LMFDB - Embedded newform 4030.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 15x^{6} + 31x^{5} + 79x^{4} - 85x^{3} - 162x^{2} + 45x + 81 $ x^8 - 3x^7 - 15x^6 + 31x^5 + 79x^4 - 85x^3 - 162x^2 + 45*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.49715$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.49715 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.49715 q^{6} +2.81472 q^{7} -1.00000 q^{8} -0.758530 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.49715 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.49715 q^{6} +2.81472 q^{7} -1.00000 q^{8} -0.758530 q^{9} -1.00000 q^{10} +2.95950 q^{11} -1.49715 q^{12} +1.00000 q^{13} -2.81472 q^{14} -1.49715 q^{15} +1.00000 q^{16} +2.99456 q^{17} +0.758530 q^{18} +2.33488 q^{19} +1.00000 q^{20} -4.21407 q^{21} -2.95950 q^{22} +3.19084 q^{23} +1.49715 q^{24} +1.00000 q^{25} -1.00000 q^{26} +5.62710 q^{27} +2.81472 q^{28} -2.38142 q^{29} +1.49715 q^{30} +1.00000 q^{31} -1.00000 q^{32} -4.43083 q^{33} -2.99456 q^{34} +2.81472 q^{35} -0.758530 q^{36} -7.48320 q^{37} -2.33488 q^{38} -1.49715 q^{39} -1.00000 q^{40} +6.76903 q^{41} +4.21407 q^{42} +5.25457 q^{43} +2.95950 q^{44} -0.758530 q^{45} -3.19084 q^{46} +10.1282 q^{47} -1.49715 q^{48} +0.922665 q^{49} -1.00000 q^{50} -4.48331 q^{51} +1.00000 q^{52} -11.1215 q^{53} -5.62710 q^{54} +2.95950 q^{55} -2.81472 q^{56} -3.49567 q^{57} +2.38142 q^{58} +4.39835 q^{59} -1.49715 q^{60} +13.5705 q^{61} -1.00000 q^{62} -2.13505 q^{63} +1.00000 q^{64} +1.00000 q^{65} +4.43083 q^{66} -6.95307 q^{67} +2.99456 q^{68} -4.77718 q^{69} -2.81472 q^{70} -8.34019 q^{71} +0.758530 q^{72} +11.4507 q^{73} +7.48320 q^{74} -1.49715 q^{75} +2.33488 q^{76} +8.33017 q^{77} +1.49715 q^{78} -7.66425 q^{79} +1.00000 q^{80} -6.14904 q^{81} -6.76903 q^{82} -0.750736 q^{83} -4.21407 q^{84} +2.99456 q^{85} -5.25457 q^{86} +3.56536 q^{87} -2.95950 q^{88} +0.992853 q^{89} +0.758530 q^{90} +2.81472 q^{91} +3.19084 q^{92} -1.49715 q^{93} -10.1282 q^{94} +2.33488 q^{95} +1.49715 q^{96} +5.46530 q^{97} -0.922665 q^{98} -2.24487 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 5 q^{3} + 8 q^{4} + 8 q^{5} - 5 q^{6} + 5 q^{7} - 8 q^{8} + 17 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 5 * q^3 + 8 * q^4 + 8 * q^5 - 5 * q^6 + 5 * q^7 - 8 * q^8 + 17 * q^9	\( 8 q - 8 q^{2} + 5 q^{3} + 8 q^{4} + 8 q^{5} - 5 q^{6} + 5 q^{7} - 8 q^{8} + 17 q^{9} - 8 q^{10} + 4 q^{11} + 5 q^{12} + 8 q^{13} - 5 q^{14} + 5 q^{15} + 8 q^{16} + 19 q^{17} - 17 q^{18} - 14 q^{19} + 8 q^{20} + 3 q^{21} - 4 q^{22} + 8 q^{23} - 5 q^{24} + 8 q^{25} - 8 q^{26} + 17 q^{27} + 5 q^{28} + 21 q^{29} - 5 q^{30} + 8 q^{31} - 8 q^{32} + 4 q^{33} - 19 q^{34} + 5 q^{35} + 17 q^{36} - 3 q^{37} + 14 q^{38} + 5 q^{39} - 8 q^{40} - 8 q^{41} - 3 q^{42} + 25 q^{43} + 4 q^{44} + 17 q^{45} - 8 q^{46} + 11 q^{47} + 5 q^{48} + 15 q^{49} - 8 q^{50} + 7 q^{51} + 8 q^{52} + 2 q^{53} - 17 q^{54} + 4 q^{55} - 5 q^{56} + 9 q^{57} - 21 q^{58} - 22 q^{59} + 5 q^{60} - 2 q^{61} - 8 q^{62} + 30 q^{63} + 8 q^{64} + 8 q^{65} - 4 q^{66} - 14 q^{67} + 19 q^{68} + 36 q^{69} - 5 q^{70} + 4 q^{71} - 17 q^{72} + 17 q^{73} + 3 q^{74} + 5 q^{75} - 14 q^{76} + 17 q^{77} - 5 q^{78} - 12 q^{79} + 8 q^{80} + 40 q^{81} + 8 q^{82} + 21 q^{83} + 3 q^{84} + 19 q^{85} - 25 q^{86} + 25 q^{87} - 4 q^{88} - 25 q^{89} - 17 q^{90} + 5 q^{91} + 8 q^{92} + 5 q^{93} - 11 q^{94} - 14 q^{95} - 5 q^{96} + 8 q^{97} - 15 q^{98} - 57 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 5 * q^3 + 8 * q^4 + 8 * q^5 - 5 * q^6 + 5 * q^7 - 8 * q^8 + 17 * q^9 - 8 * q^10 + 4 * q^11 + 5 * q^12 + 8 * q^13 - 5 * q^14 + 5 * q^15 + 8 * q^16 + 19 * q^17 - 17 * q^18 - 14 * q^19 + 8 * q^20 + 3 * q^21 - 4 * q^22 + 8 * q^23 - 5 * q^24 + 8 * q^25 - 8 * q^26 + 17 * q^27 + 5 * q^28 + 21 * q^29 - 5 * q^30 + 8 * q^31 - 8 * q^32 + 4 * q^33 - 19 * q^34 + 5 * q^35 + 17 * q^36 - 3 * q^37 + 14 * q^38 + 5 * q^39 - 8 * q^40 - 8 * q^41 - 3 * q^42 + 25 * q^43 + 4 * q^44 + 17 * q^45 - 8 * q^46 + 11 * q^47 + 5 * q^48 + 15 * q^49 - 8 * q^50 + 7 * q^51 + 8 * q^52 + 2 * q^53 - 17 * q^54 + 4 * q^55 - 5 * q^56 + 9 * q^57 - 21 * q^58 - 22 * q^59 + 5 * q^60 - 2 * q^61 - 8 * q^62 + 30 * q^63 + 8 * q^64 + 8 * q^65 - 4 * q^66 - 14 * q^67 + 19 * q^68 + 36 * q^69 - 5 * q^70 + 4 * q^71 - 17 * q^72 + 17 * q^73 + 3 * q^74 + 5 * q^75 - 14 * q^76 + 17 * q^77 - 5 * q^78 - 12 * q^79 + 8 * q^80 + 40 * q^81 + 8 * q^82 + 21 * q^83 + 3 * q^84 + 19 * q^85 - 25 * q^86 + 25 * q^87 - 4 * q^88 - 25 * q^89 - 17 * q^90 + 5 * q^91 + 8 * q^92 + 5 * q^93 - 11 * q^94 - 14 * q^95 - 5 * q^96 + 8 * q^97 - 15 * q^98 - 57 * 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
$3$	−1.49715	−0.864382	−0.432191	−	0.901782i	$-0.642259\pi$
$3$	−1.49715	−0.864382	−0.432191	+	0.901782i	$0.642259\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.49715	0.611211
$7$	2.81472	1.06387	0.531933	−	0.846787i	$-0.321466\pi$
$7$	2.81472	1.06387	0.531933	+	0.846787i	$0.321466\pi$
$8$	−1.00000	−0.353553
$9$	−0.758530	−0.252843
$10$	−1.00000	−0.316228
$11$	2.95950	0.892323	0.446161	−	0.894952i	$-0.352791\pi$
$11$	2.95950	0.892323	0.446161	+	0.894952i	$0.352791\pi$
$12$	−1.49715	−0.432191
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.81472	−0.752266
$15$	−1.49715	−0.386563
$16$	1.00000	0.250000
$17$	2.99456	0.726287	0.363143	−	0.931733i	$-0.381703\pi$
$17$	2.99456	0.726287	0.363143	+	0.931733i	$0.381703\pi$
$18$	0.758530	0.178787
$19$	2.33488	0.535658	0.267829	−	0.963466i	$-0.413694\pi$
$19$	2.33488	0.535658	0.267829	+	0.963466i	$0.413694\pi$
$20$	1.00000	0.223607
$21$	−4.21407	−0.919586
$22$	−2.95950	−0.630968
$23$	3.19084	0.665337	0.332668	−	0.943044i	$-0.392051\pi$
$23$	3.19084	0.665337	0.332668	+	0.943044i	$0.392051\pi$
$24$	1.49715	0.305605
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	5.62710	1.08294
$28$	2.81472	0.531933
$29$	−2.38142	−0.442219	−0.221110	−	0.975249i	$-0.570968\pi$
$29$	−2.38142	−0.442219	−0.221110	+	0.975249i	$0.570968\pi$
$30$	1.49715	0.273342
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	−4.43083	−0.771308
$34$	−2.99456	−0.513562
$35$	2.81472	0.475775
$36$	−0.758530	−0.126422
$37$	−7.48320	−1.23023	−0.615115	−	0.788438i	$-0.710890\pi$
$37$	−7.48320	−1.23023	−0.615115	+	0.788438i	$0.710890\pi$
$38$	−2.33488	−0.378767
$39$	−1.49715	−0.239736
$40$	−1.00000	−0.158114
$41$	6.76903	1.05714	0.528572	−	0.848888i	$-0.322728\pi$
$41$	6.76903	1.05714	0.528572	+	0.848888i	$0.322728\pi$
$42$	4.21407	0.650246
$43$	5.25457	0.801315	0.400657	−	0.916228i	$-0.368782\pi$
$43$	5.25457	0.801315	0.400657	+	0.916228i	$0.368782\pi$
$44$	2.95950	0.446161
$45$	−0.758530	−0.113075
$46$	−3.19084	−0.470464
$47$	10.1282	1.47735	0.738676	−	0.674061i	$-0.235451\pi$
$47$	10.1282	1.47735	0.738676	+	0.674061i	$0.235451\pi$
$48$	−1.49715	−0.216096
$49$	0.922665	0.131809
$50$	−1.00000	−0.141421
$51$	−4.48331	−0.627790
$52$	1.00000	0.138675
$53$	−11.1215	−1.52766	−0.763830	−	0.645417i	$-0.776684\pi$
$53$	−11.1215	−1.52766	−0.763830	+	0.645417i	$0.776684\pi$
$54$	−5.62710	−0.765751
$55$	2.95950	0.399059
$56$	−2.81472	−0.376133
$57$	−3.49567	−0.463013
$58$	2.38142	0.312696
$59$	4.39835	0.572616	0.286308	−	0.958138i	$-0.407572\pi$
$59$	4.39835	0.572616	0.286308	+	0.958138i	$0.407572\pi$
$60$	−1.49715	−0.193282
$61$	13.5705	1.73753	0.868765	−	0.495225i	$-0.164914\pi$
$61$	13.5705	1.73753	0.868765	+	0.495225i	$0.164914\pi$
$62$	−1.00000	−0.127000
$63$	−2.13505	−0.268991
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	4.43083	0.545397
$67$	−6.95307	−0.849453	−0.424726	−	0.905322i	$-0.639630\pi$
$67$	−6.95307	−0.849453	−0.424726	+	0.905322i	$0.639630\pi$
$68$	2.99456	0.363143
$69$	−4.77718	−0.575105
$70$	−2.81472	−0.336424
$71$	−8.34019	−0.989799	−0.494899	−	0.868950i	$-0.664795\pi$
$71$	−8.34019	−0.989799	−0.494899	+	0.868950i	$0.664795\pi$
$72$	0.758530	0.0893937
$73$	11.4507	1.34021	0.670103	−	0.742269i	$-0.266250\pi$
$73$	11.4507	1.34021	0.670103	+	0.742269i	$0.266250\pi$
$74$	7.48320	0.869904
$75$	−1.49715	−0.172876
$76$	2.33488	0.267829
$77$	8.33017	0.949311
$78$	1.49715	0.169519
$79$	−7.66425	−0.862296	−0.431148	−	0.902281i	$-0.641891\pi$
$79$	−7.66425	−0.862296	−0.431148	+	0.902281i	$0.641891\pi$
$80$	1.00000	0.111803
$81$	−6.14904	−0.683227
$82$	−6.76903	−0.747514
$83$	−0.750736	−0.0824040	−0.0412020	−	0.999151i	$-0.513119\pi$
$83$	−0.750736	−0.0824040	−0.0412020	+	0.999151i	$0.513119\pi$
$84$	−4.21407	−0.459793
$85$	2.99456	0.324805
$86$	−5.25457	−0.566615
$87$	3.56536	0.382246
$88$	−2.95950	−0.315484
$89$	0.992853	0.105242	0.0526211	−	0.998615i	$-0.483242\pi$
$89$	0.992853	0.105242	0.0526211	+	0.998615i	$0.483242\pi$
$90$	0.758530	0.0799561
$91$	2.81472	0.295063
$92$	3.19084	0.332668
$93$	−1.49715	−0.155248
$94$	−10.1282	−1.04465
$95$	2.33488	0.239554
$96$	1.49715	0.152803
$97$	5.46530	0.554917	0.277459	−	0.960738i	$-0.410508\pi$
$97$	5.46530	0.554917	0.277459	+	0.960738i	$0.410508\pi$
$98$	−0.922665	−0.0932032
$99$	−2.24487	−0.225618
$100$	1.00000	0.100000
$101$	−5.13258	−0.510711	−0.255355	−	0.966847i	$-0.582193\pi$
$101$	−5.13258	−0.510711	−0.255355	+	0.966847i	$0.582193\pi$
$102$	4.48331	0.443914
$103$	−2.73405	−0.269394	−0.134697	−	0.990887i	$-0.543006\pi$
$103$	−2.73405	−0.269394	−0.134697	+	0.990887i	$0.543006\pi$
$104$	−1.00000	−0.0980581
$105$	−4.21407	−0.411251
$106$	11.1215	1.08022
$107$	12.4714	1.20565	0.602826	−	0.797872i	$-0.294041\pi$
$107$	12.4714	1.20565	0.602826	+	0.797872i	$0.294041\pi$
$108$	5.62710	0.541468
$109$	−13.7146	−1.31362	−0.656809	−	0.754057i	$-0.728094\pi$
$109$	−13.7146	−1.31362	−0.656809	+	0.754057i	$0.728094\pi$
$110$	−2.95950	−0.282177
$111$	11.2035	1.06339
$112$	2.81472	0.265966
$113$	−6.07708	−0.571683	−0.285842	−	0.958277i	$-0.592273\pi$
$113$	−6.07708	−0.571683	−0.285842	+	0.958277i	$0.592273\pi$
$114$	3.49567	0.327400
$115$	3.19084	0.297548
$116$	−2.38142	−0.221110
$117$	−0.758530	−0.0701262
$118$	−4.39835	−0.404901
$119$	8.42885	0.772671
$120$	1.49715	0.136671
$121$	−2.24136	−0.203760
$122$	−13.5705	−1.22862
$123$	−10.1343	−0.913777
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	2.13505	0.190206
$127$	−11.6515	−1.03390	−0.516950	−	0.856016i	$-0.672933\pi$
$127$	−11.6515	−1.03390	−0.516950	+	0.856016i	$0.672933\pi$
$128$	−1.00000	−0.0883883
$129$	−7.86690	−0.692642
$130$	−1.00000	−0.0877058
$131$	10.3169	0.901391	0.450695	−	0.892678i	$-0.351176\pi$
$131$	10.3169	0.901391	0.450695	+	0.892678i	$0.351176\pi$
$132$	−4.43083	−0.385654
$133$	6.57204	0.569868
$134$	6.95307	0.600654
$135$	5.62710	0.484304
$136$	−2.99456	−0.256781
$137$	23.3356	1.99370	0.996848	−	0.0793359i	$-0.0252800\pi$
$137$	23.3356	1.99370	0.996848	+	0.0793359i	$0.0252800\pi$
$138$	4.77718	0.406661
$139$	−17.1680	−1.45617	−0.728086	−	0.685486i	$-0.759590\pi$
$139$	−17.1680	−1.45617	−0.728086	+	0.685486i	$0.759590\pi$
$140$	2.81472	0.237887
$141$	−15.1635	−1.27700
$142$	8.34019	0.699893
$143$	2.95950	0.247486
$144$	−0.758530	−0.0632109
$145$	−2.38142	−0.197766
$146$	−11.4507	−0.947668
$147$	−1.38137	−0.113934
$148$	−7.48320	−0.615115
$149$	−17.9443	−1.47005	−0.735027	−	0.678038i	$-0.762831\pi$
$149$	−17.9443	−1.47005	−0.735027	+	0.678038i	$0.762831\pi$
$150$	1.49715	0.122242
$151$	−19.5620	−1.59193	−0.795967	−	0.605340i	$-0.793037\pi$
$151$	−19.5620	−1.59193	−0.795967	+	0.605340i	$0.793037\pi$
$152$	−2.33488	−0.189384
$153$	−2.27146	−0.183637
$154$	−8.33017	−0.671265
$155$	1.00000	0.0803219
$156$	−1.49715	−0.119868
$157$	−0.705421	−0.0562988	−0.0281494	−	0.999604i	$-0.508961\pi$
$157$	−0.705421	−0.0562988	−0.0281494	+	0.999604i	$0.508961\pi$
$158$	7.66425	0.609735
$159$	16.6507	1.32048
$160$	−1.00000	−0.0790569
$161$	8.98134	0.707829
$162$	6.14904	0.483114
$163$	21.6973	1.69946	0.849731	−	0.527217i	$-0.176764\pi$
$163$	21.6973	1.69946	0.849731	+	0.527217i	$0.176764\pi$
$164$	6.76903	0.528572
$165$	−4.43083	−0.344939
$166$	0.750736	0.0582684
$167$	3.59100	0.277880	0.138940	−	0.990301i	$-0.455630\pi$
$167$	3.59100	0.277880	0.138940	+	0.990301i	$0.455630\pi$
$168$	4.21407	0.325123
$169$	1.00000	0.0769231
$170$	−2.99456	−0.229672
$171$	−1.77108	−0.135438
$172$	5.25457	0.400657
$173$	−12.3950	−0.942377	−0.471188	−	0.882033i	$-0.656175\pi$
$173$	−12.3950	−0.942377	−0.471188	+	0.882033i	$0.656175\pi$
$174$	−3.56536	−0.270289
$175$	2.81472	0.212773
$176$	2.95950	0.223081
$177$	−6.58501	−0.494959
$178$	−0.992853	−0.0744175
$179$	−23.2090	−1.73472	−0.867362	−	0.497678i	$-0.834186\pi$
$179$	−23.2090	−1.73472	−0.867362	+	0.497678i	$0.834186\pi$
$180$	−0.758530	−0.0565375
$181$	17.1833	1.27723	0.638614	−	0.769527i	$-0.279508\pi$
$181$	17.1833	1.27723	0.638614	+	0.769527i	$0.279508\pi$
$182$	−2.81472	−0.208641
$183$	−20.3172	−1.50189
$184$	−3.19084	−0.235232
$185$	−7.48320	−0.550175
$186$	1.49715	0.109777
$187$	8.86240	0.648083
$188$	10.1282	0.738676
$189$	15.8387	1.15210
$190$	−2.33488	−0.169390
$191$	−11.7531	−0.850428	−0.425214	−	0.905093i	$-0.639801\pi$
$191$	−11.7531	−0.850428	−0.425214	+	0.905093i	$0.639801\pi$
$192$	−1.49715	−0.108048
$193$	17.2256	1.23992	0.619962	−	0.784631i	$-0.287148\pi$
$193$	17.2256	1.23992	0.619962	+	0.784631i	$0.287148\pi$
$194$	−5.46530	−0.392386
$195$	−1.49715	−0.107213
$196$	0.922665	0.0659046
$197$	15.6628	1.11593	0.557963	−	0.829866i	$-0.311583\pi$
$197$	15.6628	1.11593	0.557963	+	0.829866i	$0.311583\pi$
$198$	2.24487	0.159536
$199$	−8.59364	−0.609187	−0.304594	−	0.952482i	$-0.598521\pi$
$199$	−8.59364	−0.609187	−0.304594	+	0.952482i	$0.598521\pi$
$200$	−1.00000	−0.0707107
$201$	10.4098	0.734252
$202$	5.13258	0.361127
$203$	−6.70304	−0.470461
$204$	−4.48331	−0.313895
$205$	6.76903	0.472770
$206$	2.73405	0.190490
$207$	−2.42035	−0.168226
$208$	1.00000	0.0693375
$209$	6.91008	0.477980
$210$	4.21407	0.290799
$211$	−26.0056	−1.79030	−0.895149	−	0.445768i	$-0.852931\pi$
$211$	−26.0056	−1.79030	−0.895149	+	0.445768i	$0.852931\pi$
$212$	−11.1215	−0.763830
$213$	12.4866	0.855564
$214$	−12.4714	−0.852525
$215$	5.25457	0.358359
$216$	−5.62710	−0.382876
$217$	2.81472	0.191076
$218$	13.7146	0.928868
$219$	−17.1435	−1.15845
$220$	2.95950	0.199529
$221$	2.99456	0.201436
$222$	−11.2035	−0.751929
$223$	−2.70445	−0.181103	−0.0905517	−	0.995892i	$-0.528863\pi$
$223$	−2.70445	−0.181103	−0.0905517	+	0.995892i	$0.528863\pi$
$224$	−2.81472	−0.188067
$225$	−0.758530	−0.0505687
$226$	6.07708	0.404241
$227$	13.2238	0.877691	0.438846	−	0.898562i	$-0.355387\pi$
$227$	13.2238	0.877691	0.438846	+	0.898562i	$0.355387\pi$
$228$	−3.49567	−0.231507
$229$	−2.49450	−0.164841	−0.0824207	−	0.996598i	$-0.526265\pi$
$229$	−2.49450	−0.164841	−0.0824207	+	0.996598i	$0.526265\pi$
$230$	−3.19084	−0.210398
$231$	−12.4716	−0.820568
$232$	2.38142	0.156348
$233$	15.6255	1.02366	0.511831	−	0.859086i	$-0.328967\pi$
$233$	15.6255	1.02366	0.511831	+	0.859086i	$0.328967\pi$
$234$	0.758530	0.0495867
$235$	10.1282	0.660692
$236$	4.39835	0.286308
$237$	11.4746	0.745353
$238$	−8.42885	−0.546361
$239$	−7.44874	−0.481819	−0.240910	−	0.970548i	$-0.577446\pi$
$239$	−7.44874	−0.481819	−0.240910	+	0.970548i	$0.577446\pi$
$240$	−1.49715	−0.0966409
$241$	20.3723	1.31229	0.656146	−	0.754634i	$-0.272185\pi$
$241$	20.3723	1.31229	0.656146	+	0.754634i	$0.272185\pi$
$242$	2.24136	0.144080
$243$	−7.67523	−0.492367
$244$	13.5705	0.868765
$245$	0.922665	0.0589469
$246$	10.1343	0.646138
$247$	2.33488	0.148565
$248$	−1.00000	−0.0635001
$249$	1.12397	0.0712285
$250$	−1.00000	−0.0632456
$251$	29.0433	1.83319	0.916597	−	0.399813i	$-0.130925\pi$
$251$	29.0433	1.83319	0.916597	+	0.399813i	$0.130925\pi$
$252$	−2.13505	−0.134496
$253$	9.44330	0.593695
$254$	11.6515	0.731077
$255$	−4.48331	−0.280756
$256$	1.00000	0.0625000
$257$	19.8324	1.23711	0.618555	−	0.785741i	$-0.287718\pi$
$257$	19.8324	1.23711	0.618555	+	0.785741i	$0.287718\pi$
$258$	7.86690	0.489772
$259$	−21.0631	−1.30880
$260$	1.00000	0.0620174
$261$	1.80638	0.111812
$262$	−10.3169	−0.637379
$263$	24.9639	1.53934	0.769670	−	0.638442i	$-0.220421\pi$
$263$	24.9639	1.53934	0.769670	+	0.638442i	$0.220421\pi$
$264$	4.43083	0.272699
$265$	−11.1215	−0.683191
$266$	−6.57204	−0.402958
$267$	−1.48645	−0.0909695
$268$	−6.95307	−0.424726
$269$	21.0577	1.28391	0.641955	−	0.766742i	$-0.278124\pi$
$269$	21.0577	1.28391	0.641955	+	0.766742i	$0.278124\pi$
$270$	−5.62710	−0.342454
$271$	−30.8155	−1.87191	−0.935955	−	0.352119i	$-0.885461\pi$
$271$	−30.8155	−1.87191	−0.935955	+	0.352119i	$0.885461\pi$
$272$	2.99456	0.181572
$273$	−4.21407	−0.255047
$274$	−23.3356	−1.40976
$275$	2.95950	0.178465
$276$	−4.77718	−0.287553
$277$	24.1572	1.45146	0.725731	−	0.687978i	$-0.241502\pi$
$277$	24.1572	1.45146	0.725731	+	0.687978i	$0.241502\pi$
$278$	17.1680	1.02967
$279$	−0.758530	−0.0454120
$280$	−2.81472	−0.168212
$281$	26.1605	1.56061	0.780303	−	0.625401i	$-0.215065\pi$
$281$	26.1605	1.56061	0.780303	+	0.625401i	$0.215065\pi$
$282$	15.1635	0.902973
$283$	24.5218	1.45767	0.728834	−	0.684691i	$-0.240063\pi$
$283$	24.5218	1.45767	0.728834	+	0.684691i	$0.240063\pi$
$284$	−8.34019	−0.494899
$285$	−3.49567	−0.207066
$286$	−2.95950	−0.174999
$287$	19.0529	1.12466
$288$	0.758530	0.0446968
$289$	−8.03262	−0.472507
$290$	2.38142	0.139842
$291$	−8.18240	−0.479661
$292$	11.4507	0.670103
$293$	−5.46263	−0.319130	−0.159565	−	0.987187i	$-0.551009\pi$
$293$	−5.46263	−0.319130	−0.159565	+	0.987187i	$0.551009\pi$
$294$	1.38137	0.0805632
$295$	4.39835	0.256082
$296$	7.48320	0.434952
$297$	16.6534	0.966328
$298$	17.9443	1.03949
$299$	3.19084	0.184531
$300$	−1.49715	−0.0864382
$301$	14.7902	0.852491
$302$	19.5620	1.12567
$303$	7.68426	0.441449
$304$	2.33488	0.133915
$305$	13.5705	0.777047
$306$	2.27146	0.129851
$307$	−2.10122	−0.119923	−0.0599616	−	0.998201i	$-0.519098\pi$
$307$	−2.10122	−0.119923	−0.0599616	+	0.998201i	$0.519098\pi$
$308$	8.33017	0.474656
$309$	4.09329	0.232859
$310$	−1.00000	−0.0567962
$311$	14.5908	0.827367	0.413683	−	0.910421i	$-0.364242\pi$
$311$	14.5908	0.827367	0.413683	+	0.910421i	$0.364242\pi$
$312$	1.49715	0.0847596
$313$	−31.4146	−1.77566	−0.887828	−	0.460175i	$-0.847787\pi$
$313$	−31.4146	−1.77566	−0.887828	+	0.460175i	$0.847787\pi$
$314$	0.705421	0.0398092
$315$	−2.13505	−0.120297
$316$	−7.66425	−0.431148
$317$	−28.4661	−1.59881	−0.799407	−	0.600790i	$-0.794853\pi$
$317$	−28.4661	−1.59881	−0.799407	+	0.600790i	$0.794853\pi$
$318$	−16.6507	−0.933722
$319$	−7.04782	−0.394602
$320$	1.00000	0.0559017
$321$	−18.6716	−1.04214
$322$	−8.98134	−0.500511
$323$	6.99193	0.389041
$324$	−6.14904	−0.341613
$325$	1.00000	0.0554700
$326$	−21.6973	−1.20170
$327$	20.5328	1.13547
$328$	−6.76903	−0.373757
$329$	28.5081	1.57170
$330$	4.43083	0.243909
$331$	35.3026	1.94041	0.970204	−	0.242290i	$-0.0778987\pi$
$331$	35.3026	1.94041	0.970204	+	0.242290i	$0.0778987\pi$
$332$	−0.750736	−0.0412020
$333$	5.67623	0.311056
$334$	−3.59100	−0.196491
$335$	−6.95307	−0.379887
$336$	−4.21407	−0.229897
$337$	−12.4720	−0.679391	−0.339695	−	0.940536i	$-0.610324\pi$
$337$	−12.4720	−0.679391	−0.339695	+	0.940536i	$0.610324\pi$
$338$	−1.00000	−0.0543928
$339$	9.09832	0.494153
$340$	2.99456	0.162403
$341$	2.95950	0.160266
$342$	1.77108	0.0957689
$343$	−17.1060	−0.923638
$344$	−5.25457	−0.283308
$345$	−4.77718	−0.257195
$346$	12.3950	0.666361
$347$	−12.1676	−0.653188	−0.326594	−	0.945165i	$-0.605901\pi$
$347$	−12.1676	−0.653188	−0.326594	+	0.945165i	$0.605901\pi$
$348$	3.56536	0.191123
$349$	−0.308320	−0.0165040	−0.00825199	−	0.999966i	$-0.502627\pi$
$349$	−0.308320	−0.0165040	−0.00825199	+	0.999966i	$0.502627\pi$
$350$	−2.81472	−0.150453
$351$	5.62710	0.300352
$352$	−2.95950	−0.157742
$353$	21.3769	1.13778	0.568890	−	0.822414i	$-0.307373\pi$
$353$	21.3769	1.13778	0.568890	+	0.822414i	$0.307373\pi$
$354$	6.58501	0.349989
$355$	−8.34019	−0.442651
$356$	0.992853	0.0526211
$357$	−12.6193	−0.667883
$358$	23.2090	1.22664
$359$	−0.00165198	−8.71882e−5 0	−4.35941e−5	−	1.00000i	$-0.500014\pi$
$359$	−0.00165198	−8.71882e−5 0	−4.35941e−5		1.00000i	$0.500014\pi$
$360$	0.758530	0.0399781
$361$	−13.5483	−0.713070
$362$	−17.1833	−0.903136
$363$	3.35566	0.176126
$364$	2.81472	0.147532
$365$	11.4507	0.599358
$366$	20.3172	1.06200
$367$	10.3573	0.540648	0.270324	−	0.962769i	$-0.412869\pi$
$367$	10.3573	0.540648	0.270324	+	0.962769i	$0.412869\pi$
$368$	3.19084	0.166334
$369$	−5.13451	−0.267292
$370$	7.48320	0.389033
$371$	−31.3040	−1.62523
$372$	−1.49715	−0.0776238
$373$	−32.2168	−1.66813	−0.834063	−	0.551670i	$-0.813991\pi$
$373$	−32.2168	−1.66813	−0.834063	+	0.551670i	$0.813991\pi$
$374$	−8.86240	−0.458264
$375$	−1.49715	−0.0773127
$376$	−10.1282	−0.522323
$377$	−2.38142	−0.122649
$378$	−15.8387	−0.814656
$379$	37.3530	1.91870	0.959349	−	0.282224i	$-0.0910721\pi$
$379$	37.3530	1.91870	0.959349	+	0.282224i	$0.0910721\pi$
$380$	2.33488	0.119777
$381$	17.4440	0.893684
$382$	11.7531	0.601343
$383$	33.5870	1.71622	0.858108	−	0.513470i	$-0.171640\pi$
$383$	33.5870	1.71622	0.858108	+	0.513470i	$0.171640\pi$
$384$	1.49715	0.0764013
$385$	8.33017	0.424545
$386$	−17.2256	−0.876759
$387$	−3.98575	−0.202607
$388$	5.46530	0.277459
$389$	15.2666	0.774047	0.387024	−	0.922070i	$-0.373503\pi$
$389$	15.2666	0.774047	0.387024	+	0.922070i	$0.373503\pi$
$390$	1.49715	0.0758113
$391$	9.55517	0.483226
$392$	−0.922665	−0.0466016
$393$	−15.4460	−0.779146
$394$	−15.6628	−0.789079
$395$	−7.66425	−0.385630
$396$	−2.24487	−0.112809
$397$	16.1231	0.809196	0.404598	−	0.914495i	$-0.367412\pi$
$397$	16.1231	0.809196	0.404598	+	0.914495i	$0.367412\pi$
$398$	8.59364	0.430760
$399$	−9.83935	−0.492584
$400$	1.00000	0.0500000
$401$	17.3105	0.864446	0.432223	−	0.901767i	$-0.357729\pi$
$401$	17.3105	0.864446	0.432223	+	0.901767i	$0.357729\pi$
$402$	−10.4098	−0.519195
$403$	1.00000	0.0498135
$404$	−5.13258	−0.255355
$405$	−6.14904	−0.305548
$406$	6.70304	0.332666
$407$	−22.1465	−1.09776
$408$	4.48331	0.221957
$409$	34.6394	1.71281	0.856403	−	0.516307i	$-0.172694\pi$
$409$	34.6394	1.71281	0.856403	+	0.516307i	$0.172694\pi$
$410$	−6.76903	−0.334299
$411$	−34.9370	−1.72332
$412$	−2.73405	−0.134697
$413$	12.3801	0.609187
$414$	2.42035	0.118954
$415$	−0.750736	−0.0368522
$416$	−1.00000	−0.0490290
$417$	25.7032	1.25869
$418$	−6.91008	−0.337983
$419$	27.7663	1.35647	0.678235	−	0.734845i	$-0.262745\pi$
$419$	27.7663	1.35647	0.678235	+	0.734845i	$0.262745\pi$
$420$	−4.21407	−0.205626
$421$	−26.0364	−1.26894	−0.634468	−	0.772949i	$-0.718781\pi$
$421$	−26.0364	−1.26894	−0.634468	+	0.772949i	$0.718781\pi$
$422$	26.0056	1.26593
$423$	−7.68256	−0.373539
$424$	11.1215	0.540110
$425$	2.99456	0.145257
$426$	−12.4866	−0.604975
$427$	38.1973	1.84850
$428$	12.4714	0.602826
$429$	−4.43083	−0.213922
$430$	−5.25457	−0.253398
$431$	−3.03697	−0.146286	−0.0731428	−	0.997321i	$-0.523303\pi$
$431$	−3.03697	−0.146286	−0.0731428	+	0.997321i	$0.523303\pi$
$432$	5.62710	0.270734
$433$	−8.52570	−0.409719	−0.204860	−	0.978791i	$-0.565674\pi$
$433$	−8.52570	−0.409719	−0.204860	+	0.978791i	$0.565674\pi$
$434$	−2.81472	−0.135111
$435$	3.56536	0.170946
$436$	−13.7146	−0.656809
$437$	7.45024	0.356393
$438$	17.1435	0.819147
$439$	27.2551	1.30082	0.650408	−	0.759585i	$-0.274598\pi$
$439$	27.2551	1.30082	0.650408	+	0.759585i	$0.274598\pi$
$440$	−2.95950	−0.141089
$441$	−0.699869	−0.0333271
$442$	−2.99456	−0.142437
$443$	−12.7540	−0.605961	−0.302980	−	0.952997i	$-0.597982\pi$
$443$	−12.7540	−0.605961	−0.302980	+	0.952997i	$0.597982\pi$
$444$	11.2035	0.531694
$445$	0.992853	0.0470657
$446$	2.70445	0.128059
$447$	26.8654	1.27069
$448$	2.81472	0.132983
$449$	22.7459	1.07344	0.536722	−	0.843759i	$-0.319662\pi$
$449$	22.7459	1.07344	0.536722	+	0.843759i	$0.319662\pi$
$450$	0.758530	0.0357575
$451$	20.0329	0.943315
$452$	−6.07708	−0.285842
$453$	29.2873	1.37604
$454$	−13.2238	−0.620622
$455$	2.81472	0.131956
$456$	3.49567	0.163700
$457$	32.9744	1.54248	0.771239	−	0.636546i	$-0.219637\pi$
$457$	32.9744	1.54248	0.771239	+	0.636546i	$0.219637\pi$
$458$	2.49450	0.116561
$459$	16.8507	0.786522
$460$	3.19084	0.148774
$461$	−26.1865	−1.21963	−0.609813	−	0.792545i	$-0.708756\pi$
$461$	−26.1865	−1.21963	−0.609813	+	0.792545i	$0.708756\pi$
$462$	12.4716	0.580229
$463$	19.7299	0.916924	0.458462	−	0.888714i	$-0.348400\pi$
$463$	19.7299	0.916924	0.458462	+	0.888714i	$0.348400\pi$
$464$	−2.38142	−0.110555
$465$	−1.49715	−0.0694288
$466$	−15.6255	−0.723839
$467$	−7.68433	−0.355589	−0.177794	−	0.984068i	$-0.556896\pi$
$467$	−7.68433	−0.355589	−0.177794	+	0.984068i	$0.556896\pi$
$468$	−0.758530	−0.0350631
$469$	−19.5710	−0.903703
$470$	−10.1282	−0.467180
$471$	1.05612	0.0486636
$472$	−4.39835	−0.202450
$473$	15.5509	0.715032
$474$	−11.4746	−0.527044
$475$	2.33488	0.107132
$476$	8.42885	0.386336
$477$	8.43602	0.386259
$478$	7.44874	0.340697
$479$	7.71402	0.352463	0.176231	−	0.984349i	$-0.443609\pi$
$479$	7.71402	0.352463	0.176231	+	0.984349i	$0.443609\pi$
$480$	1.49715	0.0683354
$481$	−7.48320	−0.341204
$482$	−20.3723	−0.927931
$483$	−13.4465	−0.611835
$484$	−2.24136	−0.101880
$485$	5.46530	0.248167
$486$	7.67523	0.348156
$487$	−38.6193	−1.75001	−0.875004	−	0.484116i	$-0.839141\pi$
$487$	−38.6193	−1.75001	−0.875004	+	0.484116i	$0.839141\pi$
$488$	−13.5705	−0.614310
$489$	−32.4842	−1.46898
$490$	−0.922665	−0.0416817
$491$	−2.58033	−0.116449	−0.0582244	−	0.998304i	$-0.518544\pi$
$491$	−2.58033	−0.116449	−0.0582244	+	0.998304i	$0.518544\pi$
$492$	−10.1343	−0.456889
$493$	−7.13131	−0.321178
$494$	−2.33488	−0.105051
$495$	−2.24487	−0.100899
$496$	1.00000	0.0449013
$497$	−23.4753	−1.05301
$498$	−1.12397	−0.0503662
$499$	−16.8636	−0.754916	−0.377458	−	0.926027i	$-0.623202\pi$
$499$	−16.8636	−0.754916	−0.377458	+	0.926027i	$0.623202\pi$
$500$	1.00000	0.0447214
$501$	−5.37628	−0.240194
$502$	−29.0433	−1.29626
$503$	3.02960	0.135083	0.0675416	−	0.997716i	$-0.478484\pi$
$503$	3.02960	0.135083	0.0675416	+	0.997716i	$0.478484\pi$
$504$	2.13505	0.0951028
$505$	−5.13258	−0.228397
$506$	−9.44330	−0.419806
$507$	−1.49715	−0.0664909
$508$	−11.6515	−0.516950
$509$	0.760739	0.0337192	0.0168596	−	0.999858i	$-0.494633\pi$
$509$	0.760739	0.0337192	0.0168596	+	0.999858i	$0.494633\pi$
$510$	4.48331	0.198524
$511$	32.2306	1.42580
$512$	−1.00000	−0.0441942
$513$	13.1386	0.580083
$514$	−19.8324	−0.874770
$515$	−2.73405	−0.120477
$516$	−7.86690	−0.346321
$517$	29.9745	1.31828
$518$	21.0631	0.925460
$519$	18.5573	0.814574
$520$	−1.00000	−0.0438529
$521$	3.37004	0.147644	0.0738221	−	0.997271i	$-0.476480\pi$
$521$	3.37004	0.147644	0.0738221	+	0.997271i	$0.476480\pi$
$522$	−1.80638	−0.0790632
$523$	−3.29610	−0.144129	−0.0720643	−	0.997400i	$-0.522959\pi$
$523$	−3.29610	−0.144129	−0.0720643	+	0.997400i	$0.522959\pi$
$524$	10.3169	0.450695
$525$	−4.21407	−0.183917
$526$	−24.9639	−1.08848
$527$	2.99456	0.130445
$528$	−4.43083	−0.192827
$529$	−12.8185	−0.557327
$530$	11.1215	0.483089
$531$	−3.33628	−0.144782
$532$	6.57204	0.284934
$533$	6.76903	0.293199
$534$	1.48645	0.0643251
$535$	12.4714	0.539184
$536$	6.95307	0.300327
$537$	34.7475	1.49946
$538$	−21.0577	−0.907861
$539$	2.73063	0.117616
$540$	5.62710	0.242152
$541$	5.58394	0.240072	0.120036	−	0.992770i	$-0.461699\pi$
$541$	5.58394	0.240072	0.120036	+	0.992770i	$0.461699\pi$
$542$	30.8155	1.32364
$543$	−25.7261	−1.10401
$544$	−2.99456	−0.128391
$545$	−13.7146	−0.587467
$546$	4.21407	0.180346
$547$	4.77425	0.204132	0.102066	−	0.994778i	$-0.467455\pi$
$547$	4.77425	0.204132	0.102066	+	0.994778i	$0.467455\pi$
$548$	23.3356	0.996848
$549$	−10.2937	−0.439323
$550$	−2.95950	−0.126194
$551$	−5.56033	−0.236878
$552$	4.77718	0.203330
$553$	−21.5727	−0.917367
$554$	−24.1572	−1.02634
$555$	11.2035	0.475562
$556$	−17.1680	−0.728086
$557$	−7.43080	−0.314853	−0.157427	−	0.987531i	$-0.550320\pi$
$557$	−7.43080	−0.314853	−0.157427	+	0.987531i	$0.550320\pi$
$558$	0.758530	0.0321111
$559$	5.25457	0.222245
$560$	2.81472	0.118944
$561$	−13.2684	−0.560191
$562$	−26.1605	−1.10352
$563$	−24.6328	−1.03815	−0.519074	−	0.854730i	$-0.673723\pi$
$563$	−24.6328	−1.03815	−0.519074	+	0.854730i	$0.673723\pi$
$564$	−15.1635	−0.638498
$565$	−6.07708	−0.255665
$566$	−24.5218	−1.03073
$567$	−17.3078	−0.726861
$568$	8.34019	0.349947
$569$	8.95566	0.375441	0.187720	−	0.982223i	$-0.439890\pi$
$569$	8.95566	0.375441	0.187720	+	0.982223i	$0.439890\pi$
$570$	3.49567	0.146418
$571$	−45.5807	−1.90749	−0.953747	−	0.300609i	$-0.902810\pi$
$571$	−45.5807	−1.90749	−0.953747	+	0.300609i	$0.902810\pi$
$572$	2.95950	0.123743
$573$	17.5963	0.735094
$574$	−19.0529	−0.795254
$575$	3.19084	0.133067
$576$	−0.758530	−0.0316054
$577$	24.8049	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$577$	24.8049	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$578$	8.03262	0.334113
$579$	−25.7894	−1.07177
$580$	−2.38142	−0.0988832
$581$	−2.11311	−0.0876667
$582$	8.18240	0.339171
$583$	−32.9142	−1.36317
$584$	−11.4507	−0.473834
$585$	−0.758530	−0.0313614
$586$	5.46263	0.225659
$587$	6.77749	0.279737	0.139868	−	0.990170i	$-0.455332\pi$
$587$	6.77749	0.279737	0.139868	+	0.990170i	$0.455332\pi$
$588$	−1.38137	−0.0569668
$589$	2.33488	0.0962070
$590$	−4.39835	−0.181077
$591$	−23.4496	−0.964586
$592$	−7.48320	−0.307557
$593$	−28.2480	−1.16001	−0.580003	−	0.814614i	$-0.696949\pi$
$593$	−28.2480	−1.16001	−0.580003	+	0.814614i	$0.696949\pi$
$594$	−16.6534	−0.683297
$595$	8.42885	0.345549
$596$	−17.9443	−0.735027
$597$	12.8660	0.526571
$598$	−3.19084	−0.130483
$599$	−35.9201	−1.46766	−0.733828	−	0.679336i	$-0.762268\pi$
$599$	−35.9201	−1.46766	−0.733828	+	0.679336i	$0.762268\pi$
$600$	1.49715	0.0611211
$601$	−8.83103	−0.360225	−0.180113	−	0.983646i	$-0.557646\pi$
$601$	−8.83103	−0.360225	−0.180113	+	0.983646i	$0.557646\pi$
$602$	−14.7902	−0.602802
$603$	5.27412	0.214779
$604$	−19.5620	−0.795967
$605$	−2.24136	−0.0911241
$606$	−7.68426	−0.312152
$607$	−30.4603	−1.23635	−0.618173	−	0.786042i	$-0.712127\pi$
$607$	−30.4603	−1.23635	−0.618173	+	0.786042i	$0.712127\pi$
$608$	−2.33488	−0.0946919
$609$	10.0355	0.406658
$610$	−13.5705	−0.549455
$611$	10.1282	0.409744
$612$	−2.27146	−0.0918184
$613$	−16.8637	−0.681119	−0.340559	−	0.940223i	$-0.610616\pi$
$613$	−16.8637	−0.681119	−0.340559	+	0.940223i	$0.610616\pi$
$614$	2.10122	0.0847985
$615$	−10.1343	−0.408654
$616$	−8.33017	−0.335632
$617$	−9.70000	−0.390507	−0.195254	−	0.980753i	$-0.562553\pi$
$617$	−9.70000	−0.390507	−0.195254	+	0.980753i	$0.562553\pi$
$618$	−4.09329	−0.164656
$619$	−3.43286	−0.137978	−0.0689892	−	0.997617i	$-0.521977\pi$
$619$	−3.43286	−0.137978	−0.0689892	+	0.997617i	$0.521977\pi$
$620$	1.00000	0.0401610
$621$	17.9552	0.720517
$622$	−14.5908	−0.585037
$623$	2.79461	0.111964
$624$	−1.49715	−0.0599341
$625$	1.00000	0.0400000
$626$	31.4146	1.25558
$627$	−10.3454	−0.413157
$628$	−0.705421	−0.0281494
$629$	−22.4089	−0.893500
$630$	2.13505	0.0850625
$631$	39.0572	1.55484	0.777421	−	0.628980i	$-0.216527\pi$
$631$	39.0572	1.55484	0.777421	+	0.628980i	$0.216527\pi$
$632$	7.66425	0.304868
$633$	38.9343	1.54750
$634$	28.4661	1.13053
$635$	−11.6515	−0.462374
$636$	16.6507	0.660241
$637$	0.922665	0.0365573
$638$	7.04782	0.279026
$639$	6.32629	0.250264
$640$	−1.00000	−0.0395285
$641$	−14.3556	−0.567013	−0.283507	−	0.958970i	$-0.591498\pi$
$641$	−14.3556	−0.567013	−0.283507	+	0.958970i	$0.591498\pi$
$642$	18.6716	0.736908
$643$	−42.7873	−1.68737	−0.843684	−	0.536841i	$-0.819618\pi$
$643$	−42.7873	−1.68737	−0.843684	+	0.536841i	$0.819618\pi$
$644$	8.98134	0.353914
$645$	−7.86690	−0.309759
$646$	−6.99193	−0.275094
$647$	−29.3572	−1.15415	−0.577076	−	0.816690i	$-0.695806\pi$
$647$	−29.3572	−1.15415	−0.577076	+	0.816690i	$0.695806\pi$
$648$	6.14904	0.241557
$649$	13.0169	0.510959
$650$	−1.00000	−0.0392232
$651$	−4.21407	−0.165163
$652$	21.6973	0.849731
$653$	8.76271	0.342911	0.171456	−	0.985192i	$-0.445153\pi$
$653$	8.76271	0.342911	0.171456	+	0.985192i	$0.445153\pi$
$654$	−20.5328	−0.802897
$655$	10.3169	0.403114
$656$	6.76903	0.264286
$657$	−8.68572	−0.338862
$658$	−28.5081	−1.11136
$659$	35.2212	1.37202	0.686012	−	0.727590i	$-0.259360\pi$
$659$	35.2212	1.37202	0.686012	+	0.727590i	$0.259360\pi$
$660$	−4.43083	−0.172470
$661$	11.6112	0.451623	0.225811	−	0.974171i	$-0.427497\pi$
$661$	11.6112	0.451623	0.225811	+	0.974171i	$0.427497\pi$
$662$	−35.3026	−1.37208
$663$	−4.48331	−0.174117
$664$	0.750736	0.0291342
$665$	6.57204	0.254853
$666$	−5.67623	−0.219949
$667$	−7.59875	−0.294225
$668$	3.59100	0.138940
$669$	4.04898	0.156543
$670$	6.95307	0.268621
$671$	40.1620	1.55044
$672$	4.21407	0.162561
$673$	41.0086	1.58077	0.790383	−	0.612613i	$-0.209882\pi$
$673$	41.0086	1.58077	0.790383	+	0.612613i	$0.209882\pi$
$674$	12.4720	0.480402
$675$	5.62710	0.216587
$676$	1.00000	0.0384615
$677$	44.2608	1.70108	0.850540	−	0.525910i	$-0.176275\pi$
$677$	44.2608	1.70108	0.850540	+	0.525910i	$0.176275\pi$
$678$	−9.09832	−0.349419
$679$	15.3833	0.590357
$680$	−2.99456	−0.114836
$681$	−19.7980	−0.758661
$682$	−2.95950	−0.113325
$683$	35.3801	1.35378	0.676892	−	0.736083i	$-0.263327\pi$
$683$	35.3801	1.35378	0.676892	+	0.736083i	$0.263327\pi$
$684$	−1.77108	−0.0677188
$685$	23.3356	0.891608
$686$	17.1060	0.653111
$687$	3.73466	0.142486
$688$	5.25457	0.200329
$689$	−11.1215	−0.423697
$690$	4.77718	0.181864
$691$	−17.3187	−0.658835	−0.329417	−	0.944184i	$-0.606852\pi$
$691$	−17.3187	−0.658835	−0.329417	+	0.944184i	$0.606852\pi$
$692$	−12.3950	−0.471188
$693$	−6.31869	−0.240027
$694$	12.1676	0.461874
$695$	−17.1680	−0.651220
$696$	−3.56536	−0.135144
$697$	20.2702	0.767790
$698$	0.308320	0.0116701
$699$	−23.3938	−0.884836
$700$	2.81472	0.106387
$701$	5.78090	0.218342	0.109171	−	0.994023i	$-0.465180\pi$
$701$	5.78090	0.218342	0.109171	+	0.994023i	$0.465180\pi$
$702$	−5.62710	−0.212381
$703$	−17.4724	−0.658982
$704$	2.95950	0.111540
$705$	−15.1635	−0.571090
$706$	−21.3769	−0.804531
$707$	−14.4468	−0.543328
$708$	−6.58501	−0.247480
$709$	47.2359	1.77398	0.886991	−	0.461786i	$-0.152791\pi$
$709$	47.2359	1.77398	0.886991	+	0.461786i	$0.152791\pi$
$710$	8.34019	0.313002
$711$	5.81357	0.218026
$712$	−0.992853	−0.0372087
$713$	3.19084	0.119498
$714$	12.6193	0.472265
$715$	2.95950	0.110679
$716$	−23.2090	−0.867362
$717$	11.1519	0.416476
$718$	0.00165198	6.16514e−5 0
$719$	−46.3358	−1.72804	−0.864018	−	0.503462i	$-0.832059\pi$
$719$	−46.3358	−1.72804	−0.864018	+	0.503462i	$0.832059\pi$
$720$	−0.758530	−0.0282688
$721$	−7.69559	−0.286599
$722$	13.5483	0.504217
$723$	−30.5004	−1.13432
$724$	17.1833	0.638614
$725$	−2.38142	−0.0884438
$726$	−3.35566	−0.124540
$727$	−0.0295827	−0.00109716	−0.000548581	−	1.00000i	$-0.500175\pi$
$727$	−0.0295827	−0.00109716	−0.000548581		1.00000i	$0.500175\pi$
$728$	−2.81472	−0.104321
$729$	29.9381	1.10882
$730$	−11.4507	−0.423810
$731$	15.7351	0.581985
$732$	−20.3172	−0.750945
$733$	−35.3807	−1.30682	−0.653409	−	0.757005i	$-0.726662\pi$
$733$	−35.3807	−1.30682	−0.653409	+	0.757005i	$0.726662\pi$
$734$	−10.3573	−0.382296
$735$	−1.38137	−0.0509526
$736$	−3.19084	−0.117616
$737$	−20.5776	−0.757986
$738$	5.13451	0.189004
$739$	−43.6882	−1.60709	−0.803547	−	0.595241i	$-0.797057\pi$
$739$	−43.6882	−1.60709	−0.803547	+	0.595241i	$0.797057\pi$
$740$	−7.48320	−0.275088
$741$	−3.49567	−0.128417
$742$	31.3040	1.14921
$743$	−20.5777	−0.754923	−0.377461	−	0.926025i	$-0.623203\pi$
$743$	−20.5777	−0.754923	−0.377461	+	0.926025i	$0.623203\pi$
$744$	1.49715	0.0548883
$745$	−17.9443	−0.657428
$746$	32.2168	1.17954
$747$	0.569456	0.0208353
$748$	8.86240	0.324041
$749$	35.1034	1.28265
$750$	1.49715	0.0546683
$751$	19.3297	0.705351	0.352676	−	0.935746i	$-0.385272\pi$
$751$	19.3297	0.705351	0.352676	+	0.935746i	$0.385272\pi$
$752$	10.1282	0.369338
$753$	−43.4822	−1.58458
$754$	2.38142	0.0867263
$755$	−19.5620	−0.711934
$756$	15.8387	0.576049
$757$	16.5005	0.599721	0.299861	−	0.953983i	$-0.403060\pi$
$757$	16.5005	0.599721	0.299861	+	0.953983i	$0.403060\pi$
$758$	−37.3530	−1.35672
$759$	−14.1381	−0.513180
$760$	−2.33488	−0.0846950
$761$	−33.8036	−1.22538	−0.612690	−	0.790323i	$-0.709913\pi$
$761$	−33.8036	−1.22538	−0.612690	+	0.790323i	$0.709913\pi$
$762$	−17.4440	−0.631930
$763$	−38.6027	−1.39751
$764$	−11.7531	−0.425214
$765$	−2.27146	−0.0821249
$766$	−33.5870	−1.21355
$767$	4.39835	0.158815
$768$	−1.49715	−0.0540239
$769$	−36.5085	−1.31653	−0.658266	−	0.752786i	$-0.728710\pi$
$769$	−36.5085	−1.31653	−0.658266	+	0.752786i	$0.728710\pi$
$770$	−8.33017	−0.300199
$771$	−29.6922	−1.06934
$772$	17.2256	0.619962
$773$	−16.4308	−0.590973	−0.295487	−	0.955347i	$-0.595482\pi$
$773$	−16.4308	−0.590973	−0.295487	+	0.955347i	$0.595482\pi$
$774$	3.98575	0.143265
$775$	1.00000	0.0359211
$776$	−5.46530	−0.196193
$777$	31.5347	1.13130
$778$	−15.2666	−0.547334
$779$	15.8049	0.566268
$780$	−1.49715	−0.0536067
$781$	−24.6828	−0.883220
$782$	−9.55517	−0.341692
$783$	−13.4005	−0.478895
$784$	0.922665	0.0329523
$785$	−0.705421	−0.0251776
$786$	15.4460	0.550939
$787$	−9.42931	−0.336119	−0.168059	−	0.985777i	$-0.553750\pi$
$787$	−9.42931	−0.336119	−0.168059	+	0.985777i	$0.553750\pi$
$788$	15.6628	0.557963
$789$	−37.3748	−1.33058
$790$	7.66425	0.272682
$791$	−17.1053	−0.608194
$792$	2.24487	0.0797680
$793$	13.5705	0.481904
$794$	−16.1231	−0.572188
$795$	16.6507	0.590538
$796$	−8.59364	−0.304594
$797$	49.0979	1.73914	0.869568	−	0.493813i	$-0.164397\pi$
$797$	49.0979	1.73914	0.869568	+	0.493813i	$0.164397\pi$
$798$	9.83935	0.348309
$799$	30.3295	1.07298
$800$	−1.00000	−0.0353553
$801$	−0.753109	−0.0266098
$802$	−17.3105	−0.611256
$803$	33.8884	1.19590
$804$	10.4098	0.367126
$805$	8.98134	0.316551
$806$	−1.00000	−0.0352235
$807$	−31.5266	−1.10979
$808$	5.13258	0.180564
$809$	37.7526	1.32731	0.663655	−	0.748039i	$-0.269004\pi$
$809$	37.7526	1.32731	0.663655	+	0.748039i	$0.269004\pi$
$810$	6.14904	0.216055
$811$	−48.6702	−1.70904	−0.854521	−	0.519416i	$-0.826149\pi$
$811$	−48.6702	−1.70904	−0.854521	+	0.519416i	$0.826149\pi$
$812$	−6.70304	−0.235231
$813$	46.1356	1.61805
$814$	22.1465	0.776235
$815$	21.6973	0.760022
$816$	−4.48331	−0.156947
$817$	12.2688	0.429231
$818$	−34.6394	−1.21114
$819$	−2.13505	−0.0746048
$820$	6.76903	0.236385
$821$	6.68820	0.233420	0.116710	−	0.993166i	$-0.462765\pi$
$821$	6.68820	0.233420	0.116710	+	0.993166i	$0.462765\pi$
$822$	34.9370	1.21857
$823$	7.30698	0.254705	0.127353	−	0.991858i	$-0.459352\pi$
$823$	7.30698	0.254705	0.127353	+	0.991858i	$0.459352\pi$
$824$	2.73405	0.0952451
$825$	−4.43083	−0.154262
$826$	−12.3801	−0.430760
$827$	−16.1481	−0.561526	−0.280763	−	0.959777i	$-0.590587\pi$
$827$	−16.1481	−0.561526	−0.280763	+	0.959777i	$0.590587\pi$
$828$	−2.42035	−0.0841130
$829$	−43.8942	−1.52451	−0.762254	−	0.647278i	$-0.775907\pi$
$829$	−43.8942	−1.52451	−0.762254	+	0.647278i	$0.775907\pi$
$830$	0.750736	0.0260584
$831$	−36.1670	−1.25462
$832$	1.00000	0.0346688
$833$	2.76297	0.0957313
$834$	−25.7032	−0.890027
$835$	3.59100	0.124272
$836$	6.91008	0.238990
$837$	5.62710	0.194501
$838$	−27.7663	−0.959170
$839$	−49.7163	−1.71640	−0.858198	−	0.513319i	$-0.828416\pi$
$839$	−49.7163	−1.71640	−0.858198	+	0.513319i	$0.828416\pi$
$840$	4.21407	0.145399
$841$	−23.3288	−0.804442
$842$	26.0364	0.897273
$843$	−39.1663	−1.34896
$844$	−26.0056	−0.895149
$845$	1.00000	0.0344010
$846$	7.68256	0.264132
$847$	−6.30880	−0.216773
$848$	−11.1215	−0.381915
$849$	−36.7128	−1.25998
$850$	−2.99456	−0.102712
$851$	−23.8777	−0.818517
$852$	12.4866	0.427782
$853$	−5.22013	−0.178734	−0.0893668	−	0.995999i	$-0.528484\pi$
$853$	−5.22013	−0.178734	−0.0893668	+	0.995999i	$0.528484\pi$
$854$	−38.1973	−1.30709
$855$	−1.77108	−0.0605695
$856$	−12.4714	−0.426263
$857$	19.1407	0.653834	0.326917	−	0.945053i	$-0.393990\pi$
$857$	19.1407	0.653834	0.326917	+	0.945053i	$0.393990\pi$
$858$	4.43083	0.151266
$859$	−22.0781	−0.753294	−0.376647	−	0.926357i	$-0.622923\pi$
$859$	−22.0781	−0.753294	−0.376647	+	0.926357i	$0.622923\pi$
$860$	5.25457	0.179179
$861$	−28.5252	−0.972136
$862$	3.03697	0.103440
$863$	−33.8483	−1.15221	−0.576104	−	0.817376i	$-0.695428\pi$
$863$	−33.8483	−1.15221	−0.576104	+	0.817376i	$0.695428\pi$
$864$	−5.62710	−0.191438
$865$	−12.3950	−0.421444
$866$	8.52570	0.289715
$867$	12.0261	0.408427
$868$	2.81472	0.0955379
$869$	−22.6824	−0.769446
$870$	−3.56536	−0.120877
$871$	−6.95307	−0.235596
$872$	13.7146	0.464434
$873$	−4.14560	−0.140307
$874$	−7.45024	−0.252008
$875$	2.81472	0.0951550
$876$	−17.1435	−0.579225
$877$	12.2605	0.414008	0.207004	−	0.978340i	$-0.433629\pi$
$877$	12.2605	0.414008	0.207004	+	0.978340i	$0.433629\pi$
$878$	−27.2551	−0.919816
$879$	8.17839	0.275850
$880$	2.95950	0.0997647
$881$	−4.67973	−0.157664	−0.0788320	−	0.996888i	$-0.525119\pi$
$881$	−4.67973	−0.157664	−0.0788320	+	0.996888i	$0.525119\pi$
$882$	0.699869	0.0235658
$883$	0.128420	0.00432168	0.00216084	−	0.999998i	$-0.499312\pi$
$883$	0.128420	0.00432168	0.00216084	+	0.999998i	$0.499312\pi$
$884$	2.99456	0.100718
$885$	−6.58501	−0.221353
$886$	12.7540	0.428479
$887$	46.3774	1.55720	0.778601	−	0.627520i	$-0.215930\pi$
$887$	46.3774	1.55720	0.778601	+	0.627520i	$0.215930\pi$
$888$	−11.2035	−0.375965
$889$	−32.7956	−1.09993
$890$	−0.992853	−0.0332805
$891$	−18.1981	−0.609659
$892$	−2.70445	−0.0905517
$893$	23.6482	0.791356
$894$	−26.8654	−0.898512
$895$	−23.2090	−0.775792
$896$	−2.81472	−0.0940333
$897$	−4.77718	−0.159506
$898$	−22.7459	−0.759040
$899$	−2.38142	−0.0794249
$900$	−0.758530	−0.0252843
$901$	−33.3041	−1.10952
$902$	−20.0329	−0.667024
$903$	−22.1432	−0.736878
$904$	6.07708	0.202121
$905$	17.1833	0.571194
$906$	−29.2873	−0.973007
$907$	32.9157	1.09295	0.546474	−	0.837476i	$-0.315970\pi$
$907$	32.9157	1.09295	0.546474	+	0.837476i	$0.315970\pi$
$908$	13.2238	0.438846
$909$	3.89322	0.129130
$910$	−2.81472	−0.0933072
$911$	60.0018	1.98795	0.993974	−	0.109612i	$-0.0349610\pi$
$911$	60.0018	1.98795	0.993974	+	0.109612i	$0.0349610\pi$
$912$	−3.49567	−0.115753
$913$	−2.22180	−0.0735310
$914$	−32.9744	−1.09070
$915$	−20.3172	−0.671666
$916$	−2.49450	−0.0824207
$917$	29.0392	0.958958
$918$	−16.8507	−0.556155
$919$	9.66204	0.318721	0.159361	−	0.987220i	$-0.449057\pi$
$919$	9.66204	0.318721	0.159361	+	0.987220i	$0.449057\pi$
$920$	−3.19084	−0.105199
$921$	3.14586	0.103660
$922$	26.1865	0.862406
$923$	−8.34019	−0.274521
$924$	−12.4716	−0.410284
$925$	−7.48320	−0.246046
$926$	−19.7299	−0.648363
$927$	2.07386	0.0681144
$928$	2.38142	0.0781740
$929$	−10.8496	−0.355964	−0.177982	−	0.984034i	$-0.556957\pi$
$929$	−10.8496	−0.355964	−0.177982	+	0.984034i	$0.556957\pi$
$930$	1.49715	0.0490936
$931$	2.15431	0.0706047
$932$	15.6255	0.511831
$933$	−21.8446	−0.715161
$934$	7.68433	0.251439
$935$	8.86240	0.289831
$936$	0.758530	0.0247933
$937$	31.3552	1.02433	0.512165	−	0.858887i	$-0.328844\pi$
$937$	31.3552	1.02433	0.512165	+	0.858887i	$0.328844\pi$
$938$	19.5710	0.639015
$939$	47.0325	1.53485
$940$	10.1282	0.330346
$941$	23.8141	0.776318	0.388159	−	0.921592i	$-0.373111\pi$
$941$	23.8141	0.776318	0.388159	+	0.921592i	$0.373111\pi$
$942$	−1.05612	−0.0344104
$943$	21.5989	0.703358
$944$	4.39835	0.143154
$945$	15.8387	0.515234
$946$	−15.5509	−0.505604
$947$	52.7702	1.71480	0.857400	−	0.514651i	$-0.172078\pi$
$947$	52.7702	1.71480	0.857400	+	0.514651i	$0.172078\pi$
$948$	11.4746	0.372677
$949$	11.4507	0.371706
$950$	−2.33488	−0.0757535
$951$	42.6181	1.38199
$952$	−8.42885	−0.273181
$953$	−14.1712	−0.459050	−0.229525	−	0.973303i	$-0.573717\pi$
$953$	−14.1712	−0.459050	−0.229525	+	0.973303i	$0.573717\pi$
$954$	−8.43602	−0.273126
$955$	−11.7531	−0.380323
$956$	−7.44874	−0.240910
$957$	10.5517	0.341087
$958$	−7.71402	−0.249229
$959$	65.6833	2.12102
$960$	−1.49715	−0.0483204
$961$	1.00000	0.0322581
$962$	7.48320	0.241268
$963$	−9.45991	−0.304841
$964$	20.3723	0.656146
$965$	17.2256	0.554511
$966$	13.4465	0.432632
$967$	−38.4538	−1.23659	−0.618295	−	0.785946i	$-0.712176\pi$
$967$	−38.4538	−1.23659	−0.618295	+	0.785946i	$0.712176\pi$
$968$	2.24136	0.0720399
$969$	−10.4680	−0.336281
$970$	−5.46530	−0.175480
$971$	18.8459	0.604792	0.302396	−	0.953182i	$-0.402213\pi$
$971$	18.8459	0.604792	0.302396	+	0.953182i	$0.402213\pi$
$972$	−7.67523	−0.246183
$973$	−48.3232	−1.54917
$974$	38.6193	1.23744
$975$	−1.49715	−0.0479473
$976$	13.5705	0.434383
$977$	−4.77409	−0.152737	−0.0763683	−	0.997080i	$-0.524332\pi$
$977$	−4.77409	−0.152737	−0.0763683	+	0.997080i	$0.524332\pi$
$978$	32.4842	1.03873
$979$	2.93835	0.0939100
$980$	0.922665	0.0294734
$981$	10.4029	0.332140
$982$	2.58033	0.0823418
$983$	−32.3719	−1.03250	−0.516251	−	0.856437i	$-0.672673\pi$
$983$	−32.3719	−1.03250	−0.516251	+	0.856437i	$0.672673\pi$
$984$	10.1343	0.323069
$985$	15.6628	0.499057
$986$	7.13131	0.227107
$987$	−42.6810	−1.35855
$988$	2.33488	0.0742824
$989$	16.7665	0.533144
$990$	2.24487	0.0713467
$991$	55.7921	1.77229	0.886147	−	0.463404i	$-0.153372\pi$
$991$	55.7921	1.77229	0.886147	+	0.463404i	$0.153372\pi$
$992$	−1.00000	−0.0317500
$993$	−52.8535	−1.67725
$994$	23.4753	0.744592
$995$	−8.59364	−0.272437
$996$	1.12397	0.0356143
$997$	−26.0988	−0.826557	−0.413278	−	0.910605i	$-0.635616\pi$
$997$	−26.0988	−0.826557	−0.413278	+	0.910605i	$0.635616\pi$
$998$	16.8636	0.533806
$999$	−42.1087	−1.33226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.m.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.m.1.2	✓	8	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:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.49715 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.49715 q^{6} +2.81472 q^{7} -1.00000 q^{8} -0.758530 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.49715 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.49715 q^{6} +2.81472 q^{7} -1.00000 q^{8} -0.758530 q^{9} -1.00000 q^{10} +2.95950 q^{11} -1.49715 q^{12} +1.00000 q^{13} -2.81472 q^{14} -1.49715 q^{15} +1.00000 q^{16} +2.99456 q^{17} +0.758530 q^{18} +2.33488 q^{19} +1.00000 q^{20} -4.21407 q^{21} -2.95950 q^{22} +3.19084 q^{23} +1.49715 q^{24} +1.00000 q^{25} -1.00000 q^{26} +5.62710 q^{27} +2.81472 q^{28} -2.38142 q^{29} +1.49715 q^{30} +1.00000 q^{31} -1.00000 q^{32} -4.43083 q^{33} -2.99456 q^{34} +2.81472 q^{35} -0.758530 q^{36} -7.48320 q^{37} -2.33488 q^{38} -1.49715 q^{39} -1.00000 q^{40} +6.76903 q^{41} +4.21407 q^{42} +5.25457 q^{43} +2.95950 q^{44} -0.758530 q^{45} -3.19084 q^{46} +10.1282 q^{47} -1.49715 q^{48} +0.922665 q^{49} -1.00000 q^{50} -4.48331 q^{51} +1.00000 q^{52} -11.1215 q^{53} -5.62710 q^{54} +2.95950 q^{55} -2.81472 q^{56} -3.49567 q^{57} +2.38142 q^{58} +4.39835 q^{59} -1.49715 q^{60} +13.5705 q^{61} -1.00000 q^{62} -2.13505 q^{63} +1.00000 q^{64} +1.00000 q^{65} +4.43083 q^{66} -6.95307 q^{67} +2.99456 q^{68} -4.77718 q^{69} -2.81472 q^{70} -8.34019 q^{71} +0.758530 q^{72} +11.4507 q^{73} +7.48320 q^{74} -1.49715 q^{75} +2.33488 q^{76} +8.33017 q^{77} +1.49715 q^{78} -7.66425 q^{79} +1.00000 q^{80} -6.14904 q^{81} -6.76903 q^{82} -0.750736 q^{83} -4.21407 q^{84} +2.99456 q^{85} -5.25457 q^{86} +3.56536 q^{87} -2.95950 q^{88} +0.992853 q^{89} +0.758530 q^{90} +2.81472 q^{91} +3.19084 q^{92} -1.49715 q^{93} -10.1282 q^{94} +2.33488 q^{95} +1.49715 q^{96} +5.46530 q^{97} -0.922665 q^{98} -2.24487 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 5 q^{3} + 8 q^{4} + 8 q^{5} - 5 q^{6} + 5 q^{7} - 8 q^{8} + 17 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 5 * q^3 + 8 * q^4 + 8 * q^5 - 5 * q^6 + 5 * q^7 - 8 * q^8 + 17 * q^9	\( 8 q - 8 q^{2} + 5 q^{3} + 8 q^{4} + 8 q^{5} - 5 q^{6} + 5 q^{7} - 8 q^{8} + 17 q^{9} - 8 q^{10} + 4 q^{11} + 5 q^{12} + 8 q^{13} - 5 q^{14} + 5 q^{15} + 8 q^{16} + 19 q^{17} - 17 q^{18} - 14 q^{19} + 8 q^{20} + 3 q^{21} - 4 q^{22} + 8 q^{23} - 5 q^{24} + 8 q^{25} - 8 q^{26} + 17 q^{27} + 5 q^{28} + 21 q^{29} - 5 q^{30} + 8 q^{31} - 8 q^{32} + 4 q^{33} - 19 q^{34} + 5 q^{35} + 17 q^{36} - 3 q^{37} + 14 q^{38} + 5 q^{39} - 8 q^{40} - 8 q^{41} - 3 q^{42} + 25 q^{43} + 4 q^{44} + 17 q^{45} - 8 q^{46} + 11 q^{47} + 5 q^{48} + 15 q^{49} - 8 q^{50} + 7 q^{51} + 8 q^{52} + 2 q^{53} - 17 q^{54} + 4 q^{55} - 5 q^{56} + 9 q^{57} - 21 q^{58} - 22 q^{59} + 5 q^{60} - 2 q^{61} - 8 q^{62} + 30 q^{63} + 8 q^{64} + 8 q^{65} - 4 q^{66} - 14 q^{67} + 19 q^{68} + 36 q^{69} - 5 q^{70} + 4 q^{71} - 17 q^{72} + 17 q^{73} + 3 q^{74} + 5 q^{75} - 14 q^{76} + 17 q^{77} - 5 q^{78} - 12 q^{79} + 8 q^{80} + 40 q^{81} + 8 q^{82} + 21 q^{83} + 3 q^{84} + 19 q^{85} - 25 q^{86} + 25 q^{87} - 4 q^{88} - 25 q^{89} - 17 q^{90} + 5 q^{91} + 8 q^{92} + 5 q^{93} - 11 q^{94} - 14 q^{95} - 5 q^{96} + 8 q^{97} - 15 q^{98} - 57 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 5 * q^3 + 8 * q^4 + 8 * q^5 - 5 * q^6 + 5 * q^7 - 8 * q^8 + 17 * q^9 - 8 * q^10 + 4 * q^11 + 5 * q^12 + 8 * q^13 - 5 * q^14 + 5 * q^15 + 8 * q^16 + 19 * q^17 - 17 * q^18 - 14 * q^19 + 8 * q^20 + 3 * q^21 - 4 * q^22 + 8 * q^23 - 5 * q^24 + 8 * q^25 - 8 * q^26 + 17 * q^27 + 5 * q^28 + 21 * q^29 - 5 * q^30 + 8 * q^31 - 8 * q^32 + 4 * q^33 - 19 * q^34 + 5 * q^35 + 17 * q^36 - 3 * q^37 + 14 * q^38 + 5 * q^39 - 8 * q^40 - 8 * q^41 - 3 * q^42 + 25 * q^43 + 4 * q^44 + 17 * q^45 - 8 * q^46 + 11 * q^47 + 5 * q^48 + 15 * q^49 - 8 * q^50 + 7 * q^51 + 8 * q^52 + 2 * q^53 - 17 * q^54 + 4 * q^55 - 5 * q^56 + 9 * q^57 - 21 * q^58 - 22 * q^59 + 5 * q^60 - 2 * q^61 - 8 * q^62 + 30 * q^63 + 8 * q^64 + 8 * q^65 - 4 * q^66 - 14 * q^67 + 19 * q^68 + 36 * q^69 - 5 * q^70 + 4 * q^71 - 17 * q^72 + 17 * q^73 + 3 * q^74 + 5 * q^75 - 14 * q^76 + 17 * q^77 - 5 * q^78 - 12 * q^79 + 8 * q^80 + 40 * q^81 + 8 * q^82 + 21 * q^83 + 3 * q^84 + 19 * q^85 - 25 * q^86 + 25 * q^87 - 4 * q^88 - 25 * q^89 - 17 * q^90 + 5 * q^91 + 8 * q^92 + 5 * q^93 - 11 * q^94 - 14 * q^95 - 5 * q^96 + 8 * q^97 - 15 * q^98 - 57 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more