Properties

 Label 1225.2.b.c.99.2 Level $1225$ Weight $2$ Character 1225.99 Analytic conductor $9.782$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$9.78167424761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

 Embedding label 99.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1225.99 Dual form 1225.2.b.c.99.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{8} +3.00000 q^{9} +4.00000 q^{11} -1.00000 q^{16} +3.00000i q^{18} +4.00000i q^{22} -8.00000i q^{23} -2.00000 q^{29} +5.00000i q^{32} +3.00000 q^{36} -6.00000i q^{37} +12.0000i q^{43} +4.00000 q^{44} +8.00000 q^{46} +10.0000i q^{53} -2.00000i q^{58} -7.00000 q^{64} +4.00000i q^{67} +16.0000 q^{71} +9.00000i q^{72} +6.00000 q^{74} -8.00000 q^{79} +9.00000 q^{81} -12.0000 q^{86} +12.0000i q^{88} -8.00000i q^{92} +12.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 6 * q^9 $$2 q + 2 q^{4} + 6 q^{9} + 8 q^{11} - 2 q^{16} - 4 q^{29} + 6 q^{36} + 8 q^{44} + 16 q^{46} - 14 q^{64} + 32 q^{71} + 12 q^{74} - 16 q^{79} + 18 q^{81} - 24 q^{86} + 24 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 + 6 * q^9 + 8 * q^11 - 2 * q^16 - 4 * q^29 + 6 * q^36 + 8 * q^44 + 16 * q^46 - 14 * q^64 + 32 * q^71 + 12 * q^74 - 16 * q^79 + 18 * q^81 - 24 * q^86 + 24 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$$.

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

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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 3.00000i 1.06066i
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 3.00000i 0.707107i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000i 0.852803i
$$23$$ − 8.00000i − 1.66812i −0.551677 0.834058i $$-0.686012\pi$$
0.551677 0.834058i $$-0.313988\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 12.0000i 1.82998i 0.403473 + 0.914991i $$0.367803\pi$$
−0.403473 + 0.914991i $$0.632197\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 10.0000i 1.37361i 0.726844 + 0.686803i $$0.240986\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ − 2.00000i − 0.262613i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 9.00000i 1.06066i
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −12.0000 −1.29399
$$87$$ 0 0
$$88$$ 12.0000i 1.27920i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 8.00000i − 0.834058i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 0 0
$$99$$ 12.0000 1.20605
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ − 20.0000i − 1.93347i −0.255774 0.966736i $$-0.582330\pi$$
0.255774 0.966736i $$-0.417670\pi$$
$$108$$ 0 0
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 10.0000i − 0.854358i −0.904167 0.427179i $$-0.859507\pi$$
0.904167 0.427179i $$-0.140493\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 16.0000i 1.34269i
$$143$$ 0 0
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ − 6.00000i − 0.493197i
$$149$$ −22.0000 −1.80231 −0.901155 0.433497i $$-0.857280\pi$$
−0.901155 + 0.433497i $$0.857280\pi$$
$$150$$ 0 0
$$151$$ −24.0000 −1.95309 −0.976546 0.215308i $$-0.930924\pi$$
−0.976546 + 0.215308i $$0.930924\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 9.00000i 0.707107i
$$163$$ 20.0000i 1.56652i 0.621694 + 0.783260i $$0.286445\pi$$
−0.621694 + 0.783260i $$0.713555\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 12.0000i 0.914991i
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 24.0000 1.76930
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ − 18.0000i − 1.29567i −0.761781 0.647834i $$-0.775675\pi$$
0.761781 0.647834i $$-0.224325\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 26.0000i − 1.85242i −0.377004 0.926212i $$-0.623046\pi$$
0.377004 0.926212i $$-0.376954\pi$$
$$198$$ 12.0000i 0.852803i
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 24.0000i − 1.66812i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 10.0000i 0.686803i
$$213$$ 0 0
$$214$$ 20.0000 1.36717
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ − 18.0000i − 1.21911i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 22.0000i − 1.44127i −0.693316 0.720634i $$-0.743851\pi$$
0.693316 0.720634i $$-0.256149\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ − 32.0000i − 2.01182i
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ − 32.0000i − 1.97320i −0.163144 0.986602i $$-0.552164\pi$$
0.163144 0.986602i $$-0.447836\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 4.00000i 0.244339i
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 15.0000i 0.883883i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 18.0000 1.04623
$$297$$ 0 0
$$298$$ − 22.0000i − 1.27443i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 24.0000i − 1.38104i
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ − 34.0000i − 1.90963i −0.297200 0.954815i $$-0.596053\pi$$
0.297200 0.954815i $$-0.403947\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 9.00000 0.500000
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 36.0000 1.97874 0.989369 0.145424i $$-0.0464545\pi$$
0.989369 + 0.145424i $$0.0464545\pi$$
$$332$$ 0 0
$$333$$ − 18.0000i − 0.986394i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 30.0000i 1.63420i 0.576493 + 0.817102i $$0.304421\pi$$
−0.576493 + 0.817102i $$0.695579\pi$$
$$338$$ 13.0000i 0.707107i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −36.0000 −1.94099
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 20.0000i 1.06600i
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ − 4.00000i − 0.211407i
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 8.00000i 0.417029i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 22.0000i − 1.13912i −0.821951 0.569558i $$-0.807114\pi$$
0.821951 0.569558i $$-0.192886\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 8.00000i 0.409316i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 18.0000 0.916176
$$387$$ 36.0000i 1.82998i
$$388$$ 0 0
$$389$$ 38.0000 1.92668 0.963338 0.268290i $$-0.0864585\pi$$
0.963338 + 0.268290i $$0.0864585\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 26.0000 1.30986
$$395$$ 0 0
$$396$$ 12.0000 0.603023
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −34.0000 −1.69788 −0.848939 0.528490i $$-0.822758\pi$$
−0.848939 + 0.528490i $$0.822758\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 24.0000i − 1.18964i
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 24.0000 1.17954
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ 0 0
$$424$$ −30.0000 −1.45693
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ − 20.0000i − 0.966736i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 32.0000 1.54139 0.770693 0.637207i $$-0.219910\pi$$
0.770693 + 0.637207i $$0.219910\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −18.0000 −0.862044
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 20.0000i 0.950229i 0.879924 + 0.475114i $$0.157593\pi$$
−0.879924 + 0.475114i $$0.842407\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 2.00000i − 0.0940721i
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 6.00000i − 0.280668i −0.990104 0.140334i $$-0.955182\pi$$
0.990104 0.140334i $$-0.0448177\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 40.0000i 1.85896i 0.368875 + 0.929479i $$0.379743\pi$$
−0.368875 + 0.929479i $$0.620257\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ 22.0000 1.01913
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 48.0000i 2.20704i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 30.0000i 1.37361i
$$478$$ − 16.0000i − 0.731823i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 24.0000i − 1.08754i −0.839233 0.543772i $$-0.816996\pi$$
0.839233 0.543772i $$-0.183004\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 44.0000 1.98569 0.992846 0.119401i $$-0.0380974\pi$$
0.992846 + 0.119401i $$0.0380974\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 32.0000 1.42257
$$507$$ 0 0
$$508$$ 16.0000i 0.709885i
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 11.0000i − 0.486136i
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 44.0000i 1.88130i 0.339372 + 0.940652i $$0.389785\pi$$
−0.339372 + 0.940652i $$0.610215\pi$$
$$548$$ − 10.0000i − 0.427179i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 46.0000i 1.94908i 0.224208 + 0.974541i $$0.428020\pi$$
−0.224208 + 0.974541i $$0.571980\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 26.0000i − 1.09674i
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 48.0000i 2.01404i
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ 0 0
$$571$$ 4.00000 0.167395 0.0836974 0.996491i $$-0.473327\pi$$
0.0836974 + 0.996491i $$0.473327\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −21.0000 −0.875000
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 17.0000i 0.707107i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 40.0000i 1.65663i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 6.00000i 0.246598i
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −22.0000 −0.901155
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −32.0000 −1.30748 −0.653742 0.756717i $$-0.726802\pi$$
−0.653742 + 0.756717i $$0.726802\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ 12.0000i 0.488678i
$$604$$ −24.0000 −0.976546
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 38.0000i 1.53481i 0.641165 + 0.767403i $$0.278451\pi$$
−0.641165 + 0.767403i $$0.721549\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 26.0000i − 1.04672i −0.852111 0.523360i $$-0.824678\pi$$
0.852111 0.523360i $$-0.175322\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ − 24.0000i − 0.954669i
$$633$$ 0 0
$$634$$ 34.0000 1.35031
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ − 8.00000i − 0.316723i
$$639$$ 48.0000 1.89885
$$640$$ 0 0
$$641$$ 46.0000 1.81689 0.908445 0.418004i $$-0.137270\pi$$
0.908445 + 0.418004i $$0.137270\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 27.0000i 1.06066i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.0000i 0.783260i
$$653$$ − 50.0000i − 1.95665i −0.207072 0.978326i $$-0.566394\pi$$
0.207072 0.978326i $$-0.433606\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −44.0000 −1.71400 −0.856998 0.515319i $$-0.827673\pi$$
−0.856998 + 0.515319i $$0.827673\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ 36.0000i 1.39918i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 18.0000 0.697486
$$667$$ 16.0000i 0.619522i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ − 30.0000i − 1.15642i −0.815890 0.578208i $$-0.803752\pi$$
0.815890 0.578208i $$-0.196248\pi$$
$$674$$ −30.0000 −1.15556
$$675$$ 0 0
$$676$$ 13.0000 0.500000
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 52.0000i 1.98972i 0.101237 + 0.994862i $$0.467720\pi$$
−0.101237 + 0.994862i $$0.532280\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ − 12.0000i − 0.457496i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −28.0000 −1.05529
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 0 0
$$711$$ −24.0000 −0.900070
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ 0 0
$$718$$ − 8.00000i − 0.298557i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 19.0000i − 0.707107i
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 40.0000 1.47442
$$737$$ 16.0000i 0.589368i
$$738$$ 0 0
$$739$$ 52.0000 1.91285 0.956425 0.291977i $$-0.0943129\pi$$
0.956425 + 0.291977i $$0.0943129\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 40.0000i 1.46746i 0.679442 + 0.733729i $$0.262222\pi$$
−0.679442 + 0.733729i $$0.737778\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 22.0000 0.805477
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −48.0000 −1.75154 −0.875772 0.482724i $$-0.839647\pi$$
−0.875772 + 0.482724i $$0.839647\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 54.0000i − 1.96266i −0.192323 0.981332i $$-0.561602\pi$$
0.192323 0.981332i $$-0.438398\pi$$
$$758$$ 12.0000i 0.435860i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 18.0000i − 0.647834i
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ −36.0000 −1.29399
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 38.0000i 1.36237i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 64.0000 2.29010
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ − 26.0000i − 0.926212i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 36.0000i 1.27920i
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 34.0000i − 1.20058i
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 38.0000 1.33601 0.668004 0.744157i $$-0.267149\pi$$
0.668004 + 0.744157i $$0.267149\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ 0 0
$$823$$ − 32.0000i − 1.11545i −0.830026 0.557725i $$-0.811674\pi$$
0.830026 0.557725i $$-0.188326\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 44.0000i 1.53003i 0.644013 + 0.765015i $$0.277268\pi$$
−0.644013 + 0.765015i $$0.722732\pi$$
$$828$$ − 24.0000i − 0.834058i
$$829$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 26.0000i − 0.896019i
$$843$$ 0 0
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ − 10.0000i − 0.343401i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −48.0000 −1.64542
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 60.0000 2.05076
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 32.0000i 1.08992i
$$863$$ − 8.00000i − 0.272323i −0.990687 0.136162i $$-0.956523\pi$$
0.990687 0.136162i $$-0.0434766\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ 0 0
$$872$$ − 54.0000i − 1.82867i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 50.0000i 1.68838i 0.536044 + 0.844190i $$0.319918\pi$$
−0.536044 + 0.844190i $$0.680082\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 12.0000i 0.403832i 0.979403 + 0.201916i $$0.0647168\pi$$
−0.979403 + 0.201916i $$0.935283\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 36.0000 1.20605
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ − 2.00000i − 0.0667409i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 60.0000i 1.99227i 0.0878507 + 0.996134i $$0.472000\pi$$
−0.0878507 + 0.996134i $$0.528000\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 48.0000 1.58337 0.791687 0.610927i $$-0.209203\pi$$
0.791687 + 0.610927i $$0.209203\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −40.0000 −1.31448
$$927$$ 0 0
$$928$$ − 10.0000i − 0.328266i
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 22.0000i − 0.720634i
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −48.0000 −1.56061
$$947$$ − 20.0000i − 0.649913i −0.945729 0.324956i $$-0.894650\pi$$
0.945729 0.324956i $$-0.105350\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 58.0000i − 1.87880i −0.342817 0.939402i $$-0.611381\pi$$
0.342817 0.939402i $$-0.388619\pi$$
$$954$$ −30.0000 −0.971286
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ − 60.0000i − 1.93347i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 40.0000i − 1.28631i −0.765735 0.643157i $$-0.777624\pi$$
0.765735 0.643157i $$-0.222376\pi$$
$$968$$ 15.0000i 0.482118i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 24.0000 0.769010
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 46.0000i 1.47167i 0.677161 + 0.735835i $$0.263210\pi$$
−0.677161 + 0.735835i $$0.736790\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −54.0000 −1.72409
$$982$$ 44.0000i 1.40410i
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 96.0000 3.05262
$$990$$ 0 0
$$991$$ −24.0000 −0.762385 −0.381193 0.924496i $$-0.624487\pi$$
−0.381193 + 0.924496i $$0.624487\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ − 36.0000i − 1.13956i
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1225.2.b.c.99.2 2
5.2 odd 4 1225.2.a.c.1.1 1
5.3 odd 4 49.2.a.a.1.1 1
5.4 even 2 inner 1225.2.b.c.99.1 2
7.6 odd 2 CM 1225.2.b.c.99.2 2
15.8 even 4 441.2.a.c.1.1 1
20.3 even 4 784.2.a.f.1.1 1
35.3 even 12 49.2.c.a.30.1 2
35.13 even 4 49.2.a.a.1.1 1
35.18 odd 12 49.2.c.a.30.1 2
35.23 odd 12 49.2.c.a.18.1 2
35.27 even 4 1225.2.a.c.1.1 1
35.33 even 12 49.2.c.a.18.1 2
35.34 odd 2 inner 1225.2.b.c.99.1 2
40.3 even 4 3136.2.a.o.1.1 1
40.13 odd 4 3136.2.a.n.1.1 1
55.43 even 4 5929.2.a.c.1.1 1
60.23 odd 4 7056.2.a.bg.1.1 1
65.38 odd 4 8281.2.a.d.1.1 1
105.23 even 12 441.2.e.d.361.1 2
105.38 odd 12 441.2.e.d.226.1 2
105.53 even 12 441.2.e.d.226.1 2
105.68 odd 12 441.2.e.d.361.1 2
105.83 odd 4 441.2.a.c.1.1 1
140.3 odd 12 784.2.i.f.177.1 2
140.23 even 12 784.2.i.f.753.1 2
140.83 odd 4 784.2.a.f.1.1 1
140.103 odd 12 784.2.i.f.753.1 2
140.123 even 12 784.2.i.f.177.1 2
280.13 even 4 3136.2.a.n.1.1 1
280.83 odd 4 3136.2.a.o.1.1 1
385.153 odd 4 5929.2.a.c.1.1 1
420.83 even 4 7056.2.a.bg.1.1 1
455.363 even 4 8281.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
49.2.a.a.1.1 1 5.3 odd 4
49.2.a.a.1.1 1 35.13 even 4
49.2.c.a.18.1 2 35.23 odd 12
49.2.c.a.18.1 2 35.33 even 12
49.2.c.a.30.1 2 35.3 even 12
49.2.c.a.30.1 2 35.18 odd 12
441.2.a.c.1.1 1 15.8 even 4
441.2.a.c.1.1 1 105.83 odd 4
441.2.e.d.226.1 2 105.38 odd 12
441.2.e.d.226.1 2 105.53 even 12
441.2.e.d.361.1 2 105.23 even 12
441.2.e.d.361.1 2 105.68 odd 12
784.2.a.f.1.1 1 20.3 even 4
784.2.a.f.1.1 1 140.83 odd 4
784.2.i.f.177.1 2 140.3 odd 12
784.2.i.f.177.1 2 140.123 even 12
784.2.i.f.753.1 2 140.23 even 12
784.2.i.f.753.1 2 140.103 odd 12
1225.2.a.c.1.1 1 5.2 odd 4
1225.2.a.c.1.1 1 35.27 even 4
1225.2.b.c.99.1 2 5.4 even 2 inner
1225.2.b.c.99.1 2 35.34 odd 2 inner
1225.2.b.c.99.2 2 1.1 even 1 trivial
1225.2.b.c.99.2 2 7.6 odd 2 CM
3136.2.a.n.1.1 1 40.13 odd 4
3136.2.a.n.1.1 1 280.13 even 4
3136.2.a.o.1.1 1 40.3 even 4
3136.2.a.o.1.1 1 280.83 odd 4
5929.2.a.c.1.1 1 55.43 even 4
5929.2.a.c.1.1 1 385.153 odd 4
7056.2.a.bg.1.1 1 60.23 odd 4
7056.2.a.bg.1.1 1 420.83 even 4
8281.2.a.d.1.1 1 65.38 odd 4
8281.2.a.d.1.1 1 455.363 even 4