LMFDB - Embedded newform 245.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,4,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("245.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	245.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:	$14.4554679514$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.14360.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 17x - 14 $ x^3 - 17*x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.62456$ of defining polynomial
Character	$\chi$	$=$	245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.62456 q^{2} +8.38660 q^{3} +13.3866 q^{4} -5.00000 q^{5} -38.7844 q^{6} -24.9107 q^{8} +43.3350 q^{9} +23.1228 q^{10} -30.1117 q^{11} +112.268 q^{12} -88.9295 q^{13} -41.9330 q^{15} +8.10818 q^{16} +4.73699 q^{17} -200.405 q^{18} -124.818 q^{19} -66.9330 q^{20} +139.253 q^{22} +20.2680 q^{23} -208.916 q^{24} +25.0000 q^{25} +411.260 q^{26} +136.995 q^{27} +134.088 q^{29} +193.922 q^{30} +2.03767 q^{31} +161.788 q^{32} -252.534 q^{33} -21.9065 q^{34} +580.108 q^{36} -141.137 q^{37} +577.228 q^{38} -745.816 q^{39} +124.553 q^{40} -95.2784 q^{41} -298.646 q^{43} -403.093 q^{44} -216.675 q^{45} -93.7305 q^{46} +129.054 q^{47} +68.0000 q^{48} -115.614 q^{50} +39.7272 q^{51} -1190.46 q^{52} +388.429 q^{53} -633.542 q^{54} +150.558 q^{55} -1046.80 q^{57} -620.098 q^{58} -838.501 q^{59} -561.340 q^{60} -389.422 q^{61} -9.42333 q^{62} -813.067 q^{64} +444.647 q^{65} +1167.86 q^{66} +697.794 q^{67} +63.4122 q^{68} +169.979 q^{69} -523.450 q^{71} -1079.50 q^{72} -66.4684 q^{73} +652.699 q^{74} +209.665 q^{75} -1670.89 q^{76} +3449.07 q^{78} -526.982 q^{79} -40.5409 q^{80} -21.1236 q^{81} +440.621 q^{82} -70.0265 q^{83} -23.6850 q^{85} +1381.11 q^{86} +1124.54 q^{87} +750.101 q^{88} +9.27925 q^{89} +1002.03 q^{90} +271.319 q^{92} +17.0891 q^{93} -596.817 q^{94} +624.089 q^{95} +1356.85 q^{96} +4.19493 q^{97} -1304.89 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 2 q^{3} + 13 q^{4} - 15 q^{5} - 24 q^{6} - 15 q^{8} + 81 q^{9} + 15 q^{10} - 74 q^{11} + 152 q^{12} - 44 q^{13} + 10 q^{15} - 79 q^{16} + 52 q^{17} - 411 q^{18} - 168 q^{19} - 65 q^{20}+ \cdots - 3488 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 2 * q^3 + 13 * q^4 - 15 * q^5 - 24 * q^6 - 15 * q^8 + 81 * q^9 + 15 * q^10 - 74 * q^11 + 152 * q^12 - 44 * q^13 + 10 * q^15 - 79 * q^16 + 52 * q^17 - 411 * q^18 - 168 * q^19 - 65 * q^20 + 184 * q^22 - 124 * q^23 - 420 * q^24 + 75 * q^25 + 446 * q^26 - 170 * q^27 + 332 * q^29 + 120 * q^30 - 320 * q^31 - 183 * q^32 + 106 * q^33 - 582 * q^34 + 181 * q^36 - 54 * q^37 + 460 * q^38 - 982 * q^39 + 75 * q^40 - 362 * q^41 - 16 * q^43 - 264 * q^44 - 405 * q^45 - 336 * q^46 + 730 * q^47 + 204 * q^48 - 75 * q^50 - 1178 * q^51 - 1202 * q^52 + 110 * q^53 + 180 * q^54 + 370 * q^55 - 956 * q^57 + 450 * q^58 + 180 * q^59 - 760 * q^60 - 1222 * q^61 - 464 * q^62 - 391 * q^64 + 220 * q^65 + 532 * q^66 + 204 * q^67 - 918 * q^68 + 716 * q^69 - 136 * q^71 + 765 * q^72 - 310 * q^73 + 502 * q^74 - 50 * q^75 - 1796 * q^76 + 3788 * q^78 - 1034 * q^79 + 395 * q^80 + 2283 * q^81 + 6 * q^82 + 1660 * q^83 - 260 * q^85 + 764 * q^86 + 1574 * q^87 - 20 * q^88 - 242 * q^89 + 2055 * q^90 + 96 * q^92 + 1376 * q^93 + 1108 * q^94 + 840 * q^95 + 3156 * q^96 - 100 * q^97 - 3488 * 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 $

$2$	−4.62456	−1.63503	−0.817515	−	0.575907i	$-0.804649\pi$
$2$	−4.62456	−1.63503	−0.817515	+	0.575907i	$0.804649\pi$
$3$	8.38660	1.61400	0.807001	−	0.590551i	$-0.201089\pi$
$3$	8.38660	1.61400	0.807001	+	0.590551i	$0.201089\pi$
$4$	13.3866	1.67332
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−38.7844	−2.63894
$7$	0	0
$7$	0	0
$8$	−24.9107	−1.10091
$9$	43.3350	1.60500
$10$	23.1228	0.731208
$11$	−30.1117	−0.825364	−0.412682	−	0.910875i	$-0.635408\pi$
$11$	−30.1117	−0.825364	−0.412682	+	0.910875i	$0.635408\pi$
$12$	112.268	2.70075
$13$	−88.9295	−1.89728	−0.948639	−	0.316362i	$-0.897539\pi$
$13$	−88.9295	−1.89728	−0.948639	+	0.316362i	$0.897539\pi$
$14$	0	0
$15$	−41.9330	−0.721803
$16$	8.10818	0.126690
$17$	4.73699	0.0675817	0.0337909	−	0.999429i	$-0.489242\pi$
$17$	4.73699	0.0675817	0.0337909	+	0.999429i	$0.489242\pi$
$18$	−200.405	−2.62422
$19$	−124.818	−1.50711	−0.753557	−	0.657382i	$-0.771664\pi$
$19$	−124.818	−1.50711	−0.753557	+	0.657382i	$0.771664\pi$
$20$	−66.9330	−0.748333
$21$	0	0
$22$	139.253	1.34950
$23$	20.2680	0.183746	0.0918731	−	0.995771i	$-0.470715\pi$
$23$	20.2680	0.183746	0.0918731	+	0.995771i	$0.470715\pi$
$24$	−208.916	−1.77686
$25$	25.0000	0.200000
$26$	411.260	3.10211
$27$	136.995	0.976470
$28$	0	0
$29$	134.088	0.858603	0.429301	−	0.903161i	$-0.358760\pi$
$29$	134.088	0.858603	0.429301	+	0.903161i	$0.358760\pi$
$30$	193.922	1.18017
$31$	2.03767	0.0118057	0.00590284	−	0.999983i	$-0.498121\pi$
$31$	2.03767	0.0118057	0.00590284	+	0.999983i	$0.498121\pi$
$32$	161.788	0.893764
$33$	−252.534	−1.33214
$34$	−21.9065	−0.110498
$35$	0	0
$36$	580.108	2.68568
$37$	−141.137	−0.627104	−0.313552	−	0.949571i	$-0.601519\pi$
$37$	−141.137	−0.627104	−0.313552	+	0.949571i	$0.601519\pi$
$38$	577.228	2.46418
$39$	−745.816	−3.06221
$40$	124.553	0.492340
$41$	−95.2784	−0.362927	−0.181463	−	0.983398i	$-0.558083\pi$
$41$	−95.2784	−0.362927	−0.181463	+	0.983398i	$0.558083\pi$
$42$	0	0
$43$	−298.646	−1.05914	−0.529571	−	0.848266i	$-0.677647\pi$
$43$	−298.646	−1.05914	−0.529571	+	0.848266i	$0.677647\pi$
$44$	−403.093	−1.38110
$45$	−216.675	−0.717778
$46$	−93.7305	−0.300431
$47$	129.054	0.400519	0.200260	−	0.979743i	$-0.435821\pi$
$47$	129.054	0.400519	0.200260	+	0.979743i	$0.435821\pi$
$48$	68.0000	0.204478
$49$	0	0
$50$	−115.614	−0.327006
$51$	39.7272	0.109077
$52$	−1190.46	−3.17476
$53$	388.429	1.00669	0.503347	−	0.864084i	$-0.332102\pi$
$53$	388.429	1.00669	0.503347	+	0.864084i	$0.332102\pi$
$54$	−633.542	−1.59656
$55$	150.558	0.369114
$56$	0	0
$57$	−1046.80	−2.43248
$58$	−620.098	−1.40384
$59$	−838.501	−1.85023	−0.925114	−	0.379688i	$-0.876031\pi$
$59$	−838.501	−1.85023	−0.925114	+	0.379688i	$0.876031\pi$
$60$	−561.340	−1.20781
$61$	−389.422	−0.817384	−0.408692	−	0.912672i	$-0.634015\pi$
$61$	−389.422	−0.817384	−0.408692	+	0.912672i	$0.634015\pi$
$62$	−9.42333	−0.0193027
$63$	0	0
$64$	−813.067	−1.58802
$65$	444.647	0.848488
$66$	1167.86	2.17809
$67$	697.794	1.27237	0.636187	−	0.771534i	$-0.280510\pi$
$67$	697.794	1.27237	0.636187	+	0.771534i	$0.280510\pi$
$68$	63.4122	0.113086
$69$	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	$0.644129\pi$
$72$	−1079.50	−1.76695
$73$	−66.4684	−0.106569	−0.0532845	−	0.998579i	$-0.516969\pi$
$73$	−66.4684	−0.106569	−0.0532845	+	0.998579i	$0.516969\pi$
$74$	652.699	1.02533
$75$	209.665	0.322800
$76$	−1670.89	−2.52189
$77$	0	0
$78$	3449.07	5.00680
$79$	−526.982	−0.750508	−0.375254	−	0.926922i	$-0.622444\pi$
$79$	−526.982	−0.750508	−0.375254	+	0.926922i	$0.622444\pi$
$80$	−40.5409	−0.0566576
$81$	−21.1236	−0.0289762
$82$	440.621	0.593396
$83$	−70.0265	−0.0926074	−0.0463037	−	0.998927i	$-0.514744\pi$
$83$	−70.0265	−0.0926074	−0.0463037	+	0.998927i	$0.514744\pi$
$84$	0	0
$85$	−23.6850	−0.0302235
$86$	1381.11	1.73173
$87$	1124.54	1.38579
$88$	750.101	0.908649
$89$	9.27925	0.0110517	0.00552584	−	0.999985i	$-0.498241\pi$
$89$	9.27925	0.0110517	0.00552584	+	0.999985i	$0.498241\pi$
$90$	1002.03	1.17359
$91$	0	0
$92$	271.319	0.307467
$93$	17.0891	0.0190544
$94$	−596.817	−0.654861
$95$	624.089	0.674002
$96$	1356.85	1.44254
$97$	4.19493	0.00439104	0.00219552	−	0.999998i	$-0.499301\pi$
$97$	4.19493	0.00439104	0.00219552	+	0.999998i	$0.499301\pi$
$98$	0	0
$99$	−1304.89	−1.32471
$100$	334.665	0.334665
$101$	865.844	0.853016	0.426508	−	0.904484i	$-0.359744\pi$
$101$	865.844	0.853016	0.426508	+	0.904484i	$0.359744\pi$
$102$	−183.721	−0.178344
$103$	1166.12	1.11554	0.557771	−	0.829995i	$-0.311657\pi$
$103$	1166.12	1.11554	0.557771	+	0.829995i	$0.311657\pi$
$104$	2215.29	2.08872
$105$	0	0
$106$	−1796.31	−1.64598
$107$	56.9652	0.0514676	0.0257338	−	0.999669i	$-0.491808\pi$
$107$	56.9652	0.0514676	0.0257338	+	0.999669i	$0.491808\pi$
$108$	1833.90	1.63395
$109$	−1358.89	−1.19411	−0.597055	−	0.802200i	$-0.703663\pi$
$109$	−1358.89	−1.19411	−0.597055	+	0.802200i	$0.703663\pi$
$110$	−696.267	−0.603513
$111$	−1183.66	−1.01215
$112$	0	0
$113$	436.038	0.363000	0.181500	−	0.983391i	$-0.441905\pi$
$113$	436.038	0.363000	0.181500	+	0.983391i	$0.441905\pi$
$114$	4840.98	3.97719
$115$	−101.340	−0.0821738
$116$	1794.98	1.43672
$117$	−3853.76	−3.04513
$118$	3877.70	3.02518
$119$	0	0
$120$	1044.58	0.794637
$121$	−424.288	−0.318774
$122$	1800.91	1.33645
$123$	−799.062	−0.585764
$124$	27.2775	0.0197547
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1186.69	0.829144	0.414572	−	0.910017i	$-0.363931\pi$
$127$	1186.69	0.829144	0.414572	+	0.910017i	$0.363931\pi$
$128$	2465.77	1.70270
$129$	−2504.62	−1.70946
$130$	−2056.30	−1.38730
$131$	−1034.56	−0.689997	−0.344999	−	0.938603i	$-0.612121\pi$
$131$	−1034.56	−0.689997	−0.344999	+	0.938603i	$0.612121\pi$
$132$	−3380.57	−2.22910
$133$	0	0
$134$	−3226.99	−2.08037
$135$	−684.975	−0.436691
$136$	−118.002	−0.0744011
$137$	646.219	0.402994	0.201497	−	0.979489i	$-0.435419\pi$
$137$	646.219	0.402994	0.201497	+	0.979489i	$0.435419\pi$
$138$	−786.080	−0.484895
$139$	−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
$142$	2420.73	1.43058
$143$	2677.81	1.56594
$144$	351.368	0.203338
$145$	−670.439	−0.383979
$146$	307.387	0.174244
$147$	0	0
$148$	−1889.35	−1.04935
$149$	−1828.12	−1.00513	−0.502567	−	0.864538i	$-0.667611\pi$
$149$	−1828.12	−1.00513	−0.502567	+	0.864538i	$0.667611\pi$
$150$	−969.609	−0.527788
$151$	2975.17	1.60342	0.801708	−	0.597716i	$-0.203925\pi$
$151$	2975.17	1.60342	0.801708	+	0.597716i	$0.203925\pi$
$152$	3109.29	1.65919
$153$	205.278	0.108469
$154$	0	0
$155$	−10.1883	−0.00527966
$156$	−9983.93	−5.12407
$157$	2131.74	1.08364	0.541820	−	0.840495i	$-0.317736\pi$
$157$	2131.74	1.08364	0.541820	+	0.840495i	$0.317736\pi$
$158$	2437.06	1.22710
$159$	3257.59	1.62481
$160$	−808.942	−0.399703
$161$	0	0
$162$	97.6876	0.0473769
$163$	−593.939	−0.285404	−0.142702	−	0.989766i	$-0.545579\pi$
$163$	−593.939	−0.285404	−0.142702	+	0.989766i	$0.545579\pi$
$164$	−1275.45	−0.607294
$165$	1262.67	0.595751
$166$	323.842	0.151416
$167$	2936.30	1.36059	0.680293	−	0.732941i	$-0.261853\pi$
$167$	2936.30	1.36059	0.680293	+	0.732941i	$0.261853\pi$
$168$	0	0
$169$	5711.45	2.59966
$170$	109.533	0.0494163
$171$	−5408.98	−2.41892
$172$	−3997.85	−1.77229
$173$	−2347.31	−1.03158	−0.515788	−	0.856716i	$-0.672501\pi$
$173$	−2347.31	−1.03158	−0.515788	+	0.856716i	$0.672501\pi$
$174$	−5200.51	−2.26580
$175$	0	0
$176$	−244.151	−0.104566
$177$	−7032.17	−2.98627
$178$	−42.9125	−0.0180698
$179$	3036.56	1.26795	0.633975	−	0.773354i	$-0.281422\pi$
$179$	3036.56	1.26795	0.633975	+	0.773354i	$0.281422\pi$
$180$	−2900.54	−1.20107
$181$	899.776	0.369502	0.184751	−	0.982785i	$-0.440852\pi$
$181$	899.776	0.369502	0.184751	+	0.982785i	$0.440852\pi$
$182$	0	0
$183$	−3265.93	−1.31926
$184$	−504.888	−0.202287
$185$	705.687	0.280449
$186$	−79.0297	−0.0311545
$187$	−142.639	−0.0557796
$188$	1727.59	0.670199
$189$	0	0
$190$	−2886.14	−1.10201
$191$	416.168	0.157659	0.0788294	−	0.996888i	$-0.474882\pi$
$191$	416.168	0.157659	0.0788294	+	0.996888i	$0.474882\pi$
$192$	−6818.86	−2.56307
$193$	−5181.05	−1.93233	−0.966166	−	0.257922i	$-0.916962\pi$
$193$	−5181.05	−1.93233	−0.966166	+	0.257922i	$0.916962\pi$
$194$	−19.3997	−0.00717948
$195$	3729.08	1.36946
$196$	0	0
$197$	1452.34	0.525255	0.262627	−	0.964897i	$-0.415411\pi$
$197$	1452.34	0.525255	0.262627	+	0.964897i	$0.415411\pi$
$198$	6034.54	2.16594
$199$	1277.23	0.454978	0.227489	−	0.973781i	$-0.426948\pi$
$199$	1277.23	0.454978	0.227489	+	0.973781i	$0.426948\pi$
$200$	−622.766	−0.220181
$201$	5852.12	2.05361
$202$	−4004.15	−1.39471
$203$	0	0
$204$	531.813	0.182521
$205$	476.392	0.162306
$206$	−5392.78	−1.82395
$207$	878.312	0.294913
$208$	−721.056	−0.240367
$209$	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	$0.678436\pi$
$212$	5199.74	1.68453
$213$	−4389.96	−1.41218
$214$	−263.439	−0.0841511
$215$	1493.23	0.473663
$216$	−3412.63	−1.07500
$217$	0	0
$218$	6284.27	1.95241
$219$	−557.444	−0.172002
$220$	2015.46	0.617648
$221$	−421.258	−0.128221
$222$	5473.92	1.65489
$223$	−4373.35	−1.31328	−0.656639	−	0.754205i	$-0.728023\pi$
$223$	−4373.35	−1.31328	−0.656639	+	0.754205i	$0.728023\pi$
$224$	0	0
$225$	1083.37	0.321000
$226$	−2016.48	−0.593516
$227$	61.1145	0.0178692	0.00893461	−	0.999960i	$-0.497156\pi$
$227$	61.1145	0.0178692	0.00893461	+	0.999960i	$0.497156\pi$
$228$	−14013.0	−4.07034
$229$	−3019.41	−0.871302	−0.435651	−	0.900116i	$-0.643482\pi$
$229$	−3019.41	−0.871302	−0.435651	+	0.900116i	$0.643482\pi$
$230$	468.653	0.134357
$231$	0	0
$232$	−3340.22	−0.945241
$233$	−3531.17	−0.992851	−0.496426	−	0.868079i	$-0.665354\pi$
$233$	−3531.17	−0.992851	−0.496426	+	0.868079i	$0.665354\pi$
$234$	17822.0	4.97888
$235$	−645.268	−0.179118
$236$	−11224.7	−3.09603
$237$	−4419.58	−1.21132
$238$	0	0
$239$	2282.62	0.617785	0.308893	−	0.951097i	$-0.400042\pi$
$239$	2282.62	0.617785	0.308893	+	0.951097i	$0.400042\pi$
$240$	−340.000	−0.0914454
$241$	2215.68	0.592217	0.296109	−	0.955154i	$-0.404311\pi$
$241$	2215.68	0.592217	0.296109	+	0.955154i	$0.404311\pi$
$242$	1962.15	0.521205
$243$	−3876.02	−1.02324
$244$	−5213.04	−1.36775
$245$	0	0
$246$	3695.31	0.957742
$247$	11100.0	2.85941
$248$	−50.7597	−0.0129970
$249$	−587.284	−0.149468
$250$	578.071	0.146242
$251$	3082.55	0.775174	0.387587	−	0.921833i	$-0.373309\pi$
$251$	3082.55	0.775174	0.387587	+	0.921833i	$0.373309\pi$
$252$	0	0
$253$	−610.302	−0.151658
$254$	−5487.90	−1.35568
$255$	−198.636	−0.0487807
$256$	−4898.58	−1.19594
$257$	6032.40	1.46417	0.732083	−	0.681215i	$-0.238548\pi$
$257$	6032.40	1.46417	0.732083	+	0.681215i	$0.238548\pi$
$258$	11582.8	2.79501
$259$	0	0
$260$	5952.32	1.41980
$261$	5810.69	1.37806
$262$	4784.37	1.12817
$263$	5923.81	1.38889	0.694445	−	0.719546i	$-0.255650\pi$
$263$	5923.81	1.38889	0.694445	+	0.719546i	$0.255650\pi$
$264$	6290.80	1.46656
$265$	−1942.14	−0.450207
$266$	0	0
$267$	77.8213	0.0178374
$268$	9341.09	2.12910
$269$	−3252.80	−0.737273	−0.368637	−	0.929574i	$-0.620175\pi$
$269$	−3252.80	−0.737273	−0.368637	+	0.929574i	$0.620175\pi$
$270$	3167.71	0.714002
$271$	6246.26	1.40012	0.700061	−	0.714083i	$-0.253156\pi$
$271$	6246.26	1.40012	0.700061	+	0.714083i	$0.253156\pi$
$272$	38.4084	0.00856195
$273$	0	0
$274$	−2988.48	−0.658907
$275$	−752.792	−0.165073
$276$	2275.44	0.496252
$277$	−1572.17	−0.341020	−0.170510	−	0.985356i	$-0.554541\pi$
$277$	−1572.17	−0.341020	−0.170510	+	0.985356i	$0.554541\pi$
$278$	2342.27	0.505324
$279$	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	$0.813285\pi$
$282$	−5005.26	−1.05695
$283$	−6265.58	−1.31608	−0.658039	−	0.752984i	$-0.728614\pi$
$283$	−6265.58	−1.31608	−0.658039	+	0.752984i	$0.728614\pi$
$284$	−7007.21	−1.46409
$285$	5233.98	1.08784
$286$	−12383.7	−2.56037
$287$	0	0
$288$	7011.10	1.43449
$289$	−4890.56	−0.995433
$290$	3100.49	0.627817
$291$	35.1812	0.00708714
$292$	−889.785	−0.178325
$293$	7264.99	1.44855	0.724276	−	0.689511i	$-0.242174\pi$
$293$	7264.99	1.44855	0.724276	+	0.689511i	$0.242174\pi$
$294$	0	0
$295$	4192.50	0.827448
$296$	3515.83	0.690382
$297$	−4125.14	−0.805943
$298$	8454.24	1.64343
$299$	−1802.42	−0.348617
$300$	2806.70	0.540149
$301$	0	0
$302$	−13758.9	−2.62163
$303$	7261.48	1.37677
$304$	−1012.05	−0.190937
$305$	1947.11	0.365545
$306$	−949.319	−0.177350
$307$	−1328.32	−0.246943	−0.123471	−	0.992348i	$-0.539403\pi$
$307$	−1328.32	−0.246943	−0.123471	+	0.992348i	$0.539403\pi$
$308$	0	0
$309$	9779.75	1.80049
$310$	47.1167	0.00863241
$311$	−4868.68	−0.887709	−0.443855	−	0.896099i	$-0.646389\pi$
$311$	−4868.68	−0.887709	−0.443855	+	0.896099i	$0.646389\pi$
$312$	18578.8	3.37120
$313$	−7733.39	−1.39654	−0.698270	−	0.715835i	$-0.746046\pi$
$313$	−7733.39	−1.39654	−0.698270	+	0.715835i	$0.746046\pi$
$314$	−9858.37	−1.77178
$315$	0	0
$316$	−7054.49	−1.25584
$317$	−8175.03	−1.44844	−0.724220	−	0.689569i	$-0.757800\pi$
$317$	−8175.03	−1.44844	−0.724220	+	0.689569i	$0.757800\pi$
$318$	−15065.0	−2.65661
$319$	−4037.61	−0.708660
$320$	4065.33	0.710184
$321$	477.744	0.0830688
$322$	0	0
$323$	−591.261	−0.101853
$324$	−282.774	−0.0484865
$325$	−2223.24	−0.379455
$326$	2746.71	0.466644
$327$	−11396.5	−1.92729
$328$	2373.45	0.399548
$329$	0	0
$330$	−5839.31	−0.974070
$331$	−2040.76	−0.338884	−0.169442	−	0.985540i	$-0.554197\pi$
$331$	−2040.76	−0.338884	−0.169442	+	0.985540i	$0.554197\pi$
$332$	−937.417	−0.154962
$333$	−6116.19	−1.00650
$334$	−13579.1	−2.22460
$335$	−3488.97	−0.569023
$336$	0	0
$337$	7349.73	1.18803	0.594013	−	0.804455i	$-0.297543\pi$
$337$	7349.73	1.18803	0.594013	+	0.804455i	$0.297543\pi$
$338$	−26413.0	−4.25052
$339$	3656.87	0.585882
$340$	−317.061	−0.0505737
$341$	−61.3576	−0.00974399
$342$	25014.2	3.95500
$343$	0	0
$344$	7439.47	1.16602
$345$	−849.896	−0.132629
$346$	10855.3	1.68666
$347$	−12069.9	−1.86728	−0.933642	−	0.358207i	$-0.883388\pi$
$347$	−12069.9	−1.86728	−0.933642	+	0.358207i	$0.883388\pi$
$348$	15053.8	2.31887
$349$	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
$352$	−4871.72	−0.737681
$353$	12762.5	1.92430	0.962151	−	0.272517i	$-0.0878561\pi$
$353$	12762.5	1.92430	0.962151	+	0.272517i	$0.0878561\pi$
$354$	32520.7	4.88264
$355$	2617.25	0.391293
$356$	124.218	0.0184930
$357$	0	0
$358$	−14042.8	−2.07314
$359$	−2419.42	−0.355689	−0.177844	−	0.984059i	$-0.556912\pi$
$359$	−2419.42	−0.355689	−0.177844	+	0.984059i	$0.556912\pi$
$360$	5397.52	0.790206
$361$	8720.49	1.27139
$362$	−4161.07	−0.604147
$363$	−3558.33	−0.514501
$364$	0	0
$365$	332.342	0.0476591
$366$	15103.5	2.15703
$367$	7129.74	1.01409	0.507043	−	0.861921i	$-0.330739\pi$
$367$	7129.74	1.01409	0.507043	+	0.861921i	$0.330739\pi$
$368$	164.336	0.0232789
$369$	−4128.89	−0.582497
$370$	−3263.50	−0.458543
$371$	0	0
$372$	228.765	0.0318842
$373$	11596.9	1.60983	0.804914	−	0.593391i	$-0.202211\pi$
$373$	11596.9	1.60983	0.804914	+	0.593391i	$0.202211\pi$
$374$	659.642	0.0912013
$375$	−1048.32	−0.144361
$376$	−3214.81	−0.440934
$377$	−11924.4	−1.62901
$378$	0	0
$379$	−12770.8	−1.73085	−0.865424	−	0.501040i	$-0.832951\pi$
$379$	−12770.8	−1.73085	−0.865424	+	0.501040i	$0.832951\pi$
$380$	8354.43	1.12782
$381$	9952.25	1.33824
$382$	−1924.59	−0.257777
$383$	−7470.10	−0.996617	−0.498308	−	0.867000i	$-0.666045\pi$
$383$	−7470.10	−0.996617	−0.498308	+	0.867000i	$0.666045\pi$
$384$	20679.4	2.74816
$385$	0	0
$386$	23960.1	3.15942
$387$	−12941.8	−1.69992
$388$	56.1558	0.00734763
$389$	8749.77	1.14044	0.570220	−	0.821492i	$-0.306858\pi$
$389$	8749.77	1.14044	0.570220	+	0.821492i	$0.306858\pi$
$390$	−17245.4	−2.23911
$391$	96.0092	0.0124179
$392$	0	0
$393$	−8676.41	−1.11366
$394$	−6716.46	−0.858808
$395$	2634.91	0.335637
$396$	−17468.0	−2.21667
$397$	−5375.25	−0.679537	−0.339769	−	0.940509i	$-0.610349\pi$
$397$	−5375.25	−0.679537	−0.339769	+	0.940509i	$0.610349\pi$
$398$	−5906.65	−0.743903
$399$	0	0
$400$	202.704	0.0253381
$401$	7361.33	0.916727	0.458363	−	0.888765i	$-0.348436\pi$
$401$	7361.33	0.916727	0.458363	+	0.888765i	$0.348436\pi$
$402$	−27063.5	−3.35772
$403$	−181.209	−0.0223987
$404$	11590.7	1.42737
$405$	105.618	0.0129585
$406$	0	0
$407$	4249.88	0.517589
$408$	−989.632	−0.120084
$409$	2612.45	0.315837	0.157919	−	0.987452i	$-0.449522\pi$
$409$	2612.45	0.315837	0.157919	+	0.987452i	$0.449522\pi$
$410$	−2203.11	−0.265375
$411$	5419.57	0.650433
$412$	15610.3	1.86667
$413$	0	0
$414$	−4061.81	−0.482191
$415$	350.133	0.0414153
$416$	−14387.8	−1.69572
$417$	−4247.68	−0.498824
$418$	−17381.3	−2.03384
$419$	−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	$0.309720\pi$
$422$	15071.9	1.73859
$423$	5592.54	0.642833
$424$	−9676.01	−1.10828
$425$	118.425	0.0135163
$426$	20301.7	2.30896
$427$	0	0
$428$	762.570	0.0861220
$429$	22457.7	2.52744
$430$	−6905.54	−0.774453
$431$	−14314.5	−1.59978	−0.799892	−	0.600144i	$-0.795110\pi$
$431$	−14314.5	−1.59978	−0.799892	+	0.600144i	$0.795110\pi$
$432$	1110.78	0.123709
$433$	2373.62	0.263438	0.131719	−	0.991287i	$-0.457950\pi$
$433$	2373.62	0.263438	0.131719	+	0.991287i	$0.457950\pi$
$434$	0	0
$435$	−5622.70	−0.619742
$436$	−18190.9	−1.99813
$437$	−2529.80	−0.276927
$438$	2577.93	0.281229
$439$	9533.46	1.03646	0.518231	−	0.855240i	$-0.326591\pi$
$439$	9533.46	1.03646	0.518231	+	0.855240i	$0.326591\pi$
$440$	−3750.51	−0.406360
$441$	0	0
$442$	1948.14	0.209646
$443$	6647.94	0.712987	0.356493	−	0.934298i	$-0.383972\pi$
$443$	6647.94	0.712987	0.356493	+	0.934298i	$0.383972\pi$
$444$	−15845.2	−1.69365
$445$	−46.3963	−0.00494246
$446$	20224.8	2.14725
$447$	−15331.7	−1.62229
$448$	0	0
$449$	−768.256	−0.0807489	−0.0403744	−	0.999185i	$-0.512855\pi$
$449$	−768.256	−0.0807489	−0.0403744	+	0.999185i	$0.512855\pi$
$450$	−5010.14	−0.524845
$451$	2868.99	0.299547
$452$	5837.06	0.607416
$453$	24951.5	2.58791
$454$	−282.628	−0.0292167
$455$	0	0
$456$	26076.4	2.67794
$457$	−3323.50	−0.340190	−0.170095	−	0.985428i	$-0.554407\pi$
$457$	−3323.50	−0.340190	−0.170095	+	0.985428i	$0.554407\pi$
$458$	13963.5	1.42461
$459$	648.944	0.0659915
$460$	−1356.60	−0.137503
$461$	18840.7	1.90347	0.951733	−	0.306926i	$-0.0993004\pi$
$461$	18840.7	1.90347	0.951733	+	0.306926i	$0.0993004\pi$
$462$	0	0
$463$	−10759.1	−1.07995	−0.539977	−	0.841679i	$-0.681567\pi$
$463$	−10759.1	−1.07995	−0.539977	+	0.841679i	$0.681567\pi$
$464$	1087.21	0.108777
$465$	−85.4456	−0.00852138
$466$	16330.1	1.62334
$467$	−7441.70	−0.737390	−0.368695	−	0.929550i	$-0.620195\pi$
$467$	−7441.70	−0.737390	−0.368695	+	0.929550i	$0.620195\pi$
$468$	−51588.7	−5.09549
$469$	0	0
$470$	2984.08	0.292863
$471$	17878.0	1.74899
$472$	20887.6	2.03693
$473$	8992.73	0.874178
$474$	20438.7	1.98055
$475$	−3120.45	−0.301423
$476$	0	0
$477$	16832.6	1.61574
$478$	−10556.1	−1.01010
$479$	−5691.97	−0.542949	−0.271475	−	0.962446i	$-0.587511\pi$
$479$	−5691.97	−0.542949	−0.271475	+	0.962446i	$0.587511\pi$
$480$	−6784.27	−0.645121
$481$	12551.3	1.18979
$482$	−10246.5	−0.968293
$483$	0	0
$484$	−5679.77	−0.533412
$485$	−20.9746	−0.00196373
$486$	17924.9	1.67302
$487$	−2020.25	−0.187980	−0.0939899	−	0.995573i	$-0.529962\pi$
$487$	−2020.25	−0.187980	−0.0939899	+	0.995573i	$0.529962\pi$
$488$	9700.77	0.899863
$489$	−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	$0.385867\pi$
$492$	−10696.7	−0.980173
$493$	635.173	0.0580259
$494$	−51332.6	−4.67523
$495$	6524.44	0.592428
$496$	16.5218	0.00149567
$497$	0	0
$498$	2715.93	0.244385
$499$	6284.56	0.563799	0.281900	−	0.959444i	$-0.409036\pi$
$499$	6284.56	0.563799	0.281900	+	0.959444i	$0.409036\pi$
$500$	−1673.32	−0.149667
$501$	24625.6	2.19599
$502$	−14255.4	−1.26743
$503$	−11310.9	−1.00264	−0.501319	−	0.865262i	$-0.667152\pi$
$503$	−11310.9	−1.00264	−0.501319	+	0.865262i	$0.667152\pi$
$504$	0	0
$505$	−4329.22	−0.381481
$506$	2822.38	0.247965
$507$	47899.7	4.19586
$508$	15885.7	1.38743
$509$	−10712.7	−0.932876	−0.466438	−	0.884554i	$-0.654463\pi$
$509$	−10712.7	−0.932876	−0.466438	+	0.884554i	$0.654463\pi$
$510$	918.606	0.0797580
$511$	0	0
$512$	2927.65	0.252705
$513$	−17099.4	−1.47165
$514$	−27897.2	−2.39396
$515$	−5830.58	−0.498886
$516$	−33528.4	−2.86047
$517$	−3886.02	−0.330574
$518$	0	0
$519$	−19685.9	−1.66496
$520$	−11076.5	−0.934106
$521$	−17721.9	−1.49023	−0.745116	−	0.666935i	$-0.767606\pi$
$521$	−17721.9	−1.49023	−0.745116	+	0.666935i	$0.767606\pi$
$522$	−26871.9	−2.25317
$523$	−237.193	−0.0198312	−0.00991562	−	0.999951i	$-0.503156\pi$
$523$	−237.193	−0.0198312	−0.00991562	+	0.999951i	$0.503156\pi$
$524$	−13849.2	−1.15459
$525$	0	0
$526$	−27395.0	−2.27088
$527$	9.65243	0.000797849 0
$528$	−2047.59	−0.168769
$529$	−11756.2	−0.966237
$530$	8981.57	0.736103
$531$	−36336.4	−2.96962
$532$	0	0
$533$	8473.06	0.688572
$534$	−359.890	−0.0291647
$535$	−284.826	−0.0230170
$536$	−17382.5	−1.40077
$537$	25466.4	2.04647
$538$	15042.8	1.20546
$539$	0	0
$540$	−9169.48	−0.730725
$541$	−5352.94	−0.425399	−0.212699	−	0.977118i	$-0.568226\pi$
$541$	−5352.94	−0.425399	−0.212699	+	0.977118i	$0.568226\pi$
$542$	−28886.2	−2.28924
$543$	7546.06	0.596376
$544$	766.391	0.0604021
$545$	6794.45	0.534022
$546$	0	0
$547$	−192.162	−0.0150206	−0.00751030	−	0.999972i	$-0.502391\pi$
$547$	−192.162	−0.0150206	−0.00751030	+	0.999972i	$0.502391\pi$
$548$	8650.67	0.674340
$549$	−16875.6	−1.31190
$550$	3481.33	0.269899
$551$	−16736.5	−1.29401
$552$	−4234.29	−0.326492
$553$	0	0
$554$	7270.60	0.557578
$555$	5918.31	0.452646
$556$	−6780.10	−0.517159
$557$	−4850.62	−0.368990	−0.184495	−	0.982833i	$-0.559065\pi$
$557$	−4850.62	−0.368990	−0.184495	+	0.982833i	$0.559065\pi$
$558$	−408.360	−0.0309808
$559$	26558.4	2.00949
$560$	0	0
$561$	−1196.25	−0.0900283
$562$	36284.4	2.72343
$563$	−9699.11	−0.726055	−0.363027	−	0.931778i	$-0.618257\pi$
$563$	−9699.11	−0.726055	−0.363027	+	0.931778i	$0.618257\pi$
$564$	14488.6	1.08170
$565$	−2180.19	−0.162338
$566$	28975.6	2.15183
$567$	0	0
$568$	13039.5	0.963247
$569$	3109.53	0.229100	0.114550	−	0.993417i	$-0.463457\pi$
$569$	3109.53	0.229100	0.114550	+	0.993417i	$0.463457\pi$
$570$	−24204.9	−1.77865
$571$	−14476.2	−1.06097	−0.530483	−	0.847695i	$-0.677990\pi$
$571$	−14476.2	−1.06097	−0.530483	+	0.847695i	$0.677990\pi$
$572$	35846.8	2.62033
$573$	3490.23	0.254461
$574$	0	0
$575$	506.699	0.0367492
$576$	−35234.2	−2.54877
$577$	2208.23	0.159323	0.0796617	−	0.996822i	$-0.474616\pi$
$577$	2208.23	0.159323	0.0796617	+	0.996822i	$0.474616\pi$
$578$	22616.7	1.62756
$579$	−43451.4	−3.11879
$580$	−8974.90	−0.642521
$581$	0	0
$582$	−162.698	−0.0115877
$583$	−11696.2	−0.830889
$584$	1655.77	0.117322
$585$	19268.8	1.36182
$586$	−33597.4	−2.36843
$587$	23988.7	1.68675	0.843374	−	0.537327i	$-0.180566\pi$
$587$	23988.7	1.68675	0.843374	+	0.537327i	$0.180566\pi$
$588$	0	0
$589$	−254.338	−0.0177925
$590$	−19388.5	−1.35290
$591$	12180.2	0.847762
$592$	−1144.37	−0.0794480
$593$	15869.4	1.09895	0.549474	−	0.835511i	$-0.314828\pi$
$593$	15869.4	1.09895	0.549474	+	0.835511i	$0.314828\pi$
$594$	19077.0	1.31774
$595$	0	0
$596$	−24472.2	−1.68192
$597$	10711.6	0.734335
$598$	8335.41	0.570000
$599$	−15236.6	−1.03932	−0.519660	−	0.854373i	$-0.673941\pi$
$599$	−15236.6	−1.03932	−0.519660	+	0.854373i	$0.673941\pi$
$600$	−5222.89	−0.355373
$601$	−12258.8	−0.832026	−0.416013	−	0.909359i	$-0.636573\pi$
$601$	−12258.8	−0.832026	−0.416013	+	0.909359i	$0.636573\pi$
$602$	0	0
$603$	30238.9	2.04216
$604$	39827.4	2.68303
$605$	2121.44	0.142560
$606$	−33581.2	−2.25106
$607$	−23487.2	−1.57054	−0.785269	−	0.619155i	$-0.787475\pi$
$607$	−23487.2	−1.57054	−0.785269	+	0.619155i	$0.787475\pi$
$608$	−20194.1	−1.34700
$609$	0	0
$610$	−9004.54	−0.597678
$611$	−11476.7	−0.759896
$612$	2747.97	0.181503
$613$	−22305.3	−1.46966	−0.734830	−	0.678251i	$-0.762738\pi$
$613$	−22305.3	−1.46966	−0.734830	+	0.678251i	$0.762738\pi$
$614$	6142.92	0.403759
$615$	3995.31	0.261962
$616$	0	0
$617$	3285.91	0.214402	0.107201	−	0.994237i	$-0.465811\pi$
$617$	3285.91	0.214402	0.107201	+	0.994237i	$0.465811\pi$
$618$	−45227.1	−2.94385
$619$	11613.1	0.754069	0.377035	−	0.926199i	$-0.376944\pi$
$619$	11613.1	0.754069	0.377035	+	0.926199i	$0.376944\pi$
$620$	−136.387	−0.00883459
$621$	2776.61	0.179423
$622$	22515.5	1.45143
$623$	0	0
$624$	−6047.21	−0.387952
$625$	625.000	0.0400000
$626$	35763.5	2.28338
$627$	31520.8	2.00769
$628$	28536.7	1.81328
$629$	−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	$0.430252\pi$
$632$	13127.5	0.826238
$633$	−27332.7	−1.71623
$634$	37805.9	2.36824
$635$	−5933.43	−0.370804
$636$	43608.1	2.71883
$637$	0	0
$638$	18672.2	1.15868
$639$	−22683.7	−1.40431
$640$	−12328.9	−0.761470
$641$	18769.3	1.15654	0.578269	−	0.815846i	$-0.303728\pi$
$641$	18769.3	1.15654	0.578269	+	0.815846i	$0.303728\pi$
$642$	−2209.36	−0.135820
$643$	3142.30	0.192722	0.0963609	−	0.995346i	$-0.469280\pi$
$643$	3142.30	0.192722	0.0963609	+	0.995346i	$0.469280\pi$
$644$	0	0
$645$	12523.1	0.764492
$646$	2734.33	0.166533
$647$	−19038.1	−1.15683	−0.578413	−	0.815744i	$-0.696328\pi$
$647$	−19038.1	−1.15683	−0.578413	+	0.815744i	$0.696328\pi$
$648$	526.204	0.0319000
$649$	25248.7	1.52711
$650$	10281.5	0.620421
$651$	0	0
$652$	−7950.82	−0.477574
$653$	−20538.6	−1.23084	−0.615420	−	0.788199i	$-0.711014\pi$
$653$	−20538.6	−1.23084	−0.615420	+	0.788199i	$0.711014\pi$
$654$	52703.6	3.15119
$655$	5172.78	0.308576
$656$	−772.534	−0.0459793
$657$	−2880.41	−0.171043
$658$	0	0
$659$	−937.046	−0.0553902	−0.0276951	−	0.999616i	$-0.508817\pi$
$659$	−937.046	−0.0553902	−0.0276951	+	0.999616i	$0.508817\pi$
$660$	16902.9	0.996884
$661$	−21116.5	−1.24257	−0.621283	−	0.783586i	$-0.713388\pi$
$661$	−21116.5	−1.24257	−0.621283	+	0.783586i	$0.713388\pi$
$662$	9437.65	0.554085
$663$	−3532.92	−0.206949
$664$	1744.41	0.101952
$665$	0	0
$666$	28284.7	1.64566
$667$	2717.69	0.157765
$668$	39307.1	2.27670
$669$	−36677.5	−2.11963
$670$	16135.0	0.930371
$671$	11726.2	0.674640
$672$	0	0
$673$	13825.9	0.791903	0.395952	−	0.918271i	$-0.370415\pi$
$673$	13825.9	0.791903	0.395952	+	0.918271i	$0.370415\pi$
$674$	−33989.3	−1.94246
$675$	3424.87	0.195294
$676$	76456.9	4.35008
$677$	16928.4	0.961021	0.480510	−	0.876989i	$-0.340451\pi$
$677$	16928.4	0.961021	0.480510	+	0.876989i	$0.340451\pi$
$678$	−16911.4	−0.957935
$679$	0	0
$680$	590.008	0.0332732
$681$	512.543	0.0288409
$682$	283.752	0.0159317
$683$	−13817.3	−0.774091	−0.387045	−	0.922061i	$-0.626504\pi$
$683$	−13817.3	−0.774091	−0.387045	+	0.922061i	$0.626504\pi$
$684$	−72407.8	−4.04763
$685$	−3231.09	−0.180224
$686$	0	0
$687$	−25322.6	−1.40628
$688$	−2421.47	−0.134183
$689$	−34542.8	−1.90998
$690$	3930.40	0.216852
$691$	23671.6	1.30320	0.651600	−	0.758563i	$-0.274098\pi$
$691$	23671.6	1.30320	0.651600	+	0.758563i	$0.274098\pi$
$692$	−31422.5	−1.72616
$693$	0	0
$694$	55818.2	3.05307
$695$	2532.42	0.138216
$696$	−28013.0	−1.52562
$697$	−451.333	−0.0245272
$698$	−20741.0	−1.12472
$699$	−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	$0.651519\pi$
$702$	56340.6	3.02911
$703$	17616.5	0.945117
$704$	24482.8	1.31070
$705$	−5411.60	−0.289096
$706$	−59020.9	−3.14629
$707$	0	0
$708$	−94136.8	−4.99700
$709$	22038.9	1.16740	0.583701	−	0.811969i	$-0.301604\pi$
$709$	22038.9	1.16740	0.583701	+	0.811969i	$0.301604\pi$
$710$	−12103.6	−0.639777
$711$	−22836.8	−1.20456
$712$	−231.152	−0.0121669
$713$	41.2994	0.00216925
$714$	0	0
$715$	−13389.1	−0.700312
$716$	40649.2	2.12169
$717$	19143.4	0.997106
$718$	11188.8	0.581562
$719$	−7287.44	−0.377991	−0.188996	−	0.981978i	$-0.560523\pi$
$719$	−7287.44	−0.377991	−0.188996	+	0.981978i	$0.560523\pi$
$720$	−1756.84	−0.0909354
$721$	0	0
$722$	−40328.5	−2.07877
$723$	18582.0	0.955839
$724$	12044.9	0.618297
$725$	3352.20	0.171721
$726$	16455.7	0.841225
$727$	29676.7	1.51396	0.756980	−	0.653438i	$-0.226674\pi$
$727$	29676.7	1.51396	0.756980	+	0.653438i	$0.226674\pi$
$728$	0	0
$729$	−31936.3	−1.62253
$730$	−1536.94	−0.0779241
$731$	−1414.68	−0.0715786
$732$	−43719.7	−2.20755
$733$	−23111.8	−1.16460	−0.582300	−	0.812974i	$-0.697847\pi$
$733$	−23111.8	−1.16460	−0.582300	+	0.812974i	$0.697847\pi$
$734$	−32971.9	−1.65806
$735$	0	0
$736$	3279.12	0.164226
$737$	−21011.7	−1.05017
$738$	19094.3	0.952400
$739$	−31171.4	−1.55164	−0.775818	−	0.630957i	$-0.782662\pi$
$739$	−31171.4	−1.55164	−0.775818	+	0.630957i	$0.782662\pi$
$740$	9446.75	0.469283
$741$	93091.1	4.61510
$742$	0	0
$743$	−31324.4	−1.54668	−0.773338	−	0.633993i	$-0.781415\pi$
$743$	−31324.4	−1.54668	−0.773338	+	0.633993i	$0.781415\pi$
$744$	−425.701	−0.0209771
$745$	9140.58	0.449510
$746$	−53630.7	−2.63212
$747$	−3034.60	−0.148635
$748$	−1909.45	−0.0933373
$749$	0	0
$750$	4848.04	0.236034
$751$	4032.20	0.195922	0.0979608	−	0.995190i	$-0.468768\pi$
$751$	4032.20	0.195922	0.0979608	+	0.995190i	$0.468768\pi$
$752$	1046.39	0.0507419
$753$	25852.1	1.25113
$754$	55145.0	2.66348
$755$	−14875.8	−0.717069
$756$	0	0
$757$	34263.7	1.64509	0.822546	−	0.568699i	$-0.192553\pi$
$757$	34263.7	1.64509	0.822546	+	0.568699i	$0.192553\pi$
$758$	59059.3	2.82999
$759$	−5118.36	−0.244775
$760$	−15546.5	−0.742013
$761$	−7265.88	−0.346108	−0.173054	−	0.984912i	$-0.555363\pi$
$761$	−7265.88	−0.346108	−0.173054	+	0.984912i	$0.555363\pi$
$762$	−46024.8	−2.18806
$763$	0	0
$764$	5571.07	0.263814
$765$	−1026.39	−0.0485087
$766$	34546.0	1.62950
$767$	74567.5	3.51040
$768$	−41082.4	−1.93025
$769$	−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
$772$	−69356.6	−3.23342
$773$	−16158.2	−0.751838	−0.375919	−	0.926652i	$-0.622673\pi$
$773$	−16158.2	−0.751838	−0.375919	+	0.926652i	$0.622673\pi$
$774$	59850.3	2.77942
$775$	50.9417	0.00236114
$776$	−104.498	−0.00483412
$777$	0	0
$778$	−40463.9	−1.86465
$779$	11892.4	0.546972
$780$	49919.7	2.29155
$781$	15761.9	0.722160
$782$	−444.001	−0.0203036
$783$	18369.3	0.838400
$784$	0	0
$785$	−10658.7	−0.484618
$786$	40124.6	1.82086
$787$	−5092.49	−0.230658	−0.115329	−	0.993327i	$-0.536792\pi$
$787$	−5092.49	−0.230658	−0.115329	+	0.993327i	$0.536792\pi$
$788$	19441.9	0.878922
$789$	49680.6	2.24167
$790$	−12185.3	−0.548777
$791$	0	0
$792$	32505.6	1.45838
$793$	34631.1	1.55080
$794$	24858.2	1.11106
$795$	−16288.0	−0.726635
$796$	17097.8	0.761326
$797$	34666.2	1.54070	0.770350	−	0.637621i	$-0.220081\pi$
$797$	34666.2	1.54070	0.770350	+	0.637621i	$0.220081\pi$
$798$	0	0
$799$	611.326	0.0270678
$800$	4044.71	0.178753
$801$	402.116	0.0177379
$802$	−34043.0	−1.49888
$803$	2001.47	0.0879582
$804$	78339.9	3.43636
$805$	0	0
$806$	838.012	0.0366225
$807$	−27279.9	−1.18996
$808$	−21568.7	−0.939091
$809$	15126.2	0.657365	0.328683	−	0.944440i	$-0.393395\pi$
$809$	15126.2	0.657365	0.328683	+	0.944440i	$0.393395\pi$
$810$	−488.438	−0.0211876
$811$	−29416.5	−1.27368	−0.636840	−	0.770996i	$-0.719759\pi$
$811$	−29416.5	−1.27368	−0.636840	+	0.770996i	$0.719759\pi$
$812$	0	0
$813$	52384.9	2.25980
$814$	−19653.9	−0.846274
$815$	2969.69	0.127637
$816$	322.116	0.0138190
$817$	37276.3	1.59625
$818$	−12081.5	−0.516404
$819$	0	0
$820$	6377.27	0.271590
$821$	−15334.4	−0.651856	−0.325928	−	0.945395i	$-0.605677\pi$
$821$	−15334.4	−0.651856	−0.325928	+	0.945395i	$0.605677\pi$
$822$	−25063.2	−1.06348
$823$	−11003.7	−0.466056	−0.233028	−	0.972470i	$-0.574863\pi$
$823$	−11003.7	−0.466056	−0.233028	+	0.972470i	$0.574863\pi$
$824$	−29048.7	−1.22811
$825$	−6313.36	−0.266428
$826$	0	0
$827$	−3261.59	−0.137142	−0.0685711	−	0.997646i	$-0.521844\pi$
$827$	−3261.59	−0.137142	−0.0685711	+	0.997646i	$0.521844\pi$
$828$	11757.6	0.493484
$829$	−5163.30	−0.216319	−0.108160	−	0.994134i	$-0.534496\pi$
$829$	−5163.30	−0.216319	−0.108160	+	0.994134i	$0.534496\pi$
$830$	−1619.21	−0.0677152
$831$	−13185.1	−0.550406
$832$	72305.6	3.01292
$833$	0	0
$834$	19643.7	0.815593
$835$	−14681.5	−0.608472
$836$	50313.1	2.08148
$837$	279.150	0.0115279
$838$	20339.8	0.838457
$839$	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
$842$	−44966.1	−1.84042
$843$	−65801.4	−2.68840
$844$	−43628.1	−1.77931
$845$	−28557.3	−1.16260
$846$	−25863.0	−1.05105
$847$	0	0
$848$	3149.45	0.127538
$849$	−52546.8	−2.12415
$850$	−547.663	−0.0220996
$851$	−2860.57	−0.115228
$852$	−58766.7	−2.36304
$853$	7799.52	0.313072	0.156536	−	0.987672i	$-0.449967\pi$
$853$	7799.52	0.313072	0.156536	+	0.987672i	$0.449967\pi$
$854$	0	0
$855$	27044.9	1.08177
$856$	−1419.04	−0.0566610
$857$	21540.0	0.858568	0.429284	−	0.903170i	$-0.358766\pi$
$857$	21540.0	0.858568	0.429284	+	0.903170i	$0.358766\pi$
$858$	−103857.	−4.13244
$859$	−4447.97	−0.176674	−0.0883370	−	0.996091i	$-0.528155\pi$
$859$	−4447.97	−0.176674	−0.0883370	+	0.996091i	$0.528155\pi$
$860$	19989.3	0.792591
$861$	0	0
$862$	66198.5	2.61569
$863$	−9425.21	−0.371770	−0.185885	−	0.982571i	$-0.559515\pi$
$863$	−9425.21	−0.371770	−0.185885	+	0.982571i	$0.559515\pi$
$864$	22164.2	0.872733
$865$	11736.5	0.461335
$866$	−10977.0	−0.430730
$867$	−41015.2	−1.60663
$868$	0	0
$869$	15868.3	0.619442
$870$	26002.5	1.01330
$871$	−62054.5	−2.41405
$872$	33850.8	1.31460
$873$	181.787	0.00704761
$874$	11699.2	0.452783
$875$	0	0
$876$	−7462.27	−0.287816
$877$	22346.1	0.860403	0.430201	−	0.902733i	$-0.358443\pi$
$877$	22346.1	0.860403	0.430201	+	0.902733i	$0.358443\pi$
$878$	−44088.1	−1.69465
$879$	60928.6	2.33796
$880$	1220.75	0.0467632
$881$	−12074.9	−0.461762	−0.230881	−	0.972982i	$-0.574161\pi$
$881$	−12074.9	−0.461762	−0.230881	+	0.972982i	$0.574161\pi$
$882$	0	0
$883$	−30499.6	−1.16239	−0.581196	−	0.813764i	$-0.697415\pi$
$883$	−30499.6	−1.16239	−0.581196	+	0.813764i	$0.697415\pi$
$884$	−5639.22	−0.214556
$885$	35160.8	1.33550
$886$	−30743.8	−1.16576
$887$	−23344.2	−0.883675	−0.441838	−	0.897095i	$-0.645673\pi$
$887$	−23344.2	−0.883675	−0.441838	+	0.897095i	$0.645673\pi$
$888$	29485.8	1.11428
$889$	0	0
$890$	214.562	0.00808107
$891$	636.068	0.0239159
$892$	−58544.3	−2.19754
$893$	−16108.2	−0.603628
$894$	70902.3	2.65249
$895$	−15182.8	−0.567044
$896$	0	0
$897$	−15116.2	−0.562669
$898$	3552.85	0.132027
$899$	273.227	0.0101364
$900$	14502.7	0.537137
$901$	1839.98	0.0680341
$902$	−13267.8	−0.489768
$903$	0	0
$904$	−10862.0	−0.399629
$905$	−4498.88	−0.165246
$906$	−115390.	−4.23132
$907$	−15092.5	−0.552523	−0.276262	−	0.961082i	$-0.589096\pi$
$907$	−15092.5	−0.552523	−0.276262	+	0.961082i	$0.589096\pi$
$908$	818.115	0.0299010
$909$	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	$0.589188\pi$
$912$	−8487.61	−0.308172
$913$	2108.62	0.0764348
$914$	15369.7	0.556221
$915$	16329.6	0.589990
$916$	−40419.6	−1.45797
$917$	0	0
$918$	−3001.08	−0.107898
$919$	24818.1	0.890831	0.445415	−	0.895324i	$-0.353056\pi$
$919$	24818.1	0.890831	0.445415	+	0.895324i	$0.353056\pi$
$920$	2524.44	0.0904656
$921$	−11140.1	−0.398566
$922$	−87130.0	−3.11223
$923$	46550.1	1.66004
$924$	0	0
$925$	−3528.44	−0.125421
$926$	49756.3	1.76576
$927$	50533.7	1.79045
$928$	21693.9	0.767388
$929$	−39906.4	−1.40935	−0.704675	−	0.709530i	$-0.748907\pi$
$929$	−39906.4	−1.40935	−0.704675	+	0.709530i	$0.748907\pi$
$930$	395.148	0.0139327
$931$	0	0
$932$	−47270.3	−1.66136
$933$	−40831.7	−1.43276
$934$	34414.6	1.20565
$935$	713.194	0.0249454
$936$	95999.7	3.35240
$937$	−16923.0	−0.590020	−0.295010	−	0.955494i	$-0.595323\pi$
$937$	−16923.0	−0.590020	−0.295010	+	0.955494i	$0.595323\pi$
$938$	0	0
$939$	−64856.8	−2.25402
$940$	−8637.94	−0.299722
$941$	53014.1	1.83657	0.918285	−	0.395921i	$-0.129574\pi$
$941$	53014.1	1.83657	0.918285	+	0.395921i	$0.129574\pi$
$942$	−82678.1	−2.85966
$943$	−1931.10	−0.0666864
$944$	−6798.71	−0.234406
$945$	0	0
$946$	−41587.4	−1.42931
$947$	−25798.9	−0.885271	−0.442636	−	0.896702i	$-0.645957\pi$
$947$	−25798.9	−0.885271	−0.442636	+	0.896702i	$0.645957\pi$
$948$	−59163.2	−2.02693
$949$	5911.00	0.202191
$950$	14430.7	0.492836
$951$	−68560.6	−2.33778
$952$	0	0
$953$	−17942.7	−0.609885	−0.304943	−	0.952371i	$-0.598637\pi$
$953$	−17942.7	−0.609885	−0.304943	+	0.952371i	$0.598637\pi$
$954$	−77843.2	−2.64179
$955$	−2080.84	−0.0705072
$956$	30556.6	1.03375
$957$	−33861.8	−1.14378
$958$	26322.9	0.887739
$959$	0	0
$960$	34094.3	1.14624
$961$	−29786.8	−0.999861
$962$	−58044.2	−1.94534
$963$	2468.59	0.0826055
$964$	29660.4	0.990971
$965$	25905.2	0.864165
$966$	0	0
$967$	19668.3	0.654073	0.327036	−	0.945012i	$-0.393950\pi$
$967$	19668.3	0.654073	0.327036	+	0.945012i	$0.393950\pi$
$968$	10569.3	0.350940
$969$	−4958.67	−0.164392
$970$	96.9986	0.00321076
$971$	−6332.97	−0.209304	−0.104652	−	0.994509i	$-0.533373\pi$
$971$	−6332.97	−0.209304	−0.104652	+	0.994509i	$0.533373\pi$
$972$	−51886.7	−1.71221
$973$	0	0
$974$	9342.76	0.307353
$975$	−18645.4	−0.612441
$976$	−3157.51	−0.103555
$977$	−11334.1	−0.371145	−0.185573	−	0.982631i	$-0.559414\pi$
$977$	−11334.1	−0.371145	−0.185573	+	0.982631i	$0.559414\pi$
$978$	23035.5	0.753164
$979$	−279.414	−0.00912166
$980$	0	0
$981$	−58887.4	−1.91655
$982$	−35313.3	−1.14755
$983$	37654.3	1.22175	0.610877	−	0.791725i	$-0.290817\pi$
$983$	37654.3	1.22175	0.610877	+	0.791725i	$0.290817\pi$
$984$	19905.1	0.644871
$985$	−7261.72	−0.234901
$986$	−2937.40	−0.0948741
$987$	0	0
$988$	148591.	4.78473
$989$	−6052.95	−0.194613
$990$	−30172.7	−0.968638
$991$	−53441.5	−1.71304	−0.856522	−	0.516111i	$-0.827379\pi$
$991$	−53441.5	−1.71304	−0.856522	+	0.516111i	$0.827379\pi$
$992$	329.671	0.0105515
$993$	−17115.1	−0.546959
$994$	0	0
$995$	−6386.16	−0.203472
$996$	−7861.74	−0.250109
$997$	−37919.3	−1.20453	−0.602266	−	0.798296i	$-0.705735\pi$
$997$	−37919.3	−1.20453	−0.602266	+	0.798296i	$0.705735\pi$
$998$	−29063.4	−0.921829
$999$	−19335.1	−0.612348

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.4.a.l.1.1		3
3.2	odd	2		2205.4.a.bm.1.3		3
5.4	even	2		1225.4.a.y.1.3		3
7.2	even	3		245.4.e.n.116.3		6
7.3	odd	6		245.4.e.m.226.3		6
7.4	even	3		245.4.e.n.226.3		6
7.5	odd	6		245.4.e.m.116.3		6
7.6	odd	2		35.4.a.c.1.1	✓	3
21.20	even	2		315.4.a.p.1.3		3
28.27	even	2		560.4.a.u.1.3		3
35.13	even	4		175.4.b.e.99.6		6
35.27	even	4		175.4.b.e.99.1		6
35.34	odd	2		175.4.a.f.1.3		3
56.13	odd	2		2240.4.a.bt.1.3		3
56.27	even	2		2240.4.a.bv.1.1		3
105.104	even	2		1575.4.a.ba.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.c.1.1	✓	3	7.6	odd	2
175.4.a.f.1.3		3	35.34	odd	2
175.4.b.e.99.1		6	35.27	even	4
175.4.b.e.99.6		6	35.13	even	4
245.4.a.l.1.1		3	1.1	even	1	trivial
245.4.e.m.116.3		6	7.5	odd	6
245.4.e.m.226.3		6	7.3	odd	6
245.4.e.n.116.3		6	7.2	even	3
245.4.e.n.226.3		6	7.4	even	3
315.4.a.p.1.3		3	21.20	even	2
560.4.a.u.1.3		3	28.27	even	2
1225.4.a.y.1.3		3	5.4	even	2
1575.4.a.ba.1.1		3	105.104	even	2
2205.4.a.bm.1.3		3	3.2	odd	2
2240.4.a.bt.1.3		3	56.13	odd	2
2240.4.a.bv.1.1		3	56.27	even	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:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

\(f(q)\)	\(=\)	\(q-4.62456 q^{2} +8.38660 q^{3} +13.3866 q^{4} -5.00000 q^{5} -38.7844 q^{6} -24.9107 q^{8} +43.3350 q^{9} +23.1228 q^{10} -30.1117 q^{11} +112.268 q^{12} -88.9295 q^{13} -41.9330 q^{15} +8.10818 q^{16} +4.73699 q^{17} -200.405 q^{18} -124.818 q^{19} -66.9330 q^{20} +139.253 q^{22} +20.2680 q^{23} -208.916 q^{24} +25.0000 q^{25} +411.260 q^{26} +136.995 q^{27} +134.088 q^{29} +193.922 q^{30} +2.03767 q^{31} +161.788 q^{32} -252.534 q^{33} -21.9065 q^{34} +580.108 q^{36} -141.137 q^{37} +577.228 q^{38} -745.816 q^{39} +124.553 q^{40} -95.2784 q^{41} -298.646 q^{43} -403.093 q^{44} -216.675 q^{45} -93.7305 q^{46} +129.054 q^{47} +68.0000 q^{48} -115.614 q^{50} +39.7272 q^{51} -1190.46 q^{52} +388.429 q^{53} -633.542 q^{54} +150.558 q^{55} -1046.80 q^{57} -620.098 q^{58} -838.501 q^{59} -561.340 q^{60} -389.422 q^{61} -9.42333 q^{62} -813.067 q^{64} +444.647 q^{65} +1167.86 q^{66} +697.794 q^{67} +63.4122 q^{68} +169.979 q^{69} -523.450 q^{71} -1079.50 q^{72} -66.4684 q^{73} +652.699 q^{74} +209.665 q^{75} -1670.89 q^{76} +3449.07 q^{78} -526.982 q^{79} -40.5409 q^{80} -21.1236 q^{81} +440.621 q^{82} -70.0265 q^{83} -23.6850 q^{85} +1381.11 q^{86} +1124.54 q^{87} +750.101 q^{88} +9.27925 q^{89} +1002.03 q^{90} +271.319 q^{92} +17.0891 q^{93} -596.817 q^{94} +624.089 q^{95} +1356.85 q^{96} +4.19493 q^{97} -1304.89 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 2 q^{3} + 13 q^{4} - 15 q^{5} - 24 q^{6} - 15 q^{8} + 81 q^{9} + 15 q^{10} - 74 q^{11} + 152 q^{12} - 44 q^{13} + 10 q^{15} - 79 q^{16} + 52 q^{17} - 411 q^{18} - 168 q^{19} - 65 q^{20}+ \cdots - 3488 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 2 * q^3 + 13 * q^4 - 15 * q^5 - 24 * q^6 - 15 * q^8 + 81 * q^9 + 15 * q^10 - 74 * q^11 + 152 * q^12 - 44 * q^13 + 10 * q^15 - 79 * q^16 + 52 * q^17 - 411 * q^18 - 168 * q^19 - 65 * q^20 + 184 * q^22 - 124 * q^23 - 420 * q^24 + 75 * q^25 + 446 * q^26 - 170 * q^27 + 332 * q^29 + 120 * q^30 - 320 * q^31 - 183 * q^32 + 106 * q^33 - 582 * q^34 + 181 * q^36 - 54 * q^37 + 460 * q^38 - 982 * q^39 + 75 * q^40 - 362 * q^41 - 16 * q^43 - 264 * q^44 - 405 * q^45 - 336 * q^46 + 730 * q^47 + 204 * q^48 - 75 * q^50 - 1178 * q^51 - 1202 * q^52 + 110 * q^53 + 180 * q^54 + 370 * q^55 - 956 * q^57 + 450 * q^58 + 180 * q^59 - 760 * q^60 - 1222 * q^61 - 464 * q^62 - 391 * q^64 + 220 * q^65 + 532 * q^66 + 204 * q^67 - 918 * q^68 + 716 * q^69 - 136 * q^71 + 765 * q^72 - 310 * q^73 + 502 * q^74 - 50 * q^75 - 1796 * q^76 + 3788 * q^78 - 1034 * q^79 + 395 * q^80 + 2283 * q^81 + 6 * q^82 + 1660 * q^83 - 260 * q^85 + 764 * q^86 + 1574 * q^87 - 20 * q^88 - 242 * q^89 + 2055 * q^90 + 96 * q^92 + 1376 * q^93 + 1108 * q^94 + 840 * q^95 + 3156 * q^96 - 100 * q^97 - 3488 * q^99

Introduction

L-functions

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