LMFDB - Embedded newform 2300.4.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,4,Mod(1,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, 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("2300.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2300.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:	$135.704393013$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 80x^{3} + 121x^{2} + 1212x + 1044 $ x^5 - 2x^4 - 80x^3 + 121x^2 + 1212x + 1044
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.72746$ of defining polynomial
Character	$\chi$	$=$	2300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.72746 q^{3} -11.8626 q^{7} -13.1061 q^{9} +O(q^{10})$	$q+3.72746 q^{3} -11.8626 q^{7} -13.1061 q^{9} +64.0085 q^{11} +77.5549 q^{13} -25.6338 q^{17} +51.4669 q^{19} -44.2173 q^{21} +23.0000 q^{23} -149.494 q^{27} +68.3100 q^{29} -67.5033 q^{31} +238.589 q^{33} +149.346 q^{37} +289.083 q^{39} -249.799 q^{41} +534.186 q^{43} -141.452 q^{47} -202.279 q^{49} -95.5490 q^{51} +76.1827 q^{53} +191.841 q^{57} -816.330 q^{59} +356.152 q^{61} +155.472 q^{63} +671.800 q^{67} +85.7315 q^{69} -746.875 q^{71} +162.513 q^{73} -759.307 q^{77} +194.917 q^{79} -203.367 q^{81} -246.650 q^{83} +254.623 q^{87} -836.169 q^{89} -920.003 q^{91} -251.616 q^{93} +53.5771 q^{97} -838.900 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 8 q^{7} + 30 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 - 8 * q^7 + 30 * q^9	$ 5 q + 3 q^{3} - 8 q^{7} + 30 q^{9} + 7 q^{11} - 5 q^{13} + 24 q^{17} - 13 q^{19} + 115 q^{23} + 204 q^{27} - 253 q^{29} - 98 q^{31} + 473 q^{33} + 435 q^{37} - 410 q^{39} - 774 q^{41} + 498 q^{43} + 572 q^{47} - 683 q^{49} - 657 q^{51} + 665 q^{53} + 932 q^{57} - 763 q^{59} + 337 q^{61} + 1527 q^{63} + 305 q^{67} + 69 q^{69} - 1504 q^{71} + 1304 q^{73} - 182 q^{77} + 626 q^{79} - 959 q^{81} + 1703 q^{83} + 1354 q^{87} + 646 q^{89} - 767 q^{91} + 452 q^{93} + 233 q^{97} + 111 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 8 * q^7 + 30 * q^9 + 7 * q^11 - 5 * q^13 + 24 * q^17 - 13 * q^19 + 115 * q^23 + 204 * q^27 - 253 * q^29 - 98 * q^31 + 473 * q^33 + 435 * q^37 - 410 * q^39 - 774 * q^41 + 498 * q^43 + 572 * q^47 - 683 * q^49 - 657 * q^51 + 665 * q^53 + 932 * q^57 - 763 * q^59 + 337 * q^61 + 1527 * q^63 + 305 * q^67 + 69 * q^69 - 1504 * q^71 + 1304 * q^73 - 182 * q^77 + 626 * q^79 - 959 * q^81 + 1703 * q^83 + 1354 * q^87 + 646 * q^89 - 767 * q^91 + 452 * q^93 + 233 * q^97 + 111 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.72746	0.717349	0.358675	−	0.933463i	$-0.383229\pi$
$3$	3.72746	0.717349	0.358675	+	0.933463i	$0.383229\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−11.8626	−0.640520	−0.320260	−	0.947330i	$-0.603770\pi$
$7$	−11.8626	−0.640520	−0.320260	+	0.947330i	$0.603770\pi$
$8$	0	0
$9$	−13.1061	−0.485410
$10$	0	0
$11$	64.0085	1.75448	0.877241	−	0.480051i	$-0.159382\pi$
$11$	64.0085	1.75448	0.877241	+	0.480051i	$0.159382\pi$
$12$	0	0
$13$	77.5549	1.65460	0.827302	−	0.561757i	$-0.189874\pi$
$13$	77.5549	1.65460	0.827302	+	0.561757i	$0.189874\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−25.6338	−0.365713	−0.182856	−	0.983140i	$-0.558534\pi$
$17$	−25.6338	−0.365713	−0.182856	+	0.983140i	$0.558534\pi$
$18$	0	0
$19$	51.4669	0.621438	0.310719	−	0.950502i	$-0.399430\pi$
$19$	51.4669	0.621438	0.310719	+	0.950502i	$0.399430\pi$
$20$	0	0
$21$	−44.2173	−0.459477
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−149.494	−1.06556
$28$	0	0
$29$	68.3100	0.437409	0.218704	−	0.975791i	$-0.429817\pi$
$29$	68.3100	0.437409	0.218704	+	0.975791i	$0.429817\pi$
$30$	0	0
$31$	−67.5033	−0.391095	−0.195547	−	0.980694i	$-0.562648\pi$
$31$	−67.5033	−0.391095	−0.195547	+	0.980694i	$0.562648\pi$
$32$	0	0
$33$	238.589	1.25858
$34$	0	0
$35$	0	0
$36$	0	0
$37$	149.346	0.663577	0.331789	−	0.943354i	$-0.392348\pi$
$37$	149.346	0.663577	0.331789	+	0.943354i	$0.392348\pi$
$38$	0	0
$39$	289.083	1.18693
$40$	0	0
$41$	−249.799	−0.951512	−0.475756	−	0.879577i	$-0.657825\pi$
$41$	−249.799	−0.951512	−0.475756	+	0.879577i	$0.657825\pi$
$42$	0	0
$43$	534.186	1.89448	0.947240	−	0.320526i	$-0.103860\pi$
$43$	534.186	1.89448	0.947240	+	0.320526i	$0.103860\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−141.452	−0.438998	−0.219499	−	0.975613i	$-0.570442\pi$
$47$	−141.452	−0.438998	−0.219499	+	0.975613i	$0.570442\pi$
$48$	0	0
$49$	−202.279	−0.589734
$50$	0	0
$51$	−95.5490	−0.262344
$52$	0	0
$53$	76.1827	0.197443	0.0987217	−	0.995115i	$-0.468525\pi$
$53$	76.1827	0.197443	0.0987217	+	0.995115i	$0.468525\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	191.841	0.445788
$58$	0	0
$59$	−816.330	−1.80131	−0.900654	−	0.434537i	$-0.856912\pi$
$59$	−816.330	−1.80131	−0.900654	+	0.434537i	$0.856912\pi$
$60$	0	0
$61$	356.152	0.747551	0.373775	−	0.927519i	$-0.378063\pi$
$61$	356.152	0.747551	0.373775	+	0.927519i	$0.378063\pi$
$62$	0	0
$63$	155.472	0.310915
$64$	0	0
$65$	0	0
$66$	0	0
$67$	671.800	1.22498	0.612489	−	0.790479i	$-0.290168\pi$
$67$	671.800	1.22498	0.612489	+	0.790479i	$0.290168\pi$
$68$	0	0
$69$	85.7315	0.149578
$70$	0	0
$71$	−746.875	−1.24842	−0.624209	−	0.781257i	$-0.714579\pi$
$71$	−746.875	−1.24842	−0.624209	+	0.781257i	$0.714579\pi$
$72$	0	0
$73$	162.513	0.260558	0.130279	−	0.991477i	$-0.458413\pi$
$73$	162.513	0.260558	0.130279	+	0.991477i	$0.458413\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−759.307	−1.12378
$78$	0	0
$79$	194.917	0.277593	0.138797	−	0.990321i	$-0.455677\pi$
$79$	194.917	0.277593	0.138797	+	0.990321i	$0.455677\pi$
$80$	0	0
$81$	−203.367	−0.278968
$82$	0	0
$83$	−246.650	−0.326185	−0.163092	−	0.986611i	$-0.552147\pi$
$83$	−246.650	−0.326185	−0.163092	+	0.986611i	$0.552147\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	254.623	0.313775
$88$	0	0
$89$	−836.169	−0.995884	−0.497942	−	0.867210i	$-0.665911\pi$
$89$	−836.169	−0.995884	−0.497942	+	0.867210i	$0.665911\pi$
$90$	0	0
$91$	−920.003	−1.05981
$92$	0	0
$93$	−251.616	−0.280552
$94$	0	0
$95$	0	0
$96$	0	0
$97$	53.5771	0.0560817	0.0280409	−	0.999607i	$-0.491073\pi$
$97$	53.5771	0.0560817	0.0280409	+	0.999607i	$0.491073\pi$
$98$	0	0
$99$	−838.900	−0.851642
$100$	0	0
$101$	888.956	0.875787	0.437893	−	0.899027i	$-0.355725\pi$
$101$	888.956	0.875787	0.437893	+	0.899027i	$0.355725\pi$
$102$	0	0
$103$	152.873	0.146243	0.0731213	−	0.997323i	$-0.476704\pi$
$103$	152.873	0.146243	0.0731213	+	0.997323i	$0.476704\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1376.40	1.24357	0.621783	−	0.783189i	$-0.286409\pi$
$107$	1376.40	1.24357	0.621783	+	0.783189i	$0.286409\pi$
$108$	0	0
$109$	−383.902	−0.337350	−0.168675	−	0.985672i	$-0.553949\pi$
$109$	−383.902	−0.337350	−0.168675	+	0.985672i	$0.553949\pi$
$110$	0	0
$111$	556.681	0.476017
$112$	0	0
$113$	1532.24	1.27558	0.637790	−	0.770210i	$-0.279849\pi$
$113$	1532.24	1.27558	0.637790	+	0.770210i	$0.279849\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1016.44	−0.803161
$118$	0	0
$119$	304.084	0.234246
$120$	0	0
$121$	2766.09	2.07821
$122$	0	0
$123$	−931.114	−0.682567
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−897.249	−0.626913	−0.313457	−	0.949602i	$-0.601487\pi$
$127$	−897.249	−0.626913	−0.313457	+	0.949602i	$0.601487\pi$
$128$	0	0
$129$	1991.16	1.35900
$130$	0	0
$131$	1770.30	1.18070	0.590350	−	0.807147i	$-0.298990\pi$
$131$	1770.30	1.18070	0.590350	+	0.807147i	$0.298990\pi$
$132$	0	0
$133$	−610.531	−0.398043
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2984.69	1.86131	0.930655	−	0.365897i	$-0.119238\pi$
$137$	2984.69	1.86131	0.930655	+	0.365897i	$0.119238\pi$
$138$	0	0
$139$	3005.98	1.83427	0.917136	−	0.398574i	$-0.130495\pi$
$139$	3005.98	1.83427	0.917136	+	0.398574i	$0.130495\pi$
$140$	0	0
$141$	−527.256	−0.314915
$142$	0	0
$143$	4964.18	2.90297
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−753.986	−0.423045
$148$	0	0
$149$	770.968	0.423894	0.211947	−	0.977281i	$-0.432020\pi$
$149$	770.968	0.423894	0.211947	+	0.977281i	$0.432020\pi$
$150$	0	0
$151$	−2232.13	−1.20297	−0.601484	−	0.798885i	$-0.705424\pi$
$151$	−2232.13	−1.20297	−0.601484	+	0.798885i	$0.705424\pi$
$152$	0	0
$153$	335.959	0.177521
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−665.093	−0.338090	−0.169045	−	0.985608i	$-0.554068\pi$
$157$	−665.093	−0.338090	−0.169045	+	0.985608i	$0.554068\pi$
$158$	0	0
$159$	283.968	0.141636
$160$	0	0
$161$	−272.840	−0.133558
$162$	0	0
$163$	275.549	0.132409	0.0662045	−	0.997806i	$-0.478911\pi$
$163$	275.549	0.132409	0.0662045	+	0.997806i	$0.478911\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−786.620	−0.364494	−0.182247	−	0.983253i	$-0.558337\pi$
$167$	−786.620	−0.364494	−0.182247	+	0.983253i	$0.558337\pi$
$168$	0	0
$169$	3817.76	1.73772
$170$	0	0
$171$	−674.528	−0.301652
$172$	0	0
$173$	3134.38	1.37747	0.688735	−	0.725013i	$-0.258166\pi$
$173$	3134.38	1.37747	0.688735	+	0.725013i	$0.258166\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3042.84	−1.29217
$178$	0	0
$179$	51.6322	0.0215596	0.0107798	−	0.999942i	$-0.496569\pi$
$179$	51.6322	0.0215596	0.0107798	+	0.999942i	$0.496569\pi$
$180$	0	0
$181$	188.767	0.0775192	0.0387596	−	0.999249i	$-0.487659\pi$
$181$	188.767	0.0775192	0.0387596	+	0.999249i	$0.487659\pi$
$182$	0	0
$183$	1327.54	0.536255
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1640.78	−0.641636
$188$	0	0
$189$	1773.38	0.682511
$190$	0	0
$191$	1976.70	0.748842	0.374421	−	0.927259i	$-0.377842\pi$
$191$	1976.70	0.748842	0.374421	+	0.927259i	$0.377842\pi$
$192$	0	0
$193$	−89.0776	−0.0332225	−0.0166113	−	0.999862i	$-0.505288\pi$
$193$	−89.0776	−0.0332225	−0.0166113	+	0.999862i	$0.505288\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	958.216	0.346549	0.173274	−	0.984874i	$-0.444565\pi$
$197$	958.216	0.346549	0.173274	+	0.984874i	$0.444565\pi$
$198$	0	0
$199$	3706.37	1.32029	0.660145	−	0.751138i	$-0.270495\pi$
$199$	3706.37	1.32029	0.660145	+	0.751138i	$0.270495\pi$
$200$	0	0
$201$	2504.11	0.878737
$202$	0	0
$203$	−810.334	−0.280169
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−301.439	−0.101215
$208$	0	0
$209$	3294.32	1.09030
$210$	0	0
$211$	−890.496	−0.290542	−0.145271	−	0.989392i	$-0.546405\pi$
$211$	−890.496	−0.290542	−0.145271	+	0.989392i	$0.546405\pi$
$212$	0	0
$213$	−2783.94	−0.895553
$214$	0	0
$215$	0	0
$216$	0	0
$217$	800.764	0.250504
$218$	0	0
$219$	605.761	0.186911
$220$	0	0
$221$	−1988.03	−0.605110
$222$	0	0
$223$	1718.03	0.515910	0.257955	−	0.966157i	$-0.416951\pi$
$223$	1718.03	0.515910	0.257955	+	0.966157i	$0.416951\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−355.023	−0.103805	−0.0519025	−	0.998652i	$-0.516528\pi$
$227$	−355.023	−0.103805	−0.0519025	+	0.998652i	$0.516528\pi$
$228$	0	0
$229$	6681.36	1.92802	0.964011	−	0.265863i	$-0.0856570\pi$
$229$	6681.36	1.92802	0.964011	+	0.265863i	$0.0856570\pi$
$230$	0	0
$231$	−2830.29	−0.806143
$232$	0	0
$233$	−3719.27	−1.04574	−0.522870	−	0.852412i	$-0.675139\pi$
$233$	−3719.27	−1.04574	−0.522870	+	0.852412i	$0.675139\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	726.545	0.199132
$238$	0	0
$239$	−4963.49	−1.34335	−0.671677	−	0.740844i	$-0.734426\pi$
$239$	−4963.49	−1.34335	−0.671677	+	0.740844i	$0.734426\pi$
$240$	0	0
$241$	−5277.12	−1.41050	−0.705248	−	0.708961i	$-0.749164\pi$
$241$	−5277.12	−1.41050	−0.705248	+	0.708961i	$0.749164\pi$
$242$	0	0
$243$	3278.28	0.865441
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3991.51	1.02823
$248$	0	0
$249$	−919.377	−0.233988
$250$	0	0
$251$	−72.1196	−0.0181360	−0.00906802	−	0.999959i	$-0.502886\pi$
$251$	−72.1196	−0.0181360	−0.00906802	+	0.999959i	$0.502886\pi$
$252$	0	0
$253$	1472.20	0.365835
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2930.92	0.711384	0.355692	−	0.934603i	$-0.384245\pi$
$257$	2930.92	0.711384	0.355692	+	0.934603i	$0.384245\pi$
$258$	0	0
$259$	−1771.63	−0.425035
$260$	0	0
$261$	−895.275	−0.212322
$262$	0	0
$263$	1684.95	0.395050	0.197525	−	0.980298i	$-0.436710\pi$
$263$	1684.95	0.395050	0.197525	+	0.980298i	$0.436710\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3116.78	−0.714397
$268$	0	0
$269$	447.845	0.101508	0.0507538	−	0.998711i	$-0.483838\pi$
$269$	447.845	0.101508	0.0507538	+	0.998711i	$0.483838\pi$
$270$	0	0
$271$	−1659.82	−0.372055	−0.186028	−	0.982545i	$-0.559561\pi$
$271$	−1659.82	−0.372055	−0.186028	+	0.982545i	$0.559561\pi$
$272$	0	0
$273$	−3429.27	−0.760252
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1449.27	0.314362	0.157181	−	0.987570i	$-0.449759\pi$
$277$	1449.27	0.314362	0.157181	+	0.987570i	$0.449759\pi$
$278$	0	0
$279$	884.702	0.189841
$280$	0	0
$281$	−1528.94	−0.324586	−0.162293	−	0.986743i	$-0.551889\pi$
$281$	−1528.94	−0.324586	−0.162293	+	0.986743i	$0.551889\pi$
$282$	0	0
$283$	−3481.74	−0.731335	−0.365667	−	0.930746i	$-0.619159\pi$
$283$	−3481.74	−0.731335	−0.365667	+	0.930746i	$0.619159\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2963.26	0.609463
$288$	0	0
$289$	−4255.91	−0.866254
$290$	0	0
$291$	199.706	0.0402302
$292$	0	0
$293$	−4602.57	−0.917697	−0.458848	−	0.888515i	$-0.651738\pi$
$293$	−4602.57	−0.917697	−0.458848	+	0.888515i	$0.651738\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−9568.87	−1.86950
$298$	0	0
$299$	1783.76	0.345009
$300$	0	0
$301$	−6336.83	−1.21345
$302$	0	0
$303$	3313.55	0.628245
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−3004.50	−0.558554	−0.279277	−	0.960211i	$-0.590095\pi$
$307$	−3004.50	−0.558554	−0.279277	+	0.960211i	$0.590095\pi$
$308$	0	0
$309$	569.827	0.104907
$310$	0	0
$311$	9252.25	1.68697	0.843484	−	0.537155i	$-0.180501\pi$
$311$	9252.25	1.68697	0.843484	+	0.537155i	$0.180501\pi$
$312$	0	0
$313$	2524.11	0.455818	0.227909	−	0.973682i	$-0.426811\pi$
$313$	2524.11	0.455818	0.227909	+	0.973682i	$0.426811\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8948.50	−1.58548	−0.792741	−	0.609558i	$-0.791347\pi$
$317$	−8948.50	−1.58548	−0.792741	+	0.609558i	$0.791347\pi$
$318$	0	0
$319$	4372.42	0.767425
$320$	0	0
$321$	5130.47	0.892072
$322$	0	0
$323$	−1319.29	−0.227268
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1430.98	−0.241998
$328$	0	0
$329$	1677.99	0.281187
$330$	0	0
$331$	2848.38	0.472994	0.236497	−	0.971632i	$-0.424001\pi$
$331$	2848.38	0.472994	0.236497	+	0.971632i	$0.424001\pi$
$332$	0	0
$333$	−1957.34	−0.322107
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11454.1	1.85147	0.925735	−	0.378173i	$-0.123448\pi$
$337$	11454.1	1.85147	0.925735	+	0.378173i	$0.123448\pi$
$338$	0	0
$339$	5711.34	0.915037
$340$	0	0
$341$	−4320.78	−0.686169
$342$	0	0
$343$	6468.42	1.01826
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2858.52	0.442228	0.221114	−	0.975248i	$-0.429031\pi$
$347$	2858.52	0.442228	0.221114	+	0.975248i	$0.429031\pi$
$348$	0	0
$349$	2586.61	0.396729	0.198364	−	0.980128i	$-0.436437\pi$
$349$	2586.61	0.396729	0.198364	+	0.980128i	$0.436437\pi$
$350$	0	0
$351$	−11594.0	−1.76308
$352$	0	0
$353$	6506.44	0.981028	0.490514	−	0.871433i	$-0.336809\pi$
$353$	6506.44	0.981028	0.490514	+	0.871433i	$0.336809\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1133.46	0.168036
$358$	0	0
$359$	−9798.33	−1.44049	−0.720245	−	0.693720i	$-0.755971\pi$
$359$	−9798.33	−1.44049	−0.720245	+	0.693720i	$0.755971\pi$
$360$	0	0
$361$	−4210.16	−0.613815
$362$	0	0
$363$	10310.5	1.49080
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4821.10	0.685720	0.342860	−	0.939386i	$-0.388604\pi$
$367$	4821.10	0.685720	0.342860	+	0.939386i	$0.388604\pi$
$368$	0	0
$369$	3273.88	0.461873
$370$	0	0
$371$	−903.725	−0.126466
$372$	0	0
$373$	−9742.30	−1.35238	−0.676190	−	0.736728i	$-0.736370\pi$
$373$	−9742.30	−1.35238	−0.676190	+	0.736728i	$0.736370\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5297.78	0.723738
$378$	0	0
$379$	−8403.84	−1.13899	−0.569494	−	0.821996i	$-0.692861\pi$
$379$	−8403.84	−1.13899	−0.569494	+	0.821996i	$0.692861\pi$
$380$	0	0
$381$	−3344.46	−0.449716
$382$	0	0
$383$	14020.1	1.87048	0.935240	−	0.354013i	$-0.115183\pi$
$383$	14020.1	1.87048	0.935240	+	0.354013i	$0.115183\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7001.08	−0.919599
$388$	0	0
$389$	13631.3	1.77670	0.888348	−	0.459170i	$-0.151853\pi$
$389$	13631.3	1.77670	0.888348	+	0.459170i	$0.151853\pi$
$390$	0	0
$391$	−589.578	−0.0762564
$392$	0	0
$393$	6598.72	0.846975
$394$	0	0
$395$	0	0
$396$	0	0
$397$	11338.8	1.43345	0.716724	−	0.697357i	$-0.245641\pi$
$397$	11338.8	1.43345	0.716724	+	0.697357i	$0.245641\pi$
$398$	0	0
$399$	−2275.73	−0.285536
$400$	0	0
$401$	−5482.78	−0.682786	−0.341393	−	0.939921i	$-0.610899\pi$
$401$	−5482.78	−0.682786	−0.341393	+	0.939921i	$0.610899\pi$
$402$	0	0
$403$	−5235.21	−0.647108
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9559.43	1.16423
$408$	0	0
$409$	14162.6	1.71221	0.856107	−	0.516799i	$-0.172877\pi$
$409$	14162.6	1.71221	0.856107	+	0.516799i	$0.172877\pi$
$410$	0	0
$411$	11125.3	1.33521
$412$	0	0
$413$	9683.80	1.15377
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11204.7	1.31581
$418$	0	0
$419$	−9438.98	−1.10054	−0.550268	−	0.834988i	$-0.685474\pi$
$419$	−9438.98	−1.10054	−0.550268	+	0.834988i	$0.685474\pi$
$420$	0	0
$421$	−2888.29	−0.334363	−0.167181	−	0.985926i	$-0.553467\pi$
$421$	−2888.29	−0.334363	−0.167181	+	0.985926i	$0.553467\pi$
$422$	0	0
$423$	1853.88	0.213094
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4224.89	−0.478821
$428$	0	0
$429$	18503.8	2.08245
$430$	0	0
$431$	−9881.73	−1.10438	−0.552188	−	0.833719i	$-0.686207\pi$
$431$	−9881.73	−1.10438	−0.552188	+	0.833719i	$0.686207\pi$
$432$	0	0
$433$	873.860	0.0969863	0.0484931	−	0.998824i	$-0.484558\pi$
$433$	873.860	0.0969863	0.0484931	+	0.998824i	$0.484558\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1183.74	0.129579
$438$	0	0
$439$	16959.1	1.84377	0.921883	−	0.387469i	$-0.126650\pi$
$439$	16959.1	1.84377	0.921883	+	0.387469i	$0.126650\pi$
$440$	0	0
$441$	2651.08	0.286263
$442$	0	0
$443$	8914.05	0.956025	0.478013	−	0.878353i	$-0.341357\pi$
$443$	8914.05	0.956025	0.478013	+	0.878353i	$0.341357\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2873.75	0.304080
$448$	0	0
$449$	−864.502	−0.0908650	−0.0454325	−	0.998967i	$-0.514467\pi$
$449$	−864.502	−0.0908650	−0.0454325	+	0.998967i	$0.514467\pi$
$450$	0	0
$451$	−15989.2	−1.66941
$452$	0	0
$453$	−8320.17	−0.862949
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19241.7	−1.96956	−0.984779	−	0.173812i	$-0.944391\pi$
$457$	−19241.7	−1.96956	−0.984779	+	0.173812i	$0.944391\pi$
$458$	0	0
$459$	3832.09	0.389688
$460$	0	0
$461$	−3935.04	−0.397555	−0.198778	−	0.980045i	$-0.563697\pi$
$461$	−3935.04	−0.397555	−0.198778	+	0.980045i	$0.563697\pi$
$462$	0	0
$463$	1413.95	0.141926	0.0709631	−	0.997479i	$-0.477393\pi$
$463$	1413.95	0.141926	0.0709631	+	0.997479i	$0.477393\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7355.75	0.728873	0.364436	−	0.931228i	$-0.381262\pi$
$467$	7355.75	0.728873	0.364436	+	0.931228i	$0.381262\pi$
$468$	0	0
$469$	−7969.30	−0.784623
$470$	0	0
$471$	−2479.11	−0.242529
$472$	0	0
$473$	34192.5	3.32383
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−998.455	−0.0958409
$478$	0	0
$479$	−12770.7	−1.21818	−0.609090	−	0.793101i	$-0.708465\pi$
$479$	−12770.7	−1.21818	−0.609090	+	0.793101i	$0.708465\pi$
$480$	0	0
$481$	11582.5	1.09796
$482$	0	0
$483$	−1017.00	−0.0958075
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12490.4	1.16221	0.581105	−	0.813829i	$-0.302621\pi$
$487$	12490.4	1.16221	0.581105	+	0.813829i	$0.302621\pi$
$488$	0	0
$489$	1027.10	0.0949836
$490$	0	0
$491$	−13347.7	−1.22683	−0.613415	−	0.789761i	$-0.710205\pi$
$491$	−13347.7	−1.22683	−0.613415	+	0.789761i	$0.710205\pi$
$492$	0	0
$493$	−1751.05	−0.159966
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8859.87	0.799637
$498$	0	0
$499$	−15159.1	−1.35995	−0.679975	−	0.733235i	$-0.738010\pi$
$499$	−15159.1	−1.35995	−0.679975	+	0.733235i	$0.738010\pi$
$500$	0	0
$501$	−2932.09	−0.261470
$502$	0	0
$503$	−13114.9	−1.16256	−0.581278	−	0.813705i	$-0.697447\pi$
$503$	−13114.9	−1.16256	−0.581278	+	0.813705i	$0.697447\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14230.6	1.24655
$508$	0	0
$509$	14089.6	1.22694	0.613469	−	0.789719i	$-0.289774\pi$
$509$	14089.6	1.22694	0.613469	+	0.789719i	$0.289774\pi$
$510$	0	0
$511$	−1927.83	−0.166893
$512$	0	0
$513$	−7693.97	−0.662178
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9054.13	−0.770213
$518$	0	0
$519$	11683.3	0.988128
$520$	0	0
$521$	−5006.61	−0.421005	−0.210502	−	0.977593i	$-0.567510\pi$
$521$	−5006.61	−0.421005	−0.210502	+	0.977593i	$0.567510\pi$
$522$	0	0
$523$	−7991.86	−0.668183	−0.334091	−	0.942541i	$-0.608429\pi$
$523$	−7991.86	−0.668183	−0.334091	+	0.942541i	$0.608429\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1730.37	0.143028
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	10698.9	0.874372
$532$	0	0
$533$	−19373.1	−1.57438
$534$	0	0
$535$	0	0
$536$	0	0
$537$	192.457	0.0154658
$538$	0	0
$539$	−12947.6	−1.03468
$540$	0	0
$541$	−9408.67	−0.747708	−0.373854	−	0.927488i	$-0.621964\pi$
$541$	−9408.67	−0.747708	−0.373854	+	0.927488i	$0.621964\pi$
$542$	0	0
$543$	703.622	0.0556083
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7492.55	−0.585664	−0.292832	−	0.956164i	$-0.594598\pi$
$547$	−7492.55	−0.585664	−0.292832	+	0.956164i	$0.594598\pi$
$548$	0	0
$549$	−4667.75	−0.362868
$550$	0	0
$551$	3515.70	0.271822
$552$	0	0
$553$	−2312.22	−0.177804
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6822.47	−0.518990	−0.259495	−	0.965744i	$-0.583556\pi$
$557$	−6822.47	−0.518990	−0.259495	+	0.965744i	$0.583556\pi$
$558$	0	0
$559$	41428.8	3.13461
$560$	0	0
$561$	−6115.95	−0.460277
$562$	0	0
$563$	−7654.32	−0.572986	−0.286493	−	0.958082i	$-0.592490\pi$
$563$	−7654.32	−0.572986	−0.286493	+	0.958082i	$0.592490\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2412.47	0.178684
$568$	0	0
$569$	−276.732	−0.0203888	−0.0101944	−	0.999948i	$-0.503245\pi$
$569$	−276.732	−0.0203888	−0.0101944	+	0.999948i	$0.503245\pi$
$570$	0	0
$571$	15561.6	1.14051	0.570256	−	0.821467i	$-0.306844\pi$
$571$	15561.6	1.14051	0.570256	+	0.821467i	$0.306844\pi$
$572$	0	0
$573$	7368.06	0.537181
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−21913.2	−1.58103	−0.790517	−	0.612440i	$-0.790188\pi$
$577$	−21913.2	−1.58103	−0.790517	+	0.612440i	$0.790188\pi$
$578$	0	0
$579$	−332.033	−0.0238322
$580$	0	0
$581$	2925.91	0.208928
$582$	0	0
$583$	4876.34	0.346411
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23731.6	−1.66867	−0.834336	−	0.551257i	$-0.814148\pi$
$587$	−23731.6	−1.66867	−0.834336	+	0.551257i	$0.814148\pi$
$588$	0	0
$589$	−3474.18	−0.243041
$590$	0	0
$591$	3571.71	0.248596
$592$	0	0
$593$	18853.5	1.30560	0.652798	−	0.757532i	$-0.273595\pi$
$593$	18853.5	1.30560	0.652798	+	0.757532i	$0.273595\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13815.3	0.947109
$598$	0	0
$599$	15886.8	1.08367	0.541835	−	0.840485i	$-0.317730\pi$
$599$	15886.8	1.08367	0.541835	+	0.840485i	$0.317730\pi$
$600$	0	0
$601$	10870.6	0.737806	0.368903	−	0.929468i	$-0.379733\pi$
$601$	10870.6	0.737806	0.368903	+	0.929468i	$0.379733\pi$
$602$	0	0
$603$	−8804.66	−0.594616
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−26207.6	−1.75244	−0.876222	−	0.481908i	$-0.839944\pi$
$607$	−26207.6	−1.75244	−0.876222	+	0.481908i	$0.839944\pi$
$608$	0	0
$609$	−3020.49	−0.200979
$610$	0	0
$611$	−10970.3	−0.726367
$612$	0	0
$613$	−25428.3	−1.67543	−0.837714	−	0.546109i	$-0.816108\pi$
$613$	−25428.3	−1.67543	−0.837714	+	0.546109i	$0.816108\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25153.5	−1.64124	−0.820619	−	0.571476i	$-0.806371\pi$
$617$	−25153.5	−1.64124	−0.820619	+	0.571476i	$0.806371\pi$
$618$	0	0
$619$	−19039.3	−1.23628	−0.618139	−	0.786069i	$-0.712113\pi$
$619$	−19039.3	−1.23628	−0.618139	+	0.786069i	$0.712113\pi$
$620$	0	0
$621$	−3438.35	−0.222184
$622$	0	0
$623$	9919.13	0.637884
$624$	0	0
$625$	0	0
$626$	0	0
$627$	12279.4	0.782127
$628$	0	0
$629$	−3828.31	−0.242679
$630$	0	0
$631$	29970.1	1.89079	0.945396	−	0.325924i	$-0.105675\pi$
$631$	29970.1	1.89079	0.945396	+	0.325924i	$0.105675\pi$
$632$	0	0
$633$	−3319.29	−0.208420
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−15687.7	−0.975777
$638$	0	0
$639$	9788.59	0.605995
$640$	0	0
$641$	9509.35	0.585955	0.292977	−	0.956119i	$-0.405354\pi$
$641$	9509.35	0.585955	0.292977	+	0.956119i	$0.405354\pi$
$642$	0	0
$643$	22663.0	1.38996	0.694979	−	0.719030i	$-0.255414\pi$
$643$	22663.0	1.38996	0.694979	+	0.719030i	$0.255414\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25697.8	1.56149	0.780747	−	0.624848i	$-0.214839\pi$
$647$	25697.8	1.56149	0.780747	+	0.624848i	$0.214839\pi$
$648$	0	0
$649$	−52252.1	−3.16036
$650$	0	0
$651$	2984.81	0.179699
$652$	0	0
$653$	15010.3	0.899535	0.449768	−	0.893146i	$-0.351507\pi$
$653$	15010.3	0.899535	0.449768	+	0.893146i	$0.351507\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2129.91	−0.126477
$658$	0	0
$659$	23949.7	1.41570	0.707850	−	0.706362i	$-0.249665\pi$
$659$	23949.7	1.41570	0.707850	+	0.706362i	$0.249665\pi$
$660$	0	0
$661$	24381.6	1.43470	0.717348	−	0.696715i	$-0.245356\pi$
$661$	24381.6	1.43470	0.717348	+	0.696715i	$0.245356\pi$
$662$	0	0
$663$	−7410.29	−0.434075
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1571.13	0.0912060
$668$	0	0
$669$	6403.89	0.370088
$670$	0	0
$671$	22796.8	1.31156
$672$	0	0
$673$	−9221.65	−0.528185	−0.264093	−	0.964497i	$-0.585072\pi$
$673$	−9221.65	−0.528185	−0.264093	+	0.964497i	$0.585072\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18144.6	−1.03007	−0.515033	−	0.857170i	$-0.672221\pi$
$677$	−18144.6	−1.03007	−0.515033	+	0.857170i	$0.672221\pi$
$678$	0	0
$679$	−635.563	−0.0359215
$680$	0	0
$681$	−1323.33	−0.0744644
$682$	0	0
$683$	−14376.0	−0.805390	−0.402695	−	0.915334i	$-0.631926\pi$
$683$	−14376.0	−0.805390	−0.402695	+	0.915334i	$0.631926\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	24904.5	1.38307
$688$	0	0
$689$	5908.34	0.326691
$690$	0	0
$691$	29784.3	1.63972	0.819861	−	0.572563i	$-0.194051\pi$
$691$	29784.3	1.63972	0.819861	+	0.572563i	$0.194051\pi$
$692$	0	0
$693$	9951.53	0.545494
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6403.30	0.347980
$698$	0	0
$699$	−13863.4	−0.750161
$700$	0	0
$701$	5784.80	0.311682	0.155841	−	0.987782i	$-0.450191\pi$
$701$	5784.80	0.311682	0.155841	+	0.987782i	$0.450191\pi$
$702$	0	0
$703$	7686.38	0.412372
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10545.3	−0.560959
$708$	0	0
$709$	−7136.17	−0.378003	−0.189002	−	0.981977i	$-0.560525\pi$
$709$	−7136.17	−0.378003	−0.189002	+	0.981977i	$0.560525\pi$
$710$	0	0
$711$	−2554.59	−0.134747
$712$	0	0
$713$	−1552.58	−0.0815489
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−18501.2	−0.963654
$718$	0	0
$719$	−16973.2	−0.880380	−0.440190	−	0.897905i	$-0.645089\pi$
$719$	−16973.2	−0.880380	−0.440190	+	0.897905i	$0.645089\pi$
$720$	0	0
$721$	−1813.47	−0.0936714
$722$	0	0
$723$	−19670.2	−1.01182
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−18100.1	−0.923378	−0.461689	−	0.887042i	$-0.652756\pi$
$727$	−18100.1	−0.923378	−0.461689	+	0.887042i	$0.652756\pi$
$728$	0	0
$729$	17710.6	0.899791
$730$	0	0
$731$	−13693.2	−0.692835
$732$	0	0
$733$	26474.1	1.33403	0.667014	−	0.745045i	$-0.267572\pi$
$733$	26474.1	1.33403	0.667014	+	0.745045i	$0.267572\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	43001.0	2.14920
$738$	0	0
$739$	−1455.63	−0.0724575	−0.0362288	−	0.999344i	$-0.511534\pi$
$739$	−1455.63	−0.0724575	−0.0362288	+	0.999344i	$0.511534\pi$
$740$	0	0
$741$	14878.2	0.737603
$742$	0	0
$743$	4845.16	0.239235	0.119617	−	0.992820i	$-0.461833\pi$
$743$	4845.16	0.239235	0.119617	+	0.992820i	$0.461833\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3232.61	0.158333
$748$	0	0
$749$	−16327.7	−0.796529
$750$	0	0
$751$	−6411.21	−0.311516	−0.155758	−	0.987795i	$-0.549782\pi$
$751$	−6411.21	−0.311516	−0.155758	+	0.987795i	$0.549782\pi$
$752$	0	0
$753$	−268.823	−0.0130099
$754$	0	0
$755$	0	0
$756$	0	0
$757$	546.716	0.0262493	0.0131247	−	0.999914i	$-0.495822\pi$
$757$	546.716	0.0262493	0.0131247	+	0.999914i	$0.495822\pi$
$758$	0	0
$759$	5487.55	0.262431
$760$	0	0
$761$	−1981.09	−0.0943686	−0.0471843	−	0.998886i	$-0.515025\pi$
$761$	−1981.09	−0.0943686	−0.0471843	+	0.998886i	$0.515025\pi$
$762$	0	0
$763$	4554.07	0.216079
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−63310.4	−2.98045
$768$	0	0
$769$	12719.9	0.596478	0.298239	−	0.954491i	$-0.403601\pi$
$769$	12719.9	0.596478	0.298239	+	0.954491i	$0.403601\pi$
$770$	0	0
$771$	10924.9	0.510311
$772$	0	0
$773$	−6926.77	−0.322301	−0.161150	−	0.986930i	$-0.551520\pi$
$773$	−6926.77	−0.322301	−0.161150	+	0.986930i	$0.551520\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6603.69	−0.304898
$778$	0	0
$779$	−12856.4	−0.591306
$780$	0	0
$781$	−47806.4	−2.19033
$782$	0	0
$783$	−10211.9	−0.466084
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4223.75	0.191309	0.0956546	−	0.995415i	$-0.469506\pi$
$787$	4223.75	0.191309	0.0956546	+	0.995415i	$0.469506\pi$
$788$	0	0
$789$	6280.56	0.283389
$790$	0	0
$791$	−18176.3	−0.817035
$792$	0	0
$793$	27621.3	1.23690
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27988.1	1.24390	0.621951	−	0.783056i	$-0.286340\pi$
$797$	27988.1	1.24390	0.621951	+	0.783056i	$0.286340\pi$
$798$	0	0
$799$	3625.95	0.160547
$800$	0	0
$801$	10958.9	0.483412
$802$	0	0
$803$	10402.2	0.457144
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1669.32	0.0728165
$808$	0	0
$809$	30980.6	1.34638	0.673190	−	0.739469i	$-0.264924\pi$
$809$	30980.6	1.34638	0.673190	+	0.739469i	$0.264924\pi$
$810$	0	0
$811$	−389.244	−0.0168535	−0.00842677	−	0.999964i	$-0.502682\pi$
$811$	−389.244	−0.0168535	−0.00842677	+	0.999964i	$0.502682\pi$
$812$	0	0
$813$	−6186.91	−0.266894
$814$	0	0
$815$	0	0
$816$	0	0
$817$	27492.9	1.17730
$818$	0	0
$819$	12057.6	0.514441
$820$	0	0
$821$	−4469.51	−0.189996	−0.0949982	−	0.995477i	$-0.530285\pi$
$821$	−4469.51	−0.189996	−0.0949982	+	0.995477i	$0.530285\pi$
$822$	0	0
$823$	14661.6	0.620985	0.310492	−	0.950576i	$-0.399506\pi$
$823$	14661.6	0.620985	0.310492	+	0.950576i	$0.399506\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4882.54	0.205299	0.102650	−	0.994718i	$-0.467268\pi$
$827$	4882.54	0.205299	0.102650	+	0.994718i	$0.467268\pi$
$828$	0	0
$829$	−35180.8	−1.47392	−0.736960	−	0.675936i	$-0.763739\pi$
$829$	−35180.8	−1.47392	−0.736960	+	0.675936i	$0.763739\pi$
$830$	0	0
$831$	5402.10	0.225508
$832$	0	0
$833$	5185.18	0.215673
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10091.3	0.416734
$838$	0	0
$839$	−9950.73	−0.409460	−0.204730	−	0.978818i	$-0.565632\pi$
$839$	−9950.73	−0.409460	−0.204730	+	0.978818i	$0.565632\pi$
$840$	0	0
$841$	−19722.7	−0.808674
$842$	0	0
$843$	−5699.04	−0.232842
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−32813.0	−1.33113
$848$	0	0
$849$	−12978.0	−0.524622
$850$	0	0
$851$	3434.96	0.138365
$852$	0	0
$853$	−36408.0	−1.46141	−0.730707	−	0.682691i	$-0.760810\pi$
$853$	−36408.0	−1.46141	−0.730707	+	0.682691i	$0.760810\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22788.4	0.908329	0.454164	−	0.890918i	$-0.349938\pi$
$857$	22788.4	0.908329	0.454164	+	0.890918i	$0.349938\pi$
$858$	0	0
$859$	−21675.6	−0.860956	−0.430478	−	0.902601i	$-0.641655\pi$
$859$	−21675.6	−0.860956	−0.430478	+	0.902601i	$0.641655\pi$
$860$	0	0
$861$	11045.4	0.437198
$862$	0	0
$863$	16485.2	0.650248	0.325124	−	0.945671i	$-0.394594\pi$
$863$	16485.2	0.650248	0.325124	+	0.945671i	$0.394594\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−15863.7	−0.621407
$868$	0	0
$869$	12476.4	0.487033
$870$	0	0
$871$	52101.4	2.02685
$872$	0	0
$873$	−702.184	−0.0272226
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9325.23	−0.359054	−0.179527	−	0.983753i	$-0.557457\pi$
$877$	−9325.23	−0.359054	−0.179527	+	0.983753i	$0.557457\pi$
$878$	0	0
$879$	−17155.9	−0.658309
$880$	0	0
$881$	−18523.9	−0.708384	−0.354192	−	0.935173i	$-0.615244\pi$
$881$	−18523.9	−0.708384	−0.354192	+	0.935173i	$0.615244\pi$
$882$	0	0
$883$	−44162.2	−1.68310	−0.841549	−	0.540180i	$-0.818356\pi$
$883$	−44162.2	−1.68310	−0.841549	+	0.540180i	$0.818356\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31348.8	1.18669	0.593343	−	0.804950i	$-0.297808\pi$
$887$	31348.8	1.18669	0.593343	+	0.804950i	$0.297808\pi$
$888$	0	0
$889$	10643.7	0.401551
$890$	0	0
$891$	−13017.3	−0.489444
$892$	0	0
$893$	−7280.09	−0.272810
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6648.90	0.247492
$898$	0	0
$899$	−4611.15	−0.171068
$900$	0	0
$901$	−1952.85	−0.0722076
$902$	0	0
$903$	−23620.3	−0.870469
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12555.3	0.459639	0.229820	−	0.973233i	$-0.426186\pi$
$907$	12555.3	0.459639	0.229820	+	0.973233i	$0.426186\pi$
$908$	0	0
$909$	−11650.7	−0.425115
$910$	0	0
$911$	21075.9	0.766495	0.383247	−	0.923646i	$-0.374806\pi$
$911$	21075.9	0.766495	0.383247	+	0.923646i	$0.374806\pi$
$912$	0	0
$913$	−15787.7	−0.572285
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21000.4	−0.756263
$918$	0	0
$919$	11091.1	0.398108	0.199054	−	0.979989i	$-0.436213\pi$
$919$	11091.1	0.398108	0.199054	+	0.979989i	$0.436213\pi$
$920$	0	0
$921$	−11199.2	−0.400678
$922$	0	0
$923$	−57923.8	−2.06564
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−2003.56	−0.0709876
$928$	0	0
$929$	−54912.1	−1.93930	−0.969648	−	0.244504i	$-0.921375\pi$
$929$	−54912.1	−1.93930	−0.969648	+	0.244504i	$0.921375\pi$
$930$	0	0
$931$	−10410.7	−0.366483
$932$	0	0
$933$	34487.4	1.21015
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20442.3	0.712721	0.356361	−	0.934348i	$-0.384017\pi$
$937$	20442.3	0.712721	0.356361	+	0.934348i	$0.384017\pi$
$938$	0	0
$939$	9408.50	0.326980
$940$	0	0
$941$	−32786.7	−1.13583	−0.567916	−	0.823087i	$-0.692250\pi$
$941$	−32786.7	−1.13583	−0.567916	+	0.823087i	$0.692250\pi$
$942$	0	0
$943$	−5745.37	−0.198404
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24034.5	−0.824726	−0.412363	−	0.911020i	$-0.635297\pi$
$947$	−24034.5	−0.824726	−0.412363	+	0.911020i	$0.635297\pi$
$948$	0	0
$949$	12603.7	0.431121
$950$	0	0
$951$	−33355.2	−1.13735
$952$	0	0
$953$	−41931.0	−1.42527	−0.712633	−	0.701537i	$-0.752498\pi$
$953$	−41931.0	−1.42527	−0.712633	+	0.701537i	$0.752498\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16298.0	0.550512
$958$	0	0
$959$	−35406.2	−1.19221
$960$	0	0
$961$	−25234.3	−0.847045
$962$	0	0
$963$	−18039.2	−0.603639
$964$	0	0
$965$	0	0
$966$	0	0
$967$	52217.2	1.73650	0.868249	−	0.496129i	$-0.165246\pi$
$967$	52217.2	1.73650	0.868249	+	0.496129i	$0.165246\pi$
$968$	0	0
$969$	−4917.61	−0.163030
$970$	0	0
$971$	−9432.54	−0.311745	−0.155873	−	0.987777i	$-0.549819\pi$
$971$	−9432.54	−0.311745	−0.155873	+	0.987777i	$0.549819\pi$
$972$	0	0
$973$	−35658.7	−1.17489
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6562.89	−0.214908	−0.107454	−	0.994210i	$-0.534270\pi$
$977$	−6562.89	−0.214908	−0.107454	+	0.994210i	$0.534270\pi$
$978$	0	0
$979$	−53521.9	−1.74726
$980$	0	0
$981$	5031.44	0.163753
$982$	0	0
$983$	−24052.6	−0.780426	−0.390213	−	0.920725i	$-0.627599\pi$
$983$	−24052.6	−0.780426	−0.390213	+	0.920725i	$0.627599\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6254.63	0.201709
$988$	0	0
$989$	12286.3	0.395026
$990$	0	0
$991$	47681.8	1.52842	0.764209	−	0.644969i	$-0.223130\pi$
$991$	47681.8	1.52842	0.764209	+	0.644969i	$0.223130\pi$
$992$	0	0
$993$	10617.2	0.339302
$994$	0	0
$995$	0	0
$996$	0	0
$997$	6495.42	0.206331	0.103166	−	0.994664i	$-0.467103\pi$
$997$	6495.42	0.206331	0.103166	+	0.994664i	$0.467103\pi$
$998$	0	0
$999$	−22326.3	−0.707080

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.4.a.c.1.4		5
5.2	odd	4		2300.4.c.d.1749.4		10
5.3	odd	4		2300.4.c.d.1749.7		10
5.4	even	2		460.4.a.b.1.2	✓	5
20.19	odd	2		1840.4.a.o.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.b.1.2	✓	5	5.4	even	2
1840.4.a.o.1.4		5	20.19	odd	2
2300.4.a.c.1.4		5	1.1	even	1	trivial
2300.4.c.d.1749.4		10	5.2	odd	4
2300.4.c.d.1749.7		10	5.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2300.a (trivial)

\(f(q)\)	\(=\)	\(q+3.72746 q^{3} -11.8626 q^{7} -13.1061 q^{9} +O(q^{10})\)	\(q+3.72746 q^{3} -11.8626 q^{7} -13.1061 q^{9} +64.0085 q^{11} +77.5549 q^{13} -25.6338 q^{17} +51.4669 q^{19} -44.2173 q^{21} +23.0000 q^{23} -149.494 q^{27} +68.3100 q^{29} -67.5033 q^{31} +238.589 q^{33} +149.346 q^{37} +289.083 q^{39} -249.799 q^{41} +534.186 q^{43} -141.452 q^{47} -202.279 q^{49} -95.5490 q^{51} +76.1827 q^{53} +191.841 q^{57} -816.330 q^{59} +356.152 q^{61} +155.472 q^{63} +671.800 q^{67} +85.7315 q^{69} -746.875 q^{71} +162.513 q^{73} -759.307 q^{77} +194.917 q^{79} -203.367 q^{81} -246.650 q^{83} +254.623 q^{87} -836.169 q^{89} -920.003 q^{91} -251.616 q^{93} +53.5771 q^{97} -838.900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 8 q^{7} + 30 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 - 8 * q^7 + 30 * q^9	\( 5 q + 3 q^{3} - 8 q^{7} + 30 q^{9} + 7 q^{11} - 5 q^{13} + 24 q^{17} - 13 q^{19} + 115 q^{23} + 204 q^{27} - 253 q^{29} - 98 q^{31} + 473 q^{33} + 435 q^{37} - 410 q^{39} - 774 q^{41} + 498 q^{43} + 572 q^{47} - 683 q^{49} - 657 q^{51} + 665 q^{53} + 932 q^{57} - 763 q^{59} + 337 q^{61} + 1527 q^{63} + 305 q^{67} + 69 q^{69} - 1504 q^{71} + 1304 q^{73} - 182 q^{77} + 626 q^{79} - 959 q^{81} + 1703 q^{83} + 1354 q^{87} + 646 q^{89} - 767 q^{91} + 452 q^{93} + 233 q^{97} + 111 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 8 * q^7 + 30 * q^9 + 7 * q^11 - 5 * q^13 + 24 * q^17 - 13 * q^19 + 115 * q^23 + 204 * q^27 - 253 * q^29 - 98 * q^31 + 473 * q^33 + 435 * q^37 - 410 * q^39 - 774 * q^41 + 498 * q^43 + 572 * q^47 - 683 * q^49 - 657 * q^51 + 665 * q^53 + 932 * q^57 - 763 * q^59 + 337 * q^61 + 1527 * q^63 + 305 * q^67 + 69 * q^69 - 1504 * q^71 + 1304 * q^73 - 182 * q^77 + 626 * q^79 - 959 * q^81 + 1703 * q^83 + 1354 * q^87 + 646 * q^89 - 767 * q^91 + 452 * q^93 + 233 * q^97 + 111 * q^99

Introduction

L-functions

Modular forms

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Database

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