LMFDB - Embedded newform 3525.2.a.t.1.3

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			1.3
Root			$-1.74912$ 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)

$f(q)$	$=$	$q-0.665096 q^{2} +1.00000 q^{3} -1.55765 q^{4} -0.665096 q^{6} -0.526374 q^{7} +2.36618 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.665096 q^{2} +1.00000 q^{3} -1.55765 q^{4} -0.665096 q^{6} -0.526374 q^{7} +2.36618 q^{8} +1.00000 q^{9} +1.72922 q^{11} -1.55765 q^{12} -2.28216 q^{13} +0.350089 q^{14} +1.54156 q^{16} +1.49352 q^{17} -0.665096 q^{18} -3.68971 q^{19} -0.526374 q^{21} -1.15010 q^{22} -8.30205 q^{23} +2.36618 q^{24} +1.51785 q^{26} +1.00000 q^{27} +0.819905 q^{28} +0.881176 q^{29} +10.6881 q^{31} -5.75764 q^{32} +1.72922 q^{33} -0.993336 q^{34} -1.55765 q^{36} -6.72569 q^{37} +2.45401 q^{38} -2.28216 q^{39} +7.21450 q^{41} +0.350089 q^{42} -10.2869 q^{43} -2.69352 q^{44} +5.52166 q^{46} +1.00000 q^{47} +1.54156 q^{48} -6.72293 q^{49} +1.49352 q^{51} +3.55479 q^{52} +8.69284 q^{53} -0.665096 q^{54} -1.24549 q^{56} -3.68971 q^{57} -0.586066 q^{58} -2.59716 q^{59} -11.1680 q^{61} -7.10863 q^{62} -0.526374 q^{63} +0.746264 q^{64} -1.15010 q^{66} +0.466962 q^{67} -2.32638 q^{68} -8.30205 q^{69} +0.151675 q^{71} +2.36618 q^{72} +11.0313 q^{73} +4.47323 q^{74} +5.74726 q^{76} -0.910217 q^{77} +1.51785 q^{78} -3.88784 q^{79} +1.00000 q^{81} -4.79834 q^{82} -12.3267 q^{83} +0.819905 q^{84} +6.84175 q^{86} +0.881176 q^{87} +4.09164 q^{88} +16.4732 q^{89} +1.20127 q^{91} +12.9317 q^{92} +10.6881 q^{93} -0.665096 q^{94} -5.75764 q^{96} -4.86822 q^{97} +4.47139 q^{98} +1.72922 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.665096	−0.470294	−0.235147	−	0.971960i	$-0.575557\pi$
$2$	−0.665096	−0.470294	−0.235147	+	0.971960i	$0.575557\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.55765	−0.778824
$5$	0	0
$5$	0	0
$6$	−0.665096	−0.271524
$7$	−0.526374	−0.198951	−0.0994754	−	0.995040i	$-0.531716\pi$
$7$	−0.526374	−0.198951	−0.0994754	+	0.995040i	$0.531716\pi$
$8$	2.36618	0.836570
$9$	1.00000	0.333333
$10$	0	0
$11$	1.72922	0.521379	0.260690	−	0.965423i	$-0.416050\pi$
$11$	1.72922	0.521379	0.260690	+	0.965423i	$0.416050\pi$
$12$	−1.55765	−0.449654
$13$	−2.28216	−0.632956	−0.316478	−	0.948600i	$-0.602500\pi$
$13$	−2.28216	−0.632956	−0.316478	+	0.948600i	$0.602500\pi$
$14$	0.350089	0.0935653
$15$	0	0
$16$	1.54156	0.385390
$17$	1.49352	0.362233	0.181116	−	0.983462i	$-0.442029\pi$
$17$	1.49352	0.362233	0.181116	+	0.983462i	$0.442029\pi$
$18$	−0.665096	−0.156765
$19$	−3.68971	−0.846476	−0.423238	−	0.906018i	$-0.639107\pi$
$19$	−3.68971	−0.846476	−0.423238	+	0.906018i	$0.639107\pi$
$20$	0	0
$21$	−0.526374	−0.114864
$22$	−1.15010	−0.245202
$23$	−8.30205	−1.73110	−0.865549	−	0.500825i	$-0.833030\pi$
$23$	−8.30205	−1.73110	−0.865549	+	0.500825i	$0.833030\pi$
$24$	2.36618	0.482994
$25$	0	0
$26$	1.51785	0.297675
$27$	1.00000	0.192450
$28$	0.819905	0.154948
$29$	0.881176	0.163630	0.0818151	−	0.996648i	$-0.473928\pi$
$29$	0.881176	0.163630	0.0818151	+	0.996648i	$0.473928\pi$
$30$	0	0
$31$	10.6881	1.91964	0.959822	−	0.280609i	$-0.0905364\pi$
$31$	10.6881	1.91964	0.959822	+	0.280609i	$0.0905364\pi$
$32$	−5.75764	−1.01782
$33$	1.72922	0.301019
$34$	−0.993336	−0.170356
$35$	0	0
$36$	−1.55765	−0.259608
$37$	−6.72569	−1.10570	−0.552848	−	0.833282i	$-0.686459\pi$
$37$	−6.72569	−1.10570	−0.552848	+	0.833282i	$0.686459\pi$
$38$	2.45401	0.398093
$39$	−2.28216	−0.365437
$40$	0	0
$41$	7.21450	1.12672	0.563358	−	0.826213i	$-0.309509\pi$
$41$	7.21450	1.12672	0.563358	+	0.826213i	$0.309509\pi$
$42$	0.350089	0.0540200
$43$	−10.2869	−1.56873	−0.784366	−	0.620298i	$-0.787012\pi$
$43$	−10.2869	−1.56873	−0.784366	+	0.620298i	$0.787012\pi$
$44$	−2.69352	−0.406063
$45$	0	0
$46$	5.52166	0.814125
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	1.54156	0.222505
$49$	−6.72293	−0.960419
$50$	0	0
$51$	1.49352	0.209135
$52$	3.55479	0.492961
$53$	8.69284	1.19405	0.597027	−	0.802221i	$-0.296349\pi$
$53$	8.69284	1.19405	0.597027	+	0.802221i	$0.296349\pi$
$54$	−0.665096	−0.0905081
$55$	0	0
$56$	−1.24549	−0.166436
$57$	−3.68971	−0.488713
$58$	−0.586066	−0.0769543
$59$	−2.59716	−0.338122	−0.169061	−	0.985606i	$-0.554073\pi$
$59$	−2.59716	−0.338122	−0.169061	+	0.985606i	$0.554073\pi$
$60$	0	0
$61$	−11.1680	−1.42992	−0.714961	−	0.699165i	$-0.753555\pi$
$61$	−11.1680	−1.42992	−0.714961	+	0.699165i	$0.753555\pi$
$62$	−7.10863	−0.902797
$63$	−0.526374	−0.0663169
$64$	0.746264	0.0932829
$65$	0	0
$66$	−1.15010	−0.141567
$67$	0.466962	0.0570485	0.0285242	−	0.999593i	$-0.490919\pi$
$67$	0.466962	0.0570485	0.0285242	+	0.999593i	$0.490919\pi$
$68$	−2.32638	−0.282115
$69$	−8.30205	−0.999450
$70$	0	0
$71$	0.151675	0.0180006	0.00900028	−	0.999959i	$-0.497135\pi$
$71$	0.151675	0.0180006	0.00900028	+	0.999959i	$0.497135\pi$
$72$	2.36618	0.278857
$73$	11.0313	1.29111	0.645556	−	0.763713i	$-0.276626\pi$
$73$	11.0313	1.29111	0.645556	+	0.763713i	$0.276626\pi$
$74$	4.47323	0.520002
$75$	0	0
$76$	5.74726	0.659256
$77$	−0.910217	−0.103729
$78$	1.51785	0.171863
$79$	−3.88784	−0.437416	−0.218708	−	0.975790i	$-0.570184\pi$
$79$	−3.88784	−0.437416	−0.218708	+	0.975790i	$0.570184\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−4.79834	−0.529888
$83$	−12.3267	−1.35303	−0.676513	−	0.736430i	$-0.736510\pi$
$83$	−12.3267	−1.35303	−0.676513	+	0.736430i	$0.736510\pi$
$84$	0.819905	0.0894590
$85$	0	0
$86$	6.84175	0.737765
$87$	0.881176	0.0944719
$88$	4.09164	0.436170
$89$	16.4732	1.74616	0.873079	−	0.487578i	$-0.162120\pi$
$89$	16.4732	1.74616	0.873079	+	0.487578i	$0.162120\pi$
$90$	0	0
$91$	1.20127	0.125927
$92$	12.9317	1.34822
$93$	10.6881	1.10831
$94$	−0.665096	−0.0685994
$95$	0	0
$96$	−5.75764	−0.587637
$97$	−4.86822	−0.494293	−0.247147	−	0.968978i	$-0.579493\pi$
$97$	−4.86822	−0.494293	−0.247147	+	0.968978i	$0.579493\pi$
$98$	4.47139	0.451679
$99$	1.72922	0.173793
$100$	0	0
$101$	−7.69166	−0.765348	−0.382674	−	0.923883i	$-0.624997\pi$
$101$	−7.69166	−0.765348	−0.382674	+	0.923883i	$0.624997\pi$
$102$	−0.993336	−0.0983549
$103$	18.9620	1.86839	0.934193	−	0.356769i	$-0.116122\pi$
$103$	18.9620	1.86839	0.934193	+	0.356769i	$0.116122\pi$
$104$	−5.39998	−0.529512
$105$	0	0
$106$	−5.78157	−0.561556
$107$	−10.8284	−1.04682	−0.523412	−	0.852080i	$-0.675341\pi$
$107$	−10.8284	−1.04682	−0.523412	+	0.852080i	$0.675341\pi$
$108$	−1.55765	−0.149885
$109$	−15.0711	−1.44355	−0.721773	−	0.692130i	$-0.756673\pi$
$109$	−15.0711	−1.44355	−0.721773	+	0.692130i	$0.756673\pi$
$110$	0	0
$111$	−6.72569	−0.638374
$112$	−0.811437	−0.0766736
$113$	−11.1220	−1.04627	−0.523133	−	0.852251i	$-0.675237\pi$
$113$	−11.1220	−1.04627	−0.523133	+	0.852251i	$0.675237\pi$
$114$	2.45401	0.229839
$115$	0	0
$116$	−1.37256	−0.127439
$117$	−2.28216	−0.210985
$118$	1.72736	0.159017
$119$	−0.786152	−0.0720664
$120$	0	0
$121$	−8.00980	−0.728163
$122$	7.42782	0.672483
$123$	7.21450	0.650510
$124$	−16.6483	−1.49506
$125$	0	0
$126$	0.350089	0.0311884
$127$	−2.93392	−0.260344	−0.130172	−	0.991491i	$-0.541553\pi$
$127$	−2.93392	−0.260344	−0.130172	+	0.991491i	$0.541553\pi$
$128$	11.0189	0.973946
$129$	−10.2869	−0.905708
$130$	0	0
$131$	−3.91598	−0.342141	−0.171070	−	0.985259i	$-0.554723\pi$
$131$	−3.91598	−0.342141	−0.171070	+	0.985259i	$0.554723\pi$
$132$	−2.69352	−0.234440
$133$	1.94217	0.168407
$134$	−0.310575	−0.0268296
$135$	0	0
$136$	3.53394	0.303033
$137$	−2.50176	−0.213740	−0.106870	−	0.994273i	$-0.534083\pi$
$137$	−2.50176	−0.213740	−0.106870	+	0.994273i	$0.534083\pi$
$138$	5.52166	0.470035
$139$	−7.61511	−0.645905	−0.322953	−	0.946415i	$-0.604675\pi$
$139$	−7.61511	−0.645905	−0.322953	+	0.946415i	$0.604675\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	−0.100879	−0.00846556
$143$	−3.94635	−0.330010
$144$	1.54156	0.128463
$145$	0	0
$146$	−7.33686	−0.607203
$147$	−6.72293	−0.554498
$148$	10.4763	0.861143
$149$	−1.88118	−0.154112	−0.0770560	−	0.997027i	$-0.524552\pi$
$149$	−1.88118	−0.154112	−0.0770560	+	0.997027i	$0.524552\pi$
$150$	0	0
$151$	7.13872	0.580941	0.290470	−	0.956884i	$-0.406188\pi$
$151$	7.13872	0.580941	0.290470	+	0.956884i	$0.406188\pi$
$152$	−8.73050	−0.708137
$153$	1.49352	0.120744
$154$	0.605382	0.0487830
$155$	0	0
$156$	3.55479	0.284611
$157$	19.2082	1.53298	0.766491	−	0.642255i	$-0.222001\pi$
$157$	19.2082	1.53298	0.766491	+	0.642255i	$0.222001\pi$
$158$	2.58579	0.205714
$159$	8.69284	0.689387
$160$	0	0
$161$	4.36999	0.344403
$162$	−0.665096	−0.0522549
$163$	−22.0859	−1.72990	−0.864949	−	0.501860i	$-0.832649\pi$
$163$	−22.0859	−1.72990	−0.864949	+	0.501860i	$0.832649\pi$
$164$	−11.2376	−0.877513
$165$	0	0
$166$	8.19841	0.636320
$167$	−13.8334	−1.07046	−0.535231	−	0.844706i	$-0.679775\pi$
$167$	−13.8334	−1.07046	−0.535231	+	0.844706i	$0.679775\pi$
$168$	−1.24549	−0.0960920
$169$	−7.79177	−0.599367
$170$	0	0
$171$	−3.68971	−0.282159
$172$	16.0233	1.22177
$173$	−13.6655	−1.03897	−0.519483	−	0.854481i	$-0.673876\pi$
$173$	−13.6655	−1.03897	−0.519483	+	0.854481i	$0.673876\pi$
$174$	−0.586066	−0.0444296
$175$	0	0
$176$	2.66570	0.200934
$177$	−2.59716	−0.195215
$178$	−10.9563	−0.821208
$179$	−8.06255	−0.602623	−0.301311	−	0.953526i	$-0.597424\pi$
$179$	−8.06255	−0.602623	−0.301311	+	0.953526i	$0.597424\pi$
$180$	0	0
$181$	0.361465	0.0268675	0.0134337	−	0.999910i	$-0.495724\pi$
$181$	0.361465	0.0268675	0.0134337	+	0.999910i	$0.495724\pi$
$182$	−0.798958	−0.0592227
$183$	−11.1680	−0.825565
$184$	−19.6441	−1.44818
$185$	0	0
$186$	−7.10863	−0.521230
$187$	2.58263	0.188861
$188$	−1.55765	−0.113603
$189$	−0.526374	−0.0382881
$190$	0	0
$191$	−25.9456	−1.87735	−0.938677	−	0.344797i	$-0.887948\pi$
$191$	−25.9456	−1.87735	−0.938677	+	0.344797i	$0.887948\pi$
$192$	0.746264	0.0538569
$193$	11.7542	0.846086	0.423043	−	0.906110i	$-0.360962\pi$
$193$	11.7542	0.846086	0.423043	+	0.906110i	$0.360962\pi$
$194$	3.23783	0.232463
$195$	0	0
$196$	10.4720	0.747997
$197$	−15.9523	−1.13656	−0.568278	−	0.822836i	$-0.692390\pi$
$197$	−15.9523	−1.13656	−0.568278	+	0.822836i	$0.692390\pi$
$198$	−1.15010	−0.0817339
$199$	14.5409	1.03078	0.515388	−	0.856957i	$-0.327648\pi$
$199$	14.5409	1.03078	0.515388	+	0.856957i	$0.327648\pi$
$200$	0	0
$201$	0.466962	0.0329370
$202$	5.11569	0.359939
$203$	−0.463828	−0.0325544
$204$	−2.32638	−0.162879
$205$	0	0
$206$	−12.6116	−0.878690
$207$	−8.30205	−0.577033
$208$	−3.51808	−0.243935
$209$	−6.38031	−0.441335
$210$	0	0
$211$	−10.9124	−0.751244	−0.375622	−	0.926773i	$-0.622571\pi$
$211$	−10.9124	−0.751244	−0.375622	+	0.926773i	$0.622571\pi$
$212$	−13.5404	−0.929957
$213$	0.151675	0.0103926
$214$	7.20194	0.492315
$215$	0	0
$216$	2.36618	0.160998
$217$	−5.62595	−0.381915
$218$	10.0237	0.678891
$219$	11.0313	0.745424
$220$	0	0
$221$	−3.40845	−0.229277
$222$	4.47323	0.300224
$223$	5.25559	0.351941	0.175970	−	0.984395i	$-0.443694\pi$
$223$	5.25559	0.351941	0.175970	+	0.984395i	$0.443694\pi$
$224$	3.03067	0.202495
$225$	0	0
$226$	7.39717	0.492052
$227$	−25.8723	−1.71720	−0.858601	−	0.512644i	$-0.828666\pi$
$227$	−25.8723	−1.71720	−0.858601	+	0.512644i	$0.828666\pi$
$228$	5.74726	0.380622
$229$	3.74088	0.247204	0.123602	−	0.992332i	$-0.460555\pi$
$229$	3.74088	0.247204	0.123602	+	0.992332i	$0.460555\pi$
$230$	0	0
$231$	−0.910217	−0.0598879
$232$	2.08502	0.136888
$233$	−17.2645	−1.13103	−0.565517	−	0.824737i	$-0.691323\pi$
$233$	−17.2645	−1.13103	−0.565517	+	0.824737i	$0.691323\pi$
$234$	1.51785	0.0992251
$235$	0	0
$236$	4.04546	0.263337
$237$	−3.88784	−0.252542
$238$	0.522867	0.0338924
$239$	−1.90892	−0.123478	−0.0617388	−	0.998092i	$-0.519665\pi$
$239$	−1.90892	−0.123478	−0.0617388	+	0.998092i	$0.519665\pi$
$240$	0	0
$241$	14.4022	0.927725	0.463862	−	0.885907i	$-0.346463\pi$
$241$	14.4022	0.927725	0.463862	+	0.885907i	$0.346463\pi$
$242$	5.32728	0.342451
$243$	1.00000	0.0641500
$244$	17.3959	1.11366
$245$	0	0
$246$	−4.79834	−0.305931
$247$	8.42048	0.535782
$248$	25.2900	1.60592
$249$	−12.3267	−0.781170
$250$	0	0
$251$	−25.2384	−1.59304	−0.796518	−	0.604615i	$-0.793327\pi$
$251$	−25.2384	−1.59304	−0.796518	+	0.604615i	$0.793327\pi$
$252$	0.819905	0.0516492
$253$	−14.3561	−0.902559
$254$	1.95134	0.122438
$255$	0	0
$256$	−8.82118	−0.551324
$257$	18.6600	1.16398	0.581989	−	0.813197i	$-0.302275\pi$
$257$	18.6600	1.16398	0.581989	+	0.813197i	$0.302275\pi$
$258$	6.84175	0.425949
$259$	3.54023	0.219979
$260$	0	0
$261$	0.881176	0.0545434
$262$	2.60450	0.160907
$263$	−1.30245	−0.0803124	−0.0401562	−	0.999193i	$-0.512786\pi$
$263$	−1.30245	−0.0803124	−0.0401562	+	0.999193i	$0.512786\pi$
$264$	4.09164	0.251823
$265$	0	0
$266$	−1.29173	−0.0792008
$267$	16.4732	1.00815
$268$	−0.727362	−0.0444307
$269$	−22.7557	−1.38744	−0.693719	−	0.720246i	$-0.744029\pi$
$269$	−22.7557	−1.38744	−0.693719	+	0.720246i	$0.744029\pi$
$270$	0	0
$271$	20.2892	1.23248	0.616242	−	0.787557i	$-0.288655\pi$
$271$	20.2892	1.23248	0.616242	+	0.787557i	$0.288655\pi$
$272$	2.30236	0.139601
$273$	1.20127	0.0727040
$274$	1.66391	0.100521
$275$	0	0
$276$	12.9317	0.778395
$277$	−0.333326	−0.0200276	−0.0100138	−	0.999950i	$-0.503188\pi$
$277$	−0.333326	−0.0200276	−0.0100138	+	0.999950i	$0.503188\pi$
$278$	5.06478	0.303765
$279$	10.6881	0.639881
$280$	0	0
$281$	−7.82843	−0.467005	−0.233502	−	0.972356i	$-0.575019\pi$
$281$	−7.82843	−0.467005	−0.233502	+	0.972356i	$0.575019\pi$
$282$	−0.665096	−0.0396059
$283$	11.6890	0.694841	0.347420	−	0.937710i	$-0.387058\pi$
$283$	11.6890	0.694841	0.347420	+	0.937710i	$0.387058\pi$
$284$	−0.236257	−0.0140193
$285$	0	0
$286$	2.62470	0.155202
$287$	−3.79753	−0.224161
$288$	−5.75764	−0.339272
$289$	−14.7694	−0.868788
$290$	0	0
$291$	−4.86822	−0.285380
$292$	−17.1828	−1.00555
$293$	−5.64743	−0.329926	−0.164963	−	0.986300i	$-0.552751\pi$
$293$	−5.64743	−0.329926	−0.164963	+	0.986300i	$0.552751\pi$
$294$	4.47139	0.260777
$295$	0	0
$296$	−15.9142	−0.924993
$297$	1.72922	0.100340
$298$	1.25116	0.0724779
$299$	18.9466	1.09571
$300$	0	0
$301$	5.41474	0.312101
$302$	−4.74794	−0.273213
$303$	−7.69166	−0.441874
$304$	−5.68790	−0.326223
$305$	0	0
$306$	−0.993336	−0.0567853
$307$	8.07107	0.460640	0.230320	−	0.973115i	$-0.426023\pi$
$307$	8.07107	0.460640	0.230320	+	0.973115i	$0.426023\pi$
$308$	1.41780	0.0807865
$309$	18.9620	1.07871
$310$	0	0
$311$	9.61982	0.545490	0.272745	−	0.962086i	$-0.412068\pi$
$311$	9.61982	0.545490	0.272745	+	0.962086i	$0.412068\pi$
$312$	−5.39998	−0.305714
$313$	−19.5213	−1.10341	−0.551704	−	0.834040i	$-0.686022\pi$
$313$	−19.5213	−1.10341	−0.551704	+	0.834040i	$0.686022\pi$
$314$	−12.7753	−0.720952
$315$	0	0
$316$	6.05588	0.340670
$317$	−3.09881	−0.174047	−0.0870233	−	0.996206i	$-0.527735\pi$
$317$	−3.09881	−0.174047	−0.0870233	+	0.996206i	$0.527735\pi$
$318$	−5.78157	−0.324215
$319$	1.52375	0.0853134
$320$	0	0
$321$	−10.8284	−0.604384
$322$	−2.90646	−0.161971
$323$	−5.51066	−0.306621
$324$	−1.55765	−0.0865360
$325$	0	0
$326$	14.6892	0.813560
$327$	−15.0711	−0.833432
$328$	17.0708	0.942577
$329$	−0.526374	−0.0290200
$330$	0	0
$331$	2.70960	0.148933	0.0744666	−	0.997224i	$-0.476275\pi$
$331$	2.70960	0.148933	0.0744666	+	0.997224i	$0.476275\pi$
$332$	19.2006	1.05377
$333$	−6.72569	−0.368566
$334$	9.20055	0.503432
$335$	0	0
$336$	−0.811437	−0.0442675
$337$	−14.4655	−0.787985	−0.393992	−	0.919114i	$-0.628906\pi$
$337$	−14.4655	−0.787985	−0.393992	+	0.919114i	$0.628906\pi$
$338$	5.18227	0.281878
$339$	−11.1220	−0.604062
$340$	0	0
$341$	18.4821	1.00086
$342$	2.45401	0.132698
$343$	7.22340	0.390027
$344$	−24.3405	−1.31235
$345$	0	0
$346$	9.08885	0.488620
$347$	−19.4982	−1.04672	−0.523360	−	0.852112i	$-0.675322\pi$
$347$	−19.4982	−1.04672	−0.523360	+	0.852112i	$0.675322\pi$
$348$	−1.37256	−0.0735770
$349$	−18.7368	−1.00296	−0.501479	−	0.865170i	$-0.667210\pi$
$349$	−18.7368	−1.00296	−0.501479	+	0.865170i	$0.667210\pi$
$350$	0	0
$351$	−2.28216	−0.121812
$352$	−9.95623	−0.530669
$353$	23.9456	1.27449	0.637247	−	0.770660i	$-0.280073\pi$
$353$	23.9456	1.27449	0.637247	+	0.770660i	$0.280073\pi$
$354$	1.72736	0.0918083
$355$	0	0
$356$	−25.6595	−1.35995
$357$	−0.786152	−0.0416076
$358$	5.36237	0.283410
$359$	34.4688	1.81919	0.909597	−	0.415492i	$-0.136391\pi$
$359$	34.4688	1.81919	0.909597	+	0.415492i	$0.136391\pi$
$360$	0	0
$361$	−5.38607	−0.283478
$362$	−0.240409	−0.0126356
$363$	−8.00980	−0.420405
$364$	−1.87115	−0.0980750
$365$	0	0
$366$	7.42782	0.388258
$367$	−10.6761	−0.557287	−0.278643	−	0.960395i	$-0.589885\pi$
$367$	−10.6761	−0.557287	−0.278643	+	0.960395i	$0.589885\pi$
$368$	−12.7981	−0.667148
$369$	7.21450	0.375572
$370$	0	0
$371$	−4.57569	−0.237558
$372$	−16.6483	−0.863176
$373$	5.05718	0.261851	0.130925	−	0.991392i	$-0.458205\pi$
$373$	5.05718	0.261851	0.130925	+	0.991392i	$0.458205\pi$
$374$	−1.71770	−0.0888200
$375$	0	0
$376$	2.36618	0.122026
$377$	−2.01098	−0.103571
$378$	0.350089	0.0180067
$379$	−20.2467	−1.04000	−0.520001	−	0.854166i	$-0.674068\pi$
$379$	−20.2467	−1.04000	−0.520001	+	0.854166i	$0.674068\pi$
$380$	0	0
$381$	−2.93392	−0.150309
$382$	17.2563	0.882908
$383$	14.0746	0.719178	0.359589	−	0.933111i	$-0.382917\pi$
$383$	14.0746	0.719178	0.359589	+	0.933111i	$0.382917\pi$
$384$	11.0189	0.562308
$385$	0	0
$386$	−7.81767	−0.397909
$387$	−10.2869	−0.522911
$388$	7.58297	0.384967
$389$	15.2306	0.772222	0.386111	−	0.922452i	$-0.373818\pi$
$389$	15.2306	0.772222	0.386111	+	0.922452i	$0.373818\pi$
$390$	0	0
$391$	−12.3993	−0.627060
$392$	−15.9076	−0.803457
$393$	−3.91598	−0.197535
$394$	10.6098	0.534516
$395$	0	0
$396$	−2.69352	−0.135354
$397$	31.0425	1.55798	0.778990	−	0.627036i	$-0.215732\pi$
$397$	31.0425	1.55798	0.778990	+	0.627036i	$0.215732\pi$
$398$	−9.67108	−0.484768
$399$	1.94217	0.0972299
$400$	0	0
$401$	−8.53214	−0.426075	−0.213037	−	0.977044i	$-0.568336\pi$
$401$	−8.53214	−0.426075	−0.213037	+	0.977044i	$0.568336\pi$
$402$	−0.310575	−0.0154901
$403$	−24.3920	−1.21505
$404$	11.9809	0.596071
$405$	0	0
$406$	0.308490	0.0153101
$407$	−11.6302	−0.576488
$408$	3.53394	0.174956
$409$	22.6154	1.11826	0.559129	−	0.829081i	$-0.311136\pi$
$409$	22.6154	1.11826	0.559129	+	0.829081i	$0.311136\pi$
$410$	0	0
$411$	−2.50176	−0.123403
$412$	−29.5362	−1.45514
$413$	1.36708	0.0672696
$414$	5.52166	0.271375
$415$	0	0
$416$	13.1398	0.644233
$417$	−7.61511	−0.372913
$418$	4.24352	0.207557
$419$	29.2310	1.42803	0.714013	−	0.700132i	$-0.246876\pi$
$419$	29.2310	1.42803	0.714013	+	0.700132i	$0.246876\pi$
$420$	0	0
$421$	−6.69312	−0.326203	−0.163101	−	0.986609i	$-0.552150\pi$
$421$	−6.69312	−0.326203	−0.163101	+	0.986609i	$0.552150\pi$
$422$	7.25783	0.353305
$423$	1.00000	0.0486217
$424$	20.5688	0.998909
$425$	0	0
$426$	−0.100879	−0.00488759
$427$	5.87857	0.284484
$428$	16.8669	0.815291
$429$	−3.94635	−0.190532
$430$	0	0
$431$	4.26734	0.205551	0.102775	−	0.994705i	$-0.467228\pi$
$431$	4.26734	0.205551	0.102775	+	0.994705i	$0.467228\pi$
$432$	1.54156	0.0741683
$433$	19.5053	0.937365	0.468683	−	0.883367i	$-0.344729\pi$
$433$	19.5053	0.937365	0.468683	+	0.883367i	$0.344729\pi$
$434$	3.74180	0.179612
$435$	0	0
$436$	23.4754	1.12427
$437$	30.6321	1.46533
$438$	−7.33686	−0.350569
$439$	8.63668	0.412206	0.206103	−	0.978530i	$-0.433922\pi$
$439$	8.63668	0.412206	0.206103	+	0.978530i	$0.433922\pi$
$440$	0	0
$441$	−6.72293	−0.320140
$442$	2.26695	0.107828
$443$	−7.15599	−0.339991	−0.169996	−	0.985445i	$-0.554375\pi$
$443$	−7.15599	−0.339991	−0.169996	+	0.985445i	$0.554375\pi$
$444$	10.4763	0.497181
$445$	0	0
$446$	−3.49547	−0.165516
$447$	−1.88118	−0.0889766
$448$	−0.392814	−0.0185587
$449$	5.53450	0.261189	0.130595	−	0.991436i	$-0.458311\pi$
$449$	5.53450	0.261189	0.130595	+	0.991436i	$0.458311\pi$
$450$	0	0
$451$	12.4755	0.587447
$452$	17.3241	0.814857
$453$	7.13872	0.335406
$454$	17.2075	0.807590
$455$	0	0
$456$	−8.73050	−0.408843
$457$	23.8874	1.11741	0.558704	−	0.829367i	$-0.311299\pi$
$457$	23.8874	1.11741	0.558704	+	0.829367i	$0.311299\pi$
$458$	−2.48804	−0.116259
$459$	1.49352	0.0697117
$460$	0	0
$461$	−6.07587	−0.282982	−0.141491	−	0.989940i	$-0.545190\pi$
$461$	−6.07587	−0.282982	−0.141491	+	0.989940i	$0.545190\pi$
$462$	0.605382	0.0281649
$463$	27.1266	1.26068	0.630339	−	0.776320i	$-0.282916\pi$
$463$	27.1266	1.26068	0.630339	+	0.776320i	$0.282916\pi$
$464$	1.35838	0.0630614
$465$	0	0
$466$	11.4825	0.531919
$467$	16.8232	0.778486	0.389243	−	0.921135i	$-0.372737\pi$
$467$	16.8232	0.778486	0.389243	+	0.921135i	$0.372737\pi$
$468$	3.55479	0.164320
$469$	−0.245797	−0.0113498
$470$	0	0
$471$	19.2082	0.885068
$472$	−6.14535	−0.282862
$473$	−17.7883	−0.817905
$474$	2.58579	0.118769
$475$	0	0
$476$	1.22455	0.0561271
$477$	8.69284	0.398018
$478$	1.26961	0.0580708
$479$	−42.0833	−1.92284	−0.961419	−	0.275090i	$-0.911292\pi$
$479$	−42.0833	−1.92284	−0.961419	+	0.275090i	$0.911292\pi$
$480$	0	0
$481$	15.3491	0.699857
$482$	−9.57882	−0.436303
$483$	4.36999	0.198841
$484$	12.4764	0.567111
$485$	0	0
$486$	−0.665096	−0.0301694
$487$	−18.9487	−0.858648	−0.429324	−	0.903151i	$-0.641248\pi$
$487$	−18.9487	−0.858648	−0.429324	+	0.903151i	$0.641248\pi$
$488$	−26.4256	−1.19623
$489$	−22.0859	−0.998757
$490$	0	0
$491$	9.83184	0.443705	0.221852	−	0.975080i	$-0.428790\pi$
$491$	9.83184	0.443705	0.221852	+	0.975080i	$0.428790\pi$
$492$	−11.2376	−0.506632
$493$	1.31606	0.0592722
$494$	−5.60043	−0.251975
$495$	0	0
$496$	16.4764	0.739812
$497$	−0.0798381	−0.00358123
$498$	8.19841	0.367380
$499$	−21.3649	−0.956423	−0.478212	−	0.878245i	$-0.658715\pi$
$499$	−21.3649	−0.956423	−0.478212	+	0.878245i	$0.658715\pi$
$500$	0	0
$501$	−13.8334	−0.618032
$502$	16.7860	0.749195
$503$	−23.9097	−1.06608	−0.533040	−	0.846090i	$-0.678951\pi$
$503$	−23.9097	−1.06608	−0.533040	+	0.846090i	$0.678951\pi$
$504$	−1.24549	−0.0554787
$505$	0	0
$506$	9.54817	0.424468
$507$	−7.79177	−0.346044
$508$	4.57002	0.202762
$509$	6.07940	0.269465	0.134732	−	0.990882i	$-0.456983\pi$
$509$	6.07940	0.269465	0.134732	+	0.990882i	$0.456983\pi$
$510$	0	0
$511$	−5.80658	−0.256868
$512$	−16.1710	−0.714662
$513$	−3.68971	−0.162904
$514$	−12.4107	−0.547412
$515$	0	0
$516$	16.0233	0.705387
$517$	1.72922	0.0760510
$518$	−2.35459	−0.103455
$519$	−13.6655	−0.599848
$520$	0	0
$521$	−42.2861	−1.85259	−0.926293	−	0.376803i	$-0.877023\pi$
$521$	−42.2861	−1.85259	−0.926293	+	0.376803i	$0.877023\pi$
$522$	−0.586066	−0.0256514
$523$	−24.1340	−1.05531	−0.527653	−	0.849460i	$-0.676928\pi$
$523$	−24.1340	−1.05531	−0.527653	+	0.849460i	$0.676928\pi$
$524$	6.09971	0.266467
$525$	0	0
$526$	0.866253	0.0377704
$527$	15.9630	0.695358
$528$	2.66570	0.116010
$529$	45.9241	1.99670
$530$	0	0
$531$	−2.59716	−0.112707
$532$	−3.02521	−0.131159
$533$	−16.4646	−0.713162
$534$	−10.9563	−0.474125
$535$	0	0
$536$	1.10491	0.0477251
$537$	−8.06255	−0.347925
$538$	15.1347	0.652503
$539$	−11.6254	−0.500743
$540$	0	0
$541$	28.8910	1.24212	0.621060	−	0.783763i	$-0.286703\pi$
$541$	28.8910	1.24212	0.621060	+	0.783763i	$0.286703\pi$
$542$	−13.4943	−0.579629
$543$	0.361465	0.0155120
$544$	−8.59917	−0.368686
$545$	0	0
$546$	−0.798958	−0.0341923
$547$	10.3986	0.444613	0.222307	−	0.974977i	$-0.428641\pi$
$547$	10.3986	0.444613	0.222307	+	0.974977i	$0.428641\pi$
$548$	3.89687	0.166466
$549$	−11.1680	−0.476640
$550$	0	0
$551$	−3.25128	−0.138509
$552$	−19.6441	−0.836110
$553$	2.04646	0.0870243
$554$	0.221694	0.00941886
$555$	0	0
$556$	11.8617	0.503046
$557$	1.94136	0.0822580	0.0411290	−	0.999154i	$-0.486905\pi$
$557$	1.94136	0.0822580	0.0411290	+	0.999154i	$0.486905\pi$
$558$	−7.10863	−0.300932
$559$	23.4762	0.992939
$560$	0	0
$561$	2.58263	0.109039
$562$	5.20666	0.219630
$563$	42.0821	1.77355	0.886775	−	0.462201i	$-0.152940\pi$
$563$	42.0821	1.77355	0.886775	+	0.462201i	$0.152940\pi$
$564$	−1.55765	−0.0655888
$565$	0	0
$566$	−7.77433	−0.326779
$567$	−0.526374	−0.0221056
$568$	0.358891	0.0150587
$569$	−25.8874	−1.08526	−0.542629	−	0.839972i	$-0.682571\pi$
$569$	−25.8874	−1.08526	−0.542629	+	0.839972i	$0.682571\pi$
$570$	0	0
$571$	−28.7877	−1.20473	−0.602363	−	0.798222i	$-0.705774\pi$
$571$	−28.7877	−1.20473	−0.602363	+	0.798222i	$0.705774\pi$
$572$	6.14702	0.257020
$573$	−25.9456	−1.08389
$574$	2.52572	0.105422
$575$	0	0
$576$	0.746264	0.0310943
$577$	0.250509	0.0104288	0.00521441	−	0.999986i	$-0.498340\pi$
$577$	0.250509	0.0104288	0.00521441	+	0.999986i	$0.498340\pi$
$578$	9.82306	0.408585
$579$	11.7542	0.488488
$580$	0	0
$581$	6.48844	0.269186
$582$	3.23783	0.134213
$583$	15.0318	0.622555
$584$	26.1019	1.08011
$585$	0	0
$586$	3.75608	0.155162
$587$	18.8114	0.776429	0.388214	−	0.921569i	$-0.373092\pi$
$587$	18.8114	0.776429	0.388214	+	0.921569i	$0.373092\pi$
$588$	10.4720	0.431856
$589$	−39.4360	−1.62493
$590$	0	0
$591$	−15.9523	−0.656191
$592$	−10.3681	−0.426124
$593$	−1.67610	−0.0688291	−0.0344145	−	0.999408i	$-0.510957\pi$
$593$	−1.67610	−0.0688291	−0.0344145	+	0.999408i	$0.510957\pi$
$594$	−1.15010	−0.0471891
$595$	0	0
$596$	2.93021	0.120026
$597$	14.5409	0.595119
$598$	−12.6013	−0.515305
$599$	−13.5867	−0.555137	−0.277569	−	0.960706i	$-0.589529\pi$
$599$	−13.5867	−0.555137	−0.277569	+	0.960706i	$0.589529\pi$
$600$	0	0
$601$	8.46747	0.345395	0.172698	−	0.984975i	$-0.444752\pi$
$601$	8.46747	0.345395	0.172698	+	0.984975i	$0.444752\pi$
$602$	−3.60132	−0.146779
$603$	0.466962	0.0190162
$604$	−11.1196	−0.452451
$605$	0	0
$606$	5.11569	0.207811
$607$	6.17431	0.250608	0.125304	−	0.992118i	$-0.460009\pi$
$607$	6.17431	0.250608	0.125304	+	0.992118i	$0.460009\pi$
$608$	21.2440	0.861558
$609$	−0.463828	−0.0187953
$610$	0	0
$611$	−2.28216	−0.0923261
$612$	−2.32638	−0.0940384
$613$	0.550588	0.0222380	0.0111190	−	0.999938i	$-0.496461\pi$
$613$	0.550588	0.0222380	0.0111190	+	0.999938i	$0.496461\pi$
$614$	−5.36804	−0.216636
$615$	0	0
$616$	−2.15373	−0.0867764
$617$	20.3240	0.818215	0.409107	−	0.912486i	$-0.365840\pi$
$617$	20.3240	0.818215	0.409107	+	0.912486i	$0.365840\pi$
$618$	−12.6116	−0.507312
$619$	6.37978	0.256425	0.128213	−	0.991747i	$-0.459076\pi$
$619$	6.37978	0.256425	0.128213	+	0.991747i	$0.459076\pi$
$620$	0	0
$621$	−8.30205	−0.333150
$622$	−6.39810	−0.256541
$623$	−8.67108	−0.347400
$624$	−3.51808	−0.140836
$625$	0	0
$626$	12.9835	0.518926
$627$	−6.38031	−0.254805
$628$	−29.9196	−1.19392
$629$	−10.0450	−0.400519
$630$	0	0
$631$	43.1071	1.71607	0.858034	−	0.513593i	$-0.171686\pi$
$631$	43.1071	1.71607	0.858034	+	0.513593i	$0.171686\pi$
$632$	−9.19932	−0.365929
$633$	−10.9124	−0.433731
$634$	2.06101	0.0818530
$635$	0	0
$636$	−13.5404	−0.536911
$637$	15.3428	0.607903
$638$	−1.01344	−0.0401224
$639$	0.151675	0.00600019
$640$	0	0
$641$	−43.2896	−1.70984	−0.854918	−	0.518763i	$-0.826393\pi$
$641$	−43.2896	−1.70984	−0.854918	+	0.518763i	$0.826393\pi$
$642$	7.20194	0.284238
$643$	−18.5497	−0.731528	−0.365764	−	0.930708i	$-0.619192\pi$
$643$	−18.5497	−0.731528	−0.365764	+	0.930708i	$0.619192\pi$
$644$	−6.80690	−0.268229
$645$	0	0
$646$	3.66512	0.144202
$647$	16.3235	0.641743	0.320872	−	0.947123i	$-0.396024\pi$
$647$	16.3235	0.641743	0.320872	+	0.947123i	$0.396024\pi$
$648$	2.36618	0.0929522
$649$	−4.49107	−0.176290
$650$	0	0
$651$	−5.62595	−0.220499
$652$	34.4020	1.34729
$653$	−35.5264	−1.39026	−0.695129	−	0.718885i	$-0.744653\pi$
$653$	−35.5264	−1.39026	−0.695129	+	0.718885i	$0.744653\pi$
$654$	10.0237	0.391958
$655$	0	0
$656$	11.1216	0.434225
$657$	11.0313	0.430371
$658$	0.350089	0.0136479
$659$	−15.2954	−0.595824	−0.297912	−	0.954593i	$-0.596290\pi$
$659$	−15.2954	−0.595824	−0.297912	+	0.954593i	$0.596290\pi$
$660$	0	0
$661$	−30.1984	−1.17458	−0.587291	−	0.809376i	$-0.699805\pi$
$661$	−30.1984	−1.17458	−0.587291	+	0.809376i	$0.699805\pi$
$662$	−1.80215	−0.0700424
$663$	−3.40845	−0.132373
$664$	−29.1671	−1.13190
$665$	0	0
$666$	4.47323	0.173334
$667$	−7.31557	−0.283260
$668$	21.5476	0.833701
$669$	5.25559	0.203193
$670$	0	0
$671$	−19.3120	−0.745532
$672$	3.03067	0.116911
$673$	−23.8131	−0.917929	−0.458964	−	0.888455i	$-0.651779\pi$
$673$	−23.8131	−0.917929	−0.458964	+	0.888455i	$0.651779\pi$
$674$	9.62093	0.370584
$675$	0	0
$676$	12.1368	0.466801
$677$	−2.36255	−0.0908003	−0.0454002	−	0.998969i	$-0.514456\pi$
$677$	−2.36255	−0.0908003	−0.0454002	+	0.998969i	$0.514456\pi$
$678$	7.39717	0.284087
$679$	2.56251	0.0983400
$680$	0	0
$681$	−25.8723	−0.991427
$682$	−12.2924	−0.470700
$683$	1.13940	0.0435978	0.0217989	−	0.999762i	$-0.493061\pi$
$683$	1.13940	0.0435978	0.0217989	+	0.999762i	$0.493061\pi$
$684$	5.74726	0.219752
$685$	0	0
$686$	−4.80425	−0.183427
$687$	3.74088	0.142723
$688$	−15.8578	−0.604574
$689$	−19.8384	−0.755783
$690$	0	0
$691$	−43.9701	−1.67270	−0.836350	−	0.548196i	$-0.815315\pi$
$691$	−43.9701	−1.67270	−0.836350	+	0.548196i	$0.815315\pi$
$692$	21.2860	0.809172
$693$	−0.910217	−0.0345763
$694$	12.9682	0.492266
$695$	0	0
$696$	2.08502	0.0790324
$697$	10.7750	0.408133
$698$	12.4618	0.471685
$699$	−17.2645	−0.653003
$700$	0	0
$701$	37.7757	1.42677	0.713383	−	0.700774i	$-0.247162\pi$
$701$	37.7757	1.42677	0.713383	+	0.700774i	$0.247162\pi$
$702$	1.51785	0.0572877
$703$	24.8158	0.935946
$704$	1.29045	0.0486358
$705$	0	0
$706$	−15.9261	−0.599386
$707$	4.04869	0.152267
$708$	4.04546	0.152038
$709$	27.6461	1.03827	0.519135	−	0.854692i	$-0.326254\pi$
$709$	27.6461	1.03827	0.519135	+	0.854692i	$0.326254\pi$
$710$	0	0
$711$	−3.88784	−0.145805
$712$	38.9786	1.46078
$713$	−88.7334	−3.32309
$714$	0.522867	0.0195678
$715$	0	0
$716$	12.5586	0.469337
$717$	−1.90892	−0.0712899
$718$	−22.9251	−0.855556
$719$	−10.6393	−0.396780	−0.198390	−	0.980123i	$-0.563571\pi$
$719$	−10.6393	−0.396780	−0.198390	+	0.980123i	$0.563571\pi$
$720$	0	0
$721$	−9.98113	−0.371717
$722$	3.58226	0.133318
$723$	14.4022	0.535622
$724$	−0.563035	−0.0209250
$725$	0	0
$726$	5.32728	0.197714
$727$	23.0237	0.853901	0.426951	−	0.904275i	$-0.359588\pi$
$727$	23.0237	0.853901	0.426951	+	0.904275i	$0.359588\pi$
$728$	2.84241	0.105347
$729$	1.00000	0.0370370
$730$	0	0
$731$	−15.3637	−0.568246
$732$	17.3959	0.642970
$733$	17.6028	0.650173	0.325087	−	0.945684i	$-0.394607\pi$
$733$	17.6028	0.650173	0.325087	+	0.945684i	$0.394607\pi$
$734$	7.10062	0.262088
$735$	0	0
$736$	47.8002	1.76194
$737$	0.807480	0.0297439
$738$	−4.79834	−0.176629
$739$	34.9729	1.28650	0.643250	−	0.765656i	$-0.277586\pi$
$739$	34.9729	1.28650	0.643250	+	0.765656i	$0.277586\pi$
$740$	0	0
$741$	8.42048	0.309334
$742$	3.04327	0.111722
$743$	9.94598	0.364882	0.182441	−	0.983217i	$-0.441600\pi$
$743$	9.94598	0.364882	0.182441	+	0.983217i	$0.441600\pi$
$744$	25.2900	0.927176
$745$	0	0
$746$	−3.36351	−0.123147
$747$	−12.3267	−0.451009
$748$	−4.02283	−0.147089
$749$	5.69981	0.208266
$750$	0	0
$751$	20.7854	0.758469	0.379234	−	0.925301i	$-0.376187\pi$
$751$	20.7854	0.758469	0.379234	+	0.925301i	$0.376187\pi$
$752$	1.54156	0.0562149
$753$	−25.2384	−0.919740
$754$	1.33749	0.0487087
$755$	0	0
$756$	0.819905	0.0298197
$757$	48.1466	1.74992	0.874958	−	0.484198i	$-0.160889\pi$
$757$	48.1466	1.74992	0.874958	+	0.484198i	$0.160889\pi$
$758$	13.4660	0.489107
$759$	−14.3561	−0.521093
$760$	0	0
$761$	13.2600	0.480676	0.240338	−	0.970689i	$-0.422742\pi$
$761$	13.2600	0.480676	0.240338	+	0.970689i	$0.422742\pi$
$762$	1.95134	0.0706896
$763$	7.93302	0.287195
$764$	40.4140	1.46213
$765$	0	0
$766$	−9.36096	−0.338225
$767$	5.92713	0.214016
$768$	−8.82118	−0.318307
$769$	1.80865	0.0652214	0.0326107	−	0.999468i	$-0.489618\pi$
$769$	1.80865	0.0652214	0.0326107	+	0.999468i	$0.489618\pi$
$770$	0	0
$771$	18.6600	0.672023
$772$	−18.3089	−0.658952
$773$	30.7077	1.10448	0.552240	−	0.833685i	$-0.313773\pi$
$773$	30.7077	1.10448	0.552240	+	0.833685i	$0.313773\pi$
$774$	6.84175	0.245922
$775$	0	0
$776$	−11.5191	−0.413511
$777$	3.54023	0.127005
$778$	−10.1298	−0.363171
$779$	−26.6194	−0.953738
$780$	0	0
$781$	0.262280	0.00938513
$782$	8.24673	0.294902
$783$	0.881176	0.0314906
$784$	−10.3638	−0.370136
$785$	0	0
$786$	2.60450	0.0928995
$787$	50.5125	1.80058	0.900288	−	0.435295i	$-0.143356\pi$
$787$	50.5125	1.80058	0.900288	+	0.435295i	$0.143356\pi$
$788$	24.8481	0.885177
$789$	−1.30245	−0.0463684
$790$	0	0
$791$	5.85431	0.208155
$792$	4.09164	0.145390
$793$	25.4872	0.905077
$794$	−20.6463	−0.732709
$795$	0	0
$796$	−22.6496	−0.802793
$797$	38.0433	1.34756	0.673781	−	0.738931i	$-0.264669\pi$
$797$	38.0433	1.34756	0.673781	+	0.738931i	$0.264669\pi$
$798$	−1.29173	−0.0457266
$799$	1.49352	0.0528371
$800$	0	0
$801$	16.4732	0.582053
$802$	5.67469	0.200380
$803$	19.0755	0.673160
$804$	−0.727362	−0.0256521
$805$	0	0
$806$	16.2230	0.571431
$807$	−22.7557	−0.801037
$808$	−18.1998	−0.640268
$809$	46.8651	1.64769	0.823844	−	0.566817i	$-0.191825\pi$
$809$	46.8651	1.64769	0.823844	+	0.566817i	$0.191825\pi$
$810$	0	0
$811$	−25.0198	−0.878563	−0.439282	−	0.898349i	$-0.644767\pi$
$811$	−25.0198	−0.878563	−0.439282	+	0.898349i	$0.644767\pi$
$812$	0.722481	0.0253541
$813$	20.2892	0.711574
$814$	7.73520	0.271119
$815$	0	0
$816$	2.30236	0.0805985
$817$	37.9555	1.32790
$818$	−15.0414	−0.525910
$819$	1.20127	0.0419757
$820$	0	0
$821$	−15.5998	−0.544437	−0.272218	−	0.962235i	$-0.587757\pi$
$821$	−15.5998	−0.544437	−0.272218	+	0.962235i	$0.587757\pi$
$822$	1.66391	0.0580357
$823$	−33.9196	−1.18236	−0.591182	−	0.806538i	$-0.701338\pi$
$823$	−33.9196	−1.18236	−0.591182	+	0.806538i	$0.701338\pi$
$824$	44.8675	1.56304
$825$	0	0
$826$	−0.909239	−0.0316365
$827$	21.3752	0.743287	0.371644	−	0.928375i	$-0.378794\pi$
$827$	21.3752	0.743287	0.371644	+	0.928375i	$0.378794\pi$
$828$	12.9317	0.449407
$829$	−48.0921	−1.67031	−0.835154	−	0.550016i	$-0.814622\pi$
$829$	−48.0921	−1.67031	−0.835154	+	0.550016i	$0.814622\pi$
$830$	0	0
$831$	−0.333326	−0.0115629
$832$	−1.70309	−0.0590440
$833$	−10.0409	−0.347895
$834$	5.06478	0.175379
$835$	0	0
$836$	9.93828	0.343723
$837$	10.6881	0.369436
$838$	−19.4414	−0.671592
$839$	1.25743	0.0434113	0.0217057	−	0.999764i	$-0.493090\pi$
$839$	1.25743	0.0434113	0.0217057	+	0.999764i	$0.493090\pi$
$840$	0	0
$841$	−28.2235	−0.973225
$842$	4.45157	0.153411
$843$	−7.82843	−0.269625
$844$	16.9977	0.585087
$845$	0	0
$846$	−0.665096	−0.0228665
$847$	4.21615	0.144869
$848$	13.4005	0.460176
$849$	11.6890	0.401166
$850$	0	0
$851$	55.8370	1.91407
$852$	−0.236257	−0.00809403
$853$	43.7029	1.49636	0.748180	−	0.663496i	$-0.230928\pi$
$853$	43.7029	1.49636	0.748180	+	0.663496i	$0.230928\pi$
$854$	−3.90981	−0.133791
$855$	0	0
$856$	−25.6220	−0.875741
$857$	−16.7186	−0.571098	−0.285549	−	0.958364i	$-0.592176\pi$
$857$	−16.7186	−0.571098	−0.285549	+	0.958364i	$0.592176\pi$
$858$	2.62470	0.0896058
$859$	11.2233	0.382933	0.191467	−	0.981499i	$-0.438676\pi$
$859$	11.2233	0.382933	0.191467	+	0.981499i	$0.438676\pi$
$860$	0	0
$861$	−3.79753	−0.129419
$862$	−2.83819	−0.0966692
$863$	−33.6406	−1.14514	−0.572569	−	0.819857i	$-0.694053\pi$
$863$	−33.6406	−1.14514	−0.572569	+	0.819857i	$0.694053\pi$
$864$	−5.75764	−0.195879
$865$	0	0
$866$	−12.9729	−0.440837
$867$	−14.7694	−0.501595
$868$	8.76325	0.297444
$869$	−6.72293	−0.228060
$870$	0	0
$871$	−1.06568	−0.0361092
$872$	−35.6608	−1.20763
$873$	−4.86822	−0.164764
$874$	−20.3733	−0.689137
$875$	0	0
$876$	−17.1828	−0.580554
$877$	−30.3337	−1.02430	−0.512149	−	0.858897i	$-0.671150\pi$
$877$	−30.3337	−1.02430	−0.512149	+	0.858897i	$0.671150\pi$
$878$	−5.74422	−0.193858
$879$	−5.64743	−0.190483
$880$	0	0
$881$	15.2421	0.513520	0.256760	−	0.966475i	$-0.417345\pi$
$881$	15.2421	0.513520	0.256760	+	0.966475i	$0.417345\pi$
$882$	4.47139	0.150560
$883$	46.5893	1.56785	0.783927	−	0.620853i	$-0.213214\pi$
$883$	46.5893	1.56785	0.783927	+	0.620853i	$0.213214\pi$
$884$	5.30917	0.178567
$885$	0	0
$886$	4.75942	0.159896
$887$	33.5980	1.12811	0.564055	−	0.825737i	$-0.309241\pi$
$887$	33.5980	1.12811	0.564055	+	0.825737i	$0.309241\pi$
$888$	−15.9142	−0.534045
$889$	1.54434	0.0517956
$890$	0	0
$891$	1.72922	0.0579311
$892$	−8.18636	−0.274100
$893$	−3.68971	−0.123471
$894$	1.25116	0.0418451
$895$	0	0
$896$	−5.80009	−0.193767
$897$	18.9466	0.632608
$898$	−3.68097	−0.122836
$899$	9.41812	0.314112
$900$	0	0
$901$	12.9830	0.432525
$902$	−8.29738	−0.276273
$903$	5.41474	0.180191
$904$	−26.3165	−0.875275
$905$	0	0
$906$	−4.74794	−0.157740
$907$	−24.4812	−0.812886	−0.406443	−	0.913676i	$-0.633231\pi$
$907$	−24.4812	−0.812886	−0.406443	+	0.913676i	$0.633231\pi$
$908$	40.2999	1.33740
$909$	−7.69166	−0.255116
$910$	0	0
$911$	26.9524	0.892974	0.446487	−	0.894790i	$-0.352675\pi$
$911$	26.9524	0.892974	0.446487	+	0.894790i	$0.352675\pi$
$912$	−5.68790	−0.188345
$913$	−21.3155	−0.705441
$914$	−15.8874	−0.525510
$915$	0	0
$916$	−5.82697	−0.192528
$917$	2.06127	0.0680691
$918$	−0.993336	−0.0327850
$919$	39.4693	1.30197	0.650986	−	0.759090i	$-0.274356\pi$
$919$	39.4693	1.30197	0.650986	+	0.759090i	$0.274356\pi$
$920$	0	0
$921$	8.07107	0.265951
$922$	4.04104	0.133085
$923$	−0.346147	−0.0113936
$924$	1.41780	0.0466421
$925$	0	0
$926$	−18.0418	−0.592889
$927$	18.9620	0.622795
$928$	−5.07349	−0.166546
$929$	55.8490	1.83234	0.916172	−	0.400784i	$-0.131262\pi$
$929$	55.8490	1.83234	0.916172	+	0.400784i	$0.131262\pi$
$930$	0	0
$931$	24.8056	0.812972
$932$	26.8920	0.880876
$933$	9.61982	0.314939
$934$	−11.1891	−0.366117
$935$	0	0
$936$	−5.39998	−0.176504
$937$	32.6685	1.06723	0.533616	−	0.845727i	$-0.320833\pi$
$937$	32.6685	1.06723	0.533616	+	0.845727i	$0.320833\pi$
$938$	0.163478	0.00533776
$939$	−19.5213	−0.637052
$940$	0	0
$941$	−35.1817	−1.14689	−0.573446	−	0.819243i	$-0.694394\pi$
$941$	−35.1817	−1.14689	−0.573446	+	0.819243i	$0.694394\pi$
$942$	−12.7753	−0.416242
$943$	−59.8952	−1.95046
$944$	−4.00368	−0.130309
$945$	0	0
$946$	11.8309	0.384656
$947$	9.13733	0.296923	0.148462	−	0.988918i	$-0.452568\pi$
$947$	9.13733	0.296923	0.148462	+	0.988918i	$0.452568\pi$
$948$	6.05588	0.196686
$949$	−25.1751	−0.817218
$950$	0	0
$951$	−3.09881	−0.100486
$952$	−1.86018	−0.0602886
$953$	7.61627	0.246715	0.123358	−	0.992362i	$-0.460634\pi$
$953$	7.61627	0.246715	0.123358	+	0.992362i	$0.460634\pi$
$954$	−5.78157	−0.187185
$955$	0	0
$956$	2.97342	0.0961673
$957$	1.52375	0.0492557
$958$	27.9895	0.904299
$959$	1.31686	0.0425238
$960$	0	0
$961$	83.2361	2.68503
$962$	−10.2086	−0.329139
$963$	−10.8284	−0.348941
$964$	−22.4335	−0.722534
$965$	0	0
$966$	−2.90646	−0.0935138
$967$	9.90629	0.318565	0.159282	−	0.987233i	$-0.449082\pi$
$967$	9.90629	0.318565	0.159282	+	0.987233i	$0.449082\pi$
$968$	−18.9526	−0.609160
$969$	−5.51066	−0.177028
$970$	0	0
$971$	19.0684	0.611935	0.305968	−	0.952042i	$-0.401020\pi$
$971$	19.0684	0.611935	0.305968	+	0.952042i	$0.401020\pi$
$972$	−1.55765	−0.0499616
$973$	4.00840	0.128503
$974$	12.6027	0.403817
$975$	0	0
$976$	−17.2162	−0.551077
$977$	−29.6508	−0.948613	−0.474306	−	0.880360i	$-0.657301\pi$
$977$	−29.6508	−0.948613	−0.474306	+	0.880360i	$0.657301\pi$
$978$	14.6892	0.469709
$979$	28.4858	0.910411
$980$	0	0
$981$	−15.0711	−0.481182
$982$	−6.53912	−0.208672
$983$	−43.8415	−1.39833	−0.699163	−	0.714962i	$-0.746444\pi$
$983$	−43.8415	−1.39833	−0.699163	+	0.714962i	$0.746444\pi$
$984$	17.0708	0.544197
$985$	0	0
$986$	−0.875304	−0.0278753
$987$	−0.526374	−0.0167547
$988$	−13.1161	−0.417280
$989$	85.4021	2.71563
$990$	0	0
$991$	11.7094	0.371962	0.185981	−	0.982553i	$-0.440454\pi$
$991$	11.7094	0.371962	0.185981	+	0.982553i	$0.440454\pi$
$992$	−61.5384	−1.95385
$993$	2.70960	0.0859866
$994$	0.0531000	0.00168423
$995$	0	0
$996$	19.2006	0.608394
$997$	55.0051	1.74203	0.871014	−	0.491258i	$-0.163463\pi$
$997$	55.0051	1.74203	0.871014	+	0.491258i	$0.163463\pi$
$998$	14.2097	0.449800
$999$	−6.72569	−0.212791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.t.1.3		4
5.4	even	2		705.2.a.k.1.2	✓	4
15.14	odd	2		2115.2.a.o.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.k.1.2	✓	4	5.4	even	2
2115.2.a.o.1.3		4	15.14	odd	2
3525.2.a.t.1.3		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)

\(f(q)\)	\(=\)	\(q-0.665096 q^{2} +1.00000 q^{3} -1.55765 q^{4} -0.665096 q^{6} -0.526374 q^{7} +2.36618 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.665096 q^{2} +1.00000 q^{3} -1.55765 q^{4} -0.665096 q^{6} -0.526374 q^{7} +2.36618 q^{8} +1.00000 q^{9} +1.72922 q^{11} -1.55765 q^{12} -2.28216 q^{13} +0.350089 q^{14} +1.54156 q^{16} +1.49352 q^{17} -0.665096 q^{18} -3.68971 q^{19} -0.526374 q^{21} -1.15010 q^{22} -8.30205 q^{23} +2.36618 q^{24} +1.51785 q^{26} +1.00000 q^{27} +0.819905 q^{28} +0.881176 q^{29} +10.6881 q^{31} -5.75764 q^{32} +1.72922 q^{33} -0.993336 q^{34} -1.55765 q^{36} -6.72569 q^{37} +2.45401 q^{38} -2.28216 q^{39} +7.21450 q^{41} +0.350089 q^{42} -10.2869 q^{43} -2.69352 q^{44} +5.52166 q^{46} +1.00000 q^{47} +1.54156 q^{48} -6.72293 q^{49} +1.49352 q^{51} +3.55479 q^{52} +8.69284 q^{53} -0.665096 q^{54} -1.24549 q^{56} -3.68971 q^{57} -0.586066 q^{58} -2.59716 q^{59} -11.1680 q^{61} -7.10863 q^{62} -0.526374 q^{63} +0.746264 q^{64} -1.15010 q^{66} +0.466962 q^{67} -2.32638 q^{68} -8.30205 q^{69} +0.151675 q^{71} +2.36618 q^{72} +11.0313 q^{73} +4.47323 q^{74} +5.74726 q^{76} -0.910217 q^{77} +1.51785 q^{78} -3.88784 q^{79} +1.00000 q^{81} -4.79834 q^{82} -12.3267 q^{83} +0.819905 q^{84} +6.84175 q^{86} +0.881176 q^{87} +4.09164 q^{88} +16.4732 q^{89} +1.20127 q^{91} +12.9317 q^{92} +10.6881 q^{93} -0.665096 q^{94} -5.75764 q^{96} -4.86822 q^{97} +4.47139 q^{98} +1.72922 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 8 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 4 q^{9} + 4 q^{11} + 8 q^{12} + 4 q^{13} + 12 q^{16} - 4 q^{17} - 4 q^{18} - 8 q^{19} - 8 q^{21} + 16 q^{22} - 16 q^{23} - 12 q^{24} + 4 q^{27} - 20 q^{28} + 4 q^{29} - 28 q^{32} + 4 q^{33} - 16 q^{34} + 8 q^{36} + 4 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{41} - 24 q^{43} - 20 q^{44} + 32 q^{46} + 4 q^{47} + 12 q^{48} + 8 q^{49} - 4 q^{51} + 28 q^{52} - 12 q^{53} - 4 q^{54} + 40 q^{56} - 8 q^{57} + 4 q^{58} - 28 q^{61} - 12 q^{62} - 8 q^{63} + 24 q^{64} + 16 q^{66} + 8 q^{67} + 4 q^{68} - 16 q^{69} + 16 q^{71} - 12 q^{72} + 24 q^{73} - 56 q^{74} - 8 q^{76} - 4 q^{77} - 4 q^{79} + 4 q^{81} + 4 q^{82} - 24 q^{83} - 20 q^{84} - 8 q^{86} + 4 q^{87} + 40 q^{88} - 8 q^{89} - 40 q^{91} - 28 q^{92} - 4 q^{94} - 28 q^{96} + 32 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 8 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 4 * q^9 + 4 * q^11 + 8 * q^12 + 4 * q^13 + 12 * q^16 - 4 * q^17 - 4 * q^18 - 8 * q^19 - 8 * q^21 + 16 * q^22 - 16 * q^23 - 12 * q^24 + 4 * q^27 - 20 * q^28 + 4 * q^29 - 28 * q^32 + 4 * q^33 - 16 * q^34 + 8 * q^36 + 4 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^41 - 24 * q^43 - 20 * q^44 + 32 * q^46 + 4 * q^47 + 12 * q^48 + 8 * q^49 - 4 * q^51 + 28 * q^52 - 12 * q^53 - 4 * q^54 + 40 * q^56 - 8 * q^57 + 4 * q^58 - 28 * q^61 - 12 * q^62 - 8 * q^63 + 24 * q^64 + 16 * q^66 + 8 * q^67 + 4 * q^68 - 16 * q^69 + 16 * q^71 - 12 * q^72 + 24 * q^73 - 56 * q^74 - 8 * q^76 - 4 * q^77 - 4 * q^79 + 4 * q^81 + 4 * q^82 - 24 * q^83 - 20 * q^84 - 8 * q^86 + 4 * q^87 + 40 * q^88 - 8 * q^89 - 40 * q^91 - 28 * q^92 - 4 * q^94 - 28 * q^96 + 32 * q^98 + 4 * q^99

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