LMFDB - Embedded newform 2312.4.a.m.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, 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("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.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:	$136.412415933$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} - 148 x^{12} + 474 x^{11} + 8325 x^{10} - 20424 x^{9} - 224201 x^{8} + 401234 x^{7} + \cdots - 5899068 $ x^14 - 4x^13 - 148x^12 + 474x^11 + 8325x^10 - 20424x^9 - 224201x^8 + 401234x^7 + 2999018x^6 - 3680568x^5 - 18599211x^4 + 13157640x^3 + 41373828x^2 - 4758912*x - 5899068
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$5.50372$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.78344 q^{3} +11.5964 q^{5} -25.6146 q^{7} +68.7157 q^{9} +O(q^{10})$	$q+9.78344 q^{3} +11.5964 q^{5} -25.6146 q^{7} +68.7157 q^{9} -63.2802 q^{11} -9.95609 q^{13} +113.453 q^{15} -108.484 q^{19} -250.599 q^{21} +6.57019 q^{23} +9.47723 q^{25} +408.124 q^{27} +198.941 q^{29} -223.873 q^{31} -619.098 q^{33} -297.038 q^{35} -47.6360 q^{37} -97.4048 q^{39} -78.8180 q^{41} +193.194 q^{43} +796.857 q^{45} -246.840 q^{47} +313.109 q^{49} -208.770 q^{53} -733.824 q^{55} -1061.35 q^{57} -163.237 q^{59} +107.876 q^{61} -1760.13 q^{63} -115.455 q^{65} -292.899 q^{67} +64.2790 q^{69} -684.883 q^{71} +128.642 q^{73} +92.7199 q^{75} +1620.90 q^{77} +64.1162 q^{79} +2137.53 q^{81} -661.257 q^{83} +1946.33 q^{87} +657.756 q^{89} +255.021 q^{91} -2190.25 q^{93} -1258.03 q^{95} +82.1332 q^{97} -4348.34 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 190 q^{9}+O(q^{10}) $ 14 * q + 190 * q^9	$ 14 q + 190 q^{9} - 72 q^{13} - 60 q^{15} - 468 q^{19} + 60 q^{21} + 306 q^{25} - 904 q^{33} - 804 q^{35} - 896 q^{43} - 448 q^{47} + 866 q^{49} - 108 q^{53} - 1612 q^{55} - 1760 q^{59} - 284 q^{67} - 2532 q^{69} + 180 q^{77} + 950 q^{81} - 4256 q^{83} - 868 q^{87} + 4516 q^{89} - 7156 q^{93}+O(q^{100}) $ 14 * q + 190 * q^9 - 72 * q^13 - 60 * q^15 - 468 * q^19 + 60 * q^21 + 306 * q^25 - 904 * q^33 - 804 * q^35 - 896 * q^43 - 448 * q^47 + 866 * q^49 - 108 * q^53 - 1612 * q^55 - 1760 * q^59 - 284 * q^67 - 2532 * q^69 + 180 * q^77 + 950 * q^81 - 4256 * q^83 - 868 * q^87 + 4516 * q^89 - 7156 * q^93

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
$3$	9.78344	1.88282	0.941412	−	0.337258i	$-0.109500\pi$
$3$	9.78344	1.88282	0.941412	+	0.337258i	$0.109500\pi$
$4$	0	0
$5$	11.5964	1.03722	0.518608	−	0.855012i	$-0.326450\pi$
$5$	11.5964	1.03722	0.518608	+	0.855012i	$0.326450\pi$
$6$	0	0
$7$	−25.6146	−1.38306	−0.691530	−	0.722348i	$-0.743063\pi$
$7$	−25.6146	−1.38306	−0.691530	+	0.722348i	$0.743063\pi$
$8$	0	0
$9$	68.7157	2.54503
$10$	0	0
$11$	−63.2802	−1.73452	−0.867259	−	0.497858i	$-0.834120\pi$
$11$	−63.2802	−1.73452	−0.867259	+	0.497858i	$0.834120\pi$
$12$	0	0
$13$	−9.95609	−0.212409	−0.106205	−	0.994344i	$-0.533870\pi$
$13$	−9.95609	−0.212409	−0.106205	+	0.994344i	$0.533870\pi$
$14$	0	0
$15$	113.453	1.95290
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−108.484	−1.30989	−0.654946	−	0.755676i	$-0.727309\pi$
$19$	−108.484	−1.30989	−0.654946	+	0.755676i	$0.727309\pi$
$20$	0	0
$21$	−250.599	−2.60406
$22$	0	0
$23$	6.57019	0.0595643	0.0297821	−	0.999556i	$-0.490519\pi$
$23$	6.57019	0.0595643	0.0297821	+	0.999556i	$0.490519\pi$
$24$	0	0
$25$	9.47723	0.0758178
$26$	0	0
$27$	408.124	2.90902
$28$	0	0
$29$	198.941	1.27388	0.636939	−	0.770914i	$-0.280200\pi$
$29$	198.941	1.27388	0.636939	+	0.770914i	$0.280200\pi$
$30$	0	0
$31$	−223.873	−1.29706	−0.648529	−	0.761190i	$-0.724616\pi$
$31$	−223.873	−1.29706	−0.648529	+	0.761190i	$0.724616\pi$
$32$	0	0
$33$	−619.098	−3.26579
$34$	0	0
$35$	−297.038	−1.43453
$36$	0	0
$37$	−47.6360	−0.211657	−0.105829	−	0.994384i	$-0.533749\pi$
$37$	−47.6360	−0.211657	−0.105829	+	0.994384i	$0.533749\pi$
$38$	0	0
$39$	−97.4048	−0.399930
$40$	0	0
$41$	−78.8180	−0.300227	−0.150114	−	0.988669i	$-0.547964\pi$
$41$	−78.8180	−0.300227	−0.150114	+	0.988669i	$0.547964\pi$
$42$	0	0
$43$	193.194	0.685160	0.342580	−	0.939489i	$-0.388699\pi$
$43$	193.194	0.685160	0.342580	+	0.939489i	$0.388699\pi$
$44$	0	0
$45$	796.857	2.63974
$46$	0	0
$47$	−246.840	−0.766071	−0.383036	−	0.923734i	$-0.625121\pi$
$47$	−246.840	−0.766071	−0.383036	+	0.923734i	$0.625121\pi$
$48$	0	0
$49$	313.109	0.912853
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−208.770	−0.541071	−0.270535	−	0.962710i	$-0.587201\pi$
$53$	−208.770	−0.541071	−0.270535	+	0.962710i	$0.587201\pi$
$54$	0	0
$55$	−733.824	−1.79907
$56$	0	0
$57$	−1061.35	−2.46630
$58$	0	0
$59$	−163.237	−0.360197	−0.180098	−	0.983649i	$-0.557642\pi$
$59$	−163.237	−0.360197	−0.180098	+	0.983649i	$0.557642\pi$
$60$	0	0
$61$	107.876	0.226427	0.113214	−	0.993571i	$-0.463886\pi$
$61$	107.876	0.226427	0.113214	+	0.993571i	$0.463886\pi$
$62$	0	0
$63$	−1760.13	−3.51992
$64$	0	0
$65$	−115.455	−0.220315
$66$	0	0
$67$	−292.899	−0.534079	−0.267040	−	0.963686i	$-0.586045\pi$
$67$	−292.899	−0.534079	−0.267040	+	0.963686i	$0.586045\pi$
$68$	0	0
$69$	64.2790	0.112149
$70$	0	0
$71$	−684.883	−1.14480	−0.572399	−	0.819975i	$-0.693987\pi$
$71$	−684.883	−1.14480	−0.572399	+	0.819975i	$0.693987\pi$
$72$	0	0
$73$	128.642	0.206252	0.103126	−	0.994668i	$-0.467115\pi$
$73$	128.642	0.206252	0.103126	+	0.994668i	$0.467115\pi$
$74$	0	0
$75$	92.7199	0.142752
$76$	0	0
$77$	1620.90	2.39894
$78$	0	0
$79$	64.1162	0.0913119	0.0456559	−	0.998957i	$-0.485462\pi$
$79$	64.1162	0.0913119	0.0456559	+	0.998957i	$0.485462\pi$
$80$	0	0
$81$	2137.53	2.93214
$82$	0	0
$83$	−661.257	−0.874487	−0.437243	−	0.899343i	$-0.644045\pi$
$83$	−661.257	−0.874487	−0.437243	+	0.899343i	$0.644045\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1946.33	2.39849
$88$	0	0
$89$	657.756	0.783394	0.391697	−	0.920094i	$-0.371888\pi$
$89$	657.756	0.783394	0.391697	+	0.920094i	$0.371888\pi$
$90$	0	0
$91$	255.021	0.293775
$92$	0	0
$93$	−2190.25	−2.44213
$94$	0	0
$95$	−1258.03	−1.35864
$96$	0	0
$97$	82.1332	0.0859728	0.0429864	−	0.999076i	$-0.486313\pi$
$97$	82.1332	0.0859728	0.0429864	+	0.999076i	$0.486313\pi$
$98$	0	0
$99$	−4348.34	−4.41439
$100$	0	0
$101$	711.196	0.700660	0.350330	−	0.936626i	$-0.386069\pi$
$101$	711.196	0.700660	0.350330	+	0.936626i	$0.386069\pi$
$102$	0	0
$103$	−1784.81	−1.70741	−0.853703	−	0.520760i	$-0.825649\pi$
$103$	−1784.81	−1.70741	−0.853703	+	0.520760i	$0.825649\pi$
$104$	0	0
$105$	−2906.06	−2.70097
$106$	0	0
$107$	−1280.20	−1.15665	−0.578324	−	0.815807i	$-0.696293\pi$
$107$	−1280.20	−1.15665	−0.578324	+	0.815807i	$0.696293\pi$
$108$	0	0
$109$	190.674	0.167553	0.0837766	−	0.996485i	$-0.473302\pi$
$109$	190.674	0.167553	0.0837766	+	0.996485i	$0.473302\pi$
$110$	0	0
$111$	−466.045	−0.398513
$112$	0	0
$113$	−1352.70	−1.12612	−0.563059	−	0.826417i	$-0.690376\pi$
$113$	−1352.70	−1.12612	−0.563059	+	0.826417i	$0.690376\pi$
$114$	0	0
$115$	76.1907	0.0617811
$116$	0	0
$117$	−684.140	−0.540588
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2673.38	2.00855
$122$	0	0
$123$	−771.112	−0.565275
$124$	0	0
$125$	−1339.65	−0.958577
$126$	0	0
$127$	−1930.69	−1.34898	−0.674491	−	0.738283i	$-0.735637\pi$
$127$	−1930.69	−1.34898	−0.674491	+	0.738283i	$0.735637\pi$
$128$	0	0
$129$	1890.11	1.29004
$130$	0	0
$131$	2682.34	1.78898	0.894492	−	0.447084i	$-0.147538\pi$
$131$	2682.34	1.78898	0.894492	+	0.447084i	$0.147538\pi$
$132$	0	0
$133$	2778.78	1.81166
$134$	0	0
$135$	4732.78	3.01728
$136$	0	0
$137$	−1582.49	−0.986871	−0.493435	−	0.869782i	$-0.664259\pi$
$137$	−1582.49	−0.986871	−0.493435	+	0.869782i	$0.664259\pi$
$138$	0	0
$139$	−1085.15	−0.662165	−0.331083	−	0.943602i	$-0.607414\pi$
$139$	−1085.15	−0.662165	−0.331083	+	0.943602i	$0.607414\pi$
$140$	0	0
$141$	−2414.95	−1.44238
$142$	0	0
$143$	630.023	0.368428
$144$	0	0
$145$	2307.01	1.32129
$146$	0	0
$147$	3063.28	1.71874
$148$	0	0
$149$	218.582	0.120181	0.0600905	−	0.998193i	$-0.480861\pi$
$149$	218.582	0.120181	0.0600905	+	0.998193i	$0.480861\pi$
$150$	0	0
$151$	188.561	0.101622	0.0508109	−	0.998708i	$-0.483819\pi$
$151$	188.561	0.101622	0.0508109	+	0.998708i	$0.483819\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2596.13	−1.34533
$156$	0	0
$157$	−1596.67	−0.811646	−0.405823	−	0.913952i	$-0.633015\pi$
$157$	−1596.67	−0.811646	−0.405823	+	0.913952i	$0.633015\pi$
$158$	0	0
$159$	−2042.49	−1.01874
$160$	0	0
$161$	−168.293	−0.0823809
$162$	0	0
$163$	−520.158	−0.249951	−0.124975	−	0.992160i	$-0.539885\pi$
$163$	−520.158	−0.249951	−0.124975	+	0.992160i	$0.539885\pi$
$164$	0	0
$165$	−7179.33	−3.38733
$166$	0	0
$167$	1467.61	0.680044	0.340022	−	0.940417i	$-0.389565\pi$
$167$	1467.61	0.680044	0.340022	+	0.940417i	$0.389565\pi$
$168$	0	0
$169$	−2097.88	−0.954882
$170$	0	0
$171$	−7454.56	−3.33371
$172$	0	0
$173$	−810.453	−0.356171	−0.178086	−	0.984015i	$-0.556990\pi$
$173$	−810.453	−0.356171	−0.178086	+	0.984015i	$0.556990\pi$
$174$	0	0
$175$	−242.756	−0.104861
$176$	0	0
$177$	−1597.02	−0.678187
$178$	0	0
$179$	−1394.23	−0.582179	−0.291089	−	0.956696i	$-0.594018\pi$
$179$	−1394.23	−0.582179	−0.291089	+	0.956696i	$0.594018\pi$
$180$	0	0
$181$	−1244.11	−0.510905	−0.255453	−	0.966822i	$-0.582224\pi$
$181$	−1244.11	−0.510905	−0.255453	+	0.966822i	$0.582224\pi$
$182$	0	0
$183$	1055.40	0.426323
$184$	0	0
$185$	−552.408	−0.219534
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−10453.9	−4.02334
$190$	0	0
$191$	4484.76	1.69898	0.849492	−	0.527602i	$-0.176909\pi$
$191$	4484.76	1.69898	0.849492	+	0.527602i	$0.176909\pi$
$192$	0	0
$193$	−2496.16	−0.930973	−0.465486	−	0.885055i	$-0.654121\pi$
$193$	−2496.16	−0.930973	−0.465486	+	0.885055i	$0.654121\pi$
$194$	0	0
$195$	−1129.55	−0.414814
$196$	0	0
$197$	40.6043	0.0146850	0.00734249	−	0.999973i	$-0.497663\pi$
$197$	40.6043	0.0146850	0.00734249	+	0.999973i	$0.497663\pi$
$198$	0	0
$199$	3784.13	1.34799	0.673995	−	0.738736i	$-0.264577\pi$
$199$	3784.13	1.34799	0.673995	+	0.738736i	$0.264577\pi$
$200$	0	0
$201$	−2865.56	−1.00558
$202$	0	0
$203$	−5095.81	−1.76185
$204$	0	0
$205$	−914.008	−0.311400
$206$	0	0
$207$	451.475	0.151593
$208$	0	0
$209$	6864.89	2.27203
$210$	0	0
$211$	1561.86	0.509588	0.254794	−	0.966995i	$-0.417992\pi$
$211$	1561.86	0.509588	0.254794	+	0.966995i	$0.417992\pi$
$212$	0	0
$213$	−6700.51	−2.15545
$214$	0	0
$215$	2240.37	0.710659
$216$	0	0
$217$	5734.43	1.79391
$218$	0	0
$219$	1258.56	0.388337
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4998.81	1.50110	0.750550	−	0.660814i	$-0.229789\pi$
$223$	4998.81	1.50110	0.750550	+	0.660814i	$0.229789\pi$
$224$	0	0
$225$	651.235	0.192959
$226$	0	0
$227$	−1903.40	−0.556535	−0.278267	−	0.960504i	$-0.589760\pi$
$227$	−1903.40	−0.556535	−0.278267	+	0.960504i	$0.589760\pi$
$228$	0	0
$229$	3619.52	1.04447	0.522237	−	0.852801i	$-0.325098\pi$
$229$	3619.52	1.04447	0.522237	+	0.852801i	$0.325098\pi$
$230$	0	0
$231$	15858.0	4.51678
$232$	0	0
$233$	−3427.10	−0.963592	−0.481796	−	0.876283i	$-0.660015\pi$
$233$	−3427.10	−0.963592	−0.481796	+	0.876283i	$0.660015\pi$
$234$	0	0
$235$	−2862.47	−0.794582
$236$	0	0
$237$	627.277	0.171924
$238$	0	0
$239$	−5797.35	−1.56904	−0.784518	−	0.620106i	$-0.787090\pi$
$239$	−5797.35	−1.56904	−0.784518	+	0.620106i	$0.787090\pi$
$240$	0	0
$241$	4438.71	1.18640	0.593200	−	0.805055i	$-0.297864\pi$
$241$	4438.71	1.18640	0.593200	+	0.805055i	$0.297864\pi$
$242$	0	0
$243$	9893.05	2.61168
$244$	0	0
$245$	3630.94	0.946826
$246$	0	0
$247$	1080.08	0.278233
$248$	0	0
$249$	−6469.37	−1.64650
$250$	0	0
$251$	−4172.42	−1.04925	−0.524623	−	0.851335i	$-0.675794\pi$
$251$	−4172.42	−1.04925	−0.524623	+	0.851335i	$0.675794\pi$
$252$	0	0
$253$	−415.763	−0.103315
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6114.45	−1.48408	−0.742041	−	0.670354i	$-0.766142\pi$
$257$	−6114.45	−1.48408	−0.742041	+	0.670354i	$0.766142\pi$
$258$	0	0
$259$	1220.18	0.292734
$260$	0	0
$261$	13670.4	3.24206
$262$	0	0
$263$	4931.92	1.15633	0.578166	−	0.815919i	$-0.303769\pi$
$263$	4931.92	1.15633	0.578166	+	0.815919i	$0.303769\pi$
$264$	0	0
$265$	−2420.99	−0.561207
$266$	0	0
$267$	6435.12	1.47499
$268$	0	0
$269$	890.237	0.201780	0.100890	−	0.994898i	$-0.467831\pi$
$269$	890.237	0.201780	0.100890	+	0.994898i	$0.467831\pi$
$270$	0	0
$271$	7325.27	1.64199	0.820993	−	0.570938i	$-0.193420\pi$
$271$	7325.27	1.64199	0.820993	+	0.570938i	$0.193420\pi$
$272$	0	0
$273$	2494.99	0.553126
$274$	0	0
$275$	−599.721	−0.131507
$276$	0	0
$277$	2895.69	0.628105	0.314052	−	0.949406i	$-0.398313\pi$
$277$	2895.69	0.628105	0.314052	+	0.949406i	$0.398313\pi$
$278$	0	0
$279$	−15383.6	−3.30105
$280$	0	0
$281$	4743.17	1.00695	0.503476	−	0.864009i	$-0.332054\pi$
$281$	4743.17	1.00695	0.503476	+	0.864009i	$0.332054\pi$
$282$	0	0
$283$	8728.70	1.83345	0.916726	−	0.399515i	$-0.130822\pi$
$283$	8728.70	1.83345	0.916726	+	0.399515i	$0.130822\pi$
$284$	0	0
$285$	−12307.8	−2.55808
$286$	0	0
$287$	2018.89	0.415232
$288$	0	0
$289$	0	0
$290$	0	0
$291$	803.545	0.161872
$292$	0	0
$293$	8323.37	1.65958	0.829789	−	0.558077i	$-0.188461\pi$
$293$	8323.37	1.65958	0.829789	+	0.558077i	$0.188461\pi$
$294$	0	0
$295$	−1892.96	−0.373602
$296$	0	0
$297$	−25826.1	−5.04574
$298$	0	0
$299$	−65.4134	−0.0126520
$300$	0	0
$301$	−4948.60	−0.947617
$302$	0	0
$303$	6957.95	1.31922
$304$	0	0
$305$	1250.97	0.234854
$306$	0	0
$307$	8916.96	1.65771	0.828856	−	0.559462i	$-0.188992\pi$
$307$	8916.96	1.65771	0.828856	+	0.559462i	$0.188992\pi$
$308$	0	0
$309$	−17461.6	−3.21475
$310$	0	0
$311$	7455.45	1.35936	0.679678	−	0.733510i	$-0.262119\pi$
$311$	7455.45	1.35936	0.679678	+	0.733510i	$0.262119\pi$
$312$	0	0
$313$	−3773.26	−0.681396	−0.340698	−	0.940173i	$-0.610663\pi$
$313$	−3773.26	−0.681396	−0.340698	+	0.940173i	$0.610663\pi$
$314$	0	0
$315$	−20411.2	−3.65092
$316$	0	0
$317$	−3439.25	−0.609361	−0.304681	−	0.952455i	$-0.598550\pi$
$317$	−3439.25	−0.609361	−0.304681	+	0.952455i	$0.598550\pi$
$318$	0	0
$319$	−12589.0	−2.20957
$320$	0	0
$321$	−12524.7	−2.17777
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−94.3562	−0.0161044
$326$	0	0
$327$	1865.45	0.315473
$328$	0	0
$329$	6322.72	1.05952
$330$	0	0
$331$	−8940.93	−1.48471	−0.742353	−	0.670009i	$-0.766290\pi$
$331$	−8940.93	−1.48471	−0.742353	+	0.670009i	$0.766290\pi$
$332$	0	0
$333$	−3273.35	−0.538674
$334$	0	0
$335$	−3396.58	−0.553956
$336$	0	0
$337$	3333.96	0.538908	0.269454	−	0.963013i	$-0.413157\pi$
$337$	3333.96	0.538908	0.269454	+	0.963013i	$0.413157\pi$
$338$	0	0
$339$	−13234.1	−2.12028
$340$	0	0
$341$	14166.7	2.24977
$342$	0	0
$343$	765.659	0.120530
$344$	0	0
$345$	745.408	0.116323
$346$	0	0
$347$	7990.06	1.23611	0.618053	−	0.786136i	$-0.287922\pi$
$347$	7990.06	1.23611	0.618053	+	0.786136i	$0.287922\pi$
$348$	0	0
$349$	−3588.04	−0.550326	−0.275163	−	0.961398i	$-0.588732\pi$
$349$	−3588.04	−0.550326	−0.275163	+	0.961398i	$0.588732\pi$
$350$	0	0
$351$	−4063.31	−0.617902
$352$	0	0
$353$	150.268	0.0226571	0.0113286	−	0.999936i	$-0.496394\pi$
$353$	150.268	0.0226571	0.0113286	+	0.999936i	$0.496394\pi$
$354$	0	0
$355$	−7942.20	−1.18740
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2233.92	−0.328418	−0.164209	−	0.986426i	$-0.552507\pi$
$359$	−2233.92	−0.328418	−0.164209	+	0.986426i	$0.552507\pi$
$360$	0	0
$361$	4909.78	0.715816
$362$	0	0
$363$	26154.9	3.78175
$364$	0	0
$365$	1491.79	0.213928
$366$	0	0
$367$	2791.98	0.397112	0.198556	−	0.980090i	$-0.436375\pi$
$367$	2791.98	0.397112	0.198556	+	0.980090i	$0.436375\pi$
$368$	0	0
$369$	−5416.04	−0.764086
$370$	0	0
$371$	5347.56	0.748333
$372$	0	0
$373$	10409.0	1.44492	0.722461	−	0.691411i	$-0.243011\pi$
$373$	10409.0	1.44492	0.722461	+	0.691411i	$0.243011\pi$
$374$	0	0
$375$	−13106.4	−1.80483
$376$	0	0
$377$	−1980.68	−0.270584
$378$	0	0
$379$	−4731.90	−0.641323	−0.320661	−	0.947194i	$-0.603905\pi$
$379$	−4731.90	−0.641323	−0.320661	+	0.947194i	$0.603905\pi$
$380$	0	0
$381$	−18888.8	−2.53990
$382$	0	0
$383$	−4845.96	−0.646519	−0.323260	−	0.946310i	$-0.604779\pi$
$383$	−4845.96	−0.646519	−0.323260	+	0.946310i	$0.604779\pi$
$384$	0	0
$385$	18796.6	2.48822
$386$	0	0
$387$	13275.5	1.74375
$388$	0	0
$389$	8250.13	1.07532	0.537658	−	0.843163i	$-0.319309\pi$
$389$	8250.13	1.07532	0.537658	+	0.843163i	$0.319309\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	26242.5	3.36834
$394$	0	0
$395$	743.519	0.0947102
$396$	0	0
$397$	9971.59	1.26060	0.630302	−	0.776350i	$-0.282931\pi$
$397$	9971.59	1.26060	0.630302	+	0.776350i	$0.282931\pi$
$398$	0	0
$399$	27186.0	3.41103
$400$	0	0
$401$	−3302.95	−0.411325	−0.205663	−	0.978623i	$-0.565935\pi$
$401$	−3302.95	−0.411325	−0.205663	+	0.978623i	$0.565935\pi$
$402$	0	0
$403$	2228.90	0.275508
$404$	0	0
$405$	24787.7	3.04126
$406$	0	0
$407$	3014.42	0.367123
$408$	0	0
$409$	−2024.08	−0.244705	−0.122352	−	0.992487i	$-0.539044\pi$
$409$	−2024.08	−0.244705	−0.122352	+	0.992487i	$0.539044\pi$
$410$	0	0
$411$	−15482.2	−1.85810
$412$	0	0
$413$	4181.25	0.498173
$414$	0	0
$415$	−7668.22	−0.907032
$416$	0	0
$417$	−10616.5	−1.24674
$418$	0	0
$419$	−9294.25	−1.08366	−0.541830	−	0.840488i	$-0.682269\pi$
$419$	−9294.25	−1.08366	−0.541830	+	0.840488i	$0.682269\pi$
$420$	0	0
$421$	9238.43	1.06949	0.534743	−	0.845015i	$-0.320408\pi$
$421$	9238.43	1.06949	0.534743	+	0.845015i	$0.320408\pi$
$422$	0	0
$423$	−16961.8	−1.94967
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2763.19	−0.313162
$428$	0	0
$429$	6163.79	0.693685
$430$	0	0
$431$	1518.38	0.169693	0.0848464	−	0.996394i	$-0.472960\pi$
$431$	1518.38	0.169693	0.0848464	+	0.996394i	$0.472960\pi$
$432$	0	0
$433$	3810.92	0.422959	0.211479	−	0.977382i	$-0.432172\pi$
$433$	3810.92	0.422959	0.211479	+	0.977382i	$0.432172\pi$
$434$	0	0
$435$	22570.5	2.48775
$436$	0	0
$437$	−712.760	−0.0780228
$438$	0	0
$439$	−7972.81	−0.866792	−0.433396	−	0.901204i	$-0.642685\pi$
$439$	−7972.81	−0.866792	−0.433396	+	0.901204i	$0.642685\pi$
$440$	0	0
$441$	21515.5	2.32324
$442$	0	0
$443$	2524.64	0.270765	0.135383	−	0.990793i	$-0.456774\pi$
$443$	2524.64	0.270765	0.135383	+	0.990793i	$0.456774\pi$
$444$	0	0
$445$	7627.63	0.812549
$446$	0	0
$447$	2138.49	0.226280
$448$	0	0
$449$	−391.698	−0.0411701	−0.0205851	−	0.999788i	$-0.506553\pi$
$449$	−391.698	−0.0411701	−0.0205851	+	0.999788i	$0.506553\pi$
$450$	0	0
$451$	4987.62	0.520749
$452$	0	0
$453$	1844.78	0.191336
$454$	0	0
$455$	2957.34	0.304708
$456$	0	0
$457$	−7463.84	−0.763990	−0.381995	−	0.924164i	$-0.624763\pi$
$457$	−7463.84	−0.763990	−0.381995	+	0.924164i	$0.624763\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4409.10	0.445449	0.222725	−	0.974881i	$-0.428505\pi$
$461$	4409.10	0.445449	0.222725	+	0.974881i	$0.428505\pi$
$462$	0	0
$463$	−15821.7	−1.58811	−0.794055	−	0.607845i	$-0.792034\pi$
$463$	−15821.7	−1.58811	−0.794055	+	0.607845i	$0.792034\pi$
$464$	0	0
$465$	−25399.1	−2.53302
$466$	0	0
$467$	−9150.74	−0.906736	−0.453368	−	0.891323i	$-0.649778\pi$
$467$	−9150.74	−0.906736	−0.453368	+	0.891323i	$0.649778\pi$
$468$	0	0
$469$	7502.49	0.738663
$470$	0	0
$471$	−15621.0	−1.52819
$472$	0	0
$473$	−12225.4	−1.18842
$474$	0	0
$475$	−1028.13	−0.0993132
$476$	0	0
$477$	−14345.8	−1.37704
$478$	0	0
$479$	19212.9	1.83269	0.916346	−	0.400387i	$-0.131124\pi$
$479$	19212.9	1.83269	0.916346	+	0.400387i	$0.131124\pi$
$480$	0	0
$481$	474.269	0.0449580
$482$	0	0
$483$	−1646.48	−0.155109
$484$	0	0
$485$	952.452	0.0891724
$486$	0	0
$487$	5991.84	0.557528	0.278764	−	0.960360i	$-0.410075\pi$
$487$	5991.84	0.557528	0.278764	+	0.960360i	$0.410075\pi$
$488$	0	0
$489$	−5088.94	−0.470613
$490$	0	0
$491$	1595.37	0.146636	0.0733178	−	0.997309i	$-0.476641\pi$
$491$	1595.37	0.146636	0.0733178	+	0.997309i	$0.476641\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−50425.3	−4.57868
$496$	0	0
$497$	17543.0	1.58332
$498$	0	0
$499$	−5149.80	−0.461998	−0.230999	−	0.972954i	$-0.574199\pi$
$499$	−5149.80	−0.461998	−0.230999	+	0.972954i	$0.574199\pi$
$500$	0	0
$501$	14358.3	1.28040
$502$	0	0
$503$	14927.8	1.32326	0.661628	−	0.749832i	$-0.269866\pi$
$503$	14927.8	1.32326	0.661628	+	0.749832i	$0.269866\pi$
$504$	0	0
$505$	8247.34	0.726736
$506$	0	0
$507$	−20524.5	−1.79788
$508$	0	0
$509$	−18083.8	−1.57476	−0.787378	−	0.616471i	$-0.788562\pi$
$509$	−18083.8	−1.57476	−0.787378	+	0.616471i	$0.788562\pi$
$510$	0	0
$511$	−3295.12	−0.285259
$512$	0	0
$513$	−44274.9	−3.81050
$514$	0	0
$515$	−20697.5	−1.77095
$516$	0	0
$517$	15620.1	1.32876
$518$	0	0
$519$	−7929.02	−0.670608
$520$	0	0
$521$	−16390.5	−1.37827	−0.689137	−	0.724631i	$-0.742010\pi$
$521$	−16390.5	−1.37827	−0.689137	+	0.724631i	$0.742010\pi$
$522$	0	0
$523$	−17486.4	−1.46200	−0.731000	−	0.682377i	$-0.760946\pi$
$523$	−17486.4	−1.46200	−0.731000	+	0.682377i	$0.760946\pi$
$524$	0	0
$525$	−2374.99	−0.197434
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12123.8	−0.996452
$530$	0	0
$531$	−11216.9	−0.916711
$532$	0	0
$533$	784.719	0.0637710
$534$	0	0
$535$	−14845.7	−1.19969
$536$	0	0
$537$	−13640.4	−1.09614
$538$	0	0
$539$	−19813.6	−1.58336
$540$	0	0
$541$	12550.3	0.997375	0.498688	−	0.866782i	$-0.333816\pi$
$541$	12550.3	0.997375	0.498688	+	0.866782i	$0.333816\pi$
$542$	0	0
$543$	−12171.7	−0.961945
$544$	0	0
$545$	2211.14	0.173789
$546$	0	0
$547$	24692.5	1.93012	0.965061	−	0.262025i	$-0.0843904\pi$
$547$	24692.5	1.93012	0.965061	+	0.262025i	$0.0843904\pi$
$548$	0	0
$549$	7412.75	0.576263
$550$	0	0
$551$	−21582.0	−1.66864
$552$	0	0
$553$	−1642.31	−0.126290
$554$	0	0
$555$	−5404.45	−0.413345
$556$	0	0
$557$	−1130.02	−0.0859616	−0.0429808	−	0.999076i	$-0.513685\pi$
$557$	−1130.02	−0.0859616	−0.0429808	+	0.999076i	$0.513685\pi$
$558$	0	0
$559$	−1923.46	−0.145534
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14758.6	−1.10480	−0.552400	−	0.833579i	$-0.686288\pi$
$563$	−14758.6	−1.10480	−0.552400	+	0.833579i	$0.686288\pi$
$564$	0	0
$565$	−15686.5	−1.16803
$566$	0	0
$567$	−54752.0	−4.05532
$568$	0	0
$569$	−22712.9	−1.67341	−0.836707	−	0.547651i	$-0.815522\pi$
$569$	−22712.9	−1.67341	−0.836707	+	0.547651i	$0.815522\pi$
$570$	0	0
$571$	−2410.84	−0.176691	−0.0883453	−	0.996090i	$-0.528158\pi$
$571$	−2410.84	−0.176691	−0.0883453	+	0.996090i	$0.528158\pi$
$572$	0	0
$573$	43876.4	3.19889
$574$	0	0
$575$	62.2672	0.00451604
$576$	0	0
$577$	−7432.70	−0.536269	−0.268135	−	0.963381i	$-0.586407\pi$
$577$	−7432.70	−0.536269	−0.268135	+	0.963381i	$0.586407\pi$
$578$	0	0
$579$	−24421.1	−1.75286
$580$	0	0
$581$	16937.8	1.20947
$582$	0	0
$583$	13211.0	0.938497
$584$	0	0
$585$	−7933.58	−0.560707
$586$	0	0
$587$	18608.6	1.30845	0.654226	−	0.756299i	$-0.272995\pi$
$587$	18608.6	1.30845	0.654226	+	0.756299i	$0.272995\pi$
$588$	0	0
$589$	24286.7	1.69901
$590$	0	0
$591$	397.250	0.0276492
$592$	0	0
$593$	−13926.8	−0.964423	−0.482212	−	0.876055i	$-0.660166\pi$
$593$	−13926.8	−0.964423	−0.482212	+	0.876055i	$0.660166\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	37021.8	2.53803
$598$	0	0
$599$	4893.34	0.333784	0.166892	−	0.985975i	$-0.446627\pi$
$599$	4893.34	0.333784	0.166892	+	0.985975i	$0.446627\pi$
$600$	0	0
$601$	7908.38	0.536755	0.268377	−	0.963314i	$-0.413513\pi$
$601$	7908.38	0.536755	0.268377	+	0.963314i	$0.413513\pi$
$602$	0	0
$603$	−20126.8	−1.35925
$604$	0	0
$605$	31001.7	2.08330
$606$	0	0
$607$	738.635	0.0493909	0.0246954	−	0.999695i	$-0.492138\pi$
$607$	738.635	0.0493909	0.0246954	+	0.999695i	$0.492138\pi$
$608$	0	0
$609$	−49854.5	−3.31725
$610$	0	0
$611$	2457.56	0.162721
$612$	0	0
$613$	−27223.3	−1.79370	−0.896851	−	0.442333i	$-0.854151\pi$
$613$	−27223.3	−1.79370	−0.896851	+	0.442333i	$0.854151\pi$
$614$	0	0
$615$	−8942.14	−0.586312
$616$	0	0
$617$	−11842.7	−0.772722	−0.386361	−	0.922348i	$-0.626268\pi$
$617$	−11842.7	−0.772722	−0.386361	+	0.922348i	$0.626268\pi$
$618$	0	0
$619$	−6425.17	−0.417204	−0.208602	−	0.978001i	$-0.566891\pi$
$619$	−6425.17	−0.417204	−0.208602	+	0.978001i	$0.566891\pi$
$620$	0	0
$621$	2681.45	0.173273
$622$	0	0
$623$	−16848.2	−1.08348
$624$	0	0
$625$	−16719.8	−1.07007
$626$	0	0
$627$	67162.2	4.27783
$628$	0	0
$629$	0	0
$630$	0	0
$631$	24477.7	1.54428	0.772140	−	0.635453i	$-0.219187\pi$
$631$	24477.7	1.54428	0.772140	+	0.635453i	$0.219187\pi$
$632$	0	0
$633$	15280.4	0.959464
$634$	0	0
$635$	−22389.1	−1.39919
$636$	0	0
$637$	−3117.34	−0.193899
$638$	0	0
$639$	−47062.2	−2.91354
$640$	0	0
$641$	−27970.3	−1.72350	−0.861748	−	0.507337i	$-0.830630\pi$
$641$	−27970.3	−1.72350	−0.861748	+	0.507337i	$0.830630\pi$
$642$	0	0
$643$	6086.50	0.373294	0.186647	−	0.982427i	$-0.440238\pi$
$643$	6086.50	0.373294	0.186647	+	0.982427i	$0.440238\pi$
$644$	0	0
$645$	21918.5	1.33805
$646$	0	0
$647$	−18583.9	−1.12923	−0.564613	−	0.825356i	$-0.690974\pi$
$647$	−18583.9	−1.12923	−0.564613	+	0.825356i	$0.690974\pi$
$648$	0	0
$649$	10329.6	0.624768
$650$	0	0
$651$	56102.4	3.37762
$652$	0	0
$653$	25315.7	1.51712	0.758562	−	0.651601i	$-0.225902\pi$
$653$	25315.7	1.51712	0.758562	+	0.651601i	$0.225902\pi$
$654$	0	0
$655$	31105.5	1.85556
$656$	0	0
$657$	8839.74	0.524918
$658$	0	0
$659$	−4948.03	−0.292485	−0.146243	−	0.989249i	$-0.546718\pi$
$659$	−4948.03	−0.292485	−0.146243	+	0.989249i	$0.546718\pi$
$660$	0	0
$661$	1373.95	0.0808476	0.0404238	−	0.999183i	$-0.487129\pi$
$661$	1373.95	0.0808476	0.0404238	+	0.999183i	$0.487129\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	32223.9	1.87908
$666$	0	0
$667$	1307.08	0.0758777
$668$	0	0
$669$	48905.6	2.82631
$670$	0	0
$671$	−6826.39	−0.392742
$672$	0	0
$673$	−5909.61	−0.338482	−0.169241	−	0.985575i	$-0.554132\pi$
$673$	−5909.61	−0.338482	−0.169241	+	0.985575i	$0.554132\pi$
$674$	0	0
$675$	3867.88	0.220555
$676$	0	0
$677$	−16232.5	−0.921516	−0.460758	−	0.887526i	$-0.652422\pi$
$677$	−16232.5	−0.921516	−0.460758	+	0.887526i	$0.652422\pi$
$678$	0	0
$679$	−2103.81	−0.118905
$680$	0	0
$681$	−18621.8	−1.04786
$682$	0	0
$683$	−1804.37	−0.101087	−0.0505435	−	0.998722i	$-0.516095\pi$
$683$	−1804.37	−0.101087	−0.0505435	+	0.998722i	$0.516095\pi$
$684$	0	0
$685$	−18351.2	−1.02360
$686$	0	0
$687$	35411.3	1.96656
$688$	0	0
$689$	2078.53	0.114929
$690$	0	0
$691$	9644.84	0.530980	0.265490	−	0.964114i	$-0.414466\pi$
$691$	9644.84	0.530980	0.265490	+	0.964114i	$0.414466\pi$
$692$	0	0
$693$	111381.	6.10537
$694$	0	0
$695$	−12583.8	−0.686809
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−33528.8	−1.81427
$700$	0	0
$701$	−29610.2	−1.59538	−0.797689	−	0.603069i	$-0.793944\pi$
$701$	−29610.2	−1.59538	−0.797689	+	0.603069i	$0.793944\pi$
$702$	0	0
$703$	5167.75	0.277248
$704$	0	0
$705$	−28004.8	−1.49606
$706$	0	0
$707$	−18217.0	−0.969054
$708$	0	0
$709$	−8110.17	−0.429596	−0.214798	−	0.976658i	$-0.568909\pi$
$709$	−8110.17	−0.429596	−0.214798	+	0.976658i	$0.568909\pi$
$710$	0	0
$711$	4405.79	0.232391
$712$	0	0
$713$	−1470.89	−0.0772584
$714$	0	0
$715$	7306.02	0.382139
$716$	0	0
$717$	−56718.1	−2.95422
$718$	0	0
$719$	−20964.0	−1.08738	−0.543688	−	0.839287i	$-0.682973\pi$
$719$	−20964.0	−1.08738	−0.543688	+	0.839287i	$0.682973\pi$
$720$	0	0
$721$	45717.3	2.36144
$722$	0	0
$723$	43425.9	2.23378
$724$	0	0
$725$	1885.41	0.0965828
$726$	0	0
$727$	35728.7	1.82270	0.911351	−	0.411630i	$-0.135040\pi$
$727$	35728.7	1.82270	0.911351	+	0.411630i	$0.135040\pi$
$728$	0	0
$729$	39074.8	1.98521
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−22697.4	−1.14372	−0.571861	−	0.820350i	$-0.693778\pi$
$733$	−22697.4	−1.14372	−0.571861	+	0.820350i	$0.693778\pi$
$734$	0	0
$735$	35523.1	1.78271
$736$	0	0
$737$	18534.7	0.926370
$738$	0	0
$739$	5393.01	0.268451	0.134225	−	0.990951i	$-0.457145\pi$
$739$	5393.01	0.268451	0.134225	+	0.990951i	$0.457145\pi$
$740$	0	0
$741$	10566.9	0.523864
$742$	0	0
$743$	−15928.7	−0.786499	−0.393250	−	0.919432i	$-0.628649\pi$
$743$	−15928.7	−0.786499	−0.393250	+	0.919432i	$0.628649\pi$
$744$	0	0
$745$	2534.78	0.124654
$746$	0	0
$747$	−45438.8	−2.22559
$748$	0	0
$749$	32791.8	1.59971
$750$	0	0
$751$	−16896.4	−0.820983	−0.410492	−	0.911864i	$-0.634643\pi$
$751$	−16896.4	−0.820983	−0.410492	+	0.911864i	$0.634643\pi$
$752$	0	0
$753$	−40820.6	−1.97555
$754$	0	0
$755$	2186.64	0.105404
$756$	0	0
$757$	−19109.1	−0.917479	−0.458740	−	0.888571i	$-0.651699\pi$
$757$	−19109.1	−0.917479	−0.458740	+	0.888571i	$0.651699\pi$
$758$	0	0
$759$	−4067.59	−0.194525
$760$	0	0
$761$	−30188.7	−1.43803	−0.719014	−	0.694996i	$-0.755406\pi$
$761$	−30188.7	−1.43803	−0.719014	+	0.694996i	$0.755406\pi$
$762$	0	0
$763$	−4884.05	−0.231736
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1625.20	0.0765092
$768$	0	0
$769$	6235.12	0.292385	0.146193	−	0.989256i	$-0.453298\pi$
$769$	6235.12	0.292385	0.146193	+	0.989256i	$0.453298\pi$
$770$	0	0
$771$	−59820.4	−2.79427
$772$	0	0
$773$	23158.1	1.07754	0.538771	−	0.842452i	$-0.318889\pi$
$773$	23158.1	1.07754	0.538771	+	0.842452i	$0.318889\pi$
$774$	0	0
$775$	−2121.70	−0.0983402
$776$	0	0
$777$	11937.6	0.551168
$778$	0	0
$779$	8550.50	0.393265
$780$	0	0
$781$	43339.5	1.98567
$782$	0	0
$783$	81192.7	3.70573
$784$	0	0
$785$	−18515.7	−0.841853
$786$	0	0
$787$	8824.40	0.399690	0.199845	−	0.979828i	$-0.435956\pi$
$787$	8824.40	0.399690	0.199845	+	0.979828i	$0.435956\pi$
$788$	0	0
$789$	48251.2	2.17717
$790$	0	0
$791$	34648.9	1.55749
$792$	0	0
$793$	−1074.02	−0.0480953
$794$	0	0
$795$	−23685.6	−1.05666
$796$	0	0
$797$	−24954.4	−1.10907	−0.554535	−	0.832160i	$-0.687104\pi$
$797$	−24954.4	−1.10907	−0.554535	+	0.832160i	$0.687104\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	45198.2	1.99376
$802$	0	0
$803$	−8140.50	−0.357748
$804$	0	0
$805$	−1951.60	−0.0854469
$806$	0	0
$807$	8709.58	0.379916
$808$	0	0
$809$	−39222.0	−1.70454	−0.852268	−	0.523105i	$-0.824774\pi$
$809$	−39222.0	−1.70454	−0.852268	+	0.523105i	$0.824774\pi$
$810$	0	0
$811$	−18738.4	−0.811336	−0.405668	−	0.914020i	$-0.632961\pi$
$811$	−18738.4	−0.811336	−0.405668	+	0.914020i	$0.632961\pi$
$812$	0	0
$813$	71666.3	3.09157
$814$	0	0
$815$	−6031.98	−0.259253
$816$	0	0
$817$	−20958.5	−0.897485
$818$	0	0
$819$	17524.0	0.747665
$820$	0	0
$821$	13277.0	0.564397	0.282198	−	0.959356i	$-0.408936\pi$
$821$	13277.0	0.564397	0.282198	+	0.959356i	$0.408936\pi$
$822$	0	0
$823$	28984.7	1.22763	0.613817	−	0.789448i	$-0.289633\pi$
$823$	28984.7	1.22763	0.613817	+	0.789448i	$0.289633\pi$
$824$	0	0
$825$	−5867.33	−0.247605
$826$	0	0
$827$	871.224	0.0366329	0.0183165	−	0.999832i	$-0.494169\pi$
$827$	871.224	0.0366329	0.0183165	+	0.999832i	$0.494169\pi$
$828$	0	0
$829$	−25090.6	−1.05118	−0.525592	−	0.850737i	$-0.676156\pi$
$829$	−25090.6	−1.05118	−0.525592	+	0.850737i	$0.676156\pi$
$830$	0	0
$831$	28329.8	1.18261
$832$	0	0
$833$	0	0
$834$	0	0
$835$	17019.1	0.705353
$836$	0	0
$837$	−91368.0	−3.77316
$838$	0	0
$839$	−4308.76	−0.177300	−0.0886501	−	0.996063i	$-0.528255\pi$
$839$	−4308.76	−0.177300	−0.0886501	+	0.996063i	$0.528255\pi$
$840$	0	0
$841$	15188.7	0.622768
$842$	0	0
$843$	46404.5	1.89592
$844$	0	0
$845$	−24327.9	−0.990420
$846$	0	0
$847$	−68477.6	−2.77794
$848$	0	0
$849$	85396.7	3.45207
$850$	0	0
$851$	−312.978	−0.0126072
$852$	0	0
$853$	34278.3	1.37593	0.687964	−	0.725745i	$-0.258505\pi$
$853$	34278.3	1.37593	0.687964	+	0.725745i	$0.258505\pi$
$854$	0	0
$855$	−86446.3	−3.45778
$856$	0	0
$857$	−4048.60	−0.161374	−0.0806870	−	0.996739i	$-0.525711\pi$
$857$	−4048.60	−0.161374	−0.0806870	+	0.996739i	$0.525711\pi$
$858$	0	0
$859$	5033.78	0.199942	0.0999712	−	0.994990i	$-0.468125\pi$
$859$	5033.78	0.199942	0.0999712	+	0.994990i	$0.468125\pi$
$860$	0	0
$861$	19751.7	0.781808
$862$	0	0
$863$	23465.3	0.925572	0.462786	−	0.886470i	$-0.346850\pi$
$863$	23465.3	0.925572	0.462786	+	0.886470i	$0.346850\pi$
$864$	0	0
$865$	−9398.36	−0.369426
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4057.29	−0.158382
$870$	0	0
$871$	2916.13	0.113443
$872$	0	0
$873$	5643.84	0.218803
$874$	0	0
$875$	34314.7	1.32577
$876$	0	0
$877$	23612.4	0.909159	0.454580	−	0.890706i	$-0.349789\pi$
$877$	23612.4	0.909159	0.454580	+	0.890706i	$0.349789\pi$
$878$	0	0
$879$	81431.2	3.12469
$880$	0	0
$881$	28502.3	1.08997	0.544987	−	0.838445i	$-0.316535\pi$
$881$	28502.3	1.08997	0.544987	+	0.838445i	$0.316535\pi$
$882$	0	0
$883$	−39621.1	−1.51003	−0.755014	−	0.655709i	$-0.772370\pi$
$883$	−39621.1	−1.51003	−0.755014	+	0.655709i	$0.772370\pi$
$884$	0	0
$885$	−18519.7	−0.703427
$886$	0	0
$887$	−9831.89	−0.372179	−0.186089	−	0.982533i	$-0.559581\pi$
$887$	−9831.89	−0.372179	−0.186089	+	0.982533i	$0.559581\pi$
$888$	0	0
$889$	49453.8	1.86572
$890$	0	0
$891$	−135263.	−5.08584
$892$	0	0
$893$	26778.2	1.00347
$894$	0	0
$895$	−16168.1	−0.603845
$896$	0	0
$897$	−639.968	−0.0238215
$898$	0	0
$899$	−44537.7	−1.65230
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−48414.3	−1.78420
$904$	0	0
$905$	−14427.2	−0.529919
$906$	0	0
$907$	6063.91	0.221994	0.110997	−	0.993821i	$-0.464596\pi$
$907$	6063.91	0.221994	0.110997	+	0.993821i	$0.464596\pi$
$908$	0	0
$909$	48870.4	1.78320
$910$	0	0
$911$	7596.55	0.276273	0.138137	−	0.990413i	$-0.455889\pi$
$911$	7596.55	0.276273	0.138137	+	0.990413i	$0.455889\pi$
$912$	0	0
$913$	41844.5	1.51681
$914$	0	0
$915$	12238.8	0.442189
$916$	0	0
$917$	−68707.0	−2.47427
$918$	0	0
$919$	−36363.0	−1.30523	−0.652614	−	0.757691i	$-0.726327\pi$
$919$	−36363.0	−1.30523	−0.652614	+	0.757691i	$0.726327\pi$
$920$	0	0
$921$	87238.5	3.12118
$922$	0	0
$923$	6818.75	0.243166
$924$	0	0
$925$	−451.458	−0.0160474
$926$	0	0
$927$	−122645.	−4.34540
$928$	0	0
$929$	46961.3	1.65850	0.829252	−	0.558875i	$-0.188767\pi$
$929$	46961.3	1.65850	0.829252	+	0.558875i	$0.188767\pi$
$930$	0	0
$931$	−33967.3	−1.19574
$932$	0	0
$933$	72940.0	2.55943
$934$	0	0
$935$	0	0
$936$	0	0
$937$	36031.2	1.25623	0.628116	−	0.778120i	$-0.283826\pi$
$937$	36031.2	1.25623	0.628116	+	0.778120i	$0.283826\pi$
$938$	0	0
$939$	−36915.4	−1.28295
$940$	0	0
$941$	2357.76	0.0816800	0.0408400	−	0.999166i	$-0.486997\pi$
$941$	2357.76	0.0816800	0.0408400	+	0.999166i	$0.486997\pi$
$942$	0	0
$943$	−517.849	−0.0178828
$944$	0	0
$945$	−121228.	−4.17307
$946$	0	0
$947$	10541.1	0.361709	0.180855	−	0.983510i	$-0.442114\pi$
$947$	10541.1	0.361709	0.180855	+	0.983510i	$0.442114\pi$
$948$	0	0
$949$	−1280.77	−0.0438099
$950$	0	0
$951$	−33647.7	−1.14732
$952$	0	0
$953$	36771.6	1.24990	0.624948	−	0.780667i	$-0.285120\pi$
$953$	36771.6	1.24990	0.624948	+	0.780667i	$0.285120\pi$
$954$	0	0
$955$	52007.2	1.76221
$956$	0	0
$957$	−123164.	−4.16022
$958$	0	0
$959$	40534.9	1.36490
$960$	0	0
$961$	20328.2	0.682361
$962$	0	0
$963$	−87969.7	−2.94370
$964$	0	0
$965$	−28946.6	−0.965620
$966$	0	0
$967$	−38981.8	−1.29635	−0.648174	−	0.761492i	$-0.724467\pi$
$967$	−38981.8	−1.29635	−0.648174	+	0.761492i	$0.724467\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13355.0	0.441383	0.220692	−	0.975344i	$-0.429169\pi$
$971$	13355.0	0.441383	0.220692	+	0.975344i	$0.429169\pi$
$972$	0	0
$973$	27795.6	0.915814
$974$	0	0
$975$	−923.128	−0.0303218
$976$	0	0
$977$	−9868.81	−0.323164	−0.161582	−	0.986859i	$-0.551660\pi$
$977$	−9868.81	−0.323164	−0.161582	+	0.986859i	$0.551660\pi$
$978$	0	0
$979$	−41622.9	−1.35881
$980$	0	0
$981$	13102.3	0.426427
$982$	0	0
$983$	−39878.0	−1.29391	−0.646953	−	0.762530i	$-0.723957\pi$
$983$	−39878.0	−1.29391	−0.646953	+	0.762530i	$0.723957\pi$
$984$	0	0
$985$	470.866	0.0152315
$986$	0	0
$987$	61858.0	1.99489
$988$	0	0
$989$	1269.32	0.0408111
$990$	0	0
$991$	52529.3	1.68380	0.841901	−	0.539632i	$-0.181437\pi$
$991$	52529.3	1.68380	0.841901	+	0.539632i	$0.181437\pi$
$992$	0	0
$993$	−87473.0	−2.79544
$994$	0	0
$995$	43882.4	1.39816
$996$	0	0
$997$	−52390.5	−1.66422	−0.832108	−	0.554614i	$-0.812866\pi$
$997$	−52390.5	−1.66422	−0.832108	+	0.554614i	$0.812866\pi$
$998$	0	0
$999$	−19441.4	−0.615714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.m.1.14		14
17.8	even	8		136.4.k.a.81.1	✓	14
17.15	even	8		136.4.k.a.89.1	yes	14
17.16	even	2	inner	2312.4.a.m.1.1		14
68.15	odd	8		272.4.o.g.225.7		14
68.59	odd	8		272.4.o.g.81.7		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.k.a.81.1	✓	14	17.8	even	8
136.4.k.a.89.1	yes	14	17.15	even	8
272.4.o.g.81.7		14	68.59	odd	8
272.4.o.g.225.7		14	68.15	odd	8
2312.4.a.m.1.1		14	17.16	even	2	inner
2312.4.a.m.1.14		14	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q+9.78344 q^{3} +11.5964 q^{5} -25.6146 q^{7} +68.7157 q^{9} +O(q^{10})\)	\(q+9.78344 q^{3} +11.5964 q^{5} -25.6146 q^{7} +68.7157 q^{9} -63.2802 q^{11} -9.95609 q^{13} +113.453 q^{15} -108.484 q^{19} -250.599 q^{21} +6.57019 q^{23} +9.47723 q^{25} +408.124 q^{27} +198.941 q^{29} -223.873 q^{31} -619.098 q^{33} -297.038 q^{35} -47.6360 q^{37} -97.4048 q^{39} -78.8180 q^{41} +193.194 q^{43} +796.857 q^{45} -246.840 q^{47} +313.109 q^{49} -208.770 q^{53} -733.824 q^{55} -1061.35 q^{57} -163.237 q^{59} +107.876 q^{61} -1760.13 q^{63} -115.455 q^{65} -292.899 q^{67} +64.2790 q^{69} -684.883 q^{71} +128.642 q^{73} +92.7199 q^{75} +1620.90 q^{77} +64.1162 q^{79} +2137.53 q^{81} -661.257 q^{83} +1946.33 q^{87} +657.756 q^{89} +255.021 q^{91} -2190.25 q^{93} -1258.03 q^{95} +82.1332 q^{97} -4348.34 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 190 q^{9}+O(q^{10}) \) 14 * q + 190 * q^9	\( 14 q + 190 q^{9} - 72 q^{13} - 60 q^{15} - 468 q^{19} + 60 q^{21} + 306 q^{25} - 904 q^{33} - 804 q^{35} - 896 q^{43} - 448 q^{47} + 866 q^{49} - 108 q^{53} - 1612 q^{55} - 1760 q^{59} - 284 q^{67} - 2532 q^{69} + 180 q^{77} + 950 q^{81} - 4256 q^{83} - 868 q^{87} + 4516 q^{89} - 7156 q^{93}+O(q^{100}) \) 14 * q + 190 * q^9 - 72 * q^13 - 60 * q^15 - 468 * q^19 + 60 * q^21 + 306 * q^25 - 904 * q^33 - 804 * q^35 - 896 * q^43 - 448 * q^47 + 866 * q^49 - 108 * q^53 - 1612 * q^55 - 1760 * q^59 - 284 * q^67 - 2532 * q^69 + 180 * q^77 + 950 * q^81 - 4256 * q^83 - 868 * q^87 + 4516 * q^89 - 7156 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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