LMFDB - Embedded newform 2352.3.m.j.1471.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(1471,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2352.1471");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.m (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:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{37})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 10x^{2} + 9x + 81 $ x^4 - x^3 + 10x^2 + 9x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.4
Root			$-1.27069 + 2.20090i$ of defining polynomial
Character	$\chi$	$=$	2352.1471
Dual form			2352.3.m.j.1471.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +9.08276 q^{5} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +9.08276 q^{5} -3.00000 q^{9} +8.80360i q^{11} +15.7318i q^{15} -2.91724 q^{17} -17.6072i q^{19} -29.5882i q^{23} +57.4966 q^{25} -5.19615i q^{27} +48.4966 q^{29} -38.3918i q^{31} -15.2483 q^{33} -20.4966 q^{37} +26.9172 q^{41} +27.7128i q^{43} -27.2483 q^{45} +62.9272i q^{47} -5.05280i q^{51} +84.4966 q^{53} +79.9610i q^{55} +30.4966 q^{57} +42.1426i q^{59} +102.497 q^{61} +52.8216i q^{67} +51.2483 q^{69} +64.8026i q^{71} +30.4966 q^{73} +99.5870i q^{75} -149.816i q^{79} +9.00000 q^{81} -0.573396i q^{83} -26.4966 q^{85} +83.9985i q^{87} -10.0759 q^{89} +66.4966 q^{93} -159.922i q^{95} +113.503 q^{97} -26.4108i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^5 - 12 * q^9	$ 4 q + 12 q^{5} - 12 q^{9} - 36 q^{17} + 84 q^{25} + 48 q^{29} + 12 q^{33} + 64 q^{37} + 132 q^{41} - 36 q^{45} + 192 q^{53} - 24 q^{57} + 264 q^{61} + 132 q^{69} - 24 q^{73} + 36 q^{81} + 40 q^{85} + 276 q^{89} + 120 q^{93} + 600 q^{97}+O(q^{100}) $ 4 * q + 12 * q^5 - 12 * q^9 - 36 * q^17 + 84 * q^25 + 48 * q^29 + 12 * q^33 + 64 * q^37 + 132 * q^41 - 36 * q^45 + 192 * q^53 - 24 * q^57 + 264 * q^61 + 132 * q^69 - 24 * q^73 + 36 * q^81 + 40 * q^85 + 276 * q^89 + 120 * q^93 + 600 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	9.08276	1.81655	0.908276	−	0.418371i	$-0.137399\pi$
$5$	9.08276	1.81655	0.908276	+	0.418371i	$0.137399\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	8.80360i	0.800328i	0.916444	+	0.400164i	$0.131047\pi$
$11$	8.80360i	0.800328i	−0.916444	+	0.400164i	$0.868953\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	15.7318i	1.04879i
$16$	0	0
$17$	−2.91724	−0.171602	−0.0858011	−	0.996312i	$-0.527345\pi$
$17$	−2.91724	−0.171602	−0.0858011	+	0.996312i	$0.527345\pi$
$18$	0	0
$19$	− 17.6072i	− 0.926695i	−0.886177	−	0.463348i	$-0.846648\pi$
$19$	− 17.6072i	− 0.926695i	0.886177	−	0.463348i	$-0.153352\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 29.5882i	− 1.28644i	−0.765680	−	0.643222i	$-0.777597\pi$
$23$	− 29.5882i	− 1.28644i	0.765680	−	0.643222i	$-0.222403\pi$
$24$	0	0
$25$	57.4966	2.29986
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	48.4966	1.67230	0.836148	−	0.548504i	$-0.184803\pi$
$29$	48.4966	1.67230	0.836148	+	0.548504i	$0.184803\pi$
$30$	0	0
$31$	− 38.3918i	− 1.23845i	−0.785215	−	0.619223i	$-0.787448\pi$
$31$	− 38.3918i	− 1.23845i	0.785215	−	0.619223i	$-0.212552\pi$
$32$	0	0
$33$	−15.2483	−0.462069
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−20.4966	−0.553961	−0.276981	−	0.960875i	$-0.589334\pi$
$37$	−20.4966	−0.553961	−0.276981	+	0.960875i	$0.589334\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	26.9172	0.656518	0.328259	−	0.944588i	$-0.393538\pi$
$41$	26.9172	0.656518	0.328259	+	0.944588i	$0.393538\pi$
$42$	0	0
$43$	27.7128i	0.644484i	0.946657	+	0.322242i	$0.104436\pi$
$43$	27.7128i	0.644484i	−0.946657	+	0.322242i	$0.895564\pi$
$44$	0	0
$45$	−27.2483	−0.605518
$46$	0	0
$47$	62.9272i	1.33888i	0.742867	+	0.669439i	$0.233465\pi$
$47$	62.9272i	1.33888i	−0.742867	+	0.669439i	$0.766535\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 5.05280i	− 0.0990746i
$52$	0	0
$53$	84.4966	1.59428	0.797138	−	0.603798i	$-0.206347\pi$
$53$	84.4966	1.59428	0.797138	+	0.603798i	$0.206347\pi$
$54$	0	0
$55$	79.9610i	1.45384i
$56$	0	0
$57$	30.4966	0.535028
$58$	0	0
$59$	42.1426i	0.714282i	0.934051	+	0.357141i	$0.116248\pi$
$59$	42.1426i	0.714282i	−0.934051	+	0.357141i	$0.883752\pi$
$60$	0	0
$61$	102.497	1.68027	0.840136	−	0.542376i	$-0.182475\pi$
$61$	102.497	1.68027	0.840136	+	0.542376i	$0.182475\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	52.8216i	0.788382i	0.919028	+	0.394191i	$0.128975\pi$
$67$	52.8216i	0.788382i	−0.919028	+	0.394191i	$0.871025\pi$
$68$	0	0
$69$	51.2483	0.742729
$70$	0	0
$71$	64.8026i	0.912713i	0.889797	+	0.456357i	$0.150846\pi$
$71$	64.8026i	0.912713i	−0.889797	+	0.456357i	$0.849154\pi$
$72$	0	0
$73$	30.4966	0.417761	0.208881	−	0.977941i	$-0.433018\pi$
$73$	30.4966	0.417761	0.208881	+	0.977941i	$0.433018\pi$
$74$	0	0
$75$	99.5870i	1.32783i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 149.816i	− 1.89641i	−0.317658	−	0.948205i	$-0.602896\pi$
$79$	− 149.816i	− 1.89641i	0.317658	−	0.948205i	$-0.397104\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 0.573396i	− 0.00690838i	−0.999994	−	0.00345419i	$-0.998900\pi$
$83$	− 0.573396i	− 0.00690838i	0.999994	−	0.00345419i	$-0.00109951\pi$
$84$	0	0
$85$	−26.4966	−0.311724
$86$	0	0
$87$	83.9985i	0.965500i
$88$	0	0
$89$	−10.0759	−0.113213	−0.0566063	−	0.998397i	$-0.518028\pi$
$89$	−10.0759	−0.113213	−0.0566063	+	0.998397i	$0.518028\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	66.4966	0.715017
$94$	0	0
$95$	− 159.922i	− 1.68339i
$96$	0	0
$97$	113.503	1.17014	0.585069	−	0.810983i	$-0.301067\pi$
$97$	113.503	1.17014	0.585069	+	0.810983i	$0.301067\pi$
$98$	0	0
$99$	− 26.4108i	− 0.266776i
$100$	0	0
$101$	−142.076	−1.40669	−0.703346	−	0.710848i	$-0.748312\pi$
$101$	−142.076	−1.40669	−0.703346	+	0.710848i	$0.748312\pi$
$102$	0	0
$103$	− 185.604i	− 1.80198i	−0.433837	−	0.900992i	$-0.642840\pi$
$103$	− 185.604i	− 1.80198i	0.433837	−	0.900992i	$-0.357160\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	120.802i	1.12899i	0.825437	+	0.564494i	$0.190929\pi$
$107$	120.802i	1.12899i	−0.825437	+	0.564494i	$0.809071\pi$
$108$	0	0
$109$	−142.993	−1.31186	−0.655932	−	0.754820i	$-0.727724\pi$
$109$	−142.993	−1.31186	−0.655932	+	0.754820i	$0.727724\pi$
$110$	0	0
$111$	− 35.5011i	− 0.319830i
$112$	0	0
$113$	−114.993	−1.01764	−0.508819	−	0.860873i	$-0.669918\pi$
$113$	−114.993	−1.01764	−0.508819	+	0.860873i	$0.669918\pi$
$114$	0	0
$115$	− 268.743i	− 2.33689i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	43.4966	0.359476
$122$	0	0
$123$	46.6220i	0.379041i
$124$	0	0
$125$	295.159	2.36127
$126$	0	0
$127$	− 38.9652i	− 0.306813i	−0.988163	−	0.153406i	$-0.950976\pi$
$127$	− 38.9652i	− 0.306813i	0.988163	−	0.153406i	$-0.0490243\pi$
$128$	0	0
$129$	−48.0000	−0.372093
$130$	0	0
$131$	187.061i	1.42795i	0.700171	+	0.713975i	$0.253107\pi$
$131$	187.061i	1.42795i	−0.700171	+	0.713975i	$0.746893\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 47.1954i	− 0.349596i
$136$	0	0
$137$	−139.986	−1.02180	−0.510899	−	0.859641i	$-0.670687\pi$
$137$	−139.986	−1.02180	−0.510899	+	0.859641i	$0.670687\pi$
$138$	0	0
$139$	209.566i	1.50767i	0.657063	+	0.753836i	$0.271798\pi$
$139$	209.566i	1.50767i	−0.657063	+	0.753836i	$0.728202\pi$
$140$	0	0
$141$	−108.993	−0.773001
$142$	0	0
$143$	0	0
$144$	0	0
$145$	440.483	3.03781
$146$	0	0
$147$	0	0
$148$	0	0
$149$	84.4966	0.567091	0.283546	−	0.958959i	$-0.408489\pi$
$149$	84.4966	0.567091	0.283546	+	0.958959i	$0.408489\pi$
$150$	0	0
$151$	− 238.999i	− 1.58278i	−0.611314	−	0.791388i	$-0.709359\pi$
$151$	− 238.999i	− 1.58278i	0.611314	−	0.791388i	$-0.290641\pi$
$152$	0	0
$153$	8.75171	0.0572007
$154$	0	0
$155$	− 348.704i	− 2.24970i
$156$	0	0
$157$	80.4829	0.512630	0.256315	−	0.966593i	$-0.417492\pi$
$157$	80.4829	0.512630	0.256315	+	0.966593i	$0.417492\pi$
$158$	0	0
$159$	146.352i	0.920455i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	297.029i	1.82226i	0.412115	+	0.911132i	$0.364790\pi$
$163$	297.029i	1.82226i	−0.412115	+	0.911132i	$0.635210\pi$
$164$	0	0
$165$	−138.497	−0.839373
$166$	0	0
$167$	229.204i	1.37248i	0.727375	+	0.686240i	$0.240740\pi$
$167$	229.204i	1.37248i	−0.727375	+	0.686240i	$0.759260\pi$
$168$	0	0
$169$	−169.000	−1.00000
$170$	0	0
$171$	52.8216i	0.308898i
$172$	0	0
$173$	206.917	1.19605	0.598027	−	0.801476i	$-0.295952\pi$
$173$	206.917	1.19605	0.598027	+	0.801476i	$0.295952\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−72.9932	−0.412391
$178$	0	0
$179$	− 56.7276i	− 0.316914i	−0.987366	−	0.158457i	$-0.949348\pi$
$179$	− 56.7276i	− 0.316914i	0.987366	−	0.158457i	$-0.0506519\pi$
$180$	0	0
$181$	22.0137	0.121623	0.0608113	−	0.998149i	$-0.480631\pi$
$181$	22.0137	0.121623	0.0608113	+	0.998149i	$0.480631\pi$
$182$	0	0
$183$	177.529i	0.970105i
$184$	0	0
$185$	−186.166	−1.00630
$186$	0	0
$187$	− 25.6822i	− 0.137338i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 140.129i	− 0.733660i	−0.930288	−	0.366830i	$-0.880443\pi$
$191$	− 140.129i	− 0.733660i	0.930288	−	0.366830i	$-0.119557\pi$
$192$	0	0
$193$	−149.490	−0.774558	−0.387279	−	0.921963i	$-0.626585\pi$
$193$	−149.490	−0.774558	−0.387279	+	0.921963i	$0.626585\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	106.510	0.540661	0.270331	−	0.962768i	$-0.412867\pi$
$197$	106.510	0.540661	0.270331	+	0.962768i	$0.412867\pi$
$198$	0	0
$199$	− 111.998i	− 0.562804i	−0.959590	−	0.281402i	$-0.909201\pi$
$199$	− 111.998i	− 0.562804i	0.959590	−	0.281402i	$-0.0907995\pi$
$200$	0	0
$201$	−91.4897	−0.455173
$202$	0	0
$203$	0	0
$204$	0	0
$205$	244.483	1.19260
$206$	0	0
$207$	88.7646i	0.428815i
$208$	0	0
$209$	155.007	0.741660
$210$	0	0
$211$	105.643i	0.500679i	0.968158	+	0.250339i	$0.0805422\pi$
$211$	105.643i	0.500679i	−0.968158	+	0.250339i	$0.919458\pi$
$212$	0	0
$213$	−112.241	−0.526955
$214$	0	0
$215$	251.709i	1.17074i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	52.8216i	0.241195i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	255.770i	1.14695i	0.819223	+	0.573476i	$0.194405\pi$
$223$	255.770i	1.14695i	−0.819223	+	0.573476i	$0.805595\pi$
$224$	0	0
$225$	−172.490	−0.766621
$226$	0	0
$227$	− 81.4183i	− 0.358671i	−0.983788	−	0.179335i	$-0.942605\pi$
$227$	− 81.4183i	− 0.358671i	0.983788	−	0.179335i	$-0.0573947\pi$
$228$	0	0
$229$	−38.9795	−0.170216	−0.0851080	−	0.996372i	$-0.527124\pi$
$229$	−38.9795	−0.170216	−0.0851080	+	0.996372i	$0.527124\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−306.000	−1.31330	−0.656652	−	0.754193i	$-0.728028\pi$
$233$	−306.000	−1.31330	−0.656652	+	0.754193i	$0.728028\pi$
$234$	0	0
$235$	571.553i	2.43214i
$236$	0	0
$237$	259.490	1.09489
$238$	0	0
$239$	− 243.789i	− 1.02004i	−0.860163	−	0.510019i	$-0.829638\pi$
$239$	− 243.789i	− 1.02004i	0.860163	−	0.510019i	$-0.170362\pi$
$240$	0	0
$241$	−357.476	−1.48330	−0.741652	−	0.670785i	$-0.765957\pi$
$241$	−357.476	−1.48330	−0.741652	+	0.670785i	$0.765957\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.993150	0.00398856
$250$	0	0
$251$	− 189.355i	− 0.754403i	−0.926131	−	0.377201i	$-0.876886\pi$
$251$	− 189.355i	− 0.754403i	0.926131	−	0.377201i	$-0.123114\pi$
$252$	0	0
$253$	260.483	1.02958
$254$	0	0
$255$	− 45.8934i	− 0.179974i
$256$	0	0
$257$	207.910	0.808990	0.404495	−	0.914540i	$-0.367447\pi$
$257$	207.910	0.808990	0.404495	+	0.914540i	$0.367447\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−145.490	−0.557432
$262$	0	0
$263$	− 237.434i	− 0.902792i	−0.892324	−	0.451396i	$-0.850926\pi$
$263$	− 237.434i	− 0.902792i	0.892324	−	0.451396i	$-0.149074\pi$
$264$	0	0
$265$	767.462	2.89608
$266$	0	0
$267$	− 17.4520i	− 0.0653633i
$268$	0	0
$269$	83.0691	0.308807	0.154403	−	0.988008i	$-0.450654\pi$
$269$	83.0691	0.308807	0.154403	+	0.988008i	$0.450654\pi$
$270$	0	0
$271$	239.883i	0.885178i	0.896725	+	0.442589i	$0.145940\pi$
$271$	239.883i	0.885178i	−0.896725	+	0.442589i	$0.854060\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	506.177i	1.84064i
$276$	0	0
$277$	78.4829	0.283332	0.141666	−	0.989915i	$-0.454754\pi$
$277$	78.4829	0.283332	0.141666	+	0.989915i	$0.454754\pi$
$278$	0	0
$279$	115.175i	0.412815i
$280$	0	0
$281$	380.979	1.35580	0.677899	−	0.735155i	$-0.262890\pi$
$281$	380.979	1.35580	0.677899	+	0.735155i	$0.262890\pi$
$282$	0	0
$283$	− 49.3812i	− 0.174492i	−0.996187	−	0.0872460i	$-0.972193\pi$
$283$	− 49.3812i	− 0.174492i	0.996187	−	0.0872460i	$-0.0278066\pi$
$284$	0	0
$285$	276.993	0.971906
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−280.490	−0.970553
$290$	0	0
$291$	196.594i	0.675580i
$292$	0	0
$293$	217.924	0.743768	0.371884	−	0.928279i	$-0.378712\pi$
$293$	217.924	0.743768	0.371884	+	0.928279i	$0.378712\pi$
$294$	0	0
$295$	382.771i	1.29753i
$296$	0	0
$297$	45.7449	0.154023
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 246.083i	− 0.812154i
$304$	0	0
$305$	930.952	3.05230
$306$	0	0
$307$	− 164.820i	− 0.536872i	−0.963298	−	0.268436i	$-0.913493\pi$
$307$	− 164.820i	− 0.536872i	0.963298	−	0.268436i	$-0.0865068\pi$
$308$	0	0
$309$	321.476	1.04038
$310$	0	0
$311$	353.912i	1.13798i	0.822344	+	0.568990i	$0.192666\pi$
$311$	353.912i	1.13798i	−0.822344	+	0.568990i	$0.807334\pi$
$312$	0	0
$313$	265.986	0.849796	0.424898	−	0.905241i	$-0.360310\pi$
$313$	265.986	0.849796	0.424898	+	0.905241i	$0.360310\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−292.469	−0.922616	−0.461308	−	0.887240i	$-0.652620\pi$
$317$	−292.469	−0.922616	−0.461308	+	0.887240i	$0.652620\pi$
$318$	0	0
$319$	426.945i	1.33838i
$320$	0	0
$321$	−209.235	−0.651821
$322$	0	0
$323$	51.3644i	0.159023i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 247.671i	− 0.757405i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 538.632i	− 1.62729i	−0.581364	−	0.813644i	$-0.697481\pi$
$331$	− 538.632i	− 1.62729i	0.581364	−	0.813644i	$-0.302519\pi$
$332$	0	0
$333$	61.4897	0.184654
$334$	0	0
$335$	479.766i	1.43214i
$336$	0	0
$337$	135.986	0.403520	0.201760	−	0.979435i	$-0.435334\pi$
$337$	135.986	0.403520	0.201760	+	0.979435i	$0.435334\pi$
$338$	0	0
$339$	− 199.174i	− 0.587534i
$340$	0	0
$341$	337.986	0.991162
$342$	0	0
$343$	0	0
$344$	0	0
$345$	465.476	1.34921
$346$	0	0
$347$	− 312.498i	− 0.900570i	−0.892885	−	0.450285i	$-0.851322\pi$
$347$	− 312.498i	− 0.900570i	0.892885	−	0.450285i	$-0.148678\pi$
$348$	0	0
$349$	−407.462	−1.16751	−0.583757	−	0.811928i	$-0.698418\pi$
$349$	−407.462	−1.16751	−0.583757	+	0.811928i	$0.698418\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	154.076	0.436476	0.218238	−	0.975896i	$-0.429969\pi$
$353$	154.076	0.436476	0.218238	+	0.975896i	$0.429969\pi$
$354$	0	0
$355$	588.587i	1.65799i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	413.506i	1.15183i	0.817510	+	0.575914i	$0.195354\pi$
$359$	413.506i	1.15183i	−0.817510	+	0.575914i	$0.804646\pi$
$360$	0	0
$361$	50.9863	0.141236
$362$	0	0
$363$	75.3383i	0.207543i
$364$	0	0
$365$	276.993	0.758885
$366$	0	0
$367$	639.688i	1.74302i	0.490378	+	0.871510i	$0.336859\pi$
$367$	639.688i	1.74302i	−0.490378	+	0.871510i	$0.663141\pi$
$368$	0	0
$369$	−80.7517	−0.218839
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−265.973	−0.713063	−0.356532	−	0.934283i	$-0.616041\pi$
$373$	−265.973	−0.713063	−0.356532	+	0.934283i	$0.616041\pi$
$374$	0	0
$375$	511.230i	1.36328i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	32.9208i	0.0868623i	0.999056	+	0.0434312i	$0.0138289\pi$
$379$	32.9208i	0.0868623i	−0.999056	+	0.0434312i	$0.986171\pi$
$380$	0	0
$381$	67.4897	0.177138
$382$	0	0
$383$	− 673.135i	− 1.75753i	−0.477252	−	0.878766i	$-0.658367\pi$
$383$	− 673.135i	− 1.75753i	0.477252	−	0.878766i	$-0.341633\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 83.1384i	− 0.214828i
$388$	0	0
$389$	236.524	0.608031	0.304015	−	0.952667i	$-0.401673\pi$
$389$	236.524	0.608031	0.304015	+	0.952667i	$0.401673\pi$
$390$	0	0
$391$	86.3158i	0.220757i
$392$	0	0
$393$	−324.000	−0.824427
$394$	0	0
$395$	− 1360.75i	− 3.44493i
$396$	0	0
$397$	−695.462	−1.75179	−0.875897	−	0.482498i	$-0.839730\pi$
$397$	−695.462	−1.75179	−0.875897	+	0.482498i	$0.839730\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−184.014	−0.458887	−0.229444	−	0.973322i	$-0.573691\pi$
$401$	−184.014	−0.458887	−0.229444	+	0.973322i	$0.573691\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	81.7449	0.201839
$406$	0	0
$407$	− 180.444i	− 0.443351i
$408$	0	0
$409$	−257.503	−0.629593	−0.314796	−	0.949159i	$-0.601936\pi$
$409$	−257.503	−0.629593	−0.314796	+	0.949159i	$0.601936\pi$
$410$	0	0
$411$	− 242.463i	− 0.589935i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 5.20802i	− 0.0125494i
$416$	0	0
$417$	−362.979	−0.870454
$418$	0	0
$419$	361.366i	0.862448i	0.902245	+	0.431224i	$0.141918\pi$
$419$	361.366i	0.862448i	−0.902245	+	0.431224i	$0.858082\pi$
$420$	0	0
$421$	−191.476	−0.454812	−0.227406	−	0.973800i	$-0.573025\pi$
$421$	−191.476	−0.454812	−0.227406	+	0.973800i	$0.573025\pi$
$422$	0	0
$423$	− 188.782i	− 0.446292i
$424$	0	0
$425$	−167.731	−0.394662
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 299.525i	− 0.694954i	−0.937689	−	0.347477i	$-0.887038\pi$
$431$	− 299.525i	− 0.694954i	0.937689	−	0.347477i	$-0.112962\pi$
$432$	0	0
$433$	448.966	1.03687	0.518436	−	0.855116i	$-0.326514\pi$
$433$	448.966	1.03687	0.518436	+	0.855116i	$0.326514\pi$
$434$	0	0
$435$	762.939i	1.75388i
$436$	0	0
$437$	−520.966	−1.19214
$438$	0	0
$439$	486.647i	1.10854i	0.832339	+	0.554268i	$0.187002\pi$
$439$	486.647i	1.10854i	−0.832339	+	0.554268i	$0.812998\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 367.303i	− 0.829125i	−0.910021	−	0.414563i	$-0.863935\pi$
$443$	− 367.303i	− 0.829125i	0.910021	−	0.414563i	$-0.136065\pi$
$444$	0	0
$445$	−91.5171	−0.205656
$446$	0	0
$447$	146.352i	0.327410i
$448$	0	0
$449$	306.000	0.681514	0.340757	−	0.940151i	$-0.389317\pi$
$449$	306.000	0.681514	0.340757	+	0.940151i	$0.389317\pi$
$450$	0	0
$451$	236.969i	0.525429i
$452$	0	0
$453$	413.959	0.913817
$454$	0	0
$455$	0	0
$456$	0	0
$457$	152.014	0.332634	0.166317	−	0.986072i	$-0.446813\pi$
$457$	152.014	0.332634	0.166317	+	0.986072i	$0.446813\pi$
$458$	0	0
$459$	15.1584i	0.0330249i
$460$	0	0
$461$	103.883	0.225343	0.112671	−	0.993632i	$-0.464059\pi$
$461$	103.883	0.225343	0.112671	+	0.993632i	$0.464059\pi$
$462$	0	0
$463$	− 189.618i	− 0.409542i	−0.978810	−	0.204771i	$-0.934355\pi$
$463$	− 189.618i	− 0.409542i	0.978810	−	0.204771i	$-0.0656450\pi$
$464$	0	0
$465$	603.973	1.29887
$466$	0	0
$467$	187.061i	0.400560i	0.979739	+	0.200280i	$0.0641852\pi$
$467$	187.061i	0.400560i	−0.979739	+	0.200280i	$0.935815\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	139.400i	0.295967i
$472$	0	0
$473$	−243.973	−0.515798
$474$	0	0
$475$	− 1012.35i	− 2.13127i
$476$	0	0
$477$	−253.490	−0.531425
$478$	0	0
$479$	416.839i	0.870228i	0.900375	+	0.435114i	$0.143292\pi$
$479$	416.839i	0.870228i	−0.900375	+	0.435114i	$0.856708\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1030.92	2.12562
$486$	0	0
$487$	− 105.643i	− 0.216927i	−0.994100	−	0.108463i	$-0.965407\pi$
$487$	− 105.643i	− 0.216927i	0.994100	−	0.108463i	$-0.0345930\pi$
$488$	0	0
$489$	−514.469	−1.05208
$490$	0	0
$491$	− 920.412i	− 1.87457i	−0.348569	−	0.937283i	$-0.613332\pi$
$491$	− 920.412i	− 1.87457i	0.348569	−	0.937283i	$-0.386668\pi$
$492$	0	0
$493$	−141.476	−0.286970
$494$	0	0
$495$	− 239.883i	− 0.484612i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 337.762i	− 0.676877i	−0.940988	−	0.338439i	$-0.890101\pi$
$499$	− 337.762i	− 0.676877i	0.940988	−	0.338439i	$-0.109899\pi$
$500$	0	0
$501$	−396.993	−0.792401
$502$	0	0
$503$	− 612.501i	− 1.21770i	−0.793287	−	0.608848i	$-0.791632\pi$
$503$	− 612.501i	− 1.21770i	0.793287	−	0.608848i	$-0.208368\pi$
$504$	0	0
$505$	−1290.44	−2.55533
$506$	0	0
$507$	− 292.717i	− 0.577350i
$508$	0	0
$509$	181.055	0.355708	0.177854	−	0.984057i	$-0.443085\pi$
$509$	181.055	0.355708	0.177854	+	0.984057i	$0.443085\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−91.4897	−0.178343
$514$	0	0
$515$	− 1685.80i	− 3.27340i
$516$	0	0
$517$	−553.986	−1.07154
$518$	0	0
$519$	358.391i	0.690542i
$520$	0	0
$521$	878.049	1.68531	0.842657	−	0.538451i	$-0.180990\pi$
$521$	878.049	1.68531	0.842657	+	0.538451i	$0.180990\pi$
$522$	0	0
$523$	456.067i	0.872021i	0.899941	+	0.436011i	$0.143609\pi$
$523$	456.067i	0.872021i	−0.899941	+	0.436011i	$0.856391\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	111.998i	0.212520i
$528$	0	0
$529$	−346.462	−0.654938
$530$	0	0
$531$	− 126.428i	− 0.238094i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	1097.21i	2.05086i
$536$	0	0
$537$	98.2551	0.182970
$538$	0	0
$539$	0	0
$540$	0	0
$541$	558.483	1.03232	0.516158	−	0.856493i	$-0.327362\pi$
$541$	558.483	1.03232	0.516158	+	0.856493i	$0.327362\pi$
$542$	0	0
$543$	38.1288i	0.0702189i
$544$	0	0
$545$	−1298.77	−2.38307
$546$	0	0
$547$	579.365i	1.05917i	0.848257	+	0.529584i	$0.177652\pi$
$547$	579.365i	1.05917i	−0.848257	+	0.529584i	$0.822348\pi$
$548$	0	0
$549$	−307.490	−0.560091
$550$	0	0
$551$	− 853.889i	− 1.54971i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 322.448i	− 0.580988i
$556$	0	0
$557$	−15.4760	−0.0277846	−0.0138923	−	0.999903i	$-0.504422\pi$
$557$	−15.4760	−0.0277846	−0.0138923	+	0.999903i	$0.504422\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	44.4829	0.0792921
$562$	0	0
$563$	715.278i	1.27048i	0.772317	+	0.635238i	$0.219098\pi$
$563$	715.278i	1.27048i	−0.772317	+	0.635238i	$0.780902\pi$
$564$	0	0
$565$	−1044.46	−1.84859
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−660.952	−1.16160	−0.580801	−	0.814045i	$-0.697261\pi$
$569$	−660.952	−1.16160	−0.580801	+	0.814045i	$0.697261\pi$
$570$	0	0
$571$	− 281.405i	− 0.492828i	−0.969165	−	0.246414i	$-0.920748\pi$
$571$	− 281.405i	− 0.492828i	0.969165	−	0.246414i	$-0.0792523\pi$
$572$	0	0
$573$	242.711	0.423579
$574$	0	0
$575$	− 1701.22i	− 2.95865i
$576$	0	0
$577$	−570.952	−0.989518	−0.494759	−	0.869030i	$-0.664744\pi$
$577$	−570.952	−0.989518	−0.494759	+	0.869030i	$0.664744\pi$
$578$	0	0
$579$	− 258.924i	− 0.447191i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	743.874i	1.27594i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 455.541i	− 0.776050i	−0.921649	−	0.388025i	$-0.873157\pi$
$587$	− 455.541i	− 0.776050i	0.921649	−	0.388025i	$-0.126843\pi$
$588$	0	0
$589$	−675.973	−1.14766
$590$	0	0
$591$	184.481i	0.312151i
$592$	0	0
$593$	−1038.89	−1.75192	−0.875961	−	0.482382i	$-0.839772\pi$
$593$	−1038.89	−1.75192	−0.875961	+	0.482382i	$0.839772\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	193.986	0.324935
$598$	0	0
$599$	− 884.935i	− 1.47735i	−0.674060	−	0.738677i	$-0.735451\pi$
$599$	− 884.935i	− 1.47735i	0.674060	−	0.738677i	$-0.264549\pi$
$600$	0	0
$601$	282.952	0.470802	0.235401	−	0.971898i	$-0.424360\pi$
$601$	282.952	0.470802	0.235401	+	0.971898i	$0.424360\pi$
$602$	0	0
$603$	− 158.465i	− 0.262794i
$604$	0	0
$605$	395.069	0.653007
$606$	0	0
$607$	− 185.341i	− 0.305340i	−0.988277	−	0.152670i	$-0.951213\pi$
$607$	− 185.341i	− 0.305340i	0.988277	−	0.152670i	$-0.0487871\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−849.945	−1.38653	−0.693267	−	0.720681i	$-0.743829\pi$
$613$	−849.945	−1.38653	−0.693267	+	0.720681i	$0.743829\pi$
$614$	0	0
$615$	423.457i	0.688548i
$616$	0	0
$617$	37.0342	0.0600231	0.0300115	−	0.999550i	$-0.490446\pi$
$617$	37.0342	0.0600231	0.0300115	+	0.999550i	$0.490446\pi$
$618$	0	0
$619$	− 554.830i	− 0.896332i	−0.893950	−	0.448166i	$-0.852077\pi$
$619$	− 554.830i	− 0.896332i	0.893950	−	0.448166i	$-0.147923\pi$
$620$	0	0
$621$	−153.745	−0.247576
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1243.44	1.98951
$626$	0	0
$627$	268.480i	0.428197i
$628$	0	0
$629$	59.7934	0.0950610
$630$	0	0
$631$	633.023i	1.00321i	0.865098	+	0.501603i	$0.167256\pi$
$631$	633.023i	1.00321i	−0.865098	+	0.501603i	$0.832744\pi$
$632$	0	0
$633$	−182.979	−0.289067
$634$	0	0
$635$	− 353.912i	− 0.557341i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 194.408i	− 0.304238i
$640$	0	0
$641$	−228.952	−0.357179	−0.178590	−	0.983924i	$-0.557153\pi$
$641$	−228.952	−0.357179	−0.178590	+	0.983924i	$0.557153\pi$
$642$	0	0
$643$	− 724.284i	− 1.12641i	−0.826316	−	0.563207i	$-0.809567\pi$
$643$	− 724.284i	− 1.12641i	0.826316	−	0.563207i	$-0.190433\pi$
$644$	0	0
$645$	−435.973	−0.675927
$646$	0	0
$647$	− 739.503i	− 1.14297i	−0.820612	−	0.571486i	$-0.806367\pi$
$647$	− 739.503i	− 1.14297i	0.820612	−	0.571486i	$-0.193633\pi$
$648$	0	0
$649$	−371.007	−0.571659
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−195.476	−0.299351	−0.149675	−	0.988735i	$-0.547823\pi$
$653$	−195.476	−0.299351	−0.149675	+	0.988735i	$0.547823\pi$
$654$	0	0
$655$	1699.04i	2.59395i
$656$	0	0
$657$	−91.4897	−0.139254
$658$	0	0
$659$	706.737i	1.07244i	0.844079	+	0.536219i	$0.180148\pi$
$659$	706.737i	1.07244i	−0.844079	+	0.536219i	$0.819852\pi$
$660$	0	0
$661$	1210.47	1.83127	0.915635	−	0.402011i	$-0.131689\pi$
$661$	1210.47	1.83127	0.915635	+	0.402011i	$0.131689\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1434.93i	− 2.15131i
$668$	0	0
$669$	−443.007	−0.662193
$670$	0	0
$671$	902.339i	1.34477i
$672$	0	0
$673$	515.449	0.765897	0.382948	−	0.923770i	$-0.374909\pi$
$673$	515.449	0.765897	0.382948	+	0.923770i	$0.374909\pi$
$674$	0	0
$675$	− 298.761i	− 0.442609i
$676$	0	0
$677$	179.856	0.265666	0.132833	−	0.991138i	$-0.457593\pi$
$677$	179.856	0.265666	0.132833	+	0.991138i	$0.457593\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	141.021	0.207079
$682$	0	0
$683$	− 753.609i	− 1.10338i	−0.834049	−	0.551690i	$-0.813983\pi$
$683$	− 753.609i	− 1.10338i	0.834049	−	0.551690i	$-0.186017\pi$
$684$	0	0
$685$	−1271.46	−1.85615
$686$	0	0
$687$	− 67.5144i	− 0.0982742i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 235.511i	− 0.340827i	−0.985373	−	0.170413i	$-0.945490\pi$
$691$	− 235.511i	− 0.340827i	0.985373	−	0.170413i	$-0.0545103\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1903.44i	2.73876i
$696$	0	0
$697$	−78.5240	−0.112660
$698$	0	0
$699$	− 530.008i	− 0.758237i
$700$	0	0
$701$	−195.476	−0.278853	−0.139427	−	0.990232i	$-0.544526\pi$
$701$	−195.476	−0.278853	−0.139427	+	0.990232i	$0.544526\pi$
$702$	0	0
$703$	360.887i	0.513353i
$704$	0	0
$705$	−989.959	−1.40420
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−958.993	−1.35260	−0.676300	−	0.736626i	$-0.736418\pi$
$709$	−958.993	−1.35260	−0.676300	+	0.736626i	$0.736418\pi$
$710$	0	0
$711$	449.449i	0.632137i
$712$	0	0
$713$	−1135.95	−1.59319
$714$	0	0
$715$	0	0
$716$	0	0
$717$	422.255	0.588919
$718$	0	0
$719$	549.574i	0.764359i	0.924088	+	0.382180i	$0.124826\pi$
$719$	549.574i	0.764359i	−0.924088	+	0.382180i	$0.875174\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 619.167i	− 0.856385i
$724$	0	0
$725$	2788.39	3.84605
$726$	0	0
$727$	761.219i	1.04707i	0.852005	+	0.523534i	$0.175387\pi$
$727$	761.219i	1.04707i	−0.852005	+	0.523534i	$0.824613\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 80.8449i	− 0.110595i
$732$	0	0
$733$	−476.027	−0.649423	−0.324712	−	0.945813i	$-0.605267\pi$
$733$	−476.027	−0.649423	−0.324712	+	0.945813i	$0.605267\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−465.021	−0.630964
$738$	0	0
$739$	− 686.681i	− 0.929203i	−0.885520	−	0.464602i	$-0.846198\pi$
$739$	− 686.681i	− 0.929203i	0.885520	−	0.464602i	$-0.153802\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	423.302i	0.569719i	0.958569	+	0.284860i	$0.0919470\pi$
$743$	423.302i	0.569719i	−0.958569	+	0.284860i	$0.908053\pi$
$744$	0	0
$745$	767.462	1.03015
$746$	0	0
$747$	1.72019i	0.00230279i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 748.246i	− 0.996333i	−0.867081	−	0.498166i	$-0.834007\pi$
$751$	− 748.246i	− 0.996333i	0.867081	−	0.498166i	$-0.165993\pi$
$752$	0	0
$753$	327.973	0.435555
$754$	0	0
$755$	− 2170.77i	− 2.87520i
$756$	0	0
$757$	1264.90	1.67093	0.835467	−	0.549540i	$-0.185197\pi$
$757$	1264.90	1.67093	0.835467	+	0.549540i	$0.185197\pi$
$758$	0	0
$759$	451.170i	0.594426i
$760$	0	0
$761$	−28.1170	−0.0369475	−0.0184737	−	0.999829i	$-0.505881\pi$
$761$	−28.1170	−0.0369475	−0.0184737	+	0.999829i	$0.505881\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	79.4897	0.103908
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1334.98	−1.73599	−0.867997	−	0.496569i	$-0.834593\pi$
$769$	−1334.98	−1.73599	−0.867997	+	0.496569i	$0.834593\pi$
$770$	0	0
$771$	360.111i	0.467070i
$772$	0	0
$773$	969.869	1.25468	0.627341	−	0.778745i	$-0.284143\pi$
$773$	969.869	1.25468	0.627341	+	0.778745i	$0.284143\pi$
$774$	0	0
$775$	− 2207.40i	− 2.84826i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 473.937i	− 0.608392i
$780$	0	0
$781$	−570.497	−0.730469
$782$	0	0
$783$	− 251.996i	− 0.321833i
$784$	0	0
$785$	731.007	0.931219
$786$	0	0
$787$	− 1047.83i	− 1.33142i	−0.746208	−	0.665712i	$-0.768128\pi$
$787$	− 1047.83i	− 1.33142i	0.746208	−	0.665712i	$-0.231872\pi$
$788$	0	0
$789$	411.248	0.521227
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	1329.28i	1.67206i
$796$	0	0
$797$	−548.090	−0.687691	−0.343845	−	0.939026i	$-0.611730\pi$
$797$	−548.090	−0.687691	−0.343845	+	0.939026i	$0.611730\pi$
$798$	0	0
$799$	− 183.574i	− 0.229754i
$800$	0	0
$801$	30.2277	0.0377375
$802$	0	0
$803$	268.480i	0.334346i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	143.880i	0.178290i
$808$	0	0
$809$	−203.048	−0.250986	−0.125493	−	0.992094i	$-0.540051\pi$
$809$	−203.048	−0.250986	−0.125493	+	0.992094i	$0.540051\pi$
$810$	0	0
$811$	− 279.469i	− 0.344598i	−0.985045	−	0.172299i	$-0.944880\pi$
$811$	− 279.469i	− 0.344598i	0.985045	−	0.172299i	$-0.0551196\pi$
$812$	0	0
$813$	−415.490	−0.511057
$814$	0	0
$815$	2697.84i	3.31024i
$816$	0	0
$817$	487.945	0.597240
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1201.41	−1.46335	−0.731673	−	0.681655i	$-0.761260\pi$
$821$	−1201.41	−1.46335	−0.731673	+	0.681655i	$0.761260\pi$
$822$	0	0
$823$	731.786i	0.889168i	0.895737	+	0.444584i	$0.146649\pi$
$823$	731.786i	0.889168i	−0.895737	+	0.444584i	$0.853351\pi$
$824$	0	0
$825$	−876.724	−1.06270
$826$	0	0
$827$	− 1164.93i	− 1.40862i	−0.709892	−	0.704311i	$-0.751256\pi$
$827$	− 1164.93i	− 1.40862i	0.709892	−	0.704311i	$-0.248744\pi$
$828$	0	0
$829$	−22.0137	−0.0265545	−0.0132773	−	0.999912i	$-0.504226\pi$
$829$	−22.0137	−0.0265545	−0.0132773	+	0.999912i	$0.504226\pi$
$830$	0	0
$831$	135.936i	0.163582i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2081.81i	2.49318i
$836$	0	0
$837$	−199.490	−0.238339
$838$	0	0
$839$	293.278i	0.349557i	0.984608	+	0.174778i	$0.0559209\pi$
$839$	293.278i	0.349557i	−0.984608	+	0.174778i	$0.944079\pi$
$840$	0	0
$841$	1510.92	1.79657
$842$	0	0
$843$	659.876i	0.782771i
$844$	0	0
$845$	−1534.99	−1.81655
$846$	0	0
$847$	0	0
$848$	0	0
$849$	85.5308	0.100743
$850$	0	0
$851$	606.457i	0.712640i
$852$	0	0
$853$	−495.517	−0.580911	−0.290456	−	0.956888i	$-0.593807\pi$
$853$	−495.517	−0.580911	−0.290456	+	0.956888i	$0.593807\pi$
$854$	0	0
$855$	479.766i	0.561130i
$856$	0	0
$857$	−1029.99	−1.20186	−0.600930	−	0.799302i	$-0.705203\pi$
$857$	−1029.99	−1.20186	−0.600930	+	0.799302i	$0.705203\pi$
$858$	0	0
$859$	− 769.294i	− 0.895569i	−0.894142	−	0.447784i	$-0.852213\pi$
$859$	− 769.294i	− 0.895569i	0.894142	−	0.447784i	$-0.147787\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 170.972i	− 0.198113i	−0.995082	−	0.0990567i	$-0.968417\pi$
$863$	− 170.972i	− 0.198113i	0.995082	−	0.0990567i	$-0.0315825\pi$
$864$	0	0
$865$	1879.38	2.17269
$866$	0	0
$867$	− 485.822i	− 0.560349i
$868$	0	0
$869$	1318.92	1.51775
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−340.510	−0.390046
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−919.959	−1.04898	−0.524492	−	0.851415i	$-0.675745\pi$
$877$	−919.959	−1.04898	−0.524492	+	0.851415i	$0.675745\pi$
$878$	0	0
$879$	377.456i	0.429415i
$880$	0	0
$881$	1179.95	1.33933	0.669667	−	0.742662i	$-0.266437\pi$
$881$	1179.95	1.33933	0.669667	+	0.742662i	$0.266437\pi$
$882$	0	0
$883$	1453.16i	1.64570i	0.568257	+	0.822851i	$0.307618\pi$
$883$	1453.16i	1.64570i	−0.568257	+	0.822851i	$0.692382\pi$
$884$	0	0
$885$	−662.979	−0.749129
$886$	0	0
$887$	− 1017.87i	− 1.14755i	−0.819015	−	0.573773i	$-0.805479\pi$
$887$	− 1017.87i	− 1.14755i	0.819015	−	0.573773i	$-0.194521\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	79.2324i	0.0889253i
$892$	0	0
$893$	1107.97	1.24073
$894$	0	0
$895$	− 515.244i	− 0.575691i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1861.87i	− 2.07105i
$900$	0	0
$901$	−246.497	−0.273581
$902$	0	0
$903$	0	0
$904$	0	0
$905$	199.945	0.220934
$906$	0	0
$907$	720.533i	0.794414i	0.917729	+	0.397207i	$0.130020\pi$
$907$	720.533i	0.794414i	−0.917729	+	0.397207i	$0.869980\pi$
$908$	0	0
$909$	426.228	0.468897
$910$	0	0
$911$	− 1067.36i	− 1.17164i	−0.810442	−	0.585819i	$-0.800773\pi$
$911$	− 1067.36i	− 1.17164i	0.810442	−	0.585819i	$-0.199227\pi$
$912$	0	0
$913$	5.04795	0.00552897
$914$	0	0
$915$	1612.46i	1.76225i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 389.652i	− 0.423996i	−0.977270	−	0.211998i	$-0.932003\pi$
$919$	− 389.652i	− 0.423996i	0.977270	−	0.211998i	$-0.0679970\pi$
$920$	0	0
$921$	285.476	0.309963
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−1178.48	−1.27404
$926$	0	0
$927$	556.813i	0.600661i
$928$	0	0
$929$	−1268.01	−1.36492	−0.682458	−	0.730925i	$-0.739089\pi$
$929$	−1268.01	−1.36492	−0.682458	+	0.730925i	$0.739089\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−612.993	−0.657013
$934$	0	0
$935$	− 233.265i	− 0.249482i
$936$	0	0
$937$	121.986	0.130188	0.0650941	−	0.997879i	$-0.479265\pi$
$937$	121.986	0.130188	0.0650941	+	0.997879i	$0.479265\pi$
$938$	0	0
$939$	460.702i	0.490630i
$940$	0	0
$941$	−1160.79	−1.23357	−0.616787	−	0.787130i	$-0.711566\pi$
$941$	−1160.79	−1.23357	−0.616787	+	0.787130i	$0.711566\pi$
$942$	0	0
$943$	− 796.433i	− 0.844574i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	59.6421i	0.0629800i	0.999504	+	0.0314900i	$0.0100252\pi$
$947$	59.6421i	0.0629800i	−0.999504	+	0.0314900i	$0.989975\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 506.571i	− 0.532672i
$952$	0	0
$953$	959.959	1.00730	0.503651	−	0.863907i	$-0.331990\pi$
$953$	959.959	1.00730	0.503651	+	0.863907i	$0.331990\pi$
$954$	0	0
$955$	− 1272.76i	− 1.33273i
$956$	0	0
$957$	−739.490	−0.772717
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−512.932	−0.533748
$962$	0	0
$963$	− 362.405i	− 0.376329i
$964$	0	0
$965$	−1357.78	−1.40703
$966$	0	0
$967$	629.488i	0.650970i	0.945547	+	0.325485i	$0.105528\pi$
$967$	629.488i	0.650970i	−0.945547	+	0.325485i	$0.894472\pi$
$968$	0	0
$969$	−88.9658	−0.0918119
$970$	0	0
$971$	1048.98i	1.08031i	0.841566	+	0.540154i	$0.181634\pi$
$971$	1048.98i	1.08031i	−0.841566	+	0.540154i	$0.818366\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−793.945	−0.812636	−0.406318	−	0.913732i	$-0.633187\pi$
$977$	−793.945	−0.812636	−0.406318	+	0.913732i	$0.633187\pi$
$978$	0	0
$979$	− 88.7043i	− 0.0906071i
$980$	0	0
$981$	428.979	0.437288
$982$	0	0
$983$	1271.87i	1.29387i	0.762545	+	0.646935i	$0.223950\pi$
$983$	1271.87i	1.29387i	−0.762545	+	0.646935i	$0.776050\pi$
$984$	0	0
$985$	967.408	0.982140
$986$	0	0
$987$	0	0
$988$	0	0
$989$	819.973	0.829093
$990$	0	0
$991$	677.196i	0.683346i	0.939819	+	0.341673i	$0.110994\pi$
$991$	677.196i	0.683346i	−0.939819	+	0.341673i	$0.889006\pi$
$992$	0	0
$993$	932.938	0.939515
$994$	0	0
$995$	− 1017.25i	− 1.02236i
$996$	0	0
$997$	1876.35	1.88199	0.940996	−	0.338418i	$-0.109892\pi$
$997$	1876.35	1.88199	0.940996	+	0.338418i	$0.109892\pi$
$998$	0	0
$999$	106.503i	0.106610i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.m.j.1471.4	yes	4
4.3	odd	2	inner	2352.3.m.j.1471.2	yes	4
7.6	odd	2		2352.3.m.e.1471.1	✓	4
28.27	even	2		2352.3.m.e.1471.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.3.m.e.1471.1	✓	4	7.6	odd	2
2352.3.m.e.1471.3	yes	4	28.27	even	2
2352.3.m.j.1471.2	yes	4	4.3	odd	2	inner
2352.3.m.j.1471.4	yes	4	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +9.08276 q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +9.08276 q^{5} -3.00000 q^{9} +8.80360i q^{11} +15.7318i q^{15} -2.91724 q^{17} -17.6072i q^{19} -29.5882i q^{23} +57.4966 q^{25} -5.19615i q^{27} +48.4966 q^{29} -38.3918i q^{31} -15.2483 q^{33} -20.4966 q^{37} +26.9172 q^{41} +27.7128i q^{43} -27.2483 q^{45} +62.9272i q^{47} -5.05280i q^{51} +84.4966 q^{53} +79.9610i q^{55} +30.4966 q^{57} +42.1426i q^{59} +102.497 q^{61} +52.8216i q^{67} +51.2483 q^{69} +64.8026i q^{71} +30.4966 q^{73} +99.5870i q^{75} -149.816i q^{79} +9.00000 q^{81} -0.573396i q^{83} -26.4966 q^{85} +83.9985i q^{87} -10.0759 q^{89} +66.4966 q^{93} -159.922i q^{95} +113.503 q^{97} -26.4108i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^5 - 12 * q^9	\( 4 q + 12 q^{5} - 12 q^{9} - 36 q^{17} + 84 q^{25} + 48 q^{29} + 12 q^{33} + 64 q^{37} + 132 q^{41} - 36 q^{45} + 192 q^{53} - 24 q^{57} + 264 q^{61} + 132 q^{69} - 24 q^{73} + 36 q^{81} + 40 q^{85} + 276 q^{89} + 120 q^{93} + 600 q^{97}+O(q^{100}) \) 4 * q + 12 * q^5 - 12 * q^9 - 36 * q^17 + 84 * q^25 + 48 * q^29 + 12 * q^33 + 64 * q^37 + 132 * q^41 - 36 * q^45 + 192 * q^53 - 24 * q^57 + 264 * q^61 + 132 * q^69 - 24 * q^73 + 36 * q^81 + 40 * q^85 + 276 * q^89 + 120 * q^93 + 600 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more