LMFDB - Embedded newform 2175.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.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:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +3.47214 q^{11} -0.618034 q^{12} +1.76393 q^{13} +4.85410 q^{14} -4.85410 q^{16} +5.47214 q^{17} +1.61803 q^{18} -5.70820 q^{19} -3.00000 q^{21} +5.61803 q^{22} +2.23607 q^{24} +2.85410 q^{26} -1.00000 q^{27} +1.85410 q^{28} +1.00000 q^{29} -8.00000 q^{31} -3.38197 q^{32} -3.47214 q^{33} +8.85410 q^{34} +0.618034 q^{36} +8.00000 q^{37} -9.23607 q^{38} -1.76393 q^{39} +4.47214 q^{41} -4.85410 q^{42} +1.23607 q^{43} +2.14590 q^{44} +6.70820 q^{47} +4.85410 q^{48} +2.00000 q^{49} -5.47214 q^{51} +1.09017 q^{52} +11.2361 q^{53} -1.61803 q^{54} -6.70820 q^{56} +5.70820 q^{57} +1.61803 q^{58} +0.763932 q^{59} +7.70820 q^{61} -12.9443 q^{62} +3.00000 q^{63} +4.23607 q^{64} -5.61803 q^{66} +2.52786 q^{67} +3.38197 q^{68} +2.76393 q^{71} -2.23607 q^{72} +8.00000 q^{73} +12.9443 q^{74} -3.52786 q^{76} +10.4164 q^{77} -2.85410 q^{78} +16.1803 q^{79} +1.00000 q^{81} +7.23607 q^{82} -9.70820 q^{83} -1.85410 q^{84} +2.00000 q^{86} -1.00000 q^{87} -7.76393 q^{88} -11.1803 q^{89} +5.29180 q^{91} +8.00000 q^{93} +10.8541 q^{94} +3.38197 q^{96} -7.23607 q^{97} +3.23607 q^{98} +3.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9	$ 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9} - 2 q^{11} + q^{12} + 8 q^{13} + 3 q^{14} - 3 q^{16} + 2 q^{17} + q^{18} + 2 q^{19} - 6 q^{21} + 9 q^{22} - q^{26} - 2 q^{27} - 3 q^{28} + 2 q^{29} - 16 q^{31} - 9 q^{32} + 2 q^{33} + 11 q^{34} - q^{36} + 16 q^{37} - 14 q^{38} - 8 q^{39} - 3 q^{42} - 2 q^{43} + 11 q^{44} + 3 q^{48} + 4 q^{49} - 2 q^{51} - 9 q^{52} + 18 q^{53} - q^{54} - 2 q^{57} + q^{58} + 6 q^{59} + 2 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} - 9 q^{66} + 14 q^{67} + 9 q^{68} + 10 q^{71} + 16 q^{73} + 8 q^{74} - 16 q^{76} - 6 q^{77} + q^{78} + 10 q^{79} + 2 q^{81} + 10 q^{82} - 6 q^{83} + 3 q^{84} + 4 q^{86} - 2 q^{87} - 20 q^{88} + 24 q^{91} + 16 q^{93} + 15 q^{94} + 9 q^{96} - 10 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 - 2 * q^11 + q^12 + 8 * q^13 + 3 * q^14 - 3 * q^16 + 2 * q^17 + q^18 + 2 * q^19 - 6 * q^21 + 9 * q^22 - q^26 - 2 * q^27 - 3 * q^28 + 2 * q^29 - 16 * q^31 - 9 * q^32 + 2 * q^33 + 11 * q^34 - q^36 + 16 * q^37 - 14 * q^38 - 8 * q^39 - 3 * q^42 - 2 * q^43 + 11 * q^44 + 3 * q^48 + 4 * q^49 - 2 * q^51 - 9 * q^52 + 18 * q^53 - q^54 - 2 * q^57 + q^58 + 6 * q^59 + 2 * q^61 - 8 * q^62 + 6 * q^63 + 4 * q^64 - 9 * q^66 + 14 * q^67 + 9 * q^68 + 10 * q^71 + 16 * q^73 + 8 * q^74 - 16 * q^76 - 6 * q^77 + q^78 + 10 * q^79 + 2 * q^81 + 10 * q^82 - 6 * q^83 + 3 * q^84 + 4 * q^86 - 2 * q^87 - 20 * q^88 + 24 * q^91 + 16 * q^93 + 15 * q^94 + 9 * q^96 - 10 * q^97 + 2 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	3.47214	1.04689	0.523444	−	0.852060i	$-0.324647\pi$
$11$	3.47214	1.04689	0.523444	+	0.852060i	$0.324647\pi$
$12$	−0.618034	−0.178411
$13$	1.76393	0.489227	0.244613	−	0.969621i	$-0.421339\pi$
$13$	1.76393	0.489227	0.244613	+	0.969621i	$0.421339\pi$
$14$	4.85410	1.29731
$15$	0	0
$16$	−4.85410	−1.21353
$17$	5.47214	1.32719	0.663594	−	0.748093i	$-0.269030\pi$
$17$	5.47214	1.32719	0.663594	+	0.748093i	$0.269030\pi$
$18$	1.61803	0.381374
$19$	−5.70820	−1.30955	−0.654776	−	0.755823i	$-0.727237\pi$
$19$	−5.70820	−1.30955	−0.654776	+	0.755823i	$0.727237\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	5.61803	1.19777
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	2.85410	0.559735
$27$	−1.00000	−0.192450
$28$	1.85410	0.350392
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−3.38197	−0.597853
$33$	−3.47214	−0.604421
$34$	8.85410	1.51847
$35$	0	0
$36$	0.618034	0.103006
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−9.23607	−1.49829
$39$	−1.76393	−0.282455
$40$	0	0
$41$	4.47214	0.698430	0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214	0.698430	0.349215	+	0.937043i	$0.386448\pi$
$42$	−4.85410	−0.749004
$43$	1.23607	0.188499	0.0942493	−	0.995549i	$-0.469955\pi$
$43$	1.23607	0.188499	0.0942493	+	0.995549i	$0.469955\pi$
$44$	2.14590	0.323506
$45$	0	0
$46$	0	0
$47$	6.70820	0.978492	0.489246	−	0.872146i	$-0.337272\pi$
$47$	6.70820	0.978492	0.489246	+	0.872146i	$0.337272\pi$
$48$	4.85410	0.700629
$49$	2.00000	0.285714
$50$	0	0
$51$	−5.47214	−0.766252
$52$	1.09017	0.151179
$53$	11.2361	1.54339	0.771696	−	0.635991i	$-0.219409\pi$
$53$	11.2361	1.54339	0.771696	+	0.635991i	$0.219409\pi$
$54$	−1.61803	−0.220187
$55$	0	0
$56$	−6.70820	−0.896421
$57$	5.70820	0.756070
$58$	1.61803	0.212458
$59$	0.763932	0.0994555	0.0497277	−	0.998763i	$-0.484165\pi$
$59$	0.763932	0.0994555	0.0497277	+	0.998763i	$0.484165\pi$
$60$	0	0
$61$	7.70820	0.986934	0.493467	−	0.869764i	$-0.335729\pi$
$61$	7.70820	0.986934	0.493467	+	0.869764i	$0.335729\pi$
$62$	−12.9443	−1.64392
$63$	3.00000	0.377964
$64$	4.23607	0.529508
$65$	0	0
$66$	−5.61803	−0.691532
$67$	2.52786	0.308828	0.154414	−	0.988006i	$-0.450651\pi$
$67$	2.52786	0.308828	0.154414	+	0.988006i	$0.450651\pi$
$68$	3.38197	0.410124
$69$	0	0
$70$	0	0
$71$	2.76393	0.328018	0.164009	−	0.986459i	$-0.447557\pi$
$71$	2.76393	0.328018	0.164009	+	0.986459i	$0.447557\pi$
$72$	−2.23607	−0.263523
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	12.9443	1.50474
$75$	0	0
$76$	−3.52786	−0.404674
$77$	10.4164	1.18706
$78$	−2.85410	−0.323163
$79$	16.1803	1.82043	0.910215	−	0.414136i	$-0.135916\pi$
$79$	16.1803	1.82043	0.910215	+	0.414136i	$0.135916\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.23607	0.799090
$83$	−9.70820	−1.06561	−0.532807	−	0.846237i	$-0.678863\pi$
$83$	−9.70820	−1.06561	−0.532807	+	0.846237i	$0.678863\pi$
$84$	−1.85410	−0.202299
$85$	0	0
$86$	2.00000	0.215666
$87$	−1.00000	−0.107211
$88$	−7.76393	−0.827638
$89$	−11.1803	−1.18511	−0.592557	−	0.805529i	$-0.701881\pi$
$89$	−11.1803	−1.18511	−0.592557	+	0.805529i	$0.701881\pi$
$90$	0	0
$91$	5.29180	0.554731
$92$	0	0
$93$	8.00000	0.829561
$94$	10.8541	1.11952
$95$	0	0
$96$	3.38197	0.345170
$97$	−7.23607	−0.734711	−0.367356	−	0.930081i	$-0.619737\pi$
$97$	−7.23607	−0.734711	−0.367356	+	0.930081i	$0.619737\pi$
$98$	3.23607	0.326892
$99$	3.47214	0.348963
$100$	0	0
$101$	−0.236068	−0.0234896	−0.0117448	−	0.999931i	$-0.503739\pi$
$101$	−0.236068	−0.0234896	−0.0117448	+	0.999931i	$0.503739\pi$
$102$	−8.85410	−0.876687
$103$	19.4164	1.91316	0.956578	−	0.291477i	$-0.0941468\pi$
$103$	19.4164	1.91316	0.956578	+	0.291477i	$0.0941468\pi$
$104$	−3.94427	−0.386768
$105$	0	0
$106$	18.1803	1.76583
$107$	−16.4721	−1.59242	−0.796211	−	0.605019i	$-0.793165\pi$
$107$	−16.4721	−1.59242	−0.796211	+	0.605019i	$0.793165\pi$
$108$	−0.618034	−0.0594703
$109$	10.4164	0.997711	0.498855	−	0.866685i	$-0.333754\pi$
$109$	10.4164	0.997711	0.498855	+	0.866685i	$0.333754\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	−14.5623	−1.37601
$113$	−9.94427	−0.935478	−0.467739	−	0.883867i	$-0.654931\pi$
$113$	−9.94427	−0.935478	−0.467739	+	0.883867i	$0.654931\pi$
$114$	9.23607	0.865037
$115$	0	0
$116$	0.618034	0.0573830
$117$	1.76393	0.163076
$118$	1.23607	0.113789
$119$	16.4164	1.50489
$120$	0	0
$121$	1.05573	0.0959753
$122$	12.4721	1.12917
$123$	−4.47214	−0.403239
$124$	−4.94427	−0.444009
$125$	0	0
$126$	4.85410	0.432438
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	13.6180	1.20368
$129$	−1.23607	−0.108830
$130$	0	0
$131$	−5.47214	−0.478103	−0.239051	−	0.971007i	$-0.576836\pi$
$131$	−5.47214	−0.478103	−0.239051	+	0.971007i	$0.576836\pi$
$132$	−2.14590	−0.186776
$133$	−17.1246	−1.48489
$134$	4.09017	0.353337
$135$	0	0
$136$	−12.2361	−1.04923
$137$	−6.94427	−0.593289	−0.296645	−	0.954988i	$-0.595868\pi$
$137$	−6.94427	−0.593289	−0.296645	+	0.954988i	$0.595868\pi$
$138$	0	0
$139$	12.7082	1.07790	0.538948	−	0.842339i	$-0.318822\pi$
$139$	12.7082	1.07790	0.538948	+	0.842339i	$0.318822\pi$
$140$	0	0
$141$	−6.70820	−0.564933
$142$	4.47214	0.375293
$143$	6.12461	0.512166
$144$	−4.85410	−0.404508
$145$	0	0
$146$	12.9443	1.07128
$147$	−2.00000	−0.164957
$148$	4.94427	0.406417
$149$	−2.18034	−0.178620	−0.0893102	−	0.996004i	$-0.528466\pi$
$149$	−2.18034	−0.178620	−0.0893102	+	0.996004i	$0.528466\pi$
$150$	0	0
$151$	−6.47214	−0.526695	−0.263347	−	0.964701i	$-0.584827\pi$
$151$	−6.47214	−0.526695	−0.263347	+	0.964701i	$0.584827\pi$
$152$	12.7639	1.03529
$153$	5.47214	0.442396
$154$	16.8541	1.35814
$155$	0	0
$156$	−1.09017	−0.0872835
$157$	−1.70820	−0.136330	−0.0681648	−	0.997674i	$-0.521714\pi$
$157$	−1.70820	−0.136330	−0.0681648	+	0.997674i	$0.521714\pi$
$158$	26.1803	2.08280
$159$	−11.2361	−0.891078
$160$	0	0
$161$	0	0
$162$	1.61803	0.127125
$163$	−25.1246	−1.96791	−0.983956	−	0.178413i	$-0.942904\pi$
$163$	−25.1246	−1.96791	−0.983956	+	0.178413i	$0.942904\pi$
$164$	2.76393	0.215827
$165$	0	0
$166$	−15.7082	−1.21919
$167$	−17.8885	−1.38426	−0.692129	−	0.721774i	$-0.743327\pi$
$167$	−17.8885	−1.38426	−0.692129	+	0.721774i	$0.743327\pi$
$168$	6.70820	0.517549
$169$	−9.88854	−0.760657
$170$	0	0
$171$	−5.70820	−0.436517
$172$	0.763932	0.0582493
$173$	−7.41641	−0.563859	−0.281930	−	0.959435i	$-0.590974\pi$
$173$	−7.41641	−0.563859	−0.281930	+	0.959435i	$0.590974\pi$
$174$	−1.61803	−0.122663
$175$	0	0
$176$	−16.8541	−1.27043
$177$	−0.763932	−0.0574206
$178$	−18.0902	−1.35592
$179$	−18.1803	−1.35886	−0.679431	−	0.733739i	$-0.737773\pi$
$179$	−18.1803	−1.35886	−0.679431	+	0.733739i	$0.737773\pi$
$180$	0	0
$181$	−18.4164	−1.36888	−0.684440	−	0.729069i	$-0.739953\pi$
$181$	−18.4164	−1.36888	−0.684440	+	0.729069i	$0.739953\pi$
$182$	8.56231	0.634680
$183$	−7.70820	−0.569807
$184$	0	0
$185$	0	0
$186$	12.9443	0.949120
$187$	19.0000	1.38942
$188$	4.14590	0.302371
$189$	−3.00000	−0.218218
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−4.23607	−0.305712
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−11.7082	−0.840600
$195$	0	0
$196$	1.23607	0.0882906
$197$	2.94427	0.209771	0.104885	−	0.994484i	$-0.466552\pi$
$197$	2.94427	0.209771	0.104885	+	0.994484i	$0.466552\pi$
$198$	5.61803	0.399256
$199$	7.29180	0.516902	0.258451	−	0.966024i	$-0.416788\pi$
$199$	7.29180	0.516902	0.258451	+	0.966024i	$0.416788\pi$
$200$	0	0
$201$	−2.52786	−0.178302
$202$	−0.381966	−0.0268750
$203$	3.00000	0.210559
$204$	−3.38197	−0.236785
$205$	0	0
$206$	31.4164	2.18888
$207$	0	0
$208$	−8.56231	−0.593689
$209$	−19.8197	−1.37095
$210$	0	0
$211$	7.88854	0.543070	0.271535	−	0.962429i	$-0.412469\pi$
$211$	7.88854	0.543070	0.271535	+	0.962429i	$0.412469\pi$
$212$	6.94427	0.476935
$213$	−2.76393	−0.189382
$214$	−26.6525	−1.82193
$215$	0	0
$216$	2.23607	0.152145
$217$	−24.0000	−1.62923
$218$	16.8541	1.14150
$219$	−8.00000	−0.540590
$220$	0	0
$221$	9.65248	0.649296
$222$	−12.9443	−0.868763
$223$	13.0000	0.870544	0.435272	−	0.900299i	$-0.356652\pi$
$223$	13.0000	0.870544	0.435272	+	0.900299i	$0.356652\pi$
$224$	−10.1459	−0.677901
$225$	0	0
$226$	−16.0902	−1.07030
$227$	28.1803	1.87039	0.935197	−	0.354127i	$-0.115222\pi$
$227$	28.1803	1.87039	0.935197	+	0.354127i	$0.115222\pi$
$228$	3.52786	0.233639
$229$	−17.4164	−1.15091	−0.575454	−	0.817834i	$-0.695175\pi$
$229$	−17.4164	−1.15091	−0.575454	+	0.817834i	$0.695175\pi$
$230$	0	0
$231$	−10.4164	−0.685349
$232$	−2.23607	−0.146805
$233$	25.4164	1.66508	0.832542	−	0.553962i	$-0.186885\pi$
$233$	25.4164	1.66508	0.832542	+	0.553962i	$0.186885\pi$
$234$	2.85410	0.186578
$235$	0	0
$236$	0.472136	0.0307334
$237$	−16.1803	−1.05103
$238$	26.5623	1.72178
$239$	−23.8885	−1.54522	−0.772611	−	0.634880i	$-0.781050\pi$
$239$	−23.8885	−1.54522	−0.772611	+	0.634880i	$0.781050\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	1.70820	0.109808
$243$	−1.00000	−0.0641500
$244$	4.76393	0.304979
$245$	0	0
$246$	−7.23607	−0.461355
$247$	−10.0689	−0.640668
$248$	17.8885	1.13592
$249$	9.70820	0.615232
$250$	0	0
$251$	10.8885	0.687279	0.343639	−	0.939102i	$-0.388340\pi$
$251$	10.8885	0.687279	0.343639	+	0.939102i	$0.388340\pi$
$252$	1.85410	0.116797
$253$	0	0
$254$	9.70820	0.609147
$255$	0	0
$256$	13.5623	0.847644
$257$	−3.70820	−0.231311	−0.115656	−	0.993289i	$-0.536897\pi$
$257$	−3.70820	−0.231311	−0.115656	+	0.993289i	$0.536897\pi$
$258$	−2.00000	−0.124515
$259$	24.0000	1.49129
$260$	0	0
$261$	1.00000	0.0618984
$262$	−8.85410	−0.547008
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	7.76393	0.477837
$265$	0	0
$266$	−27.7082	−1.69890
$267$	11.1803	0.684226
$268$	1.56231	0.0954330
$269$	−25.7639	−1.57085	−0.785427	−	0.618954i	$-0.787557\pi$
$269$	−25.7639	−1.57085	−0.785427	+	0.618954i	$0.787557\pi$
$270$	0	0
$271$	−30.3607	−1.84428	−0.922140	−	0.386856i	$-0.873561\pi$
$271$	−30.3607	−1.84428	−0.922140	+	0.386856i	$0.873561\pi$
$272$	−26.5623	−1.61058
$273$	−5.29180	−0.320274
$274$	−11.2361	−0.678796
$275$	0	0
$276$	0	0
$277$	4.70820	0.282889	0.141444	−	0.989946i	$-0.454825\pi$
$277$	4.70820	0.282889	0.141444	+	0.989946i	$0.454825\pi$
$278$	20.5623	1.23325
$279$	−8.00000	−0.478947
$280$	0	0
$281$	17.1246	1.02157	0.510784	−	0.859709i	$-0.329355\pi$
$281$	17.1246	1.02157	0.510784	+	0.859709i	$0.329355\pi$
$282$	−10.8541	−0.646352
$283$	−9.52786	−0.566373	−0.283186	−	0.959065i	$-0.591391\pi$
$283$	−9.52786	−0.566373	−0.283186	+	0.959065i	$0.591391\pi$
$284$	1.70820	0.101363
$285$	0	0
$286$	9.90983	0.585981
$287$	13.4164	0.791946
$288$	−3.38197	−0.199284
$289$	12.9443	0.761428
$290$	0	0
$291$	7.23607	0.424186
$292$	4.94427	0.289342
$293$	0.0557281	0.00325567	0.00162783	−	0.999999i	$-0.499482\pi$
$293$	0.0557281	0.00325567	0.00162783	+	0.999999i	$0.499482\pi$
$294$	−3.23607	−0.188731
$295$	0	0
$296$	−17.8885	−1.03975
$297$	−3.47214	−0.201474
$298$	−3.52786	−0.204364
$299$	0	0
$300$	0	0
$301$	3.70820	0.213737
$302$	−10.4721	−0.602604
$303$	0.236068	0.0135618
$304$	27.7082	1.58917
$305$	0	0
$306$	8.85410	0.506155
$307$	−7.05573	−0.402692	−0.201346	−	0.979520i	$-0.564532\pi$
$307$	−7.05573	−0.402692	−0.201346	+	0.979520i	$0.564532\pi$
$308$	6.43769	0.366822
$309$	−19.4164	−1.10456
$310$	0	0
$311$	14.5279	0.823800	0.411900	−	0.911229i	$-0.364865\pi$
$311$	14.5279	0.823800	0.411900	+	0.911229i	$0.364865\pi$
$312$	3.94427	0.223300
$313$	21.1803	1.19718	0.598592	−	0.801054i	$-0.295727\pi$
$313$	21.1803	1.19718	0.598592	+	0.801054i	$0.295727\pi$
$314$	−2.76393	−0.155978
$315$	0	0
$316$	10.0000	0.562544
$317$	1.58359	0.0889434	0.0444717	−	0.999011i	$-0.485840\pi$
$317$	1.58359	0.0889434	0.0444717	+	0.999011i	$0.485840\pi$
$318$	−18.1803	−1.01950
$319$	3.47214	0.194402
$320$	0	0
$321$	16.4721	0.919385
$322$	0	0
$323$	−31.2361	−1.73802
$324$	0.618034	0.0343352
$325$	0	0
$326$	−40.6525	−2.25153
$327$	−10.4164	−0.576029
$328$	−10.0000	−0.552158
$329$	20.1246	1.10951
$330$	0	0
$331$	19.8885	1.09317	0.546587	−	0.837403i	$-0.315927\pi$
$331$	19.8885	1.09317	0.546587	+	0.837403i	$0.315927\pi$
$332$	−6.00000	−0.329293
$333$	8.00000	0.438397
$334$	−28.9443	−1.58376
$335$	0	0
$336$	14.5623	0.794439
$337$	11.5279	0.627963	0.313981	−	0.949429i	$-0.398337\pi$
$337$	11.5279	0.627963	0.313981	+	0.949429i	$0.398337\pi$
$338$	−16.0000	−0.870285
$339$	9.94427	0.540099
$340$	0	0
$341$	−27.7771	−1.50421
$342$	−9.23607	−0.499429
$343$	−15.0000	−0.809924
$344$	−2.76393	−0.149021
$345$	0	0
$346$	−12.0000	−0.645124
$347$	−26.4721	−1.42110	−0.710549	−	0.703647i	$-0.751554\pi$
$347$	−26.4721	−1.42110	−0.710549	+	0.703647i	$0.751554\pi$
$348$	−0.618034	−0.0331301
$349$	−1.05573	−0.0565118	−0.0282559	−	0.999601i	$-0.508995\pi$
$349$	−1.05573	−0.0565118	−0.0282559	+	0.999601i	$0.508995\pi$
$350$	0	0
$351$	−1.76393	−0.0941517
$352$	−11.7426	−0.625885
$353$	−12.4721	−0.663825	−0.331912	−	0.943310i	$-0.607694\pi$
$353$	−12.4721	−0.663825	−0.331912	+	0.943310i	$0.607694\pi$
$354$	−1.23607	−0.0656963
$355$	0	0
$356$	−6.90983	−0.366220
$357$	−16.4164	−0.868848
$358$	−29.4164	−1.55471
$359$	−31.4164	−1.65809	−0.829047	−	0.559178i	$-0.811117\pi$
$359$	−31.4164	−1.65809	−0.829047	+	0.559178i	$0.811117\pi$
$360$	0	0
$361$	13.5836	0.714926
$362$	−29.7984	−1.56617
$363$	−1.05573	−0.0554114
$364$	3.27051	0.171421
$365$	0	0
$366$	−12.4721	−0.651929
$367$	11.4164	0.595932	0.297966	−	0.954577i	$-0.403692\pi$
$367$	11.4164	0.595932	0.297966	+	0.954577i	$0.403692\pi$
$368$	0	0
$369$	4.47214	0.232810
$370$	0	0
$371$	33.7082	1.75004
$372$	4.94427	0.256349
$373$	19.8885	1.02979	0.514895	−	0.857253i	$-0.327831\pi$
$373$	19.8885	1.02979	0.514895	+	0.857253i	$0.327831\pi$
$374$	30.7426	1.58966
$375$	0	0
$376$	−15.0000	−0.773566
$377$	1.76393	0.0908471
$378$	−4.85410	−0.249668
$379$	−10.9443	−0.562169	−0.281085	−	0.959683i	$-0.590694\pi$
$379$	−10.9443	−0.562169	−0.281085	+	0.959683i	$0.590694\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	19.4164	0.993430
$383$	−38.0689	−1.94523	−0.972615	−	0.232424i	$-0.925334\pi$
$383$	−38.0689	−1.94523	−0.972615	+	0.232424i	$0.925334\pi$
$384$	−13.6180	−0.694942
$385$	0	0
$386$	9.70820	0.494135
$387$	1.23607	0.0628329
$388$	−4.47214	−0.227038
$389$	17.7639	0.900667	0.450334	−	0.892860i	$-0.351305\pi$
$389$	17.7639	0.900667	0.450334	+	0.892860i	$0.351305\pi$
$390$	0	0
$391$	0	0
$392$	−4.47214	−0.225877
$393$	5.47214	0.276033
$394$	4.76393	0.240003
$395$	0	0
$396$	2.14590	0.107835
$397$	33.4164	1.67712	0.838561	−	0.544808i	$-0.183398\pi$
$397$	33.4164	1.67712	0.838561	+	0.544808i	$0.183398\pi$
$398$	11.7984	0.591399
$399$	17.1246	0.857303
$400$	0	0
$401$	−5.34752	−0.267043	−0.133521	−	0.991046i	$-0.542628\pi$
$401$	−5.34752	−0.267043	−0.133521	+	0.991046i	$0.542628\pi$
$402$	−4.09017	−0.203999
$403$	−14.1115	−0.702942
$404$	−0.145898	−0.00725870
$405$	0	0
$406$	4.85410	0.240905
$407$	27.7771	1.37686
$408$	12.2361	0.605776
$409$	−22.6525	−1.12009	−0.560046	−	0.828461i	$-0.689217\pi$
$409$	−22.6525	−1.12009	−0.560046	+	0.828461i	$0.689217\pi$
$410$	0	0
$411$	6.94427	0.342536
$412$	12.0000	0.591198
$413$	2.29180	0.112772
$414$	0	0
$415$	0	0
$416$	−5.96556	−0.292486
$417$	−12.7082	−0.622323
$418$	−32.0689	−1.56854
$419$	−10.3607	−0.506152	−0.253076	−	0.967446i	$-0.581442\pi$
$419$	−10.3607	−0.506152	−0.253076	+	0.967446i	$0.581442\pi$
$420$	0	0
$421$	−24.1803	−1.17848	−0.589239	−	0.807959i	$-0.700572\pi$
$421$	−24.1803	−1.17848	−0.589239	+	0.807959i	$0.700572\pi$
$422$	12.7639	0.621338
$423$	6.70820	0.326164
$424$	−25.1246	−1.22016
$425$	0	0
$426$	−4.47214	−0.216676
$427$	23.1246	1.11908
$428$	−10.1803	−0.492085
$429$	−6.12461	−0.295699
$430$	0	0
$431$	25.5279	1.22963	0.614817	−	0.788670i	$-0.289230\pi$
$431$	25.5279	1.22963	0.614817	+	0.788670i	$0.289230\pi$
$432$	4.85410	0.233543
$433$	−26.6525	−1.28084	−0.640418	−	0.768026i	$-0.721239\pi$
$433$	−26.6525	−1.28084	−0.640418	+	0.768026i	$0.721239\pi$
$434$	−38.8328	−1.86403
$435$	0	0
$436$	6.43769	0.308310
$437$	0	0
$438$	−12.9443	−0.618501
$439$	30.1246	1.43777	0.718885	−	0.695129i	$-0.244653\pi$
$439$	30.1246	1.43777	0.718885	+	0.695129i	$0.244653\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	15.6180	0.742874
$443$	−28.2361	−1.34154	−0.670768	−	0.741667i	$-0.734035\pi$
$443$	−28.2361	−1.34154	−0.670768	+	0.741667i	$0.734035\pi$
$444$	−4.94427	−0.234645
$445$	0	0
$446$	21.0344	0.996010
$447$	2.18034	0.103127
$448$	12.7082	0.600406
$449$	6.23607	0.294298	0.147149	−	0.989114i	$-0.452990\pi$
$449$	6.23607	0.294298	0.147149	+	0.989114i	$0.452990\pi$
$450$	0	0
$451$	15.5279	0.731179
$452$	−6.14590	−0.289079
$453$	6.47214	0.304087
$454$	45.5967	2.13996
$455$	0	0
$456$	−12.7639	−0.597726
$457$	34.1246	1.59628	0.798141	−	0.602471i	$-0.205817\pi$
$457$	34.1246	1.59628	0.798141	+	0.602471i	$0.205817\pi$
$458$	−28.1803	−1.31678
$459$	−5.47214	−0.255417
$460$	0	0
$461$	−42.3607	−1.97293	−0.986467	−	0.163961i	$-0.947573\pi$
$461$	−42.3607	−1.97293	−0.986467	+	0.163961i	$0.947573\pi$
$462$	−16.8541	−0.784124
$463$	11.4721	0.533155	0.266578	−	0.963813i	$-0.414107\pi$
$463$	11.4721	0.533155	0.266578	+	0.963813i	$0.414107\pi$
$464$	−4.85410	−0.225346
$465$	0	0
$466$	41.1246	1.90506
$467$	−4.94427	−0.228794	−0.114397	−	0.993435i	$-0.536494\pi$
$467$	−4.94427	−0.228794	−0.114397	+	0.993435i	$0.536494\pi$
$468$	1.09017	0.0503931
$469$	7.58359	0.350178
$470$	0	0
$471$	1.70820	0.0787099
$472$	−1.70820	−0.0786265
$473$	4.29180	0.197337
$474$	−26.1803	−1.20250
$475$	0	0
$476$	10.1459	0.465036
$477$	11.2361	0.514464
$478$	−38.6525	−1.76792
$479$	2.47214	0.112955	0.0564774	−	0.998404i	$-0.482013\pi$
$479$	2.47214	0.112955	0.0564774	+	0.998404i	$0.482013\pi$
$480$	0	0
$481$	14.1115	0.643427
$482$	11.3262	0.515896
$483$	0	0
$484$	0.652476	0.0296580
$485$	0	0
$486$	−1.61803	−0.0733955
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	−17.2361	−0.780240
$489$	25.1246	1.13617
$490$	0	0
$491$	−9.88854	−0.446264	−0.223132	−	0.974788i	$-0.571628\pi$
$491$	−9.88854	−0.446264	−0.223132	+	0.974788i	$0.571628\pi$
$492$	−2.76393	−0.124608
$493$	5.47214	0.246453
$494$	−16.2918	−0.733003
$495$	0	0
$496$	38.8328	1.74364
$497$	8.29180	0.371938
$498$	15.7082	0.703901
$499$	−28.2361	−1.26402	−0.632010	−	0.774960i	$-0.717770\pi$
$499$	−28.2361	−1.26402	−0.632010	+	0.774960i	$0.717770\pi$
$500$	0	0
$501$	17.8885	0.799201
$502$	17.6180	0.786331
$503$	−18.5967	−0.829188	−0.414594	−	0.910006i	$-0.636076\pi$
$503$	−18.5967	−0.829188	−0.414594	+	0.910006i	$0.636076\pi$
$504$	−6.70820	−0.298807
$505$	0	0
$506$	0	0
$507$	9.88854	0.439166
$508$	3.70820	0.164525
$509$	−16.1803	−0.717181	−0.358590	−	0.933495i	$-0.616743\pi$
$509$	−16.1803	−0.717181	−0.358590	+	0.933495i	$0.616743\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	−5.29180	−0.233867
$513$	5.70820	0.252023
$514$	−6.00000	−0.264649
$515$	0	0
$516$	−0.763932	−0.0336302
$517$	23.2918	1.02437
$518$	38.8328	1.70622
$519$	7.41641	0.325544
$520$	0	0
$521$	−25.2361	−1.10561	−0.552806	−	0.833310i	$-0.686443\pi$
$521$	−25.2361	−1.10561	−0.552806	+	0.833310i	$0.686443\pi$
$522$	1.61803	0.0708194
$523$	−23.8328	−1.04214	−0.521068	−	0.853515i	$-0.674466\pi$
$523$	−23.8328	−1.04214	−0.521068	+	0.853515i	$0.674466\pi$
$524$	−3.38197	−0.147742
$525$	0	0
$526$	−38.8328	−1.69319
$527$	−43.7771	−1.90696
$528$	16.8541	0.733481
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0.763932	0.0331518
$532$	−10.5836	−0.458857
$533$	7.88854	0.341691
$534$	18.0902	0.782838
$535$	0	0
$536$	−5.65248	−0.244150
$537$	18.1803	0.784540
$538$	−41.6869	−1.79725
$539$	6.94427	0.299111
$540$	0	0
$541$	−6.36068	−0.273467	−0.136733	−	0.990608i	$-0.543660\pi$
$541$	−6.36068	−0.273467	−0.136733	+	0.990608i	$0.543660\pi$
$542$	−49.1246	−2.11008
$543$	18.4164	0.790324
$544$	−18.5066	−0.793463
$545$	0	0
$546$	−8.56231	−0.366433
$547$	−18.5279	−0.792194	−0.396097	−	0.918209i	$-0.629636\pi$
$547$	−18.5279	−0.792194	−0.396097	+	0.918209i	$0.629636\pi$
$548$	−4.29180	−0.183336
$549$	7.70820	0.328978
$550$	0	0
$551$	−5.70820	−0.243178
$552$	0	0
$553$	48.5410	2.06417
$554$	7.61803	0.323659
$555$	0	0
$556$	7.85410	0.333088
$557$	−7.23607	−0.306602	−0.153301	−	0.988180i	$-0.548990\pi$
$557$	−7.23607	−0.306602	−0.153301	+	0.988180i	$0.548990\pi$
$558$	−12.9443	−0.547975
$559$	2.18034	0.0922186
$560$	0	0
$561$	−19.0000	−0.802181
$562$	27.7082	1.16880
$563$	−3.87539	−0.163328	−0.0816641	−	0.996660i	$-0.526023\pi$
$563$	−3.87539	−0.163328	−0.0816641	+	0.996660i	$0.526023\pi$
$564$	−4.14590	−0.174574
$565$	0	0
$566$	−15.4164	−0.648000
$567$	3.00000	0.125988
$568$	−6.18034	−0.259321
$569$	17.1803	0.720237	0.360119	−	0.932907i	$-0.382736\pi$
$569$	17.1803	0.720237	0.360119	+	0.932907i	$0.382736\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	3.78522	0.158268
$573$	−12.0000	−0.501307
$574$	21.7082	0.906083
$575$	0	0
$576$	4.23607	0.176503
$577$	−33.3050	−1.38650	−0.693252	−	0.720696i	$-0.743823\pi$
$577$	−33.3050	−1.38650	−0.693252	+	0.720696i	$0.743823\pi$
$578$	20.9443	0.871167
$579$	−6.00000	−0.249351
$580$	0	0
$581$	−29.1246	−1.20829
$582$	11.7082	0.485321
$583$	39.0132	1.61576
$584$	−17.8885	−0.740233
$585$	0	0
$586$	0.0901699	0.00372489
$587$	24.1803	0.998029	0.499015	−	0.866594i	$-0.333695\pi$
$587$	24.1803	0.998029	0.499015	+	0.866594i	$0.333695\pi$
$588$	−1.23607	−0.0509746
$589$	45.6656	1.88162
$590$	0	0
$591$	−2.94427	−0.121111
$592$	−38.8328	−1.59602
$593$	−6.65248	−0.273184	−0.136592	−	0.990627i	$-0.543615\pi$
$593$	−6.65248	−0.273184	−0.136592	+	0.990627i	$0.543615\pi$
$594$	−5.61803	−0.230511
$595$	0	0
$596$	−1.34752	−0.0551967
$597$	−7.29180	−0.298433
$598$	0	0
$599$	26.8885	1.09864	0.549318	−	0.835613i	$-0.314888\pi$
$599$	26.8885	1.09864	0.549318	+	0.835613i	$0.314888\pi$
$600$	0	0
$601$	43.8885	1.79025	0.895126	−	0.445814i	$-0.147086\pi$
$601$	43.8885	1.79025	0.895126	+	0.445814i	$0.147086\pi$
$602$	6.00000	0.244542
$603$	2.52786	0.102943
$604$	−4.00000	−0.162758
$605$	0	0
$606$	0.381966	0.0155163
$607$	−9.34752	−0.379404	−0.189702	−	0.981842i	$-0.560752\pi$
$607$	−9.34752	−0.379404	−0.189702	+	0.981842i	$0.560752\pi$
$608$	19.3050	0.782919
$609$	−3.00000	−0.121566
$610$	0	0
$611$	11.8328	0.478704
$612$	3.38197	0.136708
$613$	−20.7082	−0.836396	−0.418198	−	0.908356i	$-0.637338\pi$
$613$	−20.7082	−0.836396	−0.418198	+	0.908356i	$0.637338\pi$
$614$	−11.4164	−0.460729
$615$	0	0
$616$	−23.2918	−0.938453
$617$	19.3050	0.777188	0.388594	−	0.921409i	$-0.372961\pi$
$617$	19.3050	0.777188	0.388594	+	0.921409i	$0.372961\pi$
$618$	−31.4164	−1.26375
$619$	−12.4721	−0.501297	−0.250649	−	0.968078i	$-0.580644\pi$
$619$	−12.4721	−0.501297	−0.250649	+	0.968078i	$0.580644\pi$
$620$	0	0
$621$	0	0
$622$	23.5066	0.942528
$623$	−33.5410	−1.34379
$624$	8.56231	0.342767
$625$	0	0
$626$	34.2705	1.36973
$627$	19.8197	0.791521
$628$	−1.05573	−0.0421281
$629$	43.7771	1.74551
$630$	0	0
$631$	46.4853	1.85055	0.925275	−	0.379297i	$-0.123834\pi$
$631$	46.4853	1.85055	0.925275	+	0.379297i	$0.123834\pi$
$632$	−36.1803	−1.43918
$633$	−7.88854	−0.313541
$634$	2.56231	0.101762
$635$	0	0
$636$	−6.94427	−0.275358
$637$	3.52786	0.139779
$638$	5.61803	0.222420
$639$	2.76393	0.109339
$640$	0	0
$641$	41.7639	1.64958	0.824788	−	0.565442i	$-0.191294\pi$
$641$	41.7639	1.64958	0.824788	+	0.565442i	$0.191294\pi$
$642$	26.6525	1.05189
$643$	−17.8328	−0.703258	−0.351629	−	0.936140i	$-0.614372\pi$
$643$	−17.8328	−0.703258	−0.351629	+	0.936140i	$0.614372\pi$
$644$	0	0
$645$	0	0
$646$	−50.5410	−1.98851
$647$	−6.76393	−0.265918	−0.132959	−	0.991122i	$-0.542448\pi$
$647$	−6.76393	−0.265918	−0.132959	+	0.991122i	$0.542448\pi$
$648$	−2.23607	−0.0878410
$649$	2.65248	0.104119
$650$	0	0
$651$	24.0000	0.940634
$652$	−15.5279	−0.608118
$653$	26.8885	1.05223	0.526115	−	0.850413i	$-0.323648\pi$
$653$	26.8885	1.05223	0.526115	+	0.850413i	$0.323648\pi$
$654$	−16.8541	−0.659048
$655$	0	0
$656$	−21.7082	−0.847563
$657$	8.00000	0.312110
$658$	32.5623	1.26941
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	0	0
$661$	15.8328	0.615825	0.307913	−	0.951415i	$-0.400370\pi$
$661$	15.8328	0.615825	0.307913	+	0.951415i	$0.400370\pi$
$662$	32.1803	1.25072
$663$	−9.65248	−0.374871
$664$	21.7082	0.842442
$665$	0	0
$666$	12.9443	0.501580
$667$	0	0
$668$	−11.0557	−0.427759
$669$	−13.0000	−0.502609
$670$	0	0
$671$	26.7639	1.03321
$672$	10.1459	0.391387
$673$	46.7082	1.80047	0.900234	−	0.435405i	$-0.143395\pi$
$673$	46.7082	1.80047	0.900234	+	0.435405i	$0.143395\pi$
$674$	18.6525	0.718467
$675$	0	0
$676$	−6.11146	−0.235056
$677$	−14.8885	−0.572213	−0.286107	−	0.958198i	$-0.592361\pi$
$677$	−14.8885	−0.572213	−0.286107	+	0.958198i	$0.592361\pi$
$678$	16.0902	0.617939
$679$	−21.7082	−0.833084
$680$	0	0
$681$	−28.1803	−1.07987
$682$	−44.9443	−1.72101
$683$	19.4164	0.742948	0.371474	−	0.928443i	$-0.378852\pi$
$683$	19.4164	0.742948	0.371474	+	0.928443i	$0.378852\pi$
$684$	−3.52786	−0.134891
$685$	0	0
$686$	−24.2705	−0.926652
$687$	17.4164	0.664477
$688$	−6.00000	−0.228748
$689$	19.8197	0.755069
$690$	0	0
$691$	−21.2918	−0.809978	−0.404989	−	0.914322i	$-0.632725\pi$
$691$	−21.2918	−0.809978	−0.404989	+	0.914322i	$0.632725\pi$
$692$	−4.58359	−0.174242
$693$	10.4164	0.395687
$694$	−42.8328	−1.62591
$695$	0	0
$696$	2.23607	0.0847579
$697$	24.4721	0.926948
$698$	−1.70820	−0.0646565
$699$	−25.4164	−0.961337
$700$	0	0
$701$	35.1246	1.32664	0.663319	−	0.748337i	$-0.269147\pi$
$701$	35.1246	1.32664	0.663319	+	0.748337i	$0.269147\pi$
$702$	−2.85410	−0.107721
$703$	−45.6656	−1.72231
$704$	14.7082	0.554336
$705$	0	0
$706$	−20.1803	−0.759497
$707$	−0.708204	−0.0266348
$708$	−0.472136	−0.0177440
$709$	−15.8885	−0.596707	−0.298353	−	0.954455i	$-0.596437\pi$
$709$	−15.8885	−0.596707	−0.298353	+	0.954455i	$0.596437\pi$
$710$	0	0
$711$	16.1803	0.606810
$712$	25.0000	0.936915
$713$	0	0
$714$	−26.5623	−0.994069
$715$	0	0
$716$	−11.2361	−0.419912
$717$	23.8885	0.892134
$718$	−50.8328	−1.89706
$719$	−6.76393	−0.252252	−0.126126	−	0.992014i	$-0.540254\pi$
$719$	−6.76393	−0.252252	−0.126126	+	0.992014i	$0.540254\pi$
$720$	0	0
$721$	58.2492	2.16931
$722$	21.9787	0.817963
$723$	−7.00000	−0.260333
$724$	−11.3820	−0.423007
$725$	0	0
$726$	−1.70820	−0.0633974
$727$	−37.8885	−1.40521	−0.702604	−	0.711581i	$-0.747980\pi$
$727$	−37.8885	−1.40521	−0.702604	+	0.711581i	$0.747980\pi$
$728$	−11.8328	−0.438553
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.76393	0.250173
$732$	−4.76393	−0.176080
$733$	27.7082	1.02343	0.511713	−	0.859156i	$-0.329011\pi$
$733$	27.7082	1.02343	0.511713	+	0.859156i	$0.329011\pi$
$734$	18.4721	0.681819
$735$	0	0
$736$	0	0
$737$	8.77709	0.323308
$738$	7.23607	0.266363
$739$	−43.4853	−1.59963	−0.799816	−	0.600245i	$-0.795070\pi$
$739$	−43.4853	−1.59963	−0.799816	+	0.600245i	$0.795070\pi$
$740$	0	0
$741$	10.0689	0.369890
$742$	54.5410	2.00226
$743$	33.1803	1.21727	0.608634	−	0.793451i	$-0.291718\pi$
$743$	33.1803	1.21727	0.608634	+	0.793451i	$0.291718\pi$
$744$	−17.8885	−0.655826
$745$	0	0
$746$	32.1803	1.17821
$747$	−9.70820	−0.355205
$748$	11.7426	0.429354
$749$	−49.4164	−1.80564
$750$	0	0
$751$	16.5836	0.605144	0.302572	−	0.953127i	$-0.402155\pi$
$751$	16.5836	0.605144	0.302572	+	0.953127i	$0.402155\pi$
$752$	−32.5623	−1.18743
$753$	−10.8885	−0.396801
$754$	2.85410	0.103940
$755$	0	0
$756$	−1.85410	−0.0674330
$757$	−18.8328	−0.684490	−0.342245	−	0.939611i	$-0.611187\pi$
$757$	−18.8328	−0.684490	−0.342245	+	0.939611i	$0.611187\pi$
$758$	−17.7082	−0.643191
$759$	0	0
$760$	0	0
$761$	−43.0132	−1.55923	−0.779613	−	0.626262i	$-0.784584\pi$
$761$	−43.0132	−1.55923	−0.779613	+	0.626262i	$0.784584\pi$
$762$	−9.70820	−0.351691
$763$	31.2492	1.13130
$764$	7.41641	0.268316
$765$	0	0
$766$	−61.5967	−2.22558
$767$	1.34752	0.0486563
$768$	−13.5623	−0.489388
$769$	3.34752	0.120715	0.0603574	−	0.998177i	$-0.480776\pi$
$769$	3.34752	0.120715	0.0603574	+	0.998177i	$0.480776\pi$
$770$	0	0
$771$	3.70820	0.133548
$772$	3.70820	0.133461
$773$	−28.2492	−1.01605	−0.508027	−	0.861341i	$-0.669625\pi$
$773$	−28.2492	−1.01605	−0.508027	+	0.861341i	$0.669625\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	16.1803	0.580840
$777$	−24.0000	−0.860995
$778$	28.7426	1.03047
$779$	−25.5279	−0.914631
$780$	0	0
$781$	9.59675	0.343399
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	−9.70820	−0.346722
$785$	0	0
$786$	8.85410	0.315815
$787$	33.8885	1.20800	0.603998	−	0.796986i	$-0.293573\pi$
$787$	33.8885	1.20800	0.603998	+	0.796986i	$0.293573\pi$
$788$	1.81966	0.0648227
$789$	24.0000	0.854423
$790$	0	0
$791$	−29.8328	−1.06073
$792$	−7.76393	−0.275879
$793$	13.5967	0.482835
$794$	54.0689	1.91883
$795$	0	0
$796$	4.50658	0.159731
$797$	10.9443	0.387666	0.193833	−	0.981035i	$-0.437908\pi$
$797$	10.9443	0.387666	0.193833	+	0.981035i	$0.437908\pi$
$798$	27.7082	0.980860
$799$	36.7082	1.29864
$800$	0	0
$801$	−11.1803	−0.395038
$802$	−8.65248	−0.305530
$803$	27.7771	0.980232
$804$	−1.56231	−0.0550983
$805$	0	0
$806$	−22.8328	−0.804252
$807$	25.7639	0.906933
$808$	0.527864	0.0185702
$809$	39.7639	1.39803	0.699013	−	0.715109i	$-0.253623\pi$
$809$	39.7639	1.39803	0.699013	+	0.715109i	$0.253623\pi$
$810$	0	0
$811$	−9.29180	−0.326279	−0.163140	−	0.986603i	$-0.552162\pi$
$811$	−9.29180	−0.326279	−0.163140	+	0.986603i	$0.552162\pi$
$812$	1.85410	0.0650662
$813$	30.3607	1.06480
$814$	44.9443	1.57530
$815$	0	0
$816$	26.5623	0.929867
$817$	−7.05573	−0.246849
$818$	−36.6525	−1.28152
$819$	5.29180	0.184910
$820$	0	0
$821$	−37.4164	−1.30584	−0.652921	−	0.757426i	$-0.726457\pi$
$821$	−37.4164	−1.30584	−0.652921	+	0.757426i	$0.726457\pi$
$822$	11.2361	0.391903
$823$	33.2361	1.15854	0.579268	−	0.815137i	$-0.303338\pi$
$823$	33.2361	1.15854	0.579268	+	0.815137i	$0.303338\pi$
$824$	−43.4164	−1.51248
$825$	0	0
$826$	3.70820	0.129025
$827$	−19.4164	−0.675175	−0.337587	−	0.941294i	$-0.609611\pi$
$827$	−19.4164	−0.675175	−0.337587	+	0.941294i	$0.609611\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	−4.70820	−0.163326
$832$	7.47214	0.259050
$833$	10.9443	0.379197
$834$	−20.5623	−0.712014
$835$	0	0
$836$	−12.2492	−0.423648
$837$	8.00000	0.276520
$838$	−16.7639	−0.579100
$839$	−20.8885	−0.721153	−0.360576	−	0.932730i	$-0.617420\pi$
$839$	−20.8885	−0.721153	−0.360576	+	0.932730i	$0.617420\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−39.1246	−1.34832
$843$	−17.1246	−0.589803
$844$	4.87539	0.167818
$845$	0	0
$846$	10.8541	0.373172
$847$	3.16718	0.108826
$848$	−54.5410	−1.87295
$849$	9.52786	0.326995
$850$	0	0
$851$	0	0
$852$	−1.70820	−0.0585221
$853$	33.2361	1.13798	0.568991	−	0.822344i	$-0.307334\pi$
$853$	33.2361	1.13798	0.568991	+	0.822344i	$0.307334\pi$
$854$	37.4164	1.28036
$855$	0	0
$856$	36.8328	1.25892
$857$	−11.8885	−0.406105	−0.203052	−	0.979168i	$-0.565086\pi$
$857$	−11.8885	−0.406105	−0.203052	+	0.979168i	$0.565086\pi$
$858$	−9.90983	−0.338316
$859$	−2.29180	−0.0781951	−0.0390975	−	0.999235i	$-0.512448\pi$
$859$	−2.29180	−0.0781951	−0.0390975	+	0.999235i	$0.512448\pi$
$860$	0	0
$861$	−13.4164	−0.457230
$862$	41.3050	1.40685
$863$	33.5967	1.14365	0.571823	−	0.820377i	$-0.306236\pi$
$863$	33.5967	1.14365	0.571823	+	0.820377i	$0.306236\pi$
$864$	3.38197	0.115057
$865$	0	0
$866$	−43.1246	−1.46543
$867$	−12.9443	−0.439611
$868$	−14.8328	−0.503459
$869$	56.1803	1.90579
$870$	0	0
$871$	4.45898	0.151087
$872$	−23.2918	−0.788760
$873$	−7.23607	−0.244904
$874$	0	0
$875$	0	0
$876$	−4.94427	−0.167051
$877$	4.83282	0.163193	0.0815963	−	0.996665i	$-0.473998\pi$
$877$	4.83282	0.163193	0.0815963	+	0.996665i	$0.473998\pi$
$878$	48.7426	1.64498
$879$	−0.0557281	−0.00187966
$880$	0	0
$881$	−0.708204	−0.0238600	−0.0119300	−	0.999929i	$-0.503798\pi$
$881$	−0.708204	−0.0238600	−0.0119300	+	0.999929i	$0.503798\pi$
$882$	3.23607	0.108964
$883$	32.0000	1.07689	0.538443	−	0.842662i	$-0.319013\pi$
$883$	32.0000	1.07689	0.538443	+	0.842662i	$0.319013\pi$
$884$	5.96556	0.200643
$885$	0	0
$886$	−45.6869	−1.53488
$887$	−4.59675	−0.154344	−0.0771718	−	0.997018i	$-0.524589\pi$
$887$	−4.59675	−0.154344	−0.0771718	+	0.997018i	$0.524589\pi$
$888$	17.8885	0.600300
$889$	18.0000	0.603701
$890$	0	0
$891$	3.47214	0.116321
$892$	8.03444	0.269013
$893$	−38.2918	−1.28139
$894$	3.52786	0.117989
$895$	0	0
$896$	40.8541	1.36484
$897$	0	0
$898$	10.0902	0.336713
$899$	−8.00000	−0.266815
$900$	0	0
$901$	61.4853	2.04837
$902$	25.1246	0.836558
$903$	−3.70820	−0.123401
$904$	22.2361	0.739561
$905$	0	0
$906$	10.4721	0.347913
$907$	−31.2361	−1.03718	−0.518588	−	0.855024i	$-0.673542\pi$
$907$	−31.2361	−1.03718	−0.518588	+	0.855024i	$0.673542\pi$
$908$	17.4164	0.577984
$909$	−0.236068	−0.00782988
$910$	0	0
$911$	−35.9443	−1.19089	−0.595443	−	0.803397i	$-0.703024\pi$
$911$	−35.9443	−1.19089	−0.595443	+	0.803397i	$0.703024\pi$
$912$	−27.7082	−0.917510
$913$	−33.7082	−1.11558
$914$	55.2148	1.82634
$915$	0	0
$916$	−10.7639	−0.355650
$917$	−16.4164	−0.542118
$918$	−8.85410	−0.292229
$919$	−20.1246	−0.663850	−0.331925	−	0.943306i	$-0.607698\pi$
$919$	−20.1246	−0.663850	−0.331925	+	0.943306i	$0.607698\pi$
$920$	0	0
$921$	7.05573	0.232494
$922$	−68.5410	−2.25728
$923$	4.87539	0.160475
$924$	−6.43769	−0.211785
$925$	0	0
$926$	18.5623	0.609995
$927$	19.4164	0.637719
$928$	−3.38197	−0.111018
$929$	−3.16718	−0.103912	−0.0519560	−	0.998649i	$-0.516546\pi$
$929$	−3.16718	−0.103912	−0.0519560	+	0.998649i	$0.516546\pi$
$930$	0	0
$931$	−11.4164	−0.374158
$932$	15.7082	0.514539
$933$	−14.5279	−0.475621
$934$	−8.00000	−0.261768
$935$	0	0
$936$	−3.94427	−0.128923
$937$	−24.2361	−0.791758	−0.395879	−	0.918303i	$-0.629560\pi$
$937$	−24.2361	−0.791758	−0.395879	+	0.918303i	$0.629560\pi$
$938$	12.2705	0.400646
$939$	−21.1803	−0.691194
$940$	0	0
$941$	12.0689	0.393434	0.196717	−	0.980460i	$-0.436972\pi$
$941$	12.0689	0.393434	0.196717	+	0.980460i	$0.436972\pi$
$942$	2.76393	0.0900538
$943$	0	0
$944$	−3.70820	−0.120692
$945$	0	0
$946$	6.94427	0.225778
$947$	−0.708204	−0.0230135	−0.0115068	−	0.999934i	$-0.503663\pi$
$947$	−0.708204	−0.0230135	−0.0115068	+	0.999934i	$0.503663\pi$
$948$	−10.0000	−0.324785
$949$	14.1115	0.458077
$950$	0	0
$951$	−1.58359	−0.0513515
$952$	−36.7082	−1.18972
$953$	57.0132	1.84684	0.923419	−	0.383794i	$-0.125383\pi$
$953$	57.0132	1.84684	0.923419	+	0.383794i	$0.125383\pi$
$954$	18.1803	0.588610
$955$	0	0
$956$	−14.7639	−0.477500
$957$	−3.47214	−0.112238
$958$	4.00000	0.129234
$959$	−20.8328	−0.672727
$960$	0	0
$961$	33.0000	1.06452
$962$	22.8328	0.736160
$963$	−16.4721	−0.530807
$964$	4.32624	0.139339
$965$	0	0
$966$	0	0
$967$	−5.12461	−0.164796	−0.0823982	−	0.996599i	$-0.526258\pi$
$967$	−5.12461	−0.164796	−0.0823982	+	0.996599i	$0.526258\pi$
$968$	−2.36068	−0.0758751
$969$	31.2361	1.00345
$970$	0	0
$971$	−40.9443	−1.31396	−0.656982	−	0.753906i	$-0.728167\pi$
$971$	−40.9443	−1.31396	−0.656982	+	0.753906i	$0.728167\pi$
$972$	−0.618034	−0.0198234
$973$	38.1246	1.22222
$974$	19.4164	0.622142
$975$	0	0
$976$	−37.4164	−1.19767
$977$	13.5279	0.432795	0.216397	−	0.976305i	$-0.430569\pi$
$977$	13.5279	0.432795	0.216397	+	0.976305i	$0.430569\pi$
$978$	40.6525	1.29992
$979$	−38.8197	−1.24068
$980$	0	0
$981$	10.4164	0.332570
$982$	−16.0000	−0.510581
$983$	−46.4721	−1.48223	−0.741115	−	0.671378i	$-0.765703\pi$
$983$	−46.4721	−1.48223	−0.741115	+	0.671378i	$0.765703\pi$
$984$	10.0000	0.318788
$985$	0	0
$986$	8.85410	0.281972
$987$	−20.1246	−0.640573
$988$	−6.22291	−0.197977
$989$	0	0
$990$	0	0
$991$	−62.4853	−1.98491	−0.992455	−	0.122607i	$-0.960875\pi$
$991$	−62.4853	−1.98491	−0.992455	+	0.122607i	$0.960875\pi$
$992$	27.0557	0.859020
$993$	−19.8885	−0.631144
$994$	13.4164	0.425543
$995$	0	0
$996$	6.00000	0.190117
$997$	19.4164	0.614924	0.307462	−	0.951560i	$-0.400520\pi$
$997$	19.4164	0.614924	0.307462	+	0.951560i	$0.400520\pi$
$998$	−45.6869	−1.44619
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.q.1.2		2
3.2	odd	2		6525.2.a.s.1.1		2
5.2	odd	4		2175.2.c.j.349.4		4
5.3	odd	4		2175.2.c.j.349.1		4
5.4	even	2		435.2.a.e.1.1	✓	2
15.14	odd	2		1305.2.a.k.1.2		2
20.19	odd	2		6960.2.a.bu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.e.1.1	✓	2	5.4	even	2
1305.2.a.k.1.2		2	15.14	odd	2
2175.2.a.q.1.2		2	1.1	even	1	trivial
2175.2.c.j.349.1		4	5.3	odd	4
2175.2.c.j.349.4		4	5.2	odd	4
6525.2.a.s.1.1		2	3.2	odd	2
6960.2.a.bu.1.1		2	20.19	odd	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +3.47214 q^{11} -0.618034 q^{12} +1.76393 q^{13} +4.85410 q^{14} -4.85410 q^{16} +5.47214 q^{17} +1.61803 q^{18} -5.70820 q^{19} -3.00000 q^{21} +5.61803 q^{22} +2.23607 q^{24} +2.85410 q^{26} -1.00000 q^{27} +1.85410 q^{28} +1.00000 q^{29} -8.00000 q^{31} -3.38197 q^{32} -3.47214 q^{33} +8.85410 q^{34} +0.618034 q^{36} +8.00000 q^{37} -9.23607 q^{38} -1.76393 q^{39} +4.47214 q^{41} -4.85410 q^{42} +1.23607 q^{43} +2.14590 q^{44} +6.70820 q^{47} +4.85410 q^{48} +2.00000 q^{49} -5.47214 q^{51} +1.09017 q^{52} +11.2361 q^{53} -1.61803 q^{54} -6.70820 q^{56} +5.70820 q^{57} +1.61803 q^{58} +0.763932 q^{59} +7.70820 q^{61} -12.9443 q^{62} +3.00000 q^{63} +4.23607 q^{64} -5.61803 q^{66} +2.52786 q^{67} +3.38197 q^{68} +2.76393 q^{71} -2.23607 q^{72} +8.00000 q^{73} +12.9443 q^{74} -3.52786 q^{76} +10.4164 q^{77} -2.85410 q^{78} +16.1803 q^{79} +1.00000 q^{81} +7.23607 q^{82} -9.70820 q^{83} -1.85410 q^{84} +2.00000 q^{86} -1.00000 q^{87} -7.76393 q^{88} -11.1803 q^{89} +5.29180 q^{91} +8.00000 q^{93} +10.8541 q^{94} +3.38197 q^{96} -7.23607 q^{97} +3.23607 q^{98} +3.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9} - 2 q^{11} + q^{12} + 8 q^{13} + 3 q^{14} - 3 q^{16} + 2 q^{17} + q^{18} + 2 q^{19} - 6 q^{21} + 9 q^{22} - q^{26} - 2 q^{27} - 3 q^{28} + 2 q^{29} - 16 q^{31} - 9 q^{32} + 2 q^{33} + 11 q^{34} - q^{36} + 16 q^{37} - 14 q^{38} - 8 q^{39} - 3 q^{42} - 2 q^{43} + 11 q^{44} + 3 q^{48} + 4 q^{49} - 2 q^{51} - 9 q^{52} + 18 q^{53} - q^{54} - 2 q^{57} + q^{58} + 6 q^{59} + 2 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} - 9 q^{66} + 14 q^{67} + 9 q^{68} + 10 q^{71} + 16 q^{73} + 8 q^{74} - 16 q^{76} - 6 q^{77} + q^{78} + 10 q^{79} + 2 q^{81} + 10 q^{82} - 6 q^{83} + 3 q^{84} + 4 q^{86} - 2 q^{87} - 20 q^{88} + 24 q^{91} + 16 q^{93} + 15 q^{94} + 9 q^{96} - 10 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 - 2 * q^11 + q^12 + 8 * q^13 + 3 * q^14 - 3 * q^16 + 2 * q^17 + q^18 + 2 * q^19 - 6 * q^21 + 9 * q^22 - q^26 - 2 * q^27 - 3 * q^28 + 2 * q^29 - 16 * q^31 - 9 * q^32 + 2 * q^33 + 11 * q^34 - q^36 + 16 * q^37 - 14 * q^38 - 8 * q^39 - 3 * q^42 - 2 * q^43 + 11 * q^44 + 3 * q^48 + 4 * q^49 - 2 * q^51 - 9 * q^52 + 18 * q^53 - q^54 - 2 * q^57 + q^58 + 6 * q^59 + 2 * q^61 - 8 * q^62 + 6 * q^63 + 4 * q^64 - 9 * q^66 + 14 * q^67 + 9 * q^68 + 10 * q^71 + 16 * q^73 + 8 * q^74 - 16 * q^76 - 6 * q^77 + q^78 + 10 * q^79 + 2 * q^81 + 10 * q^82 - 6 * q^83 + 3 * q^84 + 4 * q^86 - 2 * q^87 - 20 * q^88 + 24 * q^91 + 16 * q^93 + 15 * q^94 + 9 * q^96 - 10 * q^97 + 2 * q^98 - 2 * q^99

Introduction

L-functions

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