LMFDB - Embedded newform 6034.2.a.o.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6034,2,Mod(1,6034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6034, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6034 = 2 \cdot 7 \cdot 431 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6034.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:	$48.1817325796$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.42126 q^{3} +1.00000 q^{4} -0.924054 q^{5} -1.42126 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.980018 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.42126 q^{3} +1.00000 q^{4} -0.924054 q^{5} -1.42126 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.980018 q^{9} +0.924054 q^{10} -1.23484 q^{11} +1.42126 q^{12} +1.24428 q^{13} +1.00000 q^{14} -1.31332 q^{15} +1.00000 q^{16} +5.52037 q^{17} +0.980018 q^{18} -6.23981 q^{19} -0.924054 q^{20} -1.42126 q^{21} +1.23484 q^{22} +8.12384 q^{23} -1.42126 q^{24} -4.14613 q^{25} -1.24428 q^{26} -5.65664 q^{27} -1.00000 q^{28} +5.70695 q^{29} +1.31332 q^{30} -4.40527 q^{31} -1.00000 q^{32} -1.75503 q^{33} -5.52037 q^{34} +0.924054 q^{35} -0.980018 q^{36} +11.0766 q^{37} +6.23981 q^{38} +1.76844 q^{39} +0.924054 q^{40} -7.15545 q^{41} +1.42126 q^{42} +2.65760 q^{43} -1.23484 q^{44} +0.905589 q^{45} -8.12384 q^{46} -5.98316 q^{47} +1.42126 q^{48} +1.00000 q^{49} +4.14613 q^{50} +7.84588 q^{51} +1.24428 q^{52} -6.31884 q^{53} +5.65664 q^{54} +1.14106 q^{55} +1.00000 q^{56} -8.86840 q^{57} -5.70695 q^{58} -1.31252 q^{59} -1.31332 q^{60} +7.93517 q^{61} +4.40527 q^{62} +0.980018 q^{63} +1.00000 q^{64} -1.14978 q^{65} +1.75503 q^{66} -11.5559 q^{67} +5.52037 q^{68} +11.5461 q^{69} -0.924054 q^{70} +2.93087 q^{71} +0.980018 q^{72} -12.6375 q^{73} -11.0766 q^{74} -5.89272 q^{75} -6.23981 q^{76} +1.23484 q^{77} -1.76844 q^{78} +13.9685 q^{79} -0.924054 q^{80} -5.09951 q^{81} +7.15545 q^{82} -10.7419 q^{83} -1.42126 q^{84} -5.10112 q^{85} -2.65760 q^{86} +8.11107 q^{87} +1.23484 q^{88} -8.24297 q^{89} -0.905589 q^{90} -1.24428 q^{91} +8.12384 q^{92} -6.26103 q^{93} +5.98316 q^{94} +5.76592 q^{95} -1.42126 q^{96} +2.94418 q^{97} -1.00000 q^{98} +1.21016 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} + 4 q^{6} - 25 q^{7} - 25 q^{8} + 25 q^{9}+O(q^{10}) $ 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 + 4 * q^6 - 25 * q^7 - 25 * q^8 + 25 * q^9	\( 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} + 4 q^{6} - 25 q^{7} - 25 q^{8} + 25 q^{9} - 13 q^{11} - 4 q^{12} + 17 q^{13} + 25 q^{14} - 18 q^{15} + 25 q^{16} - 4 q^{17} - 25 q^{18} - 9 q^{19} + 4 q^{21} + 13 q^{22} - 14 q^{23} + 4 q^{24} + 23 q^{25} - 17 q^{26} - 7 q^{27} - 25 q^{28} - 4 q^{29} + 18 q^{30} - 15 q^{31} - 25 q^{32} - 15 q^{33} + 4 q^{34} + 25 q^{36} + 13 q^{37} + 9 q^{38} - 31 q^{39} - 31 q^{41} - 4 q^{42} + 29 q^{43} - 13 q^{44} + 10 q^{45} + 14 q^{46} - 31 q^{47} - 4 q^{48} + 25 q^{49} - 23 q^{50} - 9 q^{51} + 17 q^{52} + 23 q^{53} + 7 q^{54} - 48 q^{55} + 25 q^{56} + 32 q^{57} + 4 q^{58} - 50 q^{59} - 18 q^{60} - 2 q^{61} + 15 q^{62} - 25 q^{63} + 25 q^{64} - 4 q^{65} + 15 q^{66} - 8 q^{67} - 4 q^{68} - 57 q^{69} - 61 q^{71} - 25 q^{72} + 31 q^{73} - 13 q^{74} - 21 q^{75} - 9 q^{76} + 13 q^{77} + 31 q^{78} - 10 q^{79} + 61 q^{81} + 31 q^{82} - 47 q^{83} + 4 q^{84} + 2 q^{85} - 29 q^{86} + 17 q^{87} + 13 q^{88} - 44 q^{89} - 10 q^{90} - 17 q^{91} - 14 q^{92} - 13 q^{93} + 31 q^{94} - 7 q^{95} + 4 q^{96} + 10 q^{97} - 25 q^{98} - 47 q^{99}+O(q^{100}) \) 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 + 4 * q^6 - 25 * q^7 - 25 * q^8 + 25 * q^9 - 13 * q^11 - 4 * q^12 + 17 * q^13 + 25 * q^14 - 18 * q^15 + 25 * q^16 - 4 * q^17 - 25 * q^18 - 9 * q^19 + 4 * q^21 + 13 * q^22 - 14 * q^23 + 4 * q^24 + 23 * q^25 - 17 * q^26 - 7 * q^27 - 25 * q^28 - 4 * q^29 + 18 * q^30 - 15 * q^31 - 25 * q^32 - 15 * q^33 + 4 * q^34 + 25 * q^36 + 13 * q^37 + 9 * q^38 - 31 * q^39 - 31 * q^41 - 4 * q^42 + 29 * q^43 - 13 * q^44 + 10 * q^45 + 14 * q^46 - 31 * q^47 - 4 * q^48 + 25 * q^49 - 23 * q^50 - 9 * q^51 + 17 * q^52 + 23 * q^53 + 7 * q^54 - 48 * q^55 + 25 * q^56 + 32 * q^57 + 4 * q^58 - 50 * q^59 - 18 * q^60 - 2 * q^61 + 15 * q^62 - 25 * q^63 + 25 * q^64 - 4 * q^65 + 15 * q^66 - 8 * q^67 - 4 * q^68 - 57 * q^69 - 61 * q^71 - 25 * q^72 + 31 * q^73 - 13 * q^74 - 21 * q^75 - 9 * q^76 + 13 * q^77 + 31 * q^78 - 10 * q^79 + 61 * q^81 + 31 * q^82 - 47 * q^83 + 4 * q^84 + 2 * q^85 - 29 * q^86 + 17 * q^87 + 13 * q^88 - 44 * q^89 - 10 * q^90 - 17 * q^91 - 14 * q^92 - 13 * q^93 + 31 * q^94 - 7 * q^95 + 4 * q^96 + 10 * q^97 - 25 * q^98 - 47 * 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
$3$	1.42126	0.820565	0.410283	−	0.911958i	$-0.365430\pi$
$3$	1.42126	0.820565	0.410283	+	0.911958i	$0.365430\pi$
$4$	1.00000	0.500000
$5$	−0.924054	−0.413249	−0.206625	−	0.978420i	$-0.566248\pi$
$5$	−0.924054	−0.413249	−0.206625	+	0.978420i	$0.566248\pi$
$6$	−1.42126	−0.580227
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−0.980018	−0.326673
$10$	0.924054	0.292211
$11$	−1.23484	−0.372318	−0.186159	−	0.982520i	$-0.559604\pi$
$11$	−1.23484	−0.372318	−0.186159	+	0.982520i	$0.559604\pi$
$12$	1.42126	0.410283
$13$	1.24428	0.345101	0.172550	−	0.985001i	$-0.444799\pi$
$13$	1.24428	0.345101	0.172550	+	0.985001i	$0.444799\pi$
$14$	1.00000	0.267261
$15$	−1.31332	−0.339098
$16$	1.00000	0.250000
$17$	5.52037	1.33889	0.669443	−	0.742863i	$-0.266533\pi$
$17$	5.52037	1.33889	0.669443	+	0.742863i	$0.266533\pi$
$18$	0.980018	0.230992
$19$	−6.23981	−1.43151	−0.715756	−	0.698351i	$-0.753918\pi$
$19$	−6.23981	−1.43151	−0.715756	+	0.698351i	$0.753918\pi$
$20$	−0.924054	−0.206625
$21$	−1.42126	−0.310145
$22$	1.23484	0.263269
$23$	8.12384	1.69394	0.846969	−	0.531642i	$-0.178425\pi$
$23$	8.12384	1.69394	0.846969	+	0.531642i	$0.178425\pi$
$24$	−1.42126	−0.290114
$25$	−4.14613	−0.829225
$26$	−1.24428	−0.244023
$27$	−5.65664	−1.08862
$28$	−1.00000	−0.188982
$29$	5.70695	1.05975	0.529877	−	0.848074i	$-0.322238\pi$
$29$	5.70695	1.05975	0.529877	+	0.848074i	$0.322238\pi$
$30$	1.31332	0.239779
$31$	−4.40527	−0.791209	−0.395605	−	0.918421i	$-0.629465\pi$
$31$	−4.40527	−0.791209	−0.395605	+	0.918421i	$0.629465\pi$
$32$	−1.00000	−0.176777
$33$	−1.75503	−0.305511
$34$	−5.52037	−0.946735
$35$	0.924054	0.156194
$36$	−0.980018	−0.163336
$37$	11.0766	1.82098	0.910490	−	0.413531i	$-0.135705\pi$
$37$	11.0766	1.82098	0.910490	+	0.413531i	$0.135705\pi$
$38$	6.23981	1.01223
$39$	1.76844	0.283178
$40$	0.924054	0.146106
$41$	−7.15545	−1.11749	−0.558747	−	0.829338i	$-0.688718\pi$
$41$	−7.15545	−1.11749	−0.558747	+	0.829338i	$0.688718\pi$
$42$	1.42126	0.219305
$43$	2.65760	0.405280	0.202640	−	0.979253i	$-0.435048\pi$
$43$	2.65760	0.405280	0.202640	+	0.979253i	$0.435048\pi$
$44$	−1.23484	−0.186159
$45$	0.905589	0.134997
$46$	−8.12384	−1.19779
$47$	−5.98316	−0.872734	−0.436367	−	0.899769i	$-0.643735\pi$
$47$	−5.98316	−0.872734	−0.436367	+	0.899769i	$0.643735\pi$
$48$	1.42126	0.205141
$49$	1.00000	0.142857
$50$	4.14613	0.586351
$51$	7.84588	1.09864
$52$	1.24428	0.172550
$53$	−6.31884	−0.867959	−0.433980	−	0.900923i	$-0.642891\pi$
$53$	−6.31884	−0.867959	−0.433980	+	0.900923i	$0.642891\pi$
$54$	5.65664	0.769772
$55$	1.14106	0.153860
$56$	1.00000	0.133631
$57$	−8.86840	−1.17465
$58$	−5.70695	−0.749359
$59$	−1.31252	−0.170876	−0.0854380	−	0.996343i	$-0.527229\pi$
$59$	−1.31252	−0.170876	−0.0854380	+	0.996343i	$0.527229\pi$
$60$	−1.31332	−0.169549
$61$	7.93517	1.01599	0.507997	−	0.861359i	$-0.330386\pi$
$61$	7.93517	1.01599	0.507997	+	0.861359i	$0.330386\pi$
$62$	4.40527	0.559469
$63$	0.980018	0.123471
$64$	1.00000	0.125000
$65$	−1.14978	−0.142613
$66$	1.75503	0.216029
$67$	−11.5559	−1.41178	−0.705890	−	0.708321i	$-0.749453\pi$
$67$	−11.5559	−1.41178	−0.705890	+	0.708321i	$0.749453\pi$
$68$	5.52037	0.669443
$69$	11.5461	1.38999
$70$	−0.924054	−0.110446
$71$	2.93087	0.347830	0.173915	−	0.984761i	$-0.444358\pi$
$71$	2.93087	0.347830	0.173915	+	0.984761i	$0.444358\pi$
$72$	0.980018	0.115496
$73$	−12.6375	−1.47911	−0.739553	−	0.673099i	$-0.764963\pi$
$73$	−12.6375	−1.47911	−0.739553	+	0.673099i	$0.764963\pi$
$74$	−11.0766	−1.28763
$75$	−5.89272	−0.680433
$76$	−6.23981	−0.715756
$77$	1.23484	0.140723
$78$	−1.76844	−0.200237
$79$	13.9685	1.57158	0.785791	−	0.618492i	$-0.212256\pi$
$79$	13.9685	1.57158	0.785791	+	0.618492i	$0.212256\pi$
$80$	−0.924054	−0.103312
$81$	−5.09951	−0.566612
$82$	7.15545	0.790187
$83$	−10.7419	−1.17908	−0.589538	−	0.807741i	$-0.700690\pi$
$83$	−10.7419	−1.17908	−0.589538	+	0.807741i	$0.700690\pi$
$84$	−1.42126	−0.155072
$85$	−5.10112	−0.553294
$86$	−2.65760	−0.286576
$87$	8.11107	0.869598
$88$	1.23484	0.131634
$89$	−8.24297	−0.873753	−0.436877	−	0.899521i	$-0.643915\pi$
$89$	−8.24297	−0.873753	−0.436877	+	0.899521i	$0.643915\pi$
$90$	−0.905589	−0.0954575
$91$	−1.24428	−0.130436
$92$	8.12384	0.846969
$93$	−6.26103	−0.649239
$94$	5.98316	0.617116
$95$	5.76592	0.591571
$96$	−1.42126	−0.145057
$97$	2.94418	0.298936	0.149468	−	0.988767i	$-0.452244\pi$
$97$	2.94418	0.298936	0.149468	+	0.988767i	$0.452244\pi$
$98$	−1.00000	−0.101015
$99$	1.21016	0.121626
$100$	−4.14613	−0.414613
$101$	8.88287	0.883878	0.441939	−	0.897045i	$-0.354291\pi$
$101$	8.88287	0.883878	0.441939	+	0.897045i	$0.354291\pi$
$102$	−7.84588	−0.776858
$103$	−13.8121	−1.36095	−0.680476	−	0.732771i	$-0.738227\pi$
$103$	−13.8121	−1.36095	−0.680476	+	0.732771i	$0.738227\pi$
$104$	−1.24428	−0.122012
$105$	1.31332	0.128167
$106$	6.31884	0.613740
$107$	−2.71171	−0.262151	−0.131076	−	0.991372i	$-0.541843\pi$
$107$	−2.71171	−0.262151	−0.131076	+	0.991372i	$0.541843\pi$
$108$	−5.65664	−0.544311
$109$	12.0521	1.15438	0.577192	−	0.816609i	$-0.304148\pi$
$109$	12.0521	1.15438	0.577192	+	0.816609i	$0.304148\pi$
$110$	−1.14106	−0.108796
$111$	15.7427	1.49423
$112$	−1.00000	−0.0944911
$113$	13.3243	1.25344	0.626721	−	0.779243i	$-0.284396\pi$
$113$	13.3243	1.25344	0.626721	+	0.779243i	$0.284396\pi$
$114$	8.86840	0.830602
$115$	−7.50686	−0.700019
$116$	5.70695	0.529877
$117$	−1.21942	−0.112735
$118$	1.31252	0.120828
$119$	−5.52037	−0.506051
$120$	1.31332	0.119889
$121$	−9.47517	−0.861379
$122$	−7.93517	−0.718417
$123$	−10.1698	−0.916976
$124$	−4.40527	−0.395605
$125$	8.45151	0.755926
$126$	−0.980018	−0.0873070
$127$	−2.60840	−0.231458	−0.115729	−	0.993281i	$-0.536920\pi$
$127$	−2.60840	−0.231458	−0.115729	+	0.993281i	$0.536920\pi$
$128$	−1.00000	−0.0883883
$129$	3.77714	0.332559
$130$	1.14978	0.100842
$131$	−21.6053	−1.88766	−0.943830	−	0.330432i	$-0.892805\pi$
$131$	−21.6053	−1.88766	−0.943830	+	0.330432i	$0.892805\pi$
$132$	−1.75503	−0.152756
$133$	6.23981	0.541060
$134$	11.5559	0.998279
$135$	5.22704	0.449872
$136$	−5.52037	−0.473368
$137$	23.3581	1.99562	0.997809	−	0.0661619i	$-0.0210754\pi$
$137$	23.3581	1.99562	0.997809	+	0.0661619i	$0.0210754\pi$
$138$	−11.5461	−0.982869
$139$	−13.8595	−1.17555	−0.587774	−	0.809025i	$-0.699996\pi$
$139$	−13.8595	−1.17555	−0.587774	+	0.809025i	$0.699996\pi$
$140$	0.924054	0.0780968
$141$	−8.50363	−0.716135
$142$	−2.93087	−0.245953
$143$	−1.53648	−0.128487
$144$	−0.980018	−0.0816682
$145$	−5.27353	−0.437943
$146$	12.6375	1.04589
$147$	1.42126	0.117224
$148$	11.0766	0.910490
$149$	−18.4087	−1.50810	−0.754050	−	0.656818i	$-0.771902\pi$
$149$	−18.4087	−1.50810	−0.754050	+	0.656818i	$0.771902\pi$
$150$	5.89272	0.481139
$151$	4.99940	0.406845	0.203423	−	0.979091i	$-0.434793\pi$
$151$	4.99940	0.406845	0.203423	+	0.979091i	$0.434793\pi$
$152$	6.23981	0.506116
$153$	−5.41006	−0.437378
$154$	−1.23484	−0.0995062
$155$	4.07070	0.326967
$156$	1.76844	0.141589
$157$	−16.3985	−1.30874	−0.654370	−	0.756175i	$-0.727066\pi$
$157$	−16.3985	−1.30874	−0.654370	+	0.756175i	$0.727066\pi$
$158$	−13.9685	−1.11128
$159$	−8.98072	−0.712217
$160$	0.924054	0.0730528
$161$	−8.12384	−0.640248
$162$	5.09951	0.400655
$163$	−16.0052	−1.25363	−0.626813	−	0.779170i	$-0.715641\pi$
$163$	−16.0052	−1.25363	−0.626813	+	0.779170i	$0.715641\pi$
$164$	−7.15545	−0.558747
$165$	1.62174	0.126252
$166$	10.7419	0.833733
$167$	−17.9080	−1.38576	−0.692880	−	0.721053i	$-0.743659\pi$
$167$	−17.9080	−1.38576	−0.692880	+	0.721053i	$0.743659\pi$
$168$	1.42126	0.109653
$169$	−11.4518	−0.880905
$170$	5.10112	0.391238
$171$	6.11513	0.467636
$172$	2.65760	0.202640
$173$	−6.72829	−0.511543	−0.255771	−	0.966737i	$-0.582329\pi$
$173$	−6.72829	−0.511543	−0.255771	+	0.966737i	$0.582329\pi$
$174$	−8.11107	−0.614898
$175$	4.14613	0.313418
$176$	−1.23484	−0.0930795
$177$	−1.86544	−0.140215
$178$	8.24297	0.617837
$179$	−5.26352	−0.393414	−0.196707	−	0.980462i	$-0.563025\pi$
$179$	−5.26352	−0.393414	−0.196707	+	0.980462i	$0.563025\pi$
$180$	0.905589	0.0674986
$181$	24.8497	1.84707	0.923533	−	0.383518i	$-0.125288\pi$
$181$	24.8497	1.84707	0.923533	+	0.383518i	$0.125288\pi$
$182$	1.24428	0.0922320
$183$	11.2779	0.833690
$184$	−8.12384	−0.598897
$185$	−10.2354	−0.752519
$186$	6.26103	0.459081
$187$	−6.81677	−0.498492
$188$	−5.98316	−0.436367
$189$	5.65664	0.411460
$190$	−5.76592	−0.418304
$191$	−5.59775	−0.405039	−0.202520	−	0.979278i	$-0.564913\pi$
$191$	−5.59775	−0.405039	−0.202520	+	0.979278i	$0.564913\pi$
$192$	1.42126	0.102571
$193$	−4.22332	−0.304001	−0.152000	−	0.988380i	$-0.548572\pi$
$193$	−4.22332	−0.304001	−0.152000	+	0.988380i	$0.548572\pi$
$194$	−2.94418	−0.211379
$195$	−1.63414	−0.117023
$196$	1.00000	0.0714286
$197$	−26.1982	−1.86654	−0.933271	−	0.359173i	$-0.883059\pi$
$197$	−26.1982	−1.86654	−0.933271	+	0.359173i	$0.883059\pi$
$198$	−1.21016	−0.0860027
$199$	−20.8975	−1.48138	−0.740692	−	0.671845i	$-0.765502\pi$
$199$	−20.8975	−1.48138	−0.740692	+	0.671845i	$0.765502\pi$
$200$	4.14613	0.293175
$201$	−16.4240	−1.15846
$202$	−8.88287	−0.624996
$203$	−5.70695	−0.400549
$204$	7.84588	0.549322
$205$	6.61202	0.461803
$206$	13.8121	0.962338
$207$	−7.96151	−0.553363
$208$	1.24428	0.0862752
$209$	7.70517	0.532977
$210$	−1.31332	−0.0906278
$211$	13.3607	0.919792	0.459896	−	0.887973i	$-0.347887\pi$
$211$	13.3607	0.919792	0.459896	+	0.887973i	$0.347887\pi$
$212$	−6.31884	−0.433980
$213$	4.16552	0.285417
$214$	2.71171	0.185369
$215$	−2.45576	−0.167482
$216$	5.65664	0.384886
$217$	4.40527	0.299049
$218$	−12.0521	−0.816273
$219$	−17.9612	−1.21370
$220$	1.14106	0.0769301
$221$	6.86888	0.462051
$222$	−15.7427	−1.05658
$223$	10.4305	0.698478	0.349239	−	0.937034i	$-0.386440\pi$
$223$	10.4305	0.698478	0.349239	+	0.937034i	$0.386440\pi$
$224$	1.00000	0.0668153
$225$	4.06328	0.270885
$226$	−13.3243	−0.886318
$227$	5.33359	0.354003	0.177001	−	0.984211i	$-0.443360\pi$
$227$	5.33359	0.354003	0.177001	+	0.984211i	$0.443360\pi$
$228$	−8.86840	−0.587324
$229$	7.85457	0.519044	0.259522	−	0.965737i	$-0.416435\pi$
$229$	7.85457	0.519044	0.259522	+	0.965737i	$0.416435\pi$
$230$	7.50686	0.494988
$231$	1.75503	0.115472
$232$	−5.70695	−0.374680
$233$	−14.0689	−0.921684	−0.460842	−	0.887482i	$-0.652453\pi$
$233$	−14.0689	−0.921684	−0.460842	+	0.887482i	$0.652453\pi$
$234$	1.21942	0.0797157
$235$	5.52876	0.360657
$236$	−1.31252	−0.0854380
$237$	19.8529	1.28959
$238$	5.52037	0.357832
$239$	3.02851	0.195898	0.0979490	−	0.995191i	$-0.468772\pi$
$239$	3.02851	0.195898	0.0979490	+	0.995191i	$0.468772\pi$
$240$	−1.31332	−0.0847745
$241$	−7.22390	−0.465332	−0.232666	−	0.972557i	$-0.574745\pi$
$241$	−7.22390	−0.465332	−0.232666	+	0.972557i	$0.574745\pi$
$242$	9.47517	0.609087
$243$	9.72220	0.623679
$244$	7.93517	0.507997
$245$	−0.924054	−0.0590356
$246$	10.1698	0.648400
$247$	−7.76406	−0.494016
$248$	4.40527	0.279735
$249$	−15.2670	−0.967509
$250$	−8.45151	−0.534520
$251$	−15.3772	−0.970603	−0.485301	−	0.874347i	$-0.661290\pi$
$251$	−15.3772	−0.970603	−0.485301	+	0.874347i	$0.661290\pi$
$252$	0.980018	0.0617353
$253$	−10.0316	−0.630684
$254$	2.60840	0.163666
$255$	−7.25002	−0.454014
$256$	1.00000	0.0625000
$257$	−6.21145	−0.387460	−0.193730	−	0.981055i	$-0.562059\pi$
$257$	−6.21145	−0.387460	−0.193730	+	0.981055i	$0.562059\pi$
$258$	−3.77714	−0.235155
$259$	−11.0766	−0.688266
$260$	−1.14978	−0.0713063
$261$	−5.59292	−0.346193
$262$	21.6053	1.33478
$263$	−20.3926	−1.25746	−0.628732	−	0.777622i	$-0.716426\pi$
$263$	−20.3926	−1.25746	−0.628732	+	0.777622i	$0.716426\pi$
$264$	1.75503	0.108015
$265$	5.83894	0.358684
$266$	−6.23981	−0.382587
$267$	−11.7154	−0.716972
$268$	−11.5559	−0.705890
$269$	22.0742	1.34589	0.672945	−	0.739693i	$-0.265029\pi$
$269$	22.0742	1.34589	0.672945	+	0.739693i	$0.265029\pi$
$270$	−5.22704	−0.318108
$271$	−5.58690	−0.339380	−0.169690	−	0.985498i	$-0.554277\pi$
$271$	−5.58690	−0.339380	−0.169690	+	0.985498i	$0.554277\pi$
$272$	5.52037	0.334722
$273$	−1.76844	−0.107031
$274$	−23.3581	−1.41111
$275$	5.11980	0.308735
$276$	11.5461	0.694993
$277$	−24.0636	−1.44584	−0.722922	−	0.690930i	$-0.757201\pi$
$277$	−24.0636	−1.44584	−0.722922	+	0.690930i	$0.757201\pi$
$278$	13.8595	0.831238
$279$	4.31724	0.258466
$280$	−0.924054	−0.0552228
$281$	0.872693	0.0520605	0.0260302	−	0.999661i	$-0.491713\pi$
$281$	0.872693	0.0520605	0.0260302	+	0.999661i	$0.491713\pi$
$282$	8.50363	0.506384
$283$	−23.8804	−1.41954	−0.709771	−	0.704433i	$-0.751201\pi$
$283$	−23.8804	−1.41954	−0.709771	+	0.704433i	$0.751201\pi$
$284$	2.93087	0.173915
$285$	8.19488	0.485423
$286$	1.53648	0.0908542
$287$	7.15545	0.422373
$288$	0.980018	0.0577481
$289$	13.4745	0.792616
$290$	5.27353	0.309672
$291$	4.18444	0.245296
$292$	−12.6375	−0.739553
$293$	−16.8982	−0.987206	−0.493603	−	0.869687i	$-0.664320\pi$
$293$	−16.8982	−0.987206	−0.493603	+	0.869687i	$0.664320\pi$
$294$	−1.42126	−0.0828896
$295$	1.21284	0.0706144
$296$	−11.0766	−0.643814
$297$	6.98505	0.405313
$298$	18.4087	1.06639
$299$	10.1083	0.584579
$300$	−5.89272	−0.340217
$301$	−2.65760	−0.153182
$302$	−4.99940	−0.287683
$303$	12.6249	0.725280
$304$	−6.23981	−0.357878
$305$	−7.33252	−0.419859
$306$	5.41006	0.309273
$307$	28.2474	1.61216	0.806082	−	0.591804i	$-0.201584\pi$
$307$	28.2474	1.61216	0.806082	+	0.591804i	$0.201584\pi$
$308$	1.23484	0.0703615
$309$	−19.6307	−1.11675
$310$	−4.07070	−0.231200
$311$	7.50354	0.425487	0.212743	−	0.977108i	$-0.431760\pi$
$311$	7.50354	0.425487	0.212743	+	0.977108i	$0.431760\pi$
$312$	−1.76844	−0.100118
$313$	11.1099	0.627967	0.313983	−	0.949429i	$-0.398336\pi$
$313$	11.1099	0.627967	0.313983	+	0.949429i	$0.398336\pi$
$314$	16.3985	0.925419
$315$	−0.905589	−0.0510242
$316$	13.9685	0.785791
$317$	−31.2704	−1.75632	−0.878161	−	0.478365i	$-0.841230\pi$
$317$	−31.2704	−1.75632	−0.878161	+	0.478365i	$0.841230\pi$
$318$	8.98072	0.503614
$319$	−7.04717	−0.394566
$320$	−0.924054	−0.0516562
$321$	−3.85405	−0.215112
$322$	8.12384	0.452724
$323$	−34.4461	−1.91663
$324$	−5.09951	−0.283306
$325$	−5.15893	−0.286166
$326$	16.0052	0.886447
$327$	17.1292	0.947247
$328$	7.15545	0.395094
$329$	5.98316	0.329862
$330$	−1.62174	−0.0892739
$331$	6.59848	0.362685	0.181343	−	0.983420i	$-0.441956\pi$
$331$	6.59848	0.362685	0.181343	+	0.983420i	$0.441956\pi$
$332$	−10.7419	−0.589538
$333$	−10.8553	−0.594864
$334$	17.9080	0.979881
$335$	10.6783	0.583417
$336$	−1.42126	−0.0775361
$337$	8.69404	0.473595	0.236797	−	0.971559i	$-0.423902\pi$
$337$	8.69404	0.473595	0.236797	+	0.971559i	$0.423902\pi$
$338$	11.4518	0.622894
$339$	18.9373	1.02853
$340$	−5.10112	−0.276647
$341$	5.43980	0.294582
$342$	−6.11513	−0.330668
$343$	−1.00000	−0.0539949
$344$	−2.65760	−0.143288
$345$	−10.6692	−0.574411
$346$	6.72829	0.361715
$347$	23.6177	1.26786	0.633932	−	0.773389i	$-0.281440\pi$
$347$	23.6177	1.26786	0.633932	+	0.773389i	$0.281440\pi$
$348$	8.11107	0.434799
$349$	19.3957	1.03823	0.519115	−	0.854704i	$-0.326262\pi$
$349$	19.3957	1.03823	0.519115	+	0.854704i	$0.326262\pi$
$350$	−4.14613	−0.221620
$351$	−7.03844	−0.375684
$352$	1.23484	0.0658172
$353$	−22.9193	−1.21987	−0.609937	−	0.792450i	$-0.708805\pi$
$353$	−22.9193	−1.21987	−0.609937	+	0.792450i	$0.708805\pi$
$354$	1.86544	0.0991469
$355$	−2.70828	−0.143740
$356$	−8.24297	−0.436877
$357$	−7.84588	−0.415248
$358$	5.26352	0.278186
$359$	−21.4472	−1.13194	−0.565970	−	0.824426i	$-0.691498\pi$
$359$	−21.4472	−1.13194	−0.565970	+	0.824426i	$0.691498\pi$
$360$	−0.905589	−0.0477287
$361$	19.9353	1.04922
$362$	−24.8497	−1.30607
$363$	−13.4667	−0.706818
$364$	−1.24428	−0.0652179
$365$	11.6777	0.611239
$366$	−11.2779	−0.589508
$367$	−8.75137	−0.456818	−0.228409	−	0.973565i	$-0.573352\pi$
$367$	−8.75137	−0.456818	−0.228409	+	0.973565i	$0.573352\pi$
$368$	8.12384	0.423484
$369$	7.01247	0.365055
$370$	10.2354	0.532111
$371$	6.31884	0.328058
$372$	−6.26103	−0.324619
$373$	22.5944	1.16989	0.584947	−	0.811072i	$-0.301115\pi$
$373$	22.5944	1.16989	0.584947	+	0.811072i	$0.301115\pi$
$374$	6.81677	0.352487
$375$	12.0118	0.620287
$376$	5.98316	0.308558
$377$	7.10104	0.365722
$378$	−5.65664	−0.290946
$379$	26.2522	1.34848	0.674242	−	0.738510i	$-0.264470\pi$
$379$	26.2522	1.34848	0.674242	+	0.738510i	$0.264470\pi$
$380$	5.76592	0.295785
$381$	−3.70722	−0.189927
$382$	5.59775	0.286406
$383$	−4.88237	−0.249478	−0.124739	−	0.992190i	$-0.539809\pi$
$383$	−4.88237	−0.249478	−0.124739	+	0.992190i	$0.539809\pi$
$384$	−1.42126	−0.0725284
$385$	−1.14106	−0.0581537
$386$	4.22332	0.214961
$387$	−2.60450	−0.132394
$388$	2.94418	0.149468
$389$	−27.3518	−1.38679	−0.693396	−	0.720557i	$-0.743886\pi$
$389$	−27.3518	−1.38679	−0.693396	+	0.720557i	$0.743886\pi$
$390$	1.63414	0.0827477
$391$	44.8466	2.26799
$392$	−1.00000	−0.0505076
$393$	−30.7067	−1.54895
$394$	26.1982	1.31984
$395$	−12.9077	−0.649456
$396$	1.21016	0.0608131
$397$	−33.4410	−1.67836	−0.839179	−	0.543856i	$-0.816964\pi$
$397$	−33.4410	−1.67836	−0.839179	+	0.543856i	$0.816964\pi$
$398$	20.8975	1.04750
$399$	8.86840	0.443975
$400$	−4.14613	−0.207306
$401$	23.9072	1.19387	0.596934	−	0.802290i	$-0.296385\pi$
$401$	23.9072	1.19387	0.596934	+	0.802290i	$0.296385\pi$
$402$	16.4240	0.819153
$403$	−5.48138	−0.273047
$404$	8.88287	0.441939
$405$	4.71222	0.234152
$406$	5.70695	0.283231
$407$	−13.6778	−0.677984
$408$	−7.84588	−0.388429
$409$	20.5753	1.01738	0.508692	−	0.860949i	$-0.330129\pi$
$409$	20.5753	1.01738	0.508692	+	0.860949i	$0.330129\pi$
$410$	−6.61202	−0.326544
$411$	33.1980	1.63753
$412$	−13.8121	−0.680476
$413$	1.31252	0.0645850
$414$	7.96151	0.391287
$415$	9.92609	0.487252
$416$	−1.24428	−0.0610058
$417$	−19.6980	−0.964614
$418$	−7.70517	−0.376872
$419$	0.163933	0.00800864	0.00400432	−	0.999992i	$-0.498725\pi$
$419$	0.163933	0.00800864	0.00400432	+	0.999992i	$0.498725\pi$
$420$	1.31332	0.0640835
$421$	−20.9150	−1.01934	−0.509668	−	0.860371i	$-0.670232\pi$
$421$	−20.9150	−1.01934	−0.509668	+	0.860371i	$0.670232\pi$
$422$	−13.3607	−0.650391
$423$	5.86361	0.285098
$424$	6.31884	0.306870
$425$	−22.8881	−1.11024
$426$	−4.16552	−0.201820
$427$	−7.93517	−0.384010
$428$	−2.71171	−0.131076
$429$	−2.18374	−0.105432
$430$	2.45576	0.118427
$431$	−1.00000	−0.0481683
$431$	−1.00000	−0.0481683
$432$	−5.65664	−0.272155
$433$	34.1978	1.64344	0.821721	−	0.569890i	$-0.193014\pi$
$433$	34.1978	1.64344	0.821721	+	0.569890i	$0.193014\pi$
$434$	−4.40527	−0.211460
$435$	−7.49506	−0.359361
$436$	12.0521	0.577192
$437$	−50.6912	−2.42489
$438$	17.9612	0.858217
$439$	34.0288	1.62411	0.812054	−	0.583582i	$-0.198350\pi$
$439$	34.0288	1.62411	0.812054	+	0.583582i	$0.198350\pi$
$440$	−1.14106	−0.0543978
$441$	−0.980018	−0.0466675
$442$	−6.86888	−0.326719
$443$	12.1324	0.576426	0.288213	−	0.957566i	$-0.406939\pi$
$443$	12.1324	0.576426	0.288213	+	0.957566i	$0.406939\pi$
$444$	15.7427	0.747116
$445$	7.61695	0.361078
$446$	−10.4305	−0.493899
$447$	−26.1636	−1.23749
$448$	−1.00000	−0.0472456
$449$	−25.7063	−1.21315	−0.606577	−	0.795025i	$-0.707458\pi$
$449$	−25.7063	−1.21315	−0.606577	+	0.795025i	$0.707458\pi$
$450$	−4.06328	−0.191545
$451$	8.83583	0.416063
$452$	13.3243	0.626721
$453$	7.10545	0.333843
$454$	−5.33359	−0.250318
$455$	1.14978	0.0539025
$456$	8.86840	0.415301
$457$	−10.3282	−0.483135	−0.241567	−	0.970384i	$-0.577661\pi$
$457$	−10.3282	−0.483135	−0.241567	+	0.970384i	$0.577661\pi$
$458$	−7.85457	−0.367020
$459$	−31.2268	−1.45754
$460$	−7.50686	−0.350009
$461$	−0.368987	−0.0171855	−0.00859273	−	0.999963i	$-0.502735\pi$
$461$	−0.368987	−0.0171855	−0.00859273	+	0.999963i	$0.502735\pi$
$462$	−1.75503	−0.0816513
$463$	16.4178	0.762999	0.381500	−	0.924369i	$-0.375408\pi$
$463$	16.4178	0.762999	0.381500	+	0.924369i	$0.375408\pi$
$464$	5.70695	0.264939
$465$	5.78553	0.268298
$466$	14.0689	0.651729
$467$	−24.7403	−1.14484	−0.572422	−	0.819959i	$-0.693996\pi$
$467$	−24.7403	−1.14484	−0.572422	+	0.819959i	$0.693996\pi$
$468$	−1.21942	−0.0563675
$469$	11.5559	0.533603
$470$	−5.52876	−0.255023
$471$	−23.3065	−1.07391
$472$	1.31252	0.0604138
$473$	−3.28171	−0.150893
$474$	−19.8529	−0.911875
$475$	25.8710	1.18704
$476$	−5.52037	−0.253026
$477$	6.19258	0.283539
$478$	−3.02851	−0.138521
$479$	−17.5789	−0.803200	−0.401600	−	0.915815i	$-0.631546\pi$
$479$	−17.5789	−0.803200	−0.401600	+	0.915815i	$0.631546\pi$
$480$	1.31332	0.0599446
$481$	13.7824	0.628421
$482$	7.22390	0.329040
$483$	−11.5461	−0.525366
$484$	−9.47517	−0.430690
$485$	−2.72058	−0.123535
$486$	−9.72220	−0.441008
$487$	2.98023	0.135047	0.0675237	−	0.997718i	$-0.478490\pi$
$487$	2.98023	0.135047	0.0675237	+	0.997718i	$0.478490\pi$
$488$	−7.93517	−0.359208
$489$	−22.7476	−1.02868
$490$	0.924054	0.0417445
$491$	36.7961	1.66059	0.830293	−	0.557327i	$-0.188173\pi$
$491$	36.7961	1.66059	0.830293	+	0.557327i	$0.188173\pi$
$492$	−10.1698	−0.458488
$493$	31.5045	1.41889
$494$	7.76406	0.349322
$495$	−1.11826	−0.0502619
$496$	−4.40527	−0.197802
$497$	−2.93087	−0.131467
$498$	15.2670	0.684132
$499$	−26.1265	−1.16958	−0.584792	−	0.811183i	$-0.698824\pi$
$499$	−26.1265	−1.16958	−0.584792	+	0.811183i	$0.698824\pi$
$500$	8.45151	0.377963
$501$	−25.4519	−1.13711
$502$	15.3772	0.686320
$503$	0.0987147	0.00440147	0.00220074	−	0.999998i	$-0.499299\pi$
$503$	0.0987147	0.00440147	0.00220074	+	0.999998i	$0.499299\pi$
$504$	−0.980018	−0.0436535
$505$	−8.20824	−0.365262
$506$	10.0316	0.445961
$507$	−16.2760	−0.722840
$508$	−2.60840	−0.115729
$509$	−24.3632	−1.07988	−0.539940	−	0.841704i	$-0.681553\pi$
$509$	−24.3632	−1.07988	−0.539940	+	0.841704i	$0.681553\pi$
$510$	7.25002	0.321036
$511$	12.6375	0.559049
$512$	−1.00000	−0.0441942
$513$	35.2964	1.55837
$514$	6.21145	0.273975
$515$	12.7632	0.562412
$516$	3.77714	0.166279
$517$	7.38824	0.324935
$518$	11.0766	0.486677
$519$	−9.56266	−0.419754
$520$	1.14978	0.0504212
$521$	7.73358	0.338814	0.169407	−	0.985546i	$-0.445815\pi$
$521$	7.73358	0.338814	0.169407	+	0.985546i	$0.445815\pi$
$522$	5.59292	0.244795
$523$	−29.3531	−1.28352	−0.641761	−	0.766905i	$-0.721796\pi$
$523$	−29.3531	−1.28352	−0.641761	+	0.766905i	$0.721796\pi$
$524$	−21.6053	−0.943830
$525$	5.89272	0.257180
$526$	20.3926	0.889161
$527$	−24.3187	−1.05934
$528$	−1.75503	−0.0763778
$529$	42.9968	1.86943
$530$	−5.83894	−0.253628
$531$	1.28630	0.0558205
$532$	6.23981	0.270530
$533$	−8.90337	−0.385648
$534$	11.7154	0.506975
$535$	2.50577	0.108334
$536$	11.5559	0.499140
$537$	−7.48084	−0.322822
$538$	−22.0742	−0.951688
$539$	−1.23484	−0.0531883
$540$	5.22704	0.224936
$541$	5.54629	0.238454	0.119227	−	0.992867i	$-0.461958\pi$
$541$	5.54629	0.238454	0.119227	+	0.992867i	$0.461958\pi$
$542$	5.58690	0.239978
$543$	35.3180	1.51564
$544$	−5.52037	−0.236684
$545$	−11.1368	−0.477048
$546$	1.76844	0.0756824
$547$	18.6729	0.798393	0.399197	−	0.916865i	$-0.369289\pi$
$547$	18.6729	0.798393	0.399197	+	0.916865i	$0.369289\pi$
$548$	23.3581	0.997809
$549$	−7.77661	−0.331898
$550$	−5.11980	−0.218309
$551$	−35.6103	−1.51705
$552$	−11.5461	−0.491434
$553$	−13.9685	−0.594002
$554$	24.0636	1.02237
$555$	−14.5471	−0.617491
$556$	−13.8595	−0.587774
$557$	1.42065	0.0601949	0.0300975	−	0.999547i	$-0.490418\pi$
$557$	1.42065	0.0601949	0.0300975	+	0.999547i	$0.490418\pi$
$558$	−4.31724	−0.182763
$559$	3.30679	0.139862
$560$	0.924054	0.0390484
$561$	−9.68841	−0.409045
$562$	−0.872693	−0.0368123
$563$	29.8826	1.25940	0.629700	−	0.776839i	$-0.283178\pi$
$563$	29.8826	1.25940	0.629700	+	0.776839i	$0.283178\pi$
$564$	−8.50363	−0.358068
$565$	−12.3124	−0.517984
$566$	23.8804	1.00377
$567$	5.09951	0.214159
$568$	−2.93087	−0.122976
$569$	−19.8887	−0.833776	−0.416888	−	0.908958i	$-0.636879\pi$
$569$	−19.8887	−0.833776	−0.416888	+	0.908958i	$0.636879\pi$
$570$	−8.19488	−0.343246
$571$	−8.06987	−0.337714	−0.168857	−	0.985641i	$-0.554008\pi$
$571$	−8.06987	−0.337714	−0.168857	+	0.985641i	$0.554008\pi$
$572$	−1.53648	−0.0642436
$573$	−7.95587	−0.332361
$574$	−7.15545	−0.298663
$575$	−33.6825	−1.40466
$576$	−0.980018	−0.0408341
$577$	26.8027	1.11581	0.557906	−	0.829904i	$-0.311605\pi$
$577$	26.8027	1.11581	0.557906	+	0.829904i	$0.311605\pi$
$578$	−13.4745	−0.560464
$579$	−6.00243	−0.249453
$580$	−5.27353	−0.218971
$581$	10.7419	0.445649
$582$	−4.18444	−0.173451
$583$	7.80275	0.323157
$584$	12.6375	0.522943
$585$	1.12681	0.0465877
$586$	16.8982	0.698060
$587$	42.2819	1.74516	0.872581	−	0.488469i	$-0.162445\pi$
$587$	42.2819	1.74516	0.872581	+	0.488469i	$0.162445\pi$
$588$	1.42126	0.0586118
$589$	27.4880	1.13262
$590$	−1.21284	−0.0499319
$591$	−37.2344	−1.53162
$592$	11.0766	0.455245
$593$	−43.0879	−1.76941	−0.884703	−	0.466154i	$-0.845639\pi$
$593$	−43.0879	−1.76941	−0.884703	+	0.466154i	$0.845639\pi$
$594$	−6.98505	−0.286600
$595$	5.10112	0.209125
$596$	−18.4087	−0.754050
$597$	−29.7008	−1.21557
$598$	−10.1083	−0.413360
$599$	−10.5916	−0.432762	−0.216381	−	0.976309i	$-0.569425\pi$
$599$	−10.5916	−0.432762	−0.216381	+	0.976309i	$0.569425\pi$
$600$	5.89272	0.240569
$601$	−7.83681	−0.319670	−0.159835	−	0.987144i	$-0.551096\pi$
$601$	−7.83681	−0.319670	−0.159835	+	0.987144i	$0.551096\pi$
$602$	2.65760	0.108316
$603$	11.3250	0.461190
$604$	4.99940	0.203423
$605$	8.75557	0.355964
$606$	−12.6249	−0.512850
$607$	−3.96076	−0.160762	−0.0803812	−	0.996764i	$-0.525614\pi$
$607$	−3.96076	−0.160762	−0.0803812	+	0.996764i	$0.525614\pi$
$608$	6.23981	0.253058
$609$	−8.11107	−0.328677
$610$	7.33252	0.296885
$611$	−7.44472	−0.301181
$612$	−5.41006	−0.218689
$613$	17.8412	0.720599	0.360300	−	0.932837i	$-0.382674\pi$
$613$	17.8412	0.720599	0.360300	+	0.932837i	$0.382674\pi$
$614$	−28.2474	−1.13997
$615$	9.39740	0.378940
$616$	−1.23484	−0.0497531
$617$	38.6431	1.55571	0.777856	−	0.628443i	$-0.216307\pi$
$617$	38.6431	1.55571	0.777856	+	0.628443i	$0.216307\pi$
$618$	19.6307	0.789661
$619$	−22.2213	−0.893150	−0.446575	−	0.894746i	$-0.647356\pi$
$619$	−22.2213	−0.893150	−0.446575	+	0.894746i	$0.647356\pi$
$620$	4.07070	0.163483
$621$	−45.9537	−1.84406
$622$	−7.50354	−0.300865
$623$	8.24297	0.330248
$624$	1.76844	0.0707944
$625$	12.9210	0.516839
$626$	−11.1099	−0.444039
$627$	10.9510	0.437343
$628$	−16.3985	−0.654370
$629$	61.1469	2.43808
$630$	0.905589	0.0360795
$631$	−12.9790	−0.516686	−0.258343	−	0.966053i	$-0.583177\pi$
$631$	−12.9790	−0.516686	−0.258343	+	0.966053i	$0.583177\pi$
$632$	−13.9685	−0.555638
$633$	18.9891	0.754749
$634$	31.2704	1.24191
$635$	2.41030	0.0956499
$636$	−8.98072	−0.356109
$637$	1.24428	0.0493001
$638$	7.04717	0.279000
$639$	−2.87230	−0.113626
$640$	0.924054	0.0365264
$641$	−14.8830	−0.587843	−0.293921	−	0.955830i	$-0.594960\pi$
$641$	−14.8830	−0.587843	−0.293921	+	0.955830i	$0.594960\pi$
$642$	3.85405	0.152107
$643$	−23.5429	−0.928443	−0.464221	−	0.885719i	$-0.653666\pi$
$643$	−23.5429	−0.928443	−0.464221	+	0.885719i	$0.653666\pi$
$644$	−8.12384	−0.320124
$645$	−3.49028	−0.137430
$646$	34.4461	1.35526
$647$	−26.6251	−1.04674	−0.523371	−	0.852105i	$-0.675326\pi$
$647$	−26.6251	−1.04674	−0.523371	+	0.852105i	$0.675326\pi$
$648$	5.09951	0.200328
$649$	1.62076	0.0636202
$650$	5.15893	0.202350
$651$	6.26103	0.245389
$652$	−16.0052	−0.626813
$653$	31.8895	1.24793	0.623967	−	0.781451i	$-0.285520\pi$
$653$	31.8895	1.24793	0.623967	+	0.781451i	$0.285520\pi$
$654$	−17.1292	−0.669805
$655$	19.9644	0.780074
$656$	−7.15545	−0.279373
$657$	12.3850	0.483183
$658$	−5.98316	−0.233248
$659$	31.5749	1.22998	0.614991	−	0.788534i	$-0.289160\pi$
$659$	31.5749	1.22998	0.614991	+	0.788534i	$0.289160\pi$
$660$	1.62174	0.0631262
$661$	17.1678	0.667749	0.333874	−	0.942618i	$-0.391644\pi$
$661$	17.1678	0.667749	0.333874	+	0.942618i	$0.391644\pi$
$662$	−6.59848	−0.256457
$663$	9.76246	0.379143
$664$	10.7419	0.416866
$665$	−5.76592	−0.223593
$666$	10.8553	0.420633
$667$	46.3624	1.79516
$668$	−17.9080	−0.692880
$669$	14.8245	0.573147
$670$	−10.6783	−0.412538
$671$	−9.79866	−0.378273
$672$	1.42126	0.0548263
$673$	−26.2144	−1.01049	−0.505245	−	0.862976i	$-0.668598\pi$
$673$	−26.2144	−1.01049	−0.505245	+	0.862976i	$0.668598\pi$
$674$	−8.69404	−0.334882
$675$	23.4531	0.902712
$676$	−11.4518	−0.440453
$677$	11.6469	0.447627	0.223814	−	0.974632i	$-0.428149\pi$
$677$	11.6469	0.447627	0.223814	+	0.974632i	$0.428149\pi$
$678$	−18.9373	−0.727282
$679$	−2.94418	−0.112987
$680$	5.10112	0.195619
$681$	7.58042	0.290482
$682$	−5.43980	−0.208301
$683$	−5.05599	−0.193462	−0.0967310	−	0.995311i	$-0.530839\pi$
$683$	−5.05599	−0.193462	−0.0967310	+	0.995311i	$0.530839\pi$
$684$	6.11513	0.233818
$685$	−21.5841	−0.824688
$686$	1.00000	0.0381802
$687$	11.1634	0.425910
$688$	2.65760	0.101320
$689$	−7.86239	−0.299533
$690$	10.6692	0.406170
$691$	1.10257	0.0419436	0.0209718	−	0.999780i	$-0.493324\pi$
$691$	1.10257	0.0419436	0.0209718	+	0.999780i	$0.493324\pi$
$692$	−6.72829	−0.255771
$693$	−1.21016	−0.0459704
$694$	−23.6177	−0.896516
$695$	12.8069	0.485794
$696$	−8.11107	−0.307449
$697$	−39.5007	−1.49620
$698$	−19.3957	−0.734140
$699$	−19.9956	−0.756302
$700$	4.14613	0.156709
$701$	−12.3385	−0.466018	−0.233009	−	0.972475i	$-0.574857\pi$
$701$	−12.3385	−0.466018	−0.233009	+	0.972475i	$0.574857\pi$
$702$	7.03844	0.265649
$703$	−69.1158	−2.60675
$704$	−1.23484	−0.0465398
$705$	7.85781	0.295942
$706$	22.9193	0.862581
$707$	−8.88287	−0.334075
$708$	−1.86544	−0.0701074
$709$	−7.98375	−0.299836	−0.149918	−	0.988698i	$-0.547901\pi$
$709$	−7.98375	−0.299836	−0.149918	+	0.988698i	$0.547901\pi$
$710$	2.70828	0.101640
$711$	−13.6894	−0.513393
$712$	8.24297	0.308918
$713$	−35.7877	−1.34026
$714$	7.84588	0.293625
$715$	1.41979	0.0530973
$716$	−5.26352	−0.196707
$717$	4.30430	0.160747
$718$	21.4472	0.800402
$719$	−48.5981	−1.81240	−0.906201	−	0.422848i	$-0.861031\pi$
$719$	−48.5981	−1.81240	−0.906201	+	0.422848i	$0.861031\pi$
$720$	0.905589	0.0337493
$721$	13.8121	0.514391
$722$	−19.9353	−0.741913
$723$	−10.2670	−0.381836
$724$	24.8497	0.923533
$725$	−23.6617	−0.878775
$726$	13.4667	0.499796
$727$	23.8077	0.882980	0.441490	−	0.897266i	$-0.354450\pi$
$727$	23.8077	0.882980	0.441490	+	0.897266i	$0.354450\pi$
$728$	1.24428	0.0461160
$729$	29.1163	1.07838
$730$	−11.6777	−0.432211
$731$	14.6709	0.542624
$732$	11.2779	0.416845
$733$	21.3335	0.787972	0.393986	−	0.919116i	$-0.371096\pi$
$733$	21.3335	0.787972	0.393986	+	0.919116i	$0.371096\pi$
$734$	8.75137	0.323019
$735$	−1.31332	−0.0484426
$736$	−8.12384	−0.299449
$737$	14.2697	0.525631
$738$	−7.01247	−0.258133
$739$	−30.0418	−1.10510	−0.552552	−	0.833478i	$-0.686346\pi$
$739$	−30.0418	−1.10510	−0.552552	+	0.833478i	$0.686346\pi$
$740$	−10.2354	−0.376259
$741$	−11.0348	−0.405372
$742$	−6.31884	−0.231972
$743$	4.32023	0.158494	0.0792470	−	0.996855i	$-0.474748\pi$
$743$	4.32023	0.158494	0.0792470	+	0.996855i	$0.474748\pi$
$744$	6.26103	0.229541
$745$	17.0106	0.623221
$746$	−22.5944	−0.827240
$747$	10.5273	0.385172
$748$	−6.81677	−0.249246
$749$	2.71171	0.0990839
$750$	−12.0118	−0.438609
$751$	−4.45901	−0.162711	−0.0813557	−	0.996685i	$-0.525925\pi$
$751$	−4.45901	−0.162711	−0.0813557	+	0.996685i	$0.525925\pi$
$752$	−5.98316	−0.218183
$753$	−21.8551	−0.796443
$754$	−7.10104	−0.258604
$755$	−4.61971	−0.168128
$756$	5.65664	0.205730
$757$	−17.9887	−0.653809	−0.326905	−	0.945057i	$-0.606006\pi$
$757$	−17.9887	−0.653809	−0.326905	+	0.945057i	$0.606006\pi$
$758$	−26.2522	−0.953523
$759$	−14.2576	−0.517517
$760$	−5.76592	−0.209152
$761$	−41.8945	−1.51867	−0.759337	−	0.650697i	$-0.774477\pi$
$761$	−41.8945	−1.51867	−0.759337	+	0.650697i	$0.774477\pi$
$762$	3.70722	0.134298
$763$	−12.0521	−0.436316
$764$	−5.59775	−0.202520
$765$	4.99919	0.180746
$766$	4.88237	0.176407
$767$	−1.63314	−0.0589694
$768$	1.42126	0.0512853
$769$	3.77663	0.136189	0.0680944	−	0.997679i	$-0.478308\pi$
$769$	3.77663	0.136189	0.0680944	+	0.997679i	$0.478308\pi$
$770$	1.14106	0.0411209
$771$	−8.82809	−0.317936
$772$	−4.22332	−0.152000
$773$	−23.9875	−0.862772	−0.431386	−	0.902167i	$-0.641975\pi$
$773$	−23.9875	−0.862772	−0.431386	+	0.902167i	$0.641975\pi$
$774$	2.60450	0.0936167
$775$	18.2648	0.656091
$776$	−2.94418	−0.105690
$777$	−15.7427	−0.564767
$778$	27.3518	0.980610
$779$	44.6487	1.59970
$780$	−1.63414	−0.0585115
$781$	−3.61915	−0.129503
$782$	−44.8466	−1.60371
$783$	−32.2822	−1.15367
$784$	1.00000	0.0357143
$785$	15.1531	0.540836
$786$	30.7067	1.09527
$787$	35.5956	1.26885	0.634423	−	0.772986i	$-0.281238\pi$
$787$	35.5956	1.26885	0.634423	+	0.772986i	$0.281238\pi$
$788$	−26.1982	−0.933271
$789$	−28.9832	−1.03183
$790$	12.9077	0.459234
$791$	−13.3243	−0.473757
$792$	−1.21016	−0.0430013
$793$	9.87356	0.350620
$794$	33.4410	1.18678
$795$	8.29866	0.294323
$796$	−20.8975	−0.740692
$797$	14.3915	0.509775	0.254887	−	0.966971i	$-0.417962\pi$
$797$	14.3915	0.509775	0.254887	+	0.966971i	$0.417962\pi$
$798$	−8.86840	−0.313938
$799$	−33.0293	−1.16849
$800$	4.14613	0.146588
$801$	8.07826	0.285431
$802$	−23.9072	−0.844193
$803$	15.6053	0.550698
$804$	−16.4240	−0.579229
$805$	7.50686	0.264582
$806$	5.48138	0.193073
$807$	31.3732	1.10439
$808$	−8.88287	−0.312498
$809$	−26.5401	−0.933100	−0.466550	−	0.884495i	$-0.654503\pi$
$809$	−26.5401	−0.933100	−0.466550	+	0.884495i	$0.654503\pi$
$810$	−4.71222	−0.165571
$811$	−1.05070	−0.0368952	−0.0184476	−	0.999830i	$-0.505872\pi$
$811$	−1.05070	−0.0368952	−0.0184476	+	0.999830i	$0.505872\pi$
$812$	−5.70695	−0.200275
$813$	−7.94044	−0.278483
$814$	13.6778	0.479407
$815$	14.7897	0.518060
$816$	7.84588	0.274661
$817$	−16.5829	−0.580163
$818$	−20.5753	−0.719399
$819$	1.21942	0.0426098
$820$	6.61202	0.230902
$821$	−43.8747	−1.53124	−0.765618	−	0.643295i	$-0.777567\pi$
$821$	−43.8747	−1.53124	−0.765618	+	0.643295i	$0.777567\pi$
$822$	−33.1980	−1.15791
$823$	37.4006	1.30370	0.651852	−	0.758346i	$-0.273992\pi$
$823$	37.4006	1.30370	0.651852	+	0.758346i	$0.273992\pi$
$824$	13.8121	0.481169
$825$	7.27657	0.253338
$826$	−1.31252	−0.0456685
$827$	22.5052	0.782583	0.391292	−	0.920267i	$-0.372028\pi$
$827$	22.5052	0.782583	0.391292	+	0.920267i	$0.372028\pi$
$828$	−7.96151	−0.276682
$829$	−5.31678	−0.184659	−0.0923297	−	0.995728i	$-0.529431\pi$
$829$	−5.31678	−0.184659	−0.0923297	+	0.995728i	$0.529431\pi$
$830$	−9.92609	−0.344539
$831$	−34.2007	−1.18641
$832$	1.24428	0.0431376
$833$	5.52037	0.191269
$834$	19.6980	0.682085
$835$	16.5479	0.572665
$836$	7.70517	0.266489
$837$	24.9190	0.861328
$838$	−0.163933	−0.00566297
$839$	−22.0730	−0.762044	−0.381022	−	0.924566i	$-0.624428\pi$
$839$	−22.0730	−0.762044	−0.381022	+	0.924566i	$0.624428\pi$
$840$	−1.31332	−0.0453139
$841$	3.56930	0.123079
$842$	20.9150	0.720780
$843$	1.24032	0.0427190
$844$	13.3607	0.459896
$845$	10.5821	0.364034
$846$	−5.86361	−0.201595
$847$	9.47517	0.325571
$848$	−6.31884	−0.216990
$849$	−33.9402	−1.16483
$850$	22.8881	0.785057
$851$	89.9844	3.08463
$852$	4.16552	0.142709
$853$	16.6694	0.570751	0.285375	−	0.958416i	$-0.407882\pi$
$853$	16.6694	0.570751	0.285375	+	0.958416i	$0.407882\pi$
$854$	7.93517	0.271536
$855$	−5.65071	−0.193250
$856$	2.71171	0.0926845
$857$	−35.0786	−1.19826	−0.599132	−	0.800650i	$-0.704487\pi$
$857$	−35.0786	−1.19826	−0.599132	+	0.800650i	$0.704487\pi$
$858$	2.18374	0.0745518
$859$	39.4807	1.34706	0.673532	−	0.739158i	$-0.264776\pi$
$859$	39.4807	1.34706	0.673532	+	0.739158i	$0.264776\pi$
$860$	−2.45576	−0.0837409
$861$	10.1698	0.346584
$862$	1.00000	0.0340601
$863$	9.02143	0.307093	0.153547	−	0.988141i	$-0.450931\pi$
$863$	9.02143	0.307093	0.153547	+	0.988141i	$0.450931\pi$
$864$	5.65664	0.192443
$865$	6.21730	0.211395
$866$	−34.1978	−1.16209
$867$	19.1507	0.650393
$868$	4.40527	0.149525
$869$	−17.2489	−0.585129
$870$	7.49506	0.254106
$871$	−14.3788	−0.487206
$872$	−12.0521	−0.408136
$873$	−2.88534	−0.0976541
$874$	50.6912	1.71466
$875$	−8.45151	−0.285713
$876$	−17.9612	−0.606851
$877$	32.0080	1.08083	0.540417	−	0.841397i	$-0.318266\pi$
$877$	32.0080	1.08083	0.540417	+	0.841397i	$0.318266\pi$
$878$	−34.0288	−1.14842
$879$	−24.0168	−0.810067
$880$	1.14106	0.0384650
$881$	−39.5075	−1.33104	−0.665521	−	0.746379i	$-0.731791\pi$
$881$	−39.5075	−1.33104	−0.665521	+	0.746379i	$0.731791\pi$
$882$	0.980018	0.0329989
$883$	−48.0350	−1.61651	−0.808253	−	0.588836i	$-0.799587\pi$
$883$	−48.0350	−1.61651	−0.808253	+	0.588836i	$0.799587\pi$
$884$	6.86888	0.231025
$885$	1.72376	0.0579437
$886$	−12.1324	−0.407595
$887$	19.2344	0.645829	0.322915	−	0.946428i	$-0.395337\pi$
$887$	19.2344	0.645829	0.322915	+	0.946428i	$0.395337\pi$
$888$	−15.7427	−0.528291
$889$	2.60840	0.0874830
$890$	−7.61695	−0.255321
$891$	6.29708	0.210960
$892$	10.4305	0.349239
$893$	37.3338	1.24933
$894$	26.1636	0.875040
$895$	4.86378	0.162578
$896$	1.00000	0.0334077
$897$	14.3666	0.479685
$898$	25.7063	0.857830
$899$	−25.1406	−0.838488
$900$	4.06328	0.135443
$901$	−34.8823	−1.16210
$902$	−8.83583	−0.294201
$903$	−3.77714	−0.125695
$904$	−13.3243	−0.443159
$905$	−22.9625	−0.763299
$906$	−7.10545	−0.236063
$907$	0.937230	0.0311202	0.0155601	−	0.999879i	$-0.495047\pi$
$907$	0.937230	0.0311202	0.0155601	+	0.999879i	$0.495047\pi$
$908$	5.33359	0.177001
$909$	−8.70537	−0.288739
$910$	−1.14978	−0.0381148
$911$	−7.26591	−0.240730	−0.120365	−	0.992730i	$-0.538407\pi$
$911$	−7.26591	−0.240730	−0.120365	+	0.992730i	$0.538407\pi$
$912$	−8.86840	−0.293662
$913$	13.2645	0.438991
$914$	10.3282	0.341628
$915$	−10.4214	−0.344522
$916$	7.85457	0.259522
$917$	21.6053	0.713468
$918$	31.2268	1.03064
$919$	−48.6697	−1.60546	−0.802732	−	0.596340i	$-0.796621\pi$
$919$	−48.6697	−1.60546	−0.802732	+	0.596340i	$0.796621\pi$
$920$	7.50686	0.247494
$921$	40.1469	1.32289
$922$	0.368987	0.0121520
$923$	3.64681	0.120036
$924$	1.75503	0.0577362
$925$	−45.9249	−1.51000
$926$	−16.4178	−0.539522
$927$	13.5362	0.444586
$928$	−5.70695	−0.187340
$929$	−6.07663	−0.199368	−0.0996838	−	0.995019i	$-0.531783\pi$
$929$	−6.07663	−0.199368	−0.0996838	+	0.995019i	$0.531783\pi$
$930$	−5.78553	−0.189715
$931$	−6.23981	−0.204502
$932$	−14.0689	−0.460842
$933$	10.6645	0.349140
$934$	24.7403	0.809527
$935$	6.29906	0.206001
$936$	1.21942	0.0398578
$937$	−25.5976	−0.836238	−0.418119	−	0.908392i	$-0.637311\pi$
$937$	−25.5976	−0.836238	−0.418119	+	0.908392i	$0.637311\pi$
$938$	−11.5559	−0.377314
$939$	15.7900	0.515288
$940$	5.52876	0.180328
$941$	13.0262	0.424641	0.212320	−	0.977200i	$-0.431898\pi$
$941$	13.0262	0.424641	0.212320	+	0.977200i	$0.431898\pi$
$942$	23.3065	0.759366
$943$	−58.1297	−1.89296
$944$	−1.31252	−0.0427190
$945$	−5.22704	−0.170036
$946$	3.28171	0.106698
$947$	−52.5389	−1.70729	−0.853643	−	0.520859i	$-0.825612\pi$
$947$	−52.5389	−1.70729	−0.853643	+	0.520859i	$0.825612\pi$
$948$	19.8529	0.644793
$949$	−15.7245	−0.510440
$950$	−25.8710	−0.839367
$951$	−44.4434	−1.44118
$952$	5.52037	0.178916
$953$	−0.737425	−0.0238875	−0.0119438	−	0.999929i	$-0.503802\pi$
$953$	−0.737425	−0.0238875	−0.0119438	+	0.999929i	$0.503802\pi$
$954$	−6.19258	−0.200492
$955$	5.17263	0.167382
$956$	3.02851	0.0979490
$957$	−10.0159	−0.323767
$958$	17.5789	0.567948
$959$	−23.3581	−0.754273
$960$	−1.31332	−0.0423873
$961$	−11.5936	−0.373988
$962$	−13.7824	−0.444361
$963$	2.65753	0.0856377
$964$	−7.22390	−0.232666
$965$	3.90257	0.125628
$966$	11.5461	0.371490
$967$	−2.75959	−0.0887426	−0.0443713	−	0.999015i	$-0.514128\pi$
$967$	−2.75959	−0.0887426	−0.0443713	+	0.999015i	$0.514128\pi$
$968$	9.47517	0.304544
$969$	−48.9568	−1.57272
$970$	2.72058	0.0873524
$971$	57.5371	1.84645	0.923226	−	0.384256i	$-0.125542\pi$
$971$	57.5371	1.84645	0.923226	+	0.384256i	$0.125542\pi$
$972$	9.72220	0.311840
$973$	13.8595	0.444315
$974$	−2.98023	−0.0954929
$975$	−7.33219	−0.234818
$976$	7.93517	0.253999
$977$	−2.45581	−0.0785683	−0.0392841	−	0.999228i	$-0.512508\pi$
$977$	−2.45581	−0.0785683	−0.0392841	+	0.999228i	$0.512508\pi$
$978$	22.7476	0.727387
$979$	10.1787	0.325314
$980$	−0.924054	−0.0295178
$981$	−11.8113	−0.377106
$982$	−36.7961	−1.17421
$983$	54.1629	1.72753	0.863764	−	0.503897i	$-0.168101\pi$
$983$	54.1629	1.72753	0.863764	+	0.503897i	$0.168101\pi$
$984$	10.1698	0.324200
$985$	24.2085	0.771347
$986$	−31.5045	−1.00331
$987$	8.50363	0.270674
$988$	−7.76406	−0.247008
$989$	21.5899	0.686519
$990$	1.11826	0.0355405
$991$	29.5606	0.939024	0.469512	−	0.882926i	$-0.344430\pi$
$991$	29.5606	0.939024	0.469512	+	0.882926i	$0.344430\pi$
$992$	4.40527	0.139867
$993$	9.37816	0.297607
$994$	2.93087	0.0929614
$995$	19.3104	0.612181
$996$	−15.2670	−0.483754
$997$	54.2686	1.71870	0.859352	−	0.511385i	$-0.170867\pi$
$997$	54.2686	1.71870	0.859352	+	0.511385i	$0.170867\pi$
$998$	26.1265	0.827021
$999$	−62.6563	−1.98236

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6034.2.a.o.1.19	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6034.2.a.o.1.19	✓	25	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:	\( N \)	\(=\)	\( 6034 = 2 \cdot 7 \cdot 431 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.42126 q^{3} +1.00000 q^{4} -0.924054 q^{5} -1.42126 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.980018 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.42126 q^{3} +1.00000 q^{4} -0.924054 q^{5} -1.42126 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.980018 q^{9} +0.924054 q^{10} -1.23484 q^{11} +1.42126 q^{12} +1.24428 q^{13} +1.00000 q^{14} -1.31332 q^{15} +1.00000 q^{16} +5.52037 q^{17} +0.980018 q^{18} -6.23981 q^{19} -0.924054 q^{20} -1.42126 q^{21} +1.23484 q^{22} +8.12384 q^{23} -1.42126 q^{24} -4.14613 q^{25} -1.24428 q^{26} -5.65664 q^{27} -1.00000 q^{28} +5.70695 q^{29} +1.31332 q^{30} -4.40527 q^{31} -1.00000 q^{32} -1.75503 q^{33} -5.52037 q^{34} +0.924054 q^{35} -0.980018 q^{36} +11.0766 q^{37} +6.23981 q^{38} +1.76844 q^{39} +0.924054 q^{40} -7.15545 q^{41} +1.42126 q^{42} +2.65760 q^{43} -1.23484 q^{44} +0.905589 q^{45} -8.12384 q^{46} -5.98316 q^{47} +1.42126 q^{48} +1.00000 q^{49} +4.14613 q^{50} +7.84588 q^{51} +1.24428 q^{52} -6.31884 q^{53} +5.65664 q^{54} +1.14106 q^{55} +1.00000 q^{56} -8.86840 q^{57} -5.70695 q^{58} -1.31252 q^{59} -1.31332 q^{60} +7.93517 q^{61} +4.40527 q^{62} +0.980018 q^{63} +1.00000 q^{64} -1.14978 q^{65} +1.75503 q^{66} -11.5559 q^{67} +5.52037 q^{68} +11.5461 q^{69} -0.924054 q^{70} +2.93087 q^{71} +0.980018 q^{72} -12.6375 q^{73} -11.0766 q^{74} -5.89272 q^{75} -6.23981 q^{76} +1.23484 q^{77} -1.76844 q^{78} +13.9685 q^{79} -0.924054 q^{80} -5.09951 q^{81} +7.15545 q^{82} -10.7419 q^{83} -1.42126 q^{84} -5.10112 q^{85} -2.65760 q^{86} +8.11107 q^{87} +1.23484 q^{88} -8.24297 q^{89} -0.905589 q^{90} -1.24428 q^{91} +8.12384 q^{92} -6.26103 q^{93} +5.98316 q^{94} +5.76592 q^{95} -1.42126 q^{96} +2.94418 q^{97} -1.00000 q^{98} +1.21016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} + 4 q^{6} - 25 q^{7} - 25 q^{8} + 25 q^{9}+O(q^{10}) \) 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 + 4 * q^6 - 25 * q^7 - 25 * q^8 + 25 * q^9	\( 25 q - 25 q^{2} - 4 q^{3} + 25 q^{4} + 4 q^{6} - 25 q^{7} - 25 q^{8} + 25 q^{9} - 13 q^{11} - 4 q^{12} + 17 q^{13} + 25 q^{14} - 18 q^{15} + 25 q^{16} - 4 q^{17} - 25 q^{18} - 9 q^{19} + 4 q^{21} + 13 q^{22} - 14 q^{23} + 4 q^{24} + 23 q^{25} - 17 q^{26} - 7 q^{27} - 25 q^{28} - 4 q^{29} + 18 q^{30} - 15 q^{31} - 25 q^{32} - 15 q^{33} + 4 q^{34} + 25 q^{36} + 13 q^{37} + 9 q^{38} - 31 q^{39} - 31 q^{41} - 4 q^{42} + 29 q^{43} - 13 q^{44} + 10 q^{45} + 14 q^{46} - 31 q^{47} - 4 q^{48} + 25 q^{49} - 23 q^{50} - 9 q^{51} + 17 q^{52} + 23 q^{53} + 7 q^{54} - 48 q^{55} + 25 q^{56} + 32 q^{57} + 4 q^{58} - 50 q^{59} - 18 q^{60} - 2 q^{61} + 15 q^{62} - 25 q^{63} + 25 q^{64} - 4 q^{65} + 15 q^{66} - 8 q^{67} - 4 q^{68} - 57 q^{69} - 61 q^{71} - 25 q^{72} + 31 q^{73} - 13 q^{74} - 21 q^{75} - 9 q^{76} + 13 q^{77} + 31 q^{78} - 10 q^{79} + 61 q^{81} + 31 q^{82} - 47 q^{83} + 4 q^{84} + 2 q^{85} - 29 q^{86} + 17 q^{87} + 13 q^{88} - 44 q^{89} - 10 q^{90} - 17 q^{91} - 14 q^{92} - 13 q^{93} + 31 q^{94} - 7 q^{95} + 4 q^{96} + 10 q^{97} - 25 q^{98} - 47 q^{99}+O(q^{100}) \) 25 * q - 25 * q^2 - 4 * q^3 + 25 * q^4 + 4 * q^6 - 25 * q^7 - 25 * q^8 + 25 * q^9 - 13 * q^11 - 4 * q^12 + 17 * q^13 + 25 * q^14 - 18 * q^15 + 25 * q^16 - 4 * q^17 - 25 * q^18 - 9 * q^19 + 4 * q^21 + 13 * q^22 - 14 * q^23 + 4 * q^24 + 23 * q^25 - 17 * q^26 - 7 * q^27 - 25 * q^28 - 4 * q^29 + 18 * q^30 - 15 * q^31 - 25 * q^32 - 15 * q^33 + 4 * q^34 + 25 * q^36 + 13 * q^37 + 9 * q^38 - 31 * q^39 - 31 * q^41 - 4 * q^42 + 29 * q^43 - 13 * q^44 + 10 * q^45 + 14 * q^46 - 31 * q^47 - 4 * q^48 + 25 * q^49 - 23 * q^50 - 9 * q^51 + 17 * q^52 + 23 * q^53 + 7 * q^54 - 48 * q^55 + 25 * q^56 + 32 * q^57 + 4 * q^58 - 50 * q^59 - 18 * q^60 - 2 * q^61 + 15 * q^62 - 25 * q^63 + 25 * q^64 - 4 * q^65 + 15 * q^66 - 8 * q^67 - 4 * q^68 - 57 * q^69 - 61 * q^71 - 25 * q^72 + 31 * q^73 - 13 * q^74 - 21 * q^75 - 9 * q^76 + 13 * q^77 + 31 * q^78 - 10 * q^79 + 61 * q^81 + 31 * q^82 - 47 * q^83 + 4 * q^84 + 2 * q^85 - 29 * q^86 + 17 * q^87 + 13 * q^88 - 44 * q^89 - 10 * q^90 - 17 * q^91 - 14 * q^92 - 13 * q^93 + 31 * q^94 - 7 * q^95 + 4 * q^96 + 10 * q^97 - 25 * q^98 - 47 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more