LMFDB - Embedded newform 3525.2.a.t.1.4

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			1.4
Root			$-1.27133$ 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)

$f(q)$	$=$	$q+1.68554 q^{2} +1.00000 q^{3} +0.841058 q^{4} +1.68554 q^{6} -4.79793 q^{7} -1.95345 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.68554 q^{2} +1.00000 q^{3} +0.841058 q^{4} +1.68554 q^{6} -4.79793 q^{7} -1.95345 q^{8} +1.00000 q^{9} +4.98737 q^{11} +0.841058 q^{12} +3.91032 q^{13} -8.08713 q^{14} -4.97474 q^{16} -6.51397 q^{17} +1.68554 q^{18} -4.65505 q^{19} -4.79793 q^{21} +8.40643 q^{22} +1.62636 q^{23} -1.95345 q^{24} +6.59102 q^{26} +1.00000 q^{27} -4.03534 q^{28} +3.76744 q^{29} -7.29585 q^{31} -4.47824 q^{32} +4.98737 q^{33} -10.9796 q^{34} +0.841058 q^{36} -8.07269 q^{37} -7.84629 q^{38} +3.91032 q^{39} -6.49791 q^{41} -8.08713 q^{42} -11.1463 q^{43} +4.19467 q^{44} +2.74130 q^{46} +1.00000 q^{47} -4.97474 q^{48} +16.0202 q^{49} -6.51397 q^{51} +3.28881 q^{52} -2.23921 q^{53} +1.68554 q^{54} +9.37251 q^{56} -4.65505 q^{57} +6.35018 q^{58} -2.49126 q^{59} -14.9137 q^{61} -12.2975 q^{62} -4.79793 q^{63} +2.40120 q^{64} +8.40643 q^{66} +6.18165 q^{67} -5.47863 q^{68} +1.62636 q^{69} +9.54447 q^{71} -1.95345 q^{72} +4.36101 q^{73} -13.6069 q^{74} -3.91517 q^{76} -23.9291 q^{77} +6.59102 q^{78} +3.21215 q^{79} +1.00000 q^{81} -10.9525 q^{82} -5.71423 q^{83} -4.03534 q^{84} -18.7876 q^{86} +3.76744 q^{87} -9.74256 q^{88} -1.60687 q^{89} -18.7615 q^{91} +1.36786 q^{92} -7.29585 q^{93} +1.68554 q^{94} -4.47824 q^{96} +8.26050 q^{97} +27.0027 q^{98} +4.98737 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.68554	1.19186	0.595930	−	0.803037i	$-0.296784\pi$
$2$	1.68554	1.19186	0.595930	+	0.803037i	$0.296784\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.841058	0.420529
$5$	0	0
$5$	0	0
$6$	1.68554	0.688120
$7$	−4.79793	−1.81345	−0.906724	−	0.421724i	$-0.861425\pi$
$7$	−4.79793	−1.81345	−0.906724	+	0.421724i	$0.861425\pi$
$8$	−1.95345	−0.690648
$9$	1.00000	0.333333
$10$	0	0
$11$	4.98737	1.50375	0.751874	−	0.659307i	$-0.229150\pi$
$11$	4.98737	1.50375	0.751874	+	0.659307i	$0.229150\pi$
$12$	0.841058	0.242793
$13$	3.91032	1.08453	0.542264	−	0.840208i	$-0.317567\pi$
$13$	3.91032	1.08453	0.542264	+	0.840208i	$0.317567\pi$
$14$	−8.08713	−2.16138
$15$	0	0
$16$	−4.97474	−1.24368
$17$	−6.51397	−1.57987	−0.789935	−	0.613191i	$-0.789886\pi$
$17$	−6.51397	−1.57987	−0.789935	+	0.613191i	$0.789886\pi$
$18$	1.68554	0.397287
$19$	−4.65505	−1.06794	−0.533971	−	0.845503i	$-0.679301\pi$
$19$	−4.65505	−1.06794	−0.533971	+	0.845503i	$0.679301\pi$
$20$	0	0
$21$	−4.79793	−1.04699
$22$	8.40643	1.79226
$23$	1.62636	0.339119	0.169560	−	0.985520i	$-0.445765\pi$
$23$	1.62636	0.339119	0.169560	+	0.985520i	$0.445765\pi$
$24$	−1.95345	−0.398746
$25$	0	0
$26$	6.59102	1.29261
$27$	1.00000	0.192450
$28$	−4.03534	−0.762608
$29$	3.76744	0.699596	0.349798	−	0.936825i	$-0.386250\pi$
$29$	3.76744	0.699596	0.349798	+	0.936825i	$0.386250\pi$
$30$	0	0
$31$	−7.29585	−1.31037	−0.655186	−	0.755467i	$-0.727410\pi$
$31$	−7.29585	−1.31037	−0.655186	+	0.755467i	$0.727410\pi$
$32$	−4.47824	−0.791649
$33$	4.98737	0.868189
$34$	−10.9796	−1.88298
$35$	0	0
$36$	0.841058	0.140176
$37$	−8.07269	−1.32714	−0.663571	−	0.748113i	$-0.730960\pi$
$37$	−8.07269	−1.32714	−0.663571	+	0.748113i	$0.730960\pi$
$38$	−7.84629	−1.27284
$39$	3.91032	0.626153
$40$	0	0
$41$	−6.49791	−1.01480	−0.507402	−	0.861710i	$-0.669394\pi$
$41$	−6.49791	−1.01480	−0.507402	+	0.861710i	$0.669394\pi$
$42$	−8.08713	−1.24787
$43$	−11.1463	−1.69980	−0.849898	−	0.526947i	$-0.823337\pi$
$43$	−11.1463	−1.69980	−0.849898	+	0.526947i	$0.823337\pi$
$44$	4.19467	0.632370
$45$	0	0
$46$	2.74130	0.404183
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	−4.97474	−0.718042
$49$	16.0202	2.28859
$50$	0	0
$51$	−6.51397	−0.912138
$52$	3.28881	0.456076
$53$	−2.23921	−0.307580	−0.153790	−	0.988104i	$-0.549148\pi$
$53$	−2.23921	−0.307580	−0.153790	+	0.988104i	$0.549148\pi$
$54$	1.68554	0.229373
$55$	0	0
$56$	9.37251	1.25245
$57$	−4.65505	−0.616576
$58$	6.35018	0.833820
$59$	−2.49126	−0.324335	−0.162167	−	0.986763i	$-0.551848\pi$
$59$	−2.49126	−0.324335	−0.162167	+	0.986763i	$0.551848\pi$
$60$	0	0
$61$	−14.9137	−1.90951	−0.954755	−	0.297394i	$-0.903883\pi$
$61$	−14.9137	−1.90951	−0.954755	+	0.297394i	$0.903883\pi$
$62$	−12.2975	−1.56178
$63$	−4.79793	−0.604483
$64$	2.40120	0.300150
$65$	0	0
$66$	8.40643	1.03476
$67$	6.18165	0.755209	0.377604	−	0.925967i	$-0.376748\pi$
$67$	6.18165	0.755209	0.377604	+	0.925967i	$0.376748\pi$
$68$	−5.47863	−0.664381
$69$	1.62636	0.195791
$70$	0	0
$71$	9.54447	1.13272	0.566360	−	0.824158i	$-0.308351\pi$
$71$	9.54447	1.13272	0.566360	+	0.824158i	$0.308351\pi$
$72$	−1.95345	−0.230216
$73$	4.36101	0.510417	0.255209	−	0.966886i	$-0.417856\pi$
$73$	4.36101	0.510417	0.255209	+	0.966886i	$0.417856\pi$
$74$	−13.6069	−1.58177
$75$	0	0
$76$	−3.91517	−0.449101
$77$	−23.9291	−2.72697
$78$	6.59102	0.746286
$79$	3.21215	0.361395	0.180697	−	0.983539i	$-0.442165\pi$
$79$	3.21215	0.361395	0.180697	+	0.983539i	$0.442165\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−10.9525	−1.20950
$83$	−5.71423	−0.627219	−0.313609	−	0.949552i	$-0.601538\pi$
$83$	−5.71423	−0.627219	−0.313609	+	0.949552i	$0.601538\pi$
$84$	−4.03534	−0.440292
$85$	0	0
$86$	−18.7876	−2.02592
$87$	3.76744	0.403912
$88$	−9.74256	−1.03856
$89$	−1.60687	−0.170328	−0.0851642	−	0.996367i	$-0.527141\pi$
$89$	−1.60687	−0.170328	−0.0851642	+	0.996367i	$0.527141\pi$
$90$	0	0
$91$	−18.7615	−1.96674
$92$	1.36786	0.142610
$93$	−7.29585	−0.756544
$94$	1.68554	0.173851
$95$	0	0
$96$	−4.47824	−0.457059
$97$	8.26050	0.838727	0.419364	−	0.907818i	$-0.362253\pi$
$97$	8.26050	0.838727	0.419364	+	0.907818i	$0.362253\pi$
$98$	27.0027	2.72768
$99$	4.98737	0.501249
$100$	0	0
$101$	8.38117	0.833957	0.416979	−	0.908916i	$-0.363089\pi$
$101$	8.38117	0.833957	0.416979	+	0.908916i	$0.363089\pi$
$102$	−10.9796	−1.08714
$103$	−14.1775	−1.39695	−0.698474	−	0.715635i	$-0.746137\pi$
$103$	−14.1775	−1.39695	−0.698474	+	0.715635i	$0.746137\pi$
$104$	−7.63861	−0.749027
$105$	0	0
$106$	−3.77429	−0.366592
$107$	−5.17157	−0.499955	−0.249977	−	0.968252i	$-0.580423\pi$
$107$	−5.17157	−0.499955	−0.249977	+	0.968252i	$0.580423\pi$
$108$	0.841058	0.0809309
$109$	−0.928932	−0.0889756	−0.0444878	−	0.999010i	$-0.514166\pi$
$109$	−0.928932	−0.0889756	−0.0444878	+	0.999010i	$0.514166\pi$
$110$	0	0
$111$	−8.07269	−0.766226
$112$	23.8685	2.25536
$113$	3.66170	0.344464	0.172232	−	0.985056i	$-0.444902\pi$
$113$	3.66170	0.344464	0.172232	+	0.985056i	$0.444902\pi$
$114$	−7.84629	−0.734872
$115$	0	0
$116$	3.16864	0.294200
$117$	3.91032	0.361509
$118$	−4.19913	−0.386561
$119$	31.2536	2.86501
$120$	0	0
$121$	13.8738	1.26126
$122$	−25.1378	−2.27587
$123$	−6.49791	−0.585897
$124$	−6.13623	−0.551050
$125$	0	0
$126$	−8.08713	−0.720458
$127$	−14.3633	−1.27454	−0.637269	−	0.770642i	$-0.719936\pi$
$127$	−14.3633	−1.27454	−0.637269	+	0.770642i	$0.719936\pi$
$128$	13.0038	1.14939
$129$	−11.1463	−0.981378
$130$	0	0
$131$	−2.04313	−0.178509	−0.0892544	−	0.996009i	$-0.528448\pi$
$131$	−2.04313	−0.178509	−0.0892544	+	0.996009i	$0.528448\pi$
$132$	4.19467	0.365099
$133$	22.3346	1.93666
$134$	10.4194	0.900103
$135$	0	0
$136$	12.7247	1.09113
$137$	−3.45734	−0.295380	−0.147690	−	0.989034i	$-0.547184\pi$
$137$	−3.45734	−0.295380	−0.147690	+	0.989034i	$0.547184\pi$
$138$	2.74130	0.233355
$139$	−20.8114	−1.76520	−0.882602	−	0.470122i	$-0.844210\pi$
$139$	−20.8114	−1.76520	−0.882602	+	0.470122i	$0.844210\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	16.0876	1.35004
$143$	19.5022	1.63086
$144$	−4.97474	−0.414561
$145$	0	0
$146$	7.35067	0.608346
$147$	16.0202	1.32132
$148$	−6.78960	−0.558102
$149$	−4.76744	−0.390564	−0.195282	−	0.980747i	$-0.562562\pi$
$149$	−4.76744	−0.390564	−0.195282	+	0.980747i	$0.562562\pi$
$150$	0	0
$151$	0.516523	0.0420341	0.0210170	−	0.999779i	$-0.493310\pi$
$151$	0.516523	0.0420341	0.0210170	+	0.999779i	$0.493310\pi$
$152$	9.09339	0.737572
$153$	−6.51397	−0.526623
$154$	−40.3335	−3.25016
$155$	0	0
$156$	3.28881	0.263315
$157$	−20.5054	−1.63651	−0.818256	−	0.574854i	$-0.805059\pi$
$157$	−20.5054	−1.63651	−0.818256	+	0.574854i	$0.805059\pi$
$158$	5.41421	0.430732
$159$	−2.23921	−0.177581
$160$	0	0
$161$	−7.80316	−0.614976
$162$	1.68554	0.132429
$163$	16.6527	1.30434	0.652169	−	0.758073i	$-0.273859\pi$
$163$	16.6527	1.30434	0.652169	+	0.758073i	$0.273859\pi$
$164$	−5.46512	−0.426754
$165$	0	0
$166$	−9.63159	−0.747557
$167$	−5.46381	−0.422802	−0.211401	−	0.977399i	$-0.567803\pi$
$167$	−5.46381	−0.422802	−0.211401	+	0.977399i	$0.567803\pi$
$168$	9.37251	0.723105
$169$	2.29061	0.176201
$170$	0	0
$171$	−4.65505	−0.355981
$172$	−9.37470	−0.714814
$173$	24.6727	1.87583	0.937914	−	0.346869i	$-0.112755\pi$
$173$	24.6727	1.87583	0.937914	+	0.346869i	$0.112755\pi$
$174$	6.35018	0.481406
$175$	0	0
$176$	−24.8109	−1.87019
$177$	−2.49126	−0.187255
$178$	−2.70846	−0.203007
$179$	5.27798	0.394495	0.197247	−	0.980354i	$-0.436800\pi$
$179$	5.27798	0.394495	0.197247	+	0.980354i	$0.436800\pi$
$180$	0	0
$181$	−11.0101	−0.818373	−0.409186	−	0.912451i	$-0.634187\pi$
$181$	−11.0101	−0.818373	−0.409186	+	0.912451i	$0.634187\pi$
$182$	−31.6233	−2.34407
$183$	−14.9137	−1.10246
$184$	−3.17701	−0.234212
$185$	0	0
$186$	−12.2975	−0.901694
$187$	−32.4876	−2.37573
$188$	0.841058	0.0613405
$189$	−4.79793	−0.348998
$190$	0	0
$191$	25.1201	1.81763	0.908813	−	0.417203i	$-0.136990\pi$
$191$	25.1201	1.81763	0.908813	+	0.417203i	$0.136990\pi$
$192$	2.40120	0.173291
$193$	−17.6591	−1.27113	−0.635567	−	0.772046i	$-0.719233\pi$
$193$	−17.6591	−1.27113	−0.635567	+	0.772046i	$0.719233\pi$
$194$	13.9234	0.999645
$195$	0	0
$196$	13.4739	0.962421
$197$	21.5263	1.53369	0.766844	−	0.641833i	$-0.221826\pi$
$197$	21.5263	1.53369	0.766844	+	0.641833i	$0.221826\pi$
$198$	8.40643	0.597419
$199$	3.98072	0.282186	0.141093	−	0.989996i	$-0.454938\pi$
$199$	3.98072	0.282186	0.141093	+	0.989996i	$0.454938\pi$
$200$	0	0
$201$	6.18165	0.436020
$202$	14.1268	0.993960
$203$	−18.0759	−1.26868
$204$	−5.47863	−0.383581
$205$	0	0
$206$	−23.8968	−1.66497
$207$	1.62636	0.113040
$208$	−19.4528	−1.34881
$209$	−23.2164	−1.60592
$210$	0	0
$211$	−7.12845	−0.490743	−0.245371	−	0.969429i	$-0.578910\pi$
$211$	−7.12845	−0.490743	−0.245371	+	0.969429i	$0.578910\pi$
$212$	−1.88331	−0.129346
$213$	9.54447	0.653976
$214$	−8.71691	−0.595876
$215$	0	0
$216$	−1.95345	−0.132915
$217$	35.0050	2.37629
$218$	−1.56576	−0.106046
$219$	4.36101	0.294690
$220$	0	0
$221$	−25.4717	−1.71341
$222$	−13.6069	−0.913234
$223$	12.7853	0.856167	0.428084	−	0.903739i	$-0.359189\pi$
$223$	12.7853	0.856167	0.428084	+	0.903739i	$0.359189\pi$
$224$	21.4863	1.43561
$225$	0	0
$226$	6.17196	0.410553
$227$	−15.7516	−1.04547	−0.522735	−	0.852495i	$-0.675088\pi$
$227$	−15.7516	−1.04547	−0.522735	+	0.852495i	$0.675088\pi$
$228$	−3.91517	−0.259288
$229$	−5.69998	−0.376665	−0.188333	−	0.982105i	$-0.560308\pi$
$229$	−5.69998	−0.376665	−0.188333	+	0.982105i	$0.560308\pi$
$230$	0	0
$231$	−23.9291	−1.57442
$232$	−7.35949	−0.483174
$233$	11.9949	0.785812	0.392906	−	0.919579i	$-0.371470\pi$
$233$	11.9949	0.785812	0.392906	+	0.919579i	$0.371470\pi$
$234$	6.59102	0.430868
$235$	0	0
$236$	−2.09530	−0.136392
$237$	3.21215	0.208651
$238$	52.6793	3.41469
$239$	3.78623	0.244911	0.122455	−	0.992474i	$-0.460923\pi$
$239$	3.78623	0.244911	0.122455	+	0.992474i	$0.460923\pi$
$240$	0	0
$241$	10.4642	0.674058	0.337029	−	0.941494i	$-0.390578\pi$
$241$	10.4642	0.674058	0.337029	+	0.941494i	$0.390578\pi$
$242$	23.3850	1.50324
$243$	1.00000	0.0641500
$244$	−12.5433	−0.803005
$245$	0	0
$246$	−10.9525	−0.698307
$247$	−18.2027	−1.15821
$248$	14.2520	0.905006
$249$	−5.71423	−0.362125
$250$	0	0
$251$	−10.7981	−0.681572	−0.340786	−	0.940141i	$-0.610693\pi$
$251$	−10.7981	−0.681572	−0.340786	+	0.940141i	$0.610693\pi$
$252$	−4.03534	−0.254203
$253$	8.11126	0.509950
$254$	−24.2100	−1.51907
$255$	0	0
$256$	17.1161	1.06976
$257$	−4.55112	−0.283891	−0.141945	−	0.989874i	$-0.545336\pi$
$257$	−4.55112	−0.283891	−0.141945	+	0.989874i	$0.545336\pi$
$258$	−18.7876	−1.16966
$259$	38.7322	2.40670
$260$	0	0
$261$	3.76744	0.233199
$262$	−3.44378	−0.212757
$263$	−5.18258	−0.319572	−0.159786	−	0.987152i	$-0.551080\pi$
$263$	−5.18258	−0.319572	−0.159786	+	0.987152i	$0.551080\pi$
$264$	−9.74256	−0.599613
$265$	0	0
$266$	37.6460	2.30822
$267$	−1.60687	−0.0983391
$268$	5.19913	0.317587
$269$	11.2816	0.687851	0.343925	−	0.938997i	$-0.388243\pi$
$269$	11.2816	0.687851	0.343925	+	0.938997i	$0.388243\pi$
$270$	0	0
$271$	31.4302	1.90925	0.954625	−	0.297811i	$-0.0962565\pi$
$271$	31.4302	1.90925	0.954625	+	0.297811i	$0.0962565\pi$
$272$	32.4053	1.96486
$273$	−18.7615	−1.13550
$274$	−5.82750	−0.352052
$275$	0	0
$276$	1.36786	0.0823357
$277$	16.2654	0.977290	0.488645	−	0.872483i	$-0.337491\pi$
$277$	16.2654	0.977290	0.488645	+	0.872483i	$0.337491\pi$
$278$	−35.0786	−2.10387
$279$	−7.29585	−0.436791
$280$	0	0
$281$	−2.17157	−0.129545	−0.0647726	−	0.997900i	$-0.520632\pi$
$281$	−2.17157	−0.129545	−0.0647726	+	0.997900i	$0.520632\pi$
$282$	1.68554	0.100373
$283$	8.61050	0.511841	0.255921	−	0.966698i	$-0.417621\pi$
$283$	8.61050	0.511841	0.255921	+	0.966698i	$0.417621\pi$
$284$	8.02745	0.476342
$285$	0	0
$286$	32.8718	1.94375
$287$	31.1765	1.84029
$288$	−4.47824	−0.263883
$289$	25.4318	1.49599
$290$	0	0
$291$	8.26050	0.484239
$292$	3.66786	0.214645
$293$	19.7701	1.15498	0.577491	−	0.816397i	$-0.304032\pi$
$293$	19.7701	1.15498	0.577491	+	0.816397i	$0.304032\pi$
$294$	27.0027	1.57483
$295$	0	0
$296$	15.7696	0.916588
$297$	4.98737	0.289396
$298$	−8.03573	−0.465497
$299$	6.35959	0.367785
$300$	0	0
$301$	53.4792	3.08249
$302$	0.870623	0.0500987
$303$	8.38117	0.481485
$304$	23.1576	1.32818
$305$	0	0
$306$	−10.9796	−0.627661
$307$	−6.07107	−0.346494	−0.173247	−	0.984878i	$-0.555426\pi$
$307$	−6.07107	−0.346494	−0.173247	+	0.984878i	$0.555426\pi$
$308$	−20.1257	−1.14677
$309$	−14.1775	−0.806528
$310$	0	0
$311$	29.8681	1.69366	0.846831	−	0.531861i	$-0.178507\pi$
$311$	29.8681	1.69366	0.846831	+	0.531861i	$0.178507\pi$
$312$	−7.63861	−0.432451
$313$	−2.93236	−0.165747	−0.0828734	−	0.996560i	$-0.526410\pi$
$313$	−2.93236	−0.165747	−0.0828734	+	0.996560i	$0.526410\pi$
$314$	−34.5628	−1.95049
$315$	0	0
$316$	2.70160	0.151977
$317$	19.6247	1.10224	0.551118	−	0.834428i	$-0.314202\pi$
$317$	19.6247	1.10224	0.551118	+	0.834428i	$0.314202\pi$
$318$	−3.77429	−0.211652
$319$	18.7896	1.05202
$320$	0	0
$321$	−5.17157	−0.288649
$322$	−13.1526	−0.732964
$323$	30.3229	1.68721
$324$	0.841058	0.0467255
$325$	0	0
$326$	28.0688	1.55459
$327$	−0.928932	−0.0513701
$328$	12.6933	0.700872
$329$	−4.79793	−0.264519
$330$	0	0
$331$	−0.0609889	−0.00335226	−0.00167613	−	0.999999i	$-0.500534\pi$
$331$	−0.0609889	−0.00335226	−0.00167613	+	0.999999i	$0.500534\pi$
$332$	−4.80600	−0.263764
$333$	−8.07269	−0.442381
$334$	−9.20949	−0.503921
$335$	0	0
$336$	23.8685	1.30213
$337$	24.9920	1.36140	0.680699	−	0.732563i	$-0.261676\pi$
$337$	24.9920	1.36140	0.680699	+	0.732563i	$0.261676\pi$
$338$	3.86093	0.210007
$339$	3.66170	0.198876
$340$	0	0
$341$	−36.3871	−1.97047
$342$	−7.84629	−0.424279
$343$	−43.2781	−2.33680
$344$	21.7737	1.17396
$345$	0	0
$346$	41.5868	2.23572
$347$	−18.5427	−0.995422	−0.497711	−	0.867343i	$-0.665826\pi$
$347$	−18.5427	−0.995422	−0.497711	+	0.867343i	$0.665826\pi$
$348$	3.16864	0.169857
$349$	−26.9141	−1.44068	−0.720340	−	0.693621i	$-0.756014\pi$
$349$	−26.9141	−1.44068	−0.720340	+	0.693621i	$0.756014\pi$
$350$	0	0
$351$	3.91032	0.208718
$352$	−22.3347	−1.19044
$353$	−27.1201	−1.44346	−0.721729	−	0.692176i	$-0.756652\pi$
$353$	−27.1201	−1.44346	−0.721729	+	0.692176i	$0.756652\pi$
$354$	−4.19913	−0.223181
$355$	0	0
$356$	−1.35147	−0.0716280
$357$	31.2536	1.65412
$358$	8.89627	0.470183
$359$	−0.427902	−0.0225838	−0.0112919	−	0.999936i	$-0.503594\pi$
$359$	−0.427902	−0.0225838	−0.0112919	+	0.999936i	$0.503594\pi$
$360$	0	0
$361$	2.66949	0.140499
$362$	−18.5580	−0.975385
$363$	13.8738	0.728188
$364$	−15.7795	−0.827070
$365$	0	0
$366$	−25.1378	−1.31397
$367$	8.41744	0.439387	0.219693	−	0.975569i	$-0.429494\pi$
$367$	8.41744	0.439387	0.219693	+	0.975569i	$0.429494\pi$
$368$	−8.09071	−0.421758
$369$	−6.49791	−0.338268
$370$	0	0
$371$	10.7436	0.557780
$372$	−6.13623	−0.318149
$373$	30.4169	1.57493	0.787464	−	0.616361i	$-0.211394\pi$
$373$	30.4169	1.57493	0.787464	+	0.616361i	$0.211394\pi$
$374$	−54.7592	−2.83153
$375$	0	0
$376$	−1.95345	−0.100741
$377$	14.7319	0.758731
$378$	−8.08713	−0.415957
$379$	−14.7694	−0.758655	−0.379328	−	0.925262i	$-0.623845\pi$
$379$	−14.7694	−0.758655	−0.379328	+	0.925262i	$0.623845\pi$
$380$	0	0
$381$	−14.3633	−0.735854
$382$	42.3410	2.16636
$383$	1.84361	0.0942041	0.0471021	−	0.998890i	$-0.485001\pi$
$383$	1.84361	0.0942041	0.0471021	+	0.998890i	$0.485001\pi$
$384$	13.0038	0.663598
$385$	0	0
$386$	−29.7653	−1.51501
$387$	−11.1463	−0.566599
$388$	6.94757	0.352709
$389$	5.63577	0.285745	0.142872	−	0.989741i	$-0.454366\pi$
$389$	5.63577	0.285745	0.142872	+	0.989741i	$0.454366\pi$
$390$	0	0
$391$	−10.5941	−0.535765
$392$	−31.2945	−1.58061
$393$	−2.04313	−0.103062
$394$	36.2836	1.82794
$395$	0	0
$396$	4.19467	0.210790
$397$	−2.13528	−0.107167	−0.0535833	−	0.998563i	$-0.517064\pi$
$397$	−2.13528	−0.107167	−0.0535833	+	0.998563i	$0.517064\pi$
$398$	6.70967	0.336326
$399$	22.3346	1.11813
$400$	0	0
$401$	12.0880	0.603646	0.301823	−	0.953364i	$-0.402405\pi$
$401$	12.0880	0.603646	0.301823	+	0.953364i	$0.402405\pi$
$402$	10.4194	0.519675
$403$	−28.5291	−1.42114
$404$	7.04905	0.350703
$405$	0	0
$406$	−30.4677	−1.51209
$407$	−40.2615	−1.99569
$408$	12.7247	0.629966
$409$	−23.7490	−1.17431	−0.587156	−	0.809473i	$-0.699753\pi$
$409$	−23.7490	−1.17431	−0.587156	+	0.809473i	$0.699753\pi$
$410$	0	0
$411$	−3.45734	−0.170538
$412$	−11.9241	−0.587458
$413$	11.9529	0.588164
$414$	2.74130	0.134728
$415$	0	0
$416$	−17.5114	−0.858566
$417$	−20.8114	−1.01914
$418$	−39.1323	−1.91403
$419$	−16.3513	−0.798815	−0.399408	−	0.916773i	$-0.630784\pi$
$419$	−16.3513	−0.798815	−0.399408	+	0.916773i	$0.630784\pi$
$420$	0	0
$421$	14.0036	0.682494	0.341247	−	0.939974i	$-0.389151\pi$
$421$	14.0036	0.682494	0.341247	+	0.939974i	$0.389151\pi$
$422$	−12.0153	−0.584896
$423$	1.00000	0.0486217
$424$	4.37418	0.212429
$425$	0	0
$426$	16.0876	0.779448
$427$	71.5552	3.46280
$428$	−4.34959	−0.210246
$429$	19.5022	0.941576
$430$	0	0
$431$	−27.1248	−1.30655	−0.653277	−	0.757119i	$-0.726606\pi$
$431$	−27.1248	−1.30655	−0.653277	+	0.757119i	$0.726606\pi$
$432$	−4.97474	−0.239347
$433$	35.8832	1.72444	0.862218	−	0.506537i	$-0.169075\pi$
$433$	35.8832	1.72444	0.862218	+	0.506537i	$0.169075\pi$
$434$	59.0024	2.83221
$435$	0	0
$436$	−0.781286	−0.0374168
$437$	−7.57079	−0.362160
$438$	7.35067	0.351229
$439$	10.8236	0.516581	0.258291	−	0.966067i	$-0.416841\pi$
$439$	10.8236	0.516581	0.258291	+	0.966067i	$0.416841\pi$
$440$	0	0
$441$	16.0202	0.762865
$442$	−42.9337	−2.04215
$443$	−9.79216	−0.465239	−0.232620	−	0.972568i	$-0.574730\pi$
$443$	−9.79216	−0.465239	−0.232620	+	0.972568i	$0.574730\pi$
$444$	−6.78960	−0.322220
$445$	0	0
$446$	21.5502	1.02043
$447$	−4.76744	−0.225492
$448$	−11.5208	−0.544306
$449$	−4.80409	−0.226719	−0.113360	−	0.993554i	$-0.536161\pi$
$449$	−4.80409	−0.226719	−0.113360	+	0.993554i	$0.536161\pi$
$450$	0	0
$451$	−32.4075	−1.52601
$452$	3.07970	0.144857
$453$	0.516523	0.0242684
$454$	−26.5500	−1.24605
$455$	0	0
$456$	9.09339	0.425837
$457$	2.97891	0.139348	0.0696738	−	0.997570i	$-0.477804\pi$
$457$	2.97891	0.139348	0.0696738	+	0.997570i	$0.477804\pi$
$458$	−9.60757	−0.448932
$459$	−6.51397	−0.304046
$460$	0	0
$461$	27.2372	1.26856	0.634280	−	0.773103i	$-0.281297\pi$
$461$	27.2372	1.26856	0.634280	+	0.773103i	$0.281297\pi$
$462$	−40.3335	−1.87648
$463$	−4.17841	−0.194187	−0.0970935	−	0.995275i	$-0.530955\pi$
$463$	−4.17841	−0.194187	−0.0970935	+	0.995275i	$0.530955\pi$
$464$	−18.7420	−0.870076
$465$	0	0
$466$	20.2179	0.936577
$467$	16.5287	0.764858	0.382429	−	0.923985i	$-0.375088\pi$
$467$	16.5287	0.764858	0.382429	+	0.923985i	$0.375088\pi$
$468$	3.28881	0.152025
$469$	−29.6591	−1.36953
$470$	0	0
$471$	−20.5054	−0.944841
$472$	4.86655	0.224001
$473$	−55.5908	−2.55607
$474$	5.41421	0.248683
$475$	0	0
$476$	26.2861	1.20482
$477$	−2.23921	−0.102527
$478$	6.38186	0.291899
$479$	−39.9123	−1.82364	−0.911820	−	0.410589i	$-0.865323\pi$
$479$	−39.9123	−1.82364	−0.911820	+	0.410589i	$0.865323\pi$
$480$	0	0
$481$	−31.5668	−1.43932
$482$	17.6379	0.803382
$483$	−7.80316	−0.355056
$484$	11.6687	0.530396
$485$	0	0
$486$	1.68554	0.0764578
$487$	−5.78169	−0.261993	−0.130997	−	0.991383i	$-0.541818\pi$
$487$	−5.78169	−0.261993	−0.130997	+	0.991383i	$0.541818\pi$
$488$	29.1332	1.31880
$489$	16.6527	0.753060
$490$	0	0
$491$	−17.4871	−0.789181	−0.394591	−	0.918857i	$-0.629114\pi$
$491$	−17.4871	−0.789181	−0.394591	+	0.918857i	$0.629114\pi$
$492$	−5.46512	−0.246387
$493$	−24.5410	−1.10527
$494$	−30.6815	−1.38043
$495$	0	0
$496$	36.2949	1.62969
$497$	−45.7937	−2.05413
$498$	−9.63159	−0.431602
$499$	11.6687	0.522364	0.261182	−	0.965290i	$-0.415888\pi$
$499$	11.6687	0.522364	0.261182	+	0.965290i	$0.415888\pi$
$500$	0	0
$501$	−5.46381	−0.244105
$502$	−18.2007	−0.812338
$503$	3.96440	0.176764	0.0883819	−	0.996087i	$-0.471830\pi$
$503$	3.96440	0.176764	0.0883819	+	0.996087i	$0.471830\pi$
$504$	9.37251	0.417485
$505$	0	0
$506$	13.6719	0.607789
$507$	2.29061	0.101730
$508$	−12.0804	−0.535980
$509$	−25.3225	−1.12240	−0.561199	−	0.827681i	$-0.689660\pi$
$509$	−25.3225	−1.12240	−0.561199	+	0.827681i	$0.689660\pi$
$510$	0	0
$511$	−20.9238	−0.925616
$512$	2.84232	0.125614
$513$	−4.65505	−0.205525
$514$	−7.67111	−0.338358
$515$	0	0
$516$	−9.37470	−0.412698
$517$	4.98737	0.219344
$518$	65.2849	2.86845
$519$	24.6727	1.08301
$520$	0	0
$521$	−15.5284	−0.680313	−0.340156	−	0.940369i	$-0.610480\pi$
$521$	−15.5284	−0.680313	−0.340156	+	0.940369i	$0.610480\pi$
$522$	6.35018	0.277940
$523$	−10.4599	−0.457379	−0.228690	−	0.973499i	$-0.573444\pi$
$523$	−10.4599	−0.457379	−0.228690	+	0.973499i	$0.573444\pi$
$524$	−1.71839	−0.0750681
$525$	0	0
$526$	−8.73547	−0.380885
$527$	47.5249	2.07022
$528$	−24.8109	−1.07975
$529$	−20.3550	−0.884998
$530$	0	0
$531$	−2.49126	−0.108112
$532$	18.7847	0.814421
$533$	−25.4089	−1.10058
$534$	−2.70846	−0.117206
$535$	0	0
$536$	−12.0755	−0.521583
$537$	5.27798	0.227762
$538$	19.0156	0.819822
$539$	79.8984	3.44147
$540$	0	0
$541$	−39.9025	−1.71554	−0.857770	−	0.514033i	$-0.828151\pi$
$541$	−39.9025	−1.71554	−0.857770	+	0.514033i	$0.828151\pi$
$542$	52.9770	2.27556
$543$	−11.0101	−0.472488
$544$	29.1712	1.25070
$545$	0	0
$546$	−31.6233	−1.35335
$547$	4.54952	0.194523	0.0972616	−	0.995259i	$-0.468992\pi$
$547$	4.54952	0.194523	0.0972616	+	0.995259i	$0.468992\pi$
$548$	−2.90782	−0.124216
$549$	−14.9137	−0.636503
$550$	0	0
$551$	−17.5376	−0.747127
$552$	−3.17701	−0.135222
$553$	−15.4117	−0.655370
$554$	27.4160	1.16479
$555$	0	0
$556$	−17.5036	−0.742319
$557$	−18.7945	−0.796346	−0.398173	−	0.917310i	$-0.630356\pi$
$557$	−18.7945	−0.796346	−0.398173	+	0.917310i	$0.630356\pi$
$558$	−12.2975	−0.520593
$559$	−43.5857	−1.84348
$560$	0	0
$561$	−32.4876	−1.37163
$562$	−3.66028	−0.154400
$563$	−15.0257	−0.633257	−0.316628	−	0.948550i	$-0.602551\pi$
$563$	−15.0257	−0.633257	−0.316628	+	0.948550i	$0.602551\pi$
$564$	0.841058	0.0354149
$565$	0	0
$566$	14.5134	0.610043
$567$	−4.79793	−0.201494
$568$	−18.6446	−0.782310
$569$	−4.97891	−0.208727	−0.104363	−	0.994539i	$-0.533281\pi$
$569$	−4.97891	−0.208727	−0.104363	+	0.994539i	$0.533281\pi$
$570$	0	0
$571$	−36.3235	−1.52009	−0.760046	−	0.649870i	$-0.774823\pi$
$571$	−36.3235	−1.52009	−0.760046	+	0.649870i	$0.774823\pi$
$572$	16.4025	0.685823
$573$	25.1201	1.04941
$574$	52.5494	2.19337
$575$	0	0
$576$	2.40120	0.100050
$577$	36.7158	1.52850	0.764249	−	0.644921i	$-0.223110\pi$
$577$	36.7158	1.52850	0.764249	+	0.644921i	$0.223110\pi$
$578$	42.8664	1.78301
$579$	−17.6591	−0.733889
$580$	0	0
$581$	27.4165	1.13743
$582$	13.9234	0.577145
$583$	−11.1678	−0.462522
$584$	−8.51900	−0.352519
$585$	0	0
$586$	33.3234	1.37658
$587$	14.7577	0.609117	0.304559	−	0.952494i	$-0.401491\pi$
$587$	14.7577	0.609117	0.304559	+	0.952494i	$0.401491\pi$
$588$	13.4739	0.555654
$589$	33.9625	1.39940
$590$	0	0
$591$	21.5263	0.885475
$592$	40.1595	1.65055
$593$	−32.3786	−1.32963	−0.664815	−	0.747008i	$-0.731490\pi$
$593$	−32.3786	−1.32963	−0.664815	+	0.747008i	$0.731490\pi$
$594$	8.40643	0.344920
$595$	0	0
$596$	−4.00969	−0.164243
$597$	3.98072	0.162920
$598$	10.7194	0.438348
$599$	−31.3206	−1.27972	−0.639862	−	0.768490i	$-0.721008\pi$
$599$	−31.3206	−1.27972	−0.639862	+	0.768490i	$0.721008\pi$
$600$	0	0
$601$	15.2791	0.623246	0.311623	−	0.950206i	$-0.399127\pi$
$601$	15.2791	0.623246	0.311623	+	0.950206i	$0.399127\pi$
$602$	90.1416	3.67390
$603$	6.18165	0.251736
$604$	0.434426	0.0176766
$605$	0	0
$606$	14.1268	0.573863
$607$	−13.8748	−0.563160	−0.281580	−	0.959538i	$-0.590858\pi$
$607$	−13.8748	−0.563160	−0.281580	+	0.959538i	$0.590858\pi$
$608$	20.8464	0.845435
$609$	−18.0759	−0.732473
$610$	0	0
$611$	3.91032	0.158195
$612$	−5.47863	−0.221460
$613$	−5.67042	−0.229026	−0.114513	−	0.993422i	$-0.536531\pi$
$613$	−5.67042	−0.229026	−0.114513	+	0.993422i	$0.536531\pi$
$614$	−10.2331	−0.412972
$615$	0	0
$616$	46.7442	1.88338
$617$	26.7059	1.07514	0.537570	−	0.843219i	$-0.319343\pi$
$617$	26.7059	1.07514	0.537570	+	0.843219i	$0.319343\pi$
$618$	−23.8968	−0.961269
$619$	−27.6770	−1.11243	−0.556217	−	0.831037i	$-0.687747\pi$
$619$	−27.6770	−1.11243	−0.556217	+	0.831037i	$0.687747\pi$
$620$	0	0
$621$	1.62636	0.0652636
$622$	50.3439	2.01861
$623$	7.70967	0.308882
$624$	−19.4528	−0.778736
$625$	0	0
$626$	−4.94262	−0.197547
$627$	−23.2164	−0.927176
$628$	−17.2463	−0.688201
$629$	52.5853	2.09671
$630$	0	0
$631$	3.12386	0.124359	0.0621794	−	0.998065i	$-0.480195\pi$
$631$	3.12386	0.124359	0.0621794	+	0.998065i	$0.480195\pi$
$632$	−6.27476	−0.249596
$633$	−7.12845	−0.283330
$634$	33.0784	1.31371
$635$	0	0
$636$	−1.88331	−0.0746781
$637$	62.6440	2.48204
$638$	31.6707	1.25386
$639$	9.54447	0.377573
$640$	0	0
$641$	−18.4431	−0.728459	−0.364229	−	0.931309i	$-0.618668\pi$
$641$	−18.4431	−0.728459	−0.364229	+	0.931309i	$0.618668\pi$
$642$	−8.71691	−0.344029
$643$	2.57676	0.101618	0.0508088	−	0.998708i	$-0.483820\pi$
$643$	2.57676	0.101618	0.0508088	+	0.998708i	$0.483820\pi$
$644$	−6.56292	−0.258615
$645$	0	0
$646$	51.1105	2.01092
$647$	−28.1876	−1.10817	−0.554084	−	0.832461i	$-0.686931\pi$
$647$	−28.1876	−1.10817	−0.554084	+	0.832461i	$0.686931\pi$
$648$	−1.95345	−0.0767386
$649$	−12.4248	−0.487718
$650$	0	0
$651$	35.0050	1.37195
$652$	14.0059	0.548512
$653$	36.2208	1.41743	0.708715	−	0.705495i	$-0.249275\pi$
$653$	36.2208	1.41743	0.708715	+	0.705495i	$0.249275\pi$
$654$	−1.56576	−0.0612259
$655$	0	0
$656$	32.3254	1.26210
$657$	4.36101	0.170139
$658$	−8.08713	−0.315269
$659$	−15.3532	−0.598077	−0.299038	−	0.954241i	$-0.596666\pi$
$659$	−15.3532	−0.598077	−0.299038	+	0.954241i	$0.596666\pi$
$660$	0	0
$661$	37.4276	1.45577	0.727883	−	0.685701i	$-0.240504\pi$
$661$	37.4276	1.45577	0.727883	+	0.685701i	$0.240504\pi$
$662$	−0.102800	−0.00399542
$663$	−25.4717	−0.989240
$664$	11.1625	0.433187
$665$	0	0
$666$	−13.6069	−0.527256
$667$	6.12721	0.237247
$668$	−4.59538	−0.177801
$669$	12.7853	0.494308
$670$	0	0
$671$	−74.3804	−2.87142
$672$	21.4863	0.828853
$673$	−5.37091	−0.207033	−0.103517	−	0.994628i	$-0.533009\pi$
$673$	−5.37091	−0.207033	−0.103517	+	0.994628i	$0.533009\pi$
$674$	42.1250	1.62260
$675$	0	0
$676$	1.92654	0.0740977
$677$	−22.3546	−0.859157	−0.429578	−	0.903030i	$-0.641338\pi$
$677$	−22.3546	−0.859157	−0.429578	+	0.903030i	$0.641338\pi$
$678$	6.17196	0.237033
$679$	−39.6333	−1.52099
$680$	0	0
$681$	−15.7516	−0.603602
$682$	−61.3320	−2.34852
$683$	−1.43893	−0.0550592	−0.0275296	−	0.999621i	$-0.508764\pi$
$683$	−1.43893	−0.0550592	−0.0275296	+	0.999621i	$0.508764\pi$
$684$	−3.91517	−0.149700
$685$	0	0
$686$	−72.9472	−2.78514
$687$	−5.69998	−0.217468
$688$	55.4500	2.11401
$689$	−8.75604	−0.333579
$690$	0	0
$691$	−22.4432	−0.853780	−0.426890	−	0.904304i	$-0.640391\pi$
$691$	−22.4432	−0.853780	−0.426890	+	0.904304i	$0.640391\pi$
$692$	20.7511	0.788840
$693$	−23.9291	−0.908990
$694$	−31.2545	−1.18640
$695$	0	0
$696$	−7.35949	−0.278961
$697$	42.3272	1.60326
$698$	−45.3650	−1.71709
$699$	11.9949	0.453689
$700$	0	0
$701$	−26.2203	−0.990329	−0.495164	−	0.868799i	$-0.664892\pi$
$701$	−26.2203	−0.990329	−0.495164	+	0.868799i	$0.664892\pi$
$702$	6.59102	0.248762
$703$	37.5788	1.41731
$704$	11.9757	0.451350
$705$	0	0
$706$	−45.7121	−1.72040
$707$	−40.2123	−1.51234
$708$	−2.09530	−0.0787460
$709$	−5.85921	−0.220047	−0.110024	−	0.993929i	$-0.535093\pi$
$709$	−5.85921	−0.220047	−0.110024	+	0.993929i	$0.535093\pi$
$710$	0	0
$711$	3.21215	0.120465
$712$	3.13894	0.117637
$713$	−11.8657	−0.444373
$714$	52.6793	1.97147
$715$	0	0
$716$	4.43909	0.165897
$717$	3.78623	0.141399
$718$	−0.721248	−0.0269167
$719$	−13.3431	−0.497613	−0.248807	−	0.968553i	$-0.580038\pi$
$719$	−13.3431	−0.497613	−0.248807	+	0.968553i	$0.580038\pi$
$720$	0	0
$721$	68.0226	2.53329
$722$	4.49954	0.167455
$723$	10.4642	0.389168
$724$	−9.26012	−0.344150
$725$	0	0
$726$	23.3850	0.867898
$727$	−38.3618	−1.42276	−0.711380	−	0.702807i	$-0.751930\pi$
$727$	−38.3618	−1.42276	−0.711380	+	0.702807i	$0.751930\pi$
$728$	36.6495	1.35832
$729$	1.00000	0.0370370
$730$	0	0
$731$	72.6067	2.68546
$732$	−12.5433	−0.463615
$733$	39.4542	1.45728	0.728638	−	0.684899i	$-0.240154\pi$
$733$	39.4542	1.45728	0.728638	+	0.684899i	$0.240154\pi$
$734$	14.1880	0.523687
$735$	0	0
$736$	−7.28324	−0.268464
$737$	30.8302	1.13564
$738$	−10.9525	−0.403168
$739$	11.3133	0.416168	0.208084	−	0.978111i	$-0.433277\pi$
$739$	11.3133	0.416168	0.208084	+	0.978111i	$0.433277\pi$
$740$	0	0
$741$	−18.2027	−0.668694
$742$	18.1088	0.664795
$743$	22.4849	0.824891	0.412445	−	0.910982i	$-0.364675\pi$
$743$	22.4849	0.824891	0.412445	+	0.910982i	$0.364675\pi$
$744$	14.2520	0.522505
$745$	0	0
$746$	51.2690	1.87709
$747$	−5.71423	−0.209073
$748$	−27.3239	−0.999062
$749$	24.8129	0.906642
$750$	0	0
$751$	−38.8715	−1.41844	−0.709220	−	0.704987i	$-0.750953\pi$
$751$	−38.8715	−1.41844	−0.709220	+	0.704987i	$0.750953\pi$
$752$	−4.97474	−0.181410
$753$	−10.7981	−0.393506
$754$	24.8313	0.904301
$755$	0	0
$756$	−4.03534	−0.146764
$757$	36.6789	1.33312	0.666559	−	0.745453i	$-0.267767\pi$
$757$	36.6789	1.33312	0.666559	+	0.745453i	$0.267767\pi$
$758$	−24.8945	−0.904211
$759$	8.11126	0.294420
$760$	0	0
$761$	37.6063	1.36323	0.681614	−	0.731712i	$-0.261278\pi$
$761$	37.6063	1.36323	0.681614	+	0.731712i	$0.261278\pi$
$762$	−24.2100	−0.877035
$763$	4.45695	0.161353
$764$	21.1275	0.764365
$765$	0	0
$766$	3.10749	0.112278
$767$	−9.74163	−0.351750
$768$	17.1161	0.617624
$769$	23.4609	0.846023	0.423012	−	0.906124i	$-0.360973\pi$
$769$	23.4609	0.846023	0.423012	+	0.906124i	$0.360973\pi$
$770$	0	0
$771$	−4.55112	−0.163904
$772$	−14.8524	−0.534549
$773$	−31.0435	−1.11656	−0.558279	−	0.829653i	$-0.688538\pi$
$773$	−31.0435	−1.11656	−0.558279	+	0.829653i	$0.688538\pi$
$774$	−18.7876	−0.675306
$775$	0	0
$776$	−16.1365	−0.579265
$777$	38.7322	1.38951
$778$	9.49933	0.340568
$779$	30.2481	1.08375
$780$	0	0
$781$	47.6018	1.70333
$782$	−17.8568	−0.638556
$783$	3.76744	0.134637
$784$	−79.6961	−2.84629
$785$	0	0
$786$	−3.44378	−0.122835
$787$	−25.9324	−0.924391	−0.462195	−	0.886778i	$-0.652938\pi$
$787$	−25.9324	−0.924391	−0.462195	+	0.886778i	$0.652938\pi$
$788$	18.1049	0.644961
$789$	−5.18258	−0.184505
$790$	0	0
$791$	−17.5686	−0.624667
$792$	−9.74256	−0.346187
$793$	−58.3175	−2.07092
$794$	−3.59911	−0.127728
$795$	0	0
$796$	3.34802	0.118667
$797$	−17.3135	−0.613274	−0.306637	−	0.951827i	$-0.599204\pi$
$797$	−17.3135	−0.613274	−0.306637	+	0.951827i	$0.599204\pi$
$798$	37.6460	1.33265
$799$	−6.51397	−0.230448
$800$	0	0
$801$	−1.60687	−0.0567761
$802$	20.3749	0.719461
$803$	21.7500	0.767539
$804$	5.19913	0.183359
$805$	0	0
$806$	−48.0871	−1.69379
$807$	11.2816	0.397131
$808$	−16.3722	−0.575971
$809$	−4.16229	−0.146338	−0.0731692	−	0.997320i	$-0.523311\pi$
$809$	−4.16229	−0.146338	−0.0731692	+	0.997320i	$0.523311\pi$
$810$	0	0
$811$	2.28938	0.0803909	0.0401954	−	0.999192i	$-0.487202\pi$
$811$	2.28938	0.0803909	0.0401954	+	0.999192i	$0.487202\pi$
$812$	−15.2029	−0.533517
$813$	31.4302	1.10231
$814$	−67.8625	−2.37858
$815$	0	0
$816$	32.4053	1.13441
$817$	51.8866	1.81528
$818$	−40.0300	−1.39962
$819$	−18.7615	−0.655578
$820$	0	0
$821$	−16.0108	−0.558780	−0.279390	−	0.960178i	$-0.590132\pi$
$821$	−16.0108	−0.558780	−0.279390	+	0.960178i	$0.590132\pi$
$822$	−5.82750	−0.203257
$823$	−21.2463	−0.740598	−0.370299	−	0.928913i	$-0.620745\pi$
$823$	−21.2463	−0.740598	−0.370299	+	0.928913i	$0.620745\pi$
$824$	27.6950	0.964799
$825$	0	0
$826$	20.1471	0.701009
$827$	−45.9564	−1.59806	−0.799030	−	0.601292i	$-0.794653\pi$
$827$	−45.9564	−1.59806	−0.799030	+	0.601292i	$0.794653\pi$
$828$	1.36786	0.0475365
$829$	14.4412	0.501564	0.250782	−	0.968044i	$-0.419312\pi$
$829$	14.4412	0.501564	0.250782	+	0.968044i	$0.419312\pi$
$830$	0	0
$831$	16.2654	0.564239
$832$	9.38945	0.325521
$833$	−104.355	−3.61568
$834$	−35.0786	−1.21467
$835$	0	0
$836$	−19.5264	−0.675334
$837$	−7.29585	−0.252181
$838$	−27.5609	−0.952076
$839$	−31.8243	−1.09870	−0.549348	−	0.835594i	$-0.685124\pi$
$839$	−31.8243	−1.09870	−0.549348	+	0.835594i	$0.685124\pi$
$840$	0	0
$841$	−14.8064	−0.510566
$842$	23.6037	0.813437
$843$	−2.17157	−0.0747929
$844$	−5.99544	−0.206372
$845$	0	0
$846$	1.68554	0.0579502
$847$	−66.5658	−2.28723
$848$	11.1395	0.382532
$849$	8.61050	0.295512
$850$	0	0
$851$	−13.1291	−0.450060
$852$	8.02745	0.275016
$853$	1.12254	0.0384351	0.0192176	−	0.999815i	$-0.493882\pi$
$853$	1.12254	0.0384351	0.0192176	+	0.999815i	$0.493882\pi$
$854$	120.609	4.12717
$855$	0	0
$856$	10.1024	0.345293
$857$	−14.2433	−0.486543	−0.243271	−	0.969958i	$-0.578221\pi$
$857$	−14.2433	−0.486543	−0.243271	+	0.969958i	$0.578221\pi$
$858$	32.8718	1.12223
$859$	57.3669	1.95733	0.978666	−	0.205458i	$-0.0658683\pi$
$859$	57.3669	1.95733	0.978666	+	0.205458i	$0.0658683\pi$
$860$	0	0
$861$	31.1765	1.06249
$862$	−45.7200	−1.55723
$863$	48.0449	1.63547	0.817734	−	0.575597i	$-0.195230\pi$
$863$	48.0449	1.63547	0.817734	+	0.575597i	$0.195230\pi$
$864$	−4.47824	−0.152353
$865$	0	0
$866$	60.4827	2.05529
$867$	25.4318	0.863710
$868$	29.4412	0.999300
$869$	16.0202	0.543447
$870$	0	0
$871$	24.1722	0.819045
$872$	1.81462	0.0614508
$873$	8.26050	0.279576
$874$	−12.7609	−0.431644
$875$	0	0
$876$	3.66786	0.123926
$877$	−41.0548	−1.38632	−0.693161	−	0.720783i	$-0.743782\pi$
$877$	−41.0548	−1.38632	−0.693161	+	0.720783i	$0.743782\pi$
$878$	18.2436	0.615692
$879$	19.7701	0.666830
$880$	0	0
$881$	−44.1361	−1.48698	−0.743492	−	0.668745i	$-0.766832\pi$
$881$	−44.1361	−1.48698	−0.743492	+	0.668745i	$0.766832\pi$
$882$	27.0027	0.909228
$883$	−48.2632	−1.62419	−0.812093	−	0.583528i	$-0.801672\pi$
$883$	−48.2632	−1.62419	−0.812093	+	0.583528i	$0.801672\pi$
$884$	−21.4232	−0.720540
$885$	0	0
$886$	−16.5051	−0.554500
$887$	−45.5980	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$887$	−45.5980	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$888$	15.7696	0.529192
$889$	68.9142	2.31131
$890$	0	0
$891$	4.98737	0.167083
$892$	10.7532	0.360043
$893$	−4.65505	−0.155775
$894$	−8.03573	−0.268755
$895$	0	0
$896$	−62.3914	−2.08435
$897$	6.35959	0.212341
$898$	−8.09751	−0.270217
$899$	−27.4866	−0.916731
$900$	0	0
$901$	14.5862	0.485936
$902$	−54.6242	−1.81879
$903$	53.4792	1.77968
$904$	−7.15294	−0.237903
$905$	0	0
$906$	0.870623	0.0289245
$907$	45.2934	1.50394	0.751972	−	0.659195i	$-0.229103\pi$
$907$	45.2934	1.50394	0.751972	+	0.659195i	$0.229103\pi$
$908$	−13.2480	−0.439650
$909$	8.38117	0.277986
$910$	0	0
$911$	−36.7491	−1.21755	−0.608775	−	0.793343i	$-0.708339\pi$
$911$	−36.7491	−1.21755	−0.608775	+	0.793343i	$0.708339\pi$
$912$	23.1576	0.766826
$913$	−28.4990	−0.943179
$914$	5.02109	0.166083
$915$	0	0
$916$	−4.79402	−0.158399
$917$	9.80278	0.323716
$918$	−10.9796	−0.362380
$919$	−7.84168	−0.258673	−0.129337	−	0.991601i	$-0.541285\pi$
$919$	−7.84168	−0.258673	−0.129337	+	0.991601i	$0.541285\pi$
$920$	0	0
$921$	−6.07107	−0.200048
$922$	45.9094	1.51195
$923$	37.3219	1.22847
$924$	−20.1257	−0.662088
$925$	0	0
$926$	−7.04289	−0.231444
$927$	−14.1775	−0.465649
$928$	−16.8715	−0.553834
$929$	−49.0920	−1.61066	−0.805329	−	0.592829i	$-0.798011\pi$
$929$	−49.0920	−1.61066	−0.805329	+	0.592829i	$0.798011\pi$
$930$	0	0
$931$	−74.5746	−2.44408
$932$	10.0884	0.330457
$933$	29.8681	0.977837
$934$	27.8599	0.911603
$935$	0	0
$936$	−7.63861	−0.249676
$937$	−41.1403	−1.34399	−0.671997	−	0.740554i	$-0.734563\pi$
$937$	−41.1403	−1.34399	−0.671997	+	0.740554i	$0.734563\pi$
$938$	−49.9918	−1.63229
$939$	−2.93236	−0.0956939
$940$	0	0
$941$	17.0325	0.555245	0.277622	−	0.960690i	$-0.410454\pi$
$941$	17.0325	0.555245	0.277622	+	0.960690i	$0.410454\pi$
$942$	−34.5628	−1.12612
$943$	−10.5679	−0.344140
$944$	12.3934	0.403370
$945$	0	0
$946$	−93.7007	−3.04647
$947$	0.0239474	0.000778184 0	0.000389092	−	1.00000i	$-0.499876\pi$
$947$	0.0239474	0.000778184 0	0.000389092		1.00000i	$0.499876\pi$
$948$	2.70160	0.0877440
$949$	17.0529	0.553562
$950$	0	0
$951$	19.6247	0.636376
$952$	−61.0522	−1.97871
$953$	−23.8427	−0.772340	−0.386170	−	0.922428i	$-0.626202\pi$
$953$	−23.8427	−0.772340	−0.386170	+	0.922428i	$0.626202\pi$
$954$	−3.77429	−0.122197
$955$	0	0
$956$	3.18444	0.102992
$957$	18.7896	0.607382
$958$	−67.2740	−2.17352
$959$	16.5881	0.535657
$960$	0	0
$961$	22.2294	0.717076
$962$	−53.2073	−1.71547
$963$	−5.17157	−0.166652
$964$	8.80100	0.283461
$965$	0	0
$966$	−13.1526	−0.423177
$967$	17.2054	0.553290	0.276645	−	0.960972i	$-0.410777\pi$
$967$	17.2054	0.553290	0.276645	+	0.960972i	$0.410777\pi$
$968$	−27.1018	−0.871086
$969$	30.3229	0.974111
$970$	0	0
$971$	17.9206	0.575100	0.287550	−	0.957766i	$-0.407159\pi$
$971$	17.9206	0.575100	0.287550	+	0.957766i	$0.407159\pi$
$972$	0.841058	0.0269770
$973$	99.8519	3.20110
$974$	−9.74530	−0.312259
$975$	0	0
$976$	74.1920	2.37483
$977$	−3.19742	−0.102295	−0.0511473	−	0.998691i	$-0.516288\pi$
$977$	−3.19742	−0.102295	−0.0511473	+	0.998691i	$0.516288\pi$
$978$	28.0688	0.897542
$979$	−8.01407	−0.256131
$980$	0	0
$981$	−0.928932	−0.0296585
$982$	−29.4753	−0.940593
$983$	−41.4880	−1.32326	−0.661631	−	0.749830i	$-0.730135\pi$
$983$	−41.4880	−1.32326	−0.661631	+	0.749830i	$0.730135\pi$
$984$	12.6933	0.404649
$985$	0	0
$986$	−41.3649	−1.31733
$987$	−4.79793	−0.152720
$988$	−15.3096	−0.487062
$989$	−18.1279	−0.576434
$990$	0	0
$991$	−7.51931	−0.238859	−0.119429	−	0.992843i	$-0.538106\pi$
$991$	−7.51931	−0.238859	−0.119429	+	0.992843i	$0.538106\pi$
$992$	32.6726	1.03736
$993$	−0.0609889	−0.00193543
$994$	−77.1873	−2.44823
$995$	0	0
$996$	−4.80600	−0.152284
$997$	26.0695	0.825630	0.412815	−	0.910815i	$-0.364546\pi$
$997$	26.0695	0.825630	0.412815	+	0.910815i	$0.364546\pi$
$998$	19.6682	0.622585
$999$	−8.07269	−0.255409

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.k.1.1	✓	4	5.4	even	2
2115.2.a.o.1.4		4	15.14	odd	2
3525.2.a.t.1.4		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)

\(f(q)\)	\(=\)	\(q+1.68554 q^{2} +1.00000 q^{3} +0.841058 q^{4} +1.68554 q^{6} -4.79793 q^{7} -1.95345 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.68554 q^{2} +1.00000 q^{3} +0.841058 q^{4} +1.68554 q^{6} -4.79793 q^{7} -1.95345 q^{8} +1.00000 q^{9} +4.98737 q^{11} +0.841058 q^{12} +3.91032 q^{13} -8.08713 q^{14} -4.97474 q^{16} -6.51397 q^{17} +1.68554 q^{18} -4.65505 q^{19} -4.79793 q^{21} +8.40643 q^{22} +1.62636 q^{23} -1.95345 q^{24} +6.59102 q^{26} +1.00000 q^{27} -4.03534 q^{28} +3.76744 q^{29} -7.29585 q^{31} -4.47824 q^{32} +4.98737 q^{33} -10.9796 q^{34} +0.841058 q^{36} -8.07269 q^{37} -7.84629 q^{38} +3.91032 q^{39} -6.49791 q^{41} -8.08713 q^{42} -11.1463 q^{43} +4.19467 q^{44} +2.74130 q^{46} +1.00000 q^{47} -4.97474 q^{48} +16.0202 q^{49} -6.51397 q^{51} +3.28881 q^{52} -2.23921 q^{53} +1.68554 q^{54} +9.37251 q^{56} -4.65505 q^{57} +6.35018 q^{58} -2.49126 q^{59} -14.9137 q^{61} -12.2975 q^{62} -4.79793 q^{63} +2.40120 q^{64} +8.40643 q^{66} +6.18165 q^{67} -5.47863 q^{68} +1.62636 q^{69} +9.54447 q^{71} -1.95345 q^{72} +4.36101 q^{73} -13.6069 q^{74} -3.91517 q^{76} -23.9291 q^{77} +6.59102 q^{78} +3.21215 q^{79} +1.00000 q^{81} -10.9525 q^{82} -5.71423 q^{83} -4.03534 q^{84} -18.7876 q^{86} +3.76744 q^{87} -9.74256 q^{88} -1.60687 q^{89} -18.7615 q^{91} +1.36786 q^{92} -7.29585 q^{93} +1.68554 q^{94} -4.47824 q^{96} +8.26050 q^{97} +27.0027 q^{98} +4.98737 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more