LMFDB - Embedded newform 2005.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, 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("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.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:	$16.0100056053$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73364 q^{2} -2.38520 q^{3} +5.47281 q^{4} +1.00000 q^{5} +6.52028 q^{6} -4.95208 q^{7} -9.49342 q^{8} +2.68916 q^{9} +O(q^{10})$	\(q-2.73364 q^{2} -2.38520 q^{3} +5.47281 q^{4} +1.00000 q^{5} +6.52028 q^{6} -4.95208 q^{7} -9.49342 q^{8} +2.68916 q^{9} -2.73364 q^{10} -5.64971 q^{11} -13.0537 q^{12} -0.283481 q^{13} +13.5372 q^{14} -2.38520 q^{15} +15.0060 q^{16} +0.822708 q^{17} -7.35120 q^{18} +6.19994 q^{19} +5.47281 q^{20} +11.8117 q^{21} +15.4443 q^{22} +3.64705 q^{23} +22.6437 q^{24} +1.00000 q^{25} +0.774935 q^{26} +0.741417 q^{27} -27.1018 q^{28} -6.07545 q^{29} +6.52028 q^{30} -4.62841 q^{31} -22.0342 q^{32} +13.4757 q^{33} -2.24899 q^{34} -4.95208 q^{35} +14.7172 q^{36} +10.1548 q^{37} -16.9484 q^{38} +0.676157 q^{39} -9.49342 q^{40} +8.88371 q^{41} -32.2889 q^{42} -6.64299 q^{43} -30.9198 q^{44} +2.68916 q^{45} -9.96974 q^{46} +6.47236 q^{47} -35.7923 q^{48} +17.5231 q^{49} -2.73364 q^{50} -1.96232 q^{51} -1.55144 q^{52} -2.87920 q^{53} -2.02677 q^{54} -5.64971 q^{55} +47.0121 q^{56} -14.7881 q^{57} +16.6081 q^{58} -12.5249 q^{59} -13.0537 q^{60} -1.17427 q^{61} +12.6524 q^{62} -13.3169 q^{63} +30.2217 q^{64} -0.283481 q^{65} -36.8376 q^{66} +8.06621 q^{67} +4.50252 q^{68} -8.69893 q^{69} +13.5372 q^{70} +4.34741 q^{71} -25.5293 q^{72} +0.0694370 q^{73} -27.7595 q^{74} -2.38520 q^{75} +33.9311 q^{76} +27.9778 q^{77} -1.84837 q^{78} +5.01529 q^{79} +15.0060 q^{80} -9.83590 q^{81} -24.2849 q^{82} +10.8540 q^{83} +64.6430 q^{84} +0.822708 q^{85} +18.1596 q^{86} +14.4911 q^{87} +53.6350 q^{88} +4.58846 q^{89} -7.35120 q^{90} +1.40382 q^{91} +19.9596 q^{92} +11.0397 q^{93} -17.6931 q^{94} +6.19994 q^{95} +52.5559 q^{96} -1.74507 q^{97} -47.9018 q^{98} -15.1930 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9}+O(q^{10}) $ 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9} - 5 q^{10} - 30 q^{11} - 29 q^{12} - 18 q^{13} + 6 q^{14} - 10 q^{15} + 21 q^{16} - 18 q^{17} - 30 q^{18} - 17 q^{19} + 25 q^{20} + 6 q^{21} - 2 q^{22} - 44 q^{23} + 11 q^{24} + 25 q^{25} - 14 q^{26} - 25 q^{27} - 50 q^{28} - 9 q^{29} + 2 q^{30} - 13 q^{31} - 45 q^{32} - 21 q^{33} - 21 q^{34} - 31 q^{35} + 5 q^{36} - 28 q^{37} - 32 q^{38} + 9 q^{39} - 30 q^{40} + 28 q^{41} - 67 q^{42} - 61 q^{43} - 49 q^{44} + 17 q^{45} + 18 q^{46} - 53 q^{47} - 44 q^{48} + 28 q^{49} - 5 q^{50} - 30 q^{51} - 3 q^{52} - 36 q^{53} + 17 q^{54} - 30 q^{55} - 3 q^{56} - 13 q^{57} + 2 q^{58} - 39 q^{59} - 29 q^{60} + 10 q^{61} - 30 q^{62} - 44 q^{63} - 4 q^{64} - 18 q^{65} + 33 q^{66} - 10 q^{67} - 18 q^{68} - 6 q^{69} + 6 q^{70} - 7 q^{71} - q^{72} - 26 q^{73} - 3 q^{74} - 10 q^{75} + 12 q^{76} + 29 q^{77} - 5 q^{78} - 6 q^{79} + 21 q^{80} + 13 q^{81} - 30 q^{82} - 35 q^{83} + 117 q^{84} - 18 q^{85} + 14 q^{86} - 104 q^{87} + 53 q^{88} + 7 q^{89} - 30 q^{90} - 25 q^{91} - 31 q^{92} + 2 q^{93} + 68 q^{94} - 17 q^{95} + 92 q^{96} + 6 q^{97} + 15 q^{98} - 36 q^{99}+O(q^{100}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9 - 5 * q^10 - 30 * q^11 - 29 * q^12 - 18 * q^13 + 6 * q^14 - 10 * q^15 + 21 * q^16 - 18 * q^17 - 30 * q^18 - 17 * q^19 + 25 * q^20 + 6 * q^21 - 2 * q^22 - 44 * q^23 + 11 * q^24 + 25 * q^25 - 14 * q^26 - 25 * q^27 - 50 * q^28 - 9 * q^29 + 2 * q^30 - 13 * q^31 - 45 * q^32 - 21 * q^33 - 21 * q^34 - 31 * q^35 + 5 * q^36 - 28 * q^37 - 32 * q^38 + 9 * q^39 - 30 * q^40 + 28 * q^41 - 67 * q^42 - 61 * q^43 - 49 * q^44 + 17 * q^45 + 18 * q^46 - 53 * q^47 - 44 * q^48 + 28 * q^49 - 5 * q^50 - 30 * q^51 - 3 * q^52 - 36 * q^53 + 17 * q^54 - 30 * q^55 - 3 * q^56 - 13 * q^57 + 2 * q^58 - 39 * q^59 - 29 * q^60 + 10 * q^61 - 30 * q^62 - 44 * q^63 - 4 * q^64 - 18 * q^65 + 33 * q^66 - 10 * q^67 - 18 * q^68 - 6 * q^69 + 6 * q^70 - 7 * q^71 - q^72 - 26 * q^73 - 3 * q^74 - 10 * q^75 + 12 * q^76 + 29 * q^77 - 5 * q^78 - 6 * q^79 + 21 * q^80 + 13 * q^81 - 30 * q^82 - 35 * q^83 + 117 * q^84 - 18 * q^85 + 14 * q^86 - 104 * q^87 + 53 * q^88 + 7 * q^89 - 30 * q^90 - 25 * q^91 - 31 * q^92 + 2 * q^93 + 68 * q^94 - 17 * q^95 + 92 * q^96 + 6 * q^97 + 15 * q^98 - 36 * 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 $

$2$	−2.73364	−1.93298	−0.966489	−	0.256708i	$-0.917362\pi$
$2$	−2.73364	−1.93298	−0.966489	+	0.256708i	$0.917362\pi$
$3$	−2.38520	−1.37709	−0.688547	−	0.725192i	$-0.741751\pi$
$3$	−2.38520	−1.37709	−0.688547	+	0.725192i	$0.741751\pi$
$4$	5.47281	2.73640
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	6.52028	2.66189
$7$	−4.95208	−1.87171	−0.935854	−	0.352387i	$-0.885370\pi$
$7$	−4.95208	−1.87171	−0.935854	+	0.352387i	$0.885370\pi$
$8$	−9.49342	−3.35643
$9$	2.68916	0.896386
$10$	−2.73364	−0.864454
$11$	−5.64971	−1.70345	−0.851725	−	0.523988i	$-0.824443\pi$
$11$	−5.64971	−1.70345	−0.851725	+	0.523988i	$0.824443\pi$
$12$	−13.0537	−3.76828
$13$	−0.283481	−0.0786234	−0.0393117	−	0.999227i	$-0.512517\pi$
$13$	−0.283481	−0.0786234	−0.0393117	+	0.999227i	$0.512517\pi$
$14$	13.5372	3.61797
$15$	−2.38520	−0.615855
$16$	15.0060	3.75150
$17$	0.822708	0.199536	0.0997680	−	0.995011i	$-0.468190\pi$
$17$	0.822708	0.199536	0.0997680	+	0.995011i	$0.468190\pi$
$18$	−7.35120	−1.73269
$19$	6.19994	1.42236	0.711182	−	0.703008i	$-0.248160\pi$
$19$	6.19994	1.42236	0.711182	+	0.703008i	$0.248160\pi$
$20$	5.47281	1.22376
$21$	11.8117	2.57752
$22$	15.4443	3.29273
$23$	3.64705	0.760463	0.380231	−	0.924891i	$-0.375844\pi$
$23$	3.64705	0.760463	0.380231	+	0.924891i	$0.375844\pi$
$24$	22.6437	4.62212
$25$	1.00000	0.200000
$26$	0.774935	0.151977
$27$	0.741417	0.142686
$28$	−27.1018	−5.12175
$29$	−6.07545	−1.12818	−0.564091	−	0.825712i	$-0.690773\pi$
$29$	−6.07545	−1.12818	−0.564091	+	0.825712i	$0.690773\pi$
$30$	6.52028	1.19043
$31$	−4.62841	−0.831288	−0.415644	−	0.909527i	$-0.636444\pi$
$31$	−4.62841	−0.831288	−0.415644	+	0.909527i	$0.636444\pi$
$32$	−22.0342	−3.89514
$33$	13.4757	2.34581
$34$	−2.24899	−0.385699
$35$	−4.95208	−0.837054
$36$	14.7172	2.45287
$37$	10.1548	1.66943	0.834717	−	0.550680i	$-0.185631\pi$
$37$	10.1548	1.66943	0.834717	+	0.550680i	$0.185631\pi$
$38$	−16.9484	−2.74940
$39$	0.676157	0.108272
$40$	−9.49342	−1.50104
$41$	8.88371	1.38740	0.693701	−	0.720263i	$-0.255979\pi$
$41$	8.88371	1.38740	0.693701	+	0.720263i	$0.255979\pi$
$42$	−32.2889	−4.98228
$43$	−6.64299	−1.01305	−0.506523	−	0.862226i	$-0.669070\pi$
$43$	−6.64299	−1.01305	−0.506523	+	0.862226i	$0.669070\pi$
$44$	−30.9198	−4.66133
$45$	2.68916	0.400876
$46$	−9.96974	−1.46996
$47$	6.47236	0.944091	0.472046	−	0.881574i	$-0.343516\pi$
$47$	6.47236	0.944091	0.472046	+	0.881574i	$0.343516\pi$
$48$	−35.7923	−5.16617
$49$	17.5231	2.50329
$50$	−2.73364	−0.386596
$51$	−1.96232	−0.274780
$52$	−1.55144	−0.215145
$53$	−2.87920	−0.395489	−0.197744	−	0.980254i	$-0.563362\pi$
$53$	−2.87920	−0.395489	−0.197744	+	0.980254i	$0.563362\pi$
$54$	−2.02677	−0.275808
$55$	−5.64971	−0.761806
$56$	47.0121	6.28226
$57$	−14.7881	−1.95873
$58$	16.6081	2.18075
$59$	−12.5249	−1.63060	−0.815299	−	0.579040i	$-0.803428\pi$
$59$	−12.5249	−1.63060	−0.815299	+	0.579040i	$0.803428\pi$
$60$	−13.0537	−1.68523
$61$	−1.17427	−0.150350	−0.0751749	−	0.997170i	$-0.523952\pi$
$61$	−1.17427	−0.150350	−0.0751749	+	0.997170i	$0.523952\pi$
$62$	12.6524	1.60686
$63$	−13.3169	−1.67777
$64$	30.2217	3.77771
$65$	−0.283481	−0.0351615
$66$	−36.8376	−4.53440
$67$	8.06621	0.985445	0.492722	−	0.870187i	$-0.336002\pi$
$67$	8.06621	0.985445	0.492722	+	0.870187i	$0.336002\pi$
$68$	4.50252	0.546011
$69$	−8.69893	−1.04723
$70$	13.5372	1.61801
$71$	4.34741	0.515943	0.257971	−	0.966153i	$-0.416946\pi$
$71$	4.34741	0.515943	0.257971	+	0.966153i	$0.416946\pi$
$72$	−25.5293	−3.00866
$73$	0.0694370	0.00812699	0.00406350	−	0.999992i	$-0.498707\pi$
$73$	0.0694370	0.00812699	0.00406350	+	0.999992i	$0.498707\pi$
$74$	−27.7595	−3.22698
$75$	−2.38520	−0.275419
$76$	33.9311	3.89216
$77$	27.9778	3.18836
$78$	−1.84837	−0.209287
$79$	5.01529	0.564264	0.282132	−	0.959376i	$-0.408958\pi$
$79$	5.01529	0.564264	0.282132	+	0.959376i	$0.408958\pi$
$80$	15.0060	1.67772
$81$	−9.83590	−1.09288
$82$	−24.2849	−2.68182
$83$	10.8540	1.19138	0.595689	−	0.803215i	$-0.296879\pi$
$83$	10.8540	1.19138	0.595689	+	0.803215i	$0.296879\pi$
$84$	64.6430	7.05313
$85$	0.822708	0.0892352
$86$	18.1596	1.95820
$87$	14.4911	1.55361
$88$	53.6350	5.71751
$89$	4.58846	0.486376	0.243188	−	0.969979i	$-0.421807\pi$
$89$	4.58846	0.486376	0.243188	+	0.969979i	$0.421807\pi$
$90$	−7.35120	−0.774885
$91$	1.40382	0.147160
$92$	19.9596	2.08093
$93$	11.0397	1.14476
$94$	−17.6931	−1.82491
$95$	6.19994	0.636100
$96$	52.5559	5.36397
$97$	−1.74507	−0.177185	−0.0885923	−	0.996068i	$-0.528237\pi$
$97$	−1.74507	−0.177185	−0.0885923	+	0.996068i	$0.528237\pi$
$98$	−47.9018	−4.83881
$99$	−15.1930	−1.52695
$100$	5.47281	0.547281
$101$	−18.4551	−1.83635	−0.918176	−	0.396173i	$-0.870338\pi$
$101$	−18.4551	−1.83635	−0.918176	+	0.396173i	$0.870338\pi$
$102$	5.36428	0.531143
$103$	−2.70122	−0.266159	−0.133079	−	0.991105i	$-0.542487\pi$
$103$	−2.70122	−0.266159	−0.133079	+	0.991105i	$0.542487\pi$
$104$	2.69120	0.263894
$105$	11.8117	1.15270
$106$	7.87071	0.764471
$107$	−12.6072	−1.21879	−0.609394	−	0.792868i	$-0.708587\pi$
$107$	−12.6072	−1.21879	−0.609394	+	0.792868i	$0.708587\pi$
$108$	4.05763	0.390446
$109$	11.3691	1.08896	0.544481	−	0.838773i	$-0.316727\pi$
$109$	11.3691	1.08896	0.544481	+	0.838773i	$0.316727\pi$
$110$	15.4443	1.47255
$111$	−24.2211	−2.29897
$112$	−74.3109	−7.02172
$113$	7.87597	0.740909	0.370454	−	0.928851i	$-0.379202\pi$
$113$	7.87597	0.740909	0.370454	+	0.928851i	$0.379202\pi$
$114$	40.4253	3.78618
$115$	3.64705	0.340089
$116$	−33.2498	−3.08716
$117$	−0.762325	−0.0704770
$118$	34.2385	3.15191
$119$	−4.07411	−0.373473
$120$	22.6437	2.06707
$121$	20.9192	1.90174
$122$	3.21003	0.290623
$123$	−21.1894	−1.91058
$124$	−25.3304	−2.27474
$125$	1.00000	0.0894427
$126$	36.4037	3.24310
$127$	−12.0085	−1.06558	−0.532792	−	0.846246i	$-0.678857\pi$
$127$	−12.0085	−1.06558	−0.532792	+	0.846246i	$0.678857\pi$
$128$	−38.5469	−3.40710
$129$	15.8448	1.39506
$130$	0.774935	0.0679663
$131$	−8.37832	−0.732017	−0.366009	−	0.930611i	$-0.619276\pi$
$131$	−8.37832	−0.732017	−0.366009	+	0.930611i	$0.619276\pi$
$132$	73.7497	6.41908
$133$	−30.7026	−2.66225
$134$	−22.0501	−1.90484
$135$	0.741417	0.0638110
$136$	−7.81031	−0.669728
$137$	−0.665958	−0.0568966	−0.0284483	−	0.999595i	$-0.509057\pi$
$137$	−0.665958	−0.0568966	−0.0284483	+	0.999595i	$0.509057\pi$
$138$	23.7798	2.02427
$139$	5.24577	0.444941	0.222470	−	0.974939i	$-0.428588\pi$
$139$	5.24577	0.444941	0.222470	+	0.974939i	$0.428588\pi$
$140$	−27.1018	−2.29052
$141$	−15.4379	−1.30010
$142$	−11.8843	−0.997306
$143$	1.60158	0.133931
$144$	40.3535	3.36279
$145$	−6.07545	−0.504539
$146$	−0.189816	−0.0157093
$147$	−41.7959	−3.44727
$148$	55.5751	4.56824
$149$	3.91169	0.320458	0.160229	−	0.987080i	$-0.448777\pi$
$149$	3.91169	0.320458	0.160229	+	0.987080i	$0.448777\pi$
$150$	6.52028	0.532378
$151$	12.1923	0.992195	0.496097	−	0.868267i	$-0.334766\pi$
$151$	12.1923	0.992195	0.496097	+	0.868267i	$0.334766\pi$
$152$	−58.8586	−4.77406
$153$	2.21239	0.178861
$154$	−76.4813	−6.16304
$155$	−4.62841	−0.371763
$156$	3.70048	0.296275
$157$	2.48811	0.198573	0.0992864	−	0.995059i	$-0.468344\pi$
$157$	2.48811	0.198573	0.0992864	+	0.995059i	$0.468344\pi$
$158$	−13.7100	−1.09071
$159$	6.86746	0.544625
$160$	−22.0342	−1.74196
$161$	−18.0605	−1.42336
$162$	26.8878	2.11251
$163$	14.3643	1.12510	0.562551	−	0.826762i	$-0.309820\pi$
$163$	14.3643	1.12510	0.562551	+	0.826762i	$0.309820\pi$
$164$	48.6188	3.79649
$165$	13.4757	1.04908
$166$	−29.6709	−2.30291
$167$	−21.1094	−1.63350	−0.816749	−	0.576993i	$-0.804226\pi$
$167$	−21.1094	−1.63350	−0.816749	+	0.576993i	$0.804226\pi$
$168$	−112.133	−8.65126
$169$	−12.9196	−0.993818
$170$	−2.24899	−0.172490
$171$	16.6726	1.27499
$172$	−36.3558	−2.77210
$173$	−5.58271	−0.424445	−0.212223	−	0.977221i	$-0.568070\pi$
$173$	−5.58271	−0.424445	−0.212223	+	0.977221i	$0.568070\pi$
$174$	−39.6136	−3.00310
$175$	−4.95208	−0.374342
$176$	−84.7795	−6.39050
$177$	29.8743	2.24549
$178$	−12.5432	−0.940155
$179$	−25.5637	−1.91072	−0.955361	−	0.295440i	$-0.904534\pi$
$179$	−25.5637	−1.91072	−0.955361	+	0.295440i	$0.904534\pi$
$180$	14.7172	1.09696
$181$	7.28089	0.541184	0.270592	−	0.962694i	$-0.412781\pi$
$181$	7.28089	0.541184	0.270592	+	0.962694i	$0.412781\pi$
$182$	−3.83754	−0.284457
$183$	2.80086	0.207046
$184$	−34.6230	−2.55244
$185$	10.1548	0.746593
$186$	−30.1785	−2.21280
$187$	−4.64806	−0.339900
$188$	35.4220	2.58341
$189$	−3.67155	−0.267066
$190$	−16.9484	−1.22957
$191$	8.37301	0.605850	0.302925	−	0.953014i	$-0.402037\pi$
$191$	8.37301	0.605850	0.302925	+	0.953014i	$0.402037\pi$
$192$	−72.0847	−5.20227
$193$	0.534707	0.0384891	0.0192445	−	0.999815i	$-0.493874\pi$
$193$	0.534707	0.0384891	0.0192445	+	0.999815i	$0.493874\pi$
$194$	4.77039	0.342494
$195$	0.676157	0.0484206
$196$	95.9003	6.85002
$197$	23.4634	1.67170	0.835849	−	0.548960i	$-0.184976\pi$
$197$	23.4634	1.67170	0.835849	+	0.548960i	$0.184976\pi$
$198$	41.5321	2.95156
$199$	9.73465	0.690071	0.345035	−	0.938590i	$-0.387867\pi$
$199$	9.73465	0.690071	0.345035	+	0.938590i	$0.387867\pi$
$200$	−9.49342	−0.671286
$201$	−19.2395	−1.35705
$202$	50.4497	3.54963
$203$	30.0861	2.11163
$204$	−10.7394	−0.751908
$205$	8.88371	0.620465
$206$	7.38416	0.514479
$207$	9.80750	0.681668
$208$	−4.25391	−0.294956
$209$	−35.0278	−2.42293
$210$	−32.2889	−2.22815
$211$	19.2624	1.32608	0.663038	−	0.748586i	$-0.269267\pi$
$211$	19.2624	1.32608	0.663038	+	0.748586i	$0.269267\pi$
$212$	−15.7573	−1.08222
$213$	−10.3694	−0.710501
$214$	34.4637	2.35589
$215$	−6.64299	−0.453048
$216$	−7.03858	−0.478914
$217$	22.9203	1.55593
$218$	−31.0790	−2.10494
$219$	−0.165621	−0.0111916
$220$	−30.9198	−2.08461
$221$	−0.233222	−0.0156882
$222$	66.2119	4.44385
$223$	−6.00688	−0.402250	−0.201125	−	0.979566i	$-0.564460\pi$
$223$	−6.00688	−0.402250	−0.201125	+	0.979566i	$0.564460\pi$
$224$	109.115	7.29056
$225$	2.68916	0.179277
$226$	−21.5301	−1.43216
$227$	−2.90296	−0.192676	−0.0963381	−	0.995349i	$-0.530713\pi$
$227$	−2.90296	−0.192676	−0.0963381	+	0.995349i	$0.530713\pi$
$228$	−80.9322	−5.35987
$229$	2.82645	0.186777	0.0933886	−	0.995630i	$-0.470230\pi$
$229$	2.82645	0.186777	0.0933886	+	0.995630i	$0.470230\pi$
$230$	−9.96974	−0.657385
$231$	−66.7325	−4.39067
$232$	57.6768	3.78667
$233$	−16.5193	−1.08222	−0.541109	−	0.840953i	$-0.681995\pi$
$233$	−16.5193	−1.08222	−0.541109	+	0.840953i	$0.681995\pi$
$234$	2.08392	0.136230
$235$	6.47236	0.422210
$236$	−68.5462	−4.46198
$237$	−11.9624	−0.777044
$238$	11.1372	0.721915
$239$	−10.3183	−0.667437	−0.333718	−	0.942673i	$-0.608303\pi$
$239$	−10.3183	−0.667437	−0.333718	+	0.942673i	$0.608303\pi$
$240$	−35.7923	−2.31038
$241$	−13.6307	−0.878032	−0.439016	−	0.898479i	$-0.644673\pi$
$241$	−13.6307	−0.878032	−0.439016	+	0.898479i	$0.644673\pi$
$242$	−57.1856	−3.67603
$243$	21.2363	1.36231
$244$	−6.42655	−0.411418
$245$	17.5231	1.11951
$246$	57.9242	3.69311
$247$	−1.75756	−0.111831
$248$	43.9395	2.79016
$249$	−25.8889	−1.64064
$250$	−2.73364	−0.172891
$251$	−10.6475	−0.672065	−0.336033	−	0.941850i	$-0.609085\pi$
$251$	−10.6475	−0.672065	−0.336033	+	0.941850i	$0.609085\pi$
$252$	−72.8809	−4.59107
$253$	−20.6048	−1.29541
$254$	32.8270	2.05975
$255$	−1.96232	−0.122885
$256$	44.9302	2.80814
$257$	−25.4719	−1.58889	−0.794445	−	0.607336i	$-0.792238\pi$
$257$	−25.4719	−1.58889	−0.794445	+	0.607336i	$0.792238\pi$
$258$	−43.3141	−2.69662
$259$	−50.2872	−3.12469
$260$	−1.55144	−0.0962159
$261$	−16.3379	−1.01129
$262$	22.9033	1.41497
$263$	−6.21611	−0.383302	−0.191651	−	0.981463i	$-0.561384\pi$
$263$	−6.21611	−0.383302	−0.191651	+	0.981463i	$0.561384\pi$
$264$	−127.930	−7.87355
$265$	−2.87920	−0.176868
$266$	83.9299	5.14607
$267$	−10.9444	−0.669786
$268$	44.1448	2.69657
$269$	19.3009	1.17679	0.588397	−	0.808572i	$-0.299759\pi$
$269$	19.3009	1.17679	0.588397	+	0.808572i	$0.299759\pi$
$270$	−2.02677	−0.123345
$271$	−11.0283	−0.669921	−0.334960	−	0.942232i	$-0.608723\pi$
$271$	−11.0283	−0.669921	−0.334960	+	0.942232i	$0.608723\pi$
$272$	12.3456	0.748559
$273$	−3.34838	−0.202653
$274$	1.82049	0.109980
$275$	−5.64971	−0.340690
$276$	−47.6076	−2.86564
$277$	−10.8159	−0.649864	−0.324932	−	0.945737i	$-0.605341\pi$
$277$	−10.8159	−0.649864	−0.324932	+	0.945737i	$0.605341\pi$
$278$	−14.3401	−0.860061
$279$	−12.4465	−0.745155
$280$	47.0121	2.80951
$281$	11.8163	0.704901	0.352450	−	0.935831i	$-0.385349\pi$
$281$	11.8163	0.704901	0.352450	+	0.935831i	$0.385349\pi$
$282$	42.2016	2.51307
$283$	−1.65777	−0.0985441	−0.0492720	−	0.998785i	$-0.515690\pi$
$283$	−1.65777	−0.0985441	−0.0492720	+	0.998785i	$0.515690\pi$
$284$	23.7925	1.41183
$285$	−14.7881	−0.875969
$286$	−4.37816	−0.258886
$287$	−43.9928	−2.59681
$288$	−59.2535	−3.49155
$289$	−16.3232	−0.960185
$290$	16.6081	0.975262
$291$	4.16232	0.244000
$292$	0.380015	0.0222387
$293$	−16.7201	−0.976797	−0.488398	−	0.872621i	$-0.662419\pi$
$293$	−16.7201	−0.976797	−0.488398	+	0.872621i	$0.662419\pi$
$294$	114.255	6.66349
$295$	−12.5249	−0.729226
$296$	−96.4034	−5.60333
$297$	−4.18879	−0.243058
$298$	−10.6932	−0.619438
$299$	−1.03387	−0.0597902
$300$	−13.0537	−0.753657
$301$	32.8966	1.89613
$302$	−33.3294	−1.91789
$303$	44.0190	2.52883
$304$	93.0363	5.33600
$305$	−1.17427	−0.0672385
$306$	−6.04789	−0.345735
$307$	−13.7134	−0.782665	−0.391332	−	0.920249i	$-0.627986\pi$
$307$	−13.7134	−0.782665	−0.391332	+	0.920249i	$0.627986\pi$
$308$	153.117	8.72465
$309$	6.44293	0.366526
$310$	12.6524	0.718610
$311$	5.58536	0.316717	0.158358	−	0.987382i	$-0.449380\pi$
$311$	5.58536	0.316717	0.158358	+	0.987382i	$0.449380\pi$
$312$	−6.41904	−0.363407
$313$	18.8240	1.06399	0.531997	−	0.846746i	$-0.321442\pi$
$313$	18.8240	1.06399	0.531997	+	0.846746i	$0.321442\pi$
$314$	−6.80161	−0.383837
$315$	−13.3169	−0.750323
$316$	27.4477	1.54405
$317$	10.8566	0.609767	0.304884	−	0.952390i	$-0.401382\pi$
$317$	10.8566	0.609767	0.304884	+	0.952390i	$0.401382\pi$
$318$	−18.7732	−1.05275
$319$	34.3245	1.92180
$320$	30.2217	1.68945
$321$	30.0707	1.67839
$322$	49.3709	2.75133
$323$	5.10074	0.283813
$324$	−53.8300	−2.99055
$325$	−0.283481	−0.0157247
$326$	−39.2670	−2.17480
$327$	−27.1175	−1.49960
$328$	−84.3367	−4.65672
$329$	−32.0516	−1.76706
$330$	−36.8376	−2.02785
$331$	9.93321	0.545979	0.272989	−	0.962017i	$-0.411988\pi$
$331$	9.93321	0.545979	0.272989	+	0.962017i	$0.411988\pi$
$332$	59.4017	3.26009
$333$	27.3078	1.49646
$334$	57.7057	3.15752
$335$	8.06621	0.440704
$336$	177.246	9.66956
$337$	5.88246	0.320438	0.160219	−	0.987081i	$-0.448780\pi$
$337$	5.88246	0.320438	0.160219	+	0.987081i	$0.448780\pi$
$338$	35.3177	1.92103
$339$	−18.7857	−1.02030
$340$	4.50252	0.244183
$341$	26.1492	1.41606
$342$	−45.5770	−2.46452
$343$	−52.1109	−2.81373
$344$	63.0647	3.40022
$345$	−8.69893	−0.468335
$346$	15.2611	0.820444
$347$	−13.7066	−0.735807	−0.367904	−	0.929864i	$-0.619924\pi$
$347$	−13.7066	−0.735807	−0.367904	+	0.929864i	$0.619924\pi$
$348$	79.3072	4.25131
$349$	−27.4874	−1.47137	−0.735684	−	0.677325i	$-0.763139\pi$
$349$	−27.4874	−1.47137	−0.735684	+	0.677325i	$0.763139\pi$
$350$	13.5372	0.723594
$351$	−0.210177	−0.0112184
$352$	124.487	6.63518
$353$	−7.21175	−0.383843	−0.191921	−	0.981410i	$-0.561472\pi$
$353$	−7.21175	−0.383843	−0.191921	+	0.981410i	$0.561472\pi$
$354$	−81.6656	−4.34048
$355$	4.34741	0.230737
$356$	25.1118	1.33092
$357$	9.71755	0.514307
$358$	69.8821	3.69338
$359$	−17.2468	−0.910252	−0.455126	−	0.890427i	$-0.650406\pi$
$359$	−17.2468	−0.910252	−0.455126	+	0.890427i	$0.650406\pi$
$360$	−25.5293	−1.34551
$361$	19.4392	1.02312
$362$	−19.9033	−1.04610
$363$	−49.8964	−2.61888
$364$	7.68283	0.402689
$365$	0.0694370	0.00363450
$366$	−7.65656	−0.400215
$367$	−1.47613	−0.0770534	−0.0385267	−	0.999258i	$-0.512266\pi$
$367$	−1.47613	−0.0770534	−0.0385267	+	0.999258i	$0.512266\pi$
$368$	54.7277	2.85288
$369$	23.8897	1.24365
$370$	−27.7595	−1.44315
$371$	14.2580	0.740240
$372$	60.4180	3.13253
$373$	24.3672	1.26168	0.630842	−	0.775911i	$-0.282710\pi$
$373$	24.3672	1.26168	0.630842	+	0.775911i	$0.282710\pi$
$374$	12.7061	0.657019
$375$	−2.38520	−0.123171
$376$	−61.4448	−3.16878
$377$	1.72227	0.0887016
$378$	10.0367	0.516233
$379$	−9.99551	−0.513435	−0.256717	−	0.966487i	$-0.582641\pi$
$379$	−9.99551	−0.513435	−0.256717	+	0.966487i	$0.582641\pi$
$380$	33.9311	1.74063
$381$	28.6427	1.46741
$382$	−22.8888	−1.17109
$383$	24.0530	1.22905	0.614527	−	0.788896i	$-0.289347\pi$
$383$	24.0530	1.22905	0.614527	+	0.788896i	$0.289347\pi$
$384$	91.9420	4.69190
$385$	27.9778	1.42588
$386$	−1.46170	−0.0743986
$387$	−17.8641	−0.908081
$388$	−9.55041	−0.484848
$389$	−16.1504	−0.818858	−0.409429	−	0.912342i	$-0.634272\pi$
$389$	−16.1504	−0.818858	−0.409429	+	0.912342i	$0.634272\pi$
$390$	−1.84837	−0.0935960
$391$	3.00046	0.151740
$392$	−166.354	−8.40213
$393$	19.9839	1.00806
$394$	−64.1405	−3.23135
$395$	5.01529	0.252346
$396$	−83.1481	−4.17835
$397$	−18.6450	−0.935767	−0.467884	−	0.883790i	$-0.654983\pi$
$397$	−18.6450	−0.935767	−0.467884	+	0.883790i	$0.654983\pi$
$398$	−26.6111	−1.33389
$399$	73.2316	3.66617
$400$	15.0060	0.750300
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	52.5939	2.62315
$403$	1.31207	0.0653587
$404$	−101.001	−5.02500
$405$	−9.83590	−0.488750
$406$	−82.2446	−4.08173
$407$	−57.3715	−2.84380
$408$	18.6291	0.922279
$409$	−7.79408	−0.385392	−0.192696	−	0.981258i	$-0.561723\pi$
$409$	−7.79408	−0.385392	−0.192696	+	0.981258i	$0.561723\pi$
$410$	−24.2849	−1.19935
$411$	1.58844	0.0783520
$412$	−14.7832	−0.728318
$413$	62.0241	3.05201
$414$	−26.8102	−1.31765
$415$	10.8540	0.532801
$416$	6.24628	0.306249
$417$	−12.5122	−0.612725
$418$	95.7536	4.68346
$419$	−29.2983	−1.43132	−0.715658	−	0.698451i	$-0.753873\pi$
$419$	−29.2983	−1.43132	−0.715658	+	0.698451i	$0.753873\pi$
$420$	64.6430	3.15425
$421$	10.8241	0.527537	0.263768	−	0.964586i	$-0.415035\pi$
$421$	10.8241	0.527537	0.263768	+	0.964586i	$0.415035\pi$
$422$	−52.6565	−2.56328
$423$	17.4052	0.846270
$424$	27.3335	1.32743
$425$	0.822708	0.0399072
$426$	28.3463	1.37338
$427$	5.81507	0.281411
$428$	−68.9970	−3.33510
$429$	−3.82009	−0.184436
$430$	18.1596	0.875732
$431$	17.1681	0.826959	0.413479	−	0.910513i	$-0.364313\pi$
$431$	17.1681	0.826959	0.413479	+	0.910513i	$0.364313\pi$
$432$	11.1257	0.535285
$433$	−27.9987	−1.34553	−0.672767	−	0.739855i	$-0.734894\pi$
$433$	−27.9987	−1.34553	−0.672767	+	0.739855i	$0.734894\pi$
$434$	−62.6558	−3.00757
$435$	14.4911	0.694797
$436$	62.2208	2.97984
$437$	22.6115	1.08165
$438$	0.452749	0.0216332
$439$	−21.1566	−1.00975	−0.504875	−	0.863192i	$-0.668462\pi$
$439$	−21.1566	−1.00975	−0.504875	+	0.863192i	$0.668462\pi$
$440$	53.6350	2.55695
$441$	47.1223	2.24392
$442$	0.637545	0.0303249
$443$	−15.9739	−0.758941	−0.379471	−	0.925204i	$-0.623894\pi$
$443$	−15.9739	−0.758941	−0.379471	+	0.925204i	$0.623894\pi$
$444$	−132.557	−6.29090
$445$	4.58846	0.217514
$446$	16.4207	0.777541
$447$	−9.33014	−0.441301
$448$	−149.660	−7.07078
$449$	39.6009	1.86888	0.934441	−	0.356117i	$-0.115900\pi$
$449$	39.6009	1.86888	0.934441	+	0.356117i	$0.115900\pi$
$450$	−7.35120	−0.346539
$451$	−50.1903	−2.36337
$452$	43.1036	2.02742
$453$	−29.0810	−1.36635
$454$	7.93566	0.372439
$455$	1.40382	0.0658120
$456$	140.389	6.57433
$457$	2.52358	0.118048	0.0590241	−	0.998257i	$-0.481201\pi$
$457$	2.52358	0.118048	0.0590241	+	0.998257i	$0.481201\pi$
$458$	−7.72651	−0.361036
$459$	0.609969	0.0284709
$460$	19.9596	0.930621
$461$	27.6754	1.28897	0.644485	−	0.764617i	$-0.277072\pi$
$461$	27.6754	1.28897	0.644485	+	0.764617i	$0.277072\pi$
$462$	182.423	8.48708
$463$	−35.0580	−1.62928	−0.814642	−	0.579965i	$-0.803066\pi$
$463$	−35.0580	−1.62928	−0.814642	+	0.579965i	$0.803066\pi$
$464$	−91.1682	−4.23238
$465$	11.0397	0.511953
$466$	45.1580	2.09190
$467$	−23.0154	−1.06502	−0.532512	−	0.846423i	$-0.678752\pi$
$467$	−23.0154	−1.06502	−0.532512	+	0.846423i	$0.678752\pi$
$468$	−4.17206	−0.192853
$469$	−39.9445	−1.84447
$470$	−17.6931	−0.816123
$471$	−5.93463	−0.273453
$472$	118.904	5.47299
$473$	37.5309	1.72567
$474$	32.7010	1.50201
$475$	6.19994	0.284473
$476$	−22.2968	−1.02197
$477$	−7.74263	−0.354511
$478$	28.2066	1.29014
$479$	36.8178	1.68225	0.841124	−	0.540843i	$-0.181895\pi$
$479$	36.8178	1.68225	0.841124	+	0.540843i	$0.181895\pi$
$480$	52.5559	2.39884
$481$	−2.87868	−0.131257
$482$	37.2615	1.69722
$483$	43.0778	1.96011
$484$	114.487	5.20394
$485$	−1.74507	−0.0792393
$486$	−58.0525	−2.63331
$487$	−25.4600	−1.15370	−0.576851	−	0.816850i	$-0.695719\pi$
$487$	−25.4600	−1.15370	−0.576851	+	0.816850i	$0.695719\pi$
$488$	11.1478	0.504638
$489$	−34.2618	−1.54937
$490$	−47.9018	−2.16398
$491$	−29.4808	−1.33045	−0.665226	−	0.746642i	$-0.731665\pi$
$491$	−29.4808	−1.33045	−0.665226	+	0.746642i	$0.731665\pi$
$492$	−115.965	−5.22812
$493$	−4.99832	−0.225113
$494$	4.80455	0.216167
$495$	−15.1930	−0.682873
$496$	−69.4540	−3.11858
$497$	−21.5287	−0.965694
$498$	70.7709	3.17132
$499$	26.7660	1.19821	0.599106	−	0.800670i	$-0.295523\pi$
$499$	26.7660	1.19821	0.599106	+	0.800670i	$0.295523\pi$
$500$	5.47281	0.244751
$501$	50.3502	2.24948
$502$	29.1065	1.29909
$503$	37.1414	1.65605	0.828027	−	0.560688i	$-0.189463\pi$
$503$	37.1414	1.65605	0.828027	+	0.560688i	$0.189463\pi$
$504$	126.423	5.63133
$505$	−18.4551	−0.821242
$506$	56.3261	2.50400
$507$	30.8159	1.36858
$508$	−65.7203	−2.91587
$509$	−28.9997	−1.28539	−0.642694	−	0.766123i	$-0.722183\pi$
$509$	−28.9997	−1.28539	−0.642694	+	0.766123i	$0.722183\pi$
$510$	5.36428	0.237534
$511$	−0.343857	−0.0152114
$512$	−45.7292	−2.02096
$513$	4.59674	0.202951
$514$	69.6310	3.07129
$515$	−2.70122	−0.119030
$516$	86.7157	3.81745
$517$	−36.5670	−1.60821
$518$	137.467	6.03996
$519$	13.3159	0.584501
$520$	2.69120	0.118017
$521$	10.7038	0.468944	0.234472	−	0.972123i	$-0.424664\pi$
$521$	10.7038	0.468944	0.234472	+	0.972123i	$0.424664\pi$
$522$	44.6619	1.95480
$523$	35.0643	1.53325	0.766627	−	0.642092i	$-0.221933\pi$
$523$	35.0643	1.53325	0.766627	+	0.642092i	$0.221933\pi$
$524$	−45.8529	−2.00310
$525$	11.8117	0.515504
$526$	16.9926	0.740914
$527$	−3.80783	−0.165872
$528$	202.216	8.80031
$529$	−9.69902	−0.421696
$530$	7.87071	0.341882
$531$	−33.6814	−1.46165
$532$	−168.029	−7.28499
$533$	−2.51836	−0.109082
$534$	29.9181	1.29468
$535$	−12.6072	−0.545059
$536$	−76.5759	−3.30758
$537$	60.9745	2.63124
$538$	−52.7617	−2.27472
$539$	−99.0001	−4.26424
$540$	4.05763	0.174613
$541$	−44.8591	−1.92865	−0.964323	−	0.264729i	$-0.914717\pi$
$541$	−44.8591	−1.92865	−0.964323	+	0.264729i	$0.914717\pi$
$542$	30.1474	1.29494
$543$	−17.3663	−0.745261
$544$	−18.1277	−0.777220
$545$	11.3691	0.486998
$546$	9.15328	0.391724
$547$	−27.0072	−1.15474	−0.577372	−	0.816481i	$-0.695922\pi$
$547$	−27.0072	−1.15474	−0.577372	+	0.816481i	$0.695922\pi$
$548$	−3.64466	−0.155692
$549$	−3.15780	−0.134771
$550$	15.4443	0.658547
$551$	−37.6674	−1.60469
$552$	82.5826	3.51495
$553$	−24.8361	−1.05614
$554$	29.5668	1.25617
$555$	−24.2211	−1.02813
$556$	28.7091	1.21754
$557$	−19.6661	−0.833278	−0.416639	−	0.909072i	$-0.636792\pi$
$557$	−19.6661	−0.833278	−0.416639	+	0.909072i	$0.636792\pi$
$558$	34.0244	1.44037
$559$	1.88316	0.0796492
$560$	−74.3109	−3.14021
$561$	11.0865	0.468074
$562$	−32.3015	−1.36256
$563$	24.0392	1.01313	0.506566	−	0.862201i	$-0.330915\pi$
$563$	24.0392	1.01313	0.506566	+	0.862201i	$0.330915\pi$
$564$	−84.4884	−3.55760
$565$	7.87597	0.331344
$566$	4.53175	0.190484
$567$	48.7081	2.04555
$568$	−41.2718	−1.73173
$569$	25.6611	1.07577	0.537884	−	0.843019i	$-0.319224\pi$
$569$	25.6611	1.07577	0.537884	+	0.843019i	$0.319224\pi$
$570$	40.4253	1.69323
$571$	29.1529	1.22001	0.610006	−	0.792397i	$-0.291167\pi$
$571$	29.1529	1.22001	0.610006	+	0.792397i	$0.291167\pi$
$572$	8.76516	0.366490
$573$	−19.9713	−0.834312
$574$	120.261	5.01958
$575$	3.64705	0.152093
$576$	81.2710	3.38629
$577$	−24.5788	−1.02323	−0.511614	−	0.859215i	$-0.670952\pi$
$577$	−24.5788	−1.02323	−0.511614	+	0.859215i	$0.670952\pi$
$578$	44.6217	1.85602
$579$	−1.27538	−0.0530031
$580$	−33.2498	−1.38062
$581$	−53.7497	−2.22991
$582$	−11.3783	−0.471646
$583$	16.2667	0.673696
$584$	−0.659195	−0.0272777
$585$	−0.762325	−0.0315183
$586$	45.7067	1.88813
$587$	30.0000	1.23823	0.619115	−	0.785300i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619115	+	0.785300i	$0.287491\pi$
$588$	−228.741	−9.43312
$589$	−28.6959	−1.18239
$590$	34.2385	1.40958
$591$	−55.9648	−2.30208
$592$	152.382	6.26288
$593$	−40.1018	−1.64678	−0.823392	−	0.567473i	$-0.807921\pi$
$593$	−40.1018	−1.64678	−0.823392	+	0.567473i	$0.807921\pi$
$594$	11.4506	0.469826
$595$	−4.07411	−0.167022
$596$	21.4079	0.876902
$597$	−23.2190	−0.950292
$598$	2.82623	0.115573
$599$	21.9148	0.895413	0.447706	−	0.894181i	$-0.352241\pi$
$599$	21.9148	0.895413	0.447706	+	0.894181i	$0.352241\pi$
$600$	22.6437	0.924423
$601$	−12.2226	−0.498571	−0.249285	−	0.968430i	$-0.580196\pi$
$601$	−12.2226	−0.498571	−0.249285	+	0.968430i	$0.580196\pi$
$602$	−89.9276	−3.66517
$603$	21.6913	0.883339
$604$	66.7261	2.71505
$605$	20.9192	0.850486
$606$	−120.332	−4.88817
$607$	33.1819	1.34681	0.673405	−	0.739274i	$-0.264831\pi$
$607$	33.1819	1.34681	0.673405	+	0.739274i	$0.264831\pi$
$608$	−136.611	−5.54030
$609$	−71.7612	−2.90791
$610$	3.21003	0.129970
$611$	−1.83479	−0.0742277
$612$	12.1080	0.489437
$613$	4.62776	0.186913	0.0934567	−	0.995623i	$-0.470208\pi$
$613$	4.62776	0.186913	0.0934567	+	0.995623i	$0.470208\pi$
$614$	37.4875	1.51287
$615$	−21.1894	−0.854438
$616$	−265.605	−10.7015
$617$	25.0937	1.01023	0.505116	−	0.863051i	$-0.331450\pi$
$617$	25.0937	1.01023	0.505116	+	0.863051i	$0.331450\pi$
$618$	−17.6127	−0.708486
$619$	−19.2571	−0.774007	−0.387004	−	0.922078i	$-0.626490\pi$
$619$	−19.2571	−0.774007	−0.387004	+	0.922078i	$0.626490\pi$
$620$	−25.3304	−1.01729
$621$	2.70398	0.108507
$622$	−15.2684	−0.612206
$623$	−22.7224	−0.910355
$624$	10.1464	0.406182
$625$	1.00000	0.0400000
$626$	−51.4580	−2.05668
$627$	83.5482	3.33660
$628$	13.6169	0.543375
$629$	8.35441	0.333112
$630$	36.4037	1.45036
$631$	23.4795	0.934705	0.467353	−	0.884071i	$-0.345208\pi$
$631$	23.4795	0.934705	0.467353	+	0.884071i	$0.345208\pi$
$632$	−47.6122	−1.89391
$633$	−45.9445	−1.82613
$634$	−29.6781	−1.17867
$635$	−12.0085	−0.476544
$636$	37.5843	1.49031
$637$	−4.96745	−0.196817
$638$	−93.8310	−3.71480
$639$	11.6909	0.462484
$640$	−38.5469	−1.52370
$641$	13.4294	0.530431	0.265215	−	0.964189i	$-0.414557\pi$
$641$	13.4294	0.530431	0.265215	+	0.964189i	$0.414557\pi$
$642$	−82.2027	−3.24428
$643$	10.0627	0.396833	0.198416	−	0.980118i	$-0.436420\pi$
$643$	10.0627	0.396833	0.198416	+	0.980118i	$0.436420\pi$
$644$	−98.8415	−3.89490
$645$	15.8448	0.623890
$646$	−13.9436	−0.548603
$647$	−3.22804	−0.126907	−0.0634536	−	0.997985i	$-0.520211\pi$
$647$	−3.22804	−0.126907	−0.0634536	+	0.997985i	$0.520211\pi$
$648$	93.3763	3.66817
$649$	70.7618	2.77764
$650$	0.774935	0.0303955
$651$	−54.6693	−2.14266
$652$	78.6133	3.07873
$653$	−31.3796	−1.22798	−0.613989	−	0.789314i	$-0.710436\pi$
$653$	−31.3796	−1.22798	−0.613989	+	0.789314i	$0.710436\pi$
$654$	74.1296	2.89870
$655$	−8.37832	−0.327368
$656$	133.309	5.20484
$657$	0.186727	0.00728492
$658$	87.6177	3.41569
$659$	−7.73751	−0.301410	−0.150705	−	0.988579i	$-0.548154\pi$
$659$	−7.73751	−0.301410	−0.150705	+	0.988579i	$0.548154\pi$
$660$	73.7497	2.87070
$661$	8.39894	0.326681	0.163340	−	0.986570i	$-0.447773\pi$
$661$	8.39894	0.326681	0.163340	+	0.986570i	$0.447773\pi$
$662$	−27.1539	−1.05536
$663$	0.556280	0.0216041
$664$	−103.041	−3.99878
$665$	−30.7026	−1.19059
$666$	−74.6497	−2.89262
$667$	−22.1575	−0.857941
$668$	−115.528	−4.46991
$669$	14.3276	0.553936
$670$	−22.0501	−0.851872
$671$	6.63428	0.256113
$672$	−260.261	−10.0398
$673$	−4.74857	−0.183044	−0.0915220	−	0.995803i	$-0.529173\pi$
$673$	−4.74857	−0.183044	−0.0915220	+	0.995803i	$0.529173\pi$
$674$	−16.0806	−0.619400
$675$	0.741417	0.0285371
$676$	−70.7067	−2.71949
$677$	26.3021	1.01087	0.505436	−	0.862864i	$-0.331332\pi$
$677$	26.3021	1.01087	0.505436	+	0.862864i	$0.331332\pi$
$678$	51.3535	1.97222
$679$	8.64170	0.331638
$680$	−7.81031	−0.299512
$681$	6.92413	0.265333
$682$	−71.4825	−2.73721
$683$	−0.882758	−0.0337778	−0.0168889	−	0.999857i	$-0.505376\pi$
$683$	−0.882758	−0.0337778	−0.0168889	+	0.999857i	$0.505376\pi$
$684$	91.2460	3.48888
$685$	−0.665958	−0.0254449
$686$	142.453	5.43887
$687$	−6.74164	−0.257210
$688$	−99.6847	−3.80044
$689$	0.816199	0.0310947
$690$	23.7798	0.905281
$691$	19.1535	0.728634	0.364317	−	0.931275i	$-0.381303\pi$
$691$	19.1535	0.728634	0.364317	+	0.931275i	$0.381303\pi$
$692$	−30.5531	−1.16145
$693$	75.2367	2.85801
$694$	37.4689	1.42230
$695$	5.24577	0.198984
$696$	−137.570	−5.21459
$697$	7.30869	0.276837
$698$	75.1408	2.84412
$699$	39.4019	1.49031
$700$	−27.1018	−1.02435
$701$	−21.7707	−0.822266	−0.411133	−	0.911575i	$-0.634867\pi$
$701$	−21.7707	−0.822266	−0.411133	+	0.911575i	$0.634867\pi$
$702$	0.574550	0.0216850
$703$	62.9589	2.37454
$704$	−170.744	−6.43515
$705$	−15.4379	−0.581423
$706$	19.7144	0.741960
$707$	91.3911	3.43712
$708$	163.496	6.14456
$709$	−15.6559	−0.587970	−0.293985	−	0.955810i	$-0.594982\pi$
$709$	−15.6559	−0.587970	−0.293985	+	0.955810i	$0.594982\pi$
$710$	−11.8843	−0.446009
$711$	13.4869	0.505798
$712$	−43.5602	−1.63249
$713$	−16.8801	−0.632163
$714$	−26.5643	−0.994145
$715$	1.60158	0.0598958
$716$	−139.905	−5.22851
$717$	24.6112	0.919123
$718$	47.1466	1.75950
$719$	7.22518	0.269454	0.134727	−	0.990883i	$-0.456984\pi$
$719$	7.22518	0.269454	0.134727	+	0.990883i	$0.456984\pi$
$720$	40.3535	1.50389
$721$	13.3766	0.498172
$722$	−53.1399	−1.97766
$723$	32.5119	1.20913
$724$	39.8469	1.48090
$725$	−6.07545	−0.225637
$726$	136.399	5.06224
$727$	15.8459	0.587691	0.293846	−	0.955853i	$-0.405065\pi$
$727$	15.8459	0.587691	0.293846	+	0.955853i	$0.405065\pi$
$728$	−13.3270	−0.493933
$729$	−21.1450	−0.783149
$730$	−0.189816	−0.00702541
$731$	−5.46524	−0.202139
$732$	15.3286	0.566561
$733$	28.3724	1.04796	0.523979	−	0.851731i	$-0.324447\pi$
$733$	28.3724	1.04796	0.523979	+	0.851731i	$0.324447\pi$
$734$	4.03522	0.148943
$735$	−41.7959	−1.54167
$736$	−80.3600	−2.96211
$737$	−45.5717	−1.67866
$738$	−65.3059	−2.40394
$739$	−30.9334	−1.13790	−0.568951	−	0.822372i	$-0.692651\pi$
$739$	−30.9334	−1.13790	−0.568951	+	0.822372i	$0.692651\pi$
$740$	55.5751	2.04298
$741$	4.19213	0.154002
$742$	−38.9764	−1.43087
$743$	33.0303	1.21176	0.605882	−	0.795555i	$-0.292820\pi$
$743$	33.0303	1.21176	0.605882	+	0.795555i	$0.292820\pi$
$744$	−104.804	−3.84231
$745$	3.91169	0.143313
$746$	−66.6111	−2.43881
$747$	29.1881	1.06794
$748$	−25.4379	−0.930103
$749$	62.4320	2.28122
$750$	6.52028	0.238087
$751$	22.1290	0.807499	0.403750	−	0.914869i	$-0.367707\pi$
$751$	22.1290	0.807499	0.403750	+	0.914869i	$0.367707\pi$
$752$	97.1243	3.54176
$753$	25.3964	0.925497
$754$	−4.70808	−0.171458
$755$	12.1923	0.443723
$756$	−20.0937	−0.730800
$757$	7.93505	0.288404	0.144202	−	0.989548i	$-0.453938\pi$
$757$	7.93505	0.288404	0.144202	+	0.989548i	$0.453938\pi$
$758$	27.3242	0.992458
$759$	49.1464	1.78390
$760$	−58.8586	−2.13503
$761$	−38.7574	−1.40496	−0.702478	−	0.711705i	$-0.747923\pi$
$761$	−38.7574	−1.40496	−0.702478	+	0.711705i	$0.747923\pi$
$762$	−78.2989	−2.83647
$763$	−56.3006	−2.03822
$764$	45.8239	1.65785
$765$	2.21239	0.0799892
$766$	−65.7525	−2.37573
$767$	3.55056	0.128203
$768$	−107.167	−3.86707
$769$	−14.1090	−0.508785	−0.254393	−	0.967101i	$-0.581876\pi$
$769$	−14.1090	−0.508785	−0.254393	+	0.967101i	$0.581876\pi$
$770$	−76.4813	−2.75619
$771$	60.7554	2.18805
$772$	2.92635	0.105322
$773$	21.8044	0.784251	0.392125	−	0.919912i	$-0.371740\pi$
$773$	21.8044	0.784251	0.392125	+	0.919912i	$0.371740\pi$
$774$	48.8340	1.75530
$775$	−4.62841	−0.166258
$776$	16.5666	0.594707
$777$	119.945	4.30299
$778$	44.1494	1.58283
$779$	55.0784	1.97339
$780$	3.70048	0.132498
$781$	−24.5616	−0.878883
$782$	−8.20218	−0.293309
$783$	−4.50444	−0.160976
$784$	262.951	9.39111
$785$	2.48811	0.0888044
$786$	−54.6290	−1.94855
$787$	−29.4050	−1.04817	−0.524087	−	0.851665i	$-0.675593\pi$
$787$	−29.4050	−1.04817	−0.524087	+	0.851665i	$0.675593\pi$
$788$	128.411	4.57444
$789$	14.8266	0.527842
$790$	−13.7100	−0.487780
$791$	−39.0024	−1.38677
$792$	144.233	5.12510
$793$	0.332883	0.0118210
$794$	50.9689	1.80882
$795$	6.86746	0.243564
$796$	53.2758	1.88831
$797$	−18.1063	−0.641357	−0.320678	−	0.947188i	$-0.603911\pi$
$797$	−18.1063	−0.641357	−0.320678	+	0.947188i	$0.603911\pi$
$798$	−200.189	−7.08662
$799$	5.32486	0.188380
$800$	−22.0342	−0.779028
$801$	12.3391	0.435981
$802$	−2.73364	−0.0965283
$803$	−0.392299	−0.0138439
$804$	−105.294	−3.71343
$805$	−18.0605	−0.636548
$806$	−3.58672	−0.126337
$807$	−46.0364	−1.62056
$808$	175.202	6.16359
$809$	−24.4459	−0.859472	−0.429736	−	0.902955i	$-0.641393\pi$
$809$	−24.4459	−0.859472	−0.429736	+	0.902955i	$0.641393\pi$
$810$	26.8878	0.944743
$811$	21.4730	0.754020	0.377010	−	0.926209i	$-0.376952\pi$
$811$	21.4730	0.754020	0.377010	+	0.926209i	$0.376952\pi$
$812$	164.655	5.77827
$813$	26.3046	0.922544
$814$	156.833	5.49700
$815$	14.3643	0.503161
$816$	−29.4466	−1.03084
$817$	−41.1861	−1.44092
$818$	21.3062	0.744955
$819$	3.77509	0.131912
$820$	48.6188	1.69784
$821$	−21.2987	−0.743329	−0.371664	−	0.928367i	$-0.621213\pi$
$821$	−21.2987	−0.743329	−0.371664	+	0.928367i	$0.621213\pi$
$822$	−4.34223	−0.151453
$823$	25.5321	0.889992	0.444996	−	0.895532i	$-0.353205\pi$
$823$	25.5321	0.889992	0.444996	+	0.895532i	$0.353205\pi$
$824$	25.6438	0.893343
$825$	13.4757	0.469162
$826$	−169.552	−5.89946
$827$	−11.8327	−0.411462	−0.205731	−	0.978609i	$-0.565957\pi$
$827$	−11.8327	−0.411462	−0.205731	+	0.978609i	$0.565957\pi$
$828$	53.6746	1.86532
$829$	−14.0662	−0.488538	−0.244269	−	0.969708i	$-0.578548\pi$
$829$	−14.0662	−0.488538	−0.244269	+	0.969708i	$0.578548\pi$
$830$	−29.6709	−1.02989
$831$	25.7980	0.894924
$832$	−8.56728	−0.297017
$833$	14.4164	0.499497
$834$	34.2039	1.18438
$835$	−21.1094	−0.730523
$836$	−191.701	−6.63010
$837$	−3.43158	−0.118613
$838$	80.0911	2.76670
$839$	49.4711	1.70793	0.853965	−	0.520330i	$-0.174191\pi$
$839$	49.4711	1.70793	0.853965	+	0.520330i	$0.174191\pi$
$840$	−112.133	−3.86896
$841$	7.91110	0.272797
$842$	−29.5894	−1.01972
$843$	−28.1842	−0.970714
$844$	105.419	3.62868
$845$	−12.9196	−0.444449
$846$	−47.5797	−1.63582
$847$	−103.593	−3.55951
$848$	−43.2053	−1.48368
$849$	3.95410	0.135704
$850$	−2.24899	−0.0771397
$851$	37.0350	1.26954
$852$	−56.7499	−1.94422
$853$	−37.8751	−1.29682	−0.648410	−	0.761292i	$-0.724566\pi$
$853$	−37.8751	−1.29682	−0.648410	+	0.761292i	$0.724566\pi$
$854$	−15.8963	−0.543961
$855$	16.6726	0.570192
$856$	119.686	4.09078
$857$	20.4139	0.697327	0.348664	−	0.937248i	$-0.386636\pi$
$857$	20.4139	0.697327	0.348664	+	0.937248i	$0.386636\pi$
$858$	10.4428	0.356510
$859$	−51.3064	−1.75055	−0.875276	−	0.483624i	$-0.839320\pi$
$859$	−51.3064	−1.75055	−0.875276	+	0.483624i	$0.839320\pi$
$860$	−36.3558	−1.23972
$861$	104.931	3.57605
$862$	−46.9315	−1.59849
$863$	−46.7983	−1.59303	−0.796516	−	0.604618i	$-0.793326\pi$
$863$	−46.7983	−1.59303	−0.796516	+	0.604618i	$0.793326\pi$
$864$	−16.3365	−0.555780
$865$	−5.58271	−0.189818
$866$	76.5385	2.60089
$867$	38.9339	1.32227
$868$	125.438	4.25765
$869$	−28.3349	−0.961195
$870$	−39.6136	−1.34303
$871$	−2.28662	−0.0774790
$872$	−107.932	−3.65502
$873$	−4.69276	−0.158826
$874$	−61.8118	−2.09081
$875$	−4.95208	−0.167411
$876$	−0.906411	−0.0306248
$877$	−20.6369	−0.696858	−0.348429	−	0.937335i	$-0.613285\pi$
$877$	−20.6369	−0.696858	−0.348429	+	0.937335i	$0.613285\pi$
$878$	57.8347	1.95183
$879$	39.8806	1.34514
$880$	−84.7795	−2.85792
$881$	−27.4545	−0.924965	−0.462483	−	0.886628i	$-0.653041\pi$
$881$	−27.4545	−0.924965	−0.462483	+	0.886628i	$0.653041\pi$
$882$	−128.816	−4.33744
$883$	−16.9112	−0.569108	−0.284554	−	0.958660i	$-0.591845\pi$
$883$	−16.9112	−0.569108	−0.284554	+	0.958660i	$0.591845\pi$
$884$	−1.27638	−0.0429292
$885$	29.8743	1.00421
$886$	43.6669	1.46702
$887$	−24.3601	−0.817931	−0.408966	−	0.912550i	$-0.634110\pi$
$887$	−24.3601	−0.817931	−0.408966	+	0.912550i	$0.634110\pi$
$888$	229.941	7.71632
$889$	59.4671	1.99446
$890$	−12.5432	−0.420450
$891$	55.5700	1.86166
$892$	−32.8745	−1.10072
$893$	40.1282	1.34284
$894$	25.5053	0.853024
$895$	−25.5637	−0.854501
$896$	190.887	6.37710
$897$	2.46598	0.0823367
$898$	−108.255	−3.61251
$899$	28.1197	0.937844
$900$	14.7172	0.490575
$901$	−2.36874	−0.0789143
$902$	137.202	4.56834
$903$	−78.4648	−2.61115
$904$	−74.7698	−2.48681
$905$	7.28089	0.242025
$906$	79.4971	2.64111
$907$	−16.5819	−0.550592	−0.275296	−	0.961360i	$-0.588776\pi$
$907$	−16.5819	−0.550592	−0.275296	+	0.961360i	$0.588776\pi$
$908$	−15.8873	−0.527240
$909$	−49.6287	−1.64608
$910$	−3.83754	−0.127213
$911$	25.2697	0.837224	0.418612	−	0.908165i	$-0.362517\pi$
$911$	25.2697	0.837224	0.418612	+	0.908165i	$0.362517\pi$
$912$	−221.910	−7.34817
$913$	−61.3218	−2.02946
$914$	−6.89857	−0.228185
$915$	2.80086	0.0925936
$916$	15.4686	0.511098
$917$	41.4901	1.37012
$918$	−1.66744	−0.0550337
$919$	−11.8248	−0.390063	−0.195031	−	0.980797i	$-0.562481\pi$
$919$	−11.8248	−0.390063	−0.195031	+	0.980797i	$0.562481\pi$
$920$	−34.6230	−1.14149
$921$	32.7091	1.07780
$922$	−75.6546	−2.49155
$923$	−1.23241	−0.0405652
$924$	−365.214	−12.0147
$925$	10.1548	0.333887
$926$	95.8361	3.14937
$927$	−7.26400	−0.238581
$928$	133.868	4.39443
$929$	31.2062	1.02384	0.511921	−	0.859033i	$-0.328934\pi$
$929$	31.2062	1.02384	0.511921	+	0.859033i	$0.328934\pi$
$930$	−30.1785	−0.989593
$931$	108.642	3.56059
$932$	−90.4071	−2.96138
$933$	−13.3222	−0.436148
$934$	62.9158	2.05867
$935$	−4.64806	−0.152008
$936$	7.23707	0.236551
$937$	18.7041	0.611037	0.305519	−	0.952186i	$-0.401170\pi$
$937$	18.7041	0.611037	0.305519	+	0.952186i	$0.401170\pi$
$938$	109.194	3.56531
$939$	−44.8989	−1.46522
$940$	35.4220	1.15534
$941$	47.0710	1.53447	0.767236	−	0.641365i	$-0.221631\pi$
$941$	47.0710	1.53447	0.767236	+	0.641365i	$0.221631\pi$
$942$	16.2232	0.528579
$943$	32.3993	1.05507
$944$	−187.948	−6.11719
$945$	−3.67155	−0.119436
$946$	−102.596	−3.33569
$947$	51.6060	1.67697	0.838486	−	0.544924i	$-0.183441\pi$
$947$	51.6060	1.67697	0.838486	+	0.544924i	$0.183441\pi$
$948$	−65.4681	−2.12631
$949$	−0.0196841	−0.000638972 0
$950$	−16.9484	−0.549879
$951$	−25.8951	−0.839707
$952$	38.6772	1.25354
$953$	−29.4834	−0.955060	−0.477530	−	0.878615i	$-0.658468\pi$
$953$	−29.4834	−0.955060	−0.477530	+	0.878615i	$0.658468\pi$
$954$	21.1656	0.685262
$955$	8.37301	0.270944
$956$	−56.4702	−1.82638
$957$	−81.8707	−2.64650
$958$	−100.647	−3.25175
$959$	3.29787	0.106494
$960$	−72.0847	−2.32652
$961$	−9.57779	−0.308961
$962$	7.86929	0.253716
$963$	−33.9029	−1.09251
$964$	−74.5983	−2.40265
$965$	0.534707	0.0172128
$966$	−117.759	−3.78884
$967$	−48.2413	−1.55134	−0.775668	−	0.631141i	$-0.782587\pi$
$967$	−48.2413	−1.55134	−0.775668	+	0.631141i	$0.782587\pi$
$968$	−198.595	−6.38307
$969$	−12.1663	−0.390836
$970$	4.77039	0.153168
$971$	11.1156	0.356717	0.178358	−	0.983966i	$-0.442921\pi$
$971$	11.1156	0.356717	0.178358	+	0.983966i	$0.442921\pi$
$972$	116.222	3.72783
$973$	−25.9775	−0.832800
$974$	69.5985	2.23008
$975$	0.676157	0.0216544
$976$	−17.6211	−0.564037
$977$	−28.8307	−0.922374	−0.461187	−	0.887303i	$-0.652576\pi$
$977$	−28.8307	−0.922374	−0.461187	+	0.887303i	$0.652576\pi$
$978$	93.6595	2.99490
$979$	−25.9235	−0.828518
$980$	95.9003	3.06342
$981$	30.5733	0.976130
$982$	80.5901	2.57173
$983$	10.4445	0.333126	0.166563	−	0.986031i	$-0.446733\pi$
$983$	10.4445	0.333126	0.166563	+	0.986031i	$0.446733\pi$
$984$	201.160	6.41273
$985$	23.4634	0.747606
$986$	13.6636	0.435139
$987$	76.4494	2.43341
$988$	−9.61880	−0.306015
$989$	−24.2273	−0.770384
$990$	41.5321	1.31998
$991$	−48.6077	−1.54408	−0.772038	−	0.635577i	$-0.780762\pi$
$991$	−48.6077	−1.54408	−0.772038	+	0.635577i	$0.780762\pi$
$992$	101.984	3.23798
$993$	−23.6927	−0.751864
$994$	58.8518	1.86667
$995$	9.73465	0.308609
$996$	−141.685	−4.48945
$997$	12.1537	0.384911	0.192456	−	0.981306i	$-0.438355\pi$
$997$	12.1537	0.384911	0.192456	+	0.981306i	$0.438355\pi$
$998$	−73.1687	−2.31612
$999$	7.52891	0.238204

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.d.1.2	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.d.1.2	✓	25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73364 q^{2} -2.38520 q^{3} +5.47281 q^{4} +1.00000 q^{5} +6.52028 q^{6} -4.95208 q^{7} -9.49342 q^{8} +2.68916 q^{9} +O(q^{10})\)	\(q-2.73364 q^{2} -2.38520 q^{3} +5.47281 q^{4} +1.00000 q^{5} +6.52028 q^{6} -4.95208 q^{7} -9.49342 q^{8} +2.68916 q^{9} -2.73364 q^{10} -5.64971 q^{11} -13.0537 q^{12} -0.283481 q^{13} +13.5372 q^{14} -2.38520 q^{15} +15.0060 q^{16} +0.822708 q^{17} -7.35120 q^{18} +6.19994 q^{19} +5.47281 q^{20} +11.8117 q^{21} +15.4443 q^{22} +3.64705 q^{23} +22.6437 q^{24} +1.00000 q^{25} +0.774935 q^{26} +0.741417 q^{27} -27.1018 q^{28} -6.07545 q^{29} +6.52028 q^{30} -4.62841 q^{31} -22.0342 q^{32} +13.4757 q^{33} -2.24899 q^{34} -4.95208 q^{35} +14.7172 q^{36} +10.1548 q^{37} -16.9484 q^{38} +0.676157 q^{39} -9.49342 q^{40} +8.88371 q^{41} -32.2889 q^{42} -6.64299 q^{43} -30.9198 q^{44} +2.68916 q^{45} -9.96974 q^{46} +6.47236 q^{47} -35.7923 q^{48} +17.5231 q^{49} -2.73364 q^{50} -1.96232 q^{51} -1.55144 q^{52} -2.87920 q^{53} -2.02677 q^{54} -5.64971 q^{55} +47.0121 q^{56} -14.7881 q^{57} +16.6081 q^{58} -12.5249 q^{59} -13.0537 q^{60} -1.17427 q^{61} +12.6524 q^{62} -13.3169 q^{63} +30.2217 q^{64} -0.283481 q^{65} -36.8376 q^{66} +8.06621 q^{67} +4.50252 q^{68} -8.69893 q^{69} +13.5372 q^{70} +4.34741 q^{71} -25.5293 q^{72} +0.0694370 q^{73} -27.7595 q^{74} -2.38520 q^{75} +33.9311 q^{76} +27.9778 q^{77} -1.84837 q^{78} +5.01529 q^{79} +15.0060 q^{80} -9.83590 q^{81} -24.2849 q^{82} +10.8540 q^{83} +64.6430 q^{84} +0.822708 q^{85} +18.1596 q^{86} +14.4911 q^{87} +53.6350 q^{88} +4.58846 q^{89} -7.35120 q^{90} +1.40382 q^{91} +19.9596 q^{92} +11.0397 q^{93} -17.6931 q^{94} +6.19994 q^{95} +52.5559 q^{96} -1.74507 q^{97} -47.9018 q^{98} -15.1930 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9}+O(q^{10}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9	\( 25 q - 5 q^{2} - 10 q^{3} + 25 q^{4} + 25 q^{5} + 2 q^{6} - 31 q^{7} - 30 q^{8} + 17 q^{9} - 5 q^{10} - 30 q^{11} - 29 q^{12} - 18 q^{13} + 6 q^{14} - 10 q^{15} + 21 q^{16} - 18 q^{17} - 30 q^{18} - 17 q^{19} + 25 q^{20} + 6 q^{21} - 2 q^{22} - 44 q^{23} + 11 q^{24} + 25 q^{25} - 14 q^{26} - 25 q^{27} - 50 q^{28} - 9 q^{29} + 2 q^{30} - 13 q^{31} - 45 q^{32} - 21 q^{33} - 21 q^{34} - 31 q^{35} + 5 q^{36} - 28 q^{37} - 32 q^{38} + 9 q^{39} - 30 q^{40} + 28 q^{41} - 67 q^{42} - 61 q^{43} - 49 q^{44} + 17 q^{45} + 18 q^{46} - 53 q^{47} - 44 q^{48} + 28 q^{49} - 5 q^{50} - 30 q^{51} - 3 q^{52} - 36 q^{53} + 17 q^{54} - 30 q^{55} - 3 q^{56} - 13 q^{57} + 2 q^{58} - 39 q^{59} - 29 q^{60} + 10 q^{61} - 30 q^{62} - 44 q^{63} - 4 q^{64} - 18 q^{65} + 33 q^{66} - 10 q^{67} - 18 q^{68} - 6 q^{69} + 6 q^{70} - 7 q^{71} - q^{72} - 26 q^{73} - 3 q^{74} - 10 q^{75} + 12 q^{76} + 29 q^{77} - 5 q^{78} - 6 q^{79} + 21 q^{80} + 13 q^{81} - 30 q^{82} - 35 q^{83} + 117 q^{84} - 18 q^{85} + 14 q^{86} - 104 q^{87} + 53 q^{88} + 7 q^{89} - 30 q^{90} - 25 q^{91} - 31 q^{92} + 2 q^{93} + 68 q^{94} - 17 q^{95} + 92 q^{96} + 6 q^{97} + 15 q^{98} - 36 q^{99}+O(q^{100}) \) 25 * q - 5 * q^2 - 10 * q^3 + 25 * q^4 + 25 * q^5 + 2 * q^6 - 31 * q^7 - 30 * q^8 + 17 * q^9 - 5 * q^10 - 30 * q^11 - 29 * q^12 - 18 * q^13 + 6 * q^14 - 10 * q^15 + 21 * q^16 - 18 * q^17 - 30 * q^18 - 17 * q^19 + 25 * q^20 + 6 * q^21 - 2 * q^22 - 44 * q^23 + 11 * q^24 + 25 * q^25 - 14 * q^26 - 25 * q^27 - 50 * q^28 - 9 * q^29 + 2 * q^30 - 13 * q^31 - 45 * q^32 - 21 * q^33 - 21 * q^34 - 31 * q^35 + 5 * q^36 - 28 * q^37 - 32 * q^38 + 9 * q^39 - 30 * q^40 + 28 * q^41 - 67 * q^42 - 61 * q^43 - 49 * q^44 + 17 * q^45 + 18 * q^46 - 53 * q^47 - 44 * q^48 + 28 * q^49 - 5 * q^50 - 30 * q^51 - 3 * q^52 - 36 * q^53 + 17 * q^54 - 30 * q^55 - 3 * q^56 - 13 * q^57 + 2 * q^58 - 39 * q^59 - 29 * q^60 + 10 * q^61 - 30 * q^62 - 44 * q^63 - 4 * q^64 - 18 * q^65 + 33 * q^66 - 10 * q^67 - 18 * q^68 - 6 * q^69 + 6 * q^70 - 7 * q^71 - q^72 - 26 * q^73 - 3 * q^74 - 10 * q^75 + 12 * q^76 + 29 * q^77 - 5 * q^78 - 6 * q^79 + 21 * q^80 + 13 * q^81 - 30 * q^82 - 35 * q^83 + 117 * q^84 - 18 * q^85 + 14 * q^86 - 104 * q^87 + 53 * q^88 + 7 * q^89 - 30 * q^90 - 25 * q^91 - 31 * q^92 + 2 * q^93 + 68 * q^94 - 17 * q^95 + 92 * q^96 + 6 * q^97 + 15 * q^98 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more