LMFDB - Embedded newform 2312.4.a.m.1.8

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			1.8
Root			$-0.349380$ 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)

$f(q)$	$=$	$q+1.50590 q^{3} -0.778787 q^{5} -16.0004 q^{7} -24.7323 q^{9} +O(q^{10})$	$q+1.50590 q^{3} -0.778787 q^{5} -16.0004 q^{7} -24.7323 q^{9} +7.86611 q^{11} +64.3974 q^{13} -1.17278 q^{15} +9.88432 q^{19} -24.0950 q^{21} +190.798 q^{23} -124.393 q^{25} -77.9037 q^{27} -210.052 q^{29} -6.86249 q^{31} +11.8456 q^{33} +12.4609 q^{35} -92.4597 q^{37} +96.9761 q^{39} +318.822 q^{41} +227.690 q^{43} +19.2612 q^{45} -240.883 q^{47} -86.9871 q^{49} +247.863 q^{53} -6.12603 q^{55} +14.8848 q^{57} -321.574 q^{59} +443.802 q^{61} +395.726 q^{63} -50.1518 q^{65} -17.8872 q^{67} +287.323 q^{69} -779.552 q^{71} +575.418 q^{73} -187.324 q^{75} -125.861 q^{77} +490.715 q^{79} +550.456 q^{81} +1282.66 q^{83} -316.318 q^{87} +187.071 q^{89} -1030.38 q^{91} -10.3342 q^{93} -7.69778 q^{95} +58.9763 q^{97} -194.547 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.50590	0.289811	0.144905	−	0.989446i	$-0.453712\pi$
$3$	1.50590	0.289811	0.144905	+	0.989446i	$0.453712\pi$
$4$	0	0
$5$	−0.778787	−0.0696568	−0.0348284	−	0.999393i	$-0.511088\pi$
$5$	−0.778787	−0.0696568	−0.0348284	+	0.999393i	$0.511088\pi$
$6$	0	0
$7$	−16.0004	−0.863941	−0.431970	−	0.901888i	$-0.642181\pi$
$7$	−16.0004	−0.863941	−0.431970	+	0.901888i	$0.642181\pi$
$8$	0	0
$9$	−24.7323	−0.916010
$10$	0	0
$11$	7.86611	0.215611	0.107806	−	0.994172i	$-0.465618\pi$
$11$	7.86611	0.215611	0.107806	+	0.994172i	$0.465618\pi$
$12$	0	0
$13$	64.3974	1.37389	0.686947	−	0.726708i	$-0.258951\pi$
$13$	64.3974	1.37389	0.686947	+	0.726708i	$0.258951\pi$
$14$	0	0
$15$	−1.17278	−0.0201873
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	9.88432	0.119348	0.0596741	−	0.998218i	$-0.480994\pi$
$19$	9.88432	0.119348	0.0596741	+	0.998218i	$0.480994\pi$
$20$	0	0
$21$	−24.0950	−0.250379
$22$	0	0
$23$	190.798	1.72975	0.864873	−	0.501991i	$-0.167399\pi$
$23$	190.798	1.72975	0.864873	+	0.501991i	$0.167399\pi$
$24$	0	0
$25$	−124.393	−0.995148
$26$	0	0
$27$	−77.9037	−0.555281
$28$	0	0
$29$	−210.052	−1.34502	−0.672512	−	0.740086i	$-0.734785\pi$
$29$	−210.052	−1.34502	−0.672512	+	0.740086i	$0.734785\pi$
$30$	0	0
$31$	−6.86249	−0.0397593	−0.0198797	−	0.999802i	$-0.506328\pi$
$31$	−6.86249	−0.0397593	−0.0198797	+	0.999802i	$0.506328\pi$
$32$	0	0
$33$	11.8456	0.0624865
$34$	0	0
$35$	12.4609	0.0601794
$36$	0	0
$37$	−92.4597	−0.410819	−0.205409	−	0.978676i	$-0.565853\pi$
$37$	−92.4597	−0.410819	−0.205409	+	0.978676i	$0.565853\pi$
$38$	0	0
$39$	96.9761	0.398169
$40$	0	0
$41$	318.822	1.21443	0.607215	−	0.794538i	$-0.292287\pi$
$41$	318.822	1.21443	0.607215	+	0.794538i	$0.292287\pi$
$42$	0	0
$43$	227.690	0.807497	0.403748	−	0.914870i	$-0.367707\pi$
$43$	227.690	0.807497	0.403748	+	0.914870i	$0.367707\pi$
$44$	0	0
$45$	19.2612	0.0638063
$46$	0	0
$47$	−240.883	−0.747582	−0.373791	−	0.927513i	$-0.621942\pi$
$47$	−240.883	−0.747582	−0.373791	+	0.927513i	$0.621942\pi$
$48$	0	0
$49$	−86.9871	−0.253607
$50$	0	0
$51$	0	0
$52$	0	0
$53$	247.863	0.642389	0.321195	−	0.947013i	$-0.395916\pi$
$53$	247.863	0.642389	0.321195	+	0.947013i	$0.395916\pi$
$54$	0	0
$55$	−6.12603	−0.0150188
$56$	0	0
$57$	14.8848	0.0345884
$58$	0	0
$59$	−321.574	−0.709582	−0.354791	−	0.934946i	$-0.615448\pi$
$59$	−321.574	−0.709582	−0.354791	+	0.934946i	$0.615448\pi$
$60$	0	0
$61$	443.802	0.931525	0.465763	−	0.884910i	$-0.345780\pi$
$61$	443.802	0.931525	0.465763	+	0.884910i	$0.345780\pi$
$62$	0	0
$63$	395.726	0.791378
$64$	0	0
$65$	−50.1518	−0.0957010
$66$	0	0
$67$	−17.8872	−0.0326159	−0.0163080	−	0.999867i	$-0.505191\pi$
$67$	−17.8872	−0.0326159	−0.0163080	+	0.999867i	$0.505191\pi$
$68$	0	0
$69$	287.323	0.501299
$70$	0	0
$71$	−779.552	−1.30304	−0.651519	−	0.758632i	$-0.725868\pi$
$71$	−779.552	−1.30304	−0.651519	+	0.758632i	$0.725868\pi$
$72$	0	0
$73$	575.418	0.922569	0.461284	−	0.887252i	$-0.347389\pi$
$73$	575.418	0.922569	0.461284	+	0.887252i	$0.347389\pi$
$74$	0	0
$75$	−187.324	−0.288405
$76$	0	0
$77$	−125.861	−0.186275
$78$	0	0
$79$	490.715	0.698858	0.349429	−	0.936963i	$-0.386376\pi$
$79$	490.715	0.698858	0.349429	+	0.936963i	$0.386376\pi$
$80$	0	0
$81$	550.456	0.755083
$82$	0	0
$83$	1282.66	1.69626	0.848131	−	0.529787i	$-0.177728\pi$
$83$	1282.66	1.69626	0.848131	+	0.529787i	$0.177728\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−316.318	−0.389803
$88$	0	0
$89$	187.071	0.222803	0.111402	−	0.993775i	$-0.464466\pi$
$89$	187.071	0.222803	0.111402	+	0.993775i	$0.464466\pi$
$90$	0	0
$91$	−1030.38	−1.18696
$92$	0	0
$93$	−10.3342	−0.0115227
$94$	0	0
$95$	−7.69778	−0.00831342
$96$	0	0
$97$	58.9763	0.0617334	0.0308667	−	0.999524i	$-0.490173\pi$
$97$	58.9763	0.0617334	0.0308667	+	0.999524i	$0.490173\pi$
$98$	0	0
$99$	−194.547	−0.197502
$100$	0	0
$101$	−462.190	−0.455343	−0.227671	−	0.973738i	$-0.573111\pi$
$101$	−462.190	−0.455343	−0.227671	+	0.973738i	$0.573111\pi$
$102$	0	0
$103$	−1983.00	−1.89700	−0.948499	−	0.316781i	$-0.897398\pi$
$103$	−1983.00	−1.89700	−0.948499	+	0.316781i	$0.897398\pi$
$104$	0	0
$105$	18.7649	0.0174406
$106$	0	0
$107$	−567.158	−0.512422	−0.256211	−	0.966621i	$-0.582474\pi$
$107$	−567.158	−0.512422	−0.256211	+	0.966621i	$0.582474\pi$
$108$	0	0
$109$	−1302.92	−1.14492	−0.572462	−	0.819931i	$-0.694012\pi$
$109$	−1302.92	−1.14492	−0.572462	+	0.819931i	$0.694012\pi$
$110$	0	0
$111$	−139.235	−0.119060
$112$	0	0
$113$	−225.359	−0.187610	−0.0938052	−	0.995591i	$-0.529903\pi$
$113$	−225.359	−0.187610	−0.0938052	+	0.995591i	$0.529903\pi$
$114$	0	0
$115$	−148.591	−0.120489
$116$	0	0
$117$	−1592.69	−1.25850
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1269.12	−0.953512
$122$	0	0
$123$	480.114	0.351955
$124$	0	0
$125$	194.224	0.138976
$126$	0	0
$127$	343.077	0.239710	0.119855	−	0.992791i	$-0.461757\pi$
$127$	343.077	0.239710	0.119855	+	0.992791i	$0.461757\pi$
$128$	0	0
$129$	342.878	0.234021
$130$	0	0
$131$	−2257.87	−1.50588	−0.752942	−	0.658087i	$-0.771366\pi$
$131$	−2257.87	−1.50588	−0.752942	+	0.658087i	$0.771366\pi$
$132$	0	0
$133$	−158.153	−0.103110
$134$	0	0
$135$	60.6704	0.0386791
$136$	0	0
$137$	−2058.38	−1.28364	−0.641822	−	0.766853i	$-0.721821\pi$
$137$	−2058.38	−1.28364	−0.641822	+	0.766853i	$0.721821\pi$
$138$	0	0
$139$	1454.67	0.887650	0.443825	−	0.896114i	$-0.353621\pi$
$139$	1454.67	0.887650	0.443825	+	0.896114i	$0.353621\pi$
$140$	0	0
$141$	−362.746	−0.216657
$142$	0	0
$143$	506.557	0.296227
$144$	0	0
$145$	163.586	0.0936901
$146$	0	0
$147$	−130.994	−0.0734980
$148$	0	0
$149$	4.50965	0.00247949	0.00123975	−	0.999999i	$-0.499605\pi$
$149$	4.50965	0.00247949	0.00123975	+	0.999999i	$0.499605\pi$
$150$	0	0
$151$	−3134.62	−1.68935	−0.844674	−	0.535280i	$-0.820206\pi$
$151$	−3134.62	−1.68935	−0.844674	+	0.535280i	$0.820206\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.34441	0.00276951
$156$	0	0
$157$	−376.411	−0.191343	−0.0956716	−	0.995413i	$-0.530500\pi$
$157$	−376.411	−0.191343	−0.0956716	+	0.995413i	$0.530500\pi$
$158$	0	0
$159$	373.258	0.186171
$160$	0	0
$161$	−3052.85	−1.49440
$162$	0	0
$163$	3277.65	1.57500	0.787501	−	0.616314i	$-0.211375\pi$
$163$	3277.65	1.57500	0.787501	+	0.616314i	$0.211375\pi$
$164$	0	0
$165$	−9.22519	−0.00435261
$166$	0	0
$167$	−1769.70	−0.820022	−0.410011	−	0.912081i	$-0.634475\pi$
$167$	−1769.70	−0.820022	−0.410011	+	0.912081i	$0.634475\pi$
$168$	0	0
$169$	1950.02	0.887583
$170$	0	0
$171$	−244.461	−0.109324
$172$	0	0
$173$	−2407.97	−1.05823	−0.529117	−	0.848549i	$-0.677477\pi$
$173$	−2407.97	−1.05823	−0.529117	+	0.848549i	$0.677477\pi$
$174$	0	0
$175$	1990.35	0.859749
$176$	0	0
$177$	−484.259	−0.205645
$178$	0	0
$179$	−3072.78	−1.28308	−0.641538	−	0.767091i	$-0.721703\pi$
$179$	−3072.78	−1.28308	−0.641538	+	0.767091i	$0.721703\pi$
$180$	0	0
$181$	147.021	0.0603757	0.0301879	−	0.999544i	$-0.490389\pi$
$181$	147.021	0.0603757	0.0301879	+	0.999544i	$0.490389\pi$
$182$	0	0
$183$	668.323	0.269966
$184$	0	0
$185$	72.0064	0.0286163
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1246.49	0.479729
$190$	0	0
$191$	−383.373	−0.145235	−0.0726175	−	0.997360i	$-0.523135\pi$
$191$	−383.373	−0.145235	−0.0726175	+	0.997360i	$0.523135\pi$
$192$	0	0
$193$	−1151.44	−0.429442	−0.214721	−	0.976675i	$-0.568884\pi$
$193$	−1151.44	−0.429442	−0.214721	+	0.976675i	$0.568884\pi$
$194$	0	0
$195$	−75.5237	−0.0277352
$196$	0	0
$197$	686.045	0.248115	0.124058	−	0.992275i	$-0.460409\pi$
$197$	686.045	0.248115	0.124058	+	0.992275i	$0.460409\pi$
$198$	0	0
$199$	826.830	0.294535	0.147267	−	0.989097i	$-0.452952\pi$
$199$	826.830	0.294535	0.147267	+	0.989097i	$0.452952\pi$
$200$	0	0
$201$	−26.9363	−0.00945245
$202$	0	0
$203$	3360.92	1.16202
$204$	0	0
$205$	−248.294	−0.0845933
$206$	0	0
$207$	−4718.87	−1.58446
$208$	0	0
$209$	77.7512	0.0257328
$210$	0	0
$211$	−3576.96	−1.16705	−0.583526	−	0.812095i	$-0.698327\pi$
$211$	−3576.96	−1.16705	−0.583526	+	0.812095i	$0.698327\pi$
$212$	0	0
$213$	−1173.93	−0.377635
$214$	0	0
$215$	−177.322	−0.0562477
$216$	0	0
$217$	109.803	0.0343497
$218$	0	0
$219$	866.522	0.267371
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−3158.03	−0.948328	−0.474164	−	0.880437i	$-0.657250\pi$
$223$	−3158.03	−0.948328	−0.474164	+	0.880437i	$0.657250\pi$
$224$	0	0
$225$	3076.53	0.911565
$226$	0	0
$227$	1382.59	0.404253	0.202126	−	0.979359i	$-0.435215\pi$
$227$	1382.59	0.404253	0.202126	+	0.979359i	$0.435215\pi$
$228$	0	0
$229$	−3423.08	−0.987789	−0.493894	−	0.869522i	$-0.664427\pi$
$229$	−3423.08	−0.987789	−0.493894	+	0.869522i	$0.664427\pi$
$230$	0	0
$231$	−189.534	−0.0539846
$232$	0	0
$233$	−3707.12	−1.04232	−0.521162	−	0.853457i	$-0.674501\pi$
$233$	−3707.12	−1.04232	−0.521162	+	0.853457i	$0.674501\pi$
$234$	0	0
$235$	187.596	0.0520742
$236$	0	0
$237$	738.969	0.202537
$238$	0	0
$239$	−3375.35	−0.913528	−0.456764	−	0.889588i	$-0.650992\pi$
$239$	−3375.35	−0.913528	−0.456764	+	0.889588i	$0.650992\pi$
$240$	0	0
$241$	−6086.21	−1.62675	−0.813376	−	0.581738i	$-0.802373\pi$
$241$	−6086.21	−1.62675	−0.813376	+	0.581738i	$0.802373\pi$
$242$	0	0
$243$	2932.33	0.774112
$244$	0	0
$245$	67.7444	0.0176654
$246$	0	0
$247$	636.524	0.163972
$248$	0	0
$249$	1931.55	0.491595
$250$	0	0
$251$	−647.628	−0.162860	−0.0814301	−	0.996679i	$-0.525949\pi$
$251$	−647.628	−0.162860	−0.0814301	+	0.996679i	$0.525949\pi$
$252$	0	0
$253$	1500.84	0.372952
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4665.47	1.13239	0.566194	−	0.824272i	$-0.308415\pi$
$257$	4665.47	1.13239	0.566194	+	0.824272i	$0.308415\pi$
$258$	0	0
$259$	1479.39	0.354923
$260$	0	0
$261$	5195.06	1.23205
$262$	0	0
$263$	−4412.25	−1.03449	−0.517245	−	0.855837i	$-0.673043\pi$
$263$	−4412.25	−1.03449	−0.517245	+	0.855837i	$0.673043\pi$
$264$	0	0
$265$	−193.033	−0.0447468
$266$	0	0
$267$	281.711	0.0645709
$268$	0	0
$269$	6716.55	1.52236	0.761181	−	0.648540i	$-0.224620\pi$
$269$	6716.55	1.52236	0.761181	+	0.648540i	$0.224620\pi$
$270$	0	0
$271$	−8108.32	−1.81751	−0.908755	−	0.417330i	$-0.862966\pi$
$271$	−8108.32	−1.81751	−0.908755	+	0.417330i	$0.862966\pi$
$272$	0	0
$273$	−1551.66	−0.343995
$274$	0	0
$275$	−978.493	−0.214565
$276$	0	0
$277$	6535.22	1.41756	0.708779	−	0.705431i	$-0.249247\pi$
$277$	6535.22	1.41756	0.708779	+	0.705431i	$0.249247\pi$
$278$	0	0
$279$	169.725	0.0364199
$280$	0	0
$281$	703.922	0.149439	0.0747196	−	0.997205i	$-0.476194\pi$
$281$	703.922	0.149439	0.0747196	+	0.997205i	$0.476194\pi$
$282$	0	0
$283$	6772.25	1.42250	0.711252	−	0.702937i	$-0.248129\pi$
$283$	6772.25	1.42250	0.711252	+	0.702937i	$0.248129\pi$
$284$	0	0
$285$	−11.5921	−0.00240932
$286$	0	0
$287$	−5101.28	−1.04919
$288$	0	0
$289$	0	0
$290$	0	0
$291$	88.8126	0.0178910
$292$	0	0
$293$	8418.78	1.67860	0.839301	−	0.543668i	$-0.182965\pi$
$293$	8418.78	1.67860	0.839301	+	0.543668i	$0.182965\pi$
$294$	0	0
$295$	250.438	0.0494273
$296$	0	0
$297$	−612.799	−0.119725
$298$	0	0
$299$	12286.9	2.37649
$300$	0	0
$301$	−3643.13	−0.697629
$302$	0	0
$303$	−696.013	−0.131963
$304$	0	0
$305$	−345.627	−0.0648871
$306$	0	0
$307$	−8695.79	−1.61660	−0.808298	−	0.588773i	$-0.799611\pi$
$307$	−8695.79	−1.61660	−0.808298	+	0.588773i	$0.799611\pi$
$308$	0	0
$309$	−2986.20	−0.549771
$310$	0	0
$311$	2961.27	0.539930	0.269965	−	0.962870i	$-0.412988\pi$
$311$	2961.27	0.539930	0.269965	+	0.962870i	$0.412988\pi$
$312$	0	0
$313$	−619.356	−0.111847	−0.0559234	−	0.998435i	$-0.517810\pi$
$313$	−619.356	−0.111847	−0.0559234	+	0.998435i	$0.517810\pi$
$314$	0	0
$315$	−308.186	−0.0551249
$316$	0	0
$317$	2256.90	0.399874	0.199937	−	0.979809i	$-0.435926\pi$
$317$	2256.90	0.399874	0.199937	+	0.979809i	$0.435926\pi$
$318$	0	0
$319$	−1652.29	−0.290002
$320$	0	0
$321$	−854.084	−0.148506
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−8010.61	−1.36723
$326$	0	0
$327$	−1962.06	−0.331811
$328$	0	0
$329$	3854.22	0.645867
$330$	0	0
$331$	−6697.10	−1.11210	−0.556051	−	0.831148i	$-0.687684\pi$
$331$	−6697.10	−1.11210	−0.556051	+	0.831148i	$0.687684\pi$
$332$	0	0
$333$	2286.74	0.376314
$334$	0	0
$335$	13.9303	0.00227192
$336$	0	0
$337$	−5869.78	−0.948805	−0.474403	−	0.880308i	$-0.657336\pi$
$337$	−5869.78	−0.948805	−0.474403	+	0.880308i	$0.657336\pi$
$338$	0	0
$339$	−339.368	−0.0543715
$340$	0	0
$341$	−53.9811	−0.00857255
$342$	0	0
$343$	6879.97	1.08304
$344$	0	0
$345$	−223.763	−0.0349189
$346$	0	0
$347$	−11456.8	−1.77244	−0.886218	−	0.463269i	$-0.846677\pi$
$347$	−11456.8	−1.77244	−0.886218	+	0.463269i	$0.846677\pi$
$348$	0	0
$349$	−8598.99	−1.31889	−0.659446	−	0.751752i	$-0.729209\pi$
$349$	−8598.99	−1.31889	−0.659446	+	0.751752i	$0.729209\pi$
$350$	0	0
$351$	−5016.79	−0.762896
$352$	0	0
$353$	1740.63	0.262449	0.131224	−	0.991353i	$-0.458109\pi$
$353$	1740.63	0.262449	0.131224	+	0.991353i	$0.458109\pi$
$354$	0	0
$355$	607.105	0.0907655
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5571.88	−0.819143	−0.409572	−	0.912278i	$-0.634322\pi$
$359$	−5571.88	−0.819143	−0.409572	+	0.912278i	$0.634322\pi$
$360$	0	0
$361$	−6761.30	−0.985756
$362$	0	0
$363$	−1911.18	−0.276338
$364$	0	0
$365$	−448.128	−0.0642632
$366$	0	0
$367$	8403.08	1.19520	0.597598	−	0.801796i	$-0.296122\pi$
$367$	8403.08	1.19520	0.597598	+	0.801796i	$0.296122\pi$
$368$	0	0
$369$	−7885.18	−1.11243
$370$	0	0
$371$	−3965.91	−0.554986
$372$	0	0
$373$	12564.3	1.74411	0.872055	−	0.489408i	$-0.162787\pi$
$373$	12564.3	1.74411	0.872055	+	0.489408i	$0.162787\pi$
$374$	0	0
$375$	292.483	0.0402767
$376$	0	0
$377$	−13526.8	−1.84792
$378$	0	0
$379$	8928.35	1.21008	0.605038	−	0.796197i	$-0.293158\pi$
$379$	8928.35	1.21008	0.605038	+	0.796197i	$0.293158\pi$
$380$	0	0
$381$	516.640	0.0694705
$382$	0	0
$383$	−12369.5	−1.65026	−0.825132	−	0.564940i	$-0.808899\pi$
$383$	−12369.5	−1.65026	−0.825132	+	0.564940i	$0.808899\pi$
$384$	0	0
$385$	98.0189	0.0129753
$386$	0	0
$387$	−5631.28	−0.739675
$388$	0	0
$389$	−8052.58	−1.04957	−0.524784	−	0.851235i	$-0.675854\pi$
$389$	−8052.58	−1.04957	−0.524784	+	0.851235i	$0.675854\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−3400.13	−0.436422
$394$	0	0
$395$	−382.162	−0.0486802
$396$	0	0
$397$	6123.59	0.774141	0.387070	−	0.922050i	$-0.373487\pi$
$397$	6123.59	0.774141	0.387070	+	0.922050i	$0.373487\pi$
$398$	0	0
$399$	−238.163	−0.0298824
$400$	0	0
$401$	2617.67	0.325985	0.162993	−	0.986627i	$-0.447885\pi$
$401$	2617.67	0.325985	0.162993	+	0.986627i	$0.447885\pi$
$402$	0	0
$403$	−441.926	−0.0546251
$404$	0	0
$405$	−428.688	−0.0525967
$406$	0	0
$407$	−727.299	−0.0885771
$408$	0	0
$409$	−5790.18	−0.700015	−0.350008	−	0.936747i	$-0.613821\pi$
$409$	−5790.18	−0.700015	−0.350008	+	0.936747i	$0.613821\pi$
$410$	0	0
$411$	−3099.72	−0.372014
$412$	0	0
$413$	5145.31	0.613037
$414$	0	0
$415$	−998.915	−0.118156
$416$	0	0
$417$	2190.59	0.257251
$418$	0	0
$419$	−6687.11	−0.779682	−0.389841	−	0.920882i	$-0.627470\pi$
$419$	−6687.11	−0.779682	−0.389841	+	0.920882i	$0.627470\pi$
$420$	0	0
$421$	−327.455	−0.0379078	−0.0189539	−	0.999820i	$-0.506034\pi$
$421$	−327.455	−0.0379078	−0.0189539	+	0.999820i	$0.506034\pi$
$422$	0	0
$423$	5957.57	0.684792
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−7101.01	−0.804783
$428$	0	0
$429$	762.825	0.0858497
$430$	0	0
$431$	−9415.74	−1.05230	−0.526149	−	0.850393i	$-0.676365\pi$
$431$	−9415.74	−1.05230	−0.526149	+	0.850393i	$0.676365\pi$
$432$	0	0
$433$	−14393.2	−1.59744	−0.798722	−	0.601700i	$-0.794490\pi$
$433$	−14393.2	−1.59744	−0.798722	+	0.601700i	$0.794490\pi$
$434$	0	0
$435$	246.344	0.0271524
$436$	0	0
$437$	1885.91	0.206442
$438$	0	0
$439$	1827.38	0.198670	0.0993348	−	0.995054i	$-0.468329\pi$
$439$	1827.38	0.198670	0.0993348	+	0.995054i	$0.468329\pi$
$440$	0	0
$441$	2151.39	0.232306
$442$	0	0
$443$	−8825.03	−0.946478	−0.473239	−	0.880934i	$-0.656915\pi$
$443$	−8825.03	−0.946478	−0.473239	+	0.880934i	$0.656915\pi$
$444$	0	0
$445$	−145.689	−0.0155198
$446$	0	0
$447$	6.79109	0.000718585 0
$448$	0	0
$449$	−2789.29	−0.293173	−0.146586	−	0.989198i	$-0.546829\pi$
$449$	−2789.29	−0.293173	−0.146586	+	0.989198i	$0.546829\pi$
$450$	0	0
$451$	2507.89	0.261844
$452$	0	0
$453$	−4720.43	−0.489592
$454$	0	0
$455$	802.449	0.0826800
$456$	0	0
$457$	−14264.3	−1.46008	−0.730040	−	0.683404i	$-0.760499\pi$
$457$	−14264.3	−1.46008	−0.730040	+	0.683404i	$0.760499\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10219.4	1.03247	0.516233	−	0.856448i	$-0.327334\pi$
$461$	10219.4	1.03247	0.516233	+	0.856448i	$0.327334\pi$
$462$	0	0
$463$	12707.3	1.27550	0.637750	−	0.770244i	$-0.279865\pi$
$463$	12707.3	1.27550	0.637750	+	0.770244i	$0.279865\pi$
$464$	0	0
$465$	8.04816	0.000802634 0
$466$	0	0
$467$	10029.1	0.993773	0.496886	−	0.867816i	$-0.334477\pi$
$467$	10029.1	0.993773	0.496886	+	0.867816i	$0.334477\pi$
$468$	0	0
$469$	286.202	0.0281782
$470$	0	0
$471$	−566.838	−0.0554533
$472$	0	0
$473$	1791.03	0.174105
$474$	0	0
$475$	−1229.54	−0.118769
$476$	0	0
$477$	−6130.22	−0.588435
$478$	0	0
$479$	175.559	0.0167464	0.00837319	−	0.999965i	$-0.497335\pi$
$479$	175.559	0.0167464	0.00837319	+	0.999965i	$0.497335\pi$
$480$	0	0
$481$	−5954.16	−0.564421
$482$	0	0
$483$	−4597.29	−0.433093
$484$	0	0
$485$	−45.9300	−0.00430015
$486$	0	0
$487$	16898.9	1.57241	0.786204	−	0.617968i	$-0.212044\pi$
$487$	16898.9	1.57241	0.786204	+	0.617968i	$0.212044\pi$
$488$	0	0
$489$	4935.82	0.456453
$490$	0	0
$491$	−14377.8	−1.32151	−0.660754	−	0.750602i	$-0.729763\pi$
$491$	−14377.8	−1.32151	−0.660754	+	0.750602i	$0.729763\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	151.510	0.0137574
$496$	0	0
$497$	12473.1	1.12575
$498$	0	0
$499$	−8822.48	−0.791480	−0.395740	−	0.918363i	$-0.629512\pi$
$499$	−8822.48	−0.791480	−0.395740	+	0.918363i	$0.629512\pi$
$500$	0	0
$501$	−2665.00	−0.237651
$502$	0	0
$503$	7939.11	0.703753	0.351876	−	0.936047i	$-0.385544\pi$
$503$	7939.11	0.703753	0.351876	+	0.936047i	$0.385544\pi$
$504$	0	0
$505$	359.947	0.0317177
$506$	0	0
$507$	2936.54	0.257231
$508$	0	0
$509$	−1096.19	−0.0954575	−0.0477287	−	0.998860i	$-0.515198\pi$
$509$	−1096.19	−0.0954575	−0.0477287	+	0.998860i	$0.515198\pi$
$510$	0	0
$511$	−9206.91	−0.797045
$512$	0	0
$513$	−770.025	−0.0662718
$514$	0	0
$515$	1544.33	0.132139
$516$	0	0
$517$	−1894.81	−0.161187
$518$	0	0
$519$	−3626.16	−0.306688
$520$	0	0
$521$	8637.51	0.726326	0.363163	−	0.931726i	$-0.381697\pi$
$521$	8637.51	0.726326	0.363163	+	0.931726i	$0.381697\pi$
$522$	0	0
$523$	−1232.00	−0.103005	−0.0515027	−	0.998673i	$-0.516401\pi$
$523$	−1232.00	−0.103005	−0.0515027	+	0.998673i	$0.516401\pi$
$524$	0	0
$525$	2997.27	0.249165
$526$	0	0
$527$	0	0
$528$	0	0
$529$	24236.9	1.99202
$530$	0	0
$531$	7953.25	0.649984
$532$	0	0
$533$	20531.3	1.66850
$534$	0	0
$535$	441.695	0.0356937
$536$	0	0
$537$	−4627.31	−0.371849
$538$	0	0
$539$	−684.250	−0.0546804
$540$	0	0
$541$	19625.2	1.55962	0.779808	−	0.626019i	$-0.215317\pi$
$541$	19625.2	1.55962	0.779808	+	0.626019i	$0.215317\pi$
$542$	0	0
$543$	221.400	0.0174975
$544$	0	0
$545$	1014.69	0.0797517
$546$	0	0
$547$	−8189.05	−0.640107	−0.320054	−	0.947399i	$-0.603701\pi$
$547$	−8189.05	−0.640107	−0.320054	+	0.947399i	$0.603701\pi$
$548$	0	0
$549$	−10976.2	−0.853286
$550$	0	0
$551$	−2076.22	−0.160526
$552$	0	0
$553$	−7851.64	−0.603772
$554$	0	0
$555$	108.435	0.00829332
$556$	0	0
$557$	5282.15	0.401817	0.200908	−	0.979610i	$-0.435611\pi$
$557$	5282.15	0.401817	0.200908	+	0.979610i	$0.435611\pi$
$558$	0	0
$559$	14662.6	1.10941
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21573.9	−1.61498	−0.807489	−	0.589882i	$-0.799174\pi$
$563$	−21573.9	−1.61498	−0.807489	+	0.589882i	$0.799174\pi$
$564$	0	0
$565$	175.506	0.0130683
$566$	0	0
$567$	−8807.51	−0.652347
$568$	0	0
$569$	−10265.2	−0.756306	−0.378153	−	0.925743i	$-0.623441\pi$
$569$	−10265.2	−0.756306	−0.378153	+	0.925743i	$0.623441\pi$
$570$	0	0
$571$	−14767.9	−1.08234	−0.541171	−	0.840912i	$-0.682019\pi$
$571$	−14767.9	−1.08234	−0.541171	+	0.840912i	$0.682019\pi$
$572$	0	0
$573$	−577.322	−0.0420907
$574$	0	0
$575$	−23734.0	−1.72135
$576$	0	0
$577$	20433.6	1.47429	0.737143	−	0.675737i	$-0.236174\pi$
$577$	20433.6	1.47429	0.737143	+	0.675737i	$0.236174\pi$
$578$	0	0
$579$	−1733.95	−0.124457
$580$	0	0
$581$	−20523.0	−1.46547
$582$	0	0
$583$	1949.72	0.138506
$584$	0	0
$585$	1240.37	0.0876631
$586$	0	0
$587$	20352.5	1.43107	0.715534	−	0.698578i	$-0.246183\pi$
$587$	20352.5	1.43107	0.715534	+	0.698578i	$0.246183\pi$
$588$	0	0
$589$	−67.8310	−0.00474521
$590$	0	0
$591$	1033.12	0.0719065
$592$	0	0
$593$	16460.0	1.13985	0.569926	−	0.821696i	$-0.306972\pi$
$593$	16460.0	1.13985	0.569926	+	0.821696i	$0.306972\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1245.12	0.0853594
$598$	0	0
$599$	−9181.98	−0.626320	−0.313160	−	0.949700i	$-0.601388\pi$
$599$	−9181.98	−0.626320	−0.313160	+	0.949700i	$0.601388\pi$
$600$	0	0
$601$	4362.20	0.296070	0.148035	−	0.988982i	$-0.452705\pi$
$601$	4362.20	0.296070	0.148035	+	0.988982i	$0.452705\pi$
$602$	0	0
$603$	442.391	0.0298765
$604$	0	0
$605$	988.377	0.0664186
$606$	0	0
$607$	−17237.2	−1.15262	−0.576308	−	0.817232i	$-0.695507\pi$
$607$	−17237.2	−1.15262	−0.576308	+	0.817232i	$0.695507\pi$
$608$	0	0
$609$	5061.21	0.336766
$610$	0	0
$611$	−15512.2	−1.02710
$612$	0	0
$613$	18427.6	1.21416	0.607082	−	0.794639i	$-0.292340\pi$
$613$	18427.6	1.21416	0.607082	+	0.794639i	$0.292340\pi$
$614$	0	0
$615$	−373.907	−0.0245161
$616$	0	0
$617$	12221.8	0.797455	0.398727	−	0.917070i	$-0.369452\pi$
$617$	12221.8	0.797455	0.398727	+	0.917070i	$0.369452\pi$
$618$	0	0
$619$	178.433	0.0115861	0.00579306	−	0.999983i	$-0.498156\pi$
$619$	178.433	0.0115861	0.00579306	+	0.999983i	$0.498156\pi$
$620$	0	0
$621$	−14863.9	−0.960494
$622$	0	0
$623$	−2993.21	−0.192489
$624$	0	0
$625$	15397.9	0.985467
$626$	0	0
$627$	117.086	0.00745765
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−7037.77	−0.444008	−0.222004	−	0.975046i	$-0.571260\pi$
$631$	−7037.77	−0.444008	−0.222004	+	0.975046i	$0.571260\pi$
$632$	0	0
$633$	−5386.55	−0.338224
$634$	0	0
$635$	−267.184	−0.0166974
$636$	0	0
$637$	−5601.74	−0.348429
$638$	0	0
$639$	19280.1	1.19360
$640$	0	0
$641$	10862.3	0.669324	0.334662	−	0.942338i	$-0.391378\pi$
$641$	10862.3	0.669324	0.334662	+	0.942338i	$0.391378\pi$
$642$	0	0
$643$	−4229.21	−0.259384	−0.129692	−	0.991554i	$-0.541399\pi$
$643$	−4229.21	−0.259384	−0.129692	+	0.991554i	$0.541399\pi$
$644$	0	0
$645$	−267.029	−0.0163012
$646$	0	0
$647$	−5755.35	−0.349716	−0.174858	−	0.984594i	$-0.555947\pi$
$647$	−5755.35	−0.349716	−0.174858	+	0.984594i	$0.555947\pi$
$648$	0	0
$649$	−2529.54	−0.152994
$650$	0	0
$651$	165.352	0.00995491
$652$	0	0
$653$	12516.1	0.750066	0.375033	−	0.927011i	$-0.377631\pi$
$653$	12516.1	0.750066	0.375033	+	0.927011i	$0.377631\pi$
$654$	0	0
$655$	1758.40	0.104895
$656$	0	0
$657$	−14231.4	−0.845082
$658$	0	0
$659$	−19935.0	−1.17839	−0.589194	−	0.807992i	$-0.700555\pi$
$659$	−19935.0	−1.17839	−0.589194	+	0.807992i	$0.700555\pi$
$660$	0	0
$661$	−18848.1	−1.10909	−0.554543	−	0.832155i	$-0.687107\pi$
$661$	−18848.1	−1.10909	−0.554543	+	0.832155i	$0.687107\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	123.168	0.00718230
$666$	0	0
$667$	−40077.5	−2.32655
$668$	0	0
$669$	−4755.68	−0.274836
$670$	0	0
$671$	3491.00	0.200847
$672$	0	0
$673$	−878.969	−0.0503444	−0.0251722	−	0.999683i	$-0.508013\pi$
$673$	−878.969	−0.0503444	−0.0251722	+	0.999683i	$0.508013\pi$
$674$	0	0
$675$	9690.71	0.552586
$676$	0	0
$677$	−6276.02	−0.356288	−0.178144	−	0.984004i	$-0.557009\pi$
$677$	−6276.02	−0.356288	−0.178144	+	0.984004i	$0.557009\pi$
$678$	0	0
$679$	−943.645	−0.0533340
$680$	0	0
$681$	2082.04	0.117157
$682$	0	0
$683$	29981.3	1.67965	0.839826	−	0.542856i	$-0.182657\pi$
$683$	29981.3	1.67965	0.839826	+	0.542856i	$0.182657\pi$
$684$	0	0
$685$	1603.04	0.0894146
$686$	0	0
$687$	−5154.83	−0.286272
$688$	0	0
$689$	15961.7	0.882574
$690$	0	0
$691$	−22801.4	−1.25529	−0.627646	−	0.778499i	$-0.715981\pi$
$691$	−22801.4	−1.25529	−0.627646	+	0.778499i	$0.715981\pi$
$692$	0	0
$693$	3112.83	0.170630
$694$	0	0
$695$	−1132.88	−0.0618309
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−5582.56	−0.302077
$700$	0	0
$701$	−8900.26	−0.479541	−0.239770	−	0.970830i	$-0.577072\pi$
$701$	−8900.26	−0.479541	−0.239770	+	0.970830i	$0.577072\pi$
$702$	0	0
$703$	−913.901	−0.0490305
$704$	0	0
$705$	282.502	0.0150917
$706$	0	0
$707$	7395.22	0.393389
$708$	0	0
$709$	−7823.08	−0.414389	−0.207195	−	0.978300i	$-0.566433\pi$
$709$	−7823.08	−0.414389	−0.207195	+	0.978300i	$0.566433\pi$
$710$	0	0
$711$	−12136.5	−0.640160
$712$	0	0
$713$	−1309.35	−0.0687735
$714$	0	0
$715$	−394.500	−0.0206342
$716$	0	0
$717$	−5082.95	−0.264751
$718$	0	0
$719$	8257.73	0.428319	0.214159	−	0.976799i	$-0.431299\pi$
$719$	8257.73	0.428319	0.214159	+	0.976799i	$0.431299\pi$
$720$	0	0
$721$	31728.8	1.63889
$722$	0	0
$723$	−9165.23	−0.471451
$724$	0	0
$725$	26129.1	1.33850
$726$	0	0
$727$	−4729.65	−0.241283	−0.120642	−	0.992696i	$-0.538495\pi$
$727$	−4729.65	−0.241283	−0.120642	+	0.992696i	$0.538495\pi$
$728$	0	0
$729$	−10446.5	−0.530737
$730$	0	0
$731$	0	0
$732$	0	0
$733$	33384.2	1.68223	0.841114	−	0.540857i	$-0.181900\pi$
$733$	33384.2	1.68223	0.841114	+	0.540857i	$0.181900\pi$
$734$	0	0
$735$	102.016	0.00511964
$736$	0	0
$737$	−140.703	−0.00703236
$738$	0	0
$739$	−6988.23	−0.347857	−0.173928	−	0.984758i	$-0.555646\pi$
$739$	−6988.23	−0.347857	−0.173928	+	0.984758i	$0.555646\pi$
$740$	0	0
$741$	958.542	0.0475208
$742$	0	0
$743$	−11587.6	−0.572152	−0.286076	−	0.958207i	$-0.592351\pi$
$743$	−11587.6	−0.572152	−0.286076	+	0.958207i	$0.592351\pi$
$744$	0	0
$745$	−3.51205	−0.000172714 0
$746$	0	0
$747$	−31723.0	−1.55379
$748$	0	0
$749$	9074.75	0.442702
$750$	0	0
$751$	28272.9	1.37376	0.686879	−	0.726772i	$-0.258980\pi$
$751$	28272.9	1.37376	0.686879	+	0.726772i	$0.258980\pi$
$752$	0	0
$753$	−975.264	−0.0471987
$754$	0	0
$755$	2441.20	0.117675
$756$	0	0
$757$	35534.2	1.70609	0.853047	−	0.521834i	$-0.174752\pi$
$757$	35534.2	1.70609	0.853047	+	0.521834i	$0.174752\pi$
$758$	0	0
$759$	2260.12	0.108086
$760$	0	0
$761$	−12245.5	−0.583312	−0.291656	−	0.956523i	$-0.594206\pi$
$761$	−12245.5	−0.583312	−0.291656	+	0.956523i	$0.594206\pi$
$762$	0	0
$763$	20847.2	0.989146
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−20708.5	−0.974890
$768$	0	0
$769$	25923.6	1.21564	0.607822	−	0.794073i	$-0.292043\pi$
$769$	25923.6	1.21564	0.607822	+	0.794073i	$0.292043\pi$
$770$	0	0
$771$	7025.73	0.328178
$772$	0	0
$773$	20813.4	0.968444	0.484222	−	0.874945i	$-0.339103\pi$
$773$	20813.4	0.968444	0.484222	+	0.874945i	$0.339103\pi$
$774$	0	0
$775$	853.649	0.0395664
$776$	0	0
$777$	2227.82	0.102861
$778$	0	0
$779$	3151.33	0.144940
$780$	0	0
$781$	−6132.04	−0.280950
$782$	0	0
$783$	16363.8	0.746866
$784$	0	0
$785$	293.144	0.0133284
$786$	0	0
$787$	−10856.0	−0.491710	−0.245855	−	0.969307i	$-0.579069\pi$
$787$	−10856.0	−0.491710	−0.245855	+	0.969307i	$0.579069\pi$
$788$	0	0
$789$	−6644.42	−0.299807
$790$	0	0
$791$	3605.83	0.162084
$792$	0	0
$793$	28579.7	1.27982
$794$	0	0
$795$	−290.688	−0.0129681
$796$	0	0
$797$	35021.7	1.55650	0.778251	−	0.627953i	$-0.216107\pi$
$797$	35021.7	1.55650	0.778251	+	0.627953i	$0.216107\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−4626.69	−0.204090
$802$	0	0
$803$	4526.30	0.198916
$804$	0	0
$805$	2377.52	0.104095
$806$	0	0
$807$	10114.5	0.441197
$808$	0	0
$809$	−41252.2	−1.79277	−0.896385	−	0.443277i	$-0.853816\pi$
$809$	−41252.2	−1.79277	−0.896385	+	0.443277i	$0.853816\pi$
$810$	0	0
$811$	27277.0	1.18104	0.590520	−	0.807023i	$-0.298923\pi$
$811$	27277.0	1.18104	0.590520	+	0.807023i	$0.298923\pi$
$812$	0	0
$813$	−12210.3	−0.526734
$814$	0	0
$815$	−2552.59	−0.109710
$816$	0	0
$817$	2250.56	0.0963734
$818$	0	0
$819$	25483.7	1.08727
$820$	0	0
$821$	−33012.5	−1.40334	−0.701671	−	0.712501i	$-0.747562\pi$
$821$	−33012.5	−1.40334	−0.701671	+	0.712501i	$0.747562\pi$
$822$	0	0
$823$	−25627.7	−1.08545	−0.542726	−	0.839910i	$-0.682608\pi$
$823$	−25627.7	−1.08545	−0.542726	+	0.839910i	$0.682608\pi$
$824$	0	0
$825$	−1473.51	−0.0621833
$826$	0	0
$827$	−15915.7	−0.669217	−0.334608	−	0.942357i	$-0.608604\pi$
$827$	−15915.7	−0.669217	−0.334608	+	0.942357i	$0.608604\pi$
$828$	0	0
$829$	8519.63	0.356935	0.178467	−	0.983946i	$-0.442886\pi$
$829$	8519.63	0.356935	0.178467	+	0.983946i	$0.442886\pi$
$830$	0	0
$831$	9841.40	0.410824
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1378.22	0.0571201
$836$	0	0
$837$	534.613	0.0220776
$838$	0	0
$839$	−31495.9	−1.29602	−0.648008	−	0.761633i	$-0.724398\pi$
$839$	−31495.9	−1.29602	−0.648008	+	0.761633i	$0.724398\pi$
$840$	0	0
$841$	19732.9	0.809089
$842$	0	0
$843$	1060.04	0.0433091
$844$	0	0
$845$	−1518.65	−0.0618262
$846$	0	0
$847$	20306.5	0.823778
$848$	0	0
$849$	10198.3	0.412257
$850$	0	0
$851$	−17641.1	−0.710612
$852$	0	0
$853$	−13245.8	−0.531684	−0.265842	−	0.964017i	$-0.585650\pi$
$853$	−13245.8	−0.531684	−0.265842	+	0.964017i	$0.585650\pi$
$854$	0	0
$855$	190.383	0.00761518
$856$	0	0
$857$	3533.90	0.140859	0.0704293	−	0.997517i	$-0.477563\pi$
$857$	3533.90	0.140859	0.0704293	+	0.997517i	$0.477563\pi$
$858$	0	0
$859$	−16355.9	−0.649659	−0.324829	−	0.945773i	$-0.605307\pi$
$859$	−16355.9	−0.649659	−0.324829	+	0.945773i	$0.605307\pi$
$860$	0	0
$861$	−7682.02	−0.304068
$862$	0	0
$863$	8944.44	0.352807	0.176403	−	0.984318i	$-0.443554\pi$
$863$	8944.44	0.352807	0.176403	+	0.984318i	$0.443554\pi$
$864$	0	0
$865$	1875.29	0.0737132
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3860.02	0.150682
$870$	0	0
$871$	−1151.89	−0.0448108
$872$	0	0
$873$	−1458.62	−0.0565484
$874$	0	0
$875$	−3107.67	−0.120067
$876$	0	0
$877$	−51522.2	−1.98379	−0.991894	−	0.127065i	$-0.959444\pi$
$877$	−51522.2	−1.98379	−0.991894	+	0.127065i	$0.959444\pi$
$878$	0	0
$879$	12677.8	0.486477
$880$	0	0
$881$	7933.87	0.303404	0.151702	−	0.988426i	$-0.451525\pi$
$881$	7933.87	0.303404	0.151702	+	0.988426i	$0.451525\pi$
$882$	0	0
$883$	−18241.0	−0.695197	−0.347599	−	0.937643i	$-0.613003\pi$
$883$	−18241.0	−0.695197	−0.347599	+	0.937643i	$0.613003\pi$
$884$	0	0
$885$	377.134	0.0143246
$886$	0	0
$887$	14806.6	0.560494	0.280247	−	0.959928i	$-0.409584\pi$
$887$	14806.6	0.560494	0.280247	+	0.959928i	$0.409584\pi$
$888$	0	0
$889$	−5489.37	−0.207095
$890$	0	0
$891$	4329.95	0.162804
$892$	0	0
$893$	−2380.96	−0.0892226
$894$	0	0
$895$	2393.04	0.0893750
$896$	0	0
$897$	18502.8	0.688731
$898$	0	0
$899$	1441.48	0.0534772
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−5486.19	−0.202181
$904$	0	0
$905$	−114.498	−0.00420558
$906$	0	0
$907$	−41365.6	−1.51436	−0.757180	−	0.653207i	$-0.773423\pi$
$907$	−41365.6	−1.51436	−0.757180	+	0.653207i	$0.773423\pi$
$908$	0	0
$909$	11431.0	0.417098
$910$	0	0
$911$	−22146.1	−0.805414	−0.402707	−	0.915329i	$-0.631931\pi$
$911$	−22146.1	−0.805414	−0.402707	+	0.915329i	$0.631931\pi$
$912$	0	0
$913$	10089.5	0.365733
$914$	0	0
$915$	−520.481	−0.0188050
$916$	0	0
$917$	36126.8	1.30099
$918$	0	0
$919$	47353.8	1.69974	0.849868	−	0.526995i	$-0.176681\pi$
$919$	47353.8	1.69974	0.849868	+	0.526995i	$0.176681\pi$
$920$	0	0
$921$	−13095.0	−0.468507
$922$	0	0
$923$	−50201.1	−1.79024
$924$	0	0
$925$	11501.4	0.408825
$926$	0	0
$927$	49044.1	1.73767
$928$	0	0
$929$	−53507.3	−1.88969	−0.944843	−	0.327524i	$-0.893786\pi$
$929$	−53507.3	−1.88969	−0.944843	+	0.327524i	$0.893786\pi$
$930$	0	0
$931$	−859.808	−0.0302675
$932$	0	0
$933$	4459.39	0.156478
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38169.1	1.33077	0.665384	−	0.746501i	$-0.268268\pi$
$937$	38169.1	1.33077	0.665384	+	0.746501i	$0.268268\pi$
$938$	0	0
$939$	−932.689	−0.0324144
$940$	0	0
$941$	16963.6	0.587670	0.293835	−	0.955856i	$-0.405068\pi$
$941$	16963.6	0.587670	0.293835	+	0.955856i	$0.405068\pi$
$942$	0	0
$943$	60830.6	2.10065
$944$	0	0
$945$	−970.751	−0.0334164
$946$	0	0
$947$	27933.1	0.958506	0.479253	−	0.877677i	$-0.340908\pi$
$947$	27933.1	0.958506	0.479253	+	0.877677i	$0.340908\pi$
$948$	0	0
$949$	37055.4	1.26751
$950$	0	0
$951$	3398.67	0.115888
$952$	0	0
$953$	55587.9	1.88947	0.944737	−	0.327829i	$-0.106317\pi$
$953$	55587.9	1.88947	0.944737	+	0.327829i	$0.106317\pi$
$954$	0	0
$955$	298.566	0.0101166
$956$	0	0
$957$	−2488.19	−0.0840458
$958$	0	0
$959$	32934.9	1.10899
$960$	0	0
$961$	−29743.9	−0.998419
$962$	0	0
$963$	14027.1	0.469384
$964$	0	0
$965$	896.724	0.0299136
$966$	0	0
$967$	−31365.7	−1.04307	−0.521537	−	0.853229i	$-0.674641\pi$
$967$	−31365.7	−1.04307	−0.521537	+	0.853229i	$0.674641\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28214.6	0.932492	0.466246	−	0.884655i	$-0.345606\pi$
$971$	28214.6	0.932492	0.466246	+	0.884655i	$0.345606\pi$
$972$	0	0
$973$	−23275.3	−0.766877
$974$	0	0
$975$	−12063.2	−0.396237
$976$	0	0
$977$	−5458.76	−0.178752	−0.0893762	−	0.995998i	$-0.528487\pi$
$977$	−5458.76	−0.178752	−0.0893762	+	0.995998i	$0.528487\pi$
$978$	0	0
$979$	1471.52	0.0480389
$980$	0	0
$981$	32224.0	1.04876
$982$	0	0
$983$	5613.00	0.182123	0.0910615	−	0.995845i	$-0.470974\pi$
$983$	5613.00	0.182123	0.0910615	+	0.995845i	$0.470974\pi$
$984$	0	0
$985$	−534.283	−0.0172829
$986$	0	0
$987$	5804.08	0.187179
$988$	0	0
$989$	43442.8	1.39676
$990$	0	0
$991$	37625.7	1.20608	0.603038	−	0.797713i	$-0.293957\pi$
$991$	37625.7	1.20608	0.603038	+	0.797713i	$0.293957\pi$
$992$	0	0
$993$	−10085.2	−0.322299
$994$	0	0
$995$	−643.924	−0.0205164
$996$	0	0
$997$	−19039.6	−0.604805	−0.302402	−	0.953180i	$-0.597789\pi$
$997$	−19039.6	−0.604805	−0.302402	+	0.953180i	$0.597789\pi$
$998$	0	0
$999$	7202.96	0.228120

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.m.1.8		14
17.8	even	8		136.4.k.a.81.4	✓	14
17.15	even	8		136.4.k.a.89.4	yes	14
17.16	even	2	inner	2312.4.a.m.1.7		14
68.15	odd	8		272.4.o.g.225.4		14
68.59	odd	8		272.4.o.g.81.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.k.a.81.4	✓	14	17.8	even	8
136.4.k.a.89.4	yes	14	17.15	even	8
272.4.o.g.81.4		14	68.59	odd	8
272.4.o.g.225.4		14	68.15	odd	8
2312.4.a.m.1.7		14	17.16	even	2	inner
2312.4.a.m.1.8		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)

\(f(q)\)	\(=\)	\(q+1.50590 q^{3} -0.778787 q^{5} -16.0004 q^{7} -24.7323 q^{9} +O(q^{10})\)	\(q+1.50590 q^{3} -0.778787 q^{5} -16.0004 q^{7} -24.7323 q^{9} +7.86611 q^{11} +64.3974 q^{13} -1.17278 q^{15} +9.88432 q^{19} -24.0950 q^{21} +190.798 q^{23} -124.393 q^{25} -77.9037 q^{27} -210.052 q^{29} -6.86249 q^{31} +11.8456 q^{33} +12.4609 q^{35} -92.4597 q^{37} +96.9761 q^{39} +318.822 q^{41} +227.690 q^{43} +19.2612 q^{45} -240.883 q^{47} -86.9871 q^{49} +247.863 q^{53} -6.12603 q^{55} +14.8848 q^{57} -321.574 q^{59} +443.802 q^{61} +395.726 q^{63} -50.1518 q^{65} -17.8872 q^{67} +287.323 q^{69} -779.552 q^{71} +575.418 q^{73} -187.324 q^{75} -125.861 q^{77} +490.715 q^{79} +550.456 q^{81} +1282.66 q^{83} -316.318 q^{87} +187.071 q^{89} -1030.38 q^{91} -10.3342 q^{93} -7.69778 q^{95} +58.9763 q^{97} -194.547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 190 q^{9}+O(q^{10}) \) 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

Representations

Groups

Database

Properties

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