LMFDB - Embedded newform 1205.2.a.e.1.25

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1205,2,Mod(1,1205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1205, 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("1205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1205 = 5 \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1205.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:	$9.62197344356$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.25
Character	$\chi$	$=$	1205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.77603 q^{2} +0.0161258 q^{3} +5.70635 q^{4} +1.00000 q^{5} +0.0447656 q^{6} +4.43454 q^{7} +10.2889 q^{8} -2.99974 q^{9} +O(q^{10})$	\(q+2.77603 q^{2} +0.0161258 q^{3} +5.70635 q^{4} +1.00000 q^{5} +0.0447656 q^{6} +4.43454 q^{7} +10.2889 q^{8} -2.99974 q^{9} +2.77603 q^{10} -5.65829 q^{11} +0.0920192 q^{12} -0.883628 q^{13} +12.3104 q^{14} +0.0161258 q^{15} +17.1497 q^{16} -4.14905 q^{17} -8.32737 q^{18} +0.923282 q^{19} +5.70635 q^{20} +0.0715104 q^{21} -15.7076 q^{22} -4.69414 q^{23} +0.165917 q^{24} +1.00000 q^{25} -2.45298 q^{26} -0.0967503 q^{27} +25.3051 q^{28} +2.90139 q^{29} +0.0447656 q^{30} -8.10705 q^{31} +27.0303 q^{32} -0.0912442 q^{33} -11.5179 q^{34} +4.43454 q^{35} -17.1176 q^{36} -10.7862 q^{37} +2.56306 q^{38} -0.0142492 q^{39} +10.2889 q^{40} -0.571032 q^{41} +0.198515 q^{42} +0.399694 q^{43} -32.2882 q^{44} -2.99974 q^{45} -13.0311 q^{46} +6.27481 q^{47} +0.276552 q^{48} +12.6652 q^{49} +2.77603 q^{50} -0.0669065 q^{51} -5.04229 q^{52} +3.12140 q^{53} -0.268582 q^{54} -5.65829 q^{55} +45.6268 q^{56} +0.0148886 q^{57} +8.05434 q^{58} -0.915284 q^{59} +0.0920192 q^{60} +9.75104 q^{61} -22.5054 q^{62} -13.3025 q^{63} +40.7375 q^{64} -0.883628 q^{65} -0.253297 q^{66} +9.46280 q^{67} -23.6759 q^{68} -0.0756965 q^{69} +12.3104 q^{70} -0.00570799 q^{71} -30.8641 q^{72} +0.642389 q^{73} -29.9427 q^{74} +0.0161258 q^{75} +5.26857 q^{76} -25.0919 q^{77} -0.0395561 q^{78} +5.67130 q^{79} +17.1497 q^{80} +8.99766 q^{81} -1.58520 q^{82} +7.08995 q^{83} +0.408063 q^{84} -4.14905 q^{85} +1.10956 q^{86} +0.0467870 q^{87} -58.2178 q^{88} +0.805647 q^{89} -8.32737 q^{90} -3.91849 q^{91} -26.7864 q^{92} -0.130732 q^{93} +17.4191 q^{94} +0.923282 q^{95} +0.435884 q^{96} -17.3089 q^{97} +35.1589 q^{98} +16.9734 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 6 q^{2} + 15 q^{3} + 32 q^{4} + 25 q^{5} - q^{6} + 19 q^{7} + 15 q^{8} + 32 q^{9}+O(q^{10}) $ 25 * q + 6 * q^2 + 15 * q^3 + 32 * q^4 + 25 * q^5 - q^6 + 19 * q^7 + 15 * q^8 + 32 * q^9	\( 25 q + 6 q^{2} + 15 q^{3} + 32 q^{4} + 25 q^{5} - q^{6} + 19 q^{7} + 15 q^{8} + 32 q^{9} + 6 q^{10} + 2 q^{11} + 20 q^{12} + 14 q^{13} - 5 q^{14} + 15 q^{15} + 38 q^{16} + 7 q^{17} + 9 q^{18} + 30 q^{19} + 32 q^{20} + q^{21} + q^{22} + 43 q^{23} - 6 q^{24} + 25 q^{25} - 22 q^{26} + 42 q^{27} + 32 q^{28} - 4 q^{29} - q^{30} + 14 q^{31} + 26 q^{32} + 4 q^{33} + 7 q^{34} + 19 q^{35} + 15 q^{36} + 16 q^{37} + 14 q^{38} - 21 q^{39} + 15 q^{40} - q^{41} - 25 q^{42} + 35 q^{43} - 52 q^{44} + 32 q^{45} - 27 q^{46} + 50 q^{47} + 26 q^{48} + 46 q^{49} + 6 q^{50} - 7 q^{51} + 3 q^{52} + 4 q^{53} - 31 q^{54} + 2 q^{55} - 51 q^{56} + 2 q^{58} + 6 q^{59} + 20 q^{60} + 19 q^{61} + 28 q^{63} + 49 q^{64} + 14 q^{65} - 27 q^{66} + 65 q^{67} - 25 q^{68} + 2 q^{69} - 5 q^{70} - 34 q^{71} - 10 q^{72} + 8 q^{73} - 42 q^{74} + 15 q^{75} + 71 q^{76} + q^{77} - 59 q^{78} - 12 q^{79} + 38 q^{80} + 29 q^{81} + 11 q^{82} + 41 q^{83} - 10 q^{84} + 7 q^{85} - 13 q^{86} + 40 q^{87} - 52 q^{88} - 24 q^{89} + 9 q^{90} + 46 q^{91} + 85 q^{92} - 30 q^{93} + 14 q^{94} + 30 q^{95} - 30 q^{96} + 9 q^{97} - 64 q^{98} + 2 q^{99}+O(q^{100}) \) 25 * q + 6 * q^2 + 15 * q^3 + 32 * q^4 + 25 * q^5 - q^6 + 19 * q^7 + 15 * q^8 + 32 * q^9 + 6 * q^10 + 2 * q^11 + 20 * q^12 + 14 * q^13 - 5 * q^14 + 15 * q^15 + 38 * q^16 + 7 * q^17 + 9 * q^18 + 30 * q^19 + 32 * q^20 + q^21 + q^22 + 43 * q^23 - 6 * q^24 + 25 * q^25 - 22 * q^26 + 42 * q^27 + 32 * q^28 - 4 * q^29 - q^30 + 14 * q^31 + 26 * q^32 + 4 * q^33 + 7 * q^34 + 19 * q^35 + 15 * q^36 + 16 * q^37 + 14 * q^38 - 21 * q^39 + 15 * q^40 - q^41 - 25 * q^42 + 35 * q^43 - 52 * q^44 + 32 * q^45 - 27 * q^46 + 50 * q^47 + 26 * q^48 + 46 * q^49 + 6 * q^50 - 7 * q^51 + 3 * q^52 + 4 * q^53 - 31 * q^54 + 2 * q^55 - 51 * q^56 + 2 * q^58 + 6 * q^59 + 20 * q^60 + 19 * q^61 + 28 * q^63 + 49 * q^64 + 14 * q^65 - 27 * q^66 + 65 * q^67 - 25 * q^68 + 2 * q^69 - 5 * q^70 - 34 * q^71 - 10 * q^72 + 8 * q^73 - 42 * q^74 + 15 * q^75 + 71 * q^76 + q^77 - 59 * q^78 - 12 * q^79 + 38 * q^80 + 29 * q^81 + 11 * q^82 + 41 * q^83 - 10 * q^84 + 7 * q^85 - 13 * q^86 + 40 * q^87 - 52 * q^88 - 24 * q^89 + 9 * q^90 + 46 * q^91 + 85 * q^92 - 30 * q^93 + 14 * q^94 + 30 * q^95 - 30 * q^96 + 9 * q^97 - 64 * q^98 + 2 * 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 $

$2$	2.77603	1.96295	0.981475	−	0.191589i	$-0.0613641\pi$
$2$	2.77603	1.96295	0.981475	+	0.191589i	$0.0613641\pi$
$3$	0.0161258	0.00931021	0.00465510	−	0.999989i	$-0.498518\pi$
$3$	0.0161258	0.00931021	0.00465510	+	0.999989i	$0.498518\pi$
$4$	5.70635	2.85317
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.0447656	0.0182755
$7$	4.43454	1.67610	0.838050	−	0.545593i	$-0.183696\pi$
$7$	4.43454	1.67610	0.838050	+	0.545593i	$0.183696\pi$
$8$	10.2889	3.63769
$9$	−2.99974	−0.999913
$10$	2.77603	0.877858
$11$	−5.65829	−1.70604	−0.853019	−	0.521879i	$-0.825231\pi$
$11$	−5.65829	−1.70604	−0.853019	+	0.521879i	$0.825231\pi$
$12$	0.0920192	0.0265637
$13$	−0.883628	−0.245074	−0.122537	−	0.992464i	$-0.539103\pi$
$13$	−0.883628	−0.245074	−0.122537	+	0.992464i	$0.539103\pi$
$14$	12.3104	3.29010
$15$	0.0161258	0.00416365
$16$	17.1497	4.28743
$17$	−4.14905	−1.00629	−0.503146	−	0.864201i	$-0.667824\pi$
$17$	−4.14905	−1.00629	−0.503146	+	0.864201i	$0.667824\pi$
$18$	−8.32737	−1.96278
$19$	0.923282	0.211815	0.105908	−	0.994376i	$-0.466225\pi$
$19$	0.923282	0.211815	0.105908	+	0.994376i	$0.466225\pi$
$20$	5.70635	1.27598
$21$	0.0715104	0.0156048
$22$	−15.7076	−3.34887
$23$	−4.69414	−0.978796	−0.489398	−	0.872061i	$-0.662783\pi$
$23$	−4.69414	−0.978796	−0.489398	+	0.872061i	$0.662783\pi$
$24$	0.165917	0.0338677
$25$	1.00000	0.200000
$26$	−2.45298	−0.481069
$27$	−0.0967503	−0.0186196
$28$	25.3051	4.78221
$29$	2.90139	0.538774	0.269387	−	0.963032i	$-0.413179\pi$
$29$	2.90139	0.538774	0.269387	+	0.963032i	$0.413179\pi$
$30$	0.0447656	0.00817304
$31$	−8.10705	−1.45607	−0.728035	−	0.685540i	$-0.759566\pi$
$31$	−8.10705	−1.45607	−0.728035	+	0.685540i	$0.759566\pi$
$32$	27.0303	4.77832
$33$	−0.0912442	−0.0158836
$34$	−11.5179	−1.97530
$35$	4.43454	0.749575
$36$	−17.1176	−2.85293
$37$	−10.7862	−1.77323	−0.886616	−	0.462506i	$-0.846950\pi$
$37$	−10.7862	−1.77323	−0.886616	+	0.462506i	$0.846950\pi$
$38$	2.56306	0.415783
$39$	−0.0142492	−0.00228169
$40$	10.2889	1.62682
$41$	−0.571032	−0.0891802	−0.0445901	−	0.999005i	$-0.514198\pi$
$41$	−0.571032	−0.0891802	−0.0445901	+	0.999005i	$0.514198\pi$
$42$	0.198515	0.0306315
$43$	0.399694	0.0609528	0.0304764	−	0.999535i	$-0.490298\pi$
$43$	0.399694	0.0609528	0.0304764	+	0.999535i	$0.490298\pi$
$44$	−32.2882	−4.86763
$45$	−2.99974	−0.447175
$46$	−13.0311	−1.92133
$47$	6.27481	0.915275	0.457638	−	0.889139i	$-0.348696\pi$
$47$	6.27481	0.915275	0.457638	+	0.889139i	$0.348696\pi$
$48$	0.276552	0.0399169
$49$	12.6652	1.80931
$50$	2.77603	0.392590
$51$	−0.0669065	−0.00936879
$52$	−5.04229	−0.699240
$53$	3.12140	0.428757	0.214379	−	0.976751i	$-0.431227\pi$
$53$	3.12140	0.428757	0.214379	+	0.976751i	$0.431227\pi$
$54$	−0.268582	−0.0365494
$55$	−5.65829	−0.762964
$56$	45.6268	6.09713
$57$	0.0148886	0.00197205
$58$	8.05434	1.05759
$59$	−0.915284	−0.119160	−0.0595799	−	0.998224i	$-0.518976\pi$
$59$	−0.915284	−0.119160	−0.0595799	+	0.998224i	$0.518976\pi$
$60$	0.0920192	0.0118796
$61$	9.75104	1.24849	0.624246	−	0.781228i	$-0.285406\pi$
$61$	9.75104	1.24849	0.624246	+	0.781228i	$0.285406\pi$
$62$	−22.5054	−2.85819
$63$	−13.3025	−1.67595
$64$	40.7375	5.09218
$65$	−0.883628	−0.109601
$66$	−0.253297	−0.0311787
$67$	9.46280	1.15606	0.578032	−	0.816014i	$-0.303821\pi$
$67$	9.46280	1.15606	0.578032	+	0.816014i	$0.303821\pi$
$68$	−23.6759	−2.87113
$69$	−0.0756965	−0.00911279
$70$	12.3104	1.47138
$71$	−0.00570799	−0.000677414 0	−0.000338707	−	1.00000i	$-0.500108\pi$
$71$	−0.00570799	−0.000677414 0	−0.000338707		1.00000i	$0.500108\pi$
$72$	−30.8641	−3.63737
$73$	0.642389	0.0751859	0.0375930	−	0.999293i	$-0.488031\pi$
$73$	0.642389	0.0751859	0.0375930	+	0.999293i	$0.488031\pi$
$74$	−29.9427	−3.48077
$75$	0.0161258	0.00186204
$76$	5.26857	0.604346
$77$	−25.0919	−2.85949
$78$	−0.0395561	−0.00447885
$79$	5.67130	0.638071	0.319035	−	0.947743i	$-0.396641\pi$
$79$	5.67130	0.638071	0.319035	+	0.947743i	$0.396641\pi$
$80$	17.1497	1.91740
$81$	8.99766	0.999740
$82$	−1.58520	−0.175056
$83$	7.08995	0.778223	0.389111	−	0.921191i	$-0.372782\pi$
$83$	7.08995	0.778223	0.389111	+	0.921191i	$0.372782\pi$
$84$	0.408063	0.0445233
$85$	−4.14905	−0.450028
$86$	1.10956	0.119647
$87$	0.0467870	0.00501610
$88$	−58.2178	−6.20604
$89$	0.805647	0.0853984	0.0426992	−	0.999088i	$-0.486404\pi$
$89$	0.805647	0.0853984	0.0426992	+	0.999088i	$0.486404\pi$
$90$	−8.32737	−0.877782
$91$	−3.91849	−0.410769
$92$	−26.7864	−2.79268
$93$	−0.130732	−0.0135563
$94$	17.4191	1.79664
$95$	0.923282	0.0947267
$96$	0.435884	0.0444872
$97$	−17.3089	−1.75745	−0.878725	−	0.477328i	$-0.841606\pi$
$97$	−17.3089	−1.75745	−0.878725	+	0.477328i	$0.841606\pi$
$98$	35.1589	3.55159
$99$	16.9734	1.70589
$100$	5.70635	0.570635
$101$	13.6915	1.36235	0.681175	−	0.732120i	$-0.261469\pi$
$101$	13.6915	1.36235	0.681175	+	0.732120i	$0.261469\pi$
$102$	−0.185735	−0.0183905
$103$	18.2879	1.80196	0.900980	−	0.433860i	$-0.142849\pi$
$103$	18.2879	1.80196	0.900980	+	0.433860i	$0.142849\pi$
$104$	−9.09160	−0.891505
$105$	0.0715104	0.00697870
$106$	8.66511	0.841629
$107$	−14.0360	−1.35691	−0.678455	−	0.734642i	$-0.737350\pi$
$107$	−14.0360	−1.35691	−0.678455	+	0.734642i	$0.737350\pi$
$108$	−0.552091	−0.0531250
$109$	11.6798	1.11872	0.559361	−	0.828924i	$-0.311046\pi$
$109$	11.6798	1.11872	0.559361	+	0.828924i	$0.311046\pi$
$110$	−15.7076	−1.49766
$111$	−0.173935	−0.0165092
$112$	76.0512	7.18616
$113$	7.28826	0.685622	0.342811	−	0.939404i	$-0.388621\pi$
$113$	7.28826	0.685622	0.342811	+	0.939404i	$0.388621\pi$
$114$	0.0413313	0.00387103
$115$	−4.69414	−0.437731
$116$	16.5563	1.53722
$117$	2.65066	0.245053
$118$	−2.54086	−0.233905
$119$	−18.3991	−1.68665
$120$	0.165917	0.0151461
$121$	21.0163	1.91057
$122$	27.0692	2.45073
$123$	−0.00920832	−0.000830287 0
$124$	−46.2617	−4.15442
$125$	1.00000	0.0894427
$126$	−36.9281	−3.28982
$127$	−13.4364	−1.19229	−0.596144	−	0.802877i	$-0.703301\pi$
$127$	−13.4364	−1.19229	−0.596144	+	0.802877i	$0.703301\pi$
$128$	59.0279	5.21738
$129$	0.00644537	0.000567483 0
$130$	−2.45298	−0.215141
$131$	−17.1859	−1.50154	−0.750771	−	0.660563i	$-0.770318\pi$
$131$	−17.1859	−1.50154	−0.750771	+	0.660563i	$0.770318\pi$
$132$	−0.520671	−0.0453186
$133$	4.09433	0.355024
$134$	26.2690	2.26930
$135$	−0.0967503	−0.00832694
$136$	−42.6893	−3.66058
$137$	−5.39580	−0.460994	−0.230497	−	0.973073i	$-0.574035\pi$
$137$	−5.39580	−0.460994	−0.230497	+	0.973073i	$0.574035\pi$
$138$	−0.210136	−0.0178880
$139$	6.69869	0.568175	0.284088	−	0.958798i	$-0.408309\pi$
$139$	6.69869	0.568175	0.284088	+	0.958798i	$0.408309\pi$
$140$	25.3051	2.13867
$141$	0.101186	0.00852140
$142$	−0.0158456	−0.00132973
$143$	4.99983	0.418106
$144$	−51.4447	−4.28706
$145$	2.90139	0.240947
$146$	1.78329	0.147586
$147$	0.204236	0.0168451
$148$	−61.5496	−5.05934
$149$	−14.9888	−1.22793	−0.613965	−	0.789334i	$-0.710426\pi$
$149$	−14.9888	−1.22793	−0.613965	+	0.789334i	$0.710426\pi$
$150$	0.0447656	0.00365510
$151$	11.4441	0.931305	0.465653	−	0.884968i	$-0.345820\pi$
$151$	11.4441	0.931305	0.465653	+	0.884968i	$0.345820\pi$
$152$	9.49959	0.770519
$153$	12.4461	1.00620
$154$	−69.6560	−5.61304
$155$	−8.10705	−0.651174
$156$	−0.0813108	−0.00651007
$157$	10.8518	0.866066	0.433033	−	0.901378i	$-0.357443\pi$
$157$	10.8518	0.866066	0.433033	+	0.901378i	$0.357443\pi$
$158$	15.7437	1.25250
$159$	0.0503349	0.00399182
$160$	27.0303	2.13693
$161$	−20.8164	−1.64056
$162$	24.9778	1.96244
$163$	−23.3704	−1.83051	−0.915254	−	0.402878i	$-0.868010\pi$
$163$	−23.3704	−1.83051	−0.915254	+	0.402878i	$0.868010\pi$
$164$	−3.25851	−0.254447
$165$	−0.0912442	−0.00710335
$166$	19.6819	1.52761
$167$	8.99977	0.696423	0.348212	−	0.937416i	$-0.386789\pi$
$167$	8.99977	0.696423	0.348212	+	0.937416i	$0.386789\pi$
$168$	0.735766	0.0567656
$169$	−12.2192	−0.939939
$170$	−11.5179	−0.883382
$171$	−2.76960	−0.211797
$172$	2.28079	0.173909
$173$	−10.0497	−0.764062	−0.382031	−	0.924149i	$-0.624775\pi$
$173$	−10.0497	−0.764062	−0.382031	+	0.924149i	$0.624775\pi$
$174$	0.129882	0.00984635
$175$	4.43454	0.335220
$176$	−97.0381	−7.31452
$177$	−0.0147596	−0.00110940
$178$	2.23650	0.167633
$179$	−19.1017	−1.42773	−0.713863	−	0.700285i	$-0.753056\pi$
$179$	−19.1017	−1.42773	−0.713863	+	0.700285i	$0.753056\pi$
$180$	−17.1176	−1.27587
$181$	6.56496	0.487969	0.243985	−	0.969779i	$-0.421545\pi$
$181$	6.56496	0.487969	0.243985	+	0.969779i	$0.421545\pi$
$182$	−10.8778	−0.806320
$183$	0.157243	0.0116237
$184$	−48.2977	−3.56056
$185$	−10.7862	−0.793014
$186$	−0.362917	−0.0266104
$187$	23.4765	1.71677
$188$	35.8063	2.61144
$189$	−0.429044	−0.0312083
$190$	2.56306	0.185944
$191$	−22.0112	−1.59268	−0.796338	−	0.604852i	$-0.793232\pi$
$191$	−22.0112	−1.59268	−0.796338	+	0.604852i	$0.793232\pi$
$192$	0.656922	0.0474093
$193$	−6.85693	−0.493572	−0.246786	−	0.969070i	$-0.579375\pi$
$193$	−6.85693	−0.493572	−0.246786	+	0.969070i	$0.579375\pi$
$194$	−48.0500	−3.44979
$195$	−0.0142492	−0.00102040
$196$	72.2720	5.16228
$197$	0.690678	0.0492088	0.0246044	−	0.999697i	$-0.492167\pi$
$197$	0.690678	0.0492088	0.0246044	+	0.999697i	$0.492167\pi$
$198$	47.1187	3.34858
$199$	27.1032	1.92130	0.960648	−	0.277770i	$-0.0895952\pi$
$199$	27.1032	1.92130	0.960648	+	0.277770i	$0.0895952\pi$
$200$	10.2889	0.727538
$201$	0.152595	0.0107632
$202$	38.0079	2.67423
$203$	12.8663	0.903039
$204$	−0.381792	−0.0267308
$205$	−0.571032	−0.0398826
$206$	50.7678	3.53716
$207$	14.0812	0.978711
$208$	−15.1540	−1.05074
$209$	−5.22420	−0.361365
$210$	0.198515	0.0136988
$211$	1.93760	0.133390	0.0666951	−	0.997773i	$-0.478755\pi$
$211$	1.93760	0.133390	0.0666951	+	0.997773i	$0.478755\pi$
$212$	17.8118	1.22332
$213$	−9.20456e−5 0	−6.30686e−6 0
$214$	−38.9643	−2.66355
$215$	0.399694	0.0272589
$216$	−0.995458	−0.0677324
$217$	−35.9511	−2.44052
$218$	32.4235	2.19600
$219$	0.0103590	0.000699997 0
$220$	−32.2882	−2.17687
$221$	3.66622	0.246616
$222$	−0.482849	−0.0324067
$223$	−13.3853	−0.896343	−0.448171	−	0.893948i	$-0.647925\pi$
$223$	−13.3853	−0.896343	−0.448171	+	0.893948i	$0.647925\pi$
$224$	119.867	8.00895
$225$	−2.99974	−0.199983
$226$	20.2324	1.34584
$227$	−14.2340	−0.944745	−0.472373	−	0.881399i	$-0.656602\pi$
$227$	−14.2340	−0.944745	−0.472373	+	0.881399i	$0.656602\pi$
$228$	0.0849596	0.00562659
$229$	−6.35852	−0.420183	−0.210091	−	0.977682i	$-0.567376\pi$
$229$	−6.35852	−0.420183	−0.210091	+	0.977682i	$0.567376\pi$
$230$	−13.0311	−0.859244
$231$	−0.404626	−0.0266225
$232$	29.8522	1.95989
$233$	−10.4901	−0.687230	−0.343615	−	0.939111i	$-0.611651\pi$
$233$	−10.4901	−0.687230	−0.343615	+	0.939111i	$0.611651\pi$
$234$	7.35830	0.481027
$235$	6.27481	0.409323
$236$	−5.22293	−0.339984
$237$	0.0914539	0.00594057
$238$	−51.0766	−3.31080
$239$	3.17681	0.205491	0.102745	−	0.994708i	$-0.467237\pi$
$239$	3.17681	0.205491	0.102745	+	0.994708i	$0.467237\pi$
$240$	0.276552	0.0178514
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	58.3418	3.75035
$243$	0.435345	0.0279274
$244$	55.6428	3.56217
$245$	12.6652	0.809149
$246$	−0.0255626	−0.00162981
$247$	−0.815838	−0.0519105
$248$	−83.4130	−5.29673
$249$	0.114331	0.00724542
$250$	2.77603	0.175572
$251$	21.3922	1.35026	0.675131	−	0.737698i	$-0.264087\pi$
$251$	21.3922	1.35026	0.675131	+	0.737698i	$0.264087\pi$
$252$	−75.9086	−4.78179
$253$	26.5608	1.66986
$254$	−37.2999	−2.34040
$255$	−0.0669065	−0.00418985
$256$	82.3884	5.14927
$257$	−11.8153	−0.737019	−0.368510	−	0.929624i	$-0.620132\pi$
$257$	−11.8153	−0.737019	−0.368510	+	0.929624i	$0.620132\pi$
$258$	0.0178925	0.00111394
$259$	−47.8317	−2.97212
$260$	−5.04229	−0.312710
$261$	−8.70340	−0.538727
$262$	−47.7087	−2.94745
$263$	14.3925	0.887478	0.443739	−	0.896156i	$-0.353652\pi$
$263$	14.3925	0.887478	0.443739	+	0.896156i	$0.353652\pi$
$264$	−0.938806	−0.0577795
$265$	3.12140	0.191746
$266$	11.3660	0.696894
$267$	0.0129917	0.000795077 0
$268$	53.9980	3.29845
$269$	−14.3220	−0.873226	−0.436613	−	0.899650i	$-0.643822\pi$
$269$	−14.3220	−0.873226	−0.436613	+	0.899650i	$0.643822\pi$
$270$	−0.268582	−0.0163454
$271$	5.80389	0.352561	0.176281	−	0.984340i	$-0.443593\pi$
$271$	5.80389	0.352561	0.176281	+	0.984340i	$0.443593\pi$
$272$	−71.1550	−4.31441
$273$	−0.0631886	−0.00382435
$274$	−14.9789	−0.904909
$275$	−5.65829	−0.341208
$276$	−0.431951	−0.0260004
$277$	17.7171	1.06452	0.532259	−	0.846582i	$-0.321343\pi$
$277$	17.7171	1.06452	0.532259	+	0.846582i	$0.321343\pi$
$278$	18.5958	1.11530
$279$	24.3191	1.45594
$280$	45.6268	2.72672
$281$	−21.4662	−1.28057	−0.640284	−	0.768138i	$-0.721183\pi$
$281$	−21.4662	−1.28057	−0.640284	+	0.768138i	$0.721183\pi$
$282$	0.280896	0.0167271
$283$	11.0710	0.658100	0.329050	−	0.944312i	$-0.393271\pi$
$283$	11.0710	0.658100	0.329050	+	0.944312i	$0.393271\pi$
$284$	−0.0325718	−0.00193278
$285$	0.0148886	0.000881925 0
$286$	13.8797	0.820722
$287$	−2.53227	−0.149475
$288$	−81.0838	−4.77791
$289$	0.214608	0.0126240
$290$	8.05434	0.472967
$291$	−0.279119	−0.0163622
$292$	3.66569	0.214519
$293$	18.6857	1.09163	0.545816	−	0.837905i	$-0.316220\pi$
$293$	18.6857	1.09163	0.545816	+	0.837905i	$0.316220\pi$
$294$	0.566965	0.0330660
$295$	−0.915284	−0.0532899
$296$	−110.978	−6.45047
$297$	0.547441	0.0317658
$298$	−41.6093	−2.41036
$299$	4.14787	0.239878
$300$	0.0920192	0.00531273
$301$	1.77246	0.102163
$302$	31.7691	1.82811
$303$	0.220785	0.0126838
$304$	15.8340	0.908144
$305$	9.75104	0.558343
$306$	34.5507	1.97513
$307$	22.0973	1.26116	0.630579	−	0.776125i	$-0.282817\pi$
$307$	22.0973	1.26116	0.630579	+	0.776125i	$0.282817\pi$
$308$	−143.183	−8.15863
$309$	0.294906	0.0167766
$310$	−22.5054	−1.27822
$311$	16.8706	0.956645	0.478322	−	0.878184i	$-0.341245\pi$
$311$	16.8706	0.956645	0.478322	+	0.878184i	$0.341245\pi$
$312$	−0.146609	−0.00830009
$313$	13.5836	0.767788	0.383894	−	0.923377i	$-0.374583\pi$
$313$	13.5836	0.767788	0.383894	+	0.923377i	$0.374583\pi$
$314$	30.1249	1.70005
$315$	−13.3025	−0.749510
$316$	32.3624	1.82053
$317$	−1.54768	−0.0869266	−0.0434633	−	0.999055i	$-0.513839\pi$
$317$	−1.54768	−0.0869266	−0.0434633	+	0.999055i	$0.513839\pi$
$318$	0.139731	0.00783575
$319$	−16.4169	−0.919169
$320$	40.7375	2.27729
$321$	−0.226341	−0.0126331
$322$	−57.7869	−3.22034
$323$	−3.83074	−0.213148
$324$	51.3438	2.85243
$325$	−0.883628	−0.0490149
$326$	−64.8768	−3.59320
$327$	0.188346	0.0104155
$328$	−5.87532	−0.324410
$329$	27.8259	1.53409
$330$	−0.253297	−0.0139435
$331$	−14.5748	−0.801105	−0.400552	−	0.916274i	$-0.631182\pi$
$331$	−14.5748	−0.801105	−0.400552	+	0.916274i	$0.631182\pi$
$332$	40.4577	2.22041
$333$	32.3557	1.77308
$334$	24.9836	1.36704
$335$	9.46280	0.517008
$336$	1.22638	0.0669047
$337$	23.5637	1.28360	0.641799	−	0.766873i	$-0.278188\pi$
$337$	23.5637	1.28360	0.641799	+	0.766873i	$0.278188\pi$
$338$	−33.9209	−1.84505
$339$	0.117529	0.00638328
$340$	−23.6759	−1.28401
$341$	45.8721	2.48411
$342$	−7.68851	−0.415747
$343$	25.1225	1.35649
$344$	4.11243	0.221727
$345$	−0.0756965	−0.00407537
$346$	−27.8982	−1.49982
$347$	17.8585	0.958695	0.479347	−	0.877625i	$-0.340873\pi$
$347$	17.8585	0.958695	0.479347	+	0.877625i	$0.340873\pi$
$348$	0.266983	0.0143118
$349$	0.302469	0.0161908	0.00809540	−	0.999967i	$-0.497423\pi$
$349$	0.302469	0.0161908	0.00809540	+	0.999967i	$0.497423\pi$
$350$	12.3104	0.658020
$351$	0.0854913	0.00456319
$352$	−152.945	−8.15201
$353$	−6.66887	−0.354948	−0.177474	−	0.984125i	$-0.556793\pi$
$353$	−6.66887	−0.354948	−0.177474	+	0.984125i	$0.556793\pi$
$354$	−0.0409732	−0.00217770
$355$	−0.00570799	−0.000302949 0
$356$	4.59730	0.243656
$357$	−0.296700	−0.0157030
$358$	−53.0269	−2.80256
$359$	−18.6540	−0.984519	−0.492259	−	0.870449i	$-0.663829\pi$
$359$	−18.6540	−0.984519	−0.492259	+	0.870449i	$0.663829\pi$
$360$	−30.8641	−1.62668
$361$	−18.1476	−0.955134
$362$	18.2245	0.957860
$363$	0.338903	0.0177878
$364$	−22.3603	−1.17200
$365$	0.642389	0.0336242
$366$	0.436511	0.0228168
$367$	20.4542	1.06770	0.533851	−	0.845579i	$-0.320744\pi$
$367$	20.4542	1.06770	0.533851	+	0.845579i	$0.320744\pi$
$368$	−80.5032	−4.19652
$369$	1.71295	0.0891725
$370$	−29.9427	−1.55665
$371$	13.8420	0.718640
$372$	−0.746004	−0.0386785
$373$	8.97651	0.464786	0.232393	−	0.972622i	$-0.425344\pi$
$373$	8.97651	0.464786	0.232393	+	0.972622i	$0.425344\pi$
$374$	65.1716	3.36994
$375$	0.0161258	0.000832730 0
$376$	64.5612	3.32949
$377$	−2.56375	−0.132040
$378$	−1.19104	−0.0612604
$379$	2.77138	0.142356	0.0711780	−	0.997464i	$-0.477324\pi$
$379$	2.77138	0.142356	0.0711780	+	0.997464i	$0.477324\pi$
$380$	5.26857	0.270272
$381$	−0.216672	−0.0111005
$382$	−61.1038	−3.12634
$383$	−19.0020	−0.970957	−0.485479	−	0.874249i	$-0.661355\pi$
$383$	−19.0020	−0.970957	−0.485479	+	0.874249i	$0.661355\pi$
$384$	0.951870	0.0485749
$385$	−25.0919	−1.27880
$386$	−19.0350	−0.968858
$387$	−1.19898	−0.0609475
$388$	−98.7705	−5.01431
$389$	2.35934	0.119624	0.0598118	−	0.998210i	$-0.480950\pi$
$389$	2.35934	0.119624	0.0598118	+	0.998210i	$0.480950\pi$
$390$	−0.0395561	−0.00200300
$391$	19.4762	0.984955
$392$	130.311	6.58172
$393$	−0.277136	−0.0139797
$394$	1.91734	0.0965944
$395$	5.67130	0.285354
$396$	96.8562	4.86720
$397$	17.0741	0.856922	0.428461	−	0.903560i	$-0.359056\pi$
$397$	17.0741	0.856922	0.428461	+	0.903560i	$0.359056\pi$
$398$	75.2393	3.77141
$399$	0.0660242	0.00330535
$400$	17.1497	0.857486
$401$	−14.5099	−0.724591	−0.362296	−	0.932063i	$-0.618007\pi$
$401$	−14.5099	−0.724591	−0.362296	+	0.932063i	$0.618007\pi$
$402$	0.423608	0.0211276
$403$	7.16362	0.356845
$404$	78.1282	3.88702
$405$	8.99766	0.447097
$406$	35.7173	1.77262
$407$	61.0312	3.02520
$408$	−0.688398	−0.0340808
$409$	3.01447	0.149056	0.0745281	−	0.997219i	$-0.476255\pi$
$409$	3.01447	0.149056	0.0745281	+	0.997219i	$0.476255\pi$
$410$	−1.58520	−0.0782876
$411$	−0.0870114	−0.00429195
$412$	104.357	5.14131
$413$	−4.05887	−0.199724
$414$	39.0898	1.92116
$415$	7.08995	0.348032
$416$	−23.8847	−1.17104
$417$	0.108021	0.00528983
$418$	−14.5025	−0.709342
$419$	6.09344	0.297684	0.148842	−	0.988861i	$-0.452445\pi$
$419$	6.09344	0.297684	0.148842	+	0.988861i	$0.452445\pi$
$420$	0.408063	0.0199114
$421$	−10.8609	−0.529330	−0.264665	−	0.964340i	$-0.585261\pi$
$421$	−10.8609	−0.529330	−0.264665	+	0.964340i	$0.585261\pi$
$422$	5.37885	0.261838
$423$	−18.8228	−0.915196
$424$	32.1159	1.55969
$425$	−4.14905	−0.201258
$426$	−0.000255522 0	−1.23801e−5 0
$427$	43.2414	2.09260
$428$	−80.0942	−3.87150
$429$	0.0806260	0.00389266
$430$	1.10956	0.0535079
$431$	17.6702	0.851146	0.425573	−	0.904924i	$-0.360073\pi$
$431$	17.6702	0.851146	0.425573	+	0.904924i	$0.360073\pi$
$432$	−1.65924	−0.0798303
$433$	8.46446	0.406776	0.203388	−	0.979098i	$-0.434805\pi$
$433$	8.46446	0.406776	0.203388	+	0.979098i	$0.434805\pi$
$434$	−99.8013	−4.79062
$435$	0.0467870	0.00224327
$436$	66.6491	3.19191
$437$	−4.33401	−0.207324
$438$	0.0287569	0.00137406
$439$	26.4312	1.26149	0.630747	−	0.775988i	$-0.282748\pi$
$439$	26.4312	1.26149	0.630747	+	0.775988i	$0.282748\pi$
$440$	−58.2178	−2.77543
$441$	−37.9923	−1.80916
$442$	10.1775	0.484096
$443$	34.5605	1.64202	0.821010	−	0.570914i	$-0.193411\pi$
$443$	34.5605	1.64202	0.821010	+	0.570914i	$0.193411\pi$
$444$	−0.992533	−0.0471035
$445$	0.805647	0.0381913
$446$	−37.1579	−1.75948
$447$	−0.241705	−0.0114323
$448$	180.652	8.53501
$449$	−36.7068	−1.73230	−0.866151	−	0.499782i	$-0.833413\pi$
$449$	−36.7068	−1.73230	−0.866151	+	0.499782i	$0.833413\pi$
$450$	−8.32737	−0.392556
$451$	3.23107	0.152145
$452$	41.5894	1.95620
$453$	0.184544	0.00867064
$454$	−39.5141	−1.85449
$455$	−3.91849	−0.183702
$456$	0.153188	0.00717369
$457$	−26.3622	−1.23317	−0.616585	−	0.787288i	$-0.711484\pi$
$457$	−26.3622	−1.23317	−0.616585	+	0.787288i	$0.711484\pi$
$458$	−17.6514	−0.824798
$459$	0.401422	0.0187368
$460$	−26.7864	−1.24892
$461$	13.5540	0.631274	0.315637	−	0.948880i	$-0.397782\pi$
$461$	13.5540	0.631274	0.315637	+	0.948880i	$0.397782\pi$
$462$	−1.12326	−0.0522586
$463$	7.73657	0.359549	0.179774	−	0.983708i	$-0.442463\pi$
$463$	7.73657	0.359549	0.179774	+	0.983708i	$0.442463\pi$
$464$	49.7580	2.30996
$465$	−0.130732	−0.00606257
$466$	−29.1209	−1.34900
$467$	−2.79777	−0.129465	−0.0647327	−	0.997903i	$-0.520619\pi$
$467$	−2.79777	−0.129465	−0.0647327	+	0.997903i	$0.520619\pi$
$468$	15.1256	0.699179
$469$	41.9632	1.93768
$470$	17.4191	0.803482
$471$	0.174993	0.00806326
$472$	−9.41730	−0.433466
$473$	−2.26158	−0.103988
$474$	0.253879	0.0116610
$475$	0.923282	0.0423631
$476$	−104.992	−4.81230
$477$	−9.36339	−0.428720
$478$	8.81893	0.403368
$479$	3.87948	0.177258	0.0886289	−	0.996065i	$-0.471751\pi$
$479$	3.87948	0.177258	0.0886289	+	0.996065i	$0.471751\pi$
$480$	0.435884	0.0198953
$481$	9.53095	0.434574
$482$	−2.77603	−0.126445
$483$	−0.335680	−0.0152740
$484$	119.926	5.45119
$485$	−17.3089	−0.785956
$486$	1.20853	0.0548201
$487$	−35.2547	−1.59754	−0.798771	−	0.601635i	$-0.794516\pi$
$487$	−35.2547	−1.59754	−0.798771	+	0.601635i	$0.794516\pi$
$488$	100.328	4.54163
$489$	−0.376865	−0.0170424
$490$	35.1589	1.58832
$491$	7.47393	0.337294	0.168647	−	0.985677i	$-0.446060\pi$
$491$	7.47393	0.337294	0.168647	+	0.985677i	$0.446060\pi$
$492$	−0.0525459	−0.00236895
$493$	−12.0380	−0.542164
$494$	−2.26479	−0.101898
$495$	16.9734	0.762898
$496$	−139.034	−6.24280
$497$	−0.0253123	−0.00113541
$498$	0.317386	0.0142224
$499$	−18.6982	−0.837045	−0.418523	−	0.908206i	$-0.637452\pi$
$499$	−18.6982	−0.837045	−0.418523	+	0.908206i	$0.637452\pi$
$500$	5.70635	0.255196
$501$	0.145128	0.00648384
$502$	59.3854	2.65050
$503$	32.3364	1.44181	0.720905	−	0.693033i	$-0.243726\pi$
$503$	32.3364	1.44181	0.720905	+	0.693033i	$0.243726\pi$
$504$	−136.868	−6.09660
$505$	13.6915	0.609262
$506$	73.7336	3.27786
$507$	−0.197044	−0.00875102
$508$	−76.6728	−3.40181
$509$	−44.2128	−1.95970	−0.979850	−	0.199737i	$-0.935991\pi$
$509$	−44.2128	−1.95970	−0.979850	+	0.199737i	$0.935991\pi$
$510$	−0.185735	−0.00822447
$511$	2.84870	0.126019
$512$	110.657	4.89039
$513$	−0.0893278	−0.00394392
$514$	−32.7997	−1.44673
$515$	18.2879	0.805861
$516$	0.0367795	0.00161913
$517$	−35.5047	−1.56149
$518$	−132.782	−5.83412
$519$	−0.162059	−0.00711358
$520$	−9.09160	−0.398693
$521$	−37.7735	−1.65489	−0.827443	−	0.561549i	$-0.810206\pi$
$521$	−37.7735	−1.65489	−0.827443	+	0.561549i	$0.810206\pi$
$522$	−24.1609	−1.05749
$523$	16.1245	0.705075	0.352538	−	0.935798i	$-0.385319\pi$
$523$	16.1245	0.705075	0.352538	+	0.935798i	$0.385319\pi$
$524$	−98.0689	−4.28416
$525$	0.0715104	0.00312097
$526$	39.9540	1.74208
$527$	33.6366	1.46523
$528$	−1.56481	−0.0680997
$529$	−0.965050	−0.0419587
$530$	8.66511	0.376388
$531$	2.74561	0.119149
$532$	23.3637	1.01294
$533$	0.504580	0.0218558
$534$	0.0360653	0.00156070
$535$	−14.0360	−0.606828
$536$	97.3622	4.20541
$537$	−0.308029	−0.0132924
$538$	−39.7582	−1.71410
$539$	−71.6633	−3.08676
$540$	−0.552091	−0.0237582
$541$	−15.0277	−0.646093	−0.323046	−	0.946383i	$-0.604707\pi$
$541$	−15.0277	−0.646093	−0.323046	+	0.946383i	$0.604707\pi$
$542$	16.1118	0.692061
$543$	0.105865	0.00454310
$544$	−112.150	−4.80839
$545$	11.6798	0.500308
$546$	−0.175413	−0.00750700
$547$	18.9799	0.811522	0.405761	−	0.913979i	$-0.367006\pi$
$547$	18.9799	0.811522	0.405761	+	0.913979i	$0.367006\pi$
$548$	−30.7903	−1.31530
$549$	−29.2506	−1.24838
$550$	−15.7076	−0.669774
$551$	2.67880	0.114121
$552$	−0.778837	−0.0331495
$553$	25.1496	1.06947
$554$	49.1832	2.08959
$555$	−0.173935	−0.00738312
$556$	38.2250	1.62110
$557$	−10.2325	−0.433563	−0.216782	−	0.976220i	$-0.569556\pi$
$557$	−10.2325	−0.433563	−0.216782	+	0.976220i	$0.569556\pi$
$558$	67.5104	2.85795
$559$	−0.353181	−0.0149380
$560$	76.0512	3.21375
$561$	0.378577	0.0159835
$562$	−59.5910	−2.51369
$563$	37.8817	1.59652	0.798261	−	0.602312i	$-0.205754\pi$
$563$	37.8817	1.59652	0.798261	+	0.602312i	$0.205754\pi$
$564$	0.577403	0.0243130
$565$	7.28826	0.306619
$566$	30.7333	1.29182
$567$	39.9005	1.67566
$568$	−0.0587292	−0.00246422
$569$	5.39223	0.226054	0.113027	−	0.993592i	$-0.463945\pi$
$569$	5.39223	0.226054	0.113027	+	0.993592i	$0.463945\pi$
$570$	0.0413313	0.00173118
$571$	9.44000	0.395052	0.197526	−	0.980298i	$-0.436709\pi$
$571$	9.44000	0.395052	0.197526	+	0.980298i	$0.436709\pi$
$572$	28.5308	1.19293
$573$	−0.354947	−0.0148281
$574$	−7.02965	−0.293412
$575$	−4.69414	−0.195759
$576$	−122.202	−5.09174
$577$	−7.43829	−0.309660	−0.154830	−	0.987941i	$-0.549483\pi$
$577$	−7.43829	−0.309660	−0.154830	+	0.987941i	$0.549483\pi$
$578$	0.595759	0.0247803
$579$	−0.110573	−0.00459526
$580$	16.5563	0.687464
$581$	31.4407	1.30438
$582$	−0.774843	−0.0321183
$583$	−17.6618	−0.731477
$584$	6.60950	0.273503
$585$	2.65066	0.109591
$586$	51.8721	2.14282
$587$	−20.8168	−0.859201	−0.429601	−	0.903019i	$-0.641346\pi$
$587$	−20.8168	−0.859201	−0.429601	+	0.903019i	$0.641346\pi$
$588$	1.16544	0.0480619
$589$	−7.48509	−0.308418
$590$	−2.54086	−0.104605
$591$	0.0111377	0.000458144 0
$592$	−184.980	−7.60261
$593$	−41.5094	−1.70459	−0.852294	−	0.523063i	$-0.824789\pi$
$593$	−41.5094	−1.70459	−0.852294	+	0.523063i	$0.824789\pi$
$594$	1.51971	0.0623546
$595$	−18.3991	−0.754291
$596$	−85.5312	−3.50350
$597$	0.437060	0.0178877
$598$	11.5146	0.470868
$599$	27.3590	1.11786	0.558930	−	0.829215i	$-0.311212\pi$
$599$	27.3590	1.11786	0.558930	+	0.829215i	$0.311212\pi$
$600$	0.165917	0.00677353
$601$	−37.3396	−1.52311	−0.761557	−	0.648098i	$-0.775565\pi$
$601$	−37.3396	−1.52311	−0.761557	+	0.648098i	$0.775565\pi$
$602$	4.92041	0.200541
$603$	−28.3859	−1.15596
$604$	65.3039	2.65718
$605$	21.0163	0.854432
$606$	0.612906	0.0248976
$607$	−40.3203	−1.63655	−0.818275	−	0.574827i	$-0.805069\pi$
$607$	−40.3203	−1.63655	−0.818275	+	0.574827i	$0.805069\pi$
$608$	24.9566	1.01212
$609$	0.207479	0.00840748
$610$	27.0692	1.09600
$611$	−5.54460	−0.224310
$612$	71.0216	2.87088
$613$	−3.54674	−0.143251	−0.0716257	−	0.997432i	$-0.522819\pi$
$613$	−3.54674	−0.143251	−0.0716257	+	0.997432i	$0.522819\pi$
$614$	61.3428	2.47559
$615$	−0.00920832	−0.000371315 0
$616$	−258.170	−10.4019
$617$	−26.2593	−1.05716	−0.528579	−	0.848884i	$-0.677275\pi$
$617$	−26.2593	−1.05716	−0.528579	+	0.848884i	$0.677275\pi$
$618$	0.818669	0.0329317
$619$	11.4070	0.458485	0.229242	−	0.973369i	$-0.426375\pi$
$619$	11.4070	0.458485	0.229242	+	0.973369i	$0.426375\pi$
$620$	−46.2617	−1.85791
$621$	0.454160	0.0182248
$622$	46.8334	1.87785
$623$	3.57268	0.143136
$624$	−0.244369	−0.00978260
$625$	1.00000	0.0400000
$626$	37.7084	1.50713
$627$	−0.0842441	−0.00336439
$628$	61.9240	2.47104
$629$	44.7523	1.78439
$630$	−36.9281	−1.47125
$631$	−29.7698	−1.18512	−0.592558	−	0.805528i	$-0.701882\pi$
$631$	−29.7698	−1.18512	−0.592558	+	0.805528i	$0.701882\pi$
$632$	58.3516	2.32110
$633$	0.0312453	0.00124189
$634$	−4.29642	−0.170633
$635$	−13.4364	−0.533208
$636$	0.287229	0.0113894
$637$	−11.1913	−0.443416
$638$	−45.5738	−1.80428
$639$	0.0171225	0.000677355 0
$640$	59.0279	2.33328
$641$	−7.60820	−0.300506	−0.150253	−	0.988648i	$-0.548009\pi$
$641$	−7.60820	−0.300506	−0.150253	+	0.988648i	$0.548009\pi$
$642$	−0.628329	−0.0247982
$643$	−30.7240	−1.21164	−0.605819	−	0.795603i	$-0.707154\pi$
$643$	−30.7240	−1.21164	−0.605819	+	0.795603i	$0.707154\pi$
$644$	−118.785	−4.68080
$645$	0.00644537	0.000253786 0
$646$	−10.6343	−0.418399
$647$	29.3474	1.15377	0.576883	−	0.816827i	$-0.304269\pi$
$647$	29.3474	1.15377	0.576883	+	0.816827i	$0.304269\pi$
$648$	92.5764	3.63674
$649$	5.17894	0.203291
$650$	−2.45298	−0.0962138
$651$	−0.579738	−0.0227217
$652$	−133.359	−5.22276
$653$	36.7519	1.43821	0.719106	−	0.694901i	$-0.244552\pi$
$653$	36.7519	1.43821	0.719106	+	0.694901i	$0.244552\pi$
$654$	0.522854	0.0204452
$655$	−17.1859	−0.671510
$656$	−9.79304	−0.382354
$657$	−1.92700	−0.0751794
$658$	77.2456	3.01135
$659$	15.3739	0.598882	0.299441	−	0.954115i	$-0.403200\pi$
$659$	15.3739	0.598882	0.299441	+	0.954115i	$0.403200\pi$
$660$	−0.520671	−0.0202671
$661$	−2.79346	−0.108653	−0.0543265	−	0.998523i	$-0.517301\pi$
$661$	−2.79346	−0.108653	−0.0543265	+	0.998523i	$0.517301\pi$
$662$	−40.4602	−1.57253
$663$	0.0591205	0.00229605
$664$	72.9480	2.83093
$665$	4.09433	0.158771
$666$	89.8203	3.48047
$667$	−13.6195	−0.527350
$668$	51.3558	1.98702
$669$	−0.215847	−0.00834514
$670$	26.2690	1.01486
$671$	−55.1742	−2.12998
$672$	1.93295	0.0745650
$673$	−3.22117	−0.124167	−0.0620835	−	0.998071i	$-0.519775\pi$
$673$	−3.22117	−0.124167	−0.0620835	+	0.998071i	$0.519775\pi$
$674$	65.4137	2.51964
$675$	−0.0967503	−0.00372392
$676$	−69.7270	−2.68181
$677$	−42.0773	−1.61716	−0.808581	−	0.588385i	$-0.799764\pi$
$677$	−42.0773	−1.61716	−0.808581	+	0.588385i	$0.799764\pi$
$678$	0.326263	0.0125301
$679$	−76.7570	−2.94566
$680$	−42.6893	−1.63706
$681$	−0.229534	−0.00879577
$682$	127.342	4.87619
$683$	17.6382	0.674909	0.337454	−	0.941342i	$-0.390434\pi$
$683$	17.6382	0.674909	0.337454	+	0.941342i	$0.390434\pi$
$684$	−15.8043	−0.604294
$685$	−5.39580	−0.206163
$686$	69.7409	2.66272
$687$	−0.102536	−0.00391199
$688$	6.85464	0.261331
$689$	−2.75816	−0.105077
$690$	−0.210136	−0.00799974
$691$	−12.5486	−0.477372	−0.238686	−	0.971097i	$-0.576717\pi$
$691$	−12.5486	−0.477372	−0.238686	+	0.971097i	$0.576717\pi$
$692$	−57.3469	−2.18000
$693$	75.2693	2.85924
$694$	49.5758	1.88187
$695$	6.69869	0.254096
$696$	0.481389	0.0182470
$697$	2.36924	0.0897414
$698$	0.839664	0.0317817
$699$	−0.169161	−0.00639825
$700$	25.3051	0.956441
$701$	19.0306	0.718776	0.359388	−	0.933188i	$-0.382985\pi$
$701$	19.0306	0.718776	0.359388	+	0.933188i	$0.382985\pi$
$702$	0.237327	0.00895731
$703$	−9.95866	−0.375598
$704$	−230.504	−8.68746
$705$	0.101186	0.00381089
$706$	−18.5130	−0.696745
$707$	60.7154	2.28344
$708$	−0.0842237	−0.00316532
$709$	3.62995	0.136326	0.0681628	−	0.997674i	$-0.478286\pi$
$709$	3.62995	0.136326	0.0681628	+	0.997674i	$0.478286\pi$
$710$	−0.0158456	−0.000594673 0
$711$	−17.0124	−0.638015
$712$	8.28925	0.310653
$713$	38.0556	1.42520
$714$	−0.823649	−0.0308243
$715$	4.99983	0.186983
$716$	−109.001	−4.07355
$717$	0.0512285	0.00191316
$718$	−51.7840	−1.93256
$719$	21.0559	0.785253	0.392626	−	0.919698i	$-0.371567\pi$
$719$	21.0559	0.785253	0.392626	+	0.919698i	$0.371567\pi$
$720$	−51.4447	−1.91723
$721$	81.0985	3.02027
$722$	−50.3782	−1.87488
$723$	−0.0161258	−0.000599723 0
$724$	37.4619	1.39226
$725$	2.90139	0.107755
$726$	0.940805	0.0349166
$727$	−23.3685	−0.866688	−0.433344	−	0.901229i	$-0.642667\pi$
$727$	−23.3685	−0.866688	−0.433344	+	0.901229i	$0.642667\pi$
$728$	−40.3171	−1.49425
$729$	−26.9860	−0.999480
$730$	1.78329	0.0660026
$731$	−1.65835	−0.0613363
$732$	0.897283	0.0331645
$733$	15.4366	0.570163	0.285082	−	0.958503i	$-0.407979\pi$
$733$	15.4366	0.570163	0.285082	+	0.958503i	$0.407979\pi$
$734$	56.7815	2.09584
$735$	0.204236	0.00753335
$736$	−126.884	−4.67700
$737$	−53.5433	−1.97229
$738$	4.75520	0.175041
$739$	37.7934	1.39025	0.695127	−	0.718887i	$-0.255348\pi$
$739$	37.7934	1.39025	0.695127	+	0.718887i	$0.255348\pi$
$740$	−61.5496	−2.26261
$741$	−0.0131560	−0.000483298 0
$742$	38.4258	1.41066
$743$	32.8436	1.20492	0.602458	−	0.798151i	$-0.294188\pi$
$743$	32.8436	1.20492	0.602458	+	0.798151i	$0.294188\pi$
$744$	−1.34510	−0.0493137
$745$	−14.9888	−0.549147
$746$	24.9191	0.912352
$747$	−21.2680	−0.778155
$748$	133.965	4.89825
$749$	−62.2432	−2.27432
$750$	0.0447656	0.00163461
$751$	−25.5433	−0.932087	−0.466043	−	0.884762i	$-0.654321\pi$
$751$	−25.5433	−0.932087	−0.466043	+	0.884762i	$0.654321\pi$
$752$	107.611	3.92418
$753$	0.344965	0.0125712
$754$	−7.11704	−0.259187
$755$	11.4441	0.416492
$756$	−2.44827	−0.0890428
$757$	−2.64553	−0.0961533	−0.0480766	−	0.998844i	$-0.515309\pi$
$757$	−2.64553	−0.0961533	−0.0480766	+	0.998844i	$0.515309\pi$
$758$	7.69343	0.279438
$759$	0.428313	0.0155468
$760$	9.49959	0.344586
$761$	−17.5680	−0.636839	−0.318420	−	0.947950i	$-0.603152\pi$
$761$	−17.5680	−0.636839	−0.318420	+	0.947950i	$0.603152\pi$
$762$	−0.601489	−0.0217896
$763$	51.7946	1.87509
$764$	−125.604	−4.54418
$765$	12.4461	0.449989
$766$	−52.7502	−1.90594
$767$	0.808771	0.0292030
$768$	1.32858	0.0479408
$769$	9.05937	0.326689	0.163345	−	0.986569i	$-0.447772\pi$
$769$	9.05937	0.326689	0.163345	+	0.986569i	$0.447772\pi$
$770$	−69.6560	−2.51023
$771$	−0.190531	−0.00686180
$772$	−39.1280	−1.40825
$773$	16.3381	0.587641	0.293821	−	0.955861i	$-0.405073\pi$
$773$	16.3381	0.587641	0.293821	+	0.955861i	$0.405073\pi$
$774$	−3.32840	−0.119637
$775$	−8.10705	−0.291214
$776$	−178.090	−6.39306
$777$	−0.771322	−0.0276710
$778$	6.54961	0.234815
$779$	−0.527224	−0.0188897
$780$	−0.0813108	−0.00291139
$781$	0.0322975	0.00115569
$782$	54.0666	1.93342
$783$	−0.280710	−0.0100318
$784$	217.204	7.75730
$785$	10.8518	0.387317
$786$	−0.769338	−0.0274414
$787$	13.5353	0.482481	0.241241	−	0.970465i	$-0.422446\pi$
$787$	13.5353	0.482481	0.241241	+	0.970465i	$0.422446\pi$
$788$	3.94125	0.140401
$789$	0.232090	0.00826261
$790$	15.7437	0.560136
$791$	32.3201	1.14917
$792$	174.638	6.20550
$793$	−8.61629	−0.305974
$794$	47.3981	1.68210
$795$	0.0503349	0.00178520
$796$	154.660	5.48179
$797$	−28.9851	−1.02670	−0.513352	−	0.858178i	$-0.671597\pi$
$797$	−28.9851	−1.02670	−0.513352	+	0.858178i	$0.671597\pi$
$798$	0.183285	0.00648823
$799$	−26.0345	−0.921034
$800$	27.0303	0.955665
$801$	−2.41673	−0.0853910
$802$	−40.2800	−1.42234
$803$	−3.63482	−0.128270
$804$	0.870759	0.0307093
$805$	−20.8164	−0.733681
$806$	19.8864	0.700470
$807$	−0.230952	−0.00812991
$808$	140.871	4.95581
$809$	−28.7479	−1.01072	−0.505361	−	0.862908i	$-0.668641\pi$
$809$	−28.7479	−1.01072	−0.505361	+	0.862908i	$0.668641\pi$
$810$	24.9778	0.877630
$811$	43.8974	1.54145	0.770723	−	0.637171i	$-0.219895\pi$
$811$	43.8974	1.54145	0.770723	+	0.637171i	$0.219895\pi$
$812$	73.4198	2.57653
$813$	0.0935922	0.00328242
$814$	169.424	5.93833
$815$	−23.3704	−0.818628
$816$	−1.14743	−0.0401680
$817$	0.369030	0.0129107
$818$	8.36827	0.292590
$819$	11.7544	0.410734
$820$	−3.25851	−0.113792
$821$	43.2464	1.50931	0.754655	−	0.656122i	$-0.227804\pi$
$821$	43.2464	1.50931	0.754655	+	0.656122i	$0.227804\pi$
$822$	−0.241546	−0.00842489
$823$	9.51170	0.331557	0.165779	−	0.986163i	$-0.446986\pi$
$823$	9.51170	0.331557	0.165779	+	0.986163i	$0.446986\pi$
$824$	188.163	6.55497
$825$	−0.0912442	−0.00317672
$826$	−11.2675	−0.392048
$827$	−43.4497	−1.51089	−0.755447	−	0.655209i	$-0.772580\pi$
$827$	−43.4497	−1.51089	−0.755447	+	0.655209i	$0.772580\pi$
$828$	80.3522	2.79243
$829$	11.2767	0.391658	0.195829	−	0.980638i	$-0.437260\pi$
$829$	11.2767	0.391658	0.195829	+	0.980638i	$0.437260\pi$
$830$	19.6819	0.683169
$831$	0.285702	0.00991088
$832$	−35.9968	−1.24796
$833$	−52.5485	−1.82070
$834$	0.299871	0.0103837
$835$	8.99977	0.311450
$836$	−29.8111	−1.03104
$837$	0.784360	0.0271115
$838$	16.9156	0.584339
$839$	−35.5135	−1.22606	−0.613031	−	0.790059i	$-0.710050\pi$
$839$	−35.5135	−1.22606	−0.613031	+	0.790059i	$0.710050\pi$
$840$	0.735766	0.0253863
$841$	−20.5820	−0.709723
$842$	−30.1503	−1.03905
$843$	−0.346159	−0.0119224
$844$	11.0566	0.380585
$845$	−12.2192	−0.420353
$846$	−52.2527	−1.79648
$847$	93.1975	3.20230
$848$	53.5312	1.83827
$849$	0.178528	0.00612705
$850$	−11.5179	−0.395060
$851$	50.6317	1.73563
$852$	−0.000525245 0	−1.79946e−5 0
$853$	1.97246	0.0675357	0.0337679	−	0.999430i	$-0.489249\pi$
$853$	1.97246	0.0675357	0.0337679	+	0.999430i	$0.489249\pi$
$854$	120.040	4.10767
$855$	−2.76960	−0.0947185
$856$	−144.415	−4.93602
$857$	33.7538	1.15301	0.576503	−	0.817095i	$-0.304417\pi$
$857$	33.7538	1.15301	0.576503	+	0.817095i	$0.304417\pi$
$858$	0.223820	0.00764109
$859$	−8.50816	−0.290295	−0.145147	−	0.989410i	$-0.546366\pi$
$859$	−8.50816	−0.290295	−0.145147	+	0.989410i	$0.546366\pi$
$860$	2.28079	0.0777744
$861$	−0.0408347	−0.00139164
$862$	49.0532	1.67076
$863$	54.0198	1.83886	0.919428	−	0.393258i	$-0.128652\pi$
$863$	54.0198	1.83886	0.919428	+	0.393258i	$0.128652\pi$
$864$	−2.61519	−0.0889705
$865$	−10.0497	−0.341699
$866$	23.4976	0.798482
$867$	0.00346072	0.000117532 0
$868$	−205.149	−6.96323
$869$	−32.0898	−1.08857
$870$	0.129882	0.00440342
$871$	−8.36160	−0.283322
$872$	120.173	4.06957
$873$	51.9222	1.75730
$874$	−12.0314	−0.406967
$875$	4.43454	0.149915
$876$	0.0591121	0.00199721
$877$	−7.94864	−0.268407	−0.134203	−	0.990954i	$-0.542848\pi$
$877$	−7.94864	−0.268407	−0.134203	+	0.990954i	$0.542848\pi$
$878$	73.3740	2.47625
$879$	0.301321	0.0101633
$880$	−97.0381	−3.27115
$881$	−39.1977	−1.32060	−0.660302	−	0.751000i	$-0.729572\pi$
$881$	−39.1977	−1.32060	−0.660302	+	0.751000i	$0.729572\pi$
$882$	−105.468	−3.55128
$883$	−15.4881	−0.521217	−0.260609	−	0.965445i	$-0.583923\pi$
$883$	−15.4881	−0.521217	−0.260609	+	0.965445i	$0.583923\pi$
$884$	20.9207	0.703640
$885$	−0.0147596	−0.000496140 0
$886$	95.9411	3.22320
$887$	−47.5043	−1.59504	−0.797519	−	0.603293i	$-0.793855\pi$
$887$	−47.5043	−1.59504	−0.797519	+	0.603293i	$0.793855\pi$
$888$	−1.78961	−0.0600552
$889$	−59.5844	−1.99840
$890$	2.23650	0.0749677
$891$	−50.9114	−1.70560
$892$	−76.3809	−2.55742
$893$	5.79342	0.193869
$894$	−0.670982	−0.0224410
$895$	−19.1017	−0.638499
$896$	261.762	8.74485
$897$	0.0668876	0.00223331
$898$	−101.899	−3.40042
$899$	−23.5217	−0.784492
$900$	−17.1176	−0.570585
$901$	−12.9508	−0.431455
$902$	8.96954	0.298653
$903$	0.0285823	0.000951158 0
$904$	74.9885	2.49408
$905$	6.56496	0.218227
$906$	0.512301	0.0170200
$907$	−5.94095	−0.197266	−0.0986330	−	0.995124i	$-0.531447\pi$
$907$	−5.94095	−0.197266	−0.0986330	+	0.995124i	$0.531447\pi$
$908$	−81.2243	−2.69552
$909$	−41.0708	−1.36223
$910$	−10.8778	−0.360597
$911$	2.02127	0.0669677	0.0334839	−	0.999439i	$-0.489340\pi$
$911$	2.02127	0.0669677	0.0334839	+	0.999439i	$0.489340\pi$
$912$	0.255336	0.00845501
$913$	−40.1170	−1.32768
$914$	−73.1822	−2.42065
$915$	0.157243	0.00519829
$916$	−36.2839	−1.19885
$917$	−76.2118	−2.51673
$918$	1.11436	0.0367793
$919$	7.56466	0.249535	0.124767	−	0.992186i	$-0.460182\pi$
$919$	7.56466	0.249535	0.124767	+	0.992186i	$0.460182\pi$
$920$	−48.2977	−1.59233
$921$	0.356335	0.0117417
$922$	37.6264	1.23916
$923$	0.00504374	0.000166017 0
$924$	−2.30894	−0.0759586
$925$	−10.7862	−0.354647
$926$	21.4770	0.705777
$927$	−54.8590	−1.80180
$928$	78.4253	2.57444
$929$	−23.2165	−0.761708	−0.380854	−	0.924635i	$-0.624370\pi$
$929$	−23.2165	−0.761708	−0.380854	+	0.924635i	$0.624370\pi$
$930$	−0.362917	−0.0119005
$931$	11.6935	0.383240
$932$	−59.8602	−1.96079
$933$	0.272051	0.00890656
$934$	−7.76669	−0.254134
$935$	23.4765	0.767764
$936$	27.2724	0.891427
$937$	−1.10133	−0.0359788	−0.0179894	−	0.999838i	$-0.505727\pi$
$937$	−1.10133	−0.0359788	−0.0179894	+	0.999838i	$0.505727\pi$
$938$	116.491	3.80357
$939$	0.219045	0.00714827
$940$	35.8063	1.16787
$941$	30.2413	0.985838	0.492919	−	0.870075i	$-0.335930\pi$
$941$	30.2413	0.985838	0.492919	+	0.870075i	$0.335930\pi$
$942$	0.485786	0.0158278
$943$	2.68051	0.0872893
$944$	−15.6969	−0.510889
$945$	−0.429044	−0.0139568
$946$	−6.27823	−0.204123
$947$	10.9571	0.356058	0.178029	−	0.984025i	$-0.443028\pi$
$947$	10.9571	0.356058	0.178029	+	0.984025i	$0.443028\pi$
$948$	0.521868	0.0169495
$949$	−0.567633	−0.0184261
$950$	2.56306	0.0831566
$951$	−0.0249576	−0.000809305 0
$952$	−189.308	−6.13550
$953$	16.0223	0.519012	0.259506	−	0.965741i	$-0.416440\pi$
$953$	16.0223	0.519012	0.259506	+	0.965741i	$0.416440\pi$
$954$	−25.9931	−0.841557
$955$	−22.0112	−0.712266
$956$	18.1280	0.586301
$957$	−0.264735	−0.00855766
$958$	10.7695	0.347948
$959$	−23.9279	−0.772673
$960$	0.656922	0.0212021
$961$	34.7243	1.12014
$962$	26.4582	0.853047
$963$	42.1043	1.35679
$964$	−5.70635	−0.183789
$965$	−6.85693	−0.220732
$966$	−0.931857	−0.0299820
$967$	−40.2309	−1.29374	−0.646869	−	0.762601i	$-0.723922\pi$
$967$	−40.2309	−1.29374	−0.646869	+	0.762601i	$0.723922\pi$
$968$	216.235	6.95006
$969$	−0.0617736	−0.00198445
$970$	−48.0500	−1.54279
$971$	−31.4337	−1.00876	−0.504378	−	0.863483i	$-0.668278\pi$
$971$	−31.4337	−1.00876	−0.504378	+	0.863483i	$0.668278\pi$
$972$	2.48423	0.0796817
$973$	29.7056	0.952318
$974$	−97.8681	−3.13590
$975$	−0.0142492	−0.000456339 0
$976$	167.228	5.35283
$977$	−0.758530	−0.0242675	−0.0121338	−	0.999926i	$-0.503862\pi$
$977$	−0.758530	−0.0242675	−0.0121338	+	0.999926i	$0.503862\pi$
$978$	−1.04619	−0.0334534
$979$	−4.55858	−0.145693
$980$	72.2720	2.30864
$981$	−35.0364	−1.11863
$982$	20.7479	0.662091
$983$	16.6688	0.531652	0.265826	−	0.964021i	$-0.414355\pi$
$983$	16.6688	0.531652	0.265826	+	0.964021i	$0.414355\pi$
$984$	−0.0947439	−0.00302033
$985$	0.690678	0.0220068
$986$	−33.4178	−1.06424
$987$	0.448714	0.0142827
$988$	−4.65546	−0.148110
$989$	−1.87622	−0.0596603
$990$	47.1187	1.49753
$991$	31.3410	0.995579	0.497789	−	0.867298i	$-0.334145\pi$
$991$	31.3410	0.995579	0.497789	+	0.867298i	$0.334145\pi$
$992$	−219.136	−6.95757
$993$	−0.235030	−0.00745845
$994$	−0.0702678	−0.00222876
$995$	27.1032	0.859229
$996$	0.652411	0.0206724
$997$	60.9619	1.93068	0.965341	−	0.260991i	$-0.0840493\pi$
$997$	60.9619	1.93068	0.965341	+	0.260991i	$0.0840493\pi$
$998$	−51.9067	−1.64308
$999$	1.04356	0.0330169

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1205.2.a.e.1.25	✓	25
5.4	even	2		6025.2.a.j.1.1		25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.e.1.25	✓	25	1.1	even	1	trivial
6025.2.a.j.1.1		25	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1205 = 5 \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1205.a (trivial)

\(f(q)\)	\(=\)	\(q+2.77603 q^{2} +0.0161258 q^{3} +5.70635 q^{4} +1.00000 q^{5} +0.0447656 q^{6} +4.43454 q^{7} +10.2889 q^{8} -2.99974 q^{9} +O(q^{10})\)	\(q+2.77603 q^{2} +0.0161258 q^{3} +5.70635 q^{4} +1.00000 q^{5} +0.0447656 q^{6} +4.43454 q^{7} +10.2889 q^{8} -2.99974 q^{9} +2.77603 q^{10} -5.65829 q^{11} +0.0920192 q^{12} -0.883628 q^{13} +12.3104 q^{14} +0.0161258 q^{15} +17.1497 q^{16} -4.14905 q^{17} -8.32737 q^{18} +0.923282 q^{19} +5.70635 q^{20} +0.0715104 q^{21} -15.7076 q^{22} -4.69414 q^{23} +0.165917 q^{24} +1.00000 q^{25} -2.45298 q^{26} -0.0967503 q^{27} +25.3051 q^{28} +2.90139 q^{29} +0.0447656 q^{30} -8.10705 q^{31} +27.0303 q^{32} -0.0912442 q^{33} -11.5179 q^{34} +4.43454 q^{35} -17.1176 q^{36} -10.7862 q^{37} +2.56306 q^{38} -0.0142492 q^{39} +10.2889 q^{40} -0.571032 q^{41} +0.198515 q^{42} +0.399694 q^{43} -32.2882 q^{44} -2.99974 q^{45} -13.0311 q^{46} +6.27481 q^{47} +0.276552 q^{48} +12.6652 q^{49} +2.77603 q^{50} -0.0669065 q^{51} -5.04229 q^{52} +3.12140 q^{53} -0.268582 q^{54} -5.65829 q^{55} +45.6268 q^{56} +0.0148886 q^{57} +8.05434 q^{58} -0.915284 q^{59} +0.0920192 q^{60} +9.75104 q^{61} -22.5054 q^{62} -13.3025 q^{63} +40.7375 q^{64} -0.883628 q^{65} -0.253297 q^{66} +9.46280 q^{67} -23.6759 q^{68} -0.0756965 q^{69} +12.3104 q^{70} -0.00570799 q^{71} -30.8641 q^{72} +0.642389 q^{73} -29.9427 q^{74} +0.0161258 q^{75} +5.26857 q^{76} -25.0919 q^{77} -0.0395561 q^{78} +5.67130 q^{79} +17.1497 q^{80} +8.99766 q^{81} -1.58520 q^{82} +7.08995 q^{83} +0.408063 q^{84} -4.14905 q^{85} +1.10956 q^{86} +0.0467870 q^{87} -58.2178 q^{88} +0.805647 q^{89} -8.32737 q^{90} -3.91849 q^{91} -26.7864 q^{92} -0.130732 q^{93} +17.4191 q^{94} +0.923282 q^{95} +0.435884 q^{96} -17.3089 q^{97} +35.1589 q^{98} +16.9734 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 6 q^{2} + 15 q^{3} + 32 q^{4} + 25 q^{5} - q^{6} + 19 q^{7} + 15 q^{8} + 32 q^{9}+O(q^{10}) \) 25 * q + 6 * q^2 + 15 * q^3 + 32 * q^4 + 25 * q^5 - q^6 + 19 * q^7 + 15 * q^8 + 32 * q^9	\( 25 q + 6 q^{2} + 15 q^{3} + 32 q^{4} + 25 q^{5} - q^{6} + 19 q^{7} + 15 q^{8} + 32 q^{9} + 6 q^{10} + 2 q^{11} + 20 q^{12} + 14 q^{13} - 5 q^{14} + 15 q^{15} + 38 q^{16} + 7 q^{17} + 9 q^{18} + 30 q^{19} + 32 q^{20} + q^{21} + q^{22} + 43 q^{23} - 6 q^{24} + 25 q^{25} - 22 q^{26} + 42 q^{27} + 32 q^{28} - 4 q^{29} - q^{30} + 14 q^{31} + 26 q^{32} + 4 q^{33} + 7 q^{34} + 19 q^{35} + 15 q^{36} + 16 q^{37} + 14 q^{38} - 21 q^{39} + 15 q^{40} - q^{41} - 25 q^{42} + 35 q^{43} - 52 q^{44} + 32 q^{45} - 27 q^{46} + 50 q^{47} + 26 q^{48} + 46 q^{49} + 6 q^{50} - 7 q^{51} + 3 q^{52} + 4 q^{53} - 31 q^{54} + 2 q^{55} - 51 q^{56} + 2 q^{58} + 6 q^{59} + 20 q^{60} + 19 q^{61} + 28 q^{63} + 49 q^{64} + 14 q^{65} - 27 q^{66} + 65 q^{67} - 25 q^{68} + 2 q^{69} - 5 q^{70} - 34 q^{71} - 10 q^{72} + 8 q^{73} - 42 q^{74} + 15 q^{75} + 71 q^{76} + q^{77} - 59 q^{78} - 12 q^{79} + 38 q^{80} + 29 q^{81} + 11 q^{82} + 41 q^{83} - 10 q^{84} + 7 q^{85} - 13 q^{86} + 40 q^{87} - 52 q^{88} - 24 q^{89} + 9 q^{90} + 46 q^{91} + 85 q^{92} - 30 q^{93} + 14 q^{94} + 30 q^{95} - 30 q^{96} + 9 q^{97} - 64 q^{98} + 2 q^{99}+O(q^{100}) \) 25 * q + 6 * q^2 + 15 * q^3 + 32 * q^4 + 25 * q^5 - q^6 + 19 * q^7 + 15 * q^8 + 32 * q^9 + 6 * q^10 + 2 * q^11 + 20 * q^12 + 14 * q^13 - 5 * q^14 + 15 * q^15 + 38 * q^16 + 7 * q^17 + 9 * q^18 + 30 * q^19 + 32 * q^20 + q^21 + q^22 + 43 * q^23 - 6 * q^24 + 25 * q^25 - 22 * q^26 + 42 * q^27 + 32 * q^28 - 4 * q^29 - q^30 + 14 * q^31 + 26 * q^32 + 4 * q^33 + 7 * q^34 + 19 * q^35 + 15 * q^36 + 16 * q^37 + 14 * q^38 - 21 * q^39 + 15 * q^40 - q^41 - 25 * q^42 + 35 * q^43 - 52 * q^44 + 32 * q^45 - 27 * q^46 + 50 * q^47 + 26 * q^48 + 46 * q^49 + 6 * q^50 - 7 * q^51 + 3 * q^52 + 4 * q^53 - 31 * q^54 + 2 * q^55 - 51 * q^56 + 2 * q^58 + 6 * q^59 + 20 * q^60 + 19 * q^61 + 28 * q^63 + 49 * q^64 + 14 * q^65 - 27 * q^66 + 65 * q^67 - 25 * q^68 + 2 * q^69 - 5 * q^70 - 34 * q^71 - 10 * q^72 + 8 * q^73 - 42 * q^74 + 15 * q^75 + 71 * q^76 + q^77 - 59 * q^78 - 12 * q^79 + 38 * q^80 + 29 * q^81 + 11 * q^82 + 41 * q^83 - 10 * q^84 + 7 * q^85 - 13 * q^86 + 40 * q^87 - 52 * q^88 - 24 * q^89 + 9 * q^90 + 46 * q^91 + 85 * q^92 - 30 * q^93 + 14 * q^94 + 30 * q^95 - 30 * q^96 + 9 * q^97 - 64 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more