LMFDB - Embedded newform 1575.4.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.14360.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 17x - 14 $ x^3 - 17*x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.62456$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.62456 q^{2} +13.3866 q^{4} -7.00000 q^{7} -24.9107 q^{8} +30.1117 q^{11} -88.9295 q^{13} +32.3720 q^{14} +8.10818 q^{16} -4.73699 q^{17} +124.818 q^{19} -139.253 q^{22} +20.2680 q^{23} +411.260 q^{26} -93.7062 q^{28} -134.088 q^{29} -2.03767 q^{31} +161.788 q^{32} +21.9065 q^{34} +141.137 q^{37} -577.228 q^{38} -95.2784 q^{41} +298.646 q^{43} +403.093 q^{44} -93.7305 q^{46} -129.054 q^{47} +49.0000 q^{49} -1190.46 q^{52} +388.429 q^{53} +174.375 q^{56} +620.098 q^{58} -838.501 q^{59} +389.422 q^{61} +9.42333 q^{62} -813.067 q^{64} -697.794 q^{67} -63.4122 q^{68} +523.450 q^{71} -66.4684 q^{73} -652.699 q^{74} +1670.89 q^{76} -210.782 q^{77} -526.982 q^{79} +440.621 q^{82} +70.0265 q^{83} -1381.11 q^{86} -750.101 q^{88} +9.27925 q^{89} +622.506 q^{91} +271.319 q^{92} +596.817 q^{94} +4.19493 q^{97} -226.604 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 13 q^{4} - 21 q^{7} - 15 q^{8} + 74 q^{11} - 44 q^{13} + 21 q^{14} - 79 q^{16} - 52 q^{17} + 168 q^{19} - 184 q^{22} - 124 q^{23} + 446 q^{26} - 91 q^{28} - 332 q^{29} + 320 q^{31} - 183 q^{32}+ \cdots - 147 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 13 * q^4 - 21 * q^7 - 15 * q^8 + 74 * q^11 - 44 * q^13 + 21 * q^14 - 79 * q^16 - 52 * q^17 + 168 * q^19 - 184 * q^22 - 124 * q^23 + 446 * q^26 - 91 * q^28 - 332 * q^29 + 320 * q^31 - 183 * q^32 + 582 * q^34 + 54 * q^37 - 460 * q^38 - 362 * q^41 + 16 * q^43 + 264 * q^44 - 336 * q^46 - 730 * q^47 + 147 * q^49 - 1202 * q^52 + 110 * q^53 + 105 * q^56 - 450 * q^58 + 180 * q^59 + 1222 * q^61 + 464 * q^62 - 391 * q^64 - 204 * q^67 + 918 * q^68 + 136 * q^71 - 310 * q^73 - 502 * q^74 + 1796 * q^76 - 518 * q^77 - 1034 * q^79 + 6 * q^82 - 1660 * q^83 - 764 * q^86 + 20 * q^88 - 242 * q^89 + 308 * q^91 + 96 * q^92 - 1108 * q^94 - 100 * q^97 - 147 * q^98

Coefficient data

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

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

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

$2$	−4.62456	−1.63503	−0.817515	−	0.575907i	$-0.804649\pi$
$2$	−4.62456	−1.63503	−0.817515	+	0.575907i	$0.804649\pi$
$3$	0	0
$3$	0	0
$4$	13.3866	1.67332
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−24.9107	−1.10091
$9$	0	0
$10$	0	0
$11$	30.1117	0.825364	0.412682	−	0.910875i	$-0.364592\pi$
$11$	30.1117	0.825364	0.412682	+	0.910875i	$0.364592\pi$
$12$	0	0
$13$	−88.9295	−1.89728	−0.948639	−	0.316362i	$-0.897539\pi$
$13$	−88.9295	−1.89728	−0.948639	+	0.316362i	$0.897539\pi$
$14$	32.3720	0.617983
$15$	0	0
$16$	8.10818	0.126690
$17$	−4.73699	−0.0675817	−0.0337909	−	0.999429i	$-0.510758\pi$
$17$	−4.73699	−0.0675817	−0.0337909	+	0.999429i	$0.510758\pi$
$18$	0	0
$19$	124.818	1.50711	0.753557	−	0.657382i	$-0.228336\pi$
$19$	124.818	1.50711	0.753557	+	0.657382i	$0.228336\pi$
$20$	0	0
$21$	0	0
$22$	−139.253	−1.34950
$23$	20.2680	0.183746	0.0918731	−	0.995771i	$-0.470715\pi$
$23$	20.2680	0.183746	0.0918731	+	0.995771i	$0.470715\pi$
$24$	0	0
$25$	0	0
$26$	411.260	3.10211
$27$	0	0
$28$	−93.7062	−0.632457
$29$	−134.088	−0.858603	−0.429301	−	0.903161i	$-0.641240\pi$
$29$	−134.088	−0.858603	−0.429301	+	0.903161i	$0.641240\pi$
$30$	0	0
$31$	−2.03767	−0.0118057	−0.00590284	−	0.999983i	$-0.501879\pi$
$31$	−2.03767	−0.0118057	−0.00590284	+	0.999983i	$0.501879\pi$
$32$	161.788	0.893764
$33$	0	0
$34$	21.9065	0.110498
$35$	0	0
$36$	0	0
$37$	141.137	0.627104	0.313552	−	0.949571i	$-0.398481\pi$
$37$	141.137	0.627104	0.313552	+	0.949571i	$0.398481\pi$
$38$	−577.228	−2.46418
$39$	0	0
$40$	0	0
$41$	−95.2784	−0.362927	−0.181463	−	0.983398i	$-0.558083\pi$
$41$	−95.2784	−0.362927	−0.181463	+	0.983398i	$0.558083\pi$
$42$	0	0
$43$	298.646	1.05914	0.529571	−	0.848266i	$-0.322353\pi$
$43$	298.646	1.05914	0.529571	+	0.848266i	$0.322353\pi$
$44$	403.093	1.38110
$45$	0	0
$46$	−93.7305	−0.300431
$47$	−129.054	−0.400519	−0.200260	−	0.979743i	$-0.564179\pi$
$47$	−129.054	−0.400519	−0.200260	+	0.979743i	$0.564179\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	−1190.46	−3.17476
$53$	388.429	1.00669	0.503347	−	0.864084i	$-0.332102\pi$
$53$	388.429	1.00669	0.503347	+	0.864084i	$0.332102\pi$
$54$	0	0
$55$	0	0
$56$	174.375	0.416103
$57$	0	0
$58$	620.098	1.40384
$59$	−838.501	−1.85023	−0.925114	−	0.379688i	$-0.876031\pi$
$59$	−838.501	−1.85023	−0.925114	+	0.379688i	$0.876031\pi$
$60$	0	0
$61$	389.422	0.817384	0.408692	−	0.912672i	$-0.365985\pi$
$61$	389.422	0.817384	0.408692	+	0.912672i	$0.365985\pi$
$62$	9.42333	0.0193027
$63$	0	0
$64$	−813.067	−1.58802
$65$	0	0
$66$	0	0
$67$	−697.794	−1.27237	−0.636187	−	0.771534i	$-0.719490\pi$
$67$	−697.794	−1.27237	−0.636187	+	0.771534i	$0.719490\pi$
$68$	−63.4122	−0.113086
$69$	0	0
$70$	0	0
$71$	523.450	0.874959	0.437479	−	0.899228i	$-0.355871\pi$
$71$	523.450	0.874959	0.437479	+	0.899228i	$0.355871\pi$
$72$	0	0
$73$	−66.4684	−0.106569	−0.0532845	−	0.998579i	$-0.516969\pi$
$73$	−66.4684	−0.106569	−0.0532845	+	0.998579i	$0.516969\pi$
$74$	−652.699	−1.02533
$75$	0	0
$76$	1670.89	2.52189
$77$	−210.782	−0.311958
$78$	0	0
$79$	−526.982	−0.750508	−0.375254	−	0.926922i	$-0.622444\pi$
$79$	−526.982	−0.750508	−0.375254	+	0.926922i	$0.622444\pi$
$80$	0	0
$81$	0	0
$82$	440.621	0.593396
$83$	70.0265	0.0926074	0.0463037	−	0.998927i	$-0.485256\pi$
$83$	70.0265	0.0926074	0.0463037	+	0.998927i	$0.485256\pi$
$84$	0	0
$85$	0	0
$86$	−1381.11	−1.73173
$87$	0	0
$88$	−750.101	−0.908649
$89$	9.27925	0.0110517	0.00552584	−	0.999985i	$-0.498241\pi$
$89$	9.27925	0.0110517	0.00552584	+	0.999985i	$0.498241\pi$
$90$	0	0
$91$	622.506	0.717103
$92$	271.319	0.307467
$93$	0	0
$94$	596.817	0.654861
$95$	0	0
$96$	0	0
$97$	4.19493	0.00439104	0.00219552	−	0.999998i	$-0.499301\pi$
$97$	4.19493	0.00439104	0.00219552	+	0.999998i	$0.499301\pi$
$98$	−226.604	−0.233576
$99$	0	0
$100$	0	0
$101$	865.844	0.853016	0.426508	−	0.904484i	$-0.359744\pi$
$101$	865.844	0.853016	0.426508	+	0.904484i	$0.359744\pi$
$102$	0	0
$103$	1166.12	1.11554	0.557771	−	0.829995i	$-0.311657\pi$
$103$	1166.12	1.11554	0.557771	+	0.829995i	$0.311657\pi$
$104$	2215.29	2.08872
$105$	0	0
$106$	−1796.31	−1.64598
$107$	56.9652	0.0514676	0.0257338	−	0.999669i	$-0.491808\pi$
$107$	56.9652	0.0514676	0.0257338	+	0.999669i	$0.491808\pi$
$108$	0	0
$109$	−1358.89	−1.19411	−0.597055	−	0.802200i	$-0.703663\pi$
$109$	−1358.89	−1.19411	−0.597055	+	0.802200i	$0.703663\pi$
$110$	0	0
$111$	0	0
$112$	−56.7572	−0.0478844
$113$	436.038	0.363000	0.181500	−	0.983391i	$-0.441905\pi$
$113$	436.038	0.363000	0.181500	+	0.983391i	$0.441905\pi$
$114$	0	0
$115$	0	0
$116$	−1794.98	−1.43672
$117$	0	0
$118$	3877.70	3.02518
$119$	33.1590	0.0255435
$120$	0	0
$121$	−424.288	−0.318774
$122$	−1800.91	−1.33645
$123$	0	0
$124$	−27.2775	−0.0197547
$125$	0	0
$126$	0	0
$127$	−1186.69	−0.829144	−0.414572	−	0.910017i	$-0.636069\pi$
$127$	−1186.69	−0.829144	−0.414572	+	0.910017i	$0.636069\pi$
$128$	2465.77	1.70270
$129$	0	0
$130$	0	0
$131$	−1034.56	−0.689997	−0.344999	−	0.938603i	$-0.612121\pi$
$131$	−1034.56	−0.689997	−0.344999	+	0.938603i	$0.612121\pi$
$132$	0	0
$133$	−873.725	−0.569636
$134$	3226.99	2.08037
$135$	0	0
$136$	118.002	0.0744011
$137$	646.219	0.402994	0.201497	−	0.979489i	$-0.435419\pi$
$137$	646.219	0.402994	0.201497	+	0.979489i	$0.435419\pi$
$138$	0	0
$139$	506.484	0.309061	0.154530	−	0.987988i	$-0.450614\pi$
$139$	506.484	0.309061	0.154530	+	0.987988i	$0.450614\pi$
$140$	0	0
$141$	0	0
$142$	−2420.73	−1.43058
$143$	−2677.81	−1.56594
$144$	0	0
$145$	0	0
$146$	307.387	0.174244
$147$	0	0
$148$	1889.35	1.04935
$149$	1828.12	1.00513	0.502567	−	0.864538i	$-0.332389\pi$
$149$	1828.12	1.00513	0.502567	+	0.864538i	$0.332389\pi$
$150$	0	0
$151$	2975.17	1.60342	0.801708	−	0.597716i	$-0.203925\pi$
$151$	2975.17	1.60342	0.801708	+	0.597716i	$0.203925\pi$
$152$	−3109.29	−1.65919
$153$	0	0
$154$	974.773	0.510061
$155$	0	0
$156$	0	0
$157$	2131.74	1.08364	0.541820	−	0.840495i	$-0.317736\pi$
$157$	2131.74	1.08364	0.541820	+	0.840495i	$0.317736\pi$
$158$	2437.06	1.22710
$159$	0	0
$160$	0	0
$161$	−141.876	−0.0694495
$162$	0	0
$163$	593.939	0.285404	0.142702	−	0.989766i	$-0.454421\pi$
$163$	593.939	0.285404	0.142702	+	0.989766i	$0.454421\pi$
$164$	−1275.45	−0.607294
$165$	0	0
$166$	−323.842	−0.151416
$167$	−2936.30	−1.36059	−0.680293	−	0.732941i	$-0.738147\pi$
$167$	−2936.30	−1.36059	−0.680293	+	0.732941i	$0.738147\pi$
$168$	0	0
$169$	5711.45	2.59966
$170$	0	0
$171$	0	0
$172$	3997.85	1.77229
$173$	2347.31	1.03158	0.515788	−	0.856716i	$-0.327499\pi$
$173$	2347.31	1.03158	0.515788	+	0.856716i	$0.327499\pi$
$174$	0	0
$175$	0	0
$176$	244.151	0.104566
$177$	0	0
$178$	−42.9125	−0.0180698
$179$	−3036.56	−1.26795	−0.633975	−	0.773354i	$-0.718578\pi$
$179$	−3036.56	−1.26795	−0.633975	+	0.773354i	$0.718578\pi$
$180$	0	0
$181$	−899.776	−0.369502	−0.184751	−	0.982785i	$-0.559148\pi$
$181$	−899.776	−0.369502	−0.184751	+	0.982785i	$0.559148\pi$
$182$	−2878.82	−1.17249
$183$	0	0
$184$	−504.888	−0.202287
$185$	0	0
$186$	0	0
$187$	−142.639	−0.0557796
$188$	−1727.59	−0.670199
$189$	0	0
$190$	0	0
$191$	−416.168	−0.157659	−0.0788294	−	0.996888i	$-0.525118\pi$
$191$	−416.168	−0.157659	−0.0788294	+	0.996888i	$0.525118\pi$
$192$	0	0
$193$	5181.05	1.93233	0.966166	−	0.257922i	$-0.0830376\pi$
$193$	5181.05	1.93233	0.966166	+	0.257922i	$0.0830376\pi$
$194$	−19.3997	−0.00717948
$195$	0	0
$196$	655.943	0.239046
$197$	1452.34	0.525255	0.262627	−	0.964897i	$-0.415411\pi$
$197$	1452.34	0.525255	0.262627	+	0.964897i	$0.415411\pi$
$198$	0	0
$199$	−1277.23	−0.454978	−0.227489	−	0.973781i	$-0.573052\pi$
$199$	−1277.23	−0.454978	−0.227489	+	0.973781i	$0.573052\pi$
$200$	0	0
$201$	0	0
$202$	−4004.15	−1.39471
$203$	938.615	0.324521
$204$	0	0
$205$	0	0
$206$	−5392.78	−1.82395
$207$	0	0
$208$	−721.056	−0.240367
$209$	3758.47	1.24392
$210$	0	0
$211$	−3259.09	−1.06334	−0.531670	−	0.846951i	$-0.678436\pi$
$211$	−3259.09	−1.06334	−0.531670	+	0.846951i	$0.678436\pi$
$212$	5199.74	1.68453
$213$	0	0
$214$	−263.439	−0.0841511
$215$	0	0
$216$	0	0
$217$	14.2637	0.00446213
$218$	6284.27	1.95241
$219$	0	0
$220$	0	0
$221$	421.258	0.128221
$222$	0	0
$223$	−4373.35	−1.31328	−0.656639	−	0.754205i	$-0.728023\pi$
$223$	−4373.35	−1.31328	−0.656639	+	0.754205i	$0.728023\pi$
$224$	−1132.52	−0.337811
$225$	0	0
$226$	−2016.48	−0.593516
$227$	−61.1145	−0.0178692	−0.00893461	−	0.999960i	$-0.502844\pi$
$227$	−61.1145	−0.0178692	−0.00893461	+	0.999960i	$0.502844\pi$
$228$	0	0
$229$	3019.41	0.871302	0.435651	−	0.900116i	$-0.356518\pi$
$229$	3019.41	0.871302	0.435651	+	0.900116i	$0.356518\pi$
$230$	0	0
$231$	0	0
$232$	3340.22	0.945241
$233$	−3531.17	−0.992851	−0.496426	−	0.868079i	$-0.665354\pi$
$233$	−3531.17	−0.992851	−0.496426	+	0.868079i	$0.665354\pi$
$234$	0	0
$235$	0	0
$236$	−11224.7	−3.09603
$237$	0	0
$238$	−153.346	−0.0417644
$239$	−2282.62	−0.617785	−0.308893	−	0.951097i	$-0.599958\pi$
$239$	−2282.62	−0.617785	−0.308893	+	0.951097i	$0.599958\pi$
$240$	0	0
$241$	−2215.68	−0.592217	−0.296109	−	0.955154i	$-0.595689\pi$
$241$	−2215.68	−0.592217	−0.296109	+	0.955154i	$0.595689\pi$
$242$	1962.15	0.521205
$243$	0	0
$244$	5213.04	1.36775
$245$	0	0
$246$	0	0
$247$	−11100.0	−2.85941
$248$	50.7597	0.0129970
$249$	0	0
$250$	0	0
$251$	3082.55	0.775174	0.387587	−	0.921833i	$-0.373309\pi$
$251$	3082.55	0.775174	0.387587	+	0.921833i	$0.373309\pi$
$252$	0	0
$253$	610.302	0.151658
$254$	5487.90	1.35568
$255$	0	0
$256$	−4898.58	−1.19594
$257$	−6032.40	−1.46417	−0.732083	−	0.681215i	$-0.761452\pi$
$257$	−6032.40	−1.46417	−0.732083	+	0.681215i	$0.761452\pi$
$258$	0	0
$259$	−987.962	−0.237023
$260$	0	0
$261$	0	0
$262$	4784.37	1.12817
$263$	5923.81	1.38889	0.694445	−	0.719546i	$-0.255650\pi$
$263$	5923.81	1.38889	0.694445	+	0.719546i	$0.255650\pi$
$264$	0	0
$265$	0	0
$266$	4040.60	0.931372
$267$	0	0
$268$	−9341.09	−2.12910
$269$	−3252.80	−0.737273	−0.368637	−	0.929574i	$-0.620175\pi$
$269$	−3252.80	−0.737273	−0.368637	+	0.929574i	$0.620175\pi$
$270$	0	0
$271$	−6246.26	−1.40012	−0.700061	−	0.714083i	$-0.746844\pi$
$271$	−6246.26	−1.40012	−0.700061	+	0.714083i	$0.746844\pi$
$272$	−38.4084	−0.00856195
$273$	0	0
$274$	−2988.48	−0.658907
$275$	0	0
$276$	0	0
$277$	1572.17	0.341020	0.170510	−	0.985356i	$-0.445459\pi$
$277$	1572.17	0.341020	0.170510	+	0.985356i	$0.445459\pi$
$278$	−2342.27	−0.505324
$279$	0	0
$280$	0	0
$281$	7846.03	1.66567	0.832837	−	0.553518i	$-0.186715\pi$
$281$	7846.03	1.66567	0.832837	+	0.553518i	$0.186715\pi$
$282$	0	0
$283$	−6265.58	−1.31608	−0.658039	−	0.752984i	$-0.728614\pi$
$283$	−6265.58	−1.31608	−0.658039	+	0.752984i	$0.728614\pi$
$284$	7007.21	1.46409
$285$	0	0
$286$	12383.7	2.56037
$287$	666.949	0.137173
$288$	0	0
$289$	−4890.56	−0.995433
$290$	0	0
$291$	0	0
$292$	−889.785	−0.178325
$293$	−7264.99	−1.44855	−0.724276	−	0.689511i	$-0.757826\pi$
$293$	−7264.99	−1.44855	−0.724276	+	0.689511i	$0.757826\pi$
$294$	0	0
$295$	0	0
$296$	−3515.83	−0.690382
$297$	0	0
$298$	−8454.24	−1.64343
$299$	−1802.42	−0.348617
$300$	0	0
$301$	−2090.52	−0.400318
$302$	−13758.9	−2.62163
$303$	0	0
$304$	1012.05	0.190937
$305$	0	0
$306$	0	0
$307$	−1328.32	−0.246943	−0.123471	−	0.992348i	$-0.539403\pi$
$307$	−1328.32	−0.246943	−0.123471	+	0.992348i	$0.539403\pi$
$308$	−2821.65	−0.522008
$309$	0	0
$310$	0	0
$311$	−4868.68	−0.887709	−0.443855	−	0.896099i	$-0.646389\pi$
$311$	−4868.68	−0.887709	−0.443855	+	0.896099i	$0.646389\pi$
$312$	0	0
$313$	−7733.39	−1.39654	−0.698270	−	0.715835i	$-0.746046\pi$
$313$	−7733.39	−1.39654	−0.698270	+	0.715835i	$0.746046\pi$
$314$	−9858.37	−1.77178
$315$	0	0
$316$	−7054.49	−1.25584
$317$	−8175.03	−1.44844	−0.724220	−	0.689569i	$-0.757800\pi$
$317$	−8175.03	−1.44844	−0.724220	+	0.689569i	$0.757800\pi$
$318$	0	0
$319$	−4037.61	−0.708660
$320$	0	0
$321$	0	0
$322$	656.114	0.113552
$323$	−591.261	−0.101853
$324$	0	0
$325$	0	0
$326$	−2746.71	−0.466644
$327$	0	0
$328$	2373.45	0.399548
$329$	903.375	0.151382
$330$	0	0
$331$	−2040.76	−0.338884	−0.169442	−	0.985540i	$-0.554197\pi$
$331$	−2040.76	−0.338884	−0.169442	+	0.985540i	$0.554197\pi$
$332$	937.417	0.154962
$333$	0	0
$334$	13579.1	2.22460
$335$	0	0
$336$	0	0
$337$	−7349.73	−1.18803	−0.594013	−	0.804455i	$-0.702457\pi$
$337$	−7349.73	−1.18803	−0.594013	+	0.804455i	$0.702457\pi$
$338$	−26413.0	−4.25052
$339$	0	0
$340$	0	0
$341$	−61.3576	−0.00974399
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−7439.47	−1.16602
$345$	0	0
$346$	−10855.3	−1.68666
$347$	−12069.9	−1.86728	−0.933642	−	0.358207i	$-0.883388\pi$
$347$	−12069.9	−1.86728	−0.933642	+	0.358207i	$0.883388\pi$
$348$	0	0
$349$	−4484.96	−0.687892	−0.343946	−	0.938989i	$-0.611764\pi$
$349$	−4484.96	−0.687892	−0.343946	+	0.938989i	$0.611764\pi$
$350$	0	0
$351$	0	0
$352$	4871.72	0.737681
$353$	−12762.5	−1.92430	−0.962151	−	0.272517i	$-0.912144\pi$
$353$	−12762.5	−1.92430	−0.962151	+	0.272517i	$0.912144\pi$
$354$	0	0
$355$	0	0
$356$	124.218	0.0184930
$357$	0	0
$358$	14042.8	2.07314
$359$	2419.42	0.355689	0.177844	−	0.984059i	$-0.443088\pi$
$359$	2419.42	0.355689	0.177844	+	0.984059i	$0.443088\pi$
$360$	0	0
$361$	8720.49	1.27139
$362$	4161.07	0.604147
$363$	0	0
$364$	8333.24	1.19995
$365$	0	0
$366$	0	0
$367$	7129.74	1.01409	0.507043	−	0.861921i	$-0.330739\pi$
$367$	7129.74	1.01409	0.507043	+	0.861921i	$0.330739\pi$
$368$	164.336	0.0232789
$369$	0	0
$370$	0	0
$371$	−2719.00	−0.380495
$372$	0	0
$373$	−11596.9	−1.60983	−0.804914	−	0.593391i	$-0.797789\pi$
$373$	−11596.9	−1.60983	−0.804914	+	0.593391i	$0.797789\pi$
$374$	659.642	0.0912013
$375$	0	0
$376$	3214.81	0.440934
$377$	11924.4	1.62901
$378$	0	0
$379$	−12770.8	−1.73085	−0.865424	−	0.501040i	$-0.832951\pi$
$379$	−12770.8	−1.73085	−0.865424	+	0.501040i	$0.832951\pi$
$380$	0	0
$381$	0	0
$382$	1924.59	0.257777
$383$	7470.10	0.996617	0.498308	−	0.867000i	$-0.333955\pi$
$383$	7470.10	0.996617	0.498308	+	0.867000i	$0.333955\pi$
$384$	0	0
$385$	0	0
$386$	−23960.1	−3.15942
$387$	0	0
$388$	56.1558	0.00734763
$389$	−8749.77	−1.14044	−0.570220	−	0.821492i	$-0.693142\pi$
$389$	−8749.77	−1.14044	−0.570220	+	0.821492i	$0.693142\pi$
$390$	0	0
$391$	−96.0092	−0.0124179
$392$	−1220.62	−0.157272
$393$	0	0
$394$	−6716.46	−0.858808
$395$	0	0
$396$	0	0
$397$	−5375.25	−0.679537	−0.339769	−	0.940509i	$-0.610349\pi$
$397$	−5375.25	−0.679537	−0.339769	+	0.940509i	$0.610349\pi$
$398$	5906.65	0.743903
$399$	0	0
$400$	0	0
$401$	−7361.33	−0.916727	−0.458363	−	0.888765i	$-0.651564\pi$
$401$	−7361.33	−0.916727	−0.458363	+	0.888765i	$0.651564\pi$
$402$	0	0
$403$	181.209	0.0223987
$404$	11590.7	1.42737
$405$	0	0
$406$	−4340.68	−0.530602
$407$	4249.88	0.517589
$408$	0	0
$409$	−2612.45	−0.315837	−0.157919	−	0.987452i	$-0.550478\pi$
$409$	−2612.45	−0.315837	−0.157919	+	0.987452i	$0.550478\pi$
$410$	0	0
$411$	0	0
$412$	15610.3	1.86667
$413$	5869.51	0.699321
$414$	0	0
$415$	0	0
$416$	−14387.8	−1.69572
$417$	0	0
$418$	−17381.3	−2.03384
$419$	−4398.21	−0.512808	−0.256404	−	0.966570i	$-0.582538\pi$
$419$	−4398.21	−0.512808	−0.256404	+	0.966570i	$0.582538\pi$
$420$	0	0
$421$	9723.32	1.12562	0.562810	−	0.826587i	$-0.309720\pi$
$421$	9723.32	1.12562	0.562810	+	0.826587i	$0.309720\pi$
$422$	15071.9	1.73859
$423$	0	0
$424$	−9676.01	−1.10828
$425$	0	0
$426$	0	0
$427$	−2725.96	−0.308942
$428$	762.570	0.0861220
$429$	0	0
$430$	0	0
$431$	14314.5	1.59978	0.799892	−	0.600144i	$-0.204890\pi$
$431$	14314.5	1.59978	0.799892	+	0.600144i	$0.204890\pi$
$432$	0	0
$433$	2373.62	0.263438	0.131719	−	0.991287i	$-0.457950\pi$
$433$	2373.62	0.263438	0.131719	+	0.991287i	$0.457950\pi$
$434$	−65.9633	−0.00729572
$435$	0	0
$436$	−18190.9	−1.99813
$437$	2529.80	0.276927
$438$	0	0
$439$	−9533.46	−1.03646	−0.518231	−	0.855240i	$-0.673409\pi$
$439$	−9533.46	−1.03646	−0.518231	+	0.855240i	$0.673409\pi$
$440$	0	0
$441$	0	0
$442$	−1948.14	−0.209646
$443$	6647.94	0.712987	0.356493	−	0.934298i	$-0.383972\pi$
$443$	6647.94	0.712987	0.356493	+	0.934298i	$0.383972\pi$
$444$	0	0
$445$	0	0
$446$	20224.8	2.14725
$447$	0	0
$448$	5691.47	0.600215
$449$	768.256	0.0807489	0.0403744	−	0.999185i	$-0.487145\pi$
$449$	768.256	0.0807489	0.0403744	+	0.999185i	$0.487145\pi$
$450$	0	0
$451$	−2868.99	−0.299547
$452$	5837.06	0.607416
$453$	0	0
$454$	282.628	0.0292167
$455$	0	0
$456$	0	0
$457$	3323.50	0.340190	0.170095	−	0.985428i	$-0.445593\pi$
$457$	3323.50	0.340190	0.170095	+	0.985428i	$0.445593\pi$
$458$	−13963.5	−1.42461
$459$	0	0
$460$	0	0
$461$	18840.7	1.90347	0.951733	−	0.306926i	$-0.0993004\pi$
$461$	18840.7	1.90347	0.951733	+	0.306926i	$0.0993004\pi$
$462$	0	0
$463$	10759.1	1.07995	0.539977	−	0.841679i	$-0.318433\pi$
$463$	10759.1	1.07995	0.539977	+	0.841679i	$0.318433\pi$
$464$	−1087.21	−0.108777
$465$	0	0
$466$	16330.1	1.62334
$467$	7441.70	0.737390	0.368695	−	0.929550i	$-0.379805\pi$
$467$	7441.70	0.737390	0.368695	+	0.929550i	$0.379805\pi$
$468$	0	0
$469$	4884.56	0.480913
$470$	0	0
$471$	0	0
$472$	20887.6	2.03693
$473$	8992.73	0.874178
$474$	0	0
$475$	0	0
$476$	443.885	0.0427426
$477$	0	0
$478$	10556.1	1.01010
$479$	−5691.97	−0.542949	−0.271475	−	0.962446i	$-0.587511\pi$
$479$	−5691.97	−0.542949	−0.271475	+	0.962446i	$0.587511\pi$
$480$	0	0
$481$	−12551.3	−1.18979
$482$	10246.5	0.968293
$483$	0	0
$484$	−5679.77	−0.533412
$485$	0	0
$486$	0	0
$487$	2020.25	0.187980	0.0939899	−	0.995573i	$-0.470038\pi$
$487$	2020.25	0.187980	0.0939899	+	0.995573i	$0.470038\pi$
$488$	−9700.77	−0.899863
$489$	0	0
$490$	0	0
$491$	−7636.02	−0.701851	−0.350925	−	0.936403i	$-0.614133\pi$
$491$	−7636.02	−0.701851	−0.350925	+	0.936403i	$0.614133\pi$
$492$	0	0
$493$	635.173	0.0580259
$494$	51332.6	4.67523
$495$	0	0
$496$	−16.5218	−0.00149567
$497$	−3664.15	−0.330703
$498$	0	0
$499$	6284.56	0.563799	0.281900	−	0.959444i	$-0.409036\pi$
$499$	6284.56	0.563799	0.281900	+	0.959444i	$0.409036\pi$
$500$	0	0
$501$	0	0
$502$	−14255.4	−1.26743
$503$	11310.9	1.00264	0.501319	−	0.865262i	$-0.332848\pi$
$503$	11310.9	1.00264	0.501319	+	0.865262i	$0.332848\pi$
$504$	0	0
$505$	0	0
$506$	−2822.38	−0.247965
$507$	0	0
$508$	−15885.7	−1.38743
$509$	−10712.7	−0.932876	−0.466438	−	0.884554i	$-0.654463\pi$
$509$	−10712.7	−0.932876	−0.466438	+	0.884554i	$0.654463\pi$
$510$	0	0
$511$	465.279	0.0402793
$512$	2927.65	0.252705
$513$	0	0
$514$	27897.2	2.39396
$515$	0	0
$516$	0	0
$517$	−3886.02	−0.330574
$518$	4568.89	0.387540
$519$	0	0
$520$	0	0
$521$	−17721.9	−1.49023	−0.745116	−	0.666935i	$-0.767606\pi$
$521$	−17721.9	−1.49023	−0.745116	+	0.666935i	$0.767606\pi$
$522$	0	0
$523$	−237.193	−0.0198312	−0.00991562	−	0.999951i	$-0.503156\pi$
$523$	−237.193	−0.0198312	−0.00991562	+	0.999951i	$0.503156\pi$
$524$	−13849.2	−1.15459
$525$	0	0
$526$	−27395.0	−2.27088
$527$	9.65243	0.000797849 0
$528$	0	0
$529$	−11756.2	−0.966237
$530$	0	0
$531$	0	0
$532$	−11696.2	−0.953185
$533$	8473.06	0.688572
$534$	0	0
$535$	0	0
$536$	17382.5	1.40077
$537$	0	0
$538$	15042.8	1.20546
$539$	1475.47	0.117909
$540$	0	0
$541$	−5352.94	−0.425399	−0.212699	−	0.977118i	$-0.568226\pi$
$541$	−5352.94	−0.425399	−0.212699	+	0.977118i	$0.568226\pi$
$542$	28886.2	2.28924
$543$	0	0
$544$	−766.391	−0.0604021
$545$	0	0
$546$	0	0
$547$	192.162	0.0150206	0.00751030	−	0.999972i	$-0.497609\pi$
$547$	192.162	0.0150206	0.00751030	+	0.999972i	$0.497609\pi$
$548$	8650.67	0.674340
$549$	0	0
$550$	0	0
$551$	−16736.5	−1.29401
$552$	0	0
$553$	3688.87	0.283665
$554$	−7270.60	−0.557578
$555$	0	0
$556$	6780.10	0.517159
$557$	−4850.62	−0.368990	−0.184495	−	0.982833i	$-0.559065\pi$
$557$	−4850.62	−0.368990	−0.184495	+	0.982833i	$0.559065\pi$
$558$	0	0
$559$	−26558.4	−2.00949
$560$	0	0
$561$	0	0
$562$	−36284.4	−2.72343
$563$	9699.11	0.726055	0.363027	−	0.931778i	$-0.381743\pi$
$563$	9699.11	0.726055	0.363027	+	0.931778i	$0.381743\pi$
$564$	0	0
$565$	0	0
$566$	28975.6	2.15183
$567$	0	0
$568$	−13039.5	−0.963247
$569$	−3109.53	−0.229100	−0.114550	−	0.993417i	$-0.536543\pi$
$569$	−3109.53	−0.229100	−0.114550	+	0.993417i	$0.536543\pi$
$570$	0	0
$571$	−14476.2	−1.06097	−0.530483	−	0.847695i	$-0.677990\pi$
$571$	−14476.2	−1.06097	−0.530483	+	0.847695i	$0.677990\pi$
$572$	−35846.8	−2.62033
$573$	0	0
$574$	−3084.35	−0.224283
$575$	0	0
$576$	0	0
$577$	2208.23	0.159323	0.0796617	−	0.996822i	$-0.474616\pi$
$577$	2208.23	0.159323	0.0796617	+	0.996822i	$0.474616\pi$
$578$	22616.7	1.62756
$579$	0	0
$580$	0	0
$581$	−490.186	−0.0350023
$582$	0	0
$583$	11696.2	0.830889
$584$	1655.77	0.117322
$585$	0	0
$586$	33597.4	2.36843
$587$	−23988.7	−1.68675	−0.843374	−	0.537327i	$-0.819434\pi$
$587$	−23988.7	−1.68675	−0.843374	+	0.537327i	$0.819434\pi$
$588$	0	0
$589$	−254.338	−0.0177925
$590$	0	0
$591$	0	0
$592$	1144.37	0.0794480
$593$	−15869.4	−1.09895	−0.549474	−	0.835511i	$-0.685172\pi$
$593$	−15869.4	−1.09895	−0.549474	+	0.835511i	$0.685172\pi$
$594$	0	0
$595$	0	0
$596$	24472.2	1.68192
$597$	0	0
$598$	8335.41	0.570000
$599$	15236.6	1.03932	0.519660	−	0.854373i	$-0.326059\pi$
$599$	15236.6	1.03932	0.519660	+	0.854373i	$0.326059\pi$
$600$	0	0
$601$	12258.8	0.832026	0.416013	−	0.909359i	$-0.363427\pi$
$601$	12258.8	0.832026	0.416013	+	0.909359i	$0.363427\pi$
$602$	9667.75	0.654532
$603$	0	0
$604$	39827.4	2.68303
$605$	0	0
$606$	0	0
$607$	−23487.2	−1.57054	−0.785269	−	0.619155i	$-0.787475\pi$
$607$	−23487.2	−1.57054	−0.785269	+	0.619155i	$0.787475\pi$
$608$	20194.1	1.34700
$609$	0	0
$610$	0	0
$611$	11476.7	0.759896
$612$	0	0
$613$	22305.3	1.46966	0.734830	−	0.678251i	$-0.237262\pi$
$613$	22305.3	1.46966	0.734830	+	0.678251i	$0.237262\pi$
$614$	6142.92	0.403759
$615$	0	0
$616$	5250.71	0.343437
$617$	3285.91	0.214402	0.107201	−	0.994237i	$-0.465811\pi$
$617$	3285.91	0.214402	0.107201	+	0.994237i	$0.465811\pi$
$618$	0	0
$619$	−11613.1	−0.754069	−0.377035	−	0.926199i	$-0.623056\pi$
$619$	−11613.1	−0.754069	−0.377035	+	0.926199i	$0.623056\pi$
$620$	0	0
$621$	0	0
$622$	22515.5	1.45143
$623$	−64.9548	−0.00417714
$624$	0	0
$625$	0	0
$626$	35763.5	2.28338
$627$	0	0
$628$	28536.7	1.81328
$629$	−668.567	−0.0423808
$630$	0	0
$631$	6890.91	0.434743	0.217372	−	0.976089i	$-0.430252\pi$
$631$	6890.91	0.434743	0.217372	+	0.976089i	$0.430252\pi$
$632$	13127.5	0.826238
$633$	0	0
$634$	37805.9	2.36824
$635$	0	0
$636$	0	0
$637$	−4357.55	−0.271040
$638$	18672.2	1.15868
$639$	0	0
$640$	0	0
$641$	−18769.3	−1.15654	−0.578269	−	0.815846i	$-0.696272\pi$
$641$	−18769.3	−1.15654	−0.578269	+	0.815846i	$0.696272\pi$
$642$	0	0
$643$	3142.30	0.192722	0.0963609	−	0.995346i	$-0.469280\pi$
$643$	3142.30	0.192722	0.0963609	+	0.995346i	$0.469280\pi$
$644$	−1899.23	−0.116212
$645$	0	0
$646$	2734.33	0.166533
$647$	19038.1	1.15683	0.578413	−	0.815744i	$-0.303672\pi$
$647$	19038.1	1.15683	0.578413	+	0.815744i	$0.303672\pi$
$648$	0	0
$649$	−25248.7	−1.52711
$650$	0	0
$651$	0	0
$652$	7950.82	0.477574
$653$	−20538.6	−1.23084	−0.615420	−	0.788199i	$-0.711014\pi$
$653$	−20538.6	−1.23084	−0.615420	+	0.788199i	$0.711014\pi$
$654$	0	0
$655$	0	0
$656$	−772.534	−0.0459793
$657$	0	0
$658$	−4177.72	−0.247514
$659$	937.046	0.0553902	0.0276951	−	0.999616i	$-0.491183\pi$
$659$	937.046	0.0553902	0.0276951	+	0.999616i	$0.491183\pi$
$660$	0	0
$661$	21116.5	1.24257	0.621283	−	0.783586i	$-0.286612\pi$
$661$	21116.5	1.24257	0.621283	+	0.783586i	$0.286612\pi$
$662$	9437.65	0.554085
$663$	0	0
$664$	−1744.41	−0.101952
$665$	0	0
$666$	0	0
$667$	−2717.69	−0.157765
$668$	−39307.1	−2.27670
$669$	0	0
$670$	0	0
$671$	11726.2	0.674640
$672$	0	0
$673$	−13825.9	−0.791903	−0.395952	−	0.918271i	$-0.629585\pi$
$673$	−13825.9	−0.791903	−0.395952	+	0.918271i	$0.629585\pi$
$674$	33989.3	1.94246
$675$	0	0
$676$	76456.9	4.35008
$677$	−16928.4	−0.961021	−0.480510	−	0.876989i	$-0.659549\pi$
$677$	−16928.4	−0.961021	−0.480510	+	0.876989i	$0.659549\pi$
$678$	0	0
$679$	−29.3645	−0.00165966
$680$	0	0
$681$	0	0
$682$	283.752	0.0159317
$683$	−13817.3	−0.774091	−0.387045	−	0.922061i	$-0.626504\pi$
$683$	−13817.3	−0.774091	−0.387045	+	0.922061i	$0.626504\pi$
$684$	0	0
$685$	0	0
$686$	1586.23	0.0882833
$687$	0	0
$688$	2421.47	0.134183
$689$	−34542.8	−1.90998
$690$	0	0
$691$	−23671.6	−1.30320	−0.651600	−	0.758563i	$-0.725902\pi$
$691$	−23671.6	−1.30320	−0.651600	+	0.758563i	$0.725902\pi$
$692$	31422.5	1.72616
$693$	0	0
$694$	55818.2	3.05307
$695$	0	0
$696$	0	0
$697$	451.333	0.0245272
$698$	20741.0	1.12472
$699$	0	0
$700$	0	0
$701$	17009.7	0.916472	0.458236	−	0.888831i	$-0.348481\pi$
$701$	17009.7	0.916472	0.458236	+	0.888831i	$0.348481\pi$
$702$	0	0
$703$	17616.5	0.945117
$704$	−24482.8	−1.31070
$705$	0	0
$706$	59020.9	3.14629
$707$	−6060.91	−0.322410
$708$	0	0
$709$	22038.9	1.16740	0.583701	−	0.811969i	$-0.301604\pi$
$709$	22038.9	1.16740	0.583701	+	0.811969i	$0.301604\pi$
$710$	0	0
$711$	0	0
$712$	−231.152	−0.0121669
$713$	−41.2994	−0.00216925
$714$	0	0
$715$	0	0
$716$	−40649.2	−2.12169
$717$	0	0
$718$	−11188.8	−0.581562
$719$	−7287.44	−0.377991	−0.188996	−	0.981978i	$-0.560523\pi$
$719$	−7287.44	−0.377991	−0.188996	+	0.981978i	$0.560523\pi$
$720$	0	0
$721$	−8162.82	−0.421636
$722$	−40328.5	−2.07877
$723$	0	0
$724$	−12044.9	−0.618297
$725$	0	0
$726$	0	0
$727$	29676.7	1.51396	0.756980	−	0.653438i	$-0.226674\pi$
$727$	29676.7	1.51396	0.756980	+	0.653438i	$0.226674\pi$
$728$	−15507.0	−0.789463
$729$	0	0
$730$	0	0
$731$	−1414.68	−0.0715786
$732$	0	0
$733$	−23111.8	−1.16460	−0.582300	−	0.812974i	$-0.697847\pi$
$733$	−23111.8	−1.16460	−0.582300	+	0.812974i	$0.697847\pi$
$734$	−32971.9	−1.65806
$735$	0	0
$736$	3279.12	0.164226
$737$	−21011.7	−1.05017
$738$	0	0
$739$	−31171.4	−1.55164	−0.775818	−	0.630957i	$-0.782662\pi$
$739$	−31171.4	−1.55164	−0.775818	+	0.630957i	$0.782662\pi$
$740$	0	0
$741$	0	0
$742$	12574.2	0.622120
$743$	−31324.4	−1.54668	−0.773338	−	0.633993i	$-0.781415\pi$
$743$	−31324.4	−1.54668	−0.773338	+	0.633993i	$0.781415\pi$
$744$	0	0
$745$	0	0
$746$	53630.7	2.63212
$747$	0	0
$748$	−1909.45	−0.0933373
$749$	−398.756	−0.0194529
$750$	0	0
$751$	4032.20	0.195922	0.0979608	−	0.995190i	$-0.468768\pi$
$751$	4032.20	0.195922	0.0979608	+	0.995190i	$0.468768\pi$
$752$	−1046.39	−0.0507419
$753$	0	0
$754$	−55145.0	−2.66348
$755$	0	0
$756$	0	0
$757$	−34263.7	−1.64509	−0.822546	−	0.568699i	$-0.807447\pi$
$757$	−34263.7	−1.64509	−0.822546	+	0.568699i	$0.807447\pi$
$758$	59059.3	2.82999
$759$	0	0
$760$	0	0
$761$	−7265.88	−0.346108	−0.173054	−	0.984912i	$-0.555363\pi$
$761$	−7265.88	−0.346108	−0.173054	+	0.984912i	$0.555363\pi$
$762$	0	0
$763$	9512.22	0.451331
$764$	−5571.07	−0.263814
$765$	0	0
$766$	−34546.0	−1.62950
$767$	74567.5	3.51040
$768$	0	0
$769$	38116.2	1.78739	0.893695	−	0.448674i	$-0.148104\pi$
$769$	38116.2	1.78739	0.893695	+	0.448674i	$0.148104\pi$
$770$	0	0
$771$	0	0
$772$	69356.6	3.23342
$773$	16158.2	0.751838	0.375919	−	0.926652i	$-0.377327\pi$
$773$	16158.2	0.751838	0.375919	+	0.926652i	$0.377327\pi$
$774$	0	0
$775$	0	0
$776$	−104.498	−0.00483412
$777$	0	0
$778$	40463.9	1.86465
$779$	−11892.4	−0.546972
$780$	0	0
$781$	15761.9	0.722160
$782$	444.001	0.0203036
$783$	0	0
$784$	397.301	0.0180986
$785$	0	0
$786$	0	0
$787$	−5092.49	−0.230658	−0.115329	−	0.993327i	$-0.536792\pi$
$787$	−5092.49	−0.230658	−0.115329	+	0.993327i	$0.536792\pi$
$788$	19441.9	0.878922
$789$	0	0
$790$	0	0
$791$	−3052.26	−0.137201
$792$	0	0
$793$	−34631.1	−1.55080
$794$	24858.2	1.11106
$795$	0	0
$796$	−17097.8	−0.761326
$797$	−34666.2	−1.54070	−0.770350	−	0.637621i	$-0.779919\pi$
$797$	−34666.2	−1.54070	−0.770350	+	0.637621i	$0.779919\pi$
$798$	0	0
$799$	611.326	0.0270678
$800$	0	0
$801$	0	0
$802$	34043.0	1.49888
$803$	−2001.47	−0.0879582
$804$	0	0
$805$	0	0
$806$	−838.012	−0.0366225
$807$	0	0
$808$	−21568.7	−0.939091
$809$	−15126.2	−0.657365	−0.328683	−	0.944440i	$-0.606605\pi$
$809$	−15126.2	−0.657365	−0.328683	+	0.944440i	$0.606605\pi$
$810$	0	0
$811$	29416.5	1.27368	0.636840	−	0.770996i	$-0.280241\pi$
$811$	29416.5	1.27368	0.636840	+	0.770996i	$0.280241\pi$
$812$	12564.9	0.543030
$813$	0	0
$814$	−19653.9	−0.846274
$815$	0	0
$816$	0	0
$817$	37276.3	1.59625
$818$	12081.5	0.516404
$819$	0	0
$820$	0	0
$821$	15334.4	0.651856	0.325928	−	0.945395i	$-0.394323\pi$
$821$	15334.4	0.651856	0.325928	+	0.945395i	$0.394323\pi$
$822$	0	0
$823$	11003.7	0.466056	0.233028	−	0.972470i	$-0.425137\pi$
$823$	11003.7	0.466056	0.233028	+	0.972470i	$0.425137\pi$
$824$	−29048.7	−1.22811
$825$	0	0
$826$	−27143.9	−1.14341
$827$	−3261.59	−0.137142	−0.0685711	−	0.997646i	$-0.521844\pi$
$827$	−3261.59	−0.137142	−0.0685711	+	0.997646i	$0.521844\pi$
$828$	0	0
$829$	5163.30	0.216319	0.108160	−	0.994134i	$-0.465504\pi$
$829$	5163.30	0.216319	0.108160	+	0.994134i	$0.465504\pi$
$830$	0	0
$831$	0	0
$832$	72305.6	3.01292
$833$	−232.113	−0.00965453
$834$	0	0
$835$	0	0
$836$	50313.1	2.08148
$837$	0	0
$838$	20339.8	0.838457
$839$	5641.70	0.232149	0.116075	−	0.993241i	$-0.462969\pi$
$839$	5641.70	0.232149	0.116075	+	0.993241i	$0.462969\pi$
$840$	0	0
$841$	−6409.46	−0.262801
$842$	−44966.1	−1.84042
$843$	0	0
$844$	−43628.1	−1.77931
$845$	0	0
$846$	0	0
$847$	2970.01	0.120485
$848$	3149.45	0.127538
$849$	0	0
$850$	0	0
$851$	2860.57	0.115228
$852$	0	0
$853$	7799.52	0.313072	0.156536	−	0.987672i	$-0.449967\pi$
$853$	7799.52	0.313072	0.156536	+	0.987672i	$0.449967\pi$
$854$	12606.4	0.505130
$855$	0	0
$856$	−1419.04	−0.0566610
$857$	−21540.0	−0.858568	−0.429284	−	0.903170i	$-0.641234\pi$
$857$	−21540.0	−0.858568	−0.429284	+	0.903170i	$0.641234\pi$
$858$	0	0
$859$	4447.97	0.176674	0.0883370	−	0.996091i	$-0.471845\pi$
$859$	4447.97	0.176674	0.0883370	+	0.996091i	$0.471845\pi$
$860$	0	0
$861$	0	0
$862$	−66198.5	−2.61569
$863$	−9425.21	−0.371770	−0.185885	−	0.982571i	$-0.559515\pi$
$863$	−9425.21	−0.371770	−0.185885	+	0.982571i	$0.559515\pi$
$864$	0	0
$865$	0	0
$866$	−10977.0	−0.430730
$867$	0	0
$868$	190.942	0.00746659
$869$	−15868.3	−0.619442
$870$	0	0
$871$	62054.5	2.41405
$872$	33850.8	1.31460
$873$	0	0
$874$	−11699.2	−0.452783
$875$	0	0
$876$	0	0
$877$	−22346.1	−0.860403	−0.430201	−	0.902733i	$-0.641557\pi$
$877$	−22346.1	−0.860403	−0.430201	+	0.902733i	$0.641557\pi$
$878$	44088.1	1.69465
$879$	0	0
$880$	0	0
$881$	−12074.9	−0.461762	−0.230881	−	0.972982i	$-0.574161\pi$
$881$	−12074.9	−0.461762	−0.230881	+	0.972982i	$0.574161\pi$
$882$	0	0
$883$	30499.6	1.16239	0.581196	−	0.813764i	$-0.302585\pi$
$883$	30499.6	1.16239	0.581196	+	0.813764i	$0.302585\pi$
$884$	5639.22	0.214556
$885$	0	0
$886$	−30743.8	−1.16576
$887$	23344.2	0.883675	0.441838	−	0.897095i	$-0.354327\pi$
$887$	23344.2	0.883675	0.441838	+	0.897095i	$0.354327\pi$
$888$	0	0
$889$	8306.80	0.313387
$890$	0	0
$891$	0	0
$892$	−58544.3	−2.19754
$893$	−16108.2	−0.603628
$894$	0	0
$895$	0	0
$896$	−17260.4	−0.643560
$897$	0	0
$898$	−3552.85	−0.132027
$899$	273.227	0.0101364
$900$	0	0
$901$	−1839.98	−0.0680341
$902$	13267.8	0.489768
$903$	0	0
$904$	−10862.0	−0.399629
$905$	0	0
$906$	0	0
$907$	15092.5	0.552523	0.276262	−	0.961082i	$-0.410904\pi$
$907$	15092.5	0.552523	0.276262	+	0.961082i	$0.410904\pi$
$908$	−818.115	−0.0299010
$909$	0	0
$910$	0	0
$911$	15207.8	0.553081	0.276541	−	0.961002i	$-0.410812\pi$
$911$	15207.8	0.553081	0.276541	+	0.961002i	$0.410812\pi$
$912$	0	0
$913$	2108.62	0.0764348
$914$	−15369.7	−0.556221
$915$	0	0
$916$	40419.6	1.45797
$917$	7241.90	0.260794
$918$	0	0
$919$	24818.1	0.890831	0.445415	−	0.895324i	$-0.353056\pi$
$919$	24818.1	0.890831	0.445415	+	0.895324i	$0.353056\pi$
$920$	0	0
$921$	0	0
$922$	−87130.0	−3.11223
$923$	−46550.1	−1.66004
$924$	0	0
$925$	0	0
$926$	−49756.3	−1.76576
$927$	0	0
$928$	−21693.9	−0.767388
$929$	−39906.4	−1.40935	−0.704675	−	0.709530i	$-0.748907\pi$
$929$	−39906.4	−1.40935	−0.704675	+	0.709530i	$0.748907\pi$
$930$	0	0
$931$	6116.07	0.215302
$932$	−47270.3	−1.66136
$933$	0	0
$934$	−34414.6	−1.20565
$935$	0	0
$936$	0	0
$937$	−16923.0	−0.590020	−0.295010	−	0.955494i	$-0.595323\pi$
$937$	−16923.0	−0.590020	−0.295010	+	0.955494i	$0.595323\pi$
$938$	−22589.0	−0.786307
$939$	0	0
$940$	0	0
$941$	53014.1	1.83657	0.918285	−	0.395921i	$-0.129574\pi$
$941$	53014.1	1.83657	0.918285	+	0.395921i	$0.129574\pi$
$942$	0	0
$943$	−1931.10	−0.0666864
$944$	−6798.71	−0.234406
$945$	0	0
$946$	−41587.4	−1.42931
$947$	−25798.9	−0.885271	−0.442636	−	0.896702i	$-0.645957\pi$
$947$	−25798.9	−0.885271	−0.442636	+	0.896702i	$0.645957\pi$
$948$	0	0
$949$	5911.00	0.202191
$950$	0	0
$951$	0	0
$952$	−826.011	−0.0281210
$953$	−17942.7	−0.609885	−0.304943	−	0.952371i	$-0.598637\pi$
$953$	−17942.7	−0.609885	−0.304943	+	0.952371i	$0.598637\pi$
$954$	0	0
$955$	0	0
$956$	−30556.6	−1.03375
$957$	0	0
$958$	26322.9	0.887739
$959$	−4523.53	−0.152317
$960$	0	0
$961$	−29786.8	−0.999861
$962$	58044.2	1.94534
$963$	0	0
$964$	−29660.4	−0.990971
$965$	0	0
$966$	0	0
$967$	−19668.3	−0.654073	−0.327036	−	0.945012i	$-0.606050\pi$
$967$	−19668.3	−0.654073	−0.327036	+	0.945012i	$0.606050\pi$
$968$	10569.3	0.350940
$969$	0	0
$970$	0	0
$971$	−6332.97	−0.209304	−0.104652	−	0.994509i	$-0.533373\pi$
$971$	−6332.97	−0.209304	−0.104652	+	0.994509i	$0.533373\pi$
$972$	0	0
$973$	−3545.39	−0.116814
$974$	−9342.76	−0.307353
$975$	0	0
$976$	3157.51	0.103555
$977$	−11334.1	−0.371145	−0.185573	−	0.982631i	$-0.559414\pi$
$977$	−11334.1	−0.371145	−0.185573	+	0.982631i	$0.559414\pi$
$978$	0	0
$979$	279.414	0.00912166
$980$	0	0
$981$	0	0
$982$	35313.3	1.14755
$983$	−37654.3	−1.22175	−0.610877	−	0.791725i	$-0.709183\pi$
$983$	−37654.3	−1.22175	−0.610877	+	0.791725i	$0.709183\pi$
$984$	0	0
$985$	0	0
$986$	−2937.40	−0.0948741
$987$	0	0
$988$	−148591.	−4.78473
$989$	6052.95	0.194613
$990$	0	0
$991$	−53441.5	−1.71304	−0.856522	−	0.516111i	$-0.827379\pi$
$991$	−53441.5	−1.71304	−0.856522	+	0.516111i	$0.827379\pi$
$992$	−329.671	−0.0105515
$993$	0	0
$994$	16945.1	0.540710
$995$	0	0
$996$	0	0
$997$	−37919.3	−1.20453	−0.602266	−	0.798296i	$-0.705735\pi$
$997$	−37919.3	−1.20453	−0.602266	+	0.798296i	$0.705735\pi$
$998$	−29063.4	−0.921829
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.ba.1.1		3
3.2	odd	2		175.4.a.f.1.3		3
5.4	even	2		315.4.a.p.1.3		3
15.2	even	4		175.4.b.e.99.6		6
15.8	even	4		175.4.b.e.99.1		6
15.14	odd	2		35.4.a.c.1.1	✓	3
21.20	even	2		1225.4.a.y.1.3		3
35.34	odd	2		2205.4.a.bm.1.3		3
60.59	even	2		560.4.a.u.1.3		3
105.44	odd	6		245.4.e.m.116.3		6
105.59	even	6		245.4.e.n.226.3		6
105.74	odd	6		245.4.e.m.226.3		6
105.89	even	6		245.4.e.n.116.3		6
105.104	even	2		245.4.a.l.1.1		3
120.29	odd	2		2240.4.a.bt.1.3		3
120.59	even	2		2240.4.a.bv.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.c.1.1	✓	3	15.14	odd	2
175.4.a.f.1.3		3	3.2	odd	2
175.4.b.e.99.1		6	15.8	even	4
175.4.b.e.99.6		6	15.2	even	4
245.4.a.l.1.1		3	105.104	even	2
245.4.e.m.116.3		6	105.44	odd	6
245.4.e.m.226.3		6	105.74	odd	6
245.4.e.n.116.3		6	105.89	even	6
245.4.e.n.226.3		6	105.59	even	6
315.4.a.p.1.3		3	5.4	even	2
560.4.a.u.1.3		3	60.59	even	2
1225.4.a.y.1.3		3	21.20	even	2
1575.4.a.ba.1.1		3	1.1	even	1	trivial
2205.4.a.bm.1.3		3	35.34	odd	2
2240.4.a.bt.1.3		3	120.29	odd	2
2240.4.a.bv.1.1		3	120.59	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-4.62456 q^{2} +13.3866 q^{4} -7.00000 q^{7} -24.9107 q^{8} +30.1117 q^{11} -88.9295 q^{13} +32.3720 q^{14} +8.10818 q^{16} -4.73699 q^{17} +124.818 q^{19} -139.253 q^{22} +20.2680 q^{23} +411.260 q^{26} -93.7062 q^{28} -134.088 q^{29} -2.03767 q^{31} +161.788 q^{32} +21.9065 q^{34} +141.137 q^{37} -577.228 q^{38} -95.2784 q^{41} +298.646 q^{43} +403.093 q^{44} -93.7305 q^{46} -129.054 q^{47} +49.0000 q^{49} -1190.46 q^{52} +388.429 q^{53} +174.375 q^{56} +620.098 q^{58} -838.501 q^{59} +389.422 q^{61} +9.42333 q^{62} -813.067 q^{64} -697.794 q^{67} -63.4122 q^{68} +523.450 q^{71} -66.4684 q^{73} -652.699 q^{74} +1670.89 q^{76} -210.782 q^{77} -526.982 q^{79} +440.621 q^{82} +70.0265 q^{83} -1381.11 q^{86} -750.101 q^{88} +9.27925 q^{89} +622.506 q^{91} +271.319 q^{92} +596.817 q^{94} +4.19493 q^{97} -226.604 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 13 q^{4} - 21 q^{7} - 15 q^{8} + 74 q^{11} - 44 q^{13} + 21 q^{14} - 79 q^{16} - 52 q^{17} + 168 q^{19} - 184 q^{22} - 124 q^{23} + 446 q^{26} - 91 q^{28} - 332 q^{29} + 320 q^{31} - 183 q^{32}+ \cdots - 147 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 13 * q^4 - 21 * q^7 - 15 * q^8 + 74 * q^11 - 44 * q^13 + 21 * q^14 - 79 * q^16 - 52 * q^17 + 168 * q^19 - 184 * q^22 - 124 * q^23 + 446 * q^26 - 91 * q^28 - 332 * q^29 + 320 * q^31 - 183 * q^32 + 582 * q^34 + 54 * q^37 - 460 * q^38 - 362 * q^41 + 16 * q^43 + 264 * q^44 - 336 * q^46 - 730 * q^47 + 147 * q^49 - 1202 * q^52 + 110 * q^53 + 105 * q^56 - 450 * q^58 + 180 * q^59 + 1222 * q^61 + 464 * q^62 - 391 * q^64 - 204 * q^67 + 918 * q^68 + 136 * q^71 - 310 * q^73 - 502 * q^74 + 1796 * q^76 - 518 * q^77 - 1034 * q^79 + 6 * q^82 - 1660 * q^83 - 764 * q^86 + 20 * q^88 - 242 * q^89 + 308 * q^91 + 96 * q^92 - 1108 * q^94 - 100 * q^97 - 147 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more