LMFDB - Embedded newform 4600.2.e.w.4049.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (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:	$36.7311849298$
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} + 18x^{8} + 117x^{6} + 333x^{4} + 396x^{2} + 144 $ x^10 + 18x^8 + 117x^6 + 333x^4 + 396x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$-2.61696i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.w.4049.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61696i q^{3} +3.83744i q^{7} -3.84849 q^{9} +O(q^{10})$	$q-2.61696i q^{3} +3.83744i q^{7} -3.84849 q^{9} -0.508005 q^{11} -1.01106i q^{13} +1.44705i q^{17} +0.508005 q^{19} +10.0424 q^{21} +1.00000i q^{23} +2.22047i q^{27} -7.51040 q^{29} -0.439038 q^{31} +1.32943i q^{33} -7.02642i q^{37} -2.64590 q^{39} +5.47041 q^{41} -6.72592i q^{43} -2.64098i q^{47} -7.72592 q^{49} +3.78688 q^{51} -4.77648i q^{53} -1.32943i q^{57} -3.85345 q^{59} -9.05844 q^{61} -14.7683i q^{63} +3.45696i q^{67} +2.61696 q^{69} -2.73649 q^{71} +9.21300i q^{73} -1.94944i q^{77} -10.5504 q^{79} -5.73458 q^{81} +1.40211i q^{83} +19.6544i q^{87} -6.77086 q^{89} +3.87986 q^{91} +1.14895i q^{93} -0.313420i q^{97} +1.95506 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{9}+O(q^{10}) $ 10 * q - 6 * q^9	$ 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) $ 10 * q - 6 * q^9 - 24 * q^29 - 36 * q^31 - 18 * q^39 - 12 * q^41 - 30 * q^49 - 12 * q^51 + 2 * q^59 + 20 * q^61 + 16 * q^71 - 54 * q^81 - 28 * q^89 - 92 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 2.61696i	− 1.51090i	−0.655204	−	0.755452i	$-0.727417\pi$
$3$	− 2.61696i	− 1.51090i	0.655204	−	0.755452i	$-0.272583\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.83744i	1.45041i	0.688531	+	0.725207i	$0.258256\pi$
$7$	3.83744i	1.45041i	−0.688531	+	0.725207i	$0.741744\pi$
$8$	0	0
$9$	−3.84849	−1.28283
$10$	0	0
$11$	−0.508005	−0.153169	−0.0765847	−	0.997063i	$-0.524402\pi$
$11$	−0.508005	−0.153169	−0.0765847	+	0.997063i	$0.524402\pi$
$12$	0	0
$13$	− 1.01106i	− 0.280417i	−0.990122	−	0.140208i	$-0.955223\pi$
$13$	− 1.01106i	− 0.280417i	0.990122	−	0.140208i	$-0.0447772\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.44705i	0.350961i	0.984483	+	0.175481i	$0.0561479\pi$
$17$	1.44705i	0.350961i	−0.984483	+	0.175481i	$0.943852\pi$
$18$	0	0
$19$	0.508005	0.116544	0.0582722	−	0.998301i	$-0.481441\pi$
$19$	0.508005	0.116544	0.0582722	+	0.998301i	$0.481441\pi$
$20$	0	0
$21$	10.0424	2.19144
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.22047i	0.427330i
$28$	0	0
$29$	−7.51040	−1.39465	−0.697323	−	0.716757i	$-0.745626\pi$
$29$	−7.51040	−1.39465	−0.697323	+	0.716757i	$0.745626\pi$
$30$	0	0
$31$	−0.439038	−0.0788536	−0.0394268	−	0.999222i	$-0.512553\pi$
$31$	−0.439038	−0.0788536	−0.0394268	+	0.999222i	$0.512553\pi$
$32$	0	0
$33$	1.32943i	0.231424i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.02642i	− 1.15514i	−0.816343	−	0.577568i	$-0.804002\pi$
$37$	− 7.02642i	− 1.15514i	0.816343	−	0.577568i	$-0.195998\pi$
$38$	0	0
$39$	−2.64590	−0.423683
$40$	0	0
$41$	5.47041	0.854335	0.427167	−	0.904173i	$-0.359512\pi$
$41$	5.47041	0.854335	0.427167	+	0.904173i	$0.359512\pi$
$42$	0	0
$43$	− 6.72592i	− 1.02569i	−0.858480	−	0.512847i	$-0.828591\pi$
$43$	− 6.72592i	− 1.02569i	0.858480	−	0.512847i	$-0.171409\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.64098i	− 0.385227i	−0.981275	−	0.192614i	$-0.938304\pi$
$47$	− 2.64098i	− 0.385227i	0.981275	−	0.192614i	$-0.0616964\pi$
$48$	0	0
$49$	−7.72592	−1.10370
$50$	0	0
$51$	3.78688	0.530269
$52$	0	0
$53$	− 4.77648i	− 0.656100i	−0.944660	−	0.328050i	$-0.893609\pi$
$53$	− 4.77648i	− 0.656100i	0.944660	−	0.328050i	$-0.106391\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.32943i	− 0.176087i
$58$	0	0
$59$	−3.85345	−0.501676	−0.250838	−	0.968029i	$-0.580706\pi$
$59$	−3.85345	−0.501676	−0.250838	+	0.968029i	$0.580706\pi$
$60$	0	0
$61$	−9.05844	−1.15981	−0.579907	−	0.814683i	$-0.696911\pi$
$61$	−9.05844	−1.15981	−0.579907	+	0.814683i	$0.696911\pi$
$62$	0	0
$63$	− 14.7683i	− 1.86064i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.45696i	0.422335i	0.977450	+	0.211167i	$0.0677265\pi$
$67$	3.45696i	0.422335i	−0.977450	+	0.211167i	$0.932273\pi$
$68$	0	0
$69$	2.61696	0.315045
$70$	0	0
$71$	−2.73649	−0.324762	−0.162381	−	0.986728i	$-0.551917\pi$
$71$	−2.73649	−0.324762	−0.162381	+	0.986728i	$0.551917\pi$
$72$	0	0
$73$	9.21300i	1.07830i	0.842210	+	0.539150i	$0.181254\pi$
$73$	9.21300i	1.07830i	−0.842210	+	0.539150i	$0.818746\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.94944i	− 0.222159i
$78$	0	0
$79$	−10.5504	−1.18702	−0.593508	−	0.804828i	$-0.702258\pi$
$79$	−10.5504	−1.18702	−0.593508	+	0.804828i	$0.702258\pi$
$80$	0	0
$81$	−5.73458	−0.637176
$82$	0	0
$83$	1.40211i	0.153901i	0.997035	+	0.0769505i	$0.0245183\pi$
$83$	1.40211i	0.153901i	−0.997035	+	0.0769505i	$0.975482\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	19.6544i	2.10718i
$88$	0	0
$89$	−6.77086	−0.717710	−0.358855	−	0.933393i	$-0.616833\pi$
$89$	−6.77086	−0.717710	−0.358855	+	0.933393i	$0.616833\pi$
$90$	0	0
$91$	3.87986	0.406720
$92$	0	0
$93$	1.14895i	0.119140i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 0.313420i	− 0.0318230i	−0.999873	−	0.0159115i	$-0.994935\pi$
$97$	− 0.313420i	− 0.0318230i	0.999873	−	0.0159115i	$-0.00506500\pi$
$98$	0	0
$99$	1.95506	0.196490
$100$	0	0
$101$	−4.83182	−0.480784	−0.240392	−	0.970676i	$-0.577276\pi$
$101$	−4.83182	−0.480784	−0.240392	+	0.970676i	$0.577276\pi$
$102$	0	0
$103$	− 8.07746i	− 0.795896i	−0.917408	−	0.397948i	$-0.869722\pi$
$103$	− 8.07746i	− 0.795896i	0.917408	−	0.397948i	$-0.130278\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.1587i	− 1.07876i	−0.842064	−	0.539378i	$-0.818659\pi$
$107$	− 11.1587i	− 1.07876i	0.842064	−	0.539378i	$-0.181341\pi$
$108$	0	0
$109$	−17.6015	−1.68592	−0.842958	−	0.537979i	$-0.819188\pi$
$109$	−17.6015	−1.68592	−0.842958	+	0.537979i	$0.819188\pi$
$110$	0	0
$111$	−18.3879	−1.74530
$112$	0	0
$113$	4.06454i	0.382360i	0.981555	+	0.191180i	$0.0612314\pi$
$113$	4.06454i	0.382360i	−0.981555	+	0.191180i	$0.938769\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.89104i	0.359727i
$118$	0	0
$119$	−5.55296	−0.509039
$120$	0	0
$121$	−10.7419	−0.976539
$122$	0	0
$123$	− 14.3159i	− 1.29082i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.75246i	0.421713i	0.977517	+	0.210856i	$0.0676252\pi$
$127$	4.75246i	0.421713i	−0.977517	+	0.210856i	$0.932375\pi$
$128$	0	0
$129$	−17.6015	−1.54972
$130$	0	0
$131$	1.96851	0.171989	0.0859946	−	0.996296i	$-0.472593\pi$
$131$	1.96851	0.171989	0.0859946	+	0.996296i	$0.472593\pi$
$132$	0	0
$133$	1.94944i	0.169038i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.64285i	− 0.482101i	−0.970513	−	0.241051i	$-0.922508\pi$
$137$	− 5.64285i	− 0.482101i	0.970513	−	0.241051i	$-0.0774920\pi$
$138$	0	0
$139$	7.59724	0.644390	0.322195	−	0.946673i	$-0.395579\pi$
$139$	7.59724	0.644390	0.322195	+	0.946673i	$0.395579\pi$
$140$	0	0
$141$	−6.91136	−0.582041
$142$	0	0
$143$	0.513622i	0.0429512i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	20.2184i	1.66759i
$148$	0	0
$149$	−10.7709	−0.882384	−0.441192	−	0.897413i	$-0.645444\pi$
$149$	−10.7709	−0.882384	−0.441192	+	0.897413i	$0.645444\pi$
$150$	0	0
$151$	−16.5333	−1.34546	−0.672729	−	0.739889i	$-0.734878\pi$
$151$	−16.5333	−1.34546	−0.672729	+	0.739889i	$0.734878\pi$
$152$	0	0
$153$	− 5.56896i	− 0.450224i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 11.5436i	− 0.921281i	−0.887587	−	0.460640i	$-0.847620\pi$
$157$	− 11.5436i	− 0.921281i	0.887587	−	0.460640i	$-0.152380\pi$
$158$	0	0
$159$	−12.4999	−0.991304
$160$	0	0
$161$	−3.83744	−0.302432
$162$	0	0
$163$	− 3.32143i	− 0.260155i	−0.991504	−	0.130077i	$-0.958477\pi$
$163$	− 3.32143i	− 0.260155i	0.991504	−	0.130077i	$-0.0415226\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.6742i	0.980755i	0.871510	+	0.490378i	$0.163141\pi$
$167$	12.6742i	0.980755i	−0.871510	+	0.490378i	$0.836859\pi$
$168$	0	0
$169$	11.9778	0.921367
$170$	0	0
$171$	−1.95506	−0.149507
$172$	0	0
$173$	− 17.5733i	− 1.33607i	−0.744130	−	0.668035i	$-0.767136\pi$
$173$	− 17.5733i	− 1.33607i	0.744130	−	0.668035i	$-0.232864\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.0843i	0.757984i
$178$	0	0
$179$	14.8952	1.11332	0.556659	−	0.830741i	$-0.312083\pi$
$179$	14.8952	1.11332	0.556659	+	0.830741i	$0.312083\pi$
$180$	0	0
$181$	−23.2868	−1.73089	−0.865446	−	0.501003i	$-0.832965\pi$
$181$	−23.2868	−1.73089	−0.865446	+	0.501003i	$0.832965\pi$
$182$	0	0
$183$	23.7056i	1.75237i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.735109i	− 0.0537565i
$188$	0	0
$189$	−8.52093	−0.619806
$190$	0	0
$191$	6.92360	0.500974	0.250487	−	0.968120i	$-0.419409\pi$
$191$	6.92360	0.500974	0.250487	+	0.968120i	$0.419409\pi$
$192$	0	0
$193$	7.74318i	0.557366i	0.960383	+	0.278683i	$0.0898979\pi$
$193$	7.74318i	0.557366i	−0.960383	+	0.278683i	$0.910102\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 23.8908i	− 1.70215i	−0.525045	−	0.851074i	$-0.675952\pi$
$197$	− 23.8908i	− 1.70215i	0.525045	−	0.851074i	$-0.324048\pi$
$198$	0	0
$199$	23.9444	1.69737	0.848687	−	0.528896i	$-0.177394\pi$
$199$	23.9444	1.69737	0.848687	+	0.528896i	$0.177394\pi$
$200$	0	0
$201$	9.04673	0.638107
$202$	0	0
$203$	− 28.8207i	− 2.02282i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 3.84849i	− 0.267489i
$208$	0	0
$209$	−0.258070	−0.0178510
$210$	0	0
$211$	14.4709	0.996220	0.498110	−	0.867114i	$-0.334027\pi$
$211$	14.4709	0.996220	0.498110	+	0.867114i	$0.334027\pi$
$212$	0	0
$213$	7.16129i	0.490684i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.68478i	− 0.114370i
$218$	0	0
$219$	24.1101	1.62921
$220$	0	0
$221$	1.46305	0.0984153
$222$	0	0
$223$	4.74755i	0.317919i	0.987285	+	0.158960i	$0.0508140\pi$
$223$	4.74755i	0.317919i	−0.987285	+	0.158960i	$0.949186\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.98399i	0.596288i	0.954521	+	0.298144i	$0.0963676\pi$
$227$	8.98399i	0.596288i	−0.954521	+	0.298144i	$0.903632\pi$
$228$	0	0
$229$	−20.7290	−1.36981	−0.684906	−	0.728631i	$-0.740157\pi$
$229$	−20.7290	−1.36981	−0.684906	+	0.728631i	$0.740157\pi$
$230$	0	0
$231$	−5.10161	−0.335661
$232$	0	0
$233$	− 26.0634i	− 1.70747i	−0.520707	−	0.853735i	$-0.674332\pi$
$233$	− 26.0634i	− 1.70747i	0.520707	−	0.853735i	$-0.325668\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	27.6101i	1.79347i
$238$	0	0
$239$	2.94209	0.190308	0.0951540	−	0.995463i	$-0.469666\pi$
$239$	2.94209	0.190308	0.0951540	+	0.995463i	$0.469666\pi$
$240$	0	0
$241$	−24.9413	−1.60661	−0.803306	−	0.595567i	$-0.796927\pi$
$241$	−24.9413	−1.60661	−0.803306	+	0.595567i	$0.796927\pi$
$242$	0	0
$243$	21.6686i	1.39004i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.513622i	− 0.0326810i
$248$	0	0
$249$	3.66926	0.232530
$250$	0	0
$251$	21.1908	1.33755	0.668774	−	0.743465i	$-0.266819\pi$
$251$	21.1908	1.33755	0.668774	+	0.743465i	$0.266819\pi$
$252$	0	0
$253$	− 0.508005i	− 0.0319380i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.2110i	1.63500i	0.575929	+	0.817500i	$0.304640\pi$
$257$	26.2110i	1.63500i	−0.575929	+	0.817500i	$0.695360\pi$
$258$	0	0
$259$	26.9634	1.67543
$260$	0	0
$261$	28.9037	1.78910
$262$	0	0
$263$	5.65276i	0.348564i	0.984696	+	0.174282i	$0.0557605\pi$
$263$	5.65276i	0.348564i	−0.984696	+	0.174282i	$0.944240\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	17.7191i	1.08439i
$268$	0	0
$269$	−9.65772	−0.588841	−0.294421	−	0.955676i	$-0.595127\pi$
$269$	−9.65772	−0.588841	−0.294421	+	0.955676i	$0.595127\pi$
$270$	0	0
$271$	8.39040	0.509680	0.254840	−	0.966983i	$-0.417977\pi$
$271$	8.39040	0.509680	0.254840	+	0.966983i	$0.417977\pi$
$272$	0	0
$273$	− 10.1535i	− 0.614515i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	30.3317i	1.82246i	0.411900	+	0.911229i	$0.364865\pi$
$277$	30.3317i	1.82246i	−0.411900	+	0.911229i	$0.635135\pi$
$278$	0	0
$279$	1.68964	0.101156
$280$	0	0
$281$	−25.3132	−1.51006	−0.755028	−	0.655692i	$-0.772377\pi$
$281$	−25.3132	−1.51006	−0.755028	+	0.655692i	$0.772377\pi$
$282$	0	0
$283$	12.9629i	0.770566i	0.922798	+	0.385283i	$0.125896\pi$
$283$	12.9629i	0.770566i	−0.922798	+	0.385283i	$0.874104\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.9924i	1.23914i
$288$	0	0
$289$	14.9060	0.876826
$290$	0	0
$291$	−0.820209	−0.0480815
$292$	0	0
$293$	− 25.9469i	− 1.51584i	−0.652350	−	0.757918i	$-0.726217\pi$
$293$	− 25.9469i	− 1.51584i	0.652350	−	0.757918i	$-0.273783\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.12801i	− 0.0654539i
$298$	0	0
$299$	1.01106	0.0584709
$300$	0	0
$301$	25.8103	1.48768
$302$	0	0
$303$	12.6447i	0.726419i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.0640i	0.688531i	0.938872	+	0.344266i	$0.111872\pi$
$307$	12.0640i	0.688531i	−0.938872	+	0.344266i	$0.888128\pi$
$308$	0	0
$309$	−21.1384	−1.20252
$310$	0	0
$311$	−31.8125	−1.80392	−0.901959	−	0.431821i	$-0.857871\pi$
$311$	−31.8125	−1.80392	−0.901959	+	0.431821i	$0.857871\pi$
$312$	0	0
$313$	4.30122i	0.243119i	0.992584	+	0.121560i	$0.0387895\pi$
$313$	4.30122i	0.243119i	−0.992584	+	0.121560i	$0.961210\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.28317i	− 0.352898i	−0.984310	−	0.176449i	$-0.943539\pi$
$317$	− 6.28317i	− 0.352898i	0.984310	−	0.176449i	$-0.0564611\pi$
$318$	0	0
$319$	3.81532	0.213617
$320$	0	0
$321$	−29.2020	−1.62990
$322$	0	0
$323$	0.735109i	0.0409026i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	46.0624i	2.54726i
$328$	0	0
$329$	10.1346	0.558739
$330$	0	0
$331$	−8.00053	−0.439749	−0.219874	−	0.975528i	$-0.570565\pi$
$331$	−8.00053	−0.439749	−0.219874	+	0.975528i	$0.570565\pi$
$332$	0	0
$333$	27.0411i	1.48184i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.1436i	0.552554i	0.961078	+	0.276277i	$0.0891008\pi$
$337$	10.1436i	0.552554i	−0.961078	+	0.276277i	$0.910899\pi$
$338$	0	0
$339$	10.6367	0.577709
$340$	0	0
$341$	0.223034	0.0120780
$342$	0	0
$343$	− 2.78567i	− 0.150412i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.8747i	− 1.01325i	−0.862167	−	0.506625i	$-0.830893\pi$
$347$	− 18.8747i	− 1.01325i	0.862167	−	0.506625i	$-0.169107\pi$
$348$	0	0
$349$	−9.26611	−0.496004	−0.248002	−	0.968760i	$-0.579774\pi$
$349$	−9.26611	−0.496004	−0.248002	+	0.968760i	$0.579774\pi$
$350$	0	0
$351$	2.24502	0.119831
$352$	0	0
$353$	− 8.04442i	− 0.428161i	−0.976816	−	0.214081i	$-0.931324\pi$
$353$	− 8.04442i	− 0.428161i	0.976816	−	0.214081i	$-0.0686755\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	14.5319i	0.769109i
$358$	0	0
$359$	−20.5443	−1.08429	−0.542144	−	0.840285i	$-0.682387\pi$
$359$	−20.5443	−1.08429	−0.542144	+	0.840285i	$0.682387\pi$
$360$	0	0
$361$	−18.7419	−0.986417
$362$	0	0
$363$	28.1112i	1.47546i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 8.64032i	− 0.451021i	−0.974241	−	0.225511i	$-0.927595\pi$
$367$	− 8.64032i	− 0.451021i	0.974241	−	0.225511i	$-0.0724050\pi$
$368$	0	0
$369$	−21.0528	−1.09597
$370$	0	0
$371$	18.3294	0.951617
$372$	0	0
$373$	19.0442i	0.986073i	0.870009	+	0.493037i	$0.164113\pi$
$373$	19.0442i	0.986073i	−0.870009	+	0.493037i	$0.835887\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.59344i	0.391082i
$378$	0	0
$379$	−0.926121	−0.0475717	−0.0237858	−	0.999717i	$-0.507572\pi$
$379$	−0.926121	−0.0475717	−0.0237858	+	0.999717i	$0.507572\pi$
$380$	0	0
$381$	12.4370	0.637167
$382$	0	0
$383$	9.38526i	0.479564i	0.970827	+	0.239782i	$0.0770760\pi$
$383$	9.38526i	0.479564i	−0.970827	+	0.239782i	$0.922924\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	25.8847i	1.31579i
$388$	0	0
$389$	−2.22865	−0.112997	−0.0564985	−	0.998403i	$-0.517994\pi$
$389$	−2.22865	−0.112997	−0.0564985	+	0.998403i	$0.517994\pi$
$390$	0	0
$391$	−1.44705	−0.0731805
$392$	0	0
$393$	− 5.15151i	− 0.259859i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13.0448i	0.654701i	0.944903	+	0.327350i	$0.106156\pi$
$397$	13.0448i	0.654701i	−0.944903	+	0.327350i	$0.893844\pi$
$398$	0	0
$399$	5.10161	0.255400
$400$	0	0
$401$	−2.22221	−0.110972	−0.0554859	−	0.998459i	$-0.517671\pi$
$401$	−2.22221	−0.110972	−0.0554859	+	0.998459i	$0.517671\pi$
$402$	0	0
$403$	0.443893i	0.0221119i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.56946i	0.176931i
$408$	0	0
$409$	19.5232	0.965362	0.482681	−	0.875796i	$-0.339663\pi$
$409$	19.5232	0.965362	0.482681	+	0.875796i	$0.339663\pi$
$410$	0	0
$411$	−14.7671	−0.728409
$412$	0	0
$413$	− 14.7874i	− 0.727638i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 19.8817i	− 0.973611i
$418$	0	0
$419$	24.9180	1.21732	0.608662	−	0.793430i	$-0.291707\pi$
$419$	24.9180	1.21732	0.608662	+	0.793430i	$0.291707\pi$
$420$	0	0
$421$	6.10721	0.297647	0.148824	−	0.988864i	$-0.452451\pi$
$421$	6.10721	0.297647	0.148824	+	0.988864i	$0.452451\pi$
$422$	0	0
$423$	10.1638i	0.494181i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 34.7612i	− 1.68221i
$428$	0	0
$429$	1.34413	0.0648952
$430$	0	0
$431$	4.44095	0.213913	0.106956	−	0.994264i	$-0.465889\pi$
$431$	4.44095	0.213913	0.106956	+	0.994264i	$0.465889\pi$
$432$	0	0
$433$	− 3.25554i	− 0.156451i	−0.996936	−	0.0782257i	$-0.975075\pi$
$433$	− 3.25554i	− 0.156451i	0.996936	−	0.0782257i	$-0.0249255\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.508005i	0.0243012i
$438$	0	0
$439$	26.9611	1.28678	0.643392	−	0.765537i	$-0.277527\pi$
$439$	26.9611	1.28678	0.643392	+	0.765537i	$0.277527\pi$
$440$	0	0
$441$	29.7331	1.41586
$442$	0	0
$443$	16.0072i	0.760526i	0.924878	+	0.380263i	$0.124167\pi$
$443$	16.0072i	0.760526i	−0.924878	+	0.380263i	$0.875833\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	28.1869i	1.33320i
$448$	0	0
$449$	37.4278	1.76633	0.883163	−	0.469066i	$-0.155409\pi$
$449$	37.4278	1.76633	0.883163	+	0.469066i	$0.155409\pi$
$450$	0	0
$451$	−2.77900	−0.130858
$452$	0	0
$453$	43.2670i	2.03286i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 8.75949i	− 0.409752i	−0.978788	−	0.204876i	$-0.934321\pi$
$457$	− 8.75949i	− 0.409752i	0.978788	−	0.204876i	$-0.0656790\pi$
$458$	0	0
$459$	−3.21314	−0.149976
$460$	0	0
$461$	11.4459	0.533089	0.266545	−	0.963823i	$-0.414118\pi$
$461$	11.4459	0.533089	0.266545	+	0.963823i	$0.414118\pi$
$462$	0	0
$463$	− 8.23570i	− 0.382745i	−0.981517	−	0.191373i	$-0.938706\pi$
$463$	− 8.23570i	− 0.382745i	0.981517	−	0.191373i	$-0.0612939\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 28.9436i	− 1.33935i	−0.742654	−	0.669675i	$-0.766433\pi$
$467$	− 28.9436i	− 1.33935i	0.742654	−	0.669675i	$-0.233567\pi$
$468$	0	0
$469$	−13.2659	−0.612561
$470$	0	0
$471$	−30.2092	−1.39197
$472$	0	0
$473$	3.41680i	0.157105i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	18.3823i	0.841666i
$478$	0	0
$479$	−31.1031	−1.42114	−0.710569	−	0.703627i	$-0.751563\pi$
$479$	−31.1031	−1.42114	−0.710569	+	0.703627i	$0.751563\pi$
$480$	0	0
$481$	−7.10410	−0.323919
$482$	0	0
$483$	10.0424i	0.456946i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.90588i	0.0863635i	0.999067	+	0.0431817i	$0.0137494\pi$
$487$	1.90588i	0.0863635i	−0.999067	+	0.0431817i	$0.986251\pi$
$488$	0	0
$489$	−8.69206	−0.393069
$490$	0	0
$491$	−27.2222	−1.22852	−0.614259	−	0.789104i	$-0.710545\pi$
$491$	−27.2222	−1.22852	−0.614259	+	0.789104i	$0.710545\pi$
$492$	0	0
$493$	− 10.8679i	− 0.489467i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 10.5011i	− 0.471039i
$498$	0	0
$499$	9.38913	0.420315	0.210158	−	0.977668i	$-0.432602\pi$
$499$	9.38913	0.420315	0.210158	+	0.977668i	$0.432602\pi$
$500$	0	0
$501$	33.1678	1.48183
$502$	0	0
$503$	− 29.5613i	− 1.31807i	−0.752110	−	0.659037i	$-0.770964\pi$
$503$	− 29.5613i	− 1.31807i	0.752110	−	0.659037i	$-0.229036\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 31.3454i	− 1.39210i
$508$	0	0
$509$	−0.448890	−0.0198967	−0.00994835	−	0.999951i	$-0.503167\pi$
$509$	−0.448890	−0.0198967	−0.00994835	+	0.999951i	$0.503167\pi$
$510$	0	0
$511$	−35.3543	−1.56398
$512$	0	0
$513$	1.12801i	0.0498030i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.34163i	0.0590050i
$518$	0	0
$519$	−45.9886	−2.01867
$520$	0	0
$521$	26.8712	1.17725	0.588623	−	0.808407i	$-0.299670\pi$
$521$	26.8712	1.17725	0.588623	+	0.808407i	$0.299670\pi$
$522$	0	0
$523$	− 2.90473i	− 0.127015i	−0.997981	−	0.0635075i	$-0.979771\pi$
$523$	− 2.90473i	− 0.127015i	0.997981	−	0.0635075i	$-0.0202287\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 0.635311i	− 0.0276746i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	14.8300	0.643566
$532$	0	0
$533$	− 5.53089i	− 0.239570i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 38.9801i	− 1.68212i
$538$	0	0
$539$	3.92481	0.169053
$540$	0	0
$541$	34.0386	1.46343	0.731716	−	0.681610i	$-0.238720\pi$
$541$	34.0386	1.46343	0.731716	+	0.681610i	$0.238720\pi$
$542$	0	0
$543$	60.9406i	2.61521i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 23.8364i	− 1.01917i	−0.860420	−	0.509586i	$-0.829798\pi$
$547$	− 23.8364i	− 1.01917i	0.860420	−	0.509586i	$-0.170202\pi$
$548$	0	0
$549$	34.8613	1.48785
$550$	0	0
$551$	−3.81532	−0.162538
$552$	0	0
$553$	− 40.4866i	− 1.72167i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.4201i	1.71265i	0.516435	+	0.856326i	$0.327259\pi$
$557$	40.4201i	1.71265i	−0.516435	+	0.856326i	$0.672741\pi$
$558$	0	0
$559$	−6.80028	−0.287621
$560$	0	0
$561$	−1.92375	−0.0812209
$562$	0	0
$563$	6.54758i	0.275948i	0.990436	+	0.137974i	$0.0440590\pi$
$563$	6.54758i	0.275948i	−0.990436	+	0.137974i	$0.955941\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 22.0061i	− 0.924169i
$568$	0	0
$569$	−20.4590	−0.857686	−0.428843	−	0.903379i	$-0.641079\pi$
$569$	−20.4590	−0.857686	−0.428843	+	0.903379i	$0.641079\pi$
$570$	0	0
$571$	4.71300	0.197233	0.0986164	−	0.995126i	$-0.468558\pi$
$571$	4.71300	0.197233	0.0986164	+	0.995126i	$0.468558\pi$
$572$	0	0
$573$	− 18.1188i	− 0.756924i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 25.9331i	− 1.07961i	−0.841790	−	0.539805i	$-0.818498\pi$
$577$	− 25.9331i	− 1.07961i	0.841790	−	0.539805i	$-0.181502\pi$
$578$	0	0
$579$	20.2636	0.842127
$580$	0	0
$581$	−5.38049	−0.223220
$582$	0	0
$583$	2.42648i	0.100494i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.4711i	0.432189i	0.976372	+	0.216095i	$0.0693320\pi$
$587$	10.4711i	0.432189i	−0.976372	+	0.216095i	$0.930668\pi$
$588$	0	0
$589$	−0.223034	−0.00918995
$590$	0	0
$591$	−62.5213	−2.57178
$592$	0	0
$593$	− 20.8158i	− 0.854803i	−0.904062	−	0.427401i	$-0.859429\pi$
$593$	− 20.8158i	− 0.854803i	0.904062	−	0.427401i	$-0.140571\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 62.6616i	− 2.56457i
$598$	0	0
$599$	14.7951	0.604511	0.302256	−	0.953227i	$-0.402260\pi$
$599$	14.7951	0.604511	0.302256	+	0.953227i	$0.402260\pi$
$600$	0	0
$601$	−33.8995	−1.38279	−0.691394	−	0.722477i	$-0.743003\pi$
$601$	−33.8995	−1.38279	−0.691394	+	0.722477i	$0.743003\pi$
$602$	0	0
$603$	− 13.3041i	− 0.541784i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.9350i	1.33679i	0.743807	+	0.668394i	$0.233018\pi$
$607$	32.9350i	1.33679i	−0.743807	+	0.668394i	$0.766982\pi$
$608$	0	0
$609$	−75.4227	−3.05628
$610$	0	0
$611$	−2.67018	−0.108024
$612$	0	0
$613$	24.8664i	1.00434i	0.864768	+	0.502172i	$0.167465\pi$
$613$	24.8664i	1.00434i	−0.864768	+	0.502172i	$0.832535\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 20.1280i	− 0.810324i	−0.914245	−	0.405162i	$-0.867215\pi$
$617$	− 20.1280i	− 0.810324i	0.914245	−	0.405162i	$-0.132785\pi$
$618$	0	0
$619$	15.3719	0.617847	0.308924	−	0.951087i	$-0.400031\pi$
$619$	15.3719	0.617847	0.308924	+	0.951087i	$0.400031\pi$
$620$	0	0
$621$	−2.22047	−0.0891046
$622$	0	0
$623$	− 25.9828i	− 1.04098i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.675358i	0.0269712i
$628$	0	0
$629$	10.1676	0.405408
$630$	0	0
$631$	−25.7115	−1.02356	−0.511779	−	0.859117i	$-0.671013\pi$
$631$	−25.7115	−1.02356	−0.511779	+	0.859117i	$0.671013\pi$
$632$	0	0
$633$	− 37.8699i	− 1.50519i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	7.81134i	0.309497i
$638$	0	0
$639$	10.5314	0.416614
$640$	0	0
$641$	2.95684	0.116788	0.0583941	−	0.998294i	$-0.481402\pi$
$641$	2.95684	0.116788	0.0583941	+	0.998294i	$0.481402\pi$
$642$	0	0
$643$	16.7764i	0.661596i	0.943702	+	0.330798i	$0.107318\pi$
$643$	16.7764i	0.661596i	−0.943702	+	0.330798i	$0.892682\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.99247i	− 0.0783322i	−0.999233	−	0.0391661i	$-0.987530\pi$
$647$	− 1.99247i	− 0.0783322i	0.999233	−	0.0391661i	$-0.0124702\pi$
$648$	0	0
$649$	1.95757	0.0768414
$650$	0	0
$651$	−4.40901	−0.172803
$652$	0	0
$653$	− 46.4989i	− 1.81964i	−0.414999	−	0.909822i	$-0.636218\pi$
$653$	− 46.4989i	− 1.81964i	0.414999	−	0.909822i	$-0.363782\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 35.4562i	− 1.38328i
$658$	0	0
$659$	37.1468	1.44703	0.723517	−	0.690307i	$-0.242524\pi$
$659$	37.1468	1.44703	0.723517	+	0.690307i	$0.242524\pi$
$660$	0	0
$661$	−7.01756	−0.272952	−0.136476	−	0.990643i	$-0.543578\pi$
$661$	−7.01756	−0.272952	−0.136476	+	0.990643i	$0.543578\pi$
$662$	0	0
$663$	− 3.82874i	− 0.148696i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 7.51040i	− 0.290804i
$668$	0	0
$669$	12.4242	0.480346
$670$	0	0
$671$	4.60174	0.177648
$672$	0	0
$673$	6.81155i	0.262566i	0.991345	+	0.131283i	$0.0419096\pi$
$673$	6.81155i	0.262566i	−0.991345	+	0.131283i	$0.958090\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.522812i	0.0200933i	0.999950	+	0.0100466i	$0.00319800\pi$
$677$	0.522812i	0.0200933i	−0.999950	+	0.0100466i	$0.996802\pi$
$678$	0	0
$679$	1.20273	0.0461566
$680$	0	0
$681$	23.5108	0.900934
$682$	0	0
$683$	− 2.93412i	− 0.112271i	−0.998423	−	0.0561355i	$-0.982122\pi$
$683$	− 2.93412i	− 0.112271i	0.998423	−	0.0561355i	$-0.0178779\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	54.2471i	2.06965i
$688$	0	0
$689$	−4.82929	−0.183981
$690$	0	0
$691$	−31.8456	−1.21146	−0.605731	−	0.795669i	$-0.707119\pi$
$691$	−31.8456	−1.21146	−0.605731	+	0.795669i	$0.707119\pi$
$692$	0	0
$693$	7.50240i	0.284993i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.91596i	0.299838i
$698$	0	0
$699$	−68.2070	−2.57982
$700$	0	0
$701$	11.2070	0.423283	0.211642	−	0.977347i	$-0.432119\pi$
$701$	11.2070	0.423283	0.211642	+	0.977347i	$0.432119\pi$
$702$	0	0
$703$	− 3.56946i	− 0.134625i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 18.5418i	− 0.697336i
$708$	0	0
$709$	−23.7300	−0.891198	−0.445599	−	0.895233i	$-0.647009\pi$
$709$	−23.7300	−0.891198	−0.445599	+	0.895233i	$0.647009\pi$
$710$	0	0
$711$	40.6033	1.52274
$712$	0	0
$713$	− 0.439038i	− 0.0164421i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 7.69934i	− 0.287537i
$718$	0	0
$719$	27.8687	1.03933	0.519664	−	0.854371i	$-0.326057\pi$
$719$	27.8687	1.03933	0.519664	+	0.854371i	$0.326057\pi$
$720$	0	0
$721$	30.9968	1.15438
$722$	0	0
$723$	65.2705i	2.42744i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 47.5773i	− 1.76455i	−0.470739	−	0.882273i	$-0.656013\pi$
$727$	− 47.5773i	− 1.76455i	0.470739	−	0.882273i	$-0.343987\pi$
$728$	0	0
$729$	39.5022	1.46304
$730$	0	0
$731$	9.73274	0.359978
$732$	0	0
$733$	− 15.4086i	− 0.569129i	−0.958657	−	0.284565i	$-0.908151\pi$
$733$	− 15.4086i	− 0.569129i	0.958657	−	0.284565i	$-0.0918490\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.75615i	− 0.0646888i
$738$	0	0
$739$	−53.0212	−1.95042	−0.975208	−	0.221292i	$-0.928973\pi$
$739$	−53.0212	−1.95042	−0.975208	+	0.221292i	$0.928973\pi$
$740$	0	0
$741$	−1.34413	−0.0493778
$742$	0	0
$743$	− 15.6611i	− 0.574551i	−0.957848	−	0.287275i	$-0.907250\pi$
$743$	− 15.6611i	− 0.574551i	0.957848	−	0.287275i	$-0.0927495\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5.39599i	− 0.197429i
$748$	0	0
$749$	42.8209	1.56464
$750$	0	0
$751$	34.6216	1.26336	0.631679	−	0.775230i	$-0.282366\pi$
$751$	34.6216	1.26336	0.631679	+	0.775230i	$0.282366\pi$
$752$	0	0
$753$	− 55.4554i	− 2.02091i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 38.4389i	− 1.39709i	−0.715568	−	0.698543i	$-0.753832\pi$
$757$	− 38.4389i	− 1.39709i	0.715568	−	0.698543i	$-0.246168\pi$
$758$	0	0
$759$	−1.32943	−0.0482553
$760$	0	0
$761$	42.0932	1.52588	0.762939	−	0.646470i	$-0.223755\pi$
$761$	42.0932	1.52588	0.762939	+	0.646470i	$0.223755\pi$
$762$	0	0
$763$	− 67.5446i	− 2.44528i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.89605i	0.140678i
$768$	0	0
$769$	−34.2859	−1.23638	−0.618191	−	0.786028i	$-0.712134\pi$
$769$	−34.2859	−1.23638	−0.618191	+	0.786028i	$0.712134\pi$
$770$	0	0
$771$	68.5933	2.47033
$772$	0	0
$773$	12.7422i	0.458304i	0.973391	+	0.229152i	$0.0735953\pi$
$773$	12.7422i	0.458304i	−0.973391	+	0.229152i	$0.926405\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 70.5623i	− 2.53141i
$778$	0	0
$779$	2.77900	0.0995679
$780$	0	0
$781$	1.39015	0.0497436
$782$	0	0
$783$	− 16.6766i	− 0.595975i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	28.5534i	1.01782i	0.860820	+	0.508910i	$0.169951\pi$
$787$	28.5534i	1.01782i	−0.860820	+	0.508910i	$0.830049\pi$
$788$	0	0
$789$	14.7931	0.526647
$790$	0	0
$791$	−15.5974	−0.554580
$792$	0	0
$793$	9.15859i	0.325231i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 8.39925i	− 0.297517i	−0.988874	−	0.148758i	$-0.952472\pi$
$797$	− 8.39925i	− 0.297517i	0.988874	−	0.148758i	$-0.0475276\pi$
$798$	0	0
$799$	3.82164	0.135200
$800$	0	0
$801$	26.0576	0.920701
$802$	0	0
$803$	− 4.68026i	− 0.165163i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	25.2739i	0.889683i
$808$	0	0
$809$	28.8816	1.01542	0.507712	−	0.861527i	$-0.330491\pi$
$809$	28.8816	1.01542	0.507712	+	0.861527i	$0.330491\pi$
$810$	0	0
$811$	−46.5292	−1.63386	−0.816931	−	0.576736i	$-0.804326\pi$
$811$	−46.5292	−1.63386	−0.816931	+	0.576736i	$0.804326\pi$
$812$	0	0
$813$	− 21.9574i	− 0.770078i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.41680i	− 0.119539i
$818$	0	0
$819$	−14.9316	−0.521753
$820$	0	0
$821$	−2.99065	−0.104374	−0.0521872	−	0.998637i	$-0.516619\pi$
$821$	−2.99065	−0.104374	−0.0521872	+	0.998637i	$0.516619\pi$
$822$	0	0
$823$	− 5.14357i	− 0.179294i	−0.995974	−	0.0896468i	$-0.971426\pi$
$823$	− 5.14357i	− 0.179294i	0.995974	−	0.0896468i	$-0.0285738\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 53.4288i	− 1.85790i	−0.370205	−	0.928950i	$-0.620713\pi$
$827$	− 53.4288i	− 1.85790i	0.370205	−	0.928950i	$-0.379287\pi$
$828$	0	0
$829$	−11.5290	−0.400420	−0.200210	−	0.979753i	$-0.564162\pi$
$829$	−11.5290	−0.400420	−0.200210	+	0.979753i	$0.564162\pi$
$830$	0	0
$831$	79.3770	2.75356
$832$	0	0
$833$	− 11.1798i	− 0.387357i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 0.974874i	− 0.0336966i
$838$	0	0
$839$	−12.0983	−0.417678	−0.208839	−	0.977950i	$-0.566968\pi$
$839$	−12.0983	−0.417678	−0.208839	+	0.977950i	$0.566968\pi$
$840$	0	0
$841$	27.4061	0.945038
$842$	0	0
$843$	66.2436i	2.28155i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 41.2215i	− 1.41639i
$848$	0	0
$849$	33.9235	1.16425
$850$	0	0
$851$	7.02642	0.240862
$852$	0	0
$853$	− 3.10103i	− 0.106177i	−0.998590	−	0.0530886i	$-0.983093\pi$
$853$	− 3.10103i	− 0.106177i	0.998590	−	0.0530886i	$-0.0169066\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.39457i	0.150116i	0.997179	+	0.0750578i	$0.0239141\pi$
$857$	4.39457i	0.150116i	−0.997179	+	0.0750578i	$0.976086\pi$
$858$	0	0
$859$	29.0453	0.991014	0.495507	−	0.868604i	$-0.334982\pi$
$859$	29.0453	0.991014	0.495507	+	0.868604i	$0.334982\pi$
$860$	0	0
$861$	54.9362	1.87222
$862$	0	0
$863$	40.1020i	1.36509i	0.730845	+	0.682543i	$0.239126\pi$
$863$	40.1020i	1.36509i	−0.730845	+	0.682543i	$0.760874\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 39.0086i	− 1.32480i
$868$	0	0
$869$	5.35968	0.181815
$870$	0	0
$871$	3.49518	0.118430
$872$	0	0
$873$	1.20620i	0.0408235i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37.7503i	1.27474i	0.770559	+	0.637368i	$0.219977\pi$
$877$	37.7503i	1.27474i	−0.770559	+	0.637368i	$0.780023\pi$
$878$	0	0
$879$	−67.9021	−2.29028
$880$	0	0
$881$	−29.1541	−0.982227	−0.491113	−	0.871096i	$-0.663410\pi$
$881$	−29.1541	−0.982227	−0.491113	+	0.871096i	$0.663410\pi$
$882$	0	0
$883$	43.3971i	1.46043i	0.683218	+	0.730214i	$0.260580\pi$
$883$	43.3971i	1.46043i	−0.683218	+	0.730214i	$0.739420\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 25.4091i	− 0.853155i	−0.904451	−	0.426578i	$-0.859719\pi$
$887$	− 25.4091i	− 0.853155i	0.904451	−	0.426578i	$-0.140281\pi$
$888$	0	0
$889$	−18.2373	−0.611658
$890$	0	0
$891$	2.91320	0.0975958
$892$	0	0
$893$	− 1.34163i	− 0.0448961i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.64590i	− 0.0883439i
$898$	0	0
$899$	3.29735	0.109973
$900$	0	0
$901$	6.91181	0.230266
$902$	0	0
$903$	− 67.5446i	− 2.24774i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.00778i	0.0666672i	0.999444	+	0.0333336i	$0.0106124\pi$
$907$	2.00778i	0.0666672i	−0.999444	+	0.0333336i	$0.989388\pi$
$908$	0	0
$909$	18.5952	0.616765
$910$	0	0
$911$	−8.45792	−0.280223	−0.140112	−	0.990136i	$-0.544746\pi$
$911$	−8.45792	−0.280223	−0.140112	+	0.990136i	$0.544746\pi$
$912$	0	0
$913$	− 0.712277i	− 0.0235729i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.55402i	0.249456i
$918$	0	0
$919$	−0.840028	−0.0277100	−0.0138550	−	0.999904i	$-0.504410\pi$
$919$	−0.840028	−0.0277100	−0.0138550	+	0.999904i	$0.504410\pi$
$920$	0	0
$921$	31.5711	1.04030
$922$	0	0
$923$	2.76675i	0.0910686i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	31.0861i	1.02100i
$928$	0	0
$929$	−15.7877	−0.517979	−0.258989	−	0.965880i	$-0.583389\pi$
$929$	−15.7877	−0.517979	−0.258989	+	0.965880i	$0.583389\pi$
$930$	0	0
$931$	−3.92481	−0.128630
$932$	0	0
$933$	83.2520i	2.72555i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31.6568i	1.03418i	0.855930	+	0.517091i	$0.172985\pi$
$937$	31.6568i	1.03418i	−0.855930	+	0.517091i	$0.827015\pi$
$938$	0	0
$939$	11.2561	0.367330
$940$	0	0
$941$	−38.3932	−1.25158	−0.625792	−	0.779990i	$-0.715224\pi$
$941$	−38.3932	−1.25158	−0.625792	+	0.779990i	$0.715224\pi$
$942$	0	0
$943$	5.47041i	0.178141i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 34.3783i	− 1.11714i	−0.829456	−	0.558572i	$-0.811349\pi$
$947$	− 34.3783i	− 1.11714i	0.829456	−	0.558572i	$-0.188651\pi$
$948$	0	0
$949$	9.31486	0.302373
$950$	0	0
$951$	−16.4428	−0.533195
$952$	0	0
$953$	− 48.3782i	− 1.56712i	−0.621314	−	0.783562i	$-0.713401\pi$
$953$	− 48.3782i	− 1.56712i	0.621314	−	0.783562i	$-0.286599\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 9.98456i	− 0.322755i
$958$	0	0
$959$	21.6541	0.699247
$960$	0	0
$961$	−30.8072	−0.993782
$962$	0	0
$963$	42.9443i	1.38386i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 51.3705i	− 1.65196i	−0.563697	−	0.825982i	$-0.690621\pi$
$967$	− 51.3705i	− 1.65196i	0.563697	−	0.825982i	$-0.309379\pi$
$968$	0	0
$969$	1.92375	0.0617999
$970$	0	0
$971$	50.4587	1.61930	0.809649	−	0.586914i	$-0.199657\pi$
$971$	50.4587	1.61930	0.809649	+	0.586914i	$0.199657\pi$
$972$	0	0
$973$	29.1539i	0.934633i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.1406i	1.02827i	0.857710	+	0.514134i	$0.171887\pi$
$977$	32.1406i	1.02827i	−0.857710	+	0.514134i	$0.828113\pi$
$978$	0	0
$979$	3.43964	0.109931
$980$	0	0
$981$	67.7392	2.16275
$982$	0	0
$983$	− 24.1306i	− 0.769647i	−0.922990	−	0.384824i	$-0.874262\pi$
$983$	− 24.1306i	− 0.769647i	0.922990	−	0.384824i	$-0.125738\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 26.5219i	− 0.844201i
$988$	0	0
$989$	6.72592	0.213872
$990$	0	0
$991$	40.1267	1.27467	0.637333	−	0.770588i	$-0.280038\pi$
$991$	40.1267	1.27467	0.637333	+	0.770588i	$0.280038\pi$
$992$	0	0
$993$	20.9371i	0.664418i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 48.9890i	− 1.55150i	−0.631042	−	0.775749i	$-0.717372\pi$
$997$	− 48.9890i	− 1.55150i	0.631042	−	0.775749i	$-0.282628\pi$
$998$	0	0
$999$	15.6020	0.493625

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.w.4049.1		10
5.2	odd	4		4600.2.a.bd.1.1	✓	5
5.3	odd	4		4600.2.a.bf.1.5	yes	5
5.4	even	2	inner	4600.2.e.w.4049.10		10
20.3	even	4		9200.2.a.ct.1.1		5
20.7	even	4		9200.2.a.cv.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bd.1.1	✓	5	5.2	odd	4
4600.2.a.bf.1.5	yes	5	5.3	odd	4
4600.2.e.w.4049.1		10	1.1	even	1	trivial
4600.2.e.w.4049.10		10	5.4	even	2	inner
9200.2.a.ct.1.1		5	20.3	even	4
9200.2.a.cv.1.5		5	20.7	even	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.61696i q^{3} +3.83744i q^{7} -3.84849 q^{9} +O(q^{10})\)	\(q-2.61696i q^{3} +3.83744i q^{7} -3.84849 q^{9} -0.508005 q^{11} -1.01106i q^{13} +1.44705i q^{17} +0.508005 q^{19} +10.0424 q^{21} +1.00000i q^{23} +2.22047i q^{27} -7.51040 q^{29} -0.439038 q^{31} +1.32943i q^{33} -7.02642i q^{37} -2.64590 q^{39} +5.47041 q^{41} -6.72592i q^{43} -2.64098i q^{47} -7.72592 q^{49} +3.78688 q^{51} -4.77648i q^{53} -1.32943i q^{57} -3.85345 q^{59} -9.05844 q^{61} -14.7683i q^{63} +3.45696i q^{67} +2.61696 q^{69} -2.73649 q^{71} +9.21300i q^{73} -1.94944i q^{77} -10.5504 q^{79} -5.73458 q^{81} +1.40211i q^{83} +19.6544i q^{87} -6.77086 q^{89} +3.87986 q^{91} +1.14895i q^{93} -0.313420i q^{97} +1.95506 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{9}+O(q^{10}) \) 10 * q - 6 * q^9	\( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99}+O(q^{100}) \) 10 * q - 6 * q^9 - 24 * q^29 - 36 * q^31 - 18 * q^39 - 12 * q^41 - 30 * q^49 - 12 * q^51 + 2 * q^59 + 20 * q^61 + 16 * q^71 - 54 * q^81 - 28 * q^89 - 92 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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