LMFDB - Embedded newform 2535.2.a.bd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2535,2,Mod(1,2535)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2535, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2535.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2535 = 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2535.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:	$20.2420769124$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 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			1.3
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	2535.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.80194 q^{2} -1.00000 q^{3} +1.24698 q^{4} -1.00000 q^{5} -1.80194 q^{6} +0.801938 q^{7} -1.35690 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.80194 q^{2} -1.00000 q^{3} +1.24698 q^{4} -1.00000 q^{5} -1.80194 q^{6} +0.801938 q^{7} -1.35690 q^{8} +1.00000 q^{9} -1.80194 q^{10} +1.00000 q^{11} -1.24698 q^{12} +1.44504 q^{14} +1.00000 q^{15} -4.93900 q^{16} -0.396125 q^{17} +1.80194 q^{18} -2.74094 q^{19} -1.24698 q^{20} -0.801938 q^{21} +1.80194 q^{22} +1.55496 q^{23} +1.35690 q^{24} +1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} -7.34481 q^{29} +1.80194 q^{30} +2.38404 q^{31} -6.18598 q^{32} -1.00000 q^{33} -0.713792 q^{34} -0.801938 q^{35} +1.24698 q^{36} -5.96077 q^{37} -4.93900 q^{38} +1.35690 q^{40} +8.80194 q^{41} -1.44504 q^{42} -6.80194 q^{43} +1.24698 q^{44} -1.00000 q^{45} +2.80194 q^{46} -11.7899 q^{47} +4.93900 q^{48} -6.35690 q^{49} +1.80194 q^{50} +0.396125 q^{51} +0.868313 q^{53} -1.80194 q^{54} -1.00000 q^{55} -1.08815 q^{56} +2.74094 q^{57} -13.2349 q^{58} -8.11960 q^{59} +1.24698 q^{60} -4.11529 q^{61} +4.29590 q^{62} +0.801938 q^{63} -1.26875 q^{64} -1.80194 q^{66} +9.93900 q^{67} -0.493959 q^{68} -1.55496 q^{69} -1.44504 q^{70} -6.07069 q^{71} -1.35690 q^{72} -4.29052 q^{73} -10.7409 q^{74} -1.00000 q^{75} -3.41789 q^{76} +0.801938 q^{77} -14.1075 q^{79} +4.93900 q^{80} +1.00000 q^{81} +15.8605 q^{82} +1.83877 q^{83} -1.00000 q^{84} +0.396125 q^{85} -12.2567 q^{86} +7.34481 q^{87} -1.35690 q^{88} -6.52111 q^{89} -1.80194 q^{90} +1.93900 q^{92} -2.38404 q^{93} -21.2446 q^{94} +2.74094 q^{95} +6.18598 q^{96} -13.1521 q^{97} -11.4547 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 3 q^{3} - q^{4} - 3 q^{5} - q^{6} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 - 3 * q^3 - q^4 - 3 * q^5 - q^6 - 2 * q^7 + 3 * q^9	$ 3 q + q^{2} - 3 q^{3} - q^{4} - 3 q^{5} - q^{6} - 2 q^{7} + 3 q^{9} - q^{10} + 3 q^{11} + q^{12} + 4 q^{14} + 3 q^{15} - 5 q^{16} - 10 q^{17} + q^{18} + 6 q^{19} + q^{20} + 2 q^{21} + q^{22} + 5 q^{23} + 3 q^{25} - 3 q^{27} + 3 q^{28} + q^{29} + q^{30} - 3 q^{31} - 4 q^{32} - 3 q^{33} + 6 q^{34} + 2 q^{35} - q^{36} - 5 q^{37} - 5 q^{38} + 22 q^{41} - 4 q^{42} - 16 q^{43} - q^{44} - 3 q^{45} + 4 q^{46} - 12 q^{47} + 5 q^{48} - 15 q^{49} + q^{50} + 10 q^{51} + 5 q^{53} - q^{54} - 3 q^{55} - 7 q^{56} - 6 q^{57} - 16 q^{58} - 3 q^{59} - q^{60} - 10 q^{61} - q^{62} - 2 q^{63} + 4 q^{64} - q^{66} + 20 q^{67} + 8 q^{68} - 5 q^{69} - 4 q^{70} - 6 q^{71} - 2 q^{73} - 18 q^{74} - 3 q^{75} - 16 q^{76} - 2 q^{77} - 2 q^{79} + 5 q^{80} + 3 q^{81} + 12 q^{82} - 27 q^{83} - 3 q^{84} + 10 q^{85} - 10 q^{86} - q^{87} - 4 q^{89} - q^{90} - 4 q^{92} + 3 q^{93} - 18 q^{94} - 6 q^{95} + 4 q^{96} - 9 q^{97} - 12 q^{98} + 3 q^{99}+O(q^{100}) $ 3 * q + q^2 - 3 * q^3 - q^4 - 3 * q^5 - q^6 - 2 * q^7 + 3 * q^9 - q^10 + 3 * q^11 + q^12 + 4 * q^14 + 3 * q^15 - 5 * q^16 - 10 * q^17 + q^18 + 6 * q^19 + q^20 + 2 * q^21 + q^22 + 5 * q^23 + 3 * q^25 - 3 * q^27 + 3 * q^28 + q^29 + q^30 - 3 * q^31 - 4 * q^32 - 3 * q^33 + 6 * q^34 + 2 * q^35 - q^36 - 5 * q^37 - 5 * q^38 + 22 * q^41 - 4 * q^42 - 16 * q^43 - q^44 - 3 * q^45 + 4 * q^46 - 12 * q^47 + 5 * q^48 - 15 * q^49 + q^50 + 10 * q^51 + 5 * q^53 - q^54 - 3 * q^55 - 7 * q^56 - 6 * q^57 - 16 * q^58 - 3 * q^59 - q^60 - 10 * q^61 - q^62 - 2 * q^63 + 4 * q^64 - q^66 + 20 * q^67 + 8 * q^68 - 5 * q^69 - 4 * q^70 - 6 * q^71 - 2 * q^73 - 18 * q^74 - 3 * q^75 - 16 * q^76 - 2 * q^77 - 2 * q^79 + 5 * q^80 + 3 * q^81 + 12 * q^82 - 27 * q^83 - 3 * q^84 + 10 * q^85 - 10 * q^86 - q^87 - 4 * q^89 - q^90 - 4 * q^92 + 3 * q^93 - 18 * q^94 - 6 * q^95 + 4 * q^96 - 9 * q^97 - 12 * q^98 + 3 * 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 $

$2$	1.80194	1.27416	0.637081	−	0.770797i	$-0.280142\pi$
$2$	1.80194	1.27416	0.637081	+	0.770797i	$0.280142\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.24698	0.623490
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.80194	−0.735638
$7$	0.801938	0.303104	0.151552	−	0.988449i	$-0.451573\pi$
$7$	0.801938	0.303104	0.151552	+	0.988449i	$0.451573\pi$
$8$	−1.35690	−0.479735
$9$	1.00000	0.333333
$10$	−1.80194	−0.569823
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	−1.24698	−0.359972
$13$	0	0
$13$	0	0
$14$	1.44504	0.386204
$15$	1.00000	0.258199
$16$	−4.93900	−1.23475
$17$	−0.396125	−0.0960743	−0.0480372	−	0.998846i	$-0.515297\pi$
$17$	−0.396125	−0.0960743	−0.0480372	+	0.998846i	$0.515297\pi$
$18$	1.80194	0.424721
$19$	−2.74094	−0.628814	−0.314407	−	0.949288i	$-0.601806\pi$
$19$	−2.74094	−0.628814	−0.314407	+	0.949288i	$0.601806\pi$
$20$	−1.24698	−0.278833
$21$	−0.801938	−0.174997
$22$	1.80194	0.384174
$23$	1.55496	0.324231	0.162116	−	0.986772i	$-0.448168\pi$
$23$	1.55496	0.324231	0.162116	+	0.986772i	$0.448168\pi$
$24$	1.35690	0.276975
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−7.34481	−1.36390	−0.681949	−	0.731400i	$-0.738867\pi$
$29$	−7.34481	−1.36390	−0.681949	+	0.731400i	$0.738867\pi$
$30$	1.80194	0.328987
$31$	2.38404	0.428187	0.214093	−	0.976813i	$-0.431320\pi$
$31$	2.38404	0.428187	0.214093	+	0.976813i	$0.431320\pi$
$32$	−6.18598	−1.09354
$33$	−1.00000	−0.174078
$34$	−0.713792	−0.122414
$35$	−0.801938	−0.135552
$36$	1.24698	0.207830
$37$	−5.96077	−0.979945	−0.489972	−	0.871738i	$-0.662993\pi$
$37$	−5.96077	−0.979945	−0.489972	+	0.871738i	$0.662993\pi$
$38$	−4.93900	−0.801212
$39$	0	0
$40$	1.35690	0.214544
$41$	8.80194	1.37463	0.687316	−	0.726359i	$-0.258789\pi$
$41$	8.80194	1.37463	0.687316	+	0.726359i	$0.258789\pi$
$42$	−1.44504	−0.222975
$43$	−6.80194	−1.03729	−0.518643	−	0.854991i	$-0.673563\pi$
$43$	−6.80194	−1.03729	−0.518643	+	0.854991i	$0.673563\pi$
$44$	1.24698	0.187989
$45$	−1.00000	−0.149071
$46$	2.80194	0.413123
$47$	−11.7899	−1.71973	−0.859864	−	0.510524i	$-0.829452\pi$
$47$	−11.7899	−1.71973	−0.859864	+	0.510524i	$0.829452\pi$
$48$	4.93900	0.712883
$49$	−6.35690	−0.908128
$50$	1.80194	0.254832
$51$	0.396125	0.0554685
$52$	0	0
$53$	0.868313	0.119272	0.0596360	−	0.998220i	$-0.481006\pi$
$53$	0.868313	0.119272	0.0596360	+	0.998220i	$0.481006\pi$
$54$	−1.80194	−0.245213
$55$	−1.00000	−0.134840
$56$	−1.08815	−0.145410
$57$	2.74094	0.363046
$58$	−13.2349	−1.73783
$59$	−8.11960	−1.05708	−0.528541	−	0.848908i	$-0.677261\pi$
$59$	−8.11960	−1.05708	−0.528541	+	0.848908i	$0.677261\pi$
$60$	1.24698	0.160984
$61$	−4.11529	−0.526909	−0.263455	−	0.964672i	$-0.584862\pi$
$61$	−4.11529	−0.526909	−0.263455	+	0.964672i	$0.584862\pi$
$62$	4.29590	0.545579
$63$	0.801938	0.101035
$64$	−1.26875	−0.158594
$65$	0	0
$66$	−1.80194	−0.221803
$67$	9.93900	1.21424	0.607121	−	0.794609i	$-0.292324\pi$
$67$	9.93900	1.21424	0.607121	+	0.794609i	$0.292324\pi$
$68$	−0.493959	−0.0599014
$69$	−1.55496	−0.187195
$70$	−1.44504	−0.172716
$71$	−6.07069	−0.720458	−0.360229	−	0.932864i	$-0.617301\pi$
$71$	−6.07069	−0.720458	−0.360229	+	0.932864i	$0.617301\pi$
$72$	−1.35690	−0.159912
$73$	−4.29052	−0.502167	−0.251084	−	0.967965i	$-0.580787\pi$
$73$	−4.29052	−0.502167	−0.251084	+	0.967965i	$0.580787\pi$
$74$	−10.7409	−1.24861
$75$	−1.00000	−0.115470
$76$	−3.41789	−0.392059
$77$	0.801938	0.0913893
$78$	0	0
$79$	−14.1075	−1.58722	−0.793610	−	0.608427i	$-0.791801\pi$
$79$	−14.1075	−1.58722	−0.793610	+	0.608427i	$0.791801\pi$
$80$	4.93900	0.552197
$81$	1.00000	0.111111
$82$	15.8605	1.75150
$83$	1.83877	0.201832	0.100916	−	0.994895i	$-0.467823\pi$
$83$	1.83877	0.201832	0.100916	+	0.994895i	$0.467823\pi$
$84$	−1.00000	−0.109109
$85$	0.396125	0.0429657
$86$	−12.2567	−1.32167
$87$	7.34481	0.787447
$88$	−1.35690	−0.144646
$89$	−6.52111	−0.691236	−0.345618	−	0.938375i	$-0.612331\pi$
$89$	−6.52111	−0.691236	−0.345618	+	0.938375i	$0.612331\pi$
$90$	−1.80194	−0.189941
$91$	0	0
$92$	1.93900	0.202155
$93$	−2.38404	−0.247214
$94$	−21.2446	−2.19121
$95$	2.74094	0.281214
$96$	6.18598	0.631354
$97$	−13.1521	−1.33540	−0.667698	−	0.744432i	$-0.732720\pi$
$97$	−13.1521	−1.33540	−0.667698	+	0.744432i	$0.732720\pi$
$98$	−11.4547	−1.15710
$99$	1.00000	0.100504
$100$	1.24698	0.124698
$101$	17.8756	1.77869	0.889345	−	0.457237i	$-0.151161\pi$
$101$	17.8756	1.77869	0.889345	+	0.457237i	$0.151161\pi$
$102$	0.713792	0.0706759
$103$	6.39373	0.629993	0.314997	−	0.949093i	$-0.397997\pi$
$103$	6.39373	0.629993	0.314997	+	0.949093i	$0.397997\pi$
$104$	0	0
$105$	0.801938	0.0782611
$106$	1.56465	0.151972
$107$	15.6353	1.51152	0.755762	−	0.654846i	$-0.227266\pi$
$107$	15.6353	1.51152	0.755762	+	0.654846i	$0.227266\pi$
$108$	−1.24698	−0.119991
$109$	7.42758	0.711433	0.355717	−	0.934594i	$-0.384237\pi$
$109$	7.42758	0.711433	0.355717	+	0.934594i	$0.384237\pi$
$110$	−1.80194	−0.171808
$111$	5.96077	0.565771
$112$	−3.96077	−0.374258
$113$	−15.4765	−1.45591	−0.727953	−	0.685627i	$-0.759528\pi$
$113$	−15.4765	−1.45591	−0.727953	+	0.685627i	$0.759528\pi$
$114$	4.93900	0.462580
$115$	−1.55496	−0.145001
$116$	−9.15883	−0.850376
$117$	0	0
$118$	−14.6310	−1.34689
$119$	−0.317667	−0.0291205
$120$	−1.35690	−0.123867
$121$	−10.0000	−0.909091
$122$	−7.41550	−0.671368
$123$	−8.80194	−0.793644
$124$	2.97285	0.266970
$125$	−1.00000	−0.0894427
$126$	1.44504	0.128735
$127$	−9.97823	−0.885425	−0.442712	−	0.896664i	$-0.645984\pi$
$127$	−9.97823	−0.885425	−0.442712	+	0.896664i	$0.645984\pi$
$128$	10.0858	0.891463
$129$	6.80194	0.598877
$130$	0	0
$131$	4.71917	0.412316	0.206158	−	0.978519i	$-0.433904\pi$
$131$	4.71917	0.412316	0.206158	+	0.978519i	$0.433904\pi$
$132$	−1.24698	−0.108536
$133$	−2.19806	−0.190596
$134$	17.9095	1.54714
$135$	1.00000	0.0860663
$136$	0.537500	0.0460902
$137$	−3.97046	−0.339219	−0.169610	−	0.985511i	$-0.554251\pi$
$137$	−3.97046	−0.339219	−0.169610	+	0.985511i	$0.554251\pi$
$138$	−2.80194	−0.238517
$139$	−15.3351	−1.30071	−0.650354	−	0.759631i	$-0.725380\pi$
$139$	−15.3351	−1.30071	−0.650354	+	0.759631i	$0.725380\pi$
$140$	−1.00000	−0.0845154
$141$	11.7899	0.992885
$142$	−10.9390	−0.917981
$143$	0	0
$144$	−4.93900	−0.411583
$145$	7.34481	0.609954
$146$	−7.73125	−0.639843
$147$	6.35690	0.524308
$148$	−7.43296	−0.610986
$149$	15.1860	1.24408	0.622042	−	0.782984i	$-0.286303\pi$
$149$	15.1860	1.24408	0.622042	+	0.782984i	$0.286303\pi$
$150$	−1.80194	−0.147128
$151$	17.3448	1.41150	0.705750	−	0.708460i	$-0.250610\pi$
$151$	17.3448	1.41150	0.705750	+	0.708460i	$0.250610\pi$
$152$	3.71917	0.301664
$153$	−0.396125	−0.0320248
$154$	1.44504	0.116445
$155$	−2.38404	−0.191491
$156$	0	0
$157$	−20.0489	−1.60008	−0.800039	−	0.599948i	$-0.795188\pi$
$157$	−20.0489	−1.60008	−0.800039	+	0.599948i	$0.795188\pi$
$158$	−25.4209	−2.02238
$159$	−0.868313	−0.0688617
$160$	6.18598	0.489045
$161$	1.24698	0.0982758
$162$	1.80194	0.141574
$163$	−4.68233	−0.366749	−0.183374	−	0.983043i	$-0.558702\pi$
$163$	−4.68233	−0.366749	−0.183374	+	0.983043i	$0.558702\pi$
$164$	10.9758	0.857069
$165$	1.00000	0.0778499
$166$	3.31336	0.257166
$167$	−10.5797	−0.818683	−0.409341	−	0.912381i	$-0.634241\pi$
$167$	−10.5797	−0.818683	−0.409341	+	0.912381i	$0.634241\pi$
$168$	1.08815	0.0839523
$169$	0	0
$170$	0.713792	0.0547453
$171$	−2.74094	−0.209605
$172$	−8.48188	−0.646737
$173$	−4.97823	−0.378488	−0.189244	−	0.981930i	$-0.560604\pi$
$173$	−4.97823	−0.378488	−0.189244	+	0.981930i	$0.560604\pi$
$174$	13.2349	1.00334
$175$	0.801938	0.0606208
$176$	−4.93900	−0.372291
$177$	8.11960	0.610307
$178$	−11.7506	−0.880747
$179$	13.5754	1.01467	0.507337	−	0.861748i	$-0.330630\pi$
$179$	13.5754	1.01467	0.507337	+	0.861748i	$0.330630\pi$
$180$	−1.24698	−0.0929444
$181$	9.93661	0.738582	0.369291	−	0.929314i	$-0.379601\pi$
$181$	9.93661	0.738582	0.369291	+	0.929314i	$0.379601\pi$
$182$	0	0
$183$	4.11529	0.304211
$184$	−2.10992	−0.155545
$185$	5.96077	0.438245
$186$	−4.29590	−0.314990
$187$	−0.396125	−0.0289675
$188$	−14.7017	−1.07223
$189$	−0.801938	−0.0583324
$190$	4.93900	0.358313
$191$	−18.8756	−1.36579	−0.682896	−	0.730516i	$-0.739280\pi$
$191$	−18.8756	−1.36579	−0.682896	+	0.730516i	$0.739280\pi$
$192$	1.26875	0.0915641
$193$	11.3817	0.819269	0.409635	−	0.912250i	$-0.365656\pi$
$193$	11.3817	0.819269	0.409635	+	0.912250i	$0.365656\pi$
$194$	−23.6993	−1.70151
$195$	0	0
$196$	−7.92692	−0.566209
$197$	26.5840	1.89403	0.947017	−	0.321184i	$-0.104081\pi$
$197$	26.5840	1.89403	0.947017	+	0.321184i	$0.104081\pi$
$198$	1.80194	0.128058
$199$	−8.17523	−0.579526	−0.289763	−	0.957098i	$-0.593577\pi$
$199$	−8.17523	−0.579526	−0.289763	+	0.957098i	$0.593577\pi$
$200$	−1.35690	−0.0959470
$201$	−9.93900	−0.701043
$202$	32.2107	2.26634
$203$	−5.89008	−0.413403
$204$	0.493959	0.0345841
$205$	−8.80194	−0.614754
$206$	11.5211	0.802714
$207$	1.55496	0.108077
$208$	0	0
$209$	−2.74094	−0.189595
$210$	1.44504	0.0997174
$211$	14.6286	1.00708	0.503538	−	0.863973i	$-0.332031\pi$
$211$	14.6286	1.00708	0.503538	+	0.863973i	$0.332031\pi$
$212$	1.08277	0.0743649
$213$	6.07069	0.415957
$214$	28.1739	1.92593
$215$	6.80194	0.463888
$216$	1.35690	0.0923251
$217$	1.91185	0.129785
$218$	13.3840	0.906482
$219$	4.29052	0.289926
$220$	−1.24698	−0.0840713
$221$	0	0
$222$	10.7409	0.720885
$223$	−10.2795	−0.688366	−0.344183	−	0.938902i	$-0.611844\pi$
$223$	−10.2795	−0.688366	−0.344183	+	0.938902i	$0.611844\pi$
$224$	−4.96077	−0.331455
$225$	1.00000	0.0666667
$226$	−27.8877	−1.85506
$227$	5.43296	0.360598	0.180299	−	0.983612i	$-0.442293\pi$
$227$	5.43296	0.360598	0.180299	+	0.983612i	$0.442293\pi$
$228$	3.41789	0.226356
$229$	19.8998	1.31501	0.657507	−	0.753448i	$-0.271611\pi$
$229$	19.8998	1.31501	0.657507	+	0.753448i	$0.271611\pi$
$230$	−2.80194	−0.184754
$231$	−0.801938	−0.0527636
$232$	9.96615	0.654310
$233$	−11.4940	−0.752994	−0.376497	−	0.926418i	$-0.622872\pi$
$233$	−11.4940	−0.752994	−0.376497	+	0.926418i	$0.622872\pi$
$234$	0	0
$235$	11.7899	0.769085
$236$	−10.1250	−0.659080
$237$	14.1075	0.916382
$238$	−0.572417	−0.0371043
$239$	22.0804	1.42826	0.714130	−	0.700013i	$-0.246822\pi$
$239$	22.0804	1.42826	0.714130	+	0.700013i	$0.246822\pi$
$240$	−4.93900	−0.318811
$241$	21.3817	1.37731	0.688657	−	0.725088i	$-0.258201\pi$
$241$	21.3817	1.37731	0.688657	+	0.725088i	$0.258201\pi$
$242$	−18.0194	−1.15833
$243$	−1.00000	−0.0641500
$244$	−5.13169	−0.328523
$245$	6.35690	0.406127
$246$	−15.8605	−1.01123
$247$	0	0
$248$	−3.23490	−0.205416
$249$	−1.83877	−0.116528
$250$	−1.80194	−0.113965
$251$	0.381059	0.0240522	0.0120261	−	0.999928i	$-0.496172\pi$
$251$	0.381059	0.0240522	0.0120261	+	0.999928i	$0.496172\pi$
$252$	1.00000	0.0629941
$253$	1.55496	0.0977594
$254$	−17.9801	−1.12817
$255$	−0.396125	−0.0248063
$256$	20.7114	1.29446
$257$	−17.3207	−1.08043	−0.540216	−	0.841526i	$-0.681658\pi$
$257$	−17.3207	−1.08043	−0.540216	+	0.841526i	$0.681658\pi$
$258$	12.2567	0.763067
$259$	−4.78017	−0.297025
$260$	0	0
$261$	−7.34481	−0.454633
$262$	8.50365	0.525357
$263$	23.3545	1.44010	0.720050	−	0.693922i	$-0.244119\pi$
$263$	23.3545	1.44010	0.720050	+	0.693922i	$0.244119\pi$
$264$	1.35690	0.0835112
$265$	−0.868313	−0.0533401
$266$	−3.96077	−0.242850
$267$	6.52111	0.399085
$268$	12.3937	0.757068
$269$	−20.6310	−1.25790	−0.628948	−	0.777448i	$-0.716514\pi$
$269$	−20.6310	−1.25790	−0.628948	+	0.777448i	$0.716514\pi$
$270$	1.80194	0.109662
$271$	0.356896	0.0216799	0.0108399	−	0.999941i	$-0.496549\pi$
$271$	0.356896	0.0216799	0.0108399	+	0.999941i	$0.496549\pi$
$272$	1.95646	0.118628
$273$	0	0
$274$	−7.15452	−0.432220
$275$	1.00000	0.0603023
$276$	−1.93900	−0.116714
$277$	16.6025	0.997550	0.498775	−	0.866731i	$-0.333783\pi$
$277$	16.6025	0.997550	0.498775	+	0.866731i	$0.333783\pi$
$278$	−27.6329	−1.65731
$279$	2.38404	0.142729
$280$	1.08815	0.0650292
$281$	−8.12737	−0.484839	−0.242419	−	0.970172i	$-0.577941\pi$
$281$	−8.12737	−0.484839	−0.242419	+	0.970172i	$0.577941\pi$
$282$	21.2446	1.26510
$283$	−14.5851	−0.866994	−0.433497	−	0.901155i	$-0.642720\pi$
$283$	−14.5851	−0.866994	−0.433497	+	0.901155i	$0.642720\pi$
$284$	−7.57002	−0.449198
$285$	−2.74094	−0.162359
$286$	0	0
$287$	7.05861	0.416656
$288$	−6.18598	−0.364512
$289$	−16.8431	−0.990770
$290$	13.2349	0.777180
$291$	13.1521	0.770991
$292$	−5.35019	−0.313096
$293$	25.4456	1.48655	0.743275	−	0.668986i	$-0.233271\pi$
$293$	25.4456	1.48655	0.743275	+	0.668986i	$0.233271\pi$
$294$	11.4547	0.668053
$295$	8.11960	0.472742
$296$	8.08815	0.470114
$297$	−1.00000	−0.0580259
$298$	27.3642	1.58517
$299$	0	0
$300$	−1.24698	−0.0719944
$301$	−5.45473	−0.314405
$302$	31.2543	1.79848
$303$	−17.8756	−1.02693
$304$	13.5375	0.776429
$305$	4.11529	0.235641
$306$	−0.713792	−0.0408048
$307$	−14.8955	−0.850129	−0.425064	−	0.905163i	$-0.639749\pi$
$307$	−14.8955	−0.850129	−0.425064	+	0.905163i	$0.639749\pi$
$308$	1.00000	0.0569803
$309$	−6.39373	−0.363727
$310$	−4.29590	−0.243991
$311$	−6.21313	−0.352314	−0.176157	−	0.984362i	$-0.556367\pi$
$311$	−6.21313	−0.352314	−0.176157	+	0.984362i	$0.556367\pi$
$312$	0	0
$313$	−10.7694	−0.608723	−0.304362	−	0.952557i	$-0.598443\pi$
$313$	−10.7694	−0.608723	−0.304362	+	0.952557i	$0.598443\pi$
$314$	−36.1269	−2.03876
$315$	−0.801938	−0.0451841
$316$	−17.5918	−0.989616
$317$	12.6504	0.710517	0.355259	−	0.934768i	$-0.384393\pi$
$317$	12.6504	0.710517	0.355259	+	0.934768i	$0.384393\pi$
$318$	−1.56465	−0.0877410
$319$	−7.34481	−0.411231
$320$	1.26875	0.0709253
$321$	−15.6353	−0.872679
$322$	2.24698	0.125219
$323$	1.08575	0.0604129
$324$	1.24698	0.0692766
$325$	0	0
$326$	−8.43727	−0.467297
$327$	−7.42758	−0.410746
$328$	−11.9433	−0.659459
$329$	−9.45473	−0.521256
$330$	1.80194	0.0991934
$331$	35.7972	1.96759	0.983795	−	0.179299i	$-0.0573828\pi$
$331$	35.7972	1.96759	0.983795	+	0.179299i	$0.0573828\pi$
$332$	2.29291	0.125840
$333$	−5.96077	−0.326648
$334$	−19.0640	−1.04313
$335$	−9.93900	−0.543026
$336$	3.96077	0.216078
$337$	−5.51142	−0.300226	−0.150113	−	0.988669i	$-0.547964\pi$
$337$	−5.51142	−0.300226	−0.150113	+	0.988669i	$0.547964\pi$
$338$	0	0
$339$	15.4765	0.840568
$340$	0.493959	0.0267887
$341$	2.38404	0.129103
$342$	−4.93900	−0.267071
$343$	−10.7114	−0.578361
$344$	9.22952	0.497622
$345$	1.55496	0.0837161
$346$	−8.97046	−0.482255
$347$	18.6770	1.00263	0.501316	−	0.865264i	$-0.332850\pi$
$347$	18.6770	1.00263	0.501316	+	0.865264i	$0.332850\pi$
$348$	9.15883	0.490965
$349$	16.7084	0.894381	0.447190	−	0.894439i	$-0.352425\pi$
$349$	16.7084	0.894381	0.447190	+	0.894439i	$0.352425\pi$
$350$	1.44504	0.0772407
$351$	0	0
$352$	−6.18598	−0.329714
$353$	−16.9879	−0.904176	−0.452088	−	0.891973i	$-0.649321\pi$
$353$	−16.9879	−0.904176	−0.452088	+	0.891973i	$0.649321\pi$
$354$	14.6310	0.777630
$355$	6.07069	0.322199
$356$	−8.13169	−0.430979
$357$	0.317667	0.0168127
$358$	24.4620	1.29286
$359$	24.8213	1.31002	0.655009	−	0.755621i	$-0.272665\pi$
$359$	24.8213	1.31002	0.655009	+	0.755621i	$0.272665\pi$
$360$	1.35690	0.0715147
$361$	−11.4873	−0.604592
$362$	17.9051	0.941074
$363$	10.0000	0.524864
$364$	0	0
$365$	4.29052	0.224576
$366$	7.41550	0.387614
$367$	12.5459	0.654889	0.327444	−	0.944870i	$-0.393813\pi$
$367$	12.5459	0.654889	0.327444	+	0.944870i	$0.393813\pi$
$368$	−7.67994	−0.400345
$369$	8.80194	0.458211
$370$	10.7409	0.558395
$371$	0.696333	0.0361518
$372$	−2.97285	−0.154135
$373$	−4.19136	−0.217020	−0.108510	−	0.994095i	$-0.534608\pi$
$373$	−4.19136	−0.217020	−0.108510	+	0.994095i	$0.534608\pi$
$374$	−0.713792	−0.0369093
$375$	1.00000	0.0516398
$376$	15.9976	0.825014
$377$	0	0
$378$	−1.44504	−0.0743249
$379$	4.24027	0.217808	0.108904	−	0.994052i	$-0.465266\pi$
$379$	4.24027	0.217808	0.108904	+	0.994052i	$0.465266\pi$
$380$	3.41789	0.175334
$381$	9.97823	0.511200
$382$	−34.0127	−1.74024
$383$	26.6437	1.36143	0.680715	−	0.732549i	$-0.261669\pi$
$383$	26.6437	1.36143	0.680715	+	0.732549i	$0.261669\pi$
$384$	−10.0858	−0.514686
$385$	−0.801938	−0.0408705
$386$	20.5090	1.04388
$387$	−6.80194	−0.345762
$388$	−16.4004	−0.832606
$389$	−2.76377	−0.140129	−0.0700645	−	0.997542i	$-0.522320\pi$
$389$	−2.76377	−0.140129	−0.0700645	+	0.997542i	$0.522320\pi$
$390$	0	0
$391$	−0.615957	−0.0311503
$392$	8.62565	0.435661
$393$	−4.71917	−0.238051
$394$	47.9028	2.41331
$395$	14.1075	0.709827
$396$	1.24698	0.0626631
$397$	−7.72156	−0.387534	−0.193767	−	0.981048i	$-0.562071\pi$
$397$	−7.72156	−0.387534	−0.193767	+	0.981048i	$0.562071\pi$
$398$	−14.7313	−0.738411
$399$	2.19806	0.110041
$400$	−4.93900	−0.246950
$401$	−21.8659	−1.09193	−0.545966	−	0.837807i	$-0.683837\pi$
$401$	−21.8659	−1.09193	−0.545966	+	0.837807i	$0.683837\pi$
$402$	−17.9095	−0.893243
$403$	0	0
$404$	22.2905	1.10899
$405$	−1.00000	−0.0496904
$406$	−10.6136	−0.526742
$407$	−5.96077	−0.295464
$408$	−0.537500	−0.0266102
$409$	2.50173	0.123703	0.0618513	−	0.998085i	$-0.480300\pi$
$409$	2.50173	0.123703	0.0618513	+	0.998085i	$0.480300\pi$
$410$	−15.8605	−0.783296
$411$	3.97046	0.195848
$412$	7.97285	0.392794
$413$	−6.51142	−0.320406
$414$	2.80194	0.137708
$415$	−1.83877	−0.0902618
$416$	0	0
$417$	15.3351	0.750964
$418$	−4.93900	−0.241574
$419$	3.97392	0.194139	0.0970693	−	0.995278i	$-0.469053\pi$
$419$	3.97392	0.194139	0.0970693	+	0.995278i	$0.469053\pi$
$420$	1.00000	0.0487950
$421$	−25.1105	−1.22381	−0.611906	−	0.790931i	$-0.709597\pi$
$421$	−25.1105	−1.22381	−0.611906	+	0.790931i	$0.709597\pi$
$422$	26.3599	1.28318
$423$	−11.7899	−0.573242
$424$	−1.17821	−0.0572190
$425$	−0.396125	−0.0192149
$426$	10.9390	0.529996
$427$	−3.30021	−0.159708
$428$	19.4969	0.942420
$429$	0	0
$430$	12.2567	0.591069
$431$	7.01746	0.338019	0.169010	−	0.985614i	$-0.445943\pi$
$431$	7.01746	0.338019	0.169010	+	0.985614i	$0.445943\pi$
$432$	4.93900	0.237628
$433$	−19.8334	−0.953132	−0.476566	−	0.879139i	$-0.658119\pi$
$433$	−19.8334	−0.953132	−0.476566	+	0.879139i	$0.658119\pi$
$434$	3.44504	0.165367
$435$	−7.34481	−0.352157
$436$	9.26205	0.443572
$437$	−4.26205	−0.203881
$438$	7.73125	0.369413
$439$	−17.2392	−0.822783	−0.411391	−	0.911459i	$-0.634957\pi$
$439$	−17.2392	−0.822783	−0.411391	+	0.911459i	$0.634957\pi$
$440$	1.35690	0.0646875
$441$	−6.35690	−0.302709
$442$	0	0
$443$	−2.04115	−0.0969779	−0.0484889	−	0.998824i	$-0.515441\pi$
$443$	−2.04115	−0.0969779	−0.0484889	+	0.998824i	$0.515441\pi$
$444$	7.43296	0.352753
$445$	6.52111	0.309130
$446$	−18.5230	−0.877091
$447$	−15.1860	−0.718272
$448$	−1.01746	−0.0480704
$449$	−14.3502	−0.677227	−0.338614	−	0.940925i	$-0.609958\pi$
$449$	−14.3502	−0.677227	−0.338614	+	0.940925i	$0.609958\pi$
$450$	1.80194	0.0849442
$451$	8.80194	0.414467
$452$	−19.2989	−0.907743
$453$	−17.3448	−0.814930
$454$	9.78986	0.459461
$455$	0	0
$456$	−3.71917	−0.174166
$457$	24.8122	1.16067	0.580333	−	0.814379i	$-0.302922\pi$
$457$	24.8122	1.16067	0.580333	+	0.814379i	$0.302922\pi$
$458$	35.8582	1.67554
$459$	0.396125	0.0184895
$460$	−1.93900	−0.0904064
$461$	−19.1739	−0.893018	−0.446509	−	0.894779i	$-0.647333\pi$
$461$	−19.1739	−0.893018	−0.446509	+	0.894779i	$0.647333\pi$
$462$	−1.44504	−0.0672294
$463$	−15.8538	−0.736790	−0.368395	−	0.929669i	$-0.620093\pi$
$463$	−15.8538	−0.736790	−0.368395	+	0.929669i	$0.620093\pi$
$464$	36.2760	1.68407
$465$	2.38404	0.110557
$466$	−20.7114	−0.959437
$467$	20.2741	0.938175	0.469087	−	0.883152i	$-0.344583\pi$
$467$	20.2741	0.938175	0.469087	+	0.883152i	$0.344583\pi$
$468$	0	0
$469$	7.97046	0.368042
$470$	21.2446	0.979940
$471$	20.0489	0.923805
$472$	11.0175	0.507120
$473$	−6.80194	−0.312753
$474$	25.4209	1.16762
$475$	−2.74094	−0.125763
$476$	−0.396125	−0.0181563
$477$	0.868313	0.0397573
$478$	39.7875	1.81984
$479$	−14.4316	−0.659398	−0.329699	−	0.944086i	$-0.606947\pi$
$479$	−14.4316	−0.659398	−0.329699	+	0.944086i	$0.606947\pi$
$480$	−6.18598	−0.282350
$481$	0	0
$482$	38.5284	1.75492
$483$	−1.24698	−0.0567395
$484$	−12.4698	−0.566809
$485$	13.1521	0.597207
$486$	−1.80194	−0.0817376
$487$	−35.0901	−1.59008	−0.795041	−	0.606555i	$-0.792551\pi$
$487$	−35.0901	−1.59008	−0.795041	+	0.606555i	$0.792551\pi$
$488$	5.58402	0.252777
$489$	4.68233	0.211742
$490$	11.4547	0.517472
$491$	2.75973	0.124545	0.0622723	−	0.998059i	$-0.480165\pi$
$491$	2.75973	0.124545	0.0622723	+	0.998059i	$0.480165\pi$
$492$	−10.9758	−0.494829
$493$	2.90946	0.131036
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	−11.7748	−0.528704
$497$	−4.86831	−0.218374
$498$	−3.31336	−0.148475
$499$	17.3840	0.778217	0.389108	−	0.921192i	$-0.372783\pi$
$499$	17.3840	0.778217	0.389108	+	0.921192i	$0.372783\pi$
$500$	−1.24698	−0.0557666
$501$	10.5797	0.472667
$502$	0.686645	0.0306465
$503$	12.9987	0.579582	0.289791	−	0.957090i	$-0.406414\pi$
$503$	12.9987	0.579582	0.289791	+	0.957090i	$0.406414\pi$
$504$	−1.08815	−0.0484699
$505$	−17.8756	−0.795454
$506$	2.80194	0.124561
$507$	0	0
$508$	−12.4426	−0.552053
$509$	0.839838	0.0372252	0.0186126	−	0.999827i	$-0.494075\pi$
$509$	0.839838	0.0372252	0.0186126	+	0.999827i	$0.494075\pi$
$510$	−0.713792	−0.0316072
$511$	−3.44073	−0.152209
$512$	17.1491	0.757892
$513$	2.74094	0.121015
$514$	−31.2107	−1.37665
$515$	−6.39373	−0.281741
$516$	8.48188	0.373394
$517$	−11.7899	−0.518517
$518$	−8.61356	−0.378458
$519$	4.97823	0.218520
$520$	0	0
$521$	−26.2416	−1.14967	−0.574833	−	0.818271i	$-0.694933\pi$
$521$	−26.2416	−1.14967	−0.574833	+	0.818271i	$0.694933\pi$
$522$	−13.2349	−0.579276
$523$	−29.2349	−1.27835	−0.639176	−	0.769060i	$-0.720725\pi$
$523$	−29.2349	−1.27835	−0.639176	+	0.769060i	$0.720725\pi$
$524$	5.88471	0.257075
$525$	−0.801938	−0.0349994
$526$	42.0834	1.83492
$527$	−0.944378	−0.0411377
$528$	4.93900	0.214942
$529$	−20.5821	−0.894874
$530$	−1.56465	−0.0679639
$531$	−8.11960	−0.352361
$532$	−2.74094	−0.118835
$533$	0	0
$534$	11.7506	0.508499
$535$	−15.6353	−0.675974
$536$	−13.4862	−0.582515
$537$	−13.5754	−0.585822
$538$	−37.1758	−1.60276
$539$	−6.35690	−0.273811
$540$	1.24698	0.0536615
$541$	18.4746	0.794284	0.397142	−	0.917757i	$-0.370002\pi$
$541$	18.4746	0.794284	0.397142	+	0.917757i	$0.370002\pi$
$542$	0.643104	0.0276237
$543$	−9.93661	−0.426421
$544$	2.45042	0.105061
$545$	−7.42758	−0.318163
$546$	0	0
$547$	−0.450419	−0.0192585	−0.00962926	−	0.999954i	$-0.503065\pi$
$547$	−0.450419	−0.0192585	−0.00962926	+	0.999954i	$0.503065\pi$
$548$	−4.95108	−0.211500
$549$	−4.11529	−0.175636
$550$	1.80194	0.0768349
$551$	20.1317	0.857639
$552$	2.10992	0.0898040
$553$	−11.3134	−0.481093
$554$	29.9168	1.27104
$555$	−5.96077	−0.253021
$556$	−19.1226	−0.810978
$557$	−30.4058	−1.28834	−0.644168	−	0.764884i	$-0.722796\pi$
$557$	−30.4058	−1.28834	−0.644168	+	0.764884i	$0.722796\pi$
$558$	4.29590	0.181860
$559$	0	0
$560$	3.96077	0.167373
$561$	0.396125	0.0167244
$562$	−14.6450	−0.617763
$563$	−45.0471	−1.89851	−0.949255	−	0.314508i	$-0.898160\pi$
$563$	−45.0471	−1.89851	−0.949255	+	0.314508i	$0.898160\pi$
$564$	14.7017	0.619054
$565$	15.4765	0.651101
$566$	−26.2814	−1.10469
$567$	0.801938	0.0336782
$568$	8.23729	0.345629
$569$	16.4892	0.691262	0.345631	−	0.938370i	$-0.387665\pi$
$569$	16.4892	0.691262	0.345631	+	0.938370i	$0.387665\pi$
$570$	−4.93900	−0.206872
$571$	42.9124	1.79583	0.897915	−	0.440169i	$-0.145081\pi$
$571$	42.9124	1.79583	0.897915	+	0.440169i	$0.145081\pi$
$572$	0	0
$573$	18.8756	0.788540
$574$	12.7192	0.530888
$575$	1.55496	0.0648462
$576$	−1.26875	−0.0528646
$577$	−36.8377	−1.53357	−0.766787	−	0.641902i	$-0.778146\pi$
$577$	−36.8377	−1.53357	−0.766787	+	0.641902i	$0.778146\pi$
$578$	−30.3502	−1.26240
$579$	−11.3817	−0.473005
$580$	9.15883	0.380300
$581$	1.47458	0.0611760
$582$	23.6993	0.982368
$583$	0.868313	0.0359619
$584$	5.82179	0.240907
$585$	0	0
$586$	45.8514	1.89411
$587$	25.1860	1.03954	0.519768	−	0.854307i	$-0.326018\pi$
$587$	25.1860	1.03954	0.519768	+	0.854307i	$0.326018\pi$
$588$	7.92692	0.326901
$589$	−6.53452	−0.269250
$590$	14.6310	0.602350
$591$	−26.5840	−1.09352
$592$	29.4403	1.20999
$593$	19.1263	0.785423	0.392712	−	0.919662i	$-0.371537\pi$
$593$	19.1263	0.785423	0.392712	+	0.919662i	$0.371537\pi$
$594$	−1.80194	−0.0739344
$595$	0.317667	0.0130231
$596$	18.9366	0.775674
$597$	8.17523	0.334590
$598$	0	0
$599$	42.1825	1.72353	0.861766	−	0.507307i	$-0.169359\pi$
$599$	42.1825	1.72353	0.861766	+	0.507307i	$0.169359\pi$
$600$	1.35690	0.0553950
$601$	42.2277	1.72250	0.861252	−	0.508178i	$-0.169681\pi$
$601$	42.2277	1.72250	0.861252	+	0.508178i	$0.169681\pi$
$602$	−9.82908	−0.400604
$603$	9.93900	0.404747
$604$	21.6286	0.880056
$605$	10.0000	0.406558
$606$	−32.2107	−1.30847
$607$	−0.0284750	−0.00115577	−0.000577883	−	1.00000i	$-0.500184\pi$
$607$	−0.0284750	−0.00115577	−0.000577883		1.00000i	$0.500184\pi$
$608$	16.9554	0.687632
$609$	5.89008	0.238678
$610$	7.41550	0.300245
$611$	0	0
$612$	−0.493959	−0.0199671
$613$	22.2064	0.896909	0.448454	−	0.893806i	$-0.351975\pi$
$613$	22.2064	0.896909	0.448454	+	0.893806i	$0.351975\pi$
$614$	−26.8407	−1.08320
$615$	8.80194	0.354928
$616$	−1.08815	−0.0438427
$617$	12.3653	0.497806	0.248903	−	0.968528i	$-0.419930\pi$
$617$	12.3653	0.497806	0.248903	+	0.968528i	$0.419930\pi$
$618$	−11.5211	−0.463447
$619$	23.2586	0.934842	0.467421	−	0.884035i	$-0.345183\pi$
$619$	23.2586	0.934842	0.467421	+	0.884035i	$0.345183\pi$
$620$	−2.97285	−0.119393
$621$	−1.55496	−0.0623983
$622$	−11.1957	−0.448905
$623$	−5.22952	−0.209516
$624$	0	0
$625$	1.00000	0.0400000
$626$	−19.4058	−0.775612
$627$	2.74094	0.109463
$628$	−25.0006	−0.997632
$629$	2.36121	0.0941475
$630$	−1.44504	−0.0575718
$631$	−45.3817	−1.80661	−0.903307	−	0.428994i	$-0.858868\pi$
$631$	−45.3817	−1.80661	−0.903307	+	0.428994i	$0.858868\pi$
$632$	19.1424	0.761445
$633$	−14.6286	−0.581436
$634$	22.7952	0.905314
$635$	9.97823	0.395974
$636$	−1.08277	−0.0429346
$637$	0	0
$638$	−13.2349	−0.523975
$639$	−6.07069	−0.240153
$640$	−10.0858	−0.398674
$641$	−0.511418	−0.0201998	−0.0100999	−	0.999949i	$-0.503215\pi$
$641$	−0.511418	−0.0201998	−0.0100999	+	0.999949i	$0.503215\pi$
$642$	−28.1739	−1.11194
$643$	−20.6668	−0.815019	−0.407509	−	0.913201i	$-0.633603\pi$
$643$	−20.6668	−0.815019	−0.407509	+	0.913201i	$0.633603\pi$
$644$	1.55496	0.0612739
$645$	−6.80194	−0.267826
$646$	1.95646	0.0769759
$647$	28.3491	1.11452	0.557260	−	0.830338i	$-0.311853\pi$
$647$	28.3491	1.11452	0.557260	+	0.830338i	$0.311853\pi$
$648$	−1.35690	−0.0533039
$649$	−8.11960	−0.318722
$650$	0	0
$651$	−1.91185	−0.0749315
$652$	−5.83877	−0.228664
$653$	−38.4959	−1.50646	−0.753230	−	0.657757i	$-0.771505\pi$
$653$	−38.4959	−1.50646	−0.753230	+	0.657757i	$0.771505\pi$
$654$	−13.3840	−0.523357
$655$	−4.71917	−0.184393
$656$	−43.4728	−1.69733
$657$	−4.29052	−0.167389
$658$	−17.0368	−0.664165
$659$	36.4432	1.41963	0.709814	−	0.704390i	$-0.248779\pi$
$659$	36.4432	1.41963	0.709814	+	0.704390i	$0.248779\pi$
$660$	1.24698	0.0485386
$661$	−24.4198	−0.949821	−0.474910	−	0.880034i	$-0.657520\pi$
$661$	−24.4198	−0.949821	−0.474910	+	0.880034i	$0.657520\pi$
$662$	64.5042	2.50703
$663$	0	0
$664$	−2.49502	−0.0968257
$665$	2.19806	0.0852372
$666$	−10.7409	−0.416203
$667$	−11.4209	−0.442218
$668$	−13.1927	−0.510440
$669$	10.2795	0.397429
$670$	−17.9095	−0.691903
$671$	−4.11529	−0.158869
$672$	4.96077	0.191366
$673$	−22.9366	−0.884141	−0.442071	−	0.896980i	$-0.645756\pi$
$673$	−22.9366	−0.884141	−0.442071	+	0.896980i	$0.645756\pi$
$674$	−9.93123	−0.382537
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	12.9530	0.497824	0.248912	−	0.968526i	$-0.419927\pi$
$677$	12.9530	0.497824	0.248912	+	0.968526i	$0.419927\pi$
$678$	27.8877	1.07102
$679$	−10.5472	−0.404764
$680$	−0.537500	−0.0206122
$681$	−5.43296	−0.208191
$682$	4.29590	0.164498
$683$	38.6316	1.47820	0.739099	−	0.673597i	$-0.235252\pi$
$683$	38.6316	1.47820	0.739099	+	0.673597i	$0.235252\pi$
$684$	−3.41789	−0.130686
$685$	3.97046	0.151703
$686$	−19.3013	−0.736926
$687$	−19.8998	−0.759224
$688$	33.5948	1.28079
$689$	0	0
$690$	2.80194	0.106668
$691$	−35.1347	−1.33659	−0.668293	−	0.743898i	$-0.732975\pi$
$691$	−35.1347	−1.33659	−0.668293	+	0.743898i	$0.732975\pi$
$692$	−6.20775	−0.235983
$693$	0.801938	0.0304631
$694$	33.6547	1.27752
$695$	15.3351	0.581694
$696$	−9.96615	−0.377766
$697$	−3.48666	−0.132067
$698$	30.1075	1.13959
$699$	11.4940	0.434741
$700$	1.00000	0.0377964
$701$	19.6286	0.741363	0.370682	−	0.928760i	$-0.379124\pi$
$701$	19.6286	0.741363	0.370682	+	0.928760i	$0.379124\pi$
$702$	0	0
$703$	16.3381	0.616203
$704$	−1.26875	−0.0478178
$705$	−11.7899	−0.444032
$706$	−30.6112	−1.15207
$707$	14.3351	0.539128
$708$	10.1250	0.380520
$709$	−3.88769	−0.146005	−0.0730026	−	0.997332i	$-0.523258\pi$
$709$	−3.88769	−0.146005	−0.0730026	+	0.997332i	$0.523258\pi$
$710$	10.9390	0.410533
$711$	−14.1075	−0.529073
$712$	8.84846	0.331610
$713$	3.70709	0.138831
$714$	0.572417	0.0214222
$715$	0	0
$716$	16.9282	0.632638
$717$	−22.0804	−0.824607
$718$	44.7265	1.66918
$719$	7.42998	0.277091	0.138546	−	0.990356i	$-0.455757\pi$
$719$	7.42998	0.277091	0.138546	+	0.990356i	$0.455757\pi$
$720$	4.93900	0.184066
$721$	5.12737	0.190953
$722$	−20.6993	−0.770349
$723$	−21.3817	−0.795192
$724$	12.3907	0.460499
$725$	−7.34481	−0.272780
$726$	18.0194	0.668762
$727$	−47.0127	−1.74360	−0.871802	−	0.489859i	$-0.837048\pi$
$727$	−47.0127	−1.74360	−0.871802	+	0.489859i	$0.837048\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	7.73125	0.286146
$731$	2.69441	0.0996565
$732$	5.13169	0.189673
$733$	−24.8683	−0.918532	−0.459266	−	0.888299i	$-0.651888\pi$
$733$	−24.8683	−0.918532	−0.459266	+	0.888299i	$0.651888\pi$
$734$	22.6069	0.834434
$735$	−6.35690	−0.234478
$736$	−9.61894	−0.354559
$737$	9.93900	0.366108
$738$	15.8605	0.583835
$739$	−36.8437	−1.35532	−0.677658	−	0.735377i	$-0.737005\pi$
$739$	−36.8437	−1.35532	−0.677658	+	0.735377i	$0.737005\pi$
$740$	7.43296	0.273241
$741$	0	0
$742$	1.25475	0.0460633
$743$	−20.9028	−0.766848	−0.383424	−	0.923572i	$-0.625255\pi$
$743$	−20.9028	−0.766848	−0.383424	+	0.923572i	$0.625255\pi$
$744$	3.23490	0.118597
$745$	−15.1860	−0.556371
$746$	−7.55257	−0.276519
$747$	1.83877	0.0672772
$748$	−0.493959	−0.0180609
$749$	12.5386	0.458149
$750$	1.80194	0.0657975
$751$	3.72481	0.135920	0.0679601	−	0.997688i	$-0.478351\pi$
$751$	3.72481	0.135920	0.0679601	+	0.997688i	$0.478351\pi$
$752$	58.2301	2.12343
$753$	−0.381059	−0.0138866
$754$	0	0
$755$	−17.3448	−0.631242
$756$	−1.00000	−0.0363696
$757$	−39.4446	−1.43364	−0.716819	−	0.697260i	$-0.754402\pi$
$757$	−39.4446	−1.43364	−0.716819	+	0.697260i	$0.754402\pi$
$758$	7.64071	0.277523
$759$	−1.55496	−0.0564414
$760$	−3.71917	−0.134908
$761$	28.1396	1.02006	0.510029	−	0.860157i	$-0.329635\pi$
$761$	28.1396	1.02006	0.510029	+	0.860157i	$0.329635\pi$
$762$	17.9801	0.651352
$763$	5.95646	0.215638
$764$	−23.5375	−0.851557
$765$	0.396125	0.0143219
$766$	48.0103	1.73468
$767$	0	0
$768$	−20.7114	−0.747358
$769$	46.9909	1.69454	0.847268	−	0.531166i	$-0.178246\pi$
$769$	46.9909	1.69454	0.847268	+	0.531166i	$0.178246\pi$
$770$	−1.44504	−0.0520757
$771$	17.3207	0.623788
$772$	14.1927	0.510806
$773$	−15.9594	−0.574021	−0.287011	−	0.957927i	$-0.592662\pi$
$773$	−15.9594	−0.574021	−0.287011	+	0.957927i	$0.592662\pi$
$774$	−12.2567	−0.440557
$775$	2.38404	0.0856374
$776$	17.8461	0.640637
$777$	4.78017	0.171488
$778$	−4.98015	−0.178547
$779$	−24.1256	−0.864388
$780$	0	0
$781$	−6.07069	−0.217226
$782$	−1.10992	−0.0396905
$783$	7.34481	0.262482
$784$	31.3967	1.12131
$785$	20.0489	0.715577
$786$	−8.50365	−0.303315
$787$	−43.0465	−1.53444	−0.767221	−	0.641382i	$-0.778361\pi$
$787$	−43.0465	−1.53444	−0.767221	+	0.641382i	$0.778361\pi$
$788$	33.1497	1.18091
$789$	−23.3545	−0.831442
$790$	25.4209	0.904434
$791$	−12.4112	−0.441291
$792$	−1.35690	−0.0482152
$793$	0	0
$794$	−13.9138	−0.493781
$795$	0.868313	0.0307959
$796$	−10.1943	−0.361329
$797$	−44.7646	−1.58564	−0.792822	−	0.609453i	$-0.791389\pi$
$797$	−44.7646	−1.58564	−0.792822	+	0.609453i	$0.791389\pi$
$798$	3.96077	0.140210
$799$	4.67025	0.165222
$800$	−6.18598	−0.218707
$801$	−6.52111	−0.230412
$802$	−39.4010	−1.39130
$803$	−4.29052	−0.151409
$804$	−12.3937	−0.437093
$805$	−1.24698	−0.0439503
$806$	0	0
$807$	20.6310	0.726246
$808$	−24.2553	−0.853300
$809$	−37.5918	−1.32166	−0.660829	−	0.750537i	$-0.729795\pi$
$809$	−37.5918	−1.32166	−0.660829	+	0.750537i	$0.729795\pi$
$810$	−1.80194	−0.0633136
$811$	−16.5797	−0.582192	−0.291096	−	0.956694i	$-0.594020\pi$
$811$	−16.5797	−0.582192	−0.291096	+	0.956694i	$0.594020\pi$
$812$	−7.34481	−0.257752
$813$	−0.356896	−0.0125169
$814$	−10.7409	−0.376470
$815$	4.68233	0.164015
$816$	−1.95646	−0.0684898
$817$	18.6437	0.652260
$818$	4.50796	0.157617
$819$	0	0
$820$	−10.9758	−0.383293
$821$	−3.49263	−0.121894	−0.0609468	−	0.998141i	$-0.519412\pi$
$821$	−3.49263	−0.121894	−0.0609468	+	0.998141i	$0.519412\pi$
$822$	7.15452	0.249543
$823$	−35.9318	−1.25250	−0.626252	−	0.779620i	$-0.715412\pi$
$823$	−35.9318	−1.25250	−0.626252	+	0.779620i	$0.715412\pi$
$824$	−8.67563	−0.302230
$825$	−1.00000	−0.0348155
$826$	−11.7332	−0.408249
$827$	23.2319	0.807853	0.403926	−	0.914791i	$-0.367645\pi$
$827$	23.2319	0.807853	0.403926	+	0.914791i	$0.367645\pi$
$828$	1.93900	0.0673849
$829$	−31.9028	−1.10803	−0.554014	−	0.832507i	$-0.686905\pi$
$829$	−31.9028	−1.10803	−0.554014	+	0.832507i	$0.686905\pi$
$830$	−3.31336	−0.115008
$831$	−16.6025	−0.575936
$832$	0	0
$833$	2.51812	0.0872478
$834$	27.6329	0.956851
$835$	10.5797	0.366126
$836$	−3.41789	−0.118210
$837$	−2.38404	−0.0824046
$838$	7.16075	0.247364
$839$	34.3978	1.18754	0.593772	−	0.804634i	$-0.297638\pi$
$839$	34.3978	1.18754	0.593772	+	0.804634i	$0.297638\pi$
$840$	−1.08815	−0.0375446
$841$	24.9463	0.860217
$842$	−45.2476	−1.55933
$843$	8.12737	0.279922
$844$	18.2416	0.627902
$845$	0	0
$846$	−21.2446	−0.730404
$847$	−8.01938	−0.275549
$848$	−4.28860	−0.147271
$849$	14.5851	0.500559
$850$	−0.713792	−0.0244829
$851$	−9.26875	−0.317729
$852$	7.57002	0.259345
$853$	−49.5991	−1.69824	−0.849120	−	0.528200i	$-0.822867\pi$
$853$	−49.5991	−1.69824	−0.849120	+	0.528200i	$0.822867\pi$
$854$	−5.94677	−0.203494
$855$	2.74094	0.0937381
$856$	−21.2155	−0.725132
$857$	30.5042	1.04200	0.521002	−	0.853555i	$-0.325558\pi$
$857$	30.5042	1.04200	0.521002	+	0.853555i	$0.325558\pi$
$858$	0	0
$859$	27.8595	0.950553	0.475277	−	0.879836i	$-0.342348\pi$
$859$	27.8595	0.950553	0.475277	+	0.879836i	$0.342348\pi$
$860$	8.48188	0.289230
$861$	−7.05861	−0.240557
$862$	12.6450	0.430691
$863$	18.4413	0.627750	0.313875	−	0.949464i	$-0.398373\pi$
$863$	18.4413	0.627750	0.313875	+	0.949464i	$0.398373\pi$
$864$	6.18598	0.210451
$865$	4.97823	0.169265
$866$	−35.7385	−1.21445
$867$	16.8431	0.572021
$868$	2.38404	0.0809197
$869$	−14.1075	−0.478565
$870$	−13.2349	−0.448705
$871$	0	0
$872$	−10.0785	−0.341300
$873$	−13.1521	−0.445132
$874$	−7.67994	−0.259778
$875$	−0.801938	−0.0271104
$876$	5.35019	0.180766
$877$	−36.5163	−1.23307	−0.616534	−	0.787328i	$-0.711464\pi$
$877$	−36.5163	−1.23307	−0.616534	+	0.787328i	$0.711464\pi$
$878$	−31.0640	−1.04836
$879$	−25.4456	−0.858260
$880$	4.93900	0.166494
$881$	12.0562	0.406184	0.203092	−	0.979160i	$-0.434901\pi$
$881$	12.0562	0.406184	0.203092	+	0.979160i	$0.434901\pi$
$882$	−11.4547	−0.385701
$883$	−54.3414	−1.82873	−0.914366	−	0.404888i	$-0.867310\pi$
$883$	−54.3414	−1.82873	−0.914366	+	0.404888i	$0.867310\pi$
$884$	0	0
$885$	−8.11960	−0.272938
$886$	−3.67802	−0.123566
$887$	−5.92500	−0.198942	−0.0994710	−	0.995040i	$-0.531715\pi$
$887$	−5.92500	−0.198942	−0.0994710	+	0.995040i	$0.531715\pi$
$888$	−8.08815	−0.271420
$889$	−8.00192	−0.268376
$890$	11.7506	0.393882
$891$	1.00000	0.0335013
$892$	−12.8183	−0.429189
$893$	32.3153	1.08139
$894$	−27.3642	−0.915195
$895$	−13.5754	−0.453776
$896$	8.08815	0.270206
$897$	0	0
$898$	−25.8582	−0.862898
$899$	−17.5104	−0.584003
$900$	1.24698	0.0415660
$901$	−0.343960	−0.0114590
$902$	15.8605	0.528098
$903$	5.45473	0.181522
$904$	21.0000	0.698450
$905$	−9.93661	−0.330304
$906$	−31.2543	−1.03835
$907$	27.5784	0.915725	0.457863	−	0.889023i	$-0.348615\pi$
$907$	27.5784	0.915725	0.457863	+	0.889023i	$0.348615\pi$
$908$	6.77479	0.224829
$909$	17.8756	0.592897
$910$	0	0
$911$	−19.5670	−0.648285	−0.324142	−	0.946008i	$-0.605076\pi$
$911$	−19.5670	−0.648285	−0.324142	+	0.946008i	$0.605076\pi$
$912$	−13.5375	−0.448271
$913$	1.83877	0.0608545
$914$	44.7101	1.47888
$915$	−4.11529	−0.136047
$916$	24.8146	0.819898
$917$	3.78448	0.124975
$918$	0.713792	0.0235586
$919$	−18.3739	−0.606098	−0.303049	−	0.952975i	$-0.598005\pi$
$919$	−18.3739	−0.606098	−0.303049	+	0.952975i	$0.598005\pi$
$920$	2.10992	0.0695619
$921$	14.8955	0.490822
$922$	−34.5502	−1.13785
$923$	0	0
$924$	−1.00000	−0.0328976
$925$	−5.96077	−0.195989
$926$	−28.5676	−0.938791
$927$	6.39373	0.209998
$928$	45.4349	1.49147
$929$	−8.94305	−0.293412	−0.146706	−	0.989180i	$-0.546867\pi$
$929$	−8.94305	−0.293412	−0.146706	+	0.989180i	$0.546867\pi$
$930$	4.29590	0.140868
$931$	17.4239	0.571044
$932$	−14.3327	−0.469484
$933$	6.21313	0.203409
$934$	36.5327	1.19539
$935$	0.396125	0.0129547
$936$	0	0
$937$	−35.7047	−1.16642	−0.583211	−	0.812321i	$-0.698204\pi$
$937$	−35.7047	−1.16642	−0.583211	+	0.812321i	$0.698204\pi$
$938$	14.3623	0.468945
$939$	10.7694	0.351447
$940$	14.7017	0.479517
$941$	−1.55257	−0.0506122	−0.0253061	−	0.999680i	$-0.508056\pi$
$941$	−1.55257	−0.0506122	−0.0253061	+	0.999680i	$0.508056\pi$
$942$	36.1269	1.17708
$943$	13.6866	0.445698
$944$	40.1027	1.30523
$945$	0.801938	0.0260870
$946$	−12.2567	−0.398499
$947$	5.96940	0.193979	0.0969896	−	0.995285i	$-0.469079\pi$
$947$	5.96940	0.193979	0.0969896	+	0.995285i	$0.469079\pi$
$948$	17.5918	0.571355
$949$	0	0
$950$	−4.93900	−0.160242
$951$	−12.6504	−0.410217
$952$	0.431041	0.0139701
$953$	−2.24459	−0.0727093	−0.0363546	−	0.999339i	$-0.511575\pi$
$953$	−2.24459	−0.0727093	−0.0363546	+	0.999339i	$0.511575\pi$
$954$	1.56465	0.0506573
$955$	18.8756	0.610800
$956$	27.5338	0.890506
$957$	7.34481	0.237424
$958$	−26.0049	−0.840180
$959$	−3.18406	−0.102819
$960$	−1.26875	−0.0409487
$961$	−25.3163	−0.816656
$962$	0	0
$963$	15.6353	0.503842
$964$	26.6625	0.858741
$965$	−11.3817	−0.366388
$966$	−2.24698	−0.0722954
$967$	57.1624	1.83822	0.919110	−	0.394002i	$-0.128910\pi$
$967$	57.1624	1.83822	0.919110	+	0.394002i	$0.128910\pi$
$968$	13.5690	0.436123
$969$	−1.08575	−0.0348794
$970$	23.6993	0.760939
$971$	−13.8474	−0.444384	−0.222192	−	0.975003i	$-0.571321\pi$
$971$	−13.8474	−0.444384	−0.222192	+	0.975003i	$0.571321\pi$
$972$	−1.24698	−0.0399969
$973$	−12.2978	−0.394250
$974$	−63.2301	−2.02602
$975$	0	0
$976$	20.3254	0.650601
$977$	−40.7493	−1.30369	−0.651843	−	0.758354i	$-0.726004\pi$
$977$	−40.7493	−1.30369	−0.651843	+	0.758354i	$0.726004\pi$
$978$	8.43727	0.269794
$979$	−6.52111	−0.208415
$980$	7.92692	0.253216
$981$	7.42758	0.237144
$982$	4.97285	0.158690
$983$	−21.4644	−0.684609	−0.342304	−	0.939589i	$-0.611207\pi$
$983$	−21.4644	−0.684609	−0.342304	+	0.939589i	$0.611207\pi$
$984$	11.9433	0.380739
$985$	−26.5840	−0.847037
$986$	5.24267	0.166961
$987$	9.45473	0.300947
$988$	0	0
$989$	−10.5767	−0.336320
$990$	−1.80194	−0.0572693
$991$	44.1584	1.40274	0.701368	−	0.712799i	$-0.252573\pi$
$991$	44.1584	1.40274	0.701368	+	0.712799i	$0.252573\pi$
$992$	−14.7476	−0.468238
$993$	−35.7972	−1.13599
$994$	−8.77240	−0.278244
$995$	8.17523	0.259172
$996$	−2.29291	−0.0726537
$997$	−56.7426	−1.79706	−0.898528	−	0.438916i	$-0.855362\pi$
$997$	−56.7426	−1.79706	−0.898528	+	0.438916i	$0.855362\pi$
$998$	31.3250	0.991574
$999$	5.96077	0.188590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2535.2.a.bd.1.3	yes	3
3.2	odd	2		7605.2.a.bq.1.1		3
13.12	even	2		2535.2.a.v.1.1	✓	3
39.38	odd	2		7605.2.a.bz.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2535.2.a.v.1.1	✓	3	13.12	even	2
2535.2.a.bd.1.3	yes	3	1.1	even	1	trivial
7605.2.a.bq.1.1		3	3.2	odd	2
7605.2.a.bz.1.3		3	39.38	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2535.a (trivial)

\(f(q)\)	\(=\)	\(q+1.80194 q^{2} -1.00000 q^{3} +1.24698 q^{4} -1.00000 q^{5} -1.80194 q^{6} +0.801938 q^{7} -1.35690 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.80194 q^{2} -1.00000 q^{3} +1.24698 q^{4} -1.00000 q^{5} -1.80194 q^{6} +0.801938 q^{7} -1.35690 q^{8} +1.00000 q^{9} -1.80194 q^{10} +1.00000 q^{11} -1.24698 q^{12} +1.44504 q^{14} +1.00000 q^{15} -4.93900 q^{16} -0.396125 q^{17} +1.80194 q^{18} -2.74094 q^{19} -1.24698 q^{20} -0.801938 q^{21} +1.80194 q^{22} +1.55496 q^{23} +1.35690 q^{24} +1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} -7.34481 q^{29} +1.80194 q^{30} +2.38404 q^{31} -6.18598 q^{32} -1.00000 q^{33} -0.713792 q^{34} -0.801938 q^{35} +1.24698 q^{36} -5.96077 q^{37} -4.93900 q^{38} +1.35690 q^{40} +8.80194 q^{41} -1.44504 q^{42} -6.80194 q^{43} +1.24698 q^{44} -1.00000 q^{45} +2.80194 q^{46} -11.7899 q^{47} +4.93900 q^{48} -6.35690 q^{49} +1.80194 q^{50} +0.396125 q^{51} +0.868313 q^{53} -1.80194 q^{54} -1.00000 q^{55} -1.08815 q^{56} +2.74094 q^{57} -13.2349 q^{58} -8.11960 q^{59} +1.24698 q^{60} -4.11529 q^{61} +4.29590 q^{62} +0.801938 q^{63} -1.26875 q^{64} -1.80194 q^{66} +9.93900 q^{67} -0.493959 q^{68} -1.55496 q^{69} -1.44504 q^{70} -6.07069 q^{71} -1.35690 q^{72} -4.29052 q^{73} -10.7409 q^{74} -1.00000 q^{75} -3.41789 q^{76} +0.801938 q^{77} -14.1075 q^{79} +4.93900 q^{80} +1.00000 q^{81} +15.8605 q^{82} +1.83877 q^{83} -1.00000 q^{84} +0.396125 q^{85} -12.2567 q^{86} +7.34481 q^{87} -1.35690 q^{88} -6.52111 q^{89} -1.80194 q^{90} +1.93900 q^{92} -2.38404 q^{93} -21.2446 q^{94} +2.74094 q^{95} +6.18598 q^{96} -13.1521 q^{97} -11.4547 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 3 q^{3} - q^{4} - 3 q^{5} - q^{6} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 - 3 * q^3 - q^4 - 3 * q^5 - q^6 - 2 * q^7 + 3 * q^9	\( 3 q + q^{2} - 3 q^{3} - q^{4} - 3 q^{5} - q^{6} - 2 q^{7} + 3 q^{9} - q^{10} + 3 q^{11} + q^{12} + 4 q^{14} + 3 q^{15} - 5 q^{16} - 10 q^{17} + q^{18} + 6 q^{19} + q^{20} + 2 q^{21} + q^{22} + 5 q^{23} + 3 q^{25} - 3 q^{27} + 3 q^{28} + q^{29} + q^{30} - 3 q^{31} - 4 q^{32} - 3 q^{33} + 6 q^{34} + 2 q^{35} - q^{36} - 5 q^{37} - 5 q^{38} + 22 q^{41} - 4 q^{42} - 16 q^{43} - q^{44} - 3 q^{45} + 4 q^{46} - 12 q^{47} + 5 q^{48} - 15 q^{49} + q^{50} + 10 q^{51} + 5 q^{53} - q^{54} - 3 q^{55} - 7 q^{56} - 6 q^{57} - 16 q^{58} - 3 q^{59} - q^{60} - 10 q^{61} - q^{62} - 2 q^{63} + 4 q^{64} - q^{66} + 20 q^{67} + 8 q^{68} - 5 q^{69} - 4 q^{70} - 6 q^{71} - 2 q^{73} - 18 q^{74} - 3 q^{75} - 16 q^{76} - 2 q^{77} - 2 q^{79} + 5 q^{80} + 3 q^{81} + 12 q^{82} - 27 q^{83} - 3 q^{84} + 10 q^{85} - 10 q^{86} - q^{87} - 4 q^{89} - q^{90} - 4 q^{92} + 3 q^{93} - 18 q^{94} - 6 q^{95} + 4 q^{96} - 9 q^{97} - 12 q^{98} + 3 q^{99}+O(q^{100}) \) 3 * q + q^2 - 3 * q^3 - q^4 - 3 * q^5 - q^6 - 2 * q^7 + 3 * q^9 - q^10 + 3 * q^11 + q^12 + 4 * q^14 + 3 * q^15 - 5 * q^16 - 10 * q^17 + q^18 + 6 * q^19 + q^20 + 2 * q^21 + q^22 + 5 * q^23 + 3 * q^25 - 3 * q^27 + 3 * q^28 + q^29 + q^30 - 3 * q^31 - 4 * q^32 - 3 * q^33 + 6 * q^34 + 2 * q^35 - q^36 - 5 * q^37 - 5 * q^38 + 22 * q^41 - 4 * q^42 - 16 * q^43 - q^44 - 3 * q^45 + 4 * q^46 - 12 * q^47 + 5 * q^48 - 15 * q^49 + q^50 + 10 * q^51 + 5 * q^53 - q^54 - 3 * q^55 - 7 * q^56 - 6 * q^57 - 16 * q^58 - 3 * q^59 - q^60 - 10 * q^61 - q^62 - 2 * q^63 + 4 * q^64 - q^66 + 20 * q^67 + 8 * q^68 - 5 * q^69 - 4 * q^70 - 6 * q^71 - 2 * q^73 - 18 * q^74 - 3 * q^75 - 16 * q^76 - 2 * q^77 - 2 * q^79 + 5 * q^80 + 3 * q^81 + 12 * q^82 - 27 * q^83 - 3 * q^84 + 10 * q^85 - 10 * q^86 - q^87 - 4 * q^89 - q^90 - 4 * q^92 + 3 * q^93 - 18 * q^94 - 6 * q^95 + 4 * q^96 - 9 * q^97 - 12 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

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Database

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