LMFDB - Embedded newform 8550.2.a.cp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.993.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 3 $ x^3 - x^2 - 6*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 950)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.25342$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.0778929 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +0.0778929 q^{7} +1.00000 q^{8} +4.50684 q^{11} +5.33131 q^{13} +0.0778929 q^{14} +1.00000 q^{16} +7.33131 q^{17} +1.00000 q^{19} +4.50684 q^{22} +3.40920 q^{23} +5.33131 q^{26} +0.0778929 q^{28} +1.33131 q^{29} -2.50684 q^{31} +1.00000 q^{32} +7.33131 q^{34} +5.50684 q^{37} +1.00000 q^{38} -0.506836 q^{43} +4.50684 q^{44} +3.40920 q^{46} -5.66262 q^{47} -6.99393 q^{49} +5.33131 q^{52} -12.9358 q^{53} +0.0778929 q^{56} +1.33131 q^{58} -7.56499 q^{59} -2.15579 q^{61} -2.50684 q^{62} +1.00000 q^{64} +4.58473 q^{67} +7.33131 q^{68} +10.8579 q^{71} +5.09763 q^{73} +5.50684 q^{74} +1.00000 q^{76} +0.351050 q^{77} +17.0137 q^{79} -13.1695 q^{83} -0.506836 q^{86} +4.50684 q^{88} -15.0137 q^{89} +0.415271 q^{91} +3.40920 q^{92} -5.66262 q^{94} -7.67629 q^{97} -6.99393 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 2 q^{11} + 6 q^{13} - 2 q^{14} + 3 q^{16} + 12 q^{17} + 3 q^{19} - 2 q^{22} - 2 q^{23} + 6 q^{26} - 2 q^{28} - 6 q^{29} + 8 q^{31} + 3 q^{32} + 12 q^{34} + q^{37} + 3 q^{38} + 14 q^{43} - 2 q^{44} - 2 q^{46} + 3 q^{47} + 9 q^{49} + 6 q^{52} - 10 q^{53} - 2 q^{56} - 6 q^{58} - 6 q^{59} - 2 q^{61} + 8 q^{62} + 3 q^{64} - 4 q^{67} + 12 q^{68} + 6 q^{71} + 12 q^{73} + q^{74} + 3 q^{76} - 10 q^{77} + 20 q^{79} - 4 q^{83} + 14 q^{86} - 2 q^{88} - 14 q^{89} + 19 q^{91} - 2 q^{92} + 3 q^{94} + 28 q^{97} + 9 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 2 * q^11 + 6 * q^13 - 2 * q^14 + 3 * q^16 + 12 * q^17 + 3 * q^19 - 2 * q^22 - 2 * q^23 + 6 * q^26 - 2 * q^28 - 6 * q^29 + 8 * q^31 + 3 * q^32 + 12 * q^34 + q^37 + 3 * q^38 + 14 * q^43 - 2 * q^44 - 2 * q^46 + 3 * q^47 + 9 * q^49 + 6 * q^52 - 10 * q^53 - 2 * q^56 - 6 * q^58 - 6 * q^59 - 2 * q^61 + 8 * q^62 + 3 * q^64 - 4 * q^67 + 12 * q^68 + 6 * q^71 + 12 * q^73 + q^74 + 3 * q^76 - 10 * q^77 + 20 * q^79 - 4 * q^83 + 14 * q^86 - 2 * q^88 - 14 * q^89 + 19 * q^91 - 2 * q^92 + 3 * q^94 + 28 * q^97 + 9 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.0778929	0.0294407	0.0147204	−	0.999892i	$-0.495314\pi$
$7$	0.0778929	0.0294407	0.0147204	+	0.999892i	$0.495314\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	4.50684	1.35886	0.679431	−	0.733739i	$-0.262227\pi$
$11$	4.50684	1.35886	0.679431	+	0.733739i	$0.262227\pi$
$12$	0	0
$13$	5.33131	1.47864	0.739320	−	0.673354i	$-0.235147\pi$
$13$	5.33131	1.47864	0.739320	+	0.673354i	$0.235147\pi$
$14$	0.0778929	0.0208177
$15$	0	0
$16$	1.00000	0.250000
$17$	7.33131	1.77810	0.889052	−	0.457806i	$-0.151365\pi$
$17$	7.33131	1.77810	0.889052	+	0.457806i	$0.151365\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	4.50684	0.960861
$23$	3.40920	0.710868	0.355434	−	0.934701i	$-0.384333\pi$
$23$	3.40920	0.710868	0.355434	+	0.934701i	$0.384333\pi$
$24$	0	0
$25$	0	0
$26$	5.33131	1.04556
$27$	0	0
$28$	0.0778929	0.0147204
$29$	1.33131	0.247218	0.123609	−	0.992331i	$-0.460553\pi$
$29$	1.33131	0.247218	0.123609	+	0.992331i	$0.460553\pi$
$30$	0	0
$31$	−2.50684	−0.450241	−0.225121	−	0.974331i	$-0.572278\pi$
$31$	−2.50684	−0.450241	−0.225121	+	0.974331i	$0.572278\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.33131	1.25731
$35$	0	0
$36$	0	0
$37$	5.50684	0.905318	0.452659	−	0.891684i	$-0.350475\pi$
$37$	5.50684	0.905318	0.452659	+	0.891684i	$0.350475\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−0.506836	−0.0772918	−0.0386459	−	0.999253i	$-0.512304\pi$
$43$	−0.506836	−0.0772918	−0.0386459	+	0.999253i	$0.512304\pi$
$44$	4.50684	0.679431
$45$	0	0
$46$	3.40920	0.502660
$47$	−5.66262	−0.825978	−0.412989	−	0.910736i	$-0.635515\pi$
$47$	−5.66262	−0.825978	−0.412989	+	0.910736i	$0.635515\pi$
$48$	0	0
$49$	−6.99393	−0.999133
$50$	0	0
$51$	0	0
$52$	5.33131	0.739320
$53$	−12.9358	−1.77687	−0.888433	−	0.459006i	$-0.848206\pi$
$53$	−12.9358	−1.77687	−0.888433	+	0.459006i	$0.848206\pi$
$54$	0	0
$55$	0	0
$56$	0.0778929	0.0104089
$57$	0	0
$58$	1.33131	0.174810
$59$	−7.56499	−0.984878	−0.492439	−	0.870347i	$-0.663894\pi$
$59$	−7.56499	−0.984878	−0.492439	+	0.870347i	$0.663894\pi$
$60$	0	0
$61$	−2.15579	−0.276020	−0.138010	−	0.990431i	$-0.544071\pi$
$61$	−2.15579	−0.276020	−0.138010	+	0.990431i	$0.544071\pi$
$62$	−2.50684	−0.318369
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	4.58473	0.560114	0.280057	−	0.959983i	$-0.409647\pi$
$67$	4.58473	0.560114	0.280057	+	0.959983i	$0.409647\pi$
$68$	7.33131	0.889052
$69$	0	0
$70$	0	0
$71$	10.8579	1.28859	0.644297	−	0.764775i	$-0.277150\pi$
$71$	10.8579	1.28859	0.644297	+	0.764775i	$0.277150\pi$
$72$	0	0
$73$	5.09763	0.596633	0.298316	−	0.954467i	$-0.403575\pi$
$73$	5.09763	0.596633	0.298316	+	0.954467i	$0.403575\pi$
$74$	5.50684	0.640157
$75$	0	0
$76$	1.00000	0.114708
$77$	0.351050	0.0400059
$78$	0	0
$79$	17.0137	1.91419	0.957094	−	0.289778i	$-0.0935815\pi$
$79$	17.0137	1.91419	0.957094	+	0.289778i	$0.0935815\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.1695	−1.44554	−0.722768	−	0.691091i	$-0.757130\pi$
$83$	−13.1695	−1.44554	−0.722768	+	0.691091i	$0.757130\pi$
$84$	0	0
$85$	0	0
$86$	−0.506836	−0.0546535
$87$	0	0
$88$	4.50684	0.480430
$89$	−15.0137	−1.59145	−0.795723	−	0.605661i	$-0.792909\pi$
$89$	−15.0137	−1.59145	−0.795723	+	0.605661i	$0.792909\pi$
$90$	0	0
$91$	0.415271	0.0435322
$92$	3.40920	0.355434
$93$	0	0
$94$	−5.66262	−0.584055
$95$	0	0
$96$	0	0
$97$	−7.67629	−0.779410	−0.389705	−	0.920940i	$-0.627423\pi$
$97$	−7.67629	−0.779410	−0.389705	+	0.920940i	$0.627423\pi$
$98$	−6.99393	−0.706494
$99$	0	0
$100$	0	0
$101$	4.15579	0.413516	0.206758	−	0.978392i	$-0.433709\pi$
$101$	4.15579	0.413516	0.206758	+	0.978392i	$0.433709\pi$
$102$	0	0
$103$	−2.35105	−0.231656	−0.115828	−	0.993269i	$-0.536952\pi$
$103$	−2.35105	−0.231656	−0.115828	+	0.993269i	$0.536952\pi$
$104$	5.33131	0.522778
$105$	0	0
$106$	−12.9358	−1.25643
$107$	−14.0334	−1.35666	−0.678331	−	0.734757i	$-0.737296\pi$
$107$	−14.0334	−1.35666	−0.678331	+	0.734757i	$0.737296\pi$
$108$	0	0
$109$	−0.0778929	−0.00746078	−0.00373039	−	0.999993i	$-0.501187\pi$
$109$	−0.0778929	−0.00746078	−0.00373039	+	0.999993i	$0.501187\pi$
$110$	0	0
$111$	0	0
$112$	0.0778929	0.00736018
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	1.33131	0.123609
$117$	0	0
$118$	−7.56499	−0.696414
$119$	0.571057	0.0523487
$120$	0	0
$121$	9.31157	0.846506
$122$	−2.15579	−0.195176
$123$	0	0
$124$	−2.50684	−0.225121
$125$	0	0
$126$	0	0
$127$	−17.8321	−1.58234	−0.791171	−	0.611596i	$-0.790528\pi$
$127$	−17.8321	−1.58234	−0.791171	+	0.611596i	$0.790528\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	1.49316	0.130458	0.0652292	−	0.997870i	$-0.479222\pi$
$131$	1.49316	0.130458	0.0652292	+	0.997870i	$0.479222\pi$
$132$	0	0
$133$	0.0778929	0.00675417
$134$	4.58473	0.396060
$135$	0	0
$136$	7.33131	0.628655
$137$	−8.42894	−0.720133	−0.360067	−	0.932927i	$-0.617246\pi$
$137$	−8.42894	−0.720133	−0.360067	+	0.932927i	$0.617246\pi$
$138$	0	0
$139$	8.81841	0.747968	0.373984	−	0.927435i	$-0.377992\pi$
$139$	8.81841	0.747968	0.373984	+	0.927435i	$0.377992\pi$
$140$	0	0
$141$	0	0
$142$	10.8579	0.911174
$143$	24.0273	2.00927
$144$	0	0
$145$	0	0
$146$	5.09763	0.421883
$147$	0	0
$148$	5.50684	0.452659
$149$	−17.6763	−1.44810	−0.724049	−	0.689748i	$-0.757721\pi$
$149$	−17.6763	−1.44810	−0.724049	+	0.689748i	$0.757721\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	0.351050	0.0282884
$155$	0	0
$156$	0	0
$157$	−0.506836	−0.0404499	−0.0202250	−	0.999795i	$-0.506438\pi$
$157$	−0.506836	−0.0404499	−0.0202250	+	0.999795i	$0.506438\pi$
$158$	17.0137	1.35354
$159$	0	0
$160$	0	0
$161$	0.265553	0.0209285
$162$	0	0
$163$	−0.830542	−0.0650531	−0.0325265	−	0.999471i	$-0.510355\pi$
$163$	−0.830542	−0.0650531	−0.0325265	+	0.999471i	$0.510355\pi$
$164$	0	0
$165$	0	0
$166$	−13.1695	−1.02215
$167$	16.5068	1.27734	0.638669	−	0.769482i	$-0.279485\pi$
$167$	16.5068	1.27734	0.638669	+	0.769482i	$0.279485\pi$
$168$	0	0
$169$	15.4229	1.18638
$170$	0	0
$171$	0	0
$172$	−0.506836	−0.0386459
$173$	18.8321	1.43178	0.715888	−	0.698215i	$-0.246022\pi$
$173$	18.8321	1.43178	0.715888	+	0.698215i	$0.246022\pi$
$174$	0	0
$175$	0	0
$176$	4.50684	0.339716
$177$	0	0
$178$	−15.0137	−1.12532
$179$	7.15579	0.534849	0.267424	−	0.963579i	$-0.413827\pi$
$179$	7.15579	0.534849	0.267424	+	0.963579i	$0.413827\pi$
$180$	0	0
$181$	12.5205	0.930642	0.465321	−	0.885142i	$-0.345939\pi$
$181$	12.5205	0.930642	0.465321	+	0.885142i	$0.345939\pi$
$182$	0.415271	0.0307819
$183$	0	0
$184$	3.40920	0.251330
$185$	0	0
$186$	0	0
$187$	33.0410	2.41620
$188$	−5.66262	−0.412989
$189$	0	0
$190$	0	0
$191$	4.90237	0.354723	0.177361	−	0.984146i	$-0.443244\pi$
$191$	4.90237	0.354723	0.177361	+	0.984146i	$0.443244\pi$
$192$	0	0
$193$	−18.1558	−1.30688	−0.653441	−	0.756977i	$-0.726675\pi$
$193$	−18.1558	−1.30688	−0.653441	+	0.756977i	$0.726675\pi$
$194$	−7.67629	−0.551126
$195$	0	0
$196$	−6.99393	−0.499567
$197$	−2.98633	−0.212767	−0.106384	−	0.994325i	$-0.533927\pi$
$197$	−2.98633	−0.212767	−0.106384	+	0.994325i	$0.533927\pi$
$198$	0	0
$199$	−3.06422	−0.217217	−0.108608	−	0.994085i	$-0.534639\pi$
$199$	−3.06422	−0.217217	−0.108608	+	0.994085i	$0.534639\pi$
$200$	0	0
$201$	0	0
$202$	4.15579	0.292400
$203$	0.103700	0.00727829
$204$	0	0
$205$	0	0
$206$	−2.35105	−0.163805
$207$	0	0
$208$	5.33131	0.369660
$209$	4.50684	0.311744
$210$	0	0
$211$	19.2534	1.32546	0.662730	−	0.748858i	$-0.269398\pi$
$211$	19.2534	1.32546	0.662730	+	0.748858i	$0.269398\pi$
$212$	−12.9358	−0.888433
$213$	0	0
$214$	−14.0334	−0.959304
$215$	0	0
$216$	0	0
$217$	−0.195265	−0.0132554
$218$	−0.0778929	−0.00527557
$219$	0	0
$220$	0	0
$221$	39.0855	2.62918
$222$	0	0
$223$	14.5068	0.971450	0.485725	−	0.874112i	$-0.338556\pi$
$223$	14.5068	0.971450	0.485725	+	0.874112i	$0.338556\pi$
$224$	0.0778929	0.00520444
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−21.5984	−1.43354	−0.716768	−	0.697312i	$-0.754379\pi$
$227$	−21.5984	−1.43354	−0.716768	+	0.697312i	$0.754379\pi$
$228$	0	0
$229$	−19.0137	−1.25646	−0.628229	−	0.778028i	$-0.716220\pi$
$229$	−19.0137	−1.25646	−0.628229	+	0.778028i	$0.716220\pi$
$230$	0	0
$231$	0	0
$232$	1.33131	0.0874048
$233$	−6.01367	−0.393969	−0.196984	−	0.980407i	$-0.563115\pi$
$233$	−6.01367	−0.393969	−0.196984	+	0.980407i	$0.563115\pi$
$234$	0	0
$235$	0	0
$236$	−7.56499	−0.492439
$237$	0	0
$238$	0.571057	0.0370161
$239$	−15.4092	−0.996739	−0.498369	−	0.866965i	$-0.666068\pi$
$239$	−15.4092	−0.996739	−0.498369	+	0.866965i	$0.666068\pi$
$240$	0	0
$241$	−4.81841	−0.310381	−0.155190	−	0.987885i	$-0.549599\pi$
$241$	−4.81841	−0.310381	−0.155190	+	0.987885i	$0.549599\pi$
$242$	9.31157	0.598570
$243$	0	0
$244$	−2.15579	−0.138010
$245$	0	0
$246$	0	0
$247$	5.33131	0.339223
$248$	−2.50684	−0.159184
$249$	0	0
$250$	0	0
$251$	−1.52051	−0.0959736	−0.0479868	−	0.998848i	$-0.515281\pi$
$251$	−1.52051	−0.0959736	−0.0479868	+	0.998848i	$0.515281\pi$
$252$	0	0
$253$	15.3647	0.965972
$254$	−17.8321	−1.11888
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.5068	1.02967	0.514834	−	0.857290i	$-0.327854\pi$
$257$	16.5068	1.02967	0.514834	+	0.857290i	$0.327854\pi$
$258$	0	0
$259$	0.428943	0.0266532
$260$	0	0
$261$	0	0
$262$	1.49316	0.0922480
$263$	−18.0273	−1.11161	−0.555807	−	0.831311i	$-0.687591\pi$
$263$	−18.0273	−1.11161	−0.555807	+	0.831311i	$0.687591\pi$
$264$	0	0
$265$	0	0
$266$	0.0778929	0.00477592
$267$	0	0
$268$	4.58473	0.280057
$269$	−20.2089	−1.23216	−0.616080	−	0.787683i	$-0.711280\pi$
$269$	−20.2089	−1.23216	−0.616080	+	0.787683i	$0.711280\pi$
$270$	0	0
$271$	6.08396	0.369574	0.184787	−	0.982779i	$-0.440840\pi$
$271$	6.08396	0.369574	0.184787	+	0.982779i	$0.440840\pi$
$272$	7.33131	0.444526
$273$	0	0
$274$	−8.42894	−0.509211
$275$	0	0
$276$	0	0
$277$	31.3647	1.88452	0.942262	−	0.334877i	$-0.108695\pi$
$277$	31.3647	1.88452	0.942262	+	0.334877i	$0.108695\pi$
$278$	8.81841	0.528893
$279$	0	0
$280$	0	0
$281$	−11.3252	−0.675607	−0.337804	−	0.941217i	$-0.609684\pi$
$281$	−11.3252	−0.675607	−0.337804	+	0.941217i	$0.609684\pi$
$282$	0	0
$283$	−26.1437	−1.55408	−0.777039	−	0.629452i	$-0.783279\pi$
$283$	−26.1437	−1.55408	−0.777039	+	0.629452i	$0.783279\pi$
$284$	10.8579	0.644297
$285$	0	0
$286$	24.0273	1.42077
$287$	0	0
$288$	0	0
$289$	36.7481	2.16165
$290$	0	0
$291$	0	0
$292$	5.09763	0.298316
$293$	1.33131	0.0777760	0.0388880	−	0.999244i	$-0.487618\pi$
$293$	1.33131	0.0777760	0.0388880	+	0.999244i	$0.487618\pi$
$294$	0	0
$295$	0	0
$296$	5.50684	0.320078
$297$	0	0
$298$	−17.6763	−1.02396
$299$	18.1755	1.05112
$300$	0	0
$301$	−0.0394789	−0.00227553
$302$	20.0000	1.15087
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	2.84421	0.162328	0.0811639	−	0.996701i	$-0.474136\pi$
$307$	2.84421	0.162328	0.0811639	+	0.996701i	$0.474136\pi$
$308$	0.351050	0.0200030
$309$	0	0
$310$	0	0
$311$	16.3895	0.929361	0.464681	−	0.885478i	$-0.346169\pi$
$311$	16.3895	0.929361	0.464681	+	0.885478i	$0.346169\pi$
$312$	0	0
$313$	21.0471	1.18965	0.594826	−	0.803855i	$-0.297221\pi$
$313$	21.0471	1.18965	0.594826	+	0.803855i	$0.297221\pi$
$314$	−0.506836	−0.0286024
$315$	0	0
$316$	17.0137	0.957094
$317$	−30.8902	−1.73497	−0.867484	−	0.497465i	$-0.834264\pi$
$317$	−30.8902	−1.73497	−0.867484	+	0.497465i	$0.834264\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	0.265553	0.0147987
$323$	7.33131	0.407925
$324$	0	0
$325$	0	0
$326$	−0.830542	−0.0459995
$327$	0	0
$328$	0	0
$329$	−0.441078	−0.0243174
$330$	0	0
$331$	28.3845	1.56015	0.780076	−	0.625685i	$-0.215181\pi$
$331$	28.3845	1.56015	0.780076	+	0.625685i	$0.215181\pi$
$332$	−13.1695	−0.722768
$333$	0	0
$334$	16.5068	0.903214
$335$	0	0
$336$	0	0
$337$	17.8716	0.973526	0.486763	−	0.873534i	$-0.338178\pi$
$337$	17.8716	0.973526	0.486763	+	0.873534i	$0.338178\pi$
$338$	15.4229	0.838894
$339$	0	0
$340$	0	0
$341$	−11.2979	−0.611816
$342$	0	0
$343$	−1.09003	−0.0588560
$344$	−0.506836	−0.0273268
$345$	0	0
$346$	18.8321	1.01242
$347$	33.0410	1.77373	0.886867	−	0.462024i	$-0.152877\pi$
$347$	33.0410	1.77373	0.886867	+	0.462024i	$0.152877\pi$
$348$	0	0
$349$	−19.7158	−1.05536	−0.527681	−	0.849443i	$-0.676938\pi$
$349$	−19.7158	−1.05536	−0.527681	+	0.849443i	$0.676938\pi$
$350$	0	0
$351$	0	0
$352$	4.50684	0.240215
$353$	−5.90997	−0.314556	−0.157278	−	0.987554i	$-0.550272\pi$
$353$	−5.90997	−0.314556	−0.157278	+	0.987554i	$0.550272\pi$
$354$	0	0
$355$	0	0
$356$	−15.0137	−0.795723
$357$	0	0
$358$	7.15579	0.378195
$359$	7.00760	0.369847	0.184924	−	0.982753i	$-0.440796\pi$
$359$	7.00760	0.369847	0.184924	+	0.982753i	$0.440796\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	12.5205	0.658063
$363$	0	0
$364$	0.415271	0.0217661
$365$	0	0
$366$	0	0
$367$	−17.7158	−0.924756	−0.462378	−	0.886683i	$-0.653004\pi$
$367$	−17.7158	−0.924756	−0.462378	+	0.886683i	$0.653004\pi$
$368$	3.40920	0.177717
$369$	0	0
$370$	0	0
$371$	−1.00760	−0.0523122
$372$	0	0
$373$	−15.4487	−0.799902	−0.399951	−	0.916536i	$-0.630973\pi$
$373$	−15.4487	−0.799902	−0.399951	+	0.916536i	$0.630973\pi$
$374$	33.0410	1.70851
$375$	0	0
$376$	−5.66262	−0.292027
$377$	7.09763	0.365547
$378$	0	0
$379$	34.4563	1.76990	0.884950	−	0.465685i	$-0.154192\pi$
$379$	34.4563	1.76990	0.884950	+	0.465685i	$0.154192\pi$
$380$	0	0
$381$	0	0
$382$	4.90237	0.250827
$383$	−12.0273	−0.614569	−0.307284	−	0.951618i	$-0.599420\pi$
$383$	−12.0273	−0.614569	−0.307284	+	0.951618i	$0.599420\pi$
$384$	0	0
$385$	0	0
$386$	−18.1558	−0.924105
$387$	0	0
$388$	−7.67629	−0.389705
$389$	15.3647	0.779022	0.389511	−	0.921022i	$-0.372644\pi$
$389$	15.3647	0.779022	0.389511	+	0.921022i	$0.372644\pi$
$390$	0	0
$391$	24.9939	1.26400
$392$	−6.99393	−0.353247
$393$	0	0
$394$	−2.98633	−0.150449
$395$	0	0
$396$	0	0
$397$	−1.32524	−0.0665121	−0.0332560	−	0.999447i	$-0.510588\pi$
$397$	−1.32524	−0.0665121	−0.0332560	+	0.999447i	$0.510588\pi$
$398$	−3.06422	−0.153596
$399$	0	0
$400$	0	0
$401$	6.46736	0.322964	0.161482	−	0.986876i	$-0.448373\pi$
$401$	6.46736	0.322964	0.161482	+	0.986876i	$0.448373\pi$
$402$	0	0
$403$	−13.3647	−0.665744
$404$	4.15579	0.206758
$405$	0	0
$406$	0.103700	0.00514653
$407$	24.8184	1.23020
$408$	0	0
$409$	−1.36472	−0.0674812	−0.0337406	−	0.999431i	$-0.510742\pi$
$409$	−1.36472	−0.0674812	−0.0337406	+	0.999431i	$0.510742\pi$
$410$	0	0
$411$	0	0
$412$	−2.35105	−0.115828
$413$	−0.589259	−0.0289955
$414$	0	0
$415$	0	0
$416$	5.33131	0.261389
$417$	0	0
$418$	4.50684	0.220437
$419$	−16.6231	−0.812094	−0.406047	−	0.913852i	$-0.633093\pi$
$419$	−16.6231	−0.812094	−0.406047	+	0.913852i	$0.633093\pi$
$420$	0	0
$421$	−31.1128	−1.51635	−0.758174	−	0.652053i	$-0.773908\pi$
$421$	−31.1128	−1.51635	−0.758174	+	0.652053i	$0.773908\pi$
$422$	19.2534	0.937242
$423$	0	0
$424$	−12.9358	−0.628217
$425$	0	0
$426$	0	0
$427$	−0.167920	−0.00812623
$428$	−14.0334	−0.678331
$429$	0	0
$430$	0	0
$431$	10.1831	0.490504	0.245252	−	0.969459i	$-0.421129\pi$
$431$	10.1831	0.490504	0.245252	+	0.969459i	$0.421129\pi$
$432$	0	0
$433$	−22.1953	−1.06664	−0.533318	−	0.845915i	$-0.679055\pi$
$433$	−22.1953	−1.06664	−0.533318	+	0.845915i	$0.679055\pi$
$434$	−0.195265	−0.00937300
$435$	0	0
$436$	−0.0778929	−0.00373039
$437$	3.40920	0.163084
$438$	0	0
$439$	−9.32524	−0.445070	−0.222535	−	0.974925i	$-0.571433\pi$
$439$	−9.32524	−0.445070	−0.222535	+	0.974925i	$0.571433\pi$
$440$	0	0
$441$	0	0
$442$	39.0855	1.85911
$443$	−13.9879	−0.664584	−0.332292	−	0.943177i	$-0.607822\pi$
$443$	−13.9879	−0.664584	−0.332292	+	0.943177i	$0.607822\pi$
$444$	0	0
$445$	0	0
$446$	14.5068	0.686919
$447$	0	0
$448$	0.0778929	0.00368009
$449$	−13.4932	−0.636782	−0.318391	−	0.947960i	$-0.603142\pi$
$449$	−13.4932	−0.636782	−0.318391	+	0.947960i	$0.603142\pi$
$450$	0	0
$451$	0	0
$452$	−6.00000	−0.282216
$453$	0	0
$454$	−21.5984	−1.01366
$455$	0	0
$456$	0	0
$457$	−9.68236	−0.452922	−0.226461	−	0.974020i	$-0.572716\pi$
$457$	−9.68236	−0.452922	−0.226461	+	0.974020i	$0.572716\pi$
$458$	−19.0137	−0.888451
$459$	0	0
$460$	0	0
$461$	−8.66262	−0.403459	−0.201729	−	0.979441i	$-0.564656\pi$
$461$	−8.66262	−0.403459	−0.201729	+	0.979441i	$0.564656\pi$
$462$	0	0
$463$	−28.0015	−1.30134	−0.650671	−	0.759360i	$-0.725512\pi$
$463$	−28.0015	−1.30134	−0.650671	+	0.759360i	$0.725512\pi$
$464$	1.33131	0.0618046
$465$	0	0
$466$	−6.01367	−0.278578
$467$	16.9742	0.785472	0.392736	−	0.919651i	$-0.371529\pi$
$467$	16.9742	0.785472	0.392736	+	0.919651i	$0.371529\pi$
$468$	0	0
$469$	0.357118	0.0164902
$470$	0	0
$471$	0	0
$472$	−7.56499	−0.348207
$473$	−2.28423	−0.105029
$474$	0	0
$475$	0	0
$476$	0.571057	0.0261743
$477$	0	0
$478$	−15.4092	−0.704801
$479$	−10.0532	−0.459340	−0.229670	−	0.973269i	$-0.573765\pi$
$479$	−10.0532	−0.459340	−0.229670	+	0.973269i	$0.573765\pi$
$480$	0	0
$481$	29.3587	1.33864
$482$	−4.81841	−0.219472
$483$	0	0
$484$	9.31157	0.423253
$485$	0	0
$486$	0	0
$487$	−19.2089	−0.870440	−0.435220	−	0.900324i	$-0.643329\pi$
$487$	−19.2089	−0.870440	−0.435220	+	0.900324i	$0.643329\pi$
$488$	−2.15579	−0.0975878
$489$	0	0
$490$	0	0
$491$	−5.32524	−0.240325	−0.120162	−	0.992754i	$-0.538342\pi$
$491$	−5.32524	−0.240325	−0.120162	+	0.992754i	$0.538342\pi$
$492$	0	0
$493$	9.76025	0.439580
$494$	5.33131	0.239867
$495$	0	0
$496$	−2.50684	−0.112560
$497$	0.845752	0.0379372
$498$	0	0
$499$	−11.9605	−0.535426	−0.267713	−	0.963499i	$-0.586268\pi$
$499$	−11.9605	−0.535426	−0.267713	+	0.963499i	$0.586268\pi$
$500$	0	0
$501$	0	0
$502$	−1.52051	−0.0678636
$503$	−19.3313	−0.861941	−0.430970	−	0.902366i	$-0.641829\pi$
$503$	−19.3313	−0.861941	−0.430970	+	0.902366i	$0.641829\pi$
$504$	0	0
$505$	0	0
$506$	15.3647	0.683045
$507$	0	0
$508$	−17.8321	−0.791171
$509$	36.1573	1.60265	0.801323	−	0.598232i	$-0.204130\pi$
$509$	36.1573	1.60265	0.801323	+	0.598232i	$0.204130\pi$
$510$	0	0
$511$	0.397069	0.0175653
$512$	1.00000	0.0441942
$513$	0	0
$514$	16.5068	0.728085
$515$	0	0
$516$	0	0
$517$	−25.5205	−1.12239
$518$	0.428943	0.0188467
$519$	0	0
$520$	0	0
$521$	31.5205	1.38094	0.690469	−	0.723362i	$-0.257404\pi$
$521$	31.5205	1.38094	0.690469	+	0.723362i	$0.257404\pi$
$522$	0	0
$523$	−5.59840	−0.244801	−0.122400	−	0.992481i	$-0.539059\pi$
$523$	−5.59840	−0.244801	−0.122400	+	0.992481i	$0.539059\pi$
$524$	1.49316	0.0652292
$525$	0	0
$526$	−18.0273	−0.786030
$527$	−18.3784	−0.800575
$528$	0	0
$529$	−11.3773	−0.494667
$530$	0	0
$531$	0	0
$532$	0.0778929	0.00337708
$533$	0	0
$534$	0	0
$535$	0	0
$536$	4.58473	0.198030
$537$	0	0
$538$	−20.2089	−0.871269
$539$	−31.5205	−1.35768
$540$	0	0
$541$	15.1968	0.653362	0.326681	−	0.945135i	$-0.394070\pi$
$541$	15.1968	0.653362	0.326681	+	0.945135i	$0.394070\pi$
$542$	6.08396	0.261328
$543$	0	0
$544$	7.33131	0.314327
$545$	0	0
$546$	0	0
$547$	26.6505	1.13949	0.569746	−	0.821821i	$-0.307041\pi$
$547$	26.6505	1.13949	0.569746	+	0.821821i	$0.307041\pi$
$548$	−8.42894	−0.360067
$549$	0	0
$550$	0	0
$551$	1.33131	0.0567158
$552$	0	0
$553$	1.32524	0.0563551
$554$	31.3647	1.33256
$555$	0	0
$556$	8.81841	0.373984
$557$	12.8458	0.544292	0.272146	−	0.962256i	$-0.412267\pi$
$557$	12.8458	0.544292	0.272146	+	0.962256i	$0.412267\pi$
$558$	0	0
$559$	−2.70210	−0.114287
$560$	0	0
$561$	0	0
$562$	−11.3252	−0.477727
$563$	10.1968	0.429744	0.214872	−	0.976642i	$-0.431067\pi$
$563$	10.1968	0.429744	0.214872	+	0.976642i	$0.431067\pi$
$564$	0	0
$565$	0	0
$566$	−26.1437	−1.09890
$567$	0	0
$568$	10.8579	0.455587
$569$	24.3784	1.02200	0.510998	−	0.859582i	$-0.329276\pi$
$569$	24.3784	1.02200	0.510998	+	0.859582i	$0.329276\pi$
$570$	0	0
$571$	8.32371	0.348336	0.174168	−	0.984716i	$-0.444276\pi$
$571$	8.32371	0.348336	0.174168	+	0.984716i	$0.444276\pi$
$572$	24.0273	1.00463
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.29290	0.303607	0.151804	−	0.988411i	$-0.451492\pi$
$577$	7.29290	0.303607	0.151804	+	0.988411i	$0.451492\pi$
$578$	36.7481	1.52852
$579$	0	0
$580$	0	0
$581$	−1.02581	−0.0425576
$582$	0	0
$583$	−58.2994	−2.41452
$584$	5.09763	0.210942
$585$	0	0
$586$	1.33131	0.0549959
$587$	5.53264	0.228357	0.114178	−	0.993460i	$-0.463576\pi$
$587$	5.53264	0.228357	0.114178	+	0.993460i	$0.463576\pi$
$588$	0	0
$589$	−2.50684	−0.103292
$590$	0	0
$591$	0	0
$592$	5.50684	0.226330
$593$	−26.3252	−1.08105	−0.540524	−	0.841329i	$-0.681774\pi$
$593$	−26.3252	−1.08105	−0.540524	+	0.841329i	$0.681774\pi$
$594$	0	0
$595$	0	0
$596$	−17.6763	−0.724049
$597$	0	0
$598$	18.1755	0.743252
$599$	−22.2994	−0.911130	−0.455565	−	0.890202i	$-0.650563\pi$
$599$	−22.2994	−0.911130	−0.455565	+	0.890202i	$0.650563\pi$
$600$	0	0
$601$	32.4947	1.32549	0.662743	−	0.748847i	$-0.269392\pi$
$601$	32.4947	1.32549	0.662743	+	0.748847i	$0.269392\pi$
$602$	−0.0394789	−0.00160904
$603$	0	0
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	−8.35105	−0.338959	−0.169479	−	0.985534i	$-0.554209\pi$
$607$	−8.35105	−0.338959	−0.169479	+	0.985534i	$0.554209\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−30.1892	−1.22132
$612$	0	0
$613$	38.2994	1.54690	0.773450	−	0.633858i	$-0.218529\pi$
$613$	38.2994	1.54690	0.773450	+	0.633858i	$0.218529\pi$
$614$	2.84421	0.114783
$615$	0	0
$616$	0.351050	0.0141442
$617$	−14.3526	−0.577813	−0.288907	−	0.957357i	$-0.593292\pi$
$617$	−14.3526	−0.577813	−0.288907	+	0.957357i	$0.593292\pi$
$618$	0	0
$619$	−8.62314	−0.346593	−0.173297	−	0.984870i	$-0.555442\pi$
$619$	−8.62314	−0.346593	−0.173297	+	0.984870i	$0.555442\pi$
$620$	0	0
$621$	0	0
$622$	16.3895	0.657158
$623$	−1.16946	−0.0468533
$624$	0	0
$625$	0	0
$626$	21.0471	0.841211
$627$	0	0
$628$	−0.506836	−0.0202250
$629$	40.3723	1.60975
$630$	0	0
$631$	−10.3374	−0.411525	−0.205762	−	0.978602i	$-0.565967\pi$
$631$	−10.3374	−0.411525	−0.205762	+	0.978602i	$0.565967\pi$
$632$	17.0137	0.676768
$633$	0	0
$634$	−30.8902	−1.22681
$635$	0	0
$636$	0	0
$637$	−37.2868	−1.47736
$638$	6.00000	0.237542
$639$	0	0
$640$	0	0
$641$	−5.88369	−0.232392	−0.116196	−	0.993226i	$-0.537070\pi$
$641$	−5.88369	−0.232392	−0.116196	+	0.993226i	$0.537070\pi$
$642$	0	0
$643$	25.5084	1.00595	0.502976	−	0.864300i	$-0.332238\pi$
$643$	25.5084	1.00595	0.502976	+	0.864300i	$0.332238\pi$
$644$	0.265553	0.0104642
$645$	0	0
$646$	7.33131	0.288447
$647$	−5.09157	−0.200170	−0.100085	−	0.994979i	$-0.531911\pi$
$647$	−5.09157	−0.200170	−0.100085	+	0.994979i	$0.531911\pi$
$648$	0	0
$649$	−34.0942	−1.33831
$650$	0	0
$651$	0	0
$652$	−0.830542	−0.0325265
$653$	−11.1816	−0.437570	−0.218785	−	0.975773i	$-0.570209\pi$
$653$	−11.1816	−0.437570	−0.218785	+	0.975773i	$0.570209\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	−0.441078	−0.0171950
$659$	13.7542	0.535787	0.267894	−	0.963449i	$-0.413672\pi$
$659$	13.7542	0.535787	0.267894	+	0.963449i	$0.413672\pi$
$660$	0	0
$661$	−12.6171	−0.490747	−0.245374	−	0.969429i	$-0.578911\pi$
$661$	−12.6171	−0.490747	−0.245374	+	0.969429i	$0.578911\pi$
$662$	28.3845	1.10319
$663$	0	0
$664$	−13.1695	−0.511074
$665$	0	0
$666$	0	0
$667$	4.53871	0.175740
$668$	16.5068	0.638669
$669$	0	0
$670$	0	0
$671$	−9.71577	−0.375073
$672$	0	0
$673$	−2.35105	−0.0906263	−0.0453132	−	0.998973i	$-0.514429\pi$
$673$	−2.35105	−0.0906263	−0.0453132	+	0.998973i	$0.514429\pi$
$674$	17.8716	0.688387
$675$	0	0
$676$	15.4229	0.593188
$677$	−23.7663	−0.913414	−0.456707	−	0.889617i	$-0.650971\pi$
$677$	−23.7663	−0.913414	−0.456707	+	0.889617i	$0.650971\pi$
$678$	0	0
$679$	−0.597928	−0.0229464
$680$	0	0
$681$	0	0
$682$	−11.2979	−0.432619
$683$	28.1695	1.07787	0.538937	−	0.842346i	$-0.318826\pi$
$683$	28.1695	1.07787	0.538937	+	0.842346i	$0.318826\pi$
$684$	0	0
$685$	0	0
$686$	−1.09003	−0.0416174
$687$	0	0
$688$	−0.506836	−0.0193229
$689$	−68.9647	−2.62734
$690$	0	0
$691$	2.32371	0.0883979	0.0441990	−	0.999023i	$-0.485926\pi$
$691$	2.32371	0.0883979	0.0441990	+	0.999023i	$0.485926\pi$
$692$	18.8321	0.715888
$693$	0	0
$694$	33.0410	1.25422
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−19.7158	−0.746253
$699$	0	0
$700$	0	0
$701$	−23.7036	−0.895274	−0.447637	−	0.894215i	$-0.647734\pi$
$701$	−23.7036	−0.895274	−0.447637	+	0.894215i	$0.647734\pi$
$702$	0	0
$703$	5.50684	0.207694
$704$	4.50684	0.169858
$705$	0	0
$706$	−5.90997	−0.222425
$707$	0.323706	0.0121742
$708$	0	0
$709$	9.05315	0.339998	0.169999	−	0.985444i	$-0.445624\pi$
$709$	9.05315	0.339998	0.169999	+	0.985444i	$0.445624\pi$
$710$	0	0
$711$	0	0
$712$	−15.0137	−0.562661
$713$	−8.54631	−0.320062
$714$	0	0
$715$	0	0
$716$	7.15579	0.267424
$717$	0	0
$718$	7.00760	0.261521
$719$	−35.0734	−1.30802	−0.654008	−	0.756488i	$-0.726914\pi$
$719$	−35.0734	−1.30802	−0.654008	+	0.756488i	$0.726914\pi$
$720$	0	0
$721$	−0.183130	−0.00682012
$722$	1.00000	0.0372161
$723$	0	0
$724$	12.5205	0.465321
$725$	0	0
$726$	0	0
$727$	−30.1386	−1.11778	−0.558890	−	0.829242i	$-0.688773\pi$
$727$	−30.1386	−1.11778	−0.558890	+	0.829242i	$0.688773\pi$
$728$	0.415271	0.0153910
$729$	0	0
$730$	0	0
$731$	−3.71577	−0.137433
$732$	0	0
$733$	−47.8321	−1.76672	−0.883359	−	0.468697i	$-0.844724\pi$
$733$	−47.8321	−1.76672	−0.883359	+	0.468697i	$0.844724\pi$
$734$	−17.7158	−0.653901
$735$	0	0
$736$	3.40920	0.125665
$737$	20.6626	0.761117
$738$	0	0
$739$	−22.4674	−0.826475	−0.413238	−	0.910623i	$-0.635602\pi$
$739$	−22.4674	−0.826475	−0.413238	+	0.910623i	$0.635602\pi$
$740$	0	0
$741$	0	0
$742$	−1.00760	−0.0369903
$743$	−11.7926	−0.432629	−0.216314	−	0.976324i	$-0.569404\pi$
$743$	−11.7926	−0.432629	−0.216314	+	0.976324i	$0.569404\pi$
$744$	0	0
$745$	0	0
$746$	−15.4487	−0.565616
$747$	0	0
$748$	33.0410	1.20810
$749$	−1.09310	−0.0399411
$750$	0	0
$751$	−2.03948	−0.0744216	−0.0372108	−	0.999307i	$-0.511847\pi$
$751$	−2.03948	−0.0744216	−0.0372108	+	0.999307i	$0.511847\pi$
$752$	−5.66262	−0.206495
$753$	0	0
$754$	7.09763	0.258481
$755$	0	0
$756$	0	0
$757$	−43.3526	−1.57568	−0.787838	−	0.615882i	$-0.788800\pi$
$757$	−43.3526	−1.57568	−0.787838	+	0.615882i	$0.788800\pi$
$758$	34.4563	1.25151
$759$	0	0
$760$	0	0
$761$	−6.62921	−0.240309	−0.120154	−	0.992755i	$-0.538339\pi$
$761$	−6.62921	−0.240309	−0.120154	+	0.992755i	$0.538339\pi$
$762$	0	0
$763$	−0.00606730	−0.000219651 0
$764$	4.90237	0.177361
$765$	0	0
$766$	−12.0273	−0.434566
$767$	−40.3313	−1.45628
$768$	0	0
$769$	−19.6429	−0.708340	−0.354170	−	0.935181i	$-0.615237\pi$
$769$	−19.6429	−0.708340	−0.354170	+	0.935181i	$0.615237\pi$
$770$	0	0
$771$	0	0
$772$	−18.1558	−0.653441
$773$	14.4107	0.518318	0.259159	−	0.965835i	$-0.416555\pi$
$773$	14.4107	0.518318	0.259159	+	0.965835i	$0.416555\pi$
$774$	0	0
$775$	0	0
$776$	−7.67629	−0.275563
$777$	0	0
$778$	15.3647	0.550852
$779$	0	0
$780$	0	0
$781$	48.9347	1.75102
$782$	24.9939	0.893781
$783$	0	0
$784$	−6.99393	−0.249783
$785$	0	0
$786$	0	0
$787$	−24.3176	−0.866830	−0.433415	−	0.901194i	$-0.642692\pi$
$787$	−24.3176	−0.866830	−0.433415	+	0.901194i	$0.642692\pi$
$788$	−2.98633	−0.106384
$789$	0	0
$790$	0	0
$791$	−0.467357	−0.0166173
$792$	0	0
$793$	−11.4932	−0.408134
$794$	−1.32524	−0.0470311
$795$	0	0
$796$	−3.06422	−0.108608
$797$	−7.30397	−0.258720	−0.129360	−	0.991598i	$-0.541292\pi$
$797$	−7.30397	−0.258720	−0.129360	+	0.991598i	$0.541292\pi$
$798$	0	0
$799$	−41.5144	−1.46868
$800$	0	0
$801$	0	0
$802$	6.46736	0.228370
$803$	22.9742	0.810742
$804$	0	0
$805$	0	0
$806$	−13.3647	−0.470752
$807$	0	0
$808$	4.15579	0.146200
$809$	0.584729	0.0205580	0.0102790	−	0.999947i	$-0.496728\pi$
$809$	0.584729	0.0205580	0.0102790	+	0.999947i	$0.496728\pi$
$810$	0	0
$811$	1.90997	0.0670682	0.0335341	−	0.999438i	$-0.489324\pi$
$811$	1.90997	0.0670682	0.0335341	+	0.999438i	$0.489324\pi$
$812$	0.103700	0.00363914
$813$	0	0
$814$	24.8184	0.869885
$815$	0	0
$816$	0	0
$817$	−0.506836	−0.0177319
$818$	−1.36472	−0.0477164
$819$	0	0
$820$	0	0
$821$	38.2226	1.33398	0.666989	−	0.745067i	$-0.267583\pi$
$821$	38.2226	1.33398	0.666989	+	0.745067i	$0.267583\pi$
$822$	0	0
$823$	45.7481	1.59468	0.797340	−	0.603531i	$-0.206240\pi$
$823$	45.7481	1.59468	0.797340	+	0.603531i	$0.206240\pi$
$824$	−2.35105	−0.0819027
$825$	0	0
$826$	−0.589259	−0.0205029
$827$	−22.6687	−0.788268	−0.394134	−	0.919053i	$-0.628955\pi$
$827$	−22.6687	−0.788268	−0.394134	+	0.919053i	$0.628955\pi$
$828$	0	0
$829$	−4.63028	−0.160816	−0.0804081	−	0.996762i	$-0.525622\pi$
$829$	−4.63028	−0.160816	−0.0804081	+	0.996762i	$0.525622\pi$
$830$	0	0
$831$	0	0
$832$	5.33131	0.184830
$833$	−51.2747	−1.77656
$834$	0	0
$835$	0	0
$836$	4.50684	0.155872
$837$	0	0
$838$	−16.6231	−0.574237
$839$	50.4826	1.74285	0.871426	−	0.490527i	$-0.163196\pi$
$839$	50.4826	1.74285	0.871426	+	0.490527i	$0.163196\pi$
$840$	0	0
$841$	−27.2276	−0.938883
$842$	−31.1128	−1.07222
$843$	0	0
$844$	19.2534	0.662730
$845$	0	0
$846$	0	0
$847$	0.725305	0.0249218
$848$	−12.9358	−0.444216
$849$	0	0
$850$	0	0
$851$	18.7739	0.643562
$852$	0	0
$853$	−36.2105	−1.23982	−0.619912	−	0.784672i	$-0.712832\pi$
$853$	−36.2105	−1.23982	−0.619912	+	0.784672i	$0.712832\pi$
$854$	−0.167920	−0.00574611
$855$	0	0
$856$	−14.0334	−0.479652
$857$	25.1968	0.860706	0.430353	−	0.902661i	$-0.358389\pi$
$857$	25.1968	0.860706	0.430353	+	0.902661i	$0.358389\pi$
$858$	0	0
$859$	18.8579	0.643423	0.321711	−	0.946838i	$-0.395742\pi$
$859$	18.8579	0.643423	0.321711	+	0.946838i	$0.395742\pi$
$860$	0	0
$861$	0	0
$862$	10.1831	0.346839
$863$	15.0137	0.511071	0.255536	−	0.966800i	$-0.417748\pi$
$863$	15.0137	0.511071	0.255536	+	0.966800i	$0.417748\pi$
$864$	0	0
$865$	0	0
$866$	−22.1953	−0.754226
$867$	0	0
$868$	−0.195265	−0.00662771
$869$	76.6778	2.60112
$870$	0	0
$871$	24.4426	0.828206
$872$	−0.0778929	−0.00263779
$873$	0	0
$874$	3.40920	0.115318
$875$	0	0
$876$	0	0
$877$	−35.1128	−1.18568	−0.592838	−	0.805322i	$-0.701993\pi$
$877$	−35.1128	−1.18568	−0.592838	+	0.805322i	$0.701993\pi$
$878$	−9.32524	−0.314712
$879$	0	0
$880$	0	0
$881$	41.3389	1.39274	0.696372	−	0.717681i	$-0.254797\pi$
$881$	41.3389	1.39274	0.696372	+	0.717681i	$0.254797\pi$
$882$	0	0
$883$	12.6353	0.425211	0.212605	−	0.977138i	$-0.431805\pi$
$883$	12.6353	0.425211	0.212605	+	0.977138i	$0.431805\pi$
$884$	39.0855	1.31459
$885$	0	0
$886$	−13.9879	−0.469932
$887$	4.15579	0.139538	0.0697688	−	0.997563i	$-0.477774\pi$
$887$	4.15579	0.139538	0.0697688	+	0.997563i	$0.477774\pi$
$888$	0	0
$889$	−1.38899	−0.0465853
$890$	0	0
$891$	0	0
$892$	14.5068	0.485725
$893$	−5.66262	−0.189492
$894$	0	0
$895$	0	0
$896$	0.0778929	0.00260222
$897$	0	0
$898$	−13.4932	−0.450273
$899$	−3.33738	−0.111308
$900$	0	0
$901$	−94.8362	−3.15945
$902$	0	0
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	41.7542	1.38643	0.693213	−	0.720733i	$-0.256195\pi$
$907$	41.7542	1.38643	0.693213	+	0.720733i	$0.256195\pi$
$908$	−21.5984	−0.716768
$909$	0	0
$910$	0	0
$911$	9.12998	0.302490	0.151245	−	0.988496i	$-0.451672\pi$
$911$	9.12998	0.302490	0.151245	+	0.988496i	$0.451672\pi$
$912$	0	0
$913$	−59.3526	−1.96428
$914$	−9.68236	−0.320264
$915$	0	0
$916$	−19.0137	−0.628229
$917$	0.116307	0.00384079
$918$	0	0
$919$	19.0197	0.627403	0.313702	−	0.949522i	$-0.398431\pi$
$919$	19.0197	0.627403	0.313702	+	0.949522i	$0.398431\pi$
$920$	0	0
$921$	0	0
$922$	−8.66262	−0.285288
$923$	57.8868	1.90537
$924$	0	0
$925$	0	0
$926$	−28.0015	−0.920188
$927$	0	0
$928$	1.33131	0.0437024
$929$	7.77239	0.255004	0.127502	−	0.991838i	$-0.459304\pi$
$929$	7.77239	0.255004	0.127502	+	0.991838i	$0.459304\pi$
$930$	0	0
$931$	−6.99393	−0.229217
$932$	−6.01367	−0.196984
$933$	0	0
$934$	16.9742	0.555413
$935$	0	0
$936$	0	0
$937$	45.9818	1.50216	0.751080	−	0.660211i	$-0.229533\pi$
$937$	45.9818	1.50216	0.751080	+	0.660211i	$0.229533\pi$
$938$	0.357118	0.0116603
$939$	0	0
$940$	0	0
$941$	40.6242	1.32431	0.662156	−	0.749366i	$-0.269642\pi$
$941$	40.6242	1.32431	0.662156	+	0.749366i	$0.269642\pi$
$942$	0	0
$943$	0	0
$944$	−7.56499	−0.246219
$945$	0	0
$946$	−2.28423	−0.0742666
$947$	−2.28423	−0.0742274	−0.0371137	−	0.999311i	$-0.511816\pi$
$947$	−2.28423	−0.0742274	−0.0371137	+	0.999311i	$0.511816\pi$
$948$	0	0
$949$	27.1771	0.882205
$950$	0	0
$951$	0	0
$952$	0.571057	0.0185081
$953$	−57.5084	−1.86288	−0.931439	−	0.363896i	$-0.881446\pi$
$953$	−57.5084	−1.86288	−0.931439	+	0.363896i	$0.881446\pi$
$954$	0	0
$955$	0	0
$956$	−15.4092	−0.498369
$957$	0	0
$958$	−10.0532	−0.324803
$959$	−0.656555	−0.0212013
$960$	0	0
$961$	−24.7158	−0.797283
$962$	29.3587	0.946561
$963$	0	0
$964$	−4.81841	−0.155190
$965$	0	0
$966$	0	0
$967$	4.70210	0.151209	0.0756047	−	0.997138i	$-0.475911\pi$
$967$	4.70210	0.151209	0.0756047	+	0.997138i	$0.475911\pi$
$968$	9.31157	0.299285
$969$	0	0
$970$	0	0
$971$	−14.9863	−0.480934	−0.240467	−	0.970657i	$-0.577301\pi$
$971$	−14.9863	−0.480934	−0.240467	+	0.970657i	$0.577301\pi$
$972$	0	0
$973$	0.686891	0.0220207
$974$	−19.2089	−0.615494
$975$	0	0
$976$	−2.15579	−0.0690050
$977$	8.89737	0.284652	0.142326	−	0.989820i	$-0.454542\pi$
$977$	8.89737	0.284652	0.142326	+	0.989820i	$0.454542\pi$
$978$	0	0
$979$	−67.6642	−2.16256
$980$	0	0
$981$	0	0
$982$	−5.32524	−0.169935
$983$	−1.60947	−0.0513341	−0.0256671	−	0.999671i	$-0.508171\pi$
$983$	−1.60947	−0.0513341	−0.0256671	+	0.999671i	$0.508171\pi$
$984$	0	0
$985$	0	0
$986$	9.76025	0.310830
$987$	0	0
$988$	5.33131	0.169612
$989$	−1.72791	−0.0549443
$990$	0	0
$991$	16.8974	0.536763	0.268381	−	0.963313i	$-0.413511\pi$
$991$	16.8974	0.536763	0.268381	+	0.963313i	$0.413511\pi$
$992$	−2.50684	−0.0795921
$993$	0	0
$994$	0.845752	0.0268256
$995$	0	0
$996$	0	0
$997$	−24.1558	−0.765021	−0.382511	−	0.923951i	$-0.624940\pi$
$997$	−24.1558	−0.765021	−0.382511	+	0.923951i	$0.624940\pi$
$998$	−11.9605	−0.378604
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cp.1.2		3
3.2	odd	2		950.2.a.j.1.1	✓	3
5.4	even	2		8550.2.a.ci.1.2		3
12.11	even	2		7600.2.a.bz.1.3		3
15.2	even	4		950.2.b.h.799.3		6
15.8	even	4		950.2.b.h.799.4		6
15.14	odd	2		950.2.a.l.1.3	yes	3
60.59	even	2		7600.2.a.bk.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.j.1.1	✓	3	3.2	odd	2
950.2.a.l.1.3	yes	3	15.14	odd	2
950.2.b.h.799.3		6	15.2	even	4
950.2.b.h.799.4		6	15.8	even	4
7600.2.a.bk.1.1		3	60.59	even	2
7600.2.a.bz.1.3		3	12.11	even	2
8550.2.a.ci.1.2		3	5.4	even	2
8550.2.a.cp.1.2		3	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.0778929 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.0778929 q^{7} +1.00000 q^{8} +4.50684 q^{11} +5.33131 q^{13} +0.0778929 q^{14} +1.00000 q^{16} +7.33131 q^{17} +1.00000 q^{19} +4.50684 q^{22} +3.40920 q^{23} +5.33131 q^{26} +0.0778929 q^{28} +1.33131 q^{29} -2.50684 q^{31} +1.00000 q^{32} +7.33131 q^{34} +5.50684 q^{37} +1.00000 q^{38} -0.506836 q^{43} +4.50684 q^{44} +3.40920 q^{46} -5.66262 q^{47} -6.99393 q^{49} +5.33131 q^{52} -12.9358 q^{53} +0.0778929 q^{56} +1.33131 q^{58} -7.56499 q^{59} -2.15579 q^{61} -2.50684 q^{62} +1.00000 q^{64} +4.58473 q^{67} +7.33131 q^{68} +10.8579 q^{71} +5.09763 q^{73} +5.50684 q^{74} +1.00000 q^{76} +0.351050 q^{77} +17.0137 q^{79} -13.1695 q^{83} -0.506836 q^{86} +4.50684 q^{88} -15.0137 q^{89} +0.415271 q^{91} +3.40920 q^{92} -5.66262 q^{94} -7.67629 q^{97} -6.99393 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 2 q^{11} + 6 q^{13} - 2 q^{14} + 3 q^{16} + 12 q^{17} + 3 q^{19} - 2 q^{22} - 2 q^{23} + 6 q^{26} - 2 q^{28} - 6 q^{29} + 8 q^{31} + 3 q^{32} + 12 q^{34} + q^{37} + 3 q^{38} + 14 q^{43} - 2 q^{44} - 2 q^{46} + 3 q^{47} + 9 q^{49} + 6 q^{52} - 10 q^{53} - 2 q^{56} - 6 q^{58} - 6 q^{59} - 2 q^{61} + 8 q^{62} + 3 q^{64} - 4 q^{67} + 12 q^{68} + 6 q^{71} + 12 q^{73} + q^{74} + 3 q^{76} - 10 q^{77} + 20 q^{79} - 4 q^{83} + 14 q^{86} - 2 q^{88} - 14 q^{89} + 19 q^{91} - 2 q^{92} + 3 q^{94} + 28 q^{97} + 9 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 2 * q^11 + 6 * q^13 - 2 * q^14 + 3 * q^16 + 12 * q^17 + 3 * q^19 - 2 * q^22 - 2 * q^23 + 6 * q^26 - 2 * q^28 - 6 * q^29 + 8 * q^31 + 3 * q^32 + 12 * q^34 + q^37 + 3 * q^38 + 14 * q^43 - 2 * q^44 - 2 * q^46 + 3 * q^47 + 9 * q^49 + 6 * q^52 - 10 * q^53 - 2 * q^56 - 6 * q^58 - 6 * q^59 - 2 * q^61 + 8 * q^62 + 3 * q^64 - 4 * q^67 + 12 * q^68 + 6 * q^71 + 12 * q^73 + q^74 + 3 * q^76 - 10 * q^77 + 20 * q^79 - 4 * q^83 + 14 * q^86 - 2 * q^88 - 14 * q^89 + 19 * q^91 - 2 * q^92 + 3 * q^94 + 28 * q^97 + 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more