LMFDB - Embedded newform 175.4.b.e.99.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(99,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	175.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$10.3253342510$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3299353600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 34x^{4} + 289x^{2} + 196 $ x^6 + 34x^4 + 289x^2 + 196
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.6
Root			$3.62456i$ of defining polynomial
Character	$\chi$	$=$	175.99
Dual form			175.4.b.e.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.62456i q^{2} -8.38660i q^{3} -13.3866 q^{4} +38.7844 q^{6} -7.00000i q^{7} -24.9107i q^{8} -43.3350 q^{9} -30.1117 q^{11} +112.268i q^{12} +88.9295i q^{13} +32.3720 q^{14} +8.10818 q^{16} +4.73699i q^{17} -200.405i q^{18} -124.818 q^{19} -58.7062 q^{21} -139.253i q^{22} +20.2680i q^{23} -208.916 q^{24} -411.260 q^{26} +136.995i q^{27} +93.7062i q^{28} -134.088 q^{29} -2.03767 q^{31} -161.788i q^{32} +252.534i q^{33} -21.9065 q^{34} +580.108 q^{36} +141.137i q^{37} -577.228i q^{38} +745.816 q^{39} +95.2784 q^{41} -271.490i q^{42} -298.646i q^{43} +403.093 q^{44} -93.7305 q^{46} +129.054i q^{47} -68.0000i q^{48} -49.0000 q^{49} +39.7272 q^{51} -1190.46i q^{52} +388.429i q^{53} -633.542 q^{54} -174.375 q^{56} +1046.80i q^{57} -620.098i q^{58} -838.501 q^{59} +389.422 q^{61} -9.42333i q^{62} +303.345i q^{63} +813.067 q^{64} -1167.86 q^{66} -697.794i q^{67} -63.4122i q^{68} +169.979 q^{69} -523.450 q^{71} +1079.50i q^{72} +66.4684i q^{73} -652.699 q^{74} +1670.89 q^{76} +210.782i q^{77} +3449.07i q^{78} +526.982 q^{79} -21.1236 q^{81} +440.621i q^{82} +70.0265i q^{83} +785.876 q^{84} +1381.11 q^{86} +1124.54i q^{87} +750.101i q^{88} +9.27925 q^{89} +622.506 q^{91} -271.319i q^{92} +17.0891i q^{93} -596.817 q^{94} -1356.85 q^{96} +4.19493i q^{97} -226.604i q^{98} +1304.89 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 26 q^{4} + 48 q^{6} - 162 q^{9} - 148 q^{11} + 42 q^{14} - 158 q^{16} - 336 q^{19} + 28 q^{21} - 840 q^{24} - 892 q^{26} - 664 q^{29} + 640 q^{31} - 1164 q^{34} + 362 q^{36} + 1964 q^{39} + 724 q^{41}+ \cdots + 6976 q^{99}+O(q^{100}) $ 6 * q - 26 * q^4 + 48 * q^6 - 162 * q^9 - 148 * q^11 + 42 * q^14 - 158 * q^16 - 336 * q^19 + 28 * q^21 - 840 * q^24 - 892 * q^26 - 664 * q^29 + 640 * q^31 - 1164 * q^34 + 362 * q^36 + 1964 * q^39 + 724 * q^41 + 528 * q^44 - 672 * q^46 - 294 * q^49 - 2356 * q^51 + 360 * q^54 - 210 * q^56 + 360 * q^59 + 2444 * q^61 + 782 * q^64 - 1064 * q^66 + 1432 * q^69 - 272 * q^71 - 1004 * q^74 + 3592 * q^76 + 2068 * q^79 + 4566 * q^81 + 2128 * q^84 + 1528 * q^86 - 484 * q^89 + 616 * q^91 + 2216 * q^94 - 6312 * q^96 + 6976 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.62456i	1.63503i	0.575907	+	0.817515i	$0.304649\pi$
$2$	4.62456i	1.63503i	−0.575907	+	0.817515i	$0.695351\pi$
$3$	− 8.38660i	− 1.61400i	−0.590551	−	0.807001i	$-0.701089\pi$
$3$	− 8.38660i	− 1.61400i	0.590551	−	0.807001i	$-0.298911\pi$
$4$	−13.3866	−1.67332
$5$	0	0
$5$	0	0
$6$	38.7844	2.63894
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	− 24.9107i	− 1.10091i
$9$	−43.3350	−1.60500
$10$	0	0
$11$	−30.1117	−0.825364	−0.412682	−	0.910875i	$-0.635408\pi$
$11$	−30.1117	−0.825364	−0.412682	+	0.910875i	$0.635408\pi$
$12$	112.268i	2.70075i
$13$	88.9295i	1.89728i	0.316362	+	0.948639i	$0.397539\pi$
$13$	88.9295i	1.89728i	−0.316362	+	0.948639i	$0.602461\pi$
$14$	32.3720	0.617983
$15$	0	0
$16$	8.10818	0.126690
$17$	4.73699i	0.0675817i	0.999429	+	0.0337909i	$0.0107580\pi$
$17$	4.73699i	0.0675817i	−0.999429	+	0.0337909i	$0.989242\pi$
$18$	− 200.405i	− 2.62422i
$19$	−124.818	−1.50711	−0.753557	−	0.657382i	$-0.771664\pi$
$19$	−124.818	−1.50711	−0.753557	+	0.657382i	$0.771664\pi$
$20$	0	0
$21$	−58.7062	−0.610035
$22$	− 139.253i	− 1.34950i
$23$	20.2680i	0.183746i	0.995771	+	0.0918731i	$0.0292854\pi$
$23$	20.2680i	0.183746i	−0.995771	+	0.0918731i	$0.970715\pi$
$24$	−208.916	−1.77686
$25$	0	0
$26$	−411.260	−3.10211
$27$	136.995i	0.976470i
$28$	93.7062i	0.632457i
$29$	−134.088	−0.858603	−0.429301	−	0.903161i	$-0.641240\pi$
$29$	−134.088	−0.858603	−0.429301	+	0.903161i	$0.641240\pi$
$30$	0	0
$31$	−2.03767	−0.0118057	−0.00590284	−	0.999983i	$-0.501879\pi$
$31$	−2.03767	−0.0118057	−0.00590284	+	0.999983i	$0.501879\pi$
$32$	− 161.788i	− 0.893764i
$33$	252.534i	1.33214i
$34$	−21.9065	−0.110498
$35$	0	0
$36$	580.108	2.68568
$37$	141.137i	0.627104i	0.949571	+	0.313552i	$0.101519\pi$
$37$	141.137i	0.627104i	−0.949571	+	0.313552i	$0.898481\pi$
$38$	− 577.228i	− 2.46418i
$39$	745.816	3.06221
$40$	0	0
$41$	95.2784	0.362927	0.181463	−	0.983398i	$-0.441917\pi$
$41$	95.2784	0.362927	0.181463	+	0.983398i	$0.441917\pi$
$42$	− 271.490i	− 0.997426i
$43$	− 298.646i	− 1.05914i	−0.848266	−	0.529571i	$-0.822353\pi$
$43$	− 298.646i	− 1.05914i	0.848266	−	0.529571i	$-0.177647\pi$
$44$	403.093	1.38110
$45$	0	0
$46$	−93.7305	−0.300431
$47$	129.054i	0.400519i	0.979743	+	0.200260i	$0.0641786\pi$
$47$	129.054i	0.400519i	−0.979743	+	0.200260i	$0.935821\pi$
$48$	− 68.0000i	− 0.204478i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	39.7272	0.109077
$52$	− 1190.46i	− 3.17476i
$53$	388.429i	1.00669i	0.864084	+	0.503347i	$0.167898\pi$
$53$	388.429i	1.00669i	−0.864084	+	0.503347i	$0.832102\pi$
$54$	−633.542	−1.59656
$55$	0	0
$56$	−174.375	−0.416103
$57$	1046.80i	2.43248i
$58$	− 620.098i	− 1.40384i
$59$	−838.501	−1.85023	−0.925114	−	0.379688i	$-0.876031\pi$
$59$	−838.501	−1.85023	−0.925114	+	0.379688i	$0.876031\pi$
$60$	0	0
$61$	389.422	0.817384	0.408692	−	0.912672i	$-0.365985\pi$
$61$	389.422	0.817384	0.408692	+	0.912672i	$0.365985\pi$
$62$	− 9.42333i	− 0.0193027i
$63$	303.345i	0.606633i
$64$	813.067	1.58802
$65$	0	0
$66$	−1167.86	−2.17809
$67$	− 697.794i	− 1.27237i	−0.771534	−	0.636187i	$-0.780510\pi$
$67$	− 697.794i	− 1.27237i	0.771534	−	0.636187i	$-0.219490\pi$
$68$	− 63.4122i	− 0.113086i
$69$	169.979	0.296567
$70$	0	0
$71$	−523.450	−0.874959	−0.437479	−	0.899228i	$-0.644129\pi$
$71$	−523.450	−0.874959	−0.437479	+	0.899228i	$0.644129\pi$
$72$	1079.50i	1.76695i
$73$	66.4684i	0.106569i	0.998579	+	0.0532845i	$0.0169690\pi$
$73$	66.4684i	0.106569i	−0.998579	+	0.0532845i	$0.983031\pi$
$74$	−652.699	−1.02533
$75$	0	0
$76$	1670.89	2.52189
$77$	210.782i	0.311958i
$78$	3449.07i	5.00680i
$79$	526.982	0.750508	0.375254	−	0.926922i	$-0.377556\pi$
$79$	526.982	0.750508	0.375254	+	0.926922i	$0.377556\pi$
$80$	0	0
$81$	−21.1236	−0.0289762
$82$	440.621i	0.593396i
$83$	70.0265i	0.0926074i	0.998927	+	0.0463037i	$0.0147442\pi$
$83$	70.0265i	0.0926074i	−0.998927	+	0.0463037i	$0.985256\pi$
$84$	785.876	1.02079
$85$	0	0
$86$	1381.11	1.73173
$87$	1124.54i	1.38579i
$88$	750.101i	0.908649i
$89$	9.27925	0.0110517	0.00552584	−	0.999985i	$-0.498241\pi$
$89$	9.27925	0.0110517	0.00552584	+	0.999985i	$0.498241\pi$
$90$	0	0
$91$	622.506	0.717103
$92$	− 271.319i	− 0.307467i
$93$	17.0891i	0.0190544i
$94$	−596.817	−0.654861
$95$	0	0
$96$	−1356.85	−1.44254
$97$	4.19493i	0.00439104i	0.999998	+	0.00219552i	$0.000698856\pi$
$97$	4.19493i	0.00439104i	−0.999998	+	0.00219552i	$0.999301\pi$
$98$	− 226.604i	− 0.233576i
$99$	1304.89	1.32471
$100$	0	0
$101$	−865.844	−0.853016	−0.426508	−	0.904484i	$-0.640256\pi$
$101$	−865.844	−0.853016	−0.426508	+	0.904484i	$0.640256\pi$
$102$	183.721i	0.178344i
$103$	− 1166.12i	− 1.11554i	−0.829995	−	0.557771i	$-0.811657\pi$
$103$	− 1166.12i	− 1.11554i	0.829995	−	0.557771i	$-0.188343\pi$
$104$	2215.29	2.08872
$105$	0	0
$106$	−1796.31	−1.64598
$107$	− 56.9652i	− 0.0514676i	−0.999669	−	0.0257338i	$-0.991808\pi$
$107$	− 56.9652i	− 0.0514676i	0.999669	−	0.0257338i	$-0.00819223\pi$
$108$	− 1833.90i	− 1.63395i
$109$	1358.89	1.19411	0.597055	−	0.802200i	$-0.296337\pi$
$109$	1358.89	1.19411	0.597055	+	0.802200i	$0.296337\pi$
$110$	0	0
$111$	1183.66	1.01215
$112$	− 56.7572i	− 0.0478844i
$113$	436.038i	0.363000i	0.983391	+	0.181500i	$0.0580952\pi$
$113$	436.038i	0.363000i	−0.983391	+	0.181500i	$0.941905\pi$
$114$	−4840.98	−3.97719
$115$	0	0
$116$	1794.98	1.43672
$117$	− 3853.76i	− 3.04513i
$118$	− 3877.70i	− 3.02518i
$119$	33.1590	0.0255435
$120$	0	0
$121$	−424.288	−0.318774
$122$	1800.91i	1.33645i
$123$	− 799.062i	− 0.585764i
$124$	27.2775	0.0197547
$125$	0	0
$126$	−1402.84	−0.991863
$127$	− 1186.69i	− 0.829144i	−0.910017	−	0.414572i	$-0.863931\pi$
$127$	− 1186.69i	− 0.829144i	0.910017	−	0.414572i	$-0.136069\pi$
$128$	2465.77i	1.70270i
$129$	−2504.62	−1.70946
$130$	0	0
$131$	1034.56	0.689997	0.344999	−	0.938603i	$-0.387879\pi$
$131$	1034.56	0.689997	0.344999	+	0.938603i	$0.387879\pi$
$132$	− 3380.57i	− 2.22910i
$133$	873.725i	0.569636i
$134$	3226.99	2.08037
$135$	0	0
$136$	118.002	0.0744011
$137$	− 646.219i	− 0.402994i	−0.979489	−	0.201497i	$-0.935419\pi$
$137$	− 646.219i	− 0.402994i	0.979489	−	0.201497i	$-0.0645806\pi$
$138$	786.080i	0.484895i
$139$	−506.484	−0.309061	−0.154530	−	0.987988i	$-0.549386\pi$
$139$	−506.484	−0.309061	−0.154530	+	0.987988i	$0.549386\pi$
$140$	0	0
$141$	1082.32	0.646439
$142$	− 2420.73i	− 1.43058i
$143$	− 2677.81i	− 1.56594i
$144$	−351.368	−0.203338
$145$	0	0
$146$	−307.387	−0.174244
$147$	410.943i	0.230572i
$148$	− 1889.35i	− 1.04935i
$149$	1828.12	1.00513	0.502567	−	0.864538i	$-0.332389\pi$
$149$	1828.12	1.00513	0.502567	+	0.864538i	$0.332389\pi$
$150$	0	0
$151$	2975.17	1.60342	0.801708	−	0.597716i	$-0.203925\pi$
$151$	2975.17	1.60342	0.801708	+	0.597716i	$0.203925\pi$
$152$	3109.29i	1.65919i
$153$	− 205.278i	− 0.108469i
$154$	−974.773	−0.510061
$155$	0	0
$156$	−9983.93	−5.12407
$157$	2131.74i	1.08364i	0.840495	+	0.541820i	$0.182264\pi$
$157$	2131.74i	1.08364i	−0.840495	+	0.541820i	$0.817736\pi$
$158$	2437.06i	1.22710i
$159$	3257.59	1.62481
$160$	0	0
$161$	141.876	0.0694495
$162$	− 97.6876i	− 0.0473769i
$163$	− 593.939i	− 0.285404i	−0.989766	−	0.142702i	$-0.954421\pi$
$163$	− 593.939i	− 0.285404i	0.989766	−	0.142702i	$-0.0455791\pi$
$164$	−1275.45	−0.607294
$165$	0	0
$166$	−323.842	−0.151416
$167$	2936.30i	1.36059i	0.732941	+	0.680293i	$0.238147\pi$
$167$	2936.30i	1.36059i	−0.732941	+	0.680293i	$0.761853\pi$
$168$	1462.41i	0.671591i
$169$	−5711.45	−2.59966
$170$	0	0
$171$	5408.98	2.41892
$172$	3997.85i	1.77229i
$173$	2347.31i	1.03158i	0.856716	+	0.515788i	$0.172501\pi$
$173$	2347.31i	1.03158i	−0.856716	+	0.515788i	$0.827499\pi$
$174$	−5200.51	−2.26580
$175$	0	0
$176$	−244.151	−0.104566
$177$	7032.17i	2.98627i
$178$	42.9125i	0.0180698i
$179$	−3036.56	−1.26795	−0.633975	−	0.773354i	$-0.718578\pi$
$179$	−3036.56	−1.26795	−0.633975	+	0.773354i	$0.718578\pi$
$180$	0	0
$181$	−899.776	−0.369502	−0.184751	−	0.982785i	$-0.559148\pi$
$181$	−899.776	−0.369502	−0.184751	+	0.982785i	$0.559148\pi$
$182$	2878.82i	1.17249i
$183$	− 3265.93i	− 1.31926i
$184$	504.888	0.202287
$185$	0	0
$186$	−79.0297	−0.0311545
$187$	− 142.639i	− 0.0557796i
$188$	− 1727.59i	− 0.670199i
$189$	958.964	0.369071
$190$	0	0
$191$	416.168	0.157659	0.0788294	−	0.996888i	$-0.474882\pi$
$191$	416.168	0.157659	0.0788294	+	0.996888i	$0.474882\pi$
$192$	− 6818.86i	− 2.56307i
$193$	− 5181.05i	− 1.93233i	−0.257922	−	0.966166i	$-0.583038\pi$
$193$	− 5181.05i	− 1.93233i	0.257922	−	0.966166i	$-0.416962\pi$
$194$	−19.3997	−0.00717948
$195$	0	0
$196$	655.943	0.239046
$197$	− 1452.34i	− 0.525255i	−0.964897	−	0.262627i	$-0.915411\pi$
$197$	− 1452.34i	− 0.525255i	0.964897	−	0.262627i	$-0.0845890\pi$
$198$	6034.54i	2.16594i
$199$	1277.23	0.454978	0.227489	−	0.973781i	$-0.426948\pi$
$199$	1277.23	0.454978	0.227489	+	0.973781i	$0.426948\pi$
$200$	0	0
$201$	−5852.12	−2.05361
$202$	− 4004.15i	− 1.39471i
$203$	938.615i	0.324521i
$204$	−531.813	−0.182521
$205$	0	0
$206$	5392.78	1.82395
$207$	− 878.312i	− 0.294913i
$208$	721.056i	0.240367i
$209$	3758.47	1.24392
$210$	0	0
$211$	−3259.09	−1.06334	−0.531670	−	0.846951i	$-0.678436\pi$
$211$	−3259.09	−1.06334	−0.531670	+	0.846951i	$0.678436\pi$
$212$	− 5199.74i	− 1.68453i
$213$	4389.96i	1.41218i
$214$	263.439	0.0841511
$215$	0	0
$216$	3412.63	1.07500
$217$	14.2637i	0.00446213i
$218$	6284.27i	1.95241i
$219$	557.444	0.172002
$220$	0	0
$221$	−421.258	−0.128221
$222$	5473.92i	1.65489i
$223$	4373.35i	1.31328i	0.754205	+	0.656639i	$0.228023\pi$
$223$	4373.35i	1.31328i	−0.754205	+	0.656639i	$0.771977\pi$
$224$	−1132.52	−0.337811
$225$	0	0
$226$	−2016.48	−0.593516
$227$	61.1145i	0.0178692i	0.999960	+	0.00893461i	$0.00284401\pi$
$227$	61.1145i	0.0178692i	−0.999960	+	0.00893461i	$0.997156\pi$
$228$	− 14013.0i	− 4.07034i
$229$	−3019.41	−0.871302	−0.435651	−	0.900116i	$-0.643482\pi$
$229$	−3019.41	−0.871302	−0.435651	+	0.900116i	$0.643482\pi$
$230$	0	0
$231$	1767.74	0.503501
$232$	3340.22i	0.945241i
$233$	− 3531.17i	− 0.992851i	−0.868079	−	0.496426i	$-0.834646\pi$
$233$	− 3531.17i	− 0.992851i	0.868079	−	0.496426i	$-0.165354\pi$
$234$	17822.0	4.97888
$235$	0	0
$236$	11224.7	3.09603
$237$	− 4419.58i	− 1.21132i
$238$	153.346i	0.0417644i
$239$	−2282.62	−0.617785	−0.308893	−	0.951097i	$-0.599958\pi$
$239$	−2282.62	−0.617785	−0.308893	+	0.951097i	$0.599958\pi$
$240$	0	0
$241$	−2215.68	−0.592217	−0.296109	−	0.955154i	$-0.595689\pi$
$241$	−2215.68	−0.592217	−0.296109	+	0.955154i	$0.595689\pi$
$242$	− 1962.15i	− 0.521205i
$243$	3876.02i	1.02324i
$244$	−5213.04	−1.36775
$245$	0	0
$246$	3695.31	0.957742
$247$	− 11100.0i	− 2.85941i
$248$	50.7597i	0.0129970i
$249$	587.284	0.149468
$250$	0	0
$251$	−3082.55	−0.775174	−0.387587	−	0.921833i	$-0.626691\pi$
$251$	−3082.55	−0.775174	−0.387587	+	0.921833i	$0.626691\pi$
$252$	− 4060.76i	− 1.01509i
$253$	− 610.302i	− 0.151658i
$254$	5487.90	1.35568
$255$	0	0
$256$	−4898.58	−1.19594
$257$	6032.40i	1.46417i	0.681215	+	0.732083i	$0.261452\pi$
$257$	6032.40i	1.46417i	−0.681215	+	0.732083i	$0.738548\pi$
$258$	− 11582.8i	− 2.79501i
$259$	987.962	0.237023
$260$	0	0
$261$	5810.69	1.37806
$262$	4784.37i	1.12817i
$263$	5923.81i	1.38889i	0.719546	+	0.694445i	$0.244350\pi$
$263$	5923.81i	1.38889i	−0.719546	+	0.694445i	$0.755650\pi$
$264$	6290.80	1.46656
$265$	0	0
$266$	−4040.60	−0.931372
$267$	− 77.8213i	− 0.0178374i
$268$	9341.09i	2.12910i
$269$	−3252.80	−0.737273	−0.368637	−	0.929574i	$-0.620175\pi$
$269$	−3252.80	−0.737273	−0.368637	+	0.929574i	$0.620175\pi$
$270$	0	0
$271$	−6246.26	−1.40012	−0.700061	−	0.714083i	$-0.746844\pi$
$271$	−6246.26	−1.40012	−0.700061	+	0.714083i	$0.746844\pi$
$272$	38.4084i	0.00856195i
$273$	− 5220.71i	− 1.15741i
$274$	2988.48	0.658907
$275$	0	0
$276$	−2275.44	−0.496252
$277$	1572.17i	0.341020i	0.985356	+	0.170510i	$0.0545415\pi$
$277$	1572.17i	0.341020i	−0.985356	+	0.170510i	$0.945459\pi$
$278$	− 2342.27i	− 0.505324i
$279$	88.3024	0.0189481
$280$	0	0
$281$	−7846.03	−1.66567	−0.832837	−	0.553518i	$-0.813285\pi$
$281$	−7846.03	−1.66567	−0.832837	+	0.553518i	$0.813285\pi$
$282$	5005.26i	1.05695i
$283$	6265.58i	1.31608i	0.752984	+	0.658039i	$0.228614\pi$
$283$	6265.58i	1.31608i	−0.752984	+	0.658039i	$0.771386\pi$
$284$	7007.21	1.46409
$285$	0	0
$286$	12383.7	2.56037
$287$	− 666.949i	− 0.137173i
$288$	7011.10i	1.43449i
$289$	4890.56	0.995433
$290$	0	0
$291$	35.1812	0.00708714
$292$	− 889.785i	− 0.178325i
$293$	− 7264.99i	− 1.44855i	−0.689511	−	0.724276i	$-0.742174\pi$
$293$	− 7264.99i	− 1.44855i	0.689511	−	0.724276i	$-0.257826\pi$
$294$	−1900.43	−0.376992
$295$	0	0
$296$	3515.83	0.690382
$297$	− 4125.14i	− 0.805943i
$298$	8454.24i	1.64343i
$299$	−1802.42	−0.348617
$300$	0	0
$301$	−2090.52	−0.400318
$302$	13758.9i	2.62163i
$303$	7261.48i	1.37677i
$304$	−1012.05	−0.190937
$305$	0	0
$306$	949.319	0.177350
$307$	− 1328.32i	− 0.246943i	−0.992348	−	0.123471i	$-0.960597\pi$
$307$	− 1328.32i	− 0.246943i	0.992348	−	0.123471i	$-0.0394027\pi$
$308$	− 2821.65i	− 0.522008i
$309$	−9779.75	−1.80049
$310$	0	0
$311$	4868.68	0.887709	0.443855	−	0.896099i	$-0.353611\pi$
$311$	4868.68	0.887709	0.443855	+	0.896099i	$0.353611\pi$
$312$	− 18578.8i	− 3.37120i
$313$	7733.39i	1.39654i	0.715835	+	0.698270i	$0.246046\pi$
$313$	7733.39i	1.39654i	−0.715835	+	0.698270i	$0.753954\pi$
$314$	−9858.37	−1.77178
$315$	0	0
$316$	−7054.49	−1.25584
$317$	8175.03i	1.44844i	0.689569	+	0.724220i	$0.257800\pi$
$317$	8175.03i	1.44844i	−0.689569	+	0.724220i	$0.742200\pi$
$318$	15065.0i	2.65661i
$319$	4037.61	0.708660
$320$	0	0
$321$	−477.744	−0.0830688
$322$	656.114i	0.113552i
$323$	− 591.261i	− 0.101853i
$324$	282.774	0.0484865
$325$	0	0
$326$	2746.71	0.466644
$327$	− 11396.5i	− 1.92729i
$328$	− 2373.45i	− 0.399548i
$329$	903.375	0.151382
$330$	0	0
$331$	−2040.76	−0.338884	−0.169442	−	0.985540i	$-0.554197\pi$
$331$	−2040.76	−0.338884	−0.169442	+	0.985540i	$0.554197\pi$
$332$	− 937.417i	− 0.154962i
$333$	− 6116.19i	− 1.00650i
$334$	−13579.1	−2.22460
$335$	0	0
$336$	−476.000	−0.0772855
$337$	− 7349.73i	− 1.18803i	−0.804455	−	0.594013i	$-0.797543\pi$
$337$	− 7349.73i	− 1.18803i	0.804455	−	0.594013i	$-0.202457\pi$
$338$	− 26413.0i	− 4.25052i
$339$	3656.87	0.585882
$340$	0	0
$341$	61.3576	0.00974399
$342$	25014.2i	3.95500i
$343$	343.000i	0.0539949i
$344$	−7439.47	−1.16602
$345$	0	0
$346$	−10855.3	−1.68666
$347$	12069.9i	1.86728i	0.358207	+	0.933642i	$0.383388\pi$
$347$	12069.9i	1.86728i	−0.358207	+	0.933642i	$0.616612\pi$
$348$	− 15053.8i	− 2.31887i
$349$	4484.96	0.687892	0.343946	−	0.938989i	$-0.388236\pi$
$349$	4484.96	0.687892	0.343946	+	0.938989i	$0.388236\pi$
$350$	0	0
$351$	−12182.9	−1.85263
$352$	4871.72i	0.737681i
$353$	− 12762.5i	− 1.92430i	−0.272517	−	0.962151i	$-0.587856\pi$
$353$	− 12762.5i	− 1.92430i	0.272517	−	0.962151i	$-0.412144\pi$
$354$	−32520.7	−4.88264
$355$	0	0
$356$	−124.218	−0.0184930
$357$	− 278.091i	− 0.0412272i
$358$	− 14042.8i	− 2.07314i
$359$	2419.42	0.355689	0.177844	−	0.984059i	$-0.443088\pi$
$359$	2419.42	0.355689	0.177844	+	0.984059i	$0.443088\pi$
$360$	0	0
$361$	8720.49	1.27139
$362$	− 4161.07i	− 0.604147i
$363$	3558.33i	0.514501i
$364$	−8333.24	−1.19995
$365$	0	0
$366$	15103.5	2.15703
$367$	7129.74i	1.01409i	0.861921	+	0.507043i	$0.169261\pi$
$367$	7129.74i	1.01409i	−0.861921	+	0.507043i	$0.830739\pi$
$368$	164.336i	0.0232789i
$369$	−4128.89	−0.582497
$370$	0	0
$371$	2719.00	0.380495
$372$	− 228.765i	− 0.0318842i
$373$	11596.9i	1.60983i	0.593391	+	0.804914i	$0.297789\pi$
$373$	11596.9i	1.60983i	−0.593391	+	0.804914i	$0.702211\pi$
$374$	659.642	0.0912013
$375$	0	0
$376$	3214.81	0.440934
$377$	− 11924.4i	− 1.62901i
$378$	4434.79i	0.603442i
$379$	12770.8	1.73085	0.865424	−	0.501040i	$-0.167049\pi$
$379$	12770.8	1.73085	0.865424	+	0.501040i	$0.167049\pi$
$380$	0	0
$381$	−9952.25	−1.33824
$382$	1924.59i	0.257777i
$383$	7470.10i	0.996617i	0.867000	+	0.498308i	$0.166045\pi$
$383$	7470.10i	0.996617i	−0.867000	+	0.498308i	$0.833955\pi$
$384$	20679.4	2.74816
$385$	0	0
$386$	23960.1	3.15942
$387$	12941.8i	1.69992i
$388$	− 56.1558i	− 0.00734763i
$389$	−8749.77	−1.14044	−0.570220	−	0.821492i	$-0.693142\pi$
$389$	−8749.77	−1.14044	−0.570220	+	0.821492i	$0.693142\pi$
$390$	0	0
$391$	−96.0092	−0.0124179
$392$	1220.62i	0.157272i
$393$	− 8676.41i	− 1.11366i
$394$	6716.46	0.858808
$395$	0	0
$396$	−17468.0	−2.21667
$397$	− 5375.25i	− 0.679537i	−0.940509	−	0.339769i	$-0.889651\pi$
$397$	− 5375.25i	− 0.679537i	0.940509	−	0.339769i	$-0.110349\pi$
$398$	5906.65i	0.743903i
$399$	7327.58	0.919393
$400$	0	0
$401$	7361.33	0.916727	0.458363	−	0.888765i	$-0.348436\pi$
$401$	7361.33	0.916727	0.458363	+	0.888765i	$0.348436\pi$
$402$	− 27063.5i	− 3.35772i
$403$	− 181.209i	− 0.0223987i
$404$	11590.7	1.42737
$405$	0	0
$406$	−4340.68	−0.530602
$407$	− 4249.88i	− 0.517589i
$408$	− 989.632i	− 0.120084i
$409$	2612.45	0.315837	0.157919	−	0.987452i	$-0.449522\pi$
$409$	2612.45	0.315837	0.157919	+	0.987452i	$0.449522\pi$
$410$	0	0
$411$	−5419.57	−0.650433
$412$	15610.3i	1.86667i
$413$	5869.51i	0.699321i
$414$	4061.81	0.482191
$415$	0	0
$416$	14387.8	1.69572
$417$	4247.68i	0.498824i
$418$	17381.3i	2.03384i
$419$	−4398.21	−0.512808	−0.256404	−	0.966570i	$-0.582538\pi$
$419$	−4398.21	−0.512808	−0.256404	+	0.966570i	$0.582538\pi$
$420$	0	0
$421$	9723.32	1.12562	0.562810	−	0.826587i	$-0.309720\pi$
$421$	9723.32	1.12562	0.562810	+	0.826587i	$0.309720\pi$
$422$	− 15071.9i	− 1.73859i
$423$	− 5592.54i	− 0.642833i
$424$	9676.01	1.10828
$425$	0	0
$426$	−20301.7	−2.30896
$427$	− 2725.96i	− 0.308942i
$428$	762.570i	0.0861220i
$429$	−22457.7	−2.52744
$430$	0	0
$431$	−14314.5	−1.59978	−0.799892	−	0.600144i	$-0.795110\pi$
$431$	−14314.5	−1.59978	−0.799892	+	0.600144i	$0.795110\pi$
$432$	1110.78i	0.123709i
$433$	− 2373.62i	− 0.263438i	−0.991287	−	0.131719i	$-0.957950\pi$
$433$	− 2373.62i	− 0.263438i	0.991287	−	0.131719i	$-0.0420497\pi$
$434$	−65.9633	−0.00729572
$435$	0	0
$436$	−18190.9	−1.99813
$437$	− 2529.80i	− 0.276927i
$438$	2577.93i	0.281229i
$439$	9533.46	1.03646	0.518231	−	0.855240i	$-0.326591\pi$
$439$	9533.46	1.03646	0.518231	+	0.855240i	$0.326591\pi$
$440$	0	0
$441$	2123.41	0.229286
$442$	− 1948.14i	− 0.209646i
$443$	6647.94i	0.712987i	0.934298	+	0.356493i	$0.116028\pi$
$443$	6647.94i	0.712987i	−0.934298	+	0.356493i	$0.883972\pi$
$444$	−15845.2	−1.69365
$445$	0	0
$446$	−20224.8	−2.14725
$447$	− 15331.7i	− 1.62229i
$448$	− 5691.47i	− 0.600215i
$449$	768.256	0.0807489	0.0403744	−	0.999185i	$-0.487145\pi$
$449$	768.256	0.0807489	0.0403744	+	0.999185i	$0.487145\pi$
$450$	0	0
$451$	−2868.99	−0.299547
$452$	− 5837.06i	− 0.607416i
$453$	− 24951.5i	− 2.58791i
$454$	−282.628	−0.0292167
$455$	0	0
$456$	26076.4	2.67794
$457$	3323.50i	0.340190i	0.985428	+	0.170095i	$0.0544074\pi$
$457$	3323.50i	0.340190i	−0.985428	+	0.170095i	$0.945593\pi$
$458$	− 13963.5i	− 1.42461i
$459$	−648.944	−0.0659915
$460$	0	0
$461$	−18840.7	−1.90347	−0.951733	−	0.306926i	$-0.900700\pi$
$461$	−18840.7	−1.90347	−0.951733	+	0.306926i	$0.900700\pi$
$462$	8175.03i	0.823240i
$463$	− 10759.1i	− 1.07995i	−0.841679	−	0.539977i	$-0.818433\pi$
$463$	− 10759.1i	− 1.07995i	0.841679	−	0.539977i	$-0.181567\pi$
$464$	−1087.21	−0.108777
$465$	0	0
$466$	16330.1	1.62334
$467$	− 7441.70i	− 0.737390i	−0.929550	−	0.368695i	$-0.879805\pi$
$467$	− 7441.70i	− 0.737390i	0.929550	−	0.368695i	$-0.120195\pi$
$468$	51588.7i	5.09549i
$469$	−4884.56	−0.480913
$470$	0	0
$471$	17878.0	1.74899
$472$	20887.6i	2.03693i
$473$	8992.73i	0.874178i
$474$	20438.7	1.98055
$475$	0	0
$476$	−443.885	−0.0427426
$477$	− 16832.6i	− 1.61574i
$478$	− 10556.1i	− 1.01010i
$479$	−5691.97	−0.542949	−0.271475	−	0.962446i	$-0.587511\pi$
$479$	−5691.97	−0.542949	−0.271475	+	0.962446i	$0.587511\pi$
$480$	0	0
$481$	−12551.3	−1.18979
$482$	− 10246.5i	− 0.968293i
$483$	− 1189.85i	− 0.112092i
$484$	5679.77	0.533412
$485$	0	0
$486$	−17924.9	−1.67302
$487$	2020.25i	0.187980i	0.995573	+	0.0939899i	$0.0299621\pi$
$487$	2020.25i	0.187980i	−0.995573	+	0.0939899i	$0.970038\pi$
$488$	− 9700.77i	− 0.899863i
$489$	−4981.12	−0.460642
$490$	0	0
$491$	7636.02	0.701851	0.350925	−	0.936403i	$-0.385867\pi$
$491$	7636.02	0.701851	0.350925	+	0.936403i	$0.385867\pi$
$492$	10696.7i	0.980173i
$493$	− 635.173i	− 0.0580259i
$494$	51332.6	4.67523
$495$	0	0
$496$	−16.5218	−0.00149567
$497$	3664.15i	0.330703i
$498$	2715.93i	0.244385i
$499$	−6284.56	−0.563799	−0.281900	−	0.959444i	$-0.590964\pi$
$499$	−6284.56	−0.563799	−0.281900	+	0.959444i	$0.590964\pi$
$500$	0	0
$501$	24625.6	2.19599
$502$	− 14255.4i	− 1.26743i
$503$	11310.9i	1.00264i	0.865262	+	0.501319i	$0.167152\pi$
$503$	11310.9i	1.00264i	−0.865262	+	0.501319i	$0.832848\pi$
$504$	7556.52	0.667846
$505$	0	0
$506$	2822.38	0.247965
$507$	47899.7i	4.19586i
$508$	15885.7i	1.38743i
$509$	−10712.7	−0.932876	−0.466438	−	0.884554i	$-0.654463\pi$
$509$	−10712.7	−0.932876	−0.466438	+	0.884554i	$0.654463\pi$
$510$	0	0
$511$	465.279	0.0402793
$512$	− 2927.65i	− 0.252705i
$513$	− 17099.4i	− 1.47165i
$514$	−27897.2	−2.39396
$515$	0	0
$516$	33528.4	2.86047
$517$	− 3886.02i	− 0.330574i
$518$	4568.89i	0.387540i
$519$	19685.9	1.66496
$520$	0	0
$521$	17721.9	1.49023	0.745116	−	0.666935i	$-0.232394\pi$
$521$	17721.9	1.49023	0.745116	+	0.666935i	$0.232394\pi$
$522$	26871.9i	2.25317i
$523$	237.193i	0.0198312i	0.999951	+	0.00991562i	$0.00315629\pi$
$523$	237.193i	0.0198312i	−0.999951	+	0.00991562i	$0.996844\pi$
$524$	−13849.2	−1.15459
$525$	0	0
$526$	−27395.0	−2.27088
$527$	− 9.65243i	0 0.000797849i
$528$	2047.59i	0.168769i
$529$	11756.2	0.966237
$530$	0	0
$531$	36336.4	2.96962
$532$	− 11696.2i	− 0.953185i
$533$	8473.06i	0.688572i
$534$	359.890	0.0291647
$535$	0	0
$536$	−17382.5	−1.40077
$537$	25466.4i	2.04647i
$538$	− 15042.8i	− 1.20546i
$539$	1475.47	0.117909
$540$	0	0
$541$	−5352.94	−0.425399	−0.212699	−	0.977118i	$-0.568226\pi$
$541$	−5352.94	−0.425399	−0.212699	+	0.977118i	$0.568226\pi$
$542$	− 28886.2i	− 2.28924i
$543$	7546.06i	0.596376i
$544$	766.391	0.0604021
$545$	0	0
$546$	24143.5	1.89239
$547$	192.162i	0.0150206i	0.999972	+	0.00751030i	$0.00239063\pi$
$547$	192.162i	0.0150206i	−0.999972	+	0.00751030i	$0.997609\pi$
$548$	8650.67i	0.674340i
$549$	−16875.6	−1.31190
$550$	0	0
$551$	16736.5	1.29401
$552$	− 4234.29i	− 0.326492i
$553$	− 3688.87i	− 0.283665i
$554$	−7270.60	−0.557578
$555$	0	0
$556$	6780.10	0.517159
$557$	4850.62i	0.368990i	0.982833	+	0.184495i	$0.0590649\pi$
$557$	4850.62i	0.368990i	−0.982833	+	0.184495i	$0.940935\pi$
$558$	408.360i	0.0309808i
$559$	26558.4	2.00949
$560$	0	0
$561$	−1196.25	−0.0900283
$562$	− 36284.4i	− 2.72343i
$563$	9699.11i	0.726055i	0.931778	+	0.363027i	$0.118257\pi$
$563$	9699.11i	0.726055i	−0.931778	+	0.363027i	$0.881743\pi$
$564$	−14488.6	−1.08170
$565$	0	0
$566$	−28975.6	−2.15183
$567$	147.865i	0.0109520i
$568$	13039.5i	0.963247i
$569$	−3109.53	−0.229100	−0.114550	−	0.993417i	$-0.536543\pi$
$569$	−3109.53	−0.229100	−0.114550	+	0.993417i	$0.536543\pi$
$570$	0	0
$571$	−14476.2	−1.06097	−0.530483	−	0.847695i	$-0.677990\pi$
$571$	−14476.2	−1.06097	−0.530483	+	0.847695i	$0.677990\pi$
$572$	35846.8i	2.62033i
$573$	− 3490.23i	− 0.254461i
$574$	3084.35	0.224283
$575$	0	0
$576$	−35234.2	−2.54877
$577$	2208.23i	0.159323i	0.996822	+	0.0796617i	$0.0253840\pi$
$577$	2208.23i	0.159323i	−0.996822	+	0.0796617i	$0.974616\pi$
$578$	22616.7i	1.62756i
$579$	−43451.4	−3.11879
$580$	0	0
$581$	490.186	0.0350023
$582$	162.698i	0.0115877i
$583$	− 11696.2i	− 0.830889i
$584$	1655.77	0.117322
$585$	0	0
$586$	33597.4	2.36843
$587$	23988.7i	1.68675i	0.537327	+	0.843374i	$0.319434\pi$
$587$	23988.7i	1.68675i	−0.537327	+	0.843374i	$0.680566\pi$
$588$	− 5501.13i	− 0.385821i
$589$	254.338	0.0177925
$590$	0	0
$591$	−12180.2	−0.847762
$592$	1144.37i	0.0794480i
$593$	− 15869.4i	− 1.09895i	−0.835511	−	0.549474i	$-0.814828\pi$
$593$	− 15869.4i	− 1.09895i	0.835511	−	0.549474i	$-0.185172\pi$
$594$	19077.0	1.31774
$595$	0	0
$596$	−24472.2	−1.68192
$597$	− 10711.6i	− 0.734335i
$598$	− 8335.41i	− 0.570000i
$599$	15236.6	1.03932	0.519660	−	0.854373i	$-0.326059\pi$
$599$	15236.6	1.03932	0.519660	+	0.854373i	$0.326059\pi$
$600$	0	0
$601$	12258.8	0.832026	0.416013	−	0.909359i	$-0.363427\pi$
$601$	12258.8	0.832026	0.416013	+	0.909359i	$0.363427\pi$
$602$	− 9667.75i	− 0.654532i
$603$	30238.9i	2.04216i
$604$	−39827.4	−2.68303
$605$	0	0
$606$	−33581.2	−2.25106
$607$	− 23487.2i	− 1.57054i	−0.619155	−	0.785269i	$-0.712525\pi$
$607$	− 23487.2i	− 1.57054i	0.619155	−	0.785269i	$-0.287475\pi$
$608$	20194.1i	1.34700i
$609$	7871.78	0.523778
$610$	0	0
$611$	−11476.7	−0.759896
$612$	2747.97i	0.181503i
$613$	− 22305.3i	− 1.46966i	−0.678251	−	0.734830i	$-0.737262\pi$
$613$	− 22305.3i	− 1.46966i	0.678251	−	0.734830i	$-0.262738\pi$
$614$	6142.92	0.403759
$615$	0	0
$616$	5250.71	0.343437
$617$	− 3285.91i	− 0.214402i	−0.994237	−	0.107201i	$-0.965811\pi$
$617$	− 3285.91i	− 0.214402i	0.994237	−	0.107201i	$-0.0341888\pi$
$618$	− 45227.1i	− 2.94385i
$619$	11613.1	0.754069	0.377035	−	0.926199i	$-0.376944\pi$
$619$	11613.1	0.754069	0.377035	+	0.926199i	$0.376944\pi$
$620$	0	0
$621$	−2776.61	−0.179423
$622$	22515.5i	1.45143i
$623$	− 64.9548i	− 0.00417714i
$624$	6047.21	0.387952
$625$	0	0
$626$	−35763.5	−2.28338
$627$	− 31520.8i	− 2.00769i
$628$	− 28536.7i	− 1.81328i
$629$	−668.567	−0.0423808
$630$	0	0
$631$	6890.91	0.434743	0.217372	−	0.976089i	$-0.430252\pi$
$631$	6890.91	0.434743	0.217372	+	0.976089i	$0.430252\pi$
$632$	− 13127.5i	− 0.826238i
$633$	27332.7i	1.71623i
$634$	−37805.9	−2.36824
$635$	0	0
$636$	−43608.1	−2.71883
$637$	− 4357.55i	− 0.271040i
$638$	18672.2i	1.15868i
$639$	22683.7	1.40431
$640$	0	0
$641$	18769.3	1.15654	0.578269	−	0.815846i	$-0.303728\pi$
$641$	18769.3	1.15654	0.578269	+	0.815846i	$0.303728\pi$
$642$	− 2209.36i	− 0.135820i
$643$	− 3142.30i	− 0.192722i	−0.995346	−	0.0963609i	$-0.969280\pi$
$643$	− 3142.30i	− 0.192722i	0.995346	−	0.0963609i	$-0.0307203\pi$
$644$	−1899.23	−0.116212
$645$	0	0
$646$	2734.33	0.166533
$647$	− 19038.1i	− 1.15683i	−0.815744	−	0.578413i	$-0.803672\pi$
$647$	− 19038.1i	− 1.15683i	0.815744	−	0.578413i	$-0.196328\pi$
$648$	526.204i	0.0319000i
$649$	25248.7	1.52711
$650$	0	0
$651$	119.624	0.00720188
$652$	7950.82i	0.477574i
$653$	− 20538.6i	− 1.23084i	−0.788199	−	0.615420i	$-0.788986\pi$
$653$	− 20538.6i	− 1.23084i	0.788199	−	0.615420i	$-0.211014\pi$
$654$	52703.6	3.15119
$655$	0	0
$656$	772.534	0.0459793
$657$	− 2880.41i	− 0.171043i
$658$	4177.72i	0.247514i
$659$	937.046	0.0553902	0.0276951	−	0.999616i	$-0.491183\pi$
$659$	937.046	0.0553902	0.0276951	+	0.999616i	$0.491183\pi$
$660$	0	0
$661$	21116.5	1.24257	0.621283	−	0.783586i	$-0.286612\pi$
$661$	21116.5	1.24257	0.621283	+	0.783586i	$0.286612\pi$
$662$	− 9437.65i	− 0.554085i
$663$	3532.92i	0.206949i
$664$	1744.41	0.101952
$665$	0	0
$666$	28284.7	1.64566
$667$	− 2717.69i	− 0.157765i
$668$	− 39307.1i	− 2.27670i
$669$	36677.5	2.11963
$670$	0	0
$671$	−11726.2	−0.674640
$672$	9497.98i	0.545227i
$673$	13825.9i	0.791903i	0.918271	+	0.395952i	$0.129585\pi$
$673$	13825.9i	0.791903i	−0.918271	+	0.395952i	$0.870415\pi$
$674$	33989.3	1.94246
$675$	0	0
$676$	76456.9	4.35008
$677$	16928.4i	0.961021i	0.876989	+	0.480510i	$0.159549\pi$
$677$	16928.4i	0.961021i	−0.876989	+	0.480510i	$0.840451\pi$
$678$	16911.4i	0.957935i
$679$	29.3645	0.00165966
$680$	0	0
$681$	512.543	0.0288409
$682$	283.752i	0.0159317i
$683$	− 13817.3i	− 0.774091i	−0.922061	−	0.387045i	$-0.873496\pi$
$683$	− 13817.3i	− 0.774091i	0.922061	−	0.387045i	$-0.126504\pi$
$684$	−72407.8	−4.04763
$685$	0	0
$686$	−1586.23	−0.0882833
$687$	25322.6i	1.40628i
$688$	− 2421.47i	− 0.134183i
$689$	−34542.8	−1.90998
$690$	0	0
$691$	−23671.6	−1.30320	−0.651600	−	0.758563i	$-0.725902\pi$
$691$	−23671.6	−1.30320	−0.651600	+	0.758563i	$0.725902\pi$
$692$	− 31422.5i	− 1.72616i
$693$	− 9134.22i	− 0.500693i
$694$	−55818.2	−3.05307
$695$	0	0
$696$	28013.0	1.52562
$697$	451.333i	0.0245272i
$698$	20741.0i	1.12472i
$699$	−29614.5	−1.60246
$700$	0	0
$701$	−17009.7	−0.916472	−0.458236	−	0.888831i	$-0.651519\pi$
$701$	−17009.7	−0.916472	−0.458236	+	0.888831i	$0.651519\pi$
$702$	− 56340.6i	− 3.02911i
$703$	− 17616.5i	− 0.945117i
$704$	−24482.8	−1.31070
$705$	0	0
$706$	59020.9	3.14629
$707$	6060.91i	0.322410i
$708$	− 94136.8i	− 4.99700i
$709$	−22038.9	−1.16740	−0.583701	−	0.811969i	$-0.698396\pi$
$709$	−22038.9	−1.16740	−0.583701	+	0.811969i	$0.698396\pi$
$710$	0	0
$711$	−22836.8	−1.20456
$712$	− 231.152i	− 0.0121669i
$713$	− 41.2994i	− 0.00216925i
$714$	1286.05	0.0674078
$715$	0	0
$716$	40649.2	2.12169
$717$	19143.4i	0.997106i
$718$	11188.8i	0.581562i
$719$	−7287.44	−0.377991	−0.188996	−	0.981978i	$-0.560523\pi$
$719$	−7287.44	−0.377991	−0.188996	+	0.981978i	$0.560523\pi$
$720$	0	0
$721$	−8162.82	−0.421636
$722$	40328.5i	2.07877i
$723$	18582.0i	0.955839i
$724$	12044.9	0.618297
$725$	0	0
$726$	−16455.7	−0.841225
$727$	29676.7i	1.51396i	0.653438	+	0.756980i	$0.273326\pi$
$727$	29676.7i	1.51396i	−0.653438	+	0.756980i	$0.726674\pi$
$728$	− 15507.0i	− 0.789463i
$729$	31936.3	1.62253
$730$	0	0
$731$	1414.68	0.0715786
$732$	43719.7i	2.20755i
$733$	23111.8i	1.16460i	0.812974	+	0.582300i	$0.197847\pi$
$733$	23111.8i	1.16460i	−0.812974	+	0.582300i	$0.802153\pi$
$734$	−32971.9	−1.65806
$735$	0	0
$736$	3279.12	0.164226
$737$	21011.7i	1.05017i
$738$	− 19094.3i	− 0.952400i
$739$	31171.4	1.55164	0.775818	−	0.630957i	$-0.217338\pi$
$739$	31171.4	1.55164	0.775818	+	0.630957i	$0.217338\pi$
$740$	0	0
$741$	−93091.1	−4.61510
$742$	12574.2i	0.622120i
$743$	− 31324.4i	− 1.54668i	−0.633993	−	0.773338i	$-0.718585\pi$
$743$	− 31324.4i	− 1.54668i	0.633993	−	0.773338i	$-0.281415\pi$
$744$	425.701	0.0209771
$745$	0	0
$746$	−53630.7	−2.63212
$747$	− 3034.60i	− 0.148635i
$748$	1909.45i	0.0933373i
$749$	−398.756	−0.0194529
$750$	0	0
$751$	4032.20	0.195922	0.0979608	−	0.995190i	$-0.468768\pi$
$751$	4032.20	0.195922	0.0979608	+	0.995190i	$0.468768\pi$
$752$	1046.39i	0.0507419i
$753$	25852.1i	1.25113i
$754$	55145.0	2.66348
$755$	0	0
$756$	−12837.3	−0.617575
$757$	− 34263.7i	− 1.64509i	−0.568699	−	0.822546i	$-0.692553\pi$
$757$	− 34263.7i	− 1.64509i	0.568699	−	0.822546i	$-0.307447\pi$
$758$	59059.3i	2.82999i
$759$	−5118.36	−0.244775
$760$	0	0
$761$	7265.88	0.346108	0.173054	−	0.984912i	$-0.444637\pi$
$761$	7265.88	0.346108	0.173054	+	0.984912i	$0.444637\pi$
$762$	− 46024.8i	− 2.18806i
$763$	− 9512.22i	− 0.451331i
$764$	−5571.07	−0.263814
$765$	0	0
$766$	−34546.0	−1.62950
$767$	− 74567.5i	− 3.51040i
$768$	41082.4i	1.93025i
$769$	−38116.2	−1.78739	−0.893695	−	0.448674i	$-0.851896\pi$
$769$	−38116.2	−1.78739	−0.893695	+	0.448674i	$0.851896\pi$
$770$	0	0
$771$	50591.3	2.36317
$772$	69356.6i	3.23342i
$773$	16158.2i	0.751838i	0.926652	+	0.375919i	$0.122673\pi$
$773$	16158.2i	0.751838i	−0.926652	+	0.375919i	$0.877327\pi$
$774$	−59850.3	−2.77942
$775$	0	0
$776$	104.498	0.00483412
$777$	− 8285.64i	− 0.382555i
$778$	− 40463.9i	− 1.86465i
$779$	−11892.4	−0.546972
$780$	0	0
$781$	15761.9	0.722160
$782$	− 444.001i	− 0.0203036i
$783$	− 18369.3i	− 0.838400i
$784$	−397.301	−0.0180986
$785$	0	0
$786$	40124.6	1.82086
$787$	− 5092.49i	− 0.230658i	−0.993327	−	0.115329i	$-0.963208\pi$
$787$	− 5092.49i	− 0.230658i	0.993327	−	0.115329i	$-0.0367922\pi$
$788$	19441.9i	0.878922i
$789$	49680.6	2.24167
$790$	0	0
$791$	3052.26	0.137201
$792$	− 32505.6i	− 1.45838i
$793$	34631.1i	1.55080i
$794$	24858.2	1.11106
$795$	0	0
$796$	−17097.8	−0.761326
$797$	34666.2i	1.54070i	0.637621	+	0.770350i	$0.279919\pi$
$797$	34666.2i	1.54070i	−0.637621	+	0.770350i	$0.720081\pi$
$798$	33886.9i	1.50323i
$799$	−611.326	−0.0270678
$800$	0	0
$801$	−402.116	−0.0177379
$802$	34043.0i	1.49888i
$803$	− 2001.47i	− 0.0879582i
$804$	78339.9	3.43636
$805$	0	0
$806$	838.012	0.0366225
$807$	27279.9i	1.18996i
$808$	21568.7i	0.939091i
$809$	−15126.2	−0.657365	−0.328683	−	0.944440i	$-0.606605\pi$
$809$	−15126.2	−0.657365	−0.328683	+	0.944440i	$0.606605\pi$
$810$	0	0
$811$	29416.5	1.27368	0.636840	−	0.770996i	$-0.280241\pi$
$811$	29416.5	1.27368	0.636840	+	0.770996i	$0.280241\pi$
$812$	− 12564.9i	− 0.543030i
$813$	52384.9i	2.25980i
$814$	19653.9	0.846274
$815$	0	0
$816$	322.116	0.0138190
$817$	37276.3i	1.59625i
$818$	12081.5i	0.516404i
$819$	−26976.3	−1.15095
$820$	0	0
$821$	−15334.4	−0.651856	−0.325928	−	0.945395i	$-0.605677\pi$
$821$	−15334.4	−0.651856	−0.325928	+	0.945395i	$0.605677\pi$
$822$	− 25063.2i	− 1.06348i
$823$	− 11003.7i	− 0.466056i	−0.972470	−	0.233028i	$-0.925137\pi$
$823$	− 11003.7i	− 0.466056i	0.972470	−	0.233028i	$-0.0748634\pi$
$824$	−29048.7	−1.22811
$825$	0	0
$826$	−27143.9	−1.14341
$827$	3261.59i	0.137142i	0.997646	+	0.0685711i	$0.0218440\pi$
$827$	3261.59i	0.137142i	−0.997646	+	0.0685711i	$0.978156\pi$
$828$	11757.6i	0.493484i
$829$	−5163.30	−0.216319	−0.108160	−	0.994134i	$-0.534496\pi$
$829$	−5163.30	−0.216319	−0.108160	+	0.994134i	$0.534496\pi$
$830$	0	0
$831$	13185.1	0.550406
$832$	72305.6i	3.01292i
$833$	− 232.113i	− 0.00965453i
$834$	−19643.7	−0.815593
$835$	0	0
$836$	−50313.1	−2.08148
$837$	− 279.150i	− 0.0115279i
$838$	− 20339.8i	− 0.838457i
$839$	5641.70	0.232149	0.116075	−	0.993241i	$-0.462969\pi$
$839$	5641.70	0.232149	0.116075	+	0.993241i	$0.462969\pi$
$840$	0	0
$841$	−6409.46	−0.262801
$842$	44966.1i	1.84042i
$843$	65801.4i	2.68840i
$844$	43628.1	1.77931
$845$	0	0
$846$	25863.0	1.05105
$847$	2970.01i	0.120485i
$848$	3149.45i	0.127538i
$849$	52546.8	2.12415
$850$	0	0
$851$	−2860.57	−0.115228
$852$	− 58766.7i	− 2.36304i
$853$	− 7799.52i	− 0.313072i	−0.987672	−	0.156536i	$-0.949967\pi$
$853$	− 7799.52i	− 0.313072i	0.987672	−	0.156536i	$-0.0500327\pi$
$854$	12606.4	0.505130
$855$	0	0
$856$	−1419.04	−0.0566610
$857$	21540.0i	0.858568i	0.903170	+	0.429284i	$0.141234\pi$
$857$	21540.0i	0.858568i	−0.903170	+	0.429284i	$0.858766\pi$
$858$	− 103857.i	− 4.13244i
$859$	−4447.97	−0.176674	−0.0883370	−	0.996091i	$-0.528155\pi$
$859$	−4447.97	−0.176674	−0.0883370	+	0.996091i	$0.528155\pi$
$860$	0	0
$861$	−5593.43	−0.221398
$862$	− 66198.5i	− 2.61569i
$863$	− 9425.21i	− 0.371770i	−0.982571	−	0.185885i	$-0.940485\pi$
$863$	− 9425.21i	− 0.371770i	0.982571	−	0.185885i	$-0.0595153\pi$
$864$	22164.2	0.872733
$865$	0	0
$866$	10977.0	0.430730
$867$	− 41015.2i	− 1.60663i
$868$	− 190.942i	− 0.00746659i
$869$	−15868.3	−0.619442
$870$	0	0
$871$	62054.5	2.41405
$872$	− 33850.8i	− 1.31460i
$873$	− 181.787i	− 0.00704761i
$874$	11699.2	0.452783
$875$	0	0
$876$	−7462.27	−0.287816
$877$	− 22346.1i	− 0.860403i	−0.902733	−	0.430201i	$-0.858443\pi$
$877$	− 22346.1i	− 0.860403i	0.902733	−	0.430201i	$-0.141557\pi$
$878$	44088.1i	1.69465i
$879$	−60928.6	−2.33796
$880$	0	0
$881$	12074.9	0.461762	0.230881	−	0.972982i	$-0.425839\pi$
$881$	12074.9	0.461762	0.230881	+	0.972982i	$0.425839\pi$
$882$	9819.87i	0.374889i
$883$	− 30499.6i	− 1.16239i	−0.813764	−	0.581196i	$-0.802585\pi$
$883$	− 30499.6i	− 1.16239i	0.813764	−	0.581196i	$-0.197415\pi$
$884$	5639.22	0.214556
$885$	0	0
$886$	−30743.8	−1.16576
$887$	− 23344.2i	− 0.883675i	−0.897095	−	0.441838i	$-0.854327\pi$
$887$	− 23344.2i	− 0.883675i	0.897095	−	0.441838i	$-0.145673\pi$
$888$	− 29485.8i	− 1.11428i
$889$	−8306.80	−0.313387
$890$	0	0
$891$	636.068	0.0239159
$892$	− 58544.3i	− 2.19754i
$893$	− 16108.2i	− 0.603628i
$894$	70902.3	2.65249
$895$	0	0
$896$	17260.4	0.643560
$897$	15116.2i	0.562669i
$898$	3552.85i	0.132027i
$899$	273.227	0.0101364
$900$	0	0
$901$	−1839.98	−0.0680341
$902$	− 13267.8i	− 0.489768i
$903$	17532.4i	0.646113i
$904$	10862.0	0.399629
$905$	0	0
$906$	115390.	4.23132
$907$	15092.5i	0.552523i	0.961082	+	0.276262i	$0.0890956\pi$
$907$	15092.5i	0.552523i	−0.961082	+	0.276262i	$0.910904\pi$
$908$	− 818.115i	− 0.0299010i
$909$	37521.3	1.36909
$910$	0	0
$911$	−15207.8	−0.553081	−0.276541	−	0.961002i	$-0.589188\pi$
$911$	−15207.8	−0.553081	−0.276541	+	0.961002i	$0.589188\pi$
$912$	8487.61i	0.308172i
$913$	− 2108.62i	− 0.0764348i
$914$	−15369.7	−0.556221
$915$	0	0
$916$	40419.6	1.45797
$917$	− 7241.90i	− 0.260794i
$918$	− 3001.08i	− 0.107898i
$919$	−24818.1	−0.890831	−0.445415	−	0.895324i	$-0.646944\pi$
$919$	−24818.1	−0.890831	−0.445415	+	0.895324i	$0.646944\pi$
$920$	0	0
$921$	−11140.1	−0.398566
$922$	− 87130.0i	− 3.11223i
$923$	− 46550.1i	− 1.66004i
$924$	−23664.0	−0.842521
$925$	0	0
$926$	49756.3	1.76576
$927$	50533.7i	1.79045i
$928$	21693.9i	0.767388i
$929$	−39906.4	−1.40935	−0.704675	−	0.709530i	$-0.748907\pi$
$929$	−39906.4	−1.40935	−0.704675	+	0.709530i	$0.748907\pi$
$930$	0	0
$931$	6116.07	0.215302
$932$	47270.3i	1.66136i
$933$	− 40831.7i	− 1.43276i
$934$	34414.6	1.20565
$935$	0	0
$936$	−95999.7	−3.35240
$937$	− 16923.0i	− 0.590020i	−0.955494	−	0.295010i	$-0.904677\pi$
$937$	− 16923.0i	− 0.590020i	0.955494	−	0.295010i	$-0.0953230\pi$
$938$	− 22589.0i	− 0.786307i
$939$	64856.8	2.25402
$940$	0	0
$941$	−53014.1	−1.83657	−0.918285	−	0.395921i	$-0.870426\pi$
$941$	−53014.1	−1.83657	−0.918285	+	0.395921i	$0.870426\pi$
$942$	82678.1i	2.85966i
$943$	1931.10i	0.0666864i
$944$	−6798.71	−0.234406
$945$	0	0
$946$	−41587.4	−1.42931
$947$	25798.9i	0.885271i	0.896702	+	0.442636i	$0.145957\pi$
$947$	25798.9i	0.885271i	−0.896702	+	0.442636i	$0.854043\pi$
$948$	59163.2i	2.02693i
$949$	−5911.00	−0.202191
$950$	0	0
$951$	68560.6	2.33778
$952$	− 826.011i	− 0.0281210i
$953$	− 17942.7i	− 0.609885i	−0.952371	−	0.304943i	$-0.901363\pi$
$953$	− 17942.7i	− 0.609885i	0.952371	−	0.304943i	$-0.0986373\pi$
$954$	77843.2	2.64179
$955$	0	0
$956$	30556.6	1.03375
$957$	− 33861.8i	− 1.14378i
$958$	− 26322.9i	− 0.887739i
$959$	−4523.53	−0.152317
$960$	0	0
$961$	−29786.8	−0.999861
$962$	− 58044.2i	− 1.94534i
$963$	2468.59i	0.0826055i
$964$	29660.4	0.990971
$965$	0	0
$966$	5502.56	0.183273
$967$	− 19668.3i	− 0.654073i	−0.945012	−	0.327036i	$-0.893950\pi$
$967$	− 19668.3i	− 0.654073i	0.945012	−	0.327036i	$-0.106050\pi$
$968$	10569.3i	0.350940i
$969$	−4958.67	−0.164392
$970$	0	0
$971$	6332.97	0.209304	0.104652	−	0.994509i	$-0.466627\pi$
$971$	6332.97	0.209304	0.104652	+	0.994509i	$0.466627\pi$
$972$	− 51886.7i	− 1.71221i
$973$	3545.39i	0.116814i
$974$	−9342.76	−0.307353
$975$	0	0
$976$	3157.51	0.103555
$977$	11334.1i	0.371145i	0.982631	+	0.185573i	$0.0594140\pi$
$977$	11334.1i	0.371145i	−0.982631	+	0.185573i	$0.940586\pi$
$978$	− 23035.5i	− 0.753164i
$979$	−279.414	−0.00912166
$980$	0	0
$981$	−58887.4	−1.91655
$982$	35313.3i	1.14755i
$983$	− 37654.3i	− 1.22175i	−0.791725	−	0.610877i	$-0.790817\pi$
$983$	− 37654.3i	− 1.22175i	0.791725	−	0.610877i	$-0.209183\pi$
$984$	−19905.1	−0.644871
$985$	0	0
$986$	2937.40	0.0948741
$987$	− 7576.24i	− 0.244331i
$988$	148591.i	4.78473i
$989$	6052.95	0.194613
$990$	0	0
$991$	−53441.5	−1.71304	−0.856522	−	0.516111i	$-0.827379\pi$
$991$	−53441.5	−1.71304	−0.856522	+	0.516111i	$0.827379\pi$
$992$	329.671i	0.0105515i
$993$	17115.1i	0.546959i
$994$	−16945.1	−0.540710
$995$	0	0
$996$	−7861.74	−0.250109
$997$	− 37919.3i	− 1.20453i	−0.798296	−	0.602266i	$-0.794265\pi$
$997$	− 37919.3i	− 1.20453i	0.798296	−	0.602266i	$-0.205735\pi$
$998$	− 29063.4i	− 0.921829i
$999$	−19335.1	−0.612348

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.b.e.99.6		6
5.2	odd	4		35.4.a.c.1.1	✓	3
5.3	odd	4		175.4.a.f.1.3		3
5.4	even	2	inner	175.4.b.e.99.1		6
15.2	even	4		315.4.a.p.1.3		3
15.8	even	4		1575.4.a.ba.1.1		3
20.7	even	4		560.4.a.u.1.3		3
35.2	odd	12		245.4.e.m.116.3		6
35.12	even	12		245.4.e.n.116.3		6
35.13	even	4		1225.4.a.y.1.3		3
35.17	even	12		245.4.e.n.226.3		6
35.27	even	4		245.4.a.l.1.1		3
35.32	odd	12		245.4.e.m.226.3		6
40.27	even	4		2240.4.a.bv.1.1		3
40.37	odd	4		2240.4.a.bt.1.3		3
105.62	odd	4		2205.4.a.bm.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.c.1.1	✓	3	5.2	odd	4
175.4.a.f.1.3		3	5.3	odd	4
175.4.b.e.99.1		6	5.4	even	2	inner
175.4.b.e.99.6		6	1.1	even	1	trivial
245.4.a.l.1.1		3	35.27	even	4
245.4.e.m.116.3		6	35.2	odd	12
245.4.e.m.226.3		6	35.32	odd	12
245.4.e.n.116.3		6	35.12	even	12
245.4.e.n.226.3		6	35.17	even	12
315.4.a.p.1.3		3	15.2	even	4
560.4.a.u.1.3		3	20.7	even	4
1225.4.a.y.1.3		3	35.13	even	4
1575.4.a.ba.1.1		3	15.8	even	4
2205.4.a.bm.1.3		3	105.62	odd	4
2240.4.a.bt.1.3		3	40.37	odd	4
2240.4.a.bv.1.1		3	40.27	even	4

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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.62456i q^{2} -8.38660i q^{3} -13.3866 q^{4} +38.7844 q^{6} -7.00000i q^{7} -24.9107i q^{8} -43.3350 q^{9} -30.1117 q^{11} +112.268i q^{12} +88.9295i q^{13} +32.3720 q^{14} +8.10818 q^{16} +4.73699i q^{17} -200.405i q^{18} -124.818 q^{19} -58.7062 q^{21} -139.253i q^{22} +20.2680i q^{23} -208.916 q^{24} -411.260 q^{26} +136.995i q^{27} +93.7062i q^{28} -134.088 q^{29} -2.03767 q^{31} -161.788i q^{32} +252.534i q^{33} -21.9065 q^{34} +580.108 q^{36} +141.137i q^{37} -577.228i q^{38} +745.816 q^{39} +95.2784 q^{41} -271.490i q^{42} -298.646i q^{43} +403.093 q^{44} -93.7305 q^{46} +129.054i q^{47} -68.0000i q^{48} -49.0000 q^{49} +39.7272 q^{51} -1190.46i q^{52} +388.429i q^{53} -633.542 q^{54} -174.375 q^{56} +1046.80i q^{57} -620.098i q^{58} -838.501 q^{59} +389.422 q^{61} -9.42333i q^{62} +303.345i q^{63} +813.067 q^{64} -1167.86 q^{66} -697.794i q^{67} -63.4122i q^{68} +169.979 q^{69} -523.450 q^{71} +1079.50i q^{72} +66.4684i q^{73} -652.699 q^{74} +1670.89 q^{76} +210.782i q^{77} +3449.07i q^{78} +526.982 q^{79} -21.1236 q^{81} +440.621i q^{82} +70.0265i q^{83} +785.876 q^{84} +1381.11 q^{86} +1124.54i q^{87} +750.101i q^{88} +9.27925 q^{89} +622.506 q^{91} -271.319i q^{92} +17.0891i q^{93} -596.817 q^{94} -1356.85 q^{96} +4.19493i q^{97} -226.604i q^{98} +1304.89 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 26 q^{4} + 48 q^{6} - 162 q^{9} - 148 q^{11} + 42 q^{14} - 158 q^{16} - 336 q^{19} + 28 q^{21} - 840 q^{24} - 892 q^{26} - 664 q^{29} + 640 q^{31} - 1164 q^{34} + 362 q^{36} + 1964 q^{39} + 724 q^{41}+ \cdots + 6976 q^{99}+O(q^{100}) \) 6 * q - 26 * q^4 + 48 * q^6 - 162 * q^9 - 148 * q^11 + 42 * q^14 - 158 * q^16 - 336 * q^19 + 28 * q^21 - 840 * q^24 - 892 * q^26 - 664 * q^29 + 640 * q^31 - 1164 * q^34 + 362 * q^36 + 1964 * q^39 + 724 * q^41 + 528 * q^44 - 672 * q^46 - 294 * q^49 - 2356 * q^51 + 360 * q^54 - 210 * q^56 + 360 * q^59 + 2444 * q^61 + 782 * q^64 - 1064 * q^66 + 1432 * q^69 - 272 * q^71 - 1004 * q^74 + 3592 * q^76 + 2068 * q^79 + 4566 * q^81 + 2128 * q^84 + 1528 * q^86 - 484 * q^89 + 616 * q^91 + 2216 * q^94 - 6312 * q^96 + 6976 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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