LMFDB - Embedded newform 400.4.n.g.143.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.n (of order $4$, degree $2$, 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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.11007531417600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 $ x^16 - 7x^12 + 48x^8 - 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{36}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.5
Root			$1.56290 - 0.418778i$ of defining polynomial
Character	$\chi$	$=$	400.143
Dual form			400.4.n.g.207.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.35082 - 2.35082i) q^{3} +(-0.458991 - 0.458991i) q^{7} +15.9473i q^{9} -23.0788i q^{11} +(-1.65449 - 1.65449i) q^{13} +(63.9875 - 63.9875i) q^{17} +65.6811 q^{19} -2.15800 q^{21} +(45.2921 - 45.2921i) q^{23} +(100.961 + 100.961i) q^{27} +59.8420i q^{29} -279.285i q^{31} +(-54.2540 - 54.2540i) q^{33} +(100.494 - 100.494i) q^{37} -7.77882 q^{39} -274.789 q^{41} +(263.279 - 263.279i) q^{43} +(391.588 + 391.588i) q^{47} -342.579i q^{49} -300.846i q^{51} +(-265.404 - 265.404i) q^{53} +(154.404 - 154.404i) q^{57} +121.061 q^{59} -617.315 q^{61} +(7.31967 - 7.31967i) q^{63} +(674.536 + 674.536i) q^{67} -212.947i q^{69} -829.622i q^{71} +(110.657 + 110.657i) q^{73} +(-10.5930 + 10.5930i) q^{77} -62.7483 q^{79} +44.1046 q^{81} +(655.394 - 655.394i) q^{83} +(140.677 + 140.677i) q^{87} -92.0007i q^{89} +1.51879i q^{91} +(-656.549 - 656.549i) q^{93} +(618.474 - 618.474i) q^{97} +368.045 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 1856 q^{21} - 1968 q^{41} - 1984 q^{61} - 9616 q^{81}+O(q^{100}) $ 16 * q - 1856 * q^21 - 1968 * q^41 - 1984 * q^61 - 9616 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	2.35082	−	2.35082i	0.452415	−	0.452415i	−0.443741	−	0.896155i	$-0.646349\pi$
$3$	2.35082	−	2.35082i	0.452415	−	0.452415i	0.896155	+	0.443741i	$0.146349\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.458991	−	0.458991i	−0.0247832	−	0.0247832i	0.694607	−	0.719390i	$-0.255578\pi$
$7$	−0.458991	−	0.458991i	−0.0247832	−	0.0247832i	−0.719390	+	0.694607i	$0.755578\pi$
$8$		0			0
$9$			15.9473i			0.590642i
$10$		0			0
$11$		−	23.0788i		−	0.632593i	−0.948660	−	0.316296i	$-0.897561\pi$
$11$		−	23.0788i		−	0.632593i	0.948660	−	0.316296i	$-0.102439\pi$
$12$		0			0
$13$	−1.65449	−	1.65449i	−0.0352980	−	0.0352980i	0.689237	−	0.724536i	$-0.257946\pi$
$13$	−1.65449	−	1.65449i	−0.0352980	−	0.0352980i	−0.724536	+	0.689237i	$0.757946\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	63.9875	−	63.9875i	0.912897	−	0.912897i	−0.0836026	−	0.996499i	$-0.526643\pi$
$17$	63.9875	−	63.9875i	0.912897	−	0.912897i	0.996499	+	0.0836026i	$0.0266426\pi$
$18$		0			0
$19$	65.6811			0.793067			0.396534	−	0.918020i	$-0.370213\pi$
$19$	65.6811			0.793067			0.396534	+	0.918020i	$0.370213\pi$
$20$		0			0
$21$	−2.15800			−0.0224245
$22$		0			0
$23$	45.2921	−	45.2921i	0.410611	−	0.410611i	−0.471340	−	0.881951i	$-0.656230\pi$
$23$	45.2921	−	45.2921i	0.410611	−	0.410611i	0.881951	+	0.471340i	$0.156230\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	100.961	+	100.961i	0.719630	+	0.719630i
$28$		0			0
$29$			59.8420i			0.383186i	0.981475	+	0.191593i	$0.0613653\pi$
$29$			59.8420i			0.383186i	−0.981475	+	0.191593i	$0.938635\pi$
$30$		0			0
$31$		−	279.285i		−	1.61810i	−0.587739	−	0.809051i	$-0.699982\pi$
$31$		−	279.285i		−	1.61810i	0.587739	−	0.809051i	$-0.300018\pi$
$32$		0			0
$33$	−54.2540	−	54.2540i	−0.286194	−	0.286194i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	100.494	−	100.494i	0.446515	−	0.446515i	−0.447679	−	0.894194i	$-0.647749\pi$
$37$	100.494	−	100.494i	0.446515	−	0.446515i	0.894194	+	0.447679i	$0.147749\pi$
$38$		0			0
$39$	−7.77882			−0.0319387
$40$		0			0
$41$	−274.789			−1.04670			−0.523352	−	0.852116i	$-0.675319\pi$
$41$	−274.789			−1.04670			−0.523352	+	0.852116i	$0.675319\pi$
$42$		0			0
$43$	263.279	−	263.279i	0.933714	−	0.933714i	−0.0642215	−	0.997936i	$-0.520456\pi$
$43$	263.279	−	263.279i	0.933714	−	0.933714i	0.997936	+	0.0642215i	$0.0204564\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	391.588	+	391.588i	1.21530	+	1.21530i	0.969260	+	0.246038i	$0.0791287\pi$
$47$	391.588	+	391.588i	1.21530	+	1.21530i	0.246038	+	0.969260i	$0.420871\pi$
$48$		0			0
$49$		−	342.579i		−	0.998772i
$50$		0			0
$51$		−	300.846i		−	0.826016i
$52$		0			0
$53$	−265.404	−	265.404i	−0.687851	−	0.687851i	0.273906	−	0.961757i	$-0.411684\pi$
$53$	−265.404	−	265.404i	−0.687851	−	0.687851i	−0.961757	+	0.273906i	$0.911684\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	154.404	−	154.404i	0.358795	−	0.358795i
$58$		0			0
$59$	121.061			0.267132			0.133566	−	0.991040i	$-0.457357\pi$
$59$	121.061			0.267132			0.133566	+	0.991040i	$0.457357\pi$
$60$		0			0
$61$	−617.315			−1.29572			−0.647862	−	0.761758i	$-0.724337\pi$
$61$	−617.315			−1.29572			−0.647862	+	0.761758i	$0.724337\pi$
$62$		0			0
$63$	7.31967	−	7.31967i	0.0146380	−	0.0146380i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	674.536	+	674.536i	1.22997	+	1.22997i	0.963976	+	0.265990i	$0.0856988\pi$
$67$	674.536	+	674.536i	1.22997	+	1.22997i	0.265990	+	0.963976i	$0.414301\pi$
$68$		0			0
$69$		−	212.947i		−	0.371533i
$70$		0			0
$71$		−	829.622i		−	1.38673i	−0.720586	−	0.693366i	$-0.756127\pi$
$71$		−	829.622i		−	1.38673i	0.720586	−	0.693366i	$-0.243873\pi$
$72$		0			0
$73$	110.657	+	110.657i	0.177416	+	0.177416i	0.790229	−	0.612812i	$-0.209962\pi$
$73$	110.657	+	110.657i	0.177416	+	0.177416i	−0.612812	+	0.790229i	$0.709962\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.5930	+	10.5930i	−0.0156777	+	0.0156777i
$78$		0			0
$79$	−62.7483			−0.0893638			−0.0446819	−	0.999001i	$-0.514227\pi$
$79$	−62.7483			−0.0893638			−0.0446819	+	0.999001i	$0.514227\pi$
$80$		0			0
$81$	44.1046			0.0605002
$82$		0			0
$83$	655.394	−	655.394i	0.866733	−	0.866733i	−0.125376	−	0.992109i	$-0.540014\pi$
$83$	655.394	−	655.394i	0.866733	−	0.866733i	0.992109	+	0.125376i	$0.0400137\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	140.677	+	140.677i	0.173359	+	0.173359i
$88$		0			0
$89$		−	92.0007i		−	0.109574i	−0.998498	−	0.0547868i	$-0.982552\pi$
$89$		−	92.0007i		−	0.109574i	0.998498	−	0.0547868i	$-0.0174479\pi$
$90$		0			0
$91$			1.51879i			0.00174959i
$92$		0			0
$93$	−656.549	−	656.549i	−0.732053	−	0.732053i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	618.474	−	618.474i	0.647387	−	0.647387i	−0.304974	−	0.952361i	$-0.598648\pi$
$97$	618.474	−	618.474i	0.647387	−	0.647387i	0.952361	+	0.304974i	$0.0986476\pi$
$98$		0			0
$99$	368.045			0.373636
$100$		0			0
$101$	−246.158			−0.242511			−0.121256	−	0.992621i	$-0.538692\pi$
$101$	−246.158			−0.242511			−0.121256	+	0.992621i	$0.538692\pi$
$102$		0			0
$103$	−175.784	+	175.784i	−0.168161	+	0.168161i	−0.786170	−	0.618010i	$-0.787939\pi$
$103$	−175.784	+	175.784i	−0.168161	+	0.168161i	0.618010	+	0.786170i	$0.287939\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−446.463	−	446.463i	−0.403376	−	0.403376i	0.476045	−	0.879421i	$-0.342070\pi$
$107$	−446.463	−	446.463i	−0.403376	−	0.403376i	−0.879421	+	0.476045i	$0.842070\pi$
$108$		0			0
$109$			1523.84i			1.33906i	0.742785	+	0.669530i	$0.233504\pi$
$109$			1523.84i			1.33906i	−0.742785	+	0.669530i	$0.766496\pi$
$110$		0			0
$111$		−	472.484i		−	0.404020i
$112$		0			0
$113$	890.818	+	890.818i	0.741603	+	0.741603i	0.972886	−	0.231283i	$-0.0742924\pi$
$113$	890.818	+	890.818i	0.741603	+	0.741603i	−0.231283	+	0.972886i	$0.574292\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	26.3848	−	26.3848i	0.0208485	−	0.0208485i
$118$		0			0
$119$	−58.7393			−0.0452489
$120$		0			0
$121$	798.369			0.599826
$122$		0			0
$123$	−645.979	+	645.979i	−0.473544	+	0.473544i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−614.500	−	614.500i	−0.429355	−	0.429355i	0.459054	−	0.888409i	$-0.348189\pi$
$127$	−614.500	−	614.500i	−0.429355	−	0.429355i	−0.888409	+	0.459054i	$0.848189\pi$
$128$		0			0
$129$		−	1237.84i		−	0.844852i
$130$		0			0
$131$			1279.04i			0.853057i	0.904474	+	0.426528i	$0.140264\pi$
$131$			1279.04i			0.853057i	−0.904474	+	0.426528i	$0.859736\pi$
$132$		0			0
$133$	−30.1470	−	30.1470i	−0.0196547	−	0.0196547i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−47.6796	+	47.6796i	−0.0297339	+	0.0297339i	−0.721817	−	0.692084i	$-0.756693\pi$
$137$	−47.6796	+	47.6796i	−0.0297339	+	0.0297339i	0.692084	+	0.721817i	$0.256693\pi$
$138$		0			0
$139$	788.310			0.481033			0.240516	−	0.970645i	$-0.422683\pi$
$139$	788.310			0.481033			0.240516	+	0.970645i	$0.422683\pi$
$140$		0			0
$141$	1841.10			1.09964
$142$		0			0
$143$	−38.1838	+	38.1838i	−0.0223293	+	0.0223293i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−805.339	−	805.339i	−0.451859	−	0.451859i
$148$		0			0
$149$			2219.84i			1.22051i	0.792204	+	0.610257i	$0.208934\pi$
$149$			2219.84i			1.22051i	−0.792204	+	0.610257i	$0.791066\pi$
$150$		0			0
$151$			2166.49i			1.16759i	0.811900	+	0.583797i	$0.198433\pi$
$151$			2166.49i			1.16759i	−0.811900	+	0.583797i	$0.801567\pi$
$152$		0			0
$153$	1020.43	+	1020.43i	0.539195	+	0.539195i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−2392.68	+	2392.68i	−1.21628	+	1.21628i	−0.247359	+	0.968924i	$0.579563\pi$
$157$	−2392.68	+	2392.68i	−1.21628	+	1.21628i	−0.968924	+	0.247359i	$0.920437\pi$
$158$		0			0
$159$	−1247.83			−0.622388
$160$		0			0
$161$	−41.5773			−0.0203525
$162$		0			0
$163$	−1305.98	+	1305.98i	−0.627561	+	0.627561i	−0.947454	−	0.319893i	$-0.896353\pi$
$163$	−1305.98	+	1305.98i	−0.627561	+	0.627561i	0.319893	+	0.947454i	$0.396353\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2151.02	−	2151.02i	−0.996712	−	0.996712i	0.00328303	−	0.999995i	$-0.498955\pi$
$167$	−2151.02	−	2151.02i	−0.996712	−	0.996712i	−0.999995	+	0.00328303i	$0.998955\pi$
$168$		0			0
$169$		−	2191.53i		−	0.997508i
$170$		0			0
$171$			1047.44i			0.468419i
$172$		0			0
$173$	−1294.66	−	1294.66i	−0.568967	−	0.568967i	0.362872	−	0.931839i	$-0.381796\pi$
$173$	−1294.66	−	1294.66i	−0.568967	−	0.568967i	−0.931839	+	0.362872i	$0.881796\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	284.592	−	284.592i	0.120855	−	0.120855i
$178$		0			0
$179$	−3620.52			−1.51179			−0.755894	−	0.654694i	$-0.772797\pi$
$179$	−3620.52			−1.51179			−0.755894	+	0.654694i	$0.772797\pi$
$180$		0			0
$181$	−943.156			−0.387316			−0.193658	−	0.981069i	$-0.562035\pi$
$181$	−943.156			−0.387316			−0.193658	+	0.981069i	$0.562035\pi$
$182$		0			0
$183$	−1451.19	+	1451.19i	−0.586204	+	0.586204i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1476.75	−	1476.75i	−0.577492	−	0.577492i
$188$		0			0
$189$		−	92.6805i		−	0.0356694i
$190$		0			0
$191$			4157.37i			1.57496i	0.616342	+	0.787479i	$0.288614\pi$
$191$			4157.37i			1.57496i	−0.616342	+	0.787479i	$0.711386\pi$
$192$		0			0
$193$	2725.68	+	2725.68i	1.01657	+	1.01657i	0.999860	+	0.0167134i	$0.00532028\pi$
$193$	2725.68	+	2725.68i	1.01657	+	1.01657i	0.0167134	+	0.999860i	$0.494680\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3157.61	+	3157.61i	−1.14198	+	1.14198i	−0.153892	+	0.988088i	$0.549181\pi$
$197$	−3157.61	+	3157.61i	−1.14198	+	1.14198i	−0.988088	+	0.153892i	$0.950819\pi$
$198$		0			0
$199$	2008.14			0.715344			0.357672	−	0.933847i	$-0.383571\pi$
$199$	2008.14			0.715344			0.357672	+	0.933847i	$0.383571\pi$
$200$		0			0
$201$	3171.42			1.11291
$202$		0			0
$203$	27.4669	−	27.4669i	0.00949655	−	0.00949655i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	722.288	+	722.288i	0.242524	+	0.242524i
$208$		0			0
$209$		−	1515.84i		−	0.501689i
$210$		0			0
$211$			1687.28i			0.550506i	0.961372	+	0.275253i	$0.0887617\pi$
$211$			1687.28i			0.550506i	−0.961372	+	0.275253i	$0.911238\pi$
$212$		0			0
$213$	−1950.29	−	1950.29i	−0.627378	−	0.627378i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−128.189	+	128.189i	−0.0401017	+	0.0401017i
$218$		0			0
$219$	520.267			0.160532
$220$		0			0
$221$	−211.734			−0.0644469
$222$		0			0
$223$	−998.709	+	998.709i	−0.299903	+	0.299903i	−0.840976	−	0.541072i	$-0.818018\pi$
$223$	−998.709	+	998.709i	−0.299903	+	0.299903i	0.541072	+	0.840976i	$0.318018\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1566.54	−	1566.54i	−0.458039	−	0.458039i	0.439972	−	0.898011i	$-0.354988\pi$
$227$	−1566.54	−	1566.54i	−0.458039	−	0.458039i	−0.898011	+	0.439972i	$0.854988\pi$
$228$		0			0
$229$			2700.95i			0.779404i	0.920941	+	0.389702i	$0.127422\pi$
$229$			2700.95i			0.779404i	−0.920941	+	0.389702i	$0.872578\pi$
$230$		0			0
$231$			49.8042i			0.0141856i
$232$		0			0
$233$	2670.59	+	2670.59i	0.750884	+	0.750884i	0.974644	−	0.223761i	$-0.0718334\pi$
$233$	2670.59	+	2670.59i	0.750884	+	0.750884i	−0.223761	+	0.974644i	$0.571833\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−147.510	+	147.510i	−0.0404295	+	0.0404295i
$238$		0			0
$239$	−1786.81			−0.483594			−0.241797	−	0.970327i	$-0.577737\pi$
$239$	−1786.81			−0.483594			−0.241797	+	0.970327i	$0.577737\pi$
$240$		0			0
$241$	525.789			0.140535			0.0702677	−	0.997528i	$-0.477615\pi$
$241$	525.789			0.140535			0.0702677	+	0.997528i	$0.477615\pi$
$242$		0			0
$243$	−2622.27	+	2622.27i	−0.692259	+	0.692259i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−108.669	−	108.669i	−0.0279937	−	0.0279937i
$248$		0			0
$249$		−	3081.42i		−	0.784246i
$250$		0			0
$251$		−	3555.70i		−	0.894158i	−0.894494	−	0.447079i	$-0.852464\pi$
$251$		−	3555.70i		−	0.894158i	0.894494	−	0.447079i	$-0.147536\pi$
$252$		0			0
$253$	−1045.29	−	1045.29i	−0.259750	−	0.259750i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1775.11	−	1775.11i	0.430849	−	0.430849i	−0.458068	−	0.888917i	$-0.651458\pi$
$257$	1775.11	−	1775.11i	0.430849	−	0.430849i	0.888917	+	0.458068i	$0.151458\pi$
$258$		0			0
$259$	−92.2512			−0.0221321
$260$		0			0
$261$	−954.320			−0.226325
$262$		0			0
$263$	5778.06	−	5778.06i	1.35472	−	1.35472i	0.474417	−	0.880300i	$-0.342659\pi$
$263$	5778.06	−	5778.06i	1.35472	−	1.35472i	0.880300	−	0.474417i	$-0.157341\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−216.277	−	216.277i	−0.0495727	−	0.0495727i
$268$		0			0
$269$			7948.00i			1.80148i	0.434359	+	0.900740i	$0.356975\pi$
$269$			7948.00i			1.80148i	−0.434359	+	0.900740i	$0.643025\pi$
$270$		0			0
$271$			997.629i			0.223622i	0.993729	+	0.111811i	$0.0356652\pi$
$271$			997.629i			0.223622i	−0.993729	+	0.111811i	$0.964335\pi$
$272$		0			0
$273$	3.57041	+	3.57041i	0.000791542	+	0.000791542i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	3674.85	−	3674.85i	0.797114	−	0.797114i	−0.185526	−	0.982639i	$-0.559399\pi$
$277$	3674.85	−	3674.85i	0.797114	−	0.797114i	0.982639	+	0.185526i	$0.0593987\pi$
$278$		0			0
$279$	4453.86			0.955719
$280$		0			0
$281$	110.524			0.0234637			0.0117319	−	0.999931i	$-0.496266\pi$
$281$	110.524			0.0234637			0.0117319	+	0.999931i	$0.496266\pi$
$282$		0			0
$283$	293.182	−	293.182i	0.0615825	−	0.0615825i	−0.675645	−	0.737227i	$-0.736135\pi$
$283$	293.182	−	293.182i	0.0615825	−	0.0615825i	0.737227	+	0.675645i	$0.236135\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	126.126	+	126.126i	0.0259406	+	0.0259406i
$288$		0			0
$289$		−	3275.79i		−	0.666760i
$290$		0			0
$291$		−	2907.84i		−	0.585775i
$292$		0			0
$293$	−5842.01	−	5842.01i	−1.16483	−	1.16483i	−0.983406	−	0.181420i	$-0.941931\pi$
$293$	−5842.01	−	5842.01i	−1.16483	−	1.16483i	−0.181420	−	0.983406i	$-0.558069\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	2330.07	−	2330.07i	0.455233	−	0.455233i
$298$		0			0
$299$	−149.871			−0.0289875
$300$		0			0
$301$	−241.685			−0.0462808
$302$		0			0
$303$	−578.672	+	578.672i	−0.109716	+	0.109716i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	6610.85	+	6610.85i	1.22899	+	1.22899i	0.964344	+	0.264650i	$0.0852565\pi$
$307$	6610.85	+	6610.85i	1.22899	+	1.22899i	0.264650	+	0.964344i	$0.414743\pi$
$308$		0			0
$309$			826.473i			0.152157i
$310$		0			0
$311$		−	8080.44i		−	1.47331i	−0.676268	−	0.736655i	$-0.736404\pi$
$311$		−	8080.44i		−	1.47331i	0.676268	−	0.736655i	$-0.263596\pi$
$312$		0			0
$313$	5294.89	+	5294.89i	0.956182	+	0.956182i	0.999079	−	0.0428977i	$-0.0136589\pi$
$313$	5294.89	+	5294.89i	0.956182	+	0.956182i	−0.0428977	+	0.999079i	$0.513659\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2611.67	+	2611.67i	−0.462732	+	0.462732i	−0.899550	−	0.436818i	$-0.856105\pi$
$317$	−2611.67	+	2611.67i	−0.462732	+	0.462732i	0.436818	+	0.899550i	$0.356105\pi$
$318$		0			0
$319$	1381.08			0.242400
$320$		0			0
$321$	−2099.11			−0.364986
$322$		0			0
$323$	4202.77	−	4202.77i	0.723988	−	0.723988i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	3582.27	+	3582.27i	0.605810	+	0.605810i
$328$		0			0
$329$		−	359.471i		−	0.0602379i
$330$		0			0
$331$		−	6573.48i		−	1.09157i	−0.837924	−	0.545787i	$-0.816231\pi$
$331$		−	6573.48i		−	1.09157i	0.837924	−	0.545787i	$-0.183769\pi$
$332$		0			0
$333$	1602.60	+	1602.60i	0.263730	+	0.263730i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	6325.93	−	6325.93i	1.02254	−	1.02254i	0.0227988	−	0.999740i	$-0.492742\pi$
$337$	6325.93	−	6325.93i	1.02254	−	1.02254i	0.999740	−	0.0227988i	$-0.00725773\pi$
$338$		0			0
$339$	4188.30			0.671024
$340$		0			0
$341$	−6445.58			−1.02360
$342$		0			0
$343$	−314.674	+	314.674i	−0.0495359	+	0.0495359i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1456.40	+	1456.40i	0.225312	+	0.225312i	0.810731	−	0.585419i	$-0.199070\pi$
$347$	1456.40	+	1456.40i	0.225312	+	0.225312i	−0.585419	+	0.810731i	$0.699070\pi$
$348$		0			0
$349$		−	4287.31i		−	0.657577i	−0.944404	−	0.328789i	$-0.893360\pi$
$349$		−	4287.31i		−	0.657577i	0.944404	−	0.328789i	$-0.106640\pi$
$350$		0			0
$351$		−	334.080i		−	0.0508030i
$352$		0			0
$353$	−6611.25	−	6611.25i	−0.996832	−	0.996832i	0.00316330	−	0.999995i	$-0.498993\pi$
$353$	−6611.25	−	6611.25i	−0.996832	−	0.996832i	−0.999995	+	0.00316330i	$0.998993\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−138.085	+	138.085i	−0.0204713	+	0.0204713i
$358$		0			0
$359$	−6278.75			−0.923063			−0.461531	−	0.887124i	$-0.652700\pi$
$359$	−6278.75			−0.923063			−0.461531	+	0.887124i	$0.652700\pi$
$360$		0			0
$361$	−2544.99			−0.371044
$362$		0			0
$363$	1876.82	−	1876.82i	0.271370	−	0.271370i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	6842.24	+	6842.24i	0.973193	+	0.973193i	0.999650	−	0.0264571i	$-0.00842255\pi$
$367$	6842.24	+	6842.24i	0.973193	+	0.973193i	−0.0264571	+	0.999650i	$0.508423\pi$
$368$		0			0
$369$		−	4382.16i		−	0.618228i
$370$		0			0
$371$			243.636i			0.0340942i
$372$		0			0
$373$	−6046.84	−	6046.84i	−0.839393	−	0.839393i	0.149386	−	0.988779i	$-0.452270\pi$
$373$	−6046.84	−	6046.84i	−0.839393	−	0.839393i	−0.988779	+	0.149386i	$0.952270\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	99.0083	−	99.0083i	0.0135257	−	0.0135257i
$378$		0			0
$379$	3999.47			0.542056			0.271028	−	0.962572i	$-0.412636\pi$
$379$	3999.47			0.542056			0.271028	+	0.962572i	$0.412636\pi$
$380$		0			0
$381$	−2889.15			−0.388493
$382$		0			0
$383$	6880.48	−	6880.48i	0.917954	−	0.917954i	−0.0789268	−	0.996880i	$-0.525149\pi$
$383$	6880.48	−	6880.48i	0.917954	−	0.917954i	0.996880	+	0.0789268i	$0.0251494\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4198.60	+	4198.60i	0.551491	+	0.551491i
$388$		0			0
$389$			1069.05i			0.139340i	0.997570	+	0.0696700i	$0.0221946\pi$
$389$			1069.05i			0.139340i	−0.997570	+	0.0696700i	$0.977805\pi$
$390$		0			0
$391$		−	5796.25i		−	0.749691i
$392$		0			0
$393$	3006.79	+	3006.79i	0.385935	+	0.385935i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3539.68	+	3539.68i	−0.447485	+	0.447485i	−0.894517	−	0.447033i	$-0.852481\pi$
$397$	−3539.68	+	3539.68i	−0.447485	+	0.447485i	0.447033	+	0.894517i	$0.352481\pi$
$398$		0			0
$399$	−141.740			−0.0177842
$400$		0			0
$401$	−8202.74			−1.02151			−0.510755	−	0.859727i	$-0.670634\pi$
$401$	−8202.74			−1.02151			−0.510755	+	0.859727i	$0.670634\pi$
$402$		0			0
$403$	−462.076	+	462.076i	−0.0571158	+	0.0571158i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2319.27	−	2319.27i	−0.282462	−	0.282462i
$408$		0			0
$409$		−	4312.89i		−	0.521414i	−0.965418	−	0.260707i	$-0.916044\pi$
$409$		−	4312.89i		−	0.521414i	0.965418	−	0.260707i	$-0.0839557\pi$
$410$		0			0
$411$			224.172i			0.0269041i
$412$		0			0
$413$	−55.5659	−	55.5659i	−0.00662039	−	0.00662039i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1853.17	−	1853.17i	0.217626	−	0.217626i
$418$		0			0
$419$	−15220.7			−1.77465			−0.887326	−	0.461142i	$-0.847440\pi$
$419$	−15220.7			−1.77465			−0.887326	+	0.461142i	$0.847440\pi$
$420$		0			0
$421$	−135.795			−0.0157203			−0.00786014	−	0.999969i	$-0.502502\pi$
$421$	−135.795			−0.0157203			−0.00786014	+	0.999969i	$0.502502\pi$
$422$		0			0
$423$	−6244.79	+	6244.79i	−0.717806	+	0.717806i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	283.342	+	283.342i	0.0321121	+	0.0321121i
$428$		0			0
$429$			179.526i			0.0202042i
$430$		0			0
$431$		−	9197.09i		−	1.02786i	−0.857832	−	0.513931i	$-0.828189\pi$
$431$		−	9197.09i		−	1.02786i	0.857832	−	0.513931i	$-0.171811\pi$
$432$		0			0
$433$	373.697	+	373.697i	0.0414751	+	0.0414751i	0.727540	−	0.686065i	$-0.240663\pi$
$433$	373.697	+	373.697i	0.0414751	+	0.0414751i	−0.686065	+	0.727540i	$0.740663\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2974.83	−	2974.83i	0.325642	−	0.325642i
$438$		0			0
$439$	11961.1			1.30039			0.650194	−	0.759769i	$-0.274688\pi$
$439$	11961.1			1.30039			0.650194	+	0.759769i	$0.274688\pi$
$440$		0			0
$441$	5463.22			0.589916
$442$		0			0
$443$	7605.32	−	7605.32i	0.815665	−	0.815665i	−0.169811	−	0.985477i	$-0.554316\pi$
$443$	7605.32	−	7605.32i	0.815665	−	0.815665i	0.985477	+	0.169811i	$0.0543158\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	5218.44	+	5218.44i	0.552178	+	0.552178i
$448$		0			0
$449$			2607.05i			0.274019i	0.990570	+	0.137009i	$0.0437490\pi$
$449$			2607.05i			0.274019i	−0.990570	+	0.137009i	$0.956251\pi$
$450$		0			0
$451$			6341.81i			0.662138i
$452$		0			0
$453$	5093.02	+	5093.02i	0.528236	+	0.528236i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7349.78	+	7349.78i	−0.752315	+	0.752315i	−0.974911	−	0.222596i	$-0.928547\pi$
$457$	−7349.78	+	7349.78i	−0.752315	+	0.752315i	0.222596	+	0.974911i	$0.428547\pi$
$458$		0			0
$459$	12920.5			1.31390
$460$		0			0
$461$	11169.1			1.12841			0.564207	−	0.825633i	$-0.309182\pi$
$461$	11169.1			1.12841			0.564207	+	0.825633i	$0.309182\pi$
$462$		0			0
$463$	−8617.79	+	8617.79i	−0.865017	+	0.865017i	−0.991916	−	0.126899i	$-0.959498\pi$
$463$	−8617.79	+	8617.79i	−0.865017	+	0.865017i	0.126899	+	0.991916i	$0.459498\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5057.64	−	5057.64i	−0.501155	−	0.501155i	0.410641	−	0.911797i	$-0.365305\pi$
$467$	−5057.64	−	5057.64i	−0.501155	−	0.501155i	−0.911797	+	0.410641i	$0.865305\pi$
$468$		0			0
$469$		−	619.211i		−	0.0609649i
$470$		0			0
$471$			11249.5i			1.10053i
$472$		0			0
$473$	−6076.17	−	6076.17i	−0.590661	−	0.590661i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4232.49	−	4232.49i	0.406274	−	0.406274i
$478$		0			0
$479$	−12864.4			−1.22711			−0.613557	−	0.789650i	$-0.710262\pi$
$479$	−12864.4			−1.22711			−0.613557	+	0.789650i	$0.710262\pi$
$480$		0			0
$481$	−332.532			−0.0315222
$482$		0			0
$483$	−97.7405	+	97.7405i	−0.00920776	+	0.00920776i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−8533.44	−	8533.44i	−0.794019	−	0.794019i	0.188126	−	0.982145i	$-0.439759\pi$
$487$	−8533.44	−	8533.44i	−0.794019	−	0.794019i	−0.982145	+	0.188126i	$0.939759\pi$
$488$		0			0
$489$			6140.25i			0.567836i
$490$		0			0
$491$			6157.29i			0.565936i	0.959129	+	0.282968i	$0.0913191\pi$
$491$			6157.29i			0.565936i	−0.959129	+	0.282968i	$0.908681\pi$
$492$		0			0
$493$	3829.14	+	3829.14i	0.349809	+	0.349809i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−380.788	+	380.788i	−0.0343676	+	0.0343676i
$498$		0			0
$499$	7040.63			0.631627			0.315814	−	0.948821i	$-0.397723\pi$
$499$	7040.63			0.631627			0.315814	+	0.948821i	$0.397723\pi$
$500$		0			0
$501$	−10113.3			−0.901854
$502$		0			0
$503$	1716.19	−	1716.19i	0.152129	−	0.152129i	−0.626939	−	0.779068i	$-0.715692\pi$
$503$	1716.19	−	1716.19i	0.152129	−	0.152129i	0.779068	+	0.626939i	$0.215692\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−5151.87	−	5151.87i	−0.451287	−	0.451287i
$508$		0			0
$509$			7212.20i			0.628046i	0.949415	+	0.314023i	$0.101677\pi$
$509$			7212.20i			0.628046i	−0.949415	+	0.314023i	$0.898323\pi$
$510$		0			0
$511$		−	101.581i		−	0.00879388i
$512$		0			0
$513$	6631.25	+	6631.25i	0.570715	+	0.570715i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	9037.39	−	9037.39i	0.768789	−	0.768789i
$518$		0			0
$519$	−6087.03			−0.514818
$520$		0			0
$521$	17759.5			1.49339			0.746697	−	0.665164i	$-0.231638\pi$
$521$	17759.5			1.49339			0.746697	+	0.665164i	$0.231638\pi$
$522$		0			0
$523$	−11070.1	+	11070.1i	−0.925550	+	0.925550i	−0.997414	−	0.0718644i	$-0.977105\pi$
$523$	−11070.1	+	11070.1i	−0.925550	+	0.925550i	0.0718644	+	0.997414i	$0.477105\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−17870.8	−	17870.8i	−1.47716	−	1.47716i
$528$		0			0
$529$			8064.26i			0.662797i
$530$		0			0
$531$			1930.60i			0.157780i
$532$		0			0
$533$	454.638	+	454.638i	0.0369466	+	0.0369466i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−8511.16	+	8511.16i	−0.683955	+	0.683955i
$538$		0			0
$539$	−7906.31			−0.631816
$540$		0			0
$541$	17452.4			1.38695			0.693473	−	0.720482i	$-0.256080\pi$
$541$	17452.4			1.38695			0.693473	+	0.720482i	$0.256080\pi$
$542$		0			0
$543$	−2217.19	+	2217.19i	−0.175228	+	0.175228i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1469.79	−	1469.79i	−0.114888	−	0.114888i	0.647326	−	0.762213i	$-0.275887\pi$
$547$	−1469.79	−	1469.79i	−0.114888	−	0.114888i	−0.762213	+	0.647326i	$0.775887\pi$
$548$		0			0
$549$		−	9844.53i		−	0.765309i
$550$		0			0
$551$			3930.49i			0.303892i
$552$		0			0
$553$	28.8009	+	28.8009i	0.00221472	+	0.00221472i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18132.6	+	18132.6i	−1.37936	+	1.37936i	−0.533668	+	0.845694i	$0.679187\pi$
$557$	−18132.6	+	18132.6i	−1.37936	+	1.37936i	−0.845694	+	0.533668i	$0.820813\pi$
$558$		0			0
$559$	−871.188			−0.0659165
$560$		0			0
$561$	−6943.16			−0.522532
$562$		0			0
$563$	5041.11	−	5041.11i	0.377367	−	0.377367i	−0.492784	−	0.870151i	$-0.664021\pi$
$563$	5041.11	−	5041.11i	0.377367	−	0.377367i	0.870151	+	0.492784i	$0.164021\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−20.2436	−	20.2436i	−0.00149939	−	0.00149939i
$568$		0			0
$569$			21162.1i			1.55916i	0.626306	+	0.779578i	$0.284566\pi$
$569$			21162.1i			1.55916i	−0.626306	+	0.779578i	$0.715434\pi$
$570$		0			0
$571$			23479.4i			1.72081i	0.509613	+	0.860404i	$0.329789\pi$
$571$			23479.4i			1.72081i	−0.509613	+	0.860404i	$0.670211\pi$
$572$		0			0
$573$	9773.22	+	9773.22i	0.712534	+	0.712534i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	2307.52	−	2307.52i	0.166487	−	0.166487i	−0.618946	−	0.785433i	$-0.712440\pi$
$577$	2307.52	−	2307.52i	0.166487	−	0.166487i	0.785433	+	0.618946i	$0.212440\pi$
$578$		0			0
$579$	12815.1			0.919826
$580$		0			0
$581$	−601.640			−0.0429608
$582$		0			0
$583$	−6125.22	+	6125.22i	−0.435130	+	0.435130i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10405.2	−	10405.2i	−0.731631	−	0.731631i	0.239312	−	0.970943i	$-0.423078\pi$
$587$	−10405.2	−	10405.2i	−0.731631	−	0.731631i	−0.970943	+	0.239312i	$0.923078\pi$
$588$		0			0
$589$		−	18343.8i		−	1.28326i
$590$		0			0
$591$			14845.9i			1.03330i
$592$		0			0
$593$	−13455.2	−	13455.2i	−0.931771	−	0.931771i	0.0660453	−	0.997817i	$-0.478962\pi$
$593$	−13455.2	−	13455.2i	−0.931771	−	0.931771i	−0.997817	+	0.0660453i	$0.978962\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	4720.78	−	4720.78i	0.323632	−	0.323632i
$598$		0			0
$599$	−4714.31			−0.321572			−0.160786	−	0.986989i	$-0.551403\pi$
$599$	−4714.31			−0.321572			−0.160786	+	0.986989i	$0.551403\pi$
$600$		0			0
$601$	16938.8			1.14967			0.574833	−	0.818271i	$-0.305067\pi$
$601$	16938.8			1.14967			0.574833	+	0.818271i	$0.305067\pi$
$602$		0			0
$603$	−10757.1	+	10757.1i	−0.726469	+	0.726469i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	2394.36	+	2394.36i	0.160106	+	0.160106i	0.782613	−	0.622508i	$-0.213886\pi$
$607$	2394.36	+	2394.36i	0.160106	+	0.160106i	−0.622508	+	0.782613i	$0.713886\pi$
$608$		0			0
$609$		−	129.139i		−	0.00859275i
$610$		0			0
$611$		−	1295.76i		−	0.0857952i
$612$		0			0
$613$	12943.8	+	12943.8i	0.852845	+	0.852845i	0.990483	−	0.137637i	$-0.0439509\pi$
$613$	12943.8	+	12943.8i	0.852845	+	0.852845i	−0.137637	+	0.990483i	$0.543951\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16224.7	+	16224.7i	−1.05864	+	1.05864i	−0.0604703	+	0.998170i	$0.519260\pi$
$617$	−16224.7	+	16224.7i	−1.05864	+	1.05864i	−0.998170	+	0.0604703i	$0.980740\pi$
$618$		0			0
$619$	−18113.5			−1.17616			−0.588081	−	0.808802i	$-0.700116\pi$
$619$	−18113.5			−1.17616			−0.588081	+	0.808802i	$0.700116\pi$
$620$		0			0
$621$	9145.49			0.590976
$622$		0			0
$623$	−42.2274	+	42.2274i	−0.00271558	+	0.00271558i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−3563.46	−	3563.46i	−0.226971	−	0.226971i
$628$		0			0
$629$		−	12860.7i		−	0.815243i
$630$		0			0
$631$			245.976i			0.0155185i	0.999970	+	0.00775923i	$0.00246987\pi$
$631$			245.976i			0.0155185i	−0.999970	+	0.00775923i	$0.997530\pi$
$632$		0			0
$633$	3966.47	+	3966.47i	0.249057	+	0.249057i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−566.795	+	566.795i	−0.0352547	+	0.0352547i
$638$		0			0
$639$	13230.2			0.819062
$640$		0			0
$641$	−23632.2			−1.45619			−0.728093	−	0.685478i	$-0.759593\pi$
$641$	−23632.2			−1.45619			−0.728093	+	0.685478i	$0.759593\pi$
$642$		0			0
$643$	−12444.0	+	12444.0i	−0.763207	+	0.763207i	−0.976901	−	0.213694i	$-0.931451\pi$
$643$	−12444.0	+	12444.0i	−0.763207	+	0.763207i	0.213694	+	0.976901i	$0.431451\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6170.29	−	6170.29i	−0.374929	−	0.374929i	0.494340	−	0.869269i	$-0.335410\pi$
$647$	−6170.29	−	6170.29i	−0.374929	−	0.374929i	−0.869269	+	0.494340i	$0.835410\pi$
$648$		0			0
$649$		−	2793.95i		−	0.168986i
$650$		0			0
$651$			602.699i			0.0362852i
$652$		0			0
$653$	3951.31	+	3951.31i	0.236794	+	0.236794i	0.815521	−	0.578727i	$-0.196450\pi$
$653$	3951.31	+	3951.31i	0.236794	+	0.236794i	−0.578727	+	0.815521i	$0.696450\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1764.68	+	1764.68i	−0.104790	+	0.104790i
$658$		0			0
$659$	−21537.2			−1.27310			−0.636549	−	0.771236i	$-0.719639\pi$
$659$	−21537.2			−1.27310			−0.636549	+	0.771236i	$0.719639\pi$
$660$		0			0
$661$	−11082.9			−0.652156			−0.326078	−	0.945343i	$-0.605727\pi$
$661$	−11082.9			−0.652156			−0.326078	+	0.945343i	$0.605727\pi$
$662$		0			0
$663$	−497.747	+	497.747i	−0.0291567	+	0.0291567i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2710.37	+	2710.37i	0.157340	+	0.157340i
$668$		0			0
$669$			4695.56i			0.271361i
$670$		0			0
$671$			14246.9i			0.819665i
$672$		0			0
$673$	−16642.5	−	16642.5i	−0.953229	−	0.953229i	0.0457251	−	0.998954i	$-0.485440\pi$
$673$	−16642.5	−	16642.5i	−0.953229	−	0.953229i	−0.998954	+	0.0457251i	$0.985440\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8193.46	−	8193.46i	0.465141	−	0.465141i	−0.435195	−	0.900336i	$-0.643321\pi$
$677$	8193.46	−	8193.46i	0.465141	−	0.465141i	0.900336	+	0.435195i	$0.143321\pi$
$678$		0			0
$679$	−567.748			−0.0320886
$680$		0			0
$681$	−7365.29			−0.414447
$682$		0			0
$683$	13989.6	−	13989.6i	0.783742	−	0.783742i	−0.196718	−	0.980460i	$-0.563028\pi$
$683$	13989.6	−	13989.6i	0.783742	−	0.783742i	0.980460	+	0.196718i	$0.0630284\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	6349.42	+	6349.42i	0.352614	+	0.352614i
$688$		0			0
$689$			878.220i			0.0485596i
$690$		0			0
$691$			509.424i			0.0280454i	0.999902	+	0.0140227i	$0.00446372\pi$
$691$			509.424i			0.0280454i	−0.999902	+	0.0140227i	$0.995536\pi$
$692$		0			0
$693$	−168.929	−	168.929i	−0.00925988	−	0.00925988i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−17583.1	+	17583.1i	−0.955533	+	0.955533i
$698$		0			0
$699$	12556.1			0.679421
$700$		0			0
$701$	23230.7			1.25166			0.625828	−	0.779961i	$-0.284761\pi$
$701$	23230.7			1.25166			0.625828	+	0.779961i	$0.284761\pi$
$702$		0			0
$703$	6600.53	−	6600.53i	0.354116	−	0.354116i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	112.984	+	112.984i	0.00601020	+	0.00601020i
$708$		0			0
$709$		−	13084.4i		−	0.693083i	−0.938035	−	0.346542i	$-0.887356\pi$
$709$		−	13084.4i		−	0.693083i	0.938035	−	0.346542i	$-0.112644\pi$
$710$		0			0
$711$		−	1000.67i		−	0.0527820i
$712$		0			0
$713$	−12649.4	−	12649.4i	−0.664410	−	0.664410i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−4200.46	+	4200.46i	−0.218785	+	0.218785i
$718$		0			0
$719$	−7819.19			−0.405572			−0.202786	−	0.979223i	$-0.565000\pi$
$719$	−7819.19			−0.405572			−0.202786	+	0.979223i	$0.565000\pi$
$720$		0			0
$721$	161.367			0.00833510
$722$		0			0
$723$	1236.03	−	1236.03i	0.0635803	−	0.0635803i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	9135.95	+	9135.95i	0.466071	+	0.466071i	0.900639	−	0.434568i	$-0.143099\pi$
$727$	9135.95	+	9135.95i	0.466071	+	0.466071i	−0.434568	+	0.900639i	$0.643099\pi$
$728$		0			0
$729$			13519.8i			0.686876i
$730$		0			0
$731$		−	33693.1i		−	1.70477i
$732$		0			0
$733$	24476.0	+	24476.0i	1.23334	+	1.23334i	0.962672	+	0.270671i	$0.0872454\pi$
$733$	24476.0	+	24476.0i	1.23334	+	1.23334i	0.270671	+	0.962672i	$0.412755\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15567.5	−	15567.5i	0.778068	−	0.778068i
$738$		0			0
$739$	18652.1			0.928456			0.464228	−	0.885716i	$-0.346332\pi$
$739$	18652.1			0.928456			0.464228	+	0.885716i	$0.346332\pi$
$740$		0			0
$741$	−510.922			−0.0253295
$742$		0			0
$743$	9000.76	−	9000.76i	0.444422	−	0.444422i	−0.449073	−	0.893495i	$-0.648246\pi$
$743$	9000.76	−	9000.76i	0.444422	−	0.444422i	0.893495	+	0.449073i	$0.148246\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	10451.8	+	10451.8i	0.511929	+	0.511929i
$748$		0			0
$749$			409.845i			0.0199939i
$750$		0			0
$751$		−	14742.3i		−	0.716317i	−0.933661	−	0.358158i	$-0.883405\pi$
$751$		−	14742.3i		−	0.716317i	0.933661	−	0.358158i	$-0.116595\pi$
$752$		0			0
$753$	−8358.79	−	8358.79i	−0.404530	−	0.404530i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	25706.4	−	25706.4i	1.23423	−	1.23423i	0.271911	−	0.962322i	$-0.412344\pi$
$757$	25706.4	−	25706.4i	1.23423	−	1.23423i	0.962322	−	0.271911i	$-0.0876557\pi$
$758$		0			0
$759$	−4914.55			−0.235029
$760$		0			0
$761$	−15546.9			−0.740571			−0.370285	−	0.928918i	$-0.620740\pi$
$761$	−15546.9			−0.740571			−0.370285	+	0.928918i	$0.620740\pi$
$762$		0			0
$763$	699.428	−	699.428i	0.0331861	−	0.0331861i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−200.295	−	200.295i	−0.00942925	−	0.00942925i
$768$		0			0
$769$			12699.1i			0.595500i	0.954644	+	0.297750i	$0.0962362\pi$
$769$			12699.1i			0.595500i	−0.954644	+	0.297750i	$0.903764\pi$
$770$		0			0
$771$		−	8345.90i		−	0.389845i
$772$		0			0
$773$	−6015.01	−	6015.01i	−0.279877	−	0.279877i	0.553183	−	0.833060i	$-0.313413\pi$
$773$	−6015.01	−	6015.01i	−0.279877	−	0.279877i	−0.833060	+	0.553183i	$0.813413\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−216.866	+	216.866i	−0.0100129	+	0.0100129i
$778$		0			0
$779$	−18048.5			−0.830107
$780$		0			0
$781$	−19146.7			−0.877237
$782$		0			0
$783$	−6041.72	+	6041.72i	−0.275752	+	0.275752i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	3296.03	+	3296.03i	0.149290	+	0.149290i	0.777801	−	0.628511i	$-0.216335\pi$
$787$	3296.03	+	3296.03i	0.149290	+	0.149290i	−0.628511	+	0.777801i	$0.716335\pi$
$788$		0			0
$789$		−	27166.3i		−	1.22579i
$790$		0			0
$791$		−	817.754i		−	0.0367585i
$792$		0			0
$793$	1021.34	+	1021.34i	0.0457365	+	0.0457365i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2001.35	−	2001.35i	0.0889480	−	0.0889480i	−0.661233	−	0.750181i	$-0.729966\pi$
$797$	2001.35	−	2001.35i	0.0889480	−	0.0889480i	0.750181	+	0.661233i	$0.229966\pi$
$798$		0			0
$799$	50113.5			2.21888
$800$		0			0
$801$	1467.17			0.0647188
$802$		0			0
$803$	2553.83	−	2553.83i	0.112232	−	0.112232i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	18684.3	+	18684.3i	0.815016	+	0.815016i
$808$		0			0
$809$			36567.3i			1.58917i	0.607152	+	0.794586i	$0.292312\pi$
$809$			36567.3i			1.58917i	−0.607152	+	0.794586i	$0.707688\pi$
$810$		0			0
$811$		−	28119.3i		−	1.21751i	−0.793357	−	0.608756i	$-0.791669\pi$
$811$		−	28119.3i		−	1.21751i	0.793357	−	0.608756i	$-0.208331\pi$
$812$		0			0
$813$	2345.24	+	2345.24i	0.101170	+	0.101170i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	17292.5	−	17292.5i	0.740498	−	0.740498i
$818$		0			0
$819$	−24.2207			−0.00103338
$820$		0			0
$821$	−7379.99			−0.313719			−0.156860	−	0.987621i	$-0.550137\pi$
$821$	−7379.99			−0.313719			−0.156860	+	0.987621i	$0.550137\pi$
$822$		0			0
$823$	18103.4	−	18103.4i	0.766761	−	0.766761i	−0.210774	−	0.977535i	$-0.567598\pi$
$823$	18103.4	−	18103.4i	0.766761	−	0.766761i	0.977535	+	0.210774i	$0.0675984\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	17442.7	+	17442.7i	0.733423	+	0.733423i	0.971296	−	0.237873i	$-0.0764503\pi$
$827$	17442.7	+	17442.7i	0.733423	+	0.733423i	−0.237873	+	0.971296i	$0.576450\pi$
$828$		0			0
$829$		−	9874.48i		−	0.413697i	−0.978373	−	0.206848i	$-0.933679\pi$
$829$		−	9874.48i		−	0.413697i	0.978373	−	0.206848i	$-0.0663207\pi$
$830$		0			0
$831$		−	17277.8i		−	0.721252i
$832$		0			0
$833$	−21920.7	−	21920.7i	−0.911775	−	0.911775i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	28197.0	−	28197.0i	1.16443	−	1.16443i
$838$		0			0
$839$	14931.1			0.614398			0.307199	−	0.951645i	$-0.400608\pi$
$839$	14931.1			0.614398			0.307199	+	0.951645i	$0.400608\pi$
$840$		0			0
$841$	20807.9			0.853169
$842$		0			0
$843$	259.821	−	259.821i	0.0106153	−	0.0106153i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−366.444	−	366.444i	−0.0148656	−	0.0148656i
$848$		0			0
$849$		−	1378.43i		−	0.0557216i
$850$		0			0
$851$		−	9103.13i		−	0.366688i
$852$		0			0
$853$	−14181.4	−	14181.4i	−0.569241	−	0.569241i	0.362675	−	0.931916i	$-0.381864\pi$
$853$	−14181.4	−	14181.4i	−0.569241	−	0.569241i	−0.931916	+	0.362675i	$0.881864\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5552.32	+	5552.32i	−0.221311	+	0.221311i	−0.809050	−	0.587739i	$-0.800018\pi$
$857$	−5552.32	+	5552.32i	−0.221311	+	0.221311i	0.587739	+	0.809050i	$0.300018\pi$
$858$		0			0
$859$	−1324.77			−0.0526199			−0.0263100	−	0.999654i	$-0.508376\pi$
$859$	−1324.77			−0.0526199			−0.0263100	+	0.999654i	$0.508376\pi$
$860$		0			0
$861$	592.997			0.0234719
$862$		0			0
$863$	30628.9	−	30628.9i	1.20814	−	1.20814i	0.236505	−	0.971630i	$-0.423998\pi$
$863$	30628.9	−	30628.9i	1.20814	−	1.20814i	0.971630	−	0.236505i	$-0.0760021\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−7700.79	−	7700.79i	−0.301652	−	0.301652i
$868$		0			0
$869$			1448.16i			0.0565309i
$870$		0			0
$871$		−	2232.03i		−	0.0868307i
$872$		0			0
$873$	9863.02	+	9863.02i	0.382374	+	0.382374i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−3525.59	+	3525.59i	−0.135748	+	0.135748i	−0.771716	−	0.635968i	$-0.780601\pi$
$877$	−3525.59	+	3525.59i	−0.135748	+	0.135748i	0.635968	+	0.771716i	$0.280601\pi$
$878$		0			0
$879$	−27467.0			−1.05397
$880$		0			0
$881$	−39472.2			−1.50948			−0.754740	−	0.656023i	$-0.772237\pi$
$881$	−39472.2			−1.50948			−0.754740	+	0.656023i	$0.772237\pi$
$882$		0			0
$883$	1438.08	−	1438.08i	0.0548079	−	0.0548079i	−0.679172	−	0.733980i	$-0.737661\pi$
$883$	1438.08	−	1438.08i	0.0548079	−	0.0548079i	0.733980	+	0.679172i	$0.237661\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18472.7	+	18472.7i	0.699272	+	0.699272i	0.964253	−	0.264982i	$-0.0853659\pi$
$887$	18472.7	+	18472.7i	0.699272	+	0.699272i	−0.264982	+	0.964253i	$0.585366\pi$
$888$		0			0
$889$			564.100i			0.0212816i
$890$		0			0
$891$		−	1017.88i		−	0.0382720i
$892$		0			0
$893$	25719.9	+	25719.9i	0.963813	+	0.963813i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−352.319	+	352.319i	−0.0131144	+	0.0131144i
$898$		0			0
$899$	16713.0			0.620033
$900$		0			0
$901$	−33965.1			−1.25587
$902$		0			0
$903$	−568.158	+	568.158i	−0.0209381	+	0.0209381i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	21841.3	+	21841.3i	0.799592	+	0.799592i	0.983031	−	0.183439i	$-0.0587231\pi$
$907$	21841.3	+	21841.3i	0.799592	+	0.799592i	−0.183439	+	0.983031i	$0.558723\pi$
$908$		0			0
$909$		−	3925.56i		−	0.143237i
$910$		0			0
$911$		−	45195.1i		−	1.64367i	−0.569727	−	0.821834i	$-0.692951\pi$
$911$		−	45195.1i		−	1.64367i	0.569727	−	0.821834i	$-0.307049\pi$
$912$		0			0
$913$	−15125.7	−	15125.7i	−0.548289	−	0.548289i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	587.068	−	587.068i	0.0211414	−	0.0211414i
$918$		0			0
$919$	−35372.3			−1.26967			−0.634834	−	0.772648i	$-0.718932\pi$
$919$	−35372.3			−1.26967			−0.634834	+	0.772648i	$0.718932\pi$
$920$		0			0
$921$	31081.8			1.11203
$922$		0			0
$923$	−1372.60	+	1372.60i	−0.0489489	+	0.0489489i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−2803.29	−	2803.29i	−0.0993227	−	0.0993227i
$928$		0			0
$929$		−	8353.38i		−	0.295011i	−0.989061	−	0.147506i	$-0.952876\pi$
$929$		−	8353.38i		−	0.295011i	0.989061	−	0.147506i	$-0.0471245\pi$
$930$		0			0
$931$		−	22500.9i		−	0.792093i
$932$		0			0
$933$	−18995.6	−	18995.6i	−0.666548	−	0.666548i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−24599.3	+	24599.3i	−0.857657	+	0.857657i	−0.991062	−	0.133405i	$-0.957409\pi$
$937$	−24599.3	+	24599.3i	−0.857657	+	0.857657i	0.133405	+	0.991062i	$0.457409\pi$
$938$		0			0
$939$	24894.6			0.865181
$940$		0			0
$941$	32772.2			1.13533			0.567663	−	0.823261i	$-0.307848\pi$
$941$	32772.2			1.13533			0.567663	+	0.823261i	$0.307848\pi$
$942$		0			0
$943$	−12445.8	+	12445.8i	−0.429788	+	0.429788i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	584.174	+	584.174i	0.0200455	+	0.0200455i	0.717059	−	0.697013i	$-0.245488\pi$
$947$	584.174	+	584.174i	0.0200455	+	0.0200455i	−0.697013	+	0.717059i	$0.745488\pi$
$948$		0			0
$949$		−	366.162i		−	0.0125249i
$950$		0			0
$951$			12279.1i			0.418694i
$952$		0			0
$953$	4578.37	+	4578.37i	0.155622	+	0.155622i	0.780624	−	0.625001i	$-0.214902\pi$
$953$	4578.37	+	4578.37i	0.155622	+	0.155622i	−0.625001	+	0.780624i	$0.714902\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	3246.67	−	3246.67i	0.109666	−	0.109666i
$958$		0			0
$959$	43.7690			0.00147380
$960$		0			0
$961$	−48209.4			−1.61825
$962$		0			0
$963$	7119.90	−	7119.90i	0.238251	−	0.238251i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	11282.8	+	11282.8i	0.375214	+	0.375214i	0.869372	−	0.494158i	$-0.164524\pi$
$967$	11282.8	+	11282.8i	0.375214	+	0.375214i	−0.494158	+	0.869372i	$0.664524\pi$
$968$		0			0
$969$		−	19759.9i		−	0.655086i
$970$		0			0
$971$			56918.3i			1.88115i	0.339588	+	0.940574i	$0.389712\pi$
$971$			56918.3i			1.88115i	−0.339588	+	0.940574i	$0.610288\pi$
$972$		0			0
$973$	−361.827	−	361.827i	−0.0119215	−	0.0119215i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11582.8	−	11582.8i	0.379291	−	0.379291i	−0.491555	−	0.870846i	$-0.663571\pi$
$977$	11582.8	−	11582.8i	0.379291	−	0.379291i	0.870846	+	0.491555i	$0.163571\pi$
$978$		0			0
$979$	−2123.27			−0.0693155
$980$		0			0
$981$	−24301.2			−0.790905
$982$		0			0
$983$	−9467.95	+	9467.95i	−0.307203	+	0.307203i	−0.843824	−	0.536621i	$-0.819701\pi$
$983$	−9467.95	+	9467.95i	−0.307203	+	0.307203i	0.536621	+	0.843824i	$0.319701\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−845.049	−	845.049i	−0.0272525	−	0.0272525i
$988$		0			0
$989$		−	23848.9i		−	0.766786i
$990$		0			0
$991$			48557.7i			1.55649i	0.627958	+	0.778247i	$0.283891\pi$
$991$			48557.7i			1.55649i	−0.627958	+	0.778247i	$0.716109\pi$
$992$		0			0
$993$	−15453.0	−	15453.0i	−0.493845	−	0.493845i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	12396.0	−	12396.0i	0.393766	−	0.393766i	−0.482262	−	0.876027i	$-0.660185\pi$
$997$	12396.0	−	12396.0i	0.393766	−	0.393766i	0.876027	+	0.482262i	$0.160185\pi$
$998$		0			0
$999$	20291.9			0.642651

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.n.g.143.5	yes	16
4.3	odd	2	inner	400.4.n.g.143.4	yes	16
5.2	odd	4	inner	400.4.n.g.207.3	yes	16
5.3	odd	4	inner	400.4.n.g.207.5	yes	16
5.4	even	2	inner	400.4.n.g.143.3	✓	16
20.3	even	4	inner	400.4.n.g.207.4	yes	16
20.7	even	4	inner	400.4.n.g.207.6	yes	16
20.19	odd	2	inner	400.4.n.g.143.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.4.n.g.143.3	✓	16	5.4	even	2	inner
400.4.n.g.143.4	yes	16	4.3	odd	2	inner
400.4.n.g.143.5	yes	16	1.1	even	1	trivial
400.4.n.g.143.6	yes	16	20.19	odd	2	inner
400.4.n.g.207.3	yes	16	5.2	odd	4	inner
400.4.n.g.207.4	yes	16	20.3	even	4	inner
400.4.n.g.207.5	yes	16	5.3	odd	4	inner
400.4.n.g.207.6	yes	16	20.7	even	4	inner

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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.35082 - 2.35082i) q^{3} +(-0.458991 - 0.458991i) q^{7} +15.9473i q^{9} -23.0788i q^{11} +(-1.65449 - 1.65449i) q^{13} +(63.9875 - 63.9875i) q^{17} +65.6811 q^{19} -2.15800 q^{21} +(45.2921 - 45.2921i) q^{23} +(100.961 + 100.961i) q^{27} +59.8420i q^{29} -279.285i q^{31} +(-54.2540 - 54.2540i) q^{33} +(100.494 - 100.494i) q^{37} -7.77882 q^{39} -274.789 q^{41} +(263.279 - 263.279i) q^{43} +(391.588 + 391.588i) q^{47} -342.579i q^{49} -300.846i q^{51} +(-265.404 - 265.404i) q^{53} +(154.404 - 154.404i) q^{57} +121.061 q^{59} -617.315 q^{61} +(7.31967 - 7.31967i) q^{63} +(674.536 + 674.536i) q^{67} -212.947i q^{69} -829.622i q^{71} +(110.657 + 110.657i) q^{73} +(-10.5930 + 10.5930i) q^{77} -62.7483 q^{79} +44.1046 q^{81} +(655.394 - 655.394i) q^{83} +(140.677 + 140.677i) q^{87} -92.0007i q^{89} +1.51879i q^{91} +(-656.549 - 656.549i) q^{93} +(618.474 - 618.474i) q^{97} +368.045 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1856 q^{21} - 1968 q^{41} - 1984 q^{61} - 9616 q^{81}+O(q^{100}) \) 16 * q - 1856 * q^21 - 1968 * q^41 - 1984 * q^61 - 9616 * q^81

Introduction

L-functions

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