LMFDB - Embedded newform 381.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [381,2,Mod(1,381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(381, 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("381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 381 = 3 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	381.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:	$3.04230031701$
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			1.1
Root			$2.26835$ of defining polynomial
Character	$\chi$	$=$	381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26835 q^{2} -1.00000 q^{3} +3.14543 q^{4} -2.63584 q^{5} +2.26835 q^{6} +1.51754 q^{7} -2.59823 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.26835 q^{2} -1.00000 q^{3} +3.14543 q^{4} -2.63584 q^{5} +2.26835 q^{6} +1.51754 q^{7} -2.59823 q^{8} +1.00000 q^{9} +5.97902 q^{10} -0.0104875 q^{11} -3.14543 q^{12} +1.66550 q^{13} -3.44232 q^{14} +2.63584 q^{15} -0.397144 q^{16} +2.10782 q^{17} -2.26835 q^{18} -2.77682 q^{19} -8.29085 q^{20} -1.51754 q^{21} +0.0237894 q^{22} +2.07816 q^{23} +2.59823 q^{24} +1.94767 q^{25} -3.77794 q^{26} -1.00000 q^{27} +4.77331 q^{28} -7.41840 q^{29} -5.97902 q^{30} -4.87819 q^{31} +6.09733 q^{32} +0.0104875 q^{33} -4.78127 q^{34} -4.00000 q^{35} +3.14543 q^{36} -0.282175 q^{37} +6.29881 q^{38} -1.66550 q^{39} +6.84854 q^{40} -8.33967 q^{41} +3.44232 q^{42} -10.8276 q^{43} -0.0329877 q^{44} -2.63584 q^{45} -4.71401 q^{46} +0.613170 q^{47} +0.397144 q^{48} -4.69707 q^{49} -4.41801 q^{50} -2.10782 q^{51} +5.23870 q^{52} -4.98377 q^{53} +2.26835 q^{54} +0.0276435 q^{55} -3.94292 q^{56} +2.77682 q^{57} +16.8276 q^{58} +1.48438 q^{59} +8.29085 q^{60} -3.22612 q^{61} +11.0655 q^{62} +1.51754 q^{63} -13.0366 q^{64} -4.38999 q^{65} -0.0237894 q^{66} +1.18980 q^{67} +6.62998 q^{68} -2.07816 q^{69} +9.07341 q^{70} -8.34861 q^{71} -2.59823 q^{72} +14.7658 q^{73} +0.640073 q^{74} -1.94767 q^{75} -8.73429 q^{76} -0.0159152 q^{77} +3.77794 q^{78} +2.10307 q^{79} +1.04681 q^{80} +1.00000 q^{81} +18.9173 q^{82} -1.86476 q^{83} -4.77331 q^{84} -5.55587 q^{85} +24.5607 q^{86} +7.41840 q^{87} +0.0272490 q^{88} +15.7635 q^{89} +5.97902 q^{90} +2.52746 q^{91} +6.53671 q^{92} +4.87819 q^{93} -1.39089 q^{94} +7.31927 q^{95} -6.09733 q^{96} -11.3911 q^{97} +10.6546 q^{98} -0.0104875 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} - 5 q^{3} + q^{4} - 5 q^{5} + q^{6} - 3 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - q^2 - 5 * q^3 + q^4 - 5 * q^5 + q^6 - 3 * q^8 + 5 * q^9	\( 5 q - q^{2} - 5 q^{3} + q^{4} - 5 q^{5} + q^{6} - 3 q^{8} + 5 q^{9} - 2 q^{10} - 16 q^{11} - q^{12} + 3 q^{13} - 6 q^{14} + 5 q^{15} - 7 q^{16} - 6 q^{17} - q^{18} - 8 q^{19} - 12 q^{20} - 8 q^{22} - 9 q^{23} + 3 q^{24} + 4 q^{25} - 2 q^{26} - 5 q^{27} + 2 q^{28} - 17 q^{29} + 2 q^{30} - 9 q^{31} - 2 q^{32} + 16 q^{33} - q^{34} - 20 q^{35} + q^{36} - q^{37} + q^{38} - 3 q^{39} + 16 q^{40} - 2 q^{41} + 6 q^{42} - 4 q^{43} + 3 q^{44} - 5 q^{45} + 4 q^{46} - 8 q^{47} + 7 q^{48} + 13 q^{49} + 15 q^{50} + 6 q^{51} + 13 q^{52} - 15 q^{53} + q^{54} + 20 q^{55} + 8 q^{56} + 8 q^{57} + 34 q^{58} - 19 q^{59} + 12 q^{60} + q^{61} + 15 q^{62} + q^{64} - 5 q^{65} + 8 q^{66} - 2 q^{67} + 14 q^{68} + 9 q^{69} + 4 q^{70} - 3 q^{72} + 13 q^{73} - 17 q^{74} - 4 q^{75} + 8 q^{76} + 2 q^{77} + 2 q^{78} - 28 q^{79} + 12 q^{80} + 5 q^{81} + 30 q^{82} - q^{83} - 2 q^{84} + 6 q^{85} + 32 q^{86} + 17 q^{87} + 3 q^{88} + q^{89} - 2 q^{90} - 18 q^{91} + 12 q^{92} + 9 q^{93} + 16 q^{94} + 4 q^{95} + 2 q^{96} + 28 q^{97} + 15 q^{98} - 16 q^{99}+O(q^{100}) \) 5 * q - q^2 - 5 * q^3 + q^4 - 5 * q^5 + q^6 - 3 * q^8 + 5 * q^9 - 2 * q^10 - 16 * q^11 - q^12 + 3 * q^13 - 6 * q^14 + 5 * q^15 - 7 * q^16 - 6 * q^17 - q^18 - 8 * q^19 - 12 * q^20 - 8 * q^22 - 9 * q^23 + 3 * q^24 + 4 * q^25 - 2 * q^26 - 5 * q^27 + 2 * q^28 - 17 * q^29 + 2 * q^30 - 9 * q^31 - 2 * q^32 + 16 * q^33 - q^34 - 20 * q^35 + q^36 - q^37 + q^38 - 3 * q^39 + 16 * q^40 - 2 * q^41 + 6 * q^42 - 4 * q^43 + 3 * q^44 - 5 * q^45 + 4 * q^46 - 8 * q^47 + 7 * q^48 + 13 * q^49 + 15 * q^50 + 6 * q^51 + 13 * q^52 - 15 * q^53 + q^54 + 20 * q^55 + 8 * q^56 + 8 * q^57 + 34 * q^58 - 19 * q^59 + 12 * q^60 + q^61 + 15 * q^62 + q^64 - 5 * q^65 + 8 * q^66 - 2 * q^67 + 14 * q^68 + 9 * q^69 + 4 * q^70 - 3 * q^72 + 13 * q^73 - 17 * q^74 - 4 * q^75 + 8 * q^76 + 2 * q^77 + 2 * q^78 - 28 * q^79 + 12 * q^80 + 5 * q^81 + 30 * q^82 - q^83 - 2 * q^84 + 6 * q^85 + 32 * q^86 + 17 * q^87 + 3 * q^88 + q^89 - 2 * q^90 - 18 * q^91 + 12 * q^92 + 9 * q^93 + 16 * q^94 + 4 * q^95 + 2 * q^96 + 28 * q^97 + 15 * q^98 - 16 * 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 $

$2$	−2.26835	−1.60397	−0.801984	−	0.597346i	$-0.796222\pi$
$2$	−2.26835	−1.60397	−0.801984	+	0.597346i	$0.796222\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.14543	1.57271
$5$	−2.63584	−1.17879	−0.589393	−	0.807847i	$-0.700633\pi$
$5$	−2.63584	−1.17879	−0.589393	+	0.807847i	$0.700633\pi$
$6$	2.26835	0.926051
$7$	1.51754	0.573576	0.286788	−	0.957994i	$-0.407412\pi$
$7$	1.51754	0.573576	0.286788	+	0.957994i	$0.407412\pi$
$8$	−2.59823	−0.918614
$9$	1.00000	0.333333
$10$	5.97902	1.89073
$11$	−0.0104875	−0.00316211	−0.00158105	−	0.999999i	$-0.500503\pi$
$11$	−0.0104875	−0.00316211	−0.00158105	+	0.999999i	$0.500503\pi$
$12$	−3.14543	−0.908006
$13$	1.66550	0.461926	0.230963	−	0.972963i	$-0.425812\pi$
$13$	1.66550	0.461926	0.230963	+	0.972963i	$0.425812\pi$
$14$	−3.44232	−0.919998
$15$	2.63584	0.680572
$16$	−0.397144	−0.0992860
$17$	2.10782	0.511220	0.255610	−	0.966780i	$-0.417724\pi$
$17$	2.10782	0.511220	0.255610	+	0.966780i	$0.417724\pi$
$18$	−2.26835	−0.534656
$19$	−2.77682	−0.637046	−0.318523	−	0.947915i	$-0.603187\pi$
$19$	−2.77682	−0.637046	−0.318523	+	0.947915i	$0.603187\pi$
$20$	−8.29085	−1.85389
$21$	−1.51754	−0.331155
$22$	0.0237894	0.00507192
$23$	2.07816	0.433327	0.216663	−	0.976246i	$-0.430483\pi$
$23$	2.07816	0.433327	0.216663	+	0.976246i	$0.430483\pi$
$24$	2.59823	0.530362
$25$	1.94767	0.389534
$26$	−3.77794	−0.740914
$27$	−1.00000	−0.192450
$28$	4.77331	0.902071
$29$	−7.41840	−1.37756	−0.688781	−	0.724969i	$-0.741854\pi$
$29$	−7.41840	−1.37756	−0.688781	+	0.724969i	$0.741854\pi$
$30$	−5.97902	−1.09162
$31$	−4.87819	−0.876149	−0.438074	−	0.898939i	$-0.644339\pi$
$31$	−4.87819	−0.876149	−0.438074	+	0.898939i	$0.644339\pi$
$32$	6.09733	1.07787
$33$	0.0104875	0.00182564
$34$	−4.78127	−0.819981
$35$	−4.00000	−0.676123
$36$	3.14543	0.524238
$37$	−0.282175	−0.0463893	−0.0231946	−	0.999731i	$-0.507384\pi$
$37$	−0.282175	−0.0463893	−0.0231946	+	0.999731i	$0.507384\pi$
$38$	6.29881	1.02180
$39$	−1.66550	−0.266693
$40$	6.84854	1.08285
$41$	−8.33967	−1.30244	−0.651219	−	0.758890i	$-0.725742\pi$
$41$	−8.33967	−1.30244	−0.651219	+	0.758890i	$0.725742\pi$
$42$	3.44232	0.531161
$43$	−10.8276	−1.65119	−0.825594	−	0.564265i	$-0.809160\pi$
$43$	−10.8276	−1.65119	−0.825594	+	0.564265i	$0.809160\pi$
$44$	−0.0329877	−0.00497309
$45$	−2.63584	−0.392928
$46$	−4.71401	−0.695042
$47$	0.613170	0.0894400	0.0447200	−	0.999000i	$-0.485760\pi$
$47$	0.613170	0.0894400	0.0447200	+	0.999000i	$0.485760\pi$
$48$	0.397144	0.0573228
$49$	−4.69707	−0.671010
$50$	−4.41801	−0.624801
$51$	−2.10782	−0.295153
$52$	5.23870	0.726477
$53$	−4.98377	−0.684574	−0.342287	−	0.939595i	$-0.611202\pi$
$53$	−4.98377	−0.684574	−0.342287	+	0.939595i	$0.611202\pi$
$54$	2.26835	0.308684
$55$	0.0276435	0.00372745
$56$	−3.94292	−0.526895
$57$	2.77682	0.367799
$58$	16.8276	2.20957
$59$	1.48438	0.193250	0.0966248	−	0.995321i	$-0.469195\pi$
$59$	1.48438	0.193250	0.0966248	+	0.995321i	$0.469195\pi$
$60$	8.29085	1.07034
$61$	−3.22612	−0.413062	−0.206531	−	0.978440i	$-0.566217\pi$
$61$	−3.22612	−0.413062	−0.206531	+	0.978440i	$0.566217\pi$
$62$	11.0655	1.40531
$63$	1.51754	0.191192
$64$	−13.0366	−1.62958
$65$	−4.38999	−0.544511
$66$	−0.0237894	−0.00292827
$67$	1.18980	0.145357	0.0726784	−	0.997355i	$-0.476845\pi$
$67$	1.18980	0.145357	0.0726784	+	0.997355i	$0.476845\pi$
$68$	6.62998	0.804003
$69$	−2.07816	−0.250181
$70$	9.07341	1.08448
$71$	−8.34861	−0.990798	−0.495399	−	0.868666i	$-0.664978\pi$
$71$	−8.34861	−0.990798	−0.495399	+	0.868666i	$0.664978\pi$
$72$	−2.59823	−0.306205
$73$	14.7658	1.72820	0.864101	−	0.503318i	$-0.167888\pi$
$73$	14.7658	1.72820	0.864101	+	0.503318i	$0.167888\pi$
$74$	0.640073	0.0744069
$75$	−1.94767	−0.224898
$76$	−8.73429	−1.00189
$77$	−0.0159152	−0.00181371
$78$	3.77794	0.427767
$79$	2.10307	0.236614	0.118307	−	0.992977i	$-0.462253\pi$
$79$	2.10307	0.236614	0.118307	+	0.992977i	$0.462253\pi$
$80$	1.04681	0.117037
$81$	1.00000	0.111111
$82$	18.9173	2.08907
$83$	−1.86476	−0.204684	−0.102342	−	0.994749i	$-0.532634\pi$
$83$	−1.86476	−0.204684	−0.102342	+	0.994749i	$0.532634\pi$
$84$	−4.77331	−0.520811
$85$	−5.55587	−0.602619
$86$	24.5607	2.64845
$87$	7.41840	0.795336
$88$	0.0272490	0.00290476
$89$	15.7635	1.67093	0.835464	−	0.549545i	$-0.185199\pi$
$89$	15.7635	1.67093	0.835464	+	0.549545i	$0.185199\pi$
$90$	5.97902	0.630245
$91$	2.52746	0.264950
$92$	6.53671	0.681499
$93$	4.87819	0.505845
$94$	−1.39089	−0.143459
$95$	7.31927	0.750941
$96$	−6.09733	−0.622306
$97$	−11.3911	−1.15659	−0.578297	−	0.815827i	$-0.696282\pi$
$97$	−11.3911	−1.15659	−0.578297	+	0.815827i	$0.696282\pi$
$98$	10.6546	1.07628
$99$	−0.0104875	−0.00105404
$100$	6.12626	0.612626
$101$	7.16330	0.712775	0.356388	−	0.934338i	$-0.384008\pi$
$101$	7.16330	0.712775	0.356388	+	0.934338i	$0.384008\pi$
$102$	4.78127	0.473416
$103$	2.34669	0.231226	0.115613	−	0.993294i	$-0.463117\pi$
$103$	2.34669	0.231226	0.115613	+	0.993294i	$0.463117\pi$
$104$	−4.32735	−0.424332
$105$	4.00000	0.390360
$106$	11.3050	1.09803
$107$	−16.3301	−1.57869	−0.789346	−	0.613949i	$-0.789580\pi$
$107$	−16.3301	−1.57869	−0.789346	+	0.613949i	$0.789580\pi$
$108$	−3.14543	−0.302669
$109$	4.23155	0.405309	0.202654	−	0.979250i	$-0.435043\pi$
$109$	4.23155	0.405309	0.202654	+	0.979250i	$0.435043\pi$
$110$	−0.0627052	−0.00597870
$111$	0.282175	0.0267829
$112$	−0.602682	−0.0569481
$113$	−0.112564	−0.0105892	−0.00529459	−	0.999986i	$-0.501685\pi$
$113$	−0.112564	−0.0105892	−0.00529459	+	0.999986i	$0.501685\pi$
$114$	−6.29881	−0.589938
$115$	−5.47771	−0.510799
$116$	−23.3340	−2.16651
$117$	1.66550	0.153975
$118$	−3.36710	−0.309966
$119$	3.19870	0.293224
$120$	−6.84854	−0.625183
$121$	−10.9999	−0.999990
$122$	7.31798	0.662539
$123$	8.33967	0.751963
$124$	−15.3440	−1.37793
$125$	8.04546	0.719608
$126$	−3.44232	−0.306666
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	17.3770	1.53592
$129$	10.8276	0.953314
$130$	9.95805	0.873379
$131$	−6.66199	−0.582061	−0.291030	−	0.956714i	$-0.593998\pi$
$131$	−6.66199	−0.582061	−0.291030	+	0.956714i	$0.593998\pi$
$132$	0.0329877	0.00287121
$133$	−4.21394	−0.365395
$134$	−2.69888	−0.233148
$135$	2.63584	0.226857
$136$	−5.47660	−0.469614
$137$	9.11324	0.778597	0.389298	−	0.921112i	$-0.372717\pi$
$137$	9.11324	0.778597	0.389298	+	0.921112i	$0.372717\pi$
$138$	4.71401	0.401283
$139$	−15.9647	−1.35411	−0.677055	−	0.735933i	$-0.736744\pi$
$139$	−15.9647	−1.35411	−0.677055	+	0.735933i	$0.736744\pi$
$140$	−12.5817	−1.06335
$141$	−0.613170	−0.0516382
$142$	18.9376	1.58921
$143$	−0.0174669	−0.00146066
$144$	−0.397144	−0.0330953
$145$	19.5538	1.62385
$146$	−33.4940	−2.77198
$147$	4.69707	0.387408
$148$	−0.887561	−0.0729571
$149$	−12.2055	−0.999910	−0.499955	−	0.866051i	$-0.666650\pi$
$149$	−12.2055	−0.999910	−0.499955	+	0.866051i	$0.666650\pi$
$150$	4.41801	0.360729
$151$	−13.5603	−1.10352	−0.551761	−	0.834002i	$-0.686044\pi$
$151$	−13.5603	−1.10352	−0.551761	+	0.834002i	$0.686044\pi$
$152$	7.21483	0.585200
$153$	2.10782	0.170407
$154$	0.0361014	0.00290913
$155$	12.8581	1.03279
$156$	−5.23870	−0.419432
$157$	2.06979	0.165188	0.0825938	−	0.996583i	$-0.473680\pi$
$157$	2.06979	0.165188	0.0825938	+	0.996583i	$0.473680\pi$
$158$	−4.77050	−0.379520
$159$	4.98377	0.395239
$160$	−16.0716	−1.27057
$161$	3.15370	0.248546
$162$	−2.26835	−0.178219
$163$	−18.2821	−1.43196	−0.715981	−	0.698120i	$-0.754020\pi$
$163$	−18.2821	−1.43196	−0.715981	+	0.698120i	$0.754020\pi$
$164$	−26.2318	−2.04836
$165$	−0.0276435	−0.00215204
$166$	4.22994	0.328307
$167$	−4.51573	−0.349438	−0.174719	−	0.984618i	$-0.555902\pi$
$167$	−4.51573	−0.349438	−0.174719	+	0.984618i	$0.555902\pi$
$168$	3.94292	0.304203
$169$	−10.2261	−0.786625
$170$	12.6027	0.966582
$171$	−2.77682	−0.212349
$172$	−34.0573	−2.59684
$173$	−23.3701	−1.77680	−0.888400	−	0.459071i	$-0.848182\pi$
$173$	−23.3701	−1.77680	−0.888400	+	0.459071i	$0.848182\pi$
$174$	−16.8276	−1.27569
$175$	2.95567	0.223428
$176$	0.00416506	0.000313953 0
$177$	−1.48438	−0.111573
$178$	−35.7572	−2.68012
$179$	7.71794	0.576866	0.288433	−	0.957500i	$-0.406866\pi$
$179$	7.71794	0.576866	0.288433	+	0.957500i	$0.406866\pi$
$180$	−8.29085	−0.617964
$181$	17.5777	1.30654	0.653269	−	0.757126i	$-0.273397\pi$
$181$	17.5777	1.30654	0.653269	+	0.757126i	$0.273397\pi$
$182$	−5.73317	−0.424971
$183$	3.22612	0.238482
$184$	−5.39955	−0.398060
$185$	0.743769	0.0546830
$186$	−11.0655	−0.811359
$187$	−0.0221058	−0.00161653
$188$	1.92868	0.140663
$189$	−1.51754	−0.110385
$190$	−16.6027	−1.20449
$191$	−17.6551	−1.27748	−0.638740	−	0.769423i	$-0.720544\pi$
$191$	−17.6551	−1.27748	−0.638740	+	0.769423i	$0.720544\pi$
$192$	13.0366	0.940836
$193$	17.4512	1.25617	0.628083	−	0.778146i	$-0.283840\pi$
$193$	17.4512	1.25617	0.628083	+	0.778146i	$0.283840\pi$
$194$	25.8391	1.85514
$195$	4.38999	0.314374
$196$	−14.7743	−1.05531
$197$	23.0847	1.64472	0.822359	−	0.568969i	$-0.192657\pi$
$197$	23.0847	1.64472	0.822359	+	0.568969i	$0.192657\pi$
$198$	0.0237894	0.00169064
$199$	10.6745	0.756698	0.378349	−	0.925663i	$-0.376492\pi$
$199$	10.6745	0.756698	0.378349	+	0.925663i	$0.376492\pi$
$200$	−5.06051	−0.357832
$201$	−1.18980	−0.0839218
$202$	−16.2489	−1.14327
$203$	−11.2577	−0.790138
$204$	−6.62998	−0.464191
$205$	21.9821	1.53529
$206$	−5.32312	−0.370879
$207$	2.07816	0.144442
$208$	−0.661442	−0.0458628
$209$	0.0291220	0.00201441
$210$	−9.07341	−0.626125
$211$	−3.99830	−0.275254	−0.137627	−	0.990484i	$-0.543948\pi$
$211$	−3.99830	−0.275254	−0.137627	+	0.990484i	$0.543948\pi$
$212$	−15.6761	−1.07664
$213$	8.34861	0.572037
$214$	37.0425	2.53217
$215$	28.5398	1.94640
$216$	2.59823	0.176787
$217$	−7.40285	−0.502538
$218$	−9.59864	−0.650102
$219$	−14.7658	−0.997778
$220$	0.0869505	0.00586220
$221$	3.51056	0.236146
$222$	−0.640073	−0.0429589
$223$	−11.5449	−0.773106	−0.386553	−	0.922267i	$-0.626334\pi$
$223$	−11.5449	−0.773106	−0.386553	+	0.922267i	$0.626334\pi$
$224$	9.25294	0.618238
$225$	1.94767	0.129845
$226$	0.255336	0.0169847
$227$	−8.43126	−0.559603	−0.279801	−	0.960058i	$-0.590269\pi$
$227$	−8.43126	−0.559603	−0.279801	+	0.960058i	$0.590269\pi$
$228$	8.73429	0.578442
$229$	19.3760	1.28040	0.640201	−	0.768208i	$-0.278851\pi$
$229$	19.3760	1.28040	0.640201	+	0.768208i	$0.278851\pi$
$230$	12.4254	0.819305
$231$	0.0159152	0.00104715
$232$	19.2747	1.26545
$233$	14.3938	0.942971	0.471486	−	0.881874i	$-0.343718\pi$
$233$	14.3938	0.942971	0.471486	+	0.881874i	$0.343718\pi$
$234$	−3.77794	−0.246971
$235$	−1.61622	−0.105431
$236$	4.66901	0.303926
$237$	−2.10307	−0.136609
$238$	−7.25577	−0.470322
$239$	13.1474	0.850434	0.425217	−	0.905091i	$-0.360198\pi$
$239$	13.1474	0.850434	0.425217	+	0.905091i	$0.360198\pi$
$240$	−1.04681	−0.0675713
$241$	5.62468	0.362317	0.181159	−	0.983454i	$-0.442015\pi$
$241$	5.62468	0.362317	0.181159	+	0.983454i	$0.442015\pi$
$242$	24.9516	1.60395
$243$	−1.00000	−0.0641500
$244$	−10.1475	−0.649628
$245$	12.3807	0.790977
$246$	−18.9173	−1.20612
$247$	−4.62479	−0.294268
$248$	12.6747	0.804842
$249$	1.86476	0.118174
$250$	−18.2499	−1.15423
$251$	−0.560879	−0.0354024	−0.0177012	−	0.999843i	$-0.505635\pi$
$251$	−0.560879	−0.0354024	−0.0177012	+	0.999843i	$0.505635\pi$
$252$	4.77331	0.300690
$253$	−0.0217948	−0.00137023
$254$	2.26835	0.142329
$255$	5.55587	0.347922
$256$	−13.3439	−0.833994
$257$	25.2551	1.57537	0.787685	−	0.616079i	$-0.211280\pi$
$257$	25.2551	1.57537	0.787685	+	0.616079i	$0.211280\pi$
$258$	−24.5607	−1.52908
$259$	−0.428212	−0.0266078
$260$	−13.8084	−0.856360
$261$	−7.41840	−0.459188
$262$	15.1117	0.933607
$263$	−2.36958	−0.146115	−0.0730574	−	0.997328i	$-0.523276\pi$
$263$	−2.36958	−0.146115	−0.0730574	+	0.997328i	$0.523276\pi$
$264$	−0.0272490	−0.00167706
$265$	13.1364	0.806966
$266$	9.55870	0.586082
$267$	−15.7635	−0.964711
$268$	3.74242	0.228605
$269$	2.34194	0.142791	0.0713953	−	0.997448i	$-0.477255\pi$
$269$	2.34194	0.142791	0.0713953	+	0.997448i	$0.477255\pi$
$270$	−5.97902	−0.363872
$271$	6.17357	0.375018	0.187509	−	0.982263i	$-0.439959\pi$
$271$	6.17357	0.375018	0.187509	+	0.982263i	$0.439959\pi$
$272$	−0.837106	−0.0507570
$273$	−2.52746	−0.152969
$274$	−20.6721	−1.24884
$275$	−0.0204263	−0.00123175
$276$	−6.53671	−0.393464
$277$	7.16294	0.430379	0.215190	−	0.976572i	$-0.430963\pi$
$277$	7.16294	0.430379	0.215190	+	0.976572i	$0.430963\pi$
$278$	36.2136	2.17195
$279$	−4.87819	−0.292050
$280$	10.3929	0.621096
$281$	−23.7997	−1.41977	−0.709886	−	0.704317i	$-0.751254\pi$
$281$	−23.7997	−1.41977	−0.709886	+	0.704317i	$0.751254\pi$
$282$	1.39089	0.0828260
$283$	−29.8299	−1.77321	−0.886603	−	0.462531i	$-0.846941\pi$
$283$	−29.8299	−1.77321	−0.886603	+	0.462531i	$0.846941\pi$
$284$	−26.2599	−1.55824
$285$	−7.31927	−0.433556
$286$	0.0396212	0.00234285
$287$	−12.6558	−0.747048
$288$	6.09733	0.359289
$289$	−12.5571	−0.738654
$290$	−44.3548	−2.60460
$291$	11.3911	0.667759
$292$	46.4446	2.71797
$293$	−14.6787	−0.857540	−0.428770	−	0.903414i	$-0.641053\pi$
$293$	−14.6787	−0.857540	−0.428770	+	0.903414i	$0.641053\pi$
$294$	−10.6546	−0.621390
$295$	−3.91259	−0.227800
$296$	0.733156	0.0426139
$297$	0.0104875	0.000608548 0
$298$	27.6863	1.60382
$299$	3.46117	0.200165
$300$	−6.12626	−0.353700
$301$	−16.4313	−0.947082
$302$	30.7596	1.77001
$303$	−7.16330	−0.411521
$304$	1.10280	0.0632498
$305$	8.50354	0.486912
$306$	−4.78127	−0.273327
$307$	21.9415	1.25227	0.626135	−	0.779715i	$-0.284636\pi$
$307$	21.9415	1.25227	0.626135	+	0.779715i	$0.284636\pi$
$308$	−0.0500602	−0.00285245
$309$	−2.34669	−0.133498
$310$	−29.1668	−1.65656
$311$	−21.5563	−1.22235	−0.611174	−	0.791497i	$-0.709302\pi$
$311$	−21.5563	−1.22235	−0.611174	+	0.791497i	$0.709302\pi$
$312$	4.32735	0.244988
$313$	7.85453	0.443964	0.221982	−	0.975051i	$-0.428747\pi$
$313$	7.85453	0.443964	0.221982	+	0.975051i	$0.428747\pi$
$314$	−4.69503	−0.264956
$315$	−4.00000	−0.225374
$316$	6.61504	0.372125
$317$	−25.5438	−1.43468	−0.717341	−	0.696722i	$-0.754641\pi$
$317$	−25.5438	−1.43468	−0.717341	+	0.696722i	$0.754641\pi$
$318$	−11.3050	−0.633951
$319$	0.0778007	0.00435600
$320$	34.3625	1.92092
$321$	16.3301	0.911458
$322$	−7.15370	−0.398660
$323$	−5.85303	−0.325671
$324$	3.14543	0.174746
$325$	3.24384	0.179936
$326$	41.4702	2.29682
$327$	−4.23155	−0.234005
$328$	21.6684	1.19644
$329$	0.930510	0.0513007
$330$	0.0627052	0.00345181
$331$	−13.0625	−0.717979	−0.358989	−	0.933342i	$-0.616879\pi$
$331$	−13.0625	−0.717979	−0.358989	+	0.933342i	$0.616879\pi$
$332$	−5.86547	−0.321910
$333$	−0.282175	−0.0154631
$334$	10.2433	0.560487
$335$	−3.13612	−0.171344
$336$	0.602682	0.0328790
$337$	24.6134	1.34078	0.670388	−	0.742011i	$-0.266128\pi$
$337$	24.6134	1.34078	0.670388	+	0.742011i	$0.266128\pi$
$338$	23.1965	1.26172
$339$	0.112564	0.00611366
$340$	−17.4756	−0.947747
$341$	0.0511601	0.00277048
$342$	6.29881	0.340601
$343$	−17.7508	−0.958452
$344$	28.1325	1.51680
$345$	5.47771	0.294910
$346$	53.0117	2.84993
$347$	24.7052	1.32625	0.663123	−	0.748511i	$-0.269231\pi$
$347$	24.7052	1.32625	0.663123	+	0.748511i	$0.269231\pi$
$348$	23.3340	1.25084
$349$	32.9587	1.76424	0.882120	−	0.471025i	$-0.156116\pi$
$349$	32.9587	1.76424	0.882120	+	0.471025i	$0.156116\pi$
$350$	−6.70451	−0.358371
$351$	−1.66550	−0.0888977
$352$	−0.0639459	−0.00340833
$353$	−7.48664	−0.398474	−0.199237	−	0.979951i	$-0.563846\pi$
$353$	−7.48664	−0.398474	−0.199237	+	0.979951i	$0.563846\pi$
$354$	3.36710	0.178959
$355$	22.0056	1.16794
$356$	49.5829	2.62789
$357$	−3.19870	−0.169293
$358$	−17.5070	−0.925274
$359$	29.0163	1.53142	0.765712	−	0.643184i	$-0.222387\pi$
$359$	29.0163	1.53142	0.765712	+	0.643184i	$0.222387\pi$
$360$	6.84854	0.360949
$361$	−11.2893	−0.594172
$362$	−39.8724	−2.09564
$363$	10.9999	0.577344
$364$	7.94994	0.416690
$365$	−38.9203	−2.03718
$366$	−7.31798	−0.382517
$367$	29.7552	1.55321	0.776604	−	0.629989i	$-0.216941\pi$
$367$	29.7552	1.55321	0.776604	+	0.629989i	$0.216941\pi$
$368$	−0.825330	−0.0430233
$369$	−8.33967	−0.434146
$370$	−1.68713	−0.0877098
$371$	−7.56308	−0.392656
$372$	15.3440	0.795549
$373$	4.72350	0.244574	0.122287	−	0.992495i	$-0.460977\pi$
$373$	4.72350	0.244574	0.122287	+	0.992495i	$0.460977\pi$
$374$	0.0501437	0.00259287
$375$	−8.04546	−0.415466
$376$	−1.59316	−0.0821608
$377$	−12.3553	−0.636332
$378$	3.44232	0.177054
$379$	0.240446	0.0123509	0.00617543	−	0.999981i	$-0.498034\pi$
$379$	0.240446	0.0123509	0.00617543	+	0.999981i	$0.498034\pi$
$380$	23.0222	1.18101
$381$	1.00000	0.0512316
$382$	40.0481	2.04904
$383$	6.77614	0.346245	0.173122	−	0.984900i	$-0.444614\pi$
$383$	6.77614	0.346245	0.173122	+	0.984900i	$0.444614\pi$
$384$	−17.3770	−0.886765
$385$	0.0419501	0.00213798
$386$	−39.5855	−2.01485
$387$	−10.8276	−0.550396
$388$	−35.8299	−1.81899
$389$	22.6579	1.14880	0.574401	−	0.818574i	$-0.305235\pi$
$389$	22.6579	1.14880	0.574401	+	0.818574i	$0.305235\pi$
$390$	−9.95805	−0.504245
$391$	4.38038	0.221525
$392$	12.2041	0.616399
$393$	6.66199	0.336053
$394$	−52.3643	−2.63808
$395$	−5.54336	−0.278916
$396$	−0.0329877	−0.00165770
$397$	26.5016	1.33008	0.665040	−	0.746808i	$-0.268415\pi$
$397$	26.5016	1.33008	0.665040	+	0.746808i	$0.268415\pi$
$398$	−24.2136	−1.21372
$399$	4.21394	0.210961
$400$	−0.773506	−0.0386753
$401$	−24.8855	−1.24272	−0.621362	−	0.783523i	$-0.713420\pi$
$401$	−24.8855	−1.24272	−0.621362	+	0.783523i	$0.713420\pi$
$402$	2.69888	0.134608
$403$	−8.12461	−0.404716
$404$	22.5316	1.12099
$405$	−2.63584	−0.130976
$406$	25.5365	1.26736
$407$	0.00295932	0.000146688 0
$408$	5.47660	0.271132
$409$	37.5198	1.85523	0.927617	−	0.373532i	$-0.121853\pi$
$409$	37.5198	1.85523	0.927617	+	0.373532i	$0.121853\pi$
$410$	−49.8631	−2.46256
$411$	−9.11324	−0.449523
$412$	7.38134	0.363652
$413$	2.25261	0.110843
$414$	−4.71401	−0.231681
$415$	4.91522	0.241279
$416$	10.1551	0.497894
$417$	15.9647	0.781796
$418$	−0.0660589	−0.00323105
$419$	−28.6887	−1.40154	−0.700768	−	0.713389i	$-0.747159\pi$
$419$	−28.6887	−1.40154	−0.700768	+	0.713389i	$0.747159\pi$
$420$	12.5817	0.613924
$421$	19.1477	0.933200	0.466600	−	0.884468i	$-0.345479\pi$
$421$	19.1477	0.933200	0.466600	+	0.884468i	$0.345479\pi$
$422$	9.06956	0.441499
$423$	0.613170	0.0298133
$424$	12.9490	0.628859
$425$	4.10533	0.199138
$426$	−18.9376	−0.917529
$427$	−4.89577	−0.236923
$428$	−51.3652	−2.48283
$429$	0.0174669	0.000843312 0
$430$	−64.7383	−3.12196
$431$	1.55463	0.0748840	0.0374420	−	0.999299i	$-0.488079\pi$
$431$	1.55463	0.0748840	0.0374420	+	0.999299i	$0.488079\pi$
$432$	0.397144	0.0191076
$433$	−30.9521	−1.48746	−0.743732	−	0.668478i	$-0.766946\pi$
$433$	−30.9521	−1.48746	−0.743732	+	0.668478i	$0.766946\pi$
$434$	16.7923	0.806055
$435$	−19.5538	−0.937531
$436$	13.3100	0.637434
$437$	−5.77068	−0.276049
$438$	33.4940	1.60040
$439$	9.73798	0.464768	0.232384	−	0.972624i	$-0.425347\pi$
$439$	9.73798	0.464768	0.232384	+	0.972624i	$0.425347\pi$
$440$	−0.0718242	−0.00342408
$441$	−4.69707	−0.223670
$442$	−7.96319	−0.378770
$443$	−32.5572	−1.54684	−0.773420	−	0.633894i	$-0.781456\pi$
$443$	−32.5572	−1.54684	−0.773420	+	0.633894i	$0.781456\pi$
$444$	0.887561	0.0421218
$445$	−41.5501	−1.96967
$446$	26.1880	1.24004
$447$	12.2055	0.577298
$448$	−19.7836	−0.934686
$449$	−24.6255	−1.16215	−0.581074	−	0.813851i	$-0.697367\pi$
$449$	−24.6255	−1.16215	−0.581074	+	0.813851i	$0.697367\pi$
$450$	−4.41801	−0.208267
$451$	0.0874625	0.00411845
$452$	−0.354063	−0.0166537
$453$	13.5603	0.637119
$454$	19.1251	0.897585
$455$	−6.66199	−0.312319
$456$	−7.21483	−0.337865
$457$	17.8775	0.836275	0.418137	−	0.908384i	$-0.362683\pi$
$457$	17.8775	0.836275	0.418137	+	0.908384i	$0.362683\pi$
$458$	−43.9516	−2.05372
$459$	−2.10782	−0.0983844
$460$	−17.2297	−0.803341
$461$	−19.8380	−0.923949	−0.461974	−	0.886893i	$-0.652859\pi$
$461$	−19.8380	−0.923949	−0.461974	+	0.886893i	$0.652859\pi$
$462$	−0.0361014	−0.00167959
$463$	27.8847	1.29591	0.647955	−	0.761679i	$-0.275624\pi$
$463$	27.8847	1.29591	0.647955	+	0.761679i	$0.275624\pi$
$464$	2.94617	0.136773
$465$	−12.8581	−0.596282
$466$	−32.6503	−1.51250
$467$	13.8178	0.639409	0.319705	−	0.947517i	$-0.396416\pi$
$467$	13.8178	0.639409	0.319705	+	0.947517i	$0.396416\pi$
$468$	5.23870	0.242159
$469$	1.80556	0.0833732
$470$	3.66616	0.169107
$471$	−2.06979	−0.0953711
$472$	−3.85676	−0.177522
$473$	0.113554	0.00522123
$474$	4.77050	0.219116
$475$	−5.40834	−0.248152
$476$	10.0613	0.461157
$477$	−4.98377	−0.228191
$478$	−29.8229	−1.36407
$479$	21.1396	0.965893	0.482946	−	0.875650i	$-0.339567\pi$
$479$	21.1396	0.965893	0.482946	+	0.875650i	$0.339567\pi$
$480$	16.0716	0.733565
$481$	−0.469962	−0.0214284
$482$	−12.7588	−0.581145
$483$	−3.15370	−0.143498
$484$	−34.5993	−1.57270
$485$	30.0252	1.36337
$486$	2.26835	0.102895
$487$	−21.7539	−0.985763	−0.492882	−	0.870096i	$-0.664056\pi$
$487$	−21.7539	−0.985763	−0.492882	+	0.870096i	$0.664056\pi$
$488$	8.38221	0.379445
$489$	18.2821	0.826744
$490$	−28.0839	−1.26870
$491$	−18.3359	−0.827486	−0.413743	−	0.910394i	$-0.635779\pi$
$491$	−18.3359	−0.827486	−0.413743	+	0.910394i	$0.635779\pi$
$492$	26.2318	1.18262
$493$	−15.6366	−0.704238
$494$	10.4907	0.471997
$495$	0.0276435	0.00124248
$496$	1.93734	0.0869893
$497$	−12.6694	−0.568298
$498$	−4.22994	−0.189548
$499$	−6.88268	−0.308111	−0.154056	−	0.988062i	$-0.549233\pi$
$499$	−6.88268	−0.308111	−0.154056	+	0.988062i	$0.549233\pi$
$500$	25.3064	1.13174
$501$	4.51573	0.201748
$502$	1.27227	0.0567842
$503$	33.5050	1.49392	0.746958	−	0.664871i	$-0.231514\pi$
$503$	33.5050	1.49392	0.746958	+	0.664871i	$0.231514\pi$
$504$	−3.94292	−0.175632
$505$	−18.8813	−0.840209
$506$	0.0494383	0.00219780
$507$	10.2261	0.454158
$508$	−3.14543	−0.139556
$509$	0.0324525	0.00143843	0.000719215	−	1.00000i	$-0.499771\pi$
$509$	0.0324525	0.00143843	0.000719215		1.00000i	$0.499771\pi$
$510$	−12.6027	−0.558056
$511$	22.4076	0.991256
$512$	−4.48526	−0.198222
$513$	2.77682	0.122600
$514$	−57.2875	−2.52684
$515$	−6.18551	−0.272566
$516$	34.0573	1.49929
$517$	−0.00643063	−0.000282819 0
$518$	0.971336	0.0426781
$519$	23.3701	1.02584
$520$	11.4062	0.500196
$521$	−11.6901	−0.512151	−0.256076	−	0.966657i	$-0.582430\pi$
$521$	−11.6901	−0.512151	−0.256076	+	0.966657i	$0.582430\pi$
$522$	16.8276	0.736522
$523$	−23.2907	−1.01843	−0.509217	−	0.860638i	$-0.670065\pi$
$523$	−23.2907	−1.01843	−0.509217	+	0.860638i	$0.670065\pi$
$524$	−20.9548	−0.915415
$525$	−2.95567	−0.128996
$526$	5.37505	0.234363
$527$	−10.2823	−0.447905
$528$	−0.00416506	−0.000181261 0
$529$	−18.6812	−0.812228
$530$	−29.7981	−1.29435
$531$	1.48438	0.0644166
$532$	−13.2546	−0.574661
$533$	−13.8897	−0.601630
$534$	35.7572	1.54737
$535$	43.0436	1.86094
$536$	−3.09137	−0.133527
$537$	−7.71794	−0.333054
$538$	−5.31235	−0.229032
$539$	0.0492607	0.00212181
$540$	8.29085	0.356782
$541$	14.7322	0.633385	0.316692	−	0.948528i	$-0.397428\pi$
$541$	14.7322	0.633385	0.316692	+	0.948528i	$0.397428\pi$
$542$	−14.0038	−0.601516
$543$	−17.5777	−0.754330
$544$	12.8520	0.551027
$545$	−11.1537	−0.477772
$546$	5.73317	0.245357
$547$	−11.3142	−0.483761	−0.241880	−	0.970306i	$-0.577764\pi$
$547$	−11.3142	−0.483761	−0.241880	+	0.970306i	$0.577764\pi$
$548$	28.6650	1.22451
$549$	−3.22612	−0.137687
$550$	0.0463340	0.00197569
$551$	20.5996	0.877572
$552$	5.39955	0.229820
$553$	3.19149	0.135716
$554$	−16.2481	−0.690315
$555$	−0.743769	−0.0315713
$556$	−50.2159	−2.12963
$557$	−11.1052	−0.470544	−0.235272	−	0.971930i	$-0.575598\pi$
$557$	−11.1052	−0.470544	−0.235272	+	0.971930i	$0.575598\pi$
$558$	11.0655	0.468438
$559$	−18.0333	−0.762726
$560$	1.58858	0.0671296
$561$	0.0221058	0.000933306 0
$562$	53.9862	2.27727
$563$	−0.683837	−0.0288203	−0.0144101	−	0.999896i	$-0.504587\pi$
$563$	−0.683837	−0.0288203	−0.0144101	+	0.999896i	$0.504587\pi$
$564$	−1.92868	−0.0812121
$565$	0.296702	0.0124824
$566$	67.6648	2.84417
$567$	1.51754	0.0637307
$568$	21.6916	0.910160
$569$	27.5414	1.15460	0.577298	−	0.816533i	$-0.304107\pi$
$569$	27.5414	1.15460	0.577298	+	0.816533i	$0.304107\pi$
$570$	16.6027	0.695410
$571$	0.602431	0.0252110	0.0126055	−	0.999921i	$-0.495987\pi$
$571$	0.602431	0.0252110	0.0126055	+	0.999921i	$0.495987\pi$
$572$	−0.0549410	−0.00229720
$573$	17.6551	0.737553
$574$	28.7078	1.19824
$575$	4.04758	0.168796
$576$	−13.0366	−0.543192
$577$	−8.43262	−0.351055	−0.175527	−	0.984475i	$-0.556163\pi$
$577$	−8.43262	−0.351055	−0.175527	+	0.984475i	$0.556163\pi$
$578$	28.4840	1.18478
$579$	−17.4512	−0.725248
$580$	61.5049	2.55385
$581$	−2.82985	−0.117402
$582$	−25.8391	−1.07106
$583$	0.0522675	0.00216470
$584$	−38.3649	−1.58755
$585$	−4.38999	−0.181504
$586$	33.2965	1.37547
$587$	47.4432	1.95819	0.979096	−	0.203401i	$-0.0651994\pi$
$587$	47.4432	1.95819	0.979096	+	0.203401i	$0.0651994\pi$
$588$	14.7743	0.609281
$589$	13.5459	0.558147
$590$	8.87514	0.365384
$591$	−23.0847	−0.949579
$592$	0.112064	0.00460581
$593$	4.19118	0.172111	0.0860555	−	0.996290i	$-0.472574\pi$
$593$	4.19118	0.172111	0.0860555	+	0.996290i	$0.472574\pi$
$594$	−0.0237894	−0.000976092 0
$595$	−8.43126	−0.345648
$596$	−38.3914	−1.57257
$597$	−10.6745	−0.436880
$598$	−7.85116	−0.321058
$599$	−9.13975	−0.373440	−0.186720	−	0.982413i	$-0.559786\pi$
$599$	−9.13975	−0.373440	−0.186720	+	0.982413i	$0.559786\pi$
$600$	5.06051	0.206594
$601$	34.8399	1.42115	0.710574	−	0.703623i	$-0.248435\pi$
$601$	34.8399	1.42115	0.710574	+	0.703623i	$0.248435\pi$
$602$	37.2719	1.51909
$603$	1.18980	0.0484523
$604$	−42.6530	−1.73552
$605$	28.9940	1.17877
$606$	16.2489	0.660067
$607$	28.0143	1.13707	0.568533	−	0.822661i	$-0.307511\pi$
$607$	28.0143	1.13707	0.568533	+	0.822661i	$0.307511\pi$
$608$	−16.9312	−0.686650
$609$	11.2577	0.456186
$610$	−19.2890	−0.780991
$611$	1.02123	0.0413146
$612$	6.62998	0.268001
$613$	−24.1731	−0.976342	−0.488171	−	0.872748i	$-0.662336\pi$
$613$	−24.1731	−0.976342	−0.488171	+	0.872748i	$0.662336\pi$
$614$	−49.7712	−2.00860
$615$	−21.9821	−0.886403
$616$	0.0413515	0.00166610
$617$	−11.8934	−0.478810	−0.239405	−	0.970920i	$-0.576952\pi$
$617$	−11.8934	−0.478810	−0.239405	+	0.970920i	$0.576952\pi$
$618$	5.32312	0.214127
$619$	−20.6301	−0.829193	−0.414596	−	0.910005i	$-0.636077\pi$
$619$	−20.6301	−0.829193	−0.414596	+	0.910005i	$0.636077\pi$
$620$	40.4443	1.62428
$621$	−2.07816	−0.0833938
$622$	48.8974	1.96061
$623$	23.9218	0.958405
$624$	0.661442	0.0264789
$625$	−30.9449	−1.23780
$626$	−17.8169	−0.712105
$627$	−0.0291220	−0.00116302
$628$	6.51039	0.259793
$629$	−0.594773	−0.0237152
$630$	9.07341	0.361493
$631$	−4.13649	−0.164671	−0.0823354	−	0.996605i	$-0.526238\pi$
$631$	−4.13649	−0.164671	−0.0823354	+	0.996605i	$0.526238\pi$
$632$	−5.46426	−0.217356
$633$	3.99830	0.158918
$634$	57.9424	2.30118
$635$	2.63584	0.104600
$636$	15.6761	0.621598
$637$	−7.82296	−0.309957
$638$	−0.176479	−0.00698689
$639$	−8.34861	−0.330266
$640$	−45.8030	−1.81052
$641$	−27.4147	−1.08281	−0.541407	−	0.840760i	$-0.682108\pi$
$641$	−27.4147	−1.08281	−0.541407	+	0.840760i	$0.682108\pi$
$642$	−37.0425	−1.46195
$643$	38.0535	1.50068	0.750341	−	0.661051i	$-0.229889\pi$
$643$	38.0535	1.50068	0.750341	+	0.661051i	$0.229889\pi$
$644$	9.91972	0.390892
$645$	−28.5398	−1.12375
$646$	13.2767	0.522366
$647$	−46.9425	−1.84550	−0.922750	−	0.385400i	$-0.874064\pi$
$647$	−46.9425	−1.84550	−0.922750	+	0.385400i	$0.874064\pi$
$648$	−2.59823	−0.102068
$649$	−0.0155675	−0.000611076 0
$650$	−7.35818	−0.288612
$651$	7.40285	0.290141
$652$	−57.5049	−2.25207
$653$	−32.8959	−1.28732	−0.643659	−	0.765313i	$-0.722584\pi$
$653$	−32.8959	−1.28732	−0.643659	+	0.765313i	$0.722584\pi$
$654$	9.59864	0.375337
$655$	17.5600	0.686125
$656$	3.31205	0.129314
$657$	14.7658	0.576067
$658$	−2.11073	−0.0822846
$659$	43.1518	1.68096	0.840478	−	0.541846i	$-0.182274\pi$
$659$	43.1518	1.68096	0.840478	+	0.541846i	$0.182274\pi$
$660$	−0.0869505	−0.00338455
$661$	−8.69294	−0.338116	−0.169058	−	0.985606i	$-0.554073\pi$
$661$	−8.69294	−0.338116	−0.169058	+	0.985606i	$0.554073\pi$
$662$	29.6303	1.15161
$663$	−3.51056	−0.136339
$664$	4.84508	0.188026
$665$	11.1073	0.430722
$666$	0.640073	0.0248023
$667$	−15.4166	−0.596935
$668$	−14.2039	−0.549566
$669$	11.5449	0.446353
$670$	7.11382	0.274831
$671$	0.0338340	0.00130615
$672$	−9.25294	−0.356940
$673$	−6.03409	−0.232597	−0.116299	−	0.993214i	$-0.537103\pi$
$673$	−6.03409	−0.232597	−0.116299	+	0.993214i	$0.537103\pi$
$674$	−55.8318	−2.15056
$675$	−1.94767	−0.0749659
$676$	−32.1655	−1.23713
$677$	−3.99616	−0.153585	−0.0767925	−	0.997047i	$-0.524468\pi$
$677$	−3.99616	−0.153585	−0.0767925	+	0.997047i	$0.524468\pi$
$678$	−0.255336	−0.00980612
$679$	−17.2865	−0.663395
$680$	14.4354	0.553574
$681$	8.43126	0.323087
$682$	−0.116049	−0.00444376
$683$	37.8078	1.44668	0.723338	−	0.690494i	$-0.242607\pi$
$683$	37.8078	1.44668	0.723338	+	0.690494i	$0.242607\pi$
$684$	−8.73429	−0.333964
$685$	−24.0211	−0.917799
$686$	40.2650	1.53733
$687$	−19.3760	−0.739240
$688$	4.30010	0.163940
$689$	−8.30046	−0.316222
$690$	−12.4254	−0.473026
$691$	−25.4543	−0.968326	−0.484163	−	0.874978i	$-0.660876\pi$
$691$	−25.4543	−0.968326	−0.484163	+	0.874978i	$0.660876\pi$
$692$	−73.5091	−2.79440
$693$	−0.0159152	−0.000604570 0
$694$	−56.0402	−2.12726
$695$	42.0805	1.59620
$696$	−19.2747	−0.730607
$697$	−17.5785	−0.665833
$698$	−74.7620	−2.82978
$699$	−14.3938	−0.544425
$700$	9.29685	0.351388
$701$	42.6352	1.61031	0.805154	−	0.593065i	$-0.202082\pi$
$701$	42.6352	1.61031	0.805154	+	0.593065i	$0.202082\pi$
$702$	3.77794	0.142589
$703$	0.783550	0.0295521
$704$	0.136722	0.00515290
$705$	1.61622	0.0608703
$706$	16.9823	0.639139
$707$	10.8706	0.408831
$708$	−4.66901	−0.175472
$709$	−27.6047	−1.03672	−0.518359	−	0.855163i	$-0.673457\pi$
$709$	−27.6047	−1.03672	−0.518359	+	0.855163i	$0.673457\pi$
$710$	−49.9165	−1.87333
$711$	2.10307	0.0788712
$712$	−40.9573	−1.53494
$713$	−10.1377	−0.379659
$714$	7.25577	0.271540
$715$	0.0460401	0.00172180
$716$	24.2762	0.907244
$717$	−13.1474	−0.490998
$718$	−65.8193	−2.45635
$719$	−15.6846	−0.584936	−0.292468	−	0.956275i	$-0.594477\pi$
$719$	−15.6846	−0.584936	−0.292468	+	0.956275i	$0.594477\pi$
$720$	1.04681	0.0390123
$721$	3.56120	0.132626
$722$	25.6080	0.953033
$723$	−5.62468	−0.209184
$724$	55.2893	2.05481
$725$	−14.4486	−0.536608
$726$	−24.9516	−0.926042
$727$	47.7864	1.77230	0.886149	−	0.463400i	$-0.153371\pi$
$727$	47.7864	1.77230	0.886149	+	0.463400i	$0.153371\pi$
$728$	−6.56693	−0.243387
$729$	1.00000	0.0370370
$730$	88.2849	3.26757
$731$	−22.8225	−0.844121
$732$	10.1475	0.375063
$733$	3.47183	0.128235	0.0641175	−	0.997942i	$-0.479577\pi$
$733$	3.47183	0.128235	0.0641175	+	0.997942i	$0.479577\pi$
$734$	−67.4953	−2.49130
$735$	−12.3807	−0.456671
$736$	12.6712	0.467068
$737$	−0.0124780	−0.000459634 0
$738$	18.9173	0.696356
$739$	22.4368	0.825351	0.412676	−	0.910878i	$-0.364594\pi$
$739$	22.4368	0.825351	0.412676	+	0.910878i	$0.364594\pi$
$740$	2.33947	0.0860007
$741$	4.62479	0.169896
$742$	17.1557	0.629807
$743$	−16.2302	−0.595429	−0.297714	−	0.954655i	$-0.596224\pi$
$743$	−16.2302	−0.595429	−0.297714	+	0.954655i	$0.596224\pi$
$744$	−12.6747	−0.464676
$745$	32.1717	1.17868
$746$	−10.7146	−0.392288
$747$	−1.86476	−0.0682281
$748$	−0.0695321	−0.00254234
$749$	−24.7816	−0.905500
$750$	18.2499	0.666394
$751$	25.4121	0.927302	0.463651	−	0.886018i	$-0.346539\pi$
$751$	25.4121	0.927302	0.463651	+	0.886018i	$0.346539\pi$
$752$	−0.243517	−0.00888014
$753$	0.560879	0.0204396
$754$	28.0263	1.02066
$755$	35.7429	1.30082
$756$	−4.77331	−0.173604
$757$	38.9444	1.41546	0.707729	−	0.706484i	$-0.249720\pi$
$757$	38.9444	1.41546	0.707729	+	0.706484i	$0.249720\pi$
$758$	−0.545415	−0.0198104
$759$	0.0217948	0.000791100 0
$760$	−19.0172	−0.689825
$761$	24.0905	0.873278	0.436639	−	0.899637i	$-0.356169\pi$
$761$	24.0905	0.873278	0.436639	+	0.899637i	$0.356169\pi$
$762$	−2.26835	−0.0821738
$763$	6.42154	0.232475
$764$	−55.5329	−2.00911
$765$	−5.55587	−0.200873
$766$	−15.3707	−0.555365
$767$	2.47223	0.0892670
$768$	13.3439	0.481507
$769$	−30.4484	−1.09800	−0.548999	−	0.835823i	$-0.684991\pi$
$769$	−30.4484	−1.09800	−0.548999	+	0.835823i	$0.684991\pi$
$770$	−0.0951577	−0.00342924
$771$	−25.2551	−0.909540
$772$	54.8915	1.97559
$773$	−3.52407	−0.126752	−0.0633760	−	0.997990i	$-0.520187\pi$
$773$	−3.52407	−0.126752	−0.0633760	+	0.997990i	$0.520187\pi$
$774$	24.5607	0.882817
$775$	−9.50111	−0.341290
$776$	29.5968	1.06246
$777$	0.428212	0.0153620
$778$	−51.3962	−1.84264
$779$	23.1578	0.829714
$780$	13.8084	0.494420
$781$	0.0875563	0.00313301
$782$	−9.93626	−0.355320
$783$	7.41840	0.265112
$784$	1.86541	0.0666219
$785$	−5.45566	−0.194721
$786$	−15.1117	−0.539018
$787$	−26.7194	−0.952443	−0.476221	−	0.879325i	$-0.657994\pi$
$787$	−26.7194	−0.952443	−0.476221	+	0.879325i	$0.657994\pi$
$788$	72.6113	2.58667
$789$	2.36958	0.0843594
$790$	12.5743	0.447373
$791$	−0.170821	−0.00607370
$792$	0.0272490	0.000968252 0
$793$	−5.37309	−0.190804
$794$	−60.1151	−2.13340
$795$	−13.1364	−0.465902
$796$	33.5760	1.19007
$797$	−29.7776	−1.05478	−0.527388	−	0.849625i	$-0.676829\pi$
$797$	−29.7776	−1.05478	−0.527388	+	0.849625i	$0.676829\pi$
$798$	−9.55870	−0.338374
$799$	1.29245	0.0457235
$800$	11.8756	0.419866
$801$	15.7635	0.556976
$802$	56.4492	1.99329
$803$	−0.154856	−0.00546476
$804$	−3.74242	−0.131985
$805$	−8.31265	−0.292982
$806$	18.4295	0.649151
$807$	−2.34194	−0.0824402
$808$	−18.6119	−0.654765
$809$	23.8830	0.839680	0.419840	−	0.907598i	$-0.362086\pi$
$809$	23.8830	0.839680	0.419840	+	0.907598i	$0.362086\pi$
$810$	5.97902	0.210082
$811$	14.5688	0.511578	0.255789	−	0.966733i	$-0.417665\pi$
$811$	14.5688	0.511578	0.255789	+	0.966733i	$0.417665\pi$
$812$	−35.4104	−1.24266
$813$	−6.17357	−0.216517
$814$	−0.00671278	−0.000235283 0
$815$	48.1887	1.68798
$816$	0.837106	0.0293046
$817$	30.0662	1.05188
$818$	−85.1082	−2.97574
$819$	2.52746	0.0883166
$820$	69.1430	2.41458
$821$	46.7882	1.63292	0.816460	−	0.577402i	$-0.195934\pi$
$821$	46.7882	1.63292	0.816460	+	0.577402i	$0.195934\pi$
$822$	20.6721	0.721021
$823$	−25.0104	−0.871809	−0.435904	−	0.899993i	$-0.643571\pi$
$823$	−25.0104	−0.871809	−0.435904	+	0.899993i	$0.643571\pi$
$824$	−6.09724	−0.212408
$825$	0.0204263	0.000711151 0
$826$	−5.10970	−0.177789
$827$	−52.7563	−1.83452	−0.917258	−	0.398292i	$-0.869603\pi$
$827$	−52.7563	−1.83452	−0.917258	+	0.398292i	$0.869603\pi$
$828$	6.53671	0.227166
$829$	−27.0086	−0.938048	−0.469024	−	0.883186i	$-0.655394\pi$
$829$	−27.0086	−0.938048	−0.469024	+	0.883186i	$0.655394\pi$
$830$	−11.1495	−0.387003
$831$	−7.16294	−0.248480
$832$	−21.7124	−0.752743
$833$	−9.90056	−0.343034
$834$	−36.2136	−1.25398
$835$	11.9028	0.411912
$836$	0.0916011	0.00316809
$837$	4.87819	0.168615
$838$	65.0762	2.24802
$839$	−10.2793	−0.354882	−0.177441	−	0.984131i	$-0.556782\pi$
$839$	−10.2793	−0.354882	−0.177441	+	0.984131i	$0.556782\pi$
$840$	−10.3929	−0.358590
$841$	26.0327	0.897680
$842$	−43.4337	−1.49682
$843$	23.7997	0.819706
$844$	−12.5764	−0.432896
$845$	26.9545	0.927261
$846$	−1.39089	−0.0478196
$847$	−16.6928	−0.573571
$848$	1.97928	0.0679686
$849$	29.8299	1.02376
$850$	−9.31235	−0.319411
$851$	−0.586406	−0.0201017
$852$	26.2599	0.899651
$853$	33.1422	1.13477	0.567383	−	0.823454i	$-0.307956\pi$
$853$	33.1422	1.13477	0.567383	+	0.823454i	$0.307956\pi$
$854$	11.1053	0.380016
$855$	7.31927	0.250314
$856$	42.4294	1.45021
$857$	−0.539897	−0.0184425	−0.00922127	−	0.999957i	$-0.502935\pi$
$857$	−0.539897	−0.0184425	−0.00922127	+	0.999957i	$0.502935\pi$
$858$	−0.0396212	−0.00135265
$859$	15.7735	0.538184	0.269092	−	0.963114i	$-0.413276\pi$
$859$	15.7735	0.538184	0.269092	+	0.963114i	$0.413276\pi$
$860$	89.7697	3.06112
$861$	12.6558	0.431308
$862$	−3.52645	−0.120112
$863$	11.8283	0.402640	0.201320	−	0.979525i	$-0.435477\pi$
$863$	11.8283	0.402640	0.201320	+	0.979525i	$0.435477\pi$
$864$	−6.09733	−0.207435
$865$	61.6001	2.09446
$866$	70.2104	2.38585
$867$	12.5571	0.426462
$868$	−23.2851	−0.790349
$869$	−0.0220560	−0.000748198 0
$870$	44.3548	1.50377
$871$	1.98160	0.0671441
$872$	−10.9945	−0.372322
$873$	−11.3911	−0.385531
$874$	13.0900	0.442774
$875$	12.2093	0.412750
$876$	−46.4446	−1.56922
$877$	−54.3552	−1.83544	−0.917722	−	0.397224i	$-0.869973\pi$
$877$	−54.3552	−1.83544	−0.917722	+	0.397224i	$0.869973\pi$
$878$	−22.0892	−0.745474
$879$	14.6787	0.495101
$880$	−0.0109784	−0.000370083 0
$881$	−43.2780	−1.45807	−0.729036	−	0.684476i	$-0.760031\pi$
$881$	−43.2780	−1.45807	−0.729036	+	0.684476i	$0.760031\pi$
$882$	10.6546	0.358760
$883$	−29.0340	−0.977072	−0.488536	−	0.872544i	$-0.662469\pi$
$883$	−29.0340	−0.977072	−0.488536	+	0.872544i	$0.662469\pi$
$884$	11.0422	0.371390
$885$	3.91259	0.131520
$886$	73.8513	2.48108
$887$	−41.5669	−1.39568	−0.697840	−	0.716253i	$-0.745855\pi$
$887$	−41.5669	−1.39568	−0.697840	+	0.716253i	$0.745855\pi$
$888$	−0.733156	−0.0246031
$889$	−1.51754	−0.0508967
$890$	94.2504	3.15928
$891$	−0.0104875	−0.000351345 0
$892$	−36.3137	−1.21587
$893$	−1.70266	−0.0569774
$894$	−27.6863	−0.925968
$895$	−20.3433	−0.680001
$896$	26.3703	0.880969
$897$	−3.46117	−0.115565
$898$	55.8593	1.86405
$899$	36.1884	1.20695
$900$	6.12626	0.204209
$901$	−10.5049	−0.349968
$902$	−0.198396	−0.00660586
$903$	16.4313	0.546798
$904$	0.292469	0.00972736
$905$	−46.3320	−1.54013
$906$	−30.7596	−1.02192
$907$	−7.49326	−0.248810	−0.124405	−	0.992232i	$-0.539702\pi$
$907$	−7.49326	−0.248810	−0.124405	+	0.992232i	$0.539702\pi$
$908$	−26.5199	−0.880094
$909$	7.16330	0.237592
$910$	15.1117	0.500949
$911$	−8.26482	−0.273826	−0.136913	−	0.990583i	$-0.543718\pi$
$911$	−8.26482	−0.273826	−0.136913	+	0.990583i	$0.543718\pi$
$912$	−1.10280	−0.0365173
$913$	0.0195567	0.000647233 0
$914$	−40.5525	−1.34136
$915$	−8.50354	−0.281119
$916$	60.9458	2.01371
$917$	−10.1098	−0.333856
$918$	4.78127	0.157805
$919$	−37.8332	−1.24800	−0.624001	−	0.781423i	$-0.714494\pi$
$919$	−37.8332	−1.24800	−0.624001	+	0.781423i	$0.714494\pi$
$920$	14.2324	0.469227
$921$	−21.9415	−0.722998
$922$	44.9996	1.48198
$923$	−13.9046	−0.457675
$924$	0.0500602	0.00164686
$925$	−0.549585	−0.0180702
$926$	−63.2523	−2.07860
$927$	2.34669	0.0770754
$928$	−45.2324	−1.48483
$929$	14.3387	0.470437	0.235218	−	0.971943i	$-0.424419\pi$
$929$	14.3387	0.470437	0.235218	+	0.971943i	$0.424419\pi$
$930$	29.1668	0.956417
$931$	13.0429	0.427465
$932$	45.2747	1.48302
$933$	21.5563	0.705723
$934$	−31.3435	−1.02559
$935$	0.0582674	0.00190555
$936$	−4.32735	−0.141444
$937$	−31.5539	−1.03082	−0.515410	−	0.856944i	$-0.672360\pi$
$937$	−31.5539	−1.03082	−0.515410	+	0.856944i	$0.672360\pi$
$938$	−4.09566	−0.133728
$939$	−7.85453	−0.256323
$940$	−5.08370	−0.165812
$941$	21.6114	0.704510	0.352255	−	0.935904i	$-0.385415\pi$
$941$	21.6114	0.704510	0.352255	+	0.935904i	$0.385415\pi$
$942$	4.69503	0.152972
$943$	−17.3312	−0.564381
$944$	−0.589512	−0.0191870
$945$	4.00000	0.130120
$946$	−0.257581	−0.00837469
$947$	3.91107	0.127093	0.0635463	−	0.997979i	$-0.479759\pi$
$947$	3.91107	0.127093	0.0635463	+	0.997979i	$0.479759\pi$
$948$	−6.61504	−0.214847
$949$	24.5923	0.798301
$950$	12.2680	0.398027
$951$	25.5438	0.828314
$952$	−8.31096	−0.269360
$953$	36.7238	1.18960	0.594801	−	0.803873i	$-0.297231\pi$
$953$	36.7238	1.18960	0.594801	+	0.803873i	$0.297231\pi$
$954$	11.3050	0.366012
$955$	46.5361	1.50587
$956$	41.3541	1.33749
$957$	−0.0778007	−0.00251494
$958$	−47.9521	−1.54926
$959$	13.8297	0.446585
$960$	−34.3625	−1.10904
$961$	−7.20327	−0.232364
$962$	1.06604	0.0343705
$963$	−16.3301	−0.526231
$964$	17.6920	0.569821
$965$	−45.9987	−1.48075
$966$	7.15370	0.230166
$967$	6.81466	0.219145	0.109572	−	0.993979i	$-0.465052\pi$
$967$	6.81466	0.219145	0.109572	+	0.993979i	$0.465052\pi$
$968$	28.5803	0.918605
$969$	5.85303	0.188026
$970$	−68.1078	−2.18681
$971$	−56.2769	−1.80601	−0.903007	−	0.429627i	$-0.858645\pi$
$971$	−56.2769	−1.80601	−0.903007	+	0.429627i	$0.858645\pi$
$972$	−3.14543	−0.100890
$973$	−24.2271	−0.776685
$974$	49.3455	1.58113
$975$	−3.24384	−0.103886
$976$	1.28123	0.0410113
$977$	41.4266	1.32536	0.662678	−	0.748905i	$-0.269420\pi$
$977$	41.4266	1.32536	0.662678	+	0.748905i	$0.269420\pi$
$978$	−41.4702	−1.32607
$979$	−0.165320	−0.00528366
$980$	38.9427	1.24398
$981$	4.23155	0.135103
$982$	41.5922	1.32726
$983$	25.1118	0.800943	0.400471	−	0.916309i	$-0.368846\pi$
$983$	25.1118	0.800943	0.400471	+	0.916309i	$0.368846\pi$
$984$	−21.6684	−0.690764
$985$	−60.8477	−1.93877
$986$	35.4694	1.12958
$987$	−0.930510	−0.0296185
$988$	−14.5469	−0.462800
$989$	−22.5014	−0.715504
$990$	−0.0627052	−0.00199290
$991$	−60.9601	−1.93646	−0.968230	−	0.250060i	$-0.919550\pi$
$991$	−60.9601	−1.93646	−0.968230	+	0.250060i	$0.919550\pi$
$992$	−29.7439	−0.944370
$993$	13.0625	0.414525
$994$	28.7386	0.911532
$995$	−28.1364	−0.891985
$996$	5.86547	0.185855
$997$	51.1625	1.62033	0.810167	−	0.586199i	$-0.199376\pi$
$997$	51.1625	1.62033	0.810167	+	0.586199i	$0.199376\pi$
$998$	15.6124	0.494200
$999$	0.282175	0.00892762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	381.2.a.c.1.1	✓	5
3.2	odd	2		1143.2.a.h.1.5		5
4.3	odd	2		6096.2.a.be.1.2		5
5.4	even	2		9525.2.a.k.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
381.2.a.c.1.1	✓	5	1.1	even	1	trivial
1143.2.a.h.1.5		5	3.2	odd	2
6096.2.a.be.1.2		5	4.3	odd	2
9525.2.a.k.1.5		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:	\( N \)	\(=\)	\( 381 = 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	381.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26835 q^{2} -1.00000 q^{3} +3.14543 q^{4} -2.63584 q^{5} +2.26835 q^{6} +1.51754 q^{7} -2.59823 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.26835 q^{2} -1.00000 q^{3} +3.14543 q^{4} -2.63584 q^{5} +2.26835 q^{6} +1.51754 q^{7} -2.59823 q^{8} +1.00000 q^{9} +5.97902 q^{10} -0.0104875 q^{11} -3.14543 q^{12} +1.66550 q^{13} -3.44232 q^{14} +2.63584 q^{15} -0.397144 q^{16} +2.10782 q^{17} -2.26835 q^{18} -2.77682 q^{19} -8.29085 q^{20} -1.51754 q^{21} +0.0237894 q^{22} +2.07816 q^{23} +2.59823 q^{24} +1.94767 q^{25} -3.77794 q^{26} -1.00000 q^{27} +4.77331 q^{28} -7.41840 q^{29} -5.97902 q^{30} -4.87819 q^{31} +6.09733 q^{32} +0.0104875 q^{33} -4.78127 q^{34} -4.00000 q^{35} +3.14543 q^{36} -0.282175 q^{37} +6.29881 q^{38} -1.66550 q^{39} +6.84854 q^{40} -8.33967 q^{41} +3.44232 q^{42} -10.8276 q^{43} -0.0329877 q^{44} -2.63584 q^{45} -4.71401 q^{46} +0.613170 q^{47} +0.397144 q^{48} -4.69707 q^{49} -4.41801 q^{50} -2.10782 q^{51} +5.23870 q^{52} -4.98377 q^{53} +2.26835 q^{54} +0.0276435 q^{55} -3.94292 q^{56} +2.77682 q^{57} +16.8276 q^{58} +1.48438 q^{59} +8.29085 q^{60} -3.22612 q^{61} +11.0655 q^{62} +1.51754 q^{63} -13.0366 q^{64} -4.38999 q^{65} -0.0237894 q^{66} +1.18980 q^{67} +6.62998 q^{68} -2.07816 q^{69} +9.07341 q^{70} -8.34861 q^{71} -2.59823 q^{72} +14.7658 q^{73} +0.640073 q^{74} -1.94767 q^{75} -8.73429 q^{76} -0.0159152 q^{77} +3.77794 q^{78} +2.10307 q^{79} +1.04681 q^{80} +1.00000 q^{81} +18.9173 q^{82} -1.86476 q^{83} -4.77331 q^{84} -5.55587 q^{85} +24.5607 q^{86} +7.41840 q^{87} +0.0272490 q^{88} +15.7635 q^{89} +5.97902 q^{90} +2.52746 q^{91} +6.53671 q^{92} +4.87819 q^{93} -1.39089 q^{94} +7.31927 q^{95} -6.09733 q^{96} -11.3911 q^{97} +10.6546 q^{98} -0.0104875 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} - 5 q^{3} + q^{4} - 5 q^{5} + q^{6} - 3 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - q^2 - 5 * q^3 + q^4 - 5 * q^5 + q^6 - 3 * q^8 + 5 * q^9	\( 5 q - q^{2} - 5 q^{3} + q^{4} - 5 q^{5} + q^{6} - 3 q^{8} + 5 q^{9} - 2 q^{10} - 16 q^{11} - q^{12} + 3 q^{13} - 6 q^{14} + 5 q^{15} - 7 q^{16} - 6 q^{17} - q^{18} - 8 q^{19} - 12 q^{20} - 8 q^{22} - 9 q^{23} + 3 q^{24} + 4 q^{25} - 2 q^{26} - 5 q^{27} + 2 q^{28} - 17 q^{29} + 2 q^{30} - 9 q^{31} - 2 q^{32} + 16 q^{33} - q^{34} - 20 q^{35} + q^{36} - q^{37} + q^{38} - 3 q^{39} + 16 q^{40} - 2 q^{41} + 6 q^{42} - 4 q^{43} + 3 q^{44} - 5 q^{45} + 4 q^{46} - 8 q^{47} + 7 q^{48} + 13 q^{49} + 15 q^{50} + 6 q^{51} + 13 q^{52} - 15 q^{53} + q^{54} + 20 q^{55} + 8 q^{56} + 8 q^{57} + 34 q^{58} - 19 q^{59} + 12 q^{60} + q^{61} + 15 q^{62} + q^{64} - 5 q^{65} + 8 q^{66} - 2 q^{67} + 14 q^{68} + 9 q^{69} + 4 q^{70} - 3 q^{72} + 13 q^{73} - 17 q^{74} - 4 q^{75} + 8 q^{76} + 2 q^{77} + 2 q^{78} - 28 q^{79} + 12 q^{80} + 5 q^{81} + 30 q^{82} - q^{83} - 2 q^{84} + 6 q^{85} + 32 q^{86} + 17 q^{87} + 3 q^{88} + q^{89} - 2 q^{90} - 18 q^{91} + 12 q^{92} + 9 q^{93} + 16 q^{94} + 4 q^{95} + 2 q^{96} + 28 q^{97} + 15 q^{98} - 16 q^{99}+O(q^{100}) \) 5 * q - q^2 - 5 * q^3 + q^4 - 5 * q^5 + q^6 - 3 * q^8 + 5 * q^9 - 2 * q^10 - 16 * q^11 - q^12 + 3 * q^13 - 6 * q^14 + 5 * q^15 - 7 * q^16 - 6 * q^17 - q^18 - 8 * q^19 - 12 * q^20 - 8 * q^22 - 9 * q^23 + 3 * q^24 + 4 * q^25 - 2 * q^26 - 5 * q^27 + 2 * q^28 - 17 * q^29 + 2 * q^30 - 9 * q^31 - 2 * q^32 + 16 * q^33 - q^34 - 20 * q^35 + q^36 - q^37 + q^38 - 3 * q^39 + 16 * q^40 - 2 * q^41 + 6 * q^42 - 4 * q^43 + 3 * q^44 - 5 * q^45 + 4 * q^46 - 8 * q^47 + 7 * q^48 + 13 * q^49 + 15 * q^50 + 6 * q^51 + 13 * q^52 - 15 * q^53 + q^54 + 20 * q^55 + 8 * q^56 + 8 * q^57 + 34 * q^58 - 19 * q^59 + 12 * q^60 + q^61 + 15 * q^62 + q^64 - 5 * q^65 + 8 * q^66 - 2 * q^67 + 14 * q^68 + 9 * q^69 + 4 * q^70 - 3 * q^72 + 13 * q^73 - 17 * q^74 - 4 * q^75 + 8 * q^76 + 2 * q^77 + 2 * q^78 - 28 * q^79 + 12 * q^80 + 5 * q^81 + 30 * q^82 - q^83 - 2 * q^84 + 6 * q^85 + 32 * q^86 + 17 * q^87 + 3 * q^88 + q^89 - 2 * q^90 - 18 * q^91 + 12 * q^92 + 9 * q^93 + 16 * q^94 + 4 * q^95 + 2 * q^96 + 28 * q^97 + 15 * q^98 - 16 * q^99

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