LMFDB - Embedded newform 1148.4.a.d.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,4,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, 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("1148.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1148.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:	$67.7341926866$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 238 x^{13} + 602 x^{12} + 21013 x^{11} - 44923 x^{10} - 876344 x^{9} + \cdots - 45134496 $ x^15 - 3x^14 - 238x^13 + 602x^12 + 21013x^11 - 44923x^10 - 876344x^9 + 1512470x^8 + 18440602x^7 - 23548938x^6 - 192795335x^5 + 157072885x^4 + 930883767x^3 - 339053445x^2 - 1549251522x - 45134496
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-9.01814$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.0181 q^{3} -20.8879 q^{5} -7.00000 q^{7} +73.3631 q^{9} +O(q^{10})$	$q+10.0181 q^{3} -20.8879 q^{5} -7.00000 q^{7} +73.3631 q^{9} -48.8564 q^{11} +52.5868 q^{13} -209.257 q^{15} -51.2251 q^{17} -62.8576 q^{19} -70.1270 q^{21} -19.8570 q^{23} +311.302 q^{25} +464.473 q^{27} +71.4361 q^{29} +324.338 q^{31} -489.451 q^{33} +146.215 q^{35} +290.977 q^{37} +526.822 q^{39} +41.0000 q^{41} +434.219 q^{43} -1532.40 q^{45} +404.984 q^{47} +49.0000 q^{49} -513.180 q^{51} -24.2770 q^{53} +1020.51 q^{55} -629.716 q^{57} -185.268 q^{59} +521.179 q^{61} -513.542 q^{63} -1098.42 q^{65} -246.515 q^{67} -198.931 q^{69} -981.074 q^{71} -1222.03 q^{73} +3118.67 q^{75} +341.995 q^{77} +521.960 q^{79} +2672.35 q^{81} +527.723 q^{83} +1069.98 q^{85} +715.657 q^{87} +318.192 q^{89} -368.107 q^{91} +3249.27 q^{93} +1312.96 q^{95} +440.656 q^{97} -3584.26 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	$ 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) $ 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	10.0181	1.92799	0.963996	−	0.265917i	$-0.0856746\pi$
$3$	10.0181	1.92799	0.963996	+	0.265917i	$0.0856746\pi$
$4$	0	0
$5$	−20.8879	−1.86827	−0.934133	−	0.356925i	$-0.883825\pi$
$5$	−20.8879	−1.86827	−0.934133	+	0.356925i	$0.883825\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	73.3631	2.71715
$10$	0	0
$11$	−48.8564	−1.33916	−0.669580	−	0.742740i	$-0.733526\pi$
$11$	−48.8564	−1.33916	−0.669580	+	0.742740i	$0.733526\pi$
$12$	0	0
$13$	52.5868	1.12192	0.560960	−	0.827843i	$-0.310432\pi$
$13$	52.5868	1.12192	0.560960	+	0.827843i	$0.310432\pi$
$14$	0	0
$15$	−209.257	−3.60200
$16$	0	0
$17$	−51.2251	−0.730818	−0.365409	−	0.930847i	$-0.619071\pi$
$17$	−51.2251	−0.730818	−0.365409	+	0.930847i	$0.619071\pi$
$18$	0	0
$19$	−62.8576	−0.758974	−0.379487	−	0.925197i	$-0.623900\pi$
$19$	−62.8576	−0.758974	−0.379487	+	0.925197i	$0.623900\pi$
$20$	0	0
$21$	−70.1270	−0.728713
$22$	0	0
$23$	−19.8570	−0.180021	−0.0900104	−	0.995941i	$-0.528690\pi$
$23$	−19.8570	−0.180021	−0.0900104	+	0.995941i	$0.528690\pi$
$24$	0	0
$25$	311.302	2.49042
$26$	0	0
$27$	464.473	3.31066
$28$	0	0
$29$	71.4361	0.457426	0.228713	−	0.973494i	$-0.426548\pi$
$29$	71.4361	0.457426	0.228713	+	0.973494i	$0.426548\pi$
$30$	0	0
$31$	324.338	1.87913	0.939563	−	0.342376i	$-0.111232\pi$
$31$	324.338	1.87913	0.939563	+	0.342376i	$0.111232\pi$
$32$	0	0
$33$	−489.451	−2.58189
$34$	0	0
$35$	146.215	0.706138
$36$	0	0
$37$	290.977	1.29287	0.646436	−	0.762968i	$-0.276259\pi$
$37$	290.977	1.29287	0.646436	+	0.762968i	$0.276259\pi$
$38$	0	0
$39$	526.822	2.16305
$40$	0	0
$41$	41.0000	0.156174
$41$	41.0000	0.156174
$42$	0	0
$43$	434.219	1.53995	0.769974	−	0.638075i	$-0.220269\pi$
$43$	434.219	1.53995	0.769974	+	0.638075i	$0.220269\pi$
$44$	0	0
$45$	−1532.40	−5.07637
$46$	0	0
$47$	404.984	1.25687	0.628437	−	0.777861i	$-0.283695\pi$
$47$	404.984	1.25687	0.628437	+	0.777861i	$0.283695\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−513.180	−1.40901
$52$	0	0
$53$	−24.2770	−0.0629189	−0.0314595	−	0.999505i	$-0.510016\pi$
$53$	−24.2770	−0.0629189	−0.0314595	+	0.999505i	$0.510016\pi$
$54$	0	0
$55$	1020.51	2.50191
$56$	0	0
$57$	−629.716	−1.46330
$58$	0	0
$59$	−185.268	−0.408811	−0.204406	−	0.978886i	$-0.565526\pi$
$59$	−185.268	−0.408811	−0.204406	+	0.978886i	$0.565526\pi$
$60$	0	0
$61$	521.179	1.09394	0.546968	−	0.837153i	$-0.315782\pi$
$61$	521.179	1.09394	0.546968	+	0.837153i	$0.315782\pi$
$62$	0	0
$63$	−513.542	−1.02699
$64$	0	0
$65$	−1098.42	−2.09604
$66$	0	0
$67$	−246.515	−0.449502	−0.224751	−	0.974416i	$-0.572157\pi$
$67$	−246.515	−0.449502	−0.224751	+	0.974416i	$0.572157\pi$
$68$	0	0
$69$	−198.931	−0.347079
$70$	0	0
$71$	−981.074	−1.63989	−0.819944	−	0.572444i	$-0.805996\pi$
$71$	−981.074	−1.63989	−0.819944	+	0.572444i	$0.805996\pi$
$72$	0	0
$73$	−1222.03	−1.95928	−0.979641	−	0.200756i	$-0.935660\pi$
$73$	−1222.03	−1.95928	−0.979641	+	0.200756i	$0.935660\pi$
$74$	0	0
$75$	3118.67	4.80151
$76$	0	0
$77$	341.995	0.506155
$78$	0	0
$79$	521.960	0.743356	0.371678	−	0.928362i	$-0.378783\pi$
$79$	521.960	0.743356	0.371678	+	0.928362i	$0.378783\pi$
$80$	0	0
$81$	2672.35	3.66577
$82$	0	0
$83$	527.723	0.697893	0.348946	−	0.937143i	$-0.386539\pi$
$83$	527.723	0.697893	0.348946	+	0.937143i	$0.386539\pi$
$84$	0	0
$85$	1069.98	1.36536
$86$	0	0
$87$	715.657	0.881913
$88$	0	0
$89$	318.192	0.378969	0.189485	−	0.981884i	$-0.439318\pi$
$89$	318.192	0.378969	0.189485	+	0.981884i	$0.439318\pi$
$90$	0	0
$91$	−368.107	−0.424046
$92$	0	0
$93$	3249.27	3.62294
$94$	0	0
$95$	1312.96	1.41797
$96$	0	0
$97$	440.656	0.461256	0.230628	−	0.973042i	$-0.425922\pi$
$97$	440.656	0.461256	0.230628	+	0.973042i	$0.425922\pi$
$98$	0	0
$99$	−3584.26	−3.63871
$100$	0	0
$101$	946.510	0.932488	0.466244	−	0.884656i	$-0.345607\pi$
$101$	946.510	0.932488	0.466244	+	0.884656i	$0.345607\pi$
$102$	0	0
$103$	547.646	0.523895	0.261947	−	0.965082i	$-0.415635\pi$
$103$	547.646	0.523895	0.261947	+	0.965082i	$0.415635\pi$
$104$	0	0
$105$	1464.80	1.36143
$106$	0	0
$107$	−15.8590	−0.0143285	−0.00716426	−	0.999974i	$-0.502280\pi$
$107$	−15.8590	−0.0143285	−0.00716426	+	0.999974i	$0.502280\pi$
$108$	0	0
$109$	−942.204	−0.827952	−0.413976	−	0.910288i	$-0.635860\pi$
$109$	−942.204	−0.827952	−0.413976	+	0.910288i	$0.635860\pi$
$110$	0	0
$111$	2915.04	2.49265
$112$	0	0
$113$	−150.458	−0.125256	−0.0626280	−	0.998037i	$-0.519948\pi$
$113$	−150.458	−0.125256	−0.0626280	+	0.998037i	$0.519948\pi$
$114$	0	0
$115$	414.771	0.336327
$116$	0	0
$117$	3857.93	3.04843
$118$	0	0
$119$	358.575	0.276223
$120$	0	0
$121$	1055.95	0.793352
$122$	0	0
$123$	410.744	0.301102
$124$	0	0
$125$	−3891.46	−2.78450
$126$	0	0
$127$	1577.10	1.10193	0.550965	−	0.834528i	$-0.314260\pi$
$127$	1577.10	1.10193	0.550965	+	0.834528i	$0.314260\pi$
$128$	0	0
$129$	4350.07	2.96901
$130$	0	0
$131$	−911.469	−0.607904	−0.303952	−	0.952687i	$-0.598306\pi$
$131$	−911.469	−0.607904	−0.303952	+	0.952687i	$0.598306\pi$
$132$	0	0
$133$	440.003	0.286865
$134$	0	0
$135$	−9701.83	−6.18519
$136$	0	0
$137$	−546.607	−0.340874	−0.170437	−	0.985369i	$-0.554518\pi$
$137$	−546.607	−0.340874	−0.170437	+	0.985369i	$0.554518\pi$
$138$	0	0
$139$	−2660.04	−1.62318	−0.811589	−	0.584229i	$-0.801397\pi$
$139$	−2660.04	−1.62318	−0.811589	+	0.584229i	$0.801397\pi$
$140$	0	0
$141$	4057.19	2.42324
$142$	0	0
$143$	−2569.20	−1.50243
$144$	0	0
$145$	−1492.15	−0.854593
$146$	0	0
$147$	490.889	0.275427
$148$	0	0
$149$	1127.47	0.619908	0.309954	−	0.950752i	$-0.399686\pi$
$149$	1127.47	0.619908	0.309954	+	0.950752i	$0.399686\pi$
$150$	0	0
$151$	2161.77	1.16505	0.582523	−	0.812814i	$-0.302066\pi$
$151$	2161.77	1.16505	0.582523	+	0.812814i	$0.302066\pi$
$152$	0	0
$153$	−3758.03	−1.98574
$154$	0	0
$155$	−6774.73	−3.51071
$156$	0	0
$157$	−3371.34	−1.71377	−0.856887	−	0.515505i	$-0.827605\pi$
$157$	−3371.34	−1.71377	−0.856887	+	0.515505i	$0.827605\pi$
$158$	0	0
$159$	−243.210	−0.121307
$160$	0	0
$161$	138.999	0.0680415
$162$	0	0
$163$	2033.79	0.977292	0.488646	−	0.872482i	$-0.337491\pi$
$163$	2033.79	0.977292	0.488646	+	0.872482i	$0.337491\pi$
$164$	0	0
$165$	10223.6	4.82366
$166$	0	0
$167$	1818.19	0.842488	0.421244	−	0.906947i	$-0.361594\pi$
$167$	1818.19	0.842488	0.421244	+	0.906947i	$0.361594\pi$
$168$	0	0
$169$	568.369	0.258702
$170$	0	0
$171$	−4611.43	−2.06225
$172$	0	0
$173$	2775.88	1.21992	0.609959	−	0.792433i	$-0.291186\pi$
$173$	2775.88	1.21992	0.609959	+	0.792433i	$0.291186\pi$
$174$	0	0
$175$	−2179.12	−0.941290
$176$	0	0
$177$	−1856.04	−0.788185
$178$	0	0
$179$	−880.191	−0.367534	−0.183767	−	0.982970i	$-0.558829\pi$
$179$	−880.191	−0.367534	−0.183767	+	0.982970i	$0.558829\pi$
$180$	0	0
$181$	2442.04	1.00285	0.501424	−	0.865202i	$-0.332810\pi$
$181$	2442.04	1.00285	0.501424	+	0.865202i	$0.332810\pi$
$182$	0	0
$183$	5221.25	2.10910
$184$	0	0
$185$	−6077.88	−2.41543
$186$	0	0
$187$	2502.67	0.978683
$188$	0	0
$189$	−3251.31	−1.25131
$190$	0	0
$191$	3037.78	1.15082	0.575409	−	0.817866i	$-0.304843\pi$
$191$	3037.78	1.15082	0.575409	+	0.817866i	$0.304843\pi$
$192$	0	0
$193$	1001.79	0.373629	0.186814	−	0.982395i	$-0.440184\pi$
$193$	1001.79	0.373629	0.186814	+	0.982395i	$0.440184\pi$
$194$	0	0
$195$	−11004.2	−4.04116
$196$	0	0
$197$	438.281	0.158509	0.0792543	−	0.996854i	$-0.474746\pi$
$197$	438.281	0.158509	0.0792543	+	0.996854i	$0.474746\pi$
$198$	0	0
$199$	−883.008	−0.314547	−0.157273	−	0.987555i	$-0.550270\pi$
$199$	−883.008	−0.314547	−0.157273	+	0.987555i	$0.550270\pi$
$200$	0	0
$201$	−2469.63	−0.866637
$202$	0	0
$203$	−500.053	−0.172891
$204$	0	0
$205$	−856.402	−0.291774
$206$	0	0
$207$	−1456.77	−0.489144
$208$	0	0
$209$	3071.00	1.01639
$210$	0	0
$211$	3497.62	1.14117	0.570583	−	0.821240i	$-0.306717\pi$
$211$	3497.62	1.14117	0.570583	+	0.821240i	$0.306717\pi$
$212$	0	0
$213$	−9828.54	−3.16169
$214$	0	0
$215$	−9069.90	−2.87703
$216$	0	0
$217$	−2270.37	−0.710243
$218$	0	0
$219$	−12242.5	−3.77748
$220$	0	0
$221$	−2693.76	−0.819918
$222$	0	0
$223$	−4220.87	−1.26749	−0.633745	−	0.773542i	$-0.718483\pi$
$223$	−4220.87	−1.26749	−0.633745	+	0.773542i	$0.718483\pi$
$224$	0	0
$225$	22838.1	6.76685
$226$	0	0
$227$	−832.928	−0.243539	−0.121770	−	0.992558i	$-0.538857\pi$
$227$	−832.928	−0.243539	−0.121770	+	0.992558i	$0.538857\pi$
$228$	0	0
$229$	2188.56	0.631547	0.315773	−	0.948835i	$-0.397736\pi$
$229$	2188.56	0.631547	0.315773	+	0.948835i	$0.397736\pi$
$230$	0	0
$231$	3426.15	0.975863
$232$	0	0
$233$	471.644	0.132611	0.0663056	−	0.997799i	$-0.478879\pi$
$233$	471.644	0.132611	0.0663056	+	0.997799i	$0.478879\pi$
$234$	0	0
$235$	−8459.26	−2.34817
$236$	0	0
$237$	5229.07	1.43318
$238$	0	0
$239$	5917.67	1.60160	0.800799	−	0.598933i	$-0.204408\pi$
$239$	5917.67	1.60160	0.800799	+	0.598933i	$0.204408\pi$
$240$	0	0
$241$	−1525.38	−0.407711	−0.203855	−	0.979001i	$-0.565347\pi$
$241$	−1525.38	−0.407711	−0.203855	+	0.979001i	$0.565347\pi$
$242$	0	0
$243$	14231.2	3.75692
$244$	0	0
$245$	−1023.50	−0.266895
$246$	0	0
$247$	−3305.48	−0.851508
$248$	0	0
$249$	5286.80	1.34553
$250$	0	0
$251$	5278.44	1.32738	0.663689	−	0.748008i	$-0.268990\pi$
$251$	5278.44	1.32738	0.663689	+	0.748008i	$0.268990\pi$
$252$	0	0
$253$	970.144	0.241077
$254$	0	0
$255$	10719.2	2.63241
$256$	0	0
$257$	2536.13	0.615563	0.307782	−	0.951457i	$-0.400413\pi$
$257$	2536.13	0.615563	0.307782	+	0.951457i	$0.400413\pi$
$258$	0	0
$259$	−2036.84	−0.488660
$260$	0	0
$261$	5240.78	1.24290
$262$	0	0
$263$	−6419.74	−1.50516	−0.752582	−	0.658499i	$-0.771192\pi$
$263$	−6419.74	−1.50516	−0.752582	+	0.658499i	$0.771192\pi$
$264$	0	0
$265$	507.094	0.117549
$266$	0	0
$267$	3187.69	0.730649
$268$	0	0
$269$	5411.53	1.22657	0.613284	−	0.789863i	$-0.289848\pi$
$269$	5411.53	1.22657	0.613284	+	0.789863i	$0.289848\pi$
$270$	0	0
$271$	−923.155	−0.206929	−0.103464	−	0.994633i	$-0.532993\pi$
$271$	−923.155	−0.206929	−0.103464	+	0.994633i	$0.532993\pi$
$272$	0	0
$273$	−3687.75	−0.817556
$274$	0	0
$275$	−15209.1	−3.33507
$276$	0	0
$277$	2092.85	0.453962	0.226981	−	0.973899i	$-0.427115\pi$
$277$	2092.85	0.453962	0.226981	+	0.973899i	$0.427115\pi$
$278$	0	0
$279$	23794.5	5.10587
$280$	0	0
$281$	−1714.66	−0.364015	−0.182008	−	0.983297i	$-0.558260\pi$
$281$	−1714.66	−0.364015	−0.182008	+	0.983297i	$0.558260\pi$
$282$	0	0
$283$	4236.18	0.889805	0.444903	−	0.895579i	$-0.353238\pi$
$283$	4236.18	0.889805	0.444903	+	0.895579i	$0.353238\pi$
$284$	0	0
$285$	13153.4	2.73383
$286$	0	0
$287$	−287.000	−0.0590281
$288$	0	0
$289$	−2288.99	−0.465905
$290$	0	0
$291$	4414.55	0.889298
$292$	0	0
$293$	1182.19	0.235714	0.117857	−	0.993031i	$-0.462398\pi$
$293$	1182.19	0.235714	0.117857	+	0.993031i	$0.462398\pi$
$294$	0	0
$295$	3869.86	0.763769
$296$	0	0
$297$	−22692.5	−4.43350
$298$	0	0
$299$	−1044.22	−0.201969
$300$	0	0
$301$	−3039.53	−0.582046
$302$	0	0
$303$	9482.27	1.79783
$304$	0	0
$305$	−10886.3	−2.04377
$306$	0	0
$307$	7222.51	1.34271	0.671353	−	0.741138i	$-0.265714\pi$
$307$	7222.51	1.34271	0.671353	+	0.741138i	$0.265714\pi$
$308$	0	0
$309$	5486.39	1.01007
$310$	0	0
$311$	−9944.05	−1.81310	−0.906552	−	0.422094i	$-0.861295\pi$
$311$	−9944.05	−1.81310	−0.906552	+	0.422094i	$0.861295\pi$
$312$	0	0
$313$	−8320.48	−1.50256	−0.751280	−	0.659984i	$-0.770563\pi$
$313$	−8320.48	−1.50256	−0.751280	+	0.659984i	$0.770563\pi$
$314$	0	0
$315$	10726.8	1.91869
$316$	0	0
$317$	−5215.74	−0.924117	−0.462058	−	0.886849i	$-0.652889\pi$
$317$	−5215.74	−0.924117	−0.462058	+	0.886849i	$0.652889\pi$
$318$	0	0
$319$	−3490.11	−0.612567
$320$	0	0
$321$	−158.878	−0.0276253
$322$	0	0
$323$	3219.88	0.554672
$324$	0	0
$325$	16370.4	2.79405
$326$	0	0
$327$	−9439.13	−1.59628
$328$	0	0
$329$	−2834.89	−0.475054
$330$	0	0
$331$	4538.30	0.753618	0.376809	−	0.926291i	$-0.377021\pi$
$331$	4538.30	0.753618	0.376809	+	0.926291i	$0.377021\pi$
$332$	0	0
$333$	21347.0	3.51293
$334$	0	0
$335$	5149.18	0.839790
$336$	0	0
$337$	−6727.44	−1.08744	−0.543720	−	0.839267i	$-0.682985\pi$
$337$	−6727.44	−1.08744	−0.543720	+	0.839267i	$0.682985\pi$
$338$	0	0
$339$	−1507.31	−0.241492
$340$	0	0
$341$	−15846.0	−2.51645
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	4155.23	0.648435
$346$	0	0
$347$	−6780.04	−1.04891	−0.524455	−	0.851438i	$-0.675731\pi$
$347$	−6780.04	−1.04891	−0.524455	+	0.851438i	$0.675731\pi$
$348$	0	0
$349$	−9779.23	−1.49991	−0.749957	−	0.661486i	$-0.769926\pi$
$349$	−9779.23	−1.49991	−0.749957	+	0.661486i	$0.769926\pi$
$350$	0	0
$351$	24425.1	3.71429
$352$	0	0
$353$	12720.0	1.91790	0.958951	−	0.283573i	$-0.0915199\pi$
$353$	12720.0	1.91790	0.958951	+	0.283573i	$0.0915199\pi$
$354$	0	0
$355$	20492.5	3.06375
$356$	0	0
$357$	3592.26	0.532556
$358$	0	0
$359$	−8527.48	−1.25366	−0.626829	−	0.779157i	$-0.715648\pi$
$359$	−8527.48	−1.25366	−0.626829	+	0.779157i	$0.715648\pi$
$360$	0	0
$361$	−2907.93	−0.423958
$362$	0	0
$363$	10578.7	1.52958
$364$	0	0
$365$	25525.6	3.66046
$366$	0	0
$367$	−12460.3	−1.77227	−0.886137	−	0.463424i	$-0.846621\pi$
$367$	−12460.3	−1.77227	−0.886137	+	0.463424i	$0.846621\pi$
$368$	0	0
$369$	3007.89	0.424348
$370$	0	0
$371$	169.939	0.0237811
$372$	0	0
$373$	8137.03	1.12954	0.564772	−	0.825247i	$-0.308964\pi$
$373$	8137.03	1.12954	0.564772	+	0.825247i	$0.308964\pi$
$374$	0	0
$375$	−38985.2	−5.36849
$376$	0	0
$377$	3756.59	0.513195
$378$	0	0
$379$	2620.71	0.355190	0.177595	−	0.984104i	$-0.443168\pi$
$379$	2620.71	0.355190	0.177595	+	0.984104i	$0.443168\pi$
$380$	0	0
$381$	15799.6	2.12451
$382$	0	0
$383$	−5045.96	−0.673202	−0.336601	−	0.941647i	$-0.609277\pi$
$383$	−5045.96	−0.673202	−0.336601	+	0.941647i	$0.609277\pi$
$384$	0	0
$385$	−7143.54	−0.945633
$386$	0	0
$387$	31855.7	4.18428
$388$	0	0
$389$	−2244.14	−0.292500	−0.146250	−	0.989248i	$-0.546720\pi$
$389$	−2244.14	−0.292500	−0.146250	+	0.989248i	$0.546720\pi$
$390$	0	0
$391$	1017.18	0.131562
$392$	0	0
$393$	−9131.23	−1.17203
$394$	0	0
$395$	−10902.6	−1.38879
$396$	0	0
$397$	−9813.64	−1.24064	−0.620318	−	0.784350i	$-0.712997\pi$
$397$	−9813.64	−1.24064	−0.620318	+	0.784350i	$0.712997\pi$
$398$	0	0
$399$	4408.01	0.553074
$400$	0	0
$401$	1188.71	0.148033	0.0740167	−	0.997257i	$-0.476418\pi$
$401$	1188.71	0.148033	0.0740167	+	0.997257i	$0.476418\pi$
$402$	0	0
$403$	17055.9	2.10823
$404$	0	0
$405$	−55819.6	−6.84863
$406$	0	0
$407$	−14216.1	−1.73136
$408$	0	0
$409$	−4281.74	−0.517649	−0.258824	−	0.965924i	$-0.583335\pi$
$409$	−4281.74	−0.517649	−0.258824	+	0.965924i	$0.583335\pi$
$410$	0	0
$411$	−5475.99	−0.657203
$412$	0	0
$413$	1296.88	0.154516
$414$	0	0
$415$	−11023.0	−1.30385
$416$	0	0
$417$	−26648.7	−3.12947
$418$	0	0
$419$	4093.89	0.477326	0.238663	−	0.971102i	$-0.423291\pi$
$419$	4093.89	0.477326	0.238663	+	0.971102i	$0.423291\pi$
$420$	0	0
$421$	−11385.8	−1.31807	−0.659036	−	0.752111i	$-0.729036\pi$
$421$	−11385.8	−1.31807	−0.659036	+	0.752111i	$0.729036\pi$
$422$	0	0
$423$	29710.9	3.41512
$424$	0	0
$425$	−15946.5	−1.82004
$426$	0	0
$427$	−3648.25	−0.413469
$428$	0	0
$429$	−25738.6	−2.89667
$430$	0	0
$431$	14700.0	1.64286	0.821432	−	0.570307i	$-0.193176\pi$
$431$	14700.0	1.64286	0.821432	+	0.570307i	$0.193176\pi$
$432$	0	0
$433$	10369.8	1.15090	0.575449	−	0.817838i	$-0.304827\pi$
$433$	10369.8	1.15090	0.575449	+	0.817838i	$0.304827\pi$
$434$	0	0
$435$	−14948.5	−1.64765
$436$	0	0
$437$	1248.16	0.136631
$438$	0	0
$439$	2072.93	0.225366	0.112683	−	0.993631i	$-0.464056\pi$
$439$	2072.93	0.225366	0.112683	+	0.993631i	$0.464056\pi$
$440$	0	0
$441$	3594.79	0.388165
$442$	0	0
$443$	8302.78	0.890467	0.445233	−	0.895415i	$-0.353121\pi$
$443$	8302.78	0.890467	0.445233	+	0.895415i	$0.353121\pi$
$444$	0	0
$445$	−6646.34	−0.708015
$446$	0	0
$447$	11295.2	1.19518
$448$	0	0
$449$	−4493.90	−0.472339	−0.236170	−	0.971712i	$-0.575892\pi$
$449$	−4493.90	−0.472339	−0.236170	+	0.971712i	$0.575892\pi$
$450$	0	0
$451$	−2003.11	−0.209142
$452$	0	0
$453$	21656.9	2.24620
$454$	0	0
$455$	7688.97	0.792230
$456$	0	0
$457$	17781.3	1.82008	0.910040	−	0.414521i	$-0.136051\pi$
$457$	17781.3	1.82008	0.910040	+	0.414521i	$0.136051\pi$
$458$	0	0
$459$	−23792.6	−2.41949
$460$	0	0
$461$	17759.7	1.79426	0.897128	−	0.441771i	$-0.145650\pi$
$461$	17759.7	1.79426	0.897128	+	0.441771i	$0.145650\pi$
$462$	0	0
$463$	4251.89	0.426786	0.213393	−	0.976966i	$-0.431549\pi$
$463$	4251.89	0.426786	0.213393	+	0.976966i	$0.431549\pi$
$464$	0	0
$465$	−67870.2	−6.76862
$466$	0	0
$467$	−3422.50	−0.339131	−0.169566	−	0.985519i	$-0.554236\pi$
$467$	−3422.50	−0.339131	−0.169566	+	0.985519i	$0.554236\pi$
$468$	0	0
$469$	1725.61	0.169896
$470$	0	0
$471$	−33774.6	−3.30414
$472$	0	0
$473$	−21214.4	−2.06224
$474$	0	0
$475$	−19567.7	−1.89016
$476$	0	0
$477$	−1781.04	−0.170960
$478$	0	0
$479$	−13521.7	−1.28982	−0.644908	−	0.764260i	$-0.723104\pi$
$479$	−13521.7	−1.28982	−0.644908	+	0.764260i	$0.723104\pi$
$480$	0	0
$481$	15301.5	1.45050
$482$	0	0
$483$	1392.51	0.131183
$484$	0	0
$485$	−9204.36	−0.861749
$486$	0	0
$487$	−9595.98	−0.892886	−0.446443	−	0.894812i	$-0.647309\pi$
$487$	−9595.98	−0.892886	−0.446443	+	0.894812i	$0.647309\pi$
$488$	0	0
$489$	20374.8	1.88421
$490$	0	0
$491$	2332.00	0.214341	0.107171	−	0.994241i	$-0.465821\pi$
$491$	2332.00	0.214341	0.107171	+	0.994241i	$0.465821\pi$
$492$	0	0
$493$	−3659.32	−0.334295
$494$	0	0
$495$	74867.5	6.79807
$496$	0	0
$497$	6867.52	0.619819
$498$	0	0
$499$	−15138.3	−1.35808	−0.679042	−	0.734099i	$-0.737605\pi$
$499$	−15138.3	−1.35808	−0.679042	+	0.734099i	$0.737605\pi$
$500$	0	0
$501$	18214.8	1.62431
$502$	0	0
$503$	15487.2	1.37284	0.686421	−	0.727205i	$-0.259181\pi$
$503$	15487.2	1.37284	0.686421	+	0.727205i	$0.259181\pi$
$504$	0	0
$505$	−19770.6	−1.74214
$506$	0	0
$507$	5694.00	0.498776
$508$	0	0
$509$	−1894.00	−0.164932	−0.0824659	−	0.996594i	$-0.526280\pi$
$509$	−1894.00	−0.164932	−0.0824659	+	0.996594i	$0.526280\pi$
$510$	0	0
$511$	8554.20	0.740539
$512$	0	0
$513$	−29195.6	−2.51270
$514$	0	0
$515$	−11439.1	−0.978775
$516$	0	0
$517$	−19786.1	−1.68316
$518$	0	0
$519$	27809.1	2.35199
$520$	0	0
$521$	−9029.02	−0.759249	−0.379624	−	0.925141i	$-0.623947\pi$
$521$	−9029.02	−0.759249	−0.379624	+	0.925141i	$0.623947\pi$
$522$	0	0
$523$	−1418.53	−0.118600	−0.0593002	−	0.998240i	$-0.518887\pi$
$523$	−1418.53	−0.118600	−0.0593002	+	0.998240i	$0.518887\pi$
$524$	0	0
$525$	−21830.7	−1.81480
$526$	0	0
$527$	−16614.3	−1.37330
$528$	0	0
$529$	−11772.7	−0.967593
$530$	0	0
$531$	−13591.9	−1.11080
$532$	0	0
$533$	2156.06	0.175214
$534$	0	0
$535$	331.261	0.0267695
$536$	0	0
$537$	−8817.88	−0.708602
$538$	0	0
$539$	−2393.97	−0.191309
$540$	0	0
$541$	699.022	0.0555514	0.0277757	−	0.999614i	$-0.491158\pi$
$541$	699.022	0.0555514	0.0277757	+	0.999614i	$0.491158\pi$
$542$	0	0
$543$	24464.7	1.93348
$544$	0	0
$545$	19680.6	1.54683
$546$	0	0
$547$	5629.64	0.440048	0.220024	−	0.975495i	$-0.429387\pi$
$547$	5629.64	0.440048	0.220024	+	0.975495i	$0.429387\pi$
$548$	0	0
$549$	38235.3	2.97239
$550$	0	0
$551$	−4490.30	−0.347174
$552$	0	0
$553$	−3653.72	−0.280962
$554$	0	0
$555$	−60889.0	−4.65693
$556$	0	0
$557$	−20151.0	−1.53290	−0.766450	−	0.642304i	$-0.777979\pi$
$557$	−20151.0	−1.53290	−0.766450	+	0.642304i	$0.777979\pi$
$558$	0	0
$559$	22834.2	1.72770
$560$	0	0
$561$	25072.1	1.88689
$562$	0	0
$563$	3280.23	0.245551	0.122776	−	0.992434i	$-0.460820\pi$
$563$	3280.23	0.245551	0.122776	+	0.992434i	$0.460820\pi$
$564$	0	0
$565$	3142.75	0.234011
$566$	0	0
$567$	−18706.4	−1.38553
$568$	0	0
$569$	10019.6	0.738211	0.369106	−	0.929387i	$-0.379664\pi$
$569$	10019.6	0.738211	0.369106	+	0.929387i	$0.379664\pi$
$570$	0	0
$571$	9362.73	0.686196	0.343098	−	0.939300i	$-0.388524\pi$
$571$	9362.73	0.686196	0.343098	+	0.939300i	$0.388524\pi$
$572$	0	0
$573$	30432.9	2.21877
$574$	0	0
$575$	−6181.54	−0.448327
$576$	0	0
$577$	−5890.21	−0.424979	−0.212489	−	0.977163i	$-0.568157\pi$
$577$	−5890.21	−0.424979	−0.212489	+	0.977163i	$0.568157\pi$
$578$	0	0
$579$	10036.1	0.720354
$580$	0	0
$581$	−3694.06	−0.263779
$582$	0	0
$583$	1186.09	0.0842585
$584$	0	0
$585$	−80583.9	−5.69527
$586$	0	0
$587$	−19930.1	−1.40137	−0.700686	−	0.713470i	$-0.747122\pi$
$587$	−19930.1	−1.40137	−0.700686	+	0.713470i	$0.747122\pi$
$588$	0	0
$589$	−20387.1	−1.42621
$590$	0	0
$591$	4390.76	0.305603
$592$	0	0
$593$	1974.05	0.136702	0.0683511	−	0.997661i	$-0.478226\pi$
$593$	1974.05	0.136702	0.0683511	+	0.997661i	$0.478226\pi$
$594$	0	0
$595$	−7489.87	−0.516058
$596$	0	0
$597$	−8846.10	−0.606443
$598$	0	0
$599$	27672.8	1.88761	0.943807	−	0.330496i	$-0.107216\pi$
$599$	27672.8	1.88761	0.943807	+	0.330496i	$0.107216\pi$
$600$	0	0
$601$	1905.15	0.129306	0.0646529	−	0.997908i	$-0.479406\pi$
$601$	1905.15	0.129306	0.0646529	+	0.997908i	$0.479406\pi$
$602$	0	0
$603$	−18085.1	−1.22137
$604$	0	0
$605$	−22056.6	−1.48219
$606$	0	0
$607$	−12294.0	−0.822069	−0.411035	−	0.911620i	$-0.634832\pi$
$607$	−12294.0	−0.822069	−0.411035	+	0.911620i	$0.634832\pi$
$608$	0	0
$609$	−5009.60	−0.333332
$610$	0	0
$611$	21296.8	1.41011
$612$	0	0
$613$	17203.2	1.13349	0.566747	−	0.823892i	$-0.308202\pi$
$613$	17203.2	1.13349	0.566747	+	0.823892i	$0.308202\pi$
$614$	0	0
$615$	−8579.56	−0.562538
$616$	0	0
$617$	−3535.71	−0.230701	−0.115350	−	0.993325i	$-0.536799\pi$
$617$	−3535.71	−0.230701	−0.115350	+	0.993325i	$0.536799\pi$
$618$	0	0
$619$	−14835.8	−0.963327	−0.481663	−	0.876356i	$-0.659967\pi$
$619$	−14835.8	−0.963327	−0.481663	+	0.876356i	$0.659967\pi$
$620$	0	0
$621$	−9223.05	−0.595987
$622$	0	0
$623$	−2227.34	−0.143237
$624$	0	0
$625$	42371.4	2.71177
$626$	0	0
$627$	30765.7	1.95959
$628$	0	0
$629$	−14905.3	−0.944854
$630$	0	0
$631$	−14269.8	−0.900274	−0.450137	−	0.892960i	$-0.648625\pi$
$631$	−14269.8	−0.900274	−0.450137	+	0.892960i	$0.648625\pi$
$632$	0	0
$633$	35039.6	2.20016
$634$	0	0
$635$	−32942.3	−2.05870
$636$	0	0
$637$	2576.75	0.160274
$638$	0	0
$639$	−71974.7	−4.45583
$640$	0	0
$641$	4525.35	0.278846	0.139423	−	0.990233i	$-0.455475\pi$
$641$	4525.35	0.278846	0.139423	+	0.990233i	$0.455475\pi$
$642$	0	0
$643$	919.098	0.0563696	0.0281848	−	0.999603i	$-0.491027\pi$
$643$	919.098	0.0563696	0.0281848	+	0.999603i	$0.491027\pi$
$644$	0	0
$645$	−90863.6	−5.54690
$646$	0	0
$647$	15880.3	0.964946	0.482473	−	0.875911i	$-0.339739\pi$
$647$	15880.3	0.964946	0.482473	+	0.875911i	$0.339739\pi$
$648$	0	0
$649$	9051.55	0.547464
$650$	0	0
$651$	−22744.9	−1.36934
$652$	0	0
$653$	12247.1	0.733946	0.366973	−	0.930232i	$-0.380394\pi$
$653$	12247.1	0.733946	0.366973	+	0.930232i	$0.380394\pi$
$654$	0	0
$655$	19038.6	1.13573
$656$	0	0
$657$	−89651.9	−5.32367
$658$	0	0
$659$	6457.41	0.381707	0.190854	−	0.981619i	$-0.438874\pi$
$659$	6457.41	0.381707	0.190854	+	0.981619i	$0.438874\pi$
$660$	0	0
$661$	20222.5	1.18996	0.594979	−	0.803741i	$-0.297160\pi$
$661$	20222.5	1.18996	0.594979	+	0.803741i	$0.297160\pi$
$662$	0	0
$663$	−26986.5	−1.58080
$664$	0	0
$665$	−9190.71	−0.535941
$666$	0	0
$667$	−1418.51	−0.0823462
$668$	0	0
$669$	−42285.3	−2.44371
$670$	0	0
$671$	−25463.0	−1.46496
$672$	0	0
$673$	2586.70	0.148158	0.0740788	−	0.997252i	$-0.476398\pi$
$673$	2586.70	0.148158	0.0740788	+	0.997252i	$0.476398\pi$
$674$	0	0
$675$	144591.	8.24493
$676$	0	0
$677$	6088.54	0.345645	0.172823	−	0.984953i	$-0.444711\pi$
$677$	6088.54	0.345645	0.172823	+	0.984953i	$0.444711\pi$
$678$	0	0
$679$	−3084.59	−0.174338
$680$	0	0
$681$	−8344.39	−0.469542
$682$	0	0
$683$	−27764.1	−1.55544	−0.777719	−	0.628612i	$-0.783623\pi$
$683$	−27764.1	−1.55544	−0.777719	+	0.628612i	$0.783623\pi$
$684$	0	0
$685$	11417.4	0.636844
$686$	0	0
$687$	21925.3	1.21762
$688$	0	0
$689$	−1276.65	−0.0705899
$690$	0	0
$691$	−5819.93	−0.320406	−0.160203	−	0.987084i	$-0.551215\pi$
$691$	−5819.93	−0.320406	−0.160203	+	0.987084i	$0.551215\pi$
$692$	0	0
$693$	25089.8	1.37530
$694$	0	0
$695$	55562.5	3.03253
$696$	0	0
$697$	−2100.23	−0.114135
$698$	0	0
$699$	4724.99	0.255673
$700$	0	0
$701$	15942.3	0.858963	0.429482	−	0.903076i	$-0.358696\pi$
$701$	15942.3	0.858963	0.429482	+	0.903076i	$0.358696\pi$
$702$	0	0
$703$	−18290.1	−0.981257
$704$	0	0
$705$	−84746.0	−4.52726
$706$	0	0
$707$	−6625.57	−0.352447
$708$	0	0
$709$	22353.8	1.18408	0.592042	−	0.805907i	$-0.298322\pi$
$709$	22353.8	1.18408	0.592042	+	0.805907i	$0.298322\pi$
$710$	0	0
$711$	38292.7	2.01981
$712$	0	0
$713$	−6440.40	−0.338282
$714$	0	0
$715$	53665.1	2.80694
$716$	0	0
$717$	59284.0	3.08787
$718$	0	0
$719$	18015.5	0.934441	0.467221	−	0.884141i	$-0.345255\pi$
$719$	18015.5	0.934441	0.467221	+	0.884141i	$0.345255\pi$
$720$	0	0
$721$	−3833.52	−0.198014
$722$	0	0
$723$	−15281.5	−0.786063
$724$	0	0
$725$	22238.2	1.13918
$726$	0	0
$727$	31048.8	1.58396	0.791978	−	0.610550i	$-0.209051\pi$
$727$	31048.8	1.58396	0.791978	+	0.610550i	$0.209051\pi$
$728$	0	0
$729$	70416.6	3.57754
$730$	0	0
$731$	−22242.9	−1.12542
$732$	0	0
$733$	1993.09	0.100432	0.0502159	−	0.998738i	$-0.484009\pi$
$733$	1993.09	0.100432	0.0502159	+	0.998738i	$0.484009\pi$
$734$	0	0
$735$	−10253.6	−0.514572
$736$	0	0
$737$	12043.9	0.601956
$738$	0	0
$739$	27641.1	1.37590	0.687952	−	0.725756i	$-0.258510\pi$
$739$	27641.1	1.37590	0.687952	+	0.725756i	$0.258510\pi$
$740$	0	0
$741$	−33114.7	−1.64170
$742$	0	0
$743$	−16844.8	−0.831728	−0.415864	−	0.909427i	$-0.636521\pi$
$743$	−16844.8	−0.831728	−0.415864	+	0.909427i	$0.636521\pi$
$744$	0	0
$745$	−23550.5	−1.15815
$746$	0	0
$747$	38715.4	1.89628
$748$	0	0
$749$	111.013	0.00541567
$750$	0	0
$751$	−19398.5	−0.942559	−0.471279	−	0.881984i	$-0.656208\pi$
$751$	−19398.5	−0.942559	−0.471279	+	0.881984i	$0.656208\pi$
$752$	0	0
$753$	52880.1	2.55918
$754$	0	0
$755$	−45154.6	−2.17662
$756$	0	0
$757$	12968.3	0.622642	0.311321	−	0.950305i	$-0.399229\pi$
$757$	12968.3	0.622642	0.311321	+	0.950305i	$0.399229\pi$
$758$	0	0
$759$	9719.04	0.464794
$760$	0	0
$761$	642.510	0.0306057	0.0153029	−	0.999883i	$-0.495129\pi$
$761$	642.510	0.0306057	0.0153029	+	0.999883i	$0.495129\pi$
$762$	0	0
$763$	6595.43	0.312936
$764$	0	0
$765$	78497.2	3.70990
$766$	0	0
$767$	−9742.66	−0.458653
$768$	0	0
$769$	−18389.8	−0.862357	−0.431178	−	0.902267i	$-0.641902\pi$
$769$	−18389.8	−0.862357	−0.431178	+	0.902267i	$0.641902\pi$
$770$	0	0
$771$	25407.4	1.18680
$772$	0	0
$773$	−12851.2	−0.597961	−0.298981	−	0.954259i	$-0.596647\pi$
$773$	−12851.2	−0.597961	−0.298981	+	0.954259i	$0.596647\pi$
$774$	0	0
$775$	100967.	4.67981
$776$	0	0
$777$	−20405.3	−0.942132
$778$	0	0
$779$	−2577.16	−0.118532
$780$	0	0
$781$	47931.8	2.19607
$782$	0	0
$783$	33180.1	1.51438
$784$	0	0
$785$	70420.1	3.20179
$786$	0	0
$787$	22402.1	1.01467	0.507337	−	0.861748i	$-0.330630\pi$
$787$	22402.1	1.01467	0.507337	+	0.861748i	$0.330630\pi$
$788$	0	0
$789$	−64313.8	−2.90194
$790$	0	0
$791$	1053.21	0.0473423
$792$	0	0
$793$	27407.1	1.22731
$794$	0	0
$795$	5080.14	0.226634
$796$	0	0
$797$	5556.78	0.246965	0.123483	−	0.992347i	$-0.460594\pi$
$797$	5556.78	0.246965	0.123483	+	0.992347i	$0.460594\pi$
$798$	0	0
$799$	−20745.4	−0.918546
$800$	0	0
$801$	23343.5	1.02972
$802$	0	0
$803$	59704.0	2.62379
$804$	0	0
$805$	−2903.40	−0.127120
$806$	0	0
$807$	54213.4	2.36481
$808$	0	0
$809$	−34090.6	−1.48153	−0.740767	−	0.671762i	$-0.765538\pi$
$809$	−34090.6	−1.48153	−0.740767	+	0.671762i	$0.765538\pi$
$810$	0	0
$811$	−14865.8	−0.643660	−0.321830	−	0.946797i	$-0.604298\pi$
$811$	−14865.8	−0.643660	−0.321830	+	0.946797i	$0.604298\pi$
$812$	0	0
$813$	−9248.30	−0.398957
$814$	0	0
$815$	−42481.5	−1.82584
$816$	0	0
$817$	−27293.9	−1.16878
$818$	0	0
$819$	−27005.5	−1.15220
$820$	0	0
$821$	43050.9	1.83007	0.915035	−	0.403375i	$-0.132163\pi$
$821$	43050.9	1.83007	0.915035	+	0.403375i	$0.132163\pi$
$822$	0	0
$823$	8669.47	0.367192	0.183596	−	0.983002i	$-0.441226\pi$
$823$	8669.47	0.367192	0.183596	+	0.983002i	$0.441226\pi$
$824$	0	0
$825$	−152367.	−6.42999
$826$	0	0
$827$	19823.8	0.833544	0.416772	−	0.909011i	$-0.363161\pi$
$827$	19823.8	0.833544	0.416772	+	0.909011i	$0.363161\pi$
$828$	0	0
$829$	−22952.2	−0.961597	−0.480799	−	0.876831i	$-0.659653\pi$
$829$	−22952.2	−0.961597	−0.480799	+	0.876831i	$0.659653\pi$
$830$	0	0
$831$	20966.5	0.875234
$832$	0	0
$833$	−2510.03	−0.104403
$834$	0	0
$835$	−37978.0	−1.57399
$836$	0	0
$837$	150646.	6.22114
$838$	0	0
$839$	13953.9	0.574186	0.287093	−	0.957903i	$-0.407311\pi$
$839$	13953.9	0.574186	0.287093	+	0.957903i	$0.407311\pi$
$840$	0	0
$841$	−19285.9	−0.790762
$842$	0	0
$843$	−17177.7	−0.701819
$844$	0	0
$845$	−11872.0	−0.483325
$846$	0	0
$847$	−7391.66	−0.299859
$848$	0	0
$849$	42438.7	1.71554
$850$	0	0
$851$	−5777.93	−0.232744
$852$	0	0
$853$	−7262.79	−0.291528	−0.145764	−	0.989319i	$-0.546564\pi$
$853$	−7262.79	−0.291528	−0.145764	+	0.989319i	$0.546564\pi$
$854$	0	0
$855$	96322.8	3.85283
$856$	0	0
$857$	25473.7	1.01536	0.507681	−	0.861545i	$-0.330503\pi$
$857$	25473.7	1.01536	0.507681	+	0.861545i	$0.330503\pi$
$858$	0	0
$859$	−22839.9	−0.907204	−0.453602	−	0.891204i	$-0.649861\pi$
$859$	−22839.9	−0.907204	−0.453602	+	0.891204i	$0.649861\pi$
$860$	0	0
$861$	−2875.21	−0.113806
$862$	0	0
$863$	−45420.9	−1.79159	−0.895797	−	0.444463i	$-0.853395\pi$
$863$	−45420.9	−1.79159	−0.895797	+	0.444463i	$0.853395\pi$
$864$	0	0
$865$	−57982.1	−2.27913
$866$	0	0
$867$	−22931.5	−0.898262
$868$	0	0
$869$	−25501.1	−0.995473
$870$	0	0
$871$	−12963.4	−0.504305
$872$	0	0
$873$	32327.9	1.25330
$874$	0	0
$875$	27240.2	1.05244
$876$	0	0
$877$	33069.5	1.27329	0.636646	−	0.771156i	$-0.280321\pi$
$877$	33069.5	1.27329	0.636646	+	0.771156i	$0.280321\pi$
$878$	0	0
$879$	11843.3	0.454455
$880$	0	0
$881$	11131.5	0.425686	0.212843	−	0.977086i	$-0.431728\pi$
$881$	11131.5	0.425686	0.212843	+	0.977086i	$0.431728\pi$
$882$	0	0
$883$	−790.121	−0.0301129	−0.0150565	−	0.999887i	$-0.504793\pi$
$883$	−790.121	−0.0301129	−0.0150565	+	0.999887i	$0.504793\pi$
$884$	0	0
$885$	38768.8	1.47254
$886$	0	0
$887$	−9974.65	−0.377583	−0.188791	−	0.982017i	$-0.560457\pi$
$887$	−9974.65	−0.377583	−0.188791	+	0.982017i	$0.560457\pi$
$888$	0	0
$889$	−11039.7	−0.416491
$890$	0	0
$891$	−130561.	−4.90906
$892$	0	0
$893$	−25456.3	−0.953935
$894$	0	0
$895$	18385.3	0.686651
$896$	0	0
$897$	−10461.1	−0.389394
$898$	0	0
$899$	23169.5	0.859561
$900$	0	0
$901$	1243.59	0.0459823
$902$	0	0
$903$	−30450.5	−1.12218
$904$	0	0
$905$	−51009.0	−1.87359
$906$	0	0
$907$	−37636.0	−1.37782	−0.688910	−	0.724847i	$-0.741910\pi$
$907$	−37636.0	−1.37782	−0.688910	+	0.724847i	$0.741910\pi$
$908$	0	0
$909$	69439.0	2.53371
$910$	0	0
$911$	−3488.70	−0.126878	−0.0634390	−	0.997986i	$-0.520207\pi$
$911$	−3488.70	−0.126878	−0.0634390	+	0.997986i	$0.520207\pi$
$912$	0	0
$913$	−25782.7	−0.934591
$914$	0	0
$915$	−109061.	−3.94036
$916$	0	0
$917$	6380.28	0.229766
$918$	0	0
$919$	−24828.0	−0.891185	−0.445593	−	0.895236i	$-0.647007\pi$
$919$	−24828.0	−0.891185	−0.445593	+	0.895236i	$0.647007\pi$
$920$	0	0
$921$	72356.2	2.58873
$922$	0	0
$923$	−51591.5	−1.83982
$924$	0	0
$925$	90581.7	3.21979
$926$	0	0
$927$	40177.0	1.42350
$928$	0	0
$929$	−11528.2	−0.407133	−0.203566	−	0.979061i	$-0.565253\pi$
$929$	−11528.2	−0.407133	−0.203566	+	0.979061i	$0.565253\pi$
$930$	0	0
$931$	−3080.02	−0.108425
$932$	0	0
$933$	−99620.9	−3.49565
$934$	0	0
$935$	−52275.5	−1.82844
$936$	0	0
$937$	−26120.7	−0.910699	−0.455350	−	0.890313i	$-0.650486\pi$
$937$	−26120.7	−0.910699	−0.455350	+	0.890313i	$0.650486\pi$
$938$	0	0
$939$	−83355.7	−2.89692
$940$	0	0
$941$	−44653.1	−1.54692	−0.773458	−	0.633847i	$-0.781475\pi$
$941$	−44653.1	−1.54692	−0.773458	+	0.633847i	$0.781475\pi$
$942$	0	0
$943$	−814.139	−0.0281145
$944$	0	0
$945$	67912.8	2.33778
$946$	0	0
$947$	41721.1	1.43163	0.715816	−	0.698289i	$-0.246055\pi$
$947$	41721.1	1.43163	0.715816	+	0.698289i	$0.246055\pi$
$948$	0	0
$949$	−64262.5	−2.19816
$950$	0	0
$951$	−52252.0	−1.78169
$952$	0	0
$953$	−21087.8	−0.716789	−0.358394	−	0.933570i	$-0.616676\pi$
$953$	−21087.8	−0.716789	−0.358394	+	0.933570i	$0.616676\pi$
$954$	0	0
$955$	−63452.8	−2.15003
$956$	0	0
$957$	−34964.4	−1.18102
$958$	0	0
$959$	3826.25	0.128838
$960$	0	0
$961$	75404.4	2.53111
$962$	0	0
$963$	−1163.47	−0.0389328
$964$	0	0
$965$	−20925.2	−0.698038
$966$	0	0
$967$	−613.212	−0.0203925	−0.0101963	−	0.999948i	$-0.503246\pi$
$967$	−613.212	−0.0203925	−0.0101963	+	0.999948i	$0.503246\pi$
$968$	0	0
$969$	32257.2	1.06940
$970$	0	0
$971$	28738.8	0.949815	0.474908	−	0.880036i	$-0.342482\pi$
$971$	28738.8	0.949815	0.474908	+	0.880036i	$0.342482\pi$
$972$	0	0
$973$	18620.3	0.613504
$974$	0	0
$975$	164001.	5.38690
$976$	0	0
$977$	−36511.2	−1.19560	−0.597798	−	0.801647i	$-0.703958\pi$
$977$	−36511.2	−1.19560	−0.597798	+	0.801647i	$0.703958\pi$
$978$	0	0
$979$	−15545.7	−0.507500
$980$	0	0
$981$	−69123.0	−2.24967
$982$	0	0
$983$	36211.7	1.17495	0.587474	−	0.809243i	$-0.300122\pi$
$983$	36211.7	1.17495	0.587474	+	0.809243i	$0.300122\pi$
$984$	0	0
$985$	−9154.74	−0.296136
$986$	0	0
$987$	−28400.3	−0.915899
$988$	0	0
$989$	−8622.31	−0.277223
$990$	0	0
$991$	56617.6	1.81485	0.907425	−	0.420215i	$-0.138045\pi$
$991$	56617.6	1.81485	0.907425	+	0.420215i	$0.138045\pi$
$992$	0	0
$993$	45465.3	1.45297
$994$	0	0
$995$	18444.1	0.587657
$996$	0	0
$997$	−46439.3	−1.47517	−0.737586	−	0.675253i	$-0.764034\pi$
$997$	−46439.3	−1.47517	−0.737586	+	0.675253i	$0.764034\pi$
$998$	0	0
$999$	135151.	4.28026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.d.1.15	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.d.1.15	✓	15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q+10.0181 q^{3} -20.8879 q^{5} -7.00000 q^{7} +73.3631 q^{9} +O(q^{10})\)	\(q+10.0181 q^{3} -20.8879 q^{5} -7.00000 q^{7} +73.3631 q^{9} -48.8564 q^{11} +52.5868 q^{13} -209.257 q^{15} -51.2251 q^{17} -62.8576 q^{19} -70.1270 q^{21} -19.8570 q^{23} +311.302 q^{25} +464.473 q^{27} +71.4361 q^{29} +324.338 q^{31} -489.451 q^{33} +146.215 q^{35} +290.977 q^{37} +526.822 q^{39} +41.0000 q^{41} +434.219 q^{43} -1532.40 q^{45} +404.984 q^{47} +49.0000 q^{49} -513.180 q^{51} -24.2770 q^{53} +1020.51 q^{55} -629.716 q^{57} -185.268 q^{59} +521.179 q^{61} -513.542 q^{63} -1098.42 q^{65} -246.515 q^{67} -198.931 q^{69} -981.074 q^{71} -1222.03 q^{73} +3118.67 q^{75} +341.995 q^{77} +521.960 q^{79} +2672.35 q^{81} +527.723 q^{83} +1069.98 q^{85} +715.657 q^{87} +318.192 q^{89} -368.107 q^{91} +3249.27 q^{93} +1312.96 q^{95} +440.656 q^{97} -3584.26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9	\( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) \) 15 * q + 12 * q^3 - 4 * q^5 - 105 * q^7 + 89 * q^9 - 14 * q^11 + 34 * q^13 - 160 * q^15 - 100 * q^17 + 26 * q^19 - 84 * q^21 + 158 * q^23 + 441 * q^25 + 450 * q^27 - 156 * q^29 + 252 * q^31 - 668 * q^33 + 28 * q^35 + 182 * q^37 + 370 * q^39 + 615 * q^41 + 894 * q^43 - 158 * q^45 + 1728 * q^47 + 735 * q^49 + 630 * q^51 + 1034 * q^53 + 1944 * q^55 + 54 * q^57 + 262 * q^59 + 322 * q^61 - 623 * q^63 + 188 * q^65 + 1808 * q^67 - 168 * q^69 + 584 * q^71 - 1290 * q^73 + 5188 * q^75 + 98 * q^77 + 3726 * q^79 + 3043 * q^81 + 2484 * q^83 + 3404 * q^85 + 5448 * q^87 + 876 * q^89 - 238 * q^91 + 6174 * q^93 + 5714 * q^95 - 154 * q^97 + 2854 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more