LMFDB - Embedded newform 2304.4.f.e.1151.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,4,Mod(1151,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2304.1151");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2304.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$135.940400653$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.77720518656.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 161x^{4} + 4096 $ x^8 + 161*x^4 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.3
Root			$2.38456 - 2.38456i$ of defining polynomial
Character	$\chi$	$=$	2304.1151
Dual form			2304.4.f.e.1151.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-14.8339 q^{5} -30.9783i q^{7} +O(q^{10})$	$q-14.8339 q^{5} -30.9783i q^{7} +9.86874i q^{11} -54.9783i q^{13} +79.1652i q^{17} +43.9130 q^{19} -108.802 q^{23} +95.0435 q^{25} -231.116 q^{29} +318.978i q^{31} +459.527i q^{35} +38.1740i q^{37} +110.093i q^{41} +401.696 q^{43} -546.901 q^{47} -616.652 q^{49} +50.4660 q^{53} -146.391i q^{55} -500.693i q^{59} -865.478i q^{61} +815.540i q^{65} +306.043 q^{67} +194.024 q^{71} -867.043 q^{73} +305.716 q^{77} +816.674i q^{79} +1146.53i q^{83} -1174.33i q^{85} +468.043i q^{89} -1703.13 q^{91} -651.399 q^{95} +238.652 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 384 q^{19} + 1128 q^{25} + 640 q^{43} - 1992 q^{49} + 2816 q^{67} + 1152 q^{73} - 6272 q^{91} + 7424 q^{97}+O(q^{100}) $ 8 * q - 384 * q^19 + 1128 * q^25 + 640 * q^43 - 1992 * q^49 + 2816 * q^67 + 1152 * q^73 - 6272 * q^91 + 7424 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−14.8339	−1.32678	−0.663391	−	0.748273i	$-0.730883\pi$
$5$	−14.8339	−1.32678	−0.663391	+	0.748273i	$0.730883\pi$
$6$	0	0
$7$	− 30.9783i	− 1.67267i	−0.548220	−	0.836334i	$-0.684694\pi$
$7$	− 30.9783i	− 1.67267i	0.548220	−	0.836334i	$-0.315306\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9.86874i	0.270503i	0.990811	+	0.135252i	$0.0431843\pi$
$11$	9.86874i	0.270503i	−0.990811	+	0.135252i	$0.956816\pi$
$12$	0	0
$13$	− 54.9783i	− 1.17294i	−0.809971	−	0.586470i	$-0.800517\pi$
$13$	− 54.9783i	− 1.17294i	0.809971	−	0.586470i	$-0.199483\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	79.1652i	1.12943i	0.825285	+	0.564717i	$0.191015\pi$
$17$	79.1652i	1.12943i	−0.825285	+	0.564717i	$0.808985\pi$
$18$	0	0
$19$	43.9130	0.530228	0.265114	−	0.964217i	$-0.414590\pi$
$19$	43.9130	0.530228	0.265114	+	0.964217i	$0.414590\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−108.802	−0.986384	−0.493192	−	0.869921i	$-0.664170\pi$
$23$	−108.802	−0.986384	−0.493192	+	0.869921i	$0.664170\pi$
$24$	0	0
$25$	95.0435	0.760348
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−231.116	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$29$	−231.116	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$30$	0	0
$31$	318.978i	1.84807i	0.382307	+	0.924035i	$0.375130\pi$
$31$	318.978i	1.84807i	−0.382307	+	0.924035i	$0.624870\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	459.527i	2.21926i
$36$	0	0
$37$	38.1740i	0.169615i	0.996397	+	0.0848077i	$0.0270276\pi$
$37$	38.1740i	0.169615i	−0.996397	+	0.0848077i	$0.972972\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	110.093i	0.419358i	0.977770	+	0.209679i	$0.0672420\pi$
$41$	110.093i	0.419358i	−0.977770	+	0.209679i	$0.932758\pi$
$42$	0	0
$43$	401.696	1.42460	0.712302	−	0.701873i	$-0.247653\pi$
$43$	401.696	1.42460	0.712302	+	0.701873i	$0.247653\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−546.901	−1.69731	−0.848656	−	0.528945i	$-0.822588\pi$
$47$	−546.901	−1.69731	−0.848656	+	0.528945i	$0.822588\pi$
$48$	0	0
$49$	−616.652	−1.79782
$50$	0	0
$51$	0	0
$52$	0	0
$53$	50.4660	0.130793	0.0653966	−	0.997859i	$-0.479169\pi$
$53$	50.4660	0.130793	0.0653966	+	0.997859i	$0.479169\pi$
$54$	0	0
$55$	− 146.391i	− 0.358899i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 500.693i	− 1.10483i	−0.833571	−	0.552413i	$-0.813707\pi$
$59$	− 500.693i	− 1.10483i	0.833571	−	0.552413i	$-0.186293\pi$
$60$	0	0
$61$	− 865.478i	− 1.81661i	−0.418310	−	0.908304i	$-0.637378\pi$
$61$	− 865.478i	− 1.81661i	0.418310	−	0.908304i	$-0.362622\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	815.540i	1.55623i
$66$	0	0
$67$	306.043	0.558047	0.279024	−	0.960284i	$-0.409989\pi$
$67$	306.043	0.558047	0.279024	+	0.960284i	$0.409989\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	194.024	0.324316	0.162158	−	0.986765i	$-0.448155\pi$
$71$	194.024	0.324316	0.162158	+	0.986765i	$0.448155\pi$
$72$	0	0
$73$	−867.043	−1.39013	−0.695067	−	0.718945i	$-0.744625\pi$
$73$	−867.043	−1.39013	−0.695067	+	0.718945i	$0.744625\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	305.716	0.452462
$78$	0	0
$79$	816.674i	1.16308i	0.813519	+	0.581538i	$0.197549\pi$
$79$	816.674i	1.16308i	−0.813519	+	0.581538i	$0.802451\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1146.53i	1.51624i	0.652116	+	0.758119i	$0.273881\pi$
$83$	1146.53i	1.51624i	−0.652116	+	0.758119i	$0.726119\pi$
$84$	0	0
$85$	− 1174.33i	− 1.49851i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	468.043i	0.557444i	0.960372	+	0.278722i	$0.0899107\pi$
$89$	468.043i	0.557444i	−0.960372	+	0.278722i	$0.910089\pi$
$90$	0	0
$91$	−1703.13	−1.96194
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−651.399	−0.703497
$96$	0	0
$97$	238.652	0.249809	0.124905	−	0.992169i	$-0.460138\pi$
$97$	238.652	0.249809	0.124905	+	0.992169i	$0.460138\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	94.2144	0.0928186	0.0464093	−	0.998923i	$-0.485222\pi$
$101$	94.2144	0.0928186	0.0464093	+	0.998923i	$0.485222\pi$
$102$	0	0
$103$	− 792.152i	− 0.757796i	−0.925438	−	0.378898i	$-0.876303\pi$
$103$	− 792.152i	− 0.757796i	0.925438	−	0.378898i	$-0.123697\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	496.819i	0.448872i	0.974489	+	0.224436i	$0.0720540\pi$
$107$	496.819i	0.448872i	−0.974489	+	0.224436i	$0.927946\pi$
$108$	0	0
$109$	131.239i	0.115325i	0.998336	+	0.0576626i	$0.0183648\pi$
$109$	131.239i	0.115325i	−0.998336	+	0.0576626i	$0.981635\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1253.85i	− 1.04383i	−0.852998	−	0.521914i	$-0.825218\pi$
$113$	− 1253.85i	− 1.04383i	0.852998	−	0.521914i	$-0.174782\pi$
$114$	0	0
$115$	1613.96	1.30871
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2452.40	1.88917
$120$	0	0
$121$	1233.61	0.926828
$122$	0	0
$123$	0	0
$124$	0	0
$125$	444.371	0.317966
$126$	0	0
$127$	− 1428.76i	− 0.998284i	−0.866520	−	0.499142i	$-0.833649\pi$
$127$	− 1428.76i	− 0.998284i	0.866520	−	0.499142i	$-0.166351\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2626.19i	− 1.75154i	−0.482729	−	0.875770i	$-0.660355\pi$
$131$	− 2626.19i	− 1.75154i	0.482729	−	0.875770i	$-0.339645\pi$
$132$	0	0
$133$	− 1360.35i	− 0.886896i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2199.81i	1.37184i	0.727676	+	0.685921i	$0.240600\pi$
$137$	2199.81i	1.37184i	−0.727676	+	0.685921i	$0.759400\pi$
$138$	0	0
$139$	−686.652	−0.419001	−0.209500	−	0.977809i	$-0.567184\pi$
$139$	−686.652	−0.419001	−0.209500	+	0.977809i	$0.567184\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	542.566	0.317284
$144$	0	0
$145$	3428.35	1.96351
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−440.850	−0.242388	−0.121194	−	0.992629i	$-0.538672\pi$
$149$	−440.850	−0.242388	−0.121194	+	0.992629i	$0.538672\pi$
$150$	0	0
$151$	860.239i	0.463611i	0.972762	+	0.231805i	$0.0744633\pi$
$151$	860.239i	0.463611i	−0.972762	+	0.231805i	$0.925537\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 4731.68i	− 2.45198i
$156$	0	0
$157$	1408.78i	0.716134i	0.933696	+	0.358067i	$0.116564\pi$
$157$	1408.78i	0.716134i	−0.933696	+	0.358067i	$0.883436\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3370.50i	1.64989i
$162$	0	0
$163$	−2608.30	−1.25336	−0.626681	−	0.779276i	$-0.715587\pi$
$163$	−2608.30	−1.25336	−0.626681	+	0.779276i	$0.715587\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1473.67	0.682851	0.341426	−	0.939909i	$-0.389090\pi$
$167$	1473.67	0.682851	0.341426	+	0.939909i	$0.389090\pi$
$168$	0	0
$169$	−825.608	−0.375789
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3158.69	1.38816	0.694078	−	0.719900i	$-0.255812\pi$
$173$	3158.69	1.38816	0.694078	+	0.719900i	$0.255812\pi$
$174$	0	0
$175$	− 2944.28i	− 1.27181i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 62.1024i	− 0.0259316i	−0.999916	−	0.0129658i	$-0.995873\pi$
$179$	− 62.1024i	− 0.0259316i	0.999916	−	0.0129658i	$-0.00412725\pi$
$180$	0	0
$181$	2450.50i	1.00632i	0.864193	+	0.503161i	$0.167830\pi$
$181$	2450.50i	1.00632i	−0.864193	+	0.503161i	$0.832170\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 566.268i	− 0.225042i
$186$	0	0
$187$	−781.261	−0.305516
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1418.83	0.537501	0.268750	−	0.963210i	$-0.413389\pi$
$191$	1418.83	0.537501	0.268750	+	0.963210i	$0.413389\pi$
$192$	0	0
$193$	941.652	0.351200	0.175600	−	0.984462i	$-0.443813\pi$
$193$	941.652	0.351200	0.175600	+	0.984462i	$0.443813\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5158.88	1.86576	0.932881	−	0.360185i	$-0.117286\pi$
$197$	5158.88	1.86576	0.932881	+	0.360185i	$0.117286\pi$
$198$	0	0
$199$	44.6312i	0.0158986i	0.999968	+	0.00794930i	$0.00253037\pi$
$199$	44.6312i	0.0158986i	−0.999968	+	0.00794930i	$0.997470\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7159.58i	2.47539i
$204$	0	0
$205$	− 1633.11i	− 0.556397i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	433.366i	0.143428i
$210$	0	0
$211$	−1591.52	−0.519265	−0.259633	−	0.965707i	$-0.583601\pi$
$211$	−1591.52	−0.519265	−0.259633	+	0.965707i	$0.583601\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5958.70	−1.89014
$216$	0	0
$217$	9881.39	3.09121
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4352.36	1.32476
$222$	0	0
$223$	3940.37i	1.18326i	0.806210	+	0.591629i	$0.201515\pi$
$223$	3940.37i	1.18326i	−0.806210	+	0.591629i	$0.798485\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1123.41i	− 0.328472i	−0.986421	−	0.164236i	$-0.947484\pi$
$227$	− 1123.41i	− 0.328472i	0.986421	−	0.164236i	$-0.0525159\pi$
$228$	0	0
$229$	221.760i	0.0639927i	0.999488	+	0.0319964i	$0.0101865\pi$
$229$	221.760i	0.0639927i	−0.999488	+	0.0319964i	$0.989814\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3109.18i	0.874202i	0.899412	+	0.437101i	$0.143995\pi$
$233$	3109.18i	0.874202i	−0.899412	+	0.437101i	$0.856005\pi$
$234$	0	0
$235$	8112.65	2.25196
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2871.71	0.777221	0.388610	−	0.921402i	$-0.372955\pi$
$239$	2871.71	0.777221	0.388610	+	0.921402i	$0.372955\pi$
$240$	0	0
$241$	1802.82	0.481868	0.240934	−	0.970542i	$-0.422546\pi$
$241$	1802.82	0.481868	0.240934	+	0.970542i	$0.422546\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9147.33	2.38531
$246$	0	0
$247$	− 2414.26i	− 0.621926i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 707.015i	− 0.177794i	−0.996041	−	0.0888972i	$-0.971666\pi$
$251$	− 707.015i	− 0.177794i	0.996041	−	0.0888972i	$-0.0283343\pi$
$252$	0	0
$253$	− 1073.74i	− 0.266820i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1564.18i	− 0.379653i	−0.981818	−	0.189827i	$-0.939207\pi$
$257$	− 1564.18i	− 0.379653i	0.981818	−	0.189827i	$-0.0607926\pi$
$258$	0	0
$259$	1182.56	0.283710
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8146.79	−1.91009	−0.955043	−	0.296467i	$-0.904192\pi$
$263$	−8146.79	−1.91009	−0.955043	+	0.296467i	$0.904192\pi$
$264$	0	0
$265$	−748.606	−0.173534
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−47.3601	−0.0107346	−0.00536728	−	0.999986i	$-0.501708\pi$
$269$	−47.3601	−0.0107346	−0.00536728	+	0.999986i	$0.501708\pi$
$270$	0	0
$271$	1112.11i	0.249283i	0.992202	+	0.124642i	$0.0397781\pi$
$271$	1112.11i	0.249283i	−0.992202	+	0.124642i	$0.960222\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	937.959i	0.205677i
$276$	0	0
$277$	− 2773.59i	− 0.601620i	−0.953684	−	0.300810i	$-0.902743\pi$
$277$	− 2773.59i	− 0.601620i	0.953684	−	0.300810i	$-0.0972569\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 7526.50i	− 1.59784i	−0.601436	−	0.798921i	$-0.705405\pi$
$281$	− 7526.50i	− 1.59784i	0.601436	−	0.798921i	$-0.294595\pi$
$282$	0	0
$283$	166.175	0.0349049	0.0174524	−	0.999848i	$-0.494444\pi$
$283$	166.175	0.0349049	0.0174524	+	0.999848i	$0.494444\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3410.50	0.701447
$288$	0	0
$289$	−1354.13	−0.275622
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−740.755	−0.147698	−0.0738488	−	0.997269i	$-0.523528\pi$
$293$	−740.755	−0.147698	−0.0738488	+	0.997269i	$0.523528\pi$
$294$	0	0
$295$	7427.21i	1.46586i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5981.75i	1.15697i
$300$	0	0
$301$	− 12443.8i	− 2.38289i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12838.4i	2.41024i
$306$	0	0
$307$	−4096.30	−0.761526	−0.380763	−	0.924673i	$-0.624339\pi$
$307$	−4096.30	−0.761526	−0.380763	+	0.924673i	$0.624339\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1427.93	0.260354	0.130177	−	0.991491i	$-0.458445\pi$
$311$	1427.93	0.260354	0.130177	+	0.991491i	$0.458445\pi$
$312$	0	0
$313$	5362.17	0.968332	0.484166	−	0.874976i	$-0.339123\pi$
$313$	5362.17	0.968332	0.484166	+	0.874976i	$0.339123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2520.79	0.446630	0.223315	−	0.974746i	$-0.428312\pi$
$317$	2520.79	0.446630	0.223315	+	0.974746i	$0.428312\pi$
$318$	0	0
$319$	− 2280.83i	− 0.400319i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3476.38i	0.598858i
$324$	0	0
$325$	− 5225.33i	− 0.891843i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16942.0i	2.83904i
$330$	0	0
$331$	7083.08	1.17620	0.588099	−	0.808789i	$-0.299877\pi$
$331$	7083.08	1.17620	0.588099	+	0.808789i	$0.299877\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4539.81	−0.740406
$336$	0	0
$337$	9814.78	1.58648	0.793242	−	0.608906i	$-0.208391\pi$
$337$	9814.78	1.58648	0.793242	+	0.608906i	$0.208391\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3147.91	−0.499909
$342$	0	0
$343$	8477.26i	1.33449i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5362.98i	0.829682i	0.909894	+	0.414841i	$0.136163\pi$
$347$	5362.98i	0.829682i	−0.909894	+	0.414841i	$0.863837\pi$
$348$	0	0
$349$	752.958i	0.115487i	0.998331	+	0.0577434i	$0.0183905\pi$
$349$	752.958i	0.115487i	−0.998331	+	0.0577434i	$0.981609\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5317.93i	0.801827i	0.916116	+	0.400914i	$0.131307\pi$
$353$	5317.93i	0.801827i	−0.916116	+	0.400914i	$0.868693\pi$
$354$	0	0
$355$	−2878.13	−0.430296
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1064.38	0.156478	0.0782392	−	0.996935i	$-0.475070\pi$
$359$	1064.38	0.156478	0.0782392	+	0.996935i	$0.475070\pi$
$360$	0	0
$361$	−4930.65	−0.718858
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12861.6	1.84440
$366$	0	0
$367$	3455.59i	0.491499i	0.969333	+	0.245749i	$0.0790340\pi$
$367$	3455.59i	0.491499i	−0.969333	+	0.245749i	$0.920966\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1563.35i	− 0.218774i
$372$	0	0
$373$	− 7013.13i	− 0.973528i	−0.873533	−	0.486764i	$-0.838177\pi$
$373$	− 7013.13i	− 0.973528i	0.873533	−	0.486764i	$-0.161823\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12706.4i	1.73584i
$378$	0	0
$379$	−4644.35	−0.629457	−0.314728	−	0.949182i	$-0.601913\pi$
$379$	−4644.35	−0.629457	−0.314728	+	0.949182i	$0.601913\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12025.7	−1.60440	−0.802201	−	0.597053i	$-0.796338\pi$
$383$	−12025.7	−1.60440	−0.802201	+	0.597053i	$0.796338\pi$
$384$	0	0
$385$	−4534.95	−0.600318
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9478.32	1.23540	0.617699	−	0.786415i	$-0.288065\pi$
$389$	9478.32	1.23540	0.617699	+	0.786415i	$0.288065\pi$
$390$	0	0
$391$	− 8613.35i	− 1.11406i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 12114.4i	− 1.54315i
$396$	0	0
$397$	6747.39i	0.853002i	0.904487	+	0.426501i	$0.140254\pi$
$397$	6747.39i	0.853002i	−0.904487	+	0.426501i	$0.859746\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 524.089i	− 0.0652663i	−0.999467	−	0.0326331i	$-0.989611\pi$
$401$	− 524.089i	− 0.0652663i	0.999467	−	0.0326331i	$-0.0103893\pi$
$402$	0	0
$403$	17536.9	2.16768
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−376.729	−0.0458815
$408$	0	0
$409$	−1225.87	−0.148204	−0.0741019	−	0.997251i	$-0.523609\pi$
$409$	−1225.87	−0.148204	−0.0741019	+	0.997251i	$0.523609\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−15510.6	−1.84801
$414$	0	0
$415$	− 17007.4i	− 2.01172i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 9769.48i	− 1.13907i	−0.821967	−	0.569535i	$-0.807123\pi$
$419$	− 9769.48i	− 1.13907i	0.821967	−	0.569535i	$-0.192877\pi$
$420$	0	0
$421$	− 10097.4i	− 1.16893i	−0.811421	−	0.584463i	$-0.801305\pi$
$421$	− 10097.4i	− 1.16893i	0.811421	−	0.584463i	$-0.198695\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7524.14i	0.858763i
$426$	0	0
$427$	−26811.0	−3.03858
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9338.69	1.04369	0.521843	−	0.853041i	$-0.325245\pi$
$431$	9338.69	1.04369	0.521843	+	0.853041i	$0.325245\pi$
$432$	0	0
$433$	3872.09	0.429748	0.214874	−	0.976642i	$-0.431066\pi$
$433$	3872.09	0.429748	0.214874	+	0.976642i	$0.431066\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4777.83	−0.523008
$438$	0	0
$439$	− 10236.7i	− 1.11292i	−0.830875	−	0.556460i	$-0.812159\pi$
$439$	− 10236.7i	− 1.11292i	0.830875	−	0.556460i	$-0.187841\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2296.53i	0.246301i	0.992388	+	0.123150i	$0.0392997\pi$
$443$	2296.53i	0.246301i	−0.992388	+	0.123150i	$0.960700\pi$
$444$	0	0
$445$	− 6942.89i	− 0.739606i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14250.3i	1.49780i	0.662684	+	0.748899i	$0.269417\pi$
$449$	14250.3i	1.49780i	−0.662684	+	0.748899i	$0.730583\pi$
$450$	0	0
$451$	−1086.48	−0.113438
$452$	0	0
$453$	0	0
$454$	0	0
$455$	25264.0	2.60306
$456$	0	0
$457$	−7673.43	−0.785444	−0.392722	−	0.919657i	$-0.628467\pi$
$457$	−7673.43	−0.785444	−0.392722	+	0.919657i	$0.628467\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17198.5	1.73756	0.868781	−	0.495197i	$-0.164904\pi$
$461$	17198.5	1.73756	0.868781	+	0.495197i	$0.164904\pi$
$462$	0	0
$463$	3819.89i	0.383424i	0.981451	+	0.191712i	$0.0614040\pi$
$463$	3819.89i	0.383424i	−0.981451	+	0.191712i	$0.938596\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1280.20i	− 0.126853i	−0.997987	−	0.0634266i	$-0.979797\pi$
$467$	− 1280.20i	− 0.126853i	0.997987	−	0.0634266i	$-0.0202029\pi$
$468$	0	0
$469$	− 9480.69i	− 0.933428i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3964.23i	0.385360i
$474$	0	0
$475$	4173.65	0.403158
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13847.2	1.32087	0.660433	−	0.750885i	$-0.270373\pi$
$479$	13847.2	1.32087	0.660433	+	0.750885i	$0.270373\pi$
$480$	0	0
$481$	2098.74	0.198949
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3540.14	−0.331442
$486$	0	0
$487$	− 1663.54i	− 0.154789i	−0.997001	−	0.0773945i	$-0.975340\pi$
$487$	− 1663.54i	− 0.154789i	0.997001	−	0.0773945i	$-0.0246601\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 18521.0i	− 1.70233i	−0.524901	−	0.851163i	$-0.675898\pi$
$491$	− 18521.0i	− 1.70233i	0.524901	−	0.851163i	$-0.324102\pi$
$492$	0	0
$493$	− 18296.4i	− 1.67145i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6010.53i	− 0.542473i
$498$	0	0
$499$	−8732.34	−0.783393	−0.391697	−	0.920094i	$-0.628112\pi$
$499$	−8732.34	−0.783393	−0.391697	+	0.920094i	$0.628112\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14898.7	1.32068	0.660338	−	0.750968i	$-0.270413\pi$
$503$	14898.7	1.32068	0.660338	+	0.750968i	$0.270413\pi$
$504$	0	0
$505$	−1397.56	−0.123150
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11287.0	0.982886	0.491443	−	0.870910i	$-0.336470\pi$
$509$	11287.0	0.982886	0.491443	+	0.870910i	$0.336470\pi$
$510$	0	0
$511$	26859.5i	2.32523i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11750.7i	1.00543i
$516$	0	0
$517$	− 5397.22i	− 0.459129i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 11544.2i	− 0.970749i	−0.874306	−	0.485375i	$-0.838683\pi$
$521$	− 11544.2i	− 0.970749i	0.874306	−	0.485375i	$-0.161317\pi$
$522$	0	0
$523$	−1393.21	−0.116484	−0.0582419	−	0.998303i	$-0.518549\pi$
$523$	−1393.21	−0.116484	−0.0582419	+	0.998303i	$0.518549\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25252.0	−2.08727
$528$	0	0
$529$	−329.088	−0.0270476
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6052.74	0.491882
$534$	0	0
$535$	− 7369.74i	− 0.595555i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 6085.58i	− 0.486316i
$540$	0	0
$541$	− 9100.41i	− 0.723211i	−0.932331	−	0.361606i	$-0.882229\pi$
$541$	− 9100.41i	− 0.723211i	0.932331	−	0.361606i	$-0.117771\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1946.78i	− 0.153011i
$546$	0	0
$547$	18886.8	1.47631	0.738156	−	0.674630i	$-0.235697\pi$
$547$	18886.8	1.47631	0.738156	+	0.674630i	$0.235697\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10149.0	−0.784687
$552$	0	0
$553$	25299.1	1.94544
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4816.02	−0.366358	−0.183179	−	0.983080i	$-0.558639\pi$
$557$	−4816.02	−0.366358	−0.183179	+	0.983080i	$0.558639\pi$
$558$	0	0
$559$	− 22084.5i	− 1.67098i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4690.73i	0.351138i	0.984467	+	0.175569i	$0.0561765\pi$
$563$	4690.73i	0.351138i	−0.984467	+	0.175569i	$0.943824\pi$
$564$	0	0
$565$	18599.5i	1.38493i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 6547.96i	− 0.482434i	−0.970471	−	0.241217i	$-0.922453\pi$
$569$	− 6547.96i	− 0.482434i	0.970471	−	0.241217i	$-0.0775465\pi$
$570$	0	0
$571$	13043.0	0.955927	0.477964	−	0.878380i	$-0.341375\pi$
$571$	13043.0	0.955927	0.477964	+	0.878380i	$0.341375\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−10340.9	−0.749995
$576$	0	0
$577$	8853.82	0.638803	0.319402	−	0.947619i	$-0.396518\pi$
$577$	8853.82	0.638803	0.319402	+	0.947619i	$0.396518\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	35517.4	2.53616
$582$	0	0
$583$	498.036i	0.0353800i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18849.9i	− 1.32542i	−0.748878	−	0.662708i	$-0.769407\pi$
$587$	− 18849.9i	− 1.32542i	0.748878	−	0.662708i	$-0.230593\pi$
$588$	0	0
$589$	14007.3i	0.979899i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4673.42i	0.323633i	0.986821	+	0.161817i	$0.0517353\pi$
$593$	4673.42i	0.323633i	−0.986821	+	0.161817i	$0.948265\pi$
$594$	0	0
$595$	−36378.6	−2.50651
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12367.0	−0.843573	−0.421786	−	0.906695i	$-0.638597\pi$
$599$	−12367.0	−0.843573	−0.421786	+	0.906695i	$0.638597\pi$
$600$	0	0
$601$	14617.0	0.992077	0.496038	−	0.868301i	$-0.334788\pi$
$601$	14617.0	0.992077	0.496038	+	0.868301i	$0.334788\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−18299.2	−1.22970
$606$	0	0
$607$	4724.02i	0.315885i	0.987448	+	0.157942i	$0.0504860\pi$
$607$	4724.02i	0.315885i	−0.987448	+	0.157942i	$0.949514\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30067.6i	1.99085i
$612$	0	0
$613$	11318.3i	0.745748i	0.927882	+	0.372874i	$0.121628\pi$
$613$	11318.3i	0.745748i	−0.927882	+	0.372874i	$0.878372\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6420.71i	− 0.418943i	−0.977815	−	0.209472i	$-0.932826\pi$
$617$	− 6420.71i	− 0.418943i	0.977815	−	0.209472i	$-0.0671744\pi$
$618$	0	0
$619$	−9684.86	−0.628865	−0.314433	−	0.949280i	$-0.601814\pi$
$619$	−9684.86	−0.628865	−0.314433	+	0.949280i	$0.601814\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14499.2	0.932418
$624$	0	0
$625$	−18472.2	−1.18222
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3022.05	−0.191569
$630$	0	0
$631$	− 3642.85i	− 0.229825i	−0.993376	−	0.114912i	$-0.963341\pi$
$631$	− 3642.85i	− 0.229825i	0.993376	−	0.114912i	$-0.0366587\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	21194.0i	1.32450i
$636$	0	0
$637$	33902.4i	2.10873i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 2768.44i	− 0.170588i	−0.996356	−	0.0852939i	$-0.972817\pi$
$641$	− 2768.44i	− 0.170588i	0.996356	−	0.0852939i	$-0.0271829\pi$
$642$	0	0
$643$	8558.82	0.524925	0.262463	−	0.964942i	$-0.415465\pi$
$643$	8558.82	0.524925	0.262463	+	0.964942i	$0.415465\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12937.7	−0.786144	−0.393072	−	0.919508i	$-0.628588\pi$
$647$	−12937.7	−0.786144	−0.393072	+	0.919508i	$0.628588\pi$
$648$	0	0
$649$	4941.21	0.298859
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25595.6	1.53389	0.766946	−	0.641712i	$-0.221775\pi$
$653$	25595.6	1.53389	0.766946	+	0.641712i	$0.221775\pi$
$654$	0	0
$655$	38956.6i	2.32391i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 12541.3i	− 0.741335i	−0.928766	−	0.370668i	$-0.879129\pi$
$659$	− 12541.3i	− 0.741335i	0.928766	−	0.370668i	$-0.120871\pi$
$660$	0	0
$661$	16135.2i	0.949451i	0.880134	+	0.474726i	$0.157453\pi$
$661$	16135.2i	0.949451i	−0.880134	+	0.474726i	$0.842547\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	20179.2i	1.17672i
$666$	0	0
$667$	25146.0	1.45975
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8541.17	0.491398
$672$	0	0
$673$	25243.7	1.44588	0.722938	−	0.690913i	$-0.242791\pi$
$673$	25243.7	1.44588	0.722938	+	0.690913i	$0.242791\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−11027.7	−0.626039	−0.313019	−	0.949747i	$-0.601341\pi$
$677$	−11027.7	−0.626039	−0.313019	+	0.949747i	$0.601341\pi$
$678$	0	0
$679$	− 7393.04i	− 0.417848i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15409.7i	0.863304i	0.902040	+	0.431652i	$0.142069\pi$
$683$	15409.7i	0.863304i	−0.902040	+	0.431652i	$0.857931\pi$
$684$	0	0
$685$	− 32631.7i	− 1.82013i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 2774.53i	− 0.153413i
$690$	0	0
$691$	23759.2	1.30802	0.654010	−	0.756486i	$-0.273086\pi$
$691$	23759.2	1.30802	0.654010	+	0.756486i	$0.273086\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10185.7	0.555922
$696$	0	0
$697$	−8715.56	−0.473638
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24574.3	−1.32405	−0.662026	−	0.749481i	$-0.730303\pi$
$701$	−24574.3	−1.32405	−0.662026	+	0.749481i	$0.730303\pi$
$702$	0	0
$703$	1676.33i	0.0899348i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2918.60i	− 0.155255i
$708$	0	0
$709$	20614.2i	1.09194i	0.837806	+	0.545968i	$0.183838\pi$
$709$	20614.2i	1.09194i	−0.837806	+	0.545968i	$0.816162\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 34705.5i	− 1.82291i
$714$	0	0
$715$	−8048.35	−0.420967
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3239.50	−0.168029	−0.0840146	−	0.996465i	$-0.526774\pi$
$719$	−3239.50	−0.168029	−0.0840146	+	0.996465i	$0.526774\pi$
$720$	0	0
$721$	−24539.5	−1.26754
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−21966.1	−1.12524
$726$	0	0
$727$	27590.8i	1.40755i	0.710425	+	0.703773i	$0.248503\pi$
$727$	27590.8i	1.40755i	−0.710425	+	0.703773i	$0.751497\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	31800.3i	1.60900i
$732$	0	0
$733$	32932.1i	1.65945i	0.558173	+	0.829725i	$0.311503\pi$
$733$	32932.1i	1.65945i	−0.558173	+	0.829725i	$0.688497\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3020.26i	0.150954i
$738$	0	0
$739$	−7766.70	−0.386607	−0.193304	−	0.981139i	$-0.561920\pi$
$739$	−7766.70	−0.386607	−0.193304	+	0.981139i	$0.561920\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−32172.3	−1.58854	−0.794271	−	0.607563i	$-0.792147\pi$
$743$	−32172.3	−1.58854	−0.794271	+	0.607563i	$0.792147\pi$
$744$	0	0
$745$	6539.52	0.321596
$746$	0	0
$747$	0	0
$748$	0	0
$749$	15390.6	0.750814
$750$	0	0
$751$	− 12175.5i	− 0.591600i	−0.955250	−	0.295800i	$-0.904414\pi$
$751$	− 12175.5i	− 0.591600i	0.955250	−	0.295800i	$-0.0955862\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 12760.7i	− 0.615110i
$756$	0	0
$757$	14541.7i	0.698185i	0.937088	+	0.349092i	$0.113510\pi$
$757$	14541.7i	0.698185i	−0.937088	+	0.349092i	$0.886490\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 8669.16i	− 0.412952i	−0.978452	−	0.206476i	$-0.933800\pi$
$761$	− 8669.16i	− 0.412952i	0.978452	−	0.206476i	$-0.0661996\pi$
$762$	0	0
$763$	4065.56	0.192901
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−27527.2	−1.29589
$768$	0	0
$769$	28389.1	1.33126	0.665629	−	0.746282i	$-0.268163\pi$
$769$	28389.1	1.33126	0.665629	+	0.746282i	$0.268163\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−15782.6	−0.734362	−0.367181	−	0.930149i	$-0.619677\pi$
$773$	−15782.6	−0.734362	−0.367181	+	0.930149i	$0.619677\pi$
$774$	0	0
$775$	30316.8i	1.40518i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4834.53i	0.222356i
$780$	0	0
$781$	1914.77i	0.0877285i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 20897.7i	− 0.950153i
$786$	0	0
$787$	25460.6	1.15321	0.576603	−	0.817025i	$-0.304378\pi$
$787$	25460.6	1.15321	0.576603	+	0.817025i	$0.304378\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−38842.2	−1.74598
$792$	0	0
$793$	−47582.5	−2.13077
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19600.9	−0.871139	−0.435570	−	0.900155i	$-0.643453\pi$
$797$	−19600.9	−0.871139	−0.435570	+	0.900155i	$0.643453\pi$
$798$	0	0
$799$	− 43295.5i	− 1.91700i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 8556.62i	− 0.376036i
$804$	0	0
$805$	− 49997.5i	− 2.18905i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3065.74i	0.133233i	0.997779	+	0.0666166i	$0.0212204\pi$
$809$	3065.74i	0.133233i	−0.997779	+	0.0666166i	$0.978780\pi$
$810$	0	0
$811$	44326.1	1.91923	0.959617	−	0.281309i	$-0.0907686\pi$
$811$	44326.1	1.91923	0.959617	+	0.281309i	$0.0907686\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	38691.2	1.66294
$816$	0	0
$817$	17639.7	0.755365
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21842.2	0.928498	0.464249	−	0.885705i	$-0.346324\pi$
$821$	21842.2	0.928498	0.464249	+	0.885705i	$0.346324\pi$
$822$	0	0
$823$	5363.37i	0.227163i	0.993529	+	0.113582i	$0.0362323\pi$
$823$	5363.37i	0.227163i	−0.993529	+	0.113582i	$0.963768\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11117.1i	0.467450i	0.972303	+	0.233725i	$0.0750915\pi$
$827$	11117.1i	0.467450i	−0.972303	+	0.233725i	$0.924909\pi$
$828$	0	0
$829$	41554.2i	1.74094i	0.492224	+	0.870469i	$0.336184\pi$
$829$	41554.2i	1.74094i	−0.492224	+	0.870469i	$0.663816\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 48817.4i	− 2.03052i
$834$	0	0
$835$	−21860.2	−0.905994
$836$	0	0
$837$	0	0
$838$	0	0
$839$	34881.7	1.43534	0.717669	−	0.696384i	$-0.245209\pi$
$839$	34881.7	1.43534	0.717669	+	0.696384i	$0.245209\pi$
$840$	0	0
$841$	29025.7	1.19012
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12247.0	0.498589
$846$	0	0
$847$	− 38215.0i	− 1.55028i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 4153.41i	− 0.167306i
$852$	0	0
$853$	− 26716.2i	− 1.07239i	−0.844095	−	0.536194i	$-0.819862\pi$
$853$	− 26716.2i	− 1.07239i	0.844095	−	0.536194i	$-0.180138\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11875.9i	0.473362i	0.971587	+	0.236681i	$0.0760597\pi$
$857$	11875.9i	0.473362i	−0.971587	+	0.236681i	$0.923940\pi$
$858$	0	0
$859$	−26907.2	−1.06876	−0.534378	−	0.845246i	$-0.679454\pi$
$859$	−26907.2	−1.06876	−0.534378	+	0.845246i	$0.679454\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5839.41	0.230331	0.115166	−	0.993346i	$-0.463260\pi$
$863$	5839.41	0.230331	0.115166	+	0.993346i	$0.463260\pi$
$864$	0	0
$865$	−46855.6	−1.84178
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8059.54	−0.314616
$870$	0	0
$871$	− 16825.7i	− 0.654556i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 13765.8i	− 0.531851i
$876$	0	0
$877$	− 43994.2i	− 1.69393i	−0.531648	−	0.846965i	$-0.678427\pi$
$877$	− 43994.2i	− 1.69393i	0.531648	−	0.846965i	$-0.321573\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 12290.1i	− 0.469992i	−0.971996	−	0.234996i	$-0.924492\pi$
$881$	− 12290.1i	− 0.469992i	0.971996	−	0.234996i	$-0.0755077\pi$
$882$	0	0
$883$	−33329.9	−1.27026	−0.635131	−	0.772404i	$-0.719054\pi$
$883$	−33329.9	−1.27026	−0.635131	+	0.772404i	$0.719054\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39661.2	1.50135	0.750673	−	0.660674i	$-0.229729\pi$
$887$	39661.2	1.50135	0.750673	+	0.660674i	$0.229729\pi$
$888$	0	0
$889$	−44260.5	−1.66980
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−24016.1	−0.899963
$894$	0	0
$895$	921.218i	0.0344055i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 73721.1i	− 2.73497i
$900$	0	0
$901$	3995.15i	0.147722i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 36350.4i	− 1.33517i
$906$	0	0
$907$	−42455.0	−1.55424	−0.777120	−	0.629353i	$-0.783320\pi$
$907$	−42455.0	−1.55424	−0.777120	+	0.629353i	$0.783320\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22949.8	−0.834643	−0.417322	−	0.908759i	$-0.637031\pi$
$911$	−22949.8	−0.834643	−0.417322	+	0.908759i	$0.637031\pi$
$912$	0	0
$913$	−11314.8	−0.410147
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−81354.9	−2.92974
$918$	0	0
$919$	44604.0i	1.60103i	0.599311	+	0.800516i	$0.295441\pi$
$919$	44604.0i	1.60103i	−0.599311	+	0.800516i	$0.704559\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 10667.1i	− 0.380403i
$924$	0	0
$925$	3628.19i	0.128967i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 25063.5i	− 0.885153i	−0.896731	−	0.442577i	$-0.854064\pi$
$929$	− 25063.5i	− 0.885153i	0.896731	−	0.442577i	$-0.145936\pi$
$930$	0	0
$931$	−27079.0	−0.953254
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11589.1	0.405352
$936$	0	0
$937$	−33752.1	−1.17677	−0.588385	−	0.808581i	$-0.700236\pi$
$937$	−33752.1	−1.17677	−0.588385	+	0.808581i	$0.700236\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−39088.6	−1.35415	−0.677074	−	0.735915i	$-0.736752\pi$
$941$	−39088.6	−1.35415	−0.677074	+	0.735915i	$0.736752\pi$
$942$	0	0
$943$	− 11978.4i	− 0.413648i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7294.32i	0.250299i	0.992138	+	0.125150i	$0.0399411\pi$
$947$	7294.32i	0.250299i	−0.992138	+	0.125150i	$0.960059\pi$
$948$	0	0
$949$	47668.5i	1.63054i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16201.6i	0.550703i	0.961344	+	0.275351i	$0.0887942\pi$
$953$	16201.6i	0.550703i	−0.961344	+	0.275351i	$0.911206\pi$
$954$	0	0
$955$	−21046.7	−0.713146
$956$	0	0
$957$	0	0
$958$	0	0
$959$	68146.2	2.29464
$960$	0	0
$961$	−71956.1	−2.41536
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13968.3	−0.465965
$966$	0	0
$967$	− 15952.5i	− 0.530504i	−0.964179	−	0.265252i	$-0.914545\pi$
$967$	− 15952.5i	− 0.530504i	0.964179	−	0.265252i	$-0.0854552\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30067.2i	0.993722i	0.867830	+	0.496861i	$0.165514\pi$
$971$	30067.2i	0.993722i	−0.867830	+	0.496861i	$0.834486\pi$
$972$	0	0
$973$	21271.3i	0.700849i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	59343.9i	1.94328i	0.236474	+	0.971638i	$0.424008\pi$
$977$	59343.9i	1.94328i	−0.236474	+	0.971638i	$0.575992\pi$
$978$	0	0
$979$	−4618.99	−0.150790
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−33918.8	−1.10055	−0.550276	−	0.834983i	$-0.685477\pi$
$983$	−33918.8	−1.10055	−0.550276	+	0.834983i	$0.685477\pi$
$984$	0	0
$985$	−76526.1	−2.47546
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−43705.3	−1.40521
$990$	0	0
$991$	19427.8i	0.622748i	0.950287	+	0.311374i	$0.100789\pi$
$991$	19427.8i	0.622748i	−0.950287	+	0.311374i	$0.899211\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 662.053i	− 0.0210940i
$996$	0	0
$997$	− 2949.82i	− 0.0937028i	−0.998902	−	0.0468514i	$-0.985081\pi$
$997$	− 2949.82i	− 0.0937028i	0.998902	−	0.0468514i	$-0.0149187\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.4.f.e.1151.3		8
3.2	odd	2	inner	2304.4.f.e.1151.5		8
4.3	odd	2		2304.4.f.h.1151.4		8
8.3	odd	2	inner	2304.4.f.e.1151.6		8
8.5	even	2		2304.4.f.h.1151.5		8
12.11	even	2		2304.4.f.h.1151.6		8
16.3	odd	4		576.4.c.f.575.5		8
16.5	even	4		288.4.c.b.287.4	yes	8
16.11	odd	4		288.4.c.b.287.3	✓	8
16.13	even	4		576.4.c.f.575.6		8
24.5	odd	2		2304.4.f.h.1151.3		8
24.11	even	2	inner	2304.4.f.e.1151.4		8
48.5	odd	4		288.4.c.b.287.6	yes	8
48.11	even	4		288.4.c.b.287.5	yes	8
48.29	odd	4		576.4.c.f.575.4		8
48.35	even	4		576.4.c.f.575.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.4.c.b.287.3	✓	8	16.11	odd	4
288.4.c.b.287.4	yes	8	16.5	even	4
288.4.c.b.287.5	yes	8	48.11	even	4
288.4.c.b.287.6	yes	8	48.5	odd	4
576.4.c.f.575.3		8	48.35	even	4
576.4.c.f.575.4		8	48.29	odd	4
576.4.c.f.575.5		8	16.3	odd	4
576.4.c.f.575.6		8	16.13	even	4
2304.4.f.e.1151.3		8	1.1	even	1	trivial
2304.4.f.e.1151.4		8	24.11	even	2	inner
2304.4.f.e.1151.5		8	3.2	odd	2	inner
2304.4.f.e.1151.6		8	8.3	odd	2	inner
2304.4.f.h.1151.3		8	24.5	odd	2
2304.4.f.h.1151.4		8	4.3	odd	2
2304.4.f.h.1151.5		8	8.5	even	2
2304.4.f.h.1151.6		8	12.11	even	2

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2304.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-14.8339 q^{5} -30.9783i q^{7} +O(q^{10})\)	\(q-14.8339 q^{5} -30.9783i q^{7} +9.86874i q^{11} -54.9783i q^{13} +79.1652i q^{17} +43.9130 q^{19} -108.802 q^{23} +95.0435 q^{25} -231.116 q^{29} +318.978i q^{31} +459.527i q^{35} +38.1740i q^{37} +110.093i q^{41} +401.696 q^{43} -546.901 q^{47} -616.652 q^{49} +50.4660 q^{53} -146.391i q^{55} -500.693i q^{59} -865.478i q^{61} +815.540i q^{65} +306.043 q^{67} +194.024 q^{71} -867.043 q^{73} +305.716 q^{77} +816.674i q^{79} +1146.53i q^{83} -1174.33i q^{85} +468.043i q^{89} -1703.13 q^{91} -651.399 q^{95} +238.652 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 384 q^{19} + 1128 q^{25} + 640 q^{43} - 1992 q^{49} + 2816 q^{67} + 1152 q^{73} - 6272 q^{91} + 7424 q^{97}+O(q^{100}) \) 8 * q - 384 * q^19 + 1128 * q^25 + 640 * q^43 - 1992 * q^49 + 2816 * q^67 + 1152 * q^73 - 6272 * q^91 + 7424 * q^97

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