LMFDB - Embedded newform 6006.2.a.bw.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6006,2,Mod(1,6006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6006 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6006.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:	$47.9581514540$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.820249$ of defining polynomial
Character	$\chi$	$=$	6006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.23607 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} -1.23607 q^{15} +1.00000 q^{16} -1.00000 q^{18} -1.01388 q^{19} +1.23607 q^{20} -1.00000 q^{21} +1.00000 q^{22} -0.626615 q^{23} +1.00000 q^{24} -3.47214 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -5.01388 q^{29} +1.23607 q^{30} +5.89045 q^{31} -1.00000 q^{32} +1.00000 q^{33} +1.23607 q^{35} +1.00000 q^{36} -3.23607 q^{37} +1.01388 q^{38} -1.00000 q^{39} -1.23607 q^{40} +5.66827 q^{41} +1.00000 q^{42} +4.87657 q^{43} -1.00000 q^{44} +1.23607 q^{45} +0.626615 q^{46} -4.83164 q^{47} -1.00000 q^{48} +1.00000 q^{49} +3.47214 q^{50} +1.00000 q^{52} +1.09875 q^{53} +1.00000 q^{54} -1.23607 q^{55} -1.00000 q^{56} +1.01388 q^{57} +5.01388 q^{58} -11.1714 q^{59} -1.23607 q^{60} -14.2253 q^{61} -5.89045 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.23607 q^{65} -1.00000 q^{66} -0.432198 q^{67} +0.626615 q^{69} -1.23607 q^{70} -1.05382 q^{71} -1.00000 q^{72} -2.73925 q^{73} +3.23607 q^{74} +3.47214 q^{75} -1.01388 q^{76} -1.00000 q^{77} +1.00000 q^{78} -3.45825 q^{79} +1.23607 q^{80} +1.00000 q^{81} -5.66827 q^{82} -16.6125 q^{83} -1.00000 q^{84} -4.87657 q^{86} +5.01388 q^{87} +1.00000 q^{88} -5.01388 q^{89} -1.23607 q^{90} +1.00000 q^{91} -0.626615 q^{92} -5.89045 q^{93} +4.83164 q^{94} -1.25323 q^{95} +1.00000 q^{96} -2.33482 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} + 4 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 + 4 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} + 4 q^{7} - 4 q^{8} + 4 q^{9} + 4 q^{10} - 4 q^{11} - 4 q^{12} + 4 q^{13} - 4 q^{14} + 4 q^{15} + 4 q^{16} - 4 q^{18} + 4 q^{19} - 4 q^{20} - 4 q^{21} + 4 q^{22} - 2 q^{23} + 4 q^{24} + 4 q^{25} - 4 q^{26} - 4 q^{27} + 4 q^{28} - 12 q^{29} - 4 q^{30} - 2 q^{31} - 4 q^{32} + 4 q^{33} - 4 q^{35} + 4 q^{36} - 4 q^{37} - 4 q^{38} - 4 q^{39} + 4 q^{40} - 2 q^{41} + 4 q^{42} + 2 q^{43} - 4 q^{44} - 4 q^{45} + 2 q^{46} - 10 q^{47} - 4 q^{48} + 4 q^{49} - 4 q^{50} + 4 q^{52} - 14 q^{53} + 4 q^{54} + 4 q^{55} - 4 q^{56} - 4 q^{57} + 12 q^{58} - 2 q^{59} + 4 q^{60} - 4 q^{61} + 2 q^{62} + 4 q^{63} + 4 q^{64} - 4 q^{65} - 4 q^{66} + 14 q^{67} + 2 q^{69} + 4 q^{70} + 6 q^{71} - 4 q^{72} + 16 q^{73} + 4 q^{74} - 4 q^{75} + 4 q^{76} - 4 q^{77} + 4 q^{78} - 4 q^{79} - 4 q^{80} + 4 q^{81} + 2 q^{82} - 6 q^{83} - 4 q^{84} - 2 q^{86} + 12 q^{87} + 4 q^{88} - 12 q^{89} + 4 q^{90} + 4 q^{91} - 2 q^{92} + 2 q^{93} + 10 q^{94} - 4 q^{95} + 4 q^{96} + 18 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 + 4 * q^7 - 4 * q^8 + 4 * q^9 + 4 * q^10 - 4 * q^11 - 4 * q^12 + 4 * q^13 - 4 * q^14 + 4 * q^15 + 4 * q^16 - 4 * q^18 + 4 * q^19 - 4 * q^20 - 4 * q^21 + 4 * q^22 - 2 * q^23 + 4 * q^24 + 4 * q^25 - 4 * q^26 - 4 * q^27 + 4 * q^28 - 12 * q^29 - 4 * q^30 - 2 * q^31 - 4 * q^32 + 4 * q^33 - 4 * q^35 + 4 * q^36 - 4 * q^37 - 4 * q^38 - 4 * q^39 + 4 * q^40 - 2 * q^41 + 4 * q^42 + 2 * q^43 - 4 * q^44 - 4 * q^45 + 2 * q^46 - 10 * q^47 - 4 * q^48 + 4 * q^49 - 4 * q^50 + 4 * q^52 - 14 * q^53 + 4 * q^54 + 4 * q^55 - 4 * q^56 - 4 * q^57 + 12 * q^58 - 2 * q^59 + 4 * q^60 - 4 * q^61 + 2 * q^62 + 4 * q^63 + 4 * q^64 - 4 * q^65 - 4 * q^66 + 14 * q^67 + 2 * q^69 + 4 * q^70 + 6 * q^71 - 4 * q^72 + 16 * q^73 + 4 * q^74 - 4 * q^75 + 4 * q^76 - 4 * q^77 + 4 * q^78 - 4 * q^79 - 4 * q^80 + 4 * q^81 + 2 * q^82 - 6 * q^83 - 4 * q^84 - 2 * q^86 + 12 * q^87 + 4 * q^88 - 12 * q^89 + 4 * q^90 + 4 * q^91 - 2 * q^92 + 2 * q^93 + 10 * q^94 - 4 * q^95 + 4 * q^96 + 18 * q^97 - 4 * q^98 - 4 * 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
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.23607	−0.390879
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.00000	−0.267261
$15$	−1.23607	−0.319151
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	−1.01388	−0.232601	−0.116300	−	0.993214i	$-0.537104\pi$
$19$	−1.01388	−0.232601	−0.116300	+	0.993214i	$0.537104\pi$
$20$	1.23607	0.276393
$21$	−1.00000	−0.218218
$22$	1.00000	0.213201
$23$	−0.626615	−0.130658	−0.0653291	−	0.997864i	$-0.520810\pi$
$23$	−0.626615	−0.130658	−0.0653291	+	0.997864i	$0.520810\pi$
$24$	1.00000	0.204124
$25$	−3.47214	−0.694427
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−5.01388	−0.931055	−0.465527	−	0.885033i	$-0.654135\pi$
$29$	−5.01388	−0.931055	−0.465527	+	0.885033i	$0.654135\pi$
$30$	1.23607	0.225674
$31$	5.89045	1.05796	0.528978	−	0.848636i	$-0.322575\pi$
$31$	5.89045	1.05796	0.528978	+	0.848636i	$0.322575\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	0	0
$35$	1.23607	0.208934
$36$	1.00000	0.166667
$37$	−3.23607	−0.532006	−0.266003	−	0.963972i	$-0.585703\pi$
$37$	−3.23607	−0.532006	−0.266003	+	0.963972i	$0.585703\pi$
$38$	1.01388	0.164474
$39$	−1.00000	−0.160128
$40$	−1.23607	−0.195440
$41$	5.66827	0.885234	0.442617	−	0.896711i	$-0.354050\pi$
$41$	5.66827	0.885234	0.442617	+	0.896711i	$0.354050\pi$
$42$	1.00000	0.154303
$43$	4.87657	0.743669	0.371835	−	0.928299i	$-0.378729\pi$
$43$	4.87657	0.743669	0.371835	+	0.928299i	$0.378729\pi$
$44$	−1.00000	−0.150756
$45$	1.23607	0.184262
$46$	0.626615	0.0923893
$47$	−4.83164	−0.704767	−0.352383	−	0.935856i	$-0.614629\pi$
$47$	−4.83164	−0.704767	−0.352383	+	0.935856i	$0.614629\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	3.47214	0.491034
$51$	0	0
$52$	1.00000	0.138675
$53$	1.09875	0.150925	0.0754625	−	0.997149i	$-0.475957\pi$
$53$	1.09875	0.150925	0.0754625	+	0.997149i	$0.475957\pi$
$54$	1.00000	0.136083
$55$	−1.23607	−0.166671
$56$	−1.00000	−0.133631
$57$	1.01388	0.134292
$58$	5.01388	0.658355
$59$	−11.1714	−1.45440	−0.727199	−	0.686426i	$-0.759178\pi$
$59$	−11.1714	−1.45440	−0.727199	+	0.686426i	$0.759178\pi$
$60$	−1.23607	−0.159576
$61$	−14.2253	−1.82136	−0.910680	−	0.413114i	$-0.864441\pi$
$61$	−14.2253	−1.82136	−0.910680	+	0.413114i	$0.864441\pi$
$62$	−5.89045	−0.748088
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	1.23607	0.153315
$66$	−1.00000	−0.123091
$67$	−0.432198	−0.0528014	−0.0264007	−	0.999651i	$-0.508405\pi$
$67$	−0.432198	−0.0528014	−0.0264007	+	0.999651i	$0.508405\pi$
$68$	0	0
$69$	0.626615	0.0754355
$70$	−1.23607	−0.147738
$71$	−1.05382	−0.125066	−0.0625328	−	0.998043i	$-0.519918\pi$
$71$	−1.05382	−0.125066	−0.0625328	+	0.998043i	$0.519918\pi$
$72$	−1.00000	−0.117851
$73$	−2.73925	−0.320605	−0.160302	−	0.987068i	$-0.551247\pi$
$73$	−2.73925	−0.320605	−0.160302	+	0.987068i	$0.551247\pi$
$74$	3.23607	0.376185
$75$	3.47214	0.400928
$76$	−1.01388	−0.116300
$77$	−1.00000	−0.113961
$78$	1.00000	0.113228
$79$	−3.45825	−0.389084	−0.194542	−	0.980894i	$-0.562322\pi$
$79$	−3.45825	−0.389084	−0.194542	+	0.980894i	$0.562322\pi$
$80$	1.23607	0.138197
$81$	1.00000	0.111111
$82$	−5.66827	−0.625955
$83$	−16.6125	−1.82346	−0.911731	−	0.410787i	$-0.865254\pi$
$83$	−16.6125	−1.82346	−0.911731	+	0.410787i	$0.865254\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−4.87657	−0.525854
$87$	5.01388	0.537545
$88$	1.00000	0.106600
$89$	−5.01388	−0.531471	−0.265735	−	0.964046i	$-0.585615\pi$
$89$	−5.01388	−0.531471	−0.265735	+	0.964046i	$0.585615\pi$
$90$	−1.23607	−0.130293
$91$	1.00000	0.104828
$92$	−0.626615	−0.0653291
$93$	−5.89045	−0.610811
$94$	4.83164	0.498345
$95$	−1.25323	−0.128579
$96$	1.00000	0.102062
$97$	−2.33482	−0.237065	−0.118532	−	0.992950i	$-0.537819\pi$
$97$	−2.33482	−0.237065	−0.118532	+	0.992950i	$0.537819\pi$
$98$	−1.00000	−0.101015
$99$	−1.00000	−0.100504
$100$	−3.47214	−0.347214
$101$	6.15756	0.612700	0.306350	−	0.951919i	$-0.400892\pi$
$101$	6.15756	0.612700	0.306350	+	0.951919i	$0.400892\pi$
$102$	0	0
$103$	14.6746	1.44593	0.722967	−	0.690883i	$-0.242778\pi$
$103$	14.6746	1.44593	0.722967	+	0.690883i	$0.242778\pi$
$104$	−1.00000	−0.0980581
$105$	−1.23607	−0.120628
$106$	−1.09875	−0.106720
$107$	8.40752	0.812785	0.406393	−	0.913699i	$-0.366787\pi$
$107$	8.40752	0.812785	0.406393	+	0.913699i	$0.366787\pi$
$108$	−1.00000	−0.0962250
$109$	6.49990	0.622578	0.311289	−	0.950315i	$-0.399239\pi$
$109$	6.49990	0.622578	0.311289	+	0.950315i	$0.399239\pi$
$110$	1.23607	0.117854
$111$	3.23607	0.307154
$112$	1.00000	0.0944911
$113$	−15.8087	−1.48715	−0.743577	−	0.668650i	$-0.766872\pi$
$113$	−15.8087	−1.48715	−0.743577	+	0.668650i	$0.766872\pi$
$114$	−1.01388	−0.0949589
$115$	−0.774538	−0.0722261
$116$	−5.01388	−0.465527
$117$	1.00000	0.0924500
$118$	11.1714	1.02842
$119$	0	0
$120$	1.23607	0.112837
$121$	1.00000	0.0909091
$122$	14.2253	1.28790
$123$	−5.66827	−0.511090
$124$	5.89045	0.528978
$125$	−10.4721	−0.936656
$126$	−1.00000	−0.0890871
$127$	8.32265	0.738516	0.369258	−	0.929327i	$-0.379612\pi$
$127$	8.32265	0.738516	0.369258	+	0.929327i	$0.379612\pi$
$128$	−1.00000	−0.0883883
$129$	−4.87657	−0.429358
$130$	−1.23607	−0.108410
$131$	−10.1576	−0.887470	−0.443735	−	0.896158i	$-0.646347\pi$
$131$	−10.1576	−0.887470	−0.443735	+	0.896158i	$0.646347\pi$
$132$	1.00000	0.0870388
$133$	−1.01388	−0.0879149
$134$	0.432198	0.0373362
$135$	−1.23607	−0.106384
$136$	0	0
$137$	19.4663	1.66312	0.831560	−	0.555434i	$-0.187448\pi$
$137$	19.4663	1.66312	0.831560	+	0.555434i	$0.187448\pi$
$138$	−0.626615	−0.0533410
$139$	12.6544	1.07333	0.536665	−	0.843795i	$-0.319684\pi$
$139$	12.6544	1.07333	0.536665	+	0.843795i	$0.319684\pi$
$140$	1.23607	0.104467
$141$	4.83164	0.406897
$142$	1.05382	0.0884347
$143$	−1.00000	−0.0836242
$144$	1.00000	0.0833333
$145$	−6.19750	−0.514674
$146$	2.73925	0.226702
$147$	−1.00000	−0.0824786
$148$	−3.23607	−0.266003
$149$	3.30876	0.271065	0.135532	−	0.990773i	$-0.456726\pi$
$149$	3.30876	0.271065	0.135532	+	0.990773i	$0.456726\pi$
$150$	−3.47214	−0.283499
$151$	−10.8366	−0.881872	−0.440936	−	0.897538i	$-0.645353\pi$
$151$	−10.8366	−0.881872	−0.440936	+	0.897538i	$0.645353\pi$
$152$	1.01388	0.0822368
$153$	0	0
$154$	1.00000	0.0805823
$155$	7.28100	0.584824
$156$	−1.00000	−0.0800641
$157$	−13.8380	−1.10439	−0.552196	−	0.833714i	$-0.686210\pi$
$157$	−13.8380	−1.10439	−0.552196	+	0.833714i	$0.686210\pi$
$158$	3.45825	0.275124
$159$	−1.09875	−0.0871366
$160$	−1.23607	−0.0977198
$161$	−0.626615	−0.0493842
$162$	−1.00000	−0.0785674
$163$	−11.4386	−0.895937	−0.447969	−	0.894049i	$-0.647852\pi$
$163$	−11.4386	−0.895937	−0.447969	+	0.894049i	$0.647852\pi$
$164$	5.66827	0.442617
$165$	1.23607	0.0962278
$166$	16.6125	1.28938
$167$	−2.91650	−0.225686	−0.112843	−	0.993613i	$-0.535996\pi$
$167$	−2.91650	−0.225686	−0.112843	+	0.993613i	$0.535996\pi$
$168$	1.00000	0.0771517
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.01388	−0.0775336
$172$	4.87657	0.371835
$173$	−13.2688	−1.00881	−0.504405	−	0.863467i	$-0.668288\pi$
$173$	−13.2688	−1.00881	−0.504405	+	0.863467i	$0.668288\pi$
$174$	−5.01388	−0.380102
$175$	−3.47214	−0.262469
$176$	−1.00000	−0.0753778
$177$	11.1714	0.839697
$178$	5.01388	0.375806
$179$	5.66327	0.423293	0.211647	−	0.977346i	$-0.432117\pi$
$179$	5.66327	0.423293	0.211647	+	0.977346i	$0.432117\pi$
$180$	1.23607	0.0921311
$181$	−24.3657	−1.81109	−0.905543	−	0.424254i	$-0.860536\pi$
$181$	−24.3657	−1.81109	−0.905543	+	0.424254i	$0.860536\pi$
$182$	−1.00000	−0.0741249
$183$	14.2253	1.05156
$184$	0.626615	0.0461946
$185$	−4.00000	−0.294086
$186$	5.89045	0.431909
$187$	0	0
$188$	−4.83164	−0.352383
$189$	−1.00000	−0.0727393
$190$	1.25323	0.0909188
$191$	−6.12016	−0.442839	−0.221419	−	0.975179i	$-0.571069\pi$
$191$	−6.12016	−0.442839	−0.221419	+	0.975179i	$0.571069\pi$
$192$	−1.00000	−0.0721688
$193$	3.07098	0.221054	0.110527	−	0.993873i	$-0.464746\pi$
$193$	3.07098	0.221054	0.110527	+	0.993873i	$0.464746\pi$
$194$	2.33482	0.167630
$195$	−1.23607	−0.0885167
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	1.00000	0.0710669
$199$	14.8594	1.05336	0.526678	−	0.850065i	$-0.323437\pi$
$199$	14.8594	1.05336	0.526678	+	0.850065i	$0.323437\pi$
$200$	3.47214	0.245517
$201$	0.432198	0.0304849
$202$	−6.15756	−0.433245
$203$	−5.01388	−0.351906
$204$	0	0
$205$	7.00636	0.489346
$206$	−14.6746	−1.02243
$207$	−0.626615	−0.0435527
$208$	1.00000	0.0693375
$209$	1.01388	0.0701318
$210$	1.23607	0.0852968
$211$	10.5398	0.725593	0.362796	−	0.931868i	$-0.381822\pi$
$211$	10.5398	0.725593	0.362796	+	0.931868i	$0.381822\pi$
$212$	1.09875	0.0754625
$213$	1.05382	0.0722067
$214$	−8.40752	−0.574726
$215$	6.02777	0.411090
$216$	1.00000	0.0680414
$217$	5.89045	0.399870
$218$	−6.49990	−0.440229
$219$	2.73925	0.185101
$220$	−1.23607	−0.0833357
$221$	0	0
$222$	−3.23607	−0.217191
$223$	−11.6436	−0.779712	−0.389856	−	0.920876i	$-0.627475\pi$
$223$	−11.6436	−0.779712	−0.389856	+	0.920876i	$0.627475\pi$
$224$	−1.00000	−0.0668153
$225$	−3.47214	−0.231476
$226$	15.8087	1.05158
$227$	−22.3657	−1.48446	−0.742231	−	0.670144i	$-0.766232\pi$
$227$	−22.3657	−1.48446	−0.742231	+	0.670144i	$0.766232\pi$
$228$	1.01388	0.0671461
$229$	5.71129	0.377413	0.188706	−	0.982034i	$-0.439571\pi$
$229$	5.71129	0.377413	0.188706	+	0.982034i	$0.439571\pi$
$230$	0.774538	0.0510715
$231$	1.00000	0.0657952
$232$	5.01388	0.329178
$233$	−24.8119	−1.62548	−0.812742	−	0.582623i	$-0.802026\pi$
$233$	−24.8119	−1.62548	−0.812742	+	0.582623i	$0.802026\pi$
$234$	−1.00000	−0.0653720
$235$	−5.97223	−0.389585
$236$	−11.1714	−0.727199
$237$	3.45825	0.224638
$238$	0	0
$239$	−0.808861	−0.0523209	−0.0261604	−	0.999658i	$-0.508328\pi$
$239$	−0.808861	−0.0523209	−0.0261604	+	0.999658i	$0.508328\pi$
$240$	−1.23607	−0.0797878
$241$	5.79498	0.373287	0.186644	−	0.982428i	$-0.440239\pi$
$241$	5.79498	0.373287	0.186644	+	0.982428i	$0.440239\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−14.2253	−0.910680
$245$	1.23607	0.0789695
$246$	5.66827	0.361395
$247$	−1.01388	−0.0645119
$248$	−5.89045	−0.374044
$249$	16.6125	1.05278
$250$	10.4721	0.662316
$251$	2.19750	0.138705	0.0693525	−	0.997592i	$-0.477907\pi$
$251$	2.19750	0.138705	0.0693525	+	0.997592i	$0.477907\pi$
$252$	1.00000	0.0629941
$253$	0.626615	0.0393949
$254$	−8.32265	−0.522209
$255$	0	0
$256$	1.00000	0.0625000
$257$	25.1973	1.57176	0.785882	−	0.618376i	$-0.212209\pi$
$257$	25.1973	1.57176	0.785882	+	0.618376i	$0.212209\pi$
$258$	4.87657	0.303602
$259$	−3.23607	−0.201079
$260$	1.23607	0.0766577
$261$	−5.01388	−0.310352
$262$	10.1576	0.627536
$263$	1.05382	0.0649814	0.0324907	−	0.999472i	$-0.489656\pi$
$263$	1.05382	0.0649814	0.0324907	+	0.999472i	$0.489656\pi$
$264$	−1.00000	−0.0615457
$265$	1.35813	0.0834293
$266$	1.01388	0.0621652
$267$	5.01388	0.306845
$268$	−0.432198	−0.0264007
$269$	−18.3748	−1.12033	−0.560164	−	0.828382i	$-0.689262\pi$
$269$	−18.3748	−1.12033	−0.560164	+	0.828382i	$0.689262\pi$
$270$	1.23607	0.0752247
$271$	24.7252	1.50195	0.750974	−	0.660332i	$-0.229585\pi$
$271$	24.7252	1.50195	0.750974	+	0.660332i	$0.229585\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	−19.4663	−1.17600
$275$	3.47214	0.209378
$276$	0.626615	0.0377178
$277$	1.61582	0.0970850	0.0485425	−	0.998821i	$-0.484542\pi$
$277$	1.61582	0.0970850	0.0485425	+	0.998821i	$0.484542\pi$
$278$	−12.6544	−0.758959
$279$	5.89045	0.352652
$280$	−1.23607	−0.0738692
$281$	−27.5340	−1.64254	−0.821271	−	0.570538i	$-0.806735\pi$
$281$	−27.5340	−1.64254	−0.821271	+	0.570538i	$0.806735\pi$
$282$	−4.83164	−0.287720
$283$	20.6050	1.22484	0.612421	−	0.790532i	$-0.290196\pi$
$283$	20.6050	1.22484	0.612421	+	0.790532i	$0.290196\pi$
$284$	−1.05382	−0.0625328
$285$	1.25323	0.0742349
$286$	1.00000	0.0591312
$287$	5.66827	0.334587
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	6.19750	0.363930
$291$	2.33482	0.136869
$292$	−2.73925	−0.160302
$293$	22.1454	1.29375	0.646874	−	0.762597i	$-0.276076\pi$
$293$	22.1454	1.29375	0.646874	+	0.762597i	$0.276076\pi$
$294$	1.00000	0.0583212
$295$	−13.8087	−0.803972
$296$	3.23607	0.188093
$297$	1.00000	0.0580259
$298$	−3.30876	−0.191672
$299$	−0.626615	−0.0362381
$300$	3.47214	0.200464
$301$	4.87657	0.281081
$302$	10.8366	0.623578
$303$	−6.15756	−0.353743
$304$	−1.01388	−0.0581502
$305$	−17.5834	−1.00682
$306$	0	0
$307$	28.6556	1.63546	0.817730	−	0.575603i	$-0.195232\pi$
$307$	28.6556	1.63546	0.817730	+	0.575603i	$0.195232\pi$
$308$	−1.00000	−0.0569803
$309$	−14.6746	−0.834810
$310$	−7.28100	−0.413533
$311$	4.02025	0.227967	0.113984	−	0.993483i	$-0.463639\pi$
$311$	4.02025	0.227967	0.113984	+	0.993483i	$0.463639\pi$
$312$	1.00000	0.0566139
$313$	17.5063	0.989513	0.494756	−	0.869032i	$-0.335257\pi$
$313$	17.5063	0.989513	0.494756	+	0.869032i	$0.335257\pi$
$314$	13.8380	0.780924
$315$	1.23607	0.0696445
$316$	−3.45825	−0.194542
$317$	−18.2671	−1.02598	−0.512992	−	0.858394i	$-0.671463\pi$
$317$	−18.2671	−1.02598	−0.512992	+	0.858394i	$0.671463\pi$
$318$	1.09875	0.0616149
$319$	5.01388	0.280724
$320$	1.23607	0.0690983
$321$	−8.40752	−0.469262
$322$	0.626615	0.0349199
$323$	0	0
$324$	1.00000	0.0555556
$325$	−3.47214	−0.192599
$326$	11.4386	0.633523
$327$	−6.49990	−0.359445
$328$	−5.66827	−0.312978
$329$	−4.83164	−0.266377
$330$	−1.23607	−0.0680433
$331$	−16.2374	−0.892490	−0.446245	−	0.894911i	$-0.647239\pi$
$331$	−16.2374	−0.892490	−0.446245	+	0.894911i	$0.647239\pi$
$332$	−16.6125	−0.911731
$333$	−3.23607	−0.177335
$334$	2.91650	0.159584
$335$	−0.534226	−0.0291879
$336$	−1.00000	−0.0545545
$337$	−12.6175	−0.687321	−0.343660	−	0.939094i	$-0.611667\pi$
$337$	−12.6175	−0.687321	−0.343660	+	0.939094i	$0.611667\pi$
$338$	−1.00000	−0.0543928
$339$	15.8087	0.858609
$340$	0	0
$341$	−5.89045	−0.318986
$342$	1.01388	0.0548246
$343$	1.00000	0.0539949
$344$	−4.87657	−0.262927
$345$	0.774538	0.0416997
$346$	13.2688	0.713337
$347$	31.4351	1.68752	0.843762	−	0.536717i	$-0.180336\pi$
$347$	31.4351	1.68752	0.843762	+	0.536717i	$0.180336\pi$
$348$	5.01388	0.268772
$349$	9.78090	0.523560	0.261780	−	0.965128i	$-0.415691\pi$
$349$	9.78090	0.523560	0.261780	+	0.965128i	$0.415691\pi$
$350$	3.47214	0.185593
$351$	−1.00000	−0.0533761
$352$	1.00000	0.0533002
$353$	6.90624	0.367582	0.183791	−	0.982965i	$-0.441163\pi$
$353$	6.90624	0.367582	0.183791	+	0.982965i	$0.441163\pi$
$354$	−11.1714	−0.593756
$355$	−1.30260	−0.0691346
$356$	−5.01388	−0.265735
$357$	0	0
$358$	−5.66327	−0.299313
$359$	−7.11126	−0.375318	−0.187659	−	0.982234i	$-0.560090\pi$
$359$	−7.11126	−0.375318	−0.187659	+	0.982234i	$0.560090\pi$
$360$	−1.23607	−0.0651465
$361$	−17.9720	−0.945897
$362$	24.3657	1.28063
$363$	−1.00000	−0.0524864
$364$	1.00000	0.0524142
$365$	−3.38590	−0.177226
$366$	−14.2253	−0.743567
$367$	0.0227764	0.00118892	0.000594459	−	1.00000i	$-0.499811\pi$
$367$	0.0227764	0.00118892	0.000594459		1.00000i	$0.499811\pi$
$368$	−0.626615	−0.0326645
$369$	5.66827	0.295078
$370$	4.00000	0.207950
$371$	1.09875	0.0570443
$372$	−5.89045	−0.305406
$373$	−23.9981	−1.24257	−0.621287	−	0.783583i	$-0.713390\pi$
$373$	−23.9981	−1.24257	−0.621287	+	0.783583i	$0.713390\pi$
$374$	0	0
$375$	10.4721	0.540779
$376$	4.83164	0.249173
$377$	−5.01388	−0.258228
$378$	1.00000	0.0514344
$379$	23.3487	1.19934	0.599671	−	0.800247i	$-0.295298\pi$
$379$	23.3487	1.19934	0.599671	+	0.800247i	$0.295298\pi$
$380$	−1.25323	−0.0642893
$381$	−8.32265	−0.426382
$382$	6.12016	0.313134
$383$	−15.3593	−0.784824	−0.392412	−	0.919790i	$-0.628359\pi$
$383$	−15.3593	−0.784824	−0.392412	+	0.919790i	$0.628359\pi$
$384$	1.00000	0.0510310
$385$	−1.23607	−0.0629959
$386$	−3.07098	−0.156309
$387$	4.87657	0.247890
$388$	−2.33482	−0.118532
$389$	−22.5772	−1.14471	−0.572356	−	0.820005i	$-0.693970\pi$
$389$	−22.5772	−1.14471	−0.572356	+	0.820005i	$0.693970\pi$
$390$	1.23607	0.0625907
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	10.1576	0.512381
$394$	−6.00000	−0.302276
$395$	−4.27463	−0.215080
$396$	−1.00000	−0.0502519
$397$	−7.06942	−0.354804	−0.177402	−	0.984138i	$-0.556769\pi$
$397$	−7.06942	−0.354804	−0.177402	+	0.984138i	$0.556769\pi$
$398$	−14.8594	−0.744835
$399$	1.01388	0.0507577
$400$	−3.47214	−0.173607
$401$	−24.3551	−1.21623	−0.608117	−	0.793847i	$-0.708075\pi$
$401$	−24.3551	−1.21623	−0.608117	+	0.793847i	$0.708075\pi$
$402$	−0.432198	−0.0215561
$403$	5.89045	0.293424
$404$	6.15756	0.306350
$405$	1.23607	0.0614207
$406$	5.01388	0.248835
$407$	3.23607	0.160406
$408$	0	0
$409$	3.68352	0.182138	0.0910692	−	0.995845i	$-0.470972\pi$
$409$	3.68352	0.182138	0.0910692	+	0.995845i	$0.470972\pi$
$410$	−7.00636	−0.346020
$411$	−19.4663	−0.960203
$412$	14.6746	0.722967
$413$	−11.1714	−0.549711
$414$	0.626615	0.0307964
$415$	−20.5342	−1.00799
$416$	−1.00000	−0.0490290
$417$	−12.6544	−0.619688
$418$	−1.01388	−0.0495907
$419$	−19.7809	−0.966360	−0.483180	−	0.875521i	$-0.660518\pi$
$419$	−19.7809	−0.966360	−0.483180	+	0.875521i	$0.660518\pi$
$420$	−1.23607	−0.0603139
$421$	10.7979	0.526256	0.263128	−	0.964761i	$-0.415246\pi$
$421$	10.7979	0.526256	0.263128	+	0.964761i	$0.415246\pi$
$422$	−10.5398	−0.513071
$423$	−4.83164	−0.234922
$424$	−1.09875	−0.0533600
$425$	0	0
$426$	−1.05382	−0.0510578
$427$	−14.2253	−0.688409
$428$	8.40752	0.406393
$429$	1.00000	0.0482805
$430$	−6.02777	−0.290685
$431$	4.86440	0.234310	0.117155	−	0.993114i	$-0.462623\pi$
$431$	4.86440	0.234310	0.117155	+	0.993114i	$0.462623\pi$
$432$	−1.00000	−0.0481125
$433$	0.746771	0.0358875	0.0179438	−	0.999839i	$-0.494288\pi$
$433$	0.746771	0.0358875	0.0179438	+	0.999839i	$0.494288\pi$
$434$	−5.89045	−0.282751
$435$	6.19750	0.297147
$436$	6.49990	0.311289
$437$	0.635314	0.0303912
$438$	−2.73925	−0.130886
$439$	−38.5005	−1.83753	−0.918763	−	0.394809i	$-0.870811\pi$
$439$	−38.5005	−1.83753	−0.918763	+	0.394809i	$0.870811\pi$
$440$	1.23607	0.0589272
$441$	1.00000	0.0476190
$442$	0	0
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	3.23607	0.153577
$445$	−6.19750	−0.293790
$446$	11.6436	0.551340
$447$	−3.30876	−0.156499
$448$	1.00000	0.0472456
$449$	−4.31457	−0.203617	−0.101809	−	0.994804i	$-0.532463\pi$
$449$	−4.31457	−0.203617	−0.101809	+	0.994804i	$0.532463\pi$
$450$	3.47214	0.163678
$451$	−5.66827	−0.266908
$452$	−15.8087	−0.743577
$453$	10.8366	0.509149
$454$	22.3657	1.04967
$455$	1.23607	0.0579478
$456$	−1.01388	−0.0474795
$457$	−25.6542	−1.20005	−0.600026	−	0.799980i	$-0.704843\pi$
$457$	−25.6542	−1.20005	−0.600026	+	0.799980i	$0.704843\pi$
$458$	−5.71129	−0.266871
$459$	0	0
$460$	−0.774538	−0.0361130
$461$	−36.3086	−1.69106	−0.845529	−	0.533929i	$-0.820715\pi$
$461$	−36.3086	−1.69106	−0.845529	+	0.533929i	$0.820715\pi$
$462$	−1.00000	−0.0465242
$463$	−2.02278	−0.0940064	−0.0470032	−	0.998895i	$-0.514967\pi$
$463$	−2.02278	−0.0940064	−0.0470032	+	0.998895i	$0.514967\pi$
$464$	−5.01388	−0.232764
$465$	−7.28100	−0.337648
$466$	24.8119	1.14939
$467$	−9.38864	−0.434454	−0.217227	−	0.976121i	$-0.569701\pi$
$467$	−9.38864	−0.434454	−0.217227	+	0.976121i	$0.569701\pi$
$468$	1.00000	0.0462250
$469$	−0.432198	−0.0199571
$470$	5.97223	0.275479
$471$	13.8380	0.637621
$472$	11.1714	0.514208
$473$	−4.87657	−0.224225
$474$	−3.45825	−0.158843
$475$	3.52034	0.161524
$476$	0	0
$477$	1.09875	0.0503083
$478$	0.808861	0.0369964
$479$	16.2808	0.743889	0.371944	−	0.928255i	$-0.378691\pi$
$479$	16.2808	0.743889	0.371944	+	0.928255i	$0.378691\pi$
$480$	1.23607	0.0564185
$481$	−3.23607	−0.147552
$482$	−5.79498	−0.263954
$483$	0.626615	0.0285120
$484$	1.00000	0.0454545
$485$	−2.88599	−0.131046
$486$	1.00000	0.0453609
$487$	42.8440	1.94145	0.970723	−	0.240202i	$-0.0772137\pi$
$487$	42.8440	1.94145	0.970723	+	0.240202i	$0.0772137\pi$
$488$	14.2253	0.643948
$489$	11.4386	0.517270
$490$	−1.23607	−0.0558399
$491$	−15.0988	−0.681397	−0.340699	−	0.940173i	$-0.610664\pi$
$491$	−15.0988	−0.681397	−0.340699	+	0.940173i	$0.610664\pi$
$492$	−5.66827	−0.255545
$493$	0	0
$494$	1.01388	0.0456168
$495$	−1.23607	−0.0555571
$496$	5.89045	0.264489
$497$	−1.05382	−0.0472704
$498$	−16.6125	−0.744426
$499$	−30.5954	−1.36964	−0.684819	−	0.728714i	$-0.740119\pi$
$499$	−30.5954	−1.36964	−0.684819	+	0.728714i	$0.740119\pi$
$500$	−10.4721	−0.468328
$501$	2.91650	0.130300
$502$	−2.19750	−0.0980793
$503$	−14.3923	−0.641719	−0.320860	−	0.947127i	$-0.603972\pi$
$503$	−14.3923	−0.641719	−0.320860	+	0.947127i	$0.603972\pi$
$504$	−1.00000	−0.0445435
$505$	7.61117	0.338692
$506$	−0.626615	−0.0278564
$507$	−1.00000	−0.0444116
$508$	8.32265	0.369258
$509$	−31.5690	−1.39927	−0.699635	−	0.714500i	$-0.746654\pi$
$509$	−31.5690	−1.39927	−0.699635	+	0.714500i	$0.746654\pi$
$510$	0	0
$511$	−2.73925	−0.121177
$512$	−1.00000	−0.0441942
$513$	1.01388	0.0447641
$514$	−25.1973	−1.11141
$515$	18.1388	0.799293
$516$	−4.87657	−0.214679
$517$	4.83164	0.212495
$518$	3.23607	0.142185
$519$	13.2688	0.582437
$520$	−1.23607	−0.0542052
$521$	6.81503	0.298572	0.149286	−	0.988794i	$-0.452303\pi$
$521$	6.81503	0.298572	0.149286	+	0.988794i	$0.452303\pi$
$522$	5.01388	0.219452
$523$	16.8797	0.738096	0.369048	−	0.929410i	$-0.379684\pi$
$523$	16.8797	0.738096	0.369048	+	0.929410i	$0.379684\pi$
$524$	−10.1576	−0.443735
$525$	3.47214	0.151536
$526$	−1.05382	−0.0459488
$527$	0	0
$528$	1.00000	0.0435194
$529$	−22.6074	−0.982928
$530$	−1.35813	−0.0589934
$531$	−11.1714	−0.484800
$532$	−1.01388	−0.0439574
$533$	5.66827	0.245520
$534$	−5.01388	−0.216972
$535$	10.3923	0.449297
$536$	0.432198	0.0186681
$537$	−5.66327	−0.244388
$538$	18.3748	0.792192
$539$	−1.00000	−0.0430730
$540$	−1.23607	−0.0531919
$541$	−25.1911	−1.08305	−0.541526	−	0.840684i	$-0.682153\pi$
$541$	−25.1911	−1.08305	−0.541526	+	0.840684i	$0.682153\pi$
$542$	−24.7252	−1.06204
$543$	24.3657	1.04563
$544$	0	0
$545$	8.03432	0.344153
$546$	1.00000	0.0427960
$547$	−28.9171	−1.23640	−0.618202	−	0.786019i	$-0.712139\pi$
$547$	−28.9171	−1.23640	−0.618202	+	0.786019i	$0.712139\pi$
$548$	19.4663	0.831560
$549$	−14.2253	−0.607120
$550$	−3.47214	−0.148052
$551$	5.08350	0.216564
$552$	−0.626615	−0.0266705
$553$	−3.45825	−0.147060
$554$	−1.61582	−0.0686494
$555$	4.00000	0.169791
$556$	12.6544	0.536665
$557$	−26.8706	−1.13854	−0.569271	−	0.822150i	$-0.692775\pi$
$557$	−26.8706	−1.13854	−0.569271	+	0.822150i	$0.692775\pi$
$558$	−5.89045	−0.249363
$559$	4.87657	0.206257
$560$	1.23607	0.0522334
$561$	0	0
$562$	27.5340	1.16145
$563$	1.89797	0.0799900	0.0399950	−	0.999200i	$-0.487266\pi$
$563$	1.89797	0.0799900	0.0399950	+	0.999200i	$0.487266\pi$
$564$	4.83164	0.203449
$565$	−19.5406	−0.822079
$566$	−20.6050	−0.866093
$567$	1.00000	0.0419961
$568$	1.05382	0.0442174
$569$	−20.2222	−0.847758	−0.423879	−	0.905719i	$-0.639332\pi$
$569$	−20.2222	−0.847758	−0.423879	+	0.905719i	$0.639332\pi$
$570$	−1.25323	−0.0524920
$571$	−11.2133	−0.469262	−0.234631	−	0.972085i	$-0.575388\pi$
$571$	−11.2133	−0.469262	−0.234631	+	0.972085i	$0.575388\pi$
$572$	−1.00000	−0.0418121
$573$	6.12016	0.255673
$574$	−5.66827	−0.236589
$575$	2.17569	0.0907326
$576$	1.00000	0.0416667
$577$	−34.0914	−1.41924	−0.709621	−	0.704583i	$-0.751134\pi$
$577$	−34.0914	−1.41924	−0.709621	+	0.704583i	$0.751134\pi$
$578$	17.0000	0.707107
$579$	−3.07098	−0.127626
$580$	−6.19750	−0.257337
$581$	−16.6125	−0.689204
$582$	−2.33482	−0.0967813
$583$	−1.09875	−0.0455056
$584$	2.73925	0.113351
$585$	1.23607	0.0511051
$586$	−22.1454	−0.914818
$587$	9.28290	0.383146	0.191573	−	0.981478i	$-0.438641\pi$
$587$	9.28290	0.383146	0.191573	+	0.981478i	$0.438641\pi$
$588$	−1.00000	−0.0412393
$589$	−5.97223	−0.246082
$590$	13.8087	0.568494
$591$	−6.00000	−0.246807
$592$	−3.23607	−0.133002
$593$	36.5111	1.49933	0.749665	−	0.661818i	$-0.230215\pi$
$593$	36.5111	1.49933	0.749665	+	0.661818i	$0.230215\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	3.30876	0.135532
$597$	−14.8594	−0.608155
$598$	0.626615	0.0256242
$599$	−33.3274	−1.36172	−0.680861	−	0.732412i	$-0.738394\pi$
$599$	−33.3274	−1.36172	−0.680861	+	0.732412i	$0.738394\pi$
$600$	−3.47214	−0.141749
$601$	26.8918	1.09694	0.548470	−	0.836170i	$-0.315210\pi$
$601$	26.8918	1.09694	0.548470	+	0.836170i	$0.315210\pi$
$602$	−4.87657	−0.198754
$603$	−0.432198	−0.0176005
$604$	−10.8366	−0.440936
$605$	1.23607	0.0502533
$606$	6.15756	0.250134
$607$	−4.65747	−0.189041	−0.0945204	−	0.995523i	$-0.530132\pi$
$607$	−4.65747	−0.189041	−0.0945204	+	0.995523i	$0.530132\pi$
$608$	1.01388	0.0411184
$609$	5.01388	0.203173
$610$	17.5834	0.711931
$611$	−4.83164	−0.195467
$612$	0	0
$613$	24.8328	1.00299	0.501494	−	0.865161i	$-0.332784\pi$
$613$	24.8328	1.00299	0.501494	+	0.865161i	$0.332784\pi$
$614$	−28.6556	−1.15644
$615$	−7.00636	−0.282524
$616$	1.00000	0.0402911
$617$	−32.4106	−1.30480	−0.652401	−	0.757874i	$-0.726238\pi$
$617$	−32.4106	−1.30480	−0.652401	+	0.757874i	$0.726238\pi$
$618$	14.6746	0.590300
$619$	−49.3225	−1.98244	−0.991218	−	0.132235i	$-0.957785\pi$
$619$	−49.3225	−1.98244	−0.991218	+	0.132235i	$0.957785\pi$
$620$	7.28100	0.292412
$621$	0.626615	0.0251452
$622$	−4.02025	−0.161197
$623$	−5.01388	−0.200877
$624$	−1.00000	−0.0400320
$625$	4.41641	0.176656
$626$	−17.5063	−0.699691
$627$	−1.01388	−0.0404906
$628$	−13.8380	−0.552196
$629$	0	0
$630$	−1.23607	−0.0492461
$631$	26.7823	1.06619	0.533093	−	0.846057i	$-0.321030\pi$
$631$	26.7823	1.06619	0.533093	+	0.846057i	$0.321030\pi$
$632$	3.45825	0.137562
$633$	−10.5398	−0.418921
$634$	18.2671	0.725480
$635$	10.2874	0.408241
$636$	−1.09875	−0.0435683
$637$	1.00000	0.0396214
$638$	−5.01388	−0.198502
$639$	−1.05382	−0.0416885
$640$	−1.23607	−0.0488599
$641$	−30.3051	−1.19698	−0.598491	−	0.801130i	$-0.704233\pi$
$641$	−30.3051	−1.19698	−0.598491	+	0.801130i	$0.704233\pi$
$642$	8.40752	0.331818
$643$	−21.3225	−0.840876	−0.420438	−	0.907321i	$-0.638123\pi$
$643$	−21.3225	−0.840876	−0.420438	+	0.907321i	$0.638123\pi$
$644$	−0.626615	−0.0246921
$645$	−6.02777	−0.237343
$646$	0	0
$647$	−41.7391	−1.64093	−0.820466	−	0.571696i	$-0.806286\pi$
$647$	−41.7391	−1.64093	−0.820466	+	0.571696i	$0.806286\pi$
$648$	−1.00000	−0.0392837
$649$	11.1714	0.438518
$650$	3.47214	0.136188
$651$	−5.89045	−0.230865
$652$	−11.4386	−0.447969
$653$	−24.0153	−0.939790	−0.469895	−	0.882722i	$-0.655708\pi$
$653$	−24.0153	−0.939790	−0.469895	+	0.882722i	$0.655708\pi$
$654$	6.49990	0.254166
$655$	−12.5554	−0.490582
$656$	5.66827	0.221309
$657$	−2.73925	−0.106868
$658$	4.83164	0.188357
$659$	5.04664	0.196589	0.0982946	−	0.995157i	$-0.468661\pi$
$659$	5.04664	0.196589	0.0982946	+	0.995157i	$0.468661\pi$
$660$	1.23607	0.0481139
$661$	−15.3290	−0.596229	−0.298115	−	0.954530i	$-0.596358\pi$
$661$	−15.3290	−0.596229	−0.298115	+	0.954530i	$0.596358\pi$
$662$	16.2374	0.631086
$663$	0	0
$664$	16.6125	0.644692
$665$	−1.25323	−0.0485981
$666$	3.23607	0.125395
$667$	3.14177	0.121650
$668$	−2.91650	−0.112843
$669$	11.6436	0.450167
$670$	0.534226	0.0206390
$671$	14.2253	0.549160
$672$	1.00000	0.0385758
$673$	−13.2255	−0.509804	−0.254902	−	0.966967i	$-0.582043\pi$
$673$	−13.2255	−0.509804	−0.254902	+	0.966967i	$0.582043\pi$
$674$	12.6175	0.486009
$675$	3.47214	0.133643
$676$	1.00000	0.0384615
$677$	−17.9385	−0.689431	−0.344716	−	0.938707i	$-0.612025\pi$
$677$	−17.9385	−0.689431	−0.344716	+	0.938707i	$0.612025\pi$
$678$	−15.8087	−0.607128
$679$	−2.33482	−0.0896021
$680$	0	0
$681$	22.3657	0.857054
$682$	5.89045	0.225557
$683$	23.1911	0.887384	0.443692	−	0.896179i	$-0.353668\pi$
$683$	23.1911	0.887384	0.443692	+	0.896179i	$0.353668\pi$
$684$	−1.01388	−0.0387668
$685$	24.0617	0.919350
$686$	−1.00000	−0.0381802
$687$	−5.71129	−0.217899
$688$	4.87657	0.185917
$689$	1.09875	0.0418591
$690$	−0.774538	−0.0294862
$691$	45.5721	1.73364	0.866822	−	0.498619i	$-0.166159\pi$
$691$	45.5721	1.73364	0.866822	+	0.498619i	$0.166159\pi$
$692$	−13.2688	−0.504405
$693$	−1.00000	−0.0379869
$694$	−31.4351	−1.19326
$695$	15.6417	0.593323
$696$	−5.01388	−0.190051
$697$	0	0
$698$	−9.78090	−0.370213
$699$	24.8119	0.938474
$700$	−3.47214	−0.131234
$701$	44.1375	1.66705	0.833525	−	0.552482i	$-0.186319\pi$
$701$	44.1375	1.66705	0.833525	+	0.552482i	$0.186319\pi$
$702$	1.00000	0.0377426
$703$	3.28100	0.123745
$704$	−1.00000	−0.0376889
$705$	5.97223	0.224927
$706$	−6.90624	−0.259920
$707$	6.15756	0.231579
$708$	11.1714	0.419849
$709$	24.1865	0.908343	0.454172	−	0.890914i	$-0.349935\pi$
$709$	24.1865	0.908343	0.454172	+	0.890914i	$0.349935\pi$
$710$	1.30260	0.0488855
$711$	−3.45825	−0.129695
$712$	5.01388	0.187903
$713$	−3.69104	−0.138231
$714$	0	0
$715$	−1.23607	−0.0462263
$716$	5.66327	0.211647
$717$	0.808861	0.0302075
$718$	7.11126	0.265390
$719$	−11.0760	−0.413064	−0.206532	−	0.978440i	$-0.566218\pi$
$719$	−11.0760	−0.413064	−0.206532	+	0.978440i	$0.566218\pi$
$720$	1.23607	0.0460655
$721$	14.6746	0.546512
$722$	17.9720	0.668850
$723$	−5.79498	−0.215518
$724$	−24.3657	−0.905543
$725$	17.4089	0.646550
$726$	1.00000	0.0371135
$727$	42.9554	1.59313	0.796564	−	0.604554i	$-0.206649\pi$
$727$	42.9554	1.59313	0.796564	+	0.604554i	$0.206649\pi$
$728$	−1.00000	−0.0370625
$729$	1.00000	0.0370370
$730$	3.38590	0.125318
$731$	0	0
$732$	14.2253	0.525781
$733$	20.1632	0.744744	0.372372	−	0.928084i	$-0.378545\pi$
$733$	20.1632	0.744744	0.372372	+	0.928084i	$0.378545\pi$
$734$	−0.0227764	−0.000840692 0
$735$	−1.23607	−0.0455931
$736$	0.626615	0.0230973
$737$	0.432198	0.0159202
$738$	−5.66827	−0.208652
$739$	−28.3238	−1.04191	−0.520954	−	0.853585i	$-0.674424\pi$
$739$	−28.3238	−1.04191	−0.520954	+	0.853585i	$0.674424\pi$
$740$	−4.00000	−0.147043
$741$	1.01388	0.0372460
$742$	−1.09875	−0.0403364
$743$	11.1356	0.408526	0.204263	−	0.978916i	$-0.434520\pi$
$743$	11.1356	0.408526	0.204263	+	0.978916i	$0.434520\pi$
$744$	5.89045	0.215954
$745$	4.08986	0.149841
$746$	23.9981	0.878633
$747$	−16.6125	−0.607821
$748$	0	0
$749$	8.40752	0.307204
$750$	−10.4721	−0.382388
$751$	−46.0339	−1.67980	−0.839901	−	0.542739i	$-0.817387\pi$
$751$	−46.0339	−1.67980	−0.839901	+	0.542739i	$0.817387\pi$
$752$	−4.83164	−0.176192
$753$	−2.19750	−0.0800814
$754$	5.01388	0.182595
$755$	−13.3948	−0.487487
$756$	−1.00000	−0.0363696
$757$	24.6731	0.896758	0.448379	−	0.893844i	$-0.352002\pi$
$757$	24.6731	0.896758	0.448379	+	0.893844i	$0.352002\pi$
$758$	−23.3487	−0.848063
$759$	−0.626615	−0.0227447
$760$	1.25323	0.0454594
$761$	−31.7921	−1.15246	−0.576231	−	0.817287i	$-0.695477\pi$
$761$	−31.7921	−1.15246	−0.576231	+	0.817287i	$0.695477\pi$
$762$	8.32265	0.301498
$763$	6.49990	0.235312
$764$	−6.12016	−0.221419
$765$	0	0
$766$	15.3593	0.554954
$767$	−11.1714	−0.403378
$768$	−1.00000	−0.0360844
$769$	31.5443	1.13752	0.568758	−	0.822505i	$-0.307424\pi$
$769$	31.5443	1.13752	0.568758	+	0.822505i	$0.307424\pi$
$770$	1.23607	0.0445448
$771$	−25.1973	−0.907459
$772$	3.07098	0.110527
$773$	−15.7919	−0.567995	−0.283997	−	0.958825i	$-0.591661\pi$
$773$	−15.7919	−0.567995	−0.283997	+	0.958825i	$0.591661\pi$
$774$	−4.87657	−0.175285
$775$	−20.4524	−0.734673
$776$	2.33482	0.0838151
$777$	3.23607	0.116093
$778$	22.5772	0.809434
$779$	−5.74696	−0.205906
$780$	−1.23607	−0.0442583
$781$	1.05382	0.0377087
$782$	0	0
$783$	5.01388	0.179182
$784$	1.00000	0.0357143
$785$	−17.1047	−0.610493
$786$	−10.1576	−0.362308
$787$	−20.1213	−0.717248	−0.358624	−	0.933482i	$-0.616754\pi$
$787$	−20.1213	−0.717248	−0.358624	+	0.933482i	$0.616754\pi$
$788$	6.00000	0.213741
$789$	−1.05382	−0.0375170
$790$	4.27463	0.152085
$791$	−15.8087	−0.562092
$792$	1.00000	0.0355335
$793$	−14.2253	−0.505154
$794$	7.06942	0.250884
$795$	−1.35813	−0.0481679
$796$	14.8594	0.526678
$797$	34.3909	1.21819	0.609094	−	0.793098i	$-0.291533\pi$
$797$	34.3909	1.21819	0.609094	+	0.793098i	$0.291533\pi$
$798$	−1.01388	−0.0358911
$799$	0	0
$800$	3.47214	0.122759
$801$	−5.01388	−0.177157
$802$	24.3551	0.860007
$803$	2.73925	0.0966660
$804$	0.432198	0.0152425
$805$	−0.774538	−0.0272989
$806$	−5.89045	−0.207482
$807$	18.3748	0.646822
$808$	−6.15756	−0.216622
$809$	1.05539	0.0371054	0.0185527	−	0.999828i	$-0.494094\pi$
$809$	1.05539	0.0371054	0.0185527	+	0.999828i	$0.494094\pi$
$810$	−1.23607	−0.0434310
$811$	14.1872	0.498181	0.249091	−	0.968480i	$-0.419868\pi$
$811$	14.1872	0.498181	0.249091	+	0.968480i	$0.419868\pi$
$812$	−5.01388	−0.175953
$813$	−24.7252	−0.867150
$814$	−3.23607	−0.113424
$815$	−14.1388	−0.495262
$816$	0	0
$817$	−4.94427	−0.172978
$818$	−3.68352	−0.128791
$819$	1.00000	0.0349428
$820$	7.00636	0.244673
$821$	27.5340	0.960944	0.480472	−	0.877010i	$-0.340465\pi$
$821$	27.5340	0.960944	0.480472	+	0.877010i	$0.340465\pi$
$822$	19.4663	0.678966
$823$	24.1416	0.841523	0.420761	−	0.907171i	$-0.361763\pi$
$823$	24.1416	0.841523	0.420761	+	0.907171i	$0.361763\pi$
$824$	−14.6746	−0.511215
$825$	−3.47214	−0.120884
$826$	11.1714	0.388704
$827$	−25.2251	−0.877162	−0.438581	−	0.898692i	$-0.644519\pi$
$827$	−25.2251	−0.877162	−0.438581	+	0.898692i	$0.644519\pi$
$828$	−0.626615	−0.0217764
$829$	−11.8696	−0.412248	−0.206124	−	0.978526i	$-0.566085\pi$
$829$	−11.8696	−0.412248	−0.206124	+	0.978526i	$0.566085\pi$
$830$	20.5342	0.712753
$831$	−1.61582	−0.0560520
$832$	1.00000	0.0346688
$833$	0	0
$834$	12.6544	0.438185
$835$	−3.60500	−0.124756
$836$	1.01388	0.0350659
$837$	−5.89045	−0.203604
$838$	19.7809	0.683320
$839$	−12.8378	−0.443210	−0.221605	−	0.975136i	$-0.571130\pi$
$839$	−12.8378	−0.443210	−0.221605	+	0.975136i	$0.571130\pi$
$840$	1.23607	0.0426484
$841$	−3.86097	−0.133137
$842$	−10.7979	−0.372119
$843$	27.5340	0.948322
$844$	10.5398	0.362796
$845$	1.23607	0.0425220
$846$	4.83164	0.166115
$847$	1.00000	0.0343604
$848$	1.09875	0.0377313
$849$	−20.6050	−0.707162
$850$	0	0
$851$	2.02777	0.0695110
$852$	1.05382	0.0361033
$853$	−53.6822	−1.83804	−0.919021	−	0.394208i	$-0.871019\pi$
$853$	−53.6822	−1.83804	−0.919021	+	0.394208i	$0.871019\pi$
$854$	14.2253	0.486779
$855$	−1.25323	−0.0428595
$856$	−8.40752	−0.287363
$857$	−40.0162	−1.36693	−0.683463	−	0.729985i	$-0.739527\pi$
$857$	−40.0162	−1.36693	−0.683463	+	0.729985i	$0.739527\pi$
$858$	−1.00000	−0.0341394
$859$	51.5680	1.75948	0.879738	−	0.475458i	$-0.157718\pi$
$859$	51.5680	1.75948	0.879738	+	0.475458i	$0.157718\pi$
$860$	6.02777	0.205545
$861$	−5.66827	−0.193174
$862$	−4.86440	−0.165682
$863$	−4.00465	−0.136320	−0.0681599	−	0.997674i	$-0.521713\pi$
$863$	−4.00465	−0.136320	−0.0681599	+	0.997674i	$0.521713\pi$
$864$	1.00000	0.0340207
$865$	−16.4012	−0.557657
$866$	−0.746771	−0.0253763
$867$	17.0000	0.577350
$868$	5.89045	0.199935
$869$	3.45825	0.117313
$870$	−6.19750	−0.210115
$871$	−0.432198	−0.0146445
$872$	−6.49990	−0.220115
$873$	−2.33482	−0.0790216
$874$	−0.635314	−0.0214898
$875$	−10.4721	−0.354023
$876$	2.73925	0.0925507
$877$	−5.39520	−0.182183	−0.0910914	−	0.995843i	$-0.529036\pi$
$877$	−5.39520	−0.182183	−0.0910914	+	0.995843i	$0.529036\pi$
$878$	38.5005	1.29933
$879$	−22.1454	−0.746946
$880$	−1.23607	−0.0416678
$881$	40.8428	1.37603	0.688014	−	0.725697i	$-0.258483\pi$
$881$	40.8428	1.37603	0.688014	+	0.725697i	$0.258483\pi$
$882$	−1.00000	−0.0336718
$883$	−18.5377	−0.623842	−0.311921	−	0.950108i	$-0.600972\pi$
$883$	−18.5377	−0.623842	−0.311921	+	0.950108i	$0.600972\pi$
$884$	0	0
$885$	13.8087	0.464173
$886$	16.0000	0.537531
$887$	28.1454	0.945030	0.472515	−	0.881323i	$-0.343346\pi$
$887$	28.1454	0.945030	0.472515	+	0.881323i	$0.343346\pi$
$888$	−3.23607	−0.108595
$889$	8.32265	0.279133
$890$	6.19750	0.207741
$891$	−1.00000	−0.0335013
$892$	−11.6436	−0.389856
$893$	4.89872	0.163929
$894$	3.30876	0.110662
$895$	7.00019	0.233991
$896$	−1.00000	−0.0334077
$897$	0.626615	0.0209221
$898$	4.31457	0.143979
$899$	−29.5340	−0.985015
$900$	−3.47214	−0.115738
$901$	0	0
$902$	5.66827	0.188733
$903$	−4.87657	−0.162282
$904$	15.8087	0.525788
$905$	−30.1176	−1.00114
$906$	−10.8366	−0.360023
$907$	1.19457	0.0396649	0.0198325	−	0.999803i	$-0.493687\pi$
$907$	1.19457	0.0396649	0.0198325	+	0.999803i	$0.493687\pi$
$908$	−22.3657	−0.742231
$909$	6.15756	0.204233
$910$	−1.23607	−0.0409753
$911$	−3.95316	−0.130974	−0.0654871	−	0.997853i	$-0.520860\pi$
$911$	−3.95316	−0.130974	−0.0654871	+	0.997853i	$0.520860\pi$
$912$	1.01388	0.0335730
$913$	16.6125	0.549795
$914$	25.6542	0.848565
$915$	17.5834	0.581289
$916$	5.71129	0.188706
$917$	−10.1576	−0.335432
$918$	0	0
$919$	−52.8531	−1.74346	−0.871731	−	0.489985i	$-0.837002\pi$
$919$	−52.8531	−1.74346	−0.871731	+	0.489985i	$0.837002\pi$
$920$	0.774538	0.0255358
$921$	−28.6556	−0.944233
$922$	36.3086	1.19576
$923$	−1.05382	−0.0346870
$924$	1.00000	0.0328976
$925$	11.2361	0.369440
$926$	2.02278	0.0664726
$927$	14.6746	0.481978
$928$	5.01388	0.164589
$929$	−32.9240	−1.08020	−0.540101	−	0.841600i	$-0.681614\pi$
$929$	−32.9240	−1.08020	−0.540101	+	0.841600i	$0.681614\pi$
$930$	7.28100	0.238753
$931$	−1.01388	−0.0332287
$932$	−24.8119	−0.812742
$933$	−4.02025	−0.131617
$934$	9.38864	0.307206
$935$	0	0
$936$	−1.00000	−0.0326860
$937$	19.0310	0.621717	0.310859	−	0.950456i	$-0.399383\pi$
$937$	19.0310	0.621717	0.310859	+	0.950456i	$0.399383\pi$
$938$	0.432198	0.0141118
$939$	−17.5063	−0.571295
$940$	−5.97223	−0.194793
$941$	44.2592	1.44281	0.721404	−	0.692514i	$-0.243497\pi$
$941$	44.2592	1.44281	0.721404	+	0.692514i	$0.243497\pi$
$942$	−13.8380	−0.450866
$943$	−3.55182	−0.115663
$944$	−11.1714	−0.363600
$945$	−1.23607	−0.0402093
$946$	4.87657	0.158551
$947$	29.2594	0.950803	0.475401	−	0.879769i	$-0.342303\pi$
$947$	29.2594	0.950803	0.475401	+	0.879769i	$0.342303\pi$
$948$	3.45825	0.112319
$949$	−2.73925	−0.0889198
$950$	−3.52034	−0.114215
$951$	18.2671	0.592352
$952$	0	0
$953$	52.0548	1.68622	0.843110	−	0.537741i	$-0.180722\pi$
$953$	52.0548	1.68622	0.843110	+	0.537741i	$0.180722\pi$
$954$	−1.09875	−0.0355734
$955$	−7.56493	−0.244795
$956$	−0.808861	−0.0261604
$957$	−5.01388	−0.162076
$958$	−16.2808	−0.526009
$959$	19.4663	0.628601
$960$	−1.23607	−0.0398939
$961$	3.69740	0.119271
$962$	3.23607	0.104335
$963$	8.40752	0.270928
$964$	5.79498	0.186644
$965$	3.79594	0.122196
$966$	−0.626615	−0.0201610
$967$	29.9441	0.962937	0.481468	−	0.876463i	$-0.340104\pi$
$967$	29.9441	0.962937	0.481468	+	0.876463i	$0.340104\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	2.88599	0.0926637
$971$	−34.7186	−1.11417	−0.557087	−	0.830454i	$-0.688081\pi$
$971$	−34.7186	−1.11417	−0.557087	+	0.830454i	$0.688081\pi$
$972$	−1.00000	−0.0320750
$973$	12.6544	0.405681
$974$	−42.8440	−1.37281
$975$	3.47214	0.111197
$976$	−14.2253	−0.455340
$977$	−13.0004	−0.415918	−0.207959	−	0.978137i	$-0.566682\pi$
$977$	−13.0004	−0.415918	−0.207959	+	0.978137i	$0.566682\pi$
$978$	−11.4386	−0.365765
$979$	5.01388	0.160244
$980$	1.23607	0.0394847
$981$	6.49990	0.207526
$982$	15.0988	0.481820
$983$	33.5973	1.07159	0.535794	−	0.844349i	$-0.320012\pi$
$983$	33.5973	1.07159	0.535794	+	0.844349i	$0.320012\pi$
$984$	5.66827	0.180698
$985$	7.41641	0.236306
$986$	0	0
$987$	4.83164	0.153793
$988$	−1.01388	−0.0322559
$989$	−3.05573	−0.0971665
$990$	1.23607	0.0392848
$991$	25.2189	0.801105	0.400552	−	0.916274i	$-0.368818\pi$
$991$	25.2189	0.801105	0.400552	+	0.916274i	$0.368818\pi$
$992$	−5.89045	−0.187022
$993$	16.2374	0.515280
$994$	1.05382	0.0334252
$995$	18.3672	0.582280
$996$	16.6125	0.526388
$997$	61.7971	1.95713	0.978566	−	0.205933i	$-0.0660228\pi$
$997$	61.7971	1.95713	0.978566	+	0.205933i	$0.0660228\pi$
$998$	30.5954	0.968480
$999$	3.23607	0.102385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6006.2.a.bw.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6006.2.a.bw.1.3	✓	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6006 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.23607 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{14} -1.23607 q^{15} +1.00000 q^{16} -1.00000 q^{18} -1.01388 q^{19} +1.23607 q^{20} -1.00000 q^{21} +1.00000 q^{22} -0.626615 q^{23} +1.00000 q^{24} -3.47214 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -5.01388 q^{29} +1.23607 q^{30} +5.89045 q^{31} -1.00000 q^{32} +1.00000 q^{33} +1.23607 q^{35} +1.00000 q^{36} -3.23607 q^{37} +1.01388 q^{38} -1.00000 q^{39} -1.23607 q^{40} +5.66827 q^{41} +1.00000 q^{42} +4.87657 q^{43} -1.00000 q^{44} +1.23607 q^{45} +0.626615 q^{46} -4.83164 q^{47} -1.00000 q^{48} +1.00000 q^{49} +3.47214 q^{50} +1.00000 q^{52} +1.09875 q^{53} +1.00000 q^{54} -1.23607 q^{55} -1.00000 q^{56} +1.01388 q^{57} +5.01388 q^{58} -11.1714 q^{59} -1.23607 q^{60} -14.2253 q^{61} -5.89045 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.23607 q^{65} -1.00000 q^{66} -0.432198 q^{67} +0.626615 q^{69} -1.23607 q^{70} -1.05382 q^{71} -1.00000 q^{72} -2.73925 q^{73} +3.23607 q^{74} +3.47214 q^{75} -1.01388 q^{76} -1.00000 q^{77} +1.00000 q^{78} -3.45825 q^{79} +1.23607 q^{80} +1.00000 q^{81} -5.66827 q^{82} -16.6125 q^{83} -1.00000 q^{84} -4.87657 q^{86} +5.01388 q^{87} +1.00000 q^{88} -5.01388 q^{89} -1.23607 q^{90} +1.00000 q^{91} -0.626615 q^{92} -5.89045 q^{93} +4.83164 q^{94} -1.25323 q^{95} +1.00000 q^{96} -2.33482 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} + 4 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 + 4 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} + 4 q^{7} - 4 q^{8} + 4 q^{9} + 4 q^{10} - 4 q^{11} - 4 q^{12} + 4 q^{13} - 4 q^{14} + 4 q^{15} + 4 q^{16} - 4 q^{18} + 4 q^{19} - 4 q^{20} - 4 q^{21} + 4 q^{22} - 2 q^{23} + 4 q^{24} + 4 q^{25} - 4 q^{26} - 4 q^{27} + 4 q^{28} - 12 q^{29} - 4 q^{30} - 2 q^{31} - 4 q^{32} + 4 q^{33} - 4 q^{35} + 4 q^{36} - 4 q^{37} - 4 q^{38} - 4 q^{39} + 4 q^{40} - 2 q^{41} + 4 q^{42} + 2 q^{43} - 4 q^{44} - 4 q^{45} + 2 q^{46} - 10 q^{47} - 4 q^{48} + 4 q^{49} - 4 q^{50} + 4 q^{52} - 14 q^{53} + 4 q^{54} + 4 q^{55} - 4 q^{56} - 4 q^{57} + 12 q^{58} - 2 q^{59} + 4 q^{60} - 4 q^{61} + 2 q^{62} + 4 q^{63} + 4 q^{64} - 4 q^{65} - 4 q^{66} + 14 q^{67} + 2 q^{69} + 4 q^{70} + 6 q^{71} - 4 q^{72} + 16 q^{73} + 4 q^{74} - 4 q^{75} + 4 q^{76} - 4 q^{77} + 4 q^{78} - 4 q^{79} - 4 q^{80} + 4 q^{81} + 2 q^{82} - 6 q^{83} - 4 q^{84} - 2 q^{86} + 12 q^{87} + 4 q^{88} - 12 q^{89} + 4 q^{90} + 4 q^{91} - 2 q^{92} + 2 q^{93} + 10 q^{94} - 4 q^{95} + 4 q^{96} + 18 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 + 4 * q^7 - 4 * q^8 + 4 * q^9 + 4 * q^10 - 4 * q^11 - 4 * q^12 + 4 * q^13 - 4 * q^14 + 4 * q^15 + 4 * q^16 - 4 * q^18 + 4 * q^19 - 4 * q^20 - 4 * q^21 + 4 * q^22 - 2 * q^23 + 4 * q^24 + 4 * q^25 - 4 * q^26 - 4 * q^27 + 4 * q^28 - 12 * q^29 - 4 * q^30 - 2 * q^31 - 4 * q^32 + 4 * q^33 - 4 * q^35 + 4 * q^36 - 4 * q^37 - 4 * q^38 - 4 * q^39 + 4 * q^40 - 2 * q^41 + 4 * q^42 + 2 * q^43 - 4 * q^44 - 4 * q^45 + 2 * q^46 - 10 * q^47 - 4 * q^48 + 4 * q^49 - 4 * q^50 + 4 * q^52 - 14 * q^53 + 4 * q^54 + 4 * q^55 - 4 * q^56 - 4 * q^57 + 12 * q^58 - 2 * q^59 + 4 * q^60 - 4 * q^61 + 2 * q^62 + 4 * q^63 + 4 * q^64 - 4 * q^65 - 4 * q^66 + 14 * q^67 + 2 * q^69 + 4 * q^70 + 6 * q^71 - 4 * q^72 + 16 * q^73 + 4 * q^74 - 4 * q^75 + 4 * q^76 - 4 * q^77 + 4 * q^78 - 4 * q^79 - 4 * q^80 + 4 * q^81 + 2 * q^82 - 6 * q^83 - 4 * q^84 - 2 * q^86 + 12 * q^87 + 4 * q^88 - 12 * q^89 + 4 * q^90 + 4 * q^91 - 2 * q^92 + 2 * q^93 + 10 * q^94 - 4 * q^95 + 4 * q^96 + 18 * q^97 - 4 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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