−0.664770
$719$	49.9751	1.86376	0.931878	−	0.362773i	$-0.118170\pi$
$719$	49.9751	1.86376	0.931878	+	0.362773i	$0.118170\pi$
$720$	1.48470	0.0553314
$721$	−3.17363	−0.118192
$722$	−18.8952	−0.703205
$723$	11.1522	0.414757
$724$	−8.74842	−0.325132
$725$	8.16959	0.303411
$726$	10.9872	0.407771
$727$	−27.1960	−1.00864	−0.504322	−	0.863516i	$-0.668257\pi$
$727$	−27.1960	−1.00864	−0.504322	+	0.863516i	$0.668257\pi$
$728$	3.17363	0.117622
$729$	1.00000	0.0370370
$730$	−12.8190	−0.474453
$731$	−8.31549	−0.307559
$732$	12.7465	0.471125
$733$	−43.9063	−1.62171	−0.810857	−	0.585244i	$-0.800999\pi$
$733$	−43.9063	−1.62171	−0.810857	+	0.585244i	$0.800999\pi$
$734$	−22.0977	−0.815641
$735$	−4.56085	−0.168229
$736$	3.28852	0.121217
$737$	−1.44555	−0.0532473
$738$	10.1399	0.373257
$739$	1.33036	0.0489382	0.0244691	−	0.999701i	$-0.492210\pi$
$739$	1.33036	0.0489382	0.0244691	+	0.999701i	$0.492210\pi$
$740$	13.7048	0.503798
$741$	−0.323777	−0.0118943
$742$	34.8736	1.28025
$743$	32.2519	1.18321	0.591604	−	0.806229i	$-0.298495\pi$
$743$	32.2519	1.18321	0.591604	+	0.806229i	$0.298495\pi$
$744$	−5.22103	−0.191412
$745$	−18.2238	−0.667667
$746$	−18.0937	−0.662456
$747$	−9.53492	−0.348865
$748$	−0.440550	−0.0161081
$749$	49.1782	1.79693
$750$	11.5742	0.422631
$751$	6.42436	0.234428	0.117214	−	0.993107i	$-0.462604\pi$
$751$	6.42436	0.234428	0.117214	+	0.993107i	$0.462604\pi$
$752$	4.85813	0.177158
$753$	21.4365	0.781191
$754$	2.92223	0.106421
$755$	−3.87603	−0.141063
$756$	3.17363	0.115424
$757$	−4.27640	−0.155428	−0.0777142	−	0.996976i	$-0.524762\pi$
$757$	−4.27640	−0.155428	−0.0777142	+	0.996976i	$0.524762\pi$
$758$	31.2440	1.13483
$759$	−0.372767	−0.0135306
$760$	−0.480711	−0.0174372
$761$	37.6857	1.36611	0.683053	−	0.730368i	$-0.260652\pi$
$761$	37.6857	1.36611	0.683053	+	0.730368i	$0.260652\pi$
$762$	−3.22861	−0.116960
$763$	19.9571	0.722496
$764$	−24.0969	−0.871796
$765$	−5.77027	−0.208625
$766$	36.2653	1.31032
$767$	3.11573	0.112502
$768$	−1.00000	−0.0360844
$769$	−45.8508	−1.65342	−0.826710	−	0.562628i	$-0.809790\pi$
$769$	−45.8508	−1.65342	−0.826710	+	0.562628i	$0.809790\pi$
$770$	−0.534110	−0.0192480
$771$	2.50352	0.0901620
$772$	25.3188	0.911244
$773$	−8.20684	−0.295180	−0.147590	−	0.989049i	$-0.547152\pi$
$773$	−8.20684	−0.295180	−0.147590	+	0.989049i	$0.547152\pi$
$774$	2.13959	0.0769058
$775$	−14.5963	−0.524315
$776$	2.69171	0.0966266
$777$	29.2948	1.05095
$778$	9.58000	0.343459
$779$	−3.28308	−0.117629
$780$	1.48470	0.0531607
$781$	0.405494	0.0145097
$782$	−12.7808	−0.457042
$783$	2.92223	0.104432
$784$	3.07190	0.109711
$785$	13.5090	0.482156
$786$	7.29169	0.260086
$787$	29.7752	1.06137	0.530686	−	0.847569i	$-0.321934\pi$
$787$	29.7752	1.06137	0.530686	+	0.847569i	$0.321934\pi$
$788$	−7.34948	−0.261814
$789$	−10.3482	−0.368404
$790$	−2.94521	−0.104786
$791$	10.5828	0.376281
$792$	0.113354	0.00402786
$793$	12.7465	0.452643
$794$	−2.56093	−0.0908842
$795$	16.3147	0.578622
$796$	16.4690	0.583727
$797$	20.9013	0.740363	0.370181	−	0.928959i	$-0.379296\pi$
$797$	20.9013	0.740363	0.370181	+	0.928959i	$0.379296\pi$
$798$	−1.02755	−0.0363748
$799$	−18.8811	−0.667966
$800$	−2.79567	−0.0988420
$801$	8.30325	0.293381
$802$	−9.41791	−0.332558
$803$	−0.978710	−0.0345379
$804$	12.7525	0.449745
$805$	−15.4951	−0.546131
$806$	−5.22103	−0.183903
$807$	12.0134	0.422890
$808$	−17.5503	−0.617418
$809$	33.6191	1.18198	0.590992	−	0.806678i	$-0.298737\pi$
$809$	33.6191	1.18198	0.590992	+	0.806678i	$0.298737\pi$
$810$	1.48470	0.0521669
$811$	−36.7271	−1.28966	−0.644832	−	0.764324i	$-0.723073\pi$
$811$	−36.7271	−1.28966	−0.644832	+	0.764324i	$0.723073\pi$
$812$	9.27406	0.325456
$813$	17.2845	0.606194
$814$	1.04634	0.0366741
$815$	14.1321	0.495026
$816$	3.88650	0.136055
$817$	−0.692749	−0.0242362
$818$	−9.09171	−0.317884
$819$	3.17363	0.110895
$820$	15.0548	0.525735
$821$	−20.6978	−0.722359	−0.361180	−	0.932496i	$-0.617626\pi$
$821$	−20.6978	−0.722359	−0.361180	+	0.932496i	$0.617626\pi$
$822$	−10.0244	−0.349641
$823$	−39.6584	−1.38240	−0.691202	−	0.722662i	$-0.742919\pi$
$823$	−39.6584	−1.38240	−0.691202	+	0.722662i	$0.742919\pi$
$824$	1.00000	0.0348367
$825$	0.316901	0.0110331
$826$	9.88816	0.344053
$827$	−9.11586	−0.316990	−0.158495	−	0.987360i	$-0.550664\pi$
$827$	−9.11586	−0.316990	−0.158495	+	0.987360i	$0.550664\pi$
$828$	3.28852	0.114284
$829$	7.44086	0.258432	0.129216	−	0.991616i	$-0.458754\pi$
$829$	7.44086	0.258432	0.129216	+	0.991616i	$0.458754\pi$
$830$	−14.1565	−0.491378
$831$	−10.5495	−0.365958
$832$	−1.00000	−0.0346688
$833$	−11.9389	−0.413660
$834$	6.96503	0.241179
$835$	15.9601	0.552324
$836$	−0.0367015	−0.00126935
$837$	−5.22103	−0.180465
$838$	20.2381	0.699113
$839$	38.9381	1.34429	0.672146	−	0.740419i	$-0.265373\pi$
$839$	38.9381	1.34429	0.672146	+	0.740419i	$0.265373\pi$
$840$	4.71187	0.162575
$841$	−20.4606	−0.705538
$842$	17.8977	0.616797
$843$	15.1321	0.521178
$844$	−6.81556	−0.234601
$845$	1.48470	0.0510751
$846$	4.85813	0.167026
$847$	34.8691	1.19812
$848$	−10.9886	−0.377349
$849$	−14.4417	−0.495639
$850$	10.8654	0.372679
$851$	30.3554	1.04057
$852$	−3.57724	−0.122554
$853$	−26.5510	−0.909087	−0.454544	−	0.890724i	$-0.650198\pi$
$853$	−26.5510	−0.909087	−0.454544	+	0.890724i	$0.650198\pi$
$854$	40.4527	1.38426
$855$	−0.480711	−0.0164400
$856$	−15.4959	−0.529639
$857$	6.73667	0.230120	0.115060	−	0.993359i	$-0.463294\pi$
$857$	6.73667	0.230120	0.115060	+	0.993359i	$0.463294\pi$
$858$	0.113354	0.00386984
$859$	35.0877	1.19718	0.598589	−	0.801056i	$-0.295728\pi$
$859$	35.0877	1.19718	0.598589	+	0.801056i	$0.295728\pi$
$860$	3.17664	0.108322
$861$	32.1804	1.09671
$862$	−23.3925	−0.796750
$863$	40.5725	1.38110	0.690552	−	0.723283i	$-0.257368\pi$
$863$	40.5725	1.38110	0.690552	+	0.723283i	$0.257368\pi$
$864$	−1.00000	−0.0340207
$865$	−18.4312	−0.626680
$866$	1.86560	0.0633958
$867$	1.89515	0.0643627
$868$	−16.5696	−0.562409
$869$	−0.224861	−0.00762790
$870$	4.33862	0.147093
$871$	12.7525	0.432101
$872$	−6.28842	−0.212953
$873$	2.69171	0.0911005
$874$	−1.06475	−0.0360157
$875$	36.7322	1.24178
$876$	8.63410	0.291719
$877$	−24.7262	−0.834946	−0.417473	−	0.908689i	$-0.637084\pi$
$877$	−24.7262	−0.834946	−0.417473	+	0.908689i	$0.637084\pi$
$878$	−29.4105	−0.992557
$879$	−10.2510	−0.345756
$880$	0.168296	0.00567327
$881$	−40.7262	−1.37210	−0.686050	−	0.727555i	$-0.740657\pi$
$881$	−40.7262	−1.37210	−0.686050	+	0.727555i	$0.740657\pi$
$882$	3.07190	0.103436
$883$	4.16897	0.140297	0.0701486	−	0.997537i	$-0.477653\pi$
$883$	4.16897	0.140297	0.0701486	+	0.997537i	$0.477653\pi$
$884$	3.88650	0.130717
$885$	4.62591	0.155498
$886$	−21.4595	−0.720946
$887$	−24.7242	−0.830157	−0.415079	−	0.909786i	$-0.636246\pi$
$887$	−24.7242	−0.830157	−0.415079	+	0.909786i	$0.636246\pi$
$888$	−9.23070	−0.309762
$889$	−10.2464	−0.343654
$890$	12.3278	0.413229
$891$	0.113354	0.00379750
$892$	8.92654	0.298883
$893$	−1.57295	−0.0526369
$894$	12.2744	0.410517
$895$	−12.1436	−0.405917
$896$	−3.17363	−0.106023
$897$	3.28852	0.109801
$898$	−38.2227	−1.27551
$899$	−15.2570	−0.508851
$900$	−2.79567	−0.0931891
$901$	42.7070	1.42278
$902$	1.14940	0.0382710
$903$	6.79024	0.225965
$904$	−3.33461	−0.110908
$905$	−12.9887	−0.431761
$906$	2.61066	0.0867333
$907$	2.20119	0.0730895	0.0365447	−	0.999332i	$-0.488365\pi$
$907$	2.20119	0.0730895	0.0365447	+	0.999332i	$0.488365\pi$
$908$	22.1880	0.736336
$909$	−17.5503	−0.582108
$910$	4.71187	0.156197
$911$	−24.7417	−0.819729	−0.409864	−	0.912147i	$-0.634424\pi$
$911$	−24.7417	−0.819729	−0.409864	+	0.912147i	$0.634424\pi$
$912$	0.323777	0.0107213
$913$	−1.08082	−0.0357700
$914$	−20.7933	−0.687782
$915$	18.9247	0.625633
$916$	−14.2480	−0.470766
$917$	23.1411	0.764186
$918$	3.88650	0.128273
$919$	16.3944	0.540801	0.270400	−	0.962748i	$-0.412844\pi$
$919$	16.3944	0.540801	0.270400	+	0.962748i	$0.412844\pi$
$920$	4.88246	0.160970
$921$	−5.42553	−0.178777
$922$	−33.1573	−1.09198
$923$	−3.57724	−0.117746
$924$	0.359743	0.0118347
$925$	−25.8060	−0.848497
$926$	4.25650	0.139877
$927$	1.00000	0.0328443
$928$	−2.92223	−0.0959268
$929$	−6.98813	−0.229273	−0.114637	−	0.993407i	$-0.536570\pi$
$929$	−6.98813	−0.229273	−0.114637	+	0.993407i	$0.536570\pi$
$930$	−7.75165	−0.254187
$931$	−0.994613	−0.0325971
$932$	1.71660	0.0562293
$933$	5.69268	0.186370
$934$	−39.2985	−1.28589
$935$	−0.654083	−0.0213908
$936$	−1.00000	−0.0326860
$937$	24.9271	0.814333	0.407167	−	0.913354i	$-0.366517\pi$
$937$	24.9271	0.814333	0.407167	+	0.913354i	$0.366517\pi$
$938$	40.4716	1.32144
$939$	14.9124	0.486648
$940$	7.21286	0.235258
$941$	−43.2357	−1.40944	−0.704721	−	0.709484i	$-0.748928\pi$
$941$	−43.2357	−1.40944	−0.704721	+	0.709484i	$0.748928\pi$
$942$	−9.09881	−0.296455
$943$	33.3455	1.08588
$944$	−3.11573	−0.101408
$945$	4.71187	0.153277
$946$	0.242531	0.00788535
$947$	−23.3417	−0.758503	−0.379251	−	0.925294i	$-0.623818\pi$
$947$	−23.3417	−0.758503	−0.379251	+	0.925294i	$0.623818\pi$
$948$	1.98371	0.0644278
$949$	8.63410	0.280275
$950$	0.905176	0.0293678
$951$	2.52576	0.0819034
$952$	12.3343	0.399756
$953$	−22.0875	−0.715485	−0.357742	−	0.933820i	$-0.616453\pi$
$953$	−22.0875	−0.715485	−0.357742	+	0.933820i	$0.616453\pi$
$954$	−10.9886	−0.355768
$955$	−35.7766	−1.15770
$956$	−12.2896	−0.397474
$957$	0.331246	0.0107077
$958$	11.3113	0.365451
$959$	−31.8137	−1.02732
$960$	−1.48470	−0.0479184
$961$	−3.74082	−0.120672
$962$	−9.23070	−0.297610
$963$	−15.4959	−0.499348
$964$	−11.1522	−0.359190
$965$	37.5908	1.21009
$966$	10.4365	0.335790
$967$	−55.4979	−1.78469	−0.892347	−	0.451351i	$-0.850942\pi$
$967$	−55.4979	−1.78469	−0.892347	+	0.451351i	$0.850942\pi$
$968$	−10.9872	−0.353140
$969$	−1.25836	−0.0404243
$970$	3.99637	0.128316
$971$	19.2917	0.619100	0.309550	−	0.950883i	$-0.399822\pi$
$971$	19.2917	0.619100	0.309550	+	0.950883i	$0.399822\pi$
$972$	−1.00000	−0.0320750
$973$	22.1044	0.708634
$974$	−6.07314	−0.194596
$975$	−2.79567	−0.0895332
$976$	−12.7465	−0.408007
$977$	32.6966	1.04606	0.523029	−	0.852315i	$-0.324802\pi$
$977$	32.6966	1.04606	0.523029	+	0.852315i	$0.324802\pi$
$978$	−9.51850	−0.304368
$979$	0.941207	0.0300811
$980$	4.56085	0.145691
$981$	−6.28842	−0.200774
$982$	0.00695527	0.000221952 0
$983$	41.1113	1.31125	0.655624	−	0.755088i	$-0.272406\pi$
$983$	41.1113	1.31125	0.655624	+	0.755088i	$0.272406\pi$
$984$	−10.1399	−0.323250
$985$	−10.9118	−0.347677
$986$	11.3572	0.361688
$987$	15.4179	0.490757
$988$	0.323777	0.0103007
$989$	7.03608	0.223734
$990$	0.168296	0.00534881
$991$	−16.3776	−0.520252	−0.260126	−	0.965575i	$-0.583764\pi$
$991$	−16.3776	−0.520252	−0.260126	+	0.965575i	$0.583764\pi$
$992$	5.22103	0.165768
$993$	−19.0758	−0.605351
$994$	−11.3528	−0.360089
$995$	24.4514	0.775162
$996$	9.53492	0.302126
$997$	3.76003	0.119081	0.0595407	−	0.998226i	$-0.481036\pi$
$997$	3.76003	0.119081	0.0595407	+	0.998226i	$0.481036\pi$
$998$	12.9523	0.409999
$999$	−9.23070	−0.292047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.u.1.9	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.u.1.9	✓	11	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.48470 q^{5} -1.00000 q^{6} -3.17363 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.48470 q^{5} -1.00000 q^{6} -3.17363 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.48470 q^{10} +0.113354 q^{11} -1.00000 q^{12} -1.00000 q^{13} -3.17363 q^{14} -1.48470 q^{15} +1.00000 q^{16} -3.88650 q^{17} +1.00000 q^{18} -0.323777 q^{19} +1.48470 q^{20} +3.17363 q^{21} +0.113354 q^{22} +3.28852 q^{23} -1.00000 q^{24} -2.79567 q^{25} -1.00000 q^{26} -1.00000 q^{27} -3.17363 q^{28} -2.92223 q^{29} -1.48470 q^{30} +5.22103 q^{31} +1.00000 q^{32} -0.113354 q^{33} -3.88650 q^{34} -4.71187 q^{35} +1.00000 q^{36} +9.23070 q^{37} -0.323777 q^{38} +1.00000 q^{39} +1.48470 q^{40} +10.1399 q^{41} +3.17363 q^{42} +2.13959 q^{43} +0.113354 q^{44} +1.48470 q^{45} +3.28852 q^{46} +4.85813 q^{47} -1.00000 q^{48} +3.07190 q^{49} -2.79567 q^{50} +3.88650 q^{51} -1.00000 q^{52} -10.9886 q^{53} -1.00000 q^{54} +0.168296 q^{55} -3.17363 q^{56} +0.323777 q^{57} -2.92223 q^{58} -3.11573 q^{59} -1.48470 q^{60} -12.7465 q^{61} +5.22103 q^{62} -3.17363 q^{63} +1.00000 q^{64} -1.48470 q^{65} -0.113354 q^{66} -12.7525 q^{67} -3.88650 q^{68} -3.28852 q^{69} -4.71187 q^{70} +3.57724 q^{71} +1.00000 q^{72} -8.63410 q^{73} +9.23070 q^{74} +2.79567 q^{75} -0.323777 q^{76} -0.359743 q^{77} +1.00000 q^{78} -1.98371 q^{79} +1.48470 q^{80} +1.00000 q^{81} +10.1399 q^{82} -9.53492 q^{83} +3.17363 q^{84} -5.77027 q^{85} +2.13959 q^{86} +2.92223 q^{87} +0.113354 q^{88} +8.30325 q^{89} +1.48470 q^{90} +3.17363 q^{91} +3.28852 q^{92} -5.22103 q^{93} +4.85813 q^{94} -0.480711 q^{95} -1.00000 q^{96} +2.69171 q^{97} +3.07190 q^{98} +0.113354 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} - 2 q^{5} - 11 q^{6} - 2 q^{7} + 11 q^{8} + 11 q^{9}+O(q^{10}) \) 11 * q + 11 * q^2 - 11 * q^3 + 11 * q^4 - 2 * q^5 - 11 * q^6 - 2 * q^7 + 11 * q^8 + 11 * q^9	\( 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} - 2 q^{5} - 11 q^{6} - 2 q^{7} + 11 q^{8} + 11 q^{9} - 2 q^{10} - 10 q^{11} - 11 q^{12} - 11 q^{13} - 2 q^{14} + 2 q^{15} + 11 q^{16} + 6 q^{17} + 11 q^{18} - 3 q^{19} - 2 q^{20} + 2 q^{21} - 10 q^{22} + 3 q^{23} - 11 q^{24} - q^{25} - 11 q^{26} - 11 q^{27} - 2 q^{28} - 22 q^{29} + 2 q^{30} - 5 q^{31} + 11 q^{32} + 10 q^{33} + 6 q^{34} - 20 q^{35} + 11 q^{36} - 26 q^{37} - 3 q^{38} + 11 q^{39} - 2 q^{40} - 6 q^{41} + 2 q^{42} - 8 q^{43} - 10 q^{44} - 2 q^{45} + 3 q^{46} + 6 q^{47} - 11 q^{48} - 5 q^{49} - q^{50} - 6 q^{51} - 11 q^{52} - 25 q^{53} - 11 q^{54} - 2 q^{56} + 3 q^{57} - 22 q^{58} + 7 q^{59} + 2 q^{60} - 36 q^{61} - 5 q^{62} - 2 q^{63} + 11 q^{64} + 2 q^{65} + 10 q^{66} - 12 q^{67} + 6 q^{68} - 3 q^{69} - 20 q^{70} - 15 q^{71} + 11 q^{72} - 12 q^{73} - 26 q^{74} + q^{75} - 3 q^{76} - q^{77} + 11 q^{78} - 15 q^{79} - 2 q^{80} + 11 q^{81} - 6 q^{82} - 16 q^{83} + 2 q^{84} - 25 q^{85} - 8 q^{86} + 22 q^{87} - 10 q^{88} - 2 q^{89} - 2 q^{90} + 2 q^{91} + 3 q^{92} + 5 q^{93} + 6 q^{94} + 16 q^{95} - 11 q^{96} - 10 q^{97} - 5 q^{98} - 10 q^{99}+O(q^{100}) \) 11 * q + 11 * q^2 - 11 * q^3 + 11 * q^4 - 2 * q^5 - 11 * q^6 - 2 * q^7 + 11 * q^8 + 11 * q^9 - 2 * q^10 - 10 * q^11 - 11 * q^12 - 11 * q^13 - 2 * q^14 + 2 * q^15 + 11 * q^16 + 6 * q^17 + 11 * q^18 - 3 * q^19 - 2 * q^20 + 2 * q^21 - 10 * q^22 + 3 * q^23 - 11 * q^24 - q^25 - 11 * q^26 - 11 * q^27 - 2 * q^28 - 22 * q^29 + 2 * q^30 - 5 * q^31 + 11 * q^32 + 10 * q^33 + 6 * q^34 - 20 * q^35 + 11 * q^36 - 26 * q^37 - 3 * q^38 + 11 * q^39 - 2 * q^40 - 6 * q^41 + 2 * q^42 - 8 * q^43 - 10 * q^44 - 2 * q^45 + 3 * q^46 + 6 * q^47 - 11 * q^48 - 5 * q^49 - q^50 - 6 * q^51 - 11 * q^52 - 25 * q^53 - 11 * q^54 - 2 * q^56 + 3 * q^57 - 22 * q^58 + 7 * q^59 + 2 * q^60 - 36 * q^61 - 5 * q^62 - 2 * q^63 + 11 * q^64 + 2 * q^65 + 10 * q^66 - 12 * q^67 + 6 * q^68 - 3 * q^69 - 20 * q^70 - 15 * q^71 + 11 * q^72 - 12 * q^73 - 26 * q^74 + q^75 - 3 * q^76 - q^77 + 11 * q^78 - 15 * q^79 - 2 * q^80 + 11 * q^81 - 6 * q^82 - 16 * q^83 + 2 * q^84 - 25 * q^85 - 8 * q^86 + 22 * q^87 - 10 * q^88 - 2 * q^89 - 2 * q^90 + 2 * q^91 + 3 * q^92 + 5 * q^93 + 6 * q^94 + 16 * q^95 - 11 * q^96 - 10 * q^97 - 5 * q^98 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more