LMFDB - Embedded newform 1004.2.a.a.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1004,2,Mod(1,1004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1004 = 2^{2} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1004.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:	$8.01698036294$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 18x^{4} + 8x^{3} - 17x^{2} - 9x - 1 $ x^7 - 3x^6 - 6x^5 + 18x^4 + 8x^3 - 17x^2 - 9x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.18229$ of defining polynomial
Character	$\chi$	$=$	1004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18229 q^{3} -1.66714 q^{5} -1.81734 q^{7} +1.76240 q^{9} +O(q^{10})$	$q+2.18229 q^{3} -1.66714 q^{5} -1.81734 q^{7} +1.76240 q^{9} -3.97535 q^{11} -4.88440 q^{13} -3.63819 q^{15} -3.09643 q^{17} -3.71976 q^{19} -3.96596 q^{21} +2.52167 q^{23} -2.22063 q^{25} -2.70081 q^{27} +4.16232 q^{29} +4.17959 q^{31} -8.67538 q^{33} +3.02977 q^{35} +11.0695 q^{37} -10.6592 q^{39} -0.0987896 q^{41} -2.04789 q^{43} -2.93817 q^{45} +0.469170 q^{47} -3.69728 q^{49} -6.75731 q^{51} -3.85883 q^{53} +6.62749 q^{55} -8.11759 q^{57} +7.09924 q^{59} -0.640361 q^{61} -3.20287 q^{63} +8.14300 q^{65} +1.93468 q^{67} +5.50301 q^{69} +0.867080 q^{71} -6.51437 q^{73} -4.84606 q^{75} +7.22456 q^{77} -9.82946 q^{79} -11.1811 q^{81} +12.6647 q^{83} +5.16219 q^{85} +9.08339 q^{87} -9.81448 q^{89} +8.87662 q^{91} +9.12109 q^{93} +6.20137 q^{95} -1.98395 q^{97} -7.00615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7}+O(q^{10}) $ 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7	$ 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7} - 5 q^{11} - q^{13} - 6 q^{15} - 8 q^{17} - 15 q^{19} - 3 q^{21} - 5 q^{23} - 9 q^{25} - 9 q^{27} - 21 q^{31} - 7 q^{35} - q^{37} - 23 q^{39} - 10 q^{41} - 23 q^{43} - 4 q^{45} - 10 q^{47} - 13 q^{49} - 20 q^{51} - q^{53} - 23 q^{55} - 6 q^{57} - 4 q^{59} + 3 q^{61} - 4 q^{63} + 4 q^{65} - 28 q^{67} + 18 q^{69} - 18 q^{71} - 7 q^{73} + 11 q^{75} + 6 q^{77} - 30 q^{79} - 5 q^{81} + 13 q^{83} + q^{85} - 7 q^{87} - 18 q^{91} + 36 q^{93} - 2 q^{97} - 10 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7 - 5 * q^11 - q^13 - 6 * q^15 - 8 * q^17 - 15 * q^19 - 3 * q^21 - 5 * q^23 - 9 * q^25 - 9 * q^27 - 21 * q^31 - 7 * q^35 - q^37 - 23 * q^39 - 10 * q^41 - 23 * q^43 - 4 * q^45 - 10 * q^47 - 13 * q^49 - 20 * q^51 - q^53 - 23 * q^55 - 6 * q^57 - 4 * q^59 + 3 * q^61 - 4 * q^63 + 4 * q^65 - 28 * q^67 + 18 * q^69 - 18 * q^71 - 7 * q^73 + 11 * q^75 + 6 * q^77 - 30 * q^79 - 5 * q^81 + 13 * q^83 + q^85 - 7 * q^87 - 18 * q^91 + 36 * q^93 - 2 * q^97 - 10 * 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
$3$	2.18229	1.25995	0.629973	−	0.776617i	$-0.283066\pi$
$3$	2.18229	1.25995	0.629973	+	0.776617i	$0.283066\pi$
$4$	0	0
$5$	−1.66714	−0.745569	−0.372785	−	0.927918i	$-0.621597\pi$
$5$	−1.66714	−0.745569	−0.372785	+	0.927918i	$0.621597\pi$
$6$	0	0
$7$	−1.81734	−0.686890	−0.343445	−	0.939173i	$-0.611594\pi$
$7$	−1.81734	−0.686890	−0.343445	+	0.939173i	$0.611594\pi$
$8$	0	0
$9$	1.76240	0.587466
$10$	0	0
$11$	−3.97535	−1.19861	−0.599307	−	0.800519i	$-0.704557\pi$
$11$	−3.97535	−1.19861	−0.599307	+	0.800519i	$0.704557\pi$
$12$	0	0
$13$	−4.88440	−1.35469	−0.677345	−	0.735666i	$-0.736869\pi$
$13$	−4.88440	−1.35469	−0.677345	+	0.735666i	$0.736869\pi$
$14$	0	0
$15$	−3.63819	−0.939378
$16$	0	0
$17$	−3.09643	−0.750994	−0.375497	−	0.926824i	$-0.622528\pi$
$17$	−3.09643	−0.750994	−0.375497	+	0.926824i	$0.622528\pi$
$18$	0	0
$19$	−3.71976	−0.853370	−0.426685	−	0.904400i	$-0.640319\pi$
$19$	−3.71976	−0.853370	−0.426685	+	0.904400i	$0.640319\pi$
$20$	0	0
$21$	−3.96596	−0.865444
$22$	0	0
$23$	2.52167	0.525804	0.262902	−	0.964823i	$-0.415320\pi$
$23$	2.52167	0.525804	0.262902	+	0.964823i	$0.415320\pi$
$24$	0	0
$25$	−2.22063	−0.444126
$26$	0	0
$27$	−2.70081	−0.519771
$28$	0	0
$29$	4.16232	0.772923	0.386461	−	0.922306i	$-0.373697\pi$
$29$	4.16232	0.772923	0.386461	+	0.922306i	$0.373697\pi$
$30$	0	0
$31$	4.17959	0.750677	0.375338	−	0.926888i	$-0.377527\pi$
$31$	4.17959	0.750677	0.375338	+	0.926888i	$0.377527\pi$
$32$	0	0
$33$	−8.67538	−1.51019
$34$	0	0
$35$	3.02977	0.512124
$36$	0	0
$37$	11.0695	1.81981	0.909903	−	0.414820i	$-0.136156\pi$
$37$	11.0695	1.81981	0.909903	+	0.414820i	$0.136156\pi$
$38$	0	0
$39$	−10.6592	−1.70684
$40$	0	0
$41$	−0.0987896	−0.0154283	−0.00771417	−	0.999970i	$-0.502456\pi$
$41$	−0.0987896	−0.0154283	−0.00771417	+	0.999970i	$0.502456\pi$
$42$	0	0
$43$	−2.04789	−0.312301	−0.156150	−	0.987733i	$-0.549908\pi$
$43$	−2.04789	−0.312301	−0.156150	+	0.987733i	$0.549908\pi$
$44$	0	0
$45$	−2.93817	−0.437996
$46$	0	0
$47$	0.469170	0.0684355	0.0342177	−	0.999414i	$-0.489106\pi$
$47$	0.469170	0.0684355	0.0342177	+	0.999414i	$0.489106\pi$
$48$	0	0
$49$	−3.69728	−0.528183
$50$	0	0
$51$	−6.75731	−0.946213
$52$	0	0
$53$	−3.85883	−0.530050	−0.265025	−	0.964241i	$-0.585380\pi$
$53$	−3.85883	−0.530050	−0.265025	+	0.964241i	$0.585380\pi$
$54$	0	0
$55$	6.62749	0.893650
$56$	0	0
$57$	−8.11759	−1.07520
$58$	0	0
$59$	7.09924	0.924242	0.462121	−	0.886817i	$-0.347089\pi$
$59$	7.09924	0.924242	0.462121	+	0.886817i	$0.347089\pi$
$60$	0	0
$61$	−0.640361	−0.0819898	−0.0409949	−	0.999159i	$-0.513053\pi$
$61$	−0.640361	−0.0819898	−0.0409949	+	0.999159i	$0.513053\pi$
$62$	0	0
$63$	−3.20287	−0.403524
$64$	0	0
$65$	8.14300	1.01002
$66$	0	0
$67$	1.93468	0.236359	0.118180	−	0.992992i	$-0.462294\pi$
$67$	1.93468	0.236359	0.118180	+	0.992992i	$0.462294\pi$
$68$	0	0
$69$	5.50301	0.662485
$70$	0	0
$71$	0.867080	0.102903	0.0514517	−	0.998675i	$-0.483615\pi$
$71$	0.867080	0.102903	0.0514517	+	0.998675i	$0.483615\pi$
$72$	0	0
$73$	−6.51437	−0.762450	−0.381225	−	0.924482i	$-0.624498\pi$
$73$	−6.51437	−0.762450	−0.381225	+	0.924482i	$0.624498\pi$
$74$	0	0
$75$	−4.84606	−0.559575
$76$	0	0
$77$	7.22456	0.823315
$78$	0	0
$79$	−9.82946	−1.10590	−0.552950	−	0.833214i	$-0.686498\pi$
$79$	−9.82946	−1.10590	−0.552950	+	0.833214i	$0.686498\pi$
$80$	0	0
$81$	−11.1811	−1.24235
$82$	0	0
$83$	12.6647	1.39013	0.695064	−	0.718948i	$-0.255376\pi$
$83$	12.6647	1.39013	0.695064	+	0.718948i	$0.255376\pi$
$84$	0	0
$85$	5.16219	0.559918
$86$	0	0
$87$	9.08339	0.973842
$88$	0	0
$89$	−9.81448	−1.04033	−0.520166	−	0.854065i	$-0.674130\pi$
$89$	−9.81448	−1.04033	−0.520166	+	0.854065i	$0.674130\pi$
$90$	0	0
$91$	8.87662	0.930522
$92$	0	0
$93$	9.12109	0.945813
$94$	0	0
$95$	6.20137	0.636247
$96$	0	0
$97$	−1.98395	−0.201439	−0.100720	−	0.994915i	$-0.532115\pi$
$97$	−1.98395	−0.201439	−0.100720	+	0.994915i	$0.532115\pi$
$98$	0	0
$99$	−7.00615	−0.704145
$100$	0	0
$101$	11.6322	1.15745	0.578724	−	0.815523i	$-0.303551\pi$
$101$	11.6322	1.15745	0.578724	+	0.815523i	$0.303551\pi$
$102$	0	0
$103$	8.01880	0.790116	0.395058	−	0.918656i	$-0.370725\pi$
$103$	8.01880	0.790116	0.395058	+	0.918656i	$0.370725\pi$
$104$	0	0
$105$	6.61183	0.645249
$106$	0	0
$107$	−13.9472	−1.34833	−0.674165	−	0.738580i	$-0.735496\pi$
$107$	−13.9472	−1.34833	−0.674165	+	0.738580i	$0.735496\pi$
$108$	0	0
$109$	−7.73637	−0.741010	−0.370505	−	0.928831i	$-0.620815\pi$
$109$	−7.73637	−0.741010	−0.370505	+	0.928831i	$0.620815\pi$
$110$	0	0
$111$	24.1568	2.29286
$112$	0	0
$113$	−5.22437	−0.491468	−0.245734	−	0.969337i	$-0.579029\pi$
$113$	−5.22437	−0.491468	−0.245734	+	0.969337i	$0.579029\pi$
$114$	0	0
$115$	−4.20398	−0.392023
$116$	0	0
$117$	−8.60826	−0.795834
$118$	0	0
$119$	5.62726	0.515850
$120$	0	0
$121$	4.80343	0.436676
$122$	0	0
$123$	−0.215588	−0.0194389
$124$	0	0
$125$	12.0378	1.07670
$126$	0	0
$127$	−17.3142	−1.53639	−0.768195	−	0.640215i	$-0.778845\pi$
$127$	−17.3142	−1.53639	−0.768195	+	0.640215i	$0.778845\pi$
$128$	0	0
$129$	−4.46910	−0.393482
$130$	0	0
$131$	0.297840	0.0260224	0.0130112	−	0.999915i	$-0.495858\pi$
$131$	0.297840	0.0260224	0.0130112	+	0.999915i	$0.495858\pi$
$132$	0	0
$133$	6.76006	0.586171
$134$	0	0
$135$	4.50264	0.387526
$136$	0	0
$137$	10.9689	0.937133	0.468567	−	0.883428i	$-0.344771\pi$
$137$	10.9689	0.937133	0.468567	+	0.883428i	$0.344771\pi$
$138$	0	0
$139$	−22.5994	−1.91685	−0.958427	−	0.285338i	$-0.907894\pi$
$139$	−22.5994	−1.91685	−0.958427	+	0.285338i	$0.907894\pi$
$140$	0	0
$141$	1.02387	0.0862251
$142$	0	0
$143$	19.4172	1.62375
$144$	0	0
$145$	−6.93918	−0.576268
$146$	0	0
$147$	−8.06854	−0.665482
$148$	0	0
$149$	8.53848	0.699499	0.349750	−	0.936843i	$-0.386267\pi$
$149$	8.53848	0.699499	0.349750	+	0.936843i	$0.386267\pi$
$150$	0	0
$151$	−11.5305	−0.938341	−0.469170	−	0.883108i	$-0.655447\pi$
$151$	−11.5305	−0.938341	−0.469170	+	0.883108i	$0.655447\pi$
$152$	0	0
$153$	−5.45714	−0.441183
$154$	0	0
$155$	−6.96798	−0.559682
$156$	0	0
$157$	0.177058	0.0141308	0.00706539	−	0.999975i	$-0.497751\pi$
$157$	0.177058	0.0141308	0.00706539	+	0.999975i	$0.497751\pi$
$158$	0	0
$159$	−8.42108	−0.667835
$160$	0	0
$161$	−4.58272	−0.361169
$162$	0	0
$163$	−13.7654	−1.07819	−0.539095	−	0.842245i	$-0.681234\pi$
$163$	−13.7654	−1.07819	−0.539095	+	0.842245i	$0.681234\pi$
$164$	0	0
$165$	14.4631	1.12595
$166$	0	0
$167$	−9.58570	−0.741764	−0.370882	−	0.928680i	$-0.620945\pi$
$167$	−9.58570	−0.741764	−0.370882	+	0.928680i	$0.620945\pi$
$168$	0	0
$169$	10.8574	0.835184
$170$	0	0
$171$	−6.55568	−0.501326
$172$	0	0
$173$	−7.27316	−0.552968	−0.276484	−	0.961019i	$-0.589169\pi$
$173$	−7.27316	−0.552968	−0.276484	+	0.961019i	$0.589169\pi$
$174$	0	0
$175$	4.03564	0.305066
$176$	0	0
$177$	15.4926	1.16450
$178$	0	0
$179$	−17.2832	−1.29180	−0.645902	−	0.763420i	$-0.723519\pi$
$179$	−17.2832	−1.29180	−0.645902	+	0.763420i	$0.723519\pi$
$180$	0	0
$181$	19.8478	1.47528	0.737639	−	0.675195i	$-0.235941\pi$
$181$	19.8478	1.47528	0.737639	+	0.675195i	$0.235941\pi$
$182$	0	0
$183$	−1.39745	−0.103303
$184$	0	0
$185$	−18.4544	−1.35679
$186$	0	0
$187$	12.3094	0.900152
$188$	0	0
$189$	4.90829	0.357025
$190$	0	0
$191$	−23.3899	−1.69244	−0.846218	−	0.532837i	$-0.821126\pi$
$191$	−23.3899	−1.69244	−0.846218	+	0.532837i	$0.821126\pi$
$192$	0	0
$193$	−12.5992	−0.906909	−0.453454	−	0.891280i	$-0.649808\pi$
$193$	−12.5992	−0.906909	−0.453454	+	0.891280i	$0.649808\pi$
$194$	0	0
$195$	17.7704	1.27257
$196$	0	0
$197$	2.28151	0.162551	0.0812753	−	0.996692i	$-0.474101\pi$
$197$	2.28151	0.162551	0.0812753	+	0.996692i	$0.474101\pi$
$198$	0	0
$199$	−2.10342	−0.149108	−0.0745538	−	0.997217i	$-0.523753\pi$
$199$	−2.10342	−0.149108	−0.0745538	+	0.997217i	$0.523753\pi$
$200$	0	0
$201$	4.22204	0.297800
$202$	0	0
$203$	−7.56434	−0.530913
$204$	0	0
$205$	0.164696	0.0115029
$206$	0	0
$207$	4.44418	0.308892
$208$	0	0
$209$	14.7873	1.02286
$210$	0	0
$211$	0.501747	0.0345417	0.0172709	−	0.999851i	$-0.494502\pi$
$211$	0.501747	0.0345417	0.0172709	+	0.999851i	$0.494502\pi$
$212$	0	0
$213$	1.89222	0.129653
$214$	0	0
$215$	3.41413	0.232842
$216$	0	0
$217$	−7.59573	−0.515632
$218$	0	0
$219$	−14.2163	−0.960646
$220$	0	0
$221$	15.1242	1.01736
$222$	0	0
$223$	0.964032	0.0645563	0.0322782	−	0.999479i	$-0.489724\pi$
$223$	0.964032	0.0645563	0.0322782	+	0.999479i	$0.489724\pi$
$224$	0	0
$225$	−3.91363	−0.260909
$226$	0	0
$227$	20.6598	1.37124	0.685619	−	0.727961i	$-0.259532\pi$
$227$	20.6598	1.37124	0.685619	+	0.727961i	$0.259532\pi$
$228$	0	0
$229$	7.37963	0.487660	0.243830	−	0.969818i	$-0.421596\pi$
$229$	7.37963	0.487660	0.243830	+	0.969818i	$0.421596\pi$
$230$	0	0
$231$	15.7661	1.03733
$232$	0	0
$233$	21.1272	1.38409	0.692045	−	0.721854i	$-0.256710\pi$
$233$	21.1272	1.38409	0.692045	+	0.721854i	$0.256710\pi$
$234$	0	0
$235$	−0.782174	−0.0510234
$236$	0	0
$237$	−21.4508	−1.39338
$238$	0	0
$239$	−1.35509	−0.0876538	−0.0438269	−	0.999039i	$-0.513955\pi$
$239$	−1.35509	−0.0876538	−0.0438269	+	0.999039i	$0.513955\pi$
$240$	0	0
$241$	26.2538	1.69115	0.845577	−	0.533854i	$-0.179257\pi$
$241$	26.2538	1.69115	0.845577	+	0.533854i	$0.179257\pi$
$242$	0	0
$243$	−16.2981	−1.04552
$244$	0	0
$245$	6.16390	0.393797
$246$	0	0
$247$	18.1688	1.15605
$248$	0	0
$249$	27.6380	1.75149
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−10.0245	−0.630236
$254$	0	0
$255$	11.2654	0.705467
$256$	0	0
$257$	8.74279	0.545360	0.272680	−	0.962105i	$-0.412090\pi$
$257$	8.74279	0.545360	0.272680	+	0.962105i	$0.412090\pi$
$258$	0	0
$259$	−20.1169	−1.25001
$260$	0	0
$261$	7.33565	0.454066
$262$	0	0
$263$	19.4251	1.19780	0.598901	−	0.800823i	$-0.295604\pi$
$263$	19.4251	1.19780	0.598901	+	0.800823i	$0.295604\pi$
$264$	0	0
$265$	6.43322	0.395189
$266$	0	0
$267$	−21.4180	−1.31076
$268$	0	0
$269$	7.73284	0.471480	0.235740	−	0.971816i	$-0.424249\pi$
$269$	7.73284	0.471480	0.235740	+	0.971816i	$0.424249\pi$
$270$	0	0
$271$	−23.7030	−1.43985	−0.719927	−	0.694050i	$-0.755825\pi$
$271$	−23.7030	−1.43985	−0.719927	+	0.694050i	$0.755825\pi$
$272$	0	0
$273$	19.3714	1.17241
$274$	0	0
$275$	8.82779	0.532336
$276$	0	0
$277$	1.75856	0.105662	0.0528308	−	0.998603i	$-0.483176\pi$
$277$	1.75856	0.105662	0.0528308	+	0.998603i	$0.483176\pi$
$278$	0	0
$279$	7.36610	0.440997
$280$	0	0
$281$	20.0290	1.19483	0.597414	−	0.801933i	$-0.296195\pi$
$281$	20.0290	1.19483	0.597414	+	0.801933i	$0.296195\pi$
$282$	0	0
$283$	5.84251	0.347301	0.173651	−	0.984807i	$-0.444444\pi$
$283$	5.84251	0.347301	0.173651	+	0.984807i	$0.444444\pi$
$284$	0	0
$285$	13.5332	0.801637
$286$	0	0
$287$	0.179534	0.0105976
$288$	0	0
$289$	−7.41213	−0.436008
$290$	0	0
$291$	−4.32955	−0.253803
$292$	0	0
$293$	−3.05347	−0.178385	−0.0891927	−	0.996014i	$-0.528429\pi$
$293$	−3.05347	−0.178385	−0.0891927	+	0.996014i	$0.528429\pi$
$294$	0	0
$295$	−11.8354	−0.689087
$296$	0	0
$297$	10.7367	0.623005
$298$	0	0
$299$	−12.3168	−0.712301
$300$	0	0
$301$	3.72172	0.214516
$302$	0	0
$303$	25.3849	1.45832
$304$	0	0
$305$	1.06757	0.0611291
$306$	0	0
$307$	−14.0535	−0.802073	−0.401037	−	0.916062i	$-0.631350\pi$
$307$	−14.0535	−0.802073	−0.401037	+	0.916062i	$0.631350\pi$
$308$	0	0
$309$	17.4994	0.995504
$310$	0	0
$311$	−23.9822	−1.35990	−0.679952	−	0.733256i	$-0.738000\pi$
$311$	−23.9822	−1.35990	−0.679952	+	0.733256i	$0.738000\pi$
$312$	0	0
$313$	27.3953	1.54847	0.774237	−	0.632896i	$-0.218134\pi$
$313$	27.3953	1.54847	0.774237	+	0.632896i	$0.218134\pi$
$314$	0	0
$315$	5.33965	0.300855
$316$	0	0
$317$	−12.5013	−0.702143	−0.351072	−	0.936349i	$-0.614183\pi$
$317$	−12.5013	−0.702143	−0.351072	+	0.936349i	$0.614183\pi$
$318$	0	0
$319$	−16.5467	−0.926436
$320$	0	0
$321$	−30.4369	−1.69882
$322$	0	0
$323$	11.5180	0.640876
$324$	0	0
$325$	10.8465	0.601653
$326$	0	0
$327$	−16.8830	−0.933633
$328$	0	0
$329$	−0.852641	−0.0470076
$330$	0	0
$331$	25.6129	1.40782	0.703908	−	0.710292i	$-0.251437\pi$
$331$	25.6129	1.40782	0.703908	+	0.710292i	$0.251437\pi$
$332$	0	0
$333$	19.5088	1.06907
$334$	0	0
$335$	−3.22540	−0.176222
$336$	0	0
$337$	−9.84964	−0.536544	−0.268272	−	0.963343i	$-0.586453\pi$
$337$	−9.84964	−0.536544	−0.268272	+	0.963343i	$0.586453\pi$
$338$	0	0
$339$	−11.4011	−0.619223
$340$	0	0
$341$	−16.6154	−0.899772
$342$	0	0
$343$	19.4406	1.04969
$344$	0	0
$345$	−9.17431	−0.493928
$346$	0	0
$347$	−6.11780	−0.328421	−0.164210	−	0.986425i	$-0.552508\pi$
$347$	−6.11780	−0.328421	−0.164210	+	0.986425i	$0.552508\pi$
$348$	0	0
$349$	−9.79907	−0.524532	−0.262266	−	0.964996i	$-0.584470\pi$
$349$	−9.79907	−0.524532	−0.262266	+	0.964996i	$0.584470\pi$
$350$	0	0
$351$	13.1918	0.704129
$352$	0	0
$353$	−6.86642	−0.365463	−0.182731	−	0.983163i	$-0.558494\pi$
$353$	−6.86642	−0.365463	−0.182731	+	0.983163i	$0.558494\pi$
$354$	0	0
$355$	−1.44555	−0.0767216
$356$	0	0
$357$	12.2803	0.649944
$358$	0	0
$359$	34.4483	1.81811	0.909055	−	0.416676i	$-0.136805\pi$
$359$	34.4483	1.81811	0.909055	+	0.416676i	$0.136805\pi$
$360$	0	0
$361$	−5.16342	−0.271759
$362$	0	0
$363$	10.4825	0.550188
$364$	0	0
$365$	10.8604	0.568459
$366$	0	0
$367$	−33.8538	−1.76716	−0.883578	−	0.468285i	$-0.844872\pi$
$367$	−33.8538	−1.76716	−0.883578	+	0.468285i	$0.844872\pi$
$368$	0	0
$369$	−0.174106	−0.00906362
$370$	0	0
$371$	7.01279	0.364086
$372$	0	0
$373$	−36.7580	−1.90325	−0.951627	−	0.307254i	$-0.900590\pi$
$373$	−36.7580	−1.90325	−0.951627	+	0.307254i	$0.900590\pi$
$374$	0	0
$375$	26.2701	1.35658
$376$	0	0
$377$	−20.3304	−1.04707
$378$	0	0
$379$	−21.6407	−1.11161	−0.555803	−	0.831314i	$-0.687589\pi$
$379$	−21.6407	−1.11161	−0.555803	+	0.831314i	$0.687589\pi$
$380$	0	0
$381$	−37.7847	−1.93577
$382$	0	0
$383$	24.3327	1.24334	0.621671	−	0.783279i	$-0.286454\pi$
$383$	24.3327	1.24334	0.621671	+	0.783279i	$0.286454\pi$
$384$	0	0
$385$	−12.0444	−0.613839
$386$	0	0
$387$	−3.60920	−0.183466
$388$	0	0
$389$	25.6714	1.30159	0.650797	−	0.759252i	$-0.274435\pi$
$389$	25.6714	1.30159	0.650797	+	0.759252i	$0.274435\pi$
$390$	0	0
$391$	−7.80816	−0.394876
$392$	0	0
$393$	0.649973	0.0327868
$394$	0	0
$395$	16.3871	0.824526
$396$	0	0
$397$	18.8682	0.946970	0.473485	−	0.880802i	$-0.342996\pi$
$397$	18.8682	0.946970	0.473485	+	0.880802i	$0.342996\pi$
$398$	0	0
$399$	14.7524	0.738544
$400$	0	0
$401$	−0.851616	−0.0425277	−0.0212638	−	0.999774i	$-0.506769\pi$
$401$	−0.851616	−0.0425277	−0.0212638	+	0.999774i	$0.506769\pi$
$402$	0	0
$403$	−20.4148	−1.01693
$404$	0	0
$405$	18.6406	0.926258
$406$	0	0
$407$	−44.0050	−2.18125
$408$	0	0
$409$	4.86899	0.240756	0.120378	−	0.992728i	$-0.461589\pi$
$409$	4.86899	0.240756	0.120378	+	0.992728i	$0.461589\pi$
$410$	0	0
$411$	23.9373	1.18074
$412$	0	0
$413$	−12.9017	−0.634852
$414$	0	0
$415$	−21.1138	−1.03644
$416$	0	0
$417$	−49.3184	−2.41513
$418$	0	0
$419$	10.3339	0.504843	0.252422	−	0.967617i	$-0.418773\pi$
$419$	10.3339	0.504843	0.252422	+	0.967617i	$0.418773\pi$
$420$	0	0
$421$	24.5337	1.19570	0.597849	−	0.801608i	$-0.296022\pi$
$421$	24.5337	1.19570	0.597849	+	0.801608i	$0.296022\pi$
$422$	0	0
$423$	0.826864	0.0402035
$424$	0	0
$425$	6.87602	0.333536
$426$	0	0
$427$	1.16375	0.0563179
$428$	0	0
$429$	42.3741	2.04584
$430$	0	0
$431$	1.65100	0.0795257	0.0397628	−	0.999209i	$-0.487340\pi$
$431$	1.65100	0.0795257	0.0397628	+	0.999209i	$0.487340\pi$
$432$	0	0
$433$	19.0172	0.913907	0.456953	−	0.889491i	$-0.348941\pi$
$433$	19.0172	0.913907	0.456953	+	0.889491i	$0.348941\pi$
$434$	0	0
$435$	−15.1433	−0.726067
$436$	0	0
$437$	−9.37998	−0.448705
$438$	0	0
$439$	−31.7908	−1.51729	−0.758645	−	0.651504i	$-0.774138\pi$
$439$	−31.7908	−1.51729	−0.758645	+	0.651504i	$0.774138\pi$
$440$	0	0
$441$	−6.51607	−0.310289
$442$	0	0
$443$	7.00408	0.332774	0.166387	−	0.986061i	$-0.446790\pi$
$443$	7.00408	0.332774	0.166387	+	0.986061i	$0.446790\pi$
$444$	0	0
$445$	16.3621	0.775640
$446$	0	0
$447$	18.6334	0.881332
$448$	0	0
$449$	−39.4671	−1.86257	−0.931283	−	0.364295i	$-0.881310\pi$
$449$	−39.4671	−1.86257	−0.931283	+	0.364295i	$0.881310\pi$
$450$	0	0
$451$	0.392723	0.0184926
$452$	0	0
$453$	−25.1630	−1.18226
$454$	0	0
$455$	−14.7986	−0.693769
$456$	0	0
$457$	−14.8500	−0.694654	−0.347327	−	0.937744i	$-0.612911\pi$
$457$	−14.8500	−0.694654	−0.347327	+	0.937744i	$0.612911\pi$
$458$	0	0
$459$	8.36287	0.390345
$460$	0	0
$461$	−23.5063	−1.09480	−0.547399	−	0.836872i	$-0.684382\pi$
$461$	−23.5063	−1.09480	−0.547399	+	0.836872i	$0.684382\pi$
$462$	0	0
$463$	−7.59484	−0.352962	−0.176481	−	0.984304i	$-0.556471\pi$
$463$	−7.59484	−0.352962	−0.176481	+	0.984304i	$0.556471\pi$
$464$	0	0
$465$	−15.2062	−0.705169
$466$	0	0
$467$	1.26788	0.0586707	0.0293354	−	0.999570i	$-0.490661\pi$
$467$	1.26788	0.0586707	0.0293354	+	0.999570i	$0.490661\pi$
$468$	0	0
$469$	−3.51597	−0.162353
$470$	0	0
$471$	0.386393	0.0178040
$472$	0	0
$473$	8.14110	0.374328
$474$	0	0
$475$	8.26020	0.379004
$476$	0	0
$477$	−6.80078	−0.311386
$478$	0	0
$479$	−11.4831	−0.524674	−0.262337	−	0.964976i	$-0.584493\pi$
$479$	−11.4831	−0.524674	−0.262337	+	0.964976i	$0.584493\pi$
$480$	0	0
$481$	−54.0677	−2.46527
$482$	0	0
$483$	−10.0008	−0.455054
$484$	0	0
$485$	3.30752	0.150187
$486$	0	0
$487$	−18.5144	−0.838968	−0.419484	−	0.907763i	$-0.637789\pi$
$487$	−18.5144	−0.838968	−0.419484	+	0.907763i	$0.637789\pi$
$488$	0	0
$489$	−30.0401	−1.35846
$490$	0	0
$491$	11.3486	0.512155	0.256077	−	0.966656i	$-0.417570\pi$
$491$	11.3486	0.512155	0.256077	+	0.966656i	$0.417570\pi$
$492$	0	0
$493$	−12.8883	−0.580461
$494$	0	0
$495$	11.6803	0.524989
$496$	0	0
$497$	−1.57578	−0.0706833
$498$	0	0
$499$	−11.8934	−0.532423	−0.266212	−	0.963915i	$-0.585772\pi$
$499$	−11.8934	−0.532423	−0.266212	+	0.963915i	$0.585772\pi$
$500$	0	0
$501$	−20.9188	−0.934583
$502$	0	0
$503$	−26.9893	−1.20339	−0.601697	−	0.798725i	$-0.705508\pi$
$503$	−26.9893	−1.20339	−0.601697	+	0.798725i	$0.705508\pi$
$504$	0	0
$505$	−19.3926	−0.862958
$506$	0	0
$507$	23.6940	1.05229
$508$	0	0
$509$	0.334531	0.0148278	0.00741391	−	0.999973i	$-0.497640\pi$
$509$	0.334531	0.0148278	0.00741391	+	0.999973i	$0.497640\pi$
$510$	0	0
$511$	11.8388	0.523719
$512$	0	0
$513$	10.0464	0.443557
$514$	0	0
$515$	−13.3685	−0.589086
$516$	0	0
$517$	−1.86512	−0.0820277
$518$	0	0
$519$	−15.8721	−0.696710
$520$	0	0
$521$	22.3238	0.978023	0.489012	−	0.872277i	$-0.337358\pi$
$521$	22.3238	0.978023	0.489012	+	0.872277i	$0.337358\pi$
$522$	0	0
$523$	−14.1603	−0.619188	−0.309594	−	0.950869i	$-0.600193\pi$
$523$	−14.1603	−0.619188	−0.309594	+	0.950869i	$0.600193\pi$
$524$	0	0
$525$	8.80694	0.384366
$526$	0	0
$527$	−12.9418	−0.563754
$528$	0	0
$529$	−16.6412	−0.723530
$530$	0	0
$531$	12.5117	0.542960
$532$	0	0
$533$	0.482528	0.0209006
$534$	0	0
$535$	23.2521	1.00527
$536$	0	0
$537$	−37.7169	−1.62760
$538$	0	0
$539$	14.6980	0.633087
$540$	0	0
$541$	18.9931	0.816577	0.408289	−	0.912853i	$-0.366126\pi$
$541$	18.9931	0.816577	0.408289	+	0.912853i	$0.366126\pi$
$542$	0	0
$543$	43.3138	1.85877
$544$	0	0
$545$	12.8976	0.552474
$546$	0	0
$547$	−29.9596	−1.28098	−0.640490	−	0.767967i	$-0.721269\pi$
$547$	−29.9596	−1.28098	−0.640490	+	0.767967i	$0.721269\pi$
$548$	0	0
$549$	−1.12857	−0.0481662
$550$	0	0
$551$	−15.4828	−0.659589
$552$	0	0
$553$	17.8635	0.759632
$554$	0	0
$555$	−40.2728	−1.70949
$556$	0	0
$557$	−38.9749	−1.65142	−0.825709	−	0.564096i	$-0.809225\pi$
$557$	−38.9749	−1.65142	−0.825709	+	0.564096i	$0.809225\pi$
$558$	0	0
$559$	10.0027	0.423071
$560$	0	0
$561$	26.8627	1.13414
$562$	0	0
$563$	10.0149	0.422080	0.211040	−	0.977477i	$-0.432315\pi$
$563$	10.0149	0.422080	0.211040	+	0.977477i	$0.432315\pi$
$564$	0	0
$565$	8.70978	0.366423
$566$	0	0
$567$	20.3199	0.853357
$568$	0	0
$569$	5.81832	0.243917	0.121958	−	0.992535i	$-0.461083\pi$
$569$	5.81832	0.243917	0.121958	+	0.992535i	$0.461083\pi$
$570$	0	0
$571$	−16.0941	−0.673516	−0.336758	−	0.941591i	$-0.609330\pi$
$571$	−16.0941	−0.673516	−0.336758	+	0.941591i	$0.609330\pi$
$572$	0	0
$573$	−51.0436	−2.13238
$574$	0	0
$575$	−5.59969	−0.233523
$576$	0	0
$577$	0.178763	0.00744199	0.00372100	−	0.999993i	$-0.498816\pi$
$577$	0.178763	0.00744199	0.00372100	+	0.999993i	$0.498816\pi$
$578$	0	0
$579$	−27.4951	−1.14266
$580$	0	0
$581$	−23.0160	−0.954864
$582$	0	0
$583$	15.3402	0.635326
$584$	0	0
$585$	14.3512	0.593349
$586$	0	0
$587$	25.2001	1.04012	0.520060	−	0.854130i	$-0.325910\pi$
$587$	25.2001	1.04012	0.520060	+	0.854130i	$0.325910\pi$
$588$	0	0
$589$	−15.5471	−0.640605
$590$	0	0
$591$	4.97891	0.204805
$592$	0	0
$593$	−18.9359	−0.777605	−0.388802	−	0.921321i	$-0.627111\pi$
$593$	−18.9359	−0.777605	−0.388802	+	0.921321i	$0.627111\pi$
$594$	0	0
$595$	−9.38145	−0.384602
$596$	0	0
$597$	−4.59028	−0.187868
$598$	0	0
$599$	−20.8791	−0.853097	−0.426549	−	0.904465i	$-0.640271\pi$
$599$	−20.8791	−0.853097	−0.426549	+	0.904465i	$0.640271\pi$
$600$	0	0
$601$	−18.7344	−0.764193	−0.382097	−	0.924122i	$-0.624798\pi$
$601$	−18.7344	−0.764193	−0.382097	+	0.924122i	$0.624798\pi$
$602$	0	0
$603$	3.40968	0.138853
$604$	0	0
$605$	−8.00801	−0.325572
$606$	0	0
$607$	12.1694	0.493943	0.246971	−	0.969023i	$-0.420565\pi$
$607$	12.1694	0.493943	0.246971	+	0.969023i	$0.420565\pi$
$608$	0	0
$609$	−16.5076	−0.668922
$610$	0	0
$611$	−2.29162	−0.0927089
$612$	0	0
$613$	43.7814	1.76832	0.884158	−	0.467188i	$-0.154733\pi$
$613$	43.7814	1.76832	0.884158	+	0.467188i	$0.154733\pi$
$614$	0	0
$615$	0.359416	0.0144930
$616$	0	0
$617$	−47.6870	−1.91980	−0.959902	−	0.280335i	$-0.909555\pi$
$617$	−47.6870	−1.91980	−0.959902	+	0.280335i	$0.909555\pi$
$618$	0	0
$619$	−25.3980	−1.02083	−0.510416	−	0.859927i	$-0.670509\pi$
$619$	−25.3980	−1.02083	−0.510416	+	0.859927i	$0.670509\pi$
$620$	0	0
$621$	−6.81055	−0.273298
$622$	0	0
$623$	17.8362	0.714593
$624$	0	0
$625$	−8.96565	−0.358626
$626$	0	0
$627$	32.2703	1.28875
$628$	0	0
$629$	−34.2758	−1.36666
$630$	0	0
$631$	11.7365	0.467221	0.233611	−	0.972330i	$-0.424946\pi$
$631$	11.7365	0.467221	0.233611	+	0.972330i	$0.424946\pi$
$632$	0	0
$633$	1.09496	0.0435207
$634$	0	0
$635$	28.8653	1.14549
$636$	0	0
$637$	18.0590	0.715524
$638$	0	0
$639$	1.52814	0.0604522
$640$	0	0
$641$	33.1285	1.30850	0.654250	−	0.756279i	$-0.272985\pi$
$641$	33.1285	1.30850	0.654250	+	0.756279i	$0.272985\pi$
$642$	0	0
$643$	4.01084	0.158172	0.0790861	−	0.996868i	$-0.474800\pi$
$643$	4.01084	0.158172	0.0790861	+	0.996868i	$0.474800\pi$
$644$	0	0
$645$	7.45064	0.293368
$646$	0	0
$647$	14.5470	0.571903	0.285952	−	0.958244i	$-0.407690\pi$
$647$	14.5470	0.571903	0.285952	+	0.958244i	$0.407690\pi$
$648$	0	0
$649$	−28.2220	−1.10781
$650$	0	0
$651$	−16.5761	−0.649669
$652$	0	0
$653$	23.2618	0.910306	0.455153	−	0.890413i	$-0.349585\pi$
$653$	23.2618	0.910306	0.455153	+	0.890413i	$0.349585\pi$
$654$	0	0
$655$	−0.496542	−0.0194015
$656$	0	0
$657$	−11.4809	−0.447913
$658$	0	0
$659$	24.0497	0.936843	0.468422	−	0.883505i	$-0.344823\pi$
$659$	24.0497	0.936843	0.468422	+	0.883505i	$0.344823\pi$
$660$	0	0
$661$	23.5895	0.917526	0.458763	−	0.888559i	$-0.348293\pi$
$661$	23.5895	0.917526	0.458763	+	0.888559i	$0.348293\pi$
$662$	0	0
$663$	33.0054	1.28182
$664$	0	0
$665$	−11.2700	−0.437031
$666$	0	0
$667$	10.4960	0.406406
$668$	0	0
$669$	2.10380	0.0813375
$670$	0	0
$671$	2.54566	0.0982741
$672$	0	0
$673$	45.6895	1.76120	0.880600	−	0.473861i	$-0.157140\pi$
$673$	45.6895	1.76120	0.880600	+	0.473861i	$0.157140\pi$
$674$	0	0
$675$	5.99750	0.230844
$676$	0	0
$677$	30.6141	1.17660	0.588298	−	0.808644i	$-0.299798\pi$
$677$	30.6141	1.17660	0.588298	+	0.808644i	$0.299798\pi$
$678$	0	0
$679$	3.60550	0.138367
$680$	0	0
$681$	45.0856	1.72769
$682$	0	0
$683$	5.23237	0.200211	0.100106	−	0.994977i	$-0.468082\pi$
$683$	5.23237	0.200211	0.100106	+	0.994977i	$0.468082\pi$
$684$	0	0
$685$	−18.2867	−0.698698
$686$	0	0
$687$	16.1045	0.614425
$688$	0	0
$689$	18.8481	0.718054
$690$	0	0
$691$	−25.1535	−0.956884	−0.478442	−	0.878119i	$-0.658798\pi$
$691$	−25.1535	−0.956884	−0.478442	+	0.878119i	$0.658798\pi$
$692$	0	0
$693$	12.7325	0.483670
$694$	0	0
$695$	37.6764	1.42915
$696$	0	0
$697$	0.305895	0.0115866
$698$	0	0
$699$	46.1058	1.74388
$700$	0	0
$701$	−32.9520	−1.24458	−0.622291	−	0.782786i	$-0.713798\pi$
$701$	−32.9520	−1.24458	−0.622291	+	0.782786i	$0.713798\pi$
$702$	0	0
$703$	−41.1757	−1.55297
$704$	0	0
$705$	−1.70693	−0.0642868
$706$	0	0
$707$	−21.1397	−0.795039
$708$	0	0
$709$	39.6164	1.48782	0.743912	−	0.668278i	$-0.232968\pi$
$709$	39.6164	1.48782	0.743912	+	0.668278i	$0.232968\pi$
$710$	0	0
$711$	−17.3234	−0.649679
$712$	0	0
$713$	10.5395	0.394709
$714$	0	0
$715$	−32.3713	−1.21062
$716$	0	0
$717$	−2.95721	−0.110439
$718$	0	0
$719$	8.58402	0.320130	0.160065	−	0.987106i	$-0.448830\pi$
$719$	8.58402	0.320130	0.160065	+	0.987106i	$0.448830\pi$
$720$	0	0
$721$	−14.5729	−0.542722
$722$	0	0
$723$	57.2934	2.13076
$724$	0	0
$725$	−9.24297	−0.343275
$726$	0	0
$727$	12.3751	0.458968	0.229484	−	0.973312i	$-0.426296\pi$
$727$	12.3751	0.458968	0.229484	+	0.973312i	$0.426296\pi$
$728$	0	0
$729$	−2.02375	−0.0749536
$730$	0	0
$731$	6.34116	0.234536
$732$	0	0
$733$	27.5222	1.01656	0.508278	−	0.861193i	$-0.330282\pi$
$733$	27.5222	1.01656	0.508278	+	0.861193i	$0.330282\pi$
$734$	0	0
$735$	13.4514	0.496163
$736$	0	0
$737$	−7.69105	−0.283303
$738$	0	0
$739$	−34.1948	−1.25788	−0.628938	−	0.777455i	$-0.716510\pi$
$739$	−34.1948	−1.25788	−0.628938	+	0.777455i	$0.716510\pi$
$740$	0	0
$741$	39.6496	1.45656
$742$	0	0
$743$	−47.7219	−1.75075	−0.875373	−	0.483448i	$-0.839384\pi$
$743$	−47.7219	−1.75075	−0.875373	+	0.483448i	$0.839384\pi$
$744$	0	0
$745$	−14.2349	−0.521525
$746$	0	0
$747$	22.3202	0.816652
$748$	0	0
$749$	25.3469	0.926154
$750$	0	0
$751$	−23.0917	−0.842627	−0.421313	−	0.906915i	$-0.638431\pi$
$751$	−23.0917	−0.842627	−0.421313	+	0.906915i	$0.638431\pi$
$752$	0	0
$753$	2.18229	0.0795271
$754$	0	0
$755$	19.2230	0.699598
$756$	0	0
$757$	−40.4049	−1.46854	−0.734271	−	0.678856i	$-0.762476\pi$
$757$	−40.4049	−1.46854	−0.734271	+	0.678856i	$0.762476\pi$
$758$	0	0
$759$	−21.8764	−0.794064
$760$	0	0
$761$	−6.33788	−0.229748	−0.114874	−	0.993380i	$-0.536646\pi$
$761$	−6.33788	−0.229748	−0.114874	+	0.993380i	$0.536646\pi$
$762$	0	0
$763$	14.0596	0.508992
$764$	0	0
$765$	9.09783	0.328933
$766$	0	0
$767$	−34.6755	−1.25206
$768$	0	0
$769$	−23.8100	−0.858612	−0.429306	−	0.903159i	$-0.641242\pi$
$769$	−23.8100	−0.858612	−0.429306	+	0.903159i	$0.641242\pi$
$770$	0	0
$771$	19.0793	0.687125
$772$	0	0
$773$	2.96887	0.106783	0.0533914	−	0.998574i	$-0.482997\pi$
$773$	2.96887	0.106783	0.0533914	+	0.998574i	$0.482997\pi$
$774$	0	0
$775$	−9.28133	−0.333395
$776$	0	0
$777$	−43.9010	−1.57494
$778$	0	0
$779$	0.367473	0.0131661
$780$	0	0
$781$	−3.44695	−0.123341
$782$	0	0
$783$	−11.2416	−0.401743
$784$	0	0
$785$	−0.295182	−0.0105355
$786$	0	0
$787$	12.7084	0.453004	0.226502	−	0.974011i	$-0.427271\pi$
$787$	12.7084	0.453004	0.226502	+	0.974011i	$0.427271\pi$
$788$	0	0
$789$	42.3912	1.50917
$790$	0	0
$791$	9.49446	0.337584
$792$	0	0
$793$	3.12778	0.111071
$794$	0	0
$795$	14.0392	0.497918
$796$	0	0
$797$	−39.5534	−1.40105	−0.700526	−	0.713627i	$-0.747051\pi$
$797$	−39.5534	−1.40105	−0.700526	+	0.713627i	$0.747051\pi$
$798$	0	0
$799$	−1.45275	−0.0513947
$800$	0	0
$801$	−17.2970	−0.611159
$802$	0	0
$803$	25.8969	0.913883
$804$	0	0
$805$	7.64006	0.269277
$806$	0	0
$807$	16.8753	0.594039
$808$	0	0
$809$	31.6668	1.11335	0.556673	−	0.830732i	$-0.312078\pi$
$809$	31.6668	1.11335	0.556673	+	0.830732i	$0.312078\pi$
$810$	0	0
$811$	30.5085	1.07130	0.535650	−	0.844440i	$-0.320067\pi$
$811$	30.5085	1.07130	0.535650	+	0.844440i	$0.320067\pi$
$812$	0	0
$813$	−51.7268	−1.81414
$814$	0	0
$815$	22.9489	0.803865
$816$	0	0
$817$	7.61766	0.266508
$818$	0	0
$819$	15.6441	0.546650
$820$	0	0
$821$	−1.76561	−0.0616202	−0.0308101	−	0.999525i	$-0.509809\pi$
$821$	−1.76561	−0.0616202	−0.0308101	+	0.999525i	$0.509809\pi$
$822$	0	0
$823$	−20.9524	−0.730356	−0.365178	−	0.930938i	$-0.618992\pi$
$823$	−20.9524	−0.730356	−0.365178	+	0.930938i	$0.618992\pi$
$824$	0	0
$825$	19.2648	0.670715
$826$	0	0
$827$	−19.7468	−0.686663	−0.343331	−	0.939214i	$-0.611555\pi$
$827$	−19.7468	−0.686663	−0.343331	+	0.939214i	$0.611555\pi$
$828$	0	0
$829$	−10.1170	−0.351379	−0.175689	−	0.984446i	$-0.556215\pi$
$829$	−10.1170	−0.351379	−0.175689	+	0.984446i	$0.556215\pi$
$830$	0	0
$831$	3.83769	0.133128
$832$	0	0
$833$	11.4484	0.396662
$834$	0	0
$835$	15.9807	0.553036
$836$	0	0
$837$	−11.2883	−0.390180
$838$	0	0
$839$	−6.51976	−0.225087	−0.112544	−	0.993647i	$-0.535900\pi$
$839$	−6.51976	−0.225087	−0.112544	+	0.993647i	$0.535900\pi$
$840$	0	0
$841$	−11.6751	−0.402590
$842$	0	0
$843$	43.7091	1.50542
$844$	0	0
$845$	−18.1008	−0.622688
$846$	0	0
$847$	−8.72946	−0.299948
$848$	0	0
$849$	12.7501	0.437581
$850$	0	0
$851$	27.9135	0.956861
$852$	0	0
$853$	20.2508	0.693375	0.346688	−	0.937981i	$-0.387306\pi$
$853$	20.2508	0.693375	0.346688	+	0.937981i	$0.387306\pi$
$854$	0	0
$855$	10.9293	0.373773
$856$	0	0
$857$	11.2976	0.385919	0.192960	−	0.981207i	$-0.438191\pi$
$857$	11.2976	0.385919	0.192960	+	0.981207i	$0.438191\pi$
$858$	0	0
$859$	−14.6057	−0.498341	−0.249171	−	0.968460i	$-0.580158\pi$
$859$	−14.6057	−0.498341	−0.249171	+	0.968460i	$0.580158\pi$
$860$	0	0
$861$	0.391796	0.0133524
$862$	0	0
$863$	36.5862	1.24541	0.622704	−	0.782457i	$-0.286034\pi$
$863$	36.5862	1.24541	0.622704	+	0.782457i	$0.286034\pi$
$864$	0	0
$865$	12.1254	0.412276
$866$	0	0
$867$	−16.1754	−0.549346
$868$	0	0
$869$	39.0756	1.32555
$870$	0	0
$871$	−9.44977	−0.320193
$872$	0	0
$873$	−3.49650	−0.118339
$874$	0	0
$875$	−21.8768	−0.739571
$876$	0	0
$877$	−12.6561	−0.427366	−0.213683	−	0.976903i	$-0.568546\pi$
$877$	−12.6561	−0.427366	−0.213683	+	0.976903i	$0.568546\pi$
$878$	0	0
$879$	−6.66356	−0.224756
$880$	0	0
$881$	−49.7832	−1.67724	−0.838619	−	0.544719i	$-0.816636\pi$
$881$	−49.7832	−1.67724	−0.838619	+	0.544719i	$0.816636\pi$
$882$	0	0
$883$	−47.1957	−1.58826	−0.794132	−	0.607746i	$-0.792074\pi$
$883$	−47.1957	−1.58826	−0.794132	+	0.607746i	$0.792074\pi$
$884$	0	0
$885$	−25.8284	−0.868212
$886$	0	0
$887$	−12.6299	−0.424070	−0.212035	−	0.977262i	$-0.568009\pi$
$887$	−12.6299	−0.424070	−0.212035	+	0.977262i	$0.568009\pi$
$888$	0	0
$889$	31.4659	1.05533
$890$	0	0
$891$	44.4490	1.48910
$892$	0	0
$893$	−1.74520	−0.0584008
$894$	0	0
$895$	28.8135	0.963130
$896$	0	0
$897$	−26.8789	−0.897461
$898$	0	0
$899$	17.3968	0.580215
$900$	0	0
$901$	11.9486	0.398065
$902$	0	0
$903$	8.12187	0.270279
$904$	0	0
$905$	−33.0892	−1.09992
$906$	0	0
$907$	13.5677	0.450510	0.225255	−	0.974300i	$-0.427679\pi$
$907$	13.5677	0.450510	0.225255	+	0.974300i	$0.427679\pi$
$908$	0	0
$909$	20.5006	0.679961
$910$	0	0
$911$	−43.1151	−1.42847	−0.714233	−	0.699908i	$-0.753224\pi$
$911$	−43.1151	−1.42847	−0.714233	+	0.699908i	$0.753224\pi$
$912$	0	0
$913$	−50.3465	−1.66623
$914$	0	0
$915$	2.32976	0.0770194
$916$	0	0
$917$	−0.541276	−0.0178745
$918$	0	0
$919$	47.3464	1.56181	0.780906	−	0.624649i	$-0.214758\pi$
$919$	47.3464	1.56181	0.780906	+	0.624649i	$0.214758\pi$
$920$	0	0
$921$	−30.6687	−1.01057
$922$	0	0
$923$	−4.23517	−0.139402
$924$	0	0
$925$	−24.5812	−0.808224
$926$	0	0
$927$	14.1323	0.464166
$928$	0	0
$929$	14.3288	0.470111	0.235056	−	0.971982i	$-0.424473\pi$
$929$	14.3288	0.470111	0.235056	+	0.971982i	$0.424473\pi$
$930$	0	0
$931$	13.7530	0.450736
$932$	0	0
$933$	−52.3361	−1.71341
$934$	0	0
$935$	−20.5215	−0.671126
$936$	0	0
$937$	−4.34255	−0.141865	−0.0709324	−	0.997481i	$-0.522597\pi$
$937$	−4.34255	−0.141865	−0.0709324	+	0.997481i	$0.522597\pi$
$938$	0	0
$939$	59.7846	1.95100
$940$	0	0
$941$	24.8718	0.810798	0.405399	−	0.914140i	$-0.367133\pi$
$941$	24.8718	0.810798	0.405399	+	0.914140i	$0.367133\pi$
$942$	0	0
$943$	−0.249114	−0.00811228
$944$	0	0
$945$	−8.18282	−0.266187
$946$	0	0
$947$	19.0827	0.620105	0.310053	−	0.950719i	$-0.399653\pi$
$947$	19.0827	0.620105	0.310053	+	0.950719i	$0.399653\pi$
$948$	0	0
$949$	31.8188	1.03288
$950$	0	0
$951$	−27.2815	−0.884663
$952$	0	0
$953$	−60.9942	−1.97580	−0.987899	−	0.155102i	$-0.950430\pi$
$953$	−60.9942	−1.97580	−0.987899	+	0.155102i	$0.950430\pi$
$954$	0	0
$955$	38.9944	1.26183
$956$	0	0
$957$	−36.1097	−1.16726
$958$	0	0
$959$	−19.9341	−0.643707
$960$	0	0
$961$	−13.5310	−0.436484
$962$	0	0
$963$	−24.5806	−0.792098
$964$	0	0
$965$	21.0046	0.676163
$966$	0	0
$967$	26.1329	0.840377	0.420188	−	0.907437i	$-0.361964\pi$
$967$	26.1329	0.840377	0.420188	+	0.907437i	$0.361964\pi$
$968$	0	0
$969$	25.1355	0.807470
$970$	0	0
$971$	29.1801	0.936435	0.468217	−	0.883613i	$-0.344896\pi$
$971$	29.1801	0.936435	0.468217	+	0.883613i	$0.344896\pi$
$972$	0	0
$973$	41.0707	1.31667
$974$	0	0
$975$	23.6701	0.758051
$976$	0	0
$977$	48.9495	1.56603	0.783017	−	0.622000i	$-0.213680\pi$
$977$	48.9495	1.56603	0.783017	+	0.622000i	$0.213680\pi$
$978$	0	0
$979$	39.0160	1.24696
$980$	0	0
$981$	−13.6346	−0.435318
$982$	0	0
$983$	56.2456	1.79396	0.896979	−	0.442074i	$-0.145757\pi$
$983$	56.2456	1.79396	0.896979	+	0.442074i	$0.145757\pi$
$984$	0	0
$985$	−3.80360	−0.121193
$986$	0	0
$987$	−1.86071	−0.0592271
$988$	0	0
$989$	−5.16411	−0.164209
$990$	0	0
$991$	−9.10217	−0.289140	−0.144570	−	0.989495i	$-0.546180\pi$
$991$	−9.10217	−0.289140	−0.144570	+	0.989495i	$0.546180\pi$
$992$	0	0
$993$	55.8949	1.77377
$994$	0	0
$995$	3.50671	0.111170
$996$	0	0
$997$	39.1557	1.24007	0.620037	−	0.784573i	$-0.287118\pi$
$997$	39.1557	1.24007	0.620037	+	0.784573i	$0.287118\pi$
$998$	0	0
$999$	−29.8965	−0.945883

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1004.2.a.a.1.7	✓	7
3.2	odd	2		9036.2.a.i.1.5		7
4.3	odd	2		4016.2.a.g.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.a.1.7	✓	7	1.1	even	1	trivial
4016.2.a.g.1.1		7	4.3	odd	2
9036.2.a.i.1.5		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1004 = 2^{2} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1004.a (trivial)

\(f(q)\)	\(=\)	\(q+2.18229 q^{3} -1.66714 q^{5} -1.81734 q^{7} +1.76240 q^{9} +O(q^{10})\)	\(q+2.18229 q^{3} -1.66714 q^{5} -1.81734 q^{7} +1.76240 q^{9} -3.97535 q^{11} -4.88440 q^{13} -3.63819 q^{15} -3.09643 q^{17} -3.71976 q^{19} -3.96596 q^{21} +2.52167 q^{23} -2.22063 q^{25} -2.70081 q^{27} +4.16232 q^{29} +4.17959 q^{31} -8.67538 q^{33} +3.02977 q^{35} +11.0695 q^{37} -10.6592 q^{39} -0.0987896 q^{41} -2.04789 q^{43} -2.93817 q^{45} +0.469170 q^{47} -3.69728 q^{49} -6.75731 q^{51} -3.85883 q^{53} +6.62749 q^{55} -8.11759 q^{57} +7.09924 q^{59} -0.640361 q^{61} -3.20287 q^{63} +8.14300 q^{65} +1.93468 q^{67} +5.50301 q^{69} +0.867080 q^{71} -6.51437 q^{73} -4.84606 q^{75} +7.22456 q^{77} -9.82946 q^{79} -11.1811 q^{81} +12.6647 q^{83} +5.16219 q^{85} +9.08339 q^{87} -9.81448 q^{89} +8.87662 q^{91} +9.12109 q^{93} +6.20137 q^{95} -1.98395 q^{97} -7.00615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7}+O(q^{10}) \) 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7	\( 7 q - 3 q^{3} - 2 q^{5} - 6 q^{7} - 5 q^{11} - q^{13} - 6 q^{15} - 8 q^{17} - 15 q^{19} - 3 q^{21} - 5 q^{23} - 9 q^{25} - 9 q^{27} - 21 q^{31} - 7 q^{35} - q^{37} - 23 q^{39} - 10 q^{41} - 23 q^{43} - 4 q^{45} - 10 q^{47} - 13 q^{49} - 20 q^{51} - q^{53} - 23 q^{55} - 6 q^{57} - 4 q^{59} + 3 q^{61} - 4 q^{63} + 4 q^{65} - 28 q^{67} + 18 q^{69} - 18 q^{71} - 7 q^{73} + 11 q^{75} + 6 q^{77} - 30 q^{79} - 5 q^{81} + 13 q^{83} + q^{85} - 7 q^{87} - 18 q^{91} + 36 q^{93} - 2 q^{97} - 10 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 - 2 * q^5 - 6 * q^7 - 5 * q^11 - q^13 - 6 * q^15 - 8 * q^17 - 15 * q^19 - 3 * q^21 - 5 * q^23 - 9 * q^25 - 9 * q^27 - 21 * q^31 - 7 * q^35 - q^37 - 23 * q^39 - 10 * q^41 - 23 * q^43 - 4 * q^45 - 10 * q^47 - 13 * q^49 - 20 * q^51 - q^53 - 23 * q^55 - 6 * q^57 - 4 * q^59 + 3 * q^61 - 4 * q^63 + 4 * q^65 - 28 * q^67 + 18 * q^69 - 18 * q^71 - 7 * q^73 + 11 * q^75 + 6 * q^77 - 30 * q^79 - 5 * q^81 + 13 * q^83 + q^85 - 7 * q^87 - 18 * q^91 + 36 * q^93 - 2 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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