LMFDB - Embedded newform 4650.2.a.co.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.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:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.52892$ of defining polynomial
Character	$\chi$	$=$	4650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.133492 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.133492 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.92434 q^{11} -1.00000 q^{12} +4.92434 q^{13} -0.133492 q^{14} +1.00000 q^{16} +3.13349 q^{17} +1.00000 q^{18} -0.266984 q^{19} +0.133492 q^{21} +2.92434 q^{22} +2.00000 q^{23} -1.00000 q^{24} +4.92434 q^{26} -1.00000 q^{27} -0.133492 q^{28} +3.13349 q^{29} -1.00000 q^{31} +1.00000 q^{32} -2.92434 q^{33} +3.13349 q^{34} +1.00000 q^{36} +0.924344 q^{37} -0.266984 q^{38} -4.92434 q^{39} -11.9822 q^{41} +0.133492 q^{42} -2.05784 q^{43} +2.92434 q^{44} +2.00000 q^{46} -6.84869 q^{47} -1.00000 q^{48} -6.98218 q^{49} -3.13349 q^{51} +4.92434 q^{52} +11.1913 q^{53} -1.00000 q^{54} -0.133492 q^{56} +0.266984 q^{57} +3.13349 q^{58} +9.84869 q^{59} +4.92434 q^{61} -1.00000 q^{62} -0.133492 q^{63} +1.00000 q^{64} -2.92434 q^{66} -7.84869 q^{67} +3.13349 q^{68} -2.00000 q^{69} -15.1157 q^{71} +1.00000 q^{72} +10.0000 q^{73} +0.924344 q^{74} -0.266984 q^{76} -0.390376 q^{77} -4.92434 q^{78} -1.13349 q^{79} +1.00000 q^{81} -11.9822 q^{82} +13.1913 q^{83} +0.133492 q^{84} -2.05784 q^{86} -3.13349 q^{87} +2.92434 q^{88} -10.4482 q^{89} -0.657360 q^{91} +2.00000 q^{92} +1.00000 q^{93} -6.84869 q^{94} -1.00000 q^{96} +8.71520 q^{97} -6.98218 q^{98} +2.92434 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 6 q^{11} - 3 q^{12} + 3 q^{16} + 9 q^{17} + 3 q^{18} - 6 q^{22} + 6 q^{23} - 3 q^{24} - 3 q^{27} + 9 q^{29} - 3 q^{31} + 3 q^{32} + 6 q^{33} + 9 q^{34} + 3 q^{36} - 12 q^{37} - 6 q^{41} + 9 q^{43} - 6 q^{44} + 6 q^{46} + 9 q^{47} - 3 q^{48} + 9 q^{49} - 9 q^{51} + 18 q^{53} - 3 q^{54} + 9 q^{58} - 3 q^{62} + 3 q^{64} + 6 q^{66} + 6 q^{67} + 9 q^{68} - 6 q^{69} - 15 q^{71} + 3 q^{72} + 30 q^{73} - 12 q^{74} + 12 q^{77} - 3 q^{79} + 3 q^{81} - 6 q^{82} + 24 q^{83} + 9 q^{86} - 9 q^{87} - 6 q^{88} - 3 q^{89} + 12 q^{91} + 6 q^{92} + 3 q^{93} + 9 q^{94} - 3 q^{96} - 3 q^{97} + 9 q^{98} - 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 - 6 * q^11 - 3 * q^12 + 3 * q^16 + 9 * q^17 + 3 * q^18 - 6 * q^22 + 6 * q^23 - 3 * q^24 - 3 * q^27 + 9 * q^29 - 3 * q^31 + 3 * q^32 + 6 * q^33 + 9 * q^34 + 3 * q^36 - 12 * q^37 - 6 * q^41 + 9 * q^43 - 6 * q^44 + 6 * q^46 + 9 * q^47 - 3 * q^48 + 9 * q^49 - 9 * q^51 + 18 * q^53 - 3 * q^54 + 9 * q^58 - 3 * q^62 + 3 * q^64 + 6 * q^66 + 6 * q^67 + 9 * q^68 - 6 * q^69 - 15 * q^71 + 3 * q^72 + 30 * q^73 - 12 * q^74 + 12 * q^77 - 3 * q^79 + 3 * q^81 - 6 * q^82 + 24 * q^83 + 9 * q^86 - 9 * q^87 - 6 * q^88 - 3 * q^89 + 12 * q^91 + 6 * q^92 + 3 * q^93 + 9 * q^94 - 3 * q^96 - 3 * q^97 + 9 * q^98 - 6 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−0.133492	−0.0504552	−0.0252276	−	0.999682i	$-0.508031\pi$
$7$	−0.133492	−0.0504552	−0.0252276	+	0.999682i	$0.508031\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.92434	0.881723	0.440861	−	0.897575i	$-0.354673\pi$
$11$	2.92434	0.881723	0.440861	+	0.897575i	$0.354673\pi$
$12$	−1.00000	−0.288675
$13$	4.92434	1.36577	0.682884	−	0.730527i	$-0.260726\pi$
$13$	4.92434	1.36577	0.682884	+	0.730527i	$0.260726\pi$
$14$	−0.133492	−0.0356772
$15$	0	0
$16$	1.00000	0.250000
$17$	3.13349	0.759983	0.379992	−	0.924990i	$-0.375927\pi$
$17$	3.13349	0.759983	0.379992	+	0.924990i	$0.375927\pi$
$18$	1.00000	0.235702
$19$	−0.266984	−0.0612503	−0.0306251	−	0.999531i	$-0.509750\pi$
$19$	−0.266984	−0.0612503	−0.0306251	+	0.999531i	$0.509750\pi$
$20$	0	0
$21$	0.133492	0.0291303
$22$	2.92434	0.623472
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	4.92434	0.965743
$27$	−1.00000	−0.192450
$28$	−0.133492	−0.0252276
$29$	3.13349	0.581875	0.290937	−	0.956742i	$-0.406033\pi$
$29$	3.13349	0.581875	0.290937	+	0.956742i	$0.406033\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	−2.92434	−0.509063
$34$	3.13349	0.537389
$35$	0	0
$36$	1.00000	0.166667
$37$	0.924344	0.151961	0.0759806	−	0.997109i	$-0.475791\pi$
$37$	0.924344	0.151961	0.0759806	+	0.997109i	$0.475791\pi$
$38$	−0.266984	−0.0433105
$39$	−4.92434	−0.788526
$40$	0	0
$41$	−11.9822	−1.87130	−0.935651	−	0.352926i	$-0.885187\pi$
$41$	−11.9822	−1.87130	−0.935651	+	0.352926i	$0.885187\pi$
$42$	0.133492	0.0205983
$43$	−2.05784	−0.313817	−0.156909	−	0.987613i	$-0.550153\pi$
$43$	−2.05784	−0.313817	−0.156909	+	0.987613i	$0.550153\pi$
$44$	2.92434	0.440861
$45$	0	0
$46$	2.00000	0.294884
$47$	−6.84869	−0.998984	−0.499492	−	0.866319i	$-0.666480\pi$
$47$	−6.84869	−0.998984	−0.499492	+	0.866319i	$0.666480\pi$
$48$	−1.00000	−0.144338
$49$	−6.98218	−0.997454
$50$	0	0
$51$	−3.13349	−0.438777
$52$	4.92434	0.682884
$53$	11.1913	1.53725	0.768624	−	0.639701i	$-0.220942\pi$
$53$	11.1913	1.53725	0.768624	+	0.639701i	$0.220942\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−0.133492	−0.0178386
$57$	0.266984	0.0353629
$58$	3.13349	0.411448
$59$	9.84869	1.28219	0.641095	−	0.767462i	$-0.278480\pi$
$59$	9.84869	1.28219	0.641095	+	0.767462i	$0.278480\pi$
$60$	0	0
$61$	4.92434	0.630498	0.315249	−	0.949009i	$-0.397912\pi$
$61$	4.92434	0.630498	0.315249	+	0.949009i	$0.397912\pi$
$62$	−1.00000	−0.127000
$63$	−0.133492	−0.0168184
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.92434	−0.359962
$67$	−7.84869	−0.958870	−0.479435	−	0.877577i	$-0.659158\pi$
$67$	−7.84869	−0.958870	−0.479435	+	0.877577i	$0.659158\pi$
$68$	3.13349	0.379992
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−15.1157	−1.79390	−0.896950	−	0.442132i	$-0.854222\pi$
$71$	−15.1157	−1.79390	−0.896950	+	0.442132i	$0.854222\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0.924344	0.107453
$75$	0	0
$76$	−0.266984	−0.0306251
$77$	−0.390376	−0.0444875
$78$	−4.92434	−0.557572
$79$	−1.13349	−0.127528	−0.0637639	−	0.997965i	$-0.520310\pi$
$79$	−1.13349	−0.127528	−0.0637639	+	0.997965i	$0.520310\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−11.9822	−1.32321
$83$	13.1913	1.44794	0.723968	−	0.689833i	$-0.242316\pi$
$83$	13.1913	1.44794	0.723968	+	0.689833i	$0.242316\pi$
$84$	0.133492	0.0145652
$85$	0	0
$86$	−2.05784	−0.221902
$87$	−3.13349	−0.335946
$88$	2.92434	0.311736
$89$	−10.4482	−1.10751	−0.553754	−	0.832680i	$-0.686805\pi$
$89$	−10.4482	−1.10751	−0.553754	+	0.832680i	$0.686805\pi$
$90$	0	0
$91$	−0.657360	−0.0689101
$92$	2.00000	0.208514
$93$	1.00000	0.103695
$94$	−6.84869	−0.706388
$95$	0	0
$96$	−1.00000	−0.102062
$97$	8.71520	0.884894	0.442447	−	0.896795i	$-0.354110\pi$
$97$	8.71520	0.884894	0.442447	+	0.896795i	$0.354110\pi$
$98$	−6.98218	−0.705307
$99$	2.92434	0.293908
$100$	0	0
$101$	12.1157	1.20555	0.602777	−	0.797910i	$-0.294061\pi$
$101$	12.1157	1.20555	0.602777	+	0.797910i	$0.294061\pi$
$102$	−3.13349	−0.310262
$103$	10.1335	0.998483	0.499241	−	0.866463i	$-0.333612\pi$
$103$	10.1335	0.998483	0.499241	+	0.866463i	$0.333612\pi$
$104$	4.92434	0.482872
$105$	0	0
$106$	11.1913	1.08700
$107$	6.71520	0.649183	0.324591	−	0.945854i	$-0.394773\pi$
$107$	6.71520	0.649183	0.324591	+	0.945854i	$0.394773\pi$
$108$	−1.00000	−0.0962250
$109$	14.7152	1.40946	0.704730	−	0.709476i	$-0.251068\pi$
$109$	14.7152	1.40946	0.704730	+	0.709476i	$0.251068\pi$
$110$	0	0
$111$	−0.924344	−0.0877348
$112$	−0.133492	−0.0126138
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0.266984	0.0250053
$115$	0	0
$116$	3.13349	0.290937
$117$	4.92434	0.455256
$118$	9.84869	0.906645
$119$	−0.418296	−0.0383451
$120$	0	0
$121$	−2.44821	−0.222565
$122$	4.92434	0.445829
$123$	11.9822	1.08040
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	−0.133492	−0.0118924
$127$	−10.9822	−0.974511	−0.487255	−	0.873259i	$-0.662002\pi$
$127$	−10.9822	−0.974511	−0.487255	+	0.873259i	$0.662002\pi$
$128$	1.00000	0.0883883
$129$	2.05784	0.181182
$130$	0	0
$131$	3.24916	0.283881	0.141940	−	0.989875i	$-0.454666\pi$
$131$	3.24916	0.283881	0.141940	+	0.989875i	$0.454666\pi$
$132$	−2.92434	−0.254531
$133$	0.0356402	0.00309040
$134$	−7.84869	−0.678023
$135$	0	0
$136$	3.13349	0.268695
$137$	−1.66746	−0.142461	−0.0712303	−	0.997460i	$-0.522693\pi$
$137$	−1.66746	−0.142461	−0.0712303	+	0.997460i	$0.522693\pi$
$138$	−2.00000	−0.170251
$139$	11.9065	1.00990	0.504949	−	0.863149i	$-0.331511\pi$
$139$	11.9065	1.00990	0.504949	+	0.863149i	$0.331511\pi$
$140$	0	0
$141$	6.84869	0.576764
$142$	−15.1157	−1.26848
$143$	14.4005	1.20423
$144$	1.00000	0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	6.98218	0.575880
$148$	0.924344	0.0759806
$149$	−19.9644	−1.63554	−0.817772	−	0.575542i	$-0.804791\pi$
$149$	−19.9644	−1.63554	−0.817772	+	0.575542i	$0.804791\pi$
$150$	0	0
$151$	19.2492	1.56647	0.783237	−	0.621723i	$-0.213567\pi$
$151$	19.2492	1.56647	0.783237	+	0.621723i	$0.213567\pi$
$152$	−0.266984	−0.0216552
$153$	3.13349	0.253328
$154$	−0.390376	−0.0314574
$155$	0	0
$156$	−4.92434	−0.394263
$157$	1.40048	0.111770	0.0558851	−	0.998437i	$-0.482202\pi$
$157$	1.40048	0.111770	0.0558851	+	0.998437i	$0.482202\pi$
$158$	−1.13349	−0.0901758
$159$	−11.1913	−0.887530
$160$	0	0
$161$	−0.266984	−0.0210413
$162$	1.00000	0.0785674
$163$	−6.11567	−0.479016	−0.239508	−	0.970894i	$-0.576986\pi$
$163$	−6.11567	−0.479016	−0.239508	+	0.970894i	$0.576986\pi$
$164$	−11.9822	−0.935651
$165$	0	0
$166$	13.1913	1.02385
$167$	−0.533968	−0.0413197	−0.0206598	−	0.999787i	$-0.506577\pi$
$167$	−0.533968	−0.0413197	−0.0206598	+	0.999787i	$0.506577\pi$
$168$	0.133492	0.0102991
$169$	11.2492	0.865320
$170$	0	0
$171$	−0.266984	−0.0204168
$172$	−2.05784	−0.156909
$173$	−4.11567	−0.312909	−0.156454	−	0.987685i	$-0.550006\pi$
$173$	−4.11567	−0.312909	−0.156454	+	0.987685i	$0.550006\pi$
$174$	−3.13349	−0.237549
$175$	0	0
$176$	2.92434	0.220431
$177$	−9.84869	−0.740273
$178$	−10.4482	−0.783127
$179$	9.30700	0.695638	0.347819	−	0.937562i	$-0.386922\pi$
$179$	9.30700	0.695638	0.347819	+	0.937562i	$0.386922\pi$
$180$	0	0
$181$	−11.3070	−0.840443	−0.420221	−	0.907422i	$-0.638048\pi$
$181$	−11.3070	−0.840443	−0.420221	+	0.907422i	$0.638048\pi$
$182$	−0.657360	−0.0487268
$183$	−4.92434	−0.364018
$184$	2.00000	0.147442
$185$	0	0
$186$	1.00000	0.0733236
$187$	9.16341	0.670095
$188$	−6.84869	−0.499492
$189$	0.133492	0.00971011
$190$	0	0
$191$	−14.7152	−1.06475	−0.532377	−	0.846507i	$-0.678701\pi$
$191$	−14.7152	−1.06475	−0.532377	+	0.846507i	$0.678701\pi$
$192$	−1.00000	−0.0721688
$193$	17.9822	1.29439	0.647193	−	0.762326i	$-0.275943\pi$
$193$	17.9822	1.29439	0.647193	+	0.762326i	$0.275943\pi$
$194$	8.71520	0.625715
$195$	0	0
$196$	−6.98218	−0.498727
$197$	11.6395	0.829283	0.414641	−	0.909985i	$-0.363907\pi$
$197$	11.6395	0.829283	0.414641	+	0.909985i	$0.363907\pi$
$198$	2.92434	0.207824
$199$	−20.2313	−1.43416	−0.717081	−	0.696990i	$-0.754522\pi$
$199$	−20.2313	−1.43416	−0.717081	+	0.696990i	$0.754522\pi$
$200$	0	0
$201$	7.84869	0.553604
$202$	12.1157	0.852456
$203$	−0.418296	−0.0293586
$204$	−3.13349	−0.219388
$205$	0	0
$206$	10.1335	0.706034
$207$	2.00000	0.139010
$208$	4.92434	0.341442
$209$	−0.780753	−0.0540058
$210$	0	0
$211$	7.73302	0.532363	0.266181	−	0.963923i	$-0.414238\pi$
$211$	7.73302	0.532363	0.266181	+	0.963923i	$0.414238\pi$
$212$	11.1913	0.768624
$213$	15.1157	1.03571
$214$	6.71520	0.459041
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0.133492	0.00906202
$218$	14.7152	0.996639
$219$	−10.0000	−0.675737
$220$	0	0
$221$	15.4304	1.03796
$222$	−0.924344	−0.0620379
$223$	7.13349	0.477694	0.238847	−	0.971057i	$-0.423231\pi$
$223$	7.13349	0.477694	0.238847	+	0.971057i	$0.423231\pi$
$224$	−0.133492	−0.00891930
$225$	0	0
$226$	6.00000	0.399114
$227$	11.4005	0.756676	0.378338	−	0.925667i	$-0.376496\pi$
$227$	11.4005	0.756676	0.378338	+	0.925667i	$0.376496\pi$
$228$	0.266984	0.0176814
$229$	5.84869	0.386492	0.193246	−	0.981150i	$-0.438098\pi$
$229$	5.84869	0.386492	0.193246	+	0.981150i	$0.438098\pi$
$230$	0	0
$231$	0.390376	0.0256849
$232$	3.13349	0.205724
$233$	21.7152	1.42261	0.711305	−	0.702884i	$-0.248105\pi$
$233$	21.7152	1.42261	0.711305	+	0.702884i	$0.248105\pi$
$234$	4.92434	0.321914
$235$	0	0
$236$	9.84869	0.641095
$237$	1.13349	0.0736282
$238$	−0.418296	−0.0271141
$239$	−22.2313	−1.43803	−0.719013	−	0.694997i	$-0.755406\pi$
$239$	−22.2313	−1.43803	−0.719013	+	0.694997i	$0.755406\pi$
$240$	0	0
$241$	19.6974	1.26882	0.634410	−	0.772997i	$-0.281243\pi$
$241$	19.6974	1.26882	0.634410	+	0.772997i	$0.281243\pi$
$242$	−2.44821	−0.157377
$243$	−1.00000	−0.0641500
$244$	4.92434	0.315249
$245$	0	0
$246$	11.9822	0.763956
$247$	−1.31472	−0.0836536
$248$	−1.00000	−0.0635001
$249$	−13.1913	−0.835966
$250$	0	0
$251$	−28.2391	−1.78243	−0.891217	−	0.453577i	$-0.850148\pi$
$251$	−28.2391	−1.78243	−0.891217	+	0.453577i	$0.850148\pi$
$252$	−0.133492	−0.00840920
$253$	5.84869	0.367704
$254$	−10.9822	−0.689083
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.2492	0.888838	0.444419	−	0.895819i	$-0.353410\pi$
$257$	14.2492	0.888838	0.444419	+	0.895819i	$0.353410\pi$
$258$	2.05784	0.128115
$259$	−0.123392	−0.00766723
$260$	0	0
$261$	3.13349	0.193958
$262$	3.24916	0.200734
$263$	−18.1157	−1.11706	−0.558530	−	0.829484i	$-0.688634\pi$
$263$	−18.1157	−1.11706	−0.558530	+	0.829484i	$0.688634\pi$
$264$	−2.92434	−0.179981
$265$	0	0
$266$	0.0356402	0.00218524
$267$	10.4482	0.639420
$268$	−7.84869	−0.479435
$269$	0.866508	0.0528319	0.0264160	−	0.999651i	$-0.491591\pi$
$269$	0.866508	0.0528319	0.0264160	+	0.999651i	$0.491591\pi$
$270$	0	0
$271$	20.2969	1.23295	0.616474	−	0.787375i	$-0.288560\pi$
$271$	20.2969	1.23295	0.616474	+	0.787375i	$0.288560\pi$
$272$	3.13349	0.189996
$273$	0.657360	0.0397852
$274$	−1.66746	−0.100735
$275$	0	0
$276$	−2.00000	−0.120386
$277$	−25.3147	−1.52101	−0.760507	−	0.649330i	$-0.775050\pi$
$277$	−25.3147	−1.52101	−0.760507	+	0.649330i	$0.775050\pi$
$278$	11.9065	0.714106
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−2.24916	−0.134174	−0.0670869	−	0.997747i	$-0.521370\pi$
$281$	−2.24916	−0.134174	−0.0670869	+	0.997747i	$0.521370\pi$
$282$	6.84869	0.407833
$283$	−12.3827	−0.736072	−0.368036	−	0.929811i	$-0.619970\pi$
$283$	−12.3827	−0.736072	−0.368036	+	0.929811i	$0.619970\pi$
$284$	−15.1157	−0.896950
$285$	0	0
$286$	14.4005	0.851518
$287$	1.59952	0.0944169
$288$	1.00000	0.0589256
$289$	−7.18123	−0.422425
$290$	0	0
$291$	−8.71520	−0.510894
$292$	10.0000	0.585206
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	6.98218	0.407209
$295$	0	0
$296$	0.924344	0.0537264
$297$	−2.92434	−0.169688
$298$	−19.9644	−1.15650
$299$	9.84869	0.569564
$300$	0	0
$301$	0.274704	0.0158337
$302$	19.2492	1.10766
$303$	−12.1157	−0.696027
$304$	−0.266984	−0.0153126
$305$	0	0
$306$	3.13349	0.179130
$307$	−18.6496	−1.06439	−0.532196	−	0.846621i	$-0.678633\pi$
$307$	−18.6496	−1.06439	−0.532196	+	0.846621i	$0.678633\pi$
$308$	−0.390376	−0.0222438
$309$	−10.1335	−0.576474
$310$	0	0
$311$	15.9166	0.902549	0.451274	−	0.892385i	$-0.350970\pi$
$311$	15.9166	0.902549	0.451274	+	0.892385i	$0.350970\pi$
$312$	−4.92434	−0.278786
$313$	30.6496	1.73242	0.866210	−	0.499680i	$-0.166549\pi$
$313$	30.6496	1.73242	0.866210	+	0.499680i	$0.166549\pi$
$314$	1.40048	0.0790334
$315$	0	0
$316$	−1.13349	−0.0637639
$317$	−14.2313	−0.799312	−0.399656	−	0.916665i	$-0.630870\pi$
$317$	−14.2313	−0.799312	−0.399656	+	0.916665i	$0.630870\pi$
$318$	−11.1913	−0.627579
$319$	9.16341	0.513052
$320$	0	0
$321$	−6.71520	−0.374806
$322$	−0.266984	−0.0148784
$323$	−0.836592	−0.0465492
$324$	1.00000	0.0555556
$325$	0	0
$326$	−6.11567	−0.338716
$327$	−14.7152	−0.813752
$328$	−11.9822	−0.661605
$329$	0.914245	0.0504039
$330$	0	0
$331$	−5.19133	−0.285341	−0.142671	−	0.989770i	$-0.545569\pi$
$331$	−5.19133	−0.285341	−0.142671	+	0.989770i	$0.545569\pi$
$332$	13.1913	0.723968
$333$	0.924344	0.0506537
$334$	−0.533968	−0.0292174
$335$	0	0
$336$	0.133492	0.00728258
$337$	−7.58170	−0.413002	−0.206501	−	0.978446i	$-0.566208\pi$
$337$	−7.58170	−0.413002	−0.206501	+	0.978446i	$0.566208\pi$
$338$	11.2492	0.611874
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−2.92434	−0.158362
$342$	−0.266984	−0.0144368
$343$	1.86651	0.100782
$344$	−2.05784	−0.110951
$345$	0	0
$346$	−4.11567	−0.221260
$347$	7.45831	0.400383	0.200192	−	0.979757i	$-0.435844\pi$
$347$	7.45831	0.400383	0.200192	+	0.979757i	$0.435844\pi$
$348$	−3.13349	−0.167973
$349$	4.33254	0.231916	0.115958	−	0.993254i	$-0.463006\pi$
$349$	4.33254	0.231916	0.115958	+	0.993254i	$0.463006\pi$
$350$	0	0
$351$	−4.92434	−0.262842
$352$	2.92434	0.155868
$353$	−31.8130	−1.69324	−0.846619	−	0.532200i	$-0.821366\pi$
$353$	−31.8130	−1.69324	−0.846619	+	0.532200i	$0.821366\pi$
$354$	−9.84869	−0.523452
$355$	0	0
$356$	−10.4482	−0.553754
$357$	0.418296	0.0221386
$358$	9.30700	0.491890
$359$	28.2313	1.48999	0.744997	−	0.667068i	$-0.232451\pi$
$359$	28.2313	1.48999	0.744997	+	0.667068i	$0.232451\pi$
$360$	0	0
$361$	−18.9287	−0.996248
$362$	−11.3070	−0.594283
$363$	2.44821	0.128498
$364$	−0.657360	−0.0344550
$365$	0	0
$366$	−4.92434	−0.257400
$367$	−0.715196	−0.0373329	−0.0186665	−	0.999826i	$-0.505942\pi$
$367$	−0.715196	−0.0373329	−0.0186665	+	0.999826i	$0.505942\pi$
$368$	2.00000	0.104257
$369$	−11.9822	−0.623767
$370$	0	0
$371$	−1.49395	−0.0775621
$372$	1.00000	0.0518476
$373$	19.2492	0.996684	0.498342	−	0.866981i	$-0.333942\pi$
$373$	19.2492	0.996684	0.498342	+	0.866981i	$0.333942\pi$
$374$	9.16341	0.473829
$375$	0	0
$376$	−6.84869	−0.353194
$377$	15.4304	0.794706
$378$	0.133492	0.00686608
$379$	−5.58170	−0.286713	−0.143356	−	0.989671i	$-0.545790\pi$
$379$	−5.58170	−0.286713	−0.143356	+	0.989671i	$0.545790\pi$
$380$	0	0
$381$	10.9822	0.562634
$382$	−14.7152	−0.752895
$383$	−13.4304	−0.686261	−0.343130	−	0.939288i	$-0.611487\pi$
$383$	−13.4304	−0.686261	−0.343130	+	0.939288i	$0.611487\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	17.9822	0.915269
$387$	−2.05784	−0.104606
$388$	8.71520	0.442447
$389$	−28.1157	−1.42552	−0.712761	−	0.701407i	$-0.752555\pi$
$389$	−28.1157	−1.42552	−0.712761	+	0.701407i	$0.752555\pi$
$390$	0	0
$391$	6.26698	0.316935
$392$	−6.98218	−0.352653
$393$	−3.24916	−0.163899
$394$	11.6395	0.586392
$395$	0	0
$396$	2.92434	0.146954
$397$	−6.03564	−0.302920	−0.151460	−	0.988463i	$-0.548397\pi$
$397$	−6.03564	−0.302920	−0.151460	+	0.988463i	$0.548397\pi$
$398$	−20.2313	−1.01411
$399$	−0.0356402	−0.00178424
$400$	0	0
$401$	−10.5340	−0.526041	−0.263021	−	0.964790i	$-0.584719\pi$
$401$	−10.5340	−0.526041	−0.263021	+	0.964790i	$0.584719\pi$
$402$	7.84869	0.391457
$403$	−4.92434	−0.245299
$404$	12.1157	0.602777
$405$	0	0
$406$	−0.418296	−0.0207597
$407$	2.70310	0.133988
$408$	−3.13349	−0.155131
$409$	30.1157	1.48912	0.744562	−	0.667553i	$-0.232658\pi$
$409$	30.1157	1.48912	0.744562	+	0.667553i	$0.232658\pi$
$410$	0	0
$411$	1.66746	0.0822497
$412$	10.1335	0.499241
$413$	−1.31472	−0.0646932
$414$	2.00000	0.0982946
$415$	0	0
$416$	4.92434	0.241436
$417$	−11.9065	−0.583065
$418$	−0.780753	−0.0381879
$419$	−12.8309	−0.626829	−0.313414	−	0.949616i	$-0.601473\pi$
$419$	−12.8309	−0.626829	−0.313414	+	0.949616i	$0.601473\pi$
$420$	0	0
$421$	17.5161	0.853685	0.426842	−	0.904326i	$-0.359626\pi$
$421$	17.5161	0.853685	0.426842	+	0.904326i	$0.359626\pi$
$422$	7.73302	0.376437
$423$	−6.84869	−0.332995
$424$	11.1913	0.543499
$425$	0	0
$426$	15.1157	0.732357
$427$	−0.657360	−0.0318119
$428$	6.71520	0.324591
$429$	−14.4005	−0.695262
$430$	0	0
$431$	−9.06556	−0.436672	−0.218336	−	0.975874i	$-0.570063\pi$
$431$	−9.06556	−0.436672	−0.218336	+	0.975874i	$0.570063\pi$
$432$	−1.00000	−0.0481125
$433$	15.0477	0.723148	0.361574	−	0.932343i	$-0.382239\pi$
$433$	15.0477	0.723148	0.361574	+	0.932343i	$0.382239\pi$
$434$	0.133492	0.00640782
$435$	0	0
$436$	14.7152	0.704730
$437$	−0.533968	−0.0255431
$438$	−10.0000	−0.477818
$439$	2.80095	0.133682	0.0668411	−	0.997764i	$-0.478708\pi$
$439$	2.80095	0.133682	0.0668411	+	0.997764i	$0.478708\pi$
$440$	0	0
$441$	−6.98218	−0.332485
$442$	15.4304	0.733949
$443$	8.06556	0.383206	0.191603	−	0.981472i	$-0.438631\pi$
$443$	8.06556	0.383206	0.191603	+	0.981472i	$0.438631\pi$
$444$	−0.924344	−0.0438674
$445$	0	0
$446$	7.13349	0.337781
$447$	19.9644	0.944282
$448$	−0.133492	−0.00630690
$449$	−10.7152	−0.505681	−0.252841	−	0.967508i	$-0.581365\pi$
$449$	−10.7152	−0.505681	−0.252841	+	0.967508i	$0.581365\pi$
$450$	0	0
$451$	−35.0400	−1.64997
$452$	6.00000	0.282216
$453$	−19.2492	−0.904405
$454$	11.4005	0.535051
$455$	0	0
$456$	0.266984	0.0125027
$457$	5.31472	0.248612	0.124306	−	0.992244i	$-0.460330\pi$
$457$	5.31472	0.248612	0.124306	+	0.992244i	$0.460330\pi$
$458$	5.84869	0.273291
$459$	−3.13349	−0.146259
$460$	0	0
$461$	−29.2714	−1.36330	−0.681652	−	0.731677i	$-0.738738\pi$
$461$	−29.2714	−1.36330	−0.681652	+	0.731677i	$0.738738\pi$
$462$	0.390376	0.0181619
$463$	10.5639	0.490945	0.245473	−	0.969404i	$-0.421057\pi$
$463$	10.5639	0.490945	0.245473	+	0.969404i	$0.421057\pi$
$464$	3.13349	0.145469
$465$	0	0
$466$	21.7152	1.00594
$467$	7.55179	0.349455	0.174728	−	0.984617i	$-0.444096\pi$
$467$	7.55179	0.349455	0.174728	+	0.984617i	$0.444096\pi$
$468$	4.92434	0.227628
$469$	1.04774	0.0483800
$470$	0	0
$471$	−1.40048	−0.0645305
$472$	9.84869	0.453323
$473$	−6.01782	−0.276700
$474$	1.13349	0.0520630
$475$	0	0
$476$	−0.418296	−0.0191726
$477$	11.1913	0.512416
$478$	−22.2313	−1.01684
$479$	−30.9465	−1.41398	−0.706992	−	0.707222i	$-0.749948\pi$
$479$	−30.9465	−1.41398	−0.706992	+	0.707222i	$0.749948\pi$
$480$	0	0
$481$	4.55179	0.207544
$482$	19.6974	0.897191
$483$	0.266984	0.0121482
$484$	−2.44821	−0.111282
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−24.1812	−1.09576	−0.547878	−	0.836558i	$-0.684564\pi$
$487$	−24.1812	−1.09576	−0.547878	+	0.836558i	$0.684564\pi$
$488$	4.92434	0.222915
$489$	6.11567	0.276560
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	11.9822	0.540198
$493$	9.81877	0.442215
$494$	−1.31472	−0.0591521
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	2.01782	0.0905116
$498$	−13.1913	−0.591118
$499$	−42.6439	−1.90900	−0.954502	−	0.298205i	$-0.903612\pi$
$499$	−42.6439	−1.90900	−0.954502	+	0.298205i	$0.903612\pi$
$500$	0	0
$501$	0.533968	0.0238559
$502$	−28.2391	−1.26037
$503$	−14.3147	−0.638262	−0.319131	−	0.947711i	$-0.603391\pi$
$503$	−14.3147	−0.638262	−0.319131	+	0.947711i	$0.603391\pi$
$504$	−0.133492	−0.00594620
$505$	0	0
$506$	5.84869	0.260006
$507$	−11.2492	−0.499593
$508$	−10.9822	−0.487255
$509$	10.1234	0.448711	0.224356	−	0.974507i	$-0.427972\pi$
$509$	10.1234	0.448711	0.224356	+	0.974507i	$0.427972\pi$
$510$	0	0
$511$	−1.33492	−0.0590533
$512$	1.00000	0.0441942
$513$	0.266984	0.0117876
$514$	14.2492	0.628504
$515$	0	0
$516$	2.05784	0.0905912
$517$	−20.0279	−0.880827
$518$	−0.123392	−0.00542155
$519$	4.11567	0.180658
$520$	0	0
$521$	−12.2848	−0.538207	−0.269104	−	0.963111i	$-0.586727\pi$
$521$	−12.2848	−0.538207	−0.269104	+	0.963111i	$0.586727\pi$
$522$	3.13349	0.137149
$523$	21.5161	0.940835	0.470418	−	0.882444i	$-0.344103\pi$
$523$	21.5161	0.940835	0.470418	+	0.882444i	$0.344103\pi$
$524$	3.24916	0.141940
$525$	0	0
$526$	−18.1157	−0.789881
$527$	−3.13349	−0.136497
$528$	−2.92434	−0.127266
$529$	−19.0000	−0.826087
$530$	0	0
$531$	9.84869	0.427397
$532$	0.0356402	0.00154520
$533$	−59.0044	−2.55576
$534$	10.4482	0.452138
$535$	0	0
$536$	−7.84869	−0.339012
$537$	−9.30700	−0.401627
$538$	0.866508	0.0373578
$539$	−20.4183	−0.879478
$540$	0	0
$541$	38.6439	1.66143	0.830716	−	0.556697i	$-0.187931\pi$
$541$	38.6439	1.66143	0.830716	+	0.556697i	$0.187931\pi$
$542$	20.2969	0.871827
$543$	11.3070	0.485230
$544$	3.13349	0.134347
$545$	0	0
$546$	0.657360	0.0281324
$547$	16.2313	0.694002	0.347001	−	0.937865i	$-0.387200\pi$
$547$	16.2313	0.694002	0.347001	+	0.937865i	$0.387200\pi$
$548$	−1.66746	−0.0712303
$549$	4.92434	0.210166
$550$	0	0
$551$	−0.836592	−0.0356400
$552$	−2.00000	−0.0851257
$553$	0.151312	0.00643444
$554$	−25.3147	−1.07552
$555$	0	0
$556$	11.9065	0.504949
$557$	14.8309	0.628404	0.314202	−	0.949356i	$-0.398263\pi$
$557$	14.8309	0.628404	0.314202	+	0.949356i	$0.398263\pi$
$558$	−1.00000	−0.0423334
$559$	−10.1335	−0.428601
$560$	0	0
$561$	−9.16341	−0.386879
$562$	−2.24916	−0.0948752
$563$	−16.3325	−0.688335	−0.344167	−	0.938908i	$-0.611839\pi$
$563$	−16.3325	−0.688335	−0.344167	+	0.938908i	$0.611839\pi$
$564$	6.84869	0.288382
$565$	0	0
$566$	−12.3827	−0.520482
$567$	−0.133492	−0.00560613
$568$	−15.1157	−0.634240
$569$	6.49833	0.272424	0.136212	−	0.990680i	$-0.456507\pi$
$569$	6.49833	0.272424	0.136212	+	0.990680i	$0.456507\pi$
$570$	0	0
$571$	−3.58942	−0.150213	−0.0751064	−	0.997176i	$-0.523930\pi$
$571$	−3.58942	−0.150213	−0.0751064	+	0.997176i	$0.523930\pi$
$572$	14.4005	0.602114
$573$	14.7152	0.614736
$574$	1.59952	0.0667629
$575$	0	0
$576$	1.00000	0.0416667
$577$	22.3171	0.929073	0.464536	−	0.885554i	$-0.346221\pi$
$577$	22.3171	0.929073	0.464536	+	0.885554i	$0.346221\pi$
$578$	−7.18123	−0.298700
$579$	−17.9822	−0.747314
$580$	0	0
$581$	−1.76094	−0.0730559
$582$	−8.71520	−0.361256
$583$	32.7273	1.35543
$584$	10.0000	0.413803
$585$	0	0
$586$	6.00000	0.247858
$587$	32.4704	1.34020	0.670099	−	0.742272i	$-0.266252\pi$
$587$	32.4704	1.34020	0.670099	+	0.742272i	$0.266252\pi$
$588$	6.98218	0.287940
$589$	0.266984	0.0110009
$590$	0	0
$591$	−11.6395	−0.478787
$592$	0.924344	0.0379903
$593$	13.6796	0.561752	0.280876	−	0.959744i	$-0.409375\pi$
$593$	13.6796	0.561752	0.280876	+	0.959744i	$0.409375\pi$
$594$	−2.92434	−0.119987
$595$	0	0
$596$	−19.9644	−0.817772
$597$	20.2313	0.828014
$598$	9.84869	0.402743
$599$	25.6140	1.04656	0.523280	−	0.852161i	$-0.324708\pi$
$599$	25.6140	1.04656	0.523280	+	0.852161i	$0.324708\pi$
$600$	0	0
$601$	15.5461	0.634137	0.317069	−	0.948403i	$-0.397301\pi$
$601$	15.5461	0.634137	0.317069	+	0.948403i	$0.397301\pi$
$602$	0.274704	0.0111961
$603$	−7.84869	−0.319623
$604$	19.2492	0.783237
$605$	0	0
$606$	−12.1157	−0.492166
$607$	1.59952	0.0649227	0.0324613	−	0.999473i	$-0.489665\pi$
$607$	1.59952	0.0649227	0.0324613	+	0.999473i	$0.489665\pi$
$608$	−0.266984	−0.0108276
$609$	0.418296	0.0169502
$610$	0	0
$611$	−33.7253	−1.36438
$612$	3.13349	0.126664
$613$	2.38266	0.0962346	0.0481173	−	0.998842i	$-0.484678\pi$
$613$	2.38266	0.0962346	0.0481173	+	0.998842i	$0.484678\pi$
$614$	−18.6496	−0.752638
$615$	0	0
$616$	−0.390376	−0.0157287
$617$	−31.9109	−1.28468	−0.642342	−	0.766418i	$-0.722037\pi$
$617$	−31.9109	−1.28468	−0.642342	+	0.766418i	$0.722037\pi$
$618$	−10.1335	−0.407629
$619$	−7.25688	−0.291679	−0.145839	−	0.989308i	$-0.546588\pi$
$619$	−7.25688	−0.291679	−0.145839	+	0.989308i	$0.546588\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	15.9166	0.638198
$623$	1.39475	0.0558796
$624$	−4.92434	−0.197132
$625$	0	0
$626$	30.6496	1.22501
$627$	0.780753	0.0311803
$628$	1.40048	0.0558851
$629$	2.89642	0.115488
$630$	0	0
$631$	31.4805	1.25322	0.626610	−	0.779333i	$-0.284442\pi$
$631$	31.4805	1.25322	0.626610	+	0.779333i	$0.284442\pi$
$632$	−1.13349	−0.0450879
$633$	−7.73302	−0.307360
$634$	−14.2313	−0.565199
$635$	0	0
$636$	−11.1913	−0.443765
$637$	−34.3827	−1.36229
$638$	9.16341	0.362783
$639$	−15.1157	−0.597967
$640$	0	0
$641$	14.2313	0.562104	0.281052	−	0.959693i	$-0.409317\pi$
$641$	14.2313	0.562104	0.281052	+	0.959693i	$0.409317\pi$
$642$	−6.71520	−0.265028
$643$	8.85879	0.349356	0.174678	−	0.984626i	$-0.444111\pi$
$643$	8.85879	0.349356	0.174678	+	0.984626i	$0.444111\pi$
$644$	−0.266984	−0.0105206
$645$	0	0
$646$	−0.836592	−0.0329153
$647$	−1.06794	−0.0419849	−0.0209924	−	0.999780i	$-0.506683\pi$
$647$	−1.06794	−0.0419849	−0.0209924	+	0.999780i	$0.506683\pi$
$648$	1.00000	0.0392837
$649$	28.8010	1.13054
$650$	0	0
$651$	−0.133492	−0.00523196
$652$	−6.11567	−0.239508
$653$	0.800952	0.0313437	0.0156718	−	0.999877i	$-0.495011\pi$
$653$	0.800952	0.0313437	0.0156718	+	0.999877i	$0.495011\pi$
$654$	−14.7152	−0.575410
$655$	0	0
$656$	−11.9822	−0.467826
$657$	10.0000	0.390137
$658$	0.914245	0.0356410
$659$	33.5461	1.30677	0.653385	−	0.757026i	$-0.273348\pi$
$659$	33.5461	1.30677	0.653385	+	0.757026i	$0.273348\pi$
$660$	0	0
$661$	−21.9142	−0.852365	−0.426183	−	0.904637i	$-0.640142\pi$
$661$	−21.9142	−0.852365	−0.426183	+	0.904637i	$0.640142\pi$
$662$	−5.19133	−0.201767
$663$	−15.4304	−0.599267
$664$	13.1913	0.511923
$665$	0	0
$666$	0.924344	0.0358176
$667$	6.26698	0.242659
$668$	−0.533968	−0.0206598
$669$	−7.13349	−0.275797
$670$	0	0
$671$	14.4005	0.555924
$672$	0.133492	0.00514956
$673$	−13.1836	−0.508191	−0.254095	−	0.967179i	$-0.581778\pi$
$673$	−13.1836	−0.508191	−0.254095	+	0.967179i	$0.581778\pi$
$674$	−7.58170	−0.292036
$675$	0	0
$676$	11.2492	0.432660
$677$	−20.7229	−0.796446	−0.398223	−	0.917289i	$-0.630373\pi$
$677$	−20.7229	−0.796446	−0.398223	+	0.917289i	$0.630373\pi$
$678$	−6.00000	−0.230429
$679$	−1.16341	−0.0446475
$680$	0	0
$681$	−11.4005	−0.436867
$682$	−2.92434	−0.111979
$683$	−4.86651	−0.186212	−0.0931059	−	0.995656i	$-0.529680\pi$
$683$	−4.86651	−0.186212	−0.0931059	+	0.995656i	$0.529680\pi$
$684$	−0.266984	−0.0102084
$685$	0	0
$686$	1.86651	0.0712636
$687$	−5.84869	−0.223141
$688$	−2.05784	−0.0784543
$689$	55.1099	2.09952
$690$	0	0
$691$	−31.3147	−1.19127	−0.595634	−	0.803256i	$-0.703099\pi$
$691$	−31.3147	−1.19127	−0.595634	+	0.803256i	$0.703099\pi$
$692$	−4.11567	−0.156454
$693$	−0.390376	−0.0148292
$694$	7.45831	0.283114
$695$	0	0
$696$	−3.13349	−0.118775
$697$	−37.5461	−1.42216
$698$	4.33254	0.163989
$699$	−21.7152	−0.821344
$700$	0	0
$701$	−12.8964	−0.487091	−0.243546	−	0.969889i	$-0.578311\pi$
$701$	−12.8964	−0.487091	−0.243546	+	0.969889i	$0.578311\pi$
$702$	−4.92434	−0.185857
$703$	−0.246785	−0.00930767
$704$	2.92434	0.110215
$705$	0	0
$706$	−31.8130	−1.19730
$707$	−1.61734	−0.0608265
$708$	−9.84869	−0.370136
$709$	−7.04774	−0.264683	−0.132342	−	0.991204i	$-0.542250\pi$
$709$	−7.04774	−0.264683	−0.132342	+	0.991204i	$0.542250\pi$
$710$	0	0
$711$	−1.13349	−0.0425093
$712$	−10.4482	−0.391563
$713$	−2.00000	−0.0749006
$714$	0.418296	0.0156543
$715$	0	0
$716$	9.30700	0.347819
$717$	22.2313	0.830245
$718$	28.2313	1.05358
$719$	−24.3470	−0.907990	−0.453995	−	0.891004i	$-0.650002\pi$
$719$	−24.3470	−0.907990	−0.453995	+	0.891004i	$0.650002\pi$
$720$	0	0
$721$	−1.35274	−0.0503786
$722$	−18.9287	−0.704454
$723$	−19.6974	−0.732553
$724$	−11.3070	−0.420221
$725$	0	0
$726$	2.44821	0.0908617
$727$	40.5962	1.50563	0.752814	−	0.658233i	$-0.228696\pi$
$727$	40.5962	1.50563	0.752814	+	0.658233i	$0.228696\pi$
$728$	−0.657360	−0.0243634
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.44821	−0.238496
$732$	−4.92434	−0.182009
$733$	0.635164	0.0234603	0.0117302	−	0.999931i	$-0.496266\pi$
$733$	0.635164	0.0234603	0.0117302	+	0.999931i	$0.496266\pi$
$734$	−0.715196	−0.0263984
$735$	0	0
$736$	2.00000	0.0737210
$737$	−22.9523	−0.845457
$738$	−11.9822	−0.441070
$739$	−10.0501	−0.369699	−0.184850	−	0.982767i	$-0.559180\pi$
$739$	−10.0501	−0.369699	−0.184850	+	0.982767i	$0.559180\pi$
$740$	0	0
$741$	1.31472	0.0482975
$742$	−1.49395	−0.0548447
$743$	8.38266	0.307530	0.153765	−	0.988107i	$-0.450860\pi$
$743$	8.38266	0.307530	0.153765	+	0.988107i	$0.450860\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	19.2492	0.704762
$747$	13.1913	0.482645
$748$	9.16341	0.335047
$749$	−0.896424	−0.0327546
$750$	0	0
$751$	−45.5639	−1.66265	−0.831325	−	0.555787i	$-0.812417\pi$
$751$	−45.5639	−1.66265	−0.831325	+	0.555787i	$0.812417\pi$
$752$	−6.84869	−0.249746
$753$	28.2391	1.02909
$754$	15.4304	0.561942
$755$	0	0
$756$	0.133492	0.00485505
$757$	7.97208	0.289750	0.144875	−	0.989450i	$-0.453722\pi$
$757$	7.97208	0.289750	0.144875	+	0.989450i	$0.453722\pi$
$758$	−5.58170	−0.202737
$759$	−5.84869	−0.212294
$760$	0	0
$761$	−34.1456	−1.23778	−0.618888	−	0.785479i	$-0.712417\pi$
$761$	−34.1456	−1.23778	−0.618888	+	0.785479i	$0.712417\pi$
$762$	10.9822	0.397842
$763$	−1.96436	−0.0711146
$764$	−14.7152	−0.532377
$765$	0	0
$766$	−13.4304	−0.485260
$767$	48.4983	1.75117
$768$	−1.00000	−0.0360844
$769$	0.501672	0.0180908	0.00904539	−	0.999959i	$-0.497121\pi$
$769$	0.501672	0.0180908	0.00904539	+	0.999959i	$0.497121\pi$
$770$	0	0
$771$	−14.2492	−0.513171
$772$	17.9822	0.647193
$773$	11.6675	0.419649	0.209825	−	0.977739i	$-0.432711\pi$
$773$	11.6675	0.419649	0.209825	+	0.977739i	$0.432711\pi$
$774$	−2.05784	−0.0739674
$775$	0	0
$776$	8.71520	0.312857
$777$	0.123392	0.00442668
$778$	−28.1157	−1.00800
$779$	3.19905	0.114618
$780$	0	0
$781$	−44.2034	−1.58172
$782$	6.26698	0.224107
$783$	−3.13349	−0.111982
$784$	−6.98218	−0.249364
$785$	0	0
$786$	−3.24916	−0.115894
$787$	−29.7253	−1.05959	−0.529796	−	0.848125i	$-0.677732\pi$
$787$	−29.7253	−1.05959	−0.529796	+	0.848125i	$0.677732\pi$
$788$	11.6395	0.414641
$789$	18.1157	0.644935
$790$	0	0
$791$	−0.800952	−0.0284786
$792$	2.92434	0.103912
$793$	24.2492	0.861113
$794$	−6.03564	−0.214197
$795$	0	0
$796$	−20.2313	−0.717081
$797$	−19.2791	−0.682900	−0.341450	−	0.939900i	$-0.610918\pi$
$797$	−19.2791	−0.682900	−0.341450	+	0.939900i	$0.610918\pi$
$798$	−0.0356402	−0.00126165
$799$	−21.4603	−0.759211
$800$	0	0
$801$	−10.4482	−0.369169
$802$	−10.5340	−0.371967
$803$	29.2434	1.03198
$804$	7.84869	0.276802
$805$	0	0
$806$	−4.92434	−0.173453
$807$	−0.866508	−0.0305025
$808$	12.1157	0.426228
$809$	44.7952	1.57492	0.787458	−	0.616368i	$-0.211397\pi$
$809$	44.7952	1.57492	0.787458	+	0.616368i	$0.211397\pi$
$810$	0	0
$811$	−10.9166	−0.383334	−0.191667	−	0.981460i	$-0.561389\pi$
$811$	−10.9166	−0.383334	−0.191667	+	0.981460i	$0.561389\pi$
$812$	−0.418296	−0.0146793
$813$	−20.2969	−0.711843
$814$	2.70310	0.0947436
$815$	0	0
$816$	−3.13349	−0.109694
$817$	0.549409	0.0192214
$818$	30.1157	1.05297
$819$	−0.657360	−0.0229700
$820$	0	0
$821$	17.5740	0.613336	0.306668	−	0.951817i	$-0.400786\pi$
$821$	17.5740	0.613336	0.306668	+	0.951817i	$0.400786\pi$
$822$	1.66746	0.0581593
$823$	−36.5340	−1.27349	−0.636747	−	0.771073i	$-0.719720\pi$
$823$	−36.5340	−1.27349	−0.636747	+	0.771073i	$0.719720\pi$
$824$	10.1335	0.353017
$825$	0	0
$826$	−1.31472	−0.0457450
$827$	−53.4304	−1.85796	−0.928978	−	0.370134i	$-0.879312\pi$
$827$	−53.4304	−1.85796	−0.928978	+	0.370134i	$0.879312\pi$
$828$	2.00000	0.0695048
$829$	52.3191	1.81712	0.908559	−	0.417757i	$-0.137184\pi$
$829$	52.3191	1.81712	0.908559	+	0.417757i	$0.137184\pi$
$830$	0	0
$831$	25.3147	0.878158
$832$	4.92434	0.170721
$833$	−21.8786	−0.758049
$834$	−11.9065	−0.412289
$835$	0	0
$836$	−0.780753	−0.0270029
$837$	1.00000	0.0345651
$838$	−12.8309	−0.443235
$839$	−44.3147	−1.52991	−0.764957	−	0.644081i	$-0.777240\pi$
$839$	−44.3147	−1.52991	−0.764957	+	0.644081i	$0.777240\pi$
$840$	0	0
$841$	−19.1812	−0.661422
$842$	17.5161	0.603646
$843$	2.24916	0.0774653
$844$	7.73302	0.266181
$845$	0	0
$846$	−6.84869	−0.235463
$847$	0.326817	0.0112295
$848$	11.1913	0.384312
$849$	12.3827	0.424972
$850$	0	0
$851$	1.84869	0.0633722
$852$	15.1157	0.517854
$853$	14.9021	0.510240	0.255120	−	0.966909i	$-0.417885\pi$
$853$	14.9021	0.510240	0.255120	+	0.966909i	$0.417885\pi$
$854$	−0.657360	−0.0224944
$855$	0	0
$856$	6.71520	0.229521
$857$	−28.5161	−0.974093	−0.487047	−	0.873376i	$-0.661926\pi$
$857$	−28.5161	−0.974093	−0.487047	+	0.873376i	$0.661926\pi$
$858$	−14.4005	−0.491624
$859$	48.1379	1.64244	0.821221	−	0.570610i	$-0.193293\pi$
$859$	48.1379	1.64244	0.821221	+	0.570610i	$0.193293\pi$
$860$	0	0
$861$	−1.59952	−0.0545116
$862$	−9.06556	−0.308774
$863$	−30.7653	−1.04726	−0.523632	−	0.851945i	$-0.675423\pi$
$863$	−30.7653	−1.04726	−0.523632	+	0.851945i	$0.675423\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	15.0477	0.511343
$867$	7.18123	0.243887
$868$	0.133492	0.00453101
$869$	−3.31472	−0.112444
$870$	0	0
$871$	−38.6496	−1.30959
$872$	14.7152	0.498319
$873$	8.71520	0.294965
$874$	−0.533968	−0.0180617
$875$	0	0
$876$	−10.0000	−0.337869
$877$	−52.5282	−1.77375	−0.886876	−	0.462007i	$-0.847130\pi$
$877$	−52.5282	−1.77375	−0.886876	+	0.462007i	$0.847130\pi$
$878$	2.80095	0.0945275
$879$	−6.00000	−0.202375
$880$	0	0
$881$	31.3446	1.05603	0.528014	−	0.849236i	$-0.322937\pi$
$881$	31.3446	1.05603	0.528014	+	0.849236i	$0.322937\pi$
$882$	−6.98218	−0.235102
$883$	−56.5861	−1.90427	−0.952137	−	0.305673i	$-0.901119\pi$
$883$	−56.5861	−1.90427	−0.952137	+	0.305673i	$0.901119\pi$
$884$	15.4304	0.518980
$885$	0	0
$886$	8.06556	0.270968
$887$	7.41830	0.249082	0.124541	−	0.992214i	$-0.460254\pi$
$887$	7.41830	0.249082	0.124541	+	0.992214i	$0.460254\pi$
$888$	−0.924344	−0.0310190
$889$	1.46603	0.0491691
$890$	0	0
$891$	2.92434	0.0979692
$892$	7.13349	0.238847
$893$	1.82849	0.0611881
$894$	19.9644	0.667708
$895$	0	0
$896$	−0.133492	−0.00445965
$897$	−9.84869	−0.328838
$898$	−10.7152	−0.357571
$899$	−3.13349	−0.104508
$900$	0	0
$901$	35.0679	1.16828
$902$	−35.0400	−1.16670
$903$	−0.274704	−0.00914159
$904$	6.00000	0.199557
$905$	0	0
$906$	−19.2492	−0.639511
$907$	−23.6974	−0.786858	−0.393429	−	0.919355i	$-0.628711\pi$
$907$	−23.6974	−0.786858	−0.393429	+	0.919355i	$0.628711\pi$
$908$	11.4005	0.378338
$909$	12.1157	0.401851
$910$	0	0
$911$	10.3827	0.343993	0.171996	−	0.985098i	$-0.444978\pi$
$911$	10.3827	0.343993	0.171996	+	0.985098i	$0.444978\pi$
$912$	0.266984	0.00884072
$913$	38.5760	1.27668
$914$	5.31472	0.175795
$915$	0	0
$916$	5.84869	0.193246
$917$	−0.433737	−0.0143233
$918$	−3.13349	−0.103421
$919$	46.9943	1.55020	0.775099	−	0.631840i	$-0.217700\pi$
$919$	46.9943	1.55020	0.775099	+	0.631840i	$0.217700\pi$
$920$	0	0
$921$	18.6496	0.614527
$922$	−29.2714	−0.964001
$923$	−74.4348	−2.45005
$924$	0.390376	0.0128424
$925$	0	0
$926$	10.5639	0.347151
$927$	10.1335	0.332828
$928$	3.13349	0.102862
$929$	−38.7297	−1.27068	−0.635340	−	0.772233i	$-0.719140\pi$
$929$	−38.7297	−1.27068	−0.635340	+	0.772233i	$0.719140\pi$
$930$	0	0
$931$	1.86413	0.0610944
$932$	21.7152	0.711305
$933$	−15.9166	−0.521087
$934$	7.55179	0.247102
$935$	0	0
$936$	4.92434	0.160957
$937$	−29.1634	−0.952727	−0.476364	−	0.879248i	$-0.658045\pi$
$937$	−29.1634	−0.952727	−0.476364	+	0.879248i	$0.658045\pi$
$938$	1.04774	0.0342098
$939$	−30.6496	−1.00021
$940$	0	0
$941$	−14.8309	−0.483472	−0.241736	−	0.970342i	$-0.577717\pi$
$941$	−14.8309	−0.483472	−0.241736	+	0.970342i	$0.577717\pi$
$942$	−1.40048	−0.0456300
$943$	−23.9644	−0.780387
$944$	9.84869	0.320547
$945$	0	0
$946$	−6.01782	−0.195656
$947$	−19.5894	−0.636571	−0.318285	−	0.947995i	$-0.603107\pi$
$947$	−19.5894	−0.636571	−0.318285	+	0.947995i	$0.603107\pi$
$948$	1.13349	0.0368141
$949$	49.2434	1.59851
$950$	0	0
$951$	14.2313	0.461483
$952$	−0.418296	−0.0135570
$953$	−29.9644	−0.970641	−0.485320	−	0.874336i	$-0.661297\pi$
$953$	−29.9644	−0.970641	−0.485320	+	0.874336i	$0.661297\pi$
$954$	11.1913	0.362333
$955$	0	0
$956$	−22.2313	−0.719013
$957$	−9.16341	−0.296211
$958$	−30.9465	−0.999837
$959$	0.222592	0.00718788
$960$	0	0
$961$	1.00000	0.0322581
$962$	4.55179	0.146756
$963$	6.71520	0.216394
$964$	19.6974	0.634410
$965$	0	0
$966$	0.266984	0.00859006
$967$	4.26698	0.137217	0.0686085	−	0.997644i	$-0.478144\pi$
$967$	4.26698	0.137217	0.0686085	+	0.997644i	$0.478144\pi$
$968$	−2.44821	−0.0786885
$969$	0.836592	0.0268752
$970$	0	0
$971$	−34.7152	−1.11406	−0.557032	−	0.830491i	$-0.688060\pi$
$971$	−34.7152	−1.11406	−0.557032	+	0.830491i	$0.688060\pi$
$972$	−1.00000	−0.0320750
$973$	−1.58942	−0.0509546
$974$	−24.1812	−0.774817
$975$	0	0
$976$	4.92434	0.157624
$977$	−9.21687	−0.294874	−0.147437	−	0.989071i	$-0.547102\pi$
$977$	−9.21687	−0.294874	−0.147437	+	0.989071i	$0.547102\pi$
$978$	6.11567	0.195558
$979$	−30.5542	−0.976515
$980$	0	0
$981$	14.7152	0.469820
$982$	−20.0000	−0.638226
$983$	−29.0679	−0.927123	−0.463562	−	0.886065i	$-0.653429\pi$
$983$	−29.0679	−0.927123	−0.463562	+	0.886065i	$0.653429\pi$
$984$	11.9822	0.381978
$985$	0	0
$986$	9.81877	0.312693
$987$	−0.914245	−0.0291007
$988$	−1.31472	−0.0418268
$989$	−4.11567	−0.130871
$990$	0	0
$991$	22.2969	0.708284	0.354142	−	0.935192i	$-0.384773\pi$
$991$	22.2969	0.708284	0.354142	+	0.935192i	$0.384773\pi$
$992$	−1.00000	−0.0317500
$993$	5.19133	0.164742
$994$	2.01782	0.0640014
$995$	0	0
$996$	−13.1913	−0.417983
$997$	51.8130	1.64094	0.820468	−	0.571693i	$-0.193713\pi$
$997$	51.8130	1.64094	0.820468	+	0.571693i	$0.193713\pi$
$998$	−42.6439	−1.34987
$999$	−0.924344	−0.0292449

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.co.1.2	yes	3
5.2	odd	4		4650.2.d.bi.3349.5		6
5.3	odd	4		4650.2.d.bi.3349.2		6
5.4	even	2		4650.2.a.cj.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4650.2.a.cj.1.2	✓	3	5.4	even	2
4650.2.a.co.1.2	yes	3	1.1	even	1	trivial
4650.2.d.bi.3349.2		6	5.3	odd	4
4650.2.d.bi.3349.5		6	5.2	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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.133492 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -0.133492 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.92434 q^{11} -1.00000 q^{12} +4.92434 q^{13} -0.133492 q^{14} +1.00000 q^{16} +3.13349 q^{17} +1.00000 q^{18} -0.266984 q^{19} +0.133492 q^{21} +2.92434 q^{22} +2.00000 q^{23} -1.00000 q^{24} +4.92434 q^{26} -1.00000 q^{27} -0.133492 q^{28} +3.13349 q^{29} -1.00000 q^{31} +1.00000 q^{32} -2.92434 q^{33} +3.13349 q^{34} +1.00000 q^{36} +0.924344 q^{37} -0.266984 q^{38} -4.92434 q^{39} -11.9822 q^{41} +0.133492 q^{42} -2.05784 q^{43} +2.92434 q^{44} +2.00000 q^{46} -6.84869 q^{47} -1.00000 q^{48} -6.98218 q^{49} -3.13349 q^{51} +4.92434 q^{52} +11.1913 q^{53} -1.00000 q^{54} -0.133492 q^{56} +0.266984 q^{57} +3.13349 q^{58} +9.84869 q^{59} +4.92434 q^{61} -1.00000 q^{62} -0.133492 q^{63} +1.00000 q^{64} -2.92434 q^{66} -7.84869 q^{67} +3.13349 q^{68} -2.00000 q^{69} -15.1157 q^{71} +1.00000 q^{72} +10.0000 q^{73} +0.924344 q^{74} -0.266984 q^{76} -0.390376 q^{77} -4.92434 q^{78} -1.13349 q^{79} +1.00000 q^{81} -11.9822 q^{82} +13.1913 q^{83} +0.133492 q^{84} -2.05784 q^{86} -3.13349 q^{87} +2.92434 q^{88} -10.4482 q^{89} -0.657360 q^{91} +2.00000 q^{92} +1.00000 q^{93} -6.84869 q^{94} -1.00000 q^{96} +8.71520 q^{97} -6.98218 q^{98} +2.92434 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 6 q^{11} - 3 q^{12} + 3 q^{16} + 9 q^{17} + 3 q^{18} - 6 q^{22} + 6 q^{23} - 3 q^{24} - 3 q^{27} + 9 q^{29} - 3 q^{31} + 3 q^{32} + 6 q^{33} + 9 q^{34} + 3 q^{36} - 12 q^{37} - 6 q^{41} + 9 q^{43} - 6 q^{44} + 6 q^{46} + 9 q^{47} - 3 q^{48} + 9 q^{49} - 9 q^{51} + 18 q^{53} - 3 q^{54} + 9 q^{58} - 3 q^{62} + 3 q^{64} + 6 q^{66} + 6 q^{67} + 9 q^{68} - 6 q^{69} - 15 q^{71} + 3 q^{72} + 30 q^{73} - 12 q^{74} + 12 q^{77} - 3 q^{79} + 3 q^{81} - 6 q^{82} + 24 q^{83} + 9 q^{86} - 9 q^{87} - 6 q^{88} - 3 q^{89} + 12 q^{91} + 6 q^{92} + 3 q^{93} + 9 q^{94} - 3 q^{96} - 3 q^{97} + 9 q^{98} - 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 - 6 * q^11 - 3 * q^12 + 3 * q^16 + 9 * q^17 + 3 * q^18 - 6 * q^22 + 6 * q^23 - 3 * q^24 - 3 * q^27 + 9 * q^29 - 3 * q^31 + 3 * q^32 + 6 * q^33 + 9 * q^34 + 3 * q^36 - 12 * q^37 - 6 * q^41 + 9 * q^43 - 6 * q^44 + 6 * q^46 + 9 * q^47 - 3 * q^48 + 9 * q^49 - 9 * q^51 + 18 * q^53 - 3 * q^54 + 9 * q^58 - 3 * q^62 + 3 * q^64 + 6 * q^66 + 6 * q^67 + 9 * q^68 - 6 * q^69 - 15 * q^71 + 3 * q^72 + 30 * q^73 - 12 * q^74 + 12 * q^77 - 3 * q^79 + 3 * q^81 - 6 * q^82 + 24 * q^83 + 9 * q^86 - 9 * q^87 - 6 * q^88 - 3 * q^89 + 12 * q^91 + 6 * q^92 + 3 * q^93 + 9 * q^94 - 3 * q^96 - 3 * q^97 + 9 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

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Database

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