LMFDB - Embedded newform 6900.2.a.bd.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(1,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 11x^{4} + 46x^{3} - 32x^{2} - 30x - 2 $ x^7 - x^6 - 13x^5 + 11x^4 + 46x^3 - 32x^2 - 30*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.88373$ of defining polynomial
Character	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.31589 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.31589 q^{7} +1.00000 q^{9} -2.26617 q^{11} +2.25590 q^{13} +0.535482 q^{17} -6.91942 q^{19} +4.31589 q^{21} -1.00000 q^{23} +1.00000 q^{27} +5.64598 q^{29} -10.4240 q^{31} -2.26617 q^{33} +8.10481 q^{37} +2.25590 q^{39} +0.633143 q^{41} -1.30687 q^{43} +12.4159 q^{47} +11.6269 q^{49} +0.535482 q^{51} +5.98659 q^{53} -6.91942 q^{57} +7.01063 q^{59} +5.98705 q^{61} +4.31589 q^{63} +6.32616 q^{67} -1.00000 q^{69} +0.151090 q^{71} +8.88768 q^{73} -9.78055 q^{77} -10.3312 q^{79} +1.00000 q^{81} -0.716944 q^{83} +5.64598 q^{87} +15.5781 q^{89} +9.73623 q^{91} -10.4240 q^{93} +16.2497 q^{97} -2.26617 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} - q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 - q^7 + 7 * q^9	$ 7 q + 7 q^{3} - q^{7} + 7 q^{9} - 2 q^{13} + q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 7 q^{27} + 15 q^{29} + 3 q^{31} + 19 q^{37} - 2 q^{39} + 23 q^{41} - 4 q^{43} + 10 q^{47} + 10 q^{49} + q^{51} + 11 q^{53} - 2 q^{57} + 5 q^{59} + 32 q^{61} - q^{63} + 15 q^{67} - 7 q^{69} + 21 q^{71} - 18 q^{73} + 28 q^{77} + 16 q^{79} + 7 q^{81} + 25 q^{83} + 15 q^{87} + 26 q^{89} + 14 q^{91} + 3 q^{93} + 6 q^{97}+O(q^{100}) $ 7 * q + 7 * q^3 - q^7 + 7 * q^9 - 2 * q^13 + q^17 - 2 * q^19 - q^21 - 7 * q^23 + 7 * q^27 + 15 * q^29 + 3 * q^31 + 19 * q^37 - 2 * q^39 + 23 * q^41 - 4 * q^43 + 10 * q^47 + 10 * q^49 + q^51 + 11 * q^53 - 2 * q^57 + 5 * q^59 + 32 * q^61 - q^63 + 15 * q^67 - 7 * q^69 + 21 * q^71 - 18 * q^73 + 28 * q^77 + 16 * q^79 + 7 * q^81 + 25 * q^83 + 15 * q^87 + 26 * q^89 + 14 * q^91 + 3 * q^93 + 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.31589	1.63125	0.815626	−	0.578579i	$-0.196393\pi$
$7$	4.31589	1.63125	0.815626	+	0.578579i	$0.196393\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.26617	−0.683277	−0.341638	−	0.939831i	$-0.610982\pi$
$11$	−2.26617	−0.683277	−0.341638	+	0.939831i	$0.610982\pi$
$12$	0	0
$13$	2.25590	0.625675	0.312838	−	0.949807i	$-0.398720\pi$
$13$	2.25590	0.625675	0.312838	+	0.949807i	$0.398720\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.535482	0.129874	0.0649368	−	0.997889i	$-0.479315\pi$
$17$	0.535482	0.129874	0.0649368	+	0.997889i	$0.479315\pi$
$18$	0	0
$19$	−6.91942	−1.58742	−0.793712	−	0.608294i	$-0.791854\pi$
$19$	−6.91942	−1.58742	−0.793712	+	0.608294i	$0.791854\pi$
$20$	0	0
$21$	4.31589	0.941804
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.64598	1.04843	0.524216	−	0.851585i	$-0.324358\pi$
$29$	5.64598	1.04843	0.524216	+	0.851585i	$0.324358\pi$
$30$	0	0
$31$	−10.4240	−1.87220	−0.936099	−	0.351736i	$-0.885591\pi$
$31$	−10.4240	−1.87220	−0.936099	+	0.351736i	$0.885591\pi$
$32$	0	0
$33$	−2.26617	−0.394490
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.10481	1.33242	0.666212	−	0.745763i	$-0.267915\pi$
$37$	8.10481	1.33242	0.666212	+	0.745763i	$0.267915\pi$
$38$	0	0
$39$	2.25590	0.361234
$40$	0	0
$41$	0.633143	0.0988803	0.0494402	−	0.998777i	$-0.484256\pi$
$41$	0.633143	0.0988803	0.0494402	+	0.998777i	$0.484256\pi$
$42$	0	0
$43$	−1.30687	−0.199296	−0.0996481	−	0.995023i	$-0.531772\pi$
$43$	−1.30687	−0.199296	−0.0996481	+	0.995023i	$0.531772\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.4159	1.81104	0.905522	−	0.424299i	$-0.139479\pi$
$47$	12.4159	1.81104	0.905522	+	0.424299i	$0.139479\pi$
$48$	0	0
$49$	11.6269	1.66098
$50$	0	0
$51$	0.535482	0.0749825
$52$	0	0
$53$	5.98659	0.822322	0.411161	−	0.911563i	$-0.365123\pi$
$53$	5.98659	0.822322	0.411161	+	0.911563i	$0.365123\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.91942	−0.916499
$58$	0	0
$59$	7.01063	0.912706	0.456353	−	0.889799i	$-0.349155\pi$
$59$	7.01063	0.912706	0.456353	+	0.889799i	$0.349155\pi$
$60$	0	0
$61$	5.98705	0.766563	0.383282	−	0.923632i	$-0.374794\pi$
$61$	5.98705	0.766563	0.383282	+	0.923632i	$0.374794\pi$
$62$	0	0
$63$	4.31589	0.543751
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.32616	0.772863	0.386432	−	0.922318i	$-0.373708\pi$
$67$	6.32616	0.772863	0.386432	+	0.922318i	$0.373708\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0.151090	0.0179311	0.00896553	−	0.999960i	$-0.497146\pi$
$71$	0.151090	0.0179311	0.00896553	+	0.999960i	$0.497146\pi$
$72$	0	0
$73$	8.88768	1.04022	0.520112	−	0.854098i	$-0.325890\pi$
$73$	8.88768	1.04022	0.520112	+	0.854098i	$0.325890\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.78055	−1.11460
$78$	0	0
$79$	−10.3312	−1.16235	−0.581175	−	0.813779i	$-0.697407\pi$
$79$	−10.3312	−1.16235	−0.581175	+	0.813779i	$0.697407\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.716944	−0.0786948	−0.0393474	−	0.999226i	$-0.512528\pi$
$83$	−0.716944	−0.0786948	−0.0393474	+	0.999226i	$0.512528\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.64598	0.605313
$88$	0	0
$89$	15.5781	1.65128	0.825639	−	0.564199i	$-0.190815\pi$
$89$	15.5781	1.65128	0.825639	+	0.564199i	$0.190815\pi$
$90$	0	0
$91$	9.73623	1.02063
$92$	0	0
$93$	−10.4240	−1.08091
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.2497	1.64990	0.824951	−	0.565204i	$-0.191203\pi$
$97$	16.2497	1.64990	0.824951	+	0.565204i	$0.191203\pi$
$98$	0	0
$99$	−2.26617	−0.227759
$100$	0	0
$101$	12.8428	1.27790	0.638952	−	0.769247i	$-0.279368\pi$
$101$	12.8428	1.27790	0.638952	+	0.769247i	$0.279368\pi$
$102$	0	0
$103$	−16.1002	−1.58640	−0.793200	−	0.608961i	$-0.791587\pi$
$103$	−16.1002	−1.58640	−0.793200	+	0.608961i	$0.791587\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.3335	1.28900	0.644501	−	0.764604i	$-0.277065\pi$
$107$	13.3335	1.28900	0.644501	+	0.764604i	$0.277065\pi$
$108$	0	0
$109$	−6.61883	−0.633969	−0.316984	−	0.948431i	$-0.602670\pi$
$109$	−6.61883	−0.633969	−0.316984	+	0.948431i	$0.602670\pi$
$110$	0	0
$111$	8.10481	0.769275
$112$	0	0
$113$	17.5215	1.64828	0.824142	−	0.566383i	$-0.191658\pi$
$113$	17.5215	1.64828	0.824142	+	0.566383i	$0.191658\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.25590	0.208558
$118$	0	0
$119$	2.31108	0.211857
$120$	0	0
$121$	−5.86446	−0.533133
$122$	0	0
$123$	0.633143	0.0570886
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−13.3001	−1.18019	−0.590095	−	0.807334i	$-0.700910\pi$
$127$	−13.3001	−1.18019	−0.590095	+	0.807334i	$0.700910\pi$
$128$	0	0
$129$	−1.30687	−0.115064
$130$	0	0
$131$	−2.99386	−0.261575	−0.130787	−	0.991410i	$-0.541751\pi$
$131$	−2.99386	−0.261575	−0.130787	+	0.991410i	$0.541751\pi$
$132$	0	0
$133$	−29.8634	−2.58949
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.5528	−1.49964	−0.749819	−	0.661643i	$-0.769859\pi$
$137$	−17.5528	−1.49964	−0.749819	+	0.661643i	$0.769859\pi$
$138$	0	0
$139$	−17.8735	−1.51601	−0.758004	−	0.652250i	$-0.773825\pi$
$139$	−17.8735	−1.51601	−0.758004	+	0.652250i	$0.773825\pi$
$140$	0	0
$141$	12.4159	1.04561
$142$	0	0
$143$	−5.11227	−0.427509
$144$	0	0
$145$	0	0
$146$	0	0
$147$	11.6269	0.958970
$148$	0	0
$149$	3.71607	0.304432	0.152216	−	0.988347i	$-0.451359\pi$
$149$	3.71607	0.304432	0.152216	+	0.988347i	$0.451359\pi$
$150$	0	0
$151$	14.3527	1.16800	0.584001	−	0.811753i	$-0.301486\pi$
$151$	14.3527	1.16800	0.584001	+	0.811753i	$0.301486\pi$
$152$	0	0
$153$	0.535482	0.0432912
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−9.94566	−0.793750	−0.396875	−	0.917873i	$-0.629905\pi$
$157$	−9.94566	−0.793750	−0.396875	+	0.917873i	$0.629905\pi$
$158$	0	0
$159$	5.98659	0.474768
$160$	0	0
$161$	−4.31589	−0.340140
$162$	0	0
$163$	−9.87216	−0.773247	−0.386624	−	0.922238i	$-0.626359\pi$
$163$	−9.87216	−0.773247	−0.386624	+	0.922238i	$0.626359\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.92476	0.226325	0.113162	−	0.993577i	$-0.463902\pi$
$167$	2.92476	0.226325	0.113162	+	0.993577i	$0.463902\pi$
$168$	0	0
$169$	−7.91090	−0.608531
$170$	0	0
$171$	−6.91942	−0.529141
$172$	0	0
$173$	0.421083	0.0320144	0.0160072	−	0.999872i	$-0.494905\pi$
$173$	0.421083	0.0320144	0.0160072	+	0.999872i	$0.494905\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.01063	0.526951
$178$	0	0
$179$	14.3955	1.07597	0.537986	−	0.842954i	$-0.319185\pi$
$179$	14.3955	1.07597	0.537986	+	0.842954i	$0.319185\pi$
$180$	0	0
$181$	7.54278	0.560650	0.280325	−	0.959905i	$-0.409558\pi$
$181$	7.54278	0.560650	0.280325	+	0.959905i	$0.409558\pi$
$182$	0	0
$183$	5.98705	0.442576
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.21350	−0.0887396
$188$	0	0
$189$	4.31589	0.313935
$190$	0	0
$191$	18.9141	1.36857	0.684287	−	0.729213i	$-0.260113\pi$
$191$	18.9141	1.36857	0.684287	+	0.729213i	$0.260113\pi$
$192$	0	0
$193$	3.56998	0.256973	0.128487	−	0.991711i	$-0.458988\pi$
$193$	3.56998	0.256973	0.128487	+	0.991711i	$0.458988\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.90677	−0.135852	−0.0679258	−	0.997690i	$-0.521638\pi$
$197$	−1.90677	−0.135852	−0.0679258	+	0.997690i	$0.521638\pi$
$198$	0	0
$199$	1.84829	0.131022	0.0655110	−	0.997852i	$-0.479132\pi$
$199$	1.84829	0.131022	0.0655110	+	0.997852i	$0.479132\pi$
$200$	0	0
$201$	6.32616	0.446213
$202$	0	0
$203$	24.3674	1.71026
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	15.6806	1.08465
$210$	0	0
$211$	14.0429	0.966753	0.483377	−	0.875413i	$-0.339410\pi$
$211$	14.0429	0.966753	0.483377	+	0.875413i	$0.339410\pi$
$212$	0	0
$213$	0.151090	0.0103525
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−44.9886	−3.05403
$218$	0	0
$219$	8.88768	0.600574
$220$	0	0
$221$	1.20800	0.0812586
$222$	0	0
$223$	−11.3081	−0.757245	−0.378623	−	0.925551i	$-0.623602\pi$
$223$	−11.3081	−0.757245	−0.378623	+	0.925551i	$0.623602\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.3506	1.74895	0.874477	−	0.485067i	$-0.161205\pi$
$227$	26.3506	1.74895	0.874477	+	0.485067i	$0.161205\pi$
$228$	0	0
$229$	−13.7704	−0.909976	−0.454988	−	0.890498i	$-0.650357\pi$
$229$	−13.7704	−0.909976	−0.454988	+	0.890498i	$0.650357\pi$
$230$	0	0
$231$	−9.78055	−0.643513
$232$	0	0
$233$	6.05029	0.396368	0.198184	−	0.980165i	$-0.436496\pi$
$233$	6.05029	0.396368	0.198184	+	0.980165i	$0.436496\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−10.3312	−0.671083
$238$	0	0
$239$	−15.2468	−0.986234	−0.493117	−	0.869963i	$-0.664143\pi$
$239$	−15.2468	−0.986234	−0.493117	+	0.869963i	$0.664143\pi$
$240$	0	0
$241$	15.3001	0.985563	0.492782	−	0.870153i	$-0.335980\pi$
$241$	15.3001	0.985563	0.492782	+	0.870153i	$0.335980\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−15.6095	−0.993211
$248$	0	0
$249$	−0.716944	−0.0454345
$250$	0	0
$251$	−20.4096	−1.28825	−0.644123	−	0.764922i	$-0.722777\pi$
$251$	−20.4096	−1.28825	−0.644123	+	0.764922i	$0.722777\pi$
$252$	0	0
$253$	2.26617	0.142473
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.8072	−0.923649	−0.461825	−	0.886971i	$-0.652805\pi$
$257$	−14.8072	−0.923649	−0.461825	+	0.886971i	$0.652805\pi$
$258$	0	0
$259$	34.9795	2.17352
$260$	0	0
$261$	5.64598	0.349477
$262$	0	0
$263$	−14.5266	−0.895747	−0.447873	−	0.894097i	$-0.647818\pi$
$263$	−14.5266	−0.895747	−0.447873	+	0.894097i	$0.647818\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15.5781	0.953366
$268$	0	0
$269$	−16.4759	−1.00455	−0.502276	−	0.864708i	$-0.667504\pi$
$269$	−16.4759	−1.00455	−0.502276	+	0.864708i	$0.667504\pi$
$270$	0	0
$271$	22.0764	1.34105	0.670524	−	0.741888i	$-0.266069\pi$
$271$	22.0764	1.34105	0.670524	+	0.741888i	$0.266069\pi$
$272$	0	0
$273$	9.73623	0.589263
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.70017	−0.402574	−0.201287	−	0.979532i	$-0.564512\pi$
$277$	−6.70017	−0.402574	−0.201287	+	0.979532i	$0.564512\pi$
$278$	0	0
$279$	−10.4240	−0.624066
$280$	0	0
$281$	−22.6857	−1.35332	−0.676658	−	0.736298i	$-0.736572\pi$
$281$	−22.6857	−1.35332	−0.676658	+	0.736298i	$0.736572\pi$
$282$	0	0
$283$	19.9470	1.18572	0.592862	−	0.805304i	$-0.297998\pi$
$283$	19.9470	1.18572	0.592862	+	0.805304i	$0.297998\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.73257	0.161299
$288$	0	0
$289$	−16.7133	−0.983133
$290$	0	0
$291$	16.2497	0.952571
$292$	0	0
$293$	−14.9230	−0.871808	−0.435904	−	0.899993i	$-0.643571\pi$
$293$	−14.9230	−0.871808	−0.435904	+	0.899993i	$0.643571\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.26617	−0.131497
$298$	0	0
$299$	−2.25590	−0.130462
$300$	0	0
$301$	−5.64032	−0.325102
$302$	0	0
$303$	12.8428	0.737798
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.9612	0.739735	0.369868	−	0.929084i	$-0.379403\pi$
$307$	12.9612	0.739735	0.369868	+	0.929084i	$0.379403\pi$
$308$	0	0
$309$	−16.1002	−0.915909
$310$	0	0
$311$	−13.0709	−0.741182	−0.370591	−	0.928796i	$-0.620845\pi$
$311$	−13.0709	−0.741182	−0.370591	+	0.928796i	$0.620845\pi$
$312$	0	0
$313$	0.224783	0.0127055	0.00635273	−	0.999980i	$-0.497978\pi$
$313$	0.224783	0.0127055	0.00635273	+	0.999980i	$0.497978\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.03602	−0.395182	−0.197591	−	0.980285i	$-0.563312\pi$
$317$	−7.03602	−0.395182	−0.197591	+	0.980285i	$0.563312\pi$
$318$	0	0
$319$	−12.7948	−0.716369
$320$	0	0
$321$	13.3335	0.744205
$322$	0	0
$323$	−3.70523	−0.206164
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−6.61883	−0.366022
$328$	0	0
$329$	53.5856	2.95427
$330$	0	0
$331$	−13.7320	−0.754781	−0.377391	−	0.926054i	$-0.623179\pi$
$331$	−13.7320	−0.754781	−0.377391	+	0.926054i	$0.623179\pi$
$332$	0	0
$333$	8.10481	0.444141
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.4552	0.569533	0.284767	−	0.958597i	$-0.408084\pi$
$337$	10.4552	0.569533	0.284767	+	0.958597i	$0.408084\pi$
$338$	0	0
$339$	17.5215	0.951638
$340$	0	0
$341$	23.6225	1.27923
$342$	0	0
$343$	19.9691	1.07823
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−35.4166	−1.90126	−0.950632	−	0.310321i	$-0.899563\pi$
$347$	−35.4166	−1.90126	−0.950632	+	0.310321i	$0.899563\pi$
$348$	0	0
$349$	−10.9753	−0.587493	−0.293747	−	0.955883i	$-0.594902\pi$
$349$	−10.9753	−0.587493	−0.293747	+	0.955883i	$0.594902\pi$
$350$	0	0
$351$	2.25590	0.120411
$352$	0	0
$353$	−12.0802	−0.642963	−0.321481	−	0.946916i	$-0.604181\pi$
$353$	−12.0802	−0.642963	−0.321481	+	0.946916i	$0.604181\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.31108	0.122315
$358$	0	0
$359$	32.1356	1.69605	0.848026	−	0.529954i	$-0.177791\pi$
$359$	32.1356	1.69605	0.848026	+	0.529954i	$0.177791\pi$
$360$	0	0
$361$	28.8784	1.51991
$362$	0	0
$363$	−5.86446	−0.307804
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.7278	−0.559985	−0.279992	−	0.960002i	$-0.590332\pi$
$367$	−10.7278	−0.559985	−0.279992	+	0.960002i	$0.590332\pi$
$368$	0	0
$369$	0.633143	0.0329601
$370$	0	0
$371$	25.8375	1.34141
$372$	0	0
$373$	−23.1722	−1.19981	−0.599905	−	0.800071i	$-0.704795\pi$
$373$	−23.1722	−1.19981	−0.599905	+	0.800071i	$0.704795\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.7368	0.655978
$378$	0	0
$379$	19.2361	0.988090	0.494045	−	0.869436i	$-0.335518\pi$
$379$	19.2361	0.988090	0.494045	+	0.869436i	$0.335518\pi$
$380$	0	0
$381$	−13.3001	−0.681383
$382$	0	0
$383$	14.7618	0.754292	0.377146	−	0.926154i	$-0.376905\pi$
$383$	14.7618	0.754292	0.377146	+	0.926154i	$0.376905\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.30687	−0.0664321
$388$	0	0
$389$	−14.0615	−0.712948	−0.356474	−	0.934305i	$-0.616021\pi$
$389$	−14.0615	−0.712948	−0.356474	+	0.934305i	$0.616021\pi$
$390$	0	0
$391$	−0.535482	−0.0270805
$392$	0	0
$393$	−2.99386	−0.151020
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13.9662	−0.700945	−0.350473	−	0.936573i	$-0.613979\pi$
$397$	−13.9662	−0.700945	−0.350473	+	0.936573i	$0.613979\pi$
$398$	0	0
$399$	−29.8634	−1.49504
$400$	0	0
$401$	35.3183	1.76371	0.881855	−	0.471521i	$-0.156295\pi$
$401$	35.3183	1.76371	0.881855	+	0.471521i	$0.156295\pi$
$402$	0	0
$403$	−23.5154	−1.17139
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.3669	−0.910414
$408$	0	0
$409$	17.7067	0.875541	0.437770	−	0.899087i	$-0.355768\pi$
$409$	17.7067	0.875541	0.437770	+	0.899087i	$0.355768\pi$
$410$	0	0
$411$	−17.5528	−0.865816
$412$	0	0
$413$	30.2571	1.48885
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−17.8735	−0.875268
$418$	0	0
$419$	−27.6432	−1.35046	−0.675229	−	0.737608i	$-0.735955\pi$
$419$	−27.6432	−1.35046	−0.675229	+	0.737608i	$0.735955\pi$
$420$	0	0
$421$	−15.3867	−0.749902	−0.374951	−	0.927045i	$-0.622340\pi$
$421$	−15.3867	−0.749902	−0.374951	+	0.927045i	$0.622340\pi$
$422$	0	0
$423$	12.4159	0.603682
$424$	0	0
$425$	0	0
$426$	0	0
$427$	25.8394	1.25046
$428$	0	0
$429$	−5.11227	−0.246823
$430$	0	0
$431$	28.0437	1.35082	0.675409	−	0.737443i	$-0.263967\pi$
$431$	28.0437	1.35082	0.675409	+	0.737443i	$0.263967\pi$
$432$	0	0
$433$	10.5613	0.507544	0.253772	−	0.967264i	$-0.418329\pi$
$433$	10.5613	0.507544	0.253772	+	0.967264i	$0.418329\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.91942	0.331001
$438$	0	0
$439$	16.9599	0.809452	0.404726	−	0.914438i	$-0.367367\pi$
$439$	16.9599	0.809452	0.404726	+	0.914438i	$0.367367\pi$
$440$	0	0
$441$	11.6269	0.553661
$442$	0	0
$443$	−7.15493	−0.339941	−0.169970	−	0.985449i	$-0.554367\pi$
$443$	−7.15493	−0.339941	−0.169970	+	0.985449i	$0.554367\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.71607	0.175764
$448$	0	0
$449$	14.2411	0.672079	0.336039	−	0.941848i	$-0.390912\pi$
$449$	14.2411	0.672079	0.336039	+	0.941848i	$0.390912\pi$
$450$	0	0
$451$	−1.43481	−0.0675626
$452$	0	0
$453$	14.3527	0.674347
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.9658	1.16785	0.583926	−	0.811807i	$-0.301516\pi$
$457$	24.9658	1.16785	0.583926	+	0.811807i	$0.301516\pi$
$458$	0	0
$459$	0.535482	0.0249942
$460$	0	0
$461$	5.88444	0.274066	0.137033	−	0.990566i	$-0.456243\pi$
$461$	5.88444	0.274066	0.137033	+	0.990566i	$0.456243\pi$
$462$	0	0
$463$	10.4047	0.483546	0.241773	−	0.970333i	$-0.422271\pi$
$463$	10.4047	0.483546	0.241773	+	0.970333i	$0.422271\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−25.8260	−1.19508	−0.597541	−	0.801838i	$-0.703856\pi$
$467$	−25.8260	−1.19508	−0.597541	+	0.801838i	$0.703856\pi$
$468$	0	0
$469$	27.3030	1.26073
$470$	0	0
$471$	−9.94566	−0.458272
$472$	0	0
$473$	2.96160	0.136174
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.98659	0.274107
$478$	0	0
$479$	−19.2144	−0.877930	−0.438965	−	0.898504i	$-0.644655\pi$
$479$	−19.2144	−0.877930	−0.438965	+	0.898504i	$0.644655\pi$
$480$	0	0
$481$	18.2837	0.833664
$482$	0	0
$483$	−4.31589	−0.196380
$484$	0	0
$485$	0	0
$486$	0	0
$487$	33.4334	1.51501	0.757506	−	0.652828i	$-0.226417\pi$
$487$	33.4334	1.51501	0.757506	+	0.652828i	$0.226417\pi$
$488$	0	0
$489$	−9.87216	−0.446435
$490$	0	0
$491$	−5.17956	−0.233750	−0.116875	−	0.993147i	$-0.537288\pi$
$491$	−5.17956	−0.233750	−0.116875	+	0.993147i	$0.537288\pi$
$492$	0	0
$493$	3.02332	0.136164
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.652087	0.0292501
$498$	0	0
$499$	10.4240	0.466641	0.233320	−	0.972400i	$-0.425041\pi$
$499$	10.4240	0.466641	0.233320	+	0.972400i	$0.425041\pi$
$500$	0	0
$501$	2.92476	0.130669
$502$	0	0
$503$	33.7019	1.50269	0.751346	−	0.659908i	$-0.229405\pi$
$503$	33.7019	1.50269	0.751346	+	0.659908i	$0.229405\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7.91090	−0.351335
$508$	0	0
$509$	−23.7289	−1.05177	−0.525883	−	0.850557i	$-0.676265\pi$
$509$	−23.7289	−1.05177	−0.525883	+	0.850557i	$0.676265\pi$
$510$	0	0
$511$	38.3582	1.69687
$512$	0	0
$513$	−6.91942	−0.305500
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−28.1366	−1.23744
$518$	0	0
$519$	0.421083	0.0184835
$520$	0	0
$521$	11.0975	0.486189	0.243095	−	0.970003i	$-0.421837\pi$
$521$	11.0975	0.486189	0.243095	+	0.970003i	$0.421837\pi$
$522$	0	0
$523$	−10.9113	−0.477117	−0.238558	−	0.971128i	$-0.576675\pi$
$523$	−10.9113	−0.477117	−0.238558	+	0.971128i	$0.576675\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.58185	−0.243149
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	7.01063	0.304235
$532$	0	0
$533$	1.42831	0.0618669
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14.3955	0.621212
$538$	0	0
$539$	−26.3485	−1.13491
$540$	0	0
$541$	16.1780	0.695548	0.347774	−	0.937578i	$-0.386938\pi$
$541$	16.1780	0.695548	0.347774	+	0.937578i	$0.386938\pi$
$542$	0	0
$543$	7.54278	0.323692
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−24.2259	−1.03583	−0.517913	−	0.855434i	$-0.673291\pi$
$547$	−24.2259	−1.03583	−0.517913	+	0.855434i	$0.673291\pi$
$548$	0	0
$549$	5.98705	0.255521
$550$	0	0
$551$	−39.0669	−1.66431
$552$	0	0
$553$	−44.5882	−1.89608
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.7973	0.881211	0.440606	−	0.897701i	$-0.354764\pi$
$557$	20.7973	0.881211	0.440606	+	0.897701i	$0.354764\pi$
$558$	0	0
$559$	−2.94818	−0.124695
$560$	0	0
$561$	−1.21350	−0.0512338
$562$	0	0
$563$	−10.4382	−0.439918	−0.219959	−	0.975509i	$-0.570592\pi$
$563$	−10.4382	−0.439918	−0.219959	+	0.975509i	$0.570592\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.31589	0.181250
$568$	0	0
$569$	−12.8974	−0.540687	−0.270344	−	0.962764i	$-0.587137\pi$
$569$	−12.8974	−0.540687	−0.270344	+	0.962764i	$0.587137\pi$
$570$	0	0
$571$	−4.59496	−0.192293	−0.0961465	−	0.995367i	$-0.530652\pi$
$571$	−4.59496	−0.192293	−0.0961465	+	0.995367i	$0.530652\pi$
$572$	0	0
$573$	18.9141	0.790147
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−20.9660	−0.872825	−0.436413	−	0.899747i	$-0.643751\pi$
$577$	−20.9660	−0.872825	−0.436413	+	0.899747i	$0.643751\pi$
$578$	0	0
$579$	3.56998	0.148363
$580$	0	0
$581$	−3.09425	−0.128371
$582$	0	0
$583$	−13.5667	−0.561873
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.60723	0.272710	0.136355	−	0.990660i	$-0.456461\pi$
$587$	6.60723	0.272710	0.136355	+	0.990660i	$0.456461\pi$
$588$	0	0
$589$	72.1277	2.97197
$590$	0	0
$591$	−1.90677	−0.0784340
$592$	0	0
$593$	−4.94467	−0.203053	−0.101527	−	0.994833i	$-0.532373\pi$
$593$	−4.94467	−0.203053	−0.101527	+	0.994833i	$0.532373\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.84829	0.0756456
$598$	0	0
$599$	−21.1634	−0.864712	−0.432356	−	0.901703i	$-0.642318\pi$
$599$	−21.1634	−0.864712	−0.432356	+	0.901703i	$0.642318\pi$
$600$	0	0
$601$	−6.65198	−0.271340	−0.135670	−	0.990754i	$-0.543319\pi$
$601$	−6.65198	−0.271340	−0.135670	+	0.990754i	$0.543319\pi$
$602$	0	0
$603$	6.32616	0.257621
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.87232	0.0759953	0.0379976	−	0.999278i	$-0.487902\pi$
$607$	1.87232	0.0759953	0.0379976	+	0.999278i	$0.487902\pi$
$608$	0	0
$609$	24.3674	0.987417
$610$	0	0
$611$	28.0091	1.13313
$612$	0	0
$613$	−29.2195	−1.18017	−0.590083	−	0.807342i	$-0.700905\pi$
$613$	−29.2195	−1.18017	−0.590083	+	0.807342i	$0.700905\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19.3342	−0.778365	−0.389182	−	0.921161i	$-0.627242\pi$
$617$	−19.3342	−0.778365	−0.389182	+	0.921161i	$0.627242\pi$
$618$	0	0
$619$	−44.9507	−1.80672	−0.903360	−	0.428883i	$-0.858907\pi$
$619$	−44.9507	−1.80672	−0.903360	+	0.428883i	$0.858907\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	67.2335	2.69365
$624$	0	0
$625$	0	0
$626$	0	0
$627$	15.6806	0.626223
$628$	0	0
$629$	4.33998	0.173047
$630$	0	0
$631$	−7.86088	−0.312937	−0.156468	−	0.987683i	$-0.550011\pi$
$631$	−7.86088	−0.312937	−0.156468	+	0.987683i	$0.550011\pi$
$632$	0	0
$633$	14.0429	0.558155
$634$	0	0
$635$	0	0
$636$	0	0
$637$	26.2291	1.03924
$638$	0	0
$639$	0.151090	0.00597702
$640$	0	0
$641$	−4.20734	−0.166180	−0.0830900	−	0.996542i	$-0.526479\pi$
$641$	−4.20734	−0.166180	−0.0830900	+	0.996542i	$0.526479\pi$
$642$	0	0
$643$	−28.2418	−1.11375	−0.556874	−	0.830597i	$-0.687999\pi$
$643$	−28.2418	−1.11375	−0.556874	+	0.830597i	$0.687999\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.7760	0.895417	0.447709	−	0.894180i	$-0.352240\pi$
$647$	22.7760	0.895417	0.447709	+	0.894180i	$0.352240\pi$
$648$	0	0
$649$	−15.8873	−0.623631
$650$	0	0
$651$	−44.9886	−1.76324
$652$	0	0
$653$	−13.9316	−0.545187	−0.272593	−	0.962129i	$-0.587881\pi$
$653$	−13.9316	−0.545187	−0.272593	+	0.962129i	$0.587881\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.88768	0.346741
$658$	0	0
$659$	−47.9771	−1.86892	−0.934461	−	0.356065i	$-0.884118\pi$
$659$	−47.9771	−1.86892	−0.934461	+	0.356065i	$0.884118\pi$
$660$	0	0
$661$	−4.19509	−0.163170	−0.0815851	−	0.996666i	$-0.525998\pi$
$661$	−4.19509	−0.163170	−0.0815851	+	0.996666i	$0.525998\pi$
$662$	0	0
$663$	1.20800	0.0469147
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.64598	−0.218613
$668$	0	0
$669$	−11.3081	−0.437196
$670$	0	0
$671$	−13.5677	−0.523775
$672$	0	0
$673$	25.0494	0.965585	0.482793	−	0.875735i	$-0.339622\pi$
$673$	25.0494	0.965585	0.482793	+	0.875735i	$0.339622\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.13989	0.235975	0.117988	−	0.993015i	$-0.462356\pi$
$677$	6.13989	0.235975	0.117988	+	0.993015i	$0.462356\pi$
$678$	0	0
$679$	70.1317	2.69141
$680$	0	0
$681$	26.3506	1.00976
$682$	0	0
$683$	−28.9930	−1.10939	−0.554693	−	0.832055i	$-0.687164\pi$
$683$	−28.9930	−1.10939	−0.554693	+	0.832055i	$0.687164\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−13.7704	−0.525375
$688$	0	0
$689$	13.5052	0.514506
$690$	0	0
$691$	12.4418	0.473309	0.236654	−	0.971594i	$-0.423949\pi$
$691$	12.4418	0.473309	0.236654	+	0.971594i	$0.423949\pi$
$692$	0	0
$693$	−9.78055	−0.371532
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.339037	0.0128419
$698$	0	0
$699$	6.05029	0.228843
$700$	0	0
$701$	−49.4593	−1.86805	−0.934025	−	0.357207i	$-0.883729\pi$
$701$	−49.4593	−1.86805	−0.934025	+	0.357207i	$0.883729\pi$
$702$	0	0
$703$	−56.0806	−2.11512
$704$	0	0
$705$	0	0
$706$	0	0
$707$	55.4280	2.08458
$708$	0	0
$709$	20.5265	0.770889	0.385444	−	0.922731i	$-0.374048\pi$
$709$	20.5265	0.770889	0.385444	+	0.922731i	$0.374048\pi$
$710$	0	0
$711$	−10.3312	−0.387450
$712$	0	0
$713$	10.4240	0.390380
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.2468	−0.569403
$718$	0	0
$719$	3.25057	0.121226	0.0606129	−	0.998161i	$-0.480694\pi$
$719$	3.25057	0.121226	0.0606129	+	0.998161i	$0.480694\pi$
$720$	0	0
$721$	−69.4867	−2.58782
$722$	0	0
$723$	15.3001	0.569015
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−28.9446	−1.07350	−0.536748	−	0.843742i	$-0.680348\pi$
$727$	−28.9446	−1.07350	−0.536748	+	0.843742i	$0.680348\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.699807	−0.0258833
$732$	0	0
$733$	13.8681	0.512232	0.256116	−	0.966646i	$-0.417557\pi$
$733$	13.8681	0.512232	0.256116	+	0.966646i	$0.417557\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.3362	−0.528079
$738$	0	0
$739$	14.4772	0.532552	0.266276	−	0.963897i	$-0.414207\pi$
$739$	14.4772	0.532552	0.266276	+	0.963897i	$0.414207\pi$
$740$	0	0
$741$	−15.6095	−0.573431
$742$	0	0
$743$	−27.4335	−1.00644	−0.503218	−	0.864159i	$-0.667851\pi$
$743$	−27.4335	−1.00644	−0.503218	+	0.864159i	$0.667851\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−0.716944	−0.0262316
$748$	0	0
$749$	57.5460	2.10269
$750$	0	0
$751$	26.1414	0.953913	0.476956	−	0.878927i	$-0.341740\pi$
$751$	26.1414	0.953913	0.476956	+	0.878927i	$0.341740\pi$
$752$	0	0
$753$	−20.4096	−0.743769
$754$	0	0
$755$	0	0
$756$	0	0
$757$	14.8127	0.538379	0.269189	−	0.963087i	$-0.413244\pi$
$757$	14.8127	0.538379	0.269189	+	0.963087i	$0.413244\pi$
$758$	0	0
$759$	2.26617	0.0822569
$760$	0	0
$761$	−14.0330	−0.508695	−0.254348	−	0.967113i	$-0.581861\pi$
$761$	−14.0330	−0.508695	−0.254348	+	0.967113i	$0.581861\pi$
$762$	0	0
$763$	−28.5661	−1.03416
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15.8153	0.571058
$768$	0	0
$769$	−12.8084	−0.461881	−0.230941	−	0.972968i	$-0.574180\pi$
$769$	−12.8084	−0.461881	−0.230941	+	0.972968i	$0.574180\pi$
$770$	0	0
$771$	−14.8072	−0.533269
$772$	0	0
$773$	5.10123	0.183479	0.0917393	−	0.995783i	$-0.470757\pi$
$773$	5.10123	0.183479	0.0917393	+	0.995783i	$0.470757\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	34.9795	1.25488
$778$	0	0
$779$	−4.38098	−0.156965
$780$	0	0
$781$	−0.342396	−0.0122519
$782$	0	0
$783$	5.64598	0.201771
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−38.2717	−1.36424	−0.682119	−	0.731241i	$-0.738941\pi$
$787$	−38.2717	−1.36424	−0.682119	+	0.731241i	$0.738941\pi$
$788$	0	0
$789$	−14.5266	−0.517160
$790$	0	0
$791$	75.6209	2.68877
$792$	0	0
$793$	13.5062	0.479620
$794$	0	0
$795$	0	0
$796$	0	0
$797$	47.8787	1.69595	0.847975	−	0.530037i	$-0.177822\pi$
$797$	47.8787	1.69595	0.847975	+	0.530037i	$0.177822\pi$
$798$	0	0
$799$	6.64849	0.235207
$800$	0	0
$801$	15.5781	0.550426
$802$	0	0
$803$	−20.1410	−0.710761
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16.4759	−0.579978
$808$	0	0
$809$	29.0384	1.02093	0.510467	−	0.859897i	$-0.329472\pi$
$809$	29.0384	1.02093	0.510467	+	0.859897i	$0.329472\pi$
$810$	0	0
$811$	−54.1645	−1.90197	−0.950986	−	0.309235i	$-0.899927\pi$
$811$	−54.1645	−1.90197	−0.950986	+	0.309235i	$0.899927\pi$
$812$	0	0
$813$	22.0764	0.774255
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.04280	0.316367
$818$	0	0
$819$	9.73623	0.340211
$820$	0	0
$821$	23.7322	0.828258	0.414129	−	0.910218i	$-0.364086\pi$
$821$	23.7322	0.828258	0.414129	+	0.910218i	$0.364086\pi$
$822$	0	0
$823$	49.4716	1.72447	0.862236	−	0.506507i	$-0.169064\pi$
$823$	49.4716	1.72447	0.862236	+	0.506507i	$0.169064\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.3893	1.26538	0.632691	−	0.774405i	$-0.281951\pi$
$827$	36.3893	1.26538	0.632691	+	0.774405i	$0.281951\pi$
$828$	0	0
$829$	22.0316	0.765188	0.382594	−	0.923917i	$-0.375031\pi$
$829$	22.0316	0.765188	0.382594	+	0.923917i	$0.375031\pi$
$830$	0	0
$831$	−6.70017	−0.232426
$832$	0	0
$833$	6.22599	0.215718
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−10.4240	−0.360305
$838$	0	0
$839$	−24.4980	−0.845765	−0.422882	−	0.906185i	$-0.638982\pi$
$839$	−24.4980	−0.845765	−0.422882	+	0.906185i	$0.638982\pi$
$840$	0	0
$841$	2.87708	0.0992097
$842$	0	0
$843$	−22.6857	−0.781337
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−25.3104	−0.869674
$848$	0	0
$849$	19.9470	0.684578
$850$	0	0
$851$	−8.10481	−0.277829
$852$	0	0
$853$	−35.3227	−1.20943	−0.604713	−	0.796443i	$-0.706712\pi$
$853$	−35.3227	−1.20943	−0.604713	+	0.796443i	$0.706712\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.2075	1.44178	0.720891	−	0.693049i	$-0.243733\pi$
$857$	42.2075	1.44178	0.720891	+	0.693049i	$0.243733\pi$
$858$	0	0
$859$	−31.3797	−1.07066	−0.535331	−	0.844643i	$-0.679813\pi$
$859$	−31.3797	−1.07066	−0.535331	+	0.844643i	$0.679813\pi$
$860$	0	0
$861$	2.73257	0.0931259
$862$	0	0
$863$	−27.0104	−0.919446	−0.459723	−	0.888062i	$-0.652051\pi$
$863$	−27.0104	−0.919446	−0.459723	+	0.888062i	$0.652051\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.7133	−0.567612
$868$	0	0
$869$	23.4122	0.794206
$870$	0	0
$871$	14.2712	0.483561
$872$	0	0
$873$	16.2497	0.549967
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.74969	−0.126618	−0.0633090	−	0.997994i	$-0.520165\pi$
$877$	−3.74969	−0.126618	−0.0633090	+	0.997994i	$0.520165\pi$
$878$	0	0
$879$	−14.9230	−0.503339
$880$	0	0
$881$	48.1065	1.62075	0.810375	−	0.585912i	$-0.199263\pi$
$881$	48.1065	1.62075	0.810375	+	0.585912i	$0.199263\pi$
$882$	0	0
$883$	−48.6331	−1.63664	−0.818318	−	0.574766i	$-0.805093\pi$
$883$	−48.6331	−1.63664	−0.818318	+	0.574766i	$0.805093\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.9068	−1.23921	−0.619604	−	0.784914i	$-0.712707\pi$
$887$	−36.9068	−1.23921	−0.619604	+	0.784914i	$0.712707\pi$
$888$	0	0
$889$	−57.4016	−1.92519
$890$	0	0
$891$	−2.26617	−0.0759196
$892$	0	0
$893$	−85.9108	−2.87489
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.25590	−0.0753224
$898$	0	0
$899$	−58.8535	−1.96287
$900$	0	0
$901$	3.20572	0.106798
$902$	0	0
$903$	−5.64032	−0.187698
$904$	0	0
$905$	0	0
$906$	0	0
$907$	24.5797	0.816155	0.408078	−	0.912947i	$-0.366199\pi$
$907$	24.5797	0.816155	0.408078	+	0.912947i	$0.366199\pi$
$908$	0	0
$909$	12.8428	0.425968
$910$	0	0
$911$	−4.64580	−0.153922	−0.0769611	−	0.997034i	$-0.524522\pi$
$911$	−4.64580	−0.153922	−0.0769611	+	0.997034i	$0.524522\pi$
$912$	0	0
$913$	1.62472	0.0537703
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−12.9212	−0.426695
$918$	0	0
$919$	23.6616	0.780523	0.390261	−	0.920704i	$-0.372385\pi$
$919$	23.6616	0.780523	0.390261	+	0.920704i	$0.372385\pi$
$920$	0	0
$921$	12.9612	0.427086
$922$	0	0
$923$	0.340844	0.0112190
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.1002	−0.528800
$928$	0	0
$929$	−45.9814	−1.50860	−0.754301	−	0.656529i	$-0.772024\pi$
$929$	−45.9814	−1.50860	−0.754301	+	0.656529i	$0.772024\pi$
$930$	0	0
$931$	−80.4513	−2.63669
$932$	0	0
$933$	−13.0709	−0.427922
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.34076	0.174475	0.0872376	−	0.996188i	$-0.472196\pi$
$937$	5.34076	0.174475	0.0872376	+	0.996188i	$0.472196\pi$
$938$	0	0
$939$	0.224783	0.00733551
$940$	0	0
$941$	−57.4920	−1.87419	−0.937093	−	0.349079i	$-0.886495\pi$
$941$	−57.4920	−1.87419	−0.937093	+	0.349079i	$0.886495\pi$
$942$	0	0
$943$	−0.633143	−0.0206180
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.2375	1.11257	0.556285	−	0.830992i	$-0.312226\pi$
$947$	34.2375	1.11257	0.556285	+	0.830992i	$0.312226\pi$
$948$	0	0
$949$	20.0498	0.650842
$950$	0	0
$951$	−7.03602	−0.228159
$952$	0	0
$953$	0.00135437	4.38723e−5 0	2.19361e−5	−	1.00000i	$-0.499993\pi$
$953$	0.00135437	4.38723e−5 0	2.19361e−5		1.00000i	$0.499993\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−12.7948	−0.413596
$958$	0	0
$959$	−75.7560	−2.44629
$960$	0	0
$961$	77.6589	2.50513
$962$	0	0
$963$	13.3335	0.429667
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−36.9612	−1.18859	−0.594296	−	0.804246i	$-0.702569\pi$
$967$	−36.9612	−1.18859	−0.594296	+	0.804246i	$0.702569\pi$
$968$	0	0
$969$	−3.70523	−0.119029
$970$	0	0
$971$	39.1406	1.25608	0.628041	−	0.778180i	$-0.283857\pi$
$971$	39.1406	1.25608	0.628041	+	0.778180i	$0.283857\pi$
$972$	0	0
$973$	−77.1399	−2.47299
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.9260	0.605498	0.302749	−	0.953070i	$-0.402096\pi$
$977$	18.9260	0.605498	0.302749	+	0.953070i	$0.402096\pi$
$978$	0	0
$979$	−35.3027	−1.12828
$980$	0	0
$981$	−6.61883	−0.211323
$982$	0	0
$983$	54.1666	1.72765	0.863823	−	0.503795i	$-0.168063\pi$
$983$	54.1666	1.72765	0.863823	+	0.503795i	$0.168063\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	53.5856	1.70565
$988$	0	0
$989$	1.30687	0.0415561
$990$	0	0
$991$	−2.28568	−0.0726069	−0.0363035	−	0.999341i	$-0.511558\pi$
$991$	−2.28568	−0.0726069	−0.0363035	+	0.999341i	$0.511558\pi$
$992$	0	0
$993$	−13.7320	−0.435773
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.29171	−0.0725792	−0.0362896	−	0.999341i	$-0.511554\pi$
$997$	−2.29171	−0.0725792	−0.0362896	+	0.999341i	$0.511554\pi$
$998$	0	0
$999$	8.10481	0.256425

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.bd.1.7		7
5.2	odd	4		1380.2.f.b.829.5	✓	14
5.3	odd	4		1380.2.f.b.829.12	yes	14
5.4	even	2		6900.2.a.bc.1.1		7
15.2	even	4		4140.2.f.c.829.5		14
15.8	even	4		4140.2.f.c.829.6		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.b.829.5	✓	14	5.2	odd	4
1380.2.f.b.829.12	yes	14	5.3	odd	4
4140.2.f.c.829.5		14	15.2	even	4
4140.2.f.c.829.6		14	15.8	even	4
6900.2.a.bc.1.1		7	5.4	even	2
6900.2.a.bd.1.7		7	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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.31589 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.31589 q^{7} +1.00000 q^{9} -2.26617 q^{11} +2.25590 q^{13} +0.535482 q^{17} -6.91942 q^{19} +4.31589 q^{21} -1.00000 q^{23} +1.00000 q^{27} +5.64598 q^{29} -10.4240 q^{31} -2.26617 q^{33} +8.10481 q^{37} +2.25590 q^{39} +0.633143 q^{41} -1.30687 q^{43} +12.4159 q^{47} +11.6269 q^{49} +0.535482 q^{51} +5.98659 q^{53} -6.91942 q^{57} +7.01063 q^{59} +5.98705 q^{61} +4.31589 q^{63} +6.32616 q^{67} -1.00000 q^{69} +0.151090 q^{71} +8.88768 q^{73} -9.78055 q^{77} -10.3312 q^{79} +1.00000 q^{81} -0.716944 q^{83} +5.64598 q^{87} +15.5781 q^{89} +9.73623 q^{91} -10.4240 q^{93} +16.2497 q^{97} -2.26617 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} - q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 - q^7 + 7 * q^9	\( 7 q + 7 q^{3} - q^{7} + 7 q^{9} - 2 q^{13} + q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 7 q^{27} + 15 q^{29} + 3 q^{31} + 19 q^{37} - 2 q^{39} + 23 q^{41} - 4 q^{43} + 10 q^{47} + 10 q^{49} + q^{51} + 11 q^{53} - 2 q^{57} + 5 q^{59} + 32 q^{61} - q^{63} + 15 q^{67} - 7 q^{69} + 21 q^{71} - 18 q^{73} + 28 q^{77} + 16 q^{79} + 7 q^{81} + 25 q^{83} + 15 q^{87} + 26 q^{89} + 14 q^{91} + 3 q^{93} + 6 q^{97}+O(q^{100}) \) 7 * q + 7 * q^3 - q^7 + 7 * q^9 - 2 * q^13 + q^17 - 2 * q^19 - q^21 - 7 * q^23 + 7 * q^27 + 15 * q^29 + 3 * q^31 + 19 * q^37 - 2 * q^39 + 23 * q^41 - 4 * q^43 + 10 * q^47 + 10 * q^49 + q^51 + 11 * q^53 - 2 * q^57 + 5 * q^59 + 32 * q^61 - q^63 + 15 * q^67 - 7 * q^69 + 21 * q^71 - 18 * q^73 + 28 * q^77 + 16 * q^79 + 7 * q^81 + 25 * q^83 + 15 * q^87 + 26 * q^89 + 14 * q^91 + 3 * q^93 + 6 * q^97

Introduction

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