LMFDB - Embedded newform 675.4.b.k.649.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(649,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("675.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.2033649216.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 47x^{4} + 541x^{2} + 36 $ x^6 + 47x^4 + 541x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.5
Root			$4.45938i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.k.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.45938i q^{2} -11.8861 q^{4} -5.08123i q^{7} -17.3296i q^{8} +O(q^{10})$	$q+4.45938i q^{2} -11.8861 q^{4} -5.08123i q^{7} -17.3296i q^{8} -58.3007 q^{11} +21.2119i q^{13} +22.6592 q^{14} -17.8095 q^{16} -68.8451i q^{17} +40.8133 q^{19} -259.985i q^{22} +144.318i q^{23} -94.5921 q^{26} +60.3960i q^{28} -220.058 q^{29} +291.545 q^{31} -218.056i q^{32} +307.006 q^{34} -260.637i q^{37} +182.002i q^{38} +169.766 q^{41} -438.596i q^{43} +692.967 q^{44} -643.571 q^{46} -255.481i q^{47} +317.181 q^{49} -252.127i q^{52} -214.714i q^{53} -88.0557 q^{56} -981.322i q^{58} +331.524 q^{59} +54.9647 q^{61} +1300.11i q^{62} +829.920 q^{64} -758.179i q^{67} +818.299i q^{68} +904.348 q^{71} +866.622i q^{73} +1162.28 q^{74} -485.110 q^{76} +296.239i q^{77} -206.961 q^{79} +757.054i q^{82} +463.397i q^{83} +1955.87 q^{86} +1010.33i q^{88} +601.736 q^{89} +107.783 q^{91} -1715.38i q^{92} +1139.29 q^{94} -229.363i q^{97} +1414.43i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 46 q^{4}+O(q^{10}) $ 6 * q - 46 * q^4	$ 6 q - 46 q^{4} - 76 q^{11} - 216 q^{14} + 382 q^{16} - 374 q^{19} - 832 q^{26} + 320 q^{29} + 454 q^{31} - 34 q^{34} + 676 q^{41} + 3272 q^{44} - 2850 q^{46} + 394 q^{49} + 2508 q^{56} + 280 q^{59} + 1190 q^{61} + 1836 q^{64} + 1204 q^{71} + 5756 q^{74} + 1050 q^{76} - 1258 q^{79} + 7460 q^{86} + 4308 q^{89} - 880 q^{91} + 2216 q^{94}+O(q^{100}) $ 6 * q - 46 * q^4 - 76 * q^11 - 216 * q^14 + 382 * q^16 - 374 * q^19 - 832 * q^26 + 320 * q^29 + 454 * q^31 - 34 * q^34 + 676 * q^41 + 3272 * q^44 - 2850 * q^46 + 394 * q^49 + 2508 * q^56 + 280 * q^59 + 1190 * q^61 + 1836 * q^64 + 1204 * q^71 + 5756 * q^74 + 1050 * q^76 - 1258 * q^79 + 7460 * q^86 + 4308 * q^89 - 880 * q^91 + 2216 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.45938i	1.57663i	0.615272	+	0.788315i	$0.289046\pi$
$2$	4.45938i	1.57663i	−0.615272	+	0.788315i	$0.710954\pi$
$3$	0	0
$3$	0	0
$4$	−11.8861	−1.48576
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 5.08123i	− 0.274361i	−0.990546	−	0.137180i	$-0.956196\pi$
$7$	− 5.08123i	− 0.274361i	0.990546	−	0.137180i	$-0.0438040\pi$
$8$	− 17.3296i	− 0.765867i
$9$	0	0
$10$	0	0
$11$	−58.3007	−1.59803	−0.799014	−	0.601312i	$-0.794645\pi$
$11$	−58.3007	−1.59803	−0.799014	+	0.601312i	$0.794645\pi$
$12$	0	0
$13$	21.2119i	0.452548i	0.974064	+	0.226274i	$0.0726545\pi$
$13$	21.2119i	0.452548i	−0.974064	+	0.226274i	$0.927345\pi$
$14$	22.6592	0.432566
$15$	0	0
$16$	−17.8095	−0.278274
$17$	− 68.8451i	− 0.982199i	−0.871104	−	0.491099i	$-0.836595\pi$
$17$	− 68.8451i	− 0.982199i	0.871104	−	0.491099i	$-0.163405\pi$
$18$	0	0
$19$	40.8133	0.492800	0.246400	−	0.969168i	$-0.420752\pi$
$19$	40.8133	0.492800	0.246400	+	0.969168i	$0.420752\pi$
$20$	0	0
$21$	0	0
$22$	− 259.985i	− 2.51950i
$23$	144.318i	1.30837i	0.756336	+	0.654184i	$0.226988\pi$
$23$	144.318i	1.30837i	−0.756336	+	0.654184i	$0.773012\pi$
$24$	0	0
$25$	0	0
$26$	−94.5921	−0.713501
$27$	0	0
$28$	60.3960i	0.407635i
$29$	−220.058	−1.40909	−0.704547	−	0.709657i	$-0.748850\pi$
$29$	−220.058	−1.40909	−0.704547	+	0.709657i	$0.748850\pi$
$30$	0	0
$31$	291.545	1.68913	0.844566	−	0.535452i	$-0.179859\pi$
$31$	291.545	1.68913	0.844566	+	0.535452i	$0.179859\pi$
$32$	− 218.056i	− 1.20460i
$33$	0	0
$34$	307.006	1.54856
$35$	0	0
$36$	0	0
$37$	− 260.637i	− 1.15807i	−0.815304	−	0.579033i	$-0.803430\pi$
$37$	− 260.637i	− 1.15807i	0.815304	−	0.579033i	$-0.196570\pi$
$38$	182.002i	0.776964i
$39$	0	0
$40$	0	0
$41$	169.766	0.646660	0.323330	−	0.946286i	$-0.395198\pi$
$41$	169.766	0.646660	0.323330	+	0.946286i	$0.395198\pi$
$42$	0	0
$43$	− 438.596i	− 1.55547i	−0.628592	−	0.777735i	$-0.716369\pi$
$43$	− 438.596i	− 1.55547i	0.628592	−	0.777735i	$-0.283631\pi$
$44$	692.967	2.37429
$45$	0	0
$46$	−643.571	−2.06281
$47$	− 255.481i	− 0.792887i	−0.918059	−	0.396444i	$-0.870244\pi$
$47$	− 255.481i	− 0.792887i	0.918059	−	0.396444i	$-0.129756\pi$
$48$	0	0
$49$	317.181	0.924726
$50$	0	0
$51$	0	0
$52$	− 252.127i	− 0.672379i
$53$	− 214.714i	− 0.556477i	−0.960512	−	0.278239i	$-0.910249\pi$
$53$	− 214.714i	− 0.556477i	0.960512	−	0.278239i	$-0.0897506\pi$
$54$	0	0
$55$	0	0
$56$	−88.0557	−0.210124
$57$	0	0
$58$	− 981.322i	− 2.22162i
$59$	331.524	0.731537	0.365769	−	0.930706i	$-0.380806\pi$
$59$	331.524	0.731537	0.365769	+	0.930706i	$0.380806\pi$
$60$	0	0
$61$	54.9647	0.115369	0.0576845	−	0.998335i	$-0.481628\pi$
$61$	54.9647	0.115369	0.0576845	+	0.998335i	$0.481628\pi$
$62$	1300.11i	2.66314i
$63$	0	0
$64$	829.920	1.62094
$65$	0	0
$66$	0	0
$67$	− 758.179i	− 1.38248i	−0.722624	−	0.691241i	$-0.757064\pi$
$67$	− 758.179i	− 1.38248i	0.722624	−	0.691241i	$-0.242936\pi$
$68$	818.299i	1.45931i
$69$	0	0
$70$	0	0
$71$	904.348	1.51164	0.755819	−	0.654780i	$-0.227239\pi$
$71$	904.348	1.51164	0.755819	+	0.654780i	$0.227239\pi$
$72$	0	0
$73$	866.622i	1.38946i	0.719271	+	0.694729i	$0.244476\pi$
$73$	866.622i	1.38946i	−0.719271	+	0.694729i	$0.755524\pi$
$74$	1162.28	1.82584
$75$	0	0
$76$	−485.110	−0.732184
$77$	296.239i	0.438437i
$78$	0	0
$79$	−206.961	−0.294746	−0.147373	−	0.989081i	$-0.547082\pi$
$79$	−206.961	−0.294746	−0.147373	+	0.989081i	$0.547082\pi$
$80$	0	0
$81$	0	0
$82$	757.054i	1.01954i
$83$	463.397i	0.612825i	0.951899	+	0.306412i	$0.0991287\pi$
$83$	463.397i	0.612825i	−0.951899	+	0.306412i	$0.900871\pi$
$84$	0	0
$85$	0	0
$86$	1955.87	2.45240
$87$	0	0
$88$	1010.33i	1.22388i
$89$	601.736	0.716673	0.358337	−	0.933592i	$-0.383344\pi$
$89$	601.736	0.716673	0.358337	+	0.933592i	$0.383344\pi$
$90$	0	0
$91$	107.783	0.124162
$92$	− 1715.38i	− 1.94392i
$93$	0	0
$94$	1139.29	1.25009
$95$	0	0
$96$	0	0
$97$	− 229.363i	− 0.240086i	−0.992769	−	0.120043i	$-0.961697\pi$
$97$	− 229.363i	− 0.240086i	0.992769	−	0.120043i	$-0.0383032\pi$
$98$	1414.43i	1.45795i
$99$	0	0
$100$	0	0
$101$	−1345.66	−1.32573	−0.662863	−	0.748740i	$-0.730659\pi$
$101$	−1345.66	−1.32573	−0.662863	+	0.748740i	$0.730659\pi$
$102$	0	0
$103$	− 1596.30i	− 1.52707i	−0.645768	−	0.763534i	$-0.723463\pi$
$103$	− 1596.30i	− 1.52707i	0.645768	−	0.763534i	$-0.276537\pi$
$104$	367.594	0.346592
$105$	0	0
$106$	957.494	0.877358
$107$	958.786i	0.866256i	0.901333	+	0.433128i	$0.142590\pi$
$107$	958.786i	0.866256i	−0.901333	+	0.433128i	$0.857410\pi$
$108$	0	0
$109$	−1690.23	−1.48527	−0.742635	−	0.669696i	$-0.766424\pi$
$109$	−1690.23	−1.48527	−0.742635	+	0.669696i	$0.766424\pi$
$110$	0	0
$111$	0	0
$112$	90.4943i	0.0763474i
$113$	11.6211i	0.00967456i	0.999988	+	0.00483728i	$0.00153976\pi$
$113$	11.6211i	0.00967456i	−0.999988	+	0.00483728i	$0.998460\pi$
$114$	0	0
$115$	0	0
$116$	2615.63	2.09358
$117$	0	0
$118$	1478.39i	1.15336i
$119$	−349.818	−0.269477
$120$	0	0
$121$	2067.97	1.55370
$122$	245.109i	0.181894i
$123$	0	0
$124$	−3465.34	−2.50965
$125$	0	0
$126$	0	0
$127$	− 309.141i	− 0.215999i	−0.994151	−	0.107999i	$-0.965556\pi$
$127$	− 309.141i	− 0.215999i	0.994151	−	0.107999i	$-0.0344444\pi$
$128$	1956.48i	1.35102i
$129$	0	0
$130$	0	0
$131$	−2785.03	−1.85747	−0.928736	−	0.370742i	$-0.879103\pi$
$131$	−2785.03	−1.85747	−0.928736	+	0.370742i	$0.879103\pi$
$132$	0	0
$133$	− 207.382i	− 0.135205i
$134$	3381.01	2.17966
$135$	0	0
$136$	−1193.06	−0.752233
$137$	− 2489.41i	− 1.55244i	−0.630460	−	0.776222i	$-0.717134\pi$
$137$	− 2489.41i	− 1.55244i	0.630460	−	0.776222i	$-0.282866\pi$
$138$	0	0
$139$	−1786.05	−1.08986	−0.544931	−	0.838481i	$-0.683444\pi$
$139$	−1786.05	−1.08986	−0.544931	+	0.838481i	$0.683444\pi$
$140$	0	0
$141$	0	0
$142$	4032.83i	2.38329i
$143$	− 1236.67i	− 0.723185i
$144$	0	0
$145$	0	0
$146$	−3864.60	−2.19066
$147$	0	0
$148$	3097.95i	1.72061i
$149$	1568.83	0.862575	0.431288	−	0.902214i	$-0.358059\pi$
$149$	1568.83	0.862575	0.431288	+	0.902214i	$0.358059\pi$
$150$	0	0
$151$	−438.327	−0.236229	−0.118114	−	0.993000i	$-0.537685\pi$
$151$	−438.327	−0.236229	−0.118114	+	0.993000i	$0.537685\pi$
$152$	− 707.277i	− 0.377419i
$153$	0	0
$154$	−1321.04	−0.691252
$155$	0	0
$156$	0	0
$157$	44.7479i	0.0227469i	0.999935	+	0.0113735i	$0.00362036\pi$
$157$	44.7479i	0.0227469i	−0.999935	+	0.0113735i	$0.996380\pi$
$158$	− 922.917i	− 0.464705i
$159$	0	0
$160$	0	0
$161$	733.315	0.358965
$162$	0	0
$163$	− 2611.84i	− 1.25506i	−0.778591	−	0.627531i	$-0.784065\pi$
$163$	− 2611.84i	− 1.25506i	0.778591	−	0.627531i	$-0.215935\pi$
$164$	−2017.86	−0.960783
$165$	0	0
$166$	−2066.47	−0.966198
$167$	− 188.947i	− 0.0875516i	−0.999041	−	0.0437758i	$-0.986061\pi$
$167$	− 188.947i	− 0.0875516i	0.999041	−	0.0437758i	$-0.0139387\pi$
$168$	0	0
$169$	1747.05	0.795200
$170$	0	0
$171$	0	0
$172$	5213.19i	2.31106i
$173$	1505.02i	0.661413i	0.943734	+	0.330707i	$0.107287\pi$
$173$	1505.02i	0.661413i	−0.943734	+	0.330707i	$0.892713\pi$
$174$	0	0
$175$	0	0
$176$	1038.31	0.444689
$177$	0	0
$178$	2683.37i	1.12993i
$179$	−3136.62	−1.30973	−0.654865	−	0.755746i	$-0.727275\pi$
$179$	−3136.62	−1.30973	−0.654865	+	0.755746i	$0.727275\pi$
$180$	0	0
$181$	4512.67	1.85317	0.926586	−	0.376084i	$-0.122730\pi$
$181$	4512.67	1.85317	0.926586	+	0.376084i	$0.122730\pi$
$182$	480.644i	0.195757i
$183$	0	0
$184$	2500.98	1.00204
$185$	0	0
$186$	0	0
$187$	4013.71i	1.56958i
$188$	3036.67i	1.17804i
$189$	0	0
$190$	0	0
$191$	−1207.43	−0.457418	−0.228709	−	0.973495i	$-0.573450\pi$
$191$	−1207.43	−0.457418	−0.228709	+	0.973495i	$0.573450\pi$
$192$	0	0
$193$	923.164i	0.344305i	0.985070	+	0.172152i	$0.0550721\pi$
$193$	923.164i	0.344305i	−0.985070	+	0.172152i	$0.944928\pi$
$194$	1022.82	0.378526
$195$	0	0
$196$	−3770.04	−1.37392
$197$	− 1180.87i	− 0.427075i	−0.976935	−	0.213537i	$-0.931501\pi$
$197$	− 1180.87i	− 0.427075i	0.976935	−	0.213537i	$-0.0684985\pi$
$198$	0	0
$199$	839.805	0.299157	0.149578	−	0.988750i	$-0.452208\pi$
$199$	839.805	0.299157	0.149578	+	0.988750i	$0.452208\pi$
$200$	0	0
$201$	0	0
$202$	− 6000.82i	− 2.09018i
$203$	1118.17i	0.386600i
$204$	0	0
$205$	0	0
$206$	7118.51	2.40762
$207$	0	0
$208$	− 377.774i	− 0.125932i
$209$	−2379.44	−0.787509
$210$	0	0
$211$	−2589.65	−0.844923	−0.422461	−	0.906381i	$-0.638834\pi$
$211$	−2589.65	−0.844923	−0.422461	+	0.906381i	$0.638834\pi$
$212$	2552.12i	0.826792i
$213$	0	0
$214$	−4275.59	−1.36576
$215$	0	0
$216$	0	0
$217$	− 1481.41i	− 0.463432i
$218$	− 7537.38i	− 2.34172i
$219$	0	0
$220$	0	0
$221$	1460.34	0.444492
$222$	0	0
$223$	− 4180.76i	− 1.25544i	−0.778437	−	0.627722i	$-0.783987\pi$
$223$	− 4180.76i	− 1.25544i	0.778437	−	0.627722i	$-0.216013\pi$
$224$	−1107.99	−0.330495
$225$	0	0
$226$	−51.8232	−0.0152532
$227$	− 2602.67i	− 0.760993i	−0.924782	−	0.380497i	$-0.875753\pi$
$227$	− 2602.67i	− 0.760993i	0.924782	−	0.380497i	$-0.124247\pi$
$228$	0	0
$229$	1845.35	0.532508	0.266254	−	0.963903i	$-0.414214\pi$
$229$	1845.35	0.532508	0.266254	+	0.963903i	$0.414214\pi$
$230$	0	0
$231$	0	0
$232$	3813.51i	1.07918i
$233$	− 240.637i	− 0.0676594i	−0.999428	−	0.0338297i	$-0.989230\pi$
$233$	− 240.637i	− 0.0676594i	0.999428	−	0.0338297i	$-0.0107704\pi$
$234$	0	0
$235$	0	0
$236$	−3940.52	−1.08689
$237$	0	0
$238$	− 1559.97i	− 0.424865i
$239$	2567.64	0.694924	0.347462	−	0.937694i	$-0.387044\pi$
$239$	2567.64	0.694924	0.347462	+	0.937694i	$0.387044\pi$
$240$	0	0
$241$	3987.99	1.06593	0.532965	−	0.846137i	$-0.321078\pi$
$241$	3987.99	1.06593	0.532965	+	0.846137i	$0.321078\pi$
$242$	9221.86i	2.44960i
$243$	0	0
$244$	−653.315	−0.171411
$245$	0	0
$246$	0	0
$247$	865.728i	0.223016i
$248$	− 5052.36i	− 1.29365i
$249$	0	0
$250$	0	0
$251$	−967.393	−0.243272	−0.121636	−	0.992575i	$-0.538814\pi$
$251$	−967.393	−0.243272	−0.121636	+	0.992575i	$0.538814\pi$
$252$	0	0
$253$	− 8413.86i	− 2.09081i
$254$	1378.58	0.340550
$255$	0	0
$256$	−2085.34	−0.509116
$257$	3743.03i	0.908498i	0.890875	+	0.454249i	$0.150092\pi$
$257$	3743.03i	0.908498i	−0.890875	+	0.454249i	$0.849908\pi$
$258$	0	0
$259$	−1324.36	−0.317728
$260$	0	0
$261$	0	0
$262$	− 12419.5i	− 2.92855i
$263$	2497.47i	0.585554i	0.956181	+	0.292777i	$0.0945794\pi$
$263$	2497.47i	0.585554i	−0.956181	+	0.292777i	$0.905421\pi$
$264$	0	0
$265$	0	0
$266$	924.795	0.213168
$267$	0	0
$268$	9011.79i	2.05404i
$269$	−2142.91	−0.485709	−0.242855	−	0.970063i	$-0.578084\pi$
$269$	−2142.91	−0.485709	−0.242855	+	0.970063i	$0.578084\pi$
$270$	0	0
$271$	1540.64	0.345341	0.172671	−	0.984980i	$-0.444760\pi$
$271$	1540.64	0.345341	0.172671	+	0.984980i	$0.444760\pi$
$272$	1226.10i	0.273320i
$273$	0	0
$274$	11101.2	2.44763
$275$	0	0
$276$	0	0
$277$	− 6777.80i	− 1.47018i	−0.677972	−	0.735088i	$-0.737141\pi$
$277$	− 6777.80i	− 1.47018i	0.677972	−	0.735088i	$-0.262859\pi$
$278$	− 7964.69i	− 1.71831i
$279$	0	0
$280$	0	0
$281$	−827.653	−0.175707	−0.0878535	−	0.996133i	$-0.528001\pi$
$281$	−827.653	−0.175707	−0.0878535	+	0.996133i	$0.528001\pi$
$282$	0	0
$283$	− 3171.98i	− 0.666270i	−0.942879	−	0.333135i	$-0.891894\pi$
$283$	− 3171.98i	− 0.666270i	0.942879	−	0.333135i	$-0.108106\pi$
$284$	−10749.2	−2.24593
$285$	0	0
$286$	5514.78	1.14020
$287$	− 862.623i	− 0.177418i
$288$	0	0
$289$	173.358	0.0352856
$290$	0	0
$291$	0	0
$292$	− 10300.8i	− 2.06440i
$293$	− 1376.02i	− 0.274362i	−0.990546	−	0.137181i	$-0.956196\pi$
$293$	− 1376.02i	− 0.274362i	0.990546	−	0.137181i	$-0.0438043\pi$
$294$	0	0
$295$	0	0
$296$	−4516.73	−0.886923
$297$	0	0
$298$	6996.02i	1.35996i
$299$	−3061.27	−0.592099
$300$	0	0
$301$	−2228.61	−0.426760
$302$	− 1954.67i	− 0.372445i
$303$	0	0
$304$	−726.864	−0.137133
$305$	0	0
$306$	0	0
$307$	119.504i	0.0222165i	0.999938	+	0.0111083i	$0.00353594\pi$
$307$	119.504i	0.0222165i	−0.999938	+	0.0111083i	$0.996464\pi$
$308$	− 3521.13i	− 0.651412i
$309$	0	0
$310$	0	0
$311$	−2139.35	−0.390069	−0.195035	−	0.980796i	$-0.562482\pi$
$311$	−2139.35	−0.390069	−0.195035	+	0.980796i	$0.562482\pi$
$312$	0	0
$313$	5163.50i	0.932455i	0.884665	+	0.466227i	$0.154387\pi$
$313$	5163.50i	0.932455i	−0.884665	+	0.466227i	$0.845613\pi$
$314$	−199.548	−0.0358635
$315$	0	0
$316$	2459.95	0.437922
$317$	− 8631.69i	− 1.52935i	−0.644416	−	0.764675i	$-0.722899\pi$
$317$	− 8631.69i	− 1.52935i	0.644416	−	0.764675i	$-0.277101\pi$
$318$	0	0
$319$	12829.5	2.25177
$320$	0	0
$321$	0	0
$322$	3270.13i	0.565955i
$323$	− 2809.79i	− 0.484028i
$324$	0	0
$325$	0	0
$326$	11647.2	1.97877
$327$	0	0
$328$	− 2941.98i	− 0.495255i
$329$	−1298.16	−0.217537
$330$	0	0
$331$	−2942.34	−0.488597	−0.244298	−	0.969700i	$-0.578558\pi$
$331$	−2942.34	−0.488597	−0.244298	+	0.969700i	$0.578558\pi$
$332$	− 5507.98i	− 0.910512i
$333$	0	0
$334$	842.585	0.138037
$335$	0	0
$336$	0	0
$337$	− 9897.46i	− 1.59985i	−0.600101	−	0.799924i	$-0.704873\pi$
$337$	− 9897.46i	− 1.59985i	0.600101	−	0.799924i	$-0.295127\pi$
$338$	7790.79i	1.25374i
$339$	0	0
$340$	0	0
$341$	−16997.3	−2.69928
$342$	0	0
$343$	− 3354.53i	− 0.528070i
$344$	−7600.68	−1.19128
$345$	0	0
$346$	−6711.46	−1.04280
$347$	− 12015.7i	− 1.85890i	−0.368947	−	0.929451i	$-0.620282\pi$
$347$	− 12015.7i	− 1.85890i	0.368947	−	0.929451i	$-0.379718\pi$
$348$	0	0
$349$	−7894.62	−1.21086	−0.605428	−	0.795900i	$-0.706998\pi$
$349$	−7894.62	−1.21086	−0.605428	+	0.795900i	$0.706998\pi$
$350$	0	0
$351$	0	0
$352$	12712.8i	1.92499i
$353$	− 741.154i	− 0.111750i	−0.998438	−	0.0558748i	$-0.982205\pi$
$353$	− 741.154i	− 0.111750i	0.998438	−	0.0558748i	$-0.0177948\pi$
$354$	0	0
$355$	0	0
$356$	−7152.30	−1.06481
$357$	0	0
$358$	− 13987.4i	− 2.06496i
$359$	−11564.4	−1.70012	−0.850060	−	0.526685i	$-0.823435\pi$
$359$	−11564.4	−1.70012	−0.850060	+	0.526685i	$0.823435\pi$
$360$	0	0
$361$	−5193.28	−0.757148
$362$	20123.7i	2.92177i
$363$	0	0
$364$	−1281.12	−0.184474
$365$	0	0
$366$	0	0
$367$	1148.57i	0.163365i	0.996658	+	0.0816823i	$0.0260293\pi$
$367$	1148.57i	0.163365i	−0.996658	+	0.0816823i	$0.973971\pi$
$368$	− 2570.24i	− 0.364084i
$369$	0	0
$370$	0	0
$371$	−1091.01	−0.152676
$372$	0	0
$373$	− 4602.98i	− 0.638963i	−0.947593	−	0.319481i	$-0.896491\pi$
$373$	− 4602.98i	− 0.638963i	0.947593	−	0.319481i	$-0.103509\pi$
$374$	−17898.7	−2.47465
$375$	0	0
$376$	−4427.38	−0.607246
$377$	− 4667.85i	− 0.637683i
$378$	0	0
$379$	3988.46	0.540563	0.270282	−	0.962781i	$-0.412883\pi$
$379$	3988.46	0.540563	0.270282	+	0.962781i	$0.412883\pi$
$380$	0	0
$381$	0	0
$382$	− 5384.41i	− 0.721179i
$383$	− 7788.20i	− 1.03906i	−0.854454	−	0.519528i	$-0.826108\pi$
$383$	− 7788.20i	− 1.03906i	0.854454	−	0.519528i	$-0.173892\pi$
$384$	0	0
$385$	0	0
$386$	−4116.74	−0.542841
$387$	0	0
$388$	2726.23i	0.356710i
$389$	8524.87	1.11113	0.555563	−	0.831474i	$-0.312503\pi$
$389$	8524.87	1.11113	0.555563	+	0.831474i	$0.312503\pi$
$390$	0	0
$391$	9935.60	1.28508
$392$	− 5496.62i	− 0.708217i
$393$	0	0
$394$	5265.97	0.673339
$395$	0	0
$396$	0	0
$397$	155.729i	0.0196872i	0.999952	+	0.00984361i	$0.00313337\pi$
$397$	155.729i	0.0196872i	−0.999952	+	0.00984361i	$0.996867\pi$
$398$	3745.01i	0.471659i
$399$	0	0
$400$	0	0
$401$	−5933.96	−0.738972	−0.369486	−	0.929236i	$-0.620466\pi$
$401$	−5933.96	−0.738972	−0.369486	+	0.929236i	$0.620466\pi$
$402$	0	0
$403$	6184.23i	0.764414i
$404$	15994.7	1.96971
$405$	0	0
$406$	−4986.33	−0.609526
$407$	15195.3i	1.85062i
$408$	0	0
$409$	−14161.4	−1.71207	−0.856035	−	0.516917i	$-0.827079\pi$
$409$	−14161.4	−1.71207	−0.856035	+	0.516917i	$0.827079\pi$
$410$	0	0
$411$	0	0
$412$	18973.8i	2.26886i
$413$	− 1684.55i	− 0.200705i
$414$	0	0
$415$	0	0
$416$	4625.39	0.545140
$417$	0	0
$418$	− 10610.8i	− 1.24161i
$419$	9624.24	1.12214	0.561068	−	0.827770i	$-0.310391\pi$
$419$	9624.24	1.12214	0.561068	+	0.827770i	$0.310391\pi$
$420$	0	0
$421$	−1536.26	−0.177845	−0.0889223	−	0.996039i	$-0.528342\pi$
$421$	−1536.26	−0.177845	−0.0889223	+	0.996039i	$0.528342\pi$
$422$	− 11548.2i	− 1.33213i
$423$	0	0
$424$	−3720.91	−0.426187
$425$	0	0
$426$	0	0
$427$	− 279.288i	− 0.0316527i
$428$	− 11396.2i	− 1.28705i
$429$	0	0
$430$	0	0
$431$	−11582.2	−1.29441	−0.647207	−	0.762314i	$-0.724063\pi$
$431$	−11582.2	−1.29441	−0.647207	+	0.762314i	$0.724063\pi$
$432$	0	0
$433$	14892.6i	1.65287i	0.563029	+	0.826437i	$0.309636\pi$
$433$	14892.6i	1.65287i	−0.563029	+	0.826437i	$0.690364\pi$
$434$	6606.17	0.730660
$435$	0	0
$436$	20090.2	2.20676
$437$	5890.10i	0.644764i
$438$	0	0
$439$	1642.51	0.178571	0.0892853	−	0.996006i	$-0.471542\pi$
$439$	1642.51	0.178571	0.0892853	+	0.996006i	$0.471542\pi$
$440$	0	0
$441$	0	0
$442$	6512.20i	0.700800i
$443$	− 3916.31i	− 0.420021i	−0.977699	−	0.210011i	$-0.932650\pi$
$443$	− 3916.31i	− 0.420021i	0.977699	−	0.210011i	$-0.0673499\pi$
$444$	0	0
$445$	0	0
$446$	18643.6	1.97937
$447$	0	0
$448$	− 4217.02i	− 0.444722i
$449$	−3985.25	−0.418876	−0.209438	−	0.977822i	$-0.567163\pi$
$449$	−3985.25	−0.418876	−0.209438	+	0.977822i	$0.567163\pi$
$450$	0	0
$451$	−9897.50	−1.03338
$452$	− 138.130i	− 0.0143741i
$453$	0	0
$454$	11606.3	1.19980
$455$	0	0
$456$	0	0
$457$	14177.8i	1.45122i	0.688105	+	0.725611i	$0.258443\pi$
$457$	14177.8i	1.45122i	−0.688105	+	0.725611i	$0.741557\pi$
$458$	8229.13i	0.839567i
$459$	0	0
$460$	0	0
$461$	16394.3	1.65631	0.828154	−	0.560500i	$-0.189391\pi$
$461$	16394.3	1.65631	0.828154	+	0.560500i	$0.189391\pi$
$462$	0	0
$463$	3319.60i	0.333207i	0.986024	+	0.166603i	$0.0532800\pi$
$463$	3319.60i	0.333207i	−0.986024	+	0.166603i	$0.946720\pi$
$464$	3919.12	0.392114
$465$	0	0
$466$	1073.09	0.106674
$467$	− 2529.79i	− 0.250674i	−0.992114	−	0.125337i	$-0.959999\pi$
$467$	− 2529.79i	− 0.250674i	0.992114	−	0.125337i	$-0.0400012\pi$
$468$	0	0
$469$	−3852.49	−0.379299
$470$	0	0
$471$	0	0
$472$	− 5745.17i	− 0.560260i
$473$	25570.4i	2.48569i
$474$	0	0
$475$	0	0
$476$	4157.97	0.400379
$477$	0	0
$478$	11450.1i	1.09564i
$479$	−8646.75	−0.824802	−0.412401	−	0.911002i	$-0.635310\pi$
$479$	−8646.75	−0.824802	−0.412401	+	0.911002i	$0.635310\pi$
$480$	0	0
$481$	5528.60	0.524080
$482$	17784.0i	1.68058i
$483$	0	0
$484$	−24580.1	−2.30842
$485$	0	0
$486$	0	0
$487$	15251.7i	1.41914i	0.704636	+	0.709569i	$0.251110\pi$
$487$	15251.7i	1.41914i	−0.704636	+	0.709569i	$0.748890\pi$
$488$	− 952.515i	− 0.0883572i
$489$	0	0
$490$	0	0
$491$	1766.87	0.162398	0.0811992	−	0.996698i	$-0.474125\pi$
$491$	1766.87	0.162398	0.0811992	+	0.996698i	$0.474125\pi$
$492$	0	0
$493$	15149.9i	1.38401i
$494$	−3860.61	−0.351614
$495$	0	0
$496$	−5192.28	−0.470041
$497$	− 4595.20i	− 0.414734i
$498$	0	0
$499$	−11733.8	−1.05266	−0.526332	−	0.850279i	$-0.676433\pi$
$499$	−11733.8	−1.05266	−0.526332	+	0.850279i	$0.676433\pi$
$500$	0	0
$501$	0	0
$502$	− 4313.98i	− 0.383550i
$503$	977.608i	0.0866589i	0.999061	+	0.0433294i	$0.0137965\pi$
$503$	977.608i	0.0866589i	−0.999061	+	0.0433294i	$0.986203\pi$
$504$	0	0
$505$	0	0
$506$	37520.6	3.29643
$507$	0	0
$508$	3674.48i	0.320923i
$509$	9674.72	0.842484	0.421242	−	0.906948i	$-0.361594\pi$
$509$	9674.72	0.842484	0.421242	+	0.906948i	$0.361594\pi$
$510$	0	0
$511$	4403.51	0.381213
$512$	6352.52i	0.548329i
$513$	0	0
$514$	−16691.6	−1.43237
$515$	0	0
$516$	0	0
$517$	14894.7i	1.26706i
$518$	− 5905.81i	− 0.500939i
$519$	0	0
$520$	0	0
$521$	−5178.95	−0.435497	−0.217748	−	0.976005i	$-0.569871\pi$
$521$	−5178.95	−0.435497	−0.217748	+	0.976005i	$0.569871\pi$
$522$	0	0
$523$	14280.7i	1.19398i	0.802248	+	0.596992i	$0.203637\pi$
$523$	14280.7i	1.19398i	−0.802248	+	0.596992i	$0.796363\pi$
$524$	33103.1	2.75976
$525$	0	0
$526$	−11137.2	−0.923203
$527$	− 20071.5i	− 1.65906i
$528$	0	0
$529$	−8660.78	−0.711826
$530$	0	0
$531$	0	0
$532$	2464.96i	0.200883i
$533$	3601.07i	0.292645i
$534$	0	0
$535$	0	0
$536$	−13138.9	−1.05880
$537$	0	0
$538$	− 9556.08i	− 0.765784i
$539$	−18491.9	−1.47774
$540$	0	0
$541$	12923.8	1.02706	0.513529	−	0.858072i	$-0.328338\pi$
$541$	12923.8	1.02706	0.513529	+	0.858072i	$0.328338\pi$
$542$	6870.32i	0.544475i
$543$	0	0
$544$	−15012.1	−1.18316
$545$	0	0
$546$	0	0
$547$	− 13653.3i	− 1.06723i	−0.845728	−	0.533614i	$-0.820834\pi$
$547$	− 13653.3i	− 1.06723i	0.845728	−	0.533614i	$-0.179166\pi$
$548$	29589.4i	2.30656i
$549$	0	0
$550$	0	0
$551$	−8981.28	−0.694402
$552$	0	0
$553$	1051.62i	0.0808666i
$554$	30224.8	2.31792
$555$	0	0
$556$	21229.2	1.61928
$557$	− 9313.63i	− 0.708494i	−0.935152	−	0.354247i	$-0.884737\pi$
$557$	− 9313.63i	− 0.708494i	0.935152	−	0.354247i	$-0.115263\pi$
$558$	0	0
$559$	9303.46	0.703925
$560$	0	0
$561$	0	0
$562$	− 3690.82i	− 0.277025i
$563$	10625.4i	0.795394i	0.917517	+	0.397697i	$0.130190\pi$
$563$	10625.4i	0.795394i	−0.917517	+	0.397697i	$0.869810\pi$
$564$	0	0
$565$	0	0
$566$	14145.1	1.05046
$567$	0	0
$568$	− 15672.0i	− 1.15771i
$569$	9060.50	0.667550	0.333775	−	0.942653i	$-0.391677\pi$
$569$	9060.50	0.667550	0.333775	+	0.942653i	$0.391677\pi$
$570$	0	0
$571$	−21379.1	−1.56688	−0.783440	−	0.621467i	$-0.786537\pi$
$571$	−21379.1	−1.56688	−0.783440	+	0.621467i	$0.786537\pi$
$572$	14699.2i	1.07448i
$573$	0	0
$574$	3846.77	0.279723
$575$	0	0
$576$	0	0
$577$	− 6347.76i	− 0.457991i	−0.973427	−	0.228996i	$-0.926456\pi$
$577$	− 6347.76i	− 0.457991i	0.973427	−	0.228996i	$-0.0735441\pi$
$578$	773.071i	0.0556324i
$579$	0	0
$580$	0	0
$581$	2354.63	0.168135
$582$	0	0
$583$	12518.0i	0.889266i
$584$	15018.2	1.06414
$585$	0	0
$586$	6136.22	0.432568
$587$	14773.7i	1.03880i	0.854531	+	0.519401i	$0.173845\pi$
$587$	14773.7i	1.03880i	−0.854531	+	0.519401i	$0.826155\pi$
$588$	0	0
$589$	11898.9	0.832405
$590$	0	0
$591$	0	0
$592$	4641.81i	0.322259i
$593$	− 26868.1i	− 1.86061i	−0.366790	−	0.930304i	$-0.619543\pi$
$593$	− 26868.1i	− 1.86061i	0.366790	−	0.930304i	$-0.380457\pi$
$594$	0	0
$595$	0	0
$596$	−18647.3	−1.28158
$597$	0	0
$598$	− 13651.4i	− 0.933522i
$599$	−1749.83	−0.119359	−0.0596794	−	0.998218i	$-0.519008\pi$
$599$	−1749.83	−0.119359	−0.0596794	+	0.998218i	$0.519008\pi$
$600$	0	0
$601$	−17964.0	−1.21924	−0.609622	−	0.792692i	$-0.708679\pi$
$601$	−17964.0	−1.21924	−0.609622	+	0.792692i	$0.708679\pi$
$602$	− 9938.21i	− 0.672843i
$603$	0	0
$604$	5209.99	0.350979
$605$	0	0
$606$	0	0
$607$	4418.22i	0.295437i	0.989029	+	0.147718i	$0.0471929\pi$
$607$	4418.22i	0.295437i	−0.989029	+	0.147718i	$0.952807\pi$
$608$	− 8899.58i	− 0.593628i
$609$	0	0
$610$	0	0
$611$	5419.24	0.358820
$612$	0	0
$613$	− 20179.7i	− 1.32961i	−0.747018	−	0.664804i	$-0.768515\pi$
$613$	− 20179.7i	− 1.32961i	0.747018	−	0.664804i	$-0.231485\pi$
$614$	−532.915	−0.0350272
$615$	0	0
$616$	5133.71	0.335784
$617$	− 4673.01i	− 0.304908i	−0.988311	−	0.152454i	$-0.951282\pi$
$617$	− 4673.01i	− 0.304908i	0.988311	−	0.152454i	$-0.0487176\pi$
$618$	0	0
$619$	19976.8	1.29715	0.648574	−	0.761151i	$-0.275366\pi$
$619$	19976.8	1.29715	0.648574	+	0.761151i	$0.275366\pi$
$620$	0	0
$621$	0	0
$622$	− 9540.19i	− 0.614994i
$623$	− 3057.56i	− 0.196627i
$624$	0	0
$625$	0	0
$626$	−23026.0	−1.47014
$627$	0	0
$628$	− 531.877i	− 0.0337965i
$629$	−17943.5	−1.13745
$630$	0	0
$631$	10457.5	0.659757	0.329879	−	0.944023i	$-0.392992\pi$
$631$	10457.5	0.659757	0.329879	+	0.944023i	$0.392992\pi$
$632$	3586.54i	0.225736i
$633$	0	0
$634$	38492.0	2.41122
$635$	0	0
$636$	0	0
$637$	6728.02i	0.418483i
$638$	57211.8i	3.55021i
$639$	0	0
$640$	0	0
$641$	−4395.86	−0.270867	−0.135434	−	0.990786i	$-0.543243\pi$
$641$	−4395.86	−0.270867	−0.135434	+	0.990786i	$0.543243\pi$
$642$	0	0
$643$	5786.36i	0.354886i	0.984131	+	0.177443i	$0.0567825\pi$
$643$	5786.36i	0.354886i	−0.984131	+	0.177443i	$0.943217\pi$
$644$	−8716.26	−0.533336
$645$	0	0
$646$	12529.9	0.763133
$647$	− 25367.7i	− 1.54143i	−0.637178	−	0.770717i	$-0.719898\pi$
$647$	− 25367.7i	− 1.54143i	0.637178	−	0.770717i	$-0.280102\pi$
$648$	0	0
$649$	−19328.1	−1.16902
$650$	0	0
$651$	0	0
$652$	31044.6i	1.86472i
$653$	21633.2i	1.29644i	0.761454	+	0.648218i	$0.224486\pi$
$653$	21633.2i	1.29644i	−0.761454	+	0.648218i	$0.775514\pi$
$654$	0	0
$655$	0	0
$656$	−3023.46	−0.179948
$657$	0	0
$658$	− 5788.98i	− 0.342976i
$659$	18312.4	1.08248	0.541238	−	0.840870i	$-0.317956\pi$
$659$	18312.4	1.08248	0.541238	+	0.840870i	$0.317956\pi$
$660$	0	0
$661$	−5526.08	−0.325174	−0.162587	−	0.986694i	$-0.551984\pi$
$661$	−5526.08	−0.325174	−0.162587	+	0.986694i	$0.551984\pi$
$662$	− 13121.0i	− 0.770337i
$663$	0	0
$664$	8030.48	0.469342
$665$	0	0
$666$	0	0
$667$	− 31758.4i	− 1.84361i
$668$	2245.84i	0.130081i
$669$	0	0
$670$	0	0
$671$	−3204.48	−0.184363
$672$	0	0
$673$	1437.24i	0.0823204i	0.999153	+	0.0411602i	$0.0131054\pi$
$673$	1437.24i	0.0823204i	−0.999153	+	0.0411602i	$0.986895\pi$
$674$	44136.6	2.52237
$675$	0	0
$676$	−20765.7	−1.18148
$677$	− 23405.9i	− 1.32875i	−0.747401	−	0.664373i	$-0.768699\pi$
$677$	− 23405.9i	− 1.32875i	0.747401	−	0.664373i	$-0.231301\pi$
$678$	0	0
$679$	−1165.45	−0.0658701
$680$	0	0
$681$	0	0
$682$	− 75797.4i	− 4.25577i
$683$	− 19227.4i	− 1.07719i	−0.842566	−	0.538593i	$-0.818956\pi$
$683$	− 19227.4i	− 1.07719i	0.842566	−	0.538593i	$-0.181044\pi$
$684$	0	0
$685$	0	0
$686$	14959.2	0.832570
$687$	0	0
$688$	7811.17i	0.432846i
$689$	4554.50	0.251833
$690$	0	0
$691$	−35284.8	−1.94254	−0.971271	−	0.237975i	$-0.923516\pi$
$691$	−35284.8	−1.94254	−0.971271	+	0.237975i	$0.923516\pi$
$692$	− 17888.8i	− 0.982703i
$693$	0	0
$694$	53582.8	2.93080
$695$	0	0
$696$	0	0
$697$	− 11687.6i	− 0.635149i
$698$	− 35205.1i	− 1.90907i
$699$	0	0
$700$	0	0
$701$	−9173.00	−0.494236	−0.247118	−	0.968985i	$-0.579484\pi$
$701$	−9173.00	−0.494236	−0.247118	+	0.968985i	$0.579484\pi$
$702$	0	0
$703$	− 10637.4i	− 0.570695i
$704$	−48384.9	−2.59030
$705$	0	0
$706$	3305.09	0.176188
$707$	6837.63i	0.363728i
$708$	0	0
$709$	33951.6	1.79842	0.899210	−	0.437517i	$-0.144142\pi$
$709$	33951.6	1.79842	0.899210	+	0.437517i	$0.144142\pi$
$710$	0	0
$711$	0	0
$712$	− 10427.8i	− 0.548876i
$713$	42075.3i	2.21001i
$714$	0	0
$715$	0	0
$716$	37282.1	1.94595
$717$	0	0
$718$	− 51569.9i	− 2.68046i
$719$	−6727.90	−0.348968	−0.174484	−	0.984660i	$-0.555826\pi$
$719$	−6727.90	−0.348968	−0.174484	+	0.984660i	$0.555826\pi$
$720$	0	0
$721$	−8111.17	−0.418968
$722$	− 23158.8i	− 1.19374i
$723$	0	0
$724$	−53638.0	−2.75337
$725$	0	0
$726$	0	0
$727$	36726.1i	1.87359i	0.349885	+	0.936793i	$0.386221\pi$
$727$	36726.1i	1.87359i	−0.349885	+	0.936793i	$0.613779\pi$
$728$	− 1867.83i	− 0.0950912i
$729$	0	0
$730$	0	0
$731$	−30195.1	−1.52778
$732$	0	0
$733$	26691.4i	1.34498i	0.740108	+	0.672489i	$0.234775\pi$
$733$	26691.4i	1.34498i	−0.740108	+	0.672489i	$0.765225\pi$
$734$	−5121.91	−0.257566
$735$	0	0
$736$	31469.5	1.57606
$737$	44202.4i	2.20925i
$738$	0	0
$739$	−12207.0	−0.607634	−0.303817	−	0.952730i	$-0.598261\pi$
$739$	−12207.0	−0.607634	−0.303817	+	0.952730i	$0.598261\pi$
$740$	0	0
$741$	0	0
$742$	− 4865.25i	− 0.240713i
$743$	12473.3i	0.615882i	0.951405	+	0.307941i	$0.0996400\pi$
$743$	12473.3i	0.615882i	−0.951405	+	0.307941i	$0.900360\pi$
$744$	0	0
$745$	0	0
$746$	20526.4	1.00741
$747$	0	0
$748$	− 47707.4i	− 2.33202i
$749$	4871.82	0.237667
$750$	0	0
$751$	−15102.6	−0.733825	−0.366913	−	0.930255i	$-0.619585\pi$
$751$	−15102.6	−0.733825	−0.366913	+	0.930255i	$0.619585\pi$
$752$	4549.99i	0.220640i
$753$	0	0
$754$	20815.7	1.00539
$755$	0	0
$756$	0	0
$757$	3418.34i	0.164124i	0.996627	+	0.0820618i	$0.0261505\pi$
$757$	3418.34i	0.164124i	−0.996627	+	0.0820618i	$0.973850\pi$
$758$	17786.1i	0.852268i
$759$	0	0
$760$	0	0
$761$	12684.5	0.604224	0.302112	−	0.953272i	$-0.402308\pi$
$761$	12684.5	0.604224	0.302112	+	0.953272i	$0.402308\pi$
$762$	0	0
$763$	8588.45i	0.407500i
$764$	14351.7	0.679614
$765$	0	0
$766$	34730.6	1.63821
$767$	7032.25i	0.331056i
$768$	0	0
$769$	27580.2	1.29333	0.646663	−	0.762776i	$-0.276164\pi$
$769$	27580.2	1.29333	0.646663	+	0.762776i	$0.276164\pi$
$770$	0	0
$771$	0	0
$772$	− 10972.8i	− 0.511555i
$773$	− 17386.0i	− 0.808966i	−0.914546	−	0.404483i	$-0.867452\pi$
$773$	− 17386.0i	− 0.808966i	0.914546	−	0.404483i	$-0.132548\pi$
$774$	0	0
$775$	0	0
$776$	−3974.77	−0.183874
$777$	0	0
$778$	38015.7i	1.75183i
$779$	6928.72	0.318674
$780$	0	0
$781$	−52724.1	−2.41564
$782$	44306.7i	2.02609i
$783$	0	0
$784$	−5648.84	−0.257327
$785$	0	0
$786$	0	0
$787$	− 4680.29i	− 0.211988i	−0.994367	−	0.105994i	$-0.966198\pi$
$787$	− 4680.29i	− 0.211988i	0.994367	−	0.105994i	$-0.0338024\pi$
$788$	14036.0i	0.634531i
$789$	0	0
$790$	0	0
$791$	59.0498	0.00265432
$792$	0	0
$793$	1165.91i	0.0522100i
$794$	−694.456	−0.0310395
$795$	0	0
$796$	−9982.00	−0.444476
$797$	7278.62i	0.323490i	0.986833	+	0.161745i	$0.0517123\pi$
$797$	7278.62i	0.323490i	−0.986833	+	0.161745i	$0.948288\pi$
$798$	0	0
$799$	−17588.6	−0.778773
$800$	0	0
$801$	0	0
$802$	− 26461.8i	− 1.16508i
$803$	− 50524.7i	− 2.22039i
$804$	0	0
$805$	0	0
$806$	−27577.9	−1.20520
$807$	0	0
$808$	23319.8i	1.01533i
$809$	29212.5	1.26954	0.634770	−	0.772701i	$-0.281095\pi$
$809$	29212.5	1.26954	0.634770	+	0.772701i	$0.281095\pi$
$810$	0	0
$811$	41992.4	1.81819	0.909094	−	0.416590i	$-0.136775\pi$
$811$	41992.4	1.81819	0.909094	+	0.416590i	$0.136775\pi$
$812$	− 13290.6i	− 0.574396i
$813$	0	0
$814$	−67761.6	−2.91774
$815$	0	0
$816$	0	0
$817$	− 17900.5i	− 0.766536i
$818$	− 63151.2i	− 2.69930i
$819$	0	0
$820$	0	0
$821$	8722.99	0.370809	0.185405	−	0.982662i	$-0.440640\pi$
$821$	8722.99	0.370809	0.185405	+	0.982662i	$0.440640\pi$
$822$	0	0
$823$	13584.8i	0.575379i	0.957724	+	0.287690i	$0.0928871\pi$
$823$	13584.8i	0.575379i	−0.957724	+	0.287690i	$0.907113\pi$
$824$	−27663.2	−1.16953
$825$	0	0
$826$	7512.05	0.316438
$827$	26573.6i	1.11736i	0.829384	+	0.558679i	$0.188692\pi$
$827$	26573.6i	1.11736i	−0.829384	+	0.558679i	$0.811308\pi$
$828$	0	0
$829$	43238.4	1.81150	0.905748	−	0.423816i	$-0.139310\pi$
$829$	43238.4	1.81150	0.905748	+	0.423816i	$0.139310\pi$
$830$	0	0
$831$	0	0
$832$	17604.2i	0.733552i
$833$	− 21836.3i	− 0.908265i
$834$	0	0
$835$	0	0
$836$	28282.3	1.17005
$837$	0	0
$838$	42918.2i	1.76919i
$839$	−4093.21	−0.168431	−0.0842154	−	0.996448i	$-0.526838\pi$
$839$	−4093.21	−0.168431	−0.0842154	+	0.996448i	$0.526838\pi$
$840$	0	0
$841$	24036.5	0.985546
$842$	− 6850.75i	− 0.280395i
$843$	0	0
$844$	30780.8	1.25535
$845$	0	0
$846$	0	0
$847$	− 10507.8i	− 0.426273i
$848$	3823.96i	0.154853i
$849$	0	0
$850$	0	0
$851$	37614.7	1.51517
$852$	0	0
$853$	− 20201.5i	− 0.810886i	−0.914120	−	0.405443i	$-0.867117\pi$
$853$	− 20201.5i	− 0.810886i	0.914120	−	0.405443i	$-0.132883\pi$
$854$	1245.45	0.0499046
$855$	0	0
$856$	16615.4	0.663436
$857$	4551.65i	0.181425i	0.995877	+	0.0907126i	$0.0289145\pi$
$857$	4551.65i	0.181425i	−0.995877	+	0.0907126i	$0.971086\pi$
$858$	0	0
$859$	−11962.6	−0.475154	−0.237577	−	0.971369i	$-0.576353\pi$
$859$	−11962.6	−0.475154	−0.237577	+	0.971369i	$0.576353\pi$
$860$	0	0
$861$	0	0
$862$	− 51649.3i	− 2.04081i
$863$	7164.61i	0.282603i	0.989967	+	0.141301i	$0.0451287\pi$
$863$	7164.61i	0.282603i	−0.989967	+	0.141301i	$0.954871\pi$
$864$	0	0
$865$	0	0
$866$	−66412.0	−2.60597
$867$	0	0
$868$	17608.2i	0.688549i
$869$	12065.9	0.471012
$870$	0	0
$871$	16082.4	0.625640
$872$	29291.0i	1.13752i
$873$	0	0
$874$	−26266.2	−1.01655
$875$	0	0
$876$	0	0
$877$	19218.7i	0.739987i	0.929034	+	0.369994i	$0.120640\pi$
$877$	19218.7i	0.739987i	−0.929034	+	0.369994i	$0.879360\pi$
$878$	7324.56i	0.281540i
$879$	0	0
$880$	0	0
$881$	45914.5	1.75585	0.877923	−	0.478802i	$-0.158929\pi$
$881$	45914.5	1.75585	0.877923	+	0.478802i	$0.158929\pi$
$882$	0	0
$883$	− 44656.7i	− 1.70194i	−0.525211	−	0.850972i	$-0.676014\pi$
$883$	− 44656.7i	− 1.70194i	0.525211	−	0.850972i	$-0.323986\pi$
$884$	−17357.7	−0.660410
$885$	0	0
$886$	17464.3	0.662218
$887$	14975.2i	0.566873i	0.958991	+	0.283437i	$0.0914745\pi$
$887$	14975.2i	0.566873i	−0.958991	+	0.283437i	$0.908525\pi$
$888$	0	0
$889$	−1570.82	−0.0592616
$890$	0	0
$891$	0	0
$892$	49692.9i	1.86529i
$893$	− 10427.0i	− 0.390735i
$894$	0	0
$895$	0	0
$896$	9941.33	0.370666
$897$	0	0
$898$	− 17771.7i	− 0.660413i
$899$	−64156.8	−2.38015
$900$	0	0
$901$	−14782.0	−0.546571
$902$	− 44136.7i	− 1.62926i
$903$	0	0
$904$	201.390	0.00740943
$905$	0	0
$906$	0	0
$907$	14818.1i	0.542479i	0.962512	+	0.271240i	$0.0874336\pi$
$907$	14818.1i	0.542479i	−0.962512	+	0.271240i	$0.912566\pi$
$908$	30935.6i	1.13065i
$909$	0	0
$910$	0	0
$911$	−21846.5	−0.794518	−0.397259	−	0.917707i	$-0.630039\pi$
$911$	−21846.5	−0.794518	−0.397259	+	0.917707i	$0.630039\pi$
$912$	0	0
$913$	− 27016.4i	− 0.979312i
$914$	−63224.2	−2.28804
$915$	0	0
$916$	−21934.0	−0.791179
$917$	14151.4i	0.509618i
$918$	0	0
$919$	−28878.9	−1.03659	−0.518296	−	0.855201i	$-0.673434\pi$
$919$	−28878.9	−1.03659	−0.518296	+	0.855201i	$0.673434\pi$
$920$	0	0
$921$	0	0
$922$	73108.4i	2.61139i
$923$	19182.9i	0.684089i
$924$	0	0
$925$	0	0
$926$	−14803.4	−0.525344
$927$	0	0
$928$	47985.0i	1.69740i
$929$	4608.79	0.162766	0.0813829	−	0.996683i	$-0.474066\pi$
$929$	4608.79	0.162766	0.0813829	+	0.996683i	$0.474066\pi$
$930$	0	0
$931$	12945.2	0.455705
$932$	2860.23i	0.100526i
$933$	0	0
$934$	11281.3	0.395220
$935$	0	0
$936$	0	0
$937$	21063.8i	0.734391i	0.930144	+	0.367195i	$0.119682\pi$
$937$	21063.8i	0.734391i	−0.930144	+	0.367195i	$0.880318\pi$
$938$	− 17179.7i	− 0.598014i
$939$	0	0
$940$	0	0
$941$	20244.8	0.701339	0.350670	−	0.936499i	$-0.385954\pi$
$941$	20244.8	0.701339	0.350670	+	0.936499i	$0.385954\pi$
$942$	0	0
$943$	24500.4i	0.846069i
$944$	−5904.27	−0.203567
$945$	0	0
$946$	−114028.	−3.91901
$947$	− 29978.3i	− 1.02868i	−0.857585	−	0.514342i	$-0.828036\pi$
$947$	− 29978.3i	− 1.02868i	0.857585	−	0.514342i	$-0.171964\pi$
$948$	0	0
$949$	−18382.7	−0.628797
$950$	0	0
$951$	0	0
$952$	6062.20i	0.206383i
$953$	− 5782.65i	− 0.196556i	−0.995159	−	0.0982782i	$-0.968666\pi$
$953$	− 5782.65i	− 0.196556i	0.995159	−	0.0982782i	$-0.0313335\pi$
$954$	0	0
$955$	0	0
$956$	−30519.2	−1.03249
$957$	0	0
$958$	− 38559.2i	− 1.30041i
$959$	−12649.3	−0.425930
$960$	0	0
$961$	55207.7	1.85317
$962$	24654.2i	0.826281i
$963$	0	0
$964$	−47401.6	−1.58372
$965$	0	0
$966$	0	0
$967$	26119.2i	0.868600i	0.900768	+	0.434300i	$0.143004\pi$
$967$	26119.2i	0.868600i	−0.900768	+	0.434300i	$0.856996\pi$
$968$	− 35837.0i	− 1.18992i
$969$	0	0
$970$	0	0
$971$	−5101.93	−0.168619	−0.0843093	−	0.996440i	$-0.526868\pi$
$971$	−5101.93	−0.168619	−0.0843093	+	0.996440i	$0.526868\pi$
$972$	0	0
$973$	9075.34i	0.299016i
$974$	−68013.1	−2.23746
$975$	0	0
$976$	−978.894	−0.0321041
$977$	− 45902.4i	− 1.50312i	−0.659666	−	0.751559i	$-0.729302\pi$
$977$	− 45902.4i	− 1.50312i	0.659666	−	0.751559i	$-0.270698\pi$
$978$	0	0
$979$	−35081.6	−1.14526
$980$	0	0
$981$	0	0
$982$	7879.14i	0.256042i
$983$	− 13416.5i	− 0.435321i	−0.976025	−	0.217661i	$-0.930157\pi$
$983$	− 13416.5i	− 0.435321i	0.976025	−	0.217661i	$-0.0698426\pi$
$984$	0	0
$985$	0	0
$986$	−67559.2	−2.18207
$987$	0	0
$988$	− 10290.1i	− 0.331349i
$989$	63297.4	2.03513
$990$	0	0
$991$	3806.66	0.122021	0.0610104	−	0.998137i	$-0.480568\pi$
$991$	3806.66	0.122021	0.0610104	+	0.998137i	$0.480568\pi$
$992$	− 63573.2i	− 2.03473i
$993$	0	0
$994$	20491.8	0.653883
$995$	0	0
$996$	0	0
$997$	22523.8i	0.715483i	0.933821	+	0.357742i	$0.116453\pi$
$997$	22523.8i	0.715483i	−0.933821	+	0.357742i	$0.883547\pi$
$998$	− 52325.7i	− 1.65966i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.k.649.5		6
3.2	odd	2		675.4.b.l.649.2		6
5.2	odd	4		135.4.a.g.1.1	yes	3
5.3	odd	4		675.4.a.q.1.3		3
5.4	even	2	inner	675.4.b.k.649.2		6
15.2	even	4		135.4.a.f.1.3	✓	3
15.8	even	4		675.4.a.r.1.1		3
15.14	odd	2		675.4.b.l.649.5		6
20.7	even	4		2160.4.a.be.1.3		3
45.2	even	12		405.4.e.t.271.1		6
45.7	odd	12		405.4.e.r.271.3		6
45.22	odd	12		405.4.e.r.136.3		6
45.32	even	12		405.4.e.t.136.1		6
60.47	odd	4		2160.4.a.bm.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.f.1.3	✓	3	15.2	even	4
135.4.a.g.1.1	yes	3	5.2	odd	4
405.4.e.r.136.3		6	45.22	odd	12
405.4.e.r.271.3		6	45.7	odd	12
405.4.e.t.136.1		6	45.32	even	12
405.4.e.t.271.1		6	45.2	even	12
675.4.a.q.1.3		3	5.3	odd	4
675.4.a.r.1.1		3	15.8	even	4
675.4.b.k.649.2		6	5.4	even	2	inner
675.4.b.k.649.5		6	1.1	even	1	trivial
675.4.b.l.649.2		6	3.2	odd	2
675.4.b.l.649.5		6	15.14	odd	2
2160.4.a.be.1.3		3	20.7	even	4
2160.4.a.bm.1.3		3	60.47	odd	4

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.45938i q^{2} -11.8861 q^{4} -5.08123i q^{7} -17.3296i q^{8} +O(q^{10})\)	\(q+4.45938i q^{2} -11.8861 q^{4} -5.08123i q^{7} -17.3296i q^{8} -58.3007 q^{11} +21.2119i q^{13} +22.6592 q^{14} -17.8095 q^{16} -68.8451i q^{17} +40.8133 q^{19} -259.985i q^{22} +144.318i q^{23} -94.5921 q^{26} +60.3960i q^{28} -220.058 q^{29} +291.545 q^{31} -218.056i q^{32} +307.006 q^{34} -260.637i q^{37} +182.002i q^{38} +169.766 q^{41} -438.596i q^{43} +692.967 q^{44} -643.571 q^{46} -255.481i q^{47} +317.181 q^{49} -252.127i q^{52} -214.714i q^{53} -88.0557 q^{56} -981.322i q^{58} +331.524 q^{59} +54.9647 q^{61} +1300.11i q^{62} +829.920 q^{64} -758.179i q^{67} +818.299i q^{68} +904.348 q^{71} +866.622i q^{73} +1162.28 q^{74} -485.110 q^{76} +296.239i q^{77} -206.961 q^{79} +757.054i q^{82} +463.397i q^{83} +1955.87 q^{86} +1010.33i q^{88} +601.736 q^{89} +107.783 q^{91} -1715.38i q^{92} +1139.29 q^{94} -229.363i q^{97} +1414.43i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 46 q^{4}+O(q^{10}) \) 6 * q - 46 * q^4	\( 6 q - 46 q^{4} - 76 q^{11} - 216 q^{14} + 382 q^{16} - 374 q^{19} - 832 q^{26} + 320 q^{29} + 454 q^{31} - 34 q^{34} + 676 q^{41} + 3272 q^{44} - 2850 q^{46} + 394 q^{49} + 2508 q^{56} + 280 q^{59} + 1190 q^{61} + 1836 q^{64} + 1204 q^{71} + 5756 q^{74} + 1050 q^{76} - 1258 q^{79} + 7460 q^{86} + 4308 q^{89} - 880 q^{91} + 2216 q^{94}+O(q^{100}) \) 6 * q - 46 * q^4 - 76 * q^11 - 216 * q^14 + 382 * q^16 - 374 * q^19 - 832 * q^26 + 320 * q^29 + 454 * q^31 - 34 * q^34 + 676 * q^41 + 3272 * q^44 - 2850 * q^46 + 394 * q^49 + 2508 * q^56 + 280 * q^59 + 1190 * q^61 + 1836 * q^64 + 1204 * q^71 + 5756 * q^74 + 1050 * q^76 - 1258 * q^79 + 7460 * q^86 + 4308 * q^89 - 880 * q^91 + 2216 * q^94

Introduction

L-functions

Modular forms

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Database

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