LMFDB - Embedded newform 3525.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3525,2,Mod(1,3525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, 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("3525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3525.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:	$28.1472667125$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 705)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.334904$ of defining polynomial
Character	$\chi$	$=$	3525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.74912 q^{2} +1.00000 q^{3} +5.55765 q^{4} -2.74912 q^{6} -3.47363 q^{7} -9.78039 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.74912 q^{2} +1.00000 q^{3} +5.55765 q^{4} -2.74912 q^{6} -3.47363 q^{7} -9.78039 q^{8} +1.00000 q^{9} -5.38607 q^{11} +5.55765 q^{12} +5.69637 q^{13} +9.54941 q^{14} +15.7721 q^{16} +3.57754 q^{17} -2.74912 q^{18} -4.55294 q^{19} -3.47363 q^{21} +14.8070 q^{22} -5.35480 q^{23} -9.78039 q^{24} -15.6600 q^{26} +1.00000 q^{27} -19.3052 q^{28} +6.77568 q^{29} +0.625581 q^{31} -23.7987 q^{32} -5.38607 q^{33} -9.83509 q^{34} +5.55765 q^{36} +8.72569 q^{37} +12.5166 q^{38} +5.69637 q^{39} +0.0992070 q^{41} +9.54941 q^{42} +3.94372 q^{43} -29.9339 q^{44} +14.7210 q^{46} +1.00000 q^{47} +15.7721 q^{48} +5.06608 q^{49} +3.57754 q^{51} +31.6584 q^{52} -7.62177 q^{53} -2.74912 q^{54} +33.9734 q^{56} -4.55294 q^{57} -18.6271 q^{58} +12.4967 q^{59} -2.83196 q^{61} -1.71980 q^{62} -3.47363 q^{63} +33.8812 q^{64} +14.8070 q^{66} +6.36147 q^{67} +19.8827 q^{68} -5.35480 q^{69} -4.87960 q^{71} -9.78039 q^{72} +0.968727 q^{73} -23.9879 q^{74} -25.3036 q^{76} +18.7092 q^{77} -15.6600 q^{78} -0.940588 q^{79} +1.00000 q^{81} -0.272732 q^{82} -8.15862 q^{83} -19.3052 q^{84} -10.8418 q^{86} +6.77568 q^{87} +52.6779 q^{88} -11.9879 q^{89} -19.7871 q^{91} -29.7601 q^{92} +0.625581 q^{93} -2.74912 q^{94} -23.7987 q^{96} -14.9308 q^{97} -13.9272 q^{98} -5.38607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 4 q^{9} + 4 q^{11} + 8 q^{12} + 4 q^{13} + 12 q^{16} - 4 q^{17} - 4 q^{18} - 8 q^{19} - 8 q^{21} + 16 q^{22} - 16 q^{23} - 12 q^{24} + 4 q^{27} - 20 q^{28} + 4 q^{29} - 28 q^{32} + 4 q^{33} - 16 q^{34} + 8 q^{36} + 4 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{41} - 24 q^{43} - 20 q^{44} + 32 q^{46} + 4 q^{47} + 12 q^{48} + 8 q^{49} - 4 q^{51} + 28 q^{52} - 12 q^{53} - 4 q^{54} + 40 q^{56} - 8 q^{57} + 4 q^{58} - 28 q^{61} - 12 q^{62} - 8 q^{63} + 24 q^{64} + 16 q^{66} + 8 q^{67} + 4 q^{68} - 16 q^{69} + 16 q^{71} - 12 q^{72} + 24 q^{73} - 56 q^{74} - 8 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{81} + 4 q^{82} - 24 q^{83} - 20 q^{84} - 8 q^{86} + 4 q^{87} + 40 q^{88} - 8 q^{89} - 40 q^{91} - 28 q^{92} - 4 q^{94} - 28 q^{96} + 32 q^{98} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 4 * q^9 + 4 * q^11 + 8 * q^12 + 4 * q^13 + 12 * q^16 - 4 * q^17 - 4 * q^18 - 8 * q^19 - 8 * q^21 + 16 * q^22 - 16 * q^23 - 12 * q^24 + 4 * q^27 - 20 * q^28 + 4 * q^29 - 28 * q^32 + 4 * q^33 - 16 * q^34 + 8 * q^36 + 4 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^41 - 24 * q^43 - 20 * q^44 + 32 * q^46 + 4 * q^47 + 12 * q^48 + 8 * q^49 - 4 * q^51 + 28 * q^52 - 12 * q^53 - 4 * q^54 + 40 * q^56 - 8 * q^57 + 4 * q^58 - 28 * q^61 - 12 * q^62 - 8 * q^63 + 24 * q^64 + 16 * q^66 + 8 * q^67 + 4 * q^68 - 16 * q^69 + 16 * q^71 - 12 * q^72 + 24 * q^73 - 56 * q^74 - 8 * q^76 - 4 * q^77 - 4 * q^79 + 4 * q^81 + 4 * q^82 - 24 * q^83 - 20 * q^84 - 8 * q^86 + 4 * q^87 + 40 * q^88 - 8 * q^89 - 40 * q^91 - 28 * q^92 - 4 * q^94 - 28 * q^96 + 32 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.74912	−1.94392	−0.971960	−	0.235147i	$-0.924443\pi$
$2$	−2.74912	−1.94392	−0.971960	+	0.235147i	$0.924443\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.55765	2.77882
$5$	0	0
$5$	0	0
$6$	−2.74912	−1.12232
$7$	−3.47363	−1.31291	−0.656454	−	0.754366i	$-0.727944\pi$
$7$	−3.47363	−1.31291	−0.656454	+	0.754366i	$0.727944\pi$
$8$	−9.78039	−3.45789
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.38607	−1.62396	−0.811981	−	0.583683i	$-0.801611\pi$
$11$	−5.38607	−1.62396	−0.811981	+	0.583683i	$0.801611\pi$
$12$	5.55765	1.60435
$13$	5.69637	1.57989	0.789944	−	0.613179i	$-0.210109\pi$
$13$	5.69637	1.57989	0.789944	+	0.613179i	$0.210109\pi$
$14$	9.54941	2.55219
$15$	0	0
$16$	15.7721	3.94304
$17$	3.57754	0.867682	0.433841	−	0.900989i	$-0.357158\pi$
$17$	3.57754	0.867682	0.433841	+	0.900989i	$0.357158\pi$
$18$	−2.74912	−0.647973
$19$	−4.55294	−1.04452	−0.522258	−	0.852788i	$-0.674910\pi$
$19$	−4.55294	−1.04452	−0.522258	+	0.852788i	$0.674910\pi$
$20$	0	0
$21$	−3.47363	−0.758007
$22$	14.8070	3.15685
$23$	−5.35480	−1.11655	−0.558277	−	0.829655i	$-0.688537\pi$
$23$	−5.35480	−1.11655	−0.558277	+	0.829655i	$0.688537\pi$
$24$	−9.78039	−1.99641
$25$	0	0
$26$	−15.6600	−3.07118
$27$	1.00000	0.192450
$28$	−19.3052	−3.64834
$29$	6.77568	1.25821	0.629106	−	0.777320i	$-0.283421\pi$
$29$	6.77568	1.25821	0.629106	+	0.777320i	$0.283421\pi$
$30$	0	0
$31$	0.625581	0.112358	0.0561789	−	0.998421i	$-0.482108\pi$
$31$	0.625581	0.112358	0.0561789	+	0.998421i	$0.482108\pi$
$32$	−23.7987	−4.20706
$33$	−5.38607	−0.937595
$34$	−9.83509	−1.68670
$35$	0	0
$36$	5.55765	0.926275
$37$	8.72569	1.43449	0.717247	−	0.696819i	$-0.245402\pi$
$37$	8.72569	1.43449	0.717247	+	0.696819i	$0.245402\pi$
$38$	12.5166	2.03045
$39$	5.69637	0.912149
$40$	0	0
$41$	0.0992070	0.0154935	0.00774676	−	0.999970i	$-0.497534\pi$
$41$	0.0992070	0.0154935	0.00774676	+	0.999970i	$0.497534\pi$
$42$	9.54941	1.47351
$43$	3.94372	0.601412	0.300706	−	0.953717i	$-0.402778\pi$
$43$	3.94372	0.601412	0.300706	+	0.953717i	$0.402778\pi$
$44$	−29.9339	−4.51271
$45$	0	0
$46$	14.7210	2.17049
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	15.7721	2.27651
$49$	5.06608	0.723725
$50$	0	0
$51$	3.57754	0.500956
$52$	31.6584	4.39023
$53$	−7.62177	−1.04693	−0.523465	−	0.852047i	$-0.675361\pi$
$53$	−7.62177	−1.04693	−0.523465	+	0.852047i	$0.675361\pi$
$54$	−2.74912	−0.374108
$55$	0	0
$56$	33.9734	4.53989
$57$	−4.55294	−0.603051
$58$	−18.6271	−2.44586
$59$	12.4967	1.62693	0.813463	−	0.581617i	$-0.197580\pi$
$59$	12.4967	1.62693	0.813463	+	0.581617i	$0.197580\pi$
$60$	0	0
$61$	−2.83196	−0.362595	−0.181297	−	0.983428i	$-0.558030\pi$
$61$	−2.83196	−0.362595	−0.181297	+	0.983428i	$0.558030\pi$
$62$	−1.71980	−0.218414
$63$	−3.47363	−0.437636
$64$	33.8812	4.23514
$65$	0	0
$66$	14.8070	1.82261
$67$	6.36147	0.777177	0.388588	−	0.921411i	$-0.372963\pi$
$67$	6.36147	0.777177	0.388588	+	0.921411i	$0.372963\pi$
$68$	19.8827	2.41114
$69$	−5.35480	−0.644642
$70$	0	0
$71$	−4.87960	−0.579102	−0.289551	−	0.957163i	$-0.593506\pi$
$71$	−4.87960	−0.579102	−0.289551	+	0.957163i	$0.593506\pi$
$72$	−9.78039	−1.15263
$73$	0.968727	0.113381	0.0566905	−	0.998392i	$-0.481945\pi$
$73$	0.968727	0.113381	0.0566905	+	0.998392i	$0.481945\pi$
$74$	−23.9879	−2.78854
$75$	0	0
$76$	−25.3036	−2.90252
$77$	18.7092	2.13211
$78$	−15.6600	−1.77314
$79$	−0.940588	−0.105824	−0.0529122	−	0.998599i	$-0.516850\pi$
$79$	−0.940588	−0.105824	−0.0529122	+	0.998599i	$0.516850\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.272732	−0.0301182
$83$	−8.15862	−0.895525	−0.447762	−	0.894153i	$-0.647779\pi$
$83$	−8.15862	−0.895525	−0.447762	+	0.894153i	$0.647779\pi$
$84$	−19.3052	−2.10637
$85$	0	0
$86$	−10.8418	−1.16910
$87$	6.77568	0.726429
$88$	52.6779	5.61548
$89$	−11.9879	−1.27072	−0.635360	−	0.772216i	$-0.719148\pi$
$89$	−11.9879	−1.27072	−0.635360	+	0.772216i	$0.719148\pi$
$90$	0	0
$91$	−19.7871	−2.07425
$92$	−29.7601	−3.10270
$93$	0.625581	0.0648697
$94$	−2.74912	−0.283550
$95$	0	0
$96$	−23.7987	−2.42895
$97$	−14.9308	−1.51599	−0.757995	−	0.652260i	$-0.773821\pi$
$97$	−14.9308	−1.51599	−0.757995	+	0.652260i	$0.773821\pi$
$98$	−13.9272	−1.40686
$99$	−5.38607	−0.541321
$100$	0	0
$101$	−5.96520	−0.593559	−0.296780	−	0.954946i	$-0.595913\pi$
$101$	−5.96520	−0.593559	−0.296780	+	0.954946i	$0.595913\pi$
$102$	−9.83509	−0.973819
$103$	−1.16305	−0.114599	−0.0572994	−	0.998357i	$-0.518249\pi$
$103$	−1.16305	−0.114599	−0.0572994	+	0.998357i	$0.518249\pi$
$104$	−55.7127	−5.46308
$105$	0	0
$106$	20.9531	2.03515
$107$	−10.8284	−1.04682	−0.523412	−	0.852080i	$-0.675341\pi$
$107$	−10.8284	−1.04682	−0.523412	+	0.852080i	$0.675341\pi$
$108$	5.55765	0.534785
$109$	−15.0711	−1.44355	−0.721773	−	0.692130i	$-0.756673\pi$
$109$	−15.0711	−1.44355	−0.721773	+	0.692130i	$0.756673\pi$
$110$	0	0
$111$	8.72569	0.828206
$112$	−54.7865	−5.17684
$113$	11.9504	1.12420	0.562099	−	0.827070i	$-0.309994\pi$
$113$	11.9504	1.12420	0.562099	+	0.827070i	$0.309994\pi$
$114$	12.5166	1.17228
$115$	0	0
$116$	37.6568	3.49635
$117$	5.69637	0.526630
$118$	−34.3548	−3.16261
$119$	−12.4271	−1.13919
$120$	0	0
$121$	18.0098	1.63725
$122$	7.78538	0.704855
$123$	0.0992070	0.00894519
$124$	3.47676	0.312222
$125$	0	0
$126$	9.54941	0.850729
$127$	−14.7229	−1.30645	−0.653224	−	0.757164i	$-0.726584\pi$
$127$	−14.7229	−1.30645	−0.653224	+	0.757164i	$0.726584\pi$
$128$	−45.5459	−4.02572
$129$	3.94372	0.347225
$130$	0	0
$131$	−8.08402	−0.706304	−0.353152	−	0.935566i	$-0.614890\pi$
$131$	−8.08402	−0.706304	−0.353152	+	0.935566i	$0.614890\pi$
$132$	−29.9339	−2.60541
$133$	15.8152	1.37135
$134$	−17.4884	−1.51077
$135$	0	0
$136$	−34.9898	−3.00035
$137$	−6.66981	−0.569840	−0.284920	−	0.958551i	$-0.591967\pi$
$137$	−6.66981	−0.569840	−0.284920	+	0.958551i	$0.591967\pi$
$138$	14.7210	1.25313
$139$	−0.142252	−0.0120656	−0.00603281	−	0.999982i	$-0.501920\pi$
$139$	−0.142252	−0.0120656	−0.00603281	+	0.999982i	$0.501920\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	13.4146	1.12573
$143$	−30.6811	−2.56568
$144$	15.7721	1.31435
$145$	0	0
$146$	−2.66314	−0.220403
$147$	5.06608	0.417843
$148$	48.4943	3.98621
$149$	−7.77568	−0.637008	−0.318504	−	0.947921i	$-0.603180\pi$
$149$	−7.77568	−0.637008	−0.318504	+	0.947921i	$0.603180\pi$
$150$	0	0
$151$	6.27549	0.510692	0.255346	−	0.966850i	$-0.417811\pi$
$151$	6.27549	0.510692	0.255346	+	0.966850i	$0.417811\pi$
$152$	44.5295	3.61182
$153$	3.57754	0.289227
$154$	−51.4338	−4.14465
$155$	0	0
$156$	31.6584	2.53470
$157$	7.41921	0.592117	0.296059	−	0.955170i	$-0.404328\pi$
$157$	7.41921	0.592117	0.296059	+	0.955170i	$0.404328\pi$
$158$	2.58579	0.205714
$159$	−7.62177	−0.604446
$160$	0	0
$161$	18.6006	1.46593
$162$	−2.74912	−0.215991
$163$	−7.85527	−0.615272	−0.307636	−	0.951504i	$-0.599538\pi$
$163$	−7.85527	−0.615272	−0.307636	+	0.951504i	$0.599538\pi$
$164$	0.551357	0.0430538
$165$	0	0
$166$	22.4290	1.74083
$167$	−25.6224	−1.98272	−0.991361	−	0.131159i	$-0.958130\pi$
$167$	−25.6224	−1.98272	−0.991361	+	0.131159i	$0.958130\pi$
$168$	33.9734	2.62111
$169$	19.4486	1.49605
$170$	0	0
$171$	−4.55294	−0.348172
$172$	21.9178	1.67122
$173$	−2.23403	−0.169850	−0.0849249	−	0.996387i	$-0.527065\pi$
$173$	−2.23403	−0.169850	−0.0849249	+	0.996387i	$0.527065\pi$
$174$	−18.6271	−1.41212
$175$	0	0
$176$	−84.9500	−6.40334
$177$	12.4967	0.939306
$178$	32.9563	2.47018
$179$	12.0625	0.901597	0.450799	−	0.892626i	$-0.351139\pi$
$179$	12.0625	0.901597	0.450799	+	0.892626i	$0.351139\pi$
$180$	0	0
$181$	−5.53304	−0.411267	−0.205634	−	0.978629i	$-0.565926\pi$
$181$	−5.53304	−0.411267	−0.205634	+	0.978629i	$0.565926\pi$
$182$	54.3969	4.03217
$183$	−2.83196	−0.209344
$184$	52.3720	3.86092
$185$	0	0
$186$	−1.71980	−0.126102
$187$	−19.2689	−1.40908
$188$	5.55765	0.405333
$189$	−3.47363	−0.252669
$190$	0	0
$191$	−1.65242	−0.119565	−0.0597825	−	0.998211i	$-0.519041\pi$
$191$	−1.65242	−0.119565	−0.0597825	+	0.998211i	$0.519041\pi$
$192$	33.8812	2.44516
$193$	−10.0973	−0.726823	−0.363412	−	0.931629i	$-0.618388\pi$
$193$	−10.0973	−0.726823	−0.363412	+	0.931629i	$0.618388\pi$
$194$	41.0464	2.94696
$195$	0	0
$196$	28.1555	2.01110
$197$	9.70970	0.691787	0.345894	−	0.938274i	$-0.387576\pi$
$197$	9.70970	0.691787	0.345894	+	0.938274i	$0.387576\pi$
$198$	14.8070	1.05228
$199$	−14.7835	−1.04798	−0.523988	−	0.851726i	$-0.675556\pi$
$199$	−14.7835	−1.04798	−0.523988	+	0.851726i	$0.675556\pi$
$200$	0	0
$201$	6.36147	0.448703
$202$	16.3990	1.15383
$203$	−23.5362	−1.65192
$204$	19.8827	1.39207
$205$	0	0
$206$	3.19736	0.222771
$207$	−5.35480	−0.372184
$208$	89.8440	6.22956
$209$	24.5224	1.69625
$210$	0	0
$211$	−6.74441	−0.464304	−0.232152	−	0.972680i	$-0.574577\pi$
$211$	−6.74441	−0.464304	−0.232152	+	0.972680i	$0.574577\pi$
$212$	−42.3591	−2.90924
$213$	−4.87960	−0.334345
$214$	29.7686	2.03494
$215$	0	0
$216$	−9.78039	−0.665471
$217$	−2.17303	−0.147515
$218$	41.4321	2.80614
$219$	0.968727	0.0654605
$220$	0	0
$221$	20.3790	1.37084
$222$	−23.9879	−1.60997
$223$	1.08755	0.0728278	0.0364139	−	0.999337i	$-0.488407\pi$
$223$	1.08755	0.0728278	0.0364139	+	0.999337i	$0.488407\pi$
$224$	82.6678	5.52348
$225$	0	0
$226$	−32.8530	−2.18535
$227$	13.8723	0.920734	0.460367	−	0.887729i	$-0.347718\pi$
$227$	13.8723	0.920734	0.460367	+	0.887729i	$0.347718\pi$
$228$	−25.3036	−1.67577
$229$	−0.427167	−0.0282280	−0.0141140	−	0.999900i	$-0.504493\pi$
$229$	−0.427167	−0.0282280	−0.0141140	+	0.999900i	$0.504493\pi$
$230$	0	0
$231$	18.7092	1.23098
$232$	−66.2688	−4.35076
$233$	−19.7061	−1.29099	−0.645494	−	0.763765i	$-0.723348\pi$
$233$	−19.7061	−1.29099	−0.645494	+	0.763765i	$0.723348\pi$
$234$	−15.6600	−1.02373
$235$	0	0
$236$	69.4520	4.52094
$237$	−0.940588	−0.0610977
$238$	34.1634	2.21449
$239$	10.5952	0.685347	0.342674	−	0.939455i	$-0.388668\pi$
$239$	10.5952	0.685347	0.342674	+	0.939455i	$0.388668\pi$
$240$	0	0
$241$	−14.0590	−0.905621	−0.452810	−	0.891607i	$-0.649579\pi$
$241$	−14.0590	−0.905621	−0.452810	+	0.891607i	$0.649579\pi$
$242$	−49.5111	−3.18269
$243$	1.00000	0.0641500
$244$	−15.7390	−1.00759
$245$	0	0
$246$	−0.272732	−0.0173887
$247$	−25.9352	−1.65022
$248$	−6.11843	−0.388521
$249$	−8.15862	−0.517031
$250$	0	0
$251$	26.6526	1.68230	0.841150	−	0.540802i	$-0.181879\pi$
$251$	26.6526	1.68230	0.841150	+	0.540802i	$0.181879\pi$
$252$	−19.3052	−1.21611
$253$	28.8414	1.81324
$254$	40.4751	2.53963
$255$	0	0
$256$	57.4486	3.59054
$257$	1.48215	0.0924538	0.0462269	−	0.998931i	$-0.485280\pi$
$257$	1.48215	0.0924538	0.0462269	+	0.998931i	$0.485280\pi$
$258$	−10.8418	−0.674978
$259$	−30.3098	−1.88336
$260$	0	0
$261$	6.77568	0.419404
$262$	22.2239	1.37300
$263$	−23.8691	−1.47183	−0.735917	−	0.677072i	$-0.763248\pi$
$263$	−23.8691	−1.47183	−0.735917	+	0.677072i	$0.763248\pi$
$264$	52.6779	3.24210
$265$	0	0
$266$	−43.4778	−2.66580
$267$	−11.9879	−0.733651
$268$	35.3548	2.15964
$269$	−4.35703	−0.265653	−0.132826	−	0.991139i	$-0.542405\pi$
$269$	−4.35703	−0.265653	−0.132826	+	0.991139i	$0.542405\pi$
$270$	0	0
$271$	−23.1177	−1.40430	−0.702149	−	0.712030i	$-0.747776\pi$
$271$	−23.1177	−1.40430	−0.702149	+	0.712030i	$0.747776\pi$
$272$	56.4256	3.42130
$273$	−19.7871	−1.19757
$274$	18.3361	1.10772
$275$	0	0
$276$	−29.7601	−1.79135
$277$	12.6765	0.761655	0.380828	−	0.924646i	$-0.375639\pi$
$277$	12.6765	0.761655	0.380828	+	0.924646i	$0.375639\pi$
$278$	0.391066	0.0234546
$279$	0.625581	0.0374526
$280$	0	0
$281$	−7.82843	−0.467005	−0.233502	−	0.972356i	$-0.575019\pi$
$281$	−7.82843	−0.467005	−0.233502	+	0.972356i	$0.575019\pi$
$282$	−2.74912	−0.163708
$283$	−31.0027	−1.84292	−0.921461	−	0.388471i	$-0.873003\pi$
$283$	−31.0027	−1.84292	−0.921461	+	0.388471i	$0.873003\pi$
$284$	−27.1191	−1.60922
$285$	0	0
$286$	84.3459	4.98748
$287$	−0.344608	−0.0203416
$288$	−23.7987	−1.40235
$289$	−4.20117	−0.247128
$290$	0	0
$291$	−14.9308	−0.875257
$292$	5.38384	0.315066
$293$	−18.1516	−1.06043	−0.530213	−	0.847865i	$-0.677888\pi$
$293$	−18.1516	−1.06043	−0.530213	+	0.847865i	$0.677888\pi$
$294$	−13.9272	−0.812253
$295$	0	0
$296$	−85.3407	−4.96033
$297$	−5.38607	−0.312532
$298$	21.3763	1.23829
$299$	−30.5029	−1.76403
$300$	0	0
$301$	−13.6990	−0.789598
$302$	−17.2521	−0.992745
$303$	−5.96520	−0.342692
$304$	−71.8096	−4.11856
$305$	0	0
$306$	−9.83509	−0.562235
$307$	8.07107	0.460640	0.230320	−	0.973115i	$-0.426023\pi$
$307$	8.07107	0.460640	0.230320	+	0.973115i	$0.426023\pi$
$308$	103.979	5.92476
$309$	−1.16305	−0.0661637
$310$	0	0
$311$	−4.10510	−0.232779	−0.116389	−	0.993204i	$-0.537132\pi$
$311$	−4.10510	−0.232779	−0.116389	+	0.993204i	$0.537132\pi$
$312$	−55.7127	−3.15411
$313$	−3.20666	−0.181251	−0.0906254	−	0.995885i	$-0.528887\pi$
$313$	−3.20666	−0.181251	−0.0906254	+	0.995885i	$0.528887\pi$
$314$	−20.3963	−1.15103
$315$	0	0
$316$	−5.22746	−0.294067
$317$	15.2998	0.859324	0.429662	−	0.902990i	$-0.358633\pi$
$317$	15.2998	0.859324	0.429662	+	0.902990i	$0.358633\pi$
$318$	20.9531	1.17499
$319$	−36.4943	−2.04329
$320$	0	0
$321$	−10.8284	−0.604384
$322$	−51.1352	−2.84965
$323$	−16.2883	−0.906307
$324$	5.55765	0.308758
$325$	0	0
$326$	21.5951	1.19604
$327$	−15.0711	−0.833432
$328$	−0.970283	−0.0535749
$329$	−3.47363	−0.191507
$330$	0	0
$331$	8.60411	0.472924	0.236462	−	0.971641i	$-0.424012\pi$
$331$	8.60411	0.472924	0.236462	+	0.971641i	$0.424012\pi$
$332$	−45.3427	−2.48851
$333$	8.72569	0.478165
$334$	70.4391	3.85425
$335$	0	0
$336$	−54.7865	−2.98885
$337$	22.1223	1.20508	0.602540	−	0.798089i	$-0.294155\pi$
$337$	22.1223	1.20508	0.602540	+	0.798089i	$0.294155\pi$
$338$	−53.4665	−2.90820
$339$	11.9504	0.649056
$340$	0	0
$341$	−3.36943	−0.182465
$342$	12.5166	0.676818
$343$	6.71773	0.362723
$344$	−38.5711	−2.07962
$345$	0	0
$346$	6.14160	0.330174
$347$	−15.3302	−0.822968	−0.411484	−	0.911417i	$-0.634989\pi$
$347$	−15.3302	−0.822968	−0.411484	+	0.911417i	$0.634989\pi$
$348$	37.6568	2.01862
$349$	29.8495	1.59781	0.798903	−	0.601460i	$-0.205414\pi$
$349$	29.8495	1.59781	0.798903	+	0.601460i	$0.205414\pi$
$350$	0	0
$351$	5.69637	0.304050
$352$	128.182	6.83210
$353$	−0.347578	−0.0184997	−0.00924985	−	0.999957i	$-0.502944\pi$
$353$	−0.347578	−0.0184997	−0.00924985	+	0.999957i	$0.502944\pi$
$354$	−34.3548	−1.82594
$355$	0	0
$356$	−66.6248	−3.53111
$357$	−12.4271	−0.657709
$358$	−33.1614	−1.75263
$359$	30.3008	1.59921	0.799606	−	0.600525i	$-0.205042\pi$
$359$	30.3008	1.59921	0.799606	+	0.600525i	$0.205042\pi$
$360$	0	0
$361$	1.72922	0.0910116
$362$	15.2110	0.799471
$363$	18.0098	0.945269
$364$	−109.969	−5.76397
$365$	0	0
$366$	7.78538	0.406948
$367$	27.8476	1.45364	0.726818	−	0.686830i	$-0.240998\pi$
$367$	27.8476	1.45364	0.726818	+	0.686830i	$0.240998\pi$
$368$	−84.4567	−4.40261
$369$	0.0992070	0.00516451
$370$	0	0
$371$	26.4752	1.37452
$372$	3.47676	0.180262
$373$	−13.3415	−0.690794	−0.345397	−	0.938457i	$-0.612256\pi$
$373$	−13.3415	−0.690794	−0.345397	+	0.938457i	$0.612256\pi$
$374$	52.9725	2.73914
$375$	0	0
$376$	−9.78039	−0.504385
$377$	38.5968	1.98783
$378$	9.54941	0.491168
$379$	29.5604	1.51841	0.759207	−	0.650849i	$-0.225587\pi$
$379$	29.5604	1.51841	0.759207	+	0.650849i	$0.225587\pi$
$380$	0	0
$381$	−14.7229	−0.754279
$382$	4.54270	0.232425
$383$	22.4107	1.14513	0.572566	−	0.819858i	$-0.305948\pi$
$383$	22.4107	1.14513	0.572566	+	0.819858i	$0.305948\pi$
$384$	−45.5459	−2.32425
$385$	0	0
$386$	27.7588	1.41289
$387$	3.94372	0.200471
$388$	−82.9799	−4.21267
$389$	−13.2306	−0.670817	−0.335409	−	0.942073i	$-0.608874\pi$
$389$	−13.2306	−0.670817	−0.335409	+	0.942073i	$0.608874\pi$
$390$	0	0
$391$	−19.1570	−0.968813
$392$	−49.5482	−2.50256
$393$	−8.08402	−0.407785
$394$	−26.6931	−1.34478
$395$	0	0
$396$	−29.9339	−1.50424
$397$	−1.58669	−0.0796336	−0.0398168	−	0.999207i	$-0.512677\pi$
$397$	−1.58669	−0.0796336	−0.0398168	+	0.999207i	$0.512677\pi$
$398$	40.6416	2.03718
$399$	15.8152	0.791750
$400$	0	0
$401$	−35.2669	−1.76114	−0.880571	−	0.473914i	$-0.842841\pi$
$401$	−35.2669	−1.76114	−0.880571	+	0.473914i	$0.842841\pi$
$402$	−17.4884	−0.872243
$403$	3.56354	0.177513
$404$	−33.1525	−1.64940
$405$	0	0
$406$	64.7037	3.21119
$407$	−46.9972	−2.32957
$408$	−34.9898	−1.73225
$409$	−5.84581	−0.289057	−0.144528	−	0.989501i	$-0.546167\pi$
$409$	−5.84581	−0.289057	−0.144528	+	0.989501i	$0.546167\pi$
$410$	0	0
$411$	−6.66981	−0.328997
$412$	−6.46383	−0.318450
$413$	−43.4087	−2.13600
$414$	14.7210	0.723497
$415$	0	0
$416$	−135.566	−6.64668
$417$	−0.142252	−0.00696609
$418$	−67.4151	−3.29738
$419$	−12.7457	−0.622667	−0.311334	−	0.950301i	$-0.600776\pi$
$419$	−12.7457	−0.622667	−0.311334	+	0.950301i	$0.600776\pi$
$420$	0	0
$421$	−8.41958	−0.410345	−0.205173	−	0.978726i	$-0.565776\pi$
$421$	−8.41958	−0.410345	−0.205173	+	0.978726i	$0.565776\pi$
$422$	18.5412	0.902570
$423$	1.00000	0.0486217
$424$	74.5439	3.62017
$425$	0	0
$426$	13.4146	0.649939
$427$	9.83716	0.476053
$428$	−60.1806	−2.90894
$429$	−30.6811	−1.48130
$430$	0	0
$431$	−28.5100	−1.37328	−0.686639	−	0.726999i	$-0.740915\pi$
$431$	−28.5100	−1.37328	−0.686639	+	0.726999i	$0.740915\pi$
$432$	15.7721	0.758838
$433$	−20.0911	−0.965517	−0.482758	−	0.875754i	$-0.660365\pi$
$433$	−20.0911	−0.965517	−0.482758	+	0.875754i	$0.660365\pi$
$434$	5.97393	0.286758
$435$	0	0
$436$	−83.7597	−4.01136
$437$	24.3801	1.16626
$438$	−2.66314	−0.127250
$439$	−14.4357	−0.688977	−0.344488	−	0.938791i	$-0.611948\pi$
$439$	−14.4357	−0.688977	−0.344488	+	0.938791i	$0.611948\pi$
$440$	0	0
$441$	5.06608	0.241242
$442$	−56.0243	−2.66480
$443$	29.6413	1.40830	0.704150	−	0.710052i	$-0.251328\pi$
$443$	29.6413	1.40830	0.704150	+	0.710052i	$0.251328\pi$
$444$	48.4943	2.30144
$445$	0	0
$446$	−2.98981	−0.141571
$447$	−7.77568	−0.367777
$448$	−117.690	−5.56035
$449$	3.09292	0.145964	0.0729819	−	0.997333i	$-0.476748\pi$
$449$	3.09292	0.145964	0.0729819	+	0.997333i	$0.476748\pi$
$450$	0	0
$451$	−0.534336	−0.0251609
$452$	66.4160	3.12395
$453$	6.27549	0.294848
$454$	−38.1365	−1.78983
$455$	0	0
$456$	44.5295	2.08528
$457$	−4.57373	−0.213950	−0.106975	−	0.994262i	$-0.534117\pi$
$457$	−4.57373	−0.213950	−0.106975	+	0.994262i	$0.534117\pi$
$458$	1.17433	0.0548730
$459$	3.57754	0.166985
$460$	0	0
$461$	31.7327	1.47794	0.738970	−	0.673738i	$-0.235312\pi$
$461$	31.7327	1.47794	0.738970	+	0.673738i	$0.235312\pi$
$462$	−51.4338	−2.39292
$463$	−9.67071	−0.449436	−0.224718	−	0.974424i	$-0.572146\pi$
$463$	−9.67071	−0.449436	−0.224718	+	0.974424i	$0.572146\pi$
$464$	106.867	4.96118
$465$	0	0
$466$	54.1743	2.50958
$467$	29.1179	1.34742	0.673708	−	0.738998i	$-0.264701\pi$
$467$	29.1179	1.34742	0.673708	+	0.738998i	$0.264701\pi$
$468$	31.6584	1.46341
$469$	−22.0973	−1.02036
$470$	0	0
$471$	7.41921	0.341859
$472$	−122.222	−5.62573
$473$	−21.2412	−0.976670
$474$	2.58579	0.118769
$475$	0	0
$476$	−69.0652	−3.16560
$477$	−7.62177	−0.348977
$478$	−29.1275	−1.33226
$479$	5.63969	0.257684	0.128842	−	0.991665i	$-0.458874\pi$
$479$	5.63969	0.257684	0.128842	+	0.991665i	$0.458874\pi$
$480$	0	0
$481$	49.7048	2.26634
$482$	38.6499	1.76045
$483$	18.6006	0.846356
$484$	100.092	4.54964
$485$	0	0
$486$	−2.74912	−0.124703
$487$	−16.5071	−0.748009	−0.374005	−	0.927427i	$-0.622016\pi$
$487$	−16.5071	−0.748009	−0.374005	+	0.927427i	$0.622016\pi$
$488$	27.6976	1.25381
$489$	−7.85527	−0.355228
$490$	0	0
$491$	10.6951	0.482662	0.241331	−	0.970443i	$-0.422416\pi$
$491$	10.6951	0.482662	0.241331	+	0.970443i	$0.422416\pi$
$492$	0.551357	0.0248571
$493$	24.2403	1.09173
$494$	71.2989	3.20789
$495$	0	0
$496$	9.86676	0.443031
$497$	16.9499	0.760307
$498$	22.4290	1.00507
$499$	−16.3336	−0.731193	−0.365596	−	0.930774i	$-0.619135\pi$
$499$	−16.3336	−0.731193	−0.365596	+	0.930774i	$0.619135\pi$
$500$	0	0
$501$	−25.6224	−1.14473
$502$	−73.2713	−3.27026
$503$	−23.4040	−1.04353	−0.521767	−	0.853088i	$-0.674727\pi$
$503$	−23.4040	−1.04353	−0.521767	+	0.853088i	$0.674727\pi$
$504$	33.9734	1.51330
$505$	0	0
$506$	−79.2883	−3.52479
$507$	19.4486	0.863744
$508$	−81.8249	−3.63039
$509$	−23.3931	−1.03688	−0.518441	−	0.855114i	$-0.673487\pi$
$509$	−23.3931	−1.03688	−0.518441	+	0.855114i	$0.673487\pi$
$510$	0	0
$511$	−3.36499	−0.148859
$512$	−66.8412	−2.95399
$513$	−4.55294	−0.201017
$514$	−4.07460	−0.179723
$515$	0	0
$516$	21.9178	0.964878
$517$	−5.38607	−0.236879
$518$	83.3252	3.66110
$519$	−2.23403	−0.0980628
$520$	0	0
$521$	22.9724	1.00644	0.503219	−	0.864159i	$-0.332149\pi$
$521$	22.9724	1.00644	0.503219	+	0.864159i	$0.332149\pi$
$522$	−18.6271	−0.815288
$523$	−29.5228	−1.29094	−0.645472	−	0.763784i	$-0.723339\pi$
$523$	−29.5228	−1.29094	−0.645472	+	0.763784i	$0.723339\pi$
$524$	−44.9281	−1.96269
$525$	0	0
$526$	65.6190	2.86113
$527$	2.23804	0.0974908
$528$	−84.9500	−3.69697
$529$	5.67390	0.246691
$530$	0	0
$531$	12.4967	0.542309
$532$	87.8953	3.81074
$533$	0.565119	0.0244780
$534$	32.9563	1.42616
$535$	0	0
$536$	−62.2176	−2.68739
$537$	12.0625	0.520537
$538$	11.9780	0.516408
$539$	−27.2863	−1.17530
$540$	0	0
$541$	−30.2635	−1.30113	−0.650565	−	0.759450i	$-0.725468\pi$
$541$	−30.2635	−1.30113	−0.650565	+	0.759450i	$0.725468\pi$
$542$	63.5532	2.72984
$543$	−5.53304	−0.237445
$544$	−85.1409	−3.65039
$545$	0	0
$546$	54.3969	2.32797
$547$	−26.3986	−1.12872	−0.564362	−	0.825527i	$-0.690878\pi$
$547$	−26.3986	−1.12872	−0.564362	+	0.825527i	$0.690878\pi$
$548$	−37.0684	−1.58349
$549$	−2.83196	−0.120865
$550$	0	0
$551$	−30.8492	−1.31422
$552$	52.3720	2.22910
$553$	3.26725	0.138938
$554$	−34.8491	−1.48060
$555$	0	0
$556$	−0.790584	−0.0335282
$557$	16.8871	0.715528	0.357764	−	0.933812i	$-0.383539\pi$
$557$	16.8871	0.715528	0.357764	+	0.933812i	$0.383539\pi$
$558$	−1.71980	−0.0728048
$559$	22.4649	0.950164
$560$	0	0
$561$	−19.2689	−0.813535
$562$	21.5213	0.907820
$563$	−30.0821	−1.26781	−0.633906	−	0.773410i	$-0.718549\pi$
$563$	−30.0821	−1.26781	−0.633906	+	0.773410i	$0.718549\pi$
$564$	5.55765	0.234019
$565$	0	0
$566$	85.2302	3.58249
$567$	−3.47363	−0.145879
$568$	47.7244	2.00247
$569$	2.57373	0.107897	0.0539483	−	0.998544i	$-0.482819\pi$
$569$	2.57373	0.107897	0.0539483	+	0.998544i	$0.482819\pi$
$570$	0	0
$571$	42.8709	1.79409	0.897046	−	0.441936i	$-0.145708\pi$
$571$	42.8709	1.79409	0.897046	+	0.441936i	$0.145708\pi$
$572$	−170.515	−7.12957
$573$	−1.65242	−0.0690309
$574$	0.947367	0.0395424
$575$	0	0
$576$	33.8812	1.41171
$577$	15.8500	0.659844	0.329922	−	0.944008i	$-0.392978\pi$
$577$	15.8500	0.659844	0.329922	+	0.944008i	$0.392978\pi$
$578$	11.5495	0.480397
$579$	−10.0973	−0.419632
$580$	0	0
$581$	28.3400	1.17574
$582$	41.0464	1.70143
$583$	41.0514	1.70018
$584$	−9.47453	−0.392059
$585$	0	0
$586$	49.9008	2.06138
$587$	−21.4388	−0.884874	−0.442437	−	0.896800i	$-0.645886\pi$
$587$	−21.4388	−0.884874	−0.442437	+	0.896800i	$0.645886\pi$
$588$	28.1555	1.16111
$589$	−2.84823	−0.117359
$590$	0	0
$591$	9.70970	0.399404
$592$	137.623	5.65627
$593$	−2.18177	−0.0895944	−0.0447972	−	0.998996i	$-0.514264\pi$
$593$	−2.18177	−0.0895944	−0.0447972	+	0.998996i	$0.514264\pi$
$594$	14.8070	0.607537
$595$	0	0
$596$	−43.2145	−1.77013
$597$	−14.7835	−0.605049
$598$	83.8561	3.42913
$599$	19.0425	0.778057	0.389028	−	0.921226i	$-0.372811\pi$
$599$	19.0425	0.778057	0.389028	+	0.921226i	$0.372811\pi$
$600$	0	0
$601$	−43.7812	−1.78587	−0.892936	−	0.450184i	$-0.851358\pi$
$601$	−43.7812	−1.78587	−0.892936	+	0.450184i	$0.851358\pi$
$602$	37.6602	1.53492
$603$	6.36147	0.259059
$604$	34.8770	1.41912
$605$	0	0
$606$	16.3990	0.666165
$607$	−36.5175	−1.48220	−0.741099	−	0.671396i	$-0.765695\pi$
$607$	−36.5175	−1.48220	−0.741099	+	0.671396i	$0.765695\pi$
$608$	108.354	4.39433
$609$	−23.5362	−0.953734
$610$	0	0
$611$	5.69637	0.230450
$612$	19.8827	0.803712
$613$	−23.2369	−0.938529	−0.469264	−	0.883058i	$-0.655481\pi$
$613$	−23.2369	−0.938529	−0.469264	+	0.883058i	$0.655481\pi$
$614$	−22.1883	−0.895447
$615$	0	0
$616$	−182.983	−7.37261
$617$	−24.8093	−0.998785	−0.499393	−	0.866376i	$-0.666444\pi$
$617$	−24.8093	−0.998785	−0.499393	+	0.866376i	$0.666444\pi$
$618$	3.19736	0.128617
$619$	−5.40922	−0.217415	−0.108707	−	0.994074i	$-0.534671\pi$
$619$	−5.40922	−0.217415	−0.108707	+	0.994074i	$0.534671\pi$
$620$	0	0
$621$	−5.35480	−0.214881
$622$	11.2854	0.452504
$623$	41.6416	1.66834
$624$	89.8440	3.59664
$625$	0	0
$626$	8.81547	0.352337
$627$	24.5224	0.979332
$628$	41.2333	1.64539
$629$	31.2165	1.24469
$630$	0	0
$631$	−4.76400	−0.189652	−0.0948259	−	0.995494i	$-0.530229\pi$
$631$	−4.76400	−0.189652	−0.0948259	+	0.995494i	$0.530229\pi$
$632$	9.19932	0.365929
$633$	−6.74441	−0.268066
$634$	−42.0610	−1.67046
$635$	0	0
$636$	−42.3591	−1.67965
$637$	28.8582	1.14341
$638$	100.327	3.97199
$639$	−4.87960	−0.193034
$640$	0	0
$641$	13.6328	0.538461	0.269231	−	0.963076i	$-0.413231\pi$
$641$	13.6328	0.538461	0.269231	+	0.963076i	$0.413231\pi$
$642$	29.7686	1.17487
$643$	−27.3914	−1.08021	−0.540106	−	0.841597i	$-0.681616\pi$
$643$	−27.3914	−1.08021	−0.540106	+	0.841597i	$0.681616\pi$
$644$	103.375	4.07356
$645$	0	0
$646$	44.7785	1.76179
$647$	−9.69609	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$647$	−9.69609	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$648$	−9.78039	−0.384210
$649$	−67.3079	−2.64207
$650$	0	0
$651$	−2.17303	−0.0851680
$652$	−43.6568	−1.70973
$653$	32.1122	1.25665	0.628324	−	0.777951i	$-0.283741\pi$
$653$	32.1122	1.25665	0.628324	+	0.777951i	$0.283741\pi$
$654$	41.4321	1.62012
$655$	0	0
$656$	1.56471	0.0610915
$657$	0.968727	0.0377936
$658$	9.54941	0.372275
$659$	−21.1899	−0.825441	−0.412721	−	0.910858i	$-0.635421\pi$
$659$	−21.1899	−0.825441	−0.412721	+	0.910858i	$0.635421\pi$
$660$	0	0
$661$	−5.39959	−0.210020	−0.105010	−	0.994471i	$-0.533487\pi$
$661$	−5.39959	−0.210020	−0.105010	+	0.994471i	$0.533487\pi$
$662$	−23.6537	−0.919327
$663$	20.3790	0.791455
$664$	79.7945	3.09663
$665$	0	0
$666$	−23.9879	−0.929514
$667$	−36.2824	−1.40486
$668$	−142.400	−5.50964
$669$	1.08755	0.0420472
$670$	0	0
$671$	15.2531	0.588841
$672$	82.6678	3.18898
$673$	−5.05693	−0.194930	−0.0974652	−	0.995239i	$-0.531073\pi$
$673$	−5.05693	−0.194930	−0.0974652	+	0.995239i	$0.531073\pi$
$674$	−60.8169	−2.34258
$675$	0	0
$676$	108.089	4.15725
$677$	−13.4364	−0.516404	−0.258202	−	0.966091i	$-0.583130\pi$
$677$	−13.4364	−0.516404	−0.258202	+	0.966091i	$0.583130\pi$
$678$	−32.8530	−1.26171
$679$	51.8639	1.99035
$680$	0	0
$681$	13.8723	0.531586
$682$	9.26295	0.354697
$683$	43.8312	1.67715	0.838576	−	0.544784i	$-0.183388\pi$
$683$	43.8312	1.67715	0.838576	+	0.544784i	$0.183388\pi$
$684$	−25.3036	−0.967508
$685$	0	0
$686$	−18.4678	−0.705105
$687$	−0.427167	−0.0162974
$688$	62.2010	2.37139
$689$	−43.4164	−1.65403
$690$	0	0
$691$	−50.0127	−1.90257	−0.951286	−	0.308310i	$-0.900237\pi$
$691$	−50.0127	−1.90257	−0.951286	+	0.308310i	$0.900237\pi$
$692$	−12.4159	−0.471983
$693$	18.7092	0.710704
$694$	42.1445	1.59978
$695$	0	0
$696$	−66.2688	−2.51191
$697$	0.354917	0.0134435
$698$	−82.0597	−3.10601
$699$	−19.7061	−0.745352
$700$	0	0
$701$	−7.14824	−0.269985	−0.134993	−	0.990847i	$-0.543101\pi$
$701$	−7.14824	−0.269985	−0.134993	+	0.990847i	$0.543101\pi$
$702$	−15.6600	−0.591048
$703$	−39.7275	−1.49835
$704$	−182.486	−6.87772
$705$	0	0
$706$	0.955532	0.0359619
$707$	20.7209	0.779288
$708$	69.4520	2.61017
$709$	−46.9598	−1.76361	−0.881806	−	0.471612i	$-0.843672\pi$
$709$	−46.9598	−1.76361	−0.881806	+	0.471612i	$0.843672\pi$
$710$	0	0
$711$	−0.940588	−0.0352748
$712$	117.247	4.39401
$713$	−3.34986	−0.125453
$714$	34.1634	1.27853
$715$	0	0
$716$	67.0394	2.50538
$717$	10.5952	0.395685
$718$	−83.3003	−3.10874
$719$	23.5683	0.878948	0.439474	−	0.898255i	$-0.355165\pi$
$719$	23.5683	0.878948	0.439474	+	0.898255i	$0.355165\pi$
$720$	0	0
$721$	4.04000	0.150458
$722$	−4.75383	−0.176919
$723$	−14.0590	−0.522860
$724$	−30.7507	−1.14284
$725$	0	0
$726$	−49.5111	−1.83753
$727$	15.4027	0.571255	0.285628	−	0.958341i	$-0.407798\pi$
$727$	15.4027	0.571255	0.285628	+	0.958341i	$0.407798\pi$
$728$	193.525	7.17252
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.1088	0.521834
$732$	−15.7390	−0.581731
$733$	−36.3723	−1.34344	−0.671721	−	0.740804i	$-0.734445\pi$
$733$	−36.3723	−1.34344	−0.671721	+	0.740804i	$0.734445\pi$
$734$	−76.5565	−2.82575
$735$	0	0
$736$	127.437	4.69740
$737$	−34.2633	−1.26211
$738$	−0.272732	−0.0100394
$739$	5.79663	0.213232	0.106616	−	0.994300i	$-0.465998\pi$
$739$	5.79663	0.213232	0.106616	+	0.994300i	$0.465998\pi$
$740$	0	0
$741$	−25.9352	−0.952753
$742$	−72.7834	−2.67196
$743$	50.1962	1.84152	0.920759	−	0.390131i	$-0.127570\pi$
$743$	50.1962	1.84152	0.920759	+	0.390131i	$0.127570\pi$
$744$	−6.11843	−0.224312
$745$	0	0
$746$	36.6772	1.34285
$747$	−8.15862	−0.298508
$748$	−107.090	−3.91559
$749$	37.6139	1.37438
$750$	0	0
$751$	−18.6016	−0.678782	−0.339391	−	0.940645i	$-0.610221\pi$
$751$	−18.6016	−0.678782	−0.339391	+	0.940645i	$0.610221\pi$
$752$	15.7721	0.575151
$753$	26.6526	0.971277
$754$	−106.107	−3.86419
$755$	0	0
$756$	−19.3052	−0.702123
$757$	23.8534	0.866968	0.433484	−	0.901161i	$-0.357284\pi$
$757$	23.8534	0.866968	0.433484	+	0.901161i	$0.357284\pi$
$758$	−81.2650	−2.95168
$759$	28.8414	1.04687
$760$	0	0
$761$	−15.2012	−0.551041	−0.275521	−	0.961295i	$-0.588850\pi$
$761$	−15.2012	−0.551041	−0.275521	+	0.961295i	$0.588850\pi$
$762$	40.4751	1.46626
$763$	52.3512	1.89524
$764$	−9.18358	−0.332250
$765$	0	0
$766$	−61.6096	−2.22605
$767$	71.1856	2.57036
$768$	57.4486	2.07300
$769$	4.25023	0.153267	0.0766336	−	0.997059i	$-0.475583\pi$
$769$	4.25023	0.153267	0.0766336	+	0.997059i	$0.475583\pi$
$770$	0	0
$771$	1.48215	0.0533782
$772$	−56.1175	−2.01971
$773$	−31.3940	−1.12916	−0.564582	−	0.825377i	$-0.690963\pi$
$773$	−31.3940	−1.12916	−0.564582	+	0.825377i	$0.690963\pi$
$774$	−10.8418	−0.389699
$775$	0	0
$776$	146.029	5.24213
$777$	−30.3098	−1.08736
$778$	36.3724	1.30402
$779$	−0.451683	−0.0161832
$780$	0	0
$781$	26.2819	0.940439
$782$	52.6650	1.88330
$783$	6.77568	0.242143
$784$	79.9029	2.85368
$785$	0	0
$786$	22.2239	0.792701
$787$	51.2276	1.82607	0.913034	−	0.407884i	$-0.133733\pi$
$787$	51.2276	1.82607	0.913034	+	0.407884i	$0.133733\pi$
$788$	53.9631	1.92235
$789$	−23.8691	−0.849763
$790$	0	0
$791$	−41.5112	−1.47597
$792$	52.6779	1.87183
$793$	−16.1319	−0.572860
$794$	4.36199	0.154801
$795$	0	0
$796$	−82.1616	−2.91214
$797$	17.4125	0.616784	0.308392	−	0.951259i	$-0.400209\pi$
$797$	17.4125	0.616784	0.308392	+	0.951259i	$0.400209\pi$
$798$	−43.4778	−1.53910
$799$	3.57754	0.126564
$800$	0	0
$801$	−11.9879	−0.423573
$802$	96.9527	3.42352
$803$	−5.21764	−0.184126
$804$	35.3548	1.24687
$805$	0	0
$806$	−9.79659	−0.345070
$807$	−4.35703	−0.153375
$808$	58.3420	2.05246
$809$	35.0761	1.23321	0.616604	−	0.787273i	$-0.288508\pi$
$809$	35.0761	1.23321	0.616604	+	0.787273i	$0.288508\pi$
$810$	0	0
$811$	−22.5782	−0.792828	−0.396414	−	0.918072i	$-0.629745\pi$
$811$	−22.5782	−0.792828	−0.396414	+	0.918072i	$0.629745\pi$
$812$	−130.806	−4.59038
$813$	−23.1177	−0.810772
$814$	129.201	4.52849
$815$	0	0
$816$	56.4256	1.97529
$817$	−17.9555	−0.628184
$818$	16.0708	0.561903
$819$	−19.7871	−0.691416
$820$	0	0
$821$	10.6292	0.370963	0.185482	−	0.982648i	$-0.440615\pi$
$821$	10.6292	0.370963	0.185482	+	0.982648i	$0.440615\pi$
$822$	18.3361	0.639544
$823$	37.2333	1.29787	0.648936	−	0.760843i	$-0.275214\pi$
$823$	37.2333	1.29787	0.648936	+	0.760843i	$0.275214\pi$
$824$	11.3751	0.396270
$825$	0	0
$826$	119.336	4.15222
$827$	−15.7183	−0.546579	−0.273290	−	0.961932i	$-0.588112\pi$
$827$	−15.7183	−0.546579	−0.273290	+	0.961932i	$0.588112\pi$
$828$	−29.7601	−1.03423
$829$	0.494145	0.0171624	0.00858119	−	0.999963i	$-0.497268\pi$
$829$	0.494145	0.0171624	0.00858119	+	0.999963i	$0.497268\pi$
$830$	0	0
$831$	12.6765	0.439742
$832$	193.000	6.69106
$833$	18.1241	0.627963
$834$	0.391066	0.0135415
$835$	0	0
$836$	136.287	4.71359
$837$	0.625581	0.0216232
$838$	35.0394	1.21042
$839$	−12.9732	−0.447883	−0.223942	−	0.974603i	$-0.571892\pi$
$839$	−12.9732	−0.447883	−0.223942	+	0.974603i	$0.571892\pi$
$840$	0	0
$841$	16.9098	0.583097
$842$	23.1464	0.797678
$843$	−7.82843	−0.269625
$844$	−37.4830	−1.29022
$845$	0	0
$846$	−2.74912	−0.0945166
$847$	−62.5593	−2.14956
$848$	−120.212	−4.12809
$849$	−31.0027	−1.06401
$850$	0	0
$851$	−46.7243	−1.60169
$852$	−27.1191	−0.929084
$853$	19.4098	0.664578	0.332289	−	0.943178i	$-0.392179\pi$
$853$	19.4098	0.664578	0.332289	+	0.943178i	$0.392179\pi$
$854$	−27.0435	−0.925410
$855$	0	0
$856$	105.906	3.61980
$857$	15.4049	0.526222	0.263111	−	0.964766i	$-0.415251\pi$
$857$	15.4049	0.526222	0.263111	+	0.964766i	$0.415251\pi$
$858$	84.3459	2.87952
$859$	−12.9218	−0.440885	−0.220442	−	0.975400i	$-0.570750\pi$
$859$	−12.9218	−0.440885	−0.220442	+	0.975400i	$0.570750\pi$
$860$	0	0
$861$	−0.344608	−0.0117442
$862$	78.3773	2.66954
$863$	33.6406	1.14514	0.572569	−	0.819857i	$-0.305947\pi$
$863$	33.6406	1.14514	0.572569	+	0.819857i	$0.305947\pi$
$864$	−23.7987	−0.809649
$865$	0	0
$866$	55.2328	1.87689
$867$	−4.20117	−0.142679
$868$	−12.0770	−0.409919
$869$	5.06608	0.171855
$870$	0	0
$871$	36.2373	1.22785
$872$	147.401	4.99163
$873$	−14.9308	−0.505330
$874$	−67.0237	−2.26711
$875$	0	0
$876$	5.38384	0.181903
$877$	9.26267	0.312778	0.156389	−	0.987696i	$-0.450015\pi$
$877$	9.26267	0.312778	0.156389	+	0.987696i	$0.450015\pi$
$878$	39.6853	1.33932
$879$	−18.1516	−0.612237
$880$	0	0
$881$	34.3559	1.15748	0.578739	−	0.815513i	$-0.303545\pi$
$881$	34.3559	1.15748	0.578739	+	0.815513i	$0.303545\pi$
$882$	−13.9272	−0.468955
$883$	−23.1334	−0.778502	−0.389251	−	0.921132i	$-0.627266\pi$
$883$	−23.1334	−0.778502	−0.389251	+	0.921132i	$0.627266\pi$
$884$	113.259	3.80933
$885$	0	0
$886$	−81.4873	−2.73762
$887$	33.5980	1.12811	0.564055	−	0.825737i	$-0.309241\pi$
$887$	33.5980	1.12811	0.564055	+	0.825737i	$0.309241\pi$
$888$	−85.3407	−2.86385
$889$	51.1419	1.71525
$890$	0	0
$891$	−5.38607	−0.180440
$892$	6.04423	0.202376
$893$	−4.55294	−0.152358
$894$	21.3763	0.714929
$895$	0	0
$896$	158.209	5.28540
$897$	−30.5029	−1.01846
$898$	−8.50279	−0.283742
$899$	4.23874	0.141370
$900$	0	0
$901$	−27.2672	−0.908403
$902$	1.46895	0.0489108
$903$	−13.6990	−0.455875
$904$	−116.879	−3.88735
$905$	0	0
$906$	−17.2521	−0.573161
$907$	−35.2589	−1.17075	−0.585376	−	0.810762i	$-0.699053\pi$
$907$	−35.2589	−1.17075	−0.585376	+	0.810762i	$0.699053\pi$
$908$	77.0971	2.55856
$909$	−5.96520	−0.197853
$910$	0	0
$911$	−30.2661	−1.00276	−0.501381	−	0.865227i	$-0.667174\pi$
$911$	−30.2661	−1.00276	−0.501381	+	0.865227i	$0.667174\pi$
$912$	−71.8096	−2.37785
$913$	43.9429	1.45430
$914$	12.5737	0.415902
$915$	0	0
$916$	−2.37405	−0.0784407
$917$	28.0809	0.927312
$918$	−9.83509	−0.324606
$919$	29.2586	0.965153	0.482577	−	0.875854i	$-0.339701\pi$
$919$	29.2586	0.965153	0.482577	+	0.875854i	$0.339701\pi$
$920$	0	0
$921$	8.07107	0.265951
$922$	−87.2370	−2.87300
$923$	−27.7960	−0.914916
$924$	103.979	3.42066
$925$	0	0
$926$	26.5859	0.873668
$927$	−1.16305	−0.0381996
$928$	−161.252	−5.29337
$929$	26.3764	0.865383	0.432691	−	0.901542i	$-0.357564\pi$
$929$	26.3764	0.865383	0.432691	+	0.901542i	$0.357564\pi$
$930$	0	0
$931$	−23.0655	−0.755942
$932$	−109.519	−3.58743
$933$	−4.10510	−0.134395
$934$	−80.0485	−2.61927
$935$	0	0
$936$	−55.7127	−1.82103
$937$	−3.41365	−0.111519	−0.0557596	−	0.998444i	$-0.517758\pi$
$937$	−3.41365	−0.111519	−0.0557596	+	0.998444i	$0.517758\pi$
$938$	60.7482	1.98350
$939$	−3.20666	−0.104645
$940$	0	0
$941$	4.35332	0.141914	0.0709570	−	0.997479i	$-0.477395\pi$
$941$	4.35332	0.141914	0.0709570	+	0.997479i	$0.477395\pi$
$942$	−20.3963	−0.664546
$943$	−0.531234	−0.0172993
$944$	197.099	6.41503
$945$	0	0
$946$	58.3945	1.89857
$947$	46.9459	1.52554	0.762769	−	0.646671i	$-0.223839\pi$
$947$	46.9459	1.52554	0.762769	+	0.646671i	$0.223839\pi$
$948$	−5.22746	−0.169780
$949$	5.51823	0.179129
$950$	0	0
$951$	15.2998	0.496131
$952$	121.541	3.93918
$953$	−53.4741	−1.73220	−0.866099	−	0.499873i	$-0.833380\pi$
$953$	−53.4741	−1.73220	−0.866099	+	0.499873i	$0.833380\pi$
$954$	20.9531	0.678383
$955$	0	0
$956$	58.8844	1.90446
$957$	−36.4943	−1.17969
$958$	−15.5042	−0.500917
$959$	23.1684	0.748147
$960$	0	0
$961$	−30.6086	−0.987376
$962$	−136.644	−4.40559
$963$	−10.8284	−0.348941
$964$	−78.1351	−2.51656
$965$	0	0
$966$	−51.1352	−1.64525
$967$	−43.5631	−1.40090	−0.700448	−	0.713703i	$-0.747017\pi$
$967$	−43.5631	−1.40090	−0.700448	+	0.713703i	$0.747017\pi$
$968$	−176.143	−5.66145
$969$	−16.2883	−0.523257
$970$	0	0
$971$	−21.8969	−0.702704	−0.351352	−	0.936243i	$-0.614278\pi$
$971$	−21.8969	−0.702704	−0.351352	+	0.936243i	$0.614278\pi$
$972$	5.55765	0.178262
$973$	0.494129	0.0158410
$974$	45.3800	1.45407
$975$	0	0
$976$	−44.6660	−1.42973
$977$	51.2071	1.63826	0.819131	−	0.573607i	$-0.194456\pi$
$977$	51.2071	1.63826	0.819131	+	0.573607i	$0.194456\pi$
$978$	21.5951	0.690534
$979$	64.5680	2.06360
$980$	0	0
$981$	−15.0711	−0.481182
$982$	−29.4020	−0.938256
$983$	43.9836	1.40286	0.701430	−	0.712738i	$-0.252545\pi$
$983$	43.9836	1.40286	0.701430	+	0.712738i	$0.252545\pi$
$984$	−0.970283	−0.0309315
$985$	0	0
$986$	−66.6394	−2.12223
$987$	−3.47363	−0.110567
$988$	−144.139	−4.58566
$989$	−21.1178	−0.671508
$990$	0	0
$991$	−31.9937	−1.01631	−0.508156	−	0.861265i	$-0.669673\pi$
$991$	−31.9937	−1.01631	−0.508156	+	0.861265i	$0.669673\pi$
$992$	−14.8880	−0.472695
$993$	8.60411	0.273043
$994$	−46.5973	−1.47798
$995$	0	0
$996$	−45.3427	−1.43674
$997$	35.2376	1.11598	0.557992	−	0.829846i	$-0.311572\pi$
$997$	35.2376	1.11598	0.557992	+	0.829846i	$0.311572\pi$
$998$	44.9030	1.42138
$999$	8.72569	0.276069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.t.1.1		4
5.4	even	2		705.2.a.k.1.4	✓	4
15.14	odd	2		2115.2.a.o.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.k.1.4	✓	4	5.4	even	2
2115.2.a.o.1.1		4	15.14	odd	2
3525.2.a.t.1.1		4	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:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3525.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74912 q^{2} +1.00000 q^{3} +5.55765 q^{4} -2.74912 q^{6} -3.47363 q^{7} -9.78039 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.74912 q^{2} +1.00000 q^{3} +5.55765 q^{4} -2.74912 q^{6} -3.47363 q^{7} -9.78039 q^{8} +1.00000 q^{9} -5.38607 q^{11} +5.55765 q^{12} +5.69637 q^{13} +9.54941 q^{14} +15.7721 q^{16} +3.57754 q^{17} -2.74912 q^{18} -4.55294 q^{19} -3.47363 q^{21} +14.8070 q^{22} -5.35480 q^{23} -9.78039 q^{24} -15.6600 q^{26} +1.00000 q^{27} -19.3052 q^{28} +6.77568 q^{29} +0.625581 q^{31} -23.7987 q^{32} -5.38607 q^{33} -9.83509 q^{34} +5.55765 q^{36} +8.72569 q^{37} +12.5166 q^{38} +5.69637 q^{39} +0.0992070 q^{41} +9.54941 q^{42} +3.94372 q^{43} -29.9339 q^{44} +14.7210 q^{46} +1.00000 q^{47} +15.7721 q^{48} +5.06608 q^{49} +3.57754 q^{51} +31.6584 q^{52} -7.62177 q^{53} -2.74912 q^{54} +33.9734 q^{56} -4.55294 q^{57} -18.6271 q^{58} +12.4967 q^{59} -2.83196 q^{61} -1.71980 q^{62} -3.47363 q^{63} +33.8812 q^{64} +14.8070 q^{66} +6.36147 q^{67} +19.8827 q^{68} -5.35480 q^{69} -4.87960 q^{71} -9.78039 q^{72} +0.968727 q^{73} -23.9879 q^{74} -25.3036 q^{76} +18.7092 q^{77} -15.6600 q^{78} -0.940588 q^{79} +1.00000 q^{81} -0.272732 q^{82} -8.15862 q^{83} -19.3052 q^{84} -10.8418 q^{86} +6.77568 q^{87} +52.6779 q^{88} -11.9879 q^{89} -19.7871 q^{91} -29.7601 q^{92} +0.625581 q^{93} -2.74912 q^{94} -23.7987 q^{96} -14.9308 q^{97} -13.9272 q^{98} -5.38607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 4 q^{9} + 4 q^{11} + 8 q^{12} + 4 q^{13} + 12 q^{16} - 4 q^{17} - 4 q^{18} - 8 q^{19} - 8 q^{21} + 16 q^{22} - 16 q^{23} - 12 q^{24} + 4 q^{27} - 20 q^{28} + 4 q^{29} - 28 q^{32} + 4 q^{33} - 16 q^{34} + 8 q^{36} + 4 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{41} - 24 q^{43} - 20 q^{44} + 32 q^{46} + 4 q^{47} + 12 q^{48} + 8 q^{49} - 4 q^{51} + 28 q^{52} - 12 q^{53} - 4 q^{54} + 40 q^{56} - 8 q^{57} + 4 q^{58} - 28 q^{61} - 12 q^{62} - 8 q^{63} + 24 q^{64} + 16 q^{66} + 8 q^{67} + 4 q^{68} - 16 q^{69} + 16 q^{71} - 12 q^{72} + 24 q^{73} - 56 q^{74} - 8 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{81} + 4 q^{82} - 24 q^{83} - 20 q^{84} - 8 q^{86} + 4 q^{87} + 40 q^{88} - 8 q^{89} - 40 q^{91} - 28 q^{92} - 4 q^{94} - 28 q^{96} + 32 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 4 * q^9 + 4 * q^11 + 8 * q^12 + 4 * q^13 + 12 * q^16 - 4 * q^17 - 4 * q^18 - 8 * q^19 - 8 * q^21 + 16 * q^22 - 16 * q^23 - 12 * q^24 + 4 * q^27 - 20 * q^28 + 4 * q^29 - 28 * q^32 + 4 * q^33 - 16 * q^34 + 8 * q^36 + 4 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^41 - 24 * q^43 - 20 * q^44 + 32 * q^46 + 4 * q^47 + 12 * q^48 + 8 * q^49 - 4 * q^51 + 28 * q^52 - 12 * q^53 - 4 * q^54 + 40 * q^56 - 8 * q^57 + 4 * q^58 - 28 * q^61 - 12 * q^62 - 8 * q^63 + 24 * q^64 + 16 * q^66 + 8 * q^67 + 4 * q^68 - 16 * q^69 + 16 * q^71 - 12 * q^72 + 24 * q^73 - 56 * q^74 - 8 * q^76 - 4 * q^77 - 4 * q^79 + 4 * q^81 + 4 * q^82 - 24 * q^83 - 20 * q^84 - 8 * q^86 + 4 * q^87 + 40 * q^88 - 8 * q^89 - 40 * q^91 - 28 * q^92 - 4 * q^94 - 28 * q^96 + 32 * q^98 + 4 * q^99

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