LMFDB - Embedded newform 1216.4.a.bg.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.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:	$71.7463225670$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664 $ x^7 - 3x^6 - 121x^5 + 402x^4 + 4234x^3 - 14542x^2 - 40996x + 141664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-8.42108$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.54092 q^{3} -13.1578 q^{5} -19.9667 q^{7} +45.9472 q^{9} +O(q^{10})$	$q+8.54092 q^{3} -13.1578 q^{5} -19.9667 q^{7} +45.9472 q^{9} -25.7995 q^{11} +81.5195 q^{13} -112.379 q^{15} +115.470 q^{17} -19.0000 q^{19} -170.534 q^{21} -167.379 q^{23} +48.1264 q^{25} +161.827 q^{27} -251.521 q^{29} +189.395 q^{31} -220.351 q^{33} +262.717 q^{35} -88.3617 q^{37} +696.251 q^{39} -125.916 q^{41} -459.601 q^{43} -604.562 q^{45} -509.199 q^{47} +55.6703 q^{49} +986.219 q^{51} +604.464 q^{53} +339.463 q^{55} -162.277 q^{57} -173.618 q^{59} +271.164 q^{61} -917.416 q^{63} -1072.61 q^{65} +397.563 q^{67} -1429.57 q^{69} -762.891 q^{71} +108.403 q^{73} +411.044 q^{75} +515.131 q^{77} -230.707 q^{79} +141.573 q^{81} -412.152 q^{83} -1519.32 q^{85} -2148.22 q^{87} -1278.14 q^{89} -1627.68 q^{91} +1617.61 q^{93} +249.997 q^{95} -1333.48 q^{97} -1185.41 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} + 17 q^{5} - 42 q^{7} + 86 q^{9}+O(q^{10}) $ 7 * q + 3 * q^3 + 17 * q^5 - 42 * q^7 + 86 * q^9	$ 7 q + 3 q^{3} + 17 q^{5} - 42 q^{7} + 86 q^{9} - 33 q^{11} + 35 q^{13} - 120 q^{15} + 66 q^{17} - 133 q^{19} - 33 q^{21} - 389 q^{23} + 44 q^{25} - 39 q^{27} - 233 q^{29} - 158 q^{31} - 206 q^{33} + 123 q^{35} + 436 q^{37} - 807 q^{39} - 94 q^{41} + 645 q^{43} - 103 q^{45} - 1451 q^{47} + 93 q^{49} + 1741 q^{51} - 3 q^{53} - 1971 q^{55} - 57 q^{57} + 297 q^{59} - 93 q^{61} - 2999 q^{63} - 788 q^{65} + 1641 q^{67} - 945 q^{69} - 2392 q^{71} + 324 q^{73} + 1909 q^{75} + 711 q^{77} - 2492 q^{79} + 143 q^{81} + 310 q^{83} - 2353 q^{85} - 4795 q^{87} - 440 q^{89} - 107 q^{91} - 900 q^{93} - 323 q^{95} - 532 q^{97} - 1591 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 + 17 * q^5 - 42 * q^7 + 86 * q^9 - 33 * q^11 + 35 * q^13 - 120 * q^15 + 66 * q^17 - 133 * q^19 - 33 * q^21 - 389 * q^23 + 44 * q^25 - 39 * q^27 - 233 * q^29 - 158 * q^31 - 206 * q^33 + 123 * q^35 + 436 * q^37 - 807 * q^39 - 94 * q^41 + 645 * q^43 - 103 * q^45 - 1451 * q^47 + 93 * q^49 + 1741 * q^51 - 3 * q^53 - 1971 * q^55 - 57 * q^57 + 297 * q^59 - 93 * q^61 - 2999 * q^63 - 788 * q^65 + 1641 * q^67 - 945 * q^69 - 2392 * q^71 + 324 * q^73 + 1909 * q^75 + 711 * q^77 - 2492 * q^79 + 143 * q^81 + 310 * q^83 - 2353 * q^85 - 4795 * q^87 - 440 * q^89 - 107 * q^91 - 900 * q^93 - 323 * q^95 - 532 * q^97 - 1591 * 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
$3$	8.54092	1.64370	0.821850	−	0.569704i	$-0.192942\pi$
$3$	8.54092	1.64370	0.821850	+	0.569704i	$0.192942\pi$
$4$	0	0
$5$	−13.1578	−1.17686	−0.588432	−	0.808546i	$-0.700255\pi$
$5$	−13.1578	−1.17686	−0.588432	+	0.808546i	$0.700255\pi$
$6$	0	0
$7$	−19.9667	−1.07810	−0.539051	−	0.842273i	$-0.681217\pi$
$7$	−19.9667	−1.07810	−0.539051	+	0.842273i	$0.681217\pi$
$8$	0	0
$9$	45.9472	1.70175
$10$	0	0
$11$	−25.7995	−0.707166	−0.353583	−	0.935403i	$-0.615037\pi$
$11$	−25.7995	−0.707166	−0.353583	+	0.935403i	$0.615037\pi$
$12$	0	0
$13$	81.5195	1.73919	0.869594	−	0.493768i	$-0.164381\pi$
$13$	81.5195	1.73919	0.869594	+	0.493768i	$0.164381\pi$
$14$	0	0
$15$	−112.379	−1.93441
$16$	0	0
$17$	115.470	1.64739	0.823693	−	0.567035i	$-0.191910\pi$
$17$	115.470	1.64739	0.823693	+	0.567035i	$0.191910\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−170.534	−1.77208
$22$	0	0
$23$	−167.379	−1.51743	−0.758715	−	0.651422i	$-0.774173\pi$
$23$	−167.379	−1.51743	−0.758715	+	0.651422i	$0.774173\pi$
$24$	0	0
$25$	48.1264	0.385011
$26$	0	0
$27$	161.827	1.15347
$28$	0	0
$29$	−251.521	−1.61056	−0.805280	−	0.592895i	$-0.797985\pi$
$29$	−251.521	−1.61056	−0.805280	+	0.592895i	$0.797985\pi$
$30$	0	0
$31$	189.395	1.09730	0.548651	−	0.836051i	$-0.315142\pi$
$31$	189.395	1.09730	0.548651	+	0.836051i	$0.315142\pi$
$32$	0	0
$33$	−220.351	−1.16237
$34$	0	0
$35$	262.717	1.26878
$36$	0	0
$37$	−88.3617	−0.392610	−0.196305	−	0.980543i	$-0.562894\pi$
$37$	−88.3617	−0.392610	−0.196305	+	0.980543i	$0.562894\pi$
$38$	0	0
$39$	696.251	2.85870
$40$	0	0
$41$	−125.916	−0.479629	−0.239815	−	0.970819i	$-0.577087\pi$
$41$	−125.916	−0.479629	−0.239815	+	0.970819i	$0.577087\pi$
$42$	0	0
$43$	−459.601	−1.62996	−0.814982	−	0.579486i	$-0.803253\pi$
$43$	−459.601	−1.62996	−0.814982	+	0.579486i	$0.803253\pi$
$44$	0	0
$45$	−604.562	−2.00273
$46$	0	0
$47$	−509.199	−1.58030	−0.790152	−	0.612911i	$-0.789999\pi$
$47$	−509.199	−1.58030	−0.790152	+	0.612911i	$0.789999\pi$
$48$	0	0
$49$	55.6703	0.162304
$50$	0	0
$51$	986.219	2.70781
$52$	0	0
$53$	604.464	1.56660	0.783298	−	0.621647i	$-0.213536\pi$
$53$	604.464	1.56660	0.783298	+	0.621647i	$0.213536\pi$
$54$	0	0
$55$	339.463	0.832239
$56$	0	0
$57$	−162.277	−0.377091
$58$	0	0
$59$	−173.618	−0.383105	−0.191552	−	0.981482i	$-0.561352\pi$
$59$	−173.618	−0.383105	−0.191552	+	0.981482i	$0.561352\pi$
$60$	0	0
$61$	271.164	0.569164	0.284582	−	0.958652i	$-0.408145\pi$
$61$	271.164	0.569164	0.284582	+	0.958652i	$0.408145\pi$
$62$	0	0
$63$	−917.416	−1.83466
$64$	0	0
$65$	−1072.61	−2.04679
$66$	0	0
$67$	397.563	0.724925	0.362463	−	0.931998i	$-0.381936\pi$
$67$	397.563	0.724925	0.362463	+	0.931998i	$0.381936\pi$
$68$	0	0
$69$	−1429.57	−2.49420
$70$	0	0
$71$	−762.891	−1.27519	−0.637595	−	0.770371i	$-0.720071\pi$
$71$	−762.891	−1.27519	−0.637595	+	0.770371i	$0.720071\pi$
$72$	0	0
$73$	108.403	0.173804	0.0869018	−	0.996217i	$-0.472303\pi$
$73$	108.403	0.173804	0.0869018	+	0.996217i	$0.472303\pi$
$74$	0	0
$75$	411.044	0.632843
$76$	0	0
$77$	515.131	0.762397
$78$	0	0
$79$	−230.707	−0.328565	−0.164282	−	0.986413i	$-0.552531\pi$
$79$	−230.707	−0.328565	−0.164282	+	0.986413i	$0.552531\pi$
$80$	0	0
$81$	141.573	0.194202
$82$	0	0
$83$	−412.152	−0.545055	−0.272527	−	0.962148i	$-0.587860\pi$
$83$	−412.152	−0.545055	−0.272527	+	0.962148i	$0.587860\pi$
$84$	0	0
$85$	−1519.32	−1.93875
$86$	0	0
$87$	−2148.22	−2.64728
$88$	0	0
$89$	−1278.14	−1.52228	−0.761138	−	0.648590i	$-0.775359\pi$
$89$	−1278.14	−1.52228	−0.761138	+	0.648590i	$0.775359\pi$
$90$	0	0
$91$	−1627.68	−1.87502
$92$	0	0
$93$	1617.61	1.80364
$94$	0	0
$95$	249.997	0.269991
$96$	0	0
$97$	−1333.48	−1.39582	−0.697910	−	0.716186i	$-0.745886\pi$
$97$	−1333.48	−1.39582	−0.697910	+	0.716186i	$0.745886\pi$
$98$	0	0
$99$	−1185.41	−1.20342
$100$	0	0
$101$	−627.099	−0.617809	−0.308905	−	0.951093i	$-0.599962\pi$
$101$	−627.099	−0.617809	−0.308905	+	0.951093i	$0.599962\pi$
$102$	0	0
$103$	402.636	0.385174	0.192587	−	0.981280i	$-0.438312\pi$
$103$	402.636	0.385174	0.192587	+	0.981280i	$0.438312\pi$
$104$	0	0
$105$	2243.85	2.08549
$106$	0	0
$107$	665.703	0.601457	0.300728	−	0.953710i	$-0.402770\pi$
$107$	665.703	0.601457	0.300728	+	0.953710i	$0.402770\pi$
$108$	0	0
$109$	349.282	0.306928	0.153464	−	0.988154i	$-0.450957\pi$
$109$	349.282	0.306928	0.153464	+	0.988154i	$0.450957\pi$
$110$	0	0
$111$	−754.690	−0.645333
$112$	0	0
$113$	−994.312	−0.827761	−0.413880	−	0.910331i	$-0.635827\pi$
$113$	−994.312	−0.827761	−0.413880	+	0.910331i	$0.635827\pi$
$114$	0	0
$115$	2202.33	1.78581
$116$	0	0
$117$	3745.60	2.95966
$118$	0	0
$119$	−2305.56	−1.77605
$120$	0	0
$121$	−665.388	−0.499916
$122$	0	0
$123$	−1075.44	−0.788366
$124$	0	0
$125$	1011.48	0.723759
$126$	0	0
$127$	−2669.61	−1.86527	−0.932635	−	0.360822i	$-0.882496\pi$
$127$	−2669.61	−1.86527	−0.932635	+	0.360822i	$0.882496\pi$
$128$	0	0
$129$	−3925.41	−2.67917
$130$	0	0
$131$	126.439	0.0843287	0.0421644	−	0.999111i	$-0.486575\pi$
$131$	126.439	0.0843287	0.0421644	+	0.999111i	$0.486575\pi$
$132$	0	0
$133$	379.368	0.247334
$134$	0	0
$135$	−2129.28	−1.35747
$136$	0	0
$137$	847.326	0.528408	0.264204	−	0.964467i	$-0.414891\pi$
$137$	847.326	0.528408	0.264204	+	0.964467i	$0.414891\pi$
$138$	0	0
$139$	226.691	0.138329	0.0691643	−	0.997605i	$-0.477967\pi$
$139$	226.691	0.138329	0.0691643	+	0.997605i	$0.477967\pi$
$140$	0	0
$141$	−4349.03	−2.59755
$142$	0	0
$143$	−2103.16	−1.22990
$144$	0	0
$145$	3309.45	1.89541
$146$	0	0
$147$	475.476	0.266779
$148$	0	0
$149$	−1028.82	−0.565668	−0.282834	−	0.959169i	$-0.591275\pi$
$149$	−1028.82	−0.565668	−0.282834	+	0.959169i	$0.591275\pi$
$150$	0	0
$151$	−383.908	−0.206901	−0.103450	−	0.994635i	$-0.532988\pi$
$151$	−383.908	−0.206901	−0.103450	+	0.994635i	$0.532988\pi$
$152$	0	0
$153$	5305.53	2.80344
$154$	0	0
$155$	−2492.01	−1.29138
$156$	0	0
$157$	−341.502	−0.173597	−0.0867987	−	0.996226i	$-0.527664\pi$
$157$	−341.502	−0.173597	−0.0867987	+	0.996226i	$0.527664\pi$
$158$	0	0
$159$	5162.68	2.57501
$160$	0	0
$161$	3342.01	1.63594
$162$	0	0
$163$	79.6805	0.0382887	0.0191443	−	0.999817i	$-0.493906\pi$
$163$	79.6805	0.0382887	0.0191443	+	0.999817i	$0.493906\pi$
$164$	0	0
$165$	2899.32	1.36795
$166$	0	0
$167$	−406.653	−0.188430	−0.0942149	−	0.995552i	$-0.530034\pi$
$167$	−406.653	−0.188430	−0.0942149	+	0.995552i	$0.530034\pi$
$168$	0	0
$169$	4448.43	2.02477
$170$	0	0
$171$	−872.998	−0.390408
$172$	0	0
$173$	2246.16	0.987123	0.493562	−	0.869711i	$-0.335695\pi$
$173$	2246.16	0.987123	0.493562	+	0.869711i	$0.335695\pi$
$174$	0	0
$175$	−960.927	−0.415081
$176$	0	0
$177$	−1482.86	−0.629709
$178$	0	0
$179$	−2832.67	−1.18281	−0.591406	−	0.806374i	$-0.701427\pi$
$179$	−2832.67	−1.18281	−0.591406	+	0.806374i	$0.701427\pi$
$180$	0	0
$181$	960.875	0.394593	0.197296	−	0.980344i	$-0.436784\pi$
$181$	960.875	0.394593	0.197296	+	0.980344i	$0.436784\pi$
$182$	0	0
$183$	2315.99	0.935535
$184$	0	0
$185$	1162.64	0.462049
$186$	0	0
$187$	−2979.06	−1.16498
$188$	0	0
$189$	−3231.15	−1.24355
$190$	0	0
$191$	332.728	0.126049	0.0630245	−	0.998012i	$-0.479925\pi$
$191$	332.728	0.126049	0.0630245	+	0.998012i	$0.479925\pi$
$192$	0	0
$193$	−1500.32	−0.559563	−0.279782	−	0.960064i	$-0.590262\pi$
$193$	−1500.32	−0.559563	−0.279782	+	0.960064i	$0.590262\pi$
$194$	0	0
$195$	−9161.10	−3.36431
$196$	0	0
$197$	−610.730	−0.220877	−0.110438	−	0.993883i	$-0.535225\pi$
$197$	−610.730	−0.220877	−0.110438	+	0.993883i	$0.535225\pi$
$198$	0	0
$199$	4387.80	1.56303	0.781514	−	0.623887i	$-0.214448\pi$
$199$	4387.80	1.56303	0.781514	+	0.623887i	$0.214448\pi$
$200$	0	0
$201$	3395.55	1.19156
$202$	0	0
$203$	5022.05	1.73635
$204$	0	0
$205$	1656.77	0.564459
$206$	0	0
$207$	−7690.60	−2.58229
$208$	0	0
$209$	490.190	0.162235
$210$	0	0
$211$	5027.18	1.64021	0.820107	−	0.572210i	$-0.193914\pi$
$211$	5027.18	1.64021	0.820107	+	0.572210i	$0.193914\pi$
$212$	0	0
$213$	−6515.79	−2.09603
$214$	0	0
$215$	6047.31	1.91825
$216$	0	0
$217$	−3781.60	−1.18300
$218$	0	0
$219$	925.864	0.285681
$220$	0	0
$221$	9413.05	2.86512
$222$	0	0
$223$	3886.89	1.16720	0.583600	−	0.812041i	$-0.301644\pi$
$223$	3886.89	1.16720	0.583600	+	0.812041i	$0.301644\pi$
$224$	0	0
$225$	2211.28	0.655193
$226$	0	0
$227$	−2705.06	−0.790931	−0.395466	−	0.918481i	$-0.629417\pi$
$227$	−2705.06	−0.790931	−0.395466	+	0.918481i	$0.629417\pi$
$228$	0	0
$229$	2471.34	0.713148	0.356574	−	0.934267i	$-0.383945\pi$
$229$	2471.34	0.713148	0.356574	+	0.934267i	$0.383945\pi$
$230$	0	0
$231$	4399.69	1.25315
$232$	0	0
$233$	4029.19	1.13288	0.566439	−	0.824103i	$-0.308321\pi$
$233$	4029.19	1.13288	0.566439	+	0.824103i	$0.308321\pi$
$234$	0	0
$235$	6699.91	1.85981
$236$	0	0
$237$	−1970.45	−0.540062
$238$	0	0
$239$	−2329.50	−0.630472	−0.315236	−	0.949013i	$-0.602084\pi$
$239$	−2329.50	−0.630472	−0.315236	+	0.949013i	$0.602084\pi$
$240$	0	0
$241$	−4687.77	−1.25297	−0.626485	−	0.779434i	$-0.715507\pi$
$241$	−4687.77	−1.25297	−0.626485	+	0.779434i	$0.715507\pi$
$242$	0	0
$243$	−3160.16	−0.834256
$244$	0	0
$245$	−732.496	−0.191010
$246$	0	0
$247$	−1548.87	−0.398997
$248$	0	0
$249$	−3520.15	−0.895907
$250$	0	0
$251$	3723.91	0.936458	0.468229	−	0.883607i	$-0.344892\pi$
$251$	3723.91	0.936458	0.468229	+	0.883607i	$0.344892\pi$
$252$	0	0
$253$	4318.28	1.07308
$254$	0	0
$255$	−12976.4	−3.18673
$256$	0	0
$257$	4926.21	1.19567	0.597837	−	0.801618i	$-0.296027\pi$
$257$	4926.21	1.19567	0.597837	+	0.801618i	$0.296027\pi$
$258$	0	0
$259$	1764.29	0.423274
$260$	0	0
$261$	−11556.7	−2.74077
$262$	0	0
$263$	−2177.76	−0.510595	−0.255298	−	0.966863i	$-0.582173\pi$
$263$	−2177.76	−0.510595	−0.255298	+	0.966863i	$0.582173\pi$
$264$	0	0
$265$	−7953.39	−1.84367
$266$	0	0
$267$	−10916.5	−2.50217
$268$	0	0
$269$	5760.70	1.30571	0.652855	−	0.757483i	$-0.273571\pi$
$269$	5760.70	1.30571	0.652855	+	0.757483i	$0.273571\pi$
$270$	0	0
$271$	−528.665	−0.118502	−0.0592512	−	0.998243i	$-0.518871\pi$
$271$	−528.665	−0.118502	−0.0592512	+	0.998243i	$0.518871\pi$
$272$	0	0
$273$	−13901.9	−3.08197
$274$	0	0
$275$	−1241.64	−0.272267
$276$	0	0
$277$	−6110.07	−1.32534	−0.662669	−	0.748912i	$-0.730576\pi$
$277$	−6110.07	−1.32534	−0.662669	+	0.748912i	$0.730576\pi$
$278$	0	0
$279$	8702.18	1.86733
$280$	0	0
$281$	−4673.80	−0.992225	−0.496113	−	0.868258i	$-0.665240\pi$
$281$	−4673.80	−0.992225	−0.496113	+	0.868258i	$0.665240\pi$
$282$	0	0
$283$	−3933.39	−0.826204	−0.413102	−	0.910685i	$-0.635555\pi$
$283$	−3933.39	−0.826204	−0.413102	+	0.910685i	$0.635555\pi$
$284$	0	0
$285$	2135.21	0.443785
$286$	0	0
$287$	2514.13	0.517089
$288$	0	0
$289$	8420.31	1.71388
$290$	0	0
$291$	−11389.1	−2.29431
$292$	0	0
$293$	−843.734	−0.168230	−0.0841151	−	0.996456i	$-0.526806\pi$
$293$	−843.734	−0.168230	−0.0841151	+	0.996456i	$0.526806\pi$
$294$	0	0
$295$	2284.43	0.450863
$296$	0	0
$297$	−4175.04	−0.815692
$298$	0	0
$299$	−13644.6	−2.63910
$300$	0	0
$301$	9176.73	1.75727
$302$	0	0
$303$	−5356.00	−1.01549
$304$	0	0
$305$	−3567.91	−0.669829
$306$	0	0
$307$	−2128.89	−0.395772	−0.197886	−	0.980225i	$-0.563408\pi$
$307$	−2128.89	−0.395772	−0.197886	+	0.980225i	$0.563408\pi$
$308$	0	0
$309$	3438.88	0.633111
$310$	0	0
$311$	−8236.36	−1.50174	−0.750870	−	0.660450i	$-0.770366\pi$
$311$	−8236.36	−1.50174	−0.750870	+	0.660450i	$0.770366\pi$
$312$	0	0
$313$	−7429.56	−1.34167	−0.670836	−	0.741606i	$-0.734065\pi$
$313$	−7429.56	−1.34167	−0.670836	+	0.741606i	$0.734065\pi$
$314$	0	0
$315$	12071.1	2.15915
$316$	0	0
$317$	6298.50	1.11596	0.557980	−	0.829855i	$-0.311577\pi$
$317$	6298.50	1.11596	0.557980	+	0.829855i	$0.311577\pi$
$318$	0	0
$319$	6489.10	1.13893
$320$	0	0
$321$	5685.71	0.988615
$322$	0	0
$323$	−2193.93	−0.377936
$324$	0	0
$325$	3923.24	0.669607
$326$	0	0
$327$	2983.19	0.504497
$328$	0	0
$329$	10167.0	1.70373
$330$	0	0
$331$	−2575.03	−0.427602	−0.213801	−	0.976877i	$-0.568584\pi$
$331$	−2575.03	−0.427602	−0.213801	+	0.976877i	$0.568584\pi$
$332$	0	0
$333$	−4059.98	−0.668124
$334$	0	0
$335$	−5231.03	−0.853139
$336$	0	0
$337$	−740.602	−0.119713	−0.0598563	−	0.998207i	$-0.519064\pi$
$337$	−740.602	−0.119713	−0.0598563	+	0.998207i	$0.519064\pi$
$338$	0	0
$339$	−8492.33	−1.36059
$340$	0	0
$341$	−4886.29	−0.775975
$342$	0	0
$343$	5737.03	0.903122
$344$	0	0
$345$	18809.9	2.93534
$346$	0	0
$347$	11162.7	1.72693	0.863466	−	0.504406i	$-0.168289\pi$
$347$	11162.7	1.72693	0.863466	+	0.504406i	$0.168289\pi$
$348$	0	0
$349$	−2445.65	−0.375108	−0.187554	−	0.982254i	$-0.560056\pi$
$349$	−2445.65	−0.375108	−0.187554	+	0.982254i	$0.560056\pi$
$350$	0	0
$351$	13192.0	2.00609
$352$	0	0
$353$	−12241.3	−1.84572	−0.922858	−	0.385140i	$-0.874153\pi$
$353$	−12241.3	−1.84572	−0.922858	+	0.385140i	$0.874153\pi$
$354$	0	0
$355$	10037.9	1.50073
$356$	0	0
$357$	−19691.6	−2.91930
$358$	0	0
$359$	−996.059	−0.146435	−0.0732173	−	0.997316i	$-0.523327\pi$
$359$	−996.059	−0.146435	−0.0732173	+	0.997316i	$0.523327\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−5683.02	−0.821712
$364$	0	0
$365$	−1426.35	−0.204543
$366$	0	0
$367$	4660.35	0.662856	0.331428	−	0.943480i	$-0.392470\pi$
$367$	4660.35	0.662856	0.331428	+	0.943480i	$0.392470\pi$
$368$	0	0
$369$	−5785.50	−0.816209
$370$	0	0
$371$	−12069.2	−1.68895
$372$	0	0
$373$	2812.80	0.390459	0.195229	−	0.980758i	$-0.437455\pi$
$373$	2812.80	0.390459	0.195229	+	0.980758i	$0.437455\pi$
$374$	0	0
$375$	8639.00	1.18964
$376$	0	0
$377$	−20503.8	−2.80107
$378$	0	0
$379$	8251.88	1.11839	0.559196	−	0.829035i	$-0.311110\pi$
$379$	8251.88	1.11839	0.559196	+	0.829035i	$0.311110\pi$
$380$	0	0
$381$	−22800.9	−3.06594
$382$	0	0
$383$	2675.35	0.356930	0.178465	−	0.983946i	$-0.442887\pi$
$383$	2675.35	0.356930	0.178465	+	0.983946i	$0.442887\pi$
$384$	0	0
$385$	−6777.96	−0.897239
$386$	0	0
$387$	−21117.4	−2.77379
$388$	0	0
$389$	4063.09	0.529581	0.264791	−	0.964306i	$-0.414697\pi$
$389$	4063.09	0.529581	0.264791	+	0.964306i	$0.414697\pi$
$390$	0	0
$391$	−19327.2	−2.49980
$392$	0	0
$393$	1079.91	0.138611
$394$	0	0
$395$	3035.59	0.386676
$396$	0	0
$397$	8158.05	1.03134	0.515668	−	0.856788i	$-0.327544\pi$
$397$	8158.05	1.03134	0.515668	+	0.856788i	$0.327544\pi$
$398$	0	0
$399$	3240.15	0.406542
$400$	0	0
$401$	7411.97	0.923032	0.461516	−	0.887132i	$-0.347306\pi$
$401$	7411.97	0.923032	0.461516	+	0.887132i	$0.347306\pi$
$402$	0	0
$403$	15439.4	1.90841
$404$	0	0
$405$	−1862.79	−0.228550
$406$	0	0
$407$	2279.68	0.277641
$408$	0	0
$409$	7084.46	0.856489	0.428245	−	0.903663i	$-0.359132\pi$
$409$	7084.46	0.856489	0.428245	+	0.903663i	$0.359132\pi$
$410$	0	0
$411$	7236.94	0.868545
$412$	0	0
$413$	3466.59	0.413026
$414$	0	0
$415$	5422.99	0.641456
$416$	0	0
$417$	1936.15	0.227371
$418$	0	0
$419$	9315.37	1.08612	0.543062	−	0.839693i	$-0.317265\pi$
$419$	9315.37	1.08612	0.543062	+	0.839693i	$0.317265\pi$
$420$	0	0
$421$	1577.50	0.182619	0.0913093	−	0.995823i	$-0.470895\pi$
$421$	1577.50	0.182619	0.0913093	+	0.995823i	$0.470895\pi$
$422$	0	0
$423$	−23396.3	−2.68928
$424$	0	0
$425$	5557.15	0.634262
$426$	0	0
$427$	−5414.26	−0.613617
$428$	0	0
$429$	−17962.9	−2.02158
$430$	0	0
$431$	1798.83	0.201036	0.100518	−	0.994935i	$-0.467950\pi$
$431$	1798.83	0.201036	0.100518	+	0.994935i	$0.467950\pi$
$432$	0	0
$433$	10553.6	1.17130	0.585649	−	0.810565i	$-0.300840\pi$
$433$	10553.6	1.17130	0.585649	+	0.810565i	$0.300840\pi$
$434$	0	0
$435$	28265.7	3.11549
$436$	0	0
$437$	3180.20	0.348122
$438$	0	0
$439$	8814.06	0.958250	0.479125	−	0.877747i	$-0.340954\pi$
$439$	8814.06	0.958250	0.479125	+	0.877747i	$0.340954\pi$
$440$	0	0
$441$	2557.90	0.276201
$442$	0	0
$443$	−3323.91	−0.356487	−0.178243	−	0.983986i	$-0.557041\pi$
$443$	−3323.91	−0.356487	−0.178243	+	0.983986i	$0.557041\pi$
$444$	0	0
$445$	16817.4	1.79151
$446$	0	0
$447$	−8787.10	−0.929788
$448$	0	0
$449$	7631.27	0.802098	0.401049	−	0.916057i	$-0.368646\pi$
$449$	7631.27	0.802098	0.401049	+	0.916057i	$0.368646\pi$
$450$	0	0
$451$	3248.57	0.339178
$452$	0	0
$453$	−3278.93	−0.340083
$454$	0	0
$455$	21416.6	2.20665
$456$	0	0
$457$	105.869	0.0108366	0.00541830	−	0.999985i	$-0.498275\pi$
$457$	105.869	0.0108366	0.00541830	+	0.999985i	$0.498275\pi$
$458$	0	0
$459$	18686.1	1.90020
$460$	0	0
$461$	−15158.0	−1.53140	−0.765701	−	0.643197i	$-0.777608\pi$
$461$	−15158.0	−1.53140	−0.765701	+	0.643197i	$0.777608\pi$
$462$	0	0
$463$	−14630.3	−1.46853	−0.734264	−	0.678864i	$-0.762473\pi$
$463$	−14630.3	−1.46853	−0.734264	+	0.678864i	$0.762473\pi$
$464$	0	0
$465$	−21284.1	−2.12264
$466$	0	0
$467$	−12655.4	−1.25401	−0.627004	−	0.779016i	$-0.715719\pi$
$467$	−12655.4	−1.25401	−0.627004	+	0.779016i	$0.715719\pi$
$468$	0	0
$469$	−7938.02	−0.781543
$470$	0	0
$471$	−2916.74	−0.285342
$472$	0	0
$473$	11857.5	1.15266
$474$	0	0
$475$	−914.402	−0.0883276
$476$	0	0
$477$	27773.5	2.66595
$478$	0	0
$479$	−3774.83	−0.360075	−0.180038	−	0.983660i	$-0.557622\pi$
$479$	−3774.83	−0.360075	−0.180038	+	0.983660i	$0.557622\pi$
$480$	0	0
$481$	−7203.20	−0.682823
$482$	0	0
$483$	28543.8	2.68900
$484$	0	0
$485$	17545.6	1.64269
$486$	0	0
$487$	−15396.5	−1.43261	−0.716305	−	0.697788i	$-0.754168\pi$
$487$	−15396.5	−1.43261	−0.716305	+	0.697788i	$0.754168\pi$
$488$	0	0
$489$	680.544	0.0629351
$490$	0	0
$491$	−6166.15	−0.566750	−0.283375	−	0.959009i	$-0.591454\pi$
$491$	−6166.15	−0.566750	−0.283375	+	0.959009i	$0.591454\pi$
$492$	0	0
$493$	−29043.1	−2.65322
$494$	0	0
$495$	15597.4	1.41626
$496$	0	0
$497$	15232.4	1.37479
$498$	0	0
$499$	−4448.36	−0.399070	−0.199535	−	0.979891i	$-0.563943\pi$
$499$	−4448.36	−0.399070	−0.199535	+	0.979891i	$0.563943\pi$
$500$	0	0
$501$	−3473.19	−0.309722
$502$	0	0
$503$	15601.5	1.38297	0.691487	−	0.722389i	$-0.256956\pi$
$503$	15601.5	1.38297	0.691487	+	0.722389i	$0.256956\pi$
$504$	0	0
$505$	8251.22	0.727078
$506$	0	0
$507$	37993.7	3.32812
$508$	0	0
$509$	2056.72	0.179101	0.0895505	−	0.995982i	$-0.471457\pi$
$509$	2056.72	0.179101	0.0895505	+	0.995982i	$0.471457\pi$
$510$	0	0
$511$	−2164.46	−0.187378
$512$	0	0
$513$	−3074.71	−0.264623
$514$	0	0
$515$	−5297.79	−0.453298
$516$	0	0
$517$	13137.1	1.11754
$518$	0	0
$519$	19184.3	1.62253
$520$	0	0
$521$	6105.27	0.513391	0.256695	−	0.966492i	$-0.417366\pi$
$521$	6105.27	0.513391	0.256695	+	0.966492i	$0.417366\pi$
$522$	0	0
$523$	15629.1	1.30672	0.653358	−	0.757050i	$-0.273360\pi$
$523$	15629.1	1.30672	0.653358	+	0.757050i	$0.273360\pi$
$524$	0	0
$525$	−8207.20	−0.682269
$526$	0	0
$527$	21869.4	1.80768
$528$	0	0
$529$	15848.7	1.30260
$530$	0	0
$531$	−7977.28	−0.651948
$532$	0	0
$533$	−10264.6	−0.834165
$534$	0	0
$535$	−8759.15	−0.707834
$536$	0	0
$537$	−24193.6	−1.94419
$538$	0	0
$539$	−1436.26	−0.114776
$540$	0	0
$541$	719.686	0.0571935	0.0285968	−	0.999591i	$-0.490896\pi$
$541$	719.686	0.0571935	0.0285968	+	0.999591i	$0.490896\pi$
$542$	0	0
$543$	8206.75	0.648592
$544$	0	0
$545$	−4595.76	−0.361213
$546$	0	0
$547$	−1043.70	−0.0815820	−0.0407910	−	0.999168i	$-0.512988\pi$
$547$	−1043.70	−0.0815820	−0.0407910	+	0.999168i	$0.512988\pi$
$548$	0	0
$549$	12459.2	0.968575
$550$	0	0
$551$	4778.89	0.369488
$552$	0	0
$553$	4606.47	0.354226
$554$	0	0
$555$	9930.02	0.759470
$556$	0	0
$557$	8833.00	0.671932	0.335966	−	0.941874i	$-0.390937\pi$
$557$	8833.00	0.671932	0.335966	+	0.941874i	$0.390937\pi$
$558$	0	0
$559$	−37466.4	−2.83482
$560$	0	0
$561$	−25443.9	−1.91487
$562$	0	0
$563$	−2352.97	−0.176138	−0.0880691	−	0.996114i	$-0.528070\pi$
$563$	−2352.97	−0.176138	−0.0880691	+	0.996114i	$0.528070\pi$
$564$	0	0
$565$	13082.9	0.974163
$566$	0	0
$567$	−2826.76	−0.209370
$568$	0	0
$569$	19654.5	1.44809	0.724043	−	0.689755i	$-0.242282\pi$
$569$	19654.5	1.44809	0.724043	+	0.689755i	$0.242282\pi$
$570$	0	0
$571$	2346.30	0.171961	0.0859803	−	0.996297i	$-0.472598\pi$
$571$	2346.30	0.171961	0.0859803	+	0.996297i	$0.472598\pi$
$572$	0	0
$573$	2841.80	0.207187
$574$	0	0
$575$	−8055.34	−0.584228
$576$	0	0
$577$	21946.1	1.58341	0.791706	−	0.610903i	$-0.209193\pi$
$577$	21946.1	1.58341	0.791706	+	0.610903i	$0.209193\pi$
$578$	0	0
$579$	−12814.1	−0.919754
$580$	0	0
$581$	8229.33	0.587625
$582$	0	0
$583$	−15594.9	−1.10784
$584$	0	0
$585$	−49283.6	−3.48312
$586$	0	0
$587$	−17129.9	−1.20448	−0.602238	−	0.798316i	$-0.705724\pi$
$587$	−17129.9	−1.20448	−0.602238	+	0.798316i	$0.705724\pi$
$588$	0	0
$589$	−3598.51	−0.251738
$590$	0	0
$591$	−5216.19	−0.363055
$592$	0	0
$593$	−24042.0	−1.66490	−0.832449	−	0.554101i	$-0.813062\pi$
$593$	−24042.0	−1.66490	−0.832449	+	0.554101i	$0.813062\pi$
$594$	0	0
$595$	30336.0	2.09017
$596$	0	0
$597$	37475.8	2.56915
$598$	0	0
$599$	−24147.3	−1.64713	−0.823565	−	0.567222i	$-0.808018\pi$
$599$	−24147.3	−1.64713	−0.823565	+	0.567222i	$0.808018\pi$
$600$	0	0
$601$	−7106.93	−0.482359	−0.241179	−	0.970481i	$-0.577534\pi$
$601$	−7106.93	−0.482359	−0.241179	+	0.970481i	$0.577534\pi$
$602$	0	0
$603$	18266.9	1.23364
$604$	0	0
$605$	8755.01	0.588333
$606$	0	0
$607$	18561.0	1.24113	0.620567	−	0.784153i	$-0.286902\pi$
$607$	18561.0	1.24113	0.620567	+	0.784153i	$0.286902\pi$
$608$	0	0
$609$	42892.9	2.85403
$610$	0	0
$611$	−41509.7	−2.74845
$612$	0	0
$613$	1703.46	0.112238	0.0561192	−	0.998424i	$-0.482127\pi$
$613$	1703.46	0.112238	0.0561192	+	0.998424i	$0.482127\pi$
$614$	0	0
$615$	14150.4	0.927801
$616$	0	0
$617$	−2299.61	−0.150047	−0.0750235	−	0.997182i	$-0.523903\pi$
$617$	−2299.61	−0.150047	−0.0750235	+	0.997182i	$0.523903\pi$
$618$	0	0
$619$	27773.9	1.80343	0.901717	−	0.432327i	$-0.142307\pi$
$619$	27773.9	1.80343	0.901717	+	0.432327i	$0.142307\pi$
$620$	0	0
$621$	−27086.4	−1.75030
$622$	0	0
$623$	25520.3	1.64117
$624$	0	0
$625$	−19324.6	−1.23678
$626$	0	0
$627$	4186.67	0.266666
$628$	0	0
$629$	−10203.1	−0.646781
$630$	0	0
$631$	5059.28	0.319187	0.159593	−	0.987183i	$-0.448982\pi$
$631$	5059.28	0.319187	0.159593	+	0.987183i	$0.448982\pi$
$632$	0	0
$633$	42936.7	2.69602
$634$	0	0
$635$	35126.0	2.19517
$636$	0	0
$637$	4538.22	0.282277
$638$	0	0
$639$	−35052.8	−2.17006
$640$	0	0
$641$	−4955.44	−0.305348	−0.152674	−	0.988277i	$-0.548788\pi$
$641$	−4955.44	−0.305348	−0.152674	+	0.988277i	$0.548788\pi$
$642$	0	0
$643$	−31892.3	−1.95600	−0.978000	−	0.208604i	$-0.933108\pi$
$643$	−31892.3	−1.95600	−0.978000	+	0.208604i	$0.933108\pi$
$644$	0	0
$645$	51649.6	3.15303
$646$	0	0
$647$	−15213.1	−0.924402	−0.462201	−	0.886775i	$-0.652940\pi$
$647$	−15213.1	−0.924402	−0.462201	+	0.886775i	$0.652940\pi$
$648$	0	0
$649$	4479.26	0.270919
$650$	0	0
$651$	−32298.3	−1.94450
$652$	0	0
$653$	14131.4	0.846867	0.423434	−	0.905927i	$-0.360825\pi$
$653$	14131.4	0.846867	0.423434	+	0.905927i	$0.360825\pi$
$654$	0	0
$655$	−1663.66	−0.0992435
$656$	0	0
$657$	4980.84	0.295770
$658$	0	0
$659$	10846.1	0.641131	0.320565	−	0.947226i	$-0.396127\pi$
$659$	10846.1	0.641131	0.320565	+	0.947226i	$0.396127\pi$
$660$	0	0
$661$	−17001.0	−1.00039	−0.500197	−	0.865911i	$-0.666739\pi$
$661$	−17001.0	−1.00039	−0.500197	+	0.865911i	$0.666739\pi$
$662$	0	0
$663$	80396.1	4.70939
$664$	0	0
$665$	−4991.63	−0.291078
$666$	0	0
$667$	42099.3	2.44391
$668$	0	0
$669$	33197.6	1.91853
$670$	0	0
$671$	−6995.89	−0.402494
$672$	0	0
$673$	9819.41	0.562423	0.281211	−	0.959646i	$-0.409264\pi$
$673$	9819.41	0.562423	0.281211	+	0.959646i	$0.409264\pi$
$674$	0	0
$675$	7788.14	0.444097
$676$	0	0
$677$	−15034.0	−0.853479	−0.426739	−	0.904375i	$-0.640338\pi$
$677$	−15034.0	−0.853479	−0.426739	+	0.904375i	$0.640338\pi$
$678$	0	0
$679$	26625.3	1.50484
$680$	0	0
$681$	−23103.7	−1.30005
$682$	0	0
$683$	6423.83	0.359884	0.179942	−	0.983677i	$-0.442409\pi$
$683$	6423.83	0.359884	0.179942	+	0.983677i	$0.442409\pi$
$684$	0	0
$685$	−11148.9	−0.621865
$686$	0	0
$687$	21107.5	1.17220
$688$	0	0
$689$	49275.6	2.72460
$690$	0	0
$691$	−14553.0	−0.801191	−0.400596	−	0.916255i	$-0.631197\pi$
$691$	−14553.0	−0.801191	−0.400596	+	0.916255i	$0.631197\pi$
$692$	0	0
$693$	23668.8	1.29741
$694$	0	0
$695$	−2982.74	−0.162794
$696$	0	0
$697$	−14539.5	−0.790135
$698$	0	0
$699$	34412.9	1.86211
$700$	0	0
$701$	8367.86	0.450856	0.225428	−	0.974260i	$-0.427622\pi$
$701$	8367.86	0.450856	0.225428	+	0.974260i	$0.427622\pi$
$702$	0	0
$703$	1678.87	0.0900710
$704$	0	0
$705$	57223.4	3.05696
$706$	0	0
$707$	12521.1	0.666061
$708$	0	0
$709$	23313.7	1.23493	0.617465	−	0.786598i	$-0.288160\pi$
$709$	23313.7	1.23493	0.617465	+	0.786598i	$0.288160\pi$
$710$	0	0
$711$	−10600.4	−0.559135
$712$	0	0
$713$	−31700.7	−1.66508
$714$	0	0
$715$	27672.8	1.44742
$716$	0	0
$717$	−19896.1	−1.03631
$718$	0	0
$719$	−32486.8	−1.68505	−0.842526	−	0.538655i	$-0.818933\pi$
$719$	−32486.8	−1.68505	−0.842526	+	0.538655i	$0.818933\pi$
$720$	0	0
$721$	−8039.33	−0.415257
$722$	0	0
$723$	−40037.8	−2.05951
$724$	0	0
$725$	−12104.8	−0.620084
$726$	0	0
$727$	−12999.0	−0.663143	−0.331571	−	0.943430i	$-0.607579\pi$
$727$	−12999.0	−0.663143	−0.331571	+	0.943430i	$0.607579\pi$
$728$	0	0
$729$	−30813.1	−1.56547
$730$	0	0
$731$	−53070.1	−2.68518
$732$	0	0
$733$	−29580.1	−1.49054	−0.745269	−	0.666763i	$-0.767679\pi$
$733$	−29580.1	−1.49054	−0.745269	+	0.666763i	$0.767679\pi$
$734$	0	0
$735$	−6256.19	−0.313963
$736$	0	0
$737$	−10256.9	−0.512643
$738$	0	0
$739$	13305.9	0.662337	0.331168	−	0.943572i	$-0.392557\pi$
$739$	13305.9	0.662337	0.331168	+	0.943572i	$0.392557\pi$
$740$	0	0
$741$	−13228.8	−0.655831
$742$	0	0
$743$	11226.2	0.554308	0.277154	−	0.960825i	$-0.410609\pi$
$743$	11226.2	0.554308	0.277154	+	0.960825i	$0.410609\pi$
$744$	0	0
$745$	13537.0	0.665715
$746$	0	0
$747$	−18937.2	−0.927547
$748$	0	0
$749$	−13291.9	−0.648432
$750$	0	0
$751$	1364.07	0.0662792	0.0331396	−	0.999451i	$-0.489449\pi$
$751$	1364.07	0.0662792	0.0331396	+	0.999451i	$0.489449\pi$
$752$	0	0
$753$	31805.6	1.53926
$754$	0	0
$755$	5051.37	0.243494
$756$	0	0
$757$	−7593.24	−0.364572	−0.182286	−	0.983246i	$-0.558350\pi$
$757$	−7593.24	−0.364572	−0.182286	+	0.983246i	$0.558350\pi$
$758$	0	0
$759$	36882.1	1.76381
$760$	0	0
$761$	−22828.1	−1.08741	−0.543704	−	0.839277i	$-0.682979\pi$
$761$	−22828.1	−1.08741	−0.543704	+	0.839277i	$0.682979\pi$
$762$	0	0
$763$	−6974.01	−0.330899
$764$	0	0
$765$	−69808.8	−3.29927
$766$	0	0
$767$	−14153.3	−0.666291
$768$	0	0
$769$	19685.9	0.923137	0.461568	−	0.887105i	$-0.347287\pi$
$769$	19685.9	0.923137	0.461568	+	0.887105i	$0.347287\pi$
$770$	0	0
$771$	42074.3	1.96533
$772$	0	0
$773$	−14169.7	−0.659311	−0.329656	−	0.944101i	$-0.606933\pi$
$773$	−14169.7	−0.659311	−0.329656	+	0.944101i	$0.606933\pi$
$774$	0	0
$775$	9114.90	0.422474
$776$	0	0
$777$	15068.7	0.695735
$778$	0	0
$779$	2392.41	0.110034
$780$	0	0
$781$	19682.2	0.901772
$782$	0	0
$783$	−40702.8	−1.85773
$784$	0	0
$785$	4493.39	0.204301
$786$	0	0
$787$	−34545.9	−1.56471	−0.782355	−	0.622832i	$-0.785982\pi$
$787$	−34545.9	−1.56471	−0.782355	+	0.622832i	$0.785982\pi$
$788$	0	0
$789$	−18600.1	−0.839265
$790$	0	0
$791$	19853.2	0.892411
$792$	0	0
$793$	22105.2	0.989883
$794$	0	0
$795$	−67929.2	−3.03044
$796$	0	0
$797$	2551.15	0.113383	0.0566917	−	0.998392i	$-0.481945\pi$
$797$	2551.15	0.113383	0.0566917	+	0.998392i	$0.481945\pi$
$798$	0	0
$799$	−58797.2	−2.60337
$800$	0	0
$801$	−58727.0	−2.59053
$802$	0	0
$803$	−2796.75	−0.122908
$804$	0	0
$805$	−43973.3	−1.92529
$806$	0	0
$807$	49201.6	2.14620
$808$	0	0
$809$	19573.7	0.850648	0.425324	−	0.905041i	$-0.360160\pi$
$809$	19573.7	0.850648	0.425324	+	0.905041i	$0.360160\pi$
$810$	0	0
$811$	805.214	0.0348642	0.0174321	−	0.999848i	$-0.494451\pi$
$811$	805.214	0.0348642	0.0174321	+	0.999848i	$0.494451\pi$
$812$	0	0
$813$	−4515.29	−0.194782
$814$	0	0
$815$	−1048.42	−0.0450606
$816$	0	0
$817$	8732.42	0.373940
$818$	0	0
$819$	−74787.3	−3.19082
$820$	0	0
$821$	−37470.4	−1.59285	−0.796423	−	0.604740i	$-0.793277\pi$
$821$	−37470.4	−1.59285	−0.796423	+	0.604740i	$0.793277\pi$
$822$	0	0
$823$	−10931.7	−0.463008	−0.231504	−	0.972834i	$-0.574365\pi$
$823$	−10931.7	−0.463008	−0.231504	+	0.972834i	$0.574365\pi$
$824$	0	0
$825$	−10604.7	−0.447525
$826$	0	0
$827$	3622.95	0.152337	0.0761683	−	0.997095i	$-0.475731\pi$
$827$	3622.95	0.152337	0.0761683	+	0.997095i	$0.475731\pi$
$828$	0	0
$829$	−17057.0	−0.714615	−0.357307	−	0.933987i	$-0.616305\pi$
$829$	−17057.0	−0.714615	−0.357307	+	0.933987i	$0.616305\pi$
$830$	0	0
$831$	−52185.6	−2.17846
$832$	0	0
$833$	6428.25	0.267378
$834$	0	0
$835$	5350.64	0.221756
$836$	0	0
$837$	30649.2	1.26570
$838$	0	0
$839$	−43129.6	−1.77473	−0.887366	−	0.461066i	$-0.847467\pi$
$839$	−43129.6	−1.77473	−0.887366	+	0.461066i	$0.847467\pi$
$840$	0	0
$841$	38873.7	1.59390
$842$	0	0
$843$	−39918.5	−1.63092
$844$	0	0
$845$	−58531.3	−2.38289
$846$	0	0
$847$	13285.6	0.538960
$848$	0	0
$849$	−33594.7	−1.35803
$850$	0	0
$851$	14789.9	0.595759
$852$	0	0
$853$	−8168.69	−0.327890	−0.163945	−	0.986469i	$-0.552422\pi$
$853$	−8168.69	−0.327890	−0.163945	+	0.986469i	$0.552422\pi$
$854$	0	0
$855$	11486.7	0.459458
$856$	0	0
$857$	−39062.0	−1.55698	−0.778491	−	0.627656i	$-0.784014\pi$
$857$	−39062.0	−1.55698	−0.778491	+	0.627656i	$0.784014\pi$
$858$	0	0
$859$	483.093	0.0191885	0.00959425	−	0.999954i	$-0.496946\pi$
$859$	483.093	0.0191885	0.00959425	+	0.999954i	$0.496946\pi$
$860$	0	0
$861$	21473.0	0.849939
$862$	0	0
$863$	−23713.4	−0.935357	−0.467678	−	0.883899i	$-0.654909\pi$
$863$	−23713.4	−0.935357	−0.467678	+	0.883899i	$0.654909\pi$
$864$	0	0
$865$	−29554.4	−1.16171
$866$	0	0
$867$	71917.2	2.81711
$868$	0	0
$869$	5952.13	0.232350
$870$	0	0
$871$	32409.1	1.26078
$872$	0	0
$873$	−61269.8	−2.37534
$874$	0	0
$875$	−20196.0	−0.780286
$876$	0	0
$877$	22434.1	0.863793	0.431897	−	0.901923i	$-0.357845\pi$
$877$	22434.1	0.863793	0.431897	+	0.901923i	$0.357845\pi$
$878$	0	0
$879$	−7206.26	−0.276520
$880$	0	0
$881$	−42303.5	−1.61776	−0.808878	−	0.587977i	$-0.799925\pi$
$881$	−42303.5	−1.61776	−0.808878	+	0.587977i	$0.799925\pi$
$882$	0	0
$883$	20452.4	0.779475	0.389738	−	0.920926i	$-0.372566\pi$
$883$	20452.4	0.779475	0.389738	+	0.920926i	$0.372566\pi$
$884$	0	0
$885$	19511.1	0.741083
$886$	0	0
$887$	−23455.6	−0.887895	−0.443947	−	0.896053i	$-0.646422\pi$
$887$	−23455.6	−0.887895	−0.443947	+	0.896053i	$0.646422\pi$
$888$	0	0
$889$	53303.3	2.01095
$890$	0	0
$891$	−3652.51	−0.137333
$892$	0	0
$893$	9674.78	0.362547
$894$	0	0
$895$	37271.5	1.39201
$896$	0	0
$897$	−116538.	−4.33788
$898$	0	0
$899$	−47636.8	−1.76727
$900$	0	0
$901$	69797.5	2.58079
$902$	0	0
$903$	78377.7	2.88842
$904$	0	0
$905$	−12643.0	−0.464382
$906$	0	0
$907$	19565.8	0.716288	0.358144	−	0.933666i	$-0.383410\pi$
$907$	19565.8	0.716288	0.358144	+	0.933666i	$0.383410\pi$
$908$	0	0
$909$	−28813.5	−1.05136
$910$	0	0
$911$	50384.3	1.83239	0.916194	−	0.400735i	$-0.131245\pi$
$911$	50384.3	1.83239	0.916194	+	0.400735i	$0.131245\pi$
$912$	0	0
$913$	10633.3	0.385444
$914$	0	0
$915$	−30473.2	−1.10100
$916$	0	0
$917$	−2524.58	−0.0909150
$918$	0	0
$919$	16715.6	0.599996	0.299998	−	0.953940i	$-0.403014\pi$
$919$	16715.6	0.599996	0.299998	+	0.953940i	$0.403014\pi$
$920$	0	0
$921$	−18182.6	−0.650530
$922$	0	0
$923$	−62190.5	−2.21780
$924$	0	0
$925$	−4252.53	−0.151159
$926$	0	0
$927$	18500.0	0.655470
$928$	0	0
$929$	54313.0	1.91814	0.959069	−	0.283172i	$-0.0913869\pi$
$929$	54313.0	1.91814	0.959069	+	0.283172i	$0.0913869\pi$
$930$	0	0
$931$	−1057.74	−0.0372351
$932$	0	0
$933$	−70346.1	−2.46841
$934$	0	0
$935$	39197.8	1.37102
$936$	0	0
$937$	−39884.6	−1.39058	−0.695290	−	0.718729i	$-0.744724\pi$
$937$	−39884.6	−1.39058	−0.695290	+	0.718729i	$0.744724\pi$
$938$	0	0
$939$	−63455.2	−2.20531
$940$	0	0
$941$	−31712.3	−1.09861	−0.549305	−	0.835622i	$-0.685107\pi$
$941$	−31712.3	−1.09861	−0.549305	+	0.835622i	$0.685107\pi$
$942$	0	0
$943$	21075.7	0.727804
$944$	0	0
$945$	42514.7	1.46350
$946$	0	0
$947$	−35314.1	−1.21178	−0.605889	−	0.795549i	$-0.707182\pi$
$947$	−35314.1	−1.21178	−0.605889	+	0.795549i	$0.707182\pi$
$948$	0	0
$949$	8836.99	0.302277
$950$	0	0
$951$	53795.0	1.83430
$952$	0	0
$953$	10741.7	0.365120	0.182560	−	0.983195i	$-0.441562\pi$
$953$	10741.7	0.365120	0.182560	+	0.983195i	$0.441562\pi$
$954$	0	0
$955$	−4377.95	−0.148343
$956$	0	0
$957$	55422.8	1.87207
$958$	0	0
$959$	−16918.3	−0.569678
$960$	0	0
$961$	6079.51	0.204072
$962$	0	0
$963$	30587.2	1.02353
$964$	0	0
$965$	19740.9	0.658530
$966$	0	0
$967$	−57806.9	−1.92238	−0.961191	−	0.275883i	$-0.911030\pi$
$967$	−57806.9	−1.92238	−0.961191	+	0.275883i	$0.911030\pi$
$968$	0	0
$969$	−18738.2	−0.621214
$970$	0	0
$971$	42959.8	1.41982	0.709910	−	0.704292i	$-0.248736\pi$
$971$	42959.8	1.41982	0.709910	+	0.704292i	$0.248736\pi$
$972$	0	0
$973$	−4526.28	−0.149132
$974$	0	0
$975$	33508.1	1.10063
$976$	0	0
$977$	54273.7	1.77725	0.888623	−	0.458638i	$-0.151663\pi$
$977$	54273.7	1.77725	0.888623	+	0.458638i	$0.151663\pi$
$978$	0	0
$979$	32975.3	1.07650
$980$	0	0
$981$	16048.5	0.522314
$982$	0	0
$983$	45578.5	1.47887	0.739435	−	0.673228i	$-0.235093\pi$
$983$	45578.5	1.47887	0.739435	+	0.673228i	$0.235093\pi$
$984$	0	0
$985$	8035.84	0.259942
$986$	0	0
$987$	86835.8	2.80042
$988$	0	0
$989$	76927.5	2.47336
$990$	0	0
$991$	37734.1	1.20955	0.604774	−	0.796397i	$-0.293263\pi$
$991$	37734.1	1.20955	0.604774	+	0.796397i	$0.293263\pi$
$992$	0	0
$993$	−21993.1	−0.702849
$994$	0	0
$995$	−57733.5	−1.83947
$996$	0	0
$997$	51233.9	1.62748	0.813738	−	0.581232i	$-0.197429\pi$
$997$	51233.9	1.62748	0.813738	+	0.581232i	$0.197429\pi$
$998$	0	0
$999$	−14299.3	−0.452862

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.bg.1.7		7
4.3	odd	2		1216.4.a.bf.1.1		7
8.3	odd	2		608.4.a.k.1.7	yes	7
8.5	even	2		608.4.a.j.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.j.1.1	✓	7	8.5	even	2
608.4.a.k.1.7	yes	7	8.3	odd	2
1216.4.a.bf.1.1		7	4.3	odd	2
1216.4.a.bg.1.7		7	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q+8.54092 q^{3} -13.1578 q^{5} -19.9667 q^{7} +45.9472 q^{9} +O(q^{10})\)	\(q+8.54092 q^{3} -13.1578 q^{5} -19.9667 q^{7} +45.9472 q^{9} -25.7995 q^{11} +81.5195 q^{13} -112.379 q^{15} +115.470 q^{17} -19.0000 q^{19} -170.534 q^{21} -167.379 q^{23} +48.1264 q^{25} +161.827 q^{27} -251.521 q^{29} +189.395 q^{31} -220.351 q^{33} +262.717 q^{35} -88.3617 q^{37} +696.251 q^{39} -125.916 q^{41} -459.601 q^{43} -604.562 q^{45} -509.199 q^{47} +55.6703 q^{49} +986.219 q^{51} +604.464 q^{53} +339.463 q^{55} -162.277 q^{57} -173.618 q^{59} +271.164 q^{61} -917.416 q^{63} -1072.61 q^{65} +397.563 q^{67} -1429.57 q^{69} -762.891 q^{71} +108.403 q^{73} +411.044 q^{75} +515.131 q^{77} -230.707 q^{79} +141.573 q^{81} -412.152 q^{83} -1519.32 q^{85} -2148.22 q^{87} -1278.14 q^{89} -1627.68 q^{91} +1617.61 q^{93} +249.997 q^{95} -1333.48 q^{97} -1185.41 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} + 17 q^{5} - 42 q^{7} + 86 q^{9}+O(q^{10}) \) 7 * q + 3 * q^3 + 17 * q^5 - 42 * q^7 + 86 * q^9	\( 7 q + 3 q^{3} + 17 q^{5} - 42 q^{7} + 86 q^{9} - 33 q^{11} + 35 q^{13} - 120 q^{15} + 66 q^{17} - 133 q^{19} - 33 q^{21} - 389 q^{23} + 44 q^{25} - 39 q^{27} - 233 q^{29} - 158 q^{31} - 206 q^{33} + 123 q^{35} + 436 q^{37} - 807 q^{39} - 94 q^{41} + 645 q^{43} - 103 q^{45} - 1451 q^{47} + 93 q^{49} + 1741 q^{51} - 3 q^{53} - 1971 q^{55} - 57 q^{57} + 297 q^{59} - 93 q^{61} - 2999 q^{63} - 788 q^{65} + 1641 q^{67} - 945 q^{69} - 2392 q^{71} + 324 q^{73} + 1909 q^{75} + 711 q^{77} - 2492 q^{79} + 143 q^{81} + 310 q^{83} - 2353 q^{85} - 4795 q^{87} - 440 q^{89} - 107 q^{91} - 900 q^{93} - 323 q^{95} - 532 q^{97} - 1591 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 + 17 * q^5 - 42 * q^7 + 86 * q^9 - 33 * q^11 + 35 * q^13 - 120 * q^15 + 66 * q^17 - 133 * q^19 - 33 * q^21 - 389 * q^23 + 44 * q^25 - 39 * q^27 - 233 * q^29 - 158 * q^31 - 206 * q^33 + 123 * q^35 + 436 * q^37 - 807 * q^39 - 94 * q^41 + 645 * q^43 - 103 * q^45 - 1451 * q^47 + 93 * q^49 + 1741 * q^51 - 3 * q^53 - 1971 * q^55 - 57 * q^57 + 297 * q^59 - 93 * q^61 - 2999 * q^63 - 788 * q^65 + 1641 * q^67 - 945 * q^69 - 2392 * q^71 + 324 * q^73 + 1909 * q^75 + 711 * q^77 - 2492 * q^79 + 143 * q^81 + 310 * q^83 - 2353 * q^85 - 4795 * q^87 - 440 * q^89 - 107 * q^91 - 900 * q^93 - 323 * q^95 - 532 * q^97 - 1591 * q^99

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