LMFDB - Embedded newform 4368.2.h.r.337.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 20x^{8} + 138x^{6} + 364x^{4} + 249x^{2} + 36 $ x^10 + 20x^8 + 138x^6 + 364x^4 + 249x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.7
Root			$-0.854441i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.r.337.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.11295i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.11295i q^{5} +1.00000i q^{7} +1.00000 q^{9} -3.80539i q^{11} +(0.854441 + 3.50285i) q^{13} -2.11295i q^{15} +6.38288 q^{17} -3.24434i q^{19} -1.00000i q^{21} +9.17912 q^{23} +0.535458 q^{25} -1.00000 q^{27} -10.3993 q^{29} +2.01644i q^{31} +3.80539i q^{33} -2.11295 q^{35} -2.15698i q^{37} +(-0.854441 - 3.50285i) q^{39} -7.40847i q^{41} +1.29552 q^{43} +2.11295i q^{45} -1.12498i q^{47} -1.00000 q^{49} -6.38288 q^{51} -3.40133 q^{53} +8.04058 q^{55} +3.24434i q^{57} -10.1371i q^{59} +11.1627 q^{61} +1.00000i q^{63} +(-7.40133 + 1.80539i) q^{65} -15.9576i q^{67} -9.17912 q^{69} +2.35160i q^{71} +0.690434i q^{73} -0.535458 q^{75} +3.80539 q^{77} -11.3873 q^{79} +1.00000 q^{81} -11.0128i q^{83} +13.4867i q^{85} +10.3993 q^{87} +16.1395i q^{89} +(-3.50285 + 0.854441i) q^{91} -2.01644i q^{93} +6.85512 q^{95} -15.5108i q^{97} -3.80539i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 10 * q^9	$ 10 q - 10 q^{3} + 10 q^{9} - 4 q^{13} + 10 q^{17} - 8 q^{23} - 2 q^{25} - 10 q^{27} - 44 q^{29} + 4 q^{39} - 20 q^{43} - 10 q^{49} - 10 q^{51} + 10 q^{53} + 2 q^{55} + 18 q^{61} - 30 q^{65} + 8 q^{69} + 2 q^{75} - 2 q^{77} - 2 q^{79} + 10 q^{81} + 44 q^{87} + 6 q^{91} - 44 q^{95}+O(q^{100}) $ 10 * q - 10 * q^3 + 10 * q^9 - 4 * q^13 + 10 * q^17 - 8 * q^23 - 2 * q^25 - 10 * q^27 - 44 * q^29 + 4 * q^39 - 20 * q^43 - 10 * q^49 - 10 * q^51 + 10 * q^53 + 2 * q^55 + 18 * q^61 - 30 * q^65 + 8 * q^69 + 2 * q^75 - 2 * q^77 - 2 * q^79 + 10 * q^81 + 44 * q^87 + 6 * q^91 - 44 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			2.11295i			0.944938i	0.881347	+	0.472469i	$0.156637\pi$
$5$			2.11295i			0.944938i	−0.881347	+	0.472469i	$0.843363\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	3.80539i		−	1.14737i	−0.819077	−	0.573684i	$-0.805514\pi$
$11$		−	3.80539i		−	1.14737i	0.819077	−	0.573684i	$-0.194486\pi$
$12$		0			0
$13$	0.854441	+	3.50285i	0.236979	+	0.971515i
$13$	0.854441	+	3.50285i	0.236979	+	0.971515i
$14$		0			0
$15$		−	2.11295i		−	0.545560i
$16$		0			0
$17$	6.38288			1.54807			0.774037	−	0.633140i	$-0.218234\pi$
$17$	6.38288			1.54807			0.774037	+	0.633140i	$0.218234\pi$
$18$		0			0
$19$		−	3.24434i		−	0.744303i	−0.928172	−	0.372151i	$-0.878620\pi$
$19$		−	3.24434i		−	0.744303i	0.928172	−	0.372151i	$-0.121380\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$	9.17912			1.91398			0.956989	−	0.290124i	$-0.0936966\pi$
$23$	9.17912			1.91398			0.956989	+	0.290124i	$0.0936966\pi$
$24$		0			0
$25$	0.535458			0.107092
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−10.3993			−1.93110			−0.965552	−	0.260209i	$-0.916209\pi$
$29$	−10.3993			−1.93110			−0.965552	+	0.260209i	$0.916209\pi$
$30$		0			0
$31$			2.01644i			0.362163i	0.983468	+	0.181082i	$0.0579598\pi$
$31$			2.01644i			0.362163i	−0.983468	+	0.181082i	$0.942040\pi$
$32$		0			0
$33$			3.80539i			0.662433i
$34$		0			0
$35$	−2.11295			−0.357153
$36$		0			0
$37$		−	2.15698i		−	0.354606i	−0.984156	−	0.177303i	$-0.943263\pi$
$37$		−	2.15698i		−	0.354606i	0.984156	−	0.177303i	$-0.0567373\pi$
$38$		0			0
$39$	−0.854441	−	3.50285i	−0.136820	−	0.560904i
$40$		0			0
$41$		−	7.40847i		−	1.15701i	−0.815680	−	0.578504i	$-0.803637\pi$
$41$		−	7.40847i		−	1.15701i	0.815680	−	0.578504i	$-0.196363\pi$
$42$		0			0
$43$	1.29552			0.197565			0.0987824	−	0.995109i	$-0.468505\pi$
$43$	1.29552			0.197565			0.0987824	+	0.995109i	$0.468505\pi$
$44$		0			0
$45$			2.11295i			0.314979i
$46$		0			0
$47$		−	1.12498i		−	0.164096i	−0.996628	−	0.0820480i	$-0.973854\pi$
$47$		−	1.12498i		−	0.164096i	0.996628	−	0.0820480i	$-0.0261461\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−6.38288			−0.893781
$52$		0			0
$53$	−3.40133			−0.467208			−0.233604	−	0.972332i	$-0.575052\pi$
$53$	−3.40133			−0.467208			−0.233604	+	0.972332i	$0.575052\pi$
$54$		0			0
$55$	8.04058			1.08419
$56$		0			0
$57$			3.24434i			0.429723i
$58$		0			0
$59$		−	10.1371i		−	1.31974i	−0.751381	−	0.659868i	$-0.770612\pi$
$59$		−	10.1371i		−	1.31974i	0.751381	−	0.659868i	$-0.229388\pi$
$60$		0			0
$61$	11.1627			1.42923			0.714617	−	0.699516i	$-0.246601\pi$
$61$	11.1627			1.42923			0.714617	+	0.699516i	$0.246601\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−7.40133	+	1.80539i	−0.918021	+	0.223931i
$66$		0			0
$67$		−	15.9576i		−	1.94953i	−0.223224	−	0.974767i	$-0.571658\pi$
$67$		−	15.9576i		−	1.94953i	0.223224	−	0.974767i	$-0.428342\pi$
$68$		0			0
$69$	−9.17912			−1.10504
$70$		0			0
$71$			2.35160i			0.279083i	0.990216	+	0.139542i	$0.0445629\pi$
$71$			2.35160i			0.279083i	−0.990216	+	0.139542i	$0.955437\pi$
$72$		0			0
$73$			0.690434i			0.0808092i	0.999183	+	0.0404046i	$0.0128647\pi$
$73$			0.690434i			0.0808092i	−0.999183	+	0.0404046i	$0.987135\pi$
$74$		0			0
$75$	−0.535458			−0.0618294
$76$		0			0
$77$	3.80539			0.433664
$78$		0			0
$79$	−11.3873			−1.28117			−0.640584	−	0.767888i	$-0.721308\pi$
$79$	−11.3873			−1.28117			−0.640584	+	0.767888i	$0.721308\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	11.0128i		−	1.20882i	−0.796675	−	0.604408i	$-0.793410\pi$
$83$		−	11.0128i		−	1.20882i	0.796675	−	0.604408i	$-0.206590\pi$
$84$		0			0
$85$			13.4867i			1.46284i
$86$		0			0
$87$	10.3993			1.11492
$88$		0			0
$89$			16.1395i			1.71078i	0.517983	+	0.855391i	$0.326683\pi$
$89$			16.1395i			1.71078i	−0.517983	+	0.855391i	$0.673317\pi$
$90$		0			0
$91$	−3.50285	+	0.854441i	−0.367198	+	0.0895698i
$92$		0			0
$93$		−	2.01644i		−	0.209095i
$94$		0			0
$95$	6.85512			0.703320
$96$		0			0
$97$		−	15.5108i		−	1.57488i	−0.616388	−	0.787442i	$-0.711405\pi$
$97$		−	15.5108i		−	1.57488i	0.616388	−	0.787442i	$-0.288595\pi$
$98$		0			0
$99$		−	3.80539i		−	0.382456i
$100$		0			0
$101$	3.68603			0.366774			0.183387	−	0.983041i	$-0.441294\pi$
$101$	3.68603			0.366774			0.183387	+	0.983041i	$0.441294\pi$
$102$		0			0
$103$	1.83299			0.180609			0.0903047	−	0.995914i	$-0.471216\pi$
$103$	1.83299			0.180609			0.0903047	+	0.995914i	$0.471216\pi$
$104$		0			0
$105$	2.11295			0.206202
$106$		0			0
$107$	11.3708			1.09926			0.549630	−	0.835408i	$-0.314769\pi$
$107$	11.3708			1.09926			0.549630	+	0.835408i	$0.314769\pi$
$108$		0			0
$109$			1.22790i			0.117612i	0.998269	+	0.0588058i	$0.0187293\pi$
$109$			1.22790i			0.117612i	−0.998269	+	0.0588058i	$0.981271\pi$
$110$		0			0
$111$			2.15698i			0.204732i
$112$		0			0
$113$	20.3049			1.91012			0.955062	−	0.296406i	$-0.0957883\pi$
$113$	20.3049			1.91012			0.955062	+	0.296406i	$0.0957883\pi$
$114$		0			0
$115$			19.3950i			1.80859i
$116$		0			0
$117$	0.854441	+	3.50285i	0.0789931	+	0.323838i
$118$		0			0
$119$			6.38288i			0.585117i
$120$		0			0
$121$	−3.48098			−0.316453
$122$		0			0
$123$			7.40847i			0.667999i
$124$		0			0
$125$			11.6961i			1.04613i
$126$		0			0
$127$	−11.0938			−0.984413			−0.492206	−	0.870479i	$-0.663809\pi$
$127$	−11.0938			−0.984413			−0.492206	+	0.870479i	$0.663809\pi$
$128$		0			0
$129$	−1.29552			−0.114064
$130$		0			0
$131$	16.9044			1.47695			0.738474	−	0.674282i	$-0.235547\pi$
$131$	16.9044			1.47695			0.738474	+	0.674282i	$0.235547\pi$
$132$		0			0
$133$	3.24434			0.281320
$134$		0			0
$135$		−	2.11295i		−	0.181853i
$136$		0			0
$137$			0.424521i			0.0362693i	0.999836	+	0.0181346i	$0.00577275\pi$
$137$			0.424521i			0.0362693i	−0.999836	+	0.0181346i	$0.994227\pi$
$138$		0			0
$139$	18.8424			1.59819			0.799094	−	0.601206i	$-0.205313\pi$
$139$	18.8424			1.59819			0.799094	+	0.601206i	$0.205313\pi$
$140$		0			0
$141$			1.12498i			0.0947408i
$142$		0			0
$143$	13.3297	−	3.25148i	1.11468	−	0.271903i
$144$		0			0
$145$		−	21.9732i		−	1.82477i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$			17.5549i			1.43815i	0.694933	+	0.719075i	$0.255434\pi$
$149$			17.5549i			1.43815i	−0.694933	+	0.719075i	$0.744566\pi$
$150$		0			0
$151$			11.9754i			0.974541i	0.873251	+	0.487270i	$0.162007\pi$
$151$			11.9754i			0.974541i	−0.873251	+	0.487270i	$0.837993\pi$
$152$		0			0
$153$	6.38288			0.516025
$154$		0			0
$155$	−4.26063			−0.342222
$156$		0			0
$157$	−10.1058			−0.806531			−0.403266	−	0.915083i	$-0.632125\pi$
$157$	−10.1058			−0.806531			−0.403266	+	0.915083i	$0.632125\pi$
$158$		0			0
$159$	3.40133			0.269743
$160$		0			0
$161$			9.17912i			0.723416i
$162$		0			0
$163$			7.32969i			0.574106i	0.957915	+	0.287053i	$0.0926755\pi$
$163$			7.32969i			0.574106i	−0.957915	+	0.287053i	$0.907324\pi$
$164$		0			0
$165$	−8.04058			−0.625958
$166$		0			0
$167$			7.32767i			0.567032i	0.958967	+	0.283516i	$0.0915009\pi$
$167$			7.32767i			0.567032i	−0.958967	+	0.283516i	$0.908499\pi$
$168$		0			0
$169$	−11.5399	+	5.98595i	−0.887682	+	0.460458i
$170$		0			0
$171$		−	3.24434i		−	0.248101i
$172$		0			0
$173$	16.6634			1.26689			0.633447	−	0.773786i	$-0.281639\pi$
$173$	16.6634			1.26689			0.633447	+	0.773786i	$0.281639\pi$
$174$		0			0
$175$			0.535458i			0.0404769i
$176$		0			0
$177$			10.1371i			0.761950i
$178$		0			0
$179$	−9.55061			−0.713846			−0.356923	−	0.934134i	$-0.616174\pi$
$179$	−9.55061			−0.713846			−0.356923	+	0.934134i	$0.616174\pi$
$180$		0			0
$181$	−5.20873			−0.387162			−0.193581	−	0.981084i	$-0.562010\pi$
$181$	−5.20873			−0.387162			−0.193581	+	0.981084i	$0.562010\pi$
$182$		0			0
$183$	−11.1627			−0.825169
$184$		0			0
$185$	4.55759			0.335081
$186$		0			0
$187$		−	24.2893i		−	1.77621i
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	−0.569057			−0.0411755			−0.0205878	−	0.999788i	$-0.506554\pi$
$191$	−0.569057			−0.0411755			−0.0205878	+	0.999788i	$0.506554\pi$
$192$		0			0
$193$			10.7475i			0.773619i	0.922160	+	0.386809i	$0.126423\pi$
$193$			10.7475i			0.773619i	−0.922160	+	0.386809i	$0.873577\pi$
$194$		0			0
$195$	7.40133	−	1.80539i	0.530020	−	0.129287i
$196$		0			0
$197$		−	22.5889i		−	1.60939i	−0.593688	−	0.804695i	$-0.702329\pi$
$197$		−	22.5889i		−	1.60939i	0.593688	−	0.804695i	$-0.297671\pi$
$198$		0			0
$199$	1.75494			0.124404			0.0622022	−	0.998064i	$-0.480188\pi$
$199$	1.75494			0.124404			0.0622022	+	0.998064i	$0.480188\pi$
$200$		0			0
$201$			15.9576i			1.12556i
$202$		0			0
$203$		−	10.3993i		−	0.729889i
$204$		0			0
$205$	15.6537			1.09330
$206$		0			0
$207$	9.17912			0.637993
$208$		0			0
$209$	−12.3460			−0.853989
$210$		0			0
$211$	−7.49308			−0.515845			−0.257923	−	0.966166i	$-0.583038\pi$
$211$	−7.49308			−0.515845			−0.257923	+	0.966166i	$0.583038\pi$
$212$		0			0
$213$		−	2.35160i		−	0.161129i
$214$		0			0
$215$			2.73736i			0.186687i
$216$		0			0
$217$	−2.01644			−0.136885
$218$		0			0
$219$		−	0.690434i		−	0.0466552i
$220$		0			0
$221$	5.45379	+	22.3582i	0.366862	+	1.50398i
$222$		0			0
$223$			13.9741i			0.935773i	0.883789	+	0.467886i	$0.154984\pi$
$223$			13.9741i			0.935773i	−0.883789	+	0.467886i	$0.845016\pi$
$224$		0			0
$225$	0.535458			0.0356972
$226$		0			0
$227$			27.5508i			1.82861i	0.405023	+	0.914307i	$0.367264\pi$
$227$			27.5508i			1.82861i	−0.405023	+	0.914307i	$0.632736\pi$
$228$		0			0
$229$		−	13.4063i		−	0.885913i	−0.896543	−	0.442957i	$-0.853930\pi$
$229$		−	13.4063i		−	0.885913i	0.896543	−	0.442957i	$-0.146070\pi$
$230$		0			0
$231$	−3.80539			−0.250376
$232$		0			0
$233$	17.5432			1.14929			0.574645	−	0.818403i	$-0.305140\pi$
$233$	17.5432			1.14929			0.574645	+	0.818403i	$0.305140\pi$
$234$		0			0
$235$	2.37703			0.155061
$236$		0			0
$237$	11.3873			0.739683
$238$		0			0
$239$			6.69844i			0.433286i	0.976251	+	0.216643i	$0.0695108\pi$
$239$			6.69844i			0.433286i	−0.976251	+	0.216643i	$0.930489\pi$
$240$		0			0
$241$		−	0.763359i		−	0.0491723i	−0.999698	−	0.0245861i	$-0.992173\pi$
$241$		−	0.763359i		−	0.0491723i	0.999698	−	0.0245861i	$-0.00782680\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	2.11295i		−	0.134991i
$246$		0			0
$247$	11.3644	−	2.77210i	0.723101	−	0.176384i
$248$		0			0
$249$			11.0128i			0.697910i
$250$		0			0
$251$	−20.9739			−1.32386			−0.661931	−	0.749565i	$-0.730263\pi$
$251$	−20.9739			−1.32386			−0.661931	+	0.749565i	$0.730263\pi$
$252$		0			0
$253$		−	34.9301i		−	2.19604i
$254$		0			0
$255$		−	13.4867i		−	0.844568i
$256$		0			0
$257$	18.9076			1.17942			0.589711	−	0.807614i	$-0.299241\pi$
$257$	18.9076			1.17942			0.589711	+	0.807614i	$0.299241\pi$
$258$		0			0
$259$	2.15698			0.134028
$260$		0			0
$261$	−10.3993			−0.643702
$262$		0			0
$263$	−1.86458			−0.114975			−0.0574873	−	0.998346i	$-0.518309\pi$
$263$	−1.86458			−0.114975			−0.0574873	+	0.998346i	$0.518309\pi$
$264$		0			0
$265$		−	7.18682i		−	0.441483i
$266$		0			0
$267$		−	16.1395i		−	0.987720i
$268$		0			0
$269$	21.3963			1.30455			0.652277	−	0.757981i	$-0.273814\pi$
$269$	21.3963			1.30455			0.652277	+	0.757981i	$0.273814\pi$
$270$		0			0
$271$		−	10.2347i		−	0.621713i	−0.950457	−	0.310857i	$-0.899384\pi$
$271$		−	10.2347i		−	0.621713i	0.950457	−	0.310857i	$-0.100616\pi$
$272$		0			0
$273$	3.50285	−	0.854441i	0.212002	−	0.0517131i
$274$		0			0
$275$		−	2.03763i		−	0.122874i
$276$		0			0
$277$	−2.47224			−0.148543			−0.0742713	−	0.997238i	$-0.523663\pi$
$277$	−2.47224			−0.148543			−0.0742713	+	0.997238i	$0.523663\pi$
$278$		0			0
$279$			2.01644i			0.120721i
$280$		0			0
$281$		−	12.5446i		−	0.748349i	−0.927358	−	0.374174i	$-0.877926\pi$
$281$		−	12.5446i		−	0.748349i	0.927358	−	0.374174i	$-0.122074\pi$
$282$		0			0
$283$	−17.3480			−1.03123			−0.515615	−	0.856820i	$-0.672437\pi$
$283$	−17.3480			−1.03123			−0.515615	+	0.856820i	$0.672437\pi$
$284$		0			0
$285$	−6.85512			−0.406062
$286$		0			0
$287$	7.40847			0.437308
$288$		0			0
$289$	23.7411			1.39654
$290$		0			0
$291$			15.5108i			0.909260i
$292$		0			0
$293$			7.26559i			0.424460i	0.977220	+	0.212230i	$0.0680727\pi$
$293$			7.26559i			0.424460i	−0.977220	+	0.212230i	$0.931927\pi$
$294$		0			0
$295$	21.4191			1.24707
$296$		0			0
$297$			3.80539i			0.220811i
$298$		0			0
$299$	7.84302	+	32.1530i	0.453573	+	1.85946i
$300$		0			0
$301$			1.29552i			0.0746725i
$302$		0			0
$303$	−3.68603			−0.211757
$304$		0			0
$305$			23.5861i			1.35054i
$306$		0			0
$307$			3.67840i			0.209937i	0.994476	+	0.104969i	$0.0334742\pi$
$307$			3.67840i			0.209937i	−0.994476	+	0.104969i	$0.966526\pi$
$308$		0			0
$309$	−1.83299			−0.104275
$310$		0			0
$311$	13.3963			0.759633			0.379816	−	0.925062i	$-0.375987\pi$
$311$	13.3963			0.759633			0.379816	+	0.925062i	$0.375987\pi$
$312$		0			0
$313$	15.1380			0.855652			0.427826	−	0.903861i	$-0.359280\pi$
$313$	15.1380			0.855652			0.427826	+	0.903861i	$0.359280\pi$
$314$		0			0
$315$	−2.11295			−0.119051
$316$		0			0
$317$			24.0907i			1.35307i	0.736411	+	0.676534i	$0.236519\pi$
$317$			24.0907i			1.35307i	−0.736411	+	0.676534i	$0.763481\pi$
$318$		0			0
$319$			39.5734i			2.21569i
$320$		0			0
$321$	−11.3708			−0.634658
$322$		0			0
$323$		−	20.7082i		−	1.15224i
$324$		0			0
$325$	0.457518	+	1.87563i	0.0253785	+	0.104041i
$326$		0			0
$327$		−	1.22790i		−	0.0679031i
$328$		0			0
$329$	1.12498			0.0620224
$330$		0			0
$331$			14.5936i			0.802138i	0.916048	+	0.401069i	$0.131361\pi$
$331$			14.5936i			0.802138i	−0.916048	+	0.401069i	$0.868639\pi$
$332$		0			0
$333$		−	2.15698i		−	0.118202i
$334$		0			0
$335$	33.7176			1.84219
$336$		0			0
$337$	24.6594			1.34329			0.671643	−	0.740875i	$-0.265589\pi$
$337$	24.6594			1.34329			0.671643	+	0.740875i	$0.265589\pi$
$338$		0			0
$339$	−20.3049			−1.10281
$340$		0			0
$341$	7.67334			0.415535
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		−	19.3950i		−	1.04419i
$346$		0			0
$347$	22.9264			1.23075			0.615377	−	0.788233i	$-0.289004\pi$
$347$	22.9264			1.23075			0.615377	+	0.788233i	$0.289004\pi$
$348$		0			0
$349$		−	10.0748i		−	0.539289i	−0.962960	−	0.269645i	$-0.913094\pi$
$349$		−	10.0748i		−	0.539289i	0.962960	−	0.269645i	$-0.0869062\pi$
$350$		0			0
$351$	−0.854441	−	3.50285i	−0.0456067	−	0.186968i
$352$		0			0
$353$			31.3990i			1.67120i	0.549339	+	0.835599i	$0.314879\pi$
$353$			31.3990i			1.67120i	−0.549339	+	0.835599i	$0.685121\pi$
$354$		0			0
$355$	−4.96879			−0.263716
$356$		0			0
$357$		−	6.38288i		−	0.337818i
$358$		0			0
$359$		−	10.3187i		−	0.544601i	−0.962212	−	0.272300i	$-0.912216\pi$
$359$		−	10.3187i		−	0.544601i	0.962212	−	0.272300i	$-0.0877845\pi$
$360$		0			0
$361$	8.47425			0.446013
$362$		0			0
$363$	3.48098			0.182704
$364$		0			0
$365$	−1.45885			−0.0763597
$366$		0			0
$367$	−11.0715			−0.577925			−0.288963	−	0.957340i	$-0.593310\pi$
$367$	−11.0715			−0.577925			−0.288963	+	0.957340i	$0.593310\pi$
$368$		0			0
$369$		−	7.40847i		−	0.385669i
$370$		0			0
$371$		−	3.40133i		−	0.176588i
$372$		0			0
$373$	−6.24874			−0.323548			−0.161774	−	0.986828i	$-0.551721\pi$
$373$	−6.24874			−0.323548			−0.161774	+	0.986828i	$0.551721\pi$
$374$		0			0
$375$		−	11.6961i		−	0.603985i
$376$		0			0
$377$	−8.88561	−	36.4272i	−0.457632	−	1.87610i
$378$		0			0
$379$		−	9.85817i		−	0.506380i	−0.967417	−	0.253190i	$-0.918520\pi$
$379$		−	9.85817i		−	0.506380i	0.967417	−	0.253190i	$-0.0814798\pi$
$380$		0			0
$381$	11.0938			0.568351
$382$		0			0
$383$		−	18.2901i		−	0.934578i	−0.884105	−	0.467289i	$-0.845231\pi$
$383$		−	18.2901i		−	0.934578i	0.884105	−	0.467289i	$-0.154769\pi$
$384$		0			0
$385$			8.04058i			0.409786i
$386$		0			0
$387$	1.29552			0.0658549
$388$		0			0
$389$	−28.6393			−1.45207			−0.726036	−	0.687657i	$-0.758639\pi$
$389$	−28.6393			−1.45207			−0.726036	+	0.687657i	$0.758639\pi$
$390$		0			0
$391$	58.5892			2.96298
$392$		0			0
$393$	−16.9044			−0.852716
$394$		0			0
$395$		−	24.0607i		−	1.21063i
$396$		0			0
$397$		−	15.1640i		−	0.761058i	−0.924769	−	0.380529i	$-0.875742\pi$
$397$		−	15.1640i		−	0.761058i	0.924769	−	0.380529i	$-0.124258\pi$
$398$		0			0
$399$	−3.24434			−0.162420
$400$		0			0
$401$			6.10421i			0.304830i	0.988317	+	0.152415i	$0.0487050\pi$
$401$			6.10421i			0.304830i	−0.988317	+	0.152415i	$0.951295\pi$
$402$		0			0
$403$	−7.06328	+	1.72293i	−0.351847	+	0.0858253i
$404$		0			0
$405$			2.11295i			0.104993i
$406$		0			0
$407$	−8.20816			−0.406863
$408$		0			0
$409$		−	33.8327i		−	1.67292i	−0.548028	−	0.836460i	$-0.684621\pi$
$409$		−	33.8327i		−	1.67292i	0.548028	−	0.836460i	$-0.315379\pi$
$410$		0			0
$411$		−	0.424521i		−	0.0209401i
$412$		0			0
$413$	10.1371			0.498813
$414$		0			0
$415$	23.2695			1.14226
$416$		0			0
$417$	−18.8424			−0.922714
$418$		0			0
$419$	36.2309			1.77000			0.884998	−	0.465594i	$-0.154159\pi$
$419$	36.2309			1.77000			0.884998	+	0.465594i	$0.154159\pi$
$420$		0			0
$421$		−	7.52270i		−	0.366634i	−0.983054	−	0.183317i	$-0.941317\pi$
$421$		−	7.52270i		−	0.366634i	0.983054	−	0.183317i	$-0.0586835\pi$
$422$		0			0
$423$		−	1.12498i		−	0.0546986i
$424$		0			0
$425$	3.41777			0.165786
$426$		0			0
$427$			11.1627i			0.540200i
$428$		0			0
$429$	−13.3297	+	3.25148i	−0.643563	+	0.156983i
$430$		0			0
$431$			21.3875i			1.03020i	0.857130	+	0.515100i	$0.172245\pi$
$431$			21.3875i			1.03020i	−0.857130	+	0.515100i	$0.827755\pi$
$432$		0			0
$433$	8.49613			0.408298			0.204149	−	0.978940i	$-0.434557\pi$
$433$	8.49613			0.408298			0.204149	+	0.978940i	$0.434557\pi$
$434$		0			0
$435$			21.9732i			1.05353i
$436$		0			0
$437$		−	29.7802i		−	1.42458i
$438$		0			0
$439$	−36.8063			−1.75667			−0.878335	−	0.478045i	$-0.841346\pi$
$439$	−36.8063			−1.75667			−0.878335	+	0.478045i	$0.841346\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−20.6289			−0.980108			−0.490054	−	0.871692i	$-0.663023\pi$
$443$	−20.6289			−0.980108			−0.490054	+	0.871692i	$0.663023\pi$
$444$		0			0
$445$	−34.1019			−1.61658
$446$		0			0
$447$		−	17.5549i		−	0.830316i
$448$		0			0
$449$		−	12.3990i		−	0.585145i	−0.956243	−	0.292573i	$-0.905489\pi$
$449$		−	12.3990i		−	0.585145i	0.956243	−	0.292573i	$-0.0945113\pi$
$450$		0			0
$451$	−28.1921			−1.32751
$452$		0			0
$453$		−	11.9754i		−	0.562651i
$454$		0			0
$455$	−1.80539	−	7.40133i	−0.0846379	−	0.346979i
$456$		0			0
$457$			3.95763i			0.185130i	0.995707	+	0.0925650i	$0.0295066\pi$
$457$			3.95763i			0.185130i	−0.995707	+	0.0925650i	$0.970493\pi$
$458$		0			0
$459$	−6.38288			−0.297927
$460$		0			0
$461$			2.22294i			0.103533i	0.998659	+	0.0517663i	$0.0164851\pi$
$461$			2.22294i			0.103533i	−0.998659	+	0.0517663i	$0.983515\pi$
$462$		0			0
$463$			8.12009i			0.377372i	0.982037	+	0.188686i	$0.0604229\pi$
$463$			8.12009i			0.377372i	−0.982037	+	0.188686i	$0.939577\pi$
$464$		0			0
$465$	4.26063			0.197582
$466$		0			0
$467$	−18.4640			−0.854414			−0.427207	−	0.904154i	$-0.640502\pi$
$467$	−18.4640			−0.854414			−0.427207	+	0.904154i	$0.640502\pi$
$468$		0			0
$469$	15.9576			0.736855
$470$		0			0
$471$	10.1058			0.465651
$472$		0			0
$473$		−	4.92995i		−	0.226679i
$474$		0			0
$475$		−	1.73721i		−	0.0797087i
$476$		0			0
$477$	−3.40133			−0.155736
$478$		0			0
$479$		−	18.1870i		−	0.830984i	−0.909597	−	0.415492i	$-0.863609\pi$
$479$		−	18.1870i		−	0.830984i	0.909597	−	0.415492i	$-0.136391\pi$
$480$		0			0
$481$	7.55558	−	1.84302i	0.344505	−	0.0840343i
$482$		0			0
$483$		−	9.17912i		−	0.417664i
$484$		0			0
$485$	32.7735			1.48817
$486$		0			0
$487$			5.51390i			0.249859i	0.992166	+	0.124929i	$0.0398704\pi$
$487$			5.51390i			0.249859i	−0.992166	+	0.124929i	$0.960130\pi$
$488$		0			0
$489$		−	7.32969i		−	0.331460i
$490$		0			0
$491$	−20.5141			−0.925789			−0.462894	−	0.886413i	$-0.653189\pi$
$491$	−20.5141			−0.925789			−0.462894	+	0.886413i	$0.653189\pi$
$492$		0			0
$493$	−66.3775			−2.98949
$494$		0			0
$495$	8.04058			0.361397
$496$		0			0
$497$	−2.35160			−0.105483
$498$		0			0
$499$			36.3039i			1.62518i	0.582833	+	0.812592i	$0.301944\pi$
$499$			36.3039i			1.62518i	−0.582833	+	0.812592i	$0.698056\pi$
$500$		0			0
$501$		−	7.32767i		−	0.327376i
$502$		0			0
$503$	23.8850			1.06498			0.532489	−	0.846437i	$-0.321257\pi$
$503$	23.8850			1.06498			0.532489	+	0.846437i	$0.321257\pi$
$504$		0			0
$505$			7.78839i			0.346579i
$506$		0			0
$507$	11.5399	−	5.98595i	0.512503	−	0.265845i
$508$		0			0
$509$		−	41.8744i		−	1.85605i	−0.372520	−	0.928024i	$-0.621506\pi$
$509$		−	41.8744i		−	1.85605i	0.372520	−	0.928024i	$-0.378494\pi$
$510$		0			0
$511$	−0.690434			−0.0305430
$512$		0			0
$513$			3.24434i			0.143241i
$514$		0			0
$515$			3.87300i			0.170665i
$516$		0			0
$517$	−4.28100			−0.188278
$518$		0			0
$519$	−16.6634			−0.731442
$520$		0			0
$521$	−28.0931			−1.23078			−0.615391	−	0.788222i	$-0.711002\pi$
$521$	−28.0931			−1.23078			−0.615391	+	0.788222i	$0.711002\pi$
$522$		0			0
$523$	−1.11465			−0.0487400			−0.0243700	−	0.999703i	$-0.507758\pi$
$523$	−1.11465			−0.0487400			−0.0243700	+	0.999703i	$0.507758\pi$
$524$		0			0
$525$		−	0.535458i		−	0.0233693i
$526$		0			0
$527$			12.8707i			0.560656i
$528$		0			0
$529$	61.2562			2.66331
$530$		0			0
$531$		−	10.1371i		−	0.439912i
$532$		0			0
$533$	25.9507	−	6.33010i	1.12405	−	0.274187i
$534$		0			0
$535$			24.0260i			1.03873i
$536$		0			0
$537$	9.55061			0.412139
$538$		0			0
$539$			3.80539i			0.163910i
$540$		0			0
$541$		−	10.2476i		−	0.440580i	−0.975434	−	0.220290i	$-0.929300\pi$
$541$		−	10.2476i		−	0.440580i	0.975434	−	0.220290i	$-0.0707004\pi$
$542$		0			0
$543$	5.20873			0.223528
$544$		0			0
$545$	−2.59449			−0.111136
$546$		0			0
$547$	−13.6898			−0.585333			−0.292666	−	0.956215i	$-0.594543\pi$
$547$	−13.6898			−0.585333			−0.292666	+	0.956215i	$0.594543\pi$
$548$		0			0
$549$	11.1627			0.476411
$550$		0			0
$551$			33.7389i			1.43733i
$552$		0			0
$553$		−	11.3873i		−	0.484236i
$554$		0			0
$555$	−4.55759			−0.193459
$556$		0			0
$557$		−	26.6346i		−	1.12854i	−0.825589	−	0.564272i	$-0.809157\pi$
$557$		−	26.6346i		−	1.12854i	0.825589	−	0.564272i	$-0.190843\pi$
$558$		0			0
$559$	1.10694	+	4.53800i	0.0468188	+	0.191937i
$560$		0			0
$561$			24.2893i			1.02550i
$562$		0			0
$563$	20.5423			0.865755			0.432878	−	0.901453i	$-0.357498\pi$
$563$	20.5423			0.865755			0.432878	+	0.901453i	$0.357498\pi$
$564$		0			0
$565$			42.9031i			1.80495i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	37.9668			1.59165			0.795827	−	0.605525i	$-0.207037\pi$
$569$	37.9668			1.59165			0.795827	+	0.605525i	$0.207037\pi$
$570$		0			0
$571$	−41.7529			−1.74730			−0.873652	−	0.486552i	$-0.838255\pi$
$571$	−41.7529			−1.74730			−0.873652	+	0.486552i	$0.838255\pi$
$572$		0			0
$573$	0.569057			0.0237727
$574$		0			0
$575$	4.91504			0.204971
$576$		0			0
$577$		−	9.52725i		−	0.396625i	−0.980139	−	0.198312i	$-0.936454\pi$
$577$		−	9.52725i		−	0.396625i	0.980139	−	0.198312i	$-0.0635461\pi$
$578$		0			0
$579$		−	10.7475i		−	0.446649i
$580$		0			0
$581$	11.0128			0.456889
$582$		0			0
$583$			12.9434i			0.536059i
$584$		0			0
$585$	−7.40133	+	1.80539i	−0.306007	+	0.0746436i
$586$		0			0
$587$			35.0059i			1.44485i	0.691450	+	0.722424i	$0.256972\pi$
$587$			35.0059i			1.44485i	−0.691450	+	0.722424i	$0.743028\pi$
$588$		0			0
$589$	6.54202			0.269559
$590$		0			0
$591$			22.5889i			0.929182i
$592$		0			0
$593$		−	3.69719i		−	0.151825i	−0.997114	−	0.0759127i	$-0.975813\pi$
$593$		−	3.69719i		−	0.151825i	0.997114	−	0.0759127i	$-0.0241870\pi$
$594$		0			0
$595$	−13.4867			−0.552900
$596$		0			0
$597$	−1.75494			−0.0718249
$598$		0			0
$599$	16.8677			0.689197			0.344598	−	0.938750i	$-0.388015\pi$
$599$	16.8677			0.689197			0.344598	+	0.938750i	$0.388015\pi$
$600$		0			0
$601$	−46.1352			−1.88189			−0.940946	−	0.338556i	$-0.890061\pi$
$601$	−46.1352			−1.88189			−0.940946	+	0.338556i	$0.890061\pi$
$602$		0			0
$603$		−	15.9576i		−	0.649845i
$604$		0			0
$605$		−	7.35513i		−	0.299028i
$606$		0			0
$607$	−36.8217			−1.49455			−0.747274	−	0.664516i	$-0.768638\pi$
$607$	−36.8217			−1.49455			−0.747274	+	0.664516i	$0.768638\pi$
$608$		0			0
$609$			10.3993i			0.421402i
$610$		0			0
$611$	3.94065	−	0.961234i	0.159422	−	0.0388873i
$612$		0			0
$613$			31.1841i			1.25951i	0.776792	+	0.629757i	$0.216845\pi$
$613$			31.1841i			1.25951i	−0.776792	+	0.629757i	$0.783155\pi$
$614$		0			0
$615$	−15.6537			−0.631218
$616$		0			0
$617$			34.0032i			1.36892i	0.729052	+	0.684458i	$0.239961\pi$
$617$			34.0032i			1.36892i	−0.729052	+	0.684458i	$0.760039\pi$
$618$		0			0
$619$		−	44.1860i		−	1.77598i	−0.459859	−	0.887992i	$-0.652100\pi$
$619$		−	44.1860i		−	1.77598i	0.459859	−	0.887992i	$-0.347900\pi$
$620$		0			0
$621$	−9.17912			−0.368345
$622$		0			0
$623$	−16.1395			−0.646615
$624$		0			0
$625$	−22.0360			−0.881440
$626$		0			0
$627$	12.3460			0.493051
$628$		0			0
$629$		−	13.7678i		−	0.548957i
$630$		0			0
$631$		−	36.6659i		−	1.45964i	−0.683637	−	0.729822i	$-0.739603\pi$
$631$		−	36.6659i		−	1.45964i	0.683637	−	0.729822i	$-0.260397\pi$
$632$		0			0
$633$	7.49308			0.297823
$634$		0			0
$635$		−	23.4405i		−	0.930209i
$636$		0			0
$637$	−0.854441	−	3.50285i	−0.0338542	−	0.138788i
$638$		0			0
$639$			2.35160i			0.0930277i
$640$		0			0
$641$	−17.0706			−0.674250			−0.337125	−	0.941460i	$-0.609455\pi$
$641$	−17.0706			−0.674250			−0.337125	+	0.941460i	$0.609455\pi$
$642$		0			0
$643$			0.659946i			0.0260257i	0.999915	+	0.0130129i	$0.00414224\pi$
$643$			0.659946i			0.0260257i	−0.999915	+	0.0130129i	$0.995858\pi$
$644$		0			0
$645$		−	2.73736i		−	0.107784i
$646$		0			0
$647$	−4.34685			−0.170892			−0.0854461	−	0.996343i	$-0.527232\pi$
$647$	−4.34685			−0.170892			−0.0854461	+	0.996343i	$0.527232\pi$
$648$		0			0
$649$	−38.5756			−1.51422
$650$		0			0
$651$	2.01644			0.0790305
$652$		0			0
$653$	17.9016			0.700542			0.350271	−	0.936648i	$-0.386089\pi$
$653$	17.9016			0.700542			0.350271	+	0.936648i	$0.386089\pi$
$654$		0			0
$655$			35.7182i			1.39562i
$656$		0			0
$657$			0.690434i			0.0269364i
$658$		0			0
$659$	1.58349			0.0616840			0.0308420	−	0.999524i	$-0.490181\pi$
$659$	1.58349			0.0616840			0.0308420	+	0.999524i	$0.490181\pi$
$660$		0			0
$661$			20.7305i			0.806322i	0.915129	+	0.403161i	$0.132089\pi$
$661$			20.7305i			0.806322i	−0.915129	+	0.403161i	$0.867911\pi$
$662$		0			0
$663$	−5.45379	−	22.3582i	−0.211808	−	0.868322i
$664$		0			0
$665$			6.85512i			0.265830i
$666$		0			0
$667$	−95.4565			−3.69609
$668$		0			0
$669$		−	13.9741i		−	0.540269i
$670$		0			0
$671$		−	42.4783i		−	1.63986i
$672$		0			0
$673$	−17.2093			−0.663371			−0.331686	−	0.943390i	$-0.607617\pi$
$673$	−17.2093			−0.663371			−0.331686	+	0.943390i	$0.607617\pi$
$674$		0			0
$675$	−0.535458			−0.0206098
$676$		0			0
$677$	20.5304			0.789046			0.394523	−	0.918886i	$-0.370910\pi$
$677$	20.5304			0.789046			0.394523	+	0.918886i	$0.370910\pi$
$678$		0			0
$679$	15.5108			0.595250
$680$		0			0
$681$		−	27.5508i		−	1.05575i
$682$		0			0
$683$			26.3895i			1.00977i	0.863188	+	0.504883i	$0.168464\pi$
$683$			26.3895i			1.00977i	−0.863188	+	0.504883i	$0.831536\pi$
$684$		0			0
$685$	−0.896990			−0.0342722
$686$		0			0
$687$			13.4063i			0.511482i
$688$		0			0
$689$	−2.90623	−	11.9143i	−0.110719	−	0.453899i
$690$		0			0
$691$			16.3487i			0.621934i	0.950421	+	0.310967i	$0.100653\pi$
$691$			16.3487i			0.621934i	−0.950421	+	0.310967i	$0.899347\pi$
$692$		0			0
$693$	3.80539			0.144555
$694$		0			0
$695$			39.8129i			1.51019i
$696$		0			0
$697$		−	47.2873i		−	1.79113i
$698$		0			0
$699$	−17.5432			−0.663543
$700$		0			0
$701$	47.3265			1.78750			0.893749	−	0.448567i	$-0.148066\pi$
$701$	47.3265			1.78750			0.893749	+	0.448567i	$0.148066\pi$
$702$		0			0
$703$	−6.99799			−0.263934
$704$		0			0
$705$	−2.37703			−0.0895242
$706$		0			0
$707$			3.68603i			0.138628i
$708$		0			0
$709$			1.94882i			0.0731896i	0.999330	+	0.0365948i	$0.0116511\pi$
$709$			1.94882i			0.0731896i	−0.999330	+	0.0365948i	$0.988349\pi$
$710$		0			0
$711$	−11.3873			−0.427056
$712$		0			0
$713$			18.5091i			0.693173i
$714$		0			0
$715$	6.87020	+	28.1649i	0.256931	+	1.05331i
$716$		0			0
$717$		−	6.69844i		−	0.250158i
$718$		0			0
$719$	25.4420			0.948825			0.474413	−	0.880303i	$-0.342660\pi$
$719$	25.4420			0.948825			0.474413	+	0.880303i	$0.342660\pi$
$720$		0			0
$721$			1.83299i			0.0682640i
$722$		0			0
$723$			0.763359i			0.0283896i
$724$		0			0
$725$	−5.56840			−0.206805
$726$		0			0
$727$	−6.35478			−0.235686			−0.117843	−	0.993032i	$-0.537598\pi$
$727$	−6.35478			−0.235686			−0.117843	+	0.993032i	$0.537598\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	8.26914			0.305845
$732$		0			0
$733$		−	2.27468i		−	0.0840171i	−0.999117	−	0.0420086i	$-0.986624\pi$
$733$		−	2.27468i		−	0.0840171i	0.999117	−	0.0420086i	$-0.0133757\pi$
$734$		0			0
$735$			2.11295i			0.0779372i
$736$		0			0
$737$	−60.7250			−2.23683
$738$		0			0
$739$			43.7486i			1.60932i	0.593737	+	0.804659i	$0.297652\pi$
$739$			43.7486i			1.60932i	−0.593737	+	0.804659i	$0.702348\pi$
$740$		0			0
$741$	−11.3644	+	2.77210i	−0.417483	+	0.101836i
$742$		0			0
$743$		−	41.9157i		−	1.53774i	−0.639407	−	0.768868i	$-0.720820\pi$
$743$		−	41.9157i		−	1.53774i	0.639407	−	0.768868i	$-0.279180\pi$
$744$		0			0
$745$	−37.0925			−1.35896
$746$		0			0
$747$		−	11.0128i		−	0.402938i
$748$		0			0
$749$			11.3708i			0.415481i
$750$		0			0
$751$	−4.93672			−0.180143			−0.0900717	−	0.995935i	$-0.528710\pi$
$751$	−4.93672			−0.180143			−0.0900717	+	0.995935i	$0.528710\pi$
$752$		0			0
$753$	20.9739			0.764332
$754$		0			0
$755$	−25.3033			−0.920881
$756$		0			0
$757$	−29.4195			−1.06927			−0.534636	−	0.845083i	$-0.679551\pi$
$757$	−29.4195			−1.06927			−0.534636	+	0.845083i	$0.679551\pi$
$758$		0			0
$759$			34.9301i			1.26788i
$760$		0			0
$761$		−	20.3511i		−	0.737727i	−0.929484	−	0.368863i	$-0.879747\pi$
$761$		−	20.3511i		−	0.737727i	0.929484	−	0.368863i	$-0.120253\pi$
$762$		0			0
$763$	−1.22790			−0.0444530
$764$		0			0
$765$			13.4867i			0.487612i
$766$		0			0
$767$	35.5087	−	8.66155i	1.28214	−	0.312750i
$768$		0			0
$769$		−	12.3527i		−	0.445449i	−0.974881	−	0.222725i	$-0.928505\pi$
$769$		−	12.3527i		−	0.445449i	0.974881	−	0.222725i	$-0.0714951\pi$
$770$		0			0
$771$	−18.9076			−0.680940
$772$		0			0
$773$			22.4690i			0.808154i	0.914725	+	0.404077i	$0.132407\pi$
$773$			22.4690i			0.808154i	−0.914725	+	0.404077i	$0.867593\pi$
$774$		0			0
$775$			1.07972i			0.0387847i
$776$		0			0
$777$	−2.15698			−0.0773814
$778$		0			0
$779$	−24.0356			−0.861164
$780$		0			0
$781$	8.94873			0.320211
$782$		0			0
$783$	10.3993			0.371641
$784$		0			0
$785$		−	21.3530i		−	0.762122i
$786$		0			0
$787$			7.14199i			0.254584i	0.991865	+	0.127292i	$0.0406286\pi$
$787$			7.14199i			0.254584i	−0.991865	+	0.127292i	$0.959371\pi$
$788$		0			0
$789$	1.86458			0.0663807
$790$		0			0
$791$			20.3049i			0.721959i
$792$		0			0
$793$	9.53785	+	39.1011i	0.338699	+	1.38852i
$794$		0			0
$795$			7.18682i			0.254890i
$796$		0			0
$797$	−17.8984			−0.633994			−0.316997	−	0.948426i	$-0.602675\pi$
$797$	−17.8984			−0.633994			−0.316997	+	0.948426i	$0.602675\pi$
$798$		0			0
$799$		−	7.18064i		−	0.254033i
$800$		0			0
$801$			16.1395i			0.570260i
$802$		0			0
$803$	2.62737			0.0927178
$804$		0			0
$805$	−19.3950			−0.683583
$806$		0			0
$807$	−21.3963			−0.753184
$808$		0			0
$809$	−48.7026			−1.71229			−0.856146	−	0.516734i	$-0.827148\pi$
$809$	−48.7026			−1.71229			−0.856146	+	0.516734i	$0.827148\pi$
$810$		0			0
$811$		−	27.7791i		−	0.975458i	−0.872995	−	0.487729i	$-0.837825\pi$
$811$		−	27.7791i		−	0.975458i	0.872995	−	0.487729i	$-0.162175\pi$
$812$		0			0
$813$			10.2347i			0.358946i
$814$		0			0
$815$	−15.4872			−0.542494
$816$		0			0
$817$		−	4.20311i		−	0.147048i
$818$		0			0
$819$	−3.50285	+	0.854441i	−0.122399	+	0.0298566i
$820$		0			0
$821$			3.24232i			0.113158i	0.998398	+	0.0565789i	$0.0180192\pi$
$821$			3.24232i			0.113158i	−0.998398	+	0.0565789i	$0.981981\pi$
$822$		0			0
$823$	−7.22155			−0.251728			−0.125864	−	0.992048i	$-0.540170\pi$
$823$	−7.22155			−0.251728			−0.125864	+	0.992048i	$0.540170\pi$
$824$		0			0
$825$			2.03763i			0.0709411i
$826$		0			0
$827$		−	24.1877i		−	0.841089i	−0.907272	−	0.420544i	$-0.861839\pi$
$827$		−	24.1877i		−	0.841089i	0.907272	−	0.420544i	$-0.138161\pi$
$828$		0			0
$829$	4.90680			0.170420			0.0852101	−	0.996363i	$-0.472844\pi$
$829$	4.90680			0.170420			0.0852101	+	0.996363i	$0.472844\pi$
$830$		0			0
$831$	2.47224			0.0857611
$832$		0			0
$833$	−6.38288			−0.221154
$834$		0			0
$835$	−15.4830			−0.535810
$836$		0			0
$837$		−	2.01644i		−	0.0696984i
$838$		0			0
$839$			11.6060i			0.400685i	0.979726	+	0.200342i	$0.0642054\pi$
$839$			11.6060i			0.400685i	−0.979726	+	0.200342i	$0.935795\pi$
$840$		0			0
$841$	79.1458			2.72916
$842$		0			0
$843$			12.5446i			0.432059i
$844$		0			0
$845$	−12.6480	−	24.3831i	−0.435104	−	0.838804i
$846$		0			0
$847$		−	3.48098i		−	0.119608i
$848$		0			0
$849$	17.3480			0.595381
$850$		0			0
$851$		−	19.7992i		−	0.678708i
$852$		0			0
$853$		−	9.98152i		−	0.341761i	−0.985292	−	0.170880i	$-0.945339\pi$
$853$		−	9.98152i		−	0.341761i	0.985292	−	0.170880i	$-0.0546611\pi$
$854$		0			0
$855$	6.85512			0.234440
$856$		0			0
$857$	−9.65172			−0.329696			−0.164848	−	0.986319i	$-0.552713\pi$
$857$	−9.65172			−0.329696			−0.164848	+	0.986319i	$0.552713\pi$
$858$		0			0
$859$	22.6367			0.772356			0.386178	−	0.922424i	$-0.373795\pi$
$859$	22.6367			0.772356			0.386178	+	0.922424i	$0.373795\pi$
$860$		0			0
$861$	−7.40847			−0.252480
$862$		0			0
$863$		−	33.6262i		−	1.14465i	−0.820027	−	0.572325i	$-0.806042\pi$
$863$		−	33.6262i		−	1.14465i	0.820027	−	0.572325i	$-0.193958\pi$
$864$		0			0
$865$			35.2089i			1.19714i
$866$		0			0
$867$	−23.7411			−0.806290
$868$		0			0
$869$			43.3330i			1.46997i
$870$		0			0
$871$	55.8971	−	13.6349i	1.89400	−	0.461999i
$872$		0			0
$873$		−	15.5108i		−	0.524962i
$874$		0			0
$875$	−11.6961			−0.395401
$876$		0			0
$877$			21.2785i			0.718524i	0.933237	+	0.359262i	$0.116972\pi$
$877$			21.2785i			0.718524i	−0.933237	+	0.359262i	$0.883028\pi$
$878$		0			0
$879$		−	7.26559i		−	0.245062i
$880$		0			0
$881$	13.2385			0.446015			0.223008	−	0.974817i	$-0.428413\pi$
$881$	13.2385			0.446015			0.223008	+	0.974817i	$0.428413\pi$
$882$		0			0
$883$	45.2870			1.52403			0.762014	−	0.647561i	$-0.224211\pi$
$883$	45.2870			1.52403			0.762014	+	0.647561i	$0.224211\pi$
$884$		0			0
$885$	−21.4191			−0.719996
$886$		0			0
$887$	−15.0428			−0.505089			−0.252544	−	0.967585i	$-0.581267\pi$
$887$	−15.0428			−0.505089			−0.252544	+	0.967585i	$0.581267\pi$
$888$		0			0
$889$		−	11.0938i		−	0.372073i
$890$		0			0
$891$		−	3.80539i		−	0.127485i
$892$		0			0
$893$	−3.64983			−0.122137
$894$		0			0
$895$		−	20.1799i		−	0.674540i
$896$		0			0
$897$	−7.84302	−	32.1530i	−0.261871	−	1.07356i
$898$		0			0
$899$		−	20.9696i		−	0.699375i
$900$		0			0
$901$	−21.7102			−0.723273
$902$		0			0
$903$		−	1.29552i		−	0.0431122i
$904$		0			0
$905$		−	11.0058i		−	0.365844i
$906$		0			0
$907$	−42.9454			−1.42598			−0.712990	−	0.701174i	$-0.752660\pi$
$907$	−42.9454			−1.42598			−0.712990	+	0.701174i	$0.752660\pi$
$908$		0			0
$909$	3.68603			0.122258
$910$		0			0
$911$	2.62209			0.0868738			0.0434369	−	0.999056i	$-0.486169\pi$
$911$	2.62209			0.0868738			0.0434369	+	0.999056i	$0.486169\pi$
$912$		0			0
$913$	−41.9081			−1.38696
$914$		0			0
$915$		−	23.5861i		−	0.779734i
$916$		0			0
$917$			16.9044i			0.558234i
$918$		0			0
$919$	−32.8112			−1.08234			−0.541172	−	0.840912i	$-0.682019\pi$
$919$	−32.8112			−1.08234			−0.541172	+	0.840912i	$0.682019\pi$
$920$		0			0
$921$		−	3.67840i		−	0.121207i
$922$		0			0
$923$	−8.23728	+	2.00930i	−0.271133	+	0.0661369i
$924$		0			0
$925$		−	1.15498i		−	0.0379754i
$926$		0			0
$927$	1.83299			0.0602032
$928$		0			0
$929$		−	19.2009i		−	0.629961i	−0.949098	−	0.314981i	$-0.898002\pi$
$929$		−	19.2009i		−	0.629961i	0.949098	−	0.314981i	$-0.101998\pi$
$930$		0			0
$931$			3.24434i			0.106329i
$932$		0			0
$933$	−13.3963			−0.438574
$934$		0			0
$935$	51.3220			1.67841
$936$		0			0
$937$	−31.1077			−1.01625			−0.508123	−	0.861285i	$-0.669660\pi$
$937$	−31.1077			−1.01625			−0.508123	+	0.861285i	$0.669660\pi$
$938$		0			0
$939$	−15.1380			−0.494011
$940$		0			0
$941$			43.5317i			1.41909i	0.704658	+	0.709547i	$0.251100\pi$
$941$			43.5317i			1.41909i	−0.704658	+	0.709547i	$0.748900\pi$
$942$		0			0
$943$		−	68.0032i		−	2.21449i
$944$		0			0
$945$	2.11295			0.0687341
$946$		0			0
$947$			43.9331i			1.42763i	0.700332	+	0.713817i	$0.253035\pi$
$947$			43.9331i			1.42763i	−0.700332	+	0.713817i	$0.746965\pi$
$948$		0			0
$949$	−2.41848	+	0.589935i	−0.0785073	+	0.0191501i
$950$		0			0
$951$		−	24.0907i		−	0.781194i
$952$		0			0
$953$	−27.1404			−0.879165			−0.439582	−	0.898202i	$-0.644874\pi$
$953$	−27.1404			−0.879165			−0.439582	+	0.898202i	$0.644874\pi$
$954$		0			0
$955$		−	1.20239i		−	0.0389083i
$956$		0			0
$957$		−	39.5734i		−	1.27923i
$958$		0			0
$959$	−0.424521			−0.0137085
$960$		0			0
$961$	26.9340			0.868838
$962$		0			0
$963$	11.3708			0.366420
$964$		0			0
$965$	−22.7088			−0.731022
$966$		0			0
$967$		−	46.6088i		−	1.49884i	−0.662097	−	0.749418i	$-0.730333\pi$
$967$		−	46.6088i		−	1.49884i	0.662097	−	0.749418i	$-0.269667\pi$
$968$		0			0
$969$			20.7082i			0.665244i
$970$		0			0
$971$	−3.93424			−0.126256			−0.0631279	−	0.998005i	$-0.520108\pi$
$971$	−3.93424			−0.126256			−0.0631279	+	0.998005i	$0.520108\pi$
$972$		0			0
$973$			18.8424i			0.604058i
$974$		0			0
$975$	−0.457518	−	1.87563i	−0.0146523	−	0.0600682i
$976$		0			0
$977$			18.0127i			0.576277i	0.957589	+	0.288138i	$0.0930363\pi$
$977$			18.0127i			0.576277i	−0.957589	+	0.288138i	$0.906964\pi$
$978$		0			0
$979$	61.4170			1.96290
$980$		0			0
$981$			1.22790i			0.0392039i
$982$		0			0
$983$			49.5497i			1.58039i	0.612856	+	0.790195i	$0.290021\pi$
$983$			49.5497i			1.58039i	−0.612856	+	0.790195i	$0.709979\pi$
$984$		0			0
$985$	47.7291			1.52077
$986$		0			0
$987$	−1.12498			−0.0358087
$988$		0			0
$989$	11.8917			0.378135
$990$		0			0
$991$	7.65057			0.243028			0.121514	−	0.992590i	$-0.461225\pi$
$991$	7.65057			0.243028			0.121514	+	0.992590i	$0.461225\pi$
$992$		0			0
$993$		−	14.5936i		−	0.463114i
$994$		0			0
$995$			3.70810i			0.117555i
$996$		0			0
$997$	−30.4793			−0.965290			−0.482645	−	0.875816i	$-0.660324\pi$
$997$	−30.4793			−0.965290			−0.482645	+	0.875816i	$0.660324\pi$
$998$		0			0
$999$			2.15698i			0.0682439i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.r.337.7		10
4.3	odd	2		2184.2.h.f.337.7	yes	10
13.12	even	2	inner	4368.2.h.r.337.4		10
52.51	odd	2		2184.2.h.f.337.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.h.f.337.4	✓	10	52.51	odd	2
2184.2.h.f.337.7	yes	10	4.3	odd	2
4368.2.h.r.337.4		10	13.12	even	2	inner
4368.2.h.r.337.7		10	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.11295i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.11295i q^{5} +1.00000i q^{7} +1.00000 q^{9} -3.80539i q^{11} +(0.854441 + 3.50285i) q^{13} -2.11295i q^{15} +6.38288 q^{17} -3.24434i q^{19} -1.00000i q^{21} +9.17912 q^{23} +0.535458 q^{25} -1.00000 q^{27} -10.3993 q^{29} +2.01644i q^{31} +3.80539i q^{33} -2.11295 q^{35} -2.15698i q^{37} +(-0.854441 - 3.50285i) q^{39} -7.40847i q^{41} +1.29552 q^{43} +2.11295i q^{45} -1.12498i q^{47} -1.00000 q^{49} -6.38288 q^{51} -3.40133 q^{53} +8.04058 q^{55} +3.24434i q^{57} -10.1371i q^{59} +11.1627 q^{61} +1.00000i q^{63} +(-7.40133 + 1.80539i) q^{65} -15.9576i q^{67} -9.17912 q^{69} +2.35160i q^{71} +0.690434i q^{73} -0.535458 q^{75} +3.80539 q^{77} -11.3873 q^{79} +1.00000 q^{81} -11.0128i q^{83} +13.4867i q^{85} +10.3993 q^{87} +16.1395i q^{89} +(-3.50285 + 0.854441i) q^{91} -2.01644i q^{93} +6.85512 q^{95} -15.5108i q^{97} -3.80539i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 10 * q^9	\( 10 q - 10 q^{3} + 10 q^{9} - 4 q^{13} + 10 q^{17} - 8 q^{23} - 2 q^{25} - 10 q^{27} - 44 q^{29} + 4 q^{39} - 20 q^{43} - 10 q^{49} - 10 q^{51} + 10 q^{53} + 2 q^{55} + 18 q^{61} - 30 q^{65} + 8 q^{69} + 2 q^{75} - 2 q^{77} - 2 q^{79} + 10 q^{81} + 44 q^{87} + 6 q^{91} - 44 q^{95}+O(q^{100}) \) 10 * q - 10 * q^3 + 10 * q^9 - 4 * q^13 + 10 * q^17 - 8 * q^23 - 2 * q^25 - 10 * q^27 - 44 * q^29 + 4 * q^39 - 20 * q^43 - 10 * q^49 - 10 * q^51 + 10 * q^53 + 2 * q^55 + 18 * q^61 - 30 * q^65 + 8 * q^69 + 2 * q^75 - 2 * q^77 - 2 * q^79 + 10 * q^81 + 44 * q^87 + 6 * q^91 - 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more