LMFDB - Embedded newform 1305.4.a.n.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,4,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1305.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1305.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:	$76.9974925575$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 37x^{5} + 55x^{4} + 336x^{3} - 227x^{2} - 824x - 166 $ x^7 - 2x^6 - 37x^5 + 55x^4 + 336x^3 - 227x^2 - 824x - 166
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$5.24052$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.24052 q^{2} +19.4631 q^{4} -5.00000 q^{5} -18.4475 q^{7} +60.0724 q^{8} -26.2026 q^{10} -11.5384 q^{11} -34.4618 q^{13} -96.6746 q^{14} +159.106 q^{16} -70.7681 q^{17} +3.52127 q^{19} -97.3153 q^{20} -60.4671 q^{22} +18.3117 q^{23} +25.0000 q^{25} -180.598 q^{26} -359.045 q^{28} -29.0000 q^{29} -120.558 q^{31} +353.221 q^{32} -370.862 q^{34} +92.2376 q^{35} -182.505 q^{37} +18.4533 q^{38} -300.362 q^{40} -74.4691 q^{41} -405.899 q^{43} -224.572 q^{44} +95.9626 q^{46} +327.143 q^{47} -2.68899 q^{49} +131.013 q^{50} -670.732 q^{52} -409.067 q^{53} +57.6919 q^{55} -1108.19 q^{56} -151.975 q^{58} -120.722 q^{59} +34.6082 q^{61} -631.788 q^{62} +578.210 q^{64} +172.309 q^{65} +671.739 q^{67} -1377.36 q^{68} +483.373 q^{70} -873.524 q^{71} +432.426 q^{73} -956.419 q^{74} +68.5347 q^{76} +212.854 q^{77} -548.355 q^{79} -795.532 q^{80} -390.257 q^{82} -511.077 q^{83} +353.841 q^{85} -2127.12 q^{86} -693.138 q^{88} +788.135 q^{89} +635.734 q^{91} +356.401 q^{92} +1714.40 q^{94} -17.6063 q^{95} +967.070 q^{97} -14.0917 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 22 q^{4} - 35 q^{5} - 50 q^{7} + 33 q^{8} - 10 q^{10} - 76 q^{11} + 30 q^{13} - 89 q^{14} + 138 q^{16} + 140 q^{17} + 90 q^{19} - 110 q^{20} + 61 q^{22} - 34 q^{23} + 175 q^{25} + 241 q^{26}+ \cdots + 761 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 22 * q^4 - 35 * q^5 - 50 * q^7 + 33 * q^8 - 10 * q^10 - 76 * q^11 + 30 * q^13 - 89 * q^14 + 138 * q^16 + 140 * q^17 + 90 * q^19 - 110 * q^20 + 61 * q^22 - 34 * q^23 + 175 * q^25 + 241 * q^26 - 57 * q^28 - 203 * q^29 + 524 * q^31 + 6 * q^32 + 255 * q^34 + 250 * q^35 - 28 * q^37 - 222 * q^38 - 165 * q^40 - 1532 * q^41 - 464 * q^43 - 1475 * q^44 + 72 * q^46 + 360 * q^47 + 569 * q^49 + 50 * q^50 - 205 * q^52 - 282 * q^53 + 380 * q^55 - 1102 * q^56 - 58 * q^58 - 766 * q^59 + 1200 * q^61 - 2856 * q^62 + 701 * q^64 - 150 * q^65 + 1546 * q^67 - 1801 * q^68 + 445 * q^70 - 1802 * q^71 - 220 * q^73 - 1594 * q^74 + 1960 * q^76 - 3222 * q^77 + 1298 * q^79 - 690 * q^80 + 856 * q^82 - 1652 * q^83 - 700 * q^85 - 7628 * q^86 + 550 * q^88 - 2846 * q^89 - 816 * q^91 - 472 * q^92 + 745 * q^94 - 450 * q^95 + 1110 * q^97 + 761 * q^98

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 $

$2$	5.24052	1.85280	0.926402	−	0.376536i	$-0.122885\pi$
$2$	5.24052	1.85280	0.926402	+	0.376536i	$0.122885\pi$
$3$	0	0
$3$	0	0
$4$	19.4631	2.43288
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−18.4475	−0.996072	−0.498036	−	0.867156i	$-0.665945\pi$
$7$	−18.4475	−0.996072	−0.498036	+	0.867156i	$0.665945\pi$
$8$	60.0724	2.65485
$9$	0	0
$10$	−26.2026	−0.828599
$11$	−11.5384	−0.316268	−0.158134	−	0.987418i	$-0.550548\pi$
$11$	−11.5384	−0.316268	−0.158134	+	0.987418i	$0.550548\pi$
$12$	0	0
$13$	−34.4618	−0.735229	−0.367614	−	0.929978i	$-0.619825\pi$
$13$	−34.4618	−0.735229	−0.367614	+	0.929978i	$0.619825\pi$
$14$	−96.6746	−1.84553
$15$	0	0
$16$	159.106	2.48604
$17$	−70.7681	−1.00963	−0.504817	−	0.863226i	$-0.668440\pi$
$17$	−70.7681	−1.00963	−0.504817	+	0.863226i	$0.668440\pi$
$18$	0	0
$19$	3.52127	0.0425176	0.0212588	−	0.999774i	$-0.493233\pi$
$19$	3.52127	0.0425176	0.0212588	+	0.999774i	$0.493233\pi$
$20$	−97.3153	−1.08802
$21$	0	0
$22$	−60.4671	−0.585983
$23$	18.3117	0.166011	0.0830053	−	0.996549i	$-0.473548\pi$
$23$	18.3117	0.166011	0.0830053	+	0.996549i	$0.473548\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−180.598	−1.36224
$27$	0	0
$28$	−359.045	−2.42333
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−120.558	−0.698481	−0.349240	−	0.937033i	$-0.613560\pi$
$31$	−120.558	−0.698481	−0.349240	+	0.937033i	$0.613560\pi$
$32$	353.221	1.95129
$33$	0	0
$34$	−370.862	−1.87065
$35$	92.2376	0.445457
$36$	0	0
$37$	−182.505	−0.810907	−0.405454	−	0.914116i	$-0.632886\pi$
$37$	−182.505	−0.810907	−0.405454	+	0.914116i	$0.632886\pi$
$38$	18.4533	0.0787768
$39$	0	0
$40$	−300.362	−1.18729
$41$	−74.4691	−0.283661	−0.141831	−	0.989891i	$-0.545299\pi$
$41$	−74.4691	−0.283661	−0.141831	+	0.989891i	$0.545299\pi$
$42$	0	0
$43$	−405.899	−1.43951	−0.719756	−	0.694227i	$-0.755746\pi$
$43$	−405.899	−1.43951	−0.719756	+	0.694227i	$0.755746\pi$
$44$	−224.572	−0.769444
$45$	0	0
$46$	95.9626	0.307585
$47$	327.143	1.01529	0.507646	−	0.861566i	$-0.330516\pi$
$47$	327.143	1.01529	0.507646	+	0.861566i	$0.330516\pi$
$48$	0	0
$49$	−2.68899	−0.00783963
$50$	131.013	0.370561
$51$	0	0
$52$	−670.732	−1.78873
$53$	−409.067	−1.06018	−0.530092	−	0.847940i	$-0.677843\pi$
$53$	−409.067	−1.06018	−0.530092	+	0.847940i	$0.677843\pi$
$54$	0	0
$55$	57.6919	0.141440
$56$	−1108.19	−2.64442
$57$	0	0
$58$	−151.975	−0.344057
$59$	−120.722	−0.266384	−0.133192	−	0.991090i	$-0.542523\pi$
$59$	−120.722	−0.266384	−0.133192	+	0.991090i	$0.542523\pi$
$60$	0	0
$61$	34.6082	0.0726415	0.0363207	−	0.999340i	$-0.488436\pi$
$61$	34.6082	0.0726415	0.0363207	+	0.999340i	$0.488436\pi$
$62$	−631.788	−1.29415
$63$	0	0
$64$	578.210	1.12932
$65$	172.309	0.328804
$66$	0	0
$67$	671.739	1.22487	0.612433	−	0.790523i	$-0.290191\pi$
$67$	671.739	1.22487	0.612433	+	0.790523i	$0.290191\pi$
$68$	−1377.36	−2.45632
$69$	0	0
$70$	483.373	0.825345
$71$	−873.524	−1.46012	−0.730058	−	0.683386i	$-0.760507\pi$
$71$	−873.524	−1.46012	−0.730058	+	0.683386i	$0.760507\pi$
$72$	0	0
$73$	432.426	0.693311	0.346655	−	0.937993i	$-0.387317\pi$
$73$	432.426	0.693311	0.346655	+	0.937993i	$0.387317\pi$
$74$	−956.419	−1.50245
$75$	0	0
$76$	68.5347	0.103440
$77$	212.854	0.315026
$78$	0	0
$79$	−548.355	−0.780946	−0.390473	−	0.920614i	$-0.627689\pi$
$79$	−548.355	−0.780946	−0.390473	+	0.920614i	$0.627689\pi$
$80$	−795.532	−1.11179
$81$	0	0
$82$	−390.257	−0.525569
$83$	−511.077	−0.675879	−0.337940	−	0.941168i	$-0.609730\pi$
$83$	−511.077	−0.675879	−0.337940	+	0.941168i	$0.609730\pi$
$84$	0	0
$85$	353.841	0.451522
$86$	−2127.12	−2.66713
$87$	0	0
$88$	−693.138	−0.839646
$89$	788.135	0.938676	0.469338	−	0.883019i	$-0.344493\pi$
$89$	788.135	0.938676	0.469338	+	0.883019i	$0.344493\pi$
$90$	0	0
$91$	635.734	0.732341
$92$	356.401	0.403884
$93$	0	0
$94$	1714.40	1.88114
$95$	−17.6063	−0.0190144
$96$	0	0
$97$	967.070	1.01228	0.506140	−	0.862452i	$-0.331072\pi$
$97$	967.070	1.01228	0.506140	+	0.862452i	$0.331072\pi$
$98$	−14.0917	−0.0145253
$99$	0	0
$100$	486.577	0.486577
$101$	−510.016	−0.502460	−0.251230	−	0.967927i	$-0.580835\pi$
$101$	−510.016	−0.502460	−0.251230	+	0.967927i	$0.580835\pi$
$102$	0	0
$103$	1645.72	1.57435	0.787175	−	0.616730i	$-0.211543\pi$
$103$	1645.72	1.57435	0.787175	+	0.616730i	$0.211543\pi$
$104$	−2070.20	−1.95192
$105$	0	0
$106$	−2143.73	−1.96431
$107$	1305.35	1.17937	0.589687	−	0.807632i	$-0.299251\pi$
$107$	1305.35	1.17937	0.589687	+	0.807632i	$0.299251\pi$
$108$	0	0
$109$	932.125	0.819095	0.409548	−	0.912289i	$-0.365687\pi$
$109$	932.125	0.819095	0.409548	+	0.912289i	$0.365687\pi$
$110$	302.336	0.262060
$111$	0	0
$112$	−2935.12	−2.47627
$113$	−1719.89	−1.43180	−0.715901	−	0.698202i	$-0.753984\pi$
$113$	−1719.89	−1.43180	−0.715901	+	0.698202i	$0.753984\pi$
$114$	0	0
$115$	−91.5583	−0.0742422
$116$	−564.429	−0.451775
$117$	0	0
$118$	−632.646	−0.493558
$119$	1305.50	1.00567
$120$	0	0
$121$	−1197.87	−0.899974
$122$	181.365	0.134590
$123$	0	0
$124$	−2346.43	−1.69932
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−883.792	−0.617511	−0.308756	−	0.951141i	$-0.599912\pi$
$127$	−883.792	−0.617511	−0.308756	+	0.951141i	$0.599912\pi$
$128$	204.357	0.141116
$129$	0	0
$130$	902.988	0.609210
$131$	937.386	0.625190	0.312595	−	0.949887i	$-0.398802\pi$
$131$	937.386	0.625190	0.312595	+	0.949887i	$0.398802\pi$
$132$	0	0
$133$	−64.9587	−0.0423506
$134$	3520.26	2.26944
$135$	0	0
$136$	−4251.21	−2.68043
$137$	−237.650	−0.148203	−0.0741014	−	0.997251i	$-0.523609\pi$
$137$	−237.650	−0.148203	−0.0741014	+	0.997251i	$0.523609\pi$
$138$	0	0
$139$	1706.72	1.04146	0.520728	−	0.853723i	$-0.325661\pi$
$139$	1706.72	1.04146	0.520728	+	0.853723i	$0.325661\pi$
$140$	1795.23	1.08375
$141$	0	0
$142$	−4577.72	−2.70531
$143$	397.633	0.232530
$144$	0	0
$145$	145.000	0.0830455
$146$	2266.14	1.28457
$147$	0	0
$148$	−3552.10	−1.97284
$149$	−3546.59	−1.94999	−0.974994	−	0.222230i	$-0.928667\pi$
$149$	−3546.59	−1.94999	−0.974994	+	0.222230i	$0.928667\pi$
$150$	0	0
$151$	1818.23	0.979901	0.489951	−	0.871750i	$-0.337015\pi$
$151$	1818.23	0.979901	0.489951	+	0.871750i	$0.337015\pi$
$152$	211.531	0.112878
$153$	0	0
$154$	1115.47	0.583682
$155$	602.791	0.312370
$156$	0	0
$157$	−37.7540	−0.0191917	−0.00959585	−	0.999954i	$-0.503055\pi$
$157$	−37.7540	−0.0191917	−0.00959585	+	0.999954i	$0.503055\pi$
$158$	−2873.66	−1.44694
$159$	0	0
$160$	−1766.10	−0.872642
$161$	−337.805	−0.165359
$162$	0	0
$163$	1541.87	0.740913	0.370457	−	0.928850i	$-0.379201\pi$
$163$	1541.87	0.740913	0.370457	+	0.928850i	$0.379201\pi$
$164$	−1449.40	−0.690115
$165$	0	0
$166$	−2678.31	−1.25227
$167$	−1669.49	−0.773586	−0.386793	−	0.922167i	$-0.626417\pi$
$167$	−1669.49	−0.773586	−0.386793	+	0.922167i	$0.626417\pi$
$168$	0	0
$169$	−1009.39	−0.459438
$170$	1854.31	0.836582
$171$	0	0
$172$	−7900.04	−3.50216
$173$	728.623	0.320209	0.160105	−	0.987100i	$-0.448817\pi$
$173$	728.623	0.320209	0.160105	+	0.987100i	$0.448817\pi$
$174$	0	0
$175$	−461.188	−0.199214
$176$	−1835.83	−0.786255
$177$	0	0
$178$	4130.24	1.73918
$179$	−1926.46	−0.804416	−0.402208	−	0.915548i	$-0.631757\pi$
$179$	−1926.46	−0.804416	−0.402208	+	0.915548i	$0.631757\pi$
$180$	0	0
$181$	3695.62	1.51764	0.758821	−	0.651299i	$-0.225776\pi$
$181$	3695.62	1.51764	0.758821	+	0.651299i	$0.225776\pi$
$182$	3331.58	1.35689
$183$	0	0
$184$	1100.03	0.440733
$185$	912.523	0.362649
$186$	0	0
$187$	816.549	0.319315
$188$	6367.21	2.47009
$189$	0	0
$190$	−92.2664	−0.0352300
$191$	−2490.19	−0.943370	−0.471685	−	0.881767i	$-0.656354\pi$
$191$	−2490.19	−0.943370	−0.471685	+	0.881767i	$0.656354\pi$
$192$	0	0
$193$	4493.27	1.67582	0.837909	−	0.545809i	$-0.183778\pi$
$193$	4493.27	1.67582	0.837909	+	0.545809i	$0.183778\pi$
$194$	5067.95	1.87555
$195$	0	0
$196$	−52.3361	−0.0190729
$197$	−1990.23	−0.719787	−0.359894	−	0.932993i	$-0.617187\pi$
$197$	−1990.23	−0.719787	−0.359894	+	0.932993i	$0.617187\pi$
$198$	0	0
$199$	−83.8021	−0.0298521	−0.0149261	−	0.999889i	$-0.504751\pi$
$199$	−83.8021	−0.0298521	−0.0149261	+	0.999889i	$0.504751\pi$
$200$	1501.81	0.530970
$201$	0	0
$202$	−2672.75	−0.930960
$203$	534.978	0.184966
$204$	0	0
$205$	372.345	0.126857
$206$	8624.45	2.91696
$207$	0	0
$208$	−5483.09	−1.82781
$209$	−40.6297	−0.0134470
$210$	0	0
$211$	−4369.95	−1.42578	−0.712891	−	0.701275i	$-0.752615\pi$
$211$	−4369.95	−1.42578	−0.712891	+	0.701275i	$0.752615\pi$
$212$	−7961.71	−2.57930
$213$	0	0
$214$	6840.72	2.18515
$215$	2029.49	0.643769
$216$	0	0
$217$	2224.00	0.695738
$218$	4884.82	1.51762
$219$	0	0
$220$	1122.86	0.344106
$221$	2438.79	0.742313
$222$	0	0
$223$	3633.76	1.09119	0.545594	−	0.838050i	$-0.316304\pi$
$223$	3633.76	1.09119	0.545594	+	0.838050i	$0.316304\pi$
$224$	−6516.05	−1.94362
$225$	0	0
$226$	−9013.12	−2.65285
$227$	3692.30	1.07959	0.539794	−	0.841797i	$-0.318502\pi$
$227$	3692.30	1.07959	0.539794	+	0.841797i	$0.318502\pi$
$228$	0	0
$229$	4833.73	1.39486	0.697428	−	0.716655i	$-0.254328\pi$
$229$	4833.73	1.39486	0.697428	+	0.716655i	$0.254328\pi$
$230$	−479.813	−0.137556
$231$	0	0
$232$	−1742.10	−0.492994
$233$	4834.45	1.35929	0.679647	−	0.733539i	$-0.262133\pi$
$233$	4834.45	1.35929	0.679647	+	0.733539i	$0.262133\pi$
$234$	0	0
$235$	−1635.72	−0.454053
$236$	−2349.62	−0.648081
$237$	0	0
$238$	6841.48	1.86331
$239$	−4019.80	−1.08795	−0.543974	−	0.839102i	$-0.683081\pi$
$239$	−4019.80	−1.08795	−0.543974	+	0.839102i	$0.683081\pi$
$240$	0	0
$241$	1076.81	0.287816	0.143908	−	0.989591i	$-0.454033\pi$
$241$	1076.81	0.287816	0.143908	+	0.989591i	$0.454033\pi$
$242$	−6277.44	−1.66748
$243$	0	0
$244$	673.582	0.176728
$245$	13.4450	0.00350599
$246$	0	0
$247$	−121.349	−0.0312602
$248$	−7242.23	−1.85436
$249$	0	0
$250$	−655.065	−0.165720
$251$	3179.99	0.799677	0.399839	−	0.916586i	$-0.369066\pi$
$251$	3179.99	0.799677	0.399839	+	0.916586i	$0.369066\pi$
$252$	0	0
$253$	−211.287	−0.0525039
$254$	−4631.53	−1.14413
$255$	0	0
$256$	−3554.74	−0.867858
$257$	−3125.29	−0.758561	−0.379280	−	0.925282i	$-0.623828\pi$
$257$	−3125.29	−0.758561	−0.379280	+	0.925282i	$0.623828\pi$
$258$	0	0
$259$	3366.76	0.807722
$260$	3353.66	0.799943
$261$	0	0
$262$	4912.39	1.15835
$263$	4474.30	1.04904	0.524519	−	0.851399i	$-0.324245\pi$
$263$	4474.30	1.04904	0.524519	+	0.851399i	$0.324245\pi$
$264$	0	0
$265$	2045.34	0.474129
$266$	−340.417	−0.0784674
$267$	0	0
$268$	13074.1	2.97995
$269$	−4053.04	−0.918655	−0.459327	−	0.888267i	$-0.651909\pi$
$269$	−4053.04	−0.918655	−0.459327	+	0.888267i	$0.651909\pi$
$270$	0	0
$271$	−8045.96	−1.80353	−0.901766	−	0.432225i	$-0.857729\pi$
$271$	−8045.96	−1.80353	−0.901766	+	0.432225i	$0.857729\pi$
$272$	−11259.7	−2.50999
$273$	0	0
$274$	−1245.41	−0.274591
$275$	−288.459	−0.0632537
$276$	0	0
$277$	4625.74	1.00337	0.501685	−	0.865050i	$-0.332714\pi$
$277$	4625.74	1.00337	0.501685	+	0.865050i	$0.332714\pi$
$278$	8944.11	1.92961
$279$	0	0
$280$	5540.94	1.18262
$281$	69.7964	0.0148174	0.00740872	−	0.999973i	$-0.497642\pi$
$281$	69.7964	0.0148174	0.00740872	+	0.999973i	$0.497642\pi$
$282$	0	0
$283$	1903.58	0.399845	0.199922	−	0.979812i	$-0.435931\pi$
$283$	1903.58	0.399845	0.199922	+	0.979812i	$0.435931\pi$
$284$	−17001.4	−3.55229
$285$	0	0
$286$	2083.80	0.430832
$287$	1373.77	0.282547
$288$	0	0
$289$	95.1246	0.0193618
$290$	759.876	0.153867
$291$	0	0
$292$	8416.34	1.68674
$293$	−623.198	−0.124258	−0.0621290	−	0.998068i	$-0.519789\pi$
$293$	−623.198	−0.124258	−0.0621290	+	0.998068i	$0.519789\pi$
$294$	0	0
$295$	603.610	0.119131
$296$	−10963.5	−2.15284
$297$	0	0
$298$	−18586.0	−3.61295
$299$	−631.052	−0.122056
$300$	0	0
$301$	7487.83	1.43386
$302$	9528.45	1.81557
$303$	0	0
$304$	560.256	0.105700
$305$	−173.041	−0.0324863
$306$	0	0
$307$	−6899.90	−1.28273	−0.641365	−	0.767236i	$-0.721632\pi$
$307$	−6899.90	−1.28273	−0.641365	+	0.767236i	$0.721632\pi$
$308$	4142.80	0.766422
$309$	0	0
$310$	3158.94	0.578761
$311$	181.369	0.0330692	0.0165346	−	0.999863i	$-0.494737\pi$
$311$	181.369	0.0330692	0.0165346	+	0.999863i	$0.494737\pi$
$312$	0	0
$313$	−3839.22	−0.693308	−0.346654	−	0.937993i	$-0.612682\pi$
$313$	−3839.22	−0.693308	−0.346654	+	0.937993i	$0.612682\pi$
$314$	−197.851	−0.0355585
$315$	0	0
$316$	−10672.7	−1.89995
$317$	−7000.08	−1.24026	−0.620132	−	0.784497i	$-0.712921\pi$
$317$	−7000.08	−1.24026	−0.620132	+	0.784497i	$0.712921\pi$
$318$	0	0
$319$	334.613	0.0587296
$320$	−2891.05	−0.505046
$321$	0	0
$322$	−1770.27	−0.306377
$323$	−249.193	−0.0429272
$324$	0	0
$325$	−861.544	−0.147046
$326$	8080.22	1.37277
$327$	0	0
$328$	−4473.54	−0.753079
$329$	−6034.98	−1.01130
$330$	0	0
$331$	−1176.84	−0.195423	−0.0977115	−	0.995215i	$-0.531152\pi$
$331$	−1176.84	−0.195423	−0.0977115	+	0.995215i	$0.531152\pi$
$332$	−9947.12	−1.64433
$333$	0	0
$334$	−8748.99	−1.43330
$335$	−3358.69	−0.547776
$336$	0	0
$337$	−174.049	−0.0281336	−0.0140668	−	0.999901i	$-0.504478\pi$
$337$	−174.049	−0.0281336	−0.0140668	+	0.999901i	$0.504478\pi$
$338$	−5289.71	−0.851249
$339$	0	0
$340$	6886.82	1.09850
$341$	1391.05	0.220907
$342$	0	0
$343$	6377.10	1.00388
$344$	−24383.3	−3.82169
$345$	0	0
$346$	3818.37	0.593285
$347$	10895.5	1.68560	0.842799	−	0.538228i	$-0.180906\pi$
$347$	10895.5	1.68560	0.842799	+	0.538228i	$0.180906\pi$
$348$	0	0
$349$	−11820.3	−1.81298	−0.906488	−	0.422232i	$-0.861247\pi$
$349$	−11820.3	−1.81298	−0.906488	+	0.422232i	$0.861247\pi$
$350$	−2416.87	−0.369105
$351$	0	0
$352$	−4075.60	−0.617131
$353$	−9795.13	−1.47689	−0.738445	−	0.674314i	$-0.764439\pi$
$353$	−9795.13	−1.47689	−0.738445	+	0.674314i	$0.764439\pi$
$354$	0	0
$355$	4367.62	0.652983
$356$	15339.5	2.28369
$357$	0	0
$358$	−10095.7	−1.49042
$359$	−8894.06	−1.30755	−0.653775	−	0.756689i	$-0.726816\pi$
$359$	−8894.06	−1.30755	−0.653775	+	0.756689i	$0.726816\pi$
$360$	0	0
$361$	−6846.60	−0.998192
$362$	19367.0	2.81189
$363$	0	0
$364$	12373.3	1.78170
$365$	−2162.13	−0.310058
$366$	0	0
$367$	11113.1	1.58066	0.790329	−	0.612683i	$-0.209910\pi$
$367$	11113.1	1.58066	0.790329	+	0.612683i	$0.209910\pi$
$368$	2913.50	0.412708
$369$	0	0
$370$	4782.10	0.671917
$371$	7546.28	1.05602
$372$	0	0
$373$	13469.8	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$373$	13469.8	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$374$	4279.14	0.591629
$375$	0	0
$376$	19652.3	2.69545
$377$	999.391	0.136529
$378$	0	0
$379$	−3750.71	−0.508341	−0.254170	−	0.967159i	$-0.581802\pi$
$379$	−3750.71	−0.508341	−0.254170	+	0.967159i	$0.581802\pi$
$380$	−342.673	−0.0462599
$381$	0	0
$382$	−13049.9	−1.74788
$383$	−12404.3	−1.65490	−0.827451	−	0.561537i	$-0.810210\pi$
$383$	−12404.3	−1.65490	−0.827451	+	0.561537i	$0.810210\pi$
$384$	0	0
$385$	−1064.27	−0.140884
$386$	23547.1	3.10496
$387$	0	0
$388$	18822.1	2.46276
$389$	7718.67	1.00605	0.503023	−	0.864273i	$-0.332221\pi$
$389$	7718.67	1.00605	0.503023	+	0.864273i	$0.332221\pi$
$390$	0	0
$391$	−1295.88	−0.167610
$392$	−161.534	−0.0208131
$393$	0	0
$394$	−10429.8	−1.33362
$395$	2741.77	0.349250
$396$	0	0
$397$	−6747.40	−0.853004	−0.426502	−	0.904487i	$-0.640254\pi$
$397$	−6747.40	−0.853004	−0.426502	+	0.904487i	$0.640254\pi$
$398$	−439.167	−0.0553101
$399$	0	0
$400$	3977.66	0.497207
$401$	5047.40	0.628567	0.314283	−	0.949329i	$-0.398236\pi$
$401$	5047.40	0.628567	0.314283	+	0.949329i	$0.398236\pi$
$402$	0	0
$403$	4154.65	0.513543
$404$	−9926.47	−1.22243
$405$	0	0
$406$	2803.56	0.342706
$407$	2105.81	0.256464
$408$	0	0
$409$	5317.14	0.642826	0.321413	−	0.946939i	$-0.395842\pi$
$409$	5317.14	0.642826	0.321413	+	0.946939i	$0.395842\pi$
$410$	1951.28	0.235042
$411$	0	0
$412$	32030.8	3.83021
$413$	2227.02	0.265338
$414$	0	0
$415$	2555.38	0.302262
$416$	−12172.6	−1.43464
$417$	0	0
$418$	−212.921	−0.0249146
$419$	−2953.43	−0.344355	−0.172177	−	0.985066i	$-0.555080\pi$
$419$	−2953.43	−0.344355	−0.172177	+	0.985066i	$0.555080\pi$
$420$	0	0
$421$	7427.88	0.859888	0.429944	−	0.902855i	$-0.358533\pi$
$421$	7427.88	0.859888	0.429944	+	0.902855i	$0.358533\pi$
$422$	−22900.8	−2.64170
$423$	0	0
$424$	−24573.7	−2.81463
$425$	−1769.20	−0.201927
$426$	0	0
$427$	−638.436	−0.0723562
$428$	25406.1	2.86928
$429$	0	0
$430$	10635.6	1.19278
$431$	−3302.08	−0.369038	−0.184519	−	0.982829i	$-0.559073\pi$
$431$	−3302.08	−0.369038	−0.184519	+	0.982829i	$0.559073\pi$
$432$	0	0
$433$	9352.89	1.03804	0.519020	−	0.854762i	$-0.326297\pi$
$433$	9352.89	1.03804	0.519020	+	0.854762i	$0.326297\pi$
$434$	11654.9	1.28907
$435$	0	0
$436$	18142.0	1.99276
$437$	64.4802	0.00705837
$438$	0	0
$439$	−6109.74	−0.664241	−0.332120	−	0.943237i	$-0.607764\pi$
$439$	−6109.74	−0.664241	−0.332120	+	0.943237i	$0.607764\pi$
$440$	3465.69	0.375501
$441$	0	0
$442$	12780.6	1.37536
$443$	−14282.0	−1.53173	−0.765865	−	0.643002i	$-0.777689\pi$
$443$	−14282.0	−1.53173	−0.765865	+	0.643002i	$0.777689\pi$
$444$	0	0
$445$	−3940.68	−0.419789
$446$	19042.8	2.02176
$447$	0	0
$448$	−10666.5	−1.12488
$449$	16406.0	1.72439	0.862193	−	0.506581i	$-0.169091\pi$
$449$	16406.0	1.72439	0.862193	+	0.506581i	$0.169091\pi$
$450$	0	0
$451$	859.252	0.0897131
$452$	−33474.3	−3.48341
$453$	0	0
$454$	19349.6	2.00026
$455$	−3178.67	−0.327513
$456$	0	0
$457$	−15726.2	−1.60971	−0.804856	−	0.593470i	$-0.797757\pi$
$457$	−15726.2	−1.60971	−0.804856	+	0.593470i	$0.797757\pi$
$458$	25331.3	2.58440
$459$	0	0
$460$	−1782.00	−0.180623
$461$	14091.2	1.42363	0.711816	−	0.702366i	$-0.247873\pi$
$461$	14091.2	1.42363	0.711816	+	0.702366i	$0.247873\pi$
$462$	0	0
$463$	−8794.40	−0.882743	−0.441372	−	0.897324i	$-0.645508\pi$
$463$	−8794.40	−0.882743	−0.441372	+	0.897324i	$0.645508\pi$
$464$	−4614.08	−0.461645
$465$	0	0
$466$	25335.0	2.51850
$467$	8746.05	0.866636	0.433318	−	0.901241i	$-0.357343\pi$
$467$	8746.05	0.866636	0.433318	+	0.901241i	$0.357343\pi$
$468$	0	0
$469$	−12391.9	−1.22005
$470$	−8572.00	−0.841270
$471$	0	0
$472$	−7252.06	−0.707210
$473$	4683.41	0.455272
$474$	0	0
$475$	88.0317	0.00850352
$476$	25409.0	2.44668
$477$	0	0
$478$	−21065.9	−2.01575
$479$	11452.6	1.09245	0.546225	−	0.837638i	$-0.316064\pi$
$479$	11452.6	1.09245	0.546225	+	0.837638i	$0.316064\pi$
$480$	0	0
$481$	6289.43	0.596203
$482$	5643.06	0.533266
$483$	0	0
$484$	−23314.1	−2.18953
$485$	−4835.35	−0.452705
$486$	0	0
$487$	1635.31	0.152163	0.0760813	−	0.997102i	$-0.475759\pi$
$487$	1635.31	0.152163	0.0760813	+	0.997102i	$0.475759\pi$
$488$	2079.00	0.192852
$489$	0	0
$490$	70.4586	0.00649591
$491$	−2106.92	−0.193654	−0.0968270	−	0.995301i	$-0.530869\pi$
$491$	−2106.92	−0.193654	−0.0968270	+	0.995301i	$0.530869\pi$
$492$	0	0
$493$	2052.28	0.187484
$494$	−635.933	−0.0579190
$495$	0	0
$496$	−19181.6	−1.73645
$497$	16114.3	1.45438
$498$	0	0
$499$	−8865.74	−0.795361	−0.397681	−	0.917524i	$-0.630185\pi$
$499$	−8865.74	−0.795361	−0.397681	+	0.917524i	$0.630185\pi$
$500$	−2432.88	−0.217604
$501$	0	0
$502$	16664.8	1.48165
$503$	14591.9	1.29348	0.646738	−	0.762712i	$-0.276133\pi$
$503$	14591.9	1.29348	0.646738	+	0.762712i	$0.276133\pi$
$504$	0	0
$505$	2550.08	0.224707
$506$	−1107.25	−0.0972794
$507$	0	0
$508$	−17201.3	−1.50233
$509$	−13600.3	−1.18432	−0.592162	−	0.805819i	$-0.701726\pi$
$509$	−13600.3	−1.18432	−0.592162	+	0.805819i	$0.701726\pi$
$510$	0	0
$511$	−7977.19	−0.690588
$512$	−20263.6	−1.74909
$513$	0	0
$514$	−16378.1	−1.40546
$515$	−8228.62	−0.704070
$516$	0	0
$517$	−3774.70	−0.321105
$518$	17643.6	1.49655
$519$	0	0
$520$	10351.0	0.872927
$521$	−8355.92	−0.702647	−0.351324	−	0.936254i	$-0.614268\pi$
$521$	−8355.92	−0.702647	−0.351324	+	0.936254i	$0.614268\pi$
$522$	0	0
$523$	926.372	0.0774520	0.0387260	−	0.999250i	$-0.487670\pi$
$523$	926.372	0.0774520	0.0387260	+	0.999250i	$0.487670\pi$
$524$	18244.4	1.52101
$525$	0	0
$526$	23447.7	1.94366
$527$	8531.68	0.705210
$528$	0	0
$529$	−11831.7	−0.972440
$530$	10718.6	0.878467
$531$	0	0
$532$	−1264.29	−0.103034
$533$	2566.34	0.208556
$534$	0	0
$535$	−6526.75	−0.527432
$536$	40353.0	3.25184
$537$	0	0
$538$	−21240.0	−1.70209
$539$	31.0266	0.00247943
$540$	0	0
$541$	13503.6	1.07313	0.536565	−	0.843859i	$-0.319722\pi$
$541$	13503.6	1.07313	0.536565	+	0.843859i	$0.319722\pi$
$542$	−42165.0	−3.34159
$543$	0	0
$544$	−24996.8	−1.97009
$545$	−4660.63	−0.366311
$546$	0	0
$547$	−20425.4	−1.59658	−0.798288	−	0.602275i	$-0.794261\pi$
$547$	−20425.4	−1.59658	−0.798288	+	0.602275i	$0.794261\pi$
$548$	−4625.39	−0.360560
$549$	0	0
$550$	−1511.68	−0.117197
$551$	−102.117	−0.00789532
$552$	0	0
$553$	10115.8	0.777879
$554$	24241.3	1.85905
$555$	0	0
$556$	33218.0	2.53374
$557$	3929.12	0.298891	0.149446	−	0.988770i	$-0.452251\pi$
$557$	3929.12	0.298891	0.149446	+	0.988770i	$0.452251\pi$
$558$	0	0
$559$	13988.0	1.05837
$560$	14675.6	1.10742
$561$	0	0
$562$	365.769	0.0274538
$563$	−4452.67	−0.333318	−0.166659	−	0.986015i	$-0.553298\pi$
$563$	−4452.67	−0.333318	−0.166659	+	0.986015i	$0.553298\pi$
$564$	0	0
$565$	8599.45	0.640322
$566$	9975.75	0.740834
$567$	0	0
$568$	−52474.7	−3.87639
$569$	5462.20	0.402438	0.201219	−	0.979546i	$-0.435510\pi$
$569$	5462.20	0.402438	0.201219	+	0.979546i	$0.435510\pi$
$570$	0	0
$571$	−7228.87	−0.529805	−0.264902	−	0.964275i	$-0.585340\pi$
$571$	−7228.87	−0.529805	−0.264902	+	0.964275i	$0.585340\pi$
$572$	7739.16	0.565717
$573$	0	0
$574$	7199.27	0.523505
$575$	457.791	0.0332021
$576$	0	0
$577$	12771.6	0.921468	0.460734	−	0.887538i	$-0.347586\pi$
$577$	12771.6	0.921468	0.460734	+	0.887538i	$0.347586\pi$
$578$	498.502	0.0358736
$579$	0	0
$580$	2822.14	0.202040
$581$	9428.10	0.673224
$582$	0	0
$583$	4719.97	0.335303
$584$	25976.9	1.84064
$585$	0	0
$586$	−3265.88	−0.230226
$587$	−6986.93	−0.491280	−0.245640	−	0.969361i	$-0.578998\pi$
$587$	−6986.93	−0.491280	−0.245640	+	0.969361i	$0.578998\pi$
$588$	0	0
$589$	−424.518	−0.0296977
$590$	3163.23	0.220726
$591$	0	0
$592$	−29037.6	−2.01595
$593$	−4323.42	−0.299396	−0.149698	−	0.988732i	$-0.547830\pi$
$593$	−4323.42	−0.299396	−0.149698	+	0.988732i	$0.547830\pi$
$594$	0	0
$595$	−6527.48	−0.449749
$596$	−69027.6	−4.74409
$597$	0	0
$598$	−3307.04	−0.226145
$599$	−13443.7	−0.917019	−0.458510	−	0.888689i	$-0.651617\pi$
$599$	−13443.7	−0.917019	−0.458510	+	0.888689i	$0.651617\pi$
$600$	0	0
$601$	−7179.72	−0.487299	−0.243650	−	0.969863i	$-0.578345\pi$
$601$	−7179.72	−0.487299	−0.243650	+	0.969863i	$0.578345\pi$
$602$	39240.1	2.65666
$603$	0	0
$604$	35388.2	2.38399
$605$	5989.33	0.402481
$606$	0	0
$607$	−8560.81	−0.572442	−0.286221	−	0.958164i	$-0.592399\pi$
$607$	−8560.81	−0.572442	−0.286221	+	0.958164i	$0.592399\pi$
$608$	1243.79	0.0829641
$609$	0	0
$610$	−906.826	−0.0601907
$611$	−11273.9	−0.746472
$612$	0	0
$613$	19634.3	1.29367	0.646837	−	0.762628i	$-0.276091\pi$
$613$	19634.3	1.29367	0.646837	+	0.762628i	$0.276091\pi$
$614$	−36159.1	−2.37665
$615$	0	0
$616$	12786.7	0.836348
$617$	−14268.5	−0.931003	−0.465502	−	0.885047i	$-0.654126\pi$
$617$	−14268.5	−0.931003	−0.465502	+	0.885047i	$0.654126\pi$
$618$	0	0
$619$	1663.48	0.108014	0.0540071	−	0.998541i	$-0.482801\pi$
$619$	1663.48	0.108014	0.0540071	+	0.998541i	$0.482801\pi$
$620$	11732.2	0.759960
$621$	0	0
$622$	950.470	0.0612707
$623$	−14539.1	−0.934990
$624$	0	0
$625$	625.000	0.0400000
$626$	−20119.5	−1.28456
$627$	0	0
$628$	−734.808	−0.0466912
$629$	12915.5	0.818720
$630$	0	0
$631$	13448.3	0.848444	0.424222	−	0.905558i	$-0.360548\pi$
$631$	13448.3	0.848444	0.424222	+	0.905558i	$0.360548\pi$
$632$	−32941.0	−2.07330
$633$	0	0
$634$	−36684.1	−2.29797
$635$	4418.96	0.276159
$636$	0	0
$637$	92.6675	0.00576392
$638$	1753.55	0.108814
$639$	0	0
$640$	−1021.79	−0.0631088
$641$	−7644.92	−0.471070	−0.235535	−	0.971866i	$-0.575684\pi$
$641$	−7644.92	−0.471070	−0.235535	+	0.971866i	$0.575684\pi$
$642$	0	0
$643$	1814.60	0.111292	0.0556462	−	0.998451i	$-0.482278\pi$
$643$	1814.60	0.111292	0.0556462	+	0.998451i	$0.482278\pi$
$644$	−6574.71	−0.402298
$645$	0	0
$646$	−1305.90	−0.0795357
$647$	−18746.6	−1.13911	−0.569556	−	0.821952i	$-0.692885\pi$
$647$	−18746.6	−1.13911	−0.569556	+	0.821952i	$0.692885\pi$
$648$	0	0
$649$	1392.94	0.0842489
$650$	−4514.94	−0.272447
$651$	0	0
$652$	30009.6	1.80256
$653$	−8299.18	−0.497354	−0.248677	−	0.968586i	$-0.579996\pi$
$653$	−8299.18	−0.497354	−0.248677	+	0.968586i	$0.579996\pi$
$654$	0	0
$655$	−4686.93	−0.279593
$656$	−11848.5	−0.705192
$657$	0	0
$658$	−31626.4	−1.87375
$659$	−12624.7	−0.746262	−0.373131	−	0.927779i	$-0.621716\pi$
$659$	−12624.7	−0.746262	−0.373131	+	0.927779i	$0.621716\pi$
$660$	0	0
$661$	−10912.8	−0.642144	−0.321072	−	0.947055i	$-0.604043\pi$
$661$	−10912.8	−0.642144	−0.321072	+	0.947055i	$0.604043\pi$
$662$	−6167.26	−0.362081
$663$	0	0
$664$	−30701.6	−1.79436
$665$	324.793	0.0189398
$666$	0	0
$667$	−531.038	−0.0308274
$668$	−32493.4	−1.88204
$669$	0	0
$670$	−17601.3	−1.01492
$671$	−399.323	−0.0229742
$672$	0	0
$673$	6292.15	0.360393	0.180197	−	0.983631i	$-0.442327\pi$
$673$	6292.15	0.360393	0.180197	+	0.983631i	$0.442327\pi$
$674$	−912.105	−0.0521261
$675$	0	0
$676$	−19645.7	−1.11776
$677$	−16630.8	−0.944125	−0.472062	−	0.881565i	$-0.656490\pi$
$677$	−16630.8	−0.944125	−0.472062	+	0.881565i	$0.656490\pi$
$678$	0	0
$679$	−17840.0	−1.00830
$680$	21256.1	1.19872
$681$	0	0
$682$	7289.81	0.409298
$683$	−11843.7	−0.663521	−0.331761	−	0.943364i	$-0.607643\pi$
$683$	−11843.7	−0.663521	−0.331761	+	0.943364i	$0.607643\pi$
$684$	0	0
$685$	1188.25	0.0662783
$686$	33419.4	1.86000
$687$	0	0
$688$	−64581.1	−3.57868
$689$	14097.2	0.779478
$690$	0	0
$691$	20458.3	1.12630	0.563149	−	0.826356i	$-0.309590\pi$
$691$	20458.3	1.12630	0.563149	+	0.826356i	$0.309590\pi$
$692$	14181.2	0.779032
$693$	0	0
$694$	57098.2	3.12308
$695$	−8533.61	−0.465753
$696$	0	0
$697$	5270.03	0.286394
$698$	−61944.8	−3.35909
$699$	0	0
$700$	−8976.13	−0.484666
$701$	−23987.6	−1.29244	−0.646220	−	0.763151i	$-0.723651\pi$
$701$	−23987.6	−1.29244	−0.646220	+	0.763151i	$0.723651\pi$
$702$	0	0
$703$	−642.648	−0.0344778
$704$	−6671.61	−0.357167
$705$	0	0
$706$	−51331.6	−2.73639
$707$	9408.52	0.500487
$708$	0	0
$709$	6275.84	0.332432	0.166216	−	0.986089i	$-0.446845\pi$
$709$	6275.84	0.332432	0.166216	+	0.986089i	$0.446845\pi$
$710$	22888.6	1.20985
$711$	0	0
$712$	47345.2	2.49205
$713$	−2207.62	−0.115955
$714$	0	0
$715$	−1988.16	−0.103990
$716$	−37494.8	−1.95705
$717$	0	0
$718$	−46609.5	−2.42263
$719$	15544.4	0.806273	0.403136	−	0.915140i	$-0.367920\pi$
$719$	15544.4	0.806273	0.403136	+	0.915140i	$0.367920\pi$
$720$	0	0
$721$	−30359.5	−1.56817
$722$	−35879.8	−1.84945
$723$	0	0
$724$	71928.1	3.69225
$725$	−725.000	−0.0371391
$726$	0	0
$727$	−25002.9	−1.27552	−0.637762	−	0.770233i	$-0.720140\pi$
$727$	−25002.9	−1.27552	−0.637762	+	0.770233i	$0.720140\pi$
$728$	38190.1	1.94426
$729$	0	0
$730$	−11330.7	−0.574477
$731$	28724.7	1.45338
$732$	0	0
$733$	−16981.1	−0.855678	−0.427839	−	0.903855i	$-0.640725\pi$
$733$	−16981.1	−0.855678	−0.427839	+	0.903855i	$0.640725\pi$
$734$	58238.7	2.92865
$735$	0	0
$736$	6468.06	0.323934
$737$	−7750.78	−0.387386
$738$	0	0
$739$	920.528	0.0458216	0.0229108	−	0.999738i	$-0.492707\pi$
$739$	920.528	0.0458216	0.0229108	+	0.999738i	$0.492707\pi$
$740$	17760.5	0.882282
$741$	0	0
$742$	39546.4	1.95660
$743$	−6708.21	−0.331225	−0.165613	−	0.986191i	$-0.552960\pi$
$743$	−6708.21	−0.331225	−0.165613	+	0.986191i	$0.552960\pi$
$744$	0	0
$745$	17733.0	0.872061
$746$	70588.9	3.46440
$747$	0	0
$748$	15892.5	0.776857
$749$	−24080.5	−1.17474
$750$	0	0
$751$	34311.4	1.66717	0.833583	−	0.552394i	$-0.186286\pi$
$751$	34311.4	1.66717	0.833583	+	0.552394i	$0.186286\pi$
$752$	52050.6	2.52405
$753$	0	0
$754$	5237.33	0.252961
$755$	−9091.13	−0.438225
$756$	0	0
$757$	−12405.6	−0.595628	−0.297814	−	0.954624i	$-0.596257\pi$
$757$	−12405.6	−0.595628	−0.297814	+	0.954624i	$0.596257\pi$
$758$	−19655.7	−0.941856
$759$	0	0
$760$	−1057.66	−0.0504805
$761$	18869.8	0.898857	0.449428	−	0.893316i	$-0.351628\pi$
$761$	18869.8	0.898857	0.449428	+	0.893316i	$0.351628\pi$
$762$	0	0
$763$	−17195.4	−0.815878
$764$	−48466.7	−2.29511
$765$	0	0
$766$	−65004.8	−3.06621
$767$	4160.29	0.195853
$768$	0	0
$769$	38269.7	1.79459	0.897295	−	0.441432i	$-0.145529\pi$
$769$	38269.7	1.79459	0.897295	+	0.441432i	$0.145529\pi$
$770$	−5577.34	−0.261030
$771$	0	0
$772$	87452.9	4.07707
$773$	−37398.4	−1.74014	−0.870069	−	0.492930i	$-0.835926\pi$
$773$	−37398.4	−1.74014	−0.870069	+	0.492930i	$0.835926\pi$
$774$	0	0
$775$	−3013.96	−0.139696
$776$	58094.2	2.68745
$777$	0	0
$778$	40449.9	1.86401
$779$	−262.226	−0.0120606
$780$	0	0
$781$	10079.0	0.461788
$782$	−6791.09	−0.310549
$783$	0	0
$784$	−427.836	−0.0194896
$785$	188.770	0.00858279
$786$	0	0
$787$	−22352.2	−1.01241	−0.506206	−	0.862413i	$-0.668952\pi$
$787$	−22352.2	−1.01241	−0.506206	+	0.862413i	$0.668952\pi$
$788$	−38736.0	−1.75116
$789$	0	0
$790$	14368.3	0.647091
$791$	31727.7	1.42618
$792$	0	0
$793$	−1192.66	−0.0534081
$794$	−35359.9	−1.58045
$795$	0	0
$796$	−1631.05	−0.0726267
$797$	29993.8	1.33304	0.666521	−	0.745487i	$-0.267783\pi$
$797$	29993.8	1.33304	0.666521	+	0.745487i	$0.267783\pi$
$798$	0	0
$799$	−23151.3	−1.02507
$800$	8830.52	0.390258
$801$	0	0
$802$	26451.0	1.16461
$803$	−4989.50	−0.219272
$804$	0	0
$805$	1689.02	0.0739506
$806$	21772.5	0.951495
$807$	0	0
$808$	−30637.9	−1.33396
$809$	−28328.9	−1.23114	−0.615569	−	0.788083i	$-0.711074\pi$
$809$	−28328.9	−1.23114	−0.615569	+	0.788083i	$0.711074\pi$
$810$	0	0
$811$	4087.14	0.176965	0.0884826	−	0.996078i	$-0.471798\pi$
$811$	4087.14	0.176965	0.0884826	+	0.996078i	$0.471798\pi$
$812$	10412.3	0.450001
$813$	0	0
$814$	11035.5	0.475178
$815$	−7709.37	−0.331347
$816$	0	0
$817$	−1429.28	−0.0612046
$818$	27864.6	1.19103
$819$	0	0
$820$	7246.98	0.308629
$821$	9552.90	0.406088	0.203044	−	0.979170i	$-0.434916\pi$
$821$	9552.90	0.406088	0.203044	+	0.979170i	$0.434916\pi$
$822$	0	0
$823$	26863.5	1.13779	0.568895	−	0.822410i	$-0.307371\pi$
$823$	26863.5	1.13779	0.568895	+	0.822410i	$0.307371\pi$
$824$	98862.6	4.17966
$825$	0	0
$826$	11670.8	0.491619
$827$	43901.7	1.84596	0.922981	−	0.384846i	$-0.125746\pi$
$827$	43901.7	1.84596	0.922981	+	0.384846i	$0.125746\pi$
$828$	0	0
$829$	−9622.68	−0.403148	−0.201574	−	0.979473i	$-0.564606\pi$
$829$	−9622.68	−0.403148	−0.201574	+	0.979473i	$0.564606\pi$
$830$	13391.5	0.560033
$831$	0	0
$832$	−19926.2	−0.830307
$833$	190.295	0.00791516
$834$	0	0
$835$	8347.44	0.345958
$836$	−790.779	−0.0327149
$837$	0	0
$838$	−15477.5	−0.638022
$839$	−26742.6	−1.10043	−0.550214	−	0.835024i	$-0.685454\pi$
$839$	−26742.6	−1.10043	−0.550214	+	0.835024i	$0.685454\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	38926.0	1.59320
$843$	0	0
$844$	−85052.7	−3.46876
$845$	5046.93	0.205467
$846$	0	0
$847$	22097.7	0.896440
$848$	−65085.2	−2.63566
$849$	0	0
$850$	−9271.54	−0.374131
$851$	−3341.96	−0.134619
$852$	0	0
$853$	−16600.5	−0.666341	−0.333171	−	0.942867i	$-0.608119\pi$
$853$	−16600.5	−0.666341	−0.333171	+	0.942867i	$0.608119\pi$
$854$	−3345.74	−0.134062
$855$	0	0
$856$	78415.6	3.13106
$857$	−15752.2	−0.627869	−0.313935	−	0.949445i	$-0.601647\pi$
$857$	−15752.2	−0.627869	−0.313935	+	0.949445i	$0.601647\pi$
$858$	0	0
$859$	−35249.8	−1.40012	−0.700062	−	0.714082i	$-0.746844\pi$
$859$	−35249.8	−1.40012	−0.700062	+	0.714082i	$0.746844\pi$
$860$	39500.2	1.56622
$861$	0	0
$862$	−17304.6	−0.683755
$863$	36020.7	1.42081	0.710404	−	0.703794i	$-0.248512\pi$
$863$	36020.7	1.42081	0.710404	+	0.703794i	$0.248512\pi$
$864$	0	0
$865$	−3643.12	−0.143202
$866$	49014.0	1.92328
$867$	0	0
$868$	43285.9	1.69265
$869$	6327.12	0.246988
$870$	0	0
$871$	−23149.3	−0.900556
$872$	55995.0	2.17458
$873$	0	0
$874$	337.910	0.0130778
$875$	2305.94	0.0890914
$876$	0	0
$877$	1353.74	0.0521239	0.0260620	−	0.999660i	$-0.491703\pi$
$877$	1353.74	0.0521239	0.0260620	+	0.999660i	$0.491703\pi$
$878$	−32018.2	−1.23071
$879$	0	0
$880$	9179.15	0.351624
$881$	1774.07	0.0678435	0.0339217	−	0.999424i	$-0.489200\pi$
$881$	1774.07	0.0678435	0.0339217	+	0.999424i	$0.489200\pi$
$882$	0	0
$883$	1483.09	0.0565233	0.0282616	−	0.999601i	$-0.491003\pi$
$883$	1483.09	0.0565233	0.0282616	+	0.999601i	$0.491003\pi$
$884$	47466.4	1.80596
$885$	0	0
$886$	−74844.9	−2.83799
$887$	−18524.8	−0.701244	−0.350622	−	0.936517i	$-0.614030\pi$
$887$	−18524.8	−0.701244	−0.350622	+	0.936517i	$0.614030\pi$
$888$	0	0
$889$	16303.8	0.615086
$890$	−20651.2	−0.777786
$891$	0	0
$892$	70724.2	2.65473
$893$	1151.96	0.0431678
$894$	0	0
$895$	9632.30	0.359746
$896$	−3769.89	−0.140561
$897$	0	0
$898$	85976.2	3.19495
$899$	3496.19	0.129705
$900$	0	0
$901$	28948.9	1.07040
$902$	4502.93	0.166221
$903$	0	0
$904$	−103318.	−3.80122
$905$	−18478.1	−0.678710
$906$	0	0
$907$	−26687.1	−0.976991	−0.488495	−	0.872567i	$-0.662454\pi$
$907$	−26687.1	−0.976991	−0.488495	+	0.872567i	$0.662454\pi$
$908$	71863.4	2.62651
$909$	0	0
$910$	−16657.9	−0.606817
$911$	4443.10	0.161588	0.0807939	−	0.996731i	$-0.474254\pi$
$911$	4443.10	0.161588	0.0807939	+	0.996731i	$0.474254\pi$
$912$	0	0
$913$	5897.00	0.213759
$914$	−82413.2	−2.98248
$915$	0	0
$916$	94079.3	3.39352
$917$	−17292.5	−0.622734
$918$	0	0
$919$	17674.0	0.634398	0.317199	−	0.948359i	$-0.397258\pi$
$919$	17674.0	0.634398	0.317199	+	0.948359i	$0.397258\pi$
$920$	−5500.13	−0.197102
$921$	0	0
$922$	73845.4	2.63771
$923$	30103.2	1.07352
$924$	0	0
$925$	−4562.61	−0.162181
$926$	−46087.2	−1.63555
$927$	0	0
$928$	−10243.4	−0.362345
$929$	−10760.5	−0.380021	−0.190010	−	0.981782i	$-0.560852\pi$
$929$	−10760.5	−0.380021	−0.190010	+	0.981782i	$0.560852\pi$
$930$	0	0
$931$	−9.46867	−0.000333322 0
$932$	94093.2	3.30700
$933$	0	0
$934$	45833.9	1.60571
$935$	−4082.75	−0.142802
$936$	0	0
$937$	22210.3	0.774365	0.387182	−	0.922003i	$-0.373448\pi$
$937$	22210.3	0.774365	0.387182	+	0.922003i	$0.373448\pi$
$938$	−64940.1	−2.26052
$939$	0	0
$940$	−31836.0	−1.10466
$941$	−34977.1	−1.21171	−0.605855	−	0.795575i	$-0.707169\pi$
$941$	−34977.1	−1.21171	−0.605855	+	0.795575i	$0.707169\pi$
$942$	0	0
$943$	−1363.65	−0.0470908
$944$	−19207.6	−0.662241
$945$	0	0
$946$	24543.5	0.843530
$947$	−42798.4	−1.46860	−0.734298	−	0.678827i	$-0.762489\pi$
$947$	−42798.4	−1.46860	−0.734298	+	0.678827i	$0.762489\pi$
$948$	0	0
$949$	−14902.2	−0.509742
$950$	461.332	0.0157554
$951$	0	0
$952$	78424.3	2.66990
$953$	15363.5	0.522217	0.261108	−	0.965309i	$-0.415912\pi$
$953$	15363.5	0.522217	0.261108	+	0.965309i	$0.415912\pi$
$954$	0	0
$955$	12450.9	0.421888
$956$	−78237.7	−2.64685
$957$	0	0
$958$	60017.7	2.02410
$959$	4384.05	0.147621
$960$	0	0
$961$	−15256.7	−0.512125
$962$	32959.9	1.10465
$963$	0	0
$964$	20958.1	0.700222
$965$	−22466.4	−0.749449
$966$	0	0
$967$	−44310.2	−1.47355	−0.736774	−	0.676139i	$-0.763652\pi$
$967$	−44310.2	−1.47355	−0.736774	+	0.676139i	$0.763652\pi$
$968$	−71958.7	−2.38930
$969$	0	0
$970$	−25339.8	−0.838774
$971$	−50163.2	−1.65789	−0.828946	−	0.559328i	$-0.811059\pi$
$971$	−50163.2	−1.65789	−0.828946	+	0.559328i	$0.811059\pi$
$972$	0	0
$973$	−31484.8	−1.03736
$974$	8569.90	0.281927
$975$	0	0
$976$	5506.39	0.180589
$977$	2549.65	0.0834908	0.0417454	−	0.999128i	$-0.486708\pi$
$977$	2549.65	0.0834908	0.0417454	+	0.999128i	$0.486708\pi$
$978$	0	0
$979$	−9093.80	−0.296874
$980$	261.680	0.00852966
$981$	0	0
$982$	−11041.4	−0.358803
$983$	−25382.0	−0.823561	−0.411781	−	0.911283i	$-0.635093\pi$
$983$	−25382.0	−0.823561	−0.411781	+	0.911283i	$0.635093\pi$
$984$	0	0
$985$	9951.15	0.321899
$986$	10755.0	0.347372
$987$	0	0
$988$	−2361.83	−0.0760523
$989$	−7432.68	−0.238974
$990$	0	0
$991$	−44583.2	−1.42909	−0.714547	−	0.699588i	$-0.753367\pi$
$991$	−44583.2	−1.42909	−0.714547	+	0.699588i	$0.753367\pi$
$992$	−42583.7	−1.36294
$993$	0	0
$994$	84447.6	2.69468
$995$	419.010	0.0133503
$996$	0	0
$997$	−39668.0	−1.26008	−0.630039	−	0.776563i	$-0.716961\pi$
$997$	−39668.0	−1.26008	−0.630039	+	0.776563i	$0.716961\pi$
$998$	−46461.1	−1.47365
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.4.a.n.1.7		7
3.2	odd	2		435.4.a.i.1.1	✓	7
15.14	odd	2		2175.4.a.n.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.i.1.1	✓	7	3.2	odd	2
1305.4.a.n.1.7		7	1.1	even	1	trivial
2175.4.a.n.1.7		7	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q+5.24052 q^{2} +19.4631 q^{4} -5.00000 q^{5} -18.4475 q^{7} +60.0724 q^{8} -26.2026 q^{10} -11.5384 q^{11} -34.4618 q^{13} -96.6746 q^{14} +159.106 q^{16} -70.7681 q^{17} +3.52127 q^{19} -97.3153 q^{20} -60.4671 q^{22} +18.3117 q^{23} +25.0000 q^{25} -180.598 q^{26} -359.045 q^{28} -29.0000 q^{29} -120.558 q^{31} +353.221 q^{32} -370.862 q^{34} +92.2376 q^{35} -182.505 q^{37} +18.4533 q^{38} -300.362 q^{40} -74.4691 q^{41} -405.899 q^{43} -224.572 q^{44} +95.9626 q^{46} +327.143 q^{47} -2.68899 q^{49} +131.013 q^{50} -670.732 q^{52} -409.067 q^{53} +57.6919 q^{55} -1108.19 q^{56} -151.975 q^{58} -120.722 q^{59} +34.6082 q^{61} -631.788 q^{62} +578.210 q^{64} +172.309 q^{65} +671.739 q^{67} -1377.36 q^{68} +483.373 q^{70} -873.524 q^{71} +432.426 q^{73} -956.419 q^{74} +68.5347 q^{76} +212.854 q^{77} -548.355 q^{79} -795.532 q^{80} -390.257 q^{82} -511.077 q^{83} +353.841 q^{85} -2127.12 q^{86} -693.138 q^{88} +788.135 q^{89} +635.734 q^{91} +356.401 q^{92} +1714.40 q^{94} -17.6063 q^{95} +967.070 q^{97} -14.0917 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 22 q^{4} - 35 q^{5} - 50 q^{7} + 33 q^{8} - 10 q^{10} - 76 q^{11} + 30 q^{13} - 89 q^{14} + 138 q^{16} + 140 q^{17} + 90 q^{19} - 110 q^{20} + 61 q^{22} - 34 q^{23} + 175 q^{25} + 241 q^{26}+ \cdots + 761 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 22 * q^4 - 35 * q^5 - 50 * q^7 + 33 * q^8 - 10 * q^10 - 76 * q^11 + 30 * q^13 - 89 * q^14 + 138 * q^16 + 140 * q^17 + 90 * q^19 - 110 * q^20 + 61 * q^22 - 34 * q^23 + 175 * q^25 + 241 * q^26 - 57 * q^28 - 203 * q^29 + 524 * q^31 + 6 * q^32 + 255 * q^34 + 250 * q^35 - 28 * q^37 - 222 * q^38 - 165 * q^40 - 1532 * q^41 - 464 * q^43 - 1475 * q^44 + 72 * q^46 + 360 * q^47 + 569 * q^49 + 50 * q^50 - 205 * q^52 - 282 * q^53 + 380 * q^55 - 1102 * q^56 - 58 * q^58 - 766 * q^59 + 1200 * q^61 - 2856 * q^62 + 701 * q^64 - 150 * q^65 + 1546 * q^67 - 1801 * q^68 + 445 * q^70 - 1802 * q^71 - 220 * q^73 - 1594 * q^74 + 1960 * q^76 - 3222 * q^77 + 1298 * q^79 - 690 * q^80 + 856 * q^82 - 1652 * q^83 - 700 * q^85 - 7628 * q^86 + 550 * q^88 - 2846 * q^89 - 816 * q^91 - 472 * q^92 + 745 * q^94 - 450 * q^95 + 1110 * q^97 + 761 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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