LMFDB - Embedded newform 1815.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,2,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, 2, names="a")

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

chi := DirichletCharacter("1815.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$14.4928479669$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.456850$ 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)

$f(q)$	$=$	$q-2.18890 q^{2} +1.00000 q^{3} +2.79129 q^{4} -1.00000 q^{5} -2.18890 q^{6} -0.913701 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.18890 q^{2} +1.00000 q^{3} +2.79129 q^{4} -1.00000 q^{5} -2.18890 q^{6} -0.913701 q^{7} -1.73205 q^{8} +1.00000 q^{9} +2.18890 q^{10} +2.79129 q^{12} -4.37780 q^{13} +2.00000 q^{14} -1.00000 q^{15} -1.79129 q^{16} +1.73205 q^{17} -2.18890 q^{18} -3.46410 q^{19} -2.79129 q^{20} -0.913701 q^{21} +8.58258 q^{23} -1.73205 q^{24} +1.00000 q^{25} +9.58258 q^{26} +1.00000 q^{27} -2.55040 q^{28} -6.20520 q^{29} +2.18890 q^{30} +0.582576 q^{31} +7.38505 q^{32} -3.79129 q^{34} +0.913701 q^{35} +2.79129 q^{36} -1.58258 q^{37} +7.58258 q^{38} -4.37780 q^{39} +1.73205 q^{40} +3.46410 q^{41} +2.00000 q^{42} +6.20520 q^{43} -1.00000 q^{45} -18.7864 q^{46} +12.5826 q^{47} -1.79129 q^{48} -6.16515 q^{49} -2.18890 q^{50} +1.73205 q^{51} -12.2197 q^{52} -6.16515 q^{53} -2.18890 q^{54} +1.58258 q^{56} -3.46410 q^{57} +13.5826 q^{58} +4.41742 q^{59} -2.79129 q^{60} +7.02355 q^{61} -1.27520 q^{62} -0.913701 q^{63} -12.5826 q^{64} +4.37780 q^{65} -13.5826 q^{67} +4.83465 q^{68} +8.58258 q^{69} -2.00000 q^{70} +8.00000 q^{71} -1.73205 q^{72} +15.6838 q^{73} +3.46410 q^{74} +1.00000 q^{75} -9.66930 q^{76} +9.58258 q^{78} +6.10985 q^{79} +1.79129 q^{80} +1.00000 q^{81} -7.58258 q^{82} +15.8745 q^{83} -2.55040 q^{84} -1.73205 q^{85} -13.5826 q^{86} -6.20520 q^{87} -12.7477 q^{89} +2.18890 q^{90} +4.00000 q^{91} +23.9564 q^{92} +0.582576 q^{93} -27.5420 q^{94} +3.46410 q^{95} +7.38505 q^{96} +13.5826 q^{97} +13.4949 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{4} - 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 2 * q^4 - 4 * q^5 + 4 * q^9	$ 4 q + 4 q^{3} + 2 q^{4} - 4 q^{5} + 4 q^{9} + 2 q^{12} + 8 q^{14} - 4 q^{15} + 2 q^{16} - 2 q^{20} + 16 q^{23} + 4 q^{25} + 20 q^{26} + 4 q^{27} - 16 q^{31} - 6 q^{34} + 2 q^{36} + 12 q^{37} + 12 q^{38} + 8 q^{42} - 4 q^{45} + 32 q^{47} + 2 q^{48} + 12 q^{49} + 12 q^{53} - 12 q^{56} + 36 q^{58} + 36 q^{59} - 2 q^{60} - 32 q^{64} - 36 q^{67} + 16 q^{69} - 8 q^{70} + 32 q^{71} + 4 q^{75} + 20 q^{78} - 2 q^{80} + 4 q^{81} - 12 q^{82} - 36 q^{86} + 4 q^{89} + 16 q^{91} + 50 q^{92} - 16 q^{93} + 36 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 2 * q^4 - 4 * q^5 + 4 * q^9 + 2 * q^12 + 8 * q^14 - 4 * q^15 + 2 * q^16 - 2 * q^20 + 16 * q^23 + 4 * q^25 + 20 * q^26 + 4 * q^27 - 16 * q^31 - 6 * q^34 + 2 * q^36 + 12 * q^37 + 12 * q^38 + 8 * q^42 - 4 * q^45 + 32 * q^47 + 2 * q^48 + 12 * q^49 + 12 * q^53 - 12 * q^56 + 36 * q^58 + 36 * q^59 - 2 * q^60 - 32 * q^64 - 36 * q^67 + 16 * q^69 - 8 * q^70 + 32 * q^71 + 4 * q^75 + 20 * q^78 - 2 * q^80 + 4 * q^81 - 12 * q^82 - 36 * q^86 + 4 * q^89 + 16 * q^91 + 50 * q^92 - 16 * q^93 + 36 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.18890	−1.54779	−0.773893	−	0.633316i	$-0.781693\pi$
$2$	−2.18890	−1.54779	−0.773893	+	0.633316i	$0.781693\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.79129	1.39564
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.18890	−0.893615
$7$	−0.913701	−0.345346	−0.172673	−	0.984979i	$-0.555240\pi$
$7$	−0.913701	−0.345346	−0.172673	+	0.984979i	$0.555240\pi$
$8$	−1.73205	−0.612372
$9$	1.00000	0.333333
$10$	2.18890	0.692191
$11$	0	0
$11$	0	0
$12$	2.79129	0.805775
$13$	−4.37780	−1.21418	−0.607092	−	0.794632i	$-0.707664\pi$
$13$	−4.37780	−1.21418	−0.607092	+	0.794632i	$0.707664\pi$
$14$	2.00000	0.534522
$15$	−1.00000	−0.258199
$16$	−1.79129	−0.447822
$17$	1.73205	0.420084	0.210042	−	0.977692i	$-0.432640\pi$
$17$	1.73205	0.420084	0.210042	+	0.977692i	$0.432640\pi$
$18$	−2.18890	−0.515929
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	−2.79129	−0.624151
$21$	−0.913701	−0.199386
$22$	0	0
$23$	8.58258	1.78959	0.894795	−	0.446476i	$-0.147321\pi$
$23$	8.58258	1.78959	0.894795	+	0.446476i	$0.147321\pi$
$24$	−1.73205	−0.353553
$25$	1.00000	0.200000
$26$	9.58258	1.87930
$27$	1.00000	0.192450
$28$	−2.55040	−0.481981
$29$	−6.20520	−1.15228	−0.576139	−	0.817352i	$-0.695441\pi$
$29$	−6.20520	−1.15228	−0.576139	+	0.817352i	$0.695441\pi$
$30$	2.18890	0.399637
$31$	0.582576	0.104634	0.0523168	−	0.998631i	$-0.483339\pi$
$31$	0.582576	0.104634	0.0523168	+	0.998631i	$0.483339\pi$
$32$	7.38505	1.30551
$33$	0	0
$34$	−3.79129	−0.650201
$35$	0.913701	0.154444
$36$	2.79129	0.465215
$37$	−1.58258	−0.260174	−0.130087	−	0.991503i	$-0.541526\pi$
$37$	−1.58258	−0.260174	−0.130087	+	0.991503i	$0.541526\pi$
$38$	7.58258	1.23006
$39$	−4.37780	−0.701009
$40$	1.73205	0.273861
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	2.00000	0.308607
$43$	6.20520	0.946285	0.473142	−	0.880986i	$-0.343120\pi$
$43$	6.20520	0.946285	0.473142	+	0.880986i	$0.343120\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	−18.7864	−2.76990
$47$	12.5826	1.83536	0.917679	−	0.397323i	$-0.130061\pi$
$47$	12.5826	1.83536	0.917679	+	0.397323i	$0.130061\pi$
$48$	−1.79129	−0.258550
$49$	−6.16515	−0.880736
$50$	−2.18890	−0.309557
$51$	1.73205	0.242536
$52$	−12.2197	−1.69457
$53$	−6.16515	−0.846849	−0.423424	−	0.905931i	$-0.639172\pi$
$53$	−6.16515	−0.846849	−0.423424	+	0.905931i	$0.639172\pi$
$54$	−2.18890	−0.297872
$55$	0	0
$56$	1.58258	0.211481
$57$	−3.46410	−0.458831
$58$	13.5826	1.78348
$59$	4.41742	0.575100	0.287550	−	0.957766i	$-0.407159\pi$
$59$	4.41742	0.575100	0.287550	+	0.957766i	$0.407159\pi$
$60$	−2.79129	−0.360354
$61$	7.02355	0.899274	0.449637	−	0.893211i	$-0.351553\pi$
$61$	7.02355	0.899274	0.449637	+	0.893211i	$0.351553\pi$
$62$	−1.27520	−0.161951
$63$	−0.913701	−0.115115
$64$	−12.5826	−1.57282
$65$	4.37780	0.543000
$66$	0	0
$67$	−13.5826	−1.65938	−0.829688	−	0.558228i	$-0.811482\pi$
$67$	−13.5826	−1.65938	−0.829688	+	0.558228i	$0.811482\pi$
$68$	4.83465	0.586288
$69$	8.58258	1.03322
$70$	−2.00000	−0.239046
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.73205	−0.204124
$73$	15.6838	1.83565	0.917825	−	0.396984i	$-0.129943\pi$
$73$	15.6838	1.83565	0.917825	+	0.396984i	$0.129943\pi$
$74$	3.46410	0.402694
$75$	1.00000	0.115470
$76$	−9.66930	−1.10915
$77$	0	0
$78$	9.58258	1.08501
$79$	6.10985	0.687412	0.343706	−	0.939077i	$-0.388318\pi$
$79$	6.10985	0.687412	0.343706	+	0.939077i	$0.388318\pi$
$80$	1.79129	0.200272
$81$	1.00000	0.111111
$82$	−7.58258	−0.837355
$83$	15.8745	1.74245	0.871227	−	0.490881i	$-0.163325\pi$
$83$	15.8745	1.74245	0.871227	+	0.490881i	$0.163325\pi$
$84$	−2.55040	−0.278272
$85$	−1.73205	−0.187867
$86$	−13.5826	−1.46465
$87$	−6.20520	−0.665268
$88$	0	0
$89$	−12.7477	−1.35126	−0.675628	−	0.737243i	$-0.736128\pi$
$89$	−12.7477	−1.35126	−0.675628	+	0.737243i	$0.736128\pi$
$90$	2.18890	0.230730
$91$	4.00000	0.419314
$92$	23.9564	2.49763
$93$	0.582576	0.0604103
$94$	−27.5420	−2.84074
$95$	3.46410	0.355409
$96$	7.38505	0.753734
$97$	13.5826	1.37910	0.689551	−	0.724237i	$-0.257808\pi$
$97$	13.5826	1.37910	0.689551	+	0.724237i	$0.257808\pi$
$98$	13.4949	1.36319
$99$	0	0
$100$	2.79129	0.279129
$101$	−8.75560	−0.871215	−0.435608	−	0.900137i	$-0.643466\pi$
$101$	−8.75560	−0.871215	−0.435608	+	0.900137i	$0.643466\pi$
$102$	−3.79129	−0.375393
$103$	−11.1652	−1.10014	−0.550068	−	0.835120i	$-0.685398\pi$
$103$	−11.1652	−1.10014	−0.550068	+	0.835120i	$0.685398\pi$
$104$	7.58258	0.743533
$105$	0.913701	0.0891680
$106$	13.4949	1.31074
$107$	−7.74655	−0.748888	−0.374444	−	0.927250i	$-0.622166\pi$
$107$	−7.74655	−0.748888	−0.374444	+	0.927250i	$0.622166\pi$
$108$	2.79129	0.268592
$109$	−6.92820	−0.663602	−0.331801	−	0.943349i	$-0.607656\pi$
$109$	−6.92820	−0.663602	−0.331801	+	0.943349i	$0.607656\pi$
$110$	0	0
$111$	−1.58258	−0.150211
$112$	1.63670	0.154654
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	7.58258	0.710173
$115$	−8.58258	−0.800329
$116$	−17.3205	−1.60817
$117$	−4.37780	−0.404728
$118$	−9.66930	−0.890132
$119$	−1.58258	−0.145074
$120$	1.73205	0.158114
$121$	0	0
$122$	−15.3739	−1.39188
$123$	3.46410	0.312348
$124$	1.62614	0.146031
$125$	−1.00000	−0.0894427
$126$	2.00000	0.178174
$127$	2.55040	0.226312	0.113156	−	0.993577i	$-0.463904\pi$
$127$	2.55040	0.226312	0.113156	+	0.993577i	$0.463904\pi$
$128$	12.7719	1.12889
$129$	6.20520	0.546338
$130$	−9.58258	−0.840447
$131$	−0.190700	−0.0166616	−0.00833079	−	0.999965i	$-0.502652\pi$
$131$	−0.190700	−0.0166616	−0.00833079	+	0.999965i	$0.502652\pi$
$132$	0	0
$133$	3.16515	0.274453
$134$	29.7309	2.56836
$135$	−1.00000	−0.0860663
$136$	−3.00000	−0.257248
$137$	17.3303	1.48063	0.740314	−	0.672261i	$-0.234677\pi$
$137$	17.3303	1.48063	0.740314	+	0.672261i	$0.234677\pi$
$138$	−18.7864	−1.59921
$139$	13.0381	1.10587	0.552937	−	0.833223i	$-0.313507\pi$
$139$	13.0381	1.10587	0.552937	+	0.833223i	$0.313507\pi$
$140$	2.55040	0.215548
$141$	12.5826	1.05964
$142$	−17.5112	−1.46951
$143$	0	0
$144$	−1.79129	−0.149274
$145$	6.20520	0.515314
$146$	−34.3303	−2.84120
$147$	−6.16515	−0.508493
$148$	−4.41742	−0.363110
$149$	13.1334	1.07593	0.537965	−	0.842967i	$-0.319193\pi$
$149$	13.1334	1.07593	0.537965	+	0.842967i	$0.319193\pi$
$150$	−2.18890	−0.178723
$151$	13.0381	1.06102	0.530511	−	0.847678i	$-0.322000\pi$
$151$	13.0381	1.06102	0.530511	+	0.847678i	$0.322000\pi$
$152$	6.00000	0.486664
$153$	1.73205	0.140028
$154$	0	0
$155$	−0.582576	−0.0467936
$156$	−12.2197	−0.978359
$157$	−16.7477	−1.33661	−0.668307	−	0.743886i	$-0.732981\pi$
$157$	−16.7477	−1.33661	−0.668307	+	0.743886i	$0.732981\pi$
$158$	−13.3739	−1.06397
$159$	−6.16515	−0.488928
$160$	−7.38505	−0.583840
$161$	−7.84190	−0.618029
$162$	−2.18890	−0.171976
$163$	6.74773	0.528523	0.264261	−	0.964451i	$-0.414872\pi$
$163$	6.74773	0.528523	0.264261	+	0.964451i	$0.414872\pi$
$164$	9.66930	0.755046
$165$	0	0
$166$	−34.7477	−2.69695
$167$	−11.4014	−0.882263	−0.441132	−	0.897442i	$-0.645423\pi$
$167$	−11.4014	−0.882263	−0.441132	+	0.897442i	$0.645423\pi$
$168$	1.58258	0.122098
$169$	6.16515	0.474242
$170$	3.79129	0.290779
$171$	−3.46410	−0.264906
$172$	17.3205	1.32068
$173$	−6.92820	−0.526742	−0.263371	−	0.964695i	$-0.584834\pi$
$173$	−6.92820	−0.526742	−0.263371	+	0.964695i	$0.584834\pi$
$174$	13.5826	1.02969
$175$	−0.913701	−0.0690693
$176$	0	0
$177$	4.41742	0.332034
$178$	27.9035	2.09146
$179$	−1.58258	−0.118287	−0.0591436	−	0.998249i	$-0.518837\pi$
$179$	−1.58258	−0.118287	−0.0591436	+	0.998249i	$0.518837\pi$
$180$	−2.79129	−0.208050
$181$	13.1652	0.978558	0.489279	−	0.872127i	$-0.337260\pi$
$181$	13.1652	0.978558	0.489279	+	0.872127i	$0.337260\pi$
$182$	−8.75560	−0.649009
$183$	7.02355	0.519196
$184$	−14.8655	−1.09590
$185$	1.58258	0.116353
$186$	−1.27520	−0.0935022
$187$	0	0
$188$	35.1216	2.56151
$189$	−0.913701	−0.0664619
$190$	−7.58258	−0.550098
$191$	4.74773	0.343533	0.171767	−	0.985138i	$-0.445052\pi$
$191$	4.74773	0.343533	0.171767	+	0.985138i	$0.445052\pi$
$192$	−12.5826	−0.908069
$193$	21.6983	1.56188	0.780939	−	0.624607i	$-0.214741\pi$
$193$	21.6983	1.56188	0.780939	+	0.624607i	$0.214741\pi$
$194$	−29.7309	−2.13456
$195$	4.37780	0.313501
$196$	−17.2087	−1.22919
$197$	19.3386	1.37782	0.688909	−	0.724847i	$-0.258090\pi$
$197$	19.3386	1.37782	0.688909	+	0.724847i	$0.258090\pi$
$198$	0	0
$199$	11.7477	0.832774	0.416387	−	0.909187i	$-0.363296\pi$
$199$	11.7477	0.832774	0.416387	+	0.909187i	$0.363296\pi$
$200$	−1.73205	−0.122474
$201$	−13.5826	−0.958041
$202$	19.1652	1.34846
$203$	5.66970	0.397935
$204$	4.83465	0.338493
$205$	−3.46410	−0.241943
$206$	24.4394	1.70277
$207$	8.58258	0.596530
$208$	7.84190	0.543738
$209$	0	0
$210$	−2.00000	−0.138013
$211$	−18.1389	−1.24873	−0.624365	−	0.781133i	$-0.714642\pi$
$211$	−18.1389	−1.24873	−0.624365	+	0.781133i	$0.714642\pi$
$212$	−17.2087	−1.18190
$213$	8.00000	0.548151
$214$	16.9564	1.15912
$215$	−6.20520	−0.423191
$216$	−1.73205	−0.117851
$217$	−0.532300	−0.0361349
$218$	15.1652	1.02711
$219$	15.6838	1.05981
$220$	0	0
$221$	−7.58258	−0.510059
$222$	3.46410	0.232495
$223$	20.3303	1.36142	0.680709	−	0.732554i	$-0.261672\pi$
$223$	20.3303	1.36142	0.680709	+	0.732554i	$0.261672\pi$
$224$	−6.74773	−0.450851
$225$	1.00000	0.0666667
$226$	−19.7001	−1.31043
$227$	6.30055	0.418182	0.209091	−	0.977896i	$-0.432949\pi$
$227$	6.30055	0.418182	0.209091	+	0.977896i	$0.432949\pi$
$228$	−9.66930	−0.640365
$229$	17.0000	1.12339	0.561696	−	0.827344i	$-0.310149\pi$
$229$	17.0000	1.12339	0.561696	+	0.827344i	$0.310149\pi$
$230$	18.7864	1.23874
$231$	0	0
$232$	10.7477	0.705623
$233$	17.6066	1.15344	0.576722	−	0.816940i	$-0.304332\pi$
$233$	17.6066	1.15344	0.576722	+	0.816940i	$0.304332\pi$
$234$	9.58258	0.626433
$235$	−12.5826	−0.820797
$236$	12.3303	0.802634
$237$	6.10985	0.396878
$238$	3.46410	0.224544
$239$	26.9898	1.74583	0.872913	−	0.487876i	$-0.162228\pi$
$239$	26.9898	1.74583	0.872913	+	0.487876i	$0.162228\pi$
$240$	1.79129	0.115627
$241$	−5.00545	−0.322430	−0.161215	−	0.986919i	$-0.551541\pi$
$241$	−5.00545	−0.322430	−0.161215	+	0.986919i	$0.551541\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	19.6048	1.25507
$245$	6.16515	0.393877
$246$	−7.58258	−0.483447
$247$	15.1652	0.964935
$248$	−1.00905	−0.0640748
$249$	15.8745	1.00601
$250$	2.18890	0.138438
$251$	3.58258	0.226130	0.113065	−	0.993588i	$-0.463933\pi$
$251$	3.58258	0.226130	0.113065	+	0.993588i	$0.463933\pi$
$252$	−2.55040	−0.160660
$253$	0	0
$254$	−5.58258	−0.350282
$255$	−1.73205	−0.108465
$256$	−2.79129	−0.174455
$257$	24.1652	1.50738	0.753690	−	0.657230i	$-0.228272\pi$
$257$	24.1652	1.50738	0.753690	+	0.657230i	$0.228272\pi$
$258$	−13.5826	−0.845614
$259$	1.44600	0.0898501
$260$	12.2197	0.757834
$261$	−6.20520	−0.384092
$262$	0.417424	0.0257886
$263$	4.47315	0.275826	0.137913	−	0.990444i	$-0.455961\pi$
$263$	4.47315	0.275826	0.137913	+	0.990444i	$0.455961\pi$
$264$	0	0
$265$	6.16515	0.378722
$266$	−6.92820	−0.424795
$267$	−12.7477	−0.780148
$268$	−37.9129	−2.31590
$269$	−27.1652	−1.65629	−0.828144	−	0.560515i	$-0.810603\pi$
$269$	−27.1652	−1.65629	−0.828144	+	0.560515i	$0.810603\pi$
$270$	2.18890	0.133212
$271$	−9.76465	−0.593161	−0.296580	−	0.955008i	$-0.595846\pi$
$271$	−9.76465	−0.593161	−0.296580	+	0.955008i	$0.595846\pi$
$272$	−3.10260	−0.188123
$273$	4.00000	0.242091
$274$	−37.9343	−2.29170
$275$	0	0
$276$	23.9564	1.44201
$277$	8.75560	0.526073	0.263037	−	0.964786i	$-0.415276\pi$
$277$	8.75560	0.526073	0.263037	+	0.964786i	$0.415276\pi$
$278$	−28.5390	−1.71166
$279$	0.582576	0.0348779
$280$	−1.58258	−0.0945770
$281$	13.3241	0.794850	0.397425	−	0.917635i	$-0.369904\pi$
$281$	13.3241	0.794850	0.397425	+	0.917635i	$0.369904\pi$
$282$	−27.5420	−1.64010
$283$	31.5583	1.87595	0.937974	−	0.346707i	$-0.112700\pi$
$283$	31.5583	1.87595	0.937974	+	0.346707i	$0.112700\pi$
$284$	22.3303	1.32506
$285$	3.46410	0.205196
$286$	0	0
$287$	−3.16515	−0.186833
$288$	7.38505	0.435168
$289$	−14.0000	−0.823529
$290$	−13.5826	−0.797596
$291$	13.5826	0.796225
$292$	43.7780	2.56191
$293$	1.73205	0.101187	0.0505937	−	0.998719i	$-0.483889\pi$
$293$	1.73205	0.101187	0.0505937	+	0.998719i	$0.483889\pi$
$294$	13.4949	0.787039
$295$	−4.41742	−0.257192
$296$	2.74110	0.159323
$297$	0	0
$298$	−28.7477	−1.66531
$299$	−37.5728	−2.17289
$300$	2.79129	0.161155
$301$	−5.66970	−0.326796
$302$	−28.5390	−1.64224
$303$	−8.75560	−0.502996
$304$	6.20520	0.355893
$305$	−7.02355	−0.402167
$306$	−3.79129	−0.216734
$307$	−26.0761	−1.48824	−0.744121	−	0.668045i	$-0.767131\pi$
$307$	−26.0761	−1.48824	−0.744121	+	0.668045i	$0.767131\pi$
$308$	0	0
$309$	−11.1652	−0.635163
$310$	1.27520	0.0724265
$311$	−2.41742	−0.137080	−0.0685398	−	0.997648i	$-0.521834\pi$
$311$	−2.41742	−0.137080	−0.0685398	+	0.997648i	$0.521834\pi$
$312$	7.58258	0.429279
$313$	−3.58258	−0.202499	−0.101250	−	0.994861i	$-0.532284\pi$
$313$	−3.58258	−0.202499	−0.101250	+	0.994861i	$0.532284\pi$
$314$	36.6591	2.06879
$315$	0.913701	0.0514812
$316$	17.0544	0.959383
$317$	11.8348	0.664711	0.332356	−	0.943154i	$-0.392157\pi$
$317$	11.8348	0.664711	0.332356	+	0.943154i	$0.392157\pi$
$318$	13.4949	0.756757
$319$	0	0
$320$	12.5826	0.703387
$321$	−7.74655	−0.432370
$322$	17.1652	0.956576
$323$	−6.00000	−0.333849
$324$	2.79129	0.155072
$325$	−4.37780	−0.242837
$326$	−14.7701	−0.818041
$327$	−6.92820	−0.383131
$328$	−6.00000	−0.331295
$329$	−11.4967	−0.633834
$330$	0	0
$331$	0.252273	0.0138662	0.00693309	−	0.999976i	$-0.497793\pi$
$331$	0.252273	0.0138662	0.00693309	+	0.999976i	$0.497793\pi$
$332$	44.3103	2.43184
$333$	−1.58258	−0.0867246
$334$	24.9564	1.36556
$335$	13.5826	0.742095
$336$	1.63670	0.0892893
$337$	−20.9753	−1.14260	−0.571299	−	0.820742i	$-0.693560\pi$
$337$	−20.9753	−1.14260	−0.571299	+	0.820742i	$0.693560\pi$
$338$	−13.4949	−0.734026
$339$	9.00000	0.488813
$340$	−4.83465	−0.262196
$341$	0	0
$342$	7.58258	0.410019
$343$	12.0290	0.649505
$344$	−10.7477	−0.579479
$345$	−8.58258	−0.462070
$346$	15.1652	0.815284
$347$	−15.0562	−0.808257	−0.404128	−	0.914702i	$-0.632425\pi$
$347$	−15.0562	−0.808257	−0.404128	+	0.914702i	$0.632425\pi$
$348$	−17.3205	−0.928477
$349$	−25.9808	−1.39072	−0.695359	−	0.718662i	$-0.744755\pi$
$349$	−25.9808	−1.39072	−0.695359	+	0.718662i	$0.744755\pi$
$350$	2.00000	0.106904
$351$	−4.37780	−0.233670
$352$	0	0
$353$	36.1652	1.92488	0.962438	−	0.271500i	$-0.0875198\pi$
$353$	36.1652	1.92488	0.962438	+	0.271500i	$0.0875198\pi$
$354$	−9.66930	−0.513918
$355$	−8.00000	−0.424596
$356$	−35.5826	−1.88587
$357$	−1.58258	−0.0837588
$358$	3.46410	0.183083
$359$	−1.82740	−0.0964465	−0.0482233	−	0.998837i	$-0.515356\pi$
$359$	−1.82740	−0.0964465	−0.0482233	+	0.998837i	$0.515356\pi$
$360$	1.73205	0.0912871
$361$	−7.00000	−0.368421
$362$	−28.8172	−1.51460
$363$	0	0
$364$	11.1652	0.585213
$365$	−15.6838	−0.820928
$366$	−15.3739	−0.803605
$367$	−28.7477	−1.50062	−0.750310	−	0.661087i	$-0.770096\pi$
$367$	−28.7477	−1.50062	−0.750310	+	0.661087i	$0.770096\pi$
$368$	−15.3739	−0.801418
$369$	3.46410	0.180334
$370$	−3.46410	−0.180090
$371$	5.63310	0.292456
$372$	1.62614	0.0843112
$373$	−13.8564	−0.717458	−0.358729	−	0.933442i	$-0.616790\pi$
$373$	−13.8564	−0.717458	−0.358729	+	0.933442i	$0.616790\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−21.7937	−1.12392
$377$	27.1652	1.39908
$378$	2.00000	0.102869
$379$	11.7477	0.603440	0.301720	−	0.953397i	$-0.402439\pi$
$379$	11.7477	0.603440	0.301720	+	0.953397i	$0.402439\pi$
$380$	9.66930	0.496025
$381$	2.55040	0.130661
$382$	−10.3923	−0.531717
$383$	−30.3303	−1.54981	−0.774903	−	0.632080i	$-0.782201\pi$
$383$	−30.3303	−1.54981	−0.774903	+	0.632080i	$0.782201\pi$
$384$	12.7719	0.651764
$385$	0	0
$386$	−47.4955	−2.41745
$387$	6.20520	0.315428
$388$	37.9129	1.92473
$389$	−22.7477	−1.15336	−0.576678	−	0.816972i	$-0.695651\pi$
$389$	−22.7477	−1.15336	−0.576678	+	0.816972i	$0.695651\pi$
$390$	−9.58258	−0.485233
$391$	14.8655	0.751778
$392$	10.6784	0.539338
$393$	−0.190700	−0.00961956
$394$	−42.3303	−2.13257
$395$	−6.10985	−0.307420
$396$	0	0
$397$	−4.33030	−0.217332	−0.108666	−	0.994078i	$-0.534658\pi$
$397$	−4.33030	−0.217332	−0.108666	+	0.994078i	$0.534658\pi$
$398$	−25.7146	−1.28896
$399$	3.16515	0.158456
$400$	−1.79129	−0.0895644
$401$	−22.3303	−1.11512	−0.557561	−	0.830136i	$-0.688263\pi$
$401$	−22.3303	−1.11512	−0.557561	+	0.830136i	$0.688263\pi$
$402$	29.7309	1.48284
$403$	−2.55040	−0.127045
$404$	−24.4394	−1.21591
$405$	−1.00000	−0.0496904
$406$	−12.4104	−0.615918
$407$	0	0
$408$	−3.00000	−0.148522
$409$	−22.8981	−1.13224	−0.566118	−	0.824324i	$-0.691555\pi$
$409$	−22.8981	−1.13224	−0.566118	+	0.824324i	$0.691555\pi$
$410$	7.58258	0.374477
$411$	17.3303	0.854841
$412$	−31.1652	−1.53540
$413$	−4.03620	−0.198609
$414$	−18.7864	−0.923302
$415$	−15.8745	−0.779249
$416$	−32.3303	−1.58512
$417$	13.0381	0.638476
$418$	0	0
$419$	32.3303	1.57944	0.789719	−	0.613468i	$-0.210226\pi$
$419$	32.3303	1.57944	0.789719	+	0.613468i	$0.210226\pi$
$420$	2.55040	0.124447
$421$	−2.66970	−0.130113	−0.0650565	−	0.997882i	$-0.520723\pi$
$421$	−2.66970	−0.130113	−0.0650565	+	0.997882i	$0.520723\pi$
$422$	39.7042	1.93277
$423$	12.5826	0.611786
$424$	10.6784	0.518587
$425$	1.73205	0.0840168
$426$	−17.5112	−0.848421
$427$	−6.41742	−0.310561
$428$	−21.6229	−1.04518
$429$	0	0
$430$	13.5826	0.655010
$431$	−20.0616	−0.966334	−0.483167	−	0.875528i	$-0.660514\pi$
$431$	−20.0616	−0.966334	−0.483167	+	0.875528i	$0.660514\pi$
$432$	−1.79129	−0.0861834
$433$	−4.41742	−0.212288	−0.106144	−	0.994351i	$-0.533850\pi$
$433$	−4.41742	−0.212288	−0.106144	+	0.994351i	$0.533850\pi$
$434$	1.16515	0.0559291
$435$	6.20520	0.297517
$436$	−19.3386	−0.926151
$437$	−29.7309	−1.42222
$438$	−34.3303	−1.64037
$439$	−9.57395	−0.456940	−0.228470	−	0.973551i	$-0.573372\pi$
$439$	−9.57395	−0.456940	−0.228470	+	0.973551i	$0.573372\pi$
$440$	0	0
$441$	−6.16515	−0.293579
$442$	16.5975	0.789463
$443$	7.16515	0.340427	0.170213	−	0.985407i	$-0.445554\pi$
$443$	7.16515	0.340427	0.170213	+	0.985407i	$0.445554\pi$
$444$	−4.41742	−0.209642
$445$	12.7477	0.604300
$446$	−44.5010	−2.10718
$447$	13.1334	0.621189
$448$	11.4967	0.543168
$449$	2.83485	0.133785	0.0668924	−	0.997760i	$-0.478692\pi$
$449$	2.83485	0.133785	0.0668924	+	0.997760i	$0.478692\pi$
$450$	−2.18890	−0.103186
$451$	0	0
$452$	25.1216	1.18162
$453$	13.0381	0.612581
$454$	−13.7913	−0.647257
$455$	−4.00000	−0.187523
$456$	6.00000	0.280976
$457$	22.4213	1.04882	0.524412	−	0.851464i	$-0.324285\pi$
$457$	22.4213	1.04882	0.524412	+	0.851464i	$0.324285\pi$
$458$	−37.2113	−1.73877
$459$	1.73205	0.0808452
$460$	−23.9564	−1.11697
$461$	−8.03260	−0.374116	−0.187058	−	0.982349i	$-0.559895\pi$
$461$	−8.03260	−0.374116	−0.187058	+	0.982349i	$0.559895\pi$
$462$	0	0
$463$	−35.4955	−1.64961	−0.824807	−	0.565415i	$-0.808716\pi$
$463$	−35.4955	−1.64961	−0.824807	+	0.565415i	$0.808716\pi$
$464$	11.1153	0.516015
$465$	−0.582576	−0.0270163
$466$	−38.5390	−1.78529
$467$	−3.41742	−0.158140	−0.0790698	−	0.996869i	$-0.525195\pi$
$467$	−3.41742	−0.158140	−0.0790698	+	0.996869i	$0.525195\pi$
$468$	−12.2197	−0.564856
$469$	12.4104	0.573059
$470$	27.5420	1.27042
$471$	−16.7477	−0.771695
$472$	−7.65120	−0.352175
$473$	0	0
$474$	−13.3739	−0.614282
$475$	−3.46410	−0.158944
$476$	−4.41742	−0.202472
$477$	−6.16515	−0.282283
$478$	−59.0780	−2.70217
$479$	27.3712	1.25062	0.625311	−	0.780375i	$-0.284972\pi$
$479$	27.3712	1.25062	0.625311	+	0.780375i	$0.284972\pi$
$480$	−7.38505	−0.337080
$481$	6.92820	0.315899
$482$	10.9564	0.499052
$483$	−7.84190	−0.356819
$484$	0	0
$485$	−13.5826	−0.616753
$486$	−2.18890	−0.0992906
$487$	3.58258	0.162342	0.0811710	−	0.996700i	$-0.474134\pi$
$487$	3.58258	0.162342	0.0811710	+	0.996700i	$0.474134\pi$
$488$	−12.1652	−0.550691
$489$	6.74773	0.305143
$490$	−13.4949	−0.609638
$491$	−27.7128	−1.25066	−0.625331	−	0.780360i	$-0.715036\pi$
$491$	−27.7128	−1.25066	−0.625331	+	0.780360i	$0.715036\pi$
$492$	9.66930	0.435926
$493$	−10.7477	−0.484053
$494$	−33.1950	−1.49351
$495$	0	0
$496$	−1.04356	−0.0468573
$497$	−7.30960	−0.327881
$498$	−34.7477	−1.55708
$499$	26.3303	1.17871	0.589353	−	0.807876i	$-0.299383\pi$
$499$	26.3303	1.17871	0.589353	+	0.807876i	$0.299383\pi$
$500$	−2.79129	−0.124830
$501$	−11.4014	−0.509375
$502$	−7.84190	−0.350001
$503$	−0.818350	−0.0364884	−0.0182442	−	0.999834i	$-0.505808\pi$
$503$	−0.818350	−0.0364884	−0.0182442	+	0.999834i	$0.505808\pi$
$504$	1.58258	0.0704935
$505$	8.75560	0.389619
$506$	0	0
$507$	6.16515	0.273804
$508$	7.11890	0.315850
$509$	0.747727	0.0331424	0.0165712	−	0.999863i	$-0.494725\pi$
$509$	0.747727	0.0331424	0.0165712	+	0.999863i	$0.494725\pi$
$510$	3.79129	0.167881
$511$	−14.3303	−0.633935
$512$	−19.4340	−0.858868
$513$	−3.46410	−0.152944
$514$	−52.8951	−2.33310
$515$	11.1652	0.491995
$516$	17.3205	0.762493
$517$	0	0
$518$	−3.16515	−0.139069
$519$	−6.92820	−0.304114
$520$	−7.58258	−0.332518
$521$	9.25227	0.405349	0.202675	−	0.979246i	$-0.435037\pi$
$521$	9.25227	0.405349	0.202675	+	0.979246i	$0.435037\pi$
$522$	13.5826	0.594493
$523$	4.56850	0.199767	0.0998833	−	0.994999i	$-0.468153\pi$
$523$	4.56850	0.199767	0.0998833	+	0.994999i	$0.468153\pi$
$524$	−0.532300	−0.0232536
$525$	−0.913701	−0.0398772
$526$	−9.79129	−0.426920
$527$	1.00905	0.0439549
$528$	0	0
$529$	50.6606	2.20264
$530$	−13.4949	−0.586181
$531$	4.41742	0.191700
$532$	8.83485	0.383039
$533$	−15.1652	−0.656876
$534$	27.9035	1.20750
$535$	7.74655	0.334913
$536$	23.5257	1.01616
$537$	−1.58258	−0.0682932
$538$	59.4618	2.56358
$539$	0	0
$540$	−2.79129	−0.120118
$541$	29.9216	1.28643	0.643215	−	0.765685i	$-0.277600\pi$
$541$	29.9216	1.28643	0.643215	+	0.765685i	$0.277600\pi$
$542$	21.3739	0.918086
$543$	13.1652	0.564971
$544$	12.7913	0.548422
$545$	6.92820	0.296772
$546$	−8.75560	−0.374705
$547$	3.99640	0.170874	0.0854369	−	0.996344i	$-0.472771\pi$
$547$	3.99640	0.170874	0.0854369	+	0.996344i	$0.472771\pi$
$548$	48.3739	2.06643
$549$	7.02355	0.299758
$550$	0	0
$551$	21.4955	0.915737
$552$	−14.8655	−0.632716
$553$	−5.58258	−0.237395
$554$	−19.1652	−0.814249
$555$	1.58258	0.0671766
$556$	36.3930	1.54341
$557$	−29.6356	−1.25570	−0.627850	−	0.778335i	$-0.716065\pi$
$557$	−29.6356	−1.25570	−0.627850	+	0.778335i	$0.716065\pi$
$558$	−1.27520	−0.0539835
$559$	−27.1652	−1.14896
$560$	−1.63670	−0.0691632
$561$	0	0
$562$	−29.1652	−1.23026
$563$	−11.8383	−0.498925	−0.249463	−	0.968384i	$-0.580254\pi$
$563$	−11.8383	−0.498925	−0.249463	+	0.968384i	$0.580254\pi$
$564$	35.1216	1.47889
$565$	−9.00000	−0.378633
$566$	−69.0780	−2.90357
$567$	−0.913701	−0.0383718
$568$	−13.8564	−0.581402
$569$	−21.5076	−0.901646	−0.450823	−	0.892613i	$-0.648869\pi$
$569$	−21.5076	−0.901646	−0.450823	+	0.892613i	$0.648869\pi$
$570$	−7.58258	−0.317599
$571$	27.0852	1.13348	0.566739	−	0.823897i	$-0.308205\pi$
$571$	27.0852	1.13348	0.566739	+	0.823897i	$0.308205\pi$
$572$	0	0
$573$	4.74773	0.198339
$574$	6.92820	0.289178
$575$	8.58258	0.357918
$576$	−12.5826	−0.524274
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	30.6446	1.27465
$579$	21.6983	0.901751
$580$	17.3205	0.719195
$581$	−14.5045	−0.601750
$582$	−29.7309	−1.23239
$583$	0	0
$584$	−27.1652	−1.12410
$585$	4.37780	0.181000
$586$	−3.79129	−0.156617
$587$	−19.7477	−0.815076	−0.407538	−	0.913188i	$-0.633613\pi$
$587$	−19.7477	−0.815076	−0.407538	+	0.913188i	$0.633613\pi$
$588$	−17.2087	−0.709675
$589$	−2.01810	−0.0831544
$590$	9.66930	0.398079
$591$	19.3386	0.795484
$592$	2.83485	0.116512
$593$	−27.7128	−1.13803	−0.569014	−	0.822328i	$-0.692675\pi$
$593$	−27.7128	−1.13803	−0.569014	+	0.822328i	$0.692675\pi$
$594$	0	0
$595$	1.58258	0.0648793
$596$	36.6591	1.50162
$597$	11.7477	0.480802
$598$	82.2432	3.36317
$599$	−20.7477	−0.847729	−0.423865	−	0.905726i	$-0.639327\pi$
$599$	−20.7477	−0.847729	−0.423865	+	0.905726i	$0.639327\pi$
$600$	−1.73205	−0.0707107
$601$	33.1950	1.35405	0.677026	−	0.735959i	$-0.263268\pi$
$601$	33.1950	1.35405	0.677026	+	0.735959i	$0.263268\pi$
$602$	12.4104	0.505810
$603$	−13.5826	−0.553125
$604$	36.3930	1.48081
$605$	0	0
$606$	19.1652	0.778531
$607$	−22.0797	−0.896188	−0.448094	−	0.893986i	$-0.647897\pi$
$607$	−22.0797	−0.896188	−0.448094	+	0.893986i	$0.647897\pi$
$608$	−25.5826	−1.03751
$609$	5.66970	0.229748
$610$	15.3739	0.622470
$611$	−55.0840	−2.22846
$612$	4.83465	0.195429
$613$	−8.75560	−0.353636	−0.176818	−	0.984244i	$-0.556580\pi$
$613$	−8.75560	−0.353636	−0.176818	+	0.984244i	$0.556580\pi$
$614$	57.0780	2.30348
$615$	−3.46410	−0.139686
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	24.4394	0.983097
$619$	−41.4955	−1.66784	−0.833922	−	0.551883i	$-0.813910\pi$
$619$	−41.4955	−1.66784	−0.833922	+	0.551883i	$0.813910\pi$
$620$	−1.62614	−0.0653072
$621$	8.58258	0.344407
$622$	5.29150	0.212170
$623$	11.6476	0.466651
$624$	7.84190	0.313927
$625$	1.00000	0.0400000
$626$	7.84190	0.313426
$627$	0	0
$628$	−46.7477	−1.86544
$629$	−2.74110	−0.109295
$630$	−2.00000	−0.0796819
$631$	−30.5826	−1.21747	−0.608737	−	0.793372i	$-0.708323\pi$
$631$	−30.5826	−1.21747	−0.608737	+	0.793372i	$0.708323\pi$
$632$	−10.5826	−0.420952
$633$	−18.1389	−0.720955
$634$	−25.9053	−1.02883
$635$	−2.55040	−0.101210
$636$	−17.2087	−0.682370
$637$	26.9898	1.06938
$638$	0	0
$639$	8.00000	0.316475
$640$	−12.7719	−0.504854
$641$	20.7477	0.819486	0.409743	−	0.912201i	$-0.365618\pi$
$641$	20.7477	0.819486	0.409743	+	0.912201i	$0.365618\pi$
$642$	16.9564	0.669217
$643$	15.4955	0.611081	0.305541	−	0.952179i	$-0.401163\pi$
$643$	15.4955	0.611081	0.305541	+	0.952179i	$0.401163\pi$
$644$	−21.8890	−0.862548
$645$	−6.20520	−0.244330
$646$	13.1334	0.516727
$647$	2.25227	0.0885460	0.0442730	−	0.999019i	$-0.485903\pi$
$647$	2.25227	0.0885460	0.0442730	+	0.999019i	$0.485903\pi$
$648$	−1.73205	−0.0680414
$649$	0	0
$650$	9.58258	0.375860
$651$	−0.532300	−0.0208625
$652$	18.8348	0.737630
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	15.1652	0.593004
$655$	0.190700	0.00745128
$656$	−6.20520	−0.242272
$657$	15.6838	0.611884
$658$	25.1652	0.981040
$659$	12.9427	0.504176	0.252088	−	0.967704i	$-0.418883\pi$
$659$	12.9427	0.504176	0.252088	+	0.967704i	$0.418883\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−0.552200	−0.0214619
$663$	−7.58258	−0.294483
$664$	−27.4955	−1.06703
$665$	−3.16515	−0.122739
$666$	3.46410	0.134231
$667$	−53.2566	−2.06210
$668$	−31.8245	−1.23133
$669$	20.3303	0.786015
$670$	−29.7309	−1.14861
$671$	0	0
$672$	−6.74773	−0.260299
$673$	6.01450	0.231842	0.115921	−	0.993258i	$-0.463018\pi$
$673$	6.01450	0.231842	0.115921	+	0.993258i	$0.463018\pi$
$674$	45.9129	1.76850
$675$	1.00000	0.0384900
$676$	17.2087	0.661874
$677$	−22.6120	−0.869050	−0.434525	−	0.900660i	$-0.643084\pi$
$677$	−22.6120	−0.869050	−0.434525	+	0.900660i	$0.643084\pi$
$678$	−19.7001	−0.756578
$679$	−12.4104	−0.476268
$680$	3.00000	0.115045
$681$	6.30055	0.241438
$682$	0	0
$683$	9.66970	0.370001	0.185000	−	0.982738i	$-0.440771\pi$
$683$	9.66970	0.370001	0.185000	+	0.982738i	$0.440771\pi$
$684$	−9.66930	−0.369715
$685$	−17.3303	−0.662157
$686$	−26.3303	−1.00530
$687$	17.0000	0.648590
$688$	−11.1153	−0.423767
$689$	26.9898	1.02823
$690$	18.7864	0.715186
$691$	8.58258	0.326497	0.163248	−	0.986585i	$-0.447803\pi$
$691$	8.58258	0.326497	0.163248	+	0.986585i	$0.447803\pi$
$692$	−19.3386	−0.735144
$693$	0	0
$694$	32.9564	1.25101
$695$	−13.0381	−0.494562
$696$	10.7477	0.407392
$697$	6.00000	0.227266
$698$	56.8693	2.15254
$699$	17.6066	0.665941
$700$	−2.55040	−0.0963961
$701$	4.56850	0.172550	0.0862750	−	0.996271i	$-0.472504\pi$
$701$	4.56850	0.172550	0.0862750	+	0.996271i	$0.472504\pi$
$702$	9.58258	0.361671
$703$	5.48220	0.206765
$704$	0	0
$705$	−12.5826	−0.473887
$706$	−79.1619	−2.97930
$707$	8.00000	0.300871
$708$	12.3303	0.463401
$709$	−45.0000	−1.69001	−0.845005	−	0.534758i	$-0.820403\pi$
$709$	−45.0000	−1.69001	−0.845005	+	0.534758i	$0.820403\pi$
$710$	17.5112	0.657184
$711$	6.10985	0.229137
$712$	22.0797	0.827472
$713$	5.00000	0.187251
$714$	3.46410	0.129641
$715$	0	0
$716$	−4.41742	−0.165087
$717$	26.9898	1.00795
$718$	4.00000	0.149279
$719$	20.3303	0.758192	0.379096	−	0.925357i	$-0.376235\pi$
$719$	20.3303	0.758192	0.379096	+	0.925357i	$0.376235\pi$
$720$	1.79129	0.0667574
$721$	10.2016	0.379928
$722$	15.3223	0.570237
$723$	−5.00545	−0.186155
$724$	36.7477	1.36572
$725$	−6.20520	−0.230455
$726$	0	0
$727$	13.1652	0.488268	0.244134	−	0.969741i	$-0.421496\pi$
$727$	13.1652	0.488268	0.244134	+	0.969741i	$0.421496\pi$
$728$	−6.92820	−0.256776
$729$	1.00000	0.0370370
$730$	34.3303	1.27062
$731$	10.7477	0.397519
$732$	19.6048	0.724613
$733$	−34.8317	−1.28654	−0.643269	−	0.765640i	$-0.722422\pi$
$733$	−34.8317	−1.28654	−0.643269	+	0.765640i	$0.722422\pi$
$734$	62.9259	2.32264
$735$	6.16515	0.227405
$736$	63.3828	2.33632
$737$	0	0
$738$	−7.58258	−0.279118
$739$	15.0562	0.553850	0.276925	−	0.960892i	$-0.410685\pi$
$739$	15.0562	0.553850	0.276925	+	0.960892i	$0.410685\pi$
$740$	4.41742	0.162388
$741$	15.1652	0.557106
$742$	−12.3303	−0.452660
$743$	−49.6972	−1.82321	−0.911606	−	0.411065i	$-0.865157\pi$
$743$	−49.6972	−1.82321	−0.911606	+	0.411065i	$0.865157\pi$
$744$	−1.00905	−0.0369936
$745$	−13.1334	−0.481171
$746$	30.3303	1.11047
$747$	15.8745	0.580818
$748$	0	0
$749$	7.07803	0.258626
$750$	2.18890	0.0799274
$751$	−44.0780	−1.60843	−0.804215	−	0.594338i	$-0.797414\pi$
$751$	−44.0780	−1.60843	−0.804215	+	0.594338i	$0.797414\pi$
$752$	−22.5390	−0.821913
$753$	3.58258	0.130556
$754$	−59.4618	−2.16547
$755$	−13.0381	−0.474503
$756$	−2.55040	−0.0927572
$757$	25.1652	0.914643	0.457321	−	0.889301i	$-0.348809\pi$
$757$	25.1652	0.914643	0.457321	+	0.889301i	$0.348809\pi$
$758$	−25.7146	−0.933997
$759$	0	0
$760$	−6.00000	−0.217643
$761$	9.47860	0.343599	0.171800	−	0.985132i	$-0.445042\pi$
$761$	9.47860	0.343599	0.171800	+	0.985132i	$0.445042\pi$
$762$	−5.58258	−0.202235
$763$	6.33030	0.229172
$764$	13.2523	0.479450
$765$	−1.73205	−0.0626224
$766$	66.3900	2.39877
$767$	−19.3386	−0.698277
$768$	−2.79129	−0.100722
$769$	−7.40495	−0.267029	−0.133515	−	0.991047i	$-0.542626\pi$
$769$	−7.40495	−0.267029	−0.133515	+	0.991047i	$0.542626\pi$
$770$	0	0
$771$	24.1652	0.870287
$772$	60.5662	2.17983
$773$	−6.49545	−0.233625	−0.116813	−	0.993154i	$-0.537268\pi$
$773$	−6.49545	−0.233625	−0.116813	+	0.993154i	$0.537268\pi$
$774$	−13.5826	−0.488216
$775$	0.582576	0.0209267
$776$	−23.5257	−0.844524
$777$	1.44600	0.0518750
$778$	49.7925	1.78515
$779$	−12.0000	−0.429945
$780$	12.2197	0.437536
$781$	0	0
$782$	−32.5390	−1.16359
$783$	−6.20520	−0.221756
$784$	11.0436	0.394413
$785$	16.7477	0.597752
$786$	0.417424	0.0148890
$787$	−28.0942	−1.00145	−0.500725	−	0.865606i	$-0.666933\pi$
$787$	−28.0942	−1.00145	−0.500725	+	0.865606i	$0.666933\pi$
$788$	53.9796	1.92294
$789$	4.47315	0.159248
$790$	13.3739	0.475821
$791$	−8.22330	−0.292387
$792$	0	0
$793$	−30.7477	−1.09188
$794$	9.47860	0.336383
$795$	6.16515	0.218655
$796$	32.7913	1.16226
$797$	32.3303	1.14520	0.572599	−	0.819836i	$-0.305935\pi$
$797$	32.3303	1.14520	0.572599	+	0.819836i	$0.305935\pi$
$798$	−6.92820	−0.245256
$799$	21.7937	0.771004
$800$	7.38505	0.261101
$801$	−12.7477	−0.450419
$802$	48.8788	1.72597
$803$	0	0
$804$	−37.9129	−1.33708
$805$	7.84190	0.276391
$806$	5.58258	0.196638
$807$	−27.1652	−0.956259
$808$	15.1652	0.533508
$809$	1.82740	0.0642480	0.0321240	−	0.999484i	$-0.489773\pi$
$809$	1.82740	0.0642480	0.0321240	+	0.999484i	$0.489773\pi$
$810$	2.18890	0.0769101
$811$	28.3405	0.995168	0.497584	−	0.867416i	$-0.334221\pi$
$811$	28.3405	0.995168	0.497584	+	0.867416i	$0.334221\pi$
$812$	15.8258	0.555375
$813$	−9.76465	−0.342461
$814$	0	0
$815$	−6.74773	−0.236363
$816$	−3.10260	−0.108613
$817$	−21.4955	−0.752031
$818$	50.1216	1.75246
$819$	4.00000	0.139771
$820$	−9.66930	−0.337667
$821$	50.5155	1.76300	0.881502	−	0.472180i	$-0.156533\pi$
$821$	50.5155	1.76300	0.881502	+	0.472180i	$0.156533\pi$
$822$	−37.9343	−1.32311
$823$	−0.834849	−0.0291010	−0.0145505	−	0.999894i	$-0.504632\pi$
$823$	−0.834849	−0.0291010	−0.0145505	+	0.999894i	$0.504632\pi$
$824$	19.3386	0.673692
$825$	0	0
$826$	8.83485	0.307404
$827$	47.2421	1.64277	0.821385	−	0.570374i	$-0.193202\pi$
$827$	47.2421	1.64277	0.821385	+	0.570374i	$0.193202\pi$
$828$	23.9564	0.832544
$829$	19.0000	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$829$	19.0000	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$830$	34.7477	1.20611
$831$	8.75560	0.303729
$832$	55.0840	1.90970
$833$	−10.6784	−0.369983
$834$	−28.5390	−0.988225
$835$	11.4014	0.394560
$836$	0	0
$837$	0.582576	0.0201368
$838$	−70.7678	−2.44463
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	−1.58258	−0.0546040
$841$	9.50455	0.327743
$842$	5.84370	0.201387
$843$	13.3241	0.458907
$844$	−50.6308	−1.74278
$845$	−6.16515	−0.212088
$846$	−27.5420	−0.946914
$847$	0	0
$848$	11.0436	0.379237
$849$	31.5583	1.08308
$850$	−3.79129	−0.130040
$851$	−13.5826	−0.465605
$852$	22.3303	0.765024
$853$	42.1413	1.44289	0.721446	−	0.692471i	$-0.243478\pi$
$853$	42.1413	1.44289	0.721446	+	0.692471i	$0.243478\pi$
$854$	14.0471	0.480682
$855$	3.46410	0.118470
$856$	13.4174	0.458598
$857$	−17.4159	−0.594914	−0.297457	−	0.954735i	$-0.596139\pi$
$857$	−17.4159	−0.594914	−0.297457	+	0.954735i	$0.596139\pi$
$858$	0	0
$859$	−24.6606	−0.841409	−0.420705	−	0.907198i	$-0.638217\pi$
$859$	−24.6606	−0.841409	−0.420705	+	0.907198i	$0.638217\pi$
$860$	−17.3205	−0.590624
$861$	−3.16515	−0.107868
$862$	43.9129	1.49568
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	7.38505	0.251245
$865$	6.92820	0.235566
$866$	9.66930	0.328576
$867$	−14.0000	−0.475465
$868$	−1.48580	−0.0504314
$869$	0	0
$870$	−13.5826	−0.460492
$871$	59.4618	2.01479
$872$	12.0000	0.406371
$873$	13.5826	0.459701
$874$	65.0780	2.20130
$875$	0.913701	0.0308887
$876$	43.7780	1.47912
$877$	−41.7599	−1.41013	−0.705066	−	0.709142i	$-0.749083\pi$
$877$	−41.7599	−1.41013	−0.705066	+	0.709142i	$0.749083\pi$
$878$	20.9564	0.707246
$879$	1.73205	0.0584206
$880$	0	0
$881$	25.4955	0.858964	0.429482	−	0.903075i	$-0.358696\pi$
$881$	25.4955	0.858964	0.429482	+	0.903075i	$0.358696\pi$
$882$	13.4949	0.454397
$883$	−48.2432	−1.62351	−0.811756	−	0.583997i	$-0.801488\pi$
$883$	−48.2432	−1.62351	−0.811756	+	0.583997i	$0.801488\pi$
$884$	−21.1652	−0.711861
$885$	−4.41742	−0.148490
$886$	−15.6838	−0.526908
$887$	−2.01810	−0.0677612	−0.0338806	−	0.999426i	$-0.510787\pi$
$887$	−2.01810	−0.0677612	−0.0338806	+	0.999426i	$0.510787\pi$
$888$	2.74110	0.0919853
$889$	−2.33030	−0.0781558
$890$	−27.9035	−0.935328
$891$	0	0
$892$	56.7477	1.90005
$893$	−43.5873	−1.45859
$894$	−28.7477	−0.961468
$895$	1.58258	0.0528997
$896$	−11.6697	−0.389857
$897$	−37.5728	−1.25452
$898$	−6.20520	−0.207070
$899$	−3.61500	−0.120567
$900$	2.79129	0.0930429
$901$	−10.6784	−0.355748
$902$	0	0
$903$	−5.66970	−0.188676
$904$	−15.5885	−0.518464
$905$	−13.1652	−0.437624
$906$	−28.5390	−0.948145
$907$	29.1652	0.968413	0.484206	−	0.874954i	$-0.339108\pi$
$907$	29.1652	0.968413	0.484206	+	0.874954i	$0.339108\pi$
$908$	17.5867	0.583634
$909$	−8.75560	−0.290405
$910$	8.75560	0.290245
$911$	−44.7477	−1.48256	−0.741279	−	0.671197i	$-0.765781\pi$
$911$	−44.7477	−1.48256	−0.741279	+	0.671197i	$0.765781\pi$
$912$	6.20520	0.205475
$913$	0	0
$914$	−49.0780	−1.62336
$915$	−7.02355	−0.232192
$916$	47.4519	1.56785
$917$	0.174243	0.00575401
$918$	−3.79129	−0.125131
$919$	−27.9035	−0.920452	−0.460226	−	0.887802i	$-0.652232\pi$
$919$	−27.9035	−0.920452	−0.460226	+	0.887802i	$0.652232\pi$
$920$	14.8655	0.490100
$921$	−26.0761	−0.859237
$922$	17.5826	0.579051
$923$	−35.0224	−1.15278
$924$	0	0
$925$	−1.58258	−0.0520348
$926$	77.6960	2.55325
$927$	−11.1652	−0.366712
$928$	−45.8258	−1.50430
$929$	−16.3303	−0.535780	−0.267890	−	0.963450i	$-0.586326\pi$
$929$	−16.3303	−0.535780	−0.267890	+	0.963450i	$0.586326\pi$
$930$	1.27520	0.0418155
$931$	21.3567	0.699938
$932$	49.1450	1.60980
$933$	−2.41742	−0.0791429
$934$	7.48040	0.244766
$935$	0	0
$936$	7.58258	0.247844
$937$	−27.1805	−0.887949	−0.443974	−	0.896040i	$-0.646432\pi$
$937$	−27.1805	−0.887949	−0.443974	+	0.896040i	$0.646432\pi$
$938$	−27.1652	−0.886974
$939$	−3.58258	−0.116913
$940$	−35.1216	−1.14554
$941$	−3.99640	−0.130279	−0.0651395	−	0.997876i	$-0.520749\pi$
$941$	−3.99640	−0.130279	−0.0651395	+	0.997876i	$0.520749\pi$
$942$	36.6591	1.19442
$943$	29.7309	0.968172
$944$	−7.91288	−0.257542
$945$	0.913701	0.0297227
$946$	0	0
$947$	20.9129	0.679577	0.339789	−	0.940502i	$-0.389644\pi$
$947$	20.9129	0.679577	0.339789	+	0.940502i	$0.389644\pi$
$948$	17.0544	0.553900
$949$	−68.6606	−2.22882
$950$	7.58258	0.246011
$951$	11.8348	0.383771
$952$	2.74110	0.0888396
$953$	20.7846	0.673280	0.336640	−	0.941634i	$-0.390710\pi$
$953$	20.7846	0.673280	0.336640	+	0.941634i	$0.390710\pi$
$954$	13.4949	0.436914
$955$	−4.74773	−0.153633
$956$	75.3363	2.43655
$957$	0	0
$958$	−59.9129	−1.93570
$959$	−15.8347	−0.511329
$960$	12.5826	0.406101
$961$	−30.6606	−0.989052
$962$	−15.1652	−0.488944
$963$	−7.74655	−0.249629
$964$	−13.9717	−0.449997
$965$	−21.6983	−0.698493
$966$	17.1652	0.552280
$967$	−57.6344	−1.85340	−0.926699	−	0.375804i	$-0.877367\pi$
$967$	−57.6344	−1.85340	−0.926699	+	0.375804i	$0.877367\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	29.7309	0.954602
$971$	16.4174	0.526860	0.263430	−	0.964678i	$-0.415146\pi$
$971$	16.4174	0.526860	0.263430	+	0.964678i	$0.415146\pi$
$972$	2.79129	0.0895306
$973$	−11.9129	−0.381909
$974$	−7.84190	−0.251271
$975$	−4.37780	−0.140202
$976$	−12.5812	−0.402715
$977$	−6.16515	−0.197241	−0.0986203	−	0.995125i	$-0.531443\pi$
$977$	−6.16515	−0.197241	−0.0986203	+	0.995125i	$0.531443\pi$
$978$	−14.7701	−0.472296
$979$	0	0
$980$	17.2087	0.549712
$981$	−6.92820	−0.221201
$982$	60.6606	1.93576
$983$	30.5826	0.975433	0.487716	−	0.873002i	$-0.337830\pi$
$983$	30.5826	0.975433	0.487716	+	0.873002i	$0.337830\pi$
$984$	−6.00000	−0.191273
$985$	−19.3386	−0.616179
$986$	23.5257	0.749211
$987$	−11.4967	−0.365944
$988$	42.3303	1.34671
$989$	53.2566	1.69346
$990$	0	0
$991$	45.2432	1.43720	0.718599	−	0.695425i	$-0.244784\pi$
$991$	45.2432	1.43720	0.718599	+	0.695425i	$0.244784\pi$
$992$	4.30235	0.136600
$993$	0.252273	0.00800564
$994$	16.0000	0.507489
$995$	−11.7477	−0.372428
$996$	44.3103	1.40403
$997$	37.9542	1.20202	0.601011	−	0.799241i	$-0.294765\pi$
$997$	37.9542	1.20202	0.601011	+	0.799241i	$0.294765\pi$
$998$	−57.6344	−1.82439
$999$	−1.58258	−0.0500705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.2.a.t.1.1	✓	4
3.2	odd	2		5445.2.a.bl.1.4		4
5.4	even	2		9075.2.a.cu.1.4		4
11.10	odd	2	inner	1815.2.a.t.1.4	yes	4
33.32	even	2		5445.2.a.bl.1.1		4
55.54	odd	2		9075.2.a.cu.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.t.1.1	✓	4	1.1	even	1	trivial
1815.2.a.t.1.4	yes	4	11.10	odd	2	inner
5445.2.a.bl.1.1		4	33.32	even	2
5445.2.a.bl.1.4		4	3.2	odd	2
9075.2.a.cu.1.1		4	55.54	odd	2
9075.2.a.cu.1.4		4	5.4	even	2

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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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q-2.18890 q^{2} +1.00000 q^{3} +2.79129 q^{4} -1.00000 q^{5} -2.18890 q^{6} -0.913701 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.18890 q^{2} +1.00000 q^{3} +2.79129 q^{4} -1.00000 q^{5} -2.18890 q^{6} -0.913701 q^{7} -1.73205 q^{8} +1.00000 q^{9} +2.18890 q^{10} +2.79129 q^{12} -4.37780 q^{13} +2.00000 q^{14} -1.00000 q^{15} -1.79129 q^{16} +1.73205 q^{17} -2.18890 q^{18} -3.46410 q^{19} -2.79129 q^{20} -0.913701 q^{21} +8.58258 q^{23} -1.73205 q^{24} +1.00000 q^{25} +9.58258 q^{26} +1.00000 q^{27} -2.55040 q^{28} -6.20520 q^{29} +2.18890 q^{30} +0.582576 q^{31} +7.38505 q^{32} -3.79129 q^{34} +0.913701 q^{35} +2.79129 q^{36} -1.58258 q^{37} +7.58258 q^{38} -4.37780 q^{39} +1.73205 q^{40} +3.46410 q^{41} +2.00000 q^{42} +6.20520 q^{43} -1.00000 q^{45} -18.7864 q^{46} +12.5826 q^{47} -1.79129 q^{48} -6.16515 q^{49} -2.18890 q^{50} +1.73205 q^{51} -12.2197 q^{52} -6.16515 q^{53} -2.18890 q^{54} +1.58258 q^{56} -3.46410 q^{57} +13.5826 q^{58} +4.41742 q^{59} -2.79129 q^{60} +7.02355 q^{61} -1.27520 q^{62} -0.913701 q^{63} -12.5826 q^{64} +4.37780 q^{65} -13.5826 q^{67} +4.83465 q^{68} +8.58258 q^{69} -2.00000 q^{70} +8.00000 q^{71} -1.73205 q^{72} +15.6838 q^{73} +3.46410 q^{74} +1.00000 q^{75} -9.66930 q^{76} +9.58258 q^{78} +6.10985 q^{79} +1.79129 q^{80} +1.00000 q^{81} -7.58258 q^{82} +15.8745 q^{83} -2.55040 q^{84} -1.73205 q^{85} -13.5826 q^{86} -6.20520 q^{87} -12.7477 q^{89} +2.18890 q^{90} +4.00000 q^{91} +23.9564 q^{92} +0.582576 q^{93} -27.5420 q^{94} +3.46410 q^{95} +7.38505 q^{96} +13.5826 q^{97} +13.4949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{4} - 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 2 * q^4 - 4 * q^5 + 4 * q^9	\( 4 q + 4 q^{3} + 2 q^{4} - 4 q^{5} + 4 q^{9} + 2 q^{12} + 8 q^{14} - 4 q^{15} + 2 q^{16} - 2 q^{20} + 16 q^{23} + 4 q^{25} + 20 q^{26} + 4 q^{27} - 16 q^{31} - 6 q^{34} + 2 q^{36} + 12 q^{37} + 12 q^{38} + 8 q^{42} - 4 q^{45} + 32 q^{47} + 2 q^{48} + 12 q^{49} + 12 q^{53} - 12 q^{56} + 36 q^{58} + 36 q^{59} - 2 q^{60} - 32 q^{64} - 36 q^{67} + 16 q^{69} - 8 q^{70} + 32 q^{71} + 4 q^{75} + 20 q^{78} - 2 q^{80} + 4 q^{81} - 12 q^{82} - 36 q^{86} + 4 q^{89} + 16 q^{91} + 50 q^{92} - 16 q^{93} + 36 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 2 * q^4 - 4 * q^5 + 4 * q^9 + 2 * q^12 + 8 * q^14 - 4 * q^15 + 2 * q^16 - 2 * q^20 + 16 * q^23 + 4 * q^25 + 20 * q^26 + 4 * q^27 - 16 * q^31 - 6 * q^34 + 2 * q^36 + 12 * q^37 + 12 * q^38 + 8 * q^42 - 4 * q^45 + 32 * q^47 + 2 * q^48 + 12 * q^49 + 12 * q^53 - 12 * q^56 + 36 * q^58 + 36 * q^59 - 2 * q^60 - 32 * q^64 - 36 * q^67 + 16 * q^69 - 8 * q^70 + 32 * q^71 + 4 * q^75 + 20 * q^78 - 2 * q^80 + 4 * q^81 - 12 * q^82 - 36 * q^86 + 4 * q^89 + 16 * q^91 + 50 * q^92 - 16 * q^93 + 36 * q^97

Introduction

L-functions

Modular forms

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