LMFDB - Embedded newform 3525.2.a.w.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:	$0$
Dimension:	$6$
Coefficient field:	6.6.414764096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 29x^{2} - 42x - 11 $ x^6 - 2x^5 - 11x^4 + 20x^3 + 29x^2 - 42*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 705)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.81413$ 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)

$f(q)$	$=$	$q-2.81413 q^{2} -1.00000 q^{3} +5.91931 q^{4} +2.81413 q^{6} +3.18049 q^{7} -11.0294 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.81413 q^{2} -1.00000 q^{3} +5.91931 q^{4} +2.81413 q^{6} +3.18049 q^{7} -11.0294 q^{8} +1.00000 q^{9} -2.91931 q^{11} -5.91931 q^{12} -3.44394 q^{13} -8.95030 q^{14} +19.1996 q^{16} +3.44394 q^{17} -2.81413 q^{18} +0.147944 q^{19} -3.18049 q^{21} +8.21530 q^{22} -3.18049 q^{23} +11.0294 q^{24} +9.69168 q^{26} -1.00000 q^{27} +18.8263 q^{28} -5.95412 q^{29} -8.13944 q^{31} -31.9712 q^{32} +2.91931 q^{33} -9.69168 q^{34} +5.91931 q^{36} +0.919308 q^{37} -0.416334 q^{38} +3.44394 q^{39} -8.22169 q^{41} +8.95030 q^{42} -17.2803 q^{44} +8.95030 q^{46} +1.00000 q^{47} -19.1996 q^{48} +3.11551 q^{49} -3.44394 q^{51} -20.3857 q^{52} +11.6592 q^{53} +2.81413 q^{54} -35.0790 q^{56} -0.147944 q^{57} +16.7557 q^{58} +12.5785 q^{59} +3.88449 q^{61} +22.9054 q^{62} +3.18049 q^{63} +51.5719 q^{64} -8.21530 q^{66} -1.84979 q^{67} +20.3857 q^{68} +3.18049 q^{69} +15.1607 q^{71} -11.0294 q^{72} +10.1394 q^{73} -2.58705 q^{74} +0.875729 q^{76} -9.28483 q^{77} -9.69168 q^{78} +10.3924 q^{79} +1.00000 q^{81} +23.1369 q^{82} +3.70411 q^{83} -18.8263 q^{84} +5.95412 q^{87} +32.1983 q^{88} -13.6884 q^{89} -10.9534 q^{91} -18.8263 q^{92} +8.13944 q^{93} -2.81413 q^{94} +31.9712 q^{96} -4.36552 q^{97} -8.76743 q^{98} -2.91931 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - 6 q^{3} + 14 q^{4} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 6 * q^9	$ 6 q - 2 q^{2} - 6 q^{3} + 14 q^{4} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 6 q^{9} + 4 q^{11} - 14 q^{12} - 6 q^{13} - 2 q^{14} + 26 q^{16} + 6 q^{17} - 2 q^{18} + 10 q^{19} + 4 q^{21} + 4 q^{22} + 4 q^{23} + 6 q^{24} + 8 q^{26} - 6 q^{27} - 4 q^{28} + 8 q^{29} + 8 q^{31} - 14 q^{32} - 4 q^{33} - 8 q^{34} + 14 q^{36} - 16 q^{37} + 12 q^{38} + 6 q^{39} - 12 q^{41} + 2 q^{42} - 36 q^{44} + 2 q^{46} + 6 q^{47} - 26 q^{48} + 18 q^{49} - 6 q^{51} + 10 q^{52} + 10 q^{53} + 2 q^{54} - 26 q^{56} - 10 q^{57} + 26 q^{58} - 6 q^{59} + 24 q^{61} + 44 q^{62} - 4 q^{63} + 54 q^{64} - 4 q^{66} + 8 q^{67} - 10 q^{68} - 4 q^{69} + 26 q^{71} - 6 q^{72} + 4 q^{73} + 30 q^{76} - 8 q^{77} - 8 q^{78} + 24 q^{79} + 6 q^{81} + 82 q^{82} + 4 q^{83} + 4 q^{84} - 8 q^{87} + 16 q^{88} - 20 q^{89} - 10 q^{91} + 4 q^{92} - 8 q^{93} - 2 q^{94} + 14 q^{96} - 24 q^{98} + 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 6 * q^9 + 4 * q^11 - 14 * q^12 - 6 * q^13 - 2 * q^14 + 26 * q^16 + 6 * q^17 - 2 * q^18 + 10 * q^19 + 4 * q^21 + 4 * q^22 + 4 * q^23 + 6 * q^24 + 8 * q^26 - 6 * q^27 - 4 * q^28 + 8 * q^29 + 8 * q^31 - 14 * q^32 - 4 * q^33 - 8 * q^34 + 14 * q^36 - 16 * q^37 + 12 * q^38 + 6 * q^39 - 12 * q^41 + 2 * q^42 - 36 * q^44 + 2 * q^46 + 6 * q^47 - 26 * q^48 + 18 * q^49 - 6 * q^51 + 10 * q^52 + 10 * q^53 + 2 * q^54 - 26 * q^56 - 10 * q^57 + 26 * q^58 - 6 * q^59 + 24 * q^61 + 44 * q^62 - 4 * q^63 + 54 * q^64 - 4 * q^66 + 8 * q^67 - 10 * q^68 - 4 * q^69 + 26 * q^71 - 6 * q^72 + 4 * q^73 + 30 * q^76 - 8 * q^77 - 8 * q^78 + 24 * q^79 + 6 * q^81 + 82 * q^82 + 4 * q^83 + 4 * q^84 - 8 * q^87 + 16 * q^88 - 20 * q^89 - 10 * q^91 + 4 * q^92 - 8 * q^93 - 2 * q^94 + 14 * q^96 - 24 * 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 $

$2$	−2.81413	−1.98989	−0.994944	−	0.100431i	$-0.967978\pi$
$2$	−2.81413	−1.98989	−0.994944	+	0.100431i	$0.967978\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.91931	2.95965
$5$	0	0
$5$	0	0
$6$	2.81413	1.14886
$7$	3.18049	1.20211	0.601056	−	0.799207i	$-0.294747\pi$
$7$	3.18049	1.20211	0.601056	+	0.799207i	$0.294747\pi$
$8$	−11.0294	−3.89949
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.91931	−0.880205	−0.440102	−	0.897948i	$-0.645058\pi$
$11$	−2.91931	−0.880205	−0.440102	+	0.897948i	$0.645058\pi$
$12$	−5.91931	−1.70876
$13$	−3.44394	−0.955177	−0.477588	−	0.878584i	$-0.658489\pi$
$13$	−3.44394	−0.955177	−0.477588	+	0.878584i	$0.658489\pi$
$14$	−8.95030	−2.39207
$15$	0	0
$16$	19.1996	4.79990
$17$	3.44394	0.835278	0.417639	−	0.908613i	$-0.362858\pi$
$17$	3.44394	0.835278	0.417639	+	0.908613i	$0.362858\pi$
$18$	−2.81413	−0.663296
$19$	0.147944	0.0339408	0.0169704	−	0.999856i	$-0.494598\pi$
$19$	0.147944	0.0339408	0.0169704	+	0.999856i	$0.494598\pi$
$20$	0	0
$21$	−3.18049	−0.694039
$22$	8.21530	1.75151
$23$	−3.18049	−0.663178	−0.331589	−	0.943424i	$-0.607585\pi$
$23$	−3.18049	−0.663178	−0.331589	+	0.943424i	$0.607585\pi$
$24$	11.0294	2.25137
$25$	0	0
$26$	9.69168	1.90069
$27$	−1.00000	−0.192450
$28$	18.8263	3.55783
$29$	−5.95412	−1.10565	−0.552826	−	0.833296i	$-0.686451\pi$
$29$	−5.95412	−1.10565	−0.552826	+	0.833296i	$0.686451\pi$
$30$	0	0
$31$	−8.13944	−1.46189	−0.730944	−	0.682438i	$-0.760920\pi$
$31$	−8.13944	−1.46189	−0.730944	+	0.682438i	$0.760920\pi$
$32$	−31.9712	−5.65177
$33$	2.91931	0.508186
$34$	−9.69168	−1.66211
$35$	0	0
$36$	5.91931	0.986551
$37$	0.919308	0.151133	0.0755667	−	0.997141i	$-0.475923\pi$
$37$	0.919308	0.151133	0.0755667	+	0.997141i	$0.475923\pi$
$38$	−0.416334	−0.0675383
$39$	3.44394	0.551472
$40$	0	0
$41$	−8.22169	−1.28401	−0.642006	−	0.766699i	$-0.721898\pi$
$41$	−8.22169	−1.28401	−0.642006	+	0.766699i	$0.721898\pi$
$42$	8.95030	1.38106
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−17.2803	−2.60510
$45$	0	0
$46$	8.95030	1.31965
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	−19.1996	−2.77122
$49$	3.11551	0.445072
$50$	0	0
$51$	−3.44394	−0.482248
$52$	−20.3857	−2.82699
$53$	11.6592	1.60152	0.800760	−	0.598985i	$-0.204429\pi$
$53$	11.6592	1.60152	0.800760	+	0.598985i	$0.204429\pi$
$54$	2.81413	0.382954
$55$	0	0
$56$	−35.0790	−4.68762
$57$	−0.147944	−0.0195957
$58$	16.7557	2.20013
$59$	12.5785	1.63759	0.818794	−	0.574088i	$-0.194643\pi$
$59$	12.5785	1.63759	0.818794	+	0.574088i	$0.194643\pi$
$60$	0	0
$61$	3.88449	0.497358	0.248679	−	0.968586i	$-0.420003\pi$
$61$	3.88449	0.497358	0.248679	+	0.968586i	$0.420003\pi$
$62$	22.9054	2.90899
$63$	3.18049	0.400704
$64$	51.5719	6.44649
$65$	0	0
$66$	−8.21530	−1.01123
$67$	−1.84979	−0.225987	−0.112994	−	0.993596i	$-0.536044\pi$
$67$	−1.84979	−0.225987	−0.112994	+	0.993596i	$0.536044\pi$
$68$	20.3857	2.47213
$69$	3.18049	0.382886
$70$	0	0
$71$	15.1607	1.79924	0.899620	−	0.436674i	$-0.143844\pi$
$71$	15.1607	1.79924	0.899620	+	0.436674i	$0.143844\pi$
$72$	−11.0294	−1.29983
$73$	10.1394	1.18673	0.593366	−	0.804933i	$-0.297799\pi$
$73$	10.1394	1.18673	0.593366	+	0.804933i	$0.297799\pi$
$74$	−2.58705	−0.300738
$75$	0	0
$76$	0.875729	0.100453
$77$	−9.28483	−1.05810
$78$	−9.69168	−1.09737
$79$	10.3924	1.16924	0.584619	−	0.811308i	$-0.301244\pi$
$79$	10.3924	1.16924	0.584619	+	0.811308i	$0.301244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	23.1369	2.55504
$83$	3.70411	0.406579	0.203290	−	0.979119i	$-0.434837\pi$
$83$	3.70411	0.406579	0.203290	+	0.979119i	$0.434837\pi$
$84$	−18.8263	−2.05412
$85$	0	0
$86$	0	0
$87$	5.95412	0.638349
$88$	32.1983	3.43235
$89$	−13.6884	−1.45097	−0.725484	−	0.688239i	$-0.758384\pi$
$89$	−13.6884	−1.45097	−0.725484	+	0.688239i	$0.758384\pi$
$90$	0	0
$91$	−10.9534	−1.14823
$92$	−18.8263	−1.96278
$93$	8.13944	0.844021
$94$	−2.81413	−0.290255
$95$	0	0
$96$	31.9712	3.26305
$97$	−4.36552	−0.443251	−0.221626	−	0.975132i	$-0.571136\pi$
$97$	−4.36552	−0.443251	−0.221626	+	0.975132i	$0.571136\pi$
$98$	−8.76743	−0.885644
$99$	−2.91931	−0.293402
$100$	0	0
$101$	8.79741	0.875375	0.437688	−	0.899127i	$-0.355798\pi$
$101$	8.79741	0.875375	0.437688	+	0.899127i	$0.355798\pi$
$102$	9.69168	0.959619
$103$	−7.80080	−0.768635	−0.384318	−	0.923201i	$-0.625563\pi$
$103$	−7.80080	−0.768635	−0.384318	+	0.923201i	$0.625563\pi$
$104$	37.9847	3.72470
$105$	0	0
$106$	−32.8106	−3.18685
$107$	−0.291348	−0.0281657	−0.0140828	−	0.999901i	$-0.504483\pi$
$107$	−0.291348	−0.0281657	−0.0140828	+	0.999901i	$0.504483\pi$
$108$	−5.91931	−0.569586
$109$	17.9194	1.71637	0.858184	−	0.513342i	$-0.171593\pi$
$109$	17.9194	1.71637	0.858184	+	0.513342i	$0.171593\pi$
$110$	0	0
$111$	−0.919308	−0.0872569
$112$	61.0641	5.77001
$113$	−18.4620	−1.73676	−0.868381	−	0.495897i	$-0.834839\pi$
$113$	−18.4620	−1.73676	−0.868381	+	0.495897i	$0.834839\pi$
$114$	0.416334	0.0389933
$115$	0	0
$116$	−35.2443	−3.27235
$117$	−3.44394	−0.318392
$118$	−35.3976	−3.25862
$119$	10.9534	1.00410
$120$	0	0
$121$	−2.47764	−0.225240
$122$	−10.9315	−0.989688
$123$	8.22169	0.741325
$124$	−48.1799	−4.32668
$125$	0	0
$126$	−8.95030	−0.797356
$127$	−20.8641	−1.85139	−0.925695	−	0.378270i	$-0.876519\pi$
$127$	−20.8641	−1.85139	−0.925695	+	0.378270i	$0.876519\pi$
$128$	−81.1874	−7.17602
$129$	0	0
$130$	0	0
$131$	3.47493	0.303606	0.151803	−	0.988411i	$-0.451492\pi$
$131$	3.47493	0.303606	0.151803	+	0.988411i	$0.451492\pi$
$132$	17.2803	1.50406
$133$	0.470536	0.0408006
$134$	5.20553	0.449689
$135$	0	0
$136$	−37.9847	−3.25716
$137$	20.4230	1.74485	0.872425	−	0.488747i	$-0.162546\pi$
$137$	20.4230	1.74485	0.872425	+	0.488747i	$0.162546\pi$
$138$	−8.95030	−0.761900
$139$	17.2542	1.46349	0.731743	−	0.681581i	$-0.238707\pi$
$139$	17.2542	1.46349	0.731743	+	0.681581i	$0.238707\pi$
$140$	0	0
$141$	−1.00000	−0.0842152
$142$	−42.6640	−3.58028
$143$	10.0539	0.840751
$144$	19.1996	1.59997
$145$	0	0
$146$	−28.5337	−2.36146
$147$	−3.11551	−0.256963
$148$	5.44167	0.447302
$149$	13.7789	1.12881	0.564404	−	0.825498i	$-0.309106\pi$
$149$	13.7789	1.12881	0.564404	+	0.825498i	$0.309106\pi$
$150$	0	0
$151$	5.56583	0.452941	0.226471	−	0.974018i	$-0.427281\pi$
$151$	5.56583	0.452941	0.226471	+	0.974018i	$0.427281\pi$
$152$	−1.63174	−0.132352
$153$	3.44394	0.278426
$154$	26.1287	2.10551
$155$	0	0
$156$	20.3857	1.63217
$157$	−2.66452	−0.212652	−0.106326	−	0.994331i	$-0.533909\pi$
$157$	−2.66452	−0.212652	−0.106326	+	0.994331i	$0.533909\pi$
$158$	−29.2455	−2.32665
$159$	−11.6592	−0.924638
$160$	0	0
$161$	−10.1155	−0.797214
$162$	−2.81413	−0.221099
$163$	11.3041	0.885403	0.442701	−	0.896669i	$-0.354020\pi$
$163$	11.3041	0.885403	0.442701	+	0.896669i	$0.354020\pi$
$164$	−48.6667	−3.80023
$165$	0	0
$166$	−10.4238	−0.809047
$167$	11.8136	0.914162	0.457081	−	0.889425i	$-0.348895\pi$
$167$	11.8136	0.914162	0.457081	+	0.889425i	$0.348895\pi$
$168$	35.0790	2.70640
$169$	−1.13928	−0.0876373
$170$	0	0
$171$	0.147944	0.0113136
$172$	0	0
$173$	12.9094	0.981482	0.490741	−	0.871305i	$-0.336726\pi$
$173$	12.9094	0.981482	0.490741	+	0.871305i	$0.336726\pi$
$174$	−16.7557	−1.27024
$175$	0	0
$176$	−56.0495	−4.22489
$177$	−12.5785	−0.945462
$178$	38.5209	2.88726
$179$	−5.60760	−0.419132	−0.209566	−	0.977795i	$-0.567205\pi$
$179$	−5.60760	−0.419132	−0.209566	+	0.977795i	$0.567205\pi$
$180$	0	0
$181$	−14.4194	−1.07179	−0.535894	−	0.844285i	$-0.680025\pi$
$181$	−14.4194	−1.07179	−0.535894	+	0.844285i	$0.680025\pi$
$182$	30.8243	2.28485
$183$	−3.88449	−0.287150
$184$	35.0790	2.58606
$185$	0	0
$186$	−22.9054	−1.67951
$187$	−10.0539	−0.735215
$188$	5.91931	0.431710
$189$	−3.18049	−0.231346
$190$	0	0
$191$	−7.47493	−0.540867	−0.270433	−	0.962739i	$-0.587167\pi$
$191$	−7.47493	−0.540867	−0.270433	+	0.962739i	$0.587167\pi$
$192$	−51.5719	−3.72188
$193$	5.84810	0.420955	0.210478	−	0.977599i	$-0.432498\pi$
$193$	5.84810	0.420955	0.210478	+	0.977599i	$0.432498\pi$
$194$	12.2851	0.882020
$195$	0	0
$196$	18.4416	1.31726
$197$	−3.49604	−0.249082	−0.124541	−	0.992214i	$-0.539746\pi$
$197$	−3.49604	−0.249082	−0.124541	+	0.992214i	$0.539746\pi$
$198$	8.21530	0.583836
$199$	0.812462	0.0575939	0.0287969	−	0.999585i	$-0.490832\pi$
$199$	0.812462	0.0575939	0.0287969	+	0.999585i	$0.490832\pi$
$200$	0	0
$201$	1.84979	0.130474
$202$	−24.7570	−1.74190
$203$	−18.9370	−1.32912
$204$	−20.3857	−1.42729
$205$	0	0
$206$	21.9524	1.52950
$207$	−3.18049	−0.221059
$208$	−66.1222	−4.58475
$209$	−0.431895	−0.0298748
$210$	0	0
$211$	−1.54746	−0.106531	−0.0532656	−	0.998580i	$-0.516963\pi$
$211$	−1.54746	−0.106531	−0.0532656	+	0.998580i	$0.516963\pi$
$212$	69.0146	4.73995
$213$	−15.1607	−1.03879
$214$	0.819890	0.0560465
$215$	0	0
$216$	11.0294	0.750458
$217$	−25.8874	−1.75735
$218$	−50.4275	−3.41538
$219$	−10.1394	−0.685160
$220$	0	0
$221$	−11.8607	−0.797838
$222$	2.58705	0.173631
$223$	4.01571	0.268912	0.134456	−	0.990920i	$-0.457071\pi$
$223$	4.01571	0.268912	0.134456	+	0.990920i	$0.457071\pi$
$224$	−101.684	−6.79406
$225$	0	0
$226$	51.9545	3.45596
$227$	19.9115	1.32157	0.660787	−	0.750573i	$-0.270223\pi$
$227$	19.9115	1.32157	0.660787	+	0.750573i	$0.270223\pi$
$228$	−0.875729	−0.0579965
$229$	3.32944	0.220016	0.110008	−	0.993931i	$-0.464912\pi$
$229$	3.32944	0.220016	0.110008	+	0.993931i	$0.464912\pi$
$230$	0	0
$231$	9.28483	0.610897
$232$	65.6706	4.31148
$233$	−5.90825	−0.387062	−0.193531	−	0.981094i	$-0.561994\pi$
$233$	−5.90825	−0.387062	−0.193531	+	0.981094i	$0.561994\pi$
$234$	9.69168	0.633565
$235$	0	0
$236$	74.4563	4.84669
$237$	−10.3924	−0.675059
$238$	−30.8243	−1.99804
$239$	−15.9055	−1.02884	−0.514422	−	0.857537i	$-0.671993\pi$
$239$	−15.9055	−1.02884	−0.514422	+	0.857537i	$0.671993\pi$
$240$	0	0
$241$	16.0365	1.03300	0.516502	−	0.856286i	$-0.327234\pi$
$241$	16.0365	1.03300	0.516502	+	0.856286i	$0.327234\pi$
$242$	6.97239	0.448202
$243$	−1.00000	−0.0641500
$244$	22.9935	1.47201
$245$	0	0
$246$	−23.1369	−1.47515
$247$	−0.509512	−0.0324194
$248$	89.7734	5.70062
$249$	−3.70411	−0.234739
$250$	0	0
$251$	2.05437	0.129671	0.0648353	−	0.997896i	$-0.479348\pi$
$251$	2.05437	0.129671	0.0648353	+	0.997896i	$0.479348\pi$
$252$	18.8263	1.18594
$253$	9.28483	0.583732
$254$	58.7142	3.68406
$255$	0	0
$256$	125.328	7.83298
$257$	−1.65826	−0.103439	−0.0517197	−	0.998662i	$-0.516470\pi$
$257$	−1.65826	−0.103439	−0.0517197	+	0.998662i	$0.516470\pi$
$258$	0	0
$259$	2.92385	0.181679
$260$	0	0
$261$	−5.95412	−0.368551
$262$	−9.77889	−0.604142
$263$	23.7235	1.46285	0.731427	−	0.681920i	$-0.238855\pi$
$263$	23.7235	1.46285	0.731427	+	0.681920i	$0.238855\pi$
$264$	−32.1983	−1.98167
$265$	0	0
$266$	−1.32415	−0.0811886
$267$	13.6884	0.837717
$268$	−10.9494	−0.668844
$269$	−8.13451	−0.495970	−0.247985	−	0.968764i	$-0.579768\pi$
$269$	−8.13451	−0.495970	−0.247985	+	0.968764i	$0.579768\pi$
$270$	0	0
$271$	12.3686	0.751341	0.375670	−	0.926753i	$-0.377413\pi$
$271$	12.3686	0.751341	0.375670	+	0.926753i	$0.377413\pi$
$272$	66.1222	4.00925
$273$	10.9534	0.662930
$274$	−57.4728	−3.47206
$275$	0	0
$276$	18.8263	1.13321
$277$	−14.9889	−0.900598	−0.450299	−	0.892878i	$-0.648683\pi$
$277$	−14.9889	−0.900598	−0.450299	+	0.892878i	$0.648683\pi$
$278$	−48.5556	−2.91217
$279$	−8.13944	−0.487296
$280$	0	0
$281$	13.8624	0.826960	0.413480	−	0.910513i	$-0.364313\pi$
$281$	13.8624	0.826960	0.413480	+	0.910513i	$0.364313\pi$
$282$	2.81413	0.167579
$283$	1.99689	0.118703	0.0593514	−	0.998237i	$-0.481097\pi$
$283$	1.99689	0.118703	0.0593514	+	0.998237i	$0.481097\pi$
$284$	89.7406	5.32513
$285$	0	0
$286$	−28.2930	−1.67300
$287$	−26.1490	−1.54353
$288$	−31.9712	−1.88392
$289$	−5.13928	−0.302311
$290$	0	0
$291$	4.36552	0.255911
$292$	60.0185	3.51232
$293$	11.7639	0.687252	0.343626	−	0.939107i	$-0.388345\pi$
$293$	11.7639	0.687252	0.343626	+	0.939107i	$0.388345\pi$
$294$	8.76743	0.511327
$295$	0	0
$296$	−10.1394	−0.589343
$297$	2.91931	0.169395
$298$	−38.7755	−2.24620
$299$	10.9534	0.633452
$300$	0	0
$301$	0	0
$302$	−15.6630	−0.901302
$303$	−8.79741	−0.505398
$304$	2.84047	0.162912
$305$	0	0
$306$	−9.69168	−0.554036
$307$	10.5602	0.602701	0.301351	−	0.953513i	$-0.402563\pi$
$307$	10.5602	0.602701	0.301351	+	0.953513i	$0.402563\pi$
$308$	−54.9597	−3.13162
$309$	7.80080	0.443772
$310$	0	0
$311$	−18.4275	−1.04493	−0.522464	−	0.852661i	$-0.674987\pi$
$311$	−18.4275	−1.04493	−0.522464	+	0.852661i	$0.674987\pi$
$312$	−37.9847	−2.15046
$313$	−9.54996	−0.539795	−0.269898	−	0.962889i	$-0.586990\pi$
$313$	−9.54996	−0.539795	−0.269898	+	0.962889i	$0.586990\pi$
$314$	7.49829	0.423153
$315$	0	0
$316$	61.5159	3.46054
$317$	15.3117	0.859992	0.429996	−	0.902831i	$-0.358515\pi$
$317$	15.3117	0.859992	0.429996	+	0.902831i	$0.358515\pi$
$318$	32.8106	1.83993
$319$	17.3819	0.973201
$320$	0	0
$321$	0.291348	0.0162615
$322$	28.4663	1.58637
$323$	0.509512	0.0283500
$324$	5.91931	0.328850
$325$	0	0
$326$	−31.8111	−1.76185
$327$	−17.9194	−0.990946
$328$	90.6806	5.00700
$329$	3.18049	0.175346
$330$	0	0
$331$	−14.8482	−0.816132	−0.408066	−	0.912952i	$-0.633797\pi$
$331$	−14.8482	−0.816132	−0.408066	+	0.912952i	$0.633797\pi$
$332$	21.9258	1.20333
$333$	0.919308	0.0503778
$334$	−33.2449	−1.81908
$335$	0	0
$336$	−61.0641	−3.33132
$337$	−0.204709	−0.0111512	−0.00557559	−	0.999984i	$-0.501775\pi$
$337$	−0.204709	−0.0111512	−0.00557559	+	0.999984i	$0.501775\pi$
$338$	3.20609	0.174388
$339$	18.4620	1.00272
$340$	0	0
$341$	23.7615	1.28676
$342$	−0.416334	−0.0225128
$343$	−12.3546	−0.667085
$344$	0	0
$345$	0	0
$346$	−36.3286	−1.95304
$347$	−17.3055	−0.929008	−0.464504	−	0.885571i	$-0.653767\pi$
$347$	−17.3055	−0.929008	−0.464504	+	0.885571i	$0.653767\pi$
$348$	35.2443	1.88929
$349$	−4.57094	−0.244677	−0.122338	−	0.992488i	$-0.539039\pi$
$349$	−4.57094	−0.244677	−0.122338	+	0.992488i	$0.539039\pi$
$350$	0	0
$351$	3.44394	0.183824
$352$	93.3339	4.97471
$353$	31.3863	1.67052	0.835262	−	0.549853i	$-0.185316\pi$
$353$	31.3863	1.67052	0.835262	+	0.549853i	$0.185316\pi$
$354$	35.3976	1.88136
$355$	0	0
$356$	−81.0259	−4.29436
$357$	−10.9534	−0.579716
$358$	15.7805	0.834026
$359$	11.2610	0.594335	0.297168	−	0.954825i	$-0.403958\pi$
$359$	11.2610	0.594335	0.297168	+	0.954825i	$0.403958\pi$
$360$	0	0
$361$	−18.9781	−0.998848
$362$	40.5781	2.13274
$363$	2.47764	0.130042
$364$	−64.8366	−3.39836
$365$	0	0
$366$	10.9315	0.571396
$367$	−24.9889	−1.30441	−0.652206	−	0.758042i	$-0.726156\pi$
$367$	−24.9889	−1.30441	−0.652206	+	0.758042i	$0.726156\pi$
$368$	−61.0641	−3.18319
$369$	−8.22169	−0.428004
$370$	0	0
$371$	37.0821	1.92521
$372$	48.1799	2.49801
$373$	−19.6148	−1.01561	−0.507807	−	0.861471i	$-0.669544\pi$
$373$	−19.6148	−1.01561	−0.507807	+	0.861471i	$0.669544\pi$
$374$	28.2930	1.46300
$375$	0	0
$376$	−11.0294	−0.568799
$377$	20.5056	1.05609
$378$	8.95030	0.460354
$379$	25.7200	1.32115	0.660573	−	0.750762i	$-0.270313\pi$
$379$	25.7200	1.32115	0.660573	+	0.750762i	$0.270313\pi$
$380$	0	0
$381$	20.8641	1.06890
$382$	21.0354	1.07626
$383$	15.0143	0.767196	0.383598	−	0.923500i	$-0.374685\pi$
$383$	15.0143	0.767196	0.383598	+	0.923500i	$0.374685\pi$
$384$	81.1874	4.14308
$385$	0	0
$386$	−16.4573	−0.837654
$387$	0	0
$388$	−25.8408	−1.31187
$389$	1.08125	0.0548216	0.0274108	−	0.999624i	$-0.491274\pi$
$389$	1.08125	0.0548216	0.0274108	+	0.999624i	$0.491274\pi$
$390$	0	0
$391$	−10.9534	−0.553938
$392$	−34.3623	−1.73556
$393$	−3.47493	−0.175287
$394$	9.83829	0.495646
$395$	0	0
$396$	−17.2803	−0.868367
$397$	19.6885	0.988140	0.494070	−	0.869422i	$-0.335509\pi$
$397$	19.6885	0.988140	0.494070	+	0.869422i	$0.335509\pi$
$398$	−2.28637	−0.114605
$399$	−0.470536	−0.0235562
$400$	0	0
$401$	13.7520	0.686742	0.343371	−	0.939200i	$-0.388431\pi$
$401$	13.7520	0.686742	0.343371	+	0.939200i	$0.388431\pi$
$402$	−5.20553	−0.259628
$403$	28.0317	1.39636
$404$	52.0746	2.59081
$405$	0	0
$406$	53.2912	2.64480
$407$	−2.68374	−0.133028
$408$	37.9847	1.88052
$409$	9.81778	0.485458	0.242729	−	0.970094i	$-0.421957\pi$
$409$	9.81778	0.485458	0.242729	+	0.970094i	$0.421957\pi$
$410$	0	0
$411$	−20.4230	−1.00739
$412$	−46.1753	−2.27489
$413$	40.0059	1.96856
$414$	8.95030	0.439883
$415$	0	0
$416$	110.107	5.39844
$417$	−17.2542	−0.844944
$418$	1.21541	0.0594476
$419$	2.77380	0.135509	0.0677546	−	0.997702i	$-0.478417\pi$
$419$	2.77380	0.135509	0.0677546	+	0.997702i	$0.478417\pi$
$420$	0	0
$421$	32.0526	1.56215	0.781075	−	0.624438i	$-0.214672\pi$
$421$	32.0526	1.56215	0.781075	+	0.624438i	$0.214672\pi$
$422$	4.35474	0.211985
$423$	1.00000	0.0486217
$424$	−128.595	−6.24511
$425$	0	0
$426$	42.6640	2.06708
$427$	12.3546	0.597880
$428$	−1.72458	−0.0833607
$429$	−10.0539	−0.485408
$430$	0	0
$431$	−38.8164	−1.86972	−0.934859	−	0.355019i	$-0.884474\pi$
$431$	−38.8164	−1.86972	−0.934859	+	0.355019i	$0.884474\pi$
$432$	−19.1996	−0.923741
$433$	23.5059	1.12962	0.564811	−	0.825220i	$-0.308949\pi$
$433$	23.5059	1.12962	0.564811	+	0.825220i	$0.308949\pi$
$434$	72.8504	3.49693
$435$	0	0
$436$	106.071	5.07986
$437$	−0.470536	−0.0225088
$438$	28.5337	1.36339
$439$	28.0889	1.34061	0.670304	−	0.742086i	$-0.266164\pi$
$439$	28.0889	1.34061	0.670304	+	0.742086i	$0.266164\pi$
$440$	0	0
$441$	3.11551	0.148357
$442$	33.3776	1.58761
$443$	−4.87360	−0.231552	−0.115776	−	0.993275i	$-0.536935\pi$
$443$	−4.87360	−0.231552	−0.115776	+	0.993275i	$0.536935\pi$
$444$	−5.44167	−0.258250
$445$	0	0
$446$	−11.3007	−0.535104
$447$	−13.7789	−0.651718
$448$	164.024	7.74940
$449$	−22.6761	−1.07015	−0.535075	−	0.844805i	$-0.679717\pi$
$449$	−22.6761	−1.07015	−0.535075	+	0.844805i	$0.679717\pi$
$450$	0	0
$451$	24.0017	1.13019
$452$	−109.282	−5.14022
$453$	−5.56583	−0.261506
$454$	−56.0335	−2.62978
$455$	0	0
$456$	1.63174	0.0764133
$457$	−4.72427	−0.220992	−0.110496	−	0.993877i	$-0.535244\pi$
$457$	−4.72427	−0.220992	−0.110496	+	0.993877i	$0.535244\pi$
$458$	−9.36947	−0.437806
$459$	−3.44394	−0.160749
$460$	0	0
$461$	−0.439953	−0.0204907	−0.0102453	−	0.999948i	$-0.503261\pi$
$461$	−0.439953	−0.0204907	−0.0102453	+	0.999948i	$0.503261\pi$
$462$	−26.1287	−1.21562
$463$	17.1790	0.798377	0.399189	−	0.916869i	$-0.369292\pi$
$463$	17.1790	0.798377	0.399189	+	0.916869i	$0.369292\pi$
$464$	−114.317	−5.30702
$465$	0	0
$466$	16.6265	0.770210
$467$	20.9588	0.969856	0.484928	−	0.874554i	$-0.338846\pi$
$467$	20.9588	0.969856	0.484928	+	0.874554i	$0.338846\pi$
$468$	−20.3857	−0.942331
$469$	−5.88322	−0.271662
$470$	0	0
$471$	2.66452	0.122774
$472$	−138.734	−6.38576
$473$	0	0
$474$	29.2455	1.34329
$475$	0	0
$476$	64.8366	2.97178
$477$	11.6592	0.533840
$478$	44.7602	2.04728
$479$	38.0374	1.73797	0.868986	−	0.494837i	$-0.164772\pi$
$479$	38.0374	1.73797	0.868986	+	0.494837i	$0.164772\pi$
$480$	0	0
$481$	−3.16604	−0.144359
$482$	−45.1288	−2.05556
$483$	10.1155	0.460271
$484$	−14.6659	−0.666632
$485$	0	0
$486$	2.81413	0.127651
$487$	−12.6236	−0.572029	−0.286015	−	0.958225i	$-0.592331\pi$
$487$	−12.6236	−0.572029	−0.286015	+	0.958225i	$0.592331\pi$
$488$	−42.8438	−1.93945
$489$	−11.3041	−0.511187
$490$	0	0
$491$	17.3059	0.781006	0.390503	−	0.920602i	$-0.372301\pi$
$491$	17.3059	0.781006	0.390503	+	0.920602i	$0.372301\pi$
$492$	48.6667	2.19407
$493$	−20.5056	−0.923527
$494$	1.43383	0.0645111
$495$	0	0
$496$	−156.274	−7.01691
$497$	48.2183	2.16289
$498$	10.4238	0.467103
$499$	32.7959	1.46814	0.734072	−	0.679071i	$-0.237617\pi$
$499$	32.7959	1.46814	0.734072	+	0.679071i	$0.237617\pi$
$500$	0	0
$501$	−11.8136	−0.527792
$502$	−5.78125	−0.258030
$503$	−35.5797	−1.58642	−0.793210	−	0.608949i	$-0.791592\pi$
$503$	−35.5797	−1.58642	−0.793210	+	0.608949i	$0.791592\pi$
$504$	−35.0790	−1.56254
$505$	0	0
$506$	−26.1287	−1.16156
$507$	1.13928	0.0505974
$508$	−123.501	−5.47948
$509$	21.4767	0.951940	0.475970	−	0.879462i	$-0.342097\pi$
$509$	21.4767	0.951940	0.475970	+	0.879462i	$0.342097\pi$
$510$	0	0
$511$	32.2484	1.42658
$512$	−190.313	−8.41074
$513$	−0.147944	−0.00653191
$514$	4.66656	0.205833
$515$	0	0
$516$	0	0
$517$	−2.91931	−0.128391
$518$	−8.22808	−0.361521
$519$	−12.9094	−0.566659
$520$	0	0
$521$	−38.3389	−1.67966	−0.839828	−	0.542853i	$-0.817344\pi$
$521$	−38.3389	−1.67966	−0.839828	+	0.542853i	$0.817344\pi$
$522$	16.7557	0.733375
$523$	25.7725	1.12695	0.563476	−	0.826133i	$-0.309464\pi$
$523$	25.7725	1.12695	0.563476	+	0.826133i	$0.309464\pi$
$524$	20.5692	0.898568
$525$	0	0
$526$	−66.7609	−2.91091
$527$	−28.0317	−1.22108
$528$	56.0495	2.43924
$529$	−12.8845	−0.560195
$530$	0	0
$531$	12.5785	0.545863
$532$	2.78524	0.120756
$533$	28.3150	1.22646
$534$	−38.5209	−1.66696
$535$	0	0
$536$	20.4021	0.881235
$537$	5.60760	0.241986
$538$	22.8915	0.986924
$539$	−9.09512	−0.391755
$540$	0	0
$541$	−24.1274	−1.03732	−0.518659	−	0.854981i	$-0.673569\pi$
$541$	−24.1274	−1.03732	−0.518659	+	0.854981i	$0.673569\pi$
$542$	−34.8069	−1.49508
$543$	14.4194	0.618797
$544$	−110.107	−4.72080
$545$	0	0
$546$	−30.8243	−1.31916
$547$	41.9501	1.79366	0.896829	−	0.442378i	$-0.145865\pi$
$547$	41.9501	1.79366	0.896829	+	0.442378i	$0.145865\pi$
$548$	120.890	5.16415
$549$	3.88449	0.165786
$550$	0	0
$551$	−0.880879	−0.0375267
$552$	−35.0790	−1.49306
$553$	33.0529	1.40555
$554$	42.1808	1.79209
$555$	0	0
$556$	102.133	4.33141
$557$	18.8687	0.799490	0.399745	−	0.916626i	$-0.369099\pi$
$557$	18.8687	0.799490	0.399745	+	0.916626i	$0.369099\pi$
$558$	22.9054	0.969664
$559$	0	0
$560$	0	0
$561$	10.0539	0.424477
$562$	−39.0105	−1.64556
$563$	17.6489	0.743813	0.371906	−	0.928270i	$-0.378704\pi$
$563$	17.6489	0.743813	0.371906	+	0.928270i	$0.378704\pi$
$564$	−5.91931	−0.249248
$565$	0	0
$566$	−5.61950	−0.236205
$567$	3.18049	0.133568
$568$	−167.213	−7.01612
$569$	7.45299	0.312446	0.156223	−	0.987722i	$-0.450068\pi$
$569$	7.45299	0.312446	0.156223	+	0.987722i	$0.450068\pi$
$570$	0	0
$571$	12.5476	0.525099	0.262550	−	0.964918i	$-0.415437\pi$
$571$	12.5476	0.525099	0.262550	+	0.964918i	$0.415437\pi$
$572$	59.5122	2.48833
$573$	7.47493	0.312270
$574$	73.5866	3.07144
$575$	0	0
$576$	51.5719	2.14883
$577$	11.8733	0.494292	0.247146	−	0.968978i	$-0.420507\pi$
$577$	11.8733	0.494292	0.247146	+	0.968978i	$0.420507\pi$
$578$	14.4626	0.601565
$579$	−5.84810	−0.243039
$580$	0	0
$581$	11.7809	0.488753
$582$	−12.2851	−0.509235
$583$	−34.0369	−1.40967
$584$	−111.832	−4.62765
$585$	0	0
$586$	−33.1050	−1.36755
$587$	−34.7602	−1.43471	−0.717353	−	0.696710i	$-0.754646\pi$
$587$	−34.7602	−1.43471	−0.717353	+	0.696710i	$0.754646\pi$
$588$	−18.4416	−0.760520
$589$	−1.20419	−0.0496176
$590$	0	0
$591$	3.49604	0.143808
$592$	17.6503	0.725425
$593$	−28.1588	−1.15634	−0.578172	−	0.815915i	$-0.696234\pi$
$593$	−28.1588	−1.15634	−0.578172	+	0.815915i	$0.696234\pi$
$594$	−8.21530	−0.337078
$595$	0	0
$596$	81.5614	3.34088
$597$	−0.812462	−0.0332518
$598$	−30.8243	−1.26050
$599$	14.4156	0.589005	0.294503	−	0.955651i	$-0.404846\pi$
$599$	14.4156	0.589005	0.294503	+	0.955651i	$0.404846\pi$
$600$	0	0
$601$	29.6842	1.21085	0.605423	−	0.795904i	$-0.293004\pi$
$601$	29.6842	1.21085	0.605423	+	0.795904i	$0.293004\pi$
$602$	0	0
$603$	−1.84979	−0.0753291
$604$	32.9459	1.34055
$605$	0	0
$606$	24.7570	1.00569
$607$	1.90148	0.0771786	0.0385893	−	0.999255i	$-0.487714\pi$
$607$	1.90148	0.0771786	0.0385893	+	0.999255i	$0.487714\pi$
$608$	−4.72996	−0.191825
$609$	18.9370	0.767367
$610$	0	0
$611$	−3.44394	−0.139327
$612$	20.3857	0.824045
$613$	0.852790	0.0344439	0.0172219	−	0.999852i	$-0.494518\pi$
$613$	0.852790	0.0344439	0.0172219	+	0.999852i	$0.494518\pi$
$614$	−29.7177	−1.19931
$615$	0	0
$616$	102.406	4.12607
$617$	6.81188	0.274236	0.137118	−	0.990555i	$-0.456216\pi$
$617$	6.81188	0.274236	0.137118	+	0.990555i	$0.456216\pi$
$618$	−21.9524	−0.883056
$619$	16.3488	0.657113	0.328557	−	0.944484i	$-0.393438\pi$
$619$	16.3488	0.657113	0.328557	+	0.944484i	$0.393438\pi$
$620$	0	0
$621$	3.18049	0.127629
$622$	51.8573	2.07929
$623$	−43.5358	−1.74423
$624$	66.1222	2.64701
$625$	0	0
$626$	26.8748	1.07413
$627$	0.431895	0.0172482
$628$	−15.7721	−0.629375
$629$	3.16604	0.126238
$630$	0	0
$631$	−1.53720	−0.0611951	−0.0305975	−	0.999532i	$-0.509741\pi$
$631$	−1.53720	−0.0611951	−0.0305975	+	0.999532i	$0.509741\pi$
$632$	−114.622	−4.55943
$633$	1.54746	0.0615058
$634$	−43.0891	−1.71129
$635$	0	0
$636$	−69.0146	−2.73661
$637$	−10.7296	−0.425123
$638$	−48.9149	−1.93656
$639$	15.1607	0.599746
$640$	0	0
$641$	16.0605	0.634353	0.317177	−	0.948366i	$-0.397265\pi$
$641$	16.0605	0.634353	0.317177	+	0.948366i	$0.397265\pi$
$642$	−0.819890	−0.0323585
$643$	21.0480	0.830052	0.415026	−	0.909810i	$-0.363772\pi$
$643$	21.0480	0.830052	0.415026	+	0.909810i	$0.363772\pi$
$644$	−59.8768	−2.35948
$645$	0	0
$646$	−1.43383	−0.0564133
$647$	31.8768	1.25321	0.626604	−	0.779338i	$-0.284445\pi$
$647$	31.8768	1.25321	0.626604	+	0.779338i	$0.284445\pi$
$648$	−11.0294	−0.433277
$649$	−36.7207	−1.44141
$650$	0	0
$651$	25.8874	1.01461
$652$	66.9122	2.62049
$653$	28.8350	1.12840	0.564199	−	0.825639i	$-0.309185\pi$
$653$	28.8350	1.12840	0.564199	+	0.825639i	$0.309185\pi$
$654$	50.4275	1.97187
$655$	0	0
$656$	−157.853	−6.16313
$657$	10.1394	0.395577
$658$	−8.95030	−0.348919
$659$	28.8498	1.12383	0.561915	−	0.827195i	$-0.310065\pi$
$659$	28.8498	1.12383	0.561915	+	0.827195i	$0.310065\pi$
$660$	0	0
$661$	−29.9057	−1.16320	−0.581598	−	0.813476i	$-0.697572\pi$
$661$	−29.9057	−1.16320	−0.581598	+	0.813476i	$0.697572\pi$
$662$	41.7848	1.62401
$663$	11.8607	0.460632
$664$	−40.8542	−1.58545
$665$	0	0
$666$	−2.58705	−0.100246
$667$	18.9370	0.733244
$668$	69.9282	2.70560
$669$	−4.01571	−0.155256
$670$	0	0
$671$	−11.3400	−0.437777
$672$	101.684	3.92255
$673$	28.0662	1.08187	0.540937	−	0.841063i	$-0.318070\pi$
$673$	28.0662	1.08187	0.540937	+	0.841063i	$0.318070\pi$
$674$	0.576076	0.0221896
$675$	0	0
$676$	−6.74378	−0.259376
$677$	−0.989512	−0.0380300	−0.0190150	−	0.999819i	$-0.506053\pi$
$677$	−0.989512	−0.0380300	−0.0190150	+	0.999819i	$0.506053\pi$
$678$	−51.9545	−1.99530
$679$	−13.8845	−0.532837
$680$	0	0
$681$	−19.9115	−0.763011
$682$	−66.8680	−2.56051
$683$	18.2027	0.696507	0.348254	−	0.937400i	$-0.386775\pi$
$683$	18.2027	0.696507	0.348254	+	0.937400i	$0.386775\pi$
$684$	0.875729	0.0334843
$685$	0	0
$686$	34.7674	1.32742
$687$	−3.32944	−0.127026
$688$	0	0
$689$	−40.1537	−1.52973
$690$	0	0
$691$	−15.5338	−0.590935	−0.295467	−	0.955353i	$-0.595475\pi$
$691$	−15.5338	−0.590935	−0.295467	+	0.955353i	$0.595475\pi$
$692$	76.4146	2.90485
$693$	−9.28483	−0.352701
$694$	48.6998	1.84862
$695$	0	0
$696$	−65.6706	−2.48924
$697$	−28.3150	−1.07251
$698$	12.8632	0.486880
$699$	5.90825	0.223470
$700$	0	0
$701$	−10.6116	−0.400793	−0.200397	−	0.979715i	$-0.564223\pi$
$701$	−10.6116	−0.400793	−0.200397	+	0.979715i	$0.564223\pi$
$702$	−9.69168	−0.365789
$703$	0.136007	0.00512958
$704$	−150.554	−5.67423
$705$	0	0
$706$	−88.3250	−3.32415
$707$	27.9801	1.05230
$708$	−74.4563	−2.79824
$709$	−11.2630	−0.422989	−0.211495	−	0.977379i	$-0.567833\pi$
$709$	−11.2630	−0.422989	−0.211495	+	0.977379i	$0.567833\pi$
$710$	0	0
$711$	10.3924	0.389746
$712$	150.975	5.65804
$713$	25.8874	0.969491
$714$	30.8243	1.15357
$715$	0	0
$716$	−33.1931	−1.24049
$717$	15.9055	0.594003
$718$	−31.6900	−1.18266
$719$	30.9143	1.15291	0.576454	−	0.817129i	$-0.304436\pi$
$719$	30.9143	1.15291	0.576454	+	0.817129i	$0.304436\pi$
$720$	0	0
$721$	−24.8103	−0.923985
$722$	53.4068	1.98760
$723$	−16.0365	−0.596405
$724$	−85.3531	−3.17212
$725$	0	0
$726$	−6.97239	−0.258770
$727$	−8.21369	−0.304629	−0.152315	−	0.988332i	$-0.548673\pi$
$727$	−8.21369	−0.304629	−0.152315	+	0.988332i	$0.548673\pi$
$728$	120.810	4.47751
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−22.9935	−0.849865
$733$	21.2185	0.783724	0.391862	−	0.920024i	$-0.371831\pi$
$733$	21.2185	0.783724	0.391862	+	0.920024i	$0.371831\pi$
$734$	70.3221	2.59563
$735$	0	0
$736$	101.684	3.74813
$737$	5.40009	0.198915
$738$	23.1369	0.851680
$739$	−42.5589	−1.56555	−0.782777	−	0.622303i	$-0.786197\pi$
$739$	−42.5589	−1.56555	−0.782777	+	0.622303i	$0.786197\pi$
$740$	0	0
$741$	0.509512	0.0187174
$742$	−104.354	−3.83094
$743$	−5.69975	−0.209104	−0.104552	−	0.994519i	$-0.533341\pi$
$743$	−5.69975	−0.209104	−0.104552	+	0.994519i	$0.533341\pi$
$744$	−89.7734	−3.29125
$745$	0	0
$746$	55.1984	2.02096
$747$	3.70411	0.135526
$748$	−59.5122	−2.17598
$749$	−0.926629	−0.0338583
$750$	0	0
$751$	12.6935	0.463193	0.231596	−	0.972812i	$-0.425605\pi$
$751$	12.6935	0.463193	0.231596	+	0.972812i	$0.425605\pi$
$752$	19.1996	0.700137
$753$	−2.05437	−0.0748654
$754$	−57.7054	−2.10151
$755$	0	0
$756$	−18.8263	−0.684706
$757$	50.9440	1.85159	0.925796	−	0.378024i	$-0.123396\pi$
$757$	50.9440	1.85159	0.925796	+	0.378024i	$0.123396\pi$
$758$	−72.3793	−2.62893
$759$	−9.28483	−0.337018
$760$	0	0
$761$	−24.0473	−0.871714	−0.435857	−	0.900016i	$-0.643555\pi$
$761$	−24.0473	−0.871714	−0.435857	+	0.900016i	$0.643555\pi$
$762$	−58.7142	−2.12699
$763$	56.9925	2.06327
$764$	−44.2464	−1.60078
$765$	0	0
$766$	−42.2522	−1.52663
$767$	−43.3198	−1.56419
$768$	−125.328	−4.52238
$769$	−16.7192	−0.602908	−0.301454	−	0.953481i	$-0.597472\pi$
$769$	−16.7192	−0.602908	−0.301454	+	0.953481i	$0.597472\pi$
$770$	0	0
$771$	1.65826	0.0597208
$772$	34.6167	1.24588
$773$	−6.40846	−0.230496	−0.115248	−	0.993337i	$-0.536766\pi$
$773$	−6.40846	−0.230496	−0.115248	+	0.993337i	$0.536766\pi$
$774$	0	0
$775$	0	0
$776$	48.1492	1.72845
$777$	−2.92385	−0.104892
$778$	−3.04277	−0.109089
$779$	−1.21635	−0.0435804
$780$	0	0
$781$	−44.2586	−1.58370
$782$	30.8243	1.10227
$783$	5.95412	0.212783
$784$	59.8164	2.13630
$785$	0	0
$786$	9.77889	0.348801
$787$	−0.590559	−0.0210512	−0.0105256	−	0.999945i	$-0.503350\pi$
$787$	−0.590559	−0.0210512	−0.0105256	+	0.999945i	$0.503350\pi$
$788$	−20.6941	−0.737198
$789$	−23.7235	−0.844579
$790$	0	0
$791$	−58.7183	−2.08778
$792$	32.1983	1.14412
$793$	−13.3780	−0.475065
$794$	−55.4060	−1.96629
$795$	0	0
$796$	4.80921	0.170458
$797$	−43.2537	−1.53213	−0.766063	−	0.642766i	$-0.777787\pi$
$797$	−43.2537	−1.53213	−0.766063	+	0.642766i	$0.777787\pi$
$798$	1.32415	0.0468743
$799$	3.44394	0.121838
$800$	0	0
$801$	−13.6884	−0.483656
$802$	−38.6998	−1.36654
$803$	−29.6002	−1.04457
$804$	10.9494	0.386157
$805$	0	0
$806$	−78.8849	−2.77860
$807$	8.13451	0.286348
$808$	−97.0304	−3.41352
$809$	−18.4595	−0.649001	−0.324501	−	0.945885i	$-0.605196\pi$
$809$	−18.4595	−0.649001	−0.324501	+	0.945885i	$0.605196\pi$
$810$	0	0
$811$	−55.3614	−1.94400	−0.972002	−	0.234974i	$-0.924499\pi$
$811$	−55.3614	−1.94400	−0.972002	+	0.234974i	$0.924499\pi$
$812$	−112.094	−3.93373
$813$	−12.3686	−0.433787
$814$	7.55240	0.264711
$815$	0	0
$816$	−66.1222	−2.31474
$817$	0	0
$818$	−27.6285	−0.966007
$819$	−10.9534	−0.382743
$820$	0	0
$821$	22.4569	0.783753	0.391876	−	0.920018i	$-0.371826\pi$
$821$	22.4569	0.783753	0.391876	+	0.920018i	$0.371826\pi$
$822$	57.4728	2.00459
$823$	17.2319	0.600665	0.300332	−	0.953835i	$-0.402902\pi$
$823$	17.2319	0.600665	0.300332	+	0.953835i	$0.402902\pi$
$824$	86.0383	2.99729
$825$	0	0
$826$	−112.582	−3.91722
$827$	44.2285	1.53798	0.768989	−	0.639263i	$-0.220760\pi$
$827$	44.2285	1.53798	0.768989	+	0.639263i	$0.220760\pi$
$828$	−18.8263	−0.654259
$829$	17.4023	0.604408	0.302204	−	0.953243i	$-0.402278\pi$
$829$	17.4023	0.604408	0.302204	+	0.953243i	$0.402278\pi$
$830$	0	0
$831$	14.9889	0.519960
$832$	−177.610	−6.15753
$833$	10.7296	0.371759
$834$	48.5556	1.68134
$835$	0	0
$836$	−2.55652	−0.0884192
$837$	8.13944	0.281340
$838$	−7.80583	−0.269648
$839$	16.7438	0.578061	0.289030	−	0.957320i	$-0.406667\pi$
$839$	16.7438	0.578061	0.289030	+	0.957320i	$0.406667\pi$
$840$	0	0
$841$	6.45157	0.222468
$842$	−90.2002	−3.10850
$843$	−13.8624	−0.477445
$844$	−9.15987	−0.315296
$845$	0	0
$846$	−2.81413	−0.0967517
$847$	−7.88010	−0.270764
$848$	223.853	7.68713
$849$	−1.99689	−0.0685331
$850$	0	0
$851$	−2.92385	−0.100228
$852$	−89.7406	−3.07446
$853$	−48.6729	−1.66653	−0.833265	−	0.552874i	$-0.813531\pi$
$853$	−48.6729	−1.66653	−0.833265	+	0.552874i	$0.813531\pi$
$854$	−34.7674	−1.18971
$855$	0	0
$856$	3.21340	0.109832
$857$	−16.1605	−0.552033	−0.276016	−	0.961153i	$-0.589014\pi$
$857$	−16.1605	−0.552033	−0.276016	+	0.961153i	$0.589014\pi$
$858$	28.2930	0.965907
$859$	−17.6807	−0.603258	−0.301629	−	0.953425i	$-0.597530\pi$
$859$	−17.6807	−0.603258	−0.301629	+	0.953425i	$0.597530\pi$
$860$	0	0
$861$	26.1490	0.891155
$862$	109.234	3.72053
$863$	−29.2468	−0.995574	−0.497787	−	0.867299i	$-0.665854\pi$
$863$	−29.2468	−0.995574	−0.497787	+	0.867299i	$0.665854\pi$
$864$	31.9712	1.08768
$865$	0	0
$866$	−66.1486	−2.24782
$867$	5.13928	0.174539
$868$	−153.236	−5.20115
$869$	−30.3386	−1.02917
$870$	0	0
$871$	6.37055	0.215858
$872$	−197.641	−6.69297
$873$	−4.36552	−0.147750
$874$	1.32415	0.0447899
$875$	0	0
$876$	−60.0185	−2.02784
$877$	−30.9174	−1.04400	−0.522002	−	0.852944i	$-0.674815\pi$
$877$	−30.9174	−1.04400	−0.522002	+	0.852944i	$0.674815\pi$
$878$	−79.0456	−2.66766
$879$	−11.7639	−0.396785
$880$	0	0
$881$	44.6919	1.50571	0.752855	−	0.658186i	$-0.228676\pi$
$881$	44.6919	1.50571	0.752855	+	0.658186i	$0.228676\pi$
$882$	−8.76743	−0.295215
$883$	15.5094	0.521935	0.260967	−	0.965348i	$-0.415959\pi$
$883$	15.5094	0.521935	0.260967	+	0.965348i	$0.415959\pi$
$884$	−70.2072	−2.36132
$885$	0	0
$886$	13.7149	0.460762
$887$	49.5444	1.66354	0.831769	−	0.555122i	$-0.187328\pi$
$887$	49.5444	1.66354	0.831769	+	0.555122i	$0.187328\pi$
$888$	10.1394	0.340257
$889$	−66.3581	−2.22558
$890$	0	0
$891$	−2.91931	−0.0978005
$892$	23.7702	0.795886
$893$	0.147944	0.00495077
$894$	38.7755	1.29685
$895$	0	0
$896$	−258.215	−8.62637
$897$	−10.9534	−0.365724
$898$	63.8133	2.12948
$899$	48.4633	1.61634
$900$	0	0
$901$	40.1537	1.33771
$902$	−67.5437	−2.24896
$903$	0	0
$904$	203.626	6.77249
$905$	0	0
$906$	15.6630	0.520367
$907$	−14.9118	−0.495138	−0.247569	−	0.968870i	$-0.579632\pi$
$907$	−14.9118	−0.495138	−0.247569	+	0.968870i	$0.579632\pi$
$908$	117.862	3.91140
$909$	8.79741	0.291792
$910$	0	0
$911$	−26.3367	−0.872575	−0.436287	−	0.899807i	$-0.643707\pi$
$911$	−26.3367	−0.872575	−0.436287	+	0.899807i	$0.643707\pi$
$912$	−2.84047	−0.0940575
$913$	−10.8134	−0.357873
$914$	13.2947	0.439749
$915$	0	0
$916$	19.7080	0.651170
$917$	11.0520	0.364968
$918$	9.69168	0.319873
$919$	−10.7131	−0.353391	−0.176696	−	0.984266i	$-0.556541\pi$
$919$	−10.7131	−0.353391	−0.176696	+	0.984266i	$0.556541\pi$
$920$	0	0
$921$	−10.5602	−0.347970
$922$	1.23808	0.0407741
$923$	−52.2124	−1.71859
$924$	54.9597	1.80804
$925$	0	0
$926$	−48.3440	−1.58868
$927$	−7.80080	−0.256212
$928$	190.361	6.24889
$929$	0.885862	0.0290642	0.0145321	−	0.999894i	$-0.495374\pi$
$929$	0.885862	0.0290642	0.0145321	+	0.999894i	$0.495374\pi$
$930$	0	0
$931$	0.460922	0.0151061
$932$	−34.9727	−1.14557
$933$	18.4275	0.603289
$934$	−58.9806	−1.92991
$935$	0	0
$936$	37.9847	1.24157
$937$	25.0059	0.816906	0.408453	−	0.912779i	$-0.366068\pi$
$937$	25.0059	0.816906	0.408453	+	0.912779i	$0.366068\pi$
$938$	16.5561	0.540577
$939$	9.54996	0.311651
$940$	0	0
$941$	6.51079	0.212246	0.106123	−	0.994353i	$-0.466156\pi$
$941$	6.51079	0.212246	0.106123	+	0.994353i	$0.466156\pi$
$942$	−7.49829	−0.244307
$943$	26.1490	0.851528
$944$	241.503	7.86025
$945$	0	0
$946$	0	0
$947$	−10.9223	−0.354925	−0.177463	−	0.984128i	$-0.556789\pi$
$947$	−10.9223	−0.354925	−0.177463	+	0.984128i	$0.556789\pi$
$948$	−61.5159	−1.99794
$949$	−34.9196	−1.13354
$950$	0	0
$951$	−15.3117	−0.496516
$952$	−120.810	−3.91547
$953$	−7.72622	−0.250277	−0.125138	−	0.992139i	$-0.539937\pi$
$953$	−7.72622	−0.250277	−0.125138	+	0.992139i	$0.539937\pi$
$954$	−32.8106	−1.06228
$955$	0	0
$956$	−94.1498	−3.04502
$957$	−17.3819	−0.561878
$958$	−107.042	−3.45837
$959$	64.9550	2.09751
$960$	0	0
$961$	35.2506	1.13711
$962$	8.90964	0.287258
$963$	−0.291348	−0.00938856
$964$	94.9252	3.05733
$965$	0	0
$966$	−28.4663	−0.915889
$967$	26.6497	0.856998	0.428499	−	0.903542i	$-0.359043\pi$
$967$	26.6497	0.856998	0.428499	+	0.903542i	$0.359043\pi$
$968$	27.3270	0.878321
$969$	−0.509512	−0.0163679
$970$	0	0
$971$	6.08120	0.195155	0.0975776	−	0.995228i	$-0.468891\pi$
$971$	6.08120	0.195155	0.0975776	+	0.995228i	$0.468891\pi$
$972$	−5.91931	−0.189862
$973$	54.8769	1.75927
$974$	35.5244	1.13827
$975$	0	0
$976$	74.5807	2.38727
$977$	1.87967	0.0601360	0.0300680	−	0.999548i	$-0.490428\pi$
$977$	1.87967	0.0601360	0.0300680	+	0.999548i	$0.490428\pi$
$978$	31.8111	1.01721
$979$	39.9607	1.27715
$980$	0	0
$981$	17.9194	0.572123
$982$	−48.7011	−1.55411
$983$	4.74678	0.151399	0.0756994	−	0.997131i	$-0.475881\pi$
$983$	4.74678	0.151399	0.0756994	+	0.997131i	$0.475881\pi$
$984$	−90.6806	−2.89079
$985$	0	0
$986$	57.7054	1.83772
$987$	−3.18049	−0.101236
$988$	−3.01596	−0.0959503
$989$	0	0
$990$	0	0
$991$	47.7131	1.51566	0.757829	−	0.652454i	$-0.226260\pi$
$991$	47.7131	1.51566	0.757829	+	0.652454i	$0.226260\pi$
$992$	260.228	8.26225
$993$	14.8482	0.471194
$994$	−135.692	−4.30390
$995$	0	0
$996$	−21.9258	−0.694745
$997$	−45.9297	−1.45461	−0.727304	−	0.686315i	$-0.759227\pi$
$997$	−45.9297	−1.45461	−0.727304	+	0.686315i	$0.759227\pi$
$998$	−92.2917	−2.92144
$999$	−0.919308	−0.0290856

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.w.1.1		6
5.4	even	2		705.2.a.m.1.6	✓	6
15.14	odd	2		2115.2.a.s.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.m.1.6	✓	6	5.4	even	2
2115.2.a.s.1.1		6	15.14	odd	2
3525.2.a.w.1.1		6	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)

\(f(q)\)	\(=\)	\(q-2.81413 q^{2} -1.00000 q^{3} +5.91931 q^{4} +2.81413 q^{6} +3.18049 q^{7} -11.0294 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.81413 q^{2} -1.00000 q^{3} +5.91931 q^{4} +2.81413 q^{6} +3.18049 q^{7} -11.0294 q^{8} +1.00000 q^{9} -2.91931 q^{11} -5.91931 q^{12} -3.44394 q^{13} -8.95030 q^{14} +19.1996 q^{16} +3.44394 q^{17} -2.81413 q^{18} +0.147944 q^{19} -3.18049 q^{21} +8.21530 q^{22} -3.18049 q^{23} +11.0294 q^{24} +9.69168 q^{26} -1.00000 q^{27} +18.8263 q^{28} -5.95412 q^{29} -8.13944 q^{31} -31.9712 q^{32} +2.91931 q^{33} -9.69168 q^{34} +5.91931 q^{36} +0.919308 q^{37} -0.416334 q^{38} +3.44394 q^{39} -8.22169 q^{41} +8.95030 q^{42} -17.2803 q^{44} +8.95030 q^{46} +1.00000 q^{47} -19.1996 q^{48} +3.11551 q^{49} -3.44394 q^{51} -20.3857 q^{52} +11.6592 q^{53} +2.81413 q^{54} -35.0790 q^{56} -0.147944 q^{57} +16.7557 q^{58} +12.5785 q^{59} +3.88449 q^{61} +22.9054 q^{62} +3.18049 q^{63} +51.5719 q^{64} -8.21530 q^{66} -1.84979 q^{67} +20.3857 q^{68} +3.18049 q^{69} +15.1607 q^{71} -11.0294 q^{72} +10.1394 q^{73} -2.58705 q^{74} +0.875729 q^{76} -9.28483 q^{77} -9.69168 q^{78} +10.3924 q^{79} +1.00000 q^{81} +23.1369 q^{82} +3.70411 q^{83} -18.8263 q^{84} +5.95412 q^{87} +32.1983 q^{88} -13.6884 q^{89} -10.9534 q^{91} -18.8263 q^{92} +8.13944 q^{93} -2.81413 q^{94} +31.9712 q^{96} -4.36552 q^{97} -8.76743 q^{98} -2.91931 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - 6 q^{3} + 14 q^{4} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 6 * q^9	\( 6 q - 2 q^{2} - 6 q^{3} + 14 q^{4} + 2 q^{6} - 4 q^{7} - 6 q^{8} + 6 q^{9} + 4 q^{11} - 14 q^{12} - 6 q^{13} - 2 q^{14} + 26 q^{16} + 6 q^{17} - 2 q^{18} + 10 q^{19} + 4 q^{21} + 4 q^{22} + 4 q^{23} + 6 q^{24} + 8 q^{26} - 6 q^{27} - 4 q^{28} + 8 q^{29} + 8 q^{31} - 14 q^{32} - 4 q^{33} - 8 q^{34} + 14 q^{36} - 16 q^{37} + 12 q^{38} + 6 q^{39} - 12 q^{41} + 2 q^{42} - 36 q^{44} + 2 q^{46} + 6 q^{47} - 26 q^{48} + 18 q^{49} - 6 q^{51} + 10 q^{52} + 10 q^{53} + 2 q^{54} - 26 q^{56} - 10 q^{57} + 26 q^{58} - 6 q^{59} + 24 q^{61} + 44 q^{62} - 4 q^{63} + 54 q^{64} - 4 q^{66} + 8 q^{67} - 10 q^{68} - 4 q^{69} + 26 q^{71} - 6 q^{72} + 4 q^{73} + 30 q^{76} - 8 q^{77} - 8 q^{78} + 24 q^{79} + 6 q^{81} + 82 q^{82} + 4 q^{83} + 4 q^{84} - 8 q^{87} + 16 q^{88} - 20 q^{89} - 10 q^{91} + 4 q^{92} - 8 q^{93} - 2 q^{94} + 14 q^{96} - 24 q^{98} + 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - 6 * q^3 + 14 * q^4 + 2 * q^6 - 4 * q^7 - 6 * q^8 + 6 * q^9 + 4 * q^11 - 14 * q^12 - 6 * q^13 - 2 * q^14 + 26 * q^16 + 6 * q^17 - 2 * q^18 + 10 * q^19 + 4 * q^21 + 4 * q^22 + 4 * q^23 + 6 * q^24 + 8 * q^26 - 6 * q^27 - 4 * q^28 + 8 * q^29 + 8 * q^31 - 14 * q^32 - 4 * q^33 - 8 * q^34 + 14 * q^36 - 16 * q^37 + 12 * q^38 + 6 * q^39 - 12 * q^41 + 2 * q^42 - 36 * q^44 + 2 * q^46 + 6 * q^47 - 26 * q^48 + 18 * q^49 - 6 * q^51 + 10 * q^52 + 10 * q^53 + 2 * q^54 - 26 * q^56 - 10 * q^57 + 26 * q^58 - 6 * q^59 + 24 * q^61 + 44 * q^62 - 4 * q^63 + 54 * q^64 - 4 * q^66 + 8 * q^67 - 10 * q^68 - 4 * q^69 + 26 * q^71 - 6 * q^72 + 4 * q^73 + 30 * q^76 - 8 * q^77 - 8 * q^78 + 24 * q^79 + 6 * q^81 + 82 * q^82 + 4 * q^83 + 4 * q^84 - 8 * q^87 + 16 * q^88 - 20 * q^89 - 10 * q^91 + 4 * q^92 - 8 * q^93 - 2 * q^94 + 14 * q^96 - 24 * q^98 + 4 * q^99

Introduction

L-functions

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