LMFDB - Embedded newform 2304.2.f.e.1151.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,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, 2, names="a")

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

chi := DirichletCharacter("2304.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.1151
Dual form			2304.2.f.e.1151.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{5} +4.00000i q^{7} +O(q^{10})$	$q+1.41421 q^{5} +4.00000i q^{7} -5.65685i q^{11} +4.00000i q^{13} +4.24264i q^{17} +5.65685 q^{23} -3.00000 q^{25} +1.41421 q^{29} -4.00000i q^{31} +5.65685i q^{35} +6.00000i q^{37} +9.89949i q^{41} +8.00000 q^{43} -5.65685 q^{47} -9.00000 q^{49} +4.24264 q^{53} -8.00000i q^{55} +11.3137i q^{59} -2.00000i q^{61} +5.65685i q^{65} -8.00000 q^{67} -5.65685 q^{71} +22.6274 q^{77} +4.00000i q^{79} -5.65685i q^{83} +6.00000i q^{85} +4.24264i q^{89} -16.0000 q^{91} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 12 q^{25} + 32 q^{43} - 36 q^{49} - 32 q^{67} - 64 q^{91} - 32 q^{97}+O(q^{100}) $ 4 * q - 12 * q^25 + 32 * q^43 - 36 * q^49 - 32 * q^67 - 64 * q^91 - 32 * 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
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.65685i	− 1.70561i	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	− 5.65685i	− 1.70561i	0.522233	−	0.852803i	$-0.325099\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.24264i	1.02899i	0.857493	+	0.514496i	$0.172021\pi$
$17$	4.24264i	1.02899i	−0.857493	+	0.514496i	$0.827979\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.41421	0.262613	0.131306	−	0.991342i	$-0.458083\pi$
$29$	1.41421	0.262613	0.131306	+	0.991342i	$0.458083\pi$
$30$	0	0
$31$	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$	− 4.00000i	− 0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.65685i	0.956183i
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.89949i	1.54604i	0.634381	+	0.773021i	$0.281255\pi$
$41$	9.89949i	1.54604i	−0.634381	+	0.773021i	$0.718745\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.65685	−0.825137	−0.412568	−	0.910927i	$-0.635368\pi$
$47$	−5.65685	−0.825137	−0.412568	+	0.910927i	$0.635368\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.24264	0.582772	0.291386	−	0.956606i	$-0.405884\pi$
$53$	4.24264	0.582772	0.291386	+	0.956606i	$0.405884\pi$
$54$	0	0
$55$	− 8.00000i	− 1.07872i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137i	1.47292i	0.676481	+	0.736460i	$0.263504\pi$
$59$	11.3137i	1.47292i	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	− 2.00000i	− 0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$61$	− 2.00000i	− 0.256074i	0.991769	−	0.128037i	$-0.0408676\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.65685i	0.701646i
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	22.6274	2.57863
$78$	0	0
$79$	4.00000i	0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$	4.00000i	0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 5.65685i	− 0.620920i	−0.950586	−	0.310460i	$-0.899517\pi$
$83$	− 5.65685i	− 0.620920i	0.950586	−	0.310460i	$-0.100483\pi$
$84$	0	0
$85$	6.00000i	0.650791i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.24264i	0.449719i	0.974391	+	0.224860i	$0.0721923\pi$
$89$	4.24264i	0.449719i	−0.974391	+	0.224860i	$0.927808\pi$
$90$	0	0
$91$	−16.0000	−1.67726
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.3848	1.82935	0.914677	−	0.404186i	$-0.132445\pi$
$101$	18.3848	1.82935	0.914677	+	0.404186i	$0.132445\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	12.0000i	1.14939i	0.818367	+	0.574696i	$0.194880\pi$
$109$	12.0000i	1.14939i	−0.818367	+	0.574696i	$0.805120\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.41421i	0.133038i	0.997785	+	0.0665190i	$0.0211893\pi$
$113$	1.41421i	0.133038i	−0.997785	+	0.0665190i	$0.978811\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−16.9706	−1.55569
$120$	0	0
$121$	−21.0000	−1.90909
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 11.3137i	− 0.988483i	−0.869325	−	0.494242i	$-0.835446\pi$
$131$	− 11.3137i	− 0.988483i	0.869325	−	0.494242i	$-0.164554\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.41421i	0.120824i	0.998174	+	0.0604122i	$0.0192415\pi$
$137$	1.41421i	0.120824i	−0.998174	+	0.0604122i	$0.980758\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	22.6274	1.89220
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.41421	−0.115857	−0.0579284	−	0.998321i	$-0.518450\pi$
$149$	−1.41421	−0.115857	−0.0579284	+	0.998321i	$0.518450\pi$
$150$	0	0
$151$	20.0000i	1.62758i	0.581161	+	0.813788i	$0.302599\pi$
$151$	20.0000i	1.62758i	−0.581161	+	0.813788i	$0.697401\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 5.65685i	− 0.454369i
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	22.6274i	1.78329i
$162$	0	0
$163$	24.0000	1.87983	0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000	1.87983	0.939913	+	0.341415i	$0.110906\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.5563	1.18273	0.591364	−	0.806405i	$-0.298590\pi$
$173$	15.5563	1.18273	0.591364	+	0.806405i	$0.298590\pi$
$174$	0	0
$175$	− 12.0000i	− 0.907115i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	− 12.0000i	− 0.891953i	−0.895045	−	0.445976i	$-0.852856\pi$
$181$	− 12.0000i	− 0.891953i	0.895045	−	0.445976i	$-0.147144\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.48528i	0.623850i
$186$	0	0
$187$	24.0000	1.75505
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.07107	0.503793	0.251896	−	0.967754i	$-0.418946\pi$
$197$	7.07107	0.503793	0.251896	+	0.967754i	$0.418946\pi$
$198$	0	0
$199$	− 20.0000i	− 1.41776i	−0.705328	−	0.708881i	$-0.749200\pi$
$199$	− 20.0000i	− 1.41776i	0.705328	−	0.708881i	$-0.250800\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.65685i	0.397033i
$204$	0	0
$205$	14.0000i	0.977802i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.3137	0.771589
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−16.9706	−1.14156
$222$	0	0
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.65685i	0.375459i	0.982221	+	0.187729i	$0.0601128\pi$
$227$	5.65685i	0.375459i	−0.982221	+	0.187729i	$0.939887\pi$
$228$	0	0
$229$	28.0000i	1.85029i	0.379611	+	0.925146i	$0.376058\pi$
$229$	28.0000i	1.85029i	−0.379611	+	0.925146i	$0.623942\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.07107i	0.463241i	0.972806	+	0.231621i	$0.0744028\pi$
$233$	7.07107i	0.463241i	−0.972806	+	0.231621i	$0.925597\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.6274	−1.46365	−0.731823	−	0.681495i	$-0.761330\pi$
$239$	−22.6274	−1.46365	−0.731823	+	0.681495i	$0.761330\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−12.7279	−0.813157
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 16.9706i	− 1.07117i	−0.844481	−	0.535586i	$-0.820091\pi$
$251$	− 16.9706i	− 1.07117i	0.844481	−	0.535586i	$-0.179909\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 24.0416i	− 1.49968i	−0.661622	−	0.749838i	$-0.730131\pi$
$257$	− 24.0416i	− 1.49968i	0.661622	−	0.749838i	$-0.269869\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.3137	0.697633	0.348817	−	0.937191i	$-0.386584\pi$
$263$	11.3137	0.697633	0.348817	+	0.937191i	$0.386584\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.2132	1.29339	0.646696	−	0.762748i	$-0.276150\pi$
$269$	21.2132	1.29339	0.646696	+	0.762748i	$0.276150\pi$
$270$	0	0
$271$	12.0000i	0.728948i	0.931214	+	0.364474i	$0.118751\pi$
$271$	12.0000i	0.728948i	−0.931214	+	0.364474i	$0.881249\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	16.9706i	1.02336i
$276$	0	0
$277$	4.00000i	0.240337i	0.992754	+	0.120168i	$0.0383434\pi$
$277$	4.00000i	0.240337i	−0.992754	+	0.120168i	$0.961657\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 7.07107i	− 0.421825i	−0.977505	−	0.210912i	$-0.932357\pi$
$281$	− 7.07107i	− 0.421825i	0.977505	−	0.210912i	$-0.0676434\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−39.5980	−2.33739
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.07107	−0.413096	−0.206548	−	0.978436i	$-0.566223\pi$
$293$	−7.07107	−0.413096	−0.206548	+	0.978436i	$0.566223\pi$
$294$	0	0
$295$	16.0000i	0.931556i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	22.6274i	1.30858i
$300$	0	0
$301$	32.0000i	1.84445i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.82843i	− 0.161955i
$306$	0	0
$307$	−24.0000	−1.36975	−0.684876	−	0.728659i	$-0.740144\pi$
$307$	−24.0000	−1.36975	−0.684876	+	0.728659i	$0.740144\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.3137	−0.641542	−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137	−0.641542	−0.320771	+	0.947157i	$0.603942\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.0416	−1.35031	−0.675156	−	0.737675i	$-0.735924\pi$
$317$	−24.0416	−1.35031	−0.675156	+	0.737675i	$0.735924\pi$
$318$	0	0
$319$	− 8.00000i	− 0.447914i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 12.0000i	− 0.665640i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 22.6274i	− 1.24749i
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−22.6274	−1.22534
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 28.2843i	− 1.51838i	−0.650870	−	0.759190i	$-0.725596\pi$
$347$	− 28.2843i	− 1.51838i	0.650870	−	0.759190i	$-0.274404\pi$
$348$	0	0
$349$	− 34.0000i	− 1.81998i	−0.414632	−	0.909989i	$-0.636090\pi$
$349$	− 34.0000i	− 1.81998i	0.414632	−	0.909989i	$-0.363910\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 9.89949i	− 0.526897i	−0.964673	−	0.263448i	$-0.915140\pi$
$353$	− 9.89949i	− 0.526897i	0.964673	−	0.263448i	$-0.0848599\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.9706	−0.895672	−0.447836	−	0.894116i	$-0.647805\pi$
$359$	−16.9706	−0.895672	−0.447836	+	0.894116i	$0.647805\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 20.0000i	− 1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$	− 20.0000i	− 1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	16.9706i	0.881068i
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.65685i	0.291343i
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.6274	1.15621	0.578103	−	0.815963i	$-0.303793\pi$
$383$	22.6274	1.15621	0.578103	+	0.815963i	$0.303793\pi$
$384$	0	0
$385$	32.0000	1.63087
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.89949	0.501924	0.250962	−	0.967997i	$-0.419253\pi$
$389$	9.89949	0.501924	0.250962	+	0.967997i	$0.419253\pi$
$390$	0	0
$391$	24.0000i	1.21373i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.65685i	0.284627i
$396$	0	0
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.5563i	0.776847i	0.921481	+	0.388424i	$0.126980\pi$
$401$	15.5563i	0.776847i	−0.921481	+	0.388424i	$0.873020\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	0	0
$405$	0	0
$406$	0	0
$407$	33.9411	1.68240
$408$	0	0
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−45.2548	−2.22684
$414$	0	0
$415$	− 8.00000i	− 0.392705i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 5.65685i	− 0.276355i	−0.990407	−	0.138178i	$-0.955875\pi$
$419$	− 5.65685i	− 0.276355i	0.990407	−	0.138178i	$-0.0441245\pi$
$420$	0	0
$421$	− 28.0000i	− 1.36464i	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	− 28.0000i	− 1.36464i	0.731055	−	0.682318i	$-0.239028\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 12.7279i	− 0.617395i
$426$	0	0
$427$	8.00000	0.387147
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.9706	0.817443	0.408722	−	0.912659i	$-0.365975\pi$
$431$	16.9706	0.817443	0.408722	+	0.912659i	$0.365975\pi$
$432$	0	0
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 28.0000i	− 1.33637i	−0.743996	−	0.668184i	$-0.767072\pi$
$439$	− 28.0000i	− 1.33637i	0.743996	−	0.668184i	$-0.232928\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.65685i	0.268765i	0.990930	+	0.134383i	$0.0429051\pi$
$443$	5.65685i	0.268765i	−0.990930	+	0.134383i	$0.957095\pi$
$444$	0	0
$445$	6.00000i	0.284427i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 38.1838i	− 1.80200i	−0.433816	−	0.901002i	$-0.642833\pi$
$449$	− 38.1838i	− 1.80200i	0.433816	−	0.901002i	$-0.357167\pi$
$450$	0	0
$451$	56.0000	2.63694
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−22.6274	−1.06079
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.3848	0.856264	0.428132	−	0.903716i	$-0.359172\pi$
$461$	18.3848	0.856264	0.428132	+	0.903716i	$0.359172\pi$
$462$	0	0
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 16.9706i	− 0.785304i	−0.919687	−	0.392652i	$-0.871558\pi$
$467$	− 16.9706i	− 0.785304i	0.919687	−	0.392652i	$-0.128442\pi$
$468$	0	0
$469$	− 32.0000i	− 1.47762i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 45.2548i	− 2.08082i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	39.5980	1.80928	0.904639	−	0.426179i	$-0.140141\pi$
$479$	39.5980	1.80928	0.904639	+	0.426179i	$0.140141\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.3137	−0.513729
$486$	0	0
$487$	12.0000i	0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$	12.0000i	0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 22.6274i	− 1.02116i	−0.859830	−	0.510581i	$-0.829431\pi$
$491$	− 22.6274i	− 1.02116i	0.859830	−	0.510581i	$-0.170569\pi$
$492$	0	0
$493$	6.00000i	0.270226i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 22.6274i	− 1.01498i
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.65685	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$504$	0	0
$505$	26.0000	1.15698
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.5269	−1.44173	−0.720865	−	0.693075i	$-0.756255\pi$
$509$	−32.5269	−1.44173	−0.720865	+	0.693075i	$0.756255\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 5.65685i	− 0.249271i
$516$	0	0
$517$	32.0000i	1.40736i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 32.5269i	− 1.42503i	−0.701657	−	0.712515i	$-0.747556\pi$
$521$	− 32.5269i	− 1.42503i	0.701657	−	0.712515i	$-0.252444\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.9706	0.739249
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−39.5980	−1.71518
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	50.9117i	2.19292i
$540$	0	0
$541$	− 20.0000i	− 0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	− 20.0000i	− 0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16.9706i	0.726939i
$546$	0	0
$547$	−40.0000	−1.71028	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000	−1.71028	−0.855138	+	0.518400i	$0.826528\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.6985	1.25837	0.629183	−	0.777258i	$-0.283390\pi$
$557$	29.6985	1.25837	0.629183	+	0.777258i	$0.283390\pi$
$558$	0	0
$559$	32.0000i	1.35346i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 39.5980i	− 1.66886i	−0.551117	−	0.834428i	$-0.685798\pi$
$563$	− 39.5980i	− 1.66886i	0.551117	−	0.834428i	$-0.314202\pi$
$564$	0	0
$565$	2.00000i	0.0841406i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.0416i	1.00788i	0.863739	+	0.503939i	$0.168116\pi$
$569$	24.0416i	1.00788i	−0.863739	+	0.503939i	$0.831884\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−16.9706	−0.707721
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	22.6274	0.938743
$582$	0	0
$583$	− 24.0000i	− 0.993978i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 11.3137i	− 0.466967i	−0.972361	−	0.233483i	$-0.924988\pi$
$587$	− 11.3137i	− 0.466967i	0.972361	−	0.233483i	$-0.0750124\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 46.6690i	− 1.91647i	−0.285985	−	0.958234i	$-0.592321\pi$
$593$	− 46.6690i	− 1.91647i	0.285985	−	0.958234i	$-0.407679\pi$
$594$	0	0
$595$	−24.0000	−0.983904
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−39.5980	−1.61793	−0.808965	−	0.587857i	$-0.799972\pi$
$599$	−39.5980	−1.61793	−0.808965	+	0.587857i	$0.799972\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−29.6985	−1.20742
$606$	0	0
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 22.6274i	− 0.915407i
$612$	0	0
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.8701i	1.08175i	0.841104	+	0.540874i	$0.181906\pi$
$617$	26.8701i	1.08175i	−0.841104	+	0.540874i	$0.818094\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.9706	−0.679911
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−25.4558	−1.01499
$630$	0	0
$631$	12.0000i	0.477712i	0.971055	+	0.238856i	$0.0767725\pi$
$631$	12.0000i	0.477712i	−0.971055	+	0.238856i	$0.923228\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16.9706i	0.673456i
$636$	0	0
$637$	− 36.0000i	− 1.42637i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 7.07107i	− 0.279290i	−0.990202	−	0.139645i	$-0.955404\pi$
$641$	− 7.07107i	− 0.279290i	0.990202	−	0.139645i	$-0.0445962\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.9706	0.667182	0.333591	−	0.942718i	$-0.391740\pi$
$647$	16.9706	0.667182	0.333591	+	0.942718i	$0.391740\pi$
$648$	0	0
$649$	64.0000	2.51222
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.3848	−0.719452	−0.359726	−	0.933058i	$-0.617130\pi$
$653$	−18.3848	−0.719452	−0.359726	+	0.933058i	$0.617130\pi$
$654$	0	0
$655$	− 16.0000i	− 0.625172i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 11.3137i	− 0.440720i	−0.975419	−	0.220360i	$-0.929277\pi$
$659$	− 11.3137i	− 0.440720i	0.975419	−	0.220360i	$-0.0707231\pi$
$660$	0	0
$661$	22.0000i	0.855701i	0.903850	+	0.427850i	$0.140729\pi$
$661$	22.0000i	0.855701i	−0.903850	+	0.427850i	$0.859271\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.3137	−0.436761
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32.5269	−1.25011	−0.625055	−	0.780580i	$-0.714924\pi$
$677$	−32.5269	−1.25011	−0.625055	+	0.780580i	$0.714924\pi$
$678$	0	0
$679$	− 32.0000i	− 1.22805i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 16.9706i	− 0.649361i	−0.945824	−	0.324680i	$-0.894743\pi$
$683$	− 16.9706i	− 0.649361i	0.945824	−	0.324680i	$-0.105257\pi$
$684$	0	0
$685$	2.00000i	0.0764161i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16.9706i	0.646527i
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.3137	−0.429153
$696$	0	0
$697$	−42.0000	−1.59086
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12.7279	0.480727	0.240363	−	0.970683i	$-0.422733\pi$
$701$	12.7279	0.480727	0.240363	+	0.970683i	$0.422733\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	73.5391i	2.76572i
$708$	0	0
$709$	− 20.0000i	− 0.751116i	−0.926799	−	0.375558i	$-0.877451\pi$
$709$	− 20.0000i	− 0.751116i	0.926799	−	0.375558i	$-0.122549\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 22.6274i	− 0.847403i
$714$	0	0
$715$	32.0000	1.19673
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.9706	−0.632895	−0.316448	−	0.948610i	$-0.602490\pi$
$719$	−16.9706	−0.632895	−0.316448	+	0.948610i	$0.602490\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.24264	−0.157568
$726$	0	0
$727$	− 4.00000i	− 0.148352i	−0.997245	−	0.0741759i	$-0.976367\pi$
$727$	− 4.00000i	− 0.148352i	0.997245	−	0.0741759i	$-0.0236326\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	33.9411i	1.25536i
$732$	0	0
$733$	4.00000i	0.147743i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	4.00000i	0.147743i	−0.997268	+	0.0738717i	$0.976464\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	45.2548i	1.66698i
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11.3137	0.415060	0.207530	−	0.978229i	$-0.433458\pi$
$743$	11.3137	0.415060	0.207530	+	0.978229i	$0.433458\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 20.0000i	− 0.729810i	−0.931045	−	0.364905i	$-0.881101\pi$
$751$	− 20.0000i	− 0.729810i	0.931045	−	0.364905i	$-0.118899\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	28.2843i	1.02937i
$756$	0	0
$757$	− 20.0000i	− 0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$	− 20.0000i	− 0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.89949i	0.358856i	0.983771	+	0.179428i	$0.0574248\pi$
$761$	9.89949i	0.358856i	−0.983771	+	0.179428i	$0.942575\pi$
$762$	0	0
$763$	−48.0000	−1.73772
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−45.2548	−1.63406
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	4.24264	0.152597	0.0762986	−	0.997085i	$-0.475690\pi$
$773$	4.24264	0.152597	0.0762986	+	0.997085i	$0.475690\pi$
$774$	0	0
$775$	12.0000i	0.431053i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	32.0000i	1.14505i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.82843i	0.100951i
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.65685	−0.201135
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.07107	−0.250470	−0.125235	−	0.992127i	$-0.539968\pi$
$797$	−7.07107	−0.250470	−0.125235	+	0.992127i	$0.539968\pi$
$798$	0	0
$799$	− 24.0000i	− 0.849059i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	32.0000i	1.12785i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.2132i	0.745817i	0.927868	+	0.372908i	$0.121639\pi$
$809$	21.2132i	0.745817i	−0.927868	+	0.372908i	$0.878361\pi$
$810$	0	0
$811$	48.0000	1.68551	0.842754	−	0.538299i	$-0.180933\pi$
$811$	48.0000	1.68551	0.842754	+	0.538299i	$0.180933\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	33.9411	1.18891
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.0122	1.43134	0.715668	−	0.698441i	$-0.246123\pi$
$821$	41.0122	1.43134	0.715668	+	0.698441i	$0.246123\pi$
$822$	0	0
$823$	52.0000i	1.81261i	0.422628	+	0.906303i	$0.361108\pi$
$823$	52.0000i	1.81261i	−0.422628	+	0.906303i	$0.638892\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 33.9411i	− 1.18025i	−0.807312	−	0.590124i	$-0.799079\pi$
$827$	− 33.9411i	− 1.18025i	0.807312	−	0.590124i	$-0.200921\pi$
$828$	0	0
$829$	12.0000i	0.416777i	0.978046	+	0.208389i	$0.0668219\pi$
$829$	12.0000i	0.416777i	−0.978046	+	0.208389i	$0.933178\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 38.1838i	− 1.32299i
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.65685	0.195296	0.0976481	−	0.995221i	$-0.468868\pi$
$839$	5.65685	0.195296	0.0976481	+	0.995221i	$0.468868\pi$
$840$	0	0
$841$	−27.0000	−0.931034
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.24264	−0.145951
$846$	0	0
$847$	− 84.0000i	− 2.88627i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	33.9411i	1.16349i
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.0416i	− 0.821246i	−0.911805	−	0.410623i	$-0.865311\pi$
$857$	− 24.0416i	− 0.821246i	0.911805	−	0.410623i	$-0.134689\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.9411	1.15537	0.577685	−	0.816260i	$-0.303956\pi$
$863$	33.9411	1.15537	0.577685	+	0.816260i	$0.303956\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	0	0
$867$	0	0
$868$	0	0
$869$	22.6274	0.767583
$870$	0	0
$871$	− 32.0000i	− 1.08428i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 45.2548i	− 1.52989i
$876$	0	0
$877$	46.0000i	1.55331i	0.629926	+	0.776655i	$0.283085\pi$
$877$	46.0000i	1.55331i	−0.629926	+	0.776655i	$0.716915\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 12.7279i	− 0.428815i	−0.976744	−	0.214407i	$-0.931218\pi$
$881$	− 12.7279i	− 0.428815i	0.976744	−	0.214407i	$-0.0687820\pi$
$882$	0	0
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	56.5685	1.89939	0.949693	−	0.313183i	$-0.101395\pi$
$887$	56.5685	1.89939	0.949693	+	0.313183i	$0.101395\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 5.65685i	− 0.188667i
$900$	0	0
$901$	18.0000i	0.599667i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 16.9706i	− 0.564121i
$906$	0	0
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−11.3137	−0.374840	−0.187420	−	0.982280i	$-0.560013\pi$
$911$	−11.3137	−0.374840	−0.187420	+	0.982280i	$0.560013\pi$
$912$	0	0
$913$	−32.0000	−1.05905
$914$	0	0
$915$	0	0
$916$	0	0
$917$	45.2548	1.49445
$918$	0	0
$919$	36.0000i	1.18753i	0.804638	+	0.593765i	$0.202359\pi$
$919$	36.0000i	1.18753i	−0.804638	+	0.593765i	$0.797641\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 22.6274i	− 0.744791i
$924$	0	0
$925$	− 18.0000i	− 0.591836i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	49.4975i	1.62396i	0.583686	+	0.811980i	$0.301610\pi$
$929$	49.4975i	1.62396i	−0.583686	+	0.811980i	$0.698390\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	33.9411	1.10999
$936$	0	0
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	41.0122	1.33696	0.668480	−	0.743730i	$-0.266945\pi$
$941$	41.0122	1.33696	0.668480	+	0.743730i	$0.266945\pi$
$942$	0	0
$943$	56.0000i	1.82361i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 11.3137i	− 0.367646i	−0.982959	−	0.183823i	$-0.941153\pi$
$947$	− 11.3137i	− 0.367646i	0.982959	−	0.183823i	$-0.0588473\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 21.2132i	− 0.687163i	−0.939123	−	0.343582i	$-0.888360\pi$
$953$	− 21.2132i	− 0.687163i	0.939123	−	0.343582i	$-0.111640\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.65685	−0.182669
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.82843	−0.0910503
$966$	0	0
$967$	28.0000i	0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$	28.0000i	0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.65685i	0.181537i	0.995872	+	0.0907685i	$0.0289323\pi$
$971$	5.65685i	0.181537i	−0.995872	+	0.0907685i	$0.971068\pi$
$972$	0	0
$973$	− 32.0000i	− 1.02587i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	52.3259i	1.67405i	0.547162	+	0.837027i	$0.315708\pi$
$977$	52.3259i	1.67405i	−0.547162	+	0.837027i	$0.684292\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−56.5685	−1.80426	−0.902128	−	0.431468i	$-0.857996\pi$
$983$	−56.5685	−1.80426	−0.902128	+	0.431468i	$0.857996\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	0	0
$988$	0	0
$989$	45.2548	1.43902
$990$	0	0
$991$	12.0000i	0.381193i	0.981669	+	0.190596i	$0.0610421\pi$
$991$	12.0000i	0.381193i	−0.981669	+	0.190596i	$0.938958\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 28.2843i	− 0.896672i
$996$	0	0
$997$	− 38.0000i	− 1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$	− 38.0000i	− 1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.f.e.1151.4		4
3.2	odd	2	inner	2304.2.f.e.1151.2		4
4.3	odd	2		2304.2.f.c.1151.3		4
8.3	odd	2	inner	2304.2.f.e.1151.1		4
8.5	even	2		2304.2.f.c.1151.2		4
12.11	even	2		2304.2.f.c.1151.1		4
16.3	odd	4		576.2.c.c.575.2		4
16.5	even	4		288.2.c.a.287.3	yes	4
16.11	odd	4		288.2.c.a.287.4	yes	4
16.13	even	4		576.2.c.c.575.1		4
24.5	odd	2		2304.2.f.c.1151.4		4
24.11	even	2	inner	2304.2.f.e.1151.3		4
48.5	odd	4		288.2.c.a.287.1	✓	4
48.11	even	4		288.2.c.a.287.2	yes	4
48.29	odd	4		576.2.c.c.575.3		4
48.35	even	4		576.2.c.c.575.4		4
80.27	even	4		7200.2.o.a.7199.1		4
80.37	odd	4		7200.2.o.n.7199.3		4
80.43	even	4		7200.2.o.n.7199.2		4
80.53	odd	4		7200.2.o.a.7199.4		4
80.59	odd	4		7200.2.h.d.1151.1		4
80.69	even	4		7200.2.h.d.1151.4		4
144.5	odd	12		2592.2.s.d.1727.2		8
144.11	even	12		2592.2.s.d.863.4		8
144.43	odd	12		2592.2.s.d.863.2		8
144.59	even	12		2592.2.s.d.1727.1		8
144.85	even	12		2592.2.s.d.1727.4		8
144.101	odd	12		2592.2.s.d.863.3		8
144.133	even	12		2592.2.s.d.863.1		8
144.139	odd	12		2592.2.s.d.1727.3		8
240.53	even	4		7200.2.o.a.7199.2		4
240.59	even	4		7200.2.h.d.1151.2		4
240.107	odd	4		7200.2.o.a.7199.3		4
240.149	odd	4		7200.2.h.d.1151.3		4
240.197	even	4		7200.2.o.n.7199.1		4
240.203	odd	4		7200.2.o.n.7199.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.c.a.287.1	✓	4	48.5	odd	4
288.2.c.a.287.2	yes	4	48.11	even	4
288.2.c.a.287.3	yes	4	16.5	even	4
288.2.c.a.287.4	yes	4	16.11	odd	4
576.2.c.c.575.1		4	16.13	even	4
576.2.c.c.575.2		4	16.3	odd	4
576.2.c.c.575.3		4	48.29	odd	4
576.2.c.c.575.4		4	48.35	even	4
2304.2.f.c.1151.1		4	12.11	even	2
2304.2.f.c.1151.2		4	8.5	even	2
2304.2.f.c.1151.3		4	4.3	odd	2
2304.2.f.c.1151.4		4	24.5	odd	2
2304.2.f.e.1151.1		4	8.3	odd	2	inner
2304.2.f.e.1151.2		4	3.2	odd	2	inner
2304.2.f.e.1151.3		4	24.11	even	2	inner
2304.2.f.e.1151.4		4	1.1	even	1	trivial
2592.2.s.d.863.1		8	144.133	even	12
2592.2.s.d.863.2		8	144.43	odd	12
2592.2.s.d.863.3		8	144.101	odd	12
2592.2.s.d.863.4		8	144.11	even	12
2592.2.s.d.1727.1		8	144.59	even	12
2592.2.s.d.1727.2		8	144.5	odd	12
2592.2.s.d.1727.3		8	144.139	odd	12
2592.2.s.d.1727.4		8	144.85	even	12
7200.2.h.d.1151.1		4	80.59	odd	4
7200.2.h.d.1151.2		4	240.59	even	4
7200.2.h.d.1151.3		4	240.149	odd	4
7200.2.h.d.1151.4		4	80.69	even	4
7200.2.o.a.7199.1		4	80.27	even	4
7200.2.o.a.7199.2		4	240.53	even	4
7200.2.o.a.7199.3		4	240.107	odd	4
7200.2.o.a.7199.4		4	80.53	odd	4
7200.2.o.n.7199.1		4	240.197	even	4
7200.2.o.n.7199.2		4	80.43	even	4
7200.2.o.n.7199.3		4	80.37	odd	4
7200.2.o.n.7199.4		4	240.203	odd	4

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{5} +4.00000i q^{7} +O(q^{10})\)	\(q+1.41421 q^{5} +4.00000i q^{7} -5.65685i q^{11} +4.00000i q^{13} +4.24264i q^{17} +5.65685 q^{23} -3.00000 q^{25} +1.41421 q^{29} -4.00000i q^{31} +5.65685i q^{35} +6.00000i q^{37} +9.89949i q^{41} +8.00000 q^{43} -5.65685 q^{47} -9.00000 q^{49} +4.24264 q^{53} -8.00000i q^{55} +11.3137i q^{59} -2.00000i q^{61} +5.65685i q^{65} -8.00000 q^{67} -5.65685 q^{71} +22.6274 q^{77} +4.00000i q^{79} -5.65685i q^{83} +6.00000i q^{85} +4.24264i q^{89} -16.0000 q^{91} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{25} + 32 q^{43} - 36 q^{49} - 32 q^{67} - 64 q^{91} - 32 q^{97}+O(q^{100}) \) 4 * q - 12 * q^25 + 32 * q^43 - 36 * q^49 - 32 * q^67 - 64 * q^91 - 32 * q^97

Introduction

L-functions

Modular forms

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