LMFDB - Embedded newform 1205.2.a.c.1.2

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:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 2 x^{14} - 16 x^{13} + 31 x^{12} + 99 x^{11} - 184 x^{10} - 296 x^{9} + 519 x^{8} + 437 x^{7} + \cdots - 1 $ x^15 - 2x^14 - 16x^13 + 31x^12 + 99x^11 - 184x^10 - 296x^9 + 519x^8 + 437x^7 - 699x^6 - 297x^5 + 394x^4 + 89x^3 - 57x^2 - 17x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.06795$ of defining polynomial
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)

$f(q)$	$=$	$q-2.06795 q^{2} +1.49139 q^{3} +2.27640 q^{4} -1.00000 q^{5} -3.08411 q^{6} -2.20594 q^{7} -0.571582 q^{8} -0.775761 q^{9} +O(q^{10})$	\(q-2.06795 q^{2} +1.49139 q^{3} +2.27640 q^{4} -1.00000 q^{5} -3.08411 q^{6} -2.20594 q^{7} -0.571582 q^{8} -0.775761 q^{9} +2.06795 q^{10} +0.732051 q^{11} +3.39500 q^{12} +0.652235 q^{13} +4.56177 q^{14} -1.49139 q^{15} -3.37080 q^{16} -0.343445 q^{17} +1.60423 q^{18} +4.07805 q^{19} -2.27640 q^{20} -3.28992 q^{21} -1.51384 q^{22} +7.46680 q^{23} -0.852451 q^{24} +1.00000 q^{25} -1.34879 q^{26} -5.63113 q^{27} -5.02161 q^{28} -6.92516 q^{29} +3.08411 q^{30} -1.82292 q^{31} +8.11380 q^{32} +1.09177 q^{33} +0.710226 q^{34} +2.20594 q^{35} -1.76594 q^{36} -5.87423 q^{37} -8.43318 q^{38} +0.972735 q^{39} +0.571582 q^{40} -7.34309 q^{41} +6.80338 q^{42} -2.73880 q^{43} +1.66644 q^{44} +0.775761 q^{45} -15.4409 q^{46} +0.681256 q^{47} -5.02717 q^{48} -2.13381 q^{49} -2.06795 q^{50} -0.512210 q^{51} +1.48475 q^{52} +3.55915 q^{53} +11.6449 q^{54} -0.732051 q^{55} +1.26088 q^{56} +6.08195 q^{57} +14.3209 q^{58} -12.9845 q^{59} -3.39500 q^{60} -9.05077 q^{61} +3.76970 q^{62} +1.71129 q^{63} -10.0373 q^{64} -0.652235 q^{65} -2.25773 q^{66} -15.6566 q^{67} -0.781819 q^{68} +11.1359 q^{69} -4.56177 q^{70} +10.5344 q^{71} +0.443411 q^{72} -8.85163 q^{73} +12.1476 q^{74} +1.49139 q^{75} +9.28327 q^{76} -1.61486 q^{77} -2.01156 q^{78} +4.65671 q^{79} +3.37080 q^{80} -6.07091 q^{81} +15.1851 q^{82} +16.1391 q^{83} -7.48918 q^{84} +0.343445 q^{85} +5.66368 q^{86} -10.3281 q^{87} -0.418427 q^{88} -0.646523 q^{89} -1.60423 q^{90} -1.43879 q^{91} +16.9974 q^{92} -2.71868 q^{93} -1.40880 q^{94} -4.07805 q^{95} +12.1008 q^{96} +8.57664 q^{97} +4.41260 q^{98} -0.567897 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 2 q^{2} - 7 q^{3} + 6 q^{4} - 15 q^{5} - 5 q^{6} - 3 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10}) $ 15 * q + 2 * q^2 - 7 * q^3 + 6 * q^4 - 15 * q^5 - 5 * q^6 - 3 * q^7 + 3 * q^8 + 6 * q^9	\( 15 q + 2 q^{2} - 7 q^{3} + 6 q^{4} - 15 q^{5} - 5 q^{6} - 3 q^{7} + 3 q^{8} + 6 q^{9} - 2 q^{10} - 10 q^{11} - 6 q^{12} - 8 q^{13} - 5 q^{14} + 7 q^{15} - 16 q^{16} - q^{17} - 3 q^{18} - 30 q^{19} - 6 q^{20} - 11 q^{21} - 5 q^{22} + 19 q^{23} - 14 q^{24} + 15 q^{25} - 18 q^{26} - 22 q^{27} - 20 q^{28} - 12 q^{29} + 5 q^{30} - 22 q^{31} - 2 q^{32} + 4 q^{33} - 29 q^{34} + 3 q^{35} - 7 q^{36} - 12 q^{37} - 18 q^{38} - 17 q^{39} - 3 q^{40} - 13 q^{41} - q^{42} - 25 q^{43} - 20 q^{44} - 6 q^{45} - 7 q^{46} + 16 q^{47} - 22 q^{48} - 24 q^{49} + 2 q^{50} - 27 q^{51} - 15 q^{52} - 4 q^{53} - 43 q^{54} + 10 q^{55} - 3 q^{56} + 22 q^{57} - 20 q^{58} - 50 q^{59} + 6 q^{60} - 41 q^{61} + 12 q^{62} + 6 q^{63} - 53 q^{64} + 8 q^{65} + 5 q^{66} - 43 q^{67} + 5 q^{68} - 50 q^{69} + 5 q^{70} - 14 q^{71} + 32 q^{72} - 10 q^{73} - 26 q^{74} - 7 q^{75} - 13 q^{76} - 7 q^{77} + 3 q^{78} - 44 q^{79} + 16 q^{80} + 7 q^{81} - 19 q^{82} + 7 q^{83} - 42 q^{84} + q^{85} + 7 q^{86} + 10 q^{87} - 28 q^{88} + 4 q^{89} + 3 q^{90} - 50 q^{91} + 25 q^{92} + 22 q^{93} - 14 q^{94} + 30 q^{95} + 14 q^{96} + 9 q^{97} + 2 q^{98} - 46 q^{99}+O(q^{100}) \) 15 * q + 2 * q^2 - 7 * q^3 + 6 * q^4 - 15 * q^5 - 5 * q^6 - 3 * q^7 + 3 * q^8 + 6 * q^9 - 2 * q^10 - 10 * q^11 - 6 * q^12 - 8 * q^13 - 5 * q^14 + 7 * q^15 - 16 * q^16 - q^17 - 3 * q^18 - 30 * q^19 - 6 * q^20 - 11 * q^21 - 5 * q^22 + 19 * q^23 - 14 * q^24 + 15 * q^25 - 18 * q^26 - 22 * q^27 - 20 * q^28 - 12 * q^29 + 5 * q^30 - 22 * q^31 - 2 * q^32 + 4 * q^33 - 29 * q^34 + 3 * q^35 - 7 * q^36 - 12 * q^37 - 18 * q^38 - 17 * q^39 - 3 * q^40 - 13 * q^41 - q^42 - 25 * q^43 - 20 * q^44 - 6 * q^45 - 7 * q^46 + 16 * q^47 - 22 * q^48 - 24 * q^49 + 2 * q^50 - 27 * q^51 - 15 * q^52 - 4 * q^53 - 43 * q^54 + 10 * q^55 - 3 * q^56 + 22 * q^57 - 20 * q^58 - 50 * q^59 + 6 * q^60 - 41 * q^61 + 12 * q^62 + 6 * q^63 - 53 * q^64 + 8 * q^65 + 5 * q^66 - 43 * q^67 + 5 * q^68 - 50 * q^69 + 5 * q^70 - 14 * q^71 + 32 * q^72 - 10 * q^73 - 26 * q^74 - 7 * q^75 - 13 * q^76 - 7 * q^77 + 3 * q^78 - 44 * q^79 + 16 * q^80 + 7 * q^81 - 19 * q^82 + 7 * q^83 - 42 * q^84 + q^85 + 7 * q^86 + 10 * q^87 - 28 * q^88 + 4 * q^89 + 3 * q^90 - 50 * q^91 + 25 * q^92 + 22 * q^93 - 14 * q^94 + 30 * q^95 + 14 * q^96 + 9 * q^97 + 2 * q^98 - 46 * 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 $

$2$	−2.06795	−1.46226	−0.731129	−	0.682239i	$-0.761006\pi$
$2$	−2.06795	−1.46226	−0.731129	+	0.682239i	$0.761006\pi$
$3$	1.49139	0.861053	0.430527	−	0.902578i	$-0.358328\pi$
$3$	1.49139	0.861053	0.430527	+	0.902578i	$0.358328\pi$
$4$	2.27640	1.13820
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.08411	−1.25908
$7$	−2.20594	−0.833769	−0.416884	−	0.908960i	$-0.636878\pi$
$7$	−2.20594	−0.833769	−0.416884	+	0.908960i	$0.636878\pi$
$8$	−0.571582	−0.202085
$9$	−0.775761	−0.258587
$10$	2.06795	0.653942
$11$	0.732051	0.220722	0.110361	−	0.993892i	$-0.464799\pi$
$11$	0.732051	0.220722	0.110361	+	0.993892i	$0.464799\pi$
$12$	3.39500	0.980051
$13$	0.652235	0.180897	0.0904487	−	0.995901i	$-0.471170\pi$
$13$	0.652235	0.180897	0.0904487	+	0.995901i	$0.471170\pi$
$14$	4.56177	1.21919
$15$	−1.49139	−0.385075
$16$	−3.37080	−0.842700
$17$	−0.343445	−0.0832977	−0.0416488	−	0.999132i	$-0.513261\pi$
$17$	−0.343445	−0.0832977	−0.0416488	+	0.999132i	$0.513261\pi$
$18$	1.60423	0.378121
$19$	4.07805	0.935568	0.467784	−	0.883843i	$-0.345052\pi$
$19$	4.07805	0.935568	0.467784	+	0.883843i	$0.345052\pi$
$20$	−2.27640	−0.509019
$21$	−3.28992	−0.717919
$22$	−1.51384	−0.322752
$23$	7.46680	1.55694	0.778468	−	0.627685i	$-0.215997\pi$
$23$	7.46680	1.55694	0.778468	+	0.627685i	$0.215997\pi$
$24$	−0.852451	−0.174006
$25$	1.00000	0.200000
$26$	−1.34879	−0.264519
$27$	−5.63113	−1.08371
$28$	−5.02161	−0.948996
$29$	−6.92516	−1.28597	−0.642985	−	0.765879i	$-0.722304\pi$
$29$	−6.92516	−1.28597	−0.642985	+	0.765879i	$0.722304\pi$
$30$	3.08411	0.563079
$31$	−1.82292	−0.327406	−0.163703	−	0.986510i	$-0.552344\pi$
$31$	−1.82292	−0.327406	−0.163703	+	0.986510i	$0.552344\pi$
$32$	8.11380	1.43433
$33$	1.09177	0.190053
$34$	0.710226	0.121803
$35$	2.20594	0.372873
$36$	−1.76594	−0.294324
$37$	−5.87423	−0.965718	−0.482859	−	0.875698i	$-0.660402\pi$
$37$	−5.87423	−0.965718	−0.482859	+	0.875698i	$0.660402\pi$
$38$	−8.43318	−1.36804
$39$	0.972735	0.155762
$40$	0.571582	0.0903750
$41$	−7.34309	−1.14680	−0.573399	−	0.819276i	$-0.694376\pi$
$41$	−7.34309	−1.14680	−0.573399	+	0.819276i	$0.694376\pi$
$42$	6.80338	1.04978
$43$	−2.73880	−0.417662	−0.208831	−	0.977952i	$-0.566966\pi$
$43$	−2.73880	−0.417662	−0.208831	+	0.977952i	$0.566966\pi$
$44$	1.66644	0.251225
$45$	0.775761	0.115644
$46$	−15.4409	−2.27664
$47$	0.681256	0.0993714	0.0496857	−	0.998765i	$-0.484178\pi$
$47$	0.681256	0.0993714	0.0496857	+	0.998765i	$0.484178\pi$
$48$	−5.02717	−0.725610
$49$	−2.13381	−0.304830
$50$	−2.06795	−0.292452
$51$	−0.512210	−0.0717237
$52$	1.48475	0.205897
$53$	3.55915	0.488887	0.244443	−	0.969664i	$-0.421395\pi$
$53$	3.55915	0.488887	0.244443	+	0.969664i	$0.421395\pi$
$54$	11.6449	1.58467
$55$	−0.732051	−0.0987097
$56$	1.26088	0.168492
$57$	6.08195	0.805574
$58$	14.3209	1.88042
$59$	−12.9845	−1.69044	−0.845220	−	0.534419i	$-0.820530\pi$
$59$	−12.9845	−1.69044	−0.845220	+	0.534419i	$0.820530\pi$
$60$	−3.39500	−0.438292
$61$	−9.05077	−1.15883	−0.579416	−	0.815032i	$-0.696719\pi$
$61$	−9.05077	−1.15883	−0.579416	+	0.815032i	$0.696719\pi$
$62$	3.76970	0.478752
$63$	1.71129	0.215602
$64$	−10.0373	−1.25466
$65$	−0.652235	−0.0808998
$66$	−2.25773	−0.277907
$67$	−15.6566	−1.91276	−0.956382	−	0.292121i	$-0.905639\pi$
$67$	−15.6566	−1.91276	−0.956382	+	0.292121i	$0.905639\pi$
$68$	−0.781819	−0.0948094
$69$	11.1359	1.34060
$70$	−4.56177	−0.545236
$71$	10.5344	1.25020	0.625099	−	0.780545i	$-0.285059\pi$
$71$	10.5344	1.25020	0.625099	+	0.780545i	$0.285059\pi$
$72$	0.443411	0.0522565
$73$	−8.85163	−1.03600	−0.518002	−	0.855379i	$-0.673324\pi$
$73$	−8.85163	−1.03600	−0.518002	+	0.855379i	$0.673324\pi$
$74$	12.1476	1.41213
$75$	1.49139	0.172211
$76$	9.28327	1.06486
$77$	−1.61486	−0.184031
$78$	−2.01156	−0.227765
$79$	4.65671	0.523921	0.261960	−	0.965079i	$-0.415631\pi$
$79$	4.65671	0.523921	0.261960	+	0.965079i	$0.415631\pi$
$80$	3.37080	0.376867
$81$	−6.07091	−0.674546
$82$	15.1851	1.67692
$83$	16.1391	1.77149	0.885746	−	0.464171i	$-0.153648\pi$
$83$	16.1391	1.77149	0.885746	+	0.464171i	$0.153648\pi$
$84$	−7.48918	−0.817136
$85$	0.343445	0.0372518
$86$	5.66368	0.610731
$87$	−10.3281	−1.10729
$88$	−0.418427	−0.0446045
$89$	−0.646523	−0.0685313	−0.0342657	−	0.999413i	$-0.510909\pi$
$89$	−0.646523	−0.0685313	−0.0342657	+	0.999413i	$0.510909\pi$
$90$	−1.60423	−0.169101
$91$	−1.43879	−0.150827
$92$	16.9974	1.77210
$93$	−2.71868	−0.281914
$94$	−1.40880	−0.145307
$95$	−4.07805	−0.418399
$96$	12.1008	1.23504
$97$	8.57664	0.870825	0.435413	−	0.900231i	$-0.356602\pi$
$97$	8.57664	0.870825	0.435413	+	0.900231i	$0.356602\pi$
$98$	4.41260	0.445740
$99$	−0.567897	−0.0570758
$100$	2.27640	0.227640
$101$	−16.5698	−1.64876	−0.824379	−	0.566038i	$-0.808475\pi$
$101$	−16.5698	−1.64876	−0.824379	+	0.566038i	$0.808475\pi$
$102$	1.05922	0.104879
$103$	−12.6471	−1.24616	−0.623080	−	0.782158i	$-0.714119\pi$
$103$	−12.6471	−1.24616	−0.623080	+	0.782158i	$0.714119\pi$
$104$	−0.372806	−0.0365566
$105$	3.28992	0.321063
$106$	−7.36013	−0.714879
$107$	13.9062	1.34436	0.672182	−	0.740386i	$-0.265357\pi$
$107$	13.9062	1.34436	0.672182	+	0.740386i	$0.265357\pi$
$108$	−12.8187	−1.23348
$109$	6.79167	0.650524	0.325262	−	0.945624i	$-0.394548\pi$
$109$	6.79167	0.650524	0.325262	+	0.945624i	$0.394548\pi$
$110$	1.51384	0.144339
$111$	−8.76076	−0.831535
$112$	7.43580	0.702617
$113$	−6.91359	−0.650375	−0.325188	−	0.945649i	$-0.605427\pi$
$113$	−6.91359	−0.650375	−0.325188	+	0.945649i	$0.605427\pi$
$114$	−12.5771	−1.17796
$115$	−7.46680	−0.696283
$116$	−15.7644	−1.46369
$117$	−0.505978	−0.0467777
$118$	26.8513	2.47186
$119$	0.757621	0.0694510
$120$	0.852451	0.0778177
$121$	−10.4641	−0.951282
$122$	18.7165	1.69451
$123$	−10.9514	−0.987455
$124$	−4.14969	−0.372653
$125$	−1.00000	−0.0894427
$126$	−3.53885	−0.315266
$127$	4.16996	0.370024	0.185012	−	0.982736i	$-0.440768\pi$
$127$	4.16996	0.370024	0.185012	+	0.982736i	$0.440768\pi$
$128$	4.52899	0.400310
$129$	−4.08461	−0.359630
$130$	1.34879	0.118296
$131$	−15.5173	−1.35575	−0.677875	−	0.735177i	$-0.737099\pi$
$131$	−15.5173	−1.35575	−0.677875	+	0.735177i	$0.737099\pi$
$132$	2.48531	0.216318
$133$	−8.99595	−0.780048
$134$	32.3771	2.79695
$135$	5.63113	0.484650
$136$	0.196307	0.0168332
$137$	16.9018	1.44402	0.722011	−	0.691881i	$-0.243218\pi$
$137$	16.9018	1.44402	0.722011	+	0.691881i	$0.243218\pi$
$138$	−23.0284	−1.96031
$139$	−16.1239	−1.36761	−0.683806	−	0.729664i	$-0.739677\pi$
$139$	−16.1239	−1.36761	−0.683806	+	0.729664i	$0.739677\pi$
$140$	5.02161	0.424404
$141$	1.01602	0.0855640
$142$	−21.7845	−1.82811
$143$	0.477469	0.0399280
$144$	2.61494	0.217911
$145$	6.92516	0.575103
$146$	18.3047	1.51491
$147$	−3.18234	−0.262475
$148$	−13.3721	−1.09918
$149$	−4.36998	−0.358002	−0.179001	−	0.983849i	$-0.557287\pi$
$149$	−4.36998	−0.358002	−0.179001	+	0.983849i	$0.557287\pi$
$150$	−3.08411	−0.251817
$151$	−24.2892	−1.97663	−0.988314	−	0.152432i	$-0.951290\pi$
$151$	−24.2892	−1.97663	−0.988314	+	0.152432i	$0.951290\pi$
$152$	−2.33094	−0.189064
$153$	0.266431	0.0215397
$154$	3.33945	0.269101
$155$	1.82292	0.146420
$156$	2.21434	0.177289
$157$	−16.1400	−1.28811	−0.644054	−	0.764980i	$-0.722749\pi$
$157$	−16.1400	−1.28811	−0.644054	+	0.764980i	$0.722749\pi$
$158$	−9.62983	−0.766108
$159$	5.30808	0.420958
$160$	−8.11380	−0.641452
$161$	−16.4713	−1.29812
$162$	12.5543	0.986360
$163$	−14.3063	−1.12056	−0.560279	−	0.828304i	$-0.689306\pi$
$163$	−14.3063	−1.12056	−0.560279	+	0.828304i	$0.689306\pi$
$164$	−16.7158	−1.30529
$165$	−1.09177	−0.0849943
$166$	−33.3747	−2.59038
$167$	−3.22485	−0.249546	−0.124773	−	0.992185i	$-0.539820\pi$
$167$	−3.22485	−0.249546	−0.124773	+	0.992185i	$0.539820\pi$
$168$	1.88046	0.145081
$169$	−12.5746	−0.967276
$170$	−0.710226	−0.0544718
$171$	−3.16359	−0.241926
$172$	−6.23460	−0.475384
$173$	5.66076	0.430380	0.215190	−	0.976572i	$-0.430963\pi$
$173$	5.66076	0.430380	0.215190	+	0.976572i	$0.430963\pi$
$174$	21.3579	1.61914
$175$	−2.20594	−0.166754
$176$	−2.46760	−0.186002
$177$	−19.3649	−1.45556
$178$	1.33698	0.100211
$179$	−12.3968	−0.926579	−0.463289	−	0.886207i	$-0.653331\pi$
$179$	−12.3968	−0.926579	−0.463289	+	0.886207i	$0.653331\pi$
$180$	1.76594	0.131626
$181$	11.7735	0.875119	0.437559	−	0.899190i	$-0.355843\pi$
$181$	11.7735	0.875119	0.437559	+	0.899190i	$0.355843\pi$
$182$	2.97535	0.220547
$183$	−13.4982	−0.997816
$184$	−4.26789	−0.314633
$185$	5.87423	0.431882
$186$	5.62208	0.412231
$187$	−0.251419	−0.0183856
$188$	1.55081	0.113105
$189$	12.4220	0.903564
$190$	8.43318	0.611807
$191$	16.3589	1.18369	0.591843	−	0.806054i	$-0.298401\pi$
$191$	16.3589	1.18369	0.591843	+	0.806054i	$0.298401\pi$
$192$	−14.9695	−1.08033
$193$	15.3838	1.10735	0.553676	−	0.832732i	$-0.313225\pi$
$193$	15.3838	1.10735	0.553676	+	0.832732i	$0.313225\pi$
$194$	−17.7360	−1.27337
$195$	−0.972735	−0.0696590
$196$	−4.85740	−0.346957
$197$	7.91460	0.563892	0.281946	−	0.959430i	$-0.409020\pi$
$197$	7.91460	0.563892	0.281946	+	0.959430i	$0.409020\pi$
$198$	1.17438	0.0834595
$199$	11.6273	0.824240	0.412120	−	0.911130i	$-0.364788\pi$
$199$	11.6273	0.824240	0.412120	+	0.911130i	$0.364788\pi$
$200$	−0.571582	−0.0404169
$201$	−23.3501	−1.64699
$202$	34.2655	2.41091
$203$	15.2765	1.07220
$204$	−1.16600	−0.0816360
$205$	7.34309	0.512864
$206$	26.1536	1.82221
$207$	−5.79245	−0.402603
$208$	−2.19855	−0.152442
$209$	2.98534	0.206500
$210$	−6.80338	−0.469478
$211$	6.98323	0.480745	0.240373	−	0.970681i	$-0.422730\pi$
$211$	6.98323	0.480745	0.240373	+	0.970681i	$0.422730\pi$
$212$	8.10205	0.556451
$213$	15.7108	1.07649
$214$	−28.7573	−1.96581
$215$	2.73880	0.186784
$216$	3.21865	0.219001
$217$	4.02126	0.272981
$218$	−14.0448	−0.951234
$219$	−13.2012	−0.892055
$220$	−1.66644	−0.112351
$221$	−0.224007	−0.0150683
$222$	18.1168	1.21592
$223$	−4.95465	−0.331788	−0.165894	−	0.986144i	$-0.553051\pi$
$223$	−4.95465	−0.331788	−0.165894	+	0.986144i	$0.553051\pi$
$224$	−17.8986	−1.19590
$225$	−0.775761	−0.0517174
$226$	14.2969	0.951017
$227$	−1.47576	−0.0979494	−0.0489747	−	0.998800i	$-0.515595\pi$
$227$	−1.47576	−0.0979494	−0.0489747	+	0.998800i	$0.515595\pi$
$228$	13.8450	0.916905
$229$	1.70500	0.112669	0.0563347	−	0.998412i	$-0.482059\pi$
$229$	1.70500	0.112669	0.0563347	+	0.998412i	$0.482059\pi$
$230$	15.4409	1.01815
$231$	−2.40839	−0.158460
$232$	3.95829	0.259875
$233$	21.5622	1.41259	0.706294	−	0.707918i	$-0.250365\pi$
$233$	21.5622	1.41259	0.706294	+	0.707918i	$0.250365\pi$
$234$	1.04634	0.0684011
$235$	−0.681256	−0.0444402
$236$	−29.5579	−1.92406
$237$	6.94496	0.451124
$238$	−1.56672	−0.101555
$239$	13.8584	0.896426	0.448213	−	0.893927i	$-0.352061\pi$
$239$	13.8584	0.896426	0.448213	+	0.893927i	$0.352061\pi$
$240$	5.02717	0.324503
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	21.6392	1.39102
$243$	7.83929	0.502891
$244$	−20.6032	−1.31898
$245$	2.13381	0.136324
$246$	22.6469	1.44391
$247$	2.65984	0.169242
$248$	1.04195	0.0661637
$249$	24.0696	1.52535
$250$	2.06795	0.130788
$251$	−0.460374	−0.0290586	−0.0145293	−	0.999894i	$-0.504625\pi$
$251$	−0.460374	−0.0290586	−0.0145293	+	0.999894i	$0.504625\pi$
$252$	3.89557	0.245398
$253$	5.46608	0.343649
$254$	−8.62324	−0.541070
$255$	0.512210	0.0320758
$256$	10.7089	0.669305
$257$	−4.19809	−0.261869	−0.130935	−	0.991391i	$-0.541798\pi$
$257$	−4.19809	−0.261869	−0.130935	+	0.991391i	$0.541798\pi$
$258$	8.44675	0.525872
$259$	12.9582	0.805185
$260$	−1.48475	−0.0920801
$261$	5.37227	0.332535
$262$	32.0889	1.98246
$263$	−0.394798	−0.0243443	−0.0121721	−	0.999926i	$-0.503875\pi$
$263$	−0.394798	−0.0243443	−0.0121721	+	0.999926i	$0.503875\pi$
$264$	−0.624037	−0.0384068
$265$	−3.55915	−0.218637
$266$	18.6031	1.14063
$267$	−0.964217	−0.0590091
$268$	−35.6408	−2.17711
$269$	−3.39061	−0.206729	−0.103365	−	0.994644i	$-0.532961\pi$
$269$	−3.39061	−0.206729	−0.103365	+	0.994644i	$0.532961\pi$
$270$	−11.6449	−0.708684
$271$	17.5066	1.06345	0.531726	−	0.846916i	$-0.321544\pi$
$271$	17.5066	1.06345	0.531726	+	0.846916i	$0.321544\pi$
$272$	1.15769	0.0701950
$273$	−2.14580	−0.129870
$274$	−34.9521	−2.11153
$275$	0.732051	0.0441443
$276$	25.3498	1.52588
$277$	11.1633	0.670739	0.335370	−	0.942087i	$-0.391139\pi$
$277$	11.1633	0.670739	0.335370	+	0.942087i	$0.391139\pi$
$278$	33.3434	1.99980
$279$	1.41415	0.0846629
$280$	−1.26088	−0.0753519
$281$	−2.00628	−0.119685	−0.0598424	−	0.998208i	$-0.519060\pi$
$281$	−2.00628	−0.119685	−0.0598424	+	0.998208i	$0.519060\pi$
$282$	−2.10107	−0.125117
$283$	−15.7757	−0.937766	−0.468883	−	0.883260i	$-0.655343\pi$
$283$	−15.7757	−0.937766	−0.468883	+	0.883260i	$0.655343\pi$
$284$	23.9804	1.42298
$285$	−6.08195	−0.360264
$286$	−0.987380	−0.0583850
$287$	16.1985	0.956165
$288$	−6.29437	−0.370899
$289$	−16.8820	−0.993061
$290$	−14.3209	−0.840949
$291$	12.7911	0.749827
$292$	−20.1499	−1.17918
$293$	−3.98251	−0.232661	−0.116330	−	0.993211i	$-0.537113\pi$
$293$	−3.98251	−0.232661	−0.116330	+	0.993211i	$0.537113\pi$
$294$	6.58090	0.383806
$295$	12.9845	0.755987
$296$	3.35761	0.195157
$297$	−4.12227	−0.239198
$298$	9.03688	0.523492
$299$	4.87011	0.281645
$300$	3.39500	0.196010
$301$	6.04163	0.348234
$302$	50.2288	2.89034
$303$	−24.7120	−1.41967
$304$	−13.7463	−0.788404
$305$	9.05077	0.518245
$306$	−0.550966	−0.0314966
$307$	33.5219	1.91319	0.956597	−	0.291415i	$-0.0941261\pi$
$307$	33.5219	1.91319	0.956597	+	0.291415i	$0.0941261\pi$
$308$	−3.67608	−0.209464
$309$	−18.8618	−1.07301
$310$	−3.76970	−0.214104
$311$	−10.9257	−0.619542	−0.309771	−	0.950811i	$-0.600252\pi$
$311$	−10.9257	−0.619542	−0.309771	+	0.950811i	$0.600252\pi$
$312$	−0.555998	−0.0314772
$313$	−17.1761	−0.970849	−0.485424	−	0.874279i	$-0.661335\pi$
$313$	−17.1761	−0.970849	−0.485424	+	0.874279i	$0.661335\pi$
$314$	33.3766	1.88355
$315$	−1.71129	−0.0964201
$316$	10.6005	0.596327
$317$	4.47510	0.251347	0.125673	−	0.992072i	$-0.459891\pi$
$317$	4.47510	0.251347	0.125673	+	0.992072i	$0.459891\pi$
$318$	−10.9768	−0.615549
$319$	−5.06957	−0.283841
$320$	10.0373	0.561102
$321$	20.7395	1.15757
$322$	34.0619	1.89819
$323$	−1.40059	−0.0779307
$324$	−13.8198	−0.767768
$325$	0.652235	0.0361795
$326$	29.5847	1.63854
$327$	10.1290	0.560136
$328$	4.19718	0.231750
$329$	−1.50281	−0.0828527
$330$	2.25773	0.124284
$331$	24.8434	1.36552	0.682758	−	0.730645i	$-0.260781\pi$
$331$	24.8434	1.36552	0.682758	+	0.730645i	$0.260781\pi$
$332$	36.7390	2.01631
$333$	4.55700	0.249722
$334$	6.66881	0.364901
$335$	15.6566	0.855414
$336$	11.0897	0.604991
$337$	−31.1945	−1.69927	−0.849637	−	0.527368i	$-0.823179\pi$
$337$	−31.1945	−1.69927	−0.849637	+	0.527368i	$0.823179\pi$
$338$	26.0036	1.41441
$339$	−10.3108	−0.560008
$340$	0.781819	0.0424001
$341$	−1.33447	−0.0722655
$342$	6.54214	0.353758
$343$	20.1487	1.08793
$344$	1.56545	0.0844032
$345$	−11.1359	−0.599537
$346$	−11.7061	−0.629326
$347$	−11.1816	−0.600257	−0.300129	−	0.953899i	$-0.597030\pi$
$347$	−11.1816	−0.600257	−0.300129	+	0.953899i	$0.597030\pi$
$348$	−23.5109	−1.26032
$349$	24.5618	1.31476	0.657382	−	0.753558i	$-0.271664\pi$
$349$	24.5618	1.31476	0.657382	+	0.753558i	$0.271664\pi$
$350$	4.56177	0.243837
$351$	−3.67282	−0.196040
$352$	5.93971	0.316588
$353$	−33.1831	−1.76616	−0.883079	−	0.469225i	$-0.844533\pi$
$353$	−33.1831	−1.76616	−0.883079	+	0.469225i	$0.844533\pi$
$354$	40.0457	2.12840
$355$	−10.5344	−0.559106
$356$	−1.47175	−0.0780024
$357$	1.12991	0.0598010
$358$	25.6359	1.35490
$359$	2.62454	0.138518	0.0692590	−	0.997599i	$-0.477937\pi$
$359$	2.62454	0.138518	0.0692590	+	0.997599i	$0.477937\pi$
$360$	−0.443411	−0.0233698
$361$	−2.36953	−0.124712
$362$	−24.3470	−1.27965
$363$	−15.6060	−0.819105
$364$	−3.27527	−0.171671
$365$	8.85163	0.463315
$366$	27.9136	1.45907
$367$	30.0773	1.57002	0.785012	−	0.619481i	$-0.212657\pi$
$367$	30.0773	1.57002	0.785012	+	0.619481i	$0.212657\pi$
$368$	−25.1691	−1.31203
$369$	5.69649	0.296547
$370$	−12.1476	−0.631524
$371$	−7.85129	−0.407619
$372$	−6.18880	−0.320874
$373$	26.3157	1.36257	0.681287	−	0.732017i	$-0.261421\pi$
$373$	26.3157	1.36257	0.681287	+	0.732017i	$0.261421\pi$
$374$	0.519921	0.0268845
$375$	−1.49139	−0.0770150
$376$	−0.389393	−0.0200814
$377$	−4.51683	−0.232628
$378$	−25.6879	−1.32124
$379$	−31.0349	−1.59416	−0.797078	−	0.603877i	$-0.793622\pi$
$379$	−31.0349	−1.59416	−0.797078	+	0.603877i	$0.793622\pi$
$380$	−9.28327	−0.476222
$381$	6.21902	0.318610
$382$	−33.8292	−1.73085
$383$	30.6667	1.56699	0.783497	−	0.621396i	$-0.213434\pi$
$383$	30.6667	1.56699	0.783497	+	0.621396i	$0.213434\pi$
$384$	6.75448	0.344688
$385$	1.61486	0.0823011
$386$	−31.8129	−1.61924
$387$	2.12465	0.108002
$388$	19.5239	0.991174
$389$	1.90737	0.0967075	0.0483537	−	0.998830i	$-0.484603\pi$
$389$	1.90737	0.0967075	0.0483537	+	0.998830i	$0.484603\pi$
$390$	2.01156	0.101859
$391$	−2.56444	−0.129689
$392$	1.21965	0.0616015
$393$	−23.1423	−1.16737
$394$	−16.3670	−0.824556
$395$	−4.65671	−0.234305
$396$	−1.29276	−0.0649636
$397$	−8.16163	−0.409620	−0.204810	−	0.978802i	$-0.565658\pi$
$397$	−8.16163	−0.409620	−0.204810	+	0.978802i	$0.565658\pi$
$398$	−24.0447	−1.20525
$399$	−13.4164	−0.671663
$400$	−3.37080	−0.168540
$401$	3.42073	0.170823	0.0854114	−	0.996346i	$-0.472780\pi$
$401$	3.42073	0.170823	0.0854114	+	0.996346i	$0.472780\pi$
$402$	48.2868	2.40833
$403$	−1.18897	−0.0592268
$404$	−37.7195	−1.87662
$405$	6.07091	0.301666
$406$	−31.5910	−1.56784
$407$	−4.30024	−0.213155
$408$	0.292770	0.0144943
$409$	19.3694	0.957754	0.478877	−	0.877882i	$-0.341044\pi$
$409$	19.3694	0.957754	0.478877	+	0.877882i	$0.341044\pi$
$410$	−15.1851	−0.749940
$411$	25.2072	1.24338
$412$	−28.7900	−1.41838
$413$	28.6431	1.40944
$414$	11.9785	0.588710
$415$	−16.1391	−0.792235
$416$	5.29210	0.259467
$417$	−24.0470	−1.17759
$418$	−6.17352	−0.301957
$419$	7.11390	0.347537	0.173768	−	0.984787i	$-0.444406\pi$
$419$	7.11390	0.347537	0.173768	+	0.984787i	$0.444406\pi$
$420$	7.48918	0.365434
$421$	−4.46101	−0.217416	−0.108708	−	0.994074i	$-0.534671\pi$
$421$	−4.46101	−0.217416	−0.108708	+	0.994074i	$0.534671\pi$
$422$	−14.4409	−0.702974
$423$	−0.528492	−0.0256961
$424$	−2.03435	−0.0987966
$425$	−0.343445	−0.0166595
$426$	−32.4891	−1.57410
$427$	19.9655	0.966198
$428$	31.6561	1.53015
$429$	0.712092	0.0343801
$430$	−5.66368	−0.273127
$431$	12.4465	0.599527	0.299764	−	0.954014i	$-0.403092\pi$
$431$	12.4465	0.599527	0.299764	+	0.954014i	$0.403092\pi$
$432$	18.9814	0.913243
$433$	30.2000	1.45132	0.725659	−	0.688055i	$-0.241535\pi$
$433$	30.2000	1.45132	0.725659	+	0.688055i	$0.241535\pi$
$434$	−8.31574	−0.399168
$435$	10.3281	0.495194
$436$	15.4606	0.740427
$437$	30.4500	1.45662
$438$	27.2994	1.30442
$439$	−19.5672	−0.933893	−0.466946	−	0.884286i	$-0.654646\pi$
$439$	−19.5672	−0.933893	−0.466946	+	0.884286i	$0.654646\pi$
$440$	0.418427	0.0199477
$441$	1.65533	0.0788251
$442$	0.463234	0.0220338
$443$	−15.8536	−0.753227	−0.376614	−	0.926370i	$-0.622912\pi$
$443$	−15.8536	−0.753227	−0.376614	+	0.926370i	$0.622912\pi$
$444$	−19.9430	−0.946453
$445$	0.646523	0.0306481
$446$	10.2459	0.485159
$447$	−6.51733	−0.308259
$448$	22.1417	1.04610
$449$	−17.9764	−0.848359	−0.424180	−	0.905578i	$-0.639438\pi$
$449$	−17.9764	−0.848359	−0.424180	+	0.905578i	$0.639438\pi$
$450$	1.60423	0.0756242
$451$	−5.37552	−0.253123
$452$	−15.7381	−0.740258
$453$	−36.2246	−1.70198
$454$	3.05178	0.143227
$455$	1.43879	0.0674517
$456$	−3.47633	−0.162794
$457$	−7.22766	−0.338095	−0.169048	−	0.985608i	$-0.554069\pi$
$457$	−7.22766	−0.338095	−0.169048	+	0.985608i	$0.554069\pi$
$458$	−3.52584	−0.164752
$459$	1.93398	0.0902706
$460$	−16.9974	−0.792509
$461$	10.5925	0.493342	0.246671	−	0.969099i	$-0.420663\pi$
$461$	10.5925	0.493342	0.246671	+	0.969099i	$0.420663\pi$
$462$	4.98042	0.231710
$463$	25.3273	1.17706	0.588530	−	0.808475i	$-0.299707\pi$
$463$	25.3273	1.17706	0.588530	+	0.808475i	$0.299707\pi$
$464$	23.3433	1.08369
$465$	2.71868	0.126076
$466$	−44.5895	−2.06557
$467$	−15.6673	−0.724997	−0.362499	−	0.931984i	$-0.618076\pi$
$467$	−15.6673	−0.724997	−0.362499	+	0.931984i	$0.618076\pi$
$468$	−1.15181	−0.0532424
$469$	34.5377	1.59480
$470$	1.40880	0.0649831
$471$	−24.0709	−1.10913
$472$	7.42171	0.341612
$473$	−2.00494	−0.0921871
$474$	−14.3618	−0.659660
$475$	4.07805	0.187114
$476$	1.72465	0.0790491
$477$	−2.76105	−0.126420
$478$	−28.6584	−1.31081
$479$	11.8900	0.543269	0.271634	−	0.962401i	$-0.412436\pi$
$479$	11.8900	0.543269	0.271634	+	0.962401i	$0.412436\pi$
$480$	−12.1008	−0.552324
$481$	−3.83138	−0.174696
$482$	2.06795	0.0941924
$483$	−24.5652	−1.11775
$484$	−23.8205	−1.08275
$485$	−8.57664	−0.389445
$486$	−16.2112	−0.735356
$487$	−33.7747	−1.53048	−0.765239	−	0.643746i	$-0.777379\pi$
$487$	−33.7747	−1.53048	−0.765239	+	0.643746i	$0.777379\pi$
$488$	5.17325	0.234182
$489$	−21.3363	−0.964860
$490$	−4.41260	−0.199341
$491$	35.5069	1.60241	0.801203	−	0.598393i	$-0.204194\pi$
$491$	35.5069	1.60241	0.801203	+	0.598393i	$0.204194\pi$
$492$	−24.9298	−1.12392
$493$	2.37841	0.107118
$494$	−5.50041	−0.247475
$495$	0.567897	0.0255251
$496$	6.14469	0.275905
$497$	−23.2382	−1.04238
$498$	−49.7746	−2.23045
$499$	−30.1195	−1.34833	−0.674167	−	0.738579i	$-0.735497\pi$
$499$	−30.1195	−1.34833	−0.674167	+	0.738579i	$0.735497\pi$
$500$	−2.27640	−0.101804
$501$	−4.80950	−0.214873
$502$	0.952029	0.0424912
$503$	−36.9064	−1.64558	−0.822788	−	0.568348i	$-0.807583\pi$
$503$	−36.9064	−1.64558	−0.822788	+	0.568348i	$0.807583\pi$
$504$	−0.978140	−0.0435698
$505$	16.5698	0.737347
$506$	−11.3036	−0.502504
$507$	−18.7536	−0.832876
$508$	9.49249	0.421161
$509$	11.2012	0.496486	0.248243	−	0.968698i	$-0.420147\pi$
$509$	11.2012	0.496486	0.248243	+	0.968698i	$0.420147\pi$
$510$	−1.05922	−0.0469032
$511$	19.5262	0.863788
$512$	−31.2034	−1.37901
$513$	−22.9640	−1.01389
$514$	8.68141	0.382921
$515$	12.6471	0.557299
$516$	−9.29820	−0.409331
$517$	0.498714	0.0219334
$518$	−26.7969	−1.17739
$519$	8.44239	0.370580
$520$	0.372806	0.0163486
$521$	7.37810	0.323240	0.161620	−	0.986853i	$-0.448328\pi$
$521$	7.37810	0.323240	0.161620	+	0.986853i	$0.448328\pi$
$522$	−11.1096	−0.486252
$523$	−2.13347	−0.0932903	−0.0466452	−	0.998912i	$-0.514853\pi$
$523$	−2.13347	−0.0932903	−0.0466452	+	0.998912i	$0.514853\pi$
$524$	−35.3235	−1.54311
$525$	−3.28992	−0.143584
$526$	0.816421	0.0355977
$527$	0.626072	0.0272721
$528$	−3.68015	−0.160158
$529$	32.7531	1.42405
$530$	7.36013	0.319704
$531$	10.0729	0.437126
$532$	−20.4784	−0.887850
$533$	−4.78942	−0.207453
$534$	1.99395	0.0862866
$535$	−13.9062	−0.601217
$536$	8.94905	0.386540
$537$	−18.4884	−0.797834
$538$	7.01161	0.302292
$539$	−1.56206	−0.0672825
$540$	12.8187	0.551629
$541$	7.94556	0.341606	0.170803	−	0.985305i	$-0.445364\pi$
$541$	7.94556	0.341606	0.170803	+	0.985305i	$0.445364\pi$
$542$	−36.2028	−1.55504
$543$	17.5589	0.753524
$544$	−2.78664	−0.119476
$545$	−6.79167	−0.290923
$546$	4.43740	0.189903
$547$	−14.8181	−0.633575	−0.316788	−	0.948497i	$-0.602604\pi$
$547$	−14.8181	−0.633575	−0.316788	+	0.948497i	$0.602604\pi$
$548$	38.4754	1.64359
$549$	7.02123	0.299659
$550$	−1.51384	−0.0645504
$551$	−28.2411	−1.20311
$552$	−6.36508	−0.270916
$553$	−10.2724	−0.436829
$554$	−23.0851	−0.980794
$555$	8.76076	0.371874
$556$	−36.7045	−1.55662
$557$	27.4901	1.16479	0.582396	−	0.812905i	$-0.302115\pi$
$557$	27.4901	1.16479	0.582396	+	0.812905i	$0.302115\pi$
$558$	−2.92438	−0.123799
$559$	−1.78634	−0.0755540
$560$	−7.43580	−0.314220
$561$	−0.374964	−0.0158310
$562$	4.14889	0.175010
$563$	25.3988	1.07043	0.535217	−	0.844715i	$-0.320230\pi$
$563$	25.3988	1.07043	0.535217	+	0.844715i	$0.320230\pi$
$564$	2.31286	0.0973890
$565$	6.91359	0.290857
$566$	32.6232	1.37126
$567$	13.3921	0.562415
$568$	−6.02125	−0.252646
$569$	13.2055	0.553602	0.276801	−	0.960927i	$-0.410726\pi$
$569$	13.2055	0.553602	0.276801	+	0.960927i	$0.410726\pi$
$570$	12.5771	0.526799
$571$	−6.44669	−0.269786	−0.134893	−	0.990860i	$-0.543069\pi$
$571$	−6.44669	−0.269786	−0.134893	+	0.990860i	$0.543069\pi$
$572$	1.08691	0.0454460
$573$	24.3974	1.01922
$574$	−33.4975	−1.39816
$575$	7.46680	0.311387
$576$	7.78654	0.324439
$577$	11.7940	0.490992	0.245496	−	0.969398i	$-0.421049\pi$
$577$	11.7940	0.490992	0.245496	+	0.969398i	$0.421049\pi$
$578$	34.9112	1.45211
$579$	22.9433	0.953489
$580$	15.7644	0.654582
$581$	−35.6019	−1.47701
$582$	−26.4513	−1.09644
$583$	2.60548	0.107908
$584$	5.05943	0.209361
$585$	0.505978	0.0209196
$586$	8.23562	0.340210
$587$	−27.9429	−1.15333	−0.576664	−	0.816981i	$-0.695646\pi$
$587$	−27.9429	−1.15333	−0.576664	+	0.816981i	$0.695646\pi$
$588$	−7.24428	−0.298749
$589$	−7.43395	−0.306310
$590$	−26.8513	−1.10545
$591$	11.8037	0.485541
$592$	19.8009	0.813811
$593$	−8.22846	−0.337902	−0.168951	−	0.985624i	$-0.554038\pi$
$593$	−8.22846	−0.337902	−0.168951	+	0.985624i	$0.554038\pi$
$594$	8.52463	0.349770
$595$	−0.757621	−0.0310594
$596$	−9.94782	−0.407479
$597$	17.3409	0.709715
$598$	−10.0711	−0.411839
$599$	−42.1104	−1.72058	−0.860291	−	0.509803i	$-0.829718\pi$
$599$	−42.1104	−1.72058	−0.860291	+	0.509803i	$0.829718\pi$
$600$	−0.852451	−0.0348011
$601$	6.35438	0.259201	0.129600	−	0.991566i	$-0.458631\pi$
$601$	6.35438	0.259201	0.129600	+	0.991566i	$0.458631\pi$
$602$	−12.4938	−0.509208
$603$	12.1458	0.494616
$604$	−55.2920	−2.24980
$605$	10.4641	0.425426
$606$	51.1031	2.07592
$607$	−9.50426	−0.385766	−0.192883	−	0.981222i	$-0.561784\pi$
$607$	−9.50426	−0.385766	−0.192883	+	0.981222i	$0.561784\pi$
$608$	33.0885	1.34191
$609$	22.7832	0.923222
$610$	−18.7165	−0.757809
$611$	0.444339	0.0179760
$612$	0.606505	0.0245165
$613$	15.0625	0.608370	0.304185	−	0.952613i	$-0.401616\pi$
$613$	15.0625	0.608370	0.304185	+	0.952613i	$0.401616\pi$
$614$	−69.3214	−2.79758
$615$	10.9514	0.441603
$616$	0.923027	0.0371898
$617$	13.7503	0.553567	0.276783	−	0.960932i	$-0.410732\pi$
$617$	13.7503	0.553567	0.276783	+	0.960932i	$0.410732\pi$
$618$	39.0052	1.56902
$619$	20.7222	0.832896	0.416448	−	0.909160i	$-0.363275\pi$
$619$	20.7222	0.832896	0.416448	+	0.909160i	$0.363275\pi$
$620$	4.14969	0.166656
$621$	−42.0465	−1.68727
$622$	22.5938	0.905930
$623$	1.42619	0.0571393
$624$	−3.27890	−0.131261
$625$	1.00000	0.0400000
$626$	35.5192	1.41963
$627$	4.45230	0.177808
$628$	−36.7410	−1.46613
$629$	2.01748	0.0804421
$630$	3.53885	0.140991
$631$	−9.06039	−0.360688	−0.180344	−	0.983604i	$-0.557721\pi$
$631$	−9.06039	−0.360688	−0.180344	+	0.983604i	$0.557721\pi$
$632$	−2.66169	−0.105876
$633$	10.4147	0.413947
$634$	−9.25426	−0.367534
$635$	−4.16996	−0.165480
$636$	12.0833	0.479134
$637$	−1.39174	−0.0551429
$638$	10.4836	0.415049
$639$	−8.17215	−0.323285
$640$	−4.52899	−0.179024
$641$	−19.3832	−0.765590	−0.382795	−	0.923833i	$-0.625039\pi$
$641$	−19.3832	−0.765590	−0.382795	+	0.923833i	$0.625039\pi$
$642$	−42.8883	−1.69266
$643$	20.9974	0.828056	0.414028	−	0.910264i	$-0.364121\pi$
$643$	20.9974	0.828056	0.414028	+	0.910264i	$0.364121\pi$
$644$	−37.4954	−1.47753
$645$	4.08461	0.160831
$646$	2.89634	0.113955
$647$	−44.8987	−1.76515	−0.882575	−	0.470172i	$-0.844192\pi$
$647$	−44.8987	−1.76515	−0.882575	+	0.470172i	$0.844192\pi$
$648$	3.47002	0.136315
$649$	−9.50532	−0.373116
$650$	−1.34879	−0.0529037
$651$	5.99725	0.235051
$652$	−32.5669	−1.27542
$653$	−33.5289	−1.31209	−0.656043	−	0.754723i	$-0.727771\pi$
$653$	−33.5289	−1.31209	−0.656043	+	0.754723i	$0.727771\pi$
$654$	−20.9463	−0.819063
$655$	15.5173	0.606310
$656$	24.7521	0.966407
$657$	6.86675	0.267897
$658$	3.10773	0.121152
$659$	41.6373	1.62196	0.810980	−	0.585074i	$-0.198934\pi$
$659$	41.6373	1.62196	0.810980	+	0.585074i	$0.198934\pi$
$660$	−2.48531	−0.0967406
$661$	3.40007	0.132248	0.0661238	−	0.997811i	$-0.478937\pi$
$661$	3.40007	0.132248	0.0661238	+	0.997811i	$0.478937\pi$
$662$	−51.3747	−1.99674
$663$	−0.334081	−0.0129746
$664$	−9.22479	−0.357991
$665$	8.99595	0.348848
$666$	−9.42364	−0.365158
$667$	−51.7088	−2.00217
$668$	−7.34105	−0.284034
$669$	−7.38930	−0.285687
$670$	−32.3771	−1.25084
$671$	−6.62562	−0.255779
$672$	−26.6937	−1.02973
$673$	−11.0143	−0.424568	−0.212284	−	0.977208i	$-0.568090\pi$
$673$	−11.0143	−0.424568	−0.212284	+	0.977208i	$0.568090\pi$
$674$	64.5086	2.48478
$675$	−5.63113	−0.216742
$676$	−28.6248	−1.10095
$677$	−14.5109	−0.557698	−0.278849	−	0.960335i	$-0.589953\pi$
$677$	−14.5109	−0.557698	−0.278849	+	0.960335i	$0.589953\pi$
$678$	21.3223	0.818876
$679$	−18.9196	−0.726067
$680$	−0.196307	−0.00752803
$681$	−2.20093	−0.0843397
$682$	2.75961	0.105671
$683$	7.82061	0.299247	0.149624	−	0.988743i	$-0.452194\pi$
$683$	7.82061	0.299247	0.149624	+	0.988743i	$0.452194\pi$
$684$	−7.20160	−0.275360
$685$	−16.9018	−0.645786
$686$	−41.6664	−1.59083
$687$	2.54281	0.0970143
$688$	9.23194	0.351964
$689$	2.32140	0.0884384
$690$	23.0284	0.876677
$691$	−28.2525	−1.07477	−0.537387	−	0.843336i	$-0.680589\pi$
$691$	−28.2525	−1.07477	−0.537387	+	0.843336i	$0.680589\pi$
$692$	12.8862	0.489858
$693$	1.25275	0.0475880
$694$	23.1229	0.877732
$695$	16.1239	0.611615
$696$	5.90335	0.223766
$697$	2.52195	0.0955256
$698$	−50.7925	−1.92252
$699$	32.1577	1.21631
$700$	−5.02161	−0.189799
$701$	28.3325	1.07010	0.535052	−	0.844819i	$-0.320292\pi$
$701$	28.3325	1.07010	0.535052	+	0.844819i	$0.320292\pi$
$702$	7.59518	0.286662
$703$	−23.9554	−0.903495
$704$	−7.34781	−0.276931
$705$	−1.01602	−0.0382654
$706$	68.6208	2.58258
$707$	36.5521	1.37468
$708$	−44.0824	−1.65672
$709$	19.5017	0.732403	0.366201	−	0.930536i	$-0.380658\pi$
$709$	19.5017	0.732403	0.366201	+	0.930536i	$0.380658\pi$
$710$	21.7845	0.817557
$711$	−3.61250	−0.135479
$712$	0.369541	0.0138491
$713$	−13.6114	−0.509750
$714$	−2.33659	−0.0874445
$715$	−0.477469	−0.0178563
$716$	−28.2200	−1.05463
$717$	20.6683	0.771871
$718$	−5.42741	−0.202549
$719$	−5.36138	−0.199946	−0.0999730	−	0.994990i	$-0.531876\pi$
$719$	−5.36138	−0.199946	−0.0999730	+	0.994990i	$0.531876\pi$
$720$	−2.61494	−0.0974529
$721$	27.8989	1.03901
$722$	4.90005	0.182361
$723$	−1.49139	−0.0554653
$724$	26.8012	0.996060
$725$	−6.92516	−0.257194
$726$	32.2724	1.19774
$727$	9.02856	0.334851	0.167425	−	0.985885i	$-0.446455\pi$
$727$	9.02856	0.334851	0.167425	+	0.985885i	$0.446455\pi$
$728$	0.822388	0.0304797
$729$	29.9042	1.10756
$730$	−18.3047	−0.677487
$731$	0.940626	0.0347903
$732$	−30.7273	−1.13571
$733$	−28.6688	−1.05890	−0.529452	−	0.848340i	$-0.677603\pi$
$733$	−28.6688	−1.05890	−0.529452	+	0.848340i	$0.677603\pi$
$734$	−62.1983	−2.29578
$735$	3.18234	0.117382
$736$	60.5841	2.23316
$737$	−11.4615	−0.422188
$738$	−11.7800	−0.433629
$739$	−6.63655	−0.244129	−0.122065	−	0.992522i	$-0.538952\pi$
$739$	−6.63655	−0.244129	−0.122065	+	0.992522i	$0.538952\pi$
$740$	13.3721	0.491569
$741$	3.96686	0.145726
$742$	16.2360	0.596044
$743$	1.85960	0.0682221	0.0341110	−	0.999418i	$-0.489140\pi$
$743$	1.85960	0.0682221	0.0341110	+	0.999418i	$0.489140\pi$
$744$	1.55395	0.0569705
$745$	4.36998	0.160104
$746$	−54.4194	−1.99243
$747$	−12.5201	−0.458085
$748$	−0.572331	−0.0209265
$749$	−30.6763	−1.12089
$750$	3.08411	0.112616
$751$	1.69291	0.0617750	0.0308875	−	0.999523i	$-0.490167\pi$
$751$	1.69291	0.0617750	0.0308875	+	0.999523i	$0.490167\pi$
$752$	−2.29638	−0.0837403
$753$	−0.686597	−0.0250210
$754$	9.34056	0.340163
$755$	24.2892	0.883975
$756$	28.2773	1.02844
$757$	−14.0493	−0.510630	−0.255315	−	0.966858i	$-0.582179\pi$
$757$	−14.0493	−0.510630	−0.255315	+	0.966858i	$0.582179\pi$
$758$	64.1785	2.33107
$759$	8.15204	0.295900
$760$	2.33094	0.0845520
$761$	−11.1723	−0.404997	−0.202498	−	0.979283i	$-0.564906\pi$
$761$	−11.1723	−0.404997	−0.202498	+	0.979283i	$0.564906\pi$
$762$	−12.8606	−0.465891
$763$	−14.9820	−0.542386
$764$	37.2393	1.34727
$765$	−0.266431	−0.00963285
$766$	−63.4170	−2.29135
$767$	−8.46895	−0.305796
$768$	15.9711	0.576308
$769$	27.0072	0.973906	0.486953	−	0.873428i	$-0.338108\pi$
$769$	27.0072	0.973906	0.486953	+	0.873428i	$0.338108\pi$
$770$	−3.33945	−0.120345
$771$	−6.26098	−0.225484
$772$	35.0198	1.26039
$773$	−17.9411	−0.645297	−0.322649	−	0.946519i	$-0.604573\pi$
$773$	−17.9411	−0.645297	−0.322649	+	0.946519i	$0.604573\pi$
$774$	−4.39366	−0.157927
$775$	−1.82292	−0.0654811
$776$	−4.90225	−0.175981
$777$	19.3258	0.693308
$778$	−3.94434	−0.141411
$779$	−29.9455	−1.07291
$780$	−2.21434	−0.0792859
$781$	7.71169	0.275946
$782$	5.30311	0.189639
$783$	38.9964	1.39362
$784$	7.19264	0.256880
$785$	16.1400	0.576060
$786$	47.8570	1.70700
$787$	−35.4737	−1.26450	−0.632251	−	0.774764i	$-0.717869\pi$
$787$	−35.4737	−1.26450	−0.632251	+	0.774764i	$0.717869\pi$
$788$	18.0168	0.641822
$789$	−0.588797	−0.0209617
$790$	9.62983	0.342614
$791$	15.2510	0.542263
$792$	0.324599	0.0115341
$793$	−5.90322	−0.209630
$794$	16.8778	0.598971
$795$	−5.30808	−0.188258
$796$	26.4685	0.938150
$797$	−11.0556	−0.391609	−0.195805	−	0.980643i	$-0.562732\pi$
$797$	−11.0556	−0.391609	−0.195805	+	0.980643i	$0.562732\pi$
$798$	27.7445	0.982144
$799$	−0.233974	−0.00827740
$800$	8.11380	0.286866
$801$	0.501548	0.0177213
$802$	−7.07388	−0.249787
$803$	−6.47984	−0.228669
$804$	−53.1543	−1.87461
$805$	16.4713	0.580539
$806$	2.45873	0.0866050
$807$	−5.05672	−0.178005
$808$	9.47100	0.333189
$809$	26.4189	0.928840	0.464420	−	0.885615i	$-0.346263\pi$
$809$	26.4189	0.928840	0.464420	+	0.885615i	$0.346263\pi$
$810$	−12.5543	−0.441114
$811$	33.5518	1.17816	0.589081	−	0.808074i	$-0.299490\pi$
$811$	33.5518	1.17816	0.589081	+	0.808074i	$0.299490\pi$
$812$	34.7755	1.22038
$813$	26.1092	0.915689
$814$	8.89266	0.311688
$815$	14.3063	0.501128
$816$	1.72656	0.0604416
$817$	−11.1689	−0.390752
$818$	−40.0548	−1.40048
$819$	1.11616	0.0390018
$820$	16.7158	0.583742
$821$	3.94805	0.137788	0.0688940	−	0.997624i	$-0.478053\pi$
$821$	3.94805	0.137788	0.0688940	+	0.997624i	$0.478053\pi$
$822$	−52.1272	−1.81814
$823$	54.1724	1.88833	0.944166	−	0.329469i	$-0.106870\pi$
$823$	54.1724	1.88833	0.944166	+	0.329469i	$0.106870\pi$
$824$	7.22887	0.251830
$825$	1.09177	0.0380106
$826$	−59.2324	−2.06096
$827$	−25.8472	−0.898795	−0.449397	−	0.893332i	$-0.648361\pi$
$827$	−25.8472	−0.898795	−0.449397	+	0.893332i	$0.648361\pi$
$828$	−13.1859	−0.458243
$829$	37.8796	1.31561	0.657807	−	0.753187i	$-0.271484\pi$
$829$	37.8796	1.31561	0.657807	+	0.753187i	$0.271484\pi$
$830$	33.3747	1.15845
$831$	16.6488	0.577542
$832$	−6.54667	−0.226965
$833$	0.732846	0.0253916
$834$	49.7279	1.72194
$835$	3.22485	0.111600
$836$	6.79583	0.235039
$837$	10.2651	0.354813
$838$	−14.7112	−0.508189
$839$	10.7724	0.371905	0.185952	−	0.982559i	$-0.440463\pi$
$839$	10.7724	0.371905	0.185952	+	0.982559i	$0.440463\pi$
$840$	−1.88046	−0.0648820
$841$	18.9578	0.653717
$842$	9.22514	0.317919
$843$	−2.99215	−0.103055
$844$	15.8966	0.547184
$845$	12.5746	0.432579
$846$	1.09289	0.0375744
$847$	23.0832	0.793149
$848$	−11.9972	−0.411985
$849$	−23.5276	−0.807466
$850$	0.710226	0.0243605
$851$	−43.8617	−1.50356
$852$	35.7641	1.22526
$853$	26.0919	0.893370	0.446685	−	0.894691i	$-0.352605\pi$
$853$	26.0919	0.893370	0.446685	+	0.894691i	$0.352605\pi$
$854$	−41.2875	−1.41283
$855$	3.16359	0.108193
$856$	−7.94853	−0.271675
$857$	−28.4989	−0.973505	−0.486752	−	0.873540i	$-0.661819\pi$
$857$	−28.4989	−0.973505	−0.486752	+	0.873540i	$0.661819\pi$
$858$	−1.47257	−0.0502726
$859$	−18.2521	−0.622755	−0.311378	−	0.950286i	$-0.600790\pi$
$859$	−18.2521	−0.622755	−0.311378	+	0.950286i	$0.600790\pi$
$860$	6.23460	0.212598
$861$	24.1582	0.823309
$862$	−25.7387	−0.876664
$863$	−35.1173	−1.19541	−0.597704	−	0.801717i	$-0.703920\pi$
$863$	−35.1173	−1.19541	−0.597704	+	0.801717i	$0.703920\pi$
$864$	−45.6898	−1.55440
$865$	−5.66076	−0.192472
$866$	−62.4519	−2.12220
$867$	−25.1777	−0.855079
$868$	9.15399	0.310707
$869$	3.40895	0.115641
$870$	−21.3579	−0.724102
$871$	−10.2118	−0.346014
$872$	−3.88200	−0.131461
$873$	−6.65342	−0.225184
$874$	−62.9689	−2.12995
$875$	2.20594	0.0745745
$876$	−30.0513	−1.01534
$877$	44.5372	1.50392	0.751958	−	0.659211i	$-0.229110\pi$
$877$	44.5372	1.50392	0.751958	+	0.659211i	$0.229110\pi$
$878$	40.4640	1.36559
$879$	−5.93947	−0.200333
$880$	2.46760	0.0831827
$881$	32.7650	1.10388	0.551940	−	0.833884i	$-0.313888\pi$
$881$	32.7650	1.10388	0.551940	+	0.833884i	$0.313888\pi$
$882$	−3.42313	−0.115263
$883$	−11.6965	−0.393617	−0.196808	−	0.980442i	$-0.563058\pi$
$883$	−11.6965	−0.393617	−0.196808	+	0.980442i	$0.563058\pi$
$884$	−0.509929	−0.0171508
$885$	19.3649	0.650945
$886$	32.7844	1.10141
$887$	43.0697	1.44614	0.723070	−	0.690774i	$-0.242730\pi$
$887$	43.0697	1.44614	0.723070	+	0.690774i	$0.242730\pi$
$888$	5.00749	0.168040
$889$	−9.19869	−0.308514
$890$	−1.33698	−0.0448155
$891$	−4.44421	−0.148887
$892$	−11.2788	−0.377641
$893$	2.77819	0.0929687
$894$	13.4775	0.450755
$895$	12.3968	0.414379
$896$	−9.99070	−0.333766
$897$	7.26322	0.242512
$898$	37.1743	1.24052
$899$	12.6240	0.421034
$900$	−1.76594	−0.0588648
$901$	−1.22237	−0.0407231
$902$	11.1163	0.370132
$903$	9.01042	0.299848
$904$	3.95168	0.131431
$905$	−11.7735	−0.391365
$906$	74.9106	2.48874
$907$	1.32759	0.0440821	0.0220410	−	0.999757i	$-0.492984\pi$
$907$	1.32759	0.0440821	0.0220410	+	0.999757i	$0.492984\pi$
$908$	−3.35941	−0.111486
$909$	12.8542	0.426347
$910$	−2.97535	−0.0986318
$911$	−42.8692	−1.42032	−0.710160	−	0.704041i	$-0.751377\pi$
$911$	−42.8692	−1.42032	−0.710160	+	0.704041i	$0.751377\pi$
$912$	−20.5011	−0.678858
$913$	11.8146	0.391006
$914$	14.9464	0.494383
$915$	13.4982	0.446237
$916$	3.88126	0.128240
$917$	34.2302	1.13038
$918$	−3.99937	−0.131999
$919$	−8.18774	−0.270089	−0.135044	−	0.990840i	$-0.543118\pi$
$919$	−8.18774	−0.270089	−0.135044	+	0.990840i	$0.543118\pi$
$920$	4.26789	0.140708
$921$	49.9941	1.64736
$922$	−21.9047	−0.721394
$923$	6.87087	0.226158
$924$	−5.48246	−0.180360
$925$	−5.87423	−0.193144
$926$	−52.3755	−1.72117
$927$	9.81116	0.322241
$928$	−56.1893	−1.84451
$929$	49.3089	1.61777	0.808887	−	0.587964i	$-0.200070\pi$
$929$	49.3089	1.61777	0.808887	+	0.587964i	$0.200070\pi$
$930$	−5.62208	−0.184355
$931$	−8.70177	−0.285189
$932$	49.0843	1.60781
$933$	−16.2945	−0.533459
$934$	32.3992	1.06013
$935$	0.251419	0.00822229
$936$	0.289208	0.00945306
$937$	−48.5295	−1.58539	−0.792695	−	0.609619i	$-0.791323\pi$
$937$	−48.5295	−1.58539	−0.792695	+	0.609619i	$0.791323\pi$
$938$	−71.4221	−2.33201
$939$	−25.6162	−0.835953
$940$	−1.55081	−0.0505819
$941$	−9.22619	−0.300765	−0.150383	−	0.988628i	$-0.548051\pi$
$941$	−9.22619	−0.300765	−0.150383	+	0.988628i	$0.548051\pi$
$942$	49.7774	1.62184
$943$	−54.8294	−1.78549
$944$	43.7682	1.42453
$945$	−12.4220	−0.404086
$946$	4.14610	0.134801
$947$	19.0151	0.617907	0.308954	−	0.951077i	$-0.400021\pi$
$947$	19.0151	0.617907	0.308954	+	0.951077i	$0.400021\pi$
$948$	15.8095	0.513469
$949$	−5.77334	−0.187411
$950$	−8.43318	−0.273609
$951$	6.67411	0.216423
$952$	−0.433042	−0.0140350
$953$	27.3066	0.884547	0.442273	−	0.896880i	$-0.354172\pi$
$953$	27.3066	0.884547	0.442273	+	0.896880i	$0.354172\pi$
$954$	5.70970	0.184859
$955$	−16.3589	−0.529360
$956$	31.5473	1.02031
$957$	−7.56069	−0.244402
$958$	−24.5879	−0.794399
$959$	−37.2845	−1.20398
$960$	14.9695	0.483139
$961$	−27.6770	−0.892805
$962$	7.92309	0.255451
$963$	−10.7879	−0.347635
$964$	−2.27640	−0.0733179
$965$	−15.3838	−0.495223
$966$	50.7994	1.63445
$967$	−42.9512	−1.38122	−0.690609	−	0.723229i	$-0.742657\pi$
$967$	−42.9512	−1.38122	−0.690609	+	0.723229i	$0.742657\pi$
$968$	5.98109	0.192240
$969$	−2.08882	−0.0671025
$970$	17.7360	0.569469
$971$	−59.2007	−1.89984	−0.949920	−	0.312494i	$-0.898835\pi$
$971$	−59.2007	−1.89984	−0.949920	+	0.312494i	$0.898835\pi$
$972$	17.8454	0.572391
$973$	35.5685	1.14027
$974$	69.8443	2.23795
$975$	0.972735	0.0311525
$976$	30.5083	0.976548
$977$	27.0159	0.864316	0.432158	−	0.901798i	$-0.357752\pi$
$977$	27.0159	0.864316	0.432158	+	0.901798i	$0.357752\pi$
$978$	44.1223	1.41087
$979$	−0.473288	−0.0151263
$980$	4.85740	0.155164
$981$	−5.26871	−0.168217
$982$	−73.4264	−2.34313
$983$	2.86786	0.0914705	0.0457353	−	0.998954i	$-0.485437\pi$
$983$	2.86786	0.0914705	0.0457353	+	0.998954i	$0.485437\pi$
$984$	6.25962	0.199550
$985$	−7.91460	−0.252180
$986$	−4.91843	−0.156635
$987$	−2.24128	−0.0713406
$988$	6.05487	0.192631
$989$	−20.4500	−0.650273
$990$	−1.17438	−0.0373242
$991$	38.5167	1.22352	0.611761	−	0.791042i	$-0.290461\pi$
$991$	38.5167	1.22352	0.611761	+	0.791042i	$0.290461\pi$
$992$	−14.7908	−0.469608
$993$	37.0511	1.17578
$994$	48.0554	1.52422
$995$	−11.6273	−0.368611
$996$	54.7920	1.73615
$997$	−60.2584	−1.90840	−0.954201	−	0.299166i	$-0.903292\pi$
$997$	−60.2584	−1.90840	−0.954201	+	0.299166i	$0.903292\pi$
$998$	62.2855	1.97161
$999$	33.0785	1.04656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1205.2.a.c.1.2	✓	15
5.4	even	2		6025.2.a.i.1.14		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.c.1.2	✓	15	1.1	even	1	trivial
6025.2.a.i.1.14		15	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)

\(f(q)\)	\(=\)	\(q-2.06795 q^{2} +1.49139 q^{3} +2.27640 q^{4} -1.00000 q^{5} -3.08411 q^{6} -2.20594 q^{7} -0.571582 q^{8} -0.775761 q^{9} +O(q^{10})\)	\(q-2.06795 q^{2} +1.49139 q^{3} +2.27640 q^{4} -1.00000 q^{5} -3.08411 q^{6} -2.20594 q^{7} -0.571582 q^{8} -0.775761 q^{9} +2.06795 q^{10} +0.732051 q^{11} +3.39500 q^{12} +0.652235 q^{13} +4.56177 q^{14} -1.49139 q^{15} -3.37080 q^{16} -0.343445 q^{17} +1.60423 q^{18} +4.07805 q^{19} -2.27640 q^{20} -3.28992 q^{21} -1.51384 q^{22} +7.46680 q^{23} -0.852451 q^{24} +1.00000 q^{25} -1.34879 q^{26} -5.63113 q^{27} -5.02161 q^{28} -6.92516 q^{29} +3.08411 q^{30} -1.82292 q^{31} +8.11380 q^{32} +1.09177 q^{33} +0.710226 q^{34} +2.20594 q^{35} -1.76594 q^{36} -5.87423 q^{37} -8.43318 q^{38} +0.972735 q^{39} +0.571582 q^{40} -7.34309 q^{41} +6.80338 q^{42} -2.73880 q^{43} +1.66644 q^{44} +0.775761 q^{45} -15.4409 q^{46} +0.681256 q^{47} -5.02717 q^{48} -2.13381 q^{49} -2.06795 q^{50} -0.512210 q^{51} +1.48475 q^{52} +3.55915 q^{53} +11.6449 q^{54} -0.732051 q^{55} +1.26088 q^{56} +6.08195 q^{57} +14.3209 q^{58} -12.9845 q^{59} -3.39500 q^{60} -9.05077 q^{61} +3.76970 q^{62} +1.71129 q^{63} -10.0373 q^{64} -0.652235 q^{65} -2.25773 q^{66} -15.6566 q^{67} -0.781819 q^{68} +11.1359 q^{69} -4.56177 q^{70} +10.5344 q^{71} +0.443411 q^{72} -8.85163 q^{73} +12.1476 q^{74} +1.49139 q^{75} +9.28327 q^{76} -1.61486 q^{77} -2.01156 q^{78} +4.65671 q^{79} +3.37080 q^{80} -6.07091 q^{81} +15.1851 q^{82} +16.1391 q^{83} -7.48918 q^{84} +0.343445 q^{85} +5.66368 q^{86} -10.3281 q^{87} -0.418427 q^{88} -0.646523 q^{89} -1.60423 q^{90} -1.43879 q^{91} +16.9974 q^{92} -2.71868 q^{93} -1.40880 q^{94} -4.07805 q^{95} +12.1008 q^{96} +8.57664 q^{97} +4.41260 q^{98} -0.567897 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 2 q^{2} - 7 q^{3} + 6 q^{4} - 15 q^{5} - 5 q^{6} - 3 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10}) \) 15 * q + 2 * q^2 - 7 * q^3 + 6 * q^4 - 15 * q^5 - 5 * q^6 - 3 * q^7 + 3 * q^8 + 6 * q^9	\( 15 q + 2 q^{2} - 7 q^{3} + 6 q^{4} - 15 q^{5} - 5 q^{6} - 3 q^{7} + 3 q^{8} + 6 q^{9} - 2 q^{10} - 10 q^{11} - 6 q^{12} - 8 q^{13} - 5 q^{14} + 7 q^{15} - 16 q^{16} - q^{17} - 3 q^{18} - 30 q^{19} - 6 q^{20} - 11 q^{21} - 5 q^{22} + 19 q^{23} - 14 q^{24} + 15 q^{25} - 18 q^{26} - 22 q^{27} - 20 q^{28} - 12 q^{29} + 5 q^{30} - 22 q^{31} - 2 q^{32} + 4 q^{33} - 29 q^{34} + 3 q^{35} - 7 q^{36} - 12 q^{37} - 18 q^{38} - 17 q^{39} - 3 q^{40} - 13 q^{41} - q^{42} - 25 q^{43} - 20 q^{44} - 6 q^{45} - 7 q^{46} + 16 q^{47} - 22 q^{48} - 24 q^{49} + 2 q^{50} - 27 q^{51} - 15 q^{52} - 4 q^{53} - 43 q^{54} + 10 q^{55} - 3 q^{56} + 22 q^{57} - 20 q^{58} - 50 q^{59} + 6 q^{60} - 41 q^{61} + 12 q^{62} + 6 q^{63} - 53 q^{64} + 8 q^{65} + 5 q^{66} - 43 q^{67} + 5 q^{68} - 50 q^{69} + 5 q^{70} - 14 q^{71} + 32 q^{72} - 10 q^{73} - 26 q^{74} - 7 q^{75} - 13 q^{76} - 7 q^{77} + 3 q^{78} - 44 q^{79} + 16 q^{80} + 7 q^{81} - 19 q^{82} + 7 q^{83} - 42 q^{84} + q^{85} + 7 q^{86} + 10 q^{87} - 28 q^{88} + 4 q^{89} + 3 q^{90} - 50 q^{91} + 25 q^{92} + 22 q^{93} - 14 q^{94} + 30 q^{95} + 14 q^{96} + 9 q^{97} + 2 q^{98} - 46 q^{99}+O(q^{100}) \) 15 * q + 2 * q^2 - 7 * q^3 + 6 * q^4 - 15 * q^5 - 5 * q^6 - 3 * q^7 + 3 * q^8 + 6 * q^9 - 2 * q^10 - 10 * q^11 - 6 * q^12 - 8 * q^13 - 5 * q^14 + 7 * q^15 - 16 * q^16 - q^17 - 3 * q^18 - 30 * q^19 - 6 * q^20 - 11 * q^21 - 5 * q^22 + 19 * q^23 - 14 * q^24 + 15 * q^25 - 18 * q^26 - 22 * q^27 - 20 * q^28 - 12 * q^29 + 5 * q^30 - 22 * q^31 - 2 * q^32 + 4 * q^33 - 29 * q^34 + 3 * q^35 - 7 * q^36 - 12 * q^37 - 18 * q^38 - 17 * q^39 - 3 * q^40 - 13 * q^41 - q^42 - 25 * q^43 - 20 * q^44 - 6 * q^45 - 7 * q^46 + 16 * q^47 - 22 * q^48 - 24 * q^49 + 2 * q^50 - 27 * q^51 - 15 * q^52 - 4 * q^53 - 43 * q^54 + 10 * q^55 - 3 * q^56 + 22 * q^57 - 20 * q^58 - 50 * q^59 + 6 * q^60 - 41 * q^61 + 12 * q^62 + 6 * q^63 - 53 * q^64 + 8 * q^65 + 5 * q^66 - 43 * q^67 + 5 * q^68 - 50 * q^69 + 5 * q^70 - 14 * q^71 + 32 * q^72 - 10 * q^73 - 26 * q^74 - 7 * q^75 - 13 * q^76 - 7 * q^77 + 3 * q^78 - 44 * q^79 + 16 * q^80 + 7 * q^81 - 19 * q^82 + 7 * q^83 - 42 * q^84 + q^85 + 7 * q^86 + 10 * q^87 - 28 * q^88 + 4 * q^89 + 3 * q^90 - 50 * q^91 + 25 * q^92 + 22 * q^93 - 14 * q^94 + 30 * q^95 + 14 * q^96 + 9 * q^97 + 2 * q^98 - 46 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more