# Properties

 Label 315.5.e.b.244.1 Level $315$ Weight $5$ Character 315.244 Self dual yes Analytic conductor $32.562$ Analytic rank $0$ Dimension $1$ CM discriminant -35 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 315.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.5615383714$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 244.1 Character $$\chi$$ $$=$$ 315.244

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+16.0000 q^{4} +25.0000 q^{5} -49.0000 q^{7} +O(q^{10})$$ $$q+16.0000 q^{4} +25.0000 q^{5} -49.0000 q^{7} +73.0000 q^{11} -23.0000 q^{13} +256.000 q^{16} +263.000 q^{17} +400.000 q^{20} +625.000 q^{25} -784.000 q^{28} +1153.00 q^{29} -1225.00 q^{35} +1168.00 q^{44} -3457.00 q^{47} +2401.00 q^{49} -368.000 q^{52} +1825.00 q^{55} +4096.00 q^{64} -575.000 q^{65} +4208.00 q^{68} +10078.0 q^{71} +9502.00 q^{73} -3577.00 q^{77} +12167.0 q^{79} +6400.00 q^{80} -6382.00 q^{83} +6575.00 q^{85} +1127.00 q^{91} -3383.00 q^{97} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$136$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$3$$ 0 0
$$4$$ 16.0000 1.00000
$$5$$ 25.0000 1.00000
$$6$$ 0 0
$$7$$ −49.0000 −1.00000
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 73.0000 0.603306 0.301653 0.953418i $$-0.402462\pi$$
0.301653 + 0.953418i $$0.402462\pi$$
$$12$$ 0 0
$$13$$ −23.0000 −0.136095 −0.0680473 0.997682i $$-0.521677\pi$$
−0.0680473 + 0.997682i $$0.521677\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 256.000 1.00000
$$17$$ 263.000 0.910035 0.455017 0.890483i $$-0.349633\pi$$
0.455017 + 0.890483i $$0.349633\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 400.000 1.00000
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 625.000 1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −784.000 −1.00000
$$29$$ 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
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1225.00 −1.00000
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 1168.00 0.603306
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −3457.00 −1.56496 −0.782481 0.622675i $$-0.786046\pi$$
−0.782481 + 0.622675i $$0.786046\pi$$
$$48$$ 0 0
$$49$$ 2401.00 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −368.000 −0.136095
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 1825.00 0.603306
$$56$$ 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
$$63$$ 0 0
$$64$$ 4096.00 1.00000
$$65$$ −575.000 −0.136095
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 4208.00 0.910035
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10078.0 1.99921 0.999603 0.0281662i $$-0.00896677\pi$$
0.999603 + 0.0281662i $$0.00896677\pi$$
$$72$$ 0 0
$$73$$ 9502.00 1.78307 0.891537 0.452948i $$-0.149628\pi$$
0.891537 + 0.452948i $$0.149628\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3577.00 −0.603306
$$78$$ 0 0
$$79$$ 12167.0 1.94953 0.974764 0.223239i $$-0.0716632\pi$$
0.974764 + 0.223239i $$0.0716632\pi$$
$$80$$ 6400.00 1.00000
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −6382.00 −0.926404 −0.463202 0.886253i $$-0.653300\pi$$
−0.463202 + 0.886253i $$0.653300\pi$$
$$84$$ 0 0
$$85$$ 6575.00 0.910035
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 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
$$97$$ −3383.00 −0.359549 −0.179775 0.983708i $$-0.557537\pi$$
−0.179775 + 0.983708i $$0.557537\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 10000.0 1.00000
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ −18383.0 −1.73277 −0.866387 0.499373i $$-0.833564\pi$$
−0.866387 + 0.499373i $$0.833564\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ −14353.0 −1.20806 −0.604032 0.796960i $$-0.706440\pi$$
−0.604032 + 0.796960i $$0.706440\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −12544.0 −1.00000
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 18448.0 1.37099
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −12887.0 −0.910035
$$120$$ 0 0
$$121$$ −9312.00 −0.636022
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 15625.0 1.00000
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ −19600.0 −1.00000
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −1679.00 −0.0821067
$$144$$ 0 0
$$145$$ 28825.0 1.37099
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 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
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 31342.0 1.27153 0.635766 0.771882i $$-0.280684\pi$$
0.635766 + 0.771882i $$0.280684\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 17663.0 0.633332 0.316666 0.948537i $$-0.397437\pi$$
0.316666 + 0.948537i $$0.397437\pi$$
$$168$$ 0 0
$$169$$ −28032.0 −0.981478
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −11017.0 −0.368105 −0.184052 0.982916i $$-0.558922\pi$$
−0.184052 + 0.982916i $$0.558922\pi$$
$$174$$ 0 0
$$175$$ −30625.0 −1.00000
$$176$$ 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
$$187$$ 19199.0 0.549029
$$188$$ −55312.0 −1.56496
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −47447.0 −1.30059 −0.650297 0.759680i $$-0.725356\pi$$
−0.650297 + 0.759680i $$0.725356\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 38416.0 1.00000
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −56497.0 −1.37099
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −5888.00 −0.136095
$$209$$ 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
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 29200.0 0.603306
$$221$$ −6049.00 −0.123851
$$222$$ 0 0
$$223$$ −61343.0 −1.23355 −0.616773 0.787141i $$-0.711560\pi$$
−0.616773 + 0.787141i $$0.711560\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 49823.0 0.966892 0.483446 0.875374i $$-0.339385\pi$$
0.483446 + 0.875374i $$0.339385\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ −86425.0 −1.56496
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 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
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 60025.0 1.00000
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 65536.0 1.00000
$$257$$ 111938. 1.69477 0.847386 0.530977i $$-0.178175\pi$$
0.847386 + 0.530977i $$0.178175\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −9200.00 −0.136095
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 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 $$\pi$$
$$272$$ 67328.0 0.910035
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 45625.0 0.603306
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −119807. −1.51729 −0.758647 0.651502i $$-0.774139\pi$$
−0.758647 + 0.651502i $$0.774139\pi$$
$$282$$ 0 0
$$283$$ −152303. −1.90167 −0.950836 0.309695i $$-0.899773\pi$$
−0.950836 + 0.309695i $$0.899773\pi$$
$$284$$ 161248. 1.99921
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −14352.0 −0.171837
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 152032. 1.78307
$$293$$ −171337. −1.99579 −0.997897 0.0648123i $$-0.979355\pi$$
−0.997897 + 0.0648123i $$0.979355\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 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. −1.43517 −0.717583 0.696473i $$-0.754752\pi$$
−0.717583 + 0.696473i $$0.754752\pi$$
$$308$$ −57232.0 −0.603306
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 147097. 1.50146 0.750732 0.660606i $$-0.229701\pi$$
0.750732 + 0.660606i $$0.229701\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 194672. 1.94953
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ 84169.0 0.827124
$$320$$ 102400. 1.00000
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −14375.0 −0.136095
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 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. −0.926404
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 105200. 0.910035
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −117649. −1.00000
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −229897. −1.84495 −0.922473 0.386060i $$-0.873836\pi$$
−0.922473 + 0.386060i $$0.873836\pi$$
$$354$$ 0 0
$$355$$ 251950. 1.99921
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 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
$$363$$ 0 0
$$364$$ 18032.0 0.136095
$$365$$ 237550. 1.78307
$$366$$ 0 0
$$367$$ 116497. 0.864933 0.432467 0.901650i $$-0.357643\pi$$
0.432467 + 0.901650i $$0.357643\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −26519.0 −0.186584
$$378$$ 0 0
$$379$$ −35278.0 −0.245598 −0.122799 0.992432i $$-0.539187\pi$$
−0.122799 + 0.992432i $$0.539187\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −29182.0 −0.198938 −0.0994689 0.995041i $$-0.531714\pi$$
−0.0994689 + 0.995041i $$0.531714\pi$$
$$384$$ 0 0
$$385$$ −89425.0 −0.603306
$$386$$ 0 0
$$387$$ 0 0
$$388$$ −54128.0 −0.359549
$$389$$ −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
$$395$$ 304175. 1.94953
$$396$$ 0 0
$$397$$ 163897. 1.03990 0.519948 0.854198i $$-0.325951\pi$$
0.519948 + 0.854198i $$0.325951\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 160000. 1.00000
$$401$$ 316273. 1.96686 0.983430 0.181289i $$-0.0580270\pi$$
0.983430 + 0.181289i $$0.0580270\pi$$
$$402$$ 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. −1.73277
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −159550. −0.926404
$$416$$ 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
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 164375. 0.910035
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −356087. −1.91691 −0.958455 0.285245i $$-0.907925\pi$$
−0.958455 + 0.285245i $$0.907925\pi$$
$$432$$ 0 0
$$433$$ −193538. −1.03226 −0.516132 0.856509i $$-0.672628\pi$$
−0.516132 + 0.856509i $$0.672628\pi$$
$$434$$ 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
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ −200704. −1.00000
$$449$$ −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
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 28175.0 0.136095
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 295168. 1.37099
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 322463. 1.47858 0.739292 0.673385i $$-0.235160\pi$$
0.739292 + 0.673385i $$0.235160\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 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
$$485$$ −84575.0 −0.359549
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 470713. 1.95251 0.976255 0.216625i $$-0.0695049\pi$$
0.976255 + 0.216625i $$0.0695049\pi$$
$$492$$ 0 0
$$493$$ 303239. 1.24765
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −493822. −1.99921
$$498$$ 0 0
$$499$$ −31513.0 −0.126558 −0.0632789 0.997996i $$-0.520156\pi$$
−0.0632789 + 0.997996i $$0.520156\pi$$
$$500$$ 250000. 1.00000
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −313297. −1.23828 −0.619142 0.785279i $$-0.712519\pi$$
−0.619142 + 0.785279i $$0.712519\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 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
$$515$$ −459575. −1.73277
$$516$$ 0 0
$$517$$ −252361. −0.944150
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 440782. 1.61146 0.805732 0.592281i $$-0.201772\pi$$
0.805732 + 0.592281i $$0.201772\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 279841. 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 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
$$545$$ −358825. −1.20806
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −596183. −1.94953
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −313600. −1.00000
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 129938. 0.409939 0.204970 0.978768i $$-0.434290\pi$$
0.204970 + 0.978768i $$0.434290\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 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 $$0.934102\pi$$
$$572$$ −26864.0 −0.0821067
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 665017. 1.99747 0.998737 0.0502441i $$-0.0159999\pi$$
0.998737 + 0.0502441i $$0.0159999\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 461200. 1.37099
$$581$$ 312718. 0.926404
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 507698. 1.47343 0.736715 0.676204i $$-0.236376\pi$$
0.736715 + 0.676204i $$0.236376\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −320137. −0.910388 −0.455194 0.890392i $$-0.650430\pi$$
−0.455194 + 0.890392i $$0.650430\pi$$
$$594$$ 0 0
$$595$$ −322175. −0.910035
$$596$$ −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
$$605$$ −232800. −0.636022
$$606$$ 0 0
$$607$$ 82417.0 0.223686 0.111843 0.993726i $$-0.464325\pi$$
0.111843 + 0.993726i $$0.464325\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 79511.0 0.212983
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 390625. 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 501472. 1.27153
$$629$$ 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
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −55223.0 −0.136095
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −96002.0 −0.233649 −0.116825 0.993153i $$-0.537272\pi$$
−0.116825 + 0.993153i $$0.537272\pi$$
$$642$$ 0 0
$$643$$ −713183. −1.72496 −0.862480 0.506091i $$-0.831090\pi$$
−0.862480 + 0.506091i $$0.831090\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 111458. 0.266258 0.133129 0.991099i $$-0.457498\pi$$
0.133129 + 0.991099i $$0.457498\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 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
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 282608. 0.633332
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −448512. −0.981478
$$677$$ 750023. 1.63643 0.818215 0.574913i $$-0.194964\pi$$
0.818215 + 0.574913i $$0.194964\pi$$
$$678$$ 0 0
$$679$$ 165767. 0.359549
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 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. −0.368105
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ −490000. −1.00000
$$701$$ 884833. 1.80063 0.900317 0.435235i $$-0.143335\pi$$
0.900317 + 0.435235i $$0.143335\pi$$
$$702$$ 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 $$-0.0239102\pi$$
0.997180 + 0.0750454i $$0.0239102\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −41975.0 −0.0821067
$$716$$ 264928. 0.516775
$$717$$ 0 0
$$718$$ 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
$$725$$ 720625. 1.37099
$$726$$ 0 0
$$727$$ −976418. −1.84743 −0.923713 0.383086i $$-0.874861\pi$$
−0.923713 + 0.383086i $$0.874861\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −51143.0 −0.0951871 −0.0475936 0.998867i $$-0.515155\pi$$
−0.0475936 + 0.998867i $$0.515155\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 749207. 1.37187 0.685935 0.727663i $$-0.259393\pi$$
0.685935 + 0.727663i $$0.259393\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ −606050. −1.09193
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 307184. 0.549029
$$749$$ 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. −1.56496
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −1.13582e6 −1.99259
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 703297. 1.20806
$$764$$ −759152. −1.30059
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1.17962e6 1.97417 0.987084 0.160202i $$-0.0512144\pi$$
0.987084 + 0.160202i $$0.0512144\pi$$
$$774$$ 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
$$783$$ 0 0
$$784$$ 614656. 1.00000
$$785$$ 783550. 1.27153
$$786$$ 0 0
$$787$$ 1.03714e6 1.67451 0.837253 0.546816i $$-0.184160\pi$$
0.837253 + 0.546816i $$0.184160\pi$$
$$788$$ 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
$$797$$ 1.07354e6 1.69006 0.845031 0.534717i $$-0.179582\pi$$
0.845031 + 0.534717i $$0.179582\pi$$
$$798$$ 0 0
$$799$$ −909191. −1.42417
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 693646. 1.07574
$$804$$ 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. −1.37099
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −1.23437e6 −1.83129 −0.915647 0.401984i $$-0.868321\pi$$
−0.915647 + 0.401984i $$0.868321\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −94208.0 −0.136095
$$833$$ 631463. 0.910035
$$834$$ 0 0
$$835$$ 441575. 0.633332
$$836$$ 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
$$843$$ 0 0
$$844$$ −1.24149e6 −1.74284
$$845$$ −700800. −0.981478
$$846$$ 0 0
$$847$$ 456288. 0.636022
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 1.44782e6 1.98984 0.994918 0.100692i $$-0.0321058\pi$$
0.994918 + 0.100692i $$0.0321058\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −970462. −1.32135 −0.660674 0.750673i $$-0.729729\pi$$
−0.660674 + 0.750673i $$0.729729\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ −275425. −0.368105
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 888191. 1.17616
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −765625. −1.00000
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 467200. 0.603306
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ −96784.0 −0.123851
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −1.32950e6 −1.68983 −0.844913 0.534904i $$-0.820348\pi$$
−0.844913 + 0.534904i $$0.820348\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −981488. −1.23355
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 413950. 0.516775
$$896$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 797168. 0.966892
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −1.15584e6 −1.39271 −0.696357 0.717696i $$-0.745197\pi$$
−0.696357 + 0.717696i $$0.745197\pi$$
$$912$$ 0 0
$$913$$ −465886. −0.558905
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −695113. −0.823047 −0.411523 0.911399i $$-0.635003\pi$$
−0.411523 + 0.911399i $$0.635003\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −231794. −0.272081
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 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
$$935$$ 479975. 0.549029
$$936$$ 0 0
$$937$$ −1.58930e6 −1.81020 −0.905102 0.425195i $$-0.860206\pi$$
−0.905102 + 0.425195i $$0.860206\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −1.38280e6 −1.56496
$$941$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$948$$ 0 0
$$949$$ −218546. −0.242667
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ −1.18618e6 −1.30059
$$956$$ −693872. −0.759213
$$957$$ 0 0
$$958$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 960400. 1.00000
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −675937. −0.699518 −0.349759 0.936840i $$-0.613737\pi$$
−0.349759 + 0.936840i $$0.613737\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −737783. −0.742230 −0.371115 0.928587i $$-0.621024\pi$$
−0.371115 + 0.928587i $$0.621024\pi$$
$$998$$ 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 315.5.e.b.244.1 1
3.2 odd 2 35.5.c.b.34.1 yes 1
5.4 even 2 315.5.e.a.244.1 1
7.6 odd 2 315.5.e.a.244.1 1
12.11 even 2 560.5.p.a.209.1 1
15.2 even 4 175.5.d.c.76.1 2
15.8 even 4 175.5.d.c.76.2 2
15.14 odd 2 35.5.c.a.34.1 1
21.20 even 2 35.5.c.a.34.1 1
35.34 odd 2 CM 315.5.e.b.244.1 1
60.59 even 2 560.5.p.b.209.1 1
84.83 odd 2 560.5.p.b.209.1 1
105.62 odd 4 175.5.d.c.76.2 2
105.83 odd 4 175.5.d.c.76.1 2
105.104 even 2 35.5.c.b.34.1 yes 1
420.419 odd 2 560.5.p.a.209.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.c.a.34.1 1 15.14 odd 2
35.5.c.a.34.1 1 21.20 even 2
35.5.c.b.34.1 yes 1 3.2 odd 2
35.5.c.b.34.1 yes 1 105.104 even 2
175.5.d.c.76.1 2 15.2 even 4
175.5.d.c.76.1 2 105.83 odd 4
175.5.d.c.76.2 2 15.8 even 4
175.5.d.c.76.2 2 105.62 odd 4
315.5.e.a.244.1 1 5.4 even 2
315.5.e.a.244.1 1 7.6 odd 2
315.5.e.b.244.1 1 1.1 even 1 trivial
315.5.e.b.244.1 1 35.34 odd 2 CM
560.5.p.a.209.1 1 12.11 even 2
560.5.p.a.209.1 1 420.419 odd 2
560.5.p.b.209.1 1 60.59 even 2
560.5.p.b.209.1 1 84.83 odd 2