LMFDB - Embedded newform 1815.4.a.bl.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1815.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:	$107.088466660$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 72 x^{10} + 68 x^{9} + 1852 x^{8} - 1711 x^{7} - 20848 x^{6} + 20766 x^{5} + \cdots + 56080 $ x^12 - x^11 - 72x^10 + 68x^9 + 1852x^8 - 1711x^7 - 20848x^6 + 20766x^5 + 98931x^4 - 106938x^3 - 137664x^2 + 117840x + 56080
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 11^{3} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$5.34844$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.34844 q^{2} -3.00000 q^{3} +20.6059 q^{4} +5.00000 q^{5} -16.0453 q^{6} -17.2500 q^{7} +67.4218 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+5.34844 q^{2} -3.00000 q^{3} +20.6059 q^{4} +5.00000 q^{5} -16.0453 q^{6} -17.2500 q^{7} +67.4218 q^{8} +9.00000 q^{9} +26.7422 q^{10} -61.8176 q^{12} +8.03943 q^{13} -92.2606 q^{14} -15.0000 q^{15} +195.755 q^{16} -20.9711 q^{17} +48.1360 q^{18} +150.855 q^{19} +103.029 q^{20} +51.7499 q^{21} +150.357 q^{23} -202.265 q^{24} +25.0000 q^{25} +42.9984 q^{26} -27.0000 q^{27} -355.451 q^{28} -215.064 q^{29} -80.2267 q^{30} -11.0147 q^{31} +507.609 q^{32} -112.163 q^{34} -86.2499 q^{35} +185.453 q^{36} -131.308 q^{37} +806.842 q^{38} -24.1183 q^{39} +337.109 q^{40} +61.1300 q^{41} +276.782 q^{42} +337.107 q^{43} +45.0000 q^{45} +804.178 q^{46} -5.27318 q^{47} -587.264 q^{48} -45.4382 q^{49} +133.711 q^{50} +62.9133 q^{51} +165.659 q^{52} +747.565 q^{53} -144.408 q^{54} -1163.02 q^{56} -452.566 q^{57} -1150.26 q^{58} -492.181 q^{59} -309.088 q^{60} +766.004 q^{61} -58.9113 q^{62} -155.250 q^{63} +1148.88 q^{64} +40.1971 q^{65} +807.060 q^{67} -432.128 q^{68} -451.072 q^{69} -461.303 q^{70} -516.598 q^{71} +606.796 q^{72} +769.005 q^{73} -702.296 q^{74} -75.0000 q^{75} +3108.51 q^{76} -128.995 q^{78} +108.129 q^{79} +978.773 q^{80} +81.0000 q^{81} +326.951 q^{82} +280.093 q^{83} +1066.35 q^{84} -104.856 q^{85} +1803.00 q^{86} +645.191 q^{87} +379.836 q^{89} +240.680 q^{90} -138.680 q^{91} +3098.24 q^{92} +33.0440 q^{93} -28.2033 q^{94} +754.277 q^{95} -1522.83 q^{96} -1276.51 q^{97} -243.024 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} - 36 q^{3} + 49 q^{4} + 60 q^{5} - 3 q^{6} - 5 q^{7} - 3 q^{8} + 108 q^{9}+O(q^{10}) $ 12 * q + q^2 - 36 * q^3 + 49 * q^4 + 60 * q^5 - 3 * q^6 - 5 * q^7 - 3 * q^8 + 108 * q^9	\( 12 q + q^{2} - 36 q^{3} + 49 q^{4} + 60 q^{5} - 3 q^{6} - 5 q^{7} - 3 q^{8} + 108 q^{9} + 5 q^{10} - 147 q^{12} + 34 q^{13} - 134 q^{14} - 180 q^{15} + 265 q^{16} + 169 q^{17} + 9 q^{18} + 203 q^{19} + 245 q^{20} + 15 q^{21} + 116 q^{23} + 9 q^{24} + 300 q^{25} + 216 q^{26} - 324 q^{27} + 80 q^{28} + 409 q^{29} - 15 q^{30} - 645 q^{31} + 532 q^{32} - 1442 q^{34} - 25 q^{35} + 441 q^{36} + 40 q^{37} + 404 q^{38} - 102 q^{39} - 15 q^{40} + 1071 q^{41} + 402 q^{42} - 101 q^{43} + 540 q^{45} + 1539 q^{46} - 572 q^{47} - 795 q^{48} + 1563 q^{49} + 25 q^{50} - 507 q^{51} + 1081 q^{52} + 611 q^{53} - 27 q^{54} - 3221 q^{56} - 609 q^{57} - 354 q^{58} + 958 q^{59} - 735 q^{60} + 2620 q^{61} + 1049 q^{62} - 45 q^{63} + 2931 q^{64} + 170 q^{65} - 1027 q^{67} + 1896 q^{68} - 348 q^{69} - 670 q^{70} + 1020 q^{71} - 27 q^{72} + 2244 q^{73} - 283 q^{74} - 900 q^{75} + 4958 q^{76} - 648 q^{78} + 2199 q^{79} + 1325 q^{80} + 972 q^{81} + 362 q^{82} - 602 q^{83} - 240 q^{84} + 845 q^{85} - 1589 q^{86} - 1227 q^{87} - 2413 q^{89} + 45 q^{90} + 302 q^{91} + 3030 q^{92} + 1935 q^{93} + 2885 q^{94} + 1015 q^{95} - 1596 q^{96} + 886 q^{97} - 1236 q^{98}+O(q^{100}) \) 12 * q + q^2 - 36 * q^3 + 49 * q^4 + 60 * q^5 - 3 * q^6 - 5 * q^7 - 3 * q^8 + 108 * q^9 + 5 * q^10 - 147 * q^12 + 34 * q^13 - 134 * q^14 - 180 * q^15 + 265 * q^16 + 169 * q^17 + 9 * q^18 + 203 * q^19 + 245 * q^20 + 15 * q^21 + 116 * q^23 + 9 * q^24 + 300 * q^25 + 216 * q^26 - 324 * q^27 + 80 * q^28 + 409 * q^29 - 15 * q^30 - 645 * q^31 + 532 * q^32 - 1442 * q^34 - 25 * q^35 + 441 * q^36 + 40 * q^37 + 404 * q^38 - 102 * q^39 - 15 * q^40 + 1071 * q^41 + 402 * q^42 - 101 * q^43 + 540 * q^45 + 1539 * q^46 - 572 * q^47 - 795 * q^48 + 1563 * q^49 + 25 * q^50 - 507 * q^51 + 1081 * q^52 + 611 * q^53 - 27 * q^54 - 3221 * q^56 - 609 * q^57 - 354 * q^58 + 958 * q^59 - 735 * q^60 + 2620 * q^61 + 1049 * q^62 - 45 * q^63 + 2931 * q^64 + 170 * q^65 - 1027 * q^67 + 1896 * q^68 - 348 * q^69 - 670 * q^70 + 1020 * q^71 - 27 * q^72 + 2244 * q^73 - 283 * q^74 - 900 * q^75 + 4958 * q^76 - 648 * q^78 + 2199 * q^79 + 1325 * q^80 + 972 * q^81 + 362 * q^82 - 602 * q^83 - 240 * q^84 + 845 * q^85 - 1589 * q^86 - 1227 * q^87 - 2413 * q^89 + 45 * q^90 + 302 * q^91 + 3030 * q^92 + 1935 * q^93 + 2885 * q^94 + 1015 * q^95 - 1596 * q^96 + 886 * q^97 - 1236 * q^98

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 $

$2$	5.34844	1.89096	0.945480	−	0.325679i	$-0.105593\pi$
$2$	5.34844	1.89096	0.945480	+	0.325679i	$0.105593\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	20.6059	2.57573
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−16.0453	−1.09175
$7$	−17.2500	−0.931411	−0.465706	−	0.884940i	$-0.654199\pi$
$7$	−17.2500	−0.931411	−0.465706	+	0.884940i	$0.654199\pi$
$8$	67.4218	2.97965
$9$	9.00000	0.333333
$10$	26.7422	0.845663
$11$	0	0
$11$	0	0
$12$	−61.8176	−1.48710
$13$	8.03943	0.171518	0.0857591	−	0.996316i	$-0.472668\pi$
$13$	8.03943	0.171518	0.0857591	+	0.996316i	$0.472668\pi$
$14$	−92.2606	−1.76126
$15$	−15.0000	−0.258199
$16$	195.755	3.05867
$17$	−20.9711	−0.299191	−0.149595	−	0.988747i	$-0.547797\pi$
$17$	−20.9711	−0.299191	−0.149595	+	0.988747i	$0.547797\pi$
$18$	48.1360	0.630320
$19$	150.855	1.82151	0.910753	−	0.412951i	$-0.135502\pi$
$19$	150.855	1.82151	0.910753	+	0.412951i	$0.135502\pi$
$20$	103.029	1.15190
$21$	51.7499	0.537751
$22$	0	0
$23$	150.357	1.36312	0.681558	−	0.731764i	$-0.261303\pi$
$23$	150.357	1.36312	0.681558	+	0.731764i	$0.261303\pi$
$24$	−202.265	−1.72030
$25$	25.0000	0.200000
$26$	42.9984	0.324334
$27$	−27.0000	−0.192450
$28$	−355.451	−2.39907
$29$	−215.064	−1.37711	−0.688557	−	0.725182i	$-0.741755\pi$
$29$	−215.064	−1.37711	−0.688557	+	0.725182i	$0.741755\pi$
$30$	−80.2267	−0.488244
$31$	−11.0147	−0.0638159	−0.0319079	−	0.999491i	$-0.510158\pi$
$31$	−11.0147	−0.0638159	−0.0319079	+	0.999491i	$0.510158\pi$
$32$	507.609	2.80417
$33$	0	0
$34$	−112.163	−0.565758
$35$	−86.2499	−0.416540
$36$	185.453	0.858578
$37$	−131.308	−0.583432	−0.291716	−	0.956505i	$-0.594226\pi$
$37$	−131.308	−0.583432	−0.291716	+	0.956505i	$0.594226\pi$
$38$	806.842	3.44440
$39$	−24.1183	−0.0990260
$40$	337.109	1.33254
$41$	61.1300	0.232851	0.116426	−	0.993199i	$-0.462856\pi$
$41$	61.1300	0.232851	0.116426	+	0.993199i	$0.462856\pi$
$42$	276.782	1.01687
$43$	337.107	1.19554	0.597770	−	0.801667i	$-0.296053\pi$
$43$	337.107	1.19554	0.597770	+	0.801667i	$0.296053\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	804.178	2.57760
$47$	−5.27318	−0.0163654	−0.00818268	−	0.999967i	$-0.502605\pi$
$47$	−5.27318	−0.0163654	−0.00818268	+	0.999967i	$0.502605\pi$
$48$	−587.264	−1.76592
$49$	−45.4382	−0.132473
$50$	133.711	0.378192
$51$	62.9133	0.172738
$52$	165.659	0.441785
$53$	747.565	1.93747	0.968736	−	0.248095i	$-0.0798047\pi$
$53$	747.565	1.93747	0.968736	+	0.248095i	$0.0798047\pi$
$54$	−144.408	−0.363916
$55$	0	0
$56$	−1163.02	−2.77528
$57$	−452.566	−1.05165
$58$	−1150.26	−2.60407
$59$	−492.181	−1.08604	−0.543022	−	0.839719i	$-0.682720\pi$
$59$	−492.181	−1.08604	−0.543022	+	0.839719i	$0.682720\pi$
$60$	−309.088	−0.665051
$61$	766.004	1.60782	0.803908	−	0.594754i	$-0.202750\pi$
$61$	766.004	1.60782	0.803908	+	0.594754i	$0.202750\pi$
$62$	−58.9113	−0.120673
$63$	−155.250	−0.310470
$64$	1148.88	2.24391
$65$	40.1971	0.0767052
$66$	0	0
$67$	807.060	1.47161	0.735806	−	0.677192i	$-0.236803\pi$
$67$	807.060	1.47161	0.735806	+	0.677192i	$0.236803\pi$
$68$	−432.128	−0.770635
$69$	−451.072	−0.786995
$70$	−461.303	−0.787660
$71$	−516.598	−0.863506	−0.431753	−	0.901992i	$-0.642105\pi$
$71$	−516.598	−0.863506	−0.431753	+	0.901992i	$0.642105\pi$
$72$	606.796	0.993216
$73$	769.005	1.23295	0.616474	−	0.787375i	$-0.288561\pi$
$73$	769.005	1.23295	0.616474	+	0.787375i	$0.288561\pi$
$74$	−702.296	−1.10325
$75$	−75.0000	−0.115470
$76$	3108.51	4.69171
$77$	0	0
$78$	−128.995	−0.187254
$79$	108.129	0.153993	0.0769966	−	0.997031i	$-0.475467\pi$
$79$	108.129	0.153993	0.0769966	+	0.997031i	$0.475467\pi$
$80$	978.773	1.36788
$81$	81.0000	0.111111
$82$	326.951	0.440313
$83$	280.093	0.370412	0.185206	−	0.982700i	$-0.440705\pi$
$83$	280.093	0.370412	0.185206	+	0.982700i	$0.440705\pi$
$84$	1066.35	1.38510
$85$	−104.856	−0.133802
$86$	1803.00	2.26072
$87$	645.191	0.795077
$88$	0	0
$89$	379.836	0.452388	0.226194	−	0.974082i	$-0.427372\pi$
$89$	379.836	0.452388	0.226194	+	0.974082i	$0.427372\pi$
$90$	240.680	0.281888
$91$	−138.680	−0.159754
$92$	3098.24	3.51102
$93$	33.0440	0.0368441
$94$	−28.2033	−0.0309463
$95$	754.277	0.814602
$96$	−1522.83	−1.61899
$97$	−1276.51	−1.33619	−0.668095	−	0.744076i	$-0.732890\pi$
$97$	−1276.51	−1.33619	−0.668095	+	0.744076i	$0.732890\pi$
$98$	−243.024	−0.250501
$99$	0	0
$100$	515.147	0.515147
$101$	1124.65	1.10799	0.553994	−	0.832521i	$-0.313103\pi$
$101$	1124.65	1.10799	0.553994	+	0.832521i	$0.313103\pi$
$102$	336.489	0.326640
$103$	1421.42	1.35978	0.679889	−	0.733315i	$-0.262028\pi$
$103$	1421.42	1.35978	0.679889	+	0.733315i	$0.262028\pi$
$104$	542.032	0.511064
$105$	258.750	0.240489
$106$	3998.31	3.66368
$107$	185.204	0.167331	0.0836653	−	0.996494i	$-0.473337\pi$
$107$	185.204	0.167331	0.0836653	+	0.996494i	$0.473337\pi$
$108$	−556.358	−0.495700
$109$	1748.99	1.53690	0.768451	−	0.639908i	$-0.221028\pi$
$109$	1748.99	1.53690	0.768451	+	0.639908i	$0.221028\pi$
$110$	0	0
$111$	393.925	0.336844
$112$	−3376.76	−2.84888
$113$	−2125.94	−1.76984	−0.884920	−	0.465744i	$-0.845787\pi$
$113$	−2125.94	−1.76984	−0.884920	+	0.465744i	$0.845787\pi$
$114$	−2420.53	−1.98862
$115$	751.786	0.609604
$116$	−4431.57	−3.54708
$117$	72.3548	0.0571727
$118$	−2632.40	−2.05367
$119$	361.751	0.278670
$120$	−1011.33	−0.769342
$121$	0	0
$122$	4096.93	3.04032
$123$	−183.390	−0.134437
$124$	−226.967	−0.164373
$125$	125.000	0.0894427
$126$	−830.345	−0.587087
$127$	−1487.63	−1.03942	−0.519709	−	0.854344i	$-0.673959\pi$
$127$	−1487.63	−1.03942	−0.519709	+	0.854344i	$0.673959\pi$
$128$	2083.86	1.43897
$129$	−1011.32	−0.690246
$130$	214.992	0.145047
$131$	−400.716	−0.267258	−0.133629	−	0.991031i	$-0.542663\pi$
$131$	−400.716	−0.267258	−0.133629	+	0.991031i	$0.542663\pi$
$132$	0	0
$133$	−2602.25	−1.69657
$134$	4316.51	2.78276
$135$	−135.000	−0.0860663
$136$	−1413.91	−0.891483
$137$	−395.234	−0.246475	−0.123238	−	0.992377i	$-0.539328\pi$
$137$	−395.234	−0.246475	−0.123238	+	0.992377i	$0.539328\pi$
$138$	−2412.53	−1.48818
$139$	−2103.54	−1.28359	−0.641797	−	0.766875i	$-0.721811\pi$
$139$	−2103.54	−1.28359	−0.641797	+	0.766875i	$0.721811\pi$
$140$	−1777.25	−1.07290
$141$	15.8195	0.00944855
$142$	−2763.00	−1.63286
$143$	0	0
$144$	1761.79	1.01956
$145$	−1075.32	−0.615864
$146$	4112.98	2.33146
$147$	136.315	0.0764833
$148$	−2705.72	−1.50276
$149$	1238.80	0.681118	0.340559	−	0.940223i	$-0.389384\pi$
$149$	1238.80	0.681118	0.340559	+	0.940223i	$0.389384\pi$
$150$	−401.133	−0.218349
$151$	−1277.07	−0.688256	−0.344128	−	0.938923i	$-0.611825\pi$
$151$	−1277.07	−0.688256	−0.344128	+	0.938923i	$0.611825\pi$
$152$	10170.9	5.42745
$153$	−188.740	−0.0997302
$154$	0	0
$155$	−55.0733	−0.0285393
$156$	−496.978	−0.255065
$157$	1900.24	0.965959	0.482979	−	0.875632i	$-0.339555\pi$
$157$	1900.24	0.965959	0.482979	+	0.875632i	$0.339555\pi$
$158$	578.322	0.291195
$159$	−2242.70	−1.11860
$160$	2538.04	1.25406
$161$	−2593.66	−1.26962
$162$	433.224	0.210107
$163$	−1649.83	−0.792790	−0.396395	−	0.918080i	$-0.629739\pi$
$163$	−1649.83	−0.792790	−0.396395	+	0.918080i	$0.629739\pi$
$164$	1259.64	0.599763
$165$	0	0
$166$	1498.06	0.700434
$167$	440.155	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$167$	440.155	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$168$	3489.07	1.60231
$169$	−2132.37	−0.970582
$170$	−560.814	−0.253015
$171$	1357.70	0.607169
$172$	6946.37	3.07939
$173$	−1588.92	−0.698285	−0.349143	−	0.937070i	$-0.613527\pi$
$173$	−1588.92	−0.698285	−0.349143	+	0.937070i	$0.613527\pi$
$174$	3450.77	1.50346
$175$	−431.249	−0.186282
$176$	0	0
$177$	1476.54	0.627027
$178$	2031.53	0.855447
$179$	3316.59	1.38488	0.692441	−	0.721474i	$-0.256535\pi$
$179$	3316.59	1.38488	0.692441	+	0.721474i	$0.256535\pi$
$180$	927.264	0.383968
$181$	2248.12	0.923211	0.461606	−	0.887085i	$-0.347274\pi$
$181$	2248.12	0.923211	0.461606	+	0.887085i	$0.347274\pi$
$182$	−741.722	−0.302088
$183$	−2298.01	−0.928273
$184$	10137.4	4.06161
$185$	−656.542	−0.260919
$186$	176.734	0.0696708
$187$	0	0
$188$	−108.658	−0.0421528
$189$	465.749	0.179250
$190$	4034.21	1.54038
$191$	−1515.14	−0.573987	−0.286993	−	0.957933i	$-0.592656\pi$
$191$	−1515.14	−0.573987	−0.286993	+	0.957933i	$0.592656\pi$
$192$	−3446.64	−1.29552
$193$	−3847.19	−1.43485	−0.717426	−	0.696635i	$-0.754680\pi$
$193$	−3847.19	−1.43485	−0.717426	+	0.696635i	$0.754680\pi$
$194$	−6827.37	−2.52668
$195$	−120.591	−0.0442858
$196$	−936.294	−0.341215
$197$	−4678.76	−1.69212	−0.846060	−	0.533088i	$-0.821031\pi$
$197$	−4678.76	−1.69212	−0.846060	+	0.533088i	$0.821031\pi$
$198$	0	0
$199$	−244.631	−0.0871428	−0.0435714	−	0.999050i	$-0.513874\pi$
$199$	−244.631	−0.0871428	−0.0435714	+	0.999050i	$0.513874\pi$
$200$	1685.54	0.595930
$201$	−2421.18	−0.849636
$202$	6015.13	2.09516
$203$	3709.84	1.28266
$204$	1296.38	0.444926
$205$	305.650	0.104134
$206$	7602.41	2.57129
$207$	1353.22	0.454372
$208$	1573.76	0.524617
$209$	0	0
$210$	1383.91	0.454756
$211$	−886.564	−0.289259	−0.144629	−	0.989486i	$-0.546199\pi$
$211$	−886.564	−0.289259	−0.144629	+	0.989486i	$0.546199\pi$
$212$	15404.2	4.99041
$213$	1549.79	0.498545
$214$	990.555	0.316416
$215$	1685.53	0.534662
$216$	−1820.39	−0.573434
$217$	190.003	0.0594388
$218$	9354.35	2.90622
$219$	−2307.01	−0.711843
$220$	0	0
$221$	−168.596	−0.0513166
$222$	2106.89	0.636960
$223$	−152.653	−0.0458403	−0.0229202	−	0.999737i	$-0.507296\pi$
$223$	−152.653	−0.0458403	−0.0229202	+	0.999737i	$0.507296\pi$
$224$	−8756.24	−2.61184
$225$	225.000	0.0666667
$226$	−11370.5	−3.34670
$227$	4203.63	1.22910	0.614548	−	0.788879i	$-0.289338\pi$
$227$	4203.63	1.22910	0.614548	+	0.788879i	$0.289338\pi$
$228$	−9325.52	−2.70876
$229$	500.824	0.144521	0.0722606	−	0.997386i	$-0.476979\pi$
$229$	500.824	0.144521	0.0722606	+	0.997386i	$0.476979\pi$
$230$	4020.89	1.15274
$231$	0	0
$232$	−14500.0	−4.10332
$233$	−385.743	−0.108459	−0.0542294	−	0.998529i	$-0.517270\pi$
$233$	−385.743	−0.108459	−0.0542294	+	0.998529i	$0.517270\pi$
$234$	386.986	0.108111
$235$	−26.3659	−0.00731881
$236$	−10141.8	−2.79736
$237$	−324.387	−0.0889080
$238$	1934.81	0.526953
$239$	4298.42	1.16335	0.581677	−	0.813420i	$-0.302397\pi$
$239$	4298.42	1.16335	0.581677	+	0.813420i	$0.302397\pi$
$240$	−2936.32	−0.789744
$241$	6358.33	1.69949	0.849743	−	0.527197i	$-0.176757\pi$
$241$	6358.33	1.69949	0.849743	+	0.527197i	$0.176757\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	15784.2	4.14130
$245$	−227.191	−0.0592437
$246$	−980.852	−0.254215
$247$	1212.79	0.312421
$248$	−742.628	−0.190149
$249$	−840.278	−0.213857
$250$	668.556	0.169133
$251$	−4464.72	−1.12275	−0.561375	−	0.827561i	$-0.689728\pi$
$251$	−4464.72	−1.12275	−0.561375	+	0.827561i	$0.689728\pi$
$252$	−3199.06	−0.799689
$253$	0	0
$254$	−7956.52	−1.96550
$255$	314.567	0.0772507
$256$	1954.34	0.477134
$257$	−6951.05	−1.68714	−0.843569	−	0.537021i	$-0.819549\pi$
$257$	−6951.05	−1.68714	−0.843569	+	0.537021i	$0.819549\pi$
$258$	−5408.99	−1.30523
$259$	2265.07	0.543415
$260$	828.297	0.197572
$261$	−1935.57	−0.459038
$262$	−2143.21	−0.505374
$263$	3975.88	0.932181	0.466090	−	0.884737i	$-0.345662\pi$
$263$	3975.88	0.932181	0.466090	+	0.884737i	$0.345662\pi$
$264$	0	0
$265$	3737.83	0.866463
$266$	−13918.0	−3.20815
$267$	−1139.51	−0.261186
$268$	16630.2	3.79048
$269$	4652.53	1.05453	0.527267	−	0.849700i	$-0.323217\pi$
$269$	4652.53	1.05453	0.527267	+	0.849700i	$0.323217\pi$
$270$	−722.040	−0.162748
$271$	673.995	0.151079	0.0755393	−	0.997143i	$-0.475932\pi$
$271$	673.995	0.151079	0.0755393	+	0.997143i	$0.475932\pi$
$272$	−4105.19	−0.915125
$273$	416.040	0.0922340
$274$	−2113.89	−0.466075
$275$	0	0
$276$	−9294.73	−2.02709
$277$	−831.096	−0.180273	−0.0901367	−	0.995929i	$-0.528730\pi$
$277$	−831.096	−0.180273	−0.0901367	+	0.995929i	$0.528730\pi$
$278$	−11250.6	−2.42723
$279$	−99.1320	−0.0212720
$280$	−5815.12	−1.24114
$281$	2816.74	0.597982	0.298991	−	0.954256i	$-0.403350\pi$
$281$	2816.74	0.597982	0.298991	+	0.954256i	$0.403350\pi$
$282$	84.6099	0.0178668
$283$	−2116.14	−0.444493	−0.222246	−	0.974991i	$-0.571339\pi$
$283$	−2116.14	−0.444493	−0.222246	+	0.974991i	$0.571339\pi$
$284$	−10644.9	−2.22416
$285$	−2262.83	−0.470311
$286$	0	0
$287$	−1054.49	−0.216880
$288$	4568.48	0.934723
$289$	−4473.21	−0.910485
$290$	−5751.28	−1.16457
$291$	3829.54	0.771449
$292$	15846.0	3.17574
$293$	−6298.86	−1.25592	−0.627958	−	0.778247i	$-0.716109\pi$
$293$	−6298.86	−1.25592	−0.627958	+	0.778247i	$0.716109\pi$
$294$	729.071	0.144627
$295$	−2460.91	−0.485693
$296$	−8853.05	−1.73842
$297$	0	0
$298$	6625.66	1.28797
$299$	1208.79	0.233799
$300$	−1545.44	−0.297420
$301$	−5815.08	−1.11354
$302$	−6830.35	−1.30146
$303$	−3373.95	−0.639697
$304$	29530.7	5.57138
$305$	3830.02	0.719037
$306$	−1009.47	−0.188586
$307$	−3184.92	−0.592094	−0.296047	−	0.955173i	$-0.595669\pi$
$307$	−3184.92	−0.592094	−0.296047	+	0.955173i	$0.595669\pi$
$308$	0	0
$309$	−4264.27	−0.785068
$310$	−294.557	−0.0539668
$311$	−5890.49	−1.07402	−0.537008	−	0.843577i	$-0.680445\pi$
$311$	−5890.49	−1.07402	−0.537008	+	0.843577i	$0.680445\pi$
$312$	−1626.10	−0.295063
$313$	−2962.68	−0.535019	−0.267509	−	0.963555i	$-0.586201\pi$
$313$	−2962.68	−0.535019	−0.267509	+	0.963555i	$0.586201\pi$
$314$	10163.3	1.82659
$315$	−776.249	−0.138847
$316$	2228.09	0.396645
$317$	1253.85	0.222155	0.111077	−	0.993812i	$-0.464570\pi$
$317$	1253.85	0.222155	0.111077	+	0.993812i	$0.464570\pi$
$318$	−11994.9	−2.11523
$319$	0	0
$320$	5744.41	1.00351
$321$	−555.613	−0.0966084
$322$	−13872.0	−2.40080
$323$	−3163.61	−0.544978
$324$	1669.07	0.286193
$325$	200.986	0.0343036
$326$	−8824.04	−1.49914
$327$	−5246.96	−0.887331
$328$	4121.49	0.693815
$329$	90.9622	0.0152429
$330$	0	0
$331$	−1610.81	−0.267486	−0.133743	−	0.991016i	$-0.542700\pi$
$331$	−1610.81	−0.267486	−0.133743	+	0.991016i	$0.542700\pi$
$332$	5771.55	0.954081
$333$	−1181.78	−0.194477
$334$	2354.15	0.385668
$335$	4035.30	0.658125
$336$	10130.3	1.64480
$337$	−3792.12	−0.612968	−0.306484	−	0.951876i	$-0.599153\pi$
$337$	−3792.12	−0.612968	−0.306484	+	0.951876i	$0.599153\pi$
$338$	−11404.9	−1.83533
$339$	6377.83	1.02182
$340$	−2160.64	−0.344639
$341$	0	0
$342$	7261.58	1.14813
$343$	6700.55	1.05480
$344$	22728.3	3.56229
$345$	−2255.36	−0.351955
$346$	−8498.25	−1.32043
$347$	−10539.9	−1.63057	−0.815286	−	0.579058i	$-0.803421\pi$
$347$	−10539.9	−1.63057	−0.815286	+	0.579058i	$0.803421\pi$
$348$	13294.7	2.04791
$349$	10810.3	1.65806	0.829031	−	0.559202i	$-0.188892\pi$
$349$	10810.3	1.65806	0.829031	+	0.559202i	$0.188892\pi$
$350$	−2306.51	−0.352252
$351$	−217.064	−0.0330087
$352$	0	0
$353$	3833.35	0.577985	0.288992	−	0.957331i	$-0.406680\pi$
$353$	3833.35	0.577985	0.288992	+	0.957331i	$0.406680\pi$
$354$	7897.21	1.18568
$355$	−2582.99	−0.386171
$356$	7826.84	1.16523
$357$	−1085.25	−0.160890
$358$	17738.6	2.61876
$359$	−3995.32	−0.587367	−0.293683	−	0.955903i	$-0.594881\pi$
$359$	−3995.32	−0.587367	−0.293683	+	0.955903i	$0.594881\pi$
$360$	3033.98	0.444180
$361$	15898.4	2.31788
$362$	12023.9	1.74576
$363$	0	0
$364$	−2857.62	−0.411483
$365$	3845.02	0.551391
$366$	−12290.8	−1.75533
$367$	−3801.00	−0.540629	−0.270314	−	0.962772i	$-0.587128\pi$
$367$	−3801.00	−0.540629	−0.270314	+	0.962772i	$0.587128\pi$
$368$	29433.1	4.16932
$369$	550.170	0.0776171
$370$	−3511.48	−0.493387
$371$	−12895.5	−1.80458
$372$	680.900	0.0949006
$373$	−2481.74	−0.344502	−0.172251	−	0.985053i	$-0.555104\pi$
$373$	−2481.74	−0.344502	−0.172251	+	0.985053i	$0.555104\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	−355.527	−0.0487630
$377$	−1728.99	−0.236200
$378$	2491.04	0.338955
$379$	−6423.71	−0.870617	−0.435309	−	0.900281i	$-0.643361\pi$
$379$	−6423.71	−0.870617	−0.435309	+	0.900281i	$0.643361\pi$
$380$	15542.5	2.09820
$381$	4462.90	0.600108
$382$	−8103.63	−1.08539
$383$	−10745.2	−1.43356	−0.716778	−	0.697302i	$-0.754384\pi$
$383$	−10745.2	−1.43356	−0.716778	+	0.697302i	$0.754384\pi$
$384$	−6251.57	−0.830792
$385$	0	0
$386$	−20576.5	−2.71325
$387$	3033.96	0.398514
$388$	−26303.7	−3.44167
$389$	−5635.56	−0.734536	−0.367268	−	0.930115i	$-0.619707\pi$
$389$	−5635.56	−0.734536	−0.367268	+	0.930115i	$0.619707\pi$
$390$	−644.976	−0.0837427
$391$	−3153.16	−0.407832
$392$	−3063.52	−0.394723
$393$	1202.15	0.154301
$394$	−25024.1	−3.19973
$395$	540.645	0.0688678
$396$	0	0
$397$	−14563.2	−1.84108	−0.920540	−	0.390649i	$-0.872251\pi$
$397$	−14563.2	−1.84108	−0.920540	+	0.390649i	$0.872251\pi$
$398$	−1308.39	−0.164784
$399$	7806.76	0.979516
$400$	4893.87	0.611733
$401$	10923.5	1.36033	0.680166	−	0.733058i	$-0.261908\pi$
$401$	10923.5	1.36033	0.680166	+	0.733058i	$0.261908\pi$
$402$	−12949.5	−1.60663
$403$	−88.5516	−0.0109456
$404$	23174.4	2.85388
$405$	405.000	0.0496904
$406$	19841.9	2.42546
$407$	0	0
$408$	4241.73	0.514698
$409$	−5101.94	−0.616809	−0.308404	−	0.951255i	$-0.599795\pi$
$409$	−5101.94	−0.616809	−0.308404	+	0.951255i	$0.599795\pi$
$410$	1634.75	0.196914
$411$	1185.70	0.142303
$412$	29289.7	3.50242
$413$	8490.12	1.01155
$414$	7237.60	0.859200
$415$	1400.46	0.165653
$416$	4080.88	0.480966
$417$	6310.61	0.741083
$418$	0	0
$419$	9013.25	1.05090	0.525449	−	0.850825i	$-0.323897\pi$
$419$	9013.25	1.05090	0.525449	+	0.850825i	$0.323897\pi$
$420$	5331.76	0.619436
$421$	2082.42	0.241071	0.120535	−	0.992709i	$-0.461539\pi$
$421$	2082.42	0.241071	0.120535	+	0.992709i	$0.461539\pi$
$422$	−4741.74	−0.546977
$423$	−47.4586	−0.00545512
$424$	50402.2	5.77298
$425$	−524.278	−0.0598381
$426$	8288.99	0.942730
$427$	−13213.6	−1.49754
$428$	3816.29	0.430999
$429$	0	0
$430$	9014.98	1.01103
$431$	4749.08	0.530755	0.265377	−	0.964145i	$-0.414503\pi$
$431$	4749.08	0.530755	0.265377	+	0.964145i	$0.414503\pi$
$432$	−5285.38	−0.588641
$433$	−1182.19	−0.131207	−0.0656035	−	0.997846i	$-0.520897\pi$
$433$	−1182.19	−0.131207	−0.0656035	+	0.997846i	$0.520897\pi$
$434$	1016.22	0.112397
$435$	3225.95	0.355569
$436$	36039.4	3.95865
$437$	22682.2	2.48292
$438$	−12338.9	−1.34607
$439$	−7829.21	−0.851180	−0.425590	−	0.904916i	$-0.639933\pi$
$439$	−7829.21	−0.851180	−0.425590	+	0.904916i	$0.639933\pi$
$440$	0	0
$441$	−408.944	−0.0441576
$442$	−901.725	−0.0970377
$443$	−454.157	−0.0487080	−0.0243540	−	0.999703i	$-0.507753\pi$
$443$	−454.157	−0.0487080	−0.0243540	+	0.999703i	$0.507753\pi$
$444$	8117.17	0.867621
$445$	1899.18	0.202314
$446$	−816.455	−0.0866822
$447$	−3716.41	−0.393244
$448$	−19818.2	−2.09000
$449$	4311.48	0.453165	0.226583	−	0.973992i	$-0.427245\pi$
$449$	4311.48	0.453165	0.226583	+	0.973992i	$0.427245\pi$
$450$	1203.40	0.126064
$451$	0	0
$452$	−43806.9	−4.55863
$453$	3831.21	0.397365
$454$	22482.9	2.32417
$455$	−693.400	−0.0714441
$456$	−30512.8	−3.13354
$457$	310.370	0.0317692	0.0158846	−	0.999874i	$-0.494944\pi$
$457$	310.370	0.0317692	0.0158846	+	0.999874i	$0.494944\pi$
$458$	2678.63	0.273284
$459$	566.220	0.0575793
$460$	15491.2	1.57018
$461$	6152.94	0.621629	0.310814	−	0.950471i	$-0.399398\pi$
$461$	6152.94	0.621629	0.310814	+	0.950471i	$0.399398\pi$
$462$	0	0
$463$	−15105.6	−1.51623	−0.758116	−	0.652120i	$-0.773880\pi$
$463$	−15105.6	−1.51623	−0.758116	+	0.652120i	$0.773880\pi$
$464$	−42099.7	−4.21213
$465$	165.220	0.0164772
$466$	−2063.13	−0.205091
$467$	−1021.20	−0.101190	−0.0505948	−	0.998719i	$-0.516112\pi$
$467$	−1021.20	−0.101190	−0.0505948	+	0.998719i	$0.516112\pi$
$468$	1490.93	0.147262
$469$	−13921.8	−1.37068
$470$	−141.017	−0.0138396
$471$	−5700.71	−0.557696
$472$	−33183.7	−3.23603
$473$	0	0
$474$	−1734.97	−0.168122
$475$	3771.39	0.364301
$476$	7454.20	0.717778
$477$	6728.09	0.645824
$478$	22989.8	2.19986
$479$	−6678.02	−0.637007	−0.318504	−	0.947922i	$-0.603180\pi$
$479$	−6678.02	−0.637007	−0.318504	+	0.947922i	$0.603180\pi$
$480$	−7614.13	−0.724034
$481$	−1055.64	−0.100069
$482$	34007.2	3.21366
$483$	7780.98	0.733016
$484$	0	0
$485$	−6382.57	−0.597562
$486$	−1299.67	−0.121305
$487$	−18167.0	−1.69040	−0.845200	−	0.534451i	$-0.820518\pi$
$487$	−18167.0	−1.69040	−0.845200	+	0.534451i	$0.820518\pi$
$488$	51645.3	4.79073
$489$	4949.50	0.457718
$490$	−1215.12	−0.112028
$491$	−6833.67	−0.628104	−0.314052	−	0.949406i	$-0.601687\pi$
$491$	−6833.67	−0.628104	−0.314052	+	0.949406i	$0.601687\pi$
$492$	−3778.91	−0.346273
$493$	4510.12	0.412020
$494$	6486.55	0.590776
$495$	0	0
$496$	−2156.17	−0.195192
$497$	8911.30	0.804279
$498$	−4494.18	−0.404396
$499$	−889.858	−0.0798307	−0.0399154	−	0.999203i	$-0.512709\pi$
$499$	−889.858	−0.0798307	−0.0399154	+	0.999203i	$0.512709\pi$
$500$	2575.73	0.230381
$501$	−1320.47	−0.117753
$502$	−23879.3	−2.12308
$503$	−3116.85	−0.276289	−0.138145	−	0.990412i	$-0.544114\pi$
$503$	−3116.85	−0.276289	−0.138145	+	0.990412i	$0.544114\pi$
$504$	−10467.2	−0.925093
$505$	5623.25	0.495507
$506$	0	0
$507$	6397.10	0.560366
$508$	−30653.9	−2.67726
$509$	−5661.11	−0.492975	−0.246488	−	0.969146i	$-0.579276\pi$
$509$	−5661.11	−0.492975	−0.246488	+	0.969146i	$0.579276\pi$
$510$	1682.44	0.146078
$511$	−13265.3	−1.14838
$512$	−6218.17	−0.536732
$513$	−4073.10	−0.350549
$514$	−37177.3	−3.19031
$515$	7107.12	0.608111
$516$	−20839.1	−1.77789
$517$	0	0
$518$	12114.6	1.02758
$519$	4766.76	0.403155
$520$	2710.16	0.228555
$521$	6749.45	0.567560	0.283780	−	0.958889i	$-0.408412\pi$
$521$	6749.45	0.567560	0.283780	+	0.958889i	$0.408412\pi$
$522$	−10352.3	−0.868023
$523$	2776.28	0.232119	0.116059	−	0.993242i	$-0.462974\pi$
$523$	2776.28	0.232119	0.116059	+	0.993242i	$0.462974\pi$
$524$	−8257.11	−0.688384
$525$	1293.75	0.107550
$526$	21264.8	1.76272
$527$	230.990	0.0190931
$528$	0	0
$529$	10440.3	0.858085
$530$	19991.6	1.63845
$531$	−4429.63	−0.362014
$532$	−53621.7	−4.36991
$533$	491.450	0.0399382
$534$	−6094.59	−0.493893
$535$	926.021	0.0748325
$536$	54413.4	4.38489
$537$	−9949.78	−0.799562
$538$	24883.8	1.99408
$539$	0	0
$540$	−2781.79	−0.221684
$541$	510.573	0.0405753	0.0202877	−	0.999794i	$-0.493542\pi$
$541$	510.573	0.0405753	0.0202877	+	0.999794i	$0.493542\pi$
$542$	3604.83	0.285684
$543$	−6744.35	−0.533016
$544$	−10645.1	−0.838982
$545$	8744.93	0.687324
$546$	2225.17	0.174411
$547$	−3883.78	−0.303580	−0.151790	−	0.988413i	$-0.548504\pi$
$547$	−3883.78	−0.303580	−0.151790	+	0.988413i	$0.548504\pi$
$548$	−8144.14	−0.634855
$549$	6894.04	0.535939
$550$	0	0
$551$	−32443.5	−2.50842
$552$	−30412.1	−2.34497
$553$	−1865.22	−0.143431
$554$	−4445.07	−0.340890
$555$	1969.63	0.150641
$556$	−43345.2	−3.30619
$557$	−16086.8	−1.22373	−0.611865	−	0.790962i	$-0.709580\pi$
$557$	−16086.8	−1.22373	−0.611865	+	0.790962i	$0.709580\pi$
$558$	−530.202	−0.0402244
$559$	2710.14	0.205057
$560$	−16883.8	−1.27406
$561$	0	0
$562$	15065.2	1.13076
$563$	4675.35	0.349986	0.174993	−	0.984570i	$-0.444010\pi$
$563$	4675.35	0.349986	0.174993	+	0.984570i	$0.444010\pi$
$564$	325.975	0.0243369
$565$	−10629.7	−0.791496
$566$	−11318.1	−0.840519
$567$	−1397.25	−0.103490
$568$	−34829.9	−2.57294
$569$	15208.5	1.12052	0.560259	−	0.828317i	$-0.310702\pi$
$569$	15208.5	1.12052	0.560259	+	0.828317i	$0.310702\pi$
$570$	−12102.6	−0.889339
$571$	−6552.30	−0.480219	−0.240109	−	0.970746i	$-0.577183\pi$
$571$	−6552.30	−0.480219	−0.240109	+	0.970746i	$0.577183\pi$
$572$	0	0
$573$	4545.41	0.331391
$574$	−5639.89	−0.410112
$575$	3758.93	0.272623
$576$	10339.9	0.747970
$577$	−20103.4	−1.45046	−0.725229	−	0.688508i	$-0.758266\pi$
$577$	−20103.4	−1.45046	−0.725229	+	0.688508i	$0.758266\pi$
$578$	−23924.7	−1.72169
$579$	11541.6	0.828412
$580$	−22157.8	−1.58630
$581$	−4831.59	−0.345005
$582$	20482.1	1.45878
$583$	0	0
$584$	51847.7	3.67375
$585$	361.774	0.0255684
$586$	−33689.1	−2.37489
$587$	−16829.9	−1.18338	−0.591692	−	0.806164i	$-0.701540\pi$
$587$	−16829.9	−1.18338	−0.591692	+	0.806164i	$0.701540\pi$
$588$	2808.88	0.197001
$589$	−1661.62	−0.116241
$590$	−13162.0	−0.918427
$591$	14036.3	0.976946
$592$	−25704.2	−1.78452
$593$	−16035.1	−1.11043	−0.555214	−	0.831708i	$-0.687363\pi$
$593$	−16035.1	−1.11043	−0.555214	+	0.831708i	$0.687363\pi$
$594$	0	0
$595$	1808.76	0.124625
$596$	25526.6	1.75438
$597$	733.893	0.0503119
$598$	6465.13	0.442105
$599$	14539.1	0.991740	0.495870	−	0.868397i	$-0.334849\pi$
$599$	14539.1	0.991740	0.495870	+	0.868397i	$0.334849\pi$
$600$	−5056.63	−0.344060
$601$	14178.7	0.962332	0.481166	−	0.876629i	$-0.340213\pi$
$601$	14178.7	0.962332	0.481166	+	0.876629i	$0.340213\pi$
$602$	−31101.6	−2.10566
$603$	7263.54	0.490537
$604$	−26315.2	−1.77276
$605$	0	0
$606$	−18045.4	−1.20964
$607$	9346.10	0.624953	0.312476	−	0.949926i	$-0.398841\pi$
$607$	9346.10	0.624953	0.312476	+	0.949926i	$0.398841\pi$
$608$	76575.6	5.10781
$609$	−11129.5	−0.740544
$610$	20484.7	1.35967
$611$	−42.3933	−0.00280696
$612$	−3889.15	−0.256878
$613$	−15404.6	−1.01499	−0.507494	−	0.861655i	$-0.669428\pi$
$613$	−15404.6	−1.01499	−0.507494	+	0.861655i	$0.669428\pi$
$614$	−17034.4	−1.11963
$615$	−916.950	−0.0601220
$616$	0	0
$617$	−19278.6	−1.25790	−0.628951	−	0.777445i	$-0.716516\pi$
$617$	−19278.6	−1.25790	−0.628951	+	0.777445i	$0.716516\pi$
$618$	−22807.2	−1.48453
$619$	12174.3	0.790508	0.395254	−	0.918572i	$-0.370657\pi$
$619$	12174.3	0.790508	0.395254	+	0.918572i	$0.370657\pi$
$620$	−1134.83	−0.0735097
$621$	−4059.65	−0.262332
$622$	−31504.9	−2.03092
$623$	−6552.16	−0.421359
$624$	−4721.27	−0.302888
$625$	625.000	0.0400000
$626$	−15845.8	−1.01170
$627$	0	0
$628$	39156.0	2.48805
$629$	2753.68	0.174557
$630$	−4151.73	−0.262553
$631$	−27111.0	−1.71041	−0.855207	−	0.518287i	$-0.826570\pi$
$631$	−27111.0	−1.71041	−0.855207	+	0.518287i	$0.826570\pi$
$632$	7290.25	0.458846
$633$	2659.69	0.167003
$634$	6706.13	0.420086
$635$	−7438.16	−0.464842
$636$	−46212.7	−2.88121
$637$	−365.297	−0.0227215
$638$	0	0
$639$	−4649.38	−0.287835
$640$	10419.3	0.643529
$641$	−14163.4	−0.872731	−0.436366	−	0.899769i	$-0.643735\pi$
$641$	−14163.4	−0.872731	−0.436366	+	0.899769i	$0.643735\pi$
$642$	−2971.66	−0.182683
$643$	−11909.2	−0.730410	−0.365205	−	0.930927i	$-0.619001\pi$
$643$	−11909.2	−0.730410	−0.365205	+	0.930927i	$0.619001\pi$
$644$	−53444.6	−3.27021
$645$	−5056.60	−0.308687
$646$	−16920.4	−1.03053
$647$	−11299.7	−0.686611	−0.343306	−	0.939224i	$-0.611547\pi$
$647$	−11299.7	−0.686611	−0.343306	+	0.939224i	$0.611547\pi$
$648$	5461.16	0.331072
$649$	0	0
$650$	1074.96	0.0648668
$651$	−570.008	−0.0343170
$652$	−33996.2	−2.04202
$653$	27860.8	1.66964	0.834822	−	0.550520i	$-0.185571\pi$
$653$	27860.8	1.66964	0.834822	+	0.550520i	$0.185571\pi$
$654$	−28063.1	−1.67791
$655$	−2003.58	−0.119521
$656$	11966.5	0.712215
$657$	6921.04	0.410983
$658$	486.506	0.0288237
$659$	−25570.3	−1.51150	−0.755750	−	0.654860i	$-0.772727\pi$
$659$	−25570.3	−1.51150	−0.755750	+	0.654860i	$0.772727\pi$
$660$	0	0
$661$	15221.9	0.895711	0.447855	−	0.894106i	$-0.352188\pi$
$661$	15221.9	0.895711	0.447855	+	0.894106i	$0.352188\pi$
$662$	−8615.31	−0.505806
$663$	505.787	0.0296277
$664$	18884.3	1.10370
$665$	−13011.3	−0.758730
$666$	−6320.66	−0.367749
$667$	−32336.4	−1.87717
$668$	9069.78	0.525330
$669$	457.959	0.0264659
$670$	21582.6	1.24449
$671$	0	0
$672$	26268.7	1.50794
$673$	27201.7	1.55802	0.779012	−	0.627009i	$-0.215721\pi$
$673$	27201.7	1.55802	0.779012	+	0.627009i	$0.215721\pi$
$674$	−20282.0	−1.15910
$675$	−675.000	−0.0384900
$676$	−43939.3	−2.49996
$677$	19388.7	1.10069	0.550345	−	0.834937i	$-0.314496\pi$
$677$	19388.7	1.10069	0.550345	+	0.834937i	$0.314496\pi$
$678$	34111.5	1.93222
$679$	22019.8	1.24454
$680$	−7069.55	−0.398683
$681$	−12610.9	−0.709619
$682$	0	0
$683$	8624.66	0.483182	0.241591	−	0.970378i	$-0.422331\pi$
$683$	8624.66	0.483182	0.241591	+	0.970378i	$0.422331\pi$
$684$	27976.6	1.56390
$685$	−1976.17	−0.110227
$686$	35837.5	1.99458
$687$	−1502.47	−0.0834394
$688$	65990.2	3.65676
$689$	6009.99	0.332311
$690$	−12062.7	−0.665533
$691$	22554.8	1.24171	0.620857	−	0.783924i	$-0.286785\pi$
$691$	22554.8	1.24171	0.620857	+	0.783924i	$0.286785\pi$
$692$	−32741.1	−1.79860
$693$	0	0
$694$	−56371.8	−3.08335
$695$	−10517.7	−0.574041
$696$	43499.9	2.36905
$697$	−1281.96	−0.0696670
$698$	57818.5	3.13533
$699$	1157.23	0.0626187
$700$	−8886.27	−0.479813
$701$	29968.1	1.61466	0.807332	−	0.590098i	$-0.200911\pi$
$701$	29968.1	1.61466	0.807332	+	0.590098i	$0.200911\pi$
$702$	−1160.96	−0.0624181
$703$	−19808.6	−1.06272
$704$	0	0
$705$	79.0977	0.00422552
$706$	20502.5	1.09295
$707$	−19400.2	−1.03199
$708$	30425.5	1.61505
$709$	−7740.41	−0.410010	−0.205005	−	0.978761i	$-0.565721\pi$
$709$	−7740.41	−0.410010	−0.205005	+	0.978761i	$0.565721\pi$
$710$	−13815.0	−0.730235
$711$	973.161	0.0513311
$712$	25609.2	1.34796
$713$	−1656.14	−0.0869884
$714$	−5804.42	−0.304237
$715$	0	0
$716$	68341.3	3.56709
$717$	−12895.2	−0.671662
$718$	−21368.7	−1.11069
$719$	−3750.98	−0.194559	−0.0972794	−	0.995257i	$-0.531014\pi$
$719$	−3750.98	−0.194559	−0.0972794	+	0.995257i	$0.531014\pi$
$720$	8808.96	0.455959
$721$	−24519.5	−1.26651
$722$	85031.6	4.38303
$723$	−19075.0	−0.981199
$724$	46324.4	2.37795
$725$	−5376.59	−0.275423
$726$	0	0
$727$	−4499.18	−0.229526	−0.114763	−	0.993393i	$-0.536611\pi$
$727$	−4499.18	−0.229526	−0.114763	+	0.993393i	$0.536611\pi$
$728$	−9350.05	−0.476011
$729$	729.000	0.0370370
$730$	20564.9	1.04266
$731$	−7069.50	−0.357695
$732$	−47352.5	−2.39098
$733$	28473.3	1.43477	0.717384	−	0.696678i	$-0.245339\pi$
$733$	28473.3	1.43477	0.717384	+	0.696678i	$0.245339\pi$
$734$	−20329.5	−1.02231
$735$	681.573	0.0342044
$736$	76322.7	3.82241
$737$	0	0
$738$	2942.55	0.146771
$739$	15127.5	0.753012	0.376506	−	0.926414i	$-0.377126\pi$
$739$	15127.5	0.753012	0.376506	+	0.926414i	$0.377126\pi$
$740$	−13528.6	−0.672057
$741$	−3638.37	−0.180377
$742$	−68970.8	−3.41239
$743$	−27667.1	−1.36609	−0.683046	−	0.730375i	$-0.739345\pi$
$743$	−27667.1	−1.36609	−0.683046	+	0.730375i	$0.739345\pi$
$744$	2227.88	0.109783
$745$	6194.01	0.304605
$746$	−13273.4	−0.651440
$747$	2520.83	0.123471
$748$	0	0
$749$	−3194.77	−0.155854
$750$	−2005.67	−0.0976488
$751$	26794.2	1.30191	0.650956	−	0.759116i	$-0.274368\pi$
$751$	26794.2	1.30191	0.650956	+	0.759116i	$0.274368\pi$
$752$	−1032.25	−0.0500562
$753$	13394.2	0.648221
$754$	−9247.39	−0.446645
$755$	−6385.36	−0.307797
$756$	9597.17	0.461701
$757$	−7697.08	−0.369558	−0.184779	−	0.982780i	$-0.559157\pi$
$757$	−7697.08	−0.369558	−0.184779	+	0.982780i	$0.559157\pi$
$758$	−34356.9	−1.64630
$759$	0	0
$760$	50854.7	2.42723
$761$	9639.90	0.459193	0.229597	−	0.973286i	$-0.426259\pi$
$761$	9639.90	0.459193	0.229597	+	0.973286i	$0.426259\pi$
$762$	23869.6	1.13478
$763$	−30170.0	−1.43149
$764$	−31220.7	−1.47844
$765$	−943.700	−0.0446007
$766$	−57469.9	−2.71080
$767$	−3956.86	−0.186276
$768$	−5863.02	−0.275474
$769$	20320.9	0.952912	0.476456	−	0.879198i	$-0.341921\pi$
$769$	20320.9	0.952912	0.476456	+	0.879198i	$0.341921\pi$
$770$	0	0
$771$	20853.1	0.974069
$772$	−79274.6	−3.69580
$773$	21941.1	1.02091	0.510456	−	0.859904i	$-0.329477\pi$
$773$	21941.1	1.02091	0.510456	+	0.859904i	$0.329477\pi$
$774$	16227.0	0.753574
$775$	−275.367	−0.0127632
$776$	−86064.8	−3.98138
$777$	−6795.20	−0.313741
$778$	−30141.5	−1.38898
$779$	9221.80	0.424140
$780$	−2484.89	−0.114068
$781$	0	0
$782$	−16864.5	−0.771194
$783$	5806.72	0.265026
$784$	−8894.74	−0.405191
$785$	9501.19	0.431990
$786$	6429.63	0.291778
$787$	30770.8	1.39373	0.696863	−	0.717205i	$-0.254579\pi$
$787$	30770.8	1.39373	0.696863	+	0.717205i	$0.254579\pi$
$788$	−96409.8	−4.35845
$789$	−11927.7	−0.538195
$790$	2891.61	0.130226
$791$	36672.5	1.64845
$792$	0	0
$793$	6158.23	0.275770
$794$	−77890.7	−3.48141
$795$	−11213.5	−0.500253
$796$	−5040.83	−0.224457
$797$	−8450.40	−0.375569	−0.187785	−	0.982210i	$-0.560131\pi$
$797$	−8450.40	−0.375569	−0.187785	+	0.982210i	$0.560131\pi$
$798$	41754.0	1.85223
$799$	110.584	0.00489637
$800$	12690.2	0.560834
$801$	3418.52	0.150796
$802$	58423.7	2.57234
$803$	0	0
$804$	−49890.5	−2.18843
$805$	−12968.3	−0.567792
$806$	−473.613	−0.0206977
$807$	−13957.6	−0.608835
$808$	75825.9	3.30142
$809$	−1779.71	−0.0773441	−0.0386721	−	0.999252i	$-0.512313\pi$
$809$	−1779.71	−0.0773441	−0.0386721	+	0.999252i	$0.512313\pi$
$810$	2166.12	0.0939626
$811$	1144.23	0.0495429	0.0247715	−	0.999693i	$-0.492114\pi$
$811$	1144.23	0.0495429	0.0247715	+	0.999693i	$0.492114\pi$
$812$	76444.5	3.30379
$813$	−2021.99	−0.0872253
$814$	0	0
$815$	−8249.16	−0.354547
$816$	12315.6	0.528347
$817$	50854.4	2.17769
$818$	−27287.5	−1.16636
$819$	−1248.12	−0.0532513
$820$	6298.18	0.268222
$821$	−2258.37	−0.0960019	−0.0480010	−	0.998847i	$-0.515285\pi$
$821$	−2258.37	−0.0960019	−0.0480010	+	0.998847i	$0.515285\pi$
$822$	6341.67	0.269089
$823$	−248.144	−0.0105100	−0.00525501	−	0.999986i	$-0.501673\pi$
$823$	−248.144	−0.0105100	−0.00525501	+	0.999986i	$0.501673\pi$
$824$	95834.9	4.05166
$825$	0	0
$826$	45408.9	1.91281
$827$	−821.902	−0.0345590	−0.0172795	−	0.999851i	$-0.505501\pi$
$827$	−821.902	−0.0345590	−0.0172795	+	0.999851i	$0.505501\pi$
$828$	27884.2	1.17034
$829$	−22393.9	−0.938204	−0.469102	−	0.883144i	$-0.655422\pi$
$829$	−22393.9	−0.938204	−0.469102	+	0.883144i	$0.655422\pi$
$830$	7490.30	0.313243
$831$	2493.29	0.104081
$832$	9236.35	0.384871
$833$	952.890	0.0396347
$834$	33751.9	1.40136
$835$	2200.78	0.0912108
$836$	0	0
$837$	297.396	0.0122814
$838$	48206.9	1.98721
$839$	−31613.0	−1.30084	−0.650418	−	0.759576i	$-0.725406\pi$
$839$	−31613.0	−1.30084	−0.650418	+	0.759576i	$0.725406\pi$
$840$	17445.4	0.716574
$841$	21863.3	0.896442
$842$	11137.7	0.455855
$843$	−8450.23	−0.345245
$844$	−18268.4	−0.745053
$845$	−10661.8	−0.434057
$846$	−253.830	−0.0103154
$847$	0	0
$848$	146339.	5.92608
$849$	6348.42	0.256628
$850$	−2804.07	−0.113152
$851$	−19743.2	−0.795285
$852$	31934.8	1.28412
$853$	2685.51	0.107796	0.0538982	−	0.998546i	$-0.482835\pi$
$853$	2685.51	0.107796	0.0538982	+	0.998546i	$0.482835\pi$
$854$	−70672.0	−2.83179
$855$	6788.50	0.271534
$856$	12486.8	0.498586
$857$	29749.6	1.18579	0.592897	−	0.805278i	$-0.297984\pi$
$857$	29749.6	1.18579	0.592897	+	0.805278i	$0.297984\pi$
$858$	0	0
$859$	12305.1	0.488762	0.244381	−	0.969679i	$-0.421415\pi$
$859$	12305.1	0.488762	0.244381	+	0.969679i	$0.421415\pi$
$860$	34731.9	1.37715
$861$	3163.47	0.125216
$862$	25400.2	1.00364
$863$	−17908.0	−0.706366	−0.353183	−	0.935554i	$-0.614901\pi$
$863$	−17908.0	−0.706366	−0.353183	+	0.935554i	$0.614901\pi$
$864$	−13705.4	−0.539663
$865$	−7944.60	−0.312283
$866$	−6322.90	−0.248107
$867$	13419.6	0.525669
$868$	3915.17	0.153099
$869$	0	0
$870$	17253.8	0.672367
$871$	6488.30	0.252408
$872$	117920.	4.57943
$873$	−11488.6	−0.445397
$874$	121315.	4.69511
$875$	−2156.25	−0.0833080
$876$	−47538.0	−1.83352
$877$	−15702.6	−0.604604	−0.302302	−	0.953212i	$-0.597755\pi$
$877$	−15702.6	−0.604604	−0.302302	+	0.953212i	$0.597755\pi$
$878$	−41874.1	−1.60955
$879$	18896.6	0.725104
$880$	0	0
$881$	−47271.2	−1.80773	−0.903864	−	0.427820i	$-0.859282\pi$
$881$	−47271.2	−1.80773	−0.903864	+	0.427820i	$0.859282\pi$
$882$	−2187.21	−0.0835004
$883$	−18950.8	−0.722250	−0.361125	−	0.932517i	$-0.617607\pi$
$883$	−18950.8	−0.722250	−0.361125	+	0.932517i	$0.617607\pi$
$884$	−3474.06	−0.132178
$885$	7382.72	0.280415
$886$	−2429.03	−0.0921050
$887$	50992.3	1.93027	0.965137	−	0.261745i	$-0.0842980\pi$
$887$	50992.3	1.93027	0.965137	+	0.261745i	$0.0842980\pi$
$888$	26559.1	1.00368
$889$	25661.6	0.968125
$890$	10157.7	0.382568
$891$	0	0
$892$	−3145.54	−0.118072
$893$	−795.488	−0.0298096
$894$	−19877.0	−0.743608
$895$	16583.0	0.619338
$896$	−35946.5	−1.34028
$897$	−3626.36	−0.134984
$898$	23059.7	0.856917
$899$	2368.85	0.0878817
$900$	4636.32	0.171716
$901$	−15677.3	−0.579673
$902$	0	0
$903$	17445.2	0.642903
$904$	−143335.	−5.27350
$905$	11240.6	0.412873
$906$	20491.0	0.751401
$907$	33620.1	1.23080	0.615400	−	0.788215i	$-0.288994\pi$
$907$	33620.1	1.23080	0.615400	+	0.788215i	$0.288994\pi$
$908$	86619.5	3.16582
$909$	10121.8	0.369329
$910$	−3708.61	−0.135098
$911$	−43289.7	−1.57437	−0.787186	−	0.616716i	$-0.788463\pi$
$911$	−43289.7	−1.57437	−0.787186	+	0.616716i	$0.788463\pi$
$912$	−88592.0	−3.21664
$913$	0	0
$914$	1660.00	0.0600743
$915$	−11490.1	−0.415136
$916$	10319.9	0.372248
$917$	6912.35	0.248927
$918$	3028.40	0.108880
$919$	−31035.1	−1.11399	−0.556993	−	0.830517i	$-0.688045\pi$
$919$	−31035.1	−1.11399	−0.556993	+	0.830517i	$0.688045\pi$
$920$	50686.8	1.81641
$921$	9554.76	0.341846
$922$	32908.6	1.17548
$923$	−4153.15	−0.148107
$924$	0	0
$925$	−3282.71	−0.116686
$926$	−80791.3	−2.86713
$927$	12792.8	0.453259
$928$	−109168.	−3.86166
$929$	30300.0	1.07009	0.535043	−	0.844825i	$-0.320295\pi$
$929$	30300.0	1.07009	0.535043	+	0.844825i	$0.320295\pi$
$930$	883.670	0.0311577
$931$	−6854.60	−0.241300
$932$	−7948.58	−0.279361
$933$	17671.5	0.620083
$934$	−5461.84	−0.191346
$935$	0	0
$936$	4878.29	0.170355
$937$	25444.9	0.887138	0.443569	−	0.896240i	$-0.353712\pi$
$937$	25444.9	0.887138	0.443569	+	0.896240i	$0.353712\pi$
$938$	−74459.8	−2.59190
$939$	8888.05	0.308893
$940$	−543.292	−0.0188513
$941$	49571.3	1.71730	0.858649	−	0.512564i	$-0.171304\pi$
$941$	49571.3	1.71730	0.858649	+	0.512564i	$0.171304\pi$
$942$	−30490.0	−1.05458
$943$	9191.34	0.317403
$944$	−96346.8	−3.32184
$945$	2328.75	0.0801631
$946$	0	0
$947$	24878.5	0.853687	0.426844	−	0.904325i	$-0.359625\pi$
$947$	24878.5	0.853687	0.426844	+	0.904325i	$0.359625\pi$
$948$	−6684.27	−0.229003
$949$	6182.36	0.211473
$950$	20171.1	0.688879
$951$	−3761.54	−0.128261
$952$	24389.9	0.830338
$953$	−1124.20	−0.0382123	−0.0191062	−	0.999817i	$-0.506082\pi$
$953$	−1124.20	−0.0382123	−0.0191062	+	0.999817i	$0.506082\pi$
$954$	35984.8	1.22123
$955$	−7575.69	−0.256695
$956$	88572.6	2.99649
$957$	0	0
$958$	−35717.0	−1.20456
$959$	6817.78	0.229570
$960$	−17233.2	−0.579375
$961$	−29669.7	−0.995928
$962$	−5646.06	−0.189227
$963$	1666.84	0.0557769
$964$	131019.	4.37742
$965$	−19235.9	−0.641685
$966$	41616.1	1.38611
$967$	25594.7	0.851158	0.425579	−	0.904921i	$-0.360070\pi$
$967$	25594.7	0.851158	0.425579	+	0.904921i	$0.360070\pi$
$968$	0	0
$969$	9490.82	0.314643
$970$	−34136.8	−1.12997
$971$	−2757.32	−0.0911294	−0.0455647	−	0.998961i	$-0.514509\pi$
$971$	−2757.32	−0.0911294	−0.0455647	+	0.998961i	$0.514509\pi$
$972$	−5007.22	−0.165233
$973$	36285.9	1.19555
$974$	−97165.1	−3.19648
$975$	−602.957	−0.0198052
$976$	149949.	4.91777
$977$	−9257.70	−0.303153	−0.151576	−	0.988446i	$-0.548435\pi$
$977$	−9257.70	−0.303153	−0.151576	+	0.988446i	$0.548435\pi$
$978$	26472.1	0.865526
$979$	0	0
$980$	−4681.47	−0.152596
$981$	15740.9	0.512301
$982$	−36549.5	−1.18772
$983$	−27457.9	−0.890916	−0.445458	−	0.895303i	$-0.646959\pi$
$983$	−27457.9	−0.890916	−0.445458	+	0.895303i	$0.646959\pi$
$984$	−12364.5	−0.400574
$985$	−23393.8	−0.756739
$986$	24122.1	0.779113
$987$	−272.887	−0.00880049
$988$	24990.6	0.804714
$989$	50686.4	1.62966
$990$	0	0
$991$	−28783.9	−0.922654	−0.461327	−	0.887230i	$-0.652626\pi$
$991$	−28783.9	−0.922654	−0.461327	+	0.887230i	$0.652626\pi$
$992$	−5591.14	−0.178951
$993$	4832.42	0.154433
$994$	47661.6	1.52086
$995$	−1223.15	−0.0389715
$996$	−17314.6	−0.550839
$997$	6956.92	0.220991	0.110495	−	0.993877i	$-0.464756\pi$
$997$	6956.92	0.220991	0.110495	+	0.993877i	$0.464756\pi$
$998$	−4759.36	−0.150957
$999$	3545.33	0.112281

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.bl.1.12		12
11.7	odd	10		165.4.m.c.16.1	✓	24
11.8	odd	10		165.4.m.c.31.1	yes	24
11.10	odd	2		1815.4.a.bi.1.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.m.c.16.1	✓	24	11.7	odd	10
165.4.m.c.31.1	yes	24	11.8	odd	10
1815.4.a.bi.1.1		12	11.10	odd	2
1815.4.a.bl.1.12		12	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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q+5.34844 q^{2} -3.00000 q^{3} +20.6059 q^{4} +5.00000 q^{5} -16.0453 q^{6} -17.2500 q^{7} +67.4218 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+5.34844 q^{2} -3.00000 q^{3} +20.6059 q^{4} +5.00000 q^{5} -16.0453 q^{6} -17.2500 q^{7} +67.4218 q^{8} +9.00000 q^{9} +26.7422 q^{10} -61.8176 q^{12} +8.03943 q^{13} -92.2606 q^{14} -15.0000 q^{15} +195.755 q^{16} -20.9711 q^{17} +48.1360 q^{18} +150.855 q^{19} +103.029 q^{20} +51.7499 q^{21} +150.357 q^{23} -202.265 q^{24} +25.0000 q^{25} +42.9984 q^{26} -27.0000 q^{27} -355.451 q^{28} -215.064 q^{29} -80.2267 q^{30} -11.0147 q^{31} +507.609 q^{32} -112.163 q^{34} -86.2499 q^{35} +185.453 q^{36} -131.308 q^{37} +806.842 q^{38} -24.1183 q^{39} +337.109 q^{40} +61.1300 q^{41} +276.782 q^{42} +337.107 q^{43} +45.0000 q^{45} +804.178 q^{46} -5.27318 q^{47} -587.264 q^{48} -45.4382 q^{49} +133.711 q^{50} +62.9133 q^{51} +165.659 q^{52} +747.565 q^{53} -144.408 q^{54} -1163.02 q^{56} -452.566 q^{57} -1150.26 q^{58} -492.181 q^{59} -309.088 q^{60} +766.004 q^{61} -58.9113 q^{62} -155.250 q^{63} +1148.88 q^{64} +40.1971 q^{65} +807.060 q^{67} -432.128 q^{68} -451.072 q^{69} -461.303 q^{70} -516.598 q^{71} +606.796 q^{72} +769.005 q^{73} -702.296 q^{74} -75.0000 q^{75} +3108.51 q^{76} -128.995 q^{78} +108.129 q^{79} +978.773 q^{80} +81.0000 q^{81} +326.951 q^{82} +280.093 q^{83} +1066.35 q^{84} -104.856 q^{85} +1803.00 q^{86} +645.191 q^{87} +379.836 q^{89} +240.680 q^{90} -138.680 q^{91} +3098.24 q^{92} +33.0440 q^{93} -28.2033 q^{94} +754.277 q^{95} -1522.83 q^{96} -1276.51 q^{97} -243.024 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} - 36 q^{3} + 49 q^{4} + 60 q^{5} - 3 q^{6} - 5 q^{7} - 3 q^{8} + 108 q^{9}+O(q^{10}) \) 12 * q + q^2 - 36 * q^3 + 49 * q^4 + 60 * q^5 - 3 * q^6 - 5 * q^7 - 3 * q^8 + 108 * q^9	\( 12 q + q^{2} - 36 q^{3} + 49 q^{4} + 60 q^{5} - 3 q^{6} - 5 q^{7} - 3 q^{8} + 108 q^{9} + 5 q^{10} - 147 q^{12} + 34 q^{13} - 134 q^{14} - 180 q^{15} + 265 q^{16} + 169 q^{17} + 9 q^{18} + 203 q^{19} + 245 q^{20} + 15 q^{21} + 116 q^{23} + 9 q^{24} + 300 q^{25} + 216 q^{26} - 324 q^{27} + 80 q^{28} + 409 q^{29} - 15 q^{30} - 645 q^{31} + 532 q^{32} - 1442 q^{34} - 25 q^{35} + 441 q^{36} + 40 q^{37} + 404 q^{38} - 102 q^{39} - 15 q^{40} + 1071 q^{41} + 402 q^{42} - 101 q^{43} + 540 q^{45} + 1539 q^{46} - 572 q^{47} - 795 q^{48} + 1563 q^{49} + 25 q^{50} - 507 q^{51} + 1081 q^{52} + 611 q^{53} - 27 q^{54} - 3221 q^{56} - 609 q^{57} - 354 q^{58} + 958 q^{59} - 735 q^{60} + 2620 q^{61} + 1049 q^{62} - 45 q^{63} + 2931 q^{64} + 170 q^{65} - 1027 q^{67} + 1896 q^{68} - 348 q^{69} - 670 q^{70} + 1020 q^{71} - 27 q^{72} + 2244 q^{73} - 283 q^{74} - 900 q^{75} + 4958 q^{76} - 648 q^{78} + 2199 q^{79} + 1325 q^{80} + 972 q^{81} + 362 q^{82} - 602 q^{83} - 240 q^{84} + 845 q^{85} - 1589 q^{86} - 1227 q^{87} - 2413 q^{89} + 45 q^{90} + 302 q^{91} + 3030 q^{92} + 1935 q^{93} + 2885 q^{94} + 1015 q^{95} - 1596 q^{96} + 886 q^{97} - 1236 q^{98}+O(q^{100}) \) 12 * q + q^2 - 36 * q^3 + 49 * q^4 + 60 * q^5 - 3 * q^6 - 5 * q^7 - 3 * q^8 + 108 * q^9 + 5 * q^10 - 147 * q^12 + 34 * q^13 - 134 * q^14 - 180 * q^15 + 265 * q^16 + 169 * q^17 + 9 * q^18 + 203 * q^19 + 245 * q^20 + 15 * q^21 + 116 * q^23 + 9 * q^24 + 300 * q^25 + 216 * q^26 - 324 * q^27 + 80 * q^28 + 409 * q^29 - 15 * q^30 - 645 * q^31 + 532 * q^32 - 1442 * q^34 - 25 * q^35 + 441 * q^36 + 40 * q^37 + 404 * q^38 - 102 * q^39 - 15 * q^40 + 1071 * q^41 + 402 * q^42 - 101 * q^43 + 540 * q^45 + 1539 * q^46 - 572 * q^47 - 795 * q^48 + 1563 * q^49 + 25 * q^50 - 507 * q^51 + 1081 * q^52 + 611 * q^53 - 27 * q^54 - 3221 * q^56 - 609 * q^57 - 354 * q^58 + 958 * q^59 - 735 * q^60 + 2620 * q^61 + 1049 * q^62 - 45 * q^63 + 2931 * q^64 + 170 * q^65 - 1027 * q^67 + 1896 * q^68 - 348 * q^69 - 670 * q^70 + 1020 * q^71 - 27 * q^72 + 2244 * q^73 - 283 * q^74 - 900 * q^75 + 4958 * q^76 - 648 * q^78 + 2199 * q^79 + 1325 * q^80 + 972 * q^81 + 362 * q^82 - 602 * q^83 - 240 * q^84 + 845 * q^85 - 1589 * q^86 - 1227 * q^87 - 2413 * q^89 + 45 * q^90 + 302 * q^91 + 3030 * q^92 + 1935 * q^93 + 2885 * q^94 + 1015 * q^95 - 1596 * q^96 + 886 * q^97 - 1236 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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