LMFDB - Embedded newform 1190.2.a.j.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1190,2,Mod(1,1190)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1190, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1190.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1190 = 2 \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1190.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,-1,3,3,1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.50219784053$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.2597.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 8 $ x^3 - x^2 - 9*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.95759$ of defining polynomial
Character	$\chi$	$=$	1190.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +2.95759 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.95759 q^{6} -1.00000 q^{7} -1.00000 q^{8} +5.74732 q^{9} -1.00000 q^{10} -2.74732 q^{11} +2.95759 q^{12} +1.00000 q^{14} +2.95759 q^{15} +1.00000 q^{16} -1.00000 q^{17} -5.74732 q^{18} +6.74732 q^{19} +1.00000 q^{20} -2.95759 q^{21} +2.74732 q^{22} +2.95759 q^{23} -2.95759 q^{24} +1.00000 q^{25} +8.12544 q^{27} -1.00000 q^{28} +2.95759 q^{29} -2.95759 q^{30} +5.16785 q^{31} -1.00000 q^{32} -8.12544 q^{33} +1.00000 q^{34} -1.00000 q^{35} +5.74732 q^{36} +9.91517 q^{37} -6.74732 q^{38} -1.00000 q^{40} +1.25268 q^{41} +2.95759 q^{42} -1.04241 q^{43} -2.74732 q^{44} +5.74732 q^{45} -2.95759 q^{46} +2.95759 q^{47} +2.95759 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.95759 q^{51} -0.957587 q^{53} -8.12544 q^{54} -2.74732 q^{55} +1.00000 q^{56} +19.9558 q^{57} -2.95759 q^{58} +4.95759 q^{59} +2.95759 q^{60} -9.83035 q^{61} -5.16785 q^{62} -5.74732 q^{63} +1.00000 q^{64} +8.12544 q^{66} -11.0830 q^{67} -1.00000 q^{68} +8.74732 q^{69} +1.00000 q^{70} -0.420532 q^{71} -5.74732 q^{72} +6.00000 q^{73} -9.91517 q^{74} +2.95759 q^{75} +6.74732 q^{76} +2.74732 q^{77} -11.4098 q^{79} +1.00000 q^{80} +6.78973 q^{81} -1.25268 q^{82} -17.8303 q^{83} -2.95759 q^{84} -1.00000 q^{85} +1.04241 q^{86} +8.74732 q^{87} +2.74732 q^{88} -13.4098 q^{89} -5.74732 q^{90} +2.95759 q^{92} +15.2844 q^{93} -2.95759 q^{94} +6.74732 q^{95} -2.95759 q^{96} -9.83035 q^{97} -1.00000 q^{98} -15.7897 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 3 q^{7} - 3 q^{8} + 10 q^{9} - 3 q^{10} - q^{11} - q^{12} + 3 q^{14} - q^{15} + 3 q^{16} - 3 q^{17} - 10 q^{18} + 13 q^{19} + 3 q^{20} + q^{21}+ \cdots - 50 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + 3 * q^5 + q^6 - 3 * q^7 - 3 * q^8 + 10 * q^9 - 3 * q^10 - q^11 - q^12 + 3 * q^14 - q^15 + 3 * q^16 - 3 * q^17 - 10 * q^18 + 13 * q^19 + 3 * q^20 + q^21 + q^22 - q^23 + q^24 + 3 * q^25 + 2 * q^27 - 3 * q^28 - q^29 + q^30 + 3 * q^31 - 3 * q^32 - 2 * q^33 + 3 * q^34 - 3 * q^35 + 10 * q^36 + 10 * q^37 - 13 * q^38 - 3 * q^40 + 11 * q^41 - q^42 - 13 * q^43 - q^44 + 10 * q^45 + q^46 - q^47 - q^48 + 3 * q^49 - 3 * q^50 + q^51 + 7 * q^53 - 2 * q^54 - q^55 + 3 * q^56 - 2 * q^57 + q^58 + 5 * q^59 - q^60 + 10 * q^61 - 3 * q^62 - 10 * q^63 + 3 * q^64 + 2 * q^66 - q^67 - 3 * q^68 + 19 * q^69 + 3 * q^70 + 4 * q^71 - 10 * q^72 + 18 * q^73 - 10 * q^74 - q^75 + 13 * q^76 + q^77 + 3 * q^80 + 23 * q^81 - 11 * q^82 - 14 * q^83 + q^84 - 3 * q^85 + 13 * q^86 + 19 * q^87 + q^88 - 6 * q^89 - 10 * q^90 - q^92 + 34 * q^93 + q^94 + 13 * q^95 + q^96 + 10 * q^97 - 3 * q^98 - 50 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	2.95759	1.70756	0.853782	−	0.520631i	$-0.174303\pi$
$3$	2.95759	1.70756	0.853782	+	0.520631i	$0.174303\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.95759	−1.20743
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	5.74732	1.91577
$10$	−1.00000	−0.316228
$11$	−2.74732	−0.828348	−0.414174	−	0.910198i	$-0.635930\pi$
$11$	−2.74732	−0.828348	−0.414174	+	0.910198i	$0.635930\pi$
$12$	2.95759	0.853782
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.00000	0.267261
$15$	2.95759	0.763646
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−5.74732	−1.35466
$19$	6.74732	1.54794	0.773971	−	0.633221i	$-0.218268\pi$
$19$	6.74732	1.54794	0.773971	+	0.633221i	$0.218268\pi$
$20$	1.00000	0.223607
$21$	−2.95759	−0.645398
$22$	2.74732	0.585731
$23$	2.95759	0.616700	0.308350	−	0.951273i	$-0.400223\pi$
$23$	2.95759	0.616700	0.308350	+	0.951273i	$0.400223\pi$
$24$	−2.95759	−0.603715
$25$	1.00000	0.200000
$26$	0	0
$27$	8.12544	1.56374
$28$	−1.00000	−0.188982
$29$	2.95759	0.549210	0.274605	−	0.961557i	$-0.411453\pi$
$29$	2.95759	0.549210	0.274605	+	0.961557i	$0.411453\pi$
$30$	−2.95759	−0.539979
$31$	5.16785	0.928174	0.464087	−	0.885790i	$-0.346383\pi$
$31$	5.16785	0.928174	0.464087	+	0.885790i	$0.346383\pi$
$32$	−1.00000	−0.176777
$33$	−8.12544	−1.41446
$34$	1.00000	0.171499
$35$	−1.00000	−0.169031
$36$	5.74732	0.957887
$37$	9.91517	1.63004	0.815022	−	0.579430i	$-0.196725\pi$
$37$	9.91517	1.63004	0.815022	+	0.579430i	$0.196725\pi$
$38$	−6.74732	−1.09456
$39$	0	0
$40$	−1.00000	−0.158114
$41$	1.25268	0.195636	0.0978178	−	0.995204i	$-0.468814\pi$
$41$	1.25268	0.195636	0.0978178	+	0.995204i	$0.468814\pi$
$42$	2.95759	0.456366
$43$	−1.04241	−0.158966	−0.0794832	−	0.996836i	$-0.525327\pi$
$43$	−1.04241	−0.158966	−0.0794832	+	0.996836i	$0.525327\pi$
$44$	−2.74732	−0.414174
$45$	5.74732	0.856760
$46$	−2.95759	−0.436072
$47$	2.95759	0.431408	0.215704	−	0.976459i	$-0.430795\pi$
$47$	2.95759	0.431408	0.215704	+	0.976459i	$0.430795\pi$
$48$	2.95759	0.426891
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−2.95759	−0.414145
$52$	0	0
$53$	−0.957587	−0.131535	−0.0657673	−	0.997835i	$-0.520950\pi$
$53$	−0.957587	−0.131535	−0.0657673	+	0.997835i	$0.520950\pi$
$54$	−8.12544	−1.10573
$55$	−2.74732	−0.370449
$56$	1.00000	0.133631
$57$	19.9558	2.64321
$58$	−2.95759	−0.388350
$59$	4.95759	0.645423	0.322711	−	0.946497i	$-0.395406\pi$
$59$	4.95759	0.645423	0.322711	+	0.946497i	$0.395406\pi$
$60$	2.95759	0.381823
$61$	−9.83035	−1.25865	−0.629324	−	0.777143i	$-0.716668\pi$
$61$	−9.83035	−1.25865	−0.629324	+	0.777143i	$0.716668\pi$
$62$	−5.16785	−0.656318
$63$	−5.74732	−0.724094
$64$	1.00000	0.125000
$65$	0	0
$66$	8.12544	1.00017
$67$	−11.0830	−1.35401	−0.677004	−	0.735980i	$-0.736722\pi$
$67$	−11.0830	−1.35401	−0.677004	+	0.735980i	$0.736722\pi$
$68$	−1.00000	−0.121268
$69$	8.74732	1.05305
$70$	1.00000	0.119523
$71$	−0.420532	−0.0499080	−0.0249540	−	0.999689i	$-0.507944\pi$
$71$	−0.420532	−0.0499080	−0.0249540	+	0.999689i	$0.507944\pi$
$72$	−5.74732	−0.677328
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−9.91517	−1.15262
$75$	2.95759	0.341513
$76$	6.74732	0.773971
$77$	2.74732	0.313086
$78$	0	0
$79$	−11.4098	−1.28370	−0.641852	−	0.766828i	$-0.721834\pi$
$79$	−11.4098	−1.28370	−0.641852	+	0.766828i	$0.721834\pi$
$80$	1.00000	0.111803
$81$	6.78973	0.754415
$82$	−1.25268	−0.138335
$83$	−17.8303	−1.95713	−0.978567	−	0.205926i	$-0.933979\pi$
$83$	−17.8303	−1.95713	−0.978567	+	0.205926i	$0.933979\pi$
$84$	−2.95759	−0.322699
$85$	−1.00000	−0.108465
$86$	1.04241	0.112406
$87$	8.74732	0.937811
$88$	2.74732	0.292865
$89$	−13.4098	−1.42144	−0.710719	−	0.703476i	$-0.751630\pi$
$89$	−13.4098	−1.42144	−0.710719	+	0.703476i	$0.751630\pi$
$90$	−5.74732	−0.605821
$91$	0	0
$92$	2.95759	0.308350
$93$	15.2844	1.58492
$94$	−2.95759	−0.305052
$95$	6.74732	0.692261
$96$	−2.95759	−0.301857
$97$	−9.83035	−0.998121	−0.499060	−	0.866567i	$-0.666321\pi$
$97$	−9.83035	−0.998121	−0.499060	+	0.866567i	$0.666321\pi$
$98$	−1.00000	−0.101015
$99$	−15.7897	−1.58693
$100$	1.00000	0.100000
$101$	6.83215	0.679824	0.339912	−	0.940457i	$-0.389603\pi$
$101$	6.83215	0.679824	0.339912	+	0.940457i	$0.389603\pi$
$102$	2.95759	0.292845
$103$	−10.7879	−1.06297	−0.531483	−	0.847069i	$-0.678365\pi$
$103$	−10.7879	−1.06297	−0.531483	+	0.847069i	$0.678365\pi$
$104$	0	0
$105$	−2.95759	−0.288631
$106$	0.957587	0.0930091
$107$	−17.4098	−1.68307	−0.841535	−	0.540202i	$-0.818348\pi$
$107$	−17.4098	−1.68307	−0.841535	+	0.540202i	$0.818348\pi$
$108$	8.12544	0.781871
$109$	3.67321	0.351830	0.175915	−	0.984405i	$-0.443712\pi$
$109$	3.67321	0.351830	0.175915	+	0.984405i	$0.443712\pi$
$110$	2.74732	0.261947
$111$	29.3250	2.78340
$112$	−1.00000	−0.0944911
$113$	−12.6625	−1.19119	−0.595594	−	0.803286i	$-0.703083\pi$
$113$	−12.6625	−1.19119	−0.595594	+	0.803286i	$0.703083\pi$
$114$	−19.9558	−1.86903
$115$	2.95759	0.275796
$116$	2.95759	0.274605
$117$	0	0
$118$	−4.95759	−0.456383
$119$	1.00000	0.0916698
$120$	−2.95759	−0.269990
$121$	−3.45223	−0.313839
$122$	9.83035	0.889998
$123$	3.70491	0.334060
$124$	5.16785	0.464087
$125$	1.00000	0.0894427
$126$	5.74732	0.512012
$127$	10.3357	0.917146	0.458573	−	0.888657i	$-0.348361\pi$
$127$	10.3357	0.917146	0.458573	+	0.888657i	$0.348361\pi$
$128$	−1.00000	−0.0883883
$129$	−3.08303	−0.271445
$130$	0	0
$131$	−9.49464	−0.829551	−0.414775	−	0.909924i	$-0.636140\pi$
$131$	−9.49464	−0.829551	−0.414775	+	0.909924i	$0.636140\pi$
$132$	−8.12544	−0.707229
$133$	−6.74732	−0.585067
$134$	11.0830	0.957428
$135$	8.12544	0.699327
$136$	1.00000	0.0857493
$137$	16.3357	1.39565	0.697827	−	0.716267i	$-0.254151\pi$
$137$	16.3357	1.39565	0.697827	+	0.716267i	$0.254151\pi$
$138$	−8.74732	−0.744621
$139$	5.49464	0.466049	0.233025	−	0.972471i	$-0.425138\pi$
$139$	5.49464	0.466049	0.233025	+	0.972471i	$0.425138\pi$
$140$	−1.00000	−0.0845154
$141$	8.74732	0.736657
$142$	0.420532	0.0352903
$143$	0	0
$144$	5.74732	0.478943
$145$	2.95759	0.245614
$146$	−6.00000	−0.496564
$147$	2.95759	0.243938
$148$	9.91517	0.815022
$149$	−4.33571	−0.355195	−0.177597	−	0.984103i	$-0.556832\pi$
$149$	−4.33571	−0.355195	−0.177597	+	0.984103i	$0.556832\pi$
$150$	−2.95759	−0.241486
$151$	5.16785	0.420554	0.210277	−	0.977642i	$-0.432563\pi$
$151$	5.16785	0.420554	0.210277	+	0.977642i	$0.432563\pi$
$152$	−6.74732	−0.547280
$153$	−5.74732	−0.464643
$154$	−2.74732	−0.221385
$155$	5.16785	0.415092
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	11.4098	0.907716
$159$	−2.83215	−0.224604
$160$	−1.00000	−0.0790569
$161$	−2.95759	−0.233091
$162$	−6.78973	−0.533452
$163$	19.4946	1.52694	0.763469	−	0.645844i	$-0.223495\pi$
$163$	19.4946	1.52694	0.763469	+	0.645844i	$0.223495\pi$
$164$	1.25268	0.0978178
$165$	−8.12544	−0.632565
$166$	17.8303	1.38390
$167$	−12.4205	−0.961130	−0.480565	−	0.876959i	$-0.659568\pi$
$167$	−12.4205	−0.961130	−0.480565	+	0.876959i	$0.659568\pi$
$168$	2.95759	0.228183
$169$	−13.0000	−1.00000
$170$	1.00000	0.0766965
$171$	38.7790	2.96551
$172$	−1.04241	−0.0794832
$173$	−22.9982	−1.74852	−0.874260	−	0.485457i	$-0.838653\pi$
$173$	−22.9982	−1.74852	−0.874260	+	0.485457i	$0.838653\pi$
$174$	−8.74732	−0.663133
$175$	−1.00000	−0.0755929
$176$	−2.74732	−0.207087
$177$	14.6625	1.10210
$178$	13.4098	1.00511
$179$	−2.50536	−0.187259	−0.0936296	−	0.995607i	$-0.529847\pi$
$179$	−2.50536	−0.187259	−0.0936296	+	0.995607i	$0.529847\pi$
$180$	5.74732	0.428380
$181$	8.08483	0.600940	0.300470	−	0.953791i	$-0.402856\pi$
$181$	8.08483	0.600940	0.300470	+	0.953791i	$0.402856\pi$
$182$	0	0
$183$	−29.0741	−2.14922
$184$	−2.95759	−0.218036
$185$	9.91517	0.728978
$186$	−15.2844	−1.12070
$187$	2.74732	0.200904
$188$	2.95759	0.215704
$189$	−8.12544	−0.591039
$190$	−6.74732	−0.489502
$191$	16.1571	1.16909	0.584545	−	0.811362i	$-0.301273\pi$
$191$	16.1571	1.16909	0.584545	+	0.811362i	$0.301273\pi$
$192$	2.95759	0.213445
$193$	19.9469	1.43581	0.717904	−	0.696143i	$-0.245102\pi$
$193$	19.9469	1.43581	0.717904	+	0.696143i	$0.245102\pi$
$194$	9.83035	0.705778
$195$	0	0
$196$	1.00000	0.0714286
$197$	−2.08483	−0.148538	−0.0742689	−	0.997238i	$-0.523662\pi$
$197$	−2.08483	−0.148538	−0.0742689	+	0.997238i	$0.523662\pi$
$198$	15.7897	1.12213
$199$	−4.45223	−0.315610	−0.157805	−	0.987470i	$-0.550442\pi$
$199$	−4.45223	−0.315610	−0.157805	+	0.987470i	$0.550442\pi$
$200$	−1.00000	−0.0707107
$201$	−32.7790	−2.31205
$202$	−6.83215	−0.480708
$203$	−2.95759	−0.207582
$204$	−2.95759	−0.207073
$205$	1.25268	0.0874909
$206$	10.7879	0.751631
$207$	16.9982	1.18146
$208$	0	0
$209$	−18.5371	−1.28223
$210$	2.95759	0.204093
$211$	17.4098	1.19854	0.599271	−	0.800547i	$-0.295457\pi$
$211$	17.4098	1.19854	0.599271	+	0.800547i	$0.295457\pi$
$212$	−0.957587	−0.0657673
$213$	−1.24376	−0.0852211
$214$	17.4098	1.19011
$215$	−1.04241	−0.0710920
$216$	−8.12544	−0.552866
$217$	−5.16785	−0.350817
$218$	−3.67321	−0.248781
$219$	17.7455	1.19913
$220$	−2.74732	−0.185224
$221$	0	0
$222$	−29.3250	−1.96816
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	1.00000	0.0668153
$225$	5.74732	0.383155
$226$	12.6625	0.842297
$227$	−17.5884	−1.16738	−0.583691	−	0.811976i	$-0.698392\pi$
$227$	−17.5884	−1.16738	−0.583691	+	0.811976i	$0.698392\pi$
$228$	19.9558	1.32160
$229$	18.7879	1.24154	0.620771	−	0.783992i	$-0.286820\pi$
$229$	18.7879	1.24154	0.620771	+	0.783992i	$0.286820\pi$
$230$	−2.95759	−0.195018
$231$	8.12544	0.534615
$232$	−2.95759	−0.194175
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	2.95759	0.192932
$236$	4.95759	0.322711
$237$	−33.7455	−2.19201
$238$	−1.00000	−0.0648204
$239$	17.6518	1.14180	0.570899	−	0.821020i	$-0.306595\pi$
$239$	17.6518	1.14180	0.570899	+	0.821020i	$0.306595\pi$
$240$	2.95759	0.190911
$241$	7.46295	0.480731	0.240365	−	0.970683i	$-0.422733\pi$
$241$	7.46295	0.480731	0.240365	+	0.970683i	$0.422733\pi$
$242$	3.45223	0.221918
$243$	−4.29509	−0.275530
$244$	−9.83035	−0.629324
$245$	1.00000	0.0638877
$246$	−3.70491	−0.236216
$247$	0	0
$248$	−5.16785	−0.328159
$249$	−52.7348	−3.34193
$250$	−1.00000	−0.0632456
$251$	18.5777	1.17261	0.586306	−	0.810090i	$-0.300582\pi$
$251$	18.5777	1.17261	0.586306	+	0.810090i	$0.300582\pi$
$252$	−5.74732	−0.362047
$253$	−8.12544	−0.510842
$254$	−10.3357	−0.648520
$255$	−2.95759	−0.185211
$256$	1.00000	0.0625000
$257$	−22.8728	−1.42676	−0.713382	−	0.700776i	$-0.752837\pi$
$257$	−22.8728	−1.42676	−0.713382	+	0.700776i	$0.752837\pi$
$258$	3.08303	0.191941
$259$	−9.91517	−0.616099
$260$	0	0
$261$	16.9982	1.05216
$262$	9.49464	0.586581
$263$	−19.8303	−1.22279	−0.611396	−	0.791325i	$-0.709392\pi$
$263$	−19.8303	−1.22279	−0.611396	+	0.791325i	$0.709392\pi$
$264$	8.12544	0.500086
$265$	−0.957587	−0.0588241
$266$	6.74732	0.413705
$267$	−39.6607	−2.42720
$268$	−11.0830	−0.677004
$269$	17.6607	1.07679	0.538396	−	0.842692i	$-0.319031\pi$
$269$	17.6607	1.07679	0.538396	+	0.842692i	$0.319031\pi$
$270$	−8.12544	−0.494499
$271$	−18.1661	−1.10351	−0.551755	−	0.834007i	$-0.686041\pi$
$271$	−18.1661	−1.10351	−0.551755	+	0.834007i	$0.686041\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−16.3357	−0.986876
$275$	−2.74732	−0.165670
$276$	8.74732	0.526527
$277$	−12.2509	−0.736084	−0.368042	−	0.929809i	$-0.619972\pi$
$277$	−12.2509	−0.736084	−0.368042	+	0.929809i	$0.619972\pi$
$278$	−5.49464	−0.329547
$279$	29.7013	1.77817
$280$	1.00000	0.0597614
$281$	3.29329	0.196461	0.0982307	−	0.995164i	$-0.468682\pi$
$281$	3.29329	0.196461	0.0982307	+	0.995164i	$0.468682\pi$
$282$	−8.74732	−0.520895
$283$	−19.6607	−1.16871	−0.584354	−	0.811499i	$-0.698652\pi$
$283$	−19.6607	−1.16871	−0.584354	+	0.811499i	$0.698652\pi$
$284$	−0.420532	−0.0249540
$285$	19.9558	1.18208
$286$	0	0
$287$	−1.25268	−0.0739433
$288$	−5.74732	−0.338664
$289$	1.00000	0.0588235
$290$	−2.95759	−0.173675
$291$	−29.0741	−1.70435
$292$	6.00000	0.351123
$293$	−20.4839	−1.19668	−0.598342	−	0.801241i	$-0.704174\pi$
$293$	−20.4839	−1.19668	−0.598342	+	0.801241i	$0.704174\pi$
$294$	−2.95759	−0.172490
$295$	4.95759	0.288642
$296$	−9.91517	−0.576308
$297$	−22.3232	−1.29532
$298$	4.33571	0.251161
$299$	0	0
$300$	2.95759	0.170756
$301$	1.04241	0.0600837
$302$	−5.16785	−0.297377
$303$	20.2067	1.16084
$304$	6.74732	0.386985
$305$	−9.83035	−0.562884
$306$	5.74732	0.328552
$307$	0.0848260	0.00484128	0.00242064	−	0.999997i	$-0.499229\pi$
$307$	0.0848260	0.00484128	0.00242064	+	0.999997i	$0.499229\pi$
$308$	2.74732	0.156543
$309$	−31.9063	−1.81508
$310$	−5.16785	−0.293514
$311$	3.54777	0.201176	0.100588	−	0.994928i	$-0.467928\pi$
$311$	3.54777	0.201176	0.100588	+	0.994928i	$0.467928\pi$
$312$	0	0
$313$	−14.9045	−0.842450	−0.421225	−	0.906956i	$-0.638400\pi$
$313$	−14.9045	−0.842450	−0.421225	+	0.906956i	$0.638400\pi$
$314$	−12.0000	−0.677199
$315$	−5.74732	−0.323825
$316$	−11.4098	−0.641852
$317$	9.49464	0.533272	0.266636	−	0.963797i	$-0.414088\pi$
$317$	9.49464	0.533272	0.266636	+	0.963797i	$0.414088\pi$
$318$	2.83215	0.158819
$319$	−8.12544	−0.454937
$320$	1.00000	0.0559017
$321$	−51.4910	−2.87395
$322$	2.95759	0.164820
$323$	−6.74732	−0.375431
$324$	6.78973	0.377207
$325$	0	0
$326$	−19.4946	−1.07971
$327$	10.8638	0.600772
$328$	−1.25268	−0.0691676
$329$	−2.95759	−0.163057
$330$	8.12544	0.447291
$331$	−22.9893	−1.26361	−0.631803	−	0.775129i	$-0.717685\pi$
$331$	−22.9893	−1.26361	−0.631803	+	0.775129i	$0.717685\pi$
$332$	−17.8303	−0.978567
$333$	56.9857	3.12280
$334$	12.4205	0.679621
$335$	−11.0830	−0.605530
$336$	−2.95759	−0.161350
$337$	13.8620	0.755114	0.377557	−	0.925986i	$-0.376764\pi$
$337$	13.8620	0.755114	0.377557	+	0.925986i	$0.376764\pi$
$338$	13.0000	0.707107
$339$	−37.4504	−2.03403
$340$	−1.00000	−0.0542326
$341$	−14.1978	−0.768851
$342$	−38.7790	−2.09693
$343$	−1.00000	−0.0539949
$344$	1.04241	0.0562031
$345$	8.74732	0.470940
$346$	22.9982	1.23639
$347$	−23.7455	−1.27473	−0.637363	−	0.770564i	$-0.719975\pi$
$347$	−23.7455	−1.27473	−0.637363	+	0.770564i	$0.719975\pi$
$348$	8.74732	0.468906
$349$	35.2402	1.88636	0.943181	−	0.332278i	$-0.107817\pi$
$349$	35.2402	1.88636	0.943181	+	0.332278i	$0.107817\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	2.74732	0.146433
$353$	−10.3268	−0.549639	−0.274820	−	0.961496i	$-0.588618\pi$
$353$	−10.3268	−0.549639	−0.274820	+	0.961496i	$0.588618\pi$
$354$	−14.6625	−0.779303
$355$	−0.420532	−0.0223195
$356$	−13.4098	−0.710719
$357$	2.95759	0.156532
$358$	2.50536	0.132412
$359$	0.621881	0.0328216	0.0164108	−	0.999865i	$-0.494776\pi$
$359$	0.621881	0.0328216	0.0164108	+	0.999865i	$0.494776\pi$
$360$	−5.74732	−0.302910
$361$	26.5263	1.39612
$362$	−8.08483	−0.424929
$363$	−10.2103	−0.535900
$364$	0	0
$365$	6.00000	0.314054
$366$	29.0741	1.51973
$367$	5.01072	0.261557	0.130779	−	0.991412i	$-0.458252\pi$
$367$	5.01072	0.261557	0.130779	+	0.991412i	$0.458252\pi$
$368$	2.95759	0.154175
$369$	7.19955	0.374794
$370$	−9.91517	−0.515465
$371$	0.957587	0.0497154
$372$	15.2844	0.792458
$373$	−23.7138	−1.22786	−0.613928	−	0.789362i	$-0.710411\pi$
$373$	−23.7138	−1.22786	−0.613928	+	0.789362i	$0.710411\pi$
$374$	−2.74732	−0.142061
$375$	2.95759	0.152729
$376$	−2.95759	−0.152526
$377$	0	0
$378$	8.12544	0.417928
$379$	−1.96830	−0.101105	−0.0505525	−	0.998721i	$-0.516098\pi$
$379$	−1.96830	−0.101105	−0.0505525	+	0.998721i	$0.516098\pi$
$380$	6.74732	0.346130
$381$	30.5688	1.56608
$382$	−16.1571	−0.826671
$383$	6.24196	0.318949	0.159475	−	0.987202i	$-0.449020\pi$
$383$	6.24196	0.318949	0.159475	+	0.987202i	$0.449020\pi$
$384$	−2.95759	−0.150929
$385$	2.74732	0.140016
$386$	−19.9469	−1.01527
$387$	−5.99108	−0.304544
$388$	−9.83035	−0.499060
$389$	16.7562	0.849575	0.424788	−	0.905293i	$-0.360349\pi$
$389$	16.7562	0.849575	0.424788	+	0.905293i	$0.360349\pi$
$390$	0	0
$391$	−2.95759	−0.149572
$392$	−1.00000	−0.0505076
$393$	−28.0812	−1.41651
$394$	2.08483	0.105032
$395$	−11.4098	−0.574090
$396$	−15.7897	−0.793464
$397$	25.8620	1.29798	0.648989	−	0.760797i	$-0.275192\pi$
$397$	25.8620	1.29798	0.648989	+	0.760797i	$0.275192\pi$
$398$	4.45223	0.223170
$399$	−19.9558	−0.999039
$400$	1.00000	0.0500000
$401$	−19.4946	−0.973516	−0.486758	−	0.873537i	$-0.661821\pi$
$401$	−19.4946	−0.973516	−0.486758	+	0.873537i	$0.661821\pi$
$402$	32.7790	1.63487
$403$	0	0
$404$	6.83215	0.339912
$405$	6.78973	0.337385
$406$	2.95759	0.146783
$407$	−27.2402	−1.35024
$408$	2.95759	0.146422
$409$	20.1661	0.997147	0.498574	−	0.866847i	$-0.333857\pi$
$409$	20.1661	0.997147	0.498574	+	0.866847i	$0.333857\pi$
$410$	−1.25268	−0.0618654
$411$	48.3143	2.38317
$412$	−10.7879	−0.531483
$413$	−4.95759	−0.243947
$414$	−16.9982	−0.835416
$415$	−17.8303	−0.875257
$416$	0	0
$417$	16.2509	0.795809
$418$	18.5371	0.906677
$419$	18.5688	0.907143	0.453571	−	0.891220i	$-0.350150\pi$
$419$	18.5688	0.907143	0.453571	+	0.891220i	$0.350150\pi$
$420$	−2.95759	−0.144315
$421$	7.91517	0.385762	0.192881	−	0.981222i	$-0.438217\pi$
$421$	7.91517	0.385762	0.192881	+	0.981222i	$0.438217\pi$
$422$	−17.4098	−0.847497
$423$	16.9982	0.826481
$424$	0.957587	0.0465045
$425$	−1.00000	−0.0485071
$426$	1.24376	0.0602604
$427$	9.83035	0.475724
$428$	−17.4098	−0.841535
$429$	0	0
$430$	1.04241	0.0502696
$431$	15.6607	0.754349	0.377175	−	0.926142i	$-0.376896\pi$
$431$	15.6607	0.754349	0.377175	+	0.926142i	$0.376896\pi$
$432$	8.12544	0.390935
$433$	33.8303	1.62578	0.812891	−	0.582415i	$-0.197892\pi$
$433$	33.8303	1.62578	0.812891	+	0.582415i	$0.197892\pi$
$434$	5.16785	0.248065
$435$	8.74732	0.419402
$436$	3.67321	0.175915
$437$	19.9558	0.954615
$438$	−17.7455	−0.847914
$439$	34.0281	1.62407	0.812036	−	0.583607i	$-0.198359\pi$
$439$	34.0281	1.62407	0.812036	+	0.583607i	$0.198359\pi$
$440$	2.74732	0.130973
$441$	5.74732	0.273682
$442$	0	0
$443$	−0.0316961	−0.00150593	−0.000752964	−	1.00000i	$-0.500240\pi$
$443$	−0.0316961	−0.00150593	−0.000752964		1.00000i	$0.500240\pi$
$444$	29.3250	1.39170
$445$	−13.4098	−0.635686
$446$	4.00000	0.189405
$447$	−12.8232	−0.606518
$448$	−1.00000	−0.0472456
$449$	39.8089	1.87870	0.939349	−	0.342962i	$-0.111430\pi$
$449$	39.8089	1.87870	0.939349	+	0.342962i	$0.111430\pi$
$450$	−5.74732	−0.270931
$451$	−3.44151	−0.162054
$452$	−12.6625	−0.595594
$453$	15.2844	0.718123
$454$	17.5884	0.825464
$455$	0	0
$456$	−19.9558	−0.934515
$457$	3.32499	0.155536	0.0777682	−	0.996971i	$-0.475221\pi$
$457$	3.32499	0.155536	0.0777682	+	0.996971i	$0.475221\pi$
$458$	−18.7879	−0.877903
$459$	−8.12544	−0.379263
$460$	2.95759	0.137898
$461$	−3.67321	−0.171079	−0.0855393	−	0.996335i	$-0.527261\pi$
$461$	−3.67321	−0.171079	−0.0855393	+	0.996335i	$0.527261\pi$
$462$	−8.12544	−0.378030
$463$	−2.33571	−0.108549	−0.0542747	−	0.998526i	$-0.517285\pi$
$463$	−2.33571	−0.108549	−0.0542747	+	0.998526i	$0.517285\pi$
$464$	2.95759	0.137303
$465$	15.2844	0.708796
$466$	6.00000	0.277945
$467$	−20.3991	−0.943958	−0.471979	−	0.881610i	$-0.656460\pi$
$467$	−20.3991	−0.943958	−0.471979	+	0.881610i	$0.656460\pi$
$468$	0	0
$469$	11.0830	0.511767
$470$	−2.95759	−0.136423
$471$	35.4910	1.63534
$472$	−4.95759	−0.228191
$473$	2.86384	0.131680
$474$	33.7455	1.54998
$475$	6.74732	0.309588
$476$	1.00000	0.0458349
$477$	−5.50356	−0.251991
$478$	−17.6518	−0.807374
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	−2.95759	−0.134995
$481$	0	0
$482$	−7.46295	−0.339928
$483$	−8.74732	−0.398017
$484$	−3.45223	−0.156919
$485$	−9.83035	−0.446373
$486$	4.29509	0.194829
$487$	0.0937441	0.00424795	0.00212397	−	0.999998i	$-0.499324\pi$
$487$	0.0937441	0.00424795	0.00212397	+	0.999998i	$0.499324\pi$
$488$	9.83035	0.444999
$489$	57.6571	2.60734
$490$	−1.00000	−0.0451754
$491$	−27.6607	−1.24831	−0.624155	−	0.781301i	$-0.714557\pi$
$491$	−27.6607	−1.24831	−0.624155	+	0.781301i	$0.714557\pi$
$492$	3.70491	0.167030
$493$	−2.95759	−0.133203
$494$	0	0
$495$	−15.7897	−0.709696
$496$	5.16785	0.232043
$497$	0.420532	0.0188635
$498$	52.7348	2.36310
$499$	−41.3973	−1.85320	−0.926599	−	0.376051i	$-0.877282\pi$
$499$	−41.3973	−1.85320	−0.926599	+	0.376051i	$0.877282\pi$
$500$	1.00000	0.0447214
$501$	−36.7348	−1.64119
$502$	−18.5777	−0.829162
$503$	−29.3250	−1.30754	−0.653768	−	0.756695i	$-0.726813\pi$
$503$	−29.3250	−1.30754	−0.653768	+	0.756695i	$0.726813\pi$
$504$	5.74732	0.256006
$505$	6.83215	0.304027
$506$	8.12544	0.361220
$507$	−38.4486	−1.70756
$508$	10.3357	0.458573
$509$	7.79865	0.345669	0.172835	−	0.984951i	$-0.444707\pi$
$509$	7.79865	0.345669	0.172835	+	0.984951i	$0.444707\pi$
$510$	2.95759	0.130964
$511$	−6.00000	−0.265424
$512$	−1.00000	−0.0441942
$513$	54.8250	2.42058
$514$	22.8728	1.00887
$515$	−10.7879	−0.475373
$516$	−3.08303	−0.135723
$517$	−8.12544	−0.357356
$518$	9.91517	0.435648
$519$	−68.0192	−2.98571
$520$	0	0
$521$	−33.0388	−1.44746	−0.723728	−	0.690085i	$-0.757573\pi$
$521$	−33.0388	−1.44746	−0.723728	+	0.690085i	$0.757573\pi$
$522$	−16.9982	−0.743991
$523$	4.08483	0.178617	0.0893085	−	0.996004i	$-0.471534\pi$
$523$	4.08483	0.178617	0.0893085	+	0.996004i	$0.471534\pi$
$524$	−9.49464	−0.414775
$525$	−2.95759	−0.129080
$526$	19.8303	0.864644
$527$	−5.16785	−0.225115
$528$	−8.12544	−0.353614
$529$	−14.2527	−0.619682
$530$	0.957587	0.0415949
$531$	28.4928	1.23648
$532$	−6.74732	−0.292533
$533$	0	0
$534$	39.6607	1.71629
$535$	−17.4098	−0.752692
$536$	11.0830	0.478714
$537$	−7.40982	−0.319757
$538$	−17.6607	−0.761407
$539$	−2.74732	−0.118335
$540$	8.12544	0.349663
$541$	−22.0089	−0.946237	−0.473119	−	0.880999i	$-0.656872\pi$
$541$	−22.0089	−0.946237	−0.473119	+	0.880999i	$0.656872\pi$
$542$	18.1661	0.780299
$543$	23.9116	1.02614
$544$	1.00000	0.0428746
$545$	3.67321	0.157343
$546$	0	0
$547$	−34.7348	−1.48515	−0.742577	−	0.669761i	$-0.766396\pi$
$547$	−34.7348	−1.48515	−0.742577	+	0.669761i	$0.766396\pi$
$548$	16.3357	0.697827
$549$	−56.4982	−2.41128
$550$	2.74732	0.117146
$551$	19.9558	0.850145
$552$	−8.74732	−0.372311
$553$	11.4098	0.485195
$554$	12.2509	0.520490
$555$	29.3250	1.24478
$556$	5.49464	0.233025
$557$	32.6625	1.38395	0.691977	−	0.721919i	$-0.256740\pi$
$557$	32.6625	1.38395	0.691977	+	0.721919i	$0.256740\pi$
$558$	−29.7013	−1.25736
$559$	0	0
$560$	−1.00000	−0.0422577
$561$	8.12544	0.343056
$562$	−3.29329	−0.138919
$563$	13.6607	0.575730	0.287865	−	0.957671i	$-0.407055\pi$
$563$	13.6607	0.575730	0.287865	+	0.957671i	$0.407055\pi$
$564$	8.74732	0.368329
$565$	−12.6625	−0.532715
$566$	19.6607	0.826401
$567$	−6.78973	−0.285142
$568$	0.420532	0.0176451
$569$	32.2420	1.35165	0.675827	−	0.737061i	$-0.263787\pi$
$569$	32.2420	1.35165	0.675827	+	0.737061i	$0.263787\pi$
$570$	−19.9558	−0.835856
$571$	34.4080	1.43993	0.719965	−	0.694010i	$-0.244158\pi$
$571$	34.4080	1.43993	0.719965	+	0.694010i	$0.244158\pi$
$572$	0	0
$573$	47.7861	1.99629
$574$	1.25268	0.0522858
$575$	2.95759	0.123340
$576$	5.74732	0.239472
$577$	17.5036	0.728683	0.364341	−	0.931265i	$-0.381294\pi$
$577$	17.5036	0.728683	0.364341	+	0.931265i	$0.381294\pi$
$578$	−1.00000	−0.0415945
$579$	58.9946	2.45173
$580$	2.95759	0.122807
$581$	17.8303	0.739727
$582$	29.0741	1.20516
$583$	2.63080	0.108957
$584$	−6.00000	−0.248282
$585$	0	0
$586$	20.4839	0.846183
$587$	7.66429	0.316339	0.158170	−	0.987412i	$-0.449441\pi$
$587$	7.66429	0.316339	0.158170	+	0.987412i	$0.449441\pi$
$588$	2.95759	0.121969
$589$	34.8692	1.43676
$590$	−4.95759	−0.204101
$591$	−6.16605	−0.253638
$592$	9.91517	0.407511
$593$	31.9469	1.31190	0.655950	−	0.754804i	$-0.272268\pi$
$593$	31.9469	1.31190	0.655950	+	0.754804i	$0.272268\pi$
$594$	22.3232	0.915932
$595$	1.00000	0.0409960
$596$	−4.33571	−0.177597
$597$	−13.1679	−0.538924
$598$	0	0
$599$	−28.1129	−1.14866	−0.574331	−	0.818623i	$-0.694738\pi$
$599$	−28.1129	−1.14866	−0.574331	+	0.818623i	$0.694738\pi$
$600$	−2.95759	−0.120743
$601$	−5.90626	−0.240921	−0.120461	−	0.992718i	$-0.538437\pi$
$601$	−5.90626	−0.240921	−0.120461	+	0.992718i	$0.538437\pi$
$602$	−1.04241	−0.0424856
$603$	−63.6977	−2.59397
$604$	5.16785	0.210277
$605$	−3.45223	−0.140353
$606$	−20.2067	−0.820840
$607$	19.2402	0.780934	0.390467	−	0.920617i	$-0.372314\pi$
$607$	19.2402	0.780934	0.390467	+	0.920617i	$0.372314\pi$
$608$	−6.74732	−0.273640
$609$	−8.74732	−0.354459
$610$	9.83035	0.398019
$611$	0	0
$612$	−5.74732	−0.232322
$613$	−44.4803	−1.79654	−0.898272	−	0.439440i	$-0.855177\pi$
$613$	−44.4803	−1.79654	−0.898272	+	0.439440i	$0.855177\pi$
$614$	−0.0848260	−0.00342330
$615$	3.70491	0.149396
$616$	−2.74732	−0.110693
$617$	41.6290	1.67592	0.837960	−	0.545731i	$-0.183748\pi$
$617$	41.6290	1.67592	0.837960	+	0.545731i	$0.183748\pi$
$618$	31.9063	1.28346
$619$	−41.7455	−1.67789	−0.838947	−	0.544213i	$-0.816828\pi$
$619$	−41.7455	−1.67789	−0.838947	+	0.544213i	$0.816828\pi$
$620$	5.16785	0.207546
$621$	24.0317	0.964359
$622$	−3.54777	−0.142253
$623$	13.4098	0.537253
$624$	0	0
$625$	1.00000	0.0400000
$626$	14.9045	0.595702
$627$	−54.8250	−2.18950
$628$	12.0000	0.478852
$629$	−9.91517	−0.395344
$630$	5.74732	0.228979
$631$	1.99108	0.0792637	0.0396319	−	0.999214i	$-0.487381\pi$
$631$	1.99108	0.0792637	0.0396319	+	0.999214i	$0.487381\pi$
$632$	11.4098	0.453858
$633$	51.4910	2.04659
$634$	−9.49464	−0.377080
$635$	10.3357	0.410160
$636$	−2.83215	−0.112302
$637$	0	0
$638$	8.12544	0.321689
$639$	−2.41693	−0.0956125
$640$	−1.00000	−0.0395285
$641$	14.3143	0.565380	0.282690	−	0.959211i	$-0.408773\pi$
$641$	14.3143	0.565380	0.282690	+	0.959211i	$0.408773\pi$
$642$	51.4910	2.03219
$643$	37.9786	1.49773	0.748864	−	0.662723i	$-0.230600\pi$
$643$	37.9786	1.49773	0.748864	+	0.662723i	$0.230600\pi$
$644$	−2.95759	−0.116545
$645$	−3.08303	−0.121394
$646$	6.74732	0.265470
$647$	18.9893	0.746546	0.373273	−	0.927722i	$-0.378236\pi$
$647$	18.9893	0.746546	0.373273	+	0.927722i	$0.378236\pi$
$648$	−6.78973	−0.266726
$649$	−13.6201	−0.534635
$650$	0	0
$651$	−15.2844	−0.599042
$652$	19.4946	0.763469
$653$	−12.7348	−0.498351	−0.249176	−	0.968458i	$-0.580160\pi$
$653$	−12.7348	−0.498351	−0.249176	+	0.968458i	$0.580160\pi$
$654$	−10.8638	−0.424810
$655$	−9.49464	−0.370986
$656$	1.25268	0.0489089
$657$	34.4839	1.34535
$658$	2.95759	0.115299
$659$	−36.4839	−1.42121	−0.710606	−	0.703590i	$-0.751579\pi$
$659$	−36.4839	−1.42121	−0.710606	+	0.703590i	$0.751579\pi$
$660$	−8.12544	−0.316282
$661$	40.8285	1.58805	0.794023	−	0.607887i	$-0.207983\pi$
$661$	40.8285	1.58805	0.794023	+	0.607887i	$0.207983\pi$
$662$	22.9893	0.893504
$663$	0	0
$664$	17.8303	0.691952
$665$	−6.74732	−0.261650
$666$	−56.9857	−2.20815
$667$	8.74732	0.338698
$668$	−12.4205	−0.480565
$669$	−11.8303	−0.457388
$670$	11.0830	0.428175
$671$	27.0071	1.04260
$672$	2.95759	0.114091
$673$	22.1571	0.854095	0.427047	−	0.904229i	$-0.359554\pi$
$673$	22.1571	0.854095	0.427047	+	0.904229i	$0.359554\pi$
$674$	−13.8620	−0.533946
$675$	8.12544	0.312748
$676$	−13.0000	−0.500000
$677$	28.7879	1.10641	0.553205	−	0.833045i	$-0.313405\pi$
$677$	28.7879	1.10641	0.553205	+	0.833045i	$0.313405\pi$
$678$	37.4504	1.43828
$679$	9.83035	0.377254
$680$	1.00000	0.0383482
$681$	−52.0192	−1.99338
$682$	14.1978	0.543660
$683$	−42.6714	−1.63278	−0.816388	−	0.577504i	$-0.804027\pi$
$683$	−42.6714	−1.63278	−0.816388	+	0.577504i	$0.804027\pi$
$684$	38.7790	1.48275
$685$	16.3357	0.624155
$686$	1.00000	0.0381802
$687$	55.5670	2.12001
$688$	−1.04241	−0.0397416
$689$	0	0
$690$	−8.74732	−0.333005
$691$	−36.9045	−1.40391	−0.701956	−	0.712220i	$-0.747690\pi$
$691$	−36.9045	−1.40391	−0.701956	+	0.712220i	$0.747690\pi$
$692$	−22.9982	−0.874260
$693$	15.7897	0.599802
$694$	23.7455	0.901368
$695$	5.49464	0.208424
$696$	−8.74732	−0.331566
$697$	−1.25268	−0.0474486
$698$	−35.2402	−1.33386
$699$	−17.7455	−0.671197
$700$	−1.00000	−0.0377964
$701$	−1.40982	−0.0532480	−0.0266240	−	0.999646i	$-0.508476\pi$
$701$	−1.40982	−0.0532480	−0.0266240	+	0.999646i	$0.508476\pi$
$702$	0	0
$703$	66.9009	2.52321
$704$	−2.74732	−0.103544
$705$	8.74732	0.329443
$706$	10.3268	0.388654
$707$	−6.83215	−0.256949
$708$	14.6625	0.551050
$709$	−24.5335	−0.921373	−0.460687	−	0.887563i	$-0.652397\pi$
$709$	−24.5335	−0.921373	−0.460687	+	0.887563i	$0.652397\pi$
$710$	0.420532	0.0157823
$711$	−65.5759	−2.45929
$712$	13.4098	0.502554
$713$	15.2844	0.572404
$714$	−2.95759	−0.110685
$715$	0	0
$716$	−2.50536	−0.0936296
$717$	52.2067	1.94969
$718$	−0.621881	−0.0232084
$719$	26.4928	0.988016	0.494008	−	0.869457i	$-0.335531\pi$
$719$	26.4928	0.988016	0.494008	+	0.869457i	$0.335531\pi$
$720$	5.74732	0.214190
$721$	10.7879	0.401764
$722$	−26.5263	−0.987208
$723$	22.0723	0.820878
$724$	8.08483	0.300470
$725$	2.95759	0.109842
$726$	10.2103	0.378939
$727$	−21.8835	−0.811613	−0.405807	−	0.913959i	$-0.633009\pi$
$727$	−21.8835	−0.811613	−0.405807	+	0.913959i	$0.633009\pi$
$728$	0	0
$729$	−33.0723	−1.22490
$730$	−6.00000	−0.222070
$731$	1.04241	0.0385550
$732$	−29.0741	−1.07461
$733$	−8.48392	−0.313361	−0.156680	−	0.987649i	$-0.550079\pi$
$733$	−8.48392	−0.313361	−0.156680	+	0.987649i	$0.550079\pi$
$734$	−5.01072	−0.184949
$735$	2.95759	0.109092
$736$	−2.95759	−0.109018
$737$	30.4486	1.12159
$738$	−7.19955	−0.265019
$739$	3.57947	0.131673	0.0658364	−	0.997830i	$-0.479028\pi$
$739$	3.57947	0.131673	0.0658364	+	0.997830i	$0.479028\pi$
$740$	9.91517	0.364489
$741$	0	0
$742$	−0.957587	−0.0351541
$743$	−4.84106	−0.177601	−0.0888007	−	0.996049i	$-0.528303\pi$
$743$	−4.84106	−0.177601	−0.0888007	+	0.996049i	$0.528303\pi$
$744$	−15.2844	−0.560352
$745$	−4.33571	−0.158848
$746$	23.7138	0.868225
$747$	−102.477	−3.74943
$748$	2.74732	0.100452
$749$	17.4098	0.636141
$750$	−2.95759	−0.107996
$751$	2.14822	0.0783896	0.0391948	−	0.999232i	$-0.487521\pi$
$751$	2.14822	0.0783896	0.0391948	+	0.999232i	$0.487521\pi$
$752$	2.95759	0.107852
$753$	54.9451	2.00231
$754$	0	0
$755$	5.16785	0.188077
$756$	−8.12544	−0.295519
$757$	−35.9786	−1.30766	−0.653832	−	0.756640i	$-0.726840\pi$
$757$	−35.9786	−1.30766	−0.653832	+	0.756640i	$0.726840\pi$
$758$	1.96830	0.0714920
$759$	−24.0317	−0.872295
$760$	−6.74732	−0.244751
$761$	1.32859	0.0481612	0.0240806	−	0.999710i	$-0.492334\pi$
$761$	1.32859	0.0481612	0.0240806	+	0.999710i	$0.492334\pi$
$762$	−30.5688	−1.10739
$763$	−3.67321	−0.132979
$764$	16.1571	0.584545
$765$	−5.74732	−0.207795
$766$	−6.24196	−0.225531
$767$	0	0
$768$	2.95759	0.106723
$769$	−22.9857	−0.828885	−0.414443	−	0.910075i	$-0.636023\pi$
$769$	−22.9857	−0.828885	−0.414443	+	0.910075i	$0.636023\pi$
$770$	−2.74732	−0.0990066
$771$	−67.6482	−2.43629
$772$	19.9469	0.717904
$773$	47.3214	1.70203	0.851016	−	0.525140i	$-0.175987\pi$
$773$	47.3214	1.70203	0.851016	+	0.525140i	$0.175987\pi$
$774$	5.99108	0.215345
$775$	5.16785	0.185635
$776$	9.83035	0.352889
$777$	−29.3250	−1.05203
$778$	−16.7562	−0.600740
$779$	8.45223	0.302833
$780$	0	0
$781$	1.15534	0.0413412
$782$	2.95759	0.105763
$783$	24.0317	0.858823
$784$	1.00000	0.0357143
$785$	12.0000	0.428298
$786$	28.0812	1.00162
$787$	43.0830	1.53574	0.767872	−	0.640603i	$-0.221316\pi$
$787$	43.0830	1.53574	0.767872	+	0.640603i	$0.221316\pi$
$788$	−2.08483	−0.0742689
$789$	−58.6500	−2.08799
$790$	11.4098	0.405943
$791$	12.6625	0.450226
$792$	15.7897	0.561064
$793$	0	0
$794$	−25.8620	−0.917810
$795$	−2.83215	−0.100446
$796$	−4.45223	−0.157805
$797$	3.83035	0.135678	0.0678389	−	0.997696i	$-0.478390\pi$
$797$	3.83035	0.135678	0.0678389	+	0.997696i	$0.478390\pi$
$798$	19.9558	0.706427
$799$	−2.95759	−0.104632
$800$	−1.00000	−0.0353553
$801$	−77.0705	−2.72315
$802$	19.4946	0.688380
$803$	−16.4839	−0.581705
$804$	−32.7790	−1.15603
$805$	−2.95759	−0.104241
$806$	0	0
$807$	52.2330	1.83869
$808$	−6.83215	−0.240354
$809$	10.4205	0.366366	0.183183	−	0.983079i	$-0.441360\pi$
$809$	10.4205	0.366366	0.183183	+	0.983079i	$0.441360\pi$
$810$	−6.78973	−0.238567
$811$	9.26160	0.325219	0.162609	−	0.986691i	$-0.448009\pi$
$811$	9.26160	0.325219	0.162609	+	0.986691i	$0.448009\pi$
$812$	−2.95759	−0.103791
$813$	−53.7277	−1.88431
$814$	27.2402	0.954767
$815$	19.4946	0.682868
$816$	−2.95759	−0.103536
$817$	−7.03349	−0.246071
$818$	−20.1661	−0.705090
$819$	0	0
$820$	1.25268	0.0437455
$821$	−50.9009	−1.77645	−0.888226	−	0.459407i	$-0.848062\pi$
$821$	−50.9009	−1.77645	−0.888226	+	0.459407i	$0.848062\pi$
$822$	−48.3143	−1.68515
$823$	−27.5670	−0.960924	−0.480462	−	0.877016i	$-0.659531\pi$
$823$	−27.5670	−0.960924	−0.480462	+	0.877016i	$0.659531\pi$
$824$	10.7879	0.375816
$825$	−8.12544	−0.282892
$826$	4.95759	0.172497
$827$	39.3250	1.36746	0.683732	−	0.729733i	$-0.260356\pi$
$827$	39.3250	1.36746	0.683732	+	0.729733i	$0.260356\pi$
$828$	16.9982	0.590728
$829$	46.1357	1.60236	0.801180	−	0.598424i	$-0.204206\pi$
$829$	46.1357	1.60236	0.801180	+	0.598424i	$0.204206\pi$
$830$	17.8303	0.618900
$831$	−36.2330	−1.25691
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	−16.2509	−0.562722
$835$	−12.4205	−0.429830
$836$	−18.5371	−0.641117
$837$	41.9911	1.45142
$838$	−18.5688	−0.641447
$839$	−21.3567	−0.737315	−0.368657	−	0.929565i	$-0.620182\pi$
$839$	−21.3567	−0.737315	−0.368657	+	0.929565i	$0.620182\pi$
$840$	2.95759	0.102046
$841$	−20.2527	−0.698368
$842$	−7.91517	−0.272775
$843$	9.74020	0.335470
$844$	17.4098	0.599271
$845$	−13.0000	−0.447214
$846$	−16.9982	−0.584410
$847$	3.45223	0.118620
$848$	−0.957587	−0.0328837
$849$	−58.1482	−1.99564
$850$	1.00000	0.0342997
$851$	29.3250	1.00525
$852$	−1.24376	−0.0426106
$853$	−1.86204	−0.0637552	−0.0318776	−	0.999492i	$-0.510149\pi$
$853$	−1.86204	−0.0637552	−0.0318776	+	0.999492i	$0.510149\pi$
$854$	−9.83035	−0.336388
$855$	38.7790	1.32621
$856$	17.4098	0.595055
$857$	−43.3250	−1.47995	−0.739977	−	0.672632i	$-0.765164\pi$
$857$	−43.3250	−1.47995	−0.739977	+	0.672632i	$0.765164\pi$
$858$	0	0
$859$	16.7384	0.571107	0.285553	−	0.958363i	$-0.407823\pi$
$859$	16.7384	0.571107	0.285553	+	0.958363i	$0.407823\pi$
$860$	−1.04241	−0.0355460
$861$	−3.70491	−0.126263
$862$	−15.6607	−0.533405
$863$	−42.5866	−1.44966	−0.724832	−	0.688926i	$-0.758083\pi$
$863$	−42.5866	−1.44966	−0.724832	+	0.688926i	$0.758083\pi$
$864$	−8.12544	−0.276433
$865$	−22.9982	−0.781962
$866$	−33.8303	−1.14960
$867$	2.95759	0.100445
$868$	−5.16785	−0.175408
$869$	31.3464	1.06335
$870$	−8.74732	−0.296562
$871$	0	0
$872$	−3.67321	−0.124391
$873$	−56.4982	−1.91217
$874$	−19.9558	−0.675015
$875$	−1.00000	−0.0338062
$876$	17.7455	0.599566
$877$	13.3250	0.449953	0.224976	−	0.974364i	$-0.427769\pi$
$877$	13.3250	0.449953	0.224976	+	0.974364i	$0.427769\pi$
$878$	−34.0281	−1.14839
$879$	−60.5830	−2.04341
$880$	−2.74732	−0.0926122
$881$	−39.5263	−1.33168	−0.665838	−	0.746096i	$-0.731926\pi$
$881$	−39.5263	−1.33168	−0.665838	+	0.746096i	$0.731926\pi$
$882$	−5.74732	−0.193522
$883$	−50.6500	−1.70451	−0.852254	−	0.523129i	$-0.824765\pi$
$883$	−50.6500	−1.70451	−0.852254	+	0.523129i	$0.824765\pi$
$884$	0	0
$885$	14.6625	0.492874
$886$	0.0316961	0.00106485
$887$	52.0812	1.74872	0.874358	−	0.485281i	$-0.161283\pi$
$887$	52.0812	1.74872	0.874358	+	0.485281i	$0.161283\pi$
$888$	−29.3250	−0.984082
$889$	−10.3357	−0.346648
$890$	13.4098	0.449498
$891$	−18.6536	−0.624918
$892$	−4.00000	−0.133930
$893$	19.9558	0.667795
$894$	12.8232	0.428873
$895$	−2.50536	−0.0837449
$896$	1.00000	0.0334077
$897$	0	0
$898$	−39.8089	−1.32844
$899$	15.2844	0.509762
$900$	5.74732	0.191577
$901$	0.957587	0.0319018
$902$	3.44151	0.114590
$903$	3.08303	0.102597
$904$	12.6625	0.421148
$905$	8.08483	0.268749
$906$	−15.2844	−0.507789
$907$	8.16605	0.271149	0.135575	−	0.990767i	$-0.456712\pi$
$907$	8.16605	0.271149	0.135575	+	0.990767i	$0.456712\pi$
$908$	−17.5884	−0.583691
$909$	39.2665	1.30239
$910$	0	0
$911$	35.0705	1.16194	0.580969	−	0.813926i	$-0.302674\pi$
$911$	35.0705	1.16194	0.580969	+	0.813926i	$0.302674\pi$
$912$	19.9558	0.660802
$913$	48.9857	1.62119
$914$	−3.32499	−0.109981
$915$	−29.0741	−0.961160
$916$	18.7879	0.620771
$917$	9.49464	0.313541
$918$	8.12544	0.268179
$919$	14.1344	0.466249	0.233125	−	0.972447i	$-0.425105\pi$
$919$	14.1344	0.466249	0.233125	+	0.972447i	$0.425105\pi$
$920$	−2.95759	−0.0975088
$921$	0.250880	0.00826679
$922$	3.67321	0.120971
$923$	0	0
$924$	8.12544	0.267307
$925$	9.91517	0.326009
$926$	2.33571	0.0767561
$927$	−62.0017	−2.03640
$928$	−2.95759	−0.0970875
$929$	52.4803	1.72182	0.860912	−	0.508754i	$-0.169894\pi$
$929$	52.4803	1.72182	0.860912	+	0.508754i	$0.169894\pi$
$930$	−15.2844	−0.501194
$931$	6.74732	0.221135
$932$	−6.00000	−0.196537
$933$	10.4928	0.343520
$934$	20.3991	0.667479
$935$	2.74732	0.0898470
$936$	0	0
$937$	6.70311	0.218981	0.109491	−	0.993988i	$-0.465078\pi$
$937$	6.70311	0.218981	0.109491	+	0.993988i	$0.465078\pi$
$938$	−11.0830	−0.361874
$939$	−44.0812	−1.43854
$940$	2.95759	0.0964659
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	−35.4910	−1.15636
$943$	3.70491	0.120648
$944$	4.95759	0.161356
$945$	−8.12544	−0.264321
$946$	−2.86384	−0.0931116
$947$	−5.74912	−0.186821	−0.0934106	−	0.995628i	$-0.529777\pi$
$947$	−5.74912	−0.186821	−0.0934106	+	0.995628i	$0.529777\pi$
$948$	−33.7455	−1.09600
$949$	0	0
$950$	−6.74732	−0.218912
$951$	28.0812	0.910596
$952$	−1.00000	−0.0324102
$953$	−10.0812	−0.326563	−0.163282	−	0.986580i	$-0.552208\pi$
$953$	−10.0812	−0.326563	−0.163282	+	0.986580i	$0.552208\pi$
$954$	5.50356	0.178184
$955$	16.1571	0.522833
$956$	17.6518	0.570899
$957$	−24.0317	−0.776834
$958$	16.0000	0.516937
$959$	−16.3357	−0.527507
$960$	2.95759	0.0954557
$961$	−4.29329	−0.138493
$962$	0	0
$963$	−100.060	−3.22438
$964$	7.46295	0.240365
$965$	19.9469	0.642112
$966$	8.74732	0.281440
$967$	−40.5652	−1.30449	−0.652244	−	0.758009i	$-0.726172\pi$
$967$	−40.5652	−1.30449	−0.652244	+	0.758009i	$0.726172\pi$
$968$	3.45223	0.110959
$969$	−19.9558	−0.641072
$970$	9.83035	0.315633
$971$	−49.5848	−1.59125	−0.795626	−	0.605788i	$-0.792858\pi$
$971$	−49.5848	−1.59125	−0.795626	+	0.605788i	$0.792858\pi$
$972$	−4.29509	−0.137765
$973$	−5.49464	−0.176150
$974$	−0.0937441	−0.00300375
$975$	0	0
$976$	−9.83035	−0.314662
$977$	54.4839	1.74310	0.871548	−	0.490311i	$-0.163117\pi$
$977$	54.4839	1.74310	0.871548	+	0.490311i	$0.163117\pi$
$978$	−57.6571	−1.84367
$979$	36.8411	1.17745
$980$	1.00000	0.0319438
$981$	21.1111	0.674026
$982$	27.6607	0.882688
$983$	−7.09554	−0.226313	−0.113156	−	0.993577i	$-0.536096\pi$
$983$	−7.09554	−0.226313	−0.113156	+	0.993577i	$0.536096\pi$
$984$	−3.70491	−0.118108
$985$	−2.08483	−0.0664281
$986$	2.95759	0.0941888
$987$	−8.74732	−0.278430
$988$	0	0
$989$	−3.08303	−0.0980346
$990$	15.7897	0.501831
$991$	27.6607	0.878671	0.439335	−	0.898323i	$-0.355214\pi$
$991$	27.6607	0.878671	0.439335	+	0.898323i	$0.355214\pi$
$992$	−5.16785	−0.164080
$993$	−67.9928	−2.15769
$994$	−0.420532	−0.0133385
$995$	−4.45223	−0.141145
$996$	−52.7348	−1.67097
$997$	−9.32859	−0.295439	−0.147720	−	0.989029i	$-0.547193\pi$
$997$	−9.32859	−0.295439	−0.147720	+	0.989029i	$0.547193\pi$
$998$	41.3973	1.31041
$999$	80.5652	2.54897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1190.2.a.j.1.3	✓	3
4.3	odd	2		9520.2.a.z.1.1		3
5.4	even	2		5950.2.a.bm.1.1		3
7.6	odd	2		8330.2.a.bu.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1190.2.a.j.1.3	✓	3	1.1	even	1	trivial
5950.2.a.bm.1.1		3	5.4	even	2
8330.2.a.bu.1.1		3	7.6	odd	2
9520.2.a.z.1.1		3	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1190 = 2 \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1190.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.95759 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.95759 q^{6} -1.00000 q^{7} -1.00000 q^{8} +5.74732 q^{9} -1.00000 q^{10} -2.74732 q^{11} +2.95759 q^{12} +1.00000 q^{14} +2.95759 q^{15} +1.00000 q^{16} -1.00000 q^{17} -5.74732 q^{18} +6.74732 q^{19} +1.00000 q^{20} -2.95759 q^{21} +2.74732 q^{22} +2.95759 q^{23} -2.95759 q^{24} +1.00000 q^{25} +8.12544 q^{27} -1.00000 q^{28} +2.95759 q^{29} -2.95759 q^{30} +5.16785 q^{31} -1.00000 q^{32} -8.12544 q^{33} +1.00000 q^{34} -1.00000 q^{35} +5.74732 q^{36} +9.91517 q^{37} -6.74732 q^{38} -1.00000 q^{40} +1.25268 q^{41} +2.95759 q^{42} -1.04241 q^{43} -2.74732 q^{44} +5.74732 q^{45} -2.95759 q^{46} +2.95759 q^{47} +2.95759 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.95759 q^{51} -0.957587 q^{53} -8.12544 q^{54} -2.74732 q^{55} +1.00000 q^{56} +19.9558 q^{57} -2.95759 q^{58} +4.95759 q^{59} +2.95759 q^{60} -9.83035 q^{61} -5.16785 q^{62} -5.74732 q^{63} +1.00000 q^{64} +8.12544 q^{66} -11.0830 q^{67} -1.00000 q^{68} +8.74732 q^{69} +1.00000 q^{70} -0.420532 q^{71} -5.74732 q^{72} +6.00000 q^{73} -9.91517 q^{74} +2.95759 q^{75} +6.74732 q^{76} +2.74732 q^{77} -11.4098 q^{79} +1.00000 q^{80} +6.78973 q^{81} -1.25268 q^{82} -17.8303 q^{83} -2.95759 q^{84} -1.00000 q^{85} +1.04241 q^{86} +8.74732 q^{87} +2.74732 q^{88} -13.4098 q^{89} -5.74732 q^{90} +2.95759 q^{92} +15.2844 q^{93} -2.95759 q^{94} +6.74732 q^{95} -2.95759 q^{96} -9.83035 q^{97} -1.00000 q^{98} -15.7897 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 3 q^{7} - 3 q^{8} + 10 q^{9} - 3 q^{10} - q^{11} - q^{12} + 3 q^{14} - q^{15} + 3 q^{16} - 3 q^{17} - 10 q^{18} + 13 q^{19} + 3 q^{20} + q^{21}+ \cdots - 50 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - q^3 + 3 * q^4 + 3 * q^5 + q^6 - 3 * q^7 - 3 * q^8 + 10 * q^9 - 3 * q^10 - q^11 - q^12 + 3 * q^14 - q^15 + 3 * q^16 - 3 * q^17 - 10 * q^18 + 13 * q^19 + 3 * q^20 + q^21 + q^22 - q^23 + q^24 + 3 * q^25 + 2 * q^27 - 3 * q^28 - q^29 + q^30 + 3 * q^31 - 3 * q^32 - 2 * q^33 + 3 * q^34 - 3 * q^35 + 10 * q^36 + 10 * q^37 - 13 * q^38 - 3 * q^40 + 11 * q^41 - q^42 - 13 * q^43 - q^44 + 10 * q^45 + q^46 - q^47 - q^48 + 3 * q^49 - 3 * q^50 + q^51 + 7 * q^53 - 2 * q^54 - q^55 + 3 * q^56 - 2 * q^57 + q^58 + 5 * q^59 - q^60 + 10 * q^61 - 3 * q^62 - 10 * q^63 + 3 * q^64 + 2 * q^66 - q^67 - 3 * q^68 + 19 * q^69 + 3 * q^70 + 4 * q^71 - 10 * q^72 + 18 * q^73 - 10 * q^74 - q^75 + 13 * q^76 + q^77 + 3 * q^80 + 23 * q^81 - 11 * q^82 - 14 * q^83 + q^84 - 3 * q^85 + 13 * q^86 + 19 * q^87 + q^88 - 6 * q^89 - 10 * q^90 - q^92 + 34 * q^93 + q^94 + 13 * q^95 + q^96 + 10 * q^97 - 3 * q^98 - 50 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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