LMFDB - Embedded newform 7098.2.a.ch.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, 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("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7098.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:	$56.6778153547$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	7098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.44504 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.44504 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.44504 q^{10} -3.74094 q^{11} -1.00000 q^{12} +1.00000 q^{14} +1.44504 q^{15} +1.00000 q^{16} +0.692021 q^{17} +1.00000 q^{18} -0.753020 q^{19} -1.44504 q^{20} -1.00000 q^{21} -3.74094 q^{22} +1.82908 q^{23} -1.00000 q^{24} -2.91185 q^{25} -1.00000 q^{27} +1.00000 q^{28} +6.89977 q^{29} +1.44504 q^{30} +0.960771 q^{31} +1.00000 q^{32} +3.74094 q^{33} +0.692021 q^{34} -1.44504 q^{35} +1.00000 q^{36} -3.04892 q^{37} -0.753020 q^{38} -1.44504 q^{40} +1.44504 q^{41} -1.00000 q^{42} +5.82908 q^{43} -3.74094 q^{44} -1.44504 q^{45} +1.82908 q^{46} -3.71379 q^{47} -1.00000 q^{48} +1.00000 q^{49} -2.91185 q^{50} -0.692021 q^{51} -1.57673 q^{53} -1.00000 q^{54} +5.40581 q^{55} +1.00000 q^{56} +0.753020 q^{57} +6.89977 q^{58} +6.65279 q^{59} +1.44504 q^{60} -7.00000 q^{61} +0.960771 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.74094 q^{66} -9.11960 q^{67} +0.692021 q^{68} -1.82908 q^{69} -1.44504 q^{70} +0.219833 q^{71} +1.00000 q^{72} -4.91185 q^{73} -3.04892 q^{74} +2.91185 q^{75} -0.753020 q^{76} -3.74094 q^{77} -7.44504 q^{79} -1.44504 q^{80} +1.00000 q^{81} +1.44504 q^{82} -8.75302 q^{83} -1.00000 q^{84} -1.00000 q^{85} +5.82908 q^{86} -6.89977 q^{87} -3.74094 q^{88} -14.5579 q^{89} -1.44504 q^{90} +1.82908 q^{92} -0.960771 q^{93} -3.71379 q^{94} +1.08815 q^{95} -1.00000 q^{96} -6.06100 q^{97} +1.00000 q^{98} -3.74094 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 4 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 4 * q^5 - 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 4 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 4 q^{10} + 3 q^{11} - 3 q^{12} + 3 q^{14} + 4 q^{15} + 3 q^{16} - 3 q^{17} + 3 q^{18} - 7 q^{19} - 4 q^{20} - 3 q^{21} + 3 q^{22} - 5 q^{23} - 3 q^{24} - 5 q^{25} - 3 q^{27} + 3 q^{28} - 2 q^{29} + 4 q^{30} - 10 q^{31} + 3 q^{32} - 3 q^{33} - 3 q^{34} - 4 q^{35} + 3 q^{36} - 7 q^{38} - 4 q^{40} + 4 q^{41} - 3 q^{42} + 7 q^{43} + 3 q^{44} - 4 q^{45} - 5 q^{46} - 3 q^{47} - 3 q^{48} + 3 q^{49} - 5 q^{50} + 3 q^{51} - 2 q^{53} - 3 q^{54} + 3 q^{55} + 3 q^{56} + 7 q^{57} - 2 q^{58} + 2 q^{59} + 4 q^{60} - 21 q^{61} - 10 q^{62} + 3 q^{63} + 3 q^{64} - 3 q^{66} - 6 q^{67} - 3 q^{68} + 5 q^{69} - 4 q^{70} + 2 q^{71} + 3 q^{72} - 11 q^{73} + 5 q^{75} - 7 q^{76} + 3 q^{77} - 22 q^{79} - 4 q^{80} + 3 q^{81} + 4 q^{82} - 31 q^{83} - 3 q^{84} - 3 q^{85} + 7 q^{86} + 2 q^{87} + 3 q^{88} - 4 q^{90} - 5 q^{92} + 10 q^{93} - 3 q^{94} + 7 q^{95} - 3 q^{96} - 28 q^{97} + 3 q^{98} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 4 * q^5 - 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9 - 4 * q^10 + 3 * q^11 - 3 * q^12 + 3 * q^14 + 4 * q^15 + 3 * q^16 - 3 * q^17 + 3 * q^18 - 7 * q^19 - 4 * q^20 - 3 * q^21 + 3 * q^22 - 5 * q^23 - 3 * q^24 - 5 * q^25 - 3 * q^27 + 3 * q^28 - 2 * q^29 + 4 * q^30 - 10 * q^31 + 3 * q^32 - 3 * q^33 - 3 * q^34 - 4 * q^35 + 3 * q^36 - 7 * q^38 - 4 * q^40 + 4 * q^41 - 3 * q^42 + 7 * q^43 + 3 * q^44 - 4 * q^45 - 5 * q^46 - 3 * q^47 - 3 * q^48 + 3 * q^49 - 5 * q^50 + 3 * q^51 - 2 * q^53 - 3 * q^54 + 3 * q^55 + 3 * q^56 + 7 * q^57 - 2 * q^58 + 2 * q^59 + 4 * q^60 - 21 * q^61 - 10 * q^62 + 3 * q^63 + 3 * q^64 - 3 * q^66 - 6 * q^67 - 3 * q^68 + 5 * q^69 - 4 * q^70 + 2 * q^71 + 3 * q^72 - 11 * q^73 + 5 * q^75 - 7 * q^76 + 3 * q^77 - 22 * q^79 - 4 * q^80 + 3 * q^81 + 4 * q^82 - 31 * q^83 - 3 * q^84 - 3 * q^85 + 7 * q^86 + 2 * q^87 + 3 * q^88 - 4 * q^90 - 5 * q^92 + 10 * q^93 - 3 * q^94 + 7 * q^95 - 3 * q^96 - 28 * q^97 + 3 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.44504	−0.646242	−0.323121	−	0.946358i	$-0.604732\pi$
$5$	−1.44504	−0.646242	−0.323121	+	0.946358i	$0.604732\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.44504	−0.456962
$11$	−3.74094	−1.12794	−0.563968	−	0.825797i	$-0.690726\pi$
$11$	−3.74094	−1.12794	−0.563968	+	0.825797i	$0.690726\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	1.44504	0.373108
$16$	1.00000	0.250000
$17$	0.692021	0.167840	0.0839199	−	0.996473i	$-0.473256\pi$
$17$	0.692021	0.167840	0.0839199	+	0.996473i	$0.473256\pi$
$18$	1.00000	0.235702
$19$	−0.753020	−0.172755	−0.0863774	−	0.996262i	$-0.527529\pi$
$19$	−0.753020	−0.172755	−0.0863774	+	0.996262i	$0.527529\pi$
$20$	−1.44504	−0.323121
$21$	−1.00000	−0.218218
$22$	−3.74094	−0.797571
$23$	1.82908	0.381391	0.190695	−	0.981649i	$-0.438926\pi$
$23$	1.82908	0.381391	0.190695	+	0.981649i	$0.438926\pi$
$24$	−1.00000	−0.204124
$25$	−2.91185	−0.582371
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	6.89977	1.28126	0.640628	−	0.767852i	$-0.278674\pi$
$29$	6.89977	1.28126	0.640628	+	0.767852i	$0.278674\pi$
$30$	1.44504	0.263827
$31$	0.960771	0.172560	0.0862798	−	0.996271i	$-0.472502\pi$
$31$	0.960771	0.172560	0.0862798	+	0.996271i	$0.472502\pi$
$32$	1.00000	0.176777
$33$	3.74094	0.651214
$34$	0.692021	0.118681
$35$	−1.44504	−0.244257
$36$	1.00000	0.166667
$37$	−3.04892	−0.501239	−0.250619	−	0.968086i	$-0.580634\pi$
$37$	−3.04892	−0.501239	−0.250619	+	0.968086i	$0.580634\pi$
$38$	−0.753020	−0.122156
$39$	0	0
$40$	−1.44504	−0.228481
$41$	1.44504	0.225678	0.112839	−	0.993613i	$-0.464006\pi$
$41$	1.44504	0.225678	0.112839	+	0.993613i	$0.464006\pi$
$42$	−1.00000	−0.154303
$43$	5.82908	0.888927	0.444464	−	0.895797i	$-0.353394\pi$
$43$	5.82908	0.888927	0.444464	+	0.895797i	$0.353394\pi$
$44$	−3.74094	−0.563968
$45$	−1.44504	−0.215414
$46$	1.82908	0.269684
$47$	−3.71379	−0.541712	−0.270856	−	0.962620i	$-0.587307\pi$
$47$	−3.71379	−0.541712	−0.270856	+	0.962620i	$0.587307\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−2.91185	−0.411798
$51$	−0.692021	−0.0969024
$52$	0	0
$53$	−1.57673	−0.216580	−0.108290	−	0.994119i	$-0.534538\pi$
$53$	−1.57673	−0.216580	−0.108290	+	0.994119i	$0.534538\pi$
$54$	−1.00000	−0.136083
$55$	5.40581	0.728920
$56$	1.00000	0.133631
$57$	0.753020	0.0997400
$58$	6.89977	0.905985
$59$	6.65279	0.866120	0.433060	−	0.901365i	$-0.357434\pi$
$59$	6.65279	0.866120	0.433060	+	0.901365i	$0.357434\pi$
$60$	1.44504	0.186554
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	0.960771	0.122018
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	3.74094	0.460478
$67$	−9.11960	−1.11414	−0.557069	−	0.830467i	$-0.688074\pi$
$67$	−9.11960	−1.11414	−0.557069	+	0.830467i	$0.688074\pi$
$68$	0.692021	0.0839199
$69$	−1.82908	−0.220196
$70$	−1.44504	−0.172716
$71$	0.219833	0.0260893	0.0130447	−	0.999915i	$-0.495848\pi$
$71$	0.219833	0.0260893	0.0130447	+	0.999915i	$0.495848\pi$
$72$	1.00000	0.117851
$73$	−4.91185	−0.574889	−0.287445	−	0.957797i	$-0.592806\pi$
$73$	−4.91185	−0.574889	−0.287445	+	0.957797i	$0.592806\pi$
$74$	−3.04892	−0.354429
$75$	2.91185	0.336232
$76$	−0.753020	−0.0863774
$77$	−3.74094	−0.426320
$78$	0	0
$79$	−7.44504	−0.837633	−0.418816	−	0.908071i	$-0.637555\pi$
$79$	−7.44504	−0.837633	−0.418816	+	0.908071i	$0.637555\pi$
$80$	−1.44504	−0.161561
$81$	1.00000	0.111111
$82$	1.44504	0.159578
$83$	−8.75302	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$83$	−8.75302	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$84$	−1.00000	−0.109109
$85$	−1.00000	−0.108465
$86$	5.82908	0.628566
$87$	−6.89977	−0.739733
$88$	−3.74094	−0.398785
$89$	−14.5579	−1.54314	−0.771569	−	0.636145i	$-0.780528\pi$
$89$	−14.5579	−1.54314	−0.771569	+	0.636145i	$0.780528\pi$
$90$	−1.44504	−0.152321
$91$	0	0
$92$	1.82908	0.190695
$93$	−0.960771	−0.0996273
$94$	−3.71379	−0.383048
$95$	1.08815	0.111641
$96$	−1.00000	−0.102062
$97$	−6.06100	−0.615401	−0.307701	−	0.951483i	$-0.599560\pi$
$97$	−6.06100	−0.615401	−0.307701	+	0.951483i	$0.599560\pi$
$98$	1.00000	0.101015
$99$	−3.74094	−0.375978
$100$	−2.91185	−0.291185
$101$	−2.81402	−0.280005	−0.140003	−	0.990151i	$-0.544711\pi$
$101$	−2.81402	−0.280005	−0.140003	+	0.990151i	$0.544711\pi$
$102$	−0.692021	−0.0685203
$103$	−8.23490	−0.811409	−0.405704	−	0.914004i	$-0.632974\pi$
$103$	−8.23490	−0.811409	−0.405704	+	0.914004i	$0.632974\pi$
$104$	0	0
$105$	1.44504	0.141022
$106$	−1.57673	−0.153145
$107$	10.6039	1.02512	0.512558	−	0.858653i	$-0.328698\pi$
$107$	10.6039	1.02512	0.512558	+	0.858653i	$0.328698\pi$
$108$	−1.00000	−0.0962250
$109$	8.48858	0.813059	0.406529	−	0.913638i	$-0.366739\pi$
$109$	8.48858	0.813059	0.406529	+	0.913638i	$0.366739\pi$
$110$	5.40581	0.515424
$111$	3.04892	0.289390
$112$	1.00000	0.0944911
$113$	−5.92692	−0.557558	−0.278779	−	0.960355i	$-0.589930\pi$
$113$	−5.92692	−0.557558	−0.278779	+	0.960355i	$0.589930\pi$
$114$	0.753020	0.0705268
$115$	−2.64310	−0.246471
$116$	6.89977	0.640628
$117$	0	0
$118$	6.65279	0.612439
$119$	0.692021	0.0634375
$120$	1.44504	0.131914
$121$	2.99462	0.272238
$122$	−7.00000	−0.633750
$123$	−1.44504	−0.130295
$124$	0.960771	0.0862798
$125$	11.4330	1.02260
$126$	1.00000	0.0890871
$127$	9.67994	0.858956	0.429478	−	0.903077i	$-0.358698\pi$
$127$	9.67994	0.858956	0.429478	+	0.903077i	$0.358698\pi$
$128$	1.00000	0.0883883
$129$	−5.82908	−0.513222
$130$	0	0
$131$	2.25236	0.196789	0.0983946	−	0.995147i	$-0.468629\pi$
$131$	2.25236	0.196789	0.0983946	+	0.995147i	$0.468629\pi$
$132$	3.74094	0.325607
$133$	−0.753020	−0.0652951
$134$	−9.11960	−0.787814
$135$	1.44504	0.124369
$136$	0.692021	0.0593404
$137$	−12.0489	−1.02941	−0.514704	−	0.857368i	$-0.672098\pi$
$137$	−12.0489	−1.02941	−0.514704	+	0.857368i	$0.672098\pi$
$138$	−1.82908	−0.155702
$139$	−16.6383	−1.41124	−0.705622	−	0.708589i	$-0.749332\pi$
$139$	−16.6383	−1.41124	−0.705622	+	0.708589i	$0.749332\pi$
$140$	−1.44504	−0.122128
$141$	3.71379	0.312758
$142$	0.219833	0.0184479
$143$	0	0
$144$	1.00000	0.0833333
$145$	−9.97046	−0.828002
$146$	−4.91185	−0.406508
$147$	−1.00000	−0.0824786
$148$	−3.04892	−0.250619
$149$	8.35690	0.684624	0.342312	−	0.939586i	$-0.388790\pi$
$149$	8.35690	0.684624	0.342312	+	0.939586i	$0.388790\pi$
$150$	2.91185	0.237752
$151$	−7.63102	−0.621004	−0.310502	−	0.950573i	$-0.600497\pi$
$151$	−7.63102	−0.621004	−0.310502	+	0.950573i	$0.600497\pi$
$152$	−0.753020	−0.0610780
$153$	0.692021	0.0559466
$154$	−3.74094	−0.301453
$155$	−1.38835	−0.111515
$156$	0	0
$157$	12.2935	0.981128	0.490564	−	0.871405i	$-0.336791\pi$
$157$	12.2935	0.981128	0.490564	+	0.871405i	$0.336791\pi$
$158$	−7.44504	−0.592296
$159$	1.57673	0.125043
$160$	−1.44504	−0.114241
$161$	1.82908	0.144152
$162$	1.00000	0.0785674
$163$	−15.0151	−1.17607	−0.588035	−	0.808835i	$-0.700098\pi$
$163$	−15.0151	−1.17607	−0.588035	+	0.808835i	$0.700098\pi$
$164$	1.44504	0.112839
$165$	−5.40581	−0.420842
$166$	−8.75302	−0.679366
$167$	18.3153	1.41728	0.708639	−	0.705571i	$-0.249309\pi$
$167$	18.3153	1.41728	0.708639	+	0.705571i	$0.249309\pi$
$168$	−1.00000	−0.0771517
$169$	0	0
$170$	−1.00000	−0.0766965
$171$	−0.753020	−0.0575849
$172$	5.82908	0.444464
$173$	−16.7875	−1.27633	−0.638164	−	0.769901i	$-0.720306\pi$
$173$	−16.7875	−1.27633	−0.638164	+	0.769901i	$0.720306\pi$
$174$	−6.89977	−0.523070
$175$	−2.91185	−0.220115
$176$	−3.74094	−0.281984
$177$	−6.65279	−0.500055
$178$	−14.5579	−1.09116
$179$	4.86725	0.363795	0.181898	−	0.983317i	$-0.441776\pi$
$179$	4.86725	0.363795	0.181898	+	0.983317i	$0.441776\pi$
$180$	−1.44504	−0.107707
$181$	−13.4644	−1.00080	−0.500401	−	0.865794i	$-0.666814\pi$
$181$	−13.4644	−1.00080	−0.500401	+	0.865794i	$0.666814\pi$
$182$	0	0
$183$	7.00000	0.517455
$184$	1.82908	0.134842
$185$	4.40581	0.323922
$186$	−0.960771	−0.0704472
$187$	−2.58881	−0.189313
$188$	−3.71379	−0.270856
$189$	−1.00000	−0.0727393
$190$	1.08815	0.0789424
$191$	−5.26444	−0.380921	−0.190461	−	0.981695i	$-0.560998\pi$
$191$	−5.26444	−0.380921	−0.190461	+	0.981695i	$0.560998\pi$
$192$	−1.00000	−0.0721688
$193$	−22.8713	−1.64631	−0.823156	−	0.567815i	$-0.807789\pi$
$193$	−22.8713	−1.64631	−0.823156	+	0.567815i	$0.807789\pi$
$194$	−6.06100	−0.435154
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.8877	1.34569	0.672846	−	0.739782i	$-0.265072\pi$
$197$	18.8877	1.34569	0.672846	+	0.739782i	$0.265072\pi$
$198$	−3.74094	−0.265857
$199$	−3.36898	−0.238820	−0.119410	−	0.992845i	$-0.538100\pi$
$199$	−3.36898	−0.238820	−0.119410	+	0.992845i	$0.538100\pi$
$200$	−2.91185	−0.205899
$201$	9.11960	0.643247
$202$	−2.81402	−0.197994
$203$	6.89977	0.484269
$204$	−0.692021	−0.0484512
$205$	−2.08815	−0.145842
$206$	−8.23490	−0.573753
$207$	1.82908	0.127130
$208$	0	0
$209$	2.81700	0.194856
$210$	1.44504	0.0997174
$211$	17.1511	1.18073	0.590364	−	0.807137i	$-0.298984\pi$
$211$	17.1511	1.18073	0.590364	+	0.807137i	$0.298984\pi$
$212$	−1.57673	−0.108290
$213$	−0.219833	−0.0150627
$214$	10.6039	0.724866
$215$	−8.42327	−0.574462
$216$	−1.00000	−0.0680414
$217$	0.960771	0.0652214
$218$	8.48858	0.574919
$219$	4.91185	0.331912
$220$	5.40581	0.364460
$221$	0	0
$222$	3.04892	0.204630
$223$	−25.0911	−1.68023	−0.840113	−	0.542411i	$-0.817512\pi$
$223$	−25.0911	−1.68023	−0.840113	+	0.542411i	$0.817512\pi$
$224$	1.00000	0.0668153
$225$	−2.91185	−0.194124
$226$	−5.92692	−0.394253
$227$	5.84415	0.387890	0.193945	−	0.981012i	$-0.437872\pi$
$227$	5.84415	0.387890	0.193945	+	0.981012i	$0.437872\pi$
$228$	0.753020	0.0498700
$229$	−15.2252	−1.00611	−0.503055	−	0.864254i	$-0.667791\pi$
$229$	−15.2252	−1.00611	−0.503055	+	0.864254i	$0.667791\pi$
$230$	−2.64310	−0.174281
$231$	3.74094	0.246136
$232$	6.89977	0.452992
$233$	−25.2000	−1.65091	−0.825453	−	0.564471i	$-0.809080\pi$
$233$	−25.2000	−1.65091	−0.825453	+	0.564471i	$0.809080\pi$
$234$	0	0
$235$	5.36658	0.350077
$236$	6.65279	0.433060
$237$	7.44504	0.483607
$238$	0.692021	0.0448571
$239$	6.71379	0.434279	0.217140	−	0.976141i	$-0.430327\pi$
$239$	6.71379	0.434279	0.217140	+	0.976141i	$0.430327\pi$
$240$	1.44504	0.0932771
$241$	−5.21313	−0.335807	−0.167904	−	0.985803i	$-0.553700\pi$
$241$	−5.21313	−0.335807	−0.167904	+	0.985803i	$0.553700\pi$
$242$	2.99462	0.192502
$243$	−1.00000	−0.0641500
$244$	−7.00000	−0.448129
$245$	−1.44504	−0.0923203
$246$	−1.44504	−0.0921325
$247$	0	0
$248$	0.960771	0.0610090
$249$	8.75302	0.554700
$250$	11.4330	0.723084
$251$	9.22952	0.582562	0.291281	−	0.956638i	$-0.405919\pi$
$251$	9.22952	0.582562	0.291281	+	0.956638i	$0.405919\pi$
$252$	1.00000	0.0629941
$253$	−6.84249	−0.430184
$254$	9.67994	0.607373
$255$	1.00000	0.0626224
$256$	1.00000	0.0625000
$257$	2.51035	0.156591	0.0782957	−	0.996930i	$-0.475052\pi$
$257$	2.51035	0.156591	0.0782957	+	0.996930i	$0.475052\pi$
$258$	−5.82908	−0.362903
$259$	−3.04892	−0.189451
$260$	0	0
$261$	6.89977	0.427085
$262$	2.25236	0.139151
$263$	30.1511	1.85919	0.929597	−	0.368577i	$-0.120155\pi$
$263$	30.1511	1.85919	0.929597	+	0.368577i	$0.120155\pi$
$264$	3.74094	0.230239
$265$	2.27844	0.139963
$266$	−0.753020	−0.0461706
$267$	14.5579	0.890932
$268$	−9.11960	−0.557069
$269$	−14.6063	−0.890560	−0.445280	−	0.895391i	$-0.646896\pi$
$269$	−14.6063	−0.890560	−0.445280	+	0.895391i	$0.646896\pi$
$270$	1.44504	0.0879424
$271$	−22.0978	−1.34235	−0.671174	−	0.741300i	$-0.734210\pi$
$271$	−22.0978	−1.34235	−0.671174	+	0.741300i	$0.734210\pi$
$272$	0.692021	0.0419600
$273$	0	0
$274$	−12.0489	−0.727902
$275$	10.8931	0.656877
$276$	−1.82908	−0.110098
$277$	1.23729	0.0743416	0.0371708	−	0.999309i	$-0.488165\pi$
$277$	1.23729	0.0743416	0.0371708	+	0.999309i	$0.488165\pi$
$278$	−16.6383	−0.997900
$279$	0.960771	0.0575199
$280$	−1.44504	−0.0863578
$281$	−8.51035	−0.507685	−0.253843	−	0.967246i	$-0.581695\pi$
$281$	−8.51035	−0.507685	−0.253843	+	0.967246i	$0.581695\pi$
$282$	3.71379	0.221153
$283$	−16.3002	−0.968947	−0.484473	−	0.874806i	$-0.660989\pi$
$283$	−16.3002	−0.968947	−0.484473	+	0.874806i	$0.660989\pi$
$284$	0.219833	0.0130447
$285$	−1.08815	−0.0644562
$286$	0	0
$287$	1.44504	0.0852981
$288$	1.00000	0.0589256
$289$	−16.5211	−0.971830
$290$	−9.97046	−0.585486
$291$	6.06100	0.355302
$292$	−4.91185	−0.287445
$293$	−29.8853	−1.74592	−0.872959	−	0.487794i	$-0.837802\pi$
$293$	−29.8853	−1.74592	−0.872959	+	0.487794i	$0.837802\pi$
$294$	−1.00000	−0.0583212
$295$	−9.61356	−0.559723
$296$	−3.04892	−0.177215
$297$	3.74094	0.217071
$298$	8.35690	0.484102
$299$	0	0
$300$	2.91185	0.168116
$301$	5.82908	0.335983
$302$	−7.63102	−0.439116
$303$	2.81402	0.161661
$304$	−0.753020	−0.0431887
$305$	10.1153	0.579200
$306$	0.692021	0.0395602
$307$	−24.4547	−1.39571	−0.697853	−	0.716241i	$-0.745861\pi$
$307$	−24.4547	−1.39571	−0.697853	+	0.716241i	$0.745861\pi$
$308$	−3.74094	−0.213160
$309$	8.23490	0.468467
$310$	−1.38835	−0.0788533
$311$	−22.3177	−1.26552	−0.632759	−	0.774348i	$-0.718078\pi$
$311$	−22.3177	−1.26552	−0.632759	+	0.774348i	$0.718078\pi$
$312$	0	0
$313$	−2.47889	−0.140115	−0.0700577	−	0.997543i	$-0.522318\pi$
$313$	−2.47889	−0.140115	−0.0700577	+	0.997543i	$0.522318\pi$
$314$	12.2935	0.693763
$315$	−1.44504	−0.0814189
$316$	−7.44504	−0.418816
$317$	−5.65710	−0.317735	−0.158867	−	0.987300i	$-0.550784\pi$
$317$	−5.65710	−0.317735	−0.158867	+	0.987300i	$0.550784\pi$
$318$	1.57673	0.0884185
$319$	−25.8116	−1.44517
$320$	−1.44504	−0.0807803
$321$	−10.6039	−0.591851
$322$	1.82908	0.101931
$323$	−0.521106	−0.0289951
$324$	1.00000	0.0555556
$325$	0	0
$326$	−15.0151	−0.831608
$327$	−8.48858	−0.469420
$328$	1.44504	0.0797891
$329$	−3.71379	−0.204748
$330$	−5.40581	−0.297580
$331$	23.1879	1.27452	0.637261	−	0.770648i	$-0.280067\pi$
$331$	23.1879	1.27452	0.637261	+	0.770648i	$0.280067\pi$
$332$	−8.75302	−0.480384
$333$	−3.04892	−0.167080
$334$	18.3153	1.00217
$335$	13.1782	0.720003
$336$	−1.00000	−0.0545545
$337$	10.4873	0.571277	0.285639	−	0.958337i	$-0.407794\pi$
$337$	10.4873	0.571277	0.285639	+	0.958337i	$0.407794\pi$
$338$	0	0
$339$	5.92692	0.321906
$340$	−1.00000	−0.0542326
$341$	−3.59419	−0.194636
$342$	−0.753020	−0.0407187
$343$	1.00000	0.0539949
$344$	5.82908	0.314283
$345$	2.64310	0.142300
$346$	−16.7875	−0.902500
$347$	−5.28514	−0.283721	−0.141861	−	0.989887i	$-0.545309\pi$
$347$	−5.28514	−0.283721	−0.141861	+	0.989887i	$0.545309\pi$
$348$	−6.89977	−0.369867
$349$	8.19029	0.438416	0.219208	−	0.975678i	$-0.429653\pi$
$349$	8.19029	0.438416	0.219208	+	0.975678i	$0.429653\pi$
$350$	−2.91185	−0.155645
$351$	0	0
$352$	−3.74094	−0.199393
$353$	35.9758	1.91480	0.957400	−	0.288764i	$-0.0932443\pi$
$353$	35.9758	1.91480	0.957400	+	0.288764i	$0.0932443\pi$
$354$	−6.65279	−0.353592
$355$	−0.317667	−0.0168600
$356$	−14.5579	−0.771569
$357$	−0.692021	−0.0366257
$358$	4.86725	0.257242
$359$	−28.7603	−1.51791	−0.758956	−	0.651142i	$-0.774290\pi$
$359$	−28.7603	−1.51791	−0.758956	+	0.651142i	$0.774290\pi$
$360$	−1.44504	−0.0761604
$361$	−18.4330	−0.970156
$362$	−13.4644	−0.707674
$363$	−2.99462	−0.157177
$364$	0	0
$365$	7.09783	0.371518
$366$	7.00000	0.365896
$367$	−3.61463	−0.188682	−0.0943411	−	0.995540i	$-0.530074\pi$
$367$	−3.61463	−0.188682	−0.0943411	+	0.995540i	$0.530074\pi$
$368$	1.82908	0.0953476
$369$	1.44504	0.0752259
$370$	4.40581	0.229047
$371$	−1.57673	−0.0818597
$372$	−0.960771	−0.0498137
$373$	17.9312	0.928444	0.464222	−	0.885719i	$-0.346334\pi$
$373$	17.9312	0.928444	0.464222	+	0.885719i	$0.346334\pi$
$374$	−2.58881	−0.133864
$375$	−11.4330	−0.590396
$376$	−3.71379	−0.191524
$377$	0	0
$378$	−1.00000	−0.0514344
$379$	22.2338	1.14208	0.571038	−	0.820924i	$-0.306541\pi$
$379$	22.2338	1.14208	0.571038	+	0.820924i	$0.306541\pi$
$380$	1.08815	0.0558207
$381$	−9.67994	−0.495918
$382$	−5.26444	−0.269352
$383$	−25.4330	−1.29956	−0.649782	−	0.760121i	$-0.725140\pi$
$383$	−25.4330	−1.29956	−0.649782	+	0.760121i	$0.725140\pi$
$384$	−1.00000	−0.0510310
$385$	5.40581	0.275506
$386$	−22.8713	−1.16412
$387$	5.82908	0.296309
$388$	−6.06100	−0.307701
$389$	−14.9855	−0.759796	−0.379898	−	0.925028i	$-0.624041\pi$
$389$	−14.9855	−0.759796	−0.379898	+	0.925028i	$0.624041\pi$
$390$	0	0
$391$	1.26577	0.0640125
$392$	1.00000	0.0505076
$393$	−2.25236	−0.113616
$394$	18.8877	0.951548
$395$	10.7584	0.541314
$396$	−3.74094	−0.187989
$397$	−11.8358	−0.594021	−0.297011	−	0.954874i	$-0.595990\pi$
$397$	−11.8358	−0.594021	−0.297011	+	0.954874i	$0.595990\pi$
$398$	−3.36898	−0.168872
$399$	0.753020	0.0376982
$400$	−2.91185	−0.145593
$401$	11.6256	0.580557	0.290279	−	0.956942i	$-0.406252\pi$
$401$	11.6256	0.580557	0.290279	+	0.956942i	$0.406252\pi$
$402$	9.11960	0.454845
$403$	0	0
$404$	−2.81402	−0.140003
$405$	−1.44504	−0.0718047
$406$	6.89977	0.342430
$407$	11.4058	0.565365
$408$	−0.692021	−0.0342602
$409$	−8.09485	−0.400265	−0.200132	−	0.979769i	$-0.564137\pi$
$409$	−8.09485	−0.400265	−0.200132	+	0.979769i	$0.564137\pi$
$410$	−2.08815	−0.103126
$411$	12.0489	0.594329
$412$	−8.23490	−0.405704
$413$	6.65279	0.327363
$414$	1.82908	0.0898946
$415$	12.6485	0.620890
$416$	0	0
$417$	16.6383	0.814782
$418$	2.81700	0.137784
$419$	−5.80087	−0.283391	−0.141696	−	0.989910i	$-0.545255\pi$
$419$	−5.80087	−0.283391	−0.141696	+	0.989910i	$0.545255\pi$
$420$	1.44504	0.0705108
$421$	21.9909	1.07177	0.535885	−	0.844291i	$-0.319978\pi$
$421$	21.9909	1.07177	0.535885	+	0.844291i	$0.319978\pi$
$422$	17.1511	0.834901
$423$	−3.71379	−0.180571
$424$	−1.57673	−0.0765727
$425$	−2.01507	−0.0977450
$426$	−0.219833	−0.0106509
$427$	−7.00000	−0.338754
$428$	10.6039	0.512558
$429$	0	0
$430$	−8.42327	−0.406206
$431$	3.34242	0.160999	0.0804994	−	0.996755i	$-0.474348\pi$
$431$	3.34242	0.160999	0.0804994	+	0.996755i	$0.474348\pi$
$432$	−1.00000	−0.0481125
$433$	14.0344	0.674452	0.337226	−	0.941424i	$-0.390511\pi$
$433$	14.0344	0.674452	0.337226	+	0.941424i	$0.390511\pi$
$434$	0.960771	0.0461185
$435$	9.97046	0.478047
$436$	8.48858	0.406529
$437$	−1.37734	−0.0658870
$438$	4.91185	0.234697
$439$	25.7735	1.23010	0.615050	−	0.788488i	$-0.289136\pi$
$439$	25.7735	1.23010	0.615050	+	0.788488i	$0.289136\pi$
$440$	5.40581	0.257712
$441$	1.00000	0.0476190
$442$	0	0
$443$	5.53558	0.263003	0.131502	−	0.991316i	$-0.458020\pi$
$443$	5.53558	0.263003	0.131502	+	0.991316i	$0.458020\pi$
$444$	3.04892	0.144695
$445$	21.0368	0.997242
$446$	−25.0911	−1.18810
$447$	−8.35690	−0.395268
$448$	1.00000	0.0472456
$449$	−21.0616	−0.993958	−0.496979	−	0.867763i	$-0.665557\pi$
$449$	−21.0616	−0.993958	−0.496979	+	0.867763i	$0.665557\pi$
$450$	−2.91185	−0.137266
$451$	−5.40581	−0.254550
$452$	−5.92692	−0.278779
$453$	7.63102	0.358537
$454$	5.84415	0.274280
$455$	0	0
$456$	0.753020	0.0352634
$457$	−39.7506	−1.85946	−0.929728	−	0.368247i	$-0.879958\pi$
$457$	−39.7506	−1.85946	−0.929728	+	0.368247i	$0.879958\pi$
$458$	−15.2252	−0.711427
$459$	−0.692021	−0.0323008
$460$	−2.64310	−0.123235
$461$	−18.7851	−0.874908	−0.437454	−	0.899241i	$-0.644120\pi$
$461$	−18.7851	−0.874908	−0.437454	+	0.899241i	$0.644120\pi$
$462$	3.74094	0.174044
$463$	−30.6504	−1.42444	−0.712222	−	0.701954i	$-0.752311\pi$
$463$	−30.6504	−1.42444	−0.712222	+	0.701954i	$0.752311\pi$
$464$	6.89977	0.320314
$465$	1.38835	0.0643834
$466$	−25.2000	−1.16737
$467$	1.57673	0.0729623	0.0364811	−	0.999334i	$-0.488385\pi$
$467$	1.57673	0.0729623	0.0364811	+	0.999334i	$0.488385\pi$
$468$	0	0
$469$	−9.11960	−0.421104
$470$	5.36658	0.247542
$471$	−12.2935	−0.566455
$472$	6.65279	0.306220
$473$	−21.8062	−1.00265
$474$	7.44504	0.341962
$475$	2.19269	0.100607
$476$	0.692021	0.0317188
$477$	−1.57673	−0.0721934
$478$	6.71379	0.307082
$479$	23.5907	1.07789	0.538944	−	0.842342i	$-0.318823\pi$
$479$	23.5907	1.07789	0.538944	+	0.842342i	$0.318823\pi$
$480$	1.44504	0.0659568
$481$	0	0
$482$	−5.21313	−0.237451
$483$	−1.82908	−0.0832262
$484$	2.99462	0.136119
$485$	8.75840	0.397698
$486$	−1.00000	−0.0453609
$487$	−4.42460	−0.200498	−0.100249	−	0.994962i	$-0.531964\pi$
$487$	−4.42460	−0.200498	−0.100249	+	0.994962i	$0.531964\pi$
$488$	−7.00000	−0.316875
$489$	15.0151	0.679005
$490$	−1.44504	−0.0652803
$491$	4.27652	0.192997	0.0964983	−	0.995333i	$-0.469236\pi$
$491$	4.27652	0.192997	0.0964983	+	0.995333i	$0.469236\pi$
$492$	−1.44504	−0.0651475
$493$	4.77479	0.215046
$494$	0	0
$495$	5.40581	0.242973
$496$	0.960771	0.0431399
$497$	0.219833	0.00986084
$498$	8.75302	0.392232
$499$	−5.35152	−0.239567	−0.119783	−	0.992800i	$-0.538220\pi$
$499$	−5.35152	−0.239567	−0.119783	+	0.992800i	$0.538220\pi$
$500$	11.4330	0.511298
$501$	−18.3153	−0.818266
$502$	9.22952	0.411934
$503$	12.1806	0.543106	0.271553	−	0.962423i	$-0.412463\pi$
$503$	12.1806	0.543106	0.271553	+	0.962423i	$0.412463\pi$
$504$	1.00000	0.0445435
$505$	4.06638	0.180951
$506$	−6.84249	−0.304186
$507$	0	0
$508$	9.67994	0.429478
$509$	16.4698	0.730011	0.365005	−	0.931005i	$-0.381067\pi$
$509$	16.4698	0.730011	0.365005	+	0.931005i	$0.381067\pi$
$510$	1.00000	0.0442807
$511$	−4.91185	−0.217288
$512$	1.00000	0.0441942
$513$	0.753020	0.0332467
$514$	2.51035	0.110727
$515$	11.8998	0.524367
$516$	−5.82908	−0.256611
$517$	13.8931	0.611016
$518$	−3.04892	−0.133962
$519$	16.7875	0.736888
$520$	0	0
$521$	1.89440	0.0829950	0.0414975	−	0.999139i	$-0.486787\pi$
$521$	1.89440	0.0829950	0.0414975	+	0.999139i	$0.486787\pi$
$522$	6.89977	0.301995
$523$	31.7211	1.38707	0.693533	−	0.720425i	$-0.256053\pi$
$523$	31.7211	1.38707	0.693533	+	0.720425i	$0.256053\pi$
$524$	2.25236	0.0983946
$525$	2.91185	0.127084
$526$	30.1511	1.31465
$527$	0.664874	0.0289624
$528$	3.74094	0.162803
$529$	−19.6544	−0.854541
$530$	2.27844	0.0989690
$531$	6.65279	0.288707
$532$	−0.753020	−0.0326476
$533$	0	0
$534$	14.5579	0.629984
$535$	−15.3230	−0.662473
$536$	−9.11960	−0.393907
$537$	−4.86725	−0.210037
$538$	−14.6063	−0.629721
$539$	−3.74094	−0.161134
$540$	1.44504	0.0621847
$541$	−7.16362	−0.307988	−0.153994	−	0.988072i	$-0.549214\pi$
$541$	−7.16362	−0.307988	−0.153994	+	0.988072i	$0.549214\pi$
$542$	−22.0978	−0.949183
$543$	13.4644	0.577814
$544$	0.692021	0.0296702
$545$	−12.2664	−0.525433
$546$	0	0
$547$	−24.7754	−1.05932	−0.529659	−	0.848210i	$-0.677680\pi$
$547$	−24.7754	−1.05932	−0.529659	+	0.848210i	$0.677680\pi$
$548$	−12.0489	−0.514704
$549$	−7.00000	−0.298753
$550$	10.8931	0.464482
$551$	−5.19567	−0.221343
$552$	−1.82908	−0.0778510
$553$	−7.44504	−0.316595
$554$	1.23729	0.0525675
$555$	−4.40581	−0.187016
$556$	−16.6383	−0.705622
$557$	29.2653	1.24001	0.620005	−	0.784598i	$-0.287131\pi$
$557$	29.2653	1.24001	0.620005	+	0.784598i	$0.287131\pi$
$558$	0.960771	0.0406727
$559$	0	0
$560$	−1.44504	−0.0610642
$561$	2.58881	0.109300
$562$	−8.51035	−0.358988
$563$	−7.93230	−0.334306	−0.167153	−	0.985931i	$-0.553457\pi$
$563$	−7.93230	−0.334306	−0.167153	+	0.985931i	$0.553457\pi$
$564$	3.71379	0.156379
$565$	8.56465	0.360317
$566$	−16.3002	−0.685149
$567$	1.00000	0.0419961
$568$	0.219833	0.00922397
$569$	40.7942	1.71018	0.855090	−	0.518479i	$-0.173502\pi$
$569$	40.7942	1.71018	0.855090	+	0.518479i	$0.173502\pi$
$570$	−1.08815	−0.0455774
$571$	31.7289	1.32781	0.663906	−	0.747816i	$-0.268898\pi$
$571$	31.7289	1.32781	0.663906	+	0.747816i	$0.268898\pi$
$572$	0	0
$573$	5.26444	0.219925
$574$	1.44504	0.0603149
$575$	−5.32603	−0.222111
$576$	1.00000	0.0416667
$577$	3.88471	0.161722	0.0808612	−	0.996725i	$-0.474233\pi$
$577$	3.88471	0.161722	0.0808612	+	0.996725i	$0.474233\pi$
$578$	−16.5211	−0.687187
$579$	22.8713	0.950499
$580$	−9.97046	−0.414001
$581$	−8.75302	−0.363136
$582$	6.06100	0.251236
$583$	5.89844	0.244289
$584$	−4.91185	−0.203254
$585$	0	0
$586$	−29.8853	−1.23455
$587$	−19.2446	−0.794309	−0.397155	−	0.917752i	$-0.630002\pi$
$587$	−19.2446	−0.794309	−0.397155	+	0.917752i	$0.630002\pi$
$588$	−1.00000	−0.0412393
$589$	−0.723480	−0.0298105
$590$	−9.61356	−0.395784
$591$	−18.8877	−0.776936
$592$	−3.04892	−0.125310
$593$	−8.85086	−0.363461	−0.181731	−	0.983348i	$-0.558170\pi$
$593$	−8.85086	−0.363461	−0.181731	+	0.983348i	$0.558170\pi$
$594$	3.74094	0.153493
$595$	−1.00000	−0.0409960
$596$	8.35690	0.342312
$597$	3.36898	0.137883
$598$	0	0
$599$	24.2174	0.989498	0.494749	−	0.869036i	$-0.335260\pi$
$599$	24.2174	0.989498	0.494749	+	0.869036i	$0.335260\pi$
$600$	2.91185	0.118876
$601$	−13.3575	−0.544863	−0.272432	−	0.962175i	$-0.587828\pi$
$601$	−13.3575	−0.544863	−0.272432	+	0.962175i	$0.587828\pi$
$602$	5.82908	0.237576
$603$	−9.11960	−0.371379
$604$	−7.63102	−0.310502
$605$	−4.32736	−0.175932
$606$	2.81402	0.114312
$607$	−0.568959	−0.0230933	−0.0115467	−	0.999933i	$-0.503675\pi$
$607$	−0.568959	−0.0230933	−0.0115467	+	0.999933i	$0.503675\pi$
$608$	−0.753020	−0.0305390
$609$	−6.89977	−0.279593
$610$	10.1153	0.409556
$611$	0	0
$612$	0.692021	0.0279733
$613$	21.1008	0.852254	0.426127	−	0.904663i	$-0.359878\pi$
$613$	21.1008	0.852254	0.426127	+	0.904663i	$0.359878\pi$
$614$	−24.4547	−0.986913
$615$	2.08815	0.0842022
$616$	−3.74094	−0.150727
$617$	−15.9705	−0.642947	−0.321473	−	0.946919i	$-0.604178\pi$
$617$	−15.9705	−0.642947	−0.321473	+	0.946919i	$0.604178\pi$
$618$	8.23490	0.331256
$619$	−37.9657	−1.52597	−0.762985	−	0.646417i	$-0.776267\pi$
$619$	−37.9657	−1.52597	−0.762985	+	0.646417i	$0.776267\pi$
$620$	−1.38835	−0.0557577
$621$	−1.82908	−0.0733986
$622$	−22.3177	−0.894857
$623$	−14.5579	−0.583252
$624$	0	0
$625$	−1.96184	−0.0784735
$626$	−2.47889	−0.0990765
$627$	−2.81700	−0.112500
$628$	12.2935	0.490564
$629$	−2.10992	−0.0841279
$630$	−1.44504	−0.0575718
$631$	−16.2054	−0.645125	−0.322563	−	0.946548i	$-0.604544\pi$
$631$	−16.2054	−0.645125	−0.322563	+	0.946548i	$0.604544\pi$
$632$	−7.44504	−0.296148
$633$	−17.1511	−0.681694
$634$	−5.65710	−0.224672
$635$	−13.9879	−0.555094
$636$	1.57673	0.0625213
$637$	0	0
$638$	−25.8116	−1.02189
$639$	0.219833	0.00869644
$640$	−1.44504	−0.0571203
$641$	−33.1879	−1.31084	−0.655422	−	0.755263i	$-0.727509\pi$
$641$	−33.1879	−1.31084	−0.655422	+	0.755263i	$0.727509\pi$
$642$	−10.6039	−0.418502
$643$	−9.71486	−0.383117	−0.191558	−	0.981481i	$-0.561354\pi$
$643$	−9.71486	−0.383117	−0.191558	+	0.981481i	$0.561354\pi$
$644$	1.82908	0.0720760
$645$	8.42327	0.331666
$646$	−0.521106	−0.0205027
$647$	−15.5066	−0.609629	−0.304814	−	0.952412i	$-0.598594\pi$
$647$	−15.5066	−0.609629	−0.304814	+	0.952412i	$0.598594\pi$
$648$	1.00000	0.0392837
$649$	−24.8877	−0.976927
$650$	0	0
$651$	−0.960771	−0.0376556
$652$	−15.0151	−0.588035
$653$	−4.37627	−0.171257	−0.0856284	−	0.996327i	$-0.527290\pi$
$653$	−4.37627	−0.171257	−0.0856284	+	0.996327i	$0.527290\pi$
$654$	−8.48858	−0.331930
$655$	−3.25475	−0.127174
$656$	1.44504	0.0564194
$657$	−4.91185	−0.191630
$658$	−3.71379	−0.144779
$659$	24.9638	0.972450	0.486225	−	0.873834i	$-0.338374\pi$
$659$	24.9638	0.972450	0.486225	+	0.873834i	$0.338374\pi$
$660$	−5.40581	−0.210421
$661$	−20.2252	−0.786669	−0.393335	−	0.919395i	$-0.628679\pi$
$661$	−20.2252	−0.786669	−0.393335	+	0.919395i	$0.628679\pi$
$662$	23.1879	0.901223
$663$	0	0
$664$	−8.75302	−0.339683
$665$	1.08815	0.0421965
$666$	−3.04892	−0.118143
$667$	12.6203	0.488659
$668$	18.3153	0.708639
$669$	25.0911	0.970079
$670$	13.1782	0.509119
$671$	26.1866	1.01092
$672$	−1.00000	−0.0385758
$673$	26.9119	1.03738	0.518688	−	0.854964i	$-0.326421\pi$
$673$	26.9119	1.03738	0.518688	+	0.854964i	$0.326421\pi$
$674$	10.4873	0.403954
$675$	2.91185	0.112077
$676$	0	0
$677$	36.9571	1.42037	0.710187	−	0.704013i	$-0.248610\pi$
$677$	36.9571	1.42037	0.710187	+	0.704013i	$0.248610\pi$
$678$	5.92692	0.227622
$679$	−6.06100	−0.232600
$680$	−1.00000	−0.0383482
$681$	−5.84415	−0.223948
$682$	−3.59419	−0.137629
$683$	4.53212	0.173417	0.0867084	−	0.996234i	$-0.472365\pi$
$683$	4.53212	0.173417	0.0867084	+	0.996234i	$0.472365\pi$
$684$	−0.753020	−0.0287925
$685$	17.4112	0.665247
$686$	1.00000	0.0381802
$687$	15.2252	0.580878
$688$	5.82908	0.222232
$689$	0	0
$690$	2.64310	0.100621
$691$	−24.7670	−0.942182	−0.471091	−	0.882085i	$-0.656140\pi$
$691$	−24.7670	−0.942182	−0.471091	+	0.882085i	$0.656140\pi$
$692$	−16.7875	−0.638164
$693$	−3.74094	−0.142107
$694$	−5.28514	−0.200621
$695$	24.0431	0.912005
$696$	−6.89977	−0.261535
$697$	1.00000	0.0378777
$698$	8.19029	0.310007
$699$	25.2000	0.953151
$700$	−2.91185	−0.110058
$701$	38.2965	1.44644	0.723219	−	0.690619i	$-0.242662\pi$
$701$	38.2965	1.44644	0.723219	+	0.690619i	$0.242662\pi$
$702$	0	0
$703$	2.29590	0.0865914
$704$	−3.74094	−0.140992
$705$	−5.36658	−0.202117
$706$	35.9758	1.35397
$707$	−2.81402	−0.105832
$708$	−6.65279	−0.250027
$709$	−44.3256	−1.66468	−0.832341	−	0.554265i	$-0.813000\pi$
$709$	−44.3256	−1.66468	−0.832341	+	0.554265i	$0.813000\pi$
$710$	−0.317667	−0.0119218
$711$	−7.44504	−0.279211
$712$	−14.5579	−0.545582
$713$	1.75733	0.0658126
$714$	−0.692021	−0.0258983
$715$	0	0
$716$	4.86725	0.181898
$717$	−6.71379	−0.250731
$718$	−28.7603	−1.07333
$719$	−24.5773	−0.916579	−0.458290	−	0.888803i	$-0.651538\pi$
$719$	−24.5773	−0.916579	−0.458290	+	0.888803i	$0.651538\pi$
$720$	−1.44504	−0.0538535
$721$	−8.23490	−0.306684
$722$	−18.4330	−0.686004
$723$	5.21313	0.193878
$724$	−13.4644	−0.500401
$725$	−20.0911	−0.746166
$726$	−2.99462	−0.111141
$727$	38.8280	1.44005	0.720026	−	0.693947i	$-0.244130\pi$
$727$	38.8280	1.44005	0.720026	+	0.693947i	$0.244130\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	7.09783	0.262703
$731$	4.03385	0.149197
$732$	7.00000	0.258727
$733$	21.6142	0.798337	0.399169	−	0.916878i	$-0.369299\pi$
$733$	21.6142	0.798337	0.399169	+	0.916878i	$0.369299\pi$
$734$	−3.61463	−0.133418
$735$	1.44504	0.0533012
$736$	1.82908	0.0674210
$737$	34.1159	1.25667
$738$	1.44504	0.0531927
$739$	−13.4644	−0.495297	−0.247648	−	0.968850i	$-0.579658\pi$
$739$	−13.4644	−0.495297	−0.247648	+	0.968850i	$0.579658\pi$
$740$	4.40581	0.161961
$741$	0	0
$742$	−1.57673	−0.0578835
$743$	3.90887	0.143403	0.0717013	−	0.997426i	$-0.477157\pi$
$743$	3.90887	0.143403	0.0717013	+	0.997426i	$0.477157\pi$
$744$	−0.960771	−0.0352236
$745$	−12.0761	−0.442433
$746$	17.9312	0.656509
$747$	−8.75302	−0.320256
$748$	−2.58881	−0.0946563
$749$	10.6039	0.387457
$750$	−11.4330	−0.417473
$751$	45.2922	1.65273	0.826367	−	0.563131i	$-0.190403\pi$
$751$	45.2922	1.65273	0.826367	+	0.563131i	$0.190403\pi$
$752$	−3.71379	−0.135428
$753$	−9.22952	−0.336342
$754$	0	0
$755$	11.0271	0.401319
$756$	−1.00000	−0.0363696
$757$	23.6872	0.860927	0.430464	−	0.902608i	$-0.358350\pi$
$757$	23.6872	0.860927	0.430464	+	0.902608i	$0.358350\pi$
$758$	22.2338	0.807569
$759$	6.84249	0.248367
$760$	1.08815	0.0394712
$761$	9.25667	0.335554	0.167777	−	0.985825i	$-0.446341\pi$
$761$	9.25667	0.335554	0.167777	+	0.985825i	$0.446341\pi$
$762$	−9.67994	−0.350667
$763$	8.48858	0.307307
$764$	−5.26444	−0.190461
$765$	−1.00000	−0.0361551
$766$	−25.4330	−0.918930
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−36.1390	−1.30321	−0.651603	−	0.758561i	$-0.725903\pi$
$769$	−36.1390	−1.30321	−0.651603	+	0.758561i	$0.725903\pi$
$770$	5.40581	0.194812
$771$	−2.51035	−0.0904081
$772$	−22.8713	−0.823156
$773$	14.6552	0.527110	0.263555	−	0.964644i	$-0.415105\pi$
$773$	14.6552	0.527110	0.263555	+	0.964644i	$0.415105\pi$
$774$	5.82908	0.209522
$775$	−2.79763	−0.100494
$776$	−6.06100	−0.217577
$777$	3.04892	0.109379
$778$	−14.9855	−0.537257
$779$	−1.08815	−0.0389869
$780$	0	0
$781$	−0.822380	−0.0294271
$782$	1.26577	0.0452637
$783$	−6.89977	−0.246578
$784$	1.00000	0.0357143
$785$	−17.7646	−0.634047
$786$	−2.25236	−0.0803389
$787$	11.7724	0.419641	0.209820	−	0.977740i	$-0.432712\pi$
$787$	11.7724	0.419641	0.209820	+	0.977740i	$0.432712\pi$
$788$	18.8877	0.672846
$789$	−30.1511	−1.07341
$790$	10.7584	0.382767
$791$	−5.92692	−0.210737
$792$	−3.74094	−0.132928
$793$	0	0
$794$	−11.8358	−0.420036
$795$	−2.27844	−0.0808079
$796$	−3.36898	−0.119410
$797$	40.0431	1.41840	0.709199	−	0.705008i	$-0.249057\pi$
$797$	40.0431	1.41840	0.709199	+	0.705008i	$0.249057\pi$
$798$	0.753020	0.0266566
$799$	−2.57002	−0.0909209
$800$	−2.91185	−0.102950
$801$	−14.5579	−0.514380
$802$	11.6256	0.410516
$803$	18.3749	0.648438
$804$	9.11960	0.321624
$805$	−2.64310	−0.0931572
$806$	0	0
$807$	14.6063	0.514165
$808$	−2.81402	−0.0989969
$809$	41.8243	1.47046	0.735232	−	0.677816i	$-0.237073\pi$
$809$	41.8243	1.47046	0.735232	+	0.677816i	$0.237073\pi$
$810$	−1.44504	−0.0507736
$811$	9.53212	0.334718	0.167359	−	0.985896i	$-0.446476\pi$
$811$	9.53212	0.334718	0.167359	+	0.985896i	$0.446476\pi$
$812$	6.89977	0.242135
$813$	22.0978	0.775005
$814$	11.4058	0.399774
$815$	21.6974	0.760027
$816$	−0.692021	−0.0242256
$817$	−4.38942	−0.153566
$818$	−8.09485	−0.283030
$819$	0	0
$820$	−2.08815	−0.0729212
$821$	0.509025	0.0177651	0.00888254	−	0.999961i	$-0.497173\pi$
$821$	0.509025	0.0177651	0.00888254	+	0.999961i	$0.497173\pi$
$822$	12.0489	0.420254
$823$	35.4233	1.23478	0.617389	−	0.786658i	$-0.288191\pi$
$823$	35.4233	1.23478	0.617389	+	0.786658i	$0.288191\pi$
$824$	−8.23490	−0.286876
$825$	−10.8931	−0.379248
$826$	6.65279	0.231480
$827$	−31.6547	−1.10074	−0.550371	−	0.834920i	$-0.685514\pi$
$827$	−31.6547	−1.10074	−0.550371	+	0.834920i	$0.685514\pi$
$828$	1.82908	0.0635651
$829$	−32.3026	−1.12192	−0.560958	−	0.827844i	$-0.689567\pi$
$829$	−32.3026	−1.12192	−0.560958	+	0.827844i	$0.689567\pi$
$830$	12.6485	0.439035
$831$	−1.23729	−0.0429211
$832$	0	0
$833$	0.692021	0.0239771
$834$	16.6383	0.576138
$835$	−26.4663	−0.915905
$836$	2.81700	0.0974281
$837$	−0.960771	−0.0332091
$838$	−5.80087	−0.200388
$839$	19.4896	0.672857	0.336429	−	0.941709i	$-0.390781\pi$
$839$	19.4896	0.672857	0.336429	+	0.941709i	$0.390781\pi$
$840$	1.44504	0.0498587
$841$	18.6069	0.641616
$842$	21.9909	0.757857
$843$	8.51035	0.293112
$844$	17.1511	0.590364
$845$	0	0
$846$	−3.71379	−0.127683
$847$	2.99462	0.102896
$848$	−1.57673	−0.0541451
$849$	16.3002	0.559422
$850$	−2.01507	−0.0691162
$851$	−5.57673	−0.191168
$852$	−0.219833	−0.00753134
$853$	−6.02177	−0.206181	−0.103091	−	0.994672i	$-0.532873\pi$
$853$	−6.02177	−0.206181	−0.103091	+	0.994672i	$0.532873\pi$
$854$	−7.00000	−0.239535
$855$	1.08815	0.0372138
$856$	10.6039	0.362433
$857$	−41.1172	−1.40454	−0.702269	−	0.711912i	$-0.747829\pi$
$857$	−41.1172	−1.40454	−0.702269	+	0.711912i	$0.747829\pi$
$858$	0	0
$859$	53.3666	1.82084	0.910422	−	0.413680i	$-0.135757\pi$
$859$	53.3666	1.82084	0.910422	+	0.413680i	$0.135757\pi$
$860$	−8.42327	−0.287231
$861$	−1.44504	−0.0492469
$862$	3.34242	0.113843
$863$	−27.5056	−0.936300	−0.468150	−	0.883649i	$-0.655079\pi$
$863$	−27.5056	−0.936300	−0.468150	+	0.883649i	$0.655079\pi$
$864$	−1.00000	−0.0340207
$865$	24.2586	0.824817
$866$	14.0344	0.476910
$867$	16.5211	0.561086
$868$	0.960771	0.0326107
$869$	27.8514	0.944796
$870$	9.97046	0.338030
$871$	0	0
$872$	8.48858	0.287460
$873$	−6.06100	−0.205134
$874$	−1.37734	−0.0465892
$875$	11.4330	0.386505
$876$	4.91185	0.165956
$877$	11.8092	0.398769	0.199385	−	0.979921i	$-0.436106\pi$
$877$	11.8092	0.398769	0.199385	+	0.979921i	$0.436106\pi$
$878$	25.7735	0.869812
$879$	29.8853	1.00801
$880$	5.40581	0.182230
$881$	−35.6848	−1.20225	−0.601126	−	0.799154i	$-0.705281\pi$
$881$	−35.6848	−1.20225	−0.601126	+	0.799154i	$0.705281\pi$
$882$	1.00000	0.0336718
$883$	18.0718	0.608163	0.304081	−	0.952646i	$-0.401650\pi$
$883$	18.0718	0.608163	0.304081	+	0.952646i	$0.401650\pi$
$884$	0	0
$885$	9.61356	0.323156
$886$	5.53558	0.185971
$887$	−36.9922	−1.24208	−0.621039	−	0.783780i	$-0.713289\pi$
$887$	−36.9922	−1.24208	−0.621039	+	0.783780i	$0.713289\pi$
$888$	3.04892	0.102315
$889$	9.67994	0.324655
$890$	21.0368	0.705156
$891$	−3.74094	−0.125326
$892$	−25.0911	−0.840113
$893$	2.79656	0.0935833
$894$	−8.35690	−0.279496
$895$	−7.03338	−0.235100
$896$	1.00000	0.0334077
$897$	0	0
$898$	−21.0616	−0.702834
$899$	6.62910	0.221093
$900$	−2.91185	−0.0970618
$901$	−1.09113	−0.0363508
$902$	−5.40581	−0.179994
$903$	−5.82908	−0.193980
$904$	−5.92692	−0.197126
$905$	19.4566	0.646761
$906$	7.63102	0.253524
$907$	24.4902	0.813185	0.406593	−	0.913610i	$-0.366717\pi$
$907$	24.4902	0.813185	0.406593	+	0.913610i	$0.366717\pi$
$908$	5.84415	0.193945
$909$	−2.81402	−0.0933351
$910$	0	0
$911$	29.8407	0.988666	0.494333	−	0.869273i	$-0.335412\pi$
$911$	29.8407	0.988666	0.494333	+	0.869273i	$0.335412\pi$
$912$	0.753020	0.0249350
$913$	32.7445	1.08369
$914$	−39.7506	−1.31483
$915$	−10.1153	−0.334401
$916$	−15.2252	−0.503055
$917$	2.25236	0.0743794
$918$	−0.692021	−0.0228401
$919$	44.0756	1.45392	0.726960	−	0.686680i	$-0.240933\pi$
$919$	44.0756	1.45392	0.726960	+	0.686680i	$0.240933\pi$
$920$	−2.64310	−0.0871406
$921$	24.4547	0.805811
$922$	−18.7851	−0.618653
$923$	0	0
$924$	3.74094	0.123068
$925$	8.87800	0.291907
$926$	−30.6504	−1.00723
$927$	−8.23490	−0.270470
$928$	6.89977	0.226496
$929$	2.83340	0.0929607	0.0464804	−	0.998919i	$-0.485200\pi$
$929$	2.83340	0.0929607	0.0464804	+	0.998919i	$0.485200\pi$
$930$	1.38835	0.0455259
$931$	−0.753020	−0.0246792
$932$	−25.2000	−0.825453
$933$	22.3177	0.730648
$934$	1.57673	0.0515921
$935$	3.74094	0.122342
$936$	0	0
$937$	−40.3129	−1.31696	−0.658482	−	0.752596i	$-0.728801\pi$
$937$	−40.3129	−1.31696	−0.658482	+	0.752596i	$0.728801\pi$
$938$	−9.11960	−0.297766
$939$	2.47889	0.0808956
$940$	5.36658	0.175039
$941$	54.3223	1.77086	0.885428	−	0.464776i	$-0.153865\pi$
$941$	54.3223	1.77086	0.885428	+	0.464776i	$0.153865\pi$
$942$	−12.2935	−0.400544
$943$	2.64310	0.0860713
$944$	6.65279	0.216530
$945$	1.44504	0.0470072
$946$	−21.8062	−0.708982
$947$	50.8980	1.65396	0.826981	−	0.562230i	$-0.190056\pi$
$947$	50.8980	1.65396	0.826981	+	0.562230i	$0.190056\pi$
$948$	7.44504	0.241804
$949$	0	0
$950$	2.19269	0.0711401
$951$	5.65710	0.183444
$952$	0.692021	0.0224285
$953$	−4.62266	−0.149743	−0.0748714	−	0.997193i	$-0.523855\pi$
$953$	−4.62266	−0.149743	−0.0748714	+	0.997193i	$0.523855\pi$
$954$	−1.57673	−0.0510485
$955$	7.60733	0.246168
$956$	6.71379	0.217140
$957$	25.8116	0.834371
$958$	23.5907	0.762182
$959$	−12.0489	−0.389080
$960$	1.44504	0.0466385
$961$	−30.0769	−0.970223
$962$	0	0
$963$	10.6039	0.341705
$964$	−5.21313	−0.167904
$965$	33.0500	1.06392
$966$	−1.82908	−0.0588498
$967$	52.9235	1.70190	0.850952	−	0.525244i	$-0.176026\pi$
$967$	52.9235	1.70190	0.850952	+	0.525244i	$0.176026\pi$
$968$	2.99462	0.0962508
$969$	0.521106	0.0167403
$970$	8.75840	0.281215
$971$	2.78256	0.0892966	0.0446483	−	0.999003i	$-0.485783\pi$
$971$	2.78256	0.0892966	0.0446483	+	0.999003i	$0.485783\pi$
$972$	−1.00000	−0.0320750
$973$	−16.6383	−0.533400
$974$	−4.42460	−0.141773
$975$	0	0
$976$	−7.00000	−0.224065
$977$	−12.0922	−0.386864	−0.193432	−	0.981114i	$-0.561962\pi$
$977$	−12.0922	−0.386864	−0.193432	+	0.981114i	$0.561962\pi$
$978$	15.0151	0.480129
$979$	54.4604	1.74056
$980$	−1.44504	−0.0461602
$981$	8.48858	0.271020
$982$	4.27652	0.136469
$983$	−22.9474	−0.731907	−0.365954	−	0.930633i	$-0.619257\pi$
$983$	−22.9474	−0.731907	−0.365954	+	0.930633i	$0.619257\pi$
$984$	−1.44504	−0.0460663
$985$	−27.2935	−0.869643
$986$	4.77479	0.152060
$987$	3.71379	0.118211
$988$	0	0
$989$	10.6619	0.339028
$990$	5.40581	0.171808
$991$	−1.44876	−0.0460215	−0.0230107	−	0.999735i	$-0.507325\pi$
$991$	−1.44876	−0.0460215	−0.0230107	+	0.999735i	$0.507325\pi$
$992$	0.960771	0.0305045
$993$	−23.1879	−0.735846
$994$	0.219833	0.00697266
$995$	4.86831	0.154336
$996$	8.75302	0.277350
$997$	39.6491	1.25570	0.627849	−	0.778335i	$-0.283935\pi$
$997$	39.6491	1.25570	0.627849	+	0.778335i	$0.283935\pi$
$998$	−5.35152	−0.169399
$999$	3.04892	0.0964635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.ch.1.2	yes	3
13.12	even	2		7098.2.a.ce.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7098.2.a.ce.1.2	✓	3	13.12	even	2
7098.2.a.ch.1.2	yes	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.44504 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.44504 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.44504 q^{10} -3.74094 q^{11} -1.00000 q^{12} +1.00000 q^{14} +1.44504 q^{15} +1.00000 q^{16} +0.692021 q^{17} +1.00000 q^{18} -0.753020 q^{19} -1.44504 q^{20} -1.00000 q^{21} -3.74094 q^{22} +1.82908 q^{23} -1.00000 q^{24} -2.91185 q^{25} -1.00000 q^{27} +1.00000 q^{28} +6.89977 q^{29} +1.44504 q^{30} +0.960771 q^{31} +1.00000 q^{32} +3.74094 q^{33} +0.692021 q^{34} -1.44504 q^{35} +1.00000 q^{36} -3.04892 q^{37} -0.753020 q^{38} -1.44504 q^{40} +1.44504 q^{41} -1.00000 q^{42} +5.82908 q^{43} -3.74094 q^{44} -1.44504 q^{45} +1.82908 q^{46} -3.71379 q^{47} -1.00000 q^{48} +1.00000 q^{49} -2.91185 q^{50} -0.692021 q^{51} -1.57673 q^{53} -1.00000 q^{54} +5.40581 q^{55} +1.00000 q^{56} +0.753020 q^{57} +6.89977 q^{58} +6.65279 q^{59} +1.44504 q^{60} -7.00000 q^{61} +0.960771 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.74094 q^{66} -9.11960 q^{67} +0.692021 q^{68} -1.82908 q^{69} -1.44504 q^{70} +0.219833 q^{71} +1.00000 q^{72} -4.91185 q^{73} -3.04892 q^{74} +2.91185 q^{75} -0.753020 q^{76} -3.74094 q^{77} -7.44504 q^{79} -1.44504 q^{80} +1.00000 q^{81} +1.44504 q^{82} -8.75302 q^{83} -1.00000 q^{84} -1.00000 q^{85} +5.82908 q^{86} -6.89977 q^{87} -3.74094 q^{88} -14.5579 q^{89} -1.44504 q^{90} +1.82908 q^{92} -0.960771 q^{93} -3.71379 q^{94} +1.08815 q^{95} -1.00000 q^{96} -6.06100 q^{97} +1.00000 q^{98} -3.74094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 4 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 4 * q^5 - 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 4 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 4 q^{10} + 3 q^{11} - 3 q^{12} + 3 q^{14} + 4 q^{15} + 3 q^{16} - 3 q^{17} + 3 q^{18} - 7 q^{19} - 4 q^{20} - 3 q^{21} + 3 q^{22} - 5 q^{23} - 3 q^{24} - 5 q^{25} - 3 q^{27} + 3 q^{28} - 2 q^{29} + 4 q^{30} - 10 q^{31} + 3 q^{32} - 3 q^{33} - 3 q^{34} - 4 q^{35} + 3 q^{36} - 7 q^{38} - 4 q^{40} + 4 q^{41} - 3 q^{42} + 7 q^{43} + 3 q^{44} - 4 q^{45} - 5 q^{46} - 3 q^{47} - 3 q^{48} + 3 q^{49} - 5 q^{50} + 3 q^{51} - 2 q^{53} - 3 q^{54} + 3 q^{55} + 3 q^{56} + 7 q^{57} - 2 q^{58} + 2 q^{59} + 4 q^{60} - 21 q^{61} - 10 q^{62} + 3 q^{63} + 3 q^{64} - 3 q^{66} - 6 q^{67} - 3 q^{68} + 5 q^{69} - 4 q^{70} + 2 q^{71} + 3 q^{72} - 11 q^{73} + 5 q^{75} - 7 q^{76} + 3 q^{77} - 22 q^{79} - 4 q^{80} + 3 q^{81} + 4 q^{82} - 31 q^{83} - 3 q^{84} - 3 q^{85} + 7 q^{86} + 2 q^{87} + 3 q^{88} - 4 q^{90} - 5 q^{92} + 10 q^{93} - 3 q^{94} + 7 q^{95} - 3 q^{96} - 28 q^{97} + 3 q^{98} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 4 * q^5 - 3 * q^6 + 3 * q^7 + 3 * q^8 + 3 * q^9 - 4 * q^10 + 3 * q^11 - 3 * q^12 + 3 * q^14 + 4 * q^15 + 3 * q^16 - 3 * q^17 + 3 * q^18 - 7 * q^19 - 4 * q^20 - 3 * q^21 + 3 * q^22 - 5 * q^23 - 3 * q^24 - 5 * q^25 - 3 * q^27 + 3 * q^28 - 2 * q^29 + 4 * q^30 - 10 * q^31 + 3 * q^32 - 3 * q^33 - 3 * q^34 - 4 * q^35 + 3 * q^36 - 7 * q^38 - 4 * q^40 + 4 * q^41 - 3 * q^42 + 7 * q^43 + 3 * q^44 - 4 * q^45 - 5 * q^46 - 3 * q^47 - 3 * q^48 + 3 * q^49 - 5 * q^50 + 3 * q^51 - 2 * q^53 - 3 * q^54 + 3 * q^55 + 3 * q^56 + 7 * q^57 - 2 * q^58 + 2 * q^59 + 4 * q^60 - 21 * q^61 - 10 * q^62 + 3 * q^63 + 3 * q^64 - 3 * q^66 - 6 * q^67 - 3 * q^68 + 5 * q^69 - 4 * q^70 + 2 * q^71 + 3 * q^72 - 11 * q^73 + 5 * q^75 - 7 * q^76 + 3 * q^77 - 22 * q^79 - 4 * q^80 + 3 * q^81 + 4 * q^82 - 31 * q^83 - 3 * q^84 - 3 * q^85 + 7 * q^86 + 2 * q^87 + 3 * q^88 - 4 * q^90 - 5 * q^92 + 10 * q^93 - 3 * q^94 + 7 * q^95 - 3 * q^96 - 28 * q^97 + 3 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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