LMFDB - Embedded newform 4225.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 169)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.24698 q^{2} +0.554958 q^{3} +3.04892 q^{4} -1.24698 q^{6} +2.04892 q^{7} -2.35690 q^{8} -2.69202 q^{9} +2.55496 q^{11} +1.69202 q^{12} -4.60388 q^{14} -0.801938 q^{16} +5.29590 q^{17} +6.04892 q^{18} +5.85086 q^{19} +1.13706 q^{21} -5.74094 q^{22} +1.89008 q^{23} -1.30798 q^{24} -3.15883 q^{27} +6.24698 q^{28} +2.26875 q^{29} +4.26875 q^{31} +6.51573 q^{32} +1.41789 q^{33} -11.8998 q^{34} -8.20775 q^{36} +5.35690 q^{37} -13.1468 q^{38} -1.27413 q^{41} -2.55496 q^{42} -6.13706 q^{43} +7.78986 q^{44} -4.24698 q^{46} -2.95108 q^{47} -0.445042 q^{48} -2.80194 q^{49} +2.93900 q^{51} -5.52111 q^{53} +7.09783 q^{54} -4.82908 q^{56} +3.24698 q^{57} -5.09783 q^{58} +12.2078 q^{59} +8.56465 q^{61} -9.59179 q^{62} -5.51573 q^{63} -13.0368 q^{64} -3.18598 q^{66} +0.576728 q^{67} +16.1468 q^{68} +1.04892 q^{69} +4.59419 q^{71} +6.34481 q^{72} -10.5526 q^{73} -12.0368 q^{74} +17.8388 q^{76} +5.23490 q^{77} -15.7778 q^{79} +6.32304 q^{81} +2.86294 q^{82} +7.72348 q^{83} +3.46681 q^{84} +13.7899 q^{86} +1.25906 q^{87} -6.02177 q^{88} -6.61356 q^{89} +5.76271 q^{92} +2.36898 q^{93} +6.63102 q^{94} +3.61596 q^{96} +11.9269 q^{97} +6.29590 q^{98} -6.87800 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 2 q^{3} + q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9} + 8 q^{11} - 5 q^{14} + 2 q^{16} + 2 q^{17} + 9 q^{18} + 4 q^{19} - 2 q^{21} - 3 q^{22} + 5 q^{23} - 9 q^{24} - q^{27} + 14 q^{28} - q^{29}+ \cdots - q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 + 2 * q^3 + q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 + 8 * q^11 - 5 * q^14 + 2 * q^16 + 2 * q^17 + 9 * q^18 + 4 * q^19 - 2 * q^21 - 3 * q^22 + 5 * q^23 - 9 * q^24 - q^27 + 14 * q^28 - q^29 + 5 * q^31 + 7 * q^32 + 10 * q^33 - 13 * q^34 - 7 * q^36 + 12 * q^37 - 12 * q^38 + 7 * q^41 - 8 * q^42 - 13 * q^43 - 8 * q^46 - 18 * q^47 - q^48 - 4 * q^49 - q^51 - q^53 + 3 * q^54 - 4 * q^56 + 5 * q^57 + 3 * q^58 + 19 * q^59 + 4 * q^61 - q^62 - 4 * q^63 - 11 * q^64 + 5 * q^66 - q^67 + 21 * q^68 - 6 * q^69 + 27 * q^71 - 4 * q^72 + 9 * q^73 - 8 * q^74 + 21 * q^76 - 8 * q^77 - 5 * q^79 - q^81 + 14 * q^82 - 7 * q^83 + 7 * q^84 + 18 * q^86 + 18 * q^87 - 15 * q^88 + 11 * q^89 + 22 * q^93 + 5 * q^94 + 21 * q^96 + 7 * q^97 + 5 * q^98 - q^99

Coefficient data

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

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

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

$2$	−2.24698	−1.58885	−0.794427	−	0.607359i	$-0.792229\pi$
$2$	−2.24698	−1.58885	−0.794427	+	0.607359i	$0.792229\pi$
$3$	0.554958	0.320405	0.160203	−	0.987084i	$-0.448785\pi$
$3$	0.554958	0.320405	0.160203	+	0.987084i	$0.448785\pi$
$4$	3.04892	1.52446
$5$	0	0
$5$	0	0
$6$	−1.24698	−0.509077
$7$	2.04892	0.774418	0.387209	−	0.921992i	$-0.373439\pi$
$7$	2.04892	0.774418	0.387209	+	0.921992i	$0.373439\pi$
$8$	−2.35690	−0.833289
$9$	−2.69202	−0.897340
$10$	0	0
$11$	2.55496	0.770349	0.385174	−	0.922844i	$-0.374141\pi$
$11$	2.55496	0.770349	0.385174	+	0.922844i	$0.374141\pi$
$12$	1.69202	0.488445
$13$	0	0
$13$	0	0
$14$	−4.60388	−1.23044
$15$	0	0
$16$	−0.801938	−0.200484
$17$	5.29590	1.28444	0.642222	−	0.766519i	$-0.278013\pi$
$17$	5.29590	1.28444	0.642222	+	0.766519i	$0.278013\pi$
$18$	6.04892	1.42574
$19$	5.85086	1.34228	0.671139	−	0.741331i	$-0.265805\pi$
$19$	5.85086	1.34228	0.671139	+	0.741331i	$0.265805\pi$
$20$	0	0
$21$	1.13706	0.248128
$22$	−5.74094	−1.22397
$23$	1.89008	0.394110	0.197055	−	0.980392i	$-0.436862\pi$
$23$	1.89008	0.394110	0.197055	+	0.980392i	$0.436862\pi$
$24$	−1.30798	−0.266990
$25$	0	0
$26$	0	0
$27$	−3.15883	−0.607918
$28$	6.24698	1.18057
$29$	2.26875	0.421296	0.210648	−	0.977562i	$-0.432443\pi$
$29$	2.26875	0.421296	0.210648	+	0.977562i	$0.432443\pi$
$30$	0	0
$31$	4.26875	0.766690	0.383345	−	0.923605i	$-0.374772\pi$
$31$	4.26875	0.766690	0.383345	+	0.923605i	$0.374772\pi$
$32$	6.51573	1.15183
$33$	1.41789	0.246824
$34$	−11.8998	−2.04079
$35$	0	0
$36$	−8.20775	−1.36796
$37$	5.35690	0.880668	0.440334	−	0.897834i	$-0.354860\pi$
$37$	5.35690	0.880668	0.440334	+	0.897834i	$0.354860\pi$
$38$	−13.1468	−2.13268
$39$	0	0
$40$	0	0
$41$	−1.27413	−0.198985	−0.0994926	−	0.995038i	$-0.531722\pi$
$41$	−1.27413	−0.198985	−0.0994926	+	0.995038i	$0.531722\pi$
$42$	−2.55496	−0.394239
$43$	−6.13706	−0.935893	−0.467947	−	0.883757i	$-0.655006\pi$
$43$	−6.13706	−0.935893	−0.467947	+	0.883757i	$0.655006\pi$
$44$	7.78986	1.17437
$45$	0	0
$46$	−4.24698	−0.626183
$47$	−2.95108	−0.430460	−0.215230	−	0.976563i	$-0.569050\pi$
$47$	−2.95108	−0.430460	−0.215230	+	0.976563i	$0.569050\pi$
$48$	−0.445042	−0.0642363
$49$	−2.80194	−0.400277
$50$	0	0
$51$	2.93900	0.411542
$52$	0	0
$53$	−5.52111	−0.758382	−0.379191	−	0.925318i	$-0.623798\pi$
$53$	−5.52111	−0.758382	−0.379191	+	0.925318i	$0.623798\pi$
$54$	7.09783	0.965893
$55$	0	0
$56$	−4.82908	−0.645314
$57$	3.24698	0.430073
$58$	−5.09783	−0.669378
$59$	12.2078	1.58931	0.794657	−	0.607059i	$-0.207651\pi$
$59$	12.2078	1.58931	0.794657	+	0.607059i	$0.207651\pi$
$60$	0	0
$61$	8.56465	1.09659	0.548295	−	0.836285i	$-0.315277\pi$
$61$	8.56465	1.09659	0.548295	+	0.836285i	$0.315277\pi$
$62$	−9.59179	−1.21816
$63$	−5.51573	−0.694917
$64$	−13.0368	−1.62960
$65$	0	0
$66$	−3.18598	−0.392167
$67$	0.576728	0.0704586	0.0352293	−	0.999379i	$-0.488784\pi$
$67$	0.576728	0.0704586	0.0352293	+	0.999379i	$0.488784\pi$
$68$	16.1468	1.95808
$69$	1.04892	0.126275
$70$	0	0
$71$	4.59419	0.545230	0.272615	−	0.962123i	$-0.412112\pi$
$71$	4.59419	0.545230	0.272615	+	0.962123i	$0.412112\pi$
$72$	6.34481	0.747744
$73$	−10.5526	−1.23508	−0.617542	−	0.786538i	$-0.711872\pi$
$73$	−10.5526	−1.23508	−0.617542	+	0.786538i	$0.711872\pi$
$74$	−12.0368	−1.39925
$75$	0	0
$76$	17.8388	2.04625
$77$	5.23490	0.596572
$78$	0	0
$79$	−15.7778	−1.77514	−0.887569	−	0.460674i	$-0.847608\pi$
$79$	−15.7778	−1.77514	−0.887569	+	0.460674i	$0.847608\pi$
$80$	0	0
$81$	6.32304	0.702560
$82$	2.86294	0.316158
$83$	7.72348	0.847762	0.423881	−	0.905718i	$-0.360667\pi$
$83$	7.72348	0.847762	0.423881	+	0.905718i	$0.360667\pi$
$84$	3.46681	0.378260
$85$	0	0
$86$	13.7899	1.48700
$87$	1.25906	0.134986
$88$	−6.02177	−0.641923
$89$	−6.61356	−0.701036	−0.350518	−	0.936556i	$-0.613995\pi$
$89$	−6.61356	−0.701036	−0.350518	+	0.936556i	$0.613995\pi$
$90$	0	0
$91$	0	0
$92$	5.76271	0.600804
$93$	2.36898	0.245652
$94$	6.63102	0.683938
$95$	0	0
$96$	3.61596	0.369052
$97$	11.9269	1.21100	0.605498	−	0.795847i	$-0.292974\pi$
$97$	11.9269	1.21100	0.605498	+	0.795847i	$0.292974\pi$
$98$	6.29590	0.635982
$99$	−6.87800	−0.691265
$100$	0	0
$101$	13.0640	1.29991	0.649957	−	0.759971i	$-0.274787\pi$
$101$	13.0640	1.29991	0.649957	+	0.759971i	$0.274787\pi$
$102$	−6.60388	−0.653881
$103$	−9.16852	−0.903401	−0.451701	−	0.892170i	$-0.649182\pi$
$103$	−9.16852	−0.903401	−0.451701	+	0.892170i	$0.649182\pi$
$104$	0	0
$105$	0	0
$106$	12.4058	1.20496
$107$	6.89977	0.667026	0.333513	−	0.942745i	$-0.391766\pi$
$107$	6.89977	0.667026	0.333513	+	0.942745i	$0.391766\pi$
$108$	−9.63102	−0.926746
$109$	−0.121998	−0.0116853	−0.00584264	−	0.999983i	$-0.501860\pi$
$109$	−0.121998	−0.0116853	−0.00584264	+	0.999983i	$0.501860\pi$
$110$	0	0
$111$	2.97285	0.282171
$112$	−1.64310	−0.155259
$113$	−7.30798	−0.687477	−0.343738	−	0.939065i	$-0.611693\pi$
$113$	−7.30798	−0.687477	−0.343738	+	0.939065i	$0.611693\pi$
$114$	−7.29590	−0.683323
$115$	0	0
$116$	6.91723	0.642249
$117$	0	0
$118$	−27.4306	−2.52519
$119$	10.8509	0.994696
$120$	0	0
$121$	−4.47219	−0.406563
$122$	−19.2446	−1.74232
$123$	−0.707087	−0.0637559
$124$	13.0151	1.16879
$125$	0	0
$126$	12.3937	1.10412
$127$	18.9705	1.68336	0.841678	−	0.539980i	$-0.181568\pi$
$127$	18.9705	1.68336	0.841678	+	0.539980i	$0.181568\pi$
$128$	16.2620	1.43738
$129$	−3.40581	−0.299865
$130$	0	0
$131$	3.25667	0.284536	0.142268	−	0.989828i	$-0.454560\pi$
$131$	3.25667	0.284536	0.142268	+	0.989828i	$0.454560\pi$
$132$	4.32304	0.376273
$133$	11.9879	1.03948
$134$	−1.29590	−0.111948
$135$	0	0
$136$	−12.4819	−1.07031
$137$	0.792249	0.0676864	0.0338432	−	0.999427i	$-0.489225\pi$
$137$	0.792249	0.0676864	0.0338432	+	0.999427i	$0.489225\pi$
$138$	−2.35690	−0.200632
$139$	−11.3394	−0.961799	−0.480899	−	0.876776i	$-0.659690\pi$
$139$	−11.3394	−0.961799	−0.480899	+	0.876776i	$0.659690\pi$
$140$	0	0
$141$	−1.63773	−0.137922
$142$	−10.3230	−0.866291
$143$	0	0
$144$	2.15883	0.179903
$145$	0	0
$146$	23.7114	1.96237
$147$	−1.55496	−0.128251
$148$	16.3327	1.34254
$149$	−8.40581	−0.688631	−0.344316	−	0.938854i	$-0.611889\pi$
$149$	−8.40581	−0.688631	−0.344316	+	0.938854i	$0.611889\pi$
$150$	0	0
$151$	−14.1293	−1.14983	−0.574913	−	0.818215i	$-0.694964\pi$
$151$	−14.1293	−1.14983	−0.574913	+	0.818215i	$0.694964\pi$
$152$	−13.7899	−1.11851
$153$	−14.2567	−1.15258
$154$	−11.7627	−0.947866
$155$	0	0
$156$	0	0
$157$	9.43296	0.752832	0.376416	−	0.926451i	$-0.377156\pi$
$157$	9.43296	0.752832	0.376416	+	0.926451i	$0.377156\pi$
$158$	35.4523	2.82044
$159$	−3.06398	−0.242990
$160$	0	0
$161$	3.87263	0.305206
$162$	−14.2078	−1.11627
$163$	8.70410	0.681758	0.340879	−	0.940107i	$-0.389275\pi$
$163$	8.70410	0.681758	0.340879	+	0.940107i	$0.389275\pi$
$164$	−3.88471	−0.303345
$165$	0	0
$166$	−17.3545	−1.34697
$167$	−23.8538	−1.84587	−0.922933	−	0.384961i	$-0.874215\pi$
$167$	−23.8538	−1.84587	−0.922933	+	0.384961i	$0.874215\pi$
$168$	−2.67994	−0.206762
$169$	0	0
$170$	0	0
$171$	−15.7506	−1.20448
$172$	−18.7114	−1.42673
$173$	18.8552	1.43353	0.716766	−	0.697314i	$-0.245622\pi$
$173$	18.8552	1.43353	0.716766	+	0.697314i	$0.245622\pi$
$174$	−2.82908	−0.214472
$175$	0	0
$176$	−2.04892	−0.154443
$177$	6.77479	0.509224
$178$	14.8605	1.11384
$179$	6.02177	0.450088	0.225044	−	0.974349i	$-0.427747\pi$
$179$	6.02177	0.450088	0.225044	+	0.974349i	$0.427747\pi$
$180$	0	0
$181$	−4.77777	−0.355129	−0.177565	−	0.984109i	$-0.556822\pi$
$181$	−4.77777	−0.355129	−0.177565	+	0.984109i	$0.556822\pi$
$182$	0	0
$183$	4.75302	0.351353
$184$	−4.45473	−0.328407
$185$	0	0
$186$	−5.32304	−0.390305
$187$	13.5308	0.989470
$188$	−8.99761	−0.656218
$189$	−6.47219	−0.470782
$190$	0	0
$191$	18.4306	1.33359	0.666795	−	0.745242i	$-0.267666\pi$
$191$	18.4306	1.33359	0.666795	+	0.745242i	$0.267666\pi$
$192$	−7.23490	−0.522134
$193$	6.05429	0.435798	0.217899	−	0.975971i	$-0.430080\pi$
$193$	6.05429	0.435798	0.217899	+	0.975971i	$0.430080\pi$
$194$	−26.7995	−1.92410
$195$	0	0
$196$	−8.54288	−0.610205
$197$	−11.4155	−0.813321	−0.406660	−	0.913579i	$-0.633307\pi$
$197$	−11.4155	−0.813321	−0.406660	+	0.913579i	$0.633307\pi$
$198$	15.4547	1.09832
$199$	−13.9051	−0.985710	−0.492855	−	0.870111i	$-0.664047\pi$
$199$	−13.9051	−0.985710	−0.492855	+	0.870111i	$0.664047\pi$
$200$	0	0
$201$	0.320060	0.0225753
$202$	−29.3545	−2.06538
$203$	4.64848	0.326259
$204$	8.96077	0.627379
$205$	0	0
$206$	20.6015	1.43537
$207$	−5.08815	−0.353651
$208$	0	0
$209$	14.9487	1.03402
$210$	0	0
$211$	−13.2446	−0.911795	−0.455897	−	0.890032i	$-0.650682\pi$
$211$	−13.2446	−0.911795	−0.455897	+	0.890032i	$0.650682\pi$
$212$	−16.8334	−1.15612
$213$	2.54958	0.174694
$214$	−15.5036	−1.05981
$215$	0	0
$216$	7.44504	0.506571
$217$	8.74632	0.593739
$218$	0.274127	0.0185662
$219$	−5.85623	−0.395727
$220$	0	0
$221$	0	0
$222$	−6.67994	−0.448328
$223$	−7.33513	−0.491196	−0.245598	−	0.969372i	$-0.578984\pi$
$223$	−7.33513	−0.491196	−0.245598	+	0.969372i	$0.578984\pi$
$224$	13.3502	0.891997
$225$	0	0
$226$	16.4209	1.09230
$227$	−8.67456	−0.575751	−0.287875	−	0.957668i	$-0.592949\pi$
$227$	−8.67456	−0.575751	−0.287875	+	0.957668i	$0.592949\pi$
$228$	9.89977	0.655628
$229$	13.6866	0.904439	0.452219	−	0.891907i	$-0.350632\pi$
$229$	13.6866	0.904439	0.452219	+	0.891907i	$0.350632\pi$
$230$	0	0
$231$	2.90515	0.191145
$232$	−5.34721	−0.351061
$233$	5.08815	0.333336	0.166668	−	0.986013i	$-0.446699\pi$
$233$	5.08815	0.333336	0.166668	+	0.986013i	$0.446699\pi$
$234$	0	0
$235$	0	0
$236$	37.2204	2.42284
$237$	−8.75600	−0.568764
$238$	−24.3817	−1.58043
$239$	10.9239	0.706611	0.353305	−	0.935508i	$-0.385058\pi$
$239$	10.9239	0.706611	0.353305	+	0.935508i	$0.385058\pi$
$240$	0	0
$241$	−11.9148	−0.767502	−0.383751	−	0.923437i	$-0.625368\pi$
$241$	−11.9148	−0.767502	−0.383751	+	0.923437i	$0.625368\pi$
$242$	10.0489	0.645969
$243$	12.9855	0.833022
$244$	26.1129	1.67171
$245$	0	0
$246$	1.58881	0.101299
$247$	0	0
$248$	−10.0610	−0.638874
$249$	4.28621	0.271627
$250$	0	0
$251$	22.3478	1.41058	0.705290	−	0.708919i	$-0.250817\pi$
$251$	22.3478	1.41058	0.705290	+	0.708919i	$0.250817\pi$
$252$	−16.8170	−1.05937
$253$	4.82908	0.303602
$254$	−42.6262	−2.67461
$255$	0	0
$256$	−10.4668	−0.654176
$257$	18.6601	1.16398	0.581992	−	0.813194i	$-0.302273\pi$
$257$	18.6601	1.16398	0.581992	+	0.813194i	$0.302273\pi$
$258$	7.65279	0.476442
$259$	10.9758	0.682005
$260$	0	0
$261$	−6.10752	−0.378046
$262$	−7.31767	−0.452087
$263$	−14.3991	−0.887887	−0.443944	−	0.896055i	$-0.646421\pi$
$263$	−14.3991	−0.887887	−0.443944	+	0.896055i	$0.646421\pi$
$264$	−3.34183	−0.205675
$265$	0	0
$266$	−26.9366	−1.65159
$267$	−3.67025	−0.224616
$268$	1.75840	0.107411
$269$	0.652793	0.0398015	0.0199007	−	0.999802i	$-0.493665\pi$
$269$	0.652793	0.0398015	0.0199007	+	0.999802i	$0.493665\pi$
$270$	0	0
$271$	1.99569	0.121229	0.0606147	−	0.998161i	$-0.480694\pi$
$271$	1.99569	0.121229	0.0606147	+	0.998161i	$0.480694\pi$
$272$	−4.24698	−0.257511
$273$	0	0
$274$	−1.78017	−0.107544
$275$	0	0
$276$	3.19806	0.192501
$277$	−11.7845	−0.708061	−0.354030	−	0.935234i	$-0.615189\pi$
$277$	−11.7845	−0.708061	−0.354030	+	0.935234i	$0.615189\pi$
$278$	25.4795	1.52816
$279$	−11.4916	−0.687982
$280$	0	0
$281$	6.47219	0.386098	0.193049	−	0.981189i	$-0.438162\pi$
$281$	6.47219	0.386098	0.193049	+	0.981189i	$0.438162\pi$
$282$	3.67994	0.219137
$283$	−6.58104	−0.391202	−0.195601	−	0.980684i	$-0.562666\pi$
$283$	−6.58104	−0.391202	−0.195601	+	0.980684i	$0.562666\pi$
$284$	14.0073	0.831180
$285$	0	0
$286$	0	0
$287$	−2.61058	−0.154098
$288$	−17.5405	−1.03358
$289$	11.0465	0.649796
$290$	0	0
$291$	6.61894	0.388009
$292$	−32.1739	−1.88284
$293$	−24.3381	−1.42185	−0.710924	−	0.703269i	$-0.751723\pi$
$293$	−24.3381	−1.42185	−0.710924	+	0.703269i	$0.751723\pi$
$294$	3.49396	0.203772
$295$	0	0
$296$	−12.6256	−0.733851
$297$	−8.07069	−0.468309
$298$	18.8877	1.09413
$299$	0	0
$300$	0	0
$301$	−12.5743	−0.724773
$302$	31.7482	1.82691
$303$	7.24996	0.416500
$304$	−4.69202	−0.269106
$305$	0	0
$306$	32.0344	1.83129
$307$	−14.0737	−0.803227	−0.401613	−	0.915809i	$-0.631550\pi$
$307$	−14.0737	−0.803227	−0.401613	+	0.915809i	$0.631550\pi$
$308$	15.9608	0.909449
$309$	−5.08815	−0.289455
$310$	0	0
$311$	−29.7700	−1.68810	−0.844051	−	0.536263i	$-0.819836\pi$
$311$	−29.7700	−1.68810	−0.844051	+	0.536263i	$0.819836\pi$
$312$	0	0
$313$	7.47889	0.422732	0.211366	−	0.977407i	$-0.432209\pi$
$313$	7.47889	0.422732	0.211366	+	0.977407i	$0.432209\pi$
$314$	−21.1957	−1.19614
$315$	0	0
$316$	−48.1051	−2.70613
$317$	30.0301	1.68666	0.843330	−	0.537396i	$-0.180592\pi$
$317$	30.0301	1.68666	0.843330	+	0.537396i	$0.180592\pi$
$318$	6.88471	0.386075
$319$	5.79656	0.324545
$320$	0	0
$321$	3.82908	0.213719
$322$	−8.70171	−0.484927
$323$	30.9855	1.72408
$324$	19.2784	1.07102
$325$	0	0
$326$	−19.5579	−1.08321
$327$	−0.0677037	−0.00374402
$328$	3.00298	0.165812
$329$	−6.04652	−0.333356
$330$	0	0
$331$	15.7168	0.863872	0.431936	−	0.901904i	$-0.357831\pi$
$331$	15.7168	0.863872	0.431936	+	0.901904i	$0.357831\pi$
$332$	23.5483	1.29238
$333$	−14.4209	−0.790259
$334$	53.5991	2.93281
$335$	0	0
$336$	−0.911854	−0.0497457
$337$	−1.95407	−0.106445	−0.0532224	−	0.998583i	$-0.516949\pi$
$337$	−1.95407	−0.106445	−0.0532224	+	0.998583i	$0.516949\pi$
$338$	0	0
$339$	−4.05562	−0.220271
$340$	0	0
$341$	10.9065	0.590619
$342$	35.3913	1.91374
$343$	−20.0834	−1.08440
$344$	14.4644	0.779869
$345$	0	0
$346$	−42.3672	−2.27767
$347$	17.1250	0.919317	0.459659	−	0.888096i	$-0.347972\pi$
$347$	17.1250	0.919317	0.459659	+	0.888096i	$0.347972\pi$
$348$	3.83877	0.205780
$349$	10.4668	0.560276	0.280138	−	0.959960i	$-0.409620\pi$
$349$	10.4668	0.560276	0.280138	+	0.959960i	$0.409620\pi$
$350$	0	0
$351$	0	0
$352$	16.6474	0.887310
$353$	15.5308	0.826621	0.413310	−	0.910590i	$-0.364372\pi$
$353$	15.5308	0.826621	0.413310	+	0.910590i	$0.364372\pi$
$354$	−15.2228	−0.809084
$355$	0	0
$356$	−20.1642	−1.06870
$357$	6.02177	0.318706
$358$	−13.5308	−0.715125
$359$	21.4263	1.13083	0.565417	−	0.824805i	$-0.308715\pi$
$359$	21.4263	1.13083	0.565417	+	0.824805i	$0.308715\pi$
$360$	0	0
$361$	15.2325	0.801711
$362$	10.7356	0.564249
$363$	−2.48188	−0.130265
$364$	0	0
$365$	0	0
$366$	−10.6799	−0.558249
$367$	−34.3032	−1.79061	−0.895306	−	0.445452i	$-0.853043\pi$
$367$	−34.3032	−1.79061	−0.895306	+	0.445452i	$0.853043\pi$
$368$	−1.51573	−0.0790129
$369$	3.42998	0.178557
$370$	0	0
$371$	−11.3123	−0.587305
$372$	7.22282	0.374486
$373$	12.5961	0.652202	0.326101	−	0.945335i	$-0.394265\pi$
$373$	12.5961	0.652202	0.326101	+	0.945335i	$0.394265\pi$
$374$	−30.4034	−1.57212
$375$	0	0
$376$	6.95539	0.358697
$377$	0	0
$378$	14.5429	0.748005
$379$	−16.5386	−0.849529	−0.424765	−	0.905304i	$-0.639643\pi$
$379$	−16.5386	−0.849529	−0.424765	+	0.905304i	$0.639643\pi$
$380$	0	0
$381$	10.5278	0.539356
$382$	−41.4131	−2.11888
$383$	7.53617	0.385080	0.192540	−	0.981289i	$-0.438327\pi$
$383$	7.53617	0.385080	0.192540	+	0.981289i	$0.438327\pi$
$384$	9.02475	0.460543
$385$	0	0
$386$	−13.6039	−0.692419
$387$	16.5211	0.839815
$388$	36.3642	1.84611
$389$	35.5555	1.80274	0.901369	−	0.433052i	$-0.142563\pi$
$389$	35.5555	1.80274	0.901369	+	0.433052i	$0.142563\pi$
$390$	0	0
$391$	10.0097	0.506212
$392$	6.60388	0.333546
$393$	1.80731	0.0911670
$394$	25.6504	1.29225
$395$	0	0
$396$	−20.9705	−1.05381
$397$	1.35152	0.0678308	0.0339154	−	0.999425i	$-0.489202\pi$
$397$	1.35152	0.0678308	0.0339154	+	0.999425i	$0.489202\pi$
$398$	31.2446	1.56615
$399$	6.65279	0.333056
$400$	0	0
$401$	−0.579121	−0.0289199	−0.0144600	−	0.999895i	$-0.504603\pi$
$401$	−0.579121	−0.0289199	−0.0144600	+	0.999895i	$0.504603\pi$
$402$	−0.719169	−0.0358689
$403$	0	0
$404$	39.8310	1.98167
$405$	0	0
$406$	−10.4450	−0.518379
$407$	13.6866	0.678422
$408$	−6.92692	−0.342934
$409$	15.1575	0.749490	0.374745	−	0.927128i	$-0.377730\pi$
$409$	15.1575	0.749490	0.374745	+	0.927128i	$0.377730\pi$
$410$	0	0
$411$	0.439665	0.0216871
$412$	−27.9541	−1.37720
$413$	25.0127	1.23079
$414$	11.4330	0.561899
$415$	0	0
$416$	0	0
$417$	−6.29291	−0.308165
$418$	−33.5894	−1.64291
$419$	−35.7235	−1.74521	−0.872603	−	0.488430i	$-0.837570\pi$
$419$	−35.7235	−1.74521	−0.872603	+	0.488430i	$0.837570\pi$
$420$	0	0
$421$	35.0465	1.70806	0.854032	−	0.520221i	$-0.174151\pi$
$421$	35.0465	1.70806	0.854032	+	0.520221i	$0.174151\pi$
$422$	29.7603	1.44871
$423$	7.94438	0.386269
$424$	13.0127	0.631951
$425$	0	0
$426$	−5.72886	−0.277564
$427$	17.5483	0.849220
$428$	21.0368	1.01685
$429$	0	0
$430$	0	0
$431$	34.2814	1.65128	0.825639	−	0.564199i	$-0.190815\pi$
$431$	34.2814	1.65128	0.825639	+	0.564199i	$0.190815\pi$
$432$	2.53319	0.121878
$433$	−13.7385	−0.660232	−0.330116	−	0.943940i	$-0.607088\pi$
$433$	−13.7385	−0.660232	−0.330116	+	0.943940i	$0.607088\pi$
$434$	−19.6528	−0.943364
$435$	0	0
$436$	−0.371961	−0.0178137
$437$	11.0586	0.529005
$438$	13.1588	0.628753
$439$	10.2403	0.488742	0.244371	−	0.969682i	$-0.421419\pi$
$439$	10.2403	0.488742	0.244371	+	0.969682i	$0.421419\pi$
$440$	0	0
$441$	7.54288	0.359185
$442$	0	0
$443$	−12.1763	−0.578513	−0.289257	−	0.957252i	$-0.593408\pi$
$443$	−12.1763	−0.578513	−0.289257	+	0.957252i	$0.593408\pi$
$444$	9.06398	0.430158
$445$	0	0
$446$	16.4819	0.780440
$447$	−4.66487	−0.220641
$448$	−26.7114	−1.26199
$449$	12.9051	0.609032	0.304516	−	0.952507i	$-0.401505\pi$
$449$	12.9051	0.609032	0.304516	+	0.952507i	$0.401505\pi$
$450$	0	0
$451$	−3.25534	−0.153288
$452$	−22.2814	−1.04803
$453$	−7.84117	−0.368410
$454$	19.4916	0.914785
$455$	0	0
$456$	−7.65279	−0.358375
$457$	4.65710	0.217850	0.108925	−	0.994050i	$-0.465259\pi$
$457$	4.65710	0.217850	0.108925	+	0.994050i	$0.465259\pi$
$458$	−30.7536	−1.43702
$459$	−16.7289	−0.780836
$460$	0	0
$461$	31.5405	1.46899	0.734493	−	0.678616i	$-0.237420\pi$
$461$	31.5405	1.46899	0.734493	+	0.678616i	$0.237420\pi$
$462$	−6.52781	−0.303701
$463$	17.6504	0.820284	0.410142	−	0.912022i	$-0.365479\pi$
$463$	17.6504	0.820284	0.410142	+	0.912022i	$0.365479\pi$
$464$	−1.81940	−0.0844633
$465$	0	0
$466$	−11.4330	−0.529622
$467$	32.1726	1.48877	0.744385	−	0.667751i	$-0.232743\pi$
$467$	32.1726	1.48877	0.744385	+	0.667751i	$0.232743\pi$
$468$	0	0
$469$	1.18167	0.0545644
$470$	0	0
$471$	5.23490	0.241211
$472$	−28.7724	−1.32436
$473$	−15.6799	−0.720964
$474$	19.6746	0.903683
$475$	0	0
$476$	33.0834	1.51637
$477$	14.8629	0.680527
$478$	−24.5459	−1.12270
$479$	34.8998	1.59461	0.797306	−	0.603576i	$-0.206258\pi$
$479$	34.8998	1.59461	0.797306	+	0.603576i	$0.206258\pi$
$480$	0	0
$481$	0	0
$482$	26.7724	1.21945
$483$	2.14914	0.0977895
$484$	−13.6353	−0.619788
$485$	0	0
$486$	−29.1782	−1.32355
$487$	41.8351	1.89573	0.947864	−	0.318676i	$-0.103238\pi$
$487$	41.8351	1.89573	0.947864	+	0.318676i	$0.103238\pi$
$488$	−20.1860	−0.913776
$489$	4.83041	0.218439
$490$	0	0
$491$	21.8455	0.985873	0.492936	−	0.870065i	$-0.335924\pi$
$491$	21.8455	0.985873	0.492936	+	0.870065i	$0.335924\pi$
$492$	−2.15585	−0.0971932
$493$	12.0151	0.541131
$494$	0	0
$495$	0	0
$496$	−3.42327	−0.153709
$497$	9.41311	0.422236
$498$	−9.63102	−0.431576
$499$	−23.5472	−1.05412	−0.527058	−	0.849829i	$-0.676705\pi$
$499$	−23.5472	−1.05412	−0.527058	+	0.849829i	$0.676705\pi$
$500$	0	0
$501$	−13.2379	−0.591425
$502$	−50.2150	−2.24121
$503$	7.08682	0.315986	0.157993	−	0.987440i	$-0.449498\pi$
$503$	7.08682	0.315986	0.157993	+	0.987440i	$0.449498\pi$
$504$	13.0000	0.579066
$505$	0	0
$506$	−10.8509	−0.482379
$507$	0	0
$508$	57.8394	2.56621
$509$	−7.61894	−0.337704	−0.168852	−	0.985641i	$-0.554006\pi$
$509$	−7.61894	−0.337704	−0.168852	+	0.985641i	$0.554006\pi$
$510$	0	0
$511$	−21.6213	−0.956471
$512$	−9.00538	−0.397985
$513$	−18.4819	−0.815995
$514$	−41.9288	−1.84940
$515$	0	0
$516$	−10.3840	−0.457132
$517$	−7.53989	−0.331604
$518$	−24.6625	−1.08361
$519$	10.4638	0.459311
$520$	0	0
$521$	−39.5133	−1.73111	−0.865555	−	0.500813i	$-0.833034\pi$
$521$	−39.5133	−1.73111	−0.865555	+	0.500813i	$0.833034\pi$
$522$	13.7235	0.600660
$523$	15.8194	0.691734	0.345867	−	0.938284i	$-0.387585\pi$
$523$	15.8194	0.691734	0.345867	+	0.938284i	$0.387585\pi$
$524$	9.92931	0.433764
$525$	0	0
$526$	32.3545	1.41072
$527$	22.6069	0.984770
$528$	−1.13706	−0.0494843
$529$	−19.4276	−0.844678
$530$	0	0
$531$	−32.8635	−1.42616
$532$	36.5502	1.58465
$533$	0	0
$534$	8.24698	0.356882
$535$	0	0
$536$	−1.35929	−0.0587123
$537$	3.34183	0.144211
$538$	−1.46681	−0.0632388
$539$	−7.15883	−0.308353
$540$	0	0
$541$	−34.4819	−1.48249	−0.741246	−	0.671234i	$-0.765765\pi$
$541$	−34.4819	−1.48249	−0.741246	+	0.671234i	$0.765765\pi$
$542$	−4.48427	−0.192616
$543$	−2.65146	−0.113785
$544$	34.5066	1.47946
$545$	0	0
$546$	0	0
$547$	−36.8582	−1.57594	−0.787970	−	0.615713i	$-0.788868\pi$
$547$	−36.8582	−1.57594	−0.787970	+	0.615713i	$0.788868\pi$
$548$	2.41550	0.103185
$549$	−23.0562	−0.984015
$550$	0	0
$551$	13.2741	0.565497
$552$	−2.47219	−0.105223
$553$	−32.3274	−1.37470
$554$	26.4795	1.12501
$555$	0	0
$556$	−34.5730	−1.46622
$557$	1.27652	0.0540879	0.0270439	−	0.999634i	$-0.491391\pi$
$557$	1.27652	0.0540879	0.0270439	+	0.999634i	$0.491391\pi$
$558$	25.8213	1.09310
$559$	0	0
$560$	0	0
$561$	7.50902	0.317031
$562$	−14.5429	−0.613454
$563$	9.12737	0.384673	0.192336	−	0.981329i	$-0.438393\pi$
$563$	9.12737	0.384673	0.192336	+	0.981329i	$0.438393\pi$
$564$	−4.99330	−0.210256
$565$	0	0
$566$	14.7875	0.621563
$567$	12.9554	0.544075
$568$	−10.8280	−0.454334
$569$	−5.72156	−0.239860	−0.119930	−	0.992782i	$-0.538267\pi$
$569$	−5.72156	−0.239860	−0.119930	+	0.992782i	$0.538267\pi$
$570$	0	0
$571$	7.60148	0.318112	0.159056	−	0.987270i	$-0.449155\pi$
$571$	7.60148	0.318112	0.159056	+	0.987270i	$0.449155\pi$
$572$	0	0
$573$	10.2282	0.427289
$574$	5.86592	0.244839
$575$	0	0
$576$	35.0954	1.46231
$577$	45.1564	1.87989	0.939944	−	0.341330i	$-0.110877\pi$
$577$	45.1564	1.87989	0.939944	+	0.341330i	$0.110877\pi$
$578$	−24.8213	−1.03243
$579$	3.35988	0.139632
$580$	0	0
$581$	15.8248	0.656522
$582$	−14.8726	−0.616490
$583$	−14.1062	−0.584219
$584$	24.8713	1.02918
$585$	0	0
$586$	54.6872	2.25911
$587$	−32.4040	−1.33746	−0.668728	−	0.743507i	$-0.733161\pi$
$587$	−32.4040	−1.33746	−0.668728	+	0.743507i	$0.733161\pi$
$588$	−4.74094	−0.195513
$589$	24.9758	1.02911
$590$	0	0
$591$	−6.33513	−0.260592
$592$	−4.29590	−0.176560
$593$	36.6848	1.50647	0.753233	−	0.657754i	$-0.228493\pi$
$593$	36.6848	1.50647	0.753233	+	0.657754i	$0.228493\pi$
$594$	18.1347	0.744075
$595$	0	0
$596$	−25.6286	−1.04979
$597$	−7.71678	−0.315827
$598$	0	0
$599$	−9.99223	−0.408271	−0.204136	−	0.978943i	$-0.565438\pi$
$599$	−9.99223	−0.408271	−0.204136	+	0.978943i	$0.565438\pi$
$600$	0	0
$601$	−1.81163	−0.0738978	−0.0369489	−	0.999317i	$-0.511764\pi$
$601$	−1.81163	−0.0738978	−0.0369489	+	0.999317i	$0.511764\pi$
$602$	28.2543	1.15156
$603$	−1.55257	−0.0632253
$604$	−43.0790	−1.75286
$605$	0	0
$606$	−16.2905	−0.661757
$607$	−11.2161	−0.455248	−0.227624	−	0.973749i	$-0.573096\pi$
$607$	−11.2161	−0.455248	−0.227624	+	0.973749i	$0.573096\pi$
$608$	38.1226	1.54608
$609$	2.57971	0.104535
$610$	0	0
$611$	0	0
$612$	−43.4674	−1.75707
$613$	−20.8944	−0.843917	−0.421958	−	0.906615i	$-0.638657\pi$
$613$	−20.8944	−0.843917	−0.421958	+	0.906615i	$0.638657\pi$
$614$	31.6233	1.27621
$615$	0	0
$616$	−12.3381	−0.497117
$617$	12.0992	0.487094	0.243547	−	0.969889i	$-0.421689\pi$
$617$	12.0992	0.487094	0.243547	+	0.969889i	$0.421689\pi$
$618$	11.4330	0.459901
$619$	10.5526	0.424143	0.212072	−	0.977254i	$-0.431979\pi$
$619$	10.5526	0.424143	0.212072	+	0.977254i	$0.431979\pi$
$620$	0	0
$621$	−5.97046	−0.239586
$622$	66.8926	2.68215
$623$	−13.5506	−0.542895
$624$	0	0
$625$	0	0
$626$	−16.8049	−0.671660
$627$	8.29590	0.331306
$628$	28.7603	1.14766
$629$	28.3696	1.13117
$630$	0	0
$631$	13.8514	0.551417	0.275709	−	0.961241i	$-0.411087\pi$
$631$	13.8514	0.551417	0.275709	+	0.961241i	$0.411087\pi$
$632$	37.1866	1.47920
$633$	−7.35019	−0.292144
$634$	−67.4771	−2.67986
$635$	0	0
$636$	−9.34183	−0.370428
$637$	0	0
$638$	−13.0248	−0.515655
$639$	−12.3676	−0.489257
$640$	0	0
$641$	34.9608	1.38087	0.690434	−	0.723396i	$-0.257420\pi$
$641$	34.9608	1.38087	0.690434	+	0.723396i	$0.257420\pi$
$642$	−8.60388	−0.339568
$643$	33.3980	1.31709	0.658545	−	0.752541i	$-0.271172\pi$
$643$	33.3980	1.31709	0.658545	+	0.752541i	$0.271172\pi$
$644$	11.8073	0.465273
$645$	0	0
$646$	−69.6238	−2.73931
$647$	−2.32842	−0.0915397	−0.0457698	−	0.998952i	$-0.514574\pi$
$647$	−2.32842	−0.0915397	−0.0457698	+	0.998952i	$0.514574\pi$
$648$	−14.9028	−0.585436
$649$	31.1903	1.22433
$650$	0	0
$651$	4.85384	0.190237
$652$	26.5381	1.03931
$653$	−14.5714	−0.570221	−0.285111	−	0.958495i	$-0.592030\pi$
$653$	−14.5714	−0.570221	−0.285111	+	0.958495i	$0.592030\pi$
$654$	0.152129	0.00594871
$655$	0	0
$656$	1.02177	0.0398934
$657$	28.4077	1.10829
$658$	13.5864	0.529654
$659$	11.1395	0.433932	0.216966	−	0.976179i	$-0.430384\pi$
$659$	11.1395	0.433932	0.216966	+	0.976179i	$0.430384\pi$
$660$	0	0
$661$	13.8498	0.538694	0.269347	−	0.963043i	$-0.413192\pi$
$661$	13.8498	0.538694	0.269347	+	0.963043i	$0.413192\pi$
$662$	−35.3153	−1.37257
$663$	0	0
$664$	−18.2034	−0.706430
$665$	0	0
$666$	32.4034	1.25561
$667$	4.28813	0.166037
$668$	−72.7284	−2.81395
$669$	−4.07069	−0.157382
$670$	0	0
$671$	21.8823	0.844757
$672$	7.40880	0.285801
$673$	6.52973	0.251703	0.125851	−	0.992049i	$-0.459834\pi$
$673$	6.52973	0.251703	0.125851	+	0.992049i	$0.459834\pi$
$674$	4.39075	0.169125
$675$	0	0
$676$	0	0
$677$	11.3104	0.434693	0.217346	−	0.976095i	$-0.430260\pi$
$677$	11.3104	0.434693	0.217346	+	0.976095i	$0.430260\pi$
$678$	9.11290	0.349979
$679$	24.4373	0.937816
$680$	0	0
$681$	−4.81402	−0.184474
$682$	−24.5066	−0.938407
$683$	14.1793	0.542555	0.271277	−	0.962501i	$-0.412554\pi$
$683$	14.1793	0.542555	0.271277	+	0.962501i	$0.412554\pi$
$684$	−48.0224	−1.83618
$685$	0	0
$686$	45.1269	1.72295
$687$	7.59551	0.289787
$688$	4.92154	0.187632
$689$	0	0
$690$	0	0
$691$	−30.7952	−1.17151	−0.585753	−	0.810490i	$-0.699201\pi$
$691$	−30.7952	−1.17151	−0.585753	+	0.810490i	$0.699201\pi$
$692$	57.4878	2.18536
$693$	−14.0925	−0.535328
$694$	−38.4795	−1.46066
$695$	0	0
$696$	−2.96748	−0.112482
$697$	−6.74764	−0.255585
$698$	−23.5187	−0.890196
$699$	2.82371	0.106802
$700$	0	0
$701$	6.73184	0.254258	0.127129	−	0.991886i	$-0.459424\pi$
$701$	6.73184	0.254258	0.127129	+	0.991886i	$0.459424\pi$
$702$	0	0
$703$	31.3424	1.18210
$704$	−33.3086	−1.25536
$705$	0	0
$706$	−34.8974	−1.31338
$707$	26.7670	1.00668
$708$	20.6558	0.776292
$709$	47.6252	1.78860	0.894300	−	0.447467i	$-0.147674\pi$
$709$	47.6252	1.78860	0.894300	+	0.447467i	$0.147674\pi$
$710$	0	0
$711$	42.4741	1.59290
$712$	15.5875	0.584166
$713$	8.06829	0.302160
$714$	−13.5308	−0.506377
$715$	0	0
$716$	18.3599	0.686141
$717$	6.06233	0.226402
$718$	−48.1444	−1.79673
$719$	−5.99330	−0.223512	−0.111756	−	0.993736i	$-0.535648\pi$
$719$	−5.99330	−0.223512	−0.111756	+	0.993736i	$0.535648\pi$
$720$	0	0
$721$	−18.7855	−0.699610
$722$	−34.2271	−1.27380
$723$	−6.61224	−0.245912
$724$	−14.5670	−0.541380
$725$	0	0
$726$	5.57673	0.206972
$727$	24.1226	0.894657	0.447329	−	0.894370i	$-0.352375\pi$
$727$	24.1226	0.894657	0.447329	+	0.894370i	$0.352375\pi$
$728$	0	0
$729$	−11.7627	−0.435656
$730$	0	0
$731$	−32.5013	−1.20210
$732$	14.4916	0.535624
$733$	36.0646	1.33208	0.666038	−	0.745918i	$-0.267989\pi$
$733$	36.0646	1.33208	0.666038	+	0.745918i	$0.267989\pi$
$734$	77.0786	2.84502
$735$	0	0
$736$	12.3153	0.453947
$737$	1.47352	0.0542777
$738$	−7.70709	−0.283702
$739$	−27.5254	−1.01254	−0.506269	−	0.862375i	$-0.668976\pi$
$739$	−27.5254	−1.01254	−0.506269	+	0.862375i	$0.668976\pi$
$740$	0	0
$741$	0	0
$742$	25.4185	0.933142
$743$	−10.4692	−0.384078	−0.192039	−	0.981387i	$-0.561510\pi$
$743$	−10.4692	−0.384078	−0.192039	+	0.981387i	$0.561510\pi$
$744$	−5.58343	−0.204699
$745$	0	0
$746$	−28.3032	−1.03625
$747$	−20.7918	−0.760731
$748$	41.2543	1.50841
$749$	14.1371	0.516557
$750$	0	0
$751$	4.06770	0.148433	0.0742163	−	0.997242i	$-0.476354\pi$
$751$	4.06770	0.148433	0.0742163	+	0.997242i	$0.476354\pi$
$752$	2.36658	0.0863005
$753$	12.4021	0.451957
$754$	0	0
$755$	0	0
$756$	−19.7332	−0.717688
$757$	−20.4336	−0.742670	−0.371335	−	0.928499i	$-0.621100\pi$
$757$	−20.4336	−0.742670	−0.371335	+	0.928499i	$0.621100\pi$
$758$	37.1618	1.34978
$759$	2.67994	0.0972757
$760$	0	0
$761$	27.0237	0.979608	0.489804	−	0.871833i	$-0.337068\pi$
$761$	27.0237	0.979608	0.489804	+	0.871833i	$0.337068\pi$
$762$	−23.6558	−0.856958
$763$	−0.249964	−0.00904929
$764$	56.1933	2.03300
$765$	0	0
$766$	−16.9336	−0.611837
$767$	0	0
$768$	−5.80864	−0.209601
$769$	37.9407	1.36818	0.684088	−	0.729400i	$-0.260201\pi$
$769$	37.9407	1.36818	0.684088	+	0.729400i	$0.260201\pi$
$770$	0	0
$771$	10.3556	0.372947
$772$	18.4590	0.664355
$773$	−16.3375	−0.587620	−0.293810	−	0.955864i	$-0.594923\pi$
$773$	−16.3375	−0.587620	−0.293810	+	0.955864i	$0.594923\pi$
$774$	−37.1226	−1.33434
$775$	0	0
$776$	−28.1105	−1.00911
$777$	6.09113	0.218518
$778$	−79.8926	−2.86429
$779$	−7.45473	−0.267093
$780$	0	0
$781$	11.7380	0.420017
$782$	−22.4916	−0.804297
$783$	−7.16660	−0.256114
$784$	2.24698	0.0802493
$785$	0	0
$786$	−4.06100	−0.144851
$787$	−18.6907	−0.666251	−0.333126	−	0.942882i	$-0.608103\pi$
$787$	−18.6907	−0.666251	−0.333126	+	0.942882i	$0.608103\pi$
$788$	−34.8049	−1.23987
$789$	−7.99090	−0.284484
$790$	0	0
$791$	−14.9734	−0.532394
$792$	16.2107	0.576023
$793$	0	0
$794$	−3.03684	−0.107773
$795$	0	0
$796$	−42.3957	−1.50267
$797$	−29.2519	−1.03615	−0.518077	−	0.855334i	$-0.673352\pi$
$797$	−29.2519	−1.03615	−0.518077	+	0.855334i	$0.673352\pi$
$798$	−14.9487	−0.529178
$799$	−15.6286	−0.552901
$800$	0	0
$801$	17.8039	0.629068
$802$	1.30127	0.0459496
$803$	−26.9614	−0.951446
$804$	0.975837	0.0344151
$805$	0	0
$806$	0	0
$807$	0.362273	0.0127526
$808$	−30.7904	−1.08320
$809$	−6.65087	−0.233832	−0.116916	−	0.993142i	$-0.537301\pi$
$809$	−6.65087	−0.233832	−0.116916	+	0.993142i	$0.537301\pi$
$810$	0	0
$811$	−3.89200	−0.136667	−0.0683333	−	0.997663i	$-0.521768\pi$
$811$	−3.89200	−0.136667	−0.0683333	+	0.997663i	$0.521768\pi$
$812$	14.1728	0.497369
$813$	1.10752	0.0388425
$814$	−30.7536	−1.07791
$815$	0	0
$816$	−2.35690	−0.0825079
$817$	−35.9071	−1.25623
$818$	−34.0586	−1.19083
$819$	0	0
$820$	0	0
$821$	−45.9982	−1.60535	−0.802674	−	0.596418i	$-0.796590\pi$
$821$	−45.9982	−1.60535	−0.802674	+	0.596418i	$0.796590\pi$
$822$	−0.987918	−0.0344576
$823$	−7.95300	−0.277224	−0.138612	−	0.990347i	$-0.544264\pi$
$823$	−7.95300	−0.277224	−0.138612	+	0.990347i	$0.544264\pi$
$824$	21.6093	0.752794
$825$	0	0
$826$	−56.2030	−1.95555
$827$	27.9648	0.972432	0.486216	−	0.873839i	$-0.338377\pi$
$827$	27.9648	0.972432	0.486216	+	0.873839i	$0.338377\pi$
$828$	−15.5133	−0.539126
$829$	27.6310	0.959665	0.479833	−	0.877360i	$-0.340697\pi$
$829$	27.6310	0.959665	0.479833	+	0.877360i	$0.340697\pi$
$830$	0	0
$831$	−6.53989	−0.226866
$832$	0	0
$833$	−14.8388	−0.514133
$834$	14.1400	0.489630
$835$	0	0
$836$	45.5773	1.57632
$837$	−13.4843	−0.466085
$838$	80.2699	2.77288
$839$	−28.6848	−0.990311	−0.495155	−	0.868804i	$-0.664889\pi$
$839$	−28.6848	−0.990311	−0.495155	+	0.868804i	$0.664889\pi$
$840$	0	0
$841$	−23.8528	−0.822509
$842$	−78.7488	−2.71386
$843$	3.59179	0.123708
$844$	−40.3817	−1.38999
$845$	0	0
$846$	−17.8509	−0.613725
$847$	−9.16315	−0.314849
$848$	4.42758	0.152044
$849$	−3.65220	−0.125343
$850$	0	0
$851$	10.1250	0.347080
$852$	7.77346	0.266314
$853$	−43.2078	−1.47941	−0.739703	−	0.672934i	$-0.765034\pi$
$853$	−43.2078	−1.47941	−0.739703	+	0.672934i	$0.765034\pi$
$854$	−39.4306	−1.34929
$855$	0	0
$856$	−16.2620	−0.555825
$857$	−35.1685	−1.20133	−0.600667	−	0.799499i	$-0.705098\pi$
$857$	−35.1685	−1.20133	−0.600667	+	0.799499i	$0.705098\pi$
$858$	0	0
$859$	27.3793	0.934168	0.467084	−	0.884213i	$-0.345305\pi$
$859$	27.3793	0.934168	0.467084	+	0.884213i	$0.345305\pi$
$860$	0	0
$861$	−1.44876	−0.0493737
$862$	−77.0297	−2.62364
$863$	−41.3913	−1.40898	−0.704489	−	0.709715i	$-0.748824\pi$
$863$	−41.3913	−1.40898	−0.704489	+	0.709715i	$0.748824\pi$
$864$	−20.5821	−0.700217
$865$	0	0
$866$	30.8702	1.04901
$867$	6.13036	0.208198
$868$	26.6668	0.905130
$869$	−40.3116	−1.36748
$870$	0	0
$871$	0	0
$872$	0.287536	0.00973721
$873$	−32.1075	−1.08668
$874$	−24.8485	−0.840512
$875$	0	0
$876$	−17.8552	−0.603270
$877$	−24.7472	−0.835653	−0.417826	−	0.908527i	$-0.637208\pi$
$877$	−24.7472	−0.835653	−0.417826	+	0.908527i	$0.637208\pi$
$878$	−23.0097	−0.776539
$879$	−13.5066	−0.455567
$880$	0	0
$881$	−28.5875	−0.963137	−0.481568	−	0.876409i	$-0.659933\pi$
$881$	−28.5875	−0.963137	−0.481568	+	0.876409i	$0.659933\pi$
$882$	−16.9487	−0.570692
$883$	−9.61702	−0.323639	−0.161819	−	0.986820i	$-0.551736\pi$
$883$	−9.61702	−0.323639	−0.161819	+	0.986820i	$0.551736\pi$
$884$	0	0
$885$	0	0
$886$	27.3599	0.919173
$887$	−15.9661	−0.536091	−0.268045	−	0.963406i	$-0.586378\pi$
$887$	−15.9661	−0.536091	−0.268045	+	0.963406i	$0.586378\pi$
$888$	−7.00670	−0.235130
$889$	38.8689	1.30362
$890$	0	0
$891$	16.1551	0.541217
$892$	−22.3642	−0.748809
$893$	−17.2664	−0.577797
$894$	10.4819	0.350566
$895$	0	0
$896$	33.3196	1.11313
$897$	0	0
$898$	−28.9976	−0.967663
$899$	9.68473	0.323004
$900$	0	0
$901$	−29.2392	−0.974099
$902$	7.31468	0.243552
$903$	−6.97823	−0.232221
$904$	17.2241	0.572867
$905$	0	0
$906$	17.6189	0.585350
$907$	28.8364	0.957496	0.478748	−	0.877952i	$-0.341091\pi$
$907$	28.8364	0.957496	0.478748	+	0.877952i	$0.341091\pi$
$908$	−26.4480	−0.877709
$909$	−35.1685	−1.16647
$910$	0	0
$911$	38.5633	1.27766	0.638830	−	0.769348i	$-0.279419\pi$
$911$	38.5633	1.27766	0.638830	+	0.769348i	$0.279419\pi$
$912$	−2.60388	−0.0862229
$913$	19.7332	0.653073
$914$	−10.4644	−0.346132
$915$	0	0
$916$	41.7294	1.37878
$917$	6.67264	0.220350
$918$	37.5894	1.24064
$919$	8.87502	0.292760	0.146380	−	0.989228i	$-0.453238\pi$
$919$	8.87502	0.292760	0.146380	+	0.989228i	$0.453238\pi$
$920$	0	0
$921$	−7.81030	−0.257358
$922$	−70.8708	−2.33401
$923$	0	0
$924$	8.85756	0.291392
$925$	0	0
$926$	−39.6601	−1.30331
$927$	24.6819	0.810659
$928$	14.7826	0.485261
$929$	−24.2295	−0.794945	−0.397472	−	0.917614i	$-0.630113\pi$
$929$	−24.2295	−0.794945	−0.397472	+	0.917614i	$0.630113\pi$
$930$	0	0
$931$	−16.3937	−0.537283
$932$	15.5133	0.508156
$933$	−16.5211	−0.540877
$934$	−72.2911	−2.36544
$935$	0	0
$936$	0	0
$937$	−17.2644	−0.564005	−0.282002	−	0.959414i	$-0.590999\pi$
$937$	−17.2644	−0.564005	−0.282002	+	0.959414i	$0.590999\pi$
$938$	−2.65519	−0.0866949
$939$	4.15047	0.135446
$940$	0	0
$941$	−4.34050	−0.141496	−0.0707482	−	0.997494i	$-0.522539\pi$
$941$	−4.34050	−0.141496	−0.0707482	+	0.997494i	$0.522539\pi$
$942$	−11.7627	−0.383250
$943$	−2.40821	−0.0784220
$944$	−9.78986	−0.318633
$945$	0	0
$946$	35.2325	1.14551
$947$	−45.0146	−1.46278	−0.731389	−	0.681961i	$-0.761128\pi$
$947$	−45.0146	−1.46278	−0.731389	+	0.681961i	$0.761128\pi$
$948$	−26.6963	−0.867057
$949$	0	0
$950$	0	0
$951$	16.6655	0.540415
$952$	−25.5743	−0.828869
$953$	46.8859	1.51878	0.759391	−	0.650634i	$-0.225497\pi$
$953$	46.8859	1.51878	0.759391	+	0.650634i	$0.225497\pi$
$954$	−33.3967	−1.08126
$955$	0	0
$956$	33.3062	1.07720
$957$	3.21685	0.103986
$958$	−78.4191	−2.53361
$959$	1.62325	0.0524176
$960$	0	0
$961$	−12.7778	−0.412186
$962$	0	0
$963$	−18.5743	−0.598550
$964$	−36.3274	−1.17003
$965$	0	0
$966$	−4.82908	−0.155373
$967$	6.29457	0.202420	0.101210	−	0.994865i	$-0.467729\pi$
$967$	6.29457	0.202420	0.101210	+	0.994865i	$0.467729\pi$
$968$	10.5405	0.338784
$969$	17.1957	0.552404
$970$	0	0
$971$	−41.8068	−1.34165	−0.670823	−	0.741618i	$-0.734059\pi$
$971$	−41.8068	−1.34165	−0.670823	+	0.741618i	$0.734059\pi$
$972$	39.5918	1.26991
$973$	−23.2336	−0.744834
$974$	−94.0025	−3.01203
$975$	0	0
$976$	−6.86831	−0.219849
$977$	23.7530	0.759926	0.379963	−	0.925002i	$-0.375937\pi$
$977$	23.7530	0.759926	0.379963	+	0.925002i	$0.375937\pi$
$978$	−10.8538	−0.347067
$979$	−16.8974	−0.540043
$980$	0	0
$981$	0.328421	0.0104857
$982$	−49.0863	−1.56641
$983$	−55.7251	−1.77736	−0.888678	−	0.458532i	$-0.848375\pi$
$983$	−55.7251	−1.77736	−0.888678	+	0.458532i	$0.848375\pi$
$984$	1.66653	0.0531270
$985$	0	0
$986$	−26.9976	−0.859779
$987$	−3.35557	−0.106809
$988$	0	0
$989$	−11.5996	−0.368845
$990$	0	0
$991$	−35.5512	−1.12932	−0.564661	−	0.825323i	$-0.690993\pi$
$991$	−35.5512	−1.12932	−0.564661	+	0.825323i	$0.690993\pi$
$992$	27.8140	0.883096
$993$	8.72215	0.276789
$994$	−21.1511	−0.670871
$995$	0	0
$996$	13.0683	0.414085
$997$	−6.61058	−0.209359	−0.104680	−	0.994506i	$-0.533382\pi$
$997$	−6.61058	−0.209359	−0.104680	+	0.994506i	$0.533382\pi$
$998$	52.9101	1.67484
$999$	−16.9215	−0.535374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bb.1.1		3
5.4	even	2		169.2.a.c.1.3	yes	3
13.12	even	2		4225.2.a.bg.1.3		3
15.14	odd	2		1521.2.a.o.1.1		3
20.19	odd	2		2704.2.a.ba.1.2		3
35.34	odd	2		8281.2.a.bj.1.3		3
65.4	even	6		169.2.c.c.146.3		6
65.9	even	6		169.2.c.b.146.1		6
65.19	odd	12		169.2.e.b.23.1		12
65.24	odd	12		169.2.e.b.147.1		12
65.29	even	6		169.2.c.b.22.1		6
65.34	odd	4		169.2.b.b.168.6		6
65.44	odd	4		169.2.b.b.168.1		6
65.49	even	6		169.2.c.c.22.3		6
65.54	odd	12		169.2.e.b.147.6		12
65.59	odd	12		169.2.e.b.23.6		12
65.64	even	2		169.2.a.b.1.1	✓	3
195.44	even	4		1521.2.b.l.1351.6		6
195.164	even	4		1521.2.b.l.1351.1		6
195.194	odd	2		1521.2.a.r.1.3		3
260.99	even	4		2704.2.f.o.337.4		6
260.239	even	4		2704.2.f.o.337.3		6
260.259	odd	2		2704.2.a.z.1.2		3
455.454	odd	2		8281.2.a.bf.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.1	✓	3	65.64	even	2
169.2.a.c.1.3	yes	3	5.4	even	2
169.2.b.b.168.1		6	65.44	odd	4
169.2.b.b.168.6		6	65.34	odd	4
169.2.c.b.22.1		6	65.29	even	6
169.2.c.b.146.1		6	65.9	even	6
169.2.c.c.22.3		6	65.49	even	6
169.2.c.c.146.3		6	65.4	even	6
169.2.e.b.23.1		12	65.19	odd	12
169.2.e.b.23.6		12	65.59	odd	12
169.2.e.b.147.1		12	65.24	odd	12
169.2.e.b.147.6		12	65.54	odd	12
1521.2.a.o.1.1		3	15.14	odd	2
1521.2.a.r.1.3		3	195.194	odd	2
1521.2.b.l.1351.1		6	195.164	even	4
1521.2.b.l.1351.6		6	195.44	even	4
2704.2.a.z.1.2		3	260.259	odd	2
2704.2.a.ba.1.2		3	20.19	odd	2
2704.2.f.o.337.3		6	260.239	even	4
2704.2.f.o.337.4		6	260.99	even	4
4225.2.a.bb.1.1		3	1.1	even	1	trivial
4225.2.a.bg.1.3		3	13.12	even	2
8281.2.a.bf.1.1		3	455.454	odd	2
8281.2.a.bj.1.3		3	35.34	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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q-2.24698 q^{2} +0.554958 q^{3} +3.04892 q^{4} -1.24698 q^{6} +2.04892 q^{7} -2.35690 q^{8} -2.69202 q^{9} +2.55496 q^{11} +1.69202 q^{12} -4.60388 q^{14} -0.801938 q^{16} +5.29590 q^{17} +6.04892 q^{18} +5.85086 q^{19} +1.13706 q^{21} -5.74094 q^{22} +1.89008 q^{23} -1.30798 q^{24} -3.15883 q^{27} +6.24698 q^{28} +2.26875 q^{29} +4.26875 q^{31} +6.51573 q^{32} +1.41789 q^{33} -11.8998 q^{34} -8.20775 q^{36} +5.35690 q^{37} -13.1468 q^{38} -1.27413 q^{41} -2.55496 q^{42} -6.13706 q^{43} +7.78986 q^{44} -4.24698 q^{46} -2.95108 q^{47} -0.445042 q^{48} -2.80194 q^{49} +2.93900 q^{51} -5.52111 q^{53} +7.09783 q^{54} -4.82908 q^{56} +3.24698 q^{57} -5.09783 q^{58} +12.2078 q^{59} +8.56465 q^{61} -9.59179 q^{62} -5.51573 q^{63} -13.0368 q^{64} -3.18598 q^{66} +0.576728 q^{67} +16.1468 q^{68} +1.04892 q^{69} +4.59419 q^{71} +6.34481 q^{72} -10.5526 q^{73} -12.0368 q^{74} +17.8388 q^{76} +5.23490 q^{77} -15.7778 q^{79} +6.32304 q^{81} +2.86294 q^{82} +7.72348 q^{83} +3.46681 q^{84} +13.7899 q^{86} +1.25906 q^{87} -6.02177 q^{88} -6.61356 q^{89} +5.76271 q^{92} +2.36898 q^{93} +6.63102 q^{94} +3.61596 q^{96} +11.9269 q^{97} +6.29590 q^{98} -6.87800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 2 q^{3} + q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9} + 8 q^{11} - 5 q^{14} + 2 q^{16} + 2 q^{17} + 9 q^{18} + 4 q^{19} - 2 q^{21} - 3 q^{22} + 5 q^{23} - 9 q^{24} - q^{27} + 14 q^{28} - q^{29}+ \cdots - q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 2 * q^3 + q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 + 8 * q^11 - 5 * q^14 + 2 * q^16 + 2 * q^17 + 9 * q^18 + 4 * q^19 - 2 * q^21 - 3 * q^22 + 5 * q^23 - 9 * q^24 - q^27 + 14 * q^28 - q^29 + 5 * q^31 + 7 * q^32 + 10 * q^33 - 13 * q^34 - 7 * q^36 + 12 * q^37 - 12 * q^38 + 7 * q^41 - 8 * q^42 - 13 * q^43 - 8 * q^46 - 18 * q^47 - q^48 - 4 * q^49 - q^51 - q^53 + 3 * q^54 - 4 * q^56 + 5 * q^57 + 3 * q^58 + 19 * q^59 + 4 * q^61 - q^62 - 4 * q^63 - 11 * q^64 + 5 * q^66 - q^67 + 21 * q^68 - 6 * q^69 + 27 * q^71 - 4 * q^72 + 9 * q^73 - 8 * q^74 + 21 * q^76 - 8 * q^77 - 5 * q^79 - q^81 + 14 * q^82 - 7 * q^83 + 7 * q^84 + 18 * q^86 + 18 * q^87 - 15 * q^88 + 11 * q^89 + 22 * q^93 + 5 * q^94 + 21 * q^96 + 7 * q^97 + 5 * q^98 - q^99

Introduction

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