LMFDB - Embedded newform 3525.2.a.p.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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			$-1.73205$ 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)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} +1.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} +1.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} -2.26795 q^{13} -1.73205 q^{14} -5.00000 q^{16} +7.19615 q^{17} -1.73205 q^{18} -1.73205 q^{19} +1.00000 q^{21} +6.92820 q^{22} -2.46410 q^{23} +1.73205 q^{24} +3.92820 q^{26} +1.00000 q^{27} +1.00000 q^{28} -6.46410 q^{29} +3.46410 q^{31} +5.19615 q^{32} -4.00000 q^{33} -12.4641 q^{34} +1.00000 q^{36} -4.53590 q^{37} +3.00000 q^{38} -2.26795 q^{39} +0.464102 q^{41} -1.73205 q^{42} +11.4641 q^{43} -4.00000 q^{44} +4.26795 q^{46} +1.00000 q^{47} -5.00000 q^{48} -6.00000 q^{49} +7.19615 q^{51} -2.26795 q^{52} -14.1244 q^{53} -1.73205 q^{54} +1.73205 q^{56} -1.73205 q^{57} +11.1962 q^{58} -4.80385 q^{59} +7.92820 q^{61} -6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +6.92820 q^{66} -4.92820 q^{67} +7.19615 q^{68} -2.46410 q^{69} -7.19615 q^{71} +1.73205 q^{72} -4.00000 q^{73} +7.85641 q^{74} -1.73205 q^{76} -4.00000 q^{77} +3.92820 q^{78} -1.46410 q^{79} +1.00000 q^{81} -0.803848 q^{82} -10.9282 q^{83} +1.00000 q^{84} -19.8564 q^{86} -6.46410 q^{87} -6.92820 q^{88} -13.4641 q^{89} -2.26795 q^{91} -2.46410 q^{92} +3.46410 q^{93} -1.73205 q^{94} +5.19615 q^{96} -1.46410 q^{97} +10.3923 q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{4} + 2 q^{7} + 2 q^{9} - 8 q^{11} + 2 q^{12} - 8 q^{13} - 10 q^{16} + 4 q^{17} + 2 q^{21} + 2 q^{23} - 6 q^{26} + 2 q^{27} + 2 q^{28} - 6 q^{29} - 8 q^{33} - 18 q^{34} + 2 q^{36} - 16 q^{37} + 6 q^{38} - 8 q^{39} - 6 q^{41} + 16 q^{43} - 8 q^{44} + 12 q^{46} + 2 q^{47} - 10 q^{48} - 12 q^{49} + 4 q^{51} - 8 q^{52} - 4 q^{53} + 12 q^{58} - 20 q^{59} + 2 q^{61} - 12 q^{62} + 2 q^{63} + 2 q^{64} + 4 q^{67} + 4 q^{68} + 2 q^{69} - 4 q^{71} - 8 q^{73} - 12 q^{74} - 8 q^{77} - 6 q^{78} + 4 q^{79} + 2 q^{81} - 12 q^{82} - 8 q^{83} + 2 q^{84} - 12 q^{86} - 6 q^{87} - 20 q^{89} - 8 q^{91} + 2 q^{92} + 4 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^7 + 2 * q^9 - 8 * q^11 + 2 * q^12 - 8 * q^13 - 10 * q^16 + 4 * q^17 + 2 * q^21 + 2 * q^23 - 6 * q^26 + 2 * q^27 + 2 * q^28 - 6 * q^29 - 8 * q^33 - 18 * q^34 + 2 * q^36 - 16 * q^37 + 6 * q^38 - 8 * q^39 - 6 * q^41 + 16 * q^43 - 8 * q^44 + 12 * q^46 + 2 * q^47 - 10 * q^48 - 12 * q^49 + 4 * q^51 - 8 * q^52 - 4 * q^53 + 12 * q^58 - 20 * q^59 + 2 * q^61 - 12 * q^62 + 2 * q^63 + 2 * q^64 + 4 * q^67 + 4 * q^68 + 2 * q^69 - 4 * q^71 - 8 * q^73 - 12 * q^74 - 8 * q^77 - 6 * q^78 + 4 * q^79 + 2 * q^81 - 12 * q^82 - 8 * q^83 + 2 * q^84 - 12 * q^86 - 6 * q^87 - 20 * q^89 - 8 * q^91 + 2 * q^92 + 4 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.73205	−0.707107
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.73205	0.612372
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	−2.26795	−0.629016	−0.314508	−	0.949255i	$-0.601840\pi$
$13$	−2.26795	−0.629016	−0.314508	+	0.949255i	$0.601840\pi$
$14$	−1.73205	−0.462910
$15$	0	0
$16$	−5.00000	−1.25000
$17$	7.19615	1.74532	0.872662	−	0.488325i	$-0.162392\pi$
$17$	7.19615	1.74532	0.872662	+	0.488325i	$0.162392\pi$
$18$	−1.73205	−0.408248
$19$	−1.73205	−0.397360	−0.198680	−	0.980064i	$-0.563665\pi$
$19$	−1.73205	−0.397360	−0.198680	+	0.980064i	$0.563665\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	6.92820	1.47710
$23$	−2.46410	−0.513801	−0.256900	−	0.966438i	$-0.582701\pi$
$23$	−2.46410	−0.513801	−0.256900	+	0.966438i	$0.582701\pi$
$24$	1.73205	0.353553
$25$	0	0
$26$	3.92820	0.770384
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	−6.46410	−1.20035	−0.600177	−	0.799867i	$-0.704903\pi$
$29$	−6.46410	−1.20035	−0.600177	+	0.799867i	$0.704903\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	5.19615	0.918559
$33$	−4.00000	−0.696311
$34$	−12.4641	−2.13758
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.53590	−0.745697	−0.372849	−	0.927892i	$-0.621619\pi$
$37$	−4.53590	−0.745697	−0.372849	+	0.927892i	$0.621619\pi$
$38$	3.00000	0.486664
$39$	−2.26795	−0.363163
$40$	0	0
$41$	0.464102	0.0724805	0.0362402	−	0.999343i	$-0.488462\pi$
$41$	0.464102	0.0724805	0.0362402	+	0.999343i	$0.488462\pi$
$42$	−1.73205	−0.267261
$43$	11.4641	1.74826	0.874130	−	0.485693i	$-0.161433\pi$
$43$	11.4641	1.74826	0.874130	+	0.485693i	$0.161433\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	4.26795	0.629275
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	−5.00000	−0.721688
$49$	−6.00000	−0.857143
$50$	0	0
$51$	7.19615	1.00766
$52$	−2.26795	−0.314508
$53$	−14.1244	−1.94013	−0.970065	−	0.242846i	$-0.921919\pi$
$53$	−14.1244	−1.94013	−0.970065	+	0.242846i	$0.921919\pi$
$54$	−1.73205	−0.235702
$55$	0	0
$56$	1.73205	0.231455
$57$	−1.73205	−0.229416
$58$	11.1962	1.47013
$59$	−4.80385	−0.625408	−0.312704	−	0.949851i	$-0.601235\pi$
$59$	−4.80385	−0.625408	−0.312704	+	0.949851i	$0.601235\pi$
$60$	0	0
$61$	7.92820	1.01510	0.507551	−	0.861622i	$-0.330551\pi$
$61$	7.92820	1.01510	0.507551	+	0.861622i	$0.330551\pi$
$62$	−6.00000	−0.762001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	6.92820	0.852803
$67$	−4.92820	−0.602076	−0.301038	−	0.953612i	$-0.597333\pi$
$67$	−4.92820	−0.602076	−0.301038	+	0.953612i	$0.597333\pi$
$68$	7.19615	0.872662
$69$	−2.46410	−0.296643
$70$	0	0
$71$	−7.19615	−0.854026	−0.427013	−	0.904245i	$-0.640434\pi$
$71$	−7.19615	−0.854026	−0.427013	+	0.904245i	$0.640434\pi$
$72$	1.73205	0.204124
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	7.85641	0.913289
$75$	0	0
$76$	−1.73205	−0.198680
$77$	−4.00000	−0.455842
$78$	3.92820	0.444781
$79$	−1.46410	−0.164724	−0.0823622	−	0.996602i	$-0.526246\pi$
$79$	−1.46410	−0.164724	−0.0823622	+	0.996602i	$0.526246\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.803848	−0.0887701
$83$	−10.9282	−1.19953	−0.599763	−	0.800178i	$-0.704739\pi$
$83$	−10.9282	−1.19953	−0.599763	+	0.800178i	$0.704739\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	−19.8564	−2.14117
$87$	−6.46410	−0.693024
$88$	−6.92820	−0.738549
$89$	−13.4641	−1.42719	−0.713596	−	0.700557i	$-0.752935\pi$
$89$	−13.4641	−1.42719	−0.713596	+	0.700557i	$0.752935\pi$
$90$	0	0
$91$	−2.26795	−0.237746
$92$	−2.46410	−0.256900
$93$	3.46410	0.359211
$94$	−1.73205	−0.178647
$95$	0	0
$96$	5.19615	0.530330
$97$	−1.46410	−0.148657	−0.0743285	−	0.997234i	$-0.523681\pi$
$97$	−1.46410	−0.148657	−0.0743285	+	0.997234i	$0.523681\pi$
$98$	10.3923	1.04978
$99$	−4.00000	−0.402015
$100$	0	0
$101$	10.3923	1.03407	0.517036	−	0.855963i	$-0.327035\pi$
$101$	10.3923	1.03407	0.517036	+	0.855963i	$0.327035\pi$
$102$	−12.4641	−1.23413
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	−3.92820	−0.385192
$105$	0	0
$106$	24.4641	2.37616
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	1.00000	0.0962250
$109$	10.3923	0.995402	0.497701	−	0.867349i	$-0.334178\pi$
$109$	10.3923	0.995402	0.497701	+	0.867349i	$0.334178\pi$
$110$	0	0
$111$	−4.53590	−0.430528
$112$	−5.00000	−0.472456
$113$	−6.92820	−0.651751	−0.325875	−	0.945413i	$-0.605659\pi$
$113$	−6.92820	−0.651751	−0.325875	+	0.945413i	$0.605659\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	−6.46410	−0.600177
$117$	−2.26795	−0.209672
$118$	8.32051	0.765965
$119$	7.19615	0.659670
$120$	0	0
$121$	5.00000	0.454545
$122$	−13.7321	−1.24324
$123$	0.464102	0.0418466
$124$	3.46410	0.311086
$125$	0	0
$126$	−1.73205	−0.154303
$127$	20.3923	1.80952	0.904762	−	0.425917i	$-0.140048\pi$
$127$	20.3923	1.80952	0.904762	+	0.425917i	$0.140048\pi$
$128$	−12.1244	−1.07165
$129$	11.4641	1.00936
$130$	0	0
$131$	−3.46410	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$132$	−4.00000	−0.348155
$133$	−1.73205	−0.150188
$134$	8.53590	0.737389
$135$	0	0
$136$	12.4641	1.06879
$137$	12.3923	1.05875	0.529373	−	0.848389i	$-0.322427\pi$
$137$	12.3923	1.05875	0.529373	+	0.848389i	$0.322427\pi$
$138$	4.26795	0.363312
$139$	−11.5885	−0.982920	−0.491460	−	0.870900i	$-0.663537\pi$
$139$	−11.5885	−0.982920	−0.491460	+	0.870900i	$0.663537\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	12.4641	1.04596
$143$	9.07180	0.758622
$144$	−5.00000	−0.416667
$145$	0	0
$146$	6.92820	0.573382
$147$	−6.00000	−0.494872
$148$	−4.53590	−0.372849
$149$	−17.3205	−1.41895	−0.709476	−	0.704730i	$-0.751068\pi$
$149$	−17.3205	−1.41895	−0.709476	+	0.704730i	$0.751068\pi$
$150$	0	0
$151$	−2.26795	−0.184563	−0.0922815	−	0.995733i	$-0.529416\pi$
$151$	−2.26795	−0.184563	−0.0922815	+	0.995733i	$0.529416\pi$
$152$	−3.00000	−0.243332
$153$	7.19615	0.581774
$154$	6.92820	0.558291
$155$	0	0
$156$	−2.26795	−0.181581
$157$	−16.7846	−1.33956	−0.669779	−	0.742561i	$-0.733611\pi$
$157$	−16.7846	−1.33956	−0.669779	+	0.742561i	$0.733611\pi$
$158$	2.53590	0.201745
$159$	−14.1244	−1.12013
$160$	0	0
$161$	−2.46410	−0.194198
$162$	−1.73205	−0.136083
$163$	−9.85641	−0.772013	−0.386007	−	0.922496i	$-0.626146\pi$
$163$	−9.85641	−0.772013	−0.386007	+	0.922496i	$0.626146\pi$
$164$	0.464102	0.0362402
$165$	0	0
$166$	18.9282	1.46911
$167$	−17.5359	−1.35697	−0.678484	−	0.734615i	$-0.737363\pi$
$167$	−17.5359	−1.35697	−0.678484	+	0.734615i	$0.737363\pi$
$168$	1.73205	0.133631
$169$	−7.85641	−0.604339
$170$	0	0
$171$	−1.73205	−0.132453
$172$	11.4641	0.874130
$173$	−1.33975	−0.101859	−0.0509295	−	0.998702i	$-0.516218\pi$
$173$	−1.33975	−0.101859	−0.0509295	+	0.998702i	$0.516218\pi$
$174$	11.1962	0.848778
$175$	0	0
$176$	20.0000	1.50756
$177$	−4.80385	−0.361079
$178$	23.3205	1.74795
$179$	−13.8564	−1.03568	−0.517838	−	0.855479i	$-0.673263\pi$
$179$	−13.8564	−1.03568	−0.517838	+	0.855479i	$0.673263\pi$
$180$	0	0
$181$	11.8564	0.881280	0.440640	−	0.897684i	$-0.354752\pi$
$181$	11.8564	0.881280	0.440640	+	0.897684i	$0.354752\pi$
$182$	3.92820	0.291178
$183$	7.92820	0.586070
$184$	−4.26795	−0.314637
$185$	0	0
$186$	−6.00000	−0.439941
$187$	−28.7846	−2.10494
$188$	1.00000	0.0729325
$189$	1.00000	0.0727393
$190$	0	0
$191$	−22.3923	−1.62025	−0.810125	−	0.586257i	$-0.800601\pi$
$191$	−22.3923	−1.62025	−0.810125	+	0.586257i	$0.800601\pi$
$192$	1.00000	0.0721688
$193$	−1.07180	−0.0771496	−0.0385748	−	0.999256i	$-0.512282\pi$
$193$	−1.07180	−0.0771496	−0.0385748	+	0.999256i	$0.512282\pi$
$194$	2.53590	0.182067
$195$	0	0
$196$	−6.00000	−0.428571
$197$	25.0526	1.78492	0.892460	−	0.451126i	$-0.148977\pi$
$197$	25.0526	1.78492	0.892460	+	0.451126i	$0.148977\pi$
$198$	6.92820	0.492366
$199$	13.7321	0.973439	0.486720	−	0.873558i	$-0.338193\pi$
$199$	13.7321	0.973439	0.486720	+	0.873558i	$0.338193\pi$
$200$	0	0
$201$	−4.92820	−0.347609
$202$	−18.0000	−1.26648
$203$	−6.46410	−0.453691
$204$	7.19615	0.503831
$205$	0	0
$206$	−15.5885	−1.08610
$207$	−2.46410	−0.171267
$208$	11.3397	0.786270
$209$	6.92820	0.479234
$210$	0	0
$211$	3.46410	0.238479	0.119239	−	0.992866i	$-0.461954\pi$
$211$	3.46410	0.238479	0.119239	+	0.992866i	$0.461954\pi$
$212$	−14.1244	−0.970065
$213$	−7.19615	−0.493072
$214$	−12.0000	−0.820303
$215$	0	0
$216$	1.73205	0.117851
$217$	3.46410	0.235159
$218$	−18.0000	−1.21911
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−16.3205	−1.09784
$222$	7.85641	0.527287
$223$	−3.85641	−0.258244	−0.129122	−	0.991629i	$-0.541216\pi$
$223$	−3.85641	−0.258244	−0.129122	+	0.991629i	$0.541216\pi$
$224$	5.19615	0.347183
$225$	0	0
$226$	12.0000	0.798228
$227$	18.3205	1.21597	0.607987	−	0.793947i	$-0.291977\pi$
$227$	18.3205	1.21597	0.607987	+	0.793947i	$0.291977\pi$
$228$	−1.73205	−0.114708
$229$	1.46410	0.0967506	0.0483753	−	0.998829i	$-0.484596\pi$
$229$	1.46410	0.0967506	0.0483753	+	0.998829i	$0.484596\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	−11.1962	−0.735063
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	3.92820	0.256795
$235$	0	0
$236$	−4.80385	−0.312704
$237$	−1.46410	−0.0951036
$238$	−12.4641	−0.807928
$239$	−27.4641	−1.77651	−0.888253	−	0.459355i	$-0.848080\pi$
$239$	−27.4641	−1.77651	−0.888253	+	0.459355i	$0.848080\pi$
$240$	0	0
$241$	−25.9282	−1.67018	−0.835091	−	0.550112i	$-0.814585\pi$
$241$	−25.9282	−1.67018	−0.835091	+	0.550112i	$0.814585\pi$
$242$	−8.66025	−0.556702
$243$	1.00000	0.0641500
$244$	7.92820	0.507551
$245$	0	0
$246$	−0.803848	−0.0512514
$247$	3.92820	0.249946
$248$	6.00000	0.381000
$249$	−10.9282	−0.692547
$250$	0	0
$251$	−26.1244	−1.64895	−0.824477	−	0.565895i	$-0.808531\pi$
$251$	−26.1244	−1.64895	−0.824477	+	0.565895i	$0.808531\pi$
$252$	1.00000	0.0629941
$253$	9.85641	0.619667
$254$	−35.3205	−2.21621
$255$	0	0
$256$	19.0000	1.18750
$257$	−3.85641	−0.240556	−0.120278	−	0.992740i	$-0.538379\pi$
$257$	−3.85641	−0.240556	−0.120278	+	0.992740i	$0.538379\pi$
$258$	−19.8564	−1.23621
$259$	−4.53590	−0.281847
$260$	0	0
$261$	−6.46410	−0.400118
$262$	6.00000	0.370681
$263$	−11.8564	−0.731097	−0.365549	−	0.930792i	$-0.619119\pi$
$263$	−11.8564	−0.731097	−0.365549	+	0.930792i	$0.619119\pi$
$264$	−6.92820	−0.426401
$265$	0	0
$266$	3.00000	0.183942
$267$	−13.4641	−0.823990
$268$	−4.92820	−0.301038
$269$	14.7846	0.901434	0.450717	−	0.892667i	$-0.351168\pi$
$269$	14.7846	0.901434	0.450717	+	0.892667i	$0.351168\pi$
$270$	0	0
$271$	0.392305	0.0238308	0.0119154	−	0.999929i	$-0.496207\pi$
$271$	0.392305	0.0238308	0.0119154	+	0.999929i	$0.496207\pi$
$272$	−35.9808	−2.18165
$273$	−2.26795	−0.137263
$274$	−21.4641	−1.29669
$275$	0	0
$276$	−2.46410	−0.148321
$277$	18.3923	1.10509	0.552543	−	0.833484i	$-0.313657\pi$
$277$	18.3923	1.10509	0.552543	+	0.833484i	$0.313657\pi$
$278$	20.0718	1.20383
$279$	3.46410	0.207390
$280$	0	0
$281$	5.39230	0.321678	0.160839	−	0.986981i	$-0.448580\pi$
$281$	5.39230	0.321678	0.160839	+	0.986981i	$0.448580\pi$
$282$	−1.73205	−0.103142
$283$	5.85641	0.348127	0.174064	−	0.984734i	$-0.444310\pi$
$283$	5.85641	0.348127	0.174064	+	0.984734i	$0.444310\pi$
$284$	−7.19615	−0.427013
$285$	0	0
$286$	−15.7128	−0.929118
$287$	0.464102	0.0273951
$288$	5.19615	0.306186
$289$	34.7846	2.04615
$290$	0	0
$291$	−1.46410	−0.0858272
$292$	−4.00000	−0.234082
$293$	−15.4641	−0.903422	−0.451711	−	0.892164i	$-0.649186\pi$
$293$	−15.4641	−0.903422	−0.451711	+	0.892164i	$0.649186\pi$
$294$	10.3923	0.606092
$295$	0	0
$296$	−7.85641	−0.456644
$297$	−4.00000	−0.232104
$298$	30.0000	1.73785
$299$	5.58846	0.323189
$300$	0	0
$301$	11.4641	0.660780
$302$	3.92820	0.226043
$303$	10.3923	0.597022
$304$	8.66025	0.496700
$305$	0	0
$306$	−12.4641	−0.712525
$307$	5.00000	0.285365	0.142683	−	0.989769i	$-0.454427\pi$
$307$	5.00000	0.285365	0.142683	+	0.989769i	$0.454427\pi$
$308$	−4.00000	−0.227921
$309$	9.00000	0.511992
$310$	0	0
$311$	6.39230	0.362474	0.181237	−	0.983439i	$-0.441990\pi$
$311$	6.39230	0.362474	0.181237	+	0.983439i	$0.441990\pi$
$312$	−3.92820	−0.222391
$313$	−20.1244	−1.13750	−0.568748	−	0.822512i	$-0.692572\pi$
$313$	−20.1244	−1.13750	−0.568748	+	0.822512i	$0.692572\pi$
$314$	29.0718	1.64062
$315$	0	0
$316$	−1.46410	−0.0823622
$317$	2.53590	0.142430	0.0712151	−	0.997461i	$-0.477312\pi$
$317$	2.53590	0.142430	0.0712151	+	0.997461i	$0.477312\pi$
$318$	24.4641	1.37188
$319$	25.8564	1.44768
$320$	0	0
$321$	6.92820	0.386695
$322$	4.26795	0.237844
$323$	−12.4641	−0.693521
$324$	1.00000	0.0555556
$325$	0	0
$326$	17.0718	0.945519
$327$	10.3923	0.574696
$328$	0.803848	0.0443851
$329$	1.00000	0.0551318
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−10.9282	−0.599763
$333$	−4.53590	−0.248566
$334$	30.3731	1.66194
$335$	0	0
$336$	−5.00000	−0.272772
$337$	−27.1769	−1.48042	−0.740210	−	0.672375i	$-0.765274\pi$
$337$	−27.1769	−1.48042	−0.740210	+	0.672375i	$0.765274\pi$
$338$	13.6077	0.740161
$339$	−6.92820	−0.376288
$340$	0	0
$341$	−13.8564	−0.750366
$342$	3.00000	0.162221
$343$	−13.0000	−0.701934
$344$	19.8564	1.07059
$345$	0	0
$346$	2.32051	0.124751
$347$	−28.3923	−1.52418	−0.762089	−	0.647472i	$-0.775826\pi$
$347$	−28.3923	−1.52418	−0.762089	+	0.647472i	$0.775826\pi$
$348$	−6.46410	−0.346512
$349$	−22.7846	−1.21963	−0.609816	−	0.792543i	$-0.708757\pi$
$349$	−22.7846	−1.21963	−0.609816	+	0.792543i	$0.708757\pi$
$350$	0	0
$351$	−2.26795	−0.121054
$352$	−20.7846	−1.10782
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	8.32051	0.442230
$355$	0	0
$356$	−13.4641	−0.713596
$357$	7.19615	0.380861
$358$	24.0000	1.26844
$359$	−34.2487	−1.80758	−0.903789	−	0.427978i	$-0.859226\pi$
$359$	−34.2487	−1.80758	−0.903789	+	0.427978i	$0.859226\pi$
$360$	0	0
$361$	−16.0000	−0.842105
$362$	−20.5359	−1.07934
$363$	5.00000	0.262432
$364$	−2.26795	−0.118873
$365$	0	0
$366$	−13.7321	−0.717786
$367$	−21.7128	−1.13340	−0.566700	−	0.823924i	$-0.691780\pi$
$367$	−21.7128	−1.13340	−0.566700	+	0.823924i	$0.691780\pi$
$368$	12.3205	0.642251
$369$	0.464102	0.0241602
$370$	0	0
$371$	−14.1244	−0.733300
$372$	3.46410	0.179605
$373$	−13.0718	−0.676832	−0.338416	−	0.940997i	$-0.609891\pi$
$373$	−13.0718	−0.676832	−0.338416	+	0.940997i	$0.609891\pi$
$374$	49.8564	2.57801
$375$	0	0
$376$	1.73205	0.0893237
$377$	14.6603	0.755041
$378$	−1.73205	−0.0890871
$379$	−25.8564	−1.32815	−0.664077	−	0.747664i	$-0.731175\pi$
$379$	−25.8564	−1.32815	−0.664077	+	0.747664i	$0.731175\pi$
$380$	0	0
$381$	20.3923	1.04473
$382$	38.7846	1.98439
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	−12.1244	−0.618718
$385$	0	0
$386$	1.85641	0.0944886
$387$	11.4641	0.582753
$388$	−1.46410	−0.0743285
$389$	13.5359	0.686297	0.343149	−	0.939281i	$-0.388507\pi$
$389$	13.5359	0.686297	0.343149	+	0.939281i	$0.388507\pi$
$390$	0	0
$391$	−17.7321	−0.896748
$392$	−10.3923	−0.524891
$393$	−3.46410	−0.174741
$394$	−43.3923	−2.18607
$395$	0	0
$396$	−4.00000	−0.201008
$397$	33.1769	1.66510	0.832551	−	0.553949i	$-0.186880\pi$
$397$	33.1769	1.66510	0.832551	+	0.553949i	$0.186880\pi$
$398$	−23.7846	−1.19221
$399$	−1.73205	−0.0867110
$400$	0	0
$401$	−0.535898	−0.0267615	−0.0133807	−	0.999910i	$-0.504259\pi$
$401$	−0.535898	−0.0267615	−0.0133807	+	0.999910i	$0.504259\pi$
$402$	8.53590	0.425732
$403$	−7.85641	−0.391355
$404$	10.3923	0.517036
$405$	0	0
$406$	11.1962	0.555656
$407$	18.1436	0.899345
$408$	12.4641	0.617065
$409$	14.5359	0.718754	0.359377	−	0.933192i	$-0.382989\pi$
$409$	14.5359	0.718754	0.359377	+	0.933192i	$0.382989\pi$
$410$	0	0
$411$	12.3923	0.611267
$412$	9.00000	0.443398
$413$	−4.80385	−0.236382
$414$	4.26795	0.209758
$415$	0	0
$416$	−11.7846	−0.577788
$417$	−11.5885	−0.567489
$418$	−12.0000	−0.586939
$419$	3.60770	0.176247	0.0881237	−	0.996110i	$-0.471913\pi$
$419$	3.60770	0.176247	0.0881237	+	0.996110i	$0.471913\pi$
$420$	0	0
$421$	−5.85641	−0.285424	−0.142712	−	0.989764i	$-0.545582\pi$
$421$	−5.85641	−0.285424	−0.142712	+	0.989764i	$0.545582\pi$
$422$	−6.00000	−0.292075
$423$	1.00000	0.0486217
$424$	−24.4641	−1.18808
$425$	0	0
$426$	12.4641	0.603888
$427$	7.92820	0.383673
$428$	6.92820	0.334887
$429$	9.07180	0.437990
$430$	0	0
$431$	−23.1962	−1.11732	−0.558660	−	0.829397i	$-0.688684\pi$
$431$	−23.1962	−1.11732	−0.558660	+	0.829397i	$0.688684\pi$
$432$	−5.00000	−0.240563
$433$	−1.73205	−0.0832370	−0.0416185	−	0.999134i	$-0.513251\pi$
$433$	−1.73205	−0.0832370	−0.0416185	+	0.999134i	$0.513251\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	10.3923	0.497701
$437$	4.26795	0.204164
$438$	6.92820	0.331042
$439$	0.392305	0.0187237	0.00936184	−	0.999956i	$-0.497020\pi$
$439$	0.392305	0.0187237	0.00936184	+	0.999956i	$0.497020\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	28.2679	1.34457
$443$	−30.4641	−1.44739	−0.723697	−	0.690118i	$-0.757558\pi$
$443$	−30.4641	−1.44739	−0.723697	+	0.690118i	$0.757558\pi$
$444$	−4.53590	−0.215264
$445$	0	0
$446$	6.67949	0.316283
$447$	−17.3205	−0.819232
$448$	1.00000	0.0472456
$449$	11.3923	0.537636	0.268818	−	0.963191i	$-0.413367\pi$
$449$	11.3923	0.537636	0.268818	+	0.963191i	$0.413367\pi$
$450$	0	0
$451$	−1.85641	−0.0874148
$452$	−6.92820	−0.325875
$453$	−2.26795	−0.106558
$454$	−31.7321	−1.48926
$455$	0	0
$456$	−3.00000	−0.140488
$457$	−17.0718	−0.798585	−0.399292	−	0.916824i	$-0.630744\pi$
$457$	−17.0718	−0.798585	−0.399292	+	0.916824i	$0.630744\pi$
$458$	−2.53590	−0.118495
$459$	7.19615	0.335888
$460$	0	0
$461$	24.9282	1.16102	0.580511	−	0.814252i	$-0.302853\pi$
$461$	24.9282	1.16102	0.580511	+	0.814252i	$0.302853\pi$
$462$	6.92820	0.322329
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	32.3205	1.50044
$465$	0	0
$466$	−31.1769	−1.44424
$467$	−36.4641	−1.68736	−0.843679	−	0.536848i	$-0.819615\pi$
$467$	−36.4641	−1.68736	−0.843679	+	0.536848i	$0.819615\pi$
$468$	−2.26795	−0.104836
$469$	−4.92820	−0.227563
$470$	0	0
$471$	−16.7846	−0.773394
$472$	−8.32051	−0.382982
$473$	−45.8564	−2.10848
$474$	2.53590	0.116478
$475$	0	0
$476$	7.19615	0.329835
$477$	−14.1244	−0.646710
$478$	47.5692	2.17577
$479$	26.6603	1.21814	0.609069	−	0.793117i	$-0.291543\pi$
$479$	26.6603	1.21814	0.609069	+	0.793117i	$0.291543\pi$
$480$	0	0
$481$	10.2872	0.469055
$482$	44.9090	2.04555
$483$	−2.46410	−0.112121
$484$	5.00000	0.227273
$485$	0	0
$486$	−1.73205	−0.0785674
$487$	39.9282	1.80932	0.904660	−	0.426135i	$-0.140125\pi$
$487$	39.9282	1.80932	0.904660	+	0.426135i	$0.140125\pi$
$488$	13.7321	0.621621
$489$	−9.85641	−0.445722
$490$	0	0
$491$	−43.1962	−1.94942	−0.974708	−	0.223484i	$-0.928257\pi$
$491$	−43.1962	−1.94942	−0.974708	+	0.223484i	$0.928257\pi$
$492$	0.464102	0.0209233
$493$	−46.5167	−2.09501
$494$	−6.80385	−0.306120
$495$	0	0
$496$	−17.3205	−0.777714
$497$	−7.19615	−0.322792
$498$	18.9282	0.848193
$499$	−29.1962	−1.30700	−0.653500	−	0.756927i	$-0.726700\pi$
$499$	−29.1962	−1.30700	−0.653500	+	0.756927i	$0.726700\pi$
$500$	0	0
$501$	−17.5359	−0.783446
$502$	45.2487	2.01955
$503$	17.3923	0.775485	0.387742	−	0.921768i	$-0.373255\pi$
$503$	17.3923	0.775485	0.387742	+	0.921768i	$0.373255\pi$
$504$	1.73205	0.0771517
$505$	0	0
$506$	−17.0718	−0.758934
$507$	−7.85641	−0.348915
$508$	20.3923	0.904762
$509$	−7.39230	−0.327658	−0.163829	−	0.986489i	$-0.552385\pi$
$509$	−7.39230	−0.327658	−0.163829	+	0.986489i	$0.552385\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	−8.66025	−0.382733
$513$	−1.73205	−0.0764719
$514$	6.67949	0.294620
$515$	0	0
$516$	11.4641	0.504679
$517$	−4.00000	−0.175920
$518$	7.85641	0.345191
$519$	−1.33975	−0.0588083
$520$	0	0
$521$	13.0718	0.572686	0.286343	−	0.958127i	$-0.407560\pi$
$521$	13.0718	0.572686	0.286343	+	0.958127i	$0.407560\pi$
$522$	11.1962	0.490042
$523$	33.0000	1.44299	0.721495	−	0.692420i	$-0.243455\pi$
$523$	33.0000	1.44299	0.721495	+	0.692420i	$0.243455\pi$
$524$	−3.46410	−0.151330
$525$	0	0
$526$	20.5359	0.895408
$527$	24.9282	1.08589
$528$	20.0000	0.870388
$529$	−16.9282	−0.736009
$530$	0	0
$531$	−4.80385	−0.208469
$532$	−1.73205	−0.0750939
$533$	−1.05256	−0.0455914
$534$	23.3205	1.00918
$535$	0	0
$536$	−8.53590	−0.368695
$537$	−13.8564	−0.597948
$538$	−25.6077	−1.10403
$539$	24.0000	1.03375
$540$	0	0
$541$	6.85641	0.294780	0.147390	−	0.989078i	$-0.452913\pi$
$541$	6.85641	0.294780	0.147390	+	0.989078i	$0.452913\pi$
$542$	−0.679492	−0.0291867
$543$	11.8564	0.508807
$544$	37.3923	1.60318
$545$	0	0
$546$	3.92820	0.168112
$547$	−5.32051	−0.227488	−0.113744	−	0.993510i	$-0.536284\pi$
$547$	−5.32051	−0.227488	−0.113744	+	0.993510i	$0.536284\pi$
$548$	12.3923	0.529373
$549$	7.92820	0.338367
$550$	0	0
$551$	11.1962	0.476972
$552$	−4.26795	−0.181656
$553$	−1.46410	−0.0622599
$554$	−31.8564	−1.35345
$555$	0	0
$556$	−11.5885	−0.491460
$557$	−18.2487	−0.773223	−0.386611	−	0.922243i	$-0.626355\pi$
$557$	−18.2487	−0.773223	−0.386611	+	0.922243i	$0.626355\pi$
$558$	−6.00000	−0.254000
$559$	−26.0000	−1.09968
$560$	0	0
$561$	−28.7846	−1.21529
$562$	−9.33975	−0.393973
$563$	−25.8564	−1.08972	−0.544859	−	0.838528i	$-0.683417\pi$
$563$	−25.8564	−1.08972	−0.544859	+	0.838528i	$0.683417\pi$
$564$	1.00000	0.0421076
$565$	0	0
$566$	−10.1436	−0.426367
$567$	1.00000	0.0419961
$568$	−12.4641	−0.522982
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	18.3923	0.769694	0.384847	−	0.922980i	$-0.374254\pi$
$571$	18.3923	0.769694	0.384847	+	0.922980i	$0.374254\pi$
$572$	9.07180	0.379311
$573$	−22.3923	−0.935452
$574$	−0.803848	−0.0335519
$575$	0	0
$576$	1.00000	0.0416667
$577$	7.58846	0.315912	0.157956	−	0.987446i	$-0.449510\pi$
$577$	7.58846	0.315912	0.157956	+	0.987446i	$0.449510\pi$
$578$	−60.2487	−2.50602
$579$	−1.07180	−0.0445424
$580$	0	0
$581$	−10.9282	−0.453378
$582$	2.53590	0.105116
$583$	56.4974	2.33988
$584$	−6.92820	−0.286691
$585$	0	0
$586$	26.7846	1.10646
$587$	−22.9282	−0.946348	−0.473174	−	0.880969i	$-0.656892\pi$
$587$	−22.9282	−0.946348	−0.473174	+	0.880969i	$0.656892\pi$
$588$	−6.00000	−0.247436
$589$	−6.00000	−0.247226
$590$	0	0
$591$	25.0526	1.03052
$592$	22.6795	0.932121
$593$	26.7846	1.09991	0.549956	−	0.835194i	$-0.314644\pi$
$593$	26.7846	1.09991	0.549956	+	0.835194i	$0.314644\pi$
$594$	6.92820	0.284268
$595$	0	0
$596$	−17.3205	−0.709476
$597$	13.7321	0.562015
$598$	−9.67949	−0.395824
$599$	7.46410	0.304975	0.152487	−	0.988305i	$-0.451272\pi$
$599$	7.46410	0.304975	0.152487	+	0.988305i	$0.451272\pi$
$600$	0	0
$601$	9.14359	0.372975	0.186487	−	0.982457i	$-0.440290\pi$
$601$	9.14359	0.372975	0.186487	+	0.982457i	$0.440290\pi$
$602$	−19.8564	−0.809287
$603$	−4.92820	−0.200692
$604$	−2.26795	−0.0922815
$605$	0	0
$606$	−18.0000	−0.731200
$607$	18.3923	0.746521	0.373260	−	0.927727i	$-0.378240\pi$
$607$	18.3923	0.746521	0.373260	+	0.927727i	$0.378240\pi$
$608$	−9.00000	−0.364998
$609$	−6.46410	−0.261939
$610$	0	0
$611$	−2.26795	−0.0917514
$612$	7.19615	0.290887
$613$	−33.3205	−1.34580	−0.672901	−	0.739732i	$-0.734952\pi$
$613$	−33.3205	−1.34580	−0.672901	+	0.739732i	$0.734952\pi$
$614$	−8.66025	−0.349499
$615$	0	0
$616$	−6.92820	−0.279145
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	−15.5885	−0.627060
$619$	−17.7128	−0.711938	−0.355969	−	0.934498i	$-0.615849\pi$
$619$	−17.7128	−0.711938	−0.355969	+	0.934498i	$0.615849\pi$
$620$	0	0
$621$	−2.46410	−0.0988810
$622$	−11.0718	−0.443939
$623$	−13.4641	−0.539428
$624$	11.3397	0.453953
$625$	0	0
$626$	34.8564	1.39314
$627$	6.92820	0.276686
$628$	−16.7846	−0.669779
$629$	−32.6410	−1.30148
$630$	0	0
$631$	35.4641	1.41180	0.705902	−	0.708310i	$-0.250542\pi$
$631$	35.4641	1.41180	0.705902	+	0.708310i	$0.250542\pi$
$632$	−2.53590	−0.100873
$633$	3.46410	0.137686
$634$	−4.39230	−0.174441
$635$	0	0
$636$	−14.1244	−0.560067
$637$	13.6077	0.539157
$638$	−44.7846	−1.77304
$639$	−7.19615	−0.284675
$640$	0	0
$641$	28.6410	1.13125	0.565626	−	0.824662i	$-0.308635\pi$
$641$	28.6410	1.13125	0.565626	+	0.824662i	$0.308635\pi$
$642$	−12.0000	−0.473602
$643$	−5.14359	−0.202844	−0.101422	−	0.994844i	$-0.532339\pi$
$643$	−5.14359	−0.202844	−0.101422	+	0.994844i	$0.532339\pi$
$644$	−2.46410	−0.0970992
$645$	0	0
$646$	21.5885	0.849386
$647$	34.2487	1.34646	0.673228	−	0.739435i	$-0.264907\pi$
$647$	34.2487	1.34646	0.673228	+	0.739435i	$0.264907\pi$
$648$	1.73205	0.0680414
$649$	19.2154	0.754270
$650$	0	0
$651$	3.46410	0.135769
$652$	−9.85641	−0.386007
$653$	−20.8038	−0.814117	−0.407059	−	0.913402i	$-0.633446\pi$
$653$	−20.8038	−0.814117	−0.407059	+	0.913402i	$0.633446\pi$
$654$	−18.0000	−0.703856
$655$	0	0
$656$	−2.32051	−0.0906006
$657$	−4.00000	−0.156055
$658$	−1.73205	−0.0675224
$659$	7.46410	0.290760	0.145380	−	0.989376i	$-0.453560\pi$
$659$	7.46410	0.290760	0.145380	+	0.989376i	$0.453560\pi$
$660$	0	0
$661$	41.0000	1.59472	0.797358	−	0.603507i	$-0.206231\pi$
$661$	41.0000	1.59472	0.797358	+	0.603507i	$0.206231\pi$
$662$	6.92820	0.269272
$663$	−16.3205	−0.633836
$664$	−18.9282	−0.734557
$665$	0	0
$666$	7.85641	0.304430
$667$	15.9282	0.616742
$668$	−17.5359	−0.678484
$669$	−3.85641	−0.149097
$670$	0	0
$671$	−31.7128	−1.22426
$672$	5.19615	0.200446
$673$	21.7321	0.837709	0.418854	−	0.908053i	$-0.362432\pi$
$673$	21.7321	0.837709	0.418854	+	0.908053i	$0.362432\pi$
$674$	47.0718	1.81314
$675$	0	0
$676$	−7.85641	−0.302169
$677$	−38.1051	−1.46450	−0.732249	−	0.681037i	$-0.761529\pi$
$677$	−38.1051	−1.46450	−0.732249	+	0.681037i	$0.761529\pi$
$678$	12.0000	0.460857
$679$	−1.46410	−0.0561871
$680$	0	0
$681$	18.3205	0.702043
$682$	24.0000	0.919007
$683$	−35.4641	−1.35700	−0.678498	−	0.734602i	$-0.737369\pi$
$683$	−35.4641	−1.35700	−0.678498	+	0.734602i	$0.737369\pi$
$684$	−1.73205	−0.0662266
$685$	0	0
$686$	22.5167	0.859690
$687$	1.46410	0.0558590
$688$	−57.3205	−2.18532
$689$	32.0333	1.22037
$690$	0	0
$691$	48.3731	1.84020	0.920099	−	0.391686i	$-0.128108\pi$
$691$	48.3731	1.84020	0.920099	+	0.391686i	$0.128108\pi$
$692$	−1.33975	−0.0509295
$693$	−4.00000	−0.151947
$694$	49.1769	1.86673
$695$	0	0
$696$	−11.1962	−0.424389
$697$	3.33975	0.126502
$698$	39.4641	1.49374
$699$	18.0000	0.680823
$700$	0	0
$701$	−30.3205	−1.14519	−0.572595	−	0.819838i	$-0.694063\pi$
$701$	−30.3205	−1.14519	−0.572595	+	0.819838i	$0.694063\pi$
$702$	3.92820	0.148260
$703$	7.85641	0.296310
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−6.92820	−0.260746
$707$	10.3923	0.390843
$708$	−4.80385	−0.180540
$709$	18.7846	0.705471	0.352735	−	0.935723i	$-0.385252\pi$
$709$	18.7846	0.705471	0.352735	+	0.935723i	$0.385252\pi$
$710$	0	0
$711$	−1.46410	−0.0549081
$712$	−23.3205	−0.873973
$713$	−8.53590	−0.319672
$714$	−12.4641	−0.466457
$715$	0	0
$716$	−13.8564	−0.517838
$717$	−27.4641	−1.02567
$718$	59.3205	2.21382
$719$	35.4449	1.32187	0.660935	−	0.750443i	$-0.270160\pi$
$719$	35.4449	1.32187	0.660935	+	0.750443i	$0.270160\pi$
$720$	0	0
$721$	9.00000	0.335178
$722$	27.7128	1.03136
$723$	−25.9282	−0.964280
$724$	11.8564	0.440640
$725$	0	0
$726$	−8.66025	−0.321412
$727$	21.3205	0.790734	0.395367	−	0.918523i	$-0.370617\pi$
$727$	21.3205	0.790734	0.395367	+	0.918523i	$0.370617\pi$
$728$	−3.92820	−0.145589
$729$	1.00000	0.0370370
$730$	0	0
$731$	82.4974	3.05128
$732$	7.92820	0.293035
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	37.6077	1.38813
$735$	0	0
$736$	−12.8038	−0.471956
$737$	19.7128	0.726131
$738$	−0.803848	−0.0295900
$739$	14.3923	0.529429	0.264715	−	0.964327i	$-0.414722\pi$
$739$	14.3923	0.529429	0.264715	+	0.964327i	$0.414722\pi$
$740$	0	0
$741$	3.92820	0.144306
$742$	24.4641	0.898105
$743$	48.4641	1.77798	0.888988	−	0.457931i	$-0.151409\pi$
$743$	48.4641	1.77798	0.888988	+	0.457931i	$0.151409\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	22.6410	0.828946
$747$	−10.9282	−0.399842
$748$	−28.7846	−1.05247
$749$	6.92820	0.253151
$750$	0	0
$751$	−3.05256	−0.111389	−0.0556947	−	0.998448i	$-0.517737\pi$
$751$	−3.05256	−0.111389	−0.0556947	+	0.998448i	$0.517737\pi$
$752$	−5.00000	−0.182331
$753$	−26.1244	−0.952024
$754$	−25.3923	−0.924733
$755$	0	0
$756$	1.00000	0.0363696
$757$	−38.6410	−1.40443	−0.702216	−	0.711964i	$-0.747806\pi$
$757$	−38.6410	−1.40443	−0.702216	+	0.711964i	$0.747806\pi$
$758$	44.7846	1.62665
$759$	9.85641	0.357765
$760$	0	0
$761$	15.1769	0.550163	0.275081	−	0.961421i	$-0.411295\pi$
$761$	15.1769	0.550163	0.275081	+	0.961421i	$0.411295\pi$
$762$	−35.3205	−1.27953
$763$	10.3923	0.376227
$764$	−22.3923	−0.810125
$765$	0	0
$766$	0	0
$767$	10.8949	0.393391
$768$	19.0000	0.685603
$769$	−6.78461	−0.244659	−0.122330	−	0.992490i	$-0.539037\pi$
$769$	−6.78461	−0.244659	−0.122330	+	0.992490i	$0.539037\pi$
$770$	0	0
$771$	−3.85641	−0.138885
$772$	−1.07180	−0.0385748
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	−19.8564	−0.713724
$775$	0	0
$776$	−2.53590	−0.0910334
$777$	−4.53590	−0.162724
$778$	−23.4449	−0.840539
$779$	−0.803848	−0.0288008
$780$	0	0
$781$	28.7846	1.02999
$782$	30.7128	1.09829
$783$	−6.46410	−0.231008
$784$	30.0000	1.07143
$785$	0	0
$786$	6.00000	0.214013
$787$	−3.60770	−0.128600	−0.0643002	−	0.997931i	$-0.520482\pi$
$787$	−3.60770	−0.128600	−0.0643002	+	0.997931i	$0.520482\pi$
$788$	25.0526	0.892460
$789$	−11.8564	−0.422099
$790$	0	0
$791$	−6.92820	−0.246339
$792$	−6.92820	−0.246183
$793$	−17.9808	−0.638516
$794$	−57.4641	−2.03932
$795$	0	0
$796$	13.7321	0.486720
$797$	44.6410	1.58127	0.790633	−	0.612290i	$-0.209752\pi$
$797$	44.6410	1.58127	0.790633	+	0.612290i	$0.209752\pi$
$798$	3.00000	0.106199
$799$	7.19615	0.254582
$800$	0	0
$801$	−13.4641	−0.475731
$802$	0.928203	0.0327760
$803$	16.0000	0.564628
$804$	−4.92820	−0.173804
$805$	0	0
$806$	13.6077	0.479311
$807$	14.7846	0.520443
$808$	18.0000	0.633238
$809$	23.0718	0.811161	0.405581	−	0.914059i	$-0.367069\pi$
$809$	23.0718	0.811161	0.405581	+	0.914059i	$0.367069\pi$
$810$	0	0
$811$	0.928203	0.0325936	0.0162968	−	0.999867i	$-0.494812\pi$
$811$	0.928203	0.0325936	0.0162968	+	0.999867i	$0.494812\pi$
$812$	−6.46410	−0.226845
$813$	0.392305	0.0137587
$814$	−31.4256	−1.10147
$815$	0	0
$816$	−35.9808	−1.25958
$817$	−19.8564	−0.694688
$818$	−25.1769	−0.880290
$819$	−2.26795	−0.0792486
$820$	0	0
$821$	2.32051	0.0809863	0.0404931	−	0.999180i	$-0.487107\pi$
$821$	2.32051	0.0809863	0.0404931	+	0.999180i	$0.487107\pi$
$822$	−21.4641	−0.748647
$823$	−5.85641	−0.204141	−0.102071	−	0.994777i	$-0.532547\pi$
$823$	−5.85641	−0.204141	−0.102071	+	0.994777i	$0.532547\pi$
$824$	15.5885	0.543050
$825$	0	0
$826$	8.32051	0.289508
$827$	−8.67949	−0.301816	−0.150908	−	0.988548i	$-0.548220\pi$
$827$	−8.67949	−0.301816	−0.150908	+	0.988548i	$0.548220\pi$
$828$	−2.46410	−0.0856335
$829$	16.2487	0.564341	0.282171	−	0.959364i	$-0.408946\pi$
$829$	16.2487	0.564341	0.282171	+	0.959364i	$0.408946\pi$
$830$	0	0
$831$	18.3923	0.638022
$832$	−2.26795	−0.0786270
$833$	−43.1769	−1.49599
$834$	20.0718	0.695029
$835$	0	0
$836$	6.92820	0.239617
$837$	3.46410	0.119737
$838$	−6.24871	−0.215858
$839$	40.6410	1.40308	0.701542	−	0.712628i	$-0.252495\pi$
$839$	40.6410	1.40308	0.701542	+	0.712628i	$0.252495\pi$
$840$	0	0
$841$	12.7846	0.440849
$842$	10.1436	0.349571
$843$	5.39230	0.185721
$844$	3.46410	0.119239
$845$	0	0
$846$	−1.73205	−0.0595491
$847$	5.00000	0.171802
$848$	70.6218	2.42516
$849$	5.85641	0.200991
$850$	0	0
$851$	11.1769	0.383140
$852$	−7.19615	−0.246536
$853$	−8.67949	−0.297180	−0.148590	−	0.988899i	$-0.547473\pi$
$853$	−8.67949	−0.297180	−0.148590	+	0.988899i	$0.547473\pi$
$854$	−13.7321	−0.469901
$855$	0	0
$856$	12.0000	0.410152
$857$	43.4641	1.48471	0.742353	−	0.670009i	$-0.233710\pi$
$857$	43.4641	1.48471	0.742353	+	0.670009i	$0.233710\pi$
$858$	−15.7128	−0.536427
$859$	−3.58846	−0.122437	−0.0612183	−	0.998124i	$-0.519499\pi$
$859$	−3.58846	−0.122437	−0.0612183	+	0.998124i	$0.519499\pi$
$860$	0	0
$861$	0.464102	0.0158165
$862$	40.1769	1.36843
$863$	42.3923	1.44305	0.721525	−	0.692388i	$-0.243441\pi$
$863$	42.3923	1.44305	0.721525	+	0.692388i	$0.243441\pi$
$864$	5.19615	0.176777
$865$	0	0
$866$	3.00000	0.101944
$867$	34.7846	1.18135
$868$	3.46410	0.117579
$869$	5.85641	0.198665
$870$	0	0
$871$	11.1769	0.378715
$872$	18.0000	0.609557
$873$	−1.46410	−0.0495523
$874$	−7.39230	−0.250048
$875$	0	0
$876$	−4.00000	−0.135147
$877$	37.4449	1.26442	0.632212	−	0.774796i	$-0.282147\pi$
$877$	37.4449	1.26442	0.632212	+	0.774796i	$0.282147\pi$
$878$	−0.679492	−0.0229317
$879$	−15.4641	−0.521591
$880$	0	0
$881$	5.67949	0.191347	0.0956735	−	0.995413i	$-0.469500\pi$
$881$	5.67949	0.191347	0.0956735	+	0.995413i	$0.469500\pi$
$882$	10.3923	0.349927
$883$	−19.7846	−0.665805	−0.332903	−	0.942961i	$-0.608028\pi$
$883$	−19.7846	−0.665805	−0.332903	+	0.942961i	$0.608028\pi$
$884$	−16.3205	−0.548918
$885$	0	0
$886$	52.7654	1.77269
$887$	19.7128	0.661891	0.330946	−	0.943650i	$-0.392632\pi$
$887$	19.7128	0.661891	0.330946	+	0.943650i	$0.392632\pi$
$888$	−7.85641	−0.263644
$889$	20.3923	0.683936
$890$	0	0
$891$	−4.00000	−0.134005
$892$	−3.85641	−0.129122
$893$	−1.73205	−0.0579609
$894$	30.0000	1.00335
$895$	0	0
$896$	−12.1244	−0.405046
$897$	5.58846	0.186593
$898$	−19.7321	−0.658467
$899$	−22.3923	−0.746825
$900$	0	0
$901$	−101.641	−3.38615
$902$	3.21539	0.107061
$903$	11.4641	0.381501
$904$	−12.0000	−0.399114
$905$	0	0
$906$	3.92820	0.130506
$907$	18.7128	0.621349	0.310674	−	0.950516i	$-0.399445\pi$
$907$	18.7128	0.621349	0.310674	+	0.950516i	$0.399445\pi$
$908$	18.3205	0.607987
$909$	10.3923	0.344691
$910$	0	0
$911$	5.60770	0.185791	0.0928956	−	0.995676i	$-0.470388\pi$
$911$	5.60770	0.185791	0.0928956	+	0.995676i	$0.470388\pi$
$912$	8.66025	0.286770
$913$	43.7128	1.44668
$914$	29.5692	0.978063
$915$	0	0
$916$	1.46410	0.0483753
$917$	−3.46410	−0.114395
$918$	−12.4641	−0.411377
$919$	−13.7321	−0.452979	−0.226489	−	0.974014i	$-0.572725\pi$
$919$	−13.7321	−0.452979	−0.226489	+	0.974014i	$0.572725\pi$
$920$	0	0
$921$	5.00000	0.164756
$922$	−43.1769	−1.42196
$923$	16.3205	0.537196
$924$	−4.00000	−0.131590
$925$	0	0
$926$	−10.3923	−0.341512
$927$	9.00000	0.295599
$928$	−33.5885	−1.10260
$929$	−13.6077	−0.446454	−0.223227	−	0.974766i	$-0.571659\pi$
$929$	−13.6077	−0.446454	−0.223227	+	0.974766i	$0.571659\pi$
$930$	0	0
$931$	10.3923	0.340594
$932$	18.0000	0.589610
$933$	6.39230	0.209275
$934$	63.1577	2.06658
$935$	0	0
$936$	−3.92820	−0.128397
$937$	43.7128	1.42804	0.714018	−	0.700128i	$-0.246874\pi$
$937$	43.7128	1.42804	0.714018	+	0.700128i	$0.246874\pi$
$938$	8.53590	0.278707
$939$	−20.1244	−0.656734
$940$	0	0
$941$	27.7128	0.903412	0.451706	−	0.892167i	$-0.350816\pi$
$941$	27.7128	0.903412	0.451706	+	0.892167i	$0.350816\pi$
$942$	29.0718	0.947210
$943$	−1.14359	−0.0372405
$944$	24.0192	0.781760
$945$	0	0
$946$	79.4256	2.58235
$947$	−57.8564	−1.88008	−0.940040	−	0.341063i	$-0.889213\pi$
$947$	−57.8564	−1.88008	−0.940040	+	0.341063i	$0.889213\pi$
$948$	−1.46410	−0.0475518
$949$	9.07180	0.294483
$950$	0	0
$951$	2.53590	0.0822321
$952$	12.4641	0.403964
$953$	53.0333	1.71792	0.858959	−	0.512045i	$-0.171112\pi$
$953$	53.0333	1.71792	0.858959	+	0.512045i	$0.171112\pi$
$954$	24.4641	0.792055
$955$	0	0
$956$	−27.4641	−0.888253
$957$	25.8564	0.835819
$958$	−46.1769	−1.49191
$959$	12.3923	0.400168
$960$	0	0
$961$	−19.0000	−0.612903
$962$	−17.8179	−0.574473
$963$	6.92820	0.223258
$964$	−25.9282	−0.835091
$965$	0	0
$966$	4.26795	0.137319
$967$	−29.5692	−0.950882	−0.475441	−	0.879748i	$-0.657711\pi$
$967$	−29.5692	−0.950882	−0.475441	+	0.879748i	$0.657711\pi$
$968$	8.66025	0.278351
$969$	−12.4641	−0.400405
$970$	0	0
$971$	32.1051	1.03030	0.515151	−	0.857099i	$-0.327736\pi$
$971$	32.1051	1.03030	0.515151	+	0.857099i	$0.327736\pi$
$972$	1.00000	0.0320750
$973$	−11.5885	−0.371509
$974$	−69.1577	−2.21595
$975$	0	0
$976$	−39.6410	−1.26888
$977$	53.0526	1.69730	0.848651	−	0.528953i	$-0.177415\pi$
$977$	53.0526	1.69730	0.848651	+	0.528953i	$0.177415\pi$
$978$	17.0718	0.545896
$979$	53.8564	1.72126
$980$	0	0
$981$	10.3923	0.331801
$982$	74.8179	2.38754
$983$	36.0333	1.14928	0.574642	−	0.818405i	$-0.305141\pi$
$983$	36.0333	1.14928	0.574642	+	0.818405i	$0.305141\pi$
$984$	0.803848	0.0256257
$985$	0	0
$986$	80.5692	2.56585
$987$	1.00000	0.0318304
$988$	3.92820	0.124973
$989$	−28.2487	−0.898257
$990$	0	0
$991$	−48.6410	−1.54513	−0.772566	−	0.634934i	$-0.781027\pi$
$991$	−48.6410	−1.54513	−0.772566	+	0.634934i	$0.781027\pi$
$992$	18.0000	0.571501
$993$	−4.00000	−0.126936
$994$	12.4641	0.395337
$995$	0	0
$996$	−10.9282	−0.346273
$997$	7.58846	0.240329	0.120164	−	0.992754i	$-0.461658\pi$
$997$	7.58846	0.240329	0.120164	+	0.992754i	$0.461658\pi$
$998$	50.5692	1.60074
$999$	−4.53590	−0.143509

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.p.1.1		2
5.4	even	2		705.2.a.i.1.2	✓	2
15.14	odd	2		2115.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.i.1.2	✓	2	5.4	even	2
2115.2.a.l.1.1		2	15.14	odd	2
3525.2.a.p.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} +1.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} +1.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} -2.26795 q^{13} -1.73205 q^{14} -5.00000 q^{16} +7.19615 q^{17} -1.73205 q^{18} -1.73205 q^{19} +1.00000 q^{21} +6.92820 q^{22} -2.46410 q^{23} +1.73205 q^{24} +3.92820 q^{26} +1.00000 q^{27} +1.00000 q^{28} -6.46410 q^{29} +3.46410 q^{31} +5.19615 q^{32} -4.00000 q^{33} -12.4641 q^{34} +1.00000 q^{36} -4.53590 q^{37} +3.00000 q^{38} -2.26795 q^{39} +0.464102 q^{41} -1.73205 q^{42} +11.4641 q^{43} -4.00000 q^{44} +4.26795 q^{46} +1.00000 q^{47} -5.00000 q^{48} -6.00000 q^{49} +7.19615 q^{51} -2.26795 q^{52} -14.1244 q^{53} -1.73205 q^{54} +1.73205 q^{56} -1.73205 q^{57} +11.1962 q^{58} -4.80385 q^{59} +7.92820 q^{61} -6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +6.92820 q^{66} -4.92820 q^{67} +7.19615 q^{68} -2.46410 q^{69} -7.19615 q^{71} +1.73205 q^{72} -4.00000 q^{73} +7.85641 q^{74} -1.73205 q^{76} -4.00000 q^{77} +3.92820 q^{78} -1.46410 q^{79} +1.00000 q^{81} -0.803848 q^{82} -10.9282 q^{83} +1.00000 q^{84} -19.8564 q^{86} -6.46410 q^{87} -6.92820 q^{88} -13.4641 q^{89} -2.26795 q^{91} -2.46410 q^{92} +3.46410 q^{93} -1.73205 q^{94} +5.19615 q^{96} -1.46410 q^{97} +10.3923 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{7} + 2 q^{9} - 8 q^{11} + 2 q^{12} - 8 q^{13} - 10 q^{16} + 4 q^{17} + 2 q^{21} + 2 q^{23} - 6 q^{26} + 2 q^{27} + 2 q^{28} - 6 q^{29} - 8 q^{33} - 18 q^{34} + 2 q^{36} - 16 q^{37} + 6 q^{38} - 8 q^{39} - 6 q^{41} + 16 q^{43} - 8 q^{44} + 12 q^{46} + 2 q^{47} - 10 q^{48} - 12 q^{49} + 4 q^{51} - 8 q^{52} - 4 q^{53} + 12 q^{58} - 20 q^{59} + 2 q^{61} - 12 q^{62} + 2 q^{63} + 2 q^{64} + 4 q^{67} + 4 q^{68} + 2 q^{69} - 4 q^{71} - 8 q^{73} - 12 q^{74} - 8 q^{77} - 6 q^{78} + 4 q^{79} + 2 q^{81} - 12 q^{82} - 8 q^{83} + 2 q^{84} - 12 q^{86} - 6 q^{87} - 20 q^{89} - 8 q^{91} + 2 q^{92} + 4 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^7 + 2 * q^9 - 8 * q^11 + 2 * q^12 - 8 * q^13 - 10 * q^16 + 4 * q^17 + 2 * q^21 + 2 * q^23 - 6 * q^26 + 2 * q^27 + 2 * q^28 - 6 * q^29 - 8 * q^33 - 18 * q^34 + 2 * q^36 - 16 * q^37 + 6 * q^38 - 8 * q^39 - 6 * q^41 + 16 * q^43 - 8 * q^44 + 12 * q^46 + 2 * q^47 - 10 * q^48 - 12 * q^49 + 4 * q^51 - 8 * q^52 - 4 * q^53 + 12 * q^58 - 20 * q^59 + 2 * q^61 - 12 * q^62 + 2 * q^63 + 2 * q^64 + 4 * q^67 + 4 * q^68 + 2 * q^69 - 4 * q^71 - 8 * q^73 - 12 * q^74 - 8 * q^77 - 6 * q^78 + 4 * q^79 + 2 * q^81 - 12 * q^82 - 8 * q^83 + 2 * q^84 - 12 * q^86 - 6 * q^87 - 20 * q^89 - 8 * q^91 + 2 * q^92 + 4 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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