LMFDB - Embedded newform 445.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [445,2,Mod(1,445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(445, 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("445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 445 = 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	445.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:	$3.55334288995$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 $ x^7 - x^6 - 8x^5 + 6x^4 + 19x^3 - 10x^2 - 12*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.885013$ of defining polynomial
Character	$\chi$	$=$	445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21675 q^{2} -2.76669 q^{3} +2.91399 q^{4} +1.00000 q^{5} +6.13307 q^{6} -3.75132 q^{7} -2.02609 q^{8} +4.65459 q^{9} +O(q^{10})$	\(q-2.21675 q^{2} -2.76669 q^{3} +2.91399 q^{4} +1.00000 q^{5} +6.13307 q^{6} -3.75132 q^{7} -2.02609 q^{8} +4.65459 q^{9} -2.21675 q^{10} -1.54195 q^{11} -8.06212 q^{12} +4.15218 q^{13} +8.31575 q^{14} -2.76669 q^{15} -1.33664 q^{16} +5.74043 q^{17} -10.3181 q^{18} +7.03169 q^{19} +2.91399 q^{20} +10.3788 q^{21} +3.41813 q^{22} -6.77641 q^{23} +5.60557 q^{24} +1.00000 q^{25} -9.20436 q^{26} -4.57775 q^{27} -10.9313 q^{28} -10.1743 q^{29} +6.13307 q^{30} +0.0578180 q^{31} +7.01518 q^{32} +4.26612 q^{33} -12.7251 q^{34} -3.75132 q^{35} +13.5634 q^{36} -2.18339 q^{37} -15.5875 q^{38} -11.4878 q^{39} -2.02609 q^{40} -8.82042 q^{41} -23.0071 q^{42} -6.08017 q^{43} -4.49324 q^{44} +4.65459 q^{45} +15.0216 q^{46} +4.08434 q^{47} +3.69807 q^{48} +7.07241 q^{49} -2.21675 q^{50} -15.8820 q^{51} +12.0994 q^{52} -3.49748 q^{53} +10.1477 q^{54} -1.54195 q^{55} +7.60052 q^{56} -19.4545 q^{57} +22.5540 q^{58} -9.20908 q^{59} -8.06212 q^{60} +8.32403 q^{61} -0.128168 q^{62} -17.4609 q^{63} -12.8776 q^{64} +4.15218 q^{65} -9.45692 q^{66} +3.93525 q^{67} +16.7276 q^{68} +18.7483 q^{69} +8.31575 q^{70} -5.49996 q^{71} -9.43063 q^{72} -13.0277 q^{73} +4.84003 q^{74} -2.76669 q^{75} +20.4903 q^{76} +5.78437 q^{77} +25.4656 q^{78} +8.83960 q^{79} -1.33664 q^{80} -1.29854 q^{81} +19.5527 q^{82} -9.09443 q^{83} +30.2436 q^{84} +5.74043 q^{85} +13.4782 q^{86} +28.1493 q^{87} +3.12414 q^{88} +1.00000 q^{89} -10.3181 q^{90} -15.5762 q^{91} -19.7464 q^{92} -0.159965 q^{93} -9.05396 q^{94} +7.03169 q^{95} -19.4089 q^{96} -2.70331 q^{97} -15.6778 q^{98} -7.17717 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9}+O(q^{10}) $ 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9} - 4 q^{10} - 10 q^{11} - 11 q^{12} - 7 q^{13} + 3 q^{14} - 8 q^{15} + 10 q^{16} - 13 q^{17} + 4 q^{18} - 7 q^{19} + 8 q^{20} + 16 q^{21} + 2 q^{22} - 13 q^{23} + 4 q^{24} + 7 q^{25} + q^{26} - 23 q^{27} - 21 q^{28} - 4 q^{29} - 2 q^{30} + q^{31} - 13 q^{32} - 6 q^{33} + 10 q^{34} - 16 q^{35} + 20 q^{36} - 5 q^{37} - 40 q^{38} - 13 q^{39} - 12 q^{40} + 5 q^{41} + 30 q^{42} - 31 q^{43} - 21 q^{44} + 11 q^{45} + 16 q^{46} - 14 q^{47} - 7 q^{48} + 19 q^{49} - 4 q^{50} - q^{51} - 13 q^{53} - 17 q^{54} - 10 q^{55} - q^{56} + 21 q^{57} + 17 q^{58} - 14 q^{59} - 11 q^{60} + 3 q^{61} + 26 q^{62} - 54 q^{63} + 14 q^{64} - 7 q^{65} + 36 q^{66} + q^{67} - 35 q^{68} + 31 q^{69} + 3 q^{70} - 8 q^{71} + 53 q^{72} + 9 q^{73} - 35 q^{74} - 8 q^{75} + 40 q^{76} + 42 q^{77} + 46 q^{78} + 9 q^{79} + 10 q^{80} + 35 q^{81} + 29 q^{82} - 42 q^{83} + 55 q^{84} - 13 q^{85} + 35 q^{86} + 6 q^{87} + 30 q^{88} + 7 q^{89} + 4 q^{90} + 31 q^{91} + 19 q^{92} + 24 q^{93} + 37 q^{94} - 7 q^{95} + 44 q^{96} - 7 q^{97} + 9 q^{98} + 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9 - 4 * q^10 - 10 * q^11 - 11 * q^12 - 7 * q^13 + 3 * q^14 - 8 * q^15 + 10 * q^16 - 13 * q^17 + 4 * q^18 - 7 * q^19 + 8 * q^20 + 16 * q^21 + 2 * q^22 - 13 * q^23 + 4 * q^24 + 7 * q^25 + q^26 - 23 * q^27 - 21 * q^28 - 4 * q^29 - 2 * q^30 + q^31 - 13 * q^32 - 6 * q^33 + 10 * q^34 - 16 * q^35 + 20 * q^36 - 5 * q^37 - 40 * q^38 - 13 * q^39 - 12 * q^40 + 5 * q^41 + 30 * q^42 - 31 * q^43 - 21 * q^44 + 11 * q^45 + 16 * q^46 - 14 * q^47 - 7 * q^48 + 19 * q^49 - 4 * q^50 - q^51 - 13 * q^53 - 17 * q^54 - 10 * q^55 - q^56 + 21 * q^57 + 17 * q^58 - 14 * q^59 - 11 * q^60 + 3 * q^61 + 26 * q^62 - 54 * q^63 + 14 * q^64 - 7 * q^65 + 36 * q^66 + q^67 - 35 * q^68 + 31 * q^69 + 3 * q^70 - 8 * q^71 + 53 * q^72 + 9 * q^73 - 35 * q^74 - 8 * q^75 + 40 * q^76 + 42 * q^77 + 46 * q^78 + 9 * q^79 + 10 * q^80 + 35 * q^81 + 29 * q^82 - 42 * q^83 + 55 * q^84 - 13 * q^85 + 35 * q^86 + 6 * q^87 + 30 * q^88 + 7 * q^89 + 4 * q^90 + 31 * q^91 + 19 * q^92 + 24 * q^93 + 37 * q^94 - 7 * q^95 + 44 * q^96 - 7 * q^97 + 9 * q^98 + 5 * 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 $

$2$	−2.21675	−1.56748	−0.783740	−	0.621089i	$-0.786691\pi$
$2$	−2.21675	−1.56748	−0.783740	+	0.621089i	$0.786691\pi$
$3$	−2.76669	−1.59735	−0.798676	−	0.601762i	$-0.794466\pi$
$3$	−2.76669	−1.59735	−0.798676	+	0.601762i	$0.794466\pi$
$4$	2.91399	1.45700
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	6.13307	2.50382
$7$	−3.75132	−1.41787	−0.708933	−	0.705276i	$-0.750823\pi$
$7$	−3.75132	−1.41787	−0.708933	+	0.705276i	$0.750823\pi$
$8$	−2.02609	−0.716332
$9$	4.65459	1.55153
$10$	−2.21675	−0.700999
$11$	−1.54195	−0.464917	−0.232458	−	0.972606i	$-0.574677\pi$
$11$	−1.54195	−0.464917	−0.232458	+	0.972606i	$0.574677\pi$
$12$	−8.06212	−2.32733
$13$	4.15218	1.15161	0.575804	−	0.817588i	$-0.304689\pi$
$13$	4.15218	1.15161	0.575804	+	0.817588i	$0.304689\pi$
$14$	8.31575	2.22248
$15$	−2.76669	−0.714357
$16$	−1.33664	−0.334160
$17$	5.74043	1.39226	0.696129	−	0.717916i	$-0.254904\pi$
$17$	5.74043	1.39226	0.696129	+	0.717916i	$0.254904\pi$
$18$	−10.3181	−2.43199
$19$	7.03169	1.61318	0.806590	−	0.591111i	$-0.201310\pi$
$19$	7.03169	1.61318	0.806590	+	0.591111i	$0.201310\pi$
$20$	2.91399	0.651588
$21$	10.3788	2.26483
$22$	3.41813	0.728748
$23$	−6.77641	−1.41298	−0.706490	−	0.707723i	$-0.749722\pi$
$23$	−6.77641	−1.41298	−0.706490	+	0.707723i	$0.749722\pi$
$24$	5.60557	1.14423
$25$	1.00000	0.200000
$26$	−9.20436	−1.80512
$27$	−4.57775	−0.880989
$28$	−10.9313	−2.06582
$29$	−10.1743	−1.88933	−0.944664	−	0.328041i	$-0.893612\pi$
$29$	−10.1743	−1.88933	−0.944664	+	0.328041i	$0.893612\pi$
$30$	6.13307	1.11974
$31$	0.0578180	0.0103844	0.00519221	−	0.999987i	$-0.498347\pi$
$31$	0.0578180	0.0103844	0.00519221	+	0.999987i	$0.498347\pi$
$32$	7.01518	1.24012
$33$	4.26612	0.742636
$34$	−12.7251	−2.18234
$35$	−3.75132	−0.634089
$36$	13.5634	2.26057
$37$	−2.18339	−0.358947	−0.179473	−	0.983763i	$-0.557439\pi$
$37$	−2.18339	−0.358947	−0.179473	+	0.983763i	$0.557439\pi$
$38$	−15.5875	−2.52863
$39$	−11.4878	−1.83952
$40$	−2.02609	−0.320353
$41$	−8.82042	−1.37752	−0.688759	−	0.724990i	$-0.741844\pi$
$41$	−8.82042	−1.37752	−0.688759	+	0.724990i	$0.741844\pi$
$42$	−23.0071	−3.55008
$43$	−6.08017	−0.927218	−0.463609	−	0.886040i	$-0.653446\pi$
$43$	−6.08017	−0.927218	−0.463609	+	0.886040i	$0.653446\pi$
$44$	−4.49324	−0.677382
$45$	4.65459	0.693866
$46$	15.0216	2.21482
$47$	4.08434	0.595762	0.297881	−	0.954603i	$-0.403720\pi$
$47$	4.08434	0.595762	0.297881	+	0.954603i	$0.403720\pi$
$48$	3.69807	0.533770
$49$	7.07241	1.01034
$50$	−2.21675	−0.313496
$51$	−15.8820	−2.22393
$52$	12.0994	1.67789
$53$	−3.49748	−0.480415	−0.240208	−	0.970722i	$-0.577216\pi$
$53$	−3.49748	−0.480415	−0.240208	+	0.970722i	$0.577216\pi$
$54$	10.1477	1.38093
$55$	−1.54195	−0.207917
$56$	7.60052	1.01566
$57$	−19.4545	−2.57682
$58$	22.5540	2.96148
$59$	−9.20908	−1.19892	−0.599460	−	0.800405i	$-0.704618\pi$
$59$	−9.20908	−1.19892	−0.599460	+	0.800405i	$0.704618\pi$
$60$	−8.06212	−1.04082
$61$	8.32403	1.06578	0.532891	−	0.846184i	$-0.321105\pi$
$61$	8.32403	1.06578	0.532891	+	0.846184i	$0.321105\pi$
$62$	−0.128168	−0.0162774
$63$	−17.4609	−2.19986
$64$	−12.8776	−1.60970
$65$	4.15218	0.515015
$66$	−9.45692	−1.16407
$67$	3.93525	0.480767	0.240384	−	0.970678i	$-0.422727\pi$
$67$	3.93525	0.480767	0.240384	+	0.970678i	$0.422727\pi$
$68$	16.7276	2.02851
$69$	18.7483	2.25702
$70$	8.31575	0.993922
$71$	−5.49996	−0.652726	−0.326363	−	0.945245i	$-0.605823\pi$
$71$	−5.49996	−0.652726	−0.326363	+	0.945245i	$0.605823\pi$
$72$	−9.43063	−1.11141
$73$	−13.0277	−1.52478	−0.762388	−	0.647120i	$-0.775973\pi$
$73$	−13.0277	−1.52478	−0.762388	+	0.647120i	$0.775973\pi$
$74$	4.84003	0.562642
$75$	−2.76669	−0.319470
$76$	20.4903	2.35040
$77$	5.78437	0.659190
$78$	25.4656	2.88342
$79$	8.83960	0.994533	0.497266	−	0.867598i	$-0.334337\pi$
$79$	8.83960	0.994533	0.497266	+	0.867598i	$0.334337\pi$
$80$	−1.33664	−0.149441
$81$	−1.29854	−0.144283
$82$	19.5527	2.15923
$83$	−9.09443	−0.998243	−0.499122	−	0.866532i	$-0.666344\pi$
$83$	−9.09443	−0.998243	−0.499122	+	0.866532i	$0.666344\pi$
$84$	30.2436	3.29985
$85$	5.74043	0.622637
$86$	13.4782	1.45340
$87$	28.1493	3.01792
$88$	3.12414	0.333035
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	−10.3181	−1.08762
$91$	−15.5762	−1.63283
$92$	−19.7464	−2.05870
$93$	−0.159965	−0.0165876
$94$	−9.05396	−0.933845
$95$	7.03169	0.721436
$96$	−19.4089	−1.98091
$97$	−2.70331	−0.274479	−0.137240	−	0.990538i	$-0.543823\pi$
$97$	−2.70331	−0.274479	−0.137240	+	0.990538i	$0.543823\pi$
$98$	−15.6778	−1.58369
$99$	−7.17717	−0.721333
$100$	2.91399	0.291399
$101$	−10.6908	−1.06378	−0.531888	−	0.846815i	$-0.678517\pi$
$101$	−10.6908	−1.06378	−0.531888	+	0.846815i	$0.678517\pi$
$102$	35.2065	3.48596
$103$	13.1321	1.29395	0.646973	−	0.762513i	$-0.276034\pi$
$103$	13.1321	1.29395	0.646973	+	0.762513i	$0.276034\pi$
$104$	−8.41270	−0.824933
$105$	10.3788	1.01286
$106$	7.75304	0.753042
$107$	11.1134	1.07437	0.537187	−	0.843463i	$-0.319487\pi$
$107$	11.1134	1.07437	0.537187	+	0.843463i	$0.319487\pi$
$108$	−13.3395	−1.28360
$109$	−1.18261	−0.113274	−0.0566368	−	0.998395i	$-0.518038\pi$
$109$	−1.18261	−0.113274	−0.0566368	+	0.998395i	$0.518038\pi$
$110$	3.41813	0.325906
$111$	6.04077	0.573364
$112$	5.01416	0.473794
$113$	−7.93908	−0.746846	−0.373423	−	0.927661i	$-0.621816\pi$
$113$	−7.93908	−0.746846	−0.373423	+	0.927661i	$0.621816\pi$
$114$	43.1259	4.03911
$115$	−6.77641	−0.631904
$116$	−29.6479	−2.75274
$117$	19.3267	1.78676
$118$	20.4142	1.87928
$119$	−21.5342	−1.97404
$120$	5.60557	0.511717
$121$	−8.62238	−0.783852
$122$	−18.4523	−1.67059
$123$	24.4034	2.20038
$124$	0.168481	0.0151301
$125$	1.00000	0.0894427
$126$	38.7064	3.44824
$127$	−5.22603	−0.463735	−0.231868	−	0.972747i	$-0.574484\pi$
$127$	−5.22603	−0.463735	−0.231868	+	0.972747i	$0.574484\pi$
$128$	14.5162	1.28306
$129$	16.8220	1.48109
$130$	−9.20436	−0.807276
$131$	−3.70672	−0.323857	−0.161929	−	0.986802i	$-0.551771\pi$
$131$	−3.70672	−0.323857	−0.161929	+	0.986802i	$0.551771\pi$
$132$	12.4314	1.08202
$133$	−26.3781	−2.28727
$134$	−8.72348	−0.753594
$135$	−4.57775	−0.393990
$136$	−11.6306	−0.997319
$137$	−15.4250	−1.31785	−0.658925	−	0.752209i	$-0.728989\pi$
$137$	−15.4250	−1.31785	−0.658925	+	0.752209i	$0.728989\pi$
$138$	−41.5602	−3.53784
$139$	−7.00287	−0.593975	−0.296988	−	0.954881i	$-0.595982\pi$
$139$	−7.00287	−0.593975	−0.296988	+	0.954881i	$0.595982\pi$
$140$	−10.9313	−0.923865
$141$	−11.3001	−0.951641
$142$	12.1921	1.02313
$143$	−6.40248	−0.535402
$144$	−6.22151	−0.518459
$145$	−10.1743	−0.844933
$146$	28.8792	2.39006
$147$	−19.5672	−1.61387
$148$	−6.36237	−0.522984
$149$	12.5398	1.02730	0.513650	−	0.858000i	$-0.328293\pi$
$149$	12.5398	1.02730	0.513650	+	0.858000i	$0.328293\pi$
$150$	6.13307	0.500763
$151$	0.236140	0.0192168	0.00960840	−	0.999954i	$-0.496942\pi$
$151$	0.236140	0.0192168	0.00960840	+	0.999954i	$0.496942\pi$
$152$	−14.2469	−1.15557
$153$	26.7194	2.16013
$154$	−12.8225	−1.03327
$155$	0.0578180	0.00464406
$156$	−33.4754	−2.68018
$157$	−4.98636	−0.397955	−0.198977	−	0.980004i	$-0.563762\pi$
$157$	−4.98636	−0.397955	−0.198977	+	0.980004i	$0.563762\pi$
$158$	−19.5952	−1.55891
$159$	9.67645	0.767392
$160$	7.01518	0.554599
$161$	25.4205	2.00342
$162$	2.87855	0.226160
$163$	−10.9166	−0.855055	−0.427528	−	0.904002i	$-0.640615\pi$
$163$	−10.9166	−0.855055	−0.427528	+	0.904002i	$0.640615\pi$
$164$	−25.7026	−2.00704
$165$	4.26612	0.332117
$166$	20.1601	1.56473
$167$	19.6561	1.52104	0.760518	−	0.649317i	$-0.224945\pi$
$167$	19.6561	1.52104	0.760518	+	0.649317i	$0.224945\pi$
$168$	−21.0283	−1.62237
$169$	4.24062	0.326202
$170$	−12.7251	−0.975971
$171$	32.7297	2.50290
$172$	−17.7176	−1.35095
$173$	−13.1762	−1.00177	−0.500883	−	0.865515i	$-0.666991\pi$
$173$	−13.1762	−1.00177	−0.500883	+	0.865515i	$0.666991\pi$
$174$	−62.4000	−4.73053
$175$	−3.75132	−0.283573
$176$	2.06104	0.155356
$177$	25.4787	1.91510
$178$	−2.21675	−0.166153
$179$	9.74668	0.728501	0.364250	−	0.931301i	$-0.381325\pi$
$179$	9.74668	0.728501	0.364250	+	0.931301i	$0.381325\pi$
$180$	13.5634	1.01096
$181$	6.93270	0.515304	0.257652	−	0.966238i	$-0.417051\pi$
$181$	6.93270	0.515304	0.257652	+	0.966238i	$0.417051\pi$
$182$	34.5285	2.55942
$183$	−23.0300	−1.70243
$184$	13.7296	1.01216
$185$	−2.18339	−0.160526
$186$	0.354602	0.0260007
$187$	−8.85148	−0.647285
$188$	11.9017	0.868022
$189$	17.1726	1.24912
$190$	−15.5875	−1.13084
$191$	4.37217	0.316359	0.158180	−	0.987410i	$-0.449437\pi$
$191$	4.37217	0.316359	0.158180	+	0.987410i	$0.449437\pi$
$192$	35.6285	2.57126
$193$	2.05206	0.147711	0.0738554	−	0.997269i	$-0.476470\pi$
$193$	2.05206	0.147711	0.0738554	+	0.997269i	$0.476470\pi$
$194$	5.99256	0.430241
$195$	−11.4878	−0.822660
$196$	20.6089	1.47207
$197$	−6.48648	−0.462143	−0.231071	−	0.972937i	$-0.574223\pi$
$197$	−6.48648	−0.462143	−0.231071	+	0.972937i	$0.574223\pi$
$198$	15.9100	1.13068
$199$	13.4260	0.951745	0.475873	−	0.879514i	$-0.342132\pi$
$199$	13.4260	0.951745	0.475873	+	0.879514i	$0.342132\pi$
$200$	−2.02609	−0.143266
$201$	−10.8876	−0.767954
$202$	23.6989	1.66745
$203$	38.1672	2.67881
$204$	−46.2800	−3.24025
$205$	−8.82042	−0.616045
$206$	−29.1107	−2.02824
$207$	−31.5414	−2.19228
$208$	−5.54997	−0.384821
$209$	−10.8426	−0.749995
$210$	−23.0071	−1.58764
$211$	−20.1294	−1.38576	−0.692882	−	0.721051i	$-0.743659\pi$
$211$	−20.1294	−1.38576	−0.692882	+	0.721051i	$0.743659\pi$
$212$	−10.1916	−0.699963
$213$	15.2167	1.04263
$214$	−24.6357	−1.68406
$215$	−6.08017	−0.414664
$216$	9.27494	0.631080
$217$	−0.216894	−0.0147237
$218$	2.62155	0.177554
$219$	36.0436	2.43560
$220$	−4.49324	−0.302934
$221$	23.8353	1.60334
$222$	−13.3909	−0.898737
$223$	−8.26176	−0.553248	−0.276624	−	0.960978i	$-0.589216\pi$
$223$	−8.26176	−0.553248	−0.276624	+	0.960978i	$0.589216\pi$
$224$	−26.3162	−1.75832
$225$	4.65459	0.310306
$226$	17.5990	1.17067
$227$	−21.1024	−1.40061	−0.700306	−	0.713842i	$-0.746953\pi$
$227$	−21.1024	−1.40061	−0.700306	+	0.713842i	$0.746953\pi$
$228$	−56.6903	−3.75441
$229$	−13.3006	−0.878930	−0.439465	−	0.898260i	$-0.644832\pi$
$229$	−13.3006	−0.878930	−0.439465	+	0.898260i	$0.644832\pi$
$230$	15.0216	0.990497
$231$	−16.0036	−1.05296
$232$	20.6141	1.35338
$233$	5.42111	0.355148	0.177574	−	0.984107i	$-0.443175\pi$
$233$	5.42111	0.355148	0.177574	+	0.984107i	$0.443175\pi$
$234$	−42.8426	−2.80071
$235$	4.08434	0.266433
$236$	−26.8352	−1.74682
$237$	−24.4565	−1.58862
$238$	47.7360	3.09426
$239$	10.4763	0.677657	0.338828	−	0.940848i	$-0.389969\pi$
$239$	10.4763	0.677657	0.338828	+	0.940848i	$0.389969\pi$
$240$	3.69807	0.238709
$241$	−13.1333	−0.845991	−0.422995	−	0.906132i	$-0.639021\pi$
$241$	−13.1333	−0.845991	−0.422995	+	0.906132i	$0.639021\pi$
$242$	19.1137	1.22867
$243$	17.3259	1.11146
$244$	24.2561	1.55284
$245$	7.07241	0.451839
$246$	−54.0963	−3.44905
$247$	29.1969	1.85775
$248$	−0.117145	−0.00743869
$249$	25.1615	1.59455
$250$	−2.21675	−0.140200
$251$	−27.1469	−1.71350	−0.856748	−	0.515735i	$-0.827519\pi$
$251$	−27.1469	−1.71350	−0.856748	+	0.515735i	$0.827519\pi$
$252$	−50.8808	−3.20519
$253$	10.4489	0.656918
$254$	11.5848	0.726896
$255$	−15.8820	−0.994570
$256$	−6.42349	−0.401468
$257$	−18.1588	−1.13272	−0.566358	−	0.824159i	$-0.691648\pi$
$257$	−18.1588	−1.13272	−0.566358	+	0.824159i	$0.691648\pi$
$258$	−37.2902	−2.32158
$259$	8.19059	0.508938
$260$	12.0994	0.750374
$261$	−47.3574	−2.93135
$262$	8.21687	0.507640
$263$	−23.7946	−1.46724	−0.733619	−	0.679561i	$-0.762170\pi$
$263$	−23.7946	−1.46724	−0.733619	+	0.679561i	$0.762170\pi$
$264$	−8.64354	−0.531973
$265$	−3.49748	−0.214848
$266$	58.4738	3.58526
$267$	−2.76669	−0.169319
$268$	11.4673	0.700476
$269$	0.159138	0.00970280	0.00485140	−	0.999988i	$-0.498456\pi$
$269$	0.159138	0.00970280	0.00485140	+	0.999988i	$0.498456\pi$
$270$	10.1477	0.617572
$271$	−1.15143	−0.0699447	−0.0349723	−	0.999388i	$-0.511134\pi$
$271$	−1.15143	−0.0699447	−0.0349723	+	0.999388i	$0.511134\pi$
$272$	−7.67288	−0.465237
$273$	43.0945	2.60820
$274$	34.1935	2.06570
$275$	−1.54195	−0.0929834
$276$	54.6322	3.28847
$277$	−1.65637	−0.0995216	−0.0497608	−	0.998761i	$-0.515846\pi$
$277$	−1.65637	−0.0995216	−0.0497608	+	0.998761i	$0.515846\pi$
$278$	15.5236	0.931045
$279$	0.269119	0.0161118
$280$	7.60052	0.454218
$281$	4.08763	0.243848	0.121924	−	0.992539i	$-0.461094\pi$
$281$	4.08763	0.243848	0.121924	+	0.992539i	$0.461094\pi$
$282$	25.0495	1.49168
$283$	−14.7662	−0.877757	−0.438879	−	0.898546i	$-0.644624\pi$
$283$	−14.7662	−0.877757	−0.438879	+	0.898546i	$0.644624\pi$
$284$	−16.0268	−0.951018
$285$	−19.4545	−1.15239
$286$	14.1927	0.839232
$287$	33.0882	1.95314
$288$	32.6528	1.92409
$289$	15.9525	0.938384
$290$	22.5540	1.32442
$291$	7.47922	0.438440
$292$	−37.9626	−2.22159
$293$	3.43348	0.200586	0.100293	−	0.994958i	$-0.468022\pi$
$293$	3.43348	0.200586	0.100293	+	0.994958i	$0.468022\pi$
$294$	43.3756	2.52972
$295$	−9.20908	−0.536173
$296$	4.42374	0.257125
$297$	7.05869	0.409586
$298$	−27.7976	−1.61027
$299$	−28.1369	−1.62720
$300$	−8.06212	−0.465467
$301$	22.8087	1.31467
$302$	−0.523464	−0.0301220
$303$	29.5782	1.69922
$304$	−9.39883	−0.539060
$305$	8.32403	0.476633
$306$	−59.2302	−3.38597
$307$	−22.5178	−1.28516	−0.642581	−	0.766218i	$-0.722136\pi$
$307$	−22.5178	−1.28516	−0.642581	+	0.766218i	$0.722136\pi$
$308$	16.8556	0.960436
$309$	−36.3326	−2.06689
$310$	−0.128168	−0.00727947
$311$	11.4828	0.651129	0.325564	−	0.945520i	$-0.394446\pi$
$311$	11.4828	0.651129	0.325564	+	0.945520i	$0.394446\pi$
$312$	23.2754	1.31771
$313$	−2.22350	−0.125679	−0.0628397	−	0.998024i	$-0.520016\pi$
$313$	−2.22350	−0.125679	−0.0628397	+	0.998024i	$0.520016\pi$
$314$	11.0535	0.623786
$315$	−17.4609	−0.983809
$316$	25.7585	1.44903
$317$	32.5253	1.82680	0.913402	−	0.407059i	$-0.133446\pi$
$317$	32.5253	1.82680	0.913402	+	0.407059i	$0.133446\pi$
$318$	−21.4503	−1.20287
$319$	15.6884	0.878380
$320$	−12.8776	−0.719882
$321$	−30.7474	−1.71615
$322$	−56.3509	−3.14031
$323$	40.3649	2.24597
$324$	−3.78394	−0.210219
$325$	4.15218	0.230322
$326$	24.1994	1.34028
$327$	3.27192	0.180938
$328$	17.8710	0.986760
$329$	−15.3217	−0.844710
$330$	−9.45692	−0.520586
$331$	−27.2476	−1.49766	−0.748831	−	0.662761i	$-0.769385\pi$
$331$	−27.2476	−1.49766	−0.748831	+	0.662761i	$0.769385\pi$
$332$	−26.5011	−1.45444
$333$	−10.1628	−0.556917
$334$	−43.5727	−2.38419
$335$	3.93525	0.215006
$336$	−13.8726	−0.756815
$337$	16.5317	0.900536	0.450268	−	0.892893i	$-0.351328\pi$
$337$	16.5317	0.900536	0.450268	+	0.892893i	$0.351328\pi$
$338$	−9.40042	−0.511315
$339$	21.9650	1.19297
$340$	16.7276	0.907179
$341$	−0.0891528	−0.00482789
$342$	−72.5536	−3.92325
$343$	−0.271619	−0.0146661
$344$	12.3190	0.664195
$345$	18.7483	1.00937
$346$	29.2083	1.57025
$347$	−33.2231	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$347$	−33.2231	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$348$	82.0267	4.39709
$349$	−5.87046	−0.314239	−0.157119	−	0.987580i	$-0.550221\pi$
$349$	−5.87046	−0.314239	−0.157119	+	0.987580i	$0.550221\pi$
$350$	8.31575	0.444495
$351$	−19.0077	−1.01455
$352$	−10.8171	−0.576553
$353$	31.2340	1.66242	0.831210	−	0.555958i	$-0.187649\pi$
$353$	31.2340	1.66242	0.831210	+	0.555958i	$0.187649\pi$
$354$	−56.4800	−3.00188
$355$	−5.49996	−0.291908
$356$	2.91399	0.154441
$357$	59.5785	3.15323
$358$	−21.6060	−1.14191
$359$	−4.01656	−0.211986	−0.105993	−	0.994367i	$-0.533802\pi$
$359$	−4.01656	−0.211986	−0.105993	+	0.994367i	$0.533802\pi$
$360$	−9.43063	−0.497038
$361$	30.4447	1.60235
$362$	−15.3681	−0.807729
$363$	23.8555	1.25209
$364$	−45.3888	−2.37902
$365$	−13.0277	−0.681901
$366$	51.0519	2.66852
$367$	13.7718	0.718882	0.359441	−	0.933168i	$-0.382967\pi$
$367$	13.7718	0.718882	0.359441	+	0.933168i	$0.382967\pi$
$368$	9.05761	0.472161
$369$	−41.0555	−2.13726
$370$	4.84003	0.251621
$371$	13.1202	0.681165
$372$	−0.466136	−0.0241680
$373$	−24.2574	−1.25600	−0.628000	−	0.778214i	$-0.716126\pi$
$373$	−24.2574	−1.25600	−0.628000	+	0.778214i	$0.716126\pi$
$374$	19.6215	1.01461
$375$	−2.76669	−0.142871
$376$	−8.27524	−0.426763
$377$	−42.2457	−2.17577
$378$	−38.0674	−1.95798
$379$	29.7093	1.52606	0.763032	−	0.646361i	$-0.223710\pi$
$379$	29.7093	1.52606	0.763032	+	0.646361i	$0.223710\pi$
$380$	20.4903	1.05113
$381$	14.4588	0.740748
$382$	−9.69203	−0.495887
$383$	31.1003	1.58915	0.794575	−	0.607166i	$-0.207694\pi$
$383$	31.1003	1.58915	0.794575	+	0.607166i	$0.207694\pi$
$384$	−40.1618	−2.04950
$385$	5.78437	0.294799
$386$	−4.54892	−0.231534
$387$	−28.3007	−1.43861
$388$	−7.87741	−0.399915
$389$	−2.70571	−0.137185	−0.0685924	−	0.997645i	$-0.521851\pi$
$389$	−2.70571	−0.137185	−0.0685924	+	0.997645i	$0.521851\pi$
$390$	25.4656	1.28950
$391$	−38.8995	−1.96723
$392$	−14.3293	−0.723741
$393$	10.2553	0.517314
$394$	14.3789	0.724400
$395$	8.83960	0.444768
$396$	−20.9142	−1.05098
$397$	−22.9150	−1.15007	−0.575036	−	0.818128i	$-0.695012\pi$
$397$	−22.9150	−1.15007	−0.575036	+	0.818128i	$0.695012\pi$
$398$	−29.7622	−1.49184
$399$	72.9802	3.65358
$400$	−1.33664	−0.0668319
$401$	37.1991	1.85764	0.928818	−	0.370535i	$-0.120826\pi$
$401$	37.1991	1.85764	0.928818	+	0.370535i	$0.120826\pi$
$402$	24.1352	1.20375
$403$	0.240071	0.0119588
$404$	−31.1530	−1.54992
$405$	−1.29854	−0.0645251
$406$	−84.6072	−4.19899
$407$	3.36669	0.166880
$408$	32.1784	1.59307
$409$	25.6275	1.26720	0.633598	−	0.773663i	$-0.281577\pi$
$409$	25.6275	1.26720	0.633598	+	0.773663i	$0.281577\pi$
$410$	19.5527	0.965639
$411$	42.6764	2.10507
$412$	38.2669	1.88527
$413$	34.5462	1.69991
$414$	69.9195	3.43636
$415$	−9.09443	−0.446428
$416$	29.1283	1.42813
$417$	19.3748	0.948787
$418$	24.0353	1.17560
$419$	−28.1678	−1.37609	−0.688043	−	0.725670i	$-0.741530\pi$
$419$	−28.1678	−1.37609	−0.688043	+	0.725670i	$0.741530\pi$
$420$	30.2436	1.47574
$421$	−20.4045	−0.994453	−0.497227	−	0.867621i	$-0.665648\pi$
$421$	−20.4045	−0.994453	−0.497227	+	0.867621i	$0.665648\pi$
$422$	44.6219	2.17216
$423$	19.0109	0.924343
$424$	7.08621	0.344137
$425$	5.74043	0.278452
$426$	−33.7317	−1.63431
$427$	−31.2261	−1.51114
$428$	32.3844	1.56536
$429$	17.7137	0.855225
$430$	13.4782	0.649978
$431$	20.4146	0.983338	0.491669	−	0.870782i	$-0.336387\pi$
$431$	20.4146	0.983338	0.491669	+	0.870782i	$0.336387\pi$
$432$	6.11880	0.294391
$433$	16.1740	0.777275	0.388637	−	0.921391i	$-0.372946\pi$
$433$	16.1740	0.777275	0.388637	+	0.921391i	$0.372946\pi$
$434$	0.480800	0.0230791
$435$	28.1493	1.34965
$436$	−3.44612	−0.165039
$437$	−47.6496	−2.27939
$438$	−79.8998	−3.81776
$439$	9.85761	0.470478	0.235239	−	0.971938i	$-0.424413\pi$
$439$	9.85761	0.470478	0.235239	+	0.971938i	$0.424413\pi$
$440$	3.12414	0.148938
$441$	32.9192	1.56758
$442$	−52.8370	−2.51320
$443$	7.14208	0.339331	0.169665	−	0.985502i	$-0.445731\pi$
$443$	7.14208	0.339331	0.169665	+	0.985502i	$0.445731\pi$
$444$	17.6027	0.835389
$445$	1.00000	0.0474045
$446$	18.3143	0.867206
$447$	−34.6938	−1.64096
$448$	48.3082	2.28235
$449$	−34.4416	−1.62540	−0.812701	−	0.582681i	$-0.802004\pi$
$449$	−34.4416	−1.62540	−0.812701	+	0.582681i	$0.802004\pi$
$450$	−10.3181	−0.486399
$451$	13.6007	0.640432
$452$	−23.1344	−1.08815
$453$	−0.653327	−0.0306960
$454$	46.7787	2.19543
$455$	−15.5762	−0.730222
$456$	39.4167	1.84585
$457$	28.5729	1.33658	0.668292	−	0.743899i	$-0.267026\pi$
$457$	28.5729	1.33658	0.668292	+	0.743899i	$0.267026\pi$
$458$	29.4842	1.37771
$459$	−26.2783	−1.22656
$460$	−19.7464	−0.920681
$461$	24.3926	1.13608	0.568038	−	0.823002i	$-0.307703\pi$
$461$	24.3926	1.13608	0.568038	+	0.823002i	$0.307703\pi$
$462$	35.4760	1.65049
$463$	−9.51283	−0.442099	−0.221049	−	0.975263i	$-0.570948\pi$
$463$	−9.51283	−0.442099	−0.221049	+	0.975263i	$0.570948\pi$
$464$	13.5994	0.631337
$465$	−0.159965	−0.00741819
$466$	−12.0172	−0.556688
$467$	−11.5000	−0.532157	−0.266079	−	0.963951i	$-0.585728\pi$
$467$	−11.5000	−0.532157	−0.266079	+	0.963951i	$0.585728\pi$
$468$	56.3179	2.60330
$469$	−14.7624	−0.681664
$470$	−9.05396	−0.417628
$471$	13.7957	0.635673
$472$	18.6584	0.858824
$473$	9.37535	0.431079
$474$	54.2139	2.49013
$475$	7.03169	0.322636
$476$	−62.7504	−2.87616
$477$	−16.2793	−0.745379
$478$	−23.2234	−1.06221
$479$	−37.9454	−1.73377	−0.866884	−	0.498510i	$-0.833881\pi$
$479$	−37.9454	−1.73377	−0.866884	+	0.498510i	$0.833881\pi$
$480$	−19.4089	−0.885889
$481$	−9.06583	−0.413366
$482$	29.1133	1.32607
$483$	−70.3307	−3.20016
$484$	−25.1255	−1.14207
$485$	−2.70331	−0.122751
$486$	−38.4073	−1.74219
$487$	24.8040	1.12398	0.561988	−	0.827145i	$-0.310037\pi$
$487$	24.8040	1.12398	0.561988	+	0.827145i	$0.310037\pi$
$488$	−16.8652	−0.763454
$489$	30.2029	1.36582
$490$	−15.6778	−0.708250
$491$	−27.1395	−1.22479	−0.612395	−	0.790552i	$-0.709794\pi$
$491$	−27.1395	−1.22479	−0.612395	+	0.790552i	$0.709794\pi$
$492$	71.1113	3.20595
$493$	−58.4051	−2.63043
$494$	−64.7222	−2.91199
$495$	−7.17717	−0.322590
$496$	−0.0772818	−0.00347006
$497$	20.6321	0.925478
$498$	−55.7768	−2.49942
$499$	−33.3081	−1.49108	−0.745538	−	0.666463i	$-0.767808\pi$
$499$	−33.3081	−1.49108	−0.745538	+	0.666463i	$0.767808\pi$
$500$	2.91399	0.130318
$501$	−54.3824	−2.42963
$502$	60.1779	2.68587
$503$	−26.6651	−1.18894	−0.594469	−	0.804119i	$-0.702638\pi$
$503$	−26.6651	−1.18894	−0.594469	+	0.804119i	$0.702638\pi$
$504$	35.3773	1.57583
$505$	−10.6908	−0.475735
$506$	−23.1627	−1.02971
$507$	−11.7325	−0.521059
$508$	−15.2286	−0.675661
$509$	28.5259	1.26439	0.632193	−	0.774811i	$-0.282155\pi$
$509$	28.5259	1.26439	0.632193	+	0.774811i	$0.282155\pi$
$510$	35.2065	1.55897
$511$	48.8710	2.16193
$512$	−14.7931	−0.653767
$513$	−32.1893	−1.42119
$514$	40.2536	1.77551
$515$	13.1321	0.578671
$516$	49.0191	2.15794
$517$	−6.29786	−0.276980
$518$	−18.1565	−0.797751
$519$	36.4545	1.60017
$520$	−8.41270	−0.368921
$521$	−3.47227	−0.152123	−0.0760614	−	0.997103i	$-0.524235\pi$
$521$	−3.47227	−0.152123	−0.0760614	+	0.997103i	$0.524235\pi$
$522$	104.980	4.59483
$523$	−16.5706	−0.724584	−0.362292	−	0.932065i	$-0.618006\pi$
$523$	−16.5706	−0.724584	−0.362292	+	0.932065i	$0.618006\pi$
$524$	−10.8013	−0.471859
$525$	10.3788	0.452966
$526$	52.7467	2.29987
$527$	0.331900	0.0144578
$528$	−5.70226	−0.248159
$529$	22.9197	0.996511
$530$	7.75304	0.336771
$531$	−42.8645	−1.86016
$532$	−76.8656	−3.33255
$533$	−36.6240	−1.58636
$534$	6.13307	0.265404
$535$	11.1134	0.480475
$536$	−7.97318	−0.344389
$537$	−26.9661	−1.16367
$538$	−0.352769	−0.0152090
$539$	−10.9053	−0.469726
$540$	−13.3395	−0.574042
$541$	−29.0138	−1.24740	−0.623700	−	0.781664i	$-0.714371\pi$
$541$	−29.0138	−1.24740	−0.623700	+	0.781664i	$0.714371\pi$
$542$	2.55245	0.109637
$543$	−19.1807	−0.823121
$544$	40.2701	1.72657
$545$	−1.18261	−0.0506575
$546$	−95.5298	−4.08830
$547$	32.6731	1.39700	0.698500	−	0.715610i	$-0.253851\pi$
$547$	32.6731	1.39700	0.698500	+	0.715610i	$0.253851\pi$
$548$	−44.9484	−1.92010
$549$	38.7450	1.65359
$550$	3.41813	0.145750
$551$	−71.5428	−3.04783
$552$	−37.9857	−1.61678
$553$	−33.1602	−1.41011
$554$	3.67176	0.155998
$555$	6.04077	0.256416
$556$	−20.4063	−0.865419
$557$	−10.9818	−0.465315	−0.232658	−	0.972559i	$-0.574742\pi$
$557$	−10.9818	−0.465315	−0.232658	+	0.972559i	$0.574742\pi$
$558$	−0.596571	−0.0252549
$559$	−25.2460	−1.06779
$560$	5.01416	0.211887
$561$	24.4893	1.03394
$562$	−9.06126	−0.382226
$563$	−33.8545	−1.42680	−0.713399	−	0.700758i	$-0.752845\pi$
$563$	−33.8545	−1.42680	−0.713399	+	0.700758i	$0.752845\pi$
$564$	−32.9284	−1.38654
$565$	−7.93908	−0.333999
$566$	32.7329	1.37587
$567$	4.87125	0.204573
$568$	11.1434	0.467568
$569$	9.27784	0.388947	0.194474	−	0.980908i	$-0.437700\pi$
$569$	9.27784	0.388947	0.194474	+	0.980908i	$0.437700\pi$
$570$	43.1259	1.80634
$571$	−16.6644	−0.697383	−0.348692	−	0.937238i	$-0.613374\pi$
$571$	−16.6644	−0.697383	−0.348692	+	0.937238i	$0.613374\pi$
$572$	−18.6568	−0.780078
$573$	−12.0965	−0.505337
$574$	−73.3484	−3.06150
$575$	−6.77641	−0.282596
$576$	−59.9402	−2.49751
$577$	9.60454	0.399842	0.199921	−	0.979812i	$-0.435931\pi$
$577$	9.60454	0.399842	0.199921	+	0.979812i	$0.435931\pi$
$578$	−35.3628	−1.47090
$579$	−5.67743	−0.235946
$580$	−29.6479	−1.23106
$581$	34.1161	1.41538
$582$	−16.5796	−0.687245
$583$	5.39295	0.223353
$584$	26.3953	1.09225
$585$	19.3267	0.799062
$586$	−7.61118	−0.314415
$587$	42.5202	1.75500	0.877498	−	0.479580i	$-0.159211\pi$
$587$	42.5202	1.75500	0.877498	+	0.479580i	$0.159211\pi$
$588$	−57.0186	−2.35141
$589$	0.406559	0.0167520
$590$	20.4142	0.840441
$591$	17.9461	0.738204
$592$	2.91840	0.119946
$593$	−16.5448	−0.679412	−0.339706	−	0.940532i	$-0.610328\pi$
$593$	−16.5448	−0.679412	−0.339706	+	0.940532i	$0.610328\pi$
$594$	−15.6474	−0.642019
$595$	−21.5342	−0.882816
$596$	36.5408	1.49677
$597$	−37.1457	−1.52027
$598$	62.3725	2.55060
$599$	3.61576	0.147736	0.0738680	−	0.997268i	$-0.476466\pi$
$599$	3.61576	0.147736	0.0738680	+	0.997268i	$0.476466\pi$
$600$	5.60557	0.228847
$601$	−6.04796	−0.246701	−0.123351	−	0.992363i	$-0.539364\pi$
$601$	−6.04796	−0.246701	−0.123351	+	0.992363i	$0.539364\pi$
$602$	−50.5612	−2.06072
$603$	18.3170	0.745925
$604$	0.688110	0.0279988
$605$	−8.62238	−0.350549
$606$	−65.5676	−2.66350
$607$	−13.4200	−0.544701	−0.272350	−	0.962198i	$-0.587801\pi$
$607$	−13.4200	−0.544701	−0.272350	+	0.962198i	$0.587801\pi$
$608$	49.3286	2.00054
$609$	−105.597	−4.27900
$610$	−18.4523	−0.747112
$611$	16.9589	0.686084
$612$	77.8600	3.14730
$613$	11.6728	0.471461	0.235731	−	0.971818i	$-0.424252\pi$
$613$	11.6728	0.471461	0.235731	+	0.971818i	$0.424252\pi$
$614$	49.9165	2.01447
$615$	24.4034	0.984040
$616$	−11.7197	−0.472198
$617$	−28.0621	−1.12974	−0.564869	−	0.825180i	$-0.691073\pi$
$617$	−28.0621	−1.12974	−0.564869	+	0.825180i	$0.691073\pi$
$618$	80.5403	3.23981
$619$	20.9899	0.843655	0.421827	−	0.906676i	$-0.361389\pi$
$619$	20.9899	0.843655	0.421827	+	0.906676i	$0.361389\pi$
$620$	0.168481	0.00676637
$621$	31.0207	1.24482
$622$	−25.4545	−1.02063
$623$	−3.75132	−0.150293
$624$	15.3551	0.614695
$625$	1.00000	0.0400000
$626$	4.92894	0.197000
$627$	29.9980	1.19801
$628$	−14.5302	−0.579818
$629$	−12.5336	−0.499747
$630$	38.7064	1.54210
$631$	29.7837	1.18567	0.592836	−	0.805323i	$-0.298008\pi$
$631$	29.7837	1.18567	0.592836	+	0.805323i	$0.298008\pi$
$632$	−17.9098	−0.712415
$633$	55.6919	2.21355
$634$	−72.1006	−2.86348
$635$	−5.22603	−0.207389
$636$	28.1971	1.11809
$637$	29.3659	1.16352
$638$	−34.7772	−1.37684
$639$	−25.6001	−1.01272
$640$	14.5162	0.573802
$641$	18.9560	0.748715	0.374358	−	0.927284i	$-0.377863\pi$
$641$	18.9560	0.748715	0.374358	+	0.927284i	$0.377863\pi$
$642$	68.1594	2.69004
$643$	48.3147	1.90535	0.952673	−	0.303996i	$-0.0983211\pi$
$643$	48.3147	1.90535	0.952673	+	0.303996i	$0.0983211\pi$
$644$	74.0751	2.91897
$645$	16.8220	0.662365
$646$	−89.4791	−3.52051
$647$	−10.0752	−0.396096	−0.198048	−	0.980192i	$-0.563460\pi$
$647$	−10.0752	−0.396096	−0.198048	+	0.980192i	$0.563460\pi$
$648$	2.63097	0.103354
$649$	14.2000	0.557398
$650$	−9.20436	−0.361025
$651$	0.600079	0.0235190
$652$	−31.8109	−1.24581
$653$	10.2900	0.402678	0.201339	−	0.979522i	$-0.435471\pi$
$653$	10.2900	0.402678	0.201339	+	0.979522i	$0.435471\pi$
$654$	−7.25304	−0.283616
$655$	−3.70672	−0.144833
$656$	11.7897	0.460311
$657$	−60.6386	−2.36574
$658$	33.9643	1.32407
$659$	32.9627	1.28405	0.642023	−	0.766686i	$-0.278095\pi$
$659$	32.9627	1.28405	0.642023	+	0.766686i	$0.278095\pi$
$660$	12.4314	0.483893
$661$	40.9469	1.59265	0.796325	−	0.604869i	$-0.206775\pi$
$661$	40.9469	1.59265	0.796325	+	0.604869i	$0.206775\pi$
$662$	60.4011	2.34756
$663$	−65.9450	−2.56109
$664$	18.4261	0.715073
$665$	−26.3781	−1.02290
$666$	22.5284	0.872957
$667$	68.9455	2.66958
$668$	57.2777	2.21614
$669$	22.8578	0.883732
$670$	−8.72348	−0.337017
$671$	−12.8353	−0.495500
$672$	72.8088	2.80866
$673$	−19.3257	−0.744952	−0.372476	−	0.928042i	$-0.621491\pi$
$673$	−19.3257	−0.744952	−0.372476	+	0.928042i	$0.621491\pi$
$674$	−36.6466	−1.41157
$675$	−4.57775	−0.176198
$676$	12.3571	0.475275
$677$	13.6476	0.524518	0.262259	−	0.964998i	$-0.415533\pi$
$677$	13.6476	0.524518	0.262259	+	0.964998i	$0.415533\pi$
$678$	−48.6909	−1.86996
$679$	10.1410	0.389175
$680$	−11.6306	−0.446015
$681$	58.3838	2.23727
$682$	0.197630	0.00756763
$683$	−34.1585	−1.30704	−0.653520	−	0.756909i	$-0.726709\pi$
$683$	−34.1585	−1.30704	−0.653520	+	0.756909i	$0.726709\pi$
$684$	95.3739	3.64671
$685$	−15.4250	−0.589360
$686$	0.602113	0.0229888
$687$	36.7987	1.40396
$688$	8.12700	0.309839
$689$	−14.5222	−0.553250
$690$	−41.5602	−1.58217
$691$	−4.07591	−0.155055	−0.0775275	−	0.996990i	$-0.524703\pi$
$691$	−4.07591	−0.155055	−0.0775275	+	0.996990i	$0.524703\pi$
$692$	−38.3953	−1.45957
$693$	26.9239	1.02275
$694$	73.6475	2.79562
$695$	−7.00287	−0.265634
$696$	−57.0330	−2.16183
$697$	−50.6330	−1.91786
$698$	13.0134	0.492563
$699$	−14.9985	−0.567297
$700$	−10.9313	−0.413165
$701$	−4.59497	−0.173550	−0.0867749	−	0.996228i	$-0.527656\pi$
$701$	−4.59497	−0.173550	−0.0867749	+	0.996228i	$0.527656\pi$
$702$	42.1353	1.59029
$703$	−15.3529	−0.579046
$704$	19.8567	0.748379
$705$	−11.3001	−0.425587
$706$	−69.2381	−2.60581
$707$	40.1047	1.50829
$708$	74.2447	2.79029
$709$	−1.21881	−0.0457735	−0.0228867	−	0.999738i	$-0.507286\pi$
$709$	−1.21881	−0.0457735	−0.0228867	+	0.999738i	$0.507286\pi$
$710$	12.1921	0.457560
$711$	41.1447	1.54305
$712$	−2.02609	−0.0759310
$713$	−0.391799	−0.0146730
$714$	−132.071	−4.94263
$715$	−6.40248	−0.239439
$716$	28.4017	1.06142
$717$	−28.9848	−1.08246
$718$	8.90373	0.332284
$719$	19.6350	0.732263	0.366131	−	0.930563i	$-0.380682\pi$
$719$	19.6350	0.732263	0.366131	+	0.930563i	$0.380682\pi$
$720$	−6.22151	−0.231862
$721$	−49.2628	−1.83464
$722$	−67.4884	−2.51166
$723$	36.3358	1.35134
$724$	20.2018	0.750795
$725$	−10.1743	−0.377865
$726$	−52.8817	−1.96262
$727$	21.2231	0.787121	0.393561	−	0.919299i	$-0.371243\pi$
$727$	21.2231	0.787121	0.393561	+	0.919299i	$0.371243\pi$
$728$	31.5587	1.16964
$729$	−44.0399	−1.63111
$730$	28.8792	1.06887
$731$	−34.9028	−1.29093
$732$	−67.1093	−2.48043
$733$	18.4194	0.680338	0.340169	−	0.940364i	$-0.389516\pi$
$733$	18.4194	0.680338	0.340169	+	0.940364i	$0.389516\pi$
$734$	−30.5287	−1.12683
$735$	−19.5672	−0.721746
$736$	−47.5377	−1.75226
$737$	−6.06798	−0.223517
$738$	91.0098	3.35012
$739$	−9.08147	−0.334067	−0.167034	−	0.985951i	$-0.553419\pi$
$739$	−9.08147	−0.334067	−0.167034	+	0.985951i	$0.553419\pi$
$740$	−6.36237	−0.233885
$741$	−80.7788	−2.96748
$742$	−29.0841	−1.06771
$743$	−0.0809652	−0.00297033	−0.00148516	−	0.999999i	$-0.500473\pi$
$743$	−0.0809652	−0.00297033	−0.00148516	+	0.999999i	$0.500473\pi$
$744$	0.324103	0.0118822
$745$	12.5398	0.459422
$746$	53.7726	1.96875
$747$	−42.3309	−1.54881
$748$	−25.7931	−0.943091
$749$	−41.6900	−1.52332
$750$	6.13307	0.223948
$751$	−6.49039	−0.236838	−0.118419	−	0.992964i	$-0.537783\pi$
$751$	−6.49039	−0.236838	−0.118419	+	0.992964i	$0.537783\pi$
$752$	−5.45928	−0.199080
$753$	75.1071	2.73706
$754$	93.6483	3.41047
$755$	0.236140	0.00859402
$756$	50.0408	1.81997
$757$	−28.1185	−1.02199	−0.510993	−	0.859585i	$-0.670722\pi$
$757$	−28.1185	−1.02199	−0.510993	+	0.859585i	$0.670722\pi$
$758$	−65.8582	−2.39208
$759$	−28.9090	−1.04933
$760$	−14.2469	−0.516788
$761$	2.54060	0.0920966	0.0460483	−	0.998939i	$-0.485337\pi$
$761$	2.54060	0.0920966	0.0460483	+	0.998939i	$0.485337\pi$
$762$	−32.0517	−1.16111
$763$	4.43635	0.160607
$764$	12.7405	0.460934
$765$	26.7194	0.966041
$766$	−68.9416	−2.49096
$767$	−38.2378	−1.38069
$768$	17.7718	0.641286
$769$	0.0116365	0.000419624 0	0.000209812	−	1.00000i	$-0.499933\pi$
$769$	0.0116365	0.000419624 0	0.000209812		1.00000i	$0.499933\pi$
$770$	−12.8225	−0.462091
$771$	50.2399	1.80935
$772$	5.97970	0.215214
$773$	12.9008	0.464010	0.232005	−	0.972715i	$-0.425471\pi$
$773$	12.9008	0.464010	0.232005	+	0.972715i	$0.425471\pi$
$774$	62.7357	2.25499
$775$	0.0578180	0.00207688
$776$	5.47714	0.196618
$777$	−22.6608	−0.812953
$778$	5.99788	0.215035
$779$	−62.0225	−2.22219
$780$	−33.4754	−1.19861
$781$	8.48070	0.303463
$782$	86.2306	3.08360
$783$	46.5756	1.66448
$784$	−9.45325	−0.337616
$785$	−4.98636	−0.177971
$786$	−22.7336	−0.810880
$787$	55.1951	1.96749	0.983746	−	0.179566i	$-0.0574694\pi$
$787$	55.1951	1.96749	0.983746	+	0.179566i	$0.0574694\pi$
$788$	−18.9016	−0.673340
$789$	65.8323	2.34369
$790$	−19.5952	−0.697166
$791$	29.7820	1.05893
$792$	14.5416	0.516713
$793$	34.5629	1.22736
$794$	50.7970	1.80272
$795$	9.67645	0.343188
$796$	39.1233	1.38669
$797$	31.5728	1.11837	0.559183	−	0.829044i	$-0.311115\pi$
$797$	31.5728	1.11837	0.559183	+	0.829044i	$0.311115\pi$
$798$	−161.779	−5.72692
$799$	23.4458	0.829455
$800$	7.01518	0.248024
$801$	4.65459	0.164462
$802$	−82.4613	−2.91181
$803$	20.0881	0.708894
$804$	−31.7265	−1.11891
$805$	25.4205	0.895955
$806$	−0.532178	−0.0187452
$807$	−0.440285	−0.0154988
$808$	21.6606	0.762017
$809$	37.2161	1.30845	0.654225	−	0.756300i	$-0.272995\pi$
$809$	37.2161	1.30845	0.654225	+	0.756300i	$0.272995\pi$
$810$	2.87855	0.101142
$811$	45.2306	1.58826	0.794131	−	0.607746i	$-0.207926\pi$
$811$	45.2306	1.58826	0.794131	+	0.607746i	$0.207926\pi$
$812$	111.219	3.90302
$813$	3.18567	0.111726
$814$	−7.46311	−0.261582
$815$	−10.9166	−0.382392
$816$	21.2285	0.743147
$817$	−42.7539	−1.49577
$818$	−56.8097	−1.98630
$819$	−72.5007	−2.53338
$820$	−25.7026	−0.897575
$821$	19.6062	0.684261	0.342131	−	0.939652i	$-0.388851\pi$
$821$	19.6062	0.684261	0.342131	+	0.939652i	$0.388851\pi$
$822$	−94.6029	−3.29966
$823$	−19.4397	−0.677625	−0.338812	−	0.940854i	$-0.610025\pi$
$823$	−19.4397	−0.677625	−0.338812	+	0.940854i	$0.610025\pi$
$824$	−26.6069	−0.926895
$825$	4.26612	0.148527
$826$	−76.5804	−2.66457
$827$	−15.8859	−0.552407	−0.276203	−	0.961099i	$-0.589076\pi$
$827$	−15.8859	−0.552407	−0.276203	+	0.961099i	$0.589076\pi$
$828$	−91.9115	−3.19414
$829$	9.64772	0.335079	0.167539	−	0.985865i	$-0.446418\pi$
$829$	9.64772	0.335079	0.167539	+	0.985865i	$0.446418\pi$
$830$	20.1601	0.699767
$831$	4.58267	0.158971
$832$	−53.4703	−1.85375
$833$	40.5987	1.40666
$834$	−42.9491	−1.48721
$835$	19.6561	0.680228
$836$	−31.5951	−1.09274
$837$	−0.264677	−0.00914856
$838$	62.4410	2.15699
$839$	38.4916	1.32888	0.664439	−	0.747342i	$-0.268670\pi$
$839$	38.4916	1.32888	0.664439	+	0.747342i	$0.268670\pi$
$840$	−21.0283	−0.725545
$841$	74.5172	2.56956
$842$	45.2317	1.55879
$843$	−11.3092	−0.389510
$844$	−58.6569	−2.01905
$845$	4.24062	0.145882
$846$	−42.1425	−1.44889
$847$	32.3453	1.11140
$848$	4.67486	0.160535
$849$	40.8534	1.40209
$850$	−12.7251	−0.436468
$851$	14.7955	0.507184
$852$	44.3414	1.51911
$853$	7.63504	0.261419	0.130709	−	0.991421i	$-0.458275\pi$
$853$	7.63504	0.261419	0.130709	+	0.991421i	$0.458275\pi$
$854$	69.2205	2.36868
$855$	32.7297	1.11933
$856$	−22.5168	−0.769609
$857$	−4.22874	−0.144451	−0.0722255	−	0.997388i	$-0.523010\pi$
$857$	−4.22874	−0.144451	−0.0722255	+	0.997388i	$0.523010\pi$
$858$	−39.2669	−1.34055
$859$	−44.5369	−1.51958	−0.759789	−	0.650170i	$-0.774698\pi$
$859$	−44.5369	−1.51958	−0.759789	+	0.650170i	$0.774698\pi$
$860$	−17.7176	−0.604164
$861$	−91.5450	−3.11985
$862$	−45.2542	−1.54136
$863$	3.64429	0.124053	0.0620266	−	0.998074i	$-0.480244\pi$
$863$	3.64429	0.124053	0.0620266	+	0.998074i	$0.480244\pi$
$864$	−32.1138	−1.09253
$865$	−13.1762	−0.448004
$866$	−35.8538	−1.21836
$867$	−44.1358	−1.49893
$868$	−0.632027	−0.0214524
$869$	−13.6303	−0.462375
$870$	−62.4000	−2.11556
$871$	16.3399	0.553656
$872$	2.39608	0.0811414
$873$	−12.5828	−0.425863
$874$	105.627	3.57290
$875$	−3.75132	−0.126818
$876$	105.031	3.54866
$877$	−40.7502	−1.37604	−0.688018	−	0.725694i	$-0.741519\pi$
$877$	−40.7502	−1.37604	−0.688018	+	0.725694i	$0.741519\pi$
$878$	−21.8519	−0.737465
$879$	−9.49939	−0.320407
$880$	2.06104	0.0694775
$881$	−11.4553	−0.385940	−0.192970	−	0.981205i	$-0.561812\pi$
$881$	−11.4553	−0.385940	−0.192970	+	0.981205i	$0.561812\pi$
$882$	−72.9737	−2.45715
$883$	−5.39868	−0.181680	−0.0908401	−	0.995865i	$-0.528955\pi$
$883$	−5.39868	−0.181680	−0.0908401	+	0.995865i	$0.528955\pi$
$884$	69.4559	2.33605
$885$	25.4787	0.856457
$886$	−15.8322	−0.531894
$887$	−25.7317	−0.863987	−0.431994	−	0.901877i	$-0.642190\pi$
$887$	−25.7317	−0.863987	−0.431994	+	0.901877i	$0.642190\pi$
$888$	−12.2391	−0.410719
$889$	19.6045	0.657515
$890$	−2.21675	−0.0743057
$891$	2.00230	0.0670794
$892$	−24.0747	−0.806080
$893$	28.7198	0.961071
$894$	76.9075	2.57217
$895$	9.74668	0.325796
$896$	−54.4548	−1.81921
$897$	77.8462	2.59921
$898$	76.3486	2.54779
$899$	−0.588260	−0.0196196
$900$	13.5634	0.452115
$901$	−20.0770	−0.668862
$902$	−30.1494	−1.00386
$903$	−63.1046	−2.09999
$904$	16.0853	0.534989
$905$	6.93270	0.230451
$906$	1.44826	0.0481154
$907$	6.59646	0.219032	0.109516	−	0.993985i	$-0.465070\pi$
$907$	6.59646	0.219032	0.109516	+	0.993985i	$0.465070\pi$
$908$	−61.4921	−2.04069
$909$	−49.7614	−1.65048
$910$	34.5285	1.14461
$911$	22.1874	0.735102	0.367551	−	0.930003i	$-0.380196\pi$
$911$	22.1874	0.735102	0.367551	+	0.930003i	$0.380196\pi$
$912$	26.0037	0.861068
$913$	14.0232	0.464100
$914$	−63.3390	−2.09507
$915$	−23.0300	−0.761350
$916$	−38.7579	−1.28060
$917$	13.9051	0.459186
$918$	58.2524	1.92262
$919$	15.0284	0.495742	0.247871	−	0.968793i	$-0.420269\pi$
$919$	15.0284	0.495742	0.247871	+	0.968793i	$0.420269\pi$
$920$	13.7296	0.452652
$921$	62.3000	2.05285
$922$	−54.0723	−1.78078
$923$	−22.8369	−0.751684
$924$	−46.6343	−1.53415
$925$	−2.18339	−0.0717894
$926$	21.0876	0.692981
$927$	61.1247	2.00760
$928$	−71.3748	−2.34299
$929$	−21.3719	−0.701189	−0.350594	−	0.936527i	$-0.614020\pi$
$929$	−21.3719	−0.701189	−0.350594	+	0.936527i	$0.614020\pi$
$930$	0.354602	0.0116279
$931$	49.7310	1.62987
$932$	15.7971	0.517450
$933$	−31.7693	−1.04008
$934$	25.4927	0.834146
$935$	−8.85148	−0.289474
$936$	−39.1577	−1.27991
$937$	−9.16254	−0.299327	−0.149664	−	0.988737i	$-0.547819\pi$
$937$	−9.16254	−0.299327	−0.149664	+	0.988737i	$0.547819\pi$
$938$	32.7246	1.06849
$939$	6.15173	0.200754
$940$	11.9017	0.388191
$941$	−49.0960	−1.60048	−0.800241	−	0.599678i	$-0.795295\pi$
$941$	−49.0960	−1.60048	−0.800241	+	0.599678i	$0.795295\pi$
$942$	−30.5817	−0.996406
$943$	59.7708	1.94641
$944$	12.3092	0.400631
$945$	17.1726	0.558625
$946$	−20.7828	−0.675708
$947$	15.2650	0.496045	0.248023	−	0.968754i	$-0.420219\pi$
$947$	15.2650	0.496045	0.248023	+	0.968754i	$0.420219\pi$
$948$	−71.2659	−2.31461
$949$	−54.0934	−1.75594
$950$	−15.5875	−0.505726
$951$	−89.9876	−2.91805
$952$	43.6302	1.41406
$953$	−4.11784	−0.133390	−0.0666949	−	0.997773i	$-0.521245\pi$
$953$	−4.11784	−0.133390	−0.0666949	+	0.997773i	$0.521245\pi$
$954$	36.0872	1.16837
$955$	4.37217	0.141480
$956$	30.5279	0.987343
$957$	−43.4049	−1.40308
$958$	84.1155	2.71765
$959$	57.8643	1.86853
$960$	35.6285	1.14990
$961$	−30.9967	−0.999892
$962$	20.0967	0.647943
$963$	51.7284	1.66693
$964$	−38.2703	−1.23260
$965$	2.05206	0.0660583
$966$	155.906	5.01619
$967$	−34.1373	−1.09778	−0.548891	−	0.835894i	$-0.684950\pi$
$967$	−34.1373	−1.09778	−0.548891	+	0.835894i	$0.684950\pi$
$968$	17.4697	0.561498
$969$	−111.677	−3.58760
$970$	5.99256	0.192409
$971$	−16.7233	−0.536675	−0.268338	−	0.963325i	$-0.586474\pi$
$971$	−16.7233	−0.536675	−0.268338	+	0.963325i	$0.586474\pi$
$972$	50.4876	1.61939
$973$	26.2700	0.842177
$974$	−54.9843	−1.76181
$975$	−11.4878	−0.367905
$976$	−11.1262	−0.356142
$977$	39.4703	1.26277	0.631384	−	0.775470i	$-0.282487\pi$
$977$	39.4703	1.26277	0.631384	+	0.775470i	$0.282487\pi$
$978$	−66.9524	−2.14090
$979$	−1.54195	−0.0492811
$980$	20.6089	0.658328
$981$	−5.50457	−0.175747
$982$	60.1616	1.91983
$983$	−24.8433	−0.792380	−0.396190	−	0.918169i	$-0.629668\pi$
$983$	−24.8433	−0.792380	−0.396190	+	0.918169i	$0.629668\pi$
$984$	−49.4435	−1.57620
$985$	−6.48648	−0.206677
$986$	129.470	4.12315
$987$	42.3903	1.34930
$988$	85.0794	2.70674
$989$	41.2018	1.31014
$990$	15.9100	0.505653
$991$	−0.276840	−0.00879411	−0.00439706	−	0.999990i	$-0.501400\pi$
$991$	−0.276840	−0.00879411	−0.00439706	+	0.999990i	$0.501400\pi$
$992$	0.405604	0.0128779
$993$	75.3857	2.39229
$994$	−45.7363	−1.45067
$995$	13.4260	0.425633
$996$	73.3204	2.32324
$997$	−61.1434	−1.93643	−0.968216	−	0.250117i	$-0.919531\pi$
$997$	−61.1434	−1.93643	−0.968216	+	0.250117i	$0.919531\pi$
$998$	73.8359	2.33723
$999$	9.99501	0.316228

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	445.2.a.f.1.2	✓	7
3.2	odd	2		4005.2.a.o.1.6		7
4.3	odd	2		7120.2.a.bj.1.5		7
5.4	even	2		2225.2.a.k.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.2	✓	7	1.1	even	1	trivial
2225.2.a.k.1.6		7	5.4	even	2
4005.2.a.o.1.6		7	3.2	odd	2
7120.2.a.bj.1.5		7	4.3	odd	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:	\( N \)	\(=\)	\( 445 = 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.21675 q^{2} -2.76669 q^{3} +2.91399 q^{4} +1.00000 q^{5} +6.13307 q^{6} -3.75132 q^{7} -2.02609 q^{8} +4.65459 q^{9} +O(q^{10})\)	\(q-2.21675 q^{2} -2.76669 q^{3} +2.91399 q^{4} +1.00000 q^{5} +6.13307 q^{6} -3.75132 q^{7} -2.02609 q^{8} +4.65459 q^{9} -2.21675 q^{10} -1.54195 q^{11} -8.06212 q^{12} +4.15218 q^{13} +8.31575 q^{14} -2.76669 q^{15} -1.33664 q^{16} +5.74043 q^{17} -10.3181 q^{18} +7.03169 q^{19} +2.91399 q^{20} +10.3788 q^{21} +3.41813 q^{22} -6.77641 q^{23} +5.60557 q^{24} +1.00000 q^{25} -9.20436 q^{26} -4.57775 q^{27} -10.9313 q^{28} -10.1743 q^{29} +6.13307 q^{30} +0.0578180 q^{31} +7.01518 q^{32} +4.26612 q^{33} -12.7251 q^{34} -3.75132 q^{35} +13.5634 q^{36} -2.18339 q^{37} -15.5875 q^{38} -11.4878 q^{39} -2.02609 q^{40} -8.82042 q^{41} -23.0071 q^{42} -6.08017 q^{43} -4.49324 q^{44} +4.65459 q^{45} +15.0216 q^{46} +4.08434 q^{47} +3.69807 q^{48} +7.07241 q^{49} -2.21675 q^{50} -15.8820 q^{51} +12.0994 q^{52} -3.49748 q^{53} +10.1477 q^{54} -1.54195 q^{55} +7.60052 q^{56} -19.4545 q^{57} +22.5540 q^{58} -9.20908 q^{59} -8.06212 q^{60} +8.32403 q^{61} -0.128168 q^{62} -17.4609 q^{63} -12.8776 q^{64} +4.15218 q^{65} -9.45692 q^{66} +3.93525 q^{67} +16.7276 q^{68} +18.7483 q^{69} +8.31575 q^{70} -5.49996 q^{71} -9.43063 q^{72} -13.0277 q^{73} +4.84003 q^{74} -2.76669 q^{75} +20.4903 q^{76} +5.78437 q^{77} +25.4656 q^{78} +8.83960 q^{79} -1.33664 q^{80} -1.29854 q^{81} +19.5527 q^{82} -9.09443 q^{83} +30.2436 q^{84} +5.74043 q^{85} +13.4782 q^{86} +28.1493 q^{87} +3.12414 q^{88} +1.00000 q^{89} -10.3181 q^{90} -15.5762 q^{91} -19.7464 q^{92} -0.159965 q^{93} -9.05396 q^{94} +7.03169 q^{95} -19.4089 q^{96} -2.70331 q^{97} -15.6778 q^{98} -7.17717 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9}+O(q^{10}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9	\( 7 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 7 q^{5} - 2 q^{6} - 16 q^{7} - 12 q^{8} + 11 q^{9} - 4 q^{10} - 10 q^{11} - 11 q^{12} - 7 q^{13} + 3 q^{14} - 8 q^{15} + 10 q^{16} - 13 q^{17} + 4 q^{18} - 7 q^{19} + 8 q^{20} + 16 q^{21} + 2 q^{22} - 13 q^{23} + 4 q^{24} + 7 q^{25} + q^{26} - 23 q^{27} - 21 q^{28} - 4 q^{29} - 2 q^{30} + q^{31} - 13 q^{32} - 6 q^{33} + 10 q^{34} - 16 q^{35} + 20 q^{36} - 5 q^{37} - 40 q^{38} - 13 q^{39} - 12 q^{40} + 5 q^{41} + 30 q^{42} - 31 q^{43} - 21 q^{44} + 11 q^{45} + 16 q^{46} - 14 q^{47} - 7 q^{48} + 19 q^{49} - 4 q^{50} - q^{51} - 13 q^{53} - 17 q^{54} - 10 q^{55} - q^{56} + 21 q^{57} + 17 q^{58} - 14 q^{59} - 11 q^{60} + 3 q^{61} + 26 q^{62} - 54 q^{63} + 14 q^{64} - 7 q^{65} + 36 q^{66} + q^{67} - 35 q^{68} + 31 q^{69} + 3 q^{70} - 8 q^{71} + 53 q^{72} + 9 q^{73} - 35 q^{74} - 8 q^{75} + 40 q^{76} + 42 q^{77} + 46 q^{78} + 9 q^{79} + 10 q^{80} + 35 q^{81} + 29 q^{82} - 42 q^{83} + 55 q^{84} - 13 q^{85} + 35 q^{86} + 6 q^{87} + 30 q^{88} + 7 q^{89} + 4 q^{90} + 31 q^{91} + 19 q^{92} + 24 q^{93} + 37 q^{94} - 7 q^{95} + 44 q^{96} - 7 q^{97} + 9 q^{98} + 5 q^{99}+O(q^{100}) \) 7 * q - 4 * q^2 - 8 * q^3 + 8 * q^4 + 7 * q^5 - 2 * q^6 - 16 * q^7 - 12 * q^8 + 11 * q^9 - 4 * q^10 - 10 * q^11 - 11 * q^12 - 7 * q^13 + 3 * q^14 - 8 * q^15 + 10 * q^16 - 13 * q^17 + 4 * q^18 - 7 * q^19 + 8 * q^20 + 16 * q^21 + 2 * q^22 - 13 * q^23 + 4 * q^24 + 7 * q^25 + q^26 - 23 * q^27 - 21 * q^28 - 4 * q^29 - 2 * q^30 + q^31 - 13 * q^32 - 6 * q^33 + 10 * q^34 - 16 * q^35 + 20 * q^36 - 5 * q^37 - 40 * q^38 - 13 * q^39 - 12 * q^40 + 5 * q^41 + 30 * q^42 - 31 * q^43 - 21 * q^44 + 11 * q^45 + 16 * q^46 - 14 * q^47 - 7 * q^48 + 19 * q^49 - 4 * q^50 - q^51 - 13 * q^53 - 17 * q^54 - 10 * q^55 - q^56 + 21 * q^57 + 17 * q^58 - 14 * q^59 - 11 * q^60 + 3 * q^61 + 26 * q^62 - 54 * q^63 + 14 * q^64 - 7 * q^65 + 36 * q^66 + q^67 - 35 * q^68 + 31 * q^69 + 3 * q^70 - 8 * q^71 + 53 * q^72 + 9 * q^73 - 35 * q^74 - 8 * q^75 + 40 * q^76 + 42 * q^77 + 46 * q^78 + 9 * q^79 + 10 * q^80 + 35 * q^81 + 29 * q^82 - 42 * q^83 + 55 * q^84 - 13 * q^85 + 35 * q^86 + 6 * q^87 + 30 * q^88 + 7 * q^89 + 4 * q^90 + 31 * q^91 + 19 * q^92 + 24 * q^93 + 37 * q^94 - 7 * q^95 + 44 * q^96 - 7 * q^97 + 9 * q^98 + 5 * q^99

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