LMFDB - Embedded newform 1305.4.a.n.1.1

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			1.1
Root			$-4.88324$ 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)

$f(q)$	$=$	\(q-4.88324 q^{2} +15.8460 q^{4} -5.00000 q^{5} +5.48750 q^{7} -38.3141 q^{8} +24.4162 q^{10} -50.1324 q^{11} -20.7349 q^{13} -26.7968 q^{14} +60.3288 q^{16} +18.8027 q^{17} +78.8346 q^{19} -79.2302 q^{20} +244.809 q^{22} +6.33763 q^{23} +25.0000 q^{25} +101.254 q^{26} +86.9552 q^{28} -29.0000 q^{29} +310.950 q^{31} +11.9130 q^{32} -91.8182 q^{34} -27.4375 q^{35} -338.745 q^{37} -384.969 q^{38} +191.571 q^{40} -353.526 q^{41} +507.609 q^{43} -794.400 q^{44} -30.9482 q^{46} +112.771 q^{47} -312.887 q^{49} -122.081 q^{50} -328.566 q^{52} +144.849 q^{53} +250.662 q^{55} -210.249 q^{56} +141.614 q^{58} +342.739 q^{59} +357.486 q^{61} -1518.44 q^{62} -540.804 q^{64} +103.675 q^{65} -183.087 q^{67} +297.949 q^{68} +133.984 q^{70} +594.034 q^{71} -622.609 q^{73} +1654.17 q^{74} +1249.22 q^{76} -275.102 q^{77} -1275.18 q^{79} -301.644 q^{80} +1726.35 q^{82} +739.108 q^{83} -94.0136 q^{85} -2478.78 q^{86} +1920.78 q^{88} -906.113 q^{89} -113.783 q^{91} +100.426 q^{92} -550.688 q^{94} -394.173 q^{95} +76.0665 q^{97} +1527.90 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.88324	−1.72649	−0.863243	−	0.504788i	$-0.831571\pi$
$2$	−4.88324	−1.72649	−0.863243	+	0.504788i	$0.831571\pi$
$3$	0	0
$3$	0	0
$4$	15.8460	1.98076
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	5.48750	0.296297	0.148149	−	0.988965i	$-0.452669\pi$
$7$	5.48750	0.296297	0.148149	+	0.988965i	$0.452669\pi$
$8$	−38.3141	−1.69326
$9$	0	0
$10$	24.4162	0.772108
$11$	−50.1324	−1.37413	−0.687067	−	0.726594i	$-0.741102\pi$
$11$	−50.1324	−1.37413	−0.687067	+	0.726594i	$0.741102\pi$
$12$	0	0
$13$	−20.7349	−0.442371	−0.221186	−	0.975232i	$-0.570993\pi$
$13$	−20.7349	−0.442371	−0.221186	+	0.975232i	$0.570993\pi$
$14$	−26.7968	−0.511553
$15$	0	0
$16$	60.3288	0.942637
$17$	18.8027	0.268255	0.134127	−	0.990964i	$-0.457177\pi$
$17$	18.8027	0.268255	0.134127	+	0.990964i	$0.457177\pi$
$18$	0	0
$19$	78.8346	0.951890	0.475945	−	0.879475i	$-0.342106\pi$
$19$	78.8346	0.951890	0.475945	+	0.879475i	$0.342106\pi$
$20$	−79.2302	−0.885821
$21$	0	0
$22$	244.809	2.37243
$23$	6.33763	0.0574559	0.0287280	−	0.999587i	$-0.490854\pi$
$23$	6.33763	0.0574559	0.0287280	+	0.999587i	$0.490854\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	101.254	0.763748
$27$	0	0
$28$	86.9552	0.586892
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	310.950	1.80156	0.900778	−	0.434280i	$-0.142997\pi$
$31$	310.950	1.80156	0.900778	+	0.434280i	$0.142997\pi$
$32$	11.9130	0.0658106
$33$	0	0
$34$	−91.8182	−0.463138
$35$	−27.4375	−0.132508
$36$	0	0
$37$	−338.745	−1.50512	−0.752558	−	0.658525i	$-0.771180\pi$
$37$	−338.745	−1.50512	−0.752558	+	0.658525i	$0.771180\pi$
$38$	−384.969	−1.64343
$39$	0	0
$40$	191.571	0.757250
$41$	−353.526	−1.34662	−0.673312	−	0.739359i	$-0.735129\pi$
$41$	−353.526	−1.34662	−0.673312	+	0.739359i	$0.735129\pi$
$42$	0	0
$43$	507.609	1.80022	0.900112	−	0.435658i	$-0.143484\pi$
$43$	507.609	1.80022	0.900112	+	0.435658i	$0.143484\pi$
$44$	−794.400	−2.72183
$45$	0	0
$46$	−30.9482	−0.0991969
$47$	112.771	0.349986	0.174993	−	0.984570i	$-0.444010\pi$
$47$	112.771	0.349986	0.174993	+	0.984570i	$0.444010\pi$
$48$	0	0
$49$	−312.887	−0.912208
$50$	−122.081	−0.345297
$51$	0	0
$52$	−328.566	−0.876230
$53$	144.849	0.375406	0.187703	−	0.982226i	$-0.439896\pi$
$53$	144.849	0.375406	0.187703	+	0.982226i	$0.439896\pi$
$54$	0	0
$55$	250.662	0.614532
$56$	−210.249	−0.501709
$57$	0	0
$58$	141.614	0.320600
$59$	342.739	0.756285	0.378142	−	0.925747i	$-0.376563\pi$
$59$	342.739	0.756285	0.378142	+	0.925747i	$0.376563\pi$
$60$	0	0
$61$	357.486	0.750350	0.375175	−	0.926954i	$-0.377583\pi$
$61$	357.486	0.750350	0.375175	+	0.926954i	$0.377583\pi$
$62$	−1518.44	−3.11036
$63$	0	0
$64$	−540.804	−1.05626
$65$	103.675	0.197834
$66$	0	0
$67$	−183.087	−0.333844	−0.166922	−	0.985970i	$-0.553383\pi$
$67$	−183.087	−0.333844	−0.166922	+	0.985970i	$0.553383\pi$
$68$	297.949	0.531347
$69$	0	0
$70$	133.984	0.228773
$71$	594.034	0.992942	0.496471	−	0.868053i	$-0.334629\pi$
$71$	594.034	0.992942	0.496471	+	0.868053i	$0.334629\pi$
$72$	0	0
$73$	−622.609	−0.998231	−0.499115	−	0.866536i	$-0.666342\pi$
$73$	−622.609	−0.998231	−0.499115	+	0.866536i	$0.666342\pi$
$74$	1654.17	2.59856
$75$	0	0
$76$	1249.22	1.88546
$77$	−275.102	−0.407152
$78$	0	0
$79$	−1275.18	−1.81607	−0.908034	−	0.418897i	$-0.862417\pi$
$79$	−1275.18	−1.81607	−0.908034	+	0.418897i	$0.862417\pi$
$80$	−301.644	−0.421560
$81$	0	0
$82$	1726.35	2.32493
$83$	739.108	0.977441	0.488720	−	0.872440i	$-0.337464\pi$
$83$	739.108	0.977441	0.488720	+	0.872440i	$0.337464\pi$
$84$	0	0
$85$	−94.0136	−0.119967
$86$	−2478.78	−3.10806
$87$	0	0
$88$	1920.78	2.32677
$89$	−906.113	−1.07919	−0.539594	−	0.841925i	$-0.681422\pi$
$89$	−906.113	−1.07919	−0.539594	+	0.841925i	$0.681422\pi$
$90$	0	0
$91$	−113.783	−0.131073
$92$	100.426	0.113806
$93$	0	0
$94$	−550.688	−0.604246
$95$	−394.173	−0.425698
$96$	0	0
$97$	76.0665	0.0796225	0.0398112	−	0.999207i	$-0.487324\pi$
$97$	76.0665	0.0796225	0.0398112	+	0.999207i	$0.487324\pi$
$98$	1527.90	1.57491
$99$	0	0
$100$	396.151	0.396151
$101$	−653.553	−0.643871	−0.321935	−	0.946762i	$-0.604333\pi$
$101$	−653.553	−0.643871	−0.321935	+	0.946762i	$0.604333\pi$
$102$	0	0
$103$	969.801	0.927741	0.463871	−	0.885903i	$-0.346460\pi$
$103$	969.801	0.927741	0.463871	+	0.885903i	$0.346460\pi$
$104$	794.440	0.749050
$105$	0	0
$106$	−707.332	−0.648133
$107$	1267.53	1.14520	0.572601	−	0.819835i	$-0.305935\pi$
$107$	1267.53	1.14520	0.572601	+	0.819835i	$0.305935\pi$
$108$	0	0
$109$	−808.142	−0.710146	−0.355073	−	0.934838i	$-0.615544\pi$
$109$	−808.142	−0.710146	−0.355073	+	0.934838i	$0.615544\pi$
$110$	−1224.04	−1.06098
$111$	0	0
$112$	331.054	0.279301
$113$	1220.10	1.01572	0.507862	−	0.861438i	$-0.330436\pi$
$113$	1220.10	1.01572	0.507862	+	0.861438i	$0.330436\pi$
$114$	0	0
$115$	−31.6881	−0.0256951
$116$	−459.535	−0.367817
$117$	0	0
$118$	−1673.68	−1.30572
$119$	103.180	0.0794831
$120$	0	0
$121$	1182.26	0.888247
$122$	−1745.69	−1.29547
$123$	0	0
$124$	4927.32	3.56844
$125$	−125.000	−0.0894427
$126$	0	0
$127$	814.733	0.569258	0.284629	−	0.958638i	$-0.408130\pi$
$127$	814.733	0.569258	0.284629	+	0.958638i	$0.408130\pi$
$128$	2545.57	1.75781
$129$	0	0
$130$	−506.268	−0.341559
$131$	1493.91	0.996363	0.498181	−	0.867073i	$-0.334001\pi$
$131$	1493.91	0.996363	0.498181	+	0.867073i	$0.334001\pi$
$132$	0	0
$133$	432.605	0.282042
$134$	894.056	0.576378
$135$	0	0
$136$	−720.410	−0.454225
$137$	1608.72	1.00323	0.501615	−	0.865091i	$-0.332739\pi$
$137$	1608.72	1.00323	0.501615	+	0.865091i	$0.332739\pi$
$138$	0	0
$139$	51.5725	0.0314700	0.0157350	−	0.999876i	$-0.494991\pi$
$139$	51.5725	0.0314700	0.0157350	+	0.999876i	$0.494991\pi$
$140$	−434.776	−0.262466
$141$	0	0
$142$	−2900.81	−1.71430
$143$	1039.49	0.607878
$144$	0	0
$145$	145.000	0.0830455
$146$	3040.35	1.72343
$147$	0	0
$148$	−5367.77	−2.98127
$149$	608.330	0.334472	0.167236	−	0.985917i	$-0.446516\pi$
$149$	608.330	0.334472	0.167236	+	0.985917i	$0.446516\pi$
$150$	0	0
$151$	−341.568	−0.184082	−0.0920412	−	0.995755i	$-0.529339\pi$
$151$	−341.568	−0.184082	−0.0920412	+	0.995755i	$0.529339\pi$
$152$	−3020.48	−1.61180
$153$	0	0
$154$	1343.39	0.702943
$155$	−1554.75	−0.805680
$156$	0	0
$157$	−358.378	−0.182176	−0.0910882	−	0.995843i	$-0.529035\pi$
$157$	−358.378	−0.182176	−0.0910882	+	0.995843i	$0.529035\pi$
$158$	6227.03	3.13542
$159$	0	0
$160$	−59.5650	−0.0294314
$161$	34.7777	0.0170240
$162$	0	0
$163$	−1667.36	−0.801213	−0.400606	−	0.916250i	$-0.631201\pi$
$163$	−1667.36	−0.801213	−0.400606	+	0.916250i	$0.631201\pi$
$164$	−5602.00	−2.66733
$165$	0	0
$166$	−3609.24	−1.68754
$167$	1832.30	0.849029	0.424514	−	0.905421i	$-0.360445\pi$
$167$	1832.30	0.849029	0.424514	+	0.905421i	$0.360445\pi$
$168$	0	0
$169$	−1767.06	−0.804308
$170$	459.091	0.207122
$171$	0	0
$172$	8043.60	3.56580
$173$	−2034.77	−0.894224	−0.447112	−	0.894478i	$-0.647547\pi$
$173$	−2034.77	−0.894224	−0.447112	+	0.894478i	$0.647547\pi$
$174$	0	0
$175$	137.188	0.0592594
$176$	−3024.43	−1.29531
$177$	0	0
$178$	4424.77	1.86320
$179$	332.198	0.138713	0.0693566	−	0.997592i	$-0.477905\pi$
$179$	332.198	0.138713	0.0693566	+	0.997592i	$0.477905\pi$
$180$	0	0
$181$	−843.391	−0.346347	−0.173173	−	0.984891i	$-0.555402\pi$
$181$	−843.391	−0.346347	−0.173173	+	0.984891i	$0.555402\pi$
$182$	555.629	0.226296
$183$	0	0
$184$	−242.821	−0.0972879
$185$	1693.72	0.673109
$186$	0	0
$187$	−942.625	−0.368618
$188$	1786.97	0.693236
$189$	0	0
$190$	1924.84	0.734962
$191$	−2438.81	−0.923908	−0.461954	−	0.886904i	$-0.652852\pi$
$191$	−2438.81	−0.923908	−0.461954	+	0.886904i	$0.652852\pi$
$192$	0	0
$193$	4095.03	1.52729	0.763645	−	0.645637i	$-0.223408\pi$
$193$	4095.03	1.52729	0.763645	+	0.645637i	$0.223408\pi$
$194$	−371.451	−0.137467
$195$	0	0
$196$	−4958.03	−1.80686
$197$	−3226.76	−1.16699	−0.583495	−	0.812117i	$-0.698315\pi$
$197$	−3226.76	−1.16699	−0.583495	+	0.812117i	$0.698315\pi$
$198$	0	0
$199$	−4218.74	−1.50281	−0.751404	−	0.659842i	$-0.770623\pi$
$199$	−4218.74	−1.50281	−0.751404	+	0.659842i	$0.770623\pi$
$200$	−957.853	−0.338652
$201$	0	0
$202$	3191.46	1.11163
$203$	−159.138	−0.0550210
$204$	0	0
$205$	1767.63	0.602228
$206$	−4735.77	−1.60173
$207$	0	0
$208$	−1250.91	−0.416996
$209$	−3952.17	−1.30803
$210$	0	0
$211$	−2798.33	−0.913010	−0.456505	−	0.889721i	$-0.650899\pi$
$211$	−2798.33	−0.913010	−0.456505	+	0.889721i	$0.650899\pi$
$212$	2295.28	0.743588
$213$	0	0
$214$	−6189.64	−1.97717
$215$	−2538.05	−0.805085
$216$	0	0
$217$	1706.34	0.533796
$218$	3946.35	1.22606
$219$	0	0
$220$	3972.00	1.21724
$221$	−389.872	−0.118668
$222$	0	0
$223$	2496.01	0.749529	0.374764	−	0.927120i	$-0.377724\pi$
$223$	2496.01	0.749529	0.374764	+	0.927120i	$0.377724\pi$
$224$	65.3725	0.0194995
$225$	0	0
$226$	−5958.02	−1.75364
$227$	−3409.57	−0.996920	−0.498460	−	0.866913i	$-0.666101\pi$
$227$	−3409.57	−0.996920	−0.498460	+	0.866913i	$0.666101\pi$
$228$	0	0
$229$	4794.47	1.38353	0.691763	−	0.722125i	$-0.256834\pi$
$229$	4794.47	1.38353	0.691763	+	0.722125i	$0.256834\pi$
$230$	154.741	0.0443622
$231$	0	0
$232$	1111.11	0.314431
$233$	−1989.36	−0.559345	−0.279672	−	0.960095i	$-0.590226\pi$
$233$	−1989.36	−0.559345	−0.279672	+	0.960095i	$0.590226\pi$
$234$	0	0
$235$	−563.855	−0.156518
$236$	5431.06	1.49802
$237$	0	0
$238$	−503.852	−0.137226
$239$	−2574.69	−0.696832	−0.348416	−	0.937340i	$-0.613280\pi$
$239$	−2574.69	−0.696832	−0.348416	+	0.937340i	$0.613280\pi$
$240$	0	0
$241$	−3683.44	−0.984528	−0.492264	−	0.870446i	$-0.663831\pi$
$241$	−3683.44	−0.984528	−0.492264	+	0.870446i	$0.663831\pi$
$242$	−5773.24	−1.53355
$243$	0	0
$244$	5664.73	1.48626
$245$	1564.44	0.407952
$246$	0	0
$247$	−1634.63	−0.421089
$248$	−11913.8	−3.05050
$249$	0	0
$250$	610.405	0.154422
$251$	−4631.27	−1.16463	−0.582317	−	0.812962i	$-0.697854\pi$
$251$	−4631.27	−1.16463	−0.582317	+	0.812962i	$0.697854\pi$
$252$	0	0
$253$	−317.720	−0.0789522
$254$	−3978.54	−0.982817
$255$	0	0
$256$	−8104.22	−1.97857
$257$	1609.98	0.390770	0.195385	−	0.980727i	$-0.437404\pi$
$257$	1609.98	0.390770	0.195385	+	0.980727i	$0.437404\pi$
$258$	0	0
$259$	−1858.86	−0.445962
$260$	1642.83	0.391862
$261$	0	0
$262$	−7295.12	−1.72021
$263$	2868.22	0.672479	0.336239	−	0.941777i	$-0.390845\pi$
$263$	2868.22	0.672479	0.336239	+	0.941777i	$0.390845\pi$
$264$	0	0
$265$	−724.244	−0.167887
$266$	−2112.52	−0.486942
$267$	0	0
$268$	−2901.20	−0.661264
$269$	554.047	0.125579	0.0627896	−	0.998027i	$-0.480000\pi$
$269$	554.047	0.125579	0.0627896	+	0.998027i	$0.480000\pi$
$270$	0	0
$271$	−1826.02	−0.409309	−0.204655	−	0.978834i	$-0.565607\pi$
$271$	−1826.02	−0.409309	−0.204655	+	0.978834i	$0.565607\pi$
$272$	1134.35	0.252867
$273$	0	0
$274$	−7855.79	−1.73206
$275$	−1253.31	−0.274827
$276$	0	0
$277$	1470.00	0.318859	0.159429	−	0.987209i	$-0.449035\pi$
$277$	1470.00	0.318859	0.159429	+	0.987209i	$0.449035\pi$
$278$	−251.841	−0.0543325
$279$	0	0
$280$	1051.24	0.224371
$281$	−7826.43	−1.66151	−0.830757	−	0.556635i	$-0.812092\pi$
$281$	−7826.43	−1.66151	−0.830757	+	0.556635i	$0.812092\pi$
$282$	0	0
$283$	−6796.87	−1.42767	−0.713837	−	0.700312i	$-0.753044\pi$
$283$	−6796.87	−1.42767	−0.713837	+	0.700312i	$0.753044\pi$
$284$	9413.10	1.96678
$285$	0	0
$286$	−5076.08	−1.04949
$287$	−1939.98	−0.399001
$288$	0	0
$289$	−4559.46	−0.928039
$290$	−708.070	−0.143377
$291$	0	0
$292$	−9865.89	−1.97725
$293$	−3631.49	−0.724074	−0.362037	−	0.932164i	$-0.617919\pi$
$293$	−3631.49	−0.724074	−0.362037	+	0.932164i	$0.617919\pi$
$294$	0	0
$295$	−1713.69	−0.338221
$296$	12978.7	2.54856
$297$	0	0
$298$	−2970.62	−0.577461
$299$	−131.410	−0.0254169
$300$	0	0
$301$	2785.50	0.533401
$302$	1667.96	0.317816
$303$	0	0
$304$	4756.00	0.897287
$305$	−1787.43	−0.335567
$306$	0	0
$307$	−9080.77	−1.68817	−0.844083	−	0.536213i	$-0.819854\pi$
$307$	−9080.77	−1.68817	−0.844083	+	0.536213i	$0.819854\pi$
$308$	−4359.27	−0.806469
$309$	0	0
$310$	7592.21	1.39100
$311$	−2657.69	−0.484577	−0.242289	−	0.970204i	$-0.577898\pi$
$311$	−2657.69	−0.484577	−0.242289	+	0.970204i	$0.577898\pi$
$312$	0	0
$313$	−530.474	−0.0957960	−0.0478980	−	0.998852i	$-0.515252\pi$
$313$	−530.474	−0.0957960	−0.0478980	+	0.998852i	$0.515252\pi$
$314$	1750.05	0.314525
$315$	0	0
$316$	−20206.6	−3.59719
$317$	−2393.66	−0.424106	−0.212053	−	0.977258i	$-0.568015\pi$
$317$	−2393.66	−0.424106	−0.212053	+	0.977258i	$0.568015\pi$
$318$	0	0
$319$	1453.84	0.255170
$320$	2704.02	0.472373
$321$	0	0
$322$	−169.828	−0.0293918
$323$	1482.31	0.255349
$324$	0	0
$325$	−518.373	−0.0884743
$326$	8142.12	1.38328
$327$	0	0
$328$	13545.1	2.28019
$329$	618.830	0.103700
$330$	0	0
$331$	−7984.54	−1.32589	−0.662945	−	0.748668i	$-0.730694\pi$
$331$	−7984.54	−1.32589	−0.662945	+	0.748668i	$0.730694\pi$
$332$	11711.9	1.93607
$333$	0	0
$334$	−8947.57	−1.46584
$335$	915.433	0.149300
$336$	0	0
$337$	9704.45	1.56865	0.784325	−	0.620350i	$-0.213010\pi$
$337$	9704.45	1.56865	0.784325	+	0.620350i	$0.213010\pi$
$338$	8629.00	1.38863
$339$	0	0
$340$	−1489.74	−0.237626
$341$	−15588.7	−2.47558
$342$	0	0
$343$	−3599.18	−0.566582
$344$	−19448.6	−3.04825
$345$	0	0
$346$	9936.27	1.54387
$347$	−5514.04	−0.853053	−0.426526	−	0.904475i	$-0.640263\pi$
$347$	−5514.04	−0.853053	−0.426526	+	0.904475i	$0.640263\pi$
$348$	0	0
$349$	2521.06	0.386674	0.193337	−	0.981132i	$-0.438069\pi$
$349$	2521.06	0.386674	0.193337	+	0.981132i	$0.438069\pi$
$350$	−669.920	−0.102311
$351$	0	0
$352$	−597.227	−0.0904326
$353$	−5721.03	−0.862606	−0.431303	−	0.902207i	$-0.641946\pi$
$353$	−5721.03	−0.862606	−0.431303	+	0.902207i	$0.641946\pi$
$354$	0	0
$355$	−2970.17	−0.444057
$356$	−14358.3	−2.13761
$357$	0	0
$358$	−1622.20	−0.239486
$359$	−6403.95	−0.941470	−0.470735	−	0.882275i	$-0.656011\pi$
$359$	−6403.95	−0.941470	−0.470735	+	0.882275i	$0.656011\pi$
$360$	0	0
$361$	−644.098	−0.0939055
$362$	4118.48	0.597963
$363$	0	0
$364$	−1803.01	−0.259624
$365$	3113.04	0.446422
$366$	0	0
$367$	13350.7	1.89891	0.949456	−	0.313899i	$-0.101635\pi$
$367$	13350.7	1.89891	0.949456	+	0.313899i	$0.101635\pi$
$368$	382.341	0.0541601
$369$	0	0
$370$	−8270.87	−1.16211
$371$	794.858	0.111232
$372$	0	0
$373$	−1367.05	−0.189767	−0.0948837	−	0.995488i	$-0.530248\pi$
$373$	−1367.05	−0.189767	−0.0948837	+	0.995488i	$0.530248\pi$
$374$	4603.07	0.636414
$375$	0	0
$376$	−4320.72	−0.592617
$377$	601.312	0.0821463
$378$	0	0
$379$	9882.96	1.33945	0.669727	−	0.742607i	$-0.266411\pi$
$379$	9882.96	1.33945	0.669727	+	0.742607i	$0.266411\pi$
$380$	−6246.09	−0.843204
$381$	0	0
$382$	11909.3	1.59511
$383$	−13225.8	−1.76450	−0.882252	−	0.470777i	$-0.843974\pi$
$383$	−13225.8	−1.76450	−0.882252	+	0.470777i	$0.843974\pi$
$384$	0	0
$385$	1375.51	0.182084
$386$	−19997.0	−2.63684
$387$	0	0
$388$	1205.35	0.157713
$389$	8442.16	1.10035	0.550173	−	0.835051i	$-0.314562\pi$
$389$	8442.16	1.10035	0.550173	+	0.835051i	$0.314562\pi$
$390$	0	0
$391$	119.165	0.0154128
$392$	11988.0	1.54461
$393$	0	0
$394$	15757.0	2.01479
$395$	6375.92	0.812170
$396$	0	0
$397$	−2268.55	−0.286789	−0.143394	−	0.989666i	$-0.545802\pi$
$397$	−2268.55	−0.286789	−0.143394	+	0.989666i	$0.545802\pi$
$398$	20601.1	2.59458
$399$	0	0
$400$	1508.22	0.188527
$401$	−8294.00	−1.03287	−0.516437	−	0.856325i	$-0.672742\pi$
$401$	−8294.00	−1.03287	−0.516437	+	0.856325i	$0.672742\pi$
$402$	0	0
$403$	−6447.51	−0.796957
$404$	−10356.2	−1.27535
$405$	0	0
$406$	777.107	0.0949930
$407$	16982.1	2.06823
$408$	0	0
$409$	9913.01	1.19845	0.599226	−	0.800580i	$-0.295475\pi$
$409$	9913.01	1.19845	0.599226	+	0.800580i	$0.295475\pi$
$410$	−8631.77	−1.03974
$411$	0	0
$412$	15367.5	1.83763
$413$	1880.78	0.224085
$414$	0	0
$415$	−3695.54	−0.437125
$416$	−247.015	−0.0291127
$417$	0	0
$418$	19299.4	2.25829
$419$	13332.1	1.55446	0.777228	−	0.629219i	$-0.216625\pi$
$419$	13332.1	1.55446	0.777228	+	0.629219i	$0.216625\pi$
$420$	0	0
$421$	−2848.13	−0.329714	−0.164857	−	0.986317i	$-0.552716\pi$
$421$	−2848.13	−0.329714	−0.164857	+	0.986317i	$0.552716\pi$
$422$	13664.9	1.57630
$423$	0	0
$424$	−5549.76	−0.635661
$425$	470.068	0.0536509
$426$	0	0
$427$	1961.70	0.222326
$428$	20085.3	2.26836
$429$	0	0
$430$	12393.9	1.38997
$431$	−12669.7	−1.41595	−0.707977	−	0.706235i	$-0.750392\pi$
$431$	−12669.7	−1.41595	−0.707977	+	0.706235i	$0.750392\pi$
$432$	0	0
$433$	−14545.6	−1.61436	−0.807178	−	0.590308i	$-0.799006\pi$
$433$	−14545.6	−1.61436	−0.807178	+	0.590308i	$0.799006\pi$
$434$	−8332.45	−0.921591
$435$	0	0
$436$	−12805.9	−1.40663
$437$	499.625	0.0546917
$438$	0	0
$439$	11489.4	1.24911	0.624557	−	0.780979i	$-0.285280\pi$
$439$	11489.4	1.24911	0.624557	+	0.780979i	$0.285280\pi$
$440$	−9603.90	−1.04056
$441$	0	0
$442$	1903.84	0.204879
$443$	11125.7	1.19322	0.596611	−	0.802531i	$-0.296514\pi$
$443$	11125.7	1.19322	0.596611	+	0.802531i	$0.296514\pi$
$444$	0	0
$445$	4530.57	0.482628
$446$	−12188.6	−1.29405
$447$	0	0
$448$	−2967.66	−0.312966
$449$	6676.79	0.701776	0.350888	−	0.936418i	$-0.385880\pi$
$449$	6676.79	0.701776	0.350888	+	0.936418i	$0.385880\pi$
$450$	0	0
$451$	17723.1	1.85044
$452$	19333.7	2.01190
$453$	0	0
$454$	16649.7	1.72117
$455$	568.914	0.0586178
$456$	0	0
$457$	−10826.9	−1.10823	−0.554114	−	0.832441i	$-0.686943\pi$
$457$	−10826.9	−1.10823	−0.554114	+	0.832441i	$0.686943\pi$
$458$	−23412.6	−2.38864
$459$	0	0
$460$	−502.132	−0.0508957
$461$	4114.21	0.415657	0.207829	−	0.978165i	$-0.433360\pi$
$461$	4114.21	0.415657	0.207829	+	0.978165i	$0.433360\pi$
$462$	0	0
$463$	−6107.19	−0.613014	−0.306507	−	0.951868i	$-0.599160\pi$
$463$	−6107.19	−0.613014	−0.306507	+	0.951868i	$0.599160\pi$
$464$	−1749.54	−0.175043
$465$	0	0
$466$	9714.53	0.965701
$467$	1409.31	0.139647	0.0698235	−	0.997559i	$-0.477756\pi$
$467$	1409.31	0.139647	0.0698235	+	0.997559i	$0.477756\pi$
$468$	0	0
$469$	−1004.69	−0.0989172
$470$	2753.44	0.270227
$471$	0	0
$472$	−13131.7	−1.28059
$473$	−25447.7	−2.47375
$474$	0	0
$475$	1970.87	0.190378
$476$	1634.99	0.157437
$477$	0	0
$478$	12572.8	1.20307
$479$	−12633.9	−1.20513	−0.602566	−	0.798069i	$-0.705855\pi$
$479$	−12633.9	−1.20513	−0.602566	+	0.798069i	$0.705855\pi$
$480$	0	0
$481$	7023.84	0.665821
$482$	17987.1	1.69977
$483$	0	0
$484$	18734.1	1.75940
$485$	−380.332	−0.0356083
$486$	0	0
$487$	−17024.7	−1.58411	−0.792056	−	0.610449i	$-0.790989\pi$
$487$	−17024.7	−1.58411	−0.792056	+	0.610449i	$0.790989\pi$
$488$	−13696.7	−1.27054
$489$	0	0
$490$	−7639.52	−0.704323
$491$	3792.47	0.348578	0.174289	−	0.984695i	$-0.444237\pi$
$491$	3792.47	0.348578	0.174289	+	0.984695i	$0.444237\pi$
$492$	0	0
$493$	−545.279	−0.0498136
$494$	7982.29	0.727004
$495$	0	0
$496$	18759.2	1.69821
$497$	3259.76	0.294206
$498$	0	0
$499$	6484.19	0.581708	0.290854	−	0.956767i	$-0.406061\pi$
$499$	6484.19	0.581708	0.290854	+	0.956767i	$0.406061\pi$
$500$	−1980.76	−0.177164
$501$	0	0
$502$	22615.6	2.01073
$503$	8781.60	0.778434	0.389217	−	0.921146i	$-0.372746\pi$
$503$	8781.60	0.778434	0.389217	+	0.921146i	$0.372746\pi$
$504$	0	0
$505$	3267.76	0.287948
$506$	1551.51	0.136310
$507$	0	0
$508$	12910.3	1.12756
$509$	−21343.3	−1.85860	−0.929299	−	0.369328i	$-0.879588\pi$
$509$	−21343.3	−1.85860	−0.929299	+	0.369328i	$0.879588\pi$
$510$	0	0
$511$	−3416.57	−0.295773
$512$	19210.3	1.65817
$513$	0	0
$514$	−7861.93	−0.674659
$515$	−4849.01	−0.414898
$516$	0	0
$517$	−5653.48	−0.480928
$518$	9077.28	0.769947
$519$	0	0
$520$	−3972.20	−0.334985
$521$	−7007.06	−0.589223	−0.294611	−	0.955617i	$-0.595190\pi$
$521$	−7007.06	−0.589223	−0.294611	+	0.955617i	$0.595190\pi$
$522$	0	0
$523$	−5901.85	−0.493441	−0.246721	−	0.969087i	$-0.579353\pi$
$523$	−5901.85	−0.493441	−0.246721	+	0.969087i	$0.579353\pi$
$524$	23672.6	1.97355
$525$	0	0
$526$	−14006.2	−1.16103
$527$	5846.70	0.483276
$528$	0	0
$529$	−12126.8	−0.996699
$530$	3536.66	0.289854
$531$	0	0
$532$	6855.08	0.558657
$533$	7330.34	0.595707
$534$	0	0
$535$	−6337.64	−0.512149
$536$	7014.80	0.565286
$537$	0	0
$538$	−2705.54	−0.216811
$539$	15685.8	1.25350
$540$	0	0
$541$	7826.71	0.621990	0.310995	−	0.950412i	$-0.399338\pi$
$541$	7826.71	0.621990	0.310995	+	0.950412i	$0.399338\pi$
$542$	8916.90	0.706667
$543$	0	0
$544$	223.997	0.0176540
$545$	4040.71	0.317587
$546$	0	0
$547$	4256.78	0.332737	0.166368	−	0.986064i	$-0.446796\pi$
$547$	4256.78	0.332737	0.166368	+	0.986064i	$0.446796\pi$
$548$	25491.9	1.98716
$549$	0	0
$550$	6120.21	0.474485
$551$	−2286.20	−0.176762
$552$	0	0
$553$	−6997.57	−0.538096
$554$	−7178.38	−0.550506
$555$	0	0
$556$	817.221	0.0623343
$557$	11138.6	0.847324	0.423662	−	0.905820i	$-0.360744\pi$
$557$	11138.6	0.847324	0.423662	+	0.905820i	$0.360744\pi$
$558$	0	0
$559$	−10525.2	−0.796368
$560$	−1655.27	−0.124907
$561$	0	0
$562$	38218.3	2.86858
$563$	−16458.1	−1.23202	−0.616009	−	0.787739i	$-0.711252\pi$
$563$	−16458.1	−1.23202	−0.616009	+	0.787739i	$0.711252\pi$
$564$	0	0
$565$	−6100.48	−0.454246
$566$	33190.7	2.46486
$567$	0	0
$568$	−22759.9	−1.68131
$569$	−2449.68	−0.180485	−0.0902423	−	0.995920i	$-0.528764\pi$
$569$	−2449.68	−0.180485	−0.0902423	+	0.995920i	$0.528764\pi$
$570$	0	0
$571$	156.880	0.0114978	0.00574888	−	0.999983i	$-0.498170\pi$
$571$	156.880	0.0114978	0.00574888	+	0.999983i	$0.498170\pi$
$572$	16471.8	1.20406
$573$	0	0
$574$	9473.37	0.688869
$575$	158.441	0.0114912
$576$	0	0
$577$	−20318.3	−1.46597	−0.732984	−	0.680246i	$-0.761873\pi$
$577$	−20318.3	−1.46597	−0.732984	+	0.680246i	$0.761873\pi$
$578$	22264.9	1.60225
$579$	0	0
$580$	2297.68	0.164493
$581$	4055.85	0.289613
$582$	0	0
$583$	−7261.62	−0.515859
$584$	23854.7	1.69027
$585$	0	0
$586$	17733.4	1.25010
$587$	9510.91	0.668752	0.334376	−	0.942440i	$-0.391475\pi$
$587$	9510.91	0.668752	0.334376	+	0.942440i	$0.391475\pi$
$588$	0	0
$589$	24513.6	1.71488
$590$	8368.38	0.583934
$591$	0	0
$592$	−20436.1	−1.41878
$593$	15977.2	1.10641	0.553207	−	0.833044i	$-0.313404\pi$
$593$	15977.2	1.10641	0.553207	+	0.833044i	$0.313404\pi$
$594$	0	0
$595$	−515.900	−0.0355459
$596$	9639.62	0.662507
$597$	0	0
$598$	641.707	0.0438819
$599$	4002.17	0.272995	0.136498	−	0.990640i	$-0.456415\pi$
$599$	4002.17	0.272995	0.136498	+	0.990640i	$0.456415\pi$
$600$	0	0
$601$	−1574.81	−0.106885	−0.0534425	−	0.998571i	$-0.517019\pi$
$601$	−1574.81	−0.106885	−0.0534425	+	0.998571i	$0.517019\pi$
$602$	−13602.3	−0.920910
$603$	0	0
$604$	−5412.51	−0.364622
$605$	−5911.28	−0.397236
$606$	0	0
$607$	−8803.52	−0.588672	−0.294336	−	0.955702i	$-0.595098\pi$
$607$	−8803.52	−0.588672	−0.294336	+	0.955702i	$0.595098\pi$
$608$	939.156	0.0626444
$609$	0	0
$610$	8728.44	0.579351
$611$	−2338.29	−0.154824
$612$	0	0
$613$	16874.2	1.11181	0.555907	−	0.831245i	$-0.312371\pi$
$613$	16874.2	1.11181	0.555907	+	0.831245i	$0.312371\pi$
$614$	44343.6	2.91460
$615$	0	0
$616$	10540.3	0.689415
$617$	−4592.02	−0.299624	−0.149812	−	0.988715i	$-0.547867\pi$
$617$	−4592.02	−0.299624	−0.149812	+	0.988715i	$0.547867\pi$
$618$	0	0
$619$	4982.48	0.323526	0.161763	−	0.986830i	$-0.448282\pi$
$619$	4982.48	0.323526	0.161763	+	0.986830i	$0.448282\pi$
$620$	−24636.6	−1.59586
$621$	0	0
$622$	12978.1	0.836616
$623$	−4972.30	−0.319761
$624$	0	0
$625$	625.000	0.0400000
$626$	2590.43	0.165391
$627$	0	0
$628$	−5678.88	−0.360847
$629$	−6369.32	−0.403755
$630$	0	0
$631$	20584.3	1.29865	0.649325	−	0.760511i	$-0.275051\pi$
$631$	20584.3	1.29865	0.649325	+	0.760511i	$0.275051\pi$
$632$	48857.6	3.07508
$633$	0	0
$634$	11688.8	0.732213
$635$	−4073.66	−0.254580
$636$	0	0
$637$	6487.69	0.403535
$638$	−7099.45	−0.440548
$639$	0	0
$640$	−12727.9	−0.786115
$641$	−4840.03	−0.298237	−0.149118	−	0.988819i	$-0.547644\pi$
$641$	−4840.03	−0.298237	−0.149118	+	0.988819i	$0.547644\pi$
$642$	0	0
$643$	1340.16	0.0821941	0.0410970	−	0.999155i	$-0.486915\pi$
$643$	1340.16	0.0821941	0.0410970	+	0.999155i	$0.486915\pi$
$644$	551.090	0.0337205
$645$	0	0
$646$	−7238.46	−0.440856
$647$	−15752.3	−0.957169	−0.478585	−	0.878041i	$-0.658850\pi$
$647$	−15752.3	−0.957169	−0.478585	+	0.878041i	$0.658850\pi$
$648$	0	0
$649$	−17182.3	−1.03924
$650$	2531.34	0.152750
$651$	0	0
$652$	−26421.1	−1.58701
$653$	−2290.66	−0.137275	−0.0686373	−	0.997642i	$-0.521865\pi$
$653$	−2290.66	−0.137275	−0.0686373	+	0.997642i	$0.521865\pi$
$654$	0	0
$655$	−7469.55	−0.445587
$656$	−21327.8	−1.26938
$657$	0	0
$658$	−3021.90	−0.179036
$659$	3003.49	0.177541	0.0887704	−	0.996052i	$-0.471706\pi$
$659$	3003.49	0.177541	0.0887704	+	0.996052i	$0.471706\pi$
$660$	0	0
$661$	8104.71	0.476909	0.238454	−	0.971154i	$-0.423359\pi$
$661$	8104.71	0.476909	0.238454	+	0.971154i	$0.423359\pi$
$662$	38990.4	2.28913
$663$	0	0
$664$	−28318.3	−1.65506
$665$	−2163.03	−0.126133
$666$	0	0
$667$	−183.791	−0.0106693
$668$	29034.7	1.68172
$669$	0	0
$670$	−4470.28	−0.257764
$671$	−17921.6	−1.03108
$672$	0	0
$673$	5747.03	0.329170	0.164585	−	0.986363i	$-0.447371\pi$
$673$	5747.03	0.329170	0.164585	+	0.986363i	$0.447371\pi$
$674$	−47389.2	−2.70825
$675$	0	0
$676$	−28001.0	−1.59314
$677$	−25714.2	−1.45979	−0.729895	−	0.683559i	$-0.760431\pi$
$677$	−25714.2	−1.45979	−0.729895	+	0.683559i	$0.760431\pi$
$678$	0	0
$679$	417.415	0.0235919
$680$	3602.05	0.203136
$681$	0	0
$682$	76123.2	4.27406
$683$	−4007.09	−0.224490	−0.112245	−	0.993681i	$-0.535804\pi$
$683$	−4007.09	−0.224490	−0.112245	+	0.993681i	$0.535804\pi$
$684$	0	0
$685$	−8043.62	−0.448658
$686$	17575.7	0.978196
$687$	0	0
$688$	30623.4	1.69696
$689$	−3003.43	−0.166069
$690$	0	0
$691$	−27863.7	−1.53399	−0.766993	−	0.641655i	$-0.778248\pi$
$691$	−27863.7	−1.53399	−0.766993	+	0.641655i	$0.778248\pi$
$692$	−32243.1	−1.77124
$693$	0	0
$694$	26926.4	1.47278
$695$	−257.863	−0.0140738
$696$	0	0
$697$	−6647.26	−0.361238
$698$	−12311.0	−0.667588
$699$	0	0
$700$	2173.88	0.117378
$701$	−12916.0	−0.695906	−0.347953	−	0.937512i	$-0.613123\pi$
$701$	−12916.0	−0.695906	−0.347953	+	0.937512i	$0.613123\pi$
$702$	0	0
$703$	−26704.8	−1.43271
$704$	27111.8	1.45144
$705$	0	0
$706$	27937.2	1.48928
$707$	−3586.37	−0.190777
$708$	0	0
$709$	−30250.8	−1.60239	−0.801193	−	0.598406i	$-0.795801\pi$
$709$	−30250.8	−1.60239	−0.801193	+	0.598406i	$0.795801\pi$
$710$	14504.1	0.766659
$711$	0	0
$712$	34716.9	1.82735
$713$	1970.68	0.103510
$714$	0	0
$715$	−5197.45	−0.271851
$716$	5264.03	0.274757
$717$	0	0
$718$	31272.1	1.62544
$719$	−6161.35	−0.319582	−0.159791	−	0.987151i	$-0.551082\pi$
$719$	−6161.35	−0.319582	−0.159791	+	0.987151i	$0.551082\pi$
$720$	0	0
$721$	5321.78	0.274887
$722$	3145.29	0.162127
$723$	0	0
$724$	−13364.4	−0.686029
$725$	−725.000	−0.0371391
$726$	0	0
$727$	−35454.3	−1.80870	−0.904352	−	0.426786i	$-0.859646\pi$
$727$	−35454.3	−1.80870	−0.904352	+	0.426786i	$0.859646\pi$
$728$	4359.49	0.221941
$729$	0	0
$730$	−15201.7	−0.770742
$731$	9544.43	0.482919
$732$	0	0
$733$	−26681.1	−1.34446	−0.672231	−	0.740342i	$-0.734664\pi$
$733$	−26681.1	−1.34446	−0.672231	+	0.740342i	$0.734664\pi$
$734$	−65194.7	−3.27845
$735$	0	0
$736$	75.5001	0.00378121
$737$	9178.57	0.458747
$738$	0	0
$739$	−28886.6	−1.43791	−0.718953	−	0.695059i	$-0.755378\pi$
$739$	−28886.6	−1.43791	−0.718953	+	0.695059i	$0.755378\pi$
$740$	26838.8	1.33326
$741$	0	0
$742$	−3881.48	−0.192040
$743$	16398.0	0.809669	0.404834	−	0.914390i	$-0.367329\pi$
$743$	16398.0	0.809669	0.404834	+	0.914390i	$0.367329\pi$
$744$	0	0
$745$	−3041.65	−0.149580
$746$	6675.64	0.327631
$747$	0	0
$748$	−14936.9	−0.730142
$749$	6955.56	0.339320
$750$	0	0
$751$	−5905.21	−0.286930	−0.143465	−	0.989655i	$-0.545824\pi$
$751$	−5905.21	−0.286930	−0.143465	+	0.989655i	$0.545824\pi$
$752$	6803.33	0.329910
$753$	0	0
$754$	−2936.35	−0.141824
$755$	1707.84	0.0823241
$756$	0	0
$757$	−12439.0	−0.597229	−0.298614	−	0.954374i	$-0.596524\pi$
$757$	−12439.0	−0.597229	−0.298614	+	0.954374i	$0.596524\pi$
$758$	−48260.9	−2.31255
$759$	0	0
$760$	15102.4	0.720818
$761$	−14270.0	−0.679748	−0.339874	−	0.940471i	$-0.610384\pi$
$761$	−14270.0	−0.679748	−0.339874	+	0.940471i	$0.610384\pi$
$762$	0	0
$763$	−4434.68	−0.210414
$764$	−38645.6	−1.83004
$765$	0	0
$766$	64584.6	3.04639
$767$	−7106.66	−0.334559
$768$	0	0
$769$	−15143.9	−0.710147	−0.355074	−	0.934838i	$-0.615544\pi$
$769$	−15143.9	−0.710147	−0.355074	+	0.934838i	$0.615544\pi$
$770$	−6716.94	−0.314366
$771$	0	0
$772$	64890.1	3.02519
$773$	40617.2	1.88991	0.944955	−	0.327202i	$-0.106106\pi$
$773$	40617.2	1.88991	0.944955	+	0.327202i	$0.106106\pi$
$774$	0	0
$775$	7773.74	0.360311
$776$	−2914.42	−0.134822
$777$	0	0
$778$	−41225.1	−1.89973
$779$	−27870.1	−1.28184
$780$	0	0
$781$	−29780.4	−1.36444
$782$	−581.910	−0.0266100
$783$	0	0
$784$	−18876.1	−0.859881
$785$	1791.89	0.0814718
$786$	0	0
$787$	43878.2	1.98741	0.993703	−	0.112048i	$-0.0357412\pi$
$787$	43878.2	1.98741	0.993703	+	0.112048i	$0.0357412\pi$
$788$	−51131.3	−2.31152
$789$	0	0
$790$	−31135.1	−1.40220
$791$	6695.27	0.300956
$792$	0	0
$793$	−7412.43	−0.331933
$794$	11077.9	0.495137
$795$	0	0
$796$	−66850.4	−2.97670
$797$	−14556.0	−0.646925	−0.323463	−	0.946241i	$-0.604847\pi$
$797$	−14556.0	−0.646925	−0.323463	+	0.946241i	$0.604847\pi$
$798$	0	0
$799$	2120.40	0.0938853
$800$	297.825	0.0131621
$801$	0	0
$802$	40501.6	1.78324
$803$	31212.9	1.37170
$804$	0	0
$805$	−173.889	−0.00761338
$806$	31484.8	1.37593
$807$	0	0
$808$	25040.3	1.09024
$809$	14030.7	0.609755	0.304878	−	0.952392i	$-0.401384\pi$
$809$	14030.7	0.609755	0.304878	+	0.952392i	$0.401384\pi$
$810$	0	0
$811$	30195.7	1.30741	0.653707	−	0.756747i	$-0.273213\pi$
$811$	30195.7	1.30741	0.653707	+	0.756747i	$0.273213\pi$
$812$	−2521.70	−0.108983
$813$	0	0
$814$	−82927.7	−3.57078
$815$	8336.80	0.358313
$816$	0	0
$817$	40017.2	1.71362
$818$	−48407.6	−2.06911
$819$	0	0
$820$	28010.0	1.19287
$821$	−31237.5	−1.32789	−0.663944	−	0.747782i	$-0.731119\pi$
$821$	−31237.5	−1.32789	−0.663944	+	0.747782i	$0.731119\pi$
$822$	0	0
$823$	−22306.9	−0.944798	−0.472399	−	0.881385i	$-0.656612\pi$
$823$	−22306.9	−0.944798	−0.472399	+	0.881385i	$0.656612\pi$
$824$	−37157.1	−1.57091
$825$	0	0
$826$	−9184.30	−0.386880
$827$	−41561.9	−1.74758	−0.873790	−	0.486304i	$-0.838345\pi$
$827$	−41561.9	−1.74758	−0.873790	+	0.486304i	$0.838345\pi$
$828$	0	0
$829$	42423.6	1.77736	0.888681	−	0.458527i	$-0.151623\pi$
$829$	42423.6	1.77736	0.888681	+	0.458527i	$0.151623\pi$
$830$	18046.2	0.754690
$831$	0	0
$832$	11213.5	0.467259
$833$	−5883.13	−0.244704
$834$	0	0
$835$	−9161.51	−0.379697
$836$	−62626.3	−2.59088
$837$	0	0
$838$	−65104.0	−2.68375
$839$	31547.4	1.29814	0.649068	−	0.760730i	$-0.275159\pi$
$839$	31547.4	1.29814	0.649068	+	0.760730i	$0.275159\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	13908.1	0.569247
$843$	0	0
$844$	−44342.5	−1.80845
$845$	8835.32	0.359697
$846$	0	0
$847$	6487.63	0.263185
$848$	8738.56	0.353872
$849$	0	0
$850$	−2295.45	−0.0926276
$851$	−2146.84	−0.0864779
$852$	0	0
$853$	−5030.77	−0.201935	−0.100967	−	0.994890i	$-0.532194\pi$
$853$	−5030.77	−0.201935	−0.100967	+	0.994890i	$0.532194\pi$
$854$	−9579.46	−0.383844
$855$	0	0
$856$	−48564.2	−1.93912
$857$	16655.5	0.663875	0.331937	−	0.943301i	$-0.392298\pi$
$857$	16655.5	0.663875	0.331937	+	0.943301i	$0.392298\pi$
$858$	0	0
$859$	33206.7	1.31897	0.659486	−	0.751717i	$-0.270774\pi$
$859$	33206.7	1.31897	0.659486	+	0.751717i	$0.270774\pi$
$860$	−40218.0	−1.59468
$861$	0	0
$862$	61869.0	2.44463
$863$	12156.5	0.479504	0.239752	−	0.970834i	$-0.422934\pi$
$863$	12156.5	0.479504	0.239752	+	0.970834i	$0.422934\pi$
$864$	0	0
$865$	10173.9	0.399909
$866$	71029.6	2.78716
$867$	0	0
$868$	27038.7	1.05732
$869$	63928.0	2.49552
$870$	0	0
$871$	3796.28	0.147683
$872$	30963.3	1.20246
$873$	0	0
$874$	−2439.79	−0.0944246
$875$	−685.938	−0.0265016
$876$	0	0
$877$	10910.0	0.420075	0.210037	−	0.977693i	$-0.432641\pi$
$877$	10910.0	0.420075	0.210037	+	0.977693i	$0.432641\pi$
$878$	−56105.7	−2.15658
$879$	0	0
$880$	15122.1	0.579281
$881$	10796.4	0.412871	0.206435	−	0.978460i	$-0.433814\pi$
$881$	10796.4	0.412871	0.206435	+	0.978460i	$0.433814\pi$
$882$	0	0
$883$	−23778.3	−0.906232	−0.453116	−	0.891452i	$-0.649688\pi$
$883$	−23778.3	−0.906232	−0.453116	+	0.891452i	$0.649688\pi$
$884$	−6177.94	−0.235053
$885$	0	0
$886$	−54329.4	−2.06008
$887$	26229.5	0.992897	0.496449	−	0.868066i	$-0.334637\pi$
$887$	26229.5	0.992897	0.496449	+	0.868066i	$0.334637\pi$
$888$	0	0
$889$	4470.85	0.168670
$890$	−22123.8	−0.833251
$891$	0	0
$892$	39551.8	1.48463
$893$	8890.26	0.333148
$894$	0	0
$895$	−1660.99	−0.0620344
$896$	13968.8	0.520833
$897$	0	0
$898$	−32604.4	−1.21161
$899$	−9017.54	−0.334540
$900$	0	0
$901$	2723.55	0.100704
$902$	−86546.3	−3.19476
$903$	0	0
$904$	−46746.9	−1.71989
$905$	4216.96	0.154891
$906$	0	0
$907$	53019.1	1.94098	0.970491	−	0.241136i	$-0.0775201\pi$
$907$	53019.1	1.94098	0.970491	+	0.241136i	$0.0775201\pi$
$908$	−54028.1	−1.97465
$909$	0	0
$910$	−2778.14	−0.101203
$911$	16886.9	0.614147	0.307074	−	0.951686i	$-0.400650\pi$
$911$	16886.9	0.614147	0.307074	+	0.951686i	$0.400650\pi$
$912$	0	0
$913$	−37053.2	−1.34314
$914$	52870.3	1.91334
$915$	0	0
$916$	75973.4	2.74043
$917$	8197.83	0.295219
$918$	0	0
$919$	−54865.8	−1.96937	−0.984687	−	0.174330i	$-0.944224\pi$
$919$	−54865.8	−1.96937	−0.984687	+	0.174330i	$0.944224\pi$
$920$	1214.10	0.0435085
$921$	0	0
$922$	−20090.7	−0.717626
$923$	−12317.2	−0.439249
$924$	0	0
$925$	−8468.62	−0.301023
$926$	29822.9	1.05836
$927$	0	0
$928$	−345.477	−0.0122207
$929$	−27835.6	−0.983053	−0.491527	−	0.870863i	$-0.663561\pi$
$929$	−27835.6	−0.983053	−0.491527	+	0.870863i	$0.663561\pi$
$930$	0	0
$931$	−24666.4	−0.868322
$932$	−31523.5	−1.10793
$933$	0	0
$934$	−6882.01	−0.241099
$935$	4713.13	0.164851
$936$	0	0
$937$	−38996.7	−1.35962	−0.679811	−	0.733387i	$-0.737938\pi$
$937$	−38996.7	−1.35962	−0.679811	+	0.733387i	$0.737938\pi$
$938$	4906.13	0.170779
$939$	0	0
$940$	−8934.87	−0.310025
$941$	−15686.8	−0.543437	−0.271718	−	0.962377i	$-0.587592\pi$
$941$	−15686.8	−0.543437	−0.271718	+	0.962377i	$0.587592\pi$
$942$	0	0
$943$	−2240.52	−0.0773715
$944$	20677.0	0.712902
$945$	0	0
$946$	124267.	4.27090
$947$	54527.1	1.87106	0.935530	−	0.353248i	$-0.114923\pi$
$947$	54527.1	1.87106	0.935530	+	0.353248i	$0.114923\pi$
$948$	0	0
$949$	12909.7	0.441589
$950$	−9624.22	−0.328685
$951$	0	0
$952$	−3953.25	−0.134586
$953$	−48998.7	−1.66550	−0.832752	−	0.553647i	$-0.813236\pi$
$953$	−48998.7	−1.66550	−0.832752	+	0.553647i	$0.813236\pi$
$954$	0	0
$955$	12194.1	0.413184
$956$	−40798.6	−1.38025
$957$	0	0
$958$	61694.4	2.08064
$959$	8827.88	0.297254
$960$	0	0
$961$	66898.8	2.24560
$962$	−34299.1	−1.14953
$963$	0	0
$964$	−58368.0	−1.95011
$965$	−20475.2	−0.683025
$966$	0	0
$967$	−18866.1	−0.627395	−0.313698	−	0.949523i	$-0.601568\pi$
$967$	−18866.1	−0.627395	−0.313698	+	0.949523i	$0.601568\pi$
$968$	−45297.1	−1.50403
$969$	0	0
$970$	1857.25	0.0614772
$971$	−49921.4	−1.64990	−0.824950	−	0.565205i	$-0.808797\pi$
$971$	−49921.4	−1.64990	−0.824950	+	0.565205i	$0.808797\pi$
$972$	0	0
$973$	283.004	0.00932446
$974$	83135.6	2.73495
$975$	0	0
$976$	21566.7	0.707308
$977$	2591.52	0.0848618	0.0424309	−	0.999099i	$-0.486490\pi$
$977$	2591.52	0.0848618	0.0424309	+	0.999099i	$0.486490\pi$
$978$	0	0
$979$	45425.6	1.48295
$980$	24790.1	0.808053
$981$	0	0
$982$	−18519.6	−0.601815
$983$	−18566.6	−0.602423	−0.301211	−	0.953557i	$-0.597391\pi$
$983$	−18566.6	−0.602423	−0.301211	+	0.953557i	$0.597391\pi$
$984$	0	0
$985$	16133.8	0.521894
$986$	2662.73	0.0860026
$987$	0	0
$988$	−25902.4	−0.834074
$989$	3217.04	0.103434
$990$	0	0
$991$	−34558.3	−1.10775	−0.553875	−	0.832600i	$-0.686851\pi$
$991$	−34558.3	−1.10775	−0.553875	+	0.832600i	$0.686851\pi$
$992$	3704.34	0.118561
$993$	0	0
$994$	−15918.2	−0.507943
$995$	21093.7	0.672076
$996$	0	0
$997$	−11238.6	−0.357000	−0.178500	−	0.983940i	$-0.557124\pi$
$997$	−11238.6	−0.357000	−0.178500	+	0.983940i	$0.557124\pi$
$998$	−31663.9	−1.00431
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-4.88324 q^{2} +15.8460 q^{4} -5.00000 q^{5} +5.48750 q^{7} -38.3141 q^{8} +24.4162 q^{10} -50.1324 q^{11} -20.7349 q^{13} -26.7968 q^{14} +60.3288 q^{16} +18.8027 q^{17} +78.8346 q^{19} -79.2302 q^{20} +244.809 q^{22} +6.33763 q^{23} +25.0000 q^{25} +101.254 q^{26} +86.9552 q^{28} -29.0000 q^{29} +310.950 q^{31} +11.9130 q^{32} -91.8182 q^{34} -27.4375 q^{35} -338.745 q^{37} -384.969 q^{38} +191.571 q^{40} -353.526 q^{41} +507.609 q^{43} -794.400 q^{44} -30.9482 q^{46} +112.771 q^{47} -312.887 q^{49} -122.081 q^{50} -328.566 q^{52} +144.849 q^{53} +250.662 q^{55} -210.249 q^{56} +141.614 q^{58} +342.739 q^{59} +357.486 q^{61} -1518.44 q^{62} -540.804 q^{64} +103.675 q^{65} -183.087 q^{67} +297.949 q^{68} +133.984 q^{70} +594.034 q^{71} -622.609 q^{73} +1654.17 q^{74} +1249.22 q^{76} -275.102 q^{77} -1275.18 q^{79} -301.644 q^{80} +1726.35 q^{82} +739.108 q^{83} -94.0136 q^{85} -2478.78 q^{86} +1920.78 q^{88} -906.113 q^{89} -113.783 q^{91} +100.426 q^{92} -550.688 q^{94} -394.173 q^{95} +76.0665 q^{97} +1527.90 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 22 q^{4} - 35 q^{5} - 50 q^{7} + 33 q^{8} - 10 q^{10} - 76 q^{11} + 30 q^{13} - 89 q^{14} + 138 q^{16} + 140 q^{17} + 90 q^{19} - 110 q^{20} + 61 q^{22} - 34 q^{23} + 175 q^{25} + 241 q^{26}+ \cdots + 761 q^{98}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

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