LMFDB - Embedded newform 2704.2.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2704,2,Mod(1,2704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2704, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2704 = 2^{4} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2704.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:	$21.5915487066$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13968.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{3} - 7x^{2} + 8x + 4$ x^4 - 2x^3 - 7x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.27743$ of defining polynomial
Character	$\chi$	$=$	2704.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.27743 q^{3} -2.61023 q^{5} +1.12182 q^{7} +7.74153 q^{9} +1.12182 q^{11} -8.55485 q^{15} +5.55485 q^{17} -1.67667 q^{19} +3.67667 q^{21} -3.27743 q^{23} +1.81333 q^{25} +15.5400 q^{27} -0.334385 q^{29} -0.129717 q^{31} +3.67667 q^{33} -2.92820 q^{35} +4.53054 q^{37} +5.73205 q^{41} +4.49790 q^{43} -20.2072 q^{45} +10.7985 q^{47} -5.74153 q^{49} +18.2056 q^{51} +4.14771 q^{53} -2.92820 q^{55} -5.49516 q^{57} +0.634552 q^{59} +4.70773 q^{61} +8.68457 q^{63} +14.3612 q^{67} -10.7415 q^{69} -8.60487 q^{71} -9.94462 q^{73} +5.94304 q^{75} +1.25847 q^{77} -13.5970 q^{79} +27.7067 q^{81} +5.75637 q^{83} -14.4995 q^{85} -1.09592 q^{87} -14.3454 q^{89} -0.425137 q^{93} +4.37650 q^{95} -9.00790 q^{97} +8.68457 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 6 q^{9} + 4 q^{11} - 12 q^{15} + 16 q^{19} - 8 q^{21} - 2 q^{23} + 10 q^{25} + 14 q^{27} + 8 q^{29} + 4 q^{31} - 8 q^{33} + 16 q^{35} - 12 q^{37} + 16 q^{41} - 6 q^{43}+ \cdots + 8 q^{99}+O(q^{100})$ 4 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 6 * q^9 + 4 * q^11 - 12 * q^15 + 16 * q^19 - 8 * q^21 - 2 * q^23 + 10 * q^25 + 14 * q^27 + 8 * q^29 + 4 * q^31 - 8 * q^33 + 16 * q^35 - 12 * q^37 + 16 * q^41 - 6 * q^43 - 28 * q^45 + 20 * q^47 + 2 * q^49 + 34 * q^51 + 10 * q^53 + 16 * q^55 - 16 * q^57 + 4 * q^59 + 4 * q^61 + 8 * q^63 + 8 * q^67 - 18 * q^69 + 16 * q^71 - 24 * q^73 + 22 * q^75 + 30 * q^77 - 8 * q^79 + 20 * q^81 + 24 * q^83 - 20 * q^85 - 10 * q^87 - 16 * q^89 + 16 * q^93 - 16 * q^95 - 32 * q^97 + 8 * 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
$3$	3.27743	1.89222	0.946112	−	0.323840i	$-0.104974\pi$
$3$	3.27743	1.89222	0.946112	+	0.323840i	$0.104974\pi$
$4$	0	0
$5$	−2.61023	−1.16733	−0.583666	−	0.811994i	$-0.698382\pi$
$5$	−2.61023	−1.16733	−0.583666	+	0.811994i	$0.698382\pi$
$6$	0	0
$7$	1.12182	0.424007	0.212003	−	0.977269i	$-0.432001\pi$
$7$	1.12182	0.424007	0.212003	+	0.977269i	$0.432001\pi$
$8$	0	0
$9$	7.74153	2.58051
$10$	0	0
$11$	1.12182	0.338240	0.169120	−	0.985595i	$-0.445907\pi$
$11$	1.12182	0.338240	0.169120	+	0.985595i	$0.445907\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−8.55485	−2.20885
$16$	0	0
$17$	5.55485	1.34725	0.673625	−	0.739073i	$-0.264736\pi$
$17$	5.55485	1.34725	0.673625	+	0.739073i	$0.264736\pi$
$18$	0	0
$19$	−1.67667	−0.384655	−0.192327	−	0.981331i	$-0.561604\pi$
$19$	−1.67667	−0.384655	−0.192327	+	0.981331i	$0.561604\pi$
$20$	0	0
$21$	3.67667	0.802315
$22$	0	0
$23$	−3.27743	−0.683391	−0.341695	−	0.939811i	$-0.611001\pi$
$23$	−3.27743	−0.683391	−0.341695	+	0.939811i	$0.611001\pi$
$24$	0	0
$25$	1.81333	0.362665
$26$	0	0
$27$	15.5400	2.99068
$28$	0	0
$29$	−0.334385	−0.0620937	−0.0310468	−	0.999518i	$-0.509884\pi$
$29$	−0.334385	−0.0620937	−0.0310468	+	0.999518i	$0.509884\pi$
$30$	0	0
$31$	−0.129717	−0.0232978	−0.0116489	−	0.999932i	$-0.503708\pi$
$31$	−0.129717	−0.0232978	−0.0116489	+	0.999932i	$0.503708\pi$
$32$	0	0
$33$	3.67667	0.640026
$34$	0	0
$35$	−2.92820	−0.494957
$36$	0	0
$37$	4.53054	0.744816	0.372408	−	0.928069i	$-0.378532\pi$
$37$	4.53054	0.744816	0.372408	+	0.928069i	$0.378532\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.73205	0.895196	0.447598	−	0.894235i	$-0.352280\pi$
$41$	5.73205	0.895196	0.447598	+	0.894235i	$0.352280\pi$
$42$	0	0
$43$	4.49790	0.685923	0.342961	−	0.939349i	$-0.388570\pi$
$43$	4.49790	0.685923	0.342961	+	0.939349i	$0.388570\pi$
$44$	0	0
$45$	−20.2072	−3.01231
$46$	0	0
$47$	10.7985	1.57512	0.787561	−	0.616237i	$-0.211344\pi$
$47$	10.7985	1.57512	0.787561	+	0.616237i	$0.211344\pi$
$48$	0	0
$49$	−5.74153	−0.820218
$50$	0	0
$51$	18.2056	2.54930
$52$	0	0
$53$	4.14771	0.569732	0.284866	−	0.958567i	$-0.408051\pi$
$53$	4.14771	0.569732	0.284866	+	0.958567i	$0.408051\pi$
$54$	0	0
$55$	−2.92820	−0.394839
$56$	0	0
$57$	−5.49516	−0.727852
$58$	0	0
$59$	0.634552	0.0826116	0.0413058	−	0.999147i	$-0.486848\pi$
$59$	0.634552	0.0826116	0.0413058	+	0.999147i	$0.486848\pi$
$60$	0	0
$61$	4.70773	0.602764	0.301382	−	0.953504i	$-0.402552\pi$
$61$	4.70773	0.602764	0.301382	+	0.953504i	$0.402552\pi$
$62$	0	0
$63$	8.68457	1.09415
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.3612	1.75450	0.877252	−	0.480029i	$-0.159374\pi$
$67$	14.3612	1.75450	0.877252	+	0.480029i	$0.159374\pi$
$68$	0	0
$69$	−10.7415	−1.29313
$70$	0	0
$71$	−8.60487	−1.02121	−0.510605	−	0.859815i	$-0.670579\pi$
$71$	−8.60487	−1.02121	−0.510605	+	0.859815i	$0.670579\pi$
$72$	0	0
$73$	−9.94462	−1.16393	−0.581965	−	0.813214i	$-0.697716\pi$
$73$	−9.94462	−1.16393	−0.581965	+	0.813214i	$0.697716\pi$
$74$	0	0
$75$	5.94304	0.686243
$76$	0	0
$77$	1.25847	0.143416
$78$	0	0
$79$	−13.5970	−1.52978	−0.764889	−	0.644162i	$-0.777206\pi$
$79$	−13.5970	−1.52978	−0.764889	+	0.644162i	$0.777206\pi$
$80$	0	0
$81$	27.7067	3.07852
$82$	0	0
$83$	5.75637	0.631843	0.315922	−	0.948785i	$-0.397686\pi$
$83$	5.75637	0.631843	0.315922	+	0.948785i	$0.397686\pi$
$84$	0	0
$85$	−14.4995	−1.57269
$86$	0	0
$87$	−1.09592	−0.117495
$88$	0	0
$89$	−14.3454	−1.52061	−0.760307	−	0.649564i	$-0.774951\pi$
$89$	−14.3454	−1.52061	−0.760307	+	0.649564i	$0.774951\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.425137	−0.0440847
$94$	0	0
$95$	4.37650	0.449020
$96$	0	0
$97$	−9.00790	−0.914614	−0.457307	−	0.889309i	$-0.651186\pi$
$97$	−9.00790	−0.914614	−0.457307	+	0.889309i	$0.651186\pi$
$98$	0	0
$99$	8.68457	0.872832
$100$	0	0
$101$	−2.88924	−0.287490	−0.143745	−	0.989615i	$-0.545915\pi$
$101$	−2.88924	−0.287490	−0.143745	+	0.989615i	$0.545915\pi$
$102$	0	0
$103$	5.22047	0.514388	0.257194	−	0.966360i	$-0.417202\pi$
$103$	5.22047	0.514388	0.257194	+	0.966360i	$0.417202\pi$
$104$	0	0
$105$	−9.59697	−0.936569
$106$	0	0
$107$	−5.83228	−0.563828	−0.281914	−	0.959440i	$-0.590969\pi$
$107$	−5.83228	−0.563828	−0.281914	+	0.959440i	$0.590969\pi$
$108$	0	0
$109$	−2.92820	−0.280471	−0.140236	−	0.990118i	$-0.544786\pi$
$109$	−2.92820	−0.280471	−0.140236	+	0.990118i	$0.544786\pi$
$110$	0	0
$111$	14.8485	1.40936
$112$	0	0
$113$	4.88608	0.459644	0.229822	−	0.973233i	$-0.426186\pi$
$113$	4.88608	0.459644	0.229822	+	0.973233i	$0.426186\pi$
$114$	0	0
$115$	8.55485	0.797744
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.23152	0.571243
$120$	0	0
$121$	−9.74153	−0.885594
$122$	0	0
$123$	18.7864	1.69391
$124$	0	0
$125$	8.31797	0.743982
$126$	0	0
$127$	10.3195	0.915712	0.457856	−	0.889026i	$-0.348618\pi$
$127$	10.3195	0.915712	0.457856	+	0.889026i	$0.348618\pi$
$128$	0	0
$129$	14.7415	1.29792
$130$	0	0
$131$	0.110761	0.00967723	0.00483861	−	0.999988i	$-0.498460\pi$
$131$	0.110761	0.00967723	0.00483861	+	0.999988i	$0.498460\pi$
$132$	0	0
$133$	−1.88092	−0.163096
$134$	0	0
$135$	−40.5631	−3.49111
$136$	0	0
$137$	6.95252	0.593994	0.296997	−	0.954878i	$-0.404015\pi$
$137$	6.95252	0.593994	0.296997	+	0.954878i	$0.404015\pi$
$138$	0	0
$139$	10.4303	0.884687	0.442344	−	0.896846i	$-0.354147\pi$
$139$	10.4303	0.884687	0.442344	+	0.896846i	$0.354147\pi$
$140$	0	0
$141$	35.3913	2.98048
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.872823	0.0724840
$146$	0	0
$147$	−18.8174	−1.55204
$148$	0	0
$149$	9.60233	0.786654	0.393327	−	0.919399i	$-0.371324\pi$
$149$	9.60233	0.786654	0.393327	+	0.919399i	$0.371324\pi$
$150$	0	0
$151$	−13.7267	−1.11706	−0.558531	−	0.829484i	$-0.688635\pi$
$151$	−13.7267	−1.11706	−0.558531	+	0.829484i	$0.688635\pi$
$152$	0	0
$153$	43.0031	3.47659
$154$	0	0
$155$	0.338591	0.0271963
$156$	0	0
$157$	8.96200	0.715245	0.357623	−	0.933866i	$-0.383587\pi$
$157$	8.96200	0.715245	0.357623	+	0.933866i	$0.383587\pi$
$158$	0	0
$159$	13.5938	1.07806
$160$	0	0
$161$	−3.67667	−0.289762
$162$	0	0
$163$	2.19361	0.171817	0.0859085	−	0.996303i	$-0.472621\pi$
$163$	2.19361	0.171817	0.0859085	+	0.996303i	$0.472621\pi$
$164$	0	0
$165$	−9.59697	−0.747123
$166$	0	0
$167$	14.9782	1.15905	0.579525	−	0.814955i	$-0.303238\pi$
$167$	14.9782	1.15905	0.579525	+	0.814955i	$0.303238\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−12.9800	−0.992605
$172$	0	0
$173$	−3.99483	−0.303721	−0.151861	−	0.988402i	$-0.548527\pi$
$173$	−3.99483	−0.303721	−0.151861	+	0.988402i	$0.548527\pi$
$174$	0	0
$175$	2.03422	0.153772
$176$	0	0
$177$	2.07970	0.156320
$178$	0	0
$179$	8.61181	0.643677	0.321839	−	0.946795i	$-0.395699\pi$
$179$	8.61181	0.643677	0.321839	+	0.946795i	$0.395699\pi$
$180$	0	0
$181$	−15.6308	−1.16183	−0.580913	−	0.813966i	$-0.697304\pi$
$181$	−15.6308	−1.16183	−0.580913	+	0.813966i	$0.697304\pi$
$182$	0	0
$183$	15.4293	1.14056
$184$	0	0
$185$	−11.8258	−0.869448
$186$	0	0
$187$	6.23152	0.455694
$188$	0	0
$189$	17.4330	1.26807
$190$	0	0
$191$	−27.0559	−1.95770	−0.978848	−	0.204587i	$-0.934415\pi$
$191$	−27.0559	−1.95770	−0.978848	+	0.204587i	$0.934415\pi$
$192$	0	0
$193$	−23.9515	−1.72406	−0.862032	−	0.506854i	$-0.830808\pi$
$193$	−23.9515	−1.72406	−0.862032	+	0.506854i	$0.830808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.3454	0.737082	0.368541	−	0.929611i	$-0.379857\pi$
$197$	10.3454	0.737082	0.368541	+	0.929611i	$0.379857\pi$
$198$	0	0
$199$	−13.5728	−0.962154	−0.481077	−	0.876678i	$-0.659754\pi$
$199$	−13.5728	−0.962154	−0.481077	+	0.876678i	$0.659754\pi$
$200$	0	0
$201$	47.0679	3.31992
$202$	0	0
$203$	−0.375118	−0.0263281
$204$	0	0
$205$	−14.9620	−1.04499
$206$	0	0
$207$	−25.3723	−1.76350
$208$	0	0
$209$	−1.88092	−0.130106
$210$	0	0
$211$	1.94620	0.133982	0.0669909	−	0.997754i	$-0.478660\pi$
$211$	1.94620	0.133982	0.0669909	+	0.997754i	$0.478660\pi$
$212$	0	0
$213$	−28.2018	−1.93236
$214$	0	0
$215$	−11.7406	−0.800700
$216$	0	0
$217$	−0.145518	−0.00987843
$218$	0	0
$219$	−32.5928	−2.20242
$220$	0	0
$221$	0	0
$222$	0	0
$223$	23.9203	1.60182	0.800911	−	0.598783i	$-0.204349\pi$
$223$	23.9203	1.60182	0.800911	+	0.598783i	$0.204349\pi$
$224$	0	0
$225$	14.0379	0.935861
$226$	0	0
$227$	−17.6767	−1.17324	−0.586621	−	0.809862i	$-0.699542\pi$
$227$	−17.6767	−1.17324	−0.586621	+	0.809862i	$0.699542\pi$
$228$	0	0
$229$	−10.9282	−0.722156	−0.361078	−	0.932536i	$-0.617591\pi$
$229$	−10.9282	−0.722156	−0.361078	+	0.932536i	$0.617591\pi$
$230$	0	0
$231$	4.12455	0.271375
$232$	0	0
$233$	11.5970	0.759743	0.379871	−	0.925039i	$-0.375968\pi$
$233$	11.5970	0.759743	0.379871	+	0.925039i	$0.375968\pi$
$234$	0	0
$235$	−28.1866	−1.83869
$236$	0	0
$237$	−44.5631	−2.89468
$238$	0	0
$239$	18.0221	1.16575	0.582877	−	0.812561i	$-0.301927\pi$
$239$	18.0221	1.16575	0.582877	+	0.812561i	$0.301927\pi$
$240$	0	0
$241$	−0.819649	−0.0527982	−0.0263991	−	0.999651i	$-0.508404\pi$
$241$	−0.819649	−0.0527982	−0.0263991	+	0.999651i	$0.508404\pi$
$242$	0	0
$243$	44.1866	2.83457
$244$	0	0
$245$	14.9867	0.957468
$246$	0	0
$247$	0	0
$248$	0	0
$249$	18.8661	1.19559
$250$	0	0
$251$	−14.3575	−0.906235	−0.453117	−	0.891451i	$-0.649688\pi$
$251$	−14.3575	−0.906235	−0.453117	+	0.891451i	$0.649688\pi$
$252$	0	0
$253$	−3.67667	−0.231150
$254$	0	0
$255$	−47.5210	−2.97588
$256$	0	0
$257$	−20.5970	−1.28480	−0.642402	−	0.766368i	$-0.722062\pi$
$257$	−20.5970	−1.28480	−0.642402	+	0.766368i	$0.722062\pi$
$258$	0	0
$259$	5.08243	0.315807
$260$	0	0
$261$	−2.58865	−0.160233
$262$	0	0
$263$	29.0231	1.78964	0.894820	−	0.446428i	$-0.147304\pi$
$263$	29.0231	1.78964	0.894820	+	0.446428i	$0.147304\pi$
$264$	0	0
$265$	−10.8265	−0.665066
$266$	0	0
$267$	−47.0161	−2.87734
$268$	0	0
$269$	−8.62761	−0.526035	−0.263017	−	0.964791i	$-0.584718\pi$
$269$	−8.62761	−0.526035	−0.263017	+	0.964791i	$0.584718\pi$
$270$	0	0
$271$	−15.9879	−0.971195	−0.485598	−	0.874182i	$-0.661398\pi$
$271$	−15.9879	−0.971195	−0.485598	+	0.874182i	$0.661398\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.03422	0.122668
$276$	0	0
$277$	−28.7456	−1.72716	−0.863579	−	0.504213i	$-0.831783\pi$
$277$	−28.7456	−1.72716	−0.863579	+	0.504213i	$0.831783\pi$
$278$	0	0
$279$	−1.00421	−0.0601203
$280$	0	0
$281$	−5.83386	−0.348019	−0.174009	−	0.984744i	$-0.555672\pi$
$281$	−5.83386	−0.348019	−0.174009	+	0.984744i	$0.555672\pi$
$282$	0	0
$283$	−27.6918	−1.64611	−0.823055	−	0.567962i	$-0.807732\pi$
$283$	−27.6918	−1.64611	−0.823055	+	0.567962i	$0.807732\pi$
$284$	0	0
$285$	14.3437	0.849646
$286$	0	0
$287$	6.43031	0.379569
$288$	0	0
$289$	13.8564	0.815083
$290$	0	0
$291$	−29.5227	−1.73065
$292$	0	0
$293$	−15.9325	−0.930787	−0.465394	−	0.885104i	$-0.654087\pi$
$293$	−15.9325	−0.930787	−0.465394	+	0.885104i	$0.654087\pi$
$294$	0	0
$295$	−1.65633	−0.0964352
$296$	0	0
$297$	17.4330	1.01157
$298$	0	0
$299$	0	0
$300$	0	0
$301$	5.04581	0.290836
$302$	0	0
$303$	−9.46927	−0.543995
$304$	0	0
$305$	−12.2883	−0.703625
$306$	0	0
$307$	20.3955	1.16403	0.582015	−	0.813178i	$-0.302264\pi$
$307$	20.3955	1.16403	0.582015	+	0.813178i	$0.302264\pi$
$308$	0	0
$309$	17.1097	0.973337
$310$	0	0
$311$	12.9989	0.737103	0.368551	−	0.929607i	$-0.379854\pi$
$311$	12.9989	0.737103	0.368551	+	0.929607i	$0.379854\pi$
$312$	0	0
$313$	−9.51274	−0.537692	−0.268846	−	0.963183i	$-0.586642\pi$
$313$	−9.51274	−0.537692	−0.268846	+	0.963183i	$0.586642\pi$
$314$	0	0
$315$	−22.6688	−1.27724
$316$	0	0
$317$	19.1354	1.07475	0.537376	−	0.843343i	$-0.319416\pi$
$317$	19.1354	1.07475	0.537376	+	0.843343i	$0.319416\pi$
$318$	0	0
$319$	−0.375118	−0.0210026
$320$	0	0
$321$	−19.1149	−1.06689
$322$	0	0
$323$	−9.31366	−0.518226
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−9.59697	−0.530714
$328$	0	0
$329$	12.1139	0.667862
$330$	0	0
$331$	−11.3033	−0.621287	−0.310643	−	0.950527i	$-0.600544\pi$
$331$	−11.3033	−0.621287	−0.310643	+	0.950527i	$0.600544\pi$
$332$	0	0
$333$	35.0733	1.92200
$334$	0	0
$335$	−37.4862	−2.04809
$336$	0	0
$337$	5.14035	0.280013	0.140006	−	0.990151i	$-0.455288\pi$
$337$	5.14035	0.280013	0.140006	+	0.990151i	$0.455288\pi$
$338$	0	0
$339$	16.0138	0.869750
$340$	0	0
$341$	−0.145518	−0.00788026
$342$	0	0
$343$	−14.2937	−0.771785
$344$	0	0
$345$	28.0379	1.50951
$346$	0	0
$347$	8.72257	0.468252	0.234126	−	0.972206i	$-0.424777\pi$
$347$	8.72257	0.468252	0.234126	+	0.972206i	$0.424777\pi$
$348$	0	0
$349$	−27.0921	−1.45021	−0.725104	−	0.688639i	$-0.758209\pi$
$349$	−27.0921	−1.45021	−0.725104	+	0.688639i	$0.758209\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.0391	−1.22625	−0.613123	−	0.789988i	$-0.710087\pi$
$353$	−23.0391	−1.22625	−0.613123	+	0.789988i	$0.710087\pi$
$354$	0	0
$355$	22.4607	1.19209
$356$	0	0
$357$	20.4234	1.08092
$358$	0	0
$359$	−1.13392	−0.0598462	−0.0299231	−	0.999552i	$-0.509526\pi$
$359$	−1.13392	−0.0598462	−0.0299231	+	0.999552i	$0.509526\pi$
$360$	0	0
$361$	−16.1888	−0.852041
$362$	0	0
$363$	−31.9272	−1.67574
$364$	0	0
$365$	25.9578	1.35869
$366$	0	0
$367$	−28.7984	−1.50326	−0.751632	−	0.659583i	$-0.770733\pi$
$367$	−28.7984	−1.50326	−0.751632	+	0.659583i	$0.770733\pi$
$368$	0	0
$369$	44.3748	2.31006
$370$	0	0
$371$	4.65297	0.241570
$372$	0	0
$373$	−23.7878	−1.23168	−0.615842	−	0.787870i	$-0.711184\pi$
$373$	−23.7878	−1.23168	−0.615842	+	0.787870i	$0.711184\pi$
$374$	0	0
$375$	27.2615	1.40778
$376$	0	0
$377$	0	0
$378$	0	0
$379$	14.9458	0.767713	0.383856	−	0.923393i	$-0.374596\pi$
$379$	14.9458	0.767713	0.383856	+	0.923393i	$0.374596\pi$
$380$	0	0
$381$	33.8216	1.73273
$382$	0	0
$383$	−4.53728	−0.231844	−0.115922	−	0.993258i	$-0.536982\pi$
$383$	−4.53728	−0.231844	−0.115922	+	0.993258i	$0.536982\pi$
$384$	0	0
$385$	−3.28491	−0.167414
$386$	0	0
$387$	34.8206	1.77003
$388$	0	0
$389$	9.51685	0.482524	0.241262	−	0.970460i	$-0.422439\pi$
$389$	9.51685	0.482524	0.241262	+	0.970460i	$0.422439\pi$
$390$	0	0
$391$	−18.2056	−0.920698
$392$	0	0
$393$	0.363011	0.0183115
$394$	0	0
$395$	35.4913	1.78576
$396$	0	0
$397$	−34.4455	−1.72877	−0.864385	−	0.502831i	$-0.832292\pi$
$397$	−34.4455	−1.72877	−0.864385	+	0.502831i	$0.832292\pi$
$398$	0	0
$399$	−6.16456	−0.308614
$400$	0	0
$401$	14.1572	0.706976	0.353488	−	0.935439i	$-0.384995\pi$
$401$	14.1572	0.706976	0.353488	+	0.935439i	$0.384995\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−72.3209	−3.59366
$406$	0	0
$407$	5.08243	0.251927
$408$	0	0
$409$	0.345877	0.0171025	0.00855127	−	0.999963i	$-0.497278\pi$
$409$	0.345877	0.0171025	0.00855127	+	0.999963i	$0.497278\pi$
$410$	0	0
$411$	22.7864	1.12397
$412$	0	0
$413$	0.711850	0.0350279
$414$	0	0
$415$	−15.0255	−0.737571
$416$	0	0
$417$	34.1846	1.67403
$418$	0	0
$419$	13.9778	0.682860	0.341430	−	0.939907i	$-0.389089\pi$
$419$	13.9778	0.682860	0.341430	+	0.939907i	$0.389089\pi$
$420$	0	0
$421$	22.1609	1.08006	0.540028	−	0.841647i	$-0.318414\pi$
$421$	22.1609	1.08006	0.540028	+	0.841647i	$0.318414\pi$
$422$	0	0
$423$	83.5968	4.06462
$424$	0	0
$425$	10.0728	0.488601
$426$	0	0
$427$	5.28121	0.255576
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.6470	−1.23537	−0.617686	−	0.786425i	$-0.711930\pi$
$431$	−25.6470	−1.23537	−0.617686	+	0.786425i	$0.711930\pi$
$432$	0	0
$433$	−10.1560	−0.488068	−0.244034	−	0.969767i	$-0.578471\pi$
$433$	−10.1560	−0.488068	−0.244034	+	0.969767i	$0.578471\pi$
$434$	0	0
$435$	2.86061	0.137156
$436$	0	0
$437$	5.49516	0.262869
$438$	0	0
$439$	21.3964	1.02120	0.510598	−	0.859820i	$-0.329424\pi$
$439$	21.3964	1.02120	0.510598	+	0.859820i	$0.329424\pi$
$440$	0	0
$441$	−44.4482	−2.11658
$442$	0	0
$443$	−10.6688	−0.506889	−0.253444	−	0.967350i	$-0.581563\pi$
$443$	−10.6688	−0.506889	−0.253444	+	0.967350i	$0.581563\pi$
$444$	0	0
$445$	37.4450	1.77506
$446$	0	0
$447$	31.4710	1.48852
$448$	0	0
$449$	−9.95821	−0.469957	−0.234979	−	0.972001i	$-0.575502\pi$
$449$	−9.95821	−0.469957	−0.234979	+	0.972001i	$0.575502\pi$
$450$	0	0
$451$	6.43031	0.302791
$452$	0	0
$453$	−44.9882	−2.11373
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.7931	−1.25333	−0.626665	−	0.779289i	$-0.715580\pi$
$457$	−26.7931	−1.25333	−0.626665	+	0.779289i	$0.715580\pi$
$458$	0	0
$459$	86.3225	4.02919
$460$	0	0
$461$	−32.0326	−1.49190	−0.745952	−	0.665999i	$-0.768005\pi$
$461$	−32.0326	−1.49190	−0.745952	+	0.665999i	$0.768005\pi$
$462$	0	0
$463$	21.7564	1.01110	0.505552	−	0.862796i	$-0.331289\pi$
$463$	21.7564	1.01110	0.505552	+	0.862796i	$0.331289\pi$
$464$	0	0
$465$	1.10971	0.0514615
$466$	0	0
$467$	33.7322	1.56094	0.780469	−	0.625195i	$-0.214980\pi$
$467$	33.7322	1.56094	0.780469	+	0.625195i	$0.214980\pi$
$468$	0	0
$469$	16.1107	0.743922
$470$	0	0
$471$	29.3723	1.35340
$472$	0	0
$473$	5.04581	0.232007
$474$	0	0
$475$	−3.04035	−0.139501
$476$	0	0
$477$	32.1096	1.47020
$478$	0	0
$479$	−10.5891	−0.483827	−0.241914	−	0.970298i	$-0.577775\pi$
$479$	−10.5891	−0.483827	−0.241914	+	0.970298i	$0.577775\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−12.0500	−0.548295
$484$	0	0
$485$	23.5127	1.06766
$486$	0	0
$487$	−22.0100	−0.997368	−0.498684	−	0.866784i	$-0.666183\pi$
$487$	−22.0100	−0.997368	−0.498684	+	0.866784i	$0.666183\pi$
$488$	0	0
$489$	7.18941	0.325116
$490$	0	0
$491$	16.5145	0.745291	0.372645	−	0.927974i	$-0.378451\pi$
$491$	16.5145	0.745291	0.372645	+	0.927974i	$0.378451\pi$
$492$	0	0
$493$	−1.85746	−0.0836557
$494$	0	0
$495$	−22.6688	−1.01889
$496$	0	0
$497$	−9.65309	−0.433000
$498$	0	0
$499$	−11.1236	−0.497960	−0.248980	−	0.968509i	$-0.580095\pi$
$499$	−11.1236	−0.497960	−0.248980	+	0.968509i	$0.580095\pi$
$500$	0	0
$501$	49.0900	2.19318
$502$	0	0
$503$	29.2722	1.30518	0.652591	−	0.757711i	$-0.273682\pi$
$503$	29.2722	1.30518	0.652591	+	0.757711i	$0.273682\pi$
$504$	0	0
$505$	7.54159	0.335596
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.2406	1.25174	0.625871	−	0.779927i	$-0.284744\pi$
$509$	28.2406	1.25174	0.625871	+	0.779927i	$0.284744\pi$
$510$	0	0
$511$	−11.1560	−0.493514
$512$	0	0
$513$	−26.0555	−1.15038
$514$	0	0
$515$	−13.6267	−0.600462
$516$	0	0
$517$	12.1139	0.532769
$518$	0	0
$519$	−13.0928	−0.574709
$520$	0	0
$521$	−18.6401	−0.816636	−0.408318	−	0.912840i	$-0.633884\pi$
$521$	−18.6401	−0.816636	−0.408318	+	0.912840i	$0.633884\pi$
$522$	0	0
$523$	−35.3122	−1.54409	−0.772047	−	0.635565i	$-0.780767\pi$
$523$	−35.3122	−1.54409	−0.772047	+	0.635565i	$0.780767\pi$
$524$	0	0
$525$	6.66700	0.290972
$526$	0	0
$527$	−0.720558	−0.0313880
$528$	0	0
$529$	−12.2585	−0.532977
$530$	0	0
$531$	4.91240	0.213180
$532$	0	0
$533$	0	0
$534$	0	0
$535$	15.2236	0.658175
$536$	0	0
$537$	28.2246	1.21798
$538$	0	0
$539$	−6.44094	−0.277431
$540$	0	0
$541$	−45.2196	−1.94414	−0.972072	−	0.234682i	$-0.924595\pi$
$541$	−45.2196	−1.94414	−0.972072	+	0.234682i	$0.924595\pi$
$542$	0	0
$543$	−51.2287	−2.19843
$544$	0	0
$545$	7.64330	0.327403
$546$	0	0
$547$	37.8808	1.61967	0.809834	−	0.586660i	$-0.199557\pi$
$547$	37.8808	1.61967	0.809834	+	0.586660i	$0.199557\pi$
$548$	0	0
$549$	36.4451	1.55544
$550$	0	0
$551$	0.560653	0.0238846
$552$	0	0
$553$	−15.2533	−0.648636
$554$	0	0
$555$	−38.7581	−1.64519
$556$	0	0
$557$	20.6413	0.874600	0.437300	−	0.899316i	$-0.355935\pi$
$557$	20.6413	0.874600	0.437300	+	0.899316i	$0.355935\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	20.4234	0.862275
$562$	0	0
$563$	20.4600	0.862286	0.431143	−	0.902284i	$-0.358111\pi$
$563$	20.4600	0.862286	0.431143	+	0.902284i	$0.358111\pi$
$564$	0	0
$565$	−12.7538	−0.536558
$566$	0	0
$567$	31.0818	1.30531
$568$	0	0
$569$	−21.1361	−0.886073	−0.443037	−	0.896504i	$-0.646099\pi$
$569$	−21.1361	−0.886073	−0.443037	+	0.896504i	$0.646099\pi$
$570$	0	0
$571$	−13.9221	−0.582621	−0.291310	−	0.956629i	$-0.594091\pi$
$571$	−13.9221	−0.582621	−0.291310	+	0.956629i	$0.594091\pi$
$572$	0	0
$573$	−88.6738	−3.70440
$574$	0	0
$575$	−5.94304	−0.247842
$576$	0	0
$577$	26.2420	1.09247	0.546234	−	0.837633i	$-0.316061\pi$
$577$	26.2420	1.09247	0.546234	+	0.837633i	$0.316061\pi$
$578$	0	0
$579$	−78.4992	−3.26232
$580$	0	0
$581$	6.45759	0.267906
$582$	0	0
$583$	4.65297	0.192706
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.6725	−1.01834	−0.509171	−	0.860666i	$-0.670048\pi$
$587$	−24.6725	−1.01834	−0.509171	+	0.860666i	$0.670048\pi$
$588$	0	0
$589$	0.217492	0.00896162
$590$	0	0
$591$	33.9064	1.39472
$592$	0	0
$593$	40.3506	1.65700	0.828501	−	0.559988i	$-0.189194\pi$
$593$	40.3506	1.65700	0.828501	+	0.559988i	$0.189194\pi$
$594$	0	0
$595$	−16.2657	−0.666830
$596$	0	0
$597$	−44.4840	−1.82061
$598$	0	0
$599$	4.96104	0.202702	0.101351	−	0.994851i	$-0.467683\pi$
$599$	4.96104	0.202702	0.101351	+	0.994851i	$0.467683\pi$
$600$	0	0
$601$	−15.4769	−0.631317	−0.315658	−	0.948873i	$-0.602225\pi$
$601$	−15.4769	−0.631317	−0.315658	+	0.948873i	$0.602225\pi$
$602$	0	0
$603$	111.178	4.52752
$604$	0	0
$605$	25.4277	1.03378
$606$	0	0
$607$	−22.4335	−0.910546	−0.455273	−	0.890352i	$-0.650458\pi$
$607$	−22.4335	−0.910546	−0.455273	+	0.890352i	$0.650458\pi$
$608$	0	0
$609$	−1.22942	−0.0498187
$610$	0	0
$611$	0	0
$612$	0	0
$613$	16.5855	0.669881	0.334941	−	0.942239i	$-0.391284\pi$
$613$	16.5855	0.669881	0.334941	+	0.942239i	$0.391284\pi$
$614$	0	0
$615$	−49.0369	−1.97736
$616$	0	0
$617$	−33.0368	−1.33001	−0.665005	−	0.746839i	$-0.731571\pi$
$617$	−33.0368	−1.33001	−0.665005	+	0.746839i	$0.731571\pi$
$618$	0	0
$619$	−35.3913	−1.42249	−0.711247	−	0.702942i	$-0.751869\pi$
$619$	−35.3913	−1.42249	−0.711247	+	0.702942i	$0.751869\pi$
$620$	0	0
$621$	−50.9313	−2.04380
$622$	0	0
$623$	−16.0929	−0.644750
$624$	0	0
$625$	−30.7785	−1.23114
$626$	0	0
$627$	−6.16456	−0.246189
$628$	0	0
$629$	25.1665	1.00345
$630$	0	0
$631$	6.45305	0.256892	0.128446	−	0.991717i	$-0.459001\pi$
$631$	6.45305	0.256892	0.128446	+	0.991717i	$0.459001\pi$
$632$	0	0
$633$	6.37852	0.253523
$634$	0	0
$635$	−26.9364	−1.06894
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−66.6149	−2.63524
$640$	0	0
$641$	−21.7743	−0.860032	−0.430016	−	0.902821i	$-0.641492\pi$
$641$	−21.7743	−0.860032	−0.430016	+	0.902821i	$0.641492\pi$
$642$	0	0
$643$	40.2473	1.58720	0.793600	−	0.608440i	$-0.208204\pi$
$643$	40.2473	1.58720	0.793600	+	0.608440i	$0.208204\pi$
$644$	0	0
$645$	−38.4789	−1.51510
$646$	0	0
$647$	−37.6211	−1.47904	−0.739519	−	0.673136i	$-0.764947\pi$
$647$	−37.6211	−1.47904	−0.739519	+	0.673136i	$0.764947\pi$
$648$	0	0
$649$	0.711850	0.0279426
$650$	0	0
$651$	−0.476926	−0.0186922
$652$	0	0
$653$	−30.7119	−1.20185	−0.600924	−	0.799306i	$-0.705200\pi$
$653$	−30.7119	−1.20185	−0.600924	+	0.799306i	$0.705200\pi$
$654$	0	0
$655$	−0.289112	−0.0112965
$656$	0	0
$657$	−76.9866	−3.00353
$658$	0	0
$659$	−13.7698	−0.536394	−0.268197	−	0.963364i	$-0.586428\pi$
$659$	−13.7698	−0.536394	−0.268197	+	0.963364i	$0.586428\pi$
$660$	0	0
$661$	10.2248	0.397698	0.198849	−	0.980030i	$-0.436280\pi$
$661$	10.2248	0.397698	0.198849	+	0.980030i	$0.436280\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.90963	0.190387
$666$	0	0
$667$	1.09592	0.0424343
$668$	0	0
$669$	78.3970	3.03101
$670$	0	0
$671$	5.28121	0.203879
$672$	0	0
$673$	4.26154	0.164270	0.0821351	−	0.996621i	$-0.473826\pi$
$673$	4.26154	0.164270	0.0821351	+	0.996621i	$0.473826\pi$
$674$	0	0
$675$	28.1791	1.08461
$676$	0	0
$677$	−4.47885	−0.172136	−0.0860681	−	0.996289i	$-0.527430\pi$
$677$	−4.47885	−0.172136	−0.0860681	+	0.996289i	$0.527430\pi$
$678$	0	0
$679$	−10.1052	−0.387802
$680$	0	0
$681$	−57.9340	−2.22004
$682$	0	0
$683$	−20.0934	−0.768852	−0.384426	−	0.923156i	$-0.625601\pi$
$683$	−20.0934	−0.768852	−0.384426	+	0.923156i	$0.625601\pi$
$684$	0	0
$685$	−18.1477	−0.693388
$686$	0	0
$687$	−35.8164	−1.36648
$688$	0	0
$689$	0	0
$690$	0	0
$691$	37.7091	1.43452	0.717261	−	0.696804i	$-0.245395\pi$
$691$	37.7091	1.43452	0.717261	+	0.696804i	$0.245395\pi$
$692$	0	0
$693$	9.74249	0.370087
$694$	0	0
$695$	−27.2255	−1.03272
$696$	0	0
$697$	31.8407	1.20605
$698$	0	0
$699$	38.0082	1.43760
$700$	0	0
$701$	17.2849	0.652842	0.326421	−	0.945225i	$-0.394157\pi$
$701$	17.2849	0.652842	0.326421	+	0.945225i	$0.394157\pi$
$702$	0	0
$703$	−7.59622	−0.286497
$704$	0	0
$705$	−92.3795	−3.47921
$706$	0	0
$707$	−3.24119	−0.121898
$708$	0	0
$709$	18.1761	0.682619	0.341310	−	0.939951i	$-0.389130\pi$
$709$	18.1761	0.682619	0.341310	+	0.939951i	$0.389130\pi$
$710$	0	0
$711$	−105.261	−3.94761
$712$	0	0
$713$	0.425137	0.0159215
$714$	0	0
$715$	0	0
$716$	0	0
$717$	59.0662	2.20587
$718$	0	0
$719$	19.9134	0.742643	0.371322	−	0.928504i	$-0.378905\pi$
$719$	19.9134	0.742643	0.371322	+	0.928504i	$0.378905\pi$
$720$	0	0
$721$	5.85641	0.218104
$722$	0	0
$723$	−2.68634	−0.0999061
$724$	0	0
$725$	−0.606348	−0.0225192
$726$	0	0
$727$	38.7446	1.43696	0.718479	−	0.695549i	$-0.244839\pi$
$727$	38.7446	1.43696	0.718479	+	0.695549i	$0.244839\pi$
$728$	0	0
$729$	61.6983	2.28512
$730$	0	0
$731$	24.9852	0.924110
$732$	0	0
$733$	−3.19477	−0.118001	−0.0590007	−	0.998258i	$-0.518791\pi$
$733$	−3.19477	−0.118001	−0.0590007	+	0.998258i	$0.518791\pi$
$734$	0	0
$735$	49.1179	1.81174
$736$	0	0
$737$	16.1107	0.593444
$738$	0	0
$739$	40.6725	1.49616	0.748080	−	0.663608i	$-0.230976\pi$
$739$	40.6725	1.49616	0.748080	+	0.663608i	$0.230976\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.9100	0.510307	0.255154	−	0.966901i	$-0.417874\pi$
$743$	13.9100	0.510307	0.255154	+	0.966901i	$0.417874\pi$
$744$	0	0
$745$	−25.0643	−0.918287
$746$	0	0
$747$	44.5631	1.63048
$748$	0	0
$749$	−6.54275	−0.239067
$750$	0	0
$751$	17.5863	0.641735	0.320867	−	0.947124i	$-0.396026\pi$
$751$	17.5863	0.641735	0.320867	+	0.947124i	$0.396026\pi$
$752$	0	0
$753$	−47.0555	−1.71480
$754$	0	0
$755$	35.8299	1.30398
$756$	0	0
$757$	22.1107	0.803626	0.401813	−	0.915722i	$-0.368380\pi$
$757$	22.1107	0.803626	0.401813	+	0.915722i	$0.368380\pi$
$758$	0	0
$759$	−12.0500	−0.437388
$760$	0	0
$761$	24.3370	0.882217	0.441108	−	0.897454i	$-0.354585\pi$
$761$	24.3370	0.882217	0.441108	+	0.897454i	$0.354585\pi$
$762$	0	0
$763$	−3.28491	−0.118922
$764$	0	0
$765$	−112.248	−4.05834
$766$	0	0
$767$	0	0
$768$	0	0
$769$	7.09214	0.255749	0.127875	−	0.991790i	$-0.459185\pi$
$769$	7.09214	0.255749	0.127875	+	0.991790i	$0.459185\pi$
$770$	0	0
$771$	−67.5051	−2.43114
$772$	0	0
$773$	−0.969989	−0.0348881	−0.0174440	−	0.999848i	$-0.505553\pi$
$773$	−0.969989	−0.0348881	−0.0174440	+	0.999848i	$0.505553\pi$
$774$	0	0
$775$	−0.235219	−0.00844931
$776$	0	0
$777$	16.6573	0.597577
$778$	0	0
$779$	−9.61076	−0.344341
$780$	0	0
$781$	−9.65309	−0.345415
$782$	0	0
$783$	−5.19634	−0.185702
$784$	0	0
$785$	−23.3929	−0.834929
$786$	0	0
$787$	0.0417857	0.00148950	0.000744750	−	1.00000i	$-0.499763\pi$
$787$	0.0417857	0.00148950	0.000744750		1.00000i	$0.499763\pi$
$788$	0	0
$789$	95.1210	3.38640
$790$	0	0
$791$	5.48129	0.194892
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−35.4831	−1.25845
$796$	0	0
$797$	39.3343	1.39329	0.696646	−	0.717415i	$-0.254675\pi$
$797$	39.3343	1.39329	0.696646	+	0.717415i	$0.254675\pi$
$798$	0	0
$799$	59.9840	2.12208
$800$	0	0
$801$	−111.056	−3.92396
$802$	0	0
$803$	−11.1560	−0.393688
$804$	0	0
$805$	9.59697	0.338249
$806$	0	0
$807$	−28.2764	−0.995376
$808$	0	0
$809$	−0.808156	−0.0284133	−0.0142066	−	0.999899i	$-0.504522\pi$
$809$	−0.808156	−0.0284133	−0.0142066	+	0.999899i	$0.504522\pi$
$810$	0	0
$811$	−24.3955	−0.856640	−0.428320	−	0.903627i	$-0.640894\pi$
$811$	−24.3955	−0.856640	−0.428320	+	0.903627i	$0.640894\pi$
$812$	0	0
$813$	−52.3992	−1.83772
$814$	0	0
$815$	−5.72584	−0.200568
$816$	0	0
$817$	−7.54149	−0.263843
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.3674	−0.606125	−0.303063	−	0.952971i	$-0.598009\pi$
$821$	−17.3674	−0.606125	−0.303063	+	0.952971i	$0.598009\pi$
$822$	0	0
$823$	−5.67730	−0.197898	−0.0989491	−	0.995092i	$-0.531548\pi$
$823$	−5.67730	−0.197898	−0.0989491	+	0.995092i	$0.531548\pi$
$824$	0	0
$825$	6.66700	0.232115
$826$	0	0
$827$	−14.5094	−0.504540	−0.252270	−	0.967657i	$-0.581177\pi$
$827$	−14.5094	−0.504540	−0.252270	+	0.967657i	$0.581177\pi$
$828$	0	0
$829$	47.5033	1.64986	0.824928	−	0.565237i	$-0.191215\pi$
$829$	47.5033	1.64986	0.824928	+	0.565237i	$0.191215\pi$
$830$	0	0
$831$	−94.2118	−3.26817
$832$	0	0
$833$	−31.8934	−1.10504
$834$	0	0
$835$	−39.0967	−1.35300
$836$	0	0
$837$	−2.01580	−0.0696763
$838$	0	0
$839$	−2.05759	−0.0710358	−0.0355179	−	0.999369i	$-0.511308\pi$
$839$	−2.05759	−0.0710358	−0.0355179	+	0.999369i	$0.511308\pi$
$840$	0	0
$841$	−28.8882	−0.996144
$842$	0	0
$843$	−19.1200	−0.658529
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.9282	−0.375498
$848$	0	0
$849$	−90.7580	−3.11481
$850$	0	0
$851$	−14.8485	−0.509000
$852$	0	0
$853$	4.10363	0.140506	0.0702528	−	0.997529i	$-0.477619\pi$
$853$	4.10363	0.140506	0.0702528	+	0.997529i	$0.477619\pi$
$854$	0	0
$855$	33.8808	1.15870
$856$	0	0
$857$	−35.9776	−1.22897	−0.614486	−	0.788928i	$-0.710636\pi$
$857$	−35.9776	−1.22897	−0.614486	+	0.788928i	$0.710636\pi$
$858$	0	0
$859$	−12.7711	−0.435745	−0.217872	−	0.975977i	$-0.569912\pi$
$859$	−12.7711	−0.435745	−0.217872	+	0.975977i	$0.569912\pi$
$860$	0	0
$861$	21.0749	0.718229
$862$	0	0
$863$	−34.2815	−1.16696	−0.583479	−	0.812128i	$-0.698309\pi$
$863$	−34.2815	−1.16696	−0.583479	+	0.812128i	$0.698309\pi$
$864$	0	0
$865$	10.4274	0.354544
$866$	0	0
$867$	45.4134	1.54232
$868$	0	0
$869$	−15.2533	−0.517433
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−69.7349	−2.36017
$874$	0	0
$875$	9.33123	0.315453
$876$	0	0
$877$	45.5674	1.53870	0.769351	−	0.638827i	$-0.220580\pi$
$877$	45.5674	1.53870	0.769351	+	0.638827i	$0.220580\pi$
$878$	0	0
$879$	−52.2176	−1.76126
$880$	0	0
$881$	−13.7660	−0.463790	−0.231895	−	0.972741i	$-0.574493\pi$
$881$	−13.7660	−0.463790	−0.231895	+	0.972741i	$0.574493\pi$
$882$	0	0
$883$	−1.44199	−0.0485269	−0.0242634	−	0.999706i	$-0.507724\pi$
$883$	−1.44199	−0.0485269	−0.0242634	+	0.999706i	$0.507724\pi$
$884$	0	0
$885$	−5.42850	−0.182477
$886$	0	0
$887$	−26.5718	−0.892194	−0.446097	−	0.894985i	$-0.647186\pi$
$887$	−26.5718	−0.892194	−0.446097	+	0.894985i	$0.647186\pi$
$888$	0	0
$889$	11.5766	0.388268
$890$	0	0
$891$	31.0818	1.04128
$892$	0	0
$893$	−18.1055	−0.605878
$894$	0	0
$895$	−22.4789	−0.751385
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.0433753	0.00144665
$900$	0	0
$901$	23.0399	0.767571
$902$	0	0
$903$	16.5373	0.550326
$904$	0	0
$905$	40.8000	1.35624
$906$	0	0
$907$	10.0782	0.334642	0.167321	−	0.985902i	$-0.446488\pi$
$907$	10.0782	0.334642	0.167321	+	0.985902i	$0.446488\pi$
$908$	0	0
$909$	−22.3671	−0.741871
$910$	0	0
$911$	44.5252	1.47518	0.737592	−	0.675246i	$-0.235963\pi$
$911$	44.5252	1.47518	0.737592	+	0.675246i	$0.235963\pi$
$912$	0	0
$913$	6.45759	0.213715
$914$	0	0
$915$	−40.2740	−1.33142
$916$	0	0
$917$	0.124253	0.00410321
$918$	0	0
$919$	33.5181	1.10566	0.552830	−	0.833294i	$-0.313548\pi$
$919$	33.5181	1.10566	0.552830	+	0.833294i	$0.313548\pi$
$920$	0	0
$921$	66.8446	2.20261
$922$	0	0
$923$	0	0
$924$	0	0
$925$	8.21534	0.270119
$926$	0	0
$927$	40.4144	1.32738
$928$	0	0
$929$	−17.4449	−0.572347	−0.286174	−	0.958178i	$-0.592383\pi$
$929$	−17.4449	−0.572347	−0.286174	+	0.958178i	$0.592383\pi$
$930$	0	0
$931$	9.62665	0.315501
$932$	0	0
$933$	42.6031	1.39476
$934$	0	0
$935$	−16.2657	−0.531947
$936$	0	0
$937$	17.0052	0.555535	0.277767	−	0.960648i	$-0.410406\pi$
$937$	17.0052	0.555535	0.277767	+	0.960648i	$0.410406\pi$
$938$	0	0
$939$	−31.1773	−1.01743
$940$	0	0
$941$	−15.7406	−0.513128	−0.256564	−	0.966527i	$-0.582590\pi$
$941$	−15.7406	−0.513128	−0.256564	+	0.966527i	$0.582590\pi$
$942$	0	0
$943$	−18.7864	−0.611769
$944$	0	0
$945$	−45.5043	−1.48026
$946$	0	0
$947$	−32.5744	−1.05852	−0.529262	−	0.848458i	$-0.677531\pi$
$947$	−32.5744	−1.05852	−0.529262	+	0.848458i	$0.677531\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	62.7149	2.03367
$952$	0	0
$953$	31.2307	1.01166	0.505831	−	0.862633i	$-0.331186\pi$
$953$	31.2307	1.01166	0.505831	+	0.862633i	$0.331186\pi$
$954$	0	0
$955$	70.6223	2.28528
$956$	0	0
$957$	−1.22942	−0.0397416
$958$	0	0
$959$	7.79945	0.251857
$960$	0	0
$961$	−30.9832	−0.999457
$962$	0	0
$963$	−45.1508	−1.45496
$964$	0	0
$965$	62.5190	2.01256
$966$	0	0
$967$	−37.8998	−1.21877	−0.609387	−	0.792873i	$-0.708585\pi$
$967$	−37.8998	−1.21877	−0.609387	+	0.792873i	$0.708585\pi$
$968$	0	0
$969$	−30.5248	−0.980599
$970$	0	0
$971$	27.3257	0.876923	0.438461	−	0.898750i	$-0.355524\pi$
$971$	27.3257	0.876923	0.438461	+	0.898750i	$0.355524\pi$
$972$	0	0
$973$	11.7009	0.375113
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.8016	0.953437	0.476718	−	0.879056i	$-0.341826\pi$
$977$	29.8016	0.953437	0.476718	+	0.879056i	$0.341826\pi$
$978$	0	0
$979$	−16.0929	−0.514333
$980$	0	0
$981$	−22.6688	−0.723758
$982$	0	0
$983$	16.0704	0.512565	0.256282	−	0.966602i	$-0.417502\pi$
$983$	16.0704	0.512565	0.256282	+	0.966602i	$0.417502\pi$
$984$	0	0
$985$	−27.0040	−0.860420
$986$	0	0
$987$	39.7025	1.26374
$988$	0	0
$989$	−14.7415	−0.468753
$990$	0	0
$991$	−31.9544	−1.01507	−0.507533	−	0.861632i	$-0.669442\pi$
$991$	−31.9544	−1.01507	−0.507533	+	0.861632i	$0.669442\pi$
$992$	0	0
$993$	−37.0458	−1.17561
$994$	0	0
$995$	35.4283	1.12315
$996$	0	0
$997$	−8.79197	−0.278445	−0.139222	−	0.990261i	$-0.544460\pi$
$997$	−8.79197	−0.278445	−0.139222	+	0.990261i	$0.544460\pi$
$998$	0	0
$999$	70.4046	2.22750

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.a.bd.1.4		4
4.3	odd	2		1352.2.a.k.1.1		4
13.2	odd	12		208.2.w.c.17.1		8
13.5	odd	4		2704.2.f.q.337.8		8
13.7	odd	12		208.2.w.c.49.1		8
13.8	odd	4		2704.2.f.q.337.7		8
13.12	even	2		2704.2.a.be.1.4		4
39.2	even	12		1872.2.by.n.433.2		8
39.20	even	12		1872.2.by.n.1297.3		8
52.3	odd	6		1352.2.i.k.529.4		8
52.7	even	12		104.2.o.a.49.4	yes	8
52.11	even	12		1352.2.o.f.1161.4		8
52.15	even	12		104.2.o.a.17.4	✓	8
52.19	even	12		1352.2.o.f.361.4		8
52.23	odd	6		1352.2.i.l.529.4		8
52.31	even	4		1352.2.f.f.337.2		8
52.35	odd	6		1352.2.i.k.1329.4		8
52.43	odd	6		1352.2.i.l.1329.4		8
52.47	even	4		1352.2.f.f.337.1		8
52.51	odd	2		1352.2.a.l.1.1		4
104.59	even	12		832.2.w.g.257.1		8
104.67	even	12		832.2.w.g.641.1		8
104.85	odd	12		832.2.w.i.257.4		8
104.93	odd	12		832.2.w.i.641.4		8
156.59	odd	12		936.2.bi.b.361.3		8
156.119	odd	12		936.2.bi.b.433.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.o.a.17.4	✓	8	52.15	even	12
104.2.o.a.49.4	yes	8	52.7	even	12
208.2.w.c.17.1		8	13.2	odd	12
208.2.w.c.49.1		8	13.7	odd	12
832.2.w.g.257.1		8	104.59	even	12
832.2.w.g.641.1		8	104.67	even	12
832.2.w.i.257.4		8	104.85	odd	12
832.2.w.i.641.4		8	104.93	odd	12
936.2.bi.b.361.3		8	156.59	odd	12
936.2.bi.b.433.2		8	156.119	odd	12
1352.2.a.k.1.1		4	4.3	odd	2
1352.2.a.l.1.1		4	52.51	odd	2
1352.2.f.f.337.1		8	52.47	even	4
1352.2.f.f.337.2		8	52.31	even	4
1352.2.i.k.529.4		8	52.3	odd	6
1352.2.i.k.1329.4		8	52.35	odd	6
1352.2.i.l.529.4		8	52.23	odd	6
1352.2.i.l.1329.4		8	52.43	odd	6
1352.2.o.f.361.4		8	52.19	even	12
1352.2.o.f.1161.4		8	52.11	even	12
1872.2.by.n.433.2		8	39.2	even	12
1872.2.by.n.1297.3		8	39.20	even	12
2704.2.a.bd.1.4		4	1.1	even	1	trivial
2704.2.a.be.1.4		4	13.12	even	2
2704.2.f.q.337.7		8	13.8	odd	4
2704.2.f.q.337.8		8	13.5	odd	4

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2704.2.a.bd.1.4

Level	$2704$
Weight	$2$
Character	2704.1
Self dual	yes
Analytic conductor	$21.592$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$