LMFDB - Embedded newform 1088.4.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,4,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, 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("1088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1088.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:	$64.1940780862$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.6420.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 21x - 15 $ x^3 - x^2 - 21*x - 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.63853$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.63853 q^{3} +20.7931 q^{5} -0.399607 q^{7} +47.6243 q^{9} +O(q^{10})$	$q-8.63853 q^{3} +20.7931 q^{5} -0.399607 q^{7} +47.6243 q^{9} -0.469735 q^{11} -0.471700 q^{13} -179.622 q^{15} +17.0000 q^{17} +135.959 q^{19} +3.45202 q^{21} +40.3465 q^{23} +307.351 q^{25} -178.163 q^{27} +259.591 q^{29} -183.218 q^{31} +4.05782 q^{33} -8.30905 q^{35} +15.8821 q^{37} +4.07479 q^{39} -194.268 q^{41} +368.394 q^{43} +990.254 q^{45} -536.467 q^{47} -342.840 q^{49} -146.855 q^{51} +161.579 q^{53} -9.76723 q^{55} -1174.49 q^{57} -645.023 q^{59} +76.1410 q^{61} -19.0310 q^{63} -9.80808 q^{65} -606.759 q^{67} -348.534 q^{69} +8.25149 q^{71} +98.5778 q^{73} -2655.07 q^{75} +0.187709 q^{77} +509.842 q^{79} +253.215 q^{81} +1044.13 q^{83} +353.482 q^{85} -2242.48 q^{87} +466.756 q^{89} +0.188494 q^{91} +1582.73 q^{93} +2827.01 q^{95} +1422.77 q^{97} -22.3708 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 14 q^{3} + 18 q^{5} + 14 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q - 14 * q^3 + 18 * q^5 + 14 * q^7 + 27 * q^9	$ 3 q - 14 q^{3} + 18 q^{5} + 14 q^{7} + 27 q^{9} + 82 q^{11} + 6 q^{13} - 160 q^{15} + 51 q^{17} + 172 q^{19} + 72 q^{21} - 158 q^{23} + 69 q^{25} - 80 q^{27} + 330 q^{29} - 34 q^{31} - 84 q^{33} + 40 q^{35} + 378 q^{37} - 416 q^{39} - 282 q^{41} - 536 q^{43} + 954 q^{45} - 564 q^{47} - 321 q^{49} - 238 q^{51} + 894 q^{53} - 40 q^{55} - 984 q^{57} - 1216 q^{59} + 978 q^{61} - 742 q^{63} - 276 q^{65} + 272 q^{67} + 816 q^{69} - 158 q^{71} - 498 q^{73} - 2050 q^{75} + 1344 q^{77} + 1410 q^{79} + 111 q^{81} + 1624 q^{83} + 306 q^{85} - 1240 q^{87} + 42 q^{89} - 2224 q^{91} + 2088 q^{93} + 2960 q^{95} + 822 q^{97} - 1586 q^{99}+O(q^{100}) $ 3 * q - 14 * q^3 + 18 * q^5 + 14 * q^7 + 27 * q^9 + 82 * q^11 + 6 * q^13 - 160 * q^15 + 51 * q^17 + 172 * q^19 + 72 * q^21 - 158 * q^23 + 69 * q^25 - 80 * q^27 + 330 * q^29 - 34 * q^31 - 84 * q^33 + 40 * q^35 + 378 * q^37 - 416 * q^39 - 282 * q^41 - 536 * q^43 + 954 * q^45 - 564 * q^47 - 321 * q^49 - 238 * q^51 + 894 * q^53 - 40 * q^55 - 984 * q^57 - 1216 * q^59 + 978 * q^61 - 742 * q^63 - 276 * q^65 + 272 * q^67 + 816 * q^69 - 158 * q^71 - 498 * q^73 - 2050 * q^75 + 1344 * q^77 + 1410 * q^79 + 111 * q^81 + 1624 * q^83 + 306 * q^85 - 1240 * q^87 + 42 * q^89 - 2224 * q^91 + 2088 * q^93 + 2960 * q^95 + 822 * q^97 - 1586 * 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
$3$	−8.63853	−1.66249	−0.831243	−	0.555909i	$-0.812370\pi$
$3$	−8.63853	−1.66249	−0.831243	+	0.555909i	$0.812370\pi$
$4$	0	0
$5$	20.7931	1.85979	0.929894	−	0.367828i	$-0.119898\pi$
$5$	20.7931	1.85979	0.929894	+	0.367828i	$0.119898\pi$
$6$	0	0
$7$	−0.399607	−0.0215768	−0.0107884	−	0.999942i	$-0.503434\pi$
$7$	−0.399607	−0.0215768	−0.0107884	+	0.999942i	$0.503434\pi$
$8$	0	0
$9$	47.6243	1.76386
$10$	0	0
$11$	−0.469735	−0.0128755	−0.00643775	−	0.999979i	$-0.502049\pi$
$11$	−0.469735	−0.0128755	−0.00643775	+	0.999979i	$0.502049\pi$
$12$	0	0
$13$	−0.471700	−0.0100635	−0.00503177	−	0.999987i	$-0.501602\pi$
$13$	−0.471700	−0.0100635	−0.00503177	+	0.999987i	$0.501602\pi$
$14$	0	0
$15$	−179.622	−3.09187
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	135.959	1.64164	0.820820	−	0.571187i	$-0.193517\pi$
$19$	135.959	1.64164	0.820820	+	0.571187i	$0.193517\pi$
$20$	0	0
$21$	3.45202	0.0358711
$22$	0	0
$23$	40.3465	0.365775	0.182887	−	0.983134i	$-0.441456\pi$
$23$	40.3465	0.365775	0.182887	+	0.983134i	$0.441456\pi$
$24$	0	0
$25$	307.351	2.45881
$26$	0	0
$27$	−178.163	−1.26991
$28$	0	0
$29$	259.591	1.66223	0.831117	−	0.556097i	$-0.187702\pi$
$29$	259.591	1.66223	0.831117	+	0.556097i	$0.187702\pi$
$30$	0	0
$31$	−183.218	−1.06151	−0.530757	−	0.847524i	$-0.678092\pi$
$31$	−183.218	−1.06151	−0.530757	+	0.847524i	$0.678092\pi$
$32$	0	0
$33$	4.05782	0.0214053
$34$	0	0
$35$	−8.30905	−0.0401282
$36$	0	0
$37$	15.8821	0.0705677	0.0352839	−	0.999377i	$-0.488766\pi$
$37$	15.8821	0.0705677	0.0352839	+	0.999377i	$0.488766\pi$
$38$	0	0
$39$	4.07479	0.0167305
$40$	0	0
$41$	−194.268	−0.739990	−0.369995	−	0.929034i	$-0.620641\pi$
$41$	−194.268	−0.739990	−0.369995	+	0.929034i	$0.620641\pi$
$42$	0	0
$43$	368.394	1.30650	0.653250	−	0.757142i	$-0.273405\pi$
$43$	368.394	1.30650	0.653250	+	0.757142i	$0.273405\pi$
$44$	0	0
$45$	990.254	3.28041
$46$	0	0
$47$	−536.467	−1.66493	−0.832465	−	0.554077i	$-0.813071\pi$
$47$	−536.467	−1.66493	−0.832465	+	0.554077i	$0.813071\pi$
$48$	0	0
$49$	−342.840	−0.999534
$50$	0	0
$51$	−146.855	−0.403212
$52$	0	0
$53$	161.579	0.418766	0.209383	−	0.977834i	$-0.432854\pi$
$53$	161.579	0.418766	0.209383	+	0.977834i	$0.432854\pi$
$54$	0	0
$55$	−9.76723	−0.0239457
$56$	0	0
$57$	−1174.49	−2.72921
$58$	0	0
$59$	−645.023	−1.42330	−0.711651	−	0.702533i	$-0.752052\pi$
$59$	−645.023	−1.42330	−0.711651	+	0.702533i	$0.752052\pi$
$60$	0	0
$61$	76.1410	0.159817	0.0799087	−	0.996802i	$-0.474537\pi$
$61$	76.1410	0.159817	0.0799087	+	0.996802i	$0.474537\pi$
$62$	0	0
$63$	−19.0310	−0.0380584
$64$	0	0
$65$	−9.80808	−0.0187160
$66$	0	0
$67$	−606.759	−1.10638	−0.553190	−	0.833055i	$-0.686589\pi$
$67$	−606.759	−1.10638	−0.553190	+	0.833055i	$0.686589\pi$
$68$	0	0
$69$	−348.534	−0.608095
$70$	0	0
$71$	8.25149	0.0137926	0.00689628	−	0.999976i	$-0.497805\pi$
$71$	8.25149	0.0137926	0.00689628	+	0.999976i	$0.497805\pi$
$72$	0	0
$73$	98.5778	0.158050	0.0790250	−	0.996873i	$-0.474819\pi$
$73$	98.5778	0.158050	0.0790250	+	0.996873i	$0.474819\pi$
$74$	0	0
$75$	−2655.07	−4.08774
$76$	0	0
$77$	0.187709	0.000277811 0
$78$	0	0
$79$	509.842	0.726097	0.363048	−	0.931770i	$-0.381736\pi$
$79$	509.842	0.726097	0.363048	+	0.931770i	$0.381736\pi$
$80$	0	0
$81$	253.215	0.347346
$82$	0	0
$83$	1044.13	1.38082	0.690408	−	0.723420i	$-0.257431\pi$
$83$	1044.13	1.38082	0.690408	+	0.723420i	$0.257431\pi$
$84$	0	0
$85$	353.482	0.451065
$86$	0	0
$87$	−2242.48	−2.76344
$88$	0	0
$89$	466.756	0.555910	0.277955	−	0.960594i	$-0.410343\pi$
$89$	466.756	0.555910	0.277955	+	0.960594i	$0.410343\pi$
$90$	0	0
$91$	0.188494	0.000217138 0
$92$	0	0
$93$	1582.73	1.76475
$94$	0	0
$95$	2827.01	3.05310
$96$	0	0
$97$	1422.77	1.48928	0.744640	−	0.667467i	$-0.232621\pi$
$97$	1422.77	1.48928	0.744640	+	0.667467i	$0.232621\pi$
$98$	0	0
$99$	−22.3708	−0.0227106
$100$	0	0
$101$	−722.182	−0.711483	−0.355742	−	0.934584i	$-0.615772\pi$
$101$	−722.182	−0.711483	−0.355742	+	0.934584i	$0.615772\pi$
$102$	0	0
$103$	1508.47	1.44305	0.721524	−	0.692389i	$-0.243442\pi$
$103$	1508.47	1.44305	0.721524	+	0.692389i	$0.243442\pi$
$104$	0	0
$105$	71.7780	0.0667126
$106$	0	0
$107$	2127.78	1.92243	0.961216	−	0.275797i	$-0.0889419\pi$
$107$	2127.78	1.92243	0.961216	+	0.275797i	$0.0889419\pi$
$108$	0	0
$109$	−1552.44	−1.36419	−0.682094	−	0.731264i	$-0.738931\pi$
$109$	−1552.44	−1.36419	−0.682094	+	0.731264i	$0.738931\pi$
$110$	0	0
$111$	−137.198	−0.117318
$112$	0	0
$113$	−237.259	−0.197518	−0.0987588	−	0.995111i	$-0.531487\pi$
$113$	−237.259	−0.197518	−0.0987588	+	0.995111i	$0.531487\pi$
$114$	0	0
$115$	838.926	0.680263
$116$	0	0
$117$	−22.4643	−0.0177507
$118$	0	0
$119$	−6.79332	−0.00523313
$120$	0	0
$121$	−1330.78	−0.999834
$122$	0	0
$123$	1678.19	1.23022
$124$	0	0
$125$	3791.64	2.71308
$126$	0	0
$127$	1896.75	1.32527	0.662635	−	0.748942i	$-0.269438\pi$
$127$	1896.75	1.32527	0.662635	+	0.748942i	$0.269438\pi$
$128$	0	0
$129$	−3182.38	−2.17204
$130$	0	0
$131$	−1827.69	−1.21898	−0.609488	−	0.792796i	$-0.708625\pi$
$131$	−1827.69	−1.21898	−0.609488	+	0.792796i	$0.708625\pi$
$132$	0	0
$133$	−54.3302	−0.0354213
$134$	0	0
$135$	−3704.56	−2.36176
$136$	0	0
$137$	−487.209	−0.303833	−0.151916	−	0.988393i	$-0.548544\pi$
$137$	−487.209	−0.303833	−0.151916	+	0.988393i	$0.548544\pi$
$138$	0	0
$139$	−356.222	−0.217370	−0.108685	−	0.994076i	$-0.534664\pi$
$139$	−356.222	−0.217370	−0.108685	+	0.994076i	$0.534664\pi$
$140$	0	0
$141$	4634.29	2.76793
$142$	0	0
$143$	0.221574	0.000129573 0
$144$	0	0
$145$	5397.69	3.09140
$146$	0	0
$147$	2961.64	1.66171
$148$	0	0
$149$	2026.35	1.11413	0.557064	−	0.830469i	$-0.311928\pi$
$149$	2026.35	1.11413	0.557064	+	0.830469i	$0.311928\pi$
$150$	0	0
$151$	1401.72	0.755430	0.377715	−	0.925922i	$-0.376710\pi$
$151$	1401.72	0.755430	0.377715	+	0.925922i	$0.376710\pi$
$152$	0	0
$153$	809.612	0.427799
$154$	0	0
$155$	−3809.66	−1.97419
$156$	0	0
$157$	−3351.44	−1.70365	−0.851827	−	0.523823i	$-0.824505\pi$
$157$	−3351.44	−1.70365	−0.851827	+	0.523823i	$0.824505\pi$
$158$	0	0
$159$	−1395.81	−0.696193
$160$	0	0
$161$	−16.1227	−0.00789223
$162$	0	0
$163$	845.257	0.406170	0.203085	−	0.979161i	$-0.434903\pi$
$163$	845.257	0.406170	0.203085	+	0.979161i	$0.434903\pi$
$164$	0	0
$165$	84.3745	0.0398094
$166$	0	0
$167$	−2911.48	−1.34908	−0.674542	−	0.738237i	$-0.735659\pi$
$167$	−2911.48	−1.34908	−0.674542	+	0.738237i	$0.735659\pi$
$168$	0	0
$169$	−2196.78	−0.999899
$170$	0	0
$171$	6474.95	2.89563
$172$	0	0
$173$	1988.13	0.873727	0.436864	−	0.899528i	$-0.356089\pi$
$173$	1988.13	0.873727	0.436864	+	0.899528i	$0.356089\pi$
$174$	0	0
$175$	−122.820	−0.0530532
$176$	0	0
$177$	5572.05	2.36622
$178$	0	0
$179$	2569.83	1.07306	0.536531	−	0.843881i	$-0.319734\pi$
$179$	2569.83	1.07306	0.536531	+	0.843881i	$0.319734\pi$
$180$	0	0
$181$	−2467.73	−1.01340	−0.506698	−	0.862123i	$-0.669134\pi$
$181$	−2467.73	−1.01340	−0.506698	+	0.862123i	$0.669134\pi$
$182$	0	0
$183$	−657.747	−0.265694
$184$	0	0
$185$	330.238	0.131241
$186$	0	0
$187$	−7.98549	−0.00312277
$188$	0	0
$189$	71.1953	0.0274005
$190$	0	0
$191$	4419.70	1.67434	0.837168	−	0.546946i	$-0.184210\pi$
$191$	4419.70	1.67434	0.837168	+	0.546946i	$0.184210\pi$
$192$	0	0
$193$	−565.681	−0.210977	−0.105489	−	0.994421i	$-0.533641\pi$
$193$	−565.681	−0.210977	−0.105489	+	0.994421i	$0.533641\pi$
$194$	0	0
$195$	84.7274	0.0311152
$196$	0	0
$197$	4515.63	1.63312	0.816561	−	0.577259i	$-0.195878\pi$
$197$	4515.63	1.63312	0.816561	+	0.577259i	$0.195878\pi$
$198$	0	0
$199$	−1020.19	−0.363412	−0.181706	−	0.983353i	$-0.558162\pi$
$199$	−1020.19	−0.363412	−0.181706	+	0.983353i	$0.558162\pi$
$200$	0	0
$201$	5241.51	1.83934
$202$	0	0
$203$	−103.734	−0.0358656
$204$	0	0
$205$	−4039.43	−1.37622
$206$	0	0
$207$	1921.47	0.645176
$208$	0	0
$209$	−63.8648	−0.0211369
$210$	0	0
$211$	4255.60	1.38847	0.694236	−	0.719748i	$-0.255743\pi$
$211$	4255.60	1.38847	0.694236	+	0.719748i	$0.255743\pi$
$212$	0	0
$213$	−71.2808	−0.0229300
$214$	0	0
$215$	7660.03	2.42981
$216$	0	0
$217$	73.2152	0.0229040
$218$	0	0
$219$	−851.567	−0.262756
$220$	0	0
$221$	−8.01889	−0.00244076
$222$	0	0
$223$	4168.15	1.25166	0.625829	−	0.779960i	$-0.284761\pi$
$223$	4168.15	1.25166	0.625829	+	0.779960i	$0.284761\pi$
$224$	0	0
$225$	14637.4	4.33700
$226$	0	0
$227$	1234.57	0.360975	0.180488	−	0.983577i	$-0.442232\pi$
$227$	1234.57	0.360975	0.180488	+	0.983577i	$0.442232\pi$
$228$	0	0
$229$	3886.42	1.12149	0.560747	−	0.827987i	$-0.310514\pi$
$229$	3886.42	1.12149	0.560747	+	0.827987i	$0.310514\pi$
$230$	0	0
$231$	−1.62153	−0.000461858 0
$232$	0	0
$233$	1789.03	0.503018	0.251509	−	0.967855i	$-0.419073\pi$
$233$	1789.03	0.503018	0.251509	+	0.967855i	$0.419073\pi$
$234$	0	0
$235$	−11154.8	−3.09642
$236$	0	0
$237$	−4404.28	−1.20713
$238$	0	0
$239$	−3200.59	−0.866230	−0.433115	−	0.901339i	$-0.642586\pi$
$239$	−3200.59	−0.866230	−0.433115	+	0.901339i	$0.642586\pi$
$240$	0	0
$241$	1952.71	0.521931	0.260966	−	0.965348i	$-0.415959\pi$
$241$	1952.71	0.521931	0.260966	+	0.965348i	$0.415959\pi$
$242$	0	0
$243$	2623.00	0.692451
$244$	0	0
$245$	−7128.70	−1.85892
$246$	0	0
$247$	−64.1319	−0.0165207
$248$	0	0
$249$	−9019.72	−2.29559
$250$	0	0
$251$	−4646.54	−1.16847	−0.584237	−	0.811583i	$-0.698606\pi$
$251$	−4646.54	−1.16847	−0.584237	+	0.811583i	$0.698606\pi$
$252$	0	0
$253$	−18.9521	−0.00470953
$254$	0	0
$255$	−3053.57	−0.749889
$256$	0	0
$257$	6128.69	1.48754	0.743769	−	0.668437i	$-0.233036\pi$
$257$	6128.69	1.48754	0.743769	+	0.668437i	$0.233036\pi$
$258$	0	0
$259$	−6.34661	−0.00152262
$260$	0	0
$261$	12362.8	2.93195
$262$	0	0
$263$	1243.45	0.291537	0.145768	−	0.989319i	$-0.453435\pi$
$263$	1243.45	0.291537	0.145768	+	0.989319i	$0.453435\pi$
$264$	0	0
$265$	3359.73	0.778817
$266$	0	0
$267$	−4032.08	−0.924193
$268$	0	0
$269$	4712.14	1.06804	0.534022	−	0.845470i	$-0.320680\pi$
$269$	4712.14	1.06804	0.534022	+	0.845470i	$0.320680\pi$
$270$	0	0
$271$	−633.917	−0.142095	−0.0710474	−	0.997473i	$-0.522634\pi$
$271$	−633.917	−0.142095	−0.0710474	+	0.997473i	$0.522634\pi$
$272$	0	0
$273$	−1.62832	−0.000360990 0
$274$	0	0
$275$	−144.374	−0.0316584
$276$	0	0
$277$	−638.979	−0.138601	−0.0693006	−	0.997596i	$-0.522077\pi$
$277$	−638.979	−0.138601	−0.0693006	+	0.997596i	$0.522077\pi$
$278$	0	0
$279$	−8725.62	−1.87236
$280$	0	0
$281$	1394.65	0.296078	0.148039	−	0.988982i	$-0.452704\pi$
$281$	1394.65	0.296078	0.148039	+	0.988982i	$0.452704\pi$
$282$	0	0
$283$	−6620.84	−1.39070	−0.695350	−	0.718671i	$-0.744751\pi$
$283$	−6620.84	−1.39070	−0.695350	+	0.718671i	$0.744751\pi$
$284$	0	0
$285$	−24421.2	−5.07574
$286$	0	0
$287$	77.6310	0.0159666
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−12290.6	−2.47591
$292$	0	0
$293$	6531.85	1.30237	0.651185	−	0.758919i	$-0.274272\pi$
$293$	6531.85	1.30237	0.651185	+	0.758919i	$0.274272\pi$
$294$	0	0
$295$	−13412.0	−2.64704
$296$	0	0
$297$	83.6896	0.0163507
$298$	0	0
$299$	−19.0314	−0.00368098
$300$	0	0
$301$	−147.213	−0.0281900
$302$	0	0
$303$	6238.60	1.18283
$304$	0	0
$305$	1583.20	0.297226
$306$	0	0
$307$	5955.76	1.10721	0.553605	−	0.832779i	$-0.313252\pi$
$307$	5955.76	1.10721	0.553605	+	0.832779i	$0.313252\pi$
$308$	0	0
$309$	−13031.0	−2.39905
$310$	0	0
$311$	4405.36	0.803232	0.401616	−	0.915808i	$-0.368449\pi$
$311$	4405.36	0.803232	0.401616	+	0.915808i	$0.368449\pi$
$312$	0	0
$313$	−6387.47	−1.15349	−0.576743	−	0.816925i	$-0.695677\pi$
$313$	−6387.47	−1.15349	−0.576743	+	0.816925i	$0.695677\pi$
$314$	0	0
$315$	−395.713	−0.0707806
$316$	0	0
$317$	2509.93	0.444706	0.222353	−	0.974966i	$-0.428626\pi$
$317$	2509.93	0.444706	0.222353	+	0.974966i	$0.428626\pi$
$318$	0	0
$319$	−121.939	−0.0214021
$320$	0	0
$321$	−18380.9	−3.19602
$322$	0	0
$323$	2311.31	0.398156
$324$	0	0
$325$	−144.977	−0.0247443
$326$	0	0
$327$	13410.8	2.26794
$328$	0	0
$329$	214.376	0.0359238
$330$	0	0
$331$	−3372.66	−0.560055	−0.280028	−	0.959992i	$-0.590344\pi$
$331$	−3372.66	−0.560055	−0.280028	+	0.959992i	$0.590344\pi$
$332$	0	0
$333$	756.375	0.124472
$334$	0	0
$335$	−12616.4	−2.05763
$336$	0	0
$337$	−10571.8	−1.70885	−0.854424	−	0.519576i	$-0.826090\pi$
$337$	−10571.8	−1.70885	−0.854424	+	0.519576i	$0.826090\pi$
$338$	0	0
$339$	2049.57	0.328370
$340$	0	0
$341$	86.0639	0.0136675
$342$	0	0
$343$	274.067	0.0431435
$344$	0	0
$345$	−7247.09	−1.13093
$346$	0	0
$347$	−1158.68	−0.179254	−0.0896269	−	0.995975i	$-0.528567\pi$
$347$	−1158.68	−0.179254	−0.0896269	+	0.995975i	$0.528567\pi$
$348$	0	0
$349$	211.083	0.0323755	0.0161877	−	0.999869i	$-0.494847\pi$
$349$	211.083	0.0323755	0.0161877	+	0.999869i	$0.494847\pi$
$350$	0	0
$351$	84.0396	0.0127798
$352$	0	0
$353$	−5308.88	−0.800462	−0.400231	−	0.916414i	$-0.631070\pi$
$353$	−5308.88	−0.800462	−0.400231	+	0.916414i	$0.631070\pi$
$354$	0	0
$355$	171.574	0.0256512
$356$	0	0
$357$	58.6843	0.00870001
$358$	0	0
$359$	2254.00	0.331369	0.165684	−	0.986179i	$-0.447017\pi$
$359$	2254.00	0.331369	0.165684	+	0.986179i	$0.447017\pi$
$360$	0	0
$361$	11625.9	1.69498
$362$	0	0
$363$	11496.0	1.66221
$364$	0	0
$365$	2049.73	0.293940
$366$	0	0
$367$	4469.27	0.635679	0.317839	−	0.948145i	$-0.397043\pi$
$367$	4469.27	0.635679	0.317839	+	0.948145i	$0.397043\pi$
$368$	0	0
$369$	−9251.88	−1.30524
$370$	0	0
$371$	−64.5682	−0.00903562
$372$	0	0
$373$	5878.30	0.815996	0.407998	−	0.912983i	$-0.366227\pi$
$373$	5878.30	0.815996	0.407998	+	0.912983i	$0.366227\pi$
$374$	0	0
$375$	−32754.2	−4.51046
$376$	0	0
$377$	−122.449	−0.0167279
$378$	0	0
$379$	−1337.54	−0.181279	−0.0906396	−	0.995884i	$-0.528891\pi$
$379$	−1337.54	−0.181279	−0.0906396	+	0.995884i	$0.528891\pi$
$380$	0	0
$381$	−16385.1	−2.20324
$382$	0	0
$383$	−4394.88	−0.586339	−0.293170	−	0.956060i	$-0.594710\pi$
$383$	−4394.88	−0.586339	−0.293170	+	0.956060i	$0.594710\pi$
$384$	0	0
$385$	3.90305	0.000516670 0
$386$	0	0
$387$	17544.5	2.30449
$388$	0	0
$389$	4004.43	0.521934	0.260967	−	0.965348i	$-0.415959\pi$
$389$	4004.43	0.521934	0.260967	+	0.965348i	$0.415959\pi$
$390$	0	0
$391$	685.890	0.0887134
$392$	0	0
$393$	15788.5	2.02653
$394$	0	0
$395$	10601.2	1.35039
$396$	0	0
$397$	−1509.43	−0.190822	−0.0954110	−	0.995438i	$-0.530417\pi$
$397$	−1509.43	−0.190822	−0.0954110	+	0.995438i	$0.530417\pi$
$398$	0	0
$399$	469.334	0.0588874
$400$	0	0
$401$	−12528.9	−1.56026	−0.780128	−	0.625620i	$-0.784846\pi$
$401$	−12528.9	−1.56026	−0.780128	+	0.625620i	$0.784846\pi$
$402$	0	0
$403$	86.4239	0.0106826
$404$	0	0
$405$	5265.12	0.645990
$406$	0	0
$407$	−7.46039	−0.000908594 0
$408$	0	0
$409$	1203.93	0.145552	0.0727758	−	0.997348i	$-0.476814\pi$
$409$	1203.93	0.145552	0.0727758	+	0.997348i	$0.476814\pi$
$410$	0	0
$411$	4208.77	0.505118
$412$	0	0
$413$	257.756	0.0307102
$414$	0	0
$415$	21710.6	2.56802
$416$	0	0
$417$	3077.24	0.361374
$418$	0	0
$419$	−3730.83	−0.434995	−0.217497	−	0.976061i	$-0.569789\pi$
$419$	−3730.83	−0.434995	−0.217497	+	0.976061i	$0.569789\pi$
$420$	0	0
$421$	−4703.53	−0.544503	−0.272252	−	0.962226i	$-0.587768\pi$
$421$	−4703.53	−0.544503	−0.272252	+	0.962226i	$0.587768\pi$
$422$	0	0
$423$	−25548.8	−2.93671
$424$	0	0
$425$	5224.97	0.596349
$426$	0	0
$427$	−30.4265	−0.00344834
$428$	0	0
$429$	−1.91407	−0.000215413 0
$430$	0	0
$431$	9052.40	1.01169	0.505846	−	0.862624i	$-0.331181\pi$
$431$	9052.40	1.01169	0.505846	+	0.862624i	$0.331181\pi$
$432$	0	0
$433$	10189.2	1.13086	0.565432	−	0.824795i	$-0.308710\pi$
$433$	10189.2	1.13086	0.565432	+	0.824795i	$0.308710\pi$
$434$	0	0
$435$	−46628.1	−5.13942
$436$	0	0
$437$	5485.47	0.600470
$438$	0	0
$439$	13481.7	1.46571	0.732855	−	0.680385i	$-0.238188\pi$
$439$	13481.7	1.46571	0.732855	+	0.680385i	$0.238188\pi$
$440$	0	0
$441$	−16327.5	−1.76304
$442$	0	0
$443$	−6827.30	−0.732223	−0.366111	−	0.930571i	$-0.619311\pi$
$443$	−6827.30	−0.732223	−0.366111	+	0.930571i	$0.619311\pi$
$444$	0	0
$445$	9705.28	1.03387
$446$	0	0
$447$	−17504.7	−1.85222
$448$	0	0
$449$	−350.302	−0.0368191	−0.0184095	−	0.999831i	$-0.505860\pi$
$449$	−350.302	−0.0368191	−0.0184095	+	0.999831i	$0.505860\pi$
$450$	0	0
$451$	91.2546	0.00952774
$452$	0	0
$453$	−12108.8	−1.25589
$454$	0	0
$455$	3.91938	0.000403831 0
$456$	0	0
$457$	−15702.9	−1.60733	−0.803665	−	0.595082i	$-0.797119\pi$
$457$	−15702.9	−1.60733	−0.803665	+	0.595082i	$0.797119\pi$
$458$	0	0
$459$	−3028.78	−0.307998
$460$	0	0
$461$	−11585.0	−1.17043	−0.585215	−	0.810878i	$-0.698990\pi$
$461$	−11585.0	−1.17043	−0.585215	+	0.810878i	$0.698990\pi$
$462$	0	0
$463$	−12726.2	−1.27740	−0.638701	−	0.769455i	$-0.720528\pi$
$463$	−12726.2	−1.27740	−0.638701	+	0.769455i	$0.720528\pi$
$464$	0	0
$465$	32909.9	3.28207
$466$	0	0
$467$	−797.161	−0.0789898	−0.0394949	−	0.999220i	$-0.512575\pi$
$467$	−797.161	−0.0789898	−0.0394949	+	0.999220i	$0.512575\pi$
$468$	0	0
$469$	242.465	0.0238721
$470$	0	0
$471$	28951.5	2.83230
$472$	0	0
$473$	−173.047	−0.0168218
$474$	0	0
$475$	41787.2	4.03648
$476$	0	0
$477$	7695.09	0.738646
$478$	0	0
$479$	11302.1	1.07809	0.539046	−	0.842276i	$-0.318785\pi$
$479$	11302.1	1.07809	0.539046	+	0.842276i	$0.318785\pi$
$480$	0	0
$481$	−7.49159	−0.000710161 0
$482$	0	0
$483$	139.277	0.0131207
$484$	0	0
$485$	29583.7	2.76974
$486$	0	0
$487$	10890.1	1.01330	0.506652	−	0.862151i	$-0.330883\pi$
$487$	10890.1	1.01330	0.506652	+	0.862151i	$0.330883\pi$
$488$	0	0
$489$	−7301.78	−0.675251
$490$	0	0
$491$	−10843.9	−0.996701	−0.498350	−	0.866976i	$-0.666061\pi$
$491$	−10843.9	−0.996701	−0.498350	+	0.866976i	$0.666061\pi$
$492$	0	0
$493$	4413.04	0.403151
$494$	0	0
$495$	−465.157	−0.0422369
$496$	0	0
$497$	−3.29736	−0.000297599 0
$498$	0	0
$499$	12743.4	1.14324	0.571618	−	0.820520i	$-0.306316\pi$
$499$	12743.4	1.14324	0.571618	+	0.820520i	$0.306316\pi$
$500$	0	0
$501$	25150.9	2.24283
$502$	0	0
$503$	8514.19	0.754730	0.377365	−	0.926065i	$-0.376830\pi$
$503$	8514.19	0.754730	0.377365	+	0.926065i	$0.376830\pi$
$504$	0	0
$505$	−15016.4	−1.32321
$506$	0	0
$507$	18976.9	1.66232
$508$	0	0
$509$	−8257.03	−0.719030	−0.359515	−	0.933139i	$-0.617058\pi$
$509$	−8257.03	−0.719030	−0.359515	+	0.933139i	$0.617058\pi$
$510$	0	0
$511$	−39.3924	−0.00341021
$512$	0	0
$513$	−24222.9	−2.08473
$514$	0	0
$515$	31365.7	2.68376
$516$	0	0
$517$	251.997	0.0214368
$518$	0	0
$519$	−17174.5	−1.45256
$520$	0	0
$521$	685.063	0.0576068	0.0288034	−	0.999585i	$-0.490830\pi$
$521$	685.063	0.0576068	0.0288034	+	0.999585i	$0.490830\pi$
$522$	0	0
$523$	−11454.8	−0.957708	−0.478854	−	0.877895i	$-0.658948\pi$
$523$	−11454.8	−0.957708	−0.478854	+	0.877895i	$0.658948\pi$
$524$	0	0
$525$	1060.98	0.0882002
$526$	0	0
$527$	−3114.71	−0.257455
$528$	0	0
$529$	−10539.2	−0.866209
$530$	0	0
$531$	−30718.7	−2.51051
$532$	0	0
$533$	91.6362	0.00744691
$534$	0	0
$535$	44243.0	3.57532
$536$	0	0
$537$	−22199.6	−1.78395
$538$	0	0
$539$	161.044	0.0128695
$540$	0	0
$541$	23545.6	1.87117	0.935586	−	0.353099i	$-0.114872\pi$
$541$	23545.6	1.87117	0.935586	+	0.353099i	$0.114872\pi$
$542$	0	0
$543$	21317.6	1.68476
$544$	0	0
$545$	−32279.9	−2.53710
$546$	0	0
$547$	12192.2	0.953021	0.476510	−	0.879169i	$-0.341901\pi$
$547$	12192.2	0.953021	0.476510	+	0.879169i	$0.341901\pi$
$548$	0	0
$549$	3626.16	0.281896
$550$	0	0
$551$	35293.7	2.72879
$552$	0	0
$553$	−203.736	−0.0156668
$554$	0	0
$555$	−2852.77	−0.218186
$556$	0	0
$557$	−12246.5	−0.931596	−0.465798	−	0.884891i	$-0.654233\pi$
$557$	−12246.5	−0.931596	−0.465798	+	0.884891i	$0.654233\pi$
$558$	0	0
$559$	−173.771	−0.0131480
$560$	0	0
$561$	68.9830	0.00519156
$562$	0	0
$563$	−13507.1	−1.01111	−0.505555	−	0.862795i	$-0.668712\pi$
$563$	−13507.1	−1.01111	−0.505555	+	0.862795i	$0.668712\pi$
$564$	0	0
$565$	−4933.35	−0.367341
$566$	0	0
$567$	−101.187	−0.00749460
$568$	0	0
$569$	−8023.68	−0.591160	−0.295580	−	0.955318i	$-0.595513\pi$
$569$	−8023.68	−0.591160	−0.295580	+	0.955318i	$0.595513\pi$
$570$	0	0
$571$	−9596.13	−0.703302	−0.351651	−	0.936131i	$-0.614380\pi$
$571$	−9596.13	−0.703302	−0.351651	+	0.936131i	$0.614380\pi$
$572$	0	0
$573$	−38179.7	−2.78356
$574$	0	0
$575$	12400.5	0.899371
$576$	0	0
$577$	−14120.1	−1.01877	−0.509384	−	0.860540i	$-0.670127\pi$
$577$	−14120.1	−1.01877	−0.509384	+	0.860540i	$0.670127\pi$
$578$	0	0
$579$	4886.65	0.350747
$580$	0	0
$581$	−417.240	−0.0297935
$582$	0	0
$583$	−75.8994	−0.00539182
$584$	0	0
$585$	−467.102	−0.0330125
$586$	0	0
$587$	6586.66	0.463135	0.231568	−	0.972819i	$-0.425615\pi$
$587$	6586.66	0.463135	0.231568	+	0.972819i	$0.425615\pi$
$588$	0	0
$589$	−24910.2	−1.74262
$590$	0	0
$591$	−39008.4	−2.71504
$592$	0	0
$593$	19294.1	1.33611	0.668055	−	0.744112i	$-0.267127\pi$
$593$	19294.1	1.33611	0.668055	+	0.744112i	$0.267127\pi$
$594$	0	0
$595$	−141.254	−0.00973251
$596$	0	0
$597$	8812.90	0.604168
$598$	0	0
$599$	−21342.9	−1.45584	−0.727921	−	0.685662i	$-0.759513\pi$
$599$	−21342.9	−1.45584	−0.727921	+	0.685662i	$0.759513\pi$
$600$	0	0
$601$	10138.2	0.688094	0.344047	−	0.938953i	$-0.388202\pi$
$601$	10138.2	0.688094	0.344047	+	0.938953i	$0.388202\pi$
$602$	0	0
$603$	−28896.4	−1.95150
$604$	0	0
$605$	−27671.0	−1.85948
$606$	0	0
$607$	−22816.6	−1.52570	−0.762848	−	0.646578i	$-0.776200\pi$
$607$	−22816.6	−1.52570	−0.762848	+	0.646578i	$0.776200\pi$
$608$	0	0
$609$	896.112	0.0596261
$610$	0	0
$611$	253.051	0.0167551
$612$	0	0
$613$	5814.32	0.383096	0.191548	−	0.981483i	$-0.438649\pi$
$613$	5814.32	0.383096	0.191548	+	0.981483i	$0.438649\pi$
$614$	0	0
$615$	34894.8	2.28796
$616$	0	0
$617$	−15344.2	−1.00119	−0.500596	−	0.865681i	$-0.666886\pi$
$617$	−15344.2	−1.00119	−0.500596	+	0.865681i	$0.666886\pi$
$618$	0	0
$619$	−12742.5	−0.827407	−0.413703	−	0.910412i	$-0.635765\pi$
$619$	−12742.5	−0.827407	−0.413703	+	0.910412i	$0.635765\pi$
$620$	0	0
$621$	−7188.26	−0.464501
$622$	0	0
$623$	−186.519	−0.0119947
$624$	0	0
$625$	40420.9	2.58694
$626$	0	0
$627$	551.698	0.0351399
$628$	0	0
$629$	269.996	0.0171152
$630$	0	0
$631$	−29299.0	−1.84846	−0.924228	−	0.381841i	$-0.875290\pi$
$631$	−29299.0	−1.84846	−0.924228	+	0.381841i	$0.875290\pi$
$632$	0	0
$633$	−36762.1	−2.30832
$634$	0	0
$635$	39439.2	2.46472
$636$	0	0
$637$	161.718	0.0100588
$638$	0	0
$639$	392.971	0.0243282
$640$	0	0
$641$	−14710.5	−0.906440	−0.453220	−	0.891399i	$-0.649725\pi$
$641$	−14710.5	−0.906440	−0.453220	+	0.891399i	$0.649725\pi$
$642$	0	0
$643$	22296.3	1.36746	0.683731	−	0.729734i	$-0.260356\pi$
$643$	22296.3	1.36746	0.683731	+	0.729734i	$0.260356\pi$
$644$	0	0
$645$	−66171.4	−4.03953
$646$	0	0
$647$	−15412.4	−0.936513	−0.468256	−	0.883593i	$-0.655118\pi$
$647$	−15412.4	−0.936513	−0.468256	+	0.883593i	$0.655118\pi$
$648$	0	0
$649$	302.990	0.0183257
$650$	0	0
$651$	−632.472	−0.0380776
$652$	0	0
$653$	16010.7	0.959489	0.479744	−	0.877408i	$-0.340729\pi$
$653$	16010.7	0.959489	0.479744	+	0.877408i	$0.340729\pi$
$654$	0	0
$655$	−38003.2	−2.26704
$656$	0	0
$657$	4694.69	0.278778
$658$	0	0
$659$	12504.5	0.739162	0.369581	−	0.929198i	$-0.379501\pi$
$659$	12504.5	0.739162	0.369581	+	0.929198i	$0.379501\pi$
$660$	0	0
$661$	−15980.2	−0.940330	−0.470165	−	0.882578i	$-0.655806\pi$
$661$	−15980.2	−0.940330	−0.470165	+	0.882578i	$0.655806\pi$
$662$	0	0
$663$	69.2715	0.00405774
$664$	0	0
$665$	−1129.69	−0.0658761
$666$	0	0
$667$	10473.6	0.608003
$668$	0	0
$669$	−36006.7	−2.08087
$670$	0	0
$671$	−35.7661	−0.00205773
$672$	0	0
$673$	28971.8	1.65941	0.829704	−	0.558204i	$-0.188509\pi$
$673$	28971.8	1.65941	0.829704	+	0.558204i	$0.188509\pi$
$674$	0	0
$675$	−54758.8	−3.12247
$676$	0	0
$677$	3663.50	0.207976	0.103988	−	0.994579i	$-0.466840\pi$
$677$	3663.50	0.207976	0.103988	+	0.994579i	$0.466840\pi$
$678$	0	0
$679$	−568.548	−0.0321338
$680$	0	0
$681$	−10664.9	−0.600116
$682$	0	0
$683$	17409.5	0.975336	0.487668	−	0.873029i	$-0.337848\pi$
$683$	17409.5	0.975336	0.487668	+	0.873029i	$0.337848\pi$
$684$	0	0
$685$	−10130.6	−0.565064
$686$	0	0
$687$	−33573.0	−1.86447
$688$	0	0
$689$	−76.2168	−0.00421427
$690$	0	0
$691$	−3049.75	−0.167898	−0.0839492	−	0.996470i	$-0.526753\pi$
$691$	−3049.75	−0.167898	−0.0839492	+	0.996470i	$0.526753\pi$
$692$	0	0
$693$	8.93952	0.000490021 0
$694$	0	0
$695$	−7406.95	−0.404261
$696$	0	0
$697$	−3302.56	−0.179474
$698$	0	0
$699$	−15454.6	−0.836260
$700$	0	0
$701$	9974.54	0.537423	0.268711	−	0.963221i	$-0.413402\pi$
$701$	9974.54	0.537423	0.268711	+	0.963221i	$0.413402\pi$
$702$	0	0
$703$	2159.32	0.115847
$704$	0	0
$705$	96361.0	5.14775
$706$	0	0
$707$	288.589	0.0153515
$708$	0	0
$709$	−11613.9	−0.615187	−0.307594	−	0.951518i	$-0.599524\pi$
$709$	−11613.9	−0.615187	−0.307594	+	0.951518i	$0.599524\pi$
$710$	0	0
$711$	24280.8	1.28073
$712$	0	0
$713$	−7392.20	−0.388275
$714$	0	0
$715$	4.60720	0.000240978 0
$716$	0	0
$717$	27648.4	1.44010
$718$	0	0
$719$	−30024.0	−1.55731	−0.778656	−	0.627451i	$-0.784098\pi$
$719$	−30024.0	−1.55731	−0.778656	+	0.627451i	$0.784098\pi$
$720$	0	0
$721$	−602.795	−0.0311363
$722$	0	0
$723$	−16868.6	−0.867703
$724$	0	0
$725$	79785.6	4.08712
$726$	0	0
$727$	25373.2	1.29441	0.647207	−	0.762315i	$-0.275937\pi$
$727$	25373.2	1.29441	0.647207	+	0.762315i	$0.275937\pi$
$728$	0	0
$729$	−29495.7	−1.49854
$730$	0	0
$731$	6262.69	0.316873
$732$	0	0
$733$	25185.3	1.26909	0.634543	−	0.772888i	$-0.281188\pi$
$733$	25185.3	1.26909	0.634543	+	0.772888i	$0.281188\pi$
$734$	0	0
$735$	61581.5	3.09043
$736$	0	0
$737$	285.016	0.0142452
$738$	0	0
$739$	18027.8	0.897380	0.448690	−	0.893687i	$-0.351891\pi$
$739$	18027.8	0.897380	0.448690	+	0.893687i	$0.351891\pi$
$740$	0	0
$741$	554.005	0.0274654
$742$	0	0
$743$	−13226.8	−0.653088	−0.326544	−	0.945182i	$-0.605884\pi$
$743$	−13226.8	−0.653088	−0.326544	+	0.945182i	$0.605884\pi$
$744$	0	0
$745$	42134.0	2.07204
$746$	0	0
$747$	49725.7	2.43557
$748$	0	0
$749$	−850.276	−0.0414798
$750$	0	0
$751$	21840.7	1.06122	0.530612	−	0.847615i	$-0.321962\pi$
$751$	21840.7	1.06122	0.530612	+	0.847615i	$0.321962\pi$
$752$	0	0
$753$	40139.3	1.94257
$754$	0	0
$755$	29146.0	1.40494
$756$	0	0
$757$	−16692.3	−0.801441	−0.400721	−	0.916200i	$-0.631240\pi$
$757$	−16692.3	−0.801441	−0.400721	+	0.916200i	$0.631240\pi$
$758$	0	0
$759$	163.719	0.00782953
$760$	0	0
$761$	−20381.1	−0.970847	−0.485424	−	0.874279i	$-0.661335\pi$
$761$	−20381.1	−0.970847	−0.485424	+	0.874279i	$0.661335\pi$
$762$	0	0
$763$	620.365	0.0294348
$764$	0	0
$765$	16834.3	0.795616
$766$	0	0
$767$	304.257	0.0143234
$768$	0	0
$769$	20788.6	0.974846	0.487423	−	0.873166i	$-0.337937\pi$
$769$	20788.6	0.974846	0.487423	+	0.873166i	$0.337937\pi$
$770$	0	0
$771$	−52942.9	−2.47301
$772$	0	0
$773$	−24820.5	−1.15489	−0.577445	−	0.816430i	$-0.695950\pi$
$773$	−24820.5	−1.15489	−0.577445	+	0.816430i	$0.695950\pi$
$774$	0	0
$775$	−56312.3	−2.61006
$776$	0	0
$777$	54.8254	0.00253134
$778$	0	0
$779$	−26412.5	−1.21480
$780$	0	0
$781$	−3.87601	−0.000177586 0
$782$	0	0
$783$	−46249.6	−2.11089
$784$	0	0
$785$	−69686.6	−3.16844
$786$	0	0
$787$	13606.7	0.616297	0.308149	−	0.951338i	$-0.400291\pi$
$787$	13606.7	0.616297	0.308149	+	0.951338i	$0.400291\pi$
$788$	0	0
$789$	−10741.6	−0.484676
$790$	0	0
$791$	94.8105	0.00426179
$792$	0	0
$793$	−35.9157	−0.00160833
$794$	0	0
$795$	−29023.1	−1.29477
$796$	0	0
$797$	5310.40	0.236015	0.118007	−	0.993013i	$-0.462349\pi$
$797$	5310.40	0.236015	0.118007	+	0.993013i	$0.462349\pi$
$798$	0	0
$799$	−9119.94	−0.403805
$800$	0	0
$801$	22228.9	0.980549
$802$	0	0
$803$	−46.3054	−0.00203497
$804$	0	0
$805$	−335.241	−0.0146779
$806$	0	0
$807$	−40706.0	−1.77561
$808$	0	0
$809$	−25001.4	−1.08653	−0.543264	−	0.839562i	$-0.682812\pi$
$809$	−25001.4	−1.08653	−0.543264	+	0.839562i	$0.682812\pi$
$810$	0	0
$811$	11061.4	0.478938	0.239469	−	0.970904i	$-0.423027\pi$
$811$	11061.4	0.478938	0.239469	+	0.970904i	$0.423027\pi$
$812$	0	0
$813$	5476.11	0.236231
$814$	0	0
$815$	17575.5	0.755389
$816$	0	0
$817$	50086.5	2.14480
$818$	0	0
$819$	8.97691	0.000383002 0
$820$	0	0
$821$	−6919.25	−0.294133	−0.147067	−	0.989127i	$-0.546983\pi$
$821$	−6919.25	−0.294133	−0.147067	+	0.989127i	$0.546983\pi$
$822$	0	0
$823$	−23627.4	−1.00073	−0.500365	−	0.865815i	$-0.666801\pi$
$823$	−23627.4	−1.00073	−0.500365	+	0.865815i	$0.666801\pi$
$824$	0	0
$825$	1247.18	0.0526317
$826$	0	0
$827$	−9982.73	−0.419750	−0.209875	−	0.977728i	$-0.567306\pi$
$827$	−9982.73	−0.419750	−0.209875	+	0.977728i	$0.567306\pi$
$828$	0	0
$829$	27943.6	1.17071	0.585357	−	0.810775i	$-0.300954\pi$
$829$	27943.6	1.17071	0.585357	+	0.810775i	$0.300954\pi$
$830$	0	0
$831$	5519.84	0.230423
$832$	0	0
$833$	−5828.29	−0.242423
$834$	0	0
$835$	−60538.5	−2.50901
$836$	0	0
$837$	32642.7	1.34803
$838$	0	0
$839$	−26457.3	−1.08869	−0.544343	−	0.838862i	$-0.683221\pi$
$839$	−26457.3	−1.08869	−0.544343	+	0.838862i	$0.683221\pi$
$840$	0	0
$841$	42998.4	1.76302
$842$	0	0
$843$	−12047.7	−0.492226
$844$	0	0
$845$	−45677.7	−1.85960
$846$	0	0
$847$	531.789	0.0215732
$848$	0	0
$849$	57194.4	2.31202
$850$	0	0
$851$	640.788	0.0258119
$852$	0	0
$853$	13154.4	0.528015	0.264007	−	0.964521i	$-0.414956\pi$
$853$	13154.4	0.528015	0.264007	+	0.964521i	$0.414956\pi$
$854$	0	0
$855$	134634.	5.38525
$856$	0	0
$857$	2819.40	0.112379	0.0561894	−	0.998420i	$-0.482105\pi$
$857$	2819.40	0.112379	0.0561894	+	0.998420i	$0.482105\pi$
$858$	0	0
$859$	−8345.37	−0.331479	−0.165739	−	0.986170i	$-0.553001\pi$
$859$	−8345.37	−0.331479	−0.165739	+	0.986170i	$0.553001\pi$
$860$	0	0
$861$	−670.618	−0.0265442
$862$	0	0
$863$	−19038.3	−0.750951	−0.375476	−	0.926832i	$-0.622521\pi$
$863$	−19038.3	−0.750951	−0.375476	+	0.926832i	$0.622521\pi$
$864$	0	0
$865$	41339.3	1.62495
$866$	0	0
$867$	−2496.54	−0.0977933
$868$	0	0
$869$	−239.490	−0.00934885
$870$	0	0
$871$	286.208	0.0111341
$872$	0	0
$873$	67758.2	2.62688
$874$	0	0
$875$	−1515.17	−0.0585394
$876$	0	0
$877$	8054.02	0.310108	0.155054	−	0.987906i	$-0.450445\pi$
$877$	8054.02	0.310108	0.155054	+	0.987906i	$0.450445\pi$
$878$	0	0
$879$	−56425.6	−2.16517
$880$	0	0
$881$	−23439.4	−0.896362	−0.448181	−	0.893943i	$-0.647928\pi$
$881$	−23439.4	−0.896362	−0.448181	+	0.893943i	$0.647928\pi$
$882$	0	0
$883$	−46620.4	−1.77678	−0.888392	−	0.459085i	$-0.848177\pi$
$883$	−46620.4	−1.77678	−0.888392	+	0.459085i	$0.848177\pi$
$884$	0	0
$885$	115860.	4.40067
$886$	0	0
$887$	9014.21	0.341226	0.170613	−	0.985338i	$-0.445425\pi$
$887$	9014.21	0.341226	0.170613	+	0.985338i	$0.445425\pi$
$888$	0	0
$889$	−757.955	−0.0285950
$890$	0	0
$891$	−118.944	−0.00447225
$892$	0	0
$893$	−72937.6	−2.73322
$894$	0	0
$895$	53434.6	1.99567
$896$	0	0
$897$	164.403	0.00611959
$898$	0	0
$899$	−47561.7	−1.76448
$900$	0	0
$901$	2746.85	0.101566
$902$	0	0
$903$	1271.70	0.0468656
$904$	0	0
$905$	−51311.6	−1.88470
$906$	0	0
$907$	21960.6	0.803957	0.401978	−	0.915649i	$-0.368323\pi$
$907$	21960.6	0.803957	0.401978	+	0.915649i	$0.368323\pi$
$908$	0	0
$909$	−34393.4	−1.25496
$910$	0	0
$911$	17984.9	0.654081	0.327040	−	0.945010i	$-0.393949\pi$
$911$	17984.9	0.654081	0.327040	+	0.945010i	$0.393949\pi$
$912$	0	0
$913$	−490.462	−0.0177787
$914$	0	0
$915$	−13676.6	−0.494135
$916$	0	0
$917$	730.357	0.0263015
$918$	0	0
$919$	17490.4	0.627807	0.313903	−	0.949455i	$-0.398363\pi$
$919$	17490.4	0.627807	0.313903	+	0.949455i	$0.398363\pi$
$920$	0	0
$921$	−51449.1	−1.84072
$922$	0	0
$923$	−3.89223	−0.000138802 0
$924$	0	0
$925$	4881.40	0.173513
$926$	0	0
$927$	71839.8	2.54534
$928$	0	0
$929$	32406.0	1.14446	0.572231	−	0.820092i	$-0.306078\pi$
$929$	32406.0	1.14446	0.572231	+	0.820092i	$0.306078\pi$
$930$	0	0
$931$	−46612.3	−1.64088
$932$	0	0
$933$	−38055.9	−1.33536
$934$	0	0
$935$	−166.043	−0.00580768
$936$	0	0
$937$	−541.908	−0.0188936	−0.00944682	−	0.999955i	$-0.503007\pi$
$937$	−541.908	−0.0188936	−0.00944682	+	0.999955i	$0.503007\pi$
$938$	0	0
$939$	55178.4	1.91766
$940$	0	0
$941$	36914.3	1.27882	0.639412	−	0.768865i	$-0.279178\pi$
$941$	36914.3	1.27882	0.639412	+	0.768865i	$0.279178\pi$
$942$	0	0
$943$	−7838.03	−0.270670
$944$	0	0
$945$	1480.37	0.0509592
$946$	0	0
$947$	25996.0	0.892036	0.446018	−	0.895024i	$-0.352842\pi$
$947$	25996.0	0.892036	0.446018	+	0.895024i	$0.352842\pi$
$948$	0	0
$949$	−46.4991	−0.00159054
$950$	0	0
$951$	−21682.1	−0.739318
$952$	0	0
$953$	12352.2	0.419859	0.209930	−	0.977717i	$-0.432677\pi$
$953$	12352.2	0.419859	0.209930	+	0.977717i	$0.432677\pi$
$954$	0	0
$955$	91899.0	3.11391
$956$	0	0
$957$	1053.37	0.0355807
$958$	0	0
$959$	194.692	0.00655572
$960$	0	0
$961$	3777.84	0.126812
$962$	0	0
$963$	101334.	3.39090
$964$	0	0
$965$	−11762.2	−0.392373
$966$	0	0
$967$	799.036	0.0265721	0.0132861	−	0.999912i	$-0.495771\pi$
$967$	799.036	0.0265721	0.0132861	+	0.999912i	$0.495771\pi$
$968$	0	0
$969$	−19966.3	−0.661929
$970$	0	0
$971$	−54311.7	−1.79500	−0.897500	−	0.441014i	$-0.854619\pi$
$971$	−54311.7	−1.79500	−0.897500	+	0.441014i	$0.854619\pi$
$972$	0	0
$973$	142.349	0.00469013
$974$	0	0
$975$	1252.39	0.0411371
$976$	0	0
$977$	−59782.8	−1.95765	−0.978824	−	0.204702i	$-0.934378\pi$
$977$	−59782.8	−1.95765	−0.978824	+	0.204702i	$0.934378\pi$
$978$	0	0
$979$	−219.251	−0.00715762
$980$	0	0
$981$	−73933.7	−2.40624
$982$	0	0
$983$	−20172.1	−0.654516	−0.327258	−	0.944935i	$-0.606125\pi$
$983$	−20172.1	−0.654516	−0.327258	+	0.944935i	$0.606125\pi$
$984$	0	0
$985$	93893.7	3.03726
$986$	0	0
$987$	−1851.89	−0.0597228
$988$	0	0
$989$	14863.4	0.477885
$990$	0	0
$991$	−10917.8	−0.349964	−0.174982	−	0.984572i	$-0.555987\pi$
$991$	−10917.8	−0.349964	−0.174982	+	0.984572i	$0.555987\pi$
$992$	0	0
$993$	29134.8	0.931084
$994$	0	0
$995$	−21212.8	−0.675869
$996$	0	0
$997$	−12247.4	−0.389047	−0.194524	−	0.980898i	$-0.562316\pi$
$997$	−12247.4	−0.389047	−0.194524	+	0.980898i	$0.562316\pi$
$998$	0	0
$999$	−2829.61	−0.0896147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.r.1.1		3
4.3	odd	2		1088.4.a.ba.1.3		3
8.3	odd	2		544.4.a.f.1.1	✓	3
8.5	even	2		544.4.a.g.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.4.a.f.1.1	✓	3	8.3	odd	2
544.4.a.g.1.3	yes	3	8.5	even	2
1088.4.a.r.1.1		3	1.1	even	1	trivial
1088.4.a.ba.1.3		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q-8.63853 q^{3} +20.7931 q^{5} -0.399607 q^{7} +47.6243 q^{9} +O(q^{10})\)	\(q-8.63853 q^{3} +20.7931 q^{5} -0.399607 q^{7} +47.6243 q^{9} -0.469735 q^{11} -0.471700 q^{13} -179.622 q^{15} +17.0000 q^{17} +135.959 q^{19} +3.45202 q^{21} +40.3465 q^{23} +307.351 q^{25} -178.163 q^{27} +259.591 q^{29} -183.218 q^{31} +4.05782 q^{33} -8.30905 q^{35} +15.8821 q^{37} +4.07479 q^{39} -194.268 q^{41} +368.394 q^{43} +990.254 q^{45} -536.467 q^{47} -342.840 q^{49} -146.855 q^{51} +161.579 q^{53} -9.76723 q^{55} -1174.49 q^{57} -645.023 q^{59} +76.1410 q^{61} -19.0310 q^{63} -9.80808 q^{65} -606.759 q^{67} -348.534 q^{69} +8.25149 q^{71} +98.5778 q^{73} -2655.07 q^{75} +0.187709 q^{77} +509.842 q^{79} +253.215 q^{81} +1044.13 q^{83} +353.482 q^{85} -2242.48 q^{87} +466.756 q^{89} +0.188494 q^{91} +1582.73 q^{93} +2827.01 q^{95} +1422.77 q^{97} -22.3708 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 14 q^{3} + 18 q^{5} + 14 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q - 14 * q^3 + 18 * q^5 + 14 * q^7 + 27 * q^9	\( 3 q - 14 q^{3} + 18 q^{5} + 14 q^{7} + 27 q^{9} + 82 q^{11} + 6 q^{13} - 160 q^{15} + 51 q^{17} + 172 q^{19} + 72 q^{21} - 158 q^{23} + 69 q^{25} - 80 q^{27} + 330 q^{29} - 34 q^{31} - 84 q^{33} + 40 q^{35} + 378 q^{37} - 416 q^{39} - 282 q^{41} - 536 q^{43} + 954 q^{45} - 564 q^{47} - 321 q^{49} - 238 q^{51} + 894 q^{53} - 40 q^{55} - 984 q^{57} - 1216 q^{59} + 978 q^{61} - 742 q^{63} - 276 q^{65} + 272 q^{67} + 816 q^{69} - 158 q^{71} - 498 q^{73} - 2050 q^{75} + 1344 q^{77} + 1410 q^{79} + 111 q^{81} + 1624 q^{83} + 306 q^{85} - 1240 q^{87} + 42 q^{89} - 2224 q^{91} + 2088 q^{93} + 2960 q^{95} + 822 q^{97} - 1586 q^{99}+O(q^{100}) \) 3 * q - 14 * q^3 + 18 * q^5 + 14 * q^7 + 27 * q^9 + 82 * q^11 + 6 * q^13 - 160 * q^15 + 51 * q^17 + 172 * q^19 + 72 * q^21 - 158 * q^23 + 69 * q^25 - 80 * q^27 + 330 * q^29 - 34 * q^31 - 84 * q^33 + 40 * q^35 + 378 * q^37 - 416 * q^39 - 282 * q^41 - 536 * q^43 + 954 * q^45 - 564 * q^47 - 321 * q^49 - 238 * q^51 + 894 * q^53 - 40 * q^55 - 984 * q^57 - 1216 * q^59 + 978 * q^61 - 742 * q^63 - 276 * q^65 + 272 * q^67 + 816 * q^69 - 158 * q^71 - 498 * q^73 - 2050 * q^75 + 1344 * q^77 + 1410 * q^79 + 111 * q^81 + 1624 * q^83 + 306 * q^85 - 1240 * q^87 + 42 * q^89 - 2224 * q^91 + 2088 * q^93 + 2960 * q^95 + 822 * q^97 - 1586 * q^99

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