LMFDB - Embedded newform 2450.2.c.w.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(99,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.c (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:	$19.5633484952$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 490)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2450.99
Dual form			2450.2.c.w.99.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +3.41421i q^{3} -1.00000 q^{4} +3.41421 q^{6} +1.00000i q^{8} -8.65685 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +3.41421i q^{3} -1.00000 q^{4} +3.41421 q^{6} +1.00000i q^{8} -8.65685 q^{9} -0.828427 q^{11} -3.41421i q^{12} -4.82843i q^{13} +1.00000 q^{16} -2.58579i q^{17} +8.65685i q^{18} +0.585786 q^{19} +0.828427i q^{22} +1.17157i q^{23} -3.41421 q^{24} -4.82843 q^{26} -19.3137i q^{27} +4.82843 q^{29} +2.82843 q^{31} -1.00000i q^{32} -2.82843i q^{33} -2.58579 q^{34} +8.65685 q^{36} -7.65685i q^{37} -0.585786i q^{38} +16.4853 q^{39} +3.07107 q^{41} +8.82843i q^{43} +0.828427 q^{44} +1.17157 q^{46} -5.17157i q^{47} +3.41421i q^{48} +8.82843 q^{51} +4.82843i q^{52} -6.48528i q^{53} -19.3137 q^{54} +2.00000i q^{57} -4.82843i q^{58} +8.58579 q^{59} -9.31371 q^{61} -2.82843i q^{62} -1.00000 q^{64} -2.82843 q^{66} +1.65685i q^{67} +2.58579i q^{68} -4.00000 q^{69} -4.48528 q^{71} -8.65685i q^{72} -9.41421i q^{73} -7.65685 q^{74} -0.585786 q^{76} -16.4853i q^{78} +6.82843 q^{79} +39.9706 q^{81} -3.07107i q^{82} -2.24264i q^{83} +8.82843 q^{86} +16.4853i q^{87} -0.828427i q^{88} +12.7279 q^{89} -1.17157i q^{92} +9.65685i q^{93} -5.17157 q^{94} +3.41421 q^{96} +7.75736i q^{97} +7.17157 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{6} - 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 8 * q^6 - 12 * q^9	$ 4 q - 4 q^{4} + 8 q^{6} - 12 q^{9} + 8 q^{11} + 4 q^{16} + 8 q^{19} - 8 q^{24} - 8 q^{26} + 8 q^{29} - 16 q^{34} + 12 q^{36} + 32 q^{39} - 16 q^{41} - 8 q^{44} + 16 q^{46} + 24 q^{51} - 32 q^{54} + 40 q^{59} + 8 q^{61} - 4 q^{64} - 16 q^{69} + 16 q^{71} - 8 q^{74} - 8 q^{76} + 16 q^{79} + 92 q^{81} + 24 q^{86} - 32 q^{94} + 8 q^{96} + 40 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^6 - 12 * q^9 + 8 * q^11 + 4 * q^16 + 8 * q^19 - 8 * q^24 - 8 * q^26 + 8 * q^29 - 16 * q^34 + 12 * q^36 + 32 * q^39 - 16 * q^41 - 8 * q^44 + 16 * q^46 + 24 * q^51 - 32 * q^54 + 40 * q^59 + 8 * q^61 - 4 * q^64 - 16 * q^69 + 16 * q^71 - 8 * q^74 - 8 * q^76 + 16 * q^79 + 92 * q^81 + 24 * q^86 - 32 * q^94 + 8 * q^96 + 40 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	3.41421i	1.97120i	0.169102	+	0.985599i	$0.445913\pi$
$3$	3.41421i	1.97120i	−0.169102	+	0.985599i	$0.554087\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	3.41421	1.39385
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	−8.65685	−2.88562
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	− 3.41421i	− 0.985599i
$13$	− 4.82843i	− 1.33916i	−0.742738	−	0.669582i	$-0.766473\pi$
$13$	− 4.82843i	− 1.33916i	0.742738	−	0.669582i	$-0.233527\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.58579i	− 0.627145i	−0.949564	−	0.313573i	$-0.898474\pi$
$17$	− 2.58579i	− 0.627145i	0.949564	−	0.313573i	$-0.101526\pi$
$18$	8.65685i	2.04044i
$19$	0.585786	0.134389	0.0671943	−	0.997740i	$-0.478595\pi$
$19$	0.585786	0.134389	0.0671943	+	0.997740i	$0.478595\pi$
$20$	0	0
$21$	0	0
$22$	0.828427i	0.176621i
$23$	1.17157i	0.244290i	0.992512	+	0.122145i	$0.0389773\pi$
$23$	1.17157i	0.244290i	−0.992512	+	0.122145i	$0.961023\pi$
$24$	−3.41421	−0.696923
$25$	0	0
$26$	−4.82843	−0.946932
$27$	− 19.3137i	− 3.71692i
$28$	0	0
$29$	4.82843	0.896616	0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843	0.896616	0.448308	+	0.893879i	$0.352027\pi$
$30$	0	0
$31$	2.82843	0.508001	0.254000	−	0.967204i	$-0.418254\pi$
$31$	2.82843	0.508001	0.254000	+	0.967204i	$0.418254\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 2.82843i	− 0.492366i
$34$	−2.58579	−0.443459
$35$	0	0
$36$	8.65685	1.44281
$37$	− 7.65685i	− 1.25878i	−0.777090	−	0.629390i	$-0.783305\pi$
$37$	− 7.65685i	− 1.25878i	0.777090	−	0.629390i	$-0.216695\pi$
$38$	− 0.585786i	− 0.0950271i
$39$	16.4853	2.63976
$40$	0	0
$41$	3.07107	0.479620	0.239810	−	0.970820i	$-0.422915\pi$
$41$	3.07107	0.479620	0.239810	+	0.970820i	$0.422915\pi$
$42$	0	0
$43$	8.82843i	1.34632i	0.739496	+	0.673161i	$0.235064\pi$
$43$	8.82843i	1.34632i	−0.739496	+	0.673161i	$0.764936\pi$
$44$	0.828427	0.124890
$45$	0	0
$46$	1.17157	0.172739
$47$	− 5.17157i	− 0.754351i	−0.926142	−	0.377176i	$-0.876895\pi$
$47$	− 5.17157i	− 0.754351i	0.926142	−	0.377176i	$-0.123105\pi$
$48$	3.41421i	0.492799i
$49$	0	0
$50$	0	0
$51$	8.82843	1.23623
$52$	4.82843i	0.669582i
$53$	− 6.48528i	− 0.890822i	−0.895326	−	0.445411i	$-0.853058\pi$
$53$	− 6.48528i	− 0.890822i	0.895326	−	0.445411i	$-0.146942\pi$
$54$	−19.3137	−2.62826
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	− 4.82843i	− 0.634004i
$59$	8.58579	1.11777	0.558887	−	0.829244i	$-0.311229\pi$
$59$	8.58579	1.11777	0.558887	+	0.829244i	$0.311229\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	− 2.82843i	− 0.359211i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−2.82843	−0.348155
$67$	1.65685i	0.202417i	0.994865	+	0.101208i	$0.0322709\pi$
$67$	1.65685i	0.202417i	−0.994865	+	0.101208i	$0.967729\pi$
$68$	2.58579i	0.313573i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−4.48528	−0.532305	−0.266152	−	0.963931i	$-0.585752\pi$
$71$	−4.48528	−0.532305	−0.266152	+	0.963931i	$0.585752\pi$
$72$	− 8.65685i	− 1.02022i
$73$	− 9.41421i	− 1.10185i	−0.834555	−	0.550925i	$-0.814275\pi$
$73$	− 9.41421i	− 1.10185i	0.834555	−	0.550925i	$-0.185725\pi$
$74$	−7.65685	−0.890091
$75$	0	0
$76$	−0.585786	−0.0671943
$77$	0	0
$78$	− 16.4853i	− 1.86659i
$79$	6.82843	0.768258	0.384129	−	0.923279i	$-0.374502\pi$
$79$	6.82843	0.768258	0.384129	+	0.923279i	$0.374502\pi$
$80$	0	0
$81$	39.9706	4.44117
$82$	− 3.07107i	− 0.339143i
$83$	− 2.24264i	− 0.246162i	−0.992397	−	0.123081i	$-0.960723\pi$
$83$	− 2.24264i	− 0.246162i	0.992397	−	0.123081i	$-0.0392775\pi$
$84$	0	0
$85$	0	0
$86$	8.82843	0.951994
$87$	16.4853i	1.76741i
$88$	− 0.828427i	− 0.0883106i
$89$	12.7279	1.34916	0.674579	−	0.738203i	$-0.264325\pi$
$89$	12.7279	1.34916	0.674579	+	0.738203i	$0.264325\pi$
$90$	0	0
$91$	0	0
$92$	− 1.17157i	− 0.122145i
$93$	9.65685i	1.00137i
$94$	−5.17157	−0.533407
$95$	0	0
$96$	3.41421	0.348462
$97$	7.75736i	0.787641i	0.919187	+	0.393820i	$0.128847\pi$
$97$	7.75736i	0.787641i	−0.919187	+	0.393820i	$0.871153\pi$
$98$	0	0
$99$	7.17157	0.720770
$100$	0	0
$101$	−13.3137	−1.32476	−0.662382	−	0.749166i	$-0.730454\pi$
$101$	−13.3137	−1.32476	−0.662382	+	0.749166i	$0.730454\pi$
$102$	− 8.82843i	− 0.874145i
$103$	− 14.8284i	− 1.46109i	−0.682865	−	0.730544i	$-0.739266\pi$
$103$	− 14.8284i	− 1.46109i	0.682865	−	0.730544i	$-0.260734\pi$
$104$	4.82843	0.473466
$105$	0	0
$106$	−6.48528	−0.629906
$107$	9.65685i	0.933563i	0.884373	+	0.466782i	$0.154587\pi$
$107$	9.65685i	0.933563i	−0.884373	+	0.466782i	$0.845413\pi$
$108$	19.3137i	1.85846i
$109$	−2.48528	−0.238047	−0.119023	−	0.992891i	$-0.537976\pi$
$109$	−2.48528	−0.238047	−0.119023	+	0.992891i	$0.537976\pi$
$110$	0	0
$111$	26.1421	2.48130
$112$	0	0
$113$	15.3137i	1.44059i	0.693667	+	0.720296i	$0.255994\pi$
$113$	15.3137i	1.44059i	−0.693667	+	0.720296i	$0.744006\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	−4.82843	−0.448308
$117$	41.7990i	3.86432i
$118$	− 8.58579i	− 0.790386i
$119$	0	0
$120$	0	0
$121$	−10.3137	−0.937610
$122$	9.31371i	0.843224i
$123$	10.4853i	0.945426i
$124$	−2.82843	−0.254000
$125$	0	0
$126$	0	0
$127$	2.82843i	0.250982i	0.992095	+	0.125491i	$0.0400507\pi$
$127$	2.82843i	0.250982i	−0.992095	+	0.125491i	$0.959949\pi$
$128$	1.00000i	0.0883883i
$129$	−30.1421	−2.65387
$130$	0	0
$131$	6.24264	0.545422	0.272711	−	0.962096i	$-0.412080\pi$
$131$	6.24264	0.545422	0.272711	+	0.962096i	$0.412080\pi$
$132$	2.82843i	0.246183i
$133$	0	0
$134$	1.65685	0.143130
$135$	0	0
$136$	2.58579	0.221729
$137$	− 16.0000i	− 1.36697i	−0.729964	−	0.683486i	$-0.760463\pi$
$137$	− 16.0000i	− 1.36697i	0.729964	−	0.683486i	$-0.239537\pi$
$138$	4.00000i	0.340503i
$139$	19.8995	1.68785	0.843927	−	0.536459i	$-0.180238\pi$
$139$	19.8995	1.68785	0.843927	+	0.536459i	$0.180238\pi$
$140$	0	0
$141$	17.6569	1.48698
$142$	4.48528i	0.376396i
$143$	4.00000i	0.334497i
$144$	−8.65685	−0.721405
$145$	0	0
$146$	−9.41421	−0.779126
$147$	0	0
$148$	7.65685i	0.629390i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−11.3137	−0.920697	−0.460348	−	0.887738i	$-0.652275\pi$
$151$	−11.3137	−0.920697	−0.460348	+	0.887738i	$0.652275\pi$
$152$	0.585786i	0.0475136i
$153$	22.3848i	1.80970i
$154$	0	0
$155$	0	0
$156$	−16.4853	−1.31988
$157$	− 6.48528i	− 0.517582i	−0.965933	−	0.258791i	$-0.916676\pi$
$157$	− 6.48528i	− 0.517582i	0.965933	−	0.258791i	$-0.0833241\pi$
$158$	− 6.82843i	− 0.543240i
$159$	22.1421	1.75599
$160$	0	0
$161$	0	0
$162$	− 39.9706i	− 3.14038i
$163$	20.1421i	1.57765i	0.614615	+	0.788827i	$0.289311\pi$
$163$	20.1421i	1.57765i	−0.614615	+	0.788827i	$0.710689\pi$
$164$	−3.07107	−0.239810
$165$	0	0
$166$	−2.24264	−0.174063
$167$	− 15.7990i	− 1.22256i	−0.791413	−	0.611281i	$-0.790654\pi$
$167$	− 15.7990i	− 1.22256i	0.791413	−	0.611281i	$-0.209346\pi$
$168$	0	0
$169$	−10.3137	−0.793362
$170$	0	0
$171$	−5.07107	−0.387794
$172$	− 8.82843i	− 0.673161i
$173$	− 8.82843i	− 0.671213i	−0.942002	−	0.335606i	$-0.891059\pi$
$173$	− 8.82843i	− 0.671213i	0.942002	−	0.335606i	$-0.108941\pi$
$174$	16.4853	1.24975
$175$	0	0
$176$	−0.828427	−0.0624450
$177$	29.3137i	2.20335i
$178$	− 12.7279i	− 0.953998i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	2.48528	0.184730	0.0923648	−	0.995725i	$-0.470557\pi$
$181$	2.48528	0.184730	0.0923648	+	0.995725i	$0.470557\pi$
$182$	0	0
$183$	− 31.7990i	− 2.35065i
$184$	−1.17157	−0.0863695
$185$	0	0
$186$	9.65685	0.708075
$187$	2.14214i	0.156648i
$188$	5.17157i	0.377176i
$189$	0	0
$190$	0	0
$191$	10.1421	0.733859	0.366930	−	0.930249i	$-0.380409\pi$
$191$	10.1421	0.733859	0.366930	+	0.930249i	$0.380409\pi$
$192$	− 3.41421i	− 0.246400i
$193$	− 5.65685i	− 0.407189i	−0.979055	−	0.203595i	$-0.934738\pi$
$193$	− 5.65685i	− 0.407189i	0.979055	−	0.203595i	$-0.0652625\pi$
$194$	7.75736	0.556946
$195$	0	0
$196$	0	0
$197$	− 25.7990i	− 1.83810i	−0.394139	−	0.919051i	$-0.628957\pi$
$197$	− 25.7990i	− 1.83810i	0.394139	−	0.919051i	$-0.371043\pi$
$198$	− 7.17157i	− 0.509661i
$199$	16.4853	1.16861	0.584305	−	0.811534i	$-0.301367\pi$
$199$	16.4853	1.16861	0.584305	+	0.811534i	$0.301367\pi$
$200$	0	0
$201$	−5.65685	−0.399004
$202$	13.3137i	0.936749i
$203$	0	0
$204$	−8.82843	−0.618114
$205$	0	0
$206$	−14.8284	−1.03315
$207$	− 10.1421i	− 0.704927i
$208$	− 4.82843i	− 0.334791i
$209$	−0.485281	−0.0335676
$210$	0	0
$211$	18.6274	1.28236	0.641182	−	0.767389i	$-0.278444\pi$
$211$	18.6274	1.28236	0.641182	+	0.767389i	$0.278444\pi$
$212$	6.48528i	0.445411i
$213$	− 15.3137i	− 1.04928i
$214$	9.65685	0.660129
$215$	0	0
$216$	19.3137	1.31413
$217$	0	0
$218$	2.48528i	0.168324i
$219$	32.1421	2.17196
$220$	0	0
$221$	−12.4853	−0.839851
$222$	− 26.1421i	− 1.75455i
$223$	− 7.31371i	− 0.489762i	−0.969553	−	0.244881i	$-0.921251\pi$
$223$	− 7.31371i	− 0.489762i	0.969553	−	0.244881i	$-0.0787489\pi$
$224$	0	0
$225$	0	0
$226$	15.3137	1.01865
$227$	− 18.2426i	− 1.21081i	−0.795919	−	0.605403i	$-0.793012\pi$
$227$	− 18.2426i	− 1.21081i	0.795919	−	0.605403i	$-0.206988\pi$
$228$	− 2.00000i	− 0.132453i
$229$	−16.1421	−1.06670	−0.533351	−	0.845894i	$-0.679068\pi$
$229$	−16.1421	−1.06670	−0.533351	+	0.845894i	$0.679068\pi$
$230$	0	0
$231$	0	0
$232$	4.82843i	0.317002i
$233$	− 23.3137i	− 1.52733i	−0.645612	−	0.763666i	$-0.723397\pi$
$233$	− 23.3137i	− 1.52733i	0.645612	−	0.763666i	$-0.276603\pi$
$234$	41.7990	2.73249
$235$	0	0
$236$	−8.58579	−0.558887
$237$	23.3137i	1.51439i
$238$	0	0
$239$	−1.65685	−0.107173	−0.0535865	−	0.998563i	$-0.517065\pi$
$239$	−1.65685	−0.107173	−0.0535865	+	0.998563i	$0.517065\pi$
$240$	0	0
$241$	13.4142	0.864085	0.432043	−	0.901853i	$-0.357793\pi$
$241$	13.4142	0.864085	0.432043	+	0.901853i	$0.357793\pi$
$242$	10.3137i	0.662990i
$243$	78.5269i	5.03750i
$244$	9.31371	0.596249
$245$	0	0
$246$	10.4853	0.668517
$247$	− 2.82843i	− 0.179969i
$248$	2.82843i	0.179605i
$249$	7.65685	0.485233
$250$	0	0
$251$	−0.585786	−0.0369745	−0.0184873	−	0.999829i	$-0.505885\pi$
$251$	−0.585786	−0.0369745	−0.0184873	+	0.999829i	$0.505885\pi$
$252$	0	0
$253$	− 0.970563i	− 0.0610188i
$254$	2.82843	0.177471
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.89949i	0.617514i	0.951141	+	0.308757i	$0.0999129\pi$
$257$	9.89949i	0.617514i	−0.951141	+	0.308757i	$0.900087\pi$
$258$	30.1421i	1.87657i
$259$	0	0
$260$	0	0
$261$	−41.7990	−2.58729
$262$	− 6.24264i	− 0.385672i
$263$	− 28.0000i	− 1.72655i	−0.504730	−	0.863277i	$-0.668408\pi$
$263$	− 28.0000i	− 1.72655i	0.504730	−	0.863277i	$-0.331592\pi$
$264$	2.82843	0.174078
$265$	0	0
$266$	0	0
$267$	43.4558i	2.65945i
$268$	− 1.65685i	− 0.101208i
$269$	18.4853	1.12707	0.563534	−	0.826093i	$-0.309441\pi$
$269$	18.4853	1.12707	0.563534	+	0.826093i	$0.309441\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	− 2.58579i	− 0.156786i
$273$	0	0
$274$	−16.0000	−0.966595
$275$	0	0
$276$	4.00000	0.240772
$277$	− 8.14214i	− 0.489214i	−0.969622	−	0.244607i	$-0.921341\pi$
$277$	− 8.14214i	− 0.489214i	0.969622	−	0.244607i	$-0.0786589\pi$
$278$	− 19.8995i	− 1.19349i
$279$	−24.4853	−1.46590
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	− 17.6569i	− 1.05145i
$283$	− 2.24264i	− 0.133311i	−0.997776	−	0.0666556i	$-0.978767\pi$
$283$	− 2.24264i	− 0.133311i	0.997776	−	0.0666556i	$-0.0212329\pi$
$284$	4.48528	0.266152
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	8.65685i	0.510110i
$289$	10.3137	0.606689
$290$	0	0
$291$	−26.4853	−1.55259
$292$	9.41421i	0.550925i
$293$	− 8.34315i	− 0.487412i	−0.969849	−	0.243706i	$-0.921637\pi$
$293$	− 8.34315i	− 0.487412i	0.969849	−	0.243706i	$-0.0783632\pi$
$294$	0	0
$295$	0	0
$296$	7.65685	0.445046
$297$	16.0000i	0.928414i
$298$	− 6.00000i	− 0.347571i
$299$	5.65685	0.327144
$300$	0	0
$301$	0	0
$302$	11.3137i	0.651031i
$303$	− 45.4558i	− 2.61137i
$304$	0.585786	0.0335972
$305$	0	0
$306$	22.3848	1.27965
$307$	− 14.9289i	− 0.852039i	−0.904714	−	0.426020i	$-0.859915\pi$
$307$	− 14.9289i	− 0.852039i	0.904714	−	0.426020i	$-0.140085\pi$
$308$	0	0
$309$	50.6274	2.88009
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	16.4853i	0.933295i
$313$	14.3848i	0.813076i	0.913634	+	0.406538i	$0.133264\pi$
$313$	14.3848i	0.813076i	−0.913634	+	0.406538i	$0.866736\pi$
$314$	−6.48528	−0.365986
$315$	0	0
$316$	−6.82843	−0.384129
$317$	10.4853i	0.588912i	0.955665	+	0.294456i	$0.0951385\pi$
$317$	10.4853i	0.588912i	−0.955665	+	0.294456i	$0.904862\pi$
$318$	− 22.1421i	− 1.24167i
$319$	−4.00000	−0.223957
$320$	0	0
$321$	−32.9706	−1.84024
$322$	0	0
$323$	− 1.51472i	− 0.0842812i
$324$	−39.9706	−2.22059
$325$	0	0
$326$	20.1421	1.11557
$327$	− 8.48528i	− 0.469237i
$328$	3.07107i	0.169571i
$329$	0	0
$330$	0	0
$331$	33.7990	1.85776	0.928880	−	0.370380i	$-0.120773\pi$
$331$	33.7990	1.85776	0.928880	+	0.370380i	$0.120773\pi$
$332$	2.24264i	0.123081i
$333$	66.2843i	3.63236i
$334$	−15.7990	−0.864482
$335$	0	0
$336$	0	0
$337$	6.00000i	0.326841i	0.986557	+	0.163420i	$0.0522527\pi$
$337$	6.00000i	0.326841i	−0.986557	+	0.163420i	$0.947747\pi$
$338$	10.3137i	0.560992i
$339$	−52.2843	−2.83969
$340$	0	0
$341$	−2.34315	−0.126888
$342$	5.07107i	0.274212i
$343$	0	0
$344$	−8.82843	−0.475997
$345$	0	0
$346$	−8.82843	−0.474619
$347$	− 3.17157i	− 0.170259i	−0.996370	−	0.0851295i	$-0.972870\pi$
$347$	− 3.17157i	− 0.170259i	0.996370	−	0.0851295i	$-0.0271304\pi$
$348$	− 16.4853i	− 0.883704i
$349$	2.48528	0.133034	0.0665170	−	0.997785i	$-0.478811\pi$
$349$	2.48528	0.133034	0.0665170	+	0.997785i	$0.478811\pi$
$350$	0	0
$351$	−93.2548	−4.97757
$352$	0.828427i	0.0441553i
$353$	− 2.38478i	− 0.126929i	−0.997984	−	0.0634644i	$-0.979785\pi$
$353$	− 2.38478i	− 0.126929i	0.997984	−	0.0634644i	$-0.0202149\pi$
$354$	29.3137	1.55801
$355$	0	0
$356$	−12.7279	−0.674579
$357$	0	0
$358$	4.00000i	0.211407i
$359$	−28.2843	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$359$	−28.2843	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$360$	0	0
$361$	−18.6569	−0.981940
$362$	− 2.48528i	− 0.130623i
$363$	− 35.2132i	− 1.84821i
$364$	0	0
$365$	0	0
$366$	−31.7990	−1.66216
$367$	− 24.9706i	− 1.30345i	−0.758454	−	0.651726i	$-0.774045\pi$
$367$	− 24.9706i	− 1.30345i	0.758454	−	0.651726i	$-0.225955\pi$
$368$	1.17157i	0.0610725i
$369$	−26.5858	−1.38400
$370$	0	0
$371$	0	0
$372$	− 9.65685i	− 0.500685i
$373$	30.4853i	1.57847i	0.614093	+	0.789234i	$0.289522\pi$
$373$	30.4853i	1.57847i	−0.614093	+	0.789234i	$0.710478\pi$
$374$	2.14214	0.110767
$375$	0	0
$376$	5.17157	0.266704
$377$	− 23.3137i	− 1.20072i
$378$	0	0
$379$	−34.4853	−1.77139	−0.885695	−	0.464268i	$-0.846318\pi$
$379$	−34.4853	−1.77139	−0.885695	+	0.464268i	$0.846318\pi$
$380$	0	0
$381$	−9.65685	−0.494736
$382$	− 10.1421i	− 0.518917i
$383$	− 32.4853i	− 1.65992i	−0.557823	−	0.829960i	$-0.688363\pi$
$383$	− 32.4853i	− 1.65992i	0.557823	−	0.829960i	$-0.311637\pi$
$384$	−3.41421	−0.174231
$385$	0	0
$386$	−5.65685	−0.287926
$387$	− 76.4264i	− 3.88497i
$388$	− 7.75736i	− 0.393820i
$389$	−28.1421	−1.42686	−0.713431	−	0.700725i	$-0.752860\pi$
$389$	−28.1421	−1.42686	−0.713431	+	0.700725i	$0.752860\pi$
$390$	0	0
$391$	3.02944	0.153205
$392$	0	0
$393$	21.3137i	1.07513i
$394$	−25.7990	−1.29973
$395$	0	0
$396$	−7.17157	−0.360385
$397$	− 33.7990i	− 1.69632i	−0.529738	−	0.848161i	$-0.677710\pi$
$397$	− 33.7990i	− 1.69632i	0.529738	−	0.848161i	$-0.322290\pi$
$398$	− 16.4853i	− 0.826332i
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	5.65685i	0.282138i
$403$	− 13.6569i	− 0.680296i
$404$	13.3137	0.662382
$405$	0	0
$406$	0	0
$407$	6.34315i	0.314418i
$408$	8.82843i	0.437072i
$409$	−10.5858	−0.523433	−0.261717	−	0.965145i	$-0.584289\pi$
$409$	−10.5858	−0.523433	−0.261717	+	0.965145i	$0.584289\pi$
$410$	0	0
$411$	54.6274	2.69457
$412$	14.8284i	0.730544i
$413$	0	0
$414$	−10.1421	−0.498459
$415$	0	0
$416$	−4.82843	−0.236733
$417$	67.9411i	3.32709i
$418$	0.485281i	0.0237359i
$419$	−20.8701	−1.01957	−0.509785	−	0.860302i	$-0.670275\pi$
$419$	−20.8701	−1.01957	−0.509785	+	0.860302i	$0.670275\pi$
$420$	0	0
$421$	17.3137	0.843819	0.421909	−	0.906638i	$-0.361360\pi$
$421$	17.3137	0.843819	0.421909	+	0.906638i	$0.361360\pi$
$422$	− 18.6274i	− 0.906768i
$423$	44.7696i	2.17677i
$424$	6.48528	0.314953
$425$	0	0
$426$	−15.3137	−0.741952
$427$	0	0
$428$	− 9.65685i	− 0.466782i
$429$	−13.6569	−0.659359
$430$	0	0
$431$	−22.3431	−1.07623	−0.538116	−	0.842871i	$-0.680864\pi$
$431$	−22.3431	−1.07623	−0.538116	+	0.842871i	$0.680864\pi$
$432$	− 19.3137i	− 0.929231i
$433$	10.5858i	0.508720i	0.967110	+	0.254360i	$0.0818649\pi$
$433$	10.5858i	0.508720i	−0.967110	+	0.254360i	$0.918135\pi$
$434$	0	0
$435$	0	0
$436$	2.48528	0.119023
$437$	0.686292i	0.0328298i
$438$	− 32.1421i	− 1.53581i
$439$	−24.9706	−1.19178	−0.595890	−	0.803066i	$-0.703201\pi$
$439$	−24.9706	−1.19178	−0.595890	+	0.803066i	$0.703201\pi$
$440$	0	0
$441$	0	0
$442$	12.4853i	0.593864i
$443$	− 3.02944i	− 0.143933i	−0.997407	−	0.0719665i	$-0.977073\pi$
$443$	− 3.02944i	− 0.143933i	0.997407	−	0.0719665i	$-0.0229275\pi$
$444$	−26.1421	−1.24065
$445$	0	0
$446$	−7.31371	−0.346314
$447$	20.4853i	0.968921i
$448$	0	0
$449$	16.6274	0.784696	0.392348	−	0.919817i	$-0.371663\pi$
$449$	16.6274	0.784696	0.392348	+	0.919817i	$0.371663\pi$
$450$	0	0
$451$	−2.54416	−0.119800
$452$	− 15.3137i	− 0.720296i
$453$	− 38.6274i	− 1.81487i
$454$	−18.2426	−0.856170
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	21.6569i	1.01306i	0.862221	+	0.506532i	$0.169073\pi$
$457$	21.6569i	1.01306i	−0.862221	+	0.506532i	$0.830927\pi$
$458$	16.1421i	0.754272i
$459$	−49.9411	−2.33105
$460$	0	0
$461$	12.8284	0.597479	0.298740	−	0.954335i	$-0.403434\pi$
$461$	12.8284	0.597479	0.298740	+	0.954335i	$0.403434\pi$
$462$	0	0
$463$	16.9706i	0.788689i	0.918963	+	0.394344i	$0.129028\pi$
$463$	16.9706i	0.788689i	−0.918963	+	0.394344i	$0.870972\pi$
$464$	4.82843	0.224154
$465$	0	0
$466$	−23.3137	−1.07999
$467$	15.8995i	0.735741i	0.929877	+	0.367870i	$0.119913\pi$
$467$	15.8995i	0.735741i	−0.929877	+	0.367870i	$0.880087\pi$
$468$	− 41.7990i	− 1.93216i
$469$	0	0
$470$	0	0
$471$	22.1421	1.02026
$472$	8.58579i	0.395193i
$473$	− 7.31371i	− 0.336285i
$474$	23.3137	1.07083
$475$	0	0
$476$	0	0
$477$	56.1421i	2.57057i
$478$	1.65685i	0.0757827i
$479$	−17.1716	−0.784589	−0.392295	−	0.919840i	$-0.628319\pi$
$479$	−17.1716	−0.784589	−0.392295	+	0.919840i	$0.628319\pi$
$480$	0	0
$481$	−36.9706	−1.68571
$482$	− 13.4142i	− 0.611001i
$483$	0	0
$484$	10.3137	0.468805
$485$	0	0
$486$	78.5269	3.56205
$487$	31.7990i	1.44095i	0.693481	+	0.720475i	$0.256076\pi$
$487$	31.7990i	1.44095i	−0.693481	+	0.720475i	$0.743924\pi$
$488$	− 9.31371i	− 0.421612i
$489$	−68.7696	−3.10987
$490$	0	0
$491$	32.2843	1.45697	0.728484	−	0.685062i	$-0.240225\pi$
$491$	32.2843	1.45697	0.728484	+	0.685062i	$0.240225\pi$
$492$	− 10.4853i	− 0.472713i
$493$	− 12.4853i	− 0.562309i
$494$	−2.82843	−0.127257
$495$	0	0
$496$	2.82843	0.127000
$497$	0	0
$498$	− 7.65685i	− 0.343112i
$499$	−30.3431	−1.35835	−0.679173	−	0.733978i	$-0.737661\pi$
$499$	−30.3431	−1.35835	−0.679173	+	0.733978i	$0.737661\pi$
$500$	0	0
$501$	53.9411	2.40991
$502$	0.585786i	0.0261449i
$503$	17.6569i	0.787280i	0.919265	+	0.393640i	$0.128784\pi$
$503$	17.6569i	0.787280i	−0.919265	+	0.393640i	$0.871216\pi$
$504$	0	0
$505$	0	0
$506$	−0.970563	−0.0431468
$507$	− 35.2132i	− 1.56387i
$508$	− 2.82843i	− 0.125491i
$509$	5.79899	0.257036	0.128518	−	0.991707i	$-0.458978\pi$
$509$	5.79899	0.257036	0.128518	+	0.991707i	$0.458978\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	− 11.3137i	− 0.499512i
$514$	9.89949	0.436648
$515$	0	0
$516$	30.1421	1.32693
$517$	4.28427i	0.188422i
$518$	0	0
$519$	30.1421	1.32309
$520$	0	0
$521$	−19.0711	−0.835519	−0.417759	−	0.908558i	$-0.637184\pi$
$521$	−19.0711	−0.835519	−0.417759	+	0.908558i	$0.637184\pi$
$522$	41.7990i	1.82949i
$523$	23.8995i	1.04505i	0.852623	+	0.522526i	$0.175010\pi$
$523$	23.8995i	1.04505i	−0.852623	+	0.522526i	$0.824990\pi$
$524$	−6.24264	−0.272711
$525$	0	0
$526$	−28.0000	−1.22086
$527$	− 7.31371i	− 0.318590i
$528$	− 2.82843i	− 0.123091i
$529$	21.6274	0.940322
$530$	0	0
$531$	−74.3259	−3.22547
$532$	0	0
$533$	− 14.8284i	− 0.642290i
$534$	43.4558	1.88052
$535$	0	0
$536$	−1.65685	−0.0715652
$537$	− 13.6569i	− 0.589337i
$538$	− 18.4853i	− 0.796957i
$539$	0	0
$540$	0	0
$541$	−14.9706	−0.643635	−0.321817	−	0.946802i	$-0.604294\pi$
$541$	−14.9706	−0.643635	−0.321817	+	0.946802i	$0.604294\pi$
$542$	12.0000i	0.515444i
$543$	8.48528i	0.364138i
$544$	−2.58579	−0.110865
$545$	0	0
$546$	0	0
$547$	− 10.4853i	− 0.448318i	−0.974553	−	0.224159i	$-0.928036\pi$
$547$	− 10.4853i	− 0.448318i	0.974553	−	0.224159i	$-0.0719635\pi$
$548$	16.0000i	0.683486i
$549$	80.6274	3.44109
$550$	0	0
$551$	2.82843	0.120495
$552$	− 4.00000i	− 0.170251i
$553$	0	0
$554$	−8.14214	−0.345926
$555$	0	0
$556$	−19.8995	−0.843927
$557$	15.1716i	0.642840i	0.946937	+	0.321420i	$0.104160\pi$
$557$	15.1716i	0.642840i	−0.946937	+	0.321420i	$0.895840\pi$
$558$	24.4853i	1.03654i
$559$	42.6274	1.80295
$560$	0	0
$561$	−7.31371	−0.308785
$562$	− 8.00000i	− 0.337460i
$563$	− 36.5858i	− 1.54191i	−0.636891	−	0.770954i	$-0.719780\pi$
$563$	− 36.5858i	− 1.54191i	0.636891	−	0.770954i	$-0.280220\pi$
$564$	−17.6569	−0.743488
$565$	0	0
$566$	−2.24264	−0.0942652
$567$	0	0
$568$	− 4.48528i	− 0.188198i
$569$	29.3137	1.22889	0.614447	−	0.788958i	$-0.289379\pi$
$569$	29.3137	1.22889	0.614447	+	0.788958i	$0.289379\pi$
$570$	0	0
$571$	−2.20101	−0.0921094	−0.0460547	−	0.998939i	$-0.514665\pi$
$571$	−2.20101	−0.0921094	−0.0460547	+	0.998939i	$0.514665\pi$
$572$	− 4.00000i	− 0.167248i
$573$	34.6274i	1.44658i
$574$	0	0
$575$	0	0
$576$	8.65685	0.360702
$577$	− 6.10051i	− 0.253967i	−0.991905	−	0.126984i	$-0.959470\pi$
$577$	− 6.10051i	− 0.253967i	0.991905	−	0.126984i	$-0.0405296\pi$
$578$	− 10.3137i	− 0.428994i
$579$	19.3137	0.802650
$580$	0	0
$581$	0	0
$582$	26.4853i	1.09785i
$583$	5.37258i	0.222510i
$584$	9.41421	0.389563
$585$	0	0
$586$	−8.34315	−0.344652
$587$	− 17.0711i	− 0.704598i	−0.935887	−	0.352299i	$-0.885400\pi$
$587$	− 17.0711i	− 0.704598i	0.935887	−	0.352299i	$-0.114600\pi$
$588$	0	0
$589$	1.65685	0.0682695
$590$	0	0
$591$	88.0833	3.62326
$592$	− 7.65685i	− 0.314695i
$593$	3.27208i	0.134368i	0.997741	+	0.0671841i	$0.0214015\pi$
$593$	3.27208i	0.134368i	−0.997741	+	0.0671841i	$0.978599\pi$
$594$	16.0000	0.656488
$595$	0	0
$596$	−6.00000	−0.245770
$597$	56.2843i	2.30356i
$598$	− 5.65685i	− 0.231326i
$599$	10.8284	0.442438	0.221219	−	0.975224i	$-0.428997\pi$
$599$	10.8284	0.442438	0.221219	+	0.975224i	$0.428997\pi$
$600$	0	0
$601$	−6.58579	−0.268640	−0.134320	−	0.990938i	$-0.542885\pi$
$601$	−6.58579	−0.268640	−0.134320	+	0.990938i	$0.542885\pi$
$602$	0	0
$603$	− 14.3431i	− 0.584098i
$604$	11.3137	0.460348
$605$	0	0
$606$	−45.4558	−1.84652
$607$	− 16.2843i	− 0.660958i	−0.943813	−	0.330479i	$-0.892790\pi$
$607$	− 16.2843i	− 0.660958i	0.943813	−	0.330479i	$-0.107210\pi$
$608$	− 0.585786i	− 0.0237568i
$609$	0	0
$610$	0	0
$611$	−24.9706	−1.01020
$612$	− 22.3848i	− 0.904851i
$613$	12.3431i	0.498535i	0.968435	+	0.249267i	$0.0801898\pi$
$613$	12.3431i	0.498535i	−0.968435	+	0.249267i	$0.919810\pi$
$614$	−14.9289	−0.602483
$615$	0	0
$616$	0	0
$617$	− 33.3137i	− 1.34116i	−0.741837	−	0.670580i	$-0.766045\pi$
$617$	− 33.3137i	− 1.34116i	0.741837	−	0.670580i	$-0.233955\pi$
$618$	− 50.6274i	− 2.03653i
$619$	29.0711	1.16846	0.584232	−	0.811586i	$-0.301396\pi$
$619$	29.0711	1.16846	0.584232	+	0.811586i	$0.301396\pi$
$620$	0	0
$621$	22.6274	0.908007
$622$	4.00000i	0.160385i
$623$	0	0
$624$	16.4853	0.659939
$625$	0	0
$626$	14.3848	0.574931
$627$	− 1.65685i	− 0.0661684i
$628$	6.48528i	0.258791i
$629$	−19.7990	−0.789437
$630$	0	0
$631$	−12.4853	−0.497031	−0.248516	−	0.968628i	$-0.579943\pi$
$631$	−12.4853	−0.497031	−0.248516	+	0.968628i	$0.579943\pi$
$632$	6.82843i	0.271620i
$633$	63.5980i	2.52779i
$634$	10.4853	0.416424
$635$	0	0
$636$	−22.1421	−0.877993
$637$	0	0
$638$	4.00000i	0.158362i
$639$	38.8284	1.53603
$640$	0	0
$641$	−24.6274	−0.972724	−0.486362	−	0.873757i	$-0.661676\pi$
$641$	−24.6274	−0.972724	−0.486362	+	0.873757i	$0.661676\pi$
$642$	32.9706i	1.30124i
$643$	4.78680i	0.188773i	0.995536	+	0.0943864i	$0.0300889\pi$
$643$	4.78680i	0.188773i	−0.995536	+	0.0943864i	$0.969911\pi$
$644$	0	0
$645$	0	0
$646$	−1.51472	−0.0595958
$647$	23.1127i	0.908654i	0.890835	+	0.454327i	$0.150120\pi$
$647$	23.1127i	0.908654i	−0.890835	+	0.454327i	$0.849880\pi$
$648$	39.9706i	1.57019i
$649$	−7.11270	−0.279198
$650$	0	0
$651$	0	0
$652$	− 20.1421i	− 0.788827i
$653$	− 4.34315i	− 0.169960i	−0.996383	−	0.0849802i	$-0.972917\pi$
$653$	− 4.34315i	− 0.169960i	0.996383	−	0.0849802i	$-0.0270827\pi$
$654$	−8.48528	−0.331801
$655$	0	0
$656$	3.07107	0.119905
$657$	81.4975i	3.17952i
$658$	0	0
$659$	27.1716	1.05845	0.529227	−	0.848480i	$-0.322482\pi$
$659$	27.1716	1.05845	0.529227	+	0.848480i	$0.322482\pi$
$660$	0	0
$661$	38.2843	1.48909	0.744543	−	0.667575i	$-0.232668\pi$
$661$	38.2843	1.48909	0.744543	+	0.667575i	$0.232668\pi$
$662$	− 33.7990i	− 1.31364i
$663$	− 42.6274i	− 1.65551i
$664$	2.24264	0.0870313
$665$	0	0
$666$	66.2843	2.56846
$667$	5.65685i	0.219034i
$668$	15.7990i	0.611281i
$669$	24.9706	0.965418
$670$	0	0
$671$	7.71573	0.297862
$672$	0	0
$673$	48.0000i	1.85026i	0.379646	+	0.925132i	$0.376046\pi$
$673$	48.0000i	1.85026i	−0.379646	+	0.925132i	$0.623954\pi$
$674$	6.00000	0.231111
$675$	0	0
$676$	10.3137	0.396681
$677$	− 39.4558i	− 1.51641i	−0.652015	−	0.758206i	$-0.726076\pi$
$677$	− 39.4558i	− 1.51641i	0.652015	−	0.758206i	$-0.273924\pi$
$678$	52.2843i	2.00797i
$679$	0	0
$680$	0	0
$681$	62.2843	2.38674
$682$	2.34315i	0.0897237i
$683$	− 33.6569i	− 1.28784i	−0.765091	−	0.643922i	$-0.777306\pi$
$683$	− 33.6569i	− 1.28784i	0.765091	−	0.643922i	$-0.222694\pi$
$684$	5.07107	0.193897
$685$	0	0
$686$	0	0
$687$	− 55.1127i	− 2.10268i
$688$	8.82843i	0.336581i
$689$	−31.3137	−1.19296
$690$	0	0
$691$	−1.75736	−0.0668531	−0.0334265	−	0.999441i	$-0.510642\pi$
$691$	−1.75736	−0.0668531	−0.0334265	+	0.999441i	$0.510642\pi$
$692$	8.82843i	0.335606i
$693$	0	0
$694$	−3.17157	−0.120391
$695$	0	0
$696$	−16.4853	−0.624873
$697$	− 7.94113i	− 0.300792i
$698$	− 2.48528i	− 0.0940693i
$699$	79.5980	3.01067
$700$	0	0
$701$	2.48528	0.0938678	0.0469339	−	0.998898i	$-0.485055\pi$
$701$	2.48528	0.0938678	0.0469339	+	0.998898i	$0.485055\pi$
$702$	93.2548i	3.51968i
$703$	− 4.48528i	− 0.169166i
$704$	0.828427	0.0312225
$705$	0	0
$706$	−2.38478	−0.0897522
$707$	0	0
$708$	− 29.3137i	− 1.10168i
$709$	−45.1127	−1.69424	−0.847121	−	0.531399i	$-0.821666\pi$
$709$	−45.1127	−1.69424	−0.847121	+	0.531399i	$0.821666\pi$
$710$	0	0
$711$	−59.1127	−2.21690
$712$	12.7279i	0.476999i
$713$	3.31371i	0.124099i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	− 5.65685i	− 0.211259i
$718$	28.2843i	1.05556i
$719$	41.4558	1.54604	0.773021	−	0.634380i	$-0.218745\pi$
$719$	41.4558	1.54604	0.773021	+	0.634380i	$0.218745\pi$
$720$	0	0
$721$	0	0
$722$	18.6569i	0.694336i
$723$	45.7990i	1.70328i
$724$	−2.48528	−0.0923648
$725$	0	0
$726$	−35.2132	−1.30688
$727$	3.51472i	0.130354i	0.997874	+	0.0651768i	$0.0207612\pi$
$727$	3.51472i	0.130354i	−0.997874	+	0.0651768i	$0.979239\pi$
$728$	0	0
$729$	−148.196	−5.48874
$730$	0	0
$731$	22.8284	0.844340
$732$	31.7990i	1.17532i
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	−24.9706	−0.921680
$735$	0	0
$736$	1.17157	0.0431847
$737$	− 1.37258i	− 0.0505597i
$738$	26.5858i	0.978636i
$739$	−3.17157	−0.116668	−0.0583341	−	0.998297i	$-0.518579\pi$
$739$	−3.17157	−0.116668	−0.0583341	+	0.998297i	$0.518579\pi$
$740$	0	0
$741$	9.65685	0.354753
$742$	0	0
$743$	− 51.7990i	− 1.90032i	−0.311762	−	0.950160i	$-0.600919\pi$
$743$	− 51.7990i	− 1.90032i	0.311762	−	0.950160i	$-0.399081\pi$
$744$	−9.65685	−0.354037
$745$	0	0
$746$	30.4853	1.11615
$747$	19.4142i	0.710329i
$748$	− 2.14214i	− 0.0783242i
$749$	0	0
$750$	0	0
$751$	−39.3137	−1.43458	−0.717289	−	0.696776i	$-0.754617\pi$
$751$	−39.3137	−1.43458	−0.717289	+	0.696776i	$0.754617\pi$
$752$	− 5.17157i	− 0.188588i
$753$	− 2.00000i	− 0.0728841i
$754$	−23.3137	−0.849035
$755$	0	0
$756$	0	0
$757$	3.65685i	0.132911i	0.997789	+	0.0664553i	$0.0211690\pi$
$757$	3.65685i	0.132911i	−0.997789	+	0.0664553i	$0.978831\pi$
$758$	34.4853i	1.25256i
$759$	3.31371	0.120280
$760$	0	0
$761$	−22.3848	−0.811448	−0.405724	−	0.913996i	$-0.632980\pi$
$761$	−22.3848	−0.811448	−0.405724	+	0.913996i	$0.632980\pi$
$762$	9.65685i	0.349831i
$763$	0	0
$764$	−10.1421	−0.366930
$765$	0	0
$766$	−32.4853	−1.17374
$767$	− 41.4558i	− 1.49688i
$768$	3.41421i	0.123200i
$769$	19.5563	0.705220	0.352610	−	0.935770i	$-0.385294\pi$
$769$	19.5563	0.705220	0.352610	+	0.935770i	$0.385294\pi$
$770$	0	0
$771$	−33.7990	−1.21724
$772$	5.65685i	0.203595i
$773$	− 2.00000i	− 0.0719350i	−0.999353	−	0.0359675i	$-0.988549\pi$
$773$	− 2.00000i	− 0.0719350i	0.999353	−	0.0359675i	$-0.0114513\pi$
$774$	−76.4264	−2.74709
$775$	0	0
$776$	−7.75736	−0.278473
$777$	0	0
$778$	28.1421i	1.00894i
$779$	1.79899	0.0644555
$780$	0	0
$781$	3.71573	0.132959
$782$	− 3.02944i	− 0.108332i
$783$	− 93.2548i	− 3.33266i
$784$	0	0
$785$	0	0
$786$	21.3137	0.760235
$787$	1.27208i	0.0453447i	0.999743	+	0.0226723i	$0.00721745\pi$
$787$	1.27208i	0.0453447i	−0.999743	+	0.0226723i	$0.992783\pi$
$788$	25.7990i	0.919051i
$789$	95.5980	3.40338
$790$	0	0
$791$	0	0
$792$	7.17157i	0.254831i
$793$	44.9706i	1.59695i
$794$	−33.7990	−1.19948
$795$	0	0
$796$	−16.4853	−0.584305
$797$	41.7990i	1.48060i	0.672279	+	0.740298i	$0.265316\pi$
$797$	41.7990i	1.48060i	−0.672279	+	0.740298i	$0.734684\pi$
$798$	0	0
$799$	−13.3726	−0.473088
$800$	0	0
$801$	−110.184	−3.89315
$802$	6.00000i	0.211867i
$803$	7.79899i	0.275220i
$804$	5.65685	0.199502
$805$	0	0
$806$	−13.6569	−0.481042
$807$	63.1127i	2.22167i
$808$	− 13.3137i	− 0.468375i
$809$	−3.02944	−0.106509	−0.0532547	−	0.998581i	$-0.516960\pi$
$809$	−3.02944	−0.106509	−0.0532547	+	0.998581i	$0.516960\pi$
$810$	0	0
$811$	−32.5858	−1.14424	−0.572121	−	0.820169i	$-0.693879\pi$
$811$	−32.5858	−1.14424	−0.572121	+	0.820169i	$0.693879\pi$
$812$	0	0
$813$	− 40.9706i	− 1.43690i
$814$	6.34315	0.222327
$815$	0	0
$816$	8.82843	0.309057
$817$	5.17157i	0.180930i
$818$	10.5858i	0.370123i
$819$	0	0
$820$	0	0
$821$	17.3137	0.604253	0.302126	−	0.953268i	$-0.402304\pi$
$821$	17.3137	0.604253	0.302126	+	0.953268i	$0.402304\pi$
$822$	− 54.6274i	− 1.90535i
$823$	− 20.2843i	− 0.707065i	−0.935422	−	0.353533i	$-0.884980\pi$
$823$	− 20.2843i	− 0.707065i	0.935422	−	0.353533i	$-0.115020\pi$
$824$	14.8284	0.516573
$825$	0	0
$826$	0	0
$827$	− 5.37258i	− 0.186823i	−0.995628	−	0.0934115i	$-0.970223\pi$
$827$	− 5.37258i	− 0.186823i	0.995628	−	0.0934115i	$-0.0297772\pi$
$828$	10.1421i	0.352464i
$829$	−5.02944	−0.174680	−0.0873398	−	0.996179i	$-0.527837\pi$
$829$	−5.02944	−0.174680	−0.0873398	+	0.996179i	$0.527837\pi$
$830$	0	0
$831$	27.7990	0.964336
$832$	4.82843i	0.167396i
$833$	0	0
$834$	67.9411	2.35261
$835$	0	0
$836$	0.485281	0.0167838
$837$	− 54.6274i	− 1.88820i
$838$	20.8701i	0.720944i
$839$	−42.1421	−1.45491	−0.727454	−	0.686156i	$-0.759297\pi$
$839$	−42.1421	−1.45491	−0.727454	+	0.686156i	$0.759297\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	− 17.3137i	− 0.596670i
$843$	27.3137i	0.940734i
$844$	−18.6274	−0.641182
$845$	0	0
$846$	44.7696	1.53921
$847$	0	0
$848$	− 6.48528i	− 0.222705i
$849$	7.65685	0.262783
$850$	0	0
$851$	8.97056	0.307507
$852$	15.3137i	0.524639i
$853$	− 43.1716i	− 1.47817i	−0.673614	−	0.739083i	$-0.735259\pi$
$853$	− 43.1716i	− 1.47817i	0.673614	−	0.739083i	$-0.264741\pi$
$854$	0	0
$855$	0	0
$856$	−9.65685	−0.330064
$857$	4.92893i	0.168369i	0.996450	+	0.0841846i	$0.0268285\pi$
$857$	4.92893i	0.168369i	−0.996450	+	0.0841846i	$0.973171\pi$
$858$	13.6569i	0.466237i
$859$	−7.21320	−0.246111	−0.123056	−	0.992400i	$-0.539269\pi$
$859$	−7.21320	−0.246111	−0.123056	+	0.992400i	$0.539269\pi$
$860$	0	0
$861$	0	0
$862$	22.3431i	0.761011i
$863$	4.97056i	0.169200i	0.996415	+	0.0846000i	$0.0269612\pi$
$863$	4.97056i	0.169200i	−0.996415	+	0.0846000i	$0.973039\pi$
$864$	−19.3137	−0.657066
$865$	0	0
$866$	10.5858	0.359720
$867$	35.2132i	1.19590i
$868$	0	0
$869$	−5.65685	−0.191896
$870$	0	0
$871$	8.00000	0.271070
$872$	− 2.48528i	− 0.0841622i
$873$	− 67.1543i	− 2.27283i
$874$	0.686292	0.0232142
$875$	0	0
$876$	−32.1421	−1.08598
$877$	− 30.2843i	− 1.02263i	−0.859394	−	0.511314i	$-0.829159\pi$
$877$	− 30.2843i	− 1.02263i	0.859394	−	0.511314i	$-0.170841\pi$
$878$	24.9706i	0.842716i
$879$	28.4853	0.960785
$880$	0	0
$881$	2.38478	0.0803452	0.0401726	−	0.999193i	$-0.487209\pi$
$881$	2.38478	0.0803452	0.0401726	+	0.999193i	$0.487209\pi$
$882$	0	0
$883$	41.6569i	1.40186i	0.713228	+	0.700932i	$0.247233\pi$
$883$	41.6569i	1.40186i	−0.713228	+	0.700932i	$0.752767\pi$
$884$	12.4853	0.419925
$885$	0	0
$886$	−3.02944	−0.101776
$887$	55.1127i	1.85050i	0.379354	+	0.925252i	$0.376146\pi$
$887$	55.1127i	1.85050i	−0.379354	+	0.925252i	$0.623854\pi$
$888$	26.1421i	0.877273i
$889$	0	0
$890$	0	0
$891$	−33.1127	−1.10932
$892$	7.31371i	0.244881i
$893$	− 3.02944i	− 0.101376i
$894$	20.4853	0.685130
$895$	0	0
$896$	0	0
$897$	19.3137i	0.644866i
$898$	− 16.6274i	− 0.554864i
$899$	13.6569	0.455482
$900$	0	0
$901$	−16.7696	−0.558675
$902$	2.54416i	0.0847111i
$903$	0	0
$904$	−15.3137	−0.509326
$905$	0	0
$906$	−38.6274	−1.28331
$907$	0.284271i	0.00943907i	0.999989	+	0.00471954i	$0.00150228\pi$
$907$	0.284271i	0.00943907i	−0.999989	+	0.00471954i	$0.998498\pi$
$908$	18.2426i	0.605403i
$909$	115.255	3.82276
$910$	0	0
$911$	36.2843	1.20215	0.601076	−	0.799192i	$-0.294739\pi$
$911$	36.2843	1.20215	0.601076	+	0.799192i	$0.294739\pi$
$912$	2.00000i	0.0662266i
$913$	1.85786i	0.0614863i
$914$	21.6569	0.716345
$915$	0	0
$916$	16.1421	0.533351
$917$	0	0
$918$	49.9411i	1.64830i
$919$	−15.5147	−0.511783	−0.255892	−	0.966705i	$-0.582369\pi$
$919$	−15.5147	−0.511783	−0.255892	+	0.966705i	$0.582369\pi$
$920$	0	0
$921$	50.9706	1.67954
$922$	− 12.8284i	− 0.422482i
$923$	21.6569i	0.712844i
$924$	0	0
$925$	0	0
$926$	16.9706	0.557687
$927$	128.368i	4.21614i
$928$	− 4.82843i	− 0.158501i
$929$	−17.2132	−0.564747	−0.282373	−	0.959305i	$-0.591122\pi$
$929$	−17.2132	−0.564747	−0.282373	+	0.959305i	$0.591122\pi$
$930$	0	0
$931$	0	0
$932$	23.3137i	0.763666i
$933$	− 13.6569i	− 0.447105i
$934$	15.8995	0.520247
$935$	0	0
$936$	−41.7990	−1.36624
$937$	− 20.2426i	− 0.661298i	−0.943754	−	0.330649i	$-0.892732\pi$
$937$	− 20.2426i	− 0.661298i	0.943754	−	0.330649i	$-0.107268\pi$
$938$	0	0
$939$	−49.1127	−1.60273
$940$	0	0
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	− 22.1421i	− 0.721430i
$943$	3.59798i	0.117166i
$944$	8.58579	0.279444
$945$	0	0
$946$	−7.31371	−0.237789
$947$	− 4.82843i	− 0.156903i	−0.996918	−	0.0784514i	$-0.975002\pi$
$947$	− 4.82843i	− 0.156903i	0.996918	−	0.0784514i	$-0.0249975\pi$
$948$	− 23.3137i	− 0.757194i
$949$	−45.4558	−1.47556
$950$	0	0
$951$	−35.7990	−1.16086
$952$	0	0
$953$	− 0.343146i	− 0.0111156i	−0.999985	−	0.00555779i	$-0.998231\pi$
$953$	− 0.343146i	− 0.0111156i	0.999985	−	0.00555779i	$-0.00176911\pi$
$954$	56.1421	1.81767
$955$	0	0
$956$	1.65685	0.0535865
$957$	− 13.6569i	− 0.441463i
$958$	17.1716i	0.554788i
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	36.9706i	1.19198i
$963$	− 83.5980i	− 2.69391i
$964$	−13.4142	−0.432043
$965$	0	0
$966$	0	0
$967$	− 37.4558i	− 1.20450i	−0.798308	−	0.602249i	$-0.794271\pi$
$967$	− 37.4558i	− 1.20450i	0.798308	−	0.602249i	$-0.205729\pi$
$968$	− 10.3137i	− 0.331495i
$969$	5.17157	0.166135
$970$	0	0
$971$	33.3553	1.07042	0.535212	−	0.844718i	$-0.320232\pi$
$971$	33.3553	1.07042	0.535212	+	0.844718i	$0.320232\pi$
$972$	− 78.5269i	− 2.51875i
$973$	0	0
$974$	31.7990	1.01891
$975$	0	0
$976$	−9.31371	−0.298125
$977$	− 12.6863i	− 0.405870i	−0.979192	−	0.202935i	$-0.934952\pi$
$977$	− 12.6863i	− 0.405870i	0.979192	−	0.202935i	$-0.0650481\pi$
$978$	68.7696i	2.19901i
$979$	−10.5442	−0.336993
$980$	0	0
$981$	21.5147	0.686912
$982$	− 32.2843i	− 1.03023i
$983$	− 12.2010i	− 0.389152i	−0.980887	−	0.194576i	$-0.937667\pi$
$983$	− 12.2010i	− 0.389152i	0.980887	−	0.194576i	$-0.0623331\pi$
$984$	−10.4853	−0.334259
$985$	0	0
$986$	−12.4853	−0.397612
$987$	0	0
$988$	2.82843i	0.0899843i
$989$	−10.3431	−0.328893
$990$	0	0
$991$	44.7696	1.42215	0.711076	−	0.703115i	$-0.248208\pi$
$991$	44.7696	1.42215	0.711076	+	0.703115i	$0.248208\pi$
$992$	− 2.82843i	− 0.0898027i
$993$	115.397i	3.66201i
$994$	0	0
$995$	0	0
$996$	−7.65685	−0.242617
$997$	18.2843i	0.579069i	0.957168	+	0.289534i	$0.0935004\pi$
$997$	18.2843i	0.579069i	−0.957168	+	0.289534i	$0.906500\pi$
$998$	30.3431i	0.960495i
$999$	−147.882	−4.67879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.c.w.99.2		4
5.2	odd	4		2450.2.a.bs.1.2		2
5.3	odd	4		490.2.a.l.1.1	✓	2
5.4	even	2	inner	2450.2.c.w.99.3		4
7.6	odd	2		2450.2.c.t.99.1		4
15.8	even	4		4410.2.a.by.1.2		2
20.3	even	4		3920.2.a.ca.1.2		2
35.3	even	12		490.2.e.i.471.1		4
35.13	even	4		490.2.a.m.1.2	yes	2
35.18	odd	12		490.2.e.j.471.2		4
35.23	odd	12		490.2.e.j.361.2		4
35.27	even	4		2450.2.a.bn.1.1		2
35.33	even	12		490.2.e.i.361.1		4
35.34	odd	2		2450.2.c.t.99.4		4
105.83	odd	4		4410.2.a.bt.1.2		2
140.83	odd	4		3920.2.a.bm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.a.l.1.1	✓	2	5.3	odd	4
490.2.a.m.1.2	yes	2	35.13	even	4
490.2.e.i.361.1		4	35.33	even	12
490.2.e.i.471.1		4	35.3	even	12
490.2.e.j.361.2		4	35.23	odd	12
490.2.e.j.471.2		4	35.18	odd	12
2450.2.a.bn.1.1		2	35.27	even	4
2450.2.a.bs.1.2		2	5.2	odd	4
2450.2.c.t.99.1		4	7.6	odd	2
2450.2.c.t.99.4		4	35.34	odd	2
2450.2.c.w.99.2		4	1.1	even	1	trivial
2450.2.c.w.99.3		4	5.4	even	2	inner
3920.2.a.bm.1.1		2	140.83	odd	4
3920.2.a.ca.1.2		2	20.3	even	4
4410.2.a.bt.1.2		2	105.83	odd	4
4410.2.a.by.1.2		2	15.8	even	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +3.41421i q^{3} -1.00000 q^{4} +3.41421 q^{6} +1.00000i q^{8} -8.65685 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +3.41421i q^{3} -1.00000 q^{4} +3.41421 q^{6} +1.00000i q^{8} -8.65685 q^{9} -0.828427 q^{11} -3.41421i q^{12} -4.82843i q^{13} +1.00000 q^{16} -2.58579i q^{17} +8.65685i q^{18} +0.585786 q^{19} +0.828427i q^{22} +1.17157i q^{23} -3.41421 q^{24} -4.82843 q^{26} -19.3137i q^{27} +4.82843 q^{29} +2.82843 q^{31} -1.00000i q^{32} -2.82843i q^{33} -2.58579 q^{34} +8.65685 q^{36} -7.65685i q^{37} -0.585786i q^{38} +16.4853 q^{39} +3.07107 q^{41} +8.82843i q^{43} +0.828427 q^{44} +1.17157 q^{46} -5.17157i q^{47} +3.41421i q^{48} +8.82843 q^{51} +4.82843i q^{52} -6.48528i q^{53} -19.3137 q^{54} +2.00000i q^{57} -4.82843i q^{58} +8.58579 q^{59} -9.31371 q^{61} -2.82843i q^{62} -1.00000 q^{64} -2.82843 q^{66} +1.65685i q^{67} +2.58579i q^{68} -4.00000 q^{69} -4.48528 q^{71} -8.65685i q^{72} -9.41421i q^{73} -7.65685 q^{74} -0.585786 q^{76} -16.4853i q^{78} +6.82843 q^{79} +39.9706 q^{81} -3.07107i q^{82} -2.24264i q^{83} +8.82843 q^{86} +16.4853i q^{87} -0.828427i q^{88} +12.7279 q^{89} -1.17157i q^{92} +9.65685i q^{93} -5.17157 q^{94} +3.41421 q^{96} +7.75736i q^{97} +7.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{6} - 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 8 * q^6 - 12 * q^9	\( 4 q - 4 q^{4} + 8 q^{6} - 12 q^{9} + 8 q^{11} + 4 q^{16} + 8 q^{19} - 8 q^{24} - 8 q^{26} + 8 q^{29} - 16 q^{34} + 12 q^{36} + 32 q^{39} - 16 q^{41} - 8 q^{44} + 16 q^{46} + 24 q^{51} - 32 q^{54} + 40 q^{59} + 8 q^{61} - 4 q^{64} - 16 q^{69} + 16 q^{71} - 8 q^{74} - 8 q^{76} + 16 q^{79} + 92 q^{81} + 24 q^{86} - 32 q^{94} + 8 q^{96} + 40 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^6 - 12 * q^9 + 8 * q^11 + 4 * q^16 + 8 * q^19 - 8 * q^24 - 8 * q^26 + 8 * q^29 - 16 * q^34 + 12 * q^36 + 32 * q^39 - 16 * q^41 - 8 * q^44 + 16 * q^46 + 24 * q^51 - 32 * q^54 + 40 * q^59 + 8 * q^61 - 4 * q^64 - 16 * q^69 + 16 * q^71 - 8 * q^74 - 8 * q^76 + 16 * q^79 + 92 * q^81 + 24 * q^86 - 32 * q^94 + 8 * q^96 + 40 * q^99

Introduction

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