LMFDB - Embedded newform 3450.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 690)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.24264 q^{11} +1.00000 q^{12} +4.82843 q^{13} -2.00000 q^{14} +1.00000 q^{16} +1.17157 q^{17} +1.00000 q^{18} -2.24264 q^{19} -2.00000 q^{21} +4.24264 q^{22} -1.00000 q^{23} +1.00000 q^{24} +4.82843 q^{26} +1.00000 q^{27} -2.00000 q^{28} -7.65685 q^{29} +6.00000 q^{31} +1.00000 q^{32} +4.24264 q^{33} +1.17157 q^{34} +1.00000 q^{36} +3.41421 q^{37} -2.24264 q^{38} +4.82843 q^{39} -1.17157 q^{41} -2.00000 q^{42} +1.75736 q^{43} +4.24264 q^{44} -1.00000 q^{46} +4.82843 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.17157 q^{51} +4.82843 q^{52} +13.4142 q^{53} +1.00000 q^{54} -2.00000 q^{56} -2.24264 q^{57} -7.65685 q^{58} -8.48528 q^{59} -3.41421 q^{61} +6.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.24264 q^{66} -0.585786 q^{67} +1.17157 q^{68} -1.00000 q^{69} +5.65685 q^{71} +1.00000 q^{72} -3.65685 q^{73} +3.41421 q^{74} -2.24264 q^{76} -8.48528 q^{77} +4.82843 q^{78} +7.65685 q^{79} +1.00000 q^{81} -1.17157 q^{82} -1.41421 q^{83} -2.00000 q^{84} +1.75736 q^{86} -7.65685 q^{87} +4.24264 q^{88} +14.8284 q^{89} -9.65685 q^{91} -1.00000 q^{92} +6.00000 q^{93} +4.82843 q^{94} +1.00000 q^{96} -6.00000 q^{97} -3.00000 q^{98} +4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 4 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{12} + 4 q^{13} - 4 q^{14} + 2 q^{16} + 8 q^{17} + 2 q^{18} + 4 q^{19} - 4 q^{21} - 2 q^{23} + 2 q^{24} + 4 q^{26} + 2 q^{27} - 4 q^{28} - 4 q^{29} + 12 q^{31} + 2 q^{32} + 8 q^{34} + 2 q^{36} + 4 q^{37} + 4 q^{38} + 4 q^{39} - 8 q^{41} - 4 q^{42} + 12 q^{43} - 2 q^{46} + 4 q^{47} + 2 q^{48} - 6 q^{49} + 8 q^{51} + 4 q^{52} + 24 q^{53} + 2 q^{54} - 4 q^{56} + 4 q^{57} - 4 q^{58} - 4 q^{61} + 12 q^{62} - 4 q^{63} + 2 q^{64} - 4 q^{67} + 8 q^{68} - 2 q^{69} + 2 q^{72} + 4 q^{73} + 4 q^{74} + 4 q^{76} + 4 q^{78} + 4 q^{79} + 2 q^{81} - 8 q^{82} - 4 q^{84} + 12 q^{86} - 4 q^{87} + 24 q^{89} - 8 q^{91} - 2 q^{92} + 12 q^{93} + 4 q^{94} + 2 q^{96} - 12 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 4 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^12 + 4 * q^13 - 4 * q^14 + 2 * q^16 + 8 * q^17 + 2 * q^18 + 4 * q^19 - 4 * q^21 - 2 * q^23 + 2 * q^24 + 4 * q^26 + 2 * q^27 - 4 * q^28 - 4 * q^29 + 12 * q^31 + 2 * q^32 + 8 * q^34 + 2 * q^36 + 4 * q^37 + 4 * q^38 + 4 * q^39 - 8 * q^41 - 4 * q^42 + 12 * q^43 - 2 * q^46 + 4 * q^47 + 2 * q^48 - 6 * q^49 + 8 * q^51 + 4 * q^52 + 24 * q^53 + 2 * q^54 - 4 * q^56 + 4 * q^57 - 4 * q^58 - 4 * q^61 + 12 * q^62 - 4 * q^63 + 2 * q^64 - 4 * q^67 + 8 * q^68 - 2 * q^69 + 2 * q^72 + 4 * q^73 + 4 * q^74 + 4 * q^76 + 4 * q^78 + 4 * q^79 + 2 * q^81 - 8 * q^82 - 4 * q^84 + 12 * q^86 - 4 * q^87 + 24 * q^89 - 8 * q^91 - 2 * q^92 + 12 * q^93 + 4 * q^94 + 2 * q^96 - 12 * q^97 - 6 * 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
$3$	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
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	1.00000	0.288675
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	1.17157	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$17$	1.17157	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$18$	1.00000	0.235702
$19$	−2.24264	−0.514497	−0.257249	−	0.966345i	$-0.582816\pi$
$19$	−2.24264	−0.514497	−0.257249	+	0.966345i	$0.582816\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	4.24264	0.904534
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	0	0
$26$	4.82843	0.946932
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	4.24264	0.738549
$34$	1.17157	0.200923
$35$	0	0
$36$	1.00000	0.166667
$37$	3.41421	0.561293	0.280647	−	0.959811i	$-0.409451\pi$
$37$	3.41421	0.561293	0.280647	+	0.959811i	$0.409451\pi$
$38$	−2.24264	−0.363804
$39$	4.82843	0.773167
$40$	0	0
$41$	−1.17157	−0.182969	−0.0914845	−	0.995807i	$-0.529161\pi$
$41$	−1.17157	−0.182969	−0.0914845	+	0.995807i	$0.529161\pi$
$42$	−2.00000	−0.308607
$43$	1.75736	0.267995	0.133997	−	0.990982i	$-0.457219\pi$
$43$	1.75736	0.267995	0.133997	+	0.990982i	$0.457219\pi$
$44$	4.24264	0.639602
$45$	0	0
$46$	−1.00000	−0.147442
$47$	4.82843	0.704298	0.352149	−	0.935944i	$-0.385451\pi$
$47$	4.82843	0.704298	0.352149	+	0.935944i	$0.385451\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	1.17157	0.164053
$52$	4.82843	0.669582
$53$	13.4142	1.84258	0.921292	−	0.388872i	$-0.127135\pi$
$53$	13.4142	1.84258	0.921292	+	0.388872i	$0.127135\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	−2.24264	−0.297045
$58$	−7.65685	−1.00539
$59$	−8.48528	−1.10469	−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528	−1.10469	−0.552345	+	0.833616i	$0.686267\pi$
$60$	0	0
$61$	−3.41421	−0.437145	−0.218573	−	0.975821i	$-0.570140\pi$
$61$	−3.41421	−0.437145	−0.218573	+	0.975821i	$0.570140\pi$
$62$	6.00000	0.762001
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	4.24264	0.522233
$67$	−0.585786	−0.0715652	−0.0357826	−	0.999360i	$-0.511392\pi$
$67$	−0.585786	−0.0715652	−0.0357826	+	0.999360i	$0.511392\pi$
$68$	1.17157	0.142074
$69$	−1.00000	−0.120386
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	1.00000	0.117851
$73$	−3.65685	−0.428002	−0.214001	−	0.976833i	$-0.568650\pi$
$73$	−3.65685	−0.428002	−0.214001	+	0.976833i	$0.568650\pi$
$74$	3.41421	0.396894
$75$	0	0
$76$	−2.24264	−0.257249
$77$	−8.48528	−0.966988
$78$	4.82843	0.546712
$79$	7.65685	0.861463	0.430732	−	0.902480i	$-0.358256\pi$
$79$	7.65685	0.861463	0.430732	+	0.902480i	$0.358256\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−1.17157	−0.129379
$83$	−1.41421	−0.155230	−0.0776151	−	0.996983i	$-0.524731\pi$
$83$	−1.41421	−0.155230	−0.0776151	+	0.996983i	$0.524731\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	1.75736	0.189501
$87$	−7.65685	−0.820901
$88$	4.24264	0.452267
$89$	14.8284	1.57181	0.785905	−	0.618347i	$-0.212197\pi$
$89$	14.8284	1.57181	0.785905	+	0.618347i	$0.212197\pi$
$90$	0	0
$91$	−9.65685	−1.01231
$92$	−1.00000	−0.104257
$93$	6.00000	0.622171
$94$	4.82843	0.498014
$95$	0	0
$96$	1.00000	0.102062
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−3.00000	−0.303046
$99$	4.24264	0.426401
$100$	0	0
$101$	−9.31371	−0.926749	−0.463374	−	0.886163i	$-0.653361\pi$
$101$	−9.31371	−0.926749	−0.463374	+	0.886163i	$0.653361\pi$
$102$	1.17157	0.116003
$103$	−12.1421	−1.19640	−0.598200	−	0.801347i	$-0.704117\pi$
$103$	−12.1421	−1.19640	−0.598200	+	0.801347i	$0.704117\pi$
$104$	4.82843	0.473466
$105$	0	0
$106$	13.4142	1.30290
$107$	15.0711	1.45698	0.728488	−	0.685059i	$-0.240224\pi$
$107$	15.0711	1.45698	0.728488	+	0.685059i	$0.240224\pi$
$108$	1.00000	0.0962250
$109$	15.4142	1.47641	0.738207	−	0.674574i	$-0.235673\pi$
$109$	15.4142	1.47641	0.738207	+	0.674574i	$0.235673\pi$
$110$	0	0
$111$	3.41421	0.324063
$112$	−2.00000	−0.188982
$113$	16.9706	1.59646	0.798228	−	0.602355i	$-0.205771\pi$
$113$	16.9706	1.59646	0.798228	+	0.602355i	$0.205771\pi$
$114$	−2.24264	−0.210043
$115$	0	0
$116$	−7.65685	−0.710921
$117$	4.82843	0.446388
$118$	−8.48528	−0.781133
$119$	−2.34315	−0.214796
$120$	0	0
$121$	7.00000	0.636364
$122$	−3.41421	−0.309108
$123$	−1.17157	−0.105637
$124$	6.00000	0.538816
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	1.00000	0.0883883
$129$	1.75736	0.154727
$130$	0	0
$131$	−10.3431	−0.903685	−0.451842	−	0.892098i	$-0.649233\pi$
$131$	−10.3431	−0.903685	−0.451842	+	0.892098i	$0.649233\pi$
$132$	4.24264	0.369274
$133$	4.48528	0.388923
$134$	−0.585786	−0.0506042
$135$	0	0
$136$	1.17157	0.100462
$137$	8.48528	0.724947	0.362473	−	0.931994i	$-0.381932\pi$
$137$	8.48528	0.724947	0.362473	+	0.931994i	$0.381932\pi$
$138$	−1.00000	−0.0851257
$139$	16.9706	1.43942	0.719712	−	0.694273i	$-0.244274\pi$
$139$	16.9706	1.43942	0.719712	+	0.694273i	$0.244274\pi$
$140$	0	0
$141$	4.82843	0.406627
$142$	5.65685	0.474713
$143$	20.4853	1.71307
$144$	1.00000	0.0833333
$145$	0	0
$146$	−3.65685	−0.302643
$147$	−3.00000	−0.247436
$148$	3.41421	0.280647
$149$	−12.7279	−1.04271	−0.521356	−	0.853339i	$-0.674574\pi$
$149$	−12.7279	−1.04271	−0.521356	+	0.853339i	$0.674574\pi$
$150$	0	0
$151$	−15.6569	−1.27414	−0.637068	−	0.770807i	$-0.719853\pi$
$151$	−15.6569	−1.27414	−0.637068	+	0.770807i	$0.719853\pi$
$152$	−2.24264	−0.181902
$153$	1.17157	0.0947161
$154$	−8.48528	−0.683763
$155$	0	0
$156$	4.82843	0.386584
$157$	6.92893	0.552989	0.276494	−	0.961016i	$-0.410827\pi$
$157$	6.92893	0.552989	0.276494	+	0.961016i	$0.410827\pi$
$158$	7.65685	0.609147
$159$	13.4142	1.06382
$160$	0	0
$161$	2.00000	0.157622
$162$	1.00000	0.0785674
$163$	1.17157	0.0917647	0.0458823	−	0.998947i	$-0.485390\pi$
$163$	1.17157	0.0917647	0.0458823	+	0.998947i	$0.485390\pi$
$164$	−1.17157	−0.0914845
$165$	0	0
$166$	−1.41421	−0.109764
$167$	−8.82843	−0.683164	−0.341582	−	0.939852i	$-0.610963\pi$
$167$	−8.82843	−0.683164	−0.341582	+	0.939852i	$0.610963\pi$
$168$	−2.00000	−0.154303
$169$	10.3137	0.793362
$170$	0	0
$171$	−2.24264	−0.171499
$172$	1.75736	0.133997
$173$	3.17157	0.241130	0.120565	−	0.992705i	$-0.461529\pi$
$173$	3.17157	0.241130	0.120565	+	0.992705i	$0.461529\pi$
$174$	−7.65685	−0.580465
$175$	0	0
$176$	4.24264	0.319801
$177$	−8.48528	−0.637793
$178$	14.8284	1.11144
$179$	−10.3431	−0.773083	−0.386542	−	0.922272i	$-0.626330\pi$
$179$	−10.3431	−0.773083	−0.386542	+	0.922272i	$0.626330\pi$
$180$	0	0
$181$	16.3848	1.21787	0.608935	−	0.793220i	$-0.291597\pi$
$181$	16.3848	1.21787	0.608935	+	0.793220i	$0.291597\pi$
$182$	−9.65685	−0.715814
$183$	−3.41421	−0.252386
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	6.00000	0.439941
$187$	4.97056	0.363484
$188$	4.82843	0.352149
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−5.17157	−0.374202	−0.187101	−	0.982341i	$-0.559909\pi$
$191$	−5.17157	−0.374202	−0.187101	+	0.982341i	$0.559909\pi$
$192$	1.00000	0.0721688
$193$	12.9706	0.933642	0.466821	−	0.884352i	$-0.345399\pi$
$193$	12.9706	0.933642	0.466821	+	0.884352i	$0.345399\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−24.6274	−1.75463	−0.877315	−	0.479914i	$-0.840668\pi$
$197$	−24.6274	−1.75463	−0.877315	+	0.479914i	$0.840668\pi$
$198$	4.24264	0.301511
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	−0.585786	−0.0413182
$202$	−9.31371	−0.655310
$203$	15.3137	1.07481
$204$	1.17157	0.0820265
$205$	0	0
$206$	−12.1421	−0.845983
$207$	−1.00000	−0.0695048
$208$	4.82843	0.334791
$209$	−9.51472	−0.658147
$210$	0	0
$211$	−20.4853	−1.41026	−0.705132	−	0.709076i	$-0.749112\pi$
$211$	−20.4853	−1.41026	−0.705132	+	0.709076i	$0.749112\pi$
$212$	13.4142	0.921292
$213$	5.65685	0.387601
$214$	15.0711	1.03024
$215$	0	0
$216$	1.00000	0.0680414
$217$	−12.0000	−0.814613
$218$	15.4142	1.04398
$219$	−3.65685	−0.247107
$220$	0	0
$221$	5.65685	0.380521
$222$	3.41421	0.229147
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	16.9706	1.12887
$227$	14.3848	0.954751	0.477376	−	0.878699i	$-0.341588\pi$
$227$	14.3848	0.954751	0.477376	+	0.878699i	$0.341588\pi$
$228$	−2.24264	−0.148523
$229$	1.75736	0.116130	0.0580648	−	0.998313i	$-0.481507\pi$
$229$	1.75736	0.116130	0.0580648	+	0.998313i	$0.481507\pi$
$230$	0	0
$231$	−8.48528	−0.558291
$232$	−7.65685	−0.502697
$233$	−26.1421	−1.71263	−0.856314	−	0.516455i	$-0.827251\pi$
$233$	−26.1421	−1.71263	−0.856314	+	0.516455i	$0.827251\pi$
$234$	4.82843	0.315644
$235$	0	0
$236$	−8.48528	−0.552345
$237$	7.65685	0.497366
$238$	−2.34315	−0.151884
$239$	−11.1716	−0.722629	−0.361314	−	0.932444i	$-0.617672\pi$
$239$	−11.1716	−0.722629	−0.361314	+	0.932444i	$0.617672\pi$
$240$	0	0
$241$	−10.4853	−0.675416	−0.337708	−	0.941251i	$-0.609652\pi$
$241$	−10.4853	−0.675416	−0.337708	+	0.941251i	$0.609652\pi$
$242$	7.00000	0.449977
$243$	1.00000	0.0641500
$244$	−3.41421	−0.218573
$245$	0	0
$246$	−1.17157	−0.0746968
$247$	−10.8284	−0.688996
$248$	6.00000	0.381000
$249$	−1.41421	−0.0896221
$250$	0	0
$251$	−24.7279	−1.56081	−0.780406	−	0.625273i	$-0.784988\pi$
$251$	−24.7279	−1.56081	−0.780406	+	0.625273i	$0.784988\pi$
$252$	−2.00000	−0.125988
$253$	−4.24264	−0.266733
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	1.75736	0.109408
$259$	−6.82843	−0.424298
$260$	0	0
$261$	−7.65685	−0.473947
$262$	−10.3431	−0.639002
$263$	−17.6569	−1.08877	−0.544384	−	0.838836i	$-0.683237\pi$
$263$	−17.6569	−1.08877	−0.544384	+	0.838836i	$0.683237\pi$
$264$	4.24264	0.261116
$265$	0	0
$266$	4.48528	0.275010
$267$	14.8284	0.907485
$268$	−0.585786	−0.0357826
$269$	−24.8284	−1.51382	−0.756908	−	0.653521i	$-0.773291\pi$
$269$	−24.8284	−1.51382	−0.756908	+	0.653521i	$0.773291\pi$
$270$	0	0
$271$	−1.65685	−0.100647	−0.0503234	−	0.998733i	$-0.516025\pi$
$271$	−1.65685	−0.100647	−0.0503234	+	0.998733i	$0.516025\pi$
$272$	1.17157	0.0710370
$273$	−9.65685	−0.584459
$274$	8.48528	0.512615
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	30.4853	1.83168	0.915842	−	0.401540i	$-0.131525\pi$
$277$	30.4853	1.83168	0.915842	+	0.401540i	$0.131525\pi$
$278$	16.9706	1.01783
$279$	6.00000	0.359211
$280$	0	0
$281$	−30.1421	−1.79813	−0.899065	−	0.437816i	$-0.855752\pi$
$281$	−30.1421	−1.79813	−0.899065	+	0.437816i	$0.855752\pi$
$282$	4.82843	0.287529
$283$	−27.2132	−1.61766	−0.808829	−	0.588045i	$-0.799898\pi$
$283$	−27.2132	−1.61766	−0.808829	+	0.588045i	$0.799898\pi$
$284$	5.65685	0.335673
$285$	0	0
$286$	20.4853	1.21132
$287$	2.34315	0.138312
$288$	1.00000	0.0589256
$289$	−15.6274	−0.919260
$290$	0	0
$291$	−6.00000	−0.351726
$292$	−3.65685	−0.214001
$293$	−28.0416	−1.63821	−0.819105	−	0.573644i	$-0.805529\pi$
$293$	−28.0416	−1.63821	−0.819105	+	0.573644i	$0.805529\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	3.41421	0.198447
$297$	4.24264	0.246183
$298$	−12.7279	−0.737309
$299$	−4.82843	−0.279235
$300$	0	0
$301$	−3.51472	−0.202585
$302$	−15.6569	−0.900951
$303$	−9.31371	−0.535059
$304$	−2.24264	−0.128624
$305$	0	0
$306$	1.17157	0.0669744
$307$	−25.4558	−1.45284	−0.726421	−	0.687250i	$-0.758818\pi$
$307$	−25.4558	−1.45284	−0.726421	+	0.687250i	$0.758818\pi$
$308$	−8.48528	−0.483494
$309$	−12.1421	−0.690742
$310$	0	0
$311$	17.7990	1.00929	0.504644	−	0.863328i	$-0.331624\pi$
$311$	17.7990	1.00929	0.504644	+	0.863328i	$0.331624\pi$
$312$	4.82843	0.273356
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	6.92893	0.391022
$315$	0	0
$316$	7.65685	0.430732
$317$	2.48528	0.139587	0.0697937	−	0.997561i	$-0.477766\pi$
$317$	2.48528	0.139587	0.0697937	+	0.997561i	$0.477766\pi$
$318$	13.4142	0.752232
$319$	−32.4853	−1.81883
$320$	0	0
$321$	15.0711	0.841185
$322$	2.00000	0.111456
$323$	−2.62742	−0.146193
$324$	1.00000	0.0555556
$325$	0	0
$326$	1.17157	0.0648874
$327$	15.4142	0.852408
$328$	−1.17157	−0.0646893
$329$	−9.65685	−0.532400
$330$	0	0
$331$	−3.51472	−0.193186	−0.0965932	−	0.995324i	$-0.530795\pi$
$331$	−3.51472	−0.193186	−0.0965932	+	0.995324i	$0.530795\pi$
$332$	−1.41421	−0.0776151
$333$	3.41421	0.187098
$334$	−8.82843	−0.483070
$335$	0	0
$336$	−2.00000	−0.109109
$337$	−9.51472	−0.518300	−0.259150	−	0.965837i	$-0.583442\pi$
$337$	−9.51472	−0.518300	−0.259150	+	0.965837i	$0.583442\pi$
$338$	10.3137	0.560992
$339$	16.9706	0.921714
$340$	0	0
$341$	25.4558	1.37851
$342$	−2.24264	−0.121268
$343$	20.0000	1.07990
$344$	1.75736	0.0947505
$345$	0	0
$346$	3.17157	0.170505
$347$	7.79899	0.418672	0.209336	−	0.977844i	$-0.432870\pi$
$347$	7.79899	0.418672	0.209336	+	0.977844i	$0.432870\pi$
$348$	−7.65685	−0.410450
$349$	25.3137	1.35501	0.677506	−	0.735517i	$-0.263061\pi$
$349$	25.3137	1.35501	0.677506	+	0.735517i	$0.263061\pi$
$350$	0	0
$351$	4.82843	0.257722
$352$	4.24264	0.226134
$353$	−26.1421	−1.39141	−0.695703	−	0.718330i	$-0.744907\pi$
$353$	−26.1421	−1.39141	−0.695703	+	0.718330i	$0.744907\pi$
$354$	−8.48528	−0.450988
$355$	0	0
$356$	14.8284	0.785905
$357$	−2.34315	−0.124012
$358$	−10.3431	−0.546652
$359$	−19.7990	−1.04495	−0.522475	−	0.852654i	$-0.674991\pi$
$359$	−19.7990	−1.04495	−0.522475	+	0.852654i	$0.674991\pi$
$360$	0	0
$361$	−13.9706	−0.735293
$362$	16.3848	0.861165
$363$	7.00000	0.367405
$364$	−9.65685	−0.506157
$365$	0	0
$366$	−3.41421	−0.178464
$367$	34.9706	1.82545	0.912724	−	0.408576i	$-0.133975\pi$
$367$	34.9706	1.82545	0.912724	+	0.408576i	$0.133975\pi$
$368$	−1.00000	−0.0521286
$369$	−1.17157	−0.0609896
$370$	0	0
$371$	−26.8284	−1.39286
$372$	6.00000	0.311086
$373$	19.2132	0.994822	0.497411	−	0.867515i	$-0.334284\pi$
$373$	19.2132	0.994822	0.497411	+	0.867515i	$0.334284\pi$
$374$	4.97056	0.257022
$375$	0	0
$376$	4.82843	0.249007
$377$	−36.9706	−1.90408
$378$	−2.00000	−0.102869
$379$	−19.2132	−0.986916	−0.493458	−	0.869770i	$-0.664267\pi$
$379$	−19.2132	−0.986916	−0.493458	+	0.869770i	$0.664267\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	−5.17157	−0.264601
$383$	−18.6274	−0.951817	−0.475908	−	0.879495i	$-0.657881\pi$
$383$	−18.6274	−0.951817	−0.475908	+	0.879495i	$0.657881\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	12.9706	0.660184
$387$	1.75736	0.0893316
$388$	−6.00000	−0.304604
$389$	6.38478	0.323721	0.161861	−	0.986814i	$-0.448251\pi$
$389$	6.38478	0.323721	0.161861	+	0.986814i	$0.448251\pi$
$390$	0	0
$391$	−1.17157	−0.0592490
$392$	−3.00000	−0.151523
$393$	−10.3431	−0.521743
$394$	−24.6274	−1.24071
$395$	0	0
$396$	4.24264	0.213201
$397$	−8.34315	−0.418730	−0.209365	−	0.977838i	$-0.567140\pi$
$397$	−8.34315	−0.418730	−0.209365	+	0.977838i	$0.567140\pi$
$398$	−14.0000	−0.701757
$399$	4.48528	0.224545
$400$	0	0
$401$	−28.9706	−1.44672	−0.723360	−	0.690471i	$-0.757403\pi$
$401$	−28.9706	−1.44672	−0.723360	+	0.690471i	$0.757403\pi$
$402$	−0.585786	−0.0292164
$403$	28.9706	1.44313
$404$	−9.31371	−0.463374
$405$	0	0
$406$	15.3137	0.760007
$407$	14.4853	0.718009
$408$	1.17157	0.0580015
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	8.48528	0.418548
$412$	−12.1421	−0.598200
$413$	16.9706	0.835067
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	4.82843	0.236733
$417$	16.9706	0.831052
$418$	−9.51472	−0.465380
$419$	−24.9289	−1.21786	−0.608929	−	0.793225i	$-0.708401\pi$
$419$	−24.9289	−1.21786	−0.608929	+	0.793225i	$0.708401\pi$
$420$	0	0
$421$	10.2426	0.499196	0.249598	−	0.968350i	$-0.419702\pi$
$421$	10.2426	0.499196	0.249598	+	0.968350i	$0.419702\pi$
$422$	−20.4853	−0.997208
$423$	4.82843	0.234766
$424$	13.4142	0.651452
$425$	0	0
$426$	5.65685	0.274075
$427$	6.82843	0.330451
$428$	15.0711	0.728488
$429$	20.4853	0.989039
$430$	0	0
$431$	8.48528	0.408722	0.204361	−	0.978896i	$-0.434488\pi$
$431$	8.48528	0.408722	0.204361	+	0.978896i	$0.434488\pi$
$432$	1.00000	0.0481125
$433$	26.9706	1.29612	0.648061	−	0.761588i	$-0.275580\pi$
$433$	26.9706	1.29612	0.648061	+	0.761588i	$0.275580\pi$
$434$	−12.0000	−0.576018
$435$	0	0
$436$	15.4142	0.738207
$437$	2.24264	0.107280
$438$	−3.65685	−0.174731
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	5.65685	0.269069
$443$	−23.3137	−1.10767	−0.553834	−	0.832627i	$-0.686836\pi$
$443$	−23.3137	−1.10767	−0.553834	+	0.832627i	$0.686836\pi$
$444$	3.41421	0.162031
$445$	0	0
$446$	16.0000	0.757622
$447$	−12.7279	−0.602010
$448$	−2.00000	−0.0944911
$449$	−22.8284	−1.07734	−0.538670	−	0.842517i	$-0.681073\pi$
$449$	−22.8284	−1.07734	−0.538670	+	0.842517i	$0.681073\pi$
$450$	0	0
$451$	−4.97056	−0.234055
$452$	16.9706	0.798228
$453$	−15.6569	−0.735623
$454$	14.3848	0.675111
$455$	0	0
$456$	−2.24264	−0.105021
$457$	1.02944	0.0481550	0.0240775	−	0.999710i	$-0.492335\pi$
$457$	1.02944	0.0481550	0.0240775	+	0.999710i	$0.492335\pi$
$458$	1.75736	0.0821160
$459$	1.17157	0.0546843
$460$	0	0
$461$	−12.8284	−0.597479	−0.298740	−	0.954335i	$-0.596566\pi$
$461$	−12.8284	−0.597479	−0.298740	+	0.954335i	$0.596566\pi$
$462$	−8.48528	−0.394771
$463$	2.97056	0.138054	0.0690269	−	0.997615i	$-0.478011\pi$
$463$	2.97056	0.138054	0.0690269	+	0.997615i	$0.478011\pi$
$464$	−7.65685	−0.355461
$465$	0	0
$466$	−26.1421	−1.21101
$467$	39.3553	1.82115	0.910574	−	0.413346i	$-0.135640\pi$
$467$	39.3553	1.82115	0.910574	+	0.413346i	$0.135640\pi$
$468$	4.82843	0.223194
$469$	1.17157	0.0540982
$470$	0	0
$471$	6.92893	0.319268
$472$	−8.48528	−0.390567
$473$	7.45584	0.342820
$474$	7.65685	0.351691
$475$	0	0
$476$	−2.34315	−0.107398
$477$	13.4142	0.614195
$478$	−11.1716	−0.510976
$479$	4.68629	0.214122	0.107061	−	0.994252i	$-0.465856\pi$
$479$	4.68629	0.214122	0.107061	+	0.994252i	$0.465856\pi$
$480$	0	0
$481$	16.4853	0.751664
$482$	−10.4853	−0.477591
$483$	2.00000	0.0910032
$484$	7.00000	0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	−27.9411	−1.26613	−0.633067	−	0.774097i	$-0.718204\pi$
$487$	−27.9411	−1.26613	−0.633067	+	0.774097i	$0.718204\pi$
$488$	−3.41421	−0.154554
$489$	1.17157	0.0529804
$490$	0	0
$491$	26.6274	1.20168	0.600839	−	0.799370i	$-0.294833\pi$
$491$	26.6274	1.20168	0.600839	+	0.799370i	$0.294833\pi$
$492$	−1.17157	−0.0528186
$493$	−8.97056	−0.404014
$494$	−10.8284	−0.487194
$495$	0	0
$496$	6.00000	0.269408
$497$	−11.3137	−0.507489
$498$	−1.41421	−0.0633724
$499$	13.4558	0.602366	0.301183	−	0.953566i	$-0.402618\pi$
$499$	13.4558	0.602366	0.301183	+	0.953566i	$0.402618\pi$
$500$	0	0
$501$	−8.82843	−0.394425
$502$	−24.7279	−1.10366
$503$	22.1421	0.987269	0.493635	−	0.869669i	$-0.335668\pi$
$503$	22.1421	0.987269	0.493635	+	0.869669i	$0.335668\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	−4.24264	−0.188608
$507$	10.3137	0.458048
$508$	−2.00000	−0.0887357
$509$	22.9706	1.01815	0.509076	−	0.860721i	$-0.329987\pi$
$509$	22.9706	1.01815	0.509076	+	0.860721i	$0.329987\pi$
$510$	0	0
$511$	7.31371	0.323539
$512$	1.00000	0.0441942
$513$	−2.24264	−0.0990150
$514$	18.0000	0.793946
$515$	0	0
$516$	1.75736	0.0773634
$517$	20.4853	0.900942
$518$	−6.82843	−0.300024
$519$	3.17157	0.139217
$520$	0	0
$521$	−16.2843	−0.713427	−0.356713	−	0.934214i	$-0.616103\pi$
$521$	−16.2843	−0.713427	−0.356713	+	0.934214i	$0.616103\pi$
$522$	−7.65685	−0.335131
$523$	−2.92893	−0.128073	−0.0640366	−	0.997948i	$-0.520397\pi$
$523$	−2.92893	−0.128073	−0.0640366	+	0.997948i	$0.520397\pi$
$524$	−10.3431	−0.451842
$525$	0	0
$526$	−17.6569	−0.769875
$527$	7.02944	0.306207
$528$	4.24264	0.184637
$529$	1.00000	0.0434783
$530$	0	0
$531$	−8.48528	−0.368230
$532$	4.48528	0.194462
$533$	−5.65685	−0.245026
$534$	14.8284	0.641689
$535$	0	0
$536$	−0.585786	−0.0253021
$537$	−10.3431	−0.446340
$538$	−24.8284	−1.07043
$539$	−12.7279	−0.548230
$540$	0	0
$541$	35.9411	1.54523	0.772615	−	0.634875i	$-0.218948\pi$
$541$	35.9411	1.54523	0.772615	+	0.634875i	$0.218948\pi$
$542$	−1.65685	−0.0711680
$543$	16.3848	0.703138
$544$	1.17157	0.0502308
$545$	0	0
$546$	−9.65685	−0.413275
$547$	−4.48528	−0.191777	−0.0958884	−	0.995392i	$-0.530569\pi$
$547$	−4.48528	−0.191777	−0.0958884	+	0.995392i	$0.530569\pi$
$548$	8.48528	0.362473
$549$	−3.41421	−0.145715
$550$	0	0
$551$	17.1716	0.731534
$552$	−1.00000	−0.0425628
$553$	−15.3137	−0.651205
$554$	30.4853	1.29520
$555$	0	0
$556$	16.9706	0.719712
$557$	−14.8701	−0.630065	−0.315032	−	0.949081i	$-0.602015\pi$
$557$	−14.8701	−0.630065	−0.315032	+	0.949081i	$0.602015\pi$
$558$	6.00000	0.254000
$559$	8.48528	0.358889
$560$	0	0
$561$	4.97056	0.209857
$562$	−30.1421	−1.27147
$563$	−21.8995	−0.922954	−0.461477	−	0.887152i	$-0.652680\pi$
$563$	−21.8995	−0.922954	−0.461477	+	0.887152i	$0.652680\pi$
$564$	4.82843	0.203313
$565$	0	0
$566$	−27.2132	−1.14386
$567$	−2.00000	−0.0839921
$568$	5.65685	0.237356
$569$	25.6569	1.07559	0.537796	−	0.843075i	$-0.319257\pi$
$569$	25.6569	1.07559	0.537796	+	0.843075i	$0.319257\pi$
$570$	0	0
$571$	−42.7279	−1.78811	−0.894054	−	0.447959i	$-0.852151\pi$
$571$	−42.7279	−1.78811	−0.894054	+	0.447959i	$0.852151\pi$
$572$	20.4853	0.856533
$573$	−5.17157	−0.216046
$574$	2.34315	0.0978010
$575$	0	0
$576$	1.00000	0.0416667
$577$	−20.6274	−0.858731	−0.429365	−	0.903131i	$-0.641263\pi$
$577$	−20.6274	−0.858731	−0.429365	+	0.903131i	$0.641263\pi$
$578$	−15.6274	−0.650015
$579$	12.9706	0.539038
$580$	0	0
$581$	2.82843	0.117343
$582$	−6.00000	−0.248708
$583$	56.9117	2.35704
$584$	−3.65685	−0.151322
$585$	0	0
$586$	−28.0416	−1.15839
$587$	26.8284	1.10733	0.553664	−	0.832740i	$-0.313229\pi$
$587$	26.8284	1.10733	0.553664	+	0.832740i	$0.313229\pi$
$588$	−3.00000	−0.123718
$589$	−13.4558	−0.554438
$590$	0	0
$591$	−24.6274	−1.01304
$592$	3.41421	0.140323
$593$	−12.3431	−0.506872	−0.253436	−	0.967352i	$-0.581561\pi$
$593$	−12.3431	−0.506872	−0.253436	+	0.967352i	$0.581561\pi$
$594$	4.24264	0.174078
$595$	0	0
$596$	−12.7279	−0.521356
$597$	−14.0000	−0.572982
$598$	−4.82843	−0.197449
$599$	−40.1421	−1.64016	−0.820082	−	0.572247i	$-0.806072\pi$
$599$	−40.1421	−1.64016	−0.820082	+	0.572247i	$0.806072\pi$
$600$	0	0
$601$	−20.9706	−0.855407	−0.427704	−	0.903919i	$-0.640677\pi$
$601$	−20.9706	−0.855407	−0.427704	+	0.903919i	$0.640677\pi$
$602$	−3.51472	−0.143249
$603$	−0.585786	−0.0238551
$604$	−15.6569	−0.637068
$605$	0	0
$606$	−9.31371	−0.378344
$607$	−16.9706	−0.688814	−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706	−0.688814	−0.344407	+	0.938820i	$0.611920\pi$
$608$	−2.24264	−0.0909511
$609$	15.3137	0.620543
$610$	0	0
$611$	23.3137	0.943172
$612$	1.17157	0.0473580
$613$	−14.2426	−0.575255	−0.287627	−	0.957742i	$-0.592866\pi$
$613$	−14.2426	−0.575255	−0.287627	+	0.957742i	$0.592866\pi$
$614$	−25.4558	−1.02731
$615$	0	0
$616$	−8.48528	−0.341882
$617$	−31.3137	−1.26064	−0.630321	−	0.776334i	$-0.717077\pi$
$617$	−31.3137	−1.26064	−0.630321	+	0.776334i	$0.717077\pi$
$618$	−12.1421	−0.488428
$619$	−19.2132	−0.772244	−0.386122	−	0.922448i	$-0.626186\pi$
$619$	−19.2132	−0.772244	−0.386122	+	0.922448i	$0.626186\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	17.7990	0.713674
$623$	−29.6569	−1.18818
$624$	4.82843	0.193292
$625$	0	0
$626$	18.0000	0.719425
$627$	−9.51472	−0.379981
$628$	6.92893	0.276494
$629$	4.00000	0.159490
$630$	0	0
$631$	35.1716	1.40016	0.700079	−	0.714065i	$-0.253148\pi$
$631$	35.1716	1.40016	0.700079	+	0.714065i	$0.253148\pi$
$632$	7.65685	0.304573
$633$	−20.4853	−0.814217
$634$	2.48528	0.0987031
$635$	0	0
$636$	13.4142	0.531908
$637$	−14.4853	−0.573928
$638$	−32.4853	−1.28610
$639$	5.65685	0.223782
$640$	0	0
$641$	−8.28427	−0.327209	−0.163605	−	0.986526i	$-0.552312\pi$
$641$	−8.28427	−0.327209	−0.163605	+	0.986526i	$0.552312\pi$
$642$	15.0711	0.594808
$643$	−32.3848	−1.27713	−0.638565	−	0.769568i	$-0.720472\pi$
$643$	−32.3848	−1.27713	−0.638565	+	0.769568i	$0.720472\pi$
$644$	2.00000	0.0788110
$645$	0	0
$646$	−2.62742	−0.103374
$647$	−29.6569	−1.16593	−0.582966	−	0.812497i	$-0.698108\pi$
$647$	−29.6569	−1.16593	−0.582966	+	0.812497i	$0.698108\pi$
$648$	1.00000	0.0392837
$649$	−36.0000	−1.41312
$650$	0	0
$651$	−12.0000	−0.470317
$652$	1.17157	0.0458823
$653$	43.4558	1.70056	0.850279	−	0.526332i	$-0.176433\pi$
$653$	43.4558	1.70056	0.850279	+	0.526332i	$0.176433\pi$
$654$	15.4142	0.602743
$655$	0	0
$656$	−1.17157	−0.0457422
$657$	−3.65685	−0.142667
$658$	−9.65685	−0.376463
$659$	−26.1005	−1.01673	−0.508366	−	0.861141i	$-0.669750\pi$
$659$	−26.1005	−1.01673	−0.508366	+	0.861141i	$0.669750\pi$
$660$	0	0
$661$	9.27208	0.360642	0.180321	−	0.983608i	$-0.442286\pi$
$661$	9.27208	0.360642	0.180321	+	0.983608i	$0.442286\pi$
$662$	−3.51472	−0.136603
$663$	5.65685	0.219694
$664$	−1.41421	−0.0548821
$665$	0	0
$666$	3.41421	0.132298
$667$	7.65685	0.296475
$668$	−8.82843	−0.341582
$669$	16.0000	0.618596
$670$	0	0
$671$	−14.4853	−0.559198
$672$	−2.00000	−0.0771517
$673$	−39.9411	−1.53962	−0.769809	−	0.638275i	$-0.779648\pi$
$673$	−39.9411	−1.53962	−0.769809	+	0.638275i	$0.779648\pi$
$674$	−9.51472	−0.366493
$675$	0	0
$676$	10.3137	0.396681
$677$	24.5269	0.942646	0.471323	−	0.881961i	$-0.343777\pi$
$677$	24.5269	0.942646	0.471323	+	0.881961i	$0.343777\pi$
$678$	16.9706	0.651751
$679$	12.0000	0.460518
$680$	0	0
$681$	14.3848	0.551226
$682$	25.4558	0.974755
$683$	8.48528	0.324680	0.162340	−	0.986735i	$-0.448096\pi$
$683$	8.48528	0.324680	0.162340	+	0.986735i	$0.448096\pi$
$684$	−2.24264	−0.0857495
$685$	0	0
$686$	20.0000	0.763604
$687$	1.75736	0.0670474
$688$	1.75736	0.0669987
$689$	64.7696	2.46752
$690$	0	0
$691$	−18.8284	−0.716267	−0.358134	−	0.933670i	$-0.616587\pi$
$691$	−18.8284	−0.716267	−0.358134	+	0.933670i	$0.616587\pi$
$692$	3.17157	0.120565
$693$	−8.48528	−0.322329
$694$	7.79899	0.296046
$695$	0	0
$696$	−7.65685	−0.290232
$697$	−1.37258	−0.0519903
$698$	25.3137	0.958138
$699$	−26.1421	−0.988786
$700$	0	0
$701$	38.1838	1.44218	0.721090	−	0.692841i	$-0.243641\pi$
$701$	38.1838	1.44218	0.721090	+	0.692841i	$0.243641\pi$
$702$	4.82843	0.182237
$703$	−7.65685	−0.288784
$704$	4.24264	0.159901
$705$	0	0
$706$	−26.1421	−0.983872
$707$	18.6274	0.700556
$708$	−8.48528	−0.318896
$709$	35.4142	1.33001	0.665004	−	0.746839i	$-0.268430\pi$
$709$	35.4142	1.33001	0.665004	+	0.746839i	$0.268430\pi$
$710$	0	0
$711$	7.65685	0.287154
$712$	14.8284	0.555719
$713$	−6.00000	−0.224702
$714$	−2.34315	−0.0876900
$715$	0	0
$716$	−10.3431	−0.386542
$717$	−11.1716	−0.417210
$718$	−19.7990	−0.738892
$719$	28.1421	1.04952	0.524762	−	0.851249i	$-0.324154\pi$
$719$	28.1421	1.04952	0.524762	+	0.851249i	$0.324154\pi$
$720$	0	0
$721$	24.2843	0.904394
$722$	−13.9706	−0.519931
$723$	−10.4853	−0.389952
$724$	16.3848	0.608935
$725$	0	0
$726$	7.00000	0.259794
$727$	−31.6569	−1.17409	−0.587044	−	0.809555i	$-0.699708\pi$
$727$	−31.6569	−1.17409	−0.587044	+	0.809555i	$0.699708\pi$
$728$	−9.65685	−0.357907
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.05887	0.0761502
$732$	−3.41421	−0.126193
$733$	8.58579	0.317123	0.158562	−	0.987349i	$-0.449314\pi$
$733$	8.58579	0.317123	0.158562	+	0.987349i	$0.449314\pi$
$734$	34.9706	1.29079
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−2.48528	−0.0915465
$738$	−1.17157	−0.0431262
$739$	42.6274	1.56807	0.784037	−	0.620714i	$-0.213157\pi$
$739$	42.6274	1.56807	0.784037	+	0.620714i	$0.213157\pi$
$740$	0	0
$741$	−10.8284	−0.397792
$742$	−26.8284	−0.984903
$743$	2.34315	0.0859617	0.0429808	−	0.999076i	$-0.486315\pi$
$743$	2.34315	0.0859617	0.0429808	+	0.999076i	$0.486315\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	19.2132	0.703445
$747$	−1.41421	−0.0517434
$748$	4.97056	0.181742
$749$	−30.1421	−1.10137
$750$	0	0
$751$	−48.8284	−1.78177	−0.890887	−	0.454224i	$-0.849916\pi$
$751$	−48.8284	−1.78177	−0.890887	+	0.454224i	$0.849916\pi$
$752$	4.82843	0.176075
$753$	−24.7279	−0.901136
$754$	−36.9706	−1.34639
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	42.2426	1.53533	0.767667	−	0.640848i	$-0.221417\pi$
$757$	42.2426	1.53533	0.767667	+	0.640848i	$0.221417\pi$
$758$	−19.2132	−0.697855
$759$	−4.24264	−0.153998
$760$	0	0
$761$	−24.3431	−0.882438	−0.441219	−	0.897399i	$-0.645454\pi$
$761$	−24.3431	−0.882438	−0.441219	+	0.897399i	$0.645454\pi$
$762$	−2.00000	−0.0724524
$763$	−30.8284	−1.11606
$764$	−5.17157	−0.187101
$765$	0	0
$766$	−18.6274	−0.673036
$767$	−40.9706	−1.47936
$768$	1.00000	0.0360844
$769$	45.7990	1.65155	0.825777	−	0.563997i	$-0.190737\pi$
$769$	45.7990	1.65155	0.825777	+	0.563997i	$0.190737\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	12.9706	0.466821
$773$	−16.9289	−0.608891	−0.304446	−	0.952530i	$-0.598471\pi$
$773$	−16.9289	−0.608891	−0.304446	+	0.952530i	$0.598471\pi$
$774$	1.75736	0.0631670
$775$	0	0
$776$	−6.00000	−0.215387
$777$	−6.82843	−0.244968
$778$	6.38478	0.228905
$779$	2.62742	0.0941370
$780$	0	0
$781$	24.0000	0.858788
$782$	−1.17157	−0.0418954
$783$	−7.65685	−0.273634
$784$	−3.00000	−0.107143
$785$	0	0
$786$	−10.3431	−0.368928
$787$	2.72792	0.0972399	0.0486200	−	0.998817i	$-0.484518\pi$
$787$	2.72792	0.0972399	0.0486200	+	0.998817i	$0.484518\pi$
$788$	−24.6274	−0.877315
$789$	−17.6569	−0.628601
$790$	0	0
$791$	−33.9411	−1.20681
$792$	4.24264	0.150756
$793$	−16.4853	−0.585410
$794$	−8.34315	−0.296087
$795$	0	0
$796$	−14.0000	−0.496217
$797$	−21.4142	−0.758530	−0.379265	−	0.925288i	$-0.623823\pi$
$797$	−21.4142	−0.758530	−0.379265	+	0.925288i	$0.623823\pi$
$798$	4.48528	0.158777
$799$	5.65685	0.200125
$800$	0	0
$801$	14.8284	0.523937
$802$	−28.9706	−1.02299
$803$	−15.5147	−0.547503
$804$	−0.585786	−0.0206591
$805$	0	0
$806$	28.9706	1.02044
$807$	−24.8284	−0.874002
$808$	−9.31371	−0.327655
$809$	13.3137	0.468085	0.234043	−	0.972226i	$-0.424804\pi$
$809$	13.3137	0.468085	0.234043	+	0.972226i	$0.424804\pi$
$810$	0	0
$811$	6.82843	0.239779	0.119889	−	0.992787i	$-0.461746\pi$
$811$	6.82843	0.239779	0.119889	+	0.992787i	$0.461746\pi$
$812$	15.3137	0.537406
$813$	−1.65685	−0.0581084
$814$	14.4853	0.507709
$815$	0	0
$816$	1.17157	0.0410133
$817$	−3.94113	−0.137883
$818$	18.0000	0.629355
$819$	−9.65685	−0.337438
$820$	0	0
$821$	−6.68629	−0.233353	−0.116677	−	0.993170i	$-0.537224\pi$
$821$	−6.68629	−0.233353	−0.116677	+	0.993170i	$0.537224\pi$
$822$	8.48528	0.295958
$823$	32.2843	1.12536	0.562679	−	0.826675i	$-0.309771\pi$
$823$	32.2843	1.12536	0.562679	+	0.826675i	$0.309771\pi$
$824$	−12.1421	−0.422991
$825$	0	0
$826$	16.9706	0.590481
$827$	18.1005	0.629416	0.314708	−	0.949188i	$-0.398093\pi$
$827$	18.1005	0.629416	0.314708	+	0.949188i	$0.398093\pi$
$828$	−1.00000	−0.0347524
$829$	50.2843	1.74644	0.873222	−	0.487322i	$-0.162026\pi$
$829$	50.2843	1.74644	0.873222	+	0.487322i	$0.162026\pi$
$830$	0	0
$831$	30.4853	1.05752
$832$	4.82843	0.167396
$833$	−3.51472	−0.121778
$834$	16.9706	0.587643
$835$	0	0
$836$	−9.51472	−0.329073
$837$	6.00000	0.207390
$838$	−24.9289	−0.861156
$839$	−33.9411	−1.17178	−0.585889	−	0.810391i	$-0.699255\pi$
$839$	−33.9411	−1.17178	−0.585889	+	0.810391i	$0.699255\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	10.2426	0.352985
$843$	−30.1421	−1.03815
$844$	−20.4853	−0.705132
$845$	0	0
$846$	4.82843	0.166005
$847$	−14.0000	−0.481046
$848$	13.4142	0.460646
$849$	−27.2132	−0.933955
$850$	0	0
$851$	−3.41421	−0.117038
$852$	5.65685	0.193801
$853$	1.31371	0.0449805	0.0224903	−	0.999747i	$-0.492841\pi$
$853$	1.31371	0.0449805	0.0224903	+	0.999747i	$0.492841\pi$
$854$	6.82843	0.233664
$855$	0	0
$856$	15.0711	0.515118
$857$	−45.1716	−1.54303	−0.771516	−	0.636210i	$-0.780501\pi$
$857$	−45.1716	−1.54303	−0.771516	+	0.636210i	$0.780501\pi$
$858$	20.4853	0.699356
$859$	−32.2843	−1.10153	−0.550763	−	0.834662i	$-0.685663\pi$
$859$	−32.2843	−1.10153	−0.550763	+	0.834662i	$0.685663\pi$
$860$	0	0
$861$	2.34315	0.0798542
$862$	8.48528	0.289010
$863$	22.6274	0.770246	0.385123	−	0.922865i	$-0.374159\pi$
$863$	22.6274	0.770246	0.385123	+	0.922865i	$0.374159\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	26.9706	0.916497
$867$	−15.6274	−0.530735
$868$	−12.0000	−0.407307
$869$	32.4853	1.10199
$870$	0	0
$871$	−2.82843	−0.0958376
$872$	15.4142	0.521991
$873$	−6.00000	−0.203069
$874$	2.24264	0.0758585
$875$	0	0
$876$	−3.65685	−0.123554
$877$	−11.1716	−0.377237	−0.188619	−	0.982050i	$-0.560401\pi$
$877$	−11.1716	−0.377237	−0.188619	+	0.982050i	$0.560401\pi$
$878$	0	0
$879$	−28.0416	−0.945821
$880$	0	0
$881$	−2.14214	−0.0721704	−0.0360852	−	0.999349i	$-0.511489\pi$
$881$	−2.14214	−0.0721704	−0.0360852	+	0.999349i	$0.511489\pi$
$882$	−3.00000	−0.101015
$883$	13.1716	0.443259	0.221629	−	0.975131i	$-0.428862\pi$
$883$	13.1716	0.443259	0.221629	+	0.975131i	$0.428862\pi$
$884$	5.65685	0.190261
$885$	0	0
$886$	−23.3137	−0.783239
$887$	−48.4264	−1.62600	−0.813000	−	0.582264i	$-0.802167\pi$
$887$	−48.4264	−1.62600	−0.813000	+	0.582264i	$0.802167\pi$
$888$	3.41421	0.114574
$889$	4.00000	0.134156
$890$	0	0
$891$	4.24264	0.142134
$892$	16.0000	0.535720
$893$	−10.8284	−0.362359
$894$	−12.7279	−0.425685
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	−4.82843	−0.161216
$898$	−22.8284	−0.761794
$899$	−45.9411	−1.53222
$900$	0	0
$901$	15.7157	0.523567
$902$	−4.97056	−0.165502
$903$	−3.51472	−0.116963
$904$	16.9706	0.564433
$905$	0	0
$906$	−15.6569	−0.520164
$907$	27.2132	0.903600	0.451800	−	0.892119i	$-0.350782\pi$
$907$	27.2132	0.903600	0.451800	+	0.892119i	$0.350782\pi$
$908$	14.3848	0.477376
$909$	−9.31371	−0.308916
$910$	0	0
$911$	−0.201010	−0.00665976	−0.00332988	−	0.999994i	$-0.501060\pi$
$911$	−0.201010	−0.00665976	−0.00332988	+	0.999994i	$0.501060\pi$
$912$	−2.24264	−0.0742613
$913$	−6.00000	−0.198571
$914$	1.02944	0.0340508
$915$	0	0
$916$	1.75736	0.0580648
$917$	20.6863	0.683122
$918$	1.17157	0.0386677
$919$	5.02944	0.165906	0.0829529	−	0.996553i	$-0.473565\pi$
$919$	5.02944	0.165906	0.0829529	+	0.996553i	$0.473565\pi$
$920$	0	0
$921$	−25.4558	−0.838799
$922$	−12.8284	−0.422482
$923$	27.3137	0.899042
$924$	−8.48528	−0.279145
$925$	0	0
$926$	2.97056	0.0976187
$927$	−12.1421	−0.398800
$928$	−7.65685	−0.251349
$929$	42.1421	1.38264	0.691319	−	0.722549i	$-0.257030\pi$
$929$	42.1421	1.38264	0.691319	+	0.722549i	$0.257030\pi$
$930$	0	0
$931$	6.72792	0.220499
$932$	−26.1421	−0.856314
$933$	17.7990	0.582713
$934$	39.3553	1.28775
$935$	0	0
$936$	4.82843	0.157822
$937$	44.4264	1.45135	0.725674	−	0.688039i	$-0.241528\pi$
$937$	44.4264	1.45135	0.725674	+	0.688039i	$0.241528\pi$
$938$	1.17157	0.0382532
$939$	18.0000	0.587408
$940$	0	0
$941$	−31.5563	−1.02871	−0.514354	−	0.857578i	$-0.671968\pi$
$941$	−31.5563	−1.02871	−0.514354	+	0.857578i	$0.671968\pi$
$942$	6.92893	0.225757
$943$	1.17157	0.0381517
$944$	−8.48528	−0.276172
$945$	0	0
$946$	7.45584	0.242410
$947$	1.45584	0.0473086	0.0236543	−	0.999720i	$-0.492470\pi$
$947$	1.45584	0.0473086	0.0236543	+	0.999720i	$0.492470\pi$
$948$	7.65685	0.248683
$949$	−17.6569	−0.573166
$950$	0	0
$951$	2.48528	0.0805908
$952$	−2.34315	−0.0759418
$953$	40.0833	1.29842	0.649212	−	0.760607i	$-0.275099\pi$
$953$	40.0833	1.29842	0.649212	+	0.760607i	$0.275099\pi$
$954$	13.4142	0.434301
$955$	0	0
$956$	−11.1716	−0.361314
$957$	−32.4853	−1.05010
$958$	4.68629	0.151407
$959$	−16.9706	−0.548008
$960$	0	0
$961$	5.00000	0.161290
$962$	16.4853	0.531507
$963$	15.0711	0.485658
$964$	−10.4853	−0.337708
$965$	0	0
$966$	2.00000	0.0643489
$967$	4.97056	0.159843	0.0799213	−	0.996801i	$-0.474533\pi$
$967$	4.97056	0.159843	0.0799213	+	0.996801i	$0.474533\pi$
$968$	7.00000	0.224989
$969$	−2.62742	−0.0844048
$970$	0	0
$971$	−17.6152	−0.565299	−0.282650	−	0.959223i	$-0.591213\pi$
$971$	−17.6152	−0.565299	−0.282650	+	0.959223i	$0.591213\pi$
$972$	1.00000	0.0320750
$973$	−33.9411	−1.08810
$974$	−27.9411	−0.895291
$975$	0	0
$976$	−3.41421	−0.109286
$977$	19.7990	0.633426	0.316713	−	0.948521i	$-0.397421\pi$
$977$	19.7990	0.633426	0.316713	+	0.948521i	$0.397421\pi$
$978$	1.17157	0.0374628
$979$	62.9117	2.01067
$980$	0	0
$981$	15.4142	0.492138
$982$	26.6274	0.849715
$983$	−20.4853	−0.653379	−0.326690	−	0.945132i	$-0.605933\pi$
$983$	−20.4853	−0.653379	−0.326690	+	0.945132i	$0.605933\pi$
$984$	−1.17157	−0.0373484
$985$	0	0
$986$	−8.97056	−0.285681
$987$	−9.65685	−0.307381
$988$	−10.8284	−0.344498
$989$	−1.75736	−0.0558808
$990$	0	0
$991$	51.9411	1.64996	0.824982	−	0.565159i	$-0.191185\pi$
$991$	51.9411	1.64996	0.824982	+	0.565159i	$0.191185\pi$
$992$	6.00000	0.190500
$993$	−3.51472	−0.111536
$994$	−11.3137	−0.358849
$995$	0	0
$996$	−1.41421	−0.0448111
$997$	−46.9706	−1.48757	−0.743786	−	0.668417i	$-0.766972\pi$
$997$	−46.9706	−1.48757	−0.743786	+	0.668417i	$0.766972\pi$
$998$	13.4558	0.425937
$999$	3.41421	0.108021

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.bk.1.2		2
5.2	odd	4		690.2.d.b.139.4	yes	4
5.3	odd	4		690.2.d.b.139.2	✓	4
5.4	even	2		3450.2.a.bg.1.2		2
15.2	even	4		2070.2.d.a.829.1		4
15.8	even	4		2070.2.d.a.829.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.d.b.139.2	✓	4	5.3	odd	4
690.2.d.b.139.4	yes	4	5.2	odd	4
2070.2.d.a.829.1		4	15.2	even	4
2070.2.d.a.829.3		4	15.8	even	4
3450.2.a.bg.1.2		2	5.4	even	2
3450.2.a.bk.1.2		2	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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.24264 q^{11} +1.00000 q^{12} +4.82843 q^{13} -2.00000 q^{14} +1.00000 q^{16} +1.17157 q^{17} +1.00000 q^{18} -2.24264 q^{19} -2.00000 q^{21} +4.24264 q^{22} -1.00000 q^{23} +1.00000 q^{24} +4.82843 q^{26} +1.00000 q^{27} -2.00000 q^{28} -7.65685 q^{29} +6.00000 q^{31} +1.00000 q^{32} +4.24264 q^{33} +1.17157 q^{34} +1.00000 q^{36} +3.41421 q^{37} -2.24264 q^{38} +4.82843 q^{39} -1.17157 q^{41} -2.00000 q^{42} +1.75736 q^{43} +4.24264 q^{44} -1.00000 q^{46} +4.82843 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.17157 q^{51} +4.82843 q^{52} +13.4142 q^{53} +1.00000 q^{54} -2.00000 q^{56} -2.24264 q^{57} -7.65685 q^{58} -8.48528 q^{59} -3.41421 q^{61} +6.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.24264 q^{66} -0.585786 q^{67} +1.17157 q^{68} -1.00000 q^{69} +5.65685 q^{71} +1.00000 q^{72} -3.65685 q^{73} +3.41421 q^{74} -2.24264 q^{76} -8.48528 q^{77} +4.82843 q^{78} +7.65685 q^{79} +1.00000 q^{81} -1.17157 q^{82} -1.41421 q^{83} -2.00000 q^{84} +1.75736 q^{86} -7.65685 q^{87} +4.24264 q^{88} +14.8284 q^{89} -9.65685 q^{91} -1.00000 q^{92} +6.00000 q^{93} +4.82843 q^{94} +1.00000 q^{96} -6.00000 q^{97} -3.00000 q^{98} +4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 4 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{12} + 4 q^{13} - 4 q^{14} + 2 q^{16} + 8 q^{17} + 2 q^{18} + 4 q^{19} - 4 q^{21} - 2 q^{23} + 2 q^{24} + 4 q^{26} + 2 q^{27} - 4 q^{28} - 4 q^{29} + 12 q^{31} + 2 q^{32} + 8 q^{34} + 2 q^{36} + 4 q^{37} + 4 q^{38} + 4 q^{39} - 8 q^{41} - 4 q^{42} + 12 q^{43} - 2 q^{46} + 4 q^{47} + 2 q^{48} - 6 q^{49} + 8 q^{51} + 4 q^{52} + 24 q^{53} + 2 q^{54} - 4 q^{56} + 4 q^{57} - 4 q^{58} - 4 q^{61} + 12 q^{62} - 4 q^{63} + 2 q^{64} - 4 q^{67} + 8 q^{68} - 2 q^{69} + 2 q^{72} + 4 q^{73} + 4 q^{74} + 4 q^{76} + 4 q^{78} + 4 q^{79} + 2 q^{81} - 8 q^{82} - 4 q^{84} + 12 q^{86} - 4 q^{87} + 24 q^{89} - 8 q^{91} - 2 q^{92} + 12 q^{93} + 4 q^{94} + 2 q^{96} - 12 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 4 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^12 + 4 * q^13 - 4 * q^14 + 2 * q^16 + 8 * q^17 + 2 * q^18 + 4 * q^19 - 4 * q^21 - 2 * q^23 + 2 * q^24 + 4 * q^26 + 2 * q^27 - 4 * q^28 - 4 * q^29 + 12 * q^31 + 2 * q^32 + 8 * q^34 + 2 * q^36 + 4 * q^37 + 4 * q^38 + 4 * q^39 - 8 * q^41 - 4 * q^42 + 12 * q^43 - 2 * q^46 + 4 * q^47 + 2 * q^48 - 6 * q^49 + 8 * q^51 + 4 * q^52 + 24 * q^53 + 2 * q^54 - 4 * q^56 + 4 * q^57 - 4 * q^58 - 4 * q^61 + 12 * q^62 - 4 * q^63 + 2 * q^64 - 4 * q^67 + 8 * q^68 - 2 * q^69 + 2 * q^72 + 4 * q^73 + 4 * q^74 + 4 * q^76 + 4 * q^78 + 4 * q^79 + 2 * q^81 - 8 * q^82 - 4 * q^84 + 12 * q^86 - 4 * q^87 + 24 * q^89 - 8 * q^91 - 2 * q^92 + 12 * q^93 + 4 * q^94 + 2 * q^96 - 12 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more