# Properties

 Label 175.5.d.c.76.1 Level $175$ Weight $5$ Character 175.76 Analytic conductor $18.090$ Analytic rank $0$ Dimension $2$ CM discriminant -35 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 175.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.0897435397$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 76.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 175.76 Dual form 175.5.d.c.76.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-17.0000i q^{3} -16.0000 q^{4} -49.0000i q^{7} -208.000 q^{9} +O(q^{10})$$ $$q-17.0000i q^{3} -16.0000 q^{4} -49.0000i q^{7} -208.000 q^{9} -73.0000 q^{11} +272.000i q^{12} +23.0000i q^{13} +256.000 q^{16} -263.000i q^{17} -833.000 q^{21} +2159.00i q^{27} +784.000i q^{28} +1153.00 q^{29} +1241.00i q^{33} +3328.00 q^{36} +391.000 q^{39} +1168.00 q^{44} +3457.00i q^{47} -4352.00i q^{48} -2401.00 q^{49} -4471.00 q^{51} -368.000i q^{52} +10192.0i q^{63} -4096.00 q^{64} +4208.00i q^{68} -10078.0 q^{71} -9502.00i q^{73} +3577.00i q^{77} -12167.0 q^{79} +19855.0 q^{81} -6382.00i q^{83} +13328.0 q^{84} -19601.0i q^{87} +1127.00 q^{91} -3383.00i q^{97} +15184.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} - 416 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 - 416 * q^9 $$2 q - 32 q^{4} - 416 q^{9} - 146 q^{11} + 512 q^{16} - 1666 q^{21} + 2306 q^{29} + 6656 q^{36} + 782 q^{39} + 2336 q^{44} - 4802 q^{49} - 8942 q^{51} - 8192 q^{64} - 20156 q^{71} - 24334 q^{79} + 39710 q^{81} + 26656 q^{84} + 2254 q^{91} + 30368 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 - 416 * q^9 - 146 * q^11 + 512 * q^16 - 1666 * q^21 + 2306 * q^29 + 6656 * q^36 + 782 * q^39 + 2336 * q^44 - 4802 * q^49 - 8942 * q^51 - 8192 * q^64 - 20156 * q^71 - 24334 * q^79 + 39710 * q^81 + 26656 * q^84 + 2254 * q^91 + 30368 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ − 17.0000i − 1.88889i −0.328671 0.944444i $$-0.606601\pi$$
0.328671 0.944444i $$-0.393399\pi$$
$$4$$ −16.0000 −1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 49.0000i − 1.00000i
$$8$$ 0 0
$$9$$ −208.000 −2.56790
$$10$$ 0 0
$$11$$ −73.0000 −0.603306 −0.301653 0.953418i $$-0.597538\pi$$
−0.301653 + 0.953418i $$0.597538\pi$$
$$12$$ 272.000i 1.88889i
$$13$$ 23.0000i 0.136095i 0.997682 + 0.0680473i $$0.0216769\pi$$
−0.997682 + 0.0680473i $$0.978323\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 256.000 1.00000
$$17$$ − 263.000i − 0.910035i −0.890483 0.455017i $$-0.849633\pi$$
0.890483 0.455017i $$-0.150367\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ −833.000 −1.88889
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 2159.00i 2.96159i
$$28$$ 784.000i 1.00000i
$$29$$ 1153.00 1.37099 0.685493 0.728079i $$-0.259587\pi$$
0.685493 + 0.728079i $$0.259587\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ 1241.00i 1.13958i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 3328.00 2.56790
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ 391.000 0.257068
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 1168.00 0.603306
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3457.00i 1.56496i 0.622675 + 0.782481i $$0.286046\pi$$
−0.622675 + 0.782481i $$0.713954\pi$$
$$48$$ − 4352.00i − 1.88889i
$$49$$ −2401.00 −1.00000
$$50$$ 0 0
$$51$$ −4471.00 −1.71895
$$52$$ − 368.000i − 0.136095i
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 10192.0i 2.56790i
$$64$$ −4096.00 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 4208.00i 0.910035i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10078.0 −1.99921 −0.999603 0.0281662i $$-0.991033\pi$$
−0.999603 + 0.0281662i $$0.991033\pi$$
$$72$$ 0 0
$$73$$ − 9502.00i − 1.78307i −0.452948 0.891537i $$-0.649628\pi$$
0.452948 0.891537i $$-0.350372\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3577.00i 0.603306i
$$78$$ 0 0
$$79$$ −12167.0 −1.94953 −0.974764 0.223239i $$-0.928337\pi$$
−0.974764 + 0.223239i $$0.928337\pi$$
$$80$$ 0 0
$$81$$ 19855.0 3.02622
$$82$$ 0 0
$$83$$ − 6382.00i − 0.926404i −0.886253 0.463202i $$-0.846700\pi$$
0.886253 0.463202i $$-0.153300\pi$$
$$84$$ 13328.0 1.88889
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 19601.0i − 2.58964i
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 1127.00 0.136095
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 3383.00i − 0.359549i −0.983708 0.179775i $$-0.942463\pi$$
0.983708 0.179775i $$-0.0575369\pi$$
$$98$$ 0 0
$$99$$ 15184.0 1.54923
$$100$$ 0 0
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 18383.0i 1.73277i 0.499373 + 0.866387i $$0.333564\pi$$
−0.499373 + 0.866387i $$0.666436\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ − 34544.0i − 2.96159i
$$109$$ 14353.0 1.20806 0.604032 0.796960i $$-0.293560\pi$$
0.604032 + 0.796960i $$0.293560\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 12544.0i − 1.00000i
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −18448.0 −1.37099
$$117$$ − 4784.00i − 0.349478i
$$118$$ 0 0
$$119$$ −12887.0 −0.910035
$$120$$ 0 0
$$121$$ −9312.00 −0.636022
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ − 19856.0i − 1.13958i
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ 58769.0 2.95604
$$142$$ 0 0
$$143$$ − 1679.00i − 0.0821067i
$$144$$ −53248.0 −2.56790
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 40817.0i 1.88889i
$$148$$ 0 0
$$149$$ −24242.0 −1.09193 −0.545966 0.837807i $$-0.683837\pi$$
−0.545966 + 0.837807i $$0.683837\pi$$
$$150$$ 0 0
$$151$$ −45433.0 −1.99259 −0.996294 0.0860129i $$-0.972587\pi$$
−0.996294 + 0.0860129i $$0.972587\pi$$
$$152$$ 0 0
$$153$$ 54704.0i 2.33688i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −6256.00 −0.257068
$$157$$ 31342.0i 1.27153i 0.771882 + 0.635766i $$0.219316\pi$$
−0.771882 + 0.635766i $$0.780684\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 17663.0i − 0.633332i −0.948537 0.316666i $$-0.897437\pi$$
0.948537 0.316666i $$-0.102563\pi$$
$$168$$ 0 0
$$169$$ 28032.0 0.981478
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 11017.0i − 0.368105i −0.982916 0.184052i $$-0.941078\pi$$
0.982916 0.184052i $$-0.0589216\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −18688.0 −0.603306
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 16558.0 0.516775 0.258388 0.966041i $$-0.416809\pi$$
0.258388 + 0.966041i $$0.416809\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 19199.0i 0.549029i
$$188$$ − 55312.0i − 1.56496i
$$189$$ 105791. 2.96159
$$190$$ 0 0
$$191$$ 47447.0 1.30059 0.650297 0.759680i $$-0.274644\pi$$
0.650297 + 0.759680i $$0.274644\pi$$
$$192$$ 69632.0i 1.88889i
$$193$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 38416.0 1.00000
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 56497.0i − 1.37099i
$$204$$ 71536.0 1.71895
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 5888.00i 0.136095i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −77593.0 −1.74284 −0.871420 0.490537i $$-0.836801\pi$$
−0.871420 + 0.490537i $$0.836801\pi$$
$$212$$ 0 0
$$213$$ 171326.i 3.77628i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −161534. −3.36803
$$220$$ 0 0
$$221$$ 6049.00 0.123851
$$222$$ 0 0
$$223$$ 61343.0i 1.23355i 0.787141 + 0.616773i $$0.211560\pi$$
−0.787141 + 0.616773i $$0.788440\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 49823.0i − 0.966892i −0.875374 0.483446i $$-0.839385\pi$$
0.875374 0.483446i $$-0.160615\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 60809.0 1.13958
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 206839.i 3.68244i
$$238$$ 0 0
$$239$$ −43367.0 −0.759213 −0.379606 0.925148i $$-0.623941\pi$$
−0.379606 + 0.925148i $$0.623941\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ − 162656.i − 2.75459i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −108494. −1.74988
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ − 163072.i − 2.56790i
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 65536.0 1.00000
$$257$$ − 111938.i − 1.69477i −0.530977 0.847386i $$-0.678175\pi$$
0.530977 0.847386i $$-0.321825\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −239824. −3.52056
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ − 67328.0i − 0.910035i
$$273$$ − 19159.0i − 0.257068i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 119807. 1.51729 0.758647 0.651502i $$-0.225861\pi$$
0.758647 + 0.651502i $$0.225861\pi$$
$$282$$ 0 0
$$283$$ 152303.i 1.90167i 0.309695 + 0.950836i $$0.399773\pi$$
−0.309695 + 0.950836i $$0.600227\pi$$
$$284$$ 161248. 1.99921
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 14352.0 0.171837
$$290$$ 0 0
$$291$$ −57511.0 −0.679149
$$292$$ 152032.i 1.78307i
$$293$$ − 171337.i − 1.99579i −0.0648123 0.997897i $$-0.520645\pi$$
0.0648123 0.997897i $$-0.479355\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 157607.i − 1.78675i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 135263.i − 1.43517i −0.696473 0.717583i $$-0.745248\pi$$
0.696473 0.717583i $$-0.254752\pi$$
$$308$$ − 57232.0i − 0.603306i
$$309$$ 312511. 3.27302
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ − 147097.i − 1.50146i −0.660606 0.750732i $$-0.729701\pi$$
0.660606 0.750732i $$-0.270299\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 194672. 1.94953
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ −84169.0 −0.827124
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −317680. −3.02622
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 244001.i − 2.28190i
$$328$$ 0 0
$$329$$ 169393. 1.56496
$$330$$ 0 0
$$331$$ 138482. 1.26397 0.631986 0.774980i $$-0.282240\pi$$
0.631986 + 0.774980i $$0.282240\pi$$
$$332$$ 102112.i 0.926404i
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −213248. −1.88889
$$337$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 117649.i 1.00000i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 313616.i 2.58964i
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ −49657.0 −0.403057
$$352$$ 0 0
$$353$$ − 229897.i − 1.84495i −0.386060 0.922473i $$-0.626164\pi$$
0.386060 0.922473i $$-0.373836\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 219079.i 1.71895i
$$358$$ 0 0
$$359$$ −76322.0 −0.592190 −0.296095 0.955159i $$-0.595684\pi$$
−0.296095 + 0.955159i $$0.595684\pi$$
$$360$$ 0 0
$$361$$ 130321. 1.00000
$$362$$ 0 0
$$363$$ 158304.i 1.20138i
$$364$$ −18032.0 −0.136095
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 116497.i 0.864933i 0.901650 + 0.432467i $$0.142357\pi$$
−0.901650 + 0.432467i $$0.857643\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 26519.0i 0.186584i
$$378$$ 0 0
$$379$$ 35278.0 0.245598 0.122799 0.992432i $$-0.460813\pi$$
0.122799 + 0.992432i $$0.460813\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 29182.0i − 0.198938i −0.995041 0.0994689i $$-0.968286\pi$$
0.995041 0.0994689i $$-0.0317144\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 54128.0i 0.359549i
$$389$$ −249407. −1.64820 −0.824099 0.566446i $$-0.808318\pi$$
−0.824099 + 0.566446i $$0.808318\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −242944. −1.54923
$$397$$ 163897.i 1.03990i 0.854198 + 0.519948i $$0.174049\pi$$
−0.854198 + 0.519948i $$0.825951\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −316273. −1.96686 −0.983430 0.181289i $$-0.941973\pi$$
−0.983430 + 0.181289i $$0.941973\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 294128.i − 1.73277i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ −76753.0 −0.433043 −0.216522 0.976278i $$-0.569471\pi$$
−0.216522 + 0.976278i $$0.569471\pi$$
$$422$$ 0 0
$$423$$ − 719056.i − 4.01867i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −28543.0 −0.155090
$$430$$ 0 0
$$431$$ 356087. 1.91691 0.958455 0.285245i $$-0.0920749\pi$$
0.958455 + 0.285245i $$0.0920749\pi$$
$$432$$ 552704.i 2.96159i
$$433$$ 193538.i 1.03226i 0.856509 + 0.516132i $$0.172628\pi$$
−0.856509 + 0.516132i $$0.827372\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −229648. −1.20806
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ 499408. 2.56790
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 412114.i 2.06254i
$$448$$ 200704.i 1.00000i
$$449$$ −264287. −1.31094 −0.655470 0.755221i $$-0.727530\pi$$
−0.655470 + 0.755221i $$0.727530\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 772361.i 3.76378i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$458$$ 0 0
$$459$$ 567817. 2.69515
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 295168. 1.37099
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 322463.i − 1.47858i −0.673385 0.739292i $$-0.735160\pi$$
0.673385 0.739292i $$-0.264840\pi$$
$$468$$ 76544.0i 0.349478i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 532814. 2.40178
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 206192. 0.910035
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 148992. 0.636022
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −470713. −1.95251 −0.976255 0.216625i $$-0.930495\pi$$
−0.976255 + 0.216625i $$0.930495\pi$$
$$492$$ 0 0
$$493$$ − 303239.i − 1.24765i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 493822.i 1.99921i
$$498$$ 0 0
$$499$$ 31513.0 0.126558 0.0632789 0.997996i $$-0.479844\pi$$
0.0632789 + 0.997996i $$0.479844\pi$$
$$500$$ 0 0
$$501$$ −300271. −1.19629
$$502$$ 0 0
$$503$$ − 313297.i − 1.23828i −0.785279 0.619142i $$-0.787481\pi$$
0.785279 0.619142i $$-0.212519\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 476544.i − 1.85390i
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ −465598. −1.78307
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 252361.i − 0.944150i
$$518$$ 0 0
$$519$$ −187289. −0.695309
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ − 440782.i − 1.61146i −0.592281 0.805732i $$-0.701772\pi$$
0.592281 0.805732i $$-0.298228\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 317696.i 1.13958i
$$529$$ −279841. −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 281486.i − 0.976131i
$$538$$ 0 0
$$539$$ 175273. 0.603306
$$540$$ 0 0
$$541$$ 2927.00 0.0100006 0.00500032 0.999987i $$-0.498408\pi$$
0.00500032 + 0.999987i $$0.498408\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 596183.i 1.94953i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 326383. 1.03706
$$562$$ 0 0
$$563$$ 129938.i 0.409939i 0.978768 + 0.204970i $$0.0657095\pi$$
−0.978768 + 0.204970i $$0.934290\pi$$
$$564$$ −940304. −2.95604
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 972895.i − 3.02622i
$$568$$ 0 0
$$569$$ −566882. −1.75093 −0.875464 0.483284i $$-0.839444\pi$$
−0.875464 + 0.483284i $$0.839444\pi$$
$$570$$ 0 0
$$571$$ −638158. −1.95729 −0.978647 0.205549i $$-0.934102\pi$$
−0.978647 + 0.205549i $$0.934102\pi$$
$$572$$ 26864.0i 0.0821067i
$$573$$ − 806599.i − 2.45668i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 851968. 2.56790
$$577$$ 665017.i 1.99747i 0.0502441 + 0.998737i $$0.484000\pi$$
−0.0502441 + 0.998737i $$0.516000\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −312718. −0.926404
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 507698.i − 1.47343i −0.676204 0.736715i $$-0.736376\pi$$
0.676204 0.736715i $$-0.263624\pi$$
$$588$$ − 653072.i − 1.88889i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 320137.i − 0.910388i −0.890392 0.455194i $$-0.849570\pi$$
0.890392 0.455194i $$-0.150430\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 387872. 1.09193
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 613273. 1.70923 0.854614 0.519263i $$-0.173794\pi$$
0.854614 + 0.519263i $$0.173794\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 726928. 1.99259
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 82417.0i 0.223686i 0.993726 + 0.111843i $$0.0356754\pi$$
−0.993726 + 0.111843i $$0.964325\pi$$
$$608$$ 0 0
$$609$$ −960449. −2.58964
$$610$$ 0 0
$$611$$ −79511.0 −0.212983
$$612$$ − 875264.i − 2.33688i
$$613$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 100096. 0.257068
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ − 501472.i − 1.27153i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 100487. 0.252378 0.126189 0.992006i $$-0.459725\pi$$
0.126189 + 0.992006i $$0.459725\pi$$
$$632$$ 0 0
$$633$$ 1.31908e6i 3.29203i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 55223.0i − 0.136095i
$$638$$ 0 0
$$639$$ 2.09622e6 5.13376
$$640$$ 0 0
$$641$$ 96002.0 0.233649 0.116825 0.993153i $$-0.462728\pi$$
0.116825 + 0.993153i $$0.462728\pi$$
$$642$$ 0 0
$$643$$ 713183.i 1.72496i 0.506091 + 0.862480i $$0.331090\pi$$
−0.506091 + 0.862480i $$0.668910\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 111458.i − 0.266258i −0.991099 0.133129i $$-0.957498\pi$$
0.991099 0.133129i $$-0.0425025\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 1.97642e6i 4.57876i
$$658$$ 0 0
$$659$$ −777527. −1.79038 −0.895189 0.445687i $$-0.852959\pi$$
−0.895189 + 0.445687i $$0.852959\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ − 102833.i − 0.233941i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 282608.i 0.633332i
$$669$$ 1.04283e6 2.33003
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −448512. −0.981478
$$677$$ − 750023.i − 1.63643i −0.574913 0.818215i $$-0.694964\pi$$
0.574913 0.818215i $$-0.305036\pi$$
$$678$$ 0 0
$$679$$ −165767. −0.359549
$$680$$ 0 0
$$681$$ −846991. −1.82635
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 176272.i 0.368105i
$$693$$ − 744016.i − 1.54923i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −884833. −1.80063 −0.900317 0.435235i $$-0.856665\pi$$
−0.900317 + 0.435235i $$0.856665\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 299008. 0.603306
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1.00253e6 −1.99436 −0.997180 0.0750454i $$-0.976090\pi$$
−0.997180 + 0.0750454i $$0.976090\pi$$
$$710$$ 0 0
$$711$$ 2.53074e6 5.00619
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −264928. −0.516775
$$717$$ 737239.i 1.43407i
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 900767. 1.73277
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 976418.i − 1.84743i −0.383086 0.923713i $$-0.625139\pi$$
0.383086 0.923713i $$-0.374861\pi$$
$$728$$ 0 0
$$729$$ −1.15690e6 −2.17691
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 51143.0i 0.0951871i 0.998867 + 0.0475936i $$0.0151552\pi$$
−0.998867 + 0.0475936i $$0.984845\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −749207. −1.37187 −0.685935 0.727663i $$-0.740607\pi$$
−0.685935 + 0.727663i $$0.740607\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 1.32746e6i 2.37892i
$$748$$ − 307184.i − 0.549029i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 648887. 1.15051 0.575253 0.817975i $$-0.304903\pi$$
0.575253 + 0.817975i $$0.304903\pi$$
$$752$$ 884992.i 1.56496i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −1.69266e6 −2.96159
$$757$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ − 703297.i − 1.20806i
$$764$$ −759152. −1.30059
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ − 1.11411e6i − 1.88889i
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ −1.90295e6 −3.20124
$$772$$ 0 0
$$773$$ 1.17962e6i 1.97417i 0.160202 + 0.987084i $$0.448786\pi$$
−0.160202 + 0.987084i $$0.551214\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 735694. 1.20613
$$782$$ 0 0
$$783$$ 2.48933e6i 4.06030i
$$784$$ −614656. −1.00000
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 1.03714e6i 1.67451i 0.546816 + 0.837253i $$0.315840\pi$$
−0.546816 + 0.837253i $$0.684160\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 1.07354e6i − 1.69006i −0.534717 0.845031i $$-0.679582\pi$$
0.534717 0.845031i $$-0.320418\pi$$
$$798$$ 0 0
$$799$$ 909191. 1.42417
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 693646.i 1.07574i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 1.29955e6 1.98562 0.992812 0.119685i $$-0.0381886\pi$$
0.992812 + 0.119685i $$0.0381886\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 903952.i 1.37099i
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −1.14458e6 −1.71895
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −234416. −0.349478
$$820$$ 0 0
$$821$$ 1.23437e6 1.83129 0.915647 0.401984i $$-0.131679\pi$$
0.915647 + 0.401984i $$0.131679\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 94208.0i − 0.136095i
$$833$$ 631463.i 0.910035i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 622128. 0.879605
$$842$$ 0 0
$$843$$ − 2.03672e6i − 2.86600i
$$844$$ 1.24149e6 1.74284
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 456288.i 0.636022i
$$848$$ 0 0
$$849$$ 2.58915e6 3.59205
$$850$$ 0 0
$$851$$ 0 0
$$852$$ − 2.74122e6i − 3.77628i
$$853$$ − 1.44782e6i − 1.98984i −0.100692 0.994918i $$-0.532106\pi$$
0.100692 0.994918i $$-0.467894\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 970462.i 1.32135i 0.750673 + 0.660674i $$0.229729\pi$$
−0.750673 + 0.660674i $$0.770271\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 243984.i − 0.324581i
$$868$$ 0 0
$$869$$ 888191. 1.17616
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 703664.i 0.923287i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 2.58454e6 3.36803
$$877$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$878$$ 0 0
$$879$$ −2.91273e6 −3.76983
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ −96784.0 −0.123851
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 1.32950e6i 1.68983i 0.534904 + 0.844913i $$0.320348\pi$$
−0.534904 + 0.844913i $$0.679652\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −1.44942e6 −1.82573
$$892$$ − 981488.i − 1.23355i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$908$$ 797168.i 0.966892i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 1.15584e6 1.39271 0.696357 0.717696i $$-0.254803\pi$$
0.696357 + 0.717696i $$0.254803\pi$$
$$912$$ 0 0
$$913$$ 465886.i 0.558905i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 695113. 0.823047 0.411523 0.911399i $$-0.364997\pi$$
0.411523 + 0.911399i $$0.364997\pi$$
$$920$$ 0 0
$$921$$ −2.29947e6 −2.71087
$$922$$ 0 0
$$923$$ − 231794.i − 0.272081i
$$924$$ −972944. −1.13958
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 3.82366e6i − 4.44959i
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 1.58930e6i − 1.81020i −0.425195 0.905102i $$-0.639794\pi$$
0.425195 0.905102i $$-0.360206\pi$$
$$938$$ 0 0
$$939$$ −2.50065e6 −2.83610
$$940$$ 0 0
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ − 3.30942e6i − 3.68244i
$$949$$ 218546. 0.242667
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 693872. 0.759213
$$957$$ 1.43087e6i 1.56235i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 923521. 1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 2.60250e6i 2.75459i
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −2.98542e6 −3.10219
$$982$$ 0 0
$$983$$ − 675937.i − 0.699518i −0.936840 0.349759i $$-0.886263\pi$$
0.936840 0.349759i $$-0.113737\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 2.87968e6i − 2.95604i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −1.44288e6 −1.46920 −0.734602 0.678498i $$-0.762631\pi$$
−0.734602 + 0.678498i $$0.762631\pi$$
$$992$$ 0 0
$$993$$ − 2.35419e6i − 2.38750i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 1.73590e6 1.74988
$$997$$ − 737783.i − 0.742230i −0.928587 0.371115i $$-0.878976\pi$$
0.928587 0.371115i $$-0.121024\pi$$
$$998$$ 0 0
$$999$$ 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 175.5.d.c.76.1 2
5.2 odd 4 35.5.c.a.34.1 1
5.3 odd 4 35.5.c.b.34.1 yes 1
5.4 even 2 inner 175.5.d.c.76.2 2
7.6 odd 2 inner 175.5.d.c.76.2 2
15.2 even 4 315.5.e.a.244.1 1
15.8 even 4 315.5.e.b.244.1 1
20.3 even 4 560.5.p.a.209.1 1
20.7 even 4 560.5.p.b.209.1 1
35.13 even 4 35.5.c.a.34.1 1
35.27 even 4 35.5.c.b.34.1 yes 1
35.34 odd 2 CM 175.5.d.c.76.1 2
105.62 odd 4 315.5.e.b.244.1 1
105.83 odd 4 315.5.e.a.244.1 1
140.27 odd 4 560.5.p.a.209.1 1
140.83 odd 4 560.5.p.b.209.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.c.a.34.1 1 5.2 odd 4
35.5.c.a.34.1 1 35.13 even 4
35.5.c.b.34.1 yes 1 5.3 odd 4
35.5.c.b.34.1 yes 1 35.27 even 4
175.5.d.c.76.1 2 1.1 even 1 trivial
175.5.d.c.76.1 2 35.34 odd 2 CM
175.5.d.c.76.2 2 5.4 even 2 inner
175.5.d.c.76.2 2 7.6 odd 2 inner
315.5.e.a.244.1 1 15.2 even 4
315.5.e.a.244.1 1 105.83 odd 4
315.5.e.b.244.1 1 15.8 even 4
315.5.e.b.244.1 1 105.62 odd 4
560.5.p.a.209.1 1 20.3 even 4
560.5.p.a.209.1 1 140.27 odd 4
560.5.p.b.209.1 1 20.7 even 4
560.5.p.b.209.1 1 140.83 odd 4