LMFDB - Embedded newform 309.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [309,2,Mod(1,309)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(309, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("309.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 309 = 3 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	309.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:	$2.46737742246$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.81509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 3x^{2} + 5x - 2 $ x^5 - x^4 - 5x^3 + 3x^2 + 5*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.71377$ of defining polynomial
Character	$\chi$	$=$	309.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.82901 q^{2} -1.00000 q^{3} +1.34527 q^{4} -3.16702 q^{5} +1.82901 q^{6} +4.41676 q^{7} +1.19750 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.82901 q^{2} -1.00000 q^{3} +1.34527 q^{4} -3.16702 q^{5} +1.82901 q^{6} +4.41676 q^{7} +1.19750 q^{8} +1.00000 q^{9} +5.79251 q^{10} +1.42753 q^{11} -1.34527 q^{12} -2.72621 q^{13} -8.07829 q^{14} +3.16702 q^{15} -4.88079 q^{16} -7.27976 q^{17} -1.82901 q^{18} -4.37575 q^{19} -4.26051 q^{20} -4.41676 q^{21} -2.61097 q^{22} +4.27976 q^{23} -1.19750 q^{24} +5.03002 q^{25} +4.98626 q^{26} -1.00000 q^{27} +5.94175 q^{28} -1.24126 q^{29} -5.79251 q^{30} -4.26051 q^{31} +6.53200 q^{32} -1.42753 q^{33} +13.3147 q^{34} -13.9880 q^{35} +1.34527 q^{36} -8.74357 q^{37} +8.00329 q^{38} +2.72621 q^{39} -3.79251 q^{40} -5.17779 q^{41} +8.07829 q^{42} +2.64190 q^{43} +1.92042 q^{44} -3.16702 q^{45} -7.82771 q^{46} +0.126018 q^{47} +4.88079 q^{48} +12.5077 q^{49} -9.19996 q^{50} +7.27976 q^{51} -3.66750 q^{52} -2.10447 q^{53} +1.82901 q^{54} -4.52102 q^{55} +5.28907 q^{56} +4.37575 q^{57} +2.27028 q^{58} -6.95829 q^{59} +4.26051 q^{60} -13.1373 q^{61} +7.79251 q^{62} +4.41676 q^{63} -2.18551 q^{64} +8.63396 q^{65} +2.61097 q^{66} +1.41906 q^{67} -9.79326 q^{68} -4.27976 q^{69} +25.5841 q^{70} -2.04644 q^{71} +1.19750 q^{72} +14.6030 q^{73} +15.9921 q^{74} -5.03002 q^{75} -5.88658 q^{76} +6.30505 q^{77} -4.98626 q^{78} -15.9608 q^{79} +15.4576 q^{80} +1.00000 q^{81} +9.47023 q^{82} -1.57719 q^{83} -5.94175 q^{84} +23.0551 q^{85} -4.83206 q^{86} +1.24126 q^{87} +1.70947 q^{88} -17.1771 q^{89} +5.79251 q^{90} -12.0410 q^{91} +5.75744 q^{92} +4.26051 q^{93} -0.230487 q^{94} +13.8581 q^{95} -6.53200 q^{96} +2.07333 q^{97} -22.8768 q^{98} +1.42753 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9	\( 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9} - q^{10} - 12 q^{11} - 2 q^{12} + q^{13} - 8 q^{14} + 5 q^{15} - 4 q^{16} - 10 q^{17} - 2 q^{18} - 16 q^{19} - 13 q^{20} + 2 q^{21} + 4 q^{22} - 5 q^{23} + 6 q^{24} + 12 q^{25} - 10 q^{26} - 5 q^{27} + 7 q^{28} - 16 q^{29} + q^{30} - 13 q^{31} + 11 q^{32} + 12 q^{33} - 2 q^{34} - 22 q^{35} + 2 q^{36} + 4 q^{37} + 21 q^{38} - q^{39} + 11 q^{40} - 20 q^{41} + 8 q^{42} + 7 q^{43} + 2 q^{44} - 5 q^{45} + 8 q^{46} + 8 q^{47} + 4 q^{48} + 23 q^{49} + 13 q^{50} + 10 q^{51} + 31 q^{52} - 8 q^{53} + 2 q^{54} - 6 q^{55} + 5 q^{56} + 16 q^{57} + 20 q^{58} - 19 q^{59} + 13 q^{60} - 19 q^{61} + 9 q^{62} - 2 q^{63} - 16 q^{64} + q^{65} - 4 q^{66} + 11 q^{67} + 15 q^{68} + 5 q^{69} + 44 q^{70} - 10 q^{71} - 6 q^{72} + 20 q^{73} + 44 q^{74} - 12 q^{75} - 8 q^{76} + 8 q^{77} + 10 q^{78} - 14 q^{79} + 45 q^{80} + 5 q^{81} - 3 q^{82} - 11 q^{83} - 7 q^{84} + 12 q^{85} - 11 q^{86} + 16 q^{87} + 30 q^{88} + 22 q^{89} - q^{90} - 42 q^{91} - 21 q^{92} + 13 q^{93} - 6 q^{94} - 10 q^{95} - 11 q^{96} + 7 q^{97} - 11 q^{98} - 12 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9 - q^10 - 12 * q^11 - 2 * q^12 + q^13 - 8 * q^14 + 5 * q^15 - 4 * q^16 - 10 * q^17 - 2 * q^18 - 16 * q^19 - 13 * q^20 + 2 * q^21 + 4 * q^22 - 5 * q^23 + 6 * q^24 + 12 * q^25 - 10 * q^26 - 5 * q^27 + 7 * q^28 - 16 * q^29 + q^30 - 13 * q^31 + 11 * q^32 + 12 * q^33 - 2 * q^34 - 22 * q^35 + 2 * q^36 + 4 * q^37 + 21 * q^38 - q^39 + 11 * q^40 - 20 * q^41 + 8 * q^42 + 7 * q^43 + 2 * q^44 - 5 * q^45 + 8 * q^46 + 8 * q^47 + 4 * q^48 + 23 * q^49 + 13 * q^50 + 10 * q^51 + 31 * q^52 - 8 * q^53 + 2 * q^54 - 6 * q^55 + 5 * q^56 + 16 * q^57 + 20 * q^58 - 19 * q^59 + 13 * q^60 - 19 * q^61 + 9 * q^62 - 2 * q^63 - 16 * q^64 + q^65 - 4 * q^66 + 11 * q^67 + 15 * q^68 + 5 * q^69 + 44 * q^70 - 10 * q^71 - 6 * q^72 + 20 * q^73 + 44 * q^74 - 12 * q^75 - 8 * q^76 + 8 * q^77 + 10 * q^78 - 14 * q^79 + 45 * q^80 + 5 * q^81 - 3 * q^82 - 11 * q^83 - 7 * q^84 + 12 * q^85 - 11 * q^86 + 16 * q^87 + 30 * q^88 + 22 * q^89 - q^90 - 42 * q^91 - 21 * q^92 + 13 * q^93 - 6 * q^94 - 10 * q^95 - 11 * q^96 + 7 * q^97 - 11 * q^98 - 12 * 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 $

$2$	−1.82901	−1.29330	−0.646652	−	0.762785i	$-0.723832\pi$
$2$	−1.82901	−1.29330	−0.646652	+	0.762785i	$0.723832\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.34527	0.672637
$5$	−3.16702	−1.41633	−0.708167	−	0.706045i	$-0.750478\pi$
$5$	−3.16702	−1.41633	−0.708167	+	0.706045i	$0.750478\pi$
$6$	1.82901	0.746690
$7$	4.41676	1.66938	0.834689	−	0.550722i	$-0.185648\pi$
$7$	4.41676	1.66938	0.834689	+	0.550722i	$0.185648\pi$
$8$	1.19750	0.423380
$9$	1.00000	0.333333
$10$	5.79251	1.83175
$11$	1.42753	0.430417	0.215208	−	0.976568i	$-0.430957\pi$
$11$	1.42753	0.430417	0.215208	+	0.976568i	$0.430957\pi$
$12$	−1.34527	−0.388347
$13$	−2.72621	−0.756114	−0.378057	−	0.925782i	$-0.623408\pi$
$13$	−2.72621	−0.756114	−0.378057	+	0.925782i	$0.623408\pi$
$14$	−8.07829	−2.15901
$15$	3.16702	0.817721
$16$	−4.88079	−1.22020
$17$	−7.27976	−1.76560	−0.882800	−	0.469749i	$-0.844345\pi$
$17$	−7.27976	−1.76560	−0.882800	+	0.469749i	$0.844345\pi$
$18$	−1.82901	−0.431102
$19$	−4.37575	−1.00387	−0.501933	−	0.864906i	$-0.667378\pi$
$19$	−4.37575	−1.00387	−0.501933	+	0.864906i	$0.667378\pi$
$20$	−4.26051	−0.952679
$21$	−4.41676	−0.963815
$22$	−2.61097	−0.556660
$23$	4.27976	0.892391	0.446196	−	0.894935i	$-0.352779\pi$
$23$	4.27976	0.892391	0.446196	+	0.894935i	$0.352779\pi$
$24$	−1.19750	−0.244439
$25$	5.03002	1.00600
$26$	4.98626	0.977886
$27$	−1.00000	−0.192450
$28$	5.94175	1.12288
$29$	−1.24126	−0.230496	−0.115248	−	0.993337i	$-0.536766\pi$
$29$	−1.24126	−0.230496	−0.115248	+	0.993337i	$0.536766\pi$
$30$	−5.79251	−1.05756
$31$	−4.26051	−0.765210	−0.382605	−	0.923912i	$-0.624973\pi$
$31$	−4.26051	−0.765210	−0.382605	+	0.923912i	$0.624973\pi$
$32$	6.53200	1.15471
$33$	−1.42753	−0.248501
$34$	13.3147	2.28346
$35$	−13.9880	−2.36440
$36$	1.34527	0.224212
$37$	−8.74357	−1.43743	−0.718717	−	0.695303i	$-0.755270\pi$
$37$	−8.74357	−1.43743	−0.718717	+	0.695303i	$0.755270\pi$
$38$	8.00329	1.29831
$39$	2.72621	0.436543
$40$	−3.79251	−0.599648
$41$	−5.17779	−0.808636	−0.404318	−	0.914619i	$-0.632491\pi$
$41$	−5.17779	−0.808636	−0.404318	+	0.914619i	$0.632491\pi$
$42$	8.07829	1.24651
$43$	2.64190	0.402886	0.201443	−	0.979500i	$-0.435437\pi$
$43$	2.64190	0.402886	0.201443	+	0.979500i	$0.435437\pi$
$44$	1.92042	0.289514
$45$	−3.16702	−0.472112
$46$	−7.82771	−1.15413
$47$	0.126018	0.0183816	0.00919078	−	0.999958i	$-0.497074\pi$
$47$	0.126018	0.0183816	0.00919078	+	0.999958i	$0.497074\pi$
$48$	4.88079	0.704481
$49$	12.5077	1.78682
$50$	−9.19996	−1.30107
$51$	7.27976	1.01937
$52$	−3.66750	−0.508590
$53$	−2.10447	−0.289071	−0.144536	−	0.989500i	$-0.546169\pi$
$53$	−2.10447	−0.289071	−0.144536	+	0.989500i	$0.546169\pi$
$54$	1.82901	0.248897
$55$	−4.52102	−0.609614
$56$	5.28907	0.706782
$57$	4.37575	0.579583
$58$	2.27028	0.298102
$59$	−6.95829	−0.905892	−0.452946	−	0.891538i	$-0.649627\pi$
$59$	−6.95829	−0.905892	−0.452946	+	0.891538i	$0.649627\pi$
$60$	4.26051	0.550029
$61$	−13.1373	−1.68206	−0.841031	−	0.540987i	$-0.818051\pi$
$61$	−13.1373	−1.68206	−0.841031	+	0.540987i	$0.818051\pi$
$62$	7.79251	0.989650
$63$	4.41676	0.556459
$64$	−2.18551	−0.273189
$65$	8.63396	1.07091
$66$	2.61097	0.321388
$67$	1.41906	0.173365	0.0866827	−	0.996236i	$-0.472373\pi$
$67$	1.41906	0.173365	0.0866827	+	0.996236i	$0.472373\pi$
$68$	−9.79326	−1.18761
$69$	−4.27976	−0.515222
$70$	25.5841	3.05789
$71$	−2.04644	−0.242867	−0.121434	−	0.992600i	$-0.538749\pi$
$71$	−2.04644	−0.242867	−0.121434	+	0.992600i	$0.538749\pi$
$72$	1.19750	0.141127
$73$	14.6030	1.70915	0.854577	−	0.519324i	$-0.173816\pi$
$73$	14.6030	1.70915	0.854577	+	0.519324i	$0.173816\pi$
$74$	15.9921	1.85904
$75$	−5.03002	−0.580817
$76$	−5.88658	−0.675238
$77$	6.30505	0.718528
$78$	−4.98626	−0.564583
$79$	−15.9608	−1.79573	−0.897864	−	0.440274i	$-0.854881\pi$
$79$	−15.9608	−1.79573	−0.897864	+	0.440274i	$0.854881\pi$
$80$	15.4576	1.72821
$81$	1.00000	0.111111
$82$	9.47023	1.04581
$83$	−1.57719	−0.173120	−0.0865598	−	0.996247i	$-0.527587\pi$
$83$	−1.57719	−0.173120	−0.0865598	+	0.996247i	$0.527587\pi$
$84$	−5.94175	−0.648298
$85$	23.0551	2.50068
$86$	−4.83206	−0.521055
$87$	1.24126	0.133077
$88$	1.70947	0.182230
$89$	−17.1771	−1.82077	−0.910383	−	0.413768i	$-0.864213\pi$
$89$	−17.1771	−1.82077	−0.910383	+	0.413768i	$0.864213\pi$
$90$	5.79251	0.610584
$91$	−12.0410	−1.26224
$92$	5.75744	0.600255
$93$	4.26051	0.441794
$94$	−0.230487	−0.0237730
$95$	13.8581	1.42181
$96$	−6.53200	−0.666669
$97$	2.07333	0.210514	0.105257	−	0.994445i	$-0.466433\pi$
$97$	2.07333	0.210514	0.105257	+	0.994445i	$0.466433\pi$
$98$	−22.8768	−2.31090
$99$	1.42753	0.143472
$100$	6.76676	0.676676
$101$	8.79848	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$101$	8.79848	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$102$	−13.3147	−1.31836
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−3.26464	−0.320124
$105$	13.9880	1.36509
$106$	3.84909	0.373857
$107$	11.0290	1.06621	0.533105	−	0.846049i	$-0.321025\pi$
$107$	11.0290	1.06621	0.533105	+	0.846049i	$0.321025\pi$
$108$	−1.34527	−0.129449
$109$	−1.30945	−0.125423	−0.0627114	−	0.998032i	$-0.519975\pi$
$109$	−1.30945	−0.125423	−0.0627114	+	0.998032i	$0.519975\pi$
$110$	8.26898	0.788417
$111$	8.74357	0.829903
$112$	−21.5572	−2.03697
$113$	19.0861	1.79547	0.897734	−	0.440538i	$-0.145212\pi$
$113$	19.0861	1.79547	0.897734	+	0.440538i	$0.145212\pi$
$114$	−8.00329	−0.749577
$115$	−13.5541	−1.26392
$116$	−1.66984	−0.155040
$117$	−2.72621	−0.252038
$118$	12.7268	1.17159
$119$	−32.1529	−2.94745
$120$	3.79251	0.346207
$121$	−8.96216	−0.814742
$122$	24.0283	2.17542
$123$	5.17779	0.466866
$124$	−5.73155	−0.514708
$125$	−0.0950805	−0.00850426
$126$	−8.07829	−0.719671
$127$	−8.94403	−0.793654	−0.396827	−	0.917893i	$-0.629889\pi$
$127$	−8.94403	−0.793654	−0.396827	+	0.917893i	$0.629889\pi$
$128$	−9.06668	−0.801389
$129$	−2.64190	−0.232607
$130$	−15.7916	−1.38501
$131$	−13.4632	−1.17628	−0.588142	−	0.808757i	$-0.700141\pi$
$131$	−13.4632	−1.17628	−0.588142	+	0.808757i	$0.700141\pi$
$132$	−1.92042	−0.167151
$133$	−19.3266	−1.67583
$134$	−2.59547	−0.224214
$135$	3.16702	0.272574
$136$	−8.71751	−0.747521
$137$	−5.27817	−0.450944	−0.225472	−	0.974250i	$-0.572392\pi$
$137$	−5.27817	−0.450944	−0.225472	+	0.974250i	$0.572392\pi$
$138$	7.82771	0.666339
$139$	8.10604	0.687545	0.343773	−	0.939053i	$-0.388295\pi$
$139$	8.10604	0.687545	0.343773	+	0.939053i	$0.388295\pi$
$140$	−18.8176	−1.59038
$141$	−0.126018	−0.0106126
$142$	3.74295	0.314101
$143$	−3.89175	−0.325444
$144$	−4.88079	−0.406732
$145$	3.93110	0.326460
$146$	−26.7091	−2.21046
$147$	−12.5077	−1.03162
$148$	−11.7625	−0.966871
$149$	11.1728	0.915310	0.457655	−	0.889130i	$-0.348690\pi$
$149$	11.1728	0.915310	0.457655	+	0.889130i	$0.348690\pi$
$150$	9.19996	0.751173
$151$	16.2835	1.32513	0.662566	−	0.749003i	$-0.269467\pi$
$151$	16.2835	1.32513	0.662566	+	0.749003i	$0.269467\pi$
$152$	−5.23997	−0.425017
$153$	−7.27976	−0.588534
$154$	−11.5320	−0.929275
$155$	13.4931	1.08379
$156$	3.66750	0.293635
$157$	−16.0265	−1.27905	−0.639525	−	0.768770i	$-0.720869\pi$
$157$	−16.0265	−1.27905	−0.639525	+	0.768770i	$0.720869\pi$
$158$	29.1924	2.32242
$159$	2.10447	0.166895
$160$	−20.6870	−1.63545
$161$	18.9026	1.48974
$162$	−1.82901	−0.143701
$163$	−18.1704	−1.42321	−0.711607	−	0.702577i	$-0.752032\pi$
$163$	−18.1704	−1.42321	−0.711607	+	0.702577i	$0.752032\pi$
$164$	−6.96555	−0.543918
$165$	4.52102	0.351961
$166$	2.88470	0.223896
$167$	17.4871	1.35319	0.676597	−	0.736353i	$-0.263454\pi$
$167$	17.4871	1.35319	0.676597	+	0.736353i	$0.263454\pi$
$168$	−5.28907	−0.408060
$169$	−5.56778	−0.428291
$170$	−42.1681	−3.23414
$171$	−4.37575	−0.334622
$172$	3.55408	0.270996
$173$	21.9480	1.66868	0.834340	−	0.551251i	$-0.185849\pi$
$173$	21.9480	1.66868	0.834340	+	0.551251i	$0.185849\pi$
$174$	−2.27028	−0.172109
$175$	22.2164	1.67940
$176$	−6.96747	−0.525193
$177$	6.95829	0.523017
$178$	31.4170	2.35480
$179$	−9.01736	−0.673989	−0.336994	−	0.941507i	$-0.609410\pi$
$179$	−9.01736	−0.673989	−0.336994	+	0.941507i	$0.609410\pi$
$180$	−4.26051	−0.317560
$181$	16.9371	1.25893	0.629463	−	0.777030i	$-0.283275\pi$
$181$	16.9371	1.25893	0.629463	+	0.777030i	$0.283275\pi$
$182$	22.0231	1.63246
$183$	13.1373	0.971139
$184$	5.12501	0.377821
$185$	27.6911	2.03589
$186$	−7.79251	−0.571375
$187$	−10.3921	−0.759944
$188$	0.169528	0.0123641
$189$	−4.41676	−0.321272
$190$	−25.3466	−1.83884
$191$	10.1036	0.731068	0.365534	−	0.930798i	$-0.380886\pi$
$191$	10.1036	0.731068	0.365534	+	0.930798i	$0.380886\pi$
$192$	2.18551	0.157726
$193$	17.2149	1.23916	0.619578	−	0.784935i	$-0.287304\pi$
$193$	17.2149	1.23916	0.619578	+	0.784935i	$0.287304\pi$
$194$	−3.79213	−0.272259
$195$	−8.63396	−0.618291
$196$	16.8263	1.20188
$197$	−18.4532	−1.31473	−0.657366	−	0.753571i	$-0.728330\pi$
$197$	−18.4532	−1.31473	−0.657366	+	0.753571i	$0.728330\pi$
$198$	−2.61097	−0.185553
$199$	−1.19985	−0.0850551	−0.0425275	−	0.999095i	$-0.513541\pi$
$199$	−1.19985	−0.0850551	−0.0425275	+	0.999095i	$0.513541\pi$
$200$	6.02345	0.425923
$201$	−1.41906	−0.100093
$202$	−16.0925	−1.13226
$203$	−5.48235	−0.384785
$204$	9.79326	0.685666
$205$	16.3982	1.14530
$206$	1.82901	0.127433
$207$	4.27976	0.297464
$208$	13.3060	0.922608
$209$	−6.24652	−0.432081
$210$	−25.5841	−1.76547
$211$	−17.7695	−1.22330	−0.611652	−	0.791127i	$-0.709495\pi$
$211$	−17.7695	−1.22330	−0.611652	+	0.791127i	$0.709495\pi$
$212$	−2.83109	−0.194440
$213$	2.04644	0.140220
$214$	−20.1721	−1.37893
$215$	−8.36696	−0.570622
$216$	−1.19750	−0.0814796
$217$	−18.8176	−1.27742
$218$	2.39500	0.162210
$219$	−14.6030	−0.986781
$220$	−6.08201	−0.410049
$221$	19.8461	1.33500
$222$	−15.9921	−1.07332
$223$	9.87877	0.661531	0.330766	−	0.943713i	$-0.392693\pi$
$223$	9.87877	0.661531	0.330766	+	0.943713i	$0.392693\pi$
$224$	28.8503	1.92764
$225$	5.03002	0.335335
$226$	−34.9086	−2.32209
$227$	−9.43163	−0.625999	−0.313000	−	0.949753i	$-0.601334\pi$
$227$	−9.43163	−0.625999	−0.313000	+	0.949753i	$0.601334\pi$
$228$	5.88658	0.389849
$229$	18.8803	1.24764	0.623822	−	0.781567i	$-0.285579\pi$
$229$	18.8803	1.24764	0.623822	+	0.781567i	$0.285579\pi$
$230$	24.7905	1.63464
$231$	−6.30505	−0.414842
$232$	−1.48641	−0.0975877
$233$	−19.5676	−1.28191	−0.640957	−	0.767577i	$-0.721462\pi$
$233$	−19.5676	−1.28191	−0.640957	+	0.767577i	$0.721462\pi$
$234$	4.98626	0.325962
$235$	−0.399100	−0.0260344
$236$	−9.36080	−0.609336
$237$	15.9608	1.03676
$238$	58.8080	3.81195
$239$	−20.1742	−1.30496	−0.652481	−	0.757805i	$-0.726272\pi$
$239$	−20.1742	−1.30496	−0.652481	+	0.757805i	$0.726272\pi$
$240$	−15.4576	−0.997781
$241$	11.4515	0.737656	0.368828	−	0.929498i	$-0.379759\pi$
$241$	11.4515	0.737656	0.368828	+	0.929498i	$0.379759\pi$
$242$	16.3919	1.05371
$243$	−1.00000	−0.0641500
$244$	−17.6733	−1.13142
$245$	−39.6123	−2.53073
$246$	−9.47023	−0.603800
$247$	11.9292	0.759038
$248$	−5.10196	−0.323975
$249$	1.57719	0.0999507
$250$	0.173903	0.0109986
$251$	18.1022	1.14260	0.571300	−	0.820741i	$-0.306439\pi$
$251$	18.1022	1.14260	0.571300	+	0.820741i	$0.306439\pi$
$252$	5.94175	0.374295
$253$	6.10948	0.384100
$254$	16.3587	1.02644
$255$	−23.0551	−1.44377
$256$	20.9541	1.30963
$257$	−20.5320	−1.28075	−0.640375	−	0.768063i	$-0.721221\pi$
$257$	−20.5320	−1.28075	−0.640375	+	0.768063i	$0.721221\pi$
$258$	4.83206	0.300831
$259$	−38.6182	−2.39962
$260$	11.6150	0.720334
$261$	−1.24126	−0.0768321
$262$	24.6243	1.52129
$263$	18.7150	1.15402	0.577008	−	0.816739i	$-0.304220\pi$
$263$	18.7150	1.15402	0.577008	+	0.816739i	$0.304220\pi$
$264$	−1.70947	−0.105211
$265$	6.66490	0.409422
$266$	35.3486	2.16736
$267$	17.1771	1.05122
$268$	1.90902	0.116612
$269$	−8.78619	−0.535704	−0.267852	−	0.963460i	$-0.586314\pi$
$269$	−8.78619	−0.535704	−0.267852	+	0.963460i	$0.586314\pi$
$270$	−5.79251	−0.352521
$271$	9.29552	0.564663	0.282331	−	0.959317i	$-0.408892\pi$
$271$	9.29552	0.564663	0.282331	+	0.959317i	$0.408892\pi$
$272$	35.5309	2.15438
$273$	12.0410	0.728755
$274$	9.65381	0.583208
$275$	7.18051	0.433001
$276$	−5.75744	−0.346557
$277$	24.9131	1.49688	0.748441	−	0.663201i	$-0.230803\pi$
$277$	24.9131	1.49688	0.748441	+	0.663201i	$0.230803\pi$
$278$	−14.8260	−0.889205
$279$	−4.26051	−0.255070
$280$	−16.7506	−1.00104
$281$	−17.1640	−1.02392	−0.511959	−	0.859010i	$-0.671080\pi$
$281$	−17.1640	−1.02392	−0.511959	+	0.859010i	$0.671080\pi$
$282$	0.230487	0.0137253
$283$	−11.0224	−0.655213	−0.327606	−	0.944814i	$-0.606242\pi$
$283$	−11.0224	−0.655213	−0.327606	+	0.944814i	$0.606242\pi$
$284$	−2.75302	−0.163361
$285$	−13.8581	−0.820883
$286$	7.11804	0.420899
$287$	−22.8691	−1.34992
$288$	6.53200	0.384902
$289$	35.9949	2.11735
$290$	−7.19002	−0.422212
$291$	−2.07333	−0.121540
$292$	19.6451	1.14964
$293$	14.2854	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$293$	14.2854	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$294$	22.8768	1.33420
$295$	22.0370	1.28305
$296$	−10.4704	−0.608581
$297$	−1.42753	−0.0828337
$298$	−20.4351	−1.18377
$299$	−11.6675	−0.674750
$300$	−6.76676	−0.390679
$301$	11.6686	0.672569
$302$	−29.7827	−1.71380
$303$	−8.79848	−0.505459
$304$	21.3571	1.22491
$305$	41.6062	2.38236
$306$	13.3147	0.761153
$307$	−5.23214	−0.298614	−0.149307	−	0.988791i	$-0.547704\pi$
$307$	−5.23214	−0.298614	−0.149307	+	0.988791i	$0.547704\pi$
$308$	8.48202	0.483308
$309$	1.00000	0.0568880
$310$	−24.6790	−1.40168
$311$	−8.01727	−0.454618	−0.227309	−	0.973823i	$-0.572993\pi$
$311$	−8.01727	−0.454618	−0.227309	+	0.973823i	$0.572993\pi$
$312$	3.26464	0.184824
$313$	23.8090	1.34576	0.672881	−	0.739751i	$-0.265057\pi$
$313$	23.8090	1.34576	0.672881	+	0.739751i	$0.265057\pi$
$314$	29.3125	1.65420
$315$	−13.9880	−0.788132
$316$	−21.4716	−1.20787
$317$	−30.2243	−1.69757	−0.848784	−	0.528740i	$-0.822664\pi$
$317$	−30.2243	−1.69757	−0.848784	+	0.528740i	$0.822664\pi$
$318$	−3.84909	−0.215846
$319$	−1.77194	−0.0992095
$320$	6.92157	0.386927
$321$	−11.0290	−0.615577
$322$	−34.5731	−1.92668
$323$	31.8544	1.77243
$324$	1.34527	0.0747374
$325$	−13.7129	−0.760654
$326$	33.2338	1.84065
$327$	1.30945	0.0724129
$328$	−6.20041	−0.342361
$329$	0.556589	0.0306858
$330$	−8.26898	−0.455193
$331$	35.0598	1.92706	0.963531	−	0.267595i	$-0.0862290\pi$
$331$	35.0598	1.92706	0.963531	+	0.267595i	$0.0862290\pi$
$332$	−2.12176	−0.116447
$333$	−8.74357	−0.479144
$334$	−31.9841	−1.75009
$335$	−4.49418	−0.245543
$336$	21.5572	1.17604
$337$	−6.82350	−0.371700	−0.185850	−	0.982578i	$-0.559504\pi$
$337$	−6.82350	−0.371700	−0.185850	+	0.982578i	$0.559504\pi$
$338$	10.1835	0.553911
$339$	−19.0861	−1.03661
$340$	31.0155	1.68205
$341$	−6.08201	−0.329359
$342$	8.00329	0.432768
$343$	24.3263	1.31350
$344$	3.16368	0.170574
$345$	13.5541	0.729727
$346$	−40.1432	−2.15811
$347$	5.99549	0.321855	0.160927	−	0.986966i	$-0.448551\pi$
$347$	5.99549	0.321855	0.160927	+	0.986966i	$0.448551\pi$
$348$	1.66984	0.0895126
$349$	−4.44509	−0.237940	−0.118970	−	0.992898i	$-0.537959\pi$
$349$	−4.44509	−0.237940	−0.118970	+	0.992898i	$0.537959\pi$
$350$	−40.6340	−2.17198
$351$	2.72621	0.145514
$352$	9.32463	0.497004
$353$	−25.7195	−1.36891	−0.684456	−	0.729054i	$-0.739960\pi$
$353$	−25.7195	−1.36891	−0.684456	+	0.729054i	$0.739960\pi$
$354$	−12.7268	−0.676420
$355$	6.48111	0.343981
$356$	−23.1078	−1.22471
$357$	32.1529	1.70171
$358$	16.4928	0.871673
$359$	−2.78903	−0.147199	−0.0735996	−	0.997288i	$-0.523449\pi$
$359$	−2.78903	−0.147199	−0.0735996	+	0.997288i	$0.523449\pi$
$360$	−3.79251	−0.199883
$361$	0.147215	0.00774816
$362$	−30.9782	−1.62818
$363$	8.96216	0.470391
$364$	−16.1984	−0.849029
$365$	−46.2481	−2.42074
$366$	−24.0283	−1.25598
$367$	10.9901	0.573676	0.286838	−	0.957979i	$-0.407396\pi$
$367$	10.9901	0.573676	0.286838	+	0.957979i	$0.407396\pi$
$368$	−20.8886	−1.08889
$369$	−5.17779	−0.269545
$370$	−50.6472	−2.63302
$371$	−9.29493	−0.482569
$372$	5.73155	0.297167
$373$	−17.1168	−0.886272	−0.443136	−	0.896454i	$-0.646134\pi$
$373$	−17.1168	−0.886272	−0.443136	+	0.896454i	$0.646134\pi$
$374$	19.0072	0.982839
$375$	0.0950805	0.00490994
$376$	0.150906	0.00778239
$377$	3.38394	0.174282
$378$	8.07829	0.415502
$379$	−13.6839	−0.702897	−0.351448	−	0.936207i	$-0.614311\pi$
$379$	−13.6839	−0.702897	−0.351448	+	0.936207i	$0.614311\pi$
$380$	18.6429	0.956362
$381$	8.94403	0.458217
$382$	−18.4795	−0.945493
$383$	32.8641	1.67928	0.839638	−	0.543147i	$-0.182767\pi$
$383$	32.8641	1.67928	0.839638	+	0.543147i	$0.182767\pi$
$384$	9.06668	0.462682
$385$	−19.9682	−1.01768
$386$	−31.4862	−1.60261
$387$	2.64190	0.134295
$388$	2.78919	0.141600
$389$	−4.20058	−0.212978	−0.106489	−	0.994314i	$-0.533961\pi$
$389$	−4.20058	−0.212978	−0.106489	+	0.994314i	$0.533961\pi$
$390$	15.7916	0.799638
$391$	−31.1556	−1.57561
$392$	14.9780	0.756504
$393$	13.4632	0.679128
$394$	33.7510	1.70035
$395$	50.5481	2.54335
$396$	1.92042	0.0965047
$397$	2.53262	0.127108	0.0635542	−	0.997978i	$-0.479756\pi$
$397$	2.53262	0.127108	0.0635542	+	0.997978i	$0.479756\pi$
$398$	2.19454	0.110002
$399$	19.3266	0.967542
$400$	−24.5505	−1.22752
$401$	4.17266	0.208373	0.104186	−	0.994558i	$-0.466776\pi$
$401$	4.17266	0.208373	0.104186	+	0.994558i	$0.466776\pi$
$402$	2.59547	0.129450
$403$	11.6150	0.578586
$404$	11.8364	0.588881
$405$	−3.16702	−0.157371
$406$	10.0273	0.497645
$407$	−12.4817	−0.618695
$408$	8.71751	0.431581
$409$	−20.5484	−1.01606	−0.508028	−	0.861341i	$-0.669625\pi$
$409$	−20.5484	−1.01606	−0.508028	+	0.861341i	$0.669625\pi$
$410$	−29.9924	−1.48122
$411$	5.27817	0.260353
$412$	−1.34527	−0.0662769
$413$	−30.7331	−1.51228
$414$	−7.82771	−0.384711
$415$	4.99501	0.245195
$416$	−17.8076	−0.873089
$417$	−8.10604	−0.396954
$418$	11.4249	0.558812
$419$	−7.60178	−0.371371	−0.185686	−	0.982609i	$-0.559451\pi$
$419$	−7.60178	−0.371371	−0.185686	+	0.982609i	$0.559451\pi$
$420$	18.8176	0.918206
$421$	7.91685	0.385844	0.192922	−	0.981214i	$-0.438204\pi$
$421$	7.91685	0.385844	0.192922	+	0.981214i	$0.438204\pi$
$422$	32.5006	1.58210
$423$	0.126018	0.00612719
$424$	−2.52010	−0.122387
$425$	−36.6173	−1.77620
$426$	−3.74295	−0.181347
$427$	−58.0244	−2.80800
$428$	14.8370	0.717172
$429$	3.89175	0.187895
$430$	15.3032	0.737988
$431$	−8.36401	−0.402880	−0.201440	−	0.979501i	$-0.564562\pi$
$431$	−8.36401	−0.402880	−0.201440	+	0.979501i	$0.564562\pi$
$432$	4.88079	0.234827
$433$	−23.2493	−1.11729	−0.558645	−	0.829407i	$-0.688679\pi$
$433$	−23.2493	−1.11729	−0.558645	+	0.829407i	$0.688679\pi$
$434$	34.4176	1.65210
$435$	−3.93110	−0.188482
$436$	−1.76157	−0.0843640
$437$	−18.7272	−0.895842
$438$	26.7091	1.27621
$439$	1.28321	0.0612441	0.0306221	−	0.999531i	$-0.490251\pi$
$439$	1.28321	0.0612441	0.0306221	+	0.999531i	$0.490251\pi$
$440$	−5.41392	−0.258099
$441$	12.5077	0.595607
$442$	−36.2988	−1.72656
$443$	−12.2466	−0.581852	−0.290926	−	0.956745i	$-0.593963\pi$
$443$	−12.2466	−0.581852	−0.290926	+	0.956745i	$0.593963\pi$
$444$	11.7625	0.558223
$445$	54.4001	2.57881
$446$	−18.0683	−0.855561
$447$	−11.1728	−0.528454
$448$	−9.65288	−0.456056
$449$	0.00761676	0.000359457 0	0.000179729	−	1.00000i	$-0.499943\pi$
$449$	0.00761676	0.000359457 0	0.000179729		1.00000i	$0.499943\pi$
$450$	−9.19996	−0.433690
$451$	−7.39146	−0.348050
$452$	25.6760	1.20770
$453$	−16.2835	−0.765066
$454$	17.2505	0.809608
$455$	38.1341	1.78775
$456$	5.23997	0.245384
$457$	14.0539	0.657415	0.328708	−	0.944432i	$-0.393387\pi$
$457$	14.0539	0.657415	0.328708	+	0.944432i	$0.393387\pi$
$458$	−34.5322	−1.61358
$459$	7.27976	0.339790
$460$	−18.2339	−0.850162
$461$	33.1000	1.54162	0.770810	−	0.637066i	$-0.219852\pi$
$461$	33.1000	1.54162	0.770810	+	0.637066i	$0.219852\pi$
$462$	11.5320	0.536517
$463$	−5.72972	−0.266283	−0.133141	−	0.991097i	$-0.542506\pi$
$463$	−5.72972	−0.266283	−0.133141	+	0.991097i	$0.542506\pi$
$464$	6.05833	0.281251
$465$	−13.4931	−0.625729
$466$	35.7893	1.65791
$467$	9.18054	0.424825	0.212412	−	0.977180i	$-0.431868\pi$
$467$	9.18054	0.424825	0.212412	+	0.977180i	$0.431868\pi$
$468$	−3.66750	−0.169530
$469$	6.26763	0.289412
$470$	0.729958	0.0336705
$471$	16.0265	0.738460
$472$	−8.33256	−0.383537
$473$	3.77140	0.173409
$474$	−29.1924	−1.34085
$475$	−22.0101	−1.00989
$476$	−43.2545	−1.98257
$477$	−2.10447	−0.0963570
$478$	36.8988	1.68771
$479$	28.0251	1.28050	0.640250	−	0.768167i	$-0.278831\pi$
$479$	28.0251	1.28050	0.640250	+	0.768167i	$0.278831\pi$
$480$	20.6870	0.944227
$481$	23.8368	1.08686
$482$	−20.9449	−0.954014
$483$	−18.9026	−0.860100
$484$	−12.0566	−0.548025
$485$	−6.56626	−0.298159
$486$	1.82901	0.0829655
$487$	−11.3972	−0.516456	−0.258228	−	0.966084i	$-0.583139\pi$
$487$	−11.3972	−0.516456	−0.258228	+	0.966084i	$0.583139\pi$
$488$	−15.7320	−0.712152
$489$	18.1704	0.821693
$490$	72.4512	3.27301
$491$	−13.7857	−0.622141	−0.311070	−	0.950387i	$-0.600688\pi$
$491$	−13.7857	−0.622141	−0.311070	+	0.950387i	$0.600688\pi$
$492$	6.96555	0.314031
$493$	9.03608	0.406965
$494$	−21.8186	−0.981667
$495$	−4.52102	−0.203205
$496$	20.7946	0.933707
$497$	−9.03861	−0.405437
$498$	−2.88470	−0.129267
$499$	11.1010	0.496951	0.248476	−	0.968638i	$-0.420070\pi$
$499$	11.1010	0.496951	0.248476	+	0.968638i	$0.420070\pi$
$500$	−0.127909	−0.00572028
$501$	−17.4871	−0.781267
$502$	−33.1091	−1.47773
$503$	25.3030	1.12820	0.564102	−	0.825705i	$-0.309223\pi$
$503$	25.3030	1.12820	0.564102	+	0.825705i	$0.309223\pi$
$504$	5.28907	0.235594
$505$	−27.8650	−1.23997
$506$	−11.1743	−0.496758
$507$	5.56778	0.247274
$508$	−12.0322	−0.533841
$509$	24.7582	1.09739	0.548694	−	0.836023i	$-0.315125\pi$
$509$	24.7582	1.09739	0.548694	+	0.836023i	$0.315125\pi$
$510$	42.1681	1.86723
$511$	64.4980	2.85322
$512$	−20.1918	−0.892360
$513$	4.37575	0.193194
$514$	37.5532	1.65640
$515$	3.16702	0.139556
$516$	−3.55408	−0.156460
$517$	0.179894	0.00791173
$518$	70.6330	3.10344
$519$	−21.9480	−0.963412
$520$	10.3392	0.453403
$521$	−26.6011	−1.16542	−0.582708	−	0.812681i	$-0.698007\pi$
$521$	−26.6011	−1.16542	−0.582708	+	0.812681i	$0.698007\pi$
$522$	2.27028	0.0993674
$523$	−12.9219	−0.565035	−0.282518	−	0.959262i	$-0.591170\pi$
$523$	−12.9219	−0.565035	−0.282518	+	0.959262i	$0.591170\pi$
$524$	−18.1117	−0.791212
$525$	−22.2164	−0.969602
$526$	−34.2299	−1.49249
$527$	31.0155	1.35106
$528$	6.96747	0.303220
$529$	−4.68368	−0.203638
$530$	−12.1902	−0.529507
$531$	−6.95829	−0.301964
$532$	−25.9996	−1.12723
$533$	14.1158	0.611421
$534$	−31.4170	−1.35955
$535$	−34.9290	−1.51011
$536$	1.69932	0.0733995
$537$	9.01736	0.389128
$538$	16.0700	0.692828
$539$	17.8552	0.769077
$540$	4.26051	0.183343
$541$	−13.0408	−0.560667	−0.280333	−	0.959903i	$-0.590445\pi$
$541$	−13.0408	−0.560667	−0.280333	+	0.959903i	$0.590445\pi$
$542$	−17.0016	−0.730281
$543$	−16.9371	−0.726842
$544$	−47.5514	−2.03875
$545$	4.14707	0.177641
$546$	−22.0231	−0.942502
$547$	1.73781	0.0743032	0.0371516	−	0.999310i	$-0.488172\pi$
$547$	1.73781	0.0743032	0.0371516	+	0.999310i	$0.488172\pi$
$548$	−7.10058	−0.303322
$549$	−13.1373	−0.560687
$550$	−13.1332	−0.560002
$551$	5.43145	0.231388
$552$	−5.12501	−0.218135
$553$	−70.4948	−2.99775
$554$	−45.5663	−1.93593
$555$	−27.6911	−1.17542
$556$	10.9048	0.462468
$557$	−34.8211	−1.47542	−0.737709	−	0.675119i	$-0.764092\pi$
$557$	−34.8211	−1.47542	−0.737709	+	0.675119i	$0.764092\pi$
$558$	7.79251	0.329883
$559$	−7.20238	−0.304628
$560$	68.2722	2.88503
$561$	10.3921	0.438754
$562$	31.3931	1.32424
$563$	−2.87794	−0.121291	−0.0606453	−	0.998159i	$-0.519316\pi$
$563$	−2.87794	−0.121291	−0.0606453	+	0.998159i	$0.519316\pi$
$564$	−0.169528	−0.00713842
$565$	−60.4460	−2.54298
$566$	20.1600	0.847389
$567$	4.41676	0.185486
$568$	−2.45061	−0.102825
$569$	−6.90922	−0.289650	−0.144825	−	0.989457i	$-0.546262\pi$
$569$	−6.90922	−0.289650	−0.144825	+	0.989457i	$0.546262\pi$
$570$	25.3466	1.06165
$571$	−42.8554	−1.79344	−0.896722	−	0.442595i	$-0.854058\pi$
$571$	−42.8554	−1.79344	−0.896722	+	0.442595i	$0.854058\pi$
$572$	−5.23546	−0.218906
$573$	−10.1036	−0.422082
$574$	41.8277	1.74585
$575$	21.5273	0.897749
$576$	−2.18551	−0.0910631
$577$	25.6277	1.06689	0.533447	−	0.845834i	$-0.320896\pi$
$577$	25.6277	1.06689	0.533447	+	0.845834i	$0.320896\pi$
$578$	−65.8349	−2.73837
$579$	−17.2149	−0.715427
$580$	5.28841	0.219589
$581$	−6.96609	−0.289002
$582$	3.79213	0.157189
$583$	−3.00419	−0.124421
$584$	17.4871	0.723623
$585$	8.63396	0.356970
$586$	−26.1281	−1.07934
$587$	13.7452	0.567323	0.283662	−	0.958924i	$-0.408451\pi$
$587$	13.7452	0.567323	0.283662	+	0.958924i	$0.408451\pi$
$588$	−16.8263	−0.693906
$589$	18.6429	0.768169
$590$	−40.3060	−1.65937
$591$	18.4532	0.759061
$592$	42.6755	1.75395
$593$	−30.3854	−1.24778	−0.623889	−	0.781513i	$-0.714448\pi$
$593$	−30.3854	−1.24778	−0.623889	+	0.781513i	$0.714448\pi$
$594$	2.61097	0.107129
$595$	101.829	4.17458
$596$	15.0304	0.615671
$597$	1.19985	0.0491066
$598$	21.3400	0.872657
$599$	−23.0087	−0.940110	−0.470055	−	0.882637i	$-0.655766\pi$
$599$	−23.0087	−0.940110	−0.470055	+	0.882637i	$0.655766\pi$
$600$	−6.02345	−0.245906
$601$	23.0262	0.939257	0.469628	−	0.882864i	$-0.344388\pi$
$601$	23.0262	0.939257	0.469628	+	0.882864i	$0.344388\pi$
$602$	−21.3420	−0.869837
$603$	1.41906	0.0577884
$604$	21.9058	0.891333
$605$	28.3833	1.15395
$606$	16.0925	0.653713
$607$	−20.9378	−0.849838	−0.424919	−	0.905231i	$-0.639697\pi$
$607$	−20.9378	−0.849838	−0.424919	+	0.905231i	$0.639697\pi$
$608$	−28.5824	−1.15917
$609$	5.48235	0.222156
$610$	−76.0981	−3.08112
$611$	−0.343550	−0.0138986
$612$	−9.79326	−0.395869
$613$	34.1229	1.37821	0.689105	−	0.724662i	$-0.258004\pi$
$613$	34.1229	1.37821	0.689105	+	0.724662i	$0.258004\pi$
$614$	9.56963	0.386199
$615$	−16.3982	−0.661239
$616$	7.55031	0.304211
$617$	12.9328	0.520653	0.260326	−	0.965521i	$-0.416170\pi$
$617$	12.9328	0.520653	0.260326	+	0.965521i	$0.416170\pi$
$618$	−1.82901	−0.0735735
$619$	15.7807	0.634281	0.317141	−	0.948378i	$-0.397277\pi$
$619$	15.7807	0.634281	0.317141	+	0.948378i	$0.397277\pi$
$620$	18.1519	0.728999
$621$	−4.27976	−0.171741
$622$	14.6637	0.587959
$623$	−75.8669	−3.03954
$624$	−13.3060	−0.532668
$625$	−24.8490	−0.993960
$626$	−43.5468	−1.74048
$627$	6.24652	0.249462
$628$	−21.5600	−0.860336
$629$	63.6510	2.53793
$630$	25.5841	1.01930
$631$	35.4467	1.41111	0.705555	−	0.708655i	$-0.250698\pi$
$631$	35.4467	1.41111	0.705555	+	0.708655i	$0.250698\pi$
$632$	−19.1130	−0.760276
$633$	17.7695	0.706275
$634$	55.2806	2.19547
$635$	28.3259	1.12408
$636$	2.83109	0.112260
$637$	−34.0987	−1.35104
$638$	3.24089	0.128308
$639$	−2.04644	−0.0809558
$640$	28.7144	1.13503
$641$	−16.8473	−0.665429	−0.332714	−	0.943028i	$-0.607964\pi$
$641$	−16.8473	−0.665429	−0.332714	+	0.943028i	$0.607964\pi$
$642$	20.1721	0.796128
$643$	26.3428	1.03886	0.519429	−	0.854514i	$-0.326145\pi$
$643$	26.3428	1.03886	0.519429	+	0.854514i	$0.326145\pi$
$644$	25.4292	1.00205
$645$	8.36696	0.329449
$646$	−58.2620	−2.29229
$647$	−18.7282	−0.736282	−0.368141	−	0.929770i	$-0.620006\pi$
$647$	−18.7282	−0.736282	−0.368141	+	0.929770i	$0.620006\pi$
$648$	1.19750	0.0470423
$649$	−9.93317	−0.389911
$650$	25.0810	0.983758
$651$	18.8176	0.737521
$652$	−24.4441	−0.957306
$653$	9.59577	0.375511	0.187756	−	0.982216i	$-0.439879\pi$
$653$	9.59577	0.375511	0.187756	+	0.982216i	$0.439879\pi$
$654$	−2.39500	−0.0936520
$655$	42.6382	1.66601
$656$	25.2717	0.986695
$657$	14.6030	0.569718
$658$	−1.01801	−0.0396860
$659$	−43.8677	−1.70884	−0.854421	−	0.519581i	$-0.826088\pi$
$659$	−43.8677	−1.70884	−0.854421	+	0.519581i	$0.826088\pi$
$660$	6.08201	0.236742
$661$	−11.6033	−0.451317	−0.225659	−	0.974206i	$-0.572453\pi$
$661$	−11.6033	−0.451317	−0.225659	+	0.974206i	$0.572453\pi$
$662$	−64.1248	−2.49228
$663$	−19.8461	−0.770760
$664$	−1.88869	−0.0732955
$665$	61.2079	2.37354
$666$	15.9921	0.619680
$667$	−5.31230	−0.205693
$668$	23.5250	0.910209
$669$	−9.87877	−0.381935
$670$	8.21990	0.317562
$671$	−18.7539	−0.723988
$672$	−28.8503	−1.11292
$673$	−34.1847	−1.31772	−0.658862	−	0.752264i	$-0.728962\pi$
$673$	−34.1847	−1.31772	−0.658862	+	0.752264i	$0.728962\pi$
$674$	12.4802	0.480721
$675$	−5.03002	−0.193606
$676$	−7.49019	−0.288084
$677$	−18.3913	−0.706837	−0.353418	−	0.935465i	$-0.614981\pi$
$677$	−18.3913	−0.706837	−0.353418	+	0.935465i	$0.614981\pi$
$678$	34.9086	1.34066
$679$	9.15737	0.351428
$680$	27.6085	1.05874
$681$	9.43163	0.361421
$682$	11.1240	0.425962
$683$	9.77512	0.374035	0.187017	−	0.982357i	$-0.440118\pi$
$683$	9.77512	0.374035	0.187017	+	0.982357i	$0.440118\pi$
$684$	−5.88658	−0.225079
$685$	16.7161	0.638688
$686$	−44.4931	−1.69875
$687$	−18.8803	−0.720327
$688$	−12.8946	−0.491601
$689$	5.73723	0.218571
$690$	−24.7905	−0.943760
$691$	11.3030	0.429987	0.214994	−	0.976615i	$-0.431027\pi$
$691$	11.3030	0.429987	0.214994	+	0.976615i	$0.431027\pi$
$692$	29.5261	1.12241
$693$	6.30505	0.239509
$694$	−10.9658	−0.416256
$695$	−25.6720	−0.973794
$696$	1.48641	0.0563423
$697$	37.6931	1.42773
$698$	8.13012	0.307729
$699$	19.5676	0.740113
$700$	29.8871	1.12963
$701$	11.9209	0.450246	0.225123	−	0.974330i	$-0.427722\pi$
$701$	11.9209	0.450246	0.225123	+	0.974330i	$0.427722\pi$
$702$	−4.98626	−0.188194
$703$	38.2597	1.44299
$704$	−3.11989	−0.117585
$705$	0.399100	0.0150310
$706$	47.0412	1.77042
$707$	38.8607	1.46151
$708$	9.36080	0.351800
$709$	48.3070	1.81421	0.907103	−	0.420909i	$-0.138289\pi$
$709$	48.3070	1.81421	0.907103	+	0.420909i	$0.138289\pi$
$710$	−11.8540	−0.444873
$711$	−15.9608	−0.598576
$712$	−20.5695	−0.770876
$713$	−18.2339	−0.682867
$714$	−58.8080	−2.20083
$715$	12.3252	0.460938
$716$	−12.1308	−0.453350
$717$	20.1742	0.753420
$718$	5.10115	0.190373
$719$	−23.3762	−0.871786	−0.435893	−	0.899998i	$-0.643567\pi$
$719$	−23.3762	−0.871786	−0.435893	+	0.899998i	$0.643567\pi$
$720$	15.4576	0.576069
$721$	−4.41676	−0.164489
$722$	−0.269258	−0.0100207
$723$	−11.4515	−0.425886
$724$	22.7851	0.846800
$725$	−6.24357	−0.231880
$726$	−16.3919	−0.608359
$727$	−41.7711	−1.54920	−0.774602	−	0.632448i	$-0.782050\pi$
$727$	−41.7711	−1.54920	−0.774602	+	0.632448i	$0.782050\pi$
$728$	−14.4191	−0.534408
$729$	1.00000	0.0370370
$730$	84.5882	3.13075
$731$	−19.2324	−0.711336
$732$	17.6733	0.653224
$733$	39.7661	1.46879	0.734397	−	0.678721i	$-0.237465\pi$
$733$	39.7661	1.46879	0.734397	+	0.678721i	$0.237465\pi$
$734$	−20.1009	−0.741938
$735$	39.6123	1.46112
$736$	27.9554	1.03045
$737$	2.02575	0.0746193
$738$	9.47023	0.348604
$739$	15.8712	0.583833	0.291917	−	0.956444i	$-0.405707\pi$
$739$	15.8712	0.583833	0.291917	+	0.956444i	$0.405707\pi$
$740$	37.2520	1.36941
$741$	−11.9292	−0.438231
$742$	17.0005	0.624108
$743$	−25.2064	−0.924732	−0.462366	−	0.886689i	$-0.652999\pi$
$743$	−25.2064	−0.924732	−0.462366	+	0.886689i	$0.652999\pi$
$744$	5.10196	0.187047
$745$	−35.3844	−1.29639
$746$	31.3067	1.14622
$747$	−1.57719	−0.0577065
$748$	−13.9802	−0.511166
$749$	48.7122	1.77991
$750$	−0.173903	−0.00635004
$751$	−43.6544	−1.59297	−0.796485	−	0.604658i	$-0.793310\pi$
$751$	−43.6544	−1.59297	−0.796485	+	0.604658i	$0.793310\pi$
$752$	−0.615065	−0.0224291
$753$	−18.1022	−0.659681
$754$	−6.18925	−0.225399
$755$	−51.5702	−1.87683
$756$	−5.94175	−0.216099
$757$	−20.2987	−0.737769	−0.368885	−	0.929475i	$-0.620260\pi$
$757$	−20.2987	−0.737769	−0.368885	+	0.929475i	$0.620260\pi$
$758$	25.0280	0.909060
$759$	−6.10948	−0.221760
$760$	16.5951	0.601967
$761$	−1.78873	−0.0648413	−0.0324206	−	0.999474i	$-0.510322\pi$
$761$	−1.78873	−0.0648413	−0.0324206	+	0.999474i	$0.510322\pi$
$762$	−16.3587	−0.592614
$763$	−5.78354	−0.209378
$764$	13.5920	0.491743
$765$	23.0551	0.833561
$766$	−60.1087	−2.17181
$767$	18.9698	0.684958
$768$	−20.9541	−0.756114
$769$	−9.09667	−0.328034	−0.164017	−	0.986457i	$-0.552445\pi$
$769$	−9.09667	−0.328034	−0.164017	+	0.986457i	$0.552445\pi$
$770$	36.5221	1.31616
$771$	20.5320	0.739441
$772$	23.1588	0.833502
$773$	−24.9479	−0.897315	−0.448657	−	0.893704i	$-0.648098\pi$
$773$	−24.9479	−0.897315	−0.448657	+	0.893704i	$0.648098\pi$
$774$	−4.83206	−0.173685
$775$	−21.4305	−0.769805
$776$	2.48281	0.0891276
$777$	38.6182	1.38542
$778$	7.68291	0.275446
$779$	22.6568	0.811762
$780$	−11.6150	−0.415885
$781$	−2.92135	−0.104534
$782$	56.9839	2.03774
$783$	1.24126	0.0443591
$784$	−61.0476	−2.18027
$785$	50.7561	1.81156
$786$	−24.6243	−0.878320
$787$	−35.0202	−1.24833	−0.624167	−	0.781291i	$-0.714562\pi$
$787$	−35.0202	−1.24833	−0.624167	+	0.781291i	$0.714562\pi$
$788$	−24.8245	−0.884338
$789$	−18.7150	−0.666271
$790$	−92.4529	−3.28933
$791$	84.2986	2.99731
$792$	1.70947	0.0607433
$793$	35.8151	1.27183
$794$	−4.63218	−0.164390
$795$	−6.66490	−0.236380
$796$	−1.61413	−0.0572112
$797$	9.51403	0.337004	0.168502	−	0.985701i	$-0.446107\pi$
$797$	9.51403	0.337004	0.168502	+	0.985701i	$0.446107\pi$
$798$	−35.3486	−1.25133
$799$	−0.917378	−0.0324545
$800$	32.8561	1.16164
$801$	−17.1771	−0.606922
$802$	−7.63183	−0.269489
$803$	20.8463	0.735649
$804$	−1.90902	−0.0673259
$805$	−59.8651	−2.10997
$806$	−21.2440	−0.748288
$807$	8.78619	0.309289
$808$	10.5362	0.370662
$809$	−7.50582	−0.263891	−0.131945	−	0.991257i	$-0.542122\pi$
$809$	−7.50582	−0.263891	−0.131945	+	0.991257i	$0.542122\pi$
$810$	5.79251	0.203528
$811$	−39.0250	−1.37035	−0.685177	−	0.728377i	$-0.740275\pi$
$811$	−39.0250	−1.37035	−0.685177	+	0.728377i	$0.740275\pi$
$812$	−7.37526	−0.258821
$813$	−9.29552	−0.326008
$814$	22.8292	0.800161
$815$	57.5460	2.01575
$816$	−35.5309	−1.24383
$817$	−11.5603	−0.404444
$818$	37.5833	1.31407
$819$	−12.0410	−0.420747
$820$	22.0600	0.770370
$821$	−12.7254	−0.444119	−0.222060	−	0.975033i	$-0.571278\pi$
$821$	−12.7254	−0.444119	−0.222060	+	0.975033i	$0.571278\pi$
$822$	−9.65381	−0.336715
$823$	−32.6068	−1.13660	−0.568301	−	0.822821i	$-0.692399\pi$
$823$	−32.6068	−1.13660	−0.568301	+	0.822821i	$0.692399\pi$
$824$	−1.19750	−0.0417169
$825$	−7.18051	−0.249993
$826$	56.2111	1.95583
$827$	3.66186	0.127335	0.0636677	−	0.997971i	$-0.479720\pi$
$827$	3.66186	0.127335	0.0636677	+	0.997971i	$0.479720\pi$
$828$	5.75744	0.200085
$829$	37.3087	1.29579	0.647893	−	0.761731i	$-0.275650\pi$
$829$	37.3087	1.29579	0.647893	+	0.761731i	$0.275650\pi$
$830$	−9.13592	−0.317112
$831$	−24.9131	−0.864226
$832$	5.95817	0.206562
$833$	−91.0533	−3.15481
$834$	14.8260	0.513383
$835$	−55.3821	−1.91658
$836$	−8.40328	−0.290633
$837$	4.26051	0.147265
$838$	13.9037	0.480296
$839$	22.5620	0.778928	0.389464	−	0.921042i	$-0.372660\pi$
$839$	22.5620	0.778928	0.389464	+	0.921042i	$0.372660\pi$
$840$	16.7506	0.577950
$841$	−27.4593	−0.946871
$842$	−14.4800	−0.499014
$843$	17.1640	0.591159
$844$	−23.9049	−0.822839
$845$	17.6333	0.606603
$846$	−0.230487	−0.00792432
$847$	−39.5837	−1.36011
$848$	10.2715	0.352724
$849$	11.0224	0.378287
$850$	66.9734	2.29717
$851$	−37.4203	−1.28275
$852$	2.75302	0.0943168
$853$	29.4919	1.00978	0.504892	−	0.863183i	$-0.331532\pi$
$853$	29.4919	1.00978	0.504892	+	0.863183i	$0.331532\pi$
$854$	106.127	3.63159
$855$	13.8581	0.473937
$856$	13.2072	0.451412
$857$	33.1303	1.13171	0.565855	−	0.824505i	$-0.308546\pi$
$857$	33.1303	1.13171	0.565855	+	0.824505i	$0.308546\pi$
$858$	−7.11804	−0.243006
$859$	8.62051	0.294128	0.147064	−	0.989127i	$-0.453018\pi$
$859$	8.62051	0.294128	0.147064	+	0.989127i	$0.453018\pi$
$860$	−11.2559	−0.383821
$861$	22.8691	0.779375
$862$	15.2979	0.521047
$863$	8.97107	0.305379	0.152689	−	0.988274i	$-0.451207\pi$
$863$	8.97107	0.305379	0.152689	+	0.988274i	$0.451207\pi$
$864$	−6.53200	−0.222223
$865$	−69.5099	−2.36341
$866$	42.5232	1.44500
$867$	−35.9949	−1.22245
$868$	−25.3149	−0.859242
$869$	−22.7845	−0.772911
$870$	7.19002	0.243764
$871$	−3.86864	−0.131084
$872$	−1.56807	−0.0531016
$873$	2.07333	0.0701714
$874$	34.2521	1.15860
$875$	−0.419947	−0.0141968
$876$	−19.6451	−0.663745
$877$	−10.8353	−0.365884	−0.182942	−	0.983124i	$-0.558562\pi$
$877$	−10.8353	−0.365884	−0.182942	+	0.983124i	$0.558562\pi$
$878$	−2.34700	−0.0792073
$879$	−14.2854	−0.481835
$880$	22.0661	0.743849
$881$	27.2688	0.918710	0.459355	−	0.888253i	$-0.348081\pi$
$881$	27.2688	0.918710	0.459355	+	0.888253i	$0.348081\pi$
$882$	−22.8768	−0.770301
$883$	−3.80621	−0.128089	−0.0640446	−	0.997947i	$-0.520400\pi$
$883$	−3.80621	−0.128089	−0.0640446	+	0.997947i	$0.520400\pi$
$884$	26.6985	0.897967
$885$	−22.0370	−0.740767
$886$	22.3991	0.752512
$887$	−27.9539	−0.938601	−0.469301	−	0.883038i	$-0.655494\pi$
$887$	−27.9539	−0.938601	−0.469301	+	0.883038i	$0.655494\pi$
$888$	10.4704	0.351364
$889$	−39.5036	−1.32491
$890$	−99.4983	−3.33519
$891$	1.42753	0.0478241
$892$	13.2896	0.444970
$893$	−0.551422	−0.0184526
$894$	20.4351	0.683452
$895$	28.5582	0.954594
$896$	−40.0453	−1.33782
$897$	11.6675	0.389567
$898$	−0.0139311	−0.000464888 0
$899$	5.28841	0.176378
$900$	6.76676	0.225559
$901$	15.3200	0.510384
$902$	13.5190	0.450135
$903$	−11.6686	−0.388308
$904$	22.8556	0.760166
$905$	−53.6402	−1.78306
$906$	29.7827	0.989463
$907$	47.6567	1.58241	0.791207	−	0.611548i	$-0.209453\pi$
$907$	47.6567	1.58241	0.791207	+	0.611548i	$0.209453\pi$
$908$	−12.6881	−0.421070
$909$	8.79848	0.291827
$910$	−69.7476	−2.31211
$911$	15.7717	0.522539	0.261270	−	0.965266i	$-0.415859\pi$
$911$	15.7717	0.522539	0.261270	+	0.965266i	$0.415859\pi$
$912$	−21.3571	−0.707205
$913$	−2.25149	−0.0745136
$914$	−25.7048	−0.850238
$915$	−41.6062	−1.37546
$916$	25.3991	0.839211
$917$	−59.4636	−1.96366
$918$	−13.3147	−0.439452
$919$	−8.12466	−0.268008	−0.134004	−	0.990981i	$-0.542783\pi$
$919$	−8.12466	−0.268008	−0.134004	+	0.990981i	$0.542783\pi$
$920$	−16.2310	−0.535121
$921$	5.23214	0.172405
$922$	−60.5401	−1.99378
$923$	5.57901	0.183635
$924$	−8.48202	−0.279038
$925$	−43.9803	−1.44606
$926$	10.4797	0.344385
$927$	−1.00000	−0.0328443
$928$	−8.10792	−0.266155
$929$	17.0522	0.559463	0.279732	−	0.960078i	$-0.409754\pi$
$929$	17.0522	0.559463	0.279732	+	0.960078i	$0.409754\pi$
$930$	24.6790	0.809258
$931$	−54.7308	−1.79373
$932$	−26.3237	−0.862263
$933$	8.01727	0.262474
$934$	−16.7913	−0.549428
$935$	32.9119	1.07633
$936$	−3.26464	−0.106708
$937$	8.60190	0.281012	0.140506	−	0.990080i	$-0.455127\pi$
$937$	8.60190	0.281012	0.140506	+	0.990080i	$0.455127\pi$
$938$	−11.4635	−0.374298
$939$	−23.8090	−0.776976
$940$	−0.536899	−0.0175117
$941$	2.27901	0.0742937	0.0371468	−	0.999310i	$-0.488173\pi$
$941$	2.27901	0.0742937	0.0371468	+	0.999310i	$0.488173\pi$
$942$	−29.3125	−0.955054
$943$	−22.1597	−0.721619
$944$	33.9619	1.10537
$945$	13.9880	0.455028
$946$	−6.89792	−0.224271
$947$	−0.702644	−0.0228329	−0.0114164	−	0.999935i	$-0.503634\pi$
$947$	−0.702644	−0.0228329	−0.0114164	+	0.999935i	$0.503634\pi$
$948$	21.4716	0.697365
$949$	−39.8109	−1.29232
$950$	40.2567	1.30610
$951$	30.2243	0.980091
$952$	−38.5031	−1.24789
$953$	9.73389	0.315312	0.157656	−	0.987494i	$-0.449606\pi$
$953$	9.73389	0.315312	0.157656	+	0.987494i	$0.449606\pi$
$954$	3.84909	0.124619
$955$	−31.9982	−1.03544
$956$	−27.1399	−0.877766
$957$	1.77194	0.0572786
$958$	−51.2582	−1.65608
$959$	−23.3124	−0.752796
$960$	−6.92157	−0.223393
$961$	−12.8481	−0.414454
$962$	−43.5977	−1.40565
$963$	11.0290	0.355403
$964$	15.4054	0.496175
$965$	−54.5200	−1.75506
$966$	34.5731	1.11237
$967$	19.7175	0.634073	0.317036	−	0.948413i	$-0.397312\pi$
$967$	19.7175	0.634073	0.317036	+	0.948413i	$0.397312\pi$
$968$	−10.7322	−0.344946
$969$	−31.8544	−1.02331
$970$	12.0098	0.385610
$971$	−15.0971	−0.484490	−0.242245	−	0.970215i	$-0.577884\pi$
$971$	−15.0971	−0.484490	−0.242245	+	0.970215i	$0.577884\pi$
$972$	−1.34527	−0.0431497
$973$	35.8024	1.14777
$974$	20.8456	0.667935
$975$	13.7129	0.439164
$976$	64.1205	2.05245
$977$	27.2414	0.871529	0.435765	−	0.900061i	$-0.356478\pi$
$977$	27.2414	0.871529	0.435765	+	0.900061i	$0.356478\pi$
$978$	−33.2338	−1.06270
$979$	−24.5208	−0.783688
$980$	−53.2893	−1.70227
$981$	−1.30945	−0.0418076
$982$	25.2142	0.804618
$983$	39.4365	1.25783	0.628914	−	0.777475i	$-0.283500\pi$
$983$	39.4365	1.25783	0.628914	+	0.777475i	$0.283500\pi$
$984$	6.20041	0.197662
$985$	58.4415	1.86210
$986$	−16.5271	−0.526329
$987$	−0.556589	−0.0177164
$988$	16.0481	0.510557
$989$	11.3067	0.359532
$990$	8.26898	0.262806
$991$	−33.3721	−1.06010	−0.530050	−	0.847966i	$-0.677827\pi$
$991$	−33.3721	−1.06010	−0.530050	+	0.847966i	$0.677827\pi$
$992$	−27.8296	−0.883592
$993$	−35.0598	−1.11259
$994$	16.5317	0.524354
$995$	3.79995	0.120466
$996$	2.12176	0.0672305
$997$	−20.7588	−0.657439	−0.328720	−	0.944428i	$-0.606617\pi$
$997$	−20.7588	−0.657439	−0.328720	+	0.944428i	$0.606617\pi$
$998$	−20.3039	−0.642709
$999$	8.74357	0.276634

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	309.2.a.c.1.2	✓	5
3.2	odd	2		927.2.a.e.1.4		5
4.3	odd	2		4944.2.a.bb.1.2		5
5.4	even	2		7725.2.a.t.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
309.2.a.c.1.2	✓	5	1.1	even	1	trivial
927.2.a.e.1.4		5	3.2	odd	2
4944.2.a.bb.1.2		5	4.3	odd	2
7725.2.a.t.1.4		5	5.4	even	2

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:	\( N \)	\(=\)	\( 309 = 3 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	309.a (trivial)

\(f(q)\)	\(=\)	\(q-1.82901 q^{2} -1.00000 q^{3} +1.34527 q^{4} -3.16702 q^{5} +1.82901 q^{6} +4.41676 q^{7} +1.19750 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.82901 q^{2} -1.00000 q^{3} +1.34527 q^{4} -3.16702 q^{5} +1.82901 q^{6} +4.41676 q^{7} +1.19750 q^{8} +1.00000 q^{9} +5.79251 q^{10} +1.42753 q^{11} -1.34527 q^{12} -2.72621 q^{13} -8.07829 q^{14} +3.16702 q^{15} -4.88079 q^{16} -7.27976 q^{17} -1.82901 q^{18} -4.37575 q^{19} -4.26051 q^{20} -4.41676 q^{21} -2.61097 q^{22} +4.27976 q^{23} -1.19750 q^{24} +5.03002 q^{25} +4.98626 q^{26} -1.00000 q^{27} +5.94175 q^{28} -1.24126 q^{29} -5.79251 q^{30} -4.26051 q^{31} +6.53200 q^{32} -1.42753 q^{33} +13.3147 q^{34} -13.9880 q^{35} +1.34527 q^{36} -8.74357 q^{37} +8.00329 q^{38} +2.72621 q^{39} -3.79251 q^{40} -5.17779 q^{41} +8.07829 q^{42} +2.64190 q^{43} +1.92042 q^{44} -3.16702 q^{45} -7.82771 q^{46} +0.126018 q^{47} +4.88079 q^{48} +12.5077 q^{49} -9.19996 q^{50} +7.27976 q^{51} -3.66750 q^{52} -2.10447 q^{53} +1.82901 q^{54} -4.52102 q^{55} +5.28907 q^{56} +4.37575 q^{57} +2.27028 q^{58} -6.95829 q^{59} +4.26051 q^{60} -13.1373 q^{61} +7.79251 q^{62} +4.41676 q^{63} -2.18551 q^{64} +8.63396 q^{65} +2.61097 q^{66} +1.41906 q^{67} -9.79326 q^{68} -4.27976 q^{69} +25.5841 q^{70} -2.04644 q^{71} +1.19750 q^{72} +14.6030 q^{73} +15.9921 q^{74} -5.03002 q^{75} -5.88658 q^{76} +6.30505 q^{77} -4.98626 q^{78} -15.9608 q^{79} +15.4576 q^{80} +1.00000 q^{81} +9.47023 q^{82} -1.57719 q^{83} -5.94175 q^{84} +23.0551 q^{85} -4.83206 q^{86} +1.24126 q^{87} +1.70947 q^{88} -17.1771 q^{89} +5.79251 q^{90} -12.0410 q^{91} +5.75744 q^{92} +4.26051 q^{93} -0.230487 q^{94} +13.8581 q^{95} -6.53200 q^{96} +2.07333 q^{97} -22.8768 q^{98} +1.42753 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9	\( 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9} - q^{10} - 12 q^{11} - 2 q^{12} + q^{13} - 8 q^{14} + 5 q^{15} - 4 q^{16} - 10 q^{17} - 2 q^{18} - 16 q^{19} - 13 q^{20} + 2 q^{21} + 4 q^{22} - 5 q^{23} + 6 q^{24} + 12 q^{25} - 10 q^{26} - 5 q^{27} + 7 q^{28} - 16 q^{29} + q^{30} - 13 q^{31} + 11 q^{32} + 12 q^{33} - 2 q^{34} - 22 q^{35} + 2 q^{36} + 4 q^{37} + 21 q^{38} - q^{39} + 11 q^{40} - 20 q^{41} + 8 q^{42} + 7 q^{43} + 2 q^{44} - 5 q^{45} + 8 q^{46} + 8 q^{47} + 4 q^{48} + 23 q^{49} + 13 q^{50} + 10 q^{51} + 31 q^{52} - 8 q^{53} + 2 q^{54} - 6 q^{55} + 5 q^{56} + 16 q^{57} + 20 q^{58} - 19 q^{59} + 13 q^{60} - 19 q^{61} + 9 q^{62} - 2 q^{63} - 16 q^{64} + q^{65} - 4 q^{66} + 11 q^{67} + 15 q^{68} + 5 q^{69} + 44 q^{70} - 10 q^{71} - 6 q^{72} + 20 q^{73} + 44 q^{74} - 12 q^{75} - 8 q^{76} + 8 q^{77} + 10 q^{78} - 14 q^{79} + 45 q^{80} + 5 q^{81} - 3 q^{82} - 11 q^{83} - 7 q^{84} + 12 q^{85} - 11 q^{86} + 16 q^{87} + 30 q^{88} + 22 q^{89} - q^{90} - 42 q^{91} - 21 q^{92} + 13 q^{93} - 6 q^{94} - 10 q^{95} - 11 q^{96} + 7 q^{97} - 11 q^{98} - 12 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9 - q^10 - 12 * q^11 - 2 * q^12 + q^13 - 8 * q^14 + 5 * q^15 - 4 * q^16 - 10 * q^17 - 2 * q^18 - 16 * q^19 - 13 * q^20 + 2 * q^21 + 4 * q^22 - 5 * q^23 + 6 * q^24 + 12 * q^25 - 10 * q^26 - 5 * q^27 + 7 * q^28 - 16 * q^29 + q^30 - 13 * q^31 + 11 * q^32 + 12 * q^33 - 2 * q^34 - 22 * q^35 + 2 * q^36 + 4 * q^37 + 21 * q^38 - q^39 + 11 * q^40 - 20 * q^41 + 8 * q^42 + 7 * q^43 + 2 * q^44 - 5 * q^45 + 8 * q^46 + 8 * q^47 + 4 * q^48 + 23 * q^49 + 13 * q^50 + 10 * q^51 + 31 * q^52 - 8 * q^53 + 2 * q^54 - 6 * q^55 + 5 * q^56 + 16 * q^57 + 20 * q^58 - 19 * q^59 + 13 * q^60 - 19 * q^61 + 9 * q^62 - 2 * q^63 - 16 * q^64 + q^65 - 4 * q^66 + 11 * q^67 + 15 * q^68 + 5 * q^69 + 44 * q^70 - 10 * q^71 - 6 * q^72 + 20 * q^73 + 44 * q^74 - 12 * q^75 - 8 * q^76 + 8 * q^77 + 10 * q^78 - 14 * q^79 + 45 * q^80 + 5 * q^81 - 3 * q^82 - 11 * q^83 - 7 * q^84 + 12 * q^85 - 11 * q^86 + 16 * q^87 + 30 * q^88 + 22 * q^89 - q^90 - 42 * q^91 - 21 * q^92 + 13 * q^93 - 6 * q^94 - 10 * q^95 - 11 * q^96 + 7 * q^97 - 11 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more