LMFDB - Embedded newform 128.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,4,Mod(1,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("128.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	128.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:	$7.55224448073$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	128.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.92820 q^{3} -15.8564 q^{5} -17.8564 q^{7} -2.71281 q^{9} -52.9282 q^{11} -8.43078 q^{13} -78.1436 q^{15} +129.138 q^{17} -50.4974 q^{19} -88.0000 q^{21} +128.708 q^{23} +126.426 q^{25} -146.431 q^{27} -111.282 q^{29} -302.851 q^{31} -260.841 q^{33} +283.138 q^{35} +182.995 q^{37} -41.5486 q^{39} -94.5744 q^{41} -184.641 q^{43} +43.0155 q^{45} +296.841 q^{47} -24.1487 q^{49} +636.420 q^{51} +102.995 q^{53} +839.251 q^{55} -248.862 q^{57} +93.3693 q^{59} +338.974 q^{61} +48.4411 q^{63} +133.682 q^{65} -489.041 q^{67} +634.297 q^{69} -86.9845 q^{71} -154.267 q^{73} +623.051 q^{75} +945.108 q^{77} -449.415 q^{79} -648.395 q^{81} -383.933 q^{83} -2047.67 q^{85} -548.420 q^{87} -517.672 q^{89} +150.543 q^{91} -1492.51 q^{93} +800.708 q^{95} +1739.39 q^{97} +143.584 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 4 q^{5} - 8 q^{7} + 50 q^{9} - 92 q^{11} - 100 q^{13} - 184 q^{15} + 92 q^{17} - 4 q^{19} - 176 q^{21} + 8 q^{23} + 142 q^{25} - 376 q^{27} - 84 q^{29} - 384 q^{31} + 88 q^{33} + 400 q^{35}+ \cdots - 1916 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 - 4 * q^5 - 8 * q^7 + 50 * q^9 - 92 * q^11 - 100 * q^13 - 184 * q^15 + 92 * q^17 - 4 * q^19 - 176 * q^21 + 8 * q^23 + 142 * q^25 - 376 * q^27 - 84 * q^29 - 384 * q^31 + 88 * q^33 + 400 * q^35 + 172 * q^37 + 776 * q^39 - 300 * q^41 - 300 * q^43 + 668 * q^45 - 16 * q^47 - 270 * q^49 + 968 * q^51 + 12 * q^53 + 376 * q^55 - 664 * q^57 + 644 * q^59 - 292 * q^61 + 568 * q^63 - 952 * q^65 + 172 * q^67 + 1712 * q^69 + 408 * q^71 + 412 * q^73 + 484 * q^75 + 560 * q^77 - 400 * q^79 - 22 * q^81 - 948 * q^83 - 2488 * q^85 - 792 * q^87 + 572 * q^89 - 752 * q^91 - 768 * q^93 + 1352 * q^95 + 2204 * q^97 - 1916 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	4.92820	0.948433	0.474217	−	0.880408i	$-0.342731\pi$
$3$	4.92820	0.948433	0.474217	+	0.880408i	$0.342731\pi$
$4$	0	0
$5$	−15.8564	−1.41824	−0.709120	−	0.705088i	$-0.750908\pi$
$5$	−15.8564	−1.41824	−0.709120	+	0.705088i	$0.750908\pi$
$6$	0	0
$7$	−17.8564	−0.964155	−0.482078	−	0.876128i	$-0.660118\pi$
$7$	−17.8564	−0.964155	−0.482078	+	0.876128i	$0.660118\pi$
$8$	0	0
$9$	−2.71281	−0.100475
$10$	0	0
$11$	−52.9282	−1.45077	−0.725384	−	0.688344i	$-0.758338\pi$
$11$	−52.9282	−1.45077	−0.725384	+	0.688344i	$0.758338\pi$
$12$	0	0
$13$	−8.43078	−0.179868	−0.0899338	−	0.995948i	$-0.528666\pi$
$13$	−8.43078	−0.179868	−0.0899338	+	0.995948i	$0.528666\pi$
$14$	0	0
$15$	−78.1436	−1.34511
$16$	0	0
$17$	129.138	1.84239	0.921196	−	0.389098i	$-0.127213\pi$
$17$	129.138	1.84239	0.921196	+	0.389098i	$0.127213\pi$
$18$	0	0
$19$	−50.4974	−0.609732	−0.304866	−	0.952395i	$-0.598612\pi$
$19$	−50.4974	−0.609732	−0.304866	+	0.952395i	$0.598612\pi$
$20$	0	0
$21$	−88.0000	−0.914437
$22$	0	0
$23$	128.708	1.16684	0.583422	−	0.812169i	$-0.301713\pi$
$23$	128.708	1.16684	0.583422	+	0.812169i	$0.301713\pi$
$24$	0	0
$25$	126.426	1.01141
$26$	0	0
$27$	−146.431	−1.04373
$28$	0	0
$29$	−111.282	−0.712571	−0.356285	−	0.934377i	$-0.615957\pi$
$29$	−111.282	−0.712571	−0.356285	+	0.934377i	$0.615957\pi$
$30$	0	0
$31$	−302.851	−1.75464	−0.877318	−	0.479910i	$-0.840669\pi$
$31$	−302.851	−1.75464	−0.877318	+	0.479910i	$0.840669\pi$
$32$	0	0
$33$	−260.841	−1.37596
$34$	0	0
$35$	283.138	1.36740
$36$	0	0
$37$	182.995	0.813086	0.406543	−	0.913632i	$-0.366734\pi$
$37$	182.995	0.813086	0.406543	+	0.913632i	$0.366734\pi$
$38$	0	0
$39$	−41.5486	−0.170592
$40$	0	0
$41$	−94.5744	−0.360245	−0.180122	−	0.983644i	$-0.557649\pi$
$41$	−94.5744	−0.360245	−0.180122	+	0.983644i	$0.557649\pi$
$42$	0	0
$43$	−184.641	−0.654825	−0.327413	−	0.944881i	$-0.606177\pi$
$43$	−184.641	−0.654825	−0.327413	+	0.944881i	$0.606177\pi$
$44$	0	0
$45$	43.0155	0.142497
$46$	0	0
$47$	296.841	0.921249	0.460624	−	0.887595i	$-0.347625\pi$
$47$	296.841	0.921249	0.460624	+	0.887595i	$0.347625\pi$
$48$	0	0
$49$	−24.1487	−0.0704045
$50$	0	0
$51$	636.420	1.74739
$52$	0	0
$53$	102.995	0.266933	0.133466	−	0.991053i	$-0.457389\pi$
$53$	102.995	0.266933	0.133466	+	0.991053i	$0.457389\pi$
$54$	0	0
$55$	839.251	2.05754
$56$	0	0
$57$	−248.862	−0.578290
$58$	0	0
$59$	93.3693	0.206028	0.103014	−	0.994680i	$-0.467151\pi$
$59$	93.3693	0.206028	0.103014	+	0.994680i	$0.467151\pi$
$60$	0	0
$61$	338.974	0.711495	0.355748	−	0.934582i	$-0.384226\pi$
$61$	338.974	0.711495	0.355748	+	0.934582i	$0.384226\pi$
$62$	0	0
$63$	48.4411	0.0968731
$64$	0	0
$65$	133.682	0.255095
$66$	0	0
$67$	−489.041	−0.891729	−0.445865	−	0.895100i	$-0.647104\pi$
$67$	−489.041	−0.891729	−0.445865	+	0.895100i	$0.647104\pi$
$68$	0	0
$69$	634.297	1.10667
$70$	0	0
$71$	−86.9845	−0.145397	−0.0726983	−	0.997354i	$-0.523161\pi$
$71$	−86.9845	−0.145397	−0.0726983	+	0.997354i	$0.523161\pi$
$72$	0	0
$73$	−154.267	−0.247336	−0.123668	−	0.992324i	$-0.539466\pi$
$73$	−154.267	−0.247336	−0.123668	+	0.992324i	$0.539466\pi$
$74$	0	0
$75$	623.051	0.959250
$76$	0	0
$77$	945.108	1.39877
$78$	0	0
$79$	−449.415	−0.640040	−0.320020	−	0.947411i	$-0.603690\pi$
$79$	−449.415	−0.640040	−0.320020	+	0.947411i	$0.603690\pi$
$80$	0	0
$81$	−648.395	−0.889430
$82$	0	0
$83$	−383.933	−0.507737	−0.253868	−	0.967239i	$-0.581703\pi$
$83$	−383.933	−0.507737	−0.253868	+	0.967239i	$0.581703\pi$
$84$	0	0
$85$	−2047.67	−2.61295
$86$	0	0
$87$	−548.420	−0.675826
$88$	0	0
$89$	−517.672	−0.616551	−0.308276	−	0.951297i	$-0.599752\pi$
$89$	−517.672	−0.616551	−0.308276	+	0.951297i	$0.599752\pi$
$90$	0	0
$91$	150.543	0.173420
$92$	0	0
$93$	−1492.51	−1.66415
$94$	0	0
$95$	800.708	0.864746
$96$	0	0
$97$	1739.39	1.82071	0.910355	−	0.413829i	$-0.135809\pi$
$97$	1739.39	1.82071	0.910355	+	0.413829i	$0.135809\pi$
$98$	0	0
$99$	143.584	0.145765
$100$	0	0
$101$	1158.99	1.14182	0.570912	−	0.821011i	$-0.306590\pi$
$101$	1158.99	1.14182	0.570912	+	0.821011i	$0.306590\pi$
$102$	0	0
$103$	−1635.01	−1.56410	−0.782048	−	0.623218i	$-0.785825\pi$
$103$	−1635.01	−1.56410	−0.782048	+	0.623218i	$0.785825\pi$
$104$	0	0
$105$	1395.36	1.29689
$106$	0	0
$107$	−282.589	−0.255317	−0.127659	−	0.991818i	$-0.540746\pi$
$107$	−282.589	−0.255317	−0.127659	+	0.991818i	$0.540746\pi$
$108$	0	0
$109$	−1004.98	−0.883120	−0.441560	−	0.897232i	$-0.645575\pi$
$109$	−1004.98	−0.883120	−0.441560	+	0.897232i	$0.645575\pi$
$110$	0	0
$111$	901.836	0.771157
$112$	0	0
$113$	−567.108	−0.472115	−0.236057	−	0.971739i	$-0.575855\pi$
$113$	−567.108	−0.472115	−0.236057	+	0.971739i	$0.575855\pi$
$114$	0	0
$115$	−2040.84	−1.65486
$116$	0	0
$117$	22.8711	0.0180721
$118$	0	0
$119$	−2305.95	−1.77635
$120$	0	0
$121$	1470.39	1.10473
$122$	0	0
$123$	−466.082	−0.341668
$124$	0	0
$125$	−22.6053	−0.0161750
$126$	0	0
$127$	−496.616	−0.346988	−0.173494	−	0.984835i	$-0.555506\pi$
$127$	−496.616	−0.346988	−0.173494	+	0.984835i	$0.555506\pi$
$128$	0	0
$129$	−909.948	−0.621058
$130$	0	0
$131$	−1658.72	−1.10628	−0.553142	−	0.833087i	$-0.686571\pi$
$131$	−1658.72	−1.10628	−0.553142	+	0.833087i	$0.686571\pi$
$132$	0	0
$133$	901.703	0.587876
$134$	0	0
$135$	2321.87	1.48025
$136$	0	0
$137$	−432.872	−0.269947	−0.134974	−	0.990849i	$-0.543095\pi$
$137$	−432.872	−0.269947	−0.134974	+	0.990849i	$0.543095\pi$
$138$	0	0
$139$	146.148	0.0891809	0.0445904	−	0.999005i	$-0.485802\pi$
$139$	146.148	0.0891809	0.0445904	+	0.999005i	$0.485802\pi$
$140$	0	0
$141$	1462.89	0.873743
$142$	0	0
$143$	446.226	0.260946
$144$	0	0
$145$	1764.53	1.01060
$146$	0	0
$147$	−119.010	−0.0667740
$148$	0	0
$149$	−3556.29	−1.95532	−0.977659	−	0.210198i	$-0.932589\pi$
$149$	−3556.29	−1.95532	−0.977659	+	0.210198i	$0.932589\pi$
$150$	0	0
$151$	−1320.32	−0.711562	−0.355781	−	0.934569i	$-0.615785\pi$
$151$	−1320.32	−0.711562	−0.355781	+	0.934569i	$0.615785\pi$
$152$	0	0
$153$	−350.328	−0.185114
$154$	0	0
$155$	4802.13	2.48849
$156$	0	0
$157$	−2040.43	−1.03722	−0.518612	−	0.855010i	$-0.673551\pi$
$157$	−2040.43	−1.03722	−0.518612	+	0.855010i	$0.673551\pi$
$158$	0	0
$159$	507.580	0.253168
$160$	0	0
$161$	−2298.26	−1.12502
$162$	0	0
$163$	2392.21	1.14952	0.574762	−	0.818321i	$-0.305095\pi$
$163$	2392.21	1.14952	0.574762	+	0.818321i	$0.305095\pi$
$164$	0	0
$165$	4136.00	1.95144
$166$	0	0
$167$	2068.34	0.958400	0.479200	−	0.877706i	$-0.340927\pi$
$167$	2068.34	0.958400	0.479200	+	0.877706i	$0.340927\pi$
$168$	0	0
$169$	−2125.92	−0.967648
$170$	0	0
$171$	136.990	0.0612625
$172$	0	0
$173$	2441.87	1.07313	0.536566	−	0.843859i	$-0.319721\pi$
$173$	2441.87	1.07313	0.536566	+	0.843859i	$0.319721\pi$
$174$	0	0
$175$	−2257.51	−0.975152
$176$	0	0
$177$	460.143	0.195404
$178$	0	0
$179$	2430.84	1.01502	0.507512	−	0.861645i	$-0.330565\pi$
$179$	2430.84	1.01502	0.507512	+	0.861645i	$0.330565\pi$
$180$	0	0
$181$	−1928.96	−0.792148	−0.396074	−	0.918219i	$-0.629628\pi$
$181$	−1928.96	−0.792148	−0.396074	+	0.918219i	$0.629628\pi$
$182$	0	0
$183$	1670.53	0.674806
$184$	0	0
$185$	−2901.64	−1.15315
$186$	0	0
$187$	−6835.07	−2.67288
$188$	0	0
$189$	2614.73	1.00631
$190$	0	0
$191$	4087.34	1.54843	0.774214	−	0.632924i	$-0.218145\pi$
$191$	4087.34	1.54843	0.774214	+	0.632924i	$0.218145\pi$
$192$	0	0
$193$	−1156.52	−0.431339	−0.215669	−	0.976466i	$-0.569193\pi$
$193$	−1156.52	−0.431339	−0.215669	+	0.976466i	$0.569193\pi$
$194$	0	0
$195$	658.811	0.241941
$196$	0	0
$197$	2658.44	0.961452	0.480726	−	0.876871i	$-0.340373\pi$
$197$	2658.44	0.961452	0.480726	+	0.876871i	$0.340373\pi$
$198$	0	0
$199$	−742.369	−0.264448	−0.132224	−	0.991220i	$-0.542212\pi$
$199$	−742.369	−0.264448	−0.132224	+	0.991220i	$0.542212\pi$
$200$	0	0
$201$	−2410.09	−0.845745
$202$	0	0
$203$	1987.10	0.687029
$204$	0	0
$205$	1499.61	0.510914
$206$	0	0
$207$	−349.160	−0.117238
$208$	0	0
$209$	2672.74	0.884580
$210$	0	0
$211$	3838.30	1.25232	0.626160	−	0.779694i	$-0.284626\pi$
$211$	3838.30	1.25232	0.626160	+	0.779694i	$0.284626\pi$
$212$	0	0
$213$	−428.677	−0.137899
$214$	0	0
$215$	2927.74	0.928700
$216$	0	0
$217$	5407.84	1.69174
$218$	0	0
$219$	−760.257	−0.234582
$220$	0	0
$221$	−1088.74	−0.331387
$222$	0	0
$223$	−3418.34	−1.02650	−0.513249	−	0.858240i	$-0.671558\pi$
$223$	−3418.34	−1.02650	−0.513249	+	0.858240i	$0.671558\pi$
$224$	0	0
$225$	−342.969	−0.101620
$226$	0	0
$227$	−515.729	−0.150793	−0.0753967	−	0.997154i	$-0.524022\pi$
$227$	−515.729	−0.150793	−0.0753967	+	0.997154i	$0.524022\pi$
$228$	0	0
$229$	−1595.26	−0.460340	−0.230170	−	0.973150i	$-0.573928\pi$
$229$	−1595.26	−0.460340	−0.230170	+	0.973150i	$0.573928\pi$
$230$	0	0
$231$	4657.68	1.32664
$232$	0	0
$233$	5251.27	1.47649	0.738245	−	0.674533i	$-0.235655\pi$
$233$	5251.27	1.47649	0.738245	+	0.674533i	$0.235655\pi$
$234$	0	0
$235$	−4706.83	−1.30655
$236$	0	0
$237$	−2214.81	−0.607035
$238$	0	0
$239$	−2029.27	−0.549216	−0.274608	−	0.961556i	$-0.588548\pi$
$239$	−2029.27	−0.549216	−0.274608	+	0.961556i	$0.588548\pi$
$240$	0	0
$241$	−4615.72	−1.23371	−0.616856	−	0.787076i	$-0.711594\pi$
$241$	−4615.72	−1.23371	−0.616856	+	0.787076i	$0.711594\pi$
$242$	0	0
$243$	758.210	0.200161
$244$	0	0
$245$	382.912	0.0998505
$246$	0	0
$247$	425.733	0.109671
$248$	0	0
$249$	−1892.10	−0.481554
$250$	0	0
$251$	2702.52	0.679607	0.339804	−	0.940496i	$-0.389639\pi$
$251$	2702.52	0.679607	0.339804	+	0.940496i	$0.389639\pi$
$252$	0	0
$253$	−6812.27	−1.69282
$254$	0	0
$255$	−10091.3	−2.47821
$256$	0	0
$257$	991.785	0.240723	0.120362	−	0.992730i	$-0.461595\pi$
$257$	991.785	0.240723	0.120362	+	0.992730i	$0.461595\pi$
$258$	0	0
$259$	−3267.63	−0.783941
$260$	0	0
$261$	301.887	0.0715952
$262$	0	0
$263$	−7430.74	−1.74220	−0.871101	−	0.491105i	$-0.836593\pi$
$263$	−7430.74	−1.74220	−0.871101	+	0.491105i	$0.836593\pi$
$264$	0	0
$265$	−1633.13	−0.378575
$266$	0	0
$267$	−2551.19	−0.584758
$268$	0	0
$269$	−3281.50	−0.743778	−0.371889	−	0.928277i	$-0.621290\pi$
$269$	−3281.50	−0.743778	−0.371889	+	0.928277i	$0.621290\pi$
$270$	0	0
$271$	−221.518	−0.0496542	−0.0248271	−	0.999692i	$-0.507904\pi$
$271$	−221.518	−0.0496542	−0.0248271	+	0.999692i	$0.507904\pi$
$272$	0	0
$273$	741.909	0.164477
$274$	0	0
$275$	−6691.48	−1.46731
$276$	0	0
$277$	−1106.28	−0.239963	−0.119982	−	0.992776i	$-0.538284\pi$
$277$	−1106.28	−0.239963	−0.119982	+	0.992776i	$0.538284\pi$
$278$	0	0
$279$	821.579	0.176296
$280$	0	0
$281$	−2372.52	−0.503676	−0.251838	−	0.967769i	$-0.581035\pi$
$281$	−2372.52	−0.503676	−0.251838	+	0.967769i	$0.581035\pi$
$282$	0	0
$283$	−2549.89	−0.535602	−0.267801	−	0.963474i	$-0.586297\pi$
$283$	−2549.89	−0.535602	−0.267801	+	0.963474i	$0.586297\pi$
$284$	0	0
$285$	3946.05	0.820154
$286$	0	0
$287$	1688.76	0.347332
$288$	0	0
$289$	11763.7	2.39441
$290$	0	0
$291$	8572.09	1.72682
$292$	0	0
$293$	−130.319	−0.0259840	−0.0129920	−	0.999916i	$-0.504136\pi$
$293$	−130.319	−0.0259840	−0.0129920	+	0.999916i	$0.504136\pi$
$294$	0	0
$295$	−1480.50	−0.292197
$296$	0	0
$297$	7750.32	1.51421
$298$	0	0
$299$	−1085.11	−0.209877
$300$	0	0
$301$	3297.03	0.631353
$302$	0	0
$303$	5711.76	1.08294
$304$	0	0
$305$	−5374.91	−1.00907
$306$	0	0
$307$	62.1182	0.0115481	0.00577406	−	0.999983i	$-0.498162\pi$
$307$	62.1182	0.0115481	0.00577406	+	0.999983i	$0.498162\pi$
$308$	0	0
$309$	−8057.64	−1.48344
$310$	0	0
$311$	−593.837	−0.108275	−0.0541373	−	0.998533i	$-0.517241\pi$
$311$	−593.837	−0.108275	−0.0541373	+	0.998533i	$0.517241\pi$
$312$	0	0
$313$	3669.69	0.662694	0.331347	−	0.943509i	$-0.392497\pi$
$313$	3669.69	0.662694	0.331347	+	0.943509i	$0.392497\pi$
$314$	0	0
$315$	−768.102	−0.137389
$316$	0	0
$317$	−374.339	−0.0663249	−0.0331625	−	0.999450i	$-0.510558\pi$
$317$	−374.339	−0.0663249	−0.0331625	+	0.999450i	$0.510558\pi$
$318$	0	0
$319$	5889.96	1.03378
$320$	0	0
$321$	−1392.66	−0.242151
$322$	0	0
$323$	−6521.16	−1.12337
$324$	0	0
$325$	−1065.87	−0.181919
$326$	0	0
$327$	−4952.77	−0.837580
$328$	0	0
$329$	−5300.51	−0.888227
$330$	0	0
$331$	5871.13	0.974945	0.487472	−	0.873138i	$-0.337919\pi$
$331$	5871.13	0.974945	0.487472	+	0.873138i	$0.337919\pi$
$332$	0	0
$333$	−496.431	−0.0816944
$334$	0	0
$335$	7754.43	1.26469
$336$	0	0
$337$	1047.83	0.169373	0.0846865	−	0.996408i	$-0.473011\pi$
$337$	1047.83	0.169373	0.0846865	+	0.996408i	$0.473011\pi$
$338$	0	0
$339$	−2794.82	−0.447769
$340$	0	0
$341$	16029.4	2.54557
$342$	0	0
$343$	6555.96	1.03204
$344$	0	0
$345$	−10057.7	−1.56953
$346$	0	0
$347$	3921.88	0.606737	0.303368	−	0.952873i	$-0.401889\pi$
$347$	3921.88	0.606737	0.303368	+	0.952873i	$0.401889\pi$
$348$	0	0
$349$	11604.9	1.77993	0.889966	−	0.456027i	$-0.150728\pi$
$349$	11604.9	1.77993	0.889966	+	0.456027i	$0.150728\pi$
$350$	0	0
$351$	1234.53	0.187733
$352$	0	0
$353$	−1530.22	−0.230724	−0.115362	−	0.993324i	$-0.536803\pi$
$353$	−1530.22	−0.230724	−0.115362	+	0.993324i	$0.536803\pi$
$354$	0	0
$355$	1379.26	0.206207
$356$	0	0
$357$	−11364.2	−1.68475
$358$	0	0
$359$	−10764.0	−1.58246	−0.791231	−	0.611518i	$-0.790559\pi$
$359$	−10764.0	−1.58246	−0.791231	+	0.611518i	$0.790559\pi$
$360$	0	0
$361$	−4309.01	−0.628227
$362$	0	0
$363$	7246.40	1.04776
$364$	0	0
$365$	2446.11	0.350782
$366$	0	0
$367$	−10713.0	−1.52374	−0.761869	−	0.647731i	$-0.775718\pi$
$367$	−10713.0	−1.52374	−0.761869	+	0.647731i	$0.775718\pi$
$368$	0	0
$369$	256.563	0.0361954
$370$	0	0
$371$	−1839.12	−0.257365
$372$	0	0
$373$	9186.32	1.27520	0.637600	−	0.770368i	$-0.279927\pi$
$373$	9186.32	1.27520	0.637600	+	0.770368i	$0.279927\pi$
$374$	0	0
$375$	−111.404	−0.0153409
$376$	0	0
$377$	938.194	0.128168
$378$	0	0
$379$	14201.0	1.92468	0.962342	−	0.271842i	$-0.0876327\pi$
$379$	14201.0	1.92468	0.962342	+	0.271842i	$0.0876327\pi$
$380$	0	0
$381$	−2447.42	−0.329095
$382$	0	0
$383$	9691.24	1.29295	0.646474	−	0.762936i	$-0.276243\pi$
$383$	9691.24	1.29295	0.646474	+	0.762936i	$0.276243\pi$
$384$	0	0
$385$	−14986.0	−1.98379
$386$	0	0
$387$	500.897	0.0657933
$388$	0	0
$389$	−6358.79	−0.828801	−0.414400	−	0.910095i	$-0.636009\pi$
$389$	−6358.79	−0.828801	−0.414400	+	0.910095i	$0.636009\pi$
$390$	0	0
$391$	16621.1	2.14978
$392$	0	0
$393$	−8174.52	−1.04924
$394$	0	0
$395$	7126.11	0.907731
$396$	0	0
$397$	−13034.2	−1.64778	−0.823888	−	0.566752i	$-0.808200\pi$
$397$	−13034.2	−1.64778	−0.823888	+	0.566752i	$0.808200\pi$
$398$	0	0
$399$	4443.77	0.557561
$400$	0	0
$401$	6705.06	0.834999	0.417499	−	0.908677i	$-0.362907\pi$
$401$	6705.06	0.834999	0.417499	+	0.908677i	$0.362907\pi$
$402$	0	0
$403$	2553.27	0.315602
$404$	0	0
$405$	10281.2	1.26143
$406$	0	0
$407$	−9685.59	−1.17960
$408$	0	0
$409$	10967.8	1.32598	0.662988	−	0.748630i	$-0.269288\pi$
$409$	10967.8	1.32598	0.662988	+	0.748630i	$0.269288\pi$
$410$	0	0
$411$	−2133.28	−0.256027
$412$	0	0
$413$	−1667.24	−0.198643
$414$	0	0
$415$	6087.80	0.720093
$416$	0	0
$417$	720.249	0.0845821
$418$	0	0
$419$	−13686.0	−1.59571	−0.797856	−	0.602849i	$-0.794032\pi$
$419$	−13686.0	−1.59571	−0.797856	+	0.602849i	$0.794032\pi$
$420$	0	0
$421$	8552.92	0.990128	0.495064	−	0.868857i	$-0.335145\pi$
$421$	8552.92	0.990128	0.495064	+	0.868857i	$0.335145\pi$
$422$	0	0
$423$	−805.274	−0.0925621
$424$	0	0
$425$	16326.4	1.86340
$426$	0	0
$427$	−6052.86	−0.685992
$428$	0	0
$429$	2199.09	0.247490
$430$	0	0
$431$	12187.6	1.36208	0.681039	−	0.732247i	$-0.261528\pi$
$431$	12187.6	1.36208	0.681039	+	0.732247i	$0.261528\pi$
$432$	0	0
$433$	−8906.35	−0.988480	−0.494240	−	0.869326i	$-0.664554\pi$
$433$	−8906.35	−0.988480	−0.494240	+	0.869326i	$0.664554\pi$
$434$	0	0
$435$	8695.98	0.958483
$436$	0	0
$437$	−6499.41	−0.711462
$438$	0	0
$439$	−4655.21	−0.506107	−0.253054	−	0.967452i	$-0.581435\pi$
$439$	−4655.21	−0.506107	−0.253054	+	0.967452i	$0.581435\pi$
$440$	0	0
$441$	65.5110	0.00707386
$442$	0	0
$443$	−10008.8	−1.07343	−0.536716	−	0.843763i	$-0.680335\pi$
$443$	−10008.8	−1.07343	−0.536716	+	0.843763i	$0.680335\pi$
$444$	0	0
$445$	8208.41	0.874418
$446$	0	0
$447$	−17526.1	−1.85449
$448$	0	0
$449$	−3373.71	−0.354600	−0.177300	−	0.984157i	$-0.556736\pi$
$449$	−3373.71	−0.354600	−0.177300	+	0.984157i	$0.556736\pi$
$450$	0	0
$451$	5005.65	0.522632
$452$	0	0
$453$	−6506.79	−0.674869
$454$	0	0
$455$	−2387.08	−0.245952
$456$	0	0
$457$	−13600.4	−1.39212	−0.696060	−	0.717984i	$-0.745065\pi$
$457$	−13600.4	−1.39212	−0.696060	+	0.717984i	$0.745065\pi$
$458$	0	0
$459$	−18909.8	−1.92295
$460$	0	0
$461$	3559.45	0.359609	0.179805	−	0.983702i	$-0.442453\pi$
$461$	3559.45	0.359609	0.179805	+	0.983702i	$0.442453\pi$
$462$	0	0
$463$	−8019.14	−0.804926	−0.402463	−	0.915436i	$-0.631846\pi$
$463$	−8019.14	−0.804926	−0.402463	+	0.915436i	$0.631846\pi$
$464$	0	0
$465$	23665.9	2.36017
$466$	0	0
$467$	2066.14	0.204731	0.102365	−	0.994747i	$-0.467359\pi$
$467$	2066.14	0.204731	0.102365	+	0.994747i	$0.467359\pi$
$468$	0	0
$469$	8732.51	0.859765
$470$	0	0
$471$	−10055.7	−0.983737
$472$	0	0
$473$	9772.72	0.950000
$474$	0	0
$475$	−6384.17	−0.616686
$476$	0	0
$477$	−279.406	−0.0268199
$478$	0	0
$479$	−3679.26	−0.350960	−0.175480	−	0.984483i	$-0.556148\pi$
$479$	−3679.26	−0.350960	−0.175480	+	0.984483i	$0.556148\pi$
$480$	0	0
$481$	−1542.79	−0.146248
$482$	0	0
$483$	−11326.3	−1.06700
$484$	0	0
$485$	−27580.5	−2.58220
$486$	0	0
$487$	10884.4	1.01277	0.506386	−	0.862307i	$-0.330981\pi$
$487$	10884.4	1.01277	0.506386	+	0.862307i	$0.330981\pi$
$488$	0	0
$489$	11789.3	1.09025
$490$	0	0
$491$	14548.9	1.33723	0.668616	−	0.743608i	$-0.266887\pi$
$491$	14548.9	1.33723	0.668616	+	0.743608i	$0.266887\pi$
$492$	0	0
$493$	−14370.8	−1.31284
$494$	0	0
$495$	−2276.73	−0.206730
$496$	0	0
$497$	1553.23	0.140185
$498$	0	0
$499$	−8461.92	−0.759134	−0.379567	−	0.925164i	$-0.623927\pi$
$499$	−8461.92	−0.759134	−0.379567	+	0.925164i	$0.623927\pi$
$500$	0	0
$501$	10193.2	0.908978
$502$	0	0
$503$	−983.702	−0.0871990	−0.0435995	−	0.999049i	$-0.513883\pi$
$503$	−983.702	−0.0871990	−0.0435995	+	0.999049i	$0.513883\pi$
$504$	0	0
$505$	−18377.5	−1.61938
$506$	0	0
$507$	−10477.0	−0.917749
$508$	0	0
$509$	−14472.5	−1.26028	−0.630140	−	0.776481i	$-0.717003\pi$
$509$	−14472.5	−1.26028	−0.630140	+	0.776481i	$0.717003\pi$
$510$	0	0
$511$	2754.65	0.238470
$512$	0	0
$513$	7394.38	0.636393
$514$	0	0
$515$	25925.3	2.21826
$516$	0	0
$517$	−15711.3	−1.33652
$518$	0	0
$519$	12034.0	1.01779
$520$	0	0
$521$	−10761.8	−0.904959	−0.452480	−	0.891775i	$-0.649460\pi$
$521$	−10761.8	−0.904959	−0.452480	+	0.891775i	$0.649460\pi$
$522$	0	0
$523$	1516.67	0.126806	0.0634029	−	0.997988i	$-0.479805\pi$
$523$	1516.67	0.126806	0.0634029	+	0.997988i	$0.479805\pi$
$524$	0	0
$525$	−11125.5	−0.924866
$526$	0	0
$527$	−39109.7	−3.23273
$528$	0	0
$529$	4398.66	0.361524
$530$	0	0
$531$	−253.293	−0.0207006
$532$	0	0
$533$	797.336	0.0647963
$534$	0	0
$535$	4480.85	0.362101
$536$	0	0
$537$	11979.7	0.962682
$538$	0	0
$539$	1278.15	0.102141
$540$	0	0
$541$	22541.9	1.79141	0.895704	−	0.444652i	$-0.146673\pi$
$541$	22541.9	1.79141	0.895704	+	0.444652i	$0.146673\pi$
$542$	0	0
$543$	−9506.33	−0.751299
$544$	0	0
$545$	15935.4	1.25248
$546$	0	0
$547$	10202.0	0.797448	0.398724	−	0.917071i	$-0.369453\pi$
$547$	10202.0	0.797448	0.398724	+	0.917071i	$0.369453\pi$
$548$	0	0
$549$	−919.574	−0.0714872
$550$	0	0
$551$	5619.46	0.434477
$552$	0	0
$553$	8024.94	0.617098
$554$	0	0
$555$	−14299.9	−1.09369
$556$	0	0
$557$	7139.06	0.543073	0.271536	−	0.962428i	$-0.412468\pi$
$557$	7139.06	0.543073	0.271536	+	0.962428i	$0.412468\pi$
$558$	0	0
$559$	1556.67	0.117782
$560$	0	0
$561$	−33684.6	−2.53505
$562$	0	0
$563$	−19558.0	−1.46407	−0.732034	−	0.681268i	$-0.761429\pi$
$563$	−19558.0	−1.46407	−0.732034	+	0.681268i	$0.761429\pi$
$564$	0	0
$565$	8992.29	0.669572
$566$	0	0
$567$	11578.0	0.857549
$568$	0	0
$569$	−5835.86	−0.429969	−0.214984	−	0.976617i	$-0.568970\pi$
$569$	−5835.86	−0.429969	−0.214984	+	0.976617i	$0.568970\pi$
$570$	0	0
$571$	9820.65	0.719757	0.359879	−	0.932999i	$-0.382818\pi$
$571$	9820.65	0.719757	0.359879	+	0.932999i	$0.382818\pi$
$572$	0	0
$573$	20143.3	1.46858
$574$	0	0
$575$	16271.9	1.18015
$576$	0	0
$577$	451.231	0.0325563	0.0162782	−	0.999868i	$-0.494818\pi$
$577$	451.231	0.0325563	0.0162782	+	0.999868i	$0.494818\pi$
$578$	0	0
$579$	−5699.58	−0.409096
$580$	0	0
$581$	6855.67	0.489537
$582$	0	0
$583$	−5451.33	−0.387257
$584$	0	0
$585$	−362.654	−0.0256306
$586$	0	0
$587$	−24775.7	−1.74208	−0.871042	−	0.491209i	$-0.836555\pi$
$587$	−24775.7	−1.74208	−0.871042	+	0.491209i	$0.836555\pi$
$588$	0	0
$589$	15293.2	1.06986
$590$	0	0
$591$	13101.3	0.911873
$592$	0	0
$593$	−4207.64	−0.291378	−0.145689	−	0.989330i	$-0.546540\pi$
$593$	−4207.64	−0.291378	−0.145689	+	0.989330i	$0.546540\pi$
$594$	0	0
$595$	36564.1	2.51929
$596$	0	0
$597$	−3658.54	−0.250811
$598$	0	0
$599$	1916.32	0.130715	0.0653577	−	0.997862i	$-0.479181\pi$
$599$	1916.32	0.130715	0.0653577	+	0.997862i	$0.479181\pi$
$600$	0	0
$601$	−19361.1	−1.31407	−0.657034	−	0.753861i	$-0.728189\pi$
$601$	−19361.1	−1.31407	−0.657034	+	0.753861i	$0.728189\pi$
$602$	0	0
$603$	1326.68	0.0895961
$604$	0	0
$605$	−23315.2	−1.56677
$606$	0	0
$607$	6097.02	0.407694	0.203847	−	0.979003i	$-0.434655\pi$
$607$	6097.02	0.407694	0.203847	+	0.979003i	$0.434655\pi$
$608$	0	0
$609$	9792.82	0.651601
$610$	0	0
$611$	−2502.60	−0.165703
$612$	0	0
$613$	13837.7	0.911743	0.455872	−	0.890046i	$-0.349328\pi$
$613$	13837.7	0.911743	0.455872	+	0.890046i	$0.349328\pi$
$614$	0	0
$615$	7390.38	0.484567
$616$	0	0
$617$	11959.1	0.780318	0.390159	−	0.920747i	$-0.372420\pi$
$617$	11959.1	0.780318	0.390159	+	0.920747i	$0.372420\pi$
$618$	0	0
$619$	8385.02	0.544463	0.272231	−	0.962232i	$-0.412238\pi$
$619$	8385.02	0.544463	0.272231	+	0.962232i	$0.412238\pi$
$620$	0	0
$621$	−18846.8	−1.21787
$622$	0	0
$623$	9243.75	0.594451
$624$	0	0
$625$	−15444.8	−0.988465
$626$	0	0
$627$	13171.8	0.838965
$628$	0	0
$629$	23631.7	1.49802
$630$	0	0
$631$	13592.1	0.857513	0.428757	−	0.903420i	$-0.358952\pi$
$631$	13592.1	0.857513	0.428757	+	0.903420i	$0.358952\pi$
$632$	0	0
$633$	18915.9	1.18774
$634$	0	0
$635$	7874.54	0.492113
$636$	0	0
$637$	203.593	0.0126635
$638$	0	0
$639$	235.973	0.0146087
$640$	0	0
$641$	−18123.4	−1.11674	−0.558371	−	0.829591i	$-0.688573\pi$
$641$	−18123.4	−1.11674	−0.558371	+	0.829591i	$0.688573\pi$
$642$	0	0
$643$	25042.1	1.53587	0.767935	−	0.640528i	$-0.221284\pi$
$643$	25042.1	1.53587	0.767935	+	0.640528i	$0.221284\pi$
$644$	0	0
$645$	14428.5	0.880809
$646$	0	0
$647$	−6022.08	−0.365923	−0.182962	−	0.983120i	$-0.558568\pi$
$647$	−6022.08	−0.365923	−0.182962	+	0.983120i	$0.558568\pi$
$648$	0	0
$649$	−4941.87	−0.298899
$650$	0	0
$651$	26650.9	1.60450
$652$	0	0
$653$	14093.8	0.844617	0.422308	−	0.906452i	$-0.361220\pi$
$653$	14093.8	0.844617	0.422308	+	0.906452i	$0.361220\pi$
$654$	0	0
$655$	26301.4	1.56898
$656$	0	0
$657$	418.496	0.0248510
$658$	0	0
$659$	−17240.7	−1.01912	−0.509562	−	0.860434i	$-0.670193\pi$
$659$	−17240.7	−1.01912	−0.509562	+	0.860434i	$0.670193\pi$
$660$	0	0
$661$	−20209.5	−1.18920	−0.594598	−	0.804023i	$-0.702689\pi$
$661$	−20209.5	−1.18920	−0.594598	+	0.804023i	$0.702689\pi$
$662$	0	0
$663$	−5365.52	−0.314298
$664$	0	0
$665$	−14297.8	−0.833749
$666$	0	0
$667$	−14322.8	−0.831459
$668$	0	0
$669$	−16846.3	−0.973564
$670$	0	0
$671$	−17941.3	−1.03221
$672$	0	0
$673$	6838.55	0.391689	0.195845	−	0.980635i	$-0.437255\pi$
$673$	6838.55	0.391689	0.195845	+	0.980635i	$0.437255\pi$
$674$	0	0
$675$	−18512.6	−1.05563
$676$	0	0
$677$	187.425	0.0106401	0.00532003	−	0.999986i	$-0.498307\pi$
$677$	187.425	0.0106401	0.00532003	+	0.999986i	$0.498307\pi$
$678$	0	0
$679$	−31059.3	−1.75545
$680$	0	0
$681$	−2541.62	−0.143018
$682$	0	0
$683$	23476.0	1.31520	0.657601	−	0.753367i	$-0.271571\pi$
$683$	23476.0	1.31520	0.657601	+	0.753367i	$0.271571\pi$
$684$	0	0
$685$	6863.79	0.382850
$686$	0	0
$687$	−7861.77	−0.436602
$688$	0	0
$689$	−868.327	−0.0480125
$690$	0	0
$691$	−6309.92	−0.347382	−0.173691	−	0.984800i	$-0.555569\pi$
$691$	−6309.92	−0.347382	−0.173691	+	0.984800i	$0.555569\pi$
$692$	0	0
$693$	−2563.90	−0.140540
$694$	0	0
$695$	−2317.39	−0.126480
$696$	0	0
$697$	−12213.2	−0.663712
$698$	0	0
$699$	25879.3	1.40035
$700$	0	0
$701$	24920.4	1.34270	0.671348	−	0.741142i	$-0.265716\pi$
$701$	24920.4	1.34270	0.671348	+	0.741142i	$0.265716\pi$
$702$	0	0
$703$	−9240.77	−0.495764
$704$	0	0
$705$	−23196.2	−1.23918
$706$	0	0
$707$	−20695.5	−1.10090
$708$	0	0
$709$	−14478.9	−0.766949	−0.383474	−	0.923551i	$-0.625273\pi$
$709$	−14478.9	−0.766949	−0.383474	+	0.923551i	$0.625273\pi$
$710$	0	0
$711$	1219.18	0.0643077
$712$	0	0
$713$	−38979.3	−2.04738
$714$	0	0
$715$	−7075.54	−0.370084
$716$	0	0
$717$	−10000.7	−0.520895
$718$	0	0
$719$	6411.30	0.332547	0.166273	−	0.986080i	$-0.446827\pi$
$719$	6411.30	0.332547	0.166273	+	0.986080i	$0.446827\pi$
$720$	0	0
$721$	29195.3	1.50803
$722$	0	0
$723$	−22747.2	−1.17009
$724$	0	0
$725$	−14068.9	−0.720698
$726$	0	0
$727$	20626.3	1.05225	0.526126	−	0.850407i	$-0.323644\pi$
$727$	20626.3	1.05225	0.526126	+	0.850407i	$0.323644\pi$
$728$	0	0
$729$	21243.3	1.07927
$730$	0	0
$731$	−23844.3	−1.20645
$732$	0	0
$733$	−5732.29	−0.288850	−0.144425	−	0.989516i	$-0.546133\pi$
$733$	−5732.29	−0.288850	−0.144425	+	0.989516i	$0.546133\pi$
$734$	0	0
$735$	1887.07	0.0947015
$736$	0	0
$737$	25884.1	1.29369
$738$	0	0
$739$	−36336.0	−1.80872	−0.904359	−	0.426773i	$-0.859650\pi$
$739$	−36336.0	−1.80872	−0.904359	+	0.426773i	$0.859650\pi$
$740$	0	0
$741$	2098.10	0.104016
$742$	0	0
$743$	−4363.83	−0.215469	−0.107734	−	0.994180i	$-0.534360\pi$
$743$	−4363.83	−0.215469	−0.107734	+	0.994180i	$0.534360\pi$
$744$	0	0
$745$	56389.9	2.77311
$746$	0	0
$747$	1041.54	0.0510146
$748$	0	0
$749$	5046.03	0.246166
$750$	0	0
$751$	16162.7	0.785334	0.392667	−	0.919681i	$-0.371552\pi$
$751$	16162.7	0.785334	0.392667	+	0.919681i	$0.371552\pi$
$752$	0	0
$753$	13318.6	0.644562
$754$	0	0
$755$	20935.5	1.00917
$756$	0	0
$757$	−4411.92	−0.211828	−0.105914	−	0.994375i	$-0.533777\pi$
$757$	−4411.92	−0.211828	−0.105914	+	0.994375i	$0.533777\pi$
$758$	0	0
$759$	−33572.2	−1.60553
$760$	0	0
$761$	28281.8	1.34719	0.673597	−	0.739099i	$-0.264749\pi$
$761$	28281.8	1.34719	0.673597	+	0.739099i	$0.264749\pi$
$762$	0	0
$763$	17945.4	0.851465
$764$	0	0
$765$	5554.95	0.262535
$766$	0	0
$767$	−787.176	−0.0370577
$768$	0	0
$769$	−15016.8	−0.704186	−0.352093	−	0.935965i	$-0.614530\pi$
$769$	−15016.8	−0.704186	−0.352093	+	0.935965i	$0.614530\pi$
$770$	0	0
$771$	4887.72	0.228310
$772$	0	0
$773$	−8674.11	−0.403604	−0.201802	−	0.979426i	$-0.564680\pi$
$773$	−8674.11	−0.403604	−0.201802	+	0.979426i	$0.564680\pi$
$774$	0	0
$775$	−38288.2	−1.77465
$776$	0	0
$777$	−16103.5	−0.743515
$778$	0	0
$779$	4775.76	0.219653
$780$	0	0
$781$	4603.94	0.210937
$782$	0	0
$783$	16295.1	0.743729
$784$	0	0
$785$	32353.9	1.47103
$786$	0	0
$787$	−23660.2	−1.07166	−0.535829	−	0.844327i	$-0.680001\pi$
$787$	−23660.2	−1.07166	−0.535829	+	0.844327i	$0.680001\pi$
$788$	0	0
$789$	−36620.2	−1.65236
$790$	0	0
$791$	10126.5	0.455192
$792$	0	0
$793$	−2857.82	−0.127975
$794$	0	0
$795$	−8048.39	−0.359053
$796$	0	0
$797$	−35432.5	−1.57476	−0.787380	−	0.616468i	$-0.788563\pi$
$797$	−35432.5	−1.57476	−0.787380	+	0.616468i	$0.788563\pi$
$798$	0	0
$799$	38333.6	1.69730
$800$	0	0
$801$	1404.35	0.0619477
$802$	0	0
$803$	8165.05	0.358827
$804$	0	0
$805$	36442.1	1.59555
$806$	0	0
$807$	−16171.9	−0.705424
$808$	0	0
$809$	−14005.5	−0.608664	−0.304332	−	0.952566i	$-0.598433\pi$
$809$	−14005.5	−0.608664	−0.304332	+	0.952566i	$0.598433\pi$
$810$	0	0
$811$	−22580.8	−0.977706	−0.488853	−	0.872366i	$-0.662584\pi$
$811$	−22580.8	−0.977706	−0.488853	+	0.872366i	$0.662584\pi$
$812$	0	0
$813$	−1091.69	−0.0470937
$814$	0	0
$815$	−37931.8	−1.63030
$816$	0	0
$817$	9323.90	0.399268
$818$	0	0
$819$	−408.396	−0.0174243
$820$	0	0
$821$	−21308.9	−0.905827	−0.452914	−	0.891554i	$-0.649615\pi$
$821$	−21308.9	−0.905827	−0.452914	+	0.891554i	$0.649615\pi$
$822$	0	0
$823$	−13571.4	−0.574810	−0.287405	−	0.957809i	$-0.592793\pi$
$823$	−13571.4	−0.574810	−0.287405	+	0.957809i	$0.592793\pi$
$824$	0	0
$825$	−32977.0	−1.39165
$826$	0	0
$827$	−19295.2	−0.811318	−0.405659	−	0.914025i	$-0.632958\pi$
$827$	−19295.2	−0.811318	−0.405659	+	0.914025i	$0.632958\pi$
$828$	0	0
$829$	−28752.7	−1.20461	−0.602306	−	0.798265i	$-0.705751\pi$
$829$	−28752.7	−1.20461	−0.602306	+	0.798265i	$0.705751\pi$
$830$	0	0
$831$	−5451.96	−0.227589
$832$	0	0
$833$	−3118.53	−0.129713
$834$	0	0
$835$	−32796.4	−1.35924
$836$	0	0
$837$	44346.7	1.83136
$838$	0	0
$839$	566.927	0.0233284	0.0116642	−	0.999932i	$-0.496287\pi$
$839$	566.927	0.0233284	0.0116642	+	0.999932i	$0.496287\pi$
$840$	0	0
$841$	−12005.3	−0.492243
$842$	0	0
$843$	−11692.3	−0.477703
$844$	0	0
$845$	33709.5	1.37236
$846$	0	0
$847$	−26256.0	−1.06513
$848$	0	0
$849$	−12566.4	−0.507983
$850$	0	0
$851$	23552.8	0.948744
$852$	0	0
$853$	−34698.1	−1.39278	−0.696389	−	0.717665i	$-0.745211\pi$
$853$	−34698.1	−1.39278	−0.696389	+	0.717665i	$0.745211\pi$
$854$	0	0
$855$	−2172.17	−0.0868850
$856$	0	0
$857$	37362.0	1.48922	0.744610	−	0.667500i	$-0.232636\pi$
$857$	37362.0	1.48922	0.744610	+	0.667500i	$0.232636\pi$
$858$	0	0
$859$	29572.2	1.17461	0.587305	−	0.809366i	$-0.300189\pi$
$859$	29572.2	1.17461	0.587305	+	0.809366i	$0.300189\pi$
$860$	0	0
$861$	8322.54	0.329421
$862$	0	0
$863$	11636.5	0.458994	0.229497	−	0.973309i	$-0.426292\pi$
$863$	11636.5	0.458994	0.229497	+	0.973309i	$0.426292\pi$
$864$	0	0
$865$	−38719.2	−1.52196
$866$	0	0
$867$	57974.1	2.27094
$868$	0	0
$869$	23786.7	0.928550
$870$	0	0
$871$	4123.00	0.160393
$872$	0	0
$873$	−4718.65	−0.182935
$874$	0	0
$875$	403.649	0.0155952
$876$	0	0
$877$	−19878.2	−0.765380	−0.382690	−	0.923877i	$-0.625002\pi$
$877$	−19878.2	−0.765380	−0.382690	+	0.923877i	$0.625002\pi$
$878$	0	0
$879$	−642.238	−0.0246441
$880$	0	0
$881$	−36460.3	−1.39430	−0.697150	−	0.716926i	$-0.745549\pi$
$881$	−36460.3	−1.39430	−0.697150	+	0.716926i	$0.745549\pi$
$882$	0	0
$883$	−28119.9	−1.07170	−0.535849	−	0.844314i	$-0.680009\pi$
$883$	−28119.9	−1.07170	−0.535849	+	0.844314i	$0.680009\pi$
$884$	0	0
$885$	−7296.21	−0.277129
$886$	0	0
$887$	41888.4	1.58565	0.792826	−	0.609448i	$-0.208609\pi$
$887$	41888.4	1.58565	0.792826	+	0.609448i	$0.208609\pi$
$888$	0	0
$889$	8867.77	0.334551
$890$	0	0
$891$	34318.4	1.29036
$892$	0	0
$893$	−14989.7	−0.561715
$894$	0	0
$895$	−38544.3	−1.43955
$896$	0	0
$897$	−5347.62	−0.199055
$898$	0	0
$899$	33701.9	1.25030
$900$	0	0
$901$	13300.6	0.491795
$902$	0	0
$903$	16248.4	0.598796
$904$	0	0
$905$	30586.4	1.12346
$906$	0	0
$907$	12575.3	0.460371	0.230185	−	0.973147i	$-0.426067\pi$
$907$	12575.3	0.460371	0.230185	+	0.973147i	$0.426067\pi$
$908$	0	0
$909$	−3144.14	−0.114724
$910$	0	0
$911$	39814.7	1.44799	0.723995	−	0.689805i	$-0.242304\pi$
$911$	39814.7	1.44799	0.723995	+	0.689805i	$0.242304\pi$
$912$	0	0
$913$	20320.9	0.736609
$914$	0	0
$915$	−26488.7	−0.957036
$916$	0	0
$917$	29618.8	1.06663
$918$	0	0
$919$	−8987.93	−0.322616	−0.161308	−	0.986904i	$-0.551571\pi$
$919$	−8987.93	−0.322616	−0.161308	+	0.986904i	$0.551571\pi$
$920$	0	0
$921$	306.131	0.0109526
$922$	0	0
$923$	733.348	0.0261521
$924$	0	0
$925$	23135.2	0.822359
$926$	0	0
$927$	4435.46	0.157152
$928$	0	0
$929$	234.252	0.00827293	0.00413647	−	0.999991i	$-0.498683\pi$
$929$	234.252	0.00827293	0.00413647	+	0.999991i	$0.498683\pi$
$930$	0	0
$931$	1219.45	0.0429279
$932$	0	0
$933$	−2926.55	−0.102691
$934$	0	0
$935$	108380.	3.79079
$936$	0	0
$937$	17624.6	0.614483	0.307241	−	0.951632i	$-0.400594\pi$
$937$	17624.6	0.614483	0.307241	+	0.951632i	$0.400594\pi$
$938$	0	0
$939$	18085.0	0.628521
$940$	0	0
$941$	30532.1	1.05772	0.528861	−	0.848708i	$-0.322619\pi$
$941$	30532.1	1.05772	0.528861	+	0.848708i	$0.322619\pi$
$942$	0	0
$943$	−12172.4	−0.420349
$944$	0	0
$945$	−41460.2	−1.42720
$946$	0	0
$947$	42092.8	1.44438	0.722192	−	0.691692i	$-0.243135\pi$
$947$	42092.8	1.44438	0.722192	+	0.691692i	$0.243135\pi$
$948$	0	0
$949$	1300.59	0.0444877
$950$	0	0
$951$	−1844.82	−0.0629048
$952$	0	0
$953$	−48641.0	−1.65334	−0.826672	−	0.562683i	$-0.809769\pi$
$953$	−48641.0	−1.65334	−0.826672	+	0.562683i	$0.809769\pi$
$954$	0	0
$955$	−64810.6	−2.19604
$956$	0	0
$957$	29026.9	0.980467
$958$	0	0
$959$	7729.54	0.260271
$960$	0	0
$961$	61927.9	2.07874
$962$	0	0
$963$	766.612	0.0256529
$964$	0	0
$965$	18338.3	0.611742
$966$	0	0
$967$	53000.5	1.76254	0.881272	−	0.472609i	$-0.156688\pi$
$967$	53000.5	1.76254	0.881272	+	0.472609i	$0.156688\pi$
$968$	0	0
$969$	−32137.6	−1.06544
$970$	0	0
$971$	−26286.3	−0.868761	−0.434381	−	0.900729i	$-0.643033\pi$
$971$	−26286.3	−0.868761	−0.434381	+	0.900729i	$0.643033\pi$
$972$	0	0
$973$	−2609.68	−0.0859842
$974$	0	0
$975$	−5252.81	−0.172538
$976$	0	0
$977$	−30508.0	−0.999014	−0.499507	−	0.866310i	$-0.666486\pi$
$977$	−30508.0	−0.999014	−0.499507	+	0.866310i	$0.666486\pi$
$978$	0	0
$979$	27399.4	0.894473
$980$	0	0
$981$	2726.34	0.0887311
$982$	0	0
$983$	−37044.3	−1.20196	−0.600982	−	0.799263i	$-0.705224\pi$
$983$	−37044.3	−1.20196	−0.600982	+	0.799263i	$0.705224\pi$
$984$	0	0
$985$	−42153.3	−1.36357
$986$	0	0
$987$	−26122.0	−0.842424
$988$	0	0
$989$	−23764.7	−0.764079
$990$	0	0
$991$	−47803.5	−1.53232	−0.766160	−	0.642650i	$-0.777835\pi$
$991$	−47803.5	−1.53232	−0.766160	+	0.642650i	$0.777835\pi$
$992$	0	0
$993$	28934.1	0.924670
$994$	0	0
$995$	11771.3	0.375051
$996$	0	0
$997$	13478.1	0.428140	0.214070	−	0.976818i	$-0.431328\pi$
$997$	13478.1	0.428140	0.214070	+	0.976818i	$0.431328\pi$
$998$	0	0
$999$	−26796.1	−0.848639

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.4.a.e.1.2	✓	2
3.2	odd	2		1152.4.a.s.1.2		2
4.3	odd	2		128.4.a.g.1.1	yes	2
8.3	odd	2		128.4.a.f.1.2	yes	2
8.5	even	2		128.4.a.h.1.1	yes	2
12.11	even	2		1152.4.a.t.1.2		2
16.3	odd	4		256.4.b.h.129.2		4
16.5	even	4		256.4.b.i.129.2		4
16.11	odd	4		256.4.b.h.129.3		4
16.13	even	4		256.4.b.i.129.3		4
24.5	odd	2		1152.4.a.q.1.1		2
24.11	even	2		1152.4.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.a.e.1.2	✓	2	1.1	even	1	trivial
128.4.a.f.1.2	yes	2	8.3	odd	2
128.4.a.g.1.1	yes	2	4.3	odd	2
128.4.a.h.1.1	yes	2	8.5	even	2
256.4.b.h.129.2		4	16.3	odd	4
256.4.b.h.129.3		4	16.11	odd	4
256.4.b.i.129.2		4	16.5	even	4
256.4.b.i.129.3		4	16.13	even	4
1152.4.a.q.1.1		2	24.5	odd	2
1152.4.a.r.1.1		2	24.11	even	2
1152.4.a.s.1.2		2	3.2	odd	2
1152.4.a.t.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	128.a (trivial)

\(f(q)\)	\(=\)	\(q+4.92820 q^{3} -15.8564 q^{5} -17.8564 q^{7} -2.71281 q^{9} -52.9282 q^{11} -8.43078 q^{13} -78.1436 q^{15} +129.138 q^{17} -50.4974 q^{19} -88.0000 q^{21} +128.708 q^{23} +126.426 q^{25} -146.431 q^{27} -111.282 q^{29} -302.851 q^{31} -260.841 q^{33} +283.138 q^{35} +182.995 q^{37} -41.5486 q^{39} -94.5744 q^{41} -184.641 q^{43} +43.0155 q^{45} +296.841 q^{47} -24.1487 q^{49} +636.420 q^{51} +102.995 q^{53} +839.251 q^{55} -248.862 q^{57} +93.3693 q^{59} +338.974 q^{61} +48.4411 q^{63} +133.682 q^{65} -489.041 q^{67} +634.297 q^{69} -86.9845 q^{71} -154.267 q^{73} +623.051 q^{75} +945.108 q^{77} -449.415 q^{79} -648.395 q^{81} -383.933 q^{83} -2047.67 q^{85} -548.420 q^{87} -517.672 q^{89} +150.543 q^{91} -1492.51 q^{93} +800.708 q^{95} +1739.39 q^{97} +143.584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 4 q^{5} - 8 q^{7} + 50 q^{9} - 92 q^{11} - 100 q^{13} - 184 q^{15} + 92 q^{17} - 4 q^{19} - 176 q^{21} + 8 q^{23} + 142 q^{25} - 376 q^{27} - 84 q^{29} - 384 q^{31} + 88 q^{33} + 400 q^{35}+ \cdots - 1916 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 - 4 * q^5 - 8 * q^7 + 50 * q^9 - 92 * q^11 - 100 * q^13 - 184 * q^15 + 92 * q^17 - 4 * q^19 - 176 * q^21 + 8 * q^23 + 142 * q^25 - 376 * q^27 - 84 * q^29 - 384 * q^31 + 88 * q^33 + 400 * q^35 + 172 * q^37 + 776 * q^39 - 300 * q^41 - 300 * q^43 + 668 * q^45 - 16 * q^47 - 270 * q^49 + 968 * q^51 + 12 * q^53 + 376 * q^55 - 664 * q^57 + 644 * q^59 - 292 * q^61 + 568 * q^63 - 952 * q^65 + 172 * q^67 + 1712 * q^69 + 408 * q^71 + 412 * q^73 + 484 * q^75 + 560 * q^77 - 400 * q^79 - 22 * q^81 - 948 * q^83 - 2488 * q^85 - 792 * q^87 + 572 * q^89 - 752 * q^91 - 768 * q^93 + 1352 * q^95 + 2204 * q^97 - 1916 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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