LMFDB - Embedded newform 2312.4.a.i.1.6

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:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 87x^{4} + 46x^{3} + 1386x^{2} - 465x - 1857 $ x^6 - 3x^5 - 87x^4 + 46x^3 + 1386x^2 - 465*x - 1857
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.40297$ 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)

$f(q)$	$=$	$q+10.0775 q^{3} +10.2716 q^{5} -1.06795 q^{7} +74.5565 q^{9} +O(q^{10})$	$q+10.0775 q^{3} +10.2716 q^{5} -1.06795 q^{7} +74.5565 q^{9} -23.8200 q^{11} -64.6878 q^{13} +103.512 q^{15} +89.6338 q^{19} -10.7623 q^{21} +203.782 q^{23} -19.4945 q^{25} +479.252 q^{27} -14.3597 q^{29} +237.028 q^{31} -240.046 q^{33} -10.9696 q^{35} -21.8196 q^{37} -651.893 q^{39} -342.774 q^{41} +452.336 q^{43} +765.814 q^{45} -61.3020 q^{47} -341.859 q^{49} +268.750 q^{53} -244.669 q^{55} +903.287 q^{57} -221.572 q^{59} +69.3806 q^{61} -79.6230 q^{63} -664.447 q^{65} +922.984 q^{67} +2053.62 q^{69} +631.271 q^{71} -414.733 q^{73} -196.456 q^{75} +25.4387 q^{77} -1294.10 q^{79} +2816.65 q^{81} +910.400 q^{83} -144.710 q^{87} -27.2575 q^{89} +69.0837 q^{91} +2388.65 q^{93} +920.681 q^{95} +1316.01 q^{97} -1775.93 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{3} + 13 q^{5} + q^{7} + 63 q^{9}+O(q^{10}) $ 6 * q + q^3 + 13 * q^5 + q^7 + 63 * q^9	$ 6 q + q^{3} + 13 q^{5} + q^{7} + 63 q^{9} - 8 q^{11} - 33 q^{13} + 68 q^{15} + 75 q^{19} + 32 q^{21} + 155 q^{23} - 89 q^{25} + 322 q^{27} + 9 q^{29} + 374 q^{31} - 351 q^{33} + 13 q^{35} + 118 q^{37} - 369 q^{39} - 60 q^{41} + 230 q^{43} + 478 q^{45} + 276 q^{47} - 317 q^{49} + 569 q^{53} - 97 q^{55} + 769 q^{57} - 784 q^{59} - 28 q^{61} - 878 q^{63} + 828 q^{65} + 238 q^{67} + 1722 q^{69} + 276 q^{71} + 460 q^{73} + 1098 q^{75} + 323 q^{77} - 1812 q^{79} + 2970 q^{81} - 87 q^{83} - 423 q^{87} + 176 q^{89} - 480 q^{91} - 1011 q^{93} + 1246 q^{95} - 27 q^{97} + 1579 q^{99}+O(q^{100}) $ 6 * q + q^3 + 13 * q^5 + q^7 + 63 * q^9 - 8 * q^11 - 33 * q^13 + 68 * q^15 + 75 * q^19 + 32 * q^21 + 155 * q^23 - 89 * q^25 + 322 * q^27 + 9 * q^29 + 374 * q^31 - 351 * q^33 + 13 * q^35 + 118 * q^37 - 369 * q^39 - 60 * q^41 + 230 * q^43 + 478 * q^45 + 276 * q^47 - 317 * q^49 + 569 * q^53 - 97 * q^55 + 769 * q^57 - 784 * q^59 - 28 * q^61 - 878 * q^63 + 828 * q^65 + 238 * q^67 + 1722 * q^69 + 276 * q^71 + 460 * q^73 + 1098 * q^75 + 323 * q^77 - 1812 * q^79 + 2970 * q^81 - 87 * q^83 - 423 * q^87 + 176 * q^89 - 480 * q^91 - 1011 * q^93 + 1246 * q^95 - 27 * q^97 + 1579 * 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
$3$	10.0775	1.93942	0.969710	−	0.244258i	$-0.0785443\pi$
$3$	10.0775	1.93942	0.969710	+	0.244258i	$0.0785443\pi$
$4$	0	0
$5$	10.2716	0.918719	0.459359	−	0.888251i	$-0.348079\pi$
$5$	10.2716	0.918719	0.459359	+	0.888251i	$0.348079\pi$
$6$	0	0
$7$	−1.06795	−0.0576641	−0.0288321	−	0.999584i	$-0.509179\pi$
$7$	−1.06795	−0.0576641	−0.0288321	+	0.999584i	$0.509179\pi$
$8$	0	0
$9$	74.5565	2.76135
$10$	0	0
$11$	−23.8200	−0.652908	−0.326454	−	0.945213i	$-0.605854\pi$
$11$	−23.8200	−0.652908	−0.326454	+	0.945213i	$0.605854\pi$
$12$	0	0
$13$	−64.6878	−1.38009	−0.690045	−	0.723766i	$-0.742409\pi$
$13$	−64.6878	−1.38009	−0.690045	+	0.723766i	$0.742409\pi$
$14$	0	0
$15$	103.512	1.78178
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	89.6338	1.08228	0.541142	−	0.840931i	$-0.317992\pi$
$19$	89.6338	1.08228	0.541142	+	0.840931i	$0.317992\pi$
$20$	0	0
$21$	−10.7623	−0.111835
$22$	0	0
$23$	203.782	1.84746	0.923728	−	0.383049i	$-0.125126\pi$
$23$	203.782	1.84746	0.923728	+	0.383049i	$0.125126\pi$
$24$	0	0
$25$	−19.4945	−0.155956
$26$	0	0
$27$	479.252	3.41600
$28$	0	0
$29$	−14.3597	−0.0919490	−0.0459745	−	0.998943i	$-0.514639\pi$
$29$	−14.3597	−0.0919490	−0.0459745	+	0.998943i	$0.514639\pi$
$30$	0	0
$31$	237.028	1.37327	0.686635	−	0.727002i	$-0.259087\pi$
$31$	237.028	1.37327	0.686635	+	0.727002i	$0.259087\pi$
$32$	0	0
$33$	−240.046	−1.26626
$34$	0	0
$35$	−10.9696	−0.0529771
$36$	0	0
$37$	−21.8196	−0.0969494	−0.0484747	−	0.998824i	$-0.515436\pi$
$37$	−21.8196	−0.0969494	−0.0484747	+	0.998824i	$0.515436\pi$
$38$	0	0
$39$	−651.893	−2.67658
$40$	0	0
$41$	−342.774	−1.30567	−0.652833	−	0.757502i	$-0.726420\pi$
$41$	−342.774	−1.30567	−0.652833	+	0.757502i	$0.726420\pi$
$42$	0	0
$43$	452.336	1.60420	0.802099	−	0.597191i	$-0.203717\pi$
$43$	452.336	1.60420	0.802099	+	0.597191i	$0.203717\pi$
$44$	0	0
$45$	765.814	2.53691
$46$	0	0
$47$	−61.3020	−0.190251	−0.0951256	−	0.995465i	$-0.530325\pi$
$47$	−61.3020	−0.190251	−0.0951256	+	0.995465i	$0.530325\pi$
$48$	0	0
$49$	−341.859	−0.996675
$50$	0	0
$51$	0	0
$52$	0	0
$53$	268.750	0.696522	0.348261	−	0.937398i	$-0.386772\pi$
$53$	268.750	0.696522	0.348261	+	0.937398i	$0.386772\pi$
$54$	0	0
$55$	−244.669	−0.599839
$56$	0	0
$57$	903.287	2.09901
$58$	0	0
$59$	−221.572	−0.488920	−0.244460	−	0.969659i	$-0.578611\pi$
$59$	−221.572	−0.488920	−0.244460	+	0.969659i	$0.578611\pi$
$60$	0	0
$61$	69.3806	0.145627	0.0728137	−	0.997346i	$-0.476802\pi$
$61$	69.3806	0.145627	0.0728137	+	0.997346i	$0.476802\pi$
$62$	0	0
$63$	−79.6230	−0.159231
$64$	0	0
$65$	−664.447	−1.26792
$66$	0	0
$67$	922.984	1.68299	0.841496	−	0.540263i	$-0.181675\pi$
$67$	922.984	1.68299	0.841496	+	0.540263i	$0.181675\pi$
$68$	0	0
$69$	2053.62	3.58299
$70$	0	0
$71$	631.271	1.05518	0.527592	−	0.849498i	$-0.323095\pi$
$71$	631.271	1.05518	0.527592	+	0.849498i	$0.323095\pi$
$72$	0	0
$73$	−414.733	−0.664942	−0.332471	−	0.943113i	$-0.607882\pi$
$73$	−414.733	−0.664942	−0.332471	+	0.943113i	$0.607882\pi$
$74$	0	0
$75$	−196.456	−0.302464
$76$	0	0
$77$	25.4387	0.0376494
$78$	0	0
$79$	−1294.10	−1.84300	−0.921502	−	0.388375i	$-0.873037\pi$
$79$	−1294.10	−1.84300	−0.921502	+	0.388375i	$0.873037\pi$
$80$	0	0
$81$	2816.65	3.86372
$82$	0	0
$83$	910.400	1.20397	0.601984	−	0.798508i	$-0.294377\pi$
$83$	910.400	1.20397	0.601984	+	0.798508i	$0.294377\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−144.710	−0.178328
$88$	0	0
$89$	−27.2575	−0.0324639	−0.0162319	−	0.999868i	$-0.505167\pi$
$89$	−27.2575	−0.0324639	−0.0162319	+	0.999868i	$0.505167\pi$
$90$	0	0
$91$	69.0837	0.0795818
$92$	0	0
$93$	2388.65	2.66335
$94$	0	0
$95$	920.681	0.994315
$96$	0	0
$97$	1316.01	1.37753	0.688766	−	0.724984i	$-0.258153\pi$
$97$	1316.01	1.37753	0.688766	+	0.724984i	$0.258153\pi$
$98$	0	0
$99$	−1775.93	−1.80291
$100$	0	0
$101$	1315.43	1.29594	0.647971	−	0.761665i	$-0.275618\pi$
$101$	1315.43	1.29594	0.647971	+	0.761665i	$0.275618\pi$
$102$	0	0
$103$	227.705	0.217829	0.108915	−	0.994051i	$-0.465262\pi$
$103$	227.705	0.217829	0.108915	+	0.994051i	$0.465262\pi$
$104$	0	0
$105$	−110.546	−0.102745
$106$	0	0
$107$	−570.553	−0.515491	−0.257745	−	0.966213i	$-0.582980\pi$
$107$	−570.553	−0.515491	−0.257745	+	0.966213i	$0.582980\pi$
$108$	0	0
$109$	1436.06	1.26192	0.630961	−	0.775815i	$-0.282661\pi$
$109$	1436.06	1.26192	0.630961	+	0.775815i	$0.282661\pi$
$110$	0	0
$111$	−219.888	−0.188026
$112$	0	0
$113$	245.284	0.204198	0.102099	−	0.994774i	$-0.467444\pi$
$113$	245.284	0.204198	0.102099	+	0.994774i	$0.467444\pi$
$114$	0	0
$115$	2093.16	1.69729
$116$	0	0
$117$	−4822.90	−3.81092
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−763.609	−0.573711
$122$	0	0
$123$	−3454.31	−2.53224
$124$	0	0
$125$	−1484.19	−1.06200
$126$	0	0
$127$	498.207	0.348100	0.174050	−	0.984737i	$-0.444315\pi$
$127$	498.207	0.348100	0.174050	+	0.984737i	$0.444315\pi$
$128$	0	0
$129$	4558.42	3.11122
$130$	0	0
$131$	−1238.52	−0.826034	−0.413017	−	0.910723i	$-0.635525\pi$
$131$	−1238.52	−0.826034	−0.413017	+	0.910723i	$0.635525\pi$
$132$	0	0
$133$	−95.7249	−0.0624090
$134$	0	0
$135$	4922.68	3.13835
$136$	0	0
$137$	−2312.37	−1.44204	−0.721018	−	0.692917i	$-0.756325\pi$
$137$	−2312.37	−1.44204	−0.721018	+	0.692917i	$0.756325\pi$
$138$	0	0
$139$	−1525.32	−0.930760	−0.465380	−	0.885111i	$-0.654082\pi$
$139$	−1525.32	−0.930760	−0.465380	+	0.885111i	$0.654082\pi$
$140$	0	0
$141$	−617.772	−0.368977
$142$	0	0
$143$	1540.86	0.901073
$144$	0	0
$145$	−147.496	−0.0844753
$146$	0	0
$147$	−3445.10	−1.93297
$148$	0	0
$149$	−522.113	−0.287068	−0.143534	−	0.989645i	$-0.545847\pi$
$149$	−522.113	−0.287068	−0.143534	+	0.989645i	$0.545847\pi$
$150$	0	0
$151$	1657.03	0.893030	0.446515	−	0.894776i	$-0.352665\pi$
$151$	1657.03	0.893030	0.446515	+	0.894776i	$0.352665\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2434.65	1.26165
$156$	0	0
$157$	−1918.60	−0.975291	−0.487645	−	0.873042i	$-0.662144\pi$
$157$	−1918.60	−0.975291	−0.487645	+	0.873042i	$0.662144\pi$
$158$	0	0
$159$	2708.33	1.35085
$160$	0	0
$161$	−217.630	−0.106532
$162$	0	0
$163$	−1014.94	−0.487705	−0.243853	−	0.969812i	$-0.578411\pi$
$163$	−1014.94	−0.487705	−0.243853	+	0.969812i	$0.578411\pi$
$164$	0	0
$165$	−2465.66	−1.16334
$166$	0	0
$167$	−1332.67	−0.617514	−0.308757	−	0.951141i	$-0.599913\pi$
$167$	−1332.67	−0.617514	−0.308757	+	0.951141i	$0.599913\pi$
$168$	0	0
$169$	1987.52	0.904651
$170$	0	0
$171$	6682.79	2.98857
$172$	0	0
$173$	1439.90	0.632795	0.316398	−	0.948627i	$-0.397527\pi$
$173$	1439.90	0.632795	0.316398	+	0.948627i	$0.397527\pi$
$174$	0	0
$175$	20.8193	0.00899307
$176$	0	0
$177$	−2232.90	−0.948221
$178$	0	0
$179$	3762.85	1.57122	0.785610	−	0.618722i	$-0.212349\pi$
$179$	3762.85	1.57122	0.785610	+	0.618722i	$0.212349\pi$
$180$	0	0
$181$	1400.67	0.575200	0.287600	−	0.957751i	$-0.407143\pi$
$181$	1400.67	0.575200	0.287600	+	0.957751i	$0.407143\pi$
$182$	0	0
$183$	699.185	0.282433
$184$	0	0
$185$	−224.122	−0.0890692
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−511.820	−0.196981
$190$	0	0
$191$	383.184	0.145164	0.0725818	−	0.997362i	$-0.476876\pi$
$191$	383.184	0.145164	0.0725818	+	0.997362i	$0.476876\pi$
$192$	0	0
$193$	−522.726	−0.194957	−0.0974784	−	0.995238i	$-0.531078\pi$
$193$	−522.726	−0.194957	−0.0974784	+	0.995238i	$0.531078\pi$
$194$	0	0
$195$	−6695.98	−2.45902
$196$	0	0
$197$	3088.21	1.11688	0.558442	−	0.829544i	$-0.311399\pi$
$197$	3088.21	1.11688	0.558442	+	0.829544i	$0.311399\pi$
$198$	0	0
$199$	2.90797	0.00103588	0.000517941	−	1.00000i	$-0.499835\pi$
$199$	2.90797	0.00103588	0.000517941		1.00000i	$0.499835\pi$
$200$	0	0
$201$	9301.40	3.26403
$202$	0	0
$203$	15.3355	0.00530216
$204$	0	0
$205$	−3520.83	−1.19954
$206$	0	0
$207$	15193.3	5.10148
$208$	0	0
$209$	−2135.08	−0.706633
$210$	0	0
$211$	−3428.69	−1.11868	−0.559339	−	0.828939i	$-0.688945\pi$
$211$	−3428.69	−1.11868	−0.559339	+	0.828939i	$0.688945\pi$
$212$	0	0
$213$	6361.65	2.04645
$214$	0	0
$215$	4646.20	1.47381
$216$	0	0
$217$	−253.135	−0.0791885
$218$	0	0
$219$	−4179.48	−1.28960
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−3731.43	−1.12052	−0.560258	−	0.828318i	$-0.689298\pi$
$223$	−3731.43	−1.12052	−0.560258	+	0.828318i	$0.689298\pi$
$224$	0	0
$225$	−1453.44	−0.430650
$226$	0	0
$227$	−539.648	−0.157787	−0.0788936	−	0.996883i	$-0.525139\pi$
$227$	−539.648	−0.157787	−0.0788936	+	0.996883i	$0.525139\pi$
$228$	0	0
$229$	227.094	0.0655319	0.0327659	−	0.999463i	$-0.489568\pi$
$229$	227.094	0.0655319	0.0327659	+	0.999463i	$0.489568\pi$
$230$	0	0
$231$	256.359	0.0730180
$232$	0	0
$233$	−6043.14	−1.69914	−0.849570	−	0.527476i	$-0.823138\pi$
$233$	−6043.14	−1.69914	−0.849570	+	0.527476i	$0.823138\pi$
$234$	0	0
$235$	−629.668	−0.174787
$236$	0	0
$237$	−13041.3	−3.57436
$238$	0	0
$239$	4523.52	1.22428	0.612139	−	0.790751i	$-0.290309\pi$
$239$	4523.52	1.22428	0.612139	+	0.790751i	$0.290309\pi$
$240$	0	0
$241$	−6704.54	−1.79202	−0.896011	−	0.444032i	$-0.853548\pi$
$241$	−6704.54	−1.79202	−0.896011	+	0.444032i	$0.853548\pi$
$242$	0	0
$243$	15445.0	4.07737
$244$	0	0
$245$	−3511.44	−0.915664
$246$	0	0
$247$	−5798.22	−1.49365
$248$	0	0
$249$	9174.58	2.33500
$250$	0	0
$251$	−2980.12	−0.749416	−0.374708	−	0.927143i	$-0.622257\pi$
$251$	−2980.12	−0.749416	−0.374708	+	0.927143i	$0.622257\pi$
$252$	0	0
$253$	−4854.08	−1.20622
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1592.57	−0.386544	−0.193272	−	0.981145i	$-0.561910\pi$
$257$	−1592.57	−0.386544	−0.193272	+	0.981145i	$0.561910\pi$
$258$	0	0
$259$	23.3024	0.00559050
$260$	0	0
$261$	−1070.61	−0.253904
$262$	0	0
$263$	−4095.25	−0.960167	−0.480083	−	0.877223i	$-0.659394\pi$
$263$	−4095.25	−0.960167	−0.480083	+	0.877223i	$0.659394\pi$
$264$	0	0
$265$	2760.49	0.639907
$266$	0	0
$267$	−274.688	−0.0629611
$268$	0	0
$269$	8437.92	1.91252	0.956262	−	0.292511i	$-0.0944907\pi$
$269$	8437.92	1.91252	0.956262	+	0.292511i	$0.0944907\pi$
$270$	0	0
$271$	−312.268	−0.0699961	−0.0349980	−	0.999387i	$-0.511142\pi$
$271$	−312.268	−0.0699961	−0.0349980	+	0.999387i	$0.511142\pi$
$272$	0	0
$273$	696.193	0.154343
$274$	0	0
$275$	464.359	0.101825
$276$	0	0
$277$	−3306.05	−0.717118	−0.358559	−	0.933507i	$-0.616732\pi$
$277$	−3306.05	−0.717118	−0.358559	+	0.933507i	$0.616732\pi$
$278$	0	0
$279$	17671.9	3.79209
$280$	0	0
$281$	1848.33	0.392392	0.196196	−	0.980565i	$-0.437141\pi$
$281$	1848.33	0.392392	0.196196	+	0.980565i	$0.437141\pi$
$282$	0	0
$283$	−5314.95	−1.11640	−0.558200	−	0.829707i	$-0.688508\pi$
$283$	−5314.95	−1.11640	−0.558200	+	0.829707i	$0.688508\pi$
$284$	0	0
$285$	9278.19	1.92840
$286$	0	0
$287$	366.067	0.0752901
$288$	0	0
$289$	0	0
$290$	0	0
$291$	13262.1	2.67161
$292$	0	0
$293$	−1472.45	−0.293589	−0.146795	−	0.989167i	$-0.546896\pi$
$293$	−1472.45	−0.293589	−0.146795	+	0.989167i	$0.546896\pi$
$294$	0	0
$295$	−2275.90	−0.449179
$296$	0	0
$297$	−11415.8	−2.23034
$298$	0	0
$299$	−13182.2	−2.54966
$300$	0	0
$301$	−483.074	−0.0925047
$302$	0	0
$303$	13256.3	2.51338
$304$	0	0
$305$	712.649	0.133791
$306$	0	0
$307$	−5057.32	−0.940184	−0.470092	−	0.882617i	$-0.655779\pi$
$307$	−5057.32	−0.940184	−0.470092	+	0.882617i	$0.655779\pi$
$308$	0	0
$309$	2294.70	0.422463
$310$	0	0
$311$	739.291	0.134795	0.0673977	−	0.997726i	$-0.478530\pi$
$311$	739.291	0.134795	0.0673977	+	0.997726i	$0.478530\pi$
$312$	0	0
$313$	9334.31	1.68564	0.842822	−	0.538193i	$-0.180893\pi$
$313$	9334.31	1.68564	0.842822	+	0.538193i	$0.180893\pi$
$314$	0	0
$315$	−817.855	−0.146289
$316$	0	0
$317$	−7517.39	−1.33192	−0.665960	−	0.745988i	$-0.731978\pi$
$317$	−7517.39	−1.33192	−0.665960	+	0.745988i	$0.731978\pi$
$318$	0	0
$319$	342.047	0.0600343
$320$	0	0
$321$	−5749.77	−0.999753
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1261.06	0.215234
$326$	0	0
$327$	14471.9	2.44740
$328$	0	0
$329$	65.4677	0.0109707
$330$	0	0
$331$	−4330.23	−0.719066	−0.359533	−	0.933132i	$-0.617064\pi$
$331$	−4330.23	−0.719066	−0.359533	+	0.933132i	$0.617064\pi$
$332$	0	0
$333$	−1626.80	−0.267711
$334$	0	0
$335$	9480.51	1.54620
$336$	0	0
$337$	1037.78	0.167749	0.0838747	−	0.996476i	$-0.473270\pi$
$337$	1037.78	0.167749	0.0838747	+	0.996476i	$0.473270\pi$
$338$	0	0
$339$	2471.86	0.396026
$340$	0	0
$341$	−5645.99	−0.896620
$342$	0	0
$343$	731.399	0.115137
$344$	0	0
$345$	21093.9	3.29176
$346$	0	0
$347$	4339.40	0.671328	0.335664	−	0.941982i	$-0.391039\pi$
$347$	4339.40	0.671328	0.335664	+	0.941982i	$0.391039\pi$
$348$	0	0
$349$	−8888.55	−1.36330	−0.681652	−	0.731677i	$-0.738738\pi$
$349$	−8888.55	−1.36330	−0.681652	+	0.731677i	$0.738738\pi$
$350$	0	0
$351$	−31001.8	−4.71440
$352$	0	0
$353$	−8881.74	−1.33917	−0.669585	−	0.742735i	$-0.733528\pi$
$353$	−8881.74	−1.33917	−0.669585	+	0.742735i	$0.733528\pi$
$354$	0	0
$355$	6484.15	0.969417
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3927.50	−0.577397	−0.288699	−	0.957420i	$-0.593223\pi$
$359$	−3927.50	−0.577397	−0.288699	+	0.957420i	$0.593223\pi$
$360$	0	0
$361$	1175.22	0.171340
$362$	0	0
$363$	−7695.29	−1.11267
$364$	0	0
$365$	−4259.96	−0.610895
$366$	0	0
$367$	−1606.23	−0.228459	−0.114230	−	0.993454i	$-0.536440\pi$
$367$	−1606.23	−0.228459	−0.114230	+	0.993454i	$0.536440\pi$
$368$	0	0
$369$	−25556.0	−3.60541
$370$	0	0
$371$	−287.013	−0.0401643
$372$	0	0
$373$	−4168.00	−0.578582	−0.289291	−	0.957241i	$-0.593420\pi$
$373$	−4168.00	−0.578582	−0.289291	+	0.957241i	$0.593420\pi$
$374$	0	0
$375$	−14956.9	−2.05966
$376$	0	0
$377$	928.895	0.126898
$378$	0	0
$379$	−900.465	−0.122042	−0.0610208	−	0.998136i	$-0.519436\pi$
$379$	−900.465	−0.122042	−0.0610208	+	0.998136i	$0.519436\pi$
$380$	0	0
$381$	5020.69	0.675113
$382$	0	0
$383$	−7952.81	−1.06102	−0.530509	−	0.847679i	$-0.677999\pi$
$383$	−7952.81	−1.06102	−0.530509	+	0.847679i	$0.677999\pi$
$384$	0	0
$385$	261.295	0.0345892
$386$	0	0
$387$	33724.6	4.42976
$388$	0	0
$389$	−396.470	−0.0516756	−0.0258378	−	0.999666i	$-0.508225\pi$
$389$	−396.470	−0.0516756	−0.0258378	+	0.999666i	$0.508225\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−12481.3	−1.60203
$394$	0	0
$395$	−13292.4	−1.69320
$396$	0	0
$397$	−4264.43	−0.539107	−0.269553	−	0.962985i	$-0.586876\pi$
$397$	−4264.43	−0.539107	−0.269553	+	0.962985i	$0.586876\pi$
$398$	0	0
$399$	−964.670	−0.121037
$400$	0	0
$401$	−2184.47	−0.272038	−0.136019	−	0.990706i	$-0.543431\pi$
$401$	−2184.47	−0.272038	−0.136019	+	0.990706i	$0.543431\pi$
$402$	0	0
$403$	−15332.8	−1.89524
$404$	0	0
$405$	28931.5	3.54967
$406$	0	0
$407$	519.743	0.0632991
$408$	0	0
$409$	11126.6	1.34517	0.672587	−	0.740018i	$-0.265183\pi$
$409$	11126.6	1.34517	0.672587	+	0.740018i	$0.265183\pi$
$410$	0	0
$411$	−23302.9	−2.79671
$412$	0	0
$413$	236.629	0.0281931
$414$	0	0
$415$	9351.25	1.10611
$416$	0	0
$417$	−15371.4	−1.80514
$418$	0	0
$419$	−3165.49	−0.369080	−0.184540	−	0.982825i	$-0.559080\pi$
$419$	−3165.49	−0.369080	−0.184540	+	0.982825i	$0.559080\pi$
$420$	0	0
$421$	10160.3	1.17620	0.588102	−	0.808787i	$-0.299875\pi$
$421$	10160.3	1.17620	0.588102	+	0.808787i	$0.299875\pi$
$422$	0	0
$423$	−4570.46	−0.525351
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−74.0953	−0.00839748
$428$	0	0
$429$	15528.1	1.74756
$430$	0	0
$431$	−5630.06	−0.629212	−0.314606	−	0.949222i	$-0.601872\pi$
$431$	−5630.06	−0.629212	−0.314606	+	0.949222i	$0.601872\pi$
$432$	0	0
$433$	−4848.52	−0.538118	−0.269059	−	0.963124i	$-0.586713\pi$
$433$	−4848.52	−0.538118	−0.269059	+	0.963124i	$0.586713\pi$
$434$	0	0
$435$	−1486.40	−0.163833
$436$	0	0
$437$	18265.8	1.99947
$438$	0	0
$439$	794.804	0.0864098	0.0432049	−	0.999066i	$-0.486243\pi$
$439$	794.804	0.0864098	0.0432049	+	0.999066i	$0.486243\pi$
$440$	0	0
$441$	−25487.9	−2.75217
$442$	0	0
$443$	4849.19	0.520073	0.260036	−	0.965599i	$-0.416265\pi$
$443$	4849.19	0.520073	0.260036	+	0.965599i	$0.416265\pi$
$444$	0	0
$445$	−279.977	−0.0298252
$446$	0	0
$447$	−5261.60	−0.556746
$448$	0	0
$449$	3102.64	0.326108	0.163054	−	0.986617i	$-0.447865\pi$
$449$	3102.64	0.326108	0.163054	+	0.986617i	$0.447865\pi$
$450$	0	0
$451$	8164.87	0.852481
$452$	0	0
$453$	16698.8	1.73196
$454$	0	0
$455$	709.599	0.0731132
$456$	0	0
$457$	4277.90	0.437881	0.218941	−	0.975738i	$-0.429740\pi$
$457$	4277.90	0.437881	0.218941	+	0.975738i	$0.429740\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16367.4	1.65359	0.826795	−	0.562504i	$-0.190162\pi$
$461$	16367.4	1.65359	0.826795	+	0.562504i	$0.190162\pi$
$462$	0	0
$463$	−7249.74	−0.727698	−0.363849	−	0.931458i	$-0.618538\pi$
$463$	−7249.74	−0.727698	−0.363849	+	0.931458i	$0.618538\pi$
$464$	0	0
$465$	24535.2	2.44687
$466$	0	0
$467$	−11066.1	−1.09653	−0.548263	−	0.836306i	$-0.684711\pi$
$467$	−11066.1	−1.09653	−0.548263	+	0.836306i	$0.684711\pi$
$468$	0	0
$469$	−985.705	−0.0970483
$470$	0	0
$471$	−19334.7	−1.89150
$472$	0	0
$473$	−10774.6	−1.04739
$474$	0	0
$475$	−1747.37	−0.168789
$476$	0	0
$477$	20037.1	1.92334
$478$	0	0
$479$	11771.9	1.12291	0.561453	−	0.827508i	$-0.310242\pi$
$479$	11771.9	1.12291	0.561453	+	0.827508i	$0.310242\pi$
$480$	0	0
$481$	1411.47	0.133799
$482$	0	0
$483$	−2193.17	−0.206610
$484$	0	0
$485$	13517.5	1.26556
$486$	0	0
$487$	16081.8	1.49638	0.748190	−	0.663485i	$-0.230923\pi$
$487$	16081.8	1.49638	0.748190	+	0.663485i	$0.230923\pi$
$488$	0	0
$489$	−10228.0	−0.945865
$490$	0	0
$491$	2006.00	0.184378	0.0921890	−	0.995742i	$-0.470614\pi$
$491$	2006.00	0.184378	0.0921890	+	0.995742i	$0.470614\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−18241.7	−1.65637
$496$	0	0
$497$	−674.169	−0.0608463
$498$	0	0
$499$	−10209.7	−0.915931	−0.457966	−	0.888970i	$-0.651422\pi$
$499$	−10209.7	−0.915931	−0.457966	+	0.888970i	$0.651422\pi$
$500$	0	0
$501$	−13430.0	−1.19762
$502$	0	0
$503$	−2841.52	−0.251883	−0.125942	−	0.992038i	$-0.540195\pi$
$503$	−2841.52	−0.251883	−0.125942	+	0.992038i	$0.540195\pi$
$504$	0	0
$505$	13511.6	1.19061
$506$	0	0
$507$	20029.3	1.75450
$508$	0	0
$509$	10615.0	0.924365	0.462182	−	0.886785i	$-0.347067\pi$
$509$	10615.0	0.924365	0.462182	+	0.886785i	$0.347067\pi$
$510$	0	0
$511$	442.916	0.0383433
$512$	0	0
$513$	42957.2	3.69709
$514$	0	0
$515$	2338.89	0.200124
$516$	0	0
$517$	1460.21	0.124217
$518$	0	0
$519$	14510.6	1.22726
$520$	0	0
$521$	−1535.42	−0.129114	−0.0645568	−	0.997914i	$-0.520563\pi$
$521$	−1535.42	−0.129114	−0.0645568	+	0.997914i	$0.520563\pi$
$522$	0	0
$523$	−12236.3	−1.02305	−0.511527	−	0.859267i	$-0.670920\pi$
$523$	−12236.3	−1.02305	−0.511527	+	0.859267i	$0.670920\pi$
$524$	0	0
$525$	209.807	0.0174414
$526$	0	0
$527$	0	0
$528$	0	0
$529$	29360.1	2.41309
$530$	0	0
$531$	−16519.7	−1.35008
$532$	0	0
$533$	22173.3	1.80194
$534$	0	0
$535$	−5860.49	−0.473591
$536$	0	0
$537$	37920.2	3.04726
$538$	0	0
$539$	8143.08	0.650737
$540$	0	0
$541$	−10458.6	−0.831146	−0.415573	−	0.909560i	$-0.636419\pi$
$541$	−10458.6	−0.831146	−0.415573	+	0.909560i	$0.636419\pi$
$542$	0	0
$543$	14115.3	1.11555
$544$	0	0
$545$	14750.6	1.15935
$546$	0	0
$547$	9336.78	0.729820	0.364910	−	0.931043i	$-0.381100\pi$
$547$	9336.78	0.729820	0.364910	+	0.931043i	$0.381100\pi$
$548$	0	0
$549$	5172.78	0.402129
$550$	0	0
$551$	−1287.11	−0.0995150
$552$	0	0
$553$	1382.04	0.106275
$554$	0	0
$555$	−2258.60	−0.172743
$556$	0	0
$557$	−13959.8	−1.06193	−0.530964	−	0.847394i	$-0.678170\pi$
$557$	−13959.8	−1.06193	−0.530964	+	0.847394i	$0.678170\pi$
$558$	0	0
$559$	−29260.6	−2.21394
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10104.9	0.756430	0.378215	−	0.925718i	$-0.376538\pi$
$563$	10104.9	0.756430	0.378215	+	0.925718i	$0.376538\pi$
$564$	0	0
$565$	2519.46	0.187601
$566$	0	0
$567$	−3008.05	−0.222798
$568$	0	0
$569$	−13709.9	−1.01010	−0.505052	−	0.863089i	$-0.668527\pi$
$569$	−13709.9	−1.01010	−0.505052	+	0.863089i	$0.668527\pi$
$570$	0	0
$571$	8019.08	0.587720	0.293860	−	0.955849i	$-0.405060\pi$
$571$	8019.08	0.587720	0.293860	+	0.955849i	$0.405060\pi$
$572$	0	0
$573$	3861.55	0.281533
$574$	0	0
$575$	−3972.63	−0.288122
$576$	0	0
$577$	−9661.20	−0.697055	−0.348528	−	0.937299i	$-0.613318\pi$
$577$	−9661.20	−0.697055	−0.348528	+	0.937299i	$0.613318\pi$
$578$	0	0
$579$	−5267.79	−0.378103
$580$	0	0
$581$	−972.266	−0.0694258
$582$	0	0
$583$	−6401.62	−0.454765
$584$	0	0
$585$	−49538.8	−3.50116
$586$	0	0
$587$	−20346.6	−1.43065	−0.715326	−	0.698790i	$-0.753722\pi$
$587$	−20346.6	−1.43065	−0.715326	+	0.698790i	$0.753722\pi$
$588$	0	0
$589$	21245.7	1.48627
$590$	0	0
$591$	31121.5	2.16611
$592$	0	0
$593$	6842.74	0.473858	0.236929	−	0.971527i	$-0.423859\pi$
$593$	6842.74	0.473858	0.236929	+	0.971527i	$0.423859\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29.3051	0.00200901
$598$	0	0
$599$	23720.8	1.61804	0.809020	−	0.587780i	$-0.199998\pi$
$599$	23720.8	1.61804	0.809020	+	0.587780i	$0.199998\pi$
$600$	0	0
$601$	16660.1	1.13075	0.565375	−	0.824834i	$-0.308731\pi$
$601$	16660.1	1.13075	0.565375	+	0.824834i	$0.308731\pi$
$602$	0	0
$603$	68814.5	4.64733
$604$	0	0
$605$	−7843.47	−0.527079
$606$	0	0
$607$	8419.95	0.563023	0.281512	−	0.959558i	$-0.409164\pi$
$607$	8419.95	0.563023	0.281512	+	0.959558i	$0.409164\pi$
$608$	0	0
$609$	154.544	0.0102831
$610$	0	0
$611$	3965.49	0.262564
$612$	0	0
$613$	−15124.6	−0.996539	−0.498270	−	0.867022i	$-0.666031\pi$
$613$	−15124.6	−0.996539	−0.498270	+	0.867022i	$0.666031\pi$
$614$	0	0
$615$	−35481.3	−2.32641
$616$	0	0
$617$	−15287.2	−0.997474	−0.498737	−	0.866753i	$-0.666203\pi$
$617$	−15287.2	−0.997474	−0.498737	+	0.866753i	$0.666203\pi$
$618$	0	0
$619$	1179.64	0.0765971	0.0382986	−	0.999266i	$-0.487806\pi$
$619$	1179.64	0.0765971	0.0382986	+	0.999266i	$0.487806\pi$
$620$	0	0
$621$	97662.9	6.31092
$622$	0	0
$623$	29.1097	0.00187200
$624$	0	0
$625$	−12808.2	−0.819722
$626$	0	0
$627$	−21516.3	−1.37046
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−5009.67	−0.316057	−0.158028	−	0.987435i	$-0.550514\pi$
$631$	−5009.67	−0.316057	−0.158028	+	0.987435i	$0.550514\pi$
$632$	0	0
$633$	−34552.7	−2.16959
$634$	0	0
$635$	5117.38	0.319806
$636$	0	0
$637$	22114.2	1.37550
$638$	0	0
$639$	47065.4	2.91374
$640$	0	0
$641$	−29033.4	−1.78900	−0.894502	−	0.447064i	$-0.852470\pi$
$641$	−29033.4	−1.78900	−0.894502	+	0.447064i	$0.852470\pi$
$642$	0	0
$643$	4415.38	0.270802	0.135401	−	0.990791i	$-0.456768\pi$
$643$	4415.38	0.270802	0.135401	+	0.990791i	$0.456768\pi$
$644$	0	0
$645$	46822.2	2.85833
$646$	0	0
$647$	6127.16	0.372308	0.186154	−	0.982521i	$-0.440398\pi$
$647$	6127.16	0.372308	0.186154	+	0.982521i	$0.440398\pi$
$648$	0	0
$649$	5277.85	0.319220
$650$	0	0
$651$	−2550.97	−0.153580
$652$	0	0
$653$	14904.3	0.893184	0.446592	−	0.894738i	$-0.352637\pi$
$653$	14904.3	0.893184	0.446592	+	0.894738i	$0.352637\pi$
$654$	0	0
$655$	−12721.6	−0.758892
$656$	0	0
$657$	−30921.0	−1.83614
$658$	0	0
$659$	11562.8	0.683493	0.341747	−	0.939792i	$-0.388982\pi$
$659$	11562.8	0.683493	0.341747	+	0.939792i	$0.388982\pi$
$660$	0	0
$661$	−19701.5	−1.15930	−0.579651	−	0.814865i	$-0.696811\pi$
$661$	−19701.5	−1.15930	−0.579651	+	0.814865i	$0.696811\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−983.246	−0.0573363
$666$	0	0
$667$	−2926.24	−0.169872
$668$	0	0
$669$	−37603.6	−2.17315
$670$	0	0
$671$	−1652.64	−0.0950814
$672$	0	0
$673$	10204.4	0.584476	0.292238	−	0.956346i	$-0.405600\pi$
$673$	10204.4	0.584476	0.292238	+	0.956346i	$0.405600\pi$
$674$	0	0
$675$	−9342.79	−0.532747
$676$	0	0
$677$	−5292.23	−0.300439	−0.150219	−	0.988653i	$-0.547998\pi$
$677$	−5292.23	−0.300439	−0.150219	+	0.988653i	$0.547998\pi$
$678$	0	0
$679$	−1405.44	−0.0794342
$680$	0	0
$681$	−5438.32	−0.306016
$682$	0	0
$683$	5347.00	0.299557	0.149778	−	0.988720i	$-0.452144\pi$
$683$	5347.00	0.299557	0.149778	+	0.988720i	$0.452144\pi$
$684$	0	0
$685$	−23751.7	−1.32482
$686$	0	0
$687$	2288.55	0.127094
$688$	0	0
$689$	−17384.9	−0.961263
$690$	0	0
$691$	26263.5	1.44589	0.722945	−	0.690905i	$-0.242788\pi$
$691$	26263.5	1.44589	0.722945	+	0.690905i	$0.242788\pi$
$692$	0	0
$693$	1896.62	0.103963
$694$	0	0
$695$	−15667.4	−0.855107
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−60899.9	−3.29535
$700$	0	0
$701$	−32790.0	−1.76670	−0.883352	−	0.468710i	$-0.844719\pi$
$701$	−32790.0	−1.76670	−0.883352	+	0.468710i	$0.844719\pi$
$702$	0	0
$703$	−1955.78	−0.104927
$704$	0	0
$705$	−6345.50	−0.338986
$706$	0	0
$707$	−1404.82	−0.0747294
$708$	0	0
$709$	31413.8	1.66399	0.831995	−	0.554783i	$-0.187199\pi$
$709$	31413.8	1.66399	0.831995	+	0.554783i	$0.187199\pi$
$710$	0	0
$711$	−96483.3	−5.08918
$712$	0	0
$713$	48301.9	2.53706
$714$	0	0
$715$	15827.1	0.827833
$716$	0	0
$717$	45585.9	2.37439
$718$	0	0
$719$	5357.54	0.277889	0.138945	−	0.990300i	$-0.455629\pi$
$719$	5357.54	0.277889	0.138945	+	0.990300i	$0.455629\pi$
$720$	0	0
$721$	−243.178	−0.0125609
$722$	0	0
$723$	−67565.1	−3.47548
$724$	0	0
$725$	279.935	0.0143400
$726$	0	0
$727$	−18054.9	−0.921072	−0.460536	−	0.887641i	$-0.652343\pi$
$727$	−18054.9	−0.921072	−0.460536	+	0.887641i	$0.652343\pi$
$728$	0	0
$729$	79598.3	4.04401
$730$	0	0
$731$	0	0
$732$	0	0
$733$	2380.18	0.119937	0.0599685	−	0.998200i	$-0.480900\pi$
$733$	2380.18	0.119937	0.0599685	+	0.998200i	$0.480900\pi$
$734$	0	0
$735$	−35386.6	−1.77586
$736$	0	0
$737$	−21985.5	−1.09884
$738$	0	0
$739$	−2159.00	−0.107470	−0.0537348	−	0.998555i	$-0.517113\pi$
$739$	−2159.00	−0.107470	−0.0537348	+	0.998555i	$0.517113\pi$
$740$	0	0
$741$	−58431.7	−2.89682
$742$	0	0
$743$	6829.41	0.337210	0.168605	−	0.985684i	$-0.446074\pi$
$743$	6829.41	0.337210	0.168605	+	0.985684i	$0.446074\pi$
$744$	0	0
$745$	−5362.93	−0.263735
$746$	0	0
$747$	67876.3	3.32458
$748$	0	0
$749$	609.325	0.0297253
$750$	0	0
$751$	−36875.6	−1.79176	−0.895878	−	0.444300i	$-0.853453\pi$
$751$	−36875.6	−1.79176	−0.895878	+	0.444300i	$0.853453\pi$
$752$	0	0
$753$	−30032.2	−1.45343
$754$	0	0
$755$	17020.4	0.820443
$756$	0	0
$757$	21596.5	1.03690	0.518452	−	0.855107i	$-0.326508\pi$
$757$	21596.5	1.03690	0.518452	+	0.855107i	$0.326508\pi$
$758$	0	0
$759$	−48917.1	−2.33937
$760$	0	0
$761$	16875.8	0.803873	0.401937	−	0.915667i	$-0.368337\pi$
$761$	16875.8	0.803873	0.401937	+	0.915667i	$0.368337\pi$
$762$	0	0
$763$	−1533.65	−0.0727676
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14333.0	0.674753
$768$	0	0
$769$	−23445.8	−1.09945	−0.549725	−	0.835346i	$-0.685267\pi$
$769$	−23445.8	−1.09945	−0.549725	+	0.835346i	$0.685267\pi$
$770$	0	0
$771$	−16049.2	−0.749671
$772$	0	0
$773$	−17752.3	−0.826009	−0.413005	−	0.910729i	$-0.635521\pi$
$773$	−17752.3	−0.826009	−0.413005	+	0.910729i	$0.635521\pi$
$774$	0	0
$775$	−4620.74	−0.214170
$776$	0	0
$777$	234.830	0.0108423
$778$	0	0
$779$	−30724.2	−1.41310
$780$	0	0
$781$	−15036.9	−0.688939
$782$	0	0
$783$	−6881.90	−0.314098
$784$	0	0
$785$	−19707.0	−0.896018
$786$	0	0
$787$	−5998.38	−0.271689	−0.135845	−	0.990730i	$-0.543375\pi$
$787$	−5998.38	−0.271689	−0.135845	+	0.990730i	$0.543375\pi$
$788$	0	0
$789$	−41270.0	−1.86217
$790$	0	0
$791$	−261.953	−0.0117749
$792$	0	0
$793$	−4488.08	−0.200979
$794$	0	0
$795$	27818.9	1.24105
$796$	0	0
$797$	5161.38	0.229392	0.114696	−	0.993401i	$-0.463411\pi$
$797$	5161.38	0.229392	0.114696	+	0.993401i	$0.463411\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−2032.22	−0.0896442
$802$	0	0
$803$	9878.92	0.434146
$804$	0	0
$805$	−2235.41	−0.0978729
$806$	0	0
$807$	85033.4	3.70919
$808$	0	0
$809$	8014.13	0.348284	0.174142	−	0.984721i	$-0.444285\pi$
$809$	8014.13	0.348284	0.174142	+	0.984721i	$0.444285\pi$
$810$	0	0
$811$	15325.8	0.663579	0.331790	−	0.943353i	$-0.392348\pi$
$811$	15325.8	0.663579	0.331790	+	0.943353i	$0.392348\pi$
$812$	0	0
$813$	−3146.89	−0.135752
$814$	0	0
$815$	−10425.0	−0.448064
$816$	0	0
$817$	40544.6	1.73620
$818$	0	0
$819$	5150.64	0.219753
$820$	0	0
$821$	−20456.3	−0.869587	−0.434793	−	0.900530i	$-0.643179\pi$
$821$	−20456.3	−0.869587	−0.434793	+	0.900530i	$0.643179\pi$
$822$	0	0
$823$	−6386.95	−0.270516	−0.135258	−	0.990810i	$-0.543186\pi$
$823$	−6386.95	−0.270516	−0.135258	+	0.990810i	$0.543186\pi$
$824$	0	0
$825$	4679.59	0.197482
$826$	0	0
$827$	25663.0	1.07907	0.539535	−	0.841963i	$-0.318600\pi$
$827$	25663.0	1.07907	0.539535	+	0.841963i	$0.318600\pi$
$828$	0	0
$829$	26297.2	1.10174	0.550868	−	0.834592i	$-0.314297\pi$
$829$	26297.2	1.10174	0.550868	+	0.834592i	$0.314297\pi$
$830$	0	0
$831$	−33316.9	−1.39079
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13688.6	−0.567321
$836$	0	0
$837$	113596.	4.69110
$838$	0	0
$839$	26568.0	1.09324	0.546621	−	0.837380i	$-0.315914\pi$
$839$	26568.0	1.09324	0.546621	+	0.837380i	$0.315914\pi$
$840$	0	0
$841$	−24182.8	−0.991545
$842$	0	0
$843$	18626.6	0.761012
$844$	0	0
$845$	20415.0	0.831119
$846$	0	0
$847$	815.500	0.0330825
$848$	0	0
$849$	−53561.5	−2.16517
$850$	0	0
$851$	−4446.45	−0.179110
$852$	0	0
$853$	−4180.66	−0.167811	−0.0839057	−	0.996474i	$-0.526739\pi$
$853$	−4180.66	−0.167811	−0.0839057	+	0.996474i	$0.526739\pi$
$854$	0	0
$855$	68642.8	2.74565
$856$	0	0
$857$	29405.3	1.17207	0.586036	−	0.810285i	$-0.300688\pi$
$857$	29405.3	1.17207	0.586036	+	0.810285i	$0.300688\pi$
$858$	0	0
$859$	−43008.6	−1.70830	−0.854152	−	0.520023i	$-0.825923\pi$
$859$	−43008.6	−1.70830	−0.854152	+	0.520023i	$0.825923\pi$
$860$	0	0
$861$	3689.05	0.146019
$862$	0	0
$863$	−983.940	−0.0388108	−0.0194054	−	0.999812i	$-0.506177\pi$
$863$	−983.940	−0.0388108	−0.0194054	+	0.999812i	$0.506177\pi$
$864$	0	0
$865$	14790.1	0.581361
$866$	0	0
$867$	0	0
$868$	0	0
$869$	30825.3	1.20331
$870$	0	0
$871$	−59705.9	−2.32268
$872$	0	0
$873$	98117.1	3.80385
$874$	0	0
$875$	1585.05	0.0612392
$876$	0	0
$877$	22500.6	0.866354	0.433177	−	0.901309i	$-0.357392\pi$
$877$	22500.6	0.866354	0.433177	+	0.901309i	$0.357392\pi$
$878$	0	0
$879$	−14838.7	−0.569393
$880$	0	0
$881$	−21344.9	−0.816264	−0.408132	−	0.912923i	$-0.633820\pi$
$881$	−21344.9	−0.816264	−0.408132	+	0.912923i	$0.633820\pi$
$882$	0	0
$883$	−30289.9	−1.15440	−0.577200	−	0.816603i	$-0.695855\pi$
$883$	−30289.9	−1.15440	−0.577200	+	0.816603i	$0.695855\pi$
$884$	0	0
$885$	−22935.4	−0.871148
$886$	0	0
$887$	−40896.8	−1.54812	−0.774059	−	0.633113i	$-0.781777\pi$
$887$	−40896.8	−1.54812	−0.774059	+	0.633113i	$0.781777\pi$
$888$	0	0
$889$	−532.063	−0.0200729
$890$	0	0
$891$	−67092.5	−2.52265
$892$	0	0
$893$	−5494.73	−0.205906
$894$	0	0
$895$	38650.4	1.44351
$896$	0	0
$897$	−132844.	−4.94486
$898$	0	0
$899$	−3403.63	−0.126271
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−4868.19	−0.179406
$904$	0	0
$905$	14387.1	0.528447
$906$	0	0
$907$	−23305.7	−0.853201	−0.426601	−	0.904440i	$-0.640289\pi$
$907$	−23305.7	−0.853201	−0.426601	+	0.904440i	$0.640289\pi$
$908$	0	0
$909$	98073.9	3.57855
$910$	0	0
$911$	−33763.6	−1.22792	−0.613962	−	0.789335i	$-0.710425\pi$
$911$	−33763.6	−1.22792	−0.613962	+	0.789335i	$0.710425\pi$
$912$	0	0
$913$	−21685.7	−0.786081
$914$	0	0
$915$	7181.74	0.259476
$916$	0	0
$917$	1322.69	0.0476325
$918$	0	0
$919$	−49650.7	−1.78218	−0.891092	−	0.453823i	$-0.850060\pi$
$919$	−49650.7	−1.78218	−0.891092	+	0.453823i	$0.850060\pi$
$920$	0	0
$921$	−50965.3	−1.82341
$922$	0	0
$923$	−40835.5	−1.45625
$924$	0	0
$925$	425.363	0.0151198
$926$	0	0
$927$	16976.9	0.601503
$928$	0	0
$929$	32294.5	1.14052	0.570262	−	0.821463i	$-0.306842\pi$
$929$	32294.5	1.14052	0.570262	+	0.821463i	$0.306842\pi$
$930$	0	0
$931$	−30642.2	−1.07869
$932$	0	0
$933$	7450.22	0.261425
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−50602.5	−1.76426	−0.882129	−	0.471007i	$-0.843891\pi$
$937$	−50602.5	−1.76426	−0.882129	+	0.471007i	$0.843891\pi$
$938$	0	0
$939$	94066.7	3.26917
$940$	0	0
$941$	−29741.8	−1.03035	−0.515173	−	0.857086i	$-0.672272\pi$
$941$	−29741.8	−1.03035	−0.515173	+	0.857086i	$0.672272\pi$
$942$	0	0
$943$	−69851.2	−2.41216
$944$	0	0
$945$	−5257.20	−0.180970
$946$	0	0
$947$	−439.893	−0.0150946	−0.00754731	−	0.999972i	$-0.502402\pi$
$947$	−439.893	−0.0150946	−0.00754731	+	0.999972i	$0.502402\pi$
$948$	0	0
$949$	26828.2	0.917681
$950$	0	0
$951$	−75756.7	−2.58315
$952$	0	0
$953$	9394.04	0.319310	0.159655	−	0.987173i	$-0.448962\pi$
$953$	9394.04	0.319310	0.159655	+	0.987173i	$0.448962\pi$
$954$	0	0
$955$	3935.91	0.133364
$956$	0	0
$957$	3446.98	0.116432
$958$	0	0
$959$	2469.50	0.0831537
$960$	0	0
$961$	26391.0	0.885873
$962$	0	0
$963$	−42538.5	−1.42345
$964$	0	0
$965$	−5369.23	−0.179110
$966$	0	0
$967$	40131.3	1.33458	0.667288	−	0.744799i	$-0.267455\pi$
$967$	40131.3	1.33458	0.667288	+	0.744799i	$0.267455\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	45569.5	1.50607	0.753035	−	0.657980i	$-0.228589\pi$
$971$	45569.5	1.50607	0.753035	+	0.657980i	$0.228589\pi$
$972$	0	0
$973$	1628.97	0.0536715
$974$	0	0
$975$	12708.3	0.417428
$976$	0	0
$977$	−26661.1	−0.873046	−0.436523	−	0.899693i	$-0.643790\pi$
$977$	−26661.1	−0.873046	−0.436523	+	0.899693i	$0.643790\pi$
$978$	0	0
$979$	649.272	0.0211959
$980$	0	0
$981$	107068.	3.48461
$982$	0	0
$983$	34911.7	1.13277	0.566383	−	0.824142i	$-0.308342\pi$
$983$	34911.7	1.13277	0.566383	+	0.824142i	$0.308342\pi$
$984$	0	0
$985$	31720.8	1.02610
$986$	0	0
$987$	659.753	0.0212768
$988$	0	0
$989$	92177.8	2.96369
$990$	0	0
$991$	6897.42	0.221094	0.110547	−	0.993871i	$-0.464740\pi$
$991$	6897.42	0.221094	0.110547	+	0.993871i	$0.464740\pi$
$992$	0	0
$993$	−43638.0	−1.39457
$994$	0	0
$995$	29.8694	0.000951683 0
$996$	0	0
$997$	40166.3	1.27591	0.637954	−	0.770075i	$-0.279781\pi$
$997$	40166.3	1.27591	0.637954	+	0.770075i	$0.279781\pi$
$998$	0	0
$999$	−10457.1	−0.331179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.i.1.6	yes	6
17.16	even	2		2312.4.a.g.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.4.a.g.1.1	✓	6	17.16	even	2
2312.4.a.i.1.6	yes	6	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)

\(f(q)\)	\(=\)	\(q+10.0775 q^{3} +10.2716 q^{5} -1.06795 q^{7} +74.5565 q^{9} +O(q^{10})\)	\(q+10.0775 q^{3} +10.2716 q^{5} -1.06795 q^{7} +74.5565 q^{9} -23.8200 q^{11} -64.6878 q^{13} +103.512 q^{15} +89.6338 q^{19} -10.7623 q^{21} +203.782 q^{23} -19.4945 q^{25} +479.252 q^{27} -14.3597 q^{29} +237.028 q^{31} -240.046 q^{33} -10.9696 q^{35} -21.8196 q^{37} -651.893 q^{39} -342.774 q^{41} +452.336 q^{43} +765.814 q^{45} -61.3020 q^{47} -341.859 q^{49} +268.750 q^{53} -244.669 q^{55} +903.287 q^{57} -221.572 q^{59} +69.3806 q^{61} -79.6230 q^{63} -664.447 q^{65} +922.984 q^{67} +2053.62 q^{69} +631.271 q^{71} -414.733 q^{73} -196.456 q^{75} +25.4387 q^{77} -1294.10 q^{79} +2816.65 q^{81} +910.400 q^{83} -144.710 q^{87} -27.2575 q^{89} +69.0837 q^{91} +2388.65 q^{93} +920.681 q^{95} +1316.01 q^{97} -1775.93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{3} + 13 q^{5} + q^{7} + 63 q^{9}+O(q^{10}) \) 6 * q + q^3 + 13 * q^5 + q^7 + 63 * q^9	\( 6 q + q^{3} + 13 q^{5} + q^{7} + 63 q^{9} - 8 q^{11} - 33 q^{13} + 68 q^{15} + 75 q^{19} + 32 q^{21} + 155 q^{23} - 89 q^{25} + 322 q^{27} + 9 q^{29} + 374 q^{31} - 351 q^{33} + 13 q^{35} + 118 q^{37} - 369 q^{39} - 60 q^{41} + 230 q^{43} + 478 q^{45} + 276 q^{47} - 317 q^{49} + 569 q^{53} - 97 q^{55} + 769 q^{57} - 784 q^{59} - 28 q^{61} - 878 q^{63} + 828 q^{65} + 238 q^{67} + 1722 q^{69} + 276 q^{71} + 460 q^{73} + 1098 q^{75} + 323 q^{77} - 1812 q^{79} + 2970 q^{81} - 87 q^{83} - 423 q^{87} + 176 q^{89} - 480 q^{91} - 1011 q^{93} + 1246 q^{95} - 27 q^{97} + 1579 q^{99}+O(q^{100}) \) 6 * q + q^3 + 13 * q^5 + q^7 + 63 * q^9 - 8 * q^11 - 33 * q^13 + 68 * q^15 + 75 * q^19 + 32 * q^21 + 155 * q^23 - 89 * q^25 + 322 * q^27 + 9 * q^29 + 374 * q^31 - 351 * q^33 + 13 * q^35 + 118 * q^37 - 369 * q^39 - 60 * q^41 + 230 * q^43 + 478 * q^45 + 276 * q^47 - 317 * q^49 + 569 * q^53 - 97 * q^55 + 769 * q^57 - 784 * q^59 - 28 * q^61 - 878 * q^63 + 828 * q^65 + 238 * q^67 + 1722 * q^69 + 276 * q^71 + 460 * q^73 + 1098 * q^75 + 323 * q^77 - 1812 * q^79 + 2970 * q^81 - 87 * q^83 - 423 * q^87 + 176 * q^89 - 480 * q^91 - 1011 * q^93 + 1246 * q^95 - 27 * q^97 + 1579 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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