LMFDB - Embedded newform 4029.2.a.k.1.30

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4029,2,Mod(1,4029)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4029, 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("4029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4029 = 3 \cdot 17 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4029.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:	$32.1717269744$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.30
Character	$\chi$	$=$	4029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.59980 q^{2} +1.00000 q^{3} +4.75898 q^{4} -1.90154 q^{5} +2.59980 q^{6} +1.92356 q^{7} +7.17280 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.59980 q^{2} +1.00000 q^{3} +4.75898 q^{4} -1.90154 q^{5} +2.59980 q^{6} +1.92356 q^{7} +7.17280 q^{8} +1.00000 q^{9} -4.94363 q^{10} -0.601399 q^{11} +4.75898 q^{12} +3.49193 q^{13} +5.00088 q^{14} -1.90154 q^{15} +9.12992 q^{16} +1.00000 q^{17} +2.59980 q^{18} -2.11454 q^{19} -9.04939 q^{20} +1.92356 q^{21} -1.56352 q^{22} +6.97347 q^{23} +7.17280 q^{24} -1.38414 q^{25} +9.07834 q^{26} +1.00000 q^{27} +9.15419 q^{28} -0.266086 q^{29} -4.94363 q^{30} +4.68125 q^{31} +9.39039 q^{32} -0.601399 q^{33} +2.59980 q^{34} -3.65773 q^{35} +4.75898 q^{36} -6.70222 q^{37} -5.49738 q^{38} +3.49193 q^{39} -13.6394 q^{40} +2.69633 q^{41} +5.00088 q^{42} -2.92291 q^{43} -2.86205 q^{44} -1.90154 q^{45} +18.1296 q^{46} -7.17607 q^{47} +9.12992 q^{48} -3.29991 q^{49} -3.59849 q^{50} +1.00000 q^{51} +16.6180 q^{52} +9.70723 q^{53} +2.59980 q^{54} +1.14359 q^{55} +13.7973 q^{56} -2.11454 q^{57} -0.691773 q^{58} +0.899229 q^{59} -9.04939 q^{60} +5.08157 q^{61} +12.1703 q^{62} +1.92356 q^{63} +6.15333 q^{64} -6.64006 q^{65} -1.56352 q^{66} -1.11308 q^{67} +4.75898 q^{68} +6.97347 q^{69} -9.50939 q^{70} +2.37252 q^{71} +7.17280 q^{72} -4.92029 q^{73} -17.4245 q^{74} -1.38414 q^{75} -10.0630 q^{76} -1.15683 q^{77} +9.07834 q^{78} +1.00000 q^{79} -17.3609 q^{80} +1.00000 q^{81} +7.00993 q^{82} -8.55694 q^{83} +9.15419 q^{84} -1.90154 q^{85} -7.59898 q^{86} -0.266086 q^{87} -4.31372 q^{88} +9.51272 q^{89} -4.94363 q^{90} +6.71695 q^{91} +33.1866 q^{92} +4.68125 q^{93} -18.6564 q^{94} +4.02088 q^{95} +9.39039 q^{96} +0.679406 q^{97} -8.57912 q^{98} -0.601399 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) $ 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9} + 5 q^{10} + 26 q^{11} + 34 q^{12} + 7 q^{13} + 19 q^{14} + 11 q^{15} + 40 q^{16} + 31 q^{17} + 4 q^{18} + 32 q^{19} + 23 q^{20} + 4 q^{21} + 2 q^{22} + 29 q^{23} + 12 q^{24} + 32 q^{25} + 13 q^{26} + 31 q^{27} - 13 q^{28} + 25 q^{29} + 5 q^{30} + 22 q^{31} + 28 q^{32} + 26 q^{33} + 4 q^{34} + 20 q^{35} + 34 q^{36} - 4 q^{37} + 19 q^{38} + 7 q^{39} - 3 q^{40} + 33 q^{41} + 19 q^{42} + 6 q^{43} + 30 q^{44} + 11 q^{45} - 11 q^{46} + 23 q^{47} + 40 q^{48} + 31 q^{49} + 6 q^{50} + 31 q^{51} - 7 q^{52} + 12 q^{53} + 4 q^{54} + 40 q^{56} + 32 q^{57} + 9 q^{58} + 27 q^{59} + 23 q^{60} - 4 q^{61} + 25 q^{62} + 4 q^{63} + 10 q^{64} + 54 q^{65} + 2 q^{66} + 34 q^{68} + 29 q^{69} - 59 q^{70} + 35 q^{71} + 12 q^{72} + 5 q^{73} + 48 q^{74} + 32 q^{75} + 32 q^{76} + 42 q^{77} + 13 q^{78} + 31 q^{79} + 24 q^{80} + 31 q^{81} + 5 q^{82} + 67 q^{83} - 13 q^{84} + 11 q^{85} - 20 q^{86} + 25 q^{87} - 7 q^{88} + 22 q^{89} + 5 q^{90} + 16 q^{91} + 57 q^{92} + 22 q^{93} + 45 q^{94} + 73 q^{95} + 28 q^{96} - 13 q^{97} - 19 q^{98} + 26 q^{99}+O(q^{100}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9 + 5 * q^10 + 26 * q^11 + 34 * q^12 + 7 * q^13 + 19 * q^14 + 11 * q^15 + 40 * q^16 + 31 * q^17 + 4 * q^18 + 32 * q^19 + 23 * q^20 + 4 * q^21 + 2 * q^22 + 29 * q^23 + 12 * q^24 + 32 * q^25 + 13 * q^26 + 31 * q^27 - 13 * q^28 + 25 * q^29 + 5 * q^30 + 22 * q^31 + 28 * q^32 + 26 * q^33 + 4 * q^34 + 20 * q^35 + 34 * q^36 - 4 * q^37 + 19 * q^38 + 7 * q^39 - 3 * q^40 + 33 * q^41 + 19 * q^42 + 6 * q^43 + 30 * q^44 + 11 * q^45 - 11 * q^46 + 23 * q^47 + 40 * q^48 + 31 * q^49 + 6 * q^50 + 31 * q^51 - 7 * q^52 + 12 * q^53 + 4 * q^54 + 40 * q^56 + 32 * q^57 + 9 * q^58 + 27 * q^59 + 23 * q^60 - 4 * q^61 + 25 * q^62 + 4 * q^63 + 10 * q^64 + 54 * q^65 + 2 * q^66 + 34 * q^68 + 29 * q^69 - 59 * q^70 + 35 * q^71 + 12 * q^72 + 5 * q^73 + 48 * q^74 + 32 * q^75 + 32 * q^76 + 42 * q^77 + 13 * q^78 + 31 * q^79 + 24 * q^80 + 31 * q^81 + 5 * q^82 + 67 * q^83 - 13 * q^84 + 11 * q^85 - 20 * q^86 + 25 * q^87 - 7 * q^88 + 22 * q^89 + 5 * q^90 + 16 * q^91 + 57 * q^92 + 22 * q^93 + 45 * q^94 + 73 * q^95 + 28 * q^96 - 13 * q^97 - 19 * q^98 + 26 * 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 $

$2$	2.59980	1.83834	0.919169	−	0.393863i	$-0.128861\pi$
$2$	2.59980	1.83834	0.919169	+	0.393863i	$0.128861\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.75898	2.37949
$5$	−1.90154	−0.850395	−0.425198	−	0.905101i	$-0.639795\pi$
$5$	−1.90154	−0.850395	−0.425198	+	0.905101i	$0.639795\pi$
$6$	2.59980	1.06137
$7$	1.92356	0.727038	0.363519	−	0.931587i	$-0.381575\pi$
$7$	1.92356	0.727038	0.363519	+	0.931587i	$0.381575\pi$
$8$	7.17280	2.53597
$9$	1.00000	0.333333
$10$	−4.94363	−1.56331
$11$	−0.601399	−0.181329	−0.0906643	−	0.995882i	$-0.528899\pi$
$11$	−0.601399	−0.181329	−0.0906643	+	0.995882i	$0.528899\pi$
$12$	4.75898	1.37380
$13$	3.49193	0.968488	0.484244	−	0.874933i	$-0.339095\pi$
$13$	3.49193	0.968488	0.484244	+	0.874933i	$0.339095\pi$
$14$	5.00088	1.33654
$15$	−1.90154	−0.490976
$16$	9.12992	2.28248
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.59980	0.612780
$19$	−2.11454	−0.485108	−0.242554	−	0.970138i	$-0.577985\pi$
$19$	−2.11454	−0.485108	−0.242554	+	0.970138i	$0.577985\pi$
$20$	−9.04939	−2.02351
$21$	1.92356	0.419756
$22$	−1.56352	−0.333343
$23$	6.97347	1.45407	0.727034	−	0.686601i	$-0.240898\pi$
$23$	6.97347	1.45407	0.727034	+	0.686601i	$0.240898\pi$
$24$	7.17280	1.46414
$25$	−1.38414	−0.276828
$26$	9.07834	1.78041
$27$	1.00000	0.192450
$28$	9.15419	1.72998
$29$	−0.266086	−0.0494110	−0.0247055	−	0.999695i	$-0.507865\pi$
$29$	−0.266086	−0.0494110	−0.0247055	+	0.999695i	$0.507865\pi$
$30$	−4.94363	−0.902580
$31$	4.68125	0.840777	0.420389	−	0.907344i	$-0.361894\pi$
$31$	4.68125	0.840777	0.420389	+	0.907344i	$0.361894\pi$
$32$	9.39039	1.66000
$33$	−0.601399	−0.104690
$34$	2.59980	0.445863
$35$	−3.65773	−0.618270
$36$	4.75898	0.793163
$37$	−6.70222	−1.10184	−0.550919	−	0.834559i	$-0.685723\pi$
$37$	−6.70222	−1.10184	−0.550919	+	0.834559i	$0.685723\pi$
$38$	−5.49738	−0.891792
$39$	3.49193	0.559157
$40$	−13.6394	−2.15658
$41$	2.69633	0.421096	0.210548	−	0.977583i	$-0.432475\pi$
$41$	2.69633	0.421096	0.210548	+	0.977583i	$0.432475\pi$
$42$	5.00088	0.771653
$43$	−2.92291	−0.445739	−0.222869	−	0.974848i	$-0.571542\pi$
$43$	−2.92291	−0.445739	−0.222869	+	0.974848i	$0.571542\pi$
$44$	−2.86205	−0.431470
$45$	−1.90154	−0.283465
$46$	18.1296	2.67307
$47$	−7.17607	−1.04674	−0.523369	−	0.852106i	$-0.675325\pi$
$47$	−7.17607	−1.04674	−0.523369	+	0.852106i	$0.675325\pi$
$48$	9.12992	1.31779
$49$	−3.29991	−0.471416
$50$	−3.59849	−0.508904
$51$	1.00000	0.140028
$52$	16.6180	2.30451
$53$	9.70723	1.33339	0.666695	−	0.745331i	$-0.267708\pi$
$53$	9.70723	1.33339	0.666695	+	0.745331i	$0.267708\pi$
$54$	2.59980	0.353788
$55$	1.14359	0.154201
$56$	13.7973	1.84375
$57$	−2.11454	−0.280077
$58$	−0.691773	−0.0908342
$59$	0.899229	0.117070	0.0585348	−	0.998285i	$-0.481357\pi$
$59$	0.899229	0.117070	0.0585348	+	0.998285i	$0.481357\pi$
$60$	−9.04939	−1.16827
$61$	5.08157	0.650628	0.325314	−	0.945606i	$-0.394530\pi$
$61$	5.08157	0.650628	0.325314	+	0.945606i	$0.394530\pi$
$62$	12.1703	1.54563
$63$	1.92356	0.242346
$64$	6.15333	0.769166
$65$	−6.64006	−0.823598
$66$	−1.56352	−0.192456
$67$	−1.11308	−0.135985	−0.0679923	−	0.997686i	$-0.521659\pi$
$67$	−1.11308	−0.135985	−0.0679923	+	0.997686i	$0.521659\pi$
$68$	4.75898	0.577111
$69$	6.97347	0.839507
$70$	−9.50939	−1.13659
$71$	2.37252	0.281566	0.140783	−	0.990040i	$-0.455038\pi$
$71$	2.37252	0.281566	0.140783	+	0.990040i	$0.455038\pi$
$72$	7.17280	0.845323
$73$	−4.92029	−0.575876	−0.287938	−	0.957649i	$-0.592970\pi$
$73$	−4.92029	−0.575876	−0.287938	+	0.957649i	$0.592970\pi$
$74$	−17.4245	−2.02555
$75$	−1.38414	−0.159827
$76$	−10.0630	−1.15431
$77$	−1.15683	−0.131833
$78$	9.07834	1.02792
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	−17.3609	−1.94101
$81$	1.00000	0.111111
$82$	7.00993	0.774117
$83$	−8.55694	−0.939247	−0.469623	−	0.882867i	$-0.655610\pi$
$83$	−8.55694	−0.939247	−0.469623	+	0.882867i	$0.655610\pi$
$84$	9.15419	0.998804
$85$	−1.90154	−0.206251
$86$	−7.59898	−0.819419
$87$	−0.266086	−0.0285275
$88$	−4.31372	−0.459844
$89$	9.51272	1.00835	0.504173	−	0.863603i	$-0.331797\pi$
$89$	9.51272	1.00835	0.504173	+	0.863603i	$0.331797\pi$
$90$	−4.94363	−0.521105
$91$	6.71695	0.704128
$92$	33.1866	3.45994
$93$	4.68125	0.485423
$94$	−18.6564	−1.92426
$95$	4.02088	0.412533
$96$	9.39039	0.958403
$97$	0.679406	0.0689832	0.0344916	−	0.999405i	$-0.489019\pi$
$97$	0.679406	0.0689832	0.0344916	+	0.999405i	$0.489019\pi$
$98$	−8.57912	−0.866622
$99$	−0.601399	−0.0604429
$100$	−6.58709	−0.658709
$101$	4.24234	0.422129	0.211064	−	0.977472i	$-0.432307\pi$
$101$	4.24234	0.422129	0.211064	+	0.977472i	$0.432307\pi$
$102$	2.59980	0.257419
$103$	3.35826	0.330899	0.165450	−	0.986218i	$-0.447092\pi$
$103$	3.35826	0.330899	0.165450	+	0.986218i	$0.447092\pi$
$104$	25.0469	2.45606
$105$	−3.65773	−0.356958
$106$	25.2369	2.45122
$107$	16.7487	1.61916	0.809580	−	0.587010i	$-0.199695\pi$
$107$	16.7487	1.61916	0.809580	+	0.587010i	$0.199695\pi$
$108$	4.75898	0.457933
$109$	14.2113	1.36119	0.680596	−	0.732659i	$-0.261721\pi$
$109$	14.2113	1.36119	0.680596	+	0.732659i	$0.261721\pi$
$110$	2.97310	0.283474
$111$	−6.70222	−0.636147
$112$	17.5620	1.65945
$113$	−16.3321	−1.53640	−0.768198	−	0.640212i	$-0.778847\pi$
$113$	−16.3321	−1.53640	−0.768198	+	0.640212i	$0.778847\pi$
$114$	−5.49738	−0.514877
$115$	−13.2603	−1.23653
$116$	−1.26630	−0.117573
$117$	3.49193	0.322829
$118$	2.33782	0.215214
$119$	1.92356	0.176333
$120$	−13.6394	−1.24510
$121$	−10.6383	−0.967120
$122$	13.2111	1.19608
$123$	2.69633	0.243120
$124$	22.2780	2.00062
$125$	12.1397	1.08581
$126$	5.00088	0.445514
$127$	−11.1930	−0.993220	−0.496610	−	0.867974i	$-0.665422\pi$
$127$	−11.1930	−0.993220	−0.496610	+	0.867974i	$0.665422\pi$
$128$	−2.78333	−0.246014
$129$	−2.92291	−0.257348
$130$	−17.2628	−1.51405
$131$	−1.90163	−0.166146	−0.0830732	−	0.996543i	$-0.526474\pi$
$131$	−1.90163	−0.166146	−0.0830732	+	0.996543i	$0.526474\pi$
$132$	−2.86205	−0.249109
$133$	−4.06744	−0.352692
$134$	−2.89379	−0.249986
$135$	−1.90154	−0.163659
$136$	7.17280	0.615063
$137$	−3.91600	−0.334566	−0.167283	−	0.985909i	$-0.553499\pi$
$137$	−3.91600	−0.334566	−0.167283	+	0.985909i	$0.553499\pi$
$138$	18.1296	1.54330
$139$	−3.37630	−0.286374	−0.143187	−	0.989696i	$-0.545735\pi$
$139$	−3.37630	−0.286374	−0.143187	+	0.989696i	$0.545735\pi$
$140$	−17.4071	−1.47117
$141$	−7.17607	−0.604334
$142$	6.16809	0.517614
$143$	−2.10005	−0.175615
$144$	9.12992	0.760826
$145$	0.505974	0.0420189
$146$	−12.7918	−1.05866
$147$	−3.29991	−0.272172
$148$	−31.8957	−2.62181
$149$	−23.4255	−1.91909	−0.959544	−	0.281559i	$-0.909148\pi$
$149$	−23.4255	−1.91909	−0.959544	+	0.281559i	$0.909148\pi$
$150$	−3.59849	−0.293816
$151$	−6.06970	−0.493945	−0.246973	−	0.969022i	$-0.579436\pi$
$151$	−6.06970	−0.493945	−0.246973	+	0.969022i	$0.579436\pi$
$152$	−15.1671	−1.23022
$153$	1.00000	0.0808452
$154$	−3.00753	−0.242353
$155$	−8.90159	−0.714993
$156$	16.6180	1.33051
$157$	4.57455	0.365089	0.182544	−	0.983198i	$-0.441567\pi$
$157$	4.57455	0.365089	0.182544	+	0.983198i	$0.441567\pi$
$158$	2.59980	0.206829
$159$	9.70723	0.769833
$160$	−17.8562	−1.41166
$161$	13.4139	1.05716
$162$	2.59980	0.204260
$163$	−22.5197	−1.76388	−0.881939	−	0.471363i	$-0.843762\pi$
$163$	−22.5197	−1.76388	−0.881939	+	0.471363i	$0.843762\pi$
$164$	12.8318	1.00199
$165$	1.14359	0.0890280
$166$	−22.2464	−1.72665
$167$	19.9271	1.54200	0.771002	−	0.636832i	$-0.219756\pi$
$167$	19.9271	1.54200	0.771002	+	0.636832i	$0.219756\pi$
$168$	13.7973	1.06449
$169$	−0.806397	−0.0620305
$170$	−4.94363	−0.379159
$171$	−2.11454	−0.161703
$172$	−13.9100	−1.06063
$173$	1.98049	0.150574	0.0752869	−	0.997162i	$-0.476013\pi$
$173$	1.98049	0.150574	0.0752869	+	0.997162i	$0.476013\pi$
$174$	−0.691773	−0.0524431
$175$	−2.66248	−0.201264
$176$	−5.49072	−0.413879
$177$	0.899229	0.0675902
$178$	24.7312	1.85368
$179$	−9.55503	−0.714177	−0.357088	−	0.934071i	$-0.616231\pi$
$179$	−9.55503	−0.714177	−0.357088	+	0.934071i	$0.616231\pi$
$180$	−9.04939	−0.674502
$181$	5.96717	0.443536	0.221768	−	0.975099i	$-0.428817\pi$
$181$	5.96717	0.443536	0.221768	+	0.975099i	$0.428817\pi$
$182$	17.4628	1.29443
$183$	5.08157	0.375640
$184$	50.0193	3.68747
$185$	12.7446	0.936998
$186$	12.1703	0.892372
$187$	−0.601399	−0.0439787
$188$	−34.1508	−2.49070
$189$	1.92356	0.139919
$190$	10.4535	0.758376
$191$	−10.8475	−0.784901	−0.392450	−	0.919773i	$-0.628372\pi$
$191$	−10.8475	−0.784901	−0.392450	+	0.919773i	$0.628372\pi$
$192$	6.15333	0.444078
$193$	−24.3047	−1.74949	−0.874745	−	0.484583i	$-0.838971\pi$
$193$	−24.3047	−1.74949	−0.874745	+	0.484583i	$0.838971\pi$
$194$	1.76632	0.126815
$195$	−6.64006	−0.475504
$196$	−15.7042	−1.12173
$197$	−0.594924	−0.0423866	−0.0211933	−	0.999775i	$-0.506747\pi$
$197$	−0.594924	−0.0423866	−0.0211933	+	0.999775i	$0.506747\pi$
$198$	−1.56352	−0.111114
$199$	−5.55484	−0.393772	−0.196886	−	0.980426i	$-0.563083\pi$
$199$	−5.55484	−0.393772	−0.196886	+	0.980426i	$0.563083\pi$
$200$	−9.92816	−0.702027
$201$	−1.11308	−0.0785107
$202$	11.0293	0.776016
$203$	−0.511834	−0.0359237
$204$	4.75898	0.333195
$205$	−5.12719	−0.358098
$206$	8.73082	0.608305
$207$	6.97347	0.484690
$208$	31.8811	2.21055
$209$	1.27168	0.0879639
$210$	−9.50939	−0.656210
$211$	4.45859	0.306942	0.153471	−	0.988153i	$-0.450955\pi$
$211$	4.45859	0.306942	0.153471	+	0.988153i	$0.450955\pi$
$212$	46.1965	3.17279
$213$	2.37252	0.162562
$214$	43.5434	2.97656
$215$	5.55803	0.379054
$216$	7.17280	0.488047
$217$	9.00467	0.611277
$218$	36.9465	2.50233
$219$	−4.92029	−0.332482
$220$	5.44230	0.366920
$221$	3.49193	0.234893
$222$	−17.4245	−1.16945
$223$	−14.1334	−0.946441	−0.473221	−	0.880944i	$-0.656909\pi$
$223$	−14.1334	−0.946441	−0.473221	+	0.880944i	$0.656909\pi$
$224$	18.0630	1.20688
$225$	−1.38414	−0.0922760
$226$	−42.4603	−2.82442
$227$	−14.3418	−0.951900	−0.475950	−	0.879472i	$-0.657896\pi$
$227$	−14.3418	−0.951900	−0.475950	+	0.879472i	$0.657896\pi$
$228$	−10.0630	−0.666441
$229$	−26.1479	−1.72790	−0.863950	−	0.503578i	$-0.832017\pi$
$229$	−26.1479	−1.72790	−0.863950	+	0.503578i	$0.832017\pi$
$230$	−34.4743	−2.27317
$231$	−1.15683	−0.0761137
$232$	−1.90859	−0.125305
$233$	−22.6035	−1.48081	−0.740403	−	0.672163i	$-0.765365\pi$
$233$	−22.6035	−1.48081	−0.740403	+	0.672163i	$0.765365\pi$
$234$	9.07834	0.593470
$235$	13.6456	0.890141
$236$	4.27941	0.278566
$237$	1.00000	0.0649570
$238$	5.00088	0.324159
$239$	20.6815	1.33778	0.668888	−	0.743363i	$-0.266770\pi$
$239$	20.6815	1.33778	0.668888	+	0.743363i	$0.266770\pi$
$240$	−17.3609	−1.12064
$241$	12.5396	0.807749	0.403875	−	0.914814i	$-0.367663\pi$
$241$	12.5396	0.807749	0.403875	+	0.914814i	$0.367663\pi$
$242$	−27.6575	−1.77789
$243$	1.00000	0.0641500
$244$	24.1831	1.54816
$245$	6.27492	0.400890
$246$	7.00993	0.446937
$247$	−7.38382	−0.469821
$248$	33.5777	2.13218
$249$	−8.55694	−0.542274
$250$	31.5609	1.99608
$251$	22.3574	1.41118	0.705592	−	0.708618i	$-0.250681\pi$
$251$	22.3574	1.41118	0.705592	+	0.708618i	$0.250681\pi$
$252$	9.15419	0.576660
$253$	−4.19384	−0.263664
$254$	−29.0996	−1.82587
$255$	−1.90154	−0.119079
$256$	−19.5428	−1.22142
$257$	−13.2153	−0.824351	−0.412175	−	0.911105i	$-0.635231\pi$
$257$	−13.2153	−0.824351	−0.412175	+	0.911105i	$0.635231\pi$
$258$	−7.59898	−0.473092
$259$	−12.8921	−0.801078
$260$	−31.5999	−1.95974
$261$	−0.266086	−0.0164703
$262$	−4.94387	−0.305433
$263$	2.81471	0.173563	0.0867813	−	0.996227i	$-0.472342\pi$
$263$	2.81471	0.173563	0.0867813	+	0.996227i	$0.472342\pi$
$264$	−4.31372	−0.265491
$265$	−18.4587	−1.13391
$266$	−10.5745	−0.648367
$267$	9.51272	0.582169
$268$	−5.29713	−0.323574
$269$	0.687268	0.0419035	0.0209517	−	0.999780i	$-0.493330\pi$
$269$	0.687268	0.0419035	0.0209517	+	0.999780i	$0.493330\pi$
$270$	−4.94363	−0.300860
$271$	−0.607608	−0.0369096	−0.0184548	−	0.999830i	$-0.505875\pi$
$271$	−0.607608	−0.0369096	−0.0184548	+	0.999830i	$0.505875\pi$
$272$	9.12992	0.553583
$273$	6.71695	0.406528
$274$	−10.1808	−0.615046
$275$	0.832420	0.0501968
$276$	33.1866	1.99760
$277$	−17.5393	−1.05384	−0.526918	−	0.849916i	$-0.676653\pi$
$277$	−17.5393	−1.05384	−0.526918	+	0.849916i	$0.676653\pi$
$278$	−8.77773	−0.526453
$279$	4.68125	0.280259
$280$	−26.2362	−1.56791
$281$	12.2294	0.729546	0.364773	−	0.931097i	$-0.381147\pi$
$281$	12.2294	0.729546	0.364773	+	0.931097i	$0.381147\pi$
$282$	−18.6564	−1.11097
$283$	−15.5299	−0.923155	−0.461577	−	0.887100i	$-0.652716\pi$
$283$	−15.5299	−0.923155	−0.461577	+	0.887100i	$0.652716\pi$
$284$	11.2908	0.669984
$285$	4.02088	0.238176
$286$	−5.45971	−0.322839
$287$	5.18656	0.306153
$288$	9.39039	0.553334
$289$	1.00000	0.0588235
$290$	1.31543	0.0772450
$291$	0.679406	0.0398275
$292$	−23.4155	−1.37029
$293$	6.32152	0.369307	0.184654	−	0.982804i	$-0.440884\pi$
$293$	6.32152	0.369307	0.184654	+	0.982804i	$0.440884\pi$
$294$	−8.57912	−0.500344
$295$	−1.70992	−0.0995555
$296$	−48.0737	−2.79423
$297$	−0.601399	−0.0348967
$298$	−60.9016	−3.52793
$299$	24.3509	1.40825
$300$	−6.58709	−0.380306
$301$	−5.62239	−0.324069
$302$	−15.7800	−0.908038
$303$	4.24234	0.243716
$304$	−19.3055	−1.10725
$305$	−9.66282	−0.553291
$306$	2.59980	0.148621
$307$	−6.19070	−0.353322	−0.176661	−	0.984272i	$-0.556530\pi$
$307$	−6.19070	−0.353322	−0.176661	+	0.984272i	$0.556530\pi$
$308$	−5.50532	−0.313695
$309$	3.35826	0.191045
$310$	−23.1424	−1.31440
$311$	3.46645	0.196564	0.0982821	−	0.995159i	$-0.468665\pi$
$311$	3.46645	0.196564	0.0982821	+	0.995159i	$0.468665\pi$
$312$	25.0469	1.41800
$313$	26.9753	1.52473	0.762366	−	0.647146i	$-0.224038\pi$
$313$	26.9753	1.52473	0.762366	+	0.647146i	$0.224038\pi$
$314$	11.8929	0.671157
$315$	−3.65773	−0.206090
$316$	4.75898	0.267713
$317$	20.8973	1.17371	0.586853	−	0.809693i	$-0.300367\pi$
$317$	20.8973	1.17371	0.586853	+	0.809693i	$0.300367\pi$
$318$	25.2369	1.41521
$319$	0.160024	0.00895963
$320$	−11.7008	−0.654095
$321$	16.7487	0.934822
$322$	34.8735	1.94342
$323$	−2.11454	−0.117656
$324$	4.75898	0.264388
$325$	−4.83333	−0.268105
$326$	−58.5468	−3.24261
$327$	14.2113	0.785884
$328$	19.3403	1.06789
$329$	−13.8036	−0.761018
$330$	2.97310	0.163664
$331$	−21.4240	−1.17757	−0.588785	−	0.808290i	$-0.700394\pi$
$331$	−21.4240	−1.17757	−0.588785	+	0.808290i	$0.700394\pi$
$332$	−40.7223	−2.23493
$333$	−6.70222	−0.367279
$334$	51.8065	2.83473
$335$	2.11657	0.115641
$336$	17.5620	0.958083
$337$	−0.458943	−0.0250002	−0.0125001	−	0.999922i	$-0.503979\pi$
$337$	−0.458943	−0.0250002	−0.0125001	+	0.999922i	$0.503979\pi$
$338$	−2.09647	−0.114033
$339$	−16.3321	−0.887039
$340$	−9.04939	−0.490772
$341$	−2.81530	−0.152457
$342$	−5.49738	−0.297264
$343$	−19.8125	−1.06978
$344$	−20.9654	−1.13038
$345$	−13.2603	−0.713913
$346$	5.14888	0.276806
$347$	14.4853	0.777613	0.388806	−	0.921320i	$-0.372888\pi$
$347$	14.4853	0.777613	0.388806	+	0.921320i	$0.372888\pi$
$348$	−1.26630	−0.0678808
$349$	−20.9618	−1.12206	−0.561031	−	0.827795i	$-0.689595\pi$
$349$	−20.9618	−1.12206	−0.561031	+	0.827795i	$0.689595\pi$
$350$	−6.92192	−0.369992
$351$	3.49193	0.186386
$352$	−5.64737	−0.301006
$353$	19.8649	1.05730	0.528652	−	0.848839i	$-0.322698\pi$
$353$	19.8649	1.05730	0.528652	+	0.848839i	$0.322698\pi$
$354$	2.33782	0.124254
$355$	−4.51145	−0.239443
$356$	45.2708	2.39935
$357$	1.92356	0.101806
$358$	−24.8412	−1.31290
$359$	−30.8859	−1.63010	−0.815048	−	0.579394i	$-0.803289\pi$
$359$	−30.8859	−1.63010	−0.815048	+	0.579394i	$0.803289\pi$
$360$	−13.6394	−0.718858
$361$	−14.5287	−0.764670
$362$	15.5135	0.815370
$363$	−10.6383	−0.558367
$364$	31.9658	1.67546
$365$	9.35613	0.489722
$366$	13.2111	0.690554
$367$	−7.91530	−0.413175	−0.206588	−	0.978428i	$-0.566236\pi$
$367$	−7.91530	−0.413175	−0.206588	+	0.978428i	$0.566236\pi$
$368$	63.6672	3.31888
$369$	2.69633	0.140365
$370$	33.1333	1.72252
$371$	18.6724	0.969425
$372$	22.2780	1.15506
$373$	20.0587	1.03860	0.519301	−	0.854591i	$-0.326192\pi$
$373$	20.0587	1.03860	0.519301	+	0.854591i	$0.326192\pi$
$374$	−1.56352	−0.0808477
$375$	12.1397	0.626892
$376$	−51.4725	−2.65449
$377$	−0.929156	−0.0478540
$378$	5.00088	0.257218
$379$	0.710567	0.0364994	0.0182497	−	0.999833i	$-0.494191\pi$
$379$	0.710567	0.0364994	0.0182497	+	0.999833i	$0.494191\pi$
$380$	19.1353	0.981619
$381$	−11.1930	−0.573436
$382$	−28.2015	−1.44291
$383$	−28.8802	−1.47571	−0.737853	−	0.674961i	$-0.764160\pi$
$383$	−28.8802	−1.47571	−0.737853	+	0.674961i	$0.764160\pi$
$384$	−2.78333	−0.142036
$385$	2.19976	0.112110
$386$	−63.1874	−3.21616
$387$	−2.92291	−0.148580
$388$	3.23328	0.164145
$389$	28.6871	1.45449	0.727247	−	0.686376i	$-0.240799\pi$
$389$	28.6871	1.45449	0.727247	+	0.686376i	$0.240799\pi$
$390$	−17.2628	−0.874138
$391$	6.97347	0.352663
$392$	−23.6696	−1.19550
$393$	−1.90163	−0.0959246
$394$	−1.54669	−0.0779209
$395$	−1.90154	−0.0956769
$396$	−2.86205	−0.143823
$397$	2.22556	0.111697	0.0558487	−	0.998439i	$-0.482214\pi$
$397$	2.22556	0.111697	0.0558487	+	0.998439i	$0.482214\pi$
$398$	−14.4415	−0.723886
$399$	−4.06744	−0.203627
$400$	−12.6371	−0.631854
$401$	27.5974	1.37815	0.689075	−	0.724690i	$-0.258017\pi$
$401$	27.5974	1.37815	0.689075	+	0.724690i	$0.258017\pi$
$402$	−2.89379	−0.144329
$403$	16.3466	0.814283
$404$	20.1892	1.00445
$405$	−1.90154	−0.0944884
$406$	−1.33067	−0.0660399
$407$	4.03071	0.199795
$408$	7.17280	0.355107
$409$	−22.0903	−1.09229	−0.546147	−	0.837689i	$-0.683906\pi$
$409$	−22.0903	−1.09229	−0.546147	+	0.837689i	$0.683906\pi$
$410$	−13.3297	−0.658306
$411$	−3.91600	−0.193162
$412$	15.9819	0.787372
$413$	1.72972	0.0851141
$414$	18.1296	0.891024
$415$	16.2714	0.798731
$416$	32.7906	1.60769
$417$	−3.37630	−0.165338
$418$	3.30612	0.161708
$419$	0.0964006	0.00470948	0.00235474	−	0.999997i	$-0.499250\pi$
$419$	0.0964006	0.00470948	0.00235474	+	0.999997i	$0.499250\pi$
$420$	−17.4071	−0.849378
$421$	18.2497	0.889438	0.444719	−	0.895670i	$-0.353303\pi$
$421$	18.2497	0.889438	0.444719	+	0.895670i	$0.353303\pi$
$422$	11.5915	0.564264
$423$	−7.17607	−0.348913
$424$	69.6280	3.38143
$425$	−1.38414	−0.0671406
$426$	6.16809	0.298845
$427$	9.77471	0.473032
$428$	79.7068	3.85277
$429$	−2.10005	−0.101391
$430$	14.4498	0.696830
$431$	17.4745	0.841718	0.420859	−	0.907126i	$-0.361729\pi$
$431$	17.4745	0.841718	0.420859	+	0.907126i	$0.361729\pi$
$432$	9.12992	0.439263
$433$	22.2052	1.06711	0.533557	−	0.845764i	$-0.320855\pi$
$433$	22.2052	1.06711	0.533557	+	0.845764i	$0.320855\pi$
$434$	23.4104	1.12373
$435$	0.505974	0.0242596
$436$	67.6311	3.23894
$437$	−14.7457	−0.705380
$438$	−12.7918	−0.611215
$439$	4.95789	0.236627	0.118314	−	0.992976i	$-0.462251\pi$
$439$	4.95789	0.236627	0.118314	+	0.992976i	$0.462251\pi$
$440$	8.20271	0.391049
$441$	−3.29991	−0.157139
$442$	9.07834	0.431813
$443$	23.8399	1.13267	0.566333	−	0.824177i	$-0.308362\pi$
$443$	23.8399	1.13267	0.566333	+	0.824177i	$0.308362\pi$
$444$	−31.8957	−1.51370
$445$	−18.0888	−0.857493
$446$	−36.7440	−1.73988
$447$	−23.4255	−1.10799
$448$	11.8363	0.559213
$449$	8.75893	0.413359	0.206680	−	0.978409i	$-0.433734\pi$
$449$	8.75893	0.413359	0.206680	+	0.978409i	$0.433734\pi$
$450$	−3.59849	−0.169635
$451$	−1.62157	−0.0763568
$452$	−77.7242	−3.65584
$453$	−6.06970	−0.285179
$454$	−37.2859	−1.74991
$455$	−12.7726	−0.598787
$456$	−15.1671	−0.710267
$457$	−28.8743	−1.35068	−0.675341	−	0.737505i	$-0.736004\pi$
$457$	−28.8743	−1.35068	−0.675341	+	0.737505i	$0.736004\pi$
$458$	−67.9793	−3.17646
$459$	1.00000	0.0466760
$460$	−63.1057	−2.94232
$461$	4.75472	0.221449	0.110725	−	0.993851i	$-0.464683\pi$
$461$	4.75472	0.221449	0.110725	+	0.993851i	$0.464683\pi$
$462$	−3.00753	−0.139923
$463$	3.09374	0.143778	0.0718891	−	0.997413i	$-0.477097\pi$
$463$	3.09374	0.143778	0.0718891	+	0.997413i	$0.477097\pi$
$464$	−2.42935	−0.112780
$465$	−8.90159	−0.412801
$466$	−58.7647	−2.72222
$467$	−6.14707	−0.284453	−0.142226	−	0.989834i	$-0.545426\pi$
$467$	−6.14707	−0.284453	−0.142226	+	0.989834i	$0.545426\pi$
$468$	16.6180	0.768169
$469$	−2.14108	−0.0988659
$470$	35.4759	1.63638
$471$	4.57455	0.210784
$472$	6.44999	0.296885
$473$	1.75783	0.0808252
$474$	2.59980	0.119413
$475$	2.92681	0.134291
$476$	9.15419	0.419582
$477$	9.70723	0.444463
$478$	53.7679	2.45929
$479$	−8.27403	−0.378050	−0.189025	−	0.981972i	$-0.560533\pi$
$479$	−8.27403	−0.378050	−0.189025	+	0.981972i	$0.560533\pi$
$480$	−17.8562	−0.815021
$481$	−23.4037	−1.06712
$482$	32.6006	1.48492
$483$	13.4139	0.610354
$484$	−50.6275	−2.30125
$485$	−1.29192	−0.0586630
$486$	2.59980	0.117929
$487$	−17.6527	−0.799919	−0.399960	−	0.916533i	$-0.630976\pi$
$487$	−17.6527	−0.799919	−0.399960	+	0.916533i	$0.630976\pi$
$488$	36.4491	1.64997
$489$	−22.5197	−1.01838
$490$	16.3135	0.736971
$491$	0.378278	0.0170715	0.00853573	−	0.999964i	$-0.497283\pi$
$491$	0.378278	0.0170715	0.00853573	+	0.999964i	$0.497283\pi$
$492$	12.8318	0.578501
$493$	−0.266086	−0.0119839
$494$	−19.1965	−0.863691
$495$	1.14359	0.0514003
$496$	42.7394	1.91906
$497$	4.56369	0.204710
$498$	−22.2464	−0.996884
$499$	26.2047	1.17308	0.586542	−	0.809919i	$-0.300489\pi$
$499$	26.2047	1.17308	0.586542	+	0.809919i	$0.300489\pi$
$500$	57.7726	2.58367
$501$	19.9271	0.890277
$502$	58.1248	2.59423
$503$	−9.43420	−0.420650	−0.210325	−	0.977631i	$-0.567452\pi$
$503$	−9.43420	−0.420650	−0.210325	+	0.977631i	$0.567452\pi$
$504$	13.7973	0.614582
$505$	−8.06699	−0.358976
$506$	−10.9032	−0.484704
$507$	−0.806397	−0.0358133
$508$	−53.2673	−2.36336
$509$	−13.7613	−0.609959	−0.304980	−	0.952359i	$-0.598650\pi$
$509$	−13.7613	−0.609959	−0.304980	+	0.952359i	$0.598650\pi$
$510$	−4.94363	−0.218908
$511$	−9.46448	−0.418684
$512$	−45.2407	−1.99938
$513$	−2.11454	−0.0933590
$514$	−34.3573	−1.51544
$515$	−6.38588	−0.281395
$516$	−13.9100	−0.612356
$517$	4.31568	0.189803
$518$	−33.5170	−1.47265
$519$	1.98049	0.0869339
$520$	−47.6278	−2.08862
$521$	−12.6473	−0.554090	−0.277045	−	0.960857i	$-0.589355\pi$
$521$	−12.6473	−0.554090	−0.277045	+	0.960857i	$0.589355\pi$
$522$	−0.691773	−0.0302781
$523$	24.9151	1.08946	0.544731	−	0.838611i	$-0.316632\pi$
$523$	24.9151	1.08946	0.544731	+	0.838611i	$0.316632\pi$
$524$	−9.04983	−0.395343
$525$	−2.66248	−0.116200
$526$	7.31770	0.319067
$527$	4.68125	0.203918
$528$	−5.49072	−0.238953
$529$	25.6293	1.11432
$530$	−47.9890	−2.08451
$531$	0.899229	0.0390232
$532$	−19.3569	−0.839226
$533$	9.41541	0.407827
$534$	24.7312	1.07022
$535$	−31.8484	−1.37693
$536$	−7.98391	−0.344852
$537$	−9.55503	−0.412330
$538$	1.78676	0.0770328
$539$	1.98456	0.0854812
$540$	−9.04939	−0.389424
$541$	9.68087	0.416213	0.208106	−	0.978106i	$-0.433270\pi$
$541$	9.68087	0.416213	0.208106	+	0.978106i	$0.433270\pi$
$542$	−1.57966	−0.0678523
$543$	5.96717	0.256076
$544$	9.39039	0.402610
$545$	−27.0233	−1.15755
$546$	17.4628	0.747337
$547$	−6.63477	−0.283682	−0.141841	−	0.989889i	$-0.545302\pi$
$547$	−6.63477	−0.283682	−0.141841	+	0.989889i	$0.545302\pi$
$548$	−18.6361	−0.796096
$549$	5.08157	0.216876
$550$	2.16413	0.0922788
$551$	0.562649	0.0239697
$552$	50.0193	2.12896
$553$	1.92356	0.0817982
$554$	−45.5988	−1.93731
$555$	12.7446	0.540976
$556$	−16.0678	−0.681425
$557$	3.04568	0.129050	0.0645249	−	0.997916i	$-0.479447\pi$
$557$	3.04568	0.129050	0.0645249	+	0.997916i	$0.479447\pi$
$558$	12.1703	0.515211
$559$	−10.2066	−0.431693
$560$	−33.3948	−1.41119
$561$	−0.601399	−0.0253911
$562$	31.7941	1.34115
$563$	8.79181	0.370530	0.185265	−	0.982689i	$-0.440686\pi$
$563$	8.79181	0.370530	0.185265	+	0.982689i	$0.440686\pi$
$564$	−34.1508	−1.43801
$565$	31.0562	1.30654
$566$	−40.3746	−1.69707
$567$	1.92356	0.0807820
$568$	17.0176	0.714044
$569$	−21.4999	−0.901324	−0.450662	−	0.892695i	$-0.648812\pi$
$569$	−21.4999	−0.901324	−0.450662	+	0.892695i	$0.648812\pi$
$570$	10.4535	0.437849
$571$	28.5744	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$571$	28.5744	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$572$	−9.99407	−0.417873
$573$	−10.8475	−0.453163
$574$	13.4840	0.562813
$575$	−9.65226	−0.402527
$576$	6.15333	0.256389
$577$	5.21358	0.217044	0.108522	−	0.994094i	$-0.465388\pi$
$577$	5.21358	0.217044	0.108522	+	0.994094i	$0.465388\pi$
$578$	2.59980	0.108138
$579$	−24.3047	−1.01007
$580$	2.40792	0.0999835
$581$	−16.4598	−0.682868
$582$	1.76632	0.0732164
$583$	−5.83792	−0.241782
$584$	−35.2922	−1.46040
$585$	−6.64006	−0.274533
$586$	16.4347	0.678911
$587$	−14.4340	−0.595755	−0.297878	−	0.954604i	$-0.596279\pi$
$587$	−14.4340	−0.595755	−0.297878	+	0.954604i	$0.596279\pi$
$588$	−15.7042	−0.647630
$589$	−9.89867	−0.407868
$590$	−4.44546	−0.183017
$591$	−0.594924	−0.0244719
$592$	−61.1907	−2.51492
$593$	4.46279	0.183265	0.0916325	−	0.995793i	$-0.470791\pi$
$593$	4.46279	0.183265	0.0916325	+	0.995793i	$0.470791\pi$
$594$	−1.56352	−0.0641520
$595$	−3.65773	−0.149952
$596$	−111.481	−4.56645
$597$	−5.55484	−0.227344
$598$	63.3075	2.58884
$599$	−41.2505	−1.68545	−0.842726	−	0.538343i	$-0.819051\pi$
$599$	−41.2505	−1.68545	−0.842726	+	0.538343i	$0.819051\pi$
$600$	−9.92816	−0.405315
$601$	−12.0553	−0.491746	−0.245873	−	0.969302i	$-0.579075\pi$
$601$	−12.0553	−0.491746	−0.245873	+	0.969302i	$0.579075\pi$
$602$	−14.6171	−0.595749
$603$	−1.11308	−0.0453282
$604$	−28.8856	−1.17534
$605$	20.2292	0.822434
$606$	11.0293	0.448033
$607$	43.3323	1.75880	0.879402	−	0.476081i	$-0.157943\pi$
$607$	43.3323	1.75880	0.879402	+	0.476081i	$0.157943\pi$
$608$	−19.8563	−0.805280
$609$	−0.511834	−0.0207406
$610$	−25.1214	−1.01714
$611$	−25.0584	−1.01375
$612$	4.75898	0.192370
$613$	−7.29230	−0.294533	−0.147267	−	0.989097i	$-0.547048\pi$
$613$	−7.29230	−0.294533	−0.147267	+	0.989097i	$0.547048\pi$
$614$	−16.0946	−0.649526
$615$	−5.12719	−0.206748
$616$	−8.29770	−0.334324
$617$	−3.85716	−0.155283	−0.0776417	−	0.996981i	$-0.524739\pi$
$617$	−3.85716	−0.155283	−0.0776417	+	0.996981i	$0.524739\pi$
$618$	8.73082	0.351205
$619$	30.7765	1.23701	0.618506	−	0.785780i	$-0.287738\pi$
$619$	30.7765	1.23701	0.618506	+	0.785780i	$0.287738\pi$
$620$	−42.3625	−1.70132
$621$	6.97347	0.279836
$622$	9.01208	0.361351
$623$	18.2983	0.733106
$624$	31.8811	1.27626
$625$	−16.1635	−0.646538
$626$	70.1304	2.80297
$627$	1.27168	0.0507860
$628$	21.7702	0.868725
$629$	−6.70222	−0.267235
$630$	−9.50939	−0.378863
$631$	31.7257	1.26298	0.631491	−	0.775383i	$-0.282443\pi$
$631$	31.7257	1.26298	0.631491	+	0.775383i	$0.282443\pi$
$632$	7.17280	0.285319
$633$	4.45859	0.177213
$634$	54.3287	2.15767
$635$	21.2840	0.844629
$636$	46.1965	1.83181
$637$	−11.5231	−0.456561
$638$	0.416031	0.0164708
$639$	2.37252	0.0938555
$640$	5.29261	0.209209
$641$	−39.7667	−1.57069	−0.785345	−	0.619058i	$-0.787515\pi$
$641$	−39.7667	−1.57069	−0.785345	+	0.619058i	$0.787515\pi$
$642$	43.5434	1.71852
$643$	−11.1453	−0.439528	−0.219764	−	0.975553i	$-0.570529\pi$
$643$	−11.1453	−0.439528	−0.219764	+	0.975553i	$0.570529\pi$
$644$	63.8365	2.51551
$645$	5.55803	0.218847
$646$	−5.49738	−0.216291
$647$	5.15232	0.202559	0.101279	−	0.994858i	$-0.467706\pi$
$647$	5.15232	0.202559	0.101279	+	0.994858i	$0.467706\pi$
$648$	7.17280	0.281774
$649$	−0.540796	−0.0212281
$650$	−12.5657	−0.492867
$651$	9.00467	0.352921
$652$	−107.171	−4.19713
$653$	8.39877	0.328669	0.164335	−	0.986405i	$-0.447452\pi$
$653$	8.39877	0.328669	0.164335	+	0.986405i	$0.447452\pi$
$654$	36.9465	1.44472
$655$	3.61603	0.141290
$656$	24.6173	0.961143
$657$	−4.92029	−0.191959
$658$	−35.8867	−1.39901
$659$	−4.07650	−0.158798	−0.0793990	−	0.996843i	$-0.525300\pi$
$659$	−4.07650	−0.158798	−0.0793990	+	0.996843i	$0.525300\pi$
$660$	5.44230	0.211841
$661$	−6.83225	−0.265744	−0.132872	−	0.991133i	$-0.542420\pi$
$661$	−6.83225	−0.265744	−0.132872	+	0.991133i	$0.542420\pi$
$662$	−55.6982	−2.16477
$663$	3.49193	0.135615
$664$	−61.3773	−2.38190
$665$	7.73441	0.299927
$666$	−17.4245	−0.675184
$667$	−1.85555	−0.0718470
$668$	94.8326	3.66918
$669$	−14.1334	−0.546428
$670$	5.50267	0.212587
$671$	−3.05605	−0.117978
$672$	18.0630	0.696795
$673$	18.1685	0.700343	0.350172	−	0.936686i	$-0.386123\pi$
$673$	18.1685	0.700343	0.350172	+	0.936686i	$0.386123\pi$
$674$	−1.19316	−0.0459589
$675$	−1.38414	−0.0532756
$676$	−3.83762	−0.147601
$677$	27.5400	1.05845	0.529225	−	0.848482i	$-0.322483\pi$
$677$	27.5400	1.05845	0.529225	+	0.848482i	$0.322483\pi$
$678$	−42.4603	−1.63068
$679$	1.30688	0.0501534
$680$	−13.6394	−0.523046
$681$	−14.3418	−0.549580
$682$	−7.31922	−0.280268
$683$	−11.8293	−0.452635	−0.226318	−	0.974054i	$-0.572669\pi$
$683$	−11.8293	−0.452635	−0.226318	+	0.974054i	$0.572669\pi$
$684$	−10.0630	−0.384770
$685$	7.44643	0.284513
$686$	−51.5086	−1.96661
$687$	−26.1479	−0.997603
$688$	−26.6859	−1.01739
$689$	33.8970	1.29137
$690$	−34.4743	−1.31241
$691$	7.48404	0.284706	0.142353	−	0.989816i	$-0.454533\pi$
$691$	7.48404	0.284706	0.142353	+	0.989816i	$0.454533\pi$
$692$	9.42511	0.358289
$693$	−1.15683	−0.0439443
$694$	37.6590	1.42952
$695$	6.42018	0.243531
$696$	−1.90859	−0.0723447
$697$	2.69633	0.102131
$698$	−54.4967	−2.06273
$699$	−22.6035	−0.854944
$700$	−12.6707	−0.478907
$701$	4.19859	0.158578	0.0792892	−	0.996852i	$-0.474735\pi$
$701$	4.19859	0.158578	0.0792892	+	0.996852i	$0.474735\pi$
$702$	9.07834	0.342640
$703$	14.1721	0.534510
$704$	−3.70061	−0.139472
$705$	13.6456	0.513923
$706$	51.6450	1.94368
$707$	8.16041	0.306904
$708$	4.27941	0.160830
$709$	12.5505	0.471344	0.235672	−	0.971833i	$-0.424271\pi$
$709$	12.5505	0.471344	0.235672	+	0.971833i	$0.424271\pi$
$710$	−11.7289	−0.440177
$711$	1.00000	0.0375029
$712$	68.2328	2.55713
$713$	32.6445	1.22255
$714$	5.00088	0.187153
$715$	3.99332	0.149342
$716$	−45.4722	−1.69938
$717$	20.6815	0.772366
$718$	−80.2973	−2.99667
$719$	−50.0691	−1.86726	−0.933631	−	0.358237i	$-0.883378\pi$
$719$	−50.0691	−1.86726	−0.933631	+	0.358237i	$0.883378\pi$
$720$	−17.3609	−0.647003
$721$	6.45983	0.240577
$722$	−37.7719	−1.40572
$723$	12.5396	0.466354
$724$	28.3976	1.05539
$725$	0.368301	0.0136784
$726$	−27.6575	−1.02647
$727$	5.83137	0.216274	0.108137	−	0.994136i	$-0.465512\pi$
$727$	5.83137	0.216274	0.108137	+	0.994136i	$0.465512\pi$
$728$	48.1794	1.78565
$729$	1.00000	0.0370370
$730$	24.3241	0.900276
$731$	−2.92291	−0.108108
$732$	24.1831	0.893832
$733$	31.8669	1.17703	0.588516	−	0.808485i	$-0.299712\pi$
$733$	31.8669	1.17703	0.588516	+	0.808485i	$0.299712\pi$
$734$	−20.5782	−0.759556
$735$	6.27492	0.231454
$736$	65.4836	2.41376
$737$	0.669406	0.0246579
$738$	7.00993	0.258039
$739$	18.5598	0.682734	0.341367	−	0.939930i	$-0.389110\pi$
$739$	18.5598	0.682734	0.341367	+	0.939930i	$0.389110\pi$
$740$	60.6510	2.22958
$741$	−7.38382	−0.271251
$742$	48.5447	1.78213
$743$	7.84061	0.287644	0.143822	−	0.989604i	$-0.454061\pi$
$743$	7.84061	0.287644	0.143822	+	0.989604i	$0.454061\pi$
$744$	33.5777	1.23102
$745$	44.5445	1.63198
$746$	52.1488	1.90930
$747$	−8.55694	−0.313082
$748$	−2.86205	−0.104647
$749$	32.2172	1.17719
$750$	31.5609	1.15244
$751$	−20.9801	−0.765576	−0.382788	−	0.923836i	$-0.625036\pi$
$751$	−20.9801	−0.765576	−0.382788	+	0.923836i	$0.625036\pi$
$752$	−65.5169	−2.38916
$753$	22.3574	0.814748
$754$	−2.41562	−0.0879718
$755$	11.5418	0.420049
$756$	9.15419	0.332935
$757$	−7.84893	−0.285274	−0.142637	−	0.989775i	$-0.545558\pi$
$757$	−7.84893	−0.285274	−0.142637	+	0.989775i	$0.545558\pi$
$758$	1.84733	0.0670982
$759$	−4.19384	−0.152227
$760$	28.8410	1.04617
$761$	19.1295	0.693443	0.346721	−	0.937968i	$-0.387295\pi$
$761$	19.1295	0.693443	0.346721	+	0.937968i	$0.387295\pi$
$762$	−29.0996	−1.05417
$763$	27.3362	0.989638
$764$	−51.6232	−1.86766
$765$	−1.90154	−0.0687504
$766$	−75.0827	−2.71285
$767$	3.14005	0.113381
$768$	−19.5428	−0.705189
$769$	12.9971	0.468687	0.234343	−	0.972154i	$-0.424706\pi$
$769$	12.9971	0.468687	0.234343	+	0.972154i	$0.424706\pi$
$770$	5.71894	0.206096
$771$	−13.2153	−0.475939
$772$	−115.666	−4.16289
$773$	32.6909	1.17581	0.587905	−	0.808930i	$-0.299953\pi$
$773$	32.6909	1.17581	0.587905	+	0.808930i	$0.299953\pi$
$774$	−7.59898	−0.273140
$775$	−6.47950	−0.232751
$776$	4.87324	0.174939
$777$	−12.8921	−0.462503
$778$	74.5808	2.67385
$779$	−5.70149	−0.204277
$780$	−31.5999	−1.13146
$781$	−1.42683	−0.0510561
$782$	18.1296	0.648315
$783$	−0.266086	−0.00950915
$784$	−30.1279	−1.07600
$785$	−8.69869	−0.310470
$786$	−4.94387	−0.176342
$787$	−4.77284	−0.170133	−0.0850667	−	0.996375i	$-0.527110\pi$
$787$	−4.77284	−0.170133	−0.0850667	+	0.996375i	$0.527110\pi$
$788$	−2.83123	−0.100858
$789$	2.81471	0.100206
$790$	−4.94363	−0.175887
$791$	−31.4158	−1.11702
$792$	−4.31372	−0.153281
$793$	17.7445	0.630126
$794$	5.78601	0.205338
$795$	−18.4587	−0.654662
$796$	−26.4353	−0.936976
$797$	38.8900	1.37756	0.688778	−	0.724972i	$-0.258148\pi$
$797$	38.8900	1.37756	0.688778	+	0.724972i	$0.258148\pi$
$798$	−10.5745	−0.374335
$799$	−7.17607	−0.253871
$800$	−12.9976	−0.459535
$801$	9.51272	0.336115
$802$	71.7479	2.53351
$803$	2.95906	0.104423
$804$	−5.29713	−0.186815
$805$	−25.5071	−0.899007
$806$	42.4980	1.49693
$807$	0.687268	0.0241930
$808$	30.4295	1.07050
$809$	1.07458	0.0377803	0.0188901	−	0.999822i	$-0.493987\pi$
$809$	1.07458	0.0377803	0.0188901	+	0.999822i	$0.493987\pi$
$810$	−4.94363	−0.173702
$811$	47.9085	1.68229	0.841147	−	0.540807i	$-0.181881\pi$
$811$	47.9085	1.68229	0.841147	+	0.540807i	$0.181881\pi$
$812$	−2.43581	−0.0854800
$813$	−0.607608	−0.0213097
$814$	10.4791	0.367291
$815$	42.8221	1.49999
$816$	9.12992	0.319611
$817$	6.18059	0.216231
$818$	−57.4304	−2.00801
$819$	6.71695	0.234709
$820$	−24.4002	−0.852091
$821$	−0.00375589	−0.000131081 0	−6.55407e−5	−	1.00000i	$-0.500021\pi$
$821$	−0.00375589	−0.000131081 0	−6.55407e−5		1.00000i	$0.500021\pi$
$822$	−10.1808	−0.355097
$823$	3.91799	0.136572	0.0682862	−	0.997666i	$-0.478247\pi$
$823$	3.91799	0.136572	0.0682862	+	0.997666i	$0.478247\pi$
$824$	24.0882	0.839151
$825$	0.832420	0.0289812
$826$	4.49694	0.156469
$827$	27.3918	0.952508	0.476254	−	0.879308i	$-0.341994\pi$
$827$	27.3918	0.952508	0.476254	+	0.879308i	$0.341994\pi$
$828$	33.1866	1.15331
$829$	53.4719	1.85716	0.928578	−	0.371136i	$-0.121032\pi$
$829$	53.4719	1.85716	0.928578	+	0.371136i	$0.121032\pi$
$830$	42.3024	1.46834
$831$	−17.5393	−0.608433
$832$	21.4870	0.744929
$833$	−3.29991	−0.114335
$834$	−8.77773	−0.303948
$835$	−37.8922	−1.31131
$836$	6.05190	0.209309
$837$	4.68125	0.161808
$838$	0.250623	0.00865761
$839$	30.3556	1.04799	0.523996	−	0.851721i	$-0.324441\pi$
$839$	30.3556	1.04799	0.523996	+	0.851721i	$0.324441\pi$
$840$	−26.2362	−0.905235
$841$	−28.9292	−0.997559
$842$	47.4458	1.63509
$843$	12.2294	0.421204
$844$	21.2184	0.730366
$845$	1.53340	0.0527505
$846$	−18.6564	−0.641419
$847$	−20.4635	−0.703133
$848$	88.6262	3.04344
$849$	−15.5299	−0.532984
$850$	−3.59849	−0.123427
$851$	−46.7377	−1.60215
$852$	11.2908	0.386816
$853$	−10.5885	−0.362544	−0.181272	−	0.983433i	$-0.558021\pi$
$853$	−10.5885	−0.362544	−0.181272	+	0.983433i	$0.558021\pi$
$854$	25.4123	0.869592
$855$	4.02088	0.137511
$856$	120.135	4.10614
$857$	31.4176	1.07320	0.536602	−	0.843835i	$-0.319708\pi$
$857$	31.4176	1.07320	0.536602	+	0.843835i	$0.319708\pi$
$858$	−5.45971	−0.186391
$859$	43.1515	1.47231	0.736155	−	0.676813i	$-0.236639\pi$
$859$	43.1515	1.47231	0.736155	+	0.676813i	$0.236639\pi$
$860$	26.4505	0.901956
$861$	5.18656	0.176758
$862$	45.4303	1.54736
$863$	34.7738	1.18371	0.591857	−	0.806043i	$-0.298395\pi$
$863$	34.7738	1.18371	0.591857	+	0.806043i	$0.298395\pi$
$864$	9.39039	0.319468
$865$	−3.76598	−0.128047
$866$	57.7292	1.96172
$867$	1.00000	0.0339618
$868$	42.8530	1.45453
$869$	−0.601399	−0.0204011
$870$	1.31543	0.0445974
$871$	−3.88681	−0.131699
$872$	101.935	3.45194
$873$	0.679406	0.0229944
$874$	−38.3358	−1.29673
$875$	23.3515	0.789424
$876$	−23.4155	−0.791138
$877$	−57.1386	−1.92943	−0.964716	−	0.263293i	$-0.915192\pi$
$877$	−57.1386	−1.92943	−0.964716	+	0.263293i	$0.915192\pi$
$878$	12.8895	0.435001
$879$	6.32152	0.213220
$880$	10.4408	0.351961
$881$	15.9982	0.538994	0.269497	−	0.963001i	$-0.413143\pi$
$881$	15.9982	0.538994	0.269497	+	0.963001i	$0.413143\pi$
$882$	−8.57912	−0.288874
$883$	17.8576	0.600955	0.300478	−	0.953789i	$-0.402854\pi$
$883$	17.8576	0.600955	0.300478	+	0.953789i	$0.402854\pi$
$884$	16.6180	0.558925
$885$	−1.70992	−0.0574784
$886$	61.9789	2.08222
$887$	−3.75065	−0.125934	−0.0629672	−	0.998016i	$-0.520056\pi$
$887$	−3.75065	−0.125934	−0.0629672	+	0.998016i	$0.520056\pi$
$888$	−48.0737	−1.61325
$889$	−21.5305	−0.722108
$890$	−47.0274	−1.57636
$891$	−0.601399	−0.0201476
$892$	−67.2605	−2.25205
$893$	15.1741	0.507781
$894$	−60.9016	−2.03685
$895$	18.1693	0.607332
$896$	−5.35390	−0.178861
$897$	24.3509	0.813053
$898$	22.7715	0.759894
$899$	−1.24562	−0.0415437
$900$	−6.58709	−0.219570
$901$	9.70723	0.323395
$902$	−4.21577	−0.140370
$903$	−5.62239	−0.187101
$904$	−117.147	−3.89625
$905$	−11.3468	−0.377181
$906$	−15.7800	−0.524256
$907$	−0.702014	−0.0233100	−0.0116550	−	0.999932i	$-0.503710\pi$
$907$	−0.702014	−0.0233100	−0.0116550	+	0.999932i	$0.503710\pi$
$908$	−68.2524	−2.26504
$909$	4.24234	0.140710
$910$	−33.2061	−1.10077
$911$	−49.3118	−1.63377	−0.816886	−	0.576800i	$-0.804301\pi$
$911$	−49.3118	−1.63377	−0.816886	+	0.576800i	$0.804301\pi$
$912$	−19.3055	−0.639270
$913$	5.14614	0.170312
$914$	−75.0675	−2.48301
$915$	−9.66282	−0.319443
$916$	−124.437	−4.11152
$917$	−3.65791	−0.120795
$918$	2.59980	0.0858063
$919$	−17.4180	−0.574566	−0.287283	−	0.957846i	$-0.592752\pi$
$919$	−17.4180	−0.574566	−0.287283	+	0.957846i	$0.592752\pi$
$920$	−95.1138	−3.13581
$921$	−6.19070	−0.203991
$922$	12.3613	0.407099
$923$	8.28469	0.272694
$924$	−5.50532	−0.181112
$925$	9.27681	0.305020
$926$	8.04311	0.264313
$927$	3.35826	0.110300
$928$	−2.49866	−0.0820224
$929$	−10.3527	−0.339662	−0.169831	−	0.985473i	$-0.554322\pi$
$929$	−10.3527	−0.339662	−0.169831	+	0.985473i	$0.554322\pi$
$930$	−23.1424	−0.758869
$931$	6.97778	0.228687
$932$	−107.570	−3.52356
$933$	3.46645	0.113486
$934$	−15.9812	−0.522920
$935$	1.14359	0.0373992
$936$	25.0469	0.818685
$937$	22.2683	0.727473	0.363737	−	0.931502i	$-0.381501\pi$
$937$	22.2683	0.727473	0.363737	+	0.931502i	$0.381501\pi$
$938$	−5.56639	−0.181749
$939$	26.9753	0.880305
$940$	64.9391	2.11808
$941$	−9.75404	−0.317973	−0.158986	−	0.987281i	$-0.550823\pi$
$941$	−9.75404	−0.317973	−0.158986	+	0.987281i	$0.550823\pi$
$942$	11.8929	0.387492
$943$	18.8028	0.612303
$944$	8.20989	0.267209
$945$	−3.65773	−0.118986
$946$	4.57002	0.148584
$947$	−17.1241	−0.556459	−0.278229	−	0.960515i	$-0.589748\pi$
$947$	−17.1241	−0.556459	−0.278229	+	0.960515i	$0.589748\pi$
$948$	4.75898	0.154564
$949$	−17.1813	−0.557729
$950$	7.60914	0.246873
$951$	20.8973	0.677640
$952$	13.7973	0.447174
$953$	2.60227	0.0842957	0.0421479	−	0.999111i	$-0.486580\pi$
$953$	2.60227	0.0842957	0.0421479	+	0.999111i	$0.486580\pi$
$954$	25.2369	0.817074
$955$	20.6270	0.667476
$956$	98.4229	3.18323
$957$	0.160024	0.00517285
$958$	−21.5108	−0.694984
$959$	−7.53266	−0.243242
$960$	−11.7008	−0.377642
$961$	−9.08590	−0.293094
$962$	−60.8451	−1.96172
$963$	16.7487	0.539720
$964$	59.6759	1.92203
$965$	46.2164	1.48776
$966$	34.8735	1.12204
$967$	0.150363	0.00483536	0.00241768	−	0.999997i	$-0.499230\pi$
$967$	0.150363	0.00483536	0.00241768	+	0.999997i	$0.499230\pi$
$968$	−76.3065	−2.45259
$969$	−2.11454	−0.0679287
$970$	−3.35873	−0.107842
$971$	−35.6014	−1.14250	−0.571252	−	0.820775i	$-0.693542\pi$
$971$	−35.6014	−1.14250	−0.571252	+	0.820775i	$0.693542\pi$
$972$	4.75898	0.152644
$973$	−6.49453	−0.208205
$974$	−45.8935	−1.47052
$975$	−4.83333	−0.154790
$976$	46.3943	1.48505
$977$	34.6107	1.10729	0.553647	−	0.832751i	$-0.313236\pi$
$977$	34.6107	1.10729	0.553647	+	0.832751i	$0.313236\pi$
$978$	−58.5468	−1.87212
$979$	−5.72094	−0.182842
$980$	29.8622	0.953913
$981$	14.2113	0.453731
$982$	0.983449	0.0313831
$983$	−0.423378	−0.0135037	−0.00675184	−	0.999977i	$-0.502149\pi$
$983$	−0.423378	−0.0135037	−0.00675184	+	0.999977i	$0.502149\pi$
$984$	19.3403	0.616545
$985$	1.13127	0.0360454
$986$	−0.691773	−0.0220305
$987$	−13.8036	−0.439374
$988$	−35.1394	−1.11793
$989$	−20.3828	−0.648135
$990$	2.97310	0.0944912
$991$	4.80840	0.152744	0.0763719	−	0.997079i	$-0.475666\pi$
$991$	4.80840	0.152744	0.0763719	+	0.997079i	$0.475666\pi$
$992$	43.9588	1.39569
$993$	−21.4240	−0.679870
$994$	11.8647	0.376325
$995$	10.5628	0.334862
$996$	−40.7223	−1.29034
$997$	−0.617258	−0.0195488	−0.00977438	−	0.999952i	$-0.503111\pi$
$997$	−0.617258	−0.0195488	−0.00977438	+	0.999952i	$0.503111\pi$
$998$	68.1271	2.15653
$999$	−6.70222	−0.212049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4029.2.a.k.1.30	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4029.2.a.k.1.30	✓	31	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:	\( N \)	\(=\)	\( 4029 = 3 \cdot 17 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4029.a (trivial)

\(f(q)\)	\(=\)	\(q+2.59980 q^{2} +1.00000 q^{3} +4.75898 q^{4} -1.90154 q^{5} +2.59980 q^{6} +1.92356 q^{7} +7.17280 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.59980 q^{2} +1.00000 q^{3} +4.75898 q^{4} -1.90154 q^{5} +2.59980 q^{6} +1.92356 q^{7} +7.17280 q^{8} +1.00000 q^{9} -4.94363 q^{10} -0.601399 q^{11} +4.75898 q^{12} +3.49193 q^{13} +5.00088 q^{14} -1.90154 q^{15} +9.12992 q^{16} +1.00000 q^{17} +2.59980 q^{18} -2.11454 q^{19} -9.04939 q^{20} +1.92356 q^{21} -1.56352 q^{22} +6.97347 q^{23} +7.17280 q^{24} -1.38414 q^{25} +9.07834 q^{26} +1.00000 q^{27} +9.15419 q^{28} -0.266086 q^{29} -4.94363 q^{30} +4.68125 q^{31} +9.39039 q^{32} -0.601399 q^{33} +2.59980 q^{34} -3.65773 q^{35} +4.75898 q^{36} -6.70222 q^{37} -5.49738 q^{38} +3.49193 q^{39} -13.6394 q^{40} +2.69633 q^{41} +5.00088 q^{42} -2.92291 q^{43} -2.86205 q^{44} -1.90154 q^{45} +18.1296 q^{46} -7.17607 q^{47} +9.12992 q^{48} -3.29991 q^{49} -3.59849 q^{50} +1.00000 q^{51} +16.6180 q^{52} +9.70723 q^{53} +2.59980 q^{54} +1.14359 q^{55} +13.7973 q^{56} -2.11454 q^{57} -0.691773 q^{58} +0.899229 q^{59} -9.04939 q^{60} +5.08157 q^{61} +12.1703 q^{62} +1.92356 q^{63} +6.15333 q^{64} -6.64006 q^{65} -1.56352 q^{66} -1.11308 q^{67} +4.75898 q^{68} +6.97347 q^{69} -9.50939 q^{70} +2.37252 q^{71} +7.17280 q^{72} -4.92029 q^{73} -17.4245 q^{74} -1.38414 q^{75} -10.0630 q^{76} -1.15683 q^{77} +9.07834 q^{78} +1.00000 q^{79} -17.3609 q^{80} +1.00000 q^{81} +7.00993 q^{82} -8.55694 q^{83} +9.15419 q^{84} -1.90154 q^{85} -7.59898 q^{86} -0.266086 q^{87} -4.31372 q^{88} +9.51272 q^{89} -4.94363 q^{90} +6.71695 q^{91} +33.1866 q^{92} +4.68125 q^{93} -18.6564 q^{94} +4.02088 q^{95} +9.39039 q^{96} +0.679406 q^{97} -8.57912 q^{98} -0.601399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9}+O(q^{10}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9	\( 31 q + 4 q^{2} + 31 q^{3} + 34 q^{4} + 11 q^{5} + 4 q^{6} + 4 q^{7} + 12 q^{8} + 31 q^{9} + 5 q^{10} + 26 q^{11} + 34 q^{12} + 7 q^{13} + 19 q^{14} + 11 q^{15} + 40 q^{16} + 31 q^{17} + 4 q^{18} + 32 q^{19} + 23 q^{20} + 4 q^{21} + 2 q^{22} + 29 q^{23} + 12 q^{24} + 32 q^{25} + 13 q^{26} + 31 q^{27} - 13 q^{28} + 25 q^{29} + 5 q^{30} + 22 q^{31} + 28 q^{32} + 26 q^{33} + 4 q^{34} + 20 q^{35} + 34 q^{36} - 4 q^{37} + 19 q^{38} + 7 q^{39} - 3 q^{40} + 33 q^{41} + 19 q^{42} + 6 q^{43} + 30 q^{44} + 11 q^{45} - 11 q^{46} + 23 q^{47} + 40 q^{48} + 31 q^{49} + 6 q^{50} + 31 q^{51} - 7 q^{52} + 12 q^{53} + 4 q^{54} + 40 q^{56} + 32 q^{57} + 9 q^{58} + 27 q^{59} + 23 q^{60} - 4 q^{61} + 25 q^{62} + 4 q^{63} + 10 q^{64} + 54 q^{65} + 2 q^{66} + 34 q^{68} + 29 q^{69} - 59 q^{70} + 35 q^{71} + 12 q^{72} + 5 q^{73} + 48 q^{74} + 32 q^{75} + 32 q^{76} + 42 q^{77} + 13 q^{78} + 31 q^{79} + 24 q^{80} + 31 q^{81} + 5 q^{82} + 67 q^{83} - 13 q^{84} + 11 q^{85} - 20 q^{86} + 25 q^{87} - 7 q^{88} + 22 q^{89} + 5 q^{90} + 16 q^{91} + 57 q^{92} + 22 q^{93} + 45 q^{94} + 73 q^{95} + 28 q^{96} - 13 q^{97} - 19 q^{98} + 26 q^{99}+O(q^{100}) \) 31 * q + 4 * q^2 + 31 * q^3 + 34 * q^4 + 11 * q^5 + 4 * q^6 + 4 * q^7 + 12 * q^8 + 31 * q^9 + 5 * q^10 + 26 * q^11 + 34 * q^12 + 7 * q^13 + 19 * q^14 + 11 * q^15 + 40 * q^16 + 31 * q^17 + 4 * q^18 + 32 * q^19 + 23 * q^20 + 4 * q^21 + 2 * q^22 + 29 * q^23 + 12 * q^24 + 32 * q^25 + 13 * q^26 + 31 * q^27 - 13 * q^28 + 25 * q^29 + 5 * q^30 + 22 * q^31 + 28 * q^32 + 26 * q^33 + 4 * q^34 + 20 * q^35 + 34 * q^36 - 4 * q^37 + 19 * q^38 + 7 * q^39 - 3 * q^40 + 33 * q^41 + 19 * q^42 + 6 * q^43 + 30 * q^44 + 11 * q^45 - 11 * q^46 + 23 * q^47 + 40 * q^48 + 31 * q^49 + 6 * q^50 + 31 * q^51 - 7 * q^52 + 12 * q^53 + 4 * q^54 + 40 * q^56 + 32 * q^57 + 9 * q^58 + 27 * q^59 + 23 * q^60 - 4 * q^61 + 25 * q^62 + 4 * q^63 + 10 * q^64 + 54 * q^65 + 2 * q^66 + 34 * q^68 + 29 * q^69 - 59 * q^70 + 35 * q^71 + 12 * q^72 + 5 * q^73 + 48 * q^74 + 32 * q^75 + 32 * q^76 + 42 * q^77 + 13 * q^78 + 31 * q^79 + 24 * q^80 + 31 * q^81 + 5 * q^82 + 67 * q^83 - 13 * q^84 + 11 * q^85 - 20 * q^86 + 25 * q^87 - 7 * q^88 + 22 * q^89 + 5 * q^90 + 16 * q^91 + 57 * q^92 + 22 * q^93 + 45 * q^94 + 73 * q^95 + 28 * q^96 - 13 * q^97 - 19 * q^98 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more