LMFDB - Embedded newform 875.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [875,2,Mod(1,875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(875, 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("875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 875 = 5^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	875.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:	$6.98691017686$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1241125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} - 2x^{3} + 11x^{2} + 7x + 1 $ x^6 - 7x^4 - 2x^3 + 11x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.218811$ of defining polynomial
Character	$\chi$	$=$	875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.20648 q^{2} -2.49891 q^{3} +2.86854 q^{4} +5.51379 q^{6} +1.00000 q^{7} -1.91642 q^{8} +3.24455 q^{9} -2.20000 q^{11} -7.16823 q^{12} -5.51450 q^{13} -2.20648 q^{14} -1.50855 q^{16} +1.46051 q^{17} -7.15904 q^{18} +2.58741 q^{19} -2.49891 q^{21} +4.85426 q^{22} +8.52954 q^{23} +4.78897 q^{24} +12.1676 q^{26} -0.611121 q^{27} +2.86854 q^{28} +8.01302 q^{29} -5.09498 q^{31} +7.16141 q^{32} +5.49761 q^{33} -3.22259 q^{34} +9.30715 q^{36} -1.35264 q^{37} -5.70907 q^{38} +13.7802 q^{39} -3.94031 q^{41} +5.51379 q^{42} -8.29546 q^{43} -6.31080 q^{44} -18.8202 q^{46} +8.60911 q^{47} +3.76972 q^{48} +1.00000 q^{49} -3.64969 q^{51} -15.8186 q^{52} -4.88674 q^{53} +1.34842 q^{54} -1.91642 q^{56} -6.46571 q^{57} -17.6806 q^{58} -14.7225 q^{59} +2.64108 q^{61} +11.2420 q^{62} +3.24455 q^{63} -12.7844 q^{64} -12.1304 q^{66} +8.63974 q^{67} +4.18954 q^{68} -21.3145 q^{69} +0.518346 q^{71} -6.21794 q^{72} -11.8032 q^{73} +2.98458 q^{74} +7.42211 q^{76} -2.20000 q^{77} -30.4058 q^{78} +4.53964 q^{79} -8.20653 q^{81} +8.69420 q^{82} +11.3883 q^{83} -7.16823 q^{84} +18.3037 q^{86} -20.0238 q^{87} +4.21613 q^{88} +7.74549 q^{89} -5.51450 q^{91} +24.4673 q^{92} +12.7319 q^{93} -18.9958 q^{94} -17.8957 q^{96} -9.25710 q^{97} -2.20648 q^{98} -7.13803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - 5 q^{3} + 4 q^{4} - 2 q^{6} + 6 q^{7} - 12 q^{8} + 7 q^{9} + q^{11} - 11 q^{12} - 22 q^{13} - 2 q^{14} + 8 q^{16} - 4 q^{17} + 14 q^{18} - 17 q^{19} - 5 q^{21} - 9 q^{22} - 3 q^{23} + 7 q^{24}+ \cdots + 12 q^{99}+O(q^{100}) $ 6 * q - 2 * q^2 - 5 * q^3 + 4 * q^4 - 2 * q^6 + 6 * q^7 - 12 * q^8 + 7 * q^9 + q^11 - 11 * q^12 - 22 * q^13 - 2 * q^14 + 8 * q^16 - 4 * q^17 + 14 * q^18 - 17 * q^19 - 5 * q^21 - 9 * q^22 - 3 * q^23 + 7 * q^24 + 16 * q^26 - 26 * q^27 + 4 * q^28 + 7 * q^29 - 11 * q^31 - q^32 - 21 * q^33 - 4 * q^34 + 10 * q^36 - 33 * q^37 + 15 * q^39 + 3 * q^41 - 2 * q^42 - 2 * q^43 + 26 * q^44 - 8 * q^46 + 6 * q^47 + 20 * q^48 + 6 * q^49 + q^51 - 47 * q^52 - 33 * q^53 - 26 * q^54 - 12 * q^56 + 3 * q^57 - 43 * q^58 - 21 * q^59 - 19 * q^61 + 37 * q^62 + 7 * q^63 - 22 * q^64 - 35 * q^66 - 2 * q^67 + 30 * q^68 - 2 * q^69 + q^71 + 10 * q^72 - 16 * q^73 - 29 * q^74 - 4 * q^76 + q^77 + q^78 + 22 * q^79 + 14 * q^81 + 13 * q^82 - 3 * q^83 - 11 * q^84 + 23 * q^86 - 28 * q^88 - 8 * q^89 - 22 * q^91 + 2 * q^92 + 16 * q^93 - 26 * q^94 - 4 * q^96 - 16 * q^97 - 2 * q^98 + 12 * 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 $

$2$	−2.20648	−1.56022	−0.780108	−	0.625645i	$-0.784836\pi$
$2$	−2.20648	−1.56022	−0.780108	+	0.625645i	$0.784836\pi$
$3$	−2.49891	−1.44275	−0.721373	−	0.692546i	$-0.756489\pi$
$3$	−2.49891	−1.44275	−0.721373	+	0.692546i	$0.756489\pi$
$4$	2.86854	1.43427
$5$	0	0
$5$	0	0
$6$	5.51379	2.25100
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.91642	−0.677557
$9$	3.24455	1.08152
$10$	0	0
$11$	−2.20000	−0.663326	−0.331663	−	0.943398i	$-0.607610\pi$
$11$	−2.20000	−0.663326	−0.331663	+	0.943398i	$0.607610\pi$
$12$	−7.16823	−2.06929
$13$	−5.51450	−1.52945	−0.764724	−	0.644359i	$-0.777124\pi$
$13$	−5.51450	−1.52945	−0.764724	+	0.644359i	$0.777124\pi$
$14$	−2.20648	−0.589706
$15$	0	0
$16$	−1.50855	−0.377136
$17$	1.46051	0.354226	0.177113	−	0.984191i	$-0.443324\pi$
$17$	1.46051	0.354226	0.177113	+	0.984191i	$0.443324\pi$
$18$	−7.15904	−1.68740
$19$	2.58741	0.593593	0.296797	−	0.954941i	$-0.404082\pi$
$19$	2.58741	0.593593	0.296797	+	0.954941i	$0.404082\pi$
$20$	0	0
$21$	−2.49891	−0.545307
$22$	4.85426	1.03493
$23$	8.52954	1.77853	0.889265	−	0.457391i	$-0.151216\pi$
$23$	8.52954	1.77853	0.889265	+	0.457391i	$0.151216\pi$
$24$	4.78897	0.977544
$25$	0	0
$26$	12.1676	2.38627
$27$	−0.611121	−0.117610
$28$	2.86854	0.542104
$29$	8.01302	1.48798	0.743990	−	0.668190i	$-0.232931\pi$
$29$	8.01302	1.48798	0.743990	+	0.668190i	$0.232931\pi$
$30$	0	0
$31$	−5.09498	−0.915085	−0.457542	−	0.889188i	$-0.651270\pi$
$31$	−5.09498	−0.915085	−0.457542	+	0.889188i	$0.651270\pi$
$32$	7.16141	1.26597
$33$	5.49761	0.957011
$34$	−3.22259	−0.552669
$35$	0	0
$36$	9.30715	1.55119
$37$	−1.35264	−0.222373	−0.111187	−	0.993800i	$-0.535465\pi$
$37$	−1.35264	−0.222373	−0.111187	+	0.993800i	$0.535465\pi$
$38$	−5.70907	−0.926133
$39$	13.7802	2.20660
$40$	0	0
$41$	−3.94031	−0.615373	−0.307686	−	0.951488i	$-0.599555\pi$
$41$	−3.94031	−0.615373	−0.307686	+	0.951488i	$0.599555\pi$
$42$	5.51379	0.850796
$43$	−8.29546	−1.26505	−0.632523	−	0.774542i	$-0.717980\pi$
$43$	−8.29546	−1.26505	−0.632523	+	0.774542i	$0.717980\pi$
$44$	−6.31080	−0.951389
$45$	0	0
$46$	−18.8202	−2.77489
$47$	8.60911	1.25577	0.627884	−	0.778307i	$-0.283921\pi$
$47$	8.60911	1.25577	0.627884	+	0.778307i	$0.283921\pi$
$48$	3.76972	0.544112
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.64969	−0.511059
$52$	−15.8186	−2.19364
$53$	−4.88674	−0.671245	−0.335622	−	0.941997i	$-0.608947\pi$
$53$	−4.88674	−0.671245	−0.335622	+	0.941997i	$0.608947\pi$
$54$	1.34842	0.183497
$55$	0	0
$56$	−1.91642	−0.256093
$57$	−6.46571	−0.856405
$58$	−17.6806	−2.32157
$59$	−14.7225	−1.91671	−0.958355	−	0.285579i	$-0.907814\pi$
$59$	−14.7225	−1.91671	−0.958355	+	0.285579i	$0.907814\pi$
$60$	0	0
$61$	2.64108	0.338156	0.169078	−	0.985603i	$-0.445921\pi$
$61$	2.64108	0.338156	0.169078	+	0.985603i	$0.445921\pi$
$62$	11.2420	1.42773
$63$	3.24455	0.408775
$64$	−12.7844	−1.59805
$65$	0	0
$66$	−12.1304	−1.49314
$67$	8.63974	1.05551	0.527756	−	0.849396i	$-0.323033\pi$
$67$	8.63974	1.05551	0.527756	+	0.849396i	$0.323033\pi$
$68$	4.18954	0.508057
$69$	−21.3145	−2.56597
$70$	0	0
$71$	0.518346	0.0615164	0.0307582	−	0.999527i	$-0.490208\pi$
$71$	0.518346	0.0615164	0.0307582	+	0.999527i	$0.490208\pi$
$72$	−6.21794	−0.732791
$73$	−11.8032	−1.38146	−0.690730	−	0.723113i	$-0.742711\pi$
$73$	−11.8032	−1.38146	−0.690730	+	0.723113i	$0.742711\pi$
$74$	2.98458	0.346950
$75$	0	0
$76$	7.42211	0.851374
$77$	−2.20000	−0.250714
$78$	−30.4058	−3.44278
$79$	4.53964	0.510750	0.255375	−	0.966842i	$-0.417801\pi$
$79$	4.53964	0.510750	0.255375	+	0.966842i	$0.417801\pi$
$80$	0	0
$81$	−8.20653	−0.911836
$82$	8.69420	0.960114
$83$	11.3883	1.25003	0.625015	−	0.780612i	$-0.285093\pi$
$83$	11.3883	1.25003	0.625015	+	0.780612i	$0.285093\pi$
$84$	−7.16823	−0.782118
$85$	0	0
$86$	18.3037	1.97374
$87$	−20.0238	−2.14678
$88$	4.21613	0.449441
$89$	7.74549	0.821021	0.410510	−	0.911856i	$-0.365351\pi$
$89$	7.74549	0.821021	0.410510	+	0.911856i	$0.365351\pi$
$90$	0	0
$91$	−5.51450	−0.578077
$92$	24.4673	2.55090
$93$	12.7319	1.32024
$94$	−18.9958	−1.95927
$95$	0	0
$96$	−17.8957	−1.82648
$97$	−9.25710	−0.939916	−0.469958	−	0.882689i	$-0.655731\pi$
$97$	−9.25710	−0.939916	−0.469958	+	0.882689i	$0.655731\pi$
$98$	−2.20648	−0.222888
$99$	−7.13803	−0.717399
$100$	0	0
$101$	−8.29649	−0.825532	−0.412766	−	0.910837i	$-0.635437\pi$
$101$	−8.29649	−0.825532	−0.412766	+	0.910837i	$0.635437\pi$
$102$	8.05296	0.797362
$103$	−6.23818	−0.614666	−0.307333	−	0.951602i	$-0.599437\pi$
$103$	−6.23818	−0.614666	−0.307333	+	0.951602i	$0.599437\pi$
$104$	10.5681	1.03629
$105$	0	0
$106$	10.7825	1.04729
$107$	−18.9353	−1.83054	−0.915271	−	0.402838i	$-0.868024\pi$
$107$	−18.9353	−1.83054	−0.915271	+	0.402838i	$0.868024\pi$
$108$	−1.75303	−0.168685
$109$	−0.278409	−0.0266668	−0.0133334	−	0.999911i	$-0.504244\pi$
$109$	−0.278409	−0.0266668	−0.0133334	+	0.999911i	$0.504244\pi$
$110$	0	0
$111$	3.38014	0.320829
$112$	−1.50855	−0.142544
$113$	−11.6014	−1.09137	−0.545684	−	0.837991i	$-0.683730\pi$
$113$	−11.6014	−1.09137	−0.545684	+	0.837991i	$0.683730\pi$
$114$	14.2665	1.33618
$115$	0	0
$116$	22.9857	2.13417
$117$	−17.8921	−1.65413
$118$	32.4849	2.99048
$119$	1.46051	0.133885
$120$	0	0
$121$	−6.15999	−0.559999
$122$	−5.82748	−0.527595
$123$	9.84648	0.887827
$124$	−14.6152	−1.31248
$125$	0	0
$126$	−7.15904	−0.637778
$127$	−1.65648	−0.146988	−0.0734942	−	0.997296i	$-0.523415\pi$
$127$	−1.65648	−0.146988	−0.0734942	+	0.997296i	$0.523415\pi$
$128$	13.8857	1.22733
$129$	20.7296	1.82514
$130$	0	0
$131$	7.06316	0.617112	0.308556	−	0.951206i	$-0.400154\pi$
$131$	7.06316	0.617112	0.308556	+	0.951206i	$0.400154\pi$
$132$	15.7701	1.37261
$133$	2.58741	0.224357
$134$	−19.0634	−1.64683
$135$	0	0
$136$	−2.79896	−0.240009
$137$	−12.7607	−1.09022	−0.545112	−	0.838363i	$-0.683513\pi$
$137$	−12.7607	−1.09022	−0.545112	+	0.838363i	$0.683513\pi$
$138$	47.0301	4.00347
$139$	−16.3872	−1.38994	−0.694970	−	0.719038i	$-0.744583\pi$
$139$	−16.3872	−1.38994	−0.694970	+	0.719038i	$0.744583\pi$
$140$	0	0
$141$	−21.5134	−1.81176
$142$	−1.14372	−0.0959788
$143$	12.1319	1.01452
$144$	−4.89456	−0.407880
$145$	0	0
$146$	26.0435	2.15538
$147$	−2.49891	−0.206107
$148$	−3.88012	−0.318944
$149$	−11.8198	−0.968318	−0.484159	−	0.874980i	$-0.660874\pi$
$149$	−11.8198	−0.968318	−0.484159	+	0.874980i	$0.660874\pi$
$150$	0	0
$151$	23.9836	1.95175	0.975877	−	0.218321i	$-0.0700581\pi$
$151$	23.9836	1.95175	0.975877	+	0.218321i	$0.0700581\pi$
$152$	−4.95857	−0.402193
$153$	4.73871	0.383102
$154$	4.85426	0.391167
$155$	0	0
$156$	39.5292	3.16487
$157$	−14.5670	−1.16257	−0.581286	−	0.813699i	$-0.697450\pi$
$157$	−14.5670	−1.16257	−0.581286	+	0.813699i	$0.697450\pi$
$158$	−10.0166	−0.796880
$159$	12.2115	0.968436
$160$	0	0
$161$	8.52954	0.672222
$162$	18.1075	1.42266
$163$	2.50219	0.195986	0.0979932	−	0.995187i	$-0.468758\pi$
$163$	2.50219	0.195986	0.0979932	+	0.995187i	$0.468758\pi$
$164$	−11.3029	−0.882612
$165$	0	0
$166$	−25.1281	−1.95032
$167$	−3.92813	−0.303968	−0.151984	−	0.988383i	$-0.548566\pi$
$167$	−3.92813	−0.303968	−0.151984	+	0.988383i	$0.548566\pi$
$168$	4.78897	0.369477
$169$	17.4097	1.33921
$170$	0	0
$171$	8.39500	0.641982
$172$	−23.7959	−1.81442
$173$	−21.7981	−1.65728	−0.828638	−	0.559784i	$-0.810884\pi$
$173$	−21.7981	−1.65728	−0.828638	+	0.559784i	$0.810884\pi$
$174$	44.1821	3.34944
$175$	0	0
$176$	3.31880	0.250164
$177$	36.7903	2.76533
$178$	−17.0903	−1.28097
$179$	−19.3974	−1.44983	−0.724914	−	0.688840i	$-0.758120\pi$
$179$	−19.3974	−1.44983	−0.724914	+	0.688840i	$0.758120\pi$
$180$	0	0
$181$	−24.0082	−1.78451	−0.892257	−	0.451527i	$-0.850879\pi$
$181$	−24.0082	−1.78451	−0.892257	+	0.451527i	$0.850879\pi$
$182$	12.1676	0.901924
$183$	−6.59982	−0.487873
$184$	−16.3462	−1.20506
$185$	0	0
$186$	−28.0926	−2.05985
$187$	−3.21313	−0.234967
$188$	24.6956	1.80111
$189$	−0.611121	−0.0444525
$190$	0	0
$191$	27.3408	1.97831	0.989154	−	0.146882i	$-0.0469237\pi$
$191$	27.3408	1.97831	0.989154	+	0.146882i	$0.0469237\pi$
$192$	31.9471	2.30558
$193$	3.42208	0.246326	0.123163	−	0.992386i	$-0.460696\pi$
$193$	3.42208	0.246326	0.123163	+	0.992386i	$0.460696\pi$
$194$	20.4256	1.46647
$195$	0	0
$196$	2.86854	0.204896
$197$	3.95530	0.281803	0.140902	−	0.990024i	$-0.455000\pi$
$197$	3.95530	0.281803	0.140902	+	0.990024i	$0.455000\pi$
$198$	15.7499	1.11930
$199$	0.587532	0.0416490	0.0208245	−	0.999783i	$-0.493371\pi$
$199$	0.587532	0.0416490	0.0208245	+	0.999783i	$0.493371\pi$
$200$	0	0
$201$	−21.5899	−1.52284
$202$	18.3060	1.28801
$203$	8.01302	0.562404
$204$	−10.4693	−0.732997
$205$	0	0
$206$	13.7644	0.959011
$207$	27.6745	1.92351
$208$	8.31887	0.576810
$209$	−5.69232	−0.393746
$210$	0	0
$211$	12.7869	0.880290	0.440145	−	0.897927i	$-0.354927\pi$
$211$	12.7869	0.880290	0.440145	+	0.897927i	$0.354927\pi$
$212$	−14.0178	−0.962747
$213$	−1.29530	−0.0887526
$214$	41.7803	2.85604
$215$	0	0
$216$	1.17117	0.0796877
$217$	−5.09498	−0.345870
$218$	0.614303	0.0416059
$219$	29.4952	1.99310
$220$	0	0
$221$	−8.05399	−0.541770
$222$	−7.45820	−0.500562
$223$	−14.2942	−0.957208	−0.478604	−	0.878031i	$-0.658857\pi$
$223$	−14.2942	−0.957208	−0.478604	+	0.878031i	$0.658857\pi$
$224$	7.16141	0.478492
$225$	0	0
$226$	25.5982	1.70277
$227$	4.20386	0.279020	0.139510	−	0.990221i	$-0.455447\pi$
$227$	4.20386	0.279020	0.139510	+	0.990221i	$0.455447\pi$
$228$	−18.5472	−1.22832
$229$	−1.47983	−0.0977900	−0.0488950	−	0.998804i	$-0.515570\pi$
$229$	−1.47983	−0.0977900	−0.0488950	+	0.998804i	$0.515570\pi$
$230$	0	0
$231$	5.49761	0.361716
$232$	−15.3563	−1.00819
$233$	−5.38055	−0.352491	−0.176246	−	0.984346i	$-0.556395\pi$
$233$	−5.38055	−0.352491	−0.176246	+	0.984346i	$0.556395\pi$
$234$	39.4785	2.58079
$235$	0	0
$236$	−42.2322	−2.74908
$237$	−11.3442	−0.736883
$238$	−3.22259	−0.208889
$239$	13.8530	0.896074	0.448037	−	0.894015i	$-0.352123\pi$
$239$	13.8530	0.896074	0.448037	+	0.894015i	$0.352123\pi$
$240$	0	0
$241$	−13.7783	−0.887538	−0.443769	−	0.896141i	$-0.646359\pi$
$241$	−13.7783	−0.887538	−0.443769	+	0.896141i	$0.646359\pi$
$242$	13.5919	0.873719
$243$	22.3407	1.43316
$244$	7.57605	0.485007
$245$	0	0
$246$	−21.7260	−1.38520
$247$	−14.2683	−0.907869
$248$	9.76412	0.620022
$249$	−28.4584	−1.80348
$250$	0	0
$251$	6.65792	0.420244	0.210122	−	0.977675i	$-0.432614\pi$
$251$	6.65792	0.420244	0.210122	+	0.977675i	$0.432614\pi$
$252$	9.30715	0.586295
$253$	−18.7650	−1.17975
$254$	3.65498	0.229334
$255$	0	0
$256$	−5.06963	−0.316852
$257$	20.8599	1.30121	0.650603	−	0.759418i	$-0.274516\pi$
$257$	20.8599	1.30121	0.650603	+	0.759418i	$0.274516\pi$
$258$	−45.7394	−2.84761
$259$	−1.35264	−0.0840493
$260$	0	0
$261$	25.9987	1.60928
$262$	−15.5847	−0.962827
$263$	−5.32090	−0.328101	−0.164050	−	0.986452i	$-0.552456\pi$
$263$	−5.32090	−0.328101	−0.164050	+	0.986452i	$0.552456\pi$
$264$	−10.5357	−0.648430
$265$	0	0
$266$	−5.70907	−0.350045
$267$	−19.3553	−1.18452
$268$	24.7835	1.51389
$269$	−12.9333	−0.788558	−0.394279	−	0.918991i	$-0.629006\pi$
$269$	−12.9333	−0.788558	−0.394279	+	0.918991i	$0.629006\pi$
$270$	0	0
$271$	−25.6222	−1.55643	−0.778217	−	0.627995i	$-0.783876\pi$
$271$	−25.6222	−1.55643	−0.778217	+	0.627995i	$0.783876\pi$
$272$	−2.20325	−0.133592
$273$	13.7802	0.834018
$274$	28.1563	1.70098
$275$	0	0
$276$	−61.1417	−3.68030
$277$	−0.458679	−0.0275593	−0.0137797	−	0.999905i	$-0.504386\pi$
$277$	−0.458679	−0.0275593	−0.0137797	+	0.999905i	$0.504386\pi$
$278$	36.1579	2.16861
$279$	−16.5309	−0.989681
$280$	0	0
$281$	−3.41984	−0.204011	−0.102005	−	0.994784i	$-0.532526\pi$
$281$	−3.41984	−0.204011	−0.102005	+	0.994784i	$0.532526\pi$
$282$	47.4688	2.82673
$283$	1.57036	0.0933484	0.0466742	−	0.998910i	$-0.485138\pi$
$283$	1.57036	0.0933484	0.0466742	+	0.998910i	$0.485138\pi$
$284$	1.48690	0.0882312
$285$	0	0
$286$	−26.7688	−1.58287
$287$	−3.94031	−0.232589
$288$	23.2356	1.36917
$289$	−14.8669	−0.874524
$290$	0	0
$291$	23.1327	1.35606
$292$	−33.8580	−1.98139
$293$	−5.05836	−0.295513	−0.147756	−	0.989024i	$-0.547205\pi$
$293$	−5.05836	−0.295513	−0.147756	+	0.989024i	$0.547205\pi$
$294$	5.51379	0.321571
$295$	0	0
$296$	2.59224	0.150671
$297$	1.34447	0.0780139
$298$	26.0802	1.51079
$299$	−47.0361	−2.72017
$300$	0	0
$301$	−8.29546	−0.478142
$302$	−52.9192	−3.04516
$303$	20.7322	1.19103
$304$	−3.90323	−0.223866
$305$	0	0
$306$	−10.4559	−0.597722
$307$	−6.48905	−0.370350	−0.185175	−	0.982706i	$-0.559285\pi$
$307$	−6.48905	−0.370350	−0.185175	+	0.982706i	$0.559285\pi$
$308$	−6.31080	−0.359591
$309$	15.5886	0.886807
$310$	0	0
$311$	6.87065	0.389599	0.194799	−	0.980843i	$-0.437594\pi$
$311$	6.87065	0.389599	0.194799	+	0.980843i	$0.437594\pi$
$312$	−26.4088	−1.49510
$313$	7.45113	0.421163	0.210581	−	0.977576i	$-0.432464\pi$
$313$	7.45113	0.421163	0.210581	+	0.977576i	$0.432464\pi$
$314$	32.1418	1.81386
$315$	0	0
$316$	13.0222	0.732554
$317$	6.89000	0.386981	0.193490	−	0.981102i	$-0.438019\pi$
$317$	6.89000	0.386981	0.193490	+	0.981102i	$0.438019\pi$
$318$	−26.9444	−1.51097
$319$	−17.6287	−0.987016
$320$	0	0
$321$	47.3176	2.64101
$322$	−18.8202	−1.04881
$323$	3.77895	0.210266
$324$	−23.5408	−1.30782
$325$	0	0
$326$	−5.52102	−0.305781
$327$	0.695719	0.0384734
$328$	7.55129	0.416950
$329$	8.60911	0.474636
$330$	0	0
$331$	−18.6329	−1.02416	−0.512078	−	0.858939i	$-0.671124\pi$
$331$	−18.6329	−1.02416	−0.512078	+	0.858939i	$0.671124\pi$
$332$	32.6679	1.79288
$333$	−4.38873	−0.240501
$334$	8.66733	0.474255
$335$	0	0
$336$	3.76972	0.205655
$337$	−4.77614	−0.260173	−0.130086	−	0.991503i	$-0.541525\pi$
$337$	−4.77614	−0.260173	−0.130086	+	0.991503i	$0.541525\pi$
$338$	−38.4141	−2.08945
$339$	28.9909	1.57457
$340$	0	0
$341$	11.2090	0.606999
$342$	−18.5234	−1.00163
$343$	1.00000	0.0539949
$344$	15.8976	0.857141
$345$	0	0
$346$	48.0970	2.58571
$347$	27.0162	1.45030	0.725152	−	0.688589i	$-0.241770\pi$
$347$	27.0162	1.45030	0.725152	+	0.688589i	$0.241770\pi$
$348$	−57.4392	−3.07907
$349$	15.1923	0.813227	0.406613	−	0.913600i	$-0.366710\pi$
$349$	15.1923	0.813227	0.406613	+	0.913600i	$0.366710\pi$
$350$	0	0
$351$	3.37003	0.179879
$352$	−15.7551	−0.839751
$353$	17.0965	0.909956	0.454978	−	0.890503i	$-0.349647\pi$
$353$	17.0965	0.909956	0.454978	+	0.890503i	$0.349647\pi$
$354$	−81.1770	−4.31451
$355$	0	0
$356$	22.2183	1.17757
$357$	−3.64969	−0.193162
$358$	42.7999	2.26204
$359$	19.8353	1.04687	0.523434	−	0.852066i	$-0.324651\pi$
$359$	19.8353	1.04687	0.523434	+	0.852066i	$0.324651\pi$
$360$	0	0
$361$	−12.3053	−0.647647
$362$	52.9735	2.78423
$363$	15.3933	0.807937
$364$	−15.8186	−0.829119
$365$	0	0
$366$	14.5624	0.761187
$367$	5.79323	0.302404	0.151202	−	0.988503i	$-0.451686\pi$
$367$	5.79323	0.302404	0.151202	+	0.988503i	$0.451686\pi$
$368$	−12.8672	−0.670749
$369$	−12.7845	−0.665537
$370$	0	0
$371$	−4.88674	−0.253707
$372$	36.5220	1.89358
$373$	−20.0950	−1.04048	−0.520239	−	0.854021i	$-0.674157\pi$
$373$	−20.0950	−1.04048	−0.520239	+	0.854021i	$0.674157\pi$
$374$	7.08970	0.366600
$375$	0	0
$376$	−16.4987	−0.850855
$377$	−44.1878	−2.27579
$378$	1.34842	0.0693555
$379$	20.6237	1.05937	0.529684	−	0.848195i	$-0.322310\pi$
$379$	20.6237	1.05937	0.529684	+	0.848195i	$0.322310\pi$
$380$	0	0
$381$	4.13939	0.212067
$382$	−60.3268	−3.08659
$383$	−33.1106	−1.69187	−0.845936	−	0.533285i	$-0.820957\pi$
$383$	−33.1106	−1.69187	−0.845936	+	0.533285i	$0.820957\pi$
$384$	−34.6991	−1.77073
$385$	0	0
$386$	−7.55073	−0.384322
$387$	−26.9151	−1.36817
$388$	−26.5544	−1.34810
$389$	1.19542	0.0606101	0.0303050	−	0.999541i	$-0.490352\pi$
$389$	1.19542	0.0606101	0.0303050	+	0.999541i	$0.490352\pi$
$390$	0	0
$391$	12.4575	0.630002
$392$	−1.91642	−0.0967939
$393$	−17.6502	−0.890336
$394$	−8.72727	−0.439674
$395$	0	0
$396$	−20.4757	−1.02895
$397$	21.0786	1.05790	0.528952	−	0.848651i	$-0.322585\pi$
$397$	21.0786	1.05790	0.528952	+	0.848651i	$0.322585\pi$
$398$	−1.29638	−0.0649814
$399$	−6.46571	−0.323691
$400$	0	0
$401$	−12.4069	−0.619573	−0.309786	−	0.950806i	$-0.600258\pi$
$401$	−12.4069	−0.619573	−0.309786	+	0.950806i	$0.600258\pi$
$402$	47.6377	2.37595
$403$	28.0962	1.39957
$404$	−23.7988	−1.18404
$405$	0	0
$406$	−17.6806	−0.877471
$407$	2.97582	0.147506
$408$	6.99434	0.346272
$409$	−0.519751	−0.0257001	−0.0128500	−	0.999917i	$-0.504090\pi$
$409$	−0.519751	−0.0257001	−0.0128500	+	0.999917i	$0.504090\pi$
$410$	0	0
$411$	31.8879	1.57292
$412$	−17.8945	−0.881598
$413$	−14.7225	−0.724448
$414$	−61.0633	−3.00110
$415$	0	0
$416$	−39.4916	−1.93624
$417$	40.9501	2.00533
$418$	12.5600	0.614328
$419$	−8.52079	−0.416268	−0.208134	−	0.978100i	$-0.566739\pi$
$419$	−8.52079	−0.416268	−0.208134	+	0.978100i	$0.566739\pi$
$420$	0	0
$421$	31.2241	1.52177	0.760886	−	0.648886i	$-0.224765\pi$
$421$	31.2241	1.52177	0.760886	+	0.648886i	$0.224765\pi$
$422$	−28.2141	−1.37344
$423$	27.9327	1.35814
$424$	9.36505	0.454807
$425$	0	0
$426$	2.85805	0.138473
$427$	2.64108	0.127811
$428$	−54.3167	−2.62550
$429$	−30.3166	−1.46370
$430$	0	0
$431$	−26.8792	−1.29472	−0.647362	−	0.762182i	$-0.724128\pi$
$431$	−26.8792	−1.29472	−0.647362	+	0.762182i	$0.724128\pi$
$432$	0.921904	0.0443551
$433$	9.60392	0.461535	0.230767	−	0.973009i	$-0.425876\pi$
$433$	9.60392	0.461535	0.230767	+	0.973009i	$0.425876\pi$
$434$	11.2420	0.539631
$435$	0	0
$436$	−0.798628	−0.0382474
$437$	22.0694	1.05572
$438$	−65.0804	−3.10966
$439$	−5.50040	−0.262520	−0.131260	−	0.991348i	$-0.541902\pi$
$439$	−5.50040	−0.262520	−0.131260	+	0.991348i	$0.541902\pi$
$440$	0	0
$441$	3.24455	0.154503
$442$	17.7710	0.845278
$443$	−24.6319	−1.17029	−0.585147	−	0.810927i	$-0.698963\pi$
$443$	−24.6319	−1.17029	−0.585147	+	0.810927i	$0.698963\pi$
$444$	9.69607	0.460155
$445$	0	0
$446$	31.5397	1.49345
$447$	29.5367	1.39704
$448$	−12.7844	−0.604007
$449$	−9.58488	−0.452339	−0.226169	−	0.974088i	$-0.572620\pi$
$449$	−9.58488	−0.452339	−0.226169	+	0.974088i	$0.572620\pi$
$450$	0	0
$451$	8.66869	0.408193
$452$	−33.2791	−1.56532
$453$	−59.9328	−2.81589
$454$	−9.27572	−0.435331
$455$	0	0
$456$	12.3910	0.580263
$457$	34.0352	1.59210	0.796049	−	0.605232i	$-0.206920\pi$
$457$	34.0352	1.59210	0.796049	+	0.605232i	$0.206920\pi$
$458$	3.26521	0.152573
$459$	−0.892549	−0.0416606
$460$	0	0
$461$	−3.56421	−0.166002	−0.0830009	−	0.996549i	$-0.526450\pi$
$461$	−3.56421	−0.166002	−0.0830009	+	0.996549i	$0.526450\pi$
$462$	−12.1304	−0.564355
$463$	0.987927	0.0459129	0.0229564	−	0.999736i	$-0.492692\pi$
$463$	0.987927	0.0459129	0.0229564	+	0.999736i	$0.492692\pi$
$464$	−12.0880	−0.561172
$465$	0	0
$466$	11.8721	0.549962
$467$	−6.28063	−0.290633	−0.145317	−	0.989385i	$-0.546420\pi$
$467$	−6.28063	−0.290633	−0.145317	+	0.989385i	$0.546420\pi$
$468$	−51.3243	−2.37246
$469$	8.63974	0.398946
$470$	0	0
$471$	36.4016	1.67730
$472$	28.2146	1.29868
$473$	18.2500	0.839137
$474$	25.0306	1.14970
$475$	0	0
$476$	4.18954	0.192027
$477$	−15.8553	−0.725963
$478$	−30.5662	−1.39807
$479$	−4.84962	−0.221585	−0.110792	−	0.993844i	$-0.535339\pi$
$479$	−4.84962	−0.221585	−0.110792	+	0.993844i	$0.535339\pi$
$480$	0	0
$481$	7.45916	0.340108
$482$	30.4015	1.38475
$483$	−21.3145	−0.969845
$484$	−17.6702	−0.803190
$485$	0	0
$486$	−49.2944	−2.23604
$487$	−5.15932	−0.233791	−0.116896	−	0.993144i	$-0.537294\pi$
$487$	−5.15932	−0.233791	−0.116896	+	0.993144i	$0.537294\pi$
$488$	−5.06142	−0.229120
$489$	−6.25274	−0.282759
$490$	0	0
$491$	30.7290	1.38678	0.693391	−	0.720562i	$-0.256116\pi$
$491$	30.7290	1.38678	0.693391	+	0.720562i	$0.256116\pi$
$492$	28.2451	1.27339
$493$	11.7031	0.527082
$494$	31.4827	1.41647
$495$	0	0
$496$	7.68600	0.345112
$497$	0.518346	0.0232510
$498$	62.7928	2.81381
$499$	−34.2964	−1.53532	−0.767659	−	0.640859i	$-0.778578\pi$
$499$	−34.2964	−1.53532	−0.767659	+	0.640859i	$0.778578\pi$
$500$	0	0
$501$	9.81605	0.438549
$502$	−14.6906	−0.655672
$503$	19.4215	0.865960	0.432980	−	0.901404i	$-0.357462\pi$
$503$	19.4215	0.865960	0.432980	+	0.901404i	$0.357462\pi$
$504$	−6.21794	−0.276969
$505$	0	0
$506$	41.4046	1.84066
$507$	−43.5053	−1.93214
$508$	−4.75167	−0.210821
$509$	−3.66252	−0.162338	−0.0811691	−	0.996700i	$-0.525865\pi$
$509$	−3.66252	−0.162338	−0.0811691	+	0.996700i	$0.525865\pi$
$510$	0	0
$511$	−11.8032	−0.522143
$512$	−16.5853	−0.732975
$513$	−1.58122	−0.0698127
$514$	−46.0269	−2.03016
$515$	0	0
$516$	59.4638	2.61775
$517$	−18.9401	−0.832983
$518$	2.98458	0.131135
$519$	54.4714	2.39103
$520$	0	0
$521$	5.74690	0.251776	0.125888	−	0.992044i	$-0.459822\pi$
$521$	5.74690	0.251776	0.125888	+	0.992044i	$0.459822\pi$
$522$	−57.3655	−2.51082
$523$	−23.5769	−1.03095	−0.515473	−	0.856906i	$-0.672384\pi$
$523$	−23.5769	−1.03095	−0.515473	+	0.856906i	$0.672384\pi$
$524$	20.2610	0.885106
$525$	0	0
$526$	11.7405	0.511908
$527$	−7.44127	−0.324147
$528$	−8.29340	−0.360924
$529$	49.7530	2.16317
$530$	0	0
$531$	−47.7681	−2.07296
$532$	7.42211	0.321789
$533$	21.7288	0.941180
$534$	42.7070	1.84811
$535$	0	0
$536$	−16.5574	−0.715170
$537$	48.4723	2.09173
$538$	28.5371	1.23032
$539$	−2.20000	−0.0947608
$540$	0	0
$541$	−17.6997	−0.760969	−0.380484	−	0.924787i	$-0.624243\pi$
$541$	−17.6997	−0.760969	−0.380484	+	0.924787i	$0.624243\pi$
$542$	56.5347	2.42837
$543$	59.9943	2.57460
$544$	10.4593	0.448440
$545$	0	0
$546$	−30.4058	−1.30125
$547$	−10.7570	−0.459934	−0.229967	−	0.973198i	$-0.573862\pi$
$547$	−10.7570	−0.459934	−0.229967	+	0.973198i	$0.573862\pi$
$548$	−36.6047	−1.56368
$549$	8.56913	0.365721
$550$	0	0
$551$	20.7330	0.883255
$552$	40.8477	1.73859
$553$	4.53964	0.193045
$554$	1.01206	0.0429985
$555$	0	0
$556$	−47.0073	−1.99355
$557$	−11.9343	−0.505672	−0.252836	−	0.967509i	$-0.581363\pi$
$557$	−11.9343	−0.505672	−0.252836	+	0.967509i	$0.581363\pi$
$558$	36.4751	1.54412
$559$	45.7453	1.93482
$560$	0	0
$561$	8.02933	0.338998
$562$	7.54581	0.318301
$563$	−0.997184	−0.0420263	−0.0210132	−	0.999779i	$-0.506689\pi$
$563$	−0.997184	−0.0420263	−0.0210132	+	0.999779i	$0.506689\pi$
$564$	−61.7121	−2.59855
$565$	0	0
$566$	−3.46497	−0.145644
$567$	−8.20653	−0.344642
$568$	−0.993370	−0.0416809
$569$	−4.23527	−0.177552	−0.0887758	−	0.996052i	$-0.528295\pi$
$569$	−4.23527	−0.177552	−0.0887758	+	0.996052i	$0.528295\pi$
$570$	0	0
$571$	−7.62526	−0.319107	−0.159554	−	0.987189i	$-0.551006\pi$
$571$	−7.62526	−0.319107	−0.159554	+	0.987189i	$0.551006\pi$
$572$	34.8009	1.45510
$573$	−68.3221	−2.85420
$574$	8.69420	0.362889
$575$	0	0
$576$	−41.4797	−1.72832
$577$	−14.0812	−0.586208	−0.293104	−	0.956081i	$-0.594688\pi$
$577$	−14.0812	−0.586208	−0.293104	+	0.956081i	$0.594688\pi$
$578$	32.8035	1.36445
$579$	−8.55146	−0.355387
$580$	0	0
$581$	11.3883	0.472467
$582$	−51.0417	−2.11575
$583$	10.7508	0.445254
$584$	22.6199	0.936019
$585$	0	0
$586$	11.1612	0.461064
$587$	13.0536	0.538778	0.269389	−	0.963031i	$-0.413178\pi$
$587$	13.0536	0.538778	0.269389	+	0.963031i	$0.413178\pi$
$588$	−7.16823	−0.295613
$589$	−13.1828	−0.543188
$590$	0	0
$591$	−9.88393	−0.406571
$592$	2.04053	0.0838651
$593$	−13.9564	−0.573119	−0.286559	−	0.958062i	$-0.592512\pi$
$593$	−13.9564	−0.573119	−0.286559	+	0.958062i	$0.592512\pi$
$594$	−2.96654	−0.121719
$595$	0	0
$596$	−33.9057	−1.38883
$597$	−1.46819	−0.0600890
$598$	103.784	4.24405
$599$	23.9628	0.979095	0.489548	−	0.871977i	$-0.337162\pi$
$599$	23.9628	0.979095	0.489548	+	0.871977i	$0.337162\pi$
$600$	0	0
$601$	30.6762	1.25131	0.625655	−	0.780100i	$-0.284832\pi$
$601$	30.6762	1.25131	0.625655	+	0.780100i	$0.284832\pi$
$602$	18.3037	0.746005
$603$	28.0321	1.14156
$604$	68.7979	2.79935
$605$	0	0
$606$	−45.7451	−1.85827
$607$	−35.5284	−1.44205	−0.721027	−	0.692907i	$-0.756329\pi$
$607$	−35.5284	−1.44205	−0.721027	+	0.692907i	$0.756329\pi$
$608$	18.5295	0.751472
$609$	−20.0238	−0.811406
$610$	0	0
$611$	−47.4749	−1.92063
$612$	13.5932	0.549473
$613$	−5.02015	−0.202762	−0.101381	−	0.994848i	$-0.532326\pi$
$613$	−5.02015	−0.202762	−0.101381	+	0.994848i	$0.532326\pi$
$614$	14.3180	0.577826
$615$	0	0
$616$	4.21613	0.169873
$617$	−2.39753	−0.0965210	−0.0482605	−	0.998835i	$-0.515368\pi$
$617$	−2.39753	−0.0965210	−0.0482605	+	0.998835i	$0.515368\pi$
$618$	−34.3960	−1.38361
$619$	−31.5481	−1.26802	−0.634012	−	0.773323i	$-0.718593\pi$
$619$	−31.5481	−1.26802	−0.634012	+	0.773323i	$0.718593\pi$
$620$	0	0
$621$	−5.21258	−0.209174
$622$	−15.1599	−0.607858
$623$	7.74549	0.310317
$624$	−20.7881	−0.832191
$625$	0	0
$626$	−16.4408	−0.657105
$627$	14.2246	0.568075
$628$	−41.7861	−1.66745
$629$	−1.97555	−0.0787705
$630$	0	0
$631$	−8.79197	−0.350003	−0.175001	−	0.984568i	$-0.555993\pi$
$631$	−8.79197	−0.350003	−0.175001	+	0.984568i	$0.555993\pi$
$632$	−8.69987	−0.346062
$633$	−31.9534	−1.27004
$634$	−15.2026	−0.603774
$635$	0	0
$636$	35.0293	1.38900
$637$	−5.51450	−0.218492
$638$	38.8973	1.53996
$639$	1.68180	0.0665311
$640$	0	0
$641$	−47.2690	−1.86701	−0.933507	−	0.358558i	$-0.883268\pi$
$641$	−47.2690	−1.86701	−0.933507	+	0.358558i	$0.883268\pi$
$642$	−104.405	−4.12054
$643$	2.82283	0.111321	0.0556607	−	0.998450i	$-0.482273\pi$
$643$	2.82283	0.111321	0.0556607	+	0.998450i	$0.482273\pi$
$644$	24.4673	0.964148
$645$	0	0
$646$	−8.33816	−0.328061
$647$	−4.76624	−0.187380	−0.0936902	−	0.995601i	$-0.529866\pi$
$647$	−4.76624	−0.187380	−0.0936902	+	0.995601i	$0.529866\pi$
$648$	15.7272	0.617822
$649$	32.3896	1.27140
$650$	0	0
$651$	12.7319	0.499002
$652$	7.17763	0.281098
$653$	24.3840	0.954221	0.477110	−	0.878843i	$-0.341684\pi$
$653$	24.3840	0.954221	0.477110	+	0.878843i	$0.341684\pi$
$654$	−1.53509	−0.0600267
$655$	0	0
$656$	5.94413	0.232079
$657$	−38.2961	−1.49407
$658$	−18.9958	−0.740534
$659$	14.8136	0.577058	0.288529	−	0.957471i	$-0.406834\pi$
$659$	14.8136	0.577058	0.288529	+	0.957471i	$0.406834\pi$
$660$	0	0
$661$	24.1133	0.937898	0.468949	−	0.883225i	$-0.344633\pi$
$661$	24.1133	0.937898	0.468949	+	0.883225i	$0.344633\pi$
$662$	41.1130	1.59790
$663$	20.1262	0.781637
$664$	−21.8248	−0.846967
$665$	0	0
$666$	9.68363	0.375233
$667$	68.3474	2.64642
$668$	−11.2680	−0.435973
$669$	35.7198	1.38101
$670$	0	0
$671$	−5.81038	−0.224307
$672$	−17.8957	−0.690343
$673$	−27.3220	−1.05318	−0.526592	−	0.850118i	$-0.676530\pi$
$673$	−27.3220	−1.05318	−0.526592	+	0.850118i	$0.676530\pi$
$674$	10.5384	0.405926
$675$	0	0
$676$	49.9405	1.92079
$677$	27.6197	1.06151	0.530756	−	0.847525i	$-0.321908\pi$
$677$	27.6197	1.06151	0.530756	+	0.847525i	$0.321908\pi$
$678$	−63.9677	−2.45666
$679$	−9.25710	−0.355255
$680$	0	0
$681$	−10.5051	−0.402555
$682$	−24.7323	−0.947050
$683$	−16.1845	−0.619282	−0.309641	−	0.950854i	$-0.600209\pi$
$683$	−16.1845	−0.619282	−0.309641	+	0.950854i	$0.600209\pi$
$684$	24.0814	0.920777
$685$	0	0
$686$	−2.20648	−0.0842437
$687$	3.69796	0.141086
$688$	12.5141	0.477095
$689$	26.9479	1.02663
$690$	0	0
$691$	−29.4581	−1.12064	−0.560319	−	0.828277i	$-0.689321\pi$
$691$	−29.4581	−1.12064	−0.560319	+	0.828277i	$0.689321\pi$
$692$	−62.5287	−2.37699
$693$	−7.13803	−0.271151
$694$	−59.6105	−2.26279
$695$	0	0
$696$	38.3741	1.45457
$697$	−5.75487	−0.217981
$698$	−33.5215	−1.26881
$699$	13.4455	0.508556
$700$	0	0
$701$	−40.3478	−1.52392	−0.761958	−	0.647626i	$-0.775762\pi$
$701$	−40.3478	−1.52392	−0.761958	+	0.647626i	$0.775762\pi$
$702$	−7.43589	−0.280649
$703$	−3.49985	−0.131999
$704$	28.1257	1.06003
$705$	0	0
$706$	−37.7231	−1.41973
$707$	−8.29649	−0.312022
$708$	105.535	3.96623
$709$	48.8291	1.83382	0.916908	−	0.399099i	$-0.130677\pi$
$709$	48.8291	1.83382	0.916908	+	0.399099i	$0.130677\pi$
$710$	0	0
$711$	14.7291	0.552385
$712$	−14.8436	−0.556289
$713$	−43.4578	−1.62751
$714$	8.05296	0.301374
$715$	0	0
$716$	−55.6422	−2.07945
$717$	−34.6173	−1.29281
$718$	−43.7662	−1.63334
$719$	−15.5108	−0.578456	−0.289228	−	0.957260i	$-0.593399\pi$
$719$	−15.5108	−0.578456	−0.289228	+	0.957260i	$0.593399\pi$
$720$	0	0
$721$	−6.23818	−0.232322
$722$	27.1514	1.01047
$723$	34.4307	1.28049
$724$	−68.8685	−2.55948
$725$	0	0
$726$	−33.9649	−1.26055
$727$	34.4252	1.27676	0.638380	−	0.769721i	$-0.279605\pi$
$727$	34.4252	1.27676	0.638380	+	0.769721i	$0.279605\pi$
$728$	10.5681	0.391680
$729$	−31.2079	−1.15585
$730$	0	0
$731$	−12.1156	−0.448112
$732$	−18.9319	−0.699742
$733$	−12.0035	−0.443359	−0.221680	−	0.975120i	$-0.571154\pi$
$733$	−12.0035	−0.443359	−0.221680	+	0.975120i	$0.571154\pi$
$734$	−12.7826	−0.471815
$735$	0	0
$736$	61.0835	2.25157
$737$	−19.0075	−0.700149
$738$	28.2088	1.03838
$739$	29.0517	1.06868	0.534341	−	0.845269i	$-0.320560\pi$
$739$	29.0517	1.06868	0.534341	+	0.845269i	$0.320560\pi$
$740$	0	0
$741$	35.6552	1.30983
$742$	10.7825	0.395837
$743$	35.7292	1.31078	0.655389	−	0.755292i	$-0.272505\pi$
$743$	35.7292	1.31078	0.655389	+	0.755292i	$0.272505\pi$
$744$	−24.3997	−0.894535
$745$	0	0
$746$	44.3391	1.62337
$747$	36.9500	1.35193
$748$	−9.21700	−0.337007
$749$	−18.9353	−0.691880
$750$	0	0
$751$	5.34762	0.195138	0.0975688	−	0.995229i	$-0.468893\pi$
$751$	5.34762	0.195138	0.0975688	+	0.995229i	$0.468893\pi$
$752$	−12.9872	−0.473596
$753$	−16.6376	−0.606306
$754$	97.4994	3.55072
$755$	0	0
$756$	−1.75303	−0.0637570
$757$	−37.5413	−1.36446	−0.682232	−	0.731136i	$-0.738990\pi$
$757$	−37.5413	−1.36446	−0.682232	+	0.731136i	$0.738990\pi$
$758$	−45.5057	−1.65284
$759$	46.8921	1.70207
$760$	0	0
$761$	−1.91338	−0.0693599	−0.0346800	−	0.999398i	$-0.511041\pi$
$761$	−1.91338	−0.0693599	−0.0346800	+	0.999398i	$0.511041\pi$
$762$	−9.13346	−0.330870
$763$	−0.278409	−0.0100791
$764$	78.4282	2.83743
$765$	0	0
$766$	73.0577	2.63968
$767$	81.1874	2.93151
$768$	12.6686	0.457137
$769$	−9.89864	−0.356954	−0.178477	−	0.983944i	$-0.557117\pi$
$769$	−9.89864	−0.356954	−0.178477	+	0.983944i	$0.557117\pi$
$770$	0	0
$771$	−52.1270	−1.87731
$772$	9.81637	0.353299
$773$	6.25408	0.224944	0.112472	−	0.993655i	$-0.464123\pi$
$773$	6.25408	0.224944	0.112472	+	0.993655i	$0.464123\pi$
$774$	59.3875	2.13464
$775$	0	0
$776$	17.7405	0.636847
$777$	3.38014	0.121262
$778$	−2.63766	−0.0945647
$779$	−10.1952	−0.365281
$780$	0	0
$781$	−1.14036	−0.0408054
$782$	−27.4872	−0.982939
$783$	−4.89693	−0.175002
$784$	−1.50855	−0.0538766
$785$	0	0
$786$	38.9448	1.38912
$787$	31.6351	1.12767	0.563835	−	0.825888i	$-0.309326\pi$
$787$	31.6351	1.12767	0.563835	+	0.825888i	$0.309326\pi$
$788$	11.3459	0.404182
$789$	13.2965	0.473366
$790$	0	0
$791$	−11.6014	−0.412498
$792$	13.6795	0.486079
$793$	−14.5642	−0.517191
$794$	−46.5095	−1.65056
$795$	0	0
$796$	1.68536	0.0597360
$797$	−32.3792	−1.14693	−0.573465	−	0.819230i	$-0.694401\pi$
$797$	−32.3792	−1.14693	−0.573465	+	0.819230i	$0.694401\pi$
$798$	14.2665	0.505027
$799$	12.5737	0.444826
$800$	0	0
$801$	25.1307	0.887949
$802$	27.3756	0.966667
$803$	25.9671	0.916358
$804$	−61.9317	−2.18416
$805$	0	0
$806$	−61.9937	−2.18364
$807$	32.3192	1.13769
$808$	15.8996	0.559345
$809$	−23.0109	−0.809021	−0.404510	−	0.914533i	$-0.632558\pi$
$809$	−23.0109	−0.809021	−0.404510	+	0.914533i	$0.632558\pi$
$810$	0	0
$811$	−20.5186	−0.720504	−0.360252	−	0.932855i	$-0.617309\pi$
$811$	−20.5186	−0.720504	−0.360252	+	0.932855i	$0.617309\pi$
$812$	22.9857	0.806640
$813$	64.0275	2.24554
$814$	−6.56609	−0.230141
$815$	0	0
$816$	5.50572	0.192739
$817$	−21.4638	−0.750922
$818$	1.14682	0.0400976
$819$	−17.8921	−0.625200
$820$	0	0
$821$	32.2691	1.12620	0.563100	−	0.826389i	$-0.309608\pi$
$821$	32.2691	1.12620	0.563100	+	0.826389i	$0.309608\pi$
$822$	−70.3600	−2.45409
$823$	−13.4406	−0.468509	−0.234255	−	0.972175i	$-0.575265\pi$
$823$	−13.4406	−0.468509	−0.234255	+	0.972175i	$0.575265\pi$
$824$	11.9550	0.416471
$825$	0	0
$826$	32.4849	1.13030
$827$	17.0571	0.593134	0.296567	−	0.955012i	$-0.404158\pi$
$827$	17.0571	0.593134	0.296567	+	0.955012i	$0.404158\pi$
$828$	79.3856	2.75884
$829$	20.0961	0.697966	0.348983	−	0.937129i	$-0.386527\pi$
$829$	20.0961	0.697966	0.348983	+	0.937129i	$0.386527\pi$
$830$	0	0
$831$	1.14620	0.0397611
$832$	70.4996	2.44414
$833$	1.46051	0.0506037
$834$	−90.3554	−3.12875
$835$	0	0
$836$	−16.3287	−0.564738
$837$	3.11365	0.107623
$838$	18.8009	0.649468
$839$	2.82753	0.0976173	0.0488087	−	0.998808i	$-0.484458\pi$
$839$	2.82753	0.0976173	0.0488087	+	0.998808i	$0.484458\pi$
$840$	0	0
$841$	35.2085	1.21409
$842$	−68.8954	−2.37429
$843$	8.54589	0.294336
$844$	36.6799	1.26257
$845$	0	0
$846$	−61.6330	−2.11898
$847$	−6.15999	−0.211660
$848$	7.37186	0.253151
$849$	−3.92420	−0.134678
$850$	0	0
$851$	−11.5374	−0.395498
$852$	−3.71563	−0.127295
$853$	22.2079	0.760383	0.380192	−	0.924908i	$-0.375858\pi$
$853$	22.2079	0.760383	0.380192	+	0.924908i	$0.375858\pi$
$854$	−5.82748	−0.199412
$855$	0	0
$856$	36.2880	1.24030
$857$	45.4397	1.55219	0.776095	−	0.630617i	$-0.217198\pi$
$857$	45.4397	1.55219	0.776095	+	0.630617i	$0.217198\pi$
$858$	66.8928	2.28368
$859$	−48.4890	−1.65442	−0.827211	−	0.561891i	$-0.810074\pi$
$859$	−48.4890	−1.65442	−0.827211	+	0.561891i	$0.810074\pi$
$860$	0	0
$861$	9.84648	0.335567
$862$	59.3083	2.02005
$863$	26.9188	0.916325	0.458163	−	0.888868i	$-0.348508\pi$
$863$	26.9188	0.916325	0.458163	+	0.888868i	$0.348508\pi$
$864$	−4.37649	−0.148891
$865$	0	0
$866$	−21.1908	−0.720094
$867$	37.1511	1.26172
$868$	−14.6152	−0.496071
$869$	−9.98723	−0.338794
$870$	0	0
$871$	−47.6439	−1.61435
$872$	0.533549	0.0180683
$873$	−30.0352	−1.01654
$874$	−48.6957	−1.64716
$875$	0	0
$876$	84.6081	2.85864
$877$	36.2264	1.22328	0.611640	−	0.791136i	$-0.290510\pi$
$877$	36.2264	1.22328	0.611640	+	0.791136i	$0.290510\pi$
$878$	12.1365	0.409587
$879$	12.6404	0.426350
$880$	0	0
$881$	−2.87065	−0.0967148	−0.0483574	−	0.998830i	$-0.515399\pi$
$881$	−2.87065	−0.0967148	−0.0483574	+	0.998830i	$0.515399\pi$
$882$	−7.15904	−0.241057
$883$	34.7410	1.16913	0.584564	−	0.811347i	$-0.301265\pi$
$883$	34.7410	1.16913	0.584564	+	0.811347i	$0.301265\pi$
$884$	−23.1032	−0.777046
$885$	0	0
$886$	54.3496	1.82591
$887$	−5.05850	−0.169848	−0.0849240	−	0.996387i	$-0.527065\pi$
$887$	−5.05850	−0.169848	−0.0849240	+	0.996387i	$0.527065\pi$
$888$	−6.47777	−0.217380
$889$	−1.65648	−0.0555564
$890$	0	0
$891$	18.0544	0.604845
$892$	−41.0034	−1.37290
$893$	22.2753	0.745415
$894$	−65.1721	−2.17968
$895$	0	0
$896$	13.8857	0.463888
$897$	117.539	3.92452
$898$	21.1488	0.705746
$899$	−40.8262	−1.36163
$900$	0	0
$901$	−7.13714	−0.237772
$902$	−19.1273	−0.636868
$903$	20.7296	0.689838
$904$	22.2332	0.739464
$905$	0	0
$906$	132.240	4.39339
$907$	39.4637	1.31037	0.655185	−	0.755468i	$-0.272590\pi$
$907$	39.4637	1.31037	0.655185	+	0.755468i	$0.272590\pi$
$908$	12.0590	0.400191
$909$	−26.9184	−0.892828
$910$	0	0
$911$	−34.1840	−1.13257	−0.566283	−	0.824211i	$-0.691619\pi$
$911$	−34.1840	−1.13257	−0.566283	+	0.824211i	$0.691619\pi$
$912$	9.75382	0.322981
$913$	−25.0543	−0.829178
$914$	−75.0978	−2.48402
$915$	0	0
$916$	−4.24496	−0.140257
$917$	7.06316	0.233246
$918$	1.96939	0.0649996
$919$	−31.3281	−1.03342	−0.516710	−	0.856161i	$-0.672843\pi$
$919$	−31.3281	−1.03342	−0.516710	+	0.856161i	$0.672843\pi$
$920$	0	0
$921$	16.2156	0.534321
$922$	7.86435	0.258999
$923$	−2.85842	−0.0940861
$924$	15.7701	0.518799
$925$	0	0
$926$	−2.17984	−0.0716339
$927$	−20.2401	−0.664772
$928$	57.3846	1.88374
$929$	−26.4293	−0.867118	−0.433559	−	0.901125i	$-0.642742\pi$
$929$	−26.4293	−0.867118	−0.433559	+	0.901125i	$0.642742\pi$
$930$	0	0
$931$	2.58741	0.0847990
$932$	−15.4343	−0.505568
$933$	−17.1691	−0.562092
$934$	13.8581	0.453450
$935$	0	0
$936$	34.2888	1.12076
$937$	41.0708	1.34173	0.670863	−	0.741582i	$-0.265924\pi$
$937$	41.0708	1.34173	0.670863	+	0.741582i	$0.265924\pi$
$938$	−19.0634	−0.622442
$939$	−18.6197	−0.607631
$940$	0	0
$941$	−30.1230	−0.981982	−0.490991	−	0.871165i	$-0.663365\pi$
$941$	−30.1230	−0.981982	−0.490991	+	0.871165i	$0.663365\pi$
$942$	−80.3194	−2.61695
$943$	−33.6090	−1.09446
$944$	22.2096	0.722861
$945$	0	0
$946$	−40.2683	−1.30923
$947$	−37.9041	−1.23172	−0.615860	−	0.787856i	$-0.711191\pi$
$947$	−37.9041	−1.23172	−0.615860	+	0.787856i	$0.711191\pi$
$948$	−32.5412	−1.05689
$949$	65.0888	2.11287
$950$	0	0
$951$	−17.2175	−0.558316
$952$	−2.79896	−0.0907147
$953$	23.1391	0.749550	0.374775	−	0.927116i	$-0.377720\pi$
$953$	23.1391	0.749550	0.374775	+	0.927116i	$0.377720\pi$
$954$	34.9843	1.13266
$955$	0	0
$956$	39.7378	1.28521
$957$	44.0525	1.42401
$958$	10.7006	0.345720
$959$	−12.7607	−0.412066
$960$	0	0
$961$	−5.04121	−0.162620
$962$	−16.4585	−0.530642
$963$	−61.4366	−1.97977
$964$	−39.5237	−1.27297
$965$	0	0
$966$	47.0301	1.51317
$967$	−55.3866	−1.78111	−0.890557	−	0.454872i	$-0.849685\pi$
$967$	−55.3866	−1.78111	−0.890557	+	0.454872i	$0.849685\pi$
$968$	11.8051	0.379431
$969$	−9.44325	−0.303361
$970$	0	0
$971$	−35.7046	−1.14581	−0.572907	−	0.819620i	$-0.694184\pi$
$971$	−35.7046	−1.14581	−0.572907	+	0.819620i	$0.694184\pi$
$972$	64.0854	2.05554
$973$	−16.3872	−0.525348
$974$	11.3839	0.364765
$975$	0	0
$976$	−3.98419	−0.127531
$977$	17.8700	0.571713	0.285856	−	0.958272i	$-0.407722\pi$
$977$	17.8700	0.571713	0.285856	+	0.958272i	$0.407722\pi$
$978$	13.7965	0.441165
$979$	−17.0401	−0.544604
$980$	0	0
$981$	−0.903313	−0.0288406
$982$	−67.8029	−2.16368
$983$	46.0446	1.46859	0.734297	−	0.678828i	$-0.237512\pi$
$983$	46.0446	1.46859	0.734297	+	0.678828i	$0.237512\pi$
$984$	−18.8700	−0.601554
$985$	0	0
$986$	−25.8227	−0.822361
$987$	−21.5134	−0.684779
$988$	−40.9292	−1.30213
$989$	−70.7564	−2.24992
$990$	0	0
$991$	−37.3993	−1.18803	−0.594013	−	0.804455i	$-0.702457\pi$
$991$	−37.3993	−1.18803	−0.594013	+	0.804455i	$0.702457\pi$
$992$	−36.4872	−1.15847
$993$	46.5619	1.47760
$994$	−1.14372	−0.0362766
$995$	0	0
$996$	−81.6342	−2.58668
$997$	−21.6673	−0.686211	−0.343105	−	0.939297i	$-0.611479\pi$
$997$	−21.6673	−0.686211	−0.343105	+	0.939297i	$0.611479\pi$
$998$	75.6742	2.39543
$999$	0.826629	0.0261534

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	875.2.a.e.1.2	✓	6
3.2	odd	2		7875.2.a.t.1.5		6
5.2	odd	4		875.2.b.c.624.2		12
5.3	odd	4		875.2.b.c.624.11		12
5.4	even	2		875.2.a.h.1.5	yes	6
7.6	odd	2		6125.2.a.s.1.2		6
15.14	odd	2		7875.2.a.q.1.2		6
35.34	odd	2		6125.2.a.t.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
875.2.a.e.1.2	✓	6	1.1	even	1	trivial
875.2.a.h.1.5	yes	6	5.4	even	2
875.2.b.c.624.2		12	5.2	odd	4
875.2.b.c.624.11		12	5.3	odd	4
6125.2.a.s.1.2		6	7.6	odd	2
6125.2.a.t.1.5		6	35.34	odd	2
7875.2.a.q.1.2		6	15.14	odd	2
7875.2.a.t.1.5		6	3.2	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:	\( N \)	\(=\)	\( 875 = 5^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	875.a (trivial)

\(f(q)\)	\(=\)	\(q-2.20648 q^{2} -2.49891 q^{3} +2.86854 q^{4} +5.51379 q^{6} +1.00000 q^{7} -1.91642 q^{8} +3.24455 q^{9} -2.20000 q^{11} -7.16823 q^{12} -5.51450 q^{13} -2.20648 q^{14} -1.50855 q^{16} +1.46051 q^{17} -7.15904 q^{18} +2.58741 q^{19} -2.49891 q^{21} +4.85426 q^{22} +8.52954 q^{23} +4.78897 q^{24} +12.1676 q^{26} -0.611121 q^{27} +2.86854 q^{28} +8.01302 q^{29} -5.09498 q^{31} +7.16141 q^{32} +5.49761 q^{33} -3.22259 q^{34} +9.30715 q^{36} -1.35264 q^{37} -5.70907 q^{38} +13.7802 q^{39} -3.94031 q^{41} +5.51379 q^{42} -8.29546 q^{43} -6.31080 q^{44} -18.8202 q^{46} +8.60911 q^{47} +3.76972 q^{48} +1.00000 q^{49} -3.64969 q^{51} -15.8186 q^{52} -4.88674 q^{53} +1.34842 q^{54} -1.91642 q^{56} -6.46571 q^{57} -17.6806 q^{58} -14.7225 q^{59} +2.64108 q^{61} +11.2420 q^{62} +3.24455 q^{63} -12.7844 q^{64} -12.1304 q^{66} +8.63974 q^{67} +4.18954 q^{68} -21.3145 q^{69} +0.518346 q^{71} -6.21794 q^{72} -11.8032 q^{73} +2.98458 q^{74} +7.42211 q^{76} -2.20000 q^{77} -30.4058 q^{78} +4.53964 q^{79} -8.20653 q^{81} +8.69420 q^{82} +11.3883 q^{83} -7.16823 q^{84} +18.3037 q^{86} -20.0238 q^{87} +4.21613 q^{88} +7.74549 q^{89} -5.51450 q^{91} +24.4673 q^{92} +12.7319 q^{93} -18.9958 q^{94} -17.8957 q^{96} -9.25710 q^{97} -2.20648 q^{98} -7.13803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - 5 q^{3} + 4 q^{4} - 2 q^{6} + 6 q^{7} - 12 q^{8} + 7 q^{9} + q^{11} - 11 q^{12} - 22 q^{13} - 2 q^{14} + 8 q^{16} - 4 q^{17} + 14 q^{18} - 17 q^{19} - 5 q^{21} - 9 q^{22} - 3 q^{23} + 7 q^{24}+ \cdots + 12 q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - 5 * q^3 + 4 * q^4 - 2 * q^6 + 6 * q^7 - 12 * q^8 + 7 * q^9 + q^11 - 11 * q^12 - 22 * q^13 - 2 * q^14 + 8 * q^16 - 4 * q^17 + 14 * q^18 - 17 * q^19 - 5 * q^21 - 9 * q^22 - 3 * q^23 + 7 * q^24 + 16 * q^26 - 26 * q^27 + 4 * q^28 + 7 * q^29 - 11 * q^31 - q^32 - 21 * q^33 - 4 * q^34 + 10 * q^36 - 33 * q^37 + 15 * q^39 + 3 * q^41 - 2 * q^42 - 2 * q^43 + 26 * q^44 - 8 * q^46 + 6 * q^47 + 20 * q^48 + 6 * q^49 + q^51 - 47 * q^52 - 33 * q^53 - 26 * q^54 - 12 * q^56 + 3 * q^57 - 43 * q^58 - 21 * q^59 - 19 * q^61 + 37 * q^62 + 7 * q^63 - 22 * q^64 - 35 * q^66 - 2 * q^67 + 30 * q^68 - 2 * q^69 + q^71 + 10 * q^72 - 16 * q^73 - 29 * q^74 - 4 * q^76 + q^77 + q^78 + 22 * q^79 + 14 * q^81 + 13 * q^82 - 3 * q^83 - 11 * q^84 + 23 * q^86 - 28 * q^88 - 8 * q^89 - 22 * q^91 + 2 * q^92 + 16 * q^93 - 26 * q^94 - 4 * q^96 - 16 * q^97 - 2 * q^98 + 12 * q^99

Introduction

L-functions

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