LMFDB - Embedded newform 576.7.b.e.415.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,7,Mod(415,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("576.415");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	576.b (of order $2$, degree $1$, 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:	$132.511152165$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1301023109376.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 45x^{6} + 1541x^{4} - 21780x^{2} + 234256 $ x^8 - 45x^6 + 1541x^4 - 21780*x^2 + 234256
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 64)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			415.6
Root			$4.51805 - 2.60850i$ of defining polynomial
Character	$\chi$	$=$	576.415
Dual form			576.7.b.e.415.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+51.5041i q^{5} +316.831i q^{7} +O(q^{10})$	$q+51.5041i q^{5} +316.831i q^{7} -455.953 q^{11} -2324.48i q^{13} -5962.31 q^{17} +11298.5 q^{19} -7529.20i q^{23} +12972.3 q^{25} +48632.1i q^{29} +21236.6i q^{31} -16318.1 q^{35} +12803.2i q^{37} -19217.2 q^{41} -8941.24 q^{43} +162741. i q^{47} +17267.1 q^{49} -167076. i q^{53} -23483.5i q^{55} +136253. q^{59} +254875. i q^{61} +119720. q^{65} -291121. q^{67} -372162. i q^{71} +28637.2 q^{73} -144460. i q^{77} -747327. i q^{79} -926047. q^{83} -307084. i q^{85} -821138. q^{89} +736467. q^{91} +581918. i q^{95} +675688. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 11472 q^{17} + 89288 q^{25} + 121584 q^{41} - 847224 q^{49} + 363648 q^{65} - 517168 q^{73} - 432336 q^{89} + 1195984 q^{97}+O(q^{100}) $ 8 * q - 11472 * q^17 + 89288 * q^25 + 121584 * q^41 - 847224 * q^49 + 363648 * q^65 - 517168 * q^73 - 432336 * q^89 + 1195984 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	51.5041i	0.412033i	0.978549	+	0.206017i	$0.0660501\pi$
$5$	51.5041i	0.412033i	−0.978549	+	0.206017i	$0.933950\pi$
$6$	0	0
$7$	316.831i	0.923706i	0.886957	+	0.461853i	$0.152815\pi$
$7$	316.831i	0.923706i	−0.886957	+	0.461853i	$0.847185\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−455.953	−0.342564	−0.171282	−	0.985222i	$-0.554791\pi$
$11$	−455.953	−0.342564	−0.171282	+	0.985222i	$0.554791\pi$
$12$	0	0
$13$	− 2324.48i	− 1.05802i	−0.848614	−	0.529012i	$-0.822563\pi$
$13$	− 2324.48i	− 1.05802i	0.848614	−	0.529012i	$-0.177437\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5962.31	−1.21358	−0.606789	−	0.794863i	$-0.707543\pi$
$17$	−5962.31	−1.21358	−0.606789	+	0.794863i	$0.707543\pi$
$18$	0	0
$19$	11298.5	1.64725	0.823624	−	0.567137i	$-0.191949\pi$
$19$	11298.5	1.64725	0.823624	+	0.567137i	$0.191949\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 7529.20i	− 0.618821i	−0.950929	−	0.309411i	$-0.899868\pi$
$23$	− 7529.20i	− 0.618821i	0.950929	−	0.309411i	$-0.100132\pi$
$24$	0	0
$25$	12972.3	0.830229
$26$	0	0
$27$	0	0
$28$	0	0
$29$	48632.1i	1.99402i	0.0772777	+	0.997010i	$0.475377\pi$
$29$	48632.1i	1.99402i	−0.0772777	+	0.997010i	$0.524623\pi$
$30$	0	0
$31$	21236.6i	0.712852i	0.934324	+	0.356426i	$0.116005\pi$
$31$	21236.6i	0.712852i	−0.934324	+	0.356426i	$0.883995\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−16318.1	−0.380597
$36$	0	0
$37$	12803.2i	0.252762i	0.991982	+	0.126381i	$0.0403362\pi$
$37$	12803.2i	0.252762i	−0.991982	+	0.126381i	$0.959664\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−19217.2	−0.278829	−0.139414	−	0.990234i	$-0.544522\pi$
$41$	−19217.2	−0.278829	−0.139414	+	0.990234i	$0.544522\pi$
$42$	0	0
$43$	−8941.24	−0.112459	−0.0562293	−	0.998418i	$-0.517908\pi$
$43$	−8941.24	−0.112459	−0.0562293	+	0.998418i	$0.517908\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	162741.i	1.56749i	0.621085	+	0.783743i	$0.286692\pi$
$47$	162741.i	1.56749i	−0.621085	+	0.783743i	$0.713308\pi$
$48$	0	0
$49$	17267.1	0.146768
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 167076.i	− 1.12224i	−0.827734	−	0.561120i	$-0.810371\pi$
$53$	− 167076.i	− 1.12224i	0.827734	−	0.561120i	$-0.189629\pi$
$54$	0	0
$55$	− 23483.5i	− 0.141148i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	136253.	0.663421	0.331711	−	0.943381i	$-0.392374\pi$
$59$	136253.	0.663421	0.331711	+	0.943381i	$0.392374\pi$
$60$	0	0
$61$	254875.i	1.12289i	0.827514	+	0.561445i	$0.189754\pi$
$61$	254875.i	1.12289i	−0.827514	+	0.561445i	$0.810246\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	119720.	0.435941
$66$	0	0
$67$	−291121.	−0.967941	−0.483970	−	0.875084i	$-0.660806\pi$
$67$	−291121.	−0.967941	−0.483970	+	0.875084i	$0.660806\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 372162.i	− 1.03982i	−0.854222	−	0.519908i	$-0.825966\pi$
$71$	− 372162.i	− 1.03982i	0.854222	−	0.519908i	$-0.174034\pi$
$72$	0	0
$73$	28637.2	0.0736143	0.0368071	−	0.999322i	$-0.488281\pi$
$73$	28637.2	0.0736143	0.0368071	+	0.999322i	$0.488281\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 144460.i	− 0.316429i
$78$	0	0
$79$	− 747327.i	− 1.51576i	−0.652396	−	0.757878i	$-0.726236\pi$
$79$	− 747327.i	− 1.51576i	0.652396	−	0.757878i	$-0.273764\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−926047.	−1.61957	−0.809783	−	0.586730i	$-0.800415\pi$
$83$	−926047.	−1.61957	−0.809783	+	0.586730i	$0.800415\pi$
$84$	0	0
$85$	− 307084.i	− 0.500034i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−821138.	−1.16479	−0.582393	−	0.812907i	$-0.697883\pi$
$89$	−821138.	−1.16479	−0.582393	+	0.812907i	$0.697883\pi$
$90$	0	0
$91$	736467.	0.977303
$92$	0	0
$93$	0	0
$94$	0	0
$95$	581918.i	0.678720i
$96$	0	0
$97$	675688.	0.740339	0.370170	−	0.928964i	$-0.379300\pi$
$97$	675688.	0.740339	0.370170	+	0.928964i	$0.379300\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.26530e6i	1.22809i	0.789272	+	0.614044i	$0.210458\pi$
$101$	1.26530e6i	1.22809i	−0.789272	+	0.614044i	$0.789542\pi$
$102$	0	0
$103$	1.56235e6i	1.42977i	0.699242	+	0.714885i	$0.253521\pi$
$103$	1.56235e6i	1.42977i	−0.699242	+	0.714885i	$0.746479\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−876393.	−0.715398	−0.357699	−	0.933837i	$-0.616439\pi$
$107$	−876393.	−0.715398	−0.357699	+	0.933837i	$0.616439\pi$
$108$	0	0
$109$	− 646317.i	− 0.499075i	−0.968365	−	0.249538i	$-0.919721\pi$
$109$	− 646317.i	− 0.499075i	0.968365	−	0.249538i	$-0.0802787\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−196814.	−0.136402	−0.0682010	−	0.997672i	$-0.521726\pi$
$113$	−196814.	−0.136402	−0.0682010	+	0.997672i	$0.521726\pi$
$114$	0	0
$115$	387785.	0.254975
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1.88905e6i	− 1.12099i
$120$	0	0
$121$	−1.56367e6	−0.882650
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.47288e6i	0.754115i
$126$	0	0
$127$	− 3.31510e6i	− 1.61840i	−0.587536	−	0.809198i	$-0.699902\pi$
$127$	− 3.31510e6i	− 1.61840i	0.587536	−	0.809198i	$-0.300098\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.46157e6	0.650137	0.325069	−	0.945690i	$-0.394613\pi$
$131$	1.46157e6	0.650137	0.325069	+	0.945690i	$0.394613\pi$
$132$	0	0
$133$	3.57971e6i	1.52157i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.31324e6	−0.899621	−0.449811	−	0.893124i	$-0.648509\pi$
$137$	−2.31324e6	−0.899621	−0.449811	+	0.893124i	$0.648509\pi$
$138$	0	0
$139$	−3.08798e6	−1.14982	−0.574910	−	0.818216i	$-0.694963\pi$
$139$	−3.08798e6	−1.14982	−0.574910	+	0.818216i	$0.694963\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.05985e6i	0.362441i
$144$	0	0
$145$	−2.50476e6	−0.821602
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 506312.i	− 0.153059i	−0.997067	−	0.0765297i	$-0.975616\pi$
$149$	− 506312.i	− 0.153059i	0.997067	−	0.0765297i	$-0.0243840\pi$
$150$	0	0
$151$	540989.i	0.157129i	0.996909	+	0.0785647i	$0.0250337\pi$
$151$	540989.i	0.157129i	−0.996909	+	0.0785647i	$0.974966\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.09377e6	−0.293719
$156$	0	0
$157$	2.59485e6i	0.670522i	0.942125	+	0.335261i	$0.108824\pi$
$157$	2.59485e6i	0.670522i	−0.942125	+	0.335261i	$0.891176\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.38548e6	0.571609
$162$	0	0
$163$	−5.27225e6	−1.21740	−0.608700	−	0.793400i	$-0.708309\pi$
$163$	−5.27225e6	−1.21740	−0.608700	+	0.793400i	$0.708309\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 2.04949e6i	− 0.440043i	−0.975495	−	0.220022i	$-0.929387\pi$
$167$	− 2.04949e6i	− 0.440043i	0.975495	−	0.220022i	$-0.0706128\pi$
$168$	0	0
$169$	−576396.	−0.119416
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.44436e6i	1.24463i	0.782766	+	0.622317i	$0.213808\pi$
$173$	6.44436e6i	1.24463i	−0.782766	+	0.622317i	$0.786192\pi$
$174$	0	0
$175$	4.11004e6i	0.766887i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.00130e7	−1.74584	−0.872918	−	0.487866i	$-0.837775\pi$
$179$	−1.00130e7	−1.74584	−0.872918	+	0.487866i	$0.837775\pi$
$180$	0	0
$181$	− 3.09474e6i	− 0.521901i	−0.965352	−	0.260951i	$-0.915964\pi$
$181$	− 3.09474e6i	− 0.521901i	0.965352	−	0.260951i	$-0.0840360\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−659415.	−0.104146
$186$	0	0
$187$	2.71853e6	0.415729
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.26300e7i	− 1.81260i	−0.422634	−	0.906301i	$-0.638894\pi$
$191$	− 1.26300e7i	− 1.81260i	0.422634	−	0.906301i	$-0.361106\pi$
$192$	0	0
$193$	6.50418e6	0.904733	0.452366	−	0.891832i	$-0.350580\pi$
$193$	6.50418e6	0.904733	0.452366	+	0.891832i	$0.350580\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.04312e7i	1.36438i	0.731173	+	0.682192i	$0.238973\pi$
$197$	1.04312e7i	1.36438i	−0.731173	+	0.682192i	$0.761027\pi$
$198$	0	0
$199$	− 6.15168e6i	− 0.780611i	−0.920685	−	0.390305i	$-0.872369\pi$
$199$	− 6.15168e6i	− 0.780611i	0.920685	−	0.390305i	$-0.127631\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.54082e7	−1.84189
$204$	0	0
$205$	− 989763.i	− 0.114887i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.15157e6	−0.564288
$210$	0	0
$211$	−5.17218e6	−0.550587	−0.275294	−	0.961360i	$-0.588775\pi$
$211$	−5.17218e6	−0.550587	−0.275294	+	0.961360i	$0.588775\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 460511.i	− 0.0463367i
$216$	0	0
$217$	−6.72841e6	−0.658465
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.38593e7i	1.28400i
$222$	0	0
$223$	1.16411e7i	1.04973i	0.851185	+	0.524866i	$0.175885\pi$
$223$	1.16411e7i	1.04973i	−0.851185	+	0.524866i	$0.824115\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.01124e6	−0.684892	−0.342446	−	0.939538i	$-0.611255\pi$
$227$	−8.01124e6	−0.684892	−0.342446	+	0.939538i	$0.611255\pi$
$228$	0	0
$229$	1.11134e7i	0.925424i	0.886509	+	0.462712i	$0.153124\pi$
$229$	1.11134e7i	0.925424i	−0.886509	+	0.462712i	$0.846876\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.52724e7	−1.20737	−0.603683	−	0.797224i	$-0.706301\pi$
$233$	−1.52724e7	−1.20737	−0.603683	+	0.797224i	$0.706301\pi$
$234$	0	0
$235$	−8.38184e6	−0.645856
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 1.58629e7i	− 1.16195i	−0.813920	−	0.580976i	$-0.802671\pi$
$239$	− 1.58629e7i	− 1.16195i	0.813920	−	0.580976i	$-0.197329\pi$
$240$	0	0
$241$	−2.30600e7	−1.64743	−0.823716	−	0.567003i	$-0.808103\pi$
$241$	−2.30600e7	−1.64743	−0.823716	+	0.567003i	$0.808103\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	889325.i	0.0604731i
$246$	0	0
$247$	− 2.62631e7i	− 1.74283i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.60894e6	0.607651	0.303826	−	0.952728i	$-0.401736\pi$
$251$	9.60894e6	0.607651	0.303826	+	0.952728i	$0.401736\pi$
$252$	0	0
$253$	3.43296e6i	0.211986i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.97448e7	−1.16320	−0.581598	−	0.813477i	$-0.697572\pi$
$257$	−1.97448e7	−1.16320	−0.581598	+	0.813477i	$0.697572\pi$
$258$	0	0
$259$	−4.05644e6	−0.233478
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.10803e7i	1.15880i	0.815043	+	0.579401i	$0.196713\pi$
$263$	2.10803e7i	1.15880i	−0.815043	+	0.579401i	$0.803287\pi$
$264$	0	0
$265$	8.60509e6	0.462400
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.65879e6i	0.444836i	0.974951	+	0.222418i	$0.0713950\pi$
$269$	8.65879e6i	0.444836i	−0.974951	+	0.222418i	$0.928605\pi$
$270$	0	0
$271$	− 1.35042e7i	− 0.678518i	−0.940693	−	0.339259i	$-0.889824\pi$
$271$	− 1.35042e7i	− 0.678518i	0.940693	−	0.339259i	$-0.110176\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.91477e6	−0.284407
$276$	0	0
$277$	3.94224e7i	1.85483i	0.374036	+	0.927414i	$0.377973\pi$
$277$	3.94224e7i	1.85483i	−0.374036	+	0.927414i	$0.622027\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.52689e7	1.13885	0.569426	−	0.822043i	$-0.307165\pi$
$281$	2.52689e7	1.13885	0.569426	+	0.822043i	$0.307165\pi$
$282$	0	0
$283$	−889757.	−0.0392566	−0.0196283	−	0.999807i	$-0.506248\pi$
$283$	−889757.	−0.0392566	−0.0196283	+	0.999807i	$0.506248\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 6.08859e6i	− 0.257556i
$288$	0	0
$289$	1.14116e7	0.472773
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.86906e7i	0.743054i	0.928422	+	0.371527i	$0.121166\pi$
$293$	1.86906e7i	0.743054i	−0.928422	+	0.371527i	$0.878834\pi$
$294$	0	0
$295$	7.01758e6i	0.273352i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.75015e7	−0.654728
$300$	0	0
$301$	− 2.83286e6i	− 0.103879i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.31271e7	−0.462668
$306$	0	0
$307$	−2.12497e7	−0.734410	−0.367205	−	0.930140i	$-0.619685\pi$
$307$	−2.12497e7	−0.734410	−0.367205	+	0.930140i	$0.619685\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 2.14152e7i	− 0.711934i	−0.934498	−	0.355967i	$-0.884151\pi$
$311$	− 2.14152e7i	− 0.711934i	0.934498	−	0.355967i	$-0.115849\pi$
$312$	0	0
$313$	−1.54992e7	−0.505449	−0.252724	−	0.967538i	$-0.581327\pi$
$313$	−1.54992e7	−0.505449	−0.252724	+	0.967538i	$0.581327\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.20284e7i	0.691521i	0.938323	+	0.345761i	$0.112379\pi$
$317$	2.20284e7i	0.691521i	−0.938323	+	0.345761i	$0.887621\pi$
$318$	0	0
$319$	− 2.21740e7i	− 0.683080i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.73650e7	−1.99906
$324$	0	0
$325$	− 3.01539e7i	− 0.878402i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.15614e7	−1.44790
$330$	0	0
$331$	2.25702e7	0.622375	0.311187	−	0.950349i	$-0.399273\pi$
$331$	2.25702e7	0.622375	0.311187	+	0.950349i	$0.399273\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 1.49939e7i	− 0.398824i
$336$	0	0
$337$	−1.10276e7	−0.288132	−0.144066	−	0.989568i	$-0.546018\pi$
$337$	−1.10276e7	−0.288132	−0.144066	+	0.989568i	$0.546018\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 9.68288e6i	− 0.244198i
$342$	0	0
$343$	4.27456e7i	1.05928i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.85304e7	−0.922178	−0.461089	−	0.887354i	$-0.652541\pi$
$347$	−3.85304e7	−0.922178	−0.461089	+	0.887354i	$0.652541\pi$
$348$	0	0
$349$	− 1.67472e6i	− 0.0393972i	−0.999806	−	0.0196986i	$-0.993729\pi$
$349$	− 1.67472e6i	− 0.0393972i	0.999806	−	0.0196986i	$-0.00627067\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.39273e7	−0.998643	−0.499321	−	0.866417i	$-0.666417\pi$
$353$	−4.39273e7	−0.998643	−0.499321	+	0.866417i	$0.666417\pi$
$354$	0	0
$355$	1.91679e7	0.428439
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 7.13472e6i	− 0.154203i	−0.997023	−	0.0771017i	$-0.975433\pi$
$359$	− 7.13472e6i	− 0.154203i	0.997023	−	0.0771017i	$-0.0245666\pi$
$360$	0	0
$361$	8.06096e7	1.71342
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.47493e6i	0.0303315i
$366$	0	0
$367$	− 2.37831e7i	− 0.481139i	−0.970632	−	0.240570i	$-0.922666\pi$
$367$	− 2.37831e7i	− 0.481139i	0.970632	−	0.240570i	$-0.0773343\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.29348e7	1.03662
$372$	0	0
$373$	− 2.75669e7i	− 0.531205i	−0.964083	−	0.265602i	$-0.914429\pi$
$373$	− 2.75669e7i	− 0.531205i	0.964083	−	0.265602i	$-0.0855708\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.13044e8	2.10972
$378$	0	0
$379$	−5.70623e7	−1.04817	−0.524085	−	0.851666i	$-0.675593\pi$
$379$	−5.70623e7	−1.04817	−0.524085	+	0.851666i	$0.675593\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.45219e7i	0.258480i	0.991613	+	0.129240i	$0.0412538\pi$
$383$	1.45219e7i	0.258480i	−0.991613	+	0.129240i	$0.958746\pi$
$384$	0	0
$385$	7.44029e6	0.130379
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 3.83034e7i	− 0.650711i	−0.945592	−	0.325355i	$-0.894516\pi$
$389$	− 3.83034e7i	− 0.650711i	0.945592	−	0.325355i	$-0.105484\pi$
$390$	0	0
$391$	4.48914e7i	0.750988i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.84904e7	0.624542
$396$	0	0
$397$	2.93337e7i	0.468808i	0.972139	+	0.234404i	$0.0753139\pi$
$397$	2.93337e7i	0.468808i	−0.972139	+	0.234404i	$0.924686\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.33165e7	1.29211	0.646053	−	0.763293i	$-0.276419\pi$
$401$	8.33165e7	1.29211	0.646053	+	0.763293i	$0.276419\pi$
$402$	0	0
$403$	4.93640e7	0.754215
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5.83764e6i	− 0.0865872i
$408$	0	0
$409$	−1.28360e8	−1.87612	−0.938062	−	0.346469i	$-0.887381\pi$
$409$	−1.28360e8	−1.87612	−0.938062	+	0.346469i	$0.887381\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.31691e7i	0.612806i
$414$	0	0
$415$	− 4.76952e7i	− 0.667315i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.31253e8	1.78430	0.892148	−	0.451744i	$-0.149198\pi$
$419$	1.31253e8	1.78430	0.892148	+	0.451744i	$0.149198\pi$
$420$	0	0
$421$	1.08565e8i	1.45493i	0.686145	+	0.727465i	$0.259302\pi$
$421$	1.08565e8i	1.45493i	−0.686145	+	0.727465i	$0.740698\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.73450e7	−1.00755
$426$	0	0
$427$	−8.07522e7	−1.03722
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.30324e8i	1.62777i	0.581024	+	0.813886i	$0.302652\pi$
$431$	1.30324e8i	1.62777i	−0.581024	+	0.813886i	$0.697348\pi$
$432$	0	0
$433$	−6.54168e6	−0.0805797	−0.0402899	−	0.999188i	$-0.512828\pi$
$433$	−6.54168e6	−0.0805797	−0.0402899	+	0.999188i	$0.512828\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.50684e7i	− 1.01935i
$438$	0	0
$439$	2.19410e7i	0.259336i	0.991557	+	0.129668i	$0.0413911\pi$
$439$	2.19410e7i	0.259336i	−0.991557	+	0.129668i	$0.958609\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.92753e7	−0.221713	−0.110856	−	0.993836i	$-0.535359\pi$
$443$	−1.92753e7	−0.221713	−0.110856	+	0.993836i	$0.535359\pi$
$444$	0	0
$445$	− 4.22920e7i	− 0.479930i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.47464e7	0.273384	0.136692	−	0.990614i	$-0.456353\pi$
$449$	2.47464e7	0.273384	0.136692	+	0.990614i	$0.456353\pi$
$450$	0	0
$451$	8.76213e6	0.0955168
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.79311e7i	0.402681i
$456$	0	0
$457$	5.77220e7	0.604773	0.302387	−	0.953185i	$-0.402217\pi$
$457$	5.77220e7	0.604773	0.302387	+	0.953185i	$0.402217\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 7.06314e7i	− 0.720934i	−0.932772	−	0.360467i	$-0.882617\pi$
$461$	− 7.06314e7i	− 0.720934i	0.932772	−	0.360467i	$-0.117383\pi$
$462$	0	0
$463$	5.26034e7i	0.529994i	0.964249	+	0.264997i	$0.0853710\pi$
$463$	5.26034e7i	0.529994i	−0.964249	+	0.264997i	$0.914629\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.10121e7	0.697239	0.348620	−	0.937264i	$-0.386650\pi$
$467$	7.10121e7	0.697239	0.348620	+	0.937264i	$0.386650\pi$
$468$	0	0
$469$	− 9.22361e7i	− 0.894093i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.07679e6	0.0385243
$474$	0	0
$475$	1.46567e8	1.36759
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 1.40741e8i	− 1.28060i	−0.768125	−	0.640299i	$-0.778810\pi$
$479$	− 1.40741e8i	− 1.28060i	0.768125	−	0.640299i	$-0.221190\pi$
$480$	0	0
$481$	2.97607e7	0.267428
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.48007e7i	0.305044i
$486$	0	0
$487$	6.35775e7i	0.550448i	0.961380	+	0.275224i	$0.0887521\pi$
$487$	6.35775e7i	0.550448i	−0.961380	+	0.275224i	$0.911248\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.59327e8	1.34600	0.672999	−	0.739644i	$-0.265006\pi$
$491$	1.59327e8	1.34600	0.672999	+	0.739644i	$0.265006\pi$
$492$	0	0
$493$	− 2.89960e8i	− 2.41990i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.17912e8	0.960485
$498$	0	0
$499$	1.22399e8	0.985088	0.492544	−	0.870288i	$-0.336067\pi$
$499$	1.22399e8	0.985088	0.492544	+	0.870288i	$0.336067\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 1.79127e7i	− 0.140753i	−0.997521	−	0.0703763i	$-0.977580\pi$
$503$	− 1.79127e7i	− 0.140753i	0.997521	−	0.0703763i	$-0.0224200\pi$
$504$	0	0
$505$	−6.51682e7	−0.506013
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1.60087e8i	− 1.21395i	−0.794720	−	0.606976i	$-0.792382\pi$
$509$	− 1.60087e8i	− 1.21395i	0.794720	−	0.606976i	$-0.207618\pi$
$510$	0	0
$511$	9.07316e6i	0.0679979i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.04674e7	−0.589113
$516$	0	0
$517$	− 7.42023e7i	− 0.536965i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.93475e7	−0.207519	−0.103759	−	0.994602i	$-0.533087\pi$
$521$	−2.93475e7	−0.207519	−0.103759	+	0.994602i	$0.533087\pi$
$522$	0	0
$523$	1.69326e8	1.18363	0.591817	−	0.806072i	$-0.298411\pi$
$523$	1.69326e8	1.18363	0.591817	+	0.806072i	$0.298411\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.26619e8i	− 0.865102i
$528$	0	0
$529$	9.13471e7	0.617061
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.46699e7i	0.295008i
$534$	0	0
$535$	− 4.51379e7i	− 0.294768i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7.87297e6	−0.0502773
$540$	0	0
$541$	1.45276e8i	0.917493i	0.888567	+	0.458746i	$0.151701\pi$
$541$	1.45276e8i	0.917493i	−0.888567	+	0.458746i	$0.848299\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.32880e7	0.205636
$546$	0	0
$547$	−1.72469e8	−1.05378	−0.526889	−	0.849934i	$-0.676642\pi$
$547$	−1.72469e8	−1.05378	−0.526889	+	0.849934i	$0.676642\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.49469e8i	3.28464i
$552$	0	0
$553$	2.36776e8	1.40011
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.52417e8i	− 1.46067i	−0.683089	−	0.730335i	$-0.739364\pi$
$557$	− 2.52417e8i	− 1.46067i	0.683089	−	0.730335i	$-0.260636\pi$
$558$	0	0
$559$	2.07837e7i	0.118984i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.43419e7	−0.416589	−0.208295	−	0.978066i	$-0.566791\pi$
$563$	−7.43419e7	−0.416589	−0.208295	+	0.978066i	$0.566791\pi$
$564$	0	0
$565$	− 1.01367e7i	− 0.0562022i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.44911e8	0.786620	0.393310	−	0.919406i	$-0.371330\pi$
$569$	1.44911e8	0.786620	0.393310	+	0.919406i	$0.371330\pi$
$570$	0	0
$571$	−1.51024e8	−0.811218	−0.405609	−	0.914047i	$-0.632941\pi$
$571$	−1.51024e8	−0.811218	−0.405609	+	0.914047i	$0.632941\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 9.76712e7i	− 0.513763i
$576$	0	0
$577$	−3.65008e7	−0.190009	−0.0950046	−	0.995477i	$-0.530287\pi$
$577$	−3.65008e7	−0.190009	−0.0950046	+	0.995477i	$0.530287\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 2.93400e8i	− 1.49600i
$582$	0	0
$583$	7.61787e7i	0.384439i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.97585e8	1.47129	0.735643	−	0.677370i	$-0.236880\pi$
$587$	2.97585e8	1.47129	0.735643	+	0.677370i	$0.236880\pi$
$588$	0	0
$589$	2.39941e8i	1.17424i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.59264e8	0.763755	0.381878	−	0.924213i	$-0.375278\pi$
$593$	1.59264e8	0.763755	0.381878	+	0.924213i	$0.375278\pi$
$594$	0	0
$595$	9.72936e7	0.461885
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.93971e8i	1.36781i	0.729573	+	0.683903i	$0.239719\pi$
$599$	2.93971e8i	1.36781i	−0.729573	+	0.683903i	$0.760281\pi$
$600$	0	0
$601$	−1.17935e8	−0.543277	−0.271638	−	0.962399i	$-0.587565\pi$
$601$	−1.17935e8	−0.543277	−0.271638	+	0.962399i	$0.587565\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 8.05354e7i	− 0.363681i
$606$	0	0
$607$	− 7.36776e7i	− 0.329435i	−0.986341	−	0.164717i	$-0.947329\pi$
$607$	− 7.36776e7i	− 0.329435i	0.986341	−	0.164717i	$-0.0526712\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.78288e8	1.65844
$612$	0	0
$613$	− 1.85901e8i	− 0.807050i	−0.914969	−	0.403525i	$-0.867785\pi$
$613$	− 1.85901e8i	− 0.807050i	0.914969	−	0.403525i	$-0.132215\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.32115e7	0.141395	0.0706973	−	0.997498i	$-0.477478\pi$
$617$	3.32115e7	0.141395	0.0706973	+	0.997498i	$0.477478\pi$
$618$	0	0
$619$	−5.24234e7	−0.221031	−0.110516	−	0.993874i	$-0.535250\pi$
$619$	−5.24234e7	−0.221031	−0.110516	+	0.993874i	$0.535250\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 2.60162e8i	− 1.07592i
$624$	0	0
$625$	1.26833e8	0.519509
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 7.63364e7i	− 0.306747i
$630$	0	0
$631$	− 1.32736e8i	− 0.528326i	−0.964478	−	0.264163i	$-0.914904\pi$
$631$	− 1.32736e8i	− 0.528326i	0.964478	−	0.264163i	$-0.0850957\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.70741e8	0.666833
$636$	0	0
$637$	− 4.01369e7i	− 0.155284i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.42334e7	−0.319823	−0.159912	−	0.987131i	$-0.551121\pi$
$641$	−8.42334e7	−0.319823	−0.159912	+	0.987131i	$0.551121\pi$
$642$	0	0
$643$	3.11219e8	1.17067	0.585334	−	0.810792i	$-0.300963\pi$
$643$	3.11219e8	1.17067	0.585334	+	0.810792i	$0.300963\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.82536e8i	1.41241i	0.708010	+	0.706203i	$0.249593\pi$
$647$	3.82536e8i	1.41241i	−0.708010	+	0.706203i	$0.750407\pi$
$648$	0	0
$649$	−6.21249e7	−0.227265
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.98790e8i	1.43220i	0.697996	+	0.716102i	$0.254075\pi$
$653$	3.98790e8i	1.43220i	−0.697996	+	0.716102i	$0.745925\pi$
$654$	0	0
$655$	7.52768e7i	0.267878i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.89798e7	0.241027	0.120513	−	0.992712i	$-0.461546\pi$
$659$	6.89798e7	0.241027	0.120513	+	0.992712i	$0.461546\pi$
$660$	0	0
$661$	2.51486e8i	0.870782i	0.900241	+	0.435391i	$0.143390\pi$
$661$	2.51486e8i	0.870782i	−0.900241	+	0.435391i	$0.856610\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.84370e8	−0.626938
$666$	0	0
$667$	3.66161e8	1.23394
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1.16211e8i	− 0.384662i
$672$	0	0
$673$	2.23586e8	0.733499	0.366750	−	0.930320i	$-0.380471\pi$
$673$	2.23586e8	0.733499	0.366750	+	0.930320i	$0.380471\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.72443e8i	− 0.555751i	−0.960617	−	0.277876i	$-0.910370\pi$
$677$	− 1.72443e8i	− 0.555751i	0.960617	−	0.277876i	$-0.0896303\pi$
$678$	0	0
$679$	2.14079e8i	0.683856i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.18442e8	−1.62719	−0.813594	−	0.581433i	$-0.802492\pi$
$683$	−5.18442e8	−1.62719	−0.813594	+	0.581433i	$0.802492\pi$
$684$	0	0
$685$	− 1.19142e8i	− 0.370674i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3.88364e8	−1.18736
$690$	0	0
$691$	−4.12317e8	−1.24967	−0.624837	−	0.780755i	$-0.714834\pi$
$691$	−4.12317e8	−1.24967	−0.624837	+	0.780755i	$0.714834\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.59044e8i	− 0.473764i
$696$	0	0
$697$	1.14579e8	0.338381
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.19721e7i	0.121845i	0.998143	+	0.0609224i	$0.0194042\pi$
$701$	4.19721e7i	0.121845i	−0.998143	+	0.0609224i	$0.980596\pi$
$702$	0	0
$703$	1.44656e8i	0.416362i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.00886e8	−1.13439
$708$	0	0
$709$	− 5.94810e8i	− 1.66894i	−0.551057	−	0.834468i	$-0.685775\pi$
$709$	− 5.94810e8i	− 1.66894i	0.551057	−	0.834468i	$-0.314225\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.59894e8	0.441128
$714$	0	0
$715$	−5.45868e7	−0.149338
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 7.17679e7i	− 0.193083i	−0.995329	−	0.0965414i	$-0.969222\pi$
$719$	− 7.17679e7i	− 0.193083i	0.995329	−	0.0965414i	$-0.0307780\pi$
$720$	0	0
$721$	−4.95001e8	−1.32069
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.30872e8i	1.65549i
$726$	0	0
$727$	4.60152e7i	0.119756i	0.998206	+	0.0598781i	$0.0190712\pi$
$727$	4.60152e7i	0.119756i	−0.998206	+	0.0598781i	$0.980929\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.33105e7	0.136477
$732$	0	0
$733$	3.14277e7i	0.0797995i	0.999204	+	0.0398997i	$0.0127039\pi$
$733$	3.14277e7i	0.0797995i	−0.999204	+	0.0398997i	$0.987296\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.32737e8	0.331582
$738$	0	0
$739$	1.30671e8	0.323778	0.161889	−	0.986809i	$-0.448241\pi$
$739$	1.30671e8	0.323778	0.161889	+	0.986809i	$0.448241\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 2.39517e8i	− 0.583942i	−0.956427	−	0.291971i	$-0.905689\pi$
$743$	− 2.39517e8i	− 0.583942i	0.956427	−	0.291971i	$-0.0943111\pi$
$744$	0	0
$745$	2.60772e7	0.0630655
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 2.77669e8i	− 0.660817i
$750$	0	0
$751$	− 5.07976e8i	− 1.19929i	−0.800267	−	0.599644i	$-0.795309\pi$
$751$	− 5.07976e8i	− 1.19929i	0.800267	−	0.599644i	$-0.204691\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.78632e7	−0.0647425
$756$	0	0
$757$	3.35592e8i	0.773614i	0.922161	+	0.386807i	$0.126422\pi$
$757$	3.35592e8i	0.773614i	−0.922161	+	0.386807i	$0.873578\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.91901e8	0.662341	0.331171	−	0.943571i	$-0.392556\pi$
$761$	2.91901e8	0.662341	0.331171	+	0.943571i	$0.392556\pi$
$762$	0	0
$763$	2.04773e8	0.460999
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.16717e8i	− 0.701916i
$768$	0	0
$769$	−2.38154e8	−0.523695	−0.261847	−	0.965109i	$-0.584332\pi$
$769$	−2.38154e8	−0.523695	−0.261847	+	0.965109i	$0.584332\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.03506e8i	0.657096i	0.944487	+	0.328548i	$0.106559\pi$
$773$	3.03506e8i	0.657096i	−0.944487	+	0.328548i	$0.893441\pi$
$774$	0	0
$775$	2.75488e8i	0.591830i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.17125e8	−0.459300
$780$	0	0
$781$	1.69688e8i	0.356204i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.33645e8	−0.276277
$786$	0	0
$787$	−6.21311e8	−1.27463	−0.637316	−	0.770603i	$-0.719955\pi$
$787$	−6.21311e8	−1.27463	−0.637316	+	0.770603i	$0.719955\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 6.23568e7i	− 0.125995i
$792$	0	0
$793$	5.92451e8	1.18804
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 5.14212e8i	− 1.01570i	−0.861444	−	0.507852i	$-0.830440\pi$
$797$	− 5.14212e8i	− 1.01570i	0.861444	−	0.507852i	$-0.169560\pi$
$798$	0	0
$799$	− 9.70313e8i	− 1.90227i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.30572e7	−0.0252176
$804$	0	0
$805$	1.22862e8i	0.235522i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.15875e8	0.407715	0.203858	−	0.979001i	$-0.434652\pi$
$809$	2.15875e8	0.407715	0.203858	+	0.979001i	$0.434652\pi$
$810$	0	0
$811$	−6.81893e8	−1.27836	−0.639181	−	0.769056i	$-0.720727\pi$
$811$	−6.81893e8	−1.27836	−0.639181	+	0.769056i	$0.720727\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 2.71543e8i	− 0.501609i
$816$	0	0
$817$	−1.01022e8	−0.185247
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.70599e8i	0.669692i	0.942273	+	0.334846i	$0.108684\pi$
$821$	3.70599e8i	0.669692i	−0.942273	+	0.334846i	$0.891316\pi$
$822$	0	0
$823$	6.80700e8i	1.22111i	0.791972	+	0.610557i	$0.209054\pi$
$823$	6.80700e8i	1.22111i	−0.791972	+	0.610557i	$0.790946\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.55416e7	0.151238	0.0756189	−	0.997137i	$-0.475907\pi$
$827$	8.55416e7	0.151238	0.0756189	+	0.997137i	$0.475907\pi$
$828$	0	0
$829$	5.12614e8i	0.899761i	0.893089	+	0.449880i	$0.148533\pi$
$829$	5.12614e8i	0.899761i	−0.893089	+	0.449880i	$0.851467\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.02952e8	−0.178114
$834$	0	0
$835$	1.05557e8	0.181312
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.25162e8i	0.381249i	0.981663	+	0.190625i	$0.0610513\pi$
$839$	2.25162e8i	0.381249i	−0.981663	+	0.190625i	$0.938949\pi$
$840$	0	0
$841$	−1.77026e9	−2.97611
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2.96868e7i	− 0.0492031i
$846$	0	0
$847$	− 4.95419e8i	− 0.815309i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.63975e7	0.156414
$852$	0	0
$853$	− 1.68109e8i	− 0.270860i	−0.990787	−	0.135430i	$-0.956758\pi$
$853$	− 1.68109e8i	− 0.270860i	0.990787	−	0.135430i	$-0.0432416\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.61928e8	−1.05164	−0.525822	−	0.850595i	$-0.676242\pi$
$857$	−6.61928e8	−1.05164	−0.525822	+	0.850595i	$0.676242\pi$
$858$	0	0
$859$	−7.46051e8	−1.17703	−0.588517	−	0.808485i	$-0.700288\pi$
$859$	−7.46051e8	−1.17703	−0.588517	+	0.808485i	$0.700288\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 9.52742e8i	− 1.48232i	−0.671326	−	0.741162i	$-0.734275\pi$
$863$	− 9.52742e8i	− 1.48232i	0.671326	−	0.741162i	$-0.265725\pi$
$864$	0	0
$865$	−3.31911e8	−0.512830
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.40746e8i	0.519244i
$870$	0	0
$871$	6.76704e8i	1.02411i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.66654e8	−0.696580
$876$	0	0
$877$	− 5.38880e7i	− 0.0798902i	−0.999202	−	0.0399451i	$-0.987282\pi$
$877$	− 5.38880e7i	− 0.0798902i	0.999202	−	0.0399451i	$-0.0127183\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.06501e9	−1.55750	−0.778749	−	0.627336i	$-0.784145\pi$
$881$	−1.06501e9	−1.55750	−0.778749	+	0.627336i	$0.784145\pi$
$882$	0	0
$883$	−1.06768e8	−0.155081	−0.0775405	−	0.996989i	$-0.524707\pi$
$883$	−1.06768e8	−0.155081	−0.0775405	+	0.996989i	$0.524707\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.57318e8i	1.22849i	0.789116	+	0.614244i	$0.210539\pi$
$887$	8.57318e8i	1.22849i	−0.789116	+	0.614244i	$0.789461\pi$
$888$	0	0
$889$	1.05033e9	1.49492
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.83873e9i	2.58204i
$894$	0	0
$895$	− 5.15709e8i	− 0.719342i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.03278e9	−1.42144
$900$	0	0
$901$	9.96157e8i	1.36193i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.59392e8	0.215041
$906$	0	0
$907$	5.55579e8	0.744601	0.372301	−	0.928112i	$-0.378569\pi$
$907$	5.55579e8	0.744601	0.372301	+	0.928112i	$0.378569\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 4.58157e8i	− 0.605981i	−0.952994	−	0.302991i	$-0.902015\pi$
$911$	− 4.58157e8i	− 0.605981i	0.952994	−	0.302991i	$-0.0979851\pi$
$912$	0	0
$913$	4.22234e8	0.554805
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.63070e8i	0.600536i
$918$	0	0
$919$	8.63186e8i	1.11214i	0.831137	+	0.556068i	$0.187691\pi$
$919$	8.63186e8i	1.11214i	−0.831137	+	0.556068i	$0.812309\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.65082e8	−1.10015
$924$	0	0
$925$	1.66087e8i	0.209850i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	8.00351e8	0.998237	0.499118	−	0.866534i	$-0.333657\pi$
$929$	8.00351e8	0.998237	0.499118	+	0.866534i	$0.333657\pi$
$930$	0	0
$931$	1.95091e8	0.241763
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.40016e8i	0.171294i
$936$	0	0
$937$	5.02009e8	0.610228	0.305114	−	0.952316i	$-0.401305\pi$
$937$	5.02009e8	0.610228	0.305114	+	0.952316i	$0.401305\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.18725e9i	1.42486i	0.701744	+	0.712429i	$0.252405\pi$
$941$	1.18725e9i	1.42486i	−0.701744	+	0.712429i	$0.747595\pi$
$942$	0	0
$943$	1.44690e8i	0.172545i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.32279e9	−1.55755	−0.778775	−	0.627303i	$-0.784159\pi$
$947$	−1.32279e9	−1.55755	−0.778775	+	0.627303i	$0.784159\pi$
$948$	0	0
$949$	− 6.65666e7i	− 0.0778857i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.30974e9	−1.51324	−0.756618	−	0.653858i	$-0.773149\pi$
$953$	−1.30974e9	−1.51324	−0.756618	+	0.653858i	$0.773149\pi$
$954$	0	0
$955$	6.50496e8	0.746852
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 7.32908e8i	− 0.830986i
$960$	0	0
$961$	4.36512e8	0.491842
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.34992e8i	0.372780i
$966$	0	0
$967$	− 9.99874e8i	− 1.10577i	−0.833257	−	0.552886i	$-0.813526\pi$
$967$	− 9.99874e8i	− 1.10577i	0.833257	−	0.552886i	$-0.186474\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.22800e9	1.34135	0.670675	−	0.741751i	$-0.266004\pi$
$971$	1.22800e9	1.34135	0.670675	+	0.741751i	$0.266004\pi$
$972$	0	0
$973$	− 9.78368e8i	− 1.06210i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.12283e8	0.334861	0.167430	−	0.985884i	$-0.446453\pi$
$977$	3.12283e8	0.334861	0.167430	+	0.985884i	$0.446453\pi$
$978$	0	0
$979$	3.74400e8	0.399014
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.52567e8i	0.687013i	0.939150	+	0.343506i	$0.111615\pi$
$983$	6.52567e8i	0.687013i	−0.939150	+	0.343506i	$0.888385\pi$
$984$	0	0
$985$	−5.37251e8	−0.562171
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.73204e7i	0.0695917i
$990$	0	0
$991$	3.47356e8i	0.356906i	0.983948	+	0.178453i	$0.0571092\pi$
$991$	3.47356e8i	0.356906i	−0.983948	+	0.178453i	$0.942891\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.16837e8	0.321637
$996$	0	0
$997$	− 1.98519e8i	− 0.200317i	−0.994972	−	0.100158i	$-0.968065\pi$
$997$	− 1.98519e8i	− 0.200317i	0.994972	−	0.100158i	$-0.0319350\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.7.b.e.415.6		8
3.2	odd	2		64.7.d.b.31.3	✓	8
4.3	odd	2	inner	576.7.b.e.415.5		8
8.3	odd	2	inner	576.7.b.e.415.3		8
8.5	even	2	inner	576.7.b.e.415.4		8
12.11	even	2		64.7.d.b.31.5	yes	8
24.5	odd	2		64.7.d.b.31.6	yes	8
24.11	even	2		64.7.d.b.31.4	yes	8
48.5	odd	4		256.7.c.k.255.6		8
48.11	even	4		256.7.c.k.255.4		8
48.29	odd	4		256.7.c.k.255.3		8
48.35	even	4		256.7.c.k.255.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.7.d.b.31.3	✓	8	3.2	odd	2
64.7.d.b.31.4	yes	8	24.11	even	2
64.7.d.b.31.5	yes	8	12.11	even	2
64.7.d.b.31.6	yes	8	24.5	odd	2
256.7.c.k.255.3		8	48.29	odd	4
256.7.c.k.255.4		8	48.11	even	4
256.7.c.k.255.5		8	48.35	even	4
256.7.c.k.255.6		8	48.5	odd	4
576.7.b.e.415.3		8	8.3	odd	2	inner
576.7.b.e.415.4		8	8.5	even	2	inner
576.7.b.e.415.5		8	4.3	odd	2	inner
576.7.b.e.415.6		8	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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+51.5041i q^{5} +316.831i q^{7} +O(q^{10})\)	\(q+51.5041i q^{5} +316.831i q^{7} -455.953 q^{11} -2324.48i q^{13} -5962.31 q^{17} +11298.5 q^{19} -7529.20i q^{23} +12972.3 q^{25} +48632.1i q^{29} +21236.6i q^{31} -16318.1 q^{35} +12803.2i q^{37} -19217.2 q^{41} -8941.24 q^{43} +162741. i q^{47} +17267.1 q^{49} -167076. i q^{53} -23483.5i q^{55} +136253. q^{59} +254875. i q^{61} +119720. q^{65} -291121. q^{67} -372162. i q^{71} +28637.2 q^{73} -144460. i q^{77} -747327. i q^{79} -926047. q^{83} -307084. i q^{85} -821138. q^{89} +736467. q^{91} +581918. i q^{95} +675688. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 11472 q^{17} + 89288 q^{25} + 121584 q^{41} - 847224 q^{49} + 363648 q^{65} - 517168 q^{73} - 432336 q^{89} + 1195984 q^{97}+O(q^{100}) \) 8 * q - 11472 * q^17 + 89288 * q^25 + 121584 * q^41 - 847224 * q^49 + 363648 * q^65 - 517168 * q^73 - 432336 * q^89 + 1195984 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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