LMFDB - Embedded newform 2496.2.m.c.2209.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,2,Mod(2209,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2496.2209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.m (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$19.9306603445$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.162447943996702457856.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} - 15x^{8} - 16x^{4} + 256 $ x^16 - x^12 - 15x^8 - 16x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2209.7
Root			$0.825348 + 1.14839i$ of defining polynomial
Character	$\chi$	$=$	2496.2209
Dual form			2496.2.m.c.2209.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.09557 q^{5} -0.646084i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.09557 q^{5} -0.646084i q^{7} -1.00000 q^{9} -1.41421 q^{11} +(2.44949 + 2.64575i) q^{13} -3.09557i q^{15} +3.58258 q^{17} -1.11905 q^{19} -0.646084 q^{21} +3.46410 q^{23} +4.58258 q^{25} +1.00000i q^{27} -0.913701i q^{29} +1.93825i q^{31} +1.41421i q^{33} -2.00000i q^{35} +4.89898 q^{37} +(2.64575 - 2.44949i) q^{39} +5.95202i q^{41} -3.58258i q^{43} -3.09557 q^{45} -3.74166i q^{47} +6.58258 q^{49} -3.58258i q^{51} -10.5830i q^{53} -4.37780 q^{55} +1.11905i q^{57} +9.30917 q^{59} -0.913701i q^{61} +0.646084i q^{63} +(7.58258 + 8.19012i) q^{65} +14.6709 q^{67} -3.46410i q^{69} +2.44949i q^{71} +5.65685i q^{73} -4.58258i q^{75} +0.913701i q^{77} -14.0471 q^{79} +1.00000 q^{81} -11.5473 q^{83} +11.0901 q^{85} -0.913701 q^{87} +8.78044i q^{89} +(1.70938 - 1.58258i) q^{91} +1.93825 q^{93} -3.46410 q^{95} -7.89495i q^{97} +1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} - 16 q^{17} + 32 q^{49} + 48 q^{65} + 16 q^{81}+O(q^{100}) $ 16 * q - 16 * q^9 - 16 * q^17 + 32 * q^49 + 48 * q^65 + 16 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	3.09557			1.38438			0.692191	−	0.721714i	$-0.256645\pi$
$5$	3.09557			1.38438			0.692191	+	0.721714i	$0.256645\pi$
$6$		0			0
$7$		−	0.646084i		−	0.244197i	−0.992518	−	0.122098i	$-0.961038\pi$
$7$		−	0.646084i		−	0.244197i	0.992518	−	0.122098i	$-0.0389623\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.41421			−0.426401			−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421			−0.426401			−0.213201	+	0.977008i	$0.568389\pi$
$12$		0			0
$13$	2.44949	+	2.64575i	0.679366	+	0.733799i
$13$	2.44949	+	2.64575i	0.679366	+	0.733799i
$14$		0			0
$15$		−	3.09557i		−	0.799274i
$16$		0			0
$17$	3.58258			0.868902			0.434451	−	0.900695i	$-0.356942\pi$
$17$	3.58258			0.868902			0.434451	+	0.900695i	$0.356942\pi$
$18$		0			0
$19$	−1.11905			−0.256728			−0.128364	−	0.991727i	$-0.540973\pi$
$19$	−1.11905			−0.256728			−0.128364	+	0.991727i	$0.540973\pi$
$20$		0			0
$21$	−0.646084			−0.140987
$22$		0			0
$23$	3.46410			0.722315			0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410			0.722315			0.361158	+	0.932505i	$0.382382\pi$
$24$		0			0
$25$	4.58258			0.916515
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	0.913701i		−	0.169670i	−0.996395	−	0.0848350i	$-0.972964\pi$
$29$		−	0.913701i		−	0.169670i	0.996395	−	0.0848350i	$-0.0270363\pi$
$30$		0			0
$31$			1.93825i			0.348120i	0.984735	+	0.174060i	$0.0556887\pi$
$31$			1.93825i			0.348120i	−0.984735	+	0.174060i	$0.944311\pi$
$32$		0			0
$33$			1.41421i			0.246183i
$34$		0			0
$35$		−	2.00000i		−	0.338062i
$36$		0			0
$37$	4.89898			0.805387			0.402694	−	0.915335i	$-0.368074\pi$
$37$	4.89898			0.805387			0.402694	+	0.915335i	$0.368074\pi$
$38$		0			0
$39$	2.64575	−	2.44949i	0.423659	−	0.392232i
$40$		0			0
$41$			5.95202i			0.929549i	0.885429	+	0.464775i	$0.153865\pi$
$41$			5.95202i			0.929549i	−0.885429	+	0.464775i	$0.846135\pi$
$42$		0			0
$43$		−	3.58258i		−	0.546338i	−0.961966	−	0.273169i	$-0.911928\pi$
$43$		−	3.58258i		−	0.546338i	0.961966	−	0.273169i	$-0.0880718\pi$
$44$		0			0
$45$	−3.09557			−0.461461
$46$		0			0
$47$		−	3.74166i		−	0.545777i	−0.962046	−	0.272888i	$-0.912021\pi$
$47$		−	3.74166i		−	0.545777i	0.962046	−	0.272888i	$-0.0879790\pi$
$48$		0			0
$49$	6.58258			0.940368
$50$		0			0
$51$		−	3.58258i		−	0.501661i
$52$		0			0
$53$		−	10.5830i		−	1.45369i	−0.686803	−	0.726844i	$-0.740986\pi$
$53$		−	10.5830i		−	1.45369i	0.686803	−	0.726844i	$-0.259014\pi$
$54$		0			0
$55$	−4.37780			−0.590303
$56$		0			0
$57$			1.11905i			0.148222i
$58$		0			0
$59$	9.30917			1.21195			0.605975	−	0.795484i	$-0.292783\pi$
$59$	9.30917			1.21195			0.605975	+	0.795484i	$0.292783\pi$
$60$		0			0
$61$		−	0.913701i		−	0.116987i	−0.998288	−	0.0584937i	$-0.981370\pi$
$61$		−	0.913701i		−	0.116987i	0.998288	−	0.0584937i	$-0.0186298\pi$
$62$		0			0
$63$			0.646084i			0.0813989i
$64$		0			0
$65$	7.58258	+	8.19012i	0.940503	+	1.01586i
$66$		0			0
$67$	14.6709			1.79233			0.896165	−	0.443720i	$-0.146342\pi$
$67$	14.6709			1.79233			0.896165	+	0.443720i	$0.146342\pi$
$68$		0			0
$69$		−	3.46410i		−	0.417029i
$70$		0			0
$71$			2.44949i			0.290701i	0.989380	+	0.145350i	$0.0464310\pi$
$71$			2.44949i			0.290701i	−0.989380	+	0.145350i	$0.953569\pi$
$72$		0			0
$73$			5.65685i			0.662085i	0.943616	+	0.331042i	$0.107400\pi$
$73$			5.65685i			0.662085i	−0.943616	+	0.331042i	$0.892600\pi$
$74$		0			0
$75$		−	4.58258i		−	0.529150i
$76$		0			0
$77$			0.913701i			0.104126i
$78$		0			0
$79$	−14.0471			−1.58042			−0.790211	−	0.612834i	$-0.790029\pi$
$79$	−14.0471			−1.58042			−0.790211	+	0.612834i	$0.790029\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−11.5473			−1.26748			−0.633739	−	0.773547i	$-0.718481\pi$
$83$	−11.5473			−1.26748			−0.633739	+	0.773547i	$0.718481\pi$
$84$		0			0
$85$	11.0901			1.20289
$86$		0			0
$87$	−0.913701			−0.0979590
$88$		0			0
$89$			8.78044i			0.930725i	0.885120	+	0.465363i	$0.154076\pi$
$89$			8.78044i			0.930725i	−0.885120	+	0.465363i	$0.845924\pi$
$90$		0			0
$91$	1.70938	−	1.58258i	0.179191	−	0.165899i
$92$		0			0
$93$	1.93825			0.200987
$94$		0			0
$95$	−3.46410			−0.355409
$96$		0			0
$97$		−	7.89495i		−	0.801611i	−0.916163	−	0.400806i	$-0.868730\pi$
$97$		−	7.89495i		−	0.801611i	0.916163	−	0.400806i	$-0.131270\pi$
$98$		0			0
$99$	1.41421			0.142134
$100$		0			0
$101$		−	10.5830i		−	1.05305i	−0.850160	−	0.526524i	$-0.823495\pi$
$101$		−	10.5830i		−	1.05305i	0.850160	−	0.526524i	$-0.176505\pi$
$102$		0			0
$103$	−6.01450			−0.592627			−0.296313	−	0.955091i	$-0.595757\pi$
$103$	−6.01450			−0.592627			−0.296313	+	0.955091i	$0.595757\pi$
$104$		0			0
$105$	−2.00000			−0.195180
$106$		0			0
$107$		−	2.00000i		−	0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$		−	2.00000i		−	0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$		0			0
$109$	−9.79796			−0.938474			−0.469237	−	0.883072i	$-0.655471\pi$
$109$	−9.79796			−0.938474			−0.469237	+	0.883072i	$0.655471\pi$
$110$		0			0
$111$		−	4.89898i		−	0.464991i
$112$		0			0
$113$	10.7477			1.01106			0.505531	−	0.862809i	$-0.331297\pi$
$113$	10.7477			1.01106			0.505531	+	0.862809i	$0.331297\pi$
$114$		0			0
$115$	10.7234			0.999960
$116$		0			0
$117$	−2.44949	−	2.64575i	−0.226455	−	0.244600i
$118$		0			0
$119$		−	2.31464i		−	0.212183i
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$	5.95202			0.536675
$124$		0			0
$125$	−1.29217			−0.115575
$126$		0			0
$127$	−0.190700			−0.0169219			−0.00846096	−	0.999964i	$-0.502693\pi$
$127$	−0.190700			−0.0169219			−0.00846096	+	0.999964i	$0.502693\pi$
$128$		0			0
$129$	−3.58258			−0.315428
$130$		0			0
$131$			6.00000i			0.524222i	0.965038	+	0.262111i	$0.0844187\pi$
$131$			6.00000i			0.524222i	−0.965038	+	0.262111i	$0.915581\pi$
$132$		0			0
$133$			0.723000i			0.0626921i
$134$		0			0
$135$			3.09557i			0.266425i
$136$		0			0
$137$			18.3232i			1.56545i	0.622365	+	0.782727i	$0.286172\pi$
$137$			18.3232i			1.56545i	−0.622365	+	0.782727i	$0.713828\pi$
$138$		0			0
$139$		−	18.7477i		−	1.59016i	−0.606504	−	0.795081i	$-0.707428\pi$
$139$		−	18.7477i		−	1.59016i	0.606504	−	0.795081i	$-0.292572\pi$
$140$		0			0
$141$	−3.74166			−0.315104
$142$		0			0
$143$	−3.46410	−	3.74166i	−0.289683	−	0.312893i
$144$		0			0
$145$		−	2.82843i		−	0.234888i
$146$		0			0
$147$		−	6.58258i		−	0.542922i
$148$		0			0
$149$	−0.780929			−0.0639762			−0.0319881	−	0.999488i	$-0.510184\pi$
$149$	−0.780929			−0.0639762			−0.0319881	+	0.999488i	$0.510184\pi$
$150$		0			0
$151$			5.54506i			0.451251i	0.974214	+	0.225625i	$0.0724425\pi$
$151$			5.54506i			0.451251i	−0.974214	+	0.225625i	$0.927557\pi$
$152$		0			0
$153$	−3.58258			−0.289634
$154$		0			0
$155$			6.00000i			0.481932i
$156$		0			0
$157$		−	0.723000i		−	0.0577017i	−0.999584	−	0.0288508i	$-0.990815\pi$
$157$		−	0.723000i		−	0.0577017i	0.999584	−	0.0288508i	$-0.00918478\pi$
$158$		0			0
$159$	−10.5830			−0.839287
$160$		0			0
$161$		−	2.23810i		−	0.176387i
$162$		0			0
$163$	18.6799			1.46313			0.731563	−	0.681774i	$-0.238791\pi$
$163$	18.6799			1.46313			0.731563	+	0.681774i	$0.238791\pi$
$164$		0			0
$165$			4.37780i			0.340811i
$166$		0			0
$167$			4.76413i			0.368660i	0.982864	+	0.184330i	$0.0590115\pi$
$167$			4.76413i			0.368660i	−0.982864	+	0.184330i	$0.940989\pi$
$168$		0			0
$169$	−1.00000	+	12.9615i	−0.0769231	+	0.997037i
$170$		0			0
$171$	1.11905			0.0855759
$172$		0			0
$173$		−	13.8564i		−	1.05348i	−0.850026	−	0.526742i	$-0.823414\pi$
$173$		−	13.8564i		−	1.05348i	0.850026	−	0.526742i	$-0.176586\pi$
$174$		0			0
$175$		−	2.96073i		−	0.223810i
$176$		0			0
$177$		−	9.30917i		−	0.699720i
$178$		0			0
$179$		−	10.0000i		−	0.747435i	−0.927543	−	0.373718i	$-0.878083\pi$
$179$		−	10.0000i		−	0.747435i	0.927543	−	0.373718i	$-0.121917\pi$
$180$		0			0
$181$		−	4.37780i		−	0.325399i	−0.986676	−	0.162700i	$-0.947980\pi$
$181$		−	4.37780i		−	0.325399i	0.986676	−	0.162700i	$-0.0520202\pi$
$182$		0			0
$183$	−0.913701			−0.0675427
$184$		0			0
$185$	15.1652			1.11496
$186$		0			0
$187$	−5.06653			−0.370501
$188$		0			0
$189$	0.646084			0.0469957
$190$		0			0
$191$	10.3923			0.751961			0.375980	−	0.926628i	$-0.377306\pi$
$191$	10.3923			0.751961			0.375980	+	0.926628i	$0.377306\pi$
$192$		0			0
$193$		−	5.06653i		−	0.364697i	−0.983234	−	0.182348i	$-0.941630\pi$
$193$		−	5.06653i		−	0.364697i	0.983234	−	0.182348i	$-0.0583698\pi$
$194$		0			0
$195$	8.19012	−	7.58258i	0.586507	−	0.543000i
$196$		0			0
$197$	4.11805			0.293399			0.146699	−	0.989181i	$-0.453135\pi$
$197$	4.11805			0.293399			0.146699	+	0.989181i	$0.453135\pi$
$198$		0			0
$199$	3.46410			0.245564			0.122782	−	0.992434i	$-0.460818\pi$
$199$	3.46410			0.245564			0.122782	+	0.992434i	$0.460818\pi$
$200$		0			0
$201$		−	14.6709i		−	1.03480i
$202$		0			0
$203$	−0.590327			−0.0414328
$204$		0			0
$205$			18.4249i			1.28685i
$206$		0			0
$207$	−3.46410			−0.240772
$208$		0			0
$209$	1.58258			0.109469
$210$		0			0
$211$			0.834849i			0.0574733i	0.999587	+	0.0287367i	$0.00914843\pi$
$211$			0.834849i			0.0574733i	−0.999587	+	0.0287367i	$0.990852\pi$
$212$		0			0
$213$	2.44949			0.167836
$214$		0			0
$215$		−	11.0901i		−	0.756340i
$216$		0			0
$217$	1.25227			0.0850098
$218$		0			0
$219$	5.65685			0.382255
$220$		0			0
$221$	8.77548	+	9.47860i	0.590303	+	0.637600i
$222$		0			0
$223$		−	26.7028i		−	1.78815i	−0.447913	−	0.894077i	$-0.647833\pi$
$223$		−	26.7028i		−	1.78815i	0.447913	−	0.894077i	$-0.352167\pi$
$224$		0			0
$225$	−4.58258			−0.305505
$226$		0			0
$227$	−10.4898			−0.696234			−0.348117	−	0.937451i	$-0.613179\pi$
$227$	−10.4898			−0.696234			−0.348117	+	0.937451i	$0.613179\pi$
$228$		0			0
$229$	−6.19115			−0.409123			−0.204561	−	0.978854i	$-0.565577\pi$
$229$	−6.19115			−0.409123			−0.204561	+	0.978854i	$0.565577\pi$
$230$		0			0
$231$	0.913701			0.0601171
$232$		0			0
$233$	12.3303			0.807785			0.403892	−	0.914806i	$-0.367657\pi$
$233$	12.3303			0.807785			0.403892	+	0.914806i	$0.367657\pi$
$234$		0			0
$235$		−	11.5826i		−	0.755564i
$236$		0			0
$237$			14.0471i			0.912458i
$238$		0			0
$239$			22.3151i			1.44344i	0.692183	+	0.721722i	$0.256649\pi$
$239$			22.3151i			1.44344i	−0.692183	+	0.721722i	$0.743351\pi$
$240$		0			0
$241$			18.6183i			1.19931i	0.800258	+	0.599656i	$0.204696\pi$
$241$			18.6183i			1.19931i	−0.800258	+	0.599656i	$0.795304\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	20.3768			1.30183
$246$		0			0
$247$	−2.74110	−	2.96073i	−0.174412	−	0.188387i
$248$		0			0
$249$			11.5473i			0.731778i
$250$		0			0
$251$			11.1652i			0.704738i	0.935861	+	0.352369i	$0.114624\pi$
$251$			11.1652i			0.704738i	−0.935861	+	0.352369i	$0.885376\pi$
$252$		0			0
$253$	−4.89898			−0.307996
$254$		0			0
$255$		−	11.0901i		−	0.694491i
$256$		0			0
$257$	9.16515			0.571706			0.285853	−	0.958273i	$-0.407723\pi$
$257$	9.16515			0.571706			0.285853	+	0.958273i	$0.407723\pi$
$258$		0			0
$259$		−	3.16515i		−	0.196673i
$260$		0			0
$261$			0.913701i			0.0565566i
$262$		0			0
$263$	1.82740			0.112682			0.0563412	−	0.998412i	$-0.482057\pi$
$263$	1.82740			0.112682			0.0563412	+	0.998412i	$0.482057\pi$
$264$		0			0
$265$		−	32.7605i		−	2.01246i
$266$		0			0
$267$	8.78044			0.537355
$268$		0			0
$269$			16.5975i			1.01197i	0.862543	+	0.505984i	$0.168871\pi$
$269$			16.5975i			1.01197i	−0.862543	+	0.505984i	$0.831129\pi$
$270$		0			0
$271$		−	16.6352i		−	1.01052i	−0.862968	−	0.505258i	$-0.831397\pi$
$271$		−	16.6352i		−	1.01052i	0.862968	−	0.505258i	$-0.168603\pi$
$272$		0			0
$273$	−1.58258	−	1.70938i	−0.0957818	−	0.103456i
$274$		0			0
$275$	−6.48074			−0.390803
$276$		0			0
$277$		−	31.3676i		−	1.88470i	−0.334633	−	0.942349i	$-0.608612\pi$
$277$		−	31.3676i		−	1.88470i	0.334633	−	0.942349i	$-0.391388\pi$
$278$		0			0
$279$		−	1.93825i		−	0.116040i
$280$		0			0
$281$		−	14.9044i		−	0.889123i	−0.895748	−	0.444562i	$-0.853360\pi$
$281$		−	14.9044i		−	0.889123i	0.895748	−	0.444562i	$-0.146640\pi$
$282$		0			0
$283$		−	12.0000i		−	0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$		−	12.0000i		−	0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$		0			0
$285$			3.46410i			0.205196i
$286$		0			0
$287$	3.84550			0.226993
$288$		0			0
$289$	−4.16515			−0.245009
$290$		0			0
$291$	−7.89495			−0.462810
$292$		0			0
$293$	34.0513			1.98930			0.994649	−	0.103308i	$-0.0329429\pi$
$293$	34.0513			1.98930			0.994649	+	0.103308i	$0.0329429\pi$
$294$		0			0
$295$	28.8172			1.67780
$296$		0			0
$297$		−	1.41421i		−	0.0820610i
$298$		0			0
$299$	8.48528	+	9.16515i	0.490716	+	0.530034i
$300$		0			0
$301$	−2.31464			−0.133414
$302$		0			0
$303$	−10.5830			−0.607978
$304$		0			0
$305$		−	2.82843i		−	0.161955i
$306$		0			0
$307$	−20.3277			−1.16016			−0.580082	−	0.814558i	$-0.696979\pi$
$307$	−20.3277			−1.16016			−0.580082	+	0.814558i	$0.696979\pi$
$308$		0			0
$309$			6.01450i			0.342153i
$310$		0			0
$311$	−19.1479			−1.08578			−0.542889	−	0.839804i	$-0.682670\pi$
$311$	−19.1479			−1.08578			−0.542889	+	0.839804i	$0.682670\pi$
$312$		0			0
$313$	3.16515			0.178905			0.0894525	−	0.995991i	$-0.471488\pi$
$313$	3.16515			0.178905			0.0894525	+	0.995991i	$0.471488\pi$
$314$		0			0
$315$			2.00000i			0.112687i
$316$		0			0
$317$	−10.5789			−0.594170			−0.297085	−	0.954851i	$-0.596014\pi$
$317$	−10.5789			−0.594170			−0.297085	+	0.954851i	$0.596014\pi$
$318$		0			0
$319$			1.29217i			0.0723475i
$320$		0			0
$321$	−2.00000			−0.111629
$322$		0			0
$323$	−4.00908			−0.223071
$324$		0			0
$325$	11.2250	+	12.1244i	0.622649	+	0.672538i
$326$		0			0
$327$			9.79796i			0.541828i
$328$		0			0
$329$	−2.41742			−0.133277
$330$		0			0
$331$	−33.2892			−1.82974			−0.914870	−	0.403749i	$-0.867707\pi$
$331$	−33.2892			−1.82974			−0.914870	+	0.403749i	$0.867707\pi$
$332$		0			0
$333$	−4.89898			−0.268462
$334$		0			0
$335$	45.4147			2.48127
$336$		0			0
$337$	−22.3303			−1.21641			−0.608205	−	0.793780i	$-0.708110\pi$
$337$	−22.3303			−1.21641			−0.608205	+	0.793780i	$0.708110\pi$
$338$		0			0
$339$		−	10.7477i		−	0.583736i
$340$		0			0
$341$		−	2.74110i		−	0.148439i
$342$		0			0
$343$		−	8.77548i		−	0.473832i
$344$		0			0
$345$		−	10.7234i		−	0.577327i
$346$		0			0
$347$			25.4955i			1.36867i	0.729169	+	0.684334i	$0.239907\pi$
$347$			25.4955i			1.36867i	−0.729169	+	0.684334i	$0.760093\pi$
$348$		0			0
$349$	−27.3489			−1.46395			−0.731977	−	0.681329i	$-0.761402\pi$
$349$	−27.3489			−1.46395			−0.731977	+	0.681329i	$0.761402\pi$
$350$		0			0
$351$	−2.64575	+	2.44949i	−0.141220	+	0.130744i
$352$		0			0
$353$			30.2272i			1.60883i	0.594066	+	0.804416i	$0.297522\pi$
$353$			30.2272i			1.60883i	−0.594066	+	0.804416i	$0.702478\pi$
$354$		0			0
$355$			7.58258i			0.402441i
$356$		0			0
$357$	−2.31464			−0.122504
$358$		0			0
$359$			9.93280i			0.524233i	0.965036	+	0.262117i	$0.0844205\pi$
$359$			9.93280i			0.524233i	−0.965036	+	0.262117i	$0.915579\pi$
$360$		0			0
$361$	−17.7477			−0.934091
$362$		0			0
$363$			9.00000i			0.472377i
$364$		0			0
$365$			17.5112i			0.916579i
$366$		0			0
$367$	4.56850			0.238474			0.119237	−	0.992866i	$-0.461955\pi$
$367$	4.56850			0.238474			0.119237	+	0.992866i	$0.461955\pi$
$368$		0			0
$369$		−	5.95202i		−	0.309850i
$370$		0			0
$371$	−6.83751			−0.354986
$372$		0			0
$373$		−	8.94630i		−	0.463222i	−0.972808	−	0.231611i	$-0.925600\pi$
$373$		−	8.94630i		−	0.463222i	0.972808	−	0.231611i	$-0.0743997\pi$
$374$		0			0
$375$			1.29217i			0.0667273i
$376$		0			0
$377$	2.41742	−	2.23810i	0.124504	−	0.115268i
$378$		0			0
$379$	25.3942			1.30441			0.652207	−	0.758041i	$-0.273843\pi$
$379$	25.3942			1.30441			0.652207	+	0.758041i	$0.273843\pi$
$380$		0			0
$381$			0.190700i			0.00976988i
$382$		0			0
$383$			33.4052i			1.70693i	0.521152	+	0.853464i	$0.325502\pi$
$383$			33.4052i			1.70693i	−0.521152	+	0.853464i	$0.674498\pi$
$384$		0			0
$385$			2.82843i			0.144150i
$386$		0			0
$387$			3.58258i			0.182113i
$388$		0			0
$389$		−	32.2813i		−	1.63673i	−0.574700	−	0.818364i	$-0.694881\pi$
$389$		−	32.2813i		−	1.63673i	0.574700	−	0.818364i	$-0.305119\pi$
$390$		0			0
$391$	12.4104			0.627621
$392$		0			0
$393$	6.00000			0.302660
$394$		0			0
$395$	−43.4839			−2.18791
$396$		0			0
$397$	−30.6860			−1.54009			−0.770044	−	0.637991i	$-0.779766\pi$
$397$	−30.6860			−1.54009			−0.770044	+	0.637991i	$0.779766\pi$
$398$		0			0
$399$	0.723000			0.0361953
$400$		0			0
$401$			4.77136i			0.238271i	0.992878	+	0.119135i	$0.0380122\pi$
$401$			4.77136i			0.238271i	−0.992878	+	0.119135i	$0.961988\pi$
$402$		0			0
$403$	−5.12813	+	4.74773i	−0.255450	+	0.236501i
$404$		0			0
$405$	3.09557			0.153820
$406$		0			0
$407$	−6.92820			−0.343418
$408$		0			0
$409$			2.23810i			0.110667i	0.998468	+	0.0553335i	$0.0176222\pi$
$409$			2.23810i			0.110667i	−0.998468	+	0.0553335i	$0.982378\pi$
$410$		0			0
$411$	18.3232			0.903815
$412$		0			0
$413$		−	6.01450i		−	0.295954i
$414$		0			0
$415$	−35.7454			−1.75467
$416$		0			0
$417$	−18.7477			−0.918080
$418$		0			0
$419$			28.3303i			1.38403i	0.721885	+	0.692013i	$0.243276\pi$
$419$			28.3303i			1.38403i	−0.721885	+	0.692013i	$0.756724\pi$
$420$		0			0
$421$	−22.4499			−1.09414			−0.547072	−	0.837086i	$-0.684257\pi$
$421$	−22.4499			−1.09414			−0.547072	+	0.837086i	$0.684257\pi$
$422$		0			0
$423$			3.74166i			0.181926i
$424$		0			0
$425$	16.4174			0.796362
$426$		0			0
$427$	−0.590327			−0.0285679
$428$		0			0
$429$	−3.74166	+	3.46410i	−0.180649	+	0.167248i
$430$		0			0
$431$			26.9444i			1.29787i	0.760846	+	0.648933i	$0.224784\pi$
$431$			26.9444i			1.29787i	−0.760846	+	0.648933i	$0.775216\pi$
$432$		0			0
$433$	−30.7477			−1.47764			−0.738821	−	0.673902i	$-0.764617\pi$
$433$	−30.7477			−1.47764			−0.738821	+	0.673902i	$0.764617\pi$
$434$		0			0
$435$	−2.82843			−0.135613
$436$		0			0
$437$	−3.87650			−0.185438
$438$		0			0
$439$	6.01450			0.287057			0.143528	−	0.989646i	$-0.454155\pi$
$439$	6.01450			0.287057			0.143528	+	0.989646i	$0.454155\pi$
$440$		0			0
$441$	−6.58258			−0.313456
$442$		0			0
$443$			21.1652i			1.00559i	0.864407	+	0.502793i	$0.167694\pi$
$443$			21.1652i			1.00559i	−0.864407	+	0.502793i	$0.832306\pi$
$444$		0			0
$445$			27.1805i			1.28848i
$446$		0			0
$447$			0.780929i			0.0369367i
$448$		0			0
$449$			10.4282i			0.492138i	0.969252	+	0.246069i	$0.0791390\pi$
$449$			10.4282i			0.492138i	−0.969252	+	0.246069i	$0.920861\pi$
$450$		0			0
$451$		−	8.41742i		−	0.396361i
$452$		0			0
$453$	5.54506			0.260530
$454$		0			0
$455$	5.29150	−	4.89898i	0.248069	−	0.229668i
$456$		0			0
$457$		−	19.2087i		−	0.898543i	−0.893395	−	0.449272i	$-0.851684\pi$
$457$		−	19.2087i		−	0.898543i	0.893395	−	0.449272i	$-0.148316\pi$
$458$		0			0
$459$			3.58258i			0.167220i
$460$		0			0
$461$	−25.0061			−1.16465			−0.582326	−	0.812955i	$-0.697857\pi$
$461$	−25.0061			−1.16465			−0.582326	+	0.812955i	$0.697857\pi$
$462$		0			0
$463$			24.1185i			1.12088i	0.828194	+	0.560441i	$0.189368\pi$
$463$			24.1185i			1.12088i	−0.828194	+	0.560441i	$0.810632\pi$
$464$		0			0
$465$	6.00000			0.278243
$466$		0			0
$467$			36.6606i			1.69645i	0.529636	+	0.848225i	$0.322329\pi$
$467$			36.6606i			1.69645i	−0.529636	+	0.848225i	$0.677671\pi$
$468$		0			0
$469$		−	9.47860i		−	0.437681i
$470$		0			0
$471$	−0.723000			−0.0333141
$472$		0			0
$473$			5.06653i			0.232959i
$474$		0			0
$475$	−5.12813			−0.235295
$476$		0			0
$477$			10.5830i			0.484563i
$478$		0			0
$479$		−	4.01135i		−	0.183283i	−0.995792	−	0.0916416i	$-0.970789\pi$
$479$		−	4.01135i		−	0.183283i	0.995792	−	0.0916416i	$-0.0292114\pi$
$480$		0			0
$481$	12.0000	+	12.9615i	0.547153	+	0.590993i
$482$		0			0
$483$	−2.23810			−0.101837
$484$		0			0
$485$		−	24.4394i		−	1.10974i
$486$		0			0
$487$			31.3321i			1.41979i	0.704305	+	0.709897i	$0.251259\pi$
$487$			31.3321i			1.41979i	−0.704305	+	0.709897i	$0.748741\pi$
$488$		0			0
$489$		−	18.6799i		−	0.844736i
$490$		0			0
$491$			2.33030i			0.105165i	0.998617	+	0.0525825i	$0.0167453\pi$
$491$			2.33030i			0.105165i	−0.998617	+	0.0525825i	$0.983255\pi$
$492$		0			0
$493$		−	3.27340i		−	0.147427i
$494$		0			0
$495$	4.37780			0.196768
$496$		0			0
$497$	1.58258			0.0709882
$498$		0			0
$499$	20.9180			0.936420			0.468210	−	0.883617i	$-0.344899\pi$
$499$	20.9180			0.936420			0.468210	+	0.883617i	$0.344899\pi$
$500$		0			0
$501$	4.76413			0.212846
$502$		0			0
$503$	8.37420			0.373387			0.186694	−	0.982418i	$-0.440223\pi$
$503$	8.37420			0.373387			0.186694	+	0.982418i	$0.440223\pi$
$504$		0			0
$505$		−	32.7605i		−	1.45782i
$506$		0			0
$507$	12.9615	+	1.00000i	0.575640	+	0.0444116i
$508$		0			0
$509$	30.1748			1.33747			0.668737	−	0.743499i	$-0.266835\pi$
$509$	30.1748			1.33747			0.668737	+	0.743499i	$0.266835\pi$
$510$		0			0
$511$	3.65480			0.161679
$512$		0			0
$513$		−	1.11905i		−	0.0494073i
$514$		0			0
$515$	−18.6183			−0.820422
$516$		0			0
$517$			5.29150i			0.232720i
$518$		0			0
$519$	−13.8564			−0.608229
$520$		0			0
$521$	11.5826			0.507442			0.253721	−	0.967277i	$-0.418346\pi$
$521$	11.5826			0.507442			0.253721	+	0.967277i	$0.418346\pi$
$522$		0			0
$523$		−	7.58258i		−	0.331563i	−0.986163	−	0.165781i	$-0.946985\pi$
$523$		−	7.58258i		−	0.331563i	0.986163	−	0.165781i	$-0.0530146\pi$
$524$		0			0
$525$	−2.96073			−0.129217
$526$		0			0
$527$			6.94393i			0.302482i
$528$		0			0
$529$	−11.0000			−0.478261
$530$		0			0
$531$	−9.30917			−0.403983
$532$		0			0
$533$	−15.7476	+	14.5794i	−0.682102	+	0.631504i
$534$		0			0
$535$		−	6.19115i		−	0.267667i
$536$		0			0
$537$	−10.0000			−0.431532
$538$		0			0
$539$	−9.30917			−0.400974
$540$		0			0
$541$	13.9442			0.599506			0.299753	−	0.954017i	$-0.403096\pi$
$541$	13.9442			0.599506			0.299753	+	0.954017i	$0.403096\pi$
$542$		0			0
$543$	−4.37780			−0.187869
$544$		0			0
$545$	−30.3303			−1.29921
$546$		0			0
$547$			33.9129i			1.45001i	0.688744	+	0.725005i	$0.258163\pi$
$547$			33.9129i			1.45001i	−0.688744	+	0.725005i	$0.741837\pi$
$548$		0			0
$549$			0.913701i			0.0389958i
$550$		0			0
$551$			1.02248i			0.0435590i
$552$		0			0
$553$			9.07561i			0.385934i
$554$		0			0
$555$		−	15.1652i		−	0.643725i
$556$		0			0
$557$	−31.1973			−1.32187			−0.660936	−	0.750443i	$-0.729840\pi$
$557$	−31.1973			−1.32187			−0.660936	+	0.750443i	$0.729840\pi$
$558$		0			0
$559$	9.47860	−	8.77548i	0.400902	−	0.371163i
$560$		0			0
$561$			5.06653i			0.213909i
$562$		0			0
$563$			17.1652i			0.723425i	0.932290	+	0.361712i	$0.117808\pi$
$563$			17.1652i			0.723425i	−0.932290	+	0.361712i	$0.882192\pi$
$564$		0			0
$565$	33.2704			1.39970
$566$		0			0
$567$		−	0.646084i		−	0.0271330i
$568$		0			0
$569$	−18.0000			−0.754599			−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000			−0.754599			−0.377300	+	0.926091i	$0.623147\pi$
$570$		0			0
$571$			46.3303i			1.93886i	0.245362	+	0.969431i	$0.421093\pi$
$571$			46.3303i			1.93886i	−0.245362	+	0.969431i	$0.578907\pi$
$572$		0			0
$573$		−	10.3923i		−	0.434145i
$574$		0			0
$575$	15.8745			0.662013
$576$		0			0
$577$		−	36.1792i		−	1.50616i	−0.657928	−	0.753080i	$-0.728567\pi$
$577$		−	36.1792i		−	1.50616i	0.657928	−	0.753080i	$-0.271433\pi$
$578$		0			0
$579$	−5.06653			−0.210558
$580$		0			0
$581$			7.46050i			0.309514i
$582$		0			0
$583$			14.9666i			0.619854i
$584$		0			0
$585$	−7.58258	−	8.19012i	−0.313501	−	0.338620i
$586$		0			0
$587$	−34.1747			−1.41054			−0.705270	−	0.708939i	$-0.749174\pi$
$587$	−34.1747			−1.41054			−0.705270	+	0.708939i	$0.749174\pi$
$588$		0			0
$589$		−	2.16900i		−	0.0893721i
$590$		0			0
$591$		−	4.11805i		−	0.169394i
$592$		0			0
$593$		−	4.77136i		−	0.195936i	−0.995190	−	0.0979682i	$-0.968766\pi$
$593$		−	4.77136i		−	0.195936i	0.995190	−	0.0979682i	$-0.0312343\pi$
$594$		0			0
$595$		−	7.16515i		−	0.293743i
$596$		0			0
$597$		−	3.46410i		−	0.141776i
$598$		0			0
$599$	−41.9506			−1.71406			−0.857028	−	0.515270i	$-0.827692\pi$
$599$	−41.9506			−1.71406			−0.857028	+	0.515270i	$0.827692\pi$
$600$		0			0
$601$	4.83485			0.197218			0.0986088	−	0.995126i	$-0.468561\pi$
$601$	4.83485			0.197218			0.0986088	+	0.995126i	$0.468561\pi$
$602$		0			0
$603$	−14.6709			−0.597444
$604$		0			0
$605$	−27.8602			−1.13268
$606$		0			0
$607$	19.8709			0.806535			0.403268	−	0.915082i	$-0.367874\pi$
$607$	19.8709			0.806535			0.403268	+	0.915082i	$0.367874\pi$
$608$		0			0
$609$			0.590327i			0.0239213i
$610$		0			0
$611$	9.89949	−	9.16515i	0.400491	−	0.370782i
$612$		0			0
$613$	−34.8322			−1.40686			−0.703430	−	0.710764i	$-0.748349\pi$
$613$	−34.8322			−1.40686			−0.703430	+	0.710764i	$0.748349\pi$
$614$		0			0
$615$	18.4249			0.742964
$616$		0			0
$617$		−	18.9135i		−	0.761429i	−0.924693	−	0.380714i	$-0.875678\pi$
$617$		−	18.9135i		−	0.761429i	0.924693	−	0.380714i	$-0.124322\pi$
$618$		0			0
$619$	−16.9090			−0.679628			−0.339814	−	0.940493i	$-0.610364\pi$
$619$	−16.9090			−0.679628			−0.339814	+	0.940493i	$0.610364\pi$
$620$		0			0
$621$			3.46410i			0.139010i
$622$		0			0
$623$	5.67290			0.227280
$624$		0			0
$625$	−26.9129			−1.07652
$626$		0			0
$627$		−	1.58258i		−	0.0632020i
$628$		0			0
$629$	17.5510			0.699803
$630$		0			0
$631$		−	17.6577i		−	0.702941i	−0.936199	−	0.351470i	$-0.885682\pi$
$631$		−	17.6577i		−	0.702941i	0.936199	−	0.351470i	$-0.114318\pi$
$632$		0			0
$633$	0.834849			0.0331823
$634$		0			0
$635$	−0.590327			−0.0234264
$636$		0			0
$637$	16.1240	+	17.4159i	0.638854	+	0.690041i
$638$		0			0
$639$		−	2.44949i		−	0.0969003i
$640$		0			0
$641$	−15.5826			−0.615475			−0.307737	−	0.951471i	$-0.599572\pi$
$641$	−15.5826			−0.615475			−0.307737	+	0.951471i	$0.599572\pi$
$642$		0			0
$643$	−25.9846			−1.02473			−0.512366	−	0.858767i	$-0.671231\pi$
$643$	−25.9846			−1.02473			−0.512366	+	0.858767i	$0.671231\pi$
$644$		0			0
$645$	−11.0901			−0.436673
$646$		0			0
$647$	32.8136			1.29004			0.645018	−	0.764167i	$-0.276850\pi$
$647$	32.8136			1.29004			0.645018	+	0.764167i	$0.276850\pi$
$648$		0			0
$649$	−13.1652			−0.516777
$650$		0			0
$651$		−	1.25227i		−	0.0490804i
$652$		0			0
$653$			29.0079i			1.13517i	0.823316	+	0.567584i	$0.192122\pi$
$653$			29.0079i			1.13517i	−0.823316	+	0.567584i	$0.807878\pi$
$654$		0			0
$655$			18.5734i			0.725724i
$656$		0			0
$657$		−	5.65685i		−	0.220695i
$658$		0			0
$659$		−	43.1652i		−	1.68148i	−0.541442	−	0.840738i	$-0.682122\pi$
$659$		−	43.1652i		−	1.68148i	0.541442	−	0.840738i	$-0.317878\pi$
$660$		0			0
$661$	19.8656			0.772683			0.386341	−	0.922356i	$-0.373739\pi$
$661$	19.8656			0.772683			0.386341	+	0.922356i	$0.373739\pi$
$662$		0			0
$663$	9.47860	−	8.77548i	0.368118	−	0.340811i
$664$		0			0
$665$			2.23810i			0.0867898i
$666$		0			0
$667$		−	3.16515i		−	0.122555i
$668$		0			0
$669$	−26.7028			−1.03239
$670$		0			0
$671$			1.29217i			0.0498836i
$672$		0			0
$673$	25.5826			0.986136			0.493068	−	0.869991i	$-0.335875\pi$
$673$	25.5826			0.986136			0.493068	+	0.869991i	$0.335875\pi$
$674$		0			0
$675$			4.58258i			0.176383i
$676$		0			0
$677$			6.92820i			0.266272i	0.991098	+	0.133136i	$0.0425048\pi$
$677$			6.92820i			0.266272i	−0.991098	+	0.133136i	$0.957495\pi$
$678$		0			0
$679$	−5.10080			−0.195751
$680$		0			0
$681$			10.4898i			0.401971i
$682$		0			0
$683$	5.30009			0.202802			0.101401	−	0.994846i	$-0.467667\pi$
$683$	5.30009			0.202802			0.101401	+	0.994846i	$0.467667\pi$
$684$		0			0
$685$			56.7207i			2.16719i
$686$		0			0
$687$			6.19115i			0.236207i
$688$		0			0
$689$	28.0000	−	25.9230i	1.06672	−	0.987586i
$690$		0			0
$691$	19.1471			0.728388			0.364194	−	0.931323i	$-0.381344\pi$
$691$	19.1471			0.728388			0.364194	+	0.931323i	$0.381344\pi$
$692$		0			0
$693$		−	0.913701i		−	0.0347086i
$694$		0			0
$695$		−	58.0350i		−	2.20139i
$696$		0			0
$697$			21.3236i			0.807687i
$698$		0			0
$699$		−	12.3303i		−	0.466375i
$700$		0			0
$701$		−	33.7273i		−	1.27386i	−0.770920	−	0.636932i	$-0.780203\pi$
$701$		−	33.7273i		−	1.27386i	0.770920	−	0.636932i	$-0.219797\pi$
$702$		0			0
$703$	−5.48220			−0.206765
$704$		0			0
$705$	−11.5826			−0.436225
$706$		0			0
$707$	−6.83751			−0.257151
$708$		0			0
$709$	32.2479			1.21110			0.605548	−	0.795809i	$-0.292954\pi$
$709$	32.2479			1.21110			0.605548	+	0.795809i	$0.292954\pi$
$710$		0			0
$711$	14.0471			0.526808
$712$		0			0
$713$			6.71430i			0.251453i
$714$		0			0
$715$	−10.7234	−	11.5826i	−0.401032	−	0.433164i
$716$		0			0
$717$	22.3151			0.833373
$718$		0			0
$719$	3.84550			0.143413			0.0717065	−	0.997426i	$-0.477156\pi$
$719$	3.84550			0.143413			0.0717065	+	0.997426i	$0.477156\pi$
$720$		0			0
$721$			3.88587i			0.144717i
$722$		0			0
$723$	18.6183			0.692423
$724$		0			0
$725$		−	4.18710i		−	0.155505i
$726$		0			0
$727$	−36.2777			−1.34547			−0.672733	−	0.739885i	$-0.734880\pi$
$727$	−36.2777			−1.34547			−0.672733	+	0.739885i	$0.734880\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	12.8348i		−	0.474714i
$732$		0			0
$733$	−9.79796			−0.361896			−0.180948	−	0.983493i	$-0.557917\pi$
$733$	−9.79796			−0.361896			−0.180948	+	0.983493i	$0.557917\pi$
$734$		0			0
$735$		−	20.3768i		−	0.751611i
$736$		0			0
$737$	−20.7477			−0.764252
$738$		0			0
$739$	11.3753			0.418448			0.209224	−	0.977868i	$-0.432906\pi$
$739$	11.3753			0.418448			0.209224	+	0.977868i	$0.432906\pi$
$740$		0			0
$741$	−2.96073	+	2.74110i	−0.108765	+	0.100697i
$742$		0			0
$743$			1.42701i			0.0523520i	0.999657	+	0.0261760i	$0.00833304\pi$
$743$			1.42701i			0.0523520i	−0.999657	+	0.0261760i	$0.991667\pi$
$744$		0			0
$745$	−2.41742			−0.0885676
$746$		0			0
$747$	11.5473			0.422492
$748$		0			0
$749$	−1.29217			−0.0472148
$750$		0			0
$751$	7.84190			0.286155			0.143078	−	0.989711i	$-0.454300\pi$
$751$	7.84190			0.286155			0.143078	+	0.989711i	$0.454300\pi$
$752$		0			0
$753$	11.1652			0.406881
$754$		0			0
$755$			17.1652i			0.624704i
$756$		0			0
$757$			19.8709i			0.722220i	0.932523	+	0.361110i	$0.117602\pi$
$757$			19.8709i			0.722220i	−0.932523	+	0.361110i	$0.882398\pi$
$758$		0			0
$759$			4.89898i			0.177822i
$760$		0			0
$761$		−	0.885491i		−	0.0320990i	−0.999871	−	0.0160495i	$-0.994891\pi$
$761$		−	0.885491i		−	0.0320990i	0.999871	−	0.0160495i	$-0.00510894\pi$
$762$		0			0
$763$			6.33030i			0.229172i
$764$		0			0
$765$	−11.0901			−0.400964
$766$		0			0
$767$	22.8027	+	24.6297i	0.823358	+	0.889328i
$768$		0			0
$769$		−	49.1407i		−	1.77206i	−0.463629	−	0.886030i	$-0.653453\pi$
$769$		−	49.1407i		−	1.77206i	0.463629	−	0.886030i	$-0.346547\pi$
$770$		0			0
$771$		−	9.16515i		−	0.330075i
$772$		0			0
$773$	15.2082			0.547000			0.273500	−	0.961872i	$-0.411819\pi$
$773$	15.2082			0.547000			0.273500	+	0.961872i	$0.411819\pi$
$774$		0			0
$775$			8.88218i			0.319057i
$776$		0			0
$777$	−3.16515			−0.113549
$778$		0			0
$779$		−	6.66061i		−	0.238641i
$780$		0			0
$781$		−	3.46410i		−	0.123955i
$782$		0			0
$783$	0.913701			0.0326530
$784$		0			0
$785$		−	2.23810i		−	0.0798812i
$786$		0			0
$787$	−19.7374			−0.703562			−0.351781	−	0.936082i	$-0.614424\pi$
$787$	−19.7374			−0.703562			−0.351781	+	0.936082i	$0.614424\pi$
$788$		0			0
$789$		−	1.82740i		−	0.0650572i
$790$		0			0
$791$		−	6.94393i		−	0.246898i
$792$		0			0
$793$	2.41742	−	2.23810i	0.0858453	−	0.0794773i
$794$		0			0
$795$	−32.7605			−1.16189
$796$		0			0
$797$			2.74110i			0.0970948i	0.998821	+	0.0485474i	$0.0154592\pi$
$797$			2.74110i			0.0970948i	−0.998821	+	0.0485474i	$0.984541\pi$
$798$		0			0
$799$		−	13.4048i		−	0.474227i
$800$		0			0
$801$		−	8.78044i		−	0.310242i
$802$		0			0
$803$		−	8.00000i		−	0.282314i
$804$		0			0
$805$		−	6.92820i		−	0.244187i
$806$		0			0
$807$	16.5975			0.584260
$808$		0			0
$809$	−10.7477			−0.377870			−0.188935	−	0.981990i	$-0.560504\pi$
$809$	−10.7477			−0.377870			−0.188935	+	0.981990i	$0.560504\pi$
$810$		0			0
$811$	42.8319			1.50403			0.752016	−	0.659145i	$-0.229082\pi$
$811$	42.8319			1.50403			0.752016	+	0.659145i	$0.229082\pi$
$812$		0			0
$813$	−16.6352			−0.583422
$814$		0			0
$815$	57.8251			2.02553
$816$		0			0
$817$			4.00908i			0.140260i
$818$		0			0
$819$	−1.70938	+	1.58258i	−0.0597305	+	0.0552997i
$820$		0			0
$821$	−22.4218			−0.782526			−0.391263	−	0.920279i	$-0.627962\pi$
$821$	−22.4218			−0.782526			−0.391263	+	0.920279i	$0.627962\pi$
$822$		0			0
$823$	−34.1087			−1.18896			−0.594478	−	0.804112i	$-0.702641\pi$
$823$	−34.1087			−1.18896			−0.594478	+	0.804112i	$0.702641\pi$
$824$		0			0
$825$			6.48074i			0.225630i
$826$		0			0
$827$	5.42329			0.188586			0.0942932	−	0.995544i	$-0.469941\pi$
$827$	5.42329			0.188586			0.0942932	+	0.995544i	$0.469941\pi$
$828$		0			0
$829$			16.2161i			0.563209i	0.959531	+	0.281604i	$0.0908666\pi$
$829$			16.2161i			0.563209i	−0.959531	+	0.281604i	$0.909133\pi$
$830$		0			0
$831$	−31.3676			−1.08813
$832$		0			0
$833$	23.5826			0.817088
$834$		0			0
$835$			14.7477i			0.510366i
$836$		0			0
$837$	−1.93825			−0.0669958
$838$		0			0
$839$			5.30352i			0.183098i	0.995801	+	0.0915489i	$0.0291818\pi$
$839$			5.30352i			0.183098i	−0.995801	+	0.0915489i	$0.970818\pi$
$840$		0			0
$841$	28.1652			0.971212
$842$		0			0
$843$	−14.9044			−0.513335
$844$		0			0
$845$	−3.09557	+	40.1232i	−0.106491	+	1.38028i
$846$		0			0
$847$			5.81475i			0.199797i
$848$		0			0
$849$	−12.0000			−0.411839
$850$		0			0
$851$	16.9706			0.581743
$852$		0			0
$853$	14.6969			0.503214			0.251607	−	0.967830i	$-0.419041\pi$
$853$	14.6969			0.503214			0.251607	+	0.967830i	$0.419041\pi$
$854$		0			0
$855$	3.46410			0.118470
$856$		0			0
$857$	−45.1652			−1.54281			−0.771406	−	0.636343i	$-0.780446\pi$
$857$	−45.1652			−1.54281			−0.771406	+	0.636343i	$0.780446\pi$
$858$		0			0
$859$		−	28.6606i		−	0.977887i	−0.872315	−	0.488944i	$-0.837382\pi$
$859$		−	28.6606i		−	0.977887i	0.872315	−	0.488944i	$-0.162618\pi$
$860$		0			0
$861$		−	3.84550i		−	0.131054i
$862$		0			0
$863$		−	6.05630i		−	0.206159i	−0.994673	−	0.103079i	$-0.967130\pi$
$863$		−	6.05630i		−	0.206159i	0.994673	−	0.103079i	$-0.0328696\pi$
$864$		0			0
$865$		−	42.8935i		−	1.45842i
$866$		0			0
$867$			4.16515i			0.141456i
$868$		0			0
$869$	19.8656			0.673895
$870$		0			0
$871$	35.9361	+	38.8154i	1.21765	+	1.31521i
$872$		0			0
$873$			7.89495i			0.267204i
$874$		0			0
$875$			0.834849i			0.0282230i
$876$		0			0
$877$	−11.3598			−0.383594			−0.191797	−	0.981435i	$-0.561431\pi$
$877$	−11.3598			−0.383594			−0.191797	+	0.981435i	$0.561431\pi$
$878$		0			0
$879$		−	34.0513i		−	1.14852i
$880$		0			0
$881$	3.66970			0.123635			0.0618176	−	0.998087i	$-0.480310\pi$
$881$	3.66970			0.123635			0.0618176	+	0.998087i	$0.480310\pi$
$882$		0			0
$883$		−	5.49545i		−	0.184937i	−0.995716	−	0.0924684i	$-0.970524\pi$
$883$		−	5.49545i		−	0.184937i	0.995716	−	0.0924684i	$-0.0294757\pi$
$884$		0			0
$885$		−	28.8172i		−	0.968680i
$886$		0			0
$887$	−40.6953			−1.36642			−0.683208	−	0.730224i	$-0.739416\pi$
$887$	−40.6953			−1.36642			−0.683208	+	0.730224i	$0.739416\pi$
$888$		0			0
$889$			0.123208i			0.00413228i
$890$		0			0
$891$	−1.41421			−0.0473779
$892$		0			0
$893$			4.18710i			0.140116i
$894$		0			0
$895$		−	30.9557i		−	1.03474i
$896$		0			0
$897$	9.16515	−	8.48528i	0.306015	−	0.283315i
$898$		0			0
$899$	1.77098			0.0590655
$900$		0			0
$901$		−	37.9144i		−	1.26311i
$902$		0			0
$903$			2.31464i			0.0770265i
$904$		0			0
$905$		−	13.5518i		−	0.450477i
$906$		0			0
$907$			32.6606i			1.08448i	0.840224	+	0.542239i	$0.182423\pi$
$907$			32.6606i			1.08448i	−0.840224	+	0.542239i	$0.817577\pi$
$908$		0			0
$909$			10.5830i			0.351016i
$910$		0			0
$911$	−17.3205			−0.573854			−0.286927	−	0.957952i	$-0.592634\pi$
$911$	−17.3205			−0.573854			−0.286927	+	0.957952i	$0.592634\pi$
$912$		0			0
$913$	16.3303			0.540454
$914$		0			0
$915$	−2.82843			−0.0935049
$916$		0			0
$917$	3.87650			0.128013
$918$		0			0
$919$	1.25530			0.0414085			0.0207043	−	0.999786i	$-0.493409\pi$
$919$	1.25530			0.0414085			0.0207043	+	0.999786i	$0.493409\pi$
$920$		0			0
$921$			20.3277i			0.669821i
$922$		0			0
$923$	−6.48074	+	6.00000i	−0.213316	+	0.197492i
$924$		0			0
$925$	22.4499			0.738150
$926$		0			0
$927$	6.01450			0.197542
$928$		0			0
$929$		−	1.35261i		−	0.0443777i	−0.999754	−	0.0221888i	$-0.992936\pi$
$929$		−	1.35261i		−	0.0443777i	0.999754	−	0.0221888i	$-0.00706351\pi$
$930$		0			0
$931$	−7.36623			−0.241418
$932$		0			0
$933$			19.1479i			0.626874i
$934$		0			0
$935$	−15.6838			−0.512915
$936$		0			0
$937$	−45.9129			−1.49991			−0.749954	−	0.661490i	$-0.769924\pi$
$937$	−45.9129			−1.49991			−0.749954	+	0.661490i	$0.769924\pi$
$938$		0			0
$939$		−	3.16515i		−	0.103291i
$940$		0			0
$941$	20.1072			0.655475			0.327737	−	0.944769i	$-0.393714\pi$
$941$	20.1072			0.655475			0.327737	+	0.944769i	$0.393714\pi$
$942$		0			0
$943$			20.6184i			0.671427i
$944$		0			0
$945$	2.00000			0.0650600
$946$		0			0
$947$	−11.5473			−0.375236			−0.187618	−	0.982242i	$-0.560077\pi$
$947$	−11.5473			−0.375236			−0.187618	+	0.982242i	$0.560077\pi$
$948$		0			0
$949$	−14.9666	+	13.8564i	−0.485837	+	0.449798i
$950$		0			0
$951$			10.5789i			0.343044i
$952$		0			0
$953$	−6.83485			−0.221402			−0.110701	−	0.993854i	$-0.535310\pi$
$953$	−6.83485			−0.221402			−0.110701	+	0.993854i	$0.535310\pi$
$954$		0			0
$955$	32.1701			1.04100
$956$		0			0
$957$	1.29217			0.0417698
$958$		0			0
$959$	11.8383			0.382279
$960$		0			0
$961$	27.2432			0.878812
$962$		0			0
$963$			2.00000i			0.0644491i
$964$		0			0
$965$		−	15.6838i		−	0.504880i
$966$		0			0
$967$		−	38.0627i		−	1.22401i	−0.790853	−	0.612006i	$-0.790363\pi$
$967$		−	38.0627i		−	1.22401i	0.790853	−	0.612006i	$-0.209637\pi$
$968$		0			0
$969$			4.00908i			0.128790i
$970$		0			0
$971$		−	21.4955i		−	0.689822i	−0.938635	−	0.344911i	$-0.887909\pi$
$971$		−	21.4955i		−	0.689822i	0.938635	−	0.344911i	$-0.112091\pi$
$972$		0			0
$973$	−12.1126			−0.388312
$974$		0			0
$975$	12.1244	−	11.2250i	0.388290	−	0.359487i
$976$		0			0
$977$		−	18.9135i		−	0.605096i	−0.953134	−	0.302548i	$-0.902163\pi$
$977$		−	18.9135i		−	0.605096i	0.953134	−	0.302548i	$-0.0978373\pi$
$978$		0			0
$979$		−	12.4174i		−	0.396863i
$980$		0			0
$981$	9.79796			0.312825
$982$		0			0
$983$		−	35.7199i		−	1.13929i	−0.821892	−	0.569643i	$-0.807081\pi$
$983$		−	35.7199i		−	1.13929i	0.821892	−	0.569643i	$-0.192919\pi$
$984$		0			0
$985$	12.7477			0.406176
$986$		0			0
$987$			2.41742i			0.0769475i
$988$		0			0
$989$		−	12.4104i		−	0.394628i
$990$		0			0
$991$	−39.5909			−1.25765			−0.628824	−	0.777548i	$-0.716463\pi$
$991$	−39.5909			−1.25765			−0.628824	+	0.777548i	$0.716463\pi$
$992$		0			0
$993$			33.2892i			1.05640i
$994$		0			0
$995$	10.7234			0.339954
$996$		0			0
$997$			41.9506i			1.32859i	0.747471	+	0.664295i	$0.231268\pi$
$997$			41.9506i			1.32859i	−0.747471	+	0.664295i	$0.768732\pi$
$998$		0			0
$999$			4.89898i			0.154997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.m.c.2209.7	yes	16
4.3	odd	2	inner	2496.2.m.c.2209.16	yes	16
8.3	odd	2	inner	2496.2.m.c.2209.1	✓	16
8.5	even	2	inner	2496.2.m.c.2209.10	yes	16
13.12	even	2	inner	2496.2.m.c.2209.2	yes	16
52.51	odd	2	inner	2496.2.m.c.2209.9	yes	16
104.51	odd	2	inner	2496.2.m.c.2209.8	yes	16
104.77	even	2	inner	2496.2.m.c.2209.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2496.2.m.c.2209.1	✓	16	8.3	odd	2	inner
2496.2.m.c.2209.2	yes	16	13.12	even	2	inner
2496.2.m.c.2209.7	yes	16	1.1	even	1	trivial
2496.2.m.c.2209.8	yes	16	104.51	odd	2	inner
2496.2.m.c.2209.9	yes	16	52.51	odd	2	inner
2496.2.m.c.2209.10	yes	16	8.5	even	2	inner
2496.2.m.c.2209.15	yes	16	104.77	even	2	inner
2496.2.m.c.2209.16	yes	16	4.3	odd	2	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:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.09557 q^{5} -0.646084i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.09557 q^{5} -0.646084i q^{7} -1.00000 q^{9} -1.41421 q^{11} +(2.44949 + 2.64575i) q^{13} -3.09557i q^{15} +3.58258 q^{17} -1.11905 q^{19} -0.646084 q^{21} +3.46410 q^{23} +4.58258 q^{25} +1.00000i q^{27} -0.913701i q^{29} +1.93825i q^{31} +1.41421i q^{33} -2.00000i q^{35} +4.89898 q^{37} +(2.64575 - 2.44949i) q^{39} +5.95202i q^{41} -3.58258i q^{43} -3.09557 q^{45} -3.74166i q^{47} +6.58258 q^{49} -3.58258i q^{51} -10.5830i q^{53} -4.37780 q^{55} +1.11905i q^{57} +9.30917 q^{59} -0.913701i q^{61} +0.646084i q^{63} +(7.58258 + 8.19012i) q^{65} +14.6709 q^{67} -3.46410i q^{69} +2.44949i q^{71} +5.65685i q^{73} -4.58258i q^{75} +0.913701i q^{77} -14.0471 q^{79} +1.00000 q^{81} -11.5473 q^{83} +11.0901 q^{85} -0.913701 q^{87} +8.78044i q^{89} +(1.70938 - 1.58258i) q^{91} +1.93825 q^{93} -3.46410 q^{95} -7.89495i q^{97} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} - 16 q^{17} + 32 q^{49} + 48 q^{65} + 16 q^{81}+O(q^{100}) \) 16 * q - 16 * q^9 - 16 * q^17 + 32 * q^49 + 48 * q^65 + 16 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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