LMFDB - Embedded newform 4004.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.75080$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17963 q^{3} +0.636469 q^{5} +1.00000 q^{7} -1.60847 q^{9} +O(q^{10})$	$q+1.17963 q^{3} +0.636469 q^{5} +1.00000 q^{7} -1.60847 q^{9} +1.00000 q^{11} +1.00000 q^{13} +0.750800 q^{15} -2.63647 q^{17} -3.06530 q^{19} +1.17963 q^{21} -8.06850 q^{23} -4.59491 q^{25} -5.43630 q^{27} -2.89314 q^{29} -0.142335 q^{31} +1.17963 q^{33} +0.636469 q^{35} -4.28650 q^{37} +1.17963 q^{39} -7.94671 q^{41} -3.73907 q^{43} -1.02374 q^{45} -3.19591 q^{47} +1.00000 q^{49} -3.11007 q^{51} +7.60420 q^{53} +0.636469 q^{55} -3.61593 q^{57} +2.02374 q^{59} -0.167624 q^{61} -1.60847 q^{63} +0.636469 q^{65} +5.77454 q^{67} -9.51787 q^{69} +0.284669 q^{71} -4.08450 q^{73} -5.42030 q^{75} +1.00000 q^{77} -10.7143 q^{79} -1.58744 q^{81} +14.0760 q^{83} -1.67803 q^{85} -3.41284 q^{87} +16.8430 q^{89} +1.00000 q^{91} -0.167903 q^{93} -1.95097 q^{95} +1.04476 q^{97} -1.60847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9	$ 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9} + 4 q^{11} + 4 q^{13} - 5 q^{15} - 7 q^{17} - 9 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} - 9 q^{27} - 3 q^{29} - 3 q^{33} - q^{35} - 18 q^{37} - 3 q^{39} - 14 q^{41} - q^{43} + 11 q^{45} - 3 q^{47} + 4 q^{49} + 11 q^{51} + 4 q^{53} - q^{55} + 6 q^{57} - 7 q^{59} - 14 q^{61} + q^{63} - q^{65} - 20 q^{69} - 6 q^{73} + 16 q^{75} + 4 q^{77} + 7 q^{79} - 4 q^{81} + 8 q^{83} - 15 q^{85} + 11 q^{87} + 9 q^{89} + 4 q^{91} + 13 q^{93} - 9 q^{95} - 16 q^{97} + q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9 + 4 * q^11 + 4 * q^13 - 5 * q^15 - 7 * q^17 - 9 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^25 - 9 * q^27 - 3 * q^29 - 3 * q^33 - q^35 - 18 * q^37 - 3 * q^39 - 14 * q^41 - q^43 + 11 * q^45 - 3 * q^47 + 4 * q^49 + 11 * q^51 + 4 * q^53 - q^55 + 6 * q^57 - 7 * q^59 - 14 * q^61 + q^63 - q^65 - 20 * q^69 - 6 * q^73 + 16 * q^75 + 4 * q^77 + 7 * q^79 - 4 * q^81 + 8 * q^83 - 15 * q^85 + 11 * q^87 + 9 * q^89 + 4 * q^91 + 13 * q^93 - 9 * q^95 - 16 * q^97 + 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
$3$	1.17963	0.681061	0.340531	−	0.940233i	$-0.389393\pi$
$3$	1.17963	0.681061	0.340531	+	0.940233i	$0.389393\pi$
$4$	0	0
$5$	0.636469	0.284638	0.142319	−	0.989821i	$-0.454544\pi$
$5$	0.636469	0.284638	0.142319	+	0.989821i	$0.454544\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.60847	−0.536155
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.750800	0.193856
$16$	0	0
$17$	−2.63647	−0.639438	−0.319719	−	0.947512i	$-0.603588\pi$
$17$	−2.63647	−0.639438	−0.319719	+	0.947512i	$0.603588\pi$
$18$	0	0
$19$	−3.06530	−0.703229	−0.351614	−	0.936145i	$-0.614367\pi$
$19$	−3.06530	−0.703229	−0.351614	+	0.936145i	$0.614367\pi$
$20$	0	0
$21$	1.17963	0.257417
$22$	0	0
$23$	−8.06850	−1.68240	−0.841200	−	0.540725i	$-0.818150\pi$
$23$	−8.06850	−1.68240	−0.841200	+	0.540725i	$0.818150\pi$
$24$	0	0
$25$	−4.59491	−0.918981
$26$	0	0
$27$	−5.43630	−1.04622
$28$	0	0
$29$	−2.89314	−0.537242	−0.268621	−	0.963246i	$-0.586568\pi$
$29$	−2.89314	−0.537242	−0.268621	+	0.963246i	$0.586568\pi$
$30$	0	0
$31$	−0.142335	−0.0255641	−0.0127820	−	0.999918i	$-0.504069\pi$
$31$	−0.142335	−0.0255641	−0.0127820	+	0.999918i	$0.504069\pi$
$32$	0	0
$33$	1.17963	0.205348
$34$	0	0
$35$	0.636469	0.107583
$36$	0	0
$37$	−4.28650	−0.704696	−0.352348	−	0.935869i	$-0.614617\pi$
$37$	−4.28650	−0.704696	−0.352348	+	0.935869i	$0.614617\pi$
$38$	0	0
$39$	1.17963	0.188892
$40$	0	0
$41$	−7.94671	−1.24107	−0.620534	−	0.784180i	$-0.713084\pi$
$41$	−7.94671	−1.24107	−0.620534	+	0.784180i	$0.713084\pi$
$42$	0	0
$43$	−3.73907	−0.570203	−0.285101	−	0.958497i	$-0.592027\pi$
$43$	−3.73907	−0.570203	−0.285101	+	0.958497i	$0.592027\pi$
$44$	0	0
$45$	−1.02374	−0.152610
$46$	0	0
$47$	−3.19591	−0.466171	−0.233085	−	0.972456i	$-0.574882\pi$
$47$	−3.19591	−0.466171	−0.233085	+	0.972456i	$0.574882\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.11007	−0.435496
$52$	0	0
$53$	7.60420	1.04452	0.522259	−	0.852787i	$-0.325090\pi$
$53$	7.60420	1.04452	0.522259	+	0.852787i	$0.325090\pi$
$54$	0	0
$55$	0.636469	0.0858215
$56$	0	0
$57$	−3.61593	−0.478942
$58$	0	0
$59$	2.02374	0.263468	0.131734	−	0.991285i	$-0.457945\pi$
$59$	2.02374	0.263468	0.131734	+	0.991285i	$0.457945\pi$
$60$	0	0
$61$	−0.167624	−0.0214621	−0.0107310	−	0.999942i	$-0.503416\pi$
$61$	−0.167624	−0.0214621	−0.0107310	+	0.999942i	$0.503416\pi$
$62$	0	0
$63$	−1.60847	−0.202648
$64$	0	0
$65$	0.636469	0.0789443
$66$	0	0
$67$	5.77454	0.705472	0.352736	−	0.935723i	$-0.385251\pi$
$67$	5.77454	0.705472	0.352736	+	0.935723i	$0.385251\pi$
$68$	0	0
$69$	−9.51787	−1.14582
$70$	0	0
$71$	0.284669	0.0337840	0.0168920	−	0.999857i	$-0.494623\pi$
$71$	0.284669	0.0337840	0.0168920	+	0.999857i	$0.494623\pi$
$72$	0	0
$73$	−4.08450	−0.478054	−0.239027	−	0.971013i	$-0.576829\pi$
$73$	−4.08450	−0.478054	−0.239027	+	0.971013i	$0.576829\pi$
$74$	0	0
$75$	−5.42030	−0.625883
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−10.7143	−1.20545	−0.602725	−	0.797949i	$-0.705918\pi$
$79$	−10.7143	−1.20545	−0.602725	+	0.797949i	$0.705918\pi$
$80$	0	0
$81$	−1.58744	−0.176382
$82$	0	0
$83$	14.0760	1.54504	0.772519	−	0.634991i	$-0.218996\pi$
$83$	14.0760	1.54504	0.772519	+	0.634991i	$0.218996\pi$
$84$	0	0
$85$	−1.67803	−0.182008
$86$	0	0
$87$	−3.41284	−0.365895
$88$	0	0
$89$	16.8430	1.78536	0.892680	−	0.450692i	$-0.148823\pi$
$89$	16.8430	1.78536	0.892680	+	0.450692i	$0.148823\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−0.167903	−0.0174107
$94$	0	0
$95$	−1.95097	−0.200165
$96$	0	0
$97$	1.04476	0.106080	0.0530399	−	0.998592i	$-0.483109\pi$
$97$	1.04476	0.106080	0.0530399	+	0.998592i	$0.483109\pi$
$98$	0	0
$99$	−1.60847	−0.161657
$100$	0	0
$101$	2.94216	0.292756	0.146378	−	0.989229i	$-0.453238\pi$
$101$	2.94216	0.292756	0.146378	+	0.989229i	$0.453238\pi$
$102$	0	0
$103$	−2.10232	−0.207148	−0.103574	−	0.994622i	$-0.533028\pi$
$103$	−2.10232	−0.207148	−0.103574	+	0.994622i	$0.533028\pi$
$104$	0	0
$105$	0.750800	0.0732706
$106$	0	0
$107$	10.4366	1.00894	0.504471	−	0.863429i	$-0.331688\pi$
$107$	10.4366	1.00894	0.504471	+	0.863429i	$0.331688\pi$
$108$	0	0
$109$	−5.48987	−0.525834	−0.262917	−	0.964818i	$-0.584685\pi$
$109$	−5.48987	−0.525834	−0.262917	+	0.964818i	$0.584685\pi$
$110$	0	0
$111$	−5.05649	−0.479941
$112$	0	0
$113$	−3.12634	−0.294101	−0.147051	−	0.989129i	$-0.546978\pi$
$113$	−3.12634	−0.294101	−0.147051	+	0.989129i	$0.546978\pi$
$114$	0	0
$115$	−5.13536	−0.478874
$116$	0	0
$117$	−1.60847	−0.148703
$118$	0	0
$119$	−2.63647	−0.241685
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.37420	−0.845243
$124$	0	0
$125$	−6.10686	−0.546215
$126$	0	0
$127$	−18.7708	−1.66564	−0.832818	−	0.553547i	$-0.813274\pi$
$127$	−18.7708	−1.66564	−0.832818	+	0.553547i	$0.813274\pi$
$128$	0	0
$129$	−4.41073	−0.388343
$130$	0	0
$131$	−5.05649	−0.441788	−0.220894	−	0.975298i	$-0.570897\pi$
$131$	−5.05649	−0.441788	−0.220894	+	0.975298i	$0.570897\pi$
$132$	0	0
$133$	−3.06530	−0.265795
$134$	0	0
$135$	−3.46004	−0.297793
$136$	0	0
$137$	−3.20764	−0.274047	−0.137023	−	0.990568i	$-0.543754\pi$
$137$	−3.20764	−0.274047	−0.137023	+	0.990568i	$0.543754\pi$
$138$	0	0
$139$	−2.53862	−0.215323	−0.107662	−	0.994188i	$-0.534336\pi$
$139$	−2.53862	−0.215323	−0.107662	+	0.994188i	$0.534336\pi$
$140$	0	0
$141$	−3.77000	−0.317491
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−1.84139	−0.152919
$146$	0	0
$147$	1.17963	0.0972945
$148$	0	0
$149$	5.65323	0.463131	0.231565	−	0.972819i	$-0.425615\pi$
$149$	5.65323	0.463131	0.231565	+	0.972819i	$0.425615\pi$
$150$	0	0
$151$	18.0437	1.46838	0.734188	−	0.678946i	$-0.237563\pi$
$151$	18.0437	1.46838	0.734188	+	0.678946i	$0.237563\pi$
$152$	0	0
$153$	4.24067	0.342838
$154$	0	0
$155$	−0.0905917	−0.00727650
$156$	0	0
$157$	11.0437	0.881383	0.440692	−	0.897659i	$-0.354733\pi$
$157$	11.0437	0.881383	0.440692	+	0.897659i	$0.354733\pi$
$158$	0	0
$159$	8.97017	0.711381
$160$	0	0
$161$	−8.06850	−0.635887
$162$	0	0
$163$	0.179633	0.0140699	0.00703497	−	0.999975i	$-0.497761\pi$
$163$	0.179633	0.0140699	0.00703497	+	0.999975i	$0.497761\pi$
$164$	0	0
$165$	0.750800	0.0584497
$166$	0	0
$167$	−13.7558	−1.06446	−0.532229	−	0.846600i	$-0.678645\pi$
$167$	−13.7558	−1.06446	−0.532229	+	0.846600i	$0.678645\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.93043	0.377040
$172$	0	0
$173$	−0.909130	−0.0691199	−0.0345599	−	0.999403i	$-0.511003\pi$
$173$	−0.909130	−0.0691199	−0.0345599	+	0.999403i	$0.511003\pi$
$174$	0	0
$175$	−4.59491	−0.347342
$176$	0	0
$177$	2.38727	0.179438
$178$	0	0
$179$	−13.3159	−0.995275	−0.497638	−	0.867385i	$-0.665799\pi$
$179$	−13.3159	−0.995275	−0.497638	+	0.867385i	$0.665799\pi$
$180$	0	0
$181$	−6.25191	−0.464701	−0.232351	−	0.972632i	$-0.574642\pi$
$181$	−6.25191	−0.464701	−0.232351	+	0.972632i	$0.574642\pi$
$182$	0	0
$183$	−0.197735	−0.0146170
$184$	0	0
$185$	−2.72823	−0.200583
$186$	0	0
$187$	−2.63647	−0.192798
$188$	0	0
$189$	−5.43630	−0.395433
$190$	0	0
$191$	−15.5541	−1.12546	−0.562728	−	0.826642i	$-0.690248\pi$
$191$	−15.5541	−1.12546	−0.562728	+	0.826642i	$0.690248\pi$
$192$	0	0
$193$	−0.724138	−0.0521246	−0.0260623	−	0.999660i	$-0.508297\pi$
$193$	−0.724138	−0.0521246	−0.0260623	+	0.999660i	$0.508297\pi$
$194$	0	0
$195$	0.750800	0.0537659
$196$	0	0
$197$	−15.4973	−1.10414	−0.552070	−	0.833798i	$-0.686162\pi$
$197$	−15.4973	−1.10414	−0.552070	+	0.833798i	$0.686162\pi$
$198$	0	0
$199$	−17.2849	−1.22530	−0.612649	−	0.790355i	$-0.709896\pi$
$199$	−17.2849	−1.22530	−0.612649	+	0.790355i	$0.709896\pi$
$200$	0	0
$201$	6.81184	0.480470
$202$	0	0
$203$	−2.89314	−0.203058
$204$	0	0
$205$	−5.05784	−0.353255
$206$	0	0
$207$	12.9779	0.902027
$208$	0	0
$209$	−3.06530	−0.212031
$210$	0	0
$211$	7.75672	0.533994	0.266997	−	0.963697i	$-0.413969\pi$
$211$	7.75672	0.533994	0.266997	+	0.963697i	$0.413969\pi$
$212$	0	0
$213$	0.335805	0.0230090
$214$	0	0
$215$	−2.37980	−0.162301
$216$	0	0
$217$	−0.142335	−0.00966231
$218$	0	0
$219$	−4.81821	−0.325584
$220$	0	0
$221$	−2.63647	−0.177348
$222$	0	0
$223$	8.96347	0.600238	0.300119	−	0.953902i	$-0.402974\pi$
$223$	8.96347	0.600238	0.300119	+	0.953902i	$0.402974\pi$
$224$	0	0
$225$	7.39075	0.492717
$226$	0	0
$227$	5.53281	0.367225	0.183613	−	0.982999i	$-0.441221\pi$
$227$	5.53281	0.367225	0.183613	+	0.982999i	$0.441221\pi$
$228$	0	0
$229$	3.73054	0.246521	0.123261	−	0.992374i	$-0.460665\pi$
$229$	3.73054	0.246521	0.123261	+	0.992374i	$0.460665\pi$
$230$	0	0
$231$	1.17963	0.0776142
$232$	0	0
$233$	14.5328	0.952076	0.476038	−	0.879425i	$-0.342073\pi$
$233$	14.5328	0.952076	0.476038	+	0.879425i	$0.342073\pi$
$234$	0	0
$235$	−2.03410	−0.132690
$236$	0	0
$237$	−12.6389	−0.820985
$238$	0	0
$239$	17.4635	1.12962	0.564811	−	0.825221i	$-0.308949\pi$
$239$	17.4635	1.12962	0.564811	+	0.825221i	$0.308949\pi$
$240$	0	0
$241$	−0.495197	−0.0318985	−0.0159492	−	0.999873i	$-0.505077\pi$
$241$	−0.495197	−0.0318985	−0.0159492	+	0.999873i	$0.505077\pi$
$242$	0	0
$243$	14.4363	0.926089
$244$	0	0
$245$	0.636469	0.0406625
$246$	0	0
$247$	−3.06530	−0.195041
$248$	0	0
$249$	16.6045	1.05227
$250$	0	0
$251$	3.80486	0.240161	0.120080	−	0.992764i	$-0.461685\pi$
$251$	3.80486	0.240161	0.120080	+	0.992764i	$0.461685\pi$
$252$	0	0
$253$	−8.06850	−0.507262
$254$	0	0
$255$	−1.97946	−0.123959
$256$	0	0
$257$	−31.9227	−1.99128	−0.995641	−	0.0932711i	$-0.970268\pi$
$257$	−31.9227	−1.99128	−0.995641	+	0.0932711i	$0.970268\pi$
$258$	0	0
$259$	−4.28650	−0.266350
$260$	0	0
$261$	4.65351	0.288045
$262$	0	0
$263$	10.0186	0.617773	0.308886	−	0.951099i	$-0.400044\pi$
$263$	10.0186	0.617773	0.308886	+	0.951099i	$0.400044\pi$
$264$	0	0
$265$	4.83984	0.297309
$266$	0	0
$267$	19.8686	1.21594
$268$	0	0
$269$	11.0050	0.670989	0.335494	−	0.942042i	$-0.391097\pi$
$269$	11.0050	0.670989	0.335494	+	0.942042i	$0.391097\pi$
$270$	0	0
$271$	3.37051	0.204744	0.102372	−	0.994746i	$-0.467357\pi$
$271$	3.37051	0.204744	0.102372	+	0.994746i	$0.467357\pi$
$272$	0	0
$273$	1.17963	0.0713946
$274$	0	0
$275$	−4.59491	−0.277083
$276$	0	0
$277$	16.7260	1.00497	0.502484	−	0.864587i	$-0.332420\pi$
$277$	16.7260	1.00497	0.502484	+	0.864587i	$0.332420\pi$
$278$	0	0
$279$	0.228940	0.0137063
$280$	0	0
$281$	−16.8473	−1.00503	−0.502513	−	0.864570i	$-0.667591\pi$
$281$	−16.8473	−1.00503	−0.502513	+	0.864570i	$0.667591\pi$
$282$	0	0
$283$	15.7362	0.935419	0.467709	−	0.883882i	$-0.345079\pi$
$283$	15.7362	0.935419	0.467709	+	0.883882i	$0.345079\pi$
$284$	0	0
$285$	−2.30143	−0.136325
$286$	0	0
$287$	−7.94671	−0.469079
$288$	0	0
$289$	−10.0490	−0.591119
$290$	0	0
$291$	1.23244	0.0722468
$292$	0	0
$293$	28.6216	1.67209	0.836044	−	0.548662i	$-0.184863\pi$
$293$	28.6216	1.67209	0.836044	+	0.548662i	$0.184863\pi$
$294$	0	0
$295$	1.28805	0.0749931
$296$	0	0
$297$	−5.43630	−0.315446
$298$	0	0
$299$	−8.06850	−0.466614
$300$	0	0
$301$	−3.73907	−0.215516
$302$	0	0
$303$	3.47067	0.199385
$304$	0	0
$305$	−0.106688	−0.00610892
$306$	0	0
$307$	10.1351	0.578443	0.289222	−	0.957262i	$-0.406604\pi$
$307$	10.1351	0.578443	0.289222	+	0.957262i	$0.406604\pi$
$308$	0	0
$309$	−2.47997	−0.141080
$310$	0	0
$311$	−4.41207	−0.250186	−0.125093	−	0.992145i	$-0.539923\pi$
$311$	−4.41207	−0.250186	−0.125093	+	0.992145i	$0.539923\pi$
$312$	0	0
$313$	−18.7649	−1.06066	−0.530329	−	0.847792i	$-0.677932\pi$
$313$	−18.7649	−1.06066	−0.530329	+	0.847792i	$0.677932\pi$
$314$	0	0
$315$	−1.02374	−0.0576812
$316$	0	0
$317$	6.38379	0.358549	0.179275	−	0.983799i	$-0.442625\pi$
$317$	6.38379	0.358549	0.179275	+	0.983799i	$0.442625\pi$
$318$	0	0
$319$	−2.89314	−0.161984
$320$	0	0
$321$	12.3113	0.687152
$322$	0	0
$323$	8.08158	0.449671
$324$	0	0
$325$	−4.59491	−0.254880
$326$	0	0
$327$	−6.47603	−0.358125
$328$	0	0
$329$	−3.19591	−0.176196
$330$	0	0
$331$	−28.0414	−1.54129	−0.770647	−	0.637262i	$-0.780067\pi$
$331$	−28.0414	−1.54129	−0.770647	+	0.637262i	$0.780067\pi$
$332$	0	0
$333$	6.89469	0.377826
$334$	0	0
$335$	3.67532	0.200804
$336$	0	0
$337$	3.61941	0.197162	0.0985810	−	0.995129i	$-0.468570\pi$
$337$	3.61941	0.197162	0.0985810	+	0.995129i	$0.468570\pi$
$338$	0	0
$339$	−3.68793	−0.200301
$340$	0	0
$341$	−0.142335	−0.00770785
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−6.05784	−0.326143
$346$	0	0
$347$	−6.21538	−0.333659	−0.166830	−	0.985986i	$-0.553353\pi$
$347$	−6.21538	−0.333659	−0.166830	+	0.985986i	$0.553353\pi$
$348$	0	0
$349$	−29.2271	−1.56449	−0.782246	−	0.622970i	$-0.785926\pi$
$349$	−29.2271	−1.56449	−0.782246	+	0.622970i	$0.785926\pi$
$350$	0	0
$351$	−5.43630	−0.290168
$352$	0	0
$353$	−32.0959	−1.70829	−0.854147	−	0.520032i	$-0.825920\pi$
$353$	−32.0959	−1.70829	−0.854147	+	0.520032i	$0.825920\pi$
$354$	0	0
$355$	0.181183	0.00961621
$356$	0	0
$357$	−3.11007	−0.164602
$358$	0	0
$359$	31.6546	1.67067	0.835333	−	0.549744i	$-0.185275\pi$
$359$	31.6546	1.67067	0.835333	+	0.549744i	$0.185275\pi$
$360$	0	0
$361$	−9.60392	−0.505470
$362$	0	0
$363$	1.17963	0.0619147
$364$	0	0
$365$	−2.59966	−0.136072
$366$	0	0
$367$	−16.5704	−0.864967	−0.432483	−	0.901642i	$-0.642363\pi$
$367$	−16.5704	−0.864967	−0.432483	+	0.901642i	$0.642363\pi$
$368$	0	0
$369$	12.7820	0.665405
$370$	0	0
$371$	7.60420	0.394790
$372$	0	0
$373$	−10.3783	−0.537366	−0.268683	−	0.963229i	$-0.586588\pi$
$373$	−10.3783	−0.537366	−0.268683	+	0.963229i	$0.586588\pi$
$374$	0	0
$375$	−7.20386	−0.372006
$376$	0	0
$377$	−2.89314	−0.149004
$378$	0	0
$379$	−10.4366	−0.536091	−0.268045	−	0.963406i	$-0.586378\pi$
$379$	−10.4366	−0.536091	−0.268045	+	0.963406i	$0.586378\pi$
$380$	0	0
$381$	−22.1426	−1.13440
$382$	0	0
$383$	17.0997	0.873754	0.436877	−	0.899521i	$-0.356084\pi$
$383$	17.0997	0.873754	0.436877	+	0.899521i	$0.356084\pi$
$384$	0	0
$385$	0.636469	0.0324375
$386$	0	0
$387$	6.01417	0.305717
$388$	0	0
$389$	14.9350	0.757233	0.378617	−	0.925554i	$-0.376400\pi$
$389$	14.9350	0.757233	0.378617	+	0.925554i	$0.376400\pi$
$390$	0	0
$391$	21.2724	1.07579
$392$	0	0
$393$	−5.96481	−0.300885
$394$	0	0
$395$	−6.81930	−0.343116
$396$	0	0
$397$	−4.97306	−0.249591	−0.124795	−	0.992183i	$-0.539827\pi$
$397$	−4.97306	−0.249591	−0.124795	+	0.992183i	$0.539827\pi$
$398$	0	0
$399$	−3.61593	−0.181023
$400$	0	0
$401$	−21.0993	−1.05365	−0.526825	−	0.849974i	$-0.676618\pi$
$401$	−21.0993	−1.05365	−0.526825	+	0.849974i	$0.676618\pi$
$402$	0	0
$403$	−0.142335	−0.00709019
$404$	0	0
$405$	−1.01036	−0.0502051
$406$	0	0
$407$	−4.28650	−0.212474
$408$	0	0
$409$	4.99756	0.247114	0.123557	−	0.992337i	$-0.460570\pi$
$409$	4.99756	0.247114	0.123557	+	0.992337i	$0.460570\pi$
$410$	0	0
$411$	−3.78383	−0.186643
$412$	0	0
$413$	2.02374	0.0995817
$414$	0	0
$415$	8.95892	0.439776
$416$	0	0
$417$	−2.99464	−0.146648
$418$	0	0
$419$	27.4893	1.34294	0.671470	−	0.741032i	$-0.265663\pi$
$419$	27.4893	1.34294	0.671470	+	0.741032i	$0.265663\pi$
$420$	0	0
$421$	3.50752	0.170946	0.0854730	−	0.996340i	$-0.472760\pi$
$421$	3.50752	0.170946	0.0854730	+	0.996340i	$0.472760\pi$
$422$	0	0
$423$	5.14051	0.249940
$424$	0	0
$425$	12.1143	0.587631
$426$	0	0
$427$	−0.167624	−0.00811190
$428$	0	0
$429$	1.17963	0.0569532
$430$	0	0
$431$	−1.56109	−0.0751950	−0.0375975	−	0.999293i	$-0.511970\pi$
$431$	−1.56109	−0.0751950	−0.0375975	+	0.999293i	$0.511970\pi$
$432$	0	0
$433$	−20.5909	−0.989533	−0.494767	−	0.869026i	$-0.664746\pi$
$433$	−20.5909	−0.989533	−0.494767	+	0.869026i	$0.664746\pi$
$434$	0	0
$435$	−2.17217	−0.104147
$436$	0	0
$437$	24.7324	1.18311
$438$	0	0
$439$	32.7798	1.56450	0.782248	−	0.622967i	$-0.214073\pi$
$439$	32.7798	1.56450	0.782248	+	0.622967i	$0.214073\pi$
$440$	0	0
$441$	−1.60847	−0.0765936
$442$	0	0
$443$	−1.00244	−0.0476272	−0.0238136	−	0.999716i	$-0.507581\pi$
$443$	−1.00244	−0.0476272	−0.0238136	+	0.999716i	$0.507581\pi$
$444$	0	0
$445$	10.7201	0.508181
$446$	0	0
$447$	6.66874	0.315420
$448$	0	0
$449$	−27.0166	−1.27499	−0.637496	−	0.770454i	$-0.720030\pi$
$449$	−27.0166	−1.27499	−0.637496	+	0.770454i	$0.720030\pi$
$450$	0	0
$451$	−7.94671	−0.374196
$452$	0	0
$453$	21.2849	1.00005
$454$	0	0
$455$	0.636469	0.0298381
$456$	0	0
$457$	−22.6239	−1.05830	−0.529150	−	0.848528i	$-0.677489\pi$
$457$	−22.6239	−1.05830	−0.529150	+	0.848528i	$0.677489\pi$
$458$	0	0
$459$	14.3326	0.668990
$460$	0	0
$461$	−15.0339	−0.700199	−0.350100	−	0.936712i	$-0.613852\pi$
$461$	−15.0339	−0.700199	−0.350100	+	0.936712i	$0.613852\pi$
$462$	0	0
$463$	−3.05388	−0.141926	−0.0709630	−	0.997479i	$-0.522607\pi$
$463$	−3.05388	−0.141926	−0.0709630	+	0.997479i	$0.522607\pi$
$464$	0	0
$465$	−0.106865	−0.00495574
$466$	0	0
$467$	−2.37148	−0.109739	−0.0548696	−	0.998494i	$-0.517474\pi$
$467$	−2.37148	−0.109739	−0.0548696	+	0.998494i	$0.517474\pi$
$468$	0	0
$469$	5.77454	0.266643
$470$	0	0
$471$	13.0275	0.600276
$472$	0	0
$473$	−3.73907	−0.171923
$474$	0	0
$475$	14.0848	0.646254
$476$	0	0
$477$	−12.2311	−0.560023
$478$	0	0
$479$	−33.4683	−1.52920	−0.764602	−	0.644503i	$-0.777065\pi$
$479$	−33.4683	−1.52920	−0.764602	+	0.644503i	$0.777065\pi$
$480$	0	0
$481$	−4.28650	−0.195447
$482$	0	0
$483$	−9.51787	−0.433078
$484$	0	0
$485$	0.664961	0.0301943
$486$	0	0
$487$	21.9384	0.994123	0.497062	−	0.867715i	$-0.334412\pi$
$487$	21.9384	0.994123	0.497062	+	0.867715i	$0.334412\pi$
$488$	0	0
$489$	0.211901	0.00958250
$490$	0	0
$491$	5.59247	0.252385	0.126192	−	0.992006i	$-0.459724\pi$
$491$	5.59247	0.252385	0.126192	+	0.992006i	$0.459724\pi$
$492$	0	0
$493$	7.62766	0.343533
$494$	0	0
$495$	−1.02374	−0.0460137
$496$	0	0
$497$	0.284669	0.0127692
$498$	0	0
$499$	−33.3105	−1.49118	−0.745592	−	0.666402i	$-0.767833\pi$
$499$	−33.3105	−1.49118	−0.745592	+	0.666402i	$0.767833\pi$
$500$	0	0
$501$	−16.2268	−0.724961
$502$	0	0
$503$	29.3038	1.30659	0.653297	−	0.757102i	$-0.273385\pi$
$503$	29.3038	1.30659	0.653297	+	0.757102i	$0.273385\pi$
$504$	0	0
$505$	1.87260	0.0833295
$506$	0	0
$507$	1.17963	0.0523893
$508$	0	0
$509$	−8.25162	−0.365746	−0.182873	−	0.983136i	$-0.558540\pi$
$509$	−8.25162	−0.365746	−0.182873	+	0.983136i	$0.558540\pi$
$510$	0	0
$511$	−4.08450	−0.180688
$512$	0	0
$513$	16.6639	0.735729
$514$	0	0
$515$	−1.33806	−0.0589621
$516$	0	0
$517$	−3.19591	−0.140556
$518$	0	0
$519$	−1.07244	−0.0470749
$520$	0	0
$521$	22.7383	0.996184	0.498092	−	0.867124i	$-0.334034\pi$
$521$	22.7383	0.996184	0.498092	+	0.867124i	$0.334034\pi$
$522$	0	0
$523$	39.2306	1.71544	0.857718	−	0.514121i	$-0.171882\pi$
$523$	39.2306	1.71544	0.857718	+	0.514121i	$0.171882\pi$
$524$	0	0
$525$	−5.42030	−0.236561
$526$	0	0
$527$	0.375261	0.0163466
$528$	0	0
$529$	42.1008	1.83047
$530$	0	0
$531$	−3.25512	−0.141260
$532$	0	0
$533$	−7.94671	−0.344210
$534$	0	0
$535$	6.64256	0.287183
$536$	0	0
$537$	−15.7078	−0.677844
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−9.69568	−0.416850	−0.208425	−	0.978038i	$-0.566834\pi$
$541$	−9.69568	−0.416850	−0.208425	+	0.978038i	$0.566834\pi$
$542$	0	0
$543$	−7.37496	−0.316490
$544$	0	0
$545$	−3.49413	−0.149672
$546$	0	0
$547$	−23.7548	−1.01568	−0.507840	−	0.861451i	$-0.669556\pi$
$547$	−23.7548	−1.01568	−0.507840	+	0.861451i	$0.669556\pi$
$548$	0	0
$549$	0.269618	0.0115070
$550$	0	0
$551$	8.86833	0.377804
$552$	0	0
$553$	−10.7143	−0.455617
$554$	0	0
$555$	−3.21830	−0.136609
$556$	0	0
$557$	9.92995	0.420745	0.210373	−	0.977621i	$-0.432532\pi$
$557$	9.92995	0.420745	0.210373	+	0.977621i	$0.432532\pi$
$558$	0	0
$559$	−3.73907	−0.158146
$560$	0	0
$561$	−3.11007	−0.131307
$562$	0	0
$563$	−10.7090	−0.451329	−0.225664	−	0.974205i	$-0.572455\pi$
$563$	−10.7090	−0.451329	−0.225664	+	0.974205i	$0.572455\pi$
$564$	0	0
$565$	−1.98982	−0.0837123
$566$	0	0
$567$	−1.58744	−0.0666662
$568$	0	0
$569$	17.7569	0.744408	0.372204	−	0.928151i	$-0.378602\pi$
$569$	17.7569	0.744408	0.372204	+	0.928151i	$0.378602\pi$
$570$	0	0
$571$	46.7431	1.95614	0.978070	−	0.208278i	$-0.0667858\pi$
$571$	46.7431	1.95614	0.978070	+	0.208278i	$0.0667858\pi$
$572$	0	0
$573$	−18.3481	−0.766505
$574$	0	0
$575$	37.0740	1.54609
$576$	0	0
$577$	−34.1412	−1.42131	−0.710657	−	0.703538i	$-0.751602\pi$
$577$	−34.1412	−1.42131	−0.710657	+	0.703538i	$0.751602\pi$
$578$	0	0
$579$	−0.854217	−0.0355000
$580$	0	0
$581$	14.0760	0.583970
$582$	0	0
$583$	7.60420	0.314934
$584$	0	0
$585$	−1.02374	−0.0423264
$586$	0	0
$587$	−16.5010	−0.681069	−0.340534	−	0.940232i	$-0.610608\pi$
$587$	−16.5010	−0.681069	−0.340534	+	0.940232i	$0.610608\pi$
$588$	0	0
$589$	0.436299	0.0179774
$590$	0	0
$591$	−18.2812	−0.751987
$592$	0	0
$593$	−4.41178	−0.181170	−0.0905849	−	0.995889i	$-0.528874\pi$
$593$	−4.41178	−0.181170	−0.0905849	+	0.995889i	$0.528874\pi$
$594$	0	0
$595$	−1.67803	−0.0687926
$596$	0	0
$597$	−20.3899	−0.834503
$598$	0	0
$599$	−1.42855	−0.0583691	−0.0291846	−	0.999574i	$-0.509291\pi$
$599$	−1.42855	−0.0583691	−0.0291846	+	0.999574i	$0.509291\pi$
$600$	0	0
$601$	13.2753	0.541509	0.270754	−	0.962648i	$-0.412727\pi$
$601$	13.2753	0.541509	0.270754	+	0.962648i	$0.412727\pi$
$602$	0	0
$603$	−9.28815	−0.378243
$604$	0	0
$605$	0.636469	0.0258762
$606$	0	0
$607$	−24.7561	−1.00482	−0.502410	−	0.864629i	$-0.667553\pi$
$607$	−24.7561	−1.00482	−0.502410	+	0.864629i	$0.667553\pi$
$608$	0	0
$609$	−3.41284	−0.138295
$610$	0	0
$611$	−3.19591	−0.129293
$612$	0	0
$613$	−10.3035	−0.416153	−0.208077	−	0.978113i	$-0.566720\pi$
$613$	−10.3035	−0.416153	−0.208077	+	0.978113i	$0.566720\pi$
$614$	0	0
$615$	−5.96639	−0.240588
$616$	0	0
$617$	−3.01064	−0.121204	−0.0606018	−	0.998162i	$-0.519302\pi$
$617$	−3.01064	−0.121204	−0.0606018	+	0.998162i	$0.519302\pi$
$618$	0	0
$619$	18.6309	0.748838	0.374419	−	0.927260i	$-0.377842\pi$
$619$	18.6309	0.748838	0.374419	+	0.927260i	$0.377842\pi$
$620$	0	0
$621$	43.8628	1.76015
$622$	0	0
$623$	16.8430	0.674802
$624$	0	0
$625$	19.0877	0.763508
$626$	0	0
$627$	−3.61593	−0.144406
$628$	0	0
$629$	11.3012	0.450609
$630$	0	0
$631$	−31.4987	−1.25394	−0.626972	−	0.779042i	$-0.715706\pi$
$631$	−31.4987	−1.25394	−0.626972	+	0.779042i	$0.715706\pi$
$632$	0	0
$633$	9.15008	0.363683
$634$	0	0
$635$	−11.9470	−0.474103
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−0.457881	−0.0181135
$640$	0	0
$641$	5.34028	0.210928	0.105464	−	0.994423i	$-0.466367\pi$
$641$	5.34028	0.210928	0.105464	+	0.994423i	$0.466367\pi$
$642$	0	0
$643$	−24.4971	−0.966072	−0.483036	−	0.875600i	$-0.660466\pi$
$643$	−24.4971	−0.966072	−0.483036	+	0.875600i	$0.660466\pi$
$644$	0	0
$645$	−2.80730	−0.110537
$646$	0	0
$647$	−15.8706	−0.623940	−0.311970	−	0.950092i	$-0.600989\pi$
$647$	−15.8706	−0.623940	−0.311970	+	0.950092i	$0.600989\pi$
$648$	0	0
$649$	2.02374	0.0794387
$650$	0	0
$651$	−0.167903	−0.00658062
$652$	0	0
$653$	12.1810	0.476678	0.238339	−	0.971182i	$-0.423397\pi$
$653$	12.1810	0.476678	0.238339	+	0.971182i	$0.423397\pi$
$654$	0	0
$655$	−3.21830	−0.125750
$656$	0	0
$657$	6.56978	0.256311
$658$	0	0
$659$	6.21848	0.242238	0.121119	−	0.992638i	$-0.461352\pi$
$659$	6.21848	0.242238	0.121119	+	0.992638i	$0.461352\pi$
$660$	0	0
$661$	−23.6397	−0.919477	−0.459738	−	0.888054i	$-0.652057\pi$
$661$	−23.6397	−0.919477	−0.459738	+	0.888054i	$0.652057\pi$
$662$	0	0
$663$	−3.11007	−0.120785
$664$	0	0
$665$	−1.95097	−0.0756554
$666$	0	0
$667$	23.3433	0.903855
$668$	0	0
$669$	10.5736	0.408799
$670$	0	0
$671$	−0.167624	−0.00647106
$672$	0	0
$673$	−10.8163	−0.416937	−0.208469	−	0.978029i	$-0.566848\pi$
$673$	−10.8163	−0.416937	−0.208469	+	0.978029i	$0.566848\pi$
$674$	0	0
$675$	24.9793	0.961453
$676$	0	0
$677$	4.84935	0.186375	0.0931877	−	0.995649i	$-0.470294\pi$
$677$	4.84935	0.186375	0.0931877	+	0.995649i	$0.470294\pi$
$678$	0	0
$679$	1.04476	0.0400944
$680$	0	0
$681$	6.52668	0.250103
$682$	0	0
$683$	38.6194	1.47773	0.738866	−	0.673852i	$-0.235362\pi$
$683$	38.6194	1.47773	0.738866	+	0.673852i	$0.235362\pi$
$684$	0	0
$685$	−2.04156	−0.0780041
$686$	0	0
$687$	4.40067	0.167896
$688$	0	0
$689$	7.60420	0.289697
$690$	0	0
$691$	13.2981	0.505882	0.252941	−	0.967482i	$-0.418602\pi$
$691$	13.2981	0.505882	0.252941	+	0.967482i	$0.418602\pi$
$692$	0	0
$693$	−1.60847	−0.0611006
$694$	0	0
$695$	−1.61575	−0.0612891
$696$	0	0
$697$	20.9512	0.793585
$698$	0	0
$699$	17.1434	0.648422
$700$	0	0
$701$	−0.607700	−0.0229525	−0.0114763	−	0.999934i	$-0.503653\pi$
$701$	−0.607700	−0.0229525	−0.0114763	+	0.999934i	$0.503653\pi$
$702$	0	0
$703$	13.1394	0.495562
$704$	0	0
$705$	−2.39949	−0.0903699
$706$	0	0
$707$	2.94216	0.110651
$708$	0	0
$709$	−6.02075	−0.226114	−0.113057	−	0.993589i	$-0.536064\pi$
$709$	−6.02075	−0.226114	−0.113057	+	0.993589i	$0.536064\pi$
$710$	0	0
$711$	17.2335	0.646308
$712$	0	0
$713$	1.14843	0.0430090
$714$	0	0
$715$	0.636469	0.0238026
$716$	0	0
$717$	20.6005	0.769341
$718$	0	0
$719$	−13.4179	−0.500402	−0.250201	−	0.968194i	$-0.580497\pi$
$719$	−13.4179	−0.500402	−0.250201	+	0.968194i	$0.580497\pi$
$720$	0	0
$721$	−2.10232	−0.0782946
$722$	0	0
$723$	−0.584151	−0.0217248
$724$	0	0
$725$	13.2937	0.493715
$726$	0	0
$727$	−14.0737	−0.521963	−0.260981	−	0.965344i	$-0.584046\pi$
$727$	−14.0737	−0.521963	−0.260981	+	0.965344i	$0.584046\pi$
$728$	0	0
$729$	21.7919	0.807106
$730$	0	0
$731$	9.85794	0.364609
$732$	0	0
$733$	−13.5421	−0.500188	−0.250094	−	0.968222i	$-0.580461\pi$
$733$	−13.5421	−0.500188	−0.250094	+	0.968222i	$0.580461\pi$
$734$	0	0
$735$	0.750800	0.0276937
$736$	0	0
$737$	5.77454	0.212708
$738$	0	0
$739$	−4.49655	−0.165408	−0.0827042	−	0.996574i	$-0.526356\pi$
$739$	−4.49655	−0.165408	−0.0827042	+	0.996574i	$0.526356\pi$
$740$	0	0
$741$	−3.61593	−0.132835
$742$	0	0
$743$	−26.1725	−0.960175	−0.480088	−	0.877221i	$-0.659395\pi$
$743$	−26.1725	−0.960175	−0.480088	+	0.877221i	$0.659395\pi$
$744$	0	0
$745$	3.59811	0.131824
$746$	0	0
$747$	−22.6407	−0.828380
$748$	0	0
$749$	10.4366	0.381344
$750$	0	0
$751$	38.1543	1.39227	0.696134	−	0.717912i	$-0.254902\pi$
$751$	38.1543	1.39227	0.696134	+	0.717912i	$0.254902\pi$
$752$	0	0
$753$	4.48834	0.163564
$754$	0	0
$755$	11.4843	0.417955
$756$	0	0
$757$	34.6501	1.25938	0.629690	−	0.776846i	$-0.283182\pi$
$757$	34.6501	1.25938	0.629690	+	0.776846i	$0.283182\pi$
$758$	0	0
$759$	−9.51787	−0.345477
$760$	0	0
$761$	33.2657	1.20588	0.602940	−	0.797787i	$-0.293996\pi$
$761$	33.2657	1.20588	0.602940	+	0.797787i	$0.293996\pi$
$762$	0	0
$763$	−5.48987	−0.198747
$764$	0	0
$765$	2.69906	0.0975846
$766$	0	0
$767$	2.02374	0.0730730
$768$	0	0
$769$	22.3204	0.804896	0.402448	−	0.915443i	$-0.368159\pi$
$769$	22.3204	0.804896	0.402448	+	0.915443i	$0.368159\pi$
$770$	0	0
$771$	−37.6570	−1.35619
$772$	0	0
$773$	11.8937	0.427786	0.213893	−	0.976857i	$-0.431386\pi$
$773$	11.8937	0.427786	0.213893	+	0.976857i	$0.431386\pi$
$774$	0	0
$775$	0.654014	0.0234929
$776$	0	0
$777$	−5.05649	−0.181401
$778$	0	0
$779$	24.3591	0.872754
$780$	0	0
$781$	0.284669	0.0101863
$782$	0	0
$783$	15.7279	0.562071
$784$	0	0
$785$	7.02898	0.250875
$786$	0	0
$787$	−20.9957	−0.748417	−0.374209	−	0.927345i	$-0.622086\pi$
$787$	−20.9957	−0.748417	−0.374209	+	0.927345i	$0.622086\pi$
$788$	0	0
$789$	11.8183	0.420741
$790$	0	0
$791$	−3.12634	−0.111160
$792$	0	0
$793$	−0.167624	−0.00595251
$794$	0	0
$795$	5.70924	0.202486
$796$	0	0
$797$	−36.9984	−1.31055	−0.655276	−	0.755390i	$-0.727448\pi$
$797$	−36.9984	−1.31055	−0.655276	+	0.755390i	$0.727448\pi$
$798$	0	0
$799$	8.42591	0.298087
$800$	0	0
$801$	−27.0915	−0.957230
$802$	0	0
$803$	−4.08450	−0.144139
$804$	0	0
$805$	−5.13536	−0.180998
$806$	0	0
$807$	12.9819	0.456984
$808$	0	0
$809$	−36.2331	−1.27389	−0.636943	−	0.770911i	$-0.719801\pi$
$809$	−36.2331	−1.27389	−0.636943	+	0.770911i	$0.719801\pi$
$810$	0	0
$811$	−4.78008	−0.167851	−0.0839256	−	0.996472i	$-0.526746\pi$
$811$	−4.78008	−0.167851	−0.0839256	+	0.996472i	$0.526746\pi$
$812$	0	0
$813$	3.97596	0.139443
$814$	0	0
$815$	0.114331	0.00400484
$816$	0	0
$817$	11.4614	0.400983
$818$	0	0
$819$	−1.60847	−0.0562043
$820$	0	0
$821$	18.9667	0.661944	0.330972	−	0.943641i	$-0.392623\pi$
$821$	18.9667	0.661944	0.330972	+	0.943641i	$0.392623\pi$
$822$	0	0
$823$	51.8390	1.80699	0.903496	−	0.428596i	$-0.140991\pi$
$823$	51.8390	1.80699	0.903496	+	0.428596i	$0.140991\pi$
$824$	0	0
$825$	−5.42030	−0.188711
$826$	0	0
$827$	1.64722	0.0572796	0.0286398	−	0.999590i	$-0.490882\pi$
$827$	1.64722	0.0572796	0.0286398	+	0.999590i	$0.490882\pi$
$828$	0	0
$829$	−29.4742	−1.02368	−0.511840	−	0.859081i	$-0.671036\pi$
$829$	−29.4742	−1.02368	−0.511840	+	0.859081i	$0.671036\pi$
$830$	0	0
$831$	19.7305	0.684445
$832$	0	0
$833$	−2.63647	−0.0913483
$834$	0	0
$835$	−8.75517	−0.302985
$836$	0	0
$837$	0.773774	0.0267455
$838$	0	0
$839$	15.2113	0.525153	0.262577	−	0.964911i	$-0.415428\pi$
$839$	15.2113	0.525153	0.262577	+	0.964911i	$0.415428\pi$
$840$	0	0
$841$	−20.6298	−0.711371
$842$	0	0
$843$	−19.8736	−0.684485
$844$	0	0
$845$	0.636469	0.0218952
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	18.5629	0.637078
$850$	0	0
$851$	34.5856	1.18558
$852$	0	0
$853$	−27.4289	−0.939149	−0.469574	−	0.882893i	$-0.655593\pi$
$853$	−27.4289	−0.939149	−0.469574	+	0.882893i	$0.655593\pi$
$854$	0	0
$855$	3.13807	0.107320
$856$	0	0
$857$	4.73355	0.161695	0.0808475	−	0.996726i	$-0.474237\pi$
$857$	4.73355	0.161695	0.0808475	+	0.996726i	$0.474237\pi$
$858$	0	0
$859$	−50.3332	−1.71735	−0.858674	−	0.512523i	$-0.828711\pi$
$859$	−50.3332	−1.71735	−0.858674	+	0.512523i	$0.828711\pi$
$860$	0	0
$861$	−9.37420	−0.319472
$862$	0	0
$863$	−4.59771	−0.156508	−0.0782539	−	0.996933i	$-0.524934\pi$
$863$	−4.59771	−0.156508	−0.0782539	+	0.996933i	$0.524934\pi$
$864$	0	0
$865$	−0.578633	−0.0196741
$866$	0	0
$867$	−11.8542	−0.402589
$868$	0	0
$869$	−10.7143	−0.363457
$870$	0	0
$871$	5.77454	0.195663
$872$	0	0
$873$	−1.68047	−0.0568752
$874$	0	0
$875$	−6.10686	−0.206450
$876$	0	0
$877$	−40.0089	−1.35101	−0.675503	−	0.737357i	$-0.736073\pi$
$877$	−40.0089	−1.35101	−0.675503	+	0.737357i	$0.736073\pi$
$878$	0	0
$879$	33.7629	1.13880
$880$	0	0
$881$	−18.4535	−0.621713	−0.310856	−	0.950457i	$-0.600616\pi$
$881$	−18.4535	−0.621713	−0.310856	+	0.950457i	$0.600616\pi$
$882$	0	0
$883$	4.51991	0.152107	0.0760536	−	0.997104i	$-0.475768\pi$
$883$	4.51991	0.152107	0.0760536	+	0.997104i	$0.475768\pi$
$884$	0	0
$885$	1.51942	0.0510749
$886$	0	0
$887$	15.9291	0.534847	0.267423	−	0.963579i	$-0.413828\pi$
$887$	15.9291	0.534847	0.267423	+	0.963579i	$0.413828\pi$
$888$	0	0
$889$	−18.7708	−0.629551
$890$	0	0
$891$	−1.58744	−0.0531813
$892$	0	0
$893$	9.79642	0.327825
$894$	0	0
$895$	−8.47515	−0.283293
$896$	0	0
$897$	−9.51787	−0.317793
$898$	0	0
$899$	0.411793	0.0137341
$900$	0	0
$901$	−20.0482	−0.667904
$902$	0	0
$903$	−4.41073	−0.146780
$904$	0	0
$905$	−3.97915	−0.132271
$906$	0	0
$907$	−4.43541	−0.147275	−0.0736377	−	0.997285i	$-0.523461\pi$
$907$	−4.43541	−0.147275	−0.0736377	+	0.997285i	$0.523461\pi$
$908$	0	0
$909$	−4.73237	−0.156963
$910$	0	0
$911$	43.5799	1.44387	0.721933	−	0.691963i	$-0.243254\pi$
$911$	43.5799	1.44387	0.721933	+	0.691963i	$0.243254\pi$
$912$	0	0
$913$	14.0760	0.465847
$914$	0	0
$915$	−0.125852	−0.00416055
$916$	0	0
$917$	−5.05649	−0.166980
$918$	0	0
$919$	46.2577	1.52590	0.762951	−	0.646456i	$-0.223750\pi$
$919$	46.2577	1.52590	0.762951	+	0.646456i	$0.223750\pi$
$920$	0	0
$921$	11.9558	0.393956
$922$	0	0
$923$	0.284669	0.00937000
$924$	0	0
$925$	19.6961	0.647602
$926$	0	0
$927$	3.38151	0.111063
$928$	0	0
$929$	20.3791	0.668618	0.334309	−	0.942464i	$-0.391497\pi$
$929$	20.3791	0.668618	0.334309	+	0.942464i	$0.391497\pi$
$930$	0	0
$931$	−3.06530	−0.100461
$932$	0	0
$933$	−5.20463	−0.170392
$934$	0	0
$935$	−1.67803	−0.0548775
$936$	0	0
$937$	−15.4777	−0.505634	−0.252817	−	0.967514i	$-0.581357\pi$
$937$	−15.4777	−0.505634	−0.252817	+	0.967514i	$0.581357\pi$
$938$	0	0
$939$	−22.1358	−0.722373
$940$	0	0
$941$	36.2052	1.18026	0.590128	−	0.807310i	$-0.299077\pi$
$941$	36.2052	1.18026	0.590128	+	0.807310i	$0.299077\pi$
$942$	0	0
$943$	64.1180	2.08797
$944$	0	0
$945$	−3.46004	−0.112555
$946$	0	0
$947$	40.4970	1.31597	0.657987	−	0.753029i	$-0.271408\pi$
$947$	40.4970	1.31597	0.657987	+	0.753029i	$0.271408\pi$
$948$	0	0
$949$	−4.08450	−0.132588
$950$	0	0
$951$	7.53053	0.244194
$952$	0	0
$953$	−49.3670	−1.59915	−0.799577	−	0.600563i	$-0.794943\pi$
$953$	−49.3670	−1.59915	−0.799577	+	0.600563i	$0.794943\pi$
$954$	0	0
$955$	−9.89972	−0.320347
$956$	0	0
$957$	−3.41284	−0.110321
$958$	0	0
$959$	−3.20764	−0.103580
$960$	0	0
$961$	−30.9797	−0.999346
$962$	0	0
$963$	−16.7869	−0.540950
$964$	0	0
$965$	−0.460892	−0.0148366
$966$	0	0
$967$	−0.303657	−0.00976494	−0.00488247	−	0.999988i	$-0.501554\pi$
$967$	−0.303657	−0.00976494	−0.00488247	+	0.999988i	$0.501554\pi$
$968$	0	0
$969$	9.53329	0.306254
$970$	0	0
$971$	−9.24253	−0.296607	−0.148303	−	0.988942i	$-0.547381\pi$
$971$	−9.24253	−0.296607	−0.148303	+	0.988942i	$0.547381\pi$
$972$	0	0
$973$	−2.53862	−0.0813844
$974$	0	0
$975$	−5.42030	−0.173589
$976$	0	0
$977$	−33.5124	−1.07216	−0.536079	−	0.844168i	$-0.680095\pi$
$977$	−33.5124	−1.07216	−0.536079	+	0.844168i	$0.680095\pi$
$978$	0	0
$979$	16.8430	0.538306
$980$	0	0
$981$	8.83027	0.281929
$982$	0	0
$983$	41.6638	1.32887	0.664434	−	0.747347i	$-0.268673\pi$
$983$	41.6638	1.32887	0.664434	+	0.747347i	$0.268673\pi$
$984$	0	0
$985$	−9.86358	−0.314280
$986$	0	0
$987$	−3.77000	−0.120000
$988$	0	0
$989$	30.1687	0.959309
$990$	0	0
$991$	16.1881	0.514233	0.257117	−	0.966380i	$-0.417228\pi$
$991$	16.1881	0.514233	0.257117	+	0.966380i	$0.417228\pi$
$992$	0	0
$993$	−33.0785	−1.04972
$994$	0	0
$995$	−11.0013	−0.348766
$996$	0	0
$997$	−14.3886	−0.455693	−0.227846	−	0.973697i	$-0.573168\pi$
$997$	−14.3886	−0.455693	−0.227846	+	0.973697i	$0.573168\pi$
$998$	0	0
$999$	23.3027	0.737264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.d.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.d.1.4	✓	4	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17963 q^{3} +0.636469 q^{5} +1.00000 q^{7} -1.60847 q^{9} +O(q^{10})\)	\(q+1.17963 q^{3} +0.636469 q^{5} +1.00000 q^{7} -1.60847 q^{9} +1.00000 q^{11} +1.00000 q^{13} +0.750800 q^{15} -2.63647 q^{17} -3.06530 q^{19} +1.17963 q^{21} -8.06850 q^{23} -4.59491 q^{25} -5.43630 q^{27} -2.89314 q^{29} -0.142335 q^{31} +1.17963 q^{33} +0.636469 q^{35} -4.28650 q^{37} +1.17963 q^{39} -7.94671 q^{41} -3.73907 q^{43} -1.02374 q^{45} -3.19591 q^{47} +1.00000 q^{49} -3.11007 q^{51} +7.60420 q^{53} +0.636469 q^{55} -3.61593 q^{57} +2.02374 q^{59} -0.167624 q^{61} -1.60847 q^{63} +0.636469 q^{65} +5.77454 q^{67} -9.51787 q^{69} +0.284669 q^{71} -4.08450 q^{73} -5.42030 q^{75} +1.00000 q^{77} -10.7143 q^{79} -1.58744 q^{81} +14.0760 q^{83} -1.67803 q^{85} -3.41284 q^{87} +16.8430 q^{89} +1.00000 q^{91} -0.167903 q^{93} -1.95097 q^{95} +1.04476 q^{97} -1.60847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9	\( 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9} + 4 q^{11} + 4 q^{13} - 5 q^{15} - 7 q^{17} - 9 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} - 9 q^{27} - 3 q^{29} - 3 q^{33} - q^{35} - 18 q^{37} - 3 q^{39} - 14 q^{41} - q^{43} + 11 q^{45} - 3 q^{47} + 4 q^{49} + 11 q^{51} + 4 q^{53} - q^{55} + 6 q^{57} - 7 q^{59} - 14 q^{61} + q^{63} - q^{65} - 20 q^{69} - 6 q^{73} + 16 q^{75} + 4 q^{77} + 7 q^{79} - 4 q^{81} + 8 q^{83} - 15 q^{85} + 11 q^{87} + 9 q^{89} + 4 q^{91} + 13 q^{93} - 9 q^{95} - 16 q^{97} + q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9 + 4 * q^11 + 4 * q^13 - 5 * q^15 - 7 * q^17 - 9 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^25 - 9 * q^27 - 3 * q^29 - 3 * q^33 - q^35 - 18 * q^37 - 3 * q^39 - 14 * q^41 - q^43 + 11 * q^45 - 3 * q^47 + 4 * q^49 + 11 * q^51 + 4 * q^53 - q^55 + 6 * q^57 - 7 * q^59 - 14 * q^61 + q^63 - q^65 - 20 * q^69 - 6 * q^73 + 16 * q^75 + 4 * q^77 + 7 * q^79 - 4 * q^81 + 8 * q^83 - 15 * q^85 + 11 * q^87 + 9 * q^89 + 4 * q^91 + 13 * q^93 - 9 * q^95 - 16 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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