LMFDB - Embedded newform 2400.2.h.h.1151.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,2,Mod(1151,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2400.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.h (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:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.17
Character	$\chi$	$=$	2400.1151
Dual form			2400.2.h.h.1151.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.28241 - 1.16422i) q^{3} -4.22289i q^{7} +(0.289169 - 2.98603i) q^{9} +O(q^{10})$	$q+(1.28241 - 1.16422i) q^{3} -4.22289i q^{7} +(0.289169 - 2.98603i) q^{9} +2.42945 q^{11} +5.33021 q^{13} +3.57009i q^{17} +0.578337i q^{19} +(-4.91638 - 5.41549i) q^{21} +5.39325 q^{23} +(-3.10557 - 4.16598i) q^{27} +5.10860i q^{29} -7.83276i q^{31} +(3.11556 - 2.82843i) q^{33} +7.77246 q^{37} +(6.83553 - 6.20555i) q^{39} -0.613779i q^{41} +2.32845i q^{43} -3.38272 q^{47} -10.8328 q^{49} +(4.15639 + 4.57834i) q^{51} +1.26887i q^{53} +(0.673313 + 0.741667i) q^{57} -7.78774 q^{59} -6.67609 q^{61} +(-12.6097 - 1.22113i) q^{63} -0.113805i q^{67} +(6.91638 - 6.27895i) q^{69} +8.31277 q^{71} -13.1027 q^{73} -10.2593i q^{77} +3.42166i q^{79} +(-8.83276 - 1.72693i) q^{81} -5.68395 q^{83} +(5.94755 + 6.55133i) q^{87} -4.85891i q^{89} -22.5089i q^{91} +(-9.11908 - 10.0448i) q^{93} -9.31379 q^{97} +(0.702522 - 7.25443i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 8 q^{21} - 40 q^{49} + 32 q^{61} + 56 q^{69} + 8 q^{81}+O(q^{100}) $ 24 * q - 8 * q^21 - 40 * q^49 + 32 * q^61 + 56 * q^69 + 8 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	1.28241	−	1.16422i	0.740402	−	0.672165i
$3$	1.28241	−	1.16422i	0.740402	−	0.672165i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	4.22289i		−	1.59610i	−0.602591	−	0.798050i	$-0.705865\pi$
$7$		−	4.22289i		−	1.59610i	0.602591	−	0.798050i	$-0.294135\pi$
$8$		0			0
$9$	0.289169	−	2.98603i	0.0963895	−	0.995344i
$10$		0			0
$11$	2.42945			0.732508			0.366254	−	0.930515i	$-0.380640\pi$
$11$	2.42945			0.732508			0.366254	+	0.930515i	$0.380640\pi$
$12$		0			0
$13$	5.33021			1.47833			0.739167	−	0.673523i	$-0.235220\pi$
$13$	5.33021			1.47833			0.739167	+	0.673523i	$0.235220\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.57009i			0.865875i	0.901424	+	0.432938i	$0.142523\pi$
$17$			3.57009i			0.865875i	−0.901424	+	0.432938i	$0.857477\pi$
$18$		0			0
$19$			0.578337i			0.132680i	0.997797	+	0.0663398i	$0.0211321\pi$
$19$			0.578337i			0.132680i	−0.997797	+	0.0663398i	$0.978868\pi$
$20$		0			0
$21$	−4.91638	−	5.41549i	−1.07284	−	1.18176i
$22$		0			0
$23$	5.39325			1.12457			0.562286	−	0.826943i	$-0.309922\pi$
$23$	5.39325			1.12457			0.562286	+	0.826943i	$0.309922\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.10557	−	4.16598i	−0.597668	−	0.801744i
$28$		0			0
$29$			5.10860i			0.948642i	0.880352	+	0.474321i	$0.157306\pi$
$29$			5.10860i			0.948642i	−0.880352	+	0.474321i	$0.842694\pi$
$30$		0			0
$31$		−	7.83276i		−	1.40681i	−0.710791	−	0.703403i	$-0.751663\pi$
$31$		−	7.83276i		−	1.40681i	0.710791	−	0.703403i	$-0.248337\pi$
$32$		0			0
$33$	3.11556	−	2.82843i	0.542350	−	0.492366i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	7.77246			1.27778			0.638892	−	0.769296i	$-0.279393\pi$
$37$	7.77246			1.27778			0.638892	+	0.769296i	$0.279393\pi$
$38$		0			0
$39$	6.83553	−	6.20555i	1.09456	−	0.993683i
$40$		0			0
$41$		−	0.613779i		−	0.0958562i	−0.998851	−	0.0479281i	$-0.984738\pi$
$41$		−	0.613779i		−	0.0958562i	0.998851	−	0.0479281i	$-0.0152618\pi$
$42$		0			0
$43$			2.32845i			0.355085i	0.984113	+	0.177542i	$0.0568147\pi$
$43$			2.32845i			0.355085i	−0.984113	+	0.177542i	$0.943185\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.38272			−0.493420			−0.246710	−	0.969089i	$-0.579350\pi$
$47$	−3.38272			−0.493420			−0.246710	+	0.969089i	$0.579350\pi$
$48$		0			0
$49$	−10.8328			−1.54754
$50$		0			0
$51$	4.15639	+	4.57834i	0.582011	+	0.641095i
$52$		0			0
$53$			1.26887i			0.174292i	0.996196	+	0.0871462i	$0.0277747\pi$
$53$			1.26887i			0.174292i	−0.996196	+	0.0871462i	$0.972225\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.673313	+	0.741667i	0.0891825	+	0.0982362i
$58$		0			0
$59$	−7.78774			−1.01388			−0.506939	−	0.861982i	$-0.669223\pi$
$59$	−7.78774			−1.01388			−0.506939	+	0.861982i	$0.669223\pi$
$60$		0			0
$61$	−6.67609			−0.854786			−0.427393	−	0.904066i	$-0.640568\pi$
$61$	−6.67609			−0.854786			−0.427393	+	0.904066i	$0.640568\pi$
$62$		0			0
$63$	−12.6097	−	1.22113i	−1.58867	−	0.153847i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	0.113805i		−	0.0139035i	−0.999976	−	0.00695174i	$-0.997787\pi$
$67$		−	0.113805i		−	0.0139035i	0.999976	−	0.00695174i	$-0.00221283\pi$
$68$		0			0
$69$	6.91638	−	6.27895i	0.832634	−	0.755897i
$70$		0			0
$71$	8.31277			0.986545			0.493272	−	0.869875i	$-0.335801\pi$
$71$	8.31277			0.986545			0.493272	+	0.869875i	$0.335801\pi$
$72$		0			0
$73$	−13.1027			−1.53355			−0.766775	−	0.641915i	$-0.778140\pi$
$73$	−13.1027			−1.53355			−0.766775	+	0.641915i	$0.778140\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	10.2593i		−	1.16916i
$78$		0			0
$79$			3.42166i			0.384967i	0.981300	+	0.192484i	$0.0616542\pi$
$79$			3.42166i			0.384967i	−0.981300	+	0.192484i	$0.938346\pi$
$80$		0			0
$81$	−8.83276	−	1.72693i	−0.981418	−	0.191881i
$82$		0			0
$83$	−5.68395			−0.623894			−0.311947	−	0.950099i	$-0.600981\pi$
$83$	−5.68395			−0.623894			−0.311947	+	0.950099i	$0.600981\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	5.94755	+	6.55133i	0.637644	+	0.702377i
$88$		0			0
$89$		−	4.85891i		−	0.515043i	−0.966273	−	0.257522i	$-0.917094\pi$
$89$		−	4.85891i		−	0.515043i	0.966273	−	0.257522i	$-0.0829059\pi$
$90$		0			0
$91$		−	22.5089i		−	2.35957i
$92$		0			0
$93$	−9.11908	−	10.0448i	−0.945605	−	1.04160i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−9.31379			−0.945672			−0.472836	−	0.881151i	$-0.656770\pi$
$97$	−9.31379			−0.945672			−0.472836	+	0.881151i	$0.656770\pi$
$98$		0			0
$99$	0.702522	−	7.25443i	0.0706061	−	0.729097i
$100$		0			0
$101$		−	13.4214i		−	1.33548i	−0.744396	−	0.667738i	$-0.767263\pi$
$101$		−	13.4214i		−	1.33548i	0.744396	−	0.667738i	$-0.232737\pi$
$102$		0			0
$103$		−	1.78063i		−	0.175451i	−0.996145	−	0.0877256i	$-0.972040\pi$
$103$		−	1.78063i		−	0.175451i	0.996145	−	0.0877256i	$-0.0279599\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	16.9977			1.64323			0.821613	−	0.570046i	$-0.193075\pi$
$107$	16.9977			1.64323			0.821613	+	0.570046i	$0.193075\pi$
$108$		0			0
$109$	−7.83276			−0.750243			−0.375121	−	0.926976i	$-0.622399\pi$
$109$	−7.83276			−0.750243			−0.375121	+	0.926976i	$0.622399\pi$
$110$		0			0
$111$	9.96750	−	9.04888i	0.946074	−	0.858881i
$112$		0			0
$113$			18.0029i			1.69357i	0.531933	+	0.846786i	$0.321466\pi$
$113$			18.0029i			1.69357i	−0.531933	+	0.846786i	$0.678534\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.54133	−	15.9162i	0.142496	−	1.47145i
$118$		0			0
$119$	15.0761			1.38202
$120$		0			0
$121$	−5.09775			−0.463432
$122$		0			0
$123$	−0.714576	−	0.787118i	−0.0644311	−	0.0709721i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			8.01176i			0.710929i	0.934690	+	0.355465i	$0.115677\pi$
$127$			8.01176i			0.710929i	−0.934690	+	0.355465i	$0.884323\pi$
$128$		0			0
$129$	2.71083	+	2.98603i	0.238675	+	0.262905i
$130$		0			0
$131$	−7.78774			−0.680418			−0.340209	−	0.940350i	$-0.610498\pi$
$131$	−7.78774			−0.680418			−0.340209	+	0.940350i	$0.610498\pi$
$132$		0			0
$133$	2.44225			0.211770
$134$		0			0
$135$		0			0
$136$		0			0
$137$			15.4652i			1.32128i	0.750703	+	0.660640i	$0.229715\pi$
$137$			15.4652i			1.32128i	−0.750703	+	0.660640i	$0.770285\pi$
$138$		0			0
$139$			0.578337i			0.0490539i	0.999699	+	0.0245270i	$0.00780796\pi$
$139$			0.578337i			0.0490539i	−0.999699	+	0.0245270i	$0.992192\pi$
$140$		0			0
$141$	−4.33804	+	3.93824i	−0.365329	+	0.331660i
$142$		0			0
$143$	12.9495			1.08289
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−13.8921	+	12.6118i	−1.14580	+	1.04020i
$148$		0			0
$149$		−	15.9396i		−	1.30582i	−0.757435	−	0.652910i	$-0.773548\pi$
$149$		−	15.9396i		−	1.30582i	0.757435	−	0.652910i	$-0.226452\pi$
$150$		0			0
$151$			12.5783i			1.02361i	0.859101	+	0.511805i	$0.171023\pi$
$151$			12.5783i			1.02361i	−0.859101	+	0.511805i	$0.828977\pi$
$152$		0			0
$153$	10.6604	+	1.03236i	0.861843	+	0.0834613i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	10.8551			0.866332			0.433166	−	0.901314i	$-0.357396\pi$
$157$	10.8551			0.866332			0.433166	+	0.901314i	$0.357396\pi$
$158$		0			0
$159$	1.47725	+	1.62721i	0.117153	+	0.129046i
$160$		0			0
$161$		−	22.7751i		−	1.79493i
$162$		0			0
$163$		−	9.19998i		−	0.720598i	−0.932837	−	0.360299i	$-0.882675\pi$
$163$		−	9.19998i		−	0.720598i	0.932837	−	0.360299i	$-0.117325\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.74693			−0.135182			−0.0675909	−	0.997713i	$-0.521531\pi$
$167$	−1.74693			−0.135182			−0.0675909	+	0.997713i	$0.521531\pi$
$168$		0			0
$169$	15.4111			1.18547
$170$		0			0
$171$	1.72693	+	0.167237i	0.132062	+	0.0127889i
$172$		0			0
$173$			8.56151i			0.650919i	0.945556	+	0.325460i	$0.105519\pi$
$173$			8.56151i			0.650919i	−0.945556	+	0.325460i	$0.894481\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−9.98710	+	9.06666i	−0.750676	+	0.681492i
$178$		0			0
$179$	11.2416			0.840237			0.420118	−	0.907469i	$-0.361989\pi$
$179$	11.2416			0.840237			0.420118	+	0.907469i	$0.361989\pi$
$180$		0			0
$181$	0.843326			0.0626839			0.0313420	−	0.999509i	$-0.490022\pi$
$181$	0.843326			0.0626839			0.0313420	+	0.999509i	$0.490022\pi$
$182$		0			0
$183$	−8.56151	+	7.77246i	−0.632885	+	0.574557i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			8.67338i			0.634260i
$188$		0			0
$189$	−17.5925	+	13.1145i	−1.27966	+	0.953938i
$190$		0			0
$191$	18.5300			1.34078			0.670391	−	0.742008i	$-0.266127\pi$
$191$	18.5300			1.34078			0.670391	+	0.742008i	$0.266127\pi$
$192$		0			0
$193$	−2.44225			−0.175797			−0.0878986	−	0.996129i	$-0.528015\pi$
$193$	−2.44225			−0.175797			−0.0878986	+	0.996129i	$0.528015\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			5.44239i			0.387754i	0.981026	+	0.193877i	$0.0621063\pi$
$197$			5.44239i			0.387754i	−0.981026	+	0.193877i	$0.937894\pi$
$198$		0			0
$199$		−	6.67609i		−	0.473255i	−0.971600	−	0.236628i	$-0.923958\pi$
$199$		−	6.67609i		−	0.473255i	0.971600	−	0.236628i	$-0.0760422\pi$
$200$		0			0
$201$	−0.132494	−	0.145945i	−0.00934543	−	0.0102942i
$202$		0			0
$203$	21.5730			1.51413
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.55956	−	16.1044i	0.108397	−	1.11933i
$208$		0			0
$209$			1.40504i			0.0971889i
$210$		0			0
$211$		−	8.57834i		−	0.590557i	−0.955411	−	0.295279i	$-0.904588\pi$
$211$		−	8.57834i		−	0.590557i	0.955411	−	0.295279i	$-0.0954124\pi$
$212$		0			0
$213$	10.6604	−	9.67792i	0.730439	−	0.663120i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−33.0769			−2.24540
$218$		0			0
$219$	−16.8030	+	15.2544i	−1.13544	+	1.03080i
$220$		0			0
$221$			19.0293i			1.28005i
$222$		0			0
$223$			3.12726i			0.209417i	0.994503	+	0.104708i	$0.0333909\pi$
$223$			3.12726i			0.209417i	−0.994503	+	0.104708i	$0.966609\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.929043			−0.0616627			−0.0308314	−	0.999525i	$-0.509815\pi$
$227$	−0.929043			−0.0616627			−0.0308314	+	0.999525i	$0.509815\pi$
$228$		0			0
$229$	−6.00000			−0.396491			−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000			−0.396491			−0.198246	+	0.980152i	$0.563524\pi$
$230$		0			0
$231$	−11.9441	−	13.1567i	−0.785866	−	0.865646i
$232$		0			0
$233$			6.10783i			0.400137i	0.979782	+	0.200069i	$0.0641165\pi$
$233$			6.10783i			0.400137i	−0.979782	+	0.200069i	$0.935883\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	3.98358	+	4.38799i	0.258761	+	0.285030i
$238$		0			0
$239$	−15.0761			−0.975192			−0.487596	−	0.873069i	$-0.662126\pi$
$239$	−15.0761			−0.975192			−0.487596	+	0.873069i	$0.662126\pi$
$240$		0			0
$241$	−25.4005			−1.63619			−0.818096	−	0.575081i	$-0.804970\pi$
$241$	−25.4005			−1.63619			−0.818096	+	0.575081i	$0.804970\pi$
$242$		0			0
$243$	−13.3378	+	8.06867i	−0.855620	+	0.517605i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			3.08266i			0.196145i
$248$		0			0
$249$	−7.28917	+	6.61738i	−0.461932	+	0.419360i
$250$		0			0
$251$	22.8638			1.44315			0.721576	−	0.692335i	$-0.243418\pi$
$251$	22.8638			1.44315			0.721576	+	0.692335i	$0.243418\pi$
$252$		0			0
$253$	13.1027			0.823757
$254$		0			0
$255$		0			0
$256$		0			0
$257$			13.8294i			0.862654i	0.902196	+	0.431327i	$0.141955\pi$
$257$			13.8294i			0.862654i	−0.902196	+	0.431327i	$0.858045\pi$
$258$		0			0
$259$		−	32.8222i		−	2.03947i
$260$		0			0
$261$	15.2544	+	1.47725i	0.944225	+	0.0914392i
$262$		0			0
$263$	26.9663			1.66281			0.831406	−	0.555666i	$-0.187537\pi$
$263$	26.9663			1.66281			0.831406	+	0.555666i	$0.187537\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−5.65685	−	6.23113i	−0.346194	−	0.381339i
$268$		0			0
$269$		−	7.94868i		−	0.484640i	−0.970196	−	0.242320i	$-0.922092\pi$
$269$		−	7.94868i		−	0.484640i	0.970196	−	0.242320i	$-0.0779083\pi$
$270$		0			0
$271$			12.5783i			0.764080i	0.924146	+	0.382040i	$0.124778\pi$
$271$			12.5783i			0.764080i	−0.924146	+	0.382040i	$0.875222\pi$
$272$		0			0
$273$	−26.2053	−	28.8657i	−1.58602	−	1.74703i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4.68980			−0.281783			−0.140891	−	0.990025i	$-0.544997\pi$
$277$	−4.68980			−0.281783			−0.140891	+	0.990025i	$0.544997\pi$
$278$		0			0
$279$	−23.3889	−	2.26499i	−1.40026	−	0.135601i
$280$		0			0
$281$			15.6899i			0.935980i	0.883734	+	0.467990i	$0.155022\pi$
$281$			15.6899i			0.935980i	−0.883734	+	0.467990i	$0.844978\pi$
$282$		0			0
$283$			24.5173i			1.45740i	0.684832	+	0.728701i	$0.259875\pi$
$283$			24.5173i			1.45740i	−0.684832	+	0.728701i	$0.740125\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.59192			−0.152996
$288$		0			0
$289$	4.25443			0.250260
$290$		0			0
$291$	−11.9441	+	10.8433i	−0.700177	+	0.635647i
$292$		0			0
$293$		−	8.99044i		−	0.525227i	−0.964901	−	0.262614i	$-0.915416\pi$
$293$		−	8.99044i		−	0.525227i	0.964901	−	0.262614i	$-0.0845844\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−7.54485	−	10.1211i	−0.437796	−	0.587284i
$298$		0			0
$299$	28.7472			1.66249
$300$		0			0
$301$	9.83276			0.566751
$302$		0			0
$303$	−15.6255	−	17.2117i	−0.897660	−	0.988789i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	29.8804i		−	1.70536i	−0.522430	−	0.852682i	$-0.674974\pi$
$307$		−	29.8804i		−	1.70536i	0.522430	−	0.852682i	$-0.325026\pi$
$308$		0			0
$309$	−2.07306	−	2.28351i	−0.117932	−	0.129904i
$310$		0			0
$311$	−32.2010			−1.82595			−0.912976	−	0.408013i	$-0.866222\pi$
$311$	−32.2010			−1.82595			−0.912976	+	0.408013i	$0.866222\pi$
$312$		0			0
$313$	8.67338			0.490248			0.245124	−	0.969492i	$-0.421171\pi$
$313$	8.67338			0.490248			0.245124	+	0.969492i	$0.421171\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	23.4233i		−	1.31558i	−0.753201	−	0.657791i	$-0.771491\pi$
$317$		−	23.4233i		−	1.31558i	0.753201	−	0.657791i	$-0.228509\pi$
$318$		0			0
$319$			12.4111i			0.694888i
$320$		0			0
$321$	21.7980	−	19.7891i	1.21665	−	1.10452i
$322$		0			0
$323$	−2.06472			−0.114884
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−10.0448	+	9.11908i	−0.555481	+	0.504287i
$328$		0			0
$329$			14.2848i			0.787548i
$330$		0			0
$331$			22.7527i			1.25060i	0.780384	+	0.625301i	$0.215024\pi$
$331$			22.7527i			1.25060i	−0.780384	+	0.625301i	$0.784976\pi$
$332$		0			0
$333$	2.24755	−	23.2088i	0.123165	−	1.27183i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	14.4493			0.787103			0.393552	−	0.919303i	$-0.371246\pi$
$337$	14.4493			0.787103			0.393552	+	0.919303i	$0.371246\pi$
$338$		0			0
$339$	20.9594	+	23.0872i	1.13836	+	1.25392i
$340$		0			0
$341$		−	19.0293i		−	1.03050i
$342$		0			0
$343$			16.1853i			0.873925i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.83640			−0.313314			−0.156657	−	0.987653i	$-0.550072\pi$
$347$	−5.83640			−0.313314			−0.156657	+	0.987653i	$0.550072\pi$
$348$		0			0
$349$	31.9789			1.71179			0.855895	−	0.517150i	$-0.173007\pi$
$349$	31.9789			1.71179			0.855895	+	0.517150i	$0.173007\pi$
$350$		0			0
$351$	−16.5533	−	22.2056i	−0.883552	−	1.18524i
$352$		0			0
$353$			18.0029i			0.958199i	0.877761	+	0.479099i	$0.159037\pi$
$353$			18.0029i			0.958199i	−0.877761	+	0.479099i	$0.840963\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	19.3338	−	17.5519i	1.02325	−	0.928948i
$358$		0			0
$359$	−13.6711			−0.721531			−0.360765	−	0.932657i	$-0.617484\pi$
$359$	−13.6711			−0.721531			−0.360765	+	0.932657i	$0.617484\pi$
$360$		0			0
$361$	18.6655			0.982396
$362$		0			0
$363$	−6.53743	+	5.93492i	−0.343126	+	0.311503i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			4.45050i			0.232314i	0.993231	+	0.116157i	$0.0370576\pi$
$367$			4.45050i			0.232314i	−0.993231	+	0.116157i	$0.962942\pi$
$368$		0			0
$369$	−1.83276	−	0.177486i	−0.0954098	−	0.00923953i
$370$		0			0
$371$	5.35828			0.278188
$372$		0			0
$373$	−15.3502			−0.794804			−0.397402	−	0.917645i	$-0.630088\pi$
$373$	−15.3502			−0.794804			−0.397402	+	0.917645i	$0.630088\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			27.2299i			1.40241i
$378$		0			0
$379$			12.7738i			0.656148i	0.944652	+	0.328074i	$0.106400\pi$
$379$			12.7738i			0.656148i	−0.944652	+	0.328074i	$0.893600\pi$
$380$		0			0
$381$	9.32748	+	10.2744i	0.477861	+	0.526373i
$382$		0			0
$383$	−30.6668			−1.56700			−0.783499	−	0.621392i	$-0.786567\pi$
$383$	−30.6668			−1.56700			−0.783499	+	0.621392i	$0.786567\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	6.95281	+	0.673313i	0.353431	+	0.0342264i
$388$		0			0
$389$			14.5345i			0.736930i	0.929642	+	0.368465i	$0.120116\pi$
$389$			14.5345i			0.736930i	−0.929642	+	0.368465i	$0.879884\pi$
$390$		0			0
$391$			19.2544i			0.973738i
$392$		0			0
$393$	−9.98710	+	9.06666i	−0.503783	+	0.457353i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	28.4529			1.42801			0.714005	−	0.700141i	$-0.246880\pi$
$397$	28.4529			1.42801			0.714005	+	0.700141i	$0.246880\pi$
$398$		0			0
$399$	3.13198	−	2.84333i	0.156795	−	0.142344i
$400$		0			0
$401$			31.7016i			1.58310i	0.611101	+	0.791552i	$0.290727\pi$
$401$			31.7016i			1.58310i	−0.611101	+	0.791552i	$0.709273\pi$
$402$		0			0
$403$		−	41.7502i		−	2.07973i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.8828			0.935987
$408$		0			0
$409$	−3.83276			−0.189518			−0.0947590	−	0.995500i	$-0.530208\pi$
$409$	−3.83276			−0.189518			−0.0947590	+	0.995500i	$0.530208\pi$
$410$		0			0
$411$	18.0049	+	19.8328i	0.888118	+	0.978278i
$412$		0			0
$413$			32.8867i			1.61825i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.673313	+	0.741667i	0.0329723	+	0.0363196i
$418$		0			0
$419$	−14.1961			−0.693525			−0.346762	−	0.937953i	$-0.612719\pi$
$419$	−14.1961			−0.693525			−0.346762	+	0.937953i	$0.612719\pi$
$420$		0			0
$421$	−24.9894			−1.21791			−0.608955	−	0.793205i	$-0.708411\pi$
$421$	−24.9894			−1.21791			−0.608955	+	0.793205i	$0.708411\pi$
$422$		0			0
$423$	−0.978176	+	10.1009i	−0.0475605	+	0.491123i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			28.1924i			1.36432i
$428$		0			0
$429$	16.6066	−	15.0761i	0.801774	−	0.727881i
$430$		0			0
$431$	−22.8895			−1.10255			−0.551274	−	0.834324i	$-0.685858\pi$
$431$	−22.8895			−1.10255			−0.551274	+	0.834324i	$0.685858\pi$
$432$		0			0
$433$	16.8915			0.811756			0.405878	−	0.913927i	$-0.366966\pi$
$433$	16.8915			0.811756			0.405878	+	0.913927i	$0.366966\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			3.11912i			0.149208i
$438$		0			0
$439$		−	0.989437i		−	0.0472233i	−0.999721	−	0.0236116i	$-0.992483\pi$
$439$		−	0.989437i		−	0.0472233i	0.999721	−	0.0236116i	$-0.00751651\pi$
$440$		0			0
$441$	−3.13249	+	32.3470i	−0.149166	+	1.54033i
$442$		0			0
$443$	14.4599			0.687011			0.343506	−	0.939151i	$-0.388386\pi$
$443$	14.4599			0.687011			0.343506	+	0.939151i	$0.388386\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−18.5572	−	20.4411i	−0.877726	−	0.966832i
$448$		0			0
$449$		−	25.9071i		−	1.22263i	−0.791387	−	0.611315i	$-0.790641\pi$
$449$		−	25.9071i		−	1.22263i	0.791387	−	0.611315i	$-0.209359\pi$
$450$		0			0
$451$		−	1.49115i		−	0.0702154i
$452$		0			0
$453$	14.6440	+	16.1306i	0.688035	+	0.757883i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	39.5590			1.85049			0.925246	−	0.379368i	$-0.123859\pi$
$457$	39.5590			1.85049			0.925246	+	0.379368i	$0.123859\pi$
$458$		0			0
$459$	14.8730	−	11.0872i	0.694210	−	0.517506i
$460$		0			0
$461$			11.5170i			0.536398i	0.963364	+	0.268199i	$0.0864285\pi$
$461$			11.5170i			0.536398i	−0.963364	+	0.268199i	$0.913572\pi$
$462$		0			0
$463$		−	37.0721i		−	1.72289i	−0.507852	−	0.861444i	$-0.669560\pi$
$463$		−	37.0721i		−	1.72289i	0.507852	−	0.861444i	$-0.330440\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.4163			0.759654			0.379827	−	0.925057i	$-0.375983\pi$
$467$	16.4163			0.759654			0.379827	+	0.925057i	$0.375983\pi$
$468$		0			0
$469$	−0.480585			−0.0221914
$470$		0			0
$471$	13.9207	−	12.6378i	0.641434	−	0.582318i
$472$		0			0
$473$			5.65685i			0.260102i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	3.78888	+	0.366917i	0.173481	+	0.0168000i
$478$		0			0
$479$	−18.5300			−0.846656			−0.423328	−	0.905977i	$-0.639138\pi$
$479$	−18.5300			−0.846656			−0.423328	+	0.905977i	$0.639138\pi$
$480$		0			0
$481$	41.4288			1.88899
$482$		0			0
$483$	−26.5153	−	29.2071i	−1.20649	−	1.32897i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	21.9824i		−	0.996120i	−0.867143	−	0.498060i	$-0.834046\pi$
$487$		−	21.9824i		−	0.996120i	0.867143	−	0.498060i	$-0.165954\pi$
$488$		0			0
$489$	−10.7108	−	11.7982i	−0.484361	−	0.533532i
$490$		0			0
$491$	−42.7988			−1.93148			−0.965742	−	0.259502i	$-0.916441\pi$
$491$	−42.7988			−1.93148			−0.965742	+	0.259502i	$0.916441\pi$
$492$		0			0
$493$	−18.2382			−0.821406
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	35.1039i		−	1.57462i
$498$		0			0
$499$			23.0872i			1.03352i	0.856129	+	0.516762i	$0.172863\pi$
$499$			23.0872i			1.03352i	−0.856129	+	0.516762i	$0.827137\pi$
$500$		0			0
$501$	−2.24029	+	2.03382i	−0.100089	+	0.0908645i
$502$		0			0
$503$	33.5793			1.49723			0.748613	−	0.663008i	$-0.230720\pi$
$503$	33.5793			1.49723			0.748613	+	0.663008i	$0.230720\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	19.7634	−	17.9420i	0.877724	−	0.796830i
$508$		0			0
$509$			34.3551i			1.52276i	0.648303	+	0.761382i	$0.275479\pi$
$509$			34.3551i			1.52276i	−0.648303	+	0.761382i	$0.724521\pi$
$510$		0			0
$511$			55.3311i			2.44770i
$512$		0			0
$513$	2.40934	−	1.79607i	0.106375	−	0.0792983i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−8.21816			−0.361434
$518$		0			0
$519$	9.96750	+	10.9794i	0.437525	+	0.481942i
$520$		0			0
$521$			10.2172i			0.447623i	0.974632	+	0.223812i	$0.0718500\pi$
$521$			10.2172i			0.447623i	−0.974632	+	0.223812i	$0.928150\pi$
$522$		0			0
$523$			9.65520i			0.422192i	0.977465	+	0.211096i	$0.0677033\pi$
$523$			9.65520i			0.422192i	−0.977465	+	0.211096i	$0.932297\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	27.9637			1.21812
$528$		0			0
$529$	6.08719			0.264660
$530$		0			0
$531$	−2.25197	+	23.2544i	−0.0977271	+	1.00916i
$532$		0			0
$533$		−	3.27157i		−	0.141707i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	14.4164	−	13.0877i	0.622113	−	0.564777i
$538$		0			0
$539$	−26.3177			−1.13358
$540$		0			0
$541$	22.8222			0.981203			0.490602	−	0.871384i	$-0.336777\pi$
$541$	22.8222			0.981203			0.490602	+	0.871384i	$0.336777\pi$
$542$		0			0
$543$	1.08149	−	0.981820i	0.0464113	−	0.0421339i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	12.9889i		−	0.555364i	−0.960673	−	0.277682i	$-0.910434\pi$
$547$		−	12.9889i		−	0.555364i	0.960673	−	0.277682i	$-0.0895661\pi$
$548$		0			0
$549$	−1.93051	+	19.9350i	−0.0823924	+	0.850805i
$550$		0			0
$551$	−2.95449			−0.125866
$552$		0			0
$553$	14.4493			0.614446
$554$		0			0
$555$		0			0
$556$		0			0
$557$			14.0659i			0.595992i	0.954567	+	0.297996i	$0.0963182\pi$
$557$			14.0659i			0.595992i	−0.954567	+	0.297996i	$0.903682\pi$
$558$		0			0
$559$			12.4111i			0.524934i
$560$		0			0
$561$	10.0978	+	11.1229i	0.426327	+	0.469608i
$562$		0			0
$563$	20.7964			0.876465			0.438232	−	0.898862i	$-0.355605\pi$
$563$	20.7964			0.876465			0.438232	+	0.898862i	$0.355605\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−7.29264	+	37.2997i	−0.306262	+	1.56644i
$568$		0			0
$569$			25.7296i			1.07864i	0.842101	+	0.539320i	$0.181319\pi$
$569$			25.7296i			1.07864i	−0.842101	+	0.539320i	$0.818681\pi$
$570$		0			0
$571$		−	41.4005i		−	1.73256i	−0.499560	−	0.866279i	$-0.666505\pi$
$571$		−	41.4005i		−	1.73256i	0.499560	−	0.866279i	$-0.333495\pi$
$572$		0			0
$573$	23.7631	−	21.5730i	0.992717	−	0.901226i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−22.0270			−0.916998			−0.458499	−	0.888695i	$-0.651613\pi$
$577$	−22.0270			−0.916998			−0.458499	+	0.888695i	$0.651613\pi$
$578$		0			0
$579$	−3.13198	+	2.84333i	−0.130160	+	0.118165i
$580$		0			0
$581$			24.0027i			0.995798i
$582$		0			0
$583$			3.08266i			0.127671i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−23.5565			−0.972279			−0.486140	−	0.873881i	$-0.661595\pi$
$587$	−23.5565			−0.972279			−0.486140	+	0.873881i	$0.661595\pi$
$588$		0			0
$589$	4.52998			0.186654
$590$		0			0
$591$	6.33615	+	6.97939i	0.260635	+	0.287094i
$592$		0			0
$593$			21.2745i			0.873639i	0.899549	+	0.436819i	$0.143895\pi$
$593$			21.2745i			0.873639i	−0.899549	+	0.436819i	$0.856105\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.77246	−	8.56151i	−0.318106	−	0.350399i
$598$		0			0
$599$	−3.30946			−0.135221			−0.0676105	−	0.997712i	$-0.521538\pi$
$599$	−3.30946			−0.135221			−0.0676105	+	0.997712i	$0.521538\pi$
$600$		0			0
$601$	9.51941			0.388305			0.194153	−	0.980971i	$-0.437804\pi$
$601$	9.51941			0.388305			0.194153	+	0.980971i	$0.437804\pi$
$602$		0			0
$603$	−0.339825	−	0.0329088i	−0.0138387	−	0.00134015i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	20.6358i		−	0.837582i	−0.908083	−	0.418791i	$-0.862454\pi$
$607$		−	20.6358i		−	0.837582i	0.908083	−	0.418791i	$-0.137546\pi$
$608$		0			0
$609$	27.6655	−	25.1158i	1.12106	−	1.01774i
$610$		0			0
$611$	−18.0306			−0.729440
$612$		0			0
$613$	−2.63695			−0.106506			−0.0532528	−	0.998581i	$-0.516959\pi$
$613$	−2.63695			−0.106506			−0.0532528	+	0.998581i	$0.516959\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			5.20588i			0.209581i	0.994494	+	0.104790i	$0.0334172\pi$
$617$			5.20588i			0.209581i	−0.994494	+	0.104790i	$0.966583\pi$
$618$		0			0
$619$		−	45.9305i		−	1.84610i	−0.384676	−	0.923052i	$-0.625687\pi$
$619$		−	45.9305i		−	1.84610i	0.384676	−	0.923052i	$-0.374313\pi$
$620$		0			0
$621$	−16.7491	−	22.4682i	−0.672120	−	0.901618i
$622$		0			0
$623$	−20.5186			−0.822061
$624$		0			0
$625$		0			0
$626$		0			0
$627$	1.63578	+	1.80185i	0.0653269	+	0.0719588i
$628$		0			0
$629$			27.7484i			1.10640i
$630$		0			0
$631$			12.3627i			0.492153i	0.969250	+	0.246076i	$0.0791414\pi$
$631$			12.3627i			0.492153i	−0.969250	+	0.246076i	$0.920859\pi$
$632$		0			0
$633$	−9.98710	−	11.0010i	−0.396952	−	0.437249i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−57.7409			−2.28778
$638$		0			0
$639$	2.40379	−	24.8222i	0.0950926	−	0.981951i
$640$		0			0
$641$		−	25.9071i		−	1.02327i	−0.859204	−	0.511634i	$-0.829040\pi$
$641$		−	25.9071i		−	1.02327i	0.859204	−	0.511634i	$-0.170960\pi$
$642$		0			0
$643$			30.5866i			1.20622i	0.797659	+	0.603109i	$0.206072\pi$
$643$			30.5866i			1.20622i	−0.797659	+	0.603109i	$0.793928\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6.97486			0.274210			0.137105	−	0.990557i	$-0.456220\pi$
$647$	6.97486			0.274210			0.137105	+	0.990557i	$0.456220\pi$
$648$		0			0
$649$	−18.9200			−0.742673
$650$		0			0
$651$	−42.4182	+	38.5089i	−1.66250	+	1.50928i
$652$		0			0
$653$			7.50711i			0.293776i	0.989153	+	0.146888i	$0.0469256\pi$
$653$			7.50711i			0.293776i	−0.989153	+	0.146888i	$0.953074\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−3.78888	+	39.1250i	−0.147818	+	1.52641i
$658$		0			0
$659$	−36.5349			−1.42320			−0.711599	−	0.702586i	$-0.752029\pi$
$659$	−36.5349			−1.42320			−0.711599	+	0.702586i	$0.752029\pi$
$660$		0			0
$661$	30.3416			1.18015			0.590076	−	0.807348i	$-0.299098\pi$
$661$	30.3416			1.18015			0.590076	+	0.807348i	$0.299098\pi$
$662$		0			0
$663$	22.1544	+	24.4035i	0.860406	+	0.947753i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			27.5520i			1.06682i
$668$		0			0
$669$	3.64083	+	4.01044i	0.140763	+	0.155053i
$670$		0			0
$671$	−16.2193			−0.626137
$672$		0			0
$673$	−4.42928			−0.170736			−0.0853682	−	0.996349i	$-0.527207\pi$
$673$	−4.42928			−0.170736			−0.0853682	+	0.996349i	$0.527207\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			22.5213i			0.865564i	0.901498	+	0.432782i	$0.142468\pi$
$677$			22.5213i			0.865564i	−0.901498	+	0.432782i	$0.857532\pi$
$678$		0			0
$679$			39.3311i			1.50939i
$680$		0			0
$681$	−1.19142	+	1.08161i	−0.0456552	+	0.0414475i
$682$		0			0
$683$	31.1256			1.19099			0.595494	−	0.803360i	$-0.296956\pi$
$683$	31.1256			1.19099			0.595494	+	0.803360i	$0.296956\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−7.69448	+	6.98534i	−0.293563	+	0.266507i
$688$		0			0
$689$			6.76333i			0.257662i
$690$		0			0
$691$		−	36.7738i		−	1.39894i	−0.714661	−	0.699471i	$-0.753419\pi$
$691$		−	36.7738i		−	1.39894i	0.714661	−	0.699471i	$-0.246581\pi$
$692$		0			0
$693$	−30.6346	−	2.96667i	−1.16371	−	0.112694i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.19125			0.0829995
$698$		0			0
$699$	7.11088	+	7.83276i	0.268958	+	0.296262i
$700$		0			0
$701$			12.3082i			0.464875i	0.972611	+	0.232437i	$0.0746701\pi$
$701$			12.3082i			0.464875i	−0.972611	+	0.232437i	$0.925330\pi$
$702$		0			0
$703$			4.49510i			0.169536i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−56.6769			−2.13155
$708$		0			0
$709$	−16.8433			−0.632564			−0.316282	−	0.948665i	$-0.602435\pi$
$709$	−16.8433			−0.632564			−0.316282	+	0.948665i	$0.602435\pi$
$710$		0			0
$711$	10.2172	+	0.989437i	0.383175	+	0.0371068i
$712$		0			0
$713$		−	42.2441i		−	1.58205i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−19.3338	+	17.5519i	−0.722034	+	0.655489i
$718$		0			0
$719$	44.3226			1.65296			0.826478	−	0.562970i	$-0.190341\pi$
$719$	44.3226			1.65296			0.826478	+	0.562970i	$0.190341\pi$
$720$		0			0
$721$	−7.51941			−0.280038
$722$		0			0
$723$	−32.5740	+	29.5719i	−1.21144	+	1.09979i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			27.7350i			1.02863i	0.857601	+	0.514316i	$0.171954\pi$
$727$			27.7350i			1.02863i	−0.857601	+	0.514316i	$0.828046\pi$
$728$		0			0
$729$	−7.71083	+	25.8755i	−0.285586	+	0.958353i
$730$		0			0
$731$	−8.31277			−0.307459
$732$		0			0
$733$	−15.7396			−0.581356			−0.290678	−	0.956821i	$-0.593881\pi$
$733$	−15.7396			−0.581356			−0.290678	+	0.956821i	$0.593881\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	0.276484i		−	0.0101844i
$738$		0			0
$739$			27.7139i			1.01947i	0.860331	+	0.509736i	$0.170257\pi$
$739$			27.7139i			1.01947i	−0.860331	+	0.509736i	$0.829743\pi$
$740$		0			0
$741$	3.58890	+	3.95324i	0.131842	+	0.145226i
$742$		0			0
$743$	−31.1398			−1.14241			−0.571204	−	0.820808i	$-0.693524\pi$
$743$	−31.1398			−1.14241			−0.571204	+	0.820808i	$0.693524\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.64362	+	16.9724i	−0.0601369	+	0.620989i
$748$		0			0
$749$		−	71.7792i		−	2.62275i
$750$		0			0
$751$		−	2.14611i		−	0.0783127i	−0.999233	−	0.0391564i	$-0.987533\pi$
$751$		−	2.14611i		−	0.0783127i	0.999233	−	0.0391564i	$-0.0124670\pi$
$752$		0			0
$753$	29.3209	−	26.6186i	1.06851	−	0.970036i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	15.3502			0.557913			0.278957	−	0.960304i	$-0.410011\pi$
$757$	15.3502			0.557913			0.278957	+	0.960304i	$0.410011\pi$
$758$		0			0
$759$	16.8030	−	15.2544i	0.609911	−	0.553701i
$760$		0			0
$761$			13.6711i			0.495575i	0.968814	+	0.247788i	$0.0797035\pi$
$761$			13.6711i			0.495575i	−0.968814	+	0.247788i	$0.920296\pi$
$762$		0			0
$763$			33.0769i			1.19746i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−41.5102			−1.49885
$768$		0			0
$769$	−10.7456			−0.387495			−0.193748	−	0.981051i	$-0.562064\pi$
$769$	−10.7456			−0.387495			−0.193748	+	0.981051i	$0.562064\pi$
$770$		0			0
$771$	16.1005	+	17.7350i	0.579846	+	0.638711i
$772$		0			0
$773$		−	11.3757i		−	0.409156i	−0.978850	−	0.204578i	$-0.934418\pi$
$773$		−	11.3757i		−	0.409156i	0.978850	−	0.204578i	$-0.0655823\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−38.2124	−	42.0916i	−1.37086	−	1.51003i
$778$		0			0
$779$	0.354971			0.0127182
$780$		0			0
$781$	20.1955			0.722652
$782$		0			0
$783$	21.2823	−	15.8651i	0.760568	−	0.566973i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	7.85335i		−	0.279942i	−0.990156	−	0.139971i	$-0.955299\pi$
$787$		−	7.85335i		−	0.279942i	0.990156	−	0.139971i	$-0.0447009\pi$
$788$		0			0
$789$	34.5819	−	31.3948i	1.23115	−	1.11768i
$790$		0			0
$791$	76.0243			2.70311
$792$		0			0
$793$	−35.5849			−1.26366
$794$		0			0
$795$		0			0
$796$		0			0
$797$			17.6140i			0.623919i	0.950095	+	0.311959i	$0.100985\pi$
$797$			17.6140i			0.623919i	−0.950095	+	0.311959i	$0.899015\pi$
$798$		0			0
$799$		−	12.0766i		−	0.427240i
$800$		0			0
$801$	−14.5089	−	1.40504i	−0.512645	−	0.0496448i
$802$		0			0
$803$	−31.8323			−1.12334
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−9.25404	−	10.1935i	−0.325758	−	0.358828i
$808$		0			0
$809$			0.499375i			0.0175571i	0.999961	+	0.00877854i	$0.00279433\pi$
$809$			0.499375i			0.0175571i	−0.999961	+	0.00877854i	$0.997206\pi$
$810$		0			0
$811$			11.6172i			0.407934i	0.978978	+	0.203967i	$0.0653835\pi$
$811$			11.6172i			0.407934i	−0.978978	+	0.203967i	$0.934616\pi$
$812$		0			0
$813$	14.6440	+	16.1306i	0.513587	+	0.565726i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.34663			−0.0471125
$818$		0			0
$819$	−67.2121	−	6.50885i	−2.34858	−	0.227438i
$820$		0			0
$821$			11.2581i			0.392912i	0.980513	+	0.196456i	$0.0629433\pi$
$821$			11.2581i			0.392912i	−0.980513	+	0.196456i	$0.937057\pi$
$822$		0			0
$823$			43.0523i			1.50071i	0.661036	+	0.750354i	$0.270117\pi$
$823$			43.0523i			1.50071i	−0.661036	+	0.750354i	$0.729883\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	51.5202			1.79153			0.895766	−	0.444526i	$-0.146628\pi$
$827$	51.5202			1.79153			0.895766	+	0.444526i	$0.146628\pi$
$828$		0			0
$829$	2.48059			0.0861543			0.0430771	−	0.999072i	$-0.486284\pi$
$829$	2.48059			0.0861543			0.0430771	+	0.999072i	$0.486284\pi$
$830$		0			0
$831$	−6.01426	+	5.45998i	−0.208633	+	0.189404i
$832$		0			0
$833$		−	38.6740i		−	1.33997i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−32.6312	+	24.3252i	−1.12790	+	0.840803i
$838$		0			0
$839$	3.95324			0.136481			0.0682405	−	0.997669i	$-0.478261\pi$
$839$	3.95324			0.136481			0.0682405	+	0.997669i	$0.478261\pi$
$840$		0			0
$841$	2.90225			0.100078
$842$		0			0
$843$	18.2665	+	20.1209i	0.629132	+	0.693001i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			21.5272i			0.739684i
$848$		0			0
$849$	28.5436	+	31.4413i	0.979614	+	1.07906i
$850$		0			0
$851$	41.9188			1.43696
$852$		0			0
$853$	28.4529			0.974208			0.487104	−	0.873344i	$-0.338053\pi$
$853$	28.4529			0.974208			0.487104	+	0.873344i	$0.338053\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	27.6808i		−	0.945560i	−0.881181	−	0.472780i	$-0.843251\pi$
$857$		−	27.6808i		−	0.945560i	0.881181	−	0.472780i	$-0.156749\pi$
$858$		0			0
$859$			18.8917i			0.644576i	0.946642	+	0.322288i	$0.104452\pi$
$859$			18.8917i			0.644576i	−0.946642	+	0.322288i	$0.895548\pi$
$860$		0			0
$861$	−3.32391	+	3.01757i	−0.113279	+	0.102839i
$862$		0			0
$863$	−5.82219			−0.198190			−0.0990948	−	0.995078i	$-0.531595\pi$
$863$	−5.82219			−0.198190			−0.0990948	+	0.995078i	$0.531595\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	5.45593	−	4.95310i	0.185293	−	0.168216i
$868$		0			0
$869$			8.31277i			0.281992i
$870$		0			0
$871$		−	0.606604i		−	0.0205540i
$872$		0			0
$873$	−2.69325	+	27.8113i	−0.0911528	+	0.941268i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−33.9778			−1.14735			−0.573674	−	0.819084i	$-0.694482\pi$
$877$	−33.9778			−1.14735			−0.573674	+	0.819084i	$0.694482\pi$
$878$		0			0
$879$	−10.4669	−	11.5295i	−0.353039	−	0.388879i
$880$		0			0
$881$			5.79458i			0.195224i	0.995225	+	0.0976121i	$0.0311205\pi$
$881$			5.79458i			0.195224i	−0.995225	+	0.0976121i	$0.968880\pi$
$882$		0			0
$883$		−	55.4453i		−	1.86588i	−0.360027	−	0.932942i	$-0.617233\pi$
$883$		−	55.4453i		−	1.86588i	0.360027	−	0.932942i	$-0.382767\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	49.1749			1.65113			0.825565	−	0.564307i	$-0.190857\pi$
$887$	49.1749			1.65113			0.825565	+	0.564307i	$0.190857\pi$
$888$		0			0
$889$	33.8328			1.13471
$890$		0			0
$891$	−21.4588	−	4.19550i	−0.718897	−	0.140555i
$892$		0			0
$893$		−	1.95635i		−	0.0654668i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	36.8657	−	33.4681i	1.23091	−	1.11747i
$898$		0			0
$899$	40.0144			1.33456
$900$		0			0
$901$	−4.52998			−0.150915
$902$		0			0
$903$	12.6097	−	11.4475i	0.419623	−	0.380950i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	11.4804i		−	0.381202i	−0.981668	−	0.190601i	$-0.938956\pi$
$907$		−	11.4804i		−	0.381202i	0.981668	−	0.190601i	$-0.0610436\pi$
$908$		0			0
$909$	−40.0766	−	3.88104i	−1.32926	−	0.128726i
$910$		0			0
$911$	−26.6983			−0.884555			−0.442278	−	0.896878i	$-0.645829\pi$
$911$	−26.6983			−0.884555			−0.442278	+	0.896878i	$0.645829\pi$
$912$		0			0
$913$	−13.8089			−0.457007
$914$		0			0
$915$		0			0
$916$		0			0
$917$			32.8867i			1.08602i
$918$		0			0
$919$			38.1260i			1.25766i	0.777542	+	0.628831i	$0.216466\pi$
$919$			38.1260i			1.25766i	−0.777542	+	0.628831i	$0.783534\pi$
$920$		0			0
$921$	−34.7875	−	38.3190i	−1.14629	−	1.26265i
$922$		0			0
$923$	44.3088			1.45844
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−5.31703	−	0.514903i	−0.174634	−	0.0169116i
$928$		0			0
$929$		−	4.24513i		−	0.139278i	−0.997572	−	0.0696391i	$-0.977815\pi$
$929$		−	4.24513i		−	0.139278i	0.997572	−	0.0696391i	$-0.0221848\pi$
$930$		0			0
$931$		−	6.26499i		−	0.205327i
$932$		0			0
$933$	−41.2950	+	37.4892i	−1.35194	+	1.22734i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	4.81869			0.157420			0.0787098	−	0.996898i	$-0.474920\pi$
$937$	4.81869			0.157420			0.0787098	+	0.996898i	$0.474920\pi$
$938$		0			0
$939$	11.1229	−	10.0978i	0.362981	−	0.329528i
$940$		0			0
$941$		−	40.2641i		−	1.31257i	−0.754512	−	0.656286i	$-0.772126\pi$
$941$		−	40.2641i		−	1.31257i	0.754512	−	0.656286i	$-0.227874\pi$
$942$		0			0
$943$		−	3.31027i		−	0.107797i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.05870			0.196881			0.0984406	−	0.995143i	$-0.468615\pi$
$947$	6.05870			0.196881			0.0984406	+	0.995143i	$0.468615\pi$
$948$		0			0
$949$	−69.8399			−2.26710
$950$		0			0
$951$	−27.2699	−	30.0383i	−0.884287	−	0.974059i
$952$		0			0
$953$			16.3671i			0.530184i	0.964223	+	0.265092i	$0.0854023\pi$
$953$			16.3671i			0.530184i	−0.964223	+	0.265092i	$0.914598\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	14.4493	+	15.9162i	0.467079	+	0.514496i
$958$		0			0
$959$	65.3077			2.10890
$960$		0			0
$961$	−30.3522			−0.979103
$962$		0			0
$963$	4.91519	−	50.7555i	0.158390	−	1.63557i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	12.8304i		−	0.412599i	−0.978489	−	0.206300i	$-0.933858\pi$
$967$		−	12.8304i		−	0.412599i	0.978489	−	0.206300i	$-0.0661422\pi$
$968$		0			0
$969$	−2.64782	+	2.40379i	−0.0850603	+	0.0772209i
$970$		0			0
$971$	4.88457			0.156753			0.0783767	−	0.996924i	$-0.475026\pi$
$971$	4.88457			0.156753			0.0783767	+	0.996924i	$0.475026\pi$
$972$		0			0
$973$	2.44225			0.0782950
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	60.0946i		−	1.92260i	−0.275512	−	0.961298i	$-0.588847\pi$
$977$		−	60.0946i		−	1.92260i	0.275512	−	0.961298i	$-0.411153\pi$
$978$		0			0
$979$		−	11.8045i		−	0.377273i
$980$		0			0
$981$	−2.26499	+	23.3889i	−0.0723155	+	0.746749i
$982$		0			0
$983$	13.1690			0.420026			0.210013	−	0.977699i	$-0.432649\pi$
$983$	13.1690			0.420026			0.210013	+	0.977699i	$0.432649\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	16.6307	+	18.3191i	0.529362	+	0.583102i
$988$		0			0
$989$			12.5579i			0.399318i
$990$		0			0
$991$		−	26.7527i		−	0.849828i	−0.905234	−	0.424914i	$-0.860304\pi$
$991$		−	26.7527i		−	0.849828i	0.905234	−	0.424914i	$-0.139696\pi$
$992$		0			0
$993$	26.4892	+	29.1784i	0.840611	+	0.925948i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.13896			0.0994118			0.0497059	−	0.998764i	$-0.484172\pi$
$997$	3.13896			0.0994118			0.0497059	+	0.998764i	$0.484172\pi$
$998$		0			0
$999$	−24.1379	−	32.3799i	−0.763690	−	1.02446i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.h.h.1151.17		24
3.2	odd	2	inner	2400.2.h.h.1151.6		24
4.3	odd	2	inner	2400.2.h.h.1151.8		24
5.2	odd	4		480.2.o.a.479.5	✓	24
5.3	odd	4		480.2.o.a.479.20	yes	24
5.4	even	2	inner	2400.2.h.h.1151.7		24
12.11	even	2	inner	2400.2.h.h.1151.19		24
15.2	even	4		480.2.o.a.479.8	yes	24
15.8	even	4		480.2.o.a.479.17	yes	24
15.14	odd	2	inner	2400.2.h.h.1151.20		24
20.3	even	4		480.2.o.a.479.6	yes	24
20.7	even	4		480.2.o.a.479.19	yes	24
20.19	odd	2	inner	2400.2.h.h.1151.18		24
40.3	even	4		960.2.o.e.959.19		24
40.13	odd	4		960.2.o.e.959.5		24
40.27	even	4		960.2.o.e.959.6		24
40.37	odd	4		960.2.o.e.959.20		24
60.23	odd	4		480.2.o.a.479.7	yes	24
60.47	odd	4		480.2.o.a.479.18	yes	24
60.59	even	2	inner	2400.2.h.h.1151.5		24
120.53	even	4		960.2.o.e.959.8		24
120.77	even	4		960.2.o.e.959.17		24
120.83	odd	4		960.2.o.e.959.18		24
120.107	odd	4		960.2.o.e.959.7		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.o.a.479.5	✓	24	5.2	odd	4
480.2.o.a.479.6	yes	24	20.3	even	4
480.2.o.a.479.7	yes	24	60.23	odd	4
480.2.o.a.479.8	yes	24	15.2	even	4
480.2.o.a.479.17	yes	24	15.8	even	4
480.2.o.a.479.18	yes	24	60.47	odd	4
480.2.o.a.479.19	yes	24	20.7	even	4
480.2.o.a.479.20	yes	24	5.3	odd	4
960.2.o.e.959.5		24	40.13	odd	4
960.2.o.e.959.6		24	40.27	even	4
960.2.o.e.959.7		24	120.107	odd	4
960.2.o.e.959.8		24	120.53	even	4
960.2.o.e.959.17		24	120.77	even	4
960.2.o.e.959.18		24	120.83	odd	4
960.2.o.e.959.19		24	40.3	even	4
960.2.o.e.959.20		24	40.37	odd	4
2400.2.h.h.1151.5		24	60.59	even	2	inner
2400.2.h.h.1151.6		24	3.2	odd	2	inner
2400.2.h.h.1151.7		24	5.4	even	2	inner
2400.2.h.h.1151.8		24	4.3	odd	2	inner
2400.2.h.h.1151.17		24	1.1	even	1	trivial
2400.2.h.h.1151.18		24	20.19	odd	2	inner
2400.2.h.h.1151.19		24	12.11	even	2	inner
2400.2.h.h.1151.20		24	15.14	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.28241 - 1.16422i) q^{3} -4.22289i q^{7} +(0.289169 - 2.98603i) q^{9} +O(q^{10})\)	\(q+(1.28241 - 1.16422i) q^{3} -4.22289i q^{7} +(0.289169 - 2.98603i) q^{9} +2.42945 q^{11} +5.33021 q^{13} +3.57009i q^{17} +0.578337i q^{19} +(-4.91638 - 5.41549i) q^{21} +5.39325 q^{23} +(-3.10557 - 4.16598i) q^{27} +5.10860i q^{29} -7.83276i q^{31} +(3.11556 - 2.82843i) q^{33} +7.77246 q^{37} +(6.83553 - 6.20555i) q^{39} -0.613779i q^{41} +2.32845i q^{43} -3.38272 q^{47} -10.8328 q^{49} +(4.15639 + 4.57834i) q^{51} +1.26887i q^{53} +(0.673313 + 0.741667i) q^{57} -7.78774 q^{59} -6.67609 q^{61} +(-12.6097 - 1.22113i) q^{63} -0.113805i q^{67} +(6.91638 - 6.27895i) q^{69} +8.31277 q^{71} -13.1027 q^{73} -10.2593i q^{77} +3.42166i q^{79} +(-8.83276 - 1.72693i) q^{81} -5.68395 q^{83} +(5.94755 + 6.55133i) q^{87} -4.85891i q^{89} -22.5089i q^{91} +(-9.11908 - 10.0448i) q^{93} -9.31379 q^{97} +(0.702522 - 7.25443i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 8 q^{21} - 40 q^{49} + 32 q^{61} + 56 q^{69} + 8 q^{81}+O(q^{100}) \) 24 * q - 8 * q^21 - 40 * q^49 + 32 * q^61 + 56 * q^69 + 8 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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