LMFDB - Embedded newform 3525.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3525.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$28.1472667125$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 9x^{5} + 10x^{4} + 16x^{3} - 15x^{2} - 6x + 5 $ x^7 - x^6 - 9x^5 + 10x^4 + 16x^3 - 15x^2 - 6*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.85243$ 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)

$f(q)$	$=$	$q-1.85243 q^{2} +1.00000 q^{3} +1.43149 q^{4} -1.85243 q^{6} +2.79882 q^{7} +1.05312 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.85243 q^{2} +1.00000 q^{3} +1.43149 q^{4} -1.85243 q^{6} +2.79882 q^{7} +1.05312 q^{8} +1.00000 q^{9} +0.110799 q^{11} +1.43149 q^{12} -6.18763 q^{13} -5.18461 q^{14} -4.81382 q^{16} -4.27451 q^{17} -1.85243 q^{18} -0.326031 q^{19} +2.79882 q^{21} -0.205247 q^{22} +5.33219 q^{23} +1.05312 q^{24} +11.4621 q^{26} +1.00000 q^{27} +4.00648 q^{28} -8.69258 q^{29} -5.01264 q^{31} +6.81100 q^{32} +0.110799 q^{33} +7.91822 q^{34} +1.43149 q^{36} +1.49079 q^{37} +0.603948 q^{38} -6.18763 q^{39} +10.8442 q^{41} -5.18461 q^{42} +8.23207 q^{43} +0.158608 q^{44} -9.87749 q^{46} +1.00000 q^{47} -4.81382 q^{48} +0.833398 q^{49} -4.27451 q^{51} -8.85753 q^{52} +10.6703 q^{53} -1.85243 q^{54} +2.94750 q^{56} -0.326031 q^{57} +16.1024 q^{58} +7.90884 q^{59} +9.66552 q^{61} +9.28556 q^{62} +2.79882 q^{63} -2.98926 q^{64} -0.205247 q^{66} +2.31475 q^{67} -6.11892 q^{68} +5.33219 q^{69} -4.49207 q^{71} +1.05312 q^{72} +13.0475 q^{73} -2.76159 q^{74} -0.466710 q^{76} +0.310106 q^{77} +11.4621 q^{78} -11.7702 q^{79} +1.00000 q^{81} -20.0881 q^{82} +17.2486 q^{83} +4.00648 q^{84} -15.2493 q^{86} -8.69258 q^{87} +0.116685 q^{88} +5.04142 q^{89} -17.3181 q^{91} +7.63297 q^{92} -5.01264 q^{93} -1.85243 q^{94} +6.81100 q^{96} -7.54883 q^{97} -1.54381 q^{98} +0.110799 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 7 q^{3} + 5 q^{4} - q^{6} + 7 q^{7} + 6 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q - q^2 + 7 * q^3 + 5 * q^4 - q^6 + 7 * q^7 + 6 * q^8 + 7 * q^9	$ 7 q - q^{2} + 7 q^{3} + 5 q^{4} - q^{6} + 7 q^{7} + 6 q^{8} + 7 q^{9} + 5 q^{12} + 5 q^{13} + 7 q^{14} + 9 q^{16} + 2 q^{17} - q^{18} - 13 q^{19} + 7 q^{21} - 14 q^{22} + 6 q^{23} + 6 q^{24} + 7 q^{27} + 30 q^{28} + 9 q^{29} + 5 q^{31} + 26 q^{32} - 8 q^{34} + 5 q^{36} - 5 q^{37} - 2 q^{38} + 5 q^{39} + 18 q^{41} + 7 q^{42} + 14 q^{43} + 17 q^{44} - 27 q^{46} + 7 q^{47} + 9 q^{48} + 14 q^{49} + 2 q^{51} - 3 q^{52} + 20 q^{53} - q^{54} + 17 q^{56} - 13 q^{57} + 37 q^{58} + 10 q^{59} - 8 q^{61} - 6 q^{62} + 7 q^{63} + 18 q^{64} - 14 q^{66} + 4 q^{67} + 10 q^{68} + 6 q^{69} + 12 q^{71} + 6 q^{72} + 4 q^{73} - 25 q^{74} - 66 q^{76} + 6 q^{77} - 5 q^{79} + 7 q^{81} - 29 q^{82} + 52 q^{83} + 30 q^{84} - 17 q^{86} + 9 q^{87} + 26 q^{88} + 32 q^{89} - 26 q^{91} - 17 q^{92} + 5 q^{93} - q^{94} + 26 q^{96} - 12 q^{97} + 40 q^{98}+O(q^{100}) $ 7 * q - q^2 + 7 * q^3 + 5 * q^4 - q^6 + 7 * q^7 + 6 * q^8 + 7 * q^9 + 5 * q^12 + 5 * q^13 + 7 * q^14 + 9 * q^16 + 2 * q^17 - q^18 - 13 * q^19 + 7 * q^21 - 14 * q^22 + 6 * q^23 + 6 * q^24 + 7 * q^27 + 30 * q^28 + 9 * q^29 + 5 * q^31 + 26 * q^32 - 8 * q^34 + 5 * q^36 - 5 * q^37 - 2 * q^38 + 5 * q^39 + 18 * q^41 + 7 * q^42 + 14 * q^43 + 17 * q^44 - 27 * q^46 + 7 * q^47 + 9 * q^48 + 14 * q^49 + 2 * q^51 - 3 * q^52 + 20 * q^53 - q^54 + 17 * q^56 - 13 * q^57 + 37 * q^58 + 10 * q^59 - 8 * q^61 - 6 * q^62 + 7 * q^63 + 18 * q^64 - 14 * q^66 + 4 * q^67 + 10 * q^68 + 6 * q^69 + 12 * q^71 + 6 * q^72 + 4 * q^73 - 25 * q^74 - 66 * q^76 + 6 * q^77 - 5 * q^79 + 7 * q^81 - 29 * q^82 + 52 * q^83 + 30 * q^84 - 17 * q^86 + 9 * q^87 + 26 * q^88 + 32 * q^89 - 26 * q^91 - 17 * q^92 + 5 * q^93 - q^94 + 26 * q^96 - 12 * q^97 + 40 * q^98

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 $

$2$	−1.85243	−1.30986	−0.654932	−	0.755688i	$-0.727303\pi$
$2$	−1.85243	−1.30986	−0.654932	+	0.755688i	$0.727303\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.43149	0.715745
$5$	0	0
$5$	0	0
$6$	−1.85243	−0.756251
$7$	2.79882	1.05785	0.528927	−	0.848667i	$-0.322594\pi$
$7$	2.79882	1.05785	0.528927	+	0.848667i	$0.322594\pi$
$8$	1.05312	0.372335
$9$	1.00000	0.333333
$10$	0	0
$11$	0.110799	0.0334071	0.0167036	−	0.999860i	$-0.494683\pi$
$11$	0.110799	0.0334071	0.0167036	+	0.999860i	$0.494683\pi$
$12$	1.43149	0.413236
$13$	−6.18763	−1.71614	−0.858070	−	0.513533i	$-0.828337\pi$
$13$	−6.18763	−1.71614	−0.858070	+	0.513533i	$0.828337\pi$
$14$	−5.18461	−1.38565
$15$	0	0
$16$	−4.81382	−1.20345
$17$	−4.27451	−1.03672	−0.518361	−	0.855162i	$-0.673457\pi$
$17$	−4.27451	−1.03672	−0.518361	+	0.855162i	$0.673457\pi$
$18$	−1.85243	−0.436622
$19$	−0.326031	−0.0747965	−0.0373983	−	0.999300i	$-0.511907\pi$
$19$	−0.326031	−0.0747965	−0.0373983	+	0.999300i	$0.511907\pi$
$20$	0	0
$21$	2.79882	0.610753
$22$	−0.205247	−0.0437588
$23$	5.33219	1.11184	0.555919	−	0.831237i	$-0.312367\pi$
$23$	5.33219	1.11184	0.555919	+	0.831237i	$0.312367\pi$
$24$	1.05312	0.214968
$25$	0	0
$26$	11.4621	2.24791
$27$	1.00000	0.192450
$28$	4.00648	0.757154
$29$	−8.69258	−1.61417	−0.807086	−	0.590434i	$-0.798957\pi$
$29$	−8.69258	−1.61417	−0.807086	+	0.590434i	$0.798957\pi$
$30$	0	0
$31$	−5.01264	−0.900297	−0.450148	−	0.892954i	$-0.648629\pi$
$31$	−5.01264	−0.900297	−0.450148	+	0.892954i	$0.648629\pi$
$32$	6.81100	1.20403
$33$	0.110799	0.0192876
$34$	7.91822	1.35796
$35$	0	0
$36$	1.43149	0.238582
$37$	1.49079	0.245085	0.122542	−	0.992463i	$-0.460895\pi$
$37$	1.49079	0.245085	0.122542	+	0.992463i	$0.460895\pi$
$38$	0.603948	0.0979733
$39$	−6.18763	−0.990814
$40$	0	0
$41$	10.8442	1.69358	0.846791	−	0.531925i	$-0.178531\pi$
$41$	10.8442	1.69358	0.846791	+	0.531925i	$0.178531\pi$
$42$	−5.18461	−0.800003
$43$	8.23207	1.25538	0.627690	−	0.778464i	$-0.284001\pi$
$43$	8.23207	1.25538	0.627690	+	0.778464i	$0.284001\pi$
$44$	0.158608	0.0239110
$45$	0	0
$46$	−9.87749	−1.45636
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	−4.81382	−0.694815
$49$	0.833398	0.119057
$50$	0	0
$51$	−4.27451	−0.598551
$52$	−8.85753	−1.22832
$53$	10.6703	1.46568	0.732840	−	0.680401i	$-0.238194\pi$
$53$	10.6703	1.46568	0.732840	+	0.680401i	$0.238194\pi$
$54$	−1.85243	−0.252084
$55$	0	0
$56$	2.94750	0.393877
$57$	−0.326031	−0.0431838
$58$	16.1024	2.11435
$59$	7.90884	1.02964	0.514822	−	0.857297i	$-0.327858\pi$
$59$	7.90884	1.02964	0.514822	+	0.857297i	$0.327858\pi$
$60$	0	0
$61$	9.66552	1.23754	0.618771	−	0.785571i	$-0.287631\pi$
$61$	9.66552	1.23754	0.618771	+	0.785571i	$0.287631\pi$
$62$	9.28556	1.17927
$63$	2.79882	0.352618
$64$	−2.98926	−0.373657
$65$	0	0
$66$	−0.205247	−0.0252642
$67$	2.31475	0.282792	0.141396	−	0.989953i	$-0.454841\pi$
$67$	2.31475	0.282792	0.141396	+	0.989953i	$0.454841\pi$
$68$	−6.11892	−0.742028
$69$	5.33219	0.641920
$70$	0	0
$71$	−4.49207	−0.533111	−0.266556	−	0.963820i	$-0.585886\pi$
$71$	−4.49207	−0.533111	−0.266556	+	0.963820i	$0.585886\pi$
$72$	1.05312	0.124112
$73$	13.0475	1.52709	0.763546	−	0.645753i	$-0.223456\pi$
$73$	13.0475	1.52709	0.763546	+	0.645753i	$0.223456\pi$
$74$	−2.76159	−0.321028
$75$	0	0
$76$	−0.466710	−0.0535353
$77$	0.310106	0.0353399
$78$	11.4621	1.29783
$79$	−11.7702	−1.32425	−0.662124	−	0.749395i	$-0.730345\pi$
$79$	−11.7702	−1.32425	−0.662124	+	0.749395i	$0.730345\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−20.0881	−2.21836
$83$	17.2486	1.89328	0.946642	−	0.322286i	$-0.104451\pi$
$83$	17.2486	1.89328	0.946642	+	0.322286i	$0.104451\pi$
$84$	4.00648	0.437143
$85$	0	0
$86$	−15.2493	−1.64438
$87$	−8.69258	−0.931942
$88$	0.116685	0.0124387
$89$	5.04142	0.534389	0.267195	−	0.963643i	$-0.413903\pi$
$89$	5.04142	0.534389	0.267195	+	0.963643i	$0.413903\pi$
$90$	0	0
$91$	−17.3181	−1.81543
$92$	7.63297	0.795792
$93$	−5.01264	−0.519787
$94$	−1.85243	−0.191063
$95$	0	0
$96$	6.81100	0.695145
$97$	−7.54883	−0.766467	−0.383234	−	0.923651i	$-0.625190\pi$
$97$	−7.54883	−0.766467	−0.383234	+	0.923651i	$0.625190\pi$
$98$	−1.54381	−0.155948
$99$	0.110799	0.0111357
$100$	0	0
$101$	13.9362	1.38671	0.693354	−	0.720597i	$-0.256132\pi$
$101$	13.9362	1.38671	0.693354	+	0.720597i	$0.256132\pi$
$102$	7.91822	0.784021
$103$	9.10706	0.897345	0.448672	−	0.893696i	$-0.351897\pi$
$103$	9.10706	0.897345	0.448672	+	0.893696i	$0.351897\pi$
$104$	−6.51634	−0.638980
$105$	0	0
$106$	−19.7660	−1.91984
$107$	15.3830	1.48713	0.743563	−	0.668666i	$-0.233134\pi$
$107$	15.3830	1.48713	0.743563	+	0.668666i	$0.233134\pi$
$108$	1.43149	0.137745
$109$	−19.5500	−1.87255	−0.936276	−	0.351266i	$-0.885751\pi$
$109$	−19.5500	−1.87255	−0.936276	+	0.351266i	$0.885751\pi$
$110$	0	0
$111$	1.49079	0.141500
$112$	−13.4730	−1.27308
$113$	11.7869	1.10882	0.554410	−	0.832244i	$-0.312944\pi$
$113$	11.7869	1.10882	0.554410	+	0.832244i	$0.312944\pi$
$114$	0.603948	0.0565649
$115$	0	0
$116$	−12.4433	−1.15534
$117$	−6.18763	−0.572047
$118$	−14.6506	−1.34869
$119$	−11.9636	−1.09670
$120$	0	0
$121$	−10.9877	−0.998884
$122$	−17.9047	−1.62101
$123$	10.8442	0.977790
$124$	−7.17554	−0.644383
$125$	0	0
$126$	−5.18461	−0.461882
$127$	−6.22679	−0.552538	−0.276269	−	0.961080i	$-0.589098\pi$
$127$	−6.22679	−0.552538	−0.276269	+	0.961080i	$0.589098\pi$
$128$	−8.08462	−0.714586
$129$	8.23207	0.724794
$130$	0	0
$131$	−11.9362	−1.04287	−0.521436	−	0.853291i	$-0.674603\pi$
$131$	−11.9362	−1.04287	−0.521436	+	0.853291i	$0.674603\pi$
$132$	0.158608	0.0138050
$133$	−0.912501	−0.0791239
$134$	−4.28791	−0.370419
$135$	0	0
$136$	−4.50159	−0.386008
$137$	5.55617	0.474696	0.237348	−	0.971425i	$-0.423722\pi$
$137$	5.55617	0.474696	0.237348	+	0.971425i	$0.423722\pi$
$138$	−9.87749	−0.840828
$139$	2.86890	0.243337	0.121668	−	0.992571i	$-0.461176\pi$
$139$	2.86890	0.243337	0.121668	+	0.992571i	$0.461176\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	8.32124	0.698303
$143$	−0.685583	−0.0573313
$144$	−4.81382	−0.401151
$145$	0	0
$146$	−24.1695	−2.00028
$147$	0.833398	0.0687375
$148$	2.13406	0.175418
$149$	10.6867	0.875493	0.437746	−	0.899099i	$-0.355777\pi$
$149$	10.6867	0.875493	0.437746	+	0.899099i	$0.355777\pi$
$150$	0	0
$151$	19.8170	1.61268	0.806340	−	0.591452i	$-0.201445\pi$
$151$	19.8170	1.61268	0.806340	+	0.591452i	$0.201445\pi$
$152$	−0.343351	−0.0278494
$153$	−4.27451	−0.345574
$154$	−0.574450	−0.0462905
$155$	0	0
$156$	−8.85753	−0.709170
$157$	17.6734	1.41049	0.705246	−	0.708962i	$-0.250836\pi$
$157$	17.6734	1.41049	0.705246	+	0.708962i	$0.250836\pi$
$158$	21.8034	1.73458
$159$	10.6703	0.846211
$160$	0	0
$161$	14.9238	1.17616
$162$	−1.85243	−0.145541
$163$	−18.5616	−1.45386	−0.726928	−	0.686714i	$-0.759053\pi$
$163$	−18.5616	−1.45386	−0.726928	+	0.686714i	$0.759053\pi$
$164$	15.5234	1.21217
$165$	0	0
$166$	−31.9519	−2.47995
$167$	−11.3201	−0.875974	−0.437987	−	0.898981i	$-0.644308\pi$
$167$	−11.3201	−0.875974	−0.437987	+	0.898981i	$0.644308\pi$
$168$	2.94750	0.227405
$169$	25.2868	1.94514
$170$	0	0
$171$	−0.326031	−0.0249322
$172$	11.7841	0.898532
$173$	7.40410	0.562923	0.281462	−	0.959572i	$-0.409181\pi$
$173$	7.40410	0.562923	0.281462	+	0.959572i	$0.409181\pi$
$174$	16.1024	1.22072
$175$	0	0
$176$	−0.533366	−0.0402039
$177$	7.90884	0.594465
$178$	−9.33887	−0.699978
$179$	−7.94693	−0.593981	−0.296991	−	0.954880i	$-0.595983\pi$
$179$	−7.94693	−0.593981	−0.296991	+	0.954880i	$0.595983\pi$
$180$	0	0
$181$	−2.73162	−0.203040	−0.101520	−	0.994834i	$-0.532371\pi$
$181$	−2.73162	−0.203040	−0.101520	+	0.994834i	$0.532371\pi$
$182$	32.0805	2.37796
$183$	9.66552	0.714496
$184$	5.61545	0.413977
$185$	0	0
$186$	9.28556	0.680850
$187$	−0.473611	−0.0346339
$188$	1.43149	0.104402
$189$	2.79882	0.203584
$190$	0	0
$191$	12.3177	0.891279	0.445639	−	0.895213i	$-0.352976\pi$
$191$	12.3177	0.891279	0.445639	+	0.895213i	$0.352976\pi$
$192$	−2.98926	−0.215731
$193$	19.8624	1.42973	0.714864	−	0.699263i	$-0.246489\pi$
$193$	19.8624	1.42973	0.714864	+	0.699263i	$0.246489\pi$
$194$	13.9837	1.00397
$195$	0	0
$196$	1.19300	0.0852143
$197$	−8.64339	−0.615816	−0.307908	−	0.951416i	$-0.599629\pi$
$197$	−8.64339	−0.615816	−0.307908	+	0.951416i	$0.599629\pi$
$198$	−0.205247	−0.0145863
$199$	−6.75750	−0.479026	−0.239513	−	0.970893i	$-0.576988\pi$
$199$	−6.75750	−0.479026	−0.239513	+	0.970893i	$0.576988\pi$
$200$	0	0
$201$	2.31475	0.163270
$202$	−25.8159	−1.81640
$203$	−24.3290	−1.70756
$204$	−6.11892	−0.428410
$205$	0	0
$206$	−16.8702	−1.17540
$207$	5.33219	0.370613
$208$	29.7861	2.06530
$209$	−0.0361238	−0.00249874
$210$	0	0
$211$	10.4191	0.717281	0.358640	−	0.933476i	$-0.383240\pi$
$211$	10.4191	0.717281	0.358640	+	0.933476i	$0.383240\pi$
$212$	15.2744	1.04905
$213$	−4.49207	−0.307792
$214$	−28.4958	−1.94793
$215$	0	0
$216$	1.05312	0.0716560
$217$	−14.0295	−0.952383
$218$	36.2150	2.45279
$219$	13.0475	0.881667
$220$	0	0
$221$	26.4491	1.77916
$222$	−2.76159	−0.185346
$223$	−25.4295	−1.70288	−0.851441	−	0.524450i	$-0.824271\pi$
$223$	−25.4295	−1.70288	−0.851441	+	0.524450i	$0.824271\pi$
$224$	19.0628	1.27369
$225$	0	0
$226$	−21.8344	−1.45240
$227$	−0.237644	−0.0157730	−0.00788651	−	0.999969i	$-0.502510\pi$
$227$	−0.237644	−0.0157730	−0.00788651	+	0.999969i	$0.502510\pi$
$228$	−0.466710	−0.0309086
$229$	−1.04194	−0.0688534	−0.0344267	−	0.999407i	$-0.510961\pi$
$229$	−1.04194	−0.0688534	−0.0344267	+	0.999407i	$0.510961\pi$
$230$	0	0
$231$	0.310106	0.0204035
$232$	−9.15436	−0.601013
$233$	2.65399	0.173869	0.0869343	−	0.996214i	$-0.472293\pi$
$233$	2.65399	0.173869	0.0869343	+	0.996214i	$0.472293\pi$
$234$	11.4621	0.749304
$235$	0	0
$236$	11.3214	0.736962
$237$	−11.7702	−0.764555
$238$	22.1617	1.43653
$239$	−3.52751	−0.228176	−0.114088	−	0.993471i	$-0.536395\pi$
$239$	−3.52751	−0.228176	−0.114088	+	0.993471i	$0.536395\pi$
$240$	0	0
$241$	11.4873	0.739963	0.369981	−	0.929039i	$-0.379364\pi$
$241$	11.4873	0.739963	0.369981	+	0.929039i	$0.379364\pi$
$242$	20.3540	1.30840
$243$	1.00000	0.0641500
$244$	13.8361	0.885765
$245$	0	0
$246$	−20.0881	−1.28077
$247$	2.01736	0.128361
$248$	−5.27893	−0.335212
$249$	17.2486	1.09309
$250$	0	0
$251$	−15.6108	−0.985344	−0.492672	−	0.870215i	$-0.663980\pi$
$251$	−15.6108	−0.985344	−0.492672	+	0.870215i	$0.663980\pi$
$252$	4.00648	0.252385
$253$	0.590800	0.0371433
$254$	11.5347	0.723750
$255$	0	0
$256$	20.9547	1.30967
$257$	17.1122	1.06743	0.533715	−	0.845665i	$-0.320796\pi$
$257$	17.1122	1.06743	0.533715	+	0.845665i	$0.320796\pi$
$258$	−15.2493	−0.949382
$259$	4.17246	0.259264
$260$	0	0
$261$	−8.69258	−0.538057
$262$	22.1110	1.36602
$263$	4.66446	0.287623	0.143811	−	0.989605i	$-0.454064\pi$
$263$	4.66446	0.287623	0.143811	+	0.989605i	$0.454064\pi$
$264$	0.116685	0.00718146
$265$	0	0
$266$	1.69034	0.103642
$267$	5.04142	0.308530
$268$	3.31354	0.202407
$269$	20.9745	1.27884	0.639420	−	0.768858i	$-0.279175\pi$
$269$	20.9745	1.27884	0.639420	+	0.768858i	$0.279175\pi$
$270$	0	0
$271$	−7.73864	−0.470089	−0.235045	−	0.971985i	$-0.575524\pi$
$271$	−7.73864	−0.470089	−0.235045	+	0.971985i	$0.575524\pi$
$272$	20.5767	1.24765
$273$	−17.3181	−1.04814
$274$	−10.2924	−0.621787
$275$	0	0
$276$	7.63297	0.459451
$277$	−9.08992	−0.546161	−0.273080	−	0.961991i	$-0.588042\pi$
$277$	−9.08992	−0.546161	−0.273080	+	0.961991i	$0.588042\pi$
$278$	−5.31443	−0.318738
$279$	−5.01264	−0.300099
$280$	0	0
$281$	22.4182	1.33736	0.668678	−	0.743552i	$-0.266860\pi$
$281$	22.4182	1.33736	0.668678	+	0.743552i	$0.266860\pi$
$282$	−1.85243	−0.110310
$283$	5.26529	0.312989	0.156495	−	0.987679i	$-0.449981\pi$
$283$	5.26529	0.312989	0.156495	+	0.987679i	$0.449981\pi$
$284$	−6.43036	−0.381572
$285$	0	0
$286$	1.26999	0.0750962
$287$	30.3510	1.79156
$288$	6.81100	0.401342
$289$	1.27145	0.0747909
$290$	0	0
$291$	−7.54883	−0.442520
$292$	18.6773	1.09301
$293$	−21.1732	−1.23695	−0.618475	−	0.785804i	$-0.712249\pi$
$293$	−21.1732	−1.23695	−0.618475	+	0.785804i	$0.712249\pi$
$294$	−1.54381	−0.0900368
$295$	0	0
$296$	1.56999	0.0912538
$297$	0.110799	0.00642920
$298$	−19.7964	−1.14678
$299$	−32.9936	−1.90807
$300$	0	0
$301$	23.0401	1.32801
$302$	−36.7095	−2.11239
$303$	13.9362	0.800616
$304$	1.56945	0.0900142
$305$	0	0
$306$	7.91822	0.452655
$307$	3.64891	0.208254	0.104127	−	0.994564i	$-0.466795\pi$
$307$	3.64891	0.208254	0.104127	+	0.994564i	$0.466795\pi$
$308$	0.443914	0.0252943
$309$	9.10706	0.518082
$310$	0	0
$311$	−22.4701	−1.27416	−0.637082	−	0.770796i	$-0.719859\pi$
$311$	−22.4701	−1.27416	−0.637082	+	0.770796i	$0.719859\pi$
$312$	−6.51634	−0.368915
$313$	−15.6741	−0.885952	−0.442976	−	0.896533i	$-0.646077\pi$
$313$	−15.6741	−0.885952	−0.442976	+	0.896533i	$0.646077\pi$
$314$	−32.7387	−1.84755
$315$	0	0
$316$	−16.8489	−0.947824
$317$	−28.0756	−1.57688	−0.788441	−	0.615110i	$-0.789112\pi$
$317$	−28.0756	−1.57688	−0.788441	+	0.615110i	$0.789112\pi$
$318$	−19.7660	−1.10842
$319$	−0.963128	−0.0539248
$320$	0	0
$321$	15.3830	0.858593
$322$	−27.6453	−1.54061
$323$	1.39362	0.0775432
$324$	1.43149	0.0795272
$325$	0	0
$326$	34.3840	1.90435
$327$	−19.5500	−1.08112
$328$	11.4203	0.630581
$329$	2.79882	0.154304
$330$	0	0
$331$	14.2874	0.785305	0.392653	−	0.919687i	$-0.371557\pi$
$331$	14.2874	0.785305	0.392653	+	0.919687i	$0.371557\pi$
$332$	24.6913	1.35511
$333$	1.49079	0.0816950
$334$	20.9696	1.14741
$335$	0	0
$336$	−13.4730	−0.735013
$337$	21.5968	1.17645	0.588227	−	0.808696i	$-0.299826\pi$
$337$	21.5968	1.17645	0.588227	+	0.808696i	$0.299826\pi$
$338$	−46.8419	−2.54787
$339$	11.7869	0.640178
$340$	0	0
$341$	−0.555395	−0.0300763
$342$	0.603948	0.0326578
$343$	−17.2592	−0.931910
$344$	8.66939	0.467422
$345$	0	0
$346$	−13.7156	−0.737353
$347$	7.13924	0.383254	0.191627	−	0.981468i	$-0.438624\pi$
$347$	7.13924	0.383254	0.191627	+	0.981468i	$0.438624\pi$
$348$	−12.4433	−0.667033
$349$	28.1558	1.50715	0.753574	−	0.657364i	$-0.228328\pi$
$349$	28.1558	1.50715	0.753574	+	0.657364i	$0.228328\pi$
$350$	0	0
$351$	−6.18763	−0.330271
$352$	0.754651	0.0402231
$353$	−14.0360	−0.747061	−0.373530	−	0.927618i	$-0.621853\pi$
$353$	−14.0360	−0.747061	−0.373530	+	0.927618i	$0.621853\pi$
$354$	−14.6506	−0.778669
$355$	0	0
$356$	7.21674	0.382486
$357$	−11.9636	−0.633180
$358$	14.7211	0.778035
$359$	17.1517	0.905230	0.452615	−	0.891706i	$-0.350491\pi$
$359$	17.1517	0.905230	0.452615	+	0.891706i	$0.350491\pi$
$360$	0	0
$361$	−18.8937	−0.994405
$362$	5.06013	0.265955
$363$	−10.9877	−0.576706
$364$	−24.7906	−1.29938
$365$	0	0
$366$	−17.9047	−0.935892
$367$	−9.03906	−0.471835	−0.235918	−	0.971773i	$-0.575810\pi$
$367$	−9.03906	−0.471835	−0.235918	+	0.971773i	$0.575810\pi$
$368$	−25.6682	−1.33805
$369$	10.8442	0.564528
$370$	0	0
$371$	29.8643	1.55048
$372$	−7.17554	−0.372035
$373$	−16.8787	−0.873944	−0.436972	−	0.899475i	$-0.643949\pi$
$373$	−16.8787	−0.873944	−0.436972	+	0.899475i	$0.643949\pi$
$374$	0.877331	0.0453657
$375$	0	0
$376$	1.05312	0.0543107
$377$	53.7865	2.77014
$378$	−5.18461	−0.266668
$379$	−20.0643	−1.03064	−0.515318	−	0.856999i	$-0.672326\pi$
$379$	−20.0643	−1.03064	−0.515318	+	0.856999i	$0.672326\pi$
$380$	0	0
$381$	−6.22679	−0.319008
$382$	−22.8177	−1.16745
$383$	−5.87439	−0.300167	−0.150084	−	0.988673i	$-0.547954\pi$
$383$	−5.87439	−0.300167	−0.150084	+	0.988673i	$0.547954\pi$
$384$	−8.08462	−0.412566
$385$	0	0
$386$	−36.7937	−1.87275
$387$	8.23207	0.418460
$388$	−10.8061	−0.548595
$389$	11.7348	0.594976	0.297488	−	0.954725i	$-0.403851\pi$
$389$	11.7348	0.594976	0.297488	+	0.954725i	$0.403851\pi$
$390$	0	0
$391$	−22.7925	−1.15267
$392$	0.877671	0.0443291
$393$	−11.9362	−0.602102
$394$	16.0113	0.806635
$395$	0	0
$396$	0.158608	0.00797033
$397$	21.3662	1.07234	0.536171	−	0.844110i	$-0.319870\pi$
$397$	21.3662	1.07234	0.536171	+	0.844110i	$0.319870\pi$
$398$	12.5178	0.627460
$399$	−0.912501	−0.0456822
$400$	0	0
$401$	1.57801	0.0788023	0.0394011	−	0.999223i	$-0.487455\pi$
$401$	1.57801	0.0788023	0.0394011	+	0.999223i	$0.487455\pi$
$402$	−4.28791	−0.213862
$403$	31.0164	1.54504
$404$	19.9496	0.992529
$405$	0	0
$406$	45.0677	2.23667
$407$	0.165178	0.00818758
$408$	−4.50159	−0.222862
$409$	−32.6960	−1.61671	−0.808357	−	0.588693i	$-0.799643\pi$
$409$	−32.6960	−1.61671	−0.808357	+	0.588693i	$0.799643\pi$
$410$	0	0
$411$	5.55617	0.274066
$412$	13.0367	0.642270
$413$	22.1354	1.08921
$414$	−9.87749	−0.485452
$415$	0	0
$416$	−42.1440	−2.06628
$417$	2.86890	0.140490
$418$	0.0669168	0.00327301
$419$	26.7978	1.30916	0.654579	−	0.755993i	$-0.272846\pi$
$419$	26.7978	1.30916	0.654579	+	0.755993i	$0.272846\pi$
$420$	0	0
$421$	−34.3115	−1.67224	−0.836121	−	0.548546i	$-0.815182\pi$
$421$	−34.3115	−1.67224	−0.836121	+	0.548546i	$0.815182\pi$
$422$	−19.3006	−0.939541
$423$	1.00000	0.0486217
$424$	11.2372	0.545725
$425$	0	0
$426$	8.32124	0.403166
$427$	27.0521	1.30914
$428$	22.0206	1.06440
$429$	−0.685583	−0.0331002
$430$	0	0
$431$	30.4472	1.46659	0.733294	−	0.679912i	$-0.237982\pi$
$431$	30.4472	1.46659	0.733294	+	0.679912i	$0.237982\pi$
$432$	−4.81382	−0.231605
$433$	23.8613	1.14670	0.573350	−	0.819311i	$-0.305644\pi$
$433$	23.8613	1.14670	0.573350	+	0.819311i	$0.305644\pi$
$434$	25.9886	1.24749
$435$	0	0
$436$	−27.9856	−1.34027
$437$	−1.73846	−0.0831616
$438$	−24.1695	−1.15486
$439$	32.0758	1.53089	0.765447	−	0.643499i	$-0.222518\pi$
$439$	32.0758	1.53089	0.765447	+	0.643499i	$0.222518\pi$
$440$	0	0
$441$	0.833398	0.0396856
$442$	−48.9951	−2.33046
$443$	16.5191	0.784847	0.392424	−	0.919785i	$-0.371637\pi$
$443$	16.5191	0.784847	0.392424	+	0.919785i	$0.371637\pi$
$444$	2.13406	0.101278
$445$	0	0
$446$	47.1062	2.23055
$447$	10.6867	0.505466
$448$	−8.36640	−0.395275
$449$	−28.5393	−1.34685	−0.673427	−	0.739254i	$-0.735178\pi$
$449$	−28.5393	−1.34685	−0.673427	+	0.739254i	$0.735178\pi$
$450$	0	0
$451$	1.20153	0.0565777
$452$	16.8729	0.793633
$453$	19.8170	0.931082
$454$	0.440219	0.0206605
$455$	0	0
$456$	−0.343351	−0.0160789
$457$	11.2676	0.527076	0.263538	−	0.964649i	$-0.415111\pi$
$457$	11.2676	0.527076	0.263538	+	0.964649i	$0.415111\pi$
$458$	1.93012	0.0901886
$459$	−4.27451	−0.199517
$460$	0	0
$461$	28.5927	1.33170	0.665848	−	0.746088i	$-0.268070\pi$
$461$	28.5927	1.33170	0.665848	+	0.746088i	$0.268070\pi$
$462$	−0.574450	−0.0267258
$463$	22.9538	1.06675	0.533376	−	0.845878i	$-0.320923\pi$
$463$	22.9538	1.06675	0.533376	+	0.845878i	$0.320923\pi$
$464$	41.8445	1.94258
$465$	0	0
$466$	−4.91632	−0.227744
$467$	31.0778	1.43811	0.719054	−	0.694954i	$-0.244575\pi$
$467$	31.0778	1.43811	0.719054	+	0.694954i	$0.244575\pi$
$468$	−8.85753	−0.409440
$469$	6.47857	0.299153
$470$	0	0
$471$	17.6734	0.814348
$472$	8.32899	0.383373
$473$	0.912105	0.0419386
$474$	21.8034	1.00146
$475$	0	0
$476$	−17.1258	−0.784958
$477$	10.6703	0.488560
$478$	6.53446	0.298879
$479$	−30.2841	−1.38371	−0.691857	−	0.722034i	$-0.743207\pi$
$479$	−30.2841	−1.38371	−0.691857	+	0.722034i	$0.743207\pi$
$480$	0	0
$481$	−9.22448	−0.420600
$482$	−21.2794	−0.969251
$483$	14.9238	0.679058
$484$	−15.7288	−0.714946
$485$	0	0
$486$	−1.85243	−0.0840278
$487$	−12.6556	−0.573479	−0.286740	−	0.958009i	$-0.592571\pi$
$487$	−12.6556	−0.573479	−0.286740	+	0.958009i	$0.592571\pi$
$488$	10.1790	0.460781
$489$	−18.5616	−0.839384
$490$	0	0
$491$	16.8056	0.758426	0.379213	−	0.925309i	$-0.376195\pi$
$491$	16.8056	0.758426	0.379213	+	0.925309i	$0.376195\pi$
$492$	15.5234	0.699849
$493$	37.1565	1.67345
$494$	−3.73701	−0.168136
$495$	0	0
$496$	24.1299	1.08347
$497$	−12.5725	−0.563954
$498$	−31.9519	−1.43180
$499$	−10.2138	−0.457233	−0.228617	−	0.973517i	$-0.573420\pi$
$499$	−10.2138	−0.457233	−0.228617	+	0.973517i	$0.573420\pi$
$500$	0	0
$501$	−11.3201	−0.505744
$502$	28.9179	1.29067
$503$	8.50743	0.379327	0.189664	−	0.981849i	$-0.439260\pi$
$503$	8.50743	0.379327	0.189664	+	0.981849i	$0.439260\pi$
$504$	2.94750	0.131292
$505$	0	0
$506$	−1.09442	−0.0486527
$507$	25.2868	1.12303
$508$	−8.91358	−0.395476
$509$	−2.86892	−0.127162	−0.0635812	−	0.997977i	$-0.520252\pi$
$509$	−2.86892	−0.127162	−0.0635812	+	0.997977i	$0.520252\pi$
$510$	0	0
$511$	36.5176	1.61544
$512$	−22.6478	−1.00090
$513$	−0.326031	−0.0143946
$514$	−31.6991	−1.39819
$515$	0	0
$516$	11.7841	0.518768
$517$	0.110799	0.00487293
$518$	−7.72919	−0.339601
$519$	7.40410	0.325004
$520$	0	0
$521$	−35.3430	−1.54841	−0.774203	−	0.632937i	$-0.781849\pi$
$521$	−35.3430	−1.54841	−0.774203	+	0.632937i	$0.781849\pi$
$522$	16.1024	0.704782
$523$	4.53636	0.198361	0.0991806	−	0.995069i	$-0.468378\pi$
$523$	4.53636	0.198361	0.0991806	+	0.995069i	$0.468378\pi$
$524$	−17.0866	−0.746430
$525$	0	0
$526$	−8.64058	−0.376747
$527$	21.4266	0.933357
$528$	−0.533366	−0.0232118
$529$	5.43221	0.236183
$530$	0	0
$531$	7.90884	0.343215
$532$	−1.30624	−0.0566325
$533$	−67.1000	−2.90643
$534$	−9.33887	−0.404132
$535$	0	0
$536$	2.43772	0.105293
$537$	−7.94693	−0.342935
$538$	−38.8538	−1.67511
$539$	0.0923396	0.00397735
$540$	0	0
$541$	43.0858	1.85240	0.926201	−	0.377031i	$-0.123055\pi$
$541$	43.0858	1.85240	0.926201	+	0.377031i	$0.123055\pi$
$542$	14.3353	0.615753
$543$	−2.73162	−0.117225
$544$	−29.1137	−1.24824
$545$	0	0
$546$	32.0805	1.37292
$547$	13.2062	0.564658	0.282329	−	0.959318i	$-0.408893\pi$
$547$	13.2062	0.564658	0.282329	+	0.959318i	$0.408893\pi$
$548$	7.95360	0.339761
$549$	9.66552	0.412514
$550$	0	0
$551$	2.83405	0.120734
$552$	5.61545	0.239010
$553$	−32.9426	−1.40086
$554$	16.8384	0.715396
$555$	0	0
$556$	4.10680	0.174167
$557$	0.0705045	0.00298737	0.00149369	−	0.999999i	$-0.499525\pi$
$557$	0.0705045	0.00298737	0.00149369	+	0.999999i	$0.499525\pi$
$558$	9.28556	0.393089
$559$	−50.9370	−2.15441
$560$	0	0
$561$	−0.473611	−0.0199959
$562$	−41.5281	−1.75176
$563$	−32.3255	−1.36236	−0.681179	−	0.732117i	$-0.738532\pi$
$563$	−32.3255	−1.36236	−0.681179	+	0.732117i	$0.738532\pi$
$564$	1.43149	0.0602766
$565$	0	0
$566$	−9.75358	−0.409973
$567$	2.79882	0.117539
$568$	−4.73071	−0.198496
$569$	−11.1856	−0.468924	−0.234462	−	0.972125i	$-0.575333\pi$
$569$	−11.1856	−0.468924	−0.234462	+	0.972125i	$0.575333\pi$
$570$	0	0
$571$	−17.9528	−0.751303	−0.375651	−	0.926761i	$-0.622581\pi$
$571$	−17.9528	−0.751303	−0.375651	+	0.926761i	$0.622581\pi$
$572$	−0.981405	−0.0410346
$573$	12.3177	0.514580
$574$	−56.2231	−2.34671
$575$	0	0
$576$	−2.98926	−0.124552
$577$	−28.6521	−1.19280	−0.596402	−	0.802686i	$-0.703404\pi$
$577$	−28.6521	−1.19280	−0.596402	+	0.802686i	$0.703404\pi$
$578$	−2.35526	−0.0979660
$579$	19.8624	0.825454
$580$	0	0
$581$	48.2759	2.00282
$582$	13.9837	0.579641
$583$	1.18226	0.0489642
$584$	13.7406	0.568591
$585$	0	0
$586$	39.2218	1.62024
$587$	−13.9680	−0.576522	−0.288261	−	0.957552i	$-0.593077\pi$
$587$	−13.9680	−0.576522	−0.288261	+	0.957552i	$0.593077\pi$
$588$	1.19300	0.0491985
$589$	1.63427	0.0673391
$590$	0	0
$591$	−8.64339	−0.355541
$592$	−7.17641	−0.294949
$593$	8.88459	0.364846	0.182423	−	0.983220i	$-0.441606\pi$
$593$	8.88459	0.364846	0.182423	+	0.983220i	$0.441606\pi$
$594$	−0.205247	−0.00842139
$595$	0	0
$596$	15.2980	0.626629
$597$	−6.75750	−0.276566
$598$	61.1183	2.49931
$599$	8.06800	0.329650	0.164825	−	0.986323i	$-0.447294\pi$
$599$	8.06800	0.329650	0.164825	+	0.986323i	$0.447294\pi$
$600$	0	0
$601$	−13.6363	−0.556235	−0.278117	−	0.960547i	$-0.589710\pi$
$601$	−13.6363	−0.556235	−0.278117	+	0.960547i	$0.589710\pi$
$602$	−42.6801	−1.73951
$603$	2.31475	0.0942640
$604$	28.3678	1.15427
$605$	0	0
$606$	−25.8159	−1.04870
$607$	−18.5254	−0.751922	−0.375961	−	0.926635i	$-0.622687\pi$
$607$	−18.5254	−0.751922	−0.375961	+	0.926635i	$0.622687\pi$
$608$	−2.22059	−0.0900570
$609$	−24.3290	−0.985860
$610$	0	0
$611$	−6.18763	−0.250325
$612$	−6.11892	−0.247343
$613$	19.4600	0.785981	0.392991	−	0.919542i	$-0.371440\pi$
$613$	19.4600	0.785981	0.392991	+	0.919542i	$0.371440\pi$
$614$	−6.75935	−0.272785
$615$	0	0
$616$	0.326580	0.0131583
$617$	0.462654	0.0186258	0.00931288	−	0.999957i	$-0.497036\pi$
$617$	0.462654	0.0186258	0.00931288	+	0.999957i	$0.497036\pi$
$618$	−16.8702	−0.678618
$619$	23.6431	0.950297	0.475149	−	0.879906i	$-0.342394\pi$
$619$	23.6431	0.950297	0.475149	+	0.879906i	$0.342394\pi$
$620$	0	0
$621$	5.33219	0.213973
$622$	41.6243	1.66898
$623$	14.1100	0.565306
$624$	29.7861	1.19240
$625$	0	0
$626$	29.0351	1.16048
$627$	−0.0361238	−0.00144265
$628$	25.2993	1.00955
$629$	−6.37241	−0.254085
$630$	0	0
$631$	11.6511	0.463821	0.231911	−	0.972737i	$-0.425502\pi$
$631$	11.6511	0.463821	0.231911	+	0.972737i	$0.425502\pi$
$632$	−12.3954	−0.493064
$633$	10.4191	0.414122
$634$	52.0080	2.06550
$635$	0	0
$636$	15.2744	0.605671
$637$	−5.15676	−0.204318
$638$	1.78413	0.0706342
$639$	−4.49207	−0.177704
$640$	0	0
$641$	8.05056	0.317978	0.158989	−	0.987280i	$-0.449177\pi$
$641$	8.05056	0.317978	0.158989	+	0.987280i	$0.449177\pi$
$642$	−28.4958	−1.12464
$643$	24.4378	0.963732	0.481866	−	0.876245i	$-0.339959\pi$
$643$	24.4378	0.963732	0.481866	+	0.876245i	$0.339959\pi$
$644$	21.3633	0.841833
$645$	0	0
$646$	−2.58158	−0.101571
$647$	−49.9579	−1.96405	−0.982023	−	0.188759i	$-0.939553\pi$
$647$	−49.9579	−1.96405	−0.982023	+	0.188759i	$0.939553\pi$
$648$	1.05312	0.0413706
$649$	0.876291	0.0343974
$650$	0	0
$651$	−14.0295	−0.549859
$652$	−26.5707	−1.04059
$653$	−0.319214	−0.0124918	−0.00624591	−	0.999980i	$-0.501988\pi$
$653$	−0.319214	−0.0124918	−0.00624591	+	0.999980i	$0.501988\pi$
$654$	36.2150	1.41612
$655$	0	0
$656$	−52.2021	−2.03815
$657$	13.0475	0.509031
$658$	−5.18461	−0.202117
$659$	−34.4879	−1.34346	−0.671728	−	0.740798i	$-0.734448\pi$
$659$	−34.4879	−1.34346	−0.671728	+	0.740798i	$0.734448\pi$
$660$	0	0
$661$	−13.1530	−0.511593	−0.255797	−	0.966731i	$-0.582338\pi$
$661$	−13.1530	−0.511593	−0.255797	+	0.966731i	$0.582338\pi$
$662$	−26.4663	−1.02864
$663$	26.4491	1.02720
$664$	18.1650	0.704937
$665$	0	0
$666$	−2.76159	−0.107009
$667$	−46.3505	−1.79470
$668$	−16.2046	−0.626974
$669$	−25.4295	−0.983160
$670$	0	0
$671$	1.07093	0.0413427
$672$	19.0628	0.735362
$673$	27.9380	1.07693	0.538466	−	0.842647i	$-0.319004\pi$
$673$	27.9380	1.07693	0.538466	+	0.842647i	$0.319004\pi$
$674$	−40.0066	−1.54100
$675$	0	0
$676$	36.1978	1.39222
$677$	13.9989	0.538021	0.269010	−	0.963137i	$-0.413303\pi$
$677$	13.9989	0.538021	0.269010	+	0.963137i	$0.413303\pi$
$678$	−21.8344	−0.838546
$679$	−21.1278	−0.810811
$680$	0	0
$681$	−0.237644	−0.00910656
$682$	1.02883	0.0393959
$683$	10.4197	0.398700	0.199350	−	0.979928i	$-0.436117\pi$
$683$	10.4197	0.398700	0.199350	+	0.979928i	$0.436117\pi$
$684$	−0.466710	−0.0178451
$685$	0	0
$686$	31.9715	1.22068
$687$	−1.04194	−0.0397525
$688$	−39.6277	−1.51079
$689$	−66.0240	−2.51531
$690$	0	0
$691$	−17.0218	−0.647541	−0.323771	−	0.946136i	$-0.604951\pi$
$691$	−17.0218	−0.647541	−0.323771	+	0.946136i	$0.604951\pi$
$692$	10.5989	0.402910
$693$	0.310106	0.0117800
$694$	−13.2249	−0.502011
$695$	0	0
$696$	−9.15436	−0.346995
$697$	−46.3537	−1.75577
$698$	−52.1567	−1.97416
$699$	2.65399	0.100383
$700$	0	0
$701$	−29.7666	−1.12427	−0.562135	−	0.827045i	$-0.690020\pi$
$701$	−29.7666	−1.12427	−0.562135	+	0.827045i	$0.690020\pi$
$702$	11.4621	0.432611
$703$	−0.486044	−0.0183315
$704$	−0.331206	−0.0124828
$705$	0	0
$706$	26.0007	0.978548
$707$	39.0050	1.46694
$708$	11.3214	0.425485
$709$	−32.8707	−1.23449	−0.617243	−	0.786773i	$-0.711750\pi$
$709$	−32.8707	−1.23449	−0.617243	+	0.786773i	$0.711750\pi$
$710$	0	0
$711$	−11.7702	−0.441416
$712$	5.30924	0.198972
$713$	−26.7283	−1.00098
$714$	22.1617	0.829380
$715$	0	0
$716$	−11.3759	−0.425139
$717$	−3.52751	−0.131737
$718$	−31.7722	−1.18573
$719$	23.8857	0.890786	0.445393	−	0.895335i	$-0.353064\pi$
$719$	23.8857	0.890786	0.445393	+	0.895335i	$0.353064\pi$
$720$	0	0
$721$	25.4890	0.949261
$722$	34.9992	1.30254
$723$	11.4873	0.427218
$724$	−3.91029	−0.145325
$725$	0	0
$726$	20.3540	0.755407
$727$	−11.3100	−0.419465	−0.209732	−	0.977759i	$-0.567259\pi$
$727$	−11.3100	−0.419465	−0.209732	+	0.977759i	$0.567259\pi$
$728$	−18.2381	−0.675948
$729$	1.00000	0.0370370
$730$	0	0
$731$	−35.1881	−1.30148
$732$	13.8361	0.511397
$733$	9.60626	0.354815	0.177408	−	0.984137i	$-0.443229\pi$
$733$	9.60626	0.354815	0.177408	+	0.984137i	$0.443229\pi$
$734$	16.7442	0.618040
$735$	0	0
$736$	36.3175	1.33868
$737$	0.256472	0.00944726
$738$	−20.0881	−0.739455
$739$	25.1094	0.923665	0.461833	−	0.886967i	$-0.347192\pi$
$739$	25.1094	0.923665	0.461833	+	0.886967i	$0.347192\pi$
$740$	0	0
$741$	2.01736	0.0741095
$742$	−55.3215	−2.03091
$743$	5.32029	0.195183	0.0975913	−	0.995227i	$-0.468886\pi$
$743$	5.32029	0.195183	0.0975913	+	0.995227i	$0.468886\pi$
$744$	−5.27893	−0.193535
$745$	0	0
$746$	31.2665	1.14475
$747$	17.2486	0.631095
$748$	−0.677970	−0.0247890
$749$	43.0541	1.57316
$750$	0	0
$751$	−32.8005	−1.19691	−0.598453	−	0.801158i	$-0.704218\pi$
$751$	−32.8005	−1.19691	−0.598453	+	0.801158i	$0.704218\pi$
$752$	−4.81382	−0.175542
$753$	−15.6108	−0.568889
$754$	−99.6356	−3.62851
$755$	0	0
$756$	4.00648	0.145714
$757$	−27.8566	−1.01246	−0.506232	−	0.862397i	$-0.668962\pi$
$757$	−27.8566	−1.01246	−0.506232	+	0.862397i	$0.668962\pi$
$758$	37.1677	1.34999
$759$	0.590800	0.0214447
$760$	0	0
$761$	−38.6841	−1.40230	−0.701150	−	0.713014i	$-0.747329\pi$
$761$	−38.6841	−1.40230	−0.701150	+	0.713014i	$0.747329\pi$
$762$	11.5347	0.417857
$763$	−54.7170	−1.98089
$764$	17.6327	0.637928
$765$	0	0
$766$	10.8819	0.393179
$767$	−48.9370	−1.76701
$768$	20.9547	0.756137
$769$	21.3157	0.768663	0.384331	−	0.923195i	$-0.374432\pi$
$769$	21.3157	0.768663	0.384331	+	0.923195i	$0.374432\pi$
$770$	0	0
$771$	17.1122	0.616280
$772$	28.4329	1.02332
$773$	−43.1498	−1.55199	−0.775995	−	0.630739i	$-0.782752\pi$
$773$	−43.1498	−1.55199	−0.775995	+	0.630739i	$0.782752\pi$
$774$	−15.2493	−0.548126
$775$	0	0
$776$	−7.94985	−0.285383
$777$	4.17246	0.149686
$778$	−21.7378	−0.779339
$779$	−3.53555	−0.126674
$780$	0	0
$781$	−0.497717	−0.0178097
$782$	42.2214	1.50984
$783$	−8.69258	−0.310647
$784$	−4.01182	−0.143279
$785$	0	0
$786$	22.1110	0.788672
$787$	−24.4813	−0.872663	−0.436332	−	0.899786i	$-0.643723\pi$
$787$	−24.4813	−0.872663	−0.436332	+	0.899786i	$0.643723\pi$
$788$	−12.3729	−0.440767
$789$	4.66446	0.166059
$790$	0	0
$791$	32.9895	1.17297
$792$	0.116685	0.00414622
$793$	−59.8067	−2.12380
$794$	−39.5794	−1.40462
$795$	0	0
$796$	−9.67329	−0.342861
$797$	20.0382	0.709789	0.354894	−	0.934906i	$-0.384517\pi$
$797$	20.0382	0.709789	0.354894	+	0.934906i	$0.384517\pi$
$798$	1.69034	0.0598375
$799$	−4.27451	−0.151221
$800$	0	0
$801$	5.04142	0.178130
$802$	−2.92316	−0.103220
$803$	1.44565	0.0510158
$804$	3.31354	0.116860
$805$	0	0
$806$	−57.4556	−2.02379
$807$	20.9745	0.738338
$808$	14.6766	0.516320
$809$	11.6522	0.409670	0.204835	−	0.978796i	$-0.434334\pi$
$809$	11.6522	0.409670	0.204835	+	0.978796i	$0.434334\pi$
$810$	0	0
$811$	−10.1104	−0.355024	−0.177512	−	0.984119i	$-0.556805\pi$
$811$	−10.1104	−0.355024	−0.177512	+	0.984119i	$0.556805\pi$
$812$	−34.8267	−1.22218
$813$	−7.73864	−0.271406
$814$	−0.305981	−0.0107246
$815$	0	0
$816$	20.5767	0.720329
$817$	−2.68391	−0.0938981
$818$	60.5670	2.11768
$819$	−17.3181	−0.605142
$820$	0	0
$821$	2.48739	0.0868104	0.0434052	−	0.999058i	$-0.486179\pi$
$821$	2.48739	0.0868104	0.0434052	+	0.999058i	$0.486179\pi$
$822$	−10.2924	−0.358989
$823$	−0.648409	−0.0226021	−0.0113011	−	0.999936i	$-0.503597\pi$
$823$	−0.648409	−0.0226021	−0.0113011	+	0.999936i	$0.503597\pi$
$824$	9.59086	0.334113
$825$	0	0
$826$	−41.0043	−1.42672
$827$	−23.6590	−0.822705	−0.411352	−	0.911476i	$-0.634943\pi$
$827$	−23.6590	−0.822705	−0.411352	+	0.911476i	$0.634943\pi$
$828$	7.63297	0.265264
$829$	−9.72843	−0.337882	−0.168941	−	0.985626i	$-0.554035\pi$
$829$	−9.72843	−0.337882	−0.168941	+	0.985626i	$0.554035\pi$
$830$	0	0
$831$	−9.08992	−0.315326
$832$	18.4964	0.641248
$833$	−3.56237	−0.123429
$834$	−5.31443	−0.184024
$835$	0	0
$836$	−0.0517109	−0.00178846
$837$	−5.01264	−0.173262
$838$	−49.6410	−1.71482
$839$	18.9737	0.655045	0.327523	−	0.944843i	$-0.393786\pi$
$839$	18.9737	0.655045	0.327523	+	0.944843i	$0.393786\pi$
$840$	0	0
$841$	46.5609	1.60555
$842$	63.5596	2.19041
$843$	22.4182	0.772123
$844$	14.9148	0.513390
$845$	0	0
$846$	−1.85243	−0.0636878
$847$	−30.7527	−1.05667
$848$	−51.3649	−1.76388
$849$	5.26529	0.180704
$850$	0	0
$851$	7.94919	0.272495
$852$	−6.43036	−0.220300
$853$	42.6218	1.45934	0.729671	−	0.683799i	$-0.239673\pi$
$853$	42.6218	1.45934	0.729671	+	0.683799i	$0.239673\pi$
$854$	−50.1120	−1.71480
$855$	0	0
$856$	16.2002	0.553710
$857$	−34.4154	−1.17561	−0.587804	−	0.809003i	$-0.700007\pi$
$857$	−34.4154	−1.17561	−0.587804	+	0.809003i	$0.700007\pi$
$858$	1.26999	0.0433568
$859$	22.0619	0.752741	0.376370	−	0.926469i	$-0.377172\pi$
$859$	22.0619	0.752741	0.376370	+	0.926469i	$0.377172\pi$
$860$	0	0
$861$	30.3510	1.03436
$862$	−56.4012	−1.92103
$863$	−41.9460	−1.42786	−0.713930	−	0.700217i	$-0.753087\pi$
$863$	−41.9460	−1.42786	−0.713930	+	0.700217i	$0.753087\pi$
$864$	6.81100	0.231715
$865$	0	0
$866$	−44.2013	−1.50202
$867$	1.27145	0.0431806
$868$	−20.0831	−0.681664
$869$	−1.30412	−0.0442393
$870$	0	0
$871$	−14.3228	−0.485311
$872$	−20.5886	−0.697217
$873$	−7.54883	−0.255489
$874$	3.22036	0.108930
$875$	0	0
$876$	18.6773	0.631049
$877$	48.5215	1.63845	0.819227	−	0.573469i	$-0.194403\pi$
$877$	48.5215	1.63845	0.819227	+	0.573469i	$0.194403\pi$
$878$	−59.4181	−2.00526
$879$	−21.1732	−0.714154
$880$	0	0
$881$	54.5996	1.83951	0.919754	−	0.392495i	$-0.128388\pi$
$881$	54.5996	1.83951	0.919754	+	0.392495i	$0.128388\pi$
$882$	−1.54381	−0.0519828
$883$	−4.25858	−0.143313	−0.0716563	−	0.997429i	$-0.522828\pi$
$883$	−4.25858	−0.143313	−0.0716563	+	0.997429i	$0.522828\pi$
$884$	37.8616	1.27342
$885$	0	0
$886$	−30.6005	−1.02804
$887$	−14.5826	−0.489637	−0.244818	−	0.969569i	$-0.578728\pi$
$887$	−14.5826	−0.489637	−0.244818	+	0.969569i	$0.578728\pi$
$888$	1.56999	0.0526854
$889$	−17.4277	−0.584505
$890$	0	0
$891$	0.110799	0.00371190
$892$	−36.4020	−1.21883
$893$	−0.326031	−0.0109102
$894$	−19.7964	−0.662092
$895$	0	0
$896$	−22.6274	−0.755928
$897$	−32.9936	−1.10162
$898$	52.8670	1.76420
$899$	43.5728	1.45323
$900$	0	0
$901$	−45.6104	−1.51950
$902$	−2.22574	−0.0741091
$903$	23.0401	0.766727
$904$	12.4131	0.412853
$905$	0	0
$906$	−36.7095	−1.21959
$907$	55.6136	1.84662	0.923309	−	0.384058i	$-0.125474\pi$
$907$	55.6136	1.84662	0.923309	+	0.384058i	$0.125474\pi$
$908$	−0.340186	−0.0112895
$909$	13.9362	0.462236
$910$	0	0
$911$	16.0612	0.532132	0.266066	−	0.963955i	$-0.414276\pi$
$911$	16.0612	0.532132	0.266066	+	0.963955i	$0.414276\pi$
$912$	1.56945	0.0519697
$913$	1.91113	0.0632492
$914$	−20.8724	−0.690398
$915$	0	0
$916$	−1.49153	−0.0492815
$917$	−33.4073	−1.10321
$918$	7.91822	0.261340
$919$	−38.0066	−1.25372	−0.626861	−	0.779131i	$-0.715661\pi$
$919$	−38.0066	−1.25372	−0.626861	+	0.779131i	$0.715661\pi$
$920$	0	0
$921$	3.64891	0.120236
$922$	−52.9659	−1.74434
$923$	27.7953	0.914893
$924$	0.443914	0.0146037
$925$	0	0
$926$	−42.5202	−1.39730
$927$	9.10706	0.299115
$928$	−59.2052	−1.94351
$929$	−42.8736	−1.40664	−0.703318	−	0.710876i	$-0.748299\pi$
$929$	−42.8736	−1.40664	−0.703318	+	0.710876i	$0.748299\pi$
$930$	0	0
$931$	−0.271713	−0.00890504
$932$	3.79916	0.124446
$933$	−22.4701	−0.735638
$934$	−57.5693	−1.88373
$935$	0	0
$936$	−6.51634	−0.212993
$937$	−9.10412	−0.297419	−0.148709	−	0.988881i	$-0.547512\pi$
$937$	−9.10412	−0.297419	−0.148709	+	0.988881i	$0.547512\pi$
$938$	−12.0011	−0.391850
$939$	−15.6741	−0.511505
$940$	0	0
$941$	29.6419	0.966298	0.483149	−	0.875538i	$-0.339493\pi$
$941$	29.6419	0.966298	0.483149	+	0.875538i	$0.339493\pi$
$942$	−32.7387	−1.06669
$943$	57.8234	1.88299
$944$	−38.0717	−1.23913
$945$	0	0
$946$	−1.68961	−0.0549339
$947$	−16.9801	−0.551779	−0.275889	−	0.961189i	$-0.588972\pi$
$947$	−16.9801	−0.551779	−0.275889	+	0.961189i	$0.588972\pi$
$948$	−16.8489	−0.547226
$949$	−80.7330	−2.62071
$950$	0	0
$951$	−28.0756	−0.910414
$952$	−12.5991	−0.408341
$953$	−8.88843	−0.287924	−0.143962	−	0.989583i	$-0.545984\pi$
$953$	−8.88843	−0.287924	−0.143962	+	0.989583i	$0.545984\pi$
$954$	−19.7660	−0.639947
$955$	0	0
$956$	−5.04960	−0.163316
$957$	−0.963128	−0.0311335
$958$	56.0991	1.81248
$959$	15.5507	0.502159
$960$	0	0
$961$	−5.87344	−0.189466
$962$	17.0877	0.550929
$963$	15.3830	0.495709
$964$	16.4440	0.529625
$965$	0	0
$966$	−27.6453	−0.889474
$967$	51.1031	1.64336	0.821682	−	0.569946i	$-0.193036\pi$
$967$	51.1031	1.64336	0.821682	+	0.569946i	$0.193036\pi$
$968$	−11.5714	−0.371920
$969$	1.39362	0.0447696
$970$	0	0
$971$	37.7735	1.21221	0.606104	−	0.795385i	$-0.292731\pi$
$971$	37.7735	1.21221	0.606104	+	0.795385i	$0.292731\pi$
$972$	1.43149	0.0459151
$973$	8.02953	0.257415
$974$	23.4436	0.751180
$975$	0	0
$976$	−46.5280	−1.48933
$977$	25.3906	0.812316	0.406158	−	0.913803i	$-0.366868\pi$
$977$	25.3906	0.812316	0.406158	+	0.913803i	$0.366868\pi$
$978$	34.3840	1.09948
$979$	0.558584	0.0178524
$980$	0	0
$981$	−19.5500	−0.624184
$982$	−31.1312	−0.993436
$983$	−6.19401	−0.197558	−0.0987792	−	0.995109i	$-0.531494\pi$
$983$	−6.19401	−0.197558	−0.0987792	+	0.995109i	$0.531494\pi$
$984$	11.4203	0.364066
$985$	0	0
$986$	−68.8298	−2.19199
$987$	2.79882	0.0890874
$988$	2.88783	0.0918740
$989$	43.8950	1.39578
$990$	0	0
$991$	−18.6177	−0.591409	−0.295705	−	0.955279i	$-0.595554\pi$
$991$	−18.6177	−0.591409	−0.295705	+	0.955279i	$0.595554\pi$
$992$	−34.1411	−1.08398
$993$	14.2874	0.453396
$994$	23.2897	0.738704
$995$	0	0
$996$	24.6913	0.782373
$997$	−4.73093	−0.149830	−0.0749150	−	0.997190i	$-0.523869\pi$
$997$	−4.73093	−0.149830	−0.0749150	+	0.997190i	$0.523869\pi$
$998$	18.9204	0.598914
$999$	1.49079	0.0471666

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.z.1.2	✓	7
5.4	even	2		3525.2.a.ba.1.6	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3525.2.a.z.1.2	✓	7	1.1	even	1	trivial
3525.2.a.ba.1.6	yes	7	5.4	even	2

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)

\(f(q)\)	\(=\)	\(q-1.85243 q^{2} +1.00000 q^{3} +1.43149 q^{4} -1.85243 q^{6} +2.79882 q^{7} +1.05312 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.85243 q^{2} +1.00000 q^{3} +1.43149 q^{4} -1.85243 q^{6} +2.79882 q^{7} +1.05312 q^{8} +1.00000 q^{9} +0.110799 q^{11} +1.43149 q^{12} -6.18763 q^{13} -5.18461 q^{14} -4.81382 q^{16} -4.27451 q^{17} -1.85243 q^{18} -0.326031 q^{19} +2.79882 q^{21} -0.205247 q^{22} +5.33219 q^{23} +1.05312 q^{24} +11.4621 q^{26} +1.00000 q^{27} +4.00648 q^{28} -8.69258 q^{29} -5.01264 q^{31} +6.81100 q^{32} +0.110799 q^{33} +7.91822 q^{34} +1.43149 q^{36} +1.49079 q^{37} +0.603948 q^{38} -6.18763 q^{39} +10.8442 q^{41} -5.18461 q^{42} +8.23207 q^{43} +0.158608 q^{44} -9.87749 q^{46} +1.00000 q^{47} -4.81382 q^{48} +0.833398 q^{49} -4.27451 q^{51} -8.85753 q^{52} +10.6703 q^{53} -1.85243 q^{54} +2.94750 q^{56} -0.326031 q^{57} +16.1024 q^{58} +7.90884 q^{59} +9.66552 q^{61} +9.28556 q^{62} +2.79882 q^{63} -2.98926 q^{64} -0.205247 q^{66} +2.31475 q^{67} -6.11892 q^{68} +5.33219 q^{69} -4.49207 q^{71} +1.05312 q^{72} +13.0475 q^{73} -2.76159 q^{74} -0.466710 q^{76} +0.310106 q^{77} +11.4621 q^{78} -11.7702 q^{79} +1.00000 q^{81} -20.0881 q^{82} +17.2486 q^{83} +4.00648 q^{84} -15.2493 q^{86} -8.69258 q^{87} +0.116685 q^{88} +5.04142 q^{89} -17.3181 q^{91} +7.63297 q^{92} -5.01264 q^{93} -1.85243 q^{94} +6.81100 q^{96} -7.54883 q^{97} -1.54381 q^{98} +0.110799 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 7 q^{3} + 5 q^{4} - q^{6} + 7 q^{7} + 6 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q - q^2 + 7 * q^3 + 5 * q^4 - q^6 + 7 * q^7 + 6 * q^8 + 7 * q^9	\( 7 q - q^{2} + 7 q^{3} + 5 q^{4} - q^{6} + 7 q^{7} + 6 q^{8} + 7 q^{9} + 5 q^{12} + 5 q^{13} + 7 q^{14} + 9 q^{16} + 2 q^{17} - q^{18} - 13 q^{19} + 7 q^{21} - 14 q^{22} + 6 q^{23} + 6 q^{24} + 7 q^{27} + 30 q^{28} + 9 q^{29} + 5 q^{31} + 26 q^{32} - 8 q^{34} + 5 q^{36} - 5 q^{37} - 2 q^{38} + 5 q^{39} + 18 q^{41} + 7 q^{42} + 14 q^{43} + 17 q^{44} - 27 q^{46} + 7 q^{47} + 9 q^{48} + 14 q^{49} + 2 q^{51} - 3 q^{52} + 20 q^{53} - q^{54} + 17 q^{56} - 13 q^{57} + 37 q^{58} + 10 q^{59} - 8 q^{61} - 6 q^{62} + 7 q^{63} + 18 q^{64} - 14 q^{66} + 4 q^{67} + 10 q^{68} + 6 q^{69} + 12 q^{71} + 6 q^{72} + 4 q^{73} - 25 q^{74} - 66 q^{76} + 6 q^{77} - 5 q^{79} + 7 q^{81} - 29 q^{82} + 52 q^{83} + 30 q^{84} - 17 q^{86} + 9 q^{87} + 26 q^{88} + 32 q^{89} - 26 q^{91} - 17 q^{92} + 5 q^{93} - q^{94} + 26 q^{96} - 12 q^{97} + 40 q^{98}+O(q^{100}) \) 7 * q - q^2 + 7 * q^3 + 5 * q^4 - q^6 + 7 * q^7 + 6 * q^8 + 7 * q^9 + 5 * q^12 + 5 * q^13 + 7 * q^14 + 9 * q^16 + 2 * q^17 - q^18 - 13 * q^19 + 7 * q^21 - 14 * q^22 + 6 * q^23 + 6 * q^24 + 7 * q^27 + 30 * q^28 + 9 * q^29 + 5 * q^31 + 26 * q^32 - 8 * q^34 + 5 * q^36 - 5 * q^37 - 2 * q^38 + 5 * q^39 + 18 * q^41 + 7 * q^42 + 14 * q^43 + 17 * q^44 - 27 * q^46 + 7 * q^47 + 9 * q^48 + 14 * q^49 + 2 * q^51 - 3 * q^52 + 20 * q^53 - q^54 + 17 * q^56 - 13 * q^57 + 37 * q^58 + 10 * q^59 - 8 * q^61 - 6 * q^62 + 7 * q^63 + 18 * q^64 - 14 * q^66 + 4 * q^67 + 10 * q^68 + 6 * q^69 + 12 * q^71 + 6 * q^72 + 4 * q^73 - 25 * q^74 - 66 * q^76 + 6 * q^77 - 5 * q^79 + 7 * q^81 - 29 * q^82 + 52 * q^83 + 30 * q^84 - 17 * q^86 + 9 * q^87 + 26 * q^88 + 32 * q^89 - 26 * q^91 - 17 * q^92 + 5 * q^93 - q^94 + 26 * q^96 - 12 * q^97 + 40 * q^98

Introduction

L-functions

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